%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP139^2 : 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 : n018.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 08:55:52 AM UTC 2026
% Result : Theorem 5.88s 2.04s
% Output : Refutation 5.88s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP139^2 : TPTP v9.2.1. Released v7.5.0.
% 0.11/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n018.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 08:27:31 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.32/0.47 ---- Original DTF file ---
% 0.32/0.47 thf(spec,logic,$$dhol).
% 0.32/0.47 %------------------------------------------------------------------------------
% 0.32/0.47 % File : ITP139^2 : TPTP v9.2.1. Released v7.5.0.
% 0.32/0.47 % Domain : Interactive Theorem Proving
% 0.32/0.47 % Problem : Sledgehammer Paraconsistency problem prob_881__3274616_1
% 0.32/0.47 % Version : Especial.
% 0.32/0.47 % English :
% 0.32/0.47
% 0.32/0.47 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.32/0.47 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.32/0.47 % Source : [Des21]
% 0.32/0.47 % Names : Paraconsistency/prob_881__3274616_1 [Des21]
% 0.32/0.47
% 0.32/0.47 % Status : Theorem
% 0.32/0.47 % Rating : 0.33 v9.1.0, 0.00 v8.1.0, 0.25 v7.5.0
% 0.32/0.47 % Syntax : Number of formulae : 333 ( 61 unt; 45 typ; 0 def)
% 0.32/0.47 % Number of atoms : 982 ( 226 equ; 0 cnn)
% 0.32/0.47 % Maximal formula atoms : 13 ( 3 avg)
% 0.32/0.47 % Number of connectives : 3572 ( 61 ~; 12 |; 58 &;2921 @)
% 0.32/0.47 % ( 0 <=>; 520 =>; 0 <=; 0 <~>)
% 0.32/0.47 % Maximal formula depth : 17 ( 9 avg)
% 0.32/0.47 % Number of types : 3 ( 2 usr)
% 0.32/0.47 % Number of type conns : 182 ( 182 >; 0 *; 0 +; 0 <<)
% 0.32/0.47 % Number of symbols : 44 ( 43 usr; 0 con; 1-7 aty)
% 0.32/0.47 % Number of variables : 1161 ( 56 ^;1040 !; 15 ?;1161 :)
% 0.32/0.47 % ( 50 !>; 0 ?*; 0 @-; 0 @+)
% 0.32/0.47 % SPC : TH1_THM_EQU_NAR
% 0.32/0.47
% 0.32/0.47 % Comments : This file was generated by Sledgehammer 2021-02-23 16:24:02.098
% 0.32/0.47 %------------------------------------------------------------------------------
% 0.32/0.47 % Could-be-implicit typings (4)
% 0.32/0.47 thf(ty_t_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv,type,
% 0.32/0.47 paraco415392788lle_tv: $tType ).
% 0.32/0.47
% 0.32/0.47 thf(ty_t_Product__Type_Oprod,type,
% 0.32/0.47 product_prod: $tType > $tType > $tType ).
% 0.32/0.47
% 0.32/0.47 thf(ty_t_Set_Oset,type,
% 0.32/0.47 set: $tType > $tType ).
% 0.32/0.47
% 0.32/0.47 thf(ty_t_Nat_Onat,type,
% 0.32/0.47 nat: $tType ).
% 0.32/0.47
% 0.32/0.47 % Explicit typings (41)
% 0.32/0.47 thf(sy_cl_HOL_Otype,type,
% 0.32/0.47 type:
% 0.32/0.47 !>[A: $tType] : $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Orderings_Oord,type,
% 0.32/0.47 ord:
% 0.32/0.47 !>[A: $tType] : $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Orderings_Otop,type,
% 0.32/0.47 top:
% 0.32/0.47 !>[A: $tType] : $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Orderings_Oorder,type,
% 0.32/0.47 order:
% 0.32/0.47 !>[A: $tType] : $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Groups_Ogroup__add,type,
% 0.32/0.47 group_add:
% 0.32/0.47 !>[A: $tType] : $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Orderings_Ono__bot,type,
% 0.32/0.47 no_bot:
% 0.32/0.47 !>[A: $tType] : $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Orderings_Ono__top,type,
% 0.32/0.47 no_top:
% 0.32/0.47 !>[A: $tType] : $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Orderings_Olinorder,type,
% 0.32/0.47 linorder:
% 0.32/0.47 !>[A: $tType] : $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Orderings_Opreorder,type,
% 0.32/0.47 preorder:
% 0.32/0.47 !>[A: $tType] : $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Groups_Oab__group__add,type,
% 0.32/0.47 ab_group_add:
% 0.32/0.47 !>[A: $tType] : $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Orderings_Oorder__top,type,
% 0.32/0.47 order_top:
% 0.32/0.47 !>[A: $tType] : $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Orderings_Owellorder,type,
% 0.32/0.47 wellorder:
% 0.32/0.47 !>[A: $tType] : $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Groups_Osemigroup__add,type,
% 0.32/0.47 semigroup_add:
% 0.32/0.47 !>[A: $tType] : $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Orderings_Odense__order,type,
% 0.32/0.47 dense_order:
% 0.32/0.47 !>[A: $tType] : $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Groups_Ocomm__monoid__add,type,
% 0.32/0.47 comm_monoid_add:
% 0.32/0.47 !>[A: $tType] : $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Groups_Oab__semigroup__add,type,
% 0.32/0.47 ab_semigroup_add:
% 0.32/0.47 !>[A: $tType] : $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Orderings_Odense__linorder,type,
% 0.32/0.47 dense_linorder:
% 0.32/0.47 !>[A: $tType] : $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Groups_Ocancel__semigroup__add,type,
% 0.32/0.47 cancel_semigroup_add:
% 0.32/0.47 !>[A: $tType] : $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
% 0.32/0.47 ordere779506340up_add:
% 0.32/0.47 !>[A: $tType] : $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.32/0.47 canoni770627133id_add:
% 0.32/0.47 !>[A: $tType] : $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
% 0.32/0.47 ordere236663937imp_le:
% 0.32/0.47 !>[A: $tType] : $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
% 0.32/0.47 ordere223160158up_add:
% 0.32/0.47 !>[A: $tType] : $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add,type,
% 0.32/0.47 strict2144017051up_add:
% 0.32/0.47 !>[A: $tType] : $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Fun_Oinj__on,type,
% 0.32/0.47 inj_on:
% 0.32/0.47 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Fun_Ostrict__mono__on,type,
% 0.32/0.47 strict_mono_on:
% 0.32/0.47 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Fun_Othe__inv__into,type,
% 0.32/0.47 the_inv_into:
% 0.32/0.47 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > B > A ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Groups_Oplus__class_Oplus,type,
% 0.32/0.47 plus_plus:
% 0.32/0.47 !>[A: $tType] : ( A > A > A ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.32/0.47 ord_less:
% 0.32/0.47 !>[A: $tType] : ( A > A > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.32/0.47 ord_less_eq:
% 0.32/0.47 !>[A: $tType] : ( A > A > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Orderings_Oorder__class_Ostrict__mono,type,
% 0.32/0.47 order_strict_mono:
% 0.32/0.47 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Orderings_Otop__class_Otop,type,
% 0.32/0.47 top_top:
% 0.32/0.47 !>[A: $tType] : A ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ochange__tv,type,
% 0.32/0.47 paraco1920534163nge_tv: ( nat > nat ) > paraco415392788lle_tv > paraco415392788lle_tv ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv_OIndet,type,
% 0.32/0.47 paraco676387099_Indet: nat > paraco415392788lle_tv ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv_Orec__tv,type,
% 0.32/0.47 paraco152590079rec_tv:
% 0.32/0.47 !>[A: $tType] : ( ( $o > A ) > ( nat > A ) > paraco415392788lle_tv > A ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Product__Type_Oapfst,type,
% 0.32/0.47 product_apfst:
% 0.32/0.47 !>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( product_prod @ A @ B ) > ( product_prod @ C @ B ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Product__Type_Oapsnd,type,
% 0.32/0.47 product_apsnd:
% 0.32/0.47 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( product_prod @ A @ B ) > ( product_prod @ A @ C ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Product__Type_Omap__prod,type,
% 0.32/0.47 product_map_prod:
% 0.32/0.47 !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C ) > ( B > D ) > ( product_prod @ A @ B ) > ( product_prod @ C @ D ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Set_OCollect,type,
% 0.32/0.47 collect:
% 0.32/0.47 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Set_Oimage,type,
% 0.32/0.47 image:
% 0.32/0.47 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_member,type,
% 0.32/0.47 member:
% 0.32/0.47 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_v_f,type,
% 0.32/0.47 f: nat > nat ).
% 0.32/0.47
% 0.32/0.47 % Relevant facts (256)
% 0.32/0.47 thf(fact_0__092_060open_062_092_060And_062tv2_Atv1_O_Achange__tv_Af_Atv1_A_061_Achange__tv_Af_Atv2_A_092_060Longrightarrow_062_Atv1_A_061_Atv2_092_060close_062,axiom,
% 0.32/0.47 ! [Tv1: paraco415392788lle_tv,Tv2: paraco415392788lle_tv] :
% 0.32/0.47 ( ( ( paraco1920534163nge_tv @ f @ Tv1 )
% 0.32/0.47 = ( paraco1920534163nge_tv @ f @ Tv2 ) )
% 0.32/0.47 => ( Tv1 = Tv2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % \<open>\<And>tv2 tv1. change_tv f tv1 = change_tv f tv2 \<Longrightarrow> tv1 = tv2\<close>
% 0.32/0.47 thf(fact_1_UNIV__I,axiom,
% 0.32/0.47 ! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % UNIV_I
% 0.32/0.47 thf(fact_2_iso__tuple__UNIV__I,axiom,
% 0.32/0.47 ! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % iso_tuple_UNIV_I
% 0.32/0.47 thf(fact_3_top__apply,axiom,
% 0.32/0.47 ! [C: $tType,D: $tType] :
% 0.32/0.47 ( ( top @ C )
% 0.32/0.47 => ( ( top_top @ ( D > C ) )
% 0.32/0.47 = ( ^ [X2: D] : ( top_top @ C ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % top_apply
% 0.32/0.47 thf(fact_4_injD,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,F: A > B,X: A,Y: A] :
% 0.32/0.47 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.32/0.47 => ( ( ( F @ X )
% 0.32/0.47 = ( F @ Y ) )
% 0.32/0.47 => ( X = Y ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % injD
% 0.32/0.47 thf(fact_5_injI,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,F: A > B] :
% 0.32/0.47 ( ! [X3: A,Y2: A] :
% 0.32/0.47 ( ( ( F @ X3 )
% 0.32/0.47 = ( F @ Y2 ) )
% 0.32/0.47 => ( X3 = Y2 ) )
% 0.32/0.47 => ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % injI
% 0.32/0.47 thf(fact_6_inj__eq,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,F: A > B,X: A,Y: A] :
% 0.32/0.47 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.32/0.47 => ( ( ( F @ X )
% 0.32/0.47 = ( F @ Y ) )
% 0.32/0.47 = ( X = Y ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_eq
% 0.32/0.47 thf(fact_7_inj__def,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,F: A > B] :
% 0.32/0.47 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.32/0.47 = ( ! [X2: A,Y3: A] :
% 0.32/0.47 ( ( ( F @ X2 )
% 0.32/0.47 = ( F @ Y3 ) )
% 0.32/0.47 => ( X2 = Y3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_def
% 0.32/0.47 thf(fact_8_inj__onD,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,X: A,Y: A] :
% 0.32/0.47 ( ( inj_on @ A @ B @ F @ A2 )
% 0.32/0.47 => ( ( ( F @ X )
% 0.32/0.47 = ( F @ Y ) )
% 0.32/0.47 => ( ( member @ A @ X @ A2 )
% 0.32/0.47 => ( ( member @ A @ Y @ A2 )
% 0.32/0.47 => ( X = Y ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_onD
% 0.32/0.47 thf(fact_9_inj__onI,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,A2: set @ A,F: A > B] :
% 0.32/0.47 ( ! [X3: A,Y2: A] :
% 0.32/0.47 ( ( member @ A @ X3 @ A2 )
% 0.32/0.47 => ( ( member @ A @ Y2 @ A2 )
% 0.32/0.47 => ( ( ( F @ X3 )
% 0.32/0.47 = ( F @ Y2 ) )
% 0.32/0.47 => ( X3 = Y2 ) ) ) )
% 0.32/0.47 => ( inj_on @ A @ B @ F @ A2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_onI
% 0.32/0.47 thf(fact_10_inj__on__def,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType] :
% 0.32/0.47 ( ( inj_on @ A @ B )
% 0.32/0.47 = ( ^ [F2: A > B,A3: set @ A] :
% 0.32/0.47 ! [X2: A] :
% 0.32/0.47 ( ( member @ A @ X2 @ A3 )
% 0.32/0.47 => ! [Y3: A] :
% 0.32/0.47 ( ( member @ A @ Y3 @ A3 )
% 0.32/0.47 => ( ( ( F2 @ X2 )
% 0.32/0.47 = ( F2 @ Y3 ) )
% 0.32/0.47 => ( X2 = Y3 ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_def
% 0.32/0.47 thf(fact_11_inj__on__cong,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,A2: set @ A,F: A > B,G: A > B] :
% 0.32/0.47 ( ! [A4: A] :
% 0.32/0.47 ( ( member @ A @ A4 @ A2 )
% 0.32/0.47 => ( ( F @ A4 )
% 0.32/0.47 = ( G @ A4 ) ) )
% 0.32/0.47 => ( ( inj_on @ A @ B @ F @ A2 )
% 0.32/0.47 = ( inj_on @ A @ B @ G @ A2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_cong
% 0.32/0.47 thf(fact_12_inj__on__eq__iff,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,X: A,Y: A] :
% 0.32/0.47 ( ( inj_on @ A @ B @ F @ A2 )
% 0.32/0.47 => ( ( member @ A @ X @ A2 )
% 0.32/0.47 => ( ( member @ A @ Y @ A2 )
% 0.32/0.47 => ( ( ( F @ X )
% 0.32/0.47 = ( F @ Y ) )
% 0.32/0.47 = ( X = Y ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_eq_iff
% 0.32/0.47 thf(fact_13_assms,axiom,
% 0.32/0.47 inj_on @ nat @ nat @ f @ ( top_top @ ( set @ nat ) ) ).
% 0.32/0.47
% 0.32/0.47 % assms
% 0.32/0.47 thf(fact_14_top__set__def,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( top_top @ ( set @ A ) )
% 0.32/0.47 = ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % top_set_def
% 0.32/0.47 thf(fact_15_UNIV__witness,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ? [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % UNIV_witness
% 0.32/0.47 thf(fact_16_UNIV__eq__I,axiom,
% 0.32/0.47 ! [A: $tType,A2: set @ A] :
% 0.32/0.47 ( ! [X3: A] : ( member @ A @ X3 @ A2 )
% 0.32/0.47 => ( ( top_top @ ( set @ A ) )
% 0.32/0.47 = A2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % UNIV_eq_I
% 0.32/0.47 thf(fact_17_inj__on__inverseI,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,A2: set @ A,G: B > A,F: A > B] :
% 0.32/0.47 ( ! [X3: A] :
% 0.32/0.47 ( ( member @ A @ X3 @ A2 )
% 0.32/0.47 => ( ( G @ ( F @ X3 ) )
% 0.32/0.47 = X3 ) )
% 0.32/0.47 => ( inj_on @ A @ B @ F @ A2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_inverseI
% 0.32/0.47 thf(fact_18_inj__on__contraD,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,X: A,Y: A] :
% 0.32/0.47 ( ( inj_on @ A @ B @ F @ A2 )
% 0.32/0.47 => ( ( X != Y )
% 0.32/0.47 => ( ( member @ A @ X @ A2 )
% 0.32/0.47 => ( ( member @ A @ Y @ A2 )
% 0.32/0.47 => ( ( F @ X )
% 0.32/0.47 != ( F @ Y ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_contraD
% 0.32/0.47 thf(fact_19_top__empty__eq,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( top_top @ ( A > $o ) )
% 0.32/0.47 = ( ^ [X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % top_empty_eq
% 0.32/0.47 thf(fact_20_the__inv__f__f,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,F: A > B,X: A] :
% 0.32/0.47 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.32/0.47 => ( ( the_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F @ ( F @ X ) )
% 0.32/0.47 = X ) ) ).
% 0.32/0.47
% 0.32/0.47 % the_inv_f_f
% 0.32/0.47 thf(fact_21_strict__mono__on__imp__inj__on,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType] :
% 0.32/0.47 ( ( ( linorder @ A )
% 0.32/0.47 & ( preorder @ B ) )
% 0.32/0.47 => ! [F: A > B,A2: set @ A] :
% 0.32/0.47 ( ( strict_mono_on @ A @ B @ F @ A2 )
% 0.32/0.47 => ( inj_on @ A @ B @ F @ A2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % strict_mono_on_imp_inj_on
% 0.32/0.47 thf(fact_22_linorder__injI,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [F: A > B] :
% 0.32/0.47 ( ! [X3: A,Y2: A] :
% 0.32/0.47 ( ( ord_less @ A @ X3 @ Y2 )
% 0.32/0.47 => ( ( F @ X3 )
% 0.32/0.47 != ( F @ Y2 ) ) )
% 0.32/0.47 => ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % linorder_injI
% 0.32/0.47 thf(fact_23_inj__add__left,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( cancel_semigroup_add @ A )
% 0.32/0.47 => ! [A5: A] : ( inj_on @ A @ A @ ( plus_plus @ A @ A5 ) @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_add_left
% 0.32/0.47 thf(fact_24_change__tv_Osimps_I2_J,axiom,
% 0.32/0.47 ! [F: nat > nat,N: nat] :
% 0.32/0.47 ( ( paraco1920534163nge_tv @ F @ ( paraco676387099_Indet @ N ) )
% 0.32/0.47 = ( paraco676387099_Indet @ ( F @ N ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % change_tv.simps(2)
% 0.32/0.47 thf(fact_25_inj__apsnd,axiom,
% 0.32/0.47 ! [A: $tType,C: $tType,B: $tType,F: B > C] :
% 0.32/0.47 ( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
% 0.32/0.47 = ( inj_on @ B @ C @ F @ ( top_top @ ( set @ B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_apsnd
% 0.32/0.47 thf(fact_26_inj__image__mem__iff,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,F: A > B,A5: A,A2: set @ A] :
% 0.32/0.47 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.32/0.47 => ( ( member @ B @ ( F @ A5 ) @ ( image @ A @ B @ F @ A2 ) )
% 0.32/0.47 = ( member @ A @ A5 @ A2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_image_mem_iff
% 0.32/0.47 thf(fact_27_inj__image__eq__iff,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,B2: set @ A] :
% 0.32/0.47 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.32/0.47 => ( ( ( image @ A @ B @ F @ A2 )
% 0.32/0.47 = ( image @ A @ B @ F @ B2 ) )
% 0.32/0.47 = ( A2 = B2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_image_eq_iff
% 0.32/0.47 thf(fact_28_range__ex1__eq,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,F: A > B,B3: B] :
% 0.32/0.47 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.32/0.47 => ( ( member @ B @ B3 @ ( image @ A @ B @ F @ ( top_top @ ( set @ A ) ) ) )
% 0.32/0.47 = ( ? [X2: A] :
% 0.32/0.47 ( ( B3
% 0.32/0.47 = ( F @ X2 ) )
% 0.32/0.47 & ! [Y3: A] :
% 0.32/0.47 ( ( B3
% 0.32/0.47 = ( F @ Y3 ) )
% 0.32/0.47 => ( Y3 = X2 ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % range_ex1_eq
% 0.32/0.47 thf(fact_29_strict__mono__imp__inj__on,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType] :
% 0.32/0.47 ( ( ( linorder @ A )
% 0.32/0.47 & ( order @ B ) )
% 0.32/0.47 => ! [F: A > B,A2: set @ A] :
% 0.32/0.47 ( ( order_strict_mono @ A @ B @ F )
% 0.32/0.47 => ( inj_on @ A @ B @ F @ A2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % strict_mono_imp_inj_on
% 0.32/0.47 thf(fact_30_image__eqI,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,B3: A,F: B > A,X: B,A2: set @ B] :
% 0.32/0.47 ( ( B3
% 0.32/0.47 = ( F @ X ) )
% 0.32/0.47 => ( ( member @ B @ X @ A2 )
% 0.32/0.47 => ( member @ A @ B3 @ ( image @ B @ A @ F @ A2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % image_eqI
% 0.32/0.47 thf(fact_31_tv_Oinject_I2_J,axiom,
% 0.32/0.47 ! [X22: nat,Y22: nat] :
% 0.32/0.47 ( ( ( paraco676387099_Indet @ X22 )
% 0.32/0.47 = ( paraco676387099_Indet @ Y22 ) )
% 0.32/0.47 = ( X22 = Y22 ) ) ).
% 0.32/0.47
% 0.32/0.47 % tv.inject(2)
% 0.32/0.47 thf(fact_32_top1I,axiom,
% 0.32/0.47 ! [A: $tType,X: A] : ( top_top @ ( A > $o ) @ X ) ).
% 0.32/0.47
% 0.32/0.47 % top1I
% 0.32/0.47 thf(fact_33_surj__plus,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ab_group_add @ A )
% 0.32/0.47 => ! [A5: A] :
% 0.32/0.47 ( ( image @ A @ A @ ( plus_plus @ A @ A5 ) @ ( top_top @ ( set @ A ) ) )
% 0.32/0.47 = ( top_top @ ( set @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % surj_plus
% 0.32/0.47 thf(fact_34_the__inv__into__onto,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,F: A > B,A2: set @ A] :
% 0.32/0.47 ( ( inj_on @ A @ B @ F @ A2 )
% 0.32/0.47 => ( ( image @ B @ A @ ( the_inv_into @ A @ B @ A2 @ F ) @ ( image @ A @ B @ F @ A2 ) )
% 0.32/0.47 = A2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % the_inv_into_onto
% 0.32/0.47 thf(fact_35_imageI,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,X: A,A2: set @ A,F: A > B] :
% 0.32/0.47 ( ( member @ A @ X @ A2 )
% 0.32/0.47 => ( member @ B @ ( F @ X ) @ ( image @ A @ B @ F @ A2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % imageI
% 0.32/0.47 thf(fact_36_image__iff,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,Z: A,F: B > A,A2: set @ B] :
% 0.32/0.47 ( ( member @ A @ Z @ ( image @ B @ A @ F @ A2 ) )
% 0.32/0.47 = ( ? [X2: B] :
% 0.32/0.47 ( ( member @ B @ X2 @ A2 )
% 0.32/0.47 & ( Z
% 0.32/0.47 = ( F @ X2 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % image_iff
% 0.32/0.47 thf(fact_37_bex__imageD,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,F: B > A,A2: set @ B,P: A > $o] :
% 0.32/0.47 ( ? [X4: A] :
% 0.32/0.47 ( ( member @ A @ X4 @ ( image @ B @ A @ F @ A2 ) )
% 0.32/0.47 & ( P @ X4 ) )
% 0.32/0.47 => ? [X3: B] :
% 0.32/0.47 ( ( member @ B @ X3 @ A2 )
% 0.32/0.47 & ( P @ ( F @ X3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % bex_imageD
% 0.32/0.47 thf(fact_38_image__cong,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,M: set @ A,N2: set @ A,F: A > B,G: A > B] :
% 0.32/0.47 ( ( M = N2 )
% 0.32/0.47 => ( ! [X3: A] :
% 0.32/0.47 ( ( member @ A @ X3 @ N2 )
% 0.32/0.47 => ( ( F @ X3 )
% 0.32/0.47 = ( G @ X3 ) ) )
% 0.32/0.47 => ( ( image @ A @ B @ F @ M )
% 0.32/0.47 = ( image @ A @ B @ G @ N2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % image_cong
% 0.32/0.47 thf(fact_39_ball__imageD,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,F: B > A,A2: set @ B,P: A > $o] :
% 0.32/0.47 ( ! [X3: A] :
% 0.32/0.47 ( ( member @ A @ X3 @ ( image @ B @ A @ F @ A2 ) )
% 0.32/0.47 => ( P @ X3 ) )
% 0.32/0.47 => ! [X4: B] :
% 0.32/0.47 ( ( member @ B @ X4 @ A2 )
% 0.32/0.47 => ( P @ ( F @ X4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ball_imageD
% 0.32/0.47 thf(fact_40_rev__image__eqI,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,X: A,A2: set @ A,B3: B,F: A > B] :
% 0.32/0.47 ( ( member @ A @ X @ A2 )
% 0.32/0.47 => ( ( B3
% 0.32/0.47 = ( F @ X ) )
% 0.32/0.47 => ( member @ B @ B3 @ ( image @ A @ B @ F @ A2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % rev_image_eqI
% 0.32/0.47 thf(fact_41_strict__mono__onD,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType] :
% 0.32/0.47 ( ( ( ord @ A )
% 0.32/0.47 & ( ord @ B ) )
% 0.32/0.47 => ! [F: A > B,A2: set @ A,R: A,S: A] :
% 0.32/0.47 ( ( strict_mono_on @ A @ B @ F @ A2 )
% 0.32/0.47 => ( ( member @ A @ R @ A2 )
% 0.32/0.47 => ( ( member @ A @ S @ A2 )
% 0.32/0.47 => ( ( ord_less @ A @ R @ S )
% 0.32/0.47 => ( ord_less @ B @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % strict_mono_onD
% 0.32/0.47 thf(fact_42_strict__mono__onI,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType] :
% 0.32/0.47 ( ( ( ord @ A )
% 0.32/0.47 & ( ord @ B ) )
% 0.32/0.47 => ! [A2: set @ A,F: A > B] :
% 0.32/0.47 ( ! [R2: A,S2: A] :
% 0.32/0.47 ( ( member @ A @ R2 @ A2 )
% 0.32/0.47 => ( ( member @ A @ S2 @ A2 )
% 0.32/0.47 => ( ( ord_less @ A @ R2 @ S2 )
% 0.32/0.47 => ( ord_less @ B @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) )
% 0.32/0.47 => ( strict_mono_on @ A @ B @ F @ A2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % strict_mono_onI
% 0.32/0.47 thf(fact_43_strict__mono__on__def,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType] :
% 0.32/0.47 ( ( ( ord @ A )
% 0.32/0.47 & ( ord @ B ) )
% 0.32/0.47 => ( ( strict_mono_on @ A @ B )
% 0.32/0.47 = ( ^ [F2: A > B,A3: set @ A] :
% 0.32/0.47 ! [R3: A,S3: A] :
% 0.32/0.47 ( ( ( member @ A @ R3 @ A3 )
% 0.32/0.47 & ( member @ A @ S3 @ A3 )
% 0.32/0.47 & ( ord_less @ A @ R3 @ S3 ) )
% 0.32/0.47 => ( ord_less @ B @ ( F2 @ R3 ) @ ( F2 @ S3 ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % strict_mono_on_def
% 0.32/0.47 thf(fact_44_ord__eq__less__subst,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType] :
% 0.32/0.47 ( ( ( ord @ B )
% 0.32/0.47 & ( ord @ A ) )
% 0.32/0.47 => ! [A5: A,F: B > A,B3: B,C2: B] :
% 0.32/0.47 ( ( A5
% 0.32/0.47 = ( F @ B3 ) )
% 0.32/0.47 => ( ( ord_less @ B @ B3 @ C2 )
% 0.32/0.47 => ( ! [X3: B,Y2: B] :
% 0.32/0.47 ( ( ord_less @ B @ X3 @ Y2 )
% 0.32/0.47 => ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.32/0.47 => ( ord_less @ A @ A5 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_eq_less_subst
% 0.32/0.47 thf(fact_45_mem__Collect__eq,axiom,
% 0.32/0.47 ! [A: $tType,A5: A,P: A > $o] :
% 0.32/0.47 ( ( member @ A @ A5 @ ( collect @ A @ P ) )
% 0.32/0.47 = ( P @ A5 ) ) ).
% 0.32/0.47
% 0.32/0.47 % mem_Collect_eq
% 0.32/0.47 thf(fact_46_Collect__mem__eq,axiom,
% 0.32/0.47 ! [A: $tType,A2: set @ A] :
% 0.32/0.47 ( ( collect @ A
% 0.32/0.47 @ ^ [X2: A] : ( member @ A @ X2 @ A2 ) )
% 0.32/0.47 = A2 ) ).
% 0.32/0.47
% 0.32/0.47 % Collect_mem_eq
% 0.32/0.47 thf(fact_47_Collect__cong,axiom,
% 0.32/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.32/0.47 ( ! [X3: A] :
% 0.32/0.47 ( ( P @ X3 )
% 0.32/0.47 = ( Q @ X3 ) )
% 0.32/0.47 => ( ( collect @ A @ P )
% 0.32/0.47 = ( collect @ A @ Q ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Collect_cong
% 0.32/0.47 thf(fact_48_ext,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.32/0.47 ( ! [X3: A] :
% 0.32/0.47 ( ( F @ X3 )
% 0.32/0.47 = ( G @ X3 ) )
% 0.32/0.47 => ( F = G ) ) ).
% 0.32/0.47
% 0.32/0.47 % ext
% 0.32/0.47 thf(fact_49_ord__less__eq__subst,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType] :
% 0.32/0.47 ( ( ( ord @ B )
% 0.32/0.47 & ( ord @ A ) )
% 0.32/0.47 => ! [A5: A,B3: A,F: A > B,C2: B] :
% 0.32/0.47 ( ( ord_less @ A @ A5 @ B3 )
% 0.32/0.47 => ( ( ( F @ B3 )
% 0.32/0.47 = C2 )
% 0.32/0.47 => ( ! [X3: A,Y2: A] :
% 0.32/0.47 ( ( ord_less @ A @ X3 @ Y2 )
% 0.32/0.47 => ( ord_less @ B @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.32/0.47 => ( ord_less @ B @ ( F @ A5 ) @ C2 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_less_eq_subst
% 0.32/0.47 thf(fact_50_order__less__subst1,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType] :
% 0.32/0.47 ( ( ( order @ B )
% 0.32/0.47 & ( order @ A ) )
% 0.32/0.47 => ! [A5: A,F: B > A,B3: B,C2: B] :
% 0.32/0.47 ( ( ord_less @ A @ A5 @ ( F @ B3 ) )
% 0.32/0.47 => ( ( ord_less @ B @ B3 @ C2 )
% 0.32/0.47 => ( ! [X3: B,Y2: B] :
% 0.32/0.47 ( ( ord_less @ B @ X3 @ Y2 )
% 0.32/0.47 => ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.32/0.47 => ( ord_less @ A @ A5 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_less_subst1
% 0.32/0.47 thf(fact_51_order__less__subst2,axiom,
% 0.32/0.47 ! [A: $tType,C: $tType] :
% 0.32/0.47 ( ( ( order @ C )
% 0.32/0.47 & ( order @ A ) )
% 0.32/0.47 => ! [A5: A,B3: A,F: A > C,C2: C] :
% 0.32/0.47 ( ( ord_less @ A @ A5 @ B3 )
% 0.32/0.47 => ( ( ord_less @ C @ ( F @ B3 ) @ C2 )
% 0.32/0.47 => ( ! [X3: A,Y2: A] :
% 0.32/0.47 ( ( ord_less @ A @ X3 @ Y2 )
% 0.32/0.47 => ( ord_less @ C @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.32/0.47 => ( ord_less @ C @ ( F @ A5 ) @ C2 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_less_subst2
% 0.32/0.47 thf(fact_52_lt__ex,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( no_bot @ A )
% 0.32/0.47 => ! [X: A] :
% 0.32/0.47 ? [Y2: A] : ( ord_less @ A @ Y2 @ X ) ) ).
% 0.32/0.47
% 0.32/0.47 % lt_ex
% 0.32/0.47 thf(fact_53_gt__ex,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( no_top @ A )
% 0.32/0.47 => ! [X: A] :
% 0.32/0.47 ? [X_1: A] : ( ord_less @ A @ X @ X_1 ) ) ).
% 0.32/0.47
% 0.32/0.47 % gt_ex
% 0.32/0.47 thf(fact_54_neqE,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [X: A,Y: A] :
% 0.32/0.47 ( ( X != Y )
% 0.32/0.47 => ( ~ ( ord_less @ A @ X @ Y )
% 0.32/0.47 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % neqE
% 0.32/0.47 thf(fact_55_neq__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [X: A,Y: A] :
% 0.32/0.47 ( ( X != Y )
% 0.32/0.47 = ( ( ord_less @ A @ X @ Y )
% 0.32/0.47 | ( ord_less @ A @ Y @ X ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % neq_iff
% 0.32/0.47 thf(fact_56_order_Oasym,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [A5: A,B3: A] :
% 0.32/0.47 ( ( ord_less @ A @ A5 @ B3 )
% 0.32/0.47 => ~ ( ord_less @ A @ B3 @ A5 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order.asym
% 0.32/0.47 thf(fact_57_dense,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( dense_order @ A )
% 0.32/0.47 => ! [X: A,Y: A] :
% 0.32/0.47 ( ( ord_less @ A @ X @ Y )
% 0.32/0.47 => ? [Z2: A] :
% 0.32/0.47 ( ( ord_less @ A @ X @ Z2 )
% 0.32/0.47 & ( ord_less @ A @ Z2 @ Y ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dense
% 0.32/0.47 thf(fact_58_less__imp__neq,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [X: A,Y: A] :
% 0.32/0.47 ( ( ord_less @ A @ X @ Y )
% 0.32/0.47 => ( X != Y ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_imp_neq
% 0.32/0.47 thf(fact_59_strict__monoD,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType] :
% 0.32/0.47 ( ( ( order @ A )
% 0.32/0.47 & ( order @ B ) )
% 0.32/0.47 => ! [F: A > B,X: A,Y: A] :
% 0.32/0.47 ( ( order_strict_mono @ A @ B @ F )
% 0.32/0.47 => ( ( ord_less @ A @ X @ Y )
% 0.32/0.47 => ( ord_less @ B @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % strict_monoD
% 0.32/0.47 thf(fact_60_strict__monoI,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType] :
% 0.32/0.47 ( ( ( order @ A )
% 0.32/0.47 & ( order @ B ) )
% 0.32/0.47 => ! [F: A > B] :
% 0.32/0.47 ( ! [X3: A,Y2: A] :
% 0.32/0.47 ( ( ord_less @ A @ X3 @ Y2 )
% 0.32/0.47 => ( ord_less @ B @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.32/0.47 => ( order_strict_mono @ A @ B @ F ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % strict_monoI
% 0.32/0.47 thf(fact_61_less__asym,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( preorder @ A )
% 0.32/0.47 => ! [X: A,Y: A] :
% 0.32/0.47 ( ( ord_less @ A @ X @ Y )
% 0.32/0.47 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_asym
% 0.32/0.47 thf(fact_62_less__asym_H,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( preorder @ A )
% 0.32/0.47 => ! [A5: A,B3: A] :
% 0.32/0.47 ( ( ord_less @ A @ A5 @ B3 )
% 0.32/0.47 => ~ ( ord_less @ A @ B3 @ A5 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_asym'
% 0.32/0.47 thf(fact_63_less__trans,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( preorder @ A )
% 0.32/0.47 => ! [X: A,Y: A,Z: A] :
% 0.32/0.47 ( ( ord_less @ A @ X @ Y )
% 0.32/0.47 => ( ( ord_less @ A @ Y @ Z )
% 0.32/0.47 => ( ord_less @ A @ X @ Z ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_trans
% 0.32/0.47 thf(fact_64_less__linear,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [X: A,Y: A] :
% 0.32/0.47 ( ( ord_less @ A @ X @ Y )
% 0.32/0.47 | ( X = Y )
% 0.32/0.47 | ( ord_less @ A @ Y @ X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_linear
% 0.32/0.47 thf(fact_65_less__irrefl,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( preorder @ A )
% 0.32/0.47 => ! [X: A] :
% 0.32/0.47 ~ ( ord_less @ A @ X @ X ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_irrefl
% 0.32/0.47 thf(fact_66_ord__eq__less__trans,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ord @ A )
% 0.32/0.47 => ! [A5: A,B3: A,C2: A] :
% 0.32/0.47 ( ( A5 = B3 )
% 0.32/0.47 => ( ( ord_less @ A @ B3 @ C2 )
% 0.32/0.47 => ( ord_less @ A @ A5 @ C2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_eq_less_trans
% 0.32/0.47 thf(fact_67_ord__less__eq__trans,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ord @ A )
% 0.32/0.47 => ! [A5: A,B3: A,C2: A] :
% 0.32/0.47 ( ( ord_less @ A @ A5 @ B3 )
% 0.32/0.47 => ( ( B3 = C2 )
% 0.32/0.47 => ( ord_less @ A @ A5 @ C2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_less_eq_trans
% 0.32/0.47 thf(fact_68_dual__order_Oasym,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [B3: A,A5: A] :
% 0.32/0.47 ( ( ord_less @ A @ B3 @ A5 )
% 0.32/0.47 => ~ ( ord_less @ A @ A5 @ B3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.asym
% 0.32/0.47 thf(fact_69_less__imp__not__eq,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [X: A,Y: A] :
% 0.32/0.47 ( ( ord_less @ A @ X @ Y )
% 0.32/0.47 => ( X != Y ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_imp_not_eq
% 0.32/0.47 thf(fact_70_strict__mono__def,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType] :
% 0.32/0.47 ( ( ( order @ A )
% 0.32/0.47 & ( order @ B ) )
% 0.32/0.47 => ( ( order_strict_mono @ A @ B )
% 0.32/0.47 = ( ^ [F2: A > B] :
% 0.32/0.47 ! [X2: A,Y3: A] :
% 0.32/0.47 ( ( ord_less @ A @ X2 @ Y3 )
% 0.32/0.47 => ( ord_less @ B @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % strict_mono_def
% 0.32/0.47 thf(fact_71_less__not__sym,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( preorder @ A )
% 0.32/0.47 => ! [X: A,Y: A] :
% 0.32/0.47 ( ( ord_less @ A @ X @ Y )
% 0.32/0.47 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_not_sym
% 0.32/0.47 thf(fact_72_less__induct,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( wellorder @ A )
% 0.32/0.47 => ! [P: A > $o,A5: A] :
% 0.32/0.47 ( ! [X3: A] :
% 0.32/0.47 ( ! [Y4: A] :
% 0.32/0.47 ( ( ord_less @ A @ Y4 @ X3 )
% 0.32/0.47 => ( P @ Y4 ) )
% 0.32/0.47 => ( P @ X3 ) )
% 0.32/0.47 => ( P @ A5 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_induct
% 0.32/0.47 thf(fact_73_antisym__conv3,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [Y: A,X: A] :
% 0.32/0.47 ( ~ ( ord_less @ A @ Y @ X )
% 0.32/0.47 => ( ( ~ ( ord_less @ A @ X @ Y ) )
% 0.32/0.47 = ( X = Y ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antisym_conv3
% 0.32/0.47 thf(fact_74_less__imp__not__eq2,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [X: A,Y: A] :
% 0.32/0.47 ( ( ord_less @ A @ X @ Y )
% 0.32/0.47 => ( Y != X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_imp_not_eq2
% 0.32/0.47 thf(fact_75_less__imp__triv,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( preorder @ A )
% 0.32/0.47 => ! [X: A,Y: A,P: $o] :
% 0.32/0.47 ( ( ord_less @ A @ X @ Y )
% 0.32/0.47 => ( ( ord_less @ A @ Y @ X )
% 0.32/0.47 => P ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_imp_triv
% 0.32/0.47 thf(fact_76_linorder__cases,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [X: A,Y: A] :
% 0.32/0.47 ( ~ ( ord_less @ A @ X @ Y )
% 0.32/0.47 => ( ( X != Y )
% 0.32/0.47 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % linorder_cases
% 0.32/0.47 thf(fact_77_strict__mono__eq,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType] :
% 0.32/0.47 ( ( ( linorder @ A )
% 0.32/0.47 & ( order @ B ) )
% 0.32/0.47 => ! [F: A > B,X: A,Y: A] :
% 0.32/0.47 ( ( order_strict_mono @ A @ B @ F )
% 0.32/0.47 => ( ( ( F @ X )
% 0.32/0.47 = ( F @ Y ) )
% 0.32/0.47 = ( X = Y ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % strict_mono_eq
% 0.32/0.47 thf(fact_78_dual__order_Oirrefl,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [A5: A] :
% 0.32/0.47 ~ ( ord_less @ A @ A5 @ A5 ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.irrefl
% 0.32/0.47 thf(fact_79_order_Ostrict__trans,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [A5: A,B3: A,C2: A] :
% 0.32/0.47 ( ( ord_less @ A @ A5 @ B3 )
% 0.32/0.47 => ( ( ord_less @ A @ B3 @ C2 )
% 0.32/0.47 => ( ord_less @ A @ A5 @ C2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order.strict_trans
% 0.32/0.47 thf(fact_80_strict__mono__less,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType] :
% 0.32/0.47 ( ( ( linorder @ A )
% 0.32/0.47 & ( order @ B ) )
% 0.32/0.47 => ! [F: A > B,X: A,Y: A] :
% 0.32/0.47 ( ( order_strict_mono @ A @ B @ F )
% 0.32/0.47 => ( ( ord_less @ B @ ( F @ X ) @ ( F @ Y ) )
% 0.32/0.47 = ( ord_less @ A @ X @ Y ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % strict_mono_less
% 0.32/0.47 thf(fact_81_less__imp__not__less,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( preorder @ A )
% 0.32/0.47 => ! [X: A,Y: A] :
% 0.32/0.47 ( ( ord_less @ A @ X @ Y )
% 0.32/0.47 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_imp_not_less
% 0.32/0.47 thf(fact_82_exists__least__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( wellorder @ A )
% 0.32/0.47 => ( ( ^ [P2: A > $o] :
% 0.32/0.47 ? [X5: A] : ( P2 @ X5 ) )
% 0.32/0.47 = ( ^ [P3: A > $o] :
% 0.32/0.47 ? [N3: A] :
% 0.32/0.47 ( ( P3 @ N3 )
% 0.32/0.47 & ! [M2: A] :
% 0.32/0.47 ( ( ord_less @ A @ M2 @ N3 )
% 0.32/0.47 => ~ ( P3 @ M2 ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % exists_least_iff
% 0.32/0.47 thf(fact_83_linorder__less__wlog,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [P: A > A > $o,A5: A,B3: A] :
% 0.32/0.47 ( ! [A4: A,B4: A] :
% 0.32/0.47 ( ( ord_less @ A @ A4 @ B4 )
% 0.32/0.47 => ( P @ A4 @ B4 ) )
% 0.32/0.47 => ( ! [A4: A] : ( P @ A4 @ A4 )
% 0.32/0.47 => ( ! [A4: A,B4: A] :
% 0.32/0.47 ( ( P @ B4 @ A4 )
% 0.32/0.47 => ( P @ A4 @ B4 ) )
% 0.32/0.47 => ( P @ A5 @ B3 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % linorder_less_wlog
% 0.32/0.47 thf(fact_84_dual__order_Ostrict__trans,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [B3: A,A5: A,C2: A] :
% 0.32/0.47 ( ( ord_less @ A @ B3 @ A5 )
% 0.32/0.47 => ( ( ord_less @ A @ C2 @ B3 )
% 0.32/0.47 => ( ord_less @ A @ C2 @ A5 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.strict_trans
% 0.32/0.47 thf(fact_85_not__less__iff__gr__or__eq,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [X: A,Y: A] :
% 0.32/0.47 ( ( ~ ( ord_less @ A @ X @ Y ) )
% 0.32/0.47 = ( ( ord_less @ A @ Y @ X )
% 0.32/0.47 | ( X = Y ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % not_less_iff_gr_or_eq
% 0.32/0.47 thf(fact_86_order_Ostrict__implies__not__eq,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [A5: A,B3: A] :
% 0.32/0.47 ( ( ord_less @ A @ A5 @ B3 )
% 0.32/0.47 => ( A5 != B3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order.strict_implies_not_eq
% 0.32/0.47 thf(fact_87_dual__order_Ostrict__implies__not__eq,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [B3: A,A5: A] :
% 0.32/0.47 ( ( ord_less @ A @ B3 @ A5 )
% 0.32/0.47 => ( A5 != B3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.strict_implies_not_eq
% 0.32/0.47 thf(fact_88_strict__mono__on__eqD,axiom,
% 0.32/0.47 ! [D: $tType,C: $tType] :
% 0.32/0.47 ( ( ( linorder @ C )
% 0.32/0.47 & ( preorder @ D ) )
% 0.32/0.47 => ! [F: C > D,A2: set @ C,X: C,Y: C] :
% 0.32/0.47 ( ( strict_mono_on @ C @ D @ F @ A2 )
% 0.32/0.47 => ( ( ( F @ X )
% 0.32/0.47 = ( F @ Y ) )
% 0.32/0.47 => ( ( member @ C @ X @ A2 )
% 0.32/0.47 => ( ( member @ C @ Y @ A2 )
% 0.32/0.47 => ( Y = X ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % strict_mono_on_eqD
% 0.32/0.47 thf(fact_89_inj__on__strict__subset,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,F: A > B,B2: set @ A,A2: set @ A] :
% 0.32/0.47 ( ( inj_on @ A @ B @ F @ B2 )
% 0.32/0.47 => ( ( ord_less @ ( set @ A ) @ A2 @ B2 )
% 0.32/0.47 => ( ord_less @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) @ ( image @ A @ B @ F @ B2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_strict_subset
% 0.32/0.47 thf(fact_90_inj__on__the__inv__into,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,F: A > B,A2: set @ A] :
% 0.32/0.47 ( ( inj_on @ A @ B @ F @ A2 )
% 0.32/0.47 => ( inj_on @ B @ A @ ( the_inv_into @ A @ B @ A2 @ F ) @ ( image @ A @ B @ F @ A2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_the_inv_into
% 0.32/0.47 thf(fact_91_f__the__inv__into__f,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,F: A > B,A2: set @ A,Y: B] :
% 0.32/0.47 ( ( inj_on @ A @ B @ F @ A2 )
% 0.32/0.47 => ( ( member @ B @ Y @ ( image @ A @ B @ F @ A2 ) )
% 0.32/0.47 => ( ( F @ ( the_inv_into @ A @ B @ A2 @ F @ Y ) )
% 0.32/0.47 = Y ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % f_the_inv_into_f
% 0.32/0.47 thf(fact_92_strict__mono__inv,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType] :
% 0.32/0.47 ( ( ( linorder @ B )
% 0.32/0.47 & ( linorder @ A ) )
% 0.32/0.47 => ! [F: A > B,G: B > A] :
% 0.32/0.47 ( ( order_strict_mono @ A @ B @ F )
% 0.32/0.47 => ( ( ( image @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.32/0.47 = ( top_top @ ( set @ B ) ) )
% 0.32/0.47 => ( ! [X3: A] :
% 0.32/0.47 ( ( G @ ( F @ X3 ) )
% 0.32/0.47 = X3 )
% 0.32/0.47 => ( order_strict_mono @ B @ A @ G ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % strict_mono_inv
% 0.32/0.47 thf(fact_93_surjD,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,F: B > A,Y: A] :
% 0.32/0.47 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.32/0.47 = ( top_top @ ( set @ A ) ) )
% 0.32/0.47 => ? [X3: B] :
% 0.32/0.47 ( Y
% 0.32/0.47 = ( F @ X3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % surjD
% 0.32/0.47 thf(fact_94_surjE,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,F: B > A,Y: A] :
% 0.32/0.47 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.32/0.47 = ( top_top @ ( set @ A ) ) )
% 0.32/0.47 => ~ ! [X3: B] :
% 0.32/0.47 ( Y
% 0.32/0.47 != ( F @ X3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % surjE
% 0.32/0.47 thf(fact_95_surjI,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,G: B > A,F: A > B] :
% 0.32/0.47 ( ! [X3: A] :
% 0.32/0.47 ( ( G @ ( F @ X3 ) )
% 0.32/0.47 = X3 )
% 0.32/0.47 => ( ( image @ B @ A @ G @ ( top_top @ ( set @ B ) ) )
% 0.32/0.47 = ( top_top @ ( set @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % surjI
% 0.32/0.47 thf(fact_96_rangeI,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,F: B > A,X: B] : ( member @ A @ ( F @ X ) @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % rangeI
% 0.32/0.47 thf(fact_97_surj__def,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,F: B > A] :
% 0.32/0.47 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.32/0.47 = ( top_top @ ( set @ A ) ) )
% 0.32/0.47 = ( ! [Y3: A] :
% 0.32/0.47 ? [X2: B] :
% 0.32/0.47 ( Y3
% 0.32/0.47 = ( F @ X2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % surj_def
% 0.32/0.47 thf(fact_98_range__eqI,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,B3: A,F: B > A,X: B] :
% 0.32/0.47 ( ( B3
% 0.32/0.47 = ( F @ X ) )
% 0.32/0.47 => ( member @ A @ B3 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % range_eqI
% 0.32/0.47 thf(fact_99_inj__on__image__iff,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,A2: set @ A,G: A > B,F: A > A] :
% 0.32/0.47 ( ! [X3: A] :
% 0.32/0.47 ( ( member @ A @ X3 @ A2 )
% 0.32/0.47 => ! [Xa: A] :
% 0.32/0.47 ( ( member @ A @ Xa @ A2 )
% 0.32/0.47 => ( ( ( G @ ( F @ X3 ) )
% 0.32/0.47 = ( G @ ( F @ Xa ) ) )
% 0.32/0.47 = ( ( G @ X3 )
% 0.32/0.47 = ( G @ Xa ) ) ) ) )
% 0.32/0.47 => ( ( inj_on @ A @ A @ F @ A2 )
% 0.32/0.47 => ( ( inj_on @ A @ B @ G @ ( image @ A @ A @ F @ A2 ) )
% 0.32/0.47 = ( inj_on @ A @ B @ G @ A2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_image_iff
% 0.32/0.47 thf(fact_100_top_Oextremum__strict,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order_top @ A )
% 0.32/0.47 => ! [A5: A] :
% 0.32/0.47 ~ ( ord_less @ A @ ( top_top @ A ) @ A5 ) ) ).
