TPTP Problem File: ITP159^1.p
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%------------------------------------------------------------------------------
% File : ITP159^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Refine_Basic problem prob_1005__3594596_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 : Refine_Basic/prob_1005__3594596_1 [Des21]
% Status : Theorem
% Rating : 0.30 v8.2.0, 0.15 v8.1.0, 0.18 v7.5.0
% Syntax : Number of formulae : 454 ( 150 unt; 91 typ; 0 def)
% Number of atoms : 1010 ( 290 equ; 0 cnn)
% Maximal formula atoms : 6 ( 2 avg)
% Number of connectives : 2611 ( 27 ~; 1 |; 33 &;2082 @)
% ( 0 <=>; 468 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 7 avg)
% Number of types : 21 ( 20 usr)
% Number of type conns : 354 ( 354 >; 0 *; 0 +; 0 <<)
% Number of symbols : 74 ( 71 usr; 12 con; 0-4 aty)
% Number of variables : 995 ( 108 ^; 881 !; 6 ?; 995 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:34:37.553
%------------------------------------------------------------------------------
% Could-be-implicit typings (20)
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Ounit_J_J,type,
set_Pr451126599t_unit: $tType ).
thf(ty_n_t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_It__Product____Type__Ounit_J,type,
refine787176636t_unit: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__b_Mt__Product____Type__Ounit_J_J,type,
set_Pr1707992533t_unit: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Ounit_J_J,type,
set_Pr1628433942t_unit: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Ounit_Mtf__b_J_J,type,
set_Pr712638521unit_b: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Ounit_Mtf__a_J_J,type,
set_Pr1720557880unit_a: $tType ).
thf(ty_n_t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__b_J,type,
refine432681134nres_b: $tType ).
thf(ty_n_t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J,type,
refine424419629nres_a: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
set_Product_prod_b_b: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__b_Mtf__a_J_J,type,
set_Product_prod_b_a: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__b_J_J,type,
set_Product_prod_a_b: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
set_Product_prod_a_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
product_prod_b_b: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__a_Mtf__b_J,type,
product_prod_a_b: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
product_prod_a_a: $tType ).
thf(ty_n_t__Product____Type__Ounit,type,
product_unit: $tType ).
thf(ty_n_t__Set__Oset_Itf__b_J,type,
set_b: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_tf__b,type,
b: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (71)
thf(sy_c_If_001_062_I_Eo_Mt__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J_J,type,
if_o_R1705037948nres_a: $o > ( $o > refine424419629nres_a ) > ( $o > refine424419629nres_a ) > $o > refine424419629nres_a ).
thf(sy_c_If_001_062_I_Eo_Mt__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__b_J_J,type,
if_o_R697118589nres_b: $o > ( $o > refine432681134nres_b ) > ( $o > refine432681134nres_b ) > $o > refine432681134nres_b ).
thf(sy_c_If_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J,type,
if_Ref1724547303nres_a: $o > refine424419629nres_a > refine424419629nres_a > refine424419629nres_a ).
thf(sy_c_If_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__b_J,type,
if_Ref1732808808nres_b: $o > refine432681134nres_b > refine432681134nres_b > refine432681134nres_b ).
thf(sy_c_If_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
if_set1550076877od_a_a: $o > set_Product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_a ).
thf(sy_c_If_001t__Set__Oset_Itf__a_J,type,
if_set_a: $o > set_a > set_a > set_a ).
thf(sy_c_If_001t__Set__Oset_Itf__b_J,type,
if_set_b: $o > set_b > set_b > set_b ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J,type,
sup_su2034729561nres_a: refine424419629nres_a > refine424419629nres_a > refine424419629nres_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__b_J,type,
sup_su2042991066nres_b: refine432681134nres_b > refine432681134nres_b > refine432681134nres_b ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J_J,type,
ord_le516598614nres_a: ( $o > refine424419629nres_a ) > ( $o > refine424419629nres_a ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__b_J_J,type,
ord_le1656162903nres_b: ( $o > refine432681134nres_b ) > ( $o > refine432681134nres_b ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_It__Product____Type__Ounit_J,type,
ord_le1051254044t_unit: refine787176636t_unit > refine787176636t_unit > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J,type,
ord_le519537037nres_a: refine424419629nres_a > refine424419629nres_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__b_J,type,
ord_le527798542nres_b: refine432681134nres_b > refine432681134nres_b > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
ord_le1824328871od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__b_J_J,type,
ord_le816409512od_a_b: set_Product_prod_a_b > set_Product_prod_a_b > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_Itf__b_Mtf__a_J_J,type,
ord_le2044239718od_b_a: set_Product_prod_b_a > set_Product_prod_b_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
ord_le1036320359od_b_b: set_Product_prod_b_b > set_Product_prod_b_b > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__b_J,type,
ord_less_eq_set_b: set_b > set_b > $o ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J,type,
order_1714329108nres_a: ( refine424419629nres_a > $o ) > refine424419629nres_a ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__b_J,type,
order_1722590613nres_b: ( refine432681134nres_b > $o ) > refine432681134nres_b ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_I_Eo_Mt__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J_J,type,
top_to2046902534nres_a: $o > refine424419629nres_a ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_I_Eo_Mt__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__b_J_J,type,
top_to1038983175nres_b: $o > refine432681134nres_b ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_It__Product____Type__Ounit_J,type,
top_to177290092t_unit: refine787176636t_unit ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J,type,
top_to231829469nres_a: refine424419629nres_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__b_J,type,
top_to240090974nres_b: refine432681134nres_b ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
top_to398455383od_a_a: set_Product_prod_a_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__a_J,type,
top_top_set_a: set_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__b_J,type,
top_top_set_b: set_b ).
thf(sy_c_Product__Type_OPair_001tf__a_001tf__a,type,
product_Pair_a_a: a > a > product_prod_a_a ).
thf(sy_c_Product__Type_OPair_001tf__a_001tf__b,type,
product_Pair_a_b: a > b > product_prod_a_b ).
thf(sy_c_Product__Type_OPair_001tf__b_001tf__b,type,
product_Pair_b_b: b > b > product_prod_b_b ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_ORETURN_001tf__a,type,
refine2063221604TURN_a: a > refine424419629nres_a ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_ORETURN_001tf__b,type,
refine2063221605TURN_b: b > refine432681134nres_b ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Oabs__fun_001tf__a_001tf__a,type,
refine1136779702un_a_a: set_Product_prod_a_a > refine424419629nres_a > refine424419629nres_a ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Oabs__fun_001tf__a_001tf__b,type,
refine1136779703un_a_b: set_Product_prod_a_b > refine424419629nres_a > refine432681134nres_b ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Oabs__fun_001tf__b_001tf__a,type,
refine1964549109un_b_a: set_Product_prod_b_a > refine432681134nres_b > refine424419629nres_a ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Oabs__fun_001tf__b_001tf__b,type,
refine1964549110un_b_b: set_Product_prod_b_b > refine432681134nres_b > refine432681134nres_b ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Obind_001t__Product____Type__Ounit_001tf__a,type,
refine119808503unit_a: refine787176636t_unit > ( product_unit > refine424419629nres_a ) > refine424419629nres_a ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Obind_001t__Product____Type__Ounit_001tf__b,type,
refine119808504unit_b: refine787176636t_unit > ( product_unit > refine432681134nres_b ) > refine432681134nres_b ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Obind_001tf__a_001tf__a,type,
refine436832838nd_a_a: refine424419629nres_a > ( a > refine424419629nres_a ) > refine424419629nres_a ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Obind_001tf__a_001tf__b,type,
refine436832839nd_a_b: refine424419629nres_a > ( a > refine432681134nres_b ) > refine432681134nres_b ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Obind_001tf__b_001tf__a,type,
refine1264602245nd_b_a: refine432681134nres_b > ( b > refine424419629nres_a ) > refine424419629nres_a ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Obind_001tf__b_001tf__b,type,
refine1264602246nd_b_b: refine432681134nres_b > ( b > refine432681134nres_b ) > refine432681134nres_b ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Oconc__fun_001t__Product____Type__Ounit_001t__Product____Type__Ounit,type,
refine944483349t_unit: set_Pr451126599t_unit > refine787176636t_unit > refine787176636t_unit ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Oconc__fun_001t__Product____Type__Ounit_001tf__a,type,
refine2043866374unit_a: set_Pr1720557880unit_a > refine424419629nres_a > refine787176636t_unit ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Oconc__fun_001t__Product____Type__Ounit_001tf__b,type,
refine2043866375unit_b: set_Pr712638521unit_b > refine432681134nres_b > refine787176636t_unit ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Oconc__fun_001tf__a_001t__Product____Type__Ounit,type,
refine2021053540t_unit: set_Pr1628433942t_unit > refine787176636t_unit > refine424419629nres_a ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Oconc__fun_001tf__a_001tf__a,type,
refine1441824853un_a_a: set_Product_prod_a_a > refine424419629nres_a > refine424419629nres_a ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Oconc__fun_001tf__a_001tf__b,type,
refine1441824854un_a_b: set_Product_prod_a_b > refine432681134nres_b > refine424419629nres_a ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Oconc__fun_001tf__b_001t__Product____Type__Ounit,type,
refine2140306339t_unit: set_Pr1707992533t_unit > refine787176636t_unit > refine432681134nres_b ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Oconc__fun_001tf__b_001tf__a,type,
refine122110612un_b_a: set_Product_prod_b_a > refine424419629nres_a > refine432681134nres_b ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Oconc__fun_001tf__b_001tf__b,type,
refine122110613un_b_b: set_Product_prod_b_b > refine432681134nres_b > refine432681134nres_b ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Oinres_001tf__a,type,
refine1001002027nres_a: refine424419629nres_a > a > $o ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Oinres_001tf__b,type,
refine1001002028nres_b: refine432681134nres_b > b > $o ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Onofail_001tf__a,type,
refine412683989fail_a: refine424419629nres_a > $o ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Onofail_001tf__b,type,
refine412683990fail_b: refine432681134nres_b > $o ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Onres_ORES_001tf__a,type,
refine1198353288_RES_a: set_a > refine424419629nres_a ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Onres_ORES_001tf__b,type,
refine1198353289_RES_b: set_b > refine432681134nres_b ).
thf(sy_c_Relation_OId_001tf__a,type,
id_a: set_Product_prod_a_a ).
thf(sy_c_Relation_OId_001tf__b,type,
id_b: set_Product_prod_b_b ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_OCollect_001tf__b,type,
collect_b: ( b > $o ) > set_b ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
member449909584od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__b_J,type,
member458171089od_a_b: product_prod_a_b > set_Product_prod_a_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
member1285940496od_b_b: product_prod_b_b > set_Product_prod_b_b > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_c_member_001tf__b,type,
member_b: b > set_b > $o ).
thf(sy_v_R,type,
r: set_Product_prod_a_b ).
thf(sy_v_X,type,
x: refine424419629nres_a ).
% Relevant facts (347)
thf(fact_0_nres__order__simps_I4_J,axiom,
! [M: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ top_to177290092t_unit @ M )
= ( M = top_to177290092t_unit ) ) ).
% nres_order_simps(4)
thf(fact_1_nres__order__simps_I4_J,axiom,
! [M: refine432681134nres_b] :
( ( ord_le527798542nres_b @ top_to240090974nres_b @ M )
= ( M = top_to240090974nres_b ) ) ).
% nres_order_simps(4)
thf(fact_2_nres__order__simps_I4_J,axiom,
! [M: refine424419629nres_a] :
( ( ord_le519537037nres_a @ top_to231829469nres_a @ M )
= ( M = top_to231829469nres_a ) ) ).
% nres_order_simps(4)
thf(fact_3_conc__fun__fail__iff_I2_J,axiom,
! [R: set_Pr1707992533t_unit,S: refine787176636t_unit] :
( ( top_to240090974nres_b
= ( refine2140306339t_unit @ R @ S ) )
= ( S = top_to177290092t_unit ) ) ).
% conc_fun_fail_iff(2)
thf(fact_4_conc__fun__fail__iff_I2_J,axiom,
! [R: set_Pr1628433942t_unit,S: refine787176636t_unit] :
( ( top_to231829469nres_a
= ( refine2021053540t_unit @ R @ S ) )
= ( S = top_to177290092t_unit ) ) ).
% conc_fun_fail_iff(2)
thf(fact_5_conc__fun__fail__iff_I2_J,axiom,
! [R: set_Pr712638521unit_b,S: refine432681134nres_b] :
( ( top_to177290092t_unit
= ( refine2043866375unit_b @ R @ S ) )
= ( S = top_to240090974nres_b ) ) ).
% conc_fun_fail_iff(2)
thf(fact_6_conc__fun__fail__iff_I2_J,axiom,
! [R: set_Pr1720557880unit_a,S: refine424419629nres_a] :
( ( top_to177290092t_unit
= ( refine2043866374unit_a @ R @ S ) )
= ( S = top_to231829469nres_a ) ) ).
% conc_fun_fail_iff(2)
thf(fact_7_conc__fun__fail__iff_I2_J,axiom,
! [R: set_Pr451126599t_unit,S: refine787176636t_unit] :
( ( top_to177290092t_unit
= ( refine944483349t_unit @ R @ S ) )
= ( S = top_to177290092t_unit ) ) ).
% conc_fun_fail_iff(2)
thf(fact_8_conc__fun__fail__iff_I2_J,axiom,
! [R: set_Product_prod_b_a,S: refine424419629nres_a] :
( ( top_to240090974nres_b
= ( refine122110612un_b_a @ R @ S ) )
= ( S = top_to231829469nres_a ) ) ).
% conc_fun_fail_iff(2)
thf(fact_9_conc__fun__fail__iff_I2_J,axiom,
! [R: set_Product_prod_a_a,S: refine424419629nres_a] :
( ( top_to231829469nres_a
= ( refine1441824853un_a_a @ R @ S ) )
= ( S = top_to231829469nres_a ) ) ).
% conc_fun_fail_iff(2)
thf(fact_10_conc__fun__fail__iff_I2_J,axiom,
! [R: set_Product_prod_b_b,S: refine432681134nres_b] :
( ( top_to240090974nres_b
= ( refine122110613un_b_b @ R @ S ) )
= ( S = top_to240090974nres_b ) ) ).
% conc_fun_fail_iff(2)
thf(fact_11_conc__fun__fail__iff_I2_J,axiom,
! [R: set_Product_prod_a_b,S: refine432681134nres_b] :
( ( top_to231829469nres_a
= ( refine1441824854un_a_b @ R @ S ) )
= ( S = top_to240090974nres_b ) ) ).
% conc_fun_fail_iff(2)
thf(fact_12_conc__fun__fail__iff_I1_J,axiom,
! [R: set_Pr1707992533t_unit,S: refine787176636t_unit] :
( ( ( refine2140306339t_unit @ R @ S )
= top_to240090974nres_b )
= ( S = top_to177290092t_unit ) ) ).
% conc_fun_fail_iff(1)
thf(fact_13_conc__fun__fail__iff_I1_J,axiom,
! [R: set_Pr1628433942t_unit,S: refine787176636t_unit] :
( ( ( refine2021053540t_unit @ R @ S )
= top_to231829469nres_a )
= ( S = top_to177290092t_unit ) ) ).
% conc_fun_fail_iff(1)
thf(fact_14_conc__fun__fail__iff_I1_J,axiom,
! [R: set_Pr712638521unit_b,S: refine432681134nres_b] :
( ( ( refine2043866375unit_b @ R @ S )
= top_to177290092t_unit )
= ( S = top_to240090974nres_b ) ) ).
% conc_fun_fail_iff(1)
thf(fact_15_conc__fun__fail__iff_I1_J,axiom,
! [R: set_Pr1720557880unit_a,S: refine424419629nres_a] :
( ( ( refine2043866374unit_a @ R @ S )
= top_to177290092t_unit )
= ( S = top_to231829469nres_a ) ) ).
% conc_fun_fail_iff(1)
thf(fact_16_conc__fun__fail__iff_I1_J,axiom,
! [R: set_Pr451126599t_unit,S: refine787176636t_unit] :
( ( ( refine944483349t_unit @ R @ S )
= top_to177290092t_unit )
= ( S = top_to177290092t_unit ) ) ).
% conc_fun_fail_iff(1)
thf(fact_17_conc__fun__fail__iff_I1_J,axiom,
! [R: set_Product_prod_b_a,S: refine424419629nres_a] :
( ( ( refine122110612un_b_a @ R @ S )
= top_to240090974nres_b )
= ( S = top_to231829469nres_a ) ) ).
