TPTP Problem File: ITP201^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP201^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer USubst problem prob_1580__6355392_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : USubst/prob_1580__6355392_1 [Des21]
% Status : Theorem
% Rating : 0.40 v8.2.0, 0.38 v8.1.0, 0.36 v7.5.0
% Syntax : Number of formulae : 438 ( 187 unt; 83 typ; 0 def)
% Number of atoms : 872 ( 335 equ; 0 cnn)
% Maximal formula atoms : 6 ( 2 avg)
% Number of connectives : 2594 ( 63 ~; 1 |; 42 &;2168 @)
% ( 0 <=>; 320 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 6 avg)
% Number of types : 16 ( 15 usr)
% Number of type conns : 268 ( 268 >; 0 *; 0 +; 0 <<)
% Number of symbols : 70 ( 68 usr; 9 con; 0-3 aty)
% Number of variables : 972 ( 142 ^; 819 !; 11 ?; 972 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:39:17.179
%------------------------------------------------------------------------------
% Could-be-implicit typings (15)
thf(ty_n_t__Product____Type__Oprod_I_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Otrm_J_J_Mt__Product____Type__Oprod_I_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Otrm_J_J_Mt__Product____Type__Oprod_I_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ofml_J_J_M_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ogame_J_J_J_J_J,type,
produc1418842292n_game: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_I_062_It__Syntax__Ovariable_Mt__Real__Oreal_J_J_Mt__Set__Oset_I_062_It__Syntax__Ovariable_Mt__Real__Oreal_J_J_J,type,
produc2038871085e_real: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Syntax__Ovariable_J_Mt__Set__Oset_I_062_It__Syntax__Ovariable_Mt__Real__Oreal_J_J_J,type,
produc1794985442e_real: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_I_062_It__Syntax__Ovariable_Mt__Real__Oreal_J_J_Mt__Set__Oset_It__Syntax__Ovariable_J_J,type,
produc1755325794riable: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Syntax__Ovariable_J_Mt__Option__Ooption_It__Syntax__Ogame_J_J_J,type,
set_Pr166476775n_game: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Syntax__Ovariable_J_Mt__Option__Ooption_It__Syntax__Ogame_J_J,type,
produc1078154247n_game: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Option__Ooption_It__Syntax__Ogame_J_Mt__Set__Oset_It__Syntax__Ovariable_J_J,type,
produc735959047riable: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Syntax__Ovariable_J_Mt__Set__Oset_It__Syntax__Ovariable_J_J,type,
produc432717079riable: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Syntax__Ovariable_Mt__Real__Oreal_J_J,type,
set_variable_real: $tType ).
thf(ty_n_t__Option__Ooption_It__Syntax__Ogame_J,type,
option_game: $tType ).
thf(ty_n_t__Denotational____Semantics__Ointerp,type,
denotational_interp: $tType ).
thf(ty_n_t__Set__Oset_It__Syntax__Ovariable_J,type,
set_variable: $tType ).
thf(ty_n_t__Syntax__Ovariable,type,
variable: $tType ).
thf(ty_n_t__Syntax__Ogame,type,
game: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
% Explicit typings (68)
thf(sy_c_Denotational__Semantics_OUvariation,type,
denota1419872369iation: ( variable > real ) > ( variable > real ) > set_variable > $o ).
thf(sy_c_Denotational__Semantics_Ogame__sem,type,
denota1245701238me_sem: denotational_interp > game > set_variable_real > set_variable_real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_I_062_It__Syntax__Ovariable_Mt__Real__Oreal_J_M_Eo_J,type,
uminus970116630real_o: ( ( variable > real ) > $o ) > ( variable > real ) > $o ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Product____Type__Oprod_It__Set__Oset_It__Syntax__Ovariable_J_Mt__Option__Ooption_It__Syntax__Ogame_J_J_M_Eo_J,type,
uminus679091071game_o: ( produc1078154247n_game > $o ) > produc1078154247n_game > $o ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Syntax__Ovariable_M_Eo_J,type,
uminus1666842273able_o: ( variable > $o ) > variable > $o ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001_Eo,type,
uminus_uminus_o: $o > $o ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Product____Type__Oprod_It__Set__Oset_I_062_It__Syntax__Ovariable_Mt__Real__Oreal_J_J_Mt__Set__Oset_I_062_It__Syntax__Ovariable_Mt__Real__Oreal_J_J_J,type,
uminus501400228e_real: produc2038871085e_real > produc2038871085e_real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Product____Type__Oprod_It__Set__Oset_I_062_It__Syntax__Ovariable_Mt__Real__Oreal_J_J_Mt__Set__Oset_It__Syntax__Ovariable_J_J,type,
uminus2115124697riable: produc1755325794riable > produc1755325794riable ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Product____Type__Oprod_It__Set__Oset_It__Syntax__Ovariable_J_Mt__Set__Oset_I_062_It__Syntax__Ovariable_Mt__Real__Oreal_J_J_J,type,
uminus7300697e_real: produc1794985442e_real > produc1794985442e_real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Product____Type__Oprod_It__Set__Oset_It__Syntax__Ovariable_J_Mt__Set__Oset_It__Syntax__Ovariable_J_J,type,
uminus269410190riable: produc432717079riable > produc432717079riable ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_I_062_It__Syntax__Ovariable_Mt__Real__Oreal_J_J,type,
uminus430703407e_real: set_variable_real > set_variable_real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Syntax__Ovariable_J_Mt__Option__Ooption_It__Syntax__Ogame_J_J_J,type,
uminus2030647006n_game: set_Pr166476775n_game > set_Pr166476775n_game ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Syntax__Ovariable_J,type,
uminus1851247844riable: set_variable > set_variable ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_062_It__Syntax__Ovariable_Mt__Real__Oreal_J_J,type,
sup_su1685293586e_real: set_variable_real > set_variable_real > set_variable_real ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Syntax__Ovariable_J,type,
sup_sup_set_variable: set_variable > set_variable > set_variable ).
thf(sy_c_Option_Ooption_ONone_001t__Syntax__Ogame,type,
none_game: option_game ).
thf(sy_c_Option_Ooption_Othe_001t__Syntax__Ogame,type,
the_game: option_game > game ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Syntax__Ovariable_Mt__Real__Oreal_J_M_Eo_J,type,
ord_le1354144447real_o: ( ( variable > real ) > $o ) > ( ( variable > real ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_It__Syntax__Ovariable_J_M_062_It__Option__Ooption_It__Syntax__Ogame_J_M_Eo_J_J,type,
ord_le2134856704game_o: ( set_variable > option_game > $o ) > ( set_variable > option_game > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Syntax__Ovariable_M_Eo_J,type,
ord_le1407353162able_o: ( variable > $o ) > ( variable > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Syntax__Ovariable_Mt__Real__Oreal_J_J,type,
ord_le1113654598e_real: set_variable_real > set_variable_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Syntax__Ovariable_J_Mt__Option__Ooption_It__Syntax__Ogame_J_J_J,type,
ord_le17855367n_game: set_Pr166476775n_game > set_Pr166476775n_game > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Syntax__Ovariable_J,type,
ord_le282106107riable: set_variable > set_variable > $o ).
thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Syntax__Ogame_J_001t__Set__Oset_It__Syntax__Ovariable_J,type,
produc1140431679riable: option_game > set_variable > produc735959047riable ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_I_062_It__Syntax__Ovariable_Mt__Real__Oreal_J_J_001t__Set__Oset_I_062_It__Syntax__Ovariable_Mt__Real__Oreal_J_J,type,
produc86945893e_real: set_variable_real > set_variable_real > produc2038871085e_real ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_I_062_It__Syntax__Ovariable_Mt__Real__Oreal_J_J_001t__Set__Oset_It__Syntax__Ovariable_J,type,
produc544095770riable: set_variable_real > set_variable > produc1755325794riable ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Syntax__Ovariable_J_001t__Option__Ooption_It__Syntax__Ogame_J,type,
produc1149443391n_game: set_variable > option_game > produc1078154247n_game ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Syntax__Ovariable_J_001t__Set__Oset_I_062_It__Syntax__Ovariable_Mt__Real__Oreal_J_J,type,
produc1400669210e_real: set_variable > set_variable_real > produc1794985442e_real ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Syntax__Ovariable_J_001t__Set__Oset_It__Syntax__Ovariable_J,type,
produc1766592463riable: set_variable > set_variable > produc432717079riable ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Option__Ooption_It__Syntax__Ogame_J_001t__Set__Oset_It__Syntax__Ovariable_J,type,
produc884810027riable: produc735959047riable > option_game ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Set__Oset_I_062_It__Syntax__Ovariable_Mt__Real__Oreal_J_J_001t__Set__Oset_I_062_It__Syntax__Ovariable_Mt__Real__Oreal_J_J,type,
produc702045265e_real: produc2038871085e_real > set_variable_real ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Set__Oset_I_062_It__Syntax__Ovariable_Mt__Real__Oreal_J_J_001t__Set__Oset_It__Syntax__Ovariable_J,type,
produc944271878riable: produc1755325794riable > set_variable_real ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Set__Oset_It__Syntax__Ovariable_J_001t__Option__Ooption_It__Syntax__Ogame_J,type,
produc893821739n_game: produc1078154247n_game > set_variable ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Set__Oset_It__Syntax__Ovariable_J_001t__Set__Oset_I_062_It__Syntax__Ovariable_Mt__Real__Oreal_J_J,type,
produc1800845318e_real: produc1794985442e_real > set_variable ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Set__Oset_It__Syntax__Ovariable_J_001t__Set__Oset_It__Syntax__Ovariable_J,type,
produc1300679611riable: produc432717079riable > set_variable ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Option__Ooption_It__Syntax__Ogame_J_001t__Set__Oset_It__Syntax__Ovariable_J,type,
produc284475501riable: produc735959047riable > set_variable ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Set__Oset_I_062_It__Syntax__Ovariable_Mt__Real__Oreal_J_J_001t__Set__Oset_I_062_It__Syntax__Ovariable_Mt__Real__Oreal_J_J,type,
produc220849299e_real: produc2038871085e_real > set_variable_real ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Set__Oset_I_062_It__Syntax__Ovariable_Mt__Real__Oreal_J_J_001t__Set__Oset_It__Syntax__Ovariable_J,type,
produc1192298056riable: produc1755325794riable > set_variable ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Set__Oset_It__Syntax__Ovariable_J_001t__Option__Ooption_It__Syntax__Ogame_J,type,
produc293487213n_game: produc1078154247n_game > option_game ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Set__Oset_It__Syntax__Ovariable_J_001t__Set__Oset_I_062_It__Syntax__Ovariable_Mt__Real__Oreal_J_J,type,
produc2048871496e_real: produc1794985442e_real > set_variable_real ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Set__Oset_It__Syntax__Ovariable_J_001t__Set__Oset_It__Syntax__Ovariable_J,type,
produc319712253riable: produc432717079riable > set_variable ).
thf(sy_c_Product__Type_Oprod_Oswap_001t__Option__Ooption_It__Syntax__Ogame_J_001t__Set__Oset_It__Syntax__Ovariable_J,type,
produc345397471riable: produc735959047riable > produc1078154247n_game ).
thf(sy_c_Product__Type_Oprod_Oswap_001t__Set__Oset_It__Syntax__Ovariable_J_001t__Option__Ooption_It__Syntax__Ogame_J,type,
produc354409183n_game: produc1078154247n_game > produc735959047riable ).
thf(sy_c_Set_OCollect_001_062_It__Syntax__Ovariable_Mt__Real__Oreal_J,type,
collec633296133e_real: ( ( variable > real ) > $o ) > set_variable_real ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Set__Oset_It__Syntax__Ovariable_J_Mt__Option__Ooption_It__Syntax__Ogame_J_J,type,
collec1702522994n_game: ( produc1078154247n_game > $o ) > set_Pr166476775n_game ).
thf(sy_c_Set_OCollect_001t__Syntax__Ovariable,type,
collect_variable: ( variable > $o ) > set_variable ).
thf(sy_c_Syntax_OSkip,type,
skip: game ).
thf(sy_c_Syntax_Ogame_OChoice,type,
choice: game > game > game ).
thf(sy_c_Syntax_Ogame_ODual,type,
dual: game > game ).
thf(sy_c_Syntax_Ogame_OLoop,type,
loop: game > game ).
thf(sy_c_USubst__Mirabelle__vidvnmlwwz_OChoiceo,type,
uSubst1484167963hoiceo: option_game > option_game > option_game ).
thf(sy_c_USubst__Mirabelle__vidvnmlwwz_ODualo,type,
uSubst1916713664_Dualo: option_game > option_game ).
thf(sy_c_USubst__Mirabelle__vidvnmlwwz_OLoopo,type,
uSubst23177304_Loopo: option_game > option_game ).
thf(sy_c_USubst__Mirabelle__vidvnmlwwz_Oadjoint,type,
uSubst1599435252djoint: produc1418842292n_game > denotational_interp > ( variable > real ) > denotational_interp ).
thf(sy_c_USubst__Mirabelle__vidvnmlwwz_Ousubstappp,type,
uSubst516392814stappp: produc1418842292n_game > set_variable > game > produc1078154247n_game ).
thf(sy_c_member_001_062_It__Syntax__Ovariable_Mt__Real__Oreal_J,type,
member_variable_real: ( variable > real ) > set_variable_real > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Syntax__Ovariable_J_Mt__Option__Ooption_It__Syntax__Ogame_J_J,type,
member171223600n_game: produc1078154247n_game > set_Pr166476775n_game > $o ).
thf(sy_c_member_001t__Syntax__Ovariable,type,
member_variable: variable > set_variable > $o ).
thf(sy_v_I,type,
i: denotational_interp ).
thf(sy_v_U,type,
u: set_variable ).
thf(sy_v_Ua____,type,
ua: set_variable ).
thf(sy_v_Xa____,type,
xa: set_variable_real ).
thf(sy_v__092_060alpha_062_H_H____,type,
alpha: game ).
thf(sy_v__092_060nu_062,type,
nu: variable > real ).
thf(sy_v__092_060nu_062_H____,type,
nu2: variable > real ).
thf(sy_v__092_060omega_062,type,
omega: variable > real ).
thf(sy_v__092_060omega_062_H____,type,
omega2: variable > real ).
thf(sy_v__092_060sigma_062_H_H____,type,
sigma: produc1418842292n_game ).
% Relevant facts (354)
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__vidvnmlwwz_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 @ ( produc293487213n_game @ ( uSubst516392814stappp @ sigma @ ua @ alpha ) ) ) @ X ) )
= ( member_variable_real @ nu2 @ ( denota1245701238me_sem @ ( uSubst1599435252djoint @ sigma @ i @ omega2 ) @ alpha @ X ) ) ) ).
% \<open>\<And>X. (\<nu> \<in> game_sem I (the (snd (usubstappp \<sigma> U \<alpha>))) X) = (\<nu> \<in> game_sem (USubst_Mirabelle_vidvnmlwwz.adjoint \<sigma> I \<omega>) \<alpha> X)\<close>
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 @ ( produc293487213n_game @ ( uSubst516392814stappp @ sigma @ ua @ alpha ) ) ) @ X ) )
= ( member_variable_real @ nu2 @ ( denota1245701238me_sem @ ( uSubst1599435252djoint @ sigma @ i @ omega2 ) @ alpha @ X ) ) ) ) ).
% IH\<alpha>
thf(fact_2_uv,axiom,
denota1419872369iation @ nu2 @ omega2 @ ua ).
% uv
thf(fact_3_snd__uminus,axiom,
! [X2: produc432717079riable] :
( ( produc319712253riable @ ( uminus269410190riable @ X2 ) )
= ( uminus1851247844riable @ ( produc319712253riable @ X2 ) ) ) ).
% snd_uminus
thf(fact_4_snd__uminus,axiom,
! [X2: produc1794985442e_real] :
( ( produc2048871496e_real @ ( uminus7300697e_real @ X2 ) )
= ( uminus430703407e_real @ ( produc2048871496e_real @ X2 ) ) ) ).
