%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP201^2 : 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 : n031.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 08:56:22 AM UTC 2026
% Result : Theorem 5.29s 2.04s
% Output : Refutation 5.29s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP201^2 : 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
% 0.16/0.34 % Computer : n031.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 08:30:33 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.27/0.46 ---- Original DTF file ---
% 0.27/0.46 thf(spec,logic,$$dhol).
% 0.27/0.46 %------------------------------------------------------------------------------
% 0.27/0.46 % File : ITP201^2 : TPTP v9.2.1. Released v7.5.0.
% 0.27/0.46 % Domain : Interactive Theorem Proving
% 0.27/0.46 % Problem : Sledgehammer USubst problem prob_1580__6355392_1
% 0.27/0.46 % Version : Especial.
% 0.27/0.46 % English :
% 0.27/0.46
% 0.27/0.46 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.27/0.46 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.27/0.46 % Source : [Des21]
% 0.27/0.46 % Names : USubst/prob_1580__6355392_1 [Des21]
% 0.27/0.46
% 0.27/0.46 % Status : Theorem
% 0.27/0.46 % Rating : 0.33 v8.1.0, 0.25 v7.5.0
% 0.27/0.46 % Syntax : Number of formulae : 346 ( 119 unt; 57 typ; 0 def)
% 0.27/0.46 % Number of atoms : 714 ( 253 equ; 0 cnn)
% 0.27/0.46 % Maximal formula atoms : 13 ( 2 avg)
% 0.27/0.46 % Number of connectives : 4059 ( 60 ~; 5 |; 37 &;3645 @)
% 0.27/0.46 % ( 0 <=>; 312 =>; 0 <=; 0 <~>)
% 0.27/0.46 % Maximal formula depth : 28 ( 7 avg)
% 0.27/0.46 % Number of types : 8 ( 7 usr)
% 0.27/0.46 % Number of type conns : 305 ( 305 >; 0 *; 0 +; 0 <<)
% 0.27/0.46 % Number of symbols : 52 ( 50 usr; 8 con; 0-7 aty)
% 0.27/0.46 % Number of variables : 1135 ( 102 ^; 986 !; 9 ?;1135 :)
% 0.27/0.46 % ( 38 !>; 0 ?*; 0 @-; 0 @+)
% 0.27/0.46 % SPC : TH1_THM_EQU_NAR
% 0.27/0.46
% 0.27/0.46 % Comments : This file was generated by Sledgehammer 2021-02-23 16:24:22.378
% 0.27/0.46 %------------------------------------------------------------------------------
% 0.27/0.46 % Could-be-implicit typings (10)
% 0.27/0.46 thf(ty_t_Denotational__Semantics_Ointerp,type,
% 0.27/0.46 denotational_interp: $tType ).
% 0.27/0.46
% 0.27/0.46 thf(ty_t_Product__Type_Oprod,type,
% 0.27/0.46 product_prod: $tType > $tType > $tType ).
% 0.27/0.46
% 0.27/0.46 thf(ty_t_Syntax_Ovariable,type,
% 0.27/0.46 variable: $tType ).
% 0.27/0.46
% 0.27/0.46 thf(ty_t_Option_Ooption,type,
% 0.27/0.46 option: $tType > $tType ).
% 0.27/0.46
% 0.27/0.46 thf(ty_t_Syntax_Ogame,type,
% 0.27/0.46 game: $tType ).
% 0.27/0.46
% 0.27/0.46 thf(ty_t_String_Ochar,type,
% 0.27/0.46 char: $tType ).
% 0.27/0.46
% 0.27/0.46 thf(ty_t_Syntax_Otrm,type,
% 0.27/0.46 trm: $tType ).
% 0.27/0.46
% 0.27/0.46 thf(ty_t_Syntax_Ofml,type,
% 0.27/0.46 fml: $tType ).
% 0.27/0.46
% 0.27/0.46 thf(ty_t_Real_Oreal,type,
% 0.27/0.46 real: $tType ).
% 0.27/0.46
% 0.27/0.46 thf(ty_t_Set_Oset,type,
% 0.27/0.46 set: $tType > $tType ).
% 0.27/0.46
% 0.27/0.46 % Explicit typings (47)
% 0.27/0.46 thf(sy_cl_HOL_Otype,type,
% 0.27/0.46 type:
% 0.27/0.46 !>[A: $tType] : $o ).
% 0.27/0.46
% 0.27/0.46 thf(sy_cl_Groups_Ouminus,type,
% 0.27/0.46 uminus:
% 0.27/0.46 !>[A: $tType] : $o ).
% 0.27/0.46
% 0.27/0.46 thf(sy_cl_Orderings_Oord,type,
% 0.27/0.46 ord:
% 0.27/0.46 !>[A: $tType] : $o ).
% 0.27/0.46
% 0.27/0.46 thf(sy_cl_Orderings_Oorder,type,
% 0.27/0.46 order:
% 0.27/0.46 !>[A: $tType] : $o ).
% 0.27/0.46
% 0.27/0.46 thf(sy_cl_Groups_Ogroup__add,type,
% 0.27/0.46 group_add:
% 0.27/0.46 !>[A: $tType] : $o ).
% 0.27/0.46
% 0.27/0.46 thf(sy_cl_Lattices_Olattice,type,
% 0.27/0.46 lattice:
% 0.27/0.46 !>[A: $tType] : $o ).
% 0.27/0.46
% 0.27/0.46 thf(sy_cl_Orderings_Olinorder,type,
% 0.27/0.46 linorder:
% 0.27/0.46 !>[A: $tType] : $o ).
% 0.27/0.46
% 0.27/0.46 thf(sy_cl_Orderings_Opreorder,type,
% 0.27/0.46 preorder:
% 0.27/0.46 !>[A: $tType] : $o ).
% 0.27/0.46
% 0.27/0.46 thf(sy_cl_Lattices_Oboolean__algebra,type,
% 0.27/0.46 boolean_algebra:
% 0.27/0.46 !>[A: $tType] : $o ).
% 0.27/0.46
% 0.27/0.46 thf(sy_cl_Lattices_Osemilattice__sup,type,
% 0.27/0.46 semilattice_sup:
% 0.27/0.46 !>[A: $tType] : $o ).
% 0.27/0.46
% 0.27/0.46 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 0.27/0.46 ordered_ab_group_add:
% 0.27/0.46 !>[A: $tType] : $o ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
% 0.27/0.46 bNF_Ca1785829860lChain:
% 0.27/0.46 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Denotational__Semantics_OUvariation,type,
% 0.27/0.46 denota1419872369iation: ( variable > real ) > ( variable > real ) > ( set @ variable ) > $o ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Denotational__Semantics_Ogame__sem,type,
% 0.27/0.46 denota1245701238me_sem: denotational_interp > game > ( set @ ( variable > real ) ) > ( set @ ( variable > real ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Groups_Ouminus__class_Ouminus,type,
% 0.27/0.46 uminus_uminus:
% 0.27/0.46 !>[A: $tType] : ( A > A ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Lattices_Osup__class_Osup,type,
% 0.27/0.46 sup_sup:
% 0.27/0.46 !>[A: $tType] : ( A > A > A ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Option_Ooption_ONone,type,
% 0.27/0.46 none:
% 0.27/0.46 !>[A: $tType] : ( option @ A ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Option_Ooption_Othe,type,
% 0.27/0.46 the:
% 0.27/0.46 !>[A: $tType] : ( ( option @ A ) > A ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.27/0.46 ord_less_eq:
% 0.27/0.46 !>[A: $tType] : ( A > A > $o ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Product__Type_OPair,type,
% 0.27/0.46 product_Pair:
% 0.27/0.46 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 0.27/0.46 produc2004651681e_prod:
% 0.27/0.46 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 0.27/0.46 product_rec_prod:
% 0.27/0.46 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Product__Type_Oprod_Ofst,type,
% 0.27/0.46 product_fst:
% 0.27/0.46 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Product__Type_Oprod_Osnd,type,
% 0.27/0.46 product_snd:
% 0.27/0.46 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Product__Type_Oprod_Oswap,type,
% 0.27/0.46 product_swap:
% 0.27/0.46 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Product__Type_Oscomp,type,
% 0.27/0.46 product_scomp:
% 0.27/0.46 !>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( A > ( product_prod @ B @ C ) ) > ( B > C > D ) > A > D ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Set_OCollect,type,
% 0.27/0.46 collect:
% 0.27/0.46 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Syntax_OSkip,type,
% 0.27/0.46 skip: game ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Syntax_Ogame_OChoice,type,
% 0.27/0.46 choice: game > game > game ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Syntax_Ogame_ODual,type,
% 0.27/0.46 dual: game > game ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Syntax_Ogame_OLoop,type,
% 0.27/0.46 loop: game > game ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_USubst__Mirabelle__nnnzepxswx_OChoiceo,type,
% 0.27/0.46 uSubst1976112797hoiceo: ( option @ game ) > ( option @ game ) > ( option @ game ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_USubst__Mirabelle__nnnzepxswx_ODualo,type,
% 0.27/0.46 uSubst739989314_Dualo: ( option @ game ) > ( option @ game ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_USubst__Mirabelle__nnnzepxswx_OLoopo,type,
% 0.27/0.46 uSubst993936602_Loopo: ( option @ game ) > ( option @ game ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_USubst__Mirabelle__nnnzepxswx_Oadjoint,type,
% 0.27/0.46 uSubst2091380086djoint: ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) > denotational_interp > ( variable > real ) > denotational_interp ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_USubst__Mirabelle__nnnzepxswx_Ousubstappp,type,
% 0.27/0.46 uSubst95898988stappp: ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) > ( set @ variable ) > game > ( product_prod @ ( set @ variable ) @ ( option @ game ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_member,type,
% 0.27/0.46 member:
% 0.27/0.46 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_v_I,type,
% 0.27/0.46 i: denotational_interp ).
% 0.27/0.46
% 0.27/0.46 thf(sy_v_U,type,
% 0.27/0.46 u: set @ variable ).
% 0.27/0.46
% 0.27/0.46 thf(sy_v_Ua____,type,
% 0.27/0.46 ua: set @ variable ).
% 0.27/0.46
% 0.27/0.46 thf(sy_v_Xa____,type,
% 0.27/0.46 xa: set @ ( variable > real ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_v__092_060alpha_062_H_H____,type,
% 0.27/0.46 alpha: game ).
% 0.27/0.46
% 0.27/0.46 thf(sy_v__092_060nu_062,type,
% 0.27/0.46 nu: variable > real ).
% 0.27/0.46
% 0.27/0.46 thf(sy_v__092_060nu_062_H____,type,
% 0.27/0.46 nu2: variable > real ).
% 0.27/0.46
% 0.27/0.46 thf(sy_v__092_060omega_062,type,
% 0.27/0.46 omega: variable > real ).
% 0.27/0.46
% 0.27/0.46 thf(sy_v__092_060omega_062_H____,type,
% 0.27/0.46 omega2: variable > real ).
% 0.27/0.46
% 0.27/0.46 thf(sy_v__092_060sigma_062_H_H____,type,
% 0.27/0.46 sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % Relevant facts (255)
% 0.27/0.46 thf(fact_0__092_060open_062_092_060And_062X_O_A_I_092_060nu_062_A_092_060in_062_Agame__sem_AI_A_Ithe_A_Isnd_A_Iusubstappp_A_092_060sigma_062_AU_A_092_060alpha_062_J_J_J_AX_J_A_061_A_I_092_060nu_062_A_092_060in_062_Agame__sem_A_IUSubst__Mirabelle__nnnzepxswx_Oadjoint_A_092_060sigma_062_AI_A_092_060omega_062_J_A_092_060alpha_062_AX_J_092_060close_062,axiom,
% 0.27/0.46 ! [X: set @ ( variable > real )] :
% 0.27/0.46 ( ( member @ ( variable > real ) @ nu2 @ ( denota1245701238me_sem @ i @ ( the @ game @ ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ sigma @ ua @ alpha ) ) ) @ X ) )
% 0.27/0.46 = ( member @ ( variable > real ) @ nu2 @ ( denota1245701238me_sem @ ( uSubst2091380086djoint @ sigma @ i @ omega2 ) @ alpha @ X ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % \<open>\<And>X. (\<nu> \<in> game_sem I (the (snd (usubstappp \<sigma> U \<alpha>))) X) = (\<nu> \<in> game_sem (USubst_Mirabelle_nnnzepxswx.adjoint \<sigma> I \<omega>) \<alpha> X)\<close>
% 0.27/0.46 thf(fact_1_IH_092_060alpha_062,axiom,
% 0.27/0.46 ! [X: set @ ( variable > real )] :
% 0.27/0.46 ( ( denota1419872369iation @ nu2 @ omega2 @ ua )
% 0.27/0.46 => ( ( member @ ( variable > real ) @ nu2 @ ( denota1245701238me_sem @ i @ ( the @ game @ ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ sigma @ ua @ alpha ) ) ) @ X ) )
% 0.27/0.46 = ( member @ ( variable > real ) @ nu2 @ ( denota1245701238me_sem @ ( uSubst2091380086djoint @ sigma @ i @ omega2 ) @ alpha @ X ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % IH\<alpha>
% 0.27/0.46 thf(fact_2_uv,axiom,
% 0.27/0.46 denota1419872369iation @ nu2 @ omega2 @ ua ).
% 0.27/0.46
% 0.27/0.46 % uv
% 0.27/0.46 thf(fact_3_snd__uminus,axiom,
% 0.27/0.46 ! [A: $tType,B: $tType] :
% 0.27/0.46 ( ( ( uminus @ B )
% 0.27/0.46 & ( uminus @ A ) )
% 0.27/0.46 => ! [X2: product_prod @ B @ A] :
% 0.27/0.46 ( ( product_snd @ B @ A @ ( uminus_uminus @ ( product_prod @ B @ A ) @ X2 ) )
% 0.27/0.46 = ( uminus_uminus @ A @ ( product_snd @ B @ A @ X2 ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % snd_uminus
% 0.27/0.46 thf(fact_4_def,axiom,
% 0.27/0.46 ( ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ sigma @ ua @ ( dual @ alpha ) ) )
% 0.27/0.46 != ( none @ game ) ) ).
% 0.27/0.46
% 0.27/0.46 % def
% 0.27/0.46 thf(fact_5_ComplI,axiom,
% 0.27/0.46 ! [A: $tType,C2: A,A2: set @ A] :
% 0.27/0.46 ( ~ ( member @ A @ C2 @ A2 )
% 0.27/0.46 => ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A2 ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % ComplI
% 0.27/0.46 thf(fact_6_Compl__iff,axiom,
% 0.27/0.46 ! [A: $tType,C2: A,A2: set @ A] :
% 0.27/0.46 ( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
% 0.27/0.46 = ( ~ ( member @ A @ C2 @ A2 ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % Compl_iff
% 0.27/0.46 thf(fact_7_Compl__eq__Compl__iff,axiom,
% 0.27/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.27/0.46 ( ( ( uminus_uminus @ ( set @ A ) @ A2 )
% 0.27/0.46 = ( uminus_uminus @ ( set @ A ) @ B2 ) )
% 0.27/0.46 = ( A2 = B2 ) ) ).
% 0.27/0.46
% 0.27/0.46 % Compl_eq_Compl_iff
% 0.27/0.46 thf(fact_8_verit__minus__simplify_I4_J,axiom,
% 0.27/0.46 ! [B: $tType] :
% 0.27/0.46 ( ( group_add @ B )
% 0.27/0.46 => ! [B3: B] :
% 0.27/0.46 ( ( uminus_uminus @ B @ ( uminus_uminus @ B @ B3 ) )
% 0.27/0.46 = B3 ) ) ).
% 0.27/0.46
% 0.27/0.46 % verit_minus_simplify(4)
% 0.27/0.46 thf(fact_9_uminus__apply,axiom,
% 0.27/0.46 ! [B: $tType,A: $tType] :
% 0.27/0.46 ( ( uminus @ B )
% 0.27/0.46 => ( ( uminus_uminus @ ( A > B ) )
% 0.27/0.46 = ( ^ [A3: A > B,X3: A] : ( uminus_uminus @ B @ ( A3 @ X3 ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % uminus_apply
% 0.27/0.46 thf(fact_10_add_Oinverse__inverse,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( group_add @ A )
% 0.27/0.46 => ! [A4: A] :
% 0.27/0.46 ( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A4 ) )
% 0.27/0.46 = A4 ) ) ).
% 0.27/0.46
% 0.27/0.46 % add.inverse_inverse
% 0.27/0.46 thf(fact_11_neg__equal__iff__equal,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( group_add @ A )
% 0.27/0.46 => ! [A4: A,B3: A] :
% 0.27/0.46 ( ( ( uminus_uminus @ A @ A4 )
% 0.27/0.46 = ( uminus_uminus @ A @ B3 ) )
% 0.27/0.46 = ( A4 = B3 ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % neg_equal_iff_equal
% 0.27/0.46 thf(fact_12_double__compl,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( boolean_algebra @ A )
% 0.27/0.46 => ! [X2: A] :
% 0.27/0.46 ( ( uminus_uminus @ A @ ( uminus_uminus @ A @ X2 ) )
% 0.27/0.46 = X2 ) ) ).
% 0.27/0.46
% 0.27/0.46 % double_compl
% 0.27/0.46 thf(fact_13_vaouter,axiom,
% 0.27/0.46 denota1419872369iation @ nu @ omega @ u ).
% 0.27/0.46
% 0.27/0.46 % vaouter
% 0.27/0.46 thf(fact_14_compl__eq__compl__iff,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( boolean_algebra @ A )
% 0.27/0.46 => ! [X2: A,Y: A] :
% 0.27/0.46 ( ( ( uminus_uminus @ A @ X2 )
% 0.27/0.46 = ( uminus_uminus @ A @ Y ) )
% 0.27/0.46 = ( X2 = Y ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % compl_eq_compl_iff
% 0.27/0.46 thf(fact_15_usubstappp__det,axiom,
% 0.27/0.46 ! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,Alpha: game,V: set @ variable] :
% 0.27/0.46 ( ( ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) )
% 0.27/0.46 != ( none @ game ) )
% 0.27/0.46 => ( ( ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ V @ Alpha ) )
% 0.27/0.46 != ( none @ game ) )
% 0.27/0.46 => ( ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) )
% 0.27/0.46 = ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ V @ Alpha ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % usubstappp_det
% 0.27/0.46 thf(fact_16_minus__equation__iff,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( group_add @ A )
% 0.27/0.46 => ! [A4: A,B3: A] :
% 0.27/0.46 ( ( ( uminus_uminus @ A @ A4 )
% 0.27/0.46 = B3 )
% 0.27/0.46 = ( ( uminus_uminus @ A @ B3 )
% 0.27/0.46 = A4 ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % minus_equation_iff
% 0.27/0.46 thf(fact_17_equation__minus__iff,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( group_add @ A )
% 0.27/0.46 => ! [A4: A,B3: A] :
% 0.27/0.46 ( ( A4
% 0.27/0.46 = ( uminus_uminus @ A @ B3 ) )
% 0.27/0.46 = ( B3
% 0.27/0.46 = ( uminus_uminus @ A @ A4 ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % equation_minus_iff
% 0.27/0.46 thf(fact_18_fun__Compl__def,axiom,
% 0.27/0.46 ! [B: $tType,A: $tType] :
% 0.27/0.46 ( ( uminus @ B )
% 0.27/0.46 => ( ( uminus_uminus @ ( A > B ) )
% 0.27/0.46 = ( ^ [A3: A > B,X3: A] : ( uminus_uminus @ B @ ( A3 @ X3 ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % fun_Compl_def
% 0.27/0.46 thf(fact_19_verit__negate__coefficient_I3_J,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( ordered_ab_group_add @ A )
% 0.27/0.46 => ! [A4: A,B3: A] :
% 0.27/0.46 ( ( A4 = B3 )
% 0.27/0.46 => ( ( uminus_uminus @ A @ A4 )
% 0.27/0.46 = ( uminus_uminus @ A @ B3 ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % verit_negate_coefficient(3)
% 0.27/0.46 thf(fact_20_double__complement,axiom,
% 0.27/0.46 ! [A: $tType,A2: set @ A] :
% 0.27/0.46 ( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
% 0.27/0.46 = A2 ) ).
% 0.27/0.46
% 0.27/0.46 % double_complement
% 0.27/0.46 thf(fact_21_ComplD,axiom,
% 0.27/0.46 ! [A: $tType,C2: A,A2: set @ A] :
% 0.27/0.46 ( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
% 0.27/0.46 => ~ ( member @ A @ C2 @ A2 ) ) ).
% 0.27/0.46
% 0.27/0.46 % ComplD
% 0.27/0.46 thf(fact_22_game__sem_Osimps_I7_J,axiom,
% 0.27/0.46 ! [I: denotational_interp,Alpha: game] :
% 0.27/0.46 ( ( denota1245701238me_sem @ I @ ( dual @ Alpha ) )
% 0.27/0.46 = ( ^ [X4: set @ ( variable > real )] : ( uminus_uminus @ ( set @ ( variable > real ) ) @ ( denota1245701238me_sem @ I @ Alpha @ ( uminus_uminus @ ( set @ ( variable > real ) ) @ X4 ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % game_sem.simps(7)
% 0.27/0.46 thf(fact_23_Dual_OIH,axiom,
% 0.27/0.46 ! [Nu: variable > real,Omega: variable > real,X: set @ ( variable > real )] :
% 0.27/0.46 ( ( ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ sigma @ ua @ alpha ) )
% 0.27/0.46 != ( none @ game ) )
% 0.27/0.46 => ( ( denota1419872369iation @ Nu @ Omega @ ua )
% 0.27/0.46 => ( ( member @ ( variable > real ) @ Nu @ ( denota1245701238me_sem @ i @ ( the @ game @ ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ sigma @ ua @ alpha ) ) ) @ X ) )
% 0.27/0.46 = ( member @ ( variable > real ) @ Nu @ ( denota1245701238me_sem @ ( uSubst2091380086djoint @ sigma @ i @ Omega ) @ alpha @ X ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % Dual.IH
% 0.27/0.46 thf(fact_24_game_Oinject_I7_J,axiom,
% 0.27/0.46 ! [X7: game,Y7: game] :
% 0.27/0.46 ( ( ( dual @ X7 )
% 0.27/0.46 = ( dual @ Y7 ) )
% 0.27/0.46 = ( X7 = Y7 ) ) ).
% 0.27/0.46
% 0.27/0.46 % game.inject(7)
% 0.27/0.46 thf(fact_25_Dual_Oprems_I2_J,axiom,
% 0.27/0.46 denota1419872369iation @ nu2 @ omega2 @ ua ).
% 0.27/0.46
% 0.27/0.46 % Dual.prems(2)
% 0.27/0.46 thf(fact_26_option_Oexpand,axiom,
% 0.27/0.46 ! [A: $tType,Option: option @ A,Option2: option @ A] :
% 0.27/0.46 ( ( ( Option
% 0.27/0.46 = ( none @ A ) )
% 0.27/0.46 = ( Option2
% 0.27/0.46 = ( none @ A ) ) )
% 0.27/0.46 => ( ( ( Option
% 0.27/0.46 != ( none @ A ) )
% 0.27/0.46 => ( ( Option2
% 0.27/0.46 != ( none @ A ) )
% 0.27/0.46 => ( ( the @ A @ Option )
% 0.27/0.46 = ( the @ A @ Option2 ) ) ) )
% 0.27/0.46 => ( Option = Option2 ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % option.expand
% 0.27/0.46 thf(fact_27_usubst__game__loop,axiom,
% 0.27/0.46 ! [Nu: variable > real,Omega: variable > real,U: set @ variable,Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),Alpha: game,I: denotational_interp,X: set @ ( variable > real )] :
% 0.27/0.46 ( ( denota1419872369iation @ Nu @ Omega @ U )
% 0.27/0.46 => ( ! [Nu2: variable > real,Omega2: variable > real,X5: set @ ( variable > real )] :
% 0.27/0.46 ( ( denota1419872369iation @ Nu2 @ Omega2 @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) ) )
% 0.27/0.46 => ( ( ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) ) @ Alpha ) )
% 0.27/0.46 != ( none @ game ) )
% 0.27/0.46 => ( ( member @ ( variable > real ) @ Nu2 @ ( denota1245701238me_sem @ I @ ( the @ game @ ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) ) @ Alpha ) ) ) @ X5 ) )
% 0.27/0.46 = ( member @ ( variable > real ) @ Nu2 @ ( denota1245701238me_sem @ ( uSubst2091380086djoint @ Sigma @ I @ Omega2 ) @ Alpha @ X5 ) ) ) ) )
% 0.27/0.46 => ( ( ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ ( loop @ Alpha ) ) )
% 0.27/0.46 != ( none @ game ) )
% 0.27/0.46 => ( ( member @ ( variable > real ) @ Nu @ ( denota1245701238me_sem @ I @ ( the @ game @ ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ ( loop @ Alpha ) ) ) ) @ X ) )
% 0.27/0.46 = ( member @ ( variable > real ) @ Nu @ ( denota1245701238me_sem @ ( uSubst2091380086djoint @ Sigma @ I @ Omega ) @ ( loop @ Alpha ) @ X ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % usubst_game_loop
% 0.27/0.46 thf(fact_28_Uvariation__def,axiom,
% 0.27/0.46 ( denota1419872369iation
% 0.27/0.46 = ( ^ [Nu3: variable > real,Nu4: variable > real,U2: set @ variable] :
% 0.27/0.46 ! [I2: variable] :
% 0.27/0.46 ( ~ ( member @ variable @ I2 @ U2 )
% 0.27/0.46 => ( ( Nu3 @ I2 )
% 0.27/0.46 = ( Nu4 @ I2 ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % Uvariation_def
% 0.27/0.46 thf(fact_29_Uvariation__sym,axiom,
% 0.27/0.46 ( denota1419872369iation
% 0.27/0.46 = ( ^ [Omega3: variable > real,Nu3: variable > real] : ( denota1419872369iation @ Nu3 @ Omega3 ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % Uvariation_sym
% 0.27/0.46 thf(fact_30_Uvariation__refl,axiom,
% 0.27/0.46 ! [Nu: variable > real,V: set @ variable] : ( denota1419872369iation @ Nu @ Nu @ V ) ).
% 0.27/0.46
% 0.27/0.46 % Uvariation_refl
% 0.27/0.46 thf(fact_31_Uvariation__sym__rel,axiom,
% 0.27/0.46 ! [Omega: variable > real,Nu: variable > real,U: set @ variable] :
% 0.27/0.46 ( ( denota1419872369iation @ Omega @ Nu @ U )
% 0.27/0.46 => ( denota1419872369iation @ Nu @ Omega @ U ) ) ).
% 0.27/0.46
% 0.27/0.46 % Uvariation_sym_rel
% 0.27/0.46 thf(fact_32_usubstappp__choice__conv,axiom,
% 0.27/0.46 ! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,Alpha: game,Beta: game] :
% 0.27/0.46 ( ( ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ ( choice @ Alpha @ Beta ) ) )
% 0.27/0.46 != ( none @ game ) )
% 0.27/0.46 => ( ( ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) )
% 0.27/0.46 != ( none @ game ) )
% 0.27/0.46 & ( ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Beta ) )
% 0.27/0.46 != ( none @ game ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % usubstappp_choice_conv
% 0.27/0.46 thf(fact_33_game_Oinject_I6_J,axiom,
% 0.27/0.46 ! [X6: game,Y6: game] :
% 0.27/0.46 ( ( ( loop @ X6 )
% 0.27/0.46 = ( loop @ Y6 ) )
% 0.27/0.46 = ( X6 = Y6 ) ) ).
% 0.27/0.46
% 0.27/0.46 % game.inject(6)
% 0.27/0.46 thf(fact_34_game_Oinject_I4_J,axiom,
% 0.27/0.46 ! [X41: game,X42: game,Y41: game,Y42: game] :
% 0.27/0.46 ( ( ( choice @ X41 @ X42 )
% 0.27/0.46 = ( choice @ Y41 @ Y42 ) )
% 0.27/0.46 = ( ( X41 = Y41 )
% 0.27/0.46 & ( X42 = Y42 ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % game.inject(4)
% 0.27/0.46 thf(fact_35_fst__uminus,axiom,
% 0.27/0.46 ! [B: $tType,A: $tType] :
% 0.27/0.46 ( ( ( uminus @ A )
% 0.27/0.46 & ( uminus @ B ) )
% 0.27/0.46 => ! [X2: product_prod @ A @ B] :
% 0.27/0.46 ( ( product_fst @ A @ B @ ( uminus_uminus @ ( product_prod @ A @ B ) @ X2 ) )
% 0.27/0.46 = ( uminus_uminus @ A @ ( product_fst @ A @ B @ X2 ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % fst_uminus
% 0.27/0.46 thf(fact_36_Uvariation__univ,axiom,
% 0.27/0.46 ! [Nu: variable > real,Nu5: variable > real] :
% 0.27/0.46 ( denota1419872369iation @ Nu @ Nu5
% 0.27/0.46 @ ( collect @ variable
% 0.27/0.46 @ ^ [X3: variable] : $true ) ) ).
% 0.27/0.46
% 0.27/0.46 % Uvariation_univ
% 0.27/0.46 thf(fact_37_game_Odistinct_I39_J,axiom,
% 0.27/0.46 ! [X41: game,X42: game,X6: game] :
% 0.27/0.46 ( ( choice @ X41 @ X42 )
% 0.27/0.46 != ( loop @ X6 ) ) ).
% 0.27/0.46
% 0.27/0.46 % game.distinct(39)
% 0.27/0.46 thf(fact_38_uminus__set__def,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( uminus_uminus @ ( set @ A ) )
% 0.27/0.46 = ( ^ [A3: set @ A] :
% 0.27/0.46 ( collect @ A
% 0.27/0.46 @ ( uminus_uminus @ ( A > $o )
% 0.27/0.46 @ ^ [X3: A] : ( member @ A @ X3 @ A3 ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % uminus_set_def
% 0.27/0.46 thf(fact_39_Collect__neg__eq,axiom,
% 0.27/0.46 ! [A: $tType,P: A > $o] :
% 0.27/0.46 ( ( collect @ A
% 0.27/0.46 @ ^ [X3: A] :
% 0.27/0.46 ~ ( P @ X3 ) )
% 0.27/0.46 = ( uminus_uminus @ ( set @ A ) @ ( collect @ A @ P ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % Collect_neg_eq
% 0.27/0.46 thf(fact_40_Compl__eq,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( uminus_uminus @ ( set @ A ) )
% 0.27/0.46 = ( ^ [A3: set @ A] :
% 0.27/0.46 ( collect @ A
% 0.27/0.46 @ ^ [X3: A] :
% 0.27/0.46 ~ ( member @ A @ X3 @ A3 ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % Compl_eq
% 0.27/0.46 thf(fact_41_game_Odistinct_I51_J,axiom,
% 0.27/0.46 ! [X6: game,X7: game] :
% 0.27/0.46 ( ( loop @ X6 )
% 0.27/0.46 != ( dual @ X7 ) ) ).
% 0.27/0.46
% 0.27/0.46 % game.distinct(51)
% 0.27/0.46 thf(fact_42_game_Odistinct_I41_J,axiom,
% 0.27/0.46 ! [X41: game,X42: game,X7: game] :
% 0.27/0.46 ( ( choice @ X41 @ X42 )
% 0.27/0.46 != ( dual @ X7 ) ) ).
% 0.27/0.46
% 0.27/0.46 % game.distinct(41)
% 0.27/0.46 thf(fact_43_usubstappp__loop__conv,axiom,
% 0.27/0.46 ! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,Alpha: game] :
% 0.27/0.46 ( ( ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ ( loop @ Alpha ) ) )
% 0.27/0.46 != ( none @ game ) )
% 0.27/0.46 => ( ( ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) )
% 0.27/0.47 != ( none @ game ) )
% 0.27/0.47 & ( ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) ) @ Alpha ) )
% 0.27/0.47 != ( none @ game ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % usubstappp_loop_conv
% 0.27/0.47 thf(fact_44_mem__Collect__eq,axiom,
% 0.27/0.47 ! [A: $tType,A4: A,P: A > $o] :
% 0.27/0.47 ( ( member @ A @ A4 @ ( collect @ A @ P ) )
% 0.27/0.47 = ( P @ A4 ) ) ).
% 0.27/0.47
% 0.27/0.47 % mem_Collect_eq
% 0.27/0.47 thf(fact_45_Collect__mem__eq,axiom,
% 0.27/0.47 ! [A: $tType,A2: set @ A] :
% 0.27/0.47 ( ( collect @ A
% 0.27/0.47 @ ^ [X3: A] : ( member @ A @ X3 @ A2 ) )
% 0.27/0.47 = A2 ) ).
% 0.27/0.47
% 0.27/0.47 % Collect_mem_eq
% 0.27/0.47 thf(fact_46_Collect__cong,axiom,
% 0.27/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.27/0.47 ( ! [X8: A] :
% 0.27/0.47 ( ( P @ X8 )
% 0.27/0.47 = ( Q @ X8 ) )
% 0.27/0.47 => ( ( collect @ A @ P )
% 0.27/0.47 = ( collect @ A @ Q ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Collect_cong
% 0.27/0.47 thf(fact_47_ext,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.27/0.47 ( ! [X8: A] :
% 0.27/0.47 ( ( F @ X8 )
% 0.27/0.47 = ( G @ X8 ) )
% 0.27/0.47 => ( F = G ) ) ).
% 0.27/0.47
% 0.27/0.47 % ext
% 0.27/0.47 thf(fact_48_skip__id,axiom,
% 0.27/0.47 ! [I: denotational_interp,X: set @ ( variable > real )] :
% 0.27/0.47 ( ( denota1245701238me_sem @ I @ skip @ X )
% 0.27/0.47 = X ) ).
% 0.27/0.47
% 0.27/0.47 % skip_id
% 0.27/0.47 thf(fact_49_prod__eqI,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType,P2: product_prod @ A @ B,Q2: product_prod @ A @ B] :
% 0.27/0.47 ( ( ( product_fst @ A @ B @ P2 )
% 0.27/0.47 = ( product_fst @ A @ B @ Q2 ) )
% 0.27/0.47 => ( ( ( product_snd @ A @ B @ P2 )
% 0.27/0.47 = ( product_snd @ A @ B @ Q2 ) )
% 0.27/0.47 => ( P2 = Q2 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % prod_eqI
% 0.27/0.47 thf(fact_50_exE__realizer_H,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType,P: A > B > $o,P2: product_prod @ B @ A] :
% 0.27/0.47 ( ( P @ ( product_snd @ B @ A @ P2 ) @ ( product_fst @ B @ A @ P2 ) )
% 0.27/0.47 => ~ ! [X8: B,Y2: A] :
% 0.27/0.47 ~ ( P @ Y2 @ X8 ) ) ).
% 0.27/0.47
% 0.27/0.47 % exE_realizer'
% 0.27/0.47 thf(fact_51_prod_Oexpand,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod2: product_prod @ A @ B] :
% 0.27/0.47 ( ( ( ( product_fst @ A @ B @ Prod )
% 0.27/0.47 = ( product_fst @ A @ B @ Prod2 ) )
% 0.27/0.47 & ( ( product_snd @ A @ B @ Prod )
% 0.27/0.47 = ( product_snd @ A @ B @ Prod2 ) ) )
% 0.27/0.47 => ( Prod = Prod2 ) ) ).
% 0.27/0.47
% 0.27/0.47 % prod.expand
% 0.27/0.47 thf(fact_52_prod__eq__iff,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType] :
% 0.27/0.47 ( ( ^ [Y3: product_prod @ A @ B,Z: product_prod @ A @ B] : ( Y3 = Z ) )
% 0.27/0.47 = ( ^ [S: product_prod @ A @ B,T2: product_prod @ A @ B] :
% 0.27/0.47 ( ( ( product_fst @ A @ B @ S )
% 0.27/0.47 = ( product_fst @ A @ B @ T2 ) )
% 0.27/0.47 & ( ( product_snd @ A @ B @ S )
% 0.27/0.47 = ( product_snd @ A @ B @ T2 ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % prod_eq_iff
% 0.27/0.47 thf(fact_53_uminus__prod__def,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType] :
% 0.27/0.47 ( ( ( uminus @ A )
% 0.27/0.47 & ( uminus @ B ) )
% 0.27/0.47 => ( ( uminus_uminus @ ( product_prod @ A @ B ) )
% 0.27/0.47 = ( ^ [X3: product_prod @ A @ B] : ( product_Pair @ A @ B @ ( uminus_uminus @ A @ ( product_fst @ A @ B @ X3 ) ) @ ( uminus_uminus @ B @ ( product_snd @ A @ B @ X3 ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % uminus_prod_def
% 0.27/0.47 thf(fact_54_usubstappp__antimon,axiom,
% 0.27/0.47 ! [V: set @ variable,U: set @ variable,Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),Alpha: game] :
% 0.27/0.47 ( ( ord_less_eq @ ( set @ variable ) @ V @ U )
% 0.27/0.47 => ( ( ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) )
% 0.27/0.47 != ( none @ game ) )
% 0.27/0.47 => ( ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) )
% 0.27/0.47 = ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ V @ Alpha ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % usubstappp_antimon
% 0.27/0.47 thf(fact_55_exists__diff,axiom,
% 0.27/0.47 ! [A: $tType,P: ( set @ A ) > $o] :
% 0.27/0.47 ( ( ? [S2: set @ A] : ( P @ ( uminus_uminus @ ( set @ A ) @ S2 ) ) )
% 0.27/0.47 = ( ? [X9: set @ A] : ( P @ X9 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % exists_diff
% 0.27/0.47 thf(fact_56_uminus__Pair,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType] :
% 0.27/0.47 ( ( ( uminus @ B )
% 0.27/0.47 & ( uminus @ A ) )
% 0.27/0.47 => ! [A4: A,B3: B] :
% 0.27/0.47 ( ( uminus_uminus @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) )
% 0.27/0.47 = ( product_Pair @ A @ B @ ( uminus_uminus @ A @ A4 ) @ ( uminus_uminus @ B @ B3 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % uminus_Pair
% 0.27/0.47 thf(fact_57_prod_Oinject,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y22: B] :
% 0.27/0.47 ( ( ( product_Pair @ A @ B @ X1 @ X22 )
% 0.27/0.47 = ( product_Pair @ A @ B @ Y1 @ Y22 ) )
% 0.27/0.47 = ( ( X1 = Y1 )
% 0.27/0.47 & ( X22 = Y22 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % prod.inject
% 0.27/0.47 thf(fact_58_old_Oprod_Oinject,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType,A4: A,B3: B,A5: A,B4: B] :
% 0.27/0.47 ( ( ( product_Pair @ A @ B @ A4 @ B3 )
% 0.27/0.47 = ( product_Pair @ A @ B @ A5 @ B4 ) )
% 0.27/0.47 = ( ( A4 = A5 )
% 0.27/0.47 & ( B3 = B4 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % old.prod.inject
% 0.27/0.47 thf(fact_59_subset__antisym,axiom,
% 0.27/0.47 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.27/0.47 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.27/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 0.27/0.47 => ( A2 = B2 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % subset_antisym
% 0.27/0.47 thf(fact_60_subsetI,axiom,
% 0.27/0.47 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.27/0.47 ( ! [X8: A] :
% 0.27/0.47 ( ( member @ A @ X8 @ A2 )
% 0.27/0.47 => ( member @ A @ X8 @ B2 ) )
% 0.27/0.47 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.27/0.47
% 0.27/0.47 % subsetI
% 0.27/0.47 thf(fact_61_compl__le__compl__iff,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( boolean_algebra @ A )
% 0.27/0.47 => ! [X2: A,Y: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X2 ) @ ( uminus_uminus @ A @ Y ) )
% 0.27/0.47 = ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % compl_le_compl_iff
% 0.27/0.47 thf(fact_62_neg__le__iff__le,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( ordered_ab_group_add @ A )
% 0.27/0.47 => ! [B3: A,A4: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( uminus_uminus @ A @ A4 ) )
% 0.27/0.47 = ( ord_less_eq @ A @ A4 @ B3 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % neg_le_iff_le
% 0.27/0.47 thf(fact_63_Compl__subset__Compl__iff,axiom,
% 0.27/0.47 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.27/0.47 ( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) @ ( uminus_uminus @ ( set @ A ) @ B2 ) )
% 0.27/0.47 = ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% 0.27/0.47
% 0.27/0.47 % Compl_subset_Compl_iff
% 0.27/0.47 thf(fact_64_Compl__anti__mono,axiom,
% 0.27/0.47 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.27/0.47 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.27/0.47 => ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B2 ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Compl_anti_mono
% 0.27/0.47 thf(fact_65_prod_Ocollapse,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 0.27/0.47 ( ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) )
% 0.27/0.47 = Prod ) ).
% 0.27/0.47
% 0.27/0.47 % prod.collapse
% 0.27/0.47 thf(fact_66_surj__pair,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
% 0.27/0.47 ? [X8: A,Y2: B] :
% 0.27/0.47 ( P2
% 0.27/0.47 = ( product_Pair @ A @ B @ X8 @ Y2 ) ) ).
% 0.27/0.47
% 0.27/0.47 % surj_pair
% 0.27/0.47 thf(fact_67_prod__cases,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
% 0.27/0.47 ( ! [A6: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A6 @ B5 ) )
% 0.27/0.47 => ( P @ P2 ) ) ).
% 0.27/0.47
% 0.27/0.47 % prod_cases
% 0.27/0.47 thf(fact_68_Pair__inject,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType,A4: A,B3: B,A5: A,B4: B] :
% 0.27/0.47 ( ( ( product_Pair @ A @ B @ A4 @ B3 )
% 0.27/0.47 = ( product_Pair @ A @ B @ A5 @ B4 ) )
% 0.27/0.47 => ~ ( ( A4 = A5 )
% 0.27/0.47 => ( B3 != B4 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Pair_inject
% 0.27/0.47 thf(fact_69_prod__cases3,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.27/0.47 ~ ! [A6: A,B5: B,C3: C] :
% 0.27/0.47 ( Y
% 0.27/0.47 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A6 @ ( product_Pair @ B @ C @ B5 @ C3 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % prod_cases3
% 0.27/0.47 thf(fact_70_prod__cases4,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.27/0.47 ~ ! [A6: A,B5: B,C3: C,D2: D] :
% 0.27/0.47 ( Y
% 0.27/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % prod_cases4
% 0.27/0.47 thf(fact_71_prod__cases5,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 0.27/0.47 ~ ! [A6: A,B5: B,C3: C,D2: D,E2: E] :
% 0.27/0.47 ( Y
% 0.27/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % prod_cases5
% 0.27/0.47 thf(fact_72_prod__cases6,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) )] :
% 0.27/0.47 ~ ! [A6: A,B5: B,C3: C,D2: D,E2: E,F3: F2] :
% 0.27/0.47 ( Y
% 0.27/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F2 ) @ D2 @ ( product_Pair @ E @ F2 @ E2 @ F3 ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % prod_cases6
% 0.27/0.47 thf(fact_73_prod__cases7,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F2: $tType,G2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) )] :
% 0.27/0.47 ~ ! [A6: A,B5: B,C3: C,D2: D,E2: E,F3: F2,G3: G2] :
% 0.27/0.47 ( Y
% 0.27/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F2 @ G2 ) @ E2 @ ( product_Pair @ F2 @ G2 @ F3 @ G3 ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % prod_cases7
% 0.27/0.47 thf(fact_74_prod__induct3,axiom,
% 0.27/0.47 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.27/0.47 ( ! [A6: A,B5: B,C3: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A6 @ ( product_Pair @ B @ C @ B5 @ C3 ) ) )
% 0.27/0.47 => ( P @ X2 ) ) ).
% 0.27/0.47
% 0.27/0.47 % prod_induct3
% 0.27/0.47 thf(fact_75_prod__induct4,axiom,
% 0.27/0.47 ! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.27/0.47 ( ! [A6: A,B5: B,C3: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) )
% 0.27/0.47 => ( P @ X2 ) ) ).
% 0.27/0.47
% 0.27/0.47 % prod_induct4
% 0.27/0.47 thf(fact_76_prod__induct5,axiom,
% 0.27/0.47 ! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 0.27/0.47 ( ! [A6: A,B5: B,C3: C,D2: D,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
% 0.27/0.47 => ( P @ X2 ) ) ).
% 0.27/0.47
% 0.27/0.47 % prod_induct5
% 0.27/0.47 thf(fact_77_prod__induct6,axiom,
% 0.27/0.47 ! [F2: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) )] :
% 0.27/0.47 ( ! [A6: A,B5: B,C3: C,D2: D,E2: E,F3: F2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F2 ) @ D2 @ ( product_Pair @ E @ F2 @ E2 @ F3 ) ) ) ) ) )
% 0.27/0.47 => ( P @ X2 ) ) ).
% 0.27/0.47
% 0.27/0.47 % prod_induct6
% 0.27/0.47 thf(fact_78_prod__induct7,axiom,
% 0.27/0.47 ! [G2: $tType,F2: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) )] :
% 0.27/0.47 ( ! [A6: A,B5: B,C3: C,D2: D,E2: E,F3: F2,G3: G2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F2 @ G2 ) @ E2 @ ( product_Pair @ F2 @ G2 @ F3 @ G3 ) ) ) ) ) ) )
% 0.27/0.47 => ( P @ X2 ) ) ).
% 0.27/0.47
% 0.27/0.47 % prod_induct7
% 0.27/0.47 thf(fact_79_old_Oprod_Oexhaust,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
% 0.27/0.47 ~ ! [A6: A,B5: B] :
% 0.27/0.47 ( Y
% 0.27/0.47 != ( product_Pair @ A @ B @ A6 @ B5 ) ) ).
% 0.27/0.47
% 0.27/0.47 % old.prod.exhaust
% 0.27/0.47 thf(fact_80_old_Oprod_Oinducts,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 0.27/0.47 ( ! [A6: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A6 @ B5 ) )
% 0.27/0.47 => ( P @ Prod ) ) ).
% 0.27/0.47
% 0.27/0.47 % old.prod.inducts
% 0.27/0.47 thf(fact_81_verit__la__disequality,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( linorder @ A )
% 0.27/0.47 => ! [A4: A,B3: A] :
% 0.27/0.47 ( ( A4 = B3 )
% 0.27/0.47 | ~ ( ord_less_eq @ A @ A4 @ B3 )
% 0.27/0.47 | ~ ( ord_less_eq @ A @ B3 @ A4 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % verit_la_disequality
% 0.27/0.47 thf(fact_82_Collect__mono__iff,axiom,
% 0.27/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.27/0.47 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 0.27/0.47 = ( ! [X3: A] :
% 0.27/0.47 ( ( P @ X3 )
% 0.27/0.47 => ( Q @ X3 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Collect_mono_iff
% 0.27/0.47 thf(fact_83_set__eq__subset,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( ^ [Y3: set @ A,Z: set @ A] : ( Y3 = Z ) )
% 0.27/0.47 = ( ^ [A3: set @ A,B6: set @ A] :
% 0.27/0.47 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
% 0.27/0.47 & ( ord_less_eq @ ( set @ A ) @ B6 @ A3 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % set_eq_subset
% 0.27/0.47 thf(fact_84_subset__trans,axiom,
% 0.27/0.47 ! [A: $tType,A2: set @ A,B2: set @ A,C4: set @ A] :
% 0.27/0.47 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.27/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C4 )
% 0.27/0.47 => ( ord_less_eq @ ( set @ A ) @ A2 @ C4 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % subset_trans
% 0.27/0.47 thf(fact_85_Collect__mono,axiom,
% 0.27/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.27/0.47 ( ! [X8: A] :
% 0.27/0.47 ( ( P @ X8 )
% 0.27/0.47 => ( Q @ X8 ) )
% 0.27/0.47 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Collect_mono
% 0.27/0.47 thf(fact_86_subset__refl,axiom,
% 0.27/0.47 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% 0.27/0.47
% 0.27/0.47 % subset_refl
% 0.27/0.47 thf(fact_87_subset__iff,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( ord_less_eq @ ( set @ A ) )
% 0.27/0.47 = ( ^ [A3: set @ A,B6: set @ A] :
% 0.27/0.47 ! [T2: A] :
% 0.27/0.47 ( ( member @ A @ T2 @ A3 )
% 0.27/0.47 => ( member @ A @ T2 @ B6 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % subset_iff
% 0.27/0.47 thf(fact_88_equalityD2,axiom,
% 0.27/0.47 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.27/0.47 ( ( A2 = B2 )
% 0.27/0.47 => ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% 0.27/0.47
% 0.27/0.47 % equalityD2
% 0.27/0.47 thf(fact_89_equalityD1,axiom,
% 0.27/0.47 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.27/0.47 ( ( A2 = B2 )
% 0.27/0.47 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.27/0.47
% 0.27/0.47 % equalityD1
% 0.27/0.47 thf(fact_90_subset__eq,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( ord_less_eq @ ( set @ A ) )
% 0.27/0.47 = ( ^ [A3: set @ A,B6: set @ A] :
% 0.27/0.47 ! [X3: A] :
% 0.27/0.47 ( ( member @ A @ X3 @ A3 )
% 0.27/0.47 => ( member @ A @ X3 @ B6 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % subset_eq
% 0.27/0.47 thf(fact_91_equalityE,axiom,
% 0.27/0.47 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.27/0.47 ( ( A2 = B2 )
% 0.27/0.47 => ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.27/0.47 => ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % equalityE
% 0.27/0.47 thf(fact_92_subsetD,axiom,
% 0.27/0.47 ! [A: $tType,A2: set @ A,B2: set @ A,C2: A] :
% 0.27/0.47 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.27/0.47 => ( ( member @ A @ C2 @ A2 )
% 0.27/0.47 => ( member @ A @ C2 @ B2 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % subsetD
% 0.27/0.47 thf(fact_93_in__mono,axiom,
% 0.27/0.47 ! [A: $tType,A2: set @ A,B2: set @ A,X2: A] :
% 0.27/0.47 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.27/0.47 => ( ( member @ A @ X2 @ A2 )
% 0.27/0.47 => ( member @ A @ X2 @ B2 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % in_mono
% 0.27/0.47 thf(fact_94_Collect__subset,axiom,
% 0.27/0.47 ! [A: $tType,A2: set @ A,P: A > $o] :
% 0.27/0.47 ( ord_less_eq @ ( set @ A )
% 0.27/0.47 @ ( collect @ A
% 0.27/0.47 @ ^ [X3: A] :
% 0.27/0.47 ( ( member @ A @ X3 @ A2 )
% 0.27/0.47 & ( P @ X3 ) ) )
% 0.27/0.47 @ A2 ) ).