% conc_fun_fail_iff(1)
thf(fact_18_conc__fun__fail__iff_I1_J,axiom,
! [R: set_Product_prod_a_a,S: refine424419629nres_a] :
( ( ( refine1441824853un_a_a @ R @ S )
= top_to231829469nres_a )
= ( S = top_to231829469nres_a ) ) ).
% conc_fun_fail_iff(1)
thf(fact_19_conc__fun__fail__iff_I1_J,axiom,
! [R: set_Product_prod_b_b,S: refine432681134nres_b] :
( ( ( refine122110613un_b_b @ R @ S )
= top_to240090974nres_b )
= ( S = top_to240090974nres_b ) ) ).
% conc_fun_fail_iff(1)
thf(fact_20_conc__fun__fail__iff_I1_J,axiom,
! [R: set_Product_prod_a_b,S: refine432681134nres_b] :
( ( ( refine1441824854un_a_b @ R @ S )
= top_to231829469nres_a )
= ( S = top_to240090974nres_b ) ) ).
% conc_fun_fail_iff(1)
thf(fact_21_conc__fun__FAIL,axiom,
! [R: set_Pr712638521unit_b] :
( ( refine2043866375unit_b @ R @ top_to240090974nres_b )
= top_to177290092t_unit ) ).
% conc_fun_FAIL
thf(fact_22_conc__fun__FAIL,axiom,
! [R: set_Pr1720557880unit_a] :
( ( refine2043866374unit_a @ R @ top_to231829469nres_a )
= top_to177290092t_unit ) ).
% conc_fun_FAIL
thf(fact_23_conc__fun__FAIL,axiom,
! [R: set_Pr1707992533t_unit] :
( ( refine2140306339t_unit @ R @ top_to177290092t_unit )
= top_to240090974nres_b ) ).
% conc_fun_FAIL
thf(fact_24_conc__fun__FAIL,axiom,
! [R: set_Pr1628433942t_unit] :
( ( refine2021053540t_unit @ R @ top_to177290092t_unit )
= top_to231829469nres_a ) ).
% conc_fun_FAIL
thf(fact_25_conc__fun__FAIL,axiom,
! [R: set_Pr451126599t_unit] :
( ( refine944483349t_unit @ R @ top_to177290092t_unit )
= top_to177290092t_unit ) ).
% conc_fun_FAIL
thf(fact_26_conc__fun__FAIL,axiom,
! [R: set_Product_prod_b_a] :
( ( refine122110612un_b_a @ R @ top_to231829469nres_a )
= top_to240090974nres_b ) ).
% conc_fun_FAIL
thf(fact_27_conc__fun__FAIL,axiom,
! [R: set_Product_prod_a_a] :
( ( refine1441824853un_a_a @ R @ top_to231829469nres_a )
= top_to231829469nres_a ) ).
% conc_fun_FAIL
thf(fact_28_conc__fun__FAIL,axiom,
! [R: set_Product_prod_b_b] :
( ( refine122110613un_b_b @ R @ top_to240090974nres_b )
= top_to240090974nres_b ) ).
% conc_fun_FAIL
thf(fact_29_conc__fun__FAIL,axiom,
! [R: set_Product_prod_a_b] :
( ( refine1441824854un_a_b @ R @ top_to240090974nres_b )
= top_to231829469nres_a ) ).
% conc_fun_FAIL
thf(fact_30_nres__order__simps_I3_J,axiom,
! [M: refine787176636t_unit] : ( ord_le1051254044t_unit @ M @ top_to177290092t_unit ) ).
% nres_order_simps(3)
thf(fact_31_nres__order__simps_I3_J,axiom,
! [M: refine432681134nres_b] : ( ord_le527798542nres_b @ M @ top_to240090974nres_b ) ).
% nres_order_simps(3)
thf(fact_32_nres__order__simps_I3_J,axiom,
! [M: refine424419629nres_a] : ( ord_le519537037nres_a @ M @ top_to231829469nres_a ) ).
% nres_order_simps(3)
thf(fact_33_conc__trans__additional_I1_J,axiom,
! [A: refine432681134nres_b,R: set_Product_prod_b_a,B: refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_le527798542nres_b @ A @ ( refine122110612un_b_a @ R @ B ) )
=> ( ( ord_le519537037nres_a @ B @ C )
=> ( ord_le527798542nres_b @ A @ ( refine122110612un_b_a @ R @ C ) ) ) ) ).
% conc_trans_additional(1)
thf(fact_34_conc__trans__additional_I1_J,axiom,
! [A: refine432681134nres_b,R: set_Product_prod_b_b,B: refine432681134nres_b,C: refine432681134nres_b] :
( ( ord_le527798542nres_b @ A @ ( refine122110613un_b_b @ R @ B ) )
=> ( ( ord_le527798542nres_b @ B @ C )
=> ( ord_le527798542nres_b @ A @ ( refine122110613un_b_b @ R @ C ) ) ) ) ).
% conc_trans_additional(1)
thf(fact_35_conc__trans__additional_I1_J,axiom,
! [A: refine424419629nres_a,R: set_Product_prod_a_a,B: refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A @ ( refine1441824853un_a_a @ R @ B ) )
=> ( ( ord_le519537037nres_a @ B @ C )
=> ( ord_le519537037nres_a @ A @ ( refine1441824853un_a_a @ R @ C ) ) ) ) ).
% conc_trans_additional(1)
thf(fact_36_conc__trans__additional_I1_J,axiom,
! [A: refine424419629nres_a,R: set_Product_prod_a_b,B: refine432681134nres_b,C: refine432681134nres_b] :
( ( ord_le519537037nres_a @ A @ ( refine1441824854un_a_b @ R @ B ) )
=> ( ( ord_le527798542nres_b @ B @ C )
=> ( ord_le519537037nres_a @ A @ ( refine1441824854un_a_b @ R @ C ) ) ) ) ).
% conc_trans_additional(1)
thf(fact_37_conc__trans,axiom,
! [C: refine424419629nres_a,R: set_Product_prod_a_b,B: refine432681134nres_b,R2: set_Product_prod_b_a,A: refine424419629nres_a] :
( ( ord_le519537037nres_a @ C @ ( refine1441824854un_a_b @ R @ B ) )
=> ( ( ord_le527798542nres_b @ B @ ( refine122110612un_b_a @ R2 @ A ) )
=> ( ord_le519537037nres_a @ C @ ( refine1441824854un_a_b @ R @ ( refine122110612un_b_a @ R2 @ A ) ) ) ) ) ).
% conc_trans
thf(fact_38_conc__trans,axiom,
! [C: refine432681134nres_b,R: set_Product_prod_b_b,B: refine432681134nres_b,R2: set_Product_prod_b_a,A: refine424419629nres_a] :
( ( ord_le527798542nres_b @ C @ ( refine122110613un_b_b @ R @ B ) )
=> ( ( ord_le527798542nres_b @ B @ ( refine122110612un_b_a @ R2 @ A ) )
=> ( ord_le527798542nres_b @ C @ ( refine122110613un_b_b @ R @ ( refine122110612un_b_a @ R2 @ A ) ) ) ) ) ).
% conc_trans
thf(fact_39_conc__trans,axiom,
! [C: refine432681134nres_b,R: set_Product_prod_b_a,B: refine424419629nres_a,R2: set_Product_prod_a_b,A: refine432681134nres_b] :
( ( ord_le527798542nres_b @ C @ ( refine122110612un_b_a @ R @ B ) )
=> ( ( ord_le519537037nres_a @ B @ ( refine1441824854un_a_b @ R2 @ A ) )
=> ( ord_le527798542nres_b @ C @ ( refine122110612un_b_a @ R @ ( refine1441824854un_a_b @ R2 @ A ) ) ) ) ) ).
% conc_trans
thf(fact_40_conc__trans,axiom,
! [C: refine432681134nres_b,R: set_Product_prod_b_a,B: refine424419629nres_a,R2: set_Product_prod_a_a,A: refine424419629nres_a] :
( ( ord_le527798542nres_b @ C @ ( refine122110612un_b_a @ R @ B ) )
=> ( ( ord_le519537037nres_a @ B @ ( refine1441824853un_a_a @ R2 @ A ) )
=> ( ord_le527798542nres_b @ C @ ( refine122110612un_b_a @ R @ ( refine1441824853un_a_a @ R2 @ A ) ) ) ) ) ).
% conc_trans
thf(fact_41_conc__trans,axiom,
! [C: refine424419629nres_a,R: set_Product_prod_a_b,B: refine432681134nres_b,R2: set_Product_prod_b_b,A: refine432681134nres_b] :
( ( ord_le519537037nres_a @ C @ ( refine1441824854un_a_b @ R @ B ) )
=> ( ( ord_le527798542nres_b @ B @ ( refine122110613un_b_b @ R2 @ A ) )
=> ( ord_le519537037nres_a @ C @ ( refine1441824854un_a_b @ R @ ( refine122110613un_b_b @ R2 @ A ) ) ) ) ) ).
% conc_trans
thf(fact_42_conc__trans,axiom,
! [C: refine432681134nres_b,R: set_Product_prod_b_b,B: refine432681134nres_b,R2: set_Product_prod_b_b,A: refine432681134nres_b] :
( ( ord_le527798542nres_b @ C @ ( refine122110613un_b_b @ R @ B ) )
=> ( ( ord_le527798542nres_b @ B @ ( refine122110613un_b_b @ R2 @ A ) )
=> ( ord_le527798542nres_b @ C @ ( refine122110613un_b_b @ R @ ( refine122110613un_b_b @ R2 @ A ) ) ) ) ) ).
% conc_trans
thf(fact_43_conc__trans,axiom,
! [C: refine424419629nres_a,R: set_Product_prod_a_a,B: refine424419629nres_a,R2: set_Product_prod_a_a,A: refine424419629nres_a] :
( ( ord_le519537037nres_a @ C @ ( refine1441824853un_a_a @ R @ B ) )
=> ( ( ord_le519537037nres_a @ B @ ( refine1441824853un_a_a @ R2 @ A ) )
=> ( ord_le519537037nres_a @ C @ ( refine1441824853un_a_a @ R @ ( refine1441824853un_a_a @ R2 @ A ) ) ) ) ) ).
% conc_trans
thf(fact_44_conc__trans,axiom,
! [C: refine424419629nres_a,R: set_Product_prod_a_a,B: refine424419629nres_a,R2: set_Product_prod_a_b,A: refine432681134nres_b] :
( ( ord_le519537037nres_a @ C @ ( refine1441824853un_a_a @ R @ B ) )
=> ( ( ord_le519537037nres_a @ B @ ( refine1441824854un_a_b @ R2 @ A ) )
=> ( ord_le519537037nres_a @ C @ ( refine1441824853un_a_a @ R @ ( refine1441824854un_a_b @ R2 @ A ) ) ) ) ) ).
% conc_trans
thf(fact_45_order__mono__setup_Orefl,axiom,
! [X: set_Product_prod_a_a] : ( ord_le1824328871od_a_a @ X @ X ) ).
% order_mono_setup.refl
thf(fact_46_order__mono__setup_Orefl,axiom,
! [X: set_b] : ( ord_less_eq_set_b @ X @ X ) ).
% order_mono_setup.refl
thf(fact_47_order__mono__setup_Orefl,axiom,
! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% order_mono_setup.refl
thf(fact_48_order__mono__setup_Orefl,axiom,
! [X: $o > refine432681134nres_b] : ( ord_le1656162903nres_b @ X @ X ) ).
% order_mono_setup.refl
thf(fact_49_order__mono__setup_Orefl,axiom,
! [X: $o > refine424419629nres_a] : ( ord_le516598614nres_a @ X @ X ) ).
% order_mono_setup.refl
thf(fact_50_order__mono__setup_Orefl,axiom,
! [X: refine432681134nres_b] : ( ord_le527798542nres_b @ X @ X ) ).
% order_mono_setup.refl
thf(fact_51_order__mono__setup_Orefl,axiom,
! [X: refine424419629nres_a] : ( ord_le519537037nres_a @ X @ X ) ).
% order_mono_setup.refl
thf(fact_52_top__greatest,axiom,
! [A2: refine787176636t_unit] : ( ord_le1051254044t_unit @ A2 @ top_to177290092t_unit ) ).
% top_greatest
thf(fact_53_top__greatest,axiom,
! [A2: set_Product_prod_a_a] : ( ord_le1824328871od_a_a @ A2 @ top_to398455383od_a_a ) ).
% top_greatest
thf(fact_54_top__greatest,axiom,
! [A2: set_b] : ( ord_less_eq_set_b @ A2 @ top_top_set_b ) ).
% top_greatest
thf(fact_55_top__greatest,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ top_top_set_a ) ).
% top_greatest
thf(fact_56_top__greatest,axiom,
! [A2: $o > refine432681134nres_b] : ( ord_le1656162903nres_b @ A2 @ top_to1038983175nres_b ) ).
% top_greatest
thf(fact_57_top__greatest,axiom,
! [A2: $o > refine424419629nres_a] : ( ord_le516598614nres_a @ A2 @ top_to2046902534nres_a ) ).
% top_greatest
thf(fact_58_top__greatest,axiom,
! [A2: refine432681134nres_b] : ( ord_le527798542nres_b @ A2 @ top_to240090974nres_b ) ).
% top_greatest
thf(fact_59_top__greatest,axiom,
! [A2: refine424419629nres_a] : ( ord_le519537037nres_a @ A2 @ top_to231829469nres_a ) ).
% top_greatest
thf(fact_60_meta__le__everything__if__top,axiom,
! [M2: refine787176636t_unit,X: refine787176636t_unit] :
( ( M2 = top_to177290092t_unit )
=> ( ord_le1051254044t_unit @ X @ M2 ) ) ).
% meta_le_everything_if_top
thf(fact_61_meta__le__everything__if__top,axiom,
! [M2: set_Product_prod_a_a,X: set_Product_prod_a_a] :
( ( M2 = top_to398455383od_a_a )
=> ( ord_le1824328871od_a_a @ X @ M2 ) ) ).
% meta_le_everything_if_top
thf(fact_62_meta__le__everything__if__top,axiom,
! [M2: set_b,X: set_b] :
( ( M2 = top_top_set_b )
=> ( ord_less_eq_set_b @ X @ M2 ) ) ).
% meta_le_everything_if_top
thf(fact_63_meta__le__everything__if__top,axiom,
! [M2: set_a,X: set_a] :
( ( M2 = top_top_set_a )
=> ( ord_less_eq_set_a @ X @ M2 ) ) ).
% meta_le_everything_if_top
thf(fact_64_meta__le__everything__if__top,axiom,
! [M2: $o > refine432681134nres_b,X: $o > refine432681134nres_b] :
( ( M2 = top_to1038983175nres_b )
=> ( ord_le1656162903nres_b @ X @ M2 ) ) ).
% meta_le_everything_if_top
thf(fact_65_meta__le__everything__if__top,axiom,
! [M2: $o > refine424419629nres_a,X: $o > refine424419629nres_a] :
( ( M2 = top_to2046902534nres_a )
=> ( ord_le516598614nres_a @ X @ M2 ) ) ).
% meta_le_everything_if_top
thf(fact_66_meta__le__everything__if__top,axiom,
! [M2: refine424419629nres_a,X: refine424419629nres_a] :
( ( M2 = top_to231829469nres_a )
=> ( ord_le519537037nres_a @ X @ M2 ) ) ).
% meta_le_everything_if_top
thf(fact_67_meta__le__everything__if__top,axiom,
! [M2: refine432681134nres_b,X: refine432681134nres_b] :
( ( M2 = top_to240090974nres_b )
=> ( ord_le527798542nres_b @ X @ M2 ) ) ).
% meta_le_everything_if_top
thf(fact_68_top_Oextremum__unique,axiom,
! [A2: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ top_to177290092t_unit @ A2 )
= ( A2 = top_to177290092t_unit ) ) ).
% top.extremum_unique
thf(fact_69_top_Oextremum__unique,axiom,
! [A2: set_Product_prod_a_a] :
( ( ord_le1824328871od_a_a @ top_to398455383od_a_a @ A2 )
= ( A2 = top_to398455383od_a_a ) ) ).
% top.extremum_unique
thf(fact_70_top_Oextremum__unique,axiom,
! [A2: set_b] :
( ( ord_less_eq_set_b @ top_top_set_b @ A2 )
= ( A2 = top_top_set_b ) ) ).