% 0.32/0.47
% 0.32/0.47 % top.extremum_strict
% 0.32/0.47 thf(fact_101_top_Onot__eq__extremum,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order_top @ A )
% 0.32/0.47 => ! [A5: A] :
% 0.32/0.47 ( ( A5
% 0.32/0.47 != ( top_top @ A ) )
% 0.32/0.47 = ( ord_less @ A @ A5 @ ( top_top @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % top.not_eq_extremum
% 0.32/0.47 thf(fact_102_inj__on__add,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( cancel_semigroup_add @ A )
% 0.32/0.47 => ! [A5: A,A2: set @ A] : ( inj_on @ A @ A @ ( plus_plus @ A @ A5 ) @ A2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_add
% 0.32/0.47 thf(fact_103_the__inv__into__f__f,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,X: A] :
% 0.32/0.47 ( ( inj_on @ A @ B @ F @ A2 )
% 0.32/0.47 => ( ( member @ A @ X @ A2 )
% 0.32/0.47 => ( ( the_inv_into @ A @ B @ A2 @ F @ ( F @ X ) )
% 0.32/0.47 = X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % the_inv_into_f_f
% 0.32/0.47 thf(fact_104_the__inv__into__f__eq,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,X: A,Y: B] :
% 0.32/0.47 ( ( inj_on @ A @ B @ F @ A2 )
% 0.32/0.47 => ( ( ( F @ X )
% 0.32/0.47 = Y )
% 0.32/0.47 => ( ( member @ A @ X @ A2 )
% 0.32/0.47 => ( ( the_inv_into @ A @ B @ A2 @ F @ Y )
% 0.32/0.47 = X ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % the_inv_into_f_eq
% 0.32/0.47 thf(fact_105_add__less__cancel__right,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ordere236663937imp_le @ A )
% 0.32/0.47 => ! [A5: A,C2: A,B3: A] :
% 0.32/0.47 ( ( ord_less @ A @ ( plus_plus @ A @ A5 @ C2 ) @ ( plus_plus @ A @ B3 @ C2 ) )
% 0.32/0.47 = ( ord_less @ A @ A5 @ B3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_less_cancel_right
% 0.32/0.47 thf(fact_106_add__less__cancel__left,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ordere236663937imp_le @ A )
% 0.32/0.47 => ! [C2: A,A5: A,B3: A] :
% 0.32/0.47 ( ( ord_less @ A @ ( plus_plus @ A @ C2 @ A5 ) @ ( plus_plus @ A @ C2 @ B3 ) )
% 0.32/0.47 = ( ord_less @ A @ A5 @ B3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_less_cancel_left
% 0.32/0.47 thf(fact_107_add__right__cancel,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( cancel_semigroup_add @ A )
% 0.32/0.47 => ! [B3: A,A5: A,C2: A] :
% 0.32/0.47 ( ( ( plus_plus @ A @ B3 @ A5 )
% 0.32/0.47 = ( plus_plus @ A @ C2 @ A5 ) )
% 0.32/0.47 = ( B3 = C2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_right_cancel
% 0.32/0.47 thf(fact_108_add__left__cancel,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( cancel_semigroup_add @ A )
% 0.32/0.47 => ! [A5: A,B3: A,C2: A] :
% 0.32/0.47 ( ( ( plus_plus @ A @ A5 @ B3 )
% 0.32/0.47 = ( plus_plus @ A @ A5 @ C2 ) )
% 0.32/0.47 = ( B3 = C2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_left_cancel
% 0.32/0.47 thf(fact_109_inj__apfst,axiom,
% 0.32/0.47 ! [B: $tType,C: $tType,A: $tType,F: A > C] :
% 0.32/0.47 ( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ C @ B ) @ ( product_apfst @ A @ C @ B @ F ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
% 0.32/0.47 = ( inj_on @ A @ C @ F @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_apfst
% 0.32/0.47 thf(fact_110_add__less__imp__less__right,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ordere236663937imp_le @ A )
% 0.32/0.47 => ! [A5: A,C2: A,B3: A] :
% 0.32/0.47 ( ( ord_less @ A @ ( plus_plus @ A @ A5 @ C2 ) @ ( plus_plus @ A @ B3 @ C2 ) )
% 0.32/0.47 => ( ord_less @ A @ A5 @ B3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_less_imp_less_right
% 0.32/0.47 thf(fact_111_add__less__imp__less__left,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ordere236663937imp_le @ A )
% 0.32/0.47 => ! [C2: A,A5: A,B3: A] :
% 0.32/0.47 ( ( ord_less @ A @ ( plus_plus @ A @ C2 @ A5 ) @ ( plus_plus @ A @ C2 @ B3 ) )
% 0.32/0.47 => ( ord_less @ A @ A5 @ B3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_less_imp_less_left
% 0.32/0.47 thf(fact_112_add__strict__right__mono,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ordere223160158up_add @ A )
% 0.32/0.47 => ! [A5: A,B3: A,C2: A] :
% 0.32/0.47 ( ( ord_less @ A @ A5 @ B3 )
% 0.32/0.47 => ( ord_less @ A @ ( plus_plus @ A @ A5 @ C2 ) @ ( plus_plus @ A @ B3 @ C2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_strict_right_mono
% 0.32/0.47 thf(fact_113_add__strict__left__mono,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ordere223160158up_add @ A )
% 0.32/0.47 => ! [A5: A,B3: A,C2: A] :
% 0.32/0.47 ( ( ord_less @ A @ A5 @ B3 )
% 0.32/0.47 => ( ord_less @ A @ ( plus_plus @ A @ C2 @ A5 ) @ ( plus_plus @ A @ C2 @ B3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_strict_left_mono
% 0.32/0.47 thf(fact_114_psubsetD,axiom,
% 0.32/0.47 ! [A: $tType,A2: set @ A,B2: set @ A,C2: A] :
% 0.32/0.47 ( ( ord_less @ ( set @ A ) @ A2 @ B2 )
% 0.32/0.47 => ( ( member @ A @ C2 @ A2 )
% 0.32/0.47 => ( member @ A @ C2 @ B2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % psubsetD
% 0.32/0.47 thf(fact_115_psubset__trans,axiom,
% 0.32/0.47 ! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
% 0.32/0.47 ( ( ord_less @ ( set @ A ) @ A2 @ B2 )
% 0.32/0.47 => ( ( ord_less @ ( set @ A ) @ B2 @ C3 )
% 0.32/0.47 => ( ord_less @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % psubset_trans
% 0.32/0.47 thf(fact_116_apsnd__apfst__commute,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,F: C > B,G: D > A,P4: product_prod @ D @ C] :
% 0.32/0.47 ( ( product_apsnd @ C @ B @ A @ F @ ( product_apfst @ D @ A @ C @ G @ P4 ) )
% 0.32/0.47 = ( product_apfst @ D @ A @ B @ G @ ( product_apsnd @ C @ B @ D @ F @ P4 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % apsnd_apfst_commute
% 0.32/0.47 thf(fact_117_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ab_semigroup_add @ A )
% 0.32/0.47 => ! [A5: A,B3: A,C2: A] :
% 0.32/0.47 ( ( plus_plus @ A @ ( plus_plus @ A @ A5 @ B3 ) @ C2 )
% 0.32/0.47 = ( plus_plus @ A @ A5 @ ( plus_plus @ A @ B3 @ C2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ab_semigroup_add_class.add_ac(1)
% 0.32/0.47 thf(fact_118_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ordere779506340up_add @ A )
% 0.32/0.47 => ! [I: A,J: A,K: A,L: A] :
% 0.32/0.47 ( ( ( I = J )
% 0.32/0.47 & ( K = L ) )
% 0.32/0.47 => ( ( plus_plus @ A @ I @ K )
% 0.32/0.47 = ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_mono_thms_linordered_semiring(4)
% 0.32/0.47 thf(fact_119_group__cancel_Oadd1,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( comm_monoid_add @ A )
% 0.32/0.47 => ! [A2: A,K: A,A5: A,B3: A] :
% 0.32/0.47 ( ( A2
% 0.32/0.47 = ( plus_plus @ A @ K @ A5 ) )
% 0.32/0.47 => ( ( plus_plus @ A @ A2 @ B3 )
% 0.32/0.47 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A5 @ B3 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % group_cancel.add1
% 0.32/0.47 thf(fact_120_group__cancel_Oadd2,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( comm_monoid_add @ A )
% 0.32/0.47 => ! [B2: A,K: A,B3: A,A5: A] :
% 0.32/0.47 ( ( B2
% 0.32/0.47 = ( plus_plus @ A @ K @ B3 ) )
% 0.32/0.47 => ( ( plus_plus @ A @ A5 @ B2 )
% 0.32/0.47 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A5 @ B3 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % group_cancel.add2
% 0.32/0.47 thf(fact_121_add_Oassoc,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semigroup_add @ A )
% 0.32/0.47 => ! [A5: A,B3: A,C2: A] :
% 0.32/0.47 ( ( plus_plus @ A @ ( plus_plus @ A @ A5 @ B3 ) @ C2 )
% 0.32/0.47 = ( plus_plus @ A @ A5 @ ( plus_plus @ A @ B3 @ C2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add.assoc
% 0.32/0.47 thf(fact_122_add_Oleft__cancel,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( group_add @ A )
% 0.32/0.47 => ! [A5: A,B3: A,C2: A] :
% 0.32/0.47 ( ( ( plus_plus @ A @ A5 @ B3 )
% 0.32/0.47 = ( plus_plus @ A @ A5 @ C2 ) )
% 0.32/0.47 = ( B3 = C2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add.left_cancel
% 0.32/0.47 thf(fact_123_add_Oright__cancel,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( group_add @ A )
% 0.32/0.47 => ! [B3: A,A5: A,C2: A] :
% 0.32/0.47 ( ( ( plus_plus @ A @ B3 @ A5 )
% 0.32/0.47 = ( plus_plus @ A @ C2 @ A5 ) )
% 0.32/0.47 = ( B3 = C2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add.right_cancel
% 0.32/0.47 thf(fact_124_add_Ocommute,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ab_semigroup_add @ A )
% 0.32/0.47 => ( ( plus_plus @ A )
% 0.32/0.47 = ( ^ [A6: A,B5: A] : ( plus_plus @ A @ B5 @ A6 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add.commute
% 0.32/0.47 thf(fact_125_add_Oleft__commute,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ab_semigroup_add @ A )
% 0.32/0.47 => ! [B3: A,A5: A,C2: A] :
% 0.32/0.47 ( ( plus_plus @ A @ B3 @ ( plus_plus @ A @ A5 @ C2 ) )
% 0.32/0.47 = ( plus_plus @ A @ A5 @ ( plus_plus @ A @ B3 @ C2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add.left_commute
% 0.32/0.47 thf(fact_126_add__left__imp__eq,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( cancel_semigroup_add @ A )
% 0.32/0.47 => ! [A5: A,B3: A,C2: A] :
% 0.32/0.47 ( ( ( plus_plus @ A @ A5 @ B3 )
% 0.32/0.47 = ( plus_plus @ A @ A5 @ C2 ) )
% 0.32/0.47 => ( B3 = C2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_left_imp_eq
% 0.32/0.47 thf(fact_127_add__right__imp__eq,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( cancel_semigroup_add @ A )
% 0.32/0.47 => ! [B3: A,A5: A,C2: A] :
% 0.32/0.47 ( ( ( plus_plus @ A @ B3 @ A5 )
% 0.32/0.47 = ( plus_plus @ A @ C2 @ A5 ) )
% 0.32/0.47 => ( B3 = C2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_right_imp_eq
% 0.32/0.47 thf(fact_128_add__mono__thms__linordered__field_I5_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ordere223160158up_add @ A )
% 0.32/0.47 => ! [I: A,J: A,K: A,L: A] :
% 0.32/0.47 ( ( ( ord_less @ A @ I @ J )
% 0.32/0.47 & ( ord_less @ A @ K @ L ) )
% 0.32/0.47 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_mono_thms_linordered_field(5)
% 0.32/0.47 thf(fact_129_add__mono__thms__linordered__field_I2_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ordere223160158up_add @ A )
% 0.32/0.47 => ! [I: A,J: A,K: A,L: A] :
% 0.32/0.47 ( ( ( I = J )
% 0.32/0.47 & ( ord_less @ A @ K @ L ) )
% 0.32/0.47 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_mono_thms_linordered_field(2)
% 0.32/0.47 thf(fact_130_add__mono__thms__linordered__field_I1_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ordere223160158up_add @ A )
% 0.32/0.47 => ! [I: A,J: A,K: A,L: A] :
% 0.32/0.47 ( ( ( ord_less @ A @ I @ J )
% 0.32/0.47 & ( K = L ) )
% 0.32/0.47 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_mono_thms_linordered_field(1)
% 0.32/0.47 thf(fact_131_add__strict__mono,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( strict2144017051up_add @ A )
% 0.32/0.47 => ! [A5: A,B3: A,C2: A,D2: A] :
% 0.32/0.47 ( ( ord_less @ A @ A5 @ B3 )
% 0.32/0.47 => ( ( ord_less @ A @ C2 @ D2 )
% 0.32/0.47 => ( ord_less @ A @ ( plus_plus @ A @ A5 @ C2 ) @ ( plus_plus @ A @ B3 @ D2 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_strict_mono
% 0.32/0.47 thf(fact_132_top__conj_I1_J,axiom,
% 0.32/0.47 ! [A: $tType,X: A,P: $o] :
% 0.32/0.47 ( ( ( top_top @ ( A > $o ) @ X )
% 0.32/0.47 & P )
% 0.32/0.47 = P ) ).
% 0.32/0.47
% 0.32/0.47 % top_conj(1)
% 0.32/0.47 thf(fact_133_top__conj_I2_J,axiom,
% 0.32/0.47 ! [A: $tType,P: $o,X: A] :
% 0.32/0.47 ( ( P
% 0.32/0.47 & ( top_top @ ( A > $o ) @ X ) )
% 0.32/0.47 = P ) ).
% 0.32/0.47
% 0.32/0.47 % top_conj(2)
% 0.32/0.47 thf(fact_134_tv_Osimps_I8_J,axiom,
% 0.32/0.47 ! [A: $tType,F1: $o > A,F22: nat > A,X22: nat] :
% 0.32/0.47 ( ( paraco152590079rec_tv @ A @ F1 @ F22 @ ( paraco676387099_Indet @ X22 ) )
% 0.32/0.47 = ( F22 @ X22 ) ) ).
% 0.32/0.47
% 0.32/0.47 % tv.simps(8)
% 0.32/0.47 thf(fact_135_the__inv__into__into,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,X: B,B2: set @ A] :
% 0.32/0.47 ( ( inj_on @ A @ B @ F @ A2 )
% 0.32/0.47 => ( ( member @ B @ X @ ( image @ A @ B @ F @ A2 ) )
% 0.32/0.47 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.32/0.47 => ( member @ A @ ( the_inv_into @ A @ B @ A2 @ F @ X ) @ B2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % the_inv_into_into
% 0.32/0.47 thf(fact_136_prod_Oinj__map,axiom,
% 0.32/0.47 ! [D: $tType,C: $tType,B: $tType,A: $tType,F1: A > C,F22: B > D] :
% 0.32/0.47 ( ( inj_on @ A @ C @ F1 @ ( top_top @ ( set @ A ) ) )
% 0.32/0.47 => ( ( inj_on @ B @ D @ F22 @ ( top_top @ ( set @ B ) ) )
% 0.32/0.47 => ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F1 @ F22 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % prod.inj_map
% 0.32/0.47 thf(fact_137_order__refl,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( preorder @ A )
% 0.32/0.47 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_refl
% 0.32/0.47 thf(fact_138_subsetI,axiom,
% 0.32/0.47 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.32/0.47 ( ! [X3: A] :
% 0.32/0.47 ( ( member @ A @ X3 @ A2 )
% 0.32/0.47 => ( member @ A @ X3 @ B2 ) )
% 0.32/0.47 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % subsetI
% 0.32/0.47 thf(fact_139_subset__antisym,axiom,
% 0.32/0.47 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.32/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 0.32/0.47 => ( A2 = B2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % subset_antisym
% 0.32/0.47 thf(fact_140_add__le__cancel__right,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ordere236663937imp_le @ A )
% 0.32/0.47 => ! [A5: A,C2: A,B3: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A5 @ C2 ) @ ( plus_plus @ A @ B3 @ C2 ) )
% 0.32/0.47 = ( ord_less_eq @ A @ A5 @ B3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_le_cancel_right
% 0.32/0.47 thf(fact_141_add__le__cancel__left,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ordere236663937imp_le @ A )
% 0.32/0.47 => ! [C2: A,A5: A,B3: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A5 ) @ ( plus_plus @ A @ C2 @ B3 ) )
% 0.32/0.47 = ( ord_less_eq @ A @ A5 @ B3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_le_cancel_left
% 0.32/0.47 thf(fact_142_psubsetI,axiom,
% 0.32/0.47 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.32/0.47 => ( ( A2 != B2 )
% 0.32/0.47 => ( ord_less @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % psubsetI
% 0.32/0.47 thf(fact_143_psubsetE,axiom,
% 0.32/0.47 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.32/0.47 ( ( ord_less @ ( set @ A ) @ A2 @ B2 )
% 0.32/0.47 => ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.32/0.47 => ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % psubsetE
% 0.32/0.47 thf(fact_144_psubset__eq,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ord_less @ ( set @ A ) )
% 0.32/0.47 = ( ^ [A3: set @ A,B6: set @ A] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
% 0.32/0.47 & ( A3 != B6 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % psubset_eq
% 0.32/0.47 thf(fact_145_psubset__imp__subset,axiom,
% 0.32/0.47 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.32/0.47 ( ( ord_less @ ( set @ A ) @ A2 @ B2 )
% 0.32/0.47 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % psubset_imp_subset
% 0.32/0.47 thf(fact_146_psubset__subset__trans,axiom,
% 0.32/0.47 ! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
% 0.32/0.47 ( ( ord_less @ ( set @ A ) @ A2 @ B2 )
% 0.32/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
% 0.32/0.47 => ( ord_less @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % psubset_subset_trans
% 0.32/0.47 thf(fact_147_subset__not__subset__eq,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ord_less @ ( set @ A ) )
% 0.32/0.47 = ( ^ [A3: set @ A,B6: set @ A] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
% 0.32/0.47 & ~ ( ord_less_eq @ ( set @ A ) @ B6 @ A3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % subset_not_subset_eq
% 0.32/0.47 thf(fact_148_subset__psubset__trans,axiom,
% 0.32/0.47 ! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.32/0.47 => ( ( ord_less @ ( set @ A ) @ B2 @ C3 )
% 0.32/0.47 => ( ord_less @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % subset_psubset_trans
% 0.32/0.47 thf(fact_149_subset__iff__psubset__eq,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) )
% 0.32/0.47 = ( ^ [A3: set @ A,B6: set @ A] :
% 0.32/0.47 ( ( ord_less @ ( set @ A ) @ A3 @ B6 )
% 0.32/0.47 | ( A3 = B6 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % subset_iff_psubset_eq
% 0.32/0.47 thf(fact_150_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ordere779506340up_add @ A )
% 0.32/0.47 => ! [I: A,J: A,K: A,L: A] :
% 0.32/0.47 ( ( ( ord_less_eq @ A @ I @ J )
% 0.32/0.47 & ( K = L ) )
% 0.32/0.47 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_mono_thms_linordered_semiring(3)
% 0.32/0.47 thf(fact_151_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ordere779506340up_add @ A )
% 0.32/0.47 => ! [I: A,J: A,K: A,L: A] :
% 0.32/0.47 ( ( ( I = J )
% 0.32/0.47 & ( ord_less_eq @ A @ K @ L ) )
% 0.32/0.47 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_mono_thms_linordered_semiring(2)
% 0.32/0.47 thf(fact_152_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ordere779506340up_add @ A )
% 0.32/0.47 => ! [I: A,J: A,K: A,L: A] :
% 0.32/0.47 ( ( ( ord_less_eq @ A @ I @ J )
% 0.32/0.47 & ( ord_less_eq @ A @ K @ L ) )
% 0.32/0.47 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_mono_thms_linordered_semiring(1)
% 0.32/0.47 thf(fact_153_add__mono,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ordere779506340up_add @ A )
% 0.32/0.47 => ! [A5: A,B3: A,C2: A,D2: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A5 @ B3 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ C2 @ D2 )
% 0.32/0.47 => ( ord_less_eq @ A @ ( plus_plus @ A @ A5 @ C2 ) @ ( plus_plus @ A @ B3 @ D2 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_mono
% 0.32/0.47 thf(fact_154_add__left__mono,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ordere779506340up_add @ A )
% 0.32/0.47 => ! [A5: A,B3: A,C2: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A5 @ B3 )
% 0.32/0.47 => ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A5 ) @ ( plus_plus @ A @ C2 @ B3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_left_mono
% 0.32/0.47 thf(fact_155_less__eqE,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( canoni770627133id_add @ A )
% 0.32/0.47 => ! [A5: A,B3: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A5 @ B3 )
% 0.32/0.47 => ~ ! [C4: A] :
% 0.32/0.47 ( B3
% 0.32/0.47 != ( plus_plus @ A @ A5 @ C4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_eqE
% 0.32/0.47 thf(fact_156_add__right__mono,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ordere779506340up_add @ A )
% 0.32/0.47 => ! [A5: A,B3: A,C2: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A5 @ B3 )
% 0.32/0.47 => ( ord_less_eq @ A @ ( plus_plus @ A @ A5 @ C2 ) @ ( plus_plus @ A @ B3 @ C2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_right_mono
% 0.32/0.47 thf(fact_157_le__iff__add,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( canoni770627133id_add @ A )
% 0.32/0.47 => ( ( ord_less_eq @ A )
% 0.32/0.47 = ( ^ [A6: A,B5: A] :
% 0.32/0.47 ? [C5: A] :
% 0.32/0.47 ( B5
% 0.32/0.47 = ( plus_plus @ A @ A6 @ C5 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_iff_add
% 0.32/0.47 thf(fact_158_add__le__imp__le__left,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ordere236663937imp_le @ A )
% 0.32/0.47 => ! [C2: A,A5: A,B3: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A5 ) @ ( plus_plus @ A @ C2 @ B3 ) )
% 0.32/0.47 => ( ord_less_eq @ A @ A5 @ B3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_le_imp_le_left
% 0.32/0.47 thf(fact_159_add__le__imp__le__right,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ordere236663937imp_le @ A )
% 0.32/0.47 => ! [A5: A,C2: A,B3: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A5 @ C2 ) @ ( plus_plus @ A @ B3 @ C2 ) )
% 0.32/0.47 => ( ord_less_eq @ A @ A5 @ B3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % add_le_imp_le_right
% 0.32/0.47 thf(fact_160_order_Onot__eq__order__implies__strict,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [A5: A,B3: A] :
% 0.32/0.47 ( ( A5 != B3 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ A5 @ B3 )
% 0.32/0.47 => ( ord_less @ A @ A5 @ B3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order.not_eq_order_implies_strict
% 0.32/0.47 thf(fact_161_dual__order_Ostrict__implies__order,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [B3: A,A5: A] :
% 0.32/0.47 ( ( ord_less @ A @ B3 @ A5 )
% 0.32/0.47 => ( ord_less_eq @ A @ B3 @ A5 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.strict_implies_order
% 0.32/0.47 thf(fact_162_dual__order_Ostrict__iff__order,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ( ( ord_less @ A )
% 0.32/0.47 = ( ^ [B5: A,A6: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ B5 @ A6 )
% 0.32/0.47 & ( A6 != B5 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.strict_iff_order
% 0.32/0.47 thf(fact_163_dual__order_Oorder__iff__strict,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ( ( ord_less_eq @ A )
% 0.32/0.47 = ( ^ [B5: A,A6: A] :
% 0.32/0.47 ( ( ord_less @ A @ B5 @ A6 )
% 0.32/0.47 | ( A6 = B5 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.order_iff_strict
% 0.32/0.47 thf(fact_164_order_Ostrict__implies__order,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [A5: A,B3: A] :
% 0.32/0.47 ( ( ord_less @ A @ A5 @ B3 )
% 0.32/0.47 => ( ord_less_eq @ A @ A5 @ B3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order.strict_implies_order
% 0.32/0.47 thf(fact_165_dense__le__bounded,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( dense_linorder @ A )
% 0.32/0.47 => ! [X: A,Y: A,Z: A] :
% 0.32/0.47 ( ( ord_less @ A @ X @ Y )
% 0.32/0.47 => ( ! [W: A] :
% 0.32/0.47 ( ( ord_less @ A @ X @ W )
% 0.32/0.47 => ( ( ord_less @ A @ W @ Y )
% 0.32/0.47 => ( ord_less_eq @ A @ W @ Z ) ) )
% 0.32/0.47 => ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dense_le_bounded
% 0.32/0.47 thf(fact_166_dense__ge__bounded,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( dense_linorder @ A )
% 0.32/0.47 => ! [Z: A,X: A,Y: A] :
% 0.32/0.47 ( ( ord_less @ A @ Z @ X )
% 0.32/0.47 => ( ! [W: A] :
% 0.32/0.47 ( ( ord_less @ A @ Z @ W )
% 0.32/0.47 => ( ( ord_less @ A @ W @ X )
% 0.32/0.47 => ( ord_less_eq @ A @ Y @ W ) ) )
% 0.32/0.47 => ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dense_ge_bounded
% 0.32/0.47 thf(fact_167_dual__order_Ostrict__trans2,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [B3: A,A5: A,C2: A] :
% 0.32/0.47 ( ( ord_less @ A @ B3 @ A5 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ C2 @ B3 )
% 0.32/0.47 => ( ord_less @ A @ C2 @ A5 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.strict_trans2
% 0.32/0.47 thf(fact_168_dual__order_Ostrict__trans1,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [B3: A,A5: A,C2: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ B3 @ A5 )
% 0.32/0.47 => ( ( ord_less @ A @ C2 @ B3 )
% 0.32/0.47 => ( ord_less @ A @ C2 @ A5 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.strict_trans1
% 0.32/0.47 thf(fact_169_order_Ostrict__iff__order,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ( ( ord_less @ A )
% 0.32/0.47 = ( ^ [A6: A,B5: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A6 @ B5 )
% 0.32/0.47 & ( A6 != B5 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order.strict_iff_order
% 0.32/0.47 thf(fact_170_order_Oorder__iff__strict,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ( ( ord_less_eq @ A )
% 0.32/0.47 = ( ^ [A6: A,B5: A] :
% 0.32/0.47 ( ( ord_less @ A @ A6 @ B5 )
% 0.32/0.47 | ( A6 = B5 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order.order_iff_strict
% 0.32/0.47 thf(fact_171_order_Ostrict__trans2,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [A5: A,B3: A,C2: A] :
% 0.32/0.47 ( ( ord_less @ A @ A5 @ B3 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ B3 @ C2 )
% 0.32/0.47 => ( ord_less @ A @ A5 @ C2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order.strict_trans2
% 0.32/0.47 thf(fact_172_order_Ostrict__trans1,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [A5: A,B3: A,C2: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A5 @ B3 )
% 0.32/0.47 => ( ( ord_less @ A @ B3 @ C2 )
% 0.32/0.47 => ( ord_less @ A @ A5 @ C2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order.strict_trans1
% 0.32/0.47 thf(fact_173_not__le__imp__less,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [Y: A,X: A] :
% 0.32/0.47 ( ~ ( ord_less_eq @ A @ Y @ X )
% 0.32/0.47 => ( ord_less @ A @ X @ Y ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % not_le_imp_less
% 0.32/0.47 thf(fact_174_less__le__not__le,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( preorder @ A )
% 0.32/0.47 => ( ( ord_less @ A )
% 0.32/0.47 = ( ^ [X2: A,Y3: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X2 @ Y3 )
% 0.32/0.47 & ~ ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_le_not_le
% 0.32/0.47 thf(fact_175_le__imp__less__or__eq,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [X: A,Y: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X @ Y )
% 0.32/0.47 => ( ( ord_less @ A @ X @ Y )
% 0.32/0.47 | ( X = Y ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_imp_less_or_eq
% 0.32/0.47 thf(fact_176_le__less__linear,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [X: A,Y: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X @ Y )
% 0.32/0.47 | ( ord_less @ A @ Y @ X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_less_linear
% 0.32/0.47 thf(fact_177_dense__le,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( dense_linorder @ A )
% 0.32/0.47 => ! [Y: A,Z: A] :
% 0.32/0.47 ( ! [X3: A] :
% 0.32/0.47 ( ( ord_less @ A @ X3 @ Y )
% 0.32/0.47 => ( ord_less_eq @ A @ X3 @ Z ) )
% 0.32/0.47 => ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dense_le
% 0.32/0.47 thf(fact_178_dense__ge,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( dense_linorder @ A )
% 0.32/0.47 => ! [Z: A,Y: A] :
% 0.32/0.47 ( ! [X3: A] :
% 0.32/0.47 ( ( ord_less @ A @ Z @ X3 )
% 0.32/0.47 => ( ord_less_eq @ A @ Y @ X3 ) )
% 0.32/0.47 => ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dense_ge
% 0.32/0.47 thf(fact_179_less__le__trans,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( preorder @ A )
% 0.32/0.47 => ! [X: A,Y: A,Z: A] :
% 0.32/0.47 ( ( ord_less @ A @ X @ Y )
% 0.32/0.47 => ( ( ord_less_eq @ A @ Y @ Z )
% 0.32/0.47 => ( ord_less @ A @ X @ Z ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_le_trans
% 0.32/0.47 thf(fact_180_le__less__trans,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( preorder @ A )
% 0.32/0.47 => ! [X: A,Y: A,Z: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X @ Y )
% 0.32/0.47 => ( ( ord_less @ A @ Y @ Z )
% 0.32/0.47 => ( ord_less @ A @ X @ Z ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_less_trans
% 0.32/0.47 thf(fact_181_less__imp__le,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( preorder @ A )
% 0.32/0.47 => ! [X: A,Y: A] :
% 0.32/0.47 ( ( ord_less @ A @ X @ Y )
% 0.32/0.47 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_imp_le
% 0.32/0.47 thf(fact_182_antisym__conv2,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [X: A,Y: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X @ Y )
% 0.32/0.47 => ( ( ~ ( ord_less @ A @ X @ Y ) )
% 0.32/0.47 = ( X = Y ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antisym_conv2
% 0.32/0.47 thf(fact_183_antisym__conv1,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [X: A,Y: A] :
% 0.32/0.47 ( ~ ( ord_less @ A @ X @ Y )
% 0.32/0.47 => ( ( ord_less_eq @ A @ X @ Y )
% 0.32/0.47 = ( X = Y ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antisym_conv1
% 0.32/0.47 thf(fact_184_le__neq__trans,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [A5: A,B3: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A5 @ B3 )
% 0.32/0.47 => ( ( A5 != B3 )
% 0.32/0.47 => ( ord_less @ A @ A5 @ B3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_neq_trans
% 0.32/0.47 thf(fact_185_not__less,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [X: A,Y: A] :
% 0.32/0.47 ( ( ~ ( ord_less @ A @ X @ Y ) )
% 0.32/0.47 = ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % not_less
% 0.32/0.47 thf(fact_186_not__le,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [X: A,Y: A] :
% 0.32/0.47 ( ( ~ ( ord_less_eq @ A @ X @ Y ) )
% 0.32/0.47 = ( ord_less @ A @ Y @ X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % not_le
% 0.32/0.47 thf(fact_187_order__less__le__subst2,axiom,
% 0.32/0.47 ! [A: $tType,C: $tType] :
% 0.32/0.47 ( ( ( order @ C )
% 0.32/0.47 & ( order @ A ) )
% 0.32/0.47 => ! [A5: A,B3: A,F: A > C,C2: C] :
% 0.32/0.47 ( ( ord_less @ A @ A5 @ B3 )
% 0.32/0.47 => ( ( ord_less_eq @ C @ ( F @ B3 ) @ C2 )
% 0.32/0.47 => ( ! [X3: A,Y2: A] :
% 0.32/0.47 ( ( ord_less @ A @ X3 @ Y2 )
% 0.32/0.47 => ( ord_less @ C @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.32/0.47 => ( ord_less @ C @ ( F @ A5 ) @ C2 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_less_le_subst2
% 0.32/0.47 thf(fact_188_order__less__le__subst1,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType] :
% 0.32/0.47 ( ( ( order @ B )
% 0.32/0.47 & ( order @ A ) )
% 0.32/0.47 => ! [A5: A,F: B > A,B3: B,C2: B] :
% 0.32/0.47 ( ( ord_less @ A @ A5 @ ( F @ B3 ) )
% 0.32/0.47 => ( ( ord_less_eq @ B @ B3 @ C2 )
% 0.32/0.47 => ( ! [X3: B,Y2: B] :
% 0.32/0.47 ( ( ord_less_eq @ B @ X3 @ Y2 )
% 0.32/0.47 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.32/0.47 => ( ord_less @ A @ A5 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_less_le_subst1
% 0.32/0.47 thf(fact_189_order__le__less__subst2,axiom,
% 0.32/0.47 ! [A: $tType,C: $tType] :
% 0.32/0.47 ( ( ( order @ C )
% 0.32/0.47 & ( order @ A ) )
% 0.32/0.47 => ! [A5: A,B3: A,F: A > C,C2: C] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A5 @ B3 )
% 0.32/0.47 => ( ( ord_less @ C @ ( F @ B3 ) @ C2 )
% 0.32/0.47 => ( ! [X3: A,Y2: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.32/0.47 => ( ord_less_eq @ C @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.32/0.47 => ( ord_less @ C @ ( F @ A5 ) @ C2 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_le_less_subst2
% 0.32/0.47 thf(fact_190_order__le__less__subst1,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType] :
% 0.32/0.47 ( ( ( order @ B )
% 0.32/0.47 & ( order @ A ) )
% 0.32/0.47 => ! [A5: A,F: B > A,B3: B,C2: B] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A5 @ ( F @ B3 ) )
% 0.32/0.47 => ( ( ord_less @ B @ B3 @ C2 )
% 0.32/0.47 => ( ! [X3: B,Y2: B] :
% 0.32/0.47 ( ( ord_less @ B @ X3 @ Y2 )
% 0.32/0.47 => ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.32/0.47 => ( ord_less @ A @ A5 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_le_less_subst1
% 0.32/0.47 thf(fact_191_less__le,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ( ( ord_less @ A )
% 0.32/0.47 = ( ^ [X2: A,Y3: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X2 @ Y3 )
% 0.32/0.47 & ( X2 != Y3 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_le
% 0.32/0.47 thf(fact_192_le__less,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ( ( ord_less_eq @ A )
% 0.32/0.47 = ( ^ [X2: A,Y3: A] :
% 0.32/0.47 ( ( ord_less @ A @ X2 @ Y3 )
% 0.32/0.47 | ( X2 = Y3 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_less
% 0.32/0.47 thf(fact_193_leI,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [X: A,Y: A] :
% 0.32/0.47 ( ~ ( ord_less @ A @ X @ Y )
% 0.32/0.47 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % leI
% 0.32/0.47 thf(fact_194_leD,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [Y: A,X: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ Y @ X )
% 0.32/0.47 => ~ ( ord_less @ A @ X @ Y ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % leD
% 0.32/0.47 thf(fact_195_subset__image__iff,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,B2: set @ A,F: B > A,A2: set @ B] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image @ B @ A @ F @ A2 ) )
% 0.32/0.47 = ( ? [AA: set @ B] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ B ) @ AA @ A2 )
% 0.32/0.47 & ( B2
% 0.32/0.47 = ( image @ B @ A @ F @ AA ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % subset_image_iff
% 0.32/0.47 thf(fact_196_image__subset__iff,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,F: B > A,A2: set @ B,B2: set @ A] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F @ A2 ) @ B2 )
% 0.32/0.47 = ( ! [X2: B] :
% 0.32/0.47 ( ( member @ B @ X2 @ A2 )
% 0.32/0.47 => ( member @ A @ ( F @ X2 ) @ B2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % image_subset_iff
% 0.32/0.47 thf(fact_197_subset__imageE,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,B2: set @ A,F: B > A,A2: set @ B] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image @ B @ A @ F @ A2 ) )
% 0.32/0.47 => ~ ! [C6: set @ B] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ B ) @ C6 @ A2 )
% 0.32/0.47 => ( B2
% 0.32/0.47 != ( image @ B @ A @ F @ C6 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % subset_imageE
% 0.32/0.47 thf(fact_198_image__subsetI,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,A2: set @ A,F: A > B,B2: set @ B] :
% 0.32/0.47 ( ! [X3: A] :
% 0.32/0.47 ( ( member @ A @ X3 @ A2 )
% 0.32/0.47 => ( member @ B @ ( F @ X3 ) @ B2 ) )
% 0.32/0.47 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) @ B2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % image_subsetI
% 0.32/0.47 thf(fact_199_image__mono,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,A2: set @ A,B2: set @ A,F: A > B] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.32/0.47 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) @ ( image @ A @ B @ F @ B2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % image_mono
% 0.32/0.47 thf(fact_200_top_Oextremum__uniqueI,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order_top @ A )
% 0.32/0.47 => ! [A5: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A5 )
% 0.32/0.47 => ( A5
% 0.32/0.47 = ( top_top @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % top.extremum_uniqueI
% 0.32/0.47 thf(fact_201_top_Oextremum__unique,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order_top @ A )
% 0.32/0.47 => ! [A5: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A5 )
% 0.32/0.47 = ( A5
% 0.32/0.47 = ( top_top @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % top.extremum_unique
% 0.32/0.47 thf(fact_202_top__greatest,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order_top @ A )
% 0.32/0.47 => ! [A5: A] : ( ord_less_eq @ A @ A5 @ ( top_top @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % top_greatest
% 0.32/0.47 thf(fact_203_subset__UNIV,axiom,
% 0.32/0.47 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % subset_UNIV
% 0.32/0.47 thf(fact_204_subset__inj__on,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,F: A > B,B2: set @ A,A2: set @ A] :
% 0.32/0.47 ( ( inj_on @ A @ B @ F @ B2 )
% 0.32/0.47 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.32/0.47 => ( inj_on @ A @ B @ F @ A2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % subset_inj_on
% 0.32/0.47 thf(fact_205_inj__on__subset,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,B2: set @ A] :
% 0.32/0.47 ( ( inj_on @ A @ B @ F @ A2 )
% 0.32/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 0.32/0.47 => ( inj_on @ A @ B @ F @ B2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_subset
% 0.32/0.47 thf(fact_206_in__mono,axiom,
% 0.32/0.47 ! [A: $tType,A2: set @ A,B2: set @ A,X: A] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.32/0.47 => ( ( member @ A @ X @ A2 )
% 0.32/0.47 => ( member @ A @ X @ B2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % in_mono
% 0.32/0.47 thf(fact_207_subsetD,axiom,
% 0.32/0.47 ! [A: $tType,A2: set @ A,B2: set @ A,C2: A] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.32/0.47 => ( ( member @ A @ C2 @ A2 )
% 0.32/0.47 => ( member @ A @ C2 @ B2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % subsetD
% 0.32/0.47 thf(fact_208_equalityE,axiom,
% 0.32/0.47 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.32/0.47 ( ( A2 = B2 )
% 0.32/0.47 => ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.32/0.47 => ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % equalityE
% 0.32/0.47 thf(fact_209_subset__eq,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) )
% 0.32/0.47 = ( ^ [A3: set @ A,B6: set @ A] :
% 0.32/0.47 ! [X2: A] :
% 0.32/0.47 ( ( member @ A @ X2 @ A3 )
% 0.32/0.47 => ( member @ A @ X2 @ B6 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % subset_eq
% 0.32/0.47 thf(fact_210_equalityD1,axiom,
% 0.32/0.47 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.32/0.47 ( ( A2 = B2 )
% 0.32/0.47 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % equalityD1
% 0.32/0.47 thf(fact_211_equalityD2,axiom,
% 0.32/0.47 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.32/0.47 ( ( A2 = B2 )
% 0.32/0.47 => ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % equalityD2
% 0.32/0.47 thf(fact_212_subset__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) )
% 0.32/0.47 = ( ^ [A3: set @ A,B6: set @ A] :
% 0.32/0.47 ! [T: A] :
% 0.32/0.47 ( ( member @ A @ T @ A3 )
% 0.32/0.47 => ( member @ A @ T @ B6 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % subset_iff
% 0.32/0.47 thf(fact_213_subset__refl,axiom,
% 0.32/0.47 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% 0.32/0.47
% 0.32/0.47 % subset_refl
% 0.32/0.47 thf(fact_214_Collect__mono,axiom,
% 0.32/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.32/0.47 ( ! [X3: A] :
% 0.32/0.47 ( ( P @ X3 )
% 0.32/0.47 => ( Q @ X3 ) )
% 0.32/0.47 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Collect_mono
% 0.32/0.47 thf(fact_215_subset__trans,axiom,
% 0.32/0.47 ! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.32/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
% 0.32/0.47 => ( ord_less_eq @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % subset_trans
% 0.32/0.47 thf(fact_216_le__funD,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType] :
% 0.32/0.47 ( ( ord @ B )
% 0.32/0.47 => ! [F: A > B,G: A > B,X: A] :
% 0.32/0.47 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.32/0.47 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_funD
% 0.32/0.47 thf(fact_217_le__funE,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType] :
% 0.32/0.47 ( ( ord @ B )
% 0.32/0.47 => ! [F: A > B,G: A > B,X: A] :
% 0.32/0.47 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.32/0.47 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_funE
% 0.32/0.47 thf(fact_218_le__funI,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType] :
% 0.32/0.47 ( ( ord @ B )
% 0.32/0.47 => ! [F: A > B,G: A > B] :
% 0.32/0.47 ( ! [X3: A] : ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.32/0.47 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_funI
% 0.32/0.47 thf(fact_219_set__eq__subset,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ^ [Y5: set @ A,Z3: set @ A] : ( Y5 = Z3 ) )
% 0.32/0.47 = ( ^ [A3: set @ A,B6: set @ A] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
% 0.32/0.47 & ( ord_less_eq @ ( set @ A ) @ B6 @ A3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % set_eq_subset
% 0.32/0.47 thf(fact_220_le__fun__def,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType] :
% 0.32/0.47 ( ( ord @ B )
% 0.32/0.47 => ( ( ord_less_eq @ ( A > B ) )
% 0.32/0.47 = ( ^ [F2: A > B,G2: A > B] :
% 0.32/0.47 ! [X2: A] : ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_fun_def
% 0.32/0.47 thf(fact_221_Collect__mono__iff,axiom,
% 0.32/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 0.32/0.47 = ( ! [X2: A] :
% 0.32/0.47 ( ( P @ X2 )
% 0.32/0.47 => ( Q @ X2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Collect_mono_iff
% 0.32/0.47 thf(fact_222_order__subst1,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType] :
% 0.32/0.47 ( ( ( order @ B )
% 0.32/0.47 & ( order @ A ) )
% 0.32/0.47 => ! [A5: A,F: B > A,B3: B,C2: B] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A5 @ ( F @ B3 ) )
% 0.32/0.47 => ( ( ord_less_eq @ B @ B3 @ C2 )
% 0.32/0.47 => ( ! [X3: B,Y2: B] :
% 0.32/0.47 ( ( ord_less_eq @ B @ X3 @ Y2 )
% 0.32/0.47 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.32/0.47 => ( ord_less_eq @ A @ A5 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_subst1
% 0.32/0.47 thf(fact_223_order__subst2,axiom,
% 0.32/0.47 ! [A: $tType,C: $tType] :
% 0.32/0.47 ( ( ( order @ C )
% 0.32/0.47 & ( order @ A ) )
% 0.32/0.47 => ! [A5: A,B3: A,F: A > C,C2: C] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A5 @ B3 )
% 0.32/0.47 => ( ( ord_less_eq @ C @ ( F @ B3 ) @ C2 )
% 0.32/0.47 => ( ! [X3: A,Y2: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.32/0.47 => ( ord_less_eq @ C @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.32/0.47 => ( ord_less_eq @ C @ ( F @ A5 ) @ C2 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_subst2
% 0.32/0.47 thf(fact_224_ord__eq__le__subst,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType] :
% 0.32/0.47 ( ( ( ord @ B )
% 0.32/0.47 & ( ord @ A ) )
% 0.32/0.47 => ! [A5: A,F: B > A,B3: B,C2: B] :
% 0.32/0.47 ( ( A5
% 0.32/0.47 = ( F @ B3 ) )
% 0.32/0.47 => ( ( ord_less_eq @ B @ B3 @ C2 )
% 0.32/0.47 => ( ! [X3: B,Y2: B] :
% 0.32/0.47 ( ( ord_less_eq @ B @ X3 @ Y2 )
% 0.32/0.47 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.32/0.47 => ( ord_less_eq @ A @ A5 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_eq_le_subst
% 0.32/0.47 thf(fact_225_ord__le__eq__subst,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType] :
% 0.32/0.47 ( ( ( ord @ B )
% 0.32/0.47 & ( ord @ A ) )
% 0.32/0.47 => ! [A5: A,B3: A,F: A > B,C2: B] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A5 @ B3 )
% 0.32/0.47 => ( ( ( F @ B3 )
% 0.32/0.47 = C2 )
% 0.32/0.47 => ( ! [X3: A,Y2: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.32/0.47 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.32/0.47 => ( ord_less_eq @ B @ ( F @ A5 ) @ C2 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_le_eq_subst
% 0.32/0.47 thf(fact_226_eq__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ( ( ^ [Y5: A,Z3: A] : ( Y5 = Z3 ) )
% 0.32/0.47 = ( ^ [X2: A,Y3: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X2 @ Y3 )
% 0.32/0.47 & ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % eq_iff
% 0.32/0.47 thf(fact_227_antisym,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [X: A,Y: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X @ Y )
% 0.32/0.47 => ( ( ord_less_eq @ A @ Y @ X )
% 0.32/0.47 => ( X = Y ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antisym
% 0.32/0.47 thf(fact_228_linear,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [X: A,Y: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X @ Y )
% 0.32/0.47 | ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % linear
% 0.32/0.47 thf(fact_229_eq__refl,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( preorder @ A )
% 0.32/0.47 => ! [X: A,Y: A] :
% 0.32/0.47 ( ( X = Y )
% 0.32/0.47 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % eq_refl
% 0.32/0.47 thf(fact_230_le__cases,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( linorder @ A )
% 0.32/0.48 => ! [X: A,Y: A] :
% 0.32/0.48 ( ~ ( ord_less_eq @ A @ X @ Y )
% 0.32/0.48 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % le_cases
% 0.32/0.48 thf(fact_231_order_Otrans,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( order @ A )
% 0.32/0.48 => ! [A5: A,B3: A,C2: A] :
% 0.32/0.48 ( ( ord_less_eq @ A @ A5 @ B3 )
% 0.32/0.48 => ( ( ord_less_eq @ A @ B3 @ C2 )
% 0.32/0.48 => ( ord_less_eq @ A @ A5 @ C2 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % order.trans
% 0.32/0.48 thf(fact_232_le__cases3,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( linorder @ A )
% 0.32/0.48 => ! [X: A,Y: A,Z: A] :
% 0.32/0.48 ( ( ( ord_less_eq @ A @ X @ Y )
% 0.32/0.48 => ~ ( ord_less_eq @ A @ Y @ Z ) )
% 0.32/0.48 => ( ( ( ord_less_eq @ A @ Y @ X )
% 0.32/0.48 => ~ ( ord_less_eq @ A @ X @ Z ) )
% 0.32/0.48 => ( ( ( ord_less_eq @ A @ X @ Z )
% 0.32/0.48 => ~ ( ord_less_eq @ A @ Z @ Y ) )
% 0.32/0.48 => ( ( ( ord_less_eq @ A @ Z @ Y )
% 0.32/0.48 => ~ ( ord_less_eq @ A @ Y @ X ) )
% 0.32/0.48 => ( ( ( ord_less_eq @ A @ Y @ Z )
% 0.32/0.48 => ~ ( ord_less_eq @ A @ Z @ X ) )
% 0.32/0.48 => ~ ( ( ord_less_eq @ A @ Z @ X )
% 0.32/0.48 => ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % le_cases3
% 0.32/0.48 thf(fact_233_antisym__conv,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( order @ A )
% 0.32/0.48 => ! [Y: A,X: A] :
% 0.32/0.48 ( ( ord_less_eq @ A @ Y @ X )
% 0.32/0.48 => ( ( ord_less_eq @ A @ X @ Y )
% 0.32/0.48 = ( X = Y ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % antisym_conv
% 0.32/0.48 thf(fact_234_order__class_Oorder_Oeq__iff,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( order @ A )
% 0.32/0.48 => ( ( ^ [Y5: A,Z3: A] : ( Y5 = Z3 ) )
% 0.32/0.48 = ( ^ [A6: A,B5: A] :
% 0.32/0.48 ( ( ord_less_eq @ A @ A6 @ B5 )
% 0.32/0.48 & ( ord_less_eq @ A @ B5 @ A6 ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % order_class.order.eq_iff
% 0.32/0.48 thf(fact_235_ord__eq__le__trans,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( ord @ A )
% 0.32/0.48 => ! [A5: A,B3: A,C2: A] :
% 0.32/0.48 ( ( A5 = B3 )
% 0.32/0.48 => ( ( ord_less_eq @ A @ B3 @ C2 )
% 0.32/0.48 => ( ord_less_eq @ A @ A5 @ C2 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % ord_eq_le_trans
% 0.32/0.48 thf(fact_236_ord__le__eq__trans,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( ord @ A )
% 0.32/0.48 => ! [A5: A,B3: A,C2: A] :
% 0.32/0.48 ( ( ord_less_eq @ A @ A5 @ B3 )
% 0.32/0.48 => ( ( B3 = C2 )
% 0.32/0.48 => ( ord_less_eq @ A @ A5 @ C2 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % ord_le_eq_trans
% 0.32/0.48 thf(fact_237_order__class_Oorder_Oantisym,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( order @ A )
% 0.32/0.48 => ! [A5: A,B3: A] :
% 0.32/0.48 ( ( ord_less_eq @ A @ A5 @ B3 )
% 0.32/0.48 => ( ( ord_less_eq @ A @ B3 @ A5 )
% 0.32/0.48 => ( A5 = B3 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % order_class.order.antisym
% 0.32/0.48 thf(fact_238_order__trans,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( preorder @ A )
% 0.32/0.48 => ! [X: A,Y: A,Z: A] :
% 0.32/0.48 ( ( ord_less_eq @ A @ X @ Y )
% 0.32/0.48 => ( ( ord_less_eq @ A @ Y @ Z )
% 0.32/0.48 => ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % order_trans
% 0.32/0.48 thf(fact_239_dual__order_Orefl,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( order @ A )
% 0.32/0.48 => ! [A5: A] : ( ord_less_eq @ A @ A5 @ A5 ) ) ).