% 0.27/0.47
% 0.27/0.47 % Collect_subset
% 0.27/0.47 thf(fact_95_fst__eqD,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType,X2: A,Y: B,A4: A] :
% 0.27/0.47 ( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X2 @ Y ) )
% 0.27/0.47 = A4 )
% 0.27/0.47 => ( X2 = A4 ) ) ).
% 0.27/0.47
% 0.27/0.47 % fst_eqD
% 0.27/0.47 thf(fact_96_fst__conv,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType,X1: A,X22: B] :
% 0.27/0.47 ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X22 ) )
% 0.27/0.47 = X1 ) ).
% 0.27/0.47
% 0.27/0.47 % fst_conv
% 0.27/0.47 thf(fact_97_snd__conv,axiom,
% 0.27/0.47 ! [Aa: $tType,A: $tType,X1: Aa,X22: A] :
% 0.27/0.47 ( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X22 ) )
% 0.27/0.47 = X22 ) ).
% 0.27/0.47
% 0.27/0.47 % snd_conv
% 0.27/0.47 thf(fact_98_snd__eqD,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType,X2: B,Y: A,A4: A] :
% 0.27/0.47 ( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X2 @ Y ) )
% 0.27/0.47 = A4 )
% 0.27/0.47 => ( Y = A4 ) ) ).
% 0.27/0.47
% 0.27/0.47 % snd_eqD
% 0.27/0.47 thf(fact_99_le__imp__neg__le,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( ordered_ab_group_add @ A )
% 0.27/0.47 => ! [A4: A,B3: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ A4 @ B3 )
% 0.27/0.47 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( uminus_uminus @ A @ A4 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % le_imp_neg_le
% 0.27/0.47 thf(fact_100_minus__le__iff,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( ordered_ab_group_add @ A )
% 0.27/0.47 => ! [A4: A,B3: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A4 ) @ B3 )
% 0.27/0.47 = ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ A4 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % minus_le_iff
% 0.27/0.47 thf(fact_101_le__minus__iff,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( ordered_ab_group_add @ A )
% 0.27/0.47 => ! [A4: A,B3: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ A4 @ ( uminus_uminus @ A @ B3 ) )
% 0.27/0.47 = ( ord_less_eq @ A @ B3 @ ( uminus_uminus @ A @ A4 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % le_minus_iff
% 0.27/0.47 thf(fact_102_compl__le__swap2,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( boolean_algebra @ A )
% 0.27/0.47 => ! [Y: A,X2: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ X2 )
% 0.27/0.47 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ X2 ) @ Y ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % compl_le_swap2
% 0.27/0.47 thf(fact_103_compl__le__swap1,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( boolean_algebra @ A )
% 0.27/0.47 => ! [Y: A,X2: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ Y @ ( uminus_uminus @ A @ X2 ) )
% 0.27/0.47 => ( ord_less_eq @ A @ X2 @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % compl_le_swap1
% 0.27/0.47 thf(fact_104_compl__mono,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( boolean_algebra @ A )
% 0.27/0.47 => ! [X2: A,Y: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.27/0.47 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X2 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % compl_mono
% 0.27/0.47 thf(fact_105_fst__pair,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType,A4: A,B3: B] :
% 0.27/0.47 ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ A4 @ B3 ) )
% 0.27/0.47 = A4 ) ).
% 0.27/0.47
% 0.27/0.47 % fst_pair
% 0.27/0.47 thf(fact_106_snd__pair,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType,A4: B,B3: A] :
% 0.27/0.47 ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ A4 @ B3 ) )
% 0.27/0.47 = B3 ) ).
% 0.27/0.47
% 0.27/0.47 % snd_pair
% 0.27/0.47 thf(fact_107_surjective__pairing,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType,T3: product_prod @ A @ B] :
% 0.27/0.47 ( T3
% 0.27/0.47 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T3 ) @ ( product_snd @ A @ B @ T3 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % surjective_pairing
% 0.27/0.47 thf(fact_108_prod_Oexhaust__sel,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 0.27/0.47 ( Prod
% 0.27/0.47 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % prod.exhaust_sel
% 0.27/0.47 thf(fact_109_conjI__realizer,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType,P: A > $o,P2: A,Q: B > $o,Q2: B] :
% 0.27/0.47 ( ( P @ P2 )
% 0.27/0.47 => ( ( Q @ Q2 )
% 0.27/0.47 => ( ( P @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q2 ) ) )
% 0.27/0.47 & ( Q @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q2 ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % conjI_realizer
% 0.27/0.47 thf(fact_110_exI__realizer,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType,P: A > B > $o,Y: A,X2: B] :
% 0.27/0.47 ( ( P @ Y @ X2 )
% 0.27/0.47 => ( P @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X2 @ Y ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X2 @ Y ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % exI_realizer
% 0.27/0.47 thf(fact_111_Uvariation__mon,axiom,
% 0.27/0.47 ! [U: set @ variable,V: set @ variable,Omega: variable > real,Nu: variable > real] :
% 0.27/0.47 ( ( ord_less_eq @ ( set @ variable ) @ U @ V )
% 0.27/0.47 => ( ( denota1419872369iation @ Omega @ Nu @ U )
% 0.27/0.47 => ( denota1419872369iation @ Omega @ Nu @ V ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Uvariation_mon
% 0.27/0.47 thf(fact_112_usubstappp__fst__mon,axiom,
% 0.27/0.47 ! [U: set @ variable,V: set @ variable,Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),Alpha: game] :
% 0.27/0.47 ( ( ord_less_eq @ ( set @ variable ) @ U @ V )
% 0.27/0.47 => ( ord_less_eq @ ( set @ variable ) @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) ) @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ V @ Alpha ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % usubstappp_fst_mon
% 0.27/0.47 thf(fact_113_usubst__taboos__mon,axiom,
% 0.27/0.47 ! [U: set @ variable,Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),Alpha: game] : ( ord_less_eq @ ( set @ variable ) @ U @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % usubst_taboos_mon
% 0.27/0.47 thf(fact_114_old_Oprod_Orec,axiom,
% 0.27/0.47 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A4: A,B3: B] :
% 0.27/0.47 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A4 @ B3 ) )
% 0.27/0.47 = ( F1 @ A4 @ B3 ) ) ).
% 0.27/0.47
% 0.27/0.47 % old.prod.rec
% 0.27/0.47 thf(fact_115_usubstappp__dual,axiom,
% 0.27/0.47 ! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,Alpha: game] :
% 0.27/0.47 ( ( uSubst95898988stappp @ Sigma @ U @ ( dual @ Alpha ) )
% 0.27/0.47 = ( product_Pair @ ( set @ variable ) @ ( option @ game ) @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) ) @ ( uSubst739989314_Dualo @ ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % usubstappp_dual
% 0.27/0.47 thf(fact_116_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType,P: A > B > $o,X2: A,Y: B,A4: product_prod @ A @ B] :
% 0.27/0.47 ( ( P @ X2 @ Y )
% 0.27/0.47 => ( ( A4
% 0.27/0.47 = ( product_Pair @ A @ B @ X2 @ Y ) )
% 0.27/0.47 => ( P @ ( product_fst @ A @ B @ A4 ) @ ( product_snd @ A @ B @ A4 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % BNF_Greatest_Fixpoint.subst_Pair
% 0.27/0.47 thf(fact_117_usubstappp_Osimps_I7_J,axiom,
% 0.27/0.47 ! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,Alpha: game] :
% 0.27/0.47 ( ( uSubst95898988stappp @ Sigma @ U @ ( dual @ Alpha ) )
% 0.27/0.47 = ( product_Pair @ ( set @ variable ) @ ( option @ game ) @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) ) @ ( uSubst739989314_Dualo @ ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % usubstappp.simps(7)
% 0.27/0.47 thf(fact_118_usubstappp_Osimps_I6_J,axiom,
% 0.27/0.47 ! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,Alpha: game] :
% 0.27/0.47 ( ( uSubst95898988stappp @ Sigma @ U @ ( loop @ Alpha ) )
% 0.27/0.47 = ( product_Pair @ ( set @ variable ) @ ( option @ game ) @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) ) @ ( uSubst993936602_Loopo @ ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) ) @ Alpha ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % usubstappp.simps(6)
% 0.27/0.47 thf(fact_119_usubstappp__loop,axiom,
% 0.27/0.47 ! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,Alpha: game] :
% 0.27/0.47 ( ( uSubst95898988stappp @ Sigma @ U @ ( loop @ Alpha ) )
% 0.27/0.47 = ( product_Pair @ ( set @ variable ) @ ( option @ game ) @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) ) @ ( uSubst993936602_Loopo @ ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) ) @ Alpha ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % usubstappp_loop
% 0.27/0.47 thf(fact_120_order__refl,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( preorder @ A )
% 0.27/0.47 => ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% 0.27/0.47
% 0.27/0.47 % order_refl
% 0.27/0.47 thf(fact_121_monotone,axiom,
% 0.27/0.47 ! [X: set @ ( variable > real ),Y4: set @ ( variable > real ),I: denotational_interp,Alpha: game] :
% 0.27/0.47 ( ( ord_less_eq @ ( set @ ( variable > real ) ) @ X @ Y4 )
% 0.27/0.47 => ( ord_less_eq @ ( set @ ( variable > real ) ) @ ( denota1245701238me_sem @ I @ Alpha @ X ) @ ( denota1245701238me_sem @ I @ Alpha @ Y4 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % monotone
% 0.27/0.47 thf(fact_122_less__eq__set__def,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( ord_less_eq @ ( set @ A ) )
% 0.27/0.47 = ( ^ [A3: set @ A,B6: set @ A] :
% 0.27/0.47 ( ord_less_eq @ ( A > $o )
% 0.27/0.47 @ ^ [X3: A] : ( member @ A @ X3 @ A3 )
% 0.27/0.47 @ ^ [X3: A] : ( member @ A @ X3 @ B6 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % less_eq_set_def
% 0.27/0.47 thf(fact_123_Loopo__undef,axiom,
% 0.27/0.47 ! [Alpha: option @ game] :
% 0.27/0.47 ( ( ( uSubst993936602_Loopo @ Alpha )
% 0.27/0.47 = ( none @ game ) )
% 0.27/0.47 = ( Alpha
% 0.27/0.47 = ( none @ game ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Loopo_undef
% 0.27/0.47 thf(fact_124_Dualo__undef,axiom,
% 0.27/0.47 ! [Alpha: option @ game] :
% 0.27/0.47 ( ( ( uSubst739989314_Dualo @ Alpha )
% 0.27/0.47 = ( none @ game ) )
% 0.27/0.47 = ( Alpha
% 0.27/0.47 = ( none @ game ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Dualo_undef
% 0.27/0.47 thf(fact_125_Loopo_Osimps_I2_J,axiom,
% 0.27/0.47 ( ( uSubst993936602_Loopo @ ( none @ game ) )
% 0.27/0.47 = ( none @ game ) ) ).
% 0.27/0.47
% 0.27/0.47 % Loopo.simps(2)
% 0.27/0.47 thf(fact_126_Dualo_Osimps_I2_J,axiom,
% 0.27/0.47 ( ( uSubst739989314_Dualo @ ( none @ game ) )
% 0.27/0.47 = ( none @ game ) ) ).
% 0.27/0.47
% 0.27/0.47 % Dualo.simps(2)
% 0.27/0.47 thf(fact_127_dual__order_Oantisym,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( order @ A )
% 0.27/0.47 => ! [B3: A,A4: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ B3 @ A4 )
% 0.27/0.47 => ( ( ord_less_eq @ A @ A4 @ B3 )
% 0.27/0.47 => ( A4 = B3 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % dual_order.antisym
% 0.27/0.47 thf(fact_128_dual__order_Oeq__iff,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( order @ A )
% 0.27/0.47 => ( ( ^ [Y3: A,Z: A] : ( Y3 = Z ) )
% 0.27/0.47 = ( ^ [A7: A,B7: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ B7 @ A7 )
% 0.27/0.47 & ( ord_less_eq @ A @ A7 @ B7 ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % dual_order.eq_iff
% 0.27/0.47 thf(fact_129_dual__order_Otrans,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( order @ A )
% 0.27/0.47 => ! [B3: A,A4: A,C2: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ B3 @ A4 )
% 0.27/0.47 => ( ( ord_less_eq @ A @ C2 @ B3 )
% 0.27/0.47 => ( ord_less_eq @ A @ C2 @ A4 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % dual_order.trans
% 0.27/0.47 thf(fact_130_linorder__wlog,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( linorder @ A )
% 0.27/0.47 => ! [P: A > A > $o,A4: A,B3: A] :
% 0.27/0.47 ( ! [A6: A,B5: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ A6 @ B5 )
% 0.27/0.47 => ( P @ A6 @ B5 ) )
% 0.27/0.47 => ( ! [A6: A,B5: A] :
% 0.27/0.47 ( ( P @ B5 @ A6 )
% 0.27/0.47 => ( P @ A6 @ B5 ) )
% 0.27/0.47 => ( P @ A4 @ B3 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % linorder_wlog
% 0.27/0.47 thf(fact_131_dual__order_Orefl,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( order @ A )
% 0.27/0.47 => ! [A4: A] : ( ord_less_eq @ A @ A4 @ A4 ) ) ).
% 0.27/0.47
% 0.27/0.47 % dual_order.refl
% 0.27/0.47 thf(fact_132_order__trans,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( preorder @ A )
% 0.27/0.47 => ! [X2: A,Y: A,Z2: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.27/0.47 => ( ( ord_less_eq @ A @ Y @ Z2 )
% 0.27/0.47 => ( ord_less_eq @ A @ X2 @ Z2 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % order_trans
% 0.27/0.47 thf(fact_133_order__class_Oorder_Oantisym,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( order @ A )
% 0.27/0.47 => ! [A4: A,B3: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ A4 @ B3 )
% 0.27/0.47 => ( ( ord_less_eq @ A @ B3 @ A4 )
% 0.27/0.47 => ( A4 = B3 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % order_class.order.antisym
% 0.27/0.47 thf(fact_134_ord__le__eq__trans,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( ord @ A )
% 0.27/0.47 => ! [A4: A,B3: A,C2: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ A4 @ B3 )
% 0.27/0.47 => ( ( B3 = C2 )
% 0.27/0.47 => ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % ord_le_eq_trans
% 0.27/0.47 thf(fact_135_ord__eq__le__trans,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( ord @ A )
% 0.27/0.47 => ! [A4: A,B3: A,C2: A] :
% 0.27/0.47 ( ( A4 = B3 )
% 0.27/0.47 => ( ( ord_less_eq @ A @ B3 @ C2 )
% 0.27/0.47 => ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % ord_eq_le_trans
% 0.27/0.47 thf(fact_136_order__class_Oorder_Oeq__iff,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( order @ A )
% 0.27/0.47 => ( ( ^ [Y3: A,Z: A] : ( Y3 = Z ) )
% 0.27/0.47 = ( ^ [A7: A,B7: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ A7 @ B7 )
% 0.27/0.47 & ( ord_less_eq @ A @ B7 @ A7 ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % order_class.order.eq_iff
% 0.27/0.47 thf(fact_137_antisym__conv,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( order @ A )
% 0.27/0.47 => ! [Y: A,X2: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ Y @ X2 )
% 0.27/0.47 => ( ( ord_less_eq @ A @ X2 @ Y )
% 0.27/0.47 = ( X2 = Y ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % antisym_conv
% 0.27/0.47 thf(fact_138_le__cases3,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( linorder @ A )
% 0.27/0.47 => ! [X2: A,Y: A,Z2: A] :
% 0.27/0.47 ( ( ( ord_less_eq @ A @ X2 @ Y )
% 0.27/0.47 => ~ ( ord_less_eq @ A @ Y @ Z2 ) )
% 0.27/0.47 => ( ( ( ord_less_eq @ A @ Y @ X2 )
% 0.27/0.47 => ~ ( ord_less_eq @ A @ X2 @ Z2 ) )
% 0.27/0.47 => ( ( ( ord_less_eq @ A @ X2 @ Z2 )
% 0.27/0.47 => ~ ( ord_less_eq @ A @ Z2 @ Y ) )
% 0.27/0.47 => ( ( ( ord_less_eq @ A @ Z2 @ Y )
% 0.27/0.47 => ~ ( ord_less_eq @ A @ Y @ X2 ) )
% 0.27/0.47 => ( ( ( ord_less_eq @ A @ Y @ Z2 )
% 0.27/0.47 => ~ ( ord_less_eq @ A @ Z2 @ X2 ) )
% 0.27/0.47 => ~ ( ( ord_less_eq @ A @ Z2 @ X2 )
% 0.27/0.47 => ~ ( ord_less_eq @ A @ X2 @ Y ) ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % le_cases3
% 0.27/0.47 thf(fact_139_order_Otrans,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( order @ A )
% 0.27/0.47 => ! [A4: A,B3: A,C2: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ A4 @ B3 )
% 0.27/0.47 => ( ( ord_less_eq @ A @ B3 @ C2 )
% 0.27/0.47 => ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % order.trans
% 0.27/0.47 thf(fact_140_le__cases,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( linorder @ A )
% 0.27/0.47 => ! [X2: A,Y: A] :
% 0.27/0.47 ( ~ ( ord_less_eq @ A @ X2 @ Y )
% 0.27/0.47 => ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % le_cases
% 0.27/0.47 thf(fact_141_eq__refl,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( preorder @ A )
% 0.27/0.47 => ! [X2: A,Y: A] :
% 0.27/0.47 ( ( X2 = Y )
% 0.27/0.47 => ( ord_less_eq @ A @ X2 @ Y ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % eq_refl
% 0.27/0.47 thf(fact_142_linear,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( linorder @ A )
% 0.27/0.47 => ! [X2: A,Y: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.27/0.47 | ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % linear
% 0.27/0.47 thf(fact_143_antisym,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( order @ A )
% 0.27/0.47 => ! [X2: A,Y: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.27/0.47 => ( ( ord_less_eq @ A @ Y @ X2 )
% 0.27/0.47 => ( X2 = Y ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % antisym
% 0.27/0.47 thf(fact_144_eq__iff,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( order @ A )
% 0.27/0.47 => ( ( ^ [Y3: A,Z: A] : ( Y3 = Z ) )
% 0.27/0.47 = ( ^ [X3: A,Y5: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ X3 @ Y5 )
% 0.27/0.47 & ( ord_less_eq @ A @ Y5 @ X3 ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % eq_iff
% 0.27/0.47 thf(fact_145_ord__le__eq__subst,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType] :
% 0.27/0.47 ( ( ( ord @ B )
% 0.27/0.47 & ( ord @ A ) )
% 0.27/0.47 => ! [A4: A,B3: A,F: A > B,C2: B] :
% 0.27/0.47 ( ( ord_less_eq @ A @ A4 @ B3 )
% 0.27/0.47 => ( ( ( F @ B3 )
% 0.27/0.47 = C2 )
% 0.27/0.47 => ( ! [X8: A,Y2: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ X8 @ Y2 )
% 0.27/0.47 => ( ord_less_eq @ B @ ( F @ X8 ) @ ( F @ Y2 ) ) )
% 0.27/0.47 => ( ord_less_eq @ B @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % ord_le_eq_subst
% 0.27/0.47 thf(fact_146_ord__eq__le__subst,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType] :
% 0.27/0.47 ( ( ( ord @ B )
% 0.27/0.47 & ( ord @ A ) )
% 0.27/0.47 => ! [A4: A,F: B > A,B3: B,C2: B] :
% 0.27/0.47 ( ( A4
% 0.27/0.47 = ( F @ B3 ) )
% 0.27/0.47 => ( ( ord_less_eq @ B @ B3 @ C2 )
% 0.27/0.47 => ( ! [X8: B,Y2: B] :
% 0.27/0.47 ( ( ord_less_eq @ B @ X8 @ Y2 )
% 0.27/0.47 => ( ord_less_eq @ A @ ( F @ X8 ) @ ( F @ Y2 ) ) )
% 0.27/0.47 => ( ord_less_eq @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % ord_eq_le_subst
% 0.27/0.47 thf(fact_147_order__subst2,axiom,
% 0.27/0.47 ! [A: $tType,C: $tType] :
% 0.27/0.47 ( ( ( order @ C )
% 0.27/0.47 & ( order @ A ) )
% 0.27/0.47 => ! [A4: A,B3: A,F: A > C,C2: C] :
% 0.27/0.47 ( ( ord_less_eq @ A @ A4 @ B3 )
% 0.27/0.47 => ( ( ord_less_eq @ C @ ( F @ B3 ) @ C2 )
% 0.27/0.47 => ( ! [X8: A,Y2: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ X8 @ Y2 )
% 0.27/0.47 => ( ord_less_eq @ C @ ( F @ X8 ) @ ( F @ Y2 ) ) )
% 0.27/0.47 => ( ord_less_eq @ C @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % order_subst2
% 0.27/0.47 thf(fact_148_order__subst1,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType] :
% 0.27/0.47 ( ( ( order @ B )
% 0.27/0.47 & ( order @ A ) )
% 0.27/0.47 => ! [A4: A,F: B > A,B3: B,C2: B] :
% 0.27/0.47 ( ( ord_less_eq @ A @ A4 @ ( F @ B3 ) )
% 0.27/0.47 => ( ( ord_less_eq @ B @ B3 @ C2 )
% 0.27/0.47 => ( ! [X8: B,Y2: B] :
% 0.27/0.47 ( ( ord_less_eq @ B @ X8 @ Y2 )
% 0.27/0.47 => ( ord_less_eq @ A @ ( F @ X8 ) @ ( F @ Y2 ) ) )
% 0.27/0.47 => ( ord_less_eq @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % order_subst1
% 0.27/0.47 thf(fact_149_le__fun__def,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType] :
% 0.27/0.47 ( ( ord @ B )
% 0.27/0.47 => ( ( ord_less_eq @ ( A > B ) )
% 0.27/0.47 = ( ^ [F4: A > B,G4: A > B] :
% 0.27/0.47 ! [X3: A] : ( ord_less_eq @ B @ ( F4 @ X3 ) @ ( G4 @ X3 ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % le_fun_def
% 0.27/0.47 thf(fact_150_le__funI,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType] :
% 0.27/0.47 ( ( ord @ B )
% 0.27/0.47 => ! [F: A > B,G: A > B] :
% 0.27/0.47 ( ! [X8: A] : ( ord_less_eq @ B @ ( F @ X8 ) @ ( G @ X8 ) )
% 0.27/0.47 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % le_funI
% 0.27/0.47 thf(fact_151_le__funE,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType] :
% 0.27/0.47 ( ( ord @ B )
% 0.27/0.47 => ! [F: A > B,G: A > B,X2: A] :
% 0.27/0.47 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.27/0.47 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % le_funE
% 0.27/0.47 thf(fact_152_le__funD,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType] :
% 0.27/0.47 ( ( ord @ B )
% 0.27/0.47 => ! [F: A > B,G: A > B,X2: A] :
% 0.27/0.47 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.27/0.47 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % le_funD
% 0.27/0.47 thf(fact_153_subset__CollectI,axiom,
% 0.27/0.47 ! [A: $tType,B2: set @ A,A2: set @ A,Q: A > $o,P: A > $o] :
% 0.27/0.47 ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 0.27/0.47 => ( ! [X8: A] :
% 0.27/0.47 ( ( member @ A @ X8 @ B2 )
% 0.27/0.47 => ( ( Q @ X8 )
% 0.27/0.47 => ( P @ X8 ) ) )
% 0.27/0.47 => ( ord_less_eq @ ( set @ A )
% 0.27/0.47 @ ( collect @ A
% 0.27/0.47 @ ^ [X3: A] :
% 0.27/0.47 ( ( member @ A @ X3 @ B2 )
% 0.27/0.47 & ( Q @ X3 ) ) )
% 0.27/0.47 @ ( collect @ A
% 0.27/0.47 @ ^ [X3: A] :
% 0.27/0.47 ( ( member @ A @ X3 @ A2 )
% 0.27/0.47 & ( P @ X3 ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % subset_CollectI
% 0.27/0.47 thf(fact_154_subset__Collect__iff,axiom,
% 0.27/0.47 ! [A: $tType,B2: set @ A,A2: set @ A,P: A > $o] :
% 0.27/0.47 ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 0.27/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B2
% 0.27/0.47 @ ( collect @ A
% 0.27/0.47 @ ^ [X3: A] :
% 0.27/0.47 ( ( member @ A @ X3 @ A2 )
% 0.27/0.47 & ( P @ X3 ) ) ) )
% 0.27/0.47 = ( ! [X3: A] :
% 0.27/0.47 ( ( member @ A @ X3 @ B2 )
% 0.27/0.47 => ( P @ X3 ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % subset_Collect_iff
% 0.27/0.47 thf(fact_155_sndI,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType,X2: product_prod @ A @ B,Y: A,Z2: B] :
% 0.27/0.47 ( ( X2
% 0.27/0.47 = ( product_Pair @ A @ B @ Y @ Z2 ) )
% 0.27/0.47 => ( ( product_snd @ A @ B @ X2 )
% 0.27/0.47 = Z2 ) ) ).
% 0.27/0.47
% 0.27/0.47 % sndI
% 0.27/0.47 thf(fact_156_eq__snd__iff,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType,B3: A,P2: product_prod @ B @ A] :
% 0.27/0.47 ( ( B3
% 0.27/0.47 = ( product_snd @ B @ A @ P2 ) )
% 0.27/0.47 = ( ? [A7: B] :
% 0.27/0.47 ( P2
% 0.27/0.47 = ( product_Pair @ B @ A @ A7 @ B3 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % eq_snd_iff
% 0.27/0.47 thf(fact_157_eq__fst__iff,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType,A4: A,P2: product_prod @ A @ B] :
% 0.27/0.47 ( ( A4
% 0.27/0.47 = ( product_fst @ A @ B @ P2 ) )
% 0.27/0.47 = ( ? [B7: B] :
% 0.27/0.47 ( P2
% 0.27/0.47 = ( product_Pair @ A @ B @ A4 @ B7 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % eq_fst_iff
% 0.27/0.47 thf(fact_158_fstI,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType,X2: product_prod @ A @ B,Y: A,Z2: B] :
% 0.27/0.47 ( ( X2
% 0.27/0.47 = ( product_Pair @ A @ B @ Y @ Z2 ) )
% 0.27/0.47 => ( ( product_fst @ A @ B @ X2 )
% 0.27/0.47 = Y ) ) ).
% 0.27/0.47
% 0.27/0.47 % fstI
% 0.27/0.47 thf(fact_159_predicate1I,axiom,
% 0.27/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.27/0.47 ( ! [X8: A] :
% 0.27/0.47 ( ( P @ X8 )
% 0.27/0.47 => ( Q @ X8 ) )
% 0.27/0.47 => ( ord_less_eq @ ( A > $o ) @ P @ Q ) ) ).
% 0.27/0.47
% 0.27/0.47 % predicate1I
% 0.27/0.47 thf(fact_160_predicate1D,axiom,
% 0.27/0.47 ! [A: $tType,P: A > $o,Q: A > $o,X2: A] :
% 0.27/0.47 ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
% 0.27/0.47 => ( ( P @ X2 )
% 0.27/0.47 => ( Q @ X2 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % predicate1D
% 0.27/0.47 thf(fact_161_rev__predicate1D,axiom,
% 0.27/0.47 ! [A: $tType,P: A > $o,X2: A,Q: A > $o] :
% 0.27/0.47 ( ( P @ X2 )
% 0.27/0.47 => ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
% 0.27/0.47 => ( Q @ X2 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % rev_predicate1D
% 0.27/0.47 thf(fact_162_pred__subset__eq,axiom,
% 0.27/0.47 ! [A: $tType,R: set @ A,S3: set @ A] :
% 0.27/0.47 ( ( ord_less_eq @ ( A > $o )
% 0.27/0.47 @ ^ [X3: A] : ( member @ A @ X3 @ R )
% 0.27/0.47 @ ^ [X3: A] : ( member @ A @ X3 @ S3 ) )
% 0.27/0.47 = ( ord_less_eq @ ( set @ A ) @ R @ S3 ) ) ).
% 0.27/0.47
% 0.27/0.47 % pred_subset_eq
% 0.27/0.47 thf(fact_163_scomp__unfold,axiom,
% 0.27/0.47 ! [D: $tType,C: $tType,B: $tType,A: $tType] :
% 0.27/0.47 ( ( product_scomp @ A @ B @ C @ D )
% 0.27/0.47 = ( ^ [F4: A > ( product_prod @ B @ C ),G4: B > C > D,X3: A] : ( G4 @ ( product_fst @ B @ C @ ( F4 @ X3 ) ) @ ( product_snd @ B @ C @ ( F4 @ X3 ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % scomp_unfold
% 0.27/0.47 thf(fact_164_Collect__restrict,axiom,
% 0.27/0.47 ! [A: $tType,X: set @ A,P: A > $o] :
% 0.27/0.47 ( ord_less_eq @ ( set @ A )
% 0.27/0.47 @ ( collect @ A
% 0.27/0.47 @ ^ [X3: A] :
% 0.27/0.47 ( ( member @ A @ X3 @ X )
% 0.27/0.47 & ( P @ X3 ) ) )
% 0.27/0.47 @ X ) ).
% 0.27/0.47
% 0.27/0.47 % Collect_restrict
% 0.27/0.47 thf(fact_165_scomp__Pair,axiom,
% 0.27/0.47 ! [C: $tType,B: $tType,A: $tType,X2: A > ( product_prod @ B @ C )] :
% 0.27/0.47 ( ( product_scomp @ A @ B @ C @ ( product_prod @ B @ C ) @ X2 @ ( product_Pair @ B @ C ) )
% 0.27/0.47 = X2 ) ).
% 0.27/0.47
% 0.27/0.47 % scomp_Pair
% 0.27/0.47 thf(fact_166_scomp__scomp,axiom,
% 0.27/0.47 ! [A: $tType,C: $tType,D: $tType,B: $tType,F2: $tType,E: $tType,F: A > ( product_prod @ E @ F2 ),G: E > F2 > ( product_prod @ C @ D ),H: C > D > B] :
% 0.27/0.47 ( ( product_scomp @ A @ C @ D @ B @ ( product_scomp @ A @ E @ F2 @ ( product_prod @ C @ D ) @ F @ G ) @ H )
% 0.27/0.47 = ( product_scomp @ A @ E @ F2 @ B @ F
% 0.27/0.47 @ ^ [X3: E] : ( product_scomp @ F2 @ C @ D @ B @ ( G @ X3 ) @ H ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % scomp_scomp
% 0.27/0.47 thf(fact_167_Pair__scomp,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType,C: $tType,X2: C,F: C > A > B] :
% 0.27/0.47 ( ( product_scomp @ A @ C @ A @ B @ ( product_Pair @ C @ A @ X2 ) @ F )
% 0.27/0.47 = ( F @ X2 ) ) ).
% 0.27/0.47
% 0.27/0.47 % Pair_scomp
% 0.27/0.47 thf(fact_168_subrelI,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S4: set @ ( product_prod @ A @ B )] :
% 0.27/0.47 ( ! [X8: A,Y2: B] :
% 0.27/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X8 @ Y2 ) @ R2 )
% 0.27/0.47 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X8 @ Y2 ) @ S4 ) )
% 0.27/0.47 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S4 ) ) ).
% 0.27/0.47
% 0.27/0.47 % subrelI
% 0.27/0.47 thf(fact_169_ssubst__Pair__rhs,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType,R2: A,S4: B,R: set @ ( product_prod @ A @ B ),S5: B] :
% 0.27/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S4 ) @ R )
% 0.27/0.47 => ( ( S5 = S4 )
% 0.27/0.47 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S5 ) @ R ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % ssubst_Pair_rhs
% 0.27/0.47 thf(fact_170_pred__subset__eq2,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
% 0.27/0.47 ( ( ord_less_eq @ ( A > B > $o )
% 0.27/0.47 @ ^ [X3: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y5 ) @ R )
% 0.27/0.47 @ ^ [X3: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y5 ) @ S3 ) )
% 0.27/0.47 = ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S3 ) ) ).
% 0.27/0.47
% 0.27/0.47 % pred_subset_eq2
% 0.27/0.47 thf(fact_171_pred__equals__eq2,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
% 0.27/0.47 ( ( ( ^ [X3: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y5 ) @ R ) )
% 0.27/0.47 = ( ^ [X3: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y5 ) @ S3 ) ) )
% 0.27/0.47 = ( R = S3 ) ) ).
% 0.27/0.47
% 0.27/0.47 % pred_equals_eq2
% 0.27/0.47 thf(fact_172_prop__restrict,axiom,
% 0.27/0.47 ! [A: $tType,X2: A,Z3: set @ A,X: set @ A,P: A > $o] :
% 0.27/0.47 ( ( member @ A @ X2 @ Z3 )
% 0.27/0.47 => ( ( ord_less_eq @ ( set @ A ) @ Z3
% 0.27/0.47 @ ( collect @ A
% 0.27/0.47 @ ^ [X3: A] :
% 0.27/0.47 ( ( member @ A @ X3 @ X )
% 0.27/0.47 & ( P @ X3 ) ) ) )
% 0.27/0.47 => ( P @ X2 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % prop_restrict
% 0.27/0.47 thf(fact_173_conj__subset__def,axiom,
% 0.27/0.47 ! [A: $tType,A2: set @ A,P: A > $o,Q: A > $o] :
% 0.27/0.47 ( ( ord_less_eq @ ( set @ A ) @ A2
% 0.27/0.47 @ ( collect @ A
% 0.27/0.47 @ ^ [X3: A] :
% 0.27/0.47 ( ( P @ X3 )
% 0.27/0.47 & ( Q @ X3 ) ) ) )
% 0.27/0.47 = ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( collect @ A @ P ) )
% 0.27/0.47 & ( ord_less_eq @ ( set @ A ) @ A2 @ ( collect @ A @ Q ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % conj_subset_def
% 0.27/0.47 thf(fact_174_internal__case__prod__conv,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A4: B,B3: C] :
% 0.27/0.47 ( ( produc2004651681e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A4 @ B3 ) )
% 0.27/0.47 = ( C2 @ A4 @ B3 ) ) ).
% 0.27/0.47
% 0.27/0.47 % internal_case_prod_conv
% 0.27/0.47 thf(fact_175_prod_Oswap__def,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType] :
% 0.27/0.47 ( ( product_swap @ A @ B )
% 0.27/0.47 = ( ^ [P3: product_prod @ A @ B] : ( product_Pair @ B @ A @ ( product_snd @ A @ B @ P3 ) @ ( product_fst @ A @ B @ P3 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % prod.swap_def
% 0.27/0.47 thf(fact_176_predicate2I,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType,P: A > B > $o,Q: A > B > $o] :
% 0.27/0.47 ( ! [X8: A,Y2: B] :
% 0.27/0.47 ( ( P @ X8 @ Y2 )
% 0.27/0.47 => ( Q @ X8 @ Y2 ) )
% 0.27/0.47 => ( ord_less_eq @ ( A > B > $o ) @ P @ Q ) ) ).
% 0.27/0.47
% 0.27/0.47 % predicate2I
% 0.27/0.47 thf(fact_177_swap__swap,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType,P2: product_prod @ A @ B] :
% 0.27/0.47 ( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P2 ) )
% 0.27/0.47 = P2 ) ).
% 0.27/0.47
% 0.27/0.47 % swap_swap
% 0.27/0.47 thf(fact_178_swap__simp,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType,X2: B,Y: A] :
% 0.27/0.47 ( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X2 @ Y ) )
% 0.27/0.47 = ( product_Pair @ A @ B @ Y @ X2 ) ) ).
% 0.27/0.47
% 0.27/0.47 % swap_simp
% 0.27/0.47 thf(fact_179_snd__swap,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType,X2: product_prod @ A @ B] :
% 0.27/0.47 ( ( product_snd @ B @ A @ ( product_swap @ A @ B @ X2 ) )
% 0.27/0.47 = ( product_fst @ A @ B @ X2 ) ) ).
% 0.27/0.47
% 0.27/0.47 % snd_swap
% 0.27/0.47 thf(fact_180_fst__swap,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType,X2: product_prod @ B @ A] :
% 0.27/0.47 ( ( product_fst @ A @ B @ ( product_swap @ B @ A @ X2 ) )
% 0.27/0.47 = ( product_snd @ B @ A @ X2 ) ) ).
% 0.27/0.47
% 0.27/0.47 % fst_swap
% 0.27/0.47 thf(fact_181_rev__predicate2D,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType,P: A > B > $o,X2: A,Y: B,Q: A > B > $o] :
% 0.27/0.47 ( ( P @ X2 @ Y )
% 0.27/0.47 => ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
% 0.27/0.47 => ( Q @ X2 @ Y ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % rev_predicate2D
% 0.27/0.47 thf(fact_182_predicate2D,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,X2: A,Y: B] :
% 0.27/0.47 ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
% 0.27/0.47 => ( ( P @ X2 @ Y )
% 0.27/0.47 => ( Q @ X2 @ Y ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % predicate2D
% 0.27/0.47 thf(fact_183_refl__ge__eq,axiom,
% 0.27/0.47 ! [A: $tType,R: A > A > $o] :
% 0.27/0.47 ( ! [X8: A] : ( R @ X8 @ X8 )
% 0.27/0.47 => ( ord_less_eq @ ( A > A > $o )
% 0.27/0.47 @ ^ [Y3: A,Z: A] : ( Y3 = Z )
% 0.27/0.47 @ R ) ) ).
% 0.27/0.47
% 0.27/0.47 % refl_ge_eq
% 0.27/0.47 thf(fact_184_ge__eq__refl,axiom,
% 0.27/0.47 ! [A: $tType,R: A > A > $o,X2: A] :
% 0.27/0.47 ( ( ord_less_eq @ ( A > A > $o )
% 0.27/0.47 @ ^ [Y3: A,Z: A] : ( Y3 = Z )
% 0.27/0.47 @ R )
% 0.27/0.47 => ( R @ X2 @ X2 ) ) ).
% 0.27/0.47
% 0.27/0.47 % ge_eq_refl
% 0.27/0.47 thf(fact_185_eq__subset,axiom,
% 0.27/0.47 ! [A: $tType,P: A > A > $o] :
% 0.27/0.47 ( ord_less_eq @ ( A > A > $o )
% 0.27/0.47 @ ^ [Y3: A,Z: A] : ( Y3 = Z )
% 0.27/0.47 @ ^ [A7: A,B7: A] :
% 0.27/0.47 ( ( P @ A7 @ B7 )
% 0.27/0.47 | ( A7 = B7 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % eq_subset
% 0.27/0.47 thf(fact_186_usubstappp__choice,axiom,
% 0.27/0.47 ! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,Alpha: game,Beta: game] :
% 0.27/0.47 ( ( uSubst95898988stappp @ Sigma @ U @ ( choice @ Alpha @ Beta ) )
% 0.27/0.47 = ( product_Pair @ ( set @ variable ) @ ( option @ game ) @ ( sup_sup @ ( set @ variable ) @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) ) @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Beta ) ) ) @ ( uSubst1976112797hoiceo @ ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) ) @ ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Beta ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % usubstappp_choice
% 0.27/0.47 thf(fact_187_relChain__def,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType] :
% 0.27/0.47 ( ( ord @ B )
% 0.27/0.47 => ( ( bNF_Ca1785829860lChain @ A @ B )
% 0.27/0.47 = ( ^ [R3: set @ ( product_prod @ A @ A ),As: A > B] :
% 0.27/0.47 ! [I2: A,J: A] :
% 0.27/0.47 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I2 @ J ) @ R3 )
% 0.27/0.47 => ( ord_less_eq @ B @ ( As @ I2 ) @ ( As @ J ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % relChain_def
% 0.27/0.47 thf(fact_188_sup__apply,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType] :
% 0.27/0.47 ( ( semilattice_sup @ B )
% 0.27/0.47 => ( ( sup_sup @ ( A > B ) )
% 0.27/0.47 = ( ^ [F4: A > B,G4: A > B,X3: A] : ( sup_sup @ B @ ( F4 @ X3 ) @ ( G4 @ X3 ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % sup_apply
% 0.27/0.47 thf(fact_189_sup_Oidem,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( semilattice_sup @ A )
% 0.27/0.47 => ! [A4: A] :
% 0.27/0.47 ( ( sup_sup @ A @ A4 @ A4 )
% 0.27/0.47 = A4 ) ) ).
% 0.27/0.47
% 0.27/0.47 % sup.idem
% 0.27/0.47 thf(fact_190_sup__idem,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( semilattice_sup @ A )
% 0.27/0.47 => ! [X2: A] :
% 0.27/0.47 ( ( sup_sup @ A @ X2 @ X2 )
% 0.27/0.47 = X2 ) ) ).
% 0.27/0.47
% 0.27/0.47 % sup_idem
% 0.27/0.47 thf(fact_191_sup_Oleft__idem,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( semilattice_sup @ A )
% 0.27/0.47 => ! [A4: A,B3: A] :
% 0.27/0.47 ( ( sup_sup @ A @ A4 @ ( sup_sup @ A @ A4 @ B3 ) )
% 0.27/0.47 = ( sup_sup @ A @ A4 @ B3 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % sup.left_idem
% 0.27/0.47 thf(fact_192_sup__left__idem,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( semilattice_sup @ A )
% 0.27/0.47 => ! [X2: A,Y: A] :
% 0.27/0.47 ( ( sup_sup @ A @ X2 @ ( sup_sup @ A @ X2 @ Y ) )
% 0.27/0.47 = ( sup_sup @ A @ X2 @ Y ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % sup_left_idem
% 0.27/0.47 thf(fact_193_sup_Oright__idem,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( semilattice_sup @ A )
% 0.27/0.47 => ! [A4: A,B3: A] :
% 0.27/0.47 ( ( sup_sup @ A @ ( sup_sup @ A @ A4 @ B3 ) @ B3 )
% 0.27/0.47 = ( sup_sup @ A @ A4 @ B3 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % sup.right_idem
% 0.27/0.47 thf(fact_194_UnCI,axiom,
% 0.27/0.47 ! [A: $tType,C2: A,B2: set @ A,A2: set @ A] :
% 0.27/0.47 ( ( ~ ( member @ A @ C2 @ B2 )
% 0.27/0.47 => ( member @ A @ C2 @ A2 ) )
% 0.27/0.47 => ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % UnCI
% 0.27/0.47 thf(fact_195_sup_Obounded__iff,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( semilattice_sup @ A )
% 0.27/0.47 => ! [B3: A,C2: A,A4: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C2 ) @ A4 )
% 0.27/0.47 = ( ( ord_less_eq @ A @ B3 @ A4 )
% 0.27/0.47 & ( ord_less_eq @ A @ C2 @ A4 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % sup.bounded_iff
% 0.27/0.47 thf(fact_196_le__sup__iff,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( semilattice_sup @ A )
% 0.27/0.47 => ! [X2: A,Y: A,Z2: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ ( sup_sup @ A @ X2 @ Y ) @ Z2 )
% 0.27/0.47 = ( ( ord_less_eq @ A @ X2 @ Z2 )
% 0.27/0.47 & ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % le_sup_iff
% 0.27/0.47 thf(fact_197_Un__subset__iff,axiom,
% 0.27/0.47 ! [A: $tType,A2: set @ A,B2: set @ A,C4: set @ A] :
% 0.27/0.47 ( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ C4 )
% 0.27/0.47 = ( ( ord_less_eq @ ( set @ A ) @ A2 @ C4 )
% 0.27/0.47 & ( ord_less_eq @ ( set @ A ) @ B2 @ C4 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Un_subset_iff
% 0.27/0.47 thf(fact_198_subset__Un__eq,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( ord_less_eq @ ( set @ A ) )
% 0.27/0.47 = ( ^ [A3: set @ A,B6: set @ A] :
% 0.27/0.47 ( ( sup_sup @ ( set @ A ) @ A3 @ B6 )
% 0.27/0.47 = B6 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % subset_Un_eq
% 0.27/0.47 thf(fact_199_subset__UnE,axiom,
% 0.27/0.47 ! [A: $tType,C4: set @ A,A2: set @ A,B2: set @ A] :
% 0.27/0.47 ( ( ord_less_eq @ ( set @ A ) @ C4 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
% 0.27/0.47 => ~ ! [A8: set @ A] :
% 0.27/0.47 ( ( ord_less_eq @ ( set @ A ) @ A8 @ A2 )
% 0.27/0.47 => ! [B8: set @ A] :
% 0.27/0.47 ( ( ord_less_eq @ ( set @ A ) @ B8 @ B2 )
% 0.27/0.47 => ( C4
% 0.27/0.47 != ( sup_sup @ ( set @ A ) @ A8 @ B8 ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % subset_UnE
% 0.27/0.47 thf(fact_200_Un__absorb2,axiom,
% 0.27/0.47 ! [A: $tType,B2: set @ A,A2: set @ A] :
% 0.27/0.47 ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 0.27/0.47 => ( ( sup_sup @ ( set @ A ) @ A2 @ B2 )
% 0.27/0.47 = A2 ) ) ).
% 0.27/0.47
% 0.27/0.47 % Un_absorb2
% 0.27/0.47 thf(fact_201_Un__absorb1,axiom,
% 0.27/0.47 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.27/0.47 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.27/0.47 => ( ( sup_sup @ ( set @ A ) @ A2 @ B2 )
% 0.27/0.47 = B2 ) ) ).