% snd_uminus
thf(fact_5_snd__uminus,axiom,
! [X2: produc1755325794riable] :
( ( produc1192298056riable @ ( uminus2115124697riable @ X2 ) )
= ( uminus1851247844riable @ ( produc1192298056riable @ X2 ) ) ) ).
% snd_uminus
thf(fact_6_snd__uminus,axiom,
! [X2: produc2038871085e_real] :
( ( produc220849299e_real @ ( uminus501400228e_real @ X2 ) )
= ( uminus430703407e_real @ ( produc220849299e_real @ X2 ) ) ) ).
% snd_uminus
thf(fact_7_def,axiom,
( ( produc293487213n_game @ ( uSubst516392814stappp @ sigma @ ua @ ( dual @ alpha ) ) )
!= none_game ) ).
% def
thf(fact_8_ComplI,axiom,
! [C: produc1078154247n_game,A: set_Pr166476775n_game] :
( ~ ( member171223600n_game @ C @ A )
=> ( member171223600n_game @ C @ ( uminus2030647006n_game @ A ) ) ) ).
% ComplI
thf(fact_9_ComplI,axiom,
! [C: variable,A: set_variable] :
( ~ ( member_variable @ C @ A )
=> ( member_variable @ C @ ( uminus1851247844riable @ A ) ) ) ).
% ComplI
thf(fact_10_ComplI,axiom,
! [C: variable > real,A: set_variable_real] :
( ~ ( member_variable_real @ C @ A )
=> ( member_variable_real @ C @ ( uminus430703407e_real @ A ) ) ) ).
% ComplI
thf(fact_11_Compl__iff,axiom,
! [C: produc1078154247n_game,A: set_Pr166476775n_game] :
( ( member171223600n_game @ C @ ( uminus2030647006n_game @ A ) )
= ( ~ ( member171223600n_game @ C @ A ) ) ) ).
% Compl_iff
thf(fact_12_Compl__iff,axiom,
! [C: variable,A: set_variable] :
( ( member_variable @ C @ ( uminus1851247844riable @ A ) )
= ( ~ ( member_variable @ C @ A ) ) ) ).
% Compl_iff
thf(fact_13_Compl__iff,axiom,
! [C: variable > real,A: set_variable_real] :
( ( member_variable_real @ C @ ( uminus430703407e_real @ A ) )
= ( ~ ( member_variable_real @ C @ A ) ) ) ).
% Compl_iff
thf(fact_14_Compl__eq__Compl__iff,axiom,
! [A: set_variable,B: set_variable] :
( ( ( uminus1851247844riable @ A )
= ( uminus1851247844riable @ B ) )
= ( A = B ) ) ).
% Compl_eq_Compl_iff
thf(fact_15_Compl__eq__Compl__iff,axiom,
! [A: set_variable_real,B: set_variable_real] :
( ( ( uminus430703407e_real @ A )
= ( uminus430703407e_real @ B ) )
= ( A = B ) ) ).
% Compl_eq_Compl_iff
thf(fact_16_uminus__apply,axiom,
( uminus1666842273able_o
= ( ^ [A2: variable > $o,X3: variable] : ( uminus_uminus_o @ ( A2 @ X3 ) ) ) ) ).
% uminus_apply
thf(fact_17_uminus__apply,axiom,
( uminus970116630real_o
= ( ^ [A2: ( variable > real ) > $o,X3: variable > real] : ( uminus_uminus_o @ ( A2 @ X3 ) ) ) ) ).
% uminus_apply
thf(fact_18_double__compl,axiom,
! [X2: variable > $o] :
( ( uminus1666842273able_o @ ( uminus1666842273able_o @ X2 ) )
= X2 ) ).
% double_compl
thf(fact_19_double__compl,axiom,
! [X2: ( variable > real ) > $o] :
( ( uminus970116630real_o @ ( uminus970116630real_o @ X2 ) )
= X2 ) ).
% double_compl
thf(fact_20_double__compl,axiom,
! [X2: set_variable_real] :
( ( uminus430703407e_real @ ( uminus430703407e_real @ X2 ) )
= X2 ) ).
% double_compl
thf(fact_21_double__compl,axiom,
! [X2: set_variable] :
( ( uminus1851247844riable @ ( uminus1851247844riable @ X2 ) )
= X2 ) ).
% double_compl
thf(fact_22_vaouter,axiom,
denota1419872369iation @ nu @ omega @ u ).
% vaouter
thf(fact_23_compl__eq__compl__iff,axiom,
! [X2: variable > $o,Y: variable > $o] :
( ( ( uminus1666842273able_o @ X2 )
= ( uminus1666842273able_o @ Y ) )
= ( X2 = Y ) ) ).
% compl_eq_compl_iff
thf(fact_24_compl__eq__compl__iff,axiom,
! [X2: ( variable > real ) > $o,Y: ( variable > real ) > $o] :
( ( ( uminus970116630real_o @ X2 )
= ( uminus970116630real_o @ Y ) )
= ( X2 = Y ) ) ).
% compl_eq_compl_iff
thf(fact_25_compl__eq__compl__iff,axiom,
! [X2: set_variable_real,Y: set_variable_real] :
( ( ( uminus430703407e_real @ X2 )
= ( uminus430703407e_real @ Y ) )
= ( X2 = Y ) ) ).
% compl_eq_compl_iff
thf(fact_26_compl__eq__compl__iff,axiom,
! [X2: set_variable,Y: set_variable] :
( ( ( uminus1851247844riable @ X2 )
= ( uminus1851247844riable @ Y ) )
= ( X2 = Y ) ) ).
% compl_eq_compl_iff
thf(fact_27_usubstappp__det,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,Alpha: game,V: set_variable] :
( ( ( produc293487213n_game @ ( uSubst516392814stappp @ Sigma @ U @ Alpha ) )
!= none_game )
=> ( ( ( produc293487213n_game @ ( uSubst516392814stappp @ Sigma @ V @ Alpha ) )
!= none_game )
=> ( ( produc293487213n_game @ ( uSubst516392814stappp @ Sigma @ U @ Alpha ) )
= ( produc293487213n_game @ ( uSubst516392814stappp @ Sigma @ V @ Alpha ) ) ) ) ) ).
% usubstappp_det
thf(fact_28_fun__Compl__def,axiom,
( uminus1666842273able_o
= ( ^ [A2: variable > $o,X3: variable] : ( uminus_uminus_o @ ( A2 @ X3 ) ) ) ) ).
% fun_Compl_def
thf(fact_29_fun__Compl__def,axiom,
( uminus970116630real_o
= ( ^ [A2: ( variable > real ) > $o,X3: variable > real] : ( uminus_uminus_o @ ( A2 @ X3 ) ) ) ) ).
% fun_Compl_def
thf(fact_30_double__complement,axiom,
! [A: set_variable_real] :
( ( uminus430703407e_real @ ( uminus430703407e_real @ A ) )
= A ) ).
% double_complement
thf(fact_31_double__complement,axiom,
! [A: set_variable] :
( ( uminus1851247844riable @ ( uminus1851247844riable @ A ) )
= A ) ).
% double_complement
thf(fact_32_ComplD,axiom,
! [C: produc1078154247n_game,A: set_Pr166476775n_game] :
( ( member171223600n_game @ C @ ( uminus2030647006n_game @ A ) )
=> ~ ( member171223600n_game @ C @ A ) ) ).
% ComplD
thf(fact_33_ComplD,axiom,
! [C: variable > real,A: set_variable_real] :
( ( member_variable_real @ C @ ( uminus430703407e_real @ A ) )
=> ~ ( member_variable_real @ C @ A ) ) ).
% ComplD
thf(fact_34_ComplD,axiom,
! [C: variable,A: set_variable] :
( ( member_variable @ C @ ( uminus1851247844riable @ A ) )
=> ~ ( member_variable @ C @ A ) ) ).
% ComplD
thf(fact_35_game__sem_Osimps_I7_J,axiom,
! [I: denotational_interp,Alpha: game] :
( ( denota1245701238me_sem @ I @ ( dual @ Alpha ) )
= ( ^ [X4: set_variable_real] : ( uminus430703407e_real @ ( denota1245701238me_sem @ I @ Alpha @ ( uminus430703407e_real @ X4 ) ) ) ) ) ).
% game_sem.simps(7)
thf(fact_36_Dual_OIH,axiom,
! [Nu: variable > real,Omega: variable > real,X: set_variable_real] :
( ( ( produc293487213n_game @ ( uSubst516392814stappp @ sigma @ ua @ alpha ) )
!= none_game )
=> ( ( denota1419872369iation @ Nu @ Omega @ ua )
=> ( ( member_variable_real @ Nu @ ( denota1245701238me_sem @ i @ ( the_game @ ( produc293487213n_game @ ( uSubst516392814stappp @ sigma @ ua @ alpha ) ) ) @ X ) )
= ( member_variable_real @ Nu @ ( denota1245701238me_sem @ ( uSubst1599435252djoint @ sigma @ i @ Omega ) @ alpha @ X ) ) ) ) ) ).
% Dual.IH
thf(fact_37_game_Oinject_I7_J,axiom,
! [X7: game,Y7: game] :
( ( ( dual @ X7 )
= ( dual @ Y7 ) )
= ( X7 = Y7 ) ) ).
% game.inject(7)
thf(fact_38_Dual_Oprems_I2_J,axiom,
denota1419872369iation @ nu2 @ omega2 @ ua ).
% Dual.prems(2)
thf(fact_39_option_Oexpand,axiom,
! [Option: option_game,Option2: option_game] :
( ( ( Option = none_game )
= ( Option2 = none_game ) )
=> ( ( ( Option != none_game )
=> ( ( Option2 != none_game )
=> ( ( the_game @ Option )
= ( the_game @ Option2 ) ) ) )
=> ( Option = Option2 ) ) ) ).
% option.expand
thf(fact_40_usubst__game__loop,axiom,
! [Nu: variable > real,Omega: variable > real,U: set_variable,Sigma: produc1418842292n_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 @ ( produc893821739n_game @ ( uSubst516392814stappp @ Sigma @ U @ Alpha ) ) )
=> ( ( ( produc293487213n_game @ ( uSubst516392814stappp @ Sigma @ ( produc893821739n_game @ ( uSubst516392814stappp @ Sigma @ U @ Alpha ) ) @ Alpha ) )
!= none_game )
=> ( ( member_variable_real @ Nu2 @ ( denota1245701238me_sem @ I @ ( the_game @ ( produc293487213n_game @ ( uSubst516392814stappp @ Sigma @ ( produc893821739n_game @ ( uSubst516392814stappp @ Sigma @ U @ Alpha ) ) @ Alpha ) ) ) @ X5 ) )
= ( member_variable_real @ Nu2 @ ( denota1245701238me_sem @ ( uSubst1599435252djoint @ Sigma @ I @ Omega2 ) @ Alpha @ X5 ) ) ) ) )
=> ( ( ( produc293487213n_game @ ( uSubst516392814stappp @ Sigma @ U @ ( loop @ Alpha ) ) )
!= none_game )
=> ( ( member_variable_real @ Nu @ ( denota1245701238me_sem @ I @ ( the_game @ ( produc293487213n_game @ ( uSubst516392814stappp @ Sigma @ U @ ( loop @ Alpha ) ) ) ) @ X ) )
= ( member_variable_real @ Nu @ ( denota1245701238me_sem @ ( uSubst1599435252djoint @ Sigma @ I @ Omega ) @ ( loop @ Alpha ) @ X ) ) ) ) ) ) ).
% usubst_game_loop
thf(fact_41_Uvariation__def,axiom,
( denota1419872369iation
= ( ^ [Nu3: variable > real,Nu4: variable > real,U2: set_variable] :
! [I2: variable] :
( ~ ( member_variable @ I2 @ U2 )
=> ( ( Nu3 @ I2 )
= ( Nu4 @ I2 ) ) ) ) ) ).
% Uvariation_def
thf(fact_42_Uvariation__sym,axiom,
( denota1419872369iation
= ( ^ [Omega3: variable > real,Nu3: variable > real] : ( denota1419872369iation @ Nu3 @ Omega3 ) ) ) ).
% Uvariation_sym
thf(fact_43_Uvariation__refl,axiom,
! [Nu: variable > real,V: set_variable] : ( denota1419872369iation @ Nu @ Nu @ V ) ).
% Uvariation_refl
thf(fact_44_Uvariation__sym__rel,axiom,
! [Omega: variable > real,Nu: variable > real,U: set_variable] :
( ( denota1419872369iation @ Omega @ Nu @ U )
=> ( denota1419872369iation @ Nu @ Omega @ U ) ) ).
% Uvariation_sym_rel
thf(fact_45_usubstappp__choice__conv,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,Alpha: game,Beta: game] :
( ( ( produc293487213n_game @ ( uSubst516392814stappp @ Sigma @ U @ ( choice @ Alpha @ Beta ) ) )
!= none_game )
=> ( ( ( produc293487213n_game @ ( uSubst516392814stappp @ Sigma @ U @ Alpha ) )
!= none_game )
& ( ( produc293487213n_game @ ( uSubst516392814stappp @ Sigma @ U @ Beta ) )
!= none_game ) ) ) ).
% usubstappp_choice_conv
thf(fact_46_game_Oinject_I6_J,axiom,
! [X6: game,Y6: game] :
( ( ( loop @ X6 )
= ( loop @ Y6 ) )
= ( X6 = Y6 ) ) ).
% game.inject(6)
thf(fact_47_game_Oinject_I4_J,axiom,
! [X41: game,X42: game,Y41: game,Y42: game] :
( ( ( choice @ X41 @ X42 )
= ( choice @ Y41 @ Y42 ) )
= ( ( X41 = Y41 )
& ( X42 = Y42 ) ) ) ).
% game.inject(4)
thf(fact_48_fst__uminus,axiom,
! [X2: produc432717079riable] :
( ( produc1300679611riable @ ( uminus269410190riable @ X2 ) )
= ( uminus1851247844riable @ ( produc1300679611riable @ X2 ) ) ) ).
% fst_uminus
thf(fact_49_fst__uminus,axiom,
! [X2: produc1794985442e_real] :
( ( produc1800845318e_real @ ( uminus7300697e_real @ X2 ) )
= ( uminus1851247844riable @ ( produc1800845318e_real @ X2 ) ) ) ).
% fst_uminus
thf(fact_50_fst__uminus,axiom,
! [X2: produc1755325794riable] :
( ( produc944271878riable @ ( uminus2115124697riable @ X2 ) )
= ( uminus430703407e_real @ ( produc944271878riable @ X2 ) ) ) ).
% fst_uminus
thf(fact_51_fst__uminus,axiom,
! [X2: produc2038871085e_real] :
( ( produc702045265e_real @ ( uminus501400228e_real @ X2 ) )
= ( uminus430703407e_real @ ( produc702045265e_real @ X2 ) ) ) ).
% fst_uminus
thf(fact_52_Uvariation__univ,axiom,
! [Nu: variable > real,Nu5: variable > real] :
( denota1419872369iation @ Nu @ Nu5
@ ( collect_variable
@ ^ [X3: variable] : $true ) ) ).
% Uvariation_univ
thf(fact_53_game_Odistinct_I39_J,axiom,
! [X41: game,X42: game,X6: game] :
( ( choice @ X41 @ X42 )
!= ( loop @ X6 ) ) ).
% game.distinct(39)
thf(fact_54_uminus__set__def,axiom,
( uminus2030647006n_game
= ( ^ [A2: set_Pr166476775n_game] :
( collec1702522994n_game
@ ( uminus679091071game_o
@ ^ [X3: produc1078154247n_game] : ( member171223600n_game @ X3 @ A2 ) ) ) ) ) ).
% uminus_set_def
thf(fact_55_uminus__set__def,axiom,
( uminus430703407e_real
= ( ^ [A2: set_variable_real] :
( collec633296133e_real
@ ( uminus970116630real_o
@ ^ [X3: variable > real] : ( member_variable_real @ X3 @ A2 ) ) ) ) ) ).
