TPTP Problem File: ITP152^1.p
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%------------------------------------------------------------------------------
% File : ITP152^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Preferences problem prob_107__6248646_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 : Preferences/prob_107__6248646_1 [Des21]
% Status : Theorem
% Rating : 0.30 v8.2.0, 0.23 v8.1.0, 0.27 v7.5.0
% Syntax : Number of formulae : 531 ( 129 unt; 172 typ; 0 def)
% Number of atoms : 1101 ( 269 equ; 0 cnn)
% Maximal formula atoms : 8 ( 3 avg)
% Number of connectives : 3857 ( 125 ~; 24 |; 99 &;3120 @)
% ( 0 <=>; 489 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 8 avg)
% Number of types : 23 ( 22 usr)
% Number of type conns : 445 ( 445 >; 0 *; 0 +; 0 <<)
% Number of symbols : 151 ( 150 usr; 8 con; 0-3 aty)
% Number of variables : 1192 ( 57 ^;1111 !; 24 ?;1192 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:43:28.915
%------------------------------------------------------------------------------
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thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
member1899387664od_a_a: produc1990712039od_a_a > set_Pr1295299783od_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
member2057358096od_a_a: produc1572603623od_a_a > set_Pr1948701895od_a_a > $o ).
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_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_carrier,type,
carrier: set_a ).
thf(sy_v_relation,type,
relation: set_Product_prod_a_a ).
thf(sy_v_x,type,
x: a ).
thf(sy_v_y,type,
y: a ).
% Relevant facts (355)
thf(fact_0_indiff__trans,axiom,
! [X: a,Y: a,Z: a] :
( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
& ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) )
=> ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ relation )
& ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ relation ) )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ relation )
& ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X ) @ relation ) ) ) ) ).
% indiff_trans
thf(fact_1_strict__trans,axiom,
! [X: a,Y: a,Z: a] :
( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
& ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) )
=> ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ relation )
& ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ relation ) )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ relation )
& ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X ) @ relation ) ) ) ) ).
% strict_trans
thf(fact_2_rational__preference__axioms,axiom,
prefer719835986ence_a @ carrier @ relation ).
% rational_preference_axioms
thf(fact_3_compl,axiom,
! [X2: a] :
( ( member_a @ X2 @ carrier )
=> ! [Xa: a] :
( ( member_a @ Xa @ carrier )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Xa ) @ relation )
| ( member449909584od_a_a @ ( product_Pair_a_a @ Xa @ X2 ) @ relation ) ) ) ) ).
% compl
thf(fact_4_not__outside,axiom,
! [X: a,Y: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
=> ( member_a @ X @ carrier ) ) ).
% not_outside
thf(fact_5_strict__not__refl__weak,axiom,
! [X: a,Y: a] :
( ( ( member_a @ X @ carrier )
& ( member_a @ Y @ carrier ) )
=> ( ( ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) )
= ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
& ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) ) ) ) ).
% strict_not_refl_weak
thf(fact_6_preference__axioms,axiom,
prefer1292084992ence_a @ carrier @ relation ).
% preference_axioms
thf(fact_7_trans__refl,axiom,
order_preorder_on_a @ carrier @ relation ).
% trans_refl
thf(fact_8_total,axiom,
total_on_a @ carrier @ relation ).
% total
thf(fact_9_reflexivity,axiom,
refl_on_a @ carrier @ relation ).
% reflexivity
thf(fact_10_prod_Oinject,axiom,
! [X1: product_prod_a_a,X22: product_prod_a_a,Y1: product_prod_a_a,Y2: product_prod_a_a] :
( ( ( produc1474507607od_a_a @ X1 @ X22 )
= ( produc1474507607od_a_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_11_prod_Oinject,axiom,
! [X1: multiset_a,X22: multiset_a,Y1: multiset_a,Y2: multiset_a] :
( ( ( produc2037245207iset_a @ X1 @ X22 )
= ( produc2037245207iset_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_12_prod_Oinject,axiom,
! [X1: multis599418605od_a_a,X22: multis599418605od_a_a,Y1: multis599418605od_a_a,Y2: multis599418605od_a_a] :
( ( ( produc1859144151od_a_a @ X1 @ X22 )
= ( produc1859144151od_a_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_13_prod_Oinject,axiom,
! [X1: a,X22: a,Y1: a,Y2: a] :
( ( ( product_Pair_a_a @ X1 @ X22 )
= ( product_Pair_a_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_14_old_Oprod_Oinject,axiom,
! [A: product_prod_a_a,B: product_prod_a_a,A2: product_prod_a_a,B2: product_prod_a_a] :
( ( ( produc1474507607od_a_a @ A @ B )
= ( produc1474507607od_a_a @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_15_old_Oprod_Oinject,axiom,
! [A: multiset_a,B: multiset_a,A2: multiset_a,B2: multiset_a] :
( ( ( produc2037245207iset_a @ A @ B )
= ( produc2037245207iset_a @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_16_old_Oprod_Oinject,axiom,
! [A: multis599418605od_a_a,B: multis599418605od_a_a,A2: multis599418605od_a_a,B2: multis599418605od_a_a] :
( ( ( produc1859144151od_a_a @ A @ B )
= ( produc1859144151od_a_a @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_17_old_Oprod_Oinject,axiom,
! [A: a,B: a,A2: a,B2: a] :
( ( ( product_Pair_a_a @ A @ B )
= ( product_Pair_a_a @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_18_transitivity,axiom,
trans_a @ relation ).
% transitivity
thf(fact_19_surj__pair,axiom,
! [P: produc1572603623od_a_a] :
? [X3: product_prod_a_a,Y3: product_prod_a_a] :
( P
= ( produc1474507607od_a_a @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_20_surj__pair,axiom,
! [P: produc1127127335iset_a] :
? [X3: multiset_a,Y3: multiset_a] :
( P
= ( produc2037245207iset_a @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_21_surj__pair,axiom,
! [P: produc2061001575od_a_a] :
? [X3: multis599418605od_a_a,Y3: multis599418605od_a_a] :
( P
= ( produc1859144151od_a_a @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_22_surj__pair,axiom,
! [P: product_prod_a_a] :
? [X3: a,Y3: a] :
( P
= ( product_Pair_a_a @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_23_prod__cases,axiom,
! [P2: produc1572603623od_a_a > $o,P: produc1572603623od_a_a] :
( ! [A3: product_prod_a_a,B3: product_prod_a_a] : ( P2 @ ( produc1474507607od_a_a @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_24_prod__cases,axiom,
! [P2: produc1127127335iset_a > $o,P: produc1127127335iset_a] :
( ! [A3: multiset_a,B3: multiset_a] : ( P2 @ ( produc2037245207iset_a @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_25_prod__cases,axiom,
! [P2: produc2061001575od_a_a > $o,P: produc2061001575od_a_a] :
( ! [A3: multis599418605od_a_a,B3: multis599418605od_a_a] : ( P2 @ ( produc1859144151od_a_a @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_26_prod__cases,axiom,
! [P2: product_prod_a_a > $o,P: product_prod_a_a] :
( ! [A3: a,B3: a] : ( P2 @ ( product_Pair_a_a @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_27_Pair__inject,axiom,
! [A: product_prod_a_a,B: product_prod_a_a,A2: product_prod_a_a,B2: product_prod_a_a] :
( ( ( produc1474507607od_a_a @ A @ B )
= ( produc1474507607od_a_a @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_28_Pair__inject,axiom,
! [A: multiset_a,B: multiset_a,A2: multiset_a,B2: multiset_a] :
( ( ( produc2037245207iset_a @ A @ B )
= ( produc2037245207iset_a @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_29_Pair__inject,axiom,
! [A: multis599418605od_a_a,B: multis599418605od_a_a,A2: multis599418605od_a_a,B2: multis599418605od_a_a] :
( ( ( produc1859144151od_a_a @ A @ B )
= ( produc1859144151od_a_a @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_30_Pair__inject,axiom,
! [A: a,B: a,A2: a,B2: a] :
( ( ( product_Pair_a_a @ A @ B )
= ( product_Pair_a_a @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_31_prod__cases3,axiom,
! [Y: produc1572603623od_a_a] :
~ ! [A3: product_prod_a_a,B3: a,C: a] :
( Y
!= ( produc1474507607od_a_a @ A3 @ ( product_Pair_a_a @ B3 @ C ) ) ) ).
% prod_cases3
thf(fact_32_rational__preference_Oaxioms_I1_J,axiom,
! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
( ( prefer697821563od_a_a @ Carrier @ Relation )
=> ( prefer1338892073od_a_a @ Carrier @ Relation ) ) ).
% rational_preference.axioms(1)
thf(fact_33_rational__preference_Oaxioms_I1_J,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a] :
( ( prefer719835986ence_a @ Carrier @ Relation )
=> ( prefer1292084992ence_a @ Carrier @ Relation ) ) ).
% rational_preference.axioms(1)
thf(fact_34_preference_Otrans__refl,axiom,
! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
( ( prefer1338892073od_a_a @ Carrier @ Relation )
=> ( order_359341630od_a_a @ Carrier @ Relation ) ) ).
% preference.trans_refl
thf(fact_35_preference_Otrans__refl,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a] :
( ( prefer1292084992ence_a @ Carrier @ Relation )
=> ( order_preorder_on_a @ Carrier @ Relation ) ) ).
% preference.trans_refl
thf(fact_36_preference_Oreflexivity,axiom,
! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
( ( prefer1338892073od_a_a @ Carrier @ Relation )
=> ( refl_o1298442278od_a_a @ Carrier @ Relation ) ) ).
% preference.reflexivity
thf(fact_37_preference_Oreflexivity,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a] :
( ( prefer1292084992ence_a @ Carrier @ Relation )
=> ( refl_on_a @ Carrier @ Relation ) ) ).
% preference.reflexivity
thf(fact_38_preference_Otransitivity,axiom,
! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
( ( prefer1338892073od_a_a @ Carrier @ Relation )
=> ( trans_972827014od_a_a @ Relation ) ) ).
% preference.transitivity
thf(fact_39_preference_Otransitivity,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a] :
( ( prefer1292084992ence_a @ Carrier @ Relation )
=> ( trans_a @ Relation ) ) ).
% preference.transitivity
thf(fact_40_rational__preference_Ototal,axiom,
! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
( ( prefer697821563od_a_a @ Carrier @ Relation )
=> ( total_1490170027od_a_a @ Carrier @ Relation ) ) ).
% rational_preference.total
thf(fact_41_rational__preference_Ototal,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a] :
( ( prefer719835986ence_a @ Carrier @ Relation )
=> ( total_on_a @ Carrier @ Relation ) ) ).
% rational_preference.total
thf(fact_42_preference_Ointro,axiom,
! [Relation: set_Pr1295299783od_a_a,Carrier: set_Pr1948701895od_a_a] :
( ! [X3: produc1572603623od_a_a,Y3: produc1572603623od_a_a] :
( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X3 @ Y3 ) @ Relation )
=> ( member2057358096od_a_a @ X3 @ Carrier ) )
=> ( ! [X3: produc1572603623od_a_a,Y3: produc1572603623od_a_a] :
( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X3 @ Y3 ) @ Relation )
=> ( member2057358096od_a_a @ Y3 @ Carrier ) )
=> ( ( order_491428350od_a_a @ Carrier @ Relation )
=> ( prefer1409946857od_a_a @ Carrier @ Relation ) ) ) ) ).
% preference.intro
thf(fact_43_preference_Ointro,axiom,
! [Relation: set_Pr146455751iset_a,Carrier: set_Pr158363655iset_a] :
( ! [X3: produc1127127335iset_a,Y3: produc1127127335iset_a] :
( ( member567864080iset_a @ ( produc1898392407iset_a @ X3 @ Y3 ) @ Relation )
=> ( member340150864iset_a @ X3 @ Carrier ) )
=> ( ! [X3: produc1127127335iset_a,Y3: produc1127127335iset_a] :
( ( member567864080iset_a @ ( produc1898392407iset_a @ X3 @ Y3 ) @ Relation )
=> ( member340150864iset_a @ Y3 @ Carrier ) )
=> ( ( order_761691966iset_a @ Carrier @ Relation )
=> ( prefer1608839465iset_a @ Carrier @ Relation ) ) ) ) ).
% preference.intro
thf(fact_44_preference_Ointro,axiom,
! [Relation: set_Pr1930443975od_a_a,Carrier: set_Pr1821581383od_a_a] :
( ! [X3: produc2061001575od_a_a,Y3: produc2061001575od_a_a] :
( ( member2007374608od_a_a @ ( produc978545495od_a_a @ X3 @ Y3 ) @ Relation )
=> ( member1349055120od_a_a @ X3 @ Carrier ) )
=> ( ! [X3: produc2061001575od_a_a,Y3: produc2061001575od_a_a] :
( ( member2007374608od_a_a @ ( produc978545495od_a_a @ X3 @ Y3 ) @ Relation )
=> ( member1349055120od_a_a @ Y3 @ Carrier ) )
=> ( ( order_1210379134od_a_a @ Carrier @ Relation )
=> ( prefer273553257od_a_a @ Carrier @ Relation ) ) ) ) ).
% preference.intro
thf(fact_45_preference_Ointro,axiom,
! [Relation: set_Pr158363655iset_a,Carrier: set_multiset_a] :
( ! [X3: multiset_a,Y3: multiset_a] :
( ( member340150864iset_a @ ( produc2037245207iset_a @ X3 @ Y3 ) @ Relation )
=> ( member_multiset_a @ X3 @ Carrier ) )
=> ( ! [X3: multiset_a,Y3: multiset_a] :
( ( member340150864iset_a @ ( produc2037245207iset_a @ X3 @ Y3 ) @ Relation )
=> ( member_multiset_a @ Y3 @ Carrier ) )
=> ( ( order_750350043iset_a @ Carrier @ Relation )
=> ( prefer1571048262iset_a @ Carrier @ Relation ) ) ) ) ).
