%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP159^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n025.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 08:56:02 AM UTC 2026
% Result : Theorem 3.50s 1.97s
% Output : Refutation 3.50s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : ITP159^2 : TPTP v9.2.1. Released v7.5.0.
% 0.00/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.18/0.34 % Computer : n025.cluster.edu
% 0.18/0.34 % Model : x86_64 x86_64
% 0.18/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.18/0.34 % Memory : 8042.1875MB
% 0.18/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.18/0.34 % CPULimit : 300
% 0.18/0.34 % WCLimit : 300
% 0.18/0.34 % DateTime : Tue Feb 24 08:28:27 EST 2026
% 0.18/0.34 % CPUTime :
% 0.18/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.31/0.45 ---- Original DTF file ---
% 0.31/0.46 thf(spec,logic,$$dhol).
% 0.31/0.46 %------------------------------------------------------------------------------
% 0.31/0.46 % File : ITP159^2 : TPTP v9.2.1. Released v7.5.0.
% 0.31/0.46 % Domain : Interactive Theorem Proving
% 0.31/0.46 % Problem : Sledgehammer Refine_Basic problem prob_1005__3594596_1
% 0.31/0.46 % Version : Especial.
% 0.31/0.46 % English :
% 0.31/0.46
% 0.31/0.46 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.31/0.46 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.31/0.46 % Source : [Des21]
% 0.31/0.46 % Names : Refine_Basic/prob_1005__3594596_1 [Des21]
% 0.31/0.46
% 0.31/0.46 % Status : Theorem
% 0.31/0.46 % Rating : 0.00 v7.5.0
% 0.31/0.46 % Syntax : Number of formulae : 344 ( 140 unt; 45 typ; 0 def)
% 0.31/0.46 % Number of atoms : 756 ( 180 equ; 0 cnn)
% 0.31/0.46 % Maximal formula atoms : 13 ( 2 avg)
% 0.31/0.46 % Number of connectives : 4384 ( 56 ~; 6 |; 34 &;3949 @)
% 0.31/0.46 % ( 0 <=>; 339 =>; 0 <=; 0 <~>)
% 0.31/0.46 % Maximal formula depth : 20 ( 8 avg)
% 0.31/0.46 % Number of types : 4 ( 3 usr)
% 0.31/0.46 % Number of type conns : 160 ( 160 >; 0 *; 0 +; 0 <<)
% 0.31/0.46 % Number of symbols : 45 ( 42 usr; 4 con; 0-5 aty)
% 0.31/0.46 % Number of variables : 1086 ( 53 ^; 970 !; 15 ?;1086 :)
% 0.31/0.46 % ( 48 !>; 0 ?*; 0 @-; 0 @+)
% 0.31/0.46 % SPC : TH1_THM_EQU_NAR
% 0.31/0.46
% 0.31/0.46 % Comments : This file was generated by Sledgehammer 2021-02-23 16:19:56.949
% 0.31/0.46 %------------------------------------------------------------------------------
% 0.31/0.46 % Could-be-implicit typings (6)
% 0.31/0.46 thf(ty_t_Refine__Basic__Mirabelle__tqojlsrkwy_Onres,type,
% 0.31/0.46 refine1665802226e_nres: $tType > $tType ).
% 0.31/0.46
% 0.31/0.46 thf(ty_t_Product__Type_Ounit,type,
% 0.31/0.46 product_unit: $tType ).
% 0.31/0.46
% 0.31/0.46 thf(ty_t_Product__Type_Oprod,type,
% 0.31/0.46 product_prod: $tType > $tType > $tType ).
% 0.31/0.46
% 0.31/0.46 thf(ty_t_Set_Oset,type,
% 0.31/0.46 set: $tType > $tType ).
% 0.31/0.46
% 0.31/0.46 thf(ty_tf_b,type,
% 0.31/0.46 b: $tType ).
% 0.31/0.46
% 0.31/0.46 thf(ty_tf_a,type,
% 0.31/0.46 a: $tType ).
% 0.31/0.46
% 0.31/0.46 % Explicit typings (39)
% 0.31/0.46 thf(sy_cl_Lattices_Obounded__lattice,type,
% 0.31/0.46 bounded_lattice:
% 0.31/0.46 !>[A: $tType] : $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_HOL_Otype,type,
% 0.31/0.46 type:
% 0.31/0.46 !>[A: $tType] : $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Orderings_Oord,type,
% 0.31/0.46 ord:
% 0.31/0.46 !>[A: $tType] : $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Orderings_Otop,type,
% 0.31/0.46 top:
% 0.31/0.46 !>[A: $tType] : $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Orderings_Oorder,type,
% 0.31/0.46 order:
% 0.31/0.46 !>[A: $tType] : $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Orderings_Olinorder,type,
% 0.31/0.46 linorder:
% 0.31/0.46 !>[A: $tType] : $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Orderings_Opreorder,type,
% 0.31/0.46 preorder:
% 0.31/0.46 !>[A: $tType] : $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Orderings_Oorder__top,type,
% 0.31/0.46 order_top:
% 0.31/0.46 !>[A: $tType] : $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Lattices_Obounded__lattice__top,type,
% 0.31/0.46 bounded_lattice_top:
% 0.31/0.46 !>[A: $tType] : $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
% 0.31/0.46 comple187826305attice:
% 0.31/0.46 !>[A: $tType] : $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
% 0.31/0.46 bNF_Ca1785829860lChain:
% 0.31/0.46 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_If,type,
% 0.31/0.46 if:
% 0.31/0.46 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Lattices_Osup__class_Osup,type,
% 0.31/0.46 sup_sup:
% 0.31/0.46 !>[A: $tType] : ( A > A > A ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Misc_Obijective,type,
% 0.31/0.46 bijective:
% 0.31/0.46 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Misc_Ofun__of__rel,type,
% 0.31/0.46 fun_of_rel:
% 0.31/0.46 !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ A ) ) > B > A ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Misc_Orel__restrict,type,
% 0.31/0.46 rel_restrict:
% 0.31/0.46 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Misc_Ouncurry,type,
% 0.31/0.46 uncurry:
% 0.31/0.46 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.31/0.46 ord_less_eq:
% 0.31/0.46 !>[A: $tType] : ( A > A > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Orderings_Oorder__class_OGreatest,type,
% 0.31/0.46 order_Greatest:
% 0.31/0.46 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Orderings_Otop__class_Otop,type,
% 0.31/0.46 top_top:
% 0.31/0.46 !>[A: $tType] : A ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Product__Type_OPair,type,
% 0.31/0.46 product_Pair:
% 0.31/0.46 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_ORETURN,type,
% 0.31/0.46 refine1687780735RETURN:
% 0.31/0.46 !>[A: $tType] : ( A > ( refine1665802226e_nres @ A ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Oabs__fun,type,
% 0.31/0.46 refine81118332bs_fun:
% 0.31/0.46 !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ A ) ) > ( refine1665802226e_nres @ B ) > ( refine1665802226e_nres @ A ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Obind,type,
% 0.31/0.46 refine463715084e_bind:
% 0.31/0.46 !>[B: $tType,A: $tType] : ( ( refine1665802226e_nres @ B ) > ( B > ( refine1665802226e_nres @ A ) ) > ( refine1665802226e_nres @ A ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Oconc__fun,type,
% 0.31/0.46 refine1073749519nc_fun:
% 0.31/0.46 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( refine1665802226e_nres @ B ) > ( refine1665802226e_nres @ A ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Oinres,type,
% 0.31/0.46 refine1315500908_inres:
% 0.31/0.46 !>[A: $tType] : ( ( refine1665802226e_nres @ A ) > A > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onf__inres,type,
% 0.31/0.46 refine406925620_inres:
% 0.31/0.46 !>[A: $tType] : ( ( refine1665802226e_nres @ A ) > A > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onofail,type,
% 0.31/0.46 refine1102455758nofail:
% 0.31/0.46 !>[A: $tType] : ( ( refine1665802226e_nres @ A ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onres_ORES,type,
% 0.31/0.46 refine605929679le_RES:
% 0.31/0.46 !>[A: $tType] : ( ( set @ A ) > ( refine1665802226e_nres @ A ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Othe__RES,type,
% 0.31/0.46 refine1672542526he_RES:
% 0.31/0.46 !>[A: $tType] : ( ( refine1665802226e_nres @ A ) > ( set @ A ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Relation_ODomain,type,
% 0.31/0.46 domain:
% 0.31/0.46 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Relation_OId,type,
% 0.31/0.46 id:
% 0.31/0.46 !>[A: $tType] : ( set @ ( product_prod @ A @ A ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Relation_OImage,type,
% 0.31/0.46 image:
% 0.31/0.46 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) > ( set @ B ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Set_OCollect,type,
% 0.31/0.46 collect:
% 0.31/0.46 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Transitive__Closure_Ortrancl,type,
% 0.31/0.46 transitive_rtrancl:
% 0.31/0.46 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Transitive__Closure_Otrancl,type,
% 0.31/0.46 transitive_trancl:
% 0.31/0.46 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_member,type,
% 0.31/0.46 member:
% 0.31/0.46 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_v_R,type,
% 0.31/0.46 r: set @ ( product_prod @ a @ b ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_v_X,type,
% 0.31/0.46 x: refine1665802226e_nres @ a ).
% 0.31/0.46
% 0.31/0.46 % Relevant facts (253)
% 0.31/0.46 thf(fact_0_nres__order__simps_I4_J,axiom,
% 0.31/0.46 ! [D: $tType,M: refine1665802226e_nres @ D] :
% 0.31/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ D ) @ ( top_top @ ( refine1665802226e_nres @ D ) ) @ M )
% 0.31/0.46 = ( M
% 0.31/0.46 = ( top_top @ ( refine1665802226e_nres @ D ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % nres_order_simps(4)
% 0.31/0.46 thf(fact_1_conc__fun__fail__iff_I2_J,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),S: refine1665802226e_nres @ B] :
% 0.31/0.46 ( ( ( top_top @ ( refine1665802226e_nres @ A ) )
% 0.31/0.46 = ( refine1073749519nc_fun @ A @ B @ R @ S ) )
% 0.31/0.46 = ( S
% 0.31/0.46 = ( top_top @ ( refine1665802226e_nres @ B ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % conc_fun_fail_iff(2)
% 0.31/0.46 thf(fact_2_conc__fun__fail__iff_I1_J,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),S: refine1665802226e_nres @ B] :
% 0.31/0.46 ( ( ( refine1073749519nc_fun @ A @ B @ R @ S )
% 0.31/0.46 = ( top_top @ ( refine1665802226e_nres @ A ) ) )
% 0.31/0.46 = ( S
% 0.31/0.46 = ( top_top @ ( refine1665802226e_nres @ B ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % conc_fun_fail_iff(1)
% 0.31/0.46 thf(fact_3_conc__fun__FAIL,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] :
% 0.31/0.46 ( ( refine1073749519nc_fun @ A @ B @ R @ ( top_top @ ( refine1665802226e_nres @ B ) ) )
% 0.31/0.46 = ( top_top @ ( refine1665802226e_nres @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % conc_fun_FAIL
% 0.31/0.46 thf(fact_4_nres__order__simps_I3_J,axiom,
% 0.31/0.46 ! [C: $tType,M: refine1665802226e_nres @ C] : ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ M @ ( top_top @ ( refine1665802226e_nres @ C ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % nres_order_simps(3)
% 0.31/0.46 thf(fact_5_conc__trans__additional_I1_J,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,A2: refine1665802226e_nres @ A,R: set @ ( product_prod @ A @ B ),B2: refine1665802226e_nres @ B,C2: refine1665802226e_nres @ B] :
% 0.31/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ A2 @ ( refine1073749519nc_fun @ A @ B @ R @ B2 ) )
% 0.31/0.46 => ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ B2 @ C2 )
% 0.31/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ A2 @ ( refine1073749519nc_fun @ A @ B @ R @ C2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % conc_trans_additional(1)
% 0.31/0.46 thf(fact_6_conc__trans,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,C: $tType,C2: refine1665802226e_nres @ A,R: set @ ( product_prod @ A @ B ),B2: refine1665802226e_nres @ B,R2: set @ ( product_prod @ B @ C ),A2: refine1665802226e_nres @ C] :
% 0.31/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ C2 @ ( refine1073749519nc_fun @ A @ B @ R @ B2 ) )
% 0.31/0.46 => ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ B2 @ ( refine1073749519nc_fun @ B @ C @ R2 @ A2 ) )
% 0.31/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ C2 @ ( refine1073749519nc_fun @ A @ B @ R @ ( refine1073749519nc_fun @ B @ C @ R2 @ A2 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % conc_trans
% 0.31/0.46 thf(fact_7_top__apply,axiom,
% 0.31/0.46 ! [C: $tType,D: $tType] :
% 0.31/0.46 ( ( top @ C )
% 0.31/0.46 => ( ( top_top @ ( D > C ) )
% 0.31/0.46 = ( ^ [X: D] : ( top_top @ C ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % top_apply
% 0.31/0.46 thf(fact_8_order__mono__setup_Orefl,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( preorder @ A )
% 0.31/0.46 => ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_mono_setup.refl
% 0.31/0.46 thf(fact_9_top__greatest,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order_top @ A )
% 0.31/0.46 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ ( top_top @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % top_greatest
% 0.31/0.46 thf(fact_10_meta__le__everything__if__top,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order_top @ A )
% 0.31/0.46 => ! [M2: A,X2: A] :
% 0.31/0.46 ( ( M2
% 0.31/0.46 = ( top_top @ A ) )
% 0.31/0.46 => ( ord_less_eq @ A @ X2 @ M2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % meta_le_everything_if_top
% 0.31/0.46 thf(fact_11_top_Oextremum__unique,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order_top @ A )
% 0.31/0.46 => ! [A3: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
% 0.31/0.46 = ( A3
% 0.31/0.46 = ( top_top @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % top.extremum_unique
% 0.31/0.46 thf(fact_12_top_Oextremum__uniqueI,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order_top @ A )
% 0.31/0.46 => ! [A3: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
% 0.31/0.46 => ( A3
% 0.31/0.46 = ( top_top @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % top.extremum_uniqueI
% 0.31/0.46 thf(fact_13_le__funD,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType] :
% 0.31/0.46 ( ( ord @ B )
% 0.31/0.46 => ! [F: A > B,G: A > B,X2: A] :
% 0.31/0.46 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.31/0.46 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % le_funD
% 0.31/0.46 thf(fact_14_le__funE,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType] :
% 0.31/0.46 ( ( ord @ B )
% 0.31/0.46 => ! [F: A > B,G: A > B,X2: A] :
% 0.31/0.46 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.31/0.46 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % le_funE
% 0.31/0.46 thf(fact_15_le__funI,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType] :
% 0.31/0.46 ( ( ord @ B )
% 0.31/0.46 => ! [F: A > B,G: A > B] :
% 0.31/0.46 ( ! [X3: A] : ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.31/0.46 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % le_funI
% 0.31/0.46 thf(fact_16_Refine__Misc_Oif__mono,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( ord @ A )
% 0.31/0.46 => ! [B3: $o,M1: A,M12: A,M22: A,M23: A] :
% 0.31/0.46 ( ( B3
% 0.31/0.46 => ( ord_less_eq @ A @ M1 @ M12 ) )
% 0.31/0.46 => ( ( ~ B3
% 0.31/0.46 => ( ord_less_eq @ A @ M22 @ M23 ) )
% 0.31/0.46 => ( ord_less_eq @ A @ ( if @ A @ B3 @ M1 @ M22 ) @ ( if @ A @ B3 @ M12 @ M23 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Refine_Misc.if_mono
% 0.31/0.46 thf(fact_17_le__fun__def,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType] :
% 0.31/0.46 ( ( ord @ B )
% 0.31/0.46 => ( ( ord_less_eq @ ( A > B ) )
% 0.31/0.46 = ( ^ [F2: A > B,G2: A > B] :
% 0.31/0.46 ! [X: A] : ( ord_less_eq @ B @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % le_fun_def
% 0.31/0.46 thf(fact_18_conc__fun__R__mono,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),R2: set @ ( product_prod @ A @ B ),M: refine1665802226e_nres @ B] :
% 0.31/0.46 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ R2 )
% 0.31/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1073749519nc_fun @ A @ B @ R @ M ) @ ( refine1073749519nc_fun @ A @ B @ R2 @ M ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % conc_fun_R_mono
% 0.31/0.46 thf(fact_19_order__mono__setup_Omono__let,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType] :
% 0.31/0.46 ( ( preorder @ A )
% 0.31/0.46 => ! [F: B > A,F3: B > A,X2: B] :
% 0.31/0.46 ( ! [X3: B] : ( ord_less_eq @ A @ ( F @ X3 ) @ ( F3 @ X3 ) )
% 0.31/0.46 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F3 @ X2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_mono_setup.mono_let
% 0.31/0.46 thf(fact_20_order__mono__setup_Omono__if,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( preorder @ A )
% 0.31/0.46 => ! [T: A,T2: A,E: A,E2: A,B3: $o] :
% 0.31/0.46 ( ( ord_less_eq @ A @ T @ T2 )
% 0.31/0.46 => ( ( ord_less_eq @ A @ E @ E2 )
% 0.31/0.46 => ( ord_less_eq @ A @ ( if @ A @ B3 @ T @ E ) @ ( if @ A @ B3 @ T2 @ E2 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_mono_setup.mono_if
% 0.31/0.46 thf(fact_21_dual__order_Oantisym,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A )
% 0.31/0.46 => ! [B3: A,A3: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.31/0.46 => ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.31/0.46 => ( A3 = B3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.antisym
% 0.31/0.46 thf(fact_22_dual__order_Oeq__iff,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A )
% 0.31/0.46 => ( ( ^ [Y: A,Z: A] : ( Y = Z ) )
% 0.31/0.46 = ( ^ [A4: A,B4: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ B4 @ A4 )
% 0.31/0.46 & ( ord_less_eq @ A @ A4 @ B4 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.eq_iff
% 0.31/0.46 thf(fact_23_dual__order_Otrans,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A )
% 0.31/0.46 => ! [B3: A,A3: A,C3: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.31/0.46 => ( ( ord_less_eq @ A @ C3 @ B3 )
% 0.31/0.46 => ( ord_less_eq @ A @ C3 @ A3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.trans
% 0.31/0.46 thf(fact_24_linorder__wlog,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( linorder @ A )
% 0.31/0.46 => ! [P: A > A > $o,A3: A,B3: A] :
% 0.31/0.46 ( ! [A5: A,B5: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ A5 @ B5 )
% 0.31/0.46 => ( P @ A5 @ B5 ) )
% 0.31/0.46 => ( ! [A5: A,B5: A] :
% 0.31/0.46 ( ( P @ B5 @ A5 )
% 0.31/0.46 => ( P @ A5 @ B5 ) )
% 0.31/0.46 => ( P @ A3 @ B3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % linorder_wlog
% 0.31/0.46 thf(fact_25_dual__order_Orefl,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A )
% 0.31/0.46 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.refl
% 0.31/0.46 thf(fact_26_order__trans,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( preorder @ A )
% 0.31/0.46 => ! [X2: A,Y2: A,Z2: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 0.31/0.46 => ( ( ord_less_eq @ A @ Y2 @ Z2 )
% 0.31/0.46 => ( ord_less_eq @ A @ X2 @ Z2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_trans
% 0.31/0.46 thf(fact_27_order__class_Oorder_Oantisym,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A )
% 0.31/0.46 => ! [A3: A,B3: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.31/0.46 => ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.31/0.46 => ( A3 = B3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_class.order.antisym
% 0.31/0.46 thf(fact_28_ord__le__eq__trans,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( ord @ A )
% 0.31/0.46 => ! [A3: A,B3: A,C3: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.31/0.46 => ( ( B3 = C3 )
% 0.31/0.46 => ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ord_le_eq_trans
% 0.31/0.46 thf(fact_29_ord__eq__le__trans,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( ord @ A )
% 0.31/0.46 => ! [A3: A,B3: A,C3: A] :
% 0.31/0.46 ( ( A3 = B3 )
% 0.31/0.46 => ( ( ord_less_eq @ A @ B3 @ C3 )
% 0.31/0.46 => ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ord_eq_le_trans
% 0.31/0.46 thf(fact_30_order__class_Oorder_Oeq__iff,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A )
% 0.31/0.46 => ( ( ^ [Y: A,Z: A] : ( Y = Z ) )
% 0.31/0.46 = ( ^ [A4: A,B4: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.31/0.46 & ( ord_less_eq @ A @ B4 @ A4 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_class.order.eq_iff
% 0.31/0.46 thf(fact_31_antisym__conv,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A )
% 0.31/0.46 => ! [Y2: A,X2: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ Y2 @ X2 )
% 0.31/0.46 => ( ( ord_less_eq @ A @ X2 @ Y2 )
% 0.31/0.46 = ( X2 = Y2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % antisym_conv
% 0.31/0.46 thf(fact_32_le__cases3,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( linorder @ A )
% 0.31/0.46 => ! [X2: A,Y2: A,Z2: A] :
% 0.31/0.46 ( ( ( ord_less_eq @ A @ X2 @ Y2 )
% 0.31/0.46 => ~ ( ord_less_eq @ A @ Y2 @ Z2 ) )
% 0.31/0.46 => ( ( ( ord_less_eq @ A @ Y2 @ X2 )
% 0.31/0.46 => ~ ( ord_less_eq @ A @ X2 @ Z2 ) )
% 0.31/0.46 => ( ( ( ord_less_eq @ A @ X2 @ Z2 )
% 0.31/0.46 => ~ ( ord_less_eq @ A @ Z2 @ Y2 ) )
% 0.31/0.46 => ( ( ( ord_less_eq @ A @ Z2 @ Y2 )
% 0.31/0.46 => ~ ( ord_less_eq @ A @ Y2 @ X2 ) )
% 0.31/0.46 => ( ( ( ord_less_eq @ A @ Y2 @ Z2 )
% 0.31/0.46 => ~ ( ord_less_eq @ A @ Z2 @ X2 ) )
% 0.31/0.46 => ~ ( ( ord_less_eq @ A @ Z2 @ X2 )
% 0.31/0.46 => ~ ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % le_cases3
% 0.31/0.46 thf(fact_33_order_Otrans,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A )
% 0.31/0.46 => ! [A3: A,B3: A,C3: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.31/0.46 => ( ( ord_less_eq @ A @ B3 @ C3 )
% 0.31/0.46 => ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order.trans
% 0.31/0.46 thf(fact_34_le__cases,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( linorder @ A )
% 0.31/0.46 => ! [X2: A,Y2: A] :
% 0.31/0.46 ( ~ ( ord_less_eq @ A @ X2 @ Y2 )
% 0.31/0.46 => ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % le_cases
% 0.31/0.46 thf(fact_35_eq__refl,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( preorder @ A )
% 0.31/0.46 => ! [X2: A,Y2: A] :
% 0.31/0.46 ( ( X2 = Y2 )
% 0.31/0.46 => ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % eq_refl
% 0.31/0.46 thf(fact_36_linear,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( linorder @ A )
% 0.31/0.46 => ! [X2: A,Y2: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 0.31/0.46 | ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % linear
% 0.31/0.46 thf(fact_37_antisym,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A )
% 0.31/0.46 => ! [X2: A,Y2: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 0.31/0.46 => ( ( ord_less_eq @ A @ Y2 @ X2 )
% 0.31/0.46 => ( X2 = Y2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % antisym
% 0.31/0.46 thf(fact_38_eq__iff,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A )
% 0.31/0.46 => ( ( ^ [Y: A,Z: A] : ( Y = Z ) )
% 0.31/0.46 = ( ^ [X: A,Y3: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ X @ Y3 )
% 0.31/0.46 & ( ord_less_eq @ A @ Y3 @ X ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % eq_iff
% 0.31/0.46 thf(fact_39_ord__le__eq__subst,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType] :
% 0.31/0.46 ( ( ( ord @ B )
% 0.31/0.46 & ( ord @ A ) )
% 0.31/0.46 => ! [A3: A,B3: A,F: A > B,C3: B] :
% 0.31/0.46 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.31/0.46 => ( ( ( F @ B3 )
% 0.31/0.46 = C3 )
% 0.31/0.46 => ( ! [X3: A,Y4: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ X3 @ Y4 )
% 0.31/0.46 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.31/0.46 => ( ord_less_eq @ B @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ord_le_eq_subst
% 0.31/0.46 thf(fact_40_ord__eq__le__subst,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType] :
% 0.31/0.46 ( ( ( ord @ B )
% 0.31/0.46 & ( ord @ A ) )
% 0.31/0.46 => ! [A3: A,F: B > A,B3: B,C3: B] :
% 0.31/0.46 ( ( A3
% 0.31/0.46 = ( F @ B3 ) )
% 0.31/0.46 => ( ( ord_less_eq @ B @ B3 @ C3 )
% 0.31/0.46 => ( ! [X3: B,Y4: B] :
% 0.31/0.46 ( ( ord_less_eq @ B @ X3 @ Y4 )
% 0.31/0.46 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.31/0.46 => ( ord_less_eq @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ord_eq_le_subst
% 0.31/0.46 thf(fact_41_order__subst2,axiom,
% 0.31/0.46 ! [A: $tType,C: $tType] :
% 0.31/0.46 ( ( ( order @ C )
% 0.31/0.46 & ( order @ A ) )
% 0.31/0.46 => ! [A3: A,B3: A,F: A > C,C3: C] :
% 0.31/0.46 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.31/0.46 => ( ( ord_less_eq @ C @ ( F @ B3 ) @ C3 )
% 0.31/0.46 => ( ! [X3: A,Y4: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ X3 @ Y4 )
% 0.31/0.46 => ( ord_less_eq @ C @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.31/0.46 => ( ord_less_eq @ C @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_subst2
% 0.31/0.46 thf(fact_42_order__subst1,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType] :
% 0.31/0.46 ( ( ( order @ B )
% 0.31/0.46 & ( order @ A ) )
% 0.31/0.46 => ! [A3: A,F: B > A,B3: B,C3: B] :
% 0.31/0.46 ( ( ord_less_eq @ A @ A3 @ ( F @ B3 ) )
% 0.31/0.46 => ( ( ord_less_eq @ B @ B3 @ C3 )
% 0.31/0.46 => ( ! [X3: B,Y4: B] :
% 0.31/0.46 ( ( ord_less_eq @ B @ X3 @ Y4 )
% 0.31/0.46 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.31/0.46 => ( ord_less_eq @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_subst1
% 0.31/0.46 thf(fact_43_Greatest__equality,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A )
% 0.31/0.46 => ! [P: A > $o,X2: A] :
% 0.31/0.46 ( ( P @ X2 )
% 0.31/0.46 => ( ! [Y4: A] :
% 0.31/0.46 ( ( P @ Y4 )
% 0.31/0.46 => ( ord_less_eq @ A @ Y4 @ X2 ) )
% 0.31/0.46 => ( ( order_Greatest @ A @ P )
% 0.31/0.46 = X2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Greatest_equality
% 0.31/0.46 thf(fact_44_mem__Collect__eq,axiom,
% 0.31/0.46 ! [A: $tType,A3: A,P: A > $o] :
% 0.31/0.46 ( ( member @ A @ A3 @ ( collect @ A @ P ) )
% 0.31/0.46 = ( P @ A3 ) ) ).
% 0.31/0.46
% 0.31/0.46 % mem_Collect_eq
% 0.31/0.46 thf(fact_45_Collect__mem__eq,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ A] :
% 0.31/0.46 ( ( collect @ A
% 0.31/0.46 @ ^ [X: A] : ( member @ A @ X @ A2 ) )
% 0.31/0.46 = A2 ) ).
% 0.31/0.46
% 0.31/0.46 % Collect_mem_eq
% 0.31/0.46 thf(fact_46_Collect__cong,axiom,
% 0.31/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.31/0.46 ( ! [X3: A] :
% 0.31/0.46 ( ( P @ X3 )
% 0.31/0.46 = ( Q @ X3 ) )
% 0.31/0.46 => ( ( collect @ A @ P )
% 0.31/0.46 = ( collect @ A @ Q ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Collect_cong
% 0.31/0.46 thf(fact_47_ext,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.31/0.46 ( ! [X3: A] :
% 0.31/0.46 ( ( F @ X3 )
% 0.31/0.46 = ( G @ X3 ) )
% 0.31/0.46 => ( F = G ) ) ).
% 0.31/0.46
% 0.31/0.46 % ext
% 0.31/0.46 thf(fact_48_GreatestI2__order,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A )
% 0.31/0.46 => ! [P: A > $o,X2: A,Q: A > $o] :
% 0.31/0.46 ( ( P @ X2 )
% 0.31/0.46 => ( ! [Y4: A] :
% 0.31/0.46 ( ( P @ Y4 )
% 0.31/0.46 => ( ord_less_eq @ A @ Y4 @ X2 ) )
% 0.31/0.46 => ( ! [X3: A] :
% 0.31/0.46 ( ( P @ X3 )
% 0.31/0.46 => ( ! [Y5: A] :
% 0.31/0.46 ( ( P @ Y5 )
% 0.31/0.46 => ( ord_less_eq @ A @ Y5 @ X3 ) )
% 0.31/0.46 => ( Q @ X3 ) ) )
% 0.31/0.46 => ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % GreatestI2_order
% 0.31/0.46 thf(fact_49_transfer_Otransfer__Let,axiom,
% 0.31/0.46 ! [C: $tType,A: $tType,B: $tType] :
% 0.31/0.46 ( ( comple187826305attice @ A )
% 0.31/0.46 => ! [Alpha: C > A,F: B > C,F4: B > A,X2: B] :
% 0.31/0.46 ( ! [X3: B] : ( ord_less_eq @ A @ ( Alpha @ ( F @ X3 ) ) @ ( F4 @ X3 ) )
% 0.31/0.46 => ( ord_less_eq @ A @ ( Alpha @ ( F @ X2 ) ) @ ( F4 @ X2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % transfer.transfer_Let
% 0.31/0.46 thf(fact_50_transfer_Otransfer__if,axiom,
% 0.31/0.46 ! [C: $tType,A: $tType] :
% 0.31/0.46 ( ( comple187826305attice @ A )
% 0.31/0.46 => ! [B3: $o,Alpha: C > A,S1: C,S12: A,S2: C,S22: A] :
% 0.31/0.46 ( ( B3
% 0.31/0.46 => ( ord_less_eq @ A @ ( Alpha @ S1 ) @ S12 ) )
% 0.31/0.46 => ( ( ~ B3
% 0.31/0.46 => ( ord_less_eq @ A @ ( Alpha @ S2 ) @ S22 ) )
% 0.31/0.46 => ( ord_less_eq @ A @ ( Alpha @ ( if @ C @ B3 @ S1 @ S2 ) ) @ ( if @ A @ B3 @ S12 @ S22 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % transfer.transfer_if
% 0.31/0.46 thf(fact_51_le__rel__bool__arg__iff,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( ord @ A )
% 0.31/0.46 => ( ( ord_less_eq @ ( $o > A ) )
% 0.31/0.46 = ( ^ [X4: $o > A,Y6: $o > A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ ( X4 @ $false ) @ ( Y6 @ $false ) )
% 0.31/0.46 & ( ord_less_eq @ A @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % le_rel_bool_arg_iff
% 0.31/0.46 thf(fact_52_verit__la__disequality,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( linorder @ A )
% 0.31/0.46 => ! [A3: A,B3: A] :
% 0.31/0.46 ( ( A3 = B3 )
% 0.31/0.46 | ~ ( ord_less_eq @ A @ A3 @ B3 )
% 0.31/0.46 | ~ ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % verit_la_disequality
% 0.31/0.46 thf(fact_53_ord__eq__le__eq__trans,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( ord @ A )
% 0.31/0.46 => ! [A3: A,B3: A,C3: A,D2: A] :
% 0.31/0.46 ( ( A3 = B3 )
% 0.31/0.46 => ( ( ord_less_eq @ A @ B3 @ C3 )
% 0.31/0.46 => ( ( C3 = D2 )
% 0.31/0.46 => ( ord_less_eq @ A @ A3 @ D2 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ord_eq_le_eq_trans
% 0.31/0.46 thf(fact_54_Id__refine,axiom,
% 0.31/0.46 ! [A: $tType,S: refine1665802226e_nres @ A] : ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ ( refine1073749519nc_fun @ A @ A @ ( id @ A ) @ S ) ) ).
% 0.31/0.46
% 0.31/0.46 % Id_refine
% 0.31/0.46 thf(fact_55_conc__trans__additional_I2_J,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,A2: refine1665802226e_nres @ A,B2: refine1665802226e_nres @ A,R: set @ ( product_prod @ A @ B ),C2: refine1665802226e_nres @ B] :
% 0.31/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ A2 @ ( refine1073749519nc_fun @ A @ A @ ( id @ A ) @ B2 ) )
% 0.31/0.46 => ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ B2 @ ( refine1073749519nc_fun @ A @ B @ R @ C2 ) )
% 0.31/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ A2 @ ( refine1073749519nc_fun @ A @ B @ R @ C2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % conc_trans_additional(2)
% 0.31/0.46 thf(fact_56_conc__trans__additional_I3_J,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,A2: refine1665802226e_nres @ A,R: set @ ( product_prod @ A @ B ),B2: refine1665802226e_nres @ B,C2: refine1665802226e_nres @ B] :
% 0.31/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ A2 @ ( refine1073749519nc_fun @ A @ B @ R @ B2 ) )
% 0.31/0.46 => ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ B2 @ ( refine1073749519nc_fun @ B @ B @ ( id @ B ) @ C2 ) )
% 0.31/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ A2 @ ( refine1073749519nc_fun @ A @ B @ R @ C2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % conc_trans_additional(3)
% 0.31/0.46 thf(fact_57_subset__Collect__conv,axiom,
% 0.31/0.46 ! [A: $tType,S: set @ A,P: A > $o] :
% 0.31/0.46 ( ( ord_less_eq @ ( set @ A ) @ S @ ( collect @ A @ P ) )
% 0.31/0.46 = ( ! [X: A] :
% 0.31/0.46 ( ( member @ A @ X @ S )
% 0.31/0.46 => ( P @ X ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % subset_Collect_conv
% 0.31/0.46 thf(fact_58_conc__trans__additional_I6_J,axiom,
% 0.31/0.46 ! [E3: $tType,A2: refine1665802226e_nres @ E3,B2: refine1665802226e_nres @ E3,C2: refine1665802226e_nres @ E3] :
% 0.31/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ E3 ) @ A2 @ B2 )
% 0.31/0.46 => ( ( ord_less_eq @ ( refine1665802226e_nres @ E3 ) @ B2 @ ( refine1073749519nc_fun @ E3 @ E3 @ ( id @ E3 ) @ C2 ) )
% 0.31/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ E3 ) @ A2 @ C2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % conc_trans_additional(6)
% 0.31/0.46 thf(fact_59_conc__trans__additional_I5_J,axiom,
% 0.31/0.46 ! [D: $tType,A2: refine1665802226e_nres @ D,B2: refine1665802226e_nres @ D,C2: refine1665802226e_nres @ D] :
% 0.31/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ D ) @ A2 @ ( refine1073749519nc_fun @ D @ D @ ( id @ D ) @ B2 ) )
% 0.31/0.46 => ( ( ord_less_eq @ ( refine1665802226e_nres @ D ) @ B2 @ C2 )
% 0.31/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ D ) @ A2 @ C2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % conc_trans_additional(5)
% 0.31/0.46 thf(fact_60_conc__trans__additional_I4_J,axiom,
% 0.31/0.46 ! [C: $tType,A2: refine1665802226e_nres @ C,B2: refine1665802226e_nres @ C,C2: refine1665802226e_nres @ C] :
% 0.31/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ A2 @ ( refine1073749519nc_fun @ C @ C @ ( id @ C ) @ B2 ) )
% 0.31/0.46 => ( ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ B2 @ ( refine1073749519nc_fun @ C @ C @ ( id @ C ) @ C2 ) )
% 0.31/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ A2 @ C2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % conc_trans_additional(4)
% 0.31/0.46 thf(fact_61_subsetI,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.31/0.46 ( ! [X3: A] :
% 0.31/0.46 ( ( member @ A @ X3 @ A2 )
% 0.31/0.46 => ( member @ A @ X3 @ B2 ) )
% 0.31/0.46 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % subsetI
% 0.31/0.46 thf(fact_62_subset__antisym,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.31/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.31/0.46 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 0.31/0.46 => ( A2 = B2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % subset_antisym
% 0.31/0.46 thf(fact_63_bijective__Id,axiom,
% 0.31/0.46 ! [A: $tType] : ( bijective @ A @ A @ ( id @ A ) ) ).
% 0.31/0.46
% 0.31/0.46 % bijective_Id
% 0.31/0.46 thf(fact_64_relprop__id__orient,axiom,
% 0.31/0.46 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.31/0.46 ( ( R
% 0.31/0.46 = ( id @ A ) )
% 0.31/0.46 => ( ( id @ A )
% 0.31/0.46 = R ) ) ).
% 0.31/0.46
% 0.31/0.46 % relprop_id_orient
% 0.31/0.46 thf(fact_65_relprop__triggers_I2_J,axiom,
% 0.31/0.46 ! [C: $tType,R: set @ ( product_prod @ C @ C )] :
% 0.31/0.46 ( ( R
% 0.31/0.46 = ( id @ C ) )
% 0.31/0.46 => ( R
% 0.31/0.46 = ( id @ C ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % relprop_triggers(2)
% 0.31/0.46 thf(fact_66_relprop__triggers_I3_J,axiom,
% 0.31/0.46 ! [D: $tType,R: set @ ( product_prod @ D @ D )] :
% 0.31/0.46 ( ( R
% 0.31/0.46 = ( id @ D ) )
% 0.31/0.46 => ( ( id @ D )
% 0.31/0.46 = R ) ) ).
% 0.31/0.46
% 0.31/0.46 % relprop_triggers(3)
% 0.31/0.46 thf(fact_67_relprop__triggers_I6_J,axiom,
% 0.31/0.46 ! [I: $tType,R: set @ I,R2: set @ I] :
% 0.31/0.46 ( ( ord_less_eq @ ( set @ I ) @ R @ R2 )
% 0.31/0.46 => ( ord_less_eq @ ( set @ I ) @ R @ R2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % relprop_triggers(6)
% 0.31/0.46 thf(fact_68_Collect__mono__iff,axiom,
% 0.31/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.31/0.46 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 0.31/0.46 = ( ! [X: A] :
% 0.31/0.46 ( ( P @ X )
% 0.31/0.46 => ( Q @ X ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Collect_mono_iff
% 0.31/0.46 thf(fact_69_set__eq__subset,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( ^ [Y: set @ A,Z: set @ A] : ( Y = Z ) )
% 0.31/0.46 = ( ^ [A6: set @ A,B6: set @ A] :
% 0.31/0.46 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 0.31/0.46 & ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % set_eq_subset
% 0.31/0.46 thf(fact_70_subset__trans,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ A,B2: set @ A,C2: set @ A] :
% 0.31/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.31/0.46 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C2 )
% 0.31/0.46 => ( ord_less_eq @ ( set @ A ) @ A2 @ C2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % subset_trans
% 0.31/0.46 thf(fact_71_Collect__mono,axiom,
% 0.31/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.31/0.46 ( ! [X3: A] :
% 0.31/0.46 ( ( P @ X3 )
% 0.31/0.46 => ( Q @ X3 ) )
% 0.31/0.46 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Collect_mono
% 0.31/0.46 thf(fact_72_subset__refl,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% 0.31/0.46
% 0.31/0.46 % subset_refl
% 0.31/0.46 thf(fact_73_subset__UNIV,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % subset_UNIV
% 0.31/0.46 thf(fact_74_subset__iff,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( ord_less_eq @ ( set @ A ) )
% 0.31/0.46 = ( ^ [A6: set @ A,B6: set @ A] :
% 0.31/0.46 ! [T3: A] :
% 0.31/0.46 ( ( member @ A @ T3 @ A6 )
% 0.31/0.46 => ( member @ A @ T3 @ B6 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % subset_iff
% 0.31/0.46 thf(fact_75_equalityD2,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.31/0.46 ( ( A2 = B2 )
% 0.31/0.46 => ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % equalityD2
% 0.31/0.46 thf(fact_76_equalityD1,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.31/0.46 ( ( A2 = B2 )
% 0.31/0.46 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % equalityD1
% 0.31/0.46 thf(fact_77_subset__eq,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( ord_less_eq @ ( set @ A ) )
% 0.31/0.46 = ( ^ [A6: set @ A,B6: set @ A] :
% 0.31/0.46 ! [X: A] :
% 0.31/0.46 ( ( member @ A @ X @ A6 )
% 0.31/0.46 => ( member @ A @ X @ B6 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % subset_eq
% 0.31/0.46 thf(fact_78_equalityE,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.31/0.46 ( ( A2 = B2 )
% 0.31/0.46 => ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.31/0.46 => ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % equalityE
% 0.31/0.46 thf(fact_79_subsetD,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ A,B2: set @ A,C3: A] :
% 0.31/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.31/0.46 => ( ( member @ A @ C3 @ A2 )
% 0.31/0.46 => ( member @ A @ C3 @ B2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % subsetD
% 0.31/0.46 thf(fact_80_in__mono,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ A,B2: set @ A,X2: A] :
% 0.31/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.31/0.46 => ( ( member @ A @ X2 @ A2 )
% 0.31/0.46 => ( member @ A @ X2 @ B2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % in_mono
% 0.31/0.46 thf(fact_81_Id__SPEC__refine,axiom,
% 0.31/0.46 ! [A: $tType,S: refine1665802226e_nres @ A,Phi: A > $o] :
% 0.31/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) )
% 0.31/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ ( refine1073749519nc_fun @ A @ A @ ( id @ A ) @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Id_SPEC_refine
% 0.31/0.46 thf(fact_82_abs__trans__additional_I2_J,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,A2: refine1665802226e_nres @ A,B2: refine1665802226e_nres @ A,R: set @ ( product_prod @ A @ B ),C2: refine1665802226e_nres @ B] :
% 0.31/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine81118332bs_fun @ A @ A @ ( id @ A ) @ A2 ) @ B2 )
% 0.31/0.46 => ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine81118332bs_fun @ A @ B @ R @ B2 ) @ C2 )
% 0.31/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine81118332bs_fun @ A @ B @ R @ A2 ) @ C2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % abs_trans_additional(2)
% 0.31/0.46 thf(fact_83_abs__trans__additional_I3_J,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),A2: refine1665802226e_nres @ A,B2: refine1665802226e_nres @ B,C2: refine1665802226e_nres @ B] :
% 0.31/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine81118332bs_fun @ A @ B @ R @ A2 ) @ B2 )
% 0.31/0.46 => ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine81118332bs_fun @ B @ B @ ( id @ B ) @ B2 ) @ C2 )
% 0.31/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine81118332bs_fun @ A @ B @ R @ A2 ) @ C2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % abs_trans_additional(3)
% 0.31/0.46 thf(fact_84_abs__trans__additional_I4_J,axiom,
% 0.31/0.46 ! [C: $tType,A2: refine1665802226e_nres @ C,B2: refine1665802226e_nres @ C,C2: refine1665802226e_nres @ C] :
% 0.31/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ ( refine81118332bs_fun @ C @ C @ ( id @ C ) @ A2 ) @ B2 )
% 0.31/0.46 => ( ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ ( refine81118332bs_fun @ C @ C @ ( id @ C ) @ B2 ) @ C2 )
% 0.31/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ A2 @ C2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % abs_trans_additional(4)
% 0.31/0.46 thf(fact_85_abs__trans__additional_I5_J,axiom,
% 0.31/0.46 ! [D: $tType,A2: refine1665802226e_nres @ D,B2: refine1665802226e_nres @ D,C2: refine1665802226e_nres @ D] :
% 0.31/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ D ) @ ( refine81118332bs_fun @ D @ D @ ( id @ D ) @ A2 ) @ B2 )
% 0.31/0.46 => ( ( ord_less_eq @ ( refine1665802226e_nres @ D ) @ B2 @ C2 )
% 0.31/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ D ) @ A2 @ C2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % abs_trans_additional(5)
% 0.31/0.46 thf(fact_86_abs__trans__additional_I6_J,axiom,
% 0.31/0.46 ! [E3: $tType,A2: refine1665802226e_nres @ E3,B2: refine1665802226e_nres @ E3,C2: refine1665802226e_nres @ E3] :
% 0.31/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ E3 ) @ A2 @ B2 )
% 0.31/0.46 => ( ( ord_less_eq @ ( refine1665802226e_nres @ E3 ) @ ( refine81118332bs_fun @ E3 @ E3 @ ( id @ E3 ) @ B2 ) @ C2 )
% 0.31/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ E3 ) @ A2 @ C2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % abs_trans_additional(6)
% 0.31/0.46 thf(fact_87_rel__restrict__mono2,axiom,
% 0.31/0.46 ! [A: $tType,R: set @ A,S: set @ A,A2: set @ ( product_prod @ A @ A )] :
% 0.31/0.46 ( ( ord_less_eq @ ( set @ A ) @ R @ S )
% 0.31/0.46 => ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( rel_restrict @ A @ A2 @ S ) @ ( rel_restrict @ A @ A2 @ R ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rel_restrict_mono2
% 0.31/0.46 thf(fact_88_UNIV__I,axiom,
% 0.31/0.46 ! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % UNIV_I
% 0.31/0.46 thf(fact_89_nres_Oinject,axiom,
% 0.31/0.46 ! [A: $tType,X22: set @ A,Y22: set @ A] :
% 0.31/0.46 ( ( ( refine605929679le_RES @ A @ X22 )
% 0.31/0.46 = ( refine605929679le_RES @ A @ Y22 ) )
% 0.31/0.46 = ( X22 = Y22 ) ) ).