% uminus_set_def
thf(fact_56_uminus__set__def,axiom,
( uminus1851247844riable
= ( ^ [A2: set_variable] :
( collect_variable
@ ( uminus1666842273able_o
@ ^ [X3: variable] : ( member_variable @ X3 @ A2 ) ) ) ) ) ).
% uminus_set_def
thf(fact_57_Collect__neg__eq,axiom,
! [P: ( variable > real ) > $o] :
( ( collec633296133e_real
@ ^ [X3: variable > real] :
~ ( P @ X3 ) )
= ( uminus430703407e_real @ ( collec633296133e_real @ P ) ) ) ).
% Collect_neg_eq
thf(fact_58_Collect__neg__eq,axiom,
! [P: variable > $o] :
( ( collect_variable
@ ^ [X3: variable] :
~ ( P @ X3 ) )
= ( uminus1851247844riable @ ( collect_variable @ P ) ) ) ).
% Collect_neg_eq
thf(fact_59_Compl__eq,axiom,
( uminus2030647006n_game
= ( ^ [A2: set_Pr166476775n_game] :
( collec1702522994n_game
@ ^ [X3: produc1078154247n_game] :
~ ( member171223600n_game @ X3 @ A2 ) ) ) ) ).
% Compl_eq
thf(fact_60_Compl__eq,axiom,
( uminus430703407e_real
= ( ^ [A2: set_variable_real] :
( collec633296133e_real
@ ^ [X3: variable > real] :
~ ( member_variable_real @ X3 @ A2 ) ) ) ) ).
% Compl_eq
thf(fact_61_Compl__eq,axiom,
( uminus1851247844riable
= ( ^ [A2: set_variable] :
( collect_variable
@ ^ [X3: variable] :
~ ( member_variable @ X3 @ A2 ) ) ) ) ).
% Compl_eq
thf(fact_62_game_Odistinct_I51_J,axiom,
! [X6: game,X7: game] :
( ( loop @ X6 )
!= ( dual @ X7 ) ) ).
% game.distinct(51)
thf(fact_63_game_Odistinct_I41_J,axiom,
! [X41: game,X42: game,X7: game] :
( ( choice @ X41 @ X42 )
!= ( dual @ X7 ) ) ).
% game.distinct(41)
thf(fact_64_usubstappp__loop__conv,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,Alpha: game] :
( ( ( produc293487213n_game @ ( uSubst516392814stappp @ Sigma @ U @ ( loop @ Alpha ) ) )
!= none_game )
=> ( ( ( produc293487213n_game @ ( uSubst516392814stappp @ Sigma @ U @ Alpha ) )
!= none_game )
& ( ( produc293487213n_game @ ( uSubst516392814stappp @ Sigma @ ( produc893821739n_game @ ( uSubst516392814stappp @ Sigma @ U @ Alpha ) ) @ Alpha ) )
!= none_game ) ) ) ).
% usubstappp_loop_conv
thf(fact_65_mem__Collect__eq,axiom,
! [A3: produc1078154247n_game,P: produc1078154247n_game > $o] :
( ( member171223600n_game @ A3 @ ( collec1702522994n_game @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_66_mem__Collect__eq,axiom,
! [A3: variable > real,P: ( variable > real ) > $o] :
( ( member_variable_real @ A3 @ ( collec633296133e_real @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_67_mem__Collect__eq,axiom,
! [A3: variable,P: variable > $o] :
( ( member_variable @ A3 @ ( collect_variable @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_68_Collect__mem__eq,axiom,
! [A: set_Pr166476775n_game] :
( ( collec1702522994n_game
@ ^ [X3: produc1078154247n_game] : ( member171223600n_game @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_69_Collect__mem__eq,axiom,
! [A: set_variable_real] :
( ( collec633296133e_real
@ ^ [X3: variable > real] : ( member_variable_real @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_70_Collect__mem__eq,axiom,
! [A: set_variable] :
( ( collect_variable
@ ^ [X3: variable] : ( member_variable @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_71_Collect__cong,axiom,
! [P: ( variable > real ) > $o,Q: ( variable > real ) > $o] :
( ! [X8: variable > real] :
( ( P @ X8 )
= ( Q @ X8 ) )
=> ( ( collec633296133e_real @ P )
= ( collec633296133e_real @ Q ) ) ) ).
% Collect_cong
thf(fact_72_Collect__cong,axiom,
! [P: variable > $o,Q: variable > $o] :
( ! [X8: variable] :
( ( P @ X8 )
= ( Q @ X8 ) )
=> ( ( collect_variable @ P )
= ( collect_variable @ Q ) ) ) ).
% Collect_cong
thf(fact_73_skip__id,axiom,
! [I: denotational_interp,X: set_variable_real] :
( ( denota1245701238me_sem @ I @ skip @ X )
= X ) ).
% skip_id
thf(fact_74_prod__eqI,axiom,
! [P2: produc432717079riable,Q2: produc432717079riable] :
( ( ( produc1300679611riable @ P2 )
= ( produc1300679611riable @ Q2 ) )
=> ( ( ( produc319712253riable @ P2 )
= ( produc319712253riable @ Q2 ) )
=> ( P2 = Q2 ) ) ) ).
% prod_eqI
thf(fact_75_prod__eqI,axiom,
! [P2: produc1794985442e_real,Q2: produc1794985442e_real] :
( ( ( produc1800845318e_real @ P2 )
= ( produc1800845318e_real @ Q2 ) )
=> ( ( ( produc2048871496e_real @ P2 )
= ( produc2048871496e_real @ Q2 ) )
=> ( P2 = Q2 ) ) ) ).
% prod_eqI
thf(fact_76_prod__eqI,axiom,
! [P2: produc1755325794riable,Q2: produc1755325794riable] :
( ( ( produc944271878riable @ P2 )
= ( produc944271878riable @ Q2 ) )
=> ( ( ( produc1192298056riable @ P2 )
= ( produc1192298056riable @ Q2 ) )
=> ( P2 = Q2 ) ) ) ).
% prod_eqI
thf(fact_77_prod__eqI,axiom,
! [P2: produc2038871085e_real,Q2: produc2038871085e_real] :
( ( ( produc702045265e_real @ P2 )
= ( produc702045265e_real @ Q2 ) )
=> ( ( ( produc220849299e_real @ P2 )
= ( produc220849299e_real @ Q2 ) )
=> ( P2 = Q2 ) ) ) ).
% prod_eqI
thf(fact_78_prod__eqI,axiom,
! [P2: produc735959047riable,Q2: produc735959047riable] :
( ( ( produc884810027riable @ P2 )
= ( produc884810027riable @ Q2 ) )
=> ( ( ( produc284475501riable @ P2 )
= ( produc284475501riable @ Q2 ) )
=> ( P2 = Q2 ) ) ) ).
% prod_eqI
thf(fact_79_prod__eqI,axiom,
! [P2: produc1078154247n_game,Q2: produc1078154247n_game] :
( ( ( produc893821739n_game @ P2 )
= ( produc893821739n_game @ Q2 ) )
=> ( ( ( produc293487213n_game @ P2 )
= ( produc293487213n_game @ Q2 ) )
=> ( P2 = Q2 ) ) ) ).
% prod_eqI
thf(fact_80_exE__realizer_H,axiom,
! [P: set_variable_real > set_variable_real > $o,P2: produc2038871085e_real] :
( ( P @ ( produc220849299e_real @ P2 ) @ ( produc702045265e_real @ P2 ) )
=> ~ ! [X8: set_variable_real,Y2: set_variable_real] :
~ ( P @ Y2 @ X8 ) ) ).
% exE_realizer'
thf(fact_81_exE__realizer_H,axiom,
! [P: set_variable > option_game > $o,P2: produc735959047riable] :
( ( P @ ( produc284475501riable @ P2 ) @ ( produc884810027riable @ P2 ) )
=> ~ ! [X8: option_game,Y2: set_variable] :
~ ( P @ Y2 @ X8 ) ) ).
% exE_realizer'
thf(fact_82_exE__realizer_H,axiom,
! [P: option_game > set_variable > $o,P2: produc1078154247n_game] :
( ( P @ ( produc293487213n_game @ P2 ) @ ( produc893821739n_game @ P2 ) )
=> ~ ! [X8: set_variable,Y2: option_game] :
~ ( P @ Y2 @ X8 ) ) ).
% exE_realizer'
thf(fact_83_prod_Oexpand,axiom,
! [Prod: produc1078154247n_game,Prod2: produc1078154247n_game] :
( ( ( ( produc893821739n_game @ Prod )
= ( produc893821739n_game @ Prod2 ) )
& ( ( produc293487213n_game @ Prod )
= ( produc293487213n_game @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_84_prod__eq__iff,axiom,
( ( ^ [Y3: produc1078154247n_game,Z: produc1078154247n_game] : Y3 = Z )
= ( ^ [S: produc1078154247n_game,T: produc1078154247n_game] :
( ( ( produc893821739n_game @ S )
= ( produc893821739n_game @ T ) )
& ( ( produc293487213n_game @ S )
= ( produc293487213n_game @ T ) ) ) ) ) ).
% prod_eq_iff
thf(fact_85_uminus__prod__def,axiom,
( uminus501400228e_real
= ( ^ [X3: produc2038871085e_real] : ( produc86945893e_real @ ( uminus430703407e_real @ ( produc702045265e_real @ X3 ) ) @ ( uminus430703407e_real @ ( produc220849299e_real @ X3 ) ) ) ) ) ).
% uminus_prod_def
thf(fact_86_uminus__prod__def,axiom,
( uminus2115124697riable
= ( ^ [X3: produc1755325794riable] : ( produc544095770riable @ ( uminus430703407e_real @ ( produc944271878riable @ X3 ) ) @ ( uminus1851247844riable @ ( produc1192298056riable @ X3 ) ) ) ) ) ).
% uminus_prod_def
thf(fact_87_uminus__prod__def,axiom,
( uminus7300697e_real
= ( ^ [X3: produc1794985442e_real] : ( produc1400669210e_real @ ( uminus1851247844riable @ ( produc1800845318e_real @ X3 ) ) @ ( uminus430703407e_real @ ( produc2048871496e_real @ X3 ) ) ) ) ) ).
% uminus_prod_def
thf(fact_88_uminus__prod__def,axiom,
( uminus269410190riable
= ( ^ [X3: produc432717079riable] : ( produc1766592463riable @ ( uminus1851247844riable @ ( produc1300679611riable @ X3 ) ) @ ( uminus1851247844riable @ ( produc319712253riable @ X3 ) ) ) ) ) ).
% uminus_prod_def
thf(fact_89_usubstappp__antimon,axiom,
! [V: set_variable,U: set_variable,Sigma: produc1418842292n_game,Alpha: game] :
( ( ord_le282106107riable @ V @ U )
=> ( ( ( produc293487213n_game @ ( uSubst516392814stappp @ Sigma @ U @ Alpha ) )
!= none_game )
=> ( ( produc293487213n_game @ ( uSubst516392814stappp @ Sigma @ U @ Alpha ) )
= ( produc293487213n_game @ ( uSubst516392814stappp @ Sigma @ V @ Alpha ) ) ) ) ) ).
% usubstappp_antimon
thf(fact_90_exists__diff,axiom,
! [P: set_variable_real > $o] :
( ( ? [S2: set_variable_real] : ( P @ ( uminus430703407e_real @ S2 ) ) )
= ( ? [X9: set_variable_real] : ( P @ X9 ) ) ) ).
% exists_diff
thf(fact_91_exists__diff,axiom,
! [P: set_variable > $o] :
( ( ? [S2: set_variable] : ( P @ ( uminus1851247844riable @ S2 ) ) )
= ( ? [X9: set_variable] : ( P @ X9 ) ) ) ).
% exists_diff
thf(fact_92_uminus__Pair,axiom,
! [A3: set_variable_real,B2: set_variable_real] :
( ( uminus501400228e_real @ ( produc86945893e_real @ A3 @ B2 ) )
= ( produc86945893e_real @ ( uminus430703407e_real @ A3 ) @ ( uminus430703407e_real @ B2 ) ) ) ).
% uminus_Pair
thf(fact_93_uminus__Pair,axiom,
! [A3: set_variable_real,B2: set_variable] :
( ( uminus2115124697riable @ ( produc544095770riable @ A3 @ B2 ) )
= ( produc544095770riable @ ( uminus430703407e_real @ A3 ) @ ( uminus1851247844riable @ B2 ) ) ) ).
% uminus_Pair
thf(fact_94_uminus__Pair,axiom,
! [A3: set_variable,B2: set_variable_real] :
( ( uminus7300697e_real @ ( produc1400669210e_real @ A3 @ B2 ) )
= ( produc1400669210e_real @ ( uminus1851247844riable @ A3 ) @ ( uminus430703407e_real @ B2 ) ) ) ).
% uminus_Pair
thf(fact_95_uminus__Pair,axiom,
! [A3: set_variable,B2: set_variable] :
( ( uminus269410190riable @ ( produc1766592463riable @ A3 @ B2 ) )
= ( produc1766592463riable @ ( uminus1851247844riable @ A3 ) @ ( uminus1851247844riable @ B2 ) ) ) ).
% uminus_Pair
thf(fact_96_prod_Oinject,axiom,
! [X1: set_variable,X22: option_game,Y1: set_variable,Y22: option_game] :
( ( ( produc1149443391n_game @ X1 @ X22 )
= ( produc1149443391n_game @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_97_old_Oprod_Oinject,axiom,
! [A3: set_variable,B2: option_game,A4: set_variable,B3: option_game] :
( ( ( produc1149443391n_game @ A3 @ B2 )
= ( produc1149443391n_game @ A4 @ B3 ) )
= ( ( A3 = A4 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_98_subset__antisym,axiom,
! [A: set_variable,B: set_variable] :
( ( ord_le282106107riable @ A @ B )
=> ( ( ord_le282106107riable @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_99_subset__antisym,axiom,
! [A: set_variable_real,B: set_variable_real] :
( ( ord_le1113654598e_real @ A @ B )
=> ( ( ord_le1113654598e_real @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_100_subsetI,axiom,
! [A: set_variable,B: set_variable] :
( ! [X8: variable] :
( ( member_variable @ X8 @ A )
=> ( member_variable @ X8 @ B ) )
=> ( ord_le282106107riable @ A @ B ) ) ).
% subsetI
thf(fact_101_subsetI,axiom,
! [A: set_variable_real,B: set_variable_real] :
( ! [X8: variable > real] :
( ( member_variable_real @ X8 @ A )
=> ( member_variable_real @ X8 @ B ) )
=> ( ord_le1113654598e_real @ A @ B ) ) ).
% subsetI
thf(fact_102_compl__le__compl__iff,axiom,
! [X2: set_variable,Y: set_variable] :
( ( ord_le282106107riable @ ( uminus1851247844riable @ X2 ) @ ( uminus1851247844riable @ Y ) )
= ( ord_le282106107riable @ Y @ X2 ) ) ).
% compl_le_compl_iff
thf(fact_103_compl__le__compl__iff,axiom,
! [X2: set_variable_real,Y: set_variable_real] :
( ( ord_le1113654598e_real @ ( uminus430703407e_real @ X2 ) @ ( uminus430703407e_real @ Y ) )
= ( ord_le1113654598e_real @ Y @ X2 ) ) ).
% compl_le_compl_iff
thf(fact_104_Compl__subset__Compl__iff,axiom,
! [A: set_variable,B: set_variable] :
( ( ord_le282106107riable @ ( uminus1851247844riable @ A ) @ ( uminus1851247844riable @ B ) )
= ( ord_le282106107riable @ B @ A ) ) ).
% Compl_subset_Compl_iff
thf(fact_105_Compl__subset__Compl__iff,axiom,
! [A: set_variable_real,B: set_variable_real] :
( ( ord_le1113654598e_real @ ( uminus430703407e_real @ A ) @ ( uminus430703407e_real @ B ) )
= ( ord_le1113654598e_real @ B @ A ) ) ).