% preference.intro
thf(fact_46_preference_Ointro,axiom,
! [Relation: set_Pr1821581383od_a_a,Carrier: set_mu1281098573od_a_a] :
( ! [X3: multis599418605od_a_a,Y3: multis599418605od_a_a] :
( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X3 @ Y3 ) @ Relation )
=> ( member1755255446od_a_a @ X3 @ Carrier ) )
=> ( ! [X3: multis599418605od_a_a,Y3: multis599418605od_a_a] :
( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X3 @ Y3 ) @ Relation )
=> ( member1755255446od_a_a @ Y3 @ Carrier ) )
=> ( ( order_1040740996od_a_a @ Carrier @ Relation )
=> ( prefer1340910575od_a_a @ Carrier @ Relation ) ) ) ) ).
% preference.intro
thf(fact_47_preference_Ointro,axiom,
! [Relation: set_Pr1948701895od_a_a,Carrier: set_Product_prod_a_a] :
( ! [X3: product_prod_a_a,Y3: product_prod_a_a] :
( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Y3 ) @ Relation )
=> ( member449909584od_a_a @ X3 @ Carrier ) )
=> ( ! [X3: product_prod_a_a,Y3: product_prod_a_a] :
( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Y3 ) @ Relation )
=> ( member449909584od_a_a @ Y3 @ Carrier ) )
=> ( ( order_359341630od_a_a @ Carrier @ Relation )
=> ( prefer1338892073od_a_a @ Carrier @ Relation ) ) ) ) ).
% preference.intro
thf(fact_48_preference_Ointro,axiom,
! [Relation: set_Product_prod_a_a,Carrier: set_a] :
( ! [X3: a,Y3: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ Relation )
=> ( member_a @ X3 @ Carrier ) )
=> ( ! [X3: a,Y3: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ Relation )
=> ( member_a @ Y3 @ Carrier ) )
=> ( ( order_preorder_on_a @ Carrier @ Relation )
=> ( prefer1292084992ence_a @ Carrier @ Relation ) ) ) ) ).
% preference.intro
thf(fact_49_preference__def,axiom,
( prefer1409946857od_a_a
= ( ^ [Carrier2: set_Pr1948701895od_a_a,Relation2: set_Pr1295299783od_a_a] :
( ! [X4: produc1572603623od_a_a,Y4: produc1572603623od_a_a] :
( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X4 @ Y4 ) @ Relation2 )
=> ( member2057358096od_a_a @ X4 @ Carrier2 ) )
& ! [X4: produc1572603623od_a_a,Y4: produc1572603623od_a_a] :
( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X4 @ Y4 ) @ Relation2 )
=> ( member2057358096od_a_a @ Y4 @ Carrier2 ) )
& ( order_491428350od_a_a @ Carrier2 @ Relation2 ) ) ) ) ).
% preference_def
thf(fact_50_preference__def,axiom,
( prefer1608839465iset_a
= ( ^ [Carrier2: set_Pr158363655iset_a,Relation2: set_Pr146455751iset_a] :
( ! [X4: produc1127127335iset_a,Y4: produc1127127335iset_a] :
( ( member567864080iset_a @ ( produc1898392407iset_a @ X4 @ Y4 ) @ Relation2 )
=> ( member340150864iset_a @ X4 @ Carrier2 ) )
& ! [X4: produc1127127335iset_a,Y4: produc1127127335iset_a] :
( ( member567864080iset_a @ ( produc1898392407iset_a @ X4 @ Y4 ) @ Relation2 )
=> ( member340150864iset_a @ Y4 @ Carrier2 ) )
& ( order_761691966iset_a @ Carrier2 @ Relation2 ) ) ) ) ).
% preference_def
thf(fact_51_preference__def,axiom,
( prefer273553257od_a_a
= ( ^ [Carrier2: set_Pr1821581383od_a_a,Relation2: set_Pr1930443975od_a_a] :
( ! [X4: produc2061001575od_a_a,Y4: produc2061001575od_a_a] :
( ( member2007374608od_a_a @ ( produc978545495od_a_a @ X4 @ Y4 ) @ Relation2 )
=> ( member1349055120od_a_a @ X4 @ Carrier2 ) )
& ! [X4: produc2061001575od_a_a,Y4: produc2061001575od_a_a] :
( ( member2007374608od_a_a @ ( produc978545495od_a_a @ X4 @ Y4 ) @ Relation2 )
=> ( member1349055120od_a_a @ Y4 @ Carrier2 ) )
& ( order_1210379134od_a_a @ Carrier2 @ Relation2 ) ) ) ) ).
% preference_def
thf(fact_52_preference__def,axiom,
( prefer1571048262iset_a
= ( ^ [Carrier2: set_multiset_a,Relation2: set_Pr158363655iset_a] :
( ! [X4: multiset_a,Y4: multiset_a] :
( ( member340150864iset_a @ ( produc2037245207iset_a @ X4 @ Y4 ) @ Relation2 )
=> ( member_multiset_a @ X4 @ Carrier2 ) )
& ! [X4: multiset_a,Y4: multiset_a] :
( ( member340150864iset_a @ ( produc2037245207iset_a @ X4 @ Y4 ) @ Relation2 )
=> ( member_multiset_a @ Y4 @ Carrier2 ) )
& ( order_750350043iset_a @ Carrier2 @ Relation2 ) ) ) ) ).
% preference_def
thf(fact_53_preference__def,axiom,
( prefer1340910575od_a_a
= ( ^ [Carrier2: set_mu1281098573od_a_a,Relation2: set_Pr1821581383od_a_a] :
( ! [X4: multis599418605od_a_a,Y4: multis599418605od_a_a] :
( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X4 @ Y4 ) @ Relation2 )
=> ( member1755255446od_a_a @ X4 @ Carrier2 ) )
& ! [X4: multis599418605od_a_a,Y4: multis599418605od_a_a] :
( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X4 @ Y4 ) @ Relation2 )
=> ( member1755255446od_a_a @ Y4 @ Carrier2 ) )
& ( order_1040740996od_a_a @ Carrier2 @ Relation2 ) ) ) ) ).
% preference_def
thf(fact_54_preference__def,axiom,
( prefer1338892073od_a_a
= ( ^ [Carrier2: set_Product_prod_a_a,Relation2: set_Pr1948701895od_a_a] :
( ! [X4: product_prod_a_a,Y4: product_prod_a_a] :
( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y4 ) @ Relation2 )
=> ( member449909584od_a_a @ X4 @ Carrier2 ) )
& ! [X4: product_prod_a_a,Y4: product_prod_a_a] :
( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y4 ) @ Relation2 )
=> ( member449909584od_a_a @ Y4 @ Carrier2 ) )
& ( order_359341630od_a_a @ Carrier2 @ Relation2 ) ) ) ) ).
% preference_def
thf(fact_55_preference__def,axiom,
( prefer1292084992ence_a
= ( ^ [Carrier2: set_a,Relation2: set_Product_prod_a_a] :
( ! [X4: a,Y4: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ Relation2 )
=> ( member_a @ X4 @ Carrier2 ) )
& ! [X4: a,Y4: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ Relation2 )
=> ( member_a @ Y4 @ Carrier2 ) )
& ( order_preorder_on_a @ Carrier2 @ Relation2 ) ) ) ) ).
% preference_def
thf(fact_56_preference_Oindiff__trans,axiom,
! [Carrier: set_multiset_a,Relation: set_Pr158363655iset_a,X: multiset_a,Y: multiset_a,Z: multiset_a] :
( ( prefer1571048262iset_a @ Carrier @ Relation )
=> ( ( ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Y ) @ Relation )
& ( member340150864iset_a @ ( produc2037245207iset_a @ Y @ X ) @ Relation ) )
=> ( ( ( member340150864iset_a @ ( produc2037245207iset_a @ Y @ Z ) @ Relation )
& ( member340150864iset_a @ ( produc2037245207iset_a @ Z @ Y ) @ Relation ) )
=> ( ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Z ) @ Relation )
& ( member340150864iset_a @ ( produc2037245207iset_a @ Z @ X ) @ Relation ) ) ) ) ) ).
% preference.indiff_trans
thf(fact_57_preference_Oindiff__trans,axiom,
! [Carrier: set_mu1281098573od_a_a,Relation: set_Pr1821581383od_a_a,X: multis599418605od_a_a,Y: multis599418605od_a_a,Z: multis599418605od_a_a] :
( ( prefer1340910575od_a_a @ Carrier @ Relation )
=> ( ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X @ Y ) @ Relation )
& ( member1349055120od_a_a @ ( produc1859144151od_a_a @ Y @ X ) @ Relation ) )
=> ( ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ Y @ Z ) @ Relation )
& ( member1349055120od_a_a @ ( produc1859144151od_a_a @ Z @ Y ) @ Relation ) )
=> ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X @ Z ) @ Relation )
& ( member1349055120od_a_a @ ( produc1859144151od_a_a @ Z @ X ) @ Relation ) ) ) ) ) ).
% preference.indiff_trans
thf(fact_58_preference_Oindiff__trans,axiom,
! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a,Z: product_prod_a_a] :
( ( prefer1338892073od_a_a @ Carrier @ Relation )
=> ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
& ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X ) @ Relation ) )
=> ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ Z ) @ Relation )
& ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ Y ) @ Relation ) )
=> ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Z ) @ Relation )
& ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ X ) @ Relation ) ) ) ) ) ).
% preference.indiff_trans
thf(fact_59_preference_Oindiff__trans,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a,Z: a] :
( ( prefer1292084992ence_a @ Carrier @ Relation )
=> ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
& ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) )
=> ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ Relation )
& ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ Relation ) )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ Relation )
& ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X ) @ Relation ) ) ) ) ) ).
% preference.indiff_trans
thf(fact_60_preference_Onot__outside,axiom,
! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
( ( prefer1409946857od_a_a @ Carrier @ Relation )
=> ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Y ) @ Relation )
=> ( member2057358096od_a_a @ X @ Carrier ) ) ) ).
% preference.not_outside
thf(fact_61_preference_Onot__outside,axiom,
! [Carrier: set_Pr158363655iset_a,Relation: set_Pr146455751iset_a,X: produc1127127335iset_a,Y: produc1127127335iset_a] :
( ( prefer1608839465iset_a @ Carrier @ Relation )
=> ( ( member567864080iset_a @ ( produc1898392407iset_a @ X @ Y ) @ Relation )
=> ( member340150864iset_a @ X @ Carrier ) ) ) ).
% preference.not_outside
thf(fact_62_preference_Onot__outside,axiom,
! [Carrier: set_Pr1821581383od_a_a,Relation: set_Pr1930443975od_a_a,X: produc2061001575od_a_a,Y: produc2061001575od_a_a] :
( ( prefer273553257od_a_a @ Carrier @ Relation )
=> ( ( member2007374608od_a_a @ ( produc978545495od_a_a @ X @ Y ) @ Relation )
=> ( member1349055120od_a_a @ X @ Carrier ) ) ) ).
% preference.not_outside
thf(fact_63_preference_Onot__outside,axiom,
! [Carrier: set_multiset_a,Relation: set_Pr158363655iset_a,X: multiset_a,Y: multiset_a] :
( ( prefer1571048262iset_a @ Carrier @ Relation )
=> ( ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Y ) @ Relation )
=> ( member_multiset_a @ X @ Carrier ) ) ) ).
% preference.not_outside
thf(fact_64_preference_Onot__outside,axiom,
! [Carrier: set_mu1281098573od_a_a,Relation: set_Pr1821581383od_a_a,X: multis599418605od_a_a,Y: multis599418605od_a_a] :
( ( prefer1340910575od_a_a @ Carrier @ Relation )
=> ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X @ Y ) @ Relation )
=> ( member1755255446od_a_a @ X @ Carrier ) ) ) ).
% preference.not_outside
thf(fact_65_preference_Onot__outside,axiom,
! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
( ( prefer1338892073od_a_a @ Carrier @ Relation )
=> ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
=> ( member449909584od_a_a @ X @ Carrier ) ) ) ).
% preference.not_outside
thf(fact_66_preference_Onot__outside,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
( ( prefer1292084992ence_a @ Carrier @ Relation )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
=> ( member_a @ X @ Carrier ) ) ) ).
% preference.not_outside
thf(fact_67_rational__preference_Ostrict__not__refl__weak,axiom,
! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
( ( prefer1828051771od_a_a @ Carrier @ Relation )
=> ( ( ( member2057358096od_a_a @ X @ Carrier )
& ( member2057358096od_a_a @ Y @ Carrier ) )
=> ( ( ~ ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y @ X ) @ Relation ) )
= ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Y ) @ Relation )
& ~ ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y @ X ) @ Relation ) ) ) ) ) ).
% rational_preference.strict_not_refl_weak
thf(fact_68_rational__preference_Ostrict__not__refl__weak,axiom,
! [Carrier: set_Pr158363655iset_a,Relation: set_Pr146455751iset_a,X: produc1127127335iset_a,Y: produc1127127335iset_a] :
( ( prefer841781627iset_a @ Carrier @ Relation )
=> ( ( ( member340150864iset_a @ X @ Carrier )
& ( member340150864iset_a @ Y @ Carrier ) )
=> ( ( ~ ( member567864080iset_a @ ( produc1898392407iset_a @ Y @ X ) @ Relation ) )
= ( ( member567864080iset_a @ ( produc1898392407iset_a @ X @ Y ) @ Relation )
& ~ ( member567864080iset_a @ ( produc1898392407iset_a @ Y @ X ) @ Relation ) ) ) ) ) ).
% rational_preference.strict_not_refl_weak
thf(fact_69_rational__preference_Ostrict__not__refl__weak,axiom,
! [Carrier: set_Pr1821581383od_a_a,Relation: set_Pr1930443975od_a_a,X: produc2061001575od_a_a,Y: produc2061001575od_a_a] :
( ( prefer933983163od_a_a @ Carrier @ Relation )
=> ( ( ( member1349055120od_a_a @ X @ Carrier )
& ( member1349055120od_a_a @ Y @ Carrier ) )
=> ( ( ~ ( member2007374608od_a_a @ ( produc978545495od_a_a @ Y @ X ) @ Relation ) )
= ( ( member2007374608od_a_a @ ( produc978545495od_a_a @ X @ Y ) @ Relation )
& ~ ( member2007374608od_a_a @ ( produc978545495od_a_a @ Y @ X ) @ Relation ) ) ) ) ) ).