% 0.27/0.47
% 0.27/0.47 % Un_absorb1
% 0.27/0.47 thf(fact_202_Un__upper2,axiom,
% 0.27/0.47 ! [A: $tType,B2: set @ A,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ B2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.27/0.47
% 0.27/0.47 % Un_upper2
% 0.27/0.47 thf(fact_203_Un__upper1,axiom,
% 0.27/0.47 ! [A: $tType,A2: set @ A,B2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.27/0.47
% 0.27/0.47 % Un_upper1
% 0.27/0.47 thf(fact_204_Un__least,axiom,
% 0.27/0.47 ! [A: $tType,A2: set @ A,C4: set @ A,B2: set @ A] :
% 0.27/0.47 ( ( ord_less_eq @ ( set @ A ) @ A2 @ C4 )
% 0.27/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C4 )
% 0.27/0.47 => ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ C4 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Un_least
% 0.27/0.47 thf(fact_205_Un__mono,axiom,
% 0.27/0.47 ! [A: $tType,A2: set @ A,C4: set @ A,B2: set @ A,D3: set @ A] :
% 0.27/0.47 ( ( ord_less_eq @ ( set @ A ) @ A2 @ C4 )
% 0.27/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ D3 )
% 0.27/0.47 => ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ ( sup_sup @ ( set @ A ) @ C4 @ D3 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Un_mono
% 0.27/0.47 thf(fact_206_sup_OcoboundedI2,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( semilattice_sup @ A )
% 0.27/0.47 => ! [C2: A,B3: A,A4: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ C2 @ B3 )
% 0.27/0.47 => ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A4 @ B3 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % sup.coboundedI2
% 0.27/0.47 thf(fact_207_sup_OcoboundedI1,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( semilattice_sup @ A )
% 0.27/0.47 => ! [C2: A,A4: A,B3: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ C2 @ A4 )
% 0.27/0.47 => ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A4 @ B3 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % sup.coboundedI1
% 0.27/0.47 thf(fact_208_sup_Oabsorb__iff2,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( semilattice_sup @ A )
% 0.27/0.47 => ( ( ord_less_eq @ A )
% 0.27/0.47 = ( ^ [A7: A,B7: A] :
% 0.27/0.47 ( ( sup_sup @ A @ A7 @ B7 )
% 0.27/0.47 = B7 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % sup.absorb_iff2
% 0.27/0.47 thf(fact_209_sup_Oabsorb__iff1,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( semilattice_sup @ A )
% 0.27/0.47 => ( ( ord_less_eq @ A )
% 0.27/0.47 = ( ^ [B7: A,A7: A] :
% 0.27/0.47 ( ( sup_sup @ A @ A7 @ B7 )
% 0.27/0.47 = A7 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % sup.absorb_iff1
% 0.27/0.47 thf(fact_210_sup_Ocobounded2,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( semilattice_sup @ A )
% 0.27/0.47 => ! [B3: A,A4: A] : ( ord_less_eq @ A @ B3 @ ( sup_sup @ A @ A4 @ B3 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % sup.cobounded2
% 0.27/0.47 thf(fact_211_sup_Ocobounded1,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( semilattice_sup @ A )
% 0.27/0.47 => ! [A4: A,B3: A] : ( ord_less_eq @ A @ A4 @ ( sup_sup @ A @ A4 @ B3 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % sup.cobounded1
% 0.27/0.47 thf(fact_212_sup_Oorder__iff,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( semilattice_sup @ A )
% 0.27/0.47 => ( ( ord_less_eq @ A )
% 0.27/0.47 = ( ^ [B7: A,A7: A] :
% 0.27/0.47 ( A7
% 0.27/0.47 = ( sup_sup @ A @ A7 @ B7 ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % sup.order_iff
% 0.27/0.47 thf(fact_213_sup_OboundedI,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( semilattice_sup @ A )
% 0.27/0.47 => ! [B3: A,A4: A,C2: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ B3 @ A4 )
% 0.27/0.47 => ( ( ord_less_eq @ A @ C2 @ A4 )
% 0.27/0.47 => ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C2 ) @ A4 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % sup.boundedI
% 0.27/0.47 thf(fact_214_sup_OboundedE,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( semilattice_sup @ A )
% 0.27/0.47 => ! [B3: A,C2: A,A4: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C2 ) @ A4 )
% 0.27/0.47 => ~ ( ( ord_less_eq @ A @ B3 @ A4 )
% 0.27/0.47 => ~ ( ord_less_eq @ A @ C2 @ A4 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % sup.boundedE
% 0.27/0.47 thf(fact_215_sup__absorb2,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( semilattice_sup @ A )
% 0.27/0.47 => ! [X2: A,Y: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.27/0.47 => ( ( sup_sup @ A @ X2 @ Y )
% 0.27/0.47 = Y ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % sup_absorb2
% 0.27/0.47 thf(fact_216_sup__absorb1,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( semilattice_sup @ A )
% 0.27/0.47 => ! [Y: A,X2: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ Y @ X2 )
% 0.27/0.47 => ( ( sup_sup @ A @ X2 @ Y )
% 0.27/0.47 = X2 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % sup_absorb1
% 0.27/0.47 thf(fact_217_sup_Oabsorb2,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( semilattice_sup @ A )
% 0.27/0.47 => ! [A4: A,B3: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ A4 @ B3 )
% 0.27/0.47 => ( ( sup_sup @ A @ A4 @ B3 )
% 0.27/0.47 = B3 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % sup.absorb2
% 0.27/0.47 thf(fact_218_sup_Oabsorb1,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( semilattice_sup @ A )
% 0.27/0.47 => ! [B3: A,A4: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ B3 @ A4 )
% 0.27/0.47 => ( ( sup_sup @ A @ A4 @ B3 )
% 0.27/0.47 = A4 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % sup.absorb1
% 0.27/0.47 thf(fact_219_sup__unique,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( semilattice_sup @ A )
% 0.27/0.47 => ! [F: A > A > A,X2: A,Y: A] :
% 0.27/0.47 ( ! [X8: A,Y2: A] : ( ord_less_eq @ A @ X8 @ ( F @ X8 @ Y2 ) )
% 0.27/0.47 => ( ! [X8: A,Y2: A] : ( ord_less_eq @ A @ Y2 @ ( F @ X8 @ Y2 ) )
% 0.27/0.47 => ( ! [X8: A,Y2: A,Z4: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ Y2 @ X8 )
% 0.27/0.47 => ( ( ord_less_eq @ A @ Z4 @ X8 )
% 0.27/0.47 => ( ord_less_eq @ A @ ( F @ Y2 @ Z4 ) @ X8 ) ) )
% 0.27/0.47 => ( ( sup_sup @ A @ X2 @ Y )
% 0.27/0.47 = ( F @ X2 @ Y ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % sup_unique
% 0.27/0.47 thf(fact_220_sup_OorderI,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( semilattice_sup @ A )
% 0.27/0.47 => ! [A4: A,B3: A] :
% 0.27/0.47 ( ( A4
% 0.27/0.47 = ( sup_sup @ A @ A4 @ B3 ) )
% 0.27/0.47 => ( ord_less_eq @ A @ B3 @ A4 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % sup.orderI
% 0.27/0.47 thf(fact_221_sup_OorderE,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( semilattice_sup @ A )
% 0.27/0.47 => ! [B3: A,A4: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ B3 @ A4 )
% 0.27/0.47 => ( A4
% 0.27/0.47 = ( sup_sup @ A @ A4 @ B3 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % sup.orderE
% 0.27/0.47 thf(fact_222_le__iff__sup,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( semilattice_sup @ A )
% 0.27/0.47 => ( ( ord_less_eq @ A )
% 0.27/0.47 = ( ^ [X3: A,Y5: A] :
% 0.27/0.47 ( ( sup_sup @ A @ X3 @ Y5 )
% 0.27/0.47 = Y5 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % le_iff_sup
% 0.27/0.47 thf(fact_223_sup__least,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( semilattice_sup @ A )
% 0.27/0.47 => ! [Y: A,X2: A,Z2: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ Y @ X2 )
% 0.27/0.47 => ( ( ord_less_eq @ A @ Z2 @ X2 )
% 0.27/0.47 => ( ord_less_eq @ A @ ( sup_sup @ A @ Y @ Z2 ) @ X2 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % sup_least
% 0.27/0.47 thf(fact_224_sup__mono,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( semilattice_sup @ A )
% 0.27/0.47 => ! [A4: A,C2: A,B3: A,D4: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ A4 @ C2 )
% 0.27/0.47 => ( ( ord_less_eq @ A @ B3 @ D4 )
% 0.27/0.47 => ( ord_less_eq @ A @ ( sup_sup @ A @ A4 @ B3 ) @ ( sup_sup @ A @ C2 @ D4 ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % sup_mono
% 0.27/0.47 thf(fact_225_sup_Omono,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( semilattice_sup @ A )
% 0.27/0.47 => ! [C2: A,A4: A,D4: A,B3: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ C2 @ A4 )
% 0.27/0.47 => ( ( ord_less_eq @ A @ D4 @ B3 )
% 0.27/0.47 => ( ord_less_eq @ A @ ( sup_sup @ A @ C2 @ D4 ) @ ( sup_sup @ A @ A4 @ B3 ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % sup.mono
% 0.27/0.47 thf(fact_226_le__supI2,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( semilattice_sup @ A )
% 0.27/0.47 => ! [X2: A,B3: A,A4: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ X2 @ B3 )
% 0.27/0.47 => ( ord_less_eq @ A @ X2 @ ( sup_sup @ A @ A4 @ B3 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % le_supI2
% 0.27/0.47 thf(fact_227_le__supI1,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( semilattice_sup @ A )
% 0.27/0.47 => ! [X2: A,A4: A,B3: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ X2 @ A4 )
% 0.27/0.47 => ( ord_less_eq @ A @ X2 @ ( sup_sup @ A @ A4 @ B3 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % le_supI1
% 0.27/0.47 thf(fact_228_sup__ge2,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( semilattice_sup @ A )
% 0.27/0.47 => ! [Y: A,X2: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X2 @ Y ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % sup_ge2
% 0.27/0.47 thf(fact_229_sup__ge1,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( semilattice_sup @ A )
% 0.27/0.47 => ! [X2: A,Y: A] : ( ord_less_eq @ A @ X2 @ ( sup_sup @ A @ X2 @ Y ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % sup_ge1
% 0.27/0.47 thf(fact_230_le__supI,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( semilattice_sup @ A )
% 0.27/0.47 => ! [A4: A,X2: A,B3: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ A4 @ X2 )
% 0.27/0.47 => ( ( ord_less_eq @ A @ B3 @ X2 )
% 0.27/0.47 => ( ord_less_eq @ A @ ( sup_sup @ A @ A4 @ B3 ) @ X2 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % le_supI
% 0.27/0.47 thf(fact_231_le__supE,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( semilattice_sup @ A )
% 0.27/0.47 => ! [A4: A,B3: A,X2: A] :
% 0.27/0.47 ( ( ord_less_eq @ A @ ( sup_sup @ A @ A4 @ B3 ) @ X2 )
% 0.27/0.47 => ~ ( ( ord_less_eq @ A @ A4 @ X2 )
% 0.27/0.47 => ~ ( ord_less_eq @ A @ B3 @ X2 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % le_supE
% 0.27/0.47 thf(fact_232_inf__sup__ord_I3_J,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( lattice @ A )
% 0.27/0.47 => ! [X2: A,Y: A] : ( ord_less_eq @ A @ X2 @ ( sup_sup @ A @ X2 @ Y ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % inf_sup_ord(3)
% 0.27/0.47 thf(fact_233_inf__sup__ord_I4_J,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( lattice @ A )
% 0.27/0.47 => ! [Y: A,X2: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X2 @ Y ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % inf_sup_ord(4)
% 0.27/0.47 thf(fact_234_inf__sup__aci_I8_J,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( lattice @ A )
% 0.27/0.47 => ! [X2: A,Y: A] :
% 0.27/0.47 ( ( sup_sup @ A @ X2 @ ( sup_sup @ A @ X2 @ Y ) )
% 0.27/0.47 = ( sup_sup @ A @ X2 @ Y ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % inf_sup_aci(8)
% 0.27/0.47 thf(fact_235_inf__sup__aci_I7_J,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( lattice @ A )
% 0.27/0.47 => ! [X2: A,Y: A,Z2: A] :
% 0.27/0.47 ( ( sup_sup @ A @ X2 @ ( sup_sup @ A @ Y @ Z2 ) )
% 0.27/0.47 = ( sup_sup @ A @ Y @ ( sup_sup @ A @ X2 @ Z2 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % inf_sup_aci(7)
% 0.27/0.47 thf(fact_236_inf__sup__aci_I6_J,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( lattice @ A )
% 0.27/0.47 => ! [X2: A,Y: A,Z2: A] :
% 0.27/0.47 ( ( sup_sup @ A @ ( sup_sup @ A @ X2 @ Y ) @ Z2 )
% 0.27/0.47 = ( sup_sup @ A @ X2 @ ( sup_sup @ A @ Y @ Z2 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % inf_sup_aci(6)
% 0.27/0.47 thf(fact_237_inf__sup__aci_I5_J,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( lattice @ A )
% 0.27/0.47 => ( ( sup_sup @ A )
% 0.27/0.47 = ( ^ [X3: A,Y5: A] : ( sup_sup @ A @ Y5 @ X3 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % inf_sup_aci(5)
% 0.27/0.47 thf(fact_238_UnE,axiom,
% 0.27/0.47 ! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
% 0.27/0.47 ( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
% 0.27/0.47 => ( ~ ( member @ A @ C2 @ A2 )
% 0.27/0.47 => ( member @ A @ C2 @ B2 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % UnE
% 0.27/0.47 thf(fact_239_UnI1,axiom,
% 0.27/0.47 ! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
% 0.27/0.47 ( ( member @ A @ C2 @ A2 )
% 0.27/0.47 => ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % UnI1
% 0.27/0.47 thf(fact_240_UnI2,axiom,
% 0.27/0.47 ! [A: $tType,C2: A,B2: set @ A,A2: set @ A] :
% 0.27/0.47 ( ( member @ A @ C2 @ B2 )
% 0.27/0.47 => ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % UnI2
% 0.27/0.47 thf(fact_241_bex__Un,axiom,
% 0.27/0.47 ! [A: $tType,A2: set @ A,B2: set @ A,P: A > $o] :
% 0.27/0.47 ( ( ? [X3: A] :
% 0.27/0.47 ( ( member @ A @ X3 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
% 0.27/0.47 & ( P @ X3 ) ) )
% 0.27/0.47 = ( ? [X3: A] :
% 0.27/0.47 ( ( member @ A @ X3 @ A2 )
% 0.27/0.47 & ( P @ X3 ) )
% 0.27/0.47 | ? [X3: A] :
% 0.27/0.47 ( ( member @ A @ X3 @ B2 )
% 0.27/0.47 & ( P @ X3 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % bex_Un
% 0.27/0.47 thf(fact_242_ball__Un,axiom,
% 0.27/0.47 ! [A: $tType,A2: set @ A,B2: set @ A,P: A > $o] :
% 0.27/0.47 ( ( ! [X3: A] :
% 0.27/0.47 ( ( member @ A @ X3 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
% 0.27/0.47 => ( P @ X3 ) ) )
% 0.27/0.47 = ( ! [X3: A] :
% 0.27/0.47 ( ( member @ A @ X3 @ A2 )
% 0.27/0.47 => ( P @ X3 ) )
% 0.27/0.47 & ! [X3: A] :
% 0.27/0.47 ( ( member @ A @ X3 @ B2 )
% 0.27/0.47 => ( P @ X3 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % ball_Un
% 0.27/0.47 thf(fact_243_Un__assoc,axiom,
% 0.27/0.47 ! [A: $tType,A2: set @ A,B2: set @ A,C4: set @ A] :
% 0.27/0.47 ( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ C4 )
% 0.27/0.47 = ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ B2 @ C4 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Un_assoc
% 0.27/0.47 thf(fact_244_Un__absorb,axiom,
% 0.27/0.47 ! [A: $tType,A2: set @ A] :
% 0.27/0.47 ( ( sup_sup @ ( set @ A ) @ A2 @ A2 )
% 0.27/0.47 = A2 ) ).
% 0.27/0.47
% 0.27/0.47 % Un_absorb
% 0.27/0.47 thf(fact_245_Un__left__absorb,axiom,
% 0.27/0.47 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.27/0.47 ( ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
% 0.27/0.47 = ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.27/0.47
% 0.27/0.47 % Un_left_absorb
% 0.27/0.47 thf(fact_246_Un__left__commute,axiom,
% 0.27/0.47 ! [A: $tType,A2: set @ A,B2: set @ A,C4: set @ A] :
% 0.27/0.47 ( ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ B2 @ C4 ) )
% 0.27/0.47 = ( sup_sup @ ( set @ A ) @ B2 @ ( sup_sup @ ( set @ A ) @ A2 @ C4 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Un_left_commute
% 0.27/0.47 thf(fact_247_sup__fun__def,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType] :
% 0.27/0.47 ( ( semilattice_sup @ B )
% 0.27/0.47 => ( ( sup_sup @ ( A > B ) )
% 0.27/0.47 = ( ^ [F4: A > B,G4: A > B,X3: A] : ( sup_sup @ B @ ( F4 @ X3 ) @ ( G4 @ X3 ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % sup_fun_def
% 0.27/0.47 thf(fact_248_boolean__algebra__cancel_Osup1,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( semilattice_sup @ A )
% 0.27/0.47 => ! [A2: A,K: A,A4: A,B3: A] :
% 0.27/0.47 ( ( A2
% 0.27/0.47 = ( sup_sup @ A @ K @ A4 ) )
% 0.27/0.47 => ( ( sup_sup @ A @ A2 @ B3 )
% 0.27/0.47 = ( sup_sup @ A @ K @ ( sup_sup @ A @ A4 @ B3 ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % boolean_algebra_cancel.sup1
% 0.27/0.47 thf(fact_249_boolean__algebra__cancel_Osup2,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( semilattice_sup @ A )
% 0.27/0.47 => ! [B2: A,K: A,B3: A,A4: A] :
% 0.27/0.47 ( ( B2
% 0.27/0.47 = ( sup_sup @ A @ K @ B3 ) )
% 0.27/0.47 => ( ( sup_sup @ A @ A4 @ B2 )
% 0.27/0.47 = ( sup_sup @ A @ K @ ( sup_sup @ A @ A4 @ B3 ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % boolean_algebra_cancel.sup2
% 0.27/0.47 thf(fact_250_sup_Oassoc,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( semilattice_sup @ A )
% 0.27/0.47 => ! [A4: A,B3: A,C2: A] :
% 0.27/0.47 ( ( sup_sup @ A @ ( sup_sup @ A @ A4 @ B3 ) @ C2 )
% 0.27/0.47 = ( sup_sup @ A @ A4 @ ( sup_sup @ A @ B3 @ C2 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % sup.assoc
% 0.27/0.47 thf(fact_251_sup__assoc,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( semilattice_sup @ A )
% 0.27/0.47 => ! [X2: A,Y: A,Z2: A] :
% 0.27/0.47 ( ( sup_sup @ A @ ( sup_sup @ A @ X2 @ Y ) @ Z2 )
% 0.27/0.47 = ( sup_sup @ A @ X2 @ ( sup_sup @ A @ Y @ Z2 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % sup_assoc
% 0.27/0.47 thf(fact_252_sup_Ocommute,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( semilattice_sup @ A )
% 0.27/0.47 => ( ( sup_sup @ A )
% 0.27/0.47 = ( ^ [A7: A,B7: A] : ( sup_sup @ A @ B7 @ A7 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % sup.commute
% 0.27/0.47 thf(fact_253_sup__commute,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( semilattice_sup @ A )
% 0.27/0.47 => ( ( sup_sup @ A )
% 0.27/0.47 = ( ^ [X3: A,Y5: A] : ( sup_sup @ A @ Y5 @ X3 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % sup_commute
% 0.27/0.47 thf(fact_254_sup_Oleft__commute,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( semilattice_sup @ A )
% 0.27/0.47 => ! [B3: A,A4: A,C2: A] :
% 0.27/0.47 ( ( sup_sup @ A @ B3 @ ( sup_sup @ A @ A4 @ C2 ) )
% 0.27/0.47 = ( sup_sup @ A @ A4 @ ( sup_sup @ A @ B3 @ C2 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % sup.left_commute
% 0.27/0.47
% 0.27/0.47 % Type constructors (33)
% 0.27/0.47 thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
% 0.27/0.47 ! [A9: $tType,A10: $tType] :
% 0.27/0.47 ( ( semilattice_sup @ A10 )
% 0.27/0.47 => ( semilattice_sup @ ( A9 > A10 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(tcon_fun___Lattices_Oboolean__algebra,axiom,
% 0.27/0.47 ! [A9: $tType,A10: $tType] :
% 0.27/0.47 ( ( boolean_algebra @ A10 )
% 0.27/0.47 => ( boolean_algebra @ ( A9 > A10 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.27/0.47 ! [A9: $tType,A10: $tType] :
% 0.27/0.47 ( ( preorder @ A10 )
% 0.27/0.47 => ( preorder @ ( A9 > A10 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(tcon_fun___Lattices_Olattice,axiom,
% 0.27/0.47 ! [A9: $tType,A10: $tType] :
% 0.27/0.47 ( ( lattice @ A10 )
% 0.27/0.47 => ( lattice @ ( A9 > A10 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.27/0.47 ! [A9: $tType,A10: $tType] :
% 0.27/0.47 ( ( order @ A10 )
% 0.27/0.47 => ( order @ ( A9 > A10 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(tcon_fun___Orderings_Oord,axiom,
% 0.27/0.47 ! [A9: $tType,A10: $tType] :
% 0.27/0.47 ( ( ord @ A10 )
% 0.27/0.47 => ( ord @ ( A9 > A10 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(tcon_fun___Groups_Ouminus,axiom,
% 0.27/0.47 ! [A9: $tType,A10: $tType] :
% 0.27/0.47 ( ( uminus @ A10 )
% 0.27/0.47 => ( uminus @ ( A9 > A10 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_1,axiom,
% 0.27/0.47 ! [A9: $tType] : ( semilattice_sup @ ( set @ A9 ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(tcon_Set_Oset___Lattices_Oboolean__algebra_2,axiom,
% 0.27/0.47 ! [A9: $tType] : ( boolean_algebra @ ( set @ A9 ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(tcon_Set_Oset___Orderings_Opreorder_3,axiom,
% 0.27/0.47 ! [A9: $tType] : ( preorder @ ( set @ A9 ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(tcon_Set_Oset___Lattices_Olattice_4,axiom,
% 0.27/0.47 ! [A9: $tType] : ( lattice @ ( set @ A9 ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(tcon_Set_Oset___Orderings_Oorder_5,axiom,
% 0.27/0.47 ! [A9: $tType] : ( order @ ( set @ A9 ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(tcon_Set_Oset___Orderings_Oord_6,axiom,
% 0.27/0.47 ! [A9: $tType] : ( ord @ ( set @ A9 ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(tcon_Set_Oset___Groups_Ouminus_7,axiom,
% 0.27/0.47 ! [A9: $tType] : ( uminus @ ( set @ A9 ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_8,axiom,
% 0.27/0.47 semilattice_sup @ $o ).
% 0.27/0.47
% 0.27/0.47 thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_9,axiom,
% 0.27/0.47 boolean_algebra @ $o ).
% 0.27/0.47
% 0.27/0.47 thf(tcon_HOL_Obool___Orderings_Opreorder_10,axiom,
% 0.27/0.47 preorder @ $o ).
% 0.27/0.47
% 0.27/0.47 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 0.27/0.47 linorder @ $o ).
% 0.27/0.47
% 0.27/0.47 thf(tcon_HOL_Obool___Lattices_Olattice_11,axiom,
% 0.27/0.47 lattice @ $o ).
% 0.27/0.47
% 0.27/0.47 thf(tcon_HOL_Obool___Orderings_Oorder_12,axiom,
% 0.27/0.47 order @ $o ).
% 0.27/0.47
% 0.27/0.47 thf(tcon_HOL_Obool___Orderings_Oord_13,axiom,
% 0.27/0.47 ord @ $o ).
% 0.27/0.47
% 0.27/0.47 thf(tcon_HOL_Obool___Groups_Ouminus_14,axiom,
% 0.27/0.47 uminus @ $o ).
% 0.27/0.47
% 0.27/0.47 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add,axiom,
% 0.27/0.47 ordered_ab_group_add @ real ).
% 0.27/0.47
% 0.27/0.47 thf(tcon_Real_Oreal___Lattices_Osemilattice__sup_15,axiom,
% 0.27/0.47 semilattice_sup @ real ).
% 0.27/0.47
% 0.27/0.47 thf(tcon_Real_Oreal___Orderings_Opreorder_16,axiom,
% 0.27/0.47 preorder @ real ).
% 0.27/0.47
% 0.27/0.47 thf(tcon_Real_Oreal___Orderings_Olinorder_17,axiom,
% 0.27/0.47 linorder @ real ).
% 0.27/0.47
% 0.27/0.47 thf(tcon_Real_Oreal___Lattices_Olattice_18,axiom,
% 0.27/0.47 lattice @ real ).
% 0.27/0.47
% 0.27/0.47 thf(tcon_Real_Oreal___Groups_Ogroup__add,axiom,
% 0.27/0.47 group_add @ real ).
% 0.27/0.47
% 0.27/0.47 thf(tcon_Real_Oreal___Orderings_Oorder_19,axiom,
% 0.27/0.47 order @ real ).
% 0.27/0.47
% 0.27/0.47 thf(tcon_Real_Oreal___Orderings_Oord_20,axiom,
% 0.27/0.47 ord @ real ).
% 0.27/0.47
% 0.27/0.47 thf(tcon_Real_Oreal___Groups_Ouminus_21,axiom,
% 0.27/0.47 uminus @ real ).
% 0.27/0.47
% 0.27/0.47 thf(tcon_Product__Type_Oprod___Groups_Ogroup__add_22,axiom,
% 0.27/0.47 ! [A9: $tType,A10: $tType] :
% 0.27/0.47 ( ( ( group_add @ A9 )
% 0.27/0.47 & ( group_add @ A10 ) )
% 0.27/0.47 => ( group_add @ ( product_prod @ A9 @ A10 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(tcon_Product__Type_Oprod___Groups_Ouminus_23,axiom,
% 0.27/0.47 ! [A9: $tType,A10: $tType] :
% 0.27/0.47 ( ( ( uminus @ A9 )
% 0.27/0.47 & ( uminus @ A10 ) )
% 0.27/0.47 => ( uminus @ ( product_prod @ A9 @ A10 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Conjectures (1)
% 0.27/0.47 thf(conj_0,conjecture,
% 0.27/0.47 ( ( member @ ( variable > real ) @ nu2 @ ( uminus_uminus @ ( set @ ( variable > real ) ) @ ( denota1245701238me_sem @ i @ ( the @ game @ ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ sigma @ ua @ alpha ) ) ) @ ( uminus_uminus @ ( set @ ( variable > real ) ) @ xa ) ) ) )
% 0.27/0.47 = ( member @ ( variable > real ) @ nu2 @ ( uminus_uminus @ ( set @ ( variable > real ) ) @ ( denota1245701238me_sem @ ( uSubst2091380086djoint @ sigma @ i @ omega2 ) @ alpha @ ( uminus_uminus @ ( set @ ( variable > real ) ) @ xa ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 %------------------------------------------------------------------------------
% 0.27/0.47 ------------------------
% 2.78/1.58 ---- Embedded in THF ---
% 2.78/1.58 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.78/1.58 %%% Generated on Tue Feb 24 08:30:34 EST 2026
% 2.78/1.58 %%% using '$$dhol' embedding, version 1.3.0.
% 2.78/1.58 %%% Logic specification used:
% 2.78/1.58 %%% thf(spec, logic, $$dhol).
% 2.78/1.58
% 2.78/1.58 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.lOk4iEszTx/DTF2DTF4484.p
% 2.78/1.58 thf(ty_t_Denotational__Semantics_Ointerp, type, denotational_interp: $tType).
% 2.78/1.58 thf(ty_t_Product__Type_Oprod, type, product_prod: ($tType > ($tType > $tType))).
% 2.78/1.58 thf(ty_t_Syntax_Ovariable, type, variable: $tType).
% 2.78/1.58 thf(ty_t_Option_Ooption, type, option: ($tType > $tType)).
% 2.78/1.58 thf(ty_t_Syntax_Ogame, type, game: $tType).
% 2.78/1.58 thf(ty_t_String_Ochar, type, char: $tType).
% 2.78/1.58 thf(ty_t_Syntax_Otrm, type, trm: $tType).
% 2.78/1.58 thf(ty_t_Syntax_Ofml, type, fml: $tType).
% 2.78/1.58 thf(ty_t_Real_Oreal, type, real: $tType).
% 2.78/1.58 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.78/1.58 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 2.78/1.58 thf(sy_cl_Groups_Ouminus, type, uminus: (!> [A:$tType]: ($o))).
% 2.78/1.58 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 2.78/1.58 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 2.78/1.58 thf(sy_cl_Groups_Ogroup__add, type, group_add: (!> [A:$tType]: ($o))).
% 2.78/1.58 thf(sy_cl_Lattices_Olattice, type, lattice: (!> [A:$tType]: ($o))).
% 2.78/1.58 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 2.78/1.58 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 2.78/1.58 thf(sy_cl_Lattices_Oboolean__algebra, type, boolean_algebra: (!> [A:$tType]: ($o))).
% 2.78/1.58 thf(sy_cl_Lattices_Osemilattice__sup, type, semilattice_sup: (!> [A:$tType]: ($o))).
% 2.78/1.58 thf(sy_cl_Groups_Oordered__ab__group__add, type, ordered_ab_group_add: (!> [A:$tType]: ($o))).
% 2.78/1.58 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain, type, bNF_Ca1785829860lChain: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ A)) > ((A > B) > $o))))).
% 2.78/1.58 thf(sy_c_Denotational__Semantics_OUvariation, type, denota1419872369iation: ((variable > real) > ((variable > real) > ((set @ variable) > $o)))).
% 2.78/1.58 thf(sy_c_Denotational__Semantics_Ogame__sem, type, denota1245701238me_sem: (denotational_interp > (game > ((set @ (variable > real)) > (set @ (variable > real)))))).
% 2.78/1.58 thf(sy_c_Groups_Ouminus__class_Ouminus, type, uminus_uminus: (!> [A:$tType]: ((A > A)))).
% 2.78/1.58 thf(sy_c_Lattices_Osup__class_Osup, type, sup_sup: (!> [A:$tType]: ((A > (A > A))))).
% 2.78/1.58 thf(sy_c_Option_Ooption_ONone, type, none: (!> [A:$tType]: ((option @ A)))).
% 2.78/1.58 thf(sy_c_Option_Ooption_Othe, type, the: (!> [A:$tType]: (((option @ A) > A)))).
% 2.78/1.58 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.78/1.58 thf(sy_c_Product__Type_OPair, type, product_Pair: (!> [A:$tType,B:$tType]: ((A > (B > ((product_prod @ A) @ B)))))).
% 2.78/1.58 thf(sy_c_Product__Type_Ointernal__case__prod, type, produc2004651681e_prod: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > C)) > (((product_prod @ A) @ B) > C))))).
% 2.78/1.58 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod, type, product_rec_prod: (!> [A:$tType,B:$tType,T:$tType]: (((A > (B > T)) > (((product_prod @ A) @ B) > T))))).
% 2.78/1.58 thf(sy_c_Product__Type_Oprod_Ofst, type, product_fst: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > A)))).
% 2.78/1.58 thf(sy_c_Product__Type_Oprod_Osnd, type, product_snd: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > B)))).
% 2.78/1.58 thf(sy_c_Product__Type_Oprod_Oswap, type, product_swap: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > ((product_prod @ B) @ A))))).
% 2.78/1.58 thf(sy_c_Product__Type_Oscomp, type, product_scomp: (!> [A:$tType,B:$tType,C:$tType,D:$tType]: (((A > ((product_prod @ B) @ C)) > ((B > (C > D)) > (A > D)))))).
% 2.78/1.58 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.78/1.58 thf(sy_c_Syntax_OSkip, type, skip: game).
% 2.78/1.58 thf(sy_c_Syntax_Ogame_OChoice, type, choice: (game > (game > game))).
% 2.78/1.58 thf(sy_c_Syntax_Ogame_ODual, type, dual: (game > game)).
% 2.78/1.58 thf(sy_c_Syntax_Ogame_OLoop, type, loop: (game > game)).
% 2.78/1.58 thf(sy_c_USubst__Mirabelle__nnnzepxswx_OChoiceo, type, uSubst1976112797hoiceo: ((option @ game) > ((option @ game) > (option @ game)))).
% 2.78/1.58 thf(sy_c_USubst__Mirabelle__nnnzepxswx_ODualo, type, uSubst739989314_Dualo: ((option @ game) > (option @ game))).
% 2.78/1.58 thf(sy_c_USubst__Mirabelle__nnnzepxswx_OLoopo, type, uSubst993936602_Loopo: ((option @ game) > (option @ game))).
% 2.78/1.58 thf(sy_c_USubst__Mirabelle__nnnzepxswx_Oadjoint, type, uSubst2091380086djoint: (((product_prod @ (char > (option @ trm))) @ ((product_prod @ (char > (option @ trm))) @ ((product_prod @ (char > (option @ fml))) @ (char > (option @ game))))) > (denotational_interp > ((variable > real) > denotational_interp)))).
% 2.78/1.58 thf(sy_c_USubst__Mirabelle__nnnzepxswx_Ousubstappp, type, uSubst95898988stappp: (((product_prod @ (char > (option @ trm))) @ ((product_prod @ (char > (option @ trm))) @ ((product_prod @ (char > (option @ fml))) @ (char > (option @ game))))) > ((set @ variable) > (game > ((product_prod @ (set @ variable)) @ (option @ game)))))).
% 2.78/1.58 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.78/1.58 thf(sy_v_I, type, i: denotational_interp).
% 2.78/1.58 thf(sy_v_U, type, u: (set @ variable)).
% 2.78/1.58 thf(sy_v_Ua____, type, ua: (set @ variable)).
% 2.78/1.58 thf(sy_v_Xa____, type, xa: (set @ (variable > real))).
% 2.78/1.58 thf(sy_v__092_060alpha_062_H_H____, type, alpha: game).
% 2.78/1.58 thf(sy_v__092_060nu_062, type, nu: (variable > real)).
% 2.78/1.58 thf(sy_v__092_060nu_062_H____, type, nu2: (variable > real)).
% 2.78/1.58 thf(sy_v__092_060omega_062, type, omega: (variable > real)).
% 2.78/1.58 thf(sy_v__092_060omega_062_H____, type, omega2: (variable > real)).
% 2.78/1.58 thf(sy_v__092_060sigma_062_H_H____, type, sigma: ((product_prod @ (char > (option @ trm))) @ ((product_prod @ (char > (option @ trm))) @ ((product_prod @ (char > (option @ fml))) @ (char > (option @ game)))))).
% 2.78/1.58 thf(fact_0__092_060open_062_092_060And_062X_O_A_I_092_060nu_062_A_092_060in_062_Agame__sem_AI_A_Ithe_A_Isnd_A_Iusubstappp_A_092_060sigma_062_AU_A_092_060alpha_062_J_J_J_AX_J_A_061_A_I_092_060nu_062_A_092_060in_062_Agame__sem_A_IUSubst__Mirabelle__nnnzepxswx_Oadjoint_A_092_060sigma_062_AI_A_092_060omega_062_J_A_092_060alpha_062_AX_J_092_060close_062, axiom, (! [X:(set @ (variable > real))]: (((((member @ (variable > real)) @ nu2) @ (((denota1245701238me_sem @ i) @ ((the @ game) @ (((product_snd @ (set @ variable)) @ (option @ game)) @ (((uSubst95898988stappp @ sigma) @ ua) @ alpha)))) @ X)) = (((member @ (variable > real)) @ nu2) @ (((denota1245701238me_sem @ (((uSubst2091380086djoint @ sigma) @ i) @ omega2)) @ alpha) @ X)))))).
% 2.78/1.58 thf(fact_1_IH_092_060alpha_062, axiom, (! [X:(set @ (variable > real))]: (((((denota1419872369iation @ nu2) @ omega2) @ ua) => ((((member @ (variable > real)) @ nu2) @ (((denota1245701238me_sem @ i) @ ((the @ game) @ (((product_snd @ (set @ variable)) @ (option @ game)) @ (((uSubst95898988stappp @ sigma) @ ua) @ alpha)))) @ X)) = (((member @ (variable > real)) @ nu2) @ (((denota1245701238me_sem @ (((uSubst2091380086djoint @ sigma) @ i) @ omega2)) @ alpha) @ X))))))).
% 2.78/1.58 thf(fact_2_uv, axiom, (((denota1419872369iation @ nu2) @ omega2) @ ua)).
% 2.78/1.58 thf(fact_3_snd__uminus, axiom, (! [A:$tType,B:$tType]: ((((uminus @ B) & (uminus @ A)) => (! [X2:((product_prod @ B) @ A)]: (((((product_snd @ B) @ A) @ ((uminus_uminus @ ((product_prod @ B) @ A)) @ X2)) = ((uminus_uminus @ A) @ (((product_snd @ B) @ A) @ X2))))))))).
% 2.78/1.58 thf(fact_4_def, axiom, ((((product_snd @ (set @ variable)) @ (option @ game)) @ (((uSubst95898988stappp @ sigma) @ ua) @ (dual @ alpha))) != (none @ game))).
% 2.78/1.58 thf(fact_5_ComplI, axiom, (! [A:$tType,C2:A,A2:(set @ A)]: (((~ ((((member @ A) @ C2) @ A2))) => (((member @ A) @ C2) @ ((uminus_uminus @ (set @ A)) @ A2)))))).
% 2.78/1.58 thf(fact_6_Compl__iff, axiom, (! [A:$tType,C2:A,A2:(set @ A)]: (((((member @ A) @ C2) @ ((uminus_uminus @ (set @ A)) @ A2)) = (~ ((((member @ A) @ C2) @ A2))))))).
% 2.78/1.58 thf(fact_7_Compl__eq__Compl__iff, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((uminus_uminus @ (set @ A)) @ A2) = ((uminus_uminus @ (set @ A)) @ B2)) = (A2 = B2))))).
% 2.78/1.58 thf(fact_8_verit__minus__simplify_I4_J, axiom, (! [B:$tType]: (((group_add @ B) => (! [B3:B]: ((((uminus_uminus @ B) @ ((uminus_uminus @ B) @ B3)) = B3))))))).
% 2.78/1.58 thf(fact_9_uminus__apply, axiom, (! [B:$tType,A:$tType]: (((uminus @ B) => ((uminus_uminus @ (A > B)) = (^ [A3:(A > B),X3:A]: (((uminus_uminus @ B) @ (A3 @ X3))))))))).
% 2.78/1.58 thf(fact_10_add_Oinverse__inverse, axiom, (! [A:$tType]: (((group_add @ A) => (! [A4:A]: ((((uminus_uminus @ A) @ ((uminus_uminus @ A) @ A4)) = A4))))))).
% 2.78/1.58 thf(fact_11_neg__equal__iff__equal, axiom, (! [A:$tType]: (((group_add @ A) => (! [A4:A,B3:A]: (((((uminus_uminus @ A) @ A4) = ((uminus_uminus @ A) @ B3)) = (A4 = B3)))))))).
% 2.78/1.58 thf(fact_12_double__compl, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [X2:A]: ((((uminus_uminus @ A) @ ((uminus_uminus @ A) @ X2)) = X2))))))).
% 2.78/1.58 thf(fact_13_vaouter, axiom, (((denota1419872369iation @ nu) @ omega) @ u)).
% 2.78/1.58 thf(fact_14_compl__eq__compl__iff, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [X2:A,Y:A]: (((((uminus_uminus @ A) @ X2) = ((uminus_uminus @ A) @ Y)) = (X2 = Y)))))))).
% 2.78/1.58 thf(fact_15_usubstappp__det, axiom, (! [Sigma:((product_prod @ (char > (option @ trm))) @ ((product_prod @ (char > (option @ trm))) @ ((product_prod @ (char > (option @ fml))) @ (char > (option @ game))))),U:(set @ variable),Alpha:game,V:(set @ variable)]: ((((((product_snd @ (set @ variable)) @ (option @ game)) @ (((uSubst95898988stappp @ Sigma) @ U) @ Alpha)) != (none @ game)) => (((((product_snd @ (set @ variable)) @ (option @ game)) @ (((uSubst95898988stappp @ Sigma) @ V) @ Alpha)) != (none @ game)) => ((((product_snd @ (set @ variable)) @ (option @ game)) @ (((uSubst95898988stappp @ Sigma) @ U) @ Alpha)) = (((product_snd @ (set @ variable)) @ (option @ game)) @ (((uSubst95898988stappp @ Sigma) @ V) @ Alpha)))))))).
% 2.78/1.58 thf(fact_16_minus__equation__iff, axiom, (! [A:$tType]: (((group_add @ A) => (! [A4:A,B3:A]: (((((uminus_uminus @ A) @ A4) = B3) = (((uminus_uminus @ A) @ B3) = A4)))))))).
% 2.78/1.58 thf(fact_17_equation__minus__iff, axiom, (! [A:$tType]: (((group_add @ A) => (! [A4:A,B3:A]: (((A4 = ((uminus_uminus @ A) @ B3)) = (B3 = ((uminus_uminus @ A) @ A4))))))))).
% 2.78/1.58 thf(fact_18_fun__Compl__def, axiom, (! [B:$tType,A:$tType]: (((uminus @ B) => ((uminus_uminus @ (A > B)) = (^ [A3:(A > B),X3:A]: (((uminus_uminus @ B) @ (A3 @ X3))))))))).
% 2.78/1.58 thf(fact_19_verit__negate__coefficient_I3_J, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A4:A,B3:A]: (((A4 = B3) => (((uminus_uminus @ A) @ A4) = ((uminus_uminus @ A) @ B3))))))))).
% 2.78/1.58 thf(fact_20_double__complement, axiom, (! [A:$tType,A2:(set @ A)]: ((((uminus_uminus @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ A2)) = A2)))).
% 2.78/1.58 thf(fact_21_ComplD, axiom, (! [A:$tType,C2:A,A2:(set @ A)]: (((((member @ A) @ C2) @ ((uminus_uminus @ (set @ A)) @ A2)) => (~ ((((member @ A) @ C2) @ A2))))))).
% 2.78/1.58 thf(fact_22_game__sem_Osimps_I7_J, axiom, (! [I:denotational_interp,Alpha:game]: ((((denota1245701238me_sem @ I) @ (dual @ Alpha)) = (^ [X4:(set @ (variable > real))]: (((uminus_uminus @ (set @ (variable > real))) @ (((denota1245701238me_sem @ I) @ Alpha) @ ((uminus_uminus @ (set @ (variable > real))) @ X4))))))))).
% 2.78/1.58 thf(fact_23_Dual_OIH, axiom, (! [Nu:(variable > real),Omega:(variable > real),X:(set @ (variable > real))]: ((((((product_snd @ (set @ variable)) @ (option @ game)) @ (((uSubst95898988stappp @ sigma) @ ua) @ alpha)) != (none @ game)) => ((((denota1419872369iation @ Nu) @ Omega) @ ua) => ((((member @ (variable > real)) @ Nu) @ (((denota1245701238me_sem @ i) @ ((the @ game) @ (((product_snd @ (set @ variable)) @ (option @ game)) @ (((uSubst95898988stappp @ sigma) @ ua) @ alpha)))) @ X)) = (((member @ (variable > real)) @ Nu) @ (((denota1245701238me_sem @ (((uSubst2091380086djoint @ sigma) @ i) @ Omega)) @ alpha) @ X)))))))).
% 2.78/1.58 thf(fact_24_game_Oinject_I7_J, axiom, (! [X7:game,Y7:game]: ((((dual @ X7) = (dual @ Y7)) = (X7 = Y7))))).
% 2.78/1.58 thf(fact_25_Dual_Oprems_I2_J, axiom, (((denota1419872369iation @ nu2) @ omega2) @ ua)).
% 2.78/1.58 thf(fact_26_option_Oexpand, axiom, (! [A:$tType,Option:(option @ A),Option2:(option @ A)]: ((((Option = (none @ A)) = (Option2 = (none @ A))) => (((Option != (none @ A)) => ((Option2 != (none @ A)) => (((the @ A) @ Option) = ((the @ A) @ Option2)))) => (Option = Option2)))))).
% 2.78/1.58 thf(fact_27_usubst__game__loop, axiom, (! [Nu:(variable > real),Omega:(variable > real),U:(set @ variable),Sigma:((product_prod @ (char > (option @ trm))) @ ((product_prod @ (char > (option @ trm))) @ ((product_prod @ (char > (option @ fml))) @ (char > (option @ game))))),Alpha:game,I:denotational_interp,X:(set @ (variable > real))]: (((((denota1419872369iation @ Nu) @ Omega) @ U) => ((! [Nu2:(variable > real),Omega2:(variable > real),X5:(set @ (variable > real))]: (((((denota1419872369iation @ Nu2) @ Omega2) @ (((product_fst @ (set @ variable)) @ (option @ game)) @ (((uSubst95898988stappp @ Sigma) @ U) @ Alpha))) => (((((product_snd @ (set @ variable)) @ (option @ game)) @ (((uSubst95898988stappp @ Sigma) @ (((product_fst @ (set @ variable)) @ (option @ game)) @ (((uSubst95898988stappp @ Sigma) @ U) @ Alpha))) @ Alpha)) != (none @ game)) => ((((member @ (variable > real)) @ Nu2) @ (((denota1245701238me_sem @ I) @ ((the @ game) @ (((product_snd @ (set @ variable)) @ (option @ game)) @ (((uSubst95898988stappp @ Sigma) @ (((product_fst @ (set @ variable)) @ (option @ game)) @ (((uSubst95898988stappp @ Sigma) @ U) @ Alpha))) @ Alpha)))) @ X5)) = (((member @ (variable > real)) @ Nu2) @ (((denota1245701238me_sem @ (((uSubst2091380086djoint @ Sigma) @ I) @ Omega2)) @ Alpha) @ X5))))))) => (((((product_snd @ (set @ variable)) @ (option @ game)) @ (((uSubst95898988stappp @ Sigma) @ U) @ (loop @ Alpha))) != (none @ game)) => ((((member @ (variable > real)) @ Nu) @ (((denota1245701238me_sem @ I) @ ((the @ game) @ (((product_snd @ (set @ variable)) @ (option @ game)) @ (((uSubst95898988stappp @ Sigma) @ U) @ (loop @ Alpha))))) @ X)) = (((member @ (variable > real)) @ Nu) @ (((denota1245701238me_sem @ (((uSubst2091380086djoint @ Sigma) @ I) @ Omega)) @ (loop @ Alpha)) @ X))))))))).
% 2.78/1.58 thf(fact_28_Uvariation__def, axiom, (denota1419872369iation = (^ [Nu3:(variable > real),Nu4:(variable > real),U2:(set @ variable)]: ((! [I2:variable]: (((~ ((((member @ variable) @ I2) @ U2))) => ((Nu3 @ I2) = (Nu4 @ I2))))))))).
% 2.78/1.58 thf(fact_29_Uvariation__sym, axiom, (denota1419872369iation = (^ [Omega3:(variable > real),Nu3:(variable > real)]: (((denota1419872369iation @ Nu3) @ Omega3))))).
% 2.78/1.58 thf(fact_30_Uvariation__refl, axiom, (! [Nu:(variable > real),V:(set @ variable)]: ((((denota1419872369iation @ Nu) @ Nu) @ V)))).
% 2.78/1.58 thf(fact_31_Uvariation__sym__rel, axiom, (! [Omega:(variable > real),Nu:(variable > real),U:(set @ variable)]: (((((denota1419872369iation @ Omega) @ Nu) @ U) => (((denota1419872369iation @ Nu) @ Omega) @ U))))).
% 2.78/1.58 thf(fact_32_usubstappp__choice__conv, axiom, (! [Sigma:((product_prod @ (char > (option @ trm))) @ ((product_prod @ (char > (option @ trm))) @ ((product_prod @ (char > (option @ fml))) @ (char > (option @ game))))),U:(set @ variable),Alpha:game,Beta:game]: ((((((product_snd @ (set @ variable)) @ (option @ game)) @ (((uSubst95898988stappp @ Sigma) @ U) @ ((choice @ Alpha) @ Beta))) != (none @ game)) => (((((product_snd @ (set @ variable)) @ (option @ game)) @ (((uSubst95898988stappp @ Sigma) @ U) @ Alpha)) != (none @ game)) & ((((product_snd @ (set @ variable)) @ (option @ game)) @ (((uSubst95898988stappp @ Sigma) @ U) @ Beta)) != (none @ game))))))).
% 2.78/1.58 thf(fact_33_game_Oinject_I6_J, axiom, (! [X6:game,Y6:game]: ((((loop @ X6) = (loop @ Y6)) = (X6 = Y6))))).
% 2.78/1.58 thf(fact_34_game_Oinject_I4_J, axiom, (! [X41:game,X42:game,Y41:game,Y42:game]: (((((choice @ X41) @ X42) = ((choice @ Y41) @ Y42)) = ((X41 = Y41) & (X42 = Y42)))))).
% 2.78/1.58 thf(fact_35_fst__uminus, axiom, (! [B:$tType,A:$tType]: ((((uminus @ A) & (uminus @ B)) => (! [X2:((product_prod @ A) @ B)]: (((((product_fst @ A) @ B) @ ((uminus_uminus @ ((product_prod @ A) @ B)) @ X2)) = ((uminus_uminus @ A) @ (((product_fst @ A) @ B) @ X2))))))))).
% 2.78/1.58 thf(fact_36_Uvariation__univ, axiom, (! [Nu:(variable > real),Nu5:(variable > real)]: ((((denota1419872369iation @ Nu) @ Nu5) @ ((collect @ variable) @ (^ [X3:variable]: ($true))))))).
% 2.78/1.58 thf(fact_37_game_Odistinct_I39_J, axiom, (! [X41:game,X42:game,X6:game]: ((((choice @ X41) @ X42) != (loop @ X6))))).