% 0.32/0.48
% 0.32/0.48 % dual_order.refl
% 0.32/0.48 thf(fact_240_linorder__wlog,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( linorder @ A )
% 0.32/0.48 => ! [P: A > A > $o,A5: A,B3: A] :
% 0.32/0.48 ( ! [A4: A,B4: A] :
% 0.32/0.48 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.32/0.48 => ( P @ A4 @ B4 ) )
% 0.32/0.48 => ( ! [A4: A,B4: A] :
% 0.32/0.48 ( ( P @ B4 @ A4 )
% 0.32/0.48 => ( P @ A4 @ B4 ) )
% 0.32/0.48 => ( P @ A5 @ B3 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % linorder_wlog
% 0.32/0.48 thf(fact_241_dual__order_Otrans,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( order @ A )
% 0.32/0.48 => ! [B3: A,A5: A,C2: A] :
% 0.32/0.48 ( ( ord_less_eq @ A @ B3 @ A5 )
% 0.32/0.48 => ( ( ord_less_eq @ A @ C2 @ B3 )
% 0.32/0.48 => ( ord_less_eq @ A @ C2 @ A5 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dual_order.trans
% 0.32/0.48 thf(fact_242_dual__order_Oeq__iff,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( order @ A )
% 0.32/0.48 => ( ( ^ [Y5: A,Z3: A] : ( Y5 = Z3 ) )
% 0.32/0.48 = ( ^ [A6: A,B5: A] :
% 0.32/0.48 ( ( ord_less_eq @ A @ B5 @ A6 )
% 0.32/0.48 & ( ord_less_eq @ A @ A6 @ B5 ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dual_order.eq_iff
% 0.32/0.48 thf(fact_243_dual__order_Oantisym,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( order @ A )
% 0.32/0.48 => ! [B3: A,A5: A] :
% 0.32/0.48 ( ( ord_less_eq @ A @ B3 @ A5 )
% 0.32/0.48 => ( ( ord_less_eq @ A @ A5 @ B3 )
% 0.32/0.48 => ( A5 = B3 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dual_order.antisym
% 0.32/0.48 thf(fact_244_strict__mono__less__eq,axiom,
% 0.32/0.48 ! [B: $tType,A: $tType] :
% 0.32/0.48 ( ( ( linorder @ A )
% 0.32/0.48 & ( order @ B ) )
% 0.32/0.48 => ! [F: A > B,X: A,Y: A] :
% 0.32/0.48 ( ( order_strict_mono @ A @ B @ F )
% 0.32/0.48 => ( ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y ) )
% 0.32/0.48 = ( ord_less_eq @ A @ X @ Y ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % strict_mono_less_eq
% 0.32/0.48 thf(fact_245_strict__mono__on__leD,axiom,
% 0.32/0.48 ! [B: $tType,A: $tType] :
% 0.32/0.48 ( ( ( linorder @ A )
% 0.32/0.48 & ( preorder @ B ) )
% 0.32/0.48 => ! [F: A > B,A2: set @ A,X: A,Y: A] :
% 0.32/0.48 ( ( strict_mono_on @ A @ B @ F @ A2 )
% 0.32/0.48 => ( ( member @ A @ X @ A2 )
% 0.32/0.48 => ( ( member @ A @ Y @ A2 )
% 0.32/0.48 => ( ( ord_less_eq @ A @ X @ Y )
% 0.32/0.48 => ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % strict_mono_on_leD
% 0.32/0.48 thf(fact_246_add__less__le__mono,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( ordere223160158up_add @ A )
% 0.32/0.48 => ! [A5: A,B3: A,C2: A,D2: A] :
% 0.32/0.48 ( ( ord_less @ A @ A5 @ B3 )
% 0.32/0.48 => ( ( ord_less_eq @ A @ C2 @ D2 )
% 0.32/0.48 => ( ord_less @ A @ ( plus_plus @ A @ A5 @ C2 ) @ ( plus_plus @ A @ B3 @ D2 ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_less_le_mono
% 0.32/0.48 thf(fact_247_add__le__less__mono,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( ordere223160158up_add @ A )
% 0.32/0.48 => ! [A5: A,B3: A,C2: A,D2: A] :
% 0.32/0.48 ( ( ord_less_eq @ A @ A5 @ B3 )
% 0.32/0.48 => ( ( ord_less @ A @ C2 @ D2 )
% 0.32/0.48 => ( ord_less @ A @ ( plus_plus @ A @ A5 @ C2 ) @ ( plus_plus @ A @ B3 @ D2 ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_le_less_mono
% 0.32/0.48 thf(fact_248_add__mono__thms__linordered__field_I3_J,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( ordere223160158up_add @ A )
% 0.32/0.48 => ! [I: A,J: A,K: A,L: A] :
% 0.32/0.48 ( ( ( ord_less @ A @ I @ J )
% 0.32/0.48 & ( ord_less_eq @ A @ K @ L ) )
% 0.32/0.48 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_mono_thms_linordered_field(3)
% 0.32/0.48 thf(fact_249_add__mono__thms__linordered__field_I4_J,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( ordere223160158up_add @ A )
% 0.32/0.48 => ! [I: A,J: A,K: A,L: A] :
% 0.32/0.48 ( ( ( ord_less_eq @ A @ I @ J )
% 0.32/0.48 & ( ord_less @ A @ K @ L ) )
% 0.32/0.48 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % add_mono_thms_linordered_field(4)
% 0.32/0.48 thf(fact_250_linorder__inj__onI,axiom,
% 0.32/0.48 ! [B: $tType,A: $tType] :
% 0.32/0.48 ( ( order @ A )
% 0.32/0.48 => ! [A2: set @ A,F: A > B] :
% 0.32/0.48 ( ! [X3: A,Y2: A] :
% 0.32/0.48 ( ( ord_less @ A @ X3 @ Y2 )
% 0.32/0.48 => ( ( member @ A @ X3 @ A2 )
% 0.32/0.48 => ( ( member @ A @ Y2 @ A2 )
% 0.32/0.48 => ( ( F @ X3 )
% 0.32/0.48 != ( F @ Y2 ) ) ) ) )
% 0.32/0.48 => ( ! [X3: A,Y2: A] :
% 0.32/0.48 ( ( member @ A @ X3 @ A2 )
% 0.32/0.48 => ( ( member @ A @ Y2 @ A2 )
% 0.32/0.48 => ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.32/0.48 | ( ord_less_eq @ A @ Y2 @ X3 ) ) ) )
% 0.32/0.48 => ( inj_on @ A @ B @ F @ A2 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % linorder_inj_onI
% 0.32/0.48 thf(fact_251_inj__on__image__mem__iff,axiom,
% 0.32/0.48 ! [B: $tType,A: $tType,F: A > B,B2: set @ A,A5: A,A2: set @ A] :
% 0.32/0.48 ( ( inj_on @ A @ B @ F @ B2 )
% 0.32/0.48 => ( ( member @ A @ A5 @ B2 )
% 0.32/0.48 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.32/0.48 => ( ( member @ B @ ( F @ A5 ) @ ( image @ A @ B @ F @ A2 ) )
% 0.32/0.48 = ( member @ A @ A5 @ A2 ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % inj_on_image_mem_iff
% 0.32/0.48 thf(fact_252_inj__on__image__eq__iff,axiom,
% 0.32/0.48 ! [B: $tType,A: $tType,F: A > B,C3: set @ A,A2: set @ A,B2: set @ A] :
% 0.32/0.48 ( ( inj_on @ A @ B @ F @ C3 )
% 0.32/0.48 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ C3 )
% 0.32/0.48 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
% 0.32/0.48 => ( ( ( image @ A @ B @ F @ A2 )
% 0.32/0.48 = ( image @ A @ B @ F @ B2 ) )
% 0.32/0.48 = ( A2 = B2 ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % inj_on_image_eq_iff
% 0.32/0.48 thf(fact_253_inj__image__subset__iff,axiom,
% 0.32/0.48 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,B2: set @ A] :
% 0.32/0.48 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.32/0.48 => ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) @ ( image @ A @ B @ F @ B2 ) )
% 0.32/0.48 = ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % inj_image_subset_iff
% 0.32/0.48 thf(fact_254_map__prod__surj,axiom,
% 0.32/0.48 ! [A: $tType,C: $tType,D: $tType,B: $tType,F: A > B,G: C > D] :
% 0.32/0.48 ( ( ( image @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.32/0.48 = ( top_top @ ( set @ B ) ) )
% 0.32/0.48 => ( ( ( image @ C @ D @ G @ ( top_top @ ( set @ C ) ) )
% 0.32/0.48 = ( top_top @ ( set @ D ) ) )
% 0.32/0.48 => ( ( image @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_map_prod @ A @ B @ C @ D @ F @ G ) @ ( top_top @ ( set @ ( product_prod @ A @ C ) ) ) )
% 0.32/0.48 = ( top_top @ ( set @ ( product_prod @ B @ D ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % map_prod_surj
% 0.32/0.48 thf(fact_255_subset__image__inj,axiom,
% 0.32/0.48 ! [A: $tType,B: $tType,S4: set @ A,F: B > A,T2: set @ B] :
% 0.32/0.48 ( ( ord_less_eq @ ( set @ A ) @ S4 @ ( image @ B @ A @ F @ T2 ) )
% 0.32/0.48 = ( ? [U: set @ B] :
% 0.32/0.48 ( ( ord_less_eq @ ( set @ B ) @ U @ T2 )
% 0.32/0.48 & ( inj_on @ B @ A @ F @ U )
% 0.32/0.48 & ( S4
% 0.32/0.48 = ( image @ B @ A @ F @ U ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % subset_image_inj
% 0.32/0.48
% 0.32/0.48 % Type constructors (31)
% 0.32/0.48 thf(tcon_fun___Orderings_Oorder__top,axiom,
% 0.32/0.48 ! [A7: $tType,A8: $tType] :
% 0.32/0.48 ( ( order_top @ A8 )
% 0.32/0.48 => ( order_top @ ( A7 > A8 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.32/0.48 ! [A7: $tType,A8: $tType] :
% 0.32/0.48 ( ( preorder @ A8 )
% 0.32/0.48 => ( preorder @ ( A7 > A8 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.32/0.48 ! [A7: $tType,A8: $tType] :
% 0.32/0.48 ( ( order @ A8 )
% 0.32/0.48 => ( order @ ( A7 > A8 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_fun___Orderings_Otop,axiom,
% 0.32/0.48 ! [A7: $tType,A8: $tType] :
% 0.32/0.48 ( ( top @ A8 )
% 0.32/0.48 => ( top @ ( A7 > A8 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_fun___Orderings_Oord,axiom,
% 0.32/0.48 ! [A7: $tType,A8: $tType] :
% 0.32/0.48 ( ( ord @ A8 )
% 0.32/0.48 => ( ord @ ( A7 > A8 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
% 0.32/0.48 strict2144017051up_add @ nat ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add,axiom,
% 0.32/0.48 ordere223160158up_add @ nat ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
% 0.32/0.48 ordere236663937imp_le @ nat ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 0.32/0.48 canoni770627133id_add @ nat ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add,axiom,
% 0.32/0.48 ordere779506340up_add @ nat ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
% 0.32/0.48 cancel_semigroup_add @ nat ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add,axiom,
% 0.32/0.48 ab_semigroup_add @ nat ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
% 0.32/0.48 comm_monoid_add @ nat ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Nat_Onat___Groups_Osemigroup__add,axiom,
% 0.32/0.48 semigroup_add @ nat ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 0.32/0.48 wellorder @ nat ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
% 0.32/0.48 preorder @ nat ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 0.32/0.48 linorder @ nat ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
% 0.32/0.48 no_top @ nat ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
% 0.32/0.48 order @ nat ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
% 0.32/0.48 ord @ nat ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Set_Oset___Orderings_Oorder__top_4,axiom,
% 0.32/0.48 ! [A7: $tType] : ( order_top @ ( set @ A7 ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Set_Oset___Orderings_Opreorder_5,axiom,
% 0.32/0.48 ! [A7: $tType] : ( preorder @ ( set @ A7 ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Set_Oset___Orderings_Oorder_6,axiom,
% 0.32/0.48 ! [A7: $tType] : ( order @ ( set @ A7 ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Set_Oset___Orderings_Otop_7,axiom,
% 0.32/0.48 ! [A7: $tType] : ( top @ ( set @ A7 ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Set_Oset___Orderings_Oord_8,axiom,
% 0.32/0.48 ! [A7: $tType] : ( ord @ ( set @ A7 ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_HOL_Obool___Orderings_Oorder__top_9,axiom,
% 0.32/0.48 order_top @ $o ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_HOL_Obool___Orderings_Opreorder_10,axiom,
% 0.32/0.48 preorder @ $o ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_HOL_Obool___Orderings_Olinorder_11,axiom,
% 0.32/0.48 linorder @ $o ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_HOL_Obool___Orderings_Oorder_12,axiom,
% 0.32/0.48 order @ $o ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_HOL_Obool___Orderings_Otop_13,axiom,
% 0.32/0.48 top @ $o ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_HOL_Obool___Orderings_Oord_14,axiom,
% 0.32/0.48 ord @ $o ).
% 0.32/0.48
% 0.32/0.48 % Conjectures (1)
% 0.32/0.48 thf(conj_0,conjecture,
% 0.32/0.48 inj_on @ paraco415392788lle_tv @ paraco415392788lle_tv @ ( paraco1920534163nge_tv @ f ) @ ( top_top @ ( set @ paraco415392788lle_tv ) ) ).
% 0.32/0.48
% 0.32/0.48 %------------------------------------------------------------------------------
% 0.32/0.48 ------------------------
% 2.97/1.57 ---- Embedded in THF ---
% 2.97/1.57 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.97/1.57 %%% Generated on Tue Feb 24 08:27:32 EST 2026
% 2.97/1.57 %%% using '$$dhol' embedding, version 1.3.0.
% 2.97/1.57 %%% Logic specification used:
% 2.97/1.57 %%% thf(spec, logic, $$dhol).
% 2.97/1.57
% 2.97/1.57 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.Ai5Zk59Gvc/DTF2DTF9159.p
% 2.97/1.57 thf(ty_t_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv, type, paraco415392788lle_tv: $tType).
% 2.97/1.57 thf(ty_t_Product__Type_Oprod, type, product_prod: ($tType > ($tType > $tType))).
% 2.97/1.57 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.97/1.57 thf(ty_t_Nat_Onat, type, nat: $tType).
% 2.97/1.57 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 2.97/1.57 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 2.97/1.57 thf(sy_cl_Orderings_Otop, type, top: (!> [A:$tType]: ($o))).
% 2.97/1.57 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 2.97/1.57 thf(sy_cl_Groups_Ogroup__add, type, group_add: (!> [A:$tType]: ($o))).
% 2.97/1.57 thf(sy_cl_Orderings_Ono__bot, type, no_bot: (!> [A:$tType]: ($o))).
% 2.97/1.57 thf(sy_cl_Orderings_Ono__top, type, no_top: (!> [A:$tType]: ($o))).
% 2.97/1.57 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 2.97/1.57 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 2.97/1.57 thf(sy_cl_Groups_Oab__group__add, type, ab_group_add: (!> [A:$tType]: ($o))).
% 2.97/1.57 thf(sy_cl_Orderings_Oorder__top, type, order_top: (!> [A:$tType]: ($o))).
% 2.97/1.57 thf(sy_cl_Orderings_Owellorder, type, wellorder: (!> [A:$tType]: ($o))).
% 2.97/1.57 thf(sy_cl_Groups_Osemigroup__add, type, semigroup_add: (!> [A:$tType]: ($o))).
% 2.97/1.57 thf(sy_cl_Orderings_Odense__order, type, dense_order: (!> [A:$tType]: ($o))).
% 2.97/1.57 thf(sy_cl_Groups_Ocomm__monoid__add, type, comm_monoid_add: (!> [A:$tType]: ($o))).
% 2.97/1.57 thf(sy_cl_Groups_Oab__semigroup__add, type, ab_semigroup_add: (!> [A:$tType]: ($o))).
% 2.97/1.57 thf(sy_cl_Orderings_Odense__linorder, type, dense_linorder: (!> [A:$tType]: ($o))).
% 2.97/1.57 thf(sy_cl_Groups_Ocancel__semigroup__add, type, cancel_semigroup_add: (!> [A:$tType]: ($o))).
% 2.97/1.57 thf(sy_cl_Groups_Oordered__ab__semigroup__add, type, ordere779506340up_add: (!> [A:$tType]: ($o))).
% 2.97/1.57 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add, type, canoni770627133id_add: (!> [A:$tType]: ($o))).
% 2.97/1.57 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le, type, ordere236663937imp_le: (!> [A:$tType]: ($o))).
% 2.97/1.57 thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add, type, ordere223160158up_add: (!> [A:$tType]: ($o))).
% 2.97/1.57 thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add, type, strict2144017051up_add: (!> [A:$tType]: ($o))).
% 2.97/1.57 thf(sy_c_Fun_Oinj__on, type, inj_on: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > $o))))).
% 2.97/1.57 thf(sy_c_Fun_Ostrict__mono__on, type, strict_mono_on: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > $o))))).
% 2.97/1.57 thf(sy_c_Fun_Othe__inv__into, type, the_inv_into: (!> [A:$tType,B:$tType]: (((set @ A) > ((A > B) > (B > A)))))).
% 2.97/1.57 thf(sy_c_Groups_Oplus__class_Oplus, type, plus_plus: (!> [A:$tType]: ((A > (A > A))))).
% 2.97/1.57 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 2.97/1.57 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.97/1.57 thf(sy_c_Orderings_Oorder__class_Ostrict__mono, type, order_strict_mono: (!> [A:$tType,B:$tType]: (((A > B) > $o)))).
% 2.97/1.57 thf(sy_c_Orderings_Otop__class_Otop, type, top_top: (!> [A:$tType]: (A))).
% 2.97/1.57 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ochange__tv, type, paraco1920534163nge_tv: ((nat > nat) > (paraco415392788lle_tv > paraco415392788lle_tv))).
% 2.97/1.57 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv_OIndet, type, paraco676387099_Indet: (nat > paraco415392788lle_tv)).
% 2.97/1.57 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv_Orec__tv, type, paraco152590079rec_tv: (!> [A:$tType]: ((($o > A) > ((nat > A) > (paraco415392788lle_tv > A)))))).
% 2.97/1.57 thf(sy_c_Product__Type_Oapfst, type, product_apfst: (!> [A:$tType,C:$tType,B:$tType]: (((A > C) > (((product_prod @ A) @ B) > ((product_prod @ C) @ B)))))).
% 2.97/1.57 thf(sy_c_Product__Type_Oapsnd, type, product_apsnd: (!> [B:$tType,C:$tType,A:$tType]: (((B > C) > (((product_prod @ A) @ B) > ((product_prod @ A) @ C)))))).
% 2.97/1.57 thf(sy_c_Product__Type_Omap__prod, type, product_map_prod: (!> [A:$tType,C:$tType,B:$tType,D:$tType]: (((A > C) > ((B > D) > (((product_prod @ A) @ B) > ((product_prod @ C) @ D))))))).
% 2.97/1.57 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.97/1.57 thf(sy_c_Set_Oimage, type, image: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > (set @ B)))))).
% 2.97/1.57 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.97/1.57 thf(sy_v_f, type, f: (nat > nat)).
% 2.97/1.57 thf(fact_0__092_060open_062_092_060And_062tv2_Atv1_O_Achange__tv_Af_Atv1_A_061_Achange__tv_Af_Atv2_A_092_060Longrightarrow_062_Atv1_A_061_Atv2_092_060close_062, axiom, (! [Tv1:paraco415392788lle_tv,Tv2:paraco415392788lle_tv]: (((((paraco1920534163nge_tv @ f) @ Tv1) = ((paraco1920534163nge_tv @ f) @ Tv2)) => (Tv1 = Tv2))))).
% 2.97/1.57 thf(fact_1_UNIV__I, axiom, (! [A:$tType,X:A]: ((((member @ A) @ X) @ (top_top @ (set @ A)))))).
% 2.97/1.57 thf(fact_2_iso__tuple__UNIV__I, axiom, (! [A:$tType,X:A]: ((((member @ A) @ X) @ (top_top @ (set @ A)))))).
% 2.97/1.57 thf(fact_3_top__apply, axiom, (! [C:$tType,D:$tType]: (((top @ C) => ((top_top @ (D > C)) = (^ [X2:D]: ((top_top @ C)))))))).
% 2.97/1.57 thf(fact_4_injD, axiom, (! [B:$tType,A:$tType,F:(A > B),X:A,Y:A]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => (((F @ X) = (F @ Y)) => (X = Y)))))).
% 2.97/1.57 thf(fact_5_injI, axiom, (! [B:$tType,A:$tType,F:(A > B)]: (((! [X3:A,Y2:A]: ((((F @ X3) = (F @ Y2)) => (X3 = Y2)))) => ((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))))))).
% 2.97/1.57 thf(fact_6_inj__eq, axiom, (! [B:$tType,A:$tType,F:(A > B),X:A,Y:A]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => (((F @ X) = (F @ Y)) = (X = Y)))))).
% 2.97/1.57 thf(fact_7_inj__def, axiom, (! [B:$tType,A:$tType,F:(A > B)]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) = (! [X2:A,Y3:A]: ((((F @ X2) = (F @ Y3)) => (X2 = Y3)))))))).
% 2.97/1.57 thf(fact_8_inj__onD, axiom, (! [B:$tType,A:$tType,F:(A > B),A2:(set @ A),X:A,Y:A]: ((((((inj_on @ A) @ B) @ F) @ A2) => (((F @ X) = (F @ Y)) => ((((member @ A) @ X) @ A2) => ((((member @ A) @ Y) @ A2) => (X = Y)))))))).
% 2.97/1.57 thf(fact_9_inj__onI, axiom, (! [B:$tType,A:$tType,A2:(set @ A),F:(A > B)]: (((! [X3:A,Y2:A]: (((((member @ A) @ X3) @ A2) => ((((member @ A) @ Y2) @ A2) => (((F @ X3) = (F @ Y2)) => (X3 = Y2)))))) => ((((inj_on @ A) @ B) @ F) @ A2))))).
% 2.97/1.57 thf(fact_10_inj__on__def, axiom, (! [B:$tType,A:$tType]: ((((inj_on @ A) @ B) = (^ [F2:(A > B),A3:(set @ A)]: ((! [X2:A]: (((((member @ A) @ X2) @ A3) => (! [Y3:A]: (((((member @ A) @ Y3) @ A3) => (((F2 @ X2) = (F2 @ Y3)) => (X2 = Y3)))))))))))))).
% 2.97/1.57 thf(fact_11_inj__on__cong, axiom, (! [B:$tType,A:$tType,A2:(set @ A),F:(A > B),G:(A > B)]: (((! [A4:A]: (((((member @ A) @ A4) @ A2) => ((F @ A4) = (G @ A4))))) => (((((inj_on @ A) @ B) @ F) @ A2) = ((((inj_on @ A) @ B) @ G) @ A2)))))).
% 2.97/1.57 thf(fact_12_inj__on__eq__iff, axiom, (! [B:$tType,A:$tType,F:(A > B),A2:(set @ A),X:A,Y:A]: ((((((inj_on @ A) @ B) @ F) @ A2) => ((((member @ A) @ X) @ A2) => ((((member @ A) @ Y) @ A2) => (((F @ X) = (F @ Y)) = (X = Y)))))))).
% 2.97/1.57 thf(fact_13_assms, axiom, ((((inj_on @ nat) @ nat) @ f) @ (top_top @ (set @ nat)))).
% 2.97/1.57 thf(fact_14_top__set__def, axiom, (! [A:$tType]: (((top_top @ (set @ A)) = ((collect @ A) @ (top_top @ (A > $o))))))).
% 2.97/1.57 thf(fact_15_UNIV__witness, axiom, (! [A:$tType]: ((? [X3:A]: ((((member @ A) @ X3) @ (top_top @ (set @ A)))))))).
% 2.97/1.57 thf(fact_16_UNIV__eq__I, axiom, (! [A:$tType,A2:(set @ A)]: (((! [X3:A]: ((((member @ A) @ X3) @ A2))) => ((top_top @ (set @ A)) = A2))))).
% 2.97/1.57 thf(fact_17_inj__on__inverseI, axiom, (! [B:$tType,A:$tType,A2:(set @ A),G:(B > A),F:(A > B)]: (((! [X3:A]: (((((member @ A) @ X3) @ A2) => ((G @ (F @ X3)) = X3)))) => ((((inj_on @ A) @ B) @ F) @ A2))))).
% 2.97/1.57 thf(fact_18_inj__on__contraD, axiom, (! [B:$tType,A:$tType,F:(A > B),A2:(set @ A),X:A,Y:A]: ((((((inj_on @ A) @ B) @ F) @ A2) => ((X != Y) => ((((member @ A) @ X) @ A2) => ((((member @ A) @ Y) @ A2) => ((F @ X) != (F @ Y))))))))).
% 2.97/1.57 thf(fact_19_top__empty__eq, axiom, (! [A:$tType]: (((top_top @ (A > $o)) = (^ [X2:A]: ((((member @ A) @ X2) @ (top_top @ (set @ A))))))))).
% 2.97/1.57 thf(fact_20_the__inv__f__f, axiom, (! [B:$tType,A:$tType,F:(A > B),X:A]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => ((((((the_inv_into @ A) @ B) @ (top_top @ (set @ A))) @ F) @ (F @ X)) = X))))).
% 2.97/1.57 thf(fact_21_strict__mono__on__imp__inj__on, axiom, (! [B:$tType,A:$tType]: ((((linorder @ A) & (preorder @ B)) => (! [F:(A > B),A2:(set @ A)]: ((((((strict_mono_on @ A) @ B) @ F) @ A2) => ((((inj_on @ A) @ B) @ F) @ A2)))))))).
% 2.97/1.57 thf(fact_22_linorder__injI, axiom, (! [B:$tType,A:$tType]: (((linorder @ A) => (! [F:(A > B)]: (((! [X3:A,Y2:A]: (((((ord_less @ A) @ X3) @ Y2) => ((F @ X3) != (F @ Y2))))) => ((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A)))))))))).
% 2.97/1.57 thf(fact_23_inj__add__left, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [A5:A]: (((((inj_on @ A) @ A) @ ((plus_plus @ A) @ A5)) @ (top_top @ (set @ A))))))))).
% 2.97/1.57 thf(fact_24_change__tv_Osimps_I2_J, axiom, (! [F:(nat > nat),N:nat]: ((((paraco1920534163nge_tv @ F) @ (paraco676387099_Indet @ N)) = (paraco676387099_Indet @ (F @ N)))))).
% 2.97/1.57 thf(fact_25_inj__apsnd, axiom, (! [A:$tType,C:$tType,B:$tType,F:(B > C)]: ((((((inj_on @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ C)) @ ((((product_apsnd @ B) @ C) @ A) @ F)) @ (top_top @ (set @ ((product_prod @ A) @ B)))) = ((((inj_on @ B) @ C) @ F) @ (top_top @ (set @ B))))))).
% 2.97/1.57 thf(fact_26_inj__image__mem__iff, axiom, (! [B:$tType,A:$tType,F:(A > B),A5:A,A2:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => ((((member @ B) @ (F @ A5)) @ ((((image @ A) @ B) @ F) @ A2)) = (((member @ A) @ A5) @ A2)))))).
% 2.97/1.57 thf(fact_27_inj__image__eq__iff, axiom, (! [B:$tType,A:$tType,F:(A > B),A2:(set @ A),B2:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => ((((((image @ A) @ B) @ F) @ A2) = ((((image @ A) @ B) @ F) @ B2)) = (A2 = B2)))))).
% 2.97/1.57 thf(fact_28_range__ex1__eq, axiom, (! [B:$tType,A:$tType,F:(A > B),B3:B]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => ((((member @ B) @ B3) @ ((((image @ A) @ B) @ F) @ (top_top @ (set @ A)))) = (? [X2:A]: (((B3 = (F @ X2)) & (! [Y3:A]: (((B3 = (F @ Y3)) => (Y3 = X2)))))))))))).
% 2.97/1.57 thf(fact_29_strict__mono__imp__inj__on, axiom, (! [B:$tType,A:$tType]: ((((linorder @ A) & (order @ B)) => (! [F:(A > B),A2:(set @ A)]: (((((order_strict_mono @ A) @ B) @ F) => ((((inj_on @ A) @ B) @ F) @ A2)))))))).
% 2.97/1.57 thf(fact_30_image__eqI, axiom, (! [A:$tType,B:$tType,B3:A,F:(B > A),X:B,A2:(set @ B)]: (((B3 = (F @ X)) => ((((member @ B) @ X) @ A2) => (((member @ A) @ B3) @ ((((image @ B) @ A) @ F) @ A2))))))).
% 2.97/1.57 thf(fact_31_tv_Oinject_I2_J, axiom, (! [X22:nat,Y22:nat]: ((((paraco676387099_Indet @ X22) = (paraco676387099_Indet @ Y22)) = (X22 = Y22))))).
% 2.97/1.57 thf(fact_32_top1I, axiom, (! [A:$tType,X:A]: (((top_top @ (A > $o)) @ X)))).
% 2.97/1.57 thf(fact_33_surj__plus, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A5:A]: ((((((image @ A) @ A) @ ((plus_plus @ A) @ A5)) @ (top_top @ (set @ A))) = (top_top @ (set @ A))))))))).
% 2.97/1.57 thf(fact_34_the__inv__into__onto, axiom, (! [B:$tType,A:$tType,F:(A > B),A2:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ A2) => (((((image @ B) @ A) @ ((((the_inv_into @ A) @ B) @ A2) @ F)) @ ((((image @ A) @ B) @ F) @ A2)) = A2))))).
% 2.97/1.57 thf(fact_35_imageI, axiom, (! [B:$tType,A:$tType,X:A,A2:(set @ A),F:(A > B)]: (((((member @ A) @ X) @ A2) => (((member @ B) @ (F @ X)) @ ((((image @ A) @ B) @ F) @ A2)))))).
% 2.97/1.57 thf(fact_36_image__iff, axiom, (! [A:$tType,B:$tType,Z:A,F:(B > A),A2:(set @ B)]: (((((member @ A) @ Z) @ ((((image @ B) @ A) @ F) @ A2)) = (? [X2:B]: (((((member @ B) @ X2) @ A2) & (Z = (F @ X2))))))))).
% 2.97/1.57 thf(fact_37_bex__imageD, axiom, (! [A:$tType,B:$tType,F:(B > A),A2:(set @ B),P:(A > $o)]: (((? [X4:A]: (((((member @ A) @ X4) @ ((((image @ B) @ A) @ F) @ A2)) & (P @ X4)))) => (? [X3:B]: (((((member @ B) @ X3) @ A2) & (P @ (F @ X3))))))))).
% 2.97/1.57 thf(fact_38_image__cong, axiom, (! [B:$tType,A:$tType,M:(set @ A),N2:(set @ A),F:(A > B),G:(A > B)]: (((M = N2) => ((! [X3:A]: (((((member @ A) @ X3) @ N2) => ((F @ X3) = (G @ X3))))) => (((((image @ A) @ B) @ F) @ M) = ((((image @ A) @ B) @ G) @ N2))))))).
% 2.97/1.57 thf(fact_39_ball__imageD, axiom, (! [A:$tType,B:$tType,F:(B > A),A2:(set @ B),P:(A > $o)]: (((! [X3:A]: (((((member @ A) @ X3) @ ((((image @ B) @ A) @ F) @ A2)) => (P @ X3)))) => (! [X4:B]: (((((member @ B) @ X4) @ A2) => (P @ (F @ X4))))))))).
% 2.97/1.57 thf(fact_40_rev__image__eqI, axiom, (! [B:$tType,A:$tType,X:A,A2:(set @ A),B3:B,F:(A > B)]: (((((member @ A) @ X) @ A2) => ((B3 = (F @ X)) => (((member @ B) @ B3) @ ((((image @ A) @ B) @ F) @ A2))))))).
% 2.97/1.57 thf(fact_41_strict__mono__onD, axiom, (! [B:$tType,A:$tType]: ((((ord @ A) & (ord @ B)) => (! [F:(A > B),A2:(set @ A),R:A,S:A]: ((((((strict_mono_on @ A) @ B) @ F) @ A2) => ((((member @ A) @ R) @ A2) => ((((member @ A) @ S) @ A2) => ((((ord_less @ A) @ R) @ S) => (((ord_less @ B) @ (F @ R)) @ (F @ S)))))))))))).
% 2.97/1.57 thf(fact_42_strict__mono__onI, axiom, (! [B:$tType,A:$tType]: ((((ord @ A) & (ord @ B)) => (! [A2:(set @ A),F:(A > B)]: (((! [R2:A,S2:A]: (((((member @ A) @ R2) @ A2) => ((((member @ A) @ S2) @ A2) => ((((ord_less @ A) @ R2) @ S2) => (((ord_less @ B) @ (F @ R2)) @ (F @ S2))))))) => ((((strict_mono_on @ A) @ B) @ F) @ A2)))))))).
% 2.97/1.57 thf(fact_43_strict__mono__on__def, axiom, (! [B:$tType,A:$tType]: ((((ord @ A) & (ord @ B)) => (((strict_mono_on @ A) @ B) = (^ [F2:(A > B),A3:(set @ A)]: ((! [R3:A,S3:A]: ((((((member @ A) @ R3) @ A3) & ((((member @ A) @ S3) @ A3) & (((ord_less @ A) @ R3) @ S3))) => (((ord_less @ B) @ (F2 @ R3)) @ (F2 @ S3)))))))))))).
% 2.97/1.57 thf(fact_44_ord__eq__less__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A5:A,F:(B > A),B3:B,C2:B]: (((A5 = (F @ B3)) => ((((ord_less @ B) @ B3) @ C2) => ((! [X3:B,Y2:B]: (((((ord_less @ B) @ X3) @ Y2) => (((ord_less @ A) @ (F @ X3)) @ (F @ Y2))))) => (((ord_less @ A) @ A5) @ (F @ C2))))))))))).
% 2.97/1.57 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A5:A,P:(A > $o)]: (((((member @ A) @ A5) @ ((collect @ A) @ P)) = (P @ A5))))).
% 2.97/1.57 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A2:(set @ A)]: ((((collect @ A) @ (^ [X2:A]: ((((member @ A) @ X2) @ A2)))) = A2)))).
% 2.97/1.57 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P @ X3) = (Q @ X3)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 2.97/1.57 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X3:A]: (((F @ X3) = (G @ X3)))) => (F = G))))).
% 2.97/1.57 thf(fact_49_ord__less__eq__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A5:A,B3:A,F:(A > B),C2:B]: (((((ord_less @ A) @ A5) @ B3) => (((F @ B3) = C2) => ((! [X3:A,Y2:A]: (((((ord_less @ A) @ X3) @ Y2) => (((ord_less @ B) @ (F @ X3)) @ (F @ Y2))))) => (((ord_less @ B) @ (F @ A5)) @ C2)))))))))).
% 2.97/1.57 thf(fact_50_order__less__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A5:A,F:(B > A),B3:B,C2:B]: (((((ord_less @ A) @ A5) @ (F @ B3)) => ((((ord_less @ B) @ B3) @ C2) => ((! [X3:B,Y2:B]: (((((ord_less @ B) @ X3) @ Y2) => (((ord_less @ A) @ (F @ X3)) @ (F @ Y2))))) => (((ord_less @ A) @ A5) @ (F @ C2))))))))))).
% 2.97/1.57 thf(fact_51_order__less__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A5:A,B3:A,F:(A > C),C2:C]: (((((ord_less @ A) @ A5) @ B3) => ((((ord_less @ C) @ (F @ B3)) @ C2) => ((! [X3:A,Y2:A]: (((((ord_less @ A) @ X3) @ Y2) => (((ord_less @ C) @ (F @ X3)) @ (F @ Y2))))) => (((ord_less @ C) @ (F @ A5)) @ C2)))))))))).
% 2.97/1.57 thf(fact_52_lt__ex, axiom, (! [A:$tType]: (((no_bot @ A) => (! [X:A]: ((? [Y2:A]: ((((ord_less @ A) @ Y2) @ X))))))))).
% 2.97/1.57 thf(fact_53_gt__ex, axiom, (! [A:$tType]: (((no_top @ A) => (! [X:A]: ((? [X_1:A]: ((((ord_less @ A) @ X) @ X_1))))))))).
% 2.97/1.57 thf(fact_54_neqE, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((X != Y) => ((~ ((((ord_less @ A) @ X) @ Y))) => (((ord_less @ A) @ Y) @ X))))))))).
% 2.97/1.57 thf(fact_55_neq__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((X != Y) = ((((ord_less @ A) @ X) @ Y) | (((ord_less @ A) @ Y) @ X))))))))).
% 2.97/1.57 thf(fact_56_order_Oasym, axiom, (! [A:$tType]: (((order @ A) => (! [A5:A,B3:A]: (((((ord_less @ A) @ A5) @ B3) => (~ ((((ord_less @ A) @ B3) @ A5)))))))))).
% 2.97/1.57 thf(fact_57_dense, axiom, (! [A:$tType]: (((dense_order @ A) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (? [Z2:A]: (((((ord_less @ A) @ X) @ Z2) & (((ord_less @ A) @ Z2) @ Y))))))))))).
% 2.97/1.57 thf(fact_58_less__imp__neq, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (X != Y)))))))).
% 2.97/1.57 thf(fact_59_strict__monoD, axiom, (! [B:$tType,A:$tType]: ((((order @ A) & (order @ B)) => (! [F:(A > B),X:A,Y:A]: (((((order_strict_mono @ A) @ B) @ F) => ((((ord_less @ A) @ X) @ Y) => (((ord_less @ B) @ (F @ X)) @ (F @ Y)))))))))).
% 2.97/1.57 thf(fact_60_strict__monoI, axiom, (! [B:$tType,A:$tType]: ((((order @ A) & (order @ B)) => (! [F:(A > B)]: (((! [X3:A,Y2:A]: (((((ord_less @ A) @ X3) @ Y2) => (((ord_less @ B) @ (F @ X3)) @ (F @ Y2))))) => (((order_strict_mono @ A) @ B) @ F)))))))).
% 2.97/1.57 thf(fact_61_less__asym, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (~ ((((ord_less @ A) @ Y) @ X)))))))))).
% 2.97/1.57 thf(fact_62_less__asym_H, axiom, (! [A:$tType]: (((preorder @ A) => (! [A5:A,B3:A]: (((((ord_less @ A) @ A5) @ B3) => (~ ((((ord_less @ A) @ B3) @ A5)))))))))).
% 2.97/1.57 thf(fact_63_less__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A,Z:A]: (((((ord_less @ A) @ X) @ Y) => ((((ord_less @ A) @ Y) @ Z) => (((ord_less @ A) @ X) @ Z))))))))).
% 2.97/1.57 thf(fact_64_less__linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) | ((X = Y) | (((ord_less @ A) @ Y) @ X))))))))).
% 2.97/1.57 thf(fact_65_less__irrefl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A]: ((~ ((((ord_less @ A) @ X) @ X))))))))).
% 2.97/1.57 thf(fact_66_ord__eq__less__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A5:A,B3:A,C2:A]: (((A5 = B3) => ((((ord_less @ A) @ B3) @ C2) => (((ord_less @ A) @ A5) @ C2))))))))).
% 2.97/1.57 thf(fact_67_ord__less__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A5:A,B3:A,C2:A]: (((((ord_less @ A) @ A5) @ B3) => ((B3 = C2) => (((ord_less @ A) @ A5) @ C2))))))))).