% rational_preference.strict_not_refl_weak
thf(fact_70_rational__preference_Ostrict__not__refl__weak,axiom,
! [Carrier: set_multiset_a,Relation: set_Pr158363655iset_a,X: multiset_a,Y: multiset_a] :
( ( prefer1779827352iset_a @ Carrier @ Relation )
=> ( ( ( member_multiset_a @ X @ Carrier )
& ( member_multiset_a @ Y @ Carrier ) )
=> ( ( ~ ( member340150864iset_a @ ( produc2037245207iset_a @ Y @ X ) @ Relation ) )
= ( ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Y ) @ Relation )
& ~ ( member340150864iset_a @ ( produc2037245207iset_a @ Y @ X ) @ Relation ) ) ) ) ) ).
% rational_preference.strict_not_refl_weak
thf(fact_71_rational__preference_Ostrict__not__refl__weak,axiom,
! [Carrier: set_mu1281098573od_a_a,Relation: set_Pr1821581383od_a_a,X: multis599418605od_a_a,Y: multis599418605od_a_a] :
( ( prefer1669415233od_a_a @ Carrier @ Relation )
=> ( ( ( member1755255446od_a_a @ X @ Carrier )
& ( member1755255446od_a_a @ Y @ Carrier ) )
=> ( ( ~ ( member1349055120od_a_a @ ( produc1859144151od_a_a @ Y @ X ) @ Relation ) )
= ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X @ Y ) @ Relation )
& ~ ( member1349055120od_a_a @ ( produc1859144151od_a_a @ Y @ X ) @ Relation ) ) ) ) ) ).
% rational_preference.strict_not_refl_weak
thf(fact_72_rational__preference_Ostrict__not__refl__weak,axiom,
! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
( ( prefer697821563od_a_a @ Carrier @ Relation )
=> ( ( ( member449909584od_a_a @ X @ Carrier )
& ( member449909584od_a_a @ Y @ Carrier ) )
=> ( ( ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X ) @ Relation ) )
= ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
& ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X ) @ Relation ) ) ) ) ) ).
% rational_preference.strict_not_refl_weak
thf(fact_73_rational__preference_Ostrict__not__refl__weak,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
( ( prefer719835986ence_a @ Carrier @ Relation )
=> ( ( ( member_a @ X @ Carrier )
& ( member_a @ Y @ Carrier ) )
=> ( ( ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) )
= ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
& ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) ) ) ) ) ).
% rational_preference.strict_not_refl_weak
thf(fact_74_rational__preference_Ostrict__trans,axiom,
! [Carrier: set_multiset_a,Relation: set_Pr158363655iset_a,X: multiset_a,Y: multiset_a,Z: multiset_a] :
( ( prefer1779827352iset_a @ Carrier @ Relation )
=> ( ( ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Y ) @ Relation )
& ~ ( member340150864iset_a @ ( produc2037245207iset_a @ Y @ X ) @ Relation ) )
=> ( ( ( member340150864iset_a @ ( produc2037245207iset_a @ Y @ Z ) @ Relation )
& ~ ( member340150864iset_a @ ( produc2037245207iset_a @ Z @ Y ) @ Relation ) )
=> ( ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Z ) @ Relation )
& ~ ( member340150864iset_a @ ( produc2037245207iset_a @ Z @ X ) @ Relation ) ) ) ) ) ).
% rational_preference.strict_trans
thf(fact_75_rational__preference_Ostrict__trans,axiom,
! [Carrier: set_mu1281098573od_a_a,Relation: set_Pr1821581383od_a_a,X: multis599418605od_a_a,Y: multis599418605od_a_a,Z: multis599418605od_a_a] :
( ( prefer1669415233od_a_a @ Carrier @ Relation )
=> ( ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X @ Y ) @ Relation )
& ~ ( member1349055120od_a_a @ ( produc1859144151od_a_a @ Y @ X ) @ Relation ) )
=> ( ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ Y @ Z ) @ Relation )
& ~ ( member1349055120od_a_a @ ( produc1859144151od_a_a @ Z @ Y ) @ Relation ) )
=> ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X @ Z ) @ Relation )
& ~ ( member1349055120od_a_a @ ( produc1859144151od_a_a @ Z @ X ) @ Relation ) ) ) ) ) ).
% rational_preference.strict_trans
thf(fact_76_rational__preference_Ostrict__trans,axiom,
! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a,Z: product_prod_a_a] :
( ( prefer697821563od_a_a @ Carrier @ Relation )
=> ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
& ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X ) @ Relation ) )
=> ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ Z ) @ Relation )
& ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ Y ) @ Relation ) )
=> ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Z ) @ Relation )
& ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ X ) @ Relation ) ) ) ) ) ).
% rational_preference.strict_trans
thf(fact_77_rational__preference_Ostrict__trans,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a,Z: a] :
( ( prefer719835986ence_a @ Carrier @ Relation )
=> ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
& ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) )
=> ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ Relation )
& ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ Relation ) )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ Relation )
& ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X ) @ Relation ) ) ) ) ) ).
% rational_preference.strict_trans
thf(fact_78_rational__preference_Ocompl,axiom,
! [Carrier: set_multiset_a,Relation: set_Pr158363655iset_a] :
( ( prefer1779827352iset_a @ Carrier @ Relation )
=> ! [X2: multiset_a] :
( ( member_multiset_a @ X2 @ Carrier )
=> ! [Xa: multiset_a] :
( ( member_multiset_a @ Xa @ Carrier )
=> ( ( member340150864iset_a @ ( produc2037245207iset_a @ X2 @ Xa ) @ Relation )
| ( member340150864iset_a @ ( produc2037245207iset_a @ Xa @ X2 ) @ Relation ) ) ) ) ) ).
% rational_preference.compl
thf(fact_79_rational__preference_Ocompl,axiom,
! [Carrier: set_mu1281098573od_a_a,Relation: set_Pr1821581383od_a_a] :
( ( prefer1669415233od_a_a @ Carrier @ Relation )
=> ! [X2: multis599418605od_a_a] :
( ( member1755255446od_a_a @ X2 @ Carrier )
=> ! [Xa: multis599418605od_a_a] :
( ( member1755255446od_a_a @ Xa @ Carrier )
=> ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X2 @ Xa ) @ Relation )
| ( member1349055120od_a_a @ ( produc1859144151od_a_a @ Xa @ X2 ) @ Relation ) ) ) ) ) ).
% rational_preference.compl
thf(fact_80_rational__preference_Ocompl,axiom,
! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
( ( prefer697821563od_a_a @ Carrier @ Relation )
=> ! [X2: product_prod_a_a] :
( ( member449909584od_a_a @ X2 @ Carrier )
=> ! [Xa: product_prod_a_a] :
( ( member449909584od_a_a @ Xa @ Carrier )
=> ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Xa ) @ Relation )
| ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Xa @ X2 ) @ Relation ) ) ) ) ) ).
% rational_preference.compl
thf(fact_81_rational__preference_Ocompl,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a] :
( ( prefer719835986ence_a @ Carrier @ Relation )
=> ! [X2: a] :
( ( member_a @ X2 @ Carrier )
=> ! [Xa: a] :
( ( member_a @ Xa @ Carrier )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Xa ) @ Relation )
| ( member449909584od_a_a @ ( product_Pair_a_a @ Xa @ X2 ) @ Relation ) ) ) ) ) ).
% rational_preference.compl
thf(fact_82_old_Oprod_Oinducts,axiom,
! [P2: produc1572603623od_a_a > $o,Prod: produc1572603623od_a_a] :
( ! [A3: product_prod_a_a,B3: product_prod_a_a] : ( P2 @ ( produc1474507607od_a_a @ A3 @ B3 ) )
=> ( P2 @ Prod ) ) ).
% old.prod.inducts
thf(fact_83_old_Oprod_Oinducts,axiom,
! [P2: produc1127127335iset_a > $o,Prod: produc1127127335iset_a] :
( ! [A3: multiset_a,B3: multiset_a] : ( P2 @ ( produc2037245207iset_a @ A3 @ B3 ) )
=> ( P2 @ Prod ) ) ).
% old.prod.inducts
thf(fact_84_old_Oprod_Oinducts,axiom,
! [P2: produc2061001575od_a_a > $o,Prod: produc2061001575od_a_a] :
( ! [A3: multis599418605od_a_a,B3: multis599418605od_a_a] : ( P2 @ ( produc1859144151od_a_a @ A3 @ B3 ) )
=> ( P2 @ Prod ) ) ).
% old.prod.inducts
thf(fact_85_old_Oprod_Oinducts,axiom,
! [P2: product_prod_a_a > $o,Prod: product_prod_a_a] :
( ! [A3: a,B3: a] : ( P2 @ ( product_Pair_a_a @ A3 @ B3 ) )
=> ( P2 @ Prod ) ) ).
% old.prod.inducts
thf(fact_86_old_Oprod_Oexhaust,axiom,
! [Y: produc1572603623od_a_a] :
~ ! [A3: product_prod_a_a,B3: product_prod_a_a] :
( Y
!= ( produc1474507607od_a_a @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_87_old_Oprod_Oexhaust,axiom,
! [Y: produc1127127335iset_a] :
~ ! [A3: multiset_a,B3: multiset_a] :
( Y
!= ( produc2037245207iset_a @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_88_old_Oprod_Oexhaust,axiom,
! [Y: produc2061001575od_a_a] :
~ ! [A3: multis599418605od_a_a,B3: multis599418605od_a_a] :
( Y
!= ( produc1859144151od_a_a @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_89_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_a_a] :
~ ! [A3: a,B3: a] :
( Y
!= ( product_Pair_a_a @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_90_prod__induct3,axiom,
! [P2: produc1572603623od_a_a > $o,X: produc1572603623od_a_a] :
( ! [A3: product_prod_a_a,B3: a,C: a] : ( P2 @ ( produc1474507607od_a_a @ A3 @ ( product_Pair_a_a @ B3 @ C ) ) )
=> ( P2 @ X ) ) ).
% prod_induct3
thf(fact_91_preorder__on__def,axiom,
( order_359341630od_a_a
= ( ^ [A4: set_Product_prod_a_a,R: set_Pr1948701895od_a_a] :
( ( refl_o1298442278od_a_a @ A4 @ R )
& ( trans_972827014od_a_a @ R ) ) ) ) ).
% preorder_on_def
thf(fact_92_preorder__on__def,axiom,
( order_preorder_on_a
= ( ^ [A4: set_a,R: set_Product_prod_a_a] :
( ( refl_on_a @ A4 @ R )
& ( trans_a @ R ) ) ) ) ).
% preorder_on_def
thf(fact_93_total__on__def,axiom,
( total_on_multiset_a
= ( ^ [A4: set_multiset_a,R: set_Pr158363655iset_a] :
! [X4: multiset_a] :
( ( member_multiset_a @ X4 @ A4 )
=> ! [Y4: multiset_a] :
( ( member_multiset_a @ Y4 @ A4 )
=> ( ( X4 != Y4 )
=> ( ( member340150864iset_a @ ( produc2037245207iset_a @ X4 @ Y4 ) @ R )
| ( member340150864iset_a @ ( produc2037245207iset_a @ Y4 @ X4 ) @ R ) ) ) ) ) ) ) ).
% total_on_def
thf(fact_94_total__on__def,axiom,
( total_1024184433od_a_a
= ( ^ [A4: set_mu1281098573od_a_a,R: set_Pr1821581383od_a_a] :
! [X4: multis599418605od_a_a] :
( ( member1755255446od_a_a @ X4 @ A4 )
=> ! [Y4: multis599418605od_a_a] :
( ( member1755255446od_a_a @ Y4 @ A4 )
=> ( ( X4 != Y4 )
=> ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X4 @ Y4 ) @ R )
| ( member1349055120od_a_a @ ( produc1859144151od_a_a @ Y4 @ X4 ) @ R ) ) ) ) ) ) ) ).
% total_on_def
thf(fact_95_total__on__def,axiom,
( total_1490170027od_a_a
= ( ^ [A4: set_Product_prod_a_a,R: set_Pr1948701895od_a_a] :
! [X4: product_prod_a_a] :
( ( member449909584od_a_a @ X4 @ A4 )
=> ! [Y4: product_prod_a_a] :
( ( member449909584od_a_a @ Y4 @ A4 )
=> ( ( X4 != Y4 )
=> ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y4 ) @ R )
| ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y4 @ X4 ) @ R ) ) ) ) ) ) ) ).
% total_on_def
thf(fact_96_total__on__def,axiom,
( total_on_a
= ( ^ [A4: set_a,R: set_Product_prod_a_a] :
! [X4: a] :
( ( member_a @ X4 @ A4 )
=> ! [Y4: a] :
( ( member_a @ Y4 @ A4 )
=> ( ( X4 != Y4 )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ R )
| ( member449909584od_a_a @ ( product_Pair_a_a @ Y4 @ X4 ) @ R ) ) ) ) ) ) ) ).
% total_on_def
thf(fact_97_total__onI,axiom,
! [A5: set_Pr1948701895od_a_a,R2: set_Pr1295299783od_a_a] :
( ! [X3: produc1572603623od_a_a,Y3: produc1572603623od_a_a] :
( ( member2057358096od_a_a @ X3 @ A5 )
=> ( ( member2057358096od_a_a @ Y3 @ A5 )
=> ( ( X3 != Y3 )
=> ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X3 @ Y3 ) @ R2 )
| ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y3 @ X3 ) @ R2 ) ) ) ) )
=> ( total_1340260971od_a_a @ A5 @ R2 ) ) ).
% total_onI
thf(fact_98_total__onI,axiom,
! [A5: set_Pr158363655iset_a,R2: set_Pr146455751iset_a] :
( ! [X3: produc1127127335iset_a,Y3: produc1127127335iset_a] :
( ( member340150864iset_a @ X3 @ A5 )
=> ( ( member340150864iset_a @ Y3 @ A5 )
=> ( ( X3 != Y3 )
=> ( ( member567864080iset_a @ ( produc1898392407iset_a @ X3 @ Y3 ) @ R2 )
| ( member567864080iset_a @ ( produc1898392407iset_a @ Y3 @ X3 ) @ R2 ) ) ) ) )
=> ( total_313747627iset_a @ A5 @ R2 ) ) ).