% top.extremum_unique
thf(fact_71_top_Oextremum__unique,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A2 )
= ( A2 = top_top_set_a ) ) ).
% top.extremum_unique
thf(fact_72_top_Oextremum__unique,axiom,
! [A2: $o > refine432681134nres_b] :
( ( ord_le1656162903nres_b @ top_to1038983175nres_b @ A2 )
= ( A2 = top_to1038983175nres_b ) ) ).
% top.extremum_unique
thf(fact_73_top_Oextremum__unique,axiom,
! [A2: $o > refine424419629nres_a] :
( ( ord_le516598614nres_a @ top_to2046902534nres_a @ A2 )
= ( A2 = top_to2046902534nres_a ) ) ).
% top.extremum_unique
thf(fact_74_top_Oextremum__unique,axiom,
! [A2: refine424419629nres_a] :
( ( ord_le519537037nres_a @ top_to231829469nres_a @ A2 )
= ( A2 = top_to231829469nres_a ) ) ).
% top.extremum_unique
thf(fact_75_top_Oextremum__unique,axiom,
! [A2: refine432681134nres_b] :
( ( ord_le527798542nres_b @ top_to240090974nres_b @ A2 )
= ( A2 = top_to240090974nres_b ) ) ).
% top.extremum_unique
thf(fact_76_top_Oextremum__uniqueI,axiom,
! [A2: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ top_to177290092t_unit @ A2 )
=> ( A2 = top_to177290092t_unit ) ) ).
% top.extremum_uniqueI
thf(fact_77_top_Oextremum__uniqueI,axiom,
! [A2: set_Product_prod_a_a] :
( ( ord_le1824328871od_a_a @ top_to398455383od_a_a @ A2 )
=> ( A2 = top_to398455383od_a_a ) ) ).
% top.extremum_uniqueI
thf(fact_78_top_Oextremum__uniqueI,axiom,
! [A2: set_b] :
( ( ord_less_eq_set_b @ top_top_set_b @ A2 )
=> ( A2 = top_top_set_b ) ) ).
% top.extremum_uniqueI
thf(fact_79_top_Oextremum__uniqueI,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A2 )
=> ( A2 = top_top_set_a ) ) ).
% top.extremum_uniqueI
thf(fact_80_top_Oextremum__uniqueI,axiom,
! [A2: $o > refine432681134nres_b] :
( ( ord_le1656162903nres_b @ top_to1038983175nres_b @ A2 )
=> ( A2 = top_to1038983175nres_b ) ) ).
% top.extremum_uniqueI
thf(fact_81_top_Oextremum__uniqueI,axiom,
! [A2: $o > refine424419629nres_a] :
( ( ord_le516598614nres_a @ top_to2046902534nres_a @ A2 )
=> ( A2 = top_to2046902534nres_a ) ) ).
% top.extremum_uniqueI
thf(fact_82_top_Oextremum__uniqueI,axiom,
! [A2: refine424419629nres_a] :
( ( ord_le519537037nres_a @ top_to231829469nres_a @ A2 )
=> ( A2 = top_to231829469nres_a ) ) ).
% top.extremum_uniqueI
thf(fact_83_top_Oextremum__uniqueI,axiom,
! [A2: refine432681134nres_b] :
( ( ord_le527798542nres_b @ top_to240090974nres_b @ A2 )
=> ( A2 = top_to240090974nres_b ) ) ).
% top.extremum_uniqueI
thf(fact_84_le__funD,axiom,
! [F: $o > refine432681134nres_b,G: $o > refine432681134nres_b,X: $o] :
( ( ord_le1656162903nres_b @ F @ G )
=> ( ord_le527798542nres_b @ ( F @ X ) @ ( G @ X ) ) ) ).
% le_funD
thf(fact_85_le__funD,axiom,
! [F: $o > refine424419629nres_a,G: $o > refine424419629nres_a,X: $o] :
( ( ord_le516598614nres_a @ F @ G )
=> ( ord_le519537037nres_a @ ( F @ X ) @ ( G @ X ) ) ) ).
% le_funD
thf(fact_86_le__funE,axiom,
! [F: $o > refine432681134nres_b,G: $o > refine432681134nres_b,X: $o] :
( ( ord_le1656162903nres_b @ F @ G )
=> ( ord_le527798542nres_b @ ( F @ X ) @ ( G @ X ) ) ) ).
% le_funE
thf(fact_87_le__funE,axiom,
! [F: $o > refine424419629nres_a,G: $o > refine424419629nres_a,X: $o] :
( ( ord_le516598614nres_a @ F @ G )
=> ( ord_le519537037nres_a @ ( F @ X ) @ ( G @ X ) ) ) ).
% le_funE
thf(fact_88_le__funI,axiom,
! [F: $o > refine424419629nres_a,G: $o > refine424419629nres_a] :
( ! [X2: $o] : ( ord_le519537037nres_a @ ( F @ X2 ) @ ( G @ X2 ) )
=> ( ord_le516598614nres_a @ F @ G ) ) ).
% le_funI
thf(fact_89_le__funI,axiom,
! [F: $o > refine432681134nres_b,G: $o > refine432681134nres_b] :
( ! [X2: $o] : ( ord_le527798542nres_b @ ( F @ X2 ) @ ( G @ X2 ) )
=> ( ord_le1656162903nres_b @ F @ G ) ) ).
% le_funI
thf(fact_90_Refine__Misc_Oif__mono,axiom,
! [B2: $o,M1: set_Product_prod_a_a,M12: set_Product_prod_a_a,M22: set_Product_prod_a_a,M23: set_Product_prod_a_a] :
( ( B2
=> ( ord_le1824328871od_a_a @ M1 @ M12 ) )
=> ( ( ~ B2
=> ( ord_le1824328871od_a_a @ M22 @ M23 ) )
=> ( ord_le1824328871od_a_a @ ( if_set1550076877od_a_a @ B2 @ M1 @ M22 ) @ ( if_set1550076877od_a_a @ B2 @ M12 @ M23 ) ) ) ) ).
% Refine_Misc.if_mono
thf(fact_91_Refine__Misc_Oif__mono,axiom,
! [B2: $o,M1: set_b,M12: set_b,M22: set_b,M23: set_b] :
( ( B2
=> ( ord_less_eq_set_b @ M1 @ M12 ) )
=> ( ( ~ B2
=> ( ord_less_eq_set_b @ M22 @ M23 ) )
=> ( ord_less_eq_set_b @ ( if_set_b @ B2 @ M1 @ M22 ) @ ( if_set_b @ B2 @ M12 @ M23 ) ) ) ) ).
% Refine_Misc.if_mono
thf(fact_92_Refine__Misc_Oif__mono,axiom,
! [B2: $o,M1: set_a,M12: set_a,M22: set_a,M23: set_a] :
( ( B2
=> ( ord_less_eq_set_a @ M1 @ M12 ) )
=> ( ( ~ B2
=> ( ord_less_eq_set_a @ M22 @ M23 ) )
=> ( ord_less_eq_set_a @ ( if_set_a @ B2 @ M1 @ M22 ) @ ( if_set_a @ B2 @ M12 @ M23 ) ) ) ) ).
% Refine_Misc.if_mono
thf(fact_93_Refine__Misc_Oif__mono,axiom,
! [B2: $o,M1: $o > refine432681134nres_b,M12: $o > refine432681134nres_b,M22: $o > refine432681134nres_b,M23: $o > refine432681134nres_b] :
( ( B2
=> ( ord_le1656162903nres_b @ M1 @ M12 ) )
=> ( ( ~ B2
=> ( ord_le1656162903nres_b @ M22 @ M23 ) )
=> ( ord_le1656162903nres_b @ ( if_o_R697118589nres_b @ B2 @ M1 @ M22 ) @ ( if_o_R697118589nres_b @ B2 @ M12 @ M23 ) ) ) ) ).
% Refine_Misc.if_mono
thf(fact_94_Refine__Misc_Oif__mono,axiom,
! [B2: $o,M1: $o > refine424419629nres_a,M12: $o > refine424419629nres_a,M22: $o > refine424419629nres_a,M23: $o > refine424419629nres_a] :
( ( B2
=> ( ord_le516598614nres_a @ M1 @ M12 ) )
=> ( ( ~ B2
=> ( ord_le516598614nres_a @ M22 @ M23 ) )
=> ( ord_le516598614nres_a @ ( if_o_R1705037948nres_a @ B2 @ M1 @ M22 ) @ ( if_o_R1705037948nres_a @ B2 @ M12 @ M23 ) ) ) ) ).
% Refine_Misc.if_mono
thf(fact_95_Refine__Misc_Oif__mono,axiom,
! [B2: $o,M1: refine424419629nres_a,M12: refine424419629nres_a,M22: refine424419629nres_a,M23: refine424419629nres_a] :
( ( B2
=> ( ord_le519537037nres_a @ M1 @ M12 ) )
=> ( ( ~ B2
=> ( ord_le519537037nres_a @ M22 @ M23 ) )
=> ( ord_le519537037nres_a @ ( if_Ref1724547303nres_a @ B2 @ M1 @ M22 ) @ ( if_Ref1724547303nres_a @ B2 @ M12 @ M23 ) ) ) ) ).
% Refine_Misc.if_mono
thf(fact_96_Refine__Misc_Oif__mono,axiom,
! [B2: $o,M1: refine432681134nres_b,M12: refine432681134nres_b,M22: refine432681134nres_b,M23: refine432681134nres_b] :
( ( B2
=> ( ord_le527798542nres_b @ M1 @ M12 ) )
=> ( ( ~ B2
=> ( ord_le527798542nres_b @ M22 @ M23 ) )
=> ( ord_le527798542nres_b @ ( if_Ref1732808808nres_b @ B2 @ M1 @ M22 ) @ ( if_Ref1732808808nres_b @ B2 @ M12 @ M23 ) ) ) ) ).
% Refine_Misc.if_mono
thf(fact_97_le__fun__def,axiom,
( ord_le1656162903nres_b
= ( ^ [F2: $o > refine432681134nres_b,G2: $o > refine432681134nres_b] :
! [X3: $o] : ( ord_le527798542nres_b @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ).
% le_fun_def
thf(fact_98_le__fun__def,axiom,
( ord_le516598614nres_a
= ( ^ [F2: $o > refine424419629nres_a,G2: $o > refine424419629nres_a] :
! [X3: $o] : ( ord_le519537037nres_a @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ).
% le_fun_def
thf(fact_99_conc__fun__R__mono,axiom,
! [R: set_Product_prod_b_a,R2: set_Product_prod_b_a,M: refine424419629nres_a] :
( ( ord_le2044239718od_b_a @ R @ R2 )
=> ( ord_le527798542nres_b @ ( refine122110612un_b_a @ R @ M ) @ ( refine122110612un_b_a @ R2 @ M ) ) ) ).
% conc_fun_R_mono
thf(fact_100_conc__fun__R__mono,axiom,
! [R: set_Product_prod_a_b,R2: set_Product_prod_a_b,M: refine432681134nres_b] :
( ( ord_le816409512od_a_b @ R @ R2 )
=> ( ord_le519537037nres_a @ ( refine1441824854un_a_b @ R @ M ) @ ( refine1441824854un_a_b @ R2 @ M ) ) ) ).
% conc_fun_R_mono
thf(fact_101_conc__fun__R__mono,axiom,
! [R: set_Product_prod_b_b,R2: set_Product_prod_b_b,M: refine432681134nres_b] :
( ( ord_le1036320359od_b_b @ R @ R2 )
=> ( ord_le527798542nres_b @ ( refine122110613un_b_b @ R @ M ) @ ( refine122110613un_b_b @ R2 @ M ) ) ) ).
% conc_fun_R_mono
thf(fact_102_conc__fun__R__mono,axiom,
! [R: set_Product_prod_a_a,R2: set_Product_prod_a_a,M: refine424419629nres_a] :
( ( ord_le1824328871od_a_a @ R @ R2 )
=> ( ord_le519537037nres_a @ ( refine1441824853un_a_a @ R @ M ) @ ( refine1441824853un_a_a @ R2 @ M ) ) ) ).
% conc_fun_R_mono
thf(fact_103_order__mono__setup_Omono__if,axiom,
! [T: set_Product_prod_a_a,T2: set_Product_prod_a_a,E: set_Product_prod_a_a,E2: set_Product_prod_a_a,B2: $o] :
( ( ord_le1824328871od_a_a @ T @ T2 )
=> ( ( ord_le1824328871od_a_a @ E @ E2 )
=> ( ord_le1824328871od_a_a @ ( if_set1550076877od_a_a @ B2 @ T @ E ) @ ( if_set1550076877od_a_a @ B2 @ T2 @ E2 ) ) ) ) ).
% order_mono_setup.mono_if
thf(fact_104_order__mono__setup_Omono__if,axiom,
! [T: set_b,T2: set_b,E: set_b,E2: set_b,B2: $o] :
( ( ord_less_eq_set_b @ T @ T2 )
=> ( ( ord_less_eq_set_b @ E @ E2 )
=> ( ord_less_eq_set_b @ ( if_set_b @ B2 @ T @ E ) @ ( if_set_b @ B2 @ T2 @ E2 ) ) ) ) ).
% order_mono_setup.mono_if
thf(fact_105_order__mono__setup_Omono__if,axiom,
! [T: set_a,T2: set_a,E: set_a,E2: set_a,B2: $o] :
( ( ord_less_eq_set_a @ T @ T2 )
=> ( ( ord_less_eq_set_a @ E @ E2 )
=> ( ord_less_eq_set_a @ ( if_set_a @ B2 @ T @ E ) @ ( if_set_a @ B2 @ T2 @ E2 ) ) ) ) ).
% order_mono_setup.mono_if
thf(fact_106_order__mono__setup_Omono__if,axiom,
! [T: $o > refine432681134nres_b,T2: $o > refine432681134nres_b,E: $o > refine432681134nres_b,E2: $o > refine432681134nres_b,B2: $o] :
( ( ord_le1656162903nres_b @ T @ T2 )
=> ( ( ord_le1656162903nres_b @ E @ E2 )
=> ( ord_le1656162903nres_b @ ( if_o_R697118589nres_b @ B2 @ T @ E ) @ ( if_o_R697118589nres_b @ B2 @ T2 @ E2 ) ) ) ) ).
% order_mono_setup.mono_if
thf(fact_107_order__mono__setup_Omono__if,axiom,
! [T: $o > refine424419629nres_a,T2: $o > refine424419629nres_a,E: $o > refine424419629nres_a,E2: $o > refine424419629nres_a,B2: $o] :
( ( ord_le516598614nres_a @ T @ T2 )
=> ( ( ord_le516598614nres_a @ E @ E2 )
=> ( ord_le516598614nres_a @ ( if_o_R1705037948nres_a @ B2 @ T @ E ) @ ( if_o_R1705037948nres_a @ B2 @ T2 @ E2 ) ) ) ) ).
% order_mono_setup.mono_if
thf(fact_108_order__mono__setup_Omono__if,axiom,
! [T: refine424419629nres_a,T2: refine424419629nres_a,E: refine424419629nres_a,E2: refine424419629nres_a,B2: $o] :
( ( ord_le519537037nres_a @ T @ T2 )
=> ( ( ord_le519537037nres_a @ E @ E2 )
=> ( ord_le519537037nres_a @ ( if_Ref1724547303nres_a @ B2 @ T @ E ) @ ( if_Ref1724547303nres_a @ B2 @ T2 @ E2 ) ) ) ) ).
% order_mono_setup.mono_if
thf(fact_109_order__mono__setup_Omono__if,axiom,
! [T: refine432681134nres_b,T2: refine432681134nres_b,E: refine432681134nres_b,E2: refine432681134nres_b,B2: $o] :
( ( ord_le527798542nres_b @ T @ T2 )
=> ( ( ord_le527798542nres_b @ E @ E2 )
=> ( ord_le527798542nres_b @ ( if_Ref1732808808nres_b @ B2 @ T @ E ) @ ( if_Ref1732808808nres_b @ B2 @ T2 @ E2 ) ) ) ) ).