% 2.78/1.58 thf(fact_38_uminus__set__def, axiom, (! [A:$tType]: (((uminus_uminus @ (set @ A)) = (^ [A3:(set @ A)]: (((collect @ A) @ ((uminus_uminus @ (A > $o)) @ (^ [X3:A]: ((((member @ A) @ X3) @ A3))))))))))).
% 2.78/1.58 thf(fact_39_Collect__neg__eq, axiom, (! [A:$tType,P:(A > $o)]: ((((collect @ A) @ (^ [X3:A]: ((~ ((P @ X3)))))) = ((uminus_uminus @ (set @ A)) @ ((collect @ A) @ P)))))).
% 2.78/1.58 thf(fact_40_Compl__eq, axiom, (! [A:$tType]: (((uminus_uminus @ (set @ A)) = (^ [A3:(set @ A)]: (((collect @ A) @ (^ [X3:A]: ((~ ((((member @ A) @ X3) @ A3)))))))))))).
% 2.78/1.58 thf(fact_41_game_Odistinct_I51_J, axiom, (! [X6:game,X7:game]: (((loop @ X6) != (dual @ X7))))).
% 2.78/1.58 thf(fact_42_game_Odistinct_I41_J, axiom, (! [X41:game,X42:game,X7:game]: ((((choice @ X41) @ X42) != (dual @ X7))))).
% 2.78/1.58 thf(fact_43_usubstappp__loop__conv, axiom, (! [Sigma:((product_prod @ (char > (option @ trm))) @ ((product_prod @ (char > (option @ trm))) @ ((product_prod @ (char > (option @ fml))) @ (char > (option @ game))))),U:(set @ variable),Alpha:game]: ((((((product_snd @ (set @ variable)) @ (option @ game)) @ (((uSubst95898988stappp @ Sigma) @ U) @ (loop @ Alpha))) != (none @ game)) => (((((product_snd @ (set @ variable)) @ (option @ game)) @ (((uSubst95898988stappp @ Sigma) @ U) @ Alpha)) != (none @ game)) & ((((product_snd @ (set @ variable)) @ (option @ game)) @ (((uSubst95898988stappp @ Sigma) @ (((product_fst @ (set @ variable)) @ (option @ game)) @ (((uSubst95898988stappp @ Sigma) @ U) @ Alpha))) @ Alpha)) != (none @ game))))))).
% 2.78/1.58 thf(fact_44_mem__Collect__eq, axiom, (! [A:$tType,A4:A,P:(A > $o)]: (((((member @ A) @ A4) @ ((collect @ A) @ P)) = (P @ A4))))).
% 2.78/1.58 thf(fact_45_Collect__mem__eq, axiom, (! [A:$tType,A2:(set @ A)]: ((((collect @ A) @ (^ [X3:A]: ((((member @ A) @ X3) @ A2)))) = A2)))).
% 2.78/1.58 thf(fact_46_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X8:A]: (((P @ X8) = (Q @ X8)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 2.78/1.58 thf(fact_47_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X8:A]: (((F @ X8) = (G @ X8)))) => (F = G))))).
% 2.78/1.58 thf(fact_48_skip__id, axiom, (! [I:denotational_interp,X:(set @ (variable > real))]: (((((denota1245701238me_sem @ I) @ skip) @ X) = X)))).
% 2.78/1.58 thf(fact_49_prod__eqI, axiom, (! [B:$tType,A:$tType,P2:((product_prod @ A) @ B),Q2:((product_prod @ A) @ B)]: ((((((product_fst @ A) @ B) @ P2) = (((product_fst @ A) @ B) @ Q2)) => (((((product_snd @ A) @ B) @ P2) = (((product_snd @ A) @ B) @ Q2)) => (P2 = Q2)))))).
% 2.78/1.58 thf(fact_50_exE__realizer_H, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o)),P2:((product_prod @ B) @ A)]: ((((P @ (((product_snd @ B) @ A) @ P2)) @ (((product_fst @ B) @ A) @ P2)) => (~ ((! [X8:B,Y2:A]: ((~ (((P @ Y2) @ X8))))))))))).
% 2.78/1.58 thf(fact_51_prod_Oexpand, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B),Prod2:((product_prod @ A) @ B)]: (((((((product_fst @ A) @ B) @ Prod) = (((product_fst @ A) @ B) @ Prod2)) & ((((product_snd @ A) @ B) @ Prod) = (((product_snd @ A) @ B) @ Prod2))) => (Prod = Prod2))))).
% 2.78/1.58 thf(fact_52_prod__eq__iff, axiom, (! [B:$tType,A:$tType]: (((^ [Y3:((product_prod @ A) @ B),Z:((product_prod @ A) @ B)]: ((Y3 = Z))) = (^ [S:((product_prod @ A) @ B),T2:((product_prod @ A) @ B)]: ((((((product_fst @ A) @ B) @ S) = (((product_fst @ A) @ B) @ T2)) & ((((product_snd @ A) @ B) @ S) = (((product_snd @ A) @ B) @ T2))))))))).
% 2.78/1.58 thf(fact_53_uminus__prod__def, axiom, (! [B:$tType,A:$tType]: ((((uminus @ A) & (uminus @ B)) => ((uminus_uminus @ ((product_prod @ A) @ B)) = (^ [X3:((product_prod @ A) @ B)]: (((((product_Pair @ A) @ B) @ ((uminus_uminus @ A) @ (((product_fst @ A) @ B) @ X3))) @ ((uminus_uminus @ B) @ (((product_snd @ A) @ B) @ X3)))))))))).
% 2.78/1.58 thf(fact_54_usubstappp__antimon, axiom, (! [V:(set @ variable),U:(set @ variable),Sigma:((product_prod @ (char > (option @ trm))) @ ((product_prod @ (char > (option @ trm))) @ ((product_prod @ (char > (option @ fml))) @ (char > (option @ game))))),Alpha:game]: (((((ord_less_eq @ (set @ variable)) @ V) @ U) => (((((product_snd @ (set @ variable)) @ (option @ game)) @ (((uSubst95898988stappp @ Sigma) @ U) @ Alpha)) != (none @ game)) => ((((product_snd @ (set @ variable)) @ (option @ game)) @ (((uSubst95898988stappp @ Sigma) @ U) @ Alpha)) = (((product_snd @ (set @ variable)) @ (option @ game)) @ (((uSubst95898988stappp @ Sigma) @ V) @ Alpha)))))))).
% 2.78/1.58 thf(fact_55_exists__diff, axiom, (! [A:$tType,P:((set @ A) > $o)]: (((? [S2:(set @ A)]: ((P @ ((uminus_uminus @ (set @ A)) @ S2)))) = (? [X9:(set @ A)]: ((P @ X9))))))).
% 2.78/1.58 thf(fact_56_uminus__Pair, axiom, (! [A:$tType,B:$tType]: ((((uminus @ B) & (uminus @ A)) => (! [A4:A,B3:B]: ((((uminus_uminus @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A4) @ B3)) = ((((product_Pair @ A) @ B) @ ((uminus_uminus @ A) @ A4)) @ ((uminus_uminus @ B) @ B3))))))))).
% 2.78/1.58 thf(fact_57_prod_Oinject, axiom, (! [A:$tType,B:$tType,X1:A,X22:B,Y1:A,Y22:B]: (((((((product_Pair @ A) @ B) @ X1) @ X22) = ((((product_Pair @ A) @ B) @ Y1) @ Y22)) = ((X1 = Y1) & (X22 = Y22)))))).
% 2.78/1.58 thf(fact_58_old_Oprod_Oinject, axiom, (! [A:$tType,B:$tType,A4:A,B3:B,A5:A,B4:B]: (((((((product_Pair @ A) @ B) @ A4) @ B3) = ((((product_Pair @ A) @ B) @ A5) @ B4)) = ((A4 = A5) & (B3 = B4)))))).
% 2.78/1.58 thf(fact_59_subset__antisym, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((ord_less_eq @ (set @ A)) @ B2) @ A2) => (A2 = B2)))))).
% 2.78/1.58 thf(fact_60_subsetI, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((! [X8:A]: (((((member @ A) @ X8) @ A2) => (((member @ A) @ X8) @ B2)))) => (((ord_less_eq @ (set @ A)) @ A2) @ B2))))).
% 2.78/1.58 thf(fact_61_compl__le__compl__iff, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [X2:A,Y:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ X2)) @ ((uminus_uminus @ A) @ Y)) = (((ord_less_eq @ A) @ Y) @ X2)))))))).
% 2.78/1.58 thf(fact_62_neg__le__iff__le, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [B3:A,A4:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B3)) @ ((uminus_uminus @ A) @ A4)) = (((ord_less_eq @ A) @ A4) @ B3)))))))).
% 2.78/1.58 thf(fact_63_Compl__subset__Compl__iff, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ A2)) @ ((uminus_uminus @ (set @ A)) @ B2)) = (((ord_less_eq @ (set @ A)) @ B2) @ A2))))).
% 2.78/1.58 thf(fact_64_Compl__anti__mono, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (((ord_less_eq @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ B2)) @ ((uminus_uminus @ (set @ A)) @ A2)))))).
% 2.78/1.58 thf(fact_65_prod_Ocollapse, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B)]: ((((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod)) = Prod)))).
% 2.78/1.58 thf(fact_66_surj__pair, axiom, (! [A:$tType,B:$tType,P2:((product_prod @ A) @ B)]: ((? [X8:A,Y2:B]: ((P2 = ((((product_Pair @ A) @ B) @ X8) @ Y2))))))).
% 2.78/1.58 thf(fact_67_prod__cases, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),P2:((product_prod @ A) @ B)]: (((! [A6:A,B5:B]: ((P @ ((((product_Pair @ A) @ B) @ A6) @ B5)))) => (P @ P2))))).
% 2.78/1.58 thf(fact_68_Pair__inject, axiom, (! [A:$tType,B:$tType,A4:A,B3:B,A5:A,B4:B]: (((((((product_Pair @ A) @ B) @ A4) @ B3) = ((((product_Pair @ A) @ B) @ A5) @ B4)) => (~ (((A4 = A5) => (B3 != B4)))))))).
% 2.78/1.58 thf(fact_69_prod__cases3, axiom, (! [A:$tType,B:$tType,C:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ C))]: ((~ ((! [A6:A,B5:B,C3:C]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A6) @ ((((product_Pair @ B) @ C) @ B5) @ C3)))))))))).
% 2.78/1.58 thf(fact_70_prod__cases4, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: ((~ ((! [A6:A,B5:B,C3:C,D2:D]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A6) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B5) @ ((((product_Pair @ C) @ D) @ C3) @ D2))))))))))).
% 2.78/1.58 thf(fact_71_prod__cases5, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))))]: ((~ ((! [A6:A,B5:B,C3:C,D2:D,E2:E]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) @ A6) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E)) @ C3) @ ((((product_Pair @ D) @ E) @ D2) @ E2)))))))))))).
% 2.78/1.58 thf(fact_72_prod__cases6, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,F2:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F2)))))]: ((~ ((! [A6:A,B5:B,C3:C,D2:D,E2:E,F3:F2]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F2))))) @ A6) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F2)))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F2))) @ C3) @ ((((product_Pair @ D) @ ((product_prod @ E) @ F2)) @ D2) @ ((((product_Pair @ E) @ F2) @ E2) @ F3))))))))))))).
% 2.78/1.58 thf(fact_73_prod__cases7, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,F2:$tType,G2:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F2) @ G2))))))]: ((~ ((! [A6:A,B5:B,C3:C,D2:D,E2:E,F3:F2,G3:G2]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F2) @ G2)))))) @ A6) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F2) @ G2))))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F2) @ G2)))) @ C3) @ ((((product_Pair @ D) @ ((product_prod @ E) @ ((product_prod @ F2) @ G2))) @ D2) @ ((((product_Pair @ E) @ ((product_prod @ F2) @ G2)) @ E2) @ ((((product_Pair @ F2) @ G2) @ F3) @ G3)))))))))))))).
% 2.78/1.58 thf(fact_74_prod__induct3, axiom, (! [C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ C)) > $o),X2:((product_prod @ A) @ ((product_prod @ B) @ C))]: (((! [A6:A,B5:B,C3:C]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A6) @ ((((product_Pair @ B) @ C) @ B5) @ C3))))) => (P @ X2))))).
% 2.78/1.58 thf(fact_75_prod__induct4, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) > $o),X2:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: (((! [A6:A,B5:B,C3:C,D2:D]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A6) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B5) @ ((((product_Pair @ C) @ D) @ C3) @ D2)))))) => (P @ X2))))).
% 2.78/1.58 thf(fact_76_prod__induct5, axiom, (! [E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) > $o),X2:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))))]: (((! [A6:A,B5:B,C3:C,D2:D,E2:E]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) @ A6) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E)) @ C3) @ ((((product_Pair @ D) @ E) @ D2) @ E2))))))) => (P @ X2))))).
% 2.78/1.58 thf(fact_77_prod__induct6, axiom, (! [F2:$tType,E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F2))))) > $o),X2:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F2)))))]: (((! [A6:A,B5:B,C3:C,D2:D,E2:E,F3:F2]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F2))))) @ A6) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F2)))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F2))) @ C3) @ ((((product_Pair @ D) @ ((product_prod @ E) @ F2)) @ D2) @ ((((product_Pair @ E) @ F2) @ E2) @ F3)))))))) => (P @ X2))))).
% 2.78/1.58 thf(fact_78_prod__induct7, axiom, (! [G2:$tType,F2:$tType,E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F2) @ G2)))))) > $o),X2:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F2) @ G2))))))]: (((! [A6:A,B5:B,C3:C,D2:D,E2:E,F3:F2,G3:G2]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F2) @ G2)))))) @ A6) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F2) @ G2))))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F2) @ G2)))) @ C3) @ ((((product_Pair @ D) @ ((product_prod @ E) @ ((product_prod @ F2) @ G2))) @ D2) @ ((((product_Pair @ E) @ ((product_prod @ F2) @ G2)) @ E2) @ ((((product_Pair @ F2) @ G2) @ F3) @ G3))))))))) => (P @ X2))))).
% 2.78/1.58 thf(fact_79_old_Oprod_Oexhaust, axiom, (! [A:$tType,B:$tType,Y:((product_prod @ A) @ B)]: ((~ ((! [A6:A,B5:B]: ((Y != ((((product_Pair @ A) @ B) @ A6) @ B5))))))))).
% 2.78/1.58 thf(fact_80_old_Oprod_Oinducts, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),Prod:((product_prod @ A) @ B)]: (((! [A6:A,B5:B]: ((P @ ((((product_Pair @ A) @ B) @ A6) @ B5)))) => (P @ Prod))))).
% 2.78/1.58 thf(fact_81_verit__la__disequality, axiom, (! [A:$tType]: (((linorder @ A) => (! [A4:A,B3:A]: (((A4 = B3) | ((~ ((((ord_less_eq @ A) @ A4) @ B3))) | (~ ((((ord_less_eq @ A) @ B3) @ A4))))))))))).
% 2.78/1.58 thf(fact_82_Collect__mono__iff, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)) = (! [X3:A]: (((P @ X3) => (Q @ X3)))))))).
% 2.78/1.58 thf(fact_83_set__eq__subset, axiom, (! [A:$tType]: (((^ [Y3:(set @ A),Z:(set @ A)]: ((Y3 = Z))) = (^ [A3:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B6) & (((ord_less_eq @ (set @ A)) @ B6) @ A3)))))))).
% 2.78/1.58 thf(fact_84_subset__trans, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((ord_less_eq @ (set @ A)) @ B2) @ C4) => (((ord_less_eq @ (set @ A)) @ A2) @ C4)))))).
% 2.78/1.58 thf(fact_85_Collect__mono, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X8:A]: (((P @ X8) => (Q @ X8)))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)))))).
% 2.78/1.58 thf(fact_86_subset__refl, axiom, (! [A:$tType,A2:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A2) @ A2)))).
% 2.78/1.58 thf(fact_87_subset__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A3:(set @ A),B6:(set @ A)]: ((! [T2:A]: (((((member @ A) @ T2) @ A3) => (((member @ A) @ T2) @ B6)))))))))).
% 2.78/1.58 thf(fact_88_equalityD2, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((A2 = B2) => (((ord_less_eq @ (set @ A)) @ B2) @ A2))))).
% 2.78/1.58 thf(fact_89_equalityD1, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((A2 = B2) => (((ord_less_eq @ (set @ A)) @ A2) @ B2))))).
% 2.78/1.58 thf(fact_90_subset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A3:(set @ A),B6:(set @ A)]: ((! [X3:A]: (((((member @ A) @ X3) @ A3) => (((member @ A) @ X3) @ B6)))))))))).
% 2.78/1.58 thf(fact_91_equalityE, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((A2 = B2) => (~ (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (~ ((((ord_less_eq @ (set @ A)) @ B2) @ A2)))))))))).
% 2.78/1.58 thf(fact_92_subsetD, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C2:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((member @ A) @ C2) @ A2) => (((member @ A) @ C2) @ B2)))))).
% 2.78/1.58 thf(fact_93_in__mono, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),X2:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((member @ A) @ X2) @ A2) => (((member @ A) @ X2) @ B2)))))).
% 2.78/1.58 thf(fact_94_Collect__subset, axiom, (! [A:$tType,A2:(set @ A),P:(A > $o)]: ((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ (^ [X3:A]: (((((member @ A) @ X3) @ A2) & (P @ X3)))))) @ A2)))).
% 2.78/1.58 thf(fact_95_fst__eqD, axiom, (! [B:$tType,A:$tType,X2:A,Y:B,A4:A]: ((((((product_fst @ A) @ B) @ ((((product_Pair @ A) @ B) @ X2) @ Y)) = A4) => (X2 = A4))))).
% 2.78/1.58 thf(fact_96_fst__conv, axiom, (! [B:$tType,A:$tType,X1:A,X22:B]: (((((product_fst @ A) @ B) @ ((((product_Pair @ A) @ B) @ X1) @ X22)) = X1)))).
% 2.78/1.58 thf(fact_97_snd__conv, axiom, (! [Aa:$tType,A:$tType,X1:Aa,X22:A]: (((((product_snd @ Aa) @ A) @ ((((product_Pair @ Aa) @ A) @ X1) @ X22)) = X22)))).
% 2.78/1.58 thf(fact_98_snd__eqD, axiom, (! [B:$tType,A:$tType,X2:B,Y:A,A4:A]: ((((((product_snd @ B) @ A) @ ((((product_Pair @ B) @ A) @ X2) @ Y)) = A4) => (Y = A4))))).
% 2.78/1.58 thf(fact_99_le__imp__neg__le, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A4:A,B3:A]: (((((ord_less_eq @ A) @ A4) @ B3) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B3)) @ ((uminus_uminus @ A) @ A4))))))))).
% 2.78/1.58 thf(fact_100_minus__le__iff, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A4:A,B3:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A4)) @ B3) = (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B3)) @ A4)))))))).
% 2.78/1.58 thf(fact_101_le__minus__iff, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A4:A,B3:A]: (((((ord_less_eq @ A) @ A4) @ ((uminus_uminus @ A) @ B3)) = (((ord_less_eq @ A) @ B3) @ ((uminus_uminus @ A) @ A4))))))))).
% 2.78/1.58 thf(fact_102_compl__le__swap2, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [Y:A,X2:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ Y)) @ X2) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ X2)) @ Y)))))))).
% 2.78/1.58 thf(fact_103_compl__le__swap1, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [Y:A,X2:A]: (((((ord_less_eq @ A) @ Y) @ ((uminus_uminus @ A) @ X2)) => (((ord_less_eq @ A) @ X2) @ ((uminus_uminus @ A) @ Y))))))))).
% 2.78/1.58 thf(fact_104_compl__mono, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [X2:A,Y:A]: (((((ord_less_eq @ A) @ X2) @ Y) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ Y)) @ ((uminus_uminus @ A) @ X2))))))))).
% 2.78/1.58 thf(fact_105_fst__pair, axiom, (! [B:$tType,A:$tType,A4:A,B3:B]: (((((product_fst @ A) @ B) @ ((((product_Pair @ A) @ B) @ A4) @ B3)) = A4)))).
% 2.78/1.58 thf(fact_106_snd__pair, axiom, (! [B:$tType,A:$tType,A4:B,B3:A]: (((((product_snd @ B) @ A) @ ((((product_Pair @ B) @ A) @ A4) @ B3)) = B3)))).
% 2.78/1.58 thf(fact_107_surjective__pairing, axiom, (! [B:$tType,A:$tType,T3:((product_prod @ A) @ B)]: ((T3 = ((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ T3)) @ (((product_snd @ A) @ B) @ T3)))))).
% 2.78/1.58 thf(fact_108_prod_Oexhaust__sel, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B)]: ((Prod = ((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod)))))).
% 2.78/1.58 thf(fact_109_conjI__realizer, axiom, (! [A:$tType,B:$tType,P:(A > $o),P2:A,Q:(B > $o),Q2:B]: (((P @ P2) => ((Q @ Q2) => ((P @ (((product_fst @ A) @ B) @ ((((product_Pair @ A) @ B) @ P2) @ Q2))) & (Q @ (((product_snd @ A) @ B) @ ((((product_Pair @ A) @ B) @ P2) @ Q2))))))))).
% 2.78/1.58 thf(fact_110_exI__realizer, axiom, (! [B:$tType,A:$tType,P:(A > (B > $o)),Y:A,X2:B]: ((((P @ Y) @ X2) => ((P @ (((product_snd @ B) @ A) @ ((((product_Pair @ B) @ A) @ X2) @ Y))) @ (((product_fst @ B) @ A) @ ((((product_Pair @ B) @ A) @ X2) @ Y))))))).
% 2.78/1.58 thf(fact_111_Uvariation__mon, axiom, (! [U:(set @ variable),V:(set @ variable),Omega:(variable > real),Nu:(variable > real)]: (((((ord_less_eq @ (set @ variable)) @ U) @ V) => ((((denota1419872369iation @ Omega) @ Nu) @ U) => (((denota1419872369iation @ Omega) @ Nu) @ V)))))).
% 2.78/1.58 thf(fact_112_usubstappp__fst__mon, axiom, (! [U:(set @ variable),V:(set @ variable),Sigma:((product_prod @ (char > (option @ trm))) @ ((product_prod @ (char > (option @ trm))) @ ((product_prod @ (char > (option @ fml))) @ (char > (option @ game))))),Alpha:game]: (((((ord_less_eq @ (set @ variable)) @ U) @ V) => (((ord_less_eq @ (set @ variable)) @ (((product_fst @ (set @ variable)) @ (option @ game)) @ (((uSubst95898988stappp @ Sigma) @ U) @ Alpha))) @ (((product_fst @ (set @ variable)) @ (option @ game)) @ (((uSubst95898988stappp @ Sigma) @ V) @ Alpha))))))).
% 2.78/1.58 thf(fact_113_usubst__taboos__mon, axiom, (! [U:(set @ variable),Sigma:((product_prod @ (char > (option @ trm))) @ ((product_prod @ (char > (option @ trm))) @ ((product_prod @ (char > (option @ fml))) @ (char > (option @ game))))),Alpha:game]: ((((ord_less_eq @ (set @ variable)) @ U) @ (((product_fst @ (set @ variable)) @ (option @ game)) @ (((uSubst95898988stappp @ Sigma) @ U) @ Alpha)))))).
% 2.78/1.58 thf(fact_114_old_Oprod_Orec, axiom, (! [A:$tType,T:$tType,B:$tType,F1:(A > (B > T)),A4:A,B3:B]: (((((((product_rec_prod @ A) @ B) @ T) @ F1) @ ((((product_Pair @ A) @ B) @ A4) @ B3)) = ((F1 @ A4) @ B3))))).
% 2.78/1.58 thf(fact_115_usubstappp__dual, axiom, (! [Sigma:((product_prod @ (char > (option @ trm))) @ ((product_prod @ (char > (option @ trm))) @ ((product_prod @ (char > (option @ fml))) @ (char > (option @ game))))),U:(set @ variable),Alpha:game]: (((((uSubst95898988stappp @ Sigma) @ U) @ (dual @ Alpha)) = ((((product_Pair @ (set @ variable)) @ (option @ game)) @ (((product_fst @ (set @ variable)) @ (option @ game)) @ (((uSubst95898988stappp @ Sigma) @ U) @ Alpha))) @ (uSubst739989314_Dualo @ (((product_snd @ (set @ variable)) @ (option @ game)) @ (((uSubst95898988stappp @ Sigma) @ U) @ Alpha)))))))).
% 2.78/1.58 thf(fact_116_BNF__Greatest__Fixpoint_Osubst__Pair, axiom, (! [B:$tType,A:$tType,P:(A > (B > $o)),X2:A,Y:B,A4:((product_prod @ A) @ B)]: ((((P @ X2) @ Y) => ((A4 = ((((product_Pair @ A) @ B) @ X2) @ Y)) => ((P @ (((product_fst @ A) @ B) @ A4)) @ (((product_snd @ A) @ B) @ A4))))))).
% 2.78/1.58 thf(fact_117_usubstappp_Osimps_I7_J, axiom, (! [Sigma:((product_prod @ (char > (option @ trm))) @ ((product_prod @ (char > (option @ trm))) @ ((product_prod @ (char > (option @ fml))) @ (char > (option @ game))))),U:(set @ variable),Alpha:game]: (((((uSubst95898988stappp @ Sigma) @ U) @ (dual @ Alpha)) = ((((product_Pair @ (set @ variable)) @ (option @ game)) @ (((product_fst @ (set @ variable)) @ (option @ game)) @ (((uSubst95898988stappp @ Sigma) @ U) @ Alpha))) @ (uSubst739989314_Dualo @ (((product_snd @ (set @ variable)) @ (option @ game)) @ (((uSubst95898988stappp @ Sigma) @ U) @ Alpha)))))))).
% 2.78/1.58 thf(fact_118_usubstappp_Osimps_I6_J, axiom, (! [Sigma:((product_prod @ (char > (option @ trm))) @ ((product_prod @ (char > (option @ trm))) @ ((product_prod @ (char > (option @ fml))) @ (char > (option @ game))))),U:(set @ variable),Alpha:game]: (((((uSubst95898988stappp @ Sigma) @ U) @ (loop @ Alpha)) = ((((product_Pair @ (set @ variable)) @ (option @ game)) @ (((product_fst @ (set @ variable)) @ (option @ game)) @ (((uSubst95898988stappp @ Sigma) @ U) @ Alpha))) @ (uSubst993936602_Loopo @ (((product_snd @ (set @ variable)) @ (option @ game)) @ (((uSubst95898988stappp @ Sigma) @ (((product_fst @ (set @ variable)) @ (option @ game)) @ (((uSubst95898988stappp @ Sigma) @ U) @ Alpha))) @ Alpha)))))))).
% 2.78/1.58 thf(fact_119_usubstappp__loop, axiom, (! [Sigma:((product_prod @ (char > (option @ trm))) @ ((product_prod @ (char > (option @ trm))) @ ((product_prod @ (char > (option @ fml))) @ (char > (option @ game))))),U:(set @ variable),Alpha:game]: (((((uSubst95898988stappp @ Sigma) @ U) @ (loop @ Alpha)) = ((((product_Pair @ (set @ variable)) @ (option @ game)) @ (((product_fst @ (set @ variable)) @ (option @ game)) @ (((uSubst95898988stappp @ Sigma) @ U) @ Alpha))) @ (uSubst993936602_Loopo @ (((product_snd @ (set @ variable)) @ (option @ game)) @ (((uSubst95898988stappp @ Sigma) @ (((product_fst @ (set @ variable)) @ (option @ game)) @ (((uSubst95898988stappp @ Sigma) @ U) @ Alpha))) @ Alpha)))))))).
% 2.78/1.58 thf(fact_120_order__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A]: ((((ord_less_eq @ A) @ X2) @ X2))))))).
% 2.78/1.58 thf(fact_121_monotone, axiom, (! [X:(set @ (variable > real)),Y4:(set @ (variable > real)),I:denotational_interp,Alpha:game]: (((((ord_less_eq @ (set @ (variable > real))) @ X) @ Y4) => (((ord_less_eq @ (set @ (variable > real))) @ (((denota1245701238me_sem @ I) @ Alpha) @ X)) @ (((denota1245701238me_sem @ I) @ Alpha) @ Y4)))))).
% 2.78/1.58 thf(fact_122_less__eq__set__def, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A3:(set @ A),B6:(set @ A)]: ((((ord_less_eq @ (A > $o)) @ (^ [X3:A]: ((((member @ A) @ X3) @ A3)))) @ (^ [X3:A]: ((((member @ A) @ X3) @ B6)))))))))).
% 2.78/1.58 thf(fact_123_Loopo__undef, axiom, (! [Alpha:(option @ game)]: ((((uSubst993936602_Loopo @ Alpha) = (none @ game)) = (Alpha = (none @ game)))))).
% 2.78/1.58 thf(fact_124_Dualo__undef, axiom, (! [Alpha:(option @ game)]: ((((uSubst739989314_Dualo @ Alpha) = (none @ game)) = (Alpha = (none @ game)))))).
% 2.78/1.58 thf(fact_125_Loopo_Osimps_I2_J, axiom, ((uSubst993936602_Loopo @ (none @ game)) = (none @ game))).
% 2.78/1.58 thf(fact_126_Dualo_Osimps_I2_J, axiom, ((uSubst739989314_Dualo @ (none @ game)) = (none @ game))).
% 2.78/1.58 thf(fact_127_dual__order_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A4:A]: (((((ord_less_eq @ A) @ B3) @ A4) => ((((ord_less_eq @ A) @ A4) @ B3) => (A4 = B3))))))))).
% 2.78/1.58 thf(fact_128_dual__order_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y3:A,Z:A]: ((Y3 = Z))) = (^ [A7:A,B7:A]: (((((ord_less_eq @ A) @ B7) @ A7) & (((ord_less_eq @ A) @ A7) @ B7))))))))).
% 2.78/1.58 thf(fact_129_dual__order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A4:A,C2:A]: (((((ord_less_eq @ A) @ B3) @ A4) => ((((ord_less_eq @ A) @ C2) @ B3) => (((ord_less_eq @ A) @ C2) @ A4))))))))).
% 2.78/1.58 thf(fact_130_linorder__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > (A > $o)),A4:A,B3:A]: (((! [A6:A,B5:A]: (((((ord_less_eq @ A) @ A6) @ B5) => ((P @ A6) @ B5)))) => ((! [A6:A,B5:A]: ((((P @ B5) @ A6) => ((P @ A6) @ B5)))) => ((P @ A4) @ B3))))))))).
% 2.78/1.58 thf(fact_131_dual__order_Orefl, axiom, (! [A:$tType]: (((order @ A) => (! [A4:A]: ((((ord_less_eq @ A) @ A4) @ A4))))))).
% 2.78/1.58 thf(fact_132_order__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y:A,Z2:A]: (((((ord_less_eq @ A) @ X2) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z2) => (((ord_less_eq @ A) @ X2) @ Z2))))))))).
% 2.78/1.58 thf(fact_133_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [A4:A,B3:A]: (((((ord_less_eq @ A) @ A4) @ B3) => ((((ord_less_eq @ A) @ B3) @ A4) => (A4 = B3))))))))).
% 2.78/1.58 thf(fact_134_ord__le__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A4:A,B3:A,C2:A]: (((((ord_less_eq @ A) @ A4) @ B3) => ((B3 = C2) => (((ord_less_eq @ A) @ A4) @ C2))))))))).
% 2.78/1.58 thf(fact_135_ord__eq__le__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A4:A,B3:A,C2:A]: (((A4 = B3) => ((((ord_less_eq @ A) @ B3) @ C2) => (((ord_less_eq @ A) @ A4) @ C2))))))))).
% 2.78/1.58 thf(fact_136_order__class_Oorder_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y3:A,Z:A]: ((Y3 = Z))) = (^ [A7:A,B7:A]: (((((ord_less_eq @ A) @ A7) @ B7) & (((ord_less_eq @ A) @ B7) @ A7))))))))).
% 2.78/1.58 thf(fact_137_antisym__conv, axiom, (! [A:$tType]: (((order @ A) => (! [Y:A,X2:A]: (((((ord_less_eq @ A) @ Y) @ X2) => ((((ord_less_eq @ A) @ X2) @ Y) = (X2 = Y))))))))).
% 2.78/1.58 thf(fact_138_le__cases3, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A,Z2:A]: ((((((ord_less_eq @ A) @ X2) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ Z2)))) => (((((ord_less_eq @ A) @ Y) @ X2) => (~ ((((ord_less_eq @ A) @ X2) @ Z2)))) => (((((ord_less_eq @ A) @ X2) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ Y)))) => (((((ord_less_eq @ A) @ Z2) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ X2)))) => (((((ord_less_eq @ A) @ Y) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ X2)))) => (~ (((((ord_less_eq @ A) @ Z2) @ X2) => (~ ((((ord_less_eq @ A) @ X2) @ Y))))))))))))))))).
% 2.78/1.58 thf(fact_139_order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [A4:A,B3:A,C2:A]: (((((ord_less_eq @ A) @ A4) @ B3) => ((((ord_less_eq @ A) @ B3) @ C2) => (((ord_less_eq @ A) @ A4) @ C2))))))))).
% 2.78/1.58 thf(fact_140_le__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A]: (((~ ((((ord_less_eq @ A) @ X2) @ Y))) => (((ord_less_eq @ A) @ Y) @ X2)))))))).
% 2.78/1.58 thf(fact_141_eq__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y:A]: (((X2 = Y) => (((ord_less_eq @ A) @ X2) @ Y)))))))).
% 2.78/1.58 thf(fact_142_linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A]: (((((ord_less_eq @ A) @ X2) @ Y) | (((ord_less_eq @ A) @ Y) @ X2)))))))).
% 2.78/1.58 thf(fact_143_antisym, axiom, (! [A:$tType]: (((order @ A) => (! [X2:A,Y:A]: (((((ord_less_eq @ A) @ X2) @ Y) => ((((ord_less_eq @ A) @ Y) @ X2) => (X2 = Y))))))))).
% 2.78/1.58 thf(fact_144_eq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y3:A,Z:A]: ((Y3 = Z))) = (^ [X3:A,Y5:A]: (((((ord_less_eq @ A) @ X3) @ Y5) & (((ord_less_eq @ A) @ Y5) @ X3))))))))).
% 2.78/1.58 thf(fact_145_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A4:A,B3:A,F:(A > B),C2:B]: (((((ord_less_eq @ A) @ A4) @ B3) => (((F @ B3) = C2) => ((! [X8:A,Y2:A]: (((((ord_less_eq @ A) @ X8) @ Y2) => (((ord_less_eq @ B) @ (F @ X8)) @ (F @ Y2))))) => (((ord_less_eq @ B) @ (F @ A4)) @ C2)))))))))).
% 2.78/1.58 thf(fact_146_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A4:A,F:(B > A),B3:B,C2:B]: (((A4 = (F @ B3)) => ((((ord_less_eq @ B) @ B3) @ C2) => ((! [X8:B,Y2:B]: (((((ord_less_eq @ B) @ X8) @ Y2) => (((ord_less_eq @ A) @ (F @ X8)) @ (F @ Y2))))) => (((ord_less_eq @ A) @ A4) @ (F @ C2))))))))))).
% 2.78/1.58 thf(fact_147_order__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A4:A,B3:A,F:(A > C),C2:C]: (((((ord_less_eq @ A) @ A4) @ B3) => ((((ord_less_eq @ C) @ (F @ B3)) @ C2) => ((! [X8:A,Y2:A]: (((((ord_less_eq @ A) @ X8) @ Y2) => (((ord_less_eq @ C) @ (F @ X8)) @ (F @ Y2))))) => (((ord_less_eq @ C) @ (F @ A4)) @ C2)))))))))).
% 2.78/1.58 thf(fact_148_order__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A4:A,F:(B > A),B3:B,C2:B]: (((((ord_less_eq @ A) @ A4) @ (F @ B3)) => ((((ord_less_eq @ B) @ B3) @ C2) => ((! [X8:B,Y2:B]: (((((ord_less_eq @ B) @ X8) @ Y2) => (((ord_less_eq @ A) @ (F @ X8)) @ (F @ Y2))))) => (((ord_less_eq @ A) @ A4) @ (F @ C2))))))))))).
% 2.78/1.58 thf(fact_149_le__fun__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => ((ord_less_eq @ (A > B)) = (^ [F4:(A > B),G4:(A > B)]: ((! [X3:A]: ((((ord_less_eq @ B) @ (F4 @ X3)) @ (G4 @ X3))))))))))).
% 2.78/1.58 thf(fact_150_le__funI, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B)]: (((! [X8:A]: ((((ord_less_eq @ B) @ (F @ X8)) @ (G @ X8)))) => (((ord_less_eq @ (A > B)) @ F) @ G)))))))).
% 2.78/1.58 thf(fact_151_le__funE, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B),X2:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X2)) @ (G @ X2))))))))).
% 2.78/1.58 thf(fact_152_le__funD, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B),X2:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X2)) @ (G @ X2))))))))).
% 2.78/1.58 thf(fact_153_subset__CollectI, axiom, (! [A:$tType,B2:(set @ A),A2:(set @ A),Q:(A > $o),P:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ B2) @ A2) => ((! [X8:A]: (((((member @ A) @ X8) @ B2) => ((Q @ X8) => (P @ X8))))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ (^ [X3:A]: (((((member @ A) @ X3) @ B2) & (Q @ X3)))))) @ ((collect @ A) @ (^ [X3:A]: (((((member @ A) @ X3) @ A2) & (P @ X3))))))))))).
% 2.78/1.58 thf(fact_154_subset__Collect__iff, axiom, (! [A:$tType,B2:(set @ A),A2:(set @ A),P:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ B2) @ A2) => ((((ord_less_eq @ (set @ A)) @ B2) @ ((collect @ A) @ (^ [X3:A]: (((((member @ A) @ X3) @ A2) & (P @ X3)))))) = (! [X3:A]: (((((member @ A) @ X3) @ B2) => (P @ X3))))))))).
% 2.78/1.58 thf(fact_155_sndI, axiom, (! [A:$tType,B:$tType,X2:((product_prod @ A) @ B),Y:A,Z2:B]: (((X2 = ((((product_Pair @ A) @ B) @ Y) @ Z2)) => ((((product_snd @ A) @ B) @ X2) = Z2))))).
% 2.78/1.58 thf(fact_156_eq__snd__iff, axiom, (! [A:$tType,B:$tType,B3:A,P2:((product_prod @ B) @ A)]: (((B3 = (((product_snd @ B) @ A) @ P2)) = (? [A7:B]: ((P2 = ((((product_Pair @ B) @ A) @ A7) @ B3)))))))).
% 2.78/1.58 thf(fact_157_eq__fst__iff, axiom, (! [A:$tType,B:$tType,A4:A,P2:((product_prod @ A) @ B)]: (((A4 = (((product_fst @ A) @ B) @ P2)) = (? [B7:B]: ((P2 = ((((product_Pair @ A) @ B) @ A4) @ B7)))))))).
% 2.78/1.58 thf(fact_158_fstI, axiom, (! [B:$tType,A:$tType,X2:((product_prod @ A) @ B),Y:A,Z2:B]: (((X2 = ((((product_Pair @ A) @ B) @ Y) @ Z2)) => ((((product_fst @ A) @ B) @ X2) = Y))))).
% 2.78/1.58 thf(fact_159_predicate1I, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X8:A]: (((P @ X8) => (Q @ X8)))) => (((ord_less_eq @ (A > $o)) @ P) @ Q))))).
% 2.78/1.58 thf(fact_160_predicate1D, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o),X2:A]: (((((ord_less_eq @ (A > $o)) @ P) @ Q) => ((P @ X2) => (Q @ X2)))))).
% 2.78/1.58 thf(fact_161_rev__predicate1D, axiom, (! [A:$tType,P:(A > $o),X2:A,Q:(A > $o)]: (((P @ X2) => ((((ord_less_eq @ (A > $o)) @ P) @ Q) => (Q @ X2)))))).
% 2.78/1.58 thf(fact_162_pred__subset__eq, axiom, (! [A:$tType,R:(set @ A),S3:(set @ A)]: (((((ord_less_eq @ (A > $o)) @ (^ [X3:A]: ((((member @ A) @ X3) @ R)))) @ (^ [X3:A]: ((((member @ A) @ X3) @ S3)))) = (((ord_less_eq @ (set @ A)) @ R) @ S3))))).
% 2.78/1.58 thf(fact_163_scomp__unfold, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType]: ((((((product_scomp @ A) @ B) @ C) @ D) = (^ [F4:(A > ((product_prod @ B) @ C)),G4:(B > (C > D)),X3:A]: (((G4 @ (((product_fst @ B) @ C) @ (F4 @ X3))) @ (((product_snd @ B) @ C) @ (F4 @ X3))))))))).
% 2.78/1.58 thf(fact_164_Collect__restrict, axiom, (! [A:$tType,X:(set @ A),P:(A > $o)]: ((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ (^ [X3:A]: (((((member @ A) @ X3) @ X) & (P @ X3)))))) @ X)))).
% 2.78/1.58 thf(fact_165_scomp__Pair, axiom, (! [C:$tType,B:$tType,A:$tType,X2:(A > ((product_prod @ B) @ C))]: ((((((((product_scomp @ A) @ B) @ C) @ ((product_prod @ B) @ C)) @ X2) @ ((product_Pair @ B) @ C)) = X2)))).
% 2.78/1.58 thf(fact_166_scomp__scomp, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,F2:$tType,E:$tType,F:(A > ((product_prod @ E) @ F2)),G:(E > (F2 > ((product_prod @ C) @ D))),H:(C > (D > B))]: ((((((((product_scomp @ A) @ C) @ D) @ B) @ ((((((product_scomp @ A) @ E) @ F2) @ ((product_prod @ C) @ D)) @ F) @ G)) @ H) = ((((((product_scomp @ A) @ E) @ F2) @ B) @ F) @ (^ [X3:E]: (((((((product_scomp @ F2) @ C) @ D) @ B) @ (G @ X3)) @ H)))))))).
% 2.78/1.58 thf(fact_167_Pair__scomp, axiom, (! [A:$tType,B:$tType,C:$tType,X2:C,F:(C > (A > B))]: ((((((((product_scomp @ A) @ C) @ A) @ B) @ (((product_Pair @ C) @ A) @ X2)) @ F) = (F @ X2))))).
% 2.78/1.58 thf(fact_168_subrelI, axiom, (! [B:$tType,A:$tType,R2:(set @ ((product_prod @ A) @ B)),S4:(set @ ((product_prod @ A) @ B))]: (((! [X8:A,Y2:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X8) @ Y2)) @ R2) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X8) @ Y2)) @ S4)))) => (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R2) @ S4))))).
% 2.78/1.58 thf(fact_169_ssubst__Pair__rhs, axiom, (! [B:$tType,A:$tType,R2:A,S4:B,R:(set @ ((product_prod @ A) @ B)),S5:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ R2) @ S4)) @ R) => ((S5 = S4) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ R2) @ S5)) @ R)))))).
% 2.78/1.58 thf(fact_170_pred__subset__eq2, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ B)),S3:(set @ ((product_prod @ A) @ B))]: (((((ord_less_eq @ (A > (B > $o))) @ (^ [X3:A,Y5:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y5)) @ R)))) @ (^ [X3:A,Y5:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y5)) @ S3)))) = (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R) @ S3))))).
% 2.78/1.58 thf(fact_171_pred__equals__eq2, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ B)),S3:(set @ ((product_prod @ A) @ B))]: ((((^ [X3:A,Y5:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y5)) @ R))) = (^ [X3:A,Y5:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y5)) @ S3)))) = (R = S3))))).
% 2.78/1.58 thf(fact_172_prop__restrict, axiom, (! [A:$tType,X2:A,Z3:(set @ A),X:(set @ A),P:(A > $o)]: (((((member @ A) @ X2) @ Z3) => ((((ord_less_eq @ (set @ A)) @ Z3) @ ((collect @ A) @ (^ [X3:A]: (((((member @ A) @ X3) @ X) & (P @ X3)))))) => (P @ X2)))))).
% 2.78/1.58 thf(fact_173_conj__subset__def, axiom, (! [A:$tType,A2:(set @ A),P:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ A2) @ ((collect @ A) @ (^ [X3:A]: (((P @ X3) & (Q @ X3)))))) = ((((ord_less_eq @ (set @ A)) @ A2) @ ((collect @ A) @ P)) & (((ord_less_eq @ (set @ A)) @ A2) @ ((collect @ A) @ Q))))))).
% 2.78/1.58 thf(fact_174_internal__case__prod__conv, axiom, (! [B:$tType,A:$tType,C:$tType,C2:(B > (C > A)),A4:B,B3:C]: (((((((produc2004651681e_prod @ B) @ C) @ A) @ C2) @ ((((product_Pair @ B) @ C) @ A4) @ B3)) = ((C2 @ A4) @ B3))))).
% 2.78/1.58 thf(fact_175_prod_Oswap__def, axiom, (! [B:$tType,A:$tType]: ((((product_swap @ A) @ B) = (^ [P3:((product_prod @ A) @ B)]: (((((product_Pair @ B) @ A) @ (((product_snd @ A) @ B) @ P3)) @ (((product_fst @ A) @ B) @ P3)))))))).
% 2.78/1.58 thf(fact_176_predicate2I, axiom, (! [B:$tType,A:$tType,P:(A > (B > $o)),Q:(A > (B > $o))]: (((! [X8:A,Y2:B]: ((((P @ X8) @ Y2) => ((Q @ X8) @ Y2)))) => (((ord_less_eq @ (A > (B > $o))) @ P) @ Q))))).
% 2.78/1.58 thf(fact_177_swap__swap, axiom, (! [B:$tType,A:$tType,P2:((product_prod @ A) @ B)]: (((((product_swap @ B) @ A) @ (((product_swap @ A) @ B) @ P2)) = P2)))).
% 2.78/1.58 thf(fact_178_swap__simp, axiom, (! [A:$tType,B:$tType,X2:B,Y:A]: (((((product_swap @ B) @ A) @ ((((product_Pair @ B) @ A) @ X2) @ Y)) = ((((product_Pair @ A) @ B) @ Y) @ X2))))).
% 2.78/1.58 thf(fact_179_snd__swap, axiom, (! [B:$tType,A:$tType,X2:((product_prod @ A) @ B)]: (((((product_snd @ B) @ A) @ (((product_swap @ A) @ B) @ X2)) = (((product_fst @ A) @ B) @ X2))))).
% 2.78/1.58 thf(fact_180_fst__swap, axiom, (! [A:$tType,B:$tType,X2:((product_prod @ B) @ A)]: (((((product_fst @ A) @ B) @ (((product_swap @ B) @ A) @ X2)) = (((product_snd @ B) @ A) @ X2))))).
% 2.78/1.58 thf(fact_181_rev__predicate2D, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o)),X2:A,Y:B,Q:(A > (B > $o))]: ((((P @ X2) @ Y) => ((((ord_less_eq @ (A > (B > $o))) @ P) @ Q) => ((Q @ X2) @ Y)))))).
% 2.78/1.58 thf(fact_182_predicate2D, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o)),Q:(A > (B > $o)),X2:A,Y:B]: (((((ord_less_eq @ (A > (B > $o))) @ P) @ Q) => (((P @ X2) @ Y) => ((Q @ X2) @ Y)))))).
% 2.78/1.58 thf(fact_183_refl__ge__eq, axiom, (! [A:$tType,R:(A > (A > $o))]: (((! [X8:A]: (((R @ X8) @ X8))) => (((ord_less_eq @ (A > (A > $o))) @ (^ [Y3:A,Z:A]: ((Y3 = Z)))) @ R))))).
% 2.78/1.58 thf(fact_184_ge__eq__refl, axiom, (! [A:$tType,R:(A > (A > $o)),X2:A]: (((((ord_less_eq @ (A > (A > $o))) @ (^ [Y3:A,Z:A]: ((Y3 = Z)))) @ R) => ((R @ X2) @ X2))))).
% 2.78/1.58 thf(fact_185_eq__subset, axiom, (! [A:$tType,P:(A > (A > $o))]: ((((ord_less_eq @ (A > (A > $o))) @ (^ [Y3:A,Z:A]: ((Y3 = Z)))) @ (^ [A7:A,B7:A]: ((((P @ A7) @ B7) | (A7 = B7)))))))).