% total_onI
thf(fact_99_total__onI,axiom,
! [A5: set_Pr1821581383od_a_a,R2: set_Pr1930443975od_a_a] :
( ! [X3: produc2061001575od_a_a,Y3: produc2061001575od_a_a] :
( ( member1349055120od_a_a @ X3 @ A5 )
=> ( ( member1349055120od_a_a @ Y3 @ A5 )
=> ( ( X3 != Y3 )
=> ( ( member2007374608od_a_a @ ( produc978545495od_a_a @ X3 @ Y3 ) @ R2 )
| ( member2007374608od_a_a @ ( produc978545495od_a_a @ Y3 @ X3 ) @ R2 ) ) ) ) )
=> ( total_1170533099od_a_a @ A5 @ R2 ) ) ).
% total_onI
thf(fact_100_total__onI,axiom,
! [A5: set_multiset_a,R2: set_Pr158363655iset_a] :
( ! [X3: multiset_a,Y3: multiset_a] :
( ( member_multiset_a @ X3 @ A5 )
=> ( ( member_multiset_a @ Y3 @ A5 )
=> ( ( X3 != Y3 )
=> ( ( member340150864iset_a @ ( produc2037245207iset_a @ X3 @ Y3 ) @ R2 )
| ( member340150864iset_a @ ( produc2037245207iset_a @ Y3 @ X3 ) @ R2 ) ) ) ) )
=> ( total_on_multiset_a @ A5 @ R2 ) ) ).
% total_onI
thf(fact_101_total__onI,axiom,
! [A5: set_mu1281098573od_a_a,R2: set_Pr1821581383od_a_a] :
( ! [X3: multis599418605od_a_a,Y3: multis599418605od_a_a] :
( ( member1755255446od_a_a @ X3 @ A5 )
=> ( ( member1755255446od_a_a @ Y3 @ A5 )
=> ( ( X3 != Y3 )
=> ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X3 @ Y3 ) @ R2 )
| ( member1349055120od_a_a @ ( produc1859144151od_a_a @ Y3 @ X3 ) @ R2 ) ) ) ) )
=> ( total_1024184433od_a_a @ A5 @ R2 ) ) ).
% total_onI
thf(fact_102_total__onI,axiom,
! [A5: set_Product_prod_a_a,R2: set_Pr1948701895od_a_a] :
( ! [X3: product_prod_a_a,Y3: product_prod_a_a] :
( ( member449909584od_a_a @ X3 @ A5 )
=> ( ( member449909584od_a_a @ Y3 @ A5 )
=> ( ( X3 != Y3 )
=> ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Y3 ) @ R2 )
| ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y3 @ X3 ) @ R2 ) ) ) ) )
=> ( total_1490170027od_a_a @ A5 @ R2 ) ) ).
% total_onI
thf(fact_103_total__onI,axiom,
! [A5: set_a,R2: set_Product_prod_a_a] :
( ! [X3: a,Y3: a] :
( ( member_a @ X3 @ A5 )
=> ( ( member_a @ Y3 @ A5 )
=> ( ( X3 != Y3 )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R2 )
| ( member449909584od_a_a @ ( product_Pair_a_a @ Y3 @ X3 ) @ R2 ) ) ) ) )
=> ( total_on_a @ A5 @ R2 ) ) ).
% total_onI
thf(fact_104_refl__on__domain,axiom,
! [A5: set_Pr1948701895od_a_a,R2: set_Pr1295299783od_a_a,A: produc1572603623od_a_a,B: produc1572603623od_a_a] :
( ( refl_o198693862od_a_a @ A5 @ R2 )
=> ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ A @ B ) @ R2 )
=> ( ( member2057358096od_a_a @ A @ A5 )
& ( member2057358096od_a_a @ B @ A5 ) ) ) ) ).
% refl_on_domain
thf(fact_105_refl__on__domain,axiom,
! [A5: set_Pr158363655iset_a,R2: set_Pr146455751iset_a,A: produc1127127335iset_a,B: produc1127127335iset_a] :
( ( refl_o1522155814iset_a @ A5 @ R2 )
=> ( ( member567864080iset_a @ ( produc1898392407iset_a @ A @ B ) @ R2 )
=> ( ( member340150864iset_a @ A @ A5 )
& ( member340150864iset_a @ B @ A5 ) ) ) ) ).
% refl_on_domain
thf(fact_106_refl__on__domain,axiom,
! [A5: set_Pr1821581383od_a_a,R2: set_Pr1930443975od_a_a,A: produc2061001575od_a_a,B: produc2061001575od_a_a] :
( ( refl_o102522214od_a_a @ A5 @ R2 )
=> ( ( member2007374608od_a_a @ ( produc978545495od_a_a @ A @ B ) @ R2 )
=> ( ( member1349055120od_a_a @ A @ A5 )
& ( member1349055120od_a_a @ B @ A5 ) ) ) ) ).
% refl_on_domain
thf(fact_107_refl__on__domain,axiom,
! [A5: set_multiset_a,R2: set_Pr158363655iset_a,A: multiset_a,B: multiset_a] :
( ( refl_on_multiset_a @ A5 @ R2 )
=> ( ( member340150864iset_a @ ( produc2037245207iset_a @ A @ B ) @ R2 )
=> ( ( member_multiset_a @ A @ A5 )
& ( member_multiset_a @ B @ A5 ) ) ) ) ).
% refl_on_domain
thf(fact_108_refl__on__domain,axiom,
! [A5: set_mu1281098573od_a_a,R2: set_Pr1821581383od_a_a,A: multis599418605od_a_a,B: multis599418605od_a_a] :
( ( refl_o581572716od_a_a @ A5 @ R2 )
=> ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ A @ B ) @ R2 )
=> ( ( member1755255446od_a_a @ A @ A5 )
& ( member1755255446od_a_a @ B @ A5 ) ) ) ) ).
% refl_on_domain
thf(fact_109_refl__on__domain,axiom,
! [A5: set_Product_prod_a_a,R2: set_Pr1948701895od_a_a,A: product_prod_a_a,B: product_prod_a_a] :
( ( refl_o1298442278od_a_a @ A5 @ R2 )
=> ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A @ B ) @ R2 )
=> ( ( member449909584od_a_a @ A @ A5 )
& ( member449909584od_a_a @ B @ A5 ) ) ) ) ).
% refl_on_domain
thf(fact_110_refl__on__domain,axiom,
! [A5: set_a,R2: set_Product_prod_a_a,A: a,B: a] :
( ( refl_on_a @ A5 @ R2 )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B ) @ R2 )
=> ( ( member_a @ A @ A5 )
& ( member_a @ B @ A5 ) ) ) ) ).
% refl_on_domain
thf(fact_111_mem__Collect__eq,axiom,
! [A: produc1572603623od_a_a,P2: produc1572603623od_a_a > $o] :
( ( member2057358096od_a_a @ A @ ( collec1635618130od_a_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_112_mem__Collect__eq,axiom,
! [A: produc1127127335iset_a,P2: produc1127127335iset_a > $o] :
( ( member340150864iset_a @ A @ ( collec1104713362iset_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_113_mem__Collect__eq,axiom,
! [A: produc2061001575od_a_a,P2: produc2061001575od_a_a > $o] :
( ( member1349055120od_a_a @ A @ ( collec8264402od_a_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_114_mem__Collect__eq,axiom,
! [A: multis599418605od_a_a,P2: multis599418605od_a_a > $o] :
( ( member1755255446od_a_a @ A @ ( collec1948052440od_a_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_115_mem__Collect__eq,axiom,
! [A: multiset_a,P2: multiset_a > $o] :
( ( member_multiset_a @ A @ ( collect_multiset_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_116_mem__Collect__eq,axiom,
! [A: a,P2: a > $o] :
( ( member_a @ A @ ( collect_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_117_mem__Collect__eq,axiom,
! [A: product_prod_a_a,P2: product_prod_a_a > $o] :
( ( member449909584od_a_a @ A @ ( collec645855634od_a_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_118_Collect__mem__eq,axiom,
! [A5: set_Pr1948701895od_a_a] :
( ( collec1635618130od_a_a
@ ^ [X4: produc1572603623od_a_a] : ( member2057358096od_a_a @ X4 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_119_Collect__mem__eq,axiom,
! [A5: set_Pr158363655iset_a] :
( ( collec1104713362iset_a
@ ^ [X4: produc1127127335iset_a] : ( member340150864iset_a @ X4 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_120_Collect__mem__eq,axiom,
! [A5: set_Pr1821581383od_a_a] :
( ( collec8264402od_a_a
@ ^ [X4: produc2061001575od_a_a] : ( member1349055120od_a_a @ X4 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_121_Collect__mem__eq,axiom,
! [A5: set_mu1281098573od_a_a] :
( ( collec1948052440od_a_a
@ ^ [X4: multis599418605od_a_a] : ( member1755255446od_a_a @ X4 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_122_Collect__mem__eq,axiom,
! [A5: set_multiset_a] :
( ( collect_multiset_a
@ ^ [X4: multiset_a] : ( member_multiset_a @ X4 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_123_Collect__mem__eq,axiom,
! [A5: set_a] :
( ( collect_a
@ ^ [X4: a] : ( member_a @ X4 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_124_Collect__mem__eq,axiom,
! [A5: set_Product_prod_a_a] :
( ( collec645855634od_a_a
@ ^ [X4: product_prod_a_a] : ( member449909584od_a_a @ X4 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_125_Collect__cong,axiom,
! [P2: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ! [X3: product_prod_a_a] :
( ( P2 @ X3 )
= ( Q @ X3 ) )
=> ( ( collec645855634od_a_a @ P2 )
= ( collec645855634od_a_a @ Q ) ) ) ).
% Collect_cong
thf(fact_126_Collect__cong,axiom,
! [P2: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( P2 @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_a @ P2 )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_127_refl__onD2,axiom,
! [A5: set_Pr1948701895od_a_a,R2: set_Pr1295299783od_a_a,X: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
( ( refl_o198693862od_a_a @ A5 @ R2 )
=> ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Y ) @ R2 )
=> ( member2057358096od_a_a @ Y @ A5 ) ) ) ).
% refl_onD2
thf(fact_128_refl__onD2,axiom,
! [A5: set_Pr158363655iset_a,R2: set_Pr146455751iset_a,X: produc1127127335iset_a,Y: produc1127127335iset_a] :
( ( refl_o1522155814iset_a @ A5 @ R2 )
=> ( ( member567864080iset_a @ ( produc1898392407iset_a @ X @ Y ) @ R2 )
=> ( member340150864iset_a @ Y @ A5 ) ) ) ).
% refl_onD2
thf(fact_129_refl__onD2,axiom,
! [A5: set_Pr1821581383od_a_a,R2: set_Pr1930443975od_a_a,X: produc2061001575od_a_a,Y: produc2061001575od_a_a] :
( ( refl_o102522214od_a_a @ A5 @ R2 )
=> ( ( member2007374608od_a_a @ ( produc978545495od_a_a @ X @ Y ) @ R2 )
=> ( member1349055120od_a_a @ Y @ A5 ) ) ) ).
% refl_onD2
thf(fact_130_refl__onD2,axiom,
! [A5: set_multiset_a,R2: set_Pr158363655iset_a,X: multiset_a,Y: multiset_a] :
( ( refl_on_multiset_a @ A5 @ R2 )
=> ( ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Y ) @ R2 )
=> ( member_multiset_a @ Y @ A5 ) ) ) ).
% refl_onD2
thf(fact_131_refl__onD2,axiom,
! [A5: set_mu1281098573od_a_a,R2: set_Pr1821581383od_a_a,X: multis599418605od_a_a,Y: multis599418605od_a_a] :
( ( refl_o581572716od_a_a @ A5 @ R2 )
=> ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X @ Y ) @ R2 )
=> ( member1755255446od_a_a @ Y @ A5 ) ) ) ).
% refl_onD2
thf(fact_132_refl__onD2,axiom,
! [A5: set_Product_prod_a_a,R2: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
( ( refl_o1298442278od_a_a @ A5 @ R2 )
=> ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ R2 )
=> ( member449909584od_a_a @ Y @ A5 ) ) ) ).
% refl_onD2
thf(fact_133_refl__onD2,axiom,
! [A5: set_a,R2: set_Product_prod_a_a,X: a,Y: a] :
( ( refl_on_a @ A5 @ R2 )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 )
=> ( member_a @ Y @ A5 ) ) ) ).
% refl_onD2
thf(fact_134_refl__onD1,axiom,
! [A5: set_Pr1948701895od_a_a,R2: set_Pr1295299783od_a_a,X: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
( ( refl_o198693862od_a_a @ A5 @ R2 )
=> ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Y ) @ R2 )
=> ( member2057358096od_a_a @ X @ A5 ) ) ) ).
% refl_onD1
thf(fact_135_refl__onD1,axiom,
! [A5: set_Pr158363655iset_a,R2: set_Pr146455751iset_a,X: produc1127127335iset_a,Y: produc1127127335iset_a] :
( ( refl_o1522155814iset_a @ A5 @ R2 )
=> ( ( member567864080iset_a @ ( produc1898392407iset_a @ X @ Y ) @ R2 )
=> ( member340150864iset_a @ X @ A5 ) ) ) ).
% refl_onD1
thf(fact_136_refl__onD1,axiom,
! [A5: set_Pr1821581383od_a_a,R2: set_Pr1930443975od_a_a,X: produc2061001575od_a_a,Y: produc2061001575od_a_a] :
( ( refl_o102522214od_a_a @ A5 @ R2 )
=> ( ( member2007374608od_a_a @ ( produc978545495od_a_a @ X @ Y ) @ R2 )
=> ( member1349055120od_a_a @ X @ A5 ) ) ) ).
% refl_onD1
thf(fact_137_refl__onD1,axiom,
! [A5: set_multiset_a,R2: set_Pr158363655iset_a,X: multiset_a,Y: multiset_a] :
( ( refl_on_multiset_a @ A5 @ R2 )
=> ( ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Y ) @ R2 )
=> ( member_multiset_a @ X @ A5 ) ) ) ).