% 0.31/0.46
% 0.31/0.46 % nres.inject
% 0.31/0.46 thf(fact_90_nres__more__simps_I4_J,axiom,
% 0.31/0.46 ! [A: $tType,X5: set @ A,Y7: set @ A] :
% 0.31/0.46 ( ( ( refine605929679le_RES @ A @ X5 )
% 0.31/0.46 = ( refine605929679le_RES @ A @ Y7 ) )
% 0.31/0.46 = ( X5 = Y7 ) ) ).
% 0.31/0.46
% 0.31/0.46 % nres_more_simps(4)
% 0.31/0.46 thf(fact_91_abs__fun__simps_I1_J,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,R: set @ ( product_prod @ B @ A )] :
% 0.31/0.46 ( ( refine81118332bs_fun @ B @ A @ R @ ( top_top @ ( refine1665802226e_nres @ B ) ) )
% 0.31/0.46 = ( top_top @ ( refine1665802226e_nres @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % abs_fun_simps(1)
% 0.31/0.46 thf(fact_92_relprop__UNIV__orient,axiom,
% 0.31/0.46 ! [A: $tType,R: set @ A] :
% 0.31/0.46 ( ( R
% 0.31/0.46 = ( top_top @ ( set @ A ) ) )
% 0.31/0.46 => ( ( top_top @ ( set @ A ) )
% 0.31/0.46 = R ) ) ).
% 0.31/0.46
% 0.31/0.46 % relprop_UNIV_orient
% 0.31/0.46 thf(fact_93_eq__UNIV__iff,axiom,
% 0.31/0.46 ! [A: $tType,S: set @ A] :
% 0.31/0.46 ( ( S
% 0.31/0.46 = ( top_top @ ( set @ A ) ) )
% 0.31/0.46 = ( ! [X: A] : ( member @ A @ X @ S ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % eq_UNIV_iff
% 0.31/0.46 thf(fact_94_UNIV__witness,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ? [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % UNIV_witness
% 0.31/0.46 thf(fact_95_top__set__def,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( top_top @ ( set @ A ) )
% 0.31/0.46 = ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % top_set_def
% 0.31/0.46 thf(fact_96_UNIV__eq__I,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ A] :
% 0.31/0.46 ( ! [X3: A] : ( member @ A @ X3 @ A2 )
% 0.31/0.46 => ( ( top_top @ ( set @ A ) )
% 0.31/0.46 = A2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % UNIV_eq_I
% 0.31/0.46 thf(fact_97_if__rule,axiom,
% 0.31/0.46 ! [A: $tType,B3: $o,S12: refine1665802226e_nres @ A,Phi: A > $o,S22: refine1665802226e_nres @ A] :
% 0.31/0.46 ( ( B3
% 0.31/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S12 @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) ) )
% 0.31/0.46 => ( ( ~ B3
% 0.31/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S22 @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) ) )
% 0.31/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( if @ ( refine1665802226e_nres @ A ) @ B3 @ S12 @ S22 ) @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % if_rule
% 0.31/0.46 thf(fact_98_RES__rule,axiom,
% 0.31/0.46 ! [A: $tType,S: set @ A,Phi: A > $o] :
% 0.31/0.46 ( ! [X3: A] :
% 0.31/0.46 ( ( member @ A @ X3 @ S )
% 0.31/0.46 => ( Phi @ X3 ) )
% 0.31/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ S ) @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % RES_rule
% 0.31/0.46 thf(fact_99_SPEC__rule,axiom,
% 0.31/0.46 ! [A: $tType,Phi: A > $o,Phi2: A > $o] :
% 0.31/0.46 ( ! [X3: A] :
% 0.31/0.46 ( ( Phi @ X3 )
% 0.31/0.46 => ( Phi2 @ X3 ) )
% 0.31/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % SPEC_rule
% 0.31/0.46 thf(fact_100_SPEC__trans,axiom,
% 0.31/0.46 ! [A: $tType,X2: refine1665802226e_nres @ A,Y2: refine1665802226e_nres @ A,Postcond: A > $o] :
% 0.31/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ X2 @ Y2 )
% 0.31/0.46 => ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ Y2 @ ( refine605929679le_RES @ A @ ( collect @ A @ Postcond ) ) )
% 0.31/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ X2 @ ( refine605929679le_RES @ A @ ( collect @ A @ Postcond ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % SPEC_trans
% 0.31/0.46 thf(fact_101_SPEC__cons__rule,axiom,
% 0.31/0.46 ! [A: $tType,M2: refine1665802226e_nres @ A,Phi: A > $o,Psi: A > $o] :
% 0.31/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M2 @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) )
% 0.31/0.46 => ( ! [X3: A] :
% 0.31/0.46 ( ( Phi @ X3 )
% 0.31/0.46 => ( Psi @ X3 ) )
% 0.31/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M2 @ ( refine605929679le_RES @ A @ ( collect @ A @ Psi ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % SPEC_cons_rule
% 0.31/0.46 thf(fact_102_nres__inequalities_I1_J,axiom,
% 0.31/0.46 ! [A: $tType,X5: set @ A] :
% 0.31/0.46 ( ( top_top @ ( refine1665802226e_nres @ A ) )
% 0.31/0.46 != ( refine605929679le_RES @ A @ X5 ) ) ).
% 0.31/0.46
% 0.31/0.46 % nres_inequalities(1)
% 0.31/0.46 thf(fact_103_nres__cases,axiom,
% 0.31/0.46 ! [A: $tType,M: refine1665802226e_nres @ A] :
% 0.31/0.46 ( ( M
% 0.31/0.46 != ( top_top @ ( refine1665802226e_nres @ A ) ) )
% 0.31/0.46 => ~ ! [X6: set @ A] :
% 0.31/0.46 ( M
% 0.31/0.46 != ( refine605929679le_RES @ A @ X6 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % nres_cases
% 0.31/0.46 thf(fact_104_rel__restrict__sub,axiom,
% 0.31/0.46 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A2: set @ A] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( rel_restrict @ A @ R @ A2 ) @ R ) ).
% 0.31/0.46
% 0.31/0.46 % rel_restrict_sub
% 0.31/0.46 thf(fact_105_rel__restrict__mono,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ ( product_prod @ A @ A ),B2: set @ ( product_prod @ A @ A ),R: set @ A] :
% 0.31/0.46 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ A2 @ B2 )
% 0.31/0.46 => ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( rel_restrict @ A @ A2 @ R ) @ ( rel_restrict @ A @ B2 @ R ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rel_restrict_mono
% 0.31/0.46 thf(fact_106_abs__trans__additional_I1_J,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,A2: refine1665802226e_nres @ A,B2: refine1665802226e_nres @ A,R: set @ ( product_prod @ A @ B ),C2: refine1665802226e_nres @ B] :
% 0.31/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ A2 @ B2 )
% 0.31/0.46 => ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine81118332bs_fun @ A @ B @ R @ B2 ) @ C2 )
% 0.31/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine81118332bs_fun @ A @ B @ R @ A2 ) @ C2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % abs_trans_additional(1)
% 0.31/0.46 thf(fact_107_abs__trans,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,C: $tType,R: set @ ( product_prod @ B @ A ),C2: refine1665802226e_nres @ B,B2: refine1665802226e_nres @ A,R2: set @ ( product_prod @ A @ C ),A2: refine1665802226e_nres @ C] :
% 0.31/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine81118332bs_fun @ B @ A @ R @ C2 ) @ B2 )
% 0.31/0.46 => ( ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ ( refine81118332bs_fun @ A @ C @ R2 @ B2 ) @ A2 )
% 0.31/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ ( refine81118332bs_fun @ A @ C @ R2 @ ( refine81118332bs_fun @ B @ A @ R @ C2 ) ) @ A2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % abs_trans
% 0.31/0.46 thf(fact_108_nres__order__simps_I5_J,axiom,
% 0.31/0.46 ! [E3: $tType,X5: set @ E3,Y7: set @ E3] :
% 0.31/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ E3 ) @ ( refine605929679le_RES @ E3 @ X5 ) @ ( refine605929679le_RES @ E3 @ Y7 ) )
% 0.31/0.46 = ( ord_less_eq @ ( set @ E3 ) @ X5 @ Y7 ) ) ).
% 0.31/0.46
% 0.31/0.46 % nres_order_simps(5)
% 0.31/0.46 thf(fact_109_less__eq__nres_Osimps_I2_J,axiom,
% 0.31/0.46 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.31/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ A3 ) @ ( refine605929679le_RES @ A @ B3 ) )
% 0.31/0.46 = ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_eq_nres.simps(2)
% 0.31/0.46 thf(fact_110_iso__tuple__UNIV__I,axiom,
% 0.31/0.46 ! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % iso_tuple_UNIV_I
% 0.31/0.46 thf(fact_111_nf__inres__SPEC,axiom,
% 0.31/0.46 ! [A: $tType,Phi: A > $o,X2: A] :
% 0.31/0.46 ( ( refine406925620_inres @ A @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) @ X2 )
% 0.31/0.46 = ( Phi @ X2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % nf_inres_SPEC
% 0.31/0.46 thf(fact_112_nf__inres__RES,axiom,
% 0.31/0.46 ! [A: $tType,X5: set @ A,X2: A] :
% 0.31/0.46 ( ( refine406925620_inres @ A @ ( refine605929679le_RES @ A @ X5 ) @ X2 )
% 0.31/0.46 = ( member @ A @ X2 @ X5 ) ) ).
% 0.31/0.46
% 0.31/0.46 % nf_inres_RES
% 0.31/0.46 thf(fact_113_abs__fun__simps_I3_J,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,X5: set @ B,R: set @ ( product_prod @ B @ A )] :
% 0.31/0.46 ( ~ ( ord_less_eq @ ( set @ B ) @ X5 @ ( domain @ B @ A @ R ) )
% 0.31/0.46 => ( ( refine81118332bs_fun @ B @ A @ R @ ( refine605929679le_RES @ B @ X5 ) )
% 0.31/0.46 = ( top_top @ ( refine1665802226e_nres @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % abs_fun_simps(3)
% 0.31/0.46 thf(fact_114_the__RES_Osimps,axiom,
% 0.31/0.46 ! [A: $tType,X5: set @ A] :
% 0.31/0.46 ( ( refine1672542526he_RES @ A @ ( refine605929679le_RES @ A @ X5 ) )
% 0.31/0.46 = X5 ) ).
% 0.31/0.46
% 0.31/0.46 % the_RES.simps
% 0.31/0.46 thf(fact_115_nres__order__simps_I21_J,axiom,
% 0.31/0.46 ! [X7: $tType,X2: X7,Y7: set @ X7] :
% 0.31/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ X7 ) @ ( refine1687780735RETURN @ X7 @ X2 ) @ ( refine605929679le_RES @ X7 @ Y7 ) )
% 0.31/0.46 = ( member @ X7 @ X2 @ Y7 ) ) ).
% 0.31/0.46
% 0.31/0.46 % nres_order_simps(21)
% 0.31/0.46 thf(fact_116_nres__more__simps_I6_J,axiom,
% 0.31/0.46 ! [A: $tType,X2: A,Y2: A] :
% 0.31/0.46 ( ( ( refine1687780735RETURN @ A @ X2 )
% 0.31/0.46 = ( refine1687780735RETURN @ A @ Y2 ) )
% 0.31/0.46 = ( X2 = Y2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % nres_more_simps(6)
% 0.31/0.46 thf(fact_117_top1I,axiom,
% 0.31/0.46 ! [A: $tType,X2: A] : ( top_top @ ( A > $o ) @ X2 ) ).
% 0.31/0.46
% 0.31/0.46 % top1I
% 0.31/0.46 thf(fact_118_nres__order__simps_I20_J,axiom,
% 0.31/0.46 ! [W: $tType,X2: W,Y2: W] :
% 0.31/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ W ) @ ( refine1687780735RETURN @ W @ X2 ) @ ( refine1687780735RETURN @ W @ Y2 ) )
% 0.31/0.46 = ( X2 = Y2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % nres_order_simps(20)
% 0.31/0.46 thf(fact_119_nres__inequalities_I3_J,axiom,
% 0.31/0.46 ! [C: $tType,X2: C] :
% 0.31/0.46 ( ( top_top @ ( refine1665802226e_nres @ C ) )
% 0.31/0.46 != ( refine1687780735RETURN @ C @ X2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % nres_inequalities(3)
% 0.31/0.46 thf(fact_120_RETURN__to__SPEC__rule,axiom,
% 0.31/0.46 ! [A: $tType,M2: refine1665802226e_nres @ A,V: A] :
% 0.31/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M2
% 0.31/0.46 @ ( refine605929679le_RES @ A
% 0.31/0.46 @ ( collect @ A
% 0.31/0.46 @ ( ^ [Y: A,Z: A] : ( Y = Z )
% 0.31/0.46 @ V ) ) ) )
% 0.31/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M2 @ ( refine1687780735RETURN @ A @ V ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % RETURN_to_SPEC_rule
% 0.31/0.46 thf(fact_121_RETURN__rule,axiom,
% 0.31/0.46 ! [A: $tType,Phi: A > $o,X2: A] :
% 0.31/0.46 ( ( Phi @ X2 )
% 0.31/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X2 ) @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % RETURN_rule
% 0.31/0.46 thf(fact_122_Domain__Id,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( domain @ A @ A @ ( id @ A ) )
% 0.31/0.46 = ( top_top @ ( set @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Domain_Id
% 0.31/0.46 thf(fact_123_Domain__mono,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
% 0.31/0.46 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S3 )
% 0.31/0.46 => ( ord_less_eq @ ( set @ A ) @ ( domain @ A @ B @ R3 ) @ ( domain @ A @ B @ S3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Domain_mono
% 0.31/0.46 thf(fact_124_top__empty__eq,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( top_top @ ( A > $o ) )
% 0.31/0.46 = ( ^ [X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % top_empty_eq
% 0.31/0.46 thf(fact_125_abs__fun__simps_I2_J,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,X5: set @ B,R: set @ ( product_prod @ B @ A )] :
% 0.31/0.46 ( ( ord_less_eq @ ( set @ B ) @ X5 @ ( domain @ B @ A @ R ) )
% 0.31/0.46 => ( ( refine81118332bs_fun @ B @ A @ R @ ( refine605929679le_RES @ B @ X5 ) )
% 0.31/0.46 = ( refine605929679le_RES @ A @ ( image @ B @ A @ R @ X5 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % abs_fun_simps(2)
% 0.31/0.46 thf(fact_126_Image__Id,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ A] :
% 0.31/0.46 ( ( image @ A @ A @ ( id @ A ) @ A2 )
% 0.31/0.46 = A2 ) ).
% 0.31/0.46
% 0.31/0.46 % Image_Id
% 0.31/0.46 thf(fact_127_Image__mono,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,R4: set @ ( product_prod @ A @ B ),R3: set @ ( product_prod @ A @ B ),A7: set @ A,A2: set @ A] :
% 0.31/0.46 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R4 @ R3 )
% 0.31/0.46 => ( ( ord_less_eq @ ( set @ A ) @ A7 @ A2 )
% 0.31/0.46 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ R4 @ A7 ) @ ( image @ A @ B @ R3 @ A2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Image_mono
% 0.31/0.46 thf(fact_128_top__conj_I1_J,axiom,
% 0.31/0.46 ! [A: $tType,X2: A,P: $o] :
% 0.31/0.46 ( ( ( top_top @ ( A > $o ) @ X2 )
% 0.31/0.46 & P )
% 0.31/0.46 = P ) ).
% 0.31/0.46
% 0.31/0.46 % top_conj(1)
% 0.31/0.46 thf(fact_129_top__conj_I2_J,axiom,
% 0.31/0.46 ! [A: $tType,P: $o,X2: A] :
% 0.31/0.46 ( ( P
% 0.31/0.46 & ( top_top @ ( A > $o ) @ X2 ) )
% 0.31/0.46 = P ) ).
% 0.31/0.46
% 0.31/0.46 % top_conj(2)
% 0.31/0.46 thf(fact_130_RETURN__SPEC__refine,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,X2: B,R: set @ ( product_prod @ B @ A ),Phi: A > $o] :
% 0.31/0.46 ( ? [X8: A] :
% 0.31/0.46 ( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X2 @ X8 ) @ R )
% 0.31/0.46 & ( Phi @ X8 ) )
% 0.31/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine1687780735RETURN @ B @ X2 ) @ ( refine1073749519nc_fun @ B @ A @ R @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % RETURN_SPEC_refine
% 0.31/0.46 thf(fact_131_the__RES__inv,axiom,
% 0.31/0.46 ! [A: $tType,M2: refine1665802226e_nres @ A] :
% 0.31/0.46 ( ( refine1102455758nofail @ A @ M2 )
% 0.31/0.46 => ( ( refine605929679le_RES @ A @ ( refine1672542526he_RES @ A @ M2 ) )
% 0.31/0.46 = M2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % the_RES_inv
% 0.31/0.46 thf(fact_132_pair__in__Id__conv,axiom,
% 0.31/0.46 ! [A: $tType,A3: A,B3: A] :
% 0.31/0.46 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ ( id @ A ) )
% 0.31/0.46 = ( A3 = B3 ) ) ).
% 0.31/0.46
% 0.31/0.46 % pair_in_Id_conv
% 0.31/0.46 thf(fact_133_IdI,axiom,
% 0.31/0.46 ! [A: $tType,A3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ ( id @ A ) ) ).
% 0.31/0.46
% 0.31/0.46 % IdI
% 0.31/0.46 thf(fact_134_nofail__simps_I2_J,axiom,
% 0.31/0.46 ! [B: $tType,X5: set @ B] : ( refine1102455758nofail @ B @ ( refine605929679le_RES @ B @ X5 ) ) ).
% 0.31/0.46
% 0.31/0.46 % nofail_simps(2)
% 0.31/0.46 thf(fact_135_nofail__simps_I1_J,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ~ ( refine1102455758nofail @ A @ ( top_top @ ( refine1665802226e_nres @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % nofail_simps(1)
% 0.31/0.46 thf(fact_136_nofail__simps_I3_J,axiom,
% 0.31/0.46 ! [C: $tType,X2: C] : ( refine1102455758nofail @ C @ ( refine1687780735RETURN @ C @ X2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % nofail_simps(3)
% 0.31/0.46 thf(fact_137_DomainE,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,A3: A,R3: set @ ( product_prod @ A @ B )] :
% 0.31/0.46 ( ( member @ A @ A3 @ ( domain @ A @ B @ R3 ) )
% 0.31/0.46 => ~ ! [B5: B] :
% 0.31/0.46 ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B5 ) @ R3 ) ) ).
% 0.31/0.46
% 0.31/0.46 % DomainE
% 0.31/0.46 thf(fact_138_Domain__iff,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,A3: A,R3: set @ ( product_prod @ A @ B )] :
% 0.31/0.46 ( ( member @ A @ A3 @ ( domain @ A @ B @ R3 ) )
% 0.31/0.46 = ( ? [Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ Y3 ) @ R3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Domain_iff
% 0.31/0.46 thf(fact_139_Domain_Ocases,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,A3: A,R3: set @ ( product_prod @ A @ B )] :
% 0.31/0.46 ( ( member @ A @ A3 @ ( domain @ A @ B @ R3 ) )
% 0.31/0.46 => ~ ! [B5: B] :
% 0.31/0.46 ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B5 ) @ R3 ) ) ).
% 0.31/0.46
% 0.31/0.46 % Domain.cases
% 0.31/0.46 thf(fact_140_Domain_Osimps,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,A3: A,R3: set @ ( product_prod @ A @ B )] :
% 0.31/0.46 ( ( member @ A @ A3 @ ( domain @ A @ B @ R3 ) )
% 0.31/0.46 = ( ? [A4: A,B4: B] :
% 0.31/0.46 ( ( A3 = A4 )
% 0.31/0.46 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B4 ) @ R3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Domain.simps
% 0.31/0.46 thf(fact_141_Domain_ODomainI,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,A3: A,B3: B,R3: set @ ( product_prod @ A @ B )] :
% 0.31/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R3 )
% 0.31/0.46 => ( member @ A @ A3 @ ( domain @ A @ B @ R3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Domain.DomainI
% 0.31/0.46 thf(fact_142_Domain_Oinducts,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,X2: A,R3: set @ ( product_prod @ A @ B ),P: A > $o] :
% 0.31/0.46 ( ( member @ A @ X2 @ ( domain @ A @ B @ R3 ) )
% 0.31/0.46 => ( ! [A5: A,B5: B] :
% 0.31/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B5 ) @ R3 )
% 0.31/0.46 => ( P @ A5 ) )
% 0.31/0.46 => ( P @ X2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Domain.inducts
% 0.31/0.46 thf(fact_143_subrelI,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
% 0.31/0.46 ( ! [X3: A,Y4: B] :
% 0.31/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y4 ) @ R3 )
% 0.31/0.46 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y4 ) @ S3 ) )
% 0.31/0.46 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S3 ) ) ).
% 0.31/0.46
% 0.31/0.46 % subrelI
% 0.31/0.46 thf(fact_144_IdE,axiom,
% 0.31/0.46 ! [A: $tType,P2: product_prod @ A @ A] :
% 0.31/0.46 ( ( member @ ( product_prod @ A @ A ) @ P2 @ ( id @ A ) )
% 0.31/0.46 => ~ ! [X3: A] :
% 0.31/0.46 ( P2
% 0.31/0.46 != ( product_Pair @ A @ A @ X3 @ X3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % IdE
% 0.31/0.46 thf(fact_145_rel__restrict__notR_I2_J,axiom,
% 0.31/0.46 ! [A: $tType,X2: A,Y2: A,A2: set @ ( product_prod @ A @ A ),R: set @ A] :
% 0.31/0.46 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ ( rel_restrict @ A @ A2 @ R ) )
% 0.31/0.46 => ~ ( member @ A @ Y2 @ R ) ) ).
% 0.31/0.46
% 0.31/0.46 % rel_restrict_notR(2)
% 0.31/0.46 thf(fact_146_rel__restrict__notR_I1_J,axiom,
% 0.31/0.46 ! [A: $tType,X2: A,Y2: A,A2: set @ ( product_prod @ A @ A ),R: set @ A] :
% 0.31/0.46 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ ( rel_restrict @ A @ A2 @ R ) )
% 0.31/0.46 => ~ ( member @ A @ X2 @ R ) ) ).
% 0.31/0.46
% 0.31/0.46 % rel_restrict_notR(1)
% 0.31/0.46 thf(fact_147_rel__restrictI,axiom,
% 0.31/0.46 ! [A: $tType,X2: A,R: set @ A,Y2: A,E4: set @ ( product_prod @ A @ A )] :
% 0.31/0.46 ( ~ ( member @ A @ X2 @ R )
% 0.31/0.46 => ( ~ ( member @ A @ Y2 @ R )
% 0.31/0.46 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ E4 )
% 0.31/0.46 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ ( rel_restrict @ A @ E4 @ R ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rel_restrictI
% 0.31/0.46 thf(fact_148_rel__restrict__lift,axiom,
% 0.31/0.46 ! [A: $tType,X2: A,Y2: A,E4: set @ ( product_prod @ A @ A ),R: set @ A] :
% 0.31/0.46 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ ( rel_restrict @ A @ E4 @ R ) )
% 0.31/0.46 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ E4 ) ) ).
% 0.31/0.46
% 0.31/0.46 % rel_restrict_lift
% 0.31/0.46 thf(fact_149_pairself_Oinduct,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,P: ( A > B ) > ( product_prod @ A @ A ) > $o,A0: A > B,A1: product_prod @ A @ A] :
% 0.31/0.46 ( ! [F5: A > B,A5: A,B5: A] : ( P @ F5 @ ( product_Pair @ A @ A @ A5 @ B5 ) )
% 0.31/0.46 => ( P @ A0 @ A1 ) ) ).
% 0.31/0.46
% 0.31/0.46 % pairself.induct
% 0.31/0.46 thf(fact_150_pairself_Ocases,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,X2: product_prod @ ( A > B ) @ ( product_prod @ A @ A )] :
% 0.31/0.46 ~ ! [F5: A > B,A5: A,B5: A] :
% 0.31/0.46 ( X2
% 0.31/0.46 != ( product_Pair @ ( A > B ) @ ( product_prod @ A @ A ) @ F5 @ ( product_Pair @ A @ A @ A5 @ B5 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % pairself.cases
% 0.31/0.46 thf(fact_151_bex2I,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,A3: A,B3: B,S: set @ ( product_prod @ A @ B ),P: A > B > $o] :
% 0.31/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ S )
% 0.31/0.46 => ( ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ S )
% 0.31/0.46 => ( P @ A3 @ B3 ) )
% 0.31/0.46 => ? [A5: A,B5: B] :
% 0.31/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B5 ) @ S )
% 0.31/0.46 & ( P @ A5 @ B5 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % bex2I
% 0.31/0.46 thf(fact_152_rel__cong,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,F: A > B,G: A > B,X2: A,Y2: A] :
% 0.31/0.46 ( ( member @ ( product_prod @ ( A > B ) @ ( A > B ) ) @ ( product_Pair @ ( A > B ) @ ( A > B ) @ F @ G ) @ ( id @ ( A > B ) ) )
% 0.31/0.46 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ ( id @ A ) )
% 0.31/0.46 => ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ X2 ) @ ( G @ Y2 ) ) @ ( id @ B ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rel_cong
% 0.31/0.46 thf(fact_153_rel__fun__cong,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,F: A > B,G: A > B,X2: A] :
% 0.31/0.46 ( ( member @ ( product_prod @ ( A > B ) @ ( A > B ) ) @ ( product_Pair @ ( A > B ) @ ( A > B ) @ F @ G ) @ ( id @ ( A > B ) ) )
% 0.31/0.46 => ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ X2 ) @ ( G @ X2 ) ) @ ( id @ B ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rel_fun_cong
% 0.31/0.46 thf(fact_154_nofail__antimono__fun,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,F: A > ( refine1665802226e_nres @ B ),G: A > ( refine1665802226e_nres @ B ),X2: A] :
% 0.31/0.46 ( ( ord_less_eq @ ( A > ( refine1665802226e_nres @ B ) ) @ F @ G )
% 0.31/0.46 => ( ( refine1102455758nofail @ B @ ( G @ X2 ) )
% 0.31/0.46 => ( refine1102455758nofail @ B @ ( F @ X2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % nofail_antimono_fun
% 0.31/0.46 thf(fact_155_pw__conc__nofail,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),S: refine1665802226e_nres @ B] :
% 0.31/0.46 ( ( refine1102455758nofail @ A @ ( refine1073749519nc_fun @ A @ B @ R @ S ) )
% 0.31/0.46 = ( refine1102455758nofail @ B @ S ) ) ).
% 0.31/0.46
% 0.31/0.46 % pw_conc_nofail
% 0.31/0.46 thf(fact_156_intro__nofail_I2_J,axiom,
% 0.31/0.46 ! [A: $tType,S: refine1665802226e_nres @ A] :
% 0.31/0.46 ( ( ( top_top @ ( refine1665802226e_nres @ A ) )
% 0.31/0.46 != S )
% 0.31/0.46 = ( refine1102455758nofail @ A @ S ) ) ).
% 0.31/0.46
% 0.31/0.46 % intro_nofail(2)
% 0.31/0.46 thf(fact_157_nofail__def,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( refine1102455758nofail @ A )
% 0.31/0.46 = ( ^ [S4: refine1665802226e_nres @ A] :
% 0.31/0.46 ( S4
% 0.31/0.46 != ( top_top @ ( refine1665802226e_nres @ A ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % nofail_def
% 0.31/0.46 thf(fact_158_not__nofail__iff,axiom,
% 0.31/0.46 ! [A: $tType,S: refine1665802226e_nres @ A] :
% 0.31/0.46 ( ( ~ ( refine1102455758nofail @ A @ S ) )
% 0.31/0.46 = ( S
% 0.31/0.46 = ( top_top @ ( refine1665802226e_nres @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % not_nofail_iff
% 0.31/0.46 thf(fact_159_rel__arg__cong,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,X2: A,Y2: A,F: A > B] :
% 0.31/0.46 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ ( id @ A ) )
% 0.31/0.46 => ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ X2 ) @ ( F @ Y2 ) ) @ ( id @ B ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rel_arg_cong
% 0.31/0.46 thf(fact_160_pwD1,axiom,
% 0.31/0.46 ! [A: $tType,S: refine1665802226e_nres @ A,S5: refine1665802226e_nres @ A] :
% 0.31/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ S5 )
% 0.31/0.46 => ( ( refine1102455758nofail @ A @ S5 )
% 0.31/0.46 => ( refine1102455758nofail @ A @ S ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % pwD1
% 0.31/0.46 thf(fact_161_le__nofailI,axiom,
% 0.31/0.46 ! [A: $tType,M3: refine1665802226e_nres @ A,M: refine1665802226e_nres @ A] :
% 0.31/0.46 ( ( ( refine1102455758nofail @ A @ M3 )
% 0.31/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M @ M3 ) )
% 0.31/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M @ M3 ) ) ).
% 0.31/0.46
% 0.31/0.46 % le_nofailI
% 0.31/0.46 thf(fact_162_nofail__RES__conv,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( refine1102455758nofail @ A )
% 0.31/0.46 = ( ^ [M4: refine1665802226e_nres @ A] :
% 0.31/0.46 ? [M5: set @ A] :
% 0.31/0.46 ( M4
% 0.31/0.46 = ( refine605929679le_RES @ A @ M5 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % nofail_RES_conv
% 0.31/0.46 thf(fact_163_bijective__def,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType] :
% 0.31/0.46 ( ( bijective @ A @ B )
% 0.31/0.46 = ( ^ [R5: set @ ( product_prod @ A @ B )] :
% 0.31/0.46 ( ! [X: A,Y3: B,Z3: B] :
% 0.31/0.46 ( ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y3 ) @ R5 )
% 0.31/0.46 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Z3 ) @ R5 ) )
% 0.31/0.46 => ( Y3 = Z3 ) )
% 0.31/0.46 & ! [X: A,Y3: A,Z3: B] :
% 0.31/0.46 ( ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Z3 ) @ R5 )
% 0.31/0.46 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y3 @ Z3 ) @ R5 ) )
% 0.31/0.46 => ( X = Y3 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % bijective_def
% 0.31/0.46 thf(fact_164_RES__refine,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,S: set @ A,S5: set @ B,R: set @ ( product_prod @ A @ B )] :
% 0.31/0.46 ( ! [S6: A] :
% 0.31/0.46 ( ( member @ A @ S6 @ S )
% 0.31/0.46 => ? [X9: B] :
% 0.31/0.46 ( ( member @ B @ X9 @ S5 )
% 0.31/0.46 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ S6 @ X9 ) @ R ) ) )
% 0.31/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ S ) @ ( refine1073749519nc_fun @ A @ B @ R @ ( refine605929679le_RES @ B @ S5 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % RES_refine
% 0.31/0.46 thf(fact_165_RETURN__refine,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,X2: A,X10: B,R: set @ ( product_prod @ A @ B )] :
% 0.31/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ X10 ) @ R )
% 0.31/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X2 ) @ ( refine1073749519nc_fun @ A @ B @ R @ ( refine1687780735RETURN @ B @ X10 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % RETURN_refine
% 0.31/0.46 thf(fact_166_for__in__RI,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,X2: A,R: set @ ( product_prod @ A @ B )] :
% 0.31/0.46 ( ( member @ A @ X2 @ ( domain @ A @ B @ R ) )
% 0.31/0.46 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ ( fun_of_rel @ A @ B @ R @ X2 ) ) @ R ) ) ).
% 0.31/0.46
% 0.31/0.46 % for_in_RI
% 0.31/0.46 thf(fact_167_BNF__Greatest__Fixpoint_OIdD,axiom,
% 0.31/0.46 ! [A: $tType,A3: A,B3: A] :
% 0.31/0.46 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ ( id @ A ) )
% 0.31/0.46 => ( A3 = B3 ) ) ).
% 0.31/0.46
% 0.31/0.46 % BNF_Greatest_Fixpoint.IdD
% 0.31/0.46 thf(fact_168_E__closed__restr__reach__cases,axiom,
% 0.31/0.46 ! [A: $tType,U: A,V: A,E4: set @ ( product_prod @ A @ A ),R: set @ A] :
% 0.31/0.46 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V ) @ ( transitive_rtrancl @ A @ E4 ) )
% 0.31/0.46 => ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ E4 @ R ) @ R )
% 0.31/0.46 => ( ~ ( member @ A @ V @ R )
% 0.31/0.46 => ~ ( ~ ( member @ A @ U @ R )
% 0.31/0.46 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V ) @ ( transitive_rtrancl @ A @ ( rel_restrict @ A @ E4 @ R ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % E_closed_restr_reach_cases
% 0.31/0.46 thf(fact_169_relChain__def,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType] :
% 0.31/0.46 ( ( ord @ B )
% 0.31/0.46 => ( ( bNF_Ca1785829860lChain @ A @ B )
% 0.31/0.46 = ( ^ [R6: set @ ( product_prod @ A @ A ),As: A > B] :
% 0.31/0.46 ! [I2: A,J: A] :
% 0.31/0.46 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I2 @ J ) @ R6 )
% 0.31/0.46 => ( ord_less_eq @ B @ ( As @ I2 ) @ ( As @ J ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % relChain_def
% 0.31/0.46 thf(fact_170_converse__rtranclE_H,axiom,
% 0.31/0.46 ! [A: $tType,U: A,V: A,R: set @ ( product_prod @ A @ A )] :
% 0.31/0.46 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V ) @ ( transitive_rtrancl @ A @ R ) )
% 0.31/0.46 => ( ( U != V )
% 0.31/0.46 => ~ ! [Vh: A] :
% 0.31/0.46 ( ( U != Vh )
% 0.31/0.46 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ Vh ) @ R )
% 0.31/0.46 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Vh @ V ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % converse_rtranclE'
% 0.31/0.46 thf(fact_171_rtrancl__mono__rightI,axiom,
% 0.31/0.46 ! [A: $tType,S: set @ ( product_prod @ A @ A ),S5: set @ ( product_prod @ A @ A )] :
% 0.31/0.46 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ S @ S5 )
% 0.31/0.46 => ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ S @ ( transitive_rtrancl @ A @ S5 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rtrancl_mono_rightI
% 0.31/0.46 thf(fact_172_rtrancl__mono__mp,axiom,
% 0.31/0.46 ! [A: $tType,U2: set @ ( product_prod @ A @ A ),V2: set @ ( product_prod @ A @ A ),X2: product_prod @ A @ A] :
% 0.31/0.46 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ U2 @ V2 )
% 0.31/0.46 => ( ( member @ ( product_prod @ A @ A ) @ X2 @ ( transitive_rtrancl @ A @ U2 ) )
% 0.31/0.46 => ( member @ ( product_prod @ A @ A ) @ X2 @ ( transitive_rtrancl @ A @ V2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rtrancl_mono_mp
% 0.31/0.46 thf(fact_173_r__le__rtrancl,axiom,
% 0.31/0.46 ! [A: $tType,S: set @ ( product_prod @ A @ A )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ S @ ( transitive_rtrancl @ A @ S ) ) ).
% 0.31/0.46
% 0.31/0.46 % r_le_rtrancl
% 0.31/0.46 thf(fact_174_rtrancl__image__advance__rtrancl,axiom,
% 0.31/0.46 ! [A: $tType,Q2: A,R: set @ ( product_prod @ A @ A ),Q0: set @ A,X2: A] :
% 0.31/0.46 ( ( member @ A @ Q2 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R ) @ Q0 ) )
% 0.31/0.46 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q2 @ X2 ) @ ( transitive_rtrancl @ A @ R ) )
% 0.31/0.46 => ( member @ A @ X2 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R ) @ Q0 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rtrancl_image_advance_rtrancl
% 0.31/0.46 thf(fact_175_rtrancl__image__advance,axiom,
% 0.31/0.46 ! [A: $tType,Q2: A,R: set @ ( product_prod @ A @ A ),Q0: set @ A,X2: A] :
% 0.31/0.46 ( ( member @ A @ Q2 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R ) @ Q0 ) )
% 0.31/0.46 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q2 @ X2 ) @ R )
% 0.31/0.46 => ( member @ A @ X2 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R ) @ Q0 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rtrancl_image_advance
% 0.31/0.46 thf(fact_176_rtrancl__reachable__induct,axiom,
% 0.31/0.46 ! [A: $tType,I3: set @ A,INV: set @ A,E4: set @ ( product_prod @ A @ A )] :
% 0.31/0.46 ( ( ord_less_eq @ ( set @ A ) @ I3 @ INV )
% 0.31/0.46 => ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ E4 @ INV ) @ INV )
% 0.31/0.46 => ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E4 ) @ I3 ) @ INV ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rtrancl_reachable_induct
% 0.31/0.46 thf(fact_177_rtrancl__image__unfold__right,axiom,
% 0.31/0.46 ! [A: $tType,E4: set @ ( product_prod @ A @ A ),V2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ E4 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E4 ) @ V2 ) ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E4 ) @ V2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % rtrancl_image_unfold_right
% 0.31/0.46 thf(fact_178_reachable__mono,axiom,
% 0.31/0.46 ! [A: $tType,R: set @ ( product_prod @ A @ A ),R2: set @ ( product_prod @ A @ A ),X5: set @ A,X11: set @ A] :
% 0.31/0.46 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ R2 )
% 0.31/0.46 => ( ( ord_less_eq @ ( set @ A ) @ X5 @ X11 )
% 0.31/0.46 => ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R ) @ X5 ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R2 ) @ X11 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % reachable_mono
% 0.31/0.46 thf(fact_179_Domain__rtrancl,axiom,
% 0.31/0.46 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.31/0.46 ( ( domain @ A @ A @ ( transitive_rtrancl @ A @ R ) )
% 0.31/0.46 = ( top_top @ ( set @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Domain_rtrancl
% 0.31/0.46 thf(fact_180_Not__Domain__rtrancl,axiom,
% 0.31/0.46 ! [A: $tType,X2: A,R: set @ ( product_prod @ A @ A ),Y2: A] :
% 0.31/0.46 ( ~ ( member @ A @ X2 @ ( domain @ A @ A @ R ) )
% 0.31/0.46 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ ( transitive_rtrancl @ A @ R ) )
% 0.31/0.46 = ( X2 = Y2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Not_Domain_rtrancl
% 0.31/0.46 thf(fact_181_Image__closed__trancl,axiom,
% 0.31/0.46 ! [A: $tType,R3: set @ ( product_prod @ A @ A ),X5: set @ A] :
% 0.31/0.46 ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R3 @ X5 ) @ X5 )
% 0.31/0.46 => ( ( image @ A @ A @ ( transitive_rtrancl @ A @ R3 ) @ X5 )
% 0.31/0.46 = X5 ) ) ).
% 0.31/0.46
% 0.31/0.46 % Image_closed_trancl
% 0.31/0.46 thf(fact_182_rtrancl__subset__rtrancl,axiom,
% 0.31/0.46 ! [A: $tType,R3: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
% 0.31/0.46 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( transitive_rtrancl @ A @ S3 ) )
% 0.31/0.46 => ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R3 ) @ ( transitive_rtrancl @ A @ S3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rtrancl_subset_rtrancl
% 0.31/0.46 thf(fact_183_rtrancl__subset,axiom,
% 0.31/0.46 ! [A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
% 0.31/0.46 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ S )
% 0.31/0.46 => ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ S @ ( transitive_rtrancl @ A @ R ) )
% 0.31/0.46 => ( ( transitive_rtrancl @ A @ S )
% 0.31/0.46 = ( transitive_rtrancl @ A @ R ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rtrancl_subset
% 0.31/0.46 thf(fact_184_rtrancl__mono,axiom,
% 0.31/0.46 ! [A: $tType,R3: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
% 0.31/0.46 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ S3 )
% 0.31/0.46 => ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R3 ) @ ( transitive_rtrancl @ A @ S3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rtrancl_mono
% 0.31/0.46 thf(fact_185_rel__restrict__tranclI,axiom,
% 0.31/0.46 ! [A: $tType,X2: A,Y2: A,E4: set @ ( product_prod @ A @ A ),R: set @ A] :
% 0.31/0.46 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ ( transitive_trancl @ A @ E4 ) )
% 0.31/0.46 => ( ~ ( member @ A @ X2 @ R )
% 0.31/0.46 => ( ~ ( member @ A @ Y2 @ R )
% 0.31/0.46 => ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ E4 @ R ) @ R )
% 0.31/0.46 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ E4 @ R ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rel_restrict_tranclI
% 0.31/0.46 thf(fact_186_uncurry__apply,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,C: $tType,F: B > C > A,A3: B,B3: C] :
% 0.31/0.46 ( ( uncurry @ B @ C @ A @ F @ ( product_Pair @ B @ C @ A3 @ B3 ) )
% 0.31/0.46 = ( F @ A3 @ B3 ) ) ).
% 0.31/0.46
% 0.31/0.46 % uncurry_apply
% 0.31/0.46 thf(fact_187_pw__abs__nofail,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ A ),M: refine1665802226e_nres @ B] :
% 0.31/0.46 ( ( refine1102455758nofail @ A @ ( refine81118332bs_fun @ B @ A @ R @ M ) )
% 0.31/0.46 = ( ( refine1102455758nofail @ B @ M )
% 0.31/0.46 & ! [X: B] :
% 0.31/0.46 ( ( refine1315500908_inres @ B @ M @ X )
% 0.31/0.46 => ( member @ B @ X @ ( domain @ B @ A @ R ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % pw_abs_nofail
% 0.31/0.46 thf(fact_188_inres__simps_I2_J,axiom,
% 0.31/0.46 ! [B: $tType,X5: set @ B] :
% 0.31/0.46 ( ( refine1315500908_inres @ B @ ( refine605929679le_RES @ B @ X5 ) )
% 0.31/0.46 = ( ^ [X: B] : ( member @ B @ X @ X5 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % inres_simps(2)
% 0.31/0.46 thf(fact_189_trancl__domain,axiom,
% 0.31/0.46 ! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
% 0.31/0.46 ( ( domain @ A @ A @ ( transitive_trancl @ A @ R3 ) )
% 0.31/0.46 = ( domain @ A @ A @ R3 ) ) ).
% 0.31/0.46
% 0.31/0.46 % trancl_domain
% 0.31/0.46 thf(fact_190_inres__simps_I1_J,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( refine1315500908_inres @ A @ ( top_top @ ( refine1665802226e_nres @ A ) ) )
% 0.31/0.46 = ( ^ [Uu: A] : $true ) ) ).
% 0.31/0.46
% 0.31/0.46 % inres_simps(1)
% 0.31/0.46 thf(fact_191_inres__simps_I3_J,axiom,
% 0.31/0.46 ! [C: $tType,X2: C] :
% 0.31/0.46 ( ( refine1315500908_inres @ C @ ( refine1687780735RETURN @ C @ X2 ) )
% 0.31/0.46 = ( ^ [Y: C,Z: C] : ( Y = Z )
% 0.31/0.46 @ X2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % inres_simps(3)
% 0.31/0.46 thf(fact_192_trancl__mono,axiom,
% 0.31/0.46 ! [A: $tType,P2: product_prod @ A @ A,R3: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
% 0.31/0.46 ( ( member @ ( product_prod @ A @ A ) @ P2 @ ( transitive_trancl @ A @ R3 ) )
% 0.31/0.46 => ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ S3 )
% 0.31/0.46 => ( member @ ( product_prod @ A @ A ) @ P2 @ ( transitive_trancl @ A @ S3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % trancl_mono
% 0.31/0.46 thf(fact_193_trancl__Image__unfold__left,axiom,
% 0.31/0.46 ! [A: $tType,E4: set @ ( product_prod @ A @ A ),S: set @ A] :
% 0.31/0.46 ( ( image @ A @ A @ ( transitive_trancl @ A @ E4 ) @ S )
% 0.31/0.46 = ( image @ A @ A @ ( transitive_rtrancl @ A @ E4 ) @ ( image @ A @ A @ E4 @ S ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % trancl_Image_unfold_left
% 0.31/0.46 thf(fact_194_trancl__Image__unfold__right,axiom,
% 0.31/0.46 ! [A: $tType,E4: set @ ( product_prod @ A @ A ),S: set @ A] :
% 0.31/0.46 ( ( image @ A @ A @ ( transitive_trancl @ A @ E4 ) @ S )
% 0.31/0.46 = ( image @ A @ A @ E4 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E4 ) @ S ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % trancl_Image_unfold_right
% 0.31/0.46 thf(fact_195_rel__restrict__trancl__notR_I2_J,axiom,
% 0.31/0.46 ! [A: $tType,V: A,W2: A,E4: set @ ( product_prod @ A @ A ),R: set @ A] :
% 0.31/0.46 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W2 ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ E4 @ R ) ) )
% 0.31/0.46 => ~ ( member @ A @ W2 @ R ) ) ).
% 0.31/0.46
% 0.31/0.46 % rel_restrict_trancl_notR(2)
% 0.31/0.46 thf(fact_196_rel__restrict__trancl__notR_I1_J,axiom,
% 0.31/0.46 ! [A: $tType,V: A,W2: A,E4: set @ ( product_prod @ A @ A ),R: set @ A] :
% 0.31/0.46 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W2 ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ E4 @ R ) ) )
% 0.31/0.46 => ~ ( member @ A @ V @ R ) ) ).
% 0.31/0.46
% 0.31/0.46 % rel_restrict_trancl_notR(1)
% 0.31/0.46 thf(fact_197_rel__restrict__trancl__mem,axiom,
% 0.31/0.46 ! [A: $tType,A3: A,B3: A,A2: set @ ( product_prod @ A @ A ),R: set @ A] :
% 0.31/0.46 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ A2 @ R ) ) )
% 0.31/0.46 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ ( rel_restrict @ A @ ( transitive_trancl @ A @ A2 ) @ R ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rel_restrict_trancl_mem
% 0.31/0.46 thf(fact_198_rel__restrict__trancl__sub,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ ( product_prod @ A @ A ),R: set @ A] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ A2 @ R ) ) @ ( rel_restrict @ A @ ( transitive_trancl @ A @ A2 ) @ R ) ) ).