% order_mono_setup.mono_if
thf(fact_110_dual__order_Oantisym,axiom,
! [B2: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
( ( ord_le1824328871od_a_a @ B2 @ A2 )
=> ( ( ord_le1824328871od_a_a @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_111_dual__order_Oantisym,axiom,
! [B2: set_b,A2: set_b] :
( ( ord_less_eq_set_b @ B2 @ A2 )
=> ( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_112_dual__order_Oantisym,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_113_dual__order_Oantisym,axiom,
! [B2: $o > refine432681134nres_b,A2: $o > refine432681134nres_b] :
( ( ord_le1656162903nres_b @ B2 @ A2 )
=> ( ( ord_le1656162903nres_b @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_114_dual__order_Oantisym,axiom,
! [B2: $o > refine424419629nres_a,A2: $o > refine424419629nres_a] :
( ( ord_le516598614nres_a @ B2 @ A2 )
=> ( ( ord_le516598614nres_a @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_115_dual__order_Oantisym,axiom,
! [B2: refine424419629nres_a,A2: refine424419629nres_a] :
( ( ord_le519537037nres_a @ B2 @ A2 )
=> ( ( ord_le519537037nres_a @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_116_dual__order_Oantisym,axiom,
! [B2: refine432681134nres_b,A2: refine432681134nres_b] :
( ( ord_le527798542nres_b @ B2 @ A2 )
=> ( ( ord_le527798542nres_b @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_117_dual__order_Oeq__iff,axiom,
( ( ^ [Y: set_Product_prod_a_a,Z: set_Product_prod_a_a] : Y = Z )
= ( ^ [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
( ( ord_le1824328871od_a_a @ B3 @ A3 )
& ( ord_le1824328871od_a_a @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_118_dual__order_Oeq__iff,axiom,
( ( ^ [Y: set_b,Z: set_b] : Y = Z )
= ( ^ [A3: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ B3 @ A3 )
& ( ord_less_eq_set_b @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_119_dual__order_Oeq__iff,axiom,
( ( ^ [Y: set_a,Z: set_a] : Y = Z )
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A3 )
& ( ord_less_eq_set_a @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_120_dual__order_Oeq__iff,axiom,
( ( ^ [Y: $o > refine432681134nres_b,Z: $o > refine432681134nres_b] : Y = Z )
= ( ^ [A3: $o > refine432681134nres_b,B3: $o > refine432681134nres_b] :
( ( ord_le1656162903nres_b @ B3 @ A3 )
& ( ord_le1656162903nres_b @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_121_dual__order_Oeq__iff,axiom,
( ( ^ [Y: $o > refine424419629nres_a,Z: $o > refine424419629nres_a] : Y = Z )
= ( ^ [A3: $o > refine424419629nres_a,B3: $o > refine424419629nres_a] :
( ( ord_le516598614nres_a @ B3 @ A3 )
& ( ord_le516598614nres_a @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_122_dual__order_Oeq__iff,axiom,
( ( ^ [Y: refine424419629nres_a,Z: refine424419629nres_a] : Y = Z )
= ( ^ [A3: refine424419629nres_a,B3: refine424419629nres_a] :
( ( ord_le519537037nres_a @ B3 @ A3 )
& ( ord_le519537037nres_a @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_123_dual__order_Oeq__iff,axiom,
( ( ^ [Y: refine432681134nres_b,Z: refine432681134nres_b] : Y = Z )
= ( ^ [A3: refine432681134nres_b,B3: refine432681134nres_b] :
( ( ord_le527798542nres_b @ B3 @ A3 )
& ( ord_le527798542nres_b @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_124_dual__order_Otrans,axiom,
! [B2: set_Product_prod_a_a,A2: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( ord_le1824328871od_a_a @ B2 @ A2 )
=> ( ( ord_le1824328871od_a_a @ C2 @ B2 )
=> ( ord_le1824328871od_a_a @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_125_dual__order_Otrans,axiom,
! [B2: set_b,A2: set_b,C2: set_b] :
( ( ord_less_eq_set_b @ B2 @ A2 )
=> ( ( ord_less_eq_set_b @ C2 @ B2 )
=> ( ord_less_eq_set_b @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_126_dual__order_Otrans,axiom,
! [B2: set_a,A2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( ord_less_eq_set_a @ C2 @ B2 )
=> ( ord_less_eq_set_a @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_127_dual__order_Otrans,axiom,
! [B2: $o > refine432681134nres_b,A2: $o > refine432681134nres_b,C2: $o > refine432681134nres_b] :
( ( ord_le1656162903nres_b @ B2 @ A2 )
=> ( ( ord_le1656162903nres_b @ C2 @ B2 )
=> ( ord_le1656162903nres_b @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_128_dual__order_Otrans,axiom,
! [B2: $o > refine424419629nres_a,A2: $o > refine424419629nres_a,C2: $o > refine424419629nres_a] :
( ( ord_le516598614nres_a @ B2 @ A2 )
=> ( ( ord_le516598614nres_a @ C2 @ B2 )
=> ( ord_le516598614nres_a @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_129_dual__order_Otrans,axiom,
! [B2: refine424419629nres_a,A2: refine424419629nres_a,C2: refine424419629nres_a] :
( ( ord_le519537037nres_a @ B2 @ A2 )
=> ( ( ord_le519537037nres_a @ C2 @ B2 )
=> ( ord_le519537037nres_a @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_130_dual__order_Otrans,axiom,
! [B2: refine432681134nres_b,A2: refine432681134nres_b,C2: refine432681134nres_b] :
( ( ord_le527798542nres_b @ B2 @ A2 )
=> ( ( ord_le527798542nres_b @ C2 @ B2 )
=> ( ord_le527798542nres_b @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_131_dual__order_Orefl,axiom,
! [A2: set_Product_prod_a_a] : ( ord_le1824328871od_a_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_132_dual__order_Orefl,axiom,
! [A2: set_b] : ( ord_less_eq_set_b @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_133_dual__order_Orefl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_134_dual__order_Orefl,axiom,
! [A2: $o > refine432681134nres_b] : ( ord_le1656162903nres_b @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_135_dual__order_Orefl,axiom,
! [A2: $o > refine424419629nres_a] : ( ord_le516598614nres_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_136_dual__order_Orefl,axiom,
! [A2: refine424419629nres_a] : ( ord_le519537037nres_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_137_dual__order_Orefl,axiom,
! [A2: refine432681134nres_b] : ( ord_le527798542nres_b @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_138_order__trans,axiom,
! [X: set_Product_prod_a_a,Y2: set_Product_prod_a_a,Z2: set_Product_prod_a_a] :
( ( ord_le1824328871od_a_a @ X @ Y2 )
=> ( ( ord_le1824328871od_a_a @ Y2 @ Z2 )
=> ( ord_le1824328871od_a_a @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_139_order__trans,axiom,
! [X: set_b,Y2: set_b,Z2: set_b] :
( ( ord_less_eq_set_b @ X @ Y2 )
=> ( ( ord_less_eq_set_b @ Y2 @ Z2 )
=> ( ord_less_eq_set_b @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_140_order__trans,axiom,
! [X: set_a,Y2: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ( ord_less_eq_set_a @ Y2 @ Z2 )
=> ( ord_less_eq_set_a @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_141_order__trans,axiom,
! [X: $o > refine432681134nres_b,Y2: $o > refine432681134nres_b,Z2: $o > refine432681134nres_b] :
( ( ord_le1656162903nres_b @ X @ Y2 )
=> ( ( ord_le1656162903nres_b @ Y2 @ Z2 )
=> ( ord_le1656162903nres_b @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_142_order__trans,axiom,
! [X: $o > refine424419629nres_a,Y2: $o > refine424419629nres_a,Z2: $o > refine424419629nres_a] :
( ( ord_le516598614nres_a @ X @ Y2 )
=> ( ( ord_le516598614nres_a @ Y2 @ Z2 )
=> ( ord_le516598614nres_a @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_143_order__trans,axiom,
! [X: refine424419629nres_a,Y2: refine424419629nres_a,Z2: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X @ Y2 )
=> ( ( ord_le519537037nres_a @ Y2 @ Z2 )
=> ( ord_le519537037nres_a @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_144_order__trans,axiom,
! [X: refine432681134nres_b,Y2: refine432681134nres_b,Z2: refine432681134nres_b] :
( ( ord_le527798542nres_b @ X @ Y2 )
=> ( ( ord_le527798542nres_b @ Y2 @ Z2 )
=> ( ord_le527798542nres_b @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_145_order__class_Oorder_Oantisym,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ord_le1824328871od_a_a @ A2 @ B2 )
=> ( ( ord_le1824328871od_a_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% order_class.order.antisym
thf(fact_146_order__class_Oorder_Oantisym,axiom,
! [A2: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( ord_less_eq_set_b @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% order_class.order.antisym
thf(fact_147_order__class_Oorder_Oantisym,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% order_class.order.antisym
thf(fact_148_order__class_Oorder_Oantisym,axiom,
! [A2: $o > refine432681134nres_b,B2: $o > refine432681134nres_b] :
( ( ord_le1656162903nres_b @ A2 @ B2 )
=> ( ( ord_le1656162903nres_b @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% order_class.order.antisym
thf(fact_149_order__class_Oorder_Oantisym,axiom,
! [A2: $o > refine424419629nres_a,B2: $o > refine424419629nres_a] :
( ( ord_le516598614nres_a @ A2 @ B2 )
=> ( ( ord_le516598614nres_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% order_class.order.antisym
thf(fact_150_order__class_Oorder_Oantisym,axiom,
! [A2: refine424419629nres_a,B2: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A2 @ B2 )
=> ( ( ord_le519537037nres_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% order_class.order.antisym
thf(fact_151_order__class_Oorder_Oantisym,axiom,
! [A2: refine432681134nres_b,B2: refine432681134nres_b] :
( ( ord_le527798542nres_b @ A2 @ B2 )
=> ( ( ord_le527798542nres_b @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% order_class.order.antisym
thf(fact_152_ord__le__eq__trans,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( ord_le1824328871od_a_a @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_le1824328871od_a_a @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_153_ord__le__eq__trans,axiom,
! [A2: set_b,B2: set_b,C2: set_b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq_set_b @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_154_ord__le__eq__trans,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_155_ord__le__eq__trans,axiom,
! [A2: $o > refine432681134nres_b,B2: $o > refine432681134nres_b,C2: $o > refine432681134nres_b] :
( ( ord_le1656162903nres_b @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_le1656162903nres_b @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_156_ord__le__eq__trans,axiom,
! [A2: $o > refine424419629nres_a,B2: $o > refine424419629nres_a,C2: $o > refine424419629nres_a] :
( ( ord_le516598614nres_a @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_le516598614nres_a @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_157_ord__le__eq__trans,axiom,
! [A2: refine424419629nres_a,B2: refine424419629nres_a,C2: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_le519537037nres_a @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_158_ord__le__eq__trans,axiom,
! [A2: refine432681134nres_b,B2: refine432681134nres_b,C2: refine432681134nres_b] :
( ( ord_le527798542nres_b @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_le527798542nres_b @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_159_ord__eq__le__trans,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( A2 = B2 )
=> ( ( ord_le1824328871od_a_a @ B2 @ C2 )
=> ( ord_le1824328871od_a_a @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_160_ord__eq__le__trans,axiom,
! [A2: set_b,B2: set_b,C2: set_b] :
( ( A2 = B2 )
=> ( ( ord_less_eq_set_b @ B2 @ C2 )
=> ( ord_less_eq_set_b @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_161_ord__eq__le__trans,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( A2 = B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_162_ord__eq__le__trans,axiom,
! [A2: $o > refine432681134nres_b,B2: $o > refine432681134nres_b,C2: $o > refine432681134nres_b] :
( ( A2 = B2 )
=> ( ( ord_le1656162903nres_b @ B2 @ C2 )
=> ( ord_le1656162903nres_b @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_163_ord__eq__le__trans,axiom,
! [A2: $o > refine424419629nres_a,B2: $o > refine424419629nres_a,C2: $o > refine424419629nres_a] :
( ( A2 = B2 )
=> ( ( ord_le516598614nres_a @ B2 @ C2 )
=> ( ord_le516598614nres_a @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_164_ord__eq__le__trans,axiom,
! [A2: refine424419629nres_a,B2: refine424419629nres_a,C2: refine424419629nres_a] :
( ( A2 = B2 )
=> ( ( ord_le519537037nres_a @ B2 @ C2 )
=> ( ord_le519537037nres_a @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_165_ord__eq__le__trans,axiom,
! [A2: refine432681134nres_b,B2: refine432681134nres_b,C2: refine432681134nres_b] :
( ( A2 = B2 )
=> ( ( ord_le527798542nres_b @ B2 @ C2 )
=> ( ord_le527798542nres_b @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_166_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y: set_Product_prod_a_a,Z: set_Product_prod_a_a] : Y = Z )
= ( ^ [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
( ( ord_le1824328871od_a_a @ A3 @ B3 )
& ( ord_le1824328871od_a_a @ B3 @ A3 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_167_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y: set_b,Z: set_b] : Y = Z )
= ( ^ [A3: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ A3 @ B3 )
& ( ord_less_eq_set_b @ B3 @ A3 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_168_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y: set_a,Z: set_a] : Y = Z )
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
& ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_169_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y: $o > refine432681134nres_b,Z: $o > refine432681134nres_b] : Y = Z )
= ( ^ [A3: $o > refine432681134nres_b,B3: $o > refine432681134nres_b] :
( ( ord_le1656162903nres_b @ A3 @ B3 )
& ( ord_le1656162903nres_b @ B3 @ A3 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_170_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y: $o > refine424419629nres_a,Z: $o > refine424419629nres_a] : Y = Z )
= ( ^ [A3: $o > refine424419629nres_a,B3: $o > refine424419629nres_a] :
( ( ord_le516598614nres_a @ A3 @ B3 )
& ( ord_le516598614nres_a @ B3 @ A3 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_171_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y: refine424419629nres_a,Z: refine424419629nres_a] : Y = Z )
= ( ^ [A3: refine424419629nres_a,B3: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A3 @ B3 )
& ( ord_le519537037nres_a @ B3 @ A3 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_172_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y: refine432681134nres_b,Z: refine432681134nres_b] : Y = Z )
= ( ^ [A3: refine432681134nres_b,B3: refine432681134nres_b] :
( ( ord_le527798542nres_b @ A3 @ B3 )
& ( ord_le527798542nres_b @ B3 @ A3 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_173_antisym__conv,axiom,
! [Y2: set_Product_prod_a_a,X: set_Product_prod_a_a] :
( ( ord_le1824328871od_a_a @ Y2 @ X )
=> ( ( ord_le1824328871od_a_a @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv
thf(fact_174_antisym__conv,axiom,
! [Y2: set_b,X: set_b] :
( ( ord_less_eq_set_b @ Y2 @ X )
=> ( ( ord_less_eq_set_b @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv
thf(fact_175_antisym__conv,axiom,
! [Y2: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X )
=> ( ( ord_less_eq_set_a @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv
thf(fact_176_antisym__conv,axiom,
! [Y2: $o > refine432681134nres_b,X: $o > refine432681134nres_b] :
( ( ord_le1656162903nres_b @ Y2 @ X )
=> ( ( ord_le1656162903nres_b @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv
thf(fact_177_antisym__conv,axiom,
! [Y2: $o > refine424419629nres_a,X: $o > refine424419629nres_a] :
( ( ord_le516598614nres_a @ Y2 @ X )
=> ( ( ord_le516598614nres_a @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv
thf(fact_178_antisym__conv,axiom,
! [Y2: refine424419629nres_a,X: refine424419629nres_a] :
( ( ord_le519537037nres_a @ Y2 @ X )
=> ( ( ord_le519537037nres_a @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv
thf(fact_179_antisym__conv,axiom,
! [Y2: refine432681134nres_b,X: refine432681134nres_b] :
( ( ord_le527798542nres_b @ Y2 @ X )
=> ( ( ord_le527798542nres_b @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv
thf(fact_180_order_Otrans,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( ord_le1824328871od_a_a @ A2 @ B2 )
=> ( ( ord_le1824328871od_a_a @ B2 @ C2 )
=> ( ord_le1824328871od_a_a @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_181_order_Otrans,axiom,
! [A2: set_b,B2: set_b,C2: set_b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( ord_less_eq_set_b @ B2 @ C2 )
=> ( ord_less_eq_set_b @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_182_order_Otrans,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_183_order_Otrans,axiom,
! [A2: $o > refine432681134nres_b,B2: $o > refine432681134nres_b,C2: $o > refine432681134nres_b] :
( ( ord_le1656162903nres_b @ A2 @ B2 )
=> ( ( ord_le1656162903nres_b @ B2 @ C2 )
=> ( ord_le1656162903nres_b @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_184_order_Otrans,axiom,
! [A2: $o > refine424419629nres_a,B2: $o > refine424419629nres_a,C2: $o > refine424419629nres_a] :
( ( ord_le516598614nres_a @ A2 @ B2 )
=> ( ( ord_le516598614nres_a @ B2 @ C2 )
=> ( ord_le516598614nres_a @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_185_order_Otrans,axiom,
! [A2: refine424419629nres_a,B2: refine424419629nres_a,C2: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A2 @ B2 )
=> ( ( ord_le519537037nres_a @ B2 @ C2 )
=> ( ord_le519537037nres_a @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_186_order_Otrans,axiom,
! [A2: refine432681134nres_b,B2: refine432681134nres_b,C2: refine432681134nres_b] :
( ( ord_le527798542nres_b @ A2 @ B2 )
=> ( ( ord_le527798542nres_b @ B2 @ C2 )
=> ( ord_le527798542nres_b @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_187_eq__refl,axiom,
! [X: set_Product_prod_a_a,Y2: set_Product_prod_a_a] :
( ( X = Y2 )
=> ( ord_le1824328871od_a_a @ X @ Y2 ) ) ).