% Compl_subset_Compl_iff
thf(fact_106_Compl__anti__mono,axiom,
! [A: set_variable,B: set_variable] :
( ( ord_le282106107riable @ A @ B )
=> ( ord_le282106107riable @ ( uminus1851247844riable @ B ) @ ( uminus1851247844riable @ A ) ) ) ).
% Compl_anti_mono
thf(fact_107_Compl__anti__mono,axiom,
! [A: set_variable_real,B: set_variable_real] :
( ( ord_le1113654598e_real @ A @ B )
=> ( ord_le1113654598e_real @ ( uminus430703407e_real @ B ) @ ( uminus430703407e_real @ A ) ) ) ).
% Compl_anti_mono
thf(fact_108_prod_Ocollapse,axiom,
! [Prod: produc1078154247n_game] :
( ( produc1149443391n_game @ ( produc893821739n_game @ Prod ) @ ( produc293487213n_game @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_109_surj__pair,axiom,
! [P2: produc1078154247n_game] :
? [X8: set_variable,Y2: option_game] :
( P2
= ( produc1149443391n_game @ X8 @ Y2 ) ) ).
% surj_pair
thf(fact_110_prod__cases,axiom,
! [P: produc1078154247n_game > $o,P2: produc1078154247n_game] :
( ! [A5: set_variable,B4: option_game] : ( P @ ( produc1149443391n_game @ A5 @ B4 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_111_Pair__inject,axiom,
! [A3: set_variable,B2: option_game,A4: set_variable,B3: option_game] :
( ( ( produc1149443391n_game @ A3 @ B2 )
= ( produc1149443391n_game @ A4 @ B3 ) )
=> ~ ( ( A3 = A4 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_112_old_Oprod_Oexhaust,axiom,
! [Y: produc1078154247n_game] :
~ ! [A5: set_variable,B4: option_game] :
( Y
!= ( produc1149443391n_game @ A5 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_113_old_Oprod_Oinducts,axiom,
! [P: produc1078154247n_game > $o,Prod: produc1078154247n_game] :
( ! [A5: set_variable,B4: option_game] : ( P @ ( produc1149443391n_game @ A5 @ B4 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_114_Collect__mono__iff,axiom,
! [P: variable > $o,Q: variable > $o] :
( ( ord_le282106107riable @ ( collect_variable @ P ) @ ( collect_variable @ Q ) )
= ( ! [X3: variable] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_115_Collect__mono__iff,axiom,
! [P: ( variable > real ) > $o,Q: ( variable > real ) > $o] :
( ( ord_le1113654598e_real @ ( collec633296133e_real @ P ) @ ( collec633296133e_real @ Q ) )
= ( ! [X3: variable > real] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_116_set__eq__subset,axiom,
( ( ^ [Y3: set_variable,Z: set_variable] : Y3 = Z )
= ( ^ [A2: set_variable,B5: set_variable] :
( ( ord_le282106107riable @ A2 @ B5 )
& ( ord_le282106107riable @ B5 @ A2 ) ) ) ) ).
% set_eq_subset
thf(fact_117_set__eq__subset,axiom,
( ( ^ [Y3: set_variable_real,Z: set_variable_real] : Y3 = Z )
= ( ^ [A2: set_variable_real,B5: set_variable_real] :
( ( ord_le1113654598e_real @ A2 @ B5 )
& ( ord_le1113654598e_real @ B5 @ A2 ) ) ) ) ).
% set_eq_subset
thf(fact_118_subset__trans,axiom,
! [A: set_variable,B: set_variable,C2: set_variable] :
( ( ord_le282106107riable @ A @ B )
=> ( ( ord_le282106107riable @ B @ C2 )
=> ( ord_le282106107riable @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_119_subset__trans,axiom,
! [A: set_variable_real,B: set_variable_real,C2: set_variable_real] :
( ( ord_le1113654598e_real @ A @ B )
=> ( ( ord_le1113654598e_real @ B @ C2 )
=> ( ord_le1113654598e_real @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_120_Collect__mono,axiom,
! [P: variable > $o,Q: variable > $o] :
( ! [X8: variable] :
( ( P @ X8 )
=> ( Q @ X8 ) )
=> ( ord_le282106107riable @ ( collect_variable @ P ) @ ( collect_variable @ Q ) ) ) ).
% Collect_mono
thf(fact_121_Collect__mono,axiom,
! [P: ( variable > real ) > $o,Q: ( variable > real ) > $o] :
( ! [X8: variable > real] :
( ( P @ X8 )
=> ( Q @ X8 ) )
=> ( ord_le1113654598e_real @ ( collec633296133e_real @ P ) @ ( collec633296133e_real @ Q ) ) ) ).
% Collect_mono
thf(fact_122_subset__refl,axiom,
! [A: set_variable] : ( ord_le282106107riable @ A @ A ) ).
% subset_refl
thf(fact_123_subset__refl,axiom,
! [A: set_variable_real] : ( ord_le1113654598e_real @ A @ A ) ).
% subset_refl
thf(fact_124_subset__iff,axiom,
( ord_le282106107riable
= ( ^ [A2: set_variable,B5: set_variable] :
! [T: variable] :
( ( member_variable @ T @ A2 )
=> ( member_variable @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_125_subset__iff,axiom,
( ord_le1113654598e_real
= ( ^ [A2: set_variable_real,B5: set_variable_real] :
! [T: variable > real] :
( ( member_variable_real @ T @ A2 )
=> ( member_variable_real @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_126_equalityD2,axiom,
! [A: set_variable,B: set_variable] :
( ( A = B )
=> ( ord_le282106107riable @ B @ A ) ) ).
% equalityD2
thf(fact_127_equalityD2,axiom,
! [A: set_variable_real,B: set_variable_real] :
( ( A = B )
=> ( ord_le1113654598e_real @ B @ A ) ) ).
% equalityD2
thf(fact_128_equalityD1,axiom,
! [A: set_variable,B: set_variable] :
( ( A = B )
=> ( ord_le282106107riable @ A @ B ) ) ).
% equalityD1
thf(fact_129_equalityD1,axiom,
! [A: set_variable_real,B: set_variable_real] :
( ( A = B )
=> ( ord_le1113654598e_real @ A @ B ) ) ).
% equalityD1
thf(fact_130_subset__eq,axiom,
( ord_le282106107riable
= ( ^ [A2: set_variable,B5: set_variable] :
! [X3: variable] :
( ( member_variable @ X3 @ A2 )
=> ( member_variable @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_131_subset__eq,axiom,
( ord_le1113654598e_real
= ( ^ [A2: set_variable_real,B5: set_variable_real] :
! [X3: variable > real] :
( ( member_variable_real @ X3 @ A2 )
=> ( member_variable_real @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_132_equalityE,axiom,
! [A: set_variable,B: set_variable] :
( ( A = B )
=> ~ ( ( ord_le282106107riable @ A @ B )
=> ~ ( ord_le282106107riable @ B @ A ) ) ) ).
% equalityE
thf(fact_133_equalityE,axiom,
! [A: set_variable_real,B: set_variable_real] :
( ( A = B )
=> ~ ( ( ord_le1113654598e_real @ A @ B )
=> ~ ( ord_le1113654598e_real @ B @ A ) ) ) ).
% equalityE
thf(fact_134_subsetD,axiom,
! [A: set_variable,B: set_variable,C: variable] :
( ( ord_le282106107riable @ A @ B )
=> ( ( member_variable @ C @ A )
=> ( member_variable @ C @ B ) ) ) ).
% subsetD
thf(fact_135_subsetD,axiom,
! [A: set_variable_real,B: set_variable_real,C: variable > real] :
( ( ord_le1113654598e_real @ A @ B )
=> ( ( member_variable_real @ C @ A )
=> ( member_variable_real @ C @ B ) ) ) ).
% subsetD
thf(fact_136_in__mono,axiom,
! [A: set_variable,B: set_variable,X2: variable] :
( ( ord_le282106107riable @ A @ B )
=> ( ( member_variable @ X2 @ A )
=> ( member_variable @ X2 @ B ) ) ) ).
% in_mono
thf(fact_137_in__mono,axiom,
! [A: set_variable_real,B: set_variable_real,X2: variable > real] :
( ( ord_le1113654598e_real @ A @ B )
=> ( ( member_variable_real @ X2 @ A )
=> ( member_variable_real @ X2 @ B ) ) ) ).
% in_mono
thf(fact_138_Collect__subset,axiom,
! [A: set_variable,P: variable > $o] :
( ord_le282106107riable
@ ( collect_variable
@ ^ [X3: variable] :
( ( member_variable @ X3 @ A )
& ( P @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_139_Collect__subset,axiom,
! [A: set_variable_real,P: ( variable > real ) > $o] :
( ord_le1113654598e_real
@ ( collec633296133e_real
@ ^ [X3: variable > real] :
( ( member_variable_real @ X3 @ A )
& ( P @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_140_fst__eqD,axiom,
! [X2: set_variable,Y: option_game,A3: set_variable] :
( ( ( produc893821739n_game @ ( produc1149443391n_game @ X2 @ Y ) )
= A3 )
=> ( X2 = A3 ) ) ).
% fst_eqD
thf(fact_141_fst__conv,axiom,
! [X1: set_variable,X22: option_game] :
( ( produc893821739n_game @ ( produc1149443391n_game @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_142_snd__conv,axiom,
! [X1: set_variable,X22: option_game] :
( ( produc293487213n_game @ ( produc1149443391n_game @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_143_snd__eqD,axiom,
! [X2: set_variable,Y: option_game,A3: option_game] :
( ( ( produc293487213n_game @ ( produc1149443391n_game @ X2 @ Y ) )
= A3 )
=> ( Y = A3 ) ) ).
% snd_eqD
thf(fact_144_compl__le__swap2,axiom,
! [Y: set_variable,X2: set_variable] :
( ( ord_le282106107riable @ ( uminus1851247844riable @ Y ) @ X2 )
=> ( ord_le282106107riable @ ( uminus1851247844riable @ X2 ) @ Y ) ) ).
% compl_le_swap2
thf(fact_145_compl__le__swap2,axiom,
! [Y: set_variable_real,X2: set_variable_real] :
( ( ord_le1113654598e_real @ ( uminus430703407e_real @ Y ) @ X2 )
=> ( ord_le1113654598e_real @ ( uminus430703407e_real @ X2 ) @ Y ) ) ).
% compl_le_swap2
thf(fact_146_compl__le__swap1,axiom,
! [Y: set_variable,X2: set_variable] :
( ( ord_le282106107riable @ Y @ ( uminus1851247844riable @ X2 ) )
=> ( ord_le282106107riable @ X2 @ ( uminus1851247844riable @ Y ) ) ) ).
% compl_le_swap1
thf(fact_147_compl__le__swap1,axiom,
! [Y: set_variable_real,X2: set_variable_real] :
( ( ord_le1113654598e_real @ Y @ ( uminus430703407e_real @ X2 ) )
=> ( ord_le1113654598e_real @ X2 @ ( uminus430703407e_real @ Y ) ) ) ).
% compl_le_swap1
thf(fact_148_compl__mono,axiom,
! [X2: set_variable,Y: set_variable] :
( ( ord_le282106107riable @ X2 @ Y )
=> ( ord_le282106107riable @ ( uminus1851247844riable @ Y ) @ ( uminus1851247844riable @ X2 ) ) ) ).
% compl_mono
thf(fact_149_compl__mono,axiom,
! [X2: set_variable_real,Y: set_variable_real] :
( ( ord_le1113654598e_real @ X2 @ Y )
=> ( ord_le1113654598e_real @ ( uminus430703407e_real @ Y ) @ ( uminus430703407e_real @ X2 ) ) ) ).
% compl_mono
thf(fact_150_fst__pair,axiom,
! [A3: set_variable,B2: option_game] :
( ( produc893821739n_game @ ( produc1149443391n_game @ A3 @ B2 ) )
= A3 ) ).
% fst_pair
thf(fact_151_snd__pair,axiom,
! [A3: set_variable,B2: option_game] :
( ( produc293487213n_game @ ( produc1149443391n_game @ A3 @ B2 ) )
= B2 ) ).
% snd_pair
thf(fact_152_surjective__pairing,axiom,
! [T2: produc1078154247n_game] :
( T2
= ( produc1149443391n_game @ ( produc893821739n_game @ T2 ) @ ( produc293487213n_game @ T2 ) ) ) ).
% surjective_pairing
thf(fact_153_prod_Oexhaust__sel,axiom,
! [Prod: produc1078154247n_game] :
( Prod
= ( produc1149443391n_game @ ( produc893821739n_game @ Prod ) @ ( produc293487213n_game @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_154_conjI__realizer,axiom,
! [P: set_variable > $o,P2: set_variable,Q: option_game > $o,Q2: option_game] :
( ( P @ P2 )
=> ( ( Q @ Q2 )
=> ( ( P @ ( produc893821739n_game @ ( produc1149443391n_game @ P2 @ Q2 ) ) )
& ( Q @ ( produc293487213n_game @ ( produc1149443391n_game @ P2 @ Q2 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_155_exI__realizer,axiom,
! [P: option_game > set_variable > $o,Y: option_game,X2: set_variable] :
( ( P @ Y @ X2 )
=> ( P @ ( produc293487213n_game @ ( produc1149443391n_game @ X2 @ Y ) ) @ ( produc893821739n_game @ ( produc1149443391n_game @ X2 @ Y ) ) ) ) ).
% exI_realizer
thf(fact_156_Uvariation__mon,axiom,
! [U: set_variable,V: set_variable,Omega: variable > real,Nu: variable > real] :
( ( ord_le282106107riable @ U @ V )
=> ( ( denota1419872369iation @ Omega @ Nu @ U )
=> ( denota1419872369iation @ Omega @ Nu @ V ) ) ) ).
% Uvariation_mon
thf(fact_157_usubstappp__fst__mon,axiom,
! [U: set_variable,V: set_variable,Sigma: produc1418842292n_game,Alpha: game] :
( ( ord_le282106107riable @ U @ V )
=> ( ord_le282106107riable @ ( produc893821739n_game @ ( uSubst516392814stappp @ Sigma @ U @ Alpha ) ) @ ( produc893821739n_game @ ( uSubst516392814stappp @ Sigma @ V @ Alpha ) ) ) ) ).
% usubstappp_fst_mon
thf(fact_158_usubst__taboos__mon,axiom,
! [U: set_variable,Sigma: produc1418842292n_game,Alpha: game] : ( ord_le282106107riable @ U @ ( produc893821739n_game @ ( uSubst516392814stappp @ Sigma @ U @ Alpha ) ) ) ).
% usubst_taboos_mon
thf(fact_159_usubstappp__dual,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,Alpha: game] :
( ( uSubst516392814stappp @ Sigma @ U @ ( dual @ Alpha ) )
= ( produc1149443391n_game @ ( produc893821739n_game @ ( uSubst516392814stappp @ Sigma @ U @ Alpha ) ) @ ( uSubst1916713664_Dualo @ ( produc293487213n_game @ ( uSubst516392814stappp @ Sigma @ U @ Alpha ) ) ) ) ) ).
% usubstappp_dual
thf(fact_160_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: set_variable > option_game > $o,X2: set_variable,Y: option_game,A3: produc1078154247n_game] :
( ( P @ X2 @ Y )
=> ( ( A3
= ( produc1149443391n_game @ X2 @ Y ) )
=> ( P @ ( produc893821739n_game @ A3 ) @ ( produc293487213n_game @ A3 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_161_usubstappp_Osimps_I7_J,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,Alpha: game] :
( ( uSubst516392814stappp @ Sigma @ U @ ( dual @ Alpha ) )
= ( produc1149443391n_game @ ( produc893821739n_game @ ( uSubst516392814stappp @ Sigma @ U @ Alpha ) ) @ ( uSubst1916713664_Dualo @ ( produc293487213n_game @ ( uSubst516392814stappp @ Sigma @ U @ Alpha ) ) ) ) ) ).