% 0.31/0.46
% 0.31/0.46 % rel_restrict_trancl_sub
% 0.31/0.46 thf(fact_199_pw__eqI,axiom,
% 0.31/0.46 ! [A: $tType,S: refine1665802226e_nres @ A,S5: refine1665802226e_nres @ A] :
% 0.31/0.46 ( ( ( refine1102455758nofail @ A @ S )
% 0.31/0.46 = ( refine1102455758nofail @ A @ S5 ) )
% 0.31/0.46 => ( ! [X3: A] :
% 0.31/0.46 ( ( refine1315500908_inres @ A @ S @ X3 )
% 0.31/0.46 = ( refine1315500908_inres @ A @ S5 @ X3 ) )
% 0.31/0.46 => ( S = S5 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % pw_eqI
% 0.31/0.46 thf(fact_200_pw__eq__iff,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( ^ [Y: refine1665802226e_nres @ A,Z: refine1665802226e_nres @ A] : ( Y = Z ) )
% 0.31/0.46 = ( ^ [S4: refine1665802226e_nres @ A,S7: refine1665802226e_nres @ A] :
% 0.31/0.46 ( ( ( refine1102455758nofail @ A @ S4 )
% 0.31/0.46 = ( refine1102455758nofail @ A @ S7 ) )
% 0.31/0.46 & ! [X: A] :
% 0.31/0.46 ( ( refine1315500908_inres @ A @ S4 @ X )
% 0.31/0.46 = ( refine1315500908_inres @ A @ S7 @ X ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % pw_eq_iff
% 0.31/0.46 thf(fact_201_not__nofail__inres,axiom,
% 0.31/0.46 ! [A: $tType,S: refine1665802226e_nres @ A,X2: A] :
% 0.31/0.46 ( ~ ( refine1102455758nofail @ A @ S )
% 0.31/0.46 => ( refine1315500908_inres @ A @ S @ X2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % not_nofail_inres
% 0.31/0.46 thf(fact_202_pwD2,axiom,
% 0.31/0.46 ! [A: $tType,S: refine1665802226e_nres @ A,S5: refine1665802226e_nres @ A,X2: A] :
% 0.31/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ S5 )
% 0.31/0.46 => ( ( refine1315500908_inres @ A @ S @ X2 )
% 0.31/0.46 => ( refine1315500908_inres @ A @ S5 @ X2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % pwD2
% 0.31/0.46 thf(fact_203_trancl__sub,axiom,
% 0.31/0.46 ! [A: $tType,R: set @ ( product_prod @ A @ A )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( transitive_trancl @ A @ R ) ) ).
% 0.31/0.46
% 0.31/0.46 % trancl_sub
% 0.31/0.46 thf(fact_204_trancl__mono__mp,axiom,
% 0.31/0.46 ! [A: $tType,U2: set @ ( product_prod @ A @ A ),V2: set @ ( product_prod @ A @ A ),X2: product_prod @ A @ A] :
% 0.31/0.46 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ U2 @ V2 )
% 0.31/0.46 => ( ( member @ ( product_prod @ A @ A ) @ X2 @ ( transitive_trancl @ A @ U2 ) )
% 0.31/0.46 => ( member @ ( product_prod @ A @ A ) @ X2 @ ( transitive_trancl @ A @ V2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % trancl_mono_mp
% 0.31/0.46 thf(fact_205_pw__leI,axiom,
% 0.31/0.46 ! [A: $tType,S5: refine1665802226e_nres @ A,S: refine1665802226e_nres @ A] :
% 0.31/0.46 ( ( ( refine1102455758nofail @ A @ S5 )
% 0.31/0.46 => ( ( refine1102455758nofail @ A @ S )
% 0.31/0.46 & ! [X3: A] :
% 0.31/0.46 ( ( refine1315500908_inres @ A @ S @ X3 )
% 0.31/0.46 => ( refine1315500908_inres @ A @ S5 @ X3 ) ) ) )
% 0.31/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ S5 ) ) ).
% 0.31/0.46
% 0.31/0.46 % pw_leI
% 0.31/0.46 thf(fact_206_pw__leI_H,axiom,
% 0.31/0.46 ! [A: $tType,S5: refine1665802226e_nres @ A,S: refine1665802226e_nres @ A] :
% 0.31/0.46 ( ( ( refine1102455758nofail @ A @ S5 )
% 0.31/0.46 => ( refine1102455758nofail @ A @ S ) )
% 0.31/0.46 => ( ! [X3: A] :
% 0.31/0.46 ( ( refine1102455758nofail @ A @ S5 )
% 0.31/0.46 => ( ( refine1315500908_inres @ A @ S @ X3 )
% 0.31/0.46 => ( refine1315500908_inres @ A @ S5 @ X3 ) ) )
% 0.31/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ S5 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % pw_leI'
% 0.31/0.46 thf(fact_207_pw__le__iff,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) )
% 0.31/0.46 = ( ^ [S4: refine1665802226e_nres @ A,S7: refine1665802226e_nres @ A] :
% 0.31/0.46 ( ( refine1102455758nofail @ A @ S7 )
% 0.31/0.46 => ( ( refine1102455758nofail @ A @ S4 )
% 0.31/0.46 & ! [X: A] :
% 0.31/0.46 ( ( refine1315500908_inres @ A @ S4 @ X )
% 0.31/0.46 => ( refine1315500908_inres @ A @ S7 @ X ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % pw_le_iff
% 0.31/0.46 thf(fact_208_inres__def,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( refine1315500908_inres @ A )
% 0.31/0.46 = ( ^ [S4: refine1665802226e_nres @ A,X: A] : ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X ) @ S4 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % inres_def
% 0.31/0.46 thf(fact_209_nf__inres__def,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( refine406925620_inres @ A )
% 0.31/0.46 = ( ^ [M4: refine1665802226e_nres @ A,X: A] :
% 0.31/0.46 ( ( refine1102455758nofail @ A @ M4 )
% 0.31/0.46 & ( refine1315500908_inres @ A @ M4 @ X ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % nf_inres_def
% 0.31/0.46 thf(fact_210_pw__abs__inres,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ A ),M: refine1665802226e_nres @ B,A3: A] :
% 0.31/0.46 ( ( refine1315500908_inres @ A @ ( refine81118332bs_fun @ B @ A @ R @ M ) @ A3 )
% 0.31/0.46 = ( ( refine1102455758nofail @ A @ ( refine81118332bs_fun @ B @ A @ R @ M ) )
% 0.31/0.46 => ? [C4: B] :
% 0.31/0.46 ( ( refine1315500908_inres @ B @ M @ C4 )
% 0.31/0.46 & ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ C4 @ A3 ) @ R ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % pw_abs_inres
% 0.31/0.46 thf(fact_211_trancl__reflcl,axiom,
% 0.31/0.46 ! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
% 0.31/0.46 ( ( transitive_trancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( id @ A ) ) )
% 0.31/0.46 = ( transitive_rtrancl @ A @ R3 ) ) ).
% 0.31/0.46
% 0.31/0.46 % trancl_reflcl
% 0.31/0.46 thf(fact_212_pw__bind__leI,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,S: refine1665802226e_nres @ A,M: refine1665802226e_nres @ B,F: B > ( refine1665802226e_nres @ A )] :
% 0.31/0.46 ( ( ( refine1102455758nofail @ A @ S )
% 0.31/0.46 => ( refine1102455758nofail @ B @ M ) )
% 0.31/0.46 => ( ! [X3: B] :
% 0.31/0.46 ( ( refine1102455758nofail @ B @ M )
% 0.31/0.46 => ( ( refine1315500908_inres @ B @ M @ X3 )
% 0.31/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( F @ X3 ) @ S ) ) )
% 0.31/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine463715084e_bind @ B @ A @ M @ F ) @ S ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % pw_bind_leI
% 0.31/0.46 thf(fact_213_Un__iff,axiom,
% 0.31/0.46 ! [A: $tType,C3: A,A2: set @ A,B2: set @ A] :
% 0.31/0.46 ( ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
% 0.31/0.46 = ( ( member @ A @ C3 @ A2 )
% 0.31/0.46 | ( member @ A @ C3 @ B2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Un_iff
% 0.31/0.46 thf(fact_214_UnCI,axiom,
% 0.31/0.46 ! [A: $tType,C3: A,B2: set @ A,A2: set @ A] :
% 0.31/0.46 ( ( ~ ( member @ A @ C3 @ B2 )
% 0.31/0.46 => ( member @ A @ C3 @ A2 ) )
% 0.31/0.46 => ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % UnCI
% 0.31/0.46 thf(fact_215_Un__subset__iff,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ A,B2: set @ A,C2: set @ A] :
% 0.31/0.46 ( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ C2 )
% 0.31/0.46 = ( ( ord_less_eq @ ( set @ A ) @ A2 @ C2 )
% 0.31/0.46 & ( ord_less_eq @ ( set @ A ) @ B2 @ C2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Un_subset_iff
% 0.31/0.46 thf(fact_216_bind__FAIL,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,F: B > ( refine1665802226e_nres @ A )] :
% 0.31/0.46 ( ( refine463715084e_bind @ B @ A @ ( top_top @ ( refine1665802226e_nres @ B ) ) @ F )
% 0.31/0.46 = ( top_top @ ( refine1665802226e_nres @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % bind_FAIL
% 0.31/0.46 thf(fact_217_nres__monad2,axiom,
% 0.31/0.46 ! [A: $tType,M: refine1665802226e_nres @ A] :
% 0.31/0.46 ( ( refine463715084e_bind @ A @ A @ M @ ( refine1687780735RETURN @ A ) )
% 0.31/0.46 = M ) ).
% 0.31/0.46
% 0.31/0.46 % nres_monad2
% 0.31/0.46 thf(fact_218_nres__monad1,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,X2: B,F: B > ( refine1665802226e_nres @ A )] :
% 0.31/0.46 ( ( refine463715084e_bind @ B @ A @ ( refine1687780735RETURN @ B @ X2 ) @ F )
% 0.31/0.46 = ( F @ X2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % nres_monad1
% 0.31/0.46 thf(fact_219_rtrancl__reflcl,axiom,
% 0.31/0.46 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.31/0.46 ( ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( id @ A ) ) )
% 0.31/0.46 = ( transitive_rtrancl @ A @ R ) ) ).
% 0.31/0.46
% 0.31/0.46 % rtrancl_reflcl
% 0.31/0.46 thf(fact_220_rtrancl__reflcl__absorb,axiom,
% 0.31/0.46 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.31/0.46 ( ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R ) @ ( id @ A ) )
% 0.31/0.46 = ( transitive_rtrancl @ A @ R ) ) ).
% 0.31/0.46
% 0.31/0.46 % rtrancl_reflcl_absorb
% 0.31/0.46 thf(fact_221_rtrancl__Un__subset,axiom,
% 0.31/0.46 ! [A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R ) @ ( transitive_rtrancl @ A @ S ) ) @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R @ S ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rtrancl_Un_subset
% 0.31/0.46 thf(fact_222_Un__mono,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ A,C2: set @ A,B2: set @ A,D3: set @ A] :
% 0.31/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ C2 )
% 0.31/0.46 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ D3 )
% 0.31/0.46 => ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ ( sup_sup @ ( set @ A ) @ C2 @ D3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Un_mono
% 0.31/0.46 thf(fact_223_Un__least,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ A,C2: set @ A,B2: set @ A] :
% 0.31/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ C2 )
% 0.31/0.46 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C2 )
% 0.31/0.46 => ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ C2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Un_least
% 0.31/0.46 thf(fact_224_Un__upper1,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % Un_upper1
% 0.31/0.46 thf(fact_225_Un__upper2,axiom,
% 0.31/0.46 ! [A: $tType,B2: set @ A,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ B2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % Un_upper2
% 0.31/0.46 thf(fact_226_Un__absorb1,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.31/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.31/0.46 => ( ( sup_sup @ ( set @ A ) @ A2 @ B2 )
% 0.31/0.46 = B2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % Un_absorb1
% 0.31/0.46 thf(fact_227_Un__absorb2,axiom,
% 0.31/0.46 ! [A: $tType,B2: set @ A,A2: set @ A] :
% 0.31/0.46 ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 0.31/0.46 => ( ( sup_sup @ ( set @ A ) @ A2 @ B2 )
% 0.31/0.46 = A2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % Un_absorb2
% 0.31/0.46 thf(fact_228_subset__UnE,axiom,
% 0.31/0.46 ! [A: $tType,C2: set @ A,A2: set @ A,B2: set @ A] :
% 0.31/0.46 ( ( ord_less_eq @ ( set @ A ) @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
% 0.31/0.46 => ~ ! [A8: set @ A] :
% 0.31/0.46 ( ( ord_less_eq @ ( set @ A ) @ A8 @ A2 )
% 0.31/0.46 => ! [B7: set @ A] :
% 0.31/0.46 ( ( ord_less_eq @ ( set @ A ) @ B7 @ B2 )
% 0.31/0.46 => ( C2
% 0.31/0.46 != ( sup_sup @ ( set @ A ) @ A8 @ B7 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % subset_UnE
% 0.31/0.46 thf(fact_229_subset__Un__eq,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( ord_less_eq @ ( set @ A ) )
% 0.31/0.46 = ( ^ [A6: set @ A,B6: set @ A] :
% 0.31/0.46 ( ( sup_sup @ ( set @ A ) @ A6 @ B6 )
% 0.31/0.46 = B6 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % subset_Un_eq
% 0.31/0.46 thf(fact_230_Domain__Un__eq,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,A2: set @ ( product_prod @ A @ B ),B2: set @ ( product_prod @ A @ B )] :
% 0.31/0.46 ( ( domain @ A @ B @ ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ A2 @ B2 ) )
% 0.31/0.46 = ( sup_sup @ ( set @ A ) @ ( domain @ A @ B @ A2 ) @ ( domain @ A @ B @ B2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Domain_Un_eq
% 0.31/0.46 thf(fact_231_Un__left__commute,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ A,B2: set @ A,C2: set @ A] :
% 0.31/0.46 ( ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ B2 @ C2 ) )
% 0.31/0.46 = ( sup_sup @ ( set @ A ) @ B2 @ ( sup_sup @ ( set @ A ) @ A2 @ C2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Un_left_commute
% 0.31/0.46 thf(fact_232_Un__left__absorb,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.31/0.46 ( ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
% 0.31/0.46 = ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % Un_left_absorb
% 0.31/0.46 thf(fact_233_Un__commute,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( sup_sup @ ( set @ A ) )
% 0.31/0.46 = ( ^ [A6: set @ A,B6: set @ A] : ( sup_sup @ ( set @ A ) @ B6 @ A6 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Un_commute
% 0.31/0.46 thf(fact_234_Un__absorb,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ A] :
% 0.31/0.46 ( ( sup_sup @ ( set @ A ) @ A2 @ A2 )
% 0.31/0.46 = A2 ) ).
% 0.31/0.46
% 0.31/0.46 % Un_absorb
% 0.31/0.46 thf(fact_235_Un__assoc,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ A,B2: set @ A,C2: set @ A] :
% 0.31/0.46 ( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ C2 )
% 0.31/0.46 = ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ B2 @ C2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Un_assoc
% 0.31/0.46 thf(fact_236_ball__Un,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ A,B2: set @ A,P: A > $o] :
% 0.31/0.46 ( ( ! [X: A] :
% 0.31/0.46 ( ( member @ A @ X @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
% 0.31/0.46 => ( P @ X ) ) )
% 0.31/0.46 = ( ! [X: A] :
% 0.31/0.46 ( ( member @ A @ X @ A2 )
% 0.31/0.46 => ( P @ X ) )
% 0.31/0.46 & ! [X: A] :
% 0.31/0.46 ( ( member @ A @ X @ B2 )
% 0.31/0.46 => ( P @ X ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ball_Un
% 0.31/0.46 thf(fact_237_bex__Un,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ A,B2: set @ A,P: A > $o] :
% 0.31/0.46 ( ( ? [X: A] :
% 0.31/0.46 ( ( member @ A @ X @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
% 0.31/0.46 & ( P @ X ) ) )
% 0.31/0.46 = ( ? [X: A] :
% 0.31/0.46 ( ( member @ A @ X @ A2 )
% 0.31/0.46 & ( P @ X ) )
% 0.31/0.46 | ? [X: A] :
% 0.31/0.46 ( ( member @ A @ X @ B2 )
% 0.31/0.46 & ( P @ X ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % bex_Un
% 0.31/0.46 thf(fact_238_UnI2,axiom,
% 0.31/0.46 ! [A: $tType,C3: A,B2: set @ A,A2: set @ A] :
% 0.31/0.46 ( ( member @ A @ C3 @ B2 )
% 0.31/0.46 => ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % UnI2
% 0.31/0.46 thf(fact_239_UnI1,axiom,
% 0.31/0.46 ! [A: $tType,C3: A,A2: set @ A,B2: set @ A] :
% 0.31/0.46 ( ( member @ A @ C3 @ A2 )
% 0.31/0.46 => ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % UnI1
% 0.31/0.46 thf(fact_240_UnE,axiom,
% 0.31/0.46 ! [A: $tType,C3: A,A2: set @ A,B2: set @ A] :
% 0.31/0.46 ( ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
% 0.31/0.46 => ( ~ ( member @ A @ C3 @ A2 )
% 0.31/0.46 => ( member @ A @ C3 @ B2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % UnE
% 0.31/0.46 thf(fact_241_rel__restrict__union,axiom,
% 0.31/0.46 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A2: set @ A,B2: set @ A] :
% 0.31/0.46 ( ( rel_restrict @ A @ R @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
% 0.31/0.46 = ( rel_restrict @ A @ ( rel_restrict @ A @ R @ A2 ) @ B2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % rel_restrict_union
% 0.31/0.46 thf(fact_242_ibind__strict_I2_J,axiom,
% 0.31/0.46 ! [A: $tType,F: product_unit > ( refine1665802226e_nres @ A )] :
% 0.31/0.46 ( ( refine463715084e_bind @ product_unit @ A @ ( top_top @ ( refine1665802226e_nres @ product_unit ) ) @ F )
% 0.31/0.46 = ( top_top @ ( refine1665802226e_nres @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ibind_strict(2)
% 0.31/0.46 thf(fact_243_Un__UNIV__right,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ A] :
% 0.31/0.46 ( ( sup_sup @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) )
% 0.31/0.46 = ( top_top @ ( set @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Un_UNIV_right
% 0.31/0.46 thf(fact_244_Un__UNIV__left,axiom,
% 0.31/0.46 ! [A: $tType,B2: set @ A] :
% 0.31/0.46 ( ( sup_sup @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B2 )
% 0.31/0.46 = ( top_top @ ( set @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Un_UNIV_left
% 0.31/0.46 thf(fact_245_pw__bind__nofail,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,M: refine1665802226e_nres @ B,F: B > ( refine1665802226e_nres @ A )] :
% 0.31/0.46 ( ( refine1102455758nofail @ A @ ( refine463715084e_bind @ B @ A @ M @ F ) )
% 0.31/0.46 = ( ( refine1102455758nofail @ B @ M )
% 0.31/0.46 & ! [X: B] :
% 0.31/0.46 ( ( refine1315500908_inres @ B @ M @ X )
% 0.31/0.46 => ( refine1102455758nofail @ A @ ( F @ X ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % pw_bind_nofail
% 0.31/0.46 thf(fact_246_bind__cong,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,M2: refine1665802226e_nres @ A,M6: refine1665802226e_nres @ A,F: A > ( refine1665802226e_nres @ B ),F3: A > ( refine1665802226e_nres @ B )] :
% 0.31/0.46 ( ( M2 = M6 )
% 0.31/0.46 => ( ! [X3: A] :
% 0.31/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X3 ) @ M6 )
% 0.31/0.46 => ( ( F @ X3 )
% 0.31/0.46 = ( F3 @ X3 ) ) )
% 0.31/0.46 => ( ( refine463715084e_bind @ A @ B @ M2 @ F )
% 0.31/0.46 = ( refine463715084e_bind @ A @ B @ M6 @ F3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % bind_cong
% 0.31/0.46 thf(fact_247_Refine__Basic__Mirabelle__tqojlsrkwy_Obind__mono_I1_J,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,M: refine1665802226e_nres @ A,M3: refine1665802226e_nres @ A,F: A > ( refine1665802226e_nres @ B ),F3: A > ( refine1665802226e_nres @ B )] :
% 0.31/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M @ M3 )
% 0.31/0.46 => ( ! [X3: A] :
% 0.31/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X3 ) @ M )
% 0.31/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( F @ X3 ) @ ( F3 @ X3 ) ) )
% 0.31/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine463715084e_bind @ A @ B @ M @ F ) @ ( refine463715084e_bind @ A @ B @ M3 @ F3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Refine_Basic_Mirabelle_tqojlsrkwy.bind_mono(1)
% 0.31/0.46 thf(fact_248_trancl__union__outside,axiom,
% 0.31/0.46 ! [A: $tType,V: A,W2: A,E4: set @ ( product_prod @ A @ A ),U2: set @ ( product_prod @ A @ A )] :
% 0.31/0.46 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W2 ) @ ( transitive_trancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ E4 @ U2 ) ) )
% 0.31/0.46 => ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W2 ) @ ( transitive_trancl @ A @ E4 ) )
% 0.31/0.46 => ? [X3: A,Y4: A] :
% 0.31/0.46 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ X3 ) @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ E4 @ U2 ) ) )
% 0.31/0.46 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y4 ) @ U2 )
% 0.31/0.46 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ W2 ) @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ E4 @ U2 ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % trancl_union_outside
% 0.31/0.46 thf(fact_249_rtrancl__trancl__reflcl,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( transitive_rtrancl @ A )
% 0.31/0.46 = ( ^ [R6: set @ ( product_prod @ A @ A )] : ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R6 ) @ ( id @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rtrancl_trancl_reflcl
% 0.31/0.46 thf(fact_250_trancl__image__by__rtrancl,axiom,
% 0.31/0.46 ! [A: $tType,E4: set @ ( product_prod @ A @ A ),Vi: set @ A] :
% 0.31/0.46 ( ( sup_sup @ ( set @ A ) @ ( image @ A @ A @ ( transitive_trancl @ A @ E4 ) @ Vi ) @ Vi )
% 0.31/0.46 = ( image @ A @ A @ ( transitive_rtrancl @ A @ E4 ) @ Vi ) ) ).
% 0.31/0.46
% 0.31/0.46 % trancl_image_by_rtrancl
% 0.31/0.46 thf(fact_251_pw__bind__le__iff,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,M: refine1665802226e_nres @ B,F: B > ( refine1665802226e_nres @ A ),S: refine1665802226e_nres @ A] :
% 0.31/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine463715084e_bind @ B @ A @ M @ F ) @ S )
% 0.31/0.46 = ( ( ( refine1102455758nofail @ A @ S )
% 0.31/0.46 => ( refine1102455758nofail @ B @ M ) )
% 0.31/0.46 & ! [X: B] :
% 0.31/0.46 ( ( ( refine1102455758nofail @ B @ M )
% 0.31/0.46 & ( refine1315500908_inres @ B @ M @ X ) )
% 0.31/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( F @ X ) @ S ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % pw_bind_le_iff
% 0.31/0.46 thf(fact_252_sup__top__left,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( bounded_lattice_top @ A )
% 0.31/0.46 => ! [X2: A] :
% 0.31/0.46 ( ( sup_sup @ A @ ( top_top @ A ) @ X2 )
% 0.31/0.46 = ( top_top @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % sup_top_left
% 0.31/0.46
% 0.31/0.46 % Type constructors (42)
% 0.31/0.46 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Lattices_Obounded__lattice,axiom,
% 0.31/0.46 ! [A9: $tType] : ( bounded_lattice @ ( refine1665802226e_nres @ A9 ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice_1,axiom,
% 0.31/0.46 bounded_lattice @ product_unit ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_HOL_Obool___Lattices_Obounded__lattice_2,axiom,
% 0.31/0.46 bounded_lattice @ $o ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_Set_Oset___Lattices_Obounded__lattice_3,axiom,
% 0.31/0.46 ! [A9: $tType] : ( bounded_lattice @ ( set @ A9 ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_fun___Lattices_Obounded__lattice_4,axiom,
% 0.31/0.46 ! [A9: $tType,A10: $tType] :
% 0.31/0.46 ( ( bounded_lattice @ A10 )
% 0.31/0.46 => ( bounded_lattice @ ( A9 > A10 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
% 0.31/0.46 ! [A9: $tType,A10: $tType] :
% 0.31/0.46 ( ( comple187826305attice @ A10 )
% 0.31/0.46 => ( comple187826305attice @ ( A9 > A10 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_fun___Lattices_Obounded__lattice__top,axiom,
% 0.31/0.46 ! [A9: $tType,A10: $tType] :
% 0.31/0.46 ( ( bounded_lattice @ A10 )
% 0.31/0.46 => ( bounded_lattice_top @ ( A9 > A10 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_fun___Orderings_Oorder__top,axiom,
% 0.31/0.46 ! [A9: $tType,A10: $tType] :
% 0.31/0.46 ( ( order_top @ A10 )
% 0.31/0.46 => ( order_top @ ( A9 > A10 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.31/0.46 ! [A9: $tType,A10: $tType] :
% 0.31/0.46 ( ( preorder @ A10 )
% 0.31/0.46 => ( preorder @ ( A9 > A10 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.31/0.46 ! [A9: $tType,A10: $tType] :
% 0.31/0.46 ( ( order @ A10 )
% 0.31/0.46 => ( order @ ( A9 > A10 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_fun___Orderings_Otop,axiom,
% 0.31/0.46 ! [A9: $tType,A10: $tType] :
% 0.31/0.46 ( ( top @ A10 )
% 0.31/0.46 => ( top @ ( A9 > A10 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_fun___Orderings_Oord,axiom,
% 0.31/0.46 ! [A9: $tType,A10: $tType] :
% 0.31/0.46 ( ( ord @ A10 )
% 0.31/0.46 => ( ord @ ( A9 > A10 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_5,axiom,
% 0.31/0.46 ! [A9: $tType] : ( comple187826305attice @ ( set @ A9 ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_Set_Oset___Lattices_Obounded__lattice__top_6,axiom,
% 0.31/0.46 ! [A9: $tType] : ( bounded_lattice_top @ ( set @ A9 ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_Set_Oset___Orderings_Oorder__top_7,axiom,
% 0.31/0.46 ! [A9: $tType] : ( order_top @ ( set @ A9 ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_Set_Oset___Orderings_Opreorder_8,axiom,
% 0.31/0.46 ! [A9: $tType] : ( preorder @ ( set @ A9 ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_Set_Oset___Orderings_Oorder_9,axiom,
% 0.31/0.46 ! [A9: $tType] : ( order @ ( set @ A9 ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_Set_Oset___Orderings_Otop_10,axiom,
% 0.31/0.46 ! [A9: $tType] : ( top @ ( set @ A9 ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_Set_Oset___Orderings_Oord_11,axiom,
% 0.31/0.46 ! [A9: $tType] : ( ord @ ( set @ A9 ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_12,axiom,
% 0.31/0.46 comple187826305attice @ $o ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_HOL_Obool___Lattices_Obounded__lattice__top_13,axiom,
% 0.31/0.46 bounded_lattice_top @ $o ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_HOL_Obool___Orderings_Oorder__top_14,axiom,
% 0.31/0.46 order_top @ $o ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_HOL_Obool___Orderings_Opreorder_15,axiom,
% 0.31/0.46 preorder @ $o ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 0.31/0.46 linorder @ $o ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_HOL_Obool___Orderings_Oorder_16,axiom,
% 0.31/0.46 order @ $o ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_HOL_Obool___Orderings_Otop_17,axiom,
% 0.31/0.46 top @ $o ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_HOL_Obool___Orderings_Oord_18,axiom,
% 0.31/0.46 ord @ $o ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_19,axiom,
% 0.31/0.46 comple187826305attice @ product_unit ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__top_20,axiom,
% 0.31/0.46 bounded_lattice_top @ product_unit ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_Product__Type_Ounit___Orderings_Oorder__top_21,axiom,
% 0.31/0.46 order_top @ product_unit ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_Product__Type_Ounit___Orderings_Opreorder_22,axiom,
% 0.31/0.46 preorder @ product_unit ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_Product__Type_Ounit___Orderings_Olinorder_23,axiom,
% 0.31/0.46 linorder @ product_unit ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_Product__Type_Ounit___Orderings_Oorder_24,axiom,
% 0.31/0.46 order @ product_unit ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_Product__Type_Ounit___Orderings_Otop_25,axiom,
% 0.31/0.46 top @ product_unit ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_Product__Type_Ounit___Orderings_Oord_26,axiom,
% 0.31/0.46 ord @ product_unit ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Complete__Lattices_Ocomplete__lattice_27,axiom,
% 0.31/0.46 ! [A9: $tType] : ( comple187826305attice @ ( refine1665802226e_nres @ A9 ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Lattices_Obounded__lattice__top_28,axiom,
% 0.31/0.46 ! [A9: $tType] : ( bounded_lattice_top @ ( refine1665802226e_nres @ A9 ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oorder__top_29,axiom,
% 0.31/0.46 ! [A9: $tType] : ( order_top @ ( refine1665802226e_nres @ A9 ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Opreorder_30,axiom,
% 0.31/0.46 ! [A9: $tType] : ( preorder @ ( refine1665802226e_nres @ A9 ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oorder_31,axiom,
% 0.31/0.46 ! [A9: $tType] : ( order @ ( refine1665802226e_nres @ A9 ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Otop_32,axiom,
% 0.31/0.46 ! [A9: $tType] : ( top @ ( refine1665802226e_nres @ A9 ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oord_33,axiom,
% 0.31/0.46 ! [A9: $tType] : ( ord @ ( refine1665802226e_nres @ A9 ) ) ).
% 0.31/0.46
% 0.31/0.46 % Helper facts (3)
% 0.31/0.46 thf(help_If_3_1_T,axiom,
% 0.31/0.46 ! [P: $o] :
% 0.31/0.46 ( ( P = $true )
% 0.31/0.46 | ( P = $false ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(help_If_2_1_T,axiom,
% 0.31/0.46 ! [A: $tType,X2: A,Y2: A] :
% 0.31/0.46 ( ( if @ A @ $false @ X2 @ Y2 )
% 0.31/0.46 = Y2 ) ).
% 0.31/0.46
% 0.31/0.46 thf(help_If_1_1_T,axiom,
% 0.31/0.46 ! [A: $tType,X2: A,Y2: A] :
% 0.31/0.46 ( ( if @ A @ $true @ X2 @ Y2 )
% 0.31/0.46 = X2 ) ).
% 0.31/0.46
% 0.31/0.46 % Conjectures (1)
% 0.31/0.46 thf(conj_0,conjecture,
% 0.31/0.46 ord_less_eq @ ( refine1665802226e_nres @ a ) @ x @ ( refine1073749519nc_fun @ a @ b @ r @ ( top_top @ ( refine1665802226e_nres @ b ) ) ) ).
% 0.31/0.46
% 0.31/0.46 %------------------------------------------------------------------------------
% 0.31/0.46 ------------------------
% 3.37/1.84 ---- Embedded in THF ---
% 3.37/1.84 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 3.37/1.84 %%% Generated on Tue Feb 24 08:28:29 EST 2026
% 3.37/1.84 %%% using '$$dhol' embedding, version 1.3.0.
% 3.37/1.84 %%% Logic specification used:
% 3.37/1.84 %%% thf(spec, logic, $$dhol).
% 3.37/1.84
% 3.37/1.84 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.vRzaLjzKJ7/DTF2DTF19562.p
% 3.37/1.84 thf(ty_t_Refine__Basic__Mirabelle__tqojlsrkwy_Onres, type, refine1665802226e_nres: ($tType > $tType)).
% 3.37/1.84 thf(ty_t_Product__Type_Ounit, type, product_unit: $tType).
% 3.37/1.84 thf(ty_t_Product__Type_Oprod, type, product_prod: ($tType > ($tType > $tType))).
% 3.37/1.84 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 3.37/1.84 thf(ty_tf_b, type, b: $tType).
% 3.37/1.84 thf(ty_tf_a, type, a: $tType).
% 3.37/1.84 thf(sy_cl_Lattices_Obounded__lattice, type, bounded_lattice: (!> [A:$tType]: ($o))).
% 3.37/1.84 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 3.37/1.84 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 3.37/1.84 thf(sy_cl_Orderings_Otop, type, top: (!> [A:$tType]: ($o))).
% 3.37/1.84 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 3.37/1.84 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 3.37/1.84 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 3.37/1.84 thf(sy_cl_Orderings_Oorder__top, type, order_top: (!> [A:$tType]: ($o))).
% 3.37/1.84 thf(sy_cl_Lattices_Obounded__lattice__top, type, bounded_lattice_top: (!> [A:$tType]: ($o))).
% 3.37/1.84 thf(sy_cl_Complete__Lattices_Ocomplete__lattice, type, comple187826305attice: (!> [A:$tType]: ($o))).
% 3.37/1.84 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain, type, bNF_Ca1785829860lChain: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ A)) > ((A > B) > $o))))).
% 3.37/1.84 thf(sy_c_If, type, if: (!> [A:$tType]: (($o > (A > (A > A)))))).
% 3.37/1.84 thf(sy_c_Lattices_Osup__class_Osup, type, sup_sup: (!> [A:$tType]: ((A > (A > A))))).
% 3.37/1.84 thf(sy_c_Misc_Obijective, type, bijective: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ B)) > $o)))).
% 3.37/1.84 thf(sy_c_Misc_Ofun__of__rel, type, fun_of_rel: (!> [B:$tType,A:$tType]: (((set @ ((product_prod @ B) @ A)) > (B > A))))).
% 3.37/1.84 thf(sy_c_Misc_Orel__restrict, type, rel_restrict: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > ((set @ A) > (set @ ((product_prod @ A) @ A))))))).
% 3.37/1.84 thf(sy_c_Misc_Ouncurry, type, uncurry: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > C)) > (((product_prod @ A) @ B) > C))))).
% 3.37/1.84 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 3.37/1.84 thf(sy_c_Orderings_Oorder__class_OGreatest, type, order_Greatest: (!> [A:$tType]: (((A > $o) > A)))).
% 3.37/1.84 thf(sy_c_Orderings_Otop__class_Otop, type, top_top: (!> [A:$tType]: (A))).
% 3.37/1.84 thf(sy_c_Product__Type_OPair, type, product_Pair: (!> [A:$tType,B:$tType]: ((A > (B > ((product_prod @ A) @ B)))))).
% 3.37/1.84 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_ORETURN, type, refine1687780735RETURN: (!> [A:$tType]: ((A > (refine1665802226e_nres @ A))))).
% 3.37/1.84 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Oabs__fun, type, refine81118332bs_fun: (!> [B:$tType,A:$tType]: (((set @ ((product_prod @ B) @ A)) > ((refine1665802226e_nres @ B) > (refine1665802226e_nres @ A)))))).
% 3.37/1.84 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Obind, type, refine463715084e_bind: (!> [B:$tType,A:$tType]: (((refine1665802226e_nres @ B) > ((B > (refine1665802226e_nres @ A)) > (refine1665802226e_nres @ A)))))).
% 3.37/1.84 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Oconc__fun, type, refine1073749519nc_fun: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ B)) > ((refine1665802226e_nres @ B) > (refine1665802226e_nres @ A)))))).
% 3.37/1.84 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Oinres, type, refine1315500908_inres: (!> [A:$tType]: (((refine1665802226e_nres @ A) > (A > $o))))).
% 3.37/1.84 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onf__inres, type, refine406925620_inres: (!> [A:$tType]: (((refine1665802226e_nres @ A) > (A > $o))))).
% 3.37/1.84 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onofail, type, refine1102455758nofail: (!> [A:$tType]: (((refine1665802226e_nres @ A) > $o)))).
% 3.37/1.84 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onres_ORES, type, refine605929679le_RES: (!> [A:$tType]: (((set @ A) > (refine1665802226e_nres @ A))))).
% 3.37/1.84 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Othe__RES, type, refine1672542526he_RES: (!> [A:$tType]: (((refine1665802226e_nres @ A) > (set @ A))))).
% 3.37/1.84 thf(sy_c_Relation_ODomain, type, domain: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ B)) > (set @ A))))).
% 3.37/1.84 thf(sy_c_Relation_OId, type, id: (!> [A:$tType]: ((set @ ((product_prod @ A) @ A))))).
% 3.37/1.84 thf(sy_c_Relation_OImage, type, image: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ B)) > ((set @ A) > (set @ B)))))).
% 3.37/1.84 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 3.37/1.84 thf(sy_c_Transitive__Closure_Ortrancl, type, transitive_rtrancl: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (set @ ((product_prod @ A) @ A)))))).
% 3.37/1.84 thf(sy_c_Transitive__Closure_Otrancl, type, transitive_trancl: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (set @ ((product_prod @ A) @ A)))))).
% 3.37/1.84 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 3.37/1.84 thf(sy_v_R, type, r: (set @ ((product_prod @ a) @ b))).
% 3.37/1.84 thf(sy_v_X, type, x: (refine1665802226e_nres @ a)).
% 3.37/1.84 thf(fact_0_nres__order__simps_I4_J, axiom, (! [D:$tType,M:(refine1665802226e_nres @ D)]: (((((ord_less_eq @ (refine1665802226e_nres @ D)) @ (top_top @ (refine1665802226e_nres @ D))) @ M) = (M = (top_top @ (refine1665802226e_nres @ D))))))).
% 3.37/1.84 thf(fact_1_conc__fun__fail__iff_I2_J, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ A) @ B)),S:(refine1665802226e_nres @ B)]: ((((top_top @ (refine1665802226e_nres @ A)) = ((((refine1073749519nc_fun @ A) @ B) @ R) @ S)) = (S = (top_top @ (refine1665802226e_nres @ B))))))).
% 3.37/1.84 thf(fact_2_conc__fun__fail__iff_I1_J, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ A) @ B)),S:(refine1665802226e_nres @ B)]: (((((((refine1073749519nc_fun @ A) @ B) @ R) @ S) = (top_top @ (refine1665802226e_nres @ A))) = (S = (top_top @ (refine1665802226e_nres @ B))))))).
% 3.37/1.84 thf(fact_3_conc__fun__FAIL, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ B))]: ((((((refine1073749519nc_fun @ A) @ B) @ R) @ (top_top @ (refine1665802226e_nres @ B))) = (top_top @ (refine1665802226e_nres @ A)))))).
% 3.37/1.84 thf(fact_4_nres__order__simps_I3_J, axiom, (! [C:$tType,M:(refine1665802226e_nres @ C)]: ((((ord_less_eq @ (refine1665802226e_nres @ C)) @ M) @ (top_top @ (refine1665802226e_nres @ C)))))).
% 3.37/1.84 thf(fact_5_conc__trans__additional_I1_J, axiom, (! [A:$tType,B:$tType,A2:(refine1665802226e_nres @ A),R:(set @ ((product_prod @ A) @ B)),B2:(refine1665802226e_nres @ B),C2:(refine1665802226e_nres @ B)]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ A2) @ ((((refine1073749519nc_fun @ A) @ B) @ R) @ B2)) => ((((ord_less_eq @ (refine1665802226e_nres @ B)) @ B2) @ C2) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ A2) @ ((((refine1073749519nc_fun @ A) @ B) @ R) @ C2))))))).
% 3.37/1.84 thf(fact_6_conc__trans, axiom, (! [A:$tType,B:$tType,C:$tType,C2:(refine1665802226e_nres @ A),R:(set @ ((product_prod @ A) @ B)),B2:(refine1665802226e_nres @ B),R2:(set @ ((product_prod @ B) @ C)),A2:(refine1665802226e_nres @ C)]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ C2) @ ((((refine1073749519nc_fun @ A) @ B) @ R) @ B2)) => ((((ord_less_eq @ (refine1665802226e_nres @ B)) @ B2) @ ((((refine1073749519nc_fun @ B) @ C) @ R2) @ A2)) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ C2) @ ((((refine1073749519nc_fun @ A) @ B) @ R) @ ((((refine1073749519nc_fun @ B) @ C) @ R2) @ A2)))))))).
% 3.37/1.84 thf(fact_7_top__apply, axiom, (! [C:$tType,D:$tType]: (((top @ C) => ((top_top @ (D > C)) = (^ [X:D]: ((top_top @ C)))))))).
% 3.37/1.84 thf(fact_8_order__mono__setup_Orefl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A]: ((((ord_less_eq @ A) @ X2) @ X2))))))).
% 3.37/1.84 thf(fact_9_top__greatest, axiom, (! [A:$tType]: (((order_top @ A) => (! [A3:A]: ((((ord_less_eq @ A) @ A3) @ (top_top @ A)))))))).
% 3.37/1.84 thf(fact_10_meta__le__everything__if__top, axiom, (! [A:$tType]: (((order_top @ A) => (! [M2:A,X2:A]: (((M2 = (top_top @ A)) => (((ord_less_eq @ A) @ X2) @ M2)))))))).
% 3.37/1.84 thf(fact_11_top_Oextremum__unique, axiom, (! [A:$tType]: (((order_top @ A) => (! [A3:A]: (((((ord_less_eq @ A) @ (top_top @ A)) @ A3) = (A3 = (top_top @ A))))))))).
% 3.37/1.84 thf(fact_12_top_Oextremum__uniqueI, axiom, (! [A:$tType]: (((order_top @ A) => (! [A3:A]: (((((ord_less_eq @ A) @ (top_top @ A)) @ A3) => (A3 = (top_top @ A))))))))).
% 3.37/1.84 thf(fact_13_le__funD, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B),X2:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X2)) @ (G @ X2))))))))).
% 3.37/1.84 thf(fact_14_le__funE, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B),X2:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X2)) @ (G @ X2))))))))).
% 3.37/1.84 thf(fact_15_le__funI, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B)]: (((! [X3:A]: ((((ord_less_eq @ B) @ (F @ X3)) @ (G @ X3)))) => (((ord_less_eq @ (A > B)) @ F) @ G)))))))).
% 3.37/1.84 thf(fact_16_Refine__Misc_Oif__mono, axiom, (! [A:$tType]: (((ord @ A) => (! [B3:$o,M1:A,M12:A,M22:A,M23:A]: (((B3 => (((ord_less_eq @ A) @ M1) @ M12)) => (((~ (B3)) => (((ord_less_eq @ A) @ M22) @ M23)) => (((ord_less_eq @ A) @ ((((if @ A) @ B3) @ M1) @ M22)) @ ((((if @ A) @ B3) @ M12) @ M23)))))))))).
% 3.37/1.84 thf(fact_17_le__fun__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => ((ord_less_eq @ (A > B)) = (^ [F2:(A > B),G2:(A > B)]: ((! [X:A]: ((((ord_less_eq @ B) @ (F2 @ X)) @ (G2 @ X))))))))))).
% 3.37/1.84 thf(fact_18_conc__fun__R__mono, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ A) @ B)),R2:(set @ ((product_prod @ A) @ B)),M:(refine1665802226e_nres @ B)]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R) @ R2) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ ((((refine1073749519nc_fun @ A) @ B) @ R) @ M)) @ ((((refine1073749519nc_fun @ A) @ B) @ R2) @ M)))))).
% 3.37/1.84 thf(fact_19_order__mono__setup_Omono__let, axiom, (! [A:$tType,B:$tType]: (((preorder @ A) => (! [F:(B > A),F3:(B > A),X2:B]: (((! [X3:B]: ((((ord_less_eq @ A) @ (F @ X3)) @ (F3 @ X3)))) => (((ord_less_eq @ A) @ (F @ X2)) @ (F3 @ X2))))))))).
% 3.37/1.84 thf(fact_20_order__mono__setup_Omono__if, axiom, (! [A:$tType]: (((preorder @ A) => (! [T:A,T2:A,E:A,E2:A,B3:$o]: (((((ord_less_eq @ A) @ T) @ T2) => ((((ord_less_eq @ A) @ E) @ E2) => (((ord_less_eq @ A) @ ((((if @ A) @ B3) @ T) @ E)) @ ((((if @ A) @ B3) @ T2) @ E2)))))))))).
% 3.37/1.84 thf(fact_21_dual__order_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A3:A]: (((((ord_less_eq @ A) @ B3) @ A3) => ((((ord_less_eq @ A) @ A3) @ B3) => (A3 = B3))))))))).
% 3.37/1.84 thf(fact_22_dual__order_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y:A,Z:A]: ((Y = Z))) = (^ [A4:A,B4:A]: (((((ord_less_eq @ A) @ B4) @ A4) & (((ord_less_eq @ A) @ A4) @ B4))))))))).
% 3.37/1.84 thf(fact_23_dual__order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A3:A,C3:A]: (((((ord_less_eq @ A) @ B3) @ A3) => ((((ord_less_eq @ A) @ C3) @ B3) => (((ord_less_eq @ A) @ C3) @ A3))))))))).
% 3.37/1.84 thf(fact_24_linorder__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > (A > $o)),A3:A,B3:A]: (((! [A5:A,B5:A]: (((((ord_less_eq @ A) @ A5) @ B5) => ((P @ A5) @ B5)))) => ((! [A5:A,B5:A]: ((((P @ B5) @ A5) => ((P @ A5) @ B5)))) => ((P @ A3) @ B3))))))))).
% 3.37/1.84 thf(fact_25_dual__order_Orefl, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A]: ((((ord_less_eq @ A) @ A3) @ A3))))))).