% eq_refl
thf(fact_188_eq__refl,axiom,
! [X: set_b,Y2: set_b] :
( ( X = Y2 )
=> ( ord_less_eq_set_b @ X @ Y2 ) ) ).
% eq_refl
thf(fact_189_eq__refl,axiom,
! [X: set_a,Y2: set_a] :
( ( X = Y2 )
=> ( ord_less_eq_set_a @ X @ Y2 ) ) ).
% eq_refl
thf(fact_190_eq__refl,axiom,
! [X: $o > refine432681134nres_b,Y2: $o > refine432681134nres_b] :
( ( X = Y2 )
=> ( ord_le1656162903nres_b @ X @ Y2 ) ) ).
% eq_refl
thf(fact_191_eq__refl,axiom,
! [X: $o > refine424419629nres_a,Y2: $o > refine424419629nres_a] :
( ( X = Y2 )
=> ( ord_le516598614nres_a @ X @ Y2 ) ) ).
% eq_refl
thf(fact_192_eq__refl,axiom,
! [X: refine424419629nres_a,Y2: refine424419629nres_a] :
( ( X = Y2 )
=> ( ord_le519537037nres_a @ X @ Y2 ) ) ).
% eq_refl
thf(fact_193_eq__refl,axiom,
! [X: refine432681134nres_b,Y2: refine432681134nres_b] :
( ( X = Y2 )
=> ( ord_le527798542nres_b @ X @ Y2 ) ) ).
% eq_refl
thf(fact_194_antisym,axiom,
! [X: set_Product_prod_a_a,Y2: set_Product_prod_a_a] :
( ( ord_le1824328871od_a_a @ X @ Y2 )
=> ( ( ord_le1824328871od_a_a @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% antisym
thf(fact_195_antisym,axiom,
! [X: set_b,Y2: set_b] :
( ( ord_less_eq_set_b @ X @ Y2 )
=> ( ( ord_less_eq_set_b @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% antisym
thf(fact_196_antisym,axiom,
! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ( ord_less_eq_set_a @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% antisym
thf(fact_197_antisym,axiom,
! [X: $o > refine432681134nres_b,Y2: $o > refine432681134nres_b] :
( ( ord_le1656162903nres_b @ X @ Y2 )
=> ( ( ord_le1656162903nres_b @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% antisym
thf(fact_198_antisym,axiom,
! [X: $o > refine424419629nres_a,Y2: $o > refine424419629nres_a] :
( ( ord_le516598614nres_a @ X @ Y2 )
=> ( ( ord_le516598614nres_a @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% antisym
thf(fact_199_antisym,axiom,
! [X: refine424419629nres_a,Y2: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X @ Y2 )
=> ( ( ord_le519537037nres_a @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% antisym
thf(fact_200_antisym,axiom,
! [X: refine432681134nres_b,Y2: refine432681134nres_b] :
( ( ord_le527798542nres_b @ X @ Y2 )
=> ( ( ord_le527798542nres_b @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% antisym
thf(fact_201_eq__iff,axiom,
( ( ^ [Y: set_Product_prod_a_a,Z: set_Product_prod_a_a] : Y = Z )
= ( ^ [X3: set_Product_prod_a_a,Y3: set_Product_prod_a_a] :
( ( ord_le1824328871od_a_a @ X3 @ Y3 )
& ( ord_le1824328871od_a_a @ Y3 @ X3 ) ) ) ) ).
% eq_iff
thf(fact_202_eq__iff,axiom,
( ( ^ [Y: set_b,Z: set_b] : Y = Z )
= ( ^ [X3: set_b,Y3: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y3 )
& ( ord_less_eq_set_b @ Y3 @ X3 ) ) ) ) ).
% eq_iff
thf(fact_203_eq__iff,axiom,
( ( ^ [Y: set_a,Z: set_a] : Y = Z )
= ( ^ [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
& ( ord_less_eq_set_a @ Y3 @ X3 ) ) ) ) ).
% eq_iff
thf(fact_204_eq__iff,axiom,
( ( ^ [Y: $o > refine432681134nres_b,Z: $o > refine432681134nres_b] : Y = Z )
= ( ^ [X3: $o > refine432681134nres_b,Y3: $o > refine432681134nres_b] :
( ( ord_le1656162903nres_b @ X3 @ Y3 )
& ( ord_le1656162903nres_b @ Y3 @ X3 ) ) ) ) ).
% eq_iff
thf(fact_205_eq__iff,axiom,
( ( ^ [Y: $o > refine424419629nres_a,Z: $o > refine424419629nres_a] : Y = Z )
= ( ^ [X3: $o > refine424419629nres_a,Y3: $o > refine424419629nres_a] :
( ( ord_le516598614nres_a @ X3 @ Y3 )
& ( ord_le516598614nres_a @ Y3 @ X3 ) ) ) ) ).
% eq_iff
thf(fact_206_eq__iff,axiom,
( ( ^ [Y: refine424419629nres_a,Z: refine424419629nres_a] : Y = Z )
= ( ^ [X3: refine424419629nres_a,Y3: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X3 @ Y3 )
& ( ord_le519537037nres_a @ Y3 @ X3 ) ) ) ) ).
% eq_iff
thf(fact_207_eq__iff,axiom,
( ( ^ [Y: refine432681134nres_b,Z: refine432681134nres_b] : Y = Z )
= ( ^ [X3: refine432681134nres_b,Y3: refine432681134nres_b] :
( ( ord_le527798542nres_b @ X3 @ Y3 )
& ( ord_le527798542nres_b @ Y3 @ X3 ) ) ) ) ).
% eq_iff
thf(fact_208_ord__le__eq__subst,axiom,
! [A2: $o > refine424419629nres_a,B2: $o > refine424419629nres_a,F: ( $o > refine424419629nres_a ) > set_b,C2: set_b] :
( ( ord_le516598614nres_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X2: $o > refine424419629nres_a,Y4: $o > refine424419629nres_a] :
( ( ord_le516598614nres_a @ X2 @ Y4 )
=> ( ord_less_eq_set_b @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_b @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_209_ord__le__eq__subst,axiom,
! [A2: $o > refine424419629nres_a,B2: $o > refine424419629nres_a,F: ( $o > refine424419629nres_a ) > set_a,C2: set_a] :
( ( ord_le516598614nres_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X2: $o > refine424419629nres_a,Y4: $o > refine424419629nres_a] :
( ( ord_le516598614nres_a @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_210_ord__le__eq__subst,axiom,
! [A2: $o > refine424419629nres_a,B2: $o > refine424419629nres_a,F: ( $o > refine424419629nres_a ) > $o > refine432681134nres_b,C2: $o > refine432681134nres_b] :
( ( ord_le516598614nres_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X2: $o > refine424419629nres_a,Y4: $o > refine424419629nres_a] :
( ( ord_le516598614nres_a @ X2 @ Y4 )
=> ( ord_le1656162903nres_b @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le1656162903nres_b @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_211_ord__le__eq__subst,axiom,
! [A2: $o > refine424419629nres_a,B2: $o > refine424419629nres_a,F: ( $o > refine424419629nres_a ) > $o > refine424419629nres_a,C2: $o > refine424419629nres_a] :
( ( ord_le516598614nres_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X2: $o > refine424419629nres_a,Y4: $o > refine424419629nres_a] :
( ( ord_le516598614nres_a @ X2 @ Y4 )
=> ( ord_le516598614nres_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le516598614nres_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_212_ord__le__eq__subst,axiom,
! [A2: refine424419629nres_a,B2: refine424419629nres_a,F: refine424419629nres_a > refine424419629nres_a,C2: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X2: refine424419629nres_a,Y4: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X2 @ Y4 )
=> ( ord_le519537037nres_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le519537037nres_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_213_ord__le__eq__subst,axiom,
! [A2: refine424419629nres_a,B2: refine424419629nres_a,F: refine424419629nres_a > refine432681134nres_b,C2: refine432681134nres_b] :
( ( ord_le519537037nres_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X2: refine424419629nres_a,Y4: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X2 @ Y4 )
=> ( ord_le527798542nres_b @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le527798542nres_b @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_214_ord__le__eq__subst,axiom,
! [A2: refine432681134nres_b,B2: refine432681134nres_b,F: refine432681134nres_b > refine424419629nres_a,C2: refine424419629nres_a] :
( ( ord_le527798542nres_b @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X2: refine432681134nres_b,Y4: refine432681134nres_b] :
( ( ord_le527798542nres_b @ X2 @ Y4 )
=> ( ord_le519537037nres_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le519537037nres_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_215_ord__le__eq__subst,axiom,
! [A2: refine432681134nres_b,B2: refine432681134nres_b,F: refine432681134nres_b > refine432681134nres_b,C2: refine432681134nres_b] :
( ( ord_le527798542nres_b @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X2: refine432681134nres_b,Y4: refine432681134nres_b] :
( ( ord_le527798542nres_b @ X2 @ Y4 )
=> ( ord_le527798542nres_b @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le527798542nres_b @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_216_ord__eq__le__subst,axiom,
! [A2: refine424419629nres_a,F: refine424419629nres_a > refine424419629nres_a,B2: refine424419629nres_a,C2: refine424419629nres_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le519537037nres_a @ B2 @ C2 )
=> ( ! [X2: refine424419629nres_a,Y4: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X2 @ Y4 )
=> ( ord_le519537037nres_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le519537037nres_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_217_ord__eq__le__subst,axiom,
! [A2: refine432681134nres_b,F: refine424419629nres_a > refine432681134nres_b,B2: refine424419629nres_a,C2: refine424419629nres_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le519537037nres_a @ B2 @ C2 )
=> ( ! [X2: refine424419629nres_a,Y4: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X2 @ Y4 )
=> ( ord_le527798542nres_b @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le527798542nres_b @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_218_ord__eq__le__subst,axiom,
! [A2: refine424419629nres_a,F: refine432681134nres_b > refine424419629nres_a,B2: refine432681134nres_b,C2: refine432681134nres_b] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le527798542nres_b @ B2 @ C2 )
=> ( ! [X2: refine432681134nres_b,Y4: refine432681134nres_b] :
( ( ord_le527798542nres_b @ X2 @ Y4 )
=> ( ord_le519537037nres_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le519537037nres_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_219_ord__eq__le__subst,axiom,
! [A2: refine432681134nres_b,F: refine432681134nres_b > refine432681134nres_b,B2: refine432681134nres_b,C2: refine432681134nres_b] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le527798542nres_b @ B2 @ C2 )
=> ( ! [X2: refine432681134nres_b,Y4: refine432681134nres_b] :
( ( ord_le527798542nres_b @ X2 @ Y4 )
=> ( ord_le527798542nres_b @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le527798542nres_b @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_220_order__subst2,axiom,
! [A2: refine424419629nres_a,B2: refine424419629nres_a,F: refine424419629nres_a > refine424419629nres_a,C2: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A2 @ B2 )
=> ( ( ord_le519537037nres_a @ ( F @ B2 ) @ C2 )
=> ( ! [X2: refine424419629nres_a,Y4: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X2 @ Y4 )
=> ( ord_le519537037nres_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le519537037nres_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_221_order__subst2,axiom,
! [A2: refine424419629nres_a,B2: refine424419629nres_a,F: refine424419629nres_a > refine432681134nres_b,C2: refine432681134nres_b] :
( ( ord_le519537037nres_a @ A2 @ B2 )
=> ( ( ord_le527798542nres_b @ ( F @ B2 ) @ C2 )
=> ( ! [X2: refine424419629nres_a,Y4: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X2 @ Y4 )
=> ( ord_le527798542nres_b @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le527798542nres_b @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_222_order__subst2,axiom,
! [A2: refine432681134nres_b,B2: refine432681134nres_b,F: refine432681134nres_b > refine424419629nres_a,C2: refine424419629nres_a] :
( ( ord_le527798542nres_b @ A2 @ B2 )
=> ( ( ord_le519537037nres_a @ ( F @ B2 ) @ C2 )
=> ( ! [X2: refine432681134nres_b,Y4: refine432681134nres_b] :
( ( ord_le527798542nres_b @ X2 @ Y4 )
=> ( ord_le519537037nres_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le519537037nres_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_223_order__subst2,axiom,
! [A2: refine432681134nres_b,B2: refine432681134nres_b,F: refine432681134nres_b > refine432681134nres_b,C2: refine432681134nres_b] :
( ( ord_le527798542nres_b @ A2 @ B2 )
=> ( ( ord_le527798542nres_b @ ( F @ B2 ) @ C2 )
=> ( ! [X2: refine432681134nres_b,Y4: refine432681134nres_b] :
( ( ord_le527798542nres_b @ X2 @ Y4 )
=> ( ord_le527798542nres_b @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le527798542nres_b @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_224_order__subst1,axiom,
! [A2: refine424419629nres_a,F: refine424419629nres_a > refine424419629nres_a,B2: refine424419629nres_a,C2: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_le519537037nres_a @ B2 @ C2 )
=> ( ! [X2: refine424419629nres_a,Y4: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X2 @ Y4 )
=> ( ord_le519537037nres_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le519537037nres_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_225_order__subst1,axiom,
! [A2: refine424419629nres_a,F: refine432681134nres_b > refine424419629nres_a,B2: refine432681134nres_b,C2: refine432681134nres_b] :
( ( ord_le519537037nres_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_le527798542nres_b @ B2 @ C2 )
=> ( ! [X2: refine432681134nres_b,Y4: refine432681134nres_b] :
( ( ord_le527798542nres_b @ X2 @ Y4 )
=> ( ord_le519537037nres_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le519537037nres_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_226_order__subst1,axiom,
! [A2: refine432681134nres_b,F: refine424419629nres_a > refine432681134nres_b,B2: refine424419629nres_a,C2: refine424419629nres_a] :
( ( ord_le527798542nres_b @ A2 @ ( F @ B2 ) )
=> ( ( ord_le519537037nres_a @ B2 @ C2 )
=> ( ! [X2: refine424419629nres_a,Y4: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X2 @ Y4 )
=> ( ord_le527798542nres_b @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le527798542nres_b @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_227_order__subst1,axiom,
! [A2: refine432681134nres_b,F: refine432681134nres_b > refine432681134nres_b,B2: refine432681134nres_b,C2: refine432681134nres_b] :
( ( ord_le527798542nres_b @ A2 @ ( F @ B2 ) )
=> ( ( ord_le527798542nres_b @ B2 @ C2 )
=> ( ! [X2: refine432681134nres_b,Y4: refine432681134nres_b] :
( ( ord_le527798542nres_b @ X2 @ Y4 )
=> ( ord_le527798542nres_b @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le527798542nres_b @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_228_Greatest__equality,axiom,
! [P: refine424419629nres_a > $o,X: refine424419629nres_a] :
( ( P @ X )
=> ( ! [Y4: refine424419629nres_a] :
( ( P @ Y4 )
=> ( ord_le519537037nres_a @ Y4 @ X ) )
=> ( ( order_1714329108nres_a @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_229_Greatest__equality,axiom,
! [P: refine432681134nres_b > $o,X: refine432681134nres_b] :
( ( P @ X )
=> ( ! [Y4: refine432681134nres_b] :
( ( P @ Y4 )
=> ( ord_le527798542nres_b @ Y4 @ X ) )
=> ( ( order_1722590613nres_b @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_230_GreatestI2__order,axiom,
! [P: refine424419629nres_a > $o,X: refine424419629nres_a,Q: refine424419629nres_a > $o] :
( ( P @ X )
=> ( ! [Y4: refine424419629nres_a] :
( ( P @ Y4 )
=> ( ord_le519537037nres_a @ Y4 @ X ) )
=> ( ! [X2: refine424419629nres_a] :
( ( P @ X2 )
=> ( ! [Y5: refine424419629nres_a] :
( ( P @ Y5 )
=> ( ord_le519537037nres_a @ Y5 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_1714329108nres_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_231_GreatestI2__order,axiom,
! [P: refine432681134nres_b > $o,X: refine432681134nres_b,Q: refine432681134nres_b > $o] :
( ( P @ X )
=> ( ! [Y4: refine432681134nres_b] :
( ( P @ Y4 )
=> ( ord_le527798542nres_b @ Y4 @ X ) )
=> ( ! [X2: refine432681134nres_b] :
( ( P @ X2 )
=> ( ! [Y5: refine432681134nres_b] :
( ( P @ Y5 )
=> ( ord_le527798542nres_b @ Y5 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_1722590613nres_b @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_232_le__rel__bool__arg__iff,axiom,
( ord_le516598614nres_a
= ( ^ [X4: $o > refine424419629nres_a,Y6: $o > refine424419629nres_a] :
( ( ord_le519537037nres_a @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_le519537037nres_a @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_233_le__rel__bool__arg__iff,axiom,
( ord_le1656162903nres_b
= ( ^ [X4: $o > refine432681134nres_b,Y6: $o > refine432681134nres_b] :
( ( ord_le527798542nres_b @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_le527798542nres_b @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_234_ord__eq__le__eq__trans,axiom,
! [A2: refine424419629nres_a,B2: refine424419629nres_a,C2: refine424419629nres_a,D: refine424419629nres_a] :
( ( A2 = B2 )
=> ( ( ord_le519537037nres_a @ B2 @ C2 )
=> ( ( C2 = D )
=> ( ord_le519537037nres_a @ A2 @ D ) ) ) ) ).