% usubstappp.simps(7)
thf(fact_162_usubstappp_Osimps_I6_J,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,Alpha: game] :
( ( uSubst516392814stappp @ Sigma @ U @ ( loop @ Alpha ) )
= ( produc1149443391n_game @ ( produc893821739n_game @ ( uSubst516392814stappp @ Sigma @ U @ Alpha ) ) @ ( uSubst23177304_Loopo @ ( produc293487213n_game @ ( uSubst516392814stappp @ Sigma @ ( produc893821739n_game @ ( uSubst516392814stappp @ Sigma @ U @ Alpha ) ) @ Alpha ) ) ) ) ) ).
% usubstappp.simps(6)
thf(fact_163_usubstappp__loop,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,Alpha: game] :
( ( uSubst516392814stappp @ Sigma @ U @ ( loop @ Alpha ) )
= ( produc1149443391n_game @ ( produc893821739n_game @ ( uSubst516392814stappp @ Sigma @ U @ Alpha ) ) @ ( uSubst23177304_Loopo @ ( produc293487213n_game @ ( uSubst516392814stappp @ Sigma @ ( produc893821739n_game @ ( uSubst516392814stappp @ Sigma @ U @ Alpha ) ) @ Alpha ) ) ) ) ) ).
% usubstappp_loop
thf(fact_164_order__refl,axiom,
! [X2: set_variable] : ( ord_le282106107riable @ X2 @ X2 ) ).
% order_refl
thf(fact_165_order__refl,axiom,
! [X2: set_variable_real] : ( ord_le1113654598e_real @ X2 @ X2 ) ).
% order_refl
thf(fact_166_monotone,axiom,
! [X: set_variable_real,Y4: set_variable_real,I: denotational_interp,Alpha: game] :
( ( ord_le1113654598e_real @ X @ Y4 )
=> ( ord_le1113654598e_real @ ( denota1245701238me_sem @ I @ Alpha @ X ) @ ( denota1245701238me_sem @ I @ Alpha @ Y4 ) ) ) ).
% monotone
thf(fact_167_less__eq__set__def,axiom,
( ord_le282106107riable
= ( ^ [A2: set_variable,B5: set_variable] :
( ord_le1407353162able_o
@ ^ [X3: variable] : ( member_variable @ X3 @ A2 )
@ ^ [X3: variable] : ( member_variable @ X3 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_168_less__eq__set__def,axiom,
( ord_le1113654598e_real
= ( ^ [A2: set_variable_real,B5: set_variable_real] :
( ord_le1354144447real_o
@ ^ [X3: variable > real] : ( member_variable_real @ X3 @ A2 )
@ ^ [X3: variable > real] : ( member_variable_real @ X3 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_169_Loopo__undef,axiom,
! [Alpha: option_game] :
( ( ( uSubst23177304_Loopo @ Alpha )
= none_game )
= ( Alpha = none_game ) ) ).
% Loopo_undef
thf(fact_170_Dualo__undef,axiom,
! [Alpha: option_game] :
( ( ( uSubst1916713664_Dualo @ Alpha )
= none_game )
= ( Alpha = none_game ) ) ).
% Dualo_undef
thf(fact_171_Loopo_Osimps_I2_J,axiom,
( ( uSubst23177304_Loopo @ none_game )
= none_game ) ).
% Loopo.simps(2)
thf(fact_172_Dualo_Osimps_I2_J,axiom,
( ( uSubst1916713664_Dualo @ none_game )
= none_game ) ).
% Dualo.simps(2)
thf(fact_173_dual__order_Oantisym,axiom,
! [B2: set_variable,A3: set_variable] :
( ( ord_le282106107riable @ B2 @ A3 )
=> ( ( ord_le282106107riable @ A3 @ B2 )
=> ( A3 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_174_dual__order_Oantisym,axiom,
! [B2: set_variable_real,A3: set_variable_real] :
( ( ord_le1113654598e_real @ B2 @ A3 )
=> ( ( ord_le1113654598e_real @ A3 @ B2 )
=> ( A3 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_175_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_variable,Z: set_variable] : Y3 = Z )
= ( ^ [A6: set_variable,B6: set_variable] :
( ( ord_le282106107riable @ B6 @ A6 )
& ( ord_le282106107riable @ A6 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_176_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_variable_real,Z: set_variable_real] : Y3 = Z )
= ( ^ [A6: set_variable_real,B6: set_variable_real] :
( ( ord_le1113654598e_real @ B6 @ A6 )
& ( ord_le1113654598e_real @ A6 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_177_dual__order_Otrans,axiom,
! [B2: set_variable,A3: set_variable,C: set_variable] :
( ( ord_le282106107riable @ B2 @ A3 )
=> ( ( ord_le282106107riable @ C @ B2 )
=> ( ord_le282106107riable @ C @ A3 ) ) ) ).
% dual_order.trans
thf(fact_178_dual__order_Otrans,axiom,
! [B2: set_variable_real,A3: set_variable_real,C: set_variable_real] :
( ( ord_le1113654598e_real @ B2 @ A3 )
=> ( ( ord_le1113654598e_real @ C @ B2 )
=> ( ord_le1113654598e_real @ C @ A3 ) ) ) ).
% dual_order.trans
thf(fact_179_dual__order_Orefl,axiom,
! [A3: set_variable] : ( ord_le282106107riable @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_180_dual__order_Orefl,axiom,
! [A3: set_variable_real] : ( ord_le1113654598e_real @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_181_order__trans,axiom,
! [X2: set_variable,Y: set_variable,Z2: set_variable] :
( ( ord_le282106107riable @ X2 @ Y )
=> ( ( ord_le282106107riable @ Y @ Z2 )
=> ( ord_le282106107riable @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_182_order__trans,axiom,
! [X2: set_variable_real,Y: set_variable_real,Z2: set_variable_real] :
( ( ord_le1113654598e_real @ X2 @ Y )
=> ( ( ord_le1113654598e_real @ Y @ Z2 )
=> ( ord_le1113654598e_real @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_183_order__class_Oorder_Oantisym,axiom,
! [A3: set_variable,B2: set_variable] :
( ( ord_le282106107riable @ A3 @ B2 )
=> ( ( ord_le282106107riable @ B2 @ A3 )
=> ( A3 = B2 ) ) ) ).
% order_class.order.antisym
thf(fact_184_order__class_Oorder_Oantisym,axiom,
! [A3: set_variable_real,B2: set_variable_real] :
( ( ord_le1113654598e_real @ A3 @ B2 )
=> ( ( ord_le1113654598e_real @ B2 @ A3 )
=> ( A3 = B2 ) ) ) ).
% order_class.order.antisym
thf(fact_185_ord__le__eq__trans,axiom,
! [A3: set_variable,B2: set_variable,C: set_variable] :
( ( ord_le282106107riable @ A3 @ B2 )
=> ( ( B2 = C )
=> ( ord_le282106107riable @ A3 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_186_ord__le__eq__trans,axiom,
! [A3: set_variable_real,B2: set_variable_real,C: set_variable_real] :
( ( ord_le1113654598e_real @ A3 @ B2 )
=> ( ( B2 = C )
=> ( ord_le1113654598e_real @ A3 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_187_ord__eq__le__trans,axiom,
! [A3: set_variable,B2: set_variable,C: set_variable] :
( ( A3 = B2 )
=> ( ( ord_le282106107riable @ B2 @ C )
=> ( ord_le282106107riable @ A3 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_188_ord__eq__le__trans,axiom,
! [A3: set_variable_real,B2: set_variable_real,C: set_variable_real] :
( ( A3 = B2 )
=> ( ( ord_le1113654598e_real @ B2 @ C )
=> ( ord_le1113654598e_real @ A3 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_189_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y3: set_variable,Z: set_variable] : Y3 = Z )
= ( ^ [A6: set_variable,B6: set_variable] :
( ( ord_le282106107riable @ A6 @ B6 )
& ( ord_le282106107riable @ B6 @ A6 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_190_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y3: set_variable_real,Z: set_variable_real] : Y3 = Z )
= ( ^ [A6: set_variable_real,B6: set_variable_real] :
( ( ord_le1113654598e_real @ A6 @ B6 )
& ( ord_le1113654598e_real @ B6 @ A6 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_191_antisym__conv,axiom,
! [Y: set_variable,X2: set_variable] :
( ( ord_le282106107riable @ Y @ X2 )
=> ( ( ord_le282106107riable @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv
thf(fact_192_antisym__conv,axiom,
! [Y: set_variable_real,X2: set_variable_real] :
( ( ord_le1113654598e_real @ Y @ X2 )
=> ( ( ord_le1113654598e_real @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv
thf(fact_193_order_Otrans,axiom,
! [A3: set_variable,B2: set_variable,C: set_variable] :
( ( ord_le282106107riable @ A3 @ B2 )
=> ( ( ord_le282106107riable @ B2 @ C )
=> ( ord_le282106107riable @ A3 @ C ) ) ) ).
% order.trans
thf(fact_194_order_Otrans,axiom,
! [A3: set_variable_real,B2: set_variable_real,C: set_variable_real] :
( ( ord_le1113654598e_real @ A3 @ B2 )
=> ( ( ord_le1113654598e_real @ B2 @ C )
=> ( ord_le1113654598e_real @ A3 @ C ) ) ) ).
% order.trans
thf(fact_195_eq__refl,axiom,
! [X2: set_variable,Y: set_variable] :
( ( X2 = Y )
=> ( ord_le282106107riable @ X2 @ Y ) ) ).
% eq_refl
thf(fact_196_eq__refl,axiom,
! [X2: set_variable_real,Y: set_variable_real] :
( ( X2 = Y )
=> ( ord_le1113654598e_real @ X2 @ Y ) ) ).
% eq_refl
thf(fact_197_antisym,axiom,
! [X2: set_variable,Y: set_variable] :
( ( ord_le282106107riable @ X2 @ Y )
=> ( ( ord_le282106107riable @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% antisym
thf(fact_198_antisym,axiom,
! [X2: set_variable_real,Y: set_variable_real] :
( ( ord_le1113654598e_real @ X2 @ Y )
=> ( ( ord_le1113654598e_real @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% antisym
thf(fact_199_eq__iff,axiom,
( ( ^ [Y3: set_variable,Z: set_variable] : Y3 = Z )
= ( ^ [X3: set_variable,Y5: set_variable] :
( ( ord_le282106107riable @ X3 @ Y5 )
& ( ord_le282106107riable @ Y5 @ X3 ) ) ) ) ).
% eq_iff
thf(fact_200_eq__iff,axiom,
( ( ^ [Y3: set_variable_real,Z: set_variable_real] : Y3 = Z )
= ( ^ [X3: set_variable_real,Y5: set_variable_real] :
( ( ord_le1113654598e_real @ X3 @ Y5 )
& ( ord_le1113654598e_real @ Y5 @ X3 ) ) ) ) ).
% eq_iff
thf(fact_201_ord__le__eq__subst,axiom,
! [A3: set_variable,B2: set_variable,F: set_variable > set_variable,C: set_variable] :
( ( ord_le282106107riable @ A3 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X8: set_variable,Y2: set_variable] :
( ( ord_le282106107riable @ X8 @ Y2 )
=> ( ord_le282106107riable @ ( F @ X8 ) @ ( F @ Y2 ) ) )
=> ( ord_le282106107riable @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_202_ord__le__eq__subst,axiom,
! [A3: set_variable,B2: set_variable,F: set_variable > set_variable_real,C: set_variable_real] :
( ( ord_le282106107riable @ A3 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X8: set_variable,Y2: set_variable] :
( ( ord_le282106107riable @ X8 @ Y2 )
=> ( ord_le1113654598e_real @ ( F @ X8 ) @ ( F @ Y2 ) ) )
=> ( ord_le1113654598e_real @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_203_ord__le__eq__subst,axiom,
! [A3: set_variable_real,B2: set_variable_real,F: set_variable_real > set_variable,C: set_variable] :
( ( ord_le1113654598e_real @ A3 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X8: set_variable_real,Y2: set_variable_real] :
( ( ord_le1113654598e_real @ X8 @ Y2 )
=> ( ord_le282106107riable @ ( F @ X8 ) @ ( F @ Y2 ) ) )
=> ( ord_le282106107riable @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_204_ord__le__eq__subst,axiom,
! [A3: set_variable_real,B2: set_variable_real,F: set_variable_real > set_variable_real,C: set_variable_real] :
( ( ord_le1113654598e_real @ A3 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X8: set_variable_real,Y2: set_variable_real] :
( ( ord_le1113654598e_real @ X8 @ Y2 )
=> ( ord_le1113654598e_real @ ( F @ X8 ) @ ( F @ Y2 ) ) )
=> ( ord_le1113654598e_real @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_205_ord__eq__le__subst,axiom,
! [A3: set_variable,F: set_variable > set_variable,B2: set_variable,C: set_variable] :
( ( A3
= ( F @ B2 ) )
=> ( ( ord_le282106107riable @ B2 @ C )
=> ( ! [X8: set_variable,Y2: set_variable] :
( ( ord_le282106107riable @ X8 @ Y2 )
=> ( ord_le282106107riable @ ( F @ X8 ) @ ( F @ Y2 ) ) )
=> ( ord_le282106107riable @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_206_ord__eq__le__subst,axiom,
! [A3: set_variable_real,F: set_variable > set_variable_real,B2: set_variable,C: set_variable] :
( ( A3
= ( F @ B2 ) )
=> ( ( ord_le282106107riable @ B2 @ C )
=> ( ! [X8: set_variable,Y2: set_variable] :
( ( ord_le282106107riable @ X8 @ Y2 )
=> ( ord_le1113654598e_real @ ( F @ X8 ) @ ( F @ Y2 ) ) )
=> ( ord_le1113654598e_real @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_207_ord__eq__le__subst,axiom,
! [A3: set_variable,F: set_variable_real > set_variable,B2: set_variable_real,C: set_variable_real] :
( ( A3
= ( F @ B2 ) )
=> ( ( ord_le1113654598e_real @ B2 @ C )
=> ( ! [X8: set_variable_real,Y2: set_variable_real] :
( ( ord_le1113654598e_real @ X8 @ Y2 )
=> ( ord_le282106107riable @ ( F @ X8 ) @ ( F @ Y2 ) ) )
=> ( ord_le282106107riable @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_208_ord__eq__le__subst,axiom,
! [A3: set_variable_real,F: set_variable_real > set_variable_real,B2: set_variable_real,C: set_variable_real] :
( ( A3
= ( F @ B2 ) )
=> ( ( ord_le1113654598e_real @ B2 @ C )
=> ( ! [X8: set_variable_real,Y2: set_variable_real] :
( ( ord_le1113654598e_real @ X8 @ Y2 )
=> ( ord_le1113654598e_real @ ( F @ X8 ) @ ( F @ Y2 ) ) )
=> ( ord_le1113654598e_real @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_209_order__subst2,axiom,
! [A3: set_variable,B2: set_variable,F: set_variable > set_variable,C: set_variable] :
( ( ord_le282106107riable @ A3 @ B2 )
=> ( ( ord_le282106107riable @ ( F @ B2 ) @ C )
=> ( ! [X8: set_variable,Y2: set_variable] :
( ( ord_le282106107riable @ X8 @ Y2 )
=> ( ord_le282106107riable @ ( F @ X8 ) @ ( F @ Y2 ) ) )
=> ( ord_le282106107riable @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_210_order__subst2,axiom,
! [A3: set_variable,B2: set_variable,F: set_variable > set_variable_real,C: set_variable_real] :
( ( ord_le282106107riable @ A3 @ B2 )