% 3.37/1.84 thf(fact_26_order__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y2:A,Z2:A]: (((((ord_less_eq @ A) @ X2) @ Y2) => ((((ord_less_eq @ A) @ Y2) @ Z2) => (((ord_less_eq @ A) @ X2) @ Z2))))))))).
% 3.37/1.84 thf(fact_27_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B3:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((ord_less_eq @ A) @ B3) @ A3) => (A3 = B3))))))))).
% 3.37/1.84 thf(fact_28_ord__le__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A3:A,B3:A,C3:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((B3 = C3) => (((ord_less_eq @ A) @ A3) @ C3))))))))).
% 3.37/1.84 thf(fact_29_ord__eq__le__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A3:A,B3:A,C3:A]: (((A3 = B3) => ((((ord_less_eq @ A) @ B3) @ C3) => (((ord_less_eq @ A) @ A3) @ C3))))))))).
% 3.37/1.84 thf(fact_30_order__class_Oorder_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y:A,Z:A]: ((Y = Z))) = (^ [A4:A,B4:A]: (((((ord_less_eq @ A) @ A4) @ B4) & (((ord_less_eq @ A) @ B4) @ A4))))))))).
% 3.37/1.84 thf(fact_31_antisym__conv, axiom, (! [A:$tType]: (((order @ A) => (! [Y2:A,X2:A]: (((((ord_less_eq @ A) @ Y2) @ X2) => ((((ord_less_eq @ A) @ X2) @ Y2) = (X2 = Y2))))))))).
% 3.37/1.84 thf(fact_32_le__cases3, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y2:A,Z2:A]: ((((((ord_less_eq @ A) @ X2) @ Y2) => (~ ((((ord_less_eq @ A) @ Y2) @ Z2)))) => (((((ord_less_eq @ A) @ Y2) @ X2) => (~ ((((ord_less_eq @ A) @ X2) @ Z2)))) => (((((ord_less_eq @ A) @ X2) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ Y2)))) => (((((ord_less_eq @ A) @ Z2) @ Y2) => (~ ((((ord_less_eq @ A) @ Y2) @ X2)))) => (((((ord_less_eq @ A) @ Y2) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ X2)))) => (~ (((((ord_less_eq @ A) @ Z2) @ X2) => (~ ((((ord_less_eq @ A) @ X2) @ Y2))))))))))))))))).
% 3.37/1.84 thf(fact_33_order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B3:A,C3:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((ord_less_eq @ A) @ B3) @ C3) => (((ord_less_eq @ A) @ A3) @ C3))))))))).
% 3.37/1.84 thf(fact_34_le__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y2:A]: (((~ ((((ord_less_eq @ A) @ X2) @ Y2))) => (((ord_less_eq @ A) @ Y2) @ X2)))))))).
% 3.37/1.84 thf(fact_35_eq__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y2:A]: (((X2 = Y2) => (((ord_less_eq @ A) @ X2) @ Y2)))))))).
% 3.37/1.84 thf(fact_36_linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y2:A]: (((((ord_less_eq @ A) @ X2) @ Y2) | (((ord_less_eq @ A) @ Y2) @ X2)))))))).
% 3.37/1.84 thf(fact_37_antisym, axiom, (! [A:$tType]: (((order @ A) => (! [X2:A,Y2:A]: (((((ord_less_eq @ A) @ X2) @ Y2) => ((((ord_less_eq @ A) @ Y2) @ X2) => (X2 = Y2))))))))).
% 3.37/1.84 thf(fact_38_eq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y:A,Z:A]: ((Y = Z))) = (^ [X:A,Y3:A]: (((((ord_less_eq @ A) @ X) @ Y3) & (((ord_less_eq @ A) @ Y3) @ X))))))))).
% 3.37/1.84 thf(fact_39_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A3:A,B3:A,F:(A > B),C3:B]: (((((ord_less_eq @ A) @ A3) @ B3) => (((F @ B3) = C3) => ((! [X3:A,Y4:A]: (((((ord_less_eq @ A) @ X3) @ Y4) => (((ord_less_eq @ B) @ (F @ X3)) @ (F @ Y4))))) => (((ord_less_eq @ B) @ (F @ A3)) @ C3)))))))))).
% 3.37/1.84 thf(fact_40_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A3:A,F:(B > A),B3:B,C3:B]: (((A3 = (F @ B3)) => ((((ord_less_eq @ B) @ B3) @ C3) => ((! [X3:B,Y4:B]: (((((ord_less_eq @ B) @ X3) @ Y4) => (((ord_less_eq @ A) @ (F @ X3)) @ (F @ Y4))))) => (((ord_less_eq @ A) @ A3) @ (F @ C3))))))))))).
% 3.37/1.84 thf(fact_41_order__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A3:A,B3:A,F:(A > C),C3:C]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((ord_less_eq @ C) @ (F @ B3)) @ C3) => ((! [X3:A,Y4:A]: (((((ord_less_eq @ A) @ X3) @ Y4) => (((ord_less_eq @ C) @ (F @ X3)) @ (F @ Y4))))) => (((ord_less_eq @ C) @ (F @ A3)) @ C3)))))))))).
% 3.37/1.84 thf(fact_42_order__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A3:A,F:(B > A),B3:B,C3:B]: (((((ord_less_eq @ A) @ A3) @ (F @ B3)) => ((((ord_less_eq @ B) @ B3) @ C3) => ((! [X3:B,Y4:B]: (((((ord_less_eq @ B) @ X3) @ Y4) => (((ord_less_eq @ A) @ (F @ X3)) @ (F @ Y4))))) => (((ord_less_eq @ A) @ A3) @ (F @ C3))))))))))).
% 3.37/1.84 thf(fact_43_Greatest__equality, axiom, (! [A:$tType]: (((order @ A) => (! [P:(A > $o),X2:A]: (((P @ X2) => ((! [Y4:A]: (((P @ Y4) => (((ord_less_eq @ A) @ Y4) @ X2)))) => (((order_Greatest @ A) @ P) = X2))))))))).
% 3.37/1.84 thf(fact_44_mem__Collect__eq, axiom, (! [A:$tType,A3:A,P:(A > $o)]: (((((member @ A) @ A3) @ ((collect @ A) @ P)) = (P @ A3))))).
% 3.37/1.84 thf(fact_45_Collect__mem__eq, axiom, (! [A:$tType,A2:(set @ A)]: ((((collect @ A) @ (^ [X:A]: ((((member @ A) @ X) @ A2)))) = A2)))).
% 3.37/1.84 thf(fact_46_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P @ X3) = (Q @ X3)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 3.37/1.84 thf(fact_47_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X3:A]: (((F @ X3) = (G @ X3)))) => (F = G))))).
% 3.37/1.84 thf(fact_48_GreatestI2__order, axiom, (! [A:$tType]: (((order @ A) => (! [P:(A > $o),X2:A,Q:(A > $o)]: (((P @ X2) => ((! [Y4:A]: (((P @ Y4) => (((ord_less_eq @ A) @ Y4) @ X2)))) => ((! [X3:A]: (((P @ X3) => ((! [Y5:A]: (((P @ Y5) => (((ord_less_eq @ A) @ Y5) @ X3)))) => (Q @ X3))))) => (Q @ ((order_Greatest @ A) @ P))))))))))).
% 3.37/1.84 thf(fact_49_transfer_Otransfer__Let, axiom, (! [C:$tType,A:$tType,B:$tType]: (((comple187826305attice @ A) => (! [Alpha:(C > A),F:(B > C),F4:(B > A),X2:B]: (((! [X3:B]: ((((ord_less_eq @ A) @ (Alpha @ (F @ X3))) @ (F4 @ X3)))) => (((ord_less_eq @ A) @ (Alpha @ (F @ X2))) @ (F4 @ X2))))))))).
% 3.37/1.84 thf(fact_50_transfer_Otransfer__if, axiom, (! [C:$tType,A:$tType]: (((comple187826305attice @ A) => (! [B3:$o,Alpha:(C > A),S1:C,S12:A,S2:C,S22:A]: (((B3 => (((ord_less_eq @ A) @ (Alpha @ S1)) @ S12)) => (((~ (B3)) => (((ord_less_eq @ A) @ (Alpha @ S2)) @ S22)) => (((ord_less_eq @ A) @ (Alpha @ ((((if @ C) @ B3) @ S1) @ S2))) @ ((((if @ A) @ B3) @ S12) @ S22)))))))))).
% 3.37/1.84 thf(fact_51_le__rel__bool__arg__iff, axiom, (! [A:$tType]: (((ord @ A) => ((ord_less_eq @ ($o > A)) = (^ [X4:($o > A),Y6:($o > A)]: (((((ord_less_eq @ A) @ (X4 @ $false)) @ (Y6 @ $false)) & (((ord_less_eq @ A) @ (X4 @ $true)) @ (Y6 @ $true)))))))))).
% 3.37/1.84 thf(fact_52_verit__la__disequality, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B3:A]: (((A3 = B3) | ((~ ((((ord_less_eq @ A) @ A3) @ B3))) | (~ ((((ord_less_eq @ A) @ B3) @ A3))))))))))).
% 3.37/1.84 thf(fact_53_ord__eq__le__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A3:A,B3:A,C3:A,D2:A]: (((A3 = B3) => ((((ord_less_eq @ A) @ B3) @ C3) => ((C3 = D2) => (((ord_less_eq @ A) @ A3) @ D2)))))))))).
% 3.37/1.84 thf(fact_54_Id__refine, axiom, (! [A:$tType,S:(refine1665802226e_nres @ A)]: ((((ord_less_eq @ (refine1665802226e_nres @ A)) @ S) @ ((((refine1073749519nc_fun @ A) @ A) @ (id @ A)) @ S))))).
% 3.37/1.84 thf(fact_55_conc__trans__additional_I2_J, axiom, (! [A:$tType,B:$tType,A2:(refine1665802226e_nres @ A),B2:(refine1665802226e_nres @ A),R:(set @ ((product_prod @ A) @ B)),C2:(refine1665802226e_nres @ B)]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ A2) @ ((((refine1073749519nc_fun @ A) @ A) @ (id @ A)) @ B2)) => ((((ord_less_eq @ (refine1665802226e_nres @ A)) @ B2) @ ((((refine1073749519nc_fun @ A) @ B) @ R) @ C2)) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ A2) @ ((((refine1073749519nc_fun @ A) @ B) @ R) @ C2))))))).
% 3.37/1.84 thf(fact_56_conc__trans__additional_I3_J, axiom, (! [A:$tType,B:$tType,A2:(refine1665802226e_nres @ A),R:(set @ ((product_prod @ A) @ B)),B2:(refine1665802226e_nres @ B),C2:(refine1665802226e_nres @ B)]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ A2) @ ((((refine1073749519nc_fun @ A) @ B) @ R) @ B2)) => ((((ord_less_eq @ (refine1665802226e_nres @ B)) @ B2) @ ((((refine1073749519nc_fun @ B) @ B) @ (id @ B)) @ C2)) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ A2) @ ((((refine1073749519nc_fun @ A) @ B) @ R) @ C2))))))).
% 3.37/1.84 thf(fact_57_subset__Collect__conv, axiom, (! [A:$tType,S:(set @ A),P:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ S) @ ((collect @ A) @ P)) = (! [X:A]: (((((member @ A) @ X) @ S) => (P @ X)))))))).
% 3.37/1.84 thf(fact_58_conc__trans__additional_I6_J, axiom, (! [E3:$tType,A2:(refine1665802226e_nres @ E3),B2:(refine1665802226e_nres @ E3),C2:(refine1665802226e_nres @ E3)]: (((((ord_less_eq @ (refine1665802226e_nres @ E3)) @ A2) @ B2) => ((((ord_less_eq @ (refine1665802226e_nres @ E3)) @ B2) @ ((((refine1073749519nc_fun @ E3) @ E3) @ (id @ E3)) @ C2)) => (((ord_less_eq @ (refine1665802226e_nres @ E3)) @ A2) @ C2)))))).
% 3.37/1.84 thf(fact_59_conc__trans__additional_I5_J, axiom, (! [D:$tType,A2:(refine1665802226e_nres @ D),B2:(refine1665802226e_nres @ D),C2:(refine1665802226e_nres @ D)]: (((((ord_less_eq @ (refine1665802226e_nres @ D)) @ A2) @ ((((refine1073749519nc_fun @ D) @ D) @ (id @ D)) @ B2)) => ((((ord_less_eq @ (refine1665802226e_nres @ D)) @ B2) @ C2) => (((ord_less_eq @ (refine1665802226e_nres @ D)) @ A2) @ C2)))))).
% 3.37/1.84 thf(fact_60_conc__trans__additional_I4_J, axiom, (! [C:$tType,A2:(refine1665802226e_nres @ C),B2:(refine1665802226e_nres @ C),C2:(refine1665802226e_nres @ C)]: (((((ord_less_eq @ (refine1665802226e_nres @ C)) @ A2) @ ((((refine1073749519nc_fun @ C) @ C) @ (id @ C)) @ B2)) => ((((ord_less_eq @ (refine1665802226e_nres @ C)) @ B2) @ ((((refine1073749519nc_fun @ C) @ C) @ (id @ C)) @ C2)) => (((ord_less_eq @ (refine1665802226e_nres @ C)) @ A2) @ C2)))))).
% 3.37/1.84 thf(fact_61_subsetI, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((! [X3:A]: (((((member @ A) @ X3) @ A2) => (((member @ A) @ X3) @ B2)))) => (((ord_less_eq @ (set @ A)) @ A2) @ B2))))).
% 3.37/1.84 thf(fact_62_subset__antisym, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((ord_less_eq @ (set @ A)) @ B2) @ A2) => (A2 = B2)))))).
% 3.37/1.84 thf(fact_63_bijective__Id, axiom, (! [A:$tType]: ((((bijective @ A) @ A) @ (id @ A))))).
% 3.37/1.84 thf(fact_64_relprop__id__orient, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: (((R = (id @ A)) => ((id @ A) = R))))).
% 3.37/1.84 thf(fact_65_relprop__triggers_I2_J, axiom, (! [C:$tType,R:(set @ ((product_prod @ C) @ C))]: (((R = (id @ C)) => (R = (id @ C)))))).
% 3.37/1.84 thf(fact_66_relprop__triggers_I3_J, axiom, (! [D:$tType,R:(set @ ((product_prod @ D) @ D))]: (((R = (id @ D)) => ((id @ D) = R))))).
% 3.37/1.84 thf(fact_67_relprop__triggers_I6_J, axiom, (! [I:$tType,R:(set @ I),R2:(set @ I)]: (((((ord_less_eq @ (set @ I)) @ R) @ R2) => (((ord_less_eq @ (set @ I)) @ R) @ R2))))).
% 3.37/1.84 thf(fact_68_Collect__mono__iff, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)) = (! [X:A]: (((P @ X) => (Q @ X)))))))).
% 3.37/1.84 thf(fact_69_set__eq__subset, axiom, (! [A:$tType]: (((^ [Y:(set @ A),Z:(set @ A)]: ((Y = Z))) = (^ [A6:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B6) & (((ord_less_eq @ (set @ A)) @ B6) @ A6)))))))).
% 3.37/1.84 thf(fact_70_subset__trans, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((ord_less_eq @ (set @ A)) @ B2) @ C2) => (((ord_less_eq @ (set @ A)) @ A2) @ C2)))))).
% 3.37/1.84 thf(fact_71_Collect__mono, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P @ X3) => (Q @ X3)))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)))))).
% 3.37/1.84 thf(fact_72_subset__refl, axiom, (! [A:$tType,A2:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A2) @ A2)))).
% 3.37/1.84 thf(fact_73_subset__UNIV, axiom, (! [A:$tType,A2:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A2) @ (top_top @ (set @ A)))))).
% 3.37/1.84 thf(fact_74_subset__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A6:(set @ A),B6:(set @ A)]: ((! [T3:A]: (((((member @ A) @ T3) @ A6) => (((member @ A) @ T3) @ B6)))))))))).
% 3.37/1.84 thf(fact_75_equalityD2, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((A2 = B2) => (((ord_less_eq @ (set @ A)) @ B2) @ A2))))).
% 3.37/1.84 thf(fact_76_equalityD1, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((A2 = B2) => (((ord_less_eq @ (set @ A)) @ A2) @ B2))))).
% 3.37/1.84 thf(fact_77_subset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A6:(set @ A),B6:(set @ A)]: ((! [X:A]: (((((member @ A) @ X) @ A6) => (((member @ A) @ X) @ B6)))))))))).
% 3.37/1.84 thf(fact_78_equalityE, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((A2 = B2) => (~ (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (~ ((((ord_less_eq @ (set @ A)) @ B2) @ A2)))))))))).
% 3.37/1.84 thf(fact_79_subsetD, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C3:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((member @ A) @ C3) @ A2) => (((member @ A) @ C3) @ B2)))))).
% 3.37/1.84 thf(fact_80_in__mono, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),X2:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((member @ A) @ X2) @ A2) => (((member @ A) @ X2) @ B2)))))).
% 3.37/1.84 thf(fact_81_Id__SPEC__refine, axiom, (! [A:$tType,S:(refine1665802226e_nres @ A),Phi:(A > $o)]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ S) @ ((refine605929679le_RES @ A) @ ((collect @ A) @ Phi))) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ S) @ ((((refine1073749519nc_fun @ A) @ A) @ (id @ A)) @ ((refine605929679le_RES @ A) @ ((collect @ A) @ Phi)))))))).
% 3.37/1.84 thf(fact_82_abs__trans__additional_I2_J, axiom, (! [A:$tType,B:$tType,A2:(refine1665802226e_nres @ A),B2:(refine1665802226e_nres @ A),R:(set @ ((product_prod @ A) @ B)),C2:(refine1665802226e_nres @ B)]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ ((((refine81118332bs_fun @ A) @ A) @ (id @ A)) @ A2)) @ B2) => ((((ord_less_eq @ (refine1665802226e_nres @ B)) @ ((((refine81118332bs_fun @ A) @ B) @ R) @ B2)) @ C2) => (((ord_less_eq @ (refine1665802226e_nres @ B)) @ ((((refine81118332bs_fun @ A) @ B) @ R) @ A2)) @ C2)))))).
% 3.37/1.84 thf(fact_83_abs__trans__additional_I3_J, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ A) @ B)),A2:(refine1665802226e_nres @ A),B2:(refine1665802226e_nres @ B),C2:(refine1665802226e_nres @ B)]: (((((ord_less_eq @ (refine1665802226e_nres @ B)) @ ((((refine81118332bs_fun @ A) @ B) @ R) @ A2)) @ B2) => ((((ord_less_eq @ (refine1665802226e_nres @ B)) @ ((((refine81118332bs_fun @ B) @ B) @ (id @ B)) @ B2)) @ C2) => (((ord_less_eq @ (refine1665802226e_nres @ B)) @ ((((refine81118332bs_fun @ A) @ B) @ R) @ A2)) @ C2)))))).
% 3.37/1.84 thf(fact_84_abs__trans__additional_I4_J, axiom, (! [C:$tType,A2:(refine1665802226e_nres @ C),B2:(refine1665802226e_nres @ C),C2:(refine1665802226e_nres @ C)]: (((((ord_less_eq @ (refine1665802226e_nres @ C)) @ ((((refine81118332bs_fun @ C) @ C) @ (id @ C)) @ A2)) @ B2) => ((((ord_less_eq @ (refine1665802226e_nres @ C)) @ ((((refine81118332bs_fun @ C) @ C) @ (id @ C)) @ B2)) @ C2) => (((ord_less_eq @ (refine1665802226e_nres @ C)) @ A2) @ C2)))))).
% 3.37/1.84 thf(fact_85_abs__trans__additional_I5_J, axiom, (! [D:$tType,A2:(refine1665802226e_nres @ D),B2:(refine1665802226e_nres @ D),C2:(refine1665802226e_nres @ D)]: (((((ord_less_eq @ (refine1665802226e_nres @ D)) @ ((((refine81118332bs_fun @ D) @ D) @ (id @ D)) @ A2)) @ B2) => ((((ord_less_eq @ (refine1665802226e_nres @ D)) @ B2) @ C2) => (((ord_less_eq @ (refine1665802226e_nres @ D)) @ A2) @ C2)))))).
% 3.37/1.84 thf(fact_86_abs__trans__additional_I6_J, axiom, (! [E3:$tType,A2:(refine1665802226e_nres @ E3),B2:(refine1665802226e_nres @ E3),C2:(refine1665802226e_nres @ E3)]: (((((ord_less_eq @ (refine1665802226e_nres @ E3)) @ A2) @ B2) => ((((ord_less_eq @ (refine1665802226e_nres @ E3)) @ ((((refine81118332bs_fun @ E3) @ E3) @ (id @ E3)) @ B2)) @ C2) => (((ord_less_eq @ (refine1665802226e_nres @ E3)) @ A2) @ C2)))))).
% 3.37/1.84 thf(fact_87_rel__restrict__mono2, axiom, (! [A:$tType,R:(set @ A),S:(set @ A),A2:(set @ ((product_prod @ A) @ A))]: (((((ord_less_eq @ (set @ A)) @ R) @ S) => (((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ (((rel_restrict @ A) @ A2) @ S)) @ (((rel_restrict @ A) @ A2) @ R)))))).
% 3.37/1.84 thf(fact_88_UNIV__I, axiom, (! [A:$tType,X2:A]: ((((member @ A) @ X2) @ (top_top @ (set @ A)))))).
% 3.37/1.84 thf(fact_89_nres_Oinject, axiom, (! [A:$tType,X22:(set @ A),Y22:(set @ A)]: (((((refine605929679le_RES @ A) @ X22) = ((refine605929679le_RES @ A) @ Y22)) = (X22 = Y22))))).
% 3.37/1.84 thf(fact_90_nres__more__simps_I4_J, axiom, (! [A:$tType,X5:(set @ A),Y7:(set @ A)]: (((((refine605929679le_RES @ A) @ X5) = ((refine605929679le_RES @ A) @ Y7)) = (X5 = Y7))))).
% 3.37/1.84 thf(fact_91_abs__fun__simps_I1_J, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ B) @ A))]: ((((((refine81118332bs_fun @ B) @ A) @ R) @ (top_top @ (refine1665802226e_nres @ B))) = (top_top @ (refine1665802226e_nres @ A)))))).
% 3.37/1.84 thf(fact_92_relprop__UNIV__orient, axiom, (! [A:$tType,R:(set @ A)]: (((R = (top_top @ (set @ A))) => ((top_top @ (set @ A)) = R))))).
% 3.37/1.84 thf(fact_93_eq__UNIV__iff, axiom, (! [A:$tType,S:(set @ A)]: (((S = (top_top @ (set @ A))) = (! [X:A]: ((((member @ A) @ X) @ S))))))).
% 3.37/1.84 thf(fact_94_UNIV__witness, axiom, (! [A:$tType]: ((? [X3:A]: ((((member @ A) @ X3) @ (top_top @ (set @ A)))))))).
% 3.37/1.84 thf(fact_95_top__set__def, axiom, (! [A:$tType]: (((top_top @ (set @ A)) = ((collect @ A) @ (top_top @ (A > $o))))))).
% 3.37/1.84 thf(fact_96_UNIV__eq__I, axiom, (! [A:$tType,A2:(set @ A)]: (((! [X3:A]: ((((member @ A) @ X3) @ A2))) => ((top_top @ (set @ A)) = A2))))).
% 3.37/1.84 thf(fact_97_if__rule, axiom, (! [A:$tType,B3:$o,S12:(refine1665802226e_nres @ A),Phi:(A > $o),S22:(refine1665802226e_nres @ A)]: (((B3 => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ S12) @ ((refine605929679le_RES @ A) @ ((collect @ A) @ Phi)))) => (((~ (B3)) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ S22) @ ((refine605929679le_RES @ A) @ ((collect @ A) @ Phi)))) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ ((((if @ (refine1665802226e_nres @ A)) @ B3) @ S12) @ S22)) @ ((refine605929679le_RES @ A) @ ((collect @ A) @ Phi)))))))).
% 3.37/1.84 thf(fact_98_RES__rule, axiom, (! [A:$tType,S:(set @ A),Phi:(A > $o)]: (((! [X3:A]: (((((member @ A) @ X3) @ S) => (Phi @ X3)))) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ ((refine605929679le_RES @ A) @ S)) @ ((refine605929679le_RES @ A) @ ((collect @ A) @ Phi))))))).
% 3.37/1.84 thf(fact_99_SPEC__rule, axiom, (! [A:$tType,Phi:(A > $o),Phi2:(A > $o)]: (((! [X3:A]: (((Phi @ X3) => (Phi2 @ X3)))) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ ((refine605929679le_RES @ A) @ ((collect @ A) @ Phi))) @ ((refine605929679le_RES @ A) @ ((collect @ A) @ Phi2))))))).
% 3.37/1.84 thf(fact_100_SPEC__trans, axiom, (! [A:$tType,X2:(refine1665802226e_nres @ A),Y2:(refine1665802226e_nres @ A),Postcond:(A > $o)]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ X2) @ Y2) => ((((ord_less_eq @ (refine1665802226e_nres @ A)) @ Y2) @ ((refine605929679le_RES @ A) @ ((collect @ A) @ Postcond))) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ X2) @ ((refine605929679le_RES @ A) @ ((collect @ A) @ Postcond)))))))).
% 3.37/1.84 thf(fact_101_SPEC__cons__rule, axiom, (! [A:$tType,M2:(refine1665802226e_nres @ A),Phi:(A > $o),Psi:(A > $o)]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ M2) @ ((refine605929679le_RES @ A) @ ((collect @ A) @ Phi))) => ((! [X3:A]: (((Phi @ X3) => (Psi @ X3)))) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ M2) @ ((refine605929679le_RES @ A) @ ((collect @ A) @ Psi)))))))).
% 3.37/1.84 thf(fact_102_nres__inequalities_I1_J, axiom, (! [A:$tType,X5:(set @ A)]: (((top_top @ (refine1665802226e_nres @ A)) != ((refine605929679le_RES @ A) @ X5))))).
% 3.37/1.84 thf(fact_103_nres__cases, axiom, (! [A:$tType,M:(refine1665802226e_nres @ A)]: (((M != (top_top @ (refine1665802226e_nres @ A))) => (~ ((! [X6:(set @ A)]: ((M != ((refine605929679le_RES @ A) @ X6)))))))))).
% 3.37/1.84 thf(fact_104_rel__restrict__sub, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),A2:(set @ A)]: ((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ (((rel_restrict @ A) @ R) @ A2)) @ R)))).
% 3.37/1.84 thf(fact_105_rel__restrict__mono, axiom, (! [A:$tType,A2:(set @ ((product_prod @ A) @ A)),B2:(set @ ((product_prod @ A) @ A)),R:(set @ A)]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ A2) @ B2) => (((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ (((rel_restrict @ A) @ A2) @ R)) @ (((rel_restrict @ A) @ B2) @ R)))))).
% 3.37/1.84 thf(fact_106_abs__trans__additional_I1_J, axiom, (! [A:$tType,B:$tType,A2:(refine1665802226e_nres @ A),B2:(refine1665802226e_nres @ A),R:(set @ ((product_prod @ A) @ B)),C2:(refine1665802226e_nres @ B)]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ A2) @ B2) => ((((ord_less_eq @ (refine1665802226e_nres @ B)) @ ((((refine81118332bs_fun @ A) @ B) @ R) @ B2)) @ C2) => (((ord_less_eq @ (refine1665802226e_nres @ B)) @ ((((refine81118332bs_fun @ A) @ B) @ R) @ A2)) @ C2)))))).
% 3.37/1.84 thf(fact_107_abs__trans, axiom, (! [B:$tType,A:$tType,C:$tType,R:(set @ ((product_prod @ B) @ A)),C2:(refine1665802226e_nres @ B),B2:(refine1665802226e_nres @ A),R2:(set @ ((product_prod @ A) @ C)),A2:(refine1665802226e_nres @ C)]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ ((((refine81118332bs_fun @ B) @ A) @ R) @ C2)) @ B2) => ((((ord_less_eq @ (refine1665802226e_nres @ C)) @ ((((refine81118332bs_fun @ A) @ C) @ R2) @ B2)) @ A2) => (((ord_less_eq @ (refine1665802226e_nres @ C)) @ ((((refine81118332bs_fun @ A) @ C) @ R2) @ ((((refine81118332bs_fun @ B) @ A) @ R) @ C2))) @ A2)))))).
% 3.37/1.84 thf(fact_108_nres__order__simps_I5_J, axiom, (! [E3:$tType,X5:(set @ E3),Y7:(set @ E3)]: (((((ord_less_eq @ (refine1665802226e_nres @ E3)) @ ((refine605929679le_RES @ E3) @ X5)) @ ((refine605929679le_RES @ E3) @ Y7)) = (((ord_less_eq @ (set @ E3)) @ X5) @ Y7))))).
% 3.37/1.84 thf(fact_109_less__eq__nres_Osimps_I2_J, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ ((refine605929679le_RES @ A) @ A3)) @ ((refine605929679le_RES @ A) @ B3)) = (((ord_less_eq @ (set @ A)) @ A3) @ B3))))).
% 3.37/1.84 thf(fact_110_iso__tuple__UNIV__I, axiom, (! [A:$tType,X2:A]: ((((member @ A) @ X2) @ (top_top @ (set @ A)))))).
% 3.37/1.84 thf(fact_111_nf__inres__SPEC, axiom, (! [A:$tType,Phi:(A > $o),X2:A]: (((((refine406925620_inres @ A) @ ((refine605929679le_RES @ A) @ ((collect @ A) @ Phi))) @ X2) = (Phi @ X2))))).
% 3.37/1.84 thf(fact_112_nf__inres__RES, axiom, (! [A:$tType,X5:(set @ A),X2:A]: (((((refine406925620_inres @ A) @ ((refine605929679le_RES @ A) @ X5)) @ X2) = (((member @ A) @ X2) @ X5))))).
% 3.37/1.84 thf(fact_113_abs__fun__simps_I3_J, axiom, (! [B:$tType,A:$tType,X5:(set @ B),R:(set @ ((product_prod @ B) @ A))]: (((~ ((((ord_less_eq @ (set @ B)) @ X5) @ (((domain @ B) @ A) @ R)))) => (((((refine81118332bs_fun @ B) @ A) @ R) @ ((refine605929679le_RES @ B) @ X5)) = (top_top @ (refine1665802226e_nres @ A))))))).
% 3.37/1.84 thf(fact_114_the__RES_Osimps, axiom, (! [A:$tType,X5:(set @ A)]: ((((refine1672542526he_RES @ A) @ ((refine605929679le_RES @ A) @ X5)) = X5)))).
% 3.37/1.84 thf(fact_115_nres__order__simps_I21_J, axiom, (! [X7:$tType,X2:X7,Y7:(set @ X7)]: (((((ord_less_eq @ (refine1665802226e_nres @ X7)) @ ((refine1687780735RETURN @ X7) @ X2)) @ ((refine605929679le_RES @ X7) @ Y7)) = (((member @ X7) @ X2) @ Y7))))).
% 3.37/1.84 thf(fact_116_nres__more__simps_I6_J, axiom, (! [A:$tType,X2:A,Y2:A]: (((((refine1687780735RETURN @ A) @ X2) = ((refine1687780735RETURN @ A) @ Y2)) = (X2 = Y2))))).
% 3.37/1.84 thf(fact_117_top1I, axiom, (! [A:$tType,X2:A]: (((top_top @ (A > $o)) @ X2)))).
% 3.37/1.84 thf(fact_118_nres__order__simps_I20_J, axiom, (! [W:$tType,X2:W,Y2:W]: (((((ord_less_eq @ (refine1665802226e_nres @ W)) @ ((refine1687780735RETURN @ W) @ X2)) @ ((refine1687780735RETURN @ W) @ Y2)) = (X2 = Y2))))).
% 3.37/1.84 thf(fact_119_nres__inequalities_I3_J, axiom, (! [C:$tType,X2:C]: (((top_top @ (refine1665802226e_nres @ C)) != ((refine1687780735RETURN @ C) @ X2))))).
% 3.37/1.84 thf(fact_120_RETURN__to__SPEC__rule, axiom, (! [A:$tType,M2:(refine1665802226e_nres @ A),V:A]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ M2) @ ((refine605929679le_RES @ A) @ ((collect @ A) @ ((^ [Y:A,Z:A]: ((Y = Z))) @ V)))) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ M2) @ ((refine1687780735RETURN @ A) @ V)))))).
% 3.37/1.84 thf(fact_121_RETURN__rule, axiom, (! [A:$tType,Phi:(A > $o),X2:A]: (((Phi @ X2) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ ((refine1687780735RETURN @ A) @ X2)) @ ((refine605929679le_RES @ A) @ ((collect @ A) @ Phi))))))).
% 3.37/1.84 thf(fact_122_Domain__Id, axiom, (! [A:$tType]: (((((domain @ A) @ A) @ (id @ A)) = (top_top @ (set @ A)))))).
% 3.37/1.84 thf(fact_123_Domain__mono, axiom, (! [B:$tType,A:$tType,R3:(set @ ((product_prod @ A) @ B)),S3:(set @ ((product_prod @ A) @ B))]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R3) @ S3) => (((ord_less_eq @ (set @ A)) @ (((domain @ A) @ B) @ R3)) @ (((domain @ A) @ B) @ S3)))))).
% 3.37/1.84 thf(fact_124_top__empty__eq, axiom, (! [A:$tType]: (((top_top @ (A > $o)) = (^ [X:A]: ((((member @ A) @ X) @ (top_top @ (set @ A))))))))).
% 3.37/1.84 thf(fact_125_abs__fun__simps_I2_J, axiom, (! [A:$tType,B:$tType,X5:(set @ B),R:(set @ ((product_prod @ B) @ A))]: (((((ord_less_eq @ (set @ B)) @ X5) @ (((domain @ B) @ A) @ R)) => (((((refine81118332bs_fun @ B) @ A) @ R) @ ((refine605929679le_RES @ B) @ X5)) = ((refine605929679le_RES @ A) @ ((((image @ B) @ A) @ R) @ X5))))))).
% 3.37/1.84 thf(fact_126_Image__Id, axiom, (! [A:$tType,A2:(set @ A)]: ((((((image @ A) @ A) @ (id @ A)) @ A2) = A2)))).
% 3.37/1.84 thf(fact_127_Image__mono, axiom, (! [B:$tType,A:$tType,R4:(set @ ((product_prod @ A) @ B)),R3:(set @ ((product_prod @ A) @ B)),A7:(set @ A),A2:(set @ A)]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R4) @ R3) => ((((ord_less_eq @ (set @ A)) @ A7) @ A2) => (((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ R4) @ A7)) @ ((((image @ A) @ B) @ R3) @ A2))))))).
% 3.37/1.84 thf(fact_128_top__conj_I1_J, axiom, (! [A:$tType,X2:A,P:$o]: (((((top_top @ (A > $o)) @ X2) & P) = P)))).
% 3.37/1.84 thf(fact_129_top__conj_I2_J, axiom, (! [A:$tType,P:$o,X2:A]: (((P & ((top_top @ (A > $o)) @ X2)) = P)))).
% 3.37/1.84 thf(fact_130_RETURN__SPEC__refine, axiom, (! [B:$tType,A:$tType,X2:B,R:(set @ ((product_prod @ B) @ A)),Phi:(A > $o)]: (((? [X8:A]: (((((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ X2) @ X8)) @ R) & (Phi @ X8)))) => (((ord_less_eq @ (refine1665802226e_nres @ B)) @ ((refine1687780735RETURN @ B) @ X2)) @ ((((refine1073749519nc_fun @ B) @ A) @ R) @ ((refine605929679le_RES @ A) @ ((collect @ A) @ Phi)))))))).
% 3.37/1.84 thf(fact_131_the__RES__inv, axiom, (! [A:$tType,M2:(refine1665802226e_nres @ A)]: ((((refine1102455758nofail @ A) @ M2) => (((refine605929679le_RES @ A) @ ((refine1672542526he_RES @ A) @ M2)) = M2))))).
% 3.37/1.84 thf(fact_132_pair__in__Id__conv, axiom, (! [A:$tType,A3:A,B3:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B3)) @ (id @ A)) = (A3 = B3))))).
% 3.37/1.84 thf(fact_133_IdI, axiom, (! [A:$tType,A3:A]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ A3)) @ (id @ A))))).
% 3.37/1.84 thf(fact_134_nofail__simps_I2_J, axiom, (! [B:$tType,X5:(set @ B)]: (((refine1102455758nofail @ B) @ ((refine605929679le_RES @ B) @ X5))))).
% 3.37/1.84 thf(fact_135_nofail__simps_I1_J, axiom, (! [A:$tType]: ((~ (((refine1102455758nofail @ A) @ (top_top @ (refine1665802226e_nres @ A)))))))).
% 3.37/1.84 thf(fact_136_nofail__simps_I3_J, axiom, (! [C:$tType,X2:C]: (((refine1102455758nofail @ C) @ ((refine1687780735RETURN @ C) @ X2))))).
% 3.37/1.84 thf(fact_137_DomainE, axiom, (! [B:$tType,A:$tType,A3:A,R3:(set @ ((product_prod @ A) @ B))]: (((((member @ A) @ A3) @ (((domain @ A) @ B) @ R3)) => (~ ((! [B5:B]: ((~ ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B5)) @ R3))))))))))).
% 3.37/1.84 thf(fact_138_Domain__iff, axiom, (! [A:$tType,B:$tType,A3:A,R3:(set @ ((product_prod @ A) @ B))]: (((((member @ A) @ A3) @ (((domain @ A) @ B) @ R3)) = (? [Y3:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ Y3)) @ R3))))))).
% 3.37/1.84 thf(fact_139_Domain_Ocases, axiom, (! [B:$tType,A:$tType,A3:A,R3:(set @ ((product_prod @ A) @ B))]: (((((member @ A) @ A3) @ (((domain @ A) @ B) @ R3)) => (~ ((! [B5:B]: ((~ ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B5)) @ R3))))))))))).
% 3.37/1.84 thf(fact_140_Domain_Osimps, axiom, (! [B:$tType,A:$tType,A3:A,R3:(set @ ((product_prod @ A) @ B))]: (((((member @ A) @ A3) @ (((domain @ A) @ B) @ R3)) = (? [A4:A,B4:B]: (((A3 = A4) & (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A4) @ B4)) @ R3)))))))).
% 3.37/1.84 thf(fact_141_Domain_ODomainI, axiom, (! [B:$tType,A:$tType,A3:A,B3:B,R3:(set @ ((product_prod @ A) @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B3)) @ R3) => (((member @ A) @ A3) @ (((domain @ A) @ B) @ R3)))))).
% 3.37/1.84 thf(fact_142_Domain_Oinducts, axiom, (! [B:$tType,A:$tType,X2:A,R3:(set @ ((product_prod @ A) @ B)),P:(A > $o)]: (((((member @ A) @ X2) @ (((domain @ A) @ B) @ R3)) => ((! [A5:A,B5:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A5) @ B5)) @ R3) => (P @ A5)))) => (P @ X2)))))).
% 3.37/1.84 thf(fact_143_subrelI, axiom, (! [B:$tType,A:$tType,R3:(set @ ((product_prod @ A) @ B)),S3:(set @ ((product_prod @ A) @ B))]: (((! [X3:A,Y4:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y4)) @ R3) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y4)) @ S3)))) => (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R3) @ S3))))).
% 3.37/1.84 thf(fact_144_IdE, axiom, (! [A:$tType,P2:((product_prod @ A) @ A)]: (((((member @ ((product_prod @ A) @ A)) @ P2) @ (id @ A)) => (~ ((! [X3:A]: ((P2 != ((((product_Pair @ A) @ A) @ X3) @ X3)))))))))).
% 3.37/1.84 thf(fact_145_rel__restrict__notR_I2_J, axiom, (! [A:$tType,X2:A,Y2:A,A2:(set @ ((product_prod @ A) @ A)),R:(set @ A)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Y2)) @ (((rel_restrict @ A) @ A2) @ R)) => (~ ((((member @ A) @ Y2) @ R))))))).
% 3.37/1.84 thf(fact_146_rel__restrict__notR_I1_J, axiom, (! [A:$tType,X2:A,Y2:A,A2:(set @ ((product_prod @ A) @ A)),R:(set @ A)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Y2)) @ (((rel_restrict @ A) @ A2) @ R)) => (~ ((((member @ A) @ X2) @ R))))))).
% 3.37/1.84 thf(fact_147_rel__restrictI, axiom, (! [A:$tType,X2:A,R:(set @ A),Y2:A,E4:(set @ ((product_prod @ A) @ A))]: (((~ ((((member @ A) @ X2) @ R))) => ((~ ((((member @ A) @ Y2) @ R))) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Y2)) @ E4) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Y2)) @ (((rel_restrict @ A) @ E4) @ R)))))))).
% 3.37/1.84 thf(fact_148_rel__restrict__lift, axiom, (! [A:$tType,X2:A,Y2:A,E4:(set @ ((product_prod @ A) @ A)),R:(set @ A)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Y2)) @ (((rel_restrict @ A) @ E4) @ R)) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Y2)) @ E4))))).
% 3.37/1.84 thf(fact_149_pairself_Oinduct, axiom, (! [B:$tType,A:$tType,P:((A > B) > (((product_prod @ A) @ A) > $o)),A0:(A > B),A1:((product_prod @ A) @ A)]: (((! [F5:(A > B),A5:A,B5:A]: (((P @ F5) @ ((((product_Pair @ A) @ A) @ A5) @ B5)))) => ((P @ A0) @ A1))))).
% 3.37/1.84 thf(fact_150_pairself_Ocases, axiom, (! [B:$tType,A:$tType,X2:((product_prod @ (A > B)) @ ((product_prod @ A) @ A))]: ((~ ((! [F5:(A > B),A5:A,B5:A]: ((X2 != ((((product_Pair @ (A > B)) @ ((product_prod @ A) @ A)) @ F5) @ ((((product_Pair @ A) @ A) @ A5) @ B5)))))))))).
% 3.37/1.84 thf(fact_151_bex2I, axiom, (! [A:$tType,B:$tType,A3:A,B3:B,S:(set @ ((product_prod @ A) @ B)),P:(A > (B > $o))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B3)) @ S) => (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B3)) @ S) => ((P @ A3) @ B3)) => (? [A5:A,B5:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A5) @ B5)) @ S) & ((P @ A5) @ B5))))))))).
% 3.37/1.84 thf(fact_152_rel__cong, axiom, (! [A:$tType,B:$tType,F:(A > B),G:(A > B),X2:A,Y2:A]: (((((member @ ((product_prod @ (A > B)) @ (A > B))) @ ((((product_Pair @ (A > B)) @ (A > B)) @ F) @ G)) @ (id @ (A > B))) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Y2)) @ (id @ A)) => (((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ (F @ X2)) @ (G @ Y2))) @ (id @ B))))))).
% 3.37/1.84 thf(fact_153_rel__fun__cong, axiom, (! [A:$tType,B:$tType,F:(A > B),G:(A > B),X2:A]: (((((member @ ((product_prod @ (A > B)) @ (A > B))) @ ((((product_Pair @ (A > B)) @ (A > B)) @ F) @ G)) @ (id @ (A > B))) => (((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ (F @ X2)) @ (G @ X2))) @ (id @ B)))))).
% 3.37/1.84 thf(fact_154_nofail__antimono__fun, axiom, (! [B:$tType,A:$tType,F:(A > (refine1665802226e_nres @ B)),G:(A > (refine1665802226e_nres @ B)),X2:A]: (((((ord_less_eq @ (A > (refine1665802226e_nres @ B))) @ F) @ G) => (((refine1102455758nofail @ B) @ (G @ X2)) => ((refine1102455758nofail @ B) @ (F @ X2))))))).
% 3.37/1.84 thf(fact_155_pw__conc__nofail, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ A) @ B)),S:(refine1665802226e_nres @ B)]: ((((refine1102455758nofail @ A) @ ((((refine1073749519nc_fun @ A) @ B) @ R) @ S)) = ((refine1102455758nofail @ B) @ S))))).
% 3.37/1.84 thf(fact_156_intro__nofail_I2_J, axiom, (! [A:$tType,S:(refine1665802226e_nres @ A)]: ((((top_top @ (refine1665802226e_nres @ A)) != S) = ((refine1102455758nofail @ A) @ S))))).
% 3.37/1.84 thf(fact_157_nofail__def, axiom, (! [A:$tType]: (((refine1102455758nofail @ A) = (^ [S4:(refine1665802226e_nres @ A)]: ((S4 != (top_top @ (refine1665802226e_nres @ A))))))))).
% 3.37/1.84 thf(fact_158_not__nofail__iff, axiom, (! [A:$tType,S:(refine1665802226e_nres @ A)]: (((~ (((refine1102455758nofail @ A) @ S))) = (S = (top_top @ (refine1665802226e_nres @ A))))))).
% 3.37/1.84 thf(fact_159_rel__arg__cong, axiom, (! [A:$tType,B:$tType,X2:A,Y2:A,F:(A > B)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Y2)) @ (id @ A)) => (((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ (F @ X2)) @ (F @ Y2))) @ (id @ B)))))).
% 3.37/1.84 thf(fact_160_pwD1, axiom, (! [A:$tType,S:(refine1665802226e_nres @ A),S5:(refine1665802226e_nres @ A)]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ S) @ S5) => (((refine1102455758nofail @ A) @ S5) => ((refine1102455758nofail @ A) @ S)))))).
% 3.37/1.84 thf(fact_161_le__nofailI, axiom, (! [A:$tType,M3:(refine1665802226e_nres @ A),M:(refine1665802226e_nres @ A)]: (((((refine1102455758nofail @ A) @ M3) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ M) @ M3)) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ M) @ M3))))).