% 2.78/1.58 thf(fact_186_usubstappp__choice, axiom, (! [Sigma:((product_prod @ (char > (option @ trm))) @ ((product_prod @ (char > (option @ trm))) @ ((product_prod @ (char > (option @ fml))) @ (char > (option @ game))))),U:(set @ variable),Alpha:game,Beta:game]: (((((uSubst95898988stappp @ Sigma) @ U) @ ((choice @ Alpha) @ Beta)) = ((((product_Pair @ (set @ variable)) @ (option @ game)) @ (((sup_sup @ (set @ variable)) @ (((product_fst @ (set @ variable)) @ (option @ game)) @ (((uSubst95898988stappp @ Sigma) @ U) @ Alpha))) @ (((product_fst @ (set @ variable)) @ (option @ game)) @ (((uSubst95898988stappp @ Sigma) @ U) @ Beta)))) @ ((uSubst1976112797hoiceo @ (((product_snd @ (set @ variable)) @ (option @ game)) @ (((uSubst95898988stappp @ Sigma) @ U) @ Alpha))) @ (((product_snd @ (set @ variable)) @ (option @ game)) @ (((uSubst95898988stappp @ Sigma) @ U) @ Beta)))))))).
% 2.78/1.58 thf(fact_187_relChain__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (((bNF_Ca1785829860lChain @ A) @ B) = (^ [R3:(set @ ((product_prod @ A) @ A)),As:(A > B)]: ((! [I2:A,J:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ I2) @ J)) @ R3) => (((ord_less_eq @ B) @ (As @ I2)) @ (As @ J)))))))))))).
% 2.78/1.58 thf(fact_188_sup__apply, axiom, (! [B:$tType,A:$tType]: (((semilattice_sup @ B) => ((sup_sup @ (A > B)) = (^ [F4:(A > B),G4:(A > B),X3:A]: ((((sup_sup @ B) @ (F4 @ X3)) @ (G4 @ X3))))))))).
% 2.78/1.58 thf(fact_189_sup_Oidem, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A4:A]: (((((sup_sup @ A) @ A4) @ A4) = A4))))))).
% 2.78/1.58 thf(fact_190_sup__idem, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X2:A]: (((((sup_sup @ A) @ X2) @ X2) = X2))))))).
% 2.78/1.58 thf(fact_191_sup_Oleft__idem, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A4:A,B3:A]: (((((sup_sup @ A) @ A4) @ (((sup_sup @ A) @ A4) @ B3)) = (((sup_sup @ A) @ A4) @ B3)))))))).
% 2.78/1.58 thf(fact_192_sup__left__idem, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X2:A,Y:A]: (((((sup_sup @ A) @ X2) @ (((sup_sup @ A) @ X2) @ Y)) = (((sup_sup @ A) @ X2) @ Y)))))))).
% 2.78/1.58 thf(fact_193_sup_Oright__idem, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A4:A,B3:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ A4) @ B3)) @ B3) = (((sup_sup @ A) @ A4) @ B3)))))))).
% 2.78/1.58 thf(fact_194_UnCI, axiom, (! [A:$tType,C2:A,B2:(set @ A),A2:(set @ A)]: ((((~ ((((member @ A) @ C2) @ B2))) => (((member @ A) @ C2) @ A2)) => (((member @ A) @ C2) @ (((sup_sup @ (set @ A)) @ A2) @ B2)))))).
% 2.78/1.58 thf(fact_195_sup_Obounded__iff, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B3:A,C2:A,A4:A]: (((((ord_less_eq @ A) @ (((sup_sup @ A) @ B3) @ C2)) @ A4) = ((((ord_less_eq @ A) @ B3) @ A4) & (((ord_less_eq @ A) @ C2) @ A4))))))))).
% 2.78/1.58 thf(fact_196_le__sup__iff, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X2:A,Y:A,Z2:A]: (((((ord_less_eq @ A) @ (((sup_sup @ A) @ X2) @ Y)) @ Z2) = ((((ord_less_eq @ A) @ X2) @ Z2) & (((ord_less_eq @ A) @ Y) @ Z2))))))))).
% 2.78/1.58 thf(fact_197_Un__subset__iff, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A2) @ B2)) @ C4) = ((((ord_less_eq @ (set @ A)) @ A2) @ C4) & (((ord_less_eq @ (set @ A)) @ B2) @ C4)))))).
% 2.78/1.58 thf(fact_198_subset__Un__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A3:(set @ A),B6:(set @ A)]: (((((sup_sup @ (set @ A)) @ A3) @ B6) = B6))))))).
% 2.78/1.58 thf(fact_199_subset__UnE, axiom, (! [A:$tType,C4:(set @ A),A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ C4) @ (((sup_sup @ (set @ A)) @ A2) @ B2)) => (~ ((! [A8:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A8) @ A2) => (! [B8:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ B8) @ B2) => (C4 != (((sup_sup @ (set @ A)) @ A8) @ B8)))))))))))))).
% 2.78/1.58 thf(fact_200_Un__absorb2, axiom, (! [A:$tType,B2:(set @ A),A2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ B2) @ A2) => ((((sup_sup @ (set @ A)) @ A2) @ B2) = A2))))).
% 2.78/1.58 thf(fact_201_Un__absorb1, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((sup_sup @ (set @ A)) @ A2) @ B2) = B2))))).
% 2.78/1.58 thf(fact_202_Un__upper2, axiom, (! [A:$tType,B2:(set @ A),A2:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ B2) @ (((sup_sup @ (set @ A)) @ A2) @ B2))))).
% 2.78/1.58 thf(fact_203_Un__upper1, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A2) @ (((sup_sup @ (set @ A)) @ A2) @ B2))))).
% 2.78/1.58 thf(fact_204_Un__least, axiom, (! [A:$tType,A2:(set @ A),C4:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ C4) => ((((ord_less_eq @ (set @ A)) @ B2) @ C4) => (((ord_less_eq @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A2) @ B2)) @ C4)))))).
% 2.78/1.58 thf(fact_205_Un__mono, axiom, (! [A:$tType,A2:(set @ A),C4:(set @ A),B2:(set @ A),D3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ C4) => ((((ord_less_eq @ (set @ A)) @ B2) @ D3) => (((ord_less_eq @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A2) @ B2)) @ (((sup_sup @ (set @ A)) @ C4) @ D3))))))).
% 2.78/1.58 thf(fact_206_sup_OcoboundedI2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [C2:A,B3:A,A4:A]: (((((ord_less_eq @ A) @ C2) @ B3) => (((ord_less_eq @ A) @ C2) @ (((sup_sup @ A) @ A4) @ B3))))))))).
% 2.78/1.58 thf(fact_207_sup_OcoboundedI1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [C2:A,A4:A,B3:A]: (((((ord_less_eq @ A) @ C2) @ A4) => (((ord_less_eq @ A) @ C2) @ (((sup_sup @ A) @ A4) @ B3))))))))).
% 2.78/1.58 thf(fact_208_sup_Oabsorb__iff2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((ord_less_eq @ A) = (^ [A7:A,B7:A]: (((((sup_sup @ A) @ A7) @ B7) = B7)))))))).
% 2.78/1.58 thf(fact_209_sup_Oabsorb__iff1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((ord_less_eq @ A) = (^ [B7:A,A7:A]: (((((sup_sup @ A) @ A7) @ B7) = A7)))))))).
% 2.78/1.58 thf(fact_210_sup_Ocobounded2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B3:A,A4:A]: ((((ord_less_eq @ A) @ B3) @ (((sup_sup @ A) @ A4) @ B3)))))))).
% 2.78/1.58 thf(fact_211_sup_Ocobounded1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A4:A,B3:A]: ((((ord_less_eq @ A) @ A4) @ (((sup_sup @ A) @ A4) @ B3)))))))).
% 2.78/1.58 thf(fact_212_sup_Oorder__iff, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((ord_less_eq @ A) = (^ [B7:A,A7:A]: ((A7 = (((sup_sup @ A) @ A7) @ B7))))))))).
% 2.78/1.58 thf(fact_213_sup_OboundedI, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B3:A,A4:A,C2:A]: (((((ord_less_eq @ A) @ B3) @ A4) => ((((ord_less_eq @ A) @ C2) @ A4) => (((ord_less_eq @ A) @ (((sup_sup @ A) @ B3) @ C2)) @ A4))))))))).
% 2.78/1.58 thf(fact_214_sup_OboundedE, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B3:A,C2:A,A4:A]: (((((ord_less_eq @ A) @ (((sup_sup @ A) @ B3) @ C2)) @ A4) => (~ (((((ord_less_eq @ A) @ B3) @ A4) => (~ ((((ord_less_eq @ A) @ C2) @ A4))))))))))))).
% 2.78/1.58 thf(fact_215_sup__absorb2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X2:A,Y:A]: (((((ord_less_eq @ A) @ X2) @ Y) => ((((sup_sup @ A) @ X2) @ Y) = Y)))))))).
% 2.78/1.58 thf(fact_216_sup__absorb1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [Y:A,X2:A]: (((((ord_less_eq @ A) @ Y) @ X2) => ((((sup_sup @ A) @ X2) @ Y) = X2)))))))).
% 2.78/1.58 thf(fact_217_sup_Oabsorb2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A4:A,B3:A]: (((((ord_less_eq @ A) @ A4) @ B3) => ((((sup_sup @ A) @ A4) @ B3) = B3)))))))).
% 2.78/1.58 thf(fact_218_sup_Oabsorb1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B3:A,A4:A]: (((((ord_less_eq @ A) @ B3) @ A4) => ((((sup_sup @ A) @ A4) @ B3) = A4)))))))).
% 2.78/1.58 thf(fact_219_sup__unique, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [F:(A > (A > A)),X2:A,Y:A]: (((! [X8:A,Y2:A]: ((((ord_less_eq @ A) @ X8) @ ((F @ X8) @ Y2)))) => ((! [X8:A,Y2:A]: ((((ord_less_eq @ A) @ Y2) @ ((F @ X8) @ Y2)))) => ((! [X8:A,Y2:A,Z4:A]: (((((ord_less_eq @ A) @ Y2) @ X8) => ((((ord_less_eq @ A) @ Z4) @ X8) => (((ord_less_eq @ A) @ ((F @ Y2) @ Z4)) @ X8))))) => ((((sup_sup @ A) @ X2) @ Y) = ((F @ X2) @ Y))))))))))).
% 2.78/1.58 thf(fact_220_sup_OorderI, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A4:A,B3:A]: (((A4 = (((sup_sup @ A) @ A4) @ B3)) => (((ord_less_eq @ A) @ B3) @ A4)))))))).
% 2.78/1.58 thf(fact_221_sup_OorderE, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B3:A,A4:A]: (((((ord_less_eq @ A) @ B3) @ A4) => (A4 = (((sup_sup @ A) @ A4) @ B3))))))))).
% 2.78/1.58 thf(fact_222_le__iff__sup, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((ord_less_eq @ A) = (^ [X3:A,Y5:A]: (((((sup_sup @ A) @ X3) @ Y5) = Y5)))))))).
% 2.78/1.58 thf(fact_223_sup__least, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [Y:A,X2:A,Z2:A]: (((((ord_less_eq @ A) @ Y) @ X2) => ((((ord_less_eq @ A) @ Z2) @ X2) => (((ord_less_eq @ A) @ (((sup_sup @ A) @ Y) @ Z2)) @ X2))))))))).
% 2.78/1.58 thf(fact_224_sup__mono, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A4:A,C2:A,B3:A,D4:A]: (((((ord_less_eq @ A) @ A4) @ C2) => ((((ord_less_eq @ A) @ B3) @ D4) => (((ord_less_eq @ A) @ (((sup_sup @ A) @ A4) @ B3)) @ (((sup_sup @ A) @ C2) @ D4)))))))))).
% 2.78/1.58 thf(fact_225_sup_Omono, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [C2:A,A4:A,D4:A,B3:A]: (((((ord_less_eq @ A) @ C2) @ A4) => ((((ord_less_eq @ A) @ D4) @ B3) => (((ord_less_eq @ A) @ (((sup_sup @ A) @ C2) @ D4)) @ (((sup_sup @ A) @ A4) @ B3)))))))))).
% 2.78/1.58 thf(fact_226_le__supI2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X2:A,B3:A,A4:A]: (((((ord_less_eq @ A) @ X2) @ B3) => (((ord_less_eq @ A) @ X2) @ (((sup_sup @ A) @ A4) @ B3))))))))).
% 2.78/1.58 thf(fact_227_le__supI1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X2:A,A4:A,B3:A]: (((((ord_less_eq @ A) @ X2) @ A4) => (((ord_less_eq @ A) @ X2) @ (((sup_sup @ A) @ A4) @ B3))))))))).
% 2.78/1.58 thf(fact_228_sup__ge2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [Y:A,X2:A]: ((((ord_less_eq @ A) @ Y) @ (((sup_sup @ A) @ X2) @ Y)))))))).
% 2.78/1.58 thf(fact_229_sup__ge1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X2:A,Y:A]: ((((ord_less_eq @ A) @ X2) @ (((sup_sup @ A) @ X2) @ Y)))))))).
% 2.78/1.58 thf(fact_230_le__supI, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A4:A,X2:A,B3:A]: (((((ord_less_eq @ A) @ A4) @ X2) => ((((ord_less_eq @ A) @ B3) @ X2) => (((ord_less_eq @ A) @ (((sup_sup @ A) @ A4) @ B3)) @ X2))))))))).
% 2.78/1.58 thf(fact_231_le__supE, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A4:A,B3:A,X2:A]: (((((ord_less_eq @ A) @ (((sup_sup @ A) @ A4) @ B3)) @ X2) => (~ (((((ord_less_eq @ A) @ A4) @ X2) => (~ ((((ord_less_eq @ A) @ B3) @ X2))))))))))))).
% 2.78/1.58 thf(fact_232_inf__sup__ord_I3_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X2:A,Y:A]: ((((ord_less_eq @ A) @ X2) @ (((sup_sup @ A) @ X2) @ Y)))))))).
% 2.78/1.58 thf(fact_233_inf__sup__ord_I4_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [Y:A,X2:A]: ((((ord_less_eq @ A) @ Y) @ (((sup_sup @ A) @ X2) @ Y)))))))).
% 2.78/1.58 thf(fact_234_inf__sup__aci_I8_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X2:A,Y:A]: (((((sup_sup @ A) @ X2) @ (((sup_sup @ A) @ X2) @ Y)) = (((sup_sup @ A) @ X2) @ Y)))))))).
% 2.78/1.58 thf(fact_235_inf__sup__aci_I7_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X2:A,Y:A,Z2:A]: (((((sup_sup @ A) @ X2) @ (((sup_sup @ A) @ Y) @ Z2)) = (((sup_sup @ A) @ Y) @ (((sup_sup @ A) @ X2) @ Z2))))))))).
% 2.78/1.58 thf(fact_236_inf__sup__aci_I6_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X2:A,Y:A,Z2:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ X2) @ Y)) @ Z2) = (((sup_sup @ A) @ X2) @ (((sup_sup @ A) @ Y) @ Z2))))))))).
% 2.78/1.58 thf(fact_237_inf__sup__aci_I5_J, axiom, (! [A:$tType]: (((lattice @ A) => ((sup_sup @ A) = (^ [X3:A,Y5:A]: ((((sup_sup @ A) @ Y5) @ X3)))))))).
% 2.78/1.58 thf(fact_238_UnE, axiom, (! [A:$tType,C2:A,A2:(set @ A),B2:(set @ A)]: (((((member @ A) @ C2) @ (((sup_sup @ (set @ A)) @ A2) @ B2)) => ((~ ((((member @ A) @ C2) @ A2))) => (((member @ A) @ C2) @ B2)))))).
% 2.78/1.58 thf(fact_239_UnI1, axiom, (! [A:$tType,C2:A,A2:(set @ A),B2:(set @ A)]: (((((member @ A) @ C2) @ A2) => (((member @ A) @ C2) @ (((sup_sup @ (set @ A)) @ A2) @ B2)))))).
% 2.78/1.58 thf(fact_240_UnI2, axiom, (! [A:$tType,C2:A,B2:(set @ A),A2:(set @ A)]: (((((member @ A) @ C2) @ B2) => (((member @ A) @ C2) @ (((sup_sup @ (set @ A)) @ A2) @ B2)))))).
% 2.78/1.58 thf(fact_241_bex__Un, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),P:(A > $o)]: (((? [X3:A]: (((((member @ A) @ X3) @ (((sup_sup @ (set @ A)) @ A2) @ B2)) & (P @ X3)))) = ((? [X3:A]: (((((member @ A) @ X3) @ A2) & (P @ X3)))) | (? [X3:A]: (((((member @ A) @ X3) @ B2) & (P @ X3))))))))).
% 2.78/1.58 thf(fact_242_ball__Un, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),P:(A > $o)]: (((! [X3:A]: (((((member @ A) @ X3) @ (((sup_sup @ (set @ A)) @ A2) @ B2)) => (P @ X3)))) = ((! [X3:A]: (((((member @ A) @ X3) @ A2) => (P @ X3)))) & (! [X3:A]: (((((member @ A) @ X3) @ B2) => (P @ X3))))))))).
% 2.78/1.58 thf(fact_243_Un__assoc, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C4:(set @ A)]: (((((sup_sup @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A2) @ B2)) @ C4) = (((sup_sup @ (set @ A)) @ A2) @ (((sup_sup @ (set @ A)) @ B2) @ C4)))))).
% 2.78/1.58 thf(fact_244_Un__absorb, axiom, (! [A:$tType,A2:(set @ A)]: (((((sup_sup @ (set @ A)) @ A2) @ A2) = A2)))).
% 2.78/1.58 thf(fact_245_Un__left__absorb, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((sup_sup @ (set @ A)) @ A2) @ (((sup_sup @ (set @ A)) @ A2) @ B2)) = (((sup_sup @ (set @ A)) @ A2) @ B2))))).
% 2.78/1.58 thf(fact_246_Un__left__commute, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C4:(set @ A)]: (((((sup_sup @ (set @ A)) @ A2) @ (((sup_sup @ (set @ A)) @ B2) @ C4)) = (((sup_sup @ (set @ A)) @ B2) @ (((sup_sup @ (set @ A)) @ A2) @ C4)))))).
% 2.78/1.58 thf(fact_247_sup__fun__def, axiom, (! [B:$tType,A:$tType]: (((semilattice_sup @ B) => ((sup_sup @ (A > B)) = (^ [F4:(A > B),G4:(A > B),X3:A]: ((((sup_sup @ B) @ (F4 @ X3)) @ (G4 @ X3))))))))).
% 2.78/1.58 thf(fact_248_boolean__algebra__cancel_Osup1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A2:A,K:A,A4:A,B3:A]: (((A2 = (((sup_sup @ A) @ K) @ A4)) => ((((sup_sup @ A) @ A2) @ B3) = (((sup_sup @ A) @ K) @ (((sup_sup @ A) @ A4) @ B3)))))))))).
% 2.78/1.58 thf(fact_249_boolean__algebra__cancel_Osup2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B2:A,K:A,B3:A,A4:A]: (((B2 = (((sup_sup @ A) @ K) @ B3)) => ((((sup_sup @ A) @ A4) @ B2) = (((sup_sup @ A) @ K) @ (((sup_sup @ A) @ A4) @ B3)))))))))).
% 2.78/1.58 thf(fact_250_sup_Oassoc, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A4:A,B3:A,C2:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ A4) @ B3)) @ C2) = (((sup_sup @ A) @ A4) @ (((sup_sup @ A) @ B3) @ C2))))))))).
% 2.78/1.58 thf(fact_251_sup__assoc, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X2:A,Y:A,Z2:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ X2) @ Y)) @ Z2) = (((sup_sup @ A) @ X2) @ (((sup_sup @ A) @ Y) @ Z2))))))))).
% 2.78/1.58 thf(fact_252_sup_Ocommute, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((sup_sup @ A) = (^ [A7:A,B7:A]: ((((sup_sup @ A) @ B7) @ A7)))))))).
% 2.78/1.58 thf(fact_253_sup__commute, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((sup_sup @ A) = (^ [X3:A,Y5:A]: ((((sup_sup @ A) @ Y5) @ X3)))))))).
% 2.78/1.58 thf(fact_254_sup_Oleft__commute, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B3:A,A4:A,C2:A]: (((((sup_sup @ A) @ B3) @ (((sup_sup @ A) @ A4) @ C2)) = (((sup_sup @ A) @ A4) @ (((sup_sup @ A) @ B3) @ C2))))))))).
% 2.78/1.58 thf(tcon_fun___Lattices_Osemilattice__sup, axiom, (! [A9:$tType,A10:$tType]: (((semilattice_sup @ A10) => (semilattice_sup @ (A9 > A10)))))).
% 2.78/1.58 thf(tcon_fun___Lattices_Oboolean__algebra, axiom, (! [A9:$tType,A10:$tType]: (((boolean_algebra @ A10) => (boolean_algebra @ (A9 > A10)))))).
% 2.78/1.58 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A9:$tType,A10:$tType]: (((preorder @ A10) => (preorder @ (A9 > A10)))))).
% 2.78/1.58 thf(tcon_fun___Lattices_Olattice, axiom, (! [A9:$tType,A10:$tType]: (((lattice @ A10) => (lattice @ (A9 > A10)))))).
% 2.78/1.58 thf(tcon_fun___Orderings_Oorder, axiom, (! [A9:$tType,A10:$tType]: (((order @ A10) => (order @ (A9 > A10)))))).
% 2.78/1.58 thf(tcon_fun___Orderings_Oord, axiom, (! [A9:$tType,A10:$tType]: (((ord @ A10) => (ord @ (A9 > A10)))))).
% 2.78/1.58 thf(tcon_fun___Groups_Ouminus, axiom, (! [A9:$tType,A10:$tType]: (((uminus @ A10) => (uminus @ (A9 > A10)))))).
% 2.78/1.58 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_1, axiom, (! [A9:$tType]: ((semilattice_sup @ (set @ A9))))).
% 2.78/1.58 thf(tcon_Set_Oset___Lattices_Oboolean__algebra_2, axiom, (! [A9:$tType]: ((boolean_algebra @ (set @ A9))))).
% 2.78/1.58 thf(tcon_Set_Oset___Orderings_Opreorder_3, axiom, (! [A9:$tType]: ((preorder @ (set @ A9))))).
% 2.78/1.58 thf(tcon_Set_Oset___Lattices_Olattice_4, axiom, (! [A9:$tType]: ((lattice @ (set @ A9))))).
% 2.78/1.58 thf(tcon_Set_Oset___Orderings_Oorder_5, axiom, (! [A9:$tType]: ((order @ (set @ A9))))).
% 2.78/1.58 thf(tcon_Set_Oset___Orderings_Oord_6, axiom, (! [A9:$tType]: ((ord @ (set @ A9))))).
% 2.78/1.58 thf(tcon_Set_Oset___Groups_Ouminus_7, axiom, (! [A9:$tType]: ((uminus @ (set @ A9))))).
% 2.78/1.58 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_8, axiom, (semilattice_sup @ $o)).
% 2.78/1.58 thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_9, axiom, (boolean_algebra @ $o)).
% 2.78/1.58 thf(tcon_HOL_Obool___Orderings_Opreorder_10, axiom, (preorder @ $o)).
% 2.78/1.58 thf(tcon_HOL_Obool___Orderings_Olinorder, axiom, (linorder @ $o)).
% 2.78/1.58 thf(tcon_HOL_Obool___Lattices_Olattice_11, axiom, (lattice @ $o)).
% 2.78/1.58 thf(tcon_HOL_Obool___Orderings_Oorder_12, axiom, (order @ $o)).
% 2.78/1.58 thf(tcon_HOL_Obool___Orderings_Oord_13, axiom, (ord @ $o)).
% 2.78/1.58 thf(tcon_HOL_Obool___Groups_Ouminus_14, axiom, (uminus @ $o)).
% 2.78/1.58 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add, axiom, (ordered_ab_group_add @ real)).
% 2.78/1.58 thf(tcon_Real_Oreal___Lattices_Osemilattice__sup_15, axiom, (semilattice_sup @ real)).
% 2.78/1.58 thf(tcon_Real_Oreal___Orderings_Opreorder_16, axiom, (preorder @ real)).
% 2.78/1.58 thf(tcon_Real_Oreal___Orderings_Olinorder_17, axiom, (linorder @ real)).
% 2.78/1.58 thf(tcon_Real_Oreal___Lattices_Olattice_18, axiom, (lattice @ real)).
% 2.78/1.58 thf(tcon_Real_Oreal___Groups_Ogroup__add, axiom, (group_add @ real)).
% 2.78/1.58 thf(tcon_Real_Oreal___Orderings_Oorder_19, axiom, (order @ real)).
% 2.78/1.58 thf(tcon_Real_Oreal___Orderings_Oord_20, axiom, (ord @ real)).
% 2.78/1.58 thf(tcon_Real_Oreal___Groups_Ouminus_21, axiom, (uminus @ real)).
% 2.78/1.58 thf(tcon_Product__Type_Oprod___Groups_Ogroup__add_22, axiom, (! [A9:$tType,A10:$tType]: ((((group_add @ A9) & (group_add @ A10)) => (group_add @ ((product_prod @ A9) @ A10)))))).
% 2.78/1.58 thf(tcon_Product__Type_Oprod___Groups_Ouminus_23, axiom, (! [A9:$tType,A10:$tType]: ((((uminus @ A9) & (uminus @ A10)) => (uminus @ ((product_prod @ A9) @ A10)))))).
% 2.78/1.58 thf(conj_0, conjecture, ((((member @ (variable > real)) @ nu2) @ ((uminus_uminus @ (set @ (variable > real))) @ (((denota1245701238me_sem @ i) @ ((the @ game) @ (((product_snd @ (set @ variable)) @ (option @ game)) @ (((uSubst95898988stappp @ sigma) @ ua) @ alpha)))) @ ((uminus_uminus @ (set @ (variable > real))) @ xa)))) = (((member @ (variable > real)) @ nu2) @ ((uminus_uminus @ (set @ (variable > real))) @ (((denota1245701238me_sem @ (((uSubst2091380086djoint @ sigma) @ i) @ omega2)) @ alpha) @ ((uminus_uminus @ (set @ (variable > real))) @ xa)))))).
% 2.78/1.58 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.lOk4iEszTx/DTF2DTF4484.p
% 2.78/1.58 ------------------------
% 2.98/1.63 ---- Cleaned THF ---
% 2.98/1.63 thf(ty_t_Denotational__Semantics_Ointerp,type,
% 2.98/1.63 denotational_interp: $tType ).
% 2.98/1.63
% 2.98/1.63 thf(ty_t_Product__Type_Oprod,type,
% 2.98/1.63 product_prod: $tType > $tType > $tType ).
% 2.98/1.63
% 2.98/1.63 thf(ty_t_Syntax_Ovariable,type,
% 2.98/1.63 variable: $tType ).
% 2.98/1.63
% 2.98/1.63 thf(ty_t_Option_Ooption,type,
% 2.98/1.63 option: $tType > $tType ).
% 2.98/1.63
% 2.98/1.63 thf(ty_t_Syntax_Ogame,type,
% 2.98/1.63 game: $tType ).
% 2.98/1.63
% 2.98/1.63 thf(ty_t_String_Ochar,type,
% 2.98/1.63 char: $tType ).
% 2.98/1.63
% 2.98/1.63 thf(ty_t_Syntax_Otrm,type,
% 2.98/1.63 trm: $tType ).
% 2.98/1.63
% 2.98/1.63 thf(ty_t_Syntax_Ofml,type,
% 2.98/1.63 fml: $tType ).
% 2.98/1.63
% 2.98/1.63 thf(ty_t_Real_Oreal,type,
% 2.98/1.63 real: $tType ).
% 2.98/1.63
% 2.98/1.63 thf(ty_t_Set_Oset,type,
% 2.98/1.63 set: $tType > $tType ).
% 2.98/1.63
% 2.98/1.63 thf(sy_cl_HOL_Otype,type,
% 2.98/1.63 type:
% 2.98/1.63 !>[A: $tType] : $o ).
% 2.98/1.63
% 2.98/1.63 thf(sy_cl_Groups_Ouminus,type,
% 2.98/1.63 uminus:
% 2.98/1.63 !>[A: $tType] : $o ).
% 2.98/1.63
% 2.98/1.63 thf(sy_cl_Orderings_Oord,type,
% 2.98/1.63 ord:
% 2.98/1.63 !>[A: $tType] : $o ).
% 2.98/1.63
% 2.98/1.63 thf(sy_cl_Orderings_Oorder,type,
% 2.98/1.63 order:
% 2.98/1.63 !>[A: $tType] : $o ).
% 2.98/1.63
% 2.98/1.63 thf(sy_cl_Groups_Ogroup__add,type,
% 2.98/1.63 group_add:
% 2.98/1.63 !>[A: $tType] : $o ).
% 2.98/1.63
% 2.98/1.63 thf(sy_cl_Lattices_Olattice,type,
% 2.98/1.63 lattice:
% 2.98/1.63 !>[A: $tType] : $o ).
% 2.98/1.63
% 2.98/1.63 thf(sy_cl_Orderings_Olinorder,type,
% 2.98/1.63 linorder:
% 2.98/1.63 !>[A: $tType] : $o ).
% 2.98/1.63
% 2.98/1.63 thf(sy_cl_Orderings_Opreorder,type,
% 2.98/1.63 preorder:
% 2.98/1.63 !>[A: $tType] : $o ).
% 2.98/1.63
% 2.98/1.63 thf(sy_cl_Lattices_Oboolean__algebra,type,
% 2.98/1.63 boolean_algebra:
% 2.98/1.63 !>[A: $tType] : $o ).
% 2.98/1.63
% 2.98/1.63 thf(sy_cl_Lattices_Osemilattice__sup,type,
% 2.98/1.63 semilattice_sup:
% 2.98/1.63 !>[A: $tType] : $o ).
% 2.98/1.63
% 2.98/1.63 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 2.98/1.63 ordered_ab_group_add:
% 2.98/1.63 !>[A: $tType] : $o ).
% 2.98/1.63
% 2.98/1.63 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
% 2.98/1.63 bNF_Ca1785829860lChain:
% 2.98/1.63 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
% 2.98/1.63
% 2.98/1.63 thf(sy_c_Denotational__Semantics_OUvariation,type,
% 2.98/1.63 denota1419872369iation: ( variable > real ) > ( variable > real ) > ( set @ variable ) > $o ).
% 2.98/1.63
% 2.98/1.63 thf(sy_c_Denotational__Semantics_Ogame__sem,type,
% 2.98/1.63 denota1245701238me_sem: denotational_interp > game > ( set @ ( variable > real ) ) > ( set @ ( variable > real ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(sy_c_Groups_Ouminus__class_Ouminus,type,
% 2.98/1.63 uminus_uminus:
% 2.98/1.63 !>[A: $tType] : ( A > A ) ).
% 2.98/1.63
% 2.98/1.63 thf(sy_c_Lattices_Osup__class_Osup,type,
% 2.98/1.63 sup_sup:
% 2.98/1.63 !>[A: $tType] : ( A > A > A ) ).
% 2.98/1.63
% 2.98/1.63 thf(sy_c_Option_Ooption_ONone,type,
% 2.98/1.63 none:
% 2.98/1.63 !>[A: $tType] : ( option @ A ) ).
% 2.98/1.63
% 2.98/1.63 thf(sy_c_Option_Ooption_Othe,type,
% 2.98/1.63 the:
% 2.98/1.63 !>[A: $tType] : ( ( option @ A ) > A ) ).
% 2.98/1.63
% 2.98/1.63 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.98/1.63 ord_less_eq:
% 2.98/1.63 !>[A: $tType] : ( A > A > $o ) ).
% 2.98/1.63
% 2.98/1.63 thf(sy_c_Product__Type_OPair,type,
% 2.98/1.63 product_Pair:
% 2.98/1.63 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 2.98/1.63 produc2004651681e_prod:
% 2.98/1.63 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 2.98/1.63
% 2.98/1.63 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 2.98/1.63 product_rec_prod:
% 2.98/1.63 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 2.98/1.63
% 2.98/1.63 thf(sy_c_Product__Type_Oprod_Ofst,type,
% 2.98/1.63 product_fst:
% 2.98/1.63 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
% 2.98/1.63
% 2.98/1.63 thf(sy_c_Product__Type_Oprod_Osnd,type,
% 2.98/1.63 product_snd:
% 2.98/1.63 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
% 2.98/1.63
% 2.98/1.63 thf(sy_c_Product__Type_Oprod_Oswap,type,
% 2.98/1.63 product_swap:
% 2.98/1.63 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(sy_c_Product__Type_Oscomp,type,
% 2.98/1.63 product_scomp:
% 2.98/1.63 !>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( A > ( product_prod @ B @ C ) ) > ( B > C > D ) > A > D ) ).
% 2.98/1.63
% 2.98/1.63 thf(sy_c_Set_OCollect,type,
% 2.98/1.63 collect:
% 2.98/1.63 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(sy_c_Syntax_OSkip,type,
% 2.98/1.63 skip: game ).
% 2.98/1.63
% 2.98/1.63 thf(sy_c_Syntax_Ogame_OChoice,type,
% 2.98/1.63 choice: game > game > game ).
% 2.98/1.63
% 2.98/1.63 thf(sy_c_Syntax_Ogame_ODual,type,
% 2.98/1.63 dual: game > game ).
% 2.98/1.63
% 2.98/1.63 thf(sy_c_Syntax_Ogame_OLoop,type,
% 2.98/1.63 loop: game > game ).
% 2.98/1.63
% 2.98/1.63 thf(sy_c_USubst__Mirabelle__nnnzepxswx_OChoiceo,type,
% 2.98/1.63 uSubst1976112797hoiceo: ( option @ game ) > ( option @ game ) > ( option @ game ) ).
% 2.98/1.63
% 2.98/1.63 thf(sy_c_USubst__Mirabelle__nnnzepxswx_ODualo,type,
% 2.98/1.63 uSubst739989314_Dualo: ( option @ game ) > ( option @ game ) ).
% 2.98/1.63
% 2.98/1.63 thf(sy_c_USubst__Mirabelle__nnnzepxswx_OLoopo,type,
% 2.98/1.63 uSubst993936602_Loopo: ( option @ game ) > ( option @ game ) ).
% 2.98/1.63
% 2.98/1.63 thf(sy_c_USubst__Mirabelle__nnnzepxswx_Oadjoint,type,
% 2.98/1.63 uSubst2091380086djoint: ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) > denotational_interp > ( variable > real ) > denotational_interp ).
% 2.98/1.63
% 2.98/1.63 thf(sy_c_USubst__Mirabelle__nnnzepxswx_Ousubstappp,type,
% 2.98/1.63 uSubst95898988stappp: ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) > ( set @ variable ) > game > ( product_prod @ ( set @ variable ) @ ( option @ game ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(sy_c_member,type,
% 2.98/1.63 member:
% 2.98/1.63 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.98/1.63
% 2.98/1.63 thf(sy_v_I,type,
% 2.98/1.63 i: denotational_interp ).
% 2.98/1.63
% 2.98/1.63 thf(sy_v_U,type,
% 2.98/1.63 u: set @ variable ).
% 2.98/1.63
% 2.98/1.63 thf(sy_v_Ua____,type,
% 2.98/1.63 ua: set @ variable ).
% 2.98/1.63
% 2.98/1.63 thf(sy_v_Xa____,type,
% 2.98/1.63 xa: set @ ( variable > real ) ).
% 2.98/1.63
% 2.98/1.63 thf(sy_v__092_060alpha_062_H_H____,type,
% 2.98/1.63 alpha: game ).
% 2.98/1.63
% 2.98/1.63 thf(sy_v__092_060nu_062,type,
% 2.98/1.63 nu: variable > real ).
% 2.98/1.63
% 2.98/1.63 thf(sy_v__092_060nu_062_H____,type,
% 2.98/1.63 nu2: variable > real ).
% 2.98/1.63
% 2.98/1.63 thf(sy_v__092_060omega_062,type,
% 2.98/1.63 omega: variable > real ).
% 2.98/1.63
% 2.98/1.63 thf(sy_v__092_060omega_062_H____,type,
% 2.98/1.63 omega2: variable > real ).
% 2.98/1.63
% 2.98/1.63 thf(sy_v__092_060sigma_062_H_H____,type,
% 2.98/1.63 sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_0__092_060open_062_092_060And_062X_O_A_I_092_060nu_062_A_092_060in_062_Agame__sem_AI_A_Ithe_A_Isnd_A_Iusubstappp_A_092_060sigma_062_AU_A_092_060alpha_062_J_J_J_AX_J_A_061_A_I_092_060nu_062_A_092_060in_062_Agame__sem_A_IUSubst__Mirabelle__nnnzepxswx_Oadjoint_A_092_060sigma_062_AI_A_092_060omega_062_J_A_092_060alpha_062_AX_J_092_060close_062,axiom,
% 2.98/1.63 ! [X: set @ ( variable > real )] :
% 2.98/1.63 ( ( member @ ( variable > real ) @ nu2 @ ( denota1245701238me_sem @ i @ ( the @ game @ ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ sigma @ ua @ alpha ) ) ) @ X ) )
% 2.98/1.63 = ( member @ ( variable > real ) @ nu2 @ ( denota1245701238me_sem @ ( uSubst2091380086djoint @ sigma @ i @ omega2 ) @ alpha @ X ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_1_IH_092_060alpha_062,axiom,
% 2.98/1.63 ! [X: set @ ( variable > real )] :
% 2.98/1.63 ( ( denota1419872369iation @ nu2 @ omega2 @ ua )
% 2.98/1.63 => ( ( member @ ( variable > real ) @ nu2 @ ( denota1245701238me_sem @ i @ ( the @ game @ ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ sigma @ ua @ alpha ) ) ) @ X ) )
% 2.98/1.63 = ( member @ ( variable > real ) @ nu2 @ ( denota1245701238me_sem @ ( uSubst2091380086djoint @ sigma @ i @ omega2 ) @ alpha @ X ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_2_uv,axiom,
% 2.98/1.63 denota1419872369iation @ nu2 @ omega2 @ ua ).
% 2.98/1.63
% 2.98/1.63 thf(fact_3_snd__uminus,axiom,
% 2.98/1.63 ! [A: $tType,B: $tType] :
% 2.98/1.63 ( ( ( uminus @ B )
% 2.98/1.63 & ( uminus @ A ) )
% 2.98/1.63 => ! [X2: product_prod @ B @ A] :
% 2.98/1.63 ( ( product_snd @ B @ A @ ( uminus_uminus @ ( product_prod @ B @ A ) @ X2 ) )
% 2.98/1.63 = ( uminus_uminus @ A @ ( product_snd @ B @ A @ X2 ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_4_def,axiom,
% 2.98/1.63 ( ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ sigma @ ua @ ( dual @ alpha ) ) )
% 2.98/1.63 != ( none @ game ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_5_ComplI,axiom,
% 2.98/1.63 ! [A: $tType,C2: A,A2: set @ A] :
% 2.98/1.63 ( ~ ( member @ A @ C2 @ A2 )
% 2.98/1.63 => ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A2 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_6_Compl__iff,axiom,
% 2.98/1.63 ! [A: $tType,C2: A,A2: set @ A] :
% 2.98/1.63 ( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
% 2.98/1.63 = ( ~ ( member @ A @ C2 @ A2 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_7_Compl__eq__Compl__iff,axiom,
% 2.98/1.63 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.98/1.63 ( ( ( uminus_uminus @ ( set @ A ) @ A2 )
% 2.98/1.63 = ( uminus_uminus @ ( set @ A ) @ B2 ) )
% 2.98/1.63 = ( A2 = B2 ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_8_verit__minus__simplify_I4_J,axiom,
% 2.98/1.63 ! [B: $tType] :
% 2.98/1.63 ( ( group_add @ B )
% 2.98/1.63 => ! [B3: B] :
% 2.98/1.63 ( ( uminus_uminus @ B @ ( uminus_uminus @ B @ B3 ) )
% 2.98/1.63 = B3 ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_9_uminus__apply,axiom,
% 2.98/1.63 ! [B: $tType,A: $tType] :
% 2.98/1.63 ( ( uminus @ B )
% 2.98/1.63 => ( ( uminus_uminus @ ( A > B ) )
% 2.98/1.63 = ( ^ [A3: A > B,X3: A] : ( uminus_uminus @ B @ ( A3 @ X3 ) ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_10_add_Oinverse__inverse,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( group_add @ A )
% 2.98/1.63 => ! [A4: A] :
% 2.98/1.63 ( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A4 ) )
% 2.98/1.63 = A4 ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_11_neg__equal__iff__equal,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( group_add @ A )
% 2.98/1.63 => ! [A4: A,B3: A] :
% 2.98/1.63 ( ( ( uminus_uminus @ A @ A4 )
% 2.98/1.63 = ( uminus_uminus @ A @ B3 ) )
% 2.98/1.63 = ( A4 = B3 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_12_double__compl,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( boolean_algebra @ A )
% 2.98/1.63 => ! [X2: A] :
% 2.98/1.63 ( ( uminus_uminus @ A @ ( uminus_uminus @ A @ X2 ) )
% 2.98/1.63 = X2 ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_13_vaouter,axiom,
% 2.98/1.63 denota1419872369iation @ nu @ omega @ u ).
% 2.98/1.63
% 2.98/1.63 thf(fact_14_compl__eq__compl__iff,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( boolean_algebra @ A )
% 2.98/1.63 => ! [X2: A,Y: A] :
% 2.98/1.63 ( ( ( uminus_uminus @ A @ X2 )
% 2.98/1.63 = ( uminus_uminus @ A @ Y ) )
% 2.98/1.63 = ( X2 = Y ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_15_usubstappp__det,axiom,
% 2.98/1.63 ! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,Alpha: game,V: set @ variable] :
% 2.98/1.63 ( ( ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) )
% 2.98/1.63 != ( none @ game ) )
% 2.98/1.63 => ( ( ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ V @ Alpha ) )
% 2.98/1.63 != ( none @ game ) )
% 2.98/1.63 => ( ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) )
% 2.98/1.63 = ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ V @ Alpha ) ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_16_minus__equation__iff,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( group_add @ A )
% 2.98/1.63 => ! [A4: A,B3: A] :
% 2.98/1.63 ( ( ( uminus_uminus @ A @ A4 )
% 2.98/1.63 = B3 )
% 2.98/1.63 = ( ( uminus_uminus @ A @ B3 )
% 2.98/1.63 = A4 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_17_equation__minus__iff,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( group_add @ A )
% 2.98/1.63 => ! [A4: A,B3: A] :
% 2.98/1.63 ( ( A4
% 2.98/1.63 = ( uminus_uminus @ A @ B3 ) )
% 2.98/1.63 = ( B3
% 2.98/1.63 = ( uminus_uminus @ A @ A4 ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_18_fun__Compl__def,axiom,
% 2.98/1.63 ! [B: $tType,A: $tType] :
% 2.98/1.63 ( ( uminus @ B )
% 2.98/1.63 => ( ( uminus_uminus @ ( A > B ) )
% 2.98/1.63 = ( ^ [A3: A > B,X3: A] : ( uminus_uminus @ B @ ( A3 @ X3 ) ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_19_verit__negate__coefficient_I3_J,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( ordered_ab_group_add @ A )
% 2.98/1.63 => ! [A4: A,B3: A] :
% 2.98/1.63 ( ( A4 = B3 )
% 2.98/1.63 => ( ( uminus_uminus @ A @ A4 )
% 2.98/1.63 = ( uminus_uminus @ A @ B3 ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_20_double__complement,axiom,
% 2.98/1.63 ! [A: $tType,A2: set @ A] :
% 2.98/1.63 ( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
% 2.98/1.63 = A2 ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_21_ComplD,axiom,
% 2.98/1.63 ! [A: $tType,C2: A,A2: set @ A] :
% 2.98/1.63 ( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
% 2.98/1.63 => ~ ( member @ A @ C2 @ A2 ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_22_game__sem_Osimps_I7_J,axiom,
% 2.98/1.63 ! [I: denotational_interp,Alpha: game] :
% 2.98/1.63 ( ( denota1245701238me_sem @ I @ ( dual @ Alpha ) )
% 2.98/1.63 = ( ^ [X4: set @ ( variable > real )] : ( uminus_uminus @ ( set @ ( variable > real ) ) @ ( denota1245701238me_sem @ I @ Alpha @ ( uminus_uminus @ ( set @ ( variable > real ) ) @ X4 ) ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_23_Dual_OIH,axiom,
% 2.98/1.63 ! [Nu: variable > real,Omega: variable > real,X: set @ ( variable > real )] :
% 2.98/1.63 ( ( ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ sigma @ ua @ alpha ) )
% 2.98/1.63 != ( none @ game ) )
% 2.98/1.63 => ( ( denota1419872369iation @ Nu @ Omega @ ua )
% 2.98/1.63 => ( ( member @ ( variable > real ) @ Nu @ ( denota1245701238me_sem @ i @ ( the @ game @ ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ sigma @ ua @ alpha ) ) ) @ X ) )
% 2.98/1.63 = ( member @ ( variable > real ) @ Nu @ ( denota1245701238me_sem @ ( uSubst2091380086djoint @ sigma @ i @ Omega ) @ alpha @ X ) ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_24_game_Oinject_I7_J,axiom,
% 2.98/1.63 ! [X7: game,Y7: game] :
% 2.98/1.63 ( ( ( dual @ X7 )
% 2.98/1.63 = ( dual @ Y7 ) )
% 2.98/1.63 = ( X7 = Y7 ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_25_Dual_Oprems_I2_J,axiom,
% 2.98/1.63 denota1419872369iation @ nu2 @ omega2 @ ua ).