% 2.97/1.57 thf(fact_68_dual__order_Oasym, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A5:A]: (((((ord_less @ A) @ B3) @ A5) => (~ ((((ord_less @ A) @ A5) @ B3)))))))))).
% 2.97/1.57 thf(fact_69_less__imp__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (X != Y)))))))).
% 2.97/1.57 thf(fact_70_strict__mono__def, axiom, (! [B:$tType,A:$tType]: ((((order @ A) & (order @ B)) => (((order_strict_mono @ A) @ B) = (^ [F2:(A > B)]: ((! [X2:A,Y3:A]: (((((ord_less @ A) @ X2) @ Y3) => (((ord_less @ B) @ (F2 @ X2)) @ (F2 @ Y3)))))))))))).
% 2.97/1.57 thf(fact_71_less__not__sym, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (~ ((((ord_less @ A) @ Y) @ X)))))))))).
% 2.97/1.57 thf(fact_72_less__induct, axiom, (! [A:$tType]: (((wellorder @ A) => (! [P:(A > $o),A5:A]: (((! [X3:A]: (((! [Y4:A]: (((((ord_less @ A) @ Y4) @ X3) => (P @ Y4)))) => (P @ X3)))) => (P @ A5)))))))).
% 2.97/1.57 thf(fact_73_antisym__conv3, axiom, (! [A:$tType]: (((linorder @ A) => (! [Y:A,X:A]: (((~ ((((ord_less @ A) @ Y) @ X))) => ((~ ((((ord_less @ A) @ X) @ Y))) = (X = Y))))))))).
% 2.97/1.57 thf(fact_74_less__imp__not__eq2, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (Y != X)))))))).
% 2.97/1.57 thf(fact_75_less__imp__triv, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A,P:$o]: (((((ord_less @ A) @ X) @ Y) => ((((ord_less @ A) @ Y) @ X) => P)))))))).
% 2.97/1.57 thf(fact_76_linorder__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((~ ((((ord_less @ A) @ X) @ Y))) => ((X != Y) => (((ord_less @ A) @ Y) @ X))))))))).
% 2.97/1.57 thf(fact_77_strict__mono__eq, axiom, (! [B:$tType,A:$tType]: ((((linorder @ A) & (order @ B)) => (! [F:(A > B),X:A,Y:A]: (((((order_strict_mono @ A) @ B) @ F) => (((F @ X) = (F @ Y)) = (X = Y))))))))).
% 2.97/1.57 thf(fact_78_dual__order_Oirrefl, axiom, (! [A:$tType]: (((order @ A) => (! [A5:A]: ((~ ((((ord_less @ A) @ A5) @ A5))))))))).
% 2.97/1.57 thf(fact_79_order_Ostrict__trans, axiom, (! [A:$tType]: (((order @ A) => (! [A5:A,B3:A,C2:A]: (((((ord_less @ A) @ A5) @ B3) => ((((ord_less @ A) @ B3) @ C2) => (((ord_less @ A) @ A5) @ C2))))))))).
% 2.97/1.57 thf(fact_80_strict__mono__less, axiom, (! [B:$tType,A:$tType]: ((((linorder @ A) & (order @ B)) => (! [F:(A > B),X:A,Y:A]: (((((order_strict_mono @ A) @ B) @ F) => ((((ord_less @ B) @ (F @ X)) @ (F @ Y)) = (((ord_less @ A) @ X) @ Y))))))))).
% 2.97/1.57 thf(fact_81_less__imp__not__less, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (~ ((((ord_less @ A) @ Y) @ X)))))))))).
% 2.97/1.57 thf(fact_82_exists__least__iff, axiom, (! [A:$tType]: (((wellorder @ A) => ((^ [P2:(A > $o)]: ((? [X5:A]: ((P2 @ X5))))) = (^ [P3:(A > $o)]: ((? [N3:A]: (((P3 @ N3) & (! [M2:A]: (((((ord_less @ A) @ M2) @ N3) => (~ ((P3 @ M2)))))))))))))))).
% 2.97/1.57 thf(fact_83_linorder__less__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > (A > $o)),A5:A,B3:A]: (((! [A4:A,B4:A]: (((((ord_less @ A) @ A4) @ B4) => ((P @ A4) @ B4)))) => ((! [A4:A]: (((P @ A4) @ A4))) => ((! [A4:A,B4:A]: ((((P @ B4) @ A4) => ((P @ A4) @ B4)))) => ((P @ A5) @ B3)))))))))).
% 2.97/1.57 thf(fact_84_dual__order_Ostrict__trans, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A5:A,C2:A]: (((((ord_less @ A) @ B3) @ A5) => ((((ord_less @ A) @ C2) @ B3) => (((ord_less @ A) @ C2) @ A5))))))))).
% 2.97/1.57 thf(fact_85_not__less__iff__gr__or__eq, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((~ ((((ord_less @ A) @ X) @ Y))) = ((((ord_less @ A) @ Y) @ X) | (X = Y))))))))).
% 2.97/1.57 thf(fact_86_order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [A5:A,B3:A]: (((((ord_less @ A) @ A5) @ B3) => (A5 != B3)))))))).
% 2.97/1.57 thf(fact_87_dual__order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A5:A]: (((((ord_less @ A) @ B3) @ A5) => (A5 != B3)))))))).
% 2.97/1.57 thf(fact_88_strict__mono__on__eqD, axiom, (! [D:$tType,C:$tType]: ((((linorder @ C) & (preorder @ D)) => (! [F:(C > D),A2:(set @ C),X:C,Y:C]: ((((((strict_mono_on @ C) @ D) @ F) @ A2) => (((F @ X) = (F @ Y)) => ((((member @ C) @ X) @ A2) => ((((member @ C) @ Y) @ A2) => (Y = X))))))))))).
% 2.97/1.57 thf(fact_89_inj__on__strict__subset, axiom, (! [B:$tType,A:$tType,F:(A > B),B2:(set @ A),A2:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ B2) => ((((ord_less @ (set @ A)) @ A2) @ B2) => (((ord_less @ (set @ B)) @ ((((image @ A) @ B) @ F) @ A2)) @ ((((image @ A) @ B) @ F) @ B2))))))).
% 2.97/1.57 thf(fact_90_inj__on__the__inv__into, axiom, (! [B:$tType,A:$tType,F:(A > B),A2:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ A2) => ((((inj_on @ B) @ A) @ ((((the_inv_into @ A) @ B) @ A2) @ F)) @ ((((image @ A) @ B) @ F) @ A2)))))).
% 2.97/1.57 thf(fact_91_f__the__inv__into__f, axiom, (! [A:$tType,B:$tType,F:(A > B),A2:(set @ A),Y:B]: ((((((inj_on @ A) @ B) @ F) @ A2) => ((((member @ B) @ Y) @ ((((image @ A) @ B) @ F) @ A2)) => ((F @ (((((the_inv_into @ A) @ B) @ A2) @ F) @ Y)) = Y)))))).
% 2.97/1.57 thf(fact_92_strict__mono__inv, axiom, (! [A:$tType,B:$tType]: ((((linorder @ B) & (linorder @ A)) => (! [F:(A > B),G:(B > A)]: (((((order_strict_mono @ A) @ B) @ F) => ((((((image @ A) @ B) @ F) @ (top_top @ (set @ A))) = (top_top @ (set @ B))) => ((! [X3:A]: (((G @ (F @ X3)) = X3))) => (((order_strict_mono @ B) @ A) @ G)))))))))).
% 2.97/1.57 thf(fact_93_surjD, axiom, (! [A:$tType,B:$tType,F:(B > A),Y:A]: (((((((image @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (? [X3:B]: ((Y = (F @ X3)))))))).
% 2.97/1.57 thf(fact_94_surjE, axiom, (! [A:$tType,B:$tType,F:(B > A),Y:A]: (((((((image @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (~ ((! [X3:B]: ((Y != (F @ X3)))))))))).
% 2.97/1.57 thf(fact_95_surjI, axiom, (! [B:$tType,A:$tType,G:(B > A),F:(A > B)]: (((! [X3:A]: (((G @ (F @ X3)) = X3))) => (((((image @ B) @ A) @ G) @ (top_top @ (set @ B))) = (top_top @ (set @ A))))))).
% 2.97/1.57 thf(fact_96_rangeI, axiom, (! [A:$tType,B:$tType,F:(B > A),X:B]: ((((member @ A) @ (F @ X)) @ ((((image @ B) @ A) @ F) @ (top_top @ (set @ B))))))).
% 2.97/1.57 thf(fact_97_surj__def, axiom, (! [B:$tType,A:$tType,F:(B > A)]: (((((((image @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) = (! [Y3:A]: ((? [X2:B]: ((Y3 = (F @ X2)))))))))).
% 2.97/1.57 thf(fact_98_range__eqI, axiom, (! [A:$tType,B:$tType,B3:A,F:(B > A),X:B]: (((B3 = (F @ X)) => (((member @ A) @ B3) @ ((((image @ B) @ A) @ F) @ (top_top @ (set @ B)))))))).
% 2.97/1.57 thf(fact_99_inj__on__image__iff, axiom, (! [B:$tType,A:$tType,A2:(set @ A),G:(A > B),F:(A > A)]: (((! [X3:A]: (((((member @ A) @ X3) @ A2) => (! [Xa:A]: (((((member @ A) @ Xa) @ A2) => (((G @ (F @ X3)) = (G @ (F @ Xa))) = ((G @ X3) = (G @ Xa))))))))) => (((((inj_on @ A) @ A) @ F) @ A2) => (((((inj_on @ A) @ B) @ G) @ ((((image @ A) @ A) @ F) @ A2)) = ((((inj_on @ A) @ B) @ G) @ A2))))))).
% 2.97/1.57 thf(fact_100_top_Oextremum__strict, axiom, (! [A:$tType]: (((order_top @ A) => (! [A5:A]: ((~ ((((ord_less @ A) @ (top_top @ A)) @ A5))))))))).
% 2.97/1.57 thf(fact_101_top_Onot__eq__extremum, axiom, (! [A:$tType]: (((order_top @ A) => (! [A5:A]: (((A5 != (top_top @ A)) = (((ord_less @ A) @ A5) @ (top_top @ A))))))))).
% 2.97/1.57 thf(fact_102_inj__on__add, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [A5:A,A2:(set @ A)]: (((((inj_on @ A) @ A) @ ((plus_plus @ A) @ A5)) @ A2))))))).
% 2.97/1.57 thf(fact_103_the__inv__into__f__f, axiom, (! [B:$tType,A:$tType,F:(A > B),A2:(set @ A),X:A]: ((((((inj_on @ A) @ B) @ F) @ A2) => ((((member @ A) @ X) @ A2) => ((((((the_inv_into @ A) @ B) @ A2) @ F) @ (F @ X)) = X)))))).
% 2.97/1.57 thf(fact_104_the__inv__into__f__eq, axiom, (! [B:$tType,A:$tType,F:(A > B),A2:(set @ A),X:A,Y:B]: ((((((inj_on @ A) @ B) @ F) @ A2) => (((F @ X) = Y) => ((((member @ A) @ X) @ A2) => ((((((the_inv_into @ A) @ B) @ A2) @ F) @ Y) = X))))))).
% 2.97/1.57 thf(fact_105_add__less__cancel__right, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [A5:A,C2:A,B3:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ A5) @ C2)) @ (((plus_plus @ A) @ B3) @ C2)) = (((ord_less @ A) @ A5) @ B3)))))))).
% 2.97/1.57 thf(fact_106_add__less__cancel__left, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [C2:A,A5:A,B3:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ C2) @ A5)) @ (((plus_plus @ A) @ C2) @ B3)) = (((ord_less @ A) @ A5) @ B3)))))))).
% 2.97/1.57 thf(fact_107_add__right__cancel, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [B3:A,A5:A,C2:A]: ((((((plus_plus @ A) @ B3) @ A5) = (((plus_plus @ A) @ C2) @ A5)) = (B3 = C2)))))))).
% 2.97/1.57 thf(fact_108_add__left__cancel, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [A5:A,B3:A,C2:A]: ((((((plus_plus @ A) @ A5) @ B3) = (((plus_plus @ A) @ A5) @ C2)) = (B3 = C2)))))))).
% 2.97/1.57 thf(fact_109_inj__apfst, axiom, (! [B:$tType,C:$tType,A:$tType,F:(A > C)]: ((((((inj_on @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ B)) @ ((((product_apfst @ A) @ C) @ B) @ F)) @ (top_top @ (set @ ((product_prod @ A) @ B)))) = ((((inj_on @ A) @ C) @ F) @ (top_top @ (set @ A))))))).
% 2.97/1.57 thf(fact_110_add__less__imp__less__right, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [A5:A,C2:A,B3:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ A5) @ C2)) @ (((plus_plus @ A) @ B3) @ C2)) => (((ord_less @ A) @ A5) @ B3)))))))).
% 2.97/1.57 thf(fact_111_add__less__imp__less__left, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [C2:A,A5:A,B3:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ C2) @ A5)) @ (((plus_plus @ A) @ C2) @ B3)) => (((ord_less @ A) @ A5) @ B3)))))))).
% 2.97/1.57 thf(fact_112_add__strict__right__mono, axiom, (! [A:$tType]: (((ordere223160158up_add @ A) => (! [A5:A,B3:A,C2:A]: (((((ord_less @ A) @ A5) @ B3) => (((ord_less @ A) @ (((plus_plus @ A) @ A5) @ C2)) @ (((plus_plus @ A) @ B3) @ C2))))))))).
% 2.97/1.57 thf(fact_113_add__strict__left__mono, axiom, (! [A:$tType]: (((ordere223160158up_add @ A) => (! [A5:A,B3:A,C2:A]: (((((ord_less @ A) @ A5) @ B3) => (((ord_less @ A) @ (((plus_plus @ A) @ C2) @ A5)) @ (((plus_plus @ A) @ C2) @ B3))))))))).
% 2.97/1.57 thf(fact_114_psubsetD, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C2:A]: (((((ord_less @ (set @ A)) @ A2) @ B2) => ((((member @ A) @ C2) @ A2) => (((member @ A) @ C2) @ B2)))))).
% 2.97/1.57 thf(fact_115_psubset__trans, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C3:(set @ A)]: (((((ord_less @ (set @ A)) @ A2) @ B2) => ((((ord_less @ (set @ A)) @ B2) @ C3) => (((ord_less @ (set @ A)) @ A2) @ C3)))))).
% 2.97/1.57 thf(fact_116_apsnd__apfst__commute, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,F:(C > B),G:(D > A),P4:((product_prod @ D) @ C)]: (((((((product_apsnd @ C) @ B) @ A) @ F) @ (((((product_apfst @ D) @ A) @ C) @ G) @ P4)) = (((((product_apfst @ D) @ A) @ B) @ G) @ (((((product_apsnd @ C) @ B) @ D) @ F) @ P4)))))).
% 2.97/1.57 thf(fact_117_ab__semigroup__add__class_Oadd__ac_I1_J, axiom, (! [A:$tType]: (((ab_semigroup_add @ A) => (! [A5:A,B3:A,C2:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ A5) @ B3)) @ C2) = (((plus_plus @ A) @ A5) @ (((plus_plus @ A) @ B3) @ C2))))))))).
% 2.97/1.57 thf(fact_118_add__mono__thms__linordered__semiring_I4_J, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((I = J) & (K = L)) => ((((plus_plus @ A) @ I) @ K) = (((plus_plus @ A) @ J) @ L))))))))).
% 2.97/1.57 thf(fact_119_group__cancel_Oadd1, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [A2:A,K:A,A5:A,B3:A]: (((A2 = (((plus_plus @ A) @ K) @ A5)) => ((((plus_plus @ A) @ A2) @ B3) = (((plus_plus @ A) @ K) @ (((plus_plus @ A) @ A5) @ B3)))))))))).
% 2.97/1.57 thf(fact_120_group__cancel_Oadd2, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [B2:A,K:A,B3:A,A5:A]: (((B2 = (((plus_plus @ A) @ K) @ B3)) => ((((plus_plus @ A) @ A5) @ B2) = (((plus_plus @ A) @ K) @ (((plus_plus @ A) @ A5) @ B3)))))))))).
% 2.97/1.57 thf(fact_121_add_Oassoc, axiom, (! [A:$tType]: (((semigroup_add @ A) => (! [A5:A,B3:A,C2:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ A5) @ B3)) @ C2) = (((plus_plus @ A) @ A5) @ (((plus_plus @ A) @ B3) @ C2))))))))).
% 2.97/1.57 thf(fact_122_add_Oleft__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [A5:A,B3:A,C2:A]: ((((((plus_plus @ A) @ A5) @ B3) = (((plus_plus @ A) @ A5) @ C2)) = (B3 = C2)))))))).
% 2.97/1.57 thf(fact_123_add_Oright__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [B3:A,A5:A,C2:A]: ((((((plus_plus @ A) @ B3) @ A5) = (((plus_plus @ A) @ C2) @ A5)) = (B3 = C2)))))))).
% 2.97/1.57 thf(fact_124_add_Ocommute, axiom, (! [A:$tType]: (((ab_semigroup_add @ A) => ((plus_plus @ A) = (^ [A6:A,B5:A]: ((((plus_plus @ A) @ B5) @ A6)))))))).
% 2.97/1.57 thf(fact_125_add_Oleft__commute, axiom, (! [A:$tType]: (((ab_semigroup_add @ A) => (! [B3:A,A5:A,C2:A]: (((((plus_plus @ A) @ B3) @ (((plus_plus @ A) @ A5) @ C2)) = (((plus_plus @ A) @ A5) @ (((plus_plus @ A) @ B3) @ C2))))))))).
% 2.97/1.57 thf(fact_126_add__left__imp__eq, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [A5:A,B3:A,C2:A]: ((((((plus_plus @ A) @ A5) @ B3) = (((plus_plus @ A) @ A5) @ C2)) => (B3 = C2)))))))).
% 2.97/1.57 thf(fact_127_add__right__imp__eq, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [B3:A,A5:A,C2:A]: ((((((plus_plus @ A) @ B3) @ A5) = (((plus_plus @ A) @ C2) @ A5)) => (B3 = C2)))))))).
% 2.97/1.57 thf(fact_128_add__mono__thms__linordered__field_I5_J, axiom, (! [A:$tType]: (((ordere223160158up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((((ord_less @ A) @ I) @ J) & (((ord_less @ A) @ K) @ L)) => (((ord_less @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.97/1.57 thf(fact_129_add__mono__thms__linordered__field_I2_J, axiom, (! [A:$tType]: (((ordere223160158up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((I = J) & (((ord_less @ A) @ K) @ L)) => (((ord_less @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.97/1.57 thf(fact_130_add__mono__thms__linordered__field_I1_J, axiom, (! [A:$tType]: (((ordere223160158up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((((ord_less @ A) @ I) @ J) & (K = L)) => (((ord_less @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.97/1.57 thf(fact_131_add__strict__mono, axiom, (! [A:$tType]: (((strict2144017051up_add @ A) => (! [A5:A,B3:A,C2:A,D2:A]: (((((ord_less @ A) @ A5) @ B3) => ((((ord_less @ A) @ C2) @ D2) => (((ord_less @ A) @ (((plus_plus @ A) @ A5) @ C2)) @ (((plus_plus @ A) @ B3) @ D2)))))))))).
% 2.97/1.57 thf(fact_132_top__conj_I1_J, axiom, (! [A:$tType,X:A,P:$o]: (((((top_top @ (A > $o)) @ X) & P) = P)))).
% 2.97/1.57 thf(fact_133_top__conj_I2_J, axiom, (! [A:$tType,P:$o,X:A]: (((P & ((top_top @ (A > $o)) @ X)) = P)))).
% 2.97/1.57 thf(fact_134_tv_Osimps_I8_J, axiom, (! [A:$tType,F1:($o > A),F22:(nat > A),X22:nat]: ((((((paraco152590079rec_tv @ A) @ F1) @ F22) @ (paraco676387099_Indet @ X22)) = (F22 @ X22))))).
% 2.97/1.57 thf(fact_135_the__inv__into__into, axiom, (! [B:$tType,A:$tType,F:(A > B),A2:(set @ A),X:B,B2:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ A2) => ((((member @ B) @ X) @ ((((image @ A) @ B) @ F) @ A2)) => ((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (((member @ A) @ (((((the_inv_into @ A) @ B) @ A2) @ F) @ X)) @ B2))))))).
% 2.97/1.57 thf(fact_136_prod_Oinj__map, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,F1:(A > C),F22:(B > D)]: ((((((inj_on @ A) @ C) @ F1) @ (top_top @ (set @ A))) => (((((inj_on @ B) @ D) @ F22) @ (top_top @ (set @ B))) => ((((inj_on @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ D)) @ ((((((product_map_prod @ A) @ C) @ B) @ D) @ F1) @ F22)) @ (top_top @ (set @ ((product_prod @ A) @ B))))))))).
% 2.97/1.57 thf(fact_137_order__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A]: ((((ord_less_eq @ A) @ X) @ X))))))).
% 2.97/1.57 thf(fact_138_subsetI, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((! [X3:A]: (((((member @ A) @ X3) @ A2) => (((member @ A) @ X3) @ B2)))) => (((ord_less_eq @ (set @ A)) @ A2) @ B2))))).
% 2.97/1.57 thf(fact_139_subset__antisym, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((ord_less_eq @ (set @ A)) @ B2) @ A2) => (A2 = B2)))))).
% 2.97/1.57 thf(fact_140_add__le__cancel__right, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [A5:A,C2:A,B3:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ A5) @ C2)) @ (((plus_plus @ A) @ B3) @ C2)) = (((ord_less_eq @ A) @ A5) @ B3)))))))).
% 2.97/1.57 thf(fact_141_add__le__cancel__left, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [C2:A,A5:A,B3:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ C2) @ A5)) @ (((plus_plus @ A) @ C2) @ B3)) = (((ord_less_eq @ A) @ A5) @ B3)))))))).
% 2.97/1.57 thf(fact_142_psubsetI, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((A2 != B2) => (((ord_less @ (set @ A)) @ A2) @ B2)))))).
% 2.97/1.57 thf(fact_143_psubsetE, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((ord_less @ (set @ A)) @ A2) @ B2) => (~ (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (((ord_less_eq @ (set @ A)) @ B2) @ A2)))))))).
% 2.97/1.57 thf(fact_144_psubset__eq, axiom, (! [A:$tType]: (((ord_less @ (set @ A)) = (^ [A3:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B6) & (A3 != B6)))))))).
% 2.97/1.57 thf(fact_145_psubset__imp__subset, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((ord_less @ (set @ A)) @ A2) @ B2) => (((ord_less_eq @ (set @ A)) @ A2) @ B2))))).
% 2.97/1.57 thf(fact_146_psubset__subset__trans, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C3:(set @ A)]: (((((ord_less @ (set @ A)) @ A2) @ B2) => ((((ord_less_eq @ (set @ A)) @ B2) @ C3) => (((ord_less @ (set @ A)) @ A2) @ C3)))))).
% 2.97/1.57 thf(fact_147_subset__not__subset__eq, axiom, (! [A:$tType]: (((ord_less @ (set @ A)) = (^ [A3:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B6) & (~ ((((ord_less_eq @ (set @ A)) @ B6) @ A3)))))))))).
% 2.97/1.57 thf(fact_148_subset__psubset__trans, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((ord_less @ (set @ A)) @ B2) @ C3) => (((ord_less @ (set @ A)) @ A2) @ C3)))))).
% 2.97/1.57 thf(fact_149_subset__iff__psubset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A3:(set @ A),B6:(set @ A)]: (((((ord_less @ (set @ A)) @ A3) @ B6) | (A3 = B6)))))))).
% 2.97/1.57 thf(fact_150_add__mono__thms__linordered__semiring_I3_J, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((((ord_less_eq @ A) @ I) @ J) & (K = L)) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.97/1.57 thf(fact_151_add__mono__thms__linordered__semiring_I2_J, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((I = J) & (((ord_less_eq @ A) @ K) @ L)) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.97/1.57 thf(fact_152_add__mono__thms__linordered__semiring_I1_J, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((((ord_less_eq @ A) @ I) @ J) & (((ord_less_eq @ A) @ K) @ L)) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.97/1.57 thf(fact_153_add__mono, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [A5:A,B3:A,C2:A,D2:A]: (((((ord_less_eq @ A) @ A5) @ B3) => ((((ord_less_eq @ A) @ C2) @ D2) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ A5) @ C2)) @ (((plus_plus @ A) @ B3) @ D2)))))))))).
% 2.97/1.57 thf(fact_154_add__left__mono, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [A5:A,B3:A,C2:A]: (((((ord_less_eq @ A) @ A5) @ B3) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ C2) @ A5)) @ (((plus_plus @ A) @ C2) @ B3))))))))).
% 2.97/1.57 thf(fact_155_less__eqE, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [A5:A,B3:A]: (((((ord_less_eq @ A) @ A5) @ B3) => (~ ((! [C4:A]: ((B3 != (((plus_plus @ A) @ A5) @ C4))))))))))))).
% 2.97/1.57 thf(fact_156_add__right__mono, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [A5:A,B3:A,C2:A]: (((((ord_less_eq @ A) @ A5) @ B3) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ A5) @ C2)) @ (((plus_plus @ A) @ B3) @ C2))))))))).
% 2.97/1.57 thf(fact_157_le__iff__add, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => ((ord_less_eq @ A) = (^ [A6:A,B5:A]: ((? [C5:A]: ((B5 = (((plus_plus @ A) @ A6) @ C5))))))))))).
% 2.97/1.57 thf(fact_158_add__le__imp__le__left, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [C2:A,A5:A,B3:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ C2) @ A5)) @ (((plus_plus @ A) @ C2) @ B3)) => (((ord_less_eq @ A) @ A5) @ B3)))))))).
% 2.97/1.57 thf(fact_159_add__le__imp__le__right, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [A5:A,C2:A,B3:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ A5) @ C2)) @ (((plus_plus @ A) @ B3) @ C2)) => (((ord_less_eq @ A) @ A5) @ B3)))))))).
% 2.97/1.57 thf(fact_160_order_Onot__eq__order__implies__strict, axiom, (! [A:$tType]: (((order @ A) => (! [A5:A,B3:A]: (((A5 != B3) => ((((ord_less_eq @ A) @ A5) @ B3) => (((ord_less @ A) @ A5) @ B3))))))))).
% 2.97/1.57 thf(fact_161_dual__order_Ostrict__implies__order, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A5:A]: (((((ord_less @ A) @ B3) @ A5) => (((ord_less_eq @ A) @ B3) @ A5)))))))).
% 2.97/1.57 thf(fact_162_dual__order_Ostrict__iff__order, axiom, (! [A:$tType]: (((order @ A) => ((ord_less @ A) = (^ [B5:A,A6:A]: (((((ord_less_eq @ A) @ B5) @ A6) & (A6 != B5))))))))).
% 2.97/1.57 thf(fact_163_dual__order_Oorder__iff__strict, axiom, (! [A:$tType]: (((order @ A) => ((ord_less_eq @ A) = (^ [B5:A,A6:A]: (((((ord_less @ A) @ B5) @ A6) | (A6 = B5))))))))).
% 2.97/1.57 thf(fact_164_order_Ostrict__implies__order, axiom, (! [A:$tType]: (((order @ A) => (! [A5:A,B3:A]: (((((ord_less @ A) @ A5) @ B3) => (((ord_less_eq @ A) @ A5) @ B3)))))))).
% 2.97/1.57 thf(fact_165_dense__le__bounded, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [X:A,Y:A,Z:A]: (((((ord_less @ A) @ X) @ Y) => ((! [W:A]: (((((ord_less @ A) @ X) @ W) => ((((ord_less @ A) @ W) @ Y) => (((ord_less_eq @ A) @ W) @ Z))))) => (((ord_less_eq @ A) @ Y) @ Z))))))))).
% 2.97/1.57 thf(fact_166_dense__ge__bounded, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [Z:A,X:A,Y:A]: (((((ord_less @ A) @ Z) @ X) => ((! [W:A]: (((((ord_less @ A) @ Z) @ W) => ((((ord_less @ A) @ W) @ X) => (((ord_less_eq @ A) @ Y) @ W))))) => (((ord_less_eq @ A) @ Y) @ Z))))))))).
% 2.97/1.57 thf(fact_167_dual__order_Ostrict__trans2, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A5:A,C2:A]: (((((ord_less @ A) @ B3) @ A5) => ((((ord_less_eq @ A) @ C2) @ B3) => (((ord_less @ A) @ C2) @ A5))))))))).
% 2.97/1.57 thf(fact_168_dual__order_Ostrict__trans1, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A5:A,C2:A]: (((((ord_less_eq @ A) @ B3) @ A5) => ((((ord_less @ A) @ C2) @ B3) => (((ord_less @ A) @ C2) @ A5))))))))).
% 2.97/1.57 thf(fact_169_order_Ostrict__iff__order, axiom, (! [A:$tType]: (((order @ A) => ((ord_less @ A) = (^ [A6:A,B5:A]: (((((ord_less_eq @ A) @ A6) @ B5) & (A6 != B5))))))))).
% 2.97/1.57 thf(fact_170_order_Oorder__iff__strict, axiom, (! [A:$tType]: (((order @ A) => ((ord_less_eq @ A) = (^ [A6:A,B5:A]: (((((ord_less @ A) @ A6) @ B5) | (A6 = B5))))))))).
% 2.97/1.57 thf(fact_171_order_Ostrict__trans2, axiom, (! [A:$tType]: (((order @ A) => (! [A5:A,B3:A,C2:A]: (((((ord_less @ A) @ A5) @ B3) => ((((ord_less_eq @ A) @ B3) @ C2) => (((ord_less @ A) @ A5) @ C2))))))))).
% 2.97/1.57 thf(fact_172_order_Ostrict__trans1, axiom, (! [A:$tType]: (((order @ A) => (! [A5:A,B3:A,C2:A]: (((((ord_less_eq @ A) @ A5) @ B3) => ((((ord_less @ A) @ B3) @ C2) => (((ord_less @ A) @ A5) @ C2))))))))).
% 2.97/1.57 thf(fact_173_not__le__imp__less, axiom, (! [A:$tType]: (((linorder @ A) => (! [Y:A,X:A]: (((~ ((((ord_less_eq @ A) @ Y) @ X))) => (((ord_less @ A) @ X) @ Y)))))))).
% 2.97/1.57 thf(fact_174_less__le__not__le, axiom, (! [A:$tType]: (((preorder @ A) => ((ord_less @ A) = (^ [X2:A,Y3:A]: (((((ord_less_eq @ A) @ X2) @ Y3) & (~ ((((ord_less_eq @ A) @ Y3) @ X2))))))))))).
% 2.97/1.57 thf(fact_175_le__imp__less__or__eq, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less @ A) @ X) @ Y) | (X = Y))))))))).
% 2.97/1.57 thf(fact_176_le__less__linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) | (((ord_less @ A) @ Y) @ X)))))))).
% 2.97/1.57 thf(fact_177_dense__le, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [Y:A,Z:A]: (((! [X3:A]: (((((ord_less @ A) @ X3) @ Y) => (((ord_less_eq @ A) @ X3) @ Z)))) => (((ord_less_eq @ A) @ Y) @ Z)))))))).
% 2.97/1.57 thf(fact_178_dense__ge, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [Z:A,Y:A]: (((! [X3:A]: (((((ord_less @ A) @ Z) @ X3) => (((ord_less_eq @ A) @ Y) @ X3)))) => (((ord_less_eq @ A) @ Y) @ Z)))))))).
% 2.97/1.57 thf(fact_179_less__le__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A,Z:A]: (((((ord_less @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z) => (((ord_less @ A) @ X) @ Z))))))))).
% 2.97/1.57 thf(fact_180_le__less__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A,Z:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less @ A) @ Y) @ Z) => (((ord_less @ A) @ X) @ Z))))))))).
% 2.97/1.57 thf(fact_181_less__imp__le, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (((ord_less_eq @ A) @ X) @ Y)))))))).
% 2.97/1.57 thf(fact_182_antisym__conv2, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((~ ((((ord_less @ A) @ X) @ Y))) = (X = Y))))))))).
% 2.97/1.57 thf(fact_183_antisym__conv1, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y:A]: (((~ ((((ord_less @ A) @ X) @ Y))) => ((((ord_less_eq @ A) @ X) @ Y) = (X = Y))))))))).
% 2.97/1.57 thf(fact_184_le__neq__trans, axiom, (! [A:$tType]: (((order @ A) => (! [A5:A,B3:A]: (((((ord_less_eq @ A) @ A5) @ B3) => ((A5 != B3) => (((ord_less @ A) @ A5) @ B3))))))))).
% 2.97/1.57 thf(fact_185_not__less, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((~ ((((ord_less @ A) @ X) @ Y))) = (((ord_less_eq @ A) @ Y) @ X)))))))).
% 2.97/1.57 thf(fact_186_not__le, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((~ ((((ord_less_eq @ A) @ X) @ Y))) = (((ord_less @ A) @ Y) @ X)))))))).
% 2.97/1.57 thf(fact_187_order__less__le__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A5:A,B3:A,F:(A > C),C2:C]: (((((ord_less @ A) @ A5) @ B3) => ((((ord_less_eq @ C) @ (F @ B3)) @ C2) => ((! [X3:A,Y2:A]: (((((ord_less @ A) @ X3) @ Y2) => (((ord_less @ C) @ (F @ X3)) @ (F @ Y2))))) => (((ord_less @ C) @ (F @ A5)) @ C2)))))))))).
% 2.97/1.57 thf(fact_188_order__less__le__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A5:A,F:(B > A),B3:B,C2:B]: (((((ord_less @ A) @ A5) @ (F @ B3)) => ((((ord_less_eq @ B) @ B3) @ C2) => ((! [X3:B,Y2:B]: (((((ord_less_eq @ B) @ X3) @ Y2) => (((ord_less_eq @ A) @ (F @ X3)) @ (F @ Y2))))) => (((ord_less @ A) @ A5) @ (F @ C2))))))))))).
% 2.97/1.57 thf(fact_189_order__le__less__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A5:A,B3:A,F:(A > C),C2:C]: (((((ord_less_eq @ A) @ A5) @ B3) => ((((ord_less @ C) @ (F @ B3)) @ C2) => ((! [X3:A,Y2:A]: (((((ord_less_eq @ A) @ X3) @ Y2) => (((ord_less_eq @ C) @ (F @ X3)) @ (F @ Y2))))) => (((ord_less @ C) @ (F @ A5)) @ C2)))))))))).
% 2.97/1.57 thf(fact_190_order__le__less__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A5:A,F:(B > A),B3:B,C2:B]: (((((ord_less_eq @ A) @ A5) @ (F @ B3)) => ((((ord_less @ B) @ B3) @ C2) => ((! [X3:B,Y2:B]: (((((ord_less @ B) @ X3) @ Y2) => (((ord_less @ A) @ (F @ X3)) @ (F @ Y2))))) => (((ord_less @ A) @ A5) @ (F @ C2))))))))))).
% 2.97/1.57 thf(fact_191_less__le, axiom, (! [A:$tType]: (((order @ A) => ((ord_less @ A) = (^ [X2:A,Y3:A]: (((((ord_less_eq @ A) @ X2) @ Y3) & (X2 != Y3))))))))).
% 2.97/1.57 thf(fact_192_le__less, axiom, (! [A:$tType]: (((order @ A) => ((ord_less_eq @ A) = (^ [X2:A,Y3:A]: (((((ord_less @ A) @ X2) @ Y3) | (X2 = Y3))))))))).
% 2.97/1.57 thf(fact_193_leI, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((~ ((((ord_less @ A) @ X) @ Y))) => (((ord_less_eq @ A) @ Y) @ X)))))))).
% 2.97/1.57 thf(fact_194_leD, axiom, (! [A:$tType]: (((order @ A) => (! [Y:A,X:A]: (((((ord_less_eq @ A) @ Y) @ X) => (~ ((((ord_less @ A) @ X) @ Y)))))))))).
% 2.97/1.57 thf(fact_195_subset__image__iff, axiom, (! [A:$tType,B:$tType,B2:(set @ A),F:(B > A),A2:(set @ B)]: (((((ord_less_eq @ (set @ A)) @ B2) @ ((((image @ B) @ A) @ F) @ A2)) = (? [AA:(set @ B)]: (((((ord_less_eq @ (set @ B)) @ AA) @ A2) & (B2 = ((((image @ B) @ A) @ F) @ AA))))))))).
% 2.97/1.57 thf(fact_196_image__subset__iff, axiom, (! [A:$tType,B:$tType,F:(B > A),A2:(set @ B),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((((image @ B) @ A) @ F) @ A2)) @ B2) = (! [X2:B]: (((((member @ B) @ X2) @ A2) => (((member @ A) @ (F @ X2)) @ B2)))))))).
% 2.97/1.57 thf(fact_197_subset__imageE, axiom, (! [A:$tType,B:$tType,B2:(set @ A),F:(B > A),A2:(set @ B)]: (((((ord_less_eq @ (set @ A)) @ B2) @ ((((image @ B) @ A) @ F) @ A2)) => (~ ((! [C6:(set @ B)]: (((((ord_less_eq @ (set @ B)) @ C6) @ A2) => (B2 != ((((image @ B) @ A) @ F) @ C6))))))))))).
% 2.97/1.57 thf(fact_198_image__subsetI, axiom, (! [A:$tType,B:$tType,A2:(set @ A),F:(A > B),B2:(set @ B)]: (((! [X3:A]: (((((member @ A) @ X3) @ A2) => (((member @ B) @ (F @ X3)) @ B2)))) => (((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F) @ A2)) @ B2))))).
% 2.97/1.57 thf(fact_199_image__mono, axiom, (! [B:$tType,A:$tType,A2:(set @ A),B2:(set @ A),F:(A > B)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F) @ A2)) @ ((((image @ A) @ B) @ F) @ B2)))))).
% 2.97/1.57 thf(fact_200_top_Oextremum__uniqueI, axiom, (! [A:$tType]: (((order_top @ A) => (! [A5:A]: (((((ord_less_eq @ A) @ (top_top @ A)) @ A5) => (A5 = (top_top @ A))))))))).
% 2.97/1.57 thf(fact_201_top_Oextremum__unique, axiom, (! [A:$tType]: (((order_top @ A) => (! [A5:A]: (((((ord_less_eq @ A) @ (top_top @ A)) @ A5) = (A5 = (top_top @ A))))))))).
% 2.97/1.57 thf(fact_202_top__greatest, axiom, (! [A:$tType]: (((order_top @ A) => (! [A5:A]: ((((ord_less_eq @ A) @ A5) @ (top_top @ A)))))))).
% 2.97/1.57 thf(fact_203_subset__UNIV, axiom, (! [A:$tType,A2:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A2) @ (top_top @ (set @ A)))))).
% 2.97/1.57 thf(fact_204_subset__inj__on, axiom, (! [B:$tType,A:$tType,F:(A > B),B2:(set @ A),A2:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ B2) => ((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((inj_on @ A) @ B) @ F) @ A2)))))).
% 2.97/1.57 thf(fact_205_inj__on__subset, axiom, (! [B:$tType,A:$tType,F:(A > B),A2:(set @ A),B2:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ A2) => ((((ord_less_eq @ (set @ A)) @ B2) @ A2) => ((((inj_on @ A) @ B) @ F) @ B2)))))).
% 2.97/1.57 thf(fact_206_in__mono, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),X:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((member @ A) @ X) @ A2) => (((member @ A) @ X) @ B2)))))).
% 2.97/1.57 thf(fact_207_subsetD, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C2:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((member @ A) @ C2) @ A2) => (((member @ A) @ C2) @ B2)))))).
% 2.97/1.57 thf(fact_208_equalityE, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((A2 = B2) => (~ (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (~ ((((ord_less_eq @ (set @ A)) @ B2) @ A2)))))))))).
% 2.97/1.57 thf(fact_209_subset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A3:(set @ A),B6:(set @ A)]: ((! [X2:A]: (((((member @ A) @ X2) @ A3) => (((member @ A) @ X2) @ B6)))))))))).
% 2.97/1.57 thf(fact_210_equalityD1, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((A2 = B2) => (((ord_less_eq @ (set @ A)) @ A2) @ B2))))).
% 2.97/1.57 thf(fact_211_equalityD2, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((A2 = B2) => (((ord_less_eq @ (set @ A)) @ B2) @ A2))))).
% 2.97/1.57 thf(fact_212_subset__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A3:(set @ A),B6:(set @ A)]: ((! [T:A]: (((((member @ A) @ T) @ A3) => (((member @ A) @ T) @ B6)))))))))).
% 2.97/1.57 thf(fact_213_subset__refl, axiom, (! [A:$tType,A2:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A2) @ A2)))).
% 2.97/1.57 thf(fact_214_Collect__mono, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P @ X3) => (Q @ X3)))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)))))).
% 2.97/1.57 thf(fact_215_subset__trans, axiom, (! [A:$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)))))).
% 2.97/1.57 thf(fact_216_le__funD, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B),X:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X)) @ (G @ X))))))))).
% 2.97/1.57 thf(fact_217_le__funE, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B),X:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X)) @ (G @ X))))))))).
% 2.97/1.57 thf(fact_218_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)))))))).
% 2.97/1.57 thf(fact_219_set__eq__subset, axiom, (! [A:$tType]: (((^ [Y5:(set @ A),Z3:(set @ A)]: ((Y5 = Z3))) = (^ [A3:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B6) & (((ord_less_eq @ (set @ A)) @ B6) @ A3)))))))).
% 2.97/1.57 thf(fact_220_le__fun__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => ((ord_less_eq @ (A > B)) = (^ [F2:(A > B),G2:(A > B)]: ((! [X2:A]: ((((ord_less_eq @ B) @ (F2 @ X2)) @ (G2 @ X2))))))))))).
% 2.97/1.57 thf(fact_221_Collect__mono__iff, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)) = (! [X2:A]: (((P @ X2) => (Q @ X2)))))))).
% 2.97/1.57 thf(fact_222_order__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A5:A,F:(B > A),B3:B,C2:B]: (((((ord_less_eq @ A) @ A5) @ (F @ B3)) => ((((ord_less_eq @ B) @ B3) @ C2) => ((! [X3:B,Y2:B]: (((((ord_less_eq @ B) @ X3) @ Y2) => (((ord_less_eq @ A) @ (F @ X3)) @ (F @ Y2))))) => (((ord_less_eq @ A) @ A5) @ (F @ C2))))))))))).
% 2.97/1.57 thf(fact_223_order__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A5:A,B3:A,F:(A > C),C2:C]: (((((ord_less_eq @ A) @ A5) @ B3) => ((((ord_less_eq @ C) @ (F @ B3)) @ C2) => ((! [X3:A,Y2:A]: (((((ord_less_eq @ A) @ X3) @ Y2) => (((ord_less_eq @ C) @ (F @ X3)) @ (F @ Y2))))) => (((ord_less_eq @ C) @ (F @ A5)) @ C2)))))))))).
% 2.97/1.57 thf(fact_224_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A5:A,F:(B > A),B3:B,C2:B]: (((A5 = (F @ B3)) => ((((ord_less_eq @ B) @ B3) @ C2) => ((! [X3:B,Y2:B]: (((((ord_less_eq @ B) @ X3) @ Y2) => (((ord_less_eq @ A) @ (F @ X3)) @ (F @ Y2))))) => (((ord_less_eq @ A) @ A5) @ (F @ C2))))))))))).
% 2.97/1.57 thf(fact_225_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A5:A,B3:A,F:(A > B),C2:B]: (((((ord_less_eq @ A) @ A5) @ B3) => (((F @ B3) = C2) => ((! [X3:A,Y2:A]: (((((ord_less_eq @ A) @ X3) @ Y2) => (((ord_less_eq @ B) @ (F @ X3)) @ (F @ Y2))))) => (((ord_less_eq @ B) @ (F @ A5)) @ C2)))))))))).
% 2.97/1.57 thf(fact_226_eq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y5:A,Z3:A]: ((Y5 = Z3))) = (^ [X2:A,Y3:A]: (((((ord_less_eq @ A) @ X2) @ Y3) & (((ord_less_eq @ A) @ Y3) @ X2))))))))).
% 2.97/1.57 thf(fact_227_antisym, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ X) => (X = Y))))))))).
% 2.97/1.57 thf(fact_228_linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) | (((ord_less_eq @ A) @ Y) @ X)))))))).
% 2.97/1.57 thf(fact_229_eq__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A]: (((X = Y) => (((ord_less_eq @ A) @ X) @ Y)))))))).
% 2.97/1.57 thf(fact_230_le__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((~ ((((ord_less_eq @ A) @ X) @ Y))) => (((ord_less_eq @ A) @ Y) @ X)))))))).
% 2.97/1.57 thf(fact_231_order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [A5:A,B3:A,C2:A]: (((((ord_less_eq @ A) @ A5) @ B3) => ((((ord_less_eq @ A) @ B3) @ C2) => (((ord_less_eq @ A) @ A5) @ C2))))))))).
% 2.97/1.57 thf(fact_232_le__cases3, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A,Z:A]: ((((((ord_less_eq @ A) @ X) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ Z)))) => (((((ord_less_eq @ A) @ Y) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Z)))) => (((((ord_less_eq @ A) @ X) @ Z) => (~ ((((ord_less_eq @ A) @ Z) @ Y)))) => (((((ord_less_eq @ A) @ Z) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ X)))) => (((((ord_less_eq @ A) @ Y) @ Z) => (~ ((((ord_less_eq @ A) @ Z) @ X)))) => (~ (((((ord_less_eq @ A) @ Z) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Y))))))))))))))))).
% 2.97/1.57 thf(fact_233_antisym__conv, axiom, (! [A:$tType]: (((order @ A) => (! [Y:A,X:A]: (((((ord_less_eq @ A) @ Y) @ X) => ((((ord_less_eq @ A) @ X) @ Y) = (X = Y))))))))).
% 2.97/1.57 thf(fact_234_order__class_Oorder_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y5:A,Z3:A]: ((Y5 = Z3))) = (^ [A6:A,B5:A]: (((((ord_less_eq @ A) @ A6) @ B5) & (((ord_less_eq @ A) @ B5) @ A6))))))))).
% 2.97/1.57 thf(fact_235_ord__eq__le__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A5:A,B3:A,C2:A]: (((A5 = B3) => ((((ord_less_eq @ A) @ B3) @ C2) => (((ord_less_eq @ A) @ A5) @ C2))))))))).
% 2.97/1.57 thf(fact_236_ord__le__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A5:A,B3:A,C2:A]: (((((ord_less_eq @ A) @ A5) @ B3) => ((B3 = C2) => (((ord_less_eq @ A) @ A5) @ C2))))))))).
% 2.97/1.57 thf(fact_237_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [A5:A,B3:A]: (((((ord_less_eq @ A) @ A5) @ B3) => ((((ord_less_eq @ A) @ B3) @ A5) => (A5 = B3))))))))).