% 3.37/1.84 thf(fact_162_nofail__RES__conv, axiom, (! [A:$tType]: (((refine1102455758nofail @ A) = (^ [M4:(refine1665802226e_nres @ A)]: ((? [M5:(set @ A)]: ((M4 = ((refine605929679le_RES @ A) @ M5)))))))))).
% 3.37/1.84 thf(fact_163_bijective__def, axiom, (! [B:$tType,A:$tType]: ((((bijective @ A) @ B) = (^ [R5:(set @ ((product_prod @ A) @ B))]: (((! [X:A,Y3:B,Z3:B]: ((((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y3)) @ R5) & (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Z3)) @ R5)) => (Y3 = Z3)))) & (! [X:A,Y3:A,Z3:B]: ((((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Z3)) @ R5) & (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ Y3) @ Z3)) @ R5)) => (X = Y3))))))))))).
% 3.37/1.84 thf(fact_164_RES__refine, axiom, (! [A:$tType,B:$tType,S:(set @ A),S5:(set @ B),R:(set @ ((product_prod @ A) @ B))]: (((! [S6:A]: (((((member @ A) @ S6) @ S) => (? [X9:B]: (((((member @ B) @ X9) @ S5) & (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ S6) @ X9)) @ R))))))) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ ((refine605929679le_RES @ A) @ S)) @ ((((refine1073749519nc_fun @ A) @ B) @ R) @ ((refine605929679le_RES @ B) @ S5))))))).
% 3.37/1.84 thf(fact_165_RETURN__refine, axiom, (! [A:$tType,B:$tType,X2:A,X10:B,R:(set @ ((product_prod @ A) @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X2) @ X10)) @ R) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ ((refine1687780735RETURN @ A) @ X2)) @ ((((refine1073749519nc_fun @ A) @ B) @ R) @ ((refine1687780735RETURN @ B) @ X10))))))).
% 3.37/1.84 thf(fact_166_for__in__RI, axiom, (! [B:$tType,A:$tType,X2:A,R:(set @ ((product_prod @ A) @ B))]: (((((member @ A) @ X2) @ (((domain @ A) @ B) @ R)) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X2) @ ((((fun_of_rel @ A) @ B) @ R) @ X2))) @ R))))).
% 3.37/1.84 thf(fact_167_BNF__Greatest__Fixpoint_OIdD, axiom, (! [A:$tType,A3:A,B3:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B3)) @ (id @ A)) => (A3 = B3))))).
% 3.37/1.84 thf(fact_168_E__closed__restr__reach__cases, axiom, (! [A:$tType,U:A,V:A,E4:(set @ ((product_prod @ A) @ A)),R:(set @ A)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ U) @ V)) @ ((transitive_rtrancl @ A) @ E4)) => ((((ord_less_eq @ (set @ A)) @ ((((image @ A) @ A) @ E4) @ R)) @ R) => ((~ ((((member @ A) @ V) @ R))) => (~ (((~ ((((member @ A) @ U) @ R))) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ U) @ V)) @ ((transitive_rtrancl @ A) @ (((rel_restrict @ A) @ E4) @ R)))))))))))))).
% 3.37/1.84 thf(fact_169_relChain__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (((bNF_Ca1785829860lChain @ A) @ B) = (^ [R6:(set @ ((product_prod @ A) @ A)),As:(A > B)]: ((! [I2:A,J:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ I2) @ J)) @ R6) => (((ord_less_eq @ B) @ (As @ I2)) @ (As @ J)))))))))))).
% 3.37/1.84 thf(fact_170_converse__rtranclE_H, axiom, (! [A:$tType,U:A,V:A,R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ U) @ V)) @ ((transitive_rtrancl @ A) @ R)) => ((U != V) => (~ ((! [Vh:A]: (((U != Vh) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ U) @ Vh)) @ R) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Vh) @ V)) @ ((transitive_rtrancl @ A) @ R))))))))))))))).
% 3.37/1.84 thf(fact_171_rtrancl__mono__rightI, axiom, (! [A:$tType,S:(set @ ((product_prod @ A) @ A)),S5:(set @ ((product_prod @ A) @ A))]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ S) @ S5) => (((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ S) @ ((transitive_rtrancl @ A) @ S5)))))).
% 3.37/1.84 thf(fact_172_rtrancl__mono__mp, axiom, (! [A:$tType,U2:(set @ ((product_prod @ A) @ A)),V2:(set @ ((product_prod @ A) @ A)),X2:((product_prod @ A) @ A)]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ U2) @ V2) => ((((member @ ((product_prod @ A) @ A)) @ X2) @ ((transitive_rtrancl @ A) @ U2)) => (((member @ ((product_prod @ A) @ A)) @ X2) @ ((transitive_rtrancl @ A) @ V2))))))).
% 3.37/1.84 thf(fact_173_r__le__rtrancl, axiom, (! [A:$tType,S:(set @ ((product_prod @ A) @ A))]: ((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ S) @ ((transitive_rtrancl @ A) @ S))))).
% 3.37/1.84 thf(fact_174_rtrancl__image__advance__rtrancl, axiom, (! [A:$tType,Q2:A,R:(set @ ((product_prod @ A) @ A)),Q0:(set @ A),X2:A]: (((((member @ A) @ Q2) @ ((((image @ A) @ A) @ ((transitive_rtrancl @ A) @ R)) @ Q0)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Q2) @ X2)) @ ((transitive_rtrancl @ A) @ R)) => (((member @ A) @ X2) @ ((((image @ A) @ A) @ ((transitive_rtrancl @ A) @ R)) @ Q0))))))).
% 3.37/1.84 thf(fact_175_rtrancl__image__advance, axiom, (! [A:$tType,Q2:A,R:(set @ ((product_prod @ A) @ A)),Q0:(set @ A),X2:A]: (((((member @ A) @ Q2) @ ((((image @ A) @ A) @ ((transitive_rtrancl @ A) @ R)) @ Q0)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Q2) @ X2)) @ R) => (((member @ A) @ X2) @ ((((image @ A) @ A) @ ((transitive_rtrancl @ A) @ R)) @ Q0))))))).
% 3.37/1.84 thf(fact_176_rtrancl__reachable__induct, axiom, (! [A:$tType,I3:(set @ A),INV:(set @ A),E4:(set @ ((product_prod @ A) @ A))]: (((((ord_less_eq @ (set @ A)) @ I3) @ INV) => ((((ord_less_eq @ (set @ A)) @ ((((image @ A) @ A) @ E4) @ INV)) @ INV) => (((ord_less_eq @ (set @ A)) @ ((((image @ A) @ A) @ ((transitive_rtrancl @ A) @ E4)) @ I3)) @ INV)))))).
% 3.37/1.84 thf(fact_177_rtrancl__image__unfold__right, axiom, (! [A:$tType,E4:(set @ ((product_prod @ A) @ A)),V2:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ ((((image @ A) @ A) @ E4) @ ((((image @ A) @ A) @ ((transitive_rtrancl @ A) @ E4)) @ V2))) @ ((((image @ A) @ A) @ ((transitive_rtrancl @ A) @ E4)) @ V2))))).
% 3.37/1.84 thf(fact_178_reachable__mono, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),R2:(set @ ((product_prod @ A) @ A)),X5:(set @ A),X11:(set @ A)]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ R) @ R2) => ((((ord_less_eq @ (set @ A)) @ X5) @ X11) => (((ord_less_eq @ (set @ A)) @ ((((image @ A) @ A) @ ((transitive_rtrancl @ A) @ R)) @ X5)) @ ((((image @ A) @ A) @ ((transitive_rtrancl @ A) @ R2)) @ X11))))))).
% 3.37/1.84 thf(fact_179_Domain__rtrancl, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: (((((domain @ A) @ A) @ ((transitive_rtrancl @ A) @ R)) = (top_top @ (set @ A)))))).
% 3.37/1.84 thf(fact_180_Not__Domain__rtrancl, axiom, (! [A:$tType,X2:A,R:(set @ ((product_prod @ A) @ A)),Y2:A]: (((~ ((((member @ A) @ X2) @ (((domain @ A) @ A) @ R)))) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Y2)) @ ((transitive_rtrancl @ A) @ R)) = (X2 = Y2)))))).
% 3.37/1.84 thf(fact_181_Image__closed__trancl, axiom, (! [A:$tType,R3:(set @ ((product_prod @ A) @ A)),X5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((((image @ A) @ A) @ R3) @ X5)) @ X5) => (((((image @ A) @ A) @ ((transitive_rtrancl @ A) @ R3)) @ X5) = X5))))).
% 3.37/1.84 thf(fact_182_rtrancl__subset__rtrancl, axiom, (! [A:$tType,R3:(set @ ((product_prod @ A) @ A)),S3:(set @ ((product_prod @ A) @ A))]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ R3) @ ((transitive_rtrancl @ A) @ S3)) => (((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ ((transitive_rtrancl @ A) @ R3)) @ ((transitive_rtrancl @ A) @ S3)))))).
% 3.37/1.84 thf(fact_183_rtrancl__subset, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),S:(set @ ((product_prod @ A) @ A))]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ R) @ S) => ((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ S) @ ((transitive_rtrancl @ A) @ R)) => (((transitive_rtrancl @ A) @ S) = ((transitive_rtrancl @ A) @ R))))))).
% 3.37/1.84 thf(fact_184_rtrancl__mono, axiom, (! [A:$tType,R3:(set @ ((product_prod @ A) @ A)),S3:(set @ ((product_prod @ A) @ A))]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ R3) @ S3) => (((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ ((transitive_rtrancl @ A) @ R3)) @ ((transitive_rtrancl @ A) @ S3)))))).
% 3.37/1.84 thf(fact_185_rel__restrict__tranclI, axiom, (! [A:$tType,X2:A,Y2:A,E4:(set @ ((product_prod @ A) @ A)),R:(set @ A)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Y2)) @ ((transitive_trancl @ A) @ E4)) => ((~ ((((member @ A) @ X2) @ R))) => ((~ ((((member @ A) @ Y2) @ R))) => ((((ord_less_eq @ (set @ A)) @ ((((image @ A) @ A) @ E4) @ R)) @ R) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Y2)) @ ((transitive_trancl @ A) @ (((rel_restrict @ A) @ E4) @ R)))))))))).
% 3.37/1.84 thf(fact_186_uncurry__apply, axiom, (! [B:$tType,A:$tType,C:$tType,F:(B > (C > A)),A3:B,B3:C]: (((((((uncurry @ B) @ C) @ A) @ F) @ ((((product_Pair @ B) @ C) @ A3) @ B3)) = ((F @ A3) @ B3))))).
% 3.37/1.84 thf(fact_187_pw__abs__nofail, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ B) @ A)),M:(refine1665802226e_nres @ B)]: ((((refine1102455758nofail @ A) @ ((((refine81118332bs_fun @ B) @ A) @ R) @ M)) = (((refine1102455758nofail @ B) @ M) & (! [X:B]: (((((refine1315500908_inres @ B) @ M) @ X) => (((member @ B) @ X) @ (((domain @ B) @ A) @ R)))))))))).
% 3.37/1.84 thf(fact_188_inres__simps_I2_J, axiom, (! [B:$tType,X5:(set @ B)]: ((((refine1315500908_inres @ B) @ ((refine605929679le_RES @ B) @ X5)) = (^ [X:B]: ((((member @ B) @ X) @ X5))))))).
% 3.37/1.84 thf(fact_189_trancl__domain, axiom, (! [A:$tType,R3:(set @ ((product_prod @ A) @ A))]: (((((domain @ A) @ A) @ ((transitive_trancl @ A) @ R3)) = (((domain @ A) @ A) @ R3))))).
% 3.37/1.84 thf(fact_190_inres__simps_I1_J, axiom, (! [A:$tType]: ((((refine1315500908_inres @ A) @ (top_top @ (refine1665802226e_nres @ A))) = (^ [Uu:A]: ($true)))))).
% 3.37/1.84 thf(fact_191_inres__simps_I3_J, axiom, (! [C:$tType,X2:C]: ((((refine1315500908_inres @ C) @ ((refine1687780735RETURN @ C) @ X2)) = ((^ [Y:C,Z:C]: ((Y = Z))) @ X2))))).
% 3.37/1.84 thf(fact_192_trancl__mono, axiom, (! [A:$tType,P2:((product_prod @ A) @ A),R3:(set @ ((product_prod @ A) @ A)),S3:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ P2) @ ((transitive_trancl @ A) @ R3)) => ((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ R3) @ S3) => (((member @ ((product_prod @ A) @ A)) @ P2) @ ((transitive_trancl @ A) @ S3))))))).
% 3.37/1.84 thf(fact_193_trancl__Image__unfold__left, axiom, (! [A:$tType,E4:(set @ ((product_prod @ A) @ A)),S:(set @ A)]: ((((((image @ A) @ A) @ ((transitive_trancl @ A) @ E4)) @ S) = ((((image @ A) @ A) @ ((transitive_rtrancl @ A) @ E4)) @ ((((image @ A) @ A) @ E4) @ S)))))).
% 3.37/1.84 thf(fact_194_trancl__Image__unfold__right, axiom, (! [A:$tType,E4:(set @ ((product_prod @ A) @ A)),S:(set @ A)]: ((((((image @ A) @ A) @ ((transitive_trancl @ A) @ E4)) @ S) = ((((image @ A) @ A) @ E4) @ ((((image @ A) @ A) @ ((transitive_rtrancl @ A) @ E4)) @ S)))))).
% 3.37/1.84 thf(fact_195_rel__restrict__trancl__notR_I2_J, axiom, (! [A:$tType,V:A,W2:A,E4:(set @ ((product_prod @ A) @ A)),R:(set @ A)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ V) @ W2)) @ ((transitive_trancl @ A) @ (((rel_restrict @ A) @ E4) @ R))) => (~ ((((member @ A) @ W2) @ R))))))).
% 3.37/1.84 thf(fact_196_rel__restrict__trancl__notR_I1_J, axiom, (! [A:$tType,V:A,W2:A,E4:(set @ ((product_prod @ A) @ A)),R:(set @ A)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ V) @ W2)) @ ((transitive_trancl @ A) @ (((rel_restrict @ A) @ E4) @ R))) => (~ ((((member @ A) @ V) @ R))))))).
% 3.37/1.84 thf(fact_197_rel__restrict__trancl__mem, axiom, (! [A:$tType,A3:A,B3:A,A2:(set @ ((product_prod @ A) @ A)),R:(set @ A)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B3)) @ ((transitive_trancl @ A) @ (((rel_restrict @ A) @ A2) @ R))) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B3)) @ (((rel_restrict @ A) @ ((transitive_trancl @ A) @ A2)) @ R)))))).
% 3.37/1.84 thf(fact_198_rel__restrict__trancl__sub, axiom, (! [A:$tType,A2:(set @ ((product_prod @ A) @ A)),R:(set @ A)]: ((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ ((transitive_trancl @ A) @ (((rel_restrict @ A) @ A2) @ R))) @ (((rel_restrict @ A) @ ((transitive_trancl @ A) @ A2)) @ R))))).
% 3.37/1.84 thf(fact_199_pw__eqI, axiom, (! [A:$tType,S:(refine1665802226e_nres @ A),S5:(refine1665802226e_nres @ A)]: (((((refine1102455758nofail @ A) @ S) = ((refine1102455758nofail @ A) @ S5)) => ((! [X3:A]: (((((refine1315500908_inres @ A) @ S) @ X3) = (((refine1315500908_inres @ A) @ S5) @ X3)))) => (S = S5)))))).
% 3.37/1.84 thf(fact_200_pw__eq__iff, axiom, (! [A:$tType]: (((^ [Y:(refine1665802226e_nres @ A),Z:(refine1665802226e_nres @ A)]: ((Y = Z))) = (^ [S4:(refine1665802226e_nres @ A),S7:(refine1665802226e_nres @ A)]: (((((refine1102455758nofail @ A) @ S4) = ((refine1102455758nofail @ A) @ S7)) & (! [X:A]: (((((refine1315500908_inres @ A) @ S4) @ X) = (((refine1315500908_inres @ A) @ S7) @ X))))))))))).
% 3.37/1.84 thf(fact_201_not__nofail__inres, axiom, (! [A:$tType,S:(refine1665802226e_nres @ A),X2:A]: (((~ (((refine1102455758nofail @ A) @ S))) => (((refine1315500908_inres @ A) @ S) @ X2))))).
% 3.37/1.84 thf(fact_202_pwD2, axiom, (! [A:$tType,S:(refine1665802226e_nres @ A),S5:(refine1665802226e_nres @ A),X2:A]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ S) @ S5) => ((((refine1315500908_inres @ A) @ S) @ X2) => (((refine1315500908_inres @ A) @ S5) @ X2)))))).
% 3.37/1.84 thf(fact_203_trancl__sub, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: ((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ R) @ ((transitive_trancl @ A) @ R))))).
% 3.37/1.84 thf(fact_204_trancl__mono__mp, axiom, (! [A:$tType,U2:(set @ ((product_prod @ A) @ A)),V2:(set @ ((product_prod @ A) @ A)),X2:((product_prod @ A) @ A)]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ U2) @ V2) => ((((member @ ((product_prod @ A) @ A)) @ X2) @ ((transitive_trancl @ A) @ U2)) => (((member @ ((product_prod @ A) @ A)) @ X2) @ ((transitive_trancl @ A) @ V2))))))).
% 3.37/1.84 thf(fact_205_pw__leI, axiom, (! [A:$tType,S5:(refine1665802226e_nres @ A),S:(refine1665802226e_nres @ A)]: (((((refine1102455758nofail @ A) @ S5) => (((refine1102455758nofail @ A) @ S) & (! [X3:A]: (((((refine1315500908_inres @ A) @ S) @ X3) => (((refine1315500908_inres @ A) @ S5) @ X3)))))) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ S) @ S5))))).
% 3.37/1.84 thf(fact_206_pw__leI_H, axiom, (! [A:$tType,S5:(refine1665802226e_nres @ A),S:(refine1665802226e_nres @ A)]: (((((refine1102455758nofail @ A) @ S5) => ((refine1102455758nofail @ A) @ S)) => ((! [X3:A]: ((((refine1102455758nofail @ A) @ S5) => ((((refine1315500908_inres @ A) @ S) @ X3) => (((refine1315500908_inres @ A) @ S5) @ X3))))) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ S) @ S5)))))).
% 3.37/1.84 thf(fact_207_pw__le__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (refine1665802226e_nres @ A)) = (^ [S4:(refine1665802226e_nres @ A),S7:(refine1665802226e_nres @ A)]: ((((refine1102455758nofail @ A) @ S7) => (((refine1102455758nofail @ A) @ S4) & (! [X:A]: (((((refine1315500908_inres @ A) @ S4) @ X) => (((refine1315500908_inres @ A) @ S7) @ X)))))))))))).
% 3.37/1.84 thf(fact_208_inres__def, axiom, (! [A:$tType]: (((refine1315500908_inres @ A) = (^ [S4:(refine1665802226e_nres @ A),X:A]: ((((ord_less_eq @ (refine1665802226e_nres @ A)) @ ((refine1687780735RETURN @ A) @ X)) @ S4))))))).
% 3.37/1.84 thf(fact_209_nf__inres__def, axiom, (! [A:$tType]: (((refine406925620_inres @ A) = (^ [M4:(refine1665802226e_nres @ A),X:A]: ((((refine1102455758nofail @ A) @ M4) & (((refine1315500908_inres @ A) @ M4) @ X)))))))).
% 3.37/1.84 thf(fact_210_pw__abs__inres, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ B) @ A)),M:(refine1665802226e_nres @ B),A3:A]: (((((refine1315500908_inres @ A) @ ((((refine81118332bs_fun @ B) @ A) @ R) @ M)) @ A3) = (((refine1102455758nofail @ A) @ ((((refine81118332bs_fun @ B) @ A) @ R) @ M)) => (? [C4:B]: (((((refine1315500908_inres @ B) @ M) @ C4) & (((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ C4) @ A3)) @ R))))))))).
% 3.37/1.84 thf(fact_211_trancl__reflcl, axiom, (! [A:$tType,R3:(set @ ((product_prod @ A) @ A))]: ((((transitive_trancl @ A) @ (((sup_sup @ (set @ ((product_prod @ A) @ A))) @ R3) @ (id @ A))) = ((transitive_rtrancl @ A) @ R3))))).
% 3.37/1.84 thf(fact_212_pw__bind__leI, axiom, (! [B:$tType,A:$tType,S:(refine1665802226e_nres @ A),M:(refine1665802226e_nres @ B),F:(B > (refine1665802226e_nres @ A))]: (((((refine1102455758nofail @ A) @ S) => ((refine1102455758nofail @ B) @ M)) => ((! [X3:B]: ((((refine1102455758nofail @ B) @ M) => ((((refine1315500908_inres @ B) @ M) @ X3) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ (F @ X3)) @ S))))) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ ((((refine463715084e_bind @ B) @ A) @ M) @ F)) @ S)))))).
% 3.37/1.84 thf(fact_213_Un__iff, axiom, (! [A:$tType,C3:A,A2:(set @ A),B2:(set @ A)]: (((((member @ A) @ C3) @ (((sup_sup @ (set @ A)) @ A2) @ B2)) = ((((member @ A) @ C3) @ A2) | (((member @ A) @ C3) @ B2)))))).
% 3.37/1.84 thf(fact_214_UnCI, axiom, (! [A:$tType,C3:A,B2:(set @ A),A2:(set @ A)]: ((((~ ((((member @ A) @ C3) @ B2))) => (((member @ A) @ C3) @ A2)) => (((member @ A) @ C3) @ (((sup_sup @ (set @ A)) @ A2) @ B2)))))).
% 3.37/1.84 thf(fact_215_Un__subset__iff, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A2) @ B2)) @ C2) = ((((ord_less_eq @ (set @ A)) @ A2) @ C2) & (((ord_less_eq @ (set @ A)) @ B2) @ C2)))))).
% 3.37/1.84 thf(fact_216_bind__FAIL, axiom, (! [B:$tType,A:$tType,F:(B > (refine1665802226e_nres @ A))]: ((((((refine463715084e_bind @ B) @ A) @ (top_top @ (refine1665802226e_nres @ B))) @ F) = (top_top @ (refine1665802226e_nres @ A)))))).
% 3.37/1.84 thf(fact_217_nres__monad2, axiom, (! [A:$tType,M:(refine1665802226e_nres @ A)]: ((((((refine463715084e_bind @ A) @ A) @ M) @ (refine1687780735RETURN @ A)) = M)))).
% 3.37/1.84 thf(fact_218_nres__monad1, axiom, (! [A:$tType,B:$tType,X2:B,F:(B > (refine1665802226e_nres @ A))]: ((((((refine463715084e_bind @ B) @ A) @ ((refine1687780735RETURN @ B) @ X2)) @ F) = (F @ X2))))).
% 3.37/1.84 thf(fact_219_rtrancl__reflcl, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: ((((transitive_rtrancl @ A) @ (((sup_sup @ (set @ ((product_prod @ A) @ A))) @ R) @ (id @ A))) = ((transitive_rtrancl @ A) @ R))))).
% 3.37/1.84 thf(fact_220_rtrancl__reflcl__absorb, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: (((((sup_sup @ (set @ ((product_prod @ A) @ A))) @ ((transitive_rtrancl @ A) @ R)) @ (id @ A)) = ((transitive_rtrancl @ A) @ R))))).
% 3.37/1.84 thf(fact_221_rtrancl__Un__subset, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),S:(set @ ((product_prod @ A) @ A))]: ((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ (((sup_sup @ (set @ ((product_prod @ A) @ A))) @ ((transitive_rtrancl @ A) @ R)) @ ((transitive_rtrancl @ A) @ S))) @ ((transitive_rtrancl @ A) @ (((sup_sup @ (set @ ((product_prod @ A) @ A))) @ R) @ S)))))).
% 3.37/1.84 thf(fact_222_Un__mono, axiom, (! [A:$tType,A2:(set @ A),C2:(set @ A),B2:(set @ A),D3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ C2) => ((((ord_less_eq @ (set @ A)) @ B2) @ D3) => (((ord_less_eq @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A2) @ B2)) @ (((sup_sup @ (set @ A)) @ C2) @ D3))))))).
% 3.37/1.84 thf(fact_223_Un__least, axiom, (! [A:$tType,A2:(set @ A),C2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ C2) => ((((ord_less_eq @ (set @ A)) @ B2) @ C2) => (((ord_less_eq @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A2) @ B2)) @ C2)))))).
% 3.37/1.84 thf(fact_224_Un__upper1, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A2) @ (((sup_sup @ (set @ A)) @ A2) @ B2))))).
% 3.37/1.84 thf(fact_225_Un__upper2, axiom, (! [A:$tType,B2:(set @ A),A2:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ B2) @ (((sup_sup @ (set @ A)) @ A2) @ B2))))).
% 3.37/1.84 thf(fact_226_Un__absorb1, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((sup_sup @ (set @ A)) @ A2) @ B2) = B2))))).
% 3.37/1.84 thf(fact_227_Un__absorb2, axiom, (! [A:$tType,B2:(set @ A),A2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ B2) @ A2) => ((((sup_sup @ (set @ A)) @ A2) @ B2) = A2))))).
% 3.37/1.84 thf(fact_228_subset__UnE, axiom, (! [A:$tType,C2:(set @ A),A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ C2) @ (((sup_sup @ (set @ A)) @ A2) @ B2)) => (~ ((! [A8:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A8) @ A2) => (! [B7:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ B7) @ B2) => (C2 != (((sup_sup @ (set @ A)) @ A8) @ B7)))))))))))))).
% 3.37/1.84 thf(fact_229_subset__Un__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A6:(set @ A),B6:(set @ A)]: (((((sup_sup @ (set @ A)) @ A6) @ B6) = B6))))))).
% 3.37/1.84 thf(fact_230_Domain__Un__eq, axiom, (! [B:$tType,A:$tType,A2:(set @ ((product_prod @ A) @ B)),B2:(set @ ((product_prod @ A) @ B))]: (((((domain @ A) @ B) @ (((sup_sup @ (set @ ((product_prod @ A) @ B))) @ A2) @ B2)) = (((sup_sup @ (set @ A)) @ (((domain @ A) @ B) @ A2)) @ (((domain @ A) @ B) @ B2)))))).
% 3.37/1.84 thf(fact_231_Un__left__commute, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C2:(set @ A)]: (((((sup_sup @ (set @ A)) @ A2) @ (((sup_sup @ (set @ A)) @ B2) @ C2)) = (((sup_sup @ (set @ A)) @ B2) @ (((sup_sup @ (set @ A)) @ A2) @ C2)))))).
% 3.37/1.84 thf(fact_232_Un__left__absorb, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((sup_sup @ (set @ A)) @ A2) @ (((sup_sup @ (set @ A)) @ A2) @ B2)) = (((sup_sup @ (set @ A)) @ A2) @ B2))))).
% 3.37/1.84 thf(fact_233_Un__commute, axiom, (! [A:$tType]: (((sup_sup @ (set @ A)) = (^ [A6:(set @ A),B6:(set @ A)]: ((((sup_sup @ (set @ A)) @ B6) @ A6))))))).
% 3.37/1.84 thf(fact_234_Un__absorb, axiom, (! [A:$tType,A2:(set @ A)]: (((((sup_sup @ (set @ A)) @ A2) @ A2) = A2)))).
% 3.37/1.84 thf(fact_235_Un__assoc, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C2:(set @ A)]: (((((sup_sup @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A2) @ B2)) @ C2) = (((sup_sup @ (set @ A)) @ A2) @ (((sup_sup @ (set @ A)) @ B2) @ C2)))))).
% 3.37/1.84 thf(fact_236_ball__Un, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),P:(A > $o)]: (((! [X:A]: (((((member @ A) @ X) @ (((sup_sup @ (set @ A)) @ A2) @ B2)) => (P @ X)))) = ((! [X:A]: (((((member @ A) @ X) @ A2) => (P @ X)))) & (! [X:A]: (((((member @ A) @ X) @ B2) => (P @ X))))))))).
% 3.37/1.84 thf(fact_237_bex__Un, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),P:(A > $o)]: (((? [X:A]: (((((member @ A) @ X) @ (((sup_sup @ (set @ A)) @ A2) @ B2)) & (P @ X)))) = ((? [X:A]: (((((member @ A) @ X) @ A2) & (P @ X)))) | (? [X:A]: (((((member @ A) @ X) @ B2) & (P @ X))))))))).
% 3.37/1.84 thf(fact_238_UnI2, axiom, (! [A:$tType,C3:A,B2:(set @ A),A2:(set @ A)]: (((((member @ A) @ C3) @ B2) => (((member @ A) @ C3) @ (((sup_sup @ (set @ A)) @ A2) @ B2)))))).
% 3.37/1.84 thf(fact_239_UnI1, axiom, (! [A:$tType,C3:A,A2:(set @ A),B2:(set @ A)]: (((((member @ A) @ C3) @ A2) => (((member @ A) @ C3) @ (((sup_sup @ (set @ A)) @ A2) @ B2)))))).
% 3.37/1.84 thf(fact_240_UnE, axiom, (! [A:$tType,C3:A,A2:(set @ A),B2:(set @ A)]: (((((member @ A) @ C3) @ (((sup_sup @ (set @ A)) @ A2) @ B2)) => ((~ ((((member @ A) @ C3) @ A2))) => (((member @ A) @ C3) @ B2)))))).
% 3.37/1.84 thf(fact_241_rel__restrict__union, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),A2:(set @ A),B2:(set @ A)]: (((((rel_restrict @ A) @ R) @ (((sup_sup @ (set @ A)) @ A2) @ B2)) = (((rel_restrict @ A) @ (((rel_restrict @ A) @ R) @ A2)) @ B2))))).
% 3.37/1.84 thf(fact_242_ibind__strict_I2_J, axiom, (! [A:$tType,F:(product_unit > (refine1665802226e_nres @ A))]: ((((((refine463715084e_bind @ product_unit) @ A) @ (top_top @ (refine1665802226e_nres @ product_unit))) @ F) = (top_top @ (refine1665802226e_nres @ A)))))).
% 3.37/1.84 thf(fact_243_Un__UNIV__right, axiom, (! [A:$tType,A2:(set @ A)]: (((((sup_sup @ (set @ A)) @ A2) @ (top_top @ (set @ A))) = (top_top @ (set @ A)))))).
% 3.37/1.84 thf(fact_244_Un__UNIV__left, axiom, (! [A:$tType,B2:(set @ A)]: (((((sup_sup @ (set @ A)) @ (top_top @ (set @ A))) @ B2) = (top_top @ (set @ A)))))).
% 3.37/1.84 thf(fact_245_pw__bind__nofail, axiom, (! [A:$tType,B:$tType,M:(refine1665802226e_nres @ B),F:(B > (refine1665802226e_nres @ A))]: ((((refine1102455758nofail @ A) @ ((((refine463715084e_bind @ B) @ A) @ M) @ F)) = (((refine1102455758nofail @ B) @ M) & (! [X:B]: (((((refine1315500908_inres @ B) @ M) @ X) => ((refine1102455758nofail @ A) @ (F @ X)))))))))).
% 3.37/1.84 thf(fact_246_bind__cong, axiom, (! [B:$tType,A:$tType,M2:(refine1665802226e_nres @ A),M6:(refine1665802226e_nres @ A),F:(A > (refine1665802226e_nres @ B)),F3:(A > (refine1665802226e_nres @ B))]: (((M2 = M6) => ((! [X3:A]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ ((refine1687780735RETURN @ A) @ X3)) @ M6) => ((F @ X3) = (F3 @ X3))))) => (((((refine463715084e_bind @ A) @ B) @ M2) @ F) = ((((refine463715084e_bind @ A) @ B) @ M6) @ F3))))))).
% 3.37/1.84 thf(fact_247_Refine__Basic__Mirabelle__tqojlsrkwy_Obind__mono_I1_J, axiom, (! [B:$tType,A:$tType,M:(refine1665802226e_nres @ A),M3:(refine1665802226e_nres @ A),F:(A > (refine1665802226e_nres @ B)),F3:(A > (refine1665802226e_nres @ B))]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ M) @ M3) => ((! [X3:A]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ ((refine1687780735RETURN @ A) @ X3)) @ M) => (((ord_less_eq @ (refine1665802226e_nres @ B)) @ (F @ X3)) @ (F3 @ X3))))) => (((ord_less_eq @ (refine1665802226e_nres @ B)) @ ((((refine463715084e_bind @ A) @ B) @ M) @ F)) @ ((((refine463715084e_bind @ A) @ B) @ M3) @ F3))))))).
% 3.37/1.84 thf(fact_248_trancl__union__outside, axiom, (! [A:$tType,V:A,W2:A,E4:(set @ ((product_prod @ A) @ A)),U2:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ V) @ W2)) @ ((transitive_trancl @ A) @ (((sup_sup @ (set @ ((product_prod @ A) @ A))) @ E4) @ U2))) => ((~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ V) @ W2)) @ ((transitive_trancl @ A) @ E4)))) => (? [X3:A,Y4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ V) @ X3)) @ ((transitive_rtrancl @ A) @ (((sup_sup @ (set @ ((product_prod @ A) @ A))) @ E4) @ U2))) & ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y4)) @ U2) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ W2)) @ ((transitive_rtrancl @ A) @ (((sup_sup @ (set @ ((product_prod @ A) @ A))) @ E4) @ U2)))))))))))).
% 3.37/1.84 thf(fact_249_rtrancl__trancl__reflcl, axiom, (! [A:$tType]: (((transitive_rtrancl @ A) = (^ [R6:(set @ ((product_prod @ A) @ A))]: ((((sup_sup @ (set @ ((product_prod @ A) @ A))) @ ((transitive_trancl @ A) @ R6)) @ (id @ A)))))))).
% 3.37/1.84 thf(fact_250_trancl__image__by__rtrancl, axiom, (! [A:$tType,E4:(set @ ((product_prod @ A) @ A)),Vi:(set @ A)]: (((((sup_sup @ (set @ A)) @ ((((image @ A) @ A) @ ((transitive_trancl @ A) @ E4)) @ Vi)) @ Vi) = ((((image @ A) @ A) @ ((transitive_rtrancl @ A) @ E4)) @ Vi))))).
% 3.37/1.84 thf(fact_251_pw__bind__le__iff, axiom, (! [A:$tType,B:$tType,M:(refine1665802226e_nres @ B),F:(B > (refine1665802226e_nres @ A)),S:(refine1665802226e_nres @ A)]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ ((((refine463715084e_bind @ B) @ A) @ M) @ F)) @ S) = ((((refine1102455758nofail @ A) @ S) => ((refine1102455758nofail @ B) @ M)) & (! [X:B]: (((((refine1102455758nofail @ B) @ M) & (((refine1315500908_inres @ B) @ M) @ X)) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ (F @ X)) @ S))))))))).
% 3.37/1.84 thf(fact_252_sup__top__left, axiom, (! [A:$tType]: (((bounded_lattice_top @ A) => (! [X2:A]: (((((sup_sup @ A) @ (top_top @ A)) @ X2) = (top_top @ A)))))))).
% 3.37/1.84 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Lattices_Obounded__lattice, axiom, (! [A9:$tType]: ((bounded_lattice @ (refine1665802226e_nres @ A9))))).
% 3.37/1.84 thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice_1, axiom, (bounded_lattice @ product_unit)).
% 3.37/1.84 thf(tcon_HOL_Obool___Lattices_Obounded__lattice_2, axiom, (bounded_lattice @ $o)).
% 3.37/1.84 thf(tcon_Set_Oset___Lattices_Obounded__lattice_3, axiom, (! [A9:$tType]: ((bounded_lattice @ (set @ A9))))).
% 3.37/1.84 thf(tcon_fun___Lattices_Obounded__lattice_4, axiom, (! [A9:$tType,A10:$tType]: (((bounded_lattice @ A10) => (bounded_lattice @ (A9 > A10)))))).
% 3.37/1.84 thf(tcon_fun___Complete__Lattices_Ocomplete__lattice, axiom, (! [A9:$tType,A10:$tType]: (((comple187826305attice @ A10) => (comple187826305attice @ (A9 > A10)))))).
% 3.37/1.84 thf(tcon_fun___Lattices_Obounded__lattice__top, axiom, (! [A9:$tType,A10:$tType]: (((bounded_lattice @ A10) => (bounded_lattice_top @ (A9 > A10)))))).
% 3.37/1.84 thf(tcon_fun___Orderings_Oorder__top, axiom, (! [A9:$tType,A10:$tType]: (((order_top @ A10) => (order_top @ (A9 > A10)))))).
% 3.37/1.84 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A9:$tType,A10:$tType]: (((preorder @ A10) => (preorder @ (A9 > A10)))))).
% 3.37/1.84 thf(tcon_fun___Orderings_Oorder, axiom, (! [A9:$tType,A10:$tType]: (((order @ A10) => (order @ (A9 > A10)))))).
% 3.37/1.84 thf(tcon_fun___Orderings_Otop, axiom, (! [A9:$tType,A10:$tType]: (((top @ A10) => (top @ (A9 > A10)))))).
% 3.37/1.84 thf(tcon_fun___Orderings_Oord, axiom, (! [A9:$tType,A10:$tType]: (((ord @ A10) => (ord @ (A9 > A10)))))).
% 3.37/1.84 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_5, axiom, (! [A9:$tType]: ((comple187826305attice @ (set @ A9))))).
% 3.37/1.84 thf(tcon_Set_Oset___Lattices_Obounded__lattice__top_6, axiom, (! [A9:$tType]: ((bounded_lattice_top @ (set @ A9))))).
% 3.37/1.84 thf(tcon_Set_Oset___Orderings_Oorder__top_7, axiom, (! [A9:$tType]: ((order_top @ (set @ A9))))).
% 3.37/1.84 thf(tcon_Set_Oset___Orderings_Opreorder_8, axiom, (! [A9:$tType]: ((preorder @ (set @ A9))))).
% 3.37/1.84 thf(tcon_Set_Oset___Orderings_Oorder_9, axiom, (! [A9:$tType]: ((order @ (set @ A9))))).
% 3.37/1.84 thf(tcon_Set_Oset___Orderings_Otop_10, axiom, (! [A9:$tType]: ((top @ (set @ A9))))).
% 3.37/1.84 thf(tcon_Set_Oset___Orderings_Oord_11, axiom, (! [A9:$tType]: ((ord @ (set @ A9))))).
% 3.37/1.84 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_12, axiom, (comple187826305attice @ $o)).
% 3.37/1.84 thf(tcon_HOL_Obool___Lattices_Obounded__lattice__top_13, axiom, (bounded_lattice_top @ $o)).
% 3.37/1.84 thf(tcon_HOL_Obool___Orderings_Oorder__top_14, axiom, (order_top @ $o)).
% 3.37/1.84 thf(tcon_HOL_Obool___Orderings_Opreorder_15, axiom, (preorder @ $o)).
% 3.37/1.84 thf(tcon_HOL_Obool___Orderings_Olinorder, axiom, (linorder @ $o)).
% 3.37/1.84 thf(tcon_HOL_Obool___Orderings_Oorder_16, axiom, (order @ $o)).
% 3.37/1.84 thf(tcon_HOL_Obool___Orderings_Otop_17, axiom, (top @ $o)).
% 3.37/1.84 thf(tcon_HOL_Obool___Orderings_Oord_18, axiom, (ord @ $o)).
% 3.37/1.84 thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_19, axiom, (comple187826305attice @ product_unit)).
% 3.37/1.84 thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__top_20, axiom, (bounded_lattice_top @ product_unit)).
% 3.37/1.84 thf(tcon_Product__Type_Ounit___Orderings_Oorder__top_21, axiom, (order_top @ product_unit)).
% 3.37/1.84 thf(tcon_Product__Type_Ounit___Orderings_Opreorder_22, axiom, (preorder @ product_unit)).
% 3.37/1.84 thf(tcon_Product__Type_Ounit___Orderings_Olinorder_23, axiom, (linorder @ product_unit)).
% 3.37/1.84 thf(tcon_Product__Type_Ounit___Orderings_Oorder_24, axiom, (order @ product_unit)).
% 3.37/1.84 thf(tcon_Product__Type_Ounit___Orderings_Otop_25, axiom, (top @ product_unit)).
% 3.37/1.84 thf(tcon_Product__Type_Ounit___Orderings_Oord_26, axiom, (ord @ product_unit)).
% 3.37/1.84 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Complete__Lattices_Ocomplete__lattice_27, axiom, (! [A9:$tType]: ((comple187826305attice @ (refine1665802226e_nres @ A9))))).
% 3.37/1.84 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Lattices_Obounded__lattice__top_28, axiom, (! [A9:$tType]: ((bounded_lattice_top @ (refine1665802226e_nres @ A9))))).
% 3.37/1.84 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oorder__top_29, axiom, (! [A9:$tType]: ((order_top @ (refine1665802226e_nres @ A9))))).
% 3.37/1.84 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Opreorder_30, axiom, (! [A9:$tType]: ((preorder @ (refine1665802226e_nres @ A9))))).
% 3.37/1.84 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oorder_31, axiom, (! [A9:$tType]: ((order @ (refine1665802226e_nres @ A9))))).
% 3.37/1.84 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Otop_32, axiom, (! [A9:$tType]: ((top @ (refine1665802226e_nres @ A9))))).
% 3.37/1.84 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oord_33, axiom, (! [A9:$tType]: ((ord @ (refine1665802226e_nres @ A9))))).
% 3.37/1.84 thf(help_If_3_1_T, axiom, (! [P:$o]: (((P = $true) | (P = $false))))).
% 3.37/1.84 thf(help_If_2_1_T, axiom, (! [A:$tType,X2:A,Y2:A]: ((((((if @ A) @ $false) @ X2) @ Y2) = Y2)))).
% 3.37/1.84 thf(help_If_1_1_T, axiom, (! [A:$tType,X2:A,Y2:A]: ((((((if @ A) @ $true) @ X2) @ Y2) = X2)))).
% 3.37/1.84 thf(conj_0, conjecture, (((ord_less_eq @ (refine1665802226e_nres @ a)) @ x) @ ((((refine1073749519nc_fun @ a) @ b) @ r) @ (top_top @ (refine1665802226e_nres @ b))))).
% 3.37/1.84 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.vRzaLjzKJ7/DTF2DTF19562.p
% 3.37/1.84 ------------------------
% 3.37/1.89 ---- Cleaned THF ---
% 3.37/1.90 thf(ty_t_Refine__Basic__Mirabelle__tqojlsrkwy_Onres,type,
% 3.37/1.90 refine1665802226e_nres: $tType > $tType ).
% 3.37/1.90
% 3.37/1.90 thf(ty_t_Product__Type_Ounit,type,
% 3.37/1.90 product_unit: $tType ).
% 3.37/1.90
% 3.37/1.90 thf(ty_t_Product__Type_Oprod,type,
% 3.37/1.90 product_prod: $tType > $tType > $tType ).
% 3.37/1.90
% 3.37/1.90 thf(ty_t_Set_Oset,type,
% 3.37/1.90 set: $tType > $tType ).
% 3.37/1.90
% 3.37/1.90 thf(ty_tf_b,type,
% 3.37/1.90 b: $tType ).
% 3.37/1.90
% 3.37/1.90 thf(ty_tf_a,type,
% 3.37/1.90 a: $tType ).
% 3.37/1.90
% 3.37/1.90 thf(sy_cl_Lattices_Obounded__lattice,type,
% 3.37/1.90 bounded_lattice:
% 3.37/1.90 !>[A: $tType] : $o ).
% 3.37/1.90
% 3.37/1.90 thf(sy_cl_HOL_Otype,type,
% 3.37/1.90 type:
% 3.37/1.90 !>[A: $tType] : $o ).
% 3.37/1.90
% 3.37/1.90 thf(sy_cl_Orderings_Oord,type,
% 3.37/1.90 ord:
% 3.37/1.90 !>[A: $tType] : $o ).
% 3.37/1.90
% 3.37/1.90 thf(sy_cl_Orderings_Otop,type,
% 3.37/1.90 top:
% 3.37/1.90 !>[A: $tType] : $o ).
% 3.37/1.90
% 3.37/1.90 thf(sy_cl_Orderings_Oorder,type,
% 3.37/1.90 order:
% 3.37/1.90 !>[A: $tType] : $o ).
% 3.37/1.90
% 3.37/1.90 thf(sy_cl_Orderings_Olinorder,type,
% 3.37/1.90 linorder:
% 3.37/1.90 !>[A: $tType] : $o ).
% 3.37/1.90
% 3.37/1.90 thf(sy_cl_Orderings_Opreorder,type,
% 3.37/1.90 preorder:
% 3.37/1.90 !>[A: $tType] : $o ).
% 3.37/1.90
% 3.37/1.90 thf(sy_cl_Orderings_Oorder__top,type,
% 3.37/1.90 order_top:
% 3.37/1.90 !>[A: $tType] : $o ).
% 3.37/1.90
% 3.37/1.90 thf(sy_cl_Lattices_Obounded__lattice__top,type,
% 3.37/1.90 bounded_lattice_top:
% 3.37/1.90 !>[A: $tType] : $o ).
% 3.37/1.90
% 3.37/1.90 thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
% 3.37/1.90 comple187826305attice:
% 3.37/1.90 !>[A: $tType] : $o ).
% 3.37/1.90
% 3.37/1.90 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
% 3.37/1.90 bNF_Ca1785829860lChain:
% 3.37/1.90 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
% 3.37/1.90
% 3.37/1.90 thf(sy_c_If,type,
% 3.37/1.90 if:
% 3.37/1.90 !>[A: $tType] : ( $o > A > A > A ) ).
% 3.37/1.90
% 3.37/1.90 thf(sy_c_Lattices_Osup__class_Osup,type,
% 3.37/1.90 sup_sup:
% 3.37/1.90 !>[A: $tType] : ( A > A > A ) ).