% 2.98/1.63
% 2.98/1.63 thf(fact_26_option_Oexpand,axiom,
% 2.98/1.63 ! [A: $tType,Option: option @ A,Option2: option @ A] :
% 2.98/1.63 ( ( ( Option
% 2.98/1.63 = ( none @ A ) )
% 2.98/1.63 = ( Option2
% 2.98/1.63 = ( none @ A ) ) )
% 2.98/1.63 => ( ( ( Option
% 2.98/1.63 != ( none @ A ) )
% 2.98/1.63 => ( ( Option2
% 2.98/1.63 != ( none @ A ) )
% 2.98/1.63 => ( ( the @ A @ Option )
% 2.98/1.63 = ( the @ A @ Option2 ) ) ) )
% 2.98/1.63 => ( Option = Option2 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_27_usubst__game__loop,axiom,
% 2.98/1.63 ! [Nu: variable > real,Omega: variable > real,U: set @ variable,Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),Alpha: game,I: denotational_interp,X: set @ ( variable > real )] :
% 2.98/1.63 ( ( denota1419872369iation @ Nu @ Omega @ U )
% 2.98/1.63 => ( ! [Nu2: variable > real,Omega2: variable > real,X5: set @ ( variable > real )] :
% 2.98/1.63 ( ( denota1419872369iation @ Nu2 @ Omega2 @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) ) )
% 2.98/1.63 => ( ( ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) ) @ Alpha ) )
% 2.98/1.63 != ( none @ game ) )
% 2.98/1.63 => ( ( member @ ( variable > real ) @ Nu2 @ ( denota1245701238me_sem @ I @ ( the @ game @ ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) ) @ Alpha ) ) ) @ X5 ) )
% 2.98/1.63 = ( member @ ( variable > real ) @ Nu2 @ ( denota1245701238me_sem @ ( uSubst2091380086djoint @ Sigma @ I @ Omega2 ) @ Alpha @ X5 ) ) ) ) )
% 2.98/1.63 => ( ( ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ ( loop @ Alpha ) ) )
% 2.98/1.63 != ( none @ game ) )
% 2.98/1.63 => ( ( member @ ( variable > real ) @ Nu @ ( denota1245701238me_sem @ I @ ( the @ game @ ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ ( loop @ Alpha ) ) ) ) @ X ) )
% 2.98/1.63 = ( member @ ( variable > real ) @ Nu @ ( denota1245701238me_sem @ ( uSubst2091380086djoint @ Sigma @ I @ Omega ) @ ( loop @ Alpha ) @ X ) ) ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_28_Uvariation__def,axiom,
% 2.98/1.63 ( denota1419872369iation
% 2.98/1.63 = ( ^ [Nu3: variable > real,Nu4: variable > real,U2: set @ variable] :
% 2.98/1.63 ! [I2: variable] :
% 2.98/1.63 ( ~ ( member @ variable @ I2 @ U2 )
% 2.98/1.63 => ( ( Nu3 @ I2 )
% 2.98/1.63 = ( Nu4 @ I2 ) ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_29_Uvariation__sym,axiom,
% 2.98/1.63 ( denota1419872369iation
% 2.98/1.63 = ( ^ [Omega3: variable > real,Nu3: variable > real] : ( denota1419872369iation @ Nu3 @ Omega3 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_30_Uvariation__refl,axiom,
% 2.98/1.63 ! [Nu: variable > real,V: set @ variable] : ( denota1419872369iation @ Nu @ Nu @ V ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_31_Uvariation__sym__rel,axiom,
% 2.98/1.63 ! [Omega: variable > real,Nu: variable > real,U: set @ variable] :
% 2.98/1.63 ( ( denota1419872369iation @ Omega @ Nu @ U )
% 2.98/1.63 => ( denota1419872369iation @ Nu @ Omega @ U ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_32_usubstappp__choice__conv,axiom,
% 2.98/1.63 ! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,Alpha: game,Beta: game] :
% 2.98/1.63 ( ( ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ ( choice @ Alpha @ Beta ) ) )
% 2.98/1.63 != ( none @ game ) )
% 2.98/1.63 => ( ( ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) )
% 2.98/1.63 != ( none @ game ) )
% 2.98/1.63 & ( ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Beta ) )
% 2.98/1.63 != ( none @ game ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_33_game_Oinject_I6_J,axiom,
% 2.98/1.63 ! [X6: game,Y6: game] :
% 2.98/1.63 ( ( ( loop @ X6 )
% 2.98/1.63 = ( loop @ Y6 ) )
% 2.98/1.63 = ( X6 = Y6 ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_34_game_Oinject_I4_J,axiom,
% 2.98/1.63 ! [X41: game,X42: game,Y41: game,Y42: game] :
% 2.98/1.63 ( ( ( choice @ X41 @ X42 )
% 2.98/1.63 = ( choice @ Y41 @ Y42 ) )
% 2.98/1.63 = ( ( X41 = Y41 )
% 2.98/1.63 & ( X42 = Y42 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_35_fst__uminus,axiom,
% 2.98/1.63 ! [B: $tType,A: $tType] :
% 2.98/1.63 ( ( ( uminus @ A )
% 2.98/1.63 & ( uminus @ B ) )
% 2.98/1.63 => ! [X2: product_prod @ A @ B] :
% 2.98/1.63 ( ( product_fst @ A @ B @ ( uminus_uminus @ ( product_prod @ A @ B ) @ X2 ) )
% 2.98/1.63 = ( uminus_uminus @ A @ ( product_fst @ A @ B @ X2 ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_36_Uvariation__univ,axiom,
% 2.98/1.63 ! [Nu: variable > real,Nu5: variable > real] :
% 2.98/1.63 ( denota1419872369iation @ Nu @ Nu5
% 2.98/1.63 @ ( collect @ variable
% 2.98/1.63 @ ^ [X3: variable] : $true ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_37_game_Odistinct_I39_J,axiom,
% 2.98/1.63 ! [X41: game,X42: game,X6: game] :
% 2.98/1.63 ( ( choice @ X41 @ X42 )
% 2.98/1.63 != ( loop @ X6 ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_38_uminus__set__def,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( uminus_uminus @ ( set @ A ) )
% 2.98/1.63 = ( ^ [A3: set @ A] :
% 2.98/1.63 ( collect @ A
% 2.98/1.63 @ ( uminus_uminus @ ( A > $o )
% 2.98/1.63 @ ^ [X3: A] : ( member @ A @ X3 @ A3 ) ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_39_Collect__neg__eq,axiom,
% 2.98/1.63 ! [A: $tType,P: A > $o] :
% 2.98/1.63 ( ( collect @ A
% 2.98/1.63 @ ^ [X3: A] :
% 2.98/1.63 ~ ( P @ X3 ) )
% 2.98/1.63 = ( uminus_uminus @ ( set @ A ) @ ( collect @ A @ P ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_40_Compl__eq,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( uminus_uminus @ ( set @ A ) )
% 2.98/1.63 = ( ^ [A3: set @ A] :
% 2.98/1.63 ( collect @ A
% 2.98/1.63 @ ^ [X3: A] :
% 2.98/1.63 ~ ( member @ A @ X3 @ A3 ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_41_game_Odistinct_I51_J,axiom,
% 2.98/1.63 ! [X6: game,X7: game] :
% 2.98/1.63 ( ( loop @ X6 )
% 2.98/1.63 != ( dual @ X7 ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_42_game_Odistinct_I41_J,axiom,
% 2.98/1.63 ! [X41: game,X42: game,X7: game] :
% 2.98/1.63 ( ( choice @ X41 @ X42 )
% 2.98/1.63 != ( dual @ X7 ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_43_usubstappp__loop__conv,axiom,
% 2.98/1.63 ! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,Alpha: game] :
% 2.98/1.63 ( ( ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ ( loop @ Alpha ) ) )
% 2.98/1.63 != ( none @ game ) )
% 2.98/1.63 => ( ( ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) )
% 2.98/1.63 != ( none @ game ) )
% 2.98/1.63 & ( ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) ) @ Alpha ) )
% 2.98/1.63 != ( none @ game ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_44_mem__Collect__eq,axiom,
% 2.98/1.63 ! [A: $tType,A4: A,P: A > $o] :
% 2.98/1.63 ( ( member @ A @ A4 @ ( collect @ A @ P ) )
% 2.98/1.63 = ( P @ A4 ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_45_Collect__mem__eq,axiom,
% 2.98/1.63 ! [A: $tType,A2: set @ A] :
% 2.98/1.63 ( ( collect @ A
% 2.98/1.63 @ ^ [X3: A] : ( member @ A @ X3 @ A2 ) )
% 2.98/1.63 = A2 ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_46_Collect__cong,axiom,
% 2.98/1.63 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.98/1.63 ( ! [X8: A] :
% 2.98/1.63 ( ( P @ X8 )
% 2.98/1.63 = ( Q @ X8 ) )
% 2.98/1.63 => ( ( collect @ A @ P )
% 2.98/1.63 = ( collect @ A @ Q ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_47_ext,axiom,
% 2.98/1.63 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 2.98/1.63 ( ! [X8: A] :
% 2.98/1.63 ( ( F @ X8 )
% 2.98/1.63 = ( G @ X8 ) )
% 2.98/1.63 => ( F = G ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_48_skip__id,axiom,
% 2.98/1.63 ! [I: denotational_interp,X: set @ ( variable > real )] :
% 2.98/1.63 ( ( denota1245701238me_sem @ I @ skip @ X )
% 2.98/1.63 = X ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_49_prod__eqI,axiom,
% 2.98/1.63 ! [B: $tType,A: $tType,P2: product_prod @ A @ B,Q2: product_prod @ A @ B] :
% 2.98/1.63 ( ( ( product_fst @ A @ B @ P2 )
% 2.98/1.63 = ( product_fst @ A @ B @ Q2 ) )
% 2.98/1.63 => ( ( ( product_snd @ A @ B @ P2 )
% 2.98/1.63 = ( product_snd @ A @ B @ Q2 ) )
% 2.98/1.63 => ( P2 = Q2 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_50_exE__realizer_H,axiom,
% 2.98/1.63 ! [A: $tType,B: $tType,P: A > B > $o,P2: product_prod @ B @ A] :
% 2.98/1.63 ( ( P @ ( product_snd @ B @ A @ P2 ) @ ( product_fst @ B @ A @ P2 ) )
% 2.98/1.63 => ~ ! [X8: B,Y2: A] :
% 2.98/1.63 ~ ( P @ Y2 @ X8 ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_51_prod_Oexpand,axiom,
% 2.98/1.63 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod2: product_prod @ A @ B] :
% 2.98/1.63 ( ( ( ( product_fst @ A @ B @ Prod )
% 2.98/1.63 = ( product_fst @ A @ B @ Prod2 ) )
% 2.98/1.63 & ( ( product_snd @ A @ B @ Prod )
% 2.98/1.63 = ( product_snd @ A @ B @ Prod2 ) ) )
% 2.98/1.63 => ( Prod = Prod2 ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_52_prod__eq__iff,axiom,
% 2.98/1.63 ! [B: $tType,A: $tType] :
% 2.98/1.63 ( ( ^ [Y3: product_prod @ A @ B,Z: product_prod @ A @ B] : ( Y3 = Z ) )
% 2.98/1.63 = ( ^ [S: product_prod @ A @ B,T2: product_prod @ A @ B] :
% 2.98/1.63 ( ( ( product_fst @ A @ B @ S )
% 2.98/1.63 = ( product_fst @ A @ B @ T2 ) )
% 2.98/1.63 & ( ( product_snd @ A @ B @ S )
% 2.98/1.63 = ( product_snd @ A @ B @ T2 ) ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_53_uminus__prod__def,axiom,
% 2.98/1.63 ! [B: $tType,A: $tType] :
% 2.98/1.63 ( ( ( uminus @ A )
% 2.98/1.63 & ( uminus @ B ) )
% 2.98/1.63 => ( ( uminus_uminus @ ( product_prod @ A @ B ) )
% 2.98/1.63 = ( ^ [X3: product_prod @ A @ B] : ( product_Pair @ A @ B @ ( uminus_uminus @ A @ ( product_fst @ A @ B @ X3 ) ) @ ( uminus_uminus @ B @ ( product_snd @ A @ B @ X3 ) ) ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_54_usubstappp__antimon,axiom,
% 2.98/1.63 ! [V: set @ variable,U: set @ variable,Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),Alpha: game] :
% 2.98/1.63 ( ( ord_less_eq @ ( set @ variable ) @ V @ U )
% 2.98/1.63 => ( ( ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) )
% 2.98/1.63 != ( none @ game ) )
% 2.98/1.63 => ( ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) )
% 2.98/1.63 = ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ V @ Alpha ) ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_55_exists__diff,axiom,
% 2.98/1.63 ! [A: $tType,P: ( set @ A ) > $o] :
% 2.98/1.63 ( ( ? [S2: set @ A] : ( P @ ( uminus_uminus @ ( set @ A ) @ S2 ) ) )
% 2.98/1.63 = ( ? [X9: set @ A] : ( P @ X9 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_56_uminus__Pair,axiom,
% 2.98/1.63 ! [A: $tType,B: $tType] :
% 2.98/1.63 ( ( ( uminus @ B )
% 2.98/1.63 & ( uminus @ A ) )
% 2.98/1.63 => ! [A4: A,B3: B] :
% 2.98/1.63 ( ( uminus_uminus @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) )
% 2.98/1.63 = ( product_Pair @ A @ B @ ( uminus_uminus @ A @ A4 ) @ ( uminus_uminus @ B @ B3 ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_57_prod_Oinject,axiom,
% 2.98/1.63 ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y22: B] :
% 2.98/1.63 ( ( ( product_Pair @ A @ B @ X1 @ X22 )
% 2.98/1.63 = ( product_Pair @ A @ B @ Y1 @ Y22 ) )
% 2.98/1.63 = ( ( X1 = Y1 )
% 2.98/1.63 & ( X22 = Y22 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_58_old_Oprod_Oinject,axiom,
% 2.98/1.63 ! [A: $tType,B: $tType,A4: A,B3: B,A5: A,B4: B] :
% 2.98/1.63 ( ( ( product_Pair @ A @ B @ A4 @ B3 )
% 2.98/1.63 = ( product_Pair @ A @ B @ A5 @ B4 ) )
% 2.98/1.63 = ( ( A4 = A5 )
% 2.98/1.63 & ( B3 = B4 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_59_subset__antisym,axiom,
% 2.98/1.63 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.98/1.63 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.98/1.63 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 2.98/1.63 => ( A2 = B2 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_60_subsetI,axiom,
% 2.98/1.63 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.98/1.63 ( ! [X8: A] :
% 2.98/1.63 ( ( member @ A @ X8 @ A2 )
% 2.98/1.63 => ( member @ A @ X8 @ B2 ) )
% 2.98/1.63 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_61_compl__le__compl__iff,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( boolean_algebra @ A )
% 2.98/1.63 => ! [X2: A,Y: A] :
% 2.98/1.63 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X2 ) @ ( uminus_uminus @ A @ Y ) )
% 2.98/1.63 = ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_62_neg__le__iff__le,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( ordered_ab_group_add @ A )
% 2.98/1.63 => ! [B3: A,A4: A] :
% 2.98/1.63 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( uminus_uminus @ A @ A4 ) )
% 2.98/1.63 = ( ord_less_eq @ A @ A4 @ B3 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_63_Compl__subset__Compl__iff,axiom,
% 2.98/1.63 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.98/1.63 ( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) @ ( uminus_uminus @ ( set @ A ) @ B2 ) )
% 2.98/1.63 = ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_64_Compl__anti__mono,axiom,
% 2.98/1.63 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.98/1.63 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.98/1.63 => ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B2 ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_65_prod_Ocollapse,axiom,
% 2.98/1.63 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 2.98/1.63 ( ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) )
% 2.98/1.63 = Prod ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_66_surj__pair,axiom,
% 2.98/1.63 ! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
% 2.98/1.63 ? [X8: A,Y2: B] :
% 2.98/1.63 ( P2
% 2.98/1.63 = ( product_Pair @ A @ B @ X8 @ Y2 ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_67_prod__cases,axiom,
% 2.98/1.63 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
% 2.98/1.63 ( ! [A6: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A6 @ B5 ) )
% 2.98/1.63 => ( P @ P2 ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_68_Pair__inject,axiom,
% 2.98/1.63 ! [A: $tType,B: $tType,A4: A,B3: B,A5: A,B4: B] :
% 2.98/1.63 ( ( ( product_Pair @ A @ B @ A4 @ B3 )
% 2.98/1.63 = ( product_Pair @ A @ B @ A5 @ B4 ) )
% 2.98/1.63 => ~ ( ( A4 = A5 )
% 2.98/1.63 => ( B3 != B4 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_69_prod__cases3,axiom,
% 2.98/1.63 ! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
% 2.98/1.63 ~ ! [A6: A,B5: B,C3: C] :
% 2.98/1.63 ( Y
% 2.98/1.63 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A6 @ ( product_Pair @ B @ C @ B5 @ C3 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_70_prod__cases4,axiom,
% 2.98/1.63 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 2.98/1.63 ~ ! [A6: A,B5: B,C3: C,D2: D] :
% 2.98/1.63 ( Y
% 2.98/1.63 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_71_prod__cases5,axiom,
% 2.98/1.63 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 2.98/1.63 ~ ! [A6: A,B5: B,C3: C,D2: D,E2: E] :
% 2.98/1.63 ( Y
% 2.98/1.63 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_72_prod__cases6,axiom,
% 2.98/1.63 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) )] :
% 2.98/1.63 ~ ! [A6: A,B5: B,C3: C,D2: D,E2: E,F3: F2] :
% 2.98/1.63 ( Y
% 2.98/1.63 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F2 ) @ D2 @ ( product_Pair @ E @ F2 @ E2 @ F3 ) ) ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_73_prod__cases7,axiom,
% 2.98/1.63 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F2: $tType,G2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) )] :
% 2.98/1.63 ~ ! [A6: A,B5: B,C3: C,D2: D,E2: E,F3: F2,G3: G2] :
% 2.98/1.63 ( Y
% 2.98/1.63 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F2 @ G2 ) @ E2 @ ( product_Pair @ F2 @ G2 @ F3 @ G3 ) ) ) ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_74_prod__induct3,axiom,
% 2.98/1.63 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ C )] :
% 2.98/1.63 ( ! [A6: A,B5: B,C3: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A6 @ ( product_Pair @ B @ C @ B5 @ C3 ) ) )
% 2.98/1.63 => ( P @ X2 ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_75_prod__induct4,axiom,
% 2.98/1.63 ! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 2.98/1.63 ( ! [A6: A,B5: B,C3: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) )
% 2.98/1.63 => ( P @ X2 ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_76_prod__induct5,axiom,
% 2.98/1.63 ! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 2.98/1.63 ( ! [A6: A,B5: B,C3: C,D2: D,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
% 2.98/1.63 => ( P @ X2 ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_77_prod__induct6,axiom,
% 2.98/1.63 ! [F2: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) )] :
% 2.98/1.63 ( ! [A6: A,B5: B,C3: C,D2: D,E2: E,F3: F2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F2 ) @ D2 @ ( product_Pair @ E @ F2 @ E2 @ F3 ) ) ) ) ) )
% 2.98/1.63 => ( P @ X2 ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_78_prod__induct7,axiom,
% 2.98/1.63 ! [G2: $tType,F2: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) )] :
% 2.98/1.63 ( ! [A6: A,B5: B,C3: C,D2: D,E2: E,F3: F2,G3: G2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F2 @ G2 ) @ E2 @ ( product_Pair @ F2 @ G2 @ F3 @ G3 ) ) ) ) ) ) )
% 2.98/1.63 => ( P @ X2 ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_79_old_Oprod_Oexhaust,axiom,
% 2.98/1.63 ! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
% 2.98/1.63 ~ ! [A6: A,B5: B] :
% 2.98/1.63 ( Y
% 2.98/1.63 != ( product_Pair @ A @ B @ A6 @ B5 ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_80_old_Oprod_Oinducts,axiom,
% 2.98/1.63 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 2.98/1.63 ( ! [A6: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A6 @ B5 ) )
% 2.98/1.63 => ( P @ Prod ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_81_verit__la__disequality,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( linorder @ A )
% 2.98/1.63 => ! [A4: A,B3: A] :
% 2.98/1.63 ( ( A4 = B3 )
% 2.98/1.63 | ~ ( ord_less_eq @ A @ A4 @ B3 )
% 2.98/1.63 | ~ ( ord_less_eq @ A @ B3 @ A4 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_82_Collect__mono__iff,axiom,
% 2.98/1.63 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.98/1.63 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 2.98/1.63 = ( ! [X3: A] :
% 2.98/1.63 ( ( P @ X3 )
% 2.98/1.63 => ( Q @ X3 ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_83_set__eq__subset,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( ^ [Y3: set @ A,Z: set @ A] : ( Y3 = Z ) )
% 2.98/1.63 = ( ^ [A3: set @ A,B6: set @ A] :
% 2.98/1.63 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
% 2.98/1.63 & ( ord_less_eq @ ( set @ A ) @ B6 @ A3 ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_84_subset__trans,axiom,
% 2.98/1.63 ! [A: $tType,A2: set @ A,B2: set @ A,C4: set @ A] :
% 2.98/1.63 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.98/1.63 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C4 )
% 2.98/1.63 => ( ord_less_eq @ ( set @ A ) @ A2 @ C4 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_85_Collect__mono,axiom,
% 2.98/1.63 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.98/1.63 ( ! [X8: A] :
% 2.98/1.63 ( ( P @ X8 )
% 2.98/1.63 => ( Q @ X8 ) )
% 2.98/1.63 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_86_subset__refl,axiom,
% 2.98/1.63 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_87_subset__iff,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( ord_less_eq @ ( set @ A ) )
% 2.98/1.63 = ( ^ [A3: set @ A,B6: set @ A] :
% 2.98/1.63 ! [T2: A] :
% 2.98/1.63 ( ( member @ A @ T2 @ A3 )
% 2.98/1.63 => ( member @ A @ T2 @ B6 ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_88_equalityD2,axiom,
% 2.98/1.63 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.98/1.63 ( ( A2 = B2 )
% 2.98/1.63 => ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_89_equalityD1,axiom,
% 2.98/1.63 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.98/1.63 ( ( A2 = B2 )
% 2.98/1.63 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_90_subset__eq,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( ord_less_eq @ ( set @ A ) )
% 2.98/1.63 = ( ^ [A3: set @ A,B6: set @ A] :
% 2.98/1.63 ! [X3: A] :
% 2.98/1.63 ( ( member @ A @ X3 @ A3 )
% 2.98/1.63 => ( member @ A @ X3 @ B6 ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_91_equalityE,axiom,
% 2.98/1.63 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.98/1.63 ( ( A2 = B2 )
% 2.98/1.63 => ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.98/1.63 => ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_92_subsetD,axiom,
% 2.98/1.63 ! [A: $tType,A2: set @ A,B2: set @ A,C2: A] :
% 2.98/1.63 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.98/1.63 => ( ( member @ A @ C2 @ A2 )
% 2.98/1.63 => ( member @ A @ C2 @ B2 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_93_in__mono,axiom,
% 2.98/1.63 ! [A: $tType,A2: set @ A,B2: set @ A,X2: A] :
% 2.98/1.63 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.98/1.63 => ( ( member @ A @ X2 @ A2 )
% 2.98/1.63 => ( member @ A @ X2 @ B2 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_94_Collect__subset,axiom,
% 2.98/1.63 ! [A: $tType,A2: set @ A,P: A > $o] :
% 2.98/1.63 ( ord_less_eq @ ( set @ A )
% 2.98/1.63 @ ( collect @ A
% 2.98/1.63 @ ^ [X3: A] :
% 2.98/1.63 ( ( member @ A @ X3 @ A2 )
% 2.98/1.63 & ( P @ X3 ) ) )
% 2.98/1.63 @ A2 ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_95_fst__eqD,axiom,
% 2.98/1.63 ! [B: $tType,A: $tType,X2: A,Y: B,A4: A] :
% 2.98/1.63 ( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X2 @ Y ) )
% 2.98/1.63 = A4 )
% 2.98/1.63 => ( X2 = A4 ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_96_fst__conv,axiom,
% 2.98/1.63 ! [B: $tType,A: $tType,X1: A,X22: B] :
% 2.98/1.63 ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X22 ) )
% 2.98/1.63 = X1 ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_97_snd__conv,axiom,
% 2.98/1.63 ! [Aa: $tType,A: $tType,X1: Aa,X22: A] :
% 2.98/1.63 ( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X22 ) )
% 2.98/1.63 = X22 ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_98_snd__eqD,axiom,
% 2.98/1.63 ! [B: $tType,A: $tType,X2: B,Y: A,A4: A] :
% 2.98/1.63 ( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X2 @ Y ) )
% 2.98/1.63 = A4 )
% 2.98/1.63 => ( Y = A4 ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_99_le__imp__neg__le,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( ordered_ab_group_add @ A )
% 2.98/1.63 => ! [A4: A,B3: A] :
% 2.98/1.63 ( ( ord_less_eq @ A @ A4 @ B3 )
% 2.98/1.63 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( uminus_uminus @ A @ A4 ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_100_minus__le__iff,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( ordered_ab_group_add @ A )
% 2.98/1.63 => ! [A4: A,B3: A] :
% 2.98/1.63 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A4 ) @ B3 )
% 2.98/1.63 = ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ A4 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_101_le__minus__iff,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( ordered_ab_group_add @ A )
% 2.98/1.63 => ! [A4: A,B3: A] :
% 2.98/1.63 ( ( ord_less_eq @ A @ A4 @ ( uminus_uminus @ A @ B3 ) )
% 2.98/1.63 = ( ord_less_eq @ A @ B3 @ ( uminus_uminus @ A @ A4 ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_102_compl__le__swap2,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( boolean_algebra @ A )
% 2.98/1.63 => ! [Y: A,X2: A] :
% 2.98/1.63 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ X2 )
% 2.98/1.63 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ X2 ) @ Y ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_103_compl__le__swap1,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( boolean_algebra @ A )
% 2.98/1.63 => ! [Y: A,X2: A] :
% 2.98/1.63 ( ( ord_less_eq @ A @ Y @ ( uminus_uminus @ A @ X2 ) )
% 2.98/1.63 => ( ord_less_eq @ A @ X2 @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_104_compl__mono,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( boolean_algebra @ A )
% 2.98/1.63 => ! [X2: A,Y: A] :
% 2.98/1.63 ( ( ord_less_eq @ A @ X2 @ Y )
% 2.98/1.63 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X2 ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_105_fst__pair,axiom,
% 2.98/1.63 ! [B: $tType,A: $tType,A4: A,B3: B] :
% 2.98/1.63 ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ A4 @ B3 ) )
% 2.98/1.63 = A4 ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_106_snd__pair,axiom,
% 2.98/1.63 ! [B: $tType,A: $tType,A4: B,B3: A] :
% 2.98/1.63 ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ A4 @ B3 ) )
% 2.98/1.63 = B3 ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_107_surjective__pairing,axiom,
% 2.98/1.63 ! [B: $tType,A: $tType,T3: product_prod @ A @ B] :
% 2.98/1.63 ( T3
% 2.98/1.63 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T3 ) @ ( product_snd @ A @ B @ T3 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_108_prod_Oexhaust__sel,axiom,
% 2.98/1.63 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 2.98/1.63 ( Prod
% 2.98/1.63 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_109_conjI__realizer,axiom,
% 2.98/1.63 ! [A: $tType,B: $tType,P: A > $o,P2: A,Q: B > $o,Q2: B] :
% 2.98/1.63 ( ( P @ P2 )
% 2.98/1.63 => ( ( Q @ Q2 )
% 2.98/1.63 => ( ( P @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q2 ) ) )
% 2.98/1.63 & ( Q @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q2 ) ) ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_110_exI__realizer,axiom,
% 2.98/1.63 ! [B: $tType,A: $tType,P: A > B > $o,Y: A,X2: B] :
% 2.98/1.63 ( ( P @ Y @ X2 )
% 2.98/1.63 => ( P @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X2 @ Y ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X2 @ Y ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_111_Uvariation__mon,axiom,
% 2.98/1.63 ! [U: set @ variable,V: set @ variable,Omega: variable > real,Nu: variable > real] :
% 2.98/1.63 ( ( ord_less_eq @ ( set @ variable ) @ U @ V )
% 2.98/1.63 => ( ( denota1419872369iation @ Omega @ Nu @ U )
% 2.98/1.63 => ( denota1419872369iation @ Omega @ Nu @ V ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_112_usubstappp__fst__mon,axiom,
% 2.98/1.63 ! [U: set @ variable,V: set @ variable,Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),Alpha: game] :
% 2.98/1.63 ( ( ord_less_eq @ ( set @ variable ) @ U @ V )
% 2.98/1.63 => ( ord_less_eq @ ( set @ variable ) @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) ) @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ V @ Alpha ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_113_usubst__taboos__mon,axiom,
% 2.98/1.63 ! [U: set @ variable,Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),Alpha: game] : ( ord_less_eq @ ( set @ variable ) @ U @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_114_old_Oprod_Orec,axiom,
% 2.98/1.63 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A4: A,B3: B] :
% 2.98/1.63 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A4 @ B3 ) )
% 2.98/1.63 = ( F1 @ A4 @ B3 ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_115_usubstappp__dual,axiom,
% 2.98/1.63 ! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,Alpha: game] :
% 2.98/1.63 ( ( uSubst95898988stappp @ Sigma @ U @ ( dual @ Alpha ) )
% 2.98/1.63 = ( product_Pair @ ( set @ variable ) @ ( option @ game ) @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) ) @ ( uSubst739989314_Dualo @ ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_116_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 2.98/1.63 ! [B: $tType,A: $tType,P: A > B > $o,X2: A,Y: B,A4: product_prod @ A @ B] :
% 2.98/1.63 ( ( P @ X2 @ Y )
% 2.98/1.63 => ( ( A4
% 2.98/1.63 = ( product_Pair @ A @ B @ X2 @ Y ) )
% 2.98/1.63 => ( P @ ( product_fst @ A @ B @ A4 ) @ ( product_snd @ A @ B @ A4 ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_117_usubstappp_Osimps_I7_J,axiom,
% 2.98/1.63 ! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,Alpha: game] :
% 2.98/1.63 ( ( uSubst95898988stappp @ Sigma @ U @ ( dual @ Alpha ) )
% 2.98/1.63 = ( product_Pair @ ( set @ variable ) @ ( option @ game ) @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) ) @ ( uSubst739989314_Dualo @ ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_118_usubstappp_Osimps_I6_J,axiom,
% 2.98/1.63 ! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,Alpha: game] :
% 2.98/1.63 ( ( uSubst95898988stappp @ Sigma @ U @ ( loop @ Alpha ) )
% 2.98/1.63 = ( product_Pair @ ( set @ variable ) @ ( option @ game ) @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) ) @ ( uSubst993936602_Loopo @ ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) ) @ Alpha ) ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_119_usubstappp__loop,axiom,
% 2.98/1.63 ! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,Alpha: game] :
% 2.98/1.63 ( ( uSubst95898988stappp @ Sigma @ U @ ( loop @ Alpha ) )
% 2.98/1.63 = ( product_Pair @ ( set @ variable ) @ ( option @ game ) @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) ) @ ( uSubst993936602_Loopo @ ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) ) @ Alpha ) ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_120_order__refl,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( preorder @ A )
% 2.98/1.63 => ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_121_monotone,axiom,
% 2.98/1.63 ! [X: set @ ( variable > real ),Y4: set @ ( variable > real ),I: denotational_interp,Alpha: game] :
% 2.98/1.63 ( ( ord_less_eq @ ( set @ ( variable > real ) ) @ X @ Y4 )
% 2.98/1.63 => ( ord_less_eq @ ( set @ ( variable > real ) ) @ ( denota1245701238me_sem @ I @ Alpha @ X ) @ ( denota1245701238me_sem @ I @ Alpha @ Y4 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_122_less__eq__set__def,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( ord_less_eq @ ( set @ A ) )
% 2.98/1.63 = ( ^ [A3: set @ A,B6: set @ A] :
% 2.98/1.63 ( ord_less_eq @ ( A > $o )
% 2.98/1.63 @ ^ [X3: A] : ( member @ A @ X3 @ A3 )
% 2.98/1.63 @ ^ [X3: A] : ( member @ A @ X3 @ B6 ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_123_Loopo__undef,axiom,
% 2.98/1.63 ! [Alpha: option @ game] :
% 2.98/1.63 ( ( ( uSubst993936602_Loopo @ Alpha )
% 2.98/1.63 = ( none @ game ) )
% 2.98/1.63 = ( Alpha
% 2.98/1.63 = ( none @ game ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_124_Dualo__undef,axiom,
% 2.98/1.63 ! [Alpha: option @ game] :
% 2.98/1.63 ( ( ( uSubst739989314_Dualo @ Alpha )
% 2.98/1.63 = ( none @ game ) )
% 2.98/1.63 = ( Alpha
% 2.98/1.63 = ( none @ game ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_125_Loopo_Osimps_I2_J,axiom,
% 2.98/1.63 ( ( uSubst993936602_Loopo @ ( none @ game ) )
% 2.98/1.63 = ( none @ game ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_126_Dualo_Osimps_I2_J,axiom,
% 2.98/1.63 ( ( uSubst739989314_Dualo @ ( none @ game ) )
% 2.98/1.63 = ( none @ game ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_127_dual__order_Oantisym,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( order @ A )
% 2.98/1.63 => ! [B3: A,A4: A] :
% 2.98/1.63 ( ( ord_less_eq @ A @ B3 @ A4 )
% 2.98/1.63 => ( ( ord_less_eq @ A @ A4 @ B3 )
% 2.98/1.63 => ( A4 = B3 ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_128_dual__order_Oeq__iff,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( order @ A )
% 2.98/1.63 => ( ( ^ [Y3: A,Z: A] : ( Y3 = Z ) )
% 2.98/1.63 = ( ^ [A7: A,B7: A] :
% 2.98/1.63 ( ( ord_less_eq @ A @ B7 @ A7 )
% 2.98/1.63 & ( ord_less_eq @ A @ A7 @ B7 ) ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_129_dual__order_Otrans,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( order @ A )
% 2.98/1.63 => ! [B3: A,A4: A,C2: A] :
% 2.98/1.63 ( ( ord_less_eq @ A @ B3 @ A4 )
% 2.98/1.63 => ( ( ord_less_eq @ A @ C2 @ B3 )
% 2.98/1.63 => ( ord_less_eq @ A @ C2 @ A4 ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_130_linorder__wlog,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( linorder @ A )
% 2.98/1.63 => ! [P: A > A > $o,A4: A,B3: A] :
% 2.98/1.63 ( ! [A6: A,B5: A] :
% 2.98/1.63 ( ( ord_less_eq @ A @ A6 @ B5 )
% 2.98/1.63 => ( P @ A6 @ B5 ) )
% 2.98/1.63 => ( ! [A6: A,B5: A] :
% 2.98/1.63 ( ( P @ B5 @ A6 )
% 2.98/1.63 => ( P @ A6 @ B5 ) )
% 2.98/1.63 => ( P @ A4 @ B3 ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_131_dual__order_Orefl,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( order @ A )
% 2.98/1.63 => ! [A4: A] : ( ord_less_eq @ A @ A4 @ A4 ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_132_order__trans,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( preorder @ A )
% 2.98/1.63 => ! [X2: A,Y: A,Z2: A] :
% 2.98/1.63 ( ( ord_less_eq @ A @ X2 @ Y )
% 2.98/1.63 => ( ( ord_less_eq @ A @ Y @ Z2 )
% 2.98/1.63 => ( ord_less_eq @ A @ X2 @ Z2 ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_133_order__class_Oorder_Oantisym,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( order @ A )
% 2.98/1.63 => ! [A4: A,B3: A] :
% 2.98/1.63 ( ( ord_less_eq @ A @ A4 @ B3 )
% 2.98/1.63 => ( ( ord_less_eq @ A @ B3 @ A4 )
% 2.98/1.63 => ( A4 = B3 ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_134_ord__le__eq__trans,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( ord @ A )
% 2.98/1.63 => ! [A4: A,B3: A,C2: A] :
% 2.98/1.63 ( ( ord_less_eq @ A @ A4 @ B3 )
% 2.98/1.63 => ( ( B3 = C2 )
% 2.98/1.63 => ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_135_ord__eq__le__trans,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( ord @ A )
% 2.98/1.63 => ! [A4: A,B3: A,C2: A] :
% 2.98/1.63 ( ( A4 = B3 )
% 2.98/1.63 => ( ( ord_less_eq @ A @ B3 @ C2 )
% 2.98/1.63 => ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_136_order__class_Oorder_Oeq__iff,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( order @ A )
% 2.98/1.63 => ( ( ^ [Y3: A,Z: A] : ( Y3 = Z ) )
% 2.98/1.63 = ( ^ [A7: A,B7: A] :
% 2.98/1.63 ( ( ord_less_eq @ A @ A7 @ B7 )
% 2.98/1.63 & ( ord_less_eq @ A @ B7 @ A7 ) ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_137_antisym__conv,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( order @ A )
% 2.98/1.63 => ! [Y: A,X2: A] :
% 2.98/1.63 ( ( ord_less_eq @ A @ Y @ X2 )
% 2.98/1.63 => ( ( ord_less_eq @ A @ X2 @ Y )
% 2.98/1.63 = ( X2 = Y ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_138_le__cases3,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( linorder @ A )
% 2.98/1.63 => ! [X2: A,Y: A,Z2: A] :
% 2.98/1.63 ( ( ( ord_less_eq @ A @ X2 @ Y )
% 2.98/1.63 => ~ ( ord_less_eq @ A @ Y @ Z2 ) )
% 2.98/1.63 => ( ( ( ord_less_eq @ A @ Y @ X2 )
% 2.98/1.63 => ~ ( ord_less_eq @ A @ X2 @ Z2 ) )
% 2.98/1.63 => ( ( ( ord_less_eq @ A @ X2 @ Z2 )
% 2.98/1.63 => ~ ( ord_less_eq @ A @ Z2 @ Y ) )
% 2.98/1.63 => ( ( ( ord_less_eq @ A @ Z2 @ Y )
% 2.98/1.63 => ~ ( ord_less_eq @ A @ Y @ X2 ) )
% 2.98/1.63 => ( ( ( ord_less_eq @ A @ Y @ Z2 )
% 2.98/1.63 => ~ ( ord_less_eq @ A @ Z2 @ X2 ) )
% 2.98/1.63 => ~ ( ( ord_less_eq @ A @ Z2 @ X2 )
% 2.98/1.63 => ~ ( ord_less_eq @ A @ X2 @ Y ) ) ) ) ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_139_order_Otrans,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( order @ A )
% 2.98/1.63 => ! [A4: A,B3: A,C2: A] :
% 2.98/1.63 ( ( ord_less_eq @ A @ A4 @ B3 )
% 2.98/1.63 => ( ( ord_less_eq @ A @ B3 @ C2 )
% 2.98/1.63 => ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_140_le__cases,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( linorder @ A )
% 2.98/1.63 => ! [X2: A,Y: A] :
% 2.98/1.63 ( ~ ( ord_less_eq @ A @ X2 @ Y )
% 2.98/1.63 => ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_141_eq__refl,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( preorder @ A )
% 2.98/1.63 => ! [X2: A,Y: A] :
% 2.98/1.63 ( ( X2 = Y )
% 2.98/1.63 => ( ord_less_eq @ A @ X2 @ Y ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_142_linear,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( linorder @ A )
% 2.98/1.63 => ! [X2: A,Y: A] :
% 2.98/1.63 ( ( ord_less_eq @ A @ X2 @ Y )
% 2.98/1.63 | ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_143_antisym,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( order @ A )
% 2.98/1.63 => ! [X2: A,Y: A] :
% 2.98/1.63 ( ( ord_less_eq @ A @ X2 @ Y )
% 2.98/1.63 => ( ( ord_less_eq @ A @ Y @ X2 )
% 2.98/1.63 => ( X2 = Y ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_144_eq__iff,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( order @ A )
% 2.98/1.63 => ( ( ^ [Y3: A,Z: A] : ( Y3 = Z ) )
% 2.98/1.63 = ( ^ [X3: A,Y5: A] :
% 2.98/1.63 ( ( ord_less_eq @ A @ X3 @ Y5 )
% 2.98/1.63 & ( ord_less_eq @ A @ Y5 @ X3 ) ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_145_ord__le__eq__subst,axiom,
% 2.98/1.63 ! [A: $tType,B: $tType] :
% 2.98/1.63 ( ( ( ord @ B )
% 2.98/1.63 & ( ord @ A ) )
% 2.98/1.63 => ! [A4: A,B3: A,F: A > B,C2: B] :
% 2.98/1.63 ( ( ord_less_eq @ A @ A4 @ B3 )
% 2.98/1.63 => ( ( ( F @ B3 )
% 2.98/1.63 = C2 )
% 2.98/1.63 => ( ! [X8: A,Y2: A] :
% 2.98/1.63 ( ( ord_less_eq @ A @ X8 @ Y2 )
% 2.98/1.63 => ( ord_less_eq @ B @ ( F @ X8 ) @ ( F @ Y2 ) ) )
% 2.98/1.63 => ( ord_less_eq @ B @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_146_ord__eq__le__subst,axiom,
% 2.98/1.63 ! [A: $tType,B: $tType] :
% 2.98/1.63 ( ( ( ord @ B )
% 2.98/1.63 & ( ord @ A ) )
% 2.98/1.63 => ! [A4: A,F: B > A,B3: B,C2: B] :
% 2.98/1.63 ( ( A4
% 2.98/1.63 = ( F @ B3 ) )
% 2.98/1.63 => ( ( ord_less_eq @ B @ B3 @ C2 )
% 2.98/1.63 => ( ! [X8: B,Y2: B] :
% 2.98/1.63 ( ( ord_less_eq @ B @ X8 @ Y2 )
% 2.98/1.63 => ( ord_less_eq @ A @ ( F @ X8 ) @ ( F @ Y2 ) ) )
% 2.98/1.63 => ( ord_less_eq @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_147_order__subst2,axiom,
% 2.98/1.63 ! [A: $tType,C: $tType] :
% 2.98/1.63 ( ( ( order @ C )
% 2.98/1.63 & ( order @ A ) )
% 2.98/1.63 => ! [A4: A,B3: A,F: A > C,C2: C] :
% 2.98/1.63 ( ( ord_less_eq @ A @ A4 @ B3 )
% 2.98/1.63 => ( ( ord_less_eq @ C @ ( F @ B3 ) @ C2 )
% 2.98/1.63 => ( ! [X8: A,Y2: A] :
% 2.98/1.63 ( ( ord_less_eq @ A @ X8 @ Y2 )
% 2.98/1.63 => ( ord_less_eq @ C @ ( F @ X8 ) @ ( F @ Y2 ) ) )
% 2.98/1.63 => ( ord_less_eq @ C @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_148_order__subst1,axiom,
% 2.98/1.63 ! [A: $tType,B: $tType] :
% 2.98/1.63 ( ( ( order @ B )
% 2.98/1.63 & ( order @ A ) )
% 2.98/1.63 => ! [A4: A,F: B > A,B3: B,C2: B] :
% 2.98/1.63 ( ( ord_less_eq @ A @ A4 @ ( F @ B3 ) )
% 2.98/1.63 => ( ( ord_less_eq @ B @ B3 @ C2 )
% 2.98/1.63 => ( ! [X8: B,Y2: B] :
% 2.98/1.63 ( ( ord_less_eq @ B @ X8 @ Y2 )
% 2.98/1.63 => ( ord_less_eq @ A @ ( F @ X8 ) @ ( F @ Y2 ) ) )
% 2.98/1.63 => ( ord_less_eq @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_149_le__fun__def,axiom,
% 2.98/1.63 ! [B: $tType,A: $tType] :
% 2.98/1.63 ( ( ord @ B )
% 2.98/1.63 => ( ( ord_less_eq @ ( A > B ) )
% 2.98/1.63 = ( ^ [F4: A > B,G4: A > B] :
% 2.98/1.63 ! [X3: A] : ( ord_less_eq @ B @ ( F4 @ X3 ) @ ( G4 @ X3 ) ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_150_le__funI,axiom,
% 2.98/1.63 ! [B: $tType,A: $tType] :
% 2.98/1.63 ( ( ord @ B )
% 2.98/1.63 => ! [F: A > B,G: A > B] :
% 2.98/1.63 ( ! [X8: A] : ( ord_less_eq @ B @ ( F @ X8 ) @ ( G @ X8 ) )
% 2.98/1.63 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_151_le__funE,axiom,
% 2.98/1.63 ! [B: $tType,A: $tType] :
% 2.98/1.63 ( ( ord @ B )
% 2.98/1.63 => ! [F: A > B,G: A > B,X2: A] :
% 2.98/1.63 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 2.98/1.63 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_152_le__funD,axiom,
% 2.98/1.63 ! [B: $tType,A: $tType] :
% 2.98/1.63 ( ( ord @ B )
% 2.98/1.63 => ! [F: A > B,G: A > B,X2: A] :
% 2.98/1.63 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 2.98/1.63 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_153_subset__CollectI,axiom,
% 2.98/1.63 ! [A: $tType,B2: set @ A,A2: set @ A,Q: A > $o,P: A > $o] :
% 2.98/1.63 ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 2.98/1.63 => ( ! [X8: A] :
% 2.98/1.63 ( ( member @ A @ X8 @ B2 )
% 2.98/1.63 => ( ( Q @ X8 )
% 2.98/1.63 => ( P @ X8 ) ) )
% 2.98/1.63 => ( ord_less_eq @ ( set @ A )
% 2.98/1.63 @ ( collect @ A
% 2.98/1.63 @ ^ [X3: A] :
% 2.98/1.63 ( ( member @ A @ X3 @ B2 )
% 2.98/1.63 & ( Q @ X3 ) ) )
% 2.98/1.63 @ ( collect @ A
% 2.98/1.63 @ ^ [X3: A] :
% 2.98/1.63 ( ( member @ A @ X3 @ A2 )
% 2.98/1.63 & ( P @ X3 ) ) ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_154_subset__Collect__iff,axiom,
% 2.98/1.63 ! [A: $tType,B2: set @ A,A2: set @ A,P: A > $o] :
% 2.98/1.63 ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 2.98/1.63 => ( ( ord_less_eq @ ( set @ A ) @ B2
% 2.98/1.63 @ ( collect @ A
% 2.98/1.63 @ ^ [X3: A] :
% 2.98/1.63 ( ( member @ A @ X3 @ A2 )
% 2.98/1.63 & ( P @ X3 ) ) ) )
% 2.98/1.63 = ( ! [X3: A] :
% 2.98/1.63 ( ( member @ A @ X3 @ B2 )
% 2.98/1.63 => ( P @ X3 ) ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_155_sndI,axiom,
% 2.98/1.63 ! [A: $tType,B: $tType,X2: product_prod @ A @ B,Y: A,Z2: B] :
% 2.98/1.63 ( ( X2
% 2.98/1.63 = ( product_Pair @ A @ B @ Y @ Z2 ) )
% 2.98/1.63 => ( ( product_snd @ A @ B @ X2 )
% 2.98/1.63 = Z2 ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_156_eq__snd__iff,axiom,
% 2.98/1.63 ! [A: $tType,B: $tType,B3: A,P2: product_prod @ B @ A] :
% 2.98/1.63 ( ( B3
% 2.98/1.63 = ( product_snd @ B @ A @ P2 ) )
% 2.98/1.63 = ( ? [A7: B] :
% 2.98/1.63 ( P2
% 2.98/1.63 = ( product_Pair @ B @ A @ A7 @ B3 ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_157_eq__fst__iff,axiom,
% 2.98/1.63 ! [A: $tType,B: $tType,A4: A,P2: product_prod @ A @ B] :
% 2.98/1.63 ( ( A4
% 2.98/1.63 = ( product_fst @ A @ B @ P2 ) )
% 2.98/1.63 = ( ? [B7: B] :
% 2.98/1.63 ( P2
% 2.98/1.63 = ( product_Pair @ A @ B @ A4 @ B7 ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_158_fstI,axiom,
% 2.98/1.63 ! [B: $tType,A: $tType,X2: product_prod @ A @ B,Y: A,Z2: B] :
% 2.98/1.63 ( ( X2
% 2.98/1.63 = ( product_Pair @ A @ B @ Y @ Z2 ) )
% 2.98/1.63 => ( ( product_fst @ A @ B @ X2 )
% 2.98/1.63 = Y ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_159_predicate1I,axiom,
% 2.98/1.63 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.98/1.63 ( ! [X8: A] :
% 2.98/1.63 ( ( P @ X8 )
% 2.98/1.63 => ( Q @ X8 ) )
% 2.98/1.63 => ( ord_less_eq @ ( A > $o ) @ P @ Q ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_160_predicate1D,axiom,
% 2.98/1.63 ! [A: $tType,P: A > $o,Q: A > $o,X2: A] :
% 2.98/1.63 ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
% 2.98/1.63 => ( ( P @ X2 )
% 2.98/1.63 => ( Q @ X2 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_161_rev__predicate1D,axiom,
% 2.98/1.63 ! [A: $tType,P: A > $o,X2: A,Q: A > $o] :
% 2.98/1.63 ( ( P @ X2 )
% 2.98/1.63 => ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
% 2.98/1.63 => ( Q @ X2 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_162_pred__subset__eq,axiom,
% 2.98/1.63 ! [A: $tType,R: set @ A,S3: set @ A] :
% 2.98/1.63 ( ( ord_less_eq @ ( A > $o )
% 2.98/1.63 @ ^ [X3: A] : ( member @ A @ X3 @ R )
% 2.98/1.63 @ ^ [X3: A] : ( member @ A @ X3 @ S3 ) )
% 2.98/1.63 = ( ord_less_eq @ ( set @ A ) @ R @ S3 ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_163_scomp__unfold,axiom,
% 2.98/1.63 ! [D: $tType,C: $tType,B: $tType,A: $tType] :