=> ( ( ord_le1113654598e_real @ ( F @ B2 ) @ C )
=> ( ! [X8: set_variable,Y2: set_variable] :
( ( ord_le282106107riable @ X8 @ Y2 )
=> ( ord_le1113654598e_real @ ( F @ X8 ) @ ( F @ Y2 ) ) )
=> ( ord_le1113654598e_real @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_211_order__subst2,axiom,
! [A3: set_variable_real,B2: set_variable_real,F: set_variable_real > set_variable,C: set_variable] :
( ( ord_le1113654598e_real @ A3 @ B2 )
=> ( ( ord_le282106107riable @ ( F @ B2 ) @ C )
=> ( ! [X8: set_variable_real,Y2: set_variable_real] :
( ( ord_le1113654598e_real @ X8 @ Y2 )
=> ( ord_le282106107riable @ ( F @ X8 ) @ ( F @ Y2 ) ) )
=> ( ord_le282106107riable @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_212_order__subst2,axiom,
! [A3: set_variable_real,B2: set_variable_real,F: set_variable_real > set_variable_real,C: set_variable_real] :
( ( ord_le1113654598e_real @ A3 @ B2 )
=> ( ( ord_le1113654598e_real @ ( F @ B2 ) @ C )
=> ( ! [X8: set_variable_real,Y2: set_variable_real] :
( ( ord_le1113654598e_real @ X8 @ Y2 )
=> ( ord_le1113654598e_real @ ( F @ X8 ) @ ( F @ Y2 ) ) )
=> ( ord_le1113654598e_real @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_213_order__subst1,axiom,
! [A3: set_variable,F: set_variable > set_variable,B2: set_variable,C: set_variable] :
( ( ord_le282106107riable @ A3 @ ( F @ B2 ) )
=> ( ( ord_le282106107riable @ B2 @ C )
=> ( ! [X8: set_variable,Y2: set_variable] :
( ( ord_le282106107riable @ X8 @ Y2 )
=> ( ord_le282106107riable @ ( F @ X8 ) @ ( F @ Y2 ) ) )
=> ( ord_le282106107riable @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_214_order__subst1,axiom,
! [A3: set_variable,F: set_variable_real > set_variable,B2: set_variable_real,C: set_variable_real] :
( ( ord_le282106107riable @ A3 @ ( F @ B2 ) )
=> ( ( ord_le1113654598e_real @ B2 @ C )
=> ( ! [X8: set_variable_real,Y2: set_variable_real] :
( ( ord_le1113654598e_real @ X8 @ Y2 )
=> ( ord_le282106107riable @ ( F @ X8 ) @ ( F @ Y2 ) ) )
=> ( ord_le282106107riable @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_215_order__subst1,axiom,
! [A3: set_variable_real,F: set_variable > set_variable_real,B2: set_variable,C: set_variable] :
( ( ord_le1113654598e_real @ A3 @ ( F @ B2 ) )
=> ( ( ord_le282106107riable @ B2 @ C )
=> ( ! [X8: set_variable,Y2: set_variable] :
( ( ord_le282106107riable @ X8 @ Y2 )
=> ( ord_le1113654598e_real @ ( F @ X8 ) @ ( F @ Y2 ) ) )
=> ( ord_le1113654598e_real @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_216_order__subst1,axiom,
! [A3: set_variable_real,F: set_variable_real > set_variable_real,B2: set_variable_real,C: set_variable_real] :
( ( ord_le1113654598e_real @ A3 @ ( F @ B2 ) )
=> ( ( ord_le1113654598e_real @ B2 @ C )
=> ( ! [X8: set_variable_real,Y2: set_variable_real] :
( ( ord_le1113654598e_real @ X8 @ Y2 )
=> ( ord_le1113654598e_real @ ( F @ X8 ) @ ( F @ Y2 ) ) )
=> ( ord_le1113654598e_real @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_217_subset__CollectI,axiom,
! [B: set_variable,A: set_variable,Q: variable > $o,P: variable > $o] :
( ( ord_le282106107riable @ B @ A )
=> ( ! [X8: variable] :
( ( member_variable @ X8 @ B )
=> ( ( Q @ X8 )
=> ( P @ X8 ) ) )
=> ( ord_le282106107riable
@ ( collect_variable
@ ^ [X3: variable] :
( ( member_variable @ X3 @ B )
& ( Q @ X3 ) ) )
@ ( collect_variable
@ ^ [X3: variable] :
( ( member_variable @ X3 @ A )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_218_subset__CollectI,axiom,
! [B: set_variable_real,A: set_variable_real,Q: ( variable > real ) > $o,P: ( variable > real ) > $o] :
( ( ord_le1113654598e_real @ B @ A )
=> ( ! [X8: variable > real] :
( ( member_variable_real @ X8 @ B )
=> ( ( Q @ X8 )
=> ( P @ X8 ) ) )
=> ( ord_le1113654598e_real
@ ( collec633296133e_real
@ ^ [X3: variable > real] :
( ( member_variable_real @ X3 @ B )
& ( Q @ X3 ) ) )
@ ( collec633296133e_real
@ ^ [X3: variable > real] :
( ( member_variable_real @ X3 @ A )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_219_subset__Collect__iff,axiom,
! [B: set_variable,A: set_variable,P: variable > $o] :
( ( ord_le282106107riable @ B @ A )
=> ( ( ord_le282106107riable @ B
@ ( collect_variable
@ ^ [X3: variable] :
( ( member_variable @ X3 @ A )
& ( P @ X3 ) ) ) )
= ( ! [X3: variable] :
( ( member_variable @ X3 @ B )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_220_subset__Collect__iff,axiom,
! [B: set_variable_real,A: set_variable_real,P: ( variable > real ) > $o] :
( ( ord_le1113654598e_real @ B @ A )
=> ( ( ord_le1113654598e_real @ B
@ ( collec633296133e_real
@ ^ [X3: variable > real] :
( ( member_variable_real @ X3 @ A )
& ( P @ X3 ) ) ) )
= ( ! [X3: variable > real] :
( ( member_variable_real @ X3 @ B )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_221_sndI,axiom,
! [X2: produc1078154247n_game,Y: set_variable,Z2: option_game] :
( ( X2
= ( produc1149443391n_game @ Y @ Z2 ) )
=> ( ( produc293487213n_game @ X2 )
= Z2 ) ) ).
% sndI
thf(fact_222_eq__snd__iff,axiom,
! [B2: option_game,P2: produc1078154247n_game] :
( ( B2
= ( produc293487213n_game @ P2 ) )
= ( ? [A6: set_variable] :
( P2
= ( produc1149443391n_game @ A6 @ B2 ) ) ) ) ).
% eq_snd_iff
thf(fact_223_eq__fst__iff,axiom,
! [A3: set_variable,P2: produc1078154247n_game] :
( ( A3
= ( produc893821739n_game @ P2 ) )
= ( ? [B6: option_game] :
( P2
= ( produc1149443391n_game @ A3 @ B6 ) ) ) ) ).
% eq_fst_iff
thf(fact_224_fstI,axiom,
! [X2: produc1078154247n_game,Y: set_variable,Z2: option_game] :
( ( X2
= ( produc1149443391n_game @ Y @ Z2 ) )
=> ( ( produc893821739n_game @ X2 )
= Y ) ) ).
% fstI
thf(fact_225_pred__subset__eq,axiom,
! [R: set_variable,S3: set_variable] :
( ( ord_le1407353162able_o
@ ^ [X3: variable] : ( member_variable @ X3 @ R )
@ ^ [X3: variable] : ( member_variable @ X3 @ S3 ) )
= ( ord_le282106107riable @ R @ S3 ) ) ).
% pred_subset_eq
thf(fact_226_pred__subset__eq,axiom,
! [R: set_variable_real,S3: set_variable_real] :
( ( ord_le1354144447real_o
@ ^ [X3: variable > real] : ( member_variable_real @ X3 @ R )
@ ^ [X3: variable > real] : ( member_variable_real @ X3 @ S3 ) )
= ( ord_le1113654598e_real @ R @ S3 ) ) ).
% pred_subset_eq
thf(fact_227_Collect__restrict,axiom,
! [X: set_variable,P: variable > $o] :
( ord_le282106107riable
@ ( collect_variable
@ ^ [X3: variable] :
( ( member_variable @ X3 @ X )
& ( P @ X3 ) ) )
@ X ) ).
% Collect_restrict
thf(fact_228_Collect__restrict,axiom,
! [X: set_variable_real,P: ( variable > real ) > $o] :
( ord_le1113654598e_real
@ ( collec633296133e_real
@ ^ [X3: variable > real] :
( ( member_variable_real @ X3 @ X )
& ( P @ X3 ) ) )
@ X ) ).
% Collect_restrict
thf(fact_229_subrelI,axiom,
! [R2: set_Pr166476775n_game,S4: set_Pr166476775n_game] :
( ! [X8: set_variable,Y2: option_game] :
( ( member171223600n_game @ ( produc1149443391n_game @ X8 @ Y2 ) @ R2 )
=> ( member171223600n_game @ ( produc1149443391n_game @ X8 @ Y2 ) @ S4 ) )
=> ( ord_le17855367n_game @ R2 @ S4 ) ) ).
% subrelI
thf(fact_230_ssubst__Pair__rhs,axiom,
! [R2: set_variable,S4: option_game,R: set_Pr166476775n_game,S5: option_game] :
( ( member171223600n_game @ ( produc1149443391n_game @ R2 @ S4 ) @ R )
=> ( ( S5 = S4 )
=> ( member171223600n_game @ ( produc1149443391n_game @ R2 @ S5 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_231_pred__subset__eq2,axiom,
! [R: set_Pr166476775n_game,S3: set_Pr166476775n_game] :
( ( ord_le2134856704game_o
@ ^ [X3: set_variable,Y5: option_game] : ( member171223600n_game @ ( produc1149443391n_game @ X3 @ Y5 ) @ R )
@ ^ [X3: set_variable,Y5: option_game] : ( member171223600n_game @ ( produc1149443391n_game @ X3 @ Y5 ) @ S3 ) )
= ( ord_le17855367n_game @ R @ S3 ) ) ).
% pred_subset_eq2
thf(fact_232_pred__equals__eq2,axiom,
! [R: set_Pr166476775n_game,S3: set_Pr166476775n_game] :
( ( ( ^ [X3: set_variable,Y5: option_game] : ( member171223600n_game @ ( produc1149443391n_game @ X3 @ Y5 ) @ R ) )
= ( ^ [X3: set_variable,Y5: option_game] : ( member171223600n_game @ ( produc1149443391n_game @ X3 @ Y5 ) @ S3 ) ) )
= ( R = S3 ) ) ).
% pred_equals_eq2
thf(fact_233_prop__restrict,axiom,
! [X2: variable,Z3: set_variable,X: set_variable,P: variable > $o] :
( ( member_variable @ X2 @ Z3 )
=> ( ( ord_le282106107riable @ Z3
@ ( collect_variable
@ ^ [X3: variable] :
( ( member_variable @ X3 @ X )
& ( P @ X3 ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_234_prop__restrict,axiom,
! [X2: variable > real,Z3: set_variable_real,X: set_variable_real,P: ( variable > real ) > $o] :
( ( member_variable_real @ X2 @ Z3 )
=> ( ( ord_le1113654598e_real @ Z3
@ ( collec633296133e_real
@ ^ [X3: variable > real] :
( ( member_variable_real @ X3 @ X )
& ( P @ X3 ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_235_conj__subset__def,axiom,
! [A: set_variable,P: variable > $o,Q: variable > $o] :
( ( ord_le282106107riable @ A
@ ( collect_variable
@ ^ [X3: variable] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) )
= ( ( ord_le282106107riable @ A @ ( collect_variable @ P ) )
& ( ord_le282106107riable @ A @ ( collect_variable @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_236_conj__subset__def,axiom,
! [A: set_variable_real,P: ( variable > real ) > $o,Q: ( variable > real ) > $o] :
( ( ord_le1113654598e_real @ A
@ ( collec633296133e_real
@ ^ [X3: variable > real] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) )
= ( ( ord_le1113654598e_real @ A @ ( collec633296133e_real @ P ) )
& ( ord_le1113654598e_real @ A @ ( collec633296133e_real @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_237_prod_Oswap__def,axiom,
( produc345397471riable
= ( ^ [P3: produc735959047riable] : ( produc1149443391n_game @ ( produc284475501riable @ P3 ) @ ( produc884810027riable @ P3 ) ) ) ) ).
% prod.swap_def
thf(fact_238_prod_Oswap__def,axiom,
( produc354409183n_game
= ( ^ [P3: produc1078154247n_game] : ( produc1140431679riable @ ( produc293487213n_game @ P3 ) @ ( produc893821739n_game @ P3 ) ) ) ) ).
% prod.swap_def
thf(fact_239_swap__simp,axiom,
! [X2: option_game,Y: set_variable] :
( ( produc345397471riable @ ( produc1140431679riable @ X2 @ Y ) )
= ( produc1149443391n_game @ Y @ X2 ) ) ).
% swap_simp
thf(fact_240_swap__simp,axiom,
! [X2: set_variable,Y: option_game] :
( ( produc354409183n_game @ ( produc1149443391n_game @ X2 @ Y ) )
= ( produc1140431679riable @ Y @ X2 ) ) ).
% swap_simp
thf(fact_241_snd__swap,axiom,
! [X2: produc1078154247n_game] :
( ( produc284475501riable @ ( produc354409183n_game @ X2 ) )
= ( produc893821739n_game @ X2 ) ) ).
% snd_swap
thf(fact_242_snd__swap,axiom,
! [X2: produc735959047riable] :
( ( produc293487213n_game @ ( produc345397471riable @ X2 ) )
= ( produc884810027riable @ X2 ) ) ).
% snd_swap
thf(fact_243_fst__swap,axiom,
! [X2: produc735959047riable] :
( ( produc893821739n_game @ ( produc345397471riable @ X2 ) )
= ( produc284475501riable @ X2 ) ) ).
% fst_swap
thf(fact_244_fst__swap,axiom,
! [X2: produc1078154247n_game] :
( ( produc884810027riable @ ( produc354409183n_game @ X2 ) )
= ( produc293487213n_game @ X2 ) ) ).
% fst_swap
thf(fact_245_usubstappp__choice,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,Alpha: game,Beta: game] :
( ( uSubst516392814stappp @ Sigma @ U @ ( choice @ Alpha @ Beta ) )
= ( produc1149443391n_game @ ( sup_sup_set_variable @ ( produc893821739n_game @ ( uSubst516392814stappp @ Sigma @ U @ Alpha ) ) @ ( produc893821739n_game @ ( uSubst516392814stappp @ Sigma @ U @ Beta ) ) ) @ ( uSubst1484167963hoiceo @ ( produc293487213n_game @ ( uSubst516392814stappp @ Sigma @ U @ Alpha ) ) @ ( produc293487213n_game @ ( uSubst516392814stappp @ Sigma @ U @ Beta ) ) ) ) ) ).
% usubstappp_choice
thf(fact_246_sup_Oidem,axiom,
! [A3: set_variable] :
( ( sup_sup_set_variable @ A3 @ A3 )
= A3 ) ).
% sup.idem
thf(fact_247_sup__idem,axiom,
! [X2: set_variable] :
( ( sup_sup_set_variable @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_248_sup_Oleft__idem,axiom,
! [A3: set_variable,B2: set_variable] :
( ( sup_sup_set_variable @ A3 @ ( sup_sup_set_variable @ A3 @ B2 ) )
= ( sup_sup_set_variable @ A3 @ B2 ) ) ).
% sup.left_idem
thf(fact_249_sup__left__idem,axiom,
! [X2: set_variable,Y: set_variable] :
( ( sup_sup_set_variable @ X2 @ ( sup_sup_set_variable @ X2 @ Y ) )
= ( sup_sup_set_variable @ X2 @ Y ) ) ).
% sup_left_idem
thf(fact_250_sup_Oright__idem,axiom,
! [A3: set_variable,B2: set_variable] :
( ( sup_sup_set_variable @ ( sup_sup_set_variable @ A3 @ B2 ) @ B2 )
= ( sup_sup_set_variable @ A3 @ B2 ) ) ).
% sup.right_idem
thf(fact_251_UnCI,axiom,
! [C: variable > real,B: set_variable_real,A: set_variable_real] :
( ( ~ ( member_variable_real @ C @ B )
=> ( member_variable_real @ C @ A ) )
=> ( member_variable_real @ C @ ( sup_su1685293586e_real @ A @ B ) ) ) ).
% UnCI
thf(fact_252_UnCI,axiom,
! [C: variable,B: set_variable,A: set_variable] :
( ( ~ ( member_variable @ C @ B )
=> ( member_variable @ C @ A ) )
=> ( member_variable @ C @ ( sup_sup_set_variable @ A @ B ) ) ) ).