% ord_eq_le_eq_trans
thf(fact_235_ord__eq__le__eq__trans,axiom,
! [A2: refine432681134nres_b,B2: refine432681134nres_b,C2: refine432681134nres_b,D: refine432681134nres_b] :
( ( A2 = B2 )
=> ( ( ord_le527798542nres_b @ B2 @ C2 )
=> ( ( C2 = D )
=> ( ord_le527798542nres_b @ A2 @ D ) ) ) ) ).
% ord_eq_le_eq_trans
thf(fact_236_Id__refine,axiom,
! [S: refine432681134nres_b] : ( ord_le527798542nres_b @ S @ ( refine122110613un_b_b @ id_b @ S ) ) ).
% Id_refine
thf(fact_237_Id__refine,axiom,
! [S: refine424419629nres_a] : ( ord_le519537037nres_a @ S @ ( refine1441824853un_a_a @ id_a @ S ) ) ).
% Id_refine
thf(fact_238_conc__trans__additional_I2_J,axiom,
! [A: refine432681134nres_b,B: refine432681134nres_b,R: set_Product_prod_b_b,C: refine432681134nres_b] :
( ( ord_le527798542nres_b @ A @ ( refine122110613un_b_b @ id_b @ B ) )
=> ( ( ord_le527798542nres_b @ B @ ( refine122110613un_b_b @ R @ C ) )
=> ( ord_le527798542nres_b @ A @ ( refine122110613un_b_b @ R @ C ) ) ) ) ).
% conc_trans_additional(2)
thf(fact_239_conc__trans__additional_I2_J,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a,R: set_Product_prod_a_b,C: refine432681134nres_b] :
( ( ord_le519537037nres_a @ A @ ( refine1441824853un_a_a @ id_a @ B ) )
=> ( ( ord_le519537037nres_a @ B @ ( refine1441824854un_a_b @ R @ C ) )
=> ( ord_le519537037nres_a @ A @ ( refine1441824854un_a_b @ R @ C ) ) ) ) ).
% conc_trans_additional(2)
thf(fact_240_conc__trans__additional_I2_J,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a,R: set_Product_prod_a_a,C: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A @ ( refine1441824853un_a_a @ id_a @ B ) )
=> ( ( ord_le519537037nres_a @ B @ ( refine1441824853un_a_a @ R @ C ) )
=> ( ord_le519537037nres_a @ A @ ( refine1441824853un_a_a @ R @ C ) ) ) ) ).
% conc_trans_additional(2)
thf(fact_241_conc__trans__additional_I3_J,axiom,
! [A: refine432681134nres_b,R: set_Product_prod_b_a,B: refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_le527798542nres_b @ A @ ( refine122110612un_b_a @ R @ B ) )
=> ( ( ord_le519537037nres_a @ B @ ( refine1441824853un_a_a @ id_a @ C ) )
=> ( ord_le527798542nres_b @ A @ ( refine122110612un_b_a @ R @ C ) ) ) ) ).
% conc_trans_additional(3)
thf(fact_242_conc__trans__additional_I3_J,axiom,
! [A: refine424419629nres_a,R: set_Product_prod_a_b,B: refine432681134nres_b,C: refine432681134nres_b] :
( ( ord_le519537037nres_a @ A @ ( refine1441824854un_a_b @ R @ B ) )
=> ( ( ord_le527798542nres_b @ B @ ( refine122110613un_b_b @ id_b @ C ) )
=> ( ord_le519537037nres_a @ A @ ( refine1441824854un_a_b @ R @ C ) ) ) ) ).
% conc_trans_additional(3)
thf(fact_243_conc__trans__additional_I3_J,axiom,
! [A: refine432681134nres_b,R: set_Product_prod_b_b,B: refine432681134nres_b,C: refine432681134nres_b] :
( ( ord_le527798542nres_b @ A @ ( refine122110613un_b_b @ R @ B ) )
=> ( ( ord_le527798542nres_b @ B @ ( refine122110613un_b_b @ id_b @ C ) )
=> ( ord_le527798542nres_b @ A @ ( refine122110613un_b_b @ R @ C ) ) ) ) ).
% conc_trans_additional(3)
thf(fact_244_conc__trans__additional_I3_J,axiom,
! [A: refine424419629nres_a,R: set_Product_prod_a_a,B: refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A @ ( refine1441824853un_a_a @ R @ B ) )
=> ( ( ord_le519537037nres_a @ B @ ( refine1441824853un_a_a @ id_a @ C ) )
=> ( ord_le519537037nres_a @ A @ ( refine1441824853un_a_a @ R @ C ) ) ) ) ).
% conc_trans_additional(3)
thf(fact_245_conc__trans__additional_I6_J,axiom,
! [A: refine432681134nres_b,B: refine432681134nres_b,C: refine432681134nres_b] :
( ( ord_le527798542nres_b @ A @ B )
=> ( ( ord_le527798542nres_b @ B @ ( refine122110613un_b_b @ id_b @ C ) )
=> ( ord_le527798542nres_b @ A @ C ) ) ) ).
% conc_trans_additional(6)
thf(fact_246_conc__trans__additional_I6_J,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A @ B )
=> ( ( ord_le519537037nres_a @ B @ ( refine1441824853un_a_a @ id_a @ C ) )
=> ( ord_le519537037nres_a @ A @ C ) ) ) ).
% conc_trans_additional(6)
thf(fact_247_conc__trans__additional_I5_J,axiom,
! [A: refine432681134nres_b,B: refine432681134nres_b,C: refine432681134nres_b] :
( ( ord_le527798542nres_b @ A @ ( refine122110613un_b_b @ id_b @ B ) )
=> ( ( ord_le527798542nres_b @ B @ C )
=> ( ord_le527798542nres_b @ A @ C ) ) ) ).
% conc_trans_additional(5)
thf(fact_248_conc__trans__additional_I5_J,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A @ ( refine1441824853un_a_a @ id_a @ B ) )
=> ( ( ord_le519537037nres_a @ B @ C )
=> ( ord_le519537037nres_a @ A @ C ) ) ) ).
% conc_trans_additional(5)
thf(fact_249_conc__trans__additional_I4_J,axiom,
! [A: refine432681134nres_b,B: refine432681134nres_b,C: refine432681134nres_b] :
( ( ord_le527798542nres_b @ A @ ( refine122110613un_b_b @ id_b @ B ) )
=> ( ( ord_le527798542nres_b @ B @ ( refine122110613un_b_b @ id_b @ C ) )
=> ( ord_le527798542nres_b @ A @ C ) ) ) ).
% conc_trans_additional(4)
thf(fact_250_conc__trans__additional_I4_J,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A @ ( refine1441824853un_a_a @ id_a @ B ) )
=> ( ( ord_le519537037nres_a @ B @ ( refine1441824853un_a_a @ id_a @ C ) )
=> ( ord_le519537037nres_a @ A @ C ) ) ) ).
% conc_trans_additional(4)
thf(fact_251_Id__SPEC__refine,axiom,
! [S: refine432681134nres_b,Phi: b > $o] :
( ( ord_le527798542nres_b @ S @ ( refine1198353289_RES_b @ ( collect_b @ Phi ) ) )
=> ( ord_le527798542nres_b @ S @ ( refine122110613un_b_b @ id_b @ ( refine1198353289_RES_b @ ( collect_b @ Phi ) ) ) ) ) ).
% Id_SPEC_refine
thf(fact_252_Id__SPEC__refine,axiom,
! [S: refine424419629nres_a,Phi: a > $o] :
( ( ord_le519537037nres_a @ S @ ( refine1198353288_RES_a @ ( collect_a @ Phi ) ) )
=> ( ord_le519537037nres_a @ S @ ( refine1441824853un_a_a @ id_a @ ( refine1198353288_RES_a @ ( collect_a @ Phi ) ) ) ) ) ).
% Id_SPEC_refine
thf(fact_253_abs__trans__additional_I2_J,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a,R: set_Product_prod_a_a,C: refine424419629nres_a] :
( ( ord_le519537037nres_a @ ( refine1136779702un_a_a @ id_a @ A ) @ B )
=> ( ( ord_le519537037nres_a @ ( refine1136779702un_a_a @ R @ B ) @ C )
=> ( ord_le519537037nres_a @ ( refine1136779702un_a_a @ R @ A ) @ C ) ) ) ).
% abs_trans_additional(2)
thf(fact_254_abs__trans__additional_I2_J,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a,R: set_Product_prod_a_b,C: refine432681134nres_b] :
( ( ord_le519537037nres_a @ ( refine1136779702un_a_a @ id_a @ A ) @ B )
=> ( ( ord_le527798542nres_b @ ( refine1136779703un_a_b @ R @ B ) @ C )
=> ( ord_le527798542nres_b @ ( refine1136779703un_a_b @ R @ A ) @ C ) ) ) ).
% abs_trans_additional(2)
thf(fact_255_abs__trans__additional_I2_J,axiom,
! [A: refine432681134nres_b,B: refine432681134nres_b,R: set_Product_prod_b_a,C: refine424419629nres_a] :
( ( ord_le527798542nres_b @ ( refine1964549110un_b_b @ id_b @ A ) @ B )
=> ( ( ord_le519537037nres_a @ ( refine1964549109un_b_a @ R @ B ) @ C )
=> ( ord_le519537037nres_a @ ( refine1964549109un_b_a @ R @ A ) @ C ) ) ) ).
% abs_trans_additional(2)
thf(fact_256_abs__trans__additional_I2_J,axiom,
! [A: refine432681134nres_b,B: refine432681134nres_b,R: set_Product_prod_b_b,C: refine432681134nres_b] :
( ( ord_le527798542nres_b @ ( refine1964549110un_b_b @ id_b @ A ) @ B )
=> ( ( ord_le527798542nres_b @ ( refine1964549110un_b_b @ R @ B ) @ C )
=> ( ord_le527798542nres_b @ ( refine1964549110un_b_b @ R @ A ) @ C ) ) ) ).
% abs_trans_additional(2)
thf(fact_257_abs__trans__additional_I4_J,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_le519537037nres_a @ ( refine1136779702un_a_a @ id_a @ A ) @ B )
=> ( ( ord_le519537037nres_a @ ( refine1136779702un_a_a @ id_a @ B ) @ C )
=> ( ord_le519537037nres_a @ A @ C ) ) ) ).
% abs_trans_additional(4)
thf(fact_258_abs__trans__additional_I4_J,axiom,
! [A: refine432681134nres_b,B: refine432681134nres_b,C: refine432681134nres_b] :
( ( ord_le527798542nres_b @ ( refine1964549110un_b_b @ id_b @ A ) @ B )
=> ( ( ord_le527798542nres_b @ ( refine1964549110un_b_b @ id_b @ B ) @ C )
=> ( ord_le527798542nres_b @ A @ C ) ) ) ).
% abs_trans_additional(4)
thf(fact_259_abs__trans__additional_I5_J,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_le519537037nres_a @ ( refine1136779702un_a_a @ id_a @ A ) @ B )
=> ( ( ord_le519537037nres_a @ B @ C )
=> ( ord_le519537037nres_a @ A @ C ) ) ) ).
% abs_trans_additional(5)
thf(fact_260_abs__trans__additional_I5_J,axiom,
! [A: refine432681134nres_b,B: refine432681134nres_b,C: refine432681134nres_b] :
( ( ord_le527798542nres_b @ ( refine1964549110un_b_b @ id_b @ A ) @ B )
=> ( ( ord_le527798542nres_b @ B @ C )
=> ( ord_le527798542nres_b @ A @ C ) ) ) ).
% abs_trans_additional(5)
thf(fact_261_abs__trans__additional_I6_J,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A @ B )
=> ( ( ord_le519537037nres_a @ ( refine1136779702un_a_a @ id_a @ B ) @ C )
=> ( ord_le519537037nres_a @ A @ C ) ) ) ).
% abs_trans_additional(6)
thf(fact_262_abs__trans__additional_I6_J,axiom,
! [A: refine432681134nres_b,B: refine432681134nres_b,C: refine432681134nres_b] :
( ( ord_le527798542nres_b @ A @ B )
=> ( ( ord_le527798542nres_b @ ( refine1964549110un_b_b @ id_b @ B ) @ C )
=> ( ord_le527798542nres_b @ A @ C ) ) ) ).
% abs_trans_additional(6)
thf(fact_263_abs__fun__simps_I1_J,axiom,
! [R: set_Product_prod_b_b] :
( ( refine1964549110un_b_b @ R @ top_to240090974nres_b )
= top_to240090974nres_b ) ).
% abs_fun_simps(1)
thf(fact_264_abs__fun__simps_I1_J,axiom,
! [R: set_Product_prod_b_a] :
( ( refine1964549109un_b_a @ R @ top_to240090974nres_b )
= top_to231829469nres_a ) ).
% abs_fun_simps(1)
thf(fact_265_abs__fun__simps_I1_J,axiom,
! [R: set_Product_prod_a_b] :
( ( refine1136779703un_a_b @ R @ top_to231829469nres_a )
= top_to240090974nres_b ) ).
% abs_fun_simps(1)
thf(fact_266_abs__fun__simps_I1_J,axiom,
! [R: set_Product_prod_a_a] :
( ( refine1136779702un_a_a @ R @ top_to231829469nres_a )
= top_to231829469nres_a ) ).
% abs_fun_simps(1)
thf(fact_267_if__rule,axiom,
! [B2: $o,S1: refine424419629nres_a,Phi: a > $o,S2: refine424419629nres_a] :
( ( B2
=> ( ord_le519537037nres_a @ S1 @ ( refine1198353288_RES_a @ ( collect_a @ Phi ) ) ) )
=> ( ( ~ B2
=> ( ord_le519537037nres_a @ S2 @ ( refine1198353288_RES_a @ ( collect_a @ Phi ) ) ) )
=> ( ord_le519537037nres_a @ ( if_Ref1724547303nres_a @ B2 @ S1 @ S2 ) @ ( refine1198353288_RES_a @ ( collect_a @ Phi ) ) ) ) ) ).
% if_rule
thf(fact_268_if__rule,axiom,
! [B2: $o,S1: refine432681134nres_b,Phi: b > $o,S2: refine432681134nres_b] :
( ( B2
=> ( ord_le527798542nres_b @ S1 @ ( refine1198353289_RES_b @ ( collect_b @ Phi ) ) ) )
=> ( ( ~ B2
=> ( ord_le527798542nres_b @ S2 @ ( refine1198353289_RES_b @ ( collect_b @ Phi ) ) ) )
=> ( ord_le527798542nres_b @ ( if_Ref1732808808nres_b @ B2 @ S1 @ S2 ) @ ( refine1198353289_RES_b @ ( collect_b @ Phi ) ) ) ) ) ).