% 2.98/1.63 ( ( product_scomp @ A @ B @ C @ D )
% 2.98/1.63 = ( ^ [F4: A > ( product_prod @ B @ C ),G4: B > C > D,X3: A] : ( G4 @ ( product_fst @ B @ C @ ( F4 @ X3 ) ) @ ( product_snd @ B @ C @ ( F4 @ X3 ) ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_164_Collect__restrict,axiom,
% 2.98/1.63 ! [A: $tType,X: set @ A,P: A > $o] :
% 2.98/1.63 ( ord_less_eq @ ( set @ A )
% 2.98/1.63 @ ( collect @ A
% 2.98/1.63 @ ^ [X3: A] :
% 2.98/1.63 ( ( member @ A @ X3 @ X )
% 2.98/1.63 & ( P @ X3 ) ) )
% 2.98/1.63 @ X ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_165_scomp__Pair,axiom,
% 2.98/1.63 ! [C: $tType,B: $tType,A: $tType,X2: A > ( product_prod @ B @ C )] :
% 2.98/1.63 ( ( product_scomp @ A @ B @ C @ ( product_prod @ B @ C ) @ X2 @ ( product_Pair @ B @ C ) )
% 2.98/1.63 = X2 ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_166_scomp__scomp,axiom,
% 2.98/1.63 ! [A: $tType,C: $tType,D: $tType,B: $tType,F2: $tType,E: $tType,F: A > ( product_prod @ E @ F2 ),G: E > F2 > ( product_prod @ C @ D ),H: C > D > B] :
% 2.98/1.63 ( ( product_scomp @ A @ C @ D @ B @ ( product_scomp @ A @ E @ F2 @ ( product_prod @ C @ D ) @ F @ G ) @ H )
% 2.98/1.63 = ( product_scomp @ A @ E @ F2 @ B @ F
% 2.98/1.63 @ ^ [X3: E] : ( product_scomp @ F2 @ C @ D @ B @ ( G @ X3 ) @ H ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_167_Pair__scomp,axiom,
% 2.98/1.63 ! [A: $tType,B: $tType,C: $tType,X2: C,F: C > A > B] :
% 2.98/1.63 ( ( product_scomp @ A @ C @ A @ B @ ( product_Pair @ C @ A @ X2 ) @ F )
% 2.98/1.63 = ( F @ X2 ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_168_subrelI,axiom,
% 2.98/1.63 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S4: set @ ( product_prod @ A @ B )] :
% 2.98/1.63 ( ! [X8: A,Y2: B] :
% 2.98/1.63 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X8 @ Y2 ) @ R2 )
% 2.98/1.63 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X8 @ Y2 ) @ S4 ) )
% 2.98/1.63 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S4 ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_169_ssubst__Pair__rhs,axiom,
% 2.98/1.63 ! [B: $tType,A: $tType,R2: A,S4: B,R: set @ ( product_prod @ A @ B ),S5: B] :
% 2.98/1.63 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S4 ) @ R )
% 2.98/1.63 => ( ( S5 = S4 )
% 2.98/1.63 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S5 ) @ R ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_170_pred__subset__eq2,axiom,
% 2.98/1.63 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
% 2.98/1.63 ( ( ord_less_eq @ ( A > B > $o )
% 2.98/1.63 @ ^ [X3: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y5 ) @ R )
% 2.98/1.63 @ ^ [X3: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y5 ) @ S3 ) )
% 2.98/1.63 = ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S3 ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_171_pred__equals__eq2,axiom,
% 2.98/1.63 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
% 2.98/1.63 ( ( ( ^ [X3: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y5 ) @ R ) )
% 2.98/1.63 = ( ^ [X3: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y5 ) @ S3 ) ) )
% 2.98/1.63 = ( R = S3 ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_172_prop__restrict,axiom,
% 2.98/1.63 ! [A: $tType,X2: A,Z3: set @ A,X: set @ A,P: A > $o] :
% 2.98/1.63 ( ( member @ A @ X2 @ Z3 )
% 2.98/1.63 => ( ( ord_less_eq @ ( set @ A ) @ Z3
% 2.98/1.63 @ ( collect @ A
% 2.98/1.63 @ ^ [X3: A] :
% 2.98/1.63 ( ( member @ A @ X3 @ X )
% 2.98/1.63 & ( P @ X3 ) ) ) )
% 2.98/1.63 => ( P @ X2 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_173_conj__subset__def,axiom,
% 2.98/1.63 ! [A: $tType,A2: set @ A,P: A > $o,Q: A > $o] :
% 2.98/1.63 ( ( ord_less_eq @ ( set @ A ) @ A2
% 2.98/1.63 @ ( collect @ A
% 2.98/1.63 @ ^ [X3: A] :
% 2.98/1.63 ( ( P @ X3 )
% 2.98/1.63 & ( Q @ X3 ) ) ) )
% 2.98/1.63 = ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( collect @ A @ P ) )
% 2.98/1.63 & ( ord_less_eq @ ( set @ A ) @ A2 @ ( collect @ A @ Q ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_174_internal__case__prod__conv,axiom,
% 2.98/1.63 ! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A4: B,B3: C] :
% 2.98/1.63 ( ( produc2004651681e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A4 @ B3 ) )
% 2.98/1.63 = ( C2 @ A4 @ B3 ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_175_prod_Oswap__def,axiom,
% 2.98/1.63 ! [B: $tType,A: $tType] :
% 2.98/1.63 ( ( product_swap @ A @ B )
% 2.98/1.63 = ( ^ [P3: product_prod @ A @ B] : ( product_Pair @ B @ A @ ( product_snd @ A @ B @ P3 ) @ ( product_fst @ A @ B @ P3 ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_176_predicate2I,axiom,
% 2.98/1.63 ! [B: $tType,A: $tType,P: A > B > $o,Q: A > B > $o] :
% 2.98/1.63 ( ! [X8: A,Y2: B] :
% 2.98/1.63 ( ( P @ X8 @ Y2 )
% 2.98/1.63 => ( Q @ X8 @ Y2 ) )
% 2.98/1.63 => ( ord_less_eq @ ( A > B > $o ) @ P @ Q ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_177_swap__swap,axiom,
% 2.98/1.63 ! [B: $tType,A: $tType,P2: product_prod @ A @ B] :
% 2.98/1.63 ( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P2 ) )
% 2.98/1.63 = P2 ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_178_swap__simp,axiom,
% 2.98/1.63 ! [A: $tType,B: $tType,X2: B,Y: A] :
% 2.98/1.63 ( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X2 @ Y ) )
% 2.98/1.63 = ( product_Pair @ A @ B @ Y @ X2 ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_179_snd__swap,axiom,
% 2.98/1.63 ! [B: $tType,A: $tType,X2: product_prod @ A @ B] :
% 2.98/1.63 ( ( product_snd @ B @ A @ ( product_swap @ A @ B @ X2 ) )
% 2.98/1.63 = ( product_fst @ A @ B @ X2 ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_180_fst__swap,axiom,
% 2.98/1.63 ! [A: $tType,B: $tType,X2: product_prod @ B @ A] :
% 2.98/1.63 ( ( product_fst @ A @ B @ ( product_swap @ B @ A @ X2 ) )
% 2.98/1.63 = ( product_snd @ B @ A @ X2 ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_181_rev__predicate2D,axiom,
% 2.98/1.63 ! [A: $tType,B: $tType,P: A > B > $o,X2: A,Y: B,Q: A > B > $o] :
% 2.98/1.63 ( ( P @ X2 @ Y )
% 2.98/1.63 => ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
% 2.98/1.63 => ( Q @ X2 @ Y ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_182_predicate2D,axiom,
% 2.98/1.63 ! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,X2: A,Y: B] :
% 2.98/1.63 ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
% 2.98/1.63 => ( ( P @ X2 @ Y )
% 2.98/1.63 => ( Q @ X2 @ Y ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_183_refl__ge__eq,axiom,
% 2.98/1.63 ! [A: $tType,R: A > A > $o] :
% 2.98/1.63 ( ! [X8: A] : ( R @ X8 @ X8 )
% 2.98/1.63 => ( ord_less_eq @ ( A > A > $o )
% 2.98/1.63 @ ^ [Y3: A,Z: A] : ( Y3 = Z )
% 2.98/1.63 @ R ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_184_ge__eq__refl,axiom,
% 2.98/1.63 ! [A: $tType,R: A > A > $o,X2: A] :
% 2.98/1.63 ( ( ord_less_eq @ ( A > A > $o )
% 2.98/1.63 @ ^ [Y3: A,Z: A] : ( Y3 = Z )
% 2.98/1.63 @ R )
% 2.98/1.63 => ( R @ X2 @ X2 ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_185_eq__subset,axiom,
% 2.98/1.63 ! [A: $tType,P: A > A > $o] :
% 2.98/1.63 ( ord_less_eq @ ( A > A > $o )
% 2.98/1.63 @ ^ [Y3: A,Z: A] : ( Y3 = Z )
% 2.98/1.63 @ ^ [A7: A,B7: A] :
% 2.98/1.63 ( ( P @ A7 @ B7 )
% 2.98/1.63 | ( A7 = B7 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_186_usubstappp__choice,axiom,
% 2.98/1.63 ! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,Alpha: game,Beta: game] :
% 2.98/1.63 ( ( uSubst95898988stappp @ Sigma @ U @ ( choice @ Alpha @ Beta ) )
% 2.98/1.63 = ( product_Pair @ ( set @ variable ) @ ( option @ game ) @ ( sup_sup @ ( set @ variable ) @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) ) @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Beta ) ) ) @ ( uSubst1976112797hoiceo @ ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) ) @ ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Beta ) ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_187_relChain__def,axiom,
% 2.98/1.63 ! [B: $tType,A: $tType] :
% 2.98/1.63 ( ( ord @ B )
% 2.98/1.63 => ( ( bNF_Ca1785829860lChain @ A @ B )
% 2.98/1.63 = ( ^ [R3: set @ ( product_prod @ A @ A ),As: A > B] :
% 2.98/1.63 ! [I2: A,J: A] :
% 2.98/1.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I2 @ J ) @ R3 )
% 2.98/1.63 => ( ord_less_eq @ B @ ( As @ I2 ) @ ( As @ J ) ) ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_188_sup__apply,axiom,
% 2.98/1.63 ! [B: $tType,A: $tType] :
% 2.98/1.63 ( ( semilattice_sup @ B )
% 2.98/1.63 => ( ( sup_sup @ ( A > B ) )
% 2.98/1.63 = ( ^ [F4: A > B,G4: A > B,X3: A] : ( sup_sup @ B @ ( F4 @ X3 ) @ ( G4 @ X3 ) ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_189_sup_Oidem,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( semilattice_sup @ A )
% 2.98/1.63 => ! [A4: A] :
% 2.98/1.63 ( ( sup_sup @ A @ A4 @ A4 )
% 2.98/1.63 = A4 ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_190_sup__idem,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( semilattice_sup @ A )
% 2.98/1.63 => ! [X2: A] :
% 2.98/1.63 ( ( sup_sup @ A @ X2 @ X2 )
% 2.98/1.63 = X2 ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_191_sup_Oleft__idem,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( semilattice_sup @ A )
% 2.98/1.63 => ! [A4: A,B3: A] :
% 2.98/1.63 ( ( sup_sup @ A @ A4 @ ( sup_sup @ A @ A4 @ B3 ) )
% 2.98/1.63 = ( sup_sup @ A @ A4 @ B3 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_192_sup__left__idem,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( semilattice_sup @ A )
% 2.98/1.63 => ! [X2: A,Y: A] :
% 2.98/1.63 ( ( sup_sup @ A @ X2 @ ( sup_sup @ A @ X2 @ Y ) )
% 2.98/1.63 = ( sup_sup @ A @ X2 @ Y ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_193_sup_Oright__idem,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( semilattice_sup @ A )
% 2.98/1.63 => ! [A4: A,B3: A] :
% 2.98/1.63 ( ( sup_sup @ A @ ( sup_sup @ A @ A4 @ B3 ) @ B3 )
% 2.98/1.63 = ( sup_sup @ A @ A4 @ B3 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_194_UnCI,axiom,
% 2.98/1.63 ! [A: $tType,C2: A,B2: set @ A,A2: set @ A] :
% 2.98/1.63 ( ( ~ ( member @ A @ C2 @ B2 )
% 2.98/1.63 => ( member @ A @ C2 @ A2 ) )
% 2.98/1.63 => ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_195_sup_Obounded__iff,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( semilattice_sup @ A )
% 2.98/1.63 => ! [B3: A,C2: A,A4: A] :
% 2.98/1.63 ( ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C2 ) @ A4 )
% 2.98/1.63 = ( ( ord_less_eq @ A @ B3 @ A4 )
% 2.98/1.63 & ( ord_less_eq @ A @ C2 @ A4 ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_196_le__sup__iff,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( semilattice_sup @ A )
% 2.98/1.63 => ! [X2: A,Y: A,Z2: A] :
% 2.98/1.63 ( ( ord_less_eq @ A @ ( sup_sup @ A @ X2 @ Y ) @ Z2 )
% 2.98/1.63 = ( ( ord_less_eq @ A @ X2 @ Z2 )
% 2.98/1.63 & ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_197_Un__subset__iff,axiom,
% 2.98/1.63 ! [A: $tType,A2: set @ A,B2: set @ A,C4: set @ A] :
% 2.98/1.63 ( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ C4 )
% 2.98/1.63 = ( ( ord_less_eq @ ( set @ A ) @ A2 @ C4 )
% 2.98/1.63 & ( ord_less_eq @ ( set @ A ) @ B2 @ C4 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_198_subset__Un__eq,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( ord_less_eq @ ( set @ A ) )
% 2.98/1.63 = ( ^ [A3: set @ A,B6: set @ A] :
% 2.98/1.63 ( ( sup_sup @ ( set @ A ) @ A3 @ B6 )
% 2.98/1.63 = B6 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_199_subset__UnE,axiom,
% 2.98/1.63 ! [A: $tType,C4: set @ A,A2: set @ A,B2: set @ A] :
% 2.98/1.63 ( ( ord_less_eq @ ( set @ A ) @ C4 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
% 2.98/1.63 => ~ ! [A8: set @ A] :
% 2.98/1.63 ( ( ord_less_eq @ ( set @ A ) @ A8 @ A2 )
% 2.98/1.63 => ! [B8: set @ A] :
% 2.98/1.63 ( ( ord_less_eq @ ( set @ A ) @ B8 @ B2 )
% 2.98/1.63 => ( C4
% 2.98/1.63 != ( sup_sup @ ( set @ A ) @ A8 @ B8 ) ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_200_Un__absorb2,axiom,
% 2.98/1.63 ! [A: $tType,B2: set @ A,A2: set @ A] :
% 2.98/1.63 ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 2.98/1.63 => ( ( sup_sup @ ( set @ A ) @ A2 @ B2 )
% 2.98/1.63 = A2 ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_201_Un__absorb1,axiom,
% 2.98/1.63 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.98/1.63 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.98/1.63 => ( ( sup_sup @ ( set @ A ) @ A2 @ B2 )
% 2.98/1.63 = B2 ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_202_Un__upper2,axiom,
% 2.98/1.63 ! [A: $tType,B2: set @ A,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ B2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_203_Un__upper1,axiom,
% 2.98/1.63 ! [A: $tType,A2: set @ A,B2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_204_Un__least,axiom,
% 2.98/1.63 ! [A: $tType,A2: set @ A,C4: set @ A,B2: set @ A] :
% 2.98/1.63 ( ( ord_less_eq @ ( set @ A ) @ A2 @ C4 )
% 2.98/1.63 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C4 )
% 2.98/1.63 => ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ C4 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_205_Un__mono,axiom,
% 2.98/1.63 ! [A: $tType,A2: set @ A,C4: set @ A,B2: set @ A,D3: set @ A] :
% 2.98/1.63 ( ( ord_less_eq @ ( set @ A ) @ A2 @ C4 )
% 2.98/1.63 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ D3 )
% 2.98/1.63 => ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ ( sup_sup @ ( set @ A ) @ C4 @ D3 ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_206_sup_OcoboundedI2,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( semilattice_sup @ A )
% 2.98/1.63 => ! [C2: A,B3: A,A4: A] :
% 2.98/1.63 ( ( ord_less_eq @ A @ C2 @ B3 )
% 2.98/1.63 => ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A4 @ B3 ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_207_sup_OcoboundedI1,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( semilattice_sup @ A )
% 2.98/1.63 => ! [C2: A,A4: A,B3: A] :
% 2.98/1.63 ( ( ord_less_eq @ A @ C2 @ A4 )
% 2.98/1.63 => ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A4 @ B3 ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_208_sup_Oabsorb__iff2,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( semilattice_sup @ A )
% 2.98/1.63 => ( ( ord_less_eq @ A )
% 2.98/1.63 = ( ^ [A7: A,B7: A] :
% 2.98/1.63 ( ( sup_sup @ A @ A7 @ B7 )
% 2.98/1.63 = B7 ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_209_sup_Oabsorb__iff1,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( semilattice_sup @ A )
% 2.98/1.63 => ( ( ord_less_eq @ A )
% 2.98/1.63 = ( ^ [B7: A,A7: A] :
% 2.98/1.63 ( ( sup_sup @ A @ A7 @ B7 )
% 2.98/1.63 = A7 ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_210_sup_Ocobounded2,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( semilattice_sup @ A )
% 2.98/1.63 => ! [B3: A,A4: A] : ( ord_less_eq @ A @ B3 @ ( sup_sup @ A @ A4 @ B3 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_211_sup_Ocobounded1,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( semilattice_sup @ A )
% 2.98/1.63 => ! [A4: A,B3: A] : ( ord_less_eq @ A @ A4 @ ( sup_sup @ A @ A4 @ B3 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_212_sup_Oorder__iff,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( semilattice_sup @ A )
% 2.98/1.63 => ( ( ord_less_eq @ A )
% 2.98/1.63 = ( ^ [B7: A,A7: A] :
% 2.98/1.63 ( A7
% 2.98/1.63 = ( sup_sup @ A @ A7 @ B7 ) ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_213_sup_OboundedI,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( semilattice_sup @ A )
% 2.98/1.63 => ! [B3: A,A4: A,C2: A] :
% 2.98/1.63 ( ( ord_less_eq @ A @ B3 @ A4 )
% 2.98/1.63 => ( ( ord_less_eq @ A @ C2 @ A4 )
% 2.98/1.63 => ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C2 ) @ A4 ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_214_sup_OboundedE,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( semilattice_sup @ A )
% 2.98/1.63 => ! [B3: A,C2: A,A4: A] :
% 2.98/1.63 ( ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C2 ) @ A4 )
% 2.98/1.63 => ~ ( ( ord_less_eq @ A @ B3 @ A4 )
% 2.98/1.63 => ~ ( ord_less_eq @ A @ C2 @ A4 ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_215_sup__absorb2,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( semilattice_sup @ A )
% 2.98/1.63 => ! [X2: A,Y: A] :
% 2.98/1.63 ( ( ord_less_eq @ A @ X2 @ Y )
% 2.98/1.63 => ( ( sup_sup @ A @ X2 @ Y )
% 2.98/1.63 = Y ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_216_sup__absorb1,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( semilattice_sup @ A )
% 2.98/1.63 => ! [Y: A,X2: A] :
% 2.98/1.63 ( ( ord_less_eq @ A @ Y @ X2 )
% 2.98/1.63 => ( ( sup_sup @ A @ X2 @ Y )
% 2.98/1.63 = X2 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_217_sup_Oabsorb2,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( semilattice_sup @ A )
% 2.98/1.63 => ! [A4: A,B3: A] :
% 2.98/1.63 ( ( ord_less_eq @ A @ A4 @ B3 )
% 2.98/1.63 => ( ( sup_sup @ A @ A4 @ B3 )
% 2.98/1.63 = B3 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_218_sup_Oabsorb1,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( semilattice_sup @ A )
% 2.98/1.63 => ! [B3: A,A4: A] :
% 2.98/1.63 ( ( ord_less_eq @ A @ B3 @ A4 )
% 2.98/1.63 => ( ( sup_sup @ A @ A4 @ B3 )
% 2.98/1.63 = A4 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_219_sup__unique,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( semilattice_sup @ A )
% 2.98/1.63 => ! [F: A > A > A,X2: A,Y: A] :
% 2.98/1.63 ( ! [X8: A,Y2: A] : ( ord_less_eq @ A @ X8 @ ( F @ X8 @ Y2 ) )
% 2.98/1.63 => ( ! [X8: A,Y2: A] : ( ord_less_eq @ A @ Y2 @ ( F @ X8 @ Y2 ) )
% 2.98/1.63 => ( ! [X8: A,Y2: A,Z4: A] :
% 2.98/1.63 ( ( ord_less_eq @ A @ Y2 @ X8 )
% 2.98/1.63 => ( ( ord_less_eq @ A @ Z4 @ X8 )
% 2.98/1.63 => ( ord_less_eq @ A @ ( F @ Y2 @ Z4 ) @ X8 ) ) )
% 2.98/1.63 => ( ( sup_sup @ A @ X2 @ Y )
% 2.98/1.63 = ( F @ X2 @ Y ) ) ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_220_sup_OorderI,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( semilattice_sup @ A )
% 2.98/1.63 => ! [A4: A,B3: A] :
% 2.98/1.63 ( ( A4
% 2.98/1.63 = ( sup_sup @ A @ A4 @ B3 ) )
% 2.98/1.63 => ( ord_less_eq @ A @ B3 @ A4 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_221_sup_OorderE,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( semilattice_sup @ A )
% 2.98/1.63 => ! [B3: A,A4: A] :
% 2.98/1.63 ( ( ord_less_eq @ A @ B3 @ A4 )
% 2.98/1.63 => ( A4
% 2.98/1.63 = ( sup_sup @ A @ A4 @ B3 ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_222_le__iff__sup,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( semilattice_sup @ A )
% 2.98/1.63 => ( ( ord_less_eq @ A )
% 2.98/1.63 = ( ^ [X3: A,Y5: A] :
% 2.98/1.63 ( ( sup_sup @ A @ X3 @ Y5 )
% 2.98/1.63 = Y5 ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_223_sup__least,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( semilattice_sup @ A )
% 2.98/1.63 => ! [Y: A,X2: A,Z2: A] :
% 2.98/1.63 ( ( ord_less_eq @ A @ Y @ X2 )
% 2.98/1.63 => ( ( ord_less_eq @ A @ Z2 @ X2 )
% 2.98/1.63 => ( ord_less_eq @ A @ ( sup_sup @ A @ Y @ Z2 ) @ X2 ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_224_sup__mono,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( semilattice_sup @ A )
% 2.98/1.63 => ! [A4: A,C2: A,B3: A,D4: A] :
% 2.98/1.63 ( ( ord_less_eq @ A @ A4 @ C2 )
% 2.98/1.63 => ( ( ord_less_eq @ A @ B3 @ D4 )
% 2.98/1.63 => ( ord_less_eq @ A @ ( sup_sup @ A @ A4 @ B3 ) @ ( sup_sup @ A @ C2 @ D4 ) ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_225_sup_Omono,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( semilattice_sup @ A )
% 2.98/1.63 => ! [C2: A,A4: A,D4: A,B3: A] :
% 2.98/1.63 ( ( ord_less_eq @ A @ C2 @ A4 )
% 2.98/1.63 => ( ( ord_less_eq @ A @ D4 @ B3 )
% 2.98/1.63 => ( ord_less_eq @ A @ ( sup_sup @ A @ C2 @ D4 ) @ ( sup_sup @ A @ A4 @ B3 ) ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_226_le__supI2,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( semilattice_sup @ A )
% 2.98/1.63 => ! [X2: A,B3: A,A4: A] :
% 2.98/1.63 ( ( ord_less_eq @ A @ X2 @ B3 )
% 2.98/1.63 => ( ord_less_eq @ A @ X2 @ ( sup_sup @ A @ A4 @ B3 ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_227_le__supI1,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( semilattice_sup @ A )
% 2.98/1.63 => ! [X2: A,A4: A,B3: A] :
% 2.98/1.63 ( ( ord_less_eq @ A @ X2 @ A4 )
% 2.98/1.63 => ( ord_less_eq @ A @ X2 @ ( sup_sup @ A @ A4 @ B3 ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_228_sup__ge2,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( semilattice_sup @ A )
% 2.98/1.63 => ! [Y: A,X2: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X2 @ Y ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_229_sup__ge1,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( semilattice_sup @ A )
% 2.98/1.63 => ! [X2: A,Y: A] : ( ord_less_eq @ A @ X2 @ ( sup_sup @ A @ X2 @ Y ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_230_le__supI,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( semilattice_sup @ A )
% 2.98/1.63 => ! [A4: A,X2: A,B3: A] :
% 2.98/1.63 ( ( ord_less_eq @ A @ A4 @ X2 )
% 2.98/1.63 => ( ( ord_less_eq @ A @ B3 @ X2 )
% 2.98/1.63 => ( ord_less_eq @ A @ ( sup_sup @ A @ A4 @ B3 ) @ X2 ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_231_le__supE,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( semilattice_sup @ A )
% 2.98/1.63 => ! [A4: A,B3: A,X2: A] :
% 2.98/1.63 ( ( ord_less_eq @ A @ ( sup_sup @ A @ A4 @ B3 ) @ X2 )
% 2.98/1.63 => ~ ( ( ord_less_eq @ A @ A4 @ X2 )
% 2.98/1.63 => ~ ( ord_less_eq @ A @ B3 @ X2 ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_232_inf__sup__ord_I3_J,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( lattice @ A )
% 2.98/1.63 => ! [X2: A,Y: A] : ( ord_less_eq @ A @ X2 @ ( sup_sup @ A @ X2 @ Y ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_233_inf__sup__ord_I4_J,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( lattice @ A )
% 2.98/1.63 => ! [Y: A,X2: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X2 @ Y ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_234_inf__sup__aci_I8_J,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( lattice @ A )
% 2.98/1.63 => ! [X2: A,Y: A] :
% 2.98/1.63 ( ( sup_sup @ A @ X2 @ ( sup_sup @ A @ X2 @ Y ) )
% 2.98/1.63 = ( sup_sup @ A @ X2 @ Y ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_235_inf__sup__aci_I7_J,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( lattice @ A )
% 2.98/1.63 => ! [X2: A,Y: A,Z2: A] :
% 2.98/1.63 ( ( sup_sup @ A @ X2 @ ( sup_sup @ A @ Y @ Z2 ) )
% 2.98/1.63 = ( sup_sup @ A @ Y @ ( sup_sup @ A @ X2 @ Z2 ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_236_inf__sup__aci_I6_J,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( lattice @ A )
% 2.98/1.63 => ! [X2: A,Y: A,Z2: A] :
% 2.98/1.63 ( ( sup_sup @ A @ ( sup_sup @ A @ X2 @ Y ) @ Z2 )
% 2.98/1.63 = ( sup_sup @ A @ X2 @ ( sup_sup @ A @ Y @ Z2 ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_237_inf__sup__aci_I5_J,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( lattice @ A )
% 2.98/1.63 => ( ( sup_sup @ A )
% 2.98/1.63 = ( ^ [X3: A,Y5: A] : ( sup_sup @ A @ Y5 @ X3 ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_238_UnE,axiom,
% 2.98/1.63 ! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
% 2.98/1.63 ( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
% 2.98/1.63 => ( ~ ( member @ A @ C2 @ A2 )
% 2.98/1.63 => ( member @ A @ C2 @ B2 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_239_UnI1,axiom,
% 2.98/1.63 ! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
% 2.98/1.63 ( ( member @ A @ C2 @ A2 )
% 2.98/1.63 => ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_240_UnI2,axiom,
% 2.98/1.63 ! [A: $tType,C2: A,B2: set @ A,A2: set @ A] :
% 2.98/1.63 ( ( member @ A @ C2 @ B2 )
% 2.98/1.63 => ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_241_bex__Un,axiom,
% 2.98/1.63 ! [A: $tType,A2: set @ A,B2: set @ A,P: A > $o] :
% 2.98/1.63 ( ( ? [X3: A] :
% 2.98/1.63 ( ( member @ A @ X3 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
% 2.98/1.63 & ( P @ X3 ) ) )
% 2.98/1.63 = ( ? [X3: A] :
% 2.98/1.63 ( ( member @ A @ X3 @ A2 )
% 2.98/1.63 & ( P @ X3 ) )
% 2.98/1.63 | ? [X3: A] :
% 2.98/1.63 ( ( member @ A @ X3 @ B2 )
% 2.98/1.63 & ( P @ X3 ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_242_ball__Un,axiom,
% 2.98/1.63 ! [A: $tType,A2: set @ A,B2: set @ A,P: A > $o] :
% 2.98/1.63 ( ( ! [X3: A] :
% 2.98/1.63 ( ( member @ A @ X3 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
% 2.98/1.63 => ( P @ X3 ) ) )
% 2.98/1.63 = ( ! [X3: A] :
% 2.98/1.63 ( ( member @ A @ X3 @ A2 )
% 2.98/1.63 => ( P @ X3 ) )
% 2.98/1.63 & ! [X3: A] :
% 2.98/1.63 ( ( member @ A @ X3 @ B2 )
% 2.98/1.63 => ( P @ X3 ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_243_Un__assoc,axiom,
% 2.98/1.63 ! [A: $tType,A2: set @ A,B2: set @ A,C4: set @ A] :
% 2.98/1.63 ( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ C4 )
% 2.98/1.63 = ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ B2 @ C4 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_244_Un__absorb,axiom,
% 2.98/1.63 ! [A: $tType,A2: set @ A] :
% 2.98/1.63 ( ( sup_sup @ ( set @ A ) @ A2 @ A2 )
% 2.98/1.63 = A2 ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_245_Un__left__absorb,axiom,
% 2.98/1.63 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.98/1.63 ( ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
% 2.98/1.63 = ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_246_Un__left__commute,axiom,
% 2.98/1.63 ! [A: $tType,A2: set @ A,B2: set @ A,C4: set @ A] :
% 2.98/1.63 ( ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ B2 @ C4 ) )
% 2.98/1.63 = ( sup_sup @ ( set @ A ) @ B2 @ ( sup_sup @ ( set @ A ) @ A2 @ C4 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_247_sup__fun__def,axiom,
% 2.98/1.63 ! [B: $tType,A: $tType] :
% 2.98/1.63 ( ( semilattice_sup @ B )
% 2.98/1.63 => ( ( sup_sup @ ( A > B ) )
% 2.98/1.63 = ( ^ [F4: A > B,G4: A > B,X3: A] : ( sup_sup @ B @ ( F4 @ X3 ) @ ( G4 @ X3 ) ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_248_boolean__algebra__cancel_Osup1,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( semilattice_sup @ A )
% 2.98/1.63 => ! [A2: A,K: A,A4: A,B3: A] :
% 2.98/1.63 ( ( A2
% 2.98/1.63 = ( sup_sup @ A @ K @ A4 ) )
% 2.98/1.63 => ( ( sup_sup @ A @ A2 @ B3 )
% 2.98/1.63 = ( sup_sup @ A @ K @ ( sup_sup @ A @ A4 @ B3 ) ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_249_boolean__algebra__cancel_Osup2,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( semilattice_sup @ A )
% 2.98/1.63 => ! [B2: A,K: A,B3: A,A4: A] :
% 2.98/1.63 ( ( B2
% 2.98/1.63 = ( sup_sup @ A @ K @ B3 ) )
% 2.98/1.63 => ( ( sup_sup @ A @ A4 @ B2 )
% 2.98/1.63 = ( sup_sup @ A @ K @ ( sup_sup @ A @ A4 @ B3 ) ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_250_sup_Oassoc,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( semilattice_sup @ A )
% 2.98/1.63 => ! [A4: A,B3: A,C2: A] :
% 2.98/1.63 ( ( sup_sup @ A @ ( sup_sup @ A @ A4 @ B3 ) @ C2 )
% 2.98/1.63 = ( sup_sup @ A @ A4 @ ( sup_sup @ A @ B3 @ C2 ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_251_sup__assoc,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( semilattice_sup @ A )
% 2.98/1.63 => ! [X2: A,Y: A,Z2: A] :
% 2.98/1.63 ( ( sup_sup @ A @ ( sup_sup @ A @ X2 @ Y ) @ Z2 )
% 2.98/1.63 = ( sup_sup @ A @ X2 @ ( sup_sup @ A @ Y @ Z2 ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_252_sup_Ocommute,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( semilattice_sup @ A )
% 2.98/1.63 => ( ( sup_sup @ A )
% 2.98/1.63 = ( ^ [A7: A,B7: A] : ( sup_sup @ A @ B7 @ A7 ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_253_sup__commute,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( semilattice_sup @ A )
% 2.98/1.63 => ( ( sup_sup @ A )
% 2.98/1.63 = ( ^ [X3: A,Y5: A] : ( sup_sup @ A @ Y5 @ X3 ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(fact_254_sup_Oleft__commute,axiom,
% 2.98/1.63 ! [A: $tType] :
% 2.98/1.63 ( ( semilattice_sup @ A )
% 2.98/1.63 => ! [B3: A,A4: A,C2: A] :
% 2.98/1.63 ( ( sup_sup @ A @ B3 @ ( sup_sup @ A @ A4 @ C2 ) )
% 2.98/1.63 = ( sup_sup @ A @ A4 @ ( sup_sup @ A @ B3 @ C2 ) ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
% 2.98/1.63 ! [A9: $tType,A10: $tType] :
% 2.98/1.63 ( ( semilattice_sup @ A10 )
% 2.98/1.63 => ( semilattice_sup @ ( A9 > A10 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(tcon_fun___Lattices_Oboolean__algebra,axiom,
% 2.98/1.63 ! [A9: $tType,A10: $tType] :
% 2.98/1.63 ( ( boolean_algebra @ A10 )
% 2.98/1.63 => ( boolean_algebra @ ( A9 > A10 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(tcon_fun___Orderings_Opreorder,axiom,
% 2.98/1.63 ! [A9: $tType,A10: $tType] :
% 2.98/1.63 ( ( preorder @ A10 )
% 2.98/1.63 => ( preorder @ ( A9 > A10 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(tcon_fun___Lattices_Olattice,axiom,
% 2.98/1.63 ! [A9: $tType,A10: $tType] :
% 2.98/1.63 ( ( lattice @ A10 )
% 2.98/1.63 => ( lattice @ ( A9 > A10 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.98/1.63 ! [A9: $tType,A10: $tType] :
% 2.98/1.63 ( ( order @ A10 )
% 2.98/1.63 => ( order @ ( A9 > A10 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(tcon_fun___Orderings_Oord,axiom,
% 2.98/1.63 ! [A9: $tType,A10: $tType] :
% 2.98/1.63 ( ( ord @ A10 )
% 2.98/1.63 => ( ord @ ( A9 > A10 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(tcon_fun___Groups_Ouminus,axiom,
% 2.98/1.63 ! [A9: $tType,A10: $tType] :
% 2.98/1.63 ( ( uminus @ A10 )
% 2.98/1.63 => ( uminus @ ( A9 > A10 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_1,axiom,
% 2.98/1.63 ! [A9: $tType] : ( semilattice_sup @ ( set @ A9 ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(tcon_Set_Oset___Lattices_Oboolean__algebra_2,axiom,
% 2.98/1.63 ! [A9: $tType] : ( boolean_algebra @ ( set @ A9 ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(tcon_Set_Oset___Orderings_Opreorder_3,axiom,
% 2.98/1.63 ! [A9: $tType] : ( preorder @ ( set @ A9 ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(tcon_Set_Oset___Lattices_Olattice_4,axiom,
% 2.98/1.63 ! [A9: $tType] : ( lattice @ ( set @ A9 ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(tcon_Set_Oset___Orderings_Oorder_5,axiom,
% 2.98/1.63 ! [A9: $tType] : ( order @ ( set @ A9 ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(tcon_Set_Oset___Orderings_Oord_6,axiom,
% 2.98/1.63 ! [A9: $tType] : ( ord @ ( set @ A9 ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(tcon_Set_Oset___Groups_Ouminus_7,axiom,
% 2.98/1.63 ! [A9: $tType] : ( uminus @ ( set @ A9 ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_8,axiom,
% 2.98/1.63 semilattice_sup @ $o ).
% 2.98/1.63
% 2.98/1.63 thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_9,axiom,
% 2.98/1.63 boolean_algebra @ $o ).
% 2.98/1.63
% 2.98/1.63 thf(tcon_HOL_Obool___Orderings_Opreorder_10,axiom,
% 2.98/1.63 preorder @ $o ).
% 2.98/1.63
% 2.98/1.63 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 2.98/1.63 linorder @ $o ).
% 2.98/1.63
% 2.98/1.63 thf(tcon_HOL_Obool___Lattices_Olattice_11,axiom,
% 2.98/1.63 lattice @ $o ).
% 2.98/1.63
% 2.98/1.63 thf(tcon_HOL_Obool___Orderings_Oorder_12,axiom,
% 2.98/1.63 order @ $o ).
% 2.98/1.63
% 2.98/1.63 thf(tcon_HOL_Obool___Orderings_Oord_13,axiom,
% 2.98/1.63 ord @ $o ).
% 2.98/1.63
% 2.98/1.63 thf(tcon_HOL_Obool___Groups_Ouminus_14,axiom,
% 2.98/1.63 uminus @ $o ).
% 2.98/1.63
% 2.98/1.63 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add,axiom,
% 2.98/1.63 ordered_ab_group_add @ real ).
% 2.98/1.63
% 2.98/1.63 thf(tcon_Real_Oreal___Lattices_Osemilattice__sup_15,axiom,
% 2.98/1.63 semilattice_sup @ real ).
% 2.98/1.63
% 2.98/1.63 thf(tcon_Real_Oreal___Orderings_Opreorder_16,axiom,
% 2.98/1.63 preorder @ real ).
% 2.98/1.63
% 2.98/1.63 thf(tcon_Real_Oreal___Orderings_Olinorder_17,axiom,
% 2.98/1.63 linorder @ real ).
% 2.98/1.63
% 2.98/1.63 thf(tcon_Real_Oreal___Lattices_Olattice_18,axiom,
% 2.98/1.63 lattice @ real ).
% 2.98/1.63
% 2.98/1.63 thf(tcon_Real_Oreal___Groups_Ogroup__add,axiom,
% 2.98/1.63 group_add @ real ).
% 2.98/1.63
% 2.98/1.63 thf(tcon_Real_Oreal___Orderings_Oorder_19,axiom,
% 2.98/1.63 order @ real ).
% 2.98/1.63
% 2.98/1.63 thf(tcon_Real_Oreal___Orderings_Oord_20,axiom,
% 2.98/1.63 ord @ real ).
% 2.98/1.63
% 2.98/1.63 thf(tcon_Real_Oreal___Groups_Ouminus_21,axiom,
% 2.98/1.63 uminus @ real ).
% 2.98/1.63
% 2.98/1.63 thf(tcon_Product__Type_Oprod___Groups_Ogroup__add_22,axiom,
% 2.98/1.63 ! [A9: $tType,A10: $tType] :
% 2.98/1.63 ( ( ( group_add @ A9 )
% 2.98/1.63 & ( group_add @ A10 ) )
% 2.98/1.63 => ( group_add @ ( product_prod @ A9 @ A10 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(tcon_Product__Type_Oprod___Groups_Ouminus_23,axiom,
% 2.98/1.63 ! [A9: $tType,A10: $tType] :
% 2.98/1.63 ( ( ( uminus @ A9 )
% 2.98/1.63 & ( uminus @ A10 ) )
% 2.98/1.63 => ( uminus @ ( product_prod @ A9 @ A10 ) ) ) ).
% 2.98/1.63
% 2.98/1.63 thf(conj_0,conjecture,
% 2.98/1.63 ( ( member @ ( variable > real ) @ nu2 @ ( uminus_uminus @ ( set @ ( variable > real ) ) @ ( denota1245701238me_sem @ i @ ( the @ game @ ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ sigma @ ua @ alpha ) ) ) @ ( uminus_uminus @ ( set @ ( variable > real ) ) @ xa ) ) ) )
% 2.98/1.63 = ( member @ ( variable > real ) @ nu2 @ ( uminus_uminus @ ( set @ ( variable > real ) ) @ ( denota1245701238me_sem @ ( uSubst2091380086djoint @ sigma @ i @ omega2 ) @ alpha @ ( uminus_uminus @ ( set @ ( variable > real ) ) @ xa ) ) ) ) ) ).
% 2.98/1.63 ------------------------
% 2.98/1.66 % (4752)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_4484 for (2999ds/2Mi)
% 2.98/1.66 % (4755)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_4484 for (2999ds/18Mi)
% 2.98/1.66 % (4752)Instruction limit reached!
% 2.98/1.66 % (4752)------------------------------
% 2.98/1.66 % (4752)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.98/1.66 % (4752)Termination reason: Unknown
% 2.98/1.66 % (4752)Termination phase: shuffling
% 2.98/1.66
% 2.98/1.66 % (4752)Memory used [KB]: 1407
% 2.98/1.66 % (4752)Time elapsed: 0.002 s
% 2.98/1.66 % (4752)Instructions burned: 2 (million)
% 2.98/1.66 % (4752)------------------------------
% 2.98/1.66 % (4752)------------------------------
% 2.98/1.66 % (4750)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_4484 for (2999ds/4Mi)
% 2.98/1.67 % (4750)Instruction limit reached!
% 2.98/1.67 % (4750)------------------------------
% 2.98/1.67 % (4750)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.98/1.67 % (4750)Termination reason: Unknown
% 2.98/1.67 % (4750)Termination phase: shuffling
% 2.98/1.67
% 2.98/1.67 % (4750)Memory used [KB]: 1407
% 2.98/1.67 % (4750)Time elapsed: 0.003 s
% 2.98/1.67 % (4750)Instructions burned: 6 (million)
% 2.98/1.67 % (4750)------------------------------
% 2.98/1.67 % (4750)------------------------------
% 2.98/1.67 % (4755)Instruction limit reached!
% 2.98/1.67 % (4755)------------------------------
% 2.98/1.67 % (4755)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.98/1.67 % (4755)Termination reason: Unknown
% 2.98/1.67 % (4755)Termination phase: shuffling
% 2.98/1.67
% 2.98/1.67 % (4755)Memory used [KB]: 1791
% 2.98/1.67 % (4755)Time elapsed: 0.008 s
% 2.98/1.67 % (4755)Instructions burned: 21 (million)
% 2.98/1.67 % (4755)------------------------------
% 2.98/1.67 % (4755)------------------------------
% 2.98/1.67 % (4749)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_4484 for (2999ds/183Mi)
% 2.98/1.67 % (4751)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_4484 for (2999ds/27Mi)
% 2.98/1.67 % (4753)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_4484 for (2999ds/2Mi)
% 2.98/1.67 % (4754)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_4484 for (2999ds/275Mi)
% 2.98/1.67 % (4753)Instruction limit reached!
% 2.98/1.67 % (4753)------------------------------
% 2.98/1.67 % (4753)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.98/1.67 % (4753)Termination reason: Unknown
% 2.98/1.67 % (4753)Termination phase: shuffling
% 2.98/1.67
% 2.98/1.67 % (4753)Memory used [KB]: 1407
% 2.98/1.67 % (4753)Time elapsed: 0.004 s
% 2.98/1.67 % (4753)Instructions burned: 3 (million)
% 2.98/1.67 % (4753)------------------------------
% 2.98/1.67 % (4753)------------------------------
% 2.98/1.67 % (4756)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_4484 for (2999ds/3Mi)
% 2.98/1.67 % (4756)Instruction limit reached!
% 2.98/1.67 % (4756)------------------------------
% 2.98/1.67 % (4756)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.98/1.67 % (4756)Termination reason: Unknown
% 2.98/1.67 % (4756)Termination phase: shuffling
% 2.98/1.67
% 2.98/1.67 % (4756)Memory used [KB]: 1407
% 2.98/1.67 % (4756)Time elapsed: 0.002 s
% 2.98/1.67 % (4756)Instructions burned: 3 (million)
% 2.98/1.67 % (4756)------------------------------
% 2.98/1.67 % (4756)------------------------------
% 2.98/1.68 % (4757)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_4484 for (2999ds/37Mi)
% 2.98/1.68 % (4758)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_4484 for (2999ds/15Mi)
% 2.98/1.68 % (4760)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_4484 for (2999ds/1041Mi)
% 2.98/1.69 % (4758)Instruction limit reached!
% 2.98/1.69 % (4758)------------------------------
% 2.98/1.69 % (4758)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.98/1.69 % (4758)Termination reason: Unknown
% 2.98/1.69 % (4758)Termination phase: shuffling
% 2.98/1.69
% 2.98/1.69 % (4758)Memory used [KB]: 1663
% 2.98/1.69 % (4758)Time elapsed: 0.007 s
% 2.98/1.69 % (4758)Instructions burned: 17 (million)
% 2.98/1.69 % (4758)------------------------------
% 2.98/1.69 % (4758)------------------------------
% 2.98/1.69 % (4757)Instruction limit reached!
% 2.98/1.69 % (4757)------------------------------
% 2.98/1.69 % (4757)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.98/1.69 % (4757)Termination reason: Unknown
% 2.98/1.69 % (4757)Termination phase: shuffling
% 2.98/1.69
% 2.98/1.69 % (4757)Memory used [KB]: 2174
% 2.98/1.69 % (4757)Time elapsed: 0.013 s
% 2.98/1.69 % (4757)Instructions burned: 37 (million)
% 2.98/1.69 % (4757)------------------------------
% 2.98/1.69 % (4757)------------------------------
% 2.98/1.69 % (4751)Instruction limit reached!
% 2.98/1.69 % (4751)------------------------------
% 2.98/1.69 % (4751)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.98/1.69 % (4751)Termination reason: Unknown
% 2.98/1.69 % (4751)Termination phase: Property scanning
% 2.98/1.69
% 2.98/1.69 % (4751)Memory used [KB]: 1918
% 2.98/1.69 % (4751)Time elapsed: 0.019 s
% 2.98/1.69 % (4751)Instructions burned: 28 (million)
% 2.98/1.69 % (4751)------------------------------
% 2.98/1.69 % (4751)------------------------------
% 2.98/1.69 % (4759)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_4484 for (2999ds/3Mi)
% 2.98/1.69 % (4759)Instruction limit reached!
% 2.98/1.69 % (4759)------------------------------
% 2.98/1.69 % (4759)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.98/1.69 % (4759)Termination reason: Unknown
% 2.98/1.69 % (4759)Termination phase: shuffling
% 2.98/1.69
% 2.98/1.69 % (4759)Memory used [KB]: 1407
% 2.98/1.69 % (4759)Time elapsed: 0.004 s
% 2.98/1.69 % (4759)Instructions burned: 3 (million)
% 2.98/1.69 % (4759)------------------------------
% 2.98/1.69 % (4759)------------------------------
% 2.98/1.70 % (4761)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_4484 for (2999ds/7Mi)
% 3.03/1.70 % (4762)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_4484 for (2999ds/16Mi)
% 3.03/1.70 % (4761)Instruction limit reached!
% 3.03/1.70 % (4761)------------------------------
% 3.03/1.70 % (4761)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.70 % (4761)Termination reason: Unknown
% 3.03/1.70 % (4761)Termination phase: shuffling
% 3.03/1.70
% 3.03/1.70 % (4761)Memory used [KB]: 1535
% 3.03/1.70 % (4761)Time elapsed: 0.004 s
% 3.03/1.70 % (4761)Instructions burned: 9 (million)
% 3.03/1.70 % (4761)------------------------------
% 3.03/1.70 % (4761)------------------------------
% 3.03/1.70 % (4763)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_4484 for (2999ds/3Mi)
% 3.03/1.70 % (4763)Instruction limit reached!
% 3.03/1.70 % (4763)------------------------------
% 3.03/1.70 % (4763)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.70 % (4763)Termination reason: Unknown
% 3.03/1.70 % (4763)Termination phase: shuffling
% 3.03/1.70
% 3.03/1.70 % (4763)Memory used [KB]: 1407
% 3.03/1.70 % (4763)Time elapsed: 0.003 s
% 3.03/1.70 % (4763)Instructions burned: 4 (million)
% 3.03/1.70 % (4763)------------------------------
% 3.03/1.70 % (4763)------------------------------
% 3.03/1.70 % (4762)Instruction limit reached!
% 3.03/1.70 % (4762)------------------------------
% 3.03/1.70 % (4762)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.70 % (4762)Termination reason: Unknown
% 3.03/1.70 % (4762)Termination phase: shuffling
% 3.03/1.70
% 3.03/1.70 % (4762)Memory used [KB]: 1663
% 3.03/1.70 % (4762)Time elapsed: 0.008 s
% 3.03/1.70 % (4762)Instructions burned: 18 (million)
% 3.03/1.70 % (4762)------------------------------
% 3.03/1.70 % (4762)------------------------------
% 3.03/1.71 % (4765)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_4484 for (2999ds/7Mi)
% 3.03/1.71 % (4764)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_4484 for (2999ds/3Mi)
% 3.03/1.71 % (4764)Instruction limit reached!
% 3.03/1.71 % (4764)------------------------------
% 3.03/1.71 % (4764)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.71 % (4764)Termination reason: Unknown
% 3.03/1.71 % (4764)Termination phase: shuffling
% 3.03/1.71
% 3.03/1.71 % (4764)Memory used [KB]: 1407
% 3.03/1.71 % (4764)Time elapsed: 0.003 s
% 3.03/1.71 % (4764)Instructions burned: 3 (million)
% 3.03/1.71 % (4764)------------------------------
% 3.03/1.71 % (4764)------------------------------
% 3.03/1.71 % (4765)Instruction limit reached!
% 3.03/1.71 % (4765)------------------------------
% 3.03/1.71 % (4765)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.71 % (4765)Termination reason: Unknown
% 3.03/1.71 % (4765)Termination phase: shuffling
% 3.03/1.71
% 3.03/1.71 % (4765)Memory used [KB]: 1535
% 3.03/1.71 % (4765)Time elapsed: 0.004 s
% 3.03/1.71 % (4765)Instructions burned: 9 (million)
% 3.03/1.71 % (4765)------------------------------
% 3.03/1.71 % (4765)------------------------------
% 3.03/1.71 % (4766)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_4484 for (2999ds/3Mi)
% 3.03/1.71 % (4767)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_4484 for (2999ds/4Mi)
% 3.03/1.72 % (4766)Instruction limit reached!
% 3.03/1.72 % (4766)------------------------------
% 3.03/1.72 % (4766)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.72 % (4766)Termination reason: Unknown
% 3.03/1.72 % (4766)Termination phase: shuffling
% 3.03/1.72
% 3.03/1.72 % (4766)Memory used [KB]: 1407
% 3.03/1.72 % (4766)Time elapsed: 0.003 s
% 3.03/1.72 % (4766)Instructions burned: 4 (million)
% 3.03/1.72 % (4766)------------------------------
% 3.03/1.72 % (4766)------------------------------
% 3.03/1.72 % (4767)Instruction limit reached!