% refl_onD1
thf(fact_138_refl__onD1,axiom,
! [A5: set_mu1281098573od_a_a,R2: set_Pr1821581383od_a_a,X: multis599418605od_a_a,Y: multis599418605od_a_a] :
( ( refl_o581572716od_a_a @ A5 @ R2 )
=> ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X @ Y ) @ R2 )
=> ( member1755255446od_a_a @ X @ A5 ) ) ) ).
% refl_onD1
thf(fact_139_refl__onD1,axiom,
! [A5: set_Product_prod_a_a,R2: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
( ( refl_o1298442278od_a_a @ A5 @ R2 )
=> ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ R2 )
=> ( member449909584od_a_a @ X @ A5 ) ) ) ).
% refl_onD1
thf(fact_140_refl__onD1,axiom,
! [A5: set_a,R2: set_Product_prod_a_a,X: a,Y: a] :
( ( refl_on_a @ A5 @ R2 )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 )
=> ( member_a @ X @ A5 ) ) ) ).
% refl_onD1
thf(fact_141_refl__onD,axiom,
! [A5: set_Pr1948701895od_a_a,R2: set_Pr1295299783od_a_a,A: produc1572603623od_a_a] :
( ( refl_o198693862od_a_a @ A5 @ R2 )
=> ( ( member2057358096od_a_a @ A @ A5 )
=> ( member1899387664od_a_a @ ( produc1935643479od_a_a @ A @ A ) @ R2 ) ) ) ).
% refl_onD
thf(fact_142_refl__onD,axiom,
! [A5: set_Pr158363655iset_a,R2: set_Pr146455751iset_a,A: produc1127127335iset_a] :
( ( refl_o1522155814iset_a @ A5 @ R2 )
=> ( ( member340150864iset_a @ A @ A5 )
=> ( member567864080iset_a @ ( produc1898392407iset_a @ A @ A ) @ R2 ) ) ) ).
% refl_onD
thf(fact_143_refl__onD,axiom,
! [A5: set_Pr1821581383od_a_a,R2: set_Pr1930443975od_a_a,A: produc2061001575od_a_a] :
( ( refl_o102522214od_a_a @ A5 @ R2 )
=> ( ( member1349055120od_a_a @ A @ A5 )
=> ( member2007374608od_a_a @ ( produc978545495od_a_a @ A @ A ) @ R2 ) ) ) ).
% refl_onD
thf(fact_144_refl__onD,axiom,
! [A5: set_multiset_a,R2: set_Pr158363655iset_a,A: multiset_a] :
( ( refl_on_multiset_a @ A5 @ R2 )
=> ( ( member_multiset_a @ A @ A5 )
=> ( member340150864iset_a @ ( produc2037245207iset_a @ A @ A ) @ R2 ) ) ) ).
% refl_onD
thf(fact_145_refl__onD,axiom,
! [A5: set_mu1281098573od_a_a,R2: set_Pr1821581383od_a_a,A: multis599418605od_a_a] :
( ( refl_o581572716od_a_a @ A5 @ R2 )
=> ( ( member1755255446od_a_a @ A @ A5 )
=> ( member1349055120od_a_a @ ( produc1859144151od_a_a @ A @ A ) @ R2 ) ) ) ).
% refl_onD
thf(fact_146_refl__onD,axiom,
! [A5: set_Product_prod_a_a,R2: set_Pr1948701895od_a_a,A: product_prod_a_a] :
( ( refl_o1298442278od_a_a @ A5 @ R2 )
=> ( ( member449909584od_a_a @ A @ A5 )
=> ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A @ A ) @ R2 ) ) ) ).
% refl_onD
thf(fact_147_refl__onD,axiom,
! [A5: set_a,R2: set_Product_prod_a_a,A: a] :
( ( refl_on_a @ A5 @ R2 )
=> ( ( member_a @ A @ A5 )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ A @ A ) @ R2 ) ) ) ).
% refl_onD
thf(fact_148_trans__def,axiom,
( trans_972827014od_a_a
= ( ^ [R: set_Pr1948701895od_a_a] :
! [X4: product_prod_a_a,Y4: product_prod_a_a,Z2: product_prod_a_a] :
( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y4 ) @ R )
=> ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y4 @ Z2 ) @ R )
=> ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Z2 ) @ R ) ) ) ) ) ).
% trans_def
thf(fact_149_trans__def,axiom,
( trans_multiset_a
= ( ^ [R: set_Pr158363655iset_a] :
! [X4: multiset_a,Y4: multiset_a,Z2: multiset_a] :
( ( member340150864iset_a @ ( produc2037245207iset_a @ X4 @ Y4 ) @ R )
=> ( ( member340150864iset_a @ ( produc2037245207iset_a @ Y4 @ Z2 ) @ R )
=> ( member340150864iset_a @ ( produc2037245207iset_a @ X4 @ Z2 ) @ R ) ) ) ) ) ).
% trans_def
thf(fact_150_trans__def,axiom,
( trans_1520871884od_a_a
= ( ^ [R: set_Pr1821581383od_a_a] :
! [X4: multis599418605od_a_a,Y4: multis599418605od_a_a,Z2: multis599418605od_a_a] :
( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X4 @ Y4 ) @ R )
=> ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ Y4 @ Z2 ) @ R )
=> ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X4 @ Z2 ) @ R ) ) ) ) ) ).
% trans_def
thf(fact_151_trans__def,axiom,
( trans_a
= ( ^ [R: set_Product_prod_a_a] :
! [X4: a,Y4: a,Z2: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ R )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y4 @ Z2 ) @ R )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Z2 ) @ R ) ) ) ) ) ).
% trans_def
thf(fact_152_transI,axiom,
! [R2: set_Pr1948701895od_a_a] :
( ! [X3: product_prod_a_a,Y3: product_prod_a_a,Z3: product_prod_a_a] :
( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Y3 ) @ R2 )
=> ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y3 @ Z3 ) @ R2 )
=> ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Z3 ) @ R2 ) ) )
=> ( trans_972827014od_a_a @ R2 ) ) ).
% transI
thf(fact_153_transI,axiom,
! [R2: set_Pr158363655iset_a] :
( ! [X3: multiset_a,Y3: multiset_a,Z3: multiset_a] :
( ( member340150864iset_a @ ( produc2037245207iset_a @ X3 @ Y3 ) @ R2 )
=> ( ( member340150864iset_a @ ( produc2037245207iset_a @ Y3 @ Z3 ) @ R2 )
=> ( member340150864iset_a @ ( produc2037245207iset_a @ X3 @ Z3 ) @ R2 ) ) )
=> ( trans_multiset_a @ R2 ) ) ).
% transI
thf(fact_154_transI,axiom,
! [R2: set_Pr1821581383od_a_a] :
( ! [X3: multis599418605od_a_a,Y3: multis599418605od_a_a,Z3: multis599418605od_a_a] :
( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X3 @ Y3 ) @ R2 )
=> ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ Y3 @ Z3 ) @ R2 )
=> ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X3 @ Z3 ) @ R2 ) ) )
=> ( trans_1520871884od_a_a @ R2 ) ) ).
% transI
thf(fact_155_transI,axiom,
! [R2: set_Product_prod_a_a] :
( ! [X3: a,Y3: a,Z3: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R2 )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y3 @ Z3 ) @ R2 )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Z3 ) @ R2 ) ) )
=> ( trans_a @ R2 ) ) ).
% transI
thf(fact_156_transD,axiom,
! [R2: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a,Z: product_prod_a_a] :
( ( trans_972827014od_a_a @ R2 )
=> ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ R2 )
=> ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ Z ) @ R2 )
=> ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Z ) @ R2 ) ) ) ) ).
% transD
thf(fact_157_transD,axiom,
! [R2: set_Pr158363655iset_a,X: multiset_a,Y: multiset_a,Z: multiset_a] :
( ( trans_multiset_a @ R2 )
=> ( ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Y ) @ R2 )
=> ( ( member340150864iset_a @ ( produc2037245207iset_a @ Y @ Z ) @ R2 )
=> ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Z ) @ R2 ) ) ) ) ).
% transD
thf(fact_158_transD,axiom,
! [R2: set_Pr1821581383od_a_a,X: multis599418605od_a_a,Y: multis599418605od_a_a,Z: multis599418605od_a_a] :
( ( trans_1520871884od_a_a @ R2 )
=> ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X @ Y ) @ R2 )
=> ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ Y @ Z ) @ R2 )
=> ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X @ Z ) @ R2 ) ) ) ) ).
% transD
thf(fact_159_transD,axiom,
! [R2: set_Product_prod_a_a,X: a,Y: a,Z: a] :
( ( trans_a @ R2 )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ R2 )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ R2 ) ) ) ) ).
% transD
thf(fact_160_transE,axiom,
! [R2: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a,Z: product_prod_a_a] :
( ( trans_972827014od_a_a @ R2 )
=> ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ R2 )
=> ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ Z ) @ R2 )
=> ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Z ) @ R2 ) ) ) ) ).
% transE
thf(fact_161_transE,axiom,
! [R2: set_Pr158363655iset_a,X: multiset_a,Y: multiset_a,Z: multiset_a] :
( ( trans_multiset_a @ R2 )
=> ( ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Y ) @ R2 )
=> ( ( member340150864iset_a @ ( produc2037245207iset_a @ Y @ Z ) @ R2 )
=> ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Z ) @ R2 ) ) ) ) ).
% transE
thf(fact_162_transE,axiom,
! [R2: set_Pr1821581383od_a_a,X: multis599418605od_a_a,Y: multis599418605od_a_a,Z: multis599418605od_a_a] :
( ( trans_1520871884od_a_a @ R2 )
=> ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X @ Y ) @ R2 )
=> ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ Y @ Z ) @ R2 )
=> ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X @ Z ) @ R2 ) ) ) ) ).
% transE
thf(fact_163_transE,axiom,
! [R2: set_Product_prod_a_a,X: a,Y: a,Z: a] :
( ( trans_a @ R2 )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ R2 )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ R2 ) ) ) ) ).
% transE
thf(fact_164_rational__preference_Ointro,axiom,
! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
( ( prefer1338892073od_a_a @ Carrier @ Relation )
=> ( ( prefer1064779160od_a_a @ Carrier @ Relation )
=> ( prefer697821563od_a_a @ Carrier @ Relation ) ) ) ).
% rational_preference.intro
thf(fact_165_rational__preference_Ointro,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a] :
( ( prefer1292084992ence_a @ Carrier @ Relation )
=> ( ( prefer580340847ioms_a @ Carrier @ Relation )
=> ( prefer719835986ence_a @ Carrier @ Relation ) ) ) ).
% rational_preference.intro
thf(fact_166_rational__preference__def,axiom,
( prefer697821563od_a_a
= ( ^ [Carrier2: set_Product_prod_a_a,Relation2: set_Pr1948701895od_a_a] :
( ( prefer1338892073od_a_a @ Carrier2 @ Relation2 )
& ( prefer1064779160od_a_a @ Carrier2 @ Relation2 ) ) ) ) ).
% rational_preference_def
thf(fact_167_rational__preference__def,axiom,
( prefer719835986ence_a
= ( ^ [Carrier2: set_a,Relation2: set_Product_prod_a_a] :
( ( prefer1292084992ence_a @ Carrier2 @ Relation2 )
& ( prefer580340847ioms_a @ Carrier2 @ Relation2 ) ) ) ) ).
% rational_preference_def
thf(fact_168_rational__preference_Oaxioms_I2_J,axiom,
! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
( ( prefer697821563od_a_a @ Carrier @ Relation )
=> ( prefer1064779160od_a_a @ Carrier @ Relation ) ) ).
% rational_preference.axioms(2)
thf(fact_169_rational__preference_Oaxioms_I2_J,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a] :
( ( prefer719835986ence_a @ Carrier @ Relation )
=> ( prefer580340847ioms_a @ Carrier @ Relation ) ) ).
% rational_preference.axioms(2)
thf(fact_170_rational__preference__axioms_Ointro,axiom,
! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
( ( total_1490170027od_a_a @ Carrier @ Relation )
=> ( prefer1064779160od_a_a @ Carrier @ Relation ) ) ).
% rational_preference_axioms.intro
thf(fact_171_rational__preference__axioms_Ointro,axiom,
! [Carrier: set_a,Relation: set_Product_prod_a_a] :
( ( total_on_a @ Carrier @ Relation )
=> ( prefer580340847ioms_a @ Carrier @ Relation ) ) ).
% rational_preference_axioms.intro
thf(fact_172_rational__preference__axioms__def,axiom,
prefer1064779160od_a_a = total_1490170027od_a_a ).
% rational_preference_axioms_def
thf(fact_173_rational__preference__axioms__def,axiom,
prefer580340847ioms_a = total_on_a ).
% rational_preference_axioms_def
thf(fact_174_ssubst__Pair__rhs,axiom,
! [R2: product_prod_a_a,S: product_prod_a_a,R3: set_Pr1948701895od_a_a,S2: product_prod_a_a] :
( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ R2 @ S ) @ R3 )
=> ( ( S2 = S )
=> ( member2057358096od_a_a @ ( produc1474507607od_a_a @ R2 @ S2 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_175_ssubst__Pair__rhs,axiom,
! [R2: multiset_a,S: multiset_a,R3: set_Pr158363655iset_a,S2: multiset_a] :
( ( member340150864iset_a @ ( produc2037245207iset_a @ R2 @ S ) @ R3 )
=> ( ( S2 = S )
=> ( member340150864iset_a @ ( produc2037245207iset_a @ R2 @ S2 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_176_ssubst__Pair__rhs,axiom,
! [R2: multis599418605od_a_a,S: multis599418605od_a_a,R3: set_Pr1821581383od_a_a,S2: multis599418605od_a_a] :
( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ R2 @ S ) @ R3 )
=> ( ( S2 = S )
=> ( member1349055120od_a_a @ ( produc1859144151od_a_a @ R2 @ S2 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_177_ssubst__Pair__rhs,axiom,
! [R2: a,S: a,R3: set_Product_prod_a_a,S2: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ R2 @ S ) @ R3 )
=> ( ( S2 = S )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ R2 @ S2 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_178_at__lst__asgd__ge,axiom,
! [X: produc1572603623od_a_a,Y: produc1572603623od_a_a,B4: set_Pr1948701895od_a_a,Pr: set_Pr1295299783od_a_a] :
( ( member2057358096od_a_a @ X @ ( prefer55414563od_a_a @ Y @ B4 @ Pr ) )
=> ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Y ) @ Pr ) ) ).