% if_rule
thf(fact_269_RES__rule,axiom,
! [S: set_a,Phi: a > $o] :
( ! [X2: a] :
( ( member_a @ X2 @ S )
=> ( Phi @ X2 ) )
=> ( ord_le519537037nres_a @ ( refine1198353288_RES_a @ S ) @ ( refine1198353288_RES_a @ ( collect_a @ Phi ) ) ) ) ).
% RES_rule
thf(fact_270_RES__rule,axiom,
! [S: set_b,Phi: b > $o] :
( ! [X2: b] :
( ( member_b @ X2 @ S )
=> ( Phi @ X2 ) )
=> ( ord_le527798542nres_b @ ( refine1198353289_RES_b @ S ) @ ( refine1198353289_RES_b @ ( collect_b @ Phi ) ) ) ) ).
% RES_rule
thf(fact_271_SPEC__rule,axiom,
! [Phi: a > $o,Phi2: a > $o] :
( ! [X2: a] :
( ( Phi @ X2 )
=> ( Phi2 @ X2 ) )
=> ( ord_le519537037nres_a @ ( refine1198353288_RES_a @ ( collect_a @ Phi ) ) @ ( refine1198353288_RES_a @ ( collect_a @ Phi2 ) ) ) ) ).
% SPEC_rule
thf(fact_272_SPEC__rule,axiom,
! [Phi: b > $o,Phi2: b > $o] :
( ! [X2: b] :
( ( Phi @ X2 )
=> ( Phi2 @ X2 ) )
=> ( ord_le527798542nres_b @ ( refine1198353289_RES_b @ ( collect_b @ Phi ) ) @ ( refine1198353289_RES_b @ ( collect_b @ Phi2 ) ) ) ) ).
% SPEC_rule
thf(fact_273_SPEC__trans,axiom,
! [X: refine424419629nres_a,Y2: refine424419629nres_a,Postcond: a > $o] :
( ( ord_le519537037nres_a @ X @ Y2 )
=> ( ( ord_le519537037nres_a @ Y2 @ ( refine1198353288_RES_a @ ( collect_a @ Postcond ) ) )
=> ( ord_le519537037nres_a @ X @ ( refine1198353288_RES_a @ ( collect_a @ Postcond ) ) ) ) ) ).
% SPEC_trans
thf(fact_274_SPEC__trans,axiom,
! [X: refine432681134nres_b,Y2: refine432681134nres_b,Postcond: b > $o] :
( ( ord_le527798542nres_b @ X @ Y2 )
=> ( ( ord_le527798542nres_b @ Y2 @ ( refine1198353289_RES_b @ ( collect_b @ Postcond ) ) )
=> ( ord_le527798542nres_b @ X @ ( refine1198353289_RES_b @ ( collect_b @ Postcond ) ) ) ) ) ).
% SPEC_trans
thf(fact_275_SPEC__cons__rule,axiom,
! [M2: refine424419629nres_a,Phi: a > $o,Psi: a > $o] :
( ( ord_le519537037nres_a @ M2 @ ( refine1198353288_RES_a @ ( collect_a @ Phi ) ) )
=> ( ! [X2: a] :
( ( Phi @ X2 )
=> ( Psi @ X2 ) )
=> ( ord_le519537037nres_a @ M2 @ ( refine1198353288_RES_a @ ( collect_a @ Psi ) ) ) ) ) ).
% SPEC_cons_rule
thf(fact_276_SPEC__cons__rule,axiom,
! [M2: refine432681134nres_b,Phi: b > $o,Psi: b > $o] :
( ( ord_le527798542nres_b @ M2 @ ( refine1198353289_RES_b @ ( collect_b @ Phi ) ) )
=> ( ! [X2: b] :
( ( Phi @ X2 )
=> ( Psi @ X2 ) )
=> ( ord_le527798542nres_b @ M2 @ ( refine1198353289_RES_b @ ( collect_b @ Psi ) ) ) ) ) ).
% SPEC_cons_rule
thf(fact_277_nres__inequalities_I1_J,axiom,
! [X5: set_b] :
( top_to240090974nres_b
!= ( refine1198353289_RES_b @ X5 ) ) ).
% nres_inequalities(1)
thf(fact_278_nres__inequalities_I1_J,axiom,
! [X5: set_a] :
( top_to231829469nres_a
!= ( refine1198353288_RES_a @ X5 ) ) ).
% nres_inequalities(1)
thf(fact_279_nres__cases,axiom,
! [M: refine432681134nres_b] :
( ( M != top_to240090974nres_b )
=> ~ ! [X6: set_b] :
( M
!= ( refine1198353289_RES_b @ X6 ) ) ) ).
% nres_cases
thf(fact_280_nres__cases,axiom,
! [M: refine424419629nres_a] :
( ( M != top_to231829469nres_a )
=> ~ ! [X6: set_a] :
( M
!= ( refine1198353288_RES_a @ X6 ) ) ) ).
% nres_cases
thf(fact_281_abs__trans__additional_I1_J,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a,R: set_Product_prod_a_a,C: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A @ B )
=> ( ( ord_le519537037nres_a @ ( refine1136779702un_a_a @ R @ B ) @ C )
=> ( ord_le519537037nres_a @ ( refine1136779702un_a_a @ R @ A ) @ C ) ) ) ).
% abs_trans_additional(1)
thf(fact_282_abs__trans__additional_I1_J,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a,R: set_Product_prod_a_b,C: refine432681134nres_b] :
( ( ord_le519537037nres_a @ A @ B )
=> ( ( ord_le527798542nres_b @ ( refine1136779703un_a_b @ R @ B ) @ C )
=> ( ord_le527798542nres_b @ ( refine1136779703un_a_b @ R @ A ) @ C ) ) ) ).
% abs_trans_additional(1)
thf(fact_283_abs__trans__additional_I1_J,axiom,
! [A: refine432681134nres_b,B: refine432681134nres_b,R: set_Product_prod_b_a,C: refine424419629nres_a] :
( ( ord_le527798542nres_b @ A @ B )
=> ( ( ord_le519537037nres_a @ ( refine1964549109un_b_a @ R @ B ) @ C )
=> ( ord_le519537037nres_a @ ( refine1964549109un_b_a @ R @ A ) @ C ) ) ) ).
% abs_trans_additional(1)
thf(fact_284_abs__trans__additional_I1_J,axiom,
! [A: refine432681134nres_b,B: refine432681134nres_b,R: set_Product_prod_b_b,C: refine432681134nres_b] :
( ( ord_le527798542nres_b @ A @ B )
=> ( ( ord_le527798542nres_b @ ( refine1964549110un_b_b @ R @ B ) @ C )
=> ( ord_le527798542nres_b @ ( refine1964549110un_b_b @ R @ A ) @ C ) ) ) ).
% abs_trans_additional(1)
thf(fact_285_nres__order__simps_I5_J,axiom,
! [X5: set_a,Y7: set_a] :
( ( ord_le519537037nres_a @ ( refine1198353288_RES_a @ X5 ) @ ( refine1198353288_RES_a @ Y7 ) )
= ( ord_less_eq_set_a @ X5 @ Y7 ) ) ).
% nres_order_simps(5)
thf(fact_286_nres__order__simps_I5_J,axiom,
! [X5: set_b,Y7: set_b] :
( ( ord_le527798542nres_b @ ( refine1198353289_RES_b @ X5 ) @ ( refine1198353289_RES_b @ Y7 ) )
= ( ord_less_eq_set_b @ X5 @ Y7 ) ) ).
% nres_order_simps(5)
thf(fact_287_less__eq__nres_Osimps_I2_J,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_le519537037nres_a @ ( refine1198353288_RES_a @ A2 ) @ ( refine1198353288_RES_a @ B2 ) )
= ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% less_eq_nres.simps(2)
thf(fact_288_less__eq__nres_Osimps_I2_J,axiom,
! [A2: set_b,B2: set_b] :
( ( ord_le527798542nres_b @ ( refine1198353289_RES_b @ A2 ) @ ( refine1198353289_RES_b @ B2 ) )
= ( ord_less_eq_set_b @ A2 @ B2 ) ) ).
% less_eq_nres.simps(2)
thf(fact_289_nres__order__simps_I21_J,axiom,
! [X: a,Y7: set_a] :
( ( ord_le519537037nres_a @ ( refine2063221604TURN_a @ X ) @ ( refine1198353288_RES_a @ Y7 ) )
= ( member_a @ X @ Y7 ) ) ).
% nres_order_simps(21)
thf(fact_290_nres__order__simps_I21_J,axiom,
! [X: b,Y7: set_b] :
( ( ord_le527798542nres_b @ ( refine2063221605TURN_b @ X ) @ ( refine1198353289_RES_b @ Y7 ) )
= ( member_b @ X @ Y7 ) ) ).
% nres_order_simps(21)
thf(fact_291_nres__order__simps_I20_J,axiom,
! [X: a,Y2: a] :
( ( ord_le519537037nres_a @ ( refine2063221604TURN_a @ X ) @ ( refine2063221604TURN_a @ Y2 ) )
= ( X = Y2 ) ) ).
% nres_order_simps(20)
thf(fact_292_nres__order__simps_I20_J,axiom,
! [X: b,Y2: b] :
( ( ord_le527798542nres_b @ ( refine2063221605TURN_b @ X ) @ ( refine2063221605TURN_b @ Y2 ) )
= ( X = Y2 ) ) ).
% nres_order_simps(20)
thf(fact_293_nres__inequalities_I3_J,axiom,
! [X: b] :
( top_to240090974nres_b
!= ( refine2063221605TURN_b @ X ) ) ).
% nres_inequalities(3)
thf(fact_294_nres__inequalities_I3_J,axiom,
! [X: a] :
( top_to231829469nres_a
!= ( refine2063221604TURN_a @ X ) ) ).
% nres_inequalities(3)
thf(fact_295_RETURN__to__SPEC__rule,axiom,
! [M2: refine424419629nres_a,V: a] :
( ( ord_le519537037nres_a @ M2
@ ( refine1198353288_RES_a
@ ( collect_a
@ ( ^ [Y: a,Z: a] : Y = Z
@ V ) ) ) )
=> ( ord_le519537037nres_a @ M2 @ ( refine2063221604TURN_a @ V ) ) ) ).
% RETURN_to_SPEC_rule
thf(fact_296_RETURN__to__SPEC__rule,axiom,
! [M2: refine432681134nres_b,V: b] :
( ( ord_le527798542nres_b @ M2
@ ( refine1198353289_RES_b
@ ( collect_b
@ ( ^ [Y: b,Z: b] : Y = Z
@ V ) ) ) )
=> ( ord_le527798542nres_b @ M2 @ ( refine2063221605TURN_b @ V ) ) ) ).
% RETURN_to_SPEC_rule
thf(fact_297_RETURN__rule,axiom,
! [Phi: a > $o,X: a] :
( ( Phi @ X )
=> ( ord_le519537037nres_a @ ( refine2063221604TURN_a @ X ) @ ( refine1198353288_RES_a @ ( collect_a @ Phi ) ) ) ) ).
% RETURN_rule
thf(fact_298_RETURN__rule,axiom,
! [Phi: b > $o,X: b] :
( ( Phi @ X )
=> ( ord_le527798542nres_b @ ( refine2063221605TURN_b @ X ) @ ( refine1198353289_RES_b @ ( collect_b @ Phi ) ) ) ) ).
% RETURN_rule
thf(fact_299_RETURN__SPEC__refine,axiom,
! [X: a,R: set_Product_prod_a_b,Phi: b > $o] :
( ? [X7: b] :
( ( member458171089od_a_b @ ( product_Pair_a_b @ X @ X7 ) @ R )
& ( Phi @ X7 ) )
=> ( ord_le519537037nres_a @ ( refine2063221604TURN_a @ X ) @ ( refine1441824854un_a_b @ R @ ( refine1198353289_RES_b @ ( collect_b @ Phi ) ) ) ) ) ).
% RETURN_SPEC_refine
thf(fact_300_RETURN__SPEC__refine,axiom,
! [X: b,R: set_Product_prod_b_b,Phi: b > $o] :
( ? [X7: b] :
( ( member1285940496od_b_b @ ( product_Pair_b_b @ X @ X7 ) @ R )
& ( Phi @ X7 ) )
=> ( ord_le527798542nres_b @ ( refine2063221605TURN_b @ X ) @ ( refine122110613un_b_b @ R @ ( refine1198353289_RES_b @ ( collect_b @ Phi ) ) ) ) ) ).
% RETURN_SPEC_refine
thf(fact_301_RETURN__SPEC__refine,axiom,
! [X: a,R: set_Product_prod_a_a,Phi: a > $o] :
( ? [X7: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ X7 ) @ R )
& ( Phi @ X7 ) )
=> ( ord_le519537037nres_a @ ( refine2063221604TURN_a @ X ) @ ( refine1441824853un_a_a @ R @ ( refine1198353288_RES_a @ ( collect_a @ Phi ) ) ) ) ) ).
% RETURN_SPEC_refine
thf(fact_302_nofail__simps_I1_J,axiom,
~ ( refine412683990fail_b @ top_to240090974nres_b ) ).
% nofail_simps(1)
thf(fact_303_nofail__simps_I1_J,axiom,
~ ( refine412683989fail_a @ top_to231829469nres_a ) ).
% nofail_simps(1)
thf(fact_304_pw__conc__nofail,axiom,
! [R: set_Product_prod_a_b,S: refine432681134nres_b] :
( ( refine412683989fail_a @ ( refine1441824854un_a_b @ R @ S ) )
= ( refine412683990fail_b @ S ) ) ).
% pw_conc_nofail
thf(fact_305_pw__conc__nofail,axiom,
! [R: set_Product_prod_b_b,S: refine432681134nres_b] :
( ( refine412683990fail_b @ ( refine122110613un_b_b @ R @ S ) )
= ( refine412683990fail_b @ S ) ) ).
% pw_conc_nofail
thf(fact_306_pw__conc__nofail,axiom,
! [R: set_Product_prod_a_a,S: refine424419629nres_a] :
( ( refine412683989fail_a @ ( refine1441824853un_a_a @ R @ S ) )
= ( refine412683989fail_a @ S ) ) ).
% pw_conc_nofail
thf(fact_307_intro__nofail_I2_J,axiom,
! [S: refine432681134nres_b] :
( ( top_to240090974nres_b != S )
= ( refine412683990fail_b @ S ) ) ).
% intro_nofail(2)
thf(fact_308_intro__nofail_I2_J,axiom,
! [S: refine424419629nres_a] :
( ( top_to231829469nres_a != S )
= ( refine412683989fail_a @ S ) ) ).
% intro_nofail(2)
thf(fact_309_nofail__def,axiom,
( refine412683990fail_b
= ( ^ [S3: refine432681134nres_b] : S3 != top_to240090974nres_b ) ) ).
% nofail_def
thf(fact_310_nofail__def,axiom,
( refine412683989fail_a
= ( ^ [S3: refine424419629nres_a] : S3 != top_to231829469nres_a ) ) ).
% nofail_def
thf(fact_311_not__nofail__iff,axiom,
! [S: refine432681134nres_b] :
( ( ~ ( refine412683990fail_b @ S ) )
= ( S = top_to240090974nres_b ) ) ).
% not_nofail_iff
thf(fact_312_not__nofail__iff,axiom,
! [S: refine424419629nres_a] :
( ( ~ ( refine412683989fail_a @ S ) )
= ( S = top_to231829469nres_a ) ) ).
% not_nofail_iff
thf(fact_313_pwD1,axiom,
! [S: refine424419629nres_a,S4: refine424419629nres_a] :
( ( ord_le519537037nres_a @ S @ S4 )
=> ( ( refine412683989fail_a @ S4 )
=> ( refine412683989fail_a @ S ) ) ) ).
% pwD1
thf(fact_314_pwD1,axiom,
! [S: refine432681134nres_b,S4: refine432681134nres_b] :
( ( ord_le527798542nres_b @ S @ S4 )
=> ( ( refine412683990fail_b @ S4 )
=> ( refine412683990fail_b @ S ) ) ) ).