% 2.97/1.57 thf(fact_238_order__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A,Z:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z) => (((ord_less_eq @ A) @ X) @ Z))))))))).
% 2.97/1.57 thf(fact_239_dual__order_Orefl, axiom, (! [A:$tType]: (((order @ A) => (! [A5:A]: ((((ord_less_eq @ A) @ A5) @ A5))))))).
% 2.97/1.57 thf(fact_240_linorder__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > (A > $o)),A5:A,B3:A]: (((! [A4:A,B4:A]: (((((ord_less_eq @ A) @ A4) @ B4) => ((P @ A4) @ B4)))) => ((! [A4:A,B4:A]: ((((P @ B4) @ A4) => ((P @ A4) @ B4)))) => ((P @ A5) @ B3))))))))).
% 2.97/1.57 thf(fact_241_dual__order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A5:A,C2:A]: (((((ord_less_eq @ A) @ B3) @ A5) => ((((ord_less_eq @ A) @ C2) @ B3) => (((ord_less_eq @ A) @ C2) @ A5))))))))).
% 2.97/1.57 thf(fact_242_dual__order_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y5:A,Z3:A]: ((Y5 = Z3))) = (^ [A6:A,B5:A]: (((((ord_less_eq @ A) @ B5) @ A6) & (((ord_less_eq @ A) @ A6) @ B5))))))))).
% 2.97/1.57 thf(fact_243_dual__order_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A5:A]: (((((ord_less_eq @ A) @ B3) @ A5) => ((((ord_less_eq @ A) @ A5) @ B3) => (A5 = B3))))))))).
% 2.97/1.57 thf(fact_244_strict__mono__less__eq, axiom, (! [B:$tType,A:$tType]: ((((linorder @ A) & (order @ B)) => (! [F:(A > B),X:A,Y:A]: (((((order_strict_mono @ A) @ B) @ F) => ((((ord_less_eq @ B) @ (F @ X)) @ (F @ Y)) = (((ord_less_eq @ A) @ X) @ Y))))))))).
% 2.97/1.57 thf(fact_245_strict__mono__on__leD, axiom, (! [B:$tType,A:$tType]: ((((linorder @ A) & (preorder @ B)) => (! [F:(A > B),A2:(set @ A),X:A,Y:A]: ((((((strict_mono_on @ A) @ B) @ F) @ A2) => ((((member @ A) @ X) @ A2) => ((((member @ A) @ Y) @ A2) => ((((ord_less_eq @ A) @ X) @ Y) => (((ord_less_eq @ B) @ (F @ X)) @ (F @ Y)))))))))))).
% 2.97/1.57 thf(fact_246_add__less__le__mono, axiom, (! [A:$tType]: (((ordere223160158up_add @ A) => (! [A5:A,B3:A,C2:A,D2:A]: (((((ord_less @ A) @ A5) @ B3) => ((((ord_less_eq @ A) @ C2) @ D2) => (((ord_less @ A) @ (((plus_plus @ A) @ A5) @ C2)) @ (((plus_plus @ A) @ B3) @ D2)))))))))).
% 2.97/1.57 thf(fact_247_add__le__less__mono, axiom, (! [A:$tType]: (((ordere223160158up_add @ A) => (! [A5:A,B3:A,C2:A,D2:A]: (((((ord_less_eq @ A) @ A5) @ B3) => ((((ord_less @ A) @ C2) @ D2) => (((ord_less @ A) @ (((plus_plus @ A) @ A5) @ C2)) @ (((plus_plus @ A) @ B3) @ D2)))))))))).
% 2.97/1.57 thf(fact_248_add__mono__thms__linordered__field_I3_J, axiom, (! [A:$tType]: (((ordere223160158up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((((ord_less @ A) @ I) @ J) & (((ord_less_eq @ A) @ K) @ L)) => (((ord_less @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.97/1.57 thf(fact_249_add__mono__thms__linordered__field_I4_J, axiom, (! [A:$tType]: (((ordere223160158up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((((ord_less_eq @ A) @ I) @ J) & (((ord_less @ A) @ K) @ L)) => (((ord_less @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.97/1.57 thf(fact_250_linorder__inj__onI, axiom, (! [B:$tType,A:$tType]: (((order @ A) => (! [A2:(set @ A),F:(A > B)]: (((! [X3:A,Y2:A]: (((((ord_less @ A) @ X3) @ Y2) => ((((member @ A) @ X3) @ A2) => ((((member @ A) @ Y2) @ A2) => ((F @ X3) != (F @ Y2))))))) => ((! [X3:A,Y2:A]: (((((member @ A) @ X3) @ A2) => ((((member @ A) @ Y2) @ A2) => ((((ord_less_eq @ A) @ X3) @ Y2) | (((ord_less_eq @ A) @ Y2) @ X3)))))) => ((((inj_on @ A) @ B) @ F) @ A2))))))))).
% 2.97/1.57 thf(fact_251_inj__on__image__mem__iff, axiom, (! [B:$tType,A:$tType,F:(A > B),B2:(set @ A),A5:A,A2:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ B2) => ((((member @ A) @ A5) @ B2) => ((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((member @ B) @ (F @ A5)) @ ((((image @ A) @ B) @ F) @ A2)) = (((member @ A) @ A5) @ A2)))))))).
% 2.97/1.57 thf(fact_252_inj__on__image__eq__iff, axiom, (! [B:$tType,A:$tType,F:(A > B),C3:(set @ A),A2:(set @ A),B2:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ C3) => ((((ord_less_eq @ (set @ A)) @ A2) @ C3) => ((((ord_less_eq @ (set @ A)) @ B2) @ C3) => ((((((image @ A) @ B) @ F) @ A2) = ((((image @ A) @ B) @ F) @ B2)) = (A2 = B2)))))))).
% 2.97/1.57 thf(fact_253_inj__image__subset__iff, axiom, (! [B:$tType,A:$tType,F:(A > B),A2:(set @ A),B2:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => ((((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F) @ A2)) @ ((((image @ A) @ B) @ F) @ B2)) = (((ord_less_eq @ (set @ A)) @ A2) @ B2)))))).
% 2.97/1.57 thf(fact_254_map__prod__surj, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,F:(A > B),G:(C > D)]: (((((((image @ A) @ B) @ F) @ (top_top @ (set @ A))) = (top_top @ (set @ B))) => ((((((image @ C) @ D) @ G) @ (top_top @ (set @ C))) = (top_top @ (set @ D))) => (((((image @ ((product_prod @ A) @ C)) @ ((product_prod @ B) @ D)) @ ((((((product_map_prod @ A) @ B) @ C) @ D) @ F) @ G)) @ (top_top @ (set @ ((product_prod @ A) @ C)))) = (top_top @ (set @ ((product_prod @ B) @ D))))))))).
% 2.97/1.57 thf(fact_255_subset__image__inj, axiom, (! [A:$tType,B:$tType,S4:(set @ A),F:(B > A),T2:(set @ B)]: (((((ord_less_eq @ (set @ A)) @ S4) @ ((((image @ B) @ A) @ F) @ T2)) = (? [U:(set @ B)]: (((((ord_less_eq @ (set @ B)) @ U) @ T2) & (((((inj_on @ B) @ A) @ F) @ U) & (S4 = ((((image @ B) @ A) @ F) @ U)))))))))).
% 2.97/1.57 thf(tcon_fun___Orderings_Oorder__top, axiom, (! [A7:$tType,A8:$tType]: (((order_top @ A8) => (order_top @ (A7 > A8)))))).
% 2.97/1.57 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A7:$tType,A8:$tType]: (((preorder @ A8) => (preorder @ (A7 > A8)))))).
% 2.97/1.57 thf(tcon_fun___Orderings_Oorder, axiom, (! [A7:$tType,A8:$tType]: (((order @ A8) => (order @ (A7 > A8)))))).
% 2.97/1.57 thf(tcon_fun___Orderings_Otop, axiom, (! [A7:$tType,A8:$tType]: (((top @ A8) => (top @ (A7 > A8)))))).
% 2.97/1.57 thf(tcon_fun___Orderings_Oord, axiom, (! [A7:$tType,A8:$tType]: (((ord @ A8) => (ord @ (A7 > A8)))))).
% 2.97/1.57 thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add, axiom, (strict2144017051up_add @ nat)).
% 2.97/1.57 thf(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add, axiom, (ordere223160158up_add @ nat)).
% 2.97/1.57 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le, axiom, (ordere236663937imp_le @ nat)).
% 2.97/1.57 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add, axiom, (canoni770627133id_add @ nat)).
% 2.97/1.57 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add, axiom, (ordere779506340up_add @ nat)).
% 2.97/1.57 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add, axiom, (cancel_semigroup_add @ nat)).
% 2.97/1.57 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add, axiom, (ab_semigroup_add @ nat)).
% 2.97/1.57 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add, axiom, (comm_monoid_add @ nat)).
% 2.97/1.57 thf(tcon_Nat_Onat___Groups_Osemigroup__add, axiom, (semigroup_add @ nat)).
% 2.97/1.57 thf(tcon_Nat_Onat___Orderings_Owellorder, axiom, (wellorder @ nat)).
% 2.97/1.57 thf(tcon_Nat_Onat___Orderings_Opreorder_1, axiom, (preorder @ nat)).
% 2.97/1.57 thf(tcon_Nat_Onat___Orderings_Olinorder, axiom, (linorder @ nat)).
% 2.97/1.57 thf(tcon_Nat_Onat___Orderings_Ono__top, axiom, (no_top @ nat)).
% 2.97/1.57 thf(tcon_Nat_Onat___Orderings_Oorder_2, axiom, (order @ nat)).
% 2.97/1.57 thf(tcon_Nat_Onat___Orderings_Oord_3, axiom, (ord @ nat)).
% 2.97/1.57 thf(tcon_Set_Oset___Orderings_Oorder__top_4, axiom, (! [A7:$tType]: ((order_top @ (set @ A7))))).
% 2.97/1.57 thf(tcon_Set_Oset___Orderings_Opreorder_5, axiom, (! [A7:$tType]: ((preorder @ (set @ A7))))).
% 2.97/1.57 thf(tcon_Set_Oset___Orderings_Oorder_6, axiom, (! [A7:$tType]: ((order @ (set @ A7))))).
% 2.97/1.57 thf(tcon_Set_Oset___Orderings_Otop_7, axiom, (! [A7:$tType]: ((top @ (set @ A7))))).
% 2.97/1.57 thf(tcon_Set_Oset___Orderings_Oord_8, axiom, (! [A7:$tType]: ((ord @ (set @ A7))))).
% 2.97/1.57 thf(tcon_HOL_Obool___Orderings_Oorder__top_9, axiom, (order_top @ $o)).
% 2.97/1.57 thf(tcon_HOL_Obool___Orderings_Opreorder_10, axiom, (preorder @ $o)).
% 2.97/1.57 thf(tcon_HOL_Obool___Orderings_Olinorder_11, axiom, (linorder @ $o)).
% 2.97/1.57 thf(tcon_HOL_Obool___Orderings_Oorder_12, axiom, (order @ $o)).
% 2.97/1.57 thf(tcon_HOL_Obool___Orderings_Otop_13, axiom, (top @ $o)).
% 2.97/1.57 thf(tcon_HOL_Obool___Orderings_Oord_14, axiom, (ord @ $o)).
% 2.97/1.57 thf(conj_0, conjecture, ((((inj_on @ paraco415392788lle_tv) @ paraco415392788lle_tv) @ (paraco1920534163nge_tv @ f)) @ (top_top @ (set @ paraco415392788lle_tv)))).
% 2.97/1.57 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.Ai5Zk59Gvc/DTF2DTF9159.p
% 2.97/1.57 ------------------------
% 2.97/1.60 ---- Cleaned THF ---
% 2.97/1.60 thf(ty_t_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv,type,
% 2.97/1.60 paraco415392788lle_tv: $tType ).
% 2.97/1.60
% 2.97/1.60 thf(ty_t_Product__Type_Oprod,type,
% 2.97/1.60 product_prod: $tType > $tType > $tType ).
% 2.97/1.60
% 2.97/1.60 thf(ty_t_Set_Oset,type,
% 2.97/1.60 set: $tType > $tType ).
% 2.97/1.60
% 2.97/1.60 thf(ty_t_Nat_Onat,type,
% 2.97/1.60 nat: $tType ).
% 2.97/1.60
% 2.97/1.60 thf(sy_cl_HOL_Otype,type,
% 2.97/1.60 type:
% 2.97/1.60 !>[A: $tType] : $o ).
% 2.97/1.60
% 2.97/1.60 thf(sy_cl_Orderings_Oord,type,
% 2.97/1.60 ord:
% 2.97/1.60 !>[A: $tType] : $o ).
% 2.97/1.60
% 2.97/1.60 thf(sy_cl_Orderings_Otop,type,
% 2.97/1.60 top:
% 2.97/1.60 !>[A: $tType] : $o ).
% 2.97/1.60
% 2.97/1.60 thf(sy_cl_Orderings_Oorder,type,
% 2.97/1.60 order:
% 2.97/1.60 !>[A: $tType] : $o ).
% 2.97/1.60
% 2.97/1.60 thf(sy_cl_Groups_Ogroup__add,type,
% 2.97/1.60 group_add:
% 2.97/1.60 !>[A: $tType] : $o ).
% 2.97/1.60
% 2.97/1.60 thf(sy_cl_Orderings_Ono__bot,type,
% 2.97/1.60 no_bot:
% 2.97/1.60 !>[A: $tType] : $o ).
% 2.97/1.60
% 2.97/1.60 thf(sy_cl_Orderings_Ono__top,type,
% 2.97/1.60 no_top:
% 2.97/1.60 !>[A: $tType] : $o ).
% 2.97/1.60
% 2.97/1.60 thf(sy_cl_Orderings_Olinorder,type,
% 2.97/1.60 linorder:
% 2.97/1.60 !>[A: $tType] : $o ).
% 2.97/1.60
% 2.97/1.60 thf(sy_cl_Orderings_Opreorder,type,
% 2.97/1.60 preorder:
% 2.97/1.60 !>[A: $tType] : $o ).
% 2.97/1.60
% 2.97/1.60 thf(sy_cl_Groups_Oab__group__add,type,
% 2.97/1.60 ab_group_add:
% 2.97/1.60 !>[A: $tType] : $o ).
% 2.97/1.60
% 2.97/1.60 thf(sy_cl_Orderings_Oorder__top,type,
% 2.97/1.60 order_top:
% 2.97/1.60 !>[A: $tType] : $o ).
% 2.97/1.60
% 2.97/1.60 thf(sy_cl_Orderings_Owellorder,type,
% 2.97/1.60 wellorder:
% 2.97/1.60 !>[A: $tType] : $o ).
% 2.97/1.60
% 2.97/1.60 thf(sy_cl_Groups_Osemigroup__add,type,
% 2.97/1.60 semigroup_add:
% 2.97/1.60 !>[A: $tType] : $o ).
% 2.97/1.60
% 2.97/1.60 thf(sy_cl_Orderings_Odense__order,type,
% 2.97/1.60 dense_order:
% 2.97/1.60 !>[A: $tType] : $o ).
% 2.97/1.60
% 2.97/1.60 thf(sy_cl_Groups_Ocomm__monoid__add,type,
% 2.97/1.60 comm_monoid_add:
% 2.97/1.60 !>[A: $tType] : $o ).
% 2.97/1.60
% 2.97/1.60 thf(sy_cl_Groups_Oab__semigroup__add,type,
% 2.97/1.60 ab_semigroup_add:
% 2.97/1.60 !>[A: $tType] : $o ).
% 2.97/1.60
% 2.97/1.60 thf(sy_cl_Orderings_Odense__linorder,type,
% 2.97/1.60 dense_linorder:
% 2.97/1.60 !>[A: $tType] : $o ).
% 2.97/1.60
% 2.97/1.60 thf(sy_cl_Groups_Ocancel__semigroup__add,type,
% 2.97/1.60 cancel_semigroup_add:
% 2.97/1.60 !>[A: $tType] : $o ).
% 2.97/1.60
% 2.97/1.60 thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
% 2.97/1.60 ordere779506340up_add:
% 2.97/1.60 !>[A: $tType] : $o ).
% 2.97/1.60
% 2.97/1.60 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 2.97/1.60 canoni770627133id_add:
% 2.97/1.60 !>[A: $tType] : $o ).
% 2.97/1.60
% 2.97/1.60 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
% 2.97/1.60 ordere236663937imp_le:
% 2.97/1.60 !>[A: $tType] : $o ).
% 2.97/1.60
% 2.97/1.60 thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
% 2.97/1.60 ordere223160158up_add:
% 2.97/1.60 !>[A: $tType] : $o ).
% 2.97/1.60
% 2.97/1.60 thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add,type,
% 2.97/1.60 strict2144017051up_add:
% 2.97/1.60 !>[A: $tType] : $o ).
% 2.97/1.60
% 2.97/1.60 thf(sy_c_Fun_Oinj__on,type,
% 2.97/1.60 inj_on:
% 2.97/1.60 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
% 2.97/1.60
% 2.97/1.60 thf(sy_c_Fun_Ostrict__mono__on,type,
% 2.97/1.60 strict_mono_on:
% 2.97/1.60 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
% 2.97/1.60
% 2.97/1.60 thf(sy_c_Fun_Othe__inv__into,type,
% 2.97/1.60 the_inv_into:
% 2.97/1.60 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > B > A ) ).
% 2.97/1.60
% 2.97/1.60 thf(sy_c_Groups_Oplus__class_Oplus,type,
% 2.97/1.60 plus_plus:
% 2.97/1.60 !>[A: $tType] : ( A > A > A ) ).
% 2.97/1.60
% 2.97/1.60 thf(sy_c_Orderings_Oord__class_Oless,type,
% 2.97/1.60 ord_less:
% 2.97/1.60 !>[A: $tType] : ( A > A > $o ) ).
% 2.97/1.60
% 2.97/1.60 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.97/1.60 ord_less_eq:
% 2.97/1.60 !>[A: $tType] : ( A > A > $o ) ).
% 2.97/1.60
% 2.97/1.60 thf(sy_c_Orderings_Oorder__class_Ostrict__mono,type,
% 2.97/1.60 order_strict_mono:
% 3.13/1.60 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 3.13/1.60
% 3.13/1.60 thf(sy_c_Orderings_Otop__class_Otop,type,
% 3.13/1.60 top_top:
% 3.13/1.60 !>[A: $tType] : A ).
% 3.13/1.60
% 3.13/1.60 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ochange__tv,type,
% 3.13/1.60 paraco1920534163nge_tv: ( nat > nat ) > paraco415392788lle_tv > paraco415392788lle_tv ).
% 3.13/1.60
% 3.13/1.60 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv_OIndet,type,
% 3.13/1.60 paraco676387099_Indet: nat > paraco415392788lle_tv ).
% 3.13/1.60
% 3.13/1.60 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv_Orec__tv,type,
% 3.13/1.60 paraco152590079rec_tv:
% 3.13/1.60 !>[A: $tType] : ( ( $o > A ) > ( nat > A ) > paraco415392788lle_tv > A ) ).
% 3.13/1.60
% 3.13/1.60 thf(sy_c_Product__Type_Oapfst,type,
% 3.13/1.60 product_apfst:
% 3.13/1.60 !>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( product_prod @ A @ B ) > ( product_prod @ C @ B ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(sy_c_Product__Type_Oapsnd,type,
% 3.13/1.60 product_apsnd:
% 3.13/1.60 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( product_prod @ A @ B ) > ( product_prod @ A @ C ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(sy_c_Product__Type_Omap__prod,type,
% 3.13/1.60 product_map_prod:
% 3.13/1.60 !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C ) > ( B > D ) > ( product_prod @ A @ B ) > ( product_prod @ C @ D ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(sy_c_Set_OCollect,type,
% 3.13/1.60 collect:
% 3.13/1.60 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(sy_c_Set_Oimage,type,
% 3.13/1.60 image:
% 3.13/1.60 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(sy_c_member,type,
% 3.13/1.60 member:
% 3.13/1.60 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 3.13/1.60
% 3.13/1.60 thf(sy_v_f,type,
% 3.13/1.60 f: nat > nat ).
% 3.13/1.60
% 3.13/1.60 thf(fact_0__092_060open_062_092_060And_062tv2_Atv1_O_Achange__tv_Af_Atv1_A_061_Achange__tv_Af_Atv2_A_092_060Longrightarrow_062_Atv1_A_061_Atv2_092_060close_062,axiom,
% 3.13/1.60 ! [Tv1: paraco415392788lle_tv,Tv2: paraco415392788lle_tv] :
% 3.13/1.60 ( ( ( paraco1920534163nge_tv @ f @ Tv1 )
% 3.13/1.60 = ( paraco1920534163nge_tv @ f @ Tv2 ) )
% 3.13/1.60 => ( Tv1 = Tv2 ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_1_UNIV__I,axiom,
% 3.13/1.60 ! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_2_iso__tuple__UNIV__I,axiom,
% 3.13/1.60 ! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_3_top__apply,axiom,
% 3.13/1.60 ! [C: $tType,D: $tType] :
% 3.13/1.60 ( ( top @ C )
% 3.13/1.60 => ( ( top_top @ ( D > C ) )
% 3.13/1.60 = ( ^ [X2: D] : ( top_top @ C ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_4_injD,axiom,
% 3.13/1.60 ! [B: $tType,A: $tType,F: A > B,X: A,Y: A] :
% 3.13/1.60 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 3.13/1.60 => ( ( ( F @ X )
% 3.13/1.60 = ( F @ Y ) )
% 3.13/1.60 => ( X = Y ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_5_injI,axiom,
% 3.13/1.60 ! [B: $tType,A: $tType,F: A > B] :
% 3.13/1.60 ( ! [X3: A,Y2: A] :
% 3.13/1.60 ( ( ( F @ X3 )
% 3.13/1.60 = ( F @ Y2 ) )
% 3.13/1.60 => ( X3 = Y2 ) )
% 3.13/1.60 => ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_6_inj__eq,axiom,
% 3.13/1.60 ! [B: $tType,A: $tType,F: A > B,X: A,Y: A] :
% 3.13/1.60 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 3.13/1.60 => ( ( ( F @ X )
% 3.13/1.60 = ( F @ Y ) )
% 3.13/1.60 = ( X = Y ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_7_inj__def,axiom,
% 3.13/1.60 ! [B: $tType,A: $tType,F: A > B] :
% 3.13/1.60 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 3.13/1.60 = ( ! [X2: A,Y3: A] :
% 3.13/1.60 ( ( ( F @ X2 )
% 3.13/1.60 = ( F @ Y3 ) )
% 3.13/1.60 => ( X2 = Y3 ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_8_inj__onD,axiom,
% 3.13/1.60 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,X: A,Y: A] :
% 3.13/1.60 ( ( inj_on @ A @ B @ F @ A2 )
% 3.13/1.60 => ( ( ( F @ X )
% 3.13/1.60 = ( F @ Y ) )
% 3.13/1.60 => ( ( member @ A @ X @ A2 )
% 3.13/1.60 => ( ( member @ A @ Y @ A2 )
% 3.13/1.60 => ( X = Y ) ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_9_inj__onI,axiom,
% 3.13/1.60 ! [B: $tType,A: $tType,A2: set @ A,F: A > B] :
% 3.13/1.60 ( ! [X3: A,Y2: A] :
% 3.13/1.60 ( ( member @ A @ X3 @ A2 )
% 3.13/1.60 => ( ( member @ A @ Y2 @ A2 )
% 3.13/1.60 => ( ( ( F @ X3 )
% 3.13/1.60 = ( F @ Y2 ) )
% 3.13/1.60 => ( X3 = Y2 ) ) ) )
% 3.13/1.60 => ( inj_on @ A @ B @ F @ A2 ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_10_inj__on__def,axiom,
% 3.13/1.60 ! [B: $tType,A: $tType] :
% 3.13/1.60 ( ( inj_on @ A @ B )
% 3.13/1.60 = ( ^ [F2: A > B,A3: set @ A] :
% 3.13/1.60 ! [X2: A] :
% 3.13/1.60 ( ( member @ A @ X2 @ A3 )
% 3.13/1.60 => ! [Y3: A] :
% 3.13/1.60 ( ( member @ A @ Y3 @ A3 )
% 3.13/1.60 => ( ( ( F2 @ X2 )
% 3.13/1.60 = ( F2 @ Y3 ) )
% 3.13/1.60 => ( X2 = Y3 ) ) ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_11_inj__on__cong,axiom,
% 3.13/1.60 ! [B: $tType,A: $tType,A2: set @ A,F: A > B,G: A > B] :
% 3.13/1.60 ( ! [A4: A] :
% 3.13/1.60 ( ( member @ A @ A4 @ A2 )
% 3.13/1.60 => ( ( F @ A4 )
% 3.13/1.60 = ( G @ A4 ) ) )
% 3.13/1.60 => ( ( inj_on @ A @ B @ F @ A2 )
% 3.13/1.60 = ( inj_on @ A @ B @ G @ A2 ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_12_inj__on__eq__iff,axiom,
% 3.13/1.60 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,X: A,Y: A] :
% 3.13/1.60 ( ( inj_on @ A @ B @ F @ A2 )
% 3.13/1.60 => ( ( member @ A @ X @ A2 )
% 3.13/1.60 => ( ( member @ A @ Y @ A2 )
% 3.13/1.60 => ( ( ( F @ X )
% 3.13/1.60 = ( F @ Y ) )
% 3.13/1.60 = ( X = Y ) ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_13_assms,axiom,
% 3.13/1.60 inj_on @ nat @ nat @ f @ ( top_top @ ( set @ nat ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_14_top__set__def,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( top_top @ ( set @ A ) )
% 3.13/1.60 = ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_15_UNIV__witness,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ? [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_16_UNIV__eq__I,axiom,
% 3.13/1.60 ! [A: $tType,A2: set @ A] :
% 3.13/1.60 ( ! [X3: A] : ( member @ A @ X3 @ A2 )
% 3.13/1.60 => ( ( top_top @ ( set @ A ) )
% 3.13/1.60 = A2 ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_17_inj__on__inverseI,axiom,
% 3.13/1.60 ! [B: $tType,A: $tType,A2: set @ A,G: B > A,F: A > B] :
% 3.13/1.60 ( ! [X3: A] :
% 3.13/1.60 ( ( member @ A @ X3 @ A2 )
% 3.13/1.60 => ( ( G @ ( F @ X3 ) )
% 3.13/1.60 = X3 ) )
% 3.13/1.60 => ( inj_on @ A @ B @ F @ A2 ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_18_inj__on__contraD,axiom,
% 3.13/1.60 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,X: A,Y: A] :
% 3.13/1.60 ( ( inj_on @ A @ B @ F @ A2 )
% 3.13/1.60 => ( ( X != Y )
% 3.13/1.60 => ( ( member @ A @ X @ A2 )
% 3.13/1.60 => ( ( member @ A @ Y @ A2 )
% 3.13/1.60 => ( ( F @ X )
% 3.13/1.60 != ( F @ Y ) ) ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_19_top__empty__eq,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( top_top @ ( A > $o ) )
% 3.13/1.60 = ( ^ [X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_20_the__inv__f__f,axiom,
% 3.13/1.60 ! [B: $tType,A: $tType,F: A > B,X: A] :
% 3.13/1.60 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 3.13/1.60 => ( ( the_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F @ ( F @ X ) )
% 3.13/1.60 = X ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_21_strict__mono__on__imp__inj__on,axiom,
% 3.13/1.60 ! [B: $tType,A: $tType] :
% 3.13/1.60 ( ( ( linorder @ A )
% 3.13/1.60 & ( preorder @ B ) )
% 3.13/1.60 => ! [F: A > B,A2: set @ A] :
% 3.13/1.60 ( ( strict_mono_on @ A @ B @ F @ A2 )
% 3.13/1.60 => ( inj_on @ A @ B @ F @ A2 ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_22_linorder__injI,axiom,
% 3.13/1.60 ! [B: $tType,A: $tType] :
% 3.13/1.60 ( ( linorder @ A )
% 3.13/1.60 => ! [F: A > B] :
% 3.13/1.60 ( ! [X3: A,Y2: A] :
% 3.13/1.60 ( ( ord_less @ A @ X3 @ Y2 )
% 3.13/1.60 => ( ( F @ X3 )
% 3.13/1.60 != ( F @ Y2 ) ) )
% 3.13/1.60 => ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_23_inj__add__left,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( cancel_semigroup_add @ A )
% 3.13/1.60 => ! [A5: A] : ( inj_on @ A @ A @ ( plus_plus @ A @ A5 ) @ ( top_top @ ( set @ A ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_24_change__tv_Osimps_I2_J,axiom,
% 3.13/1.60 ! [F: nat > nat,N: nat] :
% 3.13/1.60 ( ( paraco1920534163nge_tv @ F @ ( paraco676387099_Indet @ N ) )
% 3.13/1.60 = ( paraco676387099_Indet @ ( F @ N ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_25_inj__apsnd,axiom,
% 3.13/1.60 ! [A: $tType,C: $tType,B: $tType,F: B > C] :
% 3.13/1.60 ( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
% 3.13/1.60 = ( inj_on @ B @ C @ F @ ( top_top @ ( set @ B ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_26_inj__image__mem__iff,axiom,
% 3.13/1.60 ! [B: $tType,A: $tType,F: A > B,A5: A,A2: set @ A] :
% 3.13/1.60 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 3.13/1.60 => ( ( member @ B @ ( F @ A5 ) @ ( image @ A @ B @ F @ A2 ) )
% 3.13/1.60 = ( member @ A @ A5 @ A2 ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_27_inj__image__eq__iff,axiom,
% 3.13/1.60 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,B2: set @ A] :
% 3.13/1.60 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 3.13/1.60 => ( ( ( image @ A @ B @ F @ A2 )
% 3.13/1.60 = ( image @ A @ B @ F @ B2 ) )
% 3.13/1.60 = ( A2 = B2 ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_28_range__ex1__eq,axiom,
% 3.13/1.60 ! [B: $tType,A: $tType,F: A > B,B3: B] :
% 3.13/1.60 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 3.13/1.60 => ( ( member @ B @ B3 @ ( image @ A @ B @ F @ ( top_top @ ( set @ A ) ) ) )
% 3.13/1.60 = ( ? [X2: A] :
% 3.13/1.60 ( ( B3
% 3.13/1.60 = ( F @ X2 ) )
% 3.13/1.60 & ! [Y3: A] :
% 3.13/1.60 ( ( B3
% 3.13/1.60 = ( F @ Y3 ) )
% 3.13/1.60 => ( Y3 = X2 ) ) ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_29_strict__mono__imp__inj__on,axiom,
% 3.13/1.60 ! [B: $tType,A: $tType] :
% 3.13/1.60 ( ( ( linorder @ A )
% 3.13/1.60 & ( order @ B ) )
% 3.13/1.60 => ! [F: A > B,A2: set @ A] :
% 3.13/1.60 ( ( order_strict_mono @ A @ B @ F )
% 3.13/1.60 => ( inj_on @ A @ B @ F @ A2 ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_30_image__eqI,axiom,
% 3.13/1.60 ! [A: $tType,B: $tType,B3: A,F: B > A,X: B,A2: set @ B] :
% 3.13/1.60 ( ( B3
% 3.13/1.60 = ( F @ X ) )
% 3.13/1.60 => ( ( member @ B @ X @ A2 )
% 3.13/1.60 => ( member @ A @ B3 @ ( image @ B @ A @ F @ A2 ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_31_tv_Oinject_I2_J,axiom,
% 3.13/1.60 ! [X22: nat,Y22: nat] :
% 3.13/1.60 ( ( ( paraco676387099_Indet @ X22 )
% 3.13/1.60 = ( paraco676387099_Indet @ Y22 ) )
% 3.13/1.60 = ( X22 = Y22 ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_32_top1I,axiom,
% 3.13/1.60 ! [A: $tType,X: A] : ( top_top @ ( A > $o ) @ X ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_33_surj__plus,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( ab_group_add @ A )
% 3.13/1.60 => ! [A5: A] :
% 3.13/1.60 ( ( image @ A @ A @ ( plus_plus @ A @ A5 ) @ ( top_top @ ( set @ A ) ) )
% 3.13/1.60 = ( top_top @ ( set @ A ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_34_the__inv__into__onto,axiom,
% 3.13/1.60 ! [B: $tType,A: $tType,F: A > B,A2: set @ A] :
% 3.13/1.60 ( ( inj_on @ A @ B @ F @ A2 )
% 3.13/1.60 => ( ( image @ B @ A @ ( the_inv_into @ A @ B @ A2 @ F ) @ ( image @ A @ B @ F @ A2 ) )
% 3.13/1.60 = A2 ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_35_imageI,axiom,
% 3.13/1.60 ! [B: $tType,A: $tType,X: A,A2: set @ A,F: A > B] :
% 3.13/1.60 ( ( member @ A @ X @ A2 )
% 3.13/1.60 => ( member @ B @ ( F @ X ) @ ( image @ A @ B @ F @ A2 ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_36_image__iff,axiom,
% 3.13/1.60 ! [A: $tType,B: $tType,Z: A,F: B > A,A2: set @ B] :
% 3.13/1.60 ( ( member @ A @ Z @ ( image @ B @ A @ F @ A2 ) )
% 3.13/1.60 = ( ? [X2: B] :
% 3.13/1.60 ( ( member @ B @ X2 @ A2 )
% 3.13/1.60 & ( Z
% 3.13/1.60 = ( F @ X2 ) ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_37_bex__imageD,axiom,
% 3.13/1.60 ! [A: $tType,B: $tType,F: B > A,A2: set @ B,P: A > $o] :
% 3.13/1.60 ( ? [X4: A] :
% 3.13/1.60 ( ( member @ A @ X4 @ ( image @ B @ A @ F @ A2 ) )
% 3.13/1.60 & ( P @ X4 ) )
% 3.13/1.60 => ? [X3: B] :
% 3.13/1.60 ( ( member @ B @ X3 @ A2 )
% 3.13/1.60 & ( P @ ( F @ X3 ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_38_image__cong,axiom,
% 3.13/1.60 ! [B: $tType,A: $tType,M: set @ A,N2: set @ A,F: A > B,G: A > B] :
% 3.13/1.60 ( ( M = N2 )
% 3.13/1.60 => ( ! [X3: A] :
% 3.13/1.60 ( ( member @ A @ X3 @ N2 )
% 3.13/1.60 => ( ( F @ X3 )
% 3.13/1.60 = ( G @ X3 ) ) )
% 3.13/1.60 => ( ( image @ A @ B @ F @ M )
% 3.13/1.60 = ( image @ A @ B @ G @ N2 ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_39_ball__imageD,axiom,
% 3.13/1.60 ! [A: $tType,B: $tType,F: B > A,A2: set @ B,P: A > $o] :
% 3.13/1.60 ( ! [X3: A] :
% 3.13/1.60 ( ( member @ A @ X3 @ ( image @ B @ A @ F @ A2 ) )
% 3.13/1.60 => ( P @ X3 ) )
% 3.13/1.60 => ! [X4: B] :
% 3.13/1.60 ( ( member @ B @ X4 @ A2 )
% 3.13/1.60 => ( P @ ( F @ X4 ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_40_rev__image__eqI,axiom,
% 3.13/1.60 ! [B: $tType,A: $tType,X: A,A2: set @ A,B3: B,F: A > B] :
% 3.13/1.60 ( ( member @ A @ X @ A2 )
% 3.13/1.60 => ( ( B3
% 3.13/1.60 = ( F @ X ) )
% 3.13/1.60 => ( member @ B @ B3 @ ( image @ A @ B @ F @ A2 ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_41_strict__mono__onD,axiom,
% 3.13/1.60 ! [B: $tType,A: $tType] :
% 3.13/1.60 ( ( ( ord @ A )
% 3.13/1.60 & ( ord @ B ) )
% 3.13/1.60 => ! [F: A > B,A2: set @ A,R: A,S: A] :
% 3.13/1.60 ( ( strict_mono_on @ A @ B @ F @ A2 )
% 3.13/1.60 => ( ( member @ A @ R @ A2 )
% 3.13/1.60 => ( ( member @ A @ S @ A2 )
% 3.13/1.60 => ( ( ord_less @ A @ R @ S )
% 3.13/1.60 => ( ord_less @ B @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_42_strict__mono__onI,axiom,
% 3.13/1.60 ! [B: $tType,A: $tType] :
% 3.13/1.60 ( ( ( ord @ A )
% 3.13/1.60 & ( ord @ B ) )
% 3.13/1.60 => ! [A2: set @ A,F: A > B] :
% 3.13/1.60 ( ! [R2: A,S2: A] :
% 3.13/1.60 ( ( member @ A @ R2 @ A2 )
% 3.13/1.60 => ( ( member @ A @ S2 @ A2 )
% 3.13/1.60 => ( ( ord_less @ A @ R2 @ S2 )
% 3.13/1.60 => ( ord_less @ B @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) )
% 3.13/1.60 => ( strict_mono_on @ A @ B @ F @ A2 ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_43_strict__mono__on__def,axiom,
% 3.13/1.60 ! [B: $tType,A: $tType] :
% 3.13/1.60 ( ( ( ord @ A )
% 3.13/1.60 & ( ord @ B ) )
% 3.13/1.60 => ( ( strict_mono_on @ A @ B )
% 3.13/1.60 = ( ^ [F2: A > B,A3: set @ A] :
% 3.13/1.60 ! [R3: A,S3: A] :
% 3.13/1.60 ( ( ( member @ A @ R3 @ A3 )
% 3.13/1.60 & ( member @ A @ S3 @ A3 )
% 3.13/1.60 & ( ord_less @ A @ R3 @ S3 ) )
% 3.13/1.60 => ( ord_less @ B @ ( F2 @ R3 ) @ ( F2 @ S3 ) ) ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_44_ord__eq__less__subst,axiom,
% 3.13/1.60 ! [A: $tType,B: $tType] :
% 3.13/1.60 ( ( ( ord @ B )
% 3.13/1.60 & ( ord @ A ) )
% 3.13/1.60 => ! [A5: A,F: B > A,B3: B,C2: B] :
% 3.13/1.60 ( ( A5
% 3.13/1.60 = ( F @ B3 ) )
% 3.13/1.60 => ( ( ord_less @ B @ B3 @ C2 )
% 3.13/1.60 => ( ! [X3: B,Y2: B] :
% 3.13/1.60 ( ( ord_less @ B @ X3 @ Y2 )
% 3.13/1.60 => ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 3.13/1.60 => ( ord_less @ A @ A5 @ ( F @ C2 ) ) ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_45_mem__Collect__eq,axiom,
% 3.13/1.60 ! [A: $tType,A5: A,P: A > $o] :
% 3.13/1.60 ( ( member @ A @ A5 @ ( collect @ A @ P ) )
% 3.13/1.60 = ( P @ A5 ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_46_Collect__mem__eq,axiom,
% 3.13/1.60 ! [A: $tType,A2: set @ A] :
% 3.13/1.60 ( ( collect @ A
% 3.13/1.60 @ ^ [X2: A] : ( member @ A @ X2 @ A2 ) )
% 3.13/1.60 = A2 ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_47_Collect__cong,axiom,
% 3.13/1.60 ! [A: $tType,P: A > $o,Q: A > $o] :
% 3.13/1.60 ( ! [X3: A] :
% 3.13/1.60 ( ( P @ X3 )
% 3.13/1.60 = ( Q @ X3 ) )
% 3.13/1.60 => ( ( collect @ A @ P )
% 3.13/1.60 = ( collect @ A @ Q ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_48_ext,axiom,
% 3.13/1.60 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 3.13/1.60 ( ! [X3: A] :
% 3.13/1.60 ( ( F @ X3 )
% 3.13/1.60 = ( G @ X3 ) )
% 3.13/1.60 => ( F = G ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_49_ord__less__eq__subst,axiom,
% 3.13/1.60 ! [A: $tType,B: $tType] :
% 3.13/1.60 ( ( ( ord @ B )
% 3.13/1.60 & ( ord @ A ) )
% 3.13/1.60 => ! [A5: A,B3: A,F: A > B,C2: B] :
% 3.13/1.60 ( ( ord_less @ A @ A5 @ B3 )
% 3.13/1.60 => ( ( ( F @ B3 )
% 3.13/1.60 = C2 )
% 3.13/1.60 => ( ! [X3: A,Y2: A] :
% 3.13/1.60 ( ( ord_less @ A @ X3 @ Y2 )
% 3.13/1.60 => ( ord_less @ B @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 3.13/1.60 => ( ord_less @ B @ ( F @ A5 ) @ C2 ) ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_50_order__less__subst1,axiom,
% 3.13/1.60 ! [A: $tType,B: $tType] :
% 3.13/1.60 ( ( ( order @ B )
% 3.13/1.60 & ( order @ A ) )
% 3.13/1.60 => ! [A5: A,F: B > A,B3: B,C2: B] :
% 3.13/1.60 ( ( ord_less @ A @ A5 @ ( F @ B3 ) )
% 3.13/1.60 => ( ( ord_less @ B @ B3 @ C2 )
% 3.13/1.60 => ( ! [X3: B,Y2: B] :
% 3.13/1.60 ( ( ord_less @ B @ X3 @ Y2 )
% 3.13/1.60 => ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 3.13/1.60 => ( ord_less @ A @ A5 @ ( F @ C2 ) ) ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_51_order__less__subst2,axiom,
% 3.13/1.60 ! [A: $tType,C: $tType] :
% 3.13/1.60 ( ( ( order @ C )
% 3.13/1.60 & ( order @ A ) )
% 3.13/1.60 => ! [A5: A,B3: A,F: A > C,C2: C] :
% 3.13/1.60 ( ( ord_less @ A @ A5 @ B3 )
% 3.13/1.60 => ( ( ord_less @ C @ ( F @ B3 ) @ C2 )
% 3.13/1.60 => ( ! [X3: A,Y2: A] :
% 3.13/1.60 ( ( ord_less @ A @ X3 @ Y2 )
% 3.13/1.60 => ( ord_less @ C @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 3.13/1.60 => ( ord_less @ C @ ( F @ A5 ) @ C2 ) ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_52_lt__ex,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( no_bot @ A )
% 3.13/1.60 => ! [X: A] :
% 3.13/1.60 ? [Y2: A] : ( ord_less @ A @ Y2 @ X ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_53_gt__ex,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( no_top @ A )
% 3.13/1.60 => ! [X: A] :
% 3.13/1.60 ? [X_1: A] : ( ord_less @ A @ X @ X_1 ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_54_neqE,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( linorder @ A )
% 3.13/1.60 => ! [X: A,Y: A] :
% 3.13/1.60 ( ( X != Y )
% 3.13/1.60 => ( ~ ( ord_less @ A @ X @ Y )
% 3.13/1.60 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_55_neq__iff,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( linorder @ A )
% 3.13/1.60 => ! [X: A,Y: A] :
% 3.13/1.60 ( ( X != Y )
% 3.13/1.60 = ( ( ord_less @ A @ X @ Y )
% 3.13/1.60 | ( ord_less @ A @ Y @ X ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_56_order_Oasym,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( order @ A )
% 3.13/1.60 => ! [A5: A,B3: A] :
% 3.13/1.60 ( ( ord_less @ A @ A5 @ B3 )
% 3.13/1.60 => ~ ( ord_less @ A @ B3 @ A5 ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_57_dense,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( dense_order @ A )
% 3.13/1.60 => ! [X: A,Y: A] :
% 3.13/1.60 ( ( ord_less @ A @ X @ Y )
% 3.13/1.60 => ? [Z2: A] :
% 3.13/1.60 ( ( ord_less @ A @ X @ Z2 )
% 3.13/1.60 & ( ord_less @ A @ Z2 @ Y ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_58_less__imp__neq,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( order @ A )
% 3.13/1.60 => ! [X: A,Y: A] :
% 3.13/1.60 ( ( ord_less @ A @ X @ Y )
% 3.13/1.60 => ( X != Y ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_59_strict__monoD,axiom,
% 3.13/1.60 ! [B: $tType,A: $tType] :
% 3.13/1.60 ( ( ( order @ A )
% 3.13/1.60 & ( order @ B ) )
% 3.13/1.60 => ! [F: A > B,X: A,Y: A] :
% 3.13/1.60 ( ( order_strict_mono @ A @ B @ F )
% 3.13/1.60 => ( ( ord_less @ A @ X @ Y )
% 3.13/1.60 => ( ord_less @ B @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_60_strict__monoI,axiom,
% 3.13/1.60 ! [B: $tType,A: $tType] :
% 3.13/1.60 ( ( ( order @ A )
% 3.13/1.60 & ( order @ B ) )
% 3.13/1.60 => ! [F: A > B] :
% 3.13/1.60 ( ! [X3: A,Y2: A] :
% 3.13/1.60 ( ( ord_less @ A @ X3 @ Y2 )
% 3.13/1.60 => ( ord_less @ B @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 3.13/1.60 => ( order_strict_mono @ A @ B @ F ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_61_less__asym,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( preorder @ A )
% 3.13/1.60 => ! [X: A,Y: A] :
% 3.13/1.60 ( ( ord_less @ A @ X @ Y )
% 3.13/1.60 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_62_less__asym_H,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( preorder @ A )
% 3.13/1.60 => ! [A5: A,B3: A] :
% 3.13/1.60 ( ( ord_less @ A @ A5 @ B3 )
% 3.13/1.60 => ~ ( ord_less @ A @ B3 @ A5 ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_63_less__trans,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( preorder @ A )
% 3.13/1.60 => ! [X: A,Y: A,Z: A] :
% 3.13/1.60 ( ( ord_less @ A @ X @ Y )
% 3.13/1.60 => ( ( ord_less @ A @ Y @ Z )
% 3.13/1.60 => ( ord_less @ A @ X @ Z ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_64_less__linear,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( linorder @ A )
% 3.13/1.60 => ! [X: A,Y: A] :
% 3.13/1.60 ( ( ord_less @ A @ X @ Y )
% 3.13/1.60 | ( X = Y )
% 3.13/1.60 | ( ord_less @ A @ Y @ X ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_65_less__irrefl,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( preorder @ A )
% 3.13/1.60 => ! [X: A] :
% 3.13/1.60 ~ ( ord_less @ A @ X @ X ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_66_ord__eq__less__trans,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( ord @ A )
% 3.13/1.60 => ! [A5: A,B3: A,C2: A] :
% 3.13/1.60 ( ( A5 = B3 )
% 3.13/1.60 => ( ( ord_less @ A @ B3 @ C2 )
% 3.13/1.60 => ( ord_less @ A @ A5 @ C2 ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_67_ord__less__eq__trans,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( ord @ A )
% 3.13/1.60 => ! [A5: A,B3: A,C2: A] :
% 3.13/1.60 ( ( ord_less @ A @ A5 @ B3 )
% 3.13/1.60 => ( ( B3 = C2 )
% 3.13/1.60 => ( ord_less @ A @ A5 @ C2 ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_68_dual__order_Oasym,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( order @ A )
% 3.13/1.60 => ! [B3: A,A5: A] :
% 3.13/1.60 ( ( ord_less @ A @ B3 @ A5 )
% 3.13/1.60 => ~ ( ord_less @ A @ A5 @ B3 ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_69_less__imp__not__eq,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( order @ A )
% 3.13/1.60 => ! [X: A,Y: A] :
% 3.13/1.60 ( ( ord_less @ A @ X @ Y )
% 3.13/1.60 => ( X != Y ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_70_strict__mono__def,axiom,
% 3.13/1.60 ! [B: $tType,A: $tType] :
% 3.13/1.60 ( ( ( order @ A )
% 3.13/1.60 & ( order @ B ) )
% 3.13/1.60 => ( ( order_strict_mono @ A @ B )
% 3.13/1.60 = ( ^ [F2: A > B] :
% 3.13/1.60 ! [X2: A,Y3: A] :
% 3.13/1.60 ( ( ord_less @ A @ X2 @ Y3 )
% 3.13/1.60 => ( ord_less @ B @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_71_less__not__sym,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( preorder @ A )
% 3.13/1.60 => ! [X: A,Y: A] :
% 3.13/1.60 ( ( ord_less @ A @ X @ Y )
% 3.13/1.60 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_72_less__induct,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( wellorder @ A )
% 3.13/1.60 => ! [P: A > $o,A5: A] :
% 3.13/1.60 ( ! [X3: A] :
% 3.13/1.60 ( ! [Y4: A] :
% 3.13/1.60 ( ( ord_less @ A @ Y4 @ X3 )
% 3.13/1.60 => ( P @ Y4 ) )
% 3.13/1.60 => ( P @ X3 ) )
% 3.13/1.60 => ( P @ A5 ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_73_antisym__conv3,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( linorder @ A )
% 3.13/1.60 => ! [Y: A,X: A] :
% 3.13/1.60 ( ~ ( ord_less @ A @ Y @ X )
% 3.13/1.60 => ( ( ~ ( ord_less @ A @ X @ Y ) )
% 3.13/1.60 = ( X = Y ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_74_less__imp__not__eq2,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( order @ A )
% 3.13/1.60 => ! [X: A,Y: A] :
% 3.13/1.60 ( ( ord_less @ A @ X @ Y )
% 3.13/1.60 => ( Y != X ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_75_less__imp__triv,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( preorder @ A )
% 3.13/1.60 => ! [X: A,Y: A,P: $o] :
% 3.13/1.60 ( ( ord_less @ A @ X @ Y )
% 3.13/1.60 => ( ( ord_less @ A @ Y @ X )
% 3.13/1.60 => P ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_76_linorder__cases,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( linorder @ A )
% 3.13/1.60 => ! [X: A,Y: A] :
% 3.13/1.60 ( ~ ( ord_less @ A @ X @ Y )
% 3.13/1.60 => ( ( X != Y )
% 3.13/1.60 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_77_strict__mono__eq,axiom,
% 3.13/1.60 ! [B: $tType,A: $tType] :
% 3.13/1.60 ( ( ( linorder @ A )
% 3.13/1.60 & ( order @ B ) )
% 3.13/1.60 => ! [F: A > B,X: A,Y: A] :
% 3.13/1.60 ( ( order_strict_mono @ A @ B @ F )
% 3.13/1.60 => ( ( ( F @ X )
% 3.13/1.60 = ( F @ Y ) )
% 3.13/1.60 = ( X = Y ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_78_dual__order_Oirrefl,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( order @ A )
% 3.13/1.60 => ! [A5: A] :
% 3.13/1.60 ~ ( ord_less @ A @ A5 @ A5 ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_79_order_Ostrict__trans,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( order @ A )
% 3.13/1.60 => ! [A5: A,B3: A,C2: A] :
% 3.13/1.60 ( ( ord_less @ A @ A5 @ B3 )
% 3.13/1.60 => ( ( ord_less @ A @ B3 @ C2 )