% UnCI
thf(fact_253_sup_Obounded__iff,axiom,
! [B2: set_variable,C: set_variable,A3: set_variable] :
( ( ord_le282106107riable @ ( sup_sup_set_variable @ B2 @ C ) @ A3 )
= ( ( ord_le282106107riable @ B2 @ A3 )
& ( ord_le282106107riable @ C @ A3 ) ) ) ).
% sup.bounded_iff
thf(fact_254_sup_Obounded__iff,axiom,
! [B2: set_variable_real,C: set_variable_real,A3: set_variable_real] :
( ( ord_le1113654598e_real @ ( sup_su1685293586e_real @ B2 @ C ) @ A3 )
= ( ( ord_le1113654598e_real @ B2 @ A3 )
& ( ord_le1113654598e_real @ C @ A3 ) ) ) ).
% sup.bounded_iff
thf(fact_255_le__sup__iff,axiom,
! [X2: set_variable,Y: set_variable,Z2: set_variable] :
( ( ord_le282106107riable @ ( sup_sup_set_variable @ X2 @ Y ) @ Z2 )
= ( ( ord_le282106107riable @ X2 @ Z2 )
& ( ord_le282106107riable @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_256_le__sup__iff,axiom,
! [X2: set_variable_real,Y: set_variable_real,Z2: set_variable_real] :
( ( ord_le1113654598e_real @ ( sup_su1685293586e_real @ X2 @ Y ) @ Z2 )
= ( ( ord_le1113654598e_real @ X2 @ Z2 )
& ( ord_le1113654598e_real @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_257_Un__subset__iff,axiom,
! [A: set_variable,B: set_variable,C2: set_variable] :
( ( ord_le282106107riable @ ( sup_sup_set_variable @ A @ B ) @ C2 )
= ( ( ord_le282106107riable @ A @ C2 )
& ( ord_le282106107riable @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_258_Un__subset__iff,axiom,
! [A: set_variable_real,B: set_variable_real,C2: set_variable_real] :
( ( ord_le1113654598e_real @ ( sup_su1685293586e_real @ A @ B ) @ C2 )
= ( ( ord_le1113654598e_real @ A @ C2 )
& ( ord_le1113654598e_real @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_259_subset__Un__eq,axiom,
( ord_le282106107riable
= ( ^ [A2: set_variable,B5: set_variable] :
( ( sup_sup_set_variable @ A2 @ B5 )
= B5 ) ) ) ).
% subset_Un_eq
thf(fact_260_subset__Un__eq,axiom,
( ord_le1113654598e_real
= ( ^ [A2: set_variable_real,B5: set_variable_real] :
( ( sup_su1685293586e_real @ A2 @ B5 )
= B5 ) ) ) ).
% subset_Un_eq
thf(fact_261_subset__UnE,axiom,
! [C2: set_variable,A: set_variable,B: set_variable] :
( ( ord_le282106107riable @ C2 @ ( sup_sup_set_variable @ A @ B ) )
=> ~ ! [A7: set_variable] :
( ( ord_le282106107riable @ A7 @ A )
=> ! [B7: set_variable] :
( ( ord_le282106107riable @ B7 @ B )
=> ( C2
!= ( sup_sup_set_variable @ A7 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_262_subset__UnE,axiom,
! [C2: set_variable_real,A: set_variable_real,B: set_variable_real] :
( ( ord_le1113654598e_real @ C2 @ ( sup_su1685293586e_real @ A @ B ) )
=> ~ ! [A7: set_variable_real] :
( ( ord_le1113654598e_real @ A7 @ A )
=> ! [B7: set_variable_real] :
( ( ord_le1113654598e_real @ B7 @ B )
=> ( C2
!= ( sup_su1685293586e_real @ A7 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_263_Un__absorb2,axiom,
! [B: set_variable,A: set_variable] :
( ( ord_le282106107riable @ B @ A )
=> ( ( sup_sup_set_variable @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_264_Un__absorb2,axiom,
! [B: set_variable_real,A: set_variable_real] :
( ( ord_le1113654598e_real @ B @ A )
=> ( ( sup_su1685293586e_real @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_265_Un__absorb1,axiom,
! [A: set_variable,B: set_variable] :
( ( ord_le282106107riable @ A @ B )
=> ( ( sup_sup_set_variable @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_266_Un__absorb1,axiom,
! [A: set_variable_real,B: set_variable_real] :
( ( ord_le1113654598e_real @ A @ B )
=> ( ( sup_su1685293586e_real @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_267_Un__upper2,axiom,
! [B: set_variable,A: set_variable] : ( ord_le282106107riable @ B @ ( sup_sup_set_variable @ A @ B ) ) ).
% Un_upper2
thf(fact_268_Un__upper2,axiom,
! [B: set_variable_real,A: set_variable_real] : ( ord_le1113654598e_real @ B @ ( sup_su1685293586e_real @ A @ B ) ) ).
% Un_upper2
thf(fact_269_Un__upper1,axiom,
! [A: set_variable,B: set_variable] : ( ord_le282106107riable @ A @ ( sup_sup_set_variable @ A @ B ) ) ).
% Un_upper1
thf(fact_270_Un__upper1,axiom,
! [A: set_variable_real,B: set_variable_real] : ( ord_le1113654598e_real @ A @ ( sup_su1685293586e_real @ A @ B ) ) ).
% Un_upper1
thf(fact_271_Un__least,axiom,
! [A: set_variable,C2: set_variable,B: set_variable] :
( ( ord_le282106107riable @ A @ C2 )
=> ( ( ord_le282106107riable @ B @ C2 )
=> ( ord_le282106107riable @ ( sup_sup_set_variable @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_272_Un__least,axiom,
! [A: set_variable_real,C2: set_variable_real,B: set_variable_real] :
( ( ord_le1113654598e_real @ A @ C2 )
=> ( ( ord_le1113654598e_real @ B @ C2 )
=> ( ord_le1113654598e_real @ ( sup_su1685293586e_real @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_273_Un__mono,axiom,
! [A: set_variable,C2: set_variable,B: set_variable,D: set_variable] :
( ( ord_le282106107riable @ A @ C2 )
=> ( ( ord_le282106107riable @ B @ D )
=> ( ord_le282106107riable @ ( sup_sup_set_variable @ A @ B ) @ ( sup_sup_set_variable @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_274_Un__mono,axiom,
! [A: set_variable_real,C2: set_variable_real,B: set_variable_real,D: set_variable_real] :
( ( ord_le1113654598e_real @ A @ C2 )
=> ( ( ord_le1113654598e_real @ B @ D )
=> ( ord_le1113654598e_real @ ( sup_su1685293586e_real @ A @ B ) @ ( sup_su1685293586e_real @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_275_sup_OcoboundedI2,axiom,
! [C: set_variable,B2: set_variable,A3: set_variable] :
( ( ord_le282106107riable @ C @ B2 )
=> ( ord_le282106107riable @ C @ ( sup_sup_set_variable @ A3 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_276_sup_OcoboundedI2,axiom,
! [C: set_variable_real,B2: set_variable_real,A3: set_variable_real] :
( ( ord_le1113654598e_real @ C @ B2 )
=> ( ord_le1113654598e_real @ C @ ( sup_su1685293586e_real @ A3 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_277_sup_OcoboundedI1,axiom,
! [C: set_variable,A3: set_variable,B2: set_variable] :
( ( ord_le282106107riable @ C @ A3 )
=> ( ord_le282106107riable @ C @ ( sup_sup_set_variable @ A3 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_278_sup_OcoboundedI1,axiom,
! [C: set_variable_real,A3: set_variable_real,B2: set_variable_real] :
( ( ord_le1113654598e_real @ C @ A3 )
=> ( ord_le1113654598e_real @ C @ ( sup_su1685293586e_real @ A3 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_279_sup_Oabsorb__iff2,axiom,
( ord_le282106107riable
= ( ^ [A6: set_variable,B6: set_variable] :
( ( sup_sup_set_variable @ A6 @ B6 )
= B6 ) ) ) ).
% sup.absorb_iff2
thf(fact_280_sup_Oabsorb__iff2,axiom,
( ord_le1113654598e_real
= ( ^ [A6: set_variable_real,B6: set_variable_real] :
( ( sup_su1685293586e_real @ A6 @ B6 )
= B6 ) ) ) ).
% sup.absorb_iff2
thf(fact_281_sup_Oabsorb__iff1,axiom,
( ord_le282106107riable
= ( ^ [B6: set_variable,A6: set_variable] :
( ( sup_sup_set_variable @ A6 @ B6 )
= A6 ) ) ) ).
% sup.absorb_iff1
thf(fact_282_sup_Oabsorb__iff1,axiom,
( ord_le1113654598e_real
= ( ^ [B6: set_variable_real,A6: set_variable_real] :
( ( sup_su1685293586e_real @ A6 @ B6 )
= A6 ) ) ) ).
% sup.absorb_iff1
thf(fact_283_sup_Ocobounded2,axiom,
! [B2: set_variable,A3: set_variable] : ( ord_le282106107riable @ B2 @ ( sup_sup_set_variable @ A3 @ B2 ) ) ).
% sup.cobounded2
thf(fact_284_sup_Ocobounded2,axiom,
! [B2: set_variable_real,A3: set_variable_real] : ( ord_le1113654598e_real @ B2 @ ( sup_su1685293586e_real @ A3 @ B2 ) ) ).
% sup.cobounded2
thf(fact_285_sup_Ocobounded1,axiom,
! [A3: set_variable,B2: set_variable] : ( ord_le282106107riable @ A3 @ ( sup_sup_set_variable @ A3 @ B2 ) ) ).
% sup.cobounded1
thf(fact_286_sup_Ocobounded1,axiom,
! [A3: set_variable_real,B2: set_variable_real] : ( ord_le1113654598e_real @ A3 @ ( sup_su1685293586e_real @ A3 @ B2 ) ) ).
% sup.cobounded1
thf(fact_287_sup_Oorder__iff,axiom,
( ord_le282106107riable
= ( ^ [B6: set_variable,A6: set_variable] :
( A6
= ( sup_sup_set_variable @ A6 @ B6 ) ) ) ) ).
% sup.order_iff
thf(fact_288_sup_Oorder__iff,axiom,
( ord_le1113654598e_real
= ( ^ [B6: set_variable_real,A6: set_variable_real] :
( A6
= ( sup_su1685293586e_real @ A6 @ B6 ) ) ) ) ).
% sup.order_iff
thf(fact_289_sup_OboundedI,axiom,
! [B2: set_variable,A3: set_variable,C: set_variable] :
( ( ord_le282106107riable @ B2 @ A3 )
=> ( ( ord_le282106107riable @ C @ A3 )
=> ( ord_le282106107riable @ ( sup_sup_set_variable @ B2 @ C ) @ A3 ) ) ) ).
% sup.boundedI
thf(fact_290_sup_OboundedI,axiom,
! [B2: set_variable_real,A3: set_variable_real,C: set_variable_real] :
( ( ord_le1113654598e_real @ B2 @ A3 )
=> ( ( ord_le1113654598e_real @ C @ A3 )
=> ( ord_le1113654598e_real @ ( sup_su1685293586e_real @ B2 @ C ) @ A3 ) ) ) ).
% sup.boundedI
thf(fact_291_sup_OboundedE,axiom,
! [B2: set_variable,C: set_variable,A3: set_variable] :
( ( ord_le282106107riable @ ( sup_sup_set_variable @ B2 @ C ) @ A3 )
=> ~ ( ( ord_le282106107riable @ B2 @ A3 )
=> ~ ( ord_le282106107riable @ C @ A3 ) ) ) ).
% sup.boundedE
thf(fact_292_sup_OboundedE,axiom,
! [B2: set_variable_real,C: set_variable_real,A3: set_variable_real] :
( ( ord_le1113654598e_real @ ( sup_su1685293586e_real @ B2 @ C ) @ A3 )
=> ~ ( ( ord_le1113654598e_real @ B2 @ A3 )
=> ~ ( ord_le1113654598e_real @ C @ A3 ) ) ) ).
% sup.boundedE
thf(fact_293_sup__absorb2,axiom,
! [X2: set_variable,Y: set_variable] :
( ( ord_le282106107riable @ X2 @ Y )
=> ( ( sup_sup_set_variable @ X2 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_294_sup__absorb2,axiom,
! [X2: set_variable_real,Y: set_variable_real] :
( ( ord_le1113654598e_real @ X2 @ Y )
=> ( ( sup_su1685293586e_real @ X2 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_295_sup__absorb1,axiom,
! [Y: set_variable,X2: set_variable] :
( ( ord_le282106107riable @ Y @ X2 )
=> ( ( sup_sup_set_variable @ X2 @ Y )
= X2 ) ) ).
% sup_absorb1
thf(fact_296_sup__absorb1,axiom,
! [Y: set_variable_real,X2: set_variable_real] :
( ( ord_le1113654598e_real @ Y @ X2 )
=> ( ( sup_su1685293586e_real @ X2 @ Y )
= X2 ) ) ).
% sup_absorb1
thf(fact_297_sup_Oabsorb2,axiom,
! [A3: set_variable,B2: set_variable] :
( ( ord_le282106107riable @ A3 @ B2 )
=> ( ( sup_sup_set_variable @ A3 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_298_sup_Oabsorb2,axiom,
! [A3: set_variable_real,B2: set_variable_real] :
( ( ord_le1113654598e_real @ A3 @ B2 )
=> ( ( sup_su1685293586e_real @ A3 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_299_sup_Oabsorb1,axiom,
! [B2: set_variable,A3: set_variable] :
( ( ord_le282106107riable @ B2 @ A3 )
=> ( ( sup_sup_set_variable @ A3 @ B2 )
= A3 ) ) ).
% sup.absorb1
thf(fact_300_sup_Oabsorb1,axiom,
! [B2: set_variable_real,A3: set_variable_real] :
( ( ord_le1113654598e_real @ B2 @ A3 )
=> ( ( sup_su1685293586e_real @ A3 @ B2 )
= A3 ) ) ).
% sup.absorb1
thf(fact_301_sup__unique,axiom,
! [F: set_variable > set_variable > set_variable,X2: set_variable,Y: set_variable] :
( ! [X8: set_variable,Y2: set_variable] : ( ord_le282106107riable @ X8 @ ( F @ X8 @ Y2 ) )
=> ( ! [X8: set_variable,Y2: set_variable] : ( ord_le282106107riable @ Y2 @ ( F @ X8 @ Y2 ) )
=> ( ! [X8: set_variable,Y2: set_variable,Z4: set_variable] :
( ( ord_le282106107riable @ Y2 @ X8 )
=> ( ( ord_le282106107riable @ Z4 @ X8 )
=> ( ord_le282106107riable @ ( F @ Y2 @ Z4 ) @ X8 ) ) )
=> ( ( sup_sup_set_variable @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_302_sup__unique,axiom,
! [F: set_variable_real > set_variable_real > set_variable_real,X2: set_variable_real,Y: set_variable_real] :
( ! [X8: set_variable_real,Y2: set_variable_real] : ( ord_le1113654598e_real @ X8 @ ( F @ X8 @ Y2 ) )
=> ( ! [X8: set_variable_real,Y2: set_variable_real] : ( ord_le1113654598e_real @ Y2 @ ( F @ X8 @ Y2 ) )
=> ( ! [X8: set_variable_real,Y2: set_variable_real,Z4: set_variable_real] :
( ( ord_le1113654598e_real @ Y2 @ X8 )
=> ( ( ord_le1113654598e_real @ Z4 @ X8 )
=> ( ord_le1113654598e_real @ ( F @ Y2 @ Z4 ) @ X8 ) ) )
=> ( ( sup_su1685293586e_real @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_303_sup_OorderI,axiom,
! [A3: set_variable,B2: set_variable] :
( ( A3
= ( sup_sup_set_variable @ A3 @ B2 ) )
=> ( ord_le282106107riable @ B2 @ A3 ) ) ).