% 3.37/1.90
% 3.37/1.90 thf(sy_c_Misc_Obijective,type,
% 3.37/1.90 bijective:
% 3.37/1.90 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > $o ) ).
% 3.37/1.90
% 3.37/1.90 thf(sy_c_Misc_Ofun__of__rel,type,
% 3.37/1.90 fun_of_rel:
% 3.37/1.90 !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ A ) ) > B > A ) ).
% 3.37/1.90
% 3.37/1.90 thf(sy_c_Misc_Orel__restrict,type,
% 3.37/1.90 rel_restrict:
% 3.37/1.90 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 3.37/1.90
% 3.37/1.90 thf(sy_c_Misc_Ouncurry,type,
% 3.37/1.90 uncurry:
% 3.37/1.90 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 3.50/1.90
% 3.50/1.90 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 3.50/1.90 ord_less_eq:
% 3.50/1.90 !>[A: $tType] : ( A > A > $o ) ).
% 3.50/1.90
% 3.50/1.90 thf(sy_c_Orderings_Oorder__class_OGreatest,type,
% 3.50/1.90 order_Greatest:
% 3.50/1.90 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 3.50/1.90
% 3.50/1.90 thf(sy_c_Orderings_Otop__class_Otop,type,
% 3.50/1.90 top_top:
% 3.50/1.90 !>[A: $tType] : A ).
% 3.50/1.90
% 3.50/1.90 thf(sy_c_Product__Type_OPair,type,
% 3.50/1.90 product_Pair:
% 3.50/1.90 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_ORETURN,type,
% 3.50/1.90 refine1687780735RETURN:
% 3.50/1.90 !>[A: $tType] : ( A > ( refine1665802226e_nres @ A ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Oabs__fun,type,
% 3.50/1.90 refine81118332bs_fun:
% 3.50/1.90 !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ A ) ) > ( refine1665802226e_nres @ B ) > ( refine1665802226e_nres @ A ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Obind,type,
% 3.50/1.90 refine463715084e_bind:
% 3.50/1.90 !>[B: $tType,A: $tType] : ( ( refine1665802226e_nres @ B ) > ( B > ( refine1665802226e_nres @ A ) ) > ( refine1665802226e_nres @ A ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Oconc__fun,type,
% 3.50/1.90 refine1073749519nc_fun:
% 3.50/1.90 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( refine1665802226e_nres @ B ) > ( refine1665802226e_nres @ A ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Oinres,type,
% 3.50/1.90 refine1315500908_inres:
% 3.50/1.90 !>[A: $tType] : ( ( refine1665802226e_nres @ A ) > A > $o ) ).
% 3.50/1.90
% 3.50/1.90 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onf__inres,type,
% 3.50/1.90 refine406925620_inres:
% 3.50/1.90 !>[A: $tType] : ( ( refine1665802226e_nres @ A ) > A > $o ) ).
% 3.50/1.90
% 3.50/1.90 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onofail,type,
% 3.50/1.90 refine1102455758nofail:
% 3.50/1.90 !>[A: $tType] : ( ( refine1665802226e_nres @ A ) > $o ) ).
% 3.50/1.90
% 3.50/1.90 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onres_ORES,type,
% 3.50/1.90 refine605929679le_RES:
% 3.50/1.90 !>[A: $tType] : ( ( set @ A ) > ( refine1665802226e_nres @ A ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Othe__RES,type,
% 3.50/1.90 refine1672542526he_RES:
% 3.50/1.90 !>[A: $tType] : ( ( refine1665802226e_nres @ A ) > ( set @ A ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(sy_c_Relation_ODomain,type,
% 3.50/1.90 domain:
% 3.50/1.90 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(sy_c_Relation_OId,type,
% 3.50/1.90 id:
% 3.50/1.90 !>[A: $tType] : ( set @ ( product_prod @ A @ A ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(sy_c_Relation_OImage,type,
% 3.50/1.90 image:
% 3.50/1.90 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) > ( set @ B ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(sy_c_Set_OCollect,type,
% 3.50/1.90 collect:
% 3.50/1.90 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(sy_c_Transitive__Closure_Ortrancl,type,
% 3.50/1.90 transitive_rtrancl:
% 3.50/1.90 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(sy_c_Transitive__Closure_Otrancl,type,
% 3.50/1.90 transitive_trancl:
% 3.50/1.90 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(sy_c_member,type,
% 3.50/1.90 member:
% 3.50/1.90 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 3.50/1.90
% 3.50/1.90 thf(sy_v_R,type,
% 3.50/1.90 r: set @ ( product_prod @ a @ b ) ).
% 3.50/1.90
% 3.50/1.90 thf(sy_v_X,type,
% 3.50/1.90 x: refine1665802226e_nres @ a ).
% 3.50/1.90
% 3.50/1.90 thf(fact_0_nres__order__simps_I4_J,axiom,
% 3.50/1.90 ! [D: $tType,M: refine1665802226e_nres @ D] :
% 3.50/1.90 ( ( ord_less_eq @ ( refine1665802226e_nres @ D ) @ ( top_top @ ( refine1665802226e_nres @ D ) ) @ M )
% 3.50/1.90 = ( M
% 3.50/1.90 = ( top_top @ ( refine1665802226e_nres @ D ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_1_conc__fun__fail__iff_I2_J,axiom,
% 3.50/1.90 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),S: refine1665802226e_nres @ B] :
% 3.50/1.90 ( ( ( top_top @ ( refine1665802226e_nres @ A ) )
% 3.50/1.90 = ( refine1073749519nc_fun @ A @ B @ R @ S ) )
% 3.50/1.90 = ( S
% 3.50/1.90 = ( top_top @ ( refine1665802226e_nres @ B ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_2_conc__fun__fail__iff_I1_J,axiom,
% 3.50/1.90 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),S: refine1665802226e_nres @ B] :
% 3.50/1.90 ( ( ( refine1073749519nc_fun @ A @ B @ R @ S )
% 3.50/1.90 = ( top_top @ ( refine1665802226e_nres @ A ) ) )
% 3.50/1.90 = ( S
% 3.50/1.90 = ( top_top @ ( refine1665802226e_nres @ B ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_3_conc__fun__FAIL,axiom,
% 3.50/1.90 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] :
% 3.50/1.90 ( ( refine1073749519nc_fun @ A @ B @ R @ ( top_top @ ( refine1665802226e_nres @ B ) ) )
% 3.50/1.90 = ( top_top @ ( refine1665802226e_nres @ A ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_4_nres__order__simps_I3_J,axiom,
% 3.50/1.90 ! [C: $tType,M: refine1665802226e_nres @ C] : ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ M @ ( top_top @ ( refine1665802226e_nres @ C ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_5_conc__trans__additional_I1_J,axiom,
% 3.50/1.90 ! [A: $tType,B: $tType,A2: refine1665802226e_nres @ A,R: set @ ( product_prod @ A @ B ),B2: refine1665802226e_nres @ B,C2: refine1665802226e_nres @ B] :
% 3.50/1.90 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ A2 @ ( refine1073749519nc_fun @ A @ B @ R @ B2 ) )
% 3.50/1.90 => ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ B2 @ C2 )
% 3.50/1.90 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ A2 @ ( refine1073749519nc_fun @ A @ B @ R @ C2 ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_6_conc__trans,axiom,
% 3.50/1.90 ! [A: $tType,B: $tType,C: $tType,C2: refine1665802226e_nres @ A,R: set @ ( product_prod @ A @ B ),B2: refine1665802226e_nres @ B,R2: set @ ( product_prod @ B @ C ),A2: refine1665802226e_nres @ C] :
% 3.50/1.90 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ C2 @ ( refine1073749519nc_fun @ A @ B @ R @ B2 ) )
% 3.50/1.90 => ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ B2 @ ( refine1073749519nc_fun @ B @ C @ R2 @ A2 ) )
% 3.50/1.90 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ C2 @ ( refine1073749519nc_fun @ A @ B @ R @ ( refine1073749519nc_fun @ B @ C @ R2 @ A2 ) ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_7_top__apply,axiom,
% 3.50/1.90 ! [C: $tType,D: $tType] :
% 3.50/1.90 ( ( top @ C )
% 3.50/1.90 => ( ( top_top @ ( D > C ) )
% 3.50/1.90 = ( ^ [X: D] : ( top_top @ C ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_8_order__mono__setup_Orefl,axiom,
% 3.50/1.90 ! [A: $tType] :
% 3.50/1.90 ( ( preorder @ A )
% 3.50/1.90 => ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_9_top__greatest,axiom,
% 3.50/1.90 ! [A: $tType] :
% 3.50/1.90 ( ( order_top @ A )
% 3.50/1.90 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ ( top_top @ A ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_10_meta__le__everything__if__top,axiom,
% 3.50/1.90 ! [A: $tType] :
% 3.50/1.90 ( ( order_top @ A )
% 3.50/1.90 => ! [M2: A,X2: A] :
% 3.50/1.90 ( ( M2
% 3.50/1.90 = ( top_top @ A ) )
% 3.50/1.90 => ( ord_less_eq @ A @ X2 @ M2 ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_11_top_Oextremum__unique,axiom,
% 3.50/1.90 ! [A: $tType] :
% 3.50/1.90 ( ( order_top @ A )
% 3.50/1.90 => ! [A3: A] :
% 3.50/1.90 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
% 3.50/1.90 = ( A3
% 3.50/1.90 = ( top_top @ A ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_12_top_Oextremum__uniqueI,axiom,
% 3.50/1.90 ! [A: $tType] :
% 3.50/1.90 ( ( order_top @ A )
% 3.50/1.90 => ! [A3: A] :
% 3.50/1.90 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
% 3.50/1.90 => ( A3
% 3.50/1.90 = ( top_top @ A ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_13_le__funD,axiom,
% 3.50/1.90 ! [B: $tType,A: $tType] :
% 3.50/1.90 ( ( ord @ B )
% 3.50/1.90 => ! [F: A > B,G: A > B,X2: A] :
% 3.50/1.90 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 3.50/1.90 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_14_le__funE,axiom,
% 3.50/1.90 ! [B: $tType,A: $tType] :
% 3.50/1.90 ( ( ord @ B )
% 3.50/1.90 => ! [F: A > B,G: A > B,X2: A] :
% 3.50/1.90 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 3.50/1.90 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_15_le__funI,axiom,
% 3.50/1.90 ! [B: $tType,A: $tType] :
% 3.50/1.90 ( ( ord @ B )
% 3.50/1.90 => ! [F: A > B,G: A > B] :
% 3.50/1.90 ( ! [X3: A] : ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) )
% 3.50/1.90 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_16_Refine__Misc_Oif__mono,axiom,
% 3.50/1.90 ! [A: $tType] :
% 3.50/1.90 ( ( ord @ A )
% 3.50/1.90 => ! [B3: $o,M1: A,M12: A,M22: A,M23: A] :
% 3.50/1.90 ( ( B3
% 3.50/1.90 => ( ord_less_eq @ A @ M1 @ M12 ) )
% 3.50/1.90 => ( ( ~ B3
% 3.50/1.90 => ( ord_less_eq @ A @ M22 @ M23 ) )
% 3.50/1.90 => ( ord_less_eq @ A @ ( if @ A @ B3 @ M1 @ M22 ) @ ( if @ A @ B3 @ M12 @ M23 ) ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_17_le__fun__def,axiom,
% 3.50/1.90 ! [B: $tType,A: $tType] :
% 3.50/1.90 ( ( ord @ B )
% 3.50/1.90 => ( ( ord_less_eq @ ( A > B ) )
% 3.50/1.90 = ( ^ [F2: A > B,G2: A > B] :
% 3.50/1.90 ! [X: A] : ( ord_less_eq @ B @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_18_conc__fun__R__mono,axiom,
% 3.50/1.90 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),R2: set @ ( product_prod @ A @ B ),M: refine1665802226e_nres @ B] :
% 3.50/1.90 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ R2 )
% 3.50/1.90 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1073749519nc_fun @ A @ B @ R @ M ) @ ( refine1073749519nc_fun @ A @ B @ R2 @ M ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_19_order__mono__setup_Omono__let,axiom,
% 3.50/1.90 ! [A: $tType,B: $tType] :
% 3.50/1.90 ( ( preorder @ A )
% 3.50/1.90 => ! [F: B > A,F3: B > A,X2: B] :
% 3.50/1.90 ( ! [X3: B] : ( ord_less_eq @ A @ ( F @ X3 ) @ ( F3 @ X3 ) )
% 3.50/1.90 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F3 @ X2 ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_20_order__mono__setup_Omono__if,axiom,
% 3.50/1.90 ! [A: $tType] :
% 3.50/1.90 ( ( preorder @ A )
% 3.50/1.90 => ! [T: A,T2: A,E: A,E2: A,B3: $o] :
% 3.50/1.90 ( ( ord_less_eq @ A @ T @ T2 )
% 3.50/1.90 => ( ( ord_less_eq @ A @ E @ E2 )
% 3.50/1.90 => ( ord_less_eq @ A @ ( if @ A @ B3 @ T @ E ) @ ( if @ A @ B3 @ T2 @ E2 ) ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_21_dual__order_Oantisym,axiom,
% 3.50/1.90 ! [A: $tType] :
% 3.50/1.90 ( ( order @ A )
% 3.50/1.90 => ! [B3: A,A3: A] :
% 3.50/1.90 ( ( ord_less_eq @ A @ B3 @ A3 )
% 3.50/1.90 => ( ( ord_less_eq @ A @ A3 @ B3 )
% 3.50/1.90 => ( A3 = B3 ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_22_dual__order_Oeq__iff,axiom,
% 3.50/1.90 ! [A: $tType] :
% 3.50/1.90 ( ( order @ A )
% 3.50/1.90 => ( ( ^ [Y: A,Z: A] : ( Y = Z ) )
% 3.50/1.90 = ( ^ [A4: A,B4: A] :
% 3.50/1.90 ( ( ord_less_eq @ A @ B4 @ A4 )
% 3.50/1.90 & ( ord_less_eq @ A @ A4 @ B4 ) ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_23_dual__order_Otrans,axiom,
% 3.50/1.90 ! [A: $tType] :
% 3.50/1.90 ( ( order @ A )
% 3.50/1.90 => ! [B3: A,A3: A,C3: A] :
% 3.50/1.90 ( ( ord_less_eq @ A @ B3 @ A3 )
% 3.50/1.90 => ( ( ord_less_eq @ A @ C3 @ B3 )
% 3.50/1.90 => ( ord_less_eq @ A @ C3 @ A3 ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_24_linorder__wlog,axiom,
% 3.50/1.90 ! [A: $tType] :
% 3.50/1.90 ( ( linorder @ A )
% 3.50/1.90 => ! [P: A > A > $o,A3: A,B3: A] :
% 3.50/1.90 ( ! [A5: A,B5: A] :
% 3.50/1.90 ( ( ord_less_eq @ A @ A5 @ B5 )
% 3.50/1.90 => ( P @ A5 @ B5 ) )
% 3.50/1.90 => ( ! [A5: A,B5: A] :
% 3.50/1.90 ( ( P @ B5 @ A5 )
% 3.50/1.90 => ( P @ A5 @ B5 ) )
% 3.50/1.90 => ( P @ A3 @ B3 ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_25_dual__order_Orefl,axiom,
% 3.50/1.90 ! [A: $tType] :
% 3.50/1.90 ( ( order @ A )
% 3.50/1.90 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_26_order__trans,axiom,
% 3.50/1.90 ! [A: $tType] :
% 3.50/1.90 ( ( preorder @ A )
% 3.50/1.90 => ! [X2: A,Y2: A,Z2: A] :
% 3.50/1.90 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 3.50/1.90 => ( ( ord_less_eq @ A @ Y2 @ Z2 )
% 3.50/1.90 => ( ord_less_eq @ A @ X2 @ Z2 ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_27_order__class_Oorder_Oantisym,axiom,
% 3.50/1.90 ! [A: $tType] :
% 3.50/1.90 ( ( order @ A )
% 3.50/1.90 => ! [A3: A,B3: A] :
% 3.50/1.90 ( ( ord_less_eq @ A @ A3 @ B3 )
% 3.50/1.90 => ( ( ord_less_eq @ A @ B3 @ A3 )
% 3.50/1.90 => ( A3 = B3 ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_28_ord__le__eq__trans,axiom,
% 3.50/1.90 ! [A: $tType] :
% 3.50/1.90 ( ( ord @ A )
% 3.50/1.90 => ! [A3: A,B3: A,C3: A] :
% 3.50/1.90 ( ( ord_less_eq @ A @ A3 @ B3 )
% 3.50/1.90 => ( ( B3 = C3 )
% 3.50/1.90 => ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_29_ord__eq__le__trans,axiom,
% 3.50/1.90 ! [A: $tType] :
% 3.50/1.90 ( ( ord @ A )
% 3.50/1.90 => ! [A3: A,B3: A,C3: A] :
% 3.50/1.90 ( ( A3 = B3 )
% 3.50/1.90 => ( ( ord_less_eq @ A @ B3 @ C3 )
% 3.50/1.90 => ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_30_order__class_Oorder_Oeq__iff,axiom,
% 3.50/1.90 ! [A: $tType] :
% 3.50/1.90 ( ( order @ A )
% 3.50/1.90 => ( ( ^ [Y: A,Z: A] : ( Y = Z ) )
% 3.50/1.90 = ( ^ [A4: A,B4: A] :
% 3.50/1.90 ( ( ord_less_eq @ A @ A4 @ B4 )
% 3.50/1.90 & ( ord_less_eq @ A @ B4 @ A4 ) ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_31_antisym__conv,axiom,
% 3.50/1.90 ! [A: $tType] :
% 3.50/1.90 ( ( order @ A )
% 3.50/1.90 => ! [Y2: A,X2: A] :
% 3.50/1.90 ( ( ord_less_eq @ A @ Y2 @ X2 )
% 3.50/1.90 => ( ( ord_less_eq @ A @ X2 @ Y2 )
% 3.50/1.90 = ( X2 = Y2 ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_32_le__cases3,axiom,
% 3.50/1.90 ! [A: $tType] :
% 3.50/1.90 ( ( linorder @ A )
% 3.50/1.90 => ! [X2: A,Y2: A,Z2: A] :
% 3.50/1.90 ( ( ( ord_less_eq @ A @ X2 @ Y2 )
% 3.50/1.90 => ~ ( ord_less_eq @ A @ Y2 @ Z2 ) )
% 3.50/1.90 => ( ( ( ord_less_eq @ A @ Y2 @ X2 )
% 3.50/1.90 => ~ ( ord_less_eq @ A @ X2 @ Z2 ) )
% 3.50/1.90 => ( ( ( ord_less_eq @ A @ X2 @ Z2 )
% 3.50/1.90 => ~ ( ord_less_eq @ A @ Z2 @ Y2 ) )
% 3.50/1.90 => ( ( ( ord_less_eq @ A @ Z2 @ Y2 )
% 3.50/1.90 => ~ ( ord_less_eq @ A @ Y2 @ X2 ) )
% 3.50/1.90 => ( ( ( ord_less_eq @ A @ Y2 @ Z2 )
% 3.50/1.90 => ~ ( ord_less_eq @ A @ Z2 @ X2 ) )
% 3.50/1.90 => ~ ( ( ord_less_eq @ A @ Z2 @ X2 )
% 3.50/1.90 => ~ ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ) ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_33_order_Otrans,axiom,
% 3.50/1.90 ! [A: $tType] :
% 3.50/1.90 ( ( order @ A )
% 3.50/1.90 => ! [A3: A,B3: A,C3: A] :
% 3.50/1.90 ( ( ord_less_eq @ A @ A3 @ B3 )
% 3.50/1.90 => ( ( ord_less_eq @ A @ B3 @ C3 )
% 3.50/1.90 => ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_34_le__cases,axiom,
% 3.50/1.90 ! [A: $tType] :
% 3.50/1.90 ( ( linorder @ A )
% 3.50/1.90 => ! [X2: A,Y2: A] :
% 3.50/1.90 ( ~ ( ord_less_eq @ A @ X2 @ Y2 )
% 3.50/1.90 => ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_35_eq__refl,axiom,
% 3.50/1.90 ! [A: $tType] :
% 3.50/1.90 ( ( preorder @ A )
% 3.50/1.90 => ! [X2: A,Y2: A] :
% 3.50/1.90 ( ( X2 = Y2 )
% 3.50/1.90 => ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_36_linear,axiom,
% 3.50/1.90 ! [A: $tType] :
% 3.50/1.90 ( ( linorder @ A )
% 3.50/1.90 => ! [X2: A,Y2: A] :
% 3.50/1.90 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 3.50/1.90 | ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_37_antisym,axiom,
% 3.50/1.90 ! [A: $tType] :
% 3.50/1.90 ( ( order @ A )
% 3.50/1.90 => ! [X2: A,Y2: A] :
% 3.50/1.90 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 3.50/1.90 => ( ( ord_less_eq @ A @ Y2 @ X2 )
% 3.50/1.90 => ( X2 = Y2 ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_38_eq__iff,axiom,
% 3.50/1.90 ! [A: $tType] :
% 3.50/1.90 ( ( order @ A )
% 3.50/1.90 => ( ( ^ [Y: A,Z: A] : ( Y = Z ) )
% 3.50/1.90 = ( ^ [X: A,Y3: A] :
% 3.50/1.90 ( ( ord_less_eq @ A @ X @ Y3 )
% 3.50/1.90 & ( ord_less_eq @ A @ Y3 @ X ) ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_39_ord__le__eq__subst,axiom,
% 3.50/1.90 ! [A: $tType,B: $tType] :
% 3.50/1.90 ( ( ( ord @ B )
% 3.50/1.90 & ( ord @ A ) )
% 3.50/1.90 => ! [A3: A,B3: A,F: A > B,C3: B] :
% 3.50/1.90 ( ( ord_less_eq @ A @ A3 @ B3 )
% 3.50/1.90 => ( ( ( F @ B3 )
% 3.50/1.90 = C3 )
% 3.50/1.90 => ( ! [X3: A,Y4: A] :
% 3.50/1.90 ( ( ord_less_eq @ A @ X3 @ Y4 )
% 3.50/1.90 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 3.50/1.90 => ( ord_less_eq @ B @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_40_ord__eq__le__subst,axiom,
% 3.50/1.90 ! [A: $tType,B: $tType] :
% 3.50/1.90 ( ( ( ord @ B )
% 3.50/1.90 & ( ord @ A ) )
% 3.50/1.90 => ! [A3: A,F: B > A,B3: B,C3: B] :
% 3.50/1.90 ( ( A3
% 3.50/1.90 = ( F @ B3 ) )
% 3.50/1.90 => ( ( ord_less_eq @ B @ B3 @ C3 )
% 3.50/1.90 => ( ! [X3: B,Y4: B] :
% 3.50/1.90 ( ( ord_less_eq @ B @ X3 @ Y4 )
% 3.50/1.90 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 3.50/1.90 => ( ord_less_eq @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_41_order__subst2,axiom,
% 3.50/1.90 ! [A: $tType,C: $tType] :
% 3.50/1.90 ( ( ( order @ C )
% 3.50/1.90 & ( order @ A ) )
% 3.50/1.90 => ! [A3: A,B3: A,F: A > C,C3: C] :
% 3.50/1.90 ( ( ord_less_eq @ A @ A3 @ B3 )
% 3.50/1.90 => ( ( ord_less_eq @ C @ ( F @ B3 ) @ C3 )
% 3.50/1.90 => ( ! [X3: A,Y4: A] :
% 3.50/1.90 ( ( ord_less_eq @ A @ X3 @ Y4 )
% 3.50/1.90 => ( ord_less_eq @ C @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 3.50/1.90 => ( ord_less_eq @ C @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_42_order__subst1,axiom,
% 3.50/1.90 ! [A: $tType,B: $tType] :
% 3.50/1.90 ( ( ( order @ B )
% 3.50/1.90 & ( order @ A ) )
% 3.50/1.90 => ! [A3: A,F: B > A,B3: B,C3: B] :
% 3.50/1.90 ( ( ord_less_eq @ A @ A3 @ ( F @ B3 ) )
% 3.50/1.90 => ( ( ord_less_eq @ B @ B3 @ C3 )
% 3.50/1.90 => ( ! [X3: B,Y4: B] :
% 3.50/1.90 ( ( ord_less_eq @ B @ X3 @ Y4 )
% 3.50/1.90 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 3.50/1.90 => ( ord_less_eq @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_43_Greatest__equality,axiom,
% 3.50/1.90 ! [A: $tType] :
% 3.50/1.90 ( ( order @ A )
% 3.50/1.90 => ! [P: A > $o,X2: A] :
% 3.50/1.90 ( ( P @ X2 )
% 3.50/1.90 => ( ! [Y4: A] :
% 3.50/1.90 ( ( P @ Y4 )
% 3.50/1.90 => ( ord_less_eq @ A @ Y4 @ X2 ) )
% 3.50/1.90 => ( ( order_Greatest @ A @ P )
% 3.50/1.90 = X2 ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_44_mem__Collect__eq,axiom,
% 3.50/1.90 ! [A: $tType,A3: A,P: A > $o] :
% 3.50/1.90 ( ( member @ A @ A3 @ ( collect @ A @ P ) )
% 3.50/1.90 = ( P @ A3 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_45_Collect__mem__eq,axiom,
% 3.50/1.90 ! [A: $tType,A2: set @ A] :
% 3.50/1.90 ( ( collect @ A
% 3.50/1.90 @ ^ [X: A] : ( member @ A @ X @ A2 ) )
% 3.50/1.90 = A2 ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_46_Collect__cong,axiom,
% 3.50/1.90 ! [A: $tType,P: A > $o,Q: A > $o] :
% 3.50/1.90 ( ! [X3: A] :
% 3.50/1.90 ( ( P @ X3 )
% 3.50/1.90 = ( Q @ X3 ) )
% 3.50/1.90 => ( ( collect @ A @ P )
% 3.50/1.90 = ( collect @ A @ Q ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_47_ext,axiom,
% 3.50/1.90 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 3.50/1.90 ( ! [X3: A] :
% 3.50/1.90 ( ( F @ X3 )
% 3.50/1.90 = ( G @ X3 ) )
% 3.50/1.90 => ( F = G ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_48_GreatestI2__order,axiom,
% 3.50/1.90 ! [A: $tType] :
% 3.50/1.90 ( ( order @ A )
% 3.50/1.90 => ! [P: A > $o,X2: A,Q: A > $o] :
% 3.50/1.90 ( ( P @ X2 )
% 3.50/1.90 => ( ! [Y4: A] :
% 3.50/1.90 ( ( P @ Y4 )
% 3.50/1.90 => ( ord_less_eq @ A @ Y4 @ X2 ) )
% 3.50/1.90 => ( ! [X3: A] :
% 3.50/1.90 ( ( P @ X3 )
% 3.50/1.90 => ( ! [Y5: A] :
% 3.50/1.90 ( ( P @ Y5 )
% 3.50/1.90 => ( ord_less_eq @ A @ Y5 @ X3 ) )
% 3.50/1.90 => ( Q @ X3 ) ) )
% 3.50/1.90 => ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_49_transfer_Otransfer__Let,axiom,
% 3.50/1.90 ! [C: $tType,A: $tType,B: $tType] :
% 3.50/1.90 ( ( comple187826305attice @ A )
% 3.50/1.90 => ! [Alpha: C > A,F: B > C,F4: B > A,X2: B] :
% 3.50/1.90 ( ! [X3: B] : ( ord_less_eq @ A @ ( Alpha @ ( F @ X3 ) ) @ ( F4 @ X3 ) )
% 3.50/1.90 => ( ord_less_eq @ A @ ( Alpha @ ( F @ X2 ) ) @ ( F4 @ X2 ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_50_transfer_Otransfer__if,axiom,
% 3.50/1.90 ! [C: $tType,A: $tType] :
% 3.50/1.90 ( ( comple187826305attice @ A )
% 3.50/1.90 => ! [B3: $o,Alpha: C > A,S1: C,S12: A,S2: C,S22: A] :
% 3.50/1.90 ( ( B3
% 3.50/1.90 => ( ord_less_eq @ A @ ( Alpha @ S1 ) @ S12 ) )
% 3.50/1.90 => ( ( ~ B3
% 3.50/1.90 => ( ord_less_eq @ A @ ( Alpha @ S2 ) @ S22 ) )
% 3.50/1.90 => ( ord_less_eq @ A @ ( Alpha @ ( if @ C @ B3 @ S1 @ S2 ) ) @ ( if @ A @ B3 @ S12 @ S22 ) ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_51_le__rel__bool__arg__iff,axiom,
% 3.50/1.90 ! [A: $tType] :
% 3.50/1.90 ( ( ord @ A )
% 3.50/1.90 => ( ( ord_less_eq @ ( $o > A ) )
% 3.50/1.90 = ( ^ [X4: $o > A,Y6: $o > A] :
% 3.50/1.90 ( ( ord_less_eq @ A @ ( X4 @ $false ) @ ( Y6 @ $false ) )
% 3.50/1.90 & ( ord_less_eq @ A @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_52_verit__la__disequality,axiom,
% 3.50/1.90 ! [A: $tType] :
% 3.50/1.90 ( ( linorder @ A )
% 3.50/1.90 => ! [A3: A,B3: A] :
% 3.50/1.90 ( ( A3 = B3 )
% 3.50/1.90 | ~ ( ord_less_eq @ A @ A3 @ B3 )
% 3.50/1.90 | ~ ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_53_ord__eq__le__eq__trans,axiom,
% 3.50/1.90 ! [A: $tType] :
% 3.50/1.90 ( ( ord @ A )
% 3.50/1.90 => ! [A3: A,B3: A,C3: A,D2: A] :
% 3.50/1.90 ( ( A3 = B3 )
% 3.50/1.90 => ( ( ord_less_eq @ A @ B3 @ C3 )
% 3.50/1.90 => ( ( C3 = D2 )
% 3.50/1.90 => ( ord_less_eq @ A @ A3 @ D2 ) ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_54_Id__refine,axiom,
% 3.50/1.90 ! [A: $tType,S: refine1665802226e_nres @ A] : ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ ( refine1073749519nc_fun @ A @ A @ ( id @ A ) @ S ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_55_conc__trans__additional_I2_J,axiom,
% 3.50/1.90 ! [A: $tType,B: $tType,A2: refine1665802226e_nres @ A,B2: refine1665802226e_nres @ A,R: set @ ( product_prod @ A @ B ),C2: refine1665802226e_nres @ B] :
% 3.50/1.90 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ A2 @ ( refine1073749519nc_fun @ A @ A @ ( id @ A ) @ B2 ) )
% 3.50/1.90 => ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ B2 @ ( refine1073749519nc_fun @ A @ B @ R @ C2 ) )
% 3.50/1.90 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ A2 @ ( refine1073749519nc_fun @ A @ B @ R @ C2 ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_56_conc__trans__additional_I3_J,axiom,
% 3.50/1.90 ! [A: $tType,B: $tType,A2: refine1665802226e_nres @ A,R: set @ ( product_prod @ A @ B ),B2: refine1665802226e_nres @ B,C2: refine1665802226e_nres @ B] :
% 3.50/1.90 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ A2 @ ( refine1073749519nc_fun @ A @ B @ R @ B2 ) )
% 3.50/1.90 => ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ B2 @ ( refine1073749519nc_fun @ B @ B @ ( id @ B ) @ C2 ) )
% 3.50/1.90 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ A2 @ ( refine1073749519nc_fun @ A @ B @ R @ C2 ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_57_subset__Collect__conv,axiom,
% 3.50/1.90 ! [A: $tType,S: set @ A,P: A > $o] :
% 3.50/1.90 ( ( ord_less_eq @ ( set @ A ) @ S @ ( collect @ A @ P ) )
% 3.50/1.90 = ( ! [X: A] :
% 3.50/1.90 ( ( member @ A @ X @ S )
% 3.50/1.90 => ( P @ X ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_58_conc__trans__additional_I6_J,axiom,
% 3.50/1.90 ! [E3: $tType,A2: refine1665802226e_nres @ E3,B2: refine1665802226e_nres @ E3,C2: refine1665802226e_nres @ E3] :
% 3.50/1.90 ( ( ord_less_eq @ ( refine1665802226e_nres @ E3 ) @ A2 @ B2 )
% 3.50/1.90 => ( ( ord_less_eq @ ( refine1665802226e_nres @ E3 ) @ B2 @ ( refine1073749519nc_fun @ E3 @ E3 @ ( id @ E3 ) @ C2 ) )
% 3.50/1.90 => ( ord_less_eq @ ( refine1665802226e_nres @ E3 ) @ A2 @ C2 ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_59_conc__trans__additional_I5_J,axiom,
% 3.50/1.90 ! [D: $tType,A2: refine1665802226e_nres @ D,B2: refine1665802226e_nres @ D,C2: refine1665802226e_nres @ D] :
% 3.50/1.90 ( ( ord_less_eq @ ( refine1665802226e_nres @ D ) @ A2 @ ( refine1073749519nc_fun @ D @ D @ ( id @ D ) @ B2 ) )
% 3.50/1.90 => ( ( ord_less_eq @ ( refine1665802226e_nres @ D ) @ B2 @ C2 )
% 3.50/1.90 => ( ord_less_eq @ ( refine1665802226e_nres @ D ) @ A2 @ C2 ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_60_conc__trans__additional_I4_J,axiom,
% 3.50/1.90 ! [C: $tType,A2: refine1665802226e_nres @ C,B2: refine1665802226e_nres @ C,C2: refine1665802226e_nres @ C] :
% 3.50/1.90 ( ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ A2 @ ( refine1073749519nc_fun @ C @ C @ ( id @ C ) @ B2 ) )
% 3.50/1.90 => ( ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ B2 @ ( refine1073749519nc_fun @ C @ C @ ( id @ C ) @ C2 ) )
% 3.50/1.90 => ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ A2 @ C2 ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_61_subsetI,axiom,
% 3.50/1.90 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 3.50/1.90 ( ! [X3: A] :
% 3.50/1.90 ( ( member @ A @ X3 @ A2 )
% 3.50/1.90 => ( member @ A @ X3 @ B2 ) )
% 3.50/1.90 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_62_subset__antisym,axiom,
% 3.50/1.90 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 3.50/1.90 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 3.50/1.90 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 3.50/1.90 => ( A2 = B2 ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_63_bijective__Id,axiom,
% 3.50/1.90 ! [A: $tType] : ( bijective @ A @ A @ ( id @ A ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_64_relprop__id__orient,axiom,
% 3.50/1.90 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 3.50/1.90 ( ( R
% 3.50/1.90 = ( id @ A ) )
% 3.50/1.90 => ( ( id @ A )
% 3.50/1.90 = R ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_65_relprop__triggers_I2_J,axiom,
% 3.50/1.90 ! [C: $tType,R: set @ ( product_prod @ C @ C )] :
% 3.50/1.90 ( ( R
% 3.50/1.90 = ( id @ C ) )
% 3.50/1.90 => ( R
% 3.50/1.90 = ( id @ C ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_66_relprop__triggers_I3_J,axiom,
% 3.50/1.90 ! [D: $tType,R: set @ ( product_prod @ D @ D )] :
% 3.50/1.90 ( ( R
% 3.50/1.90 = ( id @ D ) )
% 3.50/1.90 => ( ( id @ D )
% 3.50/1.90 = R ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_67_relprop__triggers_I6_J,axiom,
% 3.50/1.90 ! [I: $tType,R: set @ I,R2: set @ I] :
% 3.50/1.90 ( ( ord_less_eq @ ( set @ I ) @ R @ R2 )
% 3.50/1.90 => ( ord_less_eq @ ( set @ I ) @ R @ R2 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_68_Collect__mono__iff,axiom,
% 3.50/1.90 ! [A: $tType,P: A > $o,Q: A > $o] :
% 3.50/1.90 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 3.50/1.90 = ( ! [X: A] :
% 3.50/1.90 ( ( P @ X )
% 3.50/1.90 => ( Q @ X ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_69_set__eq__subset,axiom,
% 3.50/1.90 ! [A: $tType] :
% 3.50/1.90 ( ( ^ [Y: set @ A,Z: set @ A] : ( Y = Z ) )
% 3.50/1.90 = ( ^ [A6: set @ A,B6: set @ A] :
% 3.50/1.90 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 3.50/1.90 & ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_70_subset__trans,axiom,
% 3.50/1.90 ! [A: $tType,A2: set @ A,B2: set @ A,C2: set @ A] :
% 3.50/1.90 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 3.50/1.90 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C2 )
% 3.50/1.90 => ( ord_less_eq @ ( set @ A ) @ A2 @ C2 ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_71_Collect__mono,axiom,
% 3.50/1.90 ! [A: $tType,P: A > $o,Q: A > $o] :
% 3.50/1.90 ( ! [X3: A] :
% 3.50/1.90 ( ( P @ X3 )
% 3.50/1.90 => ( Q @ X3 ) )
% 3.50/1.90 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_72_subset__refl,axiom,
% 3.50/1.90 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_73_subset__UNIV,axiom,
% 3.50/1.90 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_74_subset__iff,axiom,
% 3.50/1.90 ! [A: $tType] :
% 3.50/1.90 ( ( ord_less_eq @ ( set @ A ) )
% 3.50/1.90 = ( ^ [A6: set @ A,B6: set @ A] :
% 3.50/1.90 ! [T3: A] :
% 3.50/1.90 ( ( member @ A @ T3 @ A6 )
% 3.50/1.90 => ( member @ A @ T3 @ B6 ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_75_equalityD2,axiom,
% 3.50/1.90 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 3.50/1.90 ( ( A2 = B2 )
% 3.50/1.90 => ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_76_equalityD1,axiom,
% 3.50/1.90 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 3.50/1.90 ( ( A2 = B2 )
% 3.50/1.90 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_77_subset__eq,axiom,
% 3.50/1.90 ! [A: $tType] :
% 3.50/1.90 ( ( ord_less_eq @ ( set @ A ) )
% 3.50/1.90 = ( ^ [A6: set @ A,B6: set @ A] :
% 3.50/1.90 ! [X: A] :
% 3.50/1.90 ( ( member @ A @ X @ A6 )
% 3.50/1.90 => ( member @ A @ X @ B6 ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_78_equalityE,axiom,
% 3.50/1.90 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 3.50/1.90 ( ( A2 = B2 )
% 3.50/1.90 => ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 3.50/1.90 => ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_79_subsetD,axiom,
% 3.50/1.90 ! [A: $tType,A2: set @ A,B2: set @ A,C3: A] :
% 3.50/1.90 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 3.50/1.90 => ( ( member @ A @ C3 @ A2 )
% 3.50/1.90 => ( member @ A @ C3 @ B2 ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_80_in__mono,axiom,
% 3.50/1.90 ! [A: $tType,A2: set @ A,B2: set @ A,X2: A] :
% 3.50/1.90 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 3.50/1.90 => ( ( member @ A @ X2 @ A2 )
% 3.50/1.90 => ( member @ A @ X2 @ B2 ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_81_Id__SPEC__refine,axiom,
% 3.50/1.90 ! [A: $tType,S: refine1665802226e_nres @ A,Phi: A > $o] :
% 3.50/1.90 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) )
% 3.50/1.90 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ ( refine1073749519nc_fun @ A @ A @ ( id @ A ) @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_82_abs__trans__additional_I2_J,axiom,
% 3.50/1.90 ! [A: $tType,B: $tType,A2: refine1665802226e_nres @ A,B2: refine1665802226e_nres @ A,R: set @ ( product_prod @ A @ B ),C2: refine1665802226e_nres @ B] :
% 3.50/1.90 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine81118332bs_fun @ A @ A @ ( id @ A ) @ A2 ) @ B2 )
% 3.50/1.90 => ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine81118332bs_fun @ A @ B @ R @ B2 ) @ C2 )
% 3.50/1.90 => ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine81118332bs_fun @ A @ B @ R @ A2 ) @ C2 ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_83_abs__trans__additional_I3_J,axiom,
% 3.50/1.90 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),A2: refine1665802226e_nres @ A,B2: refine1665802226e_nres @ B,C2: refine1665802226e_nres @ B] :
% 3.50/1.90 ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine81118332bs_fun @ A @ B @ R @ A2 ) @ B2 )
% 3.50/1.90 => ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine81118332bs_fun @ B @ B @ ( id @ B ) @ B2 ) @ C2 )
% 3.50/1.90 => ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine81118332bs_fun @ A @ B @ R @ A2 ) @ C2 ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_84_abs__trans__additional_I4_J,axiom,
% 3.50/1.90 ! [C: $tType,A2: refine1665802226e_nres @ C,B2: refine1665802226e_nres @ C,C2: refine1665802226e_nres @ C] :
% 3.50/1.90 ( ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ ( refine81118332bs_fun @ C @ C @ ( id @ C ) @ A2 ) @ B2 )
% 3.50/1.90 => ( ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ ( refine81118332bs_fun @ C @ C @ ( id @ C ) @ B2 ) @ C2 )
% 3.50/1.90 => ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ A2 @ C2 ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_85_abs__trans__additional_I5_J,axiom,
% 3.50/1.90 ! [D: $tType,A2: refine1665802226e_nres @ D,B2: refine1665802226e_nres @ D,C2: refine1665802226e_nres @ D] :
% 3.50/1.90 ( ( ord_less_eq @ ( refine1665802226e_nres @ D ) @ ( refine81118332bs_fun @ D @ D @ ( id @ D ) @ A2 ) @ B2 )
% 3.50/1.90 => ( ( ord_less_eq @ ( refine1665802226e_nres @ D ) @ B2 @ C2 )
% 3.50/1.90 => ( ord_less_eq @ ( refine1665802226e_nres @ D ) @ A2 @ C2 ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_86_abs__trans__additional_I6_J,axiom,
% 3.50/1.90 ! [E3: $tType,A2: refine1665802226e_nres @ E3,B2: refine1665802226e_nres @ E3,C2: refine1665802226e_nres @ E3] :
% 3.50/1.90 ( ( ord_less_eq @ ( refine1665802226e_nres @ E3 ) @ A2 @ B2 )
% 3.50/1.90 => ( ( ord_less_eq @ ( refine1665802226e_nres @ E3 ) @ ( refine81118332bs_fun @ E3 @ E3 @ ( id @ E3 ) @ B2 ) @ C2 )
% 3.50/1.90 => ( ord_less_eq @ ( refine1665802226e_nres @ E3 ) @ A2 @ C2 ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_87_rel__restrict__mono2,axiom,
% 3.50/1.90 ! [A: $tType,R: set @ A,S: set @ A,A2: set @ ( product_prod @ A @ A )] :
% 3.50/1.90 ( ( ord_less_eq @ ( set @ A ) @ R @ S )
% 3.50/1.90 => ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( rel_restrict @ A @ A2 @ S ) @ ( rel_restrict @ A @ A2 @ R ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_88_UNIV__I,axiom,
% 3.50/1.90 ! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_89_nres_Oinject,axiom,
% 3.50/1.90 ! [A: $tType,X22: set @ A,Y22: set @ A] :
% 3.50/1.90 ( ( ( refine605929679le_RES @ A @ X22 )
% 3.50/1.90 = ( refine605929679le_RES @ A @ Y22 ) )
% 3.50/1.90 = ( X22 = Y22 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_90_nres__more__simps_I4_J,axiom,
% 3.50/1.90 ! [A: $tType,X5: set @ A,Y7: set @ A] :
% 3.50/1.90 ( ( ( refine605929679le_RES @ A @ X5 )
% 3.50/1.90 = ( refine605929679le_RES @ A @ Y7 ) )
% 3.50/1.90 = ( X5 = Y7 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_91_abs__fun__simps_I1_J,axiom,
% 3.50/1.90 ! [B: $tType,A: $tType,R: set @ ( product_prod @ B @ A )] :
% 3.50/1.90 ( ( refine81118332bs_fun @ B @ A @ R @ ( top_top @ ( refine1665802226e_nres @ B ) ) )
% 3.50/1.90 = ( top_top @ ( refine1665802226e_nres @ A ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_92_relprop__UNIV__orient,axiom,
% 3.50/1.90 ! [A: $tType,R: set @ A] :
% 3.50/1.90 ( ( R
% 3.50/1.90 = ( top_top @ ( set @ A ) ) )
% 3.50/1.90 => ( ( top_top @ ( set @ A ) )
% 3.50/1.90 = R ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_93_eq__UNIV__iff,axiom,
% 3.50/1.90 ! [A: $tType,S: set @ A] :
% 3.50/1.90 ( ( S
% 3.50/1.90 = ( top_top @ ( set @ A ) ) )
% 3.50/1.90 = ( ! [X: A] : ( member @ A @ X @ S ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_94_UNIV__witness,axiom,
% 3.50/1.90 ! [A: $tType] :
% 3.50/1.90 ? [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_95_top__set__def,axiom,
% 3.50/1.90 ! [A: $tType] :
% 3.50/1.90 ( ( top_top @ ( set @ A ) )
% 3.50/1.90 = ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_96_UNIV__eq__I,axiom,
% 3.50/1.90 ! [A: $tType,A2: set @ A] :
% 3.50/1.90 ( ! [X3: A] : ( member @ A @ X3 @ A2 )
% 3.50/1.90 => ( ( top_top @ ( set @ A ) )
% 3.50/1.90 = A2 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_97_if__rule,axiom,
% 3.50/1.90 ! [A: $tType,B3: $o,S12: refine1665802226e_nres @ A,Phi: A > $o,S22: refine1665802226e_nres @ A] :
% 3.50/1.90 ( ( B3
% 3.50/1.90 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S12 @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) ) )
% 3.50/1.90 => ( ( ~ B3
% 3.50/1.90 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S22 @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) ) )
% 3.50/1.90 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( if @ ( refine1665802226e_nres @ A ) @ B3 @ S12 @ S22 ) @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_98_RES__rule,axiom,
% 3.50/1.90 ! [A: $tType,S: set @ A,Phi: A > $o] :
% 3.50/1.90 ( ! [X3: A] :
% 3.50/1.90 ( ( member @ A @ X3 @ S )
% 3.50/1.90 => ( Phi @ X3 ) )
% 3.50/1.90 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ S ) @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_99_SPEC__rule,axiom,
% 3.50/1.90 ! [A: $tType,Phi: A > $o,Phi2: A > $o] :
% 3.50/1.90 ( ! [X3: A] :
% 3.50/1.90 ( ( Phi @ X3 )
% 3.50/1.90 => ( Phi2 @ X3 ) )
% 3.50/1.90 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi2 ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_100_SPEC__trans,axiom,
% 3.50/1.90 ! [A: $tType,X2: refine1665802226e_nres @ A,Y2: refine1665802226e_nres @ A,Postcond: A > $o] :
% 3.50/1.90 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ X2 @ Y2 )
% 3.50/1.90 => ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ Y2 @ ( refine605929679le_RES @ A @ ( collect @ A @ Postcond ) ) )
% 3.50/1.90 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ X2 @ ( refine605929679le_RES @ A @ ( collect @ A @ Postcond ) ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_101_SPEC__cons__rule,axiom,
% 3.50/1.90 ! [A: $tType,M2: refine1665802226e_nres @ A,Phi: A > $o,Psi: A > $o] :
% 3.50/1.90 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M2 @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) )
% 3.50/1.90 => ( ! [X3: A] :
% 3.50/1.90 ( ( Phi @ X3 )
% 3.50/1.90 => ( Psi @ X3 ) )
% 3.50/1.90 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M2 @ ( refine605929679le_RES @ A @ ( collect @ A @ Psi ) ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_102_nres__inequalities_I1_J,axiom,
% 3.50/1.90 ! [A: $tType,X5: set @ A] :
% 3.50/1.90 ( ( top_top @ ( refine1665802226e_nres @ A ) )