% 3.13/1.60 => ( ord_less @ A @ A5 @ C2 ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_80_strict__mono__less,axiom,
% 3.13/1.60 ! [B: $tType,A: $tType] :
% 3.13/1.60 ( ( ( linorder @ A )
% 3.13/1.60 & ( order @ B ) )
% 3.13/1.60 => ! [F: A > B,X: A,Y: A] :
% 3.13/1.60 ( ( order_strict_mono @ A @ B @ F )
% 3.13/1.60 => ( ( ord_less @ B @ ( F @ X ) @ ( F @ Y ) )
% 3.13/1.60 = ( ord_less @ A @ X @ Y ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_81_less__imp__not__less,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( preorder @ A )
% 3.13/1.60 => ! [X: A,Y: A] :
% 3.13/1.60 ( ( ord_less @ A @ X @ Y )
% 3.13/1.60 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_82_exists__least__iff,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( wellorder @ A )
% 3.13/1.60 => ( ( ^ [P2: A > $o] :
% 3.13/1.60 ? [X5: A] : ( P2 @ X5 ) )
% 3.13/1.60 = ( ^ [P3: A > $o] :
% 3.13/1.60 ? [N3: A] :
% 3.13/1.60 ( ( P3 @ N3 )
% 3.13/1.60 & ! [M2: A] :
% 3.13/1.60 ( ( ord_less @ A @ M2 @ N3 )
% 3.13/1.60 => ~ ( P3 @ M2 ) ) ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_83_linorder__less__wlog,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( linorder @ A )
% 3.13/1.60 => ! [P: A > A > $o,A5: A,B3: A] :
% 3.13/1.60 ( ! [A4: A,B4: A] :
% 3.13/1.60 ( ( ord_less @ A @ A4 @ B4 )
% 3.13/1.60 => ( P @ A4 @ B4 ) )
% 3.13/1.60 => ( ! [A4: A] : ( P @ A4 @ A4 )
% 3.13/1.60 => ( ! [A4: A,B4: A] :
% 3.13/1.60 ( ( P @ B4 @ A4 )
% 3.13/1.60 => ( P @ A4 @ B4 ) )
% 3.13/1.60 => ( P @ A5 @ B3 ) ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_84_dual__order_Ostrict__trans,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( order @ A )
% 3.13/1.60 => ! [B3: A,A5: A,C2: A] :
% 3.13/1.60 ( ( ord_less @ A @ B3 @ A5 )
% 3.13/1.60 => ( ( ord_less @ A @ C2 @ B3 )
% 3.13/1.60 => ( ord_less @ A @ C2 @ A5 ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_85_not__less__iff__gr__or__eq,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( linorder @ A )
% 3.13/1.60 => ! [X: A,Y: A] :
% 3.13/1.60 ( ( ~ ( ord_less @ A @ X @ Y ) )
% 3.13/1.60 = ( ( ord_less @ A @ Y @ X )
% 3.13/1.60 | ( X = Y ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_86_order_Ostrict__implies__not__eq,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( order @ A )
% 3.13/1.60 => ! [A5: A,B3: A] :
% 3.13/1.60 ( ( ord_less @ A @ A5 @ B3 )
% 3.13/1.60 => ( A5 != B3 ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_87_dual__order_Ostrict__implies__not__eq,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( order @ A )
% 3.13/1.60 => ! [B3: A,A5: A] :
% 3.13/1.60 ( ( ord_less @ A @ B3 @ A5 )
% 3.13/1.60 => ( A5 != B3 ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_88_strict__mono__on__eqD,axiom,
% 3.13/1.60 ! [D: $tType,C: $tType] :
% 3.13/1.60 ( ( ( linorder @ C )
% 3.13/1.60 & ( preorder @ D ) )
% 3.13/1.60 => ! [F: C > D,A2: set @ C,X: C,Y: C] :
% 3.13/1.60 ( ( strict_mono_on @ C @ D @ F @ A2 )
% 3.13/1.60 => ( ( ( F @ X )
% 3.13/1.60 = ( F @ Y ) )
% 3.13/1.60 => ( ( member @ C @ X @ A2 )
% 3.13/1.60 => ( ( member @ C @ Y @ A2 )
% 3.13/1.60 => ( Y = X ) ) ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_89_inj__on__strict__subset,axiom,
% 3.13/1.60 ! [B: $tType,A: $tType,F: A > B,B2: set @ A,A2: set @ A] :
% 3.13/1.60 ( ( inj_on @ A @ B @ F @ B2 )
% 3.13/1.60 => ( ( ord_less @ ( set @ A ) @ A2 @ B2 )
% 3.13/1.60 => ( ord_less @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) @ ( image @ A @ B @ F @ B2 ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_90_inj__on__the__inv__into,axiom,
% 3.13/1.60 ! [B: $tType,A: $tType,F: A > B,A2: set @ A] :
% 3.13/1.60 ( ( inj_on @ A @ B @ F @ A2 )
% 3.13/1.60 => ( inj_on @ B @ A @ ( the_inv_into @ A @ B @ A2 @ F ) @ ( image @ A @ B @ F @ A2 ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_91_f__the__inv__into__f,axiom,
% 3.13/1.60 ! [A: $tType,B: $tType,F: A > B,A2: set @ A,Y: B] :
% 3.13/1.60 ( ( inj_on @ A @ B @ F @ A2 )
% 3.13/1.60 => ( ( member @ B @ Y @ ( image @ A @ B @ F @ A2 ) )
% 3.13/1.60 => ( ( F @ ( the_inv_into @ A @ B @ A2 @ F @ Y ) )
% 3.13/1.60 = Y ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_92_strict__mono__inv,axiom,
% 3.13/1.60 ! [A: $tType,B: $tType] :
% 3.13/1.60 ( ( ( linorder @ B )
% 3.13/1.60 & ( linorder @ A ) )
% 3.13/1.60 => ! [F: A > B,G: B > A] :
% 3.13/1.60 ( ( order_strict_mono @ A @ B @ F )
% 3.13/1.60 => ( ( ( image @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 3.13/1.60 = ( top_top @ ( set @ B ) ) )
% 3.13/1.60 => ( ! [X3: A] :
% 3.13/1.60 ( ( G @ ( F @ X3 ) )
% 3.13/1.60 = X3 )
% 3.13/1.60 => ( order_strict_mono @ B @ A @ G ) ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_93_surjD,axiom,
% 3.13/1.60 ! [A: $tType,B: $tType,F: B > A,Y: A] :
% 3.13/1.60 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 3.13/1.60 = ( top_top @ ( set @ A ) ) )
% 3.13/1.60 => ? [X3: B] :
% 3.13/1.60 ( Y
% 3.13/1.60 = ( F @ X3 ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_94_surjE,axiom,
% 3.13/1.60 ! [A: $tType,B: $tType,F: B > A,Y: A] :
% 3.13/1.60 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 3.13/1.60 = ( top_top @ ( set @ A ) ) )
% 3.13/1.60 => ~ ! [X3: B] :
% 3.13/1.60 ( Y
% 3.13/1.60 != ( F @ X3 ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_95_surjI,axiom,
% 3.13/1.60 ! [B: $tType,A: $tType,G: B > A,F: A > B] :
% 3.13/1.60 ( ! [X3: A] :
% 3.13/1.60 ( ( G @ ( F @ X3 ) )
% 3.13/1.60 = X3 )
% 3.13/1.60 => ( ( image @ B @ A @ G @ ( top_top @ ( set @ B ) ) )
% 3.13/1.60 = ( top_top @ ( set @ A ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_96_rangeI,axiom,
% 3.13/1.60 ! [A: $tType,B: $tType,F: B > A,X: B] : ( member @ A @ ( F @ X ) @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_97_surj__def,axiom,
% 3.13/1.60 ! [B: $tType,A: $tType,F: B > A] :
% 3.13/1.60 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 3.13/1.60 = ( top_top @ ( set @ A ) ) )
% 3.13/1.60 = ( ! [Y3: A] :
% 3.13/1.60 ? [X2: B] :
% 3.13/1.60 ( Y3
% 3.13/1.60 = ( F @ X2 ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_98_range__eqI,axiom,
% 3.13/1.60 ! [A: $tType,B: $tType,B3: A,F: B > A,X: B] :
% 3.13/1.60 ( ( B3
% 3.13/1.60 = ( F @ X ) )
% 3.13/1.60 => ( member @ A @ B3 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_99_inj__on__image__iff,axiom,
% 3.13/1.60 ! [B: $tType,A: $tType,A2: set @ A,G: A > B,F: A > A] :
% 3.13/1.60 ( ! [X3: A] :
% 3.13/1.60 ( ( member @ A @ X3 @ A2 )
% 3.13/1.60 => ! [Xa: A] :
% 3.13/1.60 ( ( member @ A @ Xa @ A2 )
% 3.13/1.60 => ( ( ( G @ ( F @ X3 ) )
% 3.13/1.60 = ( G @ ( F @ Xa ) ) )
% 3.13/1.60 = ( ( G @ X3 )
% 3.13/1.60 = ( G @ Xa ) ) ) ) )
% 3.13/1.60 => ( ( inj_on @ A @ A @ F @ A2 )
% 3.13/1.60 => ( ( inj_on @ A @ B @ G @ ( image @ A @ A @ F @ A2 ) )
% 3.13/1.60 = ( inj_on @ A @ B @ G @ A2 ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_100_top_Oextremum__strict,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( order_top @ A )
% 3.13/1.60 => ! [A5: A] :
% 3.13/1.60 ~ ( ord_less @ A @ ( top_top @ A ) @ A5 ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_101_top_Onot__eq__extremum,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( order_top @ A )
% 3.13/1.60 => ! [A5: A] :
% 3.13/1.60 ( ( A5
% 3.13/1.60 != ( top_top @ A ) )
% 3.13/1.60 = ( ord_less @ A @ A5 @ ( top_top @ A ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_102_inj__on__add,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( cancel_semigroup_add @ A )
% 3.13/1.60 => ! [A5: A,A2: set @ A] : ( inj_on @ A @ A @ ( plus_plus @ A @ A5 ) @ A2 ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_103_the__inv__into__f__f,axiom,
% 3.13/1.60 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,X: A] :
% 3.13/1.60 ( ( inj_on @ A @ B @ F @ A2 )
% 3.13/1.60 => ( ( member @ A @ X @ A2 )
% 3.13/1.60 => ( ( the_inv_into @ A @ B @ A2 @ F @ ( F @ X ) )
% 3.13/1.60 = X ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_104_the__inv__into__f__eq,axiom,
% 3.13/1.60 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,X: A,Y: B] :
% 3.13/1.60 ( ( inj_on @ A @ B @ F @ A2 )
% 3.13/1.60 => ( ( ( F @ X )
% 3.13/1.60 = Y )
% 3.13/1.60 => ( ( member @ A @ X @ A2 )
% 3.13/1.60 => ( ( the_inv_into @ A @ B @ A2 @ F @ Y )
% 3.13/1.60 = X ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_105_add__less__cancel__right,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( ordere236663937imp_le @ A )
% 3.13/1.60 => ! [A5: A,C2: A,B3: A] :
% 3.13/1.60 ( ( ord_less @ A @ ( plus_plus @ A @ A5 @ C2 ) @ ( plus_plus @ A @ B3 @ C2 ) )
% 3.13/1.60 = ( ord_less @ A @ A5 @ B3 ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_106_add__less__cancel__left,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( ordere236663937imp_le @ A )
% 3.13/1.60 => ! [C2: A,A5: A,B3: A] :
% 3.13/1.60 ( ( ord_less @ A @ ( plus_plus @ A @ C2 @ A5 ) @ ( plus_plus @ A @ C2 @ B3 ) )
% 3.13/1.60 = ( ord_less @ A @ A5 @ B3 ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_107_add__right__cancel,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( cancel_semigroup_add @ A )
% 3.13/1.60 => ! [B3: A,A5: A,C2: A] :
% 3.13/1.60 ( ( ( plus_plus @ A @ B3 @ A5 )
% 3.13/1.60 = ( plus_plus @ A @ C2 @ A5 ) )
% 3.13/1.60 = ( B3 = C2 ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_108_add__left__cancel,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( cancel_semigroup_add @ A )
% 3.13/1.60 => ! [A5: A,B3: A,C2: A] :
% 3.13/1.60 ( ( ( plus_plus @ A @ A5 @ B3 )
% 3.13/1.60 = ( plus_plus @ A @ A5 @ C2 ) )
% 3.13/1.60 = ( B3 = C2 ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_109_inj__apfst,axiom,
% 3.13/1.60 ! [B: $tType,C: $tType,A: $tType,F: A > C] :
% 3.13/1.60 ( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ C @ B ) @ ( product_apfst @ A @ C @ B @ F ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
% 3.13/1.60 = ( inj_on @ A @ C @ F @ ( top_top @ ( set @ A ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_110_add__less__imp__less__right,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( ordere236663937imp_le @ A )
% 3.13/1.60 => ! [A5: A,C2: A,B3: A] :
% 3.13/1.60 ( ( ord_less @ A @ ( plus_plus @ A @ A5 @ C2 ) @ ( plus_plus @ A @ B3 @ C2 ) )
% 3.13/1.60 => ( ord_less @ A @ A5 @ B3 ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_111_add__less__imp__less__left,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( ordere236663937imp_le @ A )
% 3.13/1.60 => ! [C2: A,A5: A,B3: A] :
% 3.13/1.60 ( ( ord_less @ A @ ( plus_plus @ A @ C2 @ A5 ) @ ( plus_plus @ A @ C2 @ B3 ) )
% 3.13/1.60 => ( ord_less @ A @ A5 @ B3 ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_112_add__strict__right__mono,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( ordere223160158up_add @ A )
% 3.13/1.60 => ! [A5: A,B3: A,C2: A] :
% 3.13/1.60 ( ( ord_less @ A @ A5 @ B3 )
% 3.13/1.60 => ( ord_less @ A @ ( plus_plus @ A @ A5 @ C2 ) @ ( plus_plus @ A @ B3 @ C2 ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_113_add__strict__left__mono,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( ordere223160158up_add @ A )
% 3.13/1.60 => ! [A5: A,B3: A,C2: A] :
% 3.13/1.60 ( ( ord_less @ A @ A5 @ B3 )
% 3.13/1.60 => ( ord_less @ A @ ( plus_plus @ A @ C2 @ A5 ) @ ( plus_plus @ A @ C2 @ B3 ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_114_psubsetD,axiom,
% 3.13/1.60 ! [A: $tType,A2: set @ A,B2: set @ A,C2: A] :
% 3.13/1.60 ( ( ord_less @ ( set @ A ) @ A2 @ B2 )
% 3.13/1.60 => ( ( member @ A @ C2 @ A2 )
% 3.13/1.60 => ( member @ A @ C2 @ B2 ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_115_psubset__trans,axiom,
% 3.13/1.60 ! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
% 3.13/1.60 ( ( ord_less @ ( set @ A ) @ A2 @ B2 )
% 3.13/1.60 => ( ( ord_less @ ( set @ A ) @ B2 @ C3 )
% 3.13/1.60 => ( ord_less @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_116_apsnd__apfst__commute,axiom,
% 3.13/1.60 ! [A: $tType,B: $tType,C: $tType,D: $tType,F: C > B,G: D > A,P4: product_prod @ D @ C] :
% 3.13/1.60 ( ( product_apsnd @ C @ B @ A @ F @ ( product_apfst @ D @ A @ C @ G @ P4 ) )
% 3.13/1.60 = ( product_apfst @ D @ A @ B @ G @ ( product_apsnd @ C @ B @ D @ F @ P4 ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_117_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( ab_semigroup_add @ A )
% 3.13/1.60 => ! [A5: A,B3: A,C2: A] :
% 3.13/1.60 ( ( plus_plus @ A @ ( plus_plus @ A @ A5 @ B3 ) @ C2 )
% 3.13/1.60 = ( plus_plus @ A @ A5 @ ( plus_plus @ A @ B3 @ C2 ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_118_add__mono__thms__linordered__semiring_I4_J,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( ordere779506340up_add @ A )
% 3.13/1.60 => ! [I: A,J: A,K: A,L: A] :
% 3.13/1.60 ( ( ( I = J )
% 3.13/1.60 & ( K = L ) )
% 3.13/1.60 => ( ( plus_plus @ A @ I @ K )
% 3.13/1.60 = ( plus_plus @ A @ J @ L ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_119_group__cancel_Oadd1,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( comm_monoid_add @ A )
% 3.13/1.60 => ! [A2: A,K: A,A5: A,B3: A] :
% 3.13/1.60 ( ( A2
% 3.13/1.60 = ( plus_plus @ A @ K @ A5 ) )
% 3.13/1.60 => ( ( plus_plus @ A @ A2 @ B3 )
% 3.13/1.60 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A5 @ B3 ) ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_120_group__cancel_Oadd2,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( comm_monoid_add @ A )
% 3.13/1.60 => ! [B2: A,K: A,B3: A,A5: A] :
% 3.13/1.60 ( ( B2
% 3.13/1.60 = ( plus_plus @ A @ K @ B3 ) )
% 3.13/1.60 => ( ( plus_plus @ A @ A5 @ B2 )
% 3.13/1.60 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A5 @ B3 ) ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_121_add_Oassoc,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( semigroup_add @ A )
% 3.13/1.60 => ! [A5: A,B3: A,C2: A] :
% 3.13/1.60 ( ( plus_plus @ A @ ( plus_plus @ A @ A5 @ B3 ) @ C2 )
% 3.13/1.60 = ( plus_plus @ A @ A5 @ ( plus_plus @ A @ B3 @ C2 ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_122_add_Oleft__cancel,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( group_add @ A )
% 3.13/1.60 => ! [A5: A,B3: A,C2: A] :
% 3.13/1.60 ( ( ( plus_plus @ A @ A5 @ B3 )
% 3.13/1.60 = ( plus_plus @ A @ A5 @ C2 ) )
% 3.13/1.60 = ( B3 = C2 ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_123_add_Oright__cancel,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( group_add @ A )
% 3.13/1.60 => ! [B3: A,A5: A,C2: A] :
% 3.13/1.60 ( ( ( plus_plus @ A @ B3 @ A5 )
% 3.13/1.60 = ( plus_plus @ A @ C2 @ A5 ) )
% 3.13/1.60 = ( B3 = C2 ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_124_add_Ocommute,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( ab_semigroup_add @ A )
% 3.13/1.60 => ( ( plus_plus @ A )
% 3.13/1.60 = ( ^ [A6: A,B5: A] : ( plus_plus @ A @ B5 @ A6 ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_125_add_Oleft__commute,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( ab_semigroup_add @ A )
% 3.13/1.60 => ! [B3: A,A5: A,C2: A] :
% 3.13/1.60 ( ( plus_plus @ A @ B3 @ ( plus_plus @ A @ A5 @ C2 ) )
% 3.13/1.60 = ( plus_plus @ A @ A5 @ ( plus_plus @ A @ B3 @ C2 ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_126_add__left__imp__eq,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( cancel_semigroup_add @ A )
% 3.13/1.60 => ! [A5: A,B3: A,C2: A] :
% 3.13/1.60 ( ( ( plus_plus @ A @ A5 @ B3 )
% 3.13/1.60 = ( plus_plus @ A @ A5 @ C2 ) )
% 3.13/1.60 => ( B3 = C2 ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_127_add__right__imp__eq,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( cancel_semigroup_add @ A )
% 3.13/1.60 => ! [B3: A,A5: A,C2: A] :
% 3.13/1.60 ( ( ( plus_plus @ A @ B3 @ A5 )
% 3.13/1.60 = ( plus_plus @ A @ C2 @ A5 ) )
% 3.13/1.60 => ( B3 = C2 ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_128_add__mono__thms__linordered__field_I5_J,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( ordere223160158up_add @ A )
% 3.13/1.60 => ! [I: A,J: A,K: A,L: A] :
% 3.13/1.60 ( ( ( ord_less @ A @ I @ J )
% 3.13/1.60 & ( ord_less @ A @ K @ L ) )
% 3.13/1.60 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_129_add__mono__thms__linordered__field_I2_J,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( ordere223160158up_add @ A )
% 3.13/1.60 => ! [I: A,J: A,K: A,L: A] :
% 3.13/1.60 ( ( ( I = J )
% 3.13/1.60 & ( ord_less @ A @ K @ L ) )
% 3.13/1.60 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_130_add__mono__thms__linordered__field_I1_J,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( ordere223160158up_add @ A )
% 3.13/1.60 => ! [I: A,J: A,K: A,L: A] :
% 3.13/1.60 ( ( ( ord_less @ A @ I @ J )
% 3.13/1.60 & ( K = L ) )
% 3.13/1.60 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_131_add__strict__mono,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( strict2144017051up_add @ A )
% 3.13/1.60 => ! [A5: A,B3: A,C2: A,D2: A] :
% 3.13/1.60 ( ( ord_less @ A @ A5 @ B3 )
% 3.13/1.60 => ( ( ord_less @ A @ C2 @ D2 )
% 3.13/1.60 => ( ord_less @ A @ ( plus_plus @ A @ A5 @ C2 ) @ ( plus_plus @ A @ B3 @ D2 ) ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_132_top__conj_I1_J,axiom,
% 3.13/1.60 ! [A: $tType,X: A,P: $o] :
% 3.13/1.60 ( ( ( top_top @ ( A > $o ) @ X )
% 3.13/1.60 & P )
% 3.13/1.60 = P ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_133_top__conj_I2_J,axiom,
% 3.13/1.60 ! [A: $tType,P: $o,X: A] :
% 3.13/1.60 ( ( P
% 3.13/1.60 & ( top_top @ ( A > $o ) @ X ) )
% 3.13/1.60 = P ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_134_tv_Osimps_I8_J,axiom,
% 3.13/1.60 ! [A: $tType,F1: $o > A,F22: nat > A,X22: nat] :
% 3.13/1.60 ( ( paraco152590079rec_tv @ A @ F1 @ F22 @ ( paraco676387099_Indet @ X22 ) )
% 3.13/1.60 = ( F22 @ X22 ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_135_the__inv__into__into,axiom,
% 3.13/1.60 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,X: B,B2: set @ A] :
% 3.13/1.60 ( ( inj_on @ A @ B @ F @ A2 )
% 3.13/1.60 => ( ( member @ B @ X @ ( image @ A @ B @ F @ A2 ) )
% 3.13/1.60 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 3.13/1.60 => ( member @ A @ ( the_inv_into @ A @ B @ A2 @ F @ X ) @ B2 ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_136_prod_Oinj__map,axiom,
% 3.13/1.60 ! [D: $tType,C: $tType,B: $tType,A: $tType,F1: A > C,F22: B > D] :
% 3.13/1.60 ( ( inj_on @ A @ C @ F1 @ ( top_top @ ( set @ A ) ) )
% 3.13/1.60 => ( ( inj_on @ B @ D @ F22 @ ( top_top @ ( set @ B ) ) )
% 3.13/1.60 => ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F1 @ F22 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_137_order__refl,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( preorder @ A )
% 3.13/1.60 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_138_subsetI,axiom,
% 3.13/1.60 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 3.13/1.60 ( ! [X3: A] :
% 3.13/1.60 ( ( member @ A @ X3 @ A2 )
% 3.13/1.60 => ( member @ A @ X3 @ B2 ) )
% 3.13/1.60 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_139_subset__antisym,axiom,
% 3.13/1.60 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 3.13/1.60 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 3.13/1.60 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 3.13/1.60 => ( A2 = B2 ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_140_add__le__cancel__right,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( ordere236663937imp_le @ A )
% 3.13/1.60 => ! [A5: A,C2: A,B3: A] :
% 3.13/1.60 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A5 @ C2 ) @ ( plus_plus @ A @ B3 @ C2 ) )
% 3.13/1.60 = ( ord_less_eq @ A @ A5 @ B3 ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_141_add__le__cancel__left,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( ordere236663937imp_le @ A )
% 3.13/1.60 => ! [C2: A,A5: A,B3: A] :
% 3.13/1.60 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A5 ) @ ( plus_plus @ A @ C2 @ B3 ) )
% 3.13/1.60 = ( ord_less_eq @ A @ A5 @ B3 ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_142_psubsetI,axiom,
% 3.13/1.60 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 3.13/1.60 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 3.13/1.60 => ( ( A2 != B2 )
% 3.13/1.60 => ( ord_less @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_143_psubsetE,axiom,
% 3.13/1.60 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 3.13/1.60 ( ( ord_less @ ( set @ A ) @ A2 @ B2 )
% 3.13/1.60 => ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 3.13/1.60 => ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_144_psubset__eq,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( ord_less @ ( set @ A ) )
% 3.13/1.60 = ( ^ [A3: set @ A,B6: set @ A] :
% 3.13/1.60 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
% 3.13/1.60 & ( A3 != B6 ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_145_psubset__imp__subset,axiom,
% 3.13/1.60 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 3.13/1.60 ( ( ord_less @ ( set @ A ) @ A2 @ B2 )
% 3.13/1.60 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_146_psubset__subset__trans,axiom,
% 3.13/1.60 ! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
% 3.13/1.60 ( ( ord_less @ ( set @ A ) @ A2 @ B2 )
% 3.13/1.60 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
% 3.13/1.60 => ( ord_less @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_147_subset__not__subset__eq,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( ord_less @ ( set @ A ) )
% 3.13/1.60 = ( ^ [A3: set @ A,B6: set @ A] :
% 3.13/1.60 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
% 3.13/1.60 & ~ ( ord_less_eq @ ( set @ A ) @ B6 @ A3 ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_148_subset__psubset__trans,axiom,
% 3.13/1.60 ! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
% 3.13/1.60 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 3.13/1.60 => ( ( ord_less @ ( set @ A ) @ B2 @ C3 )
% 3.13/1.60 => ( ord_less @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_149_subset__iff__psubset__eq,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( ord_less_eq @ ( set @ A ) )
% 3.13/1.60 = ( ^ [A3: set @ A,B6: set @ A] :
% 3.13/1.60 ( ( ord_less @ ( set @ A ) @ A3 @ B6 )
% 3.13/1.60 | ( A3 = B6 ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_150_add__mono__thms__linordered__semiring_I3_J,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( ordere779506340up_add @ A )
% 3.13/1.60 => ! [I: A,J: A,K: A,L: A] :
% 3.13/1.60 ( ( ( ord_less_eq @ A @ I @ J )
% 3.13/1.60 & ( K = L ) )
% 3.13/1.60 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_151_add__mono__thms__linordered__semiring_I2_J,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( ordere779506340up_add @ A )
% 3.13/1.60 => ! [I: A,J: A,K: A,L: A] :
% 3.13/1.60 ( ( ( I = J )
% 3.13/1.60 & ( ord_less_eq @ A @ K @ L ) )
% 3.13/1.60 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_152_add__mono__thms__linordered__semiring_I1_J,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( ordere779506340up_add @ A )
% 3.13/1.60 => ! [I: A,J: A,K: A,L: A] :
% 3.13/1.60 ( ( ( ord_less_eq @ A @ I @ J )
% 3.13/1.60 & ( ord_less_eq @ A @ K @ L ) )
% 3.13/1.60 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_153_add__mono,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( ordere779506340up_add @ A )
% 3.13/1.60 => ! [A5: A,B3: A,C2: A,D2: A] :
% 3.13/1.60 ( ( ord_less_eq @ A @ A5 @ B3 )
% 3.13/1.60 => ( ( ord_less_eq @ A @ C2 @ D2 )
% 3.13/1.60 => ( ord_less_eq @ A @ ( plus_plus @ A @ A5 @ C2 ) @ ( plus_plus @ A @ B3 @ D2 ) ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_154_add__left__mono,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( ordere779506340up_add @ A )
% 3.13/1.60 => ! [A5: A,B3: A,C2: A] :
% 3.13/1.60 ( ( ord_less_eq @ A @ A5 @ B3 )
% 3.13/1.60 => ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A5 ) @ ( plus_plus @ A @ C2 @ B3 ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_155_less__eqE,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( canoni770627133id_add @ A )
% 3.13/1.60 => ! [A5: A,B3: A] :
% 3.13/1.60 ( ( ord_less_eq @ A @ A5 @ B3 )
% 3.13/1.60 => ~ ! [C4: A] :
% 3.13/1.60 ( B3
% 3.13/1.60 != ( plus_plus @ A @ A5 @ C4 ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_156_add__right__mono,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( ordere779506340up_add @ A )
% 3.13/1.60 => ! [A5: A,B3: A,C2: A] :
% 3.13/1.60 ( ( ord_less_eq @ A @ A5 @ B3 )
% 3.13/1.60 => ( ord_less_eq @ A @ ( plus_plus @ A @ A5 @ C2 ) @ ( plus_plus @ A @ B3 @ C2 ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_157_le__iff__add,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( canoni770627133id_add @ A )
% 3.13/1.60 => ( ( ord_less_eq @ A )
% 3.13/1.60 = ( ^ [A6: A,B5: A] :
% 3.13/1.60 ? [C5: A] :
% 3.13/1.60 ( B5
% 3.13/1.60 = ( plus_plus @ A @ A6 @ C5 ) ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_158_add__le__imp__le__left,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( ordere236663937imp_le @ A )
% 3.13/1.60 => ! [C2: A,A5: A,B3: A] :
% 3.13/1.60 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A5 ) @ ( plus_plus @ A @ C2 @ B3 ) )
% 3.13/1.60 => ( ord_less_eq @ A @ A5 @ B3 ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_159_add__le__imp__le__right,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( ordere236663937imp_le @ A )
% 3.13/1.60 => ! [A5: A,C2: A,B3: A] :
% 3.13/1.60 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A5 @ C2 ) @ ( plus_plus @ A @ B3 @ C2 ) )
% 3.13/1.60 => ( ord_less_eq @ A @ A5 @ B3 ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_160_order_Onot__eq__order__implies__strict,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( order @ A )
% 3.13/1.60 => ! [A5: A,B3: A] :
% 3.13/1.60 ( ( A5 != B3 )
% 3.13/1.60 => ( ( ord_less_eq @ A @ A5 @ B3 )
% 3.13/1.60 => ( ord_less @ A @ A5 @ B3 ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_161_dual__order_Ostrict__implies__order,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( order @ A )
% 3.13/1.60 => ! [B3: A,A5: A] :
% 3.13/1.60 ( ( ord_less @ A @ B3 @ A5 )
% 3.13/1.60 => ( ord_less_eq @ A @ B3 @ A5 ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_162_dual__order_Ostrict__iff__order,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( order @ A )
% 3.13/1.60 => ( ( ord_less @ A )
% 3.13/1.60 = ( ^ [B5: A,A6: A] :
% 3.13/1.60 ( ( ord_less_eq @ A @ B5 @ A6 )
% 3.13/1.60 & ( A6 != B5 ) ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_163_dual__order_Oorder__iff__strict,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( order @ A )
% 3.13/1.60 => ( ( ord_less_eq @ A )
% 3.13/1.60 = ( ^ [B5: A,A6: A] :
% 3.13/1.60 ( ( ord_less @ A @ B5 @ A6 )
% 3.13/1.60 | ( A6 = B5 ) ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_164_order_Ostrict__implies__order,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( order @ A )
% 3.13/1.60 => ! [A5: A,B3: A] :
% 3.13/1.60 ( ( ord_less @ A @ A5 @ B3 )
% 3.13/1.60 => ( ord_less_eq @ A @ A5 @ B3 ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_165_dense__le__bounded,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( dense_linorder @ A )
% 3.13/1.60 => ! [X: A,Y: A,Z: A] :
% 3.13/1.60 ( ( ord_less @ A @ X @ Y )
% 3.13/1.60 => ( ! [W: A] :
% 3.13/1.60 ( ( ord_less @ A @ X @ W )
% 3.13/1.60 => ( ( ord_less @ A @ W @ Y )
% 3.13/1.60 => ( ord_less_eq @ A @ W @ Z ) ) )
% 3.13/1.60 => ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_166_dense__ge__bounded,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( dense_linorder @ A )
% 3.13/1.60 => ! [Z: A,X: A,Y: A] :
% 3.13/1.60 ( ( ord_less @ A @ Z @ X )
% 3.13/1.60 => ( ! [W: A] :
% 3.13/1.60 ( ( ord_less @ A @ Z @ W )
% 3.13/1.60 => ( ( ord_less @ A @ W @ X )
% 3.13/1.60 => ( ord_less_eq @ A @ Y @ W ) ) )
% 3.13/1.60 => ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_167_dual__order_Ostrict__trans2,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( order @ A )
% 3.13/1.60 => ! [B3: A,A5: A,C2: A] :
% 3.13/1.60 ( ( ord_less @ A @ B3 @ A5 )
% 3.13/1.60 => ( ( ord_less_eq @ A @ C2 @ B3 )
% 3.13/1.60 => ( ord_less @ A @ C2 @ A5 ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_168_dual__order_Ostrict__trans1,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( order @ A )
% 3.13/1.60 => ! [B3: A,A5: A,C2: A] :
% 3.13/1.60 ( ( ord_less_eq @ A @ B3 @ A5 )
% 3.13/1.60 => ( ( ord_less @ A @ C2 @ B3 )
% 3.13/1.60 => ( ord_less @ A @ C2 @ A5 ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_169_order_Ostrict__iff__order,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( order @ A )
% 3.13/1.60 => ( ( ord_less @ A )
% 3.13/1.60 = ( ^ [A6: A,B5: A] :
% 3.13/1.60 ( ( ord_less_eq @ A @ A6 @ B5 )
% 3.13/1.60 & ( A6 != B5 ) ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_170_order_Oorder__iff__strict,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( order @ A )
% 3.13/1.60 => ( ( ord_less_eq @ A )
% 3.13/1.60 = ( ^ [A6: A,B5: A] :
% 3.13/1.60 ( ( ord_less @ A @ A6 @ B5 )
% 3.13/1.60 | ( A6 = B5 ) ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_171_order_Ostrict__trans2,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( order @ A )
% 3.13/1.60 => ! [A5: A,B3: A,C2: A] :
% 3.13/1.60 ( ( ord_less @ A @ A5 @ B3 )
% 3.13/1.60 => ( ( ord_less_eq @ A @ B3 @ C2 )
% 3.13/1.60 => ( ord_less @ A @ A5 @ C2 ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_172_order_Ostrict__trans1,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( order @ A )
% 3.13/1.60 => ! [A5: A,B3: A,C2: A] :
% 3.13/1.60 ( ( ord_less_eq @ A @ A5 @ B3 )
% 3.13/1.60 => ( ( ord_less @ A @ B3 @ C2 )
% 3.13/1.60 => ( ord_less @ A @ A5 @ C2 ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_173_not__le__imp__less,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( linorder @ A )
% 3.13/1.60 => ! [Y: A,X: A] :
% 3.13/1.60 ( ~ ( ord_less_eq @ A @ Y @ X )
% 3.13/1.60 => ( ord_less @ A @ X @ Y ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_174_less__le__not__le,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( preorder @ A )
% 3.13/1.60 => ( ( ord_less @ A )
% 3.13/1.60 = ( ^ [X2: A,Y3: A] :
% 3.13/1.60 ( ( ord_less_eq @ A @ X2 @ Y3 )
% 3.13/1.60 & ~ ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_175_le__imp__less__or__eq,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( order @ A )
% 3.13/1.60 => ! [X: A,Y: A] :
% 3.13/1.60 ( ( ord_less_eq @ A @ X @ Y )
% 3.13/1.60 => ( ( ord_less @ A @ X @ Y )
% 3.13/1.60 | ( X = Y ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_176_le__less__linear,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( linorder @ A )
% 3.13/1.60 => ! [X: A,Y: A] :
% 3.13/1.60 ( ( ord_less_eq @ A @ X @ Y )
% 3.13/1.60 | ( ord_less @ A @ Y @ X ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_177_dense__le,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( dense_linorder @ A )
% 3.13/1.60 => ! [Y: A,Z: A] :
% 3.13/1.60 ( ! [X3: A] :
% 3.13/1.60 ( ( ord_less @ A @ X3 @ Y )
% 3.13/1.60 => ( ord_less_eq @ A @ X3 @ Z ) )
% 3.13/1.60 => ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_178_dense__ge,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( dense_linorder @ A )
% 3.13/1.60 => ! [Z: A,Y: A] :
% 3.13/1.60 ( ! [X3: A] :
% 3.13/1.60 ( ( ord_less @ A @ Z @ X3 )
% 3.13/1.60 => ( ord_less_eq @ A @ Y @ X3 ) )
% 3.13/1.60 => ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_179_less__le__trans,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( preorder @ A )
% 3.13/1.60 => ! [X: A,Y: A,Z: A] :
% 3.13/1.60 ( ( ord_less @ A @ X @ Y )
% 3.13/1.60 => ( ( ord_less_eq @ A @ Y @ Z )
% 3.13/1.60 => ( ord_less @ A @ X @ Z ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_180_le__less__trans,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( preorder @ A )
% 3.13/1.60 => ! [X: A,Y: A,Z: A] :
% 3.13/1.60 ( ( ord_less_eq @ A @ X @ Y )
% 3.13/1.60 => ( ( ord_less @ A @ Y @ Z )
% 3.13/1.60 => ( ord_less @ A @ X @ Z ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_181_less__imp__le,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( preorder @ A )
% 3.13/1.60 => ! [X: A,Y: A] :
% 3.13/1.60 ( ( ord_less @ A @ X @ Y )
% 3.13/1.60 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_182_antisym__conv2,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( order @ A )
% 3.13/1.60 => ! [X: A,Y: A] :
% 3.13/1.60 ( ( ord_less_eq @ A @ X @ Y )
% 3.13/1.60 => ( ( ~ ( ord_less @ A @ X @ Y ) )
% 3.13/1.60 = ( X = Y ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_183_antisym__conv1,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( order @ A )
% 3.13/1.60 => ! [X: A,Y: A] :
% 3.13/1.60 ( ~ ( ord_less @ A @ X @ Y )
% 3.13/1.60 => ( ( ord_less_eq @ A @ X @ Y )
% 3.13/1.60 = ( X = Y ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_184_le__neq__trans,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( order @ A )
% 3.13/1.60 => ! [A5: A,B3: A] :
% 3.13/1.60 ( ( ord_less_eq @ A @ A5 @ B3 )
% 3.13/1.60 => ( ( A5 != B3 )
% 3.13/1.60 => ( ord_less @ A @ A5 @ B3 ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_185_not__less,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( linorder @ A )
% 3.13/1.60 => ! [X: A,Y: A] :
% 3.13/1.60 ( ( ~ ( ord_less @ A @ X @ Y ) )
% 3.13/1.60 = ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_186_not__le,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( linorder @ A )
% 3.13/1.60 => ! [X: A,Y: A] :
% 3.13/1.60 ( ( ~ ( ord_less_eq @ A @ X @ Y ) )
% 3.13/1.60 = ( ord_less @ A @ Y @ X ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_187_order__less__le__subst2,axiom,
% 3.13/1.60 ! [A: $tType,C: $tType] :
% 3.13/1.60 ( ( ( order @ C )
% 3.13/1.60 & ( order @ A ) )
% 3.13/1.60 => ! [A5: A,B3: A,F: A > C,C2: C] :
% 3.13/1.60 ( ( ord_less @ A @ A5 @ B3 )
% 3.13/1.60 => ( ( ord_less_eq @ C @ ( F @ B3 ) @ C2 )
% 3.13/1.60 => ( ! [X3: A,Y2: A] :
% 3.13/1.60 ( ( ord_less @ A @ X3 @ Y2 )
% 3.13/1.60 => ( ord_less @ C @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 3.13/1.60 => ( ord_less @ C @ ( F @ A5 ) @ C2 ) ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_188_order__less__le__subst1,axiom,
% 3.13/1.60 ! [A: $tType,B: $tType] :
% 3.13/1.60 ( ( ( order @ B )
% 3.13/1.60 & ( order @ A ) )
% 3.13/1.60 => ! [A5: A,F: B > A,B3: B,C2: B] :
% 3.13/1.60 ( ( ord_less @ A @ A5 @ ( F @ B3 ) )
% 3.13/1.60 => ( ( ord_less_eq @ B @ B3 @ C2 )
% 3.13/1.60 => ( ! [X3: B,Y2: B] :
% 3.13/1.60 ( ( ord_less_eq @ B @ X3 @ Y2 )
% 3.13/1.60 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 3.13/1.60 => ( ord_less @ A @ A5 @ ( F @ C2 ) ) ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_189_order__le__less__subst2,axiom,
% 3.13/1.60 ! [A: $tType,C: $tType] :
% 3.13/1.60 ( ( ( order @ C )
% 3.13/1.60 & ( order @ A ) )
% 3.13/1.60 => ! [A5: A,B3: A,F: A > C,C2: C] :
% 3.13/1.60 ( ( ord_less_eq @ A @ A5 @ B3 )
% 3.13/1.60 => ( ( ord_less @ C @ ( F @ B3 ) @ C2 )
% 3.13/1.60 => ( ! [X3: A,Y2: A] :
% 3.13/1.60 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 3.13/1.60 => ( ord_less_eq @ C @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 3.13/1.60 => ( ord_less @ C @ ( F @ A5 ) @ C2 ) ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_190_order__le__less__subst1,axiom,
% 3.13/1.60 ! [A: $tType,B: $tType] :
% 3.13/1.60 ( ( ( order @ B )
% 3.13/1.60 & ( order @ A ) )
% 3.13/1.60 => ! [A5: A,F: B > A,B3: B,C2: B] :
% 3.13/1.60 ( ( ord_less_eq @ A @ A5 @ ( F @ B3 ) )
% 3.13/1.60 => ( ( ord_less @ B @ B3 @ C2 )
% 3.13/1.60 => ( ! [X3: B,Y2: B] :
% 3.13/1.60 ( ( ord_less @ B @ X3 @ Y2 )
% 3.13/1.60 => ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 3.13/1.60 => ( ord_less @ A @ A5 @ ( F @ C2 ) ) ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_191_less__le,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( order @ A )
% 3.13/1.60 => ( ( ord_less @ A )
% 3.13/1.60 = ( ^ [X2: A,Y3: A] :
% 3.13/1.60 ( ( ord_less_eq @ A @ X2 @ Y3 )
% 3.13/1.60 & ( X2 != Y3 ) ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_192_le__less,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( order @ A )
% 3.13/1.60 => ( ( ord_less_eq @ A )
% 3.13/1.60 = ( ^ [X2: A,Y3: A] :
% 3.13/1.60 ( ( ord_less @ A @ X2 @ Y3 )
% 3.13/1.60 | ( X2 = Y3 ) ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_193_leI,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( linorder @ A )
% 3.13/1.60 => ! [X: A,Y: A] :
% 3.13/1.60 ( ~ ( ord_less @ A @ X @ Y )
% 3.13/1.60 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_194_leD,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( order @ A )
% 3.13/1.60 => ! [Y: A,X: A] :
% 3.13/1.60 ( ( ord_less_eq @ A @ Y @ X )
% 3.13/1.60 => ~ ( ord_less @ A @ X @ Y ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_195_subset__image__iff,axiom,
% 3.13/1.60 ! [A: $tType,B: $tType,B2: set @ A,F: B > A,A2: set @ B] :
% 3.13/1.60 ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image @ B @ A @ F @ A2 ) )
% 3.13/1.60 = ( ? [AA: set @ B] :
% 3.13/1.60 ( ( ord_less_eq @ ( set @ B ) @ AA @ A2 )
% 3.13/1.60 & ( B2
% 3.13/1.60 = ( image @ B @ A @ F @ AA ) ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_196_image__subset__iff,axiom,
% 3.13/1.60 ! [A: $tType,B: $tType,F: B > A,A2: set @ B,B2: set @ A] :
% 3.13/1.60 ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F @ A2 ) @ B2 )
% 3.13/1.60 = ( ! [X2: B] :
% 3.13/1.60 ( ( member @ B @ X2 @ A2 )
% 3.13/1.60 => ( member @ A @ ( F @ X2 ) @ B2 ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_197_subset__imageE,axiom,
% 3.13/1.60 ! [A: $tType,B: $tType,B2: set @ A,F: B > A,A2: set @ B] :
% 3.13/1.60 ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image @ B @ A @ F @ A2 ) )
% 3.13/1.60 => ~ ! [C6: set @ B] :
% 3.13/1.60 ( ( ord_less_eq @ ( set @ B ) @ C6 @ A2 )
% 3.13/1.60 => ( B2
% 3.13/1.60 != ( image @ B @ A @ F @ C6 ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_198_image__subsetI,axiom,
% 3.13/1.60 ! [A: $tType,B: $tType,A2: set @ A,F: A > B,B2: set @ B] :
% 3.13/1.60 ( ! [X3: A] :
% 3.13/1.60 ( ( member @ A @ X3 @ A2 )
% 3.13/1.60 => ( member @ B @ ( F @ X3 ) @ B2 ) )
% 3.13/1.60 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) @ B2 ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_199_image__mono,axiom,
% 3.13/1.60 ! [B: $tType,A: $tType,A2: set @ A,B2: set @ A,F: A > B] :
% 3.13/1.60 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 3.13/1.60 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) @ ( image @ A @ B @ F @ B2 ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_200_top_Oextremum__uniqueI,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( order_top @ A )
% 3.13/1.60 => ! [A5: A] :