% at_lst_asgd_ge
thf(fact_179_at__lst__asgd__ge,axiom,
! [X: produc1127127335iset_a,Y: produc1127127335iset_a,B4: set_Pr158363655iset_a,Pr: set_Pr146455751iset_a] :
( ( member340150864iset_a @ X @ ( prefer619498851iset_a @ Y @ B4 @ Pr ) )
=> ( member567864080iset_a @ ( produc1898392407iset_a @ X @ Y ) @ Pr ) ) ).
% at_lst_asgd_ge
thf(fact_180_at__lst__asgd__ge,axiom,
! [X: produc2061001575od_a_a,Y: produc2061001575od_a_a,B4: set_Pr1821581383od_a_a,Pr: set_Pr1930443975od_a_a] :
( ( member1349055120od_a_a @ X @ ( prefer737894307od_a_a @ Y @ B4 @ Pr ) )
=> ( member2007374608od_a_a @ ( produc978545495od_a_a @ X @ Y ) @ Pr ) ) ).
% at_lst_asgd_ge
thf(fact_181_at__lst__asgd__ge,axiom,
! [X: multiset_a,Y: multiset_a,B4: set_multiset_a,Pr: set_Pr158363655iset_a] :
( ( member_multiset_a @ X @ ( prefer2130004608iset_a @ Y @ B4 @ Pr ) )
=> ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Y ) @ Pr ) ) ).
% at_lst_asgd_ge
thf(fact_182_at__lst__asgd__ge,axiom,
! [X: multis599418605od_a_a,Y: multis599418605od_a_a,B4: set_mu1281098573od_a_a,Pr: set_Pr1821581383od_a_a] :
( ( member1755255446od_a_a @ X @ ( prefer1191731497od_a_a @ Y @ B4 @ Pr ) )
=> ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X @ Y ) @ Pr ) ) ).
% at_lst_asgd_ge
thf(fact_183_at__lst__asgd__ge,axiom,
! [X: product_prod_a_a,Y: product_prod_a_a,B4: set_Product_prod_a_a,Pr: set_Pr1948701895od_a_a] :
( ( member449909584od_a_a @ X @ ( prefer477097315od_a_a @ Y @ B4 @ Pr ) )
=> ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Pr ) ) ).
% at_lst_asgd_ge
thf(fact_184_at__lst__asgd__ge,axiom,
! [X: a,Y: a,B4: set_a,Pr: set_Product_prod_a_a] :
( ( member_a @ X @ ( prefer161218362good_a @ Y @ B4 @ Pr ) )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Pr ) ) ).
% at_lst_asgd_ge
thf(fact_185_at__least__as__goodD,axiom,
! [Z: produc1572603623od_a_a,Y: produc1572603623od_a_a,B4: set_Pr1948701895od_a_a,Pr: set_Pr1295299783od_a_a] :
( ( member2057358096od_a_a @ Z @ ( prefer55414563od_a_a @ Y @ B4 @ Pr ) )
=> ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Z @ Y ) @ Pr ) ) ).
% at_least_as_goodD
thf(fact_186_at__least__as__goodD,axiom,
! [Z: produc1127127335iset_a,Y: produc1127127335iset_a,B4: set_Pr158363655iset_a,Pr: set_Pr146455751iset_a] :
( ( member340150864iset_a @ Z @ ( prefer619498851iset_a @ Y @ B4 @ Pr ) )
=> ( member567864080iset_a @ ( produc1898392407iset_a @ Z @ Y ) @ Pr ) ) ).
% at_least_as_goodD
thf(fact_187_at__least__as__goodD,axiom,
! [Z: produc2061001575od_a_a,Y: produc2061001575od_a_a,B4: set_Pr1821581383od_a_a,Pr: set_Pr1930443975od_a_a] :
( ( member1349055120od_a_a @ Z @ ( prefer737894307od_a_a @ Y @ B4 @ Pr ) )
=> ( member2007374608od_a_a @ ( produc978545495od_a_a @ Z @ Y ) @ Pr ) ) ).
% at_least_as_goodD
thf(fact_188_at__least__as__goodD,axiom,
! [Z: multiset_a,Y: multiset_a,B4: set_multiset_a,Pr: set_Pr158363655iset_a] :
( ( member_multiset_a @ Z @ ( prefer2130004608iset_a @ Y @ B4 @ Pr ) )
=> ( member340150864iset_a @ ( produc2037245207iset_a @ Z @ Y ) @ Pr ) ) ).
% at_least_as_goodD
thf(fact_189_at__least__as__goodD,axiom,
! [Z: multis599418605od_a_a,Y: multis599418605od_a_a,B4: set_mu1281098573od_a_a,Pr: set_Pr1821581383od_a_a] :
( ( member1755255446od_a_a @ Z @ ( prefer1191731497od_a_a @ Y @ B4 @ Pr ) )
=> ( member1349055120od_a_a @ ( produc1859144151od_a_a @ Z @ Y ) @ Pr ) ) ).
% at_least_as_goodD
thf(fact_190_at__least__as__goodD,axiom,
! [Z: product_prod_a_a,Y: product_prod_a_a,B4: set_Product_prod_a_a,Pr: set_Pr1948701895od_a_a] :
( ( member449909584od_a_a @ Z @ ( prefer477097315od_a_a @ Y @ B4 @ Pr ) )
=> ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ Y ) @ Pr ) ) ).
% at_least_as_goodD
thf(fact_191_at__least__as__goodD,axiom,
! [Z: a,Y: a,B4: set_a,Pr: set_Product_prod_a_a] :
( ( member_a @ Z @ ( prefer161218362good_a @ Y @ B4 @ Pr ) )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ Pr ) ) ).
% at_least_as_goodD
thf(fact_192_curry__conv,axiom,
( product_curry_a_a_o
= ( ^ [F: product_prod_a_a > $o,A6: a,B5: a] : ( F @ ( product_Pair_a_a @ A6 @ B5 ) ) ) ) ).
% curry_conv
thf(fact_193_curryI,axiom,
! [F2: produc1572603623od_a_a > $o,A: product_prod_a_a,B: product_prod_a_a] :
( ( F2 @ ( produc1474507607od_a_a @ A @ B ) )
=> ( produc94438221_a_a_o @ F2 @ A @ B ) ) ).
% curryI
thf(fact_194_curryI,axiom,
! [F2: produc1127127335iset_a > $o,A: multiset_a,B: multiset_a] :
( ( F2 @ ( produc2037245207iset_a @ A @ B ) )
=> ( produc1530159757et_a_o @ F2 @ A @ B ) ) ).
% curryI
thf(fact_195_curryI,axiom,
! [F2: produc2061001575od_a_a > $o,A: multis599418605od_a_a,B: multis599418605od_a_a] :
( ( F2 @ ( produc1859144151od_a_a @ A @ B ) )
=> ( produc1548936141_a_a_o @ F2 @ A @ B ) ) ).
% curryI
thf(fact_196_curryI,axiom,
! [F2: product_prod_a_a > $o,A: a,B: a] :
( ( F2 @ ( product_Pair_a_a @ A @ B ) )
=> ( product_curry_a_a_o @ F2 @ A @ B ) ) ).
% curryI
thf(fact_197_in__inv__image,axiom,
! [X: a,Y: a,R2: set_Product_prod_a_a,F2: a > a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( inv_image_a_a @ R2 @ F2 ) )
= ( member449909584od_a_a @ ( product_Pair_a_a @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ).
% in_inv_image
thf(fact_198_in__inv__image,axiom,
! [X: a,Y: a,R2: set_Pr158363655iset_a,F2: a > multiset_a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( inv_im975322764et_a_a @ R2 @ F2 ) )
= ( member340150864iset_a @ ( produc2037245207iset_a @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ).
% in_inv_image
thf(fact_199_in__inv__image,axiom,
! [X: multiset_a,Y: multiset_a,R2: set_Product_prod_a_a,F2: multiset_a > a] :
( ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Y ) @ ( inv_im479034648iset_a @ R2 @ F2 ) )
= ( member449909584od_a_a @ ( product_Pair_a_a @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ).
% in_inv_image
thf(fact_200_in__inv__image,axiom,
! [X: a,Y: a,R2: set_Pr1948701895od_a_a,F2: a > product_prod_a_a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( inv_im777352425_a_a_a @ R2 @ F2 ) )
= ( member2057358096od_a_a @ ( produc1474507607od_a_a @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ).
% in_inv_image
thf(fact_201_in__inv__image,axiom,
! [X: product_prod_a_a,Y: product_prod_a_a,R2: set_Product_prod_a_a,F2: product_prod_a_a > a] :
( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ ( inv_im15016955od_a_a @ R2 @ F2 ) )
= ( member449909584od_a_a @ ( product_Pair_a_a @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ).
% in_inv_image
thf(fact_202_in__inv__image,axiom,
! [X: multiset_a,Y: multiset_a,R2: set_Pr158363655iset_a,F2: multiset_a > multiset_a] :
( ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Y ) @ ( inv_im343774418iset_a @ R2 @ F2 ) )
= ( member340150864iset_a @ ( produc2037245207iset_a @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ).
% in_inv_image
thf(fact_203_in__inv__image,axiom,
! [X: a,Y: a,R2: set_Pr1821581383od_a_a,F2: a > multis599418605od_a_a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( inv_im1681759523_a_a_a @ R2 @ F2 ) )
= ( member1349055120od_a_a @ ( produc1859144151od_a_a @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ).
% in_inv_image
thf(fact_204_in__inv__image,axiom,
! [X: product_prod_a_a,Y: product_prod_a_a,R2: set_Pr158363655iset_a,F2: product_prod_a_a > multiset_a] :
( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ ( inv_im1716587189od_a_a @ R2 @ F2 ) )
= ( member340150864iset_a @ ( produc2037245207iset_a @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ).
% in_inv_image
thf(fact_205_in__inv__image,axiom,
! [X: multiset_a,Y: multiset_a,R2: set_Pr1948701895od_a_a,F2: multiset_a > product_prod_a_a] :
( ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Y ) @ ( inv_im184450735iset_a @ R2 @ F2 ) )
= ( member2057358096od_a_a @ ( produc1474507607od_a_a @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ).
% in_inv_image
thf(fact_206_in__inv__image,axiom,
! [X: multis599418605od_a_a,Y: multis599418605od_a_a,R2: set_Product_prod_a_a,F2: multis599418605od_a_a > a] :
( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X @ Y ) @ ( inv_im941243329od_a_a @ R2 @ F2 ) )
= ( member449909584od_a_a @ ( product_Pair_a_a @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ).
% in_inv_image
thf(fact_207_strict__linear__order__on__def,axiom,
( order_1505515173od_a_a
= ( ^ [A4: set_Product_prod_a_a,R: set_Pr1948701895od_a_a] :
( ( trans_972827014od_a_a @ R )
& ( irrefl76447196od_a_a @ R )
& ( total_1490170027od_a_a @ A4 @ R ) ) ) ) ).
% strict_linear_order_on_def
thf(fact_208_strict__linear__order__on__def,axiom,
( order_1558116988r_on_a
= ( ^ [A4: set_a,R: set_Product_prod_a_a] :
( ( trans_a @ R )
& ( irrefl_a @ R )
& ( total_on_a @ A4 @ R ) ) ) ) ).
% strict_linear_order_on_def
thf(fact_209_swap__simp,axiom,
! [X: product_prod_a_a,Y: product_prod_a_a] :
( ( produc1641853175od_a_a @ ( produc1474507607od_a_a @ X @ Y ) )
= ( produc1474507607od_a_a @ Y @ X ) ) ).
% swap_simp
thf(fact_210_swap__simp,axiom,
! [X: multiset_a,Y: multiset_a] :
( ( produc506430135iset_a @ ( produc2037245207iset_a @ X @ Y ) )
= ( produc2037245207iset_a @ Y @ X ) ) ).
% swap_simp
thf(fact_211_swap__simp,axiom,
! [X: multis599418605od_a_a,Y: multis599418605od_a_a] :
( ( produc1913956215od_a_a @ ( produc1859144151od_a_a @ X @ Y ) )
= ( produc1859144151od_a_a @ Y @ X ) ) ).
% swap_simp
thf(fact_212_swap__simp,axiom,
! [X: a,Y: a] :
( ( product_swap_a_a @ ( product_Pair_a_a @ X @ Y ) )
= ( product_Pair_a_a @ Y @ X ) ) ).
% swap_simp
thf(fact_213_internal__case__prod__def,axiom,
produc1653225349od_a_a = produc1093061851od_a_a ).
% internal_case_prod_def
thf(fact_214_internal__case__prod__def,axiom,
produc694541532_set_a = produc961951794_set_a ).
% internal_case_prod_def
thf(fact_215_internal__case__prod__def,axiom,
produc748978966_a_a_o = produc1833107820_a_a_o ).
% internal_case_prod_def
thf(fact_216_swap__swap,axiom,
! [P: product_prod_a_a] :
( ( product_swap_a_a @ ( product_swap_a_a @ P ) )
= P ) ).
% swap_swap
thf(fact_217_case__prod__curry,axiom,
! [F2: product_prod_a_a > set_Product_prod_a_a] :
( ( produc1093061851od_a_a @ ( produc143372156od_a_a @ F2 ) )
= F2 ) ).
% case_prod_curry
thf(fact_218_case__prod__curry,axiom,
! [F2: product_prod_a_a > set_a] :
( ( produc961951794_set_a @ ( produc580150227_set_a @ F2 ) )
= F2 ) ).
% case_prod_curry
thf(fact_219_case__prod__curry,axiom,
! [F2: product_prod_a_a > $o] :
( ( produc1833107820_a_a_o @ ( product_curry_a_a_o @ F2 ) )
= F2 ) ).
% case_prod_curry
thf(fact_220_curry__case__prod,axiom,
! [F2: a > a > set_Product_prod_a_a] :
( ( produc143372156od_a_a @ ( produc1093061851od_a_a @ F2 ) )
= F2 ) ).