% pwD1
thf(fact_315_le__nofailI,axiom,
! [M3: refine424419629nres_a,M: refine424419629nres_a] :
( ( ( refine412683989fail_a @ M3 )
=> ( ord_le519537037nres_a @ M @ M3 ) )
=> ( ord_le519537037nres_a @ M @ M3 ) ) ).
% le_nofailI
thf(fact_316_le__nofailI,axiom,
! [M3: refine432681134nres_b,M: refine432681134nres_b] :
( ( ( refine412683990fail_b @ M3 )
=> ( ord_le527798542nres_b @ M @ M3 ) )
=> ( ord_le527798542nres_b @ M @ M3 ) ) ).
% le_nofailI
thf(fact_317_RES__refine,axiom,
! [S: set_a,S4: set_b,R: set_Product_prod_a_b] :
( ! [S5: a] :
( ( member_a @ S5 @ S )
=> ? [X8: b] :
( ( member_b @ X8 @ S4 )
& ( member458171089od_a_b @ ( product_Pair_a_b @ S5 @ X8 ) @ R ) ) )
=> ( ord_le519537037nres_a @ ( refine1198353288_RES_a @ S ) @ ( refine1441824854un_a_b @ R @ ( refine1198353289_RES_b @ S4 ) ) ) ) ).
% RES_refine
thf(fact_318_RES__refine,axiom,
! [S: set_b,S4: set_b,R: set_Product_prod_b_b] :
( ! [S5: b] :
( ( member_b @ S5 @ S )
=> ? [X8: b] :
( ( member_b @ X8 @ S4 )
& ( member1285940496od_b_b @ ( product_Pair_b_b @ S5 @ X8 ) @ R ) ) )
=> ( ord_le527798542nres_b @ ( refine1198353289_RES_b @ S ) @ ( refine122110613un_b_b @ R @ ( refine1198353289_RES_b @ S4 ) ) ) ) ).
% RES_refine
thf(fact_319_RES__refine,axiom,
! [S: set_a,S4: set_a,R: set_Product_prod_a_a] :
( ! [S5: a] :
( ( member_a @ S5 @ S )
=> ? [X8: a] :
( ( member_a @ X8 @ S4 )
& ( member449909584od_a_a @ ( product_Pair_a_a @ S5 @ X8 ) @ R ) ) )
=> ( ord_le519537037nres_a @ ( refine1198353288_RES_a @ S ) @ ( refine1441824853un_a_a @ R @ ( refine1198353288_RES_a @ S4 ) ) ) ) ).
% RES_refine
thf(fact_320_RETURN__refine,axiom,
! [X: a,X9: b,R: set_Product_prod_a_b] :
( ( member458171089od_a_b @ ( product_Pair_a_b @ X @ X9 ) @ R )
=> ( ord_le519537037nres_a @ ( refine2063221604TURN_a @ X ) @ ( refine1441824854un_a_b @ R @ ( refine2063221605TURN_b @ X9 ) ) ) ) ).
% RETURN_refine
thf(fact_321_RETURN__refine,axiom,
! [X: b,X9: b,R: set_Product_prod_b_b] :
( ( member1285940496od_b_b @ ( product_Pair_b_b @ X @ X9 ) @ R )
=> ( ord_le527798542nres_b @ ( refine2063221605TURN_b @ X ) @ ( refine122110613un_b_b @ R @ ( refine2063221605TURN_b @ X9 ) ) ) ) ).
% RETURN_refine
thf(fact_322_RETURN__refine,axiom,
! [X: a,X9: a,R: set_Product_prod_a_a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ X9 ) @ R )
=> ( ord_le519537037nres_a @ ( refine2063221604TURN_a @ X ) @ ( refine1441824853un_a_a @ R @ ( refine2063221604TURN_a @ X9 ) ) ) ) ).
% RETURN_refine
thf(fact_323_inres__simps_I1_J,axiom,
( ( refine1001002028nres_b @ top_to240090974nres_b )
= ( ^ [Uu: b] : $true ) ) ).
% inres_simps(1)
thf(fact_324_inres__simps_I1_J,axiom,
( ( refine1001002027nres_a @ top_to231829469nres_a )
= ( ^ [Uu: a] : $true ) ) ).
% inres_simps(1)
thf(fact_325_pwD2,axiom,
! [S: refine424419629nres_a,S4: refine424419629nres_a,X: a] :
( ( ord_le519537037nres_a @ S @ S4 )
=> ( ( refine1001002027nres_a @ S @ X )
=> ( refine1001002027nres_a @ S4 @ X ) ) ) ).
% pwD2
thf(fact_326_pwD2,axiom,
! [S: refine432681134nres_b,S4: refine432681134nres_b,X: b] :
( ( ord_le527798542nres_b @ S @ S4 )
=> ( ( refine1001002028nres_b @ S @ X )
=> ( refine1001002028nres_b @ S4 @ X ) ) ) ).
% pwD2
thf(fact_327_pw__leI,axiom,
! [S4: refine424419629nres_a,S: refine424419629nres_a] :
( ( ( refine412683989fail_a @ S4 )
=> ( ( refine412683989fail_a @ S )
& ! [X2: a] :
( ( refine1001002027nres_a @ S @ X2 )
=> ( refine1001002027nres_a @ S4 @ X2 ) ) ) )
=> ( ord_le519537037nres_a @ S @ S4 ) ) ).
% pw_leI
thf(fact_328_pw__leI,axiom,
! [S4: refine432681134nres_b,S: refine432681134nres_b] :
( ( ( refine412683990fail_b @ S4 )
=> ( ( refine412683990fail_b @ S )
& ! [X2: b] :
( ( refine1001002028nres_b @ S @ X2 )
=> ( refine1001002028nres_b @ S4 @ X2 ) ) ) )
=> ( ord_le527798542nres_b @ S @ S4 ) ) ).
% pw_leI
thf(fact_329_pw__leI_H,axiom,
! [S4: refine424419629nres_a,S: refine424419629nres_a] :
( ( ( refine412683989fail_a @ S4 )
=> ( refine412683989fail_a @ S ) )
=> ( ! [X2: a] :
( ( refine412683989fail_a @ S4 )
=> ( ( refine1001002027nres_a @ S @ X2 )
=> ( refine1001002027nres_a @ S4 @ X2 ) ) )
=> ( ord_le519537037nres_a @ S @ S4 ) ) ) ).
% pw_leI'
thf(fact_330_pw__leI_H,axiom,
! [S4: refine432681134nres_b,S: refine432681134nres_b] :
( ( ( refine412683990fail_b @ S4 )
=> ( refine412683990fail_b @ S ) )
=> ( ! [X2: b] :
( ( refine412683990fail_b @ S4 )
=> ( ( refine1001002028nres_b @ S @ X2 )
=> ( refine1001002028nres_b @ S4 @ X2 ) ) )
=> ( ord_le527798542nres_b @ S @ S4 ) ) ) ).
% pw_leI'
thf(fact_331_pw__le__iff,axiom,
( ord_le519537037nres_a
= ( ^ [S3: refine424419629nres_a,S6: refine424419629nres_a] :
( ( refine412683989fail_a @ S6 )
=> ( ( refine412683989fail_a @ S3 )
& ! [X3: a] :
( ( refine1001002027nres_a @ S3 @ X3 )
=> ( refine1001002027nres_a @ S6 @ X3 ) ) ) ) ) ) ).
% pw_le_iff
thf(fact_332_pw__le__iff,axiom,
( ord_le527798542nres_b
= ( ^ [S3: refine432681134nres_b,S6: refine432681134nres_b] :
( ( refine412683990fail_b @ S6 )
=> ( ( refine412683990fail_b @ S3 )
& ! [X3: b] :
( ( refine1001002028nres_b @ S3 @ X3 )
=> ( refine1001002028nres_b @ S6 @ X3 ) ) ) ) ) ) ).
% pw_le_iff
thf(fact_333_inres__def,axiom,
( refine1001002027nres_a
= ( ^ [S3: refine424419629nres_a,X3: a] : ( ord_le519537037nres_a @ ( refine2063221604TURN_a @ X3 ) @ S3 ) ) ) ).
% inres_def
thf(fact_334_inres__def,axiom,
( refine1001002028nres_b
= ( ^ [S3: refine432681134nres_b,X3: b] : ( ord_le527798542nres_b @ ( refine2063221605TURN_b @ X3 ) @ S3 ) ) ) ).
% inres_def
thf(fact_335_bind__FAIL,axiom,
! [F: b > refine432681134nres_b] :
( ( refine1264602246nd_b_b @ top_to240090974nres_b @ F )
= top_to240090974nres_b ) ).
% bind_FAIL
thf(fact_336_bind__FAIL,axiom,
! [F: b > refine424419629nres_a] :
( ( refine1264602245nd_b_a @ top_to240090974nres_b @ F )
= top_to231829469nres_a ) ).
% bind_FAIL
thf(fact_337_bind__FAIL,axiom,
! [F: a > refine432681134nres_b] :
( ( refine436832839nd_a_b @ top_to231829469nres_a @ F )
= top_to240090974nres_b ) ).
% bind_FAIL
thf(fact_338_bind__FAIL,axiom,
! [F: a > refine424419629nres_a] :
( ( refine436832838nd_a_a @ top_to231829469nres_a @ F )
= top_to231829469nres_a ) ).
% bind_FAIL
thf(fact_339_ibind__strict_I2_J,axiom,
! [F: product_unit > refine432681134nres_b] :
( ( refine119808504unit_b @ top_to177290092t_unit @ F )
= top_to240090974nres_b ) ).
% ibind_strict(2)
thf(fact_340_ibind__strict_I2_J,axiom,
! [F: product_unit > refine424419629nres_a] :
( ( refine119808503unit_a @ top_to177290092t_unit @ F )
= top_to231829469nres_a ) ).
% ibind_strict(2)
thf(fact_341_Refine__Basic__Mirabelle__kwjuvthmas_Obind__mono_I1_J,axiom,
! [M: refine424419629nres_a,M3: refine424419629nres_a,F: a > refine424419629nres_a,F3: a > refine424419629nres_a] :
( ( ord_le519537037nres_a @ M @ M3 )
=> ( ! [X2: a] :
( ( ord_le519537037nres_a @ ( refine2063221604TURN_a @ X2 ) @ M )
=> ( ord_le519537037nres_a @ ( F @ X2 ) @ ( F3 @ X2 ) ) )
=> ( ord_le519537037nres_a @ ( refine436832838nd_a_a @ M @ F ) @ ( refine436832838nd_a_a @ M3 @ F3 ) ) ) ) ).
% Refine_Basic_Mirabelle_kwjuvthmas.bind_mono(1)
thf(fact_342_Refine__Basic__Mirabelle__kwjuvthmas_Obind__mono_I1_J,axiom,
! [M: refine424419629nres_a,M3: refine424419629nres_a,F: a > refine432681134nres_b,F3: a > refine432681134nres_b] :
( ( ord_le519537037nres_a @ M @ M3 )
=> ( ! [X2: a] :
( ( ord_le519537037nres_a @ ( refine2063221604TURN_a @ X2 ) @ M )
=> ( ord_le527798542nres_b @ ( F @ X2 ) @ ( F3 @ X2 ) ) )
=> ( ord_le527798542nres_b @ ( refine436832839nd_a_b @ M @ F ) @ ( refine436832839nd_a_b @ M3 @ F3 ) ) ) ) ).
% Refine_Basic_Mirabelle_kwjuvthmas.bind_mono(1)
thf(fact_343_Refine__Basic__Mirabelle__kwjuvthmas_Obind__mono_I1_J,axiom,
! [M: refine432681134nres_b,M3: refine432681134nres_b,F: b > refine424419629nres_a,F3: b > refine424419629nres_a] :
( ( ord_le527798542nres_b @ M @ M3 )
=> ( ! [X2: b] :
( ( ord_le527798542nres_b @ ( refine2063221605TURN_b @ X2 ) @ M )
=> ( ord_le519537037nres_a @ ( F @ X2 ) @ ( F3 @ X2 ) ) )
=> ( ord_le519537037nres_a @ ( refine1264602245nd_b_a @ M @ F ) @ ( refine1264602245nd_b_a @ M3 @ F3 ) ) ) ) ).
% Refine_Basic_Mirabelle_kwjuvthmas.bind_mono(1)
thf(fact_344_Refine__Basic__Mirabelle__kwjuvthmas_Obind__mono_I1_J,axiom,
! [M: refine432681134nres_b,M3: refine432681134nres_b,F: b > refine432681134nres_b,F3: b > refine432681134nres_b] :
( ( ord_le527798542nres_b @ M @ M3 )
=> ( ! [X2: b] :
( ( ord_le527798542nres_b @ ( refine2063221605TURN_b @ X2 ) @ M )
=> ( ord_le527798542nres_b @ ( F @ X2 ) @ ( F3 @ X2 ) ) )
=> ( ord_le527798542nres_b @ ( refine1264602246nd_b_b @ M @ F ) @ ( refine1264602246nd_b_b @ M3 @ F3 ) ) ) ) ).
% Refine_Basic_Mirabelle_kwjuvthmas.bind_mono(1)
thf(fact_345_sup__top__left,axiom,
! [X: refine432681134nres_b] :
( ( sup_su2042991066nres_b @ top_to240090974nres_b @ X )
= top_to240090974nres_b ) ).
% sup_top_left
thf(fact_346_sup__top__left,axiom,
! [X: refine424419629nres_a] :
( ( sup_su2034729561nres_a @ top_to231829469nres_a @ X )
= top_to231829469nres_a ) ).
% sup_top_left
% Helper facts (15)
thf(help_If_2_1_If_001t__Set__Oset_Itf__a_J_T,axiom,
! [X: set_a,Y2: set_a] :
( ( if_set_a @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Set__Oset_Itf__a_J_T,axiom,
! [X: set_a,Y2: set_a] :
( ( if_set_a @ $true @ X @ Y2 )
= X ) ).
thf(help_If_2_1_If_001t__Set__Oset_Itf__b_J_T,axiom,
! [X: set_b,Y2: set_b] :
( ( if_set_b @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Set__Oset_Itf__b_J_T,axiom,
! [X: set_b,Y2: set_b] :
( ( if_set_b @ $true @ X @ Y2 )
= X ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_T,axiom,
! [X: set_Product_prod_a_a,Y2: set_Product_prod_a_a] :
( ( if_set1550076877od_a_a @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_T,axiom,
! [X: set_Product_prod_a_a,Y2: set_Product_prod_a_a] :
( ( if_set1550076877od_a_a @ $true @ X @ Y2 )
= X ) ).
thf(help_If_2_1_If_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J_T,axiom,
! [X: refine424419629nres_a,Y2: refine424419629nres_a] :
( ( if_Ref1724547303nres_a @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J_T,axiom,
! [X: refine424419629nres_a,Y2: refine424419629nres_a] :
( ( if_Ref1724547303nres_a @ $true @ X @ Y2 )
= X ) ).
thf(help_If_2_1_If_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__b_J_T,axiom,
! [X: refine432681134nres_b,Y2: refine432681134nres_b] :
( ( if_Ref1732808808nres_b @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__b_J_T,axiom,
! [X: refine432681134nres_b,Y2: refine432681134nres_b] :
( ( if_Ref1732808808nres_b @ $true @ X @ Y2 )
= X ) ).
thf(help_If_2_1_If_001_062_I_Eo_Mt__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J_J_T,axiom,
! [X: $o > refine424419629nres_a,Y2: $o > refine424419629nres_a] :
( ( if_o_R1705037948nres_a @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001_062_I_Eo_Mt__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J_J_T,axiom,
! [X: $o > refine424419629nres_a,Y2: $o > refine424419629nres_a] :
( ( if_o_R1705037948nres_a @ $true @ X @ Y2 )
= X ) ).
thf(help_If_3_1_If_001_062_I_Eo_Mt__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__b_J_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001_062_I_Eo_Mt__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__b_J_J_T,axiom,
! [X: $o > refine432681134nres_b,Y2: $o > refine432681134nres_b] :
( ( if_o_R697118589nres_b @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001_062_I_Eo_Mt__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__b_J_J_T,axiom,
! [X: $o > refine432681134nres_b,Y2: $o > refine432681134nres_b] :
( ( if_o_R697118589nres_b @ $true @ X @ Y2 )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
ord_le519537037nres_a @ x @ ( refine1441824854un_a_b @ r @ top_to240090974nres_b ) ).
%------------------------------------------------------------------------------