% 3.13/1.60 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A5 )
% 3.13/1.60 => ( A5
% 3.13/1.60 = ( top_top @ A ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_201_top_Oextremum__unique,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( order_top @ A )
% 3.13/1.60 => ! [A5: A] :
% 3.13/1.60 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A5 )
% 3.13/1.60 = ( A5
% 3.13/1.60 = ( top_top @ A ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_202_top__greatest,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( order_top @ A )
% 3.13/1.60 => ! [A5: A] : ( ord_less_eq @ A @ A5 @ ( top_top @ A ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_203_subset__UNIV,axiom,
% 3.13/1.60 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_204_subset__inj__on,axiom,
% 3.13/1.60 ! [B: $tType,A: $tType,F: A > B,B2: set @ A,A2: set @ A] :
% 3.13/1.60 ( ( inj_on @ A @ B @ F @ B2 )
% 3.13/1.60 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 3.13/1.60 => ( inj_on @ A @ B @ F @ A2 ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_205_inj__on__subset,axiom,
% 3.13/1.60 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,B2: set @ A] :
% 3.13/1.60 ( ( inj_on @ A @ B @ F @ A2 )
% 3.13/1.60 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 3.13/1.60 => ( inj_on @ A @ B @ F @ B2 ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_206_in__mono,axiom,
% 3.13/1.60 ! [A: $tType,A2: set @ A,B2: set @ A,X: A] :
% 3.13/1.60 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 3.13/1.60 => ( ( member @ A @ X @ A2 )
% 3.13/1.60 => ( member @ A @ X @ B2 ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_207_subsetD,axiom,
% 3.13/1.60 ! [A: $tType,A2: set @ A,B2: set @ A,C2: A] :
% 3.13/1.60 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 3.13/1.60 => ( ( member @ A @ C2 @ A2 )
% 3.13/1.60 => ( member @ A @ C2 @ B2 ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_208_equalityE,axiom,
% 3.13/1.60 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 3.13/1.60 ( ( A2 = B2 )
% 3.13/1.60 => ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 3.13/1.60 => ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_209_subset__eq,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( ord_less_eq @ ( set @ A ) )
% 3.13/1.60 = ( ^ [A3: set @ A,B6: set @ A] :
% 3.13/1.60 ! [X2: A] :
% 3.13/1.60 ( ( member @ A @ X2 @ A3 )
% 3.13/1.60 => ( member @ A @ X2 @ B6 ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_210_equalityD1,axiom,
% 3.13/1.60 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 3.13/1.60 ( ( A2 = B2 )
% 3.13/1.60 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_211_equalityD2,axiom,
% 3.13/1.60 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 3.13/1.60 ( ( A2 = B2 )
% 3.13/1.60 => ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_212_subset__iff,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( ord_less_eq @ ( set @ A ) )
% 3.13/1.60 = ( ^ [A3: set @ A,B6: set @ A] :
% 3.13/1.60 ! [T: A] :
% 3.13/1.60 ( ( member @ A @ T @ A3 )
% 3.13/1.60 => ( member @ A @ T @ B6 ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_213_subset__refl,axiom,
% 3.13/1.60 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_214_Collect__mono,axiom,
% 3.13/1.60 ! [A: $tType,P: A > $o,Q: A > $o] :
% 3.13/1.60 ( ! [X3: A] :
% 3.13/1.60 ( ( P @ X3 )
% 3.13/1.60 => ( Q @ X3 ) )
% 3.13/1.60 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_215_subset__trans,axiom,
% 3.13/1.60 ! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
% 3.13/1.60 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 3.13/1.60 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
% 3.13/1.60 => ( ord_less_eq @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_216_le__funD,axiom,
% 3.13/1.60 ! [B: $tType,A: $tType] :
% 3.13/1.60 ( ( ord @ B )
% 3.13/1.60 => ! [F: A > B,G: A > B,X: A] :
% 3.13/1.60 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 3.13/1.60 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_217_le__funE,axiom,
% 3.13/1.60 ! [B: $tType,A: $tType] :
% 3.13/1.60 ( ( ord @ B )
% 3.13/1.60 => ! [F: A > B,G: A > B,X: A] :
% 3.13/1.60 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 3.13/1.60 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_218_le__funI,axiom,
% 3.13/1.60 ! [B: $tType,A: $tType] :
% 3.13/1.60 ( ( ord @ B )
% 3.13/1.60 => ! [F: A > B,G: A > B] :
% 3.13/1.60 ( ! [X3: A] : ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) )
% 3.13/1.60 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_219_set__eq__subset,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( ^ [Y5: set @ A,Z3: set @ A] : ( Y5 = Z3 ) )
% 3.13/1.60 = ( ^ [A3: set @ A,B6: set @ A] :
% 3.13/1.60 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
% 3.13/1.60 & ( ord_less_eq @ ( set @ A ) @ B6 @ A3 ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_220_le__fun__def,axiom,
% 3.13/1.60 ! [B: $tType,A: $tType] :
% 3.13/1.60 ( ( ord @ B )
% 3.13/1.60 => ( ( ord_less_eq @ ( A > B ) )
% 3.13/1.60 = ( ^ [F2: A > B,G2: A > B] :
% 3.13/1.60 ! [X2: A] : ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_221_Collect__mono__iff,axiom,
% 3.13/1.60 ! [A: $tType,P: A > $o,Q: A > $o] :
% 3.13/1.60 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 3.13/1.60 = ( ! [X2: A] :
% 3.13/1.60 ( ( P @ X2 )
% 3.13/1.60 => ( Q @ X2 ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_222_order__subst1,axiom,
% 3.13/1.60 ! [A: $tType,B: $tType] :
% 3.13/1.60 ( ( ( order @ B )
% 3.13/1.60 & ( order @ A ) )
% 3.13/1.60 => ! [A5: A,F: B > A,B3: B,C2: B] :
% 3.13/1.60 ( ( ord_less_eq @ A @ A5 @ ( F @ B3 ) )
% 3.13/1.60 => ( ( ord_less_eq @ B @ B3 @ C2 )
% 3.13/1.60 => ( ! [X3: B,Y2: B] :
% 3.13/1.60 ( ( ord_less_eq @ B @ X3 @ Y2 )
% 3.13/1.60 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 3.13/1.60 => ( ord_less_eq @ A @ A5 @ ( F @ C2 ) ) ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_223_order__subst2,axiom,
% 3.13/1.60 ! [A: $tType,C: $tType] :
% 3.13/1.60 ( ( ( order @ C )
% 3.13/1.60 & ( order @ A ) )
% 3.13/1.60 => ! [A5: A,B3: A,F: A > C,C2: C] :
% 3.13/1.60 ( ( ord_less_eq @ A @ A5 @ B3 )
% 3.13/1.60 => ( ( ord_less_eq @ C @ ( F @ B3 ) @ C2 )
% 3.13/1.60 => ( ! [X3: A,Y2: A] :
% 3.13/1.60 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 3.13/1.60 => ( ord_less_eq @ C @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 3.13/1.60 => ( ord_less_eq @ C @ ( F @ A5 ) @ C2 ) ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_224_ord__eq__le__subst,axiom,
% 3.13/1.60 ! [A: $tType,B: $tType] :
% 3.13/1.60 ( ( ( ord @ B )
% 3.13/1.60 & ( ord @ A ) )
% 3.13/1.60 => ! [A5: A,F: B > A,B3: B,C2: B] :
% 3.13/1.60 ( ( A5
% 3.13/1.60 = ( F @ B3 ) )
% 3.13/1.60 => ( ( ord_less_eq @ B @ B3 @ C2 )
% 3.13/1.60 => ( ! [X3: B,Y2: B] :
% 3.13/1.60 ( ( ord_less_eq @ B @ X3 @ Y2 )
% 3.13/1.60 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 3.13/1.60 => ( ord_less_eq @ A @ A5 @ ( F @ C2 ) ) ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_225_ord__le__eq__subst,axiom,
% 3.13/1.60 ! [A: $tType,B: $tType] :
% 3.13/1.60 ( ( ( ord @ B )
% 3.13/1.60 & ( ord @ A ) )
% 3.13/1.60 => ! [A5: A,B3: A,F: A > B,C2: B] :
% 3.13/1.60 ( ( ord_less_eq @ A @ A5 @ B3 )
% 3.13/1.60 => ( ( ( F @ B3 )
% 3.13/1.60 = C2 )
% 3.13/1.60 => ( ! [X3: A,Y2: A] :
% 3.13/1.60 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 3.13/1.60 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 3.13/1.60 => ( ord_less_eq @ B @ ( F @ A5 ) @ C2 ) ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_226_eq__iff,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( order @ A )
% 3.13/1.60 => ( ( ^ [Y5: A,Z3: A] : ( Y5 = Z3 ) )
% 3.13/1.60 = ( ^ [X2: A,Y3: A] :
% 3.13/1.60 ( ( ord_less_eq @ A @ X2 @ Y3 )
% 3.13/1.60 & ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_227_antisym,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( order @ A )
% 3.13/1.60 => ! [X: A,Y: A] :
% 3.13/1.60 ( ( ord_less_eq @ A @ X @ Y )
% 3.13/1.60 => ( ( ord_less_eq @ A @ Y @ X )
% 3.13/1.60 => ( X = Y ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_228_linear,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( linorder @ A )
% 3.13/1.60 => ! [X: A,Y: A] :
% 3.13/1.60 ( ( ord_less_eq @ A @ X @ Y )
% 3.13/1.60 | ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_229_eq__refl,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( preorder @ A )
% 3.13/1.60 => ! [X: A,Y: A] :
% 3.13/1.60 ( ( X = Y )
% 3.13/1.60 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_230_le__cases,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( linorder @ A )
% 3.13/1.60 => ! [X: A,Y: A] :
% 3.13/1.60 ( ~ ( ord_less_eq @ A @ X @ Y )
% 3.13/1.60 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_231_order_Otrans,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( order @ A )
% 3.13/1.60 => ! [A5: A,B3: A,C2: A] :
% 3.13/1.60 ( ( ord_less_eq @ A @ A5 @ B3 )
% 3.13/1.60 => ( ( ord_less_eq @ A @ B3 @ C2 )
% 3.13/1.60 => ( ord_less_eq @ A @ A5 @ C2 ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_232_le__cases3,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( linorder @ A )
% 3.13/1.60 => ! [X: A,Y: A,Z: A] :
% 3.13/1.60 ( ( ( ord_less_eq @ A @ X @ Y )
% 3.13/1.60 => ~ ( ord_less_eq @ A @ Y @ Z ) )
% 3.13/1.60 => ( ( ( ord_less_eq @ A @ Y @ X )
% 3.13/1.60 => ~ ( ord_less_eq @ A @ X @ Z ) )
% 3.13/1.60 => ( ( ( ord_less_eq @ A @ X @ Z )
% 3.13/1.60 => ~ ( ord_less_eq @ A @ Z @ Y ) )
% 3.13/1.60 => ( ( ( ord_less_eq @ A @ Z @ Y )
% 3.13/1.60 => ~ ( ord_less_eq @ A @ Y @ X ) )
% 3.13/1.60 => ( ( ( ord_less_eq @ A @ Y @ Z )
% 3.13/1.60 => ~ ( ord_less_eq @ A @ Z @ X ) )
% 3.13/1.60 => ~ ( ( ord_less_eq @ A @ Z @ X )
% 3.13/1.60 => ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_233_antisym__conv,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( order @ A )
% 3.13/1.60 => ! [Y: A,X: A] :
% 3.13/1.60 ( ( ord_less_eq @ A @ Y @ X )
% 3.13/1.60 => ( ( ord_less_eq @ A @ X @ Y )
% 3.13/1.60 = ( X = Y ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_234_order__class_Oorder_Oeq__iff,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( order @ A )
% 3.13/1.60 => ( ( ^ [Y5: A,Z3: A] : ( Y5 = Z3 ) )
% 3.13/1.60 = ( ^ [A6: A,B5: A] :
% 3.13/1.60 ( ( ord_less_eq @ A @ A6 @ B5 )
% 3.13/1.60 & ( ord_less_eq @ A @ B5 @ A6 ) ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_235_ord__eq__le__trans,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( ord @ A )
% 3.13/1.60 => ! [A5: A,B3: A,C2: A] :
% 3.13/1.60 ( ( A5 = B3 )
% 3.13/1.60 => ( ( ord_less_eq @ A @ B3 @ C2 )
% 3.13/1.60 => ( ord_less_eq @ A @ A5 @ C2 ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_236_ord__le__eq__trans,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( ord @ A )
% 3.13/1.60 => ! [A5: A,B3: A,C2: A] :
% 3.13/1.60 ( ( ord_less_eq @ A @ A5 @ B3 )
% 3.13/1.60 => ( ( B3 = C2 )
% 3.13/1.60 => ( ord_less_eq @ A @ A5 @ C2 ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_237_order__class_Oorder_Oantisym,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( order @ A )
% 3.13/1.60 => ! [A5: A,B3: A] :
% 3.13/1.60 ( ( ord_less_eq @ A @ A5 @ B3 )
% 3.13/1.60 => ( ( ord_less_eq @ A @ B3 @ A5 )
% 3.13/1.60 => ( A5 = B3 ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_238_order__trans,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( preorder @ A )
% 3.13/1.60 => ! [X: A,Y: A,Z: A] :
% 3.13/1.60 ( ( ord_less_eq @ A @ X @ Y )
% 3.13/1.60 => ( ( ord_less_eq @ A @ Y @ Z )
% 3.13/1.60 => ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_239_dual__order_Orefl,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( order @ A )
% 3.13/1.60 => ! [A5: A] : ( ord_less_eq @ A @ A5 @ A5 ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_240_linorder__wlog,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( linorder @ A )
% 3.13/1.60 => ! [P: A > A > $o,A5: A,B3: A] :
% 3.13/1.60 ( ! [A4: A,B4: A] :
% 3.13/1.60 ( ( ord_less_eq @ A @ A4 @ B4 )
% 3.13/1.60 => ( P @ A4 @ B4 ) )
% 3.13/1.60 => ( ! [A4: A,B4: A] :
% 3.13/1.60 ( ( P @ B4 @ A4 )
% 3.13/1.60 => ( P @ A4 @ B4 ) )
% 3.13/1.60 => ( P @ A5 @ B3 ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_241_dual__order_Otrans,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( order @ A )
% 3.13/1.60 => ! [B3: A,A5: A,C2: A] :
% 3.13/1.60 ( ( ord_less_eq @ A @ B3 @ A5 )
% 3.13/1.60 => ( ( ord_less_eq @ A @ C2 @ B3 )
% 3.13/1.60 => ( ord_less_eq @ A @ C2 @ A5 ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_242_dual__order_Oeq__iff,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( order @ A )
% 3.13/1.60 => ( ( ^ [Y5: A,Z3: A] : ( Y5 = Z3 ) )
% 3.13/1.60 = ( ^ [A6: A,B5: A] :
% 3.13/1.60 ( ( ord_less_eq @ A @ B5 @ A6 )
% 3.13/1.60 & ( ord_less_eq @ A @ A6 @ B5 ) ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_243_dual__order_Oantisym,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( order @ A )
% 3.13/1.60 => ! [B3: A,A5: A] :
% 3.13/1.60 ( ( ord_less_eq @ A @ B3 @ A5 )
% 3.13/1.60 => ( ( ord_less_eq @ A @ A5 @ B3 )
% 3.13/1.60 => ( A5 = B3 ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_244_strict__mono__less__eq,axiom,
% 3.13/1.60 ! [B: $tType,A: $tType] :
% 3.13/1.60 ( ( ( linorder @ A )
% 3.13/1.60 & ( order @ B ) )
% 3.13/1.60 => ! [F: A > B,X: A,Y: A] :
% 3.13/1.60 ( ( order_strict_mono @ A @ B @ F )
% 3.13/1.60 => ( ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y ) )
% 3.13/1.60 = ( ord_less_eq @ A @ X @ Y ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_245_strict__mono__on__leD,axiom,
% 3.13/1.60 ! [B: $tType,A: $tType] :
% 3.13/1.60 ( ( ( linorder @ A )
% 3.13/1.60 & ( preorder @ B ) )
% 3.13/1.60 => ! [F: A > B,A2: set @ A,X: A,Y: A] :
% 3.13/1.60 ( ( strict_mono_on @ A @ B @ F @ A2 )
% 3.13/1.60 => ( ( member @ A @ X @ A2 )
% 3.13/1.60 => ( ( member @ A @ Y @ A2 )
% 3.13/1.60 => ( ( ord_less_eq @ A @ X @ Y )
% 3.13/1.60 => ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_246_add__less__le__mono,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( ordere223160158up_add @ A )
% 3.13/1.60 => ! [A5: A,B3: A,C2: A,D2: A] :
% 3.13/1.60 ( ( ord_less @ A @ A5 @ B3 )
% 3.13/1.60 => ( ( ord_less_eq @ A @ C2 @ D2 )
% 3.13/1.60 => ( ord_less @ A @ ( plus_plus @ A @ A5 @ C2 ) @ ( plus_plus @ A @ B3 @ D2 ) ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_247_add__le__less__mono,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( ordere223160158up_add @ A )
% 3.13/1.60 => ! [A5: A,B3: A,C2: A,D2: A] :
% 3.13/1.60 ( ( ord_less_eq @ A @ A5 @ B3 )
% 3.13/1.60 => ( ( ord_less @ A @ C2 @ D2 )
% 3.13/1.60 => ( ord_less @ A @ ( plus_plus @ A @ A5 @ C2 ) @ ( plus_plus @ A @ B3 @ D2 ) ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_248_add__mono__thms__linordered__field_I3_J,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( ordere223160158up_add @ A )
% 3.13/1.60 => ! [I: A,J: A,K: A,L: A] :
% 3.13/1.60 ( ( ( ord_less @ A @ I @ J )
% 3.13/1.60 & ( ord_less_eq @ A @ K @ L ) )
% 3.13/1.60 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_249_add__mono__thms__linordered__field_I4_J,axiom,
% 3.13/1.60 ! [A: $tType] :
% 3.13/1.60 ( ( ordere223160158up_add @ A )
% 3.13/1.60 => ! [I: A,J: A,K: A,L: A] :
% 3.13/1.60 ( ( ( ord_less_eq @ A @ I @ J )
% 3.13/1.60 & ( ord_less @ A @ K @ L ) )
% 3.13/1.60 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_250_linorder__inj__onI,axiom,
% 3.13/1.60 ! [B: $tType,A: $tType] :
% 3.13/1.60 ( ( order @ A )
% 3.13/1.60 => ! [A2: set @ A,F: A > B] :
% 3.13/1.60 ( ! [X3: A,Y2: A] :
% 3.13/1.60 ( ( ord_less @ A @ X3 @ Y2 )
% 3.13/1.60 => ( ( member @ A @ X3 @ A2 )
% 3.13/1.60 => ( ( member @ A @ Y2 @ A2 )
% 3.13/1.60 => ( ( F @ X3 )
% 3.13/1.60 != ( F @ Y2 ) ) ) ) )
% 3.13/1.60 => ( ! [X3: A,Y2: A] :
% 3.13/1.60 ( ( member @ A @ X3 @ A2 )
% 3.13/1.60 => ( ( member @ A @ Y2 @ A2 )
% 3.13/1.60 => ( ( ord_less_eq @ A @ X3 @ Y2 )
% 3.13/1.60 | ( ord_less_eq @ A @ Y2 @ X3 ) ) ) )
% 3.13/1.60 => ( inj_on @ A @ B @ F @ A2 ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_251_inj__on__image__mem__iff,axiom,
% 3.13/1.60 ! [B: $tType,A: $tType,F: A > B,B2: set @ A,A5: A,A2: set @ A] :
% 3.13/1.60 ( ( inj_on @ A @ B @ F @ B2 )
% 3.13/1.60 => ( ( member @ A @ A5 @ B2 )
% 3.13/1.60 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 3.13/1.60 => ( ( member @ B @ ( F @ A5 ) @ ( image @ A @ B @ F @ A2 ) )
% 3.13/1.60 = ( member @ A @ A5 @ A2 ) ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_252_inj__on__image__eq__iff,axiom,
% 3.13/1.60 ! [B: $tType,A: $tType,F: A > B,C3: set @ A,A2: set @ A,B2: set @ A] :
% 3.13/1.60 ( ( inj_on @ A @ B @ F @ C3 )
% 3.13/1.60 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ C3 )
% 3.13/1.60 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
% 3.13/1.60 => ( ( ( image @ A @ B @ F @ A2 )
% 3.13/1.60 = ( image @ A @ B @ F @ B2 ) )
% 3.13/1.60 = ( A2 = B2 ) ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_253_inj__image__subset__iff,axiom,
% 3.13/1.60 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,B2: set @ A] :
% 3.13/1.60 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 3.13/1.60 => ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) @ ( image @ A @ B @ F @ B2 ) )
% 3.13/1.60 = ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_254_map__prod__surj,axiom,
% 3.13/1.60 ! [A: $tType,C: $tType,D: $tType,B: $tType,F: A > B,G: C > D] :
% 3.13/1.60 ( ( ( image @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 3.13/1.60 = ( top_top @ ( set @ B ) ) )
% 3.13/1.60 => ( ( ( image @ C @ D @ G @ ( top_top @ ( set @ C ) ) )
% 3.13/1.60 = ( top_top @ ( set @ D ) ) )
% 3.13/1.60 => ( ( image @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_map_prod @ A @ B @ C @ D @ F @ G ) @ ( top_top @ ( set @ ( product_prod @ A @ C ) ) ) )
% 3.13/1.60 = ( top_top @ ( set @ ( product_prod @ B @ D ) ) ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(fact_255_subset__image__inj,axiom,
% 3.13/1.60 ! [A: $tType,B: $tType,S4: set @ A,F: B > A,T2: set @ B] :
% 3.13/1.60 ( ( ord_less_eq @ ( set @ A ) @ S4 @ ( image @ B @ A @ F @ T2 ) )
% 3.13/1.60 = ( ? [U: set @ B] :
% 3.13/1.60 ( ( ord_less_eq @ ( set @ B ) @ U @ T2 )
% 3.13/1.60 & ( inj_on @ B @ A @ F @ U )
% 3.13/1.60 & ( S4
% 3.13/1.60 = ( image @ B @ A @ F @ U ) ) ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(tcon_fun___Orderings_Oorder__top,axiom,
% 3.13/1.60 ! [A7: $tType,A8: $tType] :
% 3.13/1.60 ( ( order_top @ A8 )
% 3.13/1.60 => ( order_top @ ( A7 > A8 ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(tcon_fun___Orderings_Opreorder,axiom,
% 3.13/1.60 ! [A7: $tType,A8: $tType] :
% 3.13/1.60 ( ( preorder @ A8 )
% 3.13/1.60 => ( preorder @ ( A7 > A8 ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(tcon_fun___Orderings_Oorder,axiom,
% 3.13/1.60 ! [A7: $tType,A8: $tType] :
% 3.13/1.60 ( ( order @ A8 )
% 3.13/1.60 => ( order @ ( A7 > A8 ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(tcon_fun___Orderings_Otop,axiom,
% 3.13/1.60 ! [A7: $tType,A8: $tType] :
% 3.13/1.60 ( ( top @ A8 )
% 3.13/1.60 => ( top @ ( A7 > A8 ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(tcon_fun___Orderings_Oord,axiom,
% 3.13/1.60 ! [A7: $tType,A8: $tType] :
% 3.13/1.60 ( ( ord @ A8 )
% 3.13/1.60 => ( ord @ ( A7 > A8 ) ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
% 3.13/1.60 strict2144017051up_add @ nat ).
% 3.13/1.60
% 3.13/1.60 thf(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add,axiom,
% 3.13/1.60 ordere223160158up_add @ nat ).
% 3.13/1.60
% 3.13/1.60 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
% 3.13/1.60 ordere236663937imp_le @ nat ).
% 3.13/1.60
% 3.13/1.60 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 3.13/1.60 canoni770627133id_add @ nat ).
% 3.13/1.60
% 3.13/1.60 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add,axiom,
% 3.13/1.60 ordere779506340up_add @ nat ).
% 3.13/1.60
% 3.13/1.60 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
% 3.13/1.60 cancel_semigroup_add @ nat ).
% 3.13/1.60
% 3.13/1.60 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add,axiom,
% 3.13/1.60 ab_semigroup_add @ nat ).
% 3.13/1.60
% 3.13/1.60 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
% 3.13/1.60 comm_monoid_add @ nat ).
% 3.13/1.60
% 3.13/1.60 thf(tcon_Nat_Onat___Groups_Osemigroup__add,axiom,
% 3.13/1.60 semigroup_add @ nat ).
% 3.13/1.60
% 3.13/1.60 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 3.13/1.60 wellorder @ nat ).
% 3.13/1.60
% 3.13/1.60 thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
% 3.13/1.60 preorder @ nat ).
% 3.13/1.60
% 3.13/1.60 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 3.13/1.60 linorder @ nat ).
% 3.13/1.60
% 3.13/1.60 thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
% 3.13/1.60 no_top @ nat ).
% 3.13/1.60
% 3.13/1.60 thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
% 3.13/1.60 order @ nat ).
% 3.13/1.60
% 3.13/1.60 thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
% 3.13/1.60 ord @ nat ).
% 3.13/1.60
% 3.13/1.60 thf(tcon_Set_Oset___Orderings_Oorder__top_4,axiom,
% 3.13/1.60 ! [A7: $tType] : ( order_top @ ( set @ A7 ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(tcon_Set_Oset___Orderings_Opreorder_5,axiom,
% 3.13/1.60 ! [A7: $tType] : ( preorder @ ( set @ A7 ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(tcon_Set_Oset___Orderings_Oorder_6,axiom,
% 3.13/1.60 ! [A7: $tType] : ( order @ ( set @ A7 ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(tcon_Set_Oset___Orderings_Otop_7,axiom,
% 3.13/1.60 ! [A7: $tType] : ( top @ ( set @ A7 ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(tcon_Set_Oset___Orderings_Oord_8,axiom,
% 3.13/1.60 ! [A7: $tType] : ( ord @ ( set @ A7 ) ) ).
% 3.13/1.60
% 3.13/1.60 thf(tcon_HOL_Obool___Orderings_Oorder__top_9,axiom,
% 3.13/1.60 order_top @ $o ).
% 3.13/1.60
% 3.13/1.60 thf(tcon_HOL_Obool___Orderings_Opreorder_10,axiom,
% 3.13/1.60 preorder @ $o ).
% 3.13/1.60
% 3.13/1.60 thf(tcon_HOL_Obool___Orderings_Olinorder_11,axiom,
% 3.13/1.60 linorder @ $o ).
% 3.13/1.60
% 3.13/1.60 thf(tcon_HOL_Obool___Orderings_Oorder_12,axiom,
% 3.13/1.60 order @ $o ).
% 3.13/1.60
% 3.13/1.60 thf(tcon_HOL_Obool___Orderings_Otop_13,axiom,
% 3.13/1.60 top @ $o ).
% 3.13/1.60
% 3.13/1.60 thf(tcon_HOL_Obool___Orderings_Oord_14,axiom,
% 3.13/1.60 ord @ $o ).
% 3.13/1.60
% 3.13/1.60 thf(conj_0,conjecture,
% 3.13/1.60 inj_on @ paraco415392788lle_tv @ paraco415392788lle_tv @ ( paraco1920534163nge_tv @ f ) @ ( top_top @ ( set @ paraco415392788lle_tv ) ) ).
% 3.13/1.60 ------------------------
% 3.13/1.62 % (9351)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_9159 for (2999ds/183Mi)
% 3.13/1.62 % (9357)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_9159 for (2999ds/18Mi)
% 3.13/1.62 % (9355)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_9159 for (2999ds/2Mi)
% 3.13/1.62 % (9356)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_9159 for (2999ds/275Mi)
% 3.13/1.62 % (9355)Instruction limit reached!
% 3.13/1.62 % (9355)------------------------------
% 3.13/1.62 % (9355)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.62 % (9355)Termination reason: Unknown
% 3.13/1.62 % (9355)Termination phase: shuffling
% 3.13/1.62
% 3.13/1.62 % (9355)Memory used [KB]: 1407
% 3.13/1.62 % (9355)Time elapsed: 0.002 s
% 3.13/1.62 % (9355)Instructions burned: 3 (million)
% 3.13/1.62 % (9355)------------------------------
% 3.13/1.62 % (9355)------------------------------
% 3.13/1.63 % (9354)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_9159 for (2999ds/2Mi)
% 3.13/1.63 % (9357)Instruction limit reached!
% 3.13/1.63 % (9357)------------------------------
% 3.13/1.63 % (9357)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.63 % (9357)Termination reason: Unknown
% 3.13/1.63 % (9357)Termination phase: shuffling
% 3.13/1.63
% 3.13/1.63 % (9357)Memory used [KB]: 1791
% 3.13/1.63 % (9357)Time elapsed: 0.009 s
% 3.13/1.63 % (9357)Instructions burned: 19 (million)
% 3.13/1.63 % (9357)------------------------------
% 3.13/1.63 % (9357)------------------------------
% 3.13/1.63 % (9354)Instruction limit reached!
% 3.13/1.63 % (9354)------------------------------
% 3.13/1.63 % (9354)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.63 % (9354)Termination reason: Unknown
% 3.13/1.63 % (9354)Termination phase: shuffling
% 3.13/1.63
% 3.13/1.63 % (9354)Memory used [KB]: 1279
% 3.13/1.63 % (9354)Time elapsed: 0.004 s
% 3.13/1.63 % (9354)Instructions burned: 3 (million)
% 3.13/1.63 % (9354)------------------------------
% 3.13/1.63 % (9354)------------------------------
% 3.13/1.63 % (9358)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_9159 for (2999ds/3Mi)
% 3.13/1.64 % (9352)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_9159 for (2999ds/4Mi)
% 3.13/1.64 % (9358)Instruction limit reached!
% 3.13/1.64 % (9358)------------------------------
% 3.13/1.64 % (9358)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.64 % (9358)Termination reason: Unknown
% 3.13/1.64 % (9358)Termination phase: shuffling
% 3.13/1.64
% 3.13/1.64 % (9358)Memory used [KB]: 1407
% 3.13/1.64 % (9358)Time elapsed: 0.003 s
% 3.13/1.64 % (9358)Instructions burned: 4 (million)
% 3.13/1.64 % (9358)------------------------------
% 3.13/1.64 % (9358)------------------------------
% 3.13/1.64 % (9352)Instruction limit reached!
% 3.13/1.64 % (9352)------------------------------
% 3.13/1.64 % (9352)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.64 % (9352)Termination reason: Unknown
% 3.13/1.64 % (9352)Termination phase: shuffling
% 3.13/1.64
% 3.13/1.64 % (9352)Memory used [KB]: 1407
% 3.13/1.64 % (9352)Time elapsed: 0.004 s
% 3.13/1.64 % (9352)Instructions burned: 4 (million)
% 3.13/1.64 % (9352)------------------------------
% 3.13/1.64 % (9352)------------------------------
% 3.13/1.64 % (9359)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_9159 for (2999ds/37Mi)
% 3.13/1.65 % (9361)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_9159 for (2999ds/3Mi)
% 3.13/1.65 % (9361)Instruction limit reached!
% 3.13/1.65 % (9361)------------------------------
% 3.13/1.65 % (9361)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.65 % (9361)Termination reason: Unknown
% 3.13/1.65 % (9361)Termination phase: shuffling
% 3.13/1.65
% 3.13/1.65 % (9361)Memory used [KB]: 1407
% 3.13/1.65 % (9361)Time elapsed: 0.003 s
% 3.13/1.65 % (9361)Instructions burned: 4 (million)
% 3.13/1.65 % (9361)------------------------------
% 3.13/1.65 % (9361)------------------------------
% 3.13/1.65 % (9360)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_9159 for (2999ds/15Mi)
% 3.13/1.65 % (9362)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_9159 for (2999ds/1041Mi)
% 3.13/1.65 % (9353)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_9159 for (2999ds/27Mi)
% 3.13/1.65 % (9359)Instruction limit reached!
% 3.13/1.65 % (9359)------------------------------
% 3.13/1.65 % (9359)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.65 % (9359)Termination reason: Unknown
% 3.13/1.65 % (9359)Termination phase: Property scanning
% 3.13/1.65
% 3.13/1.65 % (9359)Memory used [KB]: 2174
% 3.13/1.65 % (9359)Time elapsed: 0.014 s
% 3.13/1.65 % (9359)Instructions burned: 40 (million)
% 3.13/1.65 % (9359)------------------------------
% 3.13/1.65 % (9359)------------------------------
% 3.13/1.66 % (9360)Instruction limit reached!
% 3.13/1.66 % (9360)------------------------------
% 3.13/1.66 % (9360)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.66 % (9360)Termination reason: Unknown
% 3.13/1.66 % (9360)Termination phase: shuffling
% 3.13/1.66
% 3.13/1.66 % (9360)Memory used [KB]: 1663
% 3.13/1.66 % (9360)Time elapsed: 0.012 s
% 3.13/1.66 % (9360)Instructions burned: 15 (million)
% 3.13/1.66 % (9360)------------------------------
% 3.13/1.66 % (9360)------------------------------
% 3.13/1.66 % (9363)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_9159 for (2999ds/7Mi)
% 3.13/1.66 % (9363)Instruction limit reached!
% 3.13/1.66 % (9363)------------------------------
% 3.13/1.66 % (9363)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.66 % (9363)Termination reason: Unknown
% 3.13/1.66 % (9363)Termination phase: shuffling
% 3.13/1.66
% 3.13/1.66 % (9363)Memory used [KB]: 1535
% 3.13/1.66 % (9363)Time elapsed: 0.004 s
% 3.13/1.66 % (9363)Instructions burned: 7 (million)
% 3.13/1.66 % (9363)------------------------------
% 3.13/1.66 % (9363)------------------------------
% 3.13/1.66 % (9364)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_9159 for (2999ds/16Mi)
% 3.13/1.67 % (9364)Instruction limit reached!
% 3.13/1.67 % (9364)------------------------------
% 3.13/1.67 % (9364)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.67 % (9364)Termination reason: Unknown
% 3.13/1.67 % (9364)Termination phase: shuffling
% 3.13/1.67
% 3.13/1.67 % (9364)Memory used [KB]: 1663
% 3.13/1.67 % (9364)Time elapsed: 0.007 s
% 3.13/1.67 % (9364)Instructions burned: 17 (million)
% 3.13/1.67 % (9364)------------------------------
% 3.13/1.67 % (9364)------------------------------
% 3.13/1.67 % (9366)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_9159 for (2999ds/3Mi)
% 3.13/1.67 % (9366)Instruction limit reached!
% 3.13/1.67 % (9366)------------------------------
% 3.13/1.67 % (9366)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.67 % (9366)Termination reason: Unknown
% 3.13/1.67 % (9366)Termination phase: shuffling
% 3.13/1.67
% 3.13/1.67 % (9366)Memory used [KB]: 1279
% 3.13/1.67 % (9366)Time elapsed: 0.002 s
% 3.13/1.67 % (9366)Instructions burned: 3 (million)
% 3.13/1.67 % (9366)------------------------------
% 3.13/1.67 % (9366)------------------------------
% 3.13/1.67 % (9351)Instruction limit reached!
% 3.13/1.67 % (9351)------------------------------
% 3.13/1.67 % (9351)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.67 % (9351)Termination reason: Unknown
% 3.13/1.67 % (9351)Termination phase: Saturation
% 3.13/1.67
% 3.13/1.67 % (9351)Memory used [KB]: 7291
% 3.13/1.67 % (9351)Time elapsed: 0.055 s
% 3.13/1.67 % (9351)Instructions burned: 186 (million)
% 3.13/1.67 % (9351)------------------------------
% 3.13/1.67 % (9351)------------------------------
% 3.13/1.67 % (9353)Instruction limit reached!
% 3.13/1.67 % (9353)------------------------------
% 3.13/1.67 % (9353)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.67 % (9353)Termination reason: Unknown
% 3.13/1.67 % (9353)Termination phase: Property scanning
% 3.13/1.67
% 3.13/1.67 % (9353)Memory used [KB]: 1791
% 3.13/1.67 % (9353)Time elapsed: 0.027 s
% 3.13/1.67 % (9353)Instructions burned: 27 (million)
% 3.13/1.67 % (9353)------------------------------
% 3.13/1.67 % (9353)------------------------------
% 3.13/1.68 % (9365)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_9159 for (2999ds/3Mi)
% 3.13/1.68 % (9367)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_9159 for (2999ds/7Mi)
% 3.13/1.68 % (9365)Instruction limit reached!
% 3.13/1.68 % (9365)------------------------------
% 3.13/1.68 % (9365)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.68 % (9365)Termination reason: Unknown
% 3.13/1.68 % (9365)Termination phase: shuffling
% 3.13/1.68
% 3.13/1.68 % (9365)Memory used [KB]: 1407
% 3.13/1.68 % (9365)Time elapsed: 0.004 s
% 3.13/1.68 % (9365)Instructions burned: 4 (million)
% 3.13/1.68 % (9365)------------------------------
% 3.13/1.68 % (9365)------------------------------
% 3.13/1.68 % (9367)Instruction limit reached!
% 3.13/1.68 % (9367)------------------------------
% 3.13/1.68 % (9367)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.68 % (9367)Termination reason: Unknown
% 3.13/1.68 % (9367)Termination phase: shuffling
% 3.13/1.68
% 3.13/1.68 % (9367)Memory used [KB]: 1535
% 3.13/1.68 % (9367)Time elapsed: 0.005 s
% 3.13/1.68 % (9367)Instructions burned: 9 (million)
% 3.13/1.68 % (9367)------------------------------
% 3.13/1.68 % (9367)------------------------------
% 3.13/1.68 % (9369)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_9159 for (2999ds/4Mi)
% 3.13/1.68 % (9368)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_9159 for (2999ds/3Mi)
% 3.13/1.69 % (9369)Instruction limit reached!
% 3.13/1.69 % (9369)------------------------------
% 3.13/1.69 % (9369)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.69 % (9369)Termination reason: Unknown
% 3.13/1.69 % (9369)Termination phase: shuffling
% 3.13/1.69
% 3.13/1.69 % (9369)Memory used [KB]: 1407
% 3.13/1.69 % (9369)Time elapsed: 0.003 s
% 3.13/1.69 % (9369)Instructions burned: 5 (million)
% 3.13/1.69 % (9369)------------------------------
% 3.13/1.69 % (9369)------------------------------
% 3.13/1.69 % (9370)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_9159 for (2999ds/18Mi)
% 3.13/1.69 % (9368)Instruction limit reached!
% 3.13/1.69 % (9368)------------------------------
% 3.13/1.69 % (9368)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.69 % (9368)Termination reason: Unknown
% 3.13/1.69 % (9368)Termination phase: shuffling
% 3.13/1.69
% 3.13/1.69 % (9368)Memory used [KB]: 1407
% 3.13/1.69 % (9368)Time elapsed: 0.004 s
% 3.13/1.69 % (9368)Instructions burned: 5 (million)
% 3.13/1.69 % (9368)------------------------------
% 3.13/1.69 % (9368)------------------------------
% 3.13/1.69 % (9372)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_9159 for (2999ds/6Mi)
% 3.13/1.69 % (9370)Instruction limit reached!
% 3.13/1.69 % (9370)------------------------------
% 3.13/1.69 % (9370)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.69 % (9370)Termination reason: Unknown
% 3.13/1.69 % (9370)Termination phase: shuffling
% 3.13/1.69
% 3.13/1.69 % (9370)Memory used [KB]: 1663
% 3.13/1.69 % (9370)Time elapsed: 0.007 s
% 3.13/1.69 % (9370)Instructions burned: 18 (million)
% 3.13/1.69 % (9370)------------------------------
% 3.13/1.69 % (9370)------------------------------
% 3.13/1.69 % (9372)Instruction limit reached!
% 3.13/1.69 % (9372)------------------------------
% 3.13/1.69 % (9372)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.69 % (9372)Termination reason: Unknown
% 3.13/1.69 % (9372)Termination phase: shuffling
% 3.13/1.69
% 3.13/1.69 % (9372)Memory used [KB]: 1407
% 3.13/1.69 % (9372)Time elapsed: 0.004 s
% 3.13/1.69 % (9372)Instructions burned: 6 (million)
% 3.13/1.69 % (9372)------------------------------
% 3.13/1.69 % (9372)------------------------------
% 3.13/1.69 % (9371)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_9159 for (2999ds/710Mi)
% 3.13/1.70 % (9373)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_9159 for (2999ds/902Mi)
% 3.13/1.70 % (9374)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_9159 for (2999ds/21Mi)
% 3.13/1.70 % (9375)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_9159 for (2999ds/5Mi)
% 3.13/1.70 % (9375)Instruction limit reached!
% 3.13/1.70 % (9375)------------------------------
% 3.13/1.70 % (9375)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.70 % (9375)Termination reason: Unknown
% 3.13/1.70 % (9375)Termination phase: shuffling
% 3.13/1.70
% 3.13/1.70 % (9375)Memory used [KB]: 1407
% 3.13/1.70 % (9375)Time elapsed: 0.003 s
% 3.13/1.70 % (9375)Instructions burned: 6 (million)
% 3.13/1.70 % (9375)------------------------------
% 3.13/1.70 % (9375)------------------------------
% 3.13/1.70 % (9376)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_9159 for (2999ds/6Mi)
% 3.13/1.71 % (9374)Instruction limit reached!
% 3.13/1.71 % (9374)------------------------------
% 3.13/1.71 % (9374)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.71 % (9374)Termination reason: Unknown
% 3.13/1.71 % (9374)Termination phase: shuffling
% 3.13/1.71
% 3.13/1.71 % (9374)Memory used [KB]: 1791
% 3.13/1.71 % (9374)Time elapsed: 0.011 s
% 3.13/1.71 % (9374)Instructions burned: 21 (million)
% 3.13/1.71 % (9374)------------------------------
% 3.13/1.71 % (9374)------------------------------
% 3.13/1.71 % (9376)Instruction limit reached!
% 3.13/1.71 % (9376)------------------------------
% 3.13/1.71 % (9376)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.71 % (9376)Termination reason: Unknown
% 3.13/1.71 % (9376)Termination phase: shuffling
% 3.13/1.71
% 3.13/1.71 % (9376)Memory used [KB]: 1407
% 3.13/1.71 % (9376)Time elapsed: 0.004 s
% 3.13/1.71 % (9376)Instructions burned: 7 (million)
% 3.13/1.71 % (9376)------------------------------
% 3.13/1.71 % (9376)------------------------------
% 3.13/1.71 % (9377)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_9159 for (2999ds/377Mi)
% 3.13/1.72 % (9378)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_9159 for (2999ds/779Mi)
% 3.13/1.72 % (9379)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_9159 for (2999ds/19Mi)
% 3.13/1.72 % (9379)Instruction limit reached!
% 3.13/1.72 % (9379)------------------------------
% 3.13/1.72 % (9379)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.72 % (9379)Termination reason: Unknown
% 3.13/1.72 % (9379)Termination phase: shuffling
% 3.13/1.72
% 3.13/1.72 % (9379)Memory used [KB]: 1663
% 3.13/1.72 % (9379)Time elapsed: 0.008 s
% 3.13/1.72 % (9379)Instructions burned: 20 (million)
% 3.13/1.72 % (9379)------------------------------
% 3.13/1.72 % (9379)------------------------------
% 3.13/1.73 % (9356)Instruction limit reached!
% 3.13/1.73 % (9356)------------------------------
% 3.13/1.73 % (9356)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.73 % (9356)Termination reason: Unknown
% 3.13/1.73 % (9356)Termination phase: Saturation
% 3.13/1.73
% 3.13/1.73 % (9356)Memory used [KB]: 7036
% 3.13/1.73 % (9356)Time elapsed: 0.108 s
% 3.13/1.73 % (9356)Instructions burned: 275 (million)
% 3.13/1.73 % (9356)------------------------------
% 3.13/1.73 % (9356)------------------------------
% 3.13/1.73 % (9385)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_9159 for (2998ds/879Mi)
% 3.13/1.74 % (9386)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_9159 for (2998ds/17Mi)
% 3.13/1.74 % (9386)Instruction limit reached!
% 3.13/1.74 % (9386)------------------------------
% 3.13/1.74 % (9386)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.74 % (9386)Termination reason: Unknown
% 3.13/1.74 % (9386)Termination phase: shuffling
% 3.13/1.74
% 3.13/1.74 % (9386)Memory used [KB]: 1791
% 3.13/1.74 % (9386)Time elapsed: 0.007 s
% 3.13/1.74 % (9386)Instructions burned: 18 (million)
% 3.13/1.74 % (9386)------------------------------
% 3.13/1.74 % (9386)------------------------------
% 3.13/1.75 % (9387)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_9159 for (2998ds/3Mi)
% 3.13/1.76 % (9387)Instruction limit reached!