% 3.50/1.90 != ( refine605929679le_RES @ A @ X5 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_103_nres__cases,axiom,
% 3.50/1.90 ! [A: $tType,M: refine1665802226e_nres @ A] :
% 3.50/1.90 ( ( M
% 3.50/1.90 != ( top_top @ ( refine1665802226e_nres @ A ) ) )
% 3.50/1.90 => ~ ! [X6: set @ A] :
% 3.50/1.90 ( M
% 3.50/1.90 != ( refine605929679le_RES @ A @ X6 ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_104_rel__restrict__sub,axiom,
% 3.50/1.90 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A2: set @ A] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( rel_restrict @ A @ R @ A2 ) @ R ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_105_rel__restrict__mono,axiom,
% 3.50/1.90 ! [A: $tType,A2: set @ ( product_prod @ A @ A ),B2: set @ ( product_prod @ A @ A ),R: set @ A] :
% 3.50/1.90 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ A2 @ B2 )
% 3.50/1.90 => ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( rel_restrict @ A @ A2 @ R ) @ ( rel_restrict @ A @ B2 @ R ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_106_abs__trans__additional_I1_J,axiom,
% 3.50/1.90 ! [A: $tType,B: $tType,A2: refine1665802226e_nres @ A,B2: refine1665802226e_nres @ A,R: set @ ( product_prod @ A @ B ),C2: refine1665802226e_nres @ B] :
% 3.50/1.90 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ A2 @ B2 )
% 3.50/1.90 => ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine81118332bs_fun @ A @ B @ R @ B2 ) @ C2 )
% 3.50/1.90 => ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine81118332bs_fun @ A @ B @ R @ A2 ) @ C2 ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_107_abs__trans,axiom,
% 3.50/1.90 ! [B: $tType,A: $tType,C: $tType,R: set @ ( product_prod @ B @ A ),C2: refine1665802226e_nres @ B,B2: refine1665802226e_nres @ A,R2: set @ ( product_prod @ A @ C ),A2: refine1665802226e_nres @ C] :
% 3.50/1.90 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine81118332bs_fun @ B @ A @ R @ C2 ) @ B2 )
% 3.50/1.90 => ( ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ ( refine81118332bs_fun @ A @ C @ R2 @ B2 ) @ A2 )
% 3.50/1.90 => ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ ( refine81118332bs_fun @ A @ C @ R2 @ ( refine81118332bs_fun @ B @ A @ R @ C2 ) ) @ A2 ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_108_nres__order__simps_I5_J,axiom,
% 3.50/1.90 ! [E3: $tType,X5: set @ E3,Y7: set @ E3] :
% 3.50/1.90 ( ( ord_less_eq @ ( refine1665802226e_nres @ E3 ) @ ( refine605929679le_RES @ E3 @ X5 ) @ ( refine605929679le_RES @ E3 @ Y7 ) )
% 3.50/1.90 = ( ord_less_eq @ ( set @ E3 ) @ X5 @ Y7 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_109_less__eq__nres_Osimps_I2_J,axiom,
% 3.50/1.90 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 3.50/1.90 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ A3 ) @ ( refine605929679le_RES @ A @ B3 ) )
% 3.50/1.90 = ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_110_iso__tuple__UNIV__I,axiom,
% 3.50/1.90 ! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_111_nf__inres__SPEC,axiom,
% 3.50/1.90 ! [A: $tType,Phi: A > $o,X2: A] :
% 3.50/1.90 ( ( refine406925620_inres @ A @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) @ X2 )
% 3.50/1.90 = ( Phi @ X2 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_112_nf__inres__RES,axiom,
% 3.50/1.90 ! [A: $tType,X5: set @ A,X2: A] :
% 3.50/1.90 ( ( refine406925620_inres @ A @ ( refine605929679le_RES @ A @ X5 ) @ X2 )
% 3.50/1.90 = ( member @ A @ X2 @ X5 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_113_abs__fun__simps_I3_J,axiom,
% 3.50/1.90 ! [B: $tType,A: $tType,X5: set @ B,R: set @ ( product_prod @ B @ A )] :
% 3.50/1.90 ( ~ ( ord_less_eq @ ( set @ B ) @ X5 @ ( domain @ B @ A @ R ) )
% 3.50/1.90 => ( ( refine81118332bs_fun @ B @ A @ R @ ( refine605929679le_RES @ B @ X5 ) )
% 3.50/1.90 = ( top_top @ ( refine1665802226e_nres @ A ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_114_the__RES_Osimps,axiom,
% 3.50/1.90 ! [A: $tType,X5: set @ A] :
% 3.50/1.90 ( ( refine1672542526he_RES @ A @ ( refine605929679le_RES @ A @ X5 ) )
% 3.50/1.90 = X5 ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_115_nres__order__simps_I21_J,axiom,
% 3.50/1.90 ! [X7: $tType,X2: X7,Y7: set @ X7] :
% 3.50/1.90 ( ( ord_less_eq @ ( refine1665802226e_nres @ X7 ) @ ( refine1687780735RETURN @ X7 @ X2 ) @ ( refine605929679le_RES @ X7 @ Y7 ) )
% 3.50/1.90 = ( member @ X7 @ X2 @ Y7 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_116_nres__more__simps_I6_J,axiom,
% 3.50/1.90 ! [A: $tType,X2: A,Y2: A] :
% 3.50/1.90 ( ( ( refine1687780735RETURN @ A @ X2 )
% 3.50/1.90 = ( refine1687780735RETURN @ A @ Y2 ) )
% 3.50/1.90 = ( X2 = Y2 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_117_top1I,axiom,
% 3.50/1.90 ! [A: $tType,X2: A] : ( top_top @ ( A > $o ) @ X2 ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_118_nres__order__simps_I20_J,axiom,
% 3.50/1.90 ! [W: $tType,X2: W,Y2: W] :
% 3.50/1.90 ( ( ord_less_eq @ ( refine1665802226e_nres @ W ) @ ( refine1687780735RETURN @ W @ X2 ) @ ( refine1687780735RETURN @ W @ Y2 ) )
% 3.50/1.90 = ( X2 = Y2 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_119_nres__inequalities_I3_J,axiom,
% 3.50/1.90 ! [C: $tType,X2: C] :
% 3.50/1.90 ( ( top_top @ ( refine1665802226e_nres @ C ) )
% 3.50/1.90 != ( refine1687780735RETURN @ C @ X2 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_120_RETURN__to__SPEC__rule,axiom,
% 3.50/1.90 ! [A: $tType,M2: refine1665802226e_nres @ A,V: A] :
% 3.50/1.90 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M2
% 3.50/1.90 @ ( refine605929679le_RES @ A
% 3.50/1.90 @ ( collect @ A
% 3.50/1.90 @ ( ^ [Y: A,Z: A] : ( Y = Z )
% 3.50/1.90 @ V ) ) ) )
% 3.50/1.90 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M2 @ ( refine1687780735RETURN @ A @ V ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_121_RETURN__rule,axiom,
% 3.50/1.90 ! [A: $tType,Phi: A > $o,X2: A] :
% 3.50/1.90 ( ( Phi @ X2 )
% 3.50/1.90 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X2 ) @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_122_Domain__Id,axiom,
% 3.50/1.90 ! [A: $tType] :
% 3.50/1.90 ( ( domain @ A @ A @ ( id @ A ) )
% 3.50/1.90 = ( top_top @ ( set @ A ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_123_Domain__mono,axiom,
% 3.50/1.90 ! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
% 3.50/1.90 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S3 )
% 3.50/1.90 => ( ord_less_eq @ ( set @ A ) @ ( domain @ A @ B @ R3 ) @ ( domain @ A @ B @ S3 ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_124_top__empty__eq,axiom,
% 3.50/1.90 ! [A: $tType] :
% 3.50/1.90 ( ( top_top @ ( A > $o ) )
% 3.50/1.90 = ( ^ [X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_125_abs__fun__simps_I2_J,axiom,
% 3.50/1.90 ! [A: $tType,B: $tType,X5: set @ B,R: set @ ( product_prod @ B @ A )] :
% 3.50/1.90 ( ( ord_less_eq @ ( set @ B ) @ X5 @ ( domain @ B @ A @ R ) )
% 3.50/1.90 => ( ( refine81118332bs_fun @ B @ A @ R @ ( refine605929679le_RES @ B @ X5 ) )
% 3.50/1.90 = ( refine605929679le_RES @ A @ ( image @ B @ A @ R @ X5 ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_126_Image__Id,axiom,
% 3.50/1.90 ! [A: $tType,A2: set @ A] :
% 3.50/1.90 ( ( image @ A @ A @ ( id @ A ) @ A2 )
% 3.50/1.90 = A2 ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_127_Image__mono,axiom,
% 3.50/1.90 ! [B: $tType,A: $tType,R4: set @ ( product_prod @ A @ B ),R3: set @ ( product_prod @ A @ B ),A7: set @ A,A2: set @ A] :
% 3.50/1.90 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R4 @ R3 )
% 3.50/1.90 => ( ( ord_less_eq @ ( set @ A ) @ A7 @ A2 )
% 3.50/1.90 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ R4 @ A7 ) @ ( image @ A @ B @ R3 @ A2 ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_128_top__conj_I1_J,axiom,
% 3.50/1.90 ! [A: $tType,X2: A,P: $o] :
% 3.50/1.90 ( ( ( top_top @ ( A > $o ) @ X2 )
% 3.50/1.90 & P )
% 3.50/1.90 = P ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_129_top__conj_I2_J,axiom,
% 3.50/1.90 ! [A: $tType,P: $o,X2: A] :
% 3.50/1.90 ( ( P
% 3.50/1.90 & ( top_top @ ( A > $o ) @ X2 ) )
% 3.50/1.90 = P ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_130_RETURN__SPEC__refine,axiom,
% 3.50/1.90 ! [B: $tType,A: $tType,X2: B,R: set @ ( product_prod @ B @ A ),Phi: A > $o] :
% 3.50/1.90 ( ? [X8: A] :
% 3.50/1.90 ( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X2 @ X8 ) @ R )
% 3.50/1.90 & ( Phi @ X8 ) )
% 3.50/1.90 => ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine1687780735RETURN @ B @ X2 ) @ ( refine1073749519nc_fun @ B @ A @ R @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_131_the__RES__inv,axiom,
% 3.50/1.90 ! [A: $tType,M2: refine1665802226e_nres @ A] :
% 3.50/1.90 ( ( refine1102455758nofail @ A @ M2 )
% 3.50/1.90 => ( ( refine605929679le_RES @ A @ ( refine1672542526he_RES @ A @ M2 ) )
% 3.50/1.90 = M2 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_132_pair__in__Id__conv,axiom,
% 3.50/1.90 ! [A: $tType,A3: A,B3: A] :
% 3.50/1.90 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ ( id @ A ) )
% 3.50/1.90 = ( A3 = B3 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_133_IdI,axiom,
% 3.50/1.90 ! [A: $tType,A3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ ( id @ A ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_134_nofail__simps_I2_J,axiom,
% 3.50/1.90 ! [B: $tType,X5: set @ B] : ( refine1102455758nofail @ B @ ( refine605929679le_RES @ B @ X5 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_135_nofail__simps_I1_J,axiom,
% 3.50/1.90 ! [A: $tType] :
% 3.50/1.90 ~ ( refine1102455758nofail @ A @ ( top_top @ ( refine1665802226e_nres @ A ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_136_nofail__simps_I3_J,axiom,
% 3.50/1.90 ! [C: $tType,X2: C] : ( refine1102455758nofail @ C @ ( refine1687780735RETURN @ C @ X2 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_137_DomainE,axiom,
% 3.50/1.90 ! [B: $tType,A: $tType,A3: A,R3: set @ ( product_prod @ A @ B )] :
% 3.50/1.90 ( ( member @ A @ A3 @ ( domain @ A @ B @ R3 ) )
% 3.50/1.90 => ~ ! [B5: B] :
% 3.50/1.90 ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B5 ) @ R3 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_138_Domain__iff,axiom,
% 3.50/1.90 ! [A: $tType,B: $tType,A3: A,R3: set @ ( product_prod @ A @ B )] :
% 3.50/1.90 ( ( member @ A @ A3 @ ( domain @ A @ B @ R3 ) )
% 3.50/1.90 = ( ? [Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ Y3 ) @ R3 ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_139_Domain_Ocases,axiom,
% 3.50/1.90 ! [B: $tType,A: $tType,A3: A,R3: set @ ( product_prod @ A @ B )] :
% 3.50/1.90 ( ( member @ A @ A3 @ ( domain @ A @ B @ R3 ) )
% 3.50/1.90 => ~ ! [B5: B] :
% 3.50/1.90 ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B5 ) @ R3 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_140_Domain_Osimps,axiom,
% 3.50/1.90 ! [B: $tType,A: $tType,A3: A,R3: set @ ( product_prod @ A @ B )] :
% 3.50/1.90 ( ( member @ A @ A3 @ ( domain @ A @ B @ R3 ) )
% 3.50/1.90 = ( ? [A4: A,B4: B] :
% 3.50/1.90 ( ( A3 = A4 )
% 3.50/1.90 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B4 ) @ R3 ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_141_Domain_ODomainI,axiom,
% 3.50/1.90 ! [B: $tType,A: $tType,A3: A,B3: B,R3: set @ ( product_prod @ A @ B )] :
% 3.50/1.90 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R3 )
% 3.50/1.90 => ( member @ A @ A3 @ ( domain @ A @ B @ R3 ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_142_Domain_Oinducts,axiom,
% 3.50/1.90 ! [B: $tType,A: $tType,X2: A,R3: set @ ( product_prod @ A @ B ),P: A > $o] :
% 3.50/1.90 ( ( member @ A @ X2 @ ( domain @ A @ B @ R3 ) )
% 3.50/1.90 => ( ! [A5: A,B5: B] :
% 3.50/1.90 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B5 ) @ R3 )
% 3.50/1.90 => ( P @ A5 ) )
% 3.50/1.90 => ( P @ X2 ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_143_subrelI,axiom,
% 3.50/1.90 ! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
% 3.50/1.90 ( ! [X3: A,Y4: B] :
% 3.50/1.90 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y4 ) @ R3 )
% 3.50/1.90 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y4 ) @ S3 ) )
% 3.50/1.90 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S3 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_144_IdE,axiom,
% 3.50/1.90 ! [A: $tType,P2: product_prod @ A @ A] :
% 3.50/1.90 ( ( member @ ( product_prod @ A @ A ) @ P2 @ ( id @ A ) )
% 3.50/1.90 => ~ ! [X3: A] :
% 3.50/1.90 ( P2
% 3.50/1.90 != ( product_Pair @ A @ A @ X3 @ X3 ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_145_rel__restrict__notR_I2_J,axiom,
% 3.50/1.90 ! [A: $tType,X2: A,Y2: A,A2: set @ ( product_prod @ A @ A ),R: set @ A] :
% 3.50/1.90 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ ( rel_restrict @ A @ A2 @ R ) )
% 3.50/1.90 => ~ ( member @ A @ Y2 @ R ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_146_rel__restrict__notR_I1_J,axiom,
% 3.50/1.90 ! [A: $tType,X2: A,Y2: A,A2: set @ ( product_prod @ A @ A ),R: set @ A] :
% 3.50/1.90 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ ( rel_restrict @ A @ A2 @ R ) )
% 3.50/1.90 => ~ ( member @ A @ X2 @ R ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_147_rel__restrictI,axiom,
% 3.50/1.90 ! [A: $tType,X2: A,R: set @ A,Y2: A,E4: set @ ( product_prod @ A @ A )] :
% 3.50/1.90 ( ~ ( member @ A @ X2 @ R )
% 3.50/1.90 => ( ~ ( member @ A @ Y2 @ R )
% 3.50/1.90 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ E4 )
% 3.50/1.90 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ ( rel_restrict @ A @ E4 @ R ) ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_148_rel__restrict__lift,axiom,
% 3.50/1.90 ! [A: $tType,X2: A,Y2: A,E4: set @ ( product_prod @ A @ A ),R: set @ A] :
% 3.50/1.90 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ ( rel_restrict @ A @ E4 @ R ) )
% 3.50/1.90 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ E4 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_149_pairself_Oinduct,axiom,
% 3.50/1.90 ! [B: $tType,A: $tType,P: ( A > B ) > ( product_prod @ A @ A ) > $o,A0: A > B,A1: product_prod @ A @ A] :
% 3.50/1.90 ( ! [F5: A > B,A5: A,B5: A] : ( P @ F5 @ ( product_Pair @ A @ A @ A5 @ B5 ) )
% 3.50/1.90 => ( P @ A0 @ A1 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_150_pairself_Ocases,axiom,
% 3.50/1.90 ! [B: $tType,A: $tType,X2: product_prod @ ( A > B ) @ ( product_prod @ A @ A )] :
% 3.50/1.90 ~ ! [F5: A > B,A5: A,B5: A] :
% 3.50/1.90 ( X2
% 3.50/1.90 != ( product_Pair @ ( A > B ) @ ( product_prod @ A @ A ) @ F5 @ ( product_Pair @ A @ A @ A5 @ B5 ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_151_bex2I,axiom,
% 3.50/1.90 ! [A: $tType,B: $tType,A3: A,B3: B,S: set @ ( product_prod @ A @ B ),P: A > B > $o] :
% 3.50/1.90 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ S )
% 3.50/1.90 => ( ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ S )
% 3.50/1.90 => ( P @ A3 @ B3 ) )
% 3.50/1.90 => ? [A5: A,B5: B] :
% 3.50/1.90 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B5 ) @ S )
% 3.50/1.90 & ( P @ A5 @ B5 ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_152_rel__cong,axiom,
% 3.50/1.90 ! [A: $tType,B: $tType,F: A > B,G: A > B,X2: A,Y2: A] :
% 3.50/1.90 ( ( member @ ( product_prod @ ( A > B ) @ ( A > B ) ) @ ( product_Pair @ ( A > B ) @ ( A > B ) @ F @ G ) @ ( id @ ( A > B ) ) )
% 3.50/1.90 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ ( id @ A ) )
% 3.50/1.90 => ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ X2 ) @ ( G @ Y2 ) ) @ ( id @ B ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_153_rel__fun__cong,axiom,
% 3.50/1.90 ! [A: $tType,B: $tType,F: A > B,G: A > B,X2: A] :
% 3.50/1.90 ( ( member @ ( product_prod @ ( A > B ) @ ( A > B ) ) @ ( product_Pair @ ( A > B ) @ ( A > B ) @ F @ G ) @ ( id @ ( A > B ) ) )
% 3.50/1.90 => ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ X2 ) @ ( G @ X2 ) ) @ ( id @ B ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_154_nofail__antimono__fun,axiom,
% 3.50/1.90 ! [B: $tType,A: $tType,F: A > ( refine1665802226e_nres @ B ),G: A > ( refine1665802226e_nres @ B ),X2: A] :
% 3.50/1.90 ( ( ord_less_eq @ ( A > ( refine1665802226e_nres @ B ) ) @ F @ G )
% 3.50/1.90 => ( ( refine1102455758nofail @ B @ ( G @ X2 ) )
% 3.50/1.90 => ( refine1102455758nofail @ B @ ( F @ X2 ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_155_pw__conc__nofail,axiom,
% 3.50/1.90 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),S: refine1665802226e_nres @ B] :
% 3.50/1.90 ( ( refine1102455758nofail @ A @ ( refine1073749519nc_fun @ A @ B @ R @ S ) )
% 3.50/1.90 = ( refine1102455758nofail @ B @ S ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_156_intro__nofail_I2_J,axiom,
% 3.50/1.90 ! [A: $tType,S: refine1665802226e_nres @ A] :
% 3.50/1.90 ( ( ( top_top @ ( refine1665802226e_nres @ A ) )
% 3.50/1.90 != S )
% 3.50/1.90 = ( refine1102455758nofail @ A @ S ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_157_nofail__def,axiom,
% 3.50/1.90 ! [A: $tType] :
% 3.50/1.90 ( ( refine1102455758nofail @ A )
% 3.50/1.90 = ( ^ [S4: refine1665802226e_nres @ A] :
% 3.50/1.90 ( S4
% 3.50/1.90 != ( top_top @ ( refine1665802226e_nres @ A ) ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_158_not__nofail__iff,axiom,
% 3.50/1.90 ! [A: $tType,S: refine1665802226e_nres @ A] :
% 3.50/1.90 ( ( ~ ( refine1102455758nofail @ A @ S ) )
% 3.50/1.90 = ( S
% 3.50/1.90 = ( top_top @ ( refine1665802226e_nres @ A ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_159_rel__arg__cong,axiom,
% 3.50/1.90 ! [A: $tType,B: $tType,X2: A,Y2: A,F: A > B] :
% 3.50/1.90 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ ( id @ A ) )
% 3.50/1.90 => ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ X2 ) @ ( F @ Y2 ) ) @ ( id @ B ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_160_pwD1,axiom,
% 3.50/1.90 ! [A: $tType,S: refine1665802226e_nres @ A,S5: refine1665802226e_nres @ A] :
% 3.50/1.90 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ S5 )
% 3.50/1.90 => ( ( refine1102455758nofail @ A @ S5 )
% 3.50/1.90 => ( refine1102455758nofail @ A @ S ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_161_le__nofailI,axiom,
% 3.50/1.90 ! [A: $tType,M3: refine1665802226e_nres @ A,M: refine1665802226e_nres @ A] :
% 3.50/1.90 ( ( ( refine1102455758nofail @ A @ M3 )
% 3.50/1.90 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M @ M3 ) )
% 3.50/1.90 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M @ M3 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_162_nofail__RES__conv,axiom,
% 3.50/1.90 ! [A: $tType] :
% 3.50/1.90 ( ( refine1102455758nofail @ A )
% 3.50/1.90 = ( ^ [M4: refine1665802226e_nres @ A] :
% 3.50/1.90 ? [M5: set @ A] :
% 3.50/1.90 ( M4
% 3.50/1.90 = ( refine605929679le_RES @ A @ M5 ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_163_bijective__def,axiom,
% 3.50/1.90 ! [B: $tType,A: $tType] :
% 3.50/1.90 ( ( bijective @ A @ B )
% 3.50/1.90 = ( ^ [R5: set @ ( product_prod @ A @ B )] :
% 3.50/1.90 ( ! [X: A,Y3: B,Z3: B] :
% 3.50/1.90 ( ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y3 ) @ R5 )
% 3.50/1.90 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Z3 ) @ R5 ) )
% 3.50/1.90 => ( Y3 = Z3 ) )
% 3.50/1.90 & ! [X: A,Y3: A,Z3: B] :
% 3.50/1.90 ( ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Z3 ) @ R5 )
% 3.50/1.90 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y3 @ Z3 ) @ R5 ) )
% 3.50/1.90 => ( X = Y3 ) ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_164_RES__refine,axiom,
% 3.50/1.90 ! [A: $tType,B: $tType,S: set @ A,S5: set @ B,R: set @ ( product_prod @ A @ B )] :
% 3.50/1.90 ( ! [S6: A] :
% 3.50/1.90 ( ( member @ A @ S6 @ S )
% 3.50/1.90 => ? [X9: B] :
% 3.50/1.90 ( ( member @ B @ X9 @ S5 )
% 3.50/1.90 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ S6 @ X9 ) @ R ) ) )
% 3.50/1.90 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ S ) @ ( refine1073749519nc_fun @ A @ B @ R @ ( refine605929679le_RES @ B @ S5 ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_165_RETURN__refine,axiom,
% 3.50/1.90 ! [A: $tType,B: $tType,X2: A,X10: B,R: set @ ( product_prod @ A @ B )] :
% 3.50/1.90 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ X10 ) @ R )
% 3.50/1.90 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X2 ) @ ( refine1073749519nc_fun @ A @ B @ R @ ( refine1687780735RETURN @ B @ X10 ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_166_for__in__RI,axiom,
% 3.50/1.90 ! [B: $tType,A: $tType,X2: A,R: set @ ( product_prod @ A @ B )] :
% 3.50/1.90 ( ( member @ A @ X2 @ ( domain @ A @ B @ R ) )
% 3.50/1.90 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ ( fun_of_rel @ A @ B @ R @ X2 ) ) @ R ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_167_BNF__Greatest__Fixpoint_OIdD,axiom,
% 3.50/1.90 ! [A: $tType,A3: A,B3: A] :
% 3.50/1.90 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ ( id @ A ) )
% 3.50/1.90 => ( A3 = B3 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_168_E__closed__restr__reach__cases,axiom,
% 3.50/1.90 ! [A: $tType,U: A,V: A,E4: set @ ( product_prod @ A @ A ),R: set @ A] :
% 3.50/1.90 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V ) @ ( transitive_rtrancl @ A @ E4 ) )
% 3.50/1.90 => ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ E4 @ R ) @ R )
% 3.50/1.90 => ( ~ ( member @ A @ V @ R )
% 3.50/1.90 => ~ ( ~ ( member @ A @ U @ R )
% 3.50/1.90 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V ) @ ( transitive_rtrancl @ A @ ( rel_restrict @ A @ E4 @ R ) ) ) ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_169_relChain__def,axiom,
% 3.50/1.90 ! [B: $tType,A: $tType] :
% 3.50/1.90 ( ( ord @ B )
% 3.50/1.90 => ( ( bNF_Ca1785829860lChain @ A @ B )
% 3.50/1.90 = ( ^ [R6: set @ ( product_prod @ A @ A ),As: A > B] :
% 3.50/1.90 ! [I2: A,J: A] :
% 3.50/1.90 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I2 @ J ) @ R6 )
% 3.50/1.90 => ( ord_less_eq @ B @ ( As @ I2 ) @ ( As @ J ) ) ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_170_converse__rtranclE_H,axiom,
% 3.50/1.90 ! [A: $tType,U: A,V: A,R: set @ ( product_prod @ A @ A )] :
% 3.50/1.90 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V ) @ ( transitive_rtrancl @ A @ R ) )
% 3.50/1.90 => ( ( U != V )
% 3.50/1.90 => ~ ! [Vh: A] :
% 3.50/1.90 ( ( U != Vh )
% 3.50/1.90 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ Vh ) @ R )
% 3.50/1.90 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Vh @ V ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_171_rtrancl__mono__rightI,axiom,
% 3.50/1.90 ! [A: $tType,S: set @ ( product_prod @ A @ A ),S5: set @ ( product_prod @ A @ A )] :
% 3.50/1.90 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ S @ S5 )
% 3.50/1.90 => ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ S @ ( transitive_rtrancl @ A @ S5 ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_172_rtrancl__mono__mp,axiom,
% 3.50/1.90 ! [A: $tType,U2: set @ ( product_prod @ A @ A ),V2: set @ ( product_prod @ A @ A ),X2: product_prod @ A @ A] :
% 3.50/1.90 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ U2 @ V2 )
% 3.50/1.90 => ( ( member @ ( product_prod @ A @ A ) @ X2 @ ( transitive_rtrancl @ A @ U2 ) )
% 3.50/1.90 => ( member @ ( product_prod @ A @ A ) @ X2 @ ( transitive_rtrancl @ A @ V2 ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_173_r__le__rtrancl,axiom,
% 3.50/1.90 ! [A: $tType,S: set @ ( product_prod @ A @ A )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ S @ ( transitive_rtrancl @ A @ S ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_174_rtrancl__image__advance__rtrancl,axiom,
% 3.50/1.90 ! [A: $tType,Q2: A,R: set @ ( product_prod @ A @ A ),Q0: set @ A,X2: A] :
% 3.50/1.90 ( ( member @ A @ Q2 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R ) @ Q0 ) )
% 3.50/1.90 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q2 @ X2 ) @ ( transitive_rtrancl @ A @ R ) )
% 3.50/1.90 => ( member @ A @ X2 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R ) @ Q0 ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_175_rtrancl__image__advance,axiom,
% 3.50/1.90 ! [A: $tType,Q2: A,R: set @ ( product_prod @ A @ A ),Q0: set @ A,X2: A] :
% 3.50/1.90 ( ( member @ A @ Q2 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R ) @ Q0 ) )
% 3.50/1.90 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q2 @ X2 ) @ R )
% 3.50/1.90 => ( member @ A @ X2 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R ) @ Q0 ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_176_rtrancl__reachable__induct,axiom,
% 3.50/1.90 ! [A: $tType,I3: set @ A,INV: set @ A,E4: set @ ( product_prod @ A @ A )] :
% 3.50/1.90 ( ( ord_less_eq @ ( set @ A ) @ I3 @ INV )
% 3.50/1.90 => ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ E4 @ INV ) @ INV )
% 3.50/1.90 => ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E4 ) @ I3 ) @ INV ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_177_rtrancl__image__unfold__right,axiom,
% 3.50/1.90 ! [A: $tType,E4: set @ ( product_prod @ A @ A ),V2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ E4 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E4 ) @ V2 ) ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E4 ) @ V2 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_178_reachable__mono,axiom,
% 3.50/1.90 ! [A: $tType,R: set @ ( product_prod @ A @ A ),R2: set @ ( product_prod @ A @ A ),X5: set @ A,X11: set @ A] :
% 3.50/1.90 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ R2 )
% 3.50/1.90 => ( ( ord_less_eq @ ( set @ A ) @ X5 @ X11 )
% 3.50/1.90 => ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R ) @ X5 ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R2 ) @ X11 ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_179_Domain__rtrancl,axiom,
% 3.50/1.90 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 3.50/1.90 ( ( domain @ A @ A @ ( transitive_rtrancl @ A @ R ) )
% 3.50/1.90 = ( top_top @ ( set @ A ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_180_Not__Domain__rtrancl,axiom,
% 3.50/1.90 ! [A: $tType,X2: A,R: set @ ( product_prod @ A @ A ),Y2: A] :
% 3.50/1.90 ( ~ ( member @ A @ X2 @ ( domain @ A @ A @ R ) )
% 3.50/1.90 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ ( transitive_rtrancl @ A @ R ) )
% 3.50/1.90 = ( X2 = Y2 ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_181_Image__closed__trancl,axiom,
% 3.50/1.90 ! [A: $tType,R3: set @ ( product_prod @ A @ A ),X5: set @ A] :
% 3.50/1.90 ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R3 @ X5 ) @ X5 )
% 3.50/1.90 => ( ( image @ A @ A @ ( transitive_rtrancl @ A @ R3 ) @ X5 )
% 3.50/1.90 = X5 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_182_rtrancl__subset__rtrancl,axiom,
% 3.50/1.90 ! [A: $tType,R3: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
% 3.50/1.90 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( transitive_rtrancl @ A @ S3 ) )
% 3.50/1.90 => ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R3 ) @ ( transitive_rtrancl @ A @ S3 ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_183_rtrancl__subset,axiom,
% 3.50/1.90 ! [A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
% 3.50/1.90 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ S )
% 3.50/1.90 => ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ S @ ( transitive_rtrancl @ A @ R ) )
% 3.50/1.90 => ( ( transitive_rtrancl @ A @ S )
% 3.50/1.90 = ( transitive_rtrancl @ A @ R ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_184_rtrancl__mono,axiom,
% 3.50/1.90 ! [A: $tType,R3: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
% 3.50/1.90 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ S3 )
% 3.50/1.90 => ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R3 ) @ ( transitive_rtrancl @ A @ S3 ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_185_rel__restrict__tranclI,axiom,
% 3.50/1.90 ! [A: $tType,X2: A,Y2: A,E4: set @ ( product_prod @ A @ A ),R: set @ A] :
% 3.50/1.90 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ ( transitive_trancl @ A @ E4 ) )
% 3.50/1.90 => ( ~ ( member @ A @ X2 @ R )
% 3.50/1.90 => ( ~ ( member @ A @ Y2 @ R )
% 3.50/1.90 => ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ E4 @ R ) @ R )
% 3.50/1.90 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ E4 @ R ) ) ) ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_186_uncurry__apply,axiom,
% 3.50/1.90 ! [B: $tType,A: $tType,C: $tType,F: B > C > A,A3: B,B3: C] :
% 3.50/1.90 ( ( uncurry @ B @ C @ A @ F @ ( product_Pair @ B @ C @ A3 @ B3 ) )
% 3.50/1.90 = ( F @ A3 @ B3 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_187_pw__abs__nofail,axiom,
% 3.50/1.90 ! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ A ),M: refine1665802226e_nres @ B] :
% 3.50/1.90 ( ( refine1102455758nofail @ A @ ( refine81118332bs_fun @ B @ A @ R @ M ) )
% 3.50/1.90 = ( ( refine1102455758nofail @ B @ M )
% 3.50/1.90 & ! [X: B] :
% 3.50/1.90 ( ( refine1315500908_inres @ B @ M @ X )
% 3.50/1.90 => ( member @ B @ X @ ( domain @ B @ A @ R ) ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_188_inres__simps_I2_J,axiom,
% 3.50/1.90 ! [B: $tType,X5: set @ B] :
% 3.50/1.90 ( ( refine1315500908_inres @ B @ ( refine605929679le_RES @ B @ X5 ) )
% 3.50/1.90 = ( ^ [X: B] : ( member @ B @ X @ X5 ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_189_trancl__domain,axiom,
% 3.50/1.90 ! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
% 3.50/1.90 ( ( domain @ A @ A @ ( transitive_trancl @ A @ R3 ) )
% 3.50/1.90 = ( domain @ A @ A @ R3 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_190_inres__simps_I1_J,axiom,
% 3.50/1.90 ! [A: $tType] :
% 3.50/1.90 ( ( refine1315500908_inres @ A @ ( top_top @ ( refine1665802226e_nres @ A ) ) )
% 3.50/1.90 = ( ^ [Uu: A] : $true ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_191_inres__simps_I3_J,axiom,
% 3.50/1.90 ! [C: $tType,X2: C] :
% 3.50/1.90 ( ( refine1315500908_inres @ C @ ( refine1687780735RETURN @ C @ X2 ) )
% 3.50/1.90 = ( ^ [Y: C,Z: C] : ( Y = Z )
% 3.50/1.90 @ X2 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_192_trancl__mono,axiom,
% 3.50/1.90 ! [A: $tType,P2: product_prod @ A @ A,R3: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
% 3.50/1.90 ( ( member @ ( product_prod @ A @ A ) @ P2 @ ( transitive_trancl @ A @ R3 ) )
% 3.50/1.90 => ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ S3 )
% 3.50/1.90 => ( member @ ( product_prod @ A @ A ) @ P2 @ ( transitive_trancl @ A @ S3 ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_193_trancl__Image__unfold__left,axiom,
% 3.50/1.90 ! [A: $tType,E4: set @ ( product_prod @ A @ A ),S: set @ A] :
% 3.50/1.90 ( ( image @ A @ A @ ( transitive_trancl @ A @ E4 ) @ S )
% 3.50/1.90 = ( image @ A @ A @ ( transitive_rtrancl @ A @ E4 ) @ ( image @ A @ A @ E4 @ S ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_194_trancl__Image__unfold__right,axiom,
% 3.50/1.90 ! [A: $tType,E4: set @ ( product_prod @ A @ A ),S: set @ A] :
% 3.50/1.90 ( ( image @ A @ A @ ( transitive_trancl @ A @ E4 ) @ S )
% 3.50/1.90 = ( image @ A @ A @ E4 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E4 ) @ S ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_195_rel__restrict__trancl__notR_I2_J,axiom,
% 3.50/1.90 ! [A: $tType,V: A,W2: A,E4: set @ ( product_prod @ A @ A ),R: set @ A] :
% 3.50/1.90 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W2 ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ E4 @ R ) ) )
% 3.50/1.90 => ~ ( member @ A @ W2 @ R ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_196_rel__restrict__trancl__notR_I1_J,axiom,
% 3.50/1.90 ! [A: $tType,V: A,W2: A,E4: set @ ( product_prod @ A @ A ),R: set @ A] :
% 3.50/1.90 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W2 ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ E4 @ R ) ) )
% 3.50/1.90 => ~ ( member @ A @ V @ R ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_197_rel__restrict__trancl__mem,axiom,
% 3.50/1.90 ! [A: $tType,A3: A,B3: A,A2: set @ ( product_prod @ A @ A ),R: set @ A] :
% 3.50/1.90 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ A2 @ R ) ) )
% 3.50/1.90 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ ( rel_restrict @ A @ ( transitive_trancl @ A @ A2 ) @ R ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_198_rel__restrict__trancl__sub,axiom,
% 3.50/1.90 ! [A: $tType,A2: set @ ( product_prod @ A @ A ),R: set @ A] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ A2 @ R ) ) @ ( rel_restrict @ A @ ( transitive_trancl @ A @ A2 ) @ R ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_199_pw__eqI,axiom,
% 3.50/1.90 ! [A: $tType,S: refine1665802226e_nres @ A,S5: refine1665802226e_nres @ A] :
% 3.50/1.90 ( ( ( refine1102455758nofail @ A @ S )
% 3.50/1.90 = ( refine1102455758nofail @ A @ S5 ) )
% 3.50/1.90 => ( ! [X3: A] :
% 3.50/1.90 ( ( refine1315500908_inres @ A @ S @ X3 )
% 3.50/1.90 = ( refine1315500908_inres @ A @ S5 @ X3 ) )
% 3.50/1.90 => ( S = S5 ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_200_pw__eq__iff,axiom,
% 3.50/1.90 ! [A: $tType] :
% 3.50/1.90 ( ( ^ [Y: refine1665802226e_nres @ A,Z: refine1665802226e_nres @ A] : ( Y = Z ) )
% 3.50/1.90 = ( ^ [S4: refine1665802226e_nres @ A,S7: refine1665802226e_nres @ A] :
% 3.50/1.90 ( ( ( refine1102455758nofail @ A @ S4 )
% 3.50/1.90 = ( refine1102455758nofail @ A @ S7 ) )
% 3.50/1.90 & ! [X: A] :
% 3.50/1.90 ( ( refine1315500908_inres @ A @ S4 @ X )
% 3.50/1.90 = ( refine1315500908_inres @ A @ S7 @ X ) ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_201_not__nofail__inres,axiom,
% 3.50/1.90 ! [A: $tType,S: refine1665802226e_nres @ A,X2: A] :
% 3.50/1.90 ( ~ ( refine1102455758nofail @ A @ S )
% 3.50/1.90 => ( refine1315500908_inres @ A @ S @ X2 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_202_pwD2,axiom,
% 3.50/1.90 ! [A: $tType,S: refine1665802226e_nres @ A,S5: refine1665802226e_nres @ A,X2: A] :
% 3.50/1.90 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ S5 )
% 3.50/1.90 => ( ( refine1315500908_inres @ A @ S @ X2 )
% 3.50/1.90 => ( refine1315500908_inres @ A @ S5 @ X2 ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_203_trancl__sub,axiom,
% 3.50/1.90 ! [A: $tType,R: set @ ( product_prod @ A @ A )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( transitive_trancl @ A @ R ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_204_trancl__mono__mp,axiom,
% 3.50/1.90 ! [A: $tType,U2: set @ ( product_prod @ A @ A ),V2: set @ ( product_prod @ A @ A ),X2: product_prod @ A @ A] :
% 3.50/1.90 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ U2 @ V2 )
% 3.50/1.90 => ( ( member @ ( product_prod @ A @ A ) @ X2 @ ( transitive_trancl @ A @ U2 ) )
% 3.50/1.90 => ( member @ ( product_prod @ A @ A ) @ X2 @ ( transitive_trancl @ A @ V2 ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_205_pw__leI,axiom,
% 3.50/1.90 ! [A: $tType,S5: refine1665802226e_nres @ A,S: refine1665802226e_nres @ A] :
% 3.50/1.90 ( ( ( refine1102455758nofail @ A @ S5 )
% 3.50/1.90 => ( ( refine1102455758nofail @ A @ S )
% 3.50/1.90 & ! [X3: A] :
% 3.50/1.90 ( ( refine1315500908_inres @ A @ S @ X3 )
% 3.50/1.90 => ( refine1315500908_inres @ A @ S5 @ X3 ) ) ) )
% 3.50/1.90 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ S5 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_206_pw__leI_H,axiom,
% 3.50/1.90 ! [A: $tType,S5: refine1665802226e_nres @ A,S: refine1665802226e_nres @ A] :
% 3.50/1.90 ( ( ( refine1102455758nofail @ A @ S5 )
% 3.50/1.90 => ( refine1102455758nofail @ A @ S ) )
% 3.50/1.90 => ( ! [X3: A] :
% 3.50/1.90 ( ( refine1102455758nofail @ A @ S5 )
% 3.50/1.90 => ( ( refine1315500908_inres @ A @ S @ X3 )
% 3.50/1.90 => ( refine1315500908_inres @ A @ S5 @ X3 ) ) )
% 3.50/1.90 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ S5 ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_207_pw__le__iff,axiom,
% 3.50/1.90 ! [A: $tType] :
% 3.50/1.90 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) )
% 3.50/1.90 = ( ^ [S4: refine1665802226e_nres @ A,S7: refine1665802226e_nres @ A] :
% 3.50/1.90 ( ( refine1102455758nofail @ A @ S7 )
% 3.50/1.90 => ( ( refine1102455758nofail @ A @ S4 )
% 3.50/1.90 & ! [X: A] :
% 3.50/1.90 ( ( refine1315500908_inres @ A @ S4 @ X )
% 3.50/1.90 => ( refine1315500908_inres @ A @ S7 @ X ) ) ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_208_inres__def,axiom,
% 3.50/1.90 ! [A: $tType] :
% 3.50/1.90 ( ( refine1315500908_inres @ A )
% 3.50/1.90 = ( ^ [S4: refine1665802226e_nres @ A,X: A] : ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X ) @ S4 ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_209_nf__inres__def,axiom,
% 3.50/1.90 ! [A: $tType] :
% 3.50/1.90 ( ( refine406925620_inres @ A )
% 3.50/1.90 = ( ^ [M4: refine1665802226e_nres @ A,X: A] :
% 3.50/1.90 ( ( refine1102455758nofail @ A @ M4 )
% 3.50/1.90 & ( refine1315500908_inres @ A @ M4 @ X ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_210_pw__abs__inres,axiom,
% 3.50/1.90 ! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ A ),M: refine1665802226e_nres @ B,A3: A] :
% 3.50/1.90 ( ( refine1315500908_inres @ A @ ( refine81118332bs_fun @ B @ A @ R @ M ) @ A3 )
% 3.50/1.90 = ( ( refine1102455758nofail @ A @ ( refine81118332bs_fun @ B @ A @ R @ M ) )
% 3.50/1.90 => ? [C4: B] :
% 3.50/1.90 ( ( refine1315500908_inres @ B @ M @ C4 )
% 3.50/1.90 & ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ C4 @ A3 ) @ R ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_211_trancl__reflcl,axiom,
% 3.50/1.90 ! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
% 3.50/1.90 ( ( transitive_trancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( id @ A ) ) )
% 3.50/1.90 = ( transitive_rtrancl @ A @ R3 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_212_pw__bind__leI,axiom,
% 3.50/1.90 ! [B: $tType,A: $tType,S: refine1665802226e_nres @ A,M: refine1665802226e_nres @ B,F: B > ( refine1665802226e_nres @ A )] :
% 3.50/1.90 ( ( ( refine1102455758nofail @ A @ S )
% 3.50/1.90 => ( refine1102455758nofail @ B @ M ) )
% 3.50/1.90 => ( ! [X3: B] :
% 3.50/1.90 ( ( refine1102455758nofail @ B @ M )
% 3.50/1.90 => ( ( refine1315500908_inres @ B @ M @ X3 )
% 3.50/1.90 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( F @ X3 ) @ S ) ) )