% curry_case_prod
thf(fact_221_curry__case__prod,axiom,
! [F2: a > a > set_a] :
( ( produc580150227_set_a @ ( produc961951794_set_a @ F2 ) )
= F2 ) ).
% curry_case_prod
thf(fact_222_curry__case__prod,axiom,
! [F2: a > a > $o] :
( ( product_curry_a_a_o @ ( produc1833107820_a_a_o @ F2 ) )
= F2 ) ).
% curry_case_prod
thf(fact_223_mem__case__prodE,axiom,
! [Z: a,C2: a > a > set_a,P: product_prod_a_a] :
( ( member_a @ Z @ ( produc961951794_set_a @ C2 @ P ) )
=> ~ ! [X3: a,Y3: a] :
( ( P
= ( product_Pair_a_a @ X3 @ Y3 ) )
=> ~ ( member_a @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_224_mem__case__prodE,axiom,
! [Z: product_prod_a_a,C2: a > a > set_Product_prod_a_a,P: product_prod_a_a] :
( ( member449909584od_a_a @ Z @ ( produc1093061851od_a_a @ C2 @ P ) )
=> ~ ! [X3: a,Y3: a] :
( ( P
= ( product_Pair_a_a @ X3 @ Y3 ) )
=> ~ ( member449909584od_a_a @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_225_mem__case__prodE,axiom,
! [Z: multiset_a,C2: a > a > set_multiset_a,P: product_prod_a_a] :
( ( member_multiset_a @ Z @ ( produc1828150264iset_a @ C2 @ P ) )
=> ~ ! [X3: a,Y3: a] :
( ( P
= ( product_Pair_a_a @ X3 @ Y3 ) )
=> ~ ( member_multiset_a @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_226_mem__case__prodE,axiom,
! [Z: a,C2: multiset_a > multiset_a > set_a,P: produc1127127335iset_a] :
( ( member_a @ Z @ ( produc747255346_set_a @ C2 @ P ) )
=> ~ ! [X3: multiset_a,Y3: multiset_a] :
( ( P
= ( produc2037245207iset_a @ X3 @ Y3 ) )
=> ~ ( member_a @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_227_mem__case__prodE,axiom,
! [Z: multis599418605od_a_a,C2: a > a > set_mu1281098573od_a_a,P: product_prod_a_a] :
( ( member1755255446od_a_a @ Z @ ( produc1121147169od_a_a @ C2 @ P ) )
=> ~ ! [X3: a,Y3: a] :
( ( P
= ( product_Pair_a_a @ X3 @ Y3 ) )
=> ~ ( member1755255446od_a_a @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_228_mem__case__prodE,axiom,
! [Z: multiset_a,C2: multiset_a > multiset_a > set_multiset_a,P: produc1127127335iset_a] :
( ( member_multiset_a @ Z @ ( produc429414904iset_a @ C2 @ P ) )
=> ~ ! [X3: multiset_a,Y3: multiset_a] :
( ( P
= ( produc2037245207iset_a @ X3 @ Y3 ) )
=> ~ ( member_multiset_a @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_229_mem__case__prodE,axiom,
! [Z: produc1127127335iset_a,C2: a > a > set_Pr158363655iset_a,P: product_prod_a_a] :
( ( member340150864iset_a @ Z @ ( produc93594587iset_a @ C2 @ P ) )
=> ~ ! [X3: a,Y3: a] :
( ( P
= ( product_Pair_a_a @ X3 @ Y3 ) )
=> ~ ( member340150864iset_a @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_230_mem__case__prodE,axiom,
! [Z: a,C2: product_prod_a_a > product_prod_a_a > set_a,P: produc1572603623od_a_a] :
( ( member_a @ Z @ ( produc869579762_set_a @ C2 @ P ) )
=> ~ ! [X3: product_prod_a_a,Y3: product_prod_a_a] :
( ( P
= ( produc1474507607od_a_a @ X3 @ Y3 ) )
=> ~ ( member_a @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_231_mem__case__prodE,axiom,
! [Z: product_prod_a_a,C2: multiset_a > multiset_a > set_Product_prod_a_a,P: produc1127127335iset_a] :
( ( member449909584od_a_a @ Z @ ( produc2041367259od_a_a @ C2 @ P ) )
=> ~ ! [X3: multiset_a,Y3: multiset_a] :
( ( P
= ( produc2037245207iset_a @ X3 @ Y3 ) )
=> ~ ( member449909584od_a_a @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_232_mem__case__prodE,axiom,
! [Z: multiset_a,C2: product_prod_a_a > product_prod_a_a > set_multiset_a,P: produc1572603623od_a_a] :
( ( member_multiset_a @ Z @ ( produc734625720iset_a @ C2 @ P ) )
=> ~ ! [X3: product_prod_a_a,Y3: product_prod_a_a] :
( ( P
= ( produc1474507607od_a_a @ X3 @ Y3 ) )
=> ~ ( member_multiset_a @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_233_old_Oprod_Ocase,axiom,
! [F2: a > a > set_Product_prod_a_a,X1: a,X22: a] :
( ( produc1093061851od_a_a @ F2 @ ( product_Pair_a_a @ X1 @ X22 ) )
= ( F2 @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_234_old_Oprod_Ocase,axiom,
! [F2: a > a > set_a,X1: a,X22: a] :
( ( produc961951794_set_a @ F2 @ ( product_Pair_a_a @ X1 @ X22 ) )
= ( F2 @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_235_old_Oprod_Ocase,axiom,
! [F2: a > a > $o,X1: a,X22: a] :
( ( produc1833107820_a_a_o @ F2 @ ( product_Pair_a_a @ X1 @ X22 ) )
= ( F2 @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_236_irreflI,axiom,
! [R3: set_Pr1948701895od_a_a] :
( ! [A3: product_prod_a_a] :
~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A3 @ A3 ) @ R3 )
=> ( irrefl76447196od_a_a @ R3 ) ) ).
% irreflI
thf(fact_237_irreflI,axiom,
! [R3: set_Pr158363655iset_a] :
( ! [A3: multiset_a] :
~ ( member340150864iset_a @ ( produc2037245207iset_a @ A3 @ A3 ) @ R3 )
=> ( irrefl_multiset_a @ R3 ) ) ).
% irreflI
thf(fact_238_irreflI,axiom,
! [R3: set_Pr1821581383od_a_a] :
( ! [A3: multis599418605od_a_a] :
~ ( member1349055120od_a_a @ ( produc1859144151od_a_a @ A3 @ A3 ) @ R3 )
=> ( irrefl1672195874od_a_a @ R3 ) ) ).
% irreflI
thf(fact_239_irreflI,axiom,
! [R3: set_Product_prod_a_a] :
( ! [A3: a] :
~ ( member449909584od_a_a @ ( product_Pair_a_a @ A3 @ A3 ) @ R3 )
=> ( irrefl_a @ R3 ) ) ).
% irreflI
thf(fact_240_irrefl__def,axiom,
( irrefl76447196od_a_a
= ( ^ [R: set_Pr1948701895od_a_a] :
! [A6: product_prod_a_a] :
~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A6 @ A6 ) @ R ) ) ) ).
% irrefl_def
thf(fact_241_irrefl__def,axiom,
( irrefl_multiset_a
= ( ^ [R: set_Pr158363655iset_a] :
! [A6: multiset_a] :
~ ( member340150864iset_a @ ( produc2037245207iset_a @ A6 @ A6 ) @ R ) ) ) ).
% irrefl_def
thf(fact_242_irrefl__def,axiom,
( irrefl1672195874od_a_a
= ( ^ [R: set_Pr1821581383od_a_a] :
! [A6: multis599418605od_a_a] :
~ ( member1349055120od_a_a @ ( produc1859144151od_a_a @ A6 @ A6 ) @ R ) ) ) ).
% irrefl_def
thf(fact_243_irrefl__def,axiom,
( irrefl_a
= ( ^ [R: set_Product_prod_a_a] :
! [A6: a] :
~ ( member449909584od_a_a @ ( product_Pair_a_a @ A6 @ A6 ) @ R ) ) ) ).
% irrefl_def
thf(fact_244_curryD,axiom,
! [F2: produc1127127335iset_a > $o,A: multiset_a,B: multiset_a] :
( ( produc1530159757et_a_o @ F2 @ A @ B )
=> ( F2 @ ( produc2037245207iset_a @ A @ B ) ) ) ).
% curryD
thf(fact_245_curryD,axiom,
! [F2: produc2061001575od_a_a > $o,A: multis599418605od_a_a,B: multis599418605od_a_a] :
( ( produc1548936141_a_a_o @ F2 @ A @ B )
=> ( F2 @ ( produc1859144151od_a_a @ A @ B ) ) ) ).
% curryD
thf(fact_246_curryD,axiom,
! [F2: product_prod_a_a > $o,A: a,B: a] :
( ( product_curry_a_a_o @ F2 @ A @ B )
=> ( F2 @ ( product_Pair_a_a @ A @ B ) ) ) ).
% curryD
thf(fact_247_curryE,axiom,
! [F2: product_prod_a_a > $o,A: a,B: a] :
( ( product_curry_a_a_o @ F2 @ A @ B )
=> ( F2 @ ( product_Pair_a_a @ A @ B ) ) ) ).
% curryE
thf(fact_248_trans__inv__image,axiom,
! [R2: set_Product_prod_a_a,F2: a > a] :
( ( trans_a @ R2 )
=> ( trans_a @ ( inv_image_a_a @ R2 @ F2 ) ) ) ).
% trans_inv_image
thf(fact_249_pair__in__swap__image,axiom,
! [Y: a,X: a,A5: set_Product_prod_a_a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ ( image_794773445od_a_a @ product_swap_a_a @ A5 ) )
= ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ A5 ) ) ).
% pair_in_swap_image
thf(fact_250_multp__iff,axiom,
! [R3: set_Product_prod_a_a,P2: a > a > $o,N: multiset_a,M: multiset_a] :
( ( irrefl_a @ R3 )
=> ( ( trans_a @ R3 )
=> ( ! [X3: a,Y3: a] :
( ( P2 @ X3 @ Y3 )
= ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R3 ) )
=> ( ( multp_a @ P2 @ N @ M )
= ( member340150864iset_a @ ( produc2037245207iset_a @ N @ M ) @ ( mult_a @ R3 ) ) ) ) ) ) ).
% multp_iff
thf(fact_251_asym_Oinducts,axiom,
! [X: set_Product_prod_a_a,P2: set_Product_prod_a_a > $o] :
( ( asym_a @ X )
=> ( ! [R4: set_Product_prod_a_a] :
( ! [A7: a,B6: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ A7 @ B6 ) @ R4 )
=> ~ ( member449909584od_a_a @ ( product_Pair_a_a @ B6 @ A7 ) @ R4 ) )
=> ( P2 @ R4 ) )
=> ( P2 @ X ) ) ) ).
% asym.inducts
thf(fact_252_asym_Ointros,axiom,
! [R3: set_Product_prod_a_a] :
( ! [A3: a,B3: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ A3 @ B3 ) @ R3 )
=> ~ ( member449909584od_a_a @ ( product_Pair_a_a @ B3 @ A3 ) @ R3 ) )
=> ( asym_a @ R3 ) ) ).
% asym.intros
thf(fact_253_asym_Osimps,axiom,
( asym_a
= ( ^ [A6: set_Product_prod_a_a] :
? [R5: set_Product_prod_a_a] :
( ( A6 = R5 )
& ! [X4: a,Y4: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ R5 )
=> ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y4 @ X4 ) @ R5 ) ) ) ) ) ).
% asym.simps
thf(fact_254_asym_Ocases,axiom,
! [A: set_Product_prod_a_a] :
( ( asym_a @ A )
=> ! [A7: a,B6: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ A7 @ B6 ) @ A )
=> ~ ( member449909584od_a_a @ ( product_Pair_a_a @ B6 @ A7 ) @ A ) ) ) ).
% asym.cases
thf(fact_255_asym__iff,axiom,
( asym_a
= ( ^ [R5: set_Product_prod_a_a] :
! [X4: a,Y4: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ R5 )
=> ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y4 @ X4 ) @ R5 ) ) ) ) ).
% asym_iff
thf(fact_256_asymD,axiom,
! [R3: set_Product_prod_a_a,X: a,Y: a] :
( ( asym_a @ R3 )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R3 )
=> ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ R3 ) ) ) ).
% asymD
thf(fact_257_image__eqI,axiom,
! [B: a,F2: a > a,X: a,A5: set_a] :
( ( B
= ( F2 @ X ) )
=> ( ( member_a @ X @ A5 )
=> ( member_a @ B @ ( image_a_a @ F2 @ A5 ) ) ) ) ).
% image_eqI
thf(fact_258_image__eqI,axiom,
! [B: product_prod_a_a,F2: a > product_prod_a_a,X: a,A5: set_a] :
( ( B
= ( F2 @ X ) )
=> ( ( member_a @ X @ A5 )
=> ( member449909584od_a_a @ B @ ( image_1725938350od_a_a @ F2 @ A5 ) ) ) ) ).
% image_eqI
thf(fact_259_image__eqI,axiom,
! [B: a,F2: product_prod_a_a > a,X: product_prod_a_a,A5: set_Product_prod_a_a] :
( ( B
= ( F2 @ X ) )
=> ( ( member449909584od_a_a @ X @ A5 )
=> ( member_a @ B @ ( image_340790172_a_a_a @ F2 @ A5 ) ) ) ) ).
% image_eqI
thf(fact_260_image__eqI,axiom,
! [B: product_prod_a_a,F2: product_prod_a_a > product_prod_a_a,X: product_prod_a_a,A5: set_Product_prod_a_a] :
( ( B
= ( F2 @ X ) )
=> ( ( member449909584od_a_a @ X @ A5 )
=> ( member449909584od_a_a @ B @ ( image_794773445od_a_a @ F2 @ A5 ) ) ) ) ).