% 3.13/1.76 % (9387)------------------------------
% 3.13/1.76 % (9387)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.76 % (9387)Termination reason: Unknown
% 3.13/1.76 % (9387)Termination phase: shuffling
% 3.13/1.76
% 3.13/1.76 % (9387)Memory used [KB]: 1407
% 3.13/1.76 % (9387)Time elapsed: 0.003 s
% 3.13/1.76 % (9387)Instructions burned: 5 (million)
% 3.13/1.76 % (9387)------------------------------
% 3.13/1.76 % (9387)------------------------------
% 3.13/1.77 % (9388)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_9159 for (2998ds/30Mi)
% 3.13/1.78 % (9388)Instruction limit reached!
% 3.13/1.78 % (9388)------------------------------
% 3.13/1.78 % (9388)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.78 % (9388)Termination reason: Unknown
% 3.13/1.78 % (9388)Termination phase: Property scanning
% 3.13/1.78
% 3.13/1.78 % (9388)Memory used [KB]: 1791
% 3.13/1.78 % (9388)Time elapsed: 0.011 s
% 3.13/1.78 % (9388)Instructions burned: 32 (million)
% 3.13/1.78 % (9388)------------------------------
% 3.13/1.78 % (9388)------------------------------
% 3.13/1.78 % (9389)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_9159 for (2998ds/127Mi)
% 3.13/1.83 % (9389)Instruction limit reached!
% 3.13/1.83 % (9389)------------------------------
% 3.13/1.83 % (9389)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.83 % (9389)Termination reason: Unknown
% 3.13/1.83 % (9389)Termination phase: Saturation
% 3.13/1.83
% 3.13/1.83 % (9389)Memory used [KB]: 6908
% 3.13/1.83 % (9389)Time elapsed: 0.042 s
% 3.13/1.83 % (9389)Instructions burned: 127 (million)
% 3.13/1.83 % (9389)------------------------------
% 3.13/1.83 % (9389)------------------------------
% 3.13/1.83 % (9377)Instruction limit reached!
% 3.13/1.83 % (9377)------------------------------
% 3.13/1.83 % (9377)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.83 % (9377)Termination reason: Unknown
% 3.13/1.83 % (9377)Termination phase: Saturation
% 3.13/1.83
% 3.13/1.83 % (9377)Memory used [KB]: 7931
% 3.13/1.83 % (9377)Time elapsed: 0.115 s
% 3.13/1.83 % (9377)Instructions burned: 378 (million)
% 3.13/1.83 % (9377)------------------------------
% 3.13/1.83 % (9377)------------------------------
% 3.13/1.84 % (9400)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=3:si=on:rtra=on_0 on DTF2THF_9159 for (2997ds/3Mi)
% 3.13/1.84 % (9400)Instruction limit reached!
% 3.13/1.84 % (9400)------------------------------
% 3.13/1.84 % (9400)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.84 % (9400)Termination reason: Unknown
% 3.13/1.84 % (9400)Termination phase: shuffling
% 3.13/1.84
% 3.13/1.84 % (9400)Memory used [KB]: 1407
% 3.13/1.84 % (9400)Time elapsed: 0.004 s
% 3.13/1.84 % (9400)Instructions burned: 5 (million)
% 3.13/1.84 % (9400)------------------------------
% 3.13/1.84 % (9400)------------------------------
% 3.13/1.84 % (9399)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_9159 for (2997ds/100Mi)
% 3.13/1.86 % (9406)lrs+10_8:1_au=on:avsq=on:e2e=on:ins=3:s2a=on:s2at=3.0:ss=axioms:i=20:si=on:rtra=on_0 on DTF2THF_9159 for (2997ds/20Mi)
% 3.13/1.87 % (9406)Instruction limit reached!
% 3.13/1.87 % (9406)------------------------------
% 3.13/1.87 % (9406)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.87 % (9406)Termination reason: Unknown
% 3.13/1.87 % (9406)Termination phase: shuffling
% 3.13/1.87
% 3.13/1.87 % (9406)Memory used [KB]: 1791
% 3.13/1.87 % (9406)Time elapsed: 0.013 s
% 3.13/1.87 % (9406)Instructions burned: 20 (million)
% 3.13/1.87 % (9406)------------------------------
% 3.13/1.87 % (9406)------------------------------
% 3.13/1.89 % (9399)Instruction limit reached!
% 3.13/1.89 % (9399)------------------------------
% 3.13/1.89 % (9399)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.89 % (9399)Termination reason: Unknown
% 3.13/1.89 % (9399)Termination phase: Property scanning
% 3.13/1.89
% 3.13/1.89 % (9399)Memory used [KB]: 2302
% 3.13/1.89 % (9399)Time elapsed: 0.047 s
% 3.13/1.89 % (9399)Instructions burned: 101 (million)
% 3.13/1.89 % (9399)------------------------------
% 3.13/1.89 % (9399)------------------------------
% 3.13/1.89 % (9411)dis+1002_1:1_cbe=off:hud=5:nm=4:plsq=on:plsqr=7,1:prag=on:sp=const_max:tnu=1:i=86:si=on:rtra=on_0 on DTF2THF_9159 for (2997ds/86Mi)
% 3.13/1.90 % (9415)lrs+1010_1:1_au=on:cbe=off:nm=2:ntd=on:sd=2:ss=axioms:st=5.0:i=107:si=on:rtra=on_0 on DTF2THF_9159 for (2997ds/107Mi)
% 3.13/1.92 % (9411)Instruction limit reached!
% 3.13/1.92 % (9411)------------------------------
% 3.13/1.92 % (9411)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.92 % (9411)Termination reason: Unknown
% 3.13/1.92 % (9411)Termination phase: Property scanning
% 3.13/1.92
% 3.13/1.92 % (9411)Memory used [KB]: 2302
% 3.13/1.92 % (9411)Time elapsed: 0.034 s
% 3.13/1.92 % (9411)Instructions burned: 88 (million)
% 3.13/1.92 % (9411)------------------------------
% 3.13/1.92 % (9411)------------------------------
% 3.13/1.93 % (9422)lrs+2_1:1024_cnfonf=lazy_gen:fe=off:hud=15:plsq=on:plsqc=1:plsqr=32,1:i=39:si=on:rtra=on_0 on DTF2THF_9159 for (2996ds/39Mi)
% 3.13/1.94 % (9415)Instruction limit reached!
% 3.13/1.94 % (9415)------------------------------
% 3.13/1.94 % (9415)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.94 % (9415)Termination reason: Unknown
% 3.13/1.94 % (9415)Termination phase: Saturation
% 3.13/1.94
% 3.13/1.94 % (9415)Memory used [KB]: 6780
% 3.13/1.94 % (9415)Time elapsed: 0.034 s
% 3.13/1.94 % (9415)Instructions burned: 109 (million)
% 3.13/1.94 % (9415)------------------------------
% 3.13/1.94 % (9415)------------------------------
% 3.13/1.95 % (9422)Instruction limit reached!
% 3.13/1.95 % (9422)------------------------------
% 3.13/1.95 % (9422)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.95 % (9422)Termination reason: Unknown
% 3.13/1.95 % (9422)Termination phase: Property scanning
% 3.13/1.95
% 3.13/1.95 % (9422)Memory used [KB]: 2174
% 3.13/1.95 % (9422)Time elapsed: 0.013 s
% 3.13/1.95 % (9422)Instructions burned: 41 (million)
% 3.13/1.95 % (9422)------------------------------
% 3.13/1.95 % (9422)------------------------------
% 3.13/1.95 % (9423)dis+10_1:1_cnfonf=lazy_not_gen:fsr=off:kws=precedence:nwc=5.0:s2a=on:ss=axioms:st=1.5:i=448:si=on:rtra=on_0 on DTF2THF_9159 for (2996ds/448Mi)
% 3.13/1.96 % (9424)lrs+10_1:512_au=on:fde=unused:lma=on:nm=32:plsq=on:plsqc=1:plsqr=16121663,131072:sfv=off:sp=const_max:ss=axioms:st=3.0:tgt=full:i=46:si=on:rtra=on_0 on DTF2THF_9159 for (2996ds/46Mi)
% 3.13/1.97 % (9371)Instruction limit reached!
% 3.13/1.97 % (9371)------------------------------
% 3.13/1.97 % (9371)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.97 % (9371)Termination reason: Unknown
% 3.13/1.97 % (9371)Termination phase: Saturation
% 3.13/1.97
% 3.13/1.97 % (9371)Memory used [KB]: 7931
% 3.13/1.97 % (9371)Time elapsed: 0.296 s
% 3.13/1.97 % (9371)Instructions burned: 712 (million)
% 3.13/1.97 % (9371)------------------------------
% 3.13/1.97 % (9371)------------------------------
% 3.13/1.97 % (9424)Instruction limit reached!
% 3.13/1.97 % (9424)------------------------------
% 3.13/1.97 % (9424)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.97 % (9424)Termination reason: Unknown
% 3.13/1.97 % (9424)Termination phase: Preprocessing 3
% 3.13/1.97
% 3.13/1.97 % (9424)Memory used [KB]: 2046
% 3.13/1.97 % (9424)Time elapsed: 0.016 s
% 3.13/1.97 % (9424)Instructions burned: 46 (million)
% 3.13/1.97 % (9424)------------------------------
% 3.13/1.97 % (9424)------------------------------
% 3.13/1.98 % (9425)lrs+10_1:10_au=on:av=off:cbe=off:cnfonf=lazy_pi_sigma_gen:ntd=on:plsq=on:plsqc=1:plsqr=32,1:i=98:si=on:rtra=on_0 on DTF2THF_9159 for (2996ds/98Mi)
% 3.13/1.98 % (9426)ott+1002_1:1_apa=on:au=on:bd=off:cnfonf=off:fd=off:sos=on:sp=weighted_frequency:i=507:si=on:rtra=on_0 on DTF2THF_9159 for (2996ds/507Mi)
% 3.13/1.99 % (9378)Instruction limit reached!
% 3.13/1.99 % (9378)------------------------------
% 3.13/1.99 % (9378)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/1.99 % (9378)Termination reason: Unknown
% 3.13/1.99 % (9378)Termination phase: Saturation
% 3.13/1.99
% 3.13/1.99 % (9378)Memory used [KB]: 7675
% 3.13/1.99 % (9378)Time elapsed: 0.297 s
% 3.13/1.99 % (9378)Instructions burned: 780 (million)
% 3.13/1.99 % (9378)------------------------------
% 3.13/1.99 % (9378)------------------------------
% 3.13/2.00 % (9427)dis+1010_2:3_amm=off:fd=preordered:ixr=off:nm=0:pe=on:piset=equals:prag=on:sac=on:tgt=ground:i=149:si=on:rtra=on_0 on DTF2THF_9159 for (2996ds/149Mi)
% 3.13/2.00 % (9362)Instruction limit reached!
% 3.13/2.00 % (9362)------------------------------
% 3.13/2.00 % (9362)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/2.00 % (9362)Termination reason: Unknown
% 3.13/2.00 % (9362)Termination phase: Saturation
% 3.13/2.00
% 3.13/2.00 % (9362)Memory used [KB]: 9722
% 3.13/2.00 % (9362)Time elapsed: 0.353 s
% 3.13/2.00 % (9362)Instructions burned: 1043 (million)
% 3.13/2.00 % (9362)------------------------------
% 3.13/2.00 % (9362)------------------------------
% 3.13/2.01 % (9425)Instruction limit reached!
% 3.13/2.01 % (9425)------------------------------
% 3.13/2.01 % (9425)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/2.01 % (9425)Termination reason: Unknown
% 3.13/2.01 % (9425)Termination phase: Property scanning
% 3.13/2.01
% 3.13/2.01 % (9425)Memory used [KB]: 2302
% 3.13/2.01 % (9425)Time elapsed: 0.030 s
% 3.13/2.01 % (9425)Instructions burned: 98 (million)
% 3.13/2.01 % (9425)------------------------------
% 3.13/2.01 % (9425)------------------------------
% 3.13/2.01 % (9373)Instruction limit reached!
% 3.13/2.01 % (9373)------------------------------
% 3.13/2.01 % (9373)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/2.01 % (9373)Termination reason: Unknown
% 3.13/2.01 % (9373)Termination phase: Saturation
% 3.13/2.01
% 3.13/2.01 % (9373)Memory used [KB]: 8699
% 3.13/2.01 % (9373)Time elapsed: 0.337 s
% 3.13/2.01 % (9373)Instructions burned: 902 (million)
% 3.13/2.01 % (9373)------------------------------
% 3.13/2.01 % (9373)------------------------------
% 3.13/2.01 % (9428)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_9159 for (2996ds/39Mi)
% 3.13/2.02 % (9429)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_9159 for (2996ds/6Mi)
% 3.13/2.02 % (9429)Instruction limit reached!
% 3.13/2.02 % (9429)------------------------------
% 3.13/2.02 % (9429)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.13/2.02 % (9429)Termination reason: Unknown
% 3.13/2.02 % (9429)Termination phase: shuffling
% 3.13/2.02
% 3.13/2.02 % (9429)Memory used [KB]: 1407
% 3.13/2.02 % (9429)Time elapsed: 0.004 s
% 3.13/2.02 % (9429)Instructions burned: 7 (million)
% 3.13/2.02 % (9429)------------------------------
% 3.13/2.02 % (9429)------------------------------
% 3.13/2.02 % (9430)dis+1002_1:1_aac=none:au=on:e2e=on:nm=2:nwc=10.0:sd=1:ss=axioms:st=3.0:i=1087:si=on:rtra=on_0 on DTF2THF_9159 for (2996ds/1087Mi)
% 5.88/2.02 % (9428)Instruction limit reached!
% 5.88/2.02 % (9428)------------------------------
% 5.88/2.02 % (9428)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.88/2.02 % (9428)Termination reason: Unknown
% 5.88/2.02 % (9428)Termination phase: Preprocessing 3
% 5.88/2.02
% 5.88/2.02 % (9428)Memory used [KB]: 2046
% 5.88/2.02 % (9428)Time elapsed: 0.013 s
% 5.88/2.02 % (9428)Instructions burned: 39 (million)
% 5.88/2.02 % (9428)------------------------------
% 5.88/2.02 % (9428)------------------------------
% 5.88/2.03 % (9431)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_9159 for (2995ds/47Mi)
% 5.88/2.03 % (9426)First to succeed.
% 5.88/2.03 % (9432)lrs+1010_1:128_av=off:cbe=off:cnfonf=lazy_simp:fsr=off:ntd=on:ss=axioms:st=1.5:i=44:si=on:rtra=on_0 on DTF2THF_9159 for (2995ds/44Mi)
% 5.88/2.04 % (9426)Refutation found. Thanks to Tanya!
% 5.88/2.04 % SZS status Theorem for DTF2THF_9159
% 5.88/2.04 % SZS output start Proof for DTF2THF_9159
% 5.88/2.04 thf(type_def_6, type, set: ($tType) > $tType).
% 5.88/2.04 thf(type_def_7, type, nat: $tType).
% 5.88/2.04 thf(type_def_8, type, paraco415392788lle_tv: $tType).
% 5.88/2.04 thf(type_def_9, type, product_prod: ($tType * $tType) > $tType).
% 5.88/2.04 thf(func_def_0, type, paraco415392788lle_tv: $tType).
% 5.88/2.04 thf(func_def_1, type, product_prod: ($tType * $tType) > $tType).
% 5.88/2.04 thf(func_def_2, type, set: ($tType) > $tType).
% 5.88/2.04 thf(func_def_3, type, nat: $tType).
% 5.88/2.04 thf(func_def_4, type, type: !>[X0: $tType]:($o)).
% 5.88/2.04 thf(func_def_5, type, ord: !>[X0: $tType]:($o)).
% 5.88/2.04 thf(func_def_6, type, top: !>[X0: $tType]:($o)).
% 5.88/2.04 thf(func_def_7, type, order: !>[X0: $tType]:($o)).
% 5.88/2.04 thf(func_def_8, type, group_add: !>[X0: $tType]:($o)).
% 5.88/2.04 thf(func_def_9, type, no_bot: !>[X0: $tType]:($o)).
% 5.88/2.04 thf(func_def_10, type, no_top: !>[X0: $tType]:($o)).
% 5.88/2.04 thf(func_def_11, type, linorder: !>[X0: $tType]:($o)).
% 5.88/2.04 thf(func_def_12, type, preorder: !>[X0: $tType]:($o)).
% 5.88/2.04 thf(func_def_13, type, ab_group_add: !>[X0: $tType]:($o)).
% 5.88/2.04 thf(func_def_14, type, order_top: !>[X0: $tType]:($o)).
% 5.88/2.04 thf(func_def_15, type, wellorder: !>[X0: $tType]:($o)).
% 5.88/2.04 thf(func_def_16, type, semigroup_add: !>[X0: $tType]:($o)).
% 5.88/2.04 thf(func_def_17, type, dense_order: !>[X0: $tType]:($o)).
% 5.88/2.04 thf(func_def_18, type, comm_monoid_add: !>[X0: $tType]:($o)).
% 5.88/2.04 thf(func_def_19, type, ab_semigroup_add: !>[X0: $tType]:($o)).
% 5.88/2.04 thf(func_def_20, type, dense_linorder: !>[X0: $tType]:($o)).
% 5.88/2.04 thf(func_def_21, type, cancel_semigroup_add: !>[X0: $tType]:($o)).
% 5.88/2.04 thf(func_def_22, type, ordere779506340up_add: !>[X0: $tType]:($o)).
% 5.88/2.04 thf(func_def_23, type, canoni770627133id_add: !>[X0: $tType]:($o)).
% 5.88/2.04 thf(func_def_24, type, ordere236663937imp_le: !>[X0: $tType]:($o)).
% 5.88/2.04 thf(func_def_25, type, ordere223160158up_add: !>[X0: $tType]:($o)).
% 5.88/2.04 thf(func_def_26, type, strict2144017051up_add: !>[X0: $tType]:($o)).
% 5.88/2.04 thf(func_def_27, type, inj_on: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > $o)).
% 5.88/2.04 thf(func_def_28, type, strict_mono_on: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > $o)).
% 5.88/2.04 thf(func_def_29, type, the_inv_into: !>[X0: $tType, X1: $tType]:(set @ X0 > (X0 > X1) > X1 > X0)).
% 5.88/2.04 thf(func_def_30, type, plus_plus: !>[X0: $tType]:(X0 > X0 > X0)).
% 5.88/2.04 thf(func_def_31, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 5.88/2.04 thf(func_def_32, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 5.88/2.04 thf(func_def_33, type, order_strict_mono: !>[X0: $tType, X1: $tType]:((X0 > X1) > $o)).
% 5.88/2.04 thf(func_def_34, type, top_top: !>[X0: $tType]:(X0)).
% 5.88/2.04 thf(func_def_35, type, paraco1920534163nge_tv: (nat > nat) > paraco415392788lle_tv > paraco415392788lle_tv).
% 5.88/2.04 thf(func_def_36, type, paraco676387099_Indet: nat > paraco415392788lle_tv).
% 5.88/2.04 thf(func_def_37, type, paraco152590079rec_tv: !>[X0: $tType]:(($o > X0) > (nat > X0) > paraco415392788lle_tv > X0)).
% 5.88/2.04 thf(func_def_38, type, product_apfst: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > product_prod @ X0 @ X2 > product_prod @ X1 @ X2)).
% 5.88/2.04 thf(func_def_39, type, product_apsnd: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > product_prod @ X2 @ X0 > product_prod @ X2 @ X1)).
% 5.88/2.04 thf(func_def_40, type, product_map_prod: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((X0 > X1) > (X2 > X3) > product_prod @ X0 @ X2 > product_prod @ X1 @ X3)).
% 5.88/2.04 thf(func_def_41, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 5.88/2.04 thf(func_def_42, type, image: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > set @ X1)).
% 5.88/2.04 thf(func_def_43, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 5.88/2.04 thf(func_def_44, type, f: nat > nat).
% 5.88/2.04 thf(func_def_60, type, sK0: !>[X0: $tType]:((X0 > $o) > X0)).
% 5.88/2.04 thf(func_def_61, type, sK1: !>[X0: $tType]:((X0 > $o) > X0)).
% 5.88/2.04 thf(func_def_62, type, sP2: !>[X0: $tType]:(X0 > X0 > X0 > $o)).
% 5.88/2.04 thf(func_def_63, type, sP3: !>[X0: $tType]:(X0 > X0 > X0 > $o)).
% 5.88/2.04 thf(func_def_64, type, sP4: !>[X0: $tType]:(X0 > X0 > X0 > $o)).
% 5.88/2.04 thf(func_def_65, type, sP5: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > $o)).
% 5.88/2.04 thf(func_def_66, type, sK6: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 5.88/2.04 thf(func_def_67, type, sK7: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 5.88/2.04 thf(func_def_68, type, sK8: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 5.88/2.04 thf(func_def_69, type, sK9: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 5.88/2.04 thf(func_def_70, type, sK10: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 5.88/2.04 thf(func_def_71, type, sK11: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 5.88/2.04 thf(func_def_72, type, sK12: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 5.88/2.04 thf(func_def_73, type, sK13: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 5.88/2.04 thf(func_def_74, type, sK14: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X0 > X1) > set @ X1 > X1)).
% 5.88/2.04 thf(func_def_75, type, sK15: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 5.88/2.04 thf(func_def_76, type, sK16: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 5.88/2.04 thf(func_def_77, type, sK17: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > set @ X1 > X1)).
% 5.88/2.04 thf(func_def_78, type, sK18: !>[X0: $tType]:(X0)).
% 5.88/2.04 thf(func_def_79, type, sK19: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 5.88/2.04 thf(func_def_80, type, sK20: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 5.88/2.04 thf(func_def_81, type, sK21: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 5.88/2.04 thf(func_def_82, type, sK22: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 5.88/2.04 thf(func_def_83, type, sK23: !>[X0: $tType]:(set @ X0 > X0)).
% 5.88/2.04 thf(func_def_84, type, sK24: !>[X0: $tType, X1: $tType]:((X1 > X1) > (X1 > X0) > set @ X1 > X1)).
% 5.88/2.04 thf(func_def_85, type, sK25: !>[X0: $tType, X1: $tType]:((X1 > X1) > (X1 > X0) > set @ X1 > X1)).
% 5.88/2.04 thf(func_def_86, type, sK26: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 5.88/2.04 thf(func_def_87, type, sK27: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 5.88/2.04 thf(func_def_88, type, sK28: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 5.88/2.04 thf(func_def_89, type, sK29: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 5.88/2.04 thf(func_def_90, type, sK30: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 5.88/2.04 thf(func_def_91, type, sK31: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 5.88/2.04 thf(func_def_92, type, sK32: !>[X0: $tType, X1: $tType]:((X0 > $o) > set @ X1 > (X1 > X0) > X1)).
% 5.88/2.04 thf(func_def_93, type, sK33: !>[X0: $tType]:(set @ X0 > set @ X0 > X0)).
% 5.88/2.04 thf(func_def_94, type, sK34: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 5.88/2.04 thf(func_def_95, type, sK35: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 5.88/2.04 thf(func_def_96, type, sK36: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X0 > X1) > X1)).
% 5.88/2.04 thf(func_def_97, type, sK37: !>[X0: $tType, X1: $tType]:((X1 > X0) > set @ X1 > X1)).
% 5.88/2.04 thf(func_def_98, type, sK38: !>[X0: $tType, X1: $tType]:((X1 > X0) > set @ X1 > X1)).
% 5.88/2.04 thf(func_def_99, type, sK39: !>[X0: $tType]:(X0 > X0)).
% 5.88/2.04 thf(func_def_100, type, sK40: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 5.88/2.04 thf(func_def_101, type, sK41: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 5.88/2.04 thf(func_def_102, type, sK42: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 5.88/2.04 thf(func_def_103, type, sK43: !>[X0: $tType]:(X0 > X0)).
% 5.88/2.04 thf(func_def_104, type, sK44: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 5.88/2.04 thf(func_def_105, type, sK45: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 5.88/2.04 thf(func_def_106, type, sK46: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 5.88/2.04 thf(func_def_107, type, sK47: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 5.88/2.04 thf(func_def_108, type, sK48: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 5.88/2.04 thf(func_def_109, type, sK49: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 5.88/2.04 thf(func_def_110, type, sK50: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 5.88/2.04 thf(func_def_111, type, sK51: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 5.88/2.04 thf(func_def_112, type, sK52: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 5.88/2.04 thf(func_def_113, type, sK53: !>[X0: $tType]:(X0 > X0 > X0)).
% 5.88/2.04 thf(func_def_114, type, sK54: !>[X0: $tType]:(X0 > X0 > X0 > X0)).
% 5.88/2.04 thf(func_def_115, type, sK55: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 5.88/2.04 thf(func_def_116, type, sK56: !>[X0: $tType]:(X0 > X0 > X0)).
% 5.88/2.04 thf(func_def_117, type, sK57: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X0 > X1) > X0)).
% 5.88/2.04 thf(func_def_118, type, sK58: !>[X0: $tType, X1: $tType]:((X1 > X0) > set @ X1 > X1)).
% 5.88/2.04 thf(func_def_119, type, sK59: !>[X0: $tType, X1: $tType]:((X1 > X0) > set @ X1 > X1)).
% 5.88/2.04 thf(func_def_120, type, sK60: !>[X0: $tType, X1: $tType]:((X0 > $o) > set @ X1 > (X1 > X0) > X0)).
% 5.88/2.04 thf(func_def_121, type, sK61: !>[X0: $tType]:((X0 > $o) > X0)).
% 5.88/2.04 thf(func_def_122, type, sK62: !>[X0: $tType]:(X0 > X0 > X0)).
% 5.88/2.04 thf(func_def_123, type, sK63: !>[X0: $tType, X1: $tType]:(set @ X1 > (X0 > X1) > set @ X0 > X0)).
% 5.88/2.04 thf(func_def_124, type, sK64: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 5.88/2.04 thf(func_def_125, type, sK65: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 5.88/2.04 thf(func_def_126, type, sK66: !>[X0: $tType, X1: $tType]:(set @ X1 > (X1 > X0) > set @ X0 > set @ X1)).
% 5.88/2.04 thf(func_def_127, type, sK67: !>[X0: $tType, X1: $tType]:(X0 > (X1 > X0) > X1)).
% 5.88/2.04 thf(func_def_128, type, sK68: !>[X0: $tType, X1: $tType]:(X0 > (X1 > X0) > X1)).
% 5.88/2.04 thf(func_def_129, type, sK69: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > X0)).
% 5.88/2.04 thf(func_def_130, type, sK70: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > X0)).
% 5.88/2.04 thf(func_def_131, type, sK71: !>[X0: $tType]:(set @ X0 > X0)).
% 5.88/2.04 thf(func_def_132, type, sK72: !>[X0: $tType]:(set @ X0 > X0)).
% 5.88/2.04 thf(func_def_133, type, sK73: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 5.88/2.04 thf(func_def_134, type, sK74: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 5.88/2.04 thf(func_def_135, type, sK75: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 5.88/2.04 thf(func_def_136, type, sK76: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 5.88/2.04 thf(func_def_137, type, sK77: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 5.88/2.04 thf(func_def_138, type, sK78: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 5.88/2.04 thf(func_def_139, type, sK79: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > set @ X1 > X1)).
% 5.88/2.04 thf(func_def_140, type, sK80: !>[X0: $tType]:(X0 > X0 > X0)).
% 5.88/2.04 thf(func_def_141, type, sK81: !>[X0: $tType]:(X0 > X0 > X0 > X0)).
% 5.88/2.04 thf(func_def_143, type, ph83: !>[X0: $tType]:(X0)).
% 5.88/2.04 thf(f2031,plain,(
% 5.88/2.04 $false),
% 5.88/2.04 inference(subsumption_resolution,[status(thm)],[f2021,f1893])).
% 5.88/2.04 thf(f1893,plain,(
% 5.88/2.04 ((sK76 @ paraco415392788lle_tv @ paraco415392788lle_tv @ (paraco1920534163nge_tv @ f)) != (sK77 @ paraco415392788lle_tv @ paraco415392788lle_tv @ (paraco1920534163nge_tv @ f)))),
% 5.88/2.04 inference(trivial_inequality_removal,[status(thm)],[f1869])).
% 5.88/2.04 thf(f1869,plain,(
% 5.88/2.04 ((sK76 @ paraco415392788lle_tv @ paraco415392788lle_tv @ (paraco1920534163nge_tv @ f)) != (sK77 @ paraco415392788lle_tv @ paraco415392788lle_tv @ (paraco1920534163nge_tv @ f))) | ($true != $true)),
% 5.88/2.04 inference(superposition,[status(thm)],[f1851,f1836])).
% 5.88/2.04 thf(f1836,plain,(
% 5.88/2.04 ( ! [X1 : $tType,X0 : $tType,X2 : X1 > X0] : (((!! @ X1 @ (^[Y0 : X1]: ((member @ X1 @ Y0 @ top_top @ set @ X1) => (!! @ X1 @ (^[Y1 : X1]: ((member @ X1 @ Y1 @ top_top @ set @ X1) => (((X2 @ Y1) = (X2 @ Y0)) => (Y0 = Y1)))))))) = $true) | ((sK77 @ X0 @ X1 @ X2) != (sK76 @ X0 @ X1 @ X2))) )),
% 5.88/2.04 inference(beta_eta_normalization,[status(thm)],[f1750])).
% 5.88/2.04 thf(f1750,plain,(
% 5.88/2.04 ( ! [X1 : $tType,X0 : $tType,X2 : X1 > X0] : ((((^[Y0 : X1 > X0]: ((^[Y1 : set @ X1]: (!! @ X1 @ (^[Y2 : X1]: ((member @ X1 @ Y2 @ Y1) => (!! @ X1 @ (^[Y3 : X1]: ((member @ X1 @ Y3 @ Y1) => (((Y0 @ Y3) = (Y0 @ Y2)) => (Y2 = Y3))))))))))) @ X2 @ top_top @ set @ X1) = $true) | ((sK77 @ X0 @ X1 @ X2) != (sK76 @ X0 @ X1 @ X2))) )),
% 5.88/2.04 inference(definition_unfolding,[status(thm)],[f1668,f1423])).
% 5.88/2.04 thf(f1423,plain,(
% 5.88/2.04 ( ! [X1 : $tType,X0 : $tType] : ((inj_on @ X1 @ X0 = (^[Y0 : X1 > X0]: ((^[Y1 : set @ X1]: (!! @ X1 @ (^[Y2 : X1]: ((member @ X1 @ Y2 @ Y1) => (!! @ X1 @ (^[Y3 : X1]: ((member @ X1 @ Y3 @ Y1) => (((Y0 @ Y3) = (Y0 @ Y2)) => (Y2 = Y3))))))))))))) )),
% 5.88/2.04 inference(cnf_transformation,[status(thm)],[f417])).
% 5.88/2.04 thf(f417,plain,(
% 5.88/2.04 ! [X0 : $tType,X1 : $tType] : (inj_on @ X1 @ X0 = (^[Y0 : X1 > X0]: ((^[Y1 : set @ X1]: (!! @ X1 @ (^[Y2 : X1]: ((member @ X1 @ Y2 @ Y1) => (!! @ X1 @ (^[Y3 : X1]: ((member @ X1 @ Y3 @ Y1) => (((Y0 @ Y3) = (Y0 @ Y2)) => (Y2 = Y3))))))))))))),
% 5.88/2.04 inference(fool_elimination,[status(thm)],[f416])).
% 5.88/2.04 thf(f416,plain,(
% 5.88/2.04 ! [X0 : $tType,X1 : $tType] : (inj_on @ X1 @ X0 = (^[X2 : X1 > X0, X3 : set @ X1] : (! [X4 : X1] : ((member @ X1 @ X4 @ X3) => ! [X5 : X1] : ((member @ X1 @ X5 @ X3) => (((X2 @ X5) = (X2 @ X4)) => (X4 = X5)))))))),
% 5.88/2.04 inference(rectify,[status(thm)],[f11])).
% 5.88/2.04 thf(f11,axiom,(
% 5.88/2.04 ! [X1 : $tType,X0 : $tType] : (inj_on @ X0 @ X1 = (^[X14 : X0 > X1, X15 : set @ X0] : (! [X7 : X0] : ((member @ X0 @ X7 @ X15) => ! [X12 : X0] : ((member @ X0 @ X12 @ X15) => (((X14 @ X7) = (X14 @ X12)) => (X7 = X12)))))))),
% 5.88/2.04 file('/export/starexec/sandbox/tmp/tmp.Ai5Zk59Gvc/DTF2THF_9159.p',fact_10_inj__on__def)).
% 5.88/2.04 thf(f1668,plain,(
% 5.88/2.04 ( ! [X1 : $tType,X0 : $tType,X2 : X1 > X0] : (((inj_on @ X1 @ X0 @ X2 @ top_top @ set @ X1) = $true) | ((sK77 @ X0 @ X1 @ X2) != (sK76 @ X0 @ X1 @ X2))) )),
% 5.88/2.04 inference(cnf_transformation,[status(thm)],[f1341])).
% 5.88/2.04 thf(f1341,plain,(
% 5.88/2.04 ! [X0 : $tType,X1 : $tType,X2 : X1 > X0] : (((inj_on @ X1 @ X0 @ X2 @ top_top @ set @ X1) = $true) | (((sK77 @ X0 @ X1 @ X2) != (sK76 @ X0 @ X1 @ X2)) & ((X2 @ (sK76 @ X0 @ X1 @ X2)) = (X2 @ (sK77 @ X0 @ X1 @ X2)))))),
% 5.88/2.04 inference(skolemisation,[status(esa),new_symbols(skolem,[sK76,sK77])],[f1200,f1340])).
% 5.88/2.04 thf(f1340,plain,(
% 5.88/2.04 ! [X1 : $tType,X0 : $tType,X2 : X1 > X0] : (? [X3 : X1,X4 : X1] : ((X3 != X4) & ((X2 @ X4) = (X2 @ X3))) => (((sK77 @ X0 @ X1 @ X2) != (sK76 @ X0 @ X1 @ X2)) & ((X2 @ (sK76 @ X0 @ X1 @ X2)) = (X2 @ (sK77 @ X0 @ X1 @ X2)))))),
% 5.88/2.04 introduced(definition,[],[choice_axiom])).
% 5.88/2.04 thf(f1200,plain,(
% 5.88/2.04 ! [X0 : $tType,X1 : $tType,X2 : X1 > X0] : (((inj_on @ X1 @ X0 @ X2 @ top_top @ set @ X1) = $true) | ? [X3 : X1,X4 : X1] : ((X3 != X4) & ((X2 @ X4) = (X2 @ X3))))),
% 5.88/2.04 inference(ennf_transformation,[status(thm)],[f324])).
% 5.88/2.04 thf(f324,plain,(
% 5.88/2.04 ! [X0 : $tType,X1 : $tType,X2 : X1 > X0] : (! [X3 : X1,X4 : X1] : (((X2 @ X4) = (X2 @ X3)) => (X3 = X4)) => ((inj_on @ X1 @ X0 @ X2 @ top_top @ set @ X1) = $true))),
% 5.88/2.04 inference(fool_elimination,[status(thm)],[f323])).
% 5.88/2.04 thf(f323,plain,(
% 5.88/2.04 ! [X0 : $tType,X1 : $tType,X2 : X1 > X0] : (! [X3 : X1,X4 : X1] : (((X2 @ X4) = (X2 @ X3)) => (X3 = X4)) => (inj_on @ X1 @ X0 @ X2 @ top_top @ set @ X1))),
% 5.88/2.04 inference(rectify,[status(thm)],[f6])).
% 5.88/2.04 thf(f6,axiom,(
% 5.88/2.04 ! [X1 : $tType,X0 : $tType,X8 : X0 > X1] : (! [X10 : X0,X11 : X0] : (((X8 @ X10) = (X8 @ X11)) => (X10 = X11)) => (inj_on @ X0 @ X1 @ X8 @ top_top @ set @ X0))),
% 5.88/2.04 file('/export/starexec/sandbox/tmp/tmp.Ai5Zk59Gvc/DTF2THF_9159.p',fact_5_injI)).
% 5.88/2.04 thf(f1851,plain,(
% 5.88/2.04 ((!! @ paraco415392788lle_tv @ (^[Y0 : paraco415392788lle_tv]: ((member @ paraco415392788lle_tv @ Y0 @ top_top @ set @ paraco415392788lle_tv) => (!! @ paraco415392788lle_tv @ (^[Y1 : paraco415392788lle_tv]: ((member @ paraco415392788lle_tv @ Y1 @ top_top @ set @ paraco415392788lle_tv) => (((paraco1920534163nge_tv @ f @ Y1) = (paraco1920534163nge_tv @ f @ Y0)) => (Y0 = Y1)))))))) != $true)),
% 5.88/2.04 inference(beta_eta_normalization,[status(thm)],[f1749])).
% 5.88/2.04 thf(f1749,plain,(
% 5.88/2.04 (((^[Y0 : paraco415392788lle_tv > paraco415392788lle_tv]: ((^[Y1 : set @ paraco415392788lle_tv]: (!! @ paraco415392788lle_tv @ (^[Y2 : paraco415392788lle_tv]: ((member @ paraco415392788lle_tv @ Y2 @ Y1) => (!! @ paraco415392788lle_tv @ (^[Y3 : paraco415392788lle_tv]: ((member @ paraco415392788lle_tv @ Y3 @ Y1) => (((Y0 @ Y3) = (Y0 @ Y2)) => (Y2 = Y3))))))))))) @ (paraco1920534163nge_tv @ f) @ top_top @ set @ paraco415392788lle_tv) != $true)),
% 5.88/2.04 inference(definition_unfolding,[status(thm)],[f1662,f1423])).
% 5.88/2.04 thf(f1662,plain,(
% 5.88/2.04 ((inj_on @ paraco415392788lle_tv @ paraco415392788lle_tv @ (paraco1920534163nge_tv @ f) @ top_top @ set @ paraco415392788lle_tv) != $true)),
% 5.88/2.04 inference(cnf_transformation,[status(thm)],[f889])).
% 5.88/2.04 thf(f889,plain,(
% 5.88/2.04 ((inj_on @ paraco415392788lle_tv @ paraco415392788lle_tv @ (paraco1920534163nge_tv @ f) @ top_top @ set @ paraco415392788lle_tv) != $true)),
% 5.88/2.04 inference(flattening,[status(thm)],[f525])).
% 5.88/2.04 thf(f525,plain,(
% 5.88/2.04 ~((inj_on @ paraco415392788lle_tv @ paraco415392788lle_tv @ (paraco1920534163nge_tv @ f) @ top_top @ set @ paraco415392788lle_tv) = $true)),
% 5.88/2.04 inference(fool_elimination,[status(thm)],[f524])).
% 5.88/2.04 thf(f524,plain,(
% 5.88/2.04 ~(inj_on @ paraco415392788lle_tv @ paraco415392788lle_tv @ (paraco1920534163nge_tv @ f) @ top_top @ set @ paraco415392788lle_tv)),
% 5.88/2.04 inference(rectify,[status(thm)],[f289])).
% 5.88/2.04 thf(f289,negated_conjecture,(
% 5.88/2.04 ~(inj_on @ paraco415392788lle_tv @ paraco415392788lle_tv @ (paraco1920534163nge_tv @ f) @ top_top @ set @ paraco415392788lle_tv)),
% 5.88/2.04 inference(negated_conjecture,[status(cth)],[f288])).
% 5.88/2.04 thf(f288,conjecture,(
% 5.88/2.04 (inj_on @ paraco415392788lle_tv @ paraco415392788lle_tv @ (paraco1920534163nge_tv @ f) @ top_top @ set @ paraco415392788lle_tv)),
% 5.88/2.04 file('/export/starexec/sandbox/tmp/tmp.Ai5Zk59Gvc/DTF2THF_9159.p',conj_0)).
% 5.88/2.04 thf(f2021,plain,(
% 5.88/2.04 ((sK76 @ paraco415392788lle_tv @ paraco415392788lle_tv @ (paraco1920534163nge_tv @ f)) = (sK77 @ paraco415392788lle_tv @ paraco415392788lle_tv @ (paraco1920534163nge_tv @ f)))),
% 5.88/2.04 inference(equality_resolution,[status(thm)],[f1942])).
% 5.88/2.04 thf(f1942,plain,(
% 5.88/2.04 ( ! [X0 : paraco415392788lle_tv] : (((paraco1920534163nge_tv @ f @ X0) != (paraco1920534163nge_tv @ f @ (sK76 @ paraco415392788lle_tv @ paraco415392788lle_tv @ (paraco1920534163nge_tv @ f)))) | ((sK77 @ paraco415392788lle_tv @ paraco415392788lle_tv @ (paraco1920534163nge_tv @ f)) = X0)) )),
% 5.88/2.04 inference(superposition,[status(thm)],[f1611,f1886])).
% 5.88/2.04 thf(f1886,plain,(
% 5.88/2.04 ((paraco1920534163nge_tv @ f @ (sK77 @ paraco415392788lle_tv @ paraco415392788lle_tv @ (paraco1920534163nge_tv @ f))) = (paraco1920534163nge_tv @ f @ (sK76 @ paraco415392788lle_tv @ paraco415392788lle_tv @ (paraco1920534163nge_tv @ f))))),
% 5.88/2.04 inference(trivial_inequality_removal,[status(thm)],[f1864])).
% 5.88/2.04 thf(f1864,plain,(
% 5.88/2.04 ((paraco1920534163nge_tv @ f @ (sK77 @ paraco415392788lle_tv @ paraco415392788lle_tv @ (paraco1920534163nge_tv @ f))) = (paraco1920534163nge_tv @ f @ (sK76 @ paraco415392788lle_tv @ paraco415392788lle_tv @ (paraco1920534163nge_tv @ f)))) | ($true != $true)),
% 5.88/2.04 inference(superposition,[status(thm)],[f1851,f1822])).
% 5.88/2.04 thf(f1822,plain,(
% 5.88/2.04 ( ! [X1 : $tType,X0 : $tType,X2 : X1 > X0] : (((!! @ X1 @ (^[Y0 : X1]: ((member @ X1 @ Y0 @ top_top @ set @ X1) => (!! @ X1 @ (^[Y1 : X1]: ((member @ X1 @ Y1 @ top_top @ set @ X1) => (((X2 @ Y1) = (X2 @ Y0)) => (Y0 = Y1)))))))) = $true) | ((X2 @ (sK76 @ X0 @ X1 @ X2)) = (X2 @ (sK77 @ X0 @ X1 @ X2)))) )),
% 5.88/2.04 inference(beta_eta_normalization,[status(thm)],[f1751])).
% 5.88/2.04 thf(f1751,plain,(
% 5.88/2.04 ( ! [X1 : $tType,X0 : $tType,X2 : X1 > X0] : (((X2 @ (sK76 @ X0 @ X1 @ X2)) = (X2 @ (sK77 @ X0 @ X1 @ X2))) | (((^[Y0 : X1 > X0]: ((^[Y1 : set @ X1]: (!! @ X1 @ (^[Y2 : X1]: ((member @ X1 @ Y2 @ Y1) => (!! @ X1 @ (^[Y3 : X1]: ((member @ X1 @ Y3 @ Y1) => (((Y0 @ Y3) = (Y0 @ Y2)) => (Y2 = Y3))))))))))) @ X2 @ top_top @ set @ X1) = $true)) )),
% 5.88/2.04 inference(definition_unfolding,[status(thm)],[f1667,f1423])).
% 5.88/2.04 thf(f1667,plain,(
% 5.88/2.04 ( ! [X1 : $tType,X0 : $tType,X2 : X1 > X0] : (((inj_on @ X1 @ X0 @ X2 @ top_top @ set @ X1) = $true) | ((X2 @ (sK76 @ X0 @ X1 @ X2)) = (X2 @ (sK77 @ X0 @ X1 @ X2)))) )),
% 5.88/2.04 inference(cnf_transformation,[status(thm)],[f1341])).
% 5.88/2.04 thf(f1611,plain,(
% 5.88/2.04 ( ! [X0 : paraco415392788lle_tv,X1 : paraco415392788lle_tv] : (((paraco1920534163nge_tv @ f @ X0) != (paraco1920534163nge_tv @ f @ X1)) | (X0 = X1)) )),
% 5.88/2.04 inference(cnf_transformation,[status(thm)],[f1134])).
% 5.88/2.04 thf(f1134,plain,(
% 5.88/2.04 ! [X0 : paraco415392788lle_tv,X1 : paraco415392788lle_tv] : ((X0 = X1) | ((paraco1920534163nge_tv @ f @ X0) != (paraco1920534163nge_tv @ f @ X1)))),
% 5.88/2.04 inference(ennf_transformation,[status(thm)],[f811])).
% 5.88/2.04 thf(f811,plain,(
% 5.88/2.04 ! [X0 : paraco415392788lle_tv,X1 : paraco415392788lle_tv] : (((paraco1920534163nge_tv @ f @ X0) = (paraco1920534163nge_tv @ f @ X1)) => (X0 = X1))),
% 5.88/2.04 inference(fool_elimination,[status(thm)],[f810])).
% 5.88/2.04 thf(f810,plain,(
% 5.88/2.04 ! [X0 : paraco415392788lle_tv,X1 : paraco415392788lle_tv] : (((paraco1920534163nge_tv @ f @ X0) = (paraco1920534163nge_tv @ f @ X1)) => (X0 = X1))),
% 5.88/2.04 inference(rectify,[status(thm)],[f1])).
% 5.88/2.04 thf(f1,axiom,(
% 5.88/2.04 ! [X4 : paraco415392788lle_tv,X5 : paraco415392788lle_tv] : (((paraco1920534163nge_tv @ f @ X4) = (paraco1920534163nge_tv @ f @ X5)) => (X4 = X5))),
% 5.88/2.04 file('/export/starexec/sandbox/tmp/tmp.Ai5Zk59Gvc/DTF2THF_9159.p',fact_0__092_060open_062_092_060And_062tv2_Atv1_O_Achange__tv_Af_Atv1_A_061_Achange__tv_Af_Atv2_A_092_060Longrightarrow_062_Atv1_A_061_Atv2_092_060close_062)).
% 5.88/2.04 % SZS output end Proof for DTF2THF_9159
% 5.88/2.04 % (9426)------------------------------
% 5.88/2.04 % (9426)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.88/2.04 % (9426)Termination reason: Refutation
% 5.88/2.04
% 5.88/2.04 % (9426)Memory used [KB]: 7547
% 5.88/2.04 % (9426)Time elapsed: 0.054 s
% 5.88/2.04 % (9426)Instructions burned: 172 (million)
% 5.88/2.04 % (9426)------------------------------
% 5.88/2.04 % (9426)------------------------------
% 5.88/2.04 % (9350)Success in time 0.421 s
%------------------------------------------------------------------------------