% 3.50/1.90 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine463715084e_bind @ B @ A @ M @ F ) @ S ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_213_Un__iff,axiom,
% 3.50/1.90 ! [A: $tType,C3: A,A2: set @ A,B2: set @ A] :
% 3.50/1.90 ( ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
% 3.50/1.90 = ( ( member @ A @ C3 @ A2 )
% 3.50/1.90 | ( member @ A @ C3 @ B2 ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_214_UnCI,axiom,
% 3.50/1.90 ! [A: $tType,C3: A,B2: set @ A,A2: set @ A] :
% 3.50/1.90 ( ( ~ ( member @ A @ C3 @ B2 )
% 3.50/1.90 => ( member @ A @ C3 @ A2 ) )
% 3.50/1.90 => ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_215_Un__subset__iff,axiom,
% 3.50/1.90 ! [A: $tType,A2: set @ A,B2: set @ A,C2: set @ A] :
% 3.50/1.90 ( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ C2 )
% 3.50/1.90 = ( ( ord_less_eq @ ( set @ A ) @ A2 @ C2 )
% 3.50/1.90 & ( ord_less_eq @ ( set @ A ) @ B2 @ C2 ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_216_bind__FAIL,axiom,
% 3.50/1.90 ! [B: $tType,A: $tType,F: B > ( refine1665802226e_nres @ A )] :
% 3.50/1.90 ( ( refine463715084e_bind @ B @ A @ ( top_top @ ( refine1665802226e_nres @ B ) ) @ F )
% 3.50/1.90 = ( top_top @ ( refine1665802226e_nres @ A ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_217_nres__monad2,axiom,
% 3.50/1.90 ! [A: $tType,M: refine1665802226e_nres @ A] :
% 3.50/1.90 ( ( refine463715084e_bind @ A @ A @ M @ ( refine1687780735RETURN @ A ) )
% 3.50/1.90 = M ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_218_nres__monad1,axiom,
% 3.50/1.90 ! [A: $tType,B: $tType,X2: B,F: B > ( refine1665802226e_nres @ A )] :
% 3.50/1.90 ( ( refine463715084e_bind @ B @ A @ ( refine1687780735RETURN @ B @ X2 ) @ F )
% 3.50/1.90 = ( F @ X2 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_219_rtrancl__reflcl,axiom,
% 3.50/1.90 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 3.50/1.90 ( ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( id @ A ) ) )
% 3.50/1.90 = ( transitive_rtrancl @ A @ R ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_220_rtrancl__reflcl__absorb,axiom,
% 3.50/1.90 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 3.50/1.90 ( ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R ) @ ( id @ A ) )
% 3.50/1.90 = ( transitive_rtrancl @ A @ R ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_221_rtrancl__Un__subset,axiom,
% 3.50/1.90 ! [A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R ) @ ( transitive_rtrancl @ A @ S ) ) @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R @ S ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_222_Un__mono,axiom,
% 3.50/1.90 ! [A: $tType,A2: set @ A,C2: set @ A,B2: set @ A,D3: set @ A] :
% 3.50/1.90 ( ( ord_less_eq @ ( set @ A ) @ A2 @ C2 )
% 3.50/1.90 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ D3 )
% 3.50/1.90 => ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ ( sup_sup @ ( set @ A ) @ C2 @ D3 ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_223_Un__least,axiom,
% 3.50/1.90 ! [A: $tType,A2: set @ A,C2: set @ A,B2: set @ A] :
% 3.50/1.90 ( ( ord_less_eq @ ( set @ A ) @ A2 @ C2 )
% 3.50/1.90 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C2 )
% 3.50/1.90 => ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ C2 ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_224_Un__upper1,axiom,
% 3.50/1.90 ! [A: $tType,A2: set @ A,B2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_225_Un__upper2,axiom,
% 3.50/1.90 ! [A: $tType,B2: set @ A,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ B2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_226_Un__absorb1,axiom,
% 3.50/1.90 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 3.50/1.90 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 3.50/1.90 => ( ( sup_sup @ ( set @ A ) @ A2 @ B2 )
% 3.50/1.90 = B2 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_227_Un__absorb2,axiom,
% 3.50/1.90 ! [A: $tType,B2: set @ A,A2: set @ A] :
% 3.50/1.90 ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 3.50/1.90 => ( ( sup_sup @ ( set @ A ) @ A2 @ B2 )
% 3.50/1.90 = A2 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_228_subset__UnE,axiom,
% 3.50/1.90 ! [A: $tType,C2: set @ A,A2: set @ A,B2: set @ A] :
% 3.50/1.90 ( ( ord_less_eq @ ( set @ A ) @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
% 3.50/1.90 => ~ ! [A8: set @ A] :
% 3.50/1.90 ( ( ord_less_eq @ ( set @ A ) @ A8 @ A2 )
% 3.50/1.90 => ! [B7: set @ A] :
% 3.50/1.90 ( ( ord_less_eq @ ( set @ A ) @ B7 @ B2 )
% 3.50/1.90 => ( C2
% 3.50/1.90 != ( sup_sup @ ( set @ A ) @ A8 @ B7 ) ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_229_subset__Un__eq,axiom,
% 3.50/1.90 ! [A: $tType] :
% 3.50/1.90 ( ( ord_less_eq @ ( set @ A ) )
% 3.50/1.90 = ( ^ [A6: set @ A,B6: set @ A] :
% 3.50/1.90 ( ( sup_sup @ ( set @ A ) @ A6 @ B6 )
% 3.50/1.90 = B6 ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_230_Domain__Un__eq,axiom,
% 3.50/1.90 ! [B: $tType,A: $tType,A2: set @ ( product_prod @ A @ B ),B2: set @ ( product_prod @ A @ B )] :
% 3.50/1.90 ( ( domain @ A @ B @ ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ A2 @ B2 ) )
% 3.50/1.90 = ( sup_sup @ ( set @ A ) @ ( domain @ A @ B @ A2 ) @ ( domain @ A @ B @ B2 ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_231_Un__left__commute,axiom,
% 3.50/1.90 ! [A: $tType,A2: set @ A,B2: set @ A,C2: set @ A] :
% 3.50/1.90 ( ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ B2 @ C2 ) )
% 3.50/1.90 = ( sup_sup @ ( set @ A ) @ B2 @ ( sup_sup @ ( set @ A ) @ A2 @ C2 ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_232_Un__left__absorb,axiom,
% 3.50/1.90 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 3.50/1.90 ( ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
% 3.50/1.90 = ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_233_Un__commute,axiom,
% 3.50/1.90 ! [A: $tType] :
% 3.50/1.90 ( ( sup_sup @ ( set @ A ) )
% 3.50/1.90 = ( ^ [A6: set @ A,B6: set @ A] : ( sup_sup @ ( set @ A ) @ B6 @ A6 ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_234_Un__absorb,axiom,
% 3.50/1.90 ! [A: $tType,A2: set @ A] :
% 3.50/1.90 ( ( sup_sup @ ( set @ A ) @ A2 @ A2 )
% 3.50/1.90 = A2 ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_235_Un__assoc,axiom,
% 3.50/1.90 ! [A: $tType,A2: set @ A,B2: set @ A,C2: set @ A] :
% 3.50/1.90 ( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ C2 )
% 3.50/1.90 = ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ B2 @ C2 ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_236_ball__Un,axiom,
% 3.50/1.90 ! [A: $tType,A2: set @ A,B2: set @ A,P: A > $o] :
% 3.50/1.90 ( ( ! [X: A] :
% 3.50/1.90 ( ( member @ A @ X @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
% 3.50/1.90 => ( P @ X ) ) )
% 3.50/1.90 = ( ! [X: A] :
% 3.50/1.90 ( ( member @ A @ X @ A2 )
% 3.50/1.90 => ( P @ X ) )
% 3.50/1.90 & ! [X: A] :
% 3.50/1.90 ( ( member @ A @ X @ B2 )
% 3.50/1.90 => ( P @ X ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_237_bex__Un,axiom,
% 3.50/1.90 ! [A: $tType,A2: set @ A,B2: set @ A,P: A > $o] :
% 3.50/1.90 ( ( ? [X: A] :
% 3.50/1.90 ( ( member @ A @ X @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
% 3.50/1.90 & ( P @ X ) ) )
% 3.50/1.90 = ( ? [X: A] :
% 3.50/1.90 ( ( member @ A @ X @ A2 )
% 3.50/1.90 & ( P @ X ) )
% 3.50/1.90 | ? [X: A] :
% 3.50/1.90 ( ( member @ A @ X @ B2 )
% 3.50/1.90 & ( P @ X ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_238_UnI2,axiom,
% 3.50/1.90 ! [A: $tType,C3: A,B2: set @ A,A2: set @ A] :
% 3.50/1.90 ( ( member @ A @ C3 @ B2 )
% 3.50/1.90 => ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_239_UnI1,axiom,
% 3.50/1.90 ! [A: $tType,C3: A,A2: set @ A,B2: set @ A] :
% 3.50/1.90 ( ( member @ A @ C3 @ A2 )
% 3.50/1.90 => ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_240_UnE,axiom,
% 3.50/1.90 ! [A: $tType,C3: A,A2: set @ A,B2: set @ A] :
% 3.50/1.90 ( ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
% 3.50/1.90 => ( ~ ( member @ A @ C3 @ A2 )
% 3.50/1.90 => ( member @ A @ C3 @ B2 ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_241_rel__restrict__union,axiom,
% 3.50/1.90 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A2: set @ A,B2: set @ A] :
% 3.50/1.90 ( ( rel_restrict @ A @ R @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
% 3.50/1.90 = ( rel_restrict @ A @ ( rel_restrict @ A @ R @ A2 ) @ B2 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_242_ibind__strict_I2_J,axiom,
% 3.50/1.90 ! [A: $tType,F: product_unit > ( refine1665802226e_nres @ A )] :
% 3.50/1.90 ( ( refine463715084e_bind @ product_unit @ A @ ( top_top @ ( refine1665802226e_nres @ product_unit ) ) @ F )
% 3.50/1.90 = ( top_top @ ( refine1665802226e_nres @ A ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_243_Un__UNIV__right,axiom,
% 3.50/1.90 ! [A: $tType,A2: set @ A] :
% 3.50/1.90 ( ( sup_sup @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) )
% 3.50/1.90 = ( top_top @ ( set @ A ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_244_Un__UNIV__left,axiom,
% 3.50/1.90 ! [A: $tType,B2: set @ A] :
% 3.50/1.90 ( ( sup_sup @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B2 )
% 3.50/1.90 = ( top_top @ ( set @ A ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_245_pw__bind__nofail,axiom,
% 3.50/1.90 ! [A: $tType,B: $tType,M: refine1665802226e_nres @ B,F: B > ( refine1665802226e_nres @ A )] :
% 3.50/1.90 ( ( refine1102455758nofail @ A @ ( refine463715084e_bind @ B @ A @ M @ F ) )
% 3.50/1.90 = ( ( refine1102455758nofail @ B @ M )
% 3.50/1.90 & ! [X: B] :
% 3.50/1.90 ( ( refine1315500908_inres @ B @ M @ X )
% 3.50/1.90 => ( refine1102455758nofail @ A @ ( F @ X ) ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_246_bind__cong,axiom,
% 3.50/1.90 ! [B: $tType,A: $tType,M2: refine1665802226e_nres @ A,M6: refine1665802226e_nres @ A,F: A > ( refine1665802226e_nres @ B ),F3: A > ( refine1665802226e_nres @ B )] :
% 3.50/1.90 ( ( M2 = M6 )
% 3.50/1.90 => ( ! [X3: A] :
% 3.50/1.90 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X3 ) @ M6 )
% 3.50/1.90 => ( ( F @ X3 )
% 3.50/1.90 = ( F3 @ X3 ) ) )
% 3.50/1.90 => ( ( refine463715084e_bind @ A @ B @ M2 @ F )
% 3.50/1.90 = ( refine463715084e_bind @ A @ B @ M6 @ F3 ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_247_Refine__Basic__Mirabelle__tqojlsrkwy_Obind__mono_I1_J,axiom,
% 3.50/1.90 ! [B: $tType,A: $tType,M: refine1665802226e_nres @ A,M3: refine1665802226e_nres @ A,F: A > ( refine1665802226e_nres @ B ),F3: A > ( refine1665802226e_nres @ B )] :
% 3.50/1.90 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M @ M3 )
% 3.50/1.90 => ( ! [X3: A] :
% 3.50/1.90 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X3 ) @ M )
% 3.50/1.90 => ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( F @ X3 ) @ ( F3 @ X3 ) ) )
% 3.50/1.90 => ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine463715084e_bind @ A @ B @ M @ F ) @ ( refine463715084e_bind @ A @ B @ M3 @ F3 ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_248_trancl__union__outside,axiom,
% 3.50/1.90 ! [A: $tType,V: A,W2: A,E4: set @ ( product_prod @ A @ A ),U2: set @ ( product_prod @ A @ A )] :
% 3.50/1.90 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W2 ) @ ( transitive_trancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ E4 @ U2 ) ) )
% 3.50/1.90 => ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W2 ) @ ( transitive_trancl @ A @ E4 ) )
% 3.50/1.90 => ? [X3: A,Y4: A] :
% 3.50/1.90 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ X3 ) @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ E4 @ U2 ) ) )
% 3.50/1.90 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y4 ) @ U2 )
% 3.50/1.90 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ W2 ) @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ E4 @ U2 ) ) ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_249_rtrancl__trancl__reflcl,axiom,
% 3.50/1.90 ! [A: $tType] :
% 3.50/1.90 ( ( transitive_rtrancl @ A )
% 3.50/1.90 = ( ^ [R6: set @ ( product_prod @ A @ A )] : ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R6 ) @ ( id @ A ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_250_trancl__image__by__rtrancl,axiom,
% 3.50/1.90 ! [A: $tType,E4: set @ ( product_prod @ A @ A ),Vi: set @ A] :
% 3.50/1.90 ( ( sup_sup @ ( set @ A ) @ ( image @ A @ A @ ( transitive_trancl @ A @ E4 ) @ Vi ) @ Vi )
% 3.50/1.90 = ( image @ A @ A @ ( transitive_rtrancl @ A @ E4 ) @ Vi ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_251_pw__bind__le__iff,axiom,
% 3.50/1.90 ! [A: $tType,B: $tType,M: refine1665802226e_nres @ B,F: B > ( refine1665802226e_nres @ A ),S: refine1665802226e_nres @ A] :
% 3.50/1.90 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine463715084e_bind @ B @ A @ M @ F ) @ S )
% 3.50/1.90 = ( ( ( refine1102455758nofail @ A @ S )
% 3.50/1.90 => ( refine1102455758nofail @ B @ M ) )
% 3.50/1.90 & ! [X: B] :
% 3.50/1.90 ( ( ( refine1102455758nofail @ B @ M )
% 3.50/1.90 & ( refine1315500908_inres @ B @ M @ X ) )
% 3.50/1.90 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( F @ X ) @ S ) ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(fact_252_sup__top__left,axiom,
% 3.50/1.90 ! [A: $tType] :
% 3.50/1.90 ( ( bounded_lattice_top @ A )
% 3.50/1.90 => ! [X2: A] :
% 3.50/1.90 ( ( sup_sup @ A @ ( top_top @ A ) @ X2 )
% 3.50/1.90 = ( top_top @ A ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Lattices_Obounded__lattice,axiom,
% 3.50/1.90 ! [A9: $tType] : ( bounded_lattice @ ( refine1665802226e_nres @ A9 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice_1,axiom,
% 3.50/1.90 bounded_lattice @ product_unit ).
% 3.50/1.90
% 3.50/1.90 thf(tcon_HOL_Obool___Lattices_Obounded__lattice_2,axiom,
% 3.50/1.90 bounded_lattice @ $o ).
% 3.50/1.90
% 3.50/1.90 thf(tcon_Set_Oset___Lattices_Obounded__lattice_3,axiom,
% 3.50/1.90 ! [A9: $tType] : ( bounded_lattice @ ( set @ A9 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(tcon_fun___Lattices_Obounded__lattice_4,axiom,
% 3.50/1.90 ! [A9: $tType,A10: $tType] :
% 3.50/1.90 ( ( bounded_lattice @ A10 )
% 3.50/1.90 => ( bounded_lattice @ ( A9 > A10 ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
% 3.50/1.90 ! [A9: $tType,A10: $tType] :
% 3.50/1.90 ( ( comple187826305attice @ A10 )
% 3.50/1.90 => ( comple187826305attice @ ( A9 > A10 ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(tcon_fun___Lattices_Obounded__lattice__top,axiom,
% 3.50/1.90 ! [A9: $tType,A10: $tType] :
% 3.50/1.90 ( ( bounded_lattice @ A10 )
% 3.50/1.90 => ( bounded_lattice_top @ ( A9 > A10 ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(tcon_fun___Orderings_Oorder__top,axiom,
% 3.50/1.90 ! [A9: $tType,A10: $tType] :
% 3.50/1.90 ( ( order_top @ A10 )
% 3.50/1.90 => ( order_top @ ( A9 > A10 ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(tcon_fun___Orderings_Opreorder,axiom,
% 3.50/1.90 ! [A9: $tType,A10: $tType] :
% 3.50/1.90 ( ( preorder @ A10 )
% 3.50/1.90 => ( preorder @ ( A9 > A10 ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(tcon_fun___Orderings_Oorder,axiom,
% 3.50/1.90 ! [A9: $tType,A10: $tType] :
% 3.50/1.90 ( ( order @ A10 )
% 3.50/1.90 => ( order @ ( A9 > A10 ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(tcon_fun___Orderings_Otop,axiom,
% 3.50/1.90 ! [A9: $tType,A10: $tType] :
% 3.50/1.90 ( ( top @ A10 )
% 3.50/1.90 => ( top @ ( A9 > A10 ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(tcon_fun___Orderings_Oord,axiom,
% 3.50/1.90 ! [A9: $tType,A10: $tType] :
% 3.50/1.90 ( ( ord @ A10 )
% 3.50/1.90 => ( ord @ ( A9 > A10 ) ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_5,axiom,
% 3.50/1.90 ! [A9: $tType] : ( comple187826305attice @ ( set @ A9 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(tcon_Set_Oset___Lattices_Obounded__lattice__top_6,axiom,
% 3.50/1.90 ! [A9: $tType] : ( bounded_lattice_top @ ( set @ A9 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(tcon_Set_Oset___Orderings_Oorder__top_7,axiom,
% 3.50/1.90 ! [A9: $tType] : ( order_top @ ( set @ A9 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(tcon_Set_Oset___Orderings_Opreorder_8,axiom,
% 3.50/1.90 ! [A9: $tType] : ( preorder @ ( set @ A9 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(tcon_Set_Oset___Orderings_Oorder_9,axiom,
% 3.50/1.90 ! [A9: $tType] : ( order @ ( set @ A9 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(tcon_Set_Oset___Orderings_Otop_10,axiom,
% 3.50/1.90 ! [A9: $tType] : ( top @ ( set @ A9 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(tcon_Set_Oset___Orderings_Oord_11,axiom,
% 3.50/1.90 ! [A9: $tType] : ( ord @ ( set @ A9 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_12,axiom,
% 3.50/1.90 comple187826305attice @ $o ).
% 3.50/1.90
% 3.50/1.90 thf(tcon_HOL_Obool___Lattices_Obounded__lattice__top_13,axiom,
% 3.50/1.90 bounded_lattice_top @ $o ).
% 3.50/1.90
% 3.50/1.90 thf(tcon_HOL_Obool___Orderings_Oorder__top_14,axiom,
% 3.50/1.90 order_top @ $o ).
% 3.50/1.90
% 3.50/1.90 thf(tcon_HOL_Obool___Orderings_Opreorder_15,axiom,
% 3.50/1.90 preorder @ $o ).
% 3.50/1.90
% 3.50/1.90 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 3.50/1.90 linorder @ $o ).
% 3.50/1.90
% 3.50/1.90 thf(tcon_HOL_Obool___Orderings_Oorder_16,axiom,
% 3.50/1.90 order @ $o ).
% 3.50/1.90
% 3.50/1.90 thf(tcon_HOL_Obool___Orderings_Otop_17,axiom,
% 3.50/1.90 top @ $o ).
% 3.50/1.90
% 3.50/1.90 thf(tcon_HOL_Obool___Orderings_Oord_18,axiom,
% 3.50/1.90 ord @ $o ).
% 3.50/1.90
% 3.50/1.90 thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_19,axiom,
% 3.50/1.90 comple187826305attice @ product_unit ).
% 3.50/1.90
% 3.50/1.90 thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__top_20,axiom,
% 3.50/1.90 bounded_lattice_top @ product_unit ).
% 3.50/1.90
% 3.50/1.90 thf(tcon_Product__Type_Ounit___Orderings_Oorder__top_21,axiom,
% 3.50/1.90 order_top @ product_unit ).
% 3.50/1.90
% 3.50/1.90 thf(tcon_Product__Type_Ounit___Orderings_Opreorder_22,axiom,
% 3.50/1.90 preorder @ product_unit ).
% 3.50/1.90
% 3.50/1.90 thf(tcon_Product__Type_Ounit___Orderings_Olinorder_23,axiom,
% 3.50/1.90 linorder @ product_unit ).
% 3.50/1.90
% 3.50/1.90 thf(tcon_Product__Type_Ounit___Orderings_Oorder_24,axiom,
% 3.50/1.90 order @ product_unit ).
% 3.50/1.90
% 3.50/1.90 thf(tcon_Product__Type_Ounit___Orderings_Otop_25,axiom,
% 3.50/1.90 top @ product_unit ).
% 3.50/1.90
% 3.50/1.90 thf(tcon_Product__Type_Ounit___Orderings_Oord_26,axiom,
% 3.50/1.90 ord @ product_unit ).
% 3.50/1.90
% 3.50/1.90 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Complete__Lattices_Ocomplete__lattice_27,axiom,
% 3.50/1.90 ! [A9: $tType] : ( comple187826305attice @ ( refine1665802226e_nres @ A9 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Lattices_Obounded__lattice__top_28,axiom,
% 3.50/1.90 ! [A9: $tType] : ( bounded_lattice_top @ ( refine1665802226e_nres @ A9 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oorder__top_29,axiom,
% 3.50/1.90 ! [A9: $tType] : ( order_top @ ( refine1665802226e_nres @ A9 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Opreorder_30,axiom,
% 3.50/1.90 ! [A9: $tType] : ( preorder @ ( refine1665802226e_nres @ A9 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oorder_31,axiom,
% 3.50/1.90 ! [A9: $tType] : ( order @ ( refine1665802226e_nres @ A9 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Otop_32,axiom,
% 3.50/1.90 ! [A9: $tType] : ( top @ ( refine1665802226e_nres @ A9 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oord_33,axiom,
% 3.50/1.90 ! [A9: $tType] : ( ord @ ( refine1665802226e_nres @ A9 ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(help_If_3_1_T,axiom,
% 3.50/1.90 ! [P: $o] :
% 3.50/1.90 ( ( P = $true )
% 3.50/1.90 | ( P = $false ) ) ).
% 3.50/1.90
% 3.50/1.90 thf(help_If_2_1_T,axiom,
% 3.50/1.90 ! [A: $tType,X2: A,Y2: A] :
% 3.50/1.90 ( ( if @ A @ $false @ X2 @ Y2 )
% 3.50/1.90 = Y2 ) ).
% 3.50/1.90
% 3.50/1.90 thf(help_If_1_1_T,axiom,
% 3.50/1.90 ! [A: $tType,X2: A,Y2: A] :
% 3.50/1.90 ( ( if @ A @ $true @ X2 @ Y2 )
% 3.50/1.90 = X2 ) ).
% 3.50/1.90
% 3.50/1.90 thf(conj_0,conjecture,
% 3.50/1.90 ord_less_eq @ ( refine1665802226e_nres @ a ) @ x @ ( refine1073749519nc_fun @ a @ b @ r @ ( top_top @ ( refine1665802226e_nres @ b ) ) ) ).
% 3.50/1.90 ------------------------
% 3.50/1.94 % (19669)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_19562 for (2999ds/4Mi)
% 3.50/1.94 % (19668)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_19562 for (2999ds/183Mi)
% 3.50/1.94 % (19671)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_19562 for (2999ds/2Mi)
% 3.50/1.94 % (19673)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_19562 for (2999ds/275Mi)
% 3.50/1.94 % (19674)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_19562 for (2999ds/18Mi)
% 3.50/1.94 % (19670)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_19562 for (2999ds/27Mi)
% 3.50/1.94 % (19672)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_19562 for (2999ds/2Mi)
% 3.50/1.94 % (19671)Instruction limit reached!
% 3.50/1.94 % (19671)------------------------------
% 3.50/1.94 % (19671)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.50/1.94 % (19671)Termination reason: Unknown
% 3.50/1.94 % (19672)Instruction limit reached!
% 3.50/1.94 % (19672)------------------------------
% 3.50/1.94 % (19672)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.50/1.94 % (19671)Termination phase: shuffling
% 3.50/1.94
% 3.50/1.94 % (19671)Memory used [KB]: 1407
% 3.50/1.94 % (19671)Time elapsed: 0.003 s
% 3.50/1.94 % (19671)Instructions burned: 2 (million)
% 3.50/1.94 % (19671)------------------------------
% 3.50/1.94 % (19671)------------------------------
% 3.50/1.94 % (19672)Termination reason: Unknown
% 3.50/1.94 % (19672)Termination phase: shuffling
% 3.50/1.94
% 3.50/1.94 % (19672)Memory used [KB]: 1407
% 3.50/1.94 % (19672)Time elapsed: 0.003 s
% 3.50/1.94 % (19672)Instructions burned: 2 (million)
% 3.50/1.94 % (19672)------------------------------
% 3.50/1.94 % (19672)------------------------------
% 3.50/1.94 % (19669)Instruction limit reached!
% 3.50/1.94 % (19669)------------------------------
% 3.50/1.94 % (19669)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.50/1.94 % (19669)Termination reason: Unknown
% 3.50/1.94 % (19669)Termination phase: shuffling
% 3.50/1.94
% 3.50/1.94 % (19669)Memory used [KB]: 1407
% 3.50/1.94 % (19669)Time elapsed: 0.004 s
% 3.50/1.94 % (19669)Instructions burned: 4 (million)
% 3.50/1.94 % (19669)------------------------------
% 3.50/1.94 % (19669)------------------------------
% 3.50/1.95 % (19674)Instruction limit reached!
% 3.50/1.95 % (19674)------------------------------
% 3.50/1.95 % (19674)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.50/1.95 % (19674)Termination reason: Unknown
% 3.50/1.95 % (19674)Termination phase: shuffling
% 3.50/1.95
% 3.50/1.95 % (19674)Memory used [KB]: 1791
% 3.50/1.95 % (19674)Time elapsed: 0.012 s
% 3.50/1.95 % (19674)Instructions burned: 19 (million)
% 3.50/1.95 % (19674)------------------------------
% 3.50/1.95 % (19674)------------------------------
% 3.50/1.95 % (19670)Instruction limit reached!
% 3.50/1.95 % (19670)------------------------------
% 3.50/1.95 % (19670)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.50/1.95 % (19670)Termination reason: Unknown
% 3.50/1.95 % (19670)Termination phase: Property scanning
% 3.50/1.95
% 3.50/1.95 % (19670)Memory used [KB]: 1918
% 3.50/1.95 % (19670)Time elapsed: 0.017 s
% 3.50/1.95 % (19670)Instructions burned: 28 (million)
% 3.50/1.95 % (19670)------------------------------
% 3.50/1.95 % (19670)------------------------------
% 3.50/1.96 % (19675)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_19562 for (2999ds/3Mi)
% 3.50/1.96 % (19676)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_19562 for (2999ds/37Mi)
% 3.50/1.96 % (19677)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_19562 for (2999ds/15Mi)
% 3.50/1.96 % (19675)Instruction limit reached!
% 3.50/1.96 % (19675)------------------------------
% 3.50/1.96 % (19675)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.50/1.96 % (19675)Termination reason: Unknown
% 3.50/1.96 % (19675)Termination phase: shuffling
% 3.50/1.96
% 3.50/1.96 % (19675)Memory used [KB]: 1407
% 3.50/1.96 % (19675)Time elapsed: 0.004 s
% 3.50/1.96 % (19675)Instructions burned: 3 (million)
% 3.50/1.96 % (19675)------------------------------
% 3.50/1.96 % (19675)------------------------------
% 3.50/1.97 % (19677)Instruction limit reached!
% 3.50/1.97 % (19677)------------------------------
% 3.50/1.97 % (19677)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.50/1.97 % (19677)Termination reason: Unknown
% 3.50/1.97 % (19677)Termination phase: shuffling
% 3.50/1.97
% 3.50/1.97 % (19677)Memory used [KB]: 1663
% 3.50/1.97 % (19677)Time elapsed: 0.010 s
% 3.50/1.97 % (19677)Instructions burned: 16 (million)
% 3.50/1.97 % (19677)------------------------------
% 3.50/1.97 % (19677)------------------------------
% 3.50/1.97 % (19678)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_19562 for (2999ds/3Mi)
% 3.50/1.97 % (19673)First to succeed.
% 3.50/1.97 % (19678)Instruction limit reached!
% 3.50/1.97 % (19678)------------------------------
% 3.50/1.97 % (19678)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.50/1.97 % (19678)Termination reason: Unknown
% 3.50/1.97 % (19678)Termination phase: shuffling
% 3.50/1.97
% 3.50/1.97 % (19678)Memory used [KB]: 1407
% 3.50/1.97 % (19678)Time elapsed: 0.004 s
% 3.50/1.97 % (19678)Instructions burned: 3 (million)
% 3.50/1.97 % (19678)------------------------------
% 3.50/1.97 % (19678)------------------------------
% 3.50/1.97 % (19673)Refutation found. Thanks to Tanya!
% 3.50/1.97 % SZS status Theorem for DTF2THF_19562
% 3.50/1.97 % SZS output start Proof for DTF2THF_19562
% 3.50/1.97 thf(type_def_5, type, product_prod: ($tType * $tType) > $tType).
% 3.50/1.97 thf(type_def_6, type, set: ($tType) > $tType).
% 3.50/1.97 thf(type_def_8, type, refine1665802226e_nres: ($tType) > $tType).
% 3.50/1.97 thf(type_def_9, type, a: $tType).
% 3.50/1.97 thf(type_def_10, type, b: $tType).
% 3.50/1.97 thf(type_def_11, type, product_unit: $tType).
% 3.50/1.97 thf(func_def_0, type, refine1665802226e_nres: ($tType) > $tType).
% 3.50/1.97 thf(func_def_1, type, product_unit: $tType).
% 3.50/1.97 thf(func_def_2, type, product_prod: ($tType * $tType) > $tType).
% 3.50/1.97 thf(func_def_3, type, set: ($tType) > $tType).
% 3.50/1.97 thf(func_def_4, type, b: $tType).
% 3.50/1.97 thf(func_def_5, type, a: $tType).
% 3.50/1.97 thf(func_def_6, type, bounded_lattice: !>[X0: $tType]:($o)).
% 3.50/1.97 thf(func_def_7, type, type: !>[X0: $tType]:($o)).
% 3.50/1.97 thf(func_def_8, type, ord: !>[X0: $tType]:($o)).
% 3.50/1.97 thf(func_def_9, type, top: !>[X0: $tType]:($o)).
% 3.50/1.97 thf(func_def_10, type, order: !>[X0: $tType]:($o)).
% 3.50/1.97 thf(func_def_11, type, linorder: !>[X0: $tType]:($o)).
% 3.50/1.97 thf(func_def_12, type, preorder: !>[X0: $tType]:($o)).
% 3.50/1.97 thf(func_def_13, type, order_top: !>[X0: $tType]:($o)).
% 3.50/1.97 thf(func_def_14, type, bounded_lattice_top: !>[X0: $tType]:($o)).
% 3.50/1.97 thf(func_def_15, type, comple187826305attice: !>[X0: $tType]:($o)).
% 3.50/1.97 thf(func_def_16, type, bNF_Ca1785829860lChain: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > (X0 > X1) > $o)).
% 3.50/1.97 thf(func_def_17, type, if: !>[X0: $tType]:($o > X0 > X0 > X0)).
% 3.50/1.97 thf(func_def_18, type, sup_sup: !>[X0: $tType]:(X0 > X0 > X0)).
% 3.50/1.97 thf(func_def_19, type, bijective: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > $o)).
% 3.50/1.97 thf(func_def_20, type, fun_of_rel: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > X0 > X1)).
% 3.50/1.97 thf(func_def_21, type, rel_restrict: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > set @ X0 > set @ product_prod @ X0 @ X0)).
% 3.50/1.97 thf(func_def_22, type, uncurry: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 3.50/1.97 thf(func_def_23, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 3.50/1.97 thf(func_def_24, type, order_Greatest: !>[X0: $tType]:((X0 > $o) > X0)).
% 3.50/1.97 thf(func_def_25, type, top_top: !>[X0: $tType]:(X0)).
% 3.50/1.97 thf(func_def_26, type, product_Pair: !>[X0: $tType, X1: $tType]:(X0 > X1 > product_prod @ X0 @ X1)).
% 3.50/1.97 thf(func_def_27, type, refine1687780735RETURN: !>[X0: $tType]:(X0 > refine1665802226e_nres @ X0)).
% 3.50/1.97 thf(func_def_28, type, refine81118332bs_fun: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > refine1665802226e_nres @ X0 > refine1665802226e_nres @ X1)).
% 3.50/1.97 thf(func_def_29, type, refine463715084e_bind: !>[X0: $tType, X1: $tType]:(refine1665802226e_nres @ X0 > (X0 > refine1665802226e_nres @ X1) > refine1665802226e_nres @ X1)).
% 3.50/1.97 thf(func_def_30, type, refine1073749519nc_fun: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > refine1665802226e_nres @ X1 > refine1665802226e_nres @ X0)).
% 3.50/1.97 thf(func_def_31, type, refine1315500908_inres: !>[X0: $tType]:(refine1665802226e_nres @ X0 > X0 > $o)).
% 3.50/1.97 thf(func_def_32, type, refine406925620_inres: !>[X0: $tType]:(refine1665802226e_nres @ X0 > X0 > $o)).
% 3.50/1.97 thf(func_def_33, type, refine1102455758nofail: !>[X0: $tType]:(refine1665802226e_nres @ X0 > $o)).
% 3.50/1.97 thf(func_def_34, type, refine605929679le_RES: !>[X0: $tType]:(set @ X0 > refine1665802226e_nres @ X0)).
% 3.50/1.97 thf(func_def_35, type, refine1672542526he_RES: !>[X0: $tType]:(refine1665802226e_nres @ X0 > set @ X0)).
% 3.50/1.97 thf(func_def_36, type, domain: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > set @ X0)).
% 3.50/1.97 thf(func_def_37, type, id: !>[X0: $tType]:(set @ product_prod @ X0 @ X0)).
% 3.50/1.97 thf(func_def_38, type, image: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > set @ X0 > set @ X1)).
% 3.50/1.97 thf(func_def_39, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 3.50/1.97 thf(func_def_40, type, transitive_rtrancl: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > set @ product_prod @ X0 @ X0)).
% 3.50/1.97 thf(func_def_41, type, transitive_trancl: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > set @ product_prod @ X0 @ X0)).
% 3.50/1.97 thf(func_def_42, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 3.50/1.97 thf(func_def_43, type, r: set @ product_prod @ a @ b).
% 3.50/1.97 thf(func_def_44, type, x: refine1665802226e_nres @ a).
% 3.50/1.97 thf(func_def_59, type, sK0: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > set @ X1 > set @ X0 > X0)).
% 3.50/1.97 thf(func_def_60, type, sK1: !>[X0: $tType, X1: $tType]:(((X1 > X0) > product_prod @ X1 @ X1 > $o) > X1 > X0)).
% 3.50/1.97 thf(func_def_61, type, sK2: !>[X0: $tType, X1: $tType]:(((X1 > X0) > product_prod @ X1 @ X1 > $o) > X1)).
% 3.50/1.97 thf(func_def_62, type, sK3: !>[X0: $tType, X1: $tType]:(((X1 > X0) > product_prod @ X1 @ X1 > $o) > X1)).
% 3.50/1.97 thf(func_def_63, type, sK4: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X1 @ X0 > set @ product_prod @ X1 @ X0 > X1)).
% 3.50/1.97 thf(func_def_64, type, sK5: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X1 @ X0 > set @ product_prod @ X1 @ X0 > X0)).
% 3.50/1.97 thf(func_def_65, type, sK6: !>[X0: $tType]:(X0)).
% 3.50/1.97 thf(func_def_66, type, sK7: !>[X0: $tType]:(set @ X0 > X0)).
% 3.50/1.97 thf(func_def_67, type, sK8: !>[X0: $tType]:(product_prod @ X0 @ X0 > X0)).
% 3.50/1.97 thf(func_def_68, type, sK9: !>[X0: $tType]:(set @ X0 > set @ X0 > X0)).
% 3.50/1.97 thf(func_def_69, type, sK10: !>[X0: $tType]:(set @ X0 > X0)).
% 3.50/1.97 thf(func_def_70, type, sK11: !>[X0: $tType]:(refine1665802226e_nres @ X0 > set @ X0)).
% 3.50/1.97 thf(func_def_71, type, sK12: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 3.50/1.97 thf(func_def_72, type, sK13: !>[X0: $tType, X1: $tType]:(product_prod @ (X1 > X0) @ product_prod @ X1 @ X1 > X1 > X0)).
% 3.50/1.97 thf(func_def_73, type, sK14: !>[X0: $tType, X1: $tType]:(product_prod @ (X1 > X0) @ product_prod @ X1 @ X1 > X1)).
% 3.50/1.97 thf(func_def_74, type, sK15: !>[X0: $tType, X1: $tType]:(product_prod @ (X1 > X0) @ product_prod @ X1 @ X1 > X1)).
% 3.50/1.97 thf(func_def_75, type, sK16: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > set @ product_prod @ X0 @ X1 > X0)).
% 3.50/1.97 thf(func_def_76, type, sK17: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > set @ product_prod @ X0 @ X1 > X1)).
% 3.50/1.97 thf(func_def_78, type, ph19: !>[X0: $tType]:(X0)).
% 3.50/1.97 thf(f1030,plain,(
% 3.50/1.97 $false),
% 3.50/1.97 inference(subsumption_resolution,[status(thm)],[f1029,f920])).
% 3.50/1.97 thf(f920,plain,(
% 3.50/1.97 ( ! [X0 : $tType,X1 : refine1665802226e_nres @ X0] : (((ord_less_eq @ refine1665802226e_nres @ X0 @ X1 @ top_top @ refine1665802226e_nres @ X0) = $true)) )),
% 3.50/1.97 inference(cnf_transformation,[status(thm)],[f513])).
% 3.50/1.97 thf(f513,plain,(
% 3.50/1.97 ! [X0 : $tType,X1 : refine1665802226e_nres @ X0] : ((ord_less_eq @ refine1665802226e_nres @ X0 @ X1 @ top_top @ refine1665802226e_nres @ X0) = $true)),
% 3.50/1.97 inference(fool_elimination,[status(thm)],[f512])).
% 3.50/1.97 thf(f512,plain,(
% 3.50/1.97 ! [X0 : $tType,X1 : refine1665802226e_nres @ X0] : (ord_less_eq @ refine1665802226e_nres @ X0 @ X1 @ top_top @ refine1665802226e_nres @ X0)),
% 3.50/1.97 inference(rectify,[status(thm)],[f5])).
% 3.50/1.97 thf(f5,axiom,(
% 3.50/1.97 ! [X2 : $tType,X4 : refine1665802226e_nres @ X2] : (ord_less_eq @ refine1665802226e_nres @ X2 @ X4 @ top_top @ refine1665802226e_nres @ X2)),
% 3.50/1.97 file('/export/starexec/sandbox2/tmp/tmp.vRzaLjzKJ7/DTF2THF_19562.p',fact_4_nres__order__simps_I3_J)).
% 3.50/1.97 thf(f1029,plain,(
% 3.50/1.97 ((ord_less_eq @ refine1665802226e_nres @ a @ x @ top_top @ refine1665802226e_nres @ a) != $true)),
% 3.50/1.97 inference(forward_demodulation,[status(thm)],[f966,f1021])).
% 3.50/1.97 thf(f1021,plain,(
% 3.50/1.97 ( ! [X1 : $tType,X0 : $tType,X2 : set @ product_prod @ X0 @ X1] : ((top_top @ refine1665802226e_nres @ X0 = (refine1073749519nc_fun @ X0 @ X1 @ X2 @ top_top @ refine1665802226e_nres @ X1))) )),
% 3.50/1.97 inference(equality_resolution,[status(thm)],[f963])).
% 3.50/1.97 thf(f963,plain,(
% 3.50/1.97 ( ! [X1 : $tType,X0 : $tType,X2 : set @ product_prod @ X0 @ X1,X3 : refine1665802226e_nres @ X1] : ((top_top @ refine1665802226e_nres @ X0 = (refine1073749519nc_fun @ X0 @ X1 @ X2 @ X3)) | (top_top @ refine1665802226e_nres @ X1 != X3)) )),
% 3.50/1.97 inference(cnf_transformation,[status(thm)],[f897])).
% 3.50/1.97 thf(f897,plain,(
% 3.50/1.97 ! [X0 : $tType,X1 : $tType,X2 : set @ product_prod @ X0 @ X1,X3 : refine1665802226e_nres @ X1] : (((top_top @ refine1665802226e_nres @ X1 = X3) | (top_top @ refine1665802226e_nres @ X0 != (refine1073749519nc_fun @ X0 @ X1 @ X2 @ X3))) & ((top_top @ refine1665802226e_nres @ X0 = (refine1073749519nc_fun @ X0 @ X1 @ X2 @ X3)) | (top_top @ refine1665802226e_nres @ X1 != X3)))),
% 3.50/1.97 inference(nnf_transformation,[status(thm)],[f415])).
% 3.50/1.97 thf(f415,plain,(
% 3.50/1.97 ! [X0 : $tType,X1 : $tType,X2 : set @ product_prod @ X0 @ X1,X3 : refine1665802226e_nres @ X1] : ((top_top @ refine1665802226e_nres @ X1 = X3) <=> (top_top @ refine1665802226e_nres @ X0 = (refine1073749519nc_fun @ X0 @ X1 @ X2 @ X3)))),
% 3.50/1.97 inference(fool_elimination,[status(thm)],[f414])).
% 3.50/1.97 thf(f414,plain,(
% 3.50/1.97 ! [X0 : $tType,X1 : $tType,X2 : set @ product_prod @ X0 @ X1,X3 : refine1665802226e_nres @ X1] : ((top_top @ refine1665802226e_nres @ X1 = X3) = (top_top @ refine1665802226e_nres @ X0 = (refine1073749519nc_fun @ X0 @ X1 @ X2 @ X3)))),
% 3.50/1.97 inference(rectify,[status(thm)],[f3])).
% 3.50/1.97 thf(f3,axiom,(
% 3.50/1.97 ! [X0 : $tType,X1 : $tType,X5 : set @ product_prod @ X0 @ X1,X6 : refine1665802226e_nres @ X1] : ((top_top @ refine1665802226e_nres @ X1 = X6) = (top_top @ refine1665802226e_nres @ X0 = (refine1073749519nc_fun @ X0 @ X1 @ X5 @ X6)))),
% 3.50/1.97 file('/export/starexec/sandbox2/tmp/tmp.vRzaLjzKJ7/DTF2THF_19562.p',fact_2_conc__fun__fail__iff_I1_J)).
% 3.50/1.97 thf(f966,plain,(
% 3.50/1.97 ((ord_less_eq @ refine1665802226e_nres @ a @ x @ (refine1073749519nc_fun @ a @ b @ r @ top_top @ refine1665802226e_nres @ b)) != $true)),
% 3.50/1.97 inference(cnf_transformation,[status(thm)],[f810])).
% 3.50/1.97 thf(f810,plain,(
% 3.50/1.97 ((ord_less_eq @ refine1665802226e_nres @ a @ x @ (refine1073749519nc_fun @ a @ b @ r @ top_top @ refine1665802226e_nres @ b)) != $true)),
% 3.50/1.97 inference(flattening,[status(thm)],[f739])).
% 3.50/1.97 thf(f739,plain,(
% 3.50/1.97 ~((ord_less_eq @ refine1665802226e_nres @ a @ x @ (refine1073749519nc_fun @ a @ b @ r @ top_top @ refine1665802226e_nres @ b)) = $true)),
% 3.50/1.97 inference(fool_elimination,[status(thm)],[f738])).
% 3.50/1.97 thf(f738,plain,(
% 3.50/1.97 ~(ord_less_eq @ refine1665802226e_nres @ a @ x @ (refine1073749519nc_fun @ a @ b @ r @ top_top @ refine1665802226e_nres @ b))),
% 3.50/1.97 inference(rectify,[status(thm)],[f300])).
% 3.50/1.97 thf(f300,negated_conjecture,(
% 3.50/1.97 ~(ord_less_eq @ refine1665802226e_nres @ a @ x @ (refine1073749519nc_fun @ a @ b @ r @ top_top @ refine1665802226e_nres @ b))),
% 3.50/1.97 inference(negated_conjecture,[status(cth)],[f299])).
% 3.50/1.97 thf(f299,conjecture,(
% 3.50/1.97 (ord_less_eq @ refine1665802226e_nres @ a @ x @ (refine1073749519nc_fun @ a @ b @ r @ top_top @ refine1665802226e_nres @ b))),
% 3.50/1.97 file('/export/starexec/sandbox2/tmp/tmp.vRzaLjzKJ7/DTF2THF_19562.p',conj_0)).
% 3.50/1.97 % SZS output end Proof for DTF2THF_19562
% 3.50/1.97 % (19673)------------------------------
% 3.50/1.97 % (19673)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.50/1.97 % (19673)Termination reason: Refutation
% 3.50/1.97
% 3.50/1.97 % (19673)Memory used [KB]: 6652
% 3.50/1.97 % (19673)Time elapsed: 0.032 s
% 3.50/1.97 % (19673)Instructions burned: 54 (million)
% 3.50/1.97 % (19673)------------------------------
% 3.50/1.97 % (19673)------------------------------
% 3.50/1.97 % (19667)Success in time 0.065 s
%------------------------------------------------------------------------------