% 3.03/1.72 % (4767)------------------------------
% 3.03/1.72 % (4767)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.72 % (4767)Termination reason: Unknown
% 3.03/1.72 % (4767)Termination phase: shuffling
% 3.03/1.72
% 3.03/1.72 % (4767)Memory used [KB]: 1407
% 3.03/1.72 % (4767)Time elapsed: 0.003 s
% 3.03/1.72 % (4767)Instructions burned: 4 (million)
% 3.03/1.72 % (4767)------------------------------
% 3.03/1.72 % (4767)------------------------------
% 3.03/1.72 % (4768)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_4484 for (2999ds/18Mi)
% 3.03/1.72 % (4769)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_4484 for (2999ds/710Mi)
% 3.03/1.73 % (4770)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_4484 for (2999ds/6Mi)
% 3.03/1.73 % (4771)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_4484 for (2999ds/902Mi)
% 3.03/1.73 % (4768)Instruction limit reached!
% 3.03/1.73 % (4768)------------------------------
% 3.03/1.73 % (4768)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.73 % (4768)Termination reason: Unknown
% 3.03/1.73 % (4768)Termination phase: shuffling
% 3.03/1.73
% 3.03/1.73 % (4768)Memory used [KB]: 1663
% 3.03/1.73 % (4768)Time elapsed: 0.008 s
% 3.03/1.73 % (4768)Instructions burned: 18 (million)
% 3.03/1.73 % (4768)------------------------------
% 3.03/1.73 % (4768)------------------------------
% 3.03/1.73 % (4770)Instruction limit reached!
% 3.03/1.73 % (4770)------------------------------
% 3.03/1.73 % (4770)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.73 % (4770)Termination reason: Unknown
% 3.03/1.73 % (4770)Termination phase: shuffling
% 3.03/1.73
% 3.03/1.73 % (4770)Memory used [KB]: 1535
% 3.03/1.73 % (4770)Time elapsed: 0.004 s
% 3.03/1.73 % (4770)Instructions burned: 6 (million)
% 3.03/1.73 % (4770)------------------------------
% 3.03/1.73 % (4770)------------------------------
% 3.03/1.74 % (4772)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_4484 for (2999ds/21Mi)
% 3.03/1.74 % (4773)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_4484 for (2999ds/5Mi)
% 3.03/1.74 % (4773)Instruction limit reached!
% 3.03/1.74 % (4773)------------------------------
% 3.03/1.74 % (4773)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.74 % (4773)Termination reason: Unknown
% 3.03/1.74 % (4773)Termination phase: shuffling
% 3.03/1.74
% 3.03/1.74 % (4773)Memory used [KB]: 1535
% 3.03/1.74 % (4773)Time elapsed: 0.004 s
% 3.03/1.74 % (4773)Instructions burned: 7 (million)
% 3.03/1.74 % (4773)------------------------------
% 3.03/1.74 % (4773)------------------------------
% 3.03/1.74 % (4772)Instruction limit reached!
% 3.03/1.74 % (4772)------------------------------
% 3.03/1.74 % (4772)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.74 % (4772)Termination reason: Unknown
% 3.03/1.74 % (4772)Termination phase: shuffling
% 3.03/1.74
% 3.03/1.74 % (4772)Memory used [KB]: 1791
% 3.03/1.74 % (4772)Time elapsed: 0.010 s
% 3.03/1.74 % (4772)Instructions burned: 23 (million)
% 3.03/1.74 % (4772)------------------------------
% 3.03/1.74 % (4772)------------------------------
% 3.03/1.75 % (4774)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_4484 for (2999ds/6Mi)
% 3.03/1.75 % (4774)Instruction limit reached!
% 3.03/1.75 % (4774)------------------------------
% 3.03/1.75 % (4774)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.75 % (4774)Termination reason: Unknown
% 3.03/1.75 % (4774)Termination phase: shuffling
% 3.03/1.75
% 3.03/1.75 % (4774)Memory used [KB]: 1535
% 3.03/1.75 % (4774)Time elapsed: 0.004 s
% 3.03/1.75 % (4774)Instructions burned: 8 (million)
% 3.03/1.75 % (4774)------------------------------
% 3.03/1.75 % (4774)------------------------------
% 3.03/1.75 % (4775)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_4484 for (2999ds/377Mi)
% 3.03/1.76 % (4776)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_4484 for (2998ds/779Mi)
% 3.03/1.77 % (4754)Instruction limit reached!
% 3.03/1.77 % (4754)------------------------------
% 3.03/1.77 % (4754)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.77 % (4754)Termination reason: Unknown
% 3.03/1.77 % (4754)Termination phase: Saturation
% 3.03/1.77
% 3.03/1.77 % (4754)Memory used [KB]: 7291
% 3.03/1.77 % (4754)Time elapsed: 0.096 s
% 3.03/1.77 % (4754)Instructions burned: 275 (million)
% 3.03/1.77 % (4754)------------------------------
% 3.03/1.77 % (4754)------------------------------
% 3.03/1.77 % (4749)Instruction limit reached!
% 3.03/1.77 % (4749)------------------------------
% 3.03/1.77 % (4749)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.77 % (4749)Termination reason: Unknown
% 3.03/1.77 % (4749)Termination phase: Saturation
% 3.03/1.77
% 3.03/1.77 % (4749)Memory used [KB]: 7164
% 3.03/1.77 % (4749)Time elapsed: 0.100 s
% 3.03/1.77 % (4749)Instructions burned: 184 (million)
% 3.03/1.77 % (4749)------------------------------
% 3.03/1.77 % (4749)------------------------------
% 3.03/1.78 % (4777)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_4484 for (2998ds/19Mi)
% 3.03/1.78 % (4778)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_4484 for (2998ds/879Mi)
% 3.03/1.78 % (4777)Instruction limit reached!
% 3.03/1.78 % (4777)------------------------------
% 3.03/1.78 % (4777)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.78 % (4777)Termination reason: Unknown
% 3.03/1.78 % (4777)Termination phase: shuffling
% 3.03/1.78
% 3.03/1.78 % (4777)Memory used [KB]: 1791
% 3.03/1.78 % (4777)Time elapsed: 0.008 s
% 3.03/1.78 % (4777)Instructions burned: 21 (million)
% 3.03/1.78 % (4777)------------------------------
% 3.03/1.78 % (4777)------------------------------
% 3.03/1.79 % (4779)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_4484 for (2998ds/17Mi)
% 3.03/1.80 % (4779)Instruction limit reached!
% 3.03/1.80 % (4779)------------------------------
% 3.03/1.80 % (4779)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.80 % (4779)Termination reason: Unknown
% 3.03/1.80 % (4779)Termination phase: shuffling
% 3.03/1.80
% 3.03/1.80 % (4779)Memory used [KB]: 1791
% 3.03/1.80 % (4779)Time elapsed: 0.008 s
% 3.03/1.80 % (4779)Instructions burned: 18 (million)
% 3.03/1.80 % (4779)------------------------------
% 3.03/1.80 % (4779)------------------------------
% 3.03/1.81 % (4780)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_4484 for (2998ds/3Mi)
% 3.03/1.81 % (4780)Instruction limit reached!
% 3.03/1.81 % (4780)------------------------------
% 3.03/1.81 % (4780)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.81 % (4780)Termination reason: Unknown
% 3.03/1.81 % (4780)Termination phase: shuffling
% 3.03/1.81
% 3.03/1.81 % (4780)Memory used [KB]: 1407
% 3.03/1.81 % (4780)Time elapsed: 0.003 s
% 3.03/1.81 % (4780)Instructions burned: 4 (million)
% 3.03/1.81 % (4780)------------------------------
% 3.03/1.81 % (4780)------------------------------
% 3.03/1.82 % (4781)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_4484 for (2998ds/30Mi)
% 3.03/1.83 % (4781)Instruction limit reached!
% 3.03/1.83 % (4781)------------------------------
% 3.03/1.83 % (4781)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.83 % (4781)Termination reason: Unknown
% 3.03/1.83 % (4781)Termination phase: Property scanning
% 3.03/1.83
% 3.03/1.83 % (4781)Memory used [KB]: 1918
% 3.03/1.83 % (4781)Time elapsed: 0.013 s
% 3.03/1.83 % (4781)Instructions burned: 32 (million)
% 3.03/1.83 % (4781)------------------------------
% 3.03/1.83 % (4781)------------------------------
% 3.03/1.84 % (4782)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_4484 for (2998ds/127Mi)
% 3.03/1.88 % (4775)Instruction limit reached!
% 3.03/1.88 % (4775)------------------------------
% 3.03/1.88 % (4775)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.88 % (4775)Termination reason: Unknown
% 3.03/1.88 % (4775)Termination phase: Saturation
% 3.03/1.88
% 3.03/1.88 % (4775)Memory used [KB]: 7931
% 3.03/1.88 % (4775)Time elapsed: 0.129 s
% 3.03/1.88 % (4775)Instructions burned: 379 (million)
% 3.03/1.88 % (4775)------------------------------
% 3.03/1.88 % (4775)------------------------------
% 3.03/1.88 % (4782)Instruction limit reached!
% 3.03/1.88 % (4782)------------------------------
% 3.03/1.88 % (4782)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.88 % (4782)Termination reason: Unknown
% 3.03/1.88 % (4782)Termination phase: Saturation
% 3.03/1.88
% 3.03/1.88 % (4782)Memory used [KB]: 7036
% 3.03/1.88 % (4782)Time elapsed: 0.043 s
% 3.03/1.88 % (4782)Instructions burned: 129 (million)
% 3.03/1.88 % (4782)------------------------------
% 3.03/1.88 % (4782)------------------------------
% 3.03/1.89 % (4783)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_4484 for (2997ds/100Mi)
% 3.03/1.89 % (4784)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=3:si=on:rtra=on_0 on DTF2THF_4484 for (2997ds/3Mi)
% 3.03/1.89 % (4784)Instruction limit reached!
% 3.03/1.89 % (4784)------------------------------
% 3.03/1.89 % (4784)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.89 % (4784)Termination reason: Unknown
% 3.03/1.89 % (4784)Termination phase: shuffling
% 3.03/1.89
% 3.03/1.89 % (4784)Memory used [KB]: 1407
% 3.03/1.89 % (4784)Time elapsed: 0.002 s
% 3.03/1.89 % (4784)Instructions burned: 3 (million)
% 3.03/1.89 % (4784)------------------------------
% 3.03/1.89 % (4784)------------------------------
% 3.03/1.90 % (4785)lrs+10_8:1_au=on:avsq=on:e2e=on:ins=3:s2a=on:s2at=3.0:ss=axioms:i=20:si=on:rtra=on_0 on DTF2THF_4484 for (2997ds/20Mi)
% 3.03/1.91 % (4785)Instruction limit reached!
% 3.03/1.91 % (4785)------------------------------
% 3.03/1.91 % (4785)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.91 % (4785)Termination reason: Unknown
% 3.03/1.91 % (4785)Termination phase: shuffling
% 3.03/1.91
% 3.03/1.91 % (4785)Memory used [KB]: 1791
% 3.03/1.91 % (4785)Time elapsed: 0.008 s
% 3.03/1.91 % (4785)Instructions burned: 21 (million)
% 3.03/1.91 % (4785)------------------------------
% 3.03/1.91 % (4785)------------------------------
% 3.03/1.91 % (4769)Refutation not found, non-redundant clauses discarded% (4769)------------------------------
% 3.03/1.91 % (4769)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.91 % (4769)Termination reason: Refutation not found, non-redundant clauses discarded
% 3.03/1.91
% 3.03/1.91 % (4769)Memory used [KB]: 7291
% 3.03/1.91 % (4769)Time elapsed: 0.195 s
% 3.03/1.91 % (4769)Instructions burned: 626 (million)
% 3.03/1.91 % (4769)------------------------------
% 3.03/1.91 % (4769)------------------------------
% 3.03/1.92 % (4786)dis+1002_1:1_cbe=off:hud=5:nm=4:plsq=on:plsqr=7,1:prag=on:sp=const_max:tnu=1:i=86:si=on:rtra=on_0 on DTF2THF_4484 for (2997ds/86Mi)
% 3.03/1.92 % (4783)Instruction limit reached!
% 3.03/1.92 % (4783)------------------------------
% 3.03/1.92 % (4783)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.92 % (4783)Termination reason: Unknown
% 3.03/1.92 % (4783)Termination phase: Property scanning
% 3.03/1.92
% 3.03/1.92 % (4783)Memory used [KB]: 2430
% 3.03/1.92 % (4783)Time elapsed: 0.031 s
% 3.03/1.92 % (4783)Instructions burned: 100 (million)
% 3.03/1.92 % (4783)------------------------------
% 3.03/1.92 % (4783)------------------------------
% 3.03/1.92 % (4787)lrs+1010_1:1_au=on:cbe=off:nm=2:ntd=on:sd=2:ss=axioms:st=5.0:i=107:si=on:rtra=on_0 on DTF2THF_4484 for (2997ds/107Mi)
% 3.03/1.93 % (4788)lrs+2_1:1024_cnfonf=lazy_gen:fe=off:hud=15:plsq=on:plsqc=1:plsqr=32,1:i=39:si=on:rtra=on_0 on DTF2THF_4484 for (2997ds/39Mi)
% 3.03/1.94 % (4788)Instruction limit reached!
% 3.03/1.94 % (4788)------------------------------
% 3.03/1.94 % (4788)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.94 % (4788)Termination reason: Unknown
% 3.03/1.94 % (4788)Termination phase: Property scanning
% 3.03/1.94
% 3.03/1.94 % (4788)Memory used [KB]: 2174
% 3.03/1.94 % (4788)Time elapsed: 0.014 s
% 3.03/1.94 % (4788)Instructions burned: 41 (million)
% 3.03/1.94 % (4788)------------------------------
% 3.03/1.94 % (4788)------------------------------
% 3.03/1.94 % (4786)Instruction limit reached!
% 3.03/1.94 % (4786)------------------------------
% 3.03/1.94 % (4786)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.94 % (4786)Termination reason: Unknown
% 3.03/1.94 % (4786)Termination phase: Saturation
% 3.03/1.94
% 3.03/1.94 % (4786)Memory used [KB]: 6652
% 3.03/1.94 % (4786)Time elapsed: 0.026 s
% 3.03/1.94 % (4786)Instructions burned: 86 (million)
% 3.03/1.94 % (4786)------------------------------
% 3.03/1.94 % (4786)------------------------------
% 3.03/1.95 % (4790)lrs+10_1:512_au=on:fde=unused:lma=on:nm=32:plsq=on:plsqc=1:plsqr=16121663,131072:sfv=off:sp=const_max:ss=axioms:st=3.0:tgt=full:i=46:si=on:rtra=on_0 on DTF2THF_4484 for (2996ds/46Mi)
% 3.03/1.95 % (4789)dis+10_1:1_cnfonf=lazy_not_gen:fsr=off:kws=precedence:nwc=5.0:s2a=on:ss=axioms:st=1.5:i=448:si=on:rtra=on_0 on DTF2THF_4484 for (2996ds/448Mi)
% 3.03/1.96 % (4787)Instruction limit reached!
% 3.03/1.96 % (4787)------------------------------
% 3.03/1.96 % (4787)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.96 % (4787)Termination reason: Unknown
% 3.03/1.96 % (4787)Termination phase: Saturation
% 3.03/1.96
% 3.03/1.96 % (4787)Memory used [KB]: 6780
% 3.03/1.96 % (4787)Time elapsed: 0.033 s
% 3.03/1.96 % (4787)Instructions burned: 108 (million)
% 3.03/1.96 % (4787)------------------------------
% 3.03/1.96 % (4787)------------------------------
% 3.03/1.97 % (4791)lrs+10_1:10_au=on:av=off:cbe=off:cnfonf=lazy_pi_sigma_gen:ntd=on:plsq=on:plsqc=1:plsqr=32,1:i=98:si=on:rtra=on_0 on DTF2THF_4484 for (2996ds/98Mi)
% 3.03/1.97 % (4790)Instruction limit reached!
% 3.03/1.97 % (4790)------------------------------
% 3.03/1.97 % (4790)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.97 % (4790)Termination reason: Unknown
% 3.03/1.97 % (4790)Termination phase: Preprocessing 3
% 3.03/1.97
% 3.03/1.97 % (4790)Memory used [KB]: 2046
% 3.03/1.97 % (4790)Time elapsed: 0.016 s
% 3.03/1.97 % (4790)Instructions burned: 49 (million)
% 3.03/1.97 % (4790)------------------------------
% 3.03/1.97 % (4790)------------------------------
% 3.03/1.98 % (4792)ott+1002_1:1_apa=on:au=on:bd=off:cnfonf=off:fd=off:sos=on:sp=weighted_frequency:i=507:si=on:rtra=on_0 on DTF2THF_4484 for (2996ds/507Mi)
% 3.03/1.98 % (4771)Instruction limit reached!
% 3.03/1.98 % (4771)------------------------------
% 3.03/1.98 % (4771)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.98 % (4771)Termination reason: Unknown
% 3.03/1.98 % (4771)Termination phase: Saturation
% 3.03/1.98
% 3.03/1.98 % (4771)Memory used [KB]: 7803
% 3.03/1.98 % (4771)Time elapsed: 0.260 s
% 3.03/1.98 % (4771)Instructions burned: 903 (million)
% 3.03/1.98 % (4771)------------------------------
% 3.03/1.98 % (4771)------------------------------
% 3.03/1.99 % (4791)Instruction limit reached!
% 3.03/1.99 % (4791)------------------------------
% 3.03/1.99 % (4791)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.99 % (4791)Termination reason: Unknown
% 3.03/1.99 % (4791)Termination phase: Property scanning
% 3.03/1.99
% 3.03/1.99 % (4791)Memory used [KB]: 2430
% 3.03/1.99 % (4791)Time elapsed: 0.028 s
% 3.03/1.99 % (4791)Instructions burned: 100 (million)
% 3.03/1.99 % (4791)------------------------------
% 3.03/1.99 % (4791)------------------------------
% 3.03/1.99 % (4793)dis+1010_2:3_amm=off:fd=preordered:ixr=off:nm=0:pe=on:piset=equals:prag=on:sac=on:tgt=ground:i=149:si=on:rtra=on_0 on DTF2THF_4484 for (2996ds/149Mi)
% 3.03/2.00 % (4760)Instruction limit reached!
% 3.03/2.00 % (4760)------------------------------
% 3.03/2.00 % (4760)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/2.00 % (4760)Termination reason: Unknown
% 3.03/2.00 % (4760)Termination phase: Saturation
% 3.03/2.00
% 3.03/2.00 % (4760)Memory used [KB]: 9850
% 3.03/2.00 % (4760)Time elapsed: 0.317 s
% 3.03/2.00 % (4760)Instructions burned: 1042 (million)
% 3.03/2.00 % (4760)------------------------------
% 3.03/2.00 % (4760)------------------------------
% 3.03/2.00 % (4794)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_4484 for (2996ds/39Mi)
% 3.03/2.01 % (4795)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_4484 for (2996ds/6Mi)
% 3.03/2.01 % (4795)Instruction limit reached!
% 3.03/2.01 % (4795)------------------------------
% 3.03/2.01 % (4795)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/2.01 % (4795)Termination reason: Unknown
% 3.03/2.01 % (4795)Termination phase: shuffling
% 3.03/2.01
% 3.03/2.01 % (4795)Memory used [KB]: 1535
% 3.03/2.01 % (4795)Time elapsed: 0.003 s
% 3.03/2.01 % (4795)Instructions burned: 7 (million)
% 3.03/2.01 % (4795)------------------------------
% 3.03/2.01 % (4795)------------------------------
% 3.03/2.01 % (4794)Instruction limit reached!
% 3.03/2.01 % (4794)------------------------------
% 3.03/2.01 % (4794)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/2.01 % (4794)Termination reason: Unknown
% 3.03/2.01 % (4794)Termination phase: Preprocessing 3
% 3.03/2.01
% 3.03/2.01 % (4794)Memory used [KB]: 2046
% 3.03/2.01 % (4794)Time elapsed: 0.013 s
% 3.03/2.01 % (4794)Instructions burned: 39 (million)
% 3.03/2.01 % (4794)------------------------------
% 3.03/2.01 % (4794)------------------------------
% 3.03/2.02 % (4776)Instruction limit reached!
% 3.03/2.02 % (4776)------------------------------
% 3.03/2.02 % (4776)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/2.02 % (4776)Termination reason: Unknown
% 3.03/2.02 % (4776)Termination phase: Saturation
% 3.03/2.02
% 3.03/2.02 % (4776)Memory used [KB]: 8827
% 3.03/2.02 % (4776)Time elapsed: 0.278 s
% 3.03/2.02 % (4776)Instructions burned: 779 (million)
% 3.03/2.02 % (4776)------------------------------
% 3.03/2.02 % (4776)------------------------------
% 3.03/2.02 % (4796)dis+1002_1:1_aac=none:au=on:e2e=on:nm=2:nwc=10.0:sd=1:ss=axioms:st=3.0:i=1087:si=on:rtra=on_0 on DTF2THF_4484 for (2996ds/1087Mi)
% 3.03/2.02 % (4797)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_4484 for (2996ds/47Mi)
% 5.29/2.03 % (4798)lrs+1010_1:128_av=off:cbe=off:cnfonf=lazy_simp:fsr=off:ntd=on:ss=axioms:st=1.5:i=44:si=on:rtra=on_0 on DTF2THF_4484 for (2996ds/44Mi)
% 5.29/2.04 % (4792)First to succeed.
% 5.29/2.04 % (4797)Instruction limit reached!
% 5.29/2.04 % (4797)------------------------------
% 5.29/2.04 % (4797)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.29/2.04 % (4797)Termination reason: Unknown
% 5.29/2.04 % (4797)Termination phase: Preprocessing 3
% 5.29/2.04
% 5.29/2.04 % (4797)Memory used [KB]: 2046
% 5.29/2.04 % (4797)Time elapsed: 0.016 s
% 5.29/2.04 % (4797)Instructions burned: 47 (million)
% 5.29/2.04 % (4797)------------------------------
% 5.29/2.04 % (4797)------------------------------
% 5.29/2.04 % (4793)Instruction limit reached!
% 5.29/2.04 % (4793)------------------------------
% 5.29/2.04 % (4793)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.29/2.04 % (4793)Termination reason: Unknown
% 5.29/2.04 % (4793)Termination phase: Saturation
% 5.29/2.04
% 5.29/2.04 % (4793)Memory used [KB]: 7164
% 5.29/2.04 % (4793)Time elapsed: 0.047 s
% 5.29/2.04 % (4793)Instructions burned: 151 (million)
% 5.29/2.04 % (4793)------------------------------
% 5.29/2.04 % (4793)------------------------------
% 5.29/2.04 % (4792)Refutation found. Thanks to Tanya!
% 5.29/2.04 % SZS status Theorem for DTF2THF_4484
% 5.29/2.04 % SZS output start Proof for DTF2THF_4484
% 5.29/2.04 thf(type_def_5, type, product_prod: ($tType * $tType) > $tType).
% 5.29/2.04 thf(type_def_6, type, set: ($tType) > $tType).
% 5.29/2.04 thf(type_def_8, type, variable: $tType).
% 5.29/2.04 thf(type_def_9, type, real: $tType).
% 5.29/2.04 thf(type_def_10, type, denotational_interp: $tType).
% 5.29/2.04 thf(type_def_11, type, game: $tType).
% 5.29/2.04 thf(type_def_12, type, option: ($tType) > $tType).
% 5.29/2.04 thf(type_def_13, type, char: $tType).
% 5.29/2.04 thf(type_def_14, type, trm: $tType).
% 5.29/2.04 thf(type_def_15, type, fml: $tType).
% 5.29/2.04 thf(func_def_0, type, denotational_interp: $tType).
% 5.29/2.04 thf(func_def_1, type, product_prod: ($tType * $tType) > $tType).
% 5.29/2.04 thf(func_def_2, type, variable: $tType).
% 5.29/2.04 thf(func_def_3, type, option: ($tType) > $tType).
% 5.29/2.04 thf(func_def_4, type, game: $tType).
% 5.29/2.04 thf(func_def_5, type, char: $tType).
% 5.29/2.04 thf(func_def_6, type, trm: $tType).
% 5.29/2.04 thf(func_def_7, type, fml: $tType).
% 5.29/2.04 thf(func_def_8, type, real: $tType).
% 5.29/2.04 thf(func_def_9, type, set: ($tType) > $tType).
% 5.29/2.04 thf(func_def_10, type, type: !>[X0: $tType]:($o)).
% 5.29/2.04 thf(func_def_11, type, uminus: !>[X0: $tType]:($o)).
% 5.29/2.04 thf(func_def_12, type, ord: !>[X0: $tType]:($o)).
% 5.29/2.04 thf(func_def_13, type, order: !>[X0: $tType]:($o)).
% 5.29/2.04 thf(func_def_14, type, group_add: !>[X0: $tType]:($o)).
% 5.29/2.04 thf(func_def_15, type, lattice: !>[X0: $tType]:($o)).
% 5.29/2.04 thf(func_def_16, type, linorder: !>[X0: $tType]:($o)).
% 5.29/2.04 thf(func_def_17, type, preorder: !>[X0: $tType]:($o)).
% 5.29/2.04 thf(func_def_18, type, boolean_algebra: !>[X0: $tType]:($o)).
% 5.29/2.04 thf(func_def_19, type, semilattice_sup: !>[X0: $tType]:($o)).
% 5.29/2.04 thf(func_def_20, type, ordered_ab_group_add: !>[X0: $tType]:($o)).
% 5.29/2.04 thf(func_def_21, type, bNF_Ca1785829860lChain: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > (X0 > X1) > $o)).
% 5.29/2.04 thf(func_def_22, type, denota1419872369iation: (variable > real) > (variable > real) > set @ variable > $o).
% 5.29/2.04 thf(func_def_23, type, denota1245701238me_sem: denotational_interp > game > set @ (variable > real) > set @ (variable > real)).
% 5.29/2.04 thf(func_def_24, type, uminus_uminus: !>[X0: $tType]:(X0 > X0)).
% 5.29/2.04 thf(func_def_25, type, sup_sup: !>[X0: $tType]:(X0 > X0 > X0)).
% 5.29/2.04 thf(func_def_26, type, none: !>[X0: $tType]:(option @ X0)).
% 5.29/2.04 thf(func_def_27, type, the: !>[X0: $tType]:(option @ X0 > X0)).
% 5.29/2.04 thf(func_def_28, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 5.29/2.04 thf(func_def_29, type, product_Pair: !>[X0: $tType, X1: $tType]:(X0 > X1 > product_prod @ X0 @ X1)).
% 5.29/2.04 thf(func_def_30, type, produc2004651681e_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 5.29/2.04 thf(func_def_31, type, product_rec_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 5.29/2.04 thf(func_def_32, type, product_fst: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 5.29/2.04 thf(func_def_33, type, product_snd: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 5.29/2.04 thf(func_def_34, type, product_swap: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > product_prod @ X1 @ X0)).
% 5.29/2.04 thf(func_def_35, type, product_scomp: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((X0 > product_prod @ X1 @ X2) > (X1 > X2 > X3) > X0 > X3)).
% 5.29/2.04 thf(func_def_36, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 5.29/2.04 thf(func_def_37, type, skip: game).
% 5.29/2.04 thf(func_def_38, type, choice: game > game > game).
% 5.29/2.04 thf(func_def_39, type, dual: game > game).
% 5.29/2.04 thf(func_def_40, type, loop: game > game).
% 5.29/2.04 thf(func_def_41, type, uSubst1976112797hoiceo: option @ game > option @ game > option @ game).
% 5.29/2.04 thf(func_def_42, type, uSubst739989314_Dualo: option @ game > option @ game).
% 5.29/2.04 thf(func_def_43, type, uSubst993936602_Loopo: option @ game > option @ game).
% 5.29/2.04 thf(func_def_44, type, uSubst2091380086djoint: product_prod @ (char > option @ trm) @ product_prod @ (char > option @ trm) @ product_prod @ (char > option @ fml) @ (char > option @ game) > denotational_interp > (variable > real) > denotational_interp).
% 5.29/2.04 thf(func_def_45, type, uSubst95898988stappp: product_prod @ (char > option @ trm) @ product_prod @ (char > option @ trm) @ product_prod @ (char > option @ fml) @ (char > option @ game) > set @ variable > game > product_prod @ set @ variable @ option @ game).
% 5.29/2.04 thf(func_def_46, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 5.29/2.04 thf(func_def_47, type, i: denotational_interp).
% 5.29/2.04 thf(func_def_48, type, u: set @ variable).
% 5.29/2.04 thf(func_def_49, type, ua: set @ variable).
% 5.29/2.04 thf(func_def_50, type, xa: set @ (variable > real)).
% 5.29/2.04 thf(func_def_51, type, alpha: game).
% 5.29/2.04 thf(func_def_52, type, nu: variable > real).
% 5.29/2.04 thf(func_def_53, type, nu2: variable > real).
% 5.29/2.04 thf(func_def_54, type, omega: variable > real).
% 5.29/2.04 thf(func_def_55, type, omega2: variable > real).
% 5.29/2.04 thf(func_def_56, type, sigma: product_prod @ (char > option @ trm) @ product_prod @ (char > option @ trm) @ product_prod @ (char > option @ fml) @ (char > option @ game)).
% 5.29/2.04 thf(func_def_72, type, sK0: !>[X0: $tType]:((X0 > $o) > X0)).
% 5.29/2.04 thf(func_def_73, type, sK1: !>[X0: $tType]:((X0 > $o) > X0)).
% 5.29/2.04 thf(func_def_74, type, sP2: !>[X0: $tType]:(X0 > X0 > X0 > $o)).
% 5.29/2.04 thf(func_def_75, type, sP3: !>[X0: $tType]:(X0 > X0 > X0 > $o)).
% 5.29/2.04 thf(func_def_76, type, sP4: !>[X0: $tType]:(X0 > X0 > X0 > $o)).
% 5.29/2.04 thf(func_def_77, type, sK5: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > X1)).
% 5.29/2.04 thf(func_def_78, type, sK6: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > X0)).
% 5.29/2.04 thf(func_def_79, type, sK7: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 5.29/2.04 thf(func_def_80, type, sK8: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 5.29/2.04 thf(func_def_81, type, sK9: denotational_interp > game > product_prod @ (char > option @ trm) @ product_prod @ (char > option @ trm) @ product_prod @ (char > option @ fml) @ (char > option @ game) > set @ variable > variable > real).
% 5.29/2.04 thf(func_def_82, type, sK10: denotational_interp > game > product_prod @ (char > option @ trm) @ product_prod @ (char > option @ trm) @ product_prod @ (char > option @ fml) @ (char > option @ game) > set @ variable > variable > real).
% 5.29/2.04 thf(func_def_83, type, sK11: denotational_interp > game > product_prod @ (char > option @ trm) @ product_prod @ (char > option @ trm) @ product_prod @ (char > option @ fml) @ (char > option @ game) > set @ variable > set @ (variable > real)).
% 5.29/2.04 thf(func_def_84, type, sK12: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 5.29/2.04 thf(func_def_85, type, sK13: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 5.29/2.04 thf(func_def_86, type, sK14: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 5.29/2.04 thf(func_def_87, type, sK15: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 5.29/2.04 thf(func_def_88, type, sK16: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 5.29/2.04 thf(func_def_89, type, sK17: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 5.29/2.04 thf(func_def_90, type, sK18: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X1 @ X0 > set @ product_prod @ X1 @ X0 > X1)).
% 5.29/2.04 thf(func_def_91, type, sK19: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X1 @ X0 > set @ product_prod @ X1 @ X0 > X0)).
% 5.29/2.04 thf(func_def_92, type, sK20: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 5.29/2.04 thf(func_def_93, type, sK21: !>[X0: $tType]:(set @ X0 > set @ X0 > X0)).
% 5.29/2.04 thf(func_def_94, type, sK22: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 5.29/2.04 thf(func_def_95, type, sK23: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > set @ X0 > X0)).
% 5.29/2.04 thf(func_def_96, type, sK24: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 5.29/2.04 thf(func_def_97, type, sK25: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 5.29/2.04 thf(func_def_98, type, sK26: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 5.29/2.04 thf(func_def_99, type, sK27: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 5.29/2.04 thf(func_def_100, type, sK28: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 5.29/2.04 thf(func_def_101, type, sK29: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 5.29/2.04 thf(func_def_102, type, sK30: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 5.29/2.04 thf(func_def_103, type, sK31: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 5.29/2.04 thf(func_def_104, type, sK32: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 5.29/2.04 thf(func_def_105, type, sK33: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 5.29/2.04 thf(func_def_106, type, sK34: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 5.29/2.04 thf(func_def_107, type, sK35: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 5.29/2.04 thf(func_def_108, type, sK36: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 5.29/2.04 thf(func_def_109, type, sK37: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 5.29/2.04 thf(func_def_110, type, sK38: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 5.29/2.04 thf(func_def_111, type, sK39: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 5.29/2.04 thf(func_def_112, type, sK40: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X0)).
% 5.29/2.04 thf(func_def_113, type, sK41: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X1)).
% 5.29/2.04 thf(func_def_114, type, sK42: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X2)).
% 5.29/2.04 thf(func_def_115, type, sK43: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X0)).
% 5.29/2.04 thf(func_def_116, type, sK44: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X1)).
% 5.29/2.04 thf(func_def_117, type, sK45: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X2)).
% 5.29/2.04 thf(func_def_118, type, sK46: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X3)).
% 5.29/2.04 thf(func_def_119, type, sK47: !>[X0: $tType]:((X0 > X0 > X0) > X0)).
% 5.29/2.04 thf(func_def_120, type, sK48: !>[X0: $tType]:((X0 > X0 > X0) > X0)).
% 5.29/2.04 thf(func_def_121, type, sK49: !>[X0: $tType]:((X0 > X0 > X0) > X0)).
% 5.29/2.04 thf(func_def_122, type, sK50: !>[X0: $tType]:((X0 > X0 > X0) > X0)).
% 5.29/2.04 thf(func_def_123, type, sK51: !>[X0: $tType]:((X0 > X0 > X0) > X0)).
% 5.29/2.04 thf(func_def_124, type, sK52: !>[X0: $tType]:((X0 > X0 > X0) > X0)).
% 5.29/2.04 thf(func_def_125, type, sK53: !>[X0: $tType]:((X0 > X0 > X0) > X0)).
% 5.29/2.04 thf(func_def_126, type, sK54: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 5.29/2.04 thf(func_def_127, type, sK55: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X5)).
% 5.29/2.04 thf(func_def_128, type, sK56: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 5.29/2.04 thf(func_def_129, type, sK57: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 5.29/2.04 thf(func_def_130, type, sK58: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 5.29/2.04 thf(func_def_131, type, sK59: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 5.29/2.04 thf(func_def_132, type, sK60: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 5.29/2.04 thf(func_def_133, type, sK61: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X0)).
% 5.29/2.04 thf(func_def_134, type, sK62: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X1)).
% 5.29/2.04 thf(func_def_135, type, sK63: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X2)).
% 5.29/2.04 thf(func_def_136, type, sK64: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X3)).
% 5.29/2.04 thf(func_def_137, type, sK65: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X4)).
% 5.29/2.04 thf(func_def_138, type, sK66: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0 > $o) > X1)).
% 5.29/2.04 thf(func_def_139, type, sK67: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0 > $o) > X0)).
% 5.29/2.04 thf(func_def_140, type, sK68: !>[X0: $tType]:(set @ X0 > set @ X0 > set @ X0 > set @ X0)).
% 5.29/2.04 thf(func_def_141, type, sK69: !>[X0: $tType]:(set @ X0 > set @ X0 > set @ X0 > set @ X0)).
% 5.29/2.04 thf(func_def_142, type, sK70: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 5.29/2.04 thf(func_def_143, type, sK71: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 5.29/2.04 thf(func_def_144, type, sK72: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 5.29/2.04 thf(func_def_145, type, sK73: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 5.29/2.04 thf(func_def_146, type, sK74: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 5.29/2.04 thf(func_def_147, type, sK75: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 5.29/2.04 thf(func_def_148, type, sK76: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 5.29/2.04 thf(func_def_149, type, sK77: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 5.29/2.04 thf(func_def_150, type, sK78: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 5.29/2.04 thf(func_def_151, type, sK79: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 5.29/2.04 thf(func_def_152, type, sK80: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 5.29/2.04 thf(func_def_153, type, sK81: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X0)).
% 5.29/2.04 thf(func_def_154, type, sK82: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X1)).
% 5.29/2.04 thf(func_def_155, type, sK83: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X2)).
% 5.29/2.04 thf(func_def_156, type, sK84: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X3)).
% 5.29/2.04 thf(func_def_157, type, sK85: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X4)).
% 5.29/2.04 thf(func_def_158, type, sK86: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X5)).
% 5.29/2.04 thf(func_def_159, type, sK87: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X6)).
% 5.29/2.04 thf(func_def_160, type, sK88: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X0)).
% 5.29/2.04 thf(func_def_161, type, sK89: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X1)).
% 5.29/2.04 thf(func_def_162, type, sK90: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X2)).
% 5.29/2.04 thf(func_def_163, type, sK91: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X3)).
% 5.29/2.04 thf(func_def_164, type, sK92: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X4)).
% 5.29/2.04 thf(func_def_165, type, sK93: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X5)).
% 5.29/2.04 thf(func_def_166, type, sK94: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X6)).
% 5.29/2.04 thf(func_def_167, type, sK95: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X5)).
% 5.29/2.04 thf(func_def_168, type, sK96: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 5.29/2.04 thf(func_def_169, type, sK97: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 5.29/2.04 thf(func_def_170, type, sK98: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 5.29/2.04 thf(func_def_171, type, sK99: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 5.29/2.04 thf(func_def_172, type, sK100: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 5.29/2.04 thf(func_def_174, type, ph102: !>[X0: $tType]:(X0)).
% 5.29/2.04 thf(f1734,plain,(
% 5.29/2.04 $false),
% 5.29/2.04 inference(subsumption_resolution,[status(thm)],[f1729,f1246])).
% 5.29/2.04 thf(f1246,plain,(
% 5.29/2.04 ( ! [X0 : $tType,X2 : set @ X0,X1 : X0] : (((member @ X0 @ X1 @ (uminus_uminus @ set @ X0 @ X2)) = (~ (member @ X0 @ X1 @ X2)))) )),
% 5.29/2.04 inference(cnf_transformation,[status(thm)],[f829])).
% 5.29/2.04 thf(f829,plain,(
% 5.29/2.04 ! [X0 : $tType,X1 : X0,X2 : set @ X0] : ((member @ X0 @ X1 @ (uminus_uminus @ set @ X0 @ X2)) = (~ (member @ X0 @ X1 @ X2)))),
% 5.29/2.04 inference(fool_elimination,[status(thm)],[f828])).
% 5.29/2.04 thf(f828,plain,(
% 5.29/2.04 ! [X0 : $tType,X1 : X0,X2 : set @ X0] : (~(member @ X0 @ X1 @ X2) = (member @ X0 @ X1 @ (uminus_uminus @ set @ X0 @ X2)))),
% 5.29/2.04 inference(rectify,[status(thm)],[f7])).
% 5.29/2.04 thf(f7,axiom,(
% 5.29/2.04 ! [X0 : $tType,X7 : X0,X8 : set @ X0] : (~(member @ X0 @ X7 @ X8) = (member @ X0 @ X7 @ (uminus_uminus @ set @ X0 @ X8)))),
% 5.29/2.04 file('/export/starexec/sandbox2/tmp/tmp.lOk4iEszTx/DTF2THF_4484.p',fact_6_Compl__iff)).
% 5.29/2.04 thf(f1729,plain,(
% 5.29/2.04 ((member @ (variable > real) @ nu2 @ (uminus_uminus @ set @ (variable > real) @ (denota1245701238me_sem @ (uSubst2091380086djoint @ sigma @ i @ omega2) @ alpha @ (uminus_uminus @ set @ (variable > real) @ xa)))) != (~ (member @ (variable > real) @ nu2 @ (denota1245701238me_sem @ (uSubst2091380086djoint @ sigma @ i @ omega2) @ alpha @ (uminus_uminus @ set @ (variable > real) @ xa)))))),
% 5.29/2.04 inference(superposition,[status(thm)],[f1617,f1418])).
% 5.29/2.04 thf(f1418,plain,(
% 5.29/2.04 ( ! [X0 : set @ (variable > real)] : (((member @ (variable > real) @ nu2 @ (denota1245701238me_sem @ i @ (the @ game @ (product_snd @ set @ variable @ option @ game @ (uSubst95898988stappp @ sigma @ ua @ alpha))) @ X0)) = (member @ (variable > real) @ nu2 @ (denota1245701238me_sem @ (uSubst2091380086djoint @ sigma @ i @ omega2) @ alpha @ X0)))) )),
% 5.29/2.04 inference(cnf_transformation,[status(thm)],[f714])).
% 5.29/2.04 thf(f714,plain,(
% 5.29/2.04 ! [X0 : set @ (variable > real)] : ((member @ (variable > real) @ nu2 @ (denota1245701238me_sem @ i @ (the @ game @ (product_snd @ set @ variable @ option @ game @ (uSubst95898988stappp @ sigma @ ua @ alpha))) @ X0)) = (member @ (variable > real) @ nu2 @ (denota1245701238me_sem @ (uSubst2091380086djoint @ sigma @ i @ omega2) @ alpha @ X0)))),
% 5.29/2.04 inference(fool_elimination,[status(thm)],[f713])).
% 5.29/2.04 thf(f713,plain,(
% 5.29/2.04 ! [X0 : set @ (variable > real)] : ((member @ (variable > real) @ nu2 @ (denota1245701238me_sem @ i @ (the @ game @ (product_snd @ set @ variable @ option @ game @ (uSubst95898988stappp @ sigma @ ua @ alpha))) @ X0)) = (member @ (variable > real) @ nu2 @ (denota1245701238me_sem @ (uSubst2091380086djoint @ sigma @ i @ omega2) @ alpha @ X0)))),
% 5.29/2.04 inference(rectify,[status(thm)],[f1])).
% 5.29/2.04 thf(f1,axiom,(
% 5.29/2.04 ! [X5 : set @ (variable > real)] : ((member @ (variable > real) @ nu2 @ (denota1245701238me_sem @ i @ (the @ game @ (product_snd @ set @ variable @ option @ game @ (uSubst95898988stappp @ sigma @ ua @ alpha))) @ X5)) = (member @ (variable > real) @ nu2 @ (denota1245701238me_sem @ (uSubst2091380086djoint @ sigma @ i @ omega2) @ alpha @ X5)))),
% 5.29/2.04 file('/export/starexec/sandbox2/tmp/tmp.lOk4iEszTx/DTF2THF_4484.p',fact_0__092_060open_062_092_060And_062X_O_A_I_092_060nu_062_A_092_060in_062_Agame__sem_AI_A_Ithe_A_Isnd_A_Iusubstappp_A_092_060sigma_062_AU_A_092_060alpha_062_J_J_J_AX_J_A_061_A_I_092_060nu_062_A_092_060in_062_Agame__sem_A_IUSubst__Mirabelle__nnnzepxswx_Oadjoint_A_092_060sigma_062_AI_A_092_060omega_062_J_A_092_060alpha_062_AX_J_092_060close_062)).
% 5.29/2.04 thf(f1617,plain,(
% 5.29/2.04 ((member @ (variable > real) @ nu2 @ (uminus_uminus @ set @ (variable > real) @ (denota1245701238me_sem @ (uSubst2091380086djoint @ sigma @ i @ omega2) @ alpha @ (uminus_uminus @ set @ (variable > real) @ xa)))) != (~ (member @ (variable > real) @ nu2 @ (denota1245701238me_sem @ i @ (the @ game @ (product_snd @ set @ variable @ option @ game @ (uSubst95898988stappp @ sigma @ ua @ alpha))) @ (uminus_uminus @ set @ (variable > real) @ xa)))))),
% 5.29/2.04 inference(superposition,[status(thm)],[f1461,f1246])).
% 5.29/2.04 thf(f1461,plain,(
% 5.29/2.04 ((member @ (variable > real) @ nu2 @ (uminus_uminus @ set @ (variable > real) @ (denota1245701238me_sem @ i @ (the @ game @ (product_snd @ set @ variable @ option @ game @ (uSubst95898988stappp @ sigma @ ua @ alpha))) @ (uminus_uminus @ set @ (variable > real) @ xa)))) != (member @ (variable > real) @ nu2 @ (uminus_uminus @ set @ (variable > real) @ (denota1245701238me_sem @ (uSubst2091380086djoint @ sigma @ i @ omega2) @ alpha @ (uminus_uminus @ set @ (variable > real) @ xa)))))),
% 5.29/2.04 inference(cnf_transformation,[status(thm)],[f893])).
% 5.29/2.04 thf(f893,plain,(
% 5.29/2.04 ((member @ (variable > real) @ nu2 @ (uminus_uminus @ set @ (variable > real) @ (denota1245701238me_sem @ i @ (the @ game @ (product_snd @ set @ variable @ option @ game @ (uSubst95898988stappp @ sigma @ ua @ alpha))) @ (uminus_uminus @ set @ (variable > real) @ xa)))) != (member @ (variable > real) @ nu2 @ (uminus_uminus @ set @ (variable > real) @ (denota1245701238me_sem @ (uSubst2091380086djoint @ sigma @ i @ omega2) @ alpha @ (uminus_uminus @ set @ (variable > real) @ xa)))))),
% 5.29/2.04 inference(flattening,[status(thm)],[f813])).
% 5.29/2.04 thf(f813,plain,(
% 5.29/2.04 ~((member @ (variable > real) @ nu2 @ (uminus_uminus @ set @ (variable > real) @ (denota1245701238me_sem @ i @ (the @ game @ (product_snd @ set @ variable @ option @ game @ (uSubst95898988stappp @ sigma @ ua @ alpha))) @ (uminus_uminus @ set @ (variable > real) @ xa)))) = (member @ (variable > real) @ nu2 @ (uminus_uminus @ set @ (variable > real) @ (denota1245701238me_sem @ (uSubst2091380086djoint @ sigma @ i @ omega2) @ alpha @ (uminus_uminus @ set @ (variable > real) @ xa)))))),
% 5.29/2.04 inference(fool_elimination,[status(thm)],[f290])).
% 5.29/2.04 thf(f290,negated_conjecture,(
% 5.29/2.04 ~((member @ (variable > real) @ nu2 @ (uminus_uminus @ set @ (variable > real) @ (denota1245701238me_sem @ i @ (the @ game @ (product_snd @ set @ variable @ option @ game @ (uSubst95898988stappp @ sigma @ ua @ alpha))) @ (uminus_uminus @ set @ (variable > real) @ xa)))) = (member @ (variable > real) @ nu2 @ (uminus_uminus @ set @ (variable > real) @ (denota1245701238me_sem @ (uSubst2091380086djoint @ sigma @ i @ omega2) @ alpha @ (uminus_uminus @ set @ (variable > real) @ xa)))))),
% 5.29/2.04 inference(negated_conjecture,[status(cth)],[f289])).
% 5.29/2.04 thf(f289,conjecture,(
% 5.29/2.04 ((member @ (variable > real) @ nu2 @ (uminus_uminus @ set @ (variable > real) @ (denota1245701238me_sem @ i @ (the @ game @ (product_snd @ set @ variable @ option @ game @ (uSubst95898988stappp @ sigma @ ua @ alpha))) @ (uminus_uminus @ set @ (variable > real) @ xa)))) = (member @ (variable > real) @ nu2 @ (uminus_uminus @ set @ (variable > real) @ (denota1245701238me_sem @ (uSubst2091380086djoint @ sigma @ i @ omega2) @ alpha @ (uminus_uminus @ set @ (variable > real) @ xa)))))),
% 5.29/2.04 file('/export/starexec/sandbox2/tmp/tmp.lOk4iEszTx/DTF2THF_4484.p',conj_0)).
% 5.29/2.04 % SZS output end Proof for DTF2THF_4484
% 5.29/2.04 % (4792)------------------------------
% 5.29/2.04 % (4792)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.29/2.04 % (4792)Termination reason: Refutation
% 5.29/2.04
% 5.29/2.04 % (4792)Memory used [KB]: 7547
% 5.29/2.04 % (4792)Time elapsed: 0.062 s
% 5.29/2.04 % (4792)Instructions burned: 199 (million)
% 5.29/2.04 % (4792)------------------------------
% 5.29/2.04 % (4792)------------------------------
% 5.29/2.04 % (4748)Success in time 0.398 s
%------------------------------------------------------------------------------