% sup.orderI
thf(fact_304_sup_OorderI,axiom,
! [A3: set_variable_real,B2: set_variable_real] :
( ( A3
= ( sup_su1685293586e_real @ A3 @ B2 ) )
=> ( ord_le1113654598e_real @ B2 @ A3 ) ) ).
% sup.orderI
thf(fact_305_sup_OorderE,axiom,
! [B2: set_variable,A3: set_variable] :
( ( ord_le282106107riable @ B2 @ A3 )
=> ( A3
= ( sup_sup_set_variable @ A3 @ B2 ) ) ) ).
% sup.orderE
thf(fact_306_sup_OorderE,axiom,
! [B2: set_variable_real,A3: set_variable_real] :
( ( ord_le1113654598e_real @ B2 @ A3 )
=> ( A3
= ( sup_su1685293586e_real @ A3 @ B2 ) ) ) ).
% sup.orderE
thf(fact_307_le__iff__sup,axiom,
( ord_le282106107riable
= ( ^ [X3: set_variable,Y5: set_variable] :
( ( sup_sup_set_variable @ X3 @ Y5 )
= Y5 ) ) ) ).
% le_iff_sup
thf(fact_308_le__iff__sup,axiom,
( ord_le1113654598e_real
= ( ^ [X3: set_variable_real,Y5: set_variable_real] :
( ( sup_su1685293586e_real @ X3 @ Y5 )
= Y5 ) ) ) ).
% le_iff_sup
thf(fact_309_sup__least,axiom,
! [Y: set_variable,X2: set_variable,Z2: set_variable] :
( ( ord_le282106107riable @ Y @ X2 )
=> ( ( ord_le282106107riable @ Z2 @ X2 )
=> ( ord_le282106107riable @ ( sup_sup_set_variable @ Y @ Z2 ) @ X2 ) ) ) ).
% sup_least
thf(fact_310_sup__least,axiom,
! [Y: set_variable_real,X2: set_variable_real,Z2: set_variable_real] :
( ( ord_le1113654598e_real @ Y @ X2 )
=> ( ( ord_le1113654598e_real @ Z2 @ X2 )
=> ( ord_le1113654598e_real @ ( sup_su1685293586e_real @ Y @ Z2 ) @ X2 ) ) ) ).
% sup_least
thf(fact_311_sup__mono,axiom,
! [A3: set_variable,C: set_variable,B2: set_variable,D2: set_variable] :
( ( ord_le282106107riable @ A3 @ C )
=> ( ( ord_le282106107riable @ B2 @ D2 )
=> ( ord_le282106107riable @ ( sup_sup_set_variable @ A3 @ B2 ) @ ( sup_sup_set_variable @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_312_sup__mono,axiom,
! [A3: set_variable_real,C: set_variable_real,B2: set_variable_real,D2: set_variable_real] :
( ( ord_le1113654598e_real @ A3 @ C )
=> ( ( ord_le1113654598e_real @ B2 @ D2 )
=> ( ord_le1113654598e_real @ ( sup_su1685293586e_real @ A3 @ B2 ) @ ( sup_su1685293586e_real @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_313_sup_Omono,axiom,
! [C: set_variable,A3: set_variable,D2: set_variable,B2: set_variable] :
( ( ord_le282106107riable @ C @ A3 )
=> ( ( ord_le282106107riable @ D2 @ B2 )
=> ( ord_le282106107riable @ ( sup_sup_set_variable @ C @ D2 ) @ ( sup_sup_set_variable @ A3 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_314_sup_Omono,axiom,
! [C: set_variable_real,A3: set_variable_real,D2: set_variable_real,B2: set_variable_real] :
( ( ord_le1113654598e_real @ C @ A3 )
=> ( ( ord_le1113654598e_real @ D2 @ B2 )
=> ( ord_le1113654598e_real @ ( sup_su1685293586e_real @ C @ D2 ) @ ( sup_su1685293586e_real @ A3 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_315_le__supI2,axiom,
! [X2: set_variable,B2: set_variable,A3: set_variable] :
( ( ord_le282106107riable @ X2 @ B2 )
=> ( ord_le282106107riable @ X2 @ ( sup_sup_set_variable @ A3 @ B2 ) ) ) ).
% le_supI2
thf(fact_316_le__supI2,axiom,
! [X2: set_variable_real,B2: set_variable_real,A3: set_variable_real] :
( ( ord_le1113654598e_real @ X2 @ B2 )
=> ( ord_le1113654598e_real @ X2 @ ( sup_su1685293586e_real @ A3 @ B2 ) ) ) ).
% le_supI2
thf(fact_317_le__supI1,axiom,
! [X2: set_variable,A3: set_variable,B2: set_variable] :
( ( ord_le282106107riable @ X2 @ A3 )
=> ( ord_le282106107riable @ X2 @ ( sup_sup_set_variable @ A3 @ B2 ) ) ) ).
% le_supI1
thf(fact_318_le__supI1,axiom,
! [X2: set_variable_real,A3: set_variable_real,B2: set_variable_real] :
( ( ord_le1113654598e_real @ X2 @ A3 )
=> ( ord_le1113654598e_real @ X2 @ ( sup_su1685293586e_real @ A3 @ B2 ) ) ) ).
% le_supI1
thf(fact_319_sup__ge2,axiom,
! [Y: set_variable,X2: set_variable] : ( ord_le282106107riable @ Y @ ( sup_sup_set_variable @ X2 @ Y ) ) ).
% sup_ge2
thf(fact_320_sup__ge2,axiom,
! [Y: set_variable_real,X2: set_variable_real] : ( ord_le1113654598e_real @ Y @ ( sup_su1685293586e_real @ X2 @ Y ) ) ).
% sup_ge2
thf(fact_321_sup__ge1,axiom,
! [X2: set_variable,Y: set_variable] : ( ord_le282106107riable @ X2 @ ( sup_sup_set_variable @ X2 @ Y ) ) ).
% sup_ge1
thf(fact_322_sup__ge1,axiom,
! [X2: set_variable_real,Y: set_variable_real] : ( ord_le1113654598e_real @ X2 @ ( sup_su1685293586e_real @ X2 @ Y ) ) ).
% sup_ge1
thf(fact_323_le__supI,axiom,
! [A3: set_variable,X2: set_variable,B2: set_variable] :
( ( ord_le282106107riable @ A3 @ X2 )
=> ( ( ord_le282106107riable @ B2 @ X2 )
=> ( ord_le282106107riable @ ( sup_sup_set_variable @ A3 @ B2 ) @ X2 ) ) ) ).
% le_supI
thf(fact_324_le__supI,axiom,
! [A3: set_variable_real,X2: set_variable_real,B2: set_variable_real] :
( ( ord_le1113654598e_real @ A3 @ X2 )
=> ( ( ord_le1113654598e_real @ B2 @ X2 )
=> ( ord_le1113654598e_real @ ( sup_su1685293586e_real @ A3 @ B2 ) @ X2 ) ) ) ).
% le_supI
thf(fact_325_le__supE,axiom,
! [A3: set_variable,B2: set_variable,X2: set_variable] :
( ( ord_le282106107riable @ ( sup_sup_set_variable @ A3 @ B2 ) @ X2 )
=> ~ ( ( ord_le282106107riable @ A3 @ X2 )
=> ~ ( ord_le282106107riable @ B2 @ X2 ) ) ) ).
% le_supE
thf(fact_326_le__supE,axiom,
! [A3: set_variable_real,B2: set_variable_real,X2: set_variable_real] :
( ( ord_le1113654598e_real @ ( sup_su1685293586e_real @ A3 @ B2 ) @ X2 )
=> ~ ( ( ord_le1113654598e_real @ A3 @ X2 )
=> ~ ( ord_le1113654598e_real @ B2 @ X2 ) ) ) ).
% le_supE
thf(fact_327_inf__sup__ord_I3_J,axiom,
! [X2: set_variable,Y: set_variable] : ( ord_le282106107riable @ X2 @ ( sup_sup_set_variable @ X2 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_328_inf__sup__ord_I3_J,axiom,
! [X2: set_variable_real,Y: set_variable_real] : ( ord_le1113654598e_real @ X2 @ ( sup_su1685293586e_real @ X2 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_329_inf__sup__ord_I4_J,axiom,
! [Y: set_variable,X2: set_variable] : ( ord_le282106107riable @ Y @ ( sup_sup_set_variable @ X2 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_330_inf__sup__ord_I4_J,axiom,
! [Y: set_variable_real,X2: set_variable_real] : ( ord_le1113654598e_real @ Y @ ( sup_su1685293586e_real @ X2 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_331_inf__sup__aci_I8_J,axiom,
! [X2: set_variable,Y: set_variable] :
( ( sup_sup_set_variable @ X2 @ ( sup_sup_set_variable @ X2 @ Y ) )
= ( sup_sup_set_variable @ X2 @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_332_inf__sup__aci_I7_J,axiom,
! [X2: set_variable,Y: set_variable,Z2: set_variable] :
( ( sup_sup_set_variable @ X2 @ ( sup_sup_set_variable @ Y @ Z2 ) )
= ( sup_sup_set_variable @ Y @ ( sup_sup_set_variable @ X2 @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_333_inf__sup__aci_I6_J,axiom,
! [X2: set_variable,Y: set_variable,Z2: set_variable] :
( ( sup_sup_set_variable @ ( sup_sup_set_variable @ X2 @ Y ) @ Z2 )
= ( sup_sup_set_variable @ X2 @ ( sup_sup_set_variable @ Y @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_334_inf__sup__aci_I5_J,axiom,
( sup_sup_set_variable
= ( ^ [X3: set_variable,Y5: set_variable] : ( sup_sup_set_variable @ Y5 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_335_UnE,axiom,
! [C: variable > real,A: set_variable_real,B: set_variable_real] :
( ( member_variable_real @ C @ ( sup_su1685293586e_real @ A @ B ) )
=> ( ~ ( member_variable_real @ C @ A )
=> ( member_variable_real @ C @ B ) ) ) ).
% UnE
thf(fact_336_UnE,axiom,
! [C: variable,A: set_variable,B: set_variable] :
( ( member_variable @ C @ ( sup_sup_set_variable @ A @ B ) )
=> ( ~ ( member_variable @ C @ A )
=> ( member_variable @ C @ B ) ) ) ).
% UnE
thf(fact_337_UnI1,axiom,
! [C: variable > real,A: set_variable_real,B: set_variable_real] :
( ( member_variable_real @ C @ A )
=> ( member_variable_real @ C @ ( sup_su1685293586e_real @ A @ B ) ) ) ).
% UnI1
thf(fact_338_UnI1,axiom,
! [C: variable,A: set_variable,B: set_variable] :
( ( member_variable @ C @ A )
=> ( member_variable @ C @ ( sup_sup_set_variable @ A @ B ) ) ) ).
% UnI1
thf(fact_339_UnI2,axiom,
! [C: variable > real,B: set_variable_real,A: set_variable_real] :
( ( member_variable_real @ C @ B )
=> ( member_variable_real @ C @ ( sup_su1685293586e_real @ A @ B ) ) ) ).
% UnI2
thf(fact_340_UnI2,axiom,
! [C: variable,B: set_variable,A: set_variable] :
( ( member_variable @ C @ B )
=> ( member_variable @ C @ ( sup_sup_set_variable @ A @ B ) ) ) ).
% UnI2
thf(fact_341_bex__Un,axiom,
! [A: set_variable,B: set_variable,P: variable > $o] :
( ( ? [X3: variable] :
( ( member_variable @ X3 @ ( sup_sup_set_variable @ A @ B ) )
& ( P @ X3 ) ) )
= ( ? [X3: variable] :
( ( member_variable @ X3 @ A )
& ( P @ X3 ) )
| ? [X3: variable] :
( ( member_variable @ X3 @ B )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_342_ball__Un,axiom,
! [A: set_variable,B: set_variable,P: variable > $o] :
( ( ! [X3: variable] :
( ( member_variable @ X3 @ ( sup_sup_set_variable @ A @ B ) )
=> ( P @ X3 ) ) )
= ( ! [X3: variable] :
( ( member_variable @ X3 @ A )
=> ( P @ X3 ) )
& ! [X3: variable] :
( ( member_variable @ X3 @ B )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_343_Un__assoc,axiom,
! [A: set_variable,B: set_variable,C2: set_variable] :
( ( sup_sup_set_variable @ ( sup_sup_set_variable @ A @ B ) @ C2 )
= ( sup_sup_set_variable @ A @ ( sup_sup_set_variable @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_344_Un__absorb,axiom,
! [A: set_variable] :
( ( sup_sup_set_variable @ A @ A )
= A ) ).
% Un_absorb
thf(fact_345_Un__left__absorb,axiom,
! [A: set_variable,B: set_variable] :
( ( sup_sup_set_variable @ A @ ( sup_sup_set_variable @ A @ B ) )
= ( sup_sup_set_variable @ A @ B ) ) ).
% Un_left_absorb
thf(fact_346_Un__left__commute,axiom,
! [A: set_variable,B: set_variable,C2: set_variable] :
( ( sup_sup_set_variable @ A @ ( sup_sup_set_variable @ B @ C2 ) )
= ( sup_sup_set_variable @ B @ ( sup_sup_set_variable @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_347_boolean__algebra__cancel_Osup1,axiom,
! [A: set_variable,K: set_variable,A3: set_variable,B2: set_variable] :
( ( A
= ( sup_sup_set_variable @ K @ A3 ) )
=> ( ( sup_sup_set_variable @ A @ B2 )
= ( sup_sup_set_variable @ K @ ( sup_sup_set_variable @ A3 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_348_boolean__algebra__cancel_Osup2,axiom,
! [B: set_variable,K: set_variable,B2: set_variable,A3: set_variable] :
( ( B
= ( sup_sup_set_variable @ K @ B2 ) )
=> ( ( sup_sup_set_variable @ A3 @ B )
= ( sup_sup_set_variable @ K @ ( sup_sup_set_variable @ A3 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_349_sup_Oassoc,axiom,
! [A3: set_variable,B2: set_variable,C: set_variable] :
( ( sup_sup_set_variable @ ( sup_sup_set_variable @ A3 @ B2 ) @ C )
= ( sup_sup_set_variable @ A3 @ ( sup_sup_set_variable @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_350_sup__assoc,axiom,
! [X2: set_variable,Y: set_variable,Z2: set_variable] :
( ( sup_sup_set_variable @ ( sup_sup_set_variable @ X2 @ Y ) @ Z2 )
= ( sup_sup_set_variable @ X2 @ ( sup_sup_set_variable @ Y @ Z2 ) ) ) ).
% sup_assoc
thf(fact_351_sup_Ocommute,axiom,
( sup_sup_set_variable
= ( ^ [A6: set_variable,B6: set_variable] : ( sup_sup_set_variable @ B6 @ A6 ) ) ) ).
% sup.commute
thf(fact_352_sup__commute,axiom,
( sup_sup_set_variable
= ( ^ [X3: set_variable,Y5: set_variable] : ( sup_sup_set_variable @ Y5 @ X3 ) ) ) ).
% sup_commute
thf(fact_353_sup_Oleft__commute,axiom,
! [B2: set_variable,A3: set_variable,C: set_variable] :
( ( sup_sup_set_variable @ B2 @ ( sup_sup_set_variable @ A3 @ C ) )
= ( sup_sup_set_variable @ A3 @ ( sup_sup_set_variable @ B2 @ C ) ) ) ).
% sup.left_commute
% Conjectures (1)
thf(conj_0,conjecture,
( ( member_variable_real @ nu2 @ ( uminus430703407e_real @ ( denota1245701238me_sem @ i @ ( the_game @ ( produc293487213n_game @ ( uSubst516392814stappp @ sigma @ ua @ alpha ) ) ) @ ( uminus430703407e_real @ xa ) ) ) )
= ( member_variable_real @ nu2 @ ( uminus430703407e_real @ ( denota1245701238me_sem @ ( uSubst1599435252djoint @ sigma @ i @ omega2 ) @ alpha @ ( uminus430703407e_real @ xa ) ) ) ) ) ).
%------------------------------------------------------------------------------