% image_eqI
thf(fact_261_mult__cancel,axiom,
! [S: set_Product_prod_a_a,X5: multiset_a,Z4: multiset_a,Y5: multiset_a] :
( ( trans_a @ S )
=> ( ( irrefl_a @ S )
=> ( ( member340150864iset_a @ ( produc2037245207iset_a @ ( plus_plus_multiset_a @ X5 @ Z4 ) @ ( plus_plus_multiset_a @ Y5 @ Z4 ) ) @ ( mult_a @ S ) )
= ( member340150864iset_a @ ( produc2037245207iset_a @ X5 @ Y5 ) @ ( mult_a @ S ) ) ) ) ) ).
% mult_cancel
thf(fact_262_mult__cancel__add__mset,axiom,
! [S: set_Product_prod_a_a,Uu: a,X5: multiset_a,Y5: multiset_a] :
( ( trans_a @ S )
=> ( ( irrefl_a @ S )
=> ( ( member340150864iset_a @ ( produc2037245207iset_a @ ( add_mset_a @ Uu @ X5 ) @ ( add_mset_a @ Uu @ Y5 ) ) @ ( mult_a @ S ) )
= ( member340150864iset_a @ ( produc2037245207iset_a @ X5 @ Y5 ) @ ( mult_a @ S ) ) ) ) ) ).
% mult_cancel_add_mset
thf(fact_263_imageI,axiom,
! [X: a,A5: set_a,F2: a > a] :
( ( member_a @ X @ A5 )
=> ( member_a @ ( F2 @ X ) @ ( image_a_a @ F2 @ A5 ) ) ) ).
% imageI
thf(fact_264_imageI,axiom,
! [X: a,A5: set_a,F2: a > product_prod_a_a] :
( ( member_a @ X @ A5 )
=> ( member449909584od_a_a @ ( F2 @ X ) @ ( image_1725938350od_a_a @ F2 @ A5 ) ) ) ).
% imageI
thf(fact_265_imageI,axiom,
! [X: product_prod_a_a,A5: set_Product_prod_a_a,F2: product_prod_a_a > a] :
( ( member449909584od_a_a @ X @ A5 )
=> ( member_a @ ( F2 @ X ) @ ( image_340790172_a_a_a @ F2 @ A5 ) ) ) ).
% imageI
thf(fact_266_imageI,axiom,
! [X: product_prod_a_a,A5: set_Product_prod_a_a,F2: product_prod_a_a > product_prod_a_a] :
( ( member449909584od_a_a @ X @ A5 )
=> ( member449909584od_a_a @ ( F2 @ X ) @ ( image_794773445od_a_a @ F2 @ A5 ) ) ) ).
% imageI
thf(fact_267_rev__image__eqI,axiom,
! [X: a,A5: set_a,B: a,F2: a > a] :
( ( member_a @ X @ A5 )
=> ( ( B
= ( F2 @ X ) )
=> ( member_a @ B @ ( image_a_a @ F2 @ A5 ) ) ) ) ).
% rev_image_eqI
thf(fact_268_rev__image__eqI,axiom,
! [X: a,A5: set_a,B: product_prod_a_a,F2: a > product_prod_a_a] :
( ( member_a @ X @ A5 )
=> ( ( B
= ( F2 @ X ) )
=> ( member449909584od_a_a @ B @ ( image_1725938350od_a_a @ F2 @ A5 ) ) ) ) ).
% rev_image_eqI
thf(fact_269_rev__image__eqI,axiom,
! [X: product_prod_a_a,A5: set_Product_prod_a_a,B: a,F2: product_prod_a_a > a] :
( ( member449909584od_a_a @ X @ A5 )
=> ( ( B
= ( F2 @ X ) )
=> ( member_a @ B @ ( image_340790172_a_a_a @ F2 @ A5 ) ) ) ) ).
% rev_image_eqI
thf(fact_270_rev__image__eqI,axiom,
! [X: product_prod_a_a,A5: set_Product_prod_a_a,B: product_prod_a_a,F2: product_prod_a_a > product_prod_a_a] :
( ( member449909584od_a_a @ X @ A5 )
=> ( ( B
= ( F2 @ X ) )
=> ( member449909584od_a_a @ B @ ( image_794773445od_a_a @ F2 @ A5 ) ) ) ) ).
% rev_image_eqI
thf(fact_271_top__empty__eq,axiom,
( top_top_a_o
= ( ^ [X4: a] : ( member_a @ X4 @ top_top_set_a ) ) ) ).
% top_empty_eq
thf(fact_272_top__empty__eq,axiom,
( top_to1483957574_a_a_o
= ( ^ [X4: product_prod_a_a] : ( member449909584od_a_a @ X4 @ top_to398455383od_a_a ) ) ) ).
% top_empty_eq
thf(fact_273_total__lex__prod,axiom,
! [R2: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( total_on_a @ top_top_set_a @ R2 )
=> ( ( total_on_a @ top_top_set_a @ S )
=> ( total_1490170027od_a_a @ top_to398455383od_a_a @ ( lex_prod_a_a @ R2 @ S ) ) ) ) ).
% total_lex_prod
thf(fact_274_map__prod__simp,axiom,
! [F2: a > a,G: a > a,A: a,B: a] :
( ( produc700022774_a_a_a @ F2 @ G @ ( product_Pair_a_a @ A @ B ) )
= ( product_Pair_a_a @ ( F2 @ A ) @ ( G @ B ) ) ) ).
% map_prod_simp
thf(fact_275_in__lex__prod,axiom,
! [A: a,B: a,A2: a,B2: a,R2: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( product_Pair_a_a @ A2 @ B2 ) ) @ ( lex_prod_a_a @ R2 @ S ) )
= ( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ A2 ) @ R2 )
| ( ( A = A2 )
& ( member449909584od_a_a @ ( product_Pair_a_a @ B @ B2 ) @ S ) ) ) ) ).
% in_lex_prod
thf(fact_276_trans__lex__prod,axiom,
! [R1: set_Product_prod_a_a,R22: set_Product_prod_a_a] :
( ( trans_a @ R1 )
=> ( ( trans_a @ R22 )
=> ( trans_972827014od_a_a @ ( lex_prod_a_a @ R1 @ R22 ) ) ) ) ).
% trans_lex_prod
thf(fact_277_prod_Ocollapse,axiom,
! [Prod: product_prod_a_a] :
( ( product_Pair_a_a @ ( product_fst_a_a @ Prod ) @ ( product_snd_a_a @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_278_map__prod__imageI,axiom,
! [A: a,B: a,R3: set_Product_prod_a_a,F2: a > a,G: a > a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B ) @ R3 )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ ( F2 @ A ) @ ( G @ B ) ) @ ( image_794773445od_a_a @ ( produc700022774_a_a_a @ F2 @ G ) @ R3 ) ) ) ).
% map_prod_imageI
thf(fact_279_Product__Type_OCollect__case__prodD,axiom,
! [X: product_prod_a_a,A5: a > a > $o] :
( ( member449909584od_a_a @ X @ ( collec645855634od_a_a @ ( produc1833107820_a_a_o @ A5 ) ) )
=> ( A5 @ ( product_fst_a_a @ X ) @ ( product_snd_a_a @ X ) ) ) ).
% Product_Type.Collect_case_prodD
thf(fact_280_prod_Oexhaust__sel,axiom,
! [Prod: product_prod_a_a] :
( Prod
= ( product_Pair_a_a @ ( product_fst_a_a @ Prod ) @ ( product_snd_a_a @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_281_surjective__pairing,axiom,
! [T: product_prod_a_a] :
( T
= ( product_Pair_a_a @ ( product_fst_a_a @ T ) @ ( product_snd_a_a @ T ) ) ) ).
% surjective_pairing
thf(fact_282_fst__conv,axiom,
! [X1: a,X22: a] :
( ( product_fst_a_a @ ( product_Pair_a_a @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_283_fst__eqD,axiom,
! [X: a,Y: a,A: a] :
( ( ( product_fst_a_a @ ( product_Pair_a_a @ X @ Y ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_284_snd__conv,axiom,
! [X1: a,X22: a] :
( ( product_snd_a_a @ ( product_Pair_a_a @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_285_snd__eqD,axiom,
! [X: a,Y: a,A: a] :
( ( ( product_snd_a_a @ ( product_Pair_a_a @ X @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_286_prod_Oswap__def,axiom,
( product_swap_a_a
= ( ^ [P3: product_prod_a_a] : ( product_Pair_a_a @ ( product_snd_a_a @ P3 ) @ ( product_fst_a_a @ P3 ) ) ) ) ).
% prod.swap_def
thf(fact_287_prod__fun__imageE,axiom,
! [C2: product_prod_a_a,F2: a > a,G: a > a,R3: set_Product_prod_a_a] :
( ( member449909584od_a_a @ C2 @ ( image_794773445od_a_a @ ( produc700022774_a_a_a @ F2 @ G ) @ R3 ) )
=> ~ ! [X3: a,Y3: a] :
( ( C2
= ( product_Pair_a_a @ ( F2 @ X3 ) @ ( G @ Y3 ) ) )
=> ~ ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R3 ) ) ) ).
% prod_fun_imageE
thf(fact_288_exI__realizer,axiom,
! [P2: a > a > $o,Y: a,X: a] :
( ( P2 @ Y @ X )
=> ( P2 @ ( product_snd_a_a @ ( product_Pair_a_a @ X @ Y ) ) @ ( product_fst_a_a @ ( product_Pair_a_a @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_289_conjI__realizer,axiom,
! [P2: a > $o,P: a,Q: a > $o,Q2: a] :
( ( P2 @ P )
=> ( ( Q @ Q2 )
=> ( ( P2 @ ( product_fst_a_a @ ( product_Pair_a_a @ P @ Q2 ) ) )
& ( Q @ ( product_snd_a_a @ ( product_Pair_a_a @ P @ Q2 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_290_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P2: a > a > $o,X: a,Y: a,A: product_prod_a_a] :
( ( P2 @ X @ Y )
=> ( ( A
= ( product_Pair_a_a @ X @ Y ) )
=> ( P2 @ ( product_fst_a_a @ A ) @ ( product_snd_a_a @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_291_sndI,axiom,
! [X: product_prod_a_a,Y: a,Z: a] :
( ( X
= ( product_Pair_a_a @ Y @ Z ) )
=> ( ( product_snd_a_a @ X )
= Z ) ) ).
% sndI
thf(fact_292_fstI,axiom,
! [X: product_prod_a_a,Y: a,Z: a] :
( ( X
= ( product_Pair_a_a @ Y @ Z ) )
=> ( ( product_fst_a_a @ X )
= Y ) ) ).
% fstI
thf(fact_293_eq__snd__iff,axiom,
! [B: a,P: product_prod_a_a] :
( ( B
= ( product_snd_a_a @ P ) )
= ( ? [A6: a] :
( P
= ( product_Pair_a_a @ A6 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_294_eq__fst__iff,axiom,
! [A: a,P: product_prod_a_a] :
( ( A
= ( product_fst_a_a @ P ) )
= ( ? [B5: a] :
( P
= ( product_Pair_a_a @ A @ B5 ) ) ) ) ).
% eq_fst_iff
thf(fact_295_subrelI,axiom,
! [R2: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ! [X3: a,Y3: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R2 )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ S ) )
=> ( ord_le1824328871od_a_a @ R2 @ S ) ) ).
% subrelI
thf(fact_296_image__subsetI,axiom,
! [A5: set_a,F2: a > a,B4: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A5 )
=> ( member_a @ ( F2 @ X3 ) @ B4 ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F2 @ A5 ) @ B4 ) ) ).
% image_subsetI
thf(fact_297_image__subsetI,axiom,
! [A5: set_a,F2: a > product_prod_a_a,B4: set_Product_prod_a_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A5 )
=> ( member449909584od_a_a @ ( F2 @ X3 ) @ B4 ) )
=> ( ord_le1824328871od_a_a @ ( image_1725938350od_a_a @ F2 @ A5 ) @ B4 ) ) ).
% image_subsetI
thf(fact_298_image__subsetI,axiom,
! [A5: set_Product_prod_a_a,F2: product_prod_a_a > a,B4: set_a] :
( ! [X3: product_prod_a_a] :
( ( member449909584od_a_a @ X3 @ A5 )
=> ( member_a @ ( F2 @ X3 ) @ B4 ) )
=> ( ord_less_eq_set_a @ ( image_340790172_a_a_a @ F2 @ A5 ) @ B4 ) ) ).
% image_subsetI
thf(fact_299_image__subsetI,axiom,
! [A5: set_Product_prod_a_a,F2: product_prod_a_a > product_prod_a_a,B4: set_Product_prod_a_a] :
( ! [X3: product_prod_a_a] :
( ( member449909584od_a_a @ X3 @ A5 )
=> ( member449909584od_a_a @ ( F2 @ X3 ) @ B4 ) )
=> ( ord_le1824328871od_a_a @ ( image_794773445od_a_a @ F2 @ A5 ) @ B4 ) ) ).
% image_subsetI
thf(fact_300_same__fstI,axiom,
! [P2: a > $o,X: a,Y6: a,Y: a,R3: a > set_Product_prod_a_a] :
( ( P2 @ X )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y6 @ Y ) @ ( R3 @ X ) )
=> ( member2057358096od_a_a @ ( produc1474507607od_a_a @ ( product_Pair_a_a @ X @ Y6 ) @ ( product_Pair_a_a @ X @ Y ) ) @ ( same_fst_a_a @ P2 @ R3 ) ) ) ) ).
% same_fstI
thf(fact_301_mult1E,axiom,
! [N: multis599418605od_a_a,M: multis599418605od_a_a,R2: set_Pr1948701895od_a_a] :
( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ N @ M ) @ ( mult1_1719650714od_a_a @ R2 ) )
=> ~ ! [A3: product_prod_a_a,M0: multis599418605od_a_a] :
( ( M
= ( add_ms1725977093od_a_a @ A3 @ M0 ) )
=> ! [K: multis599418605od_a_a] :
( ( N
= ( plus_p404985124od_a_a @ M0 @ K ) )
=> ~ ! [B6: product_prod_a_a] :
( ( member449909584od_a_a @ B6 @ ( set_ms1818337124od_a_a @ K ) )
=> ( member2057358096od_a_a @ ( produc1474507607od_a_a @ B6 @ A3 ) @ R2 ) ) ) ) ) ).
% mult1E
thf(fact_302_mult1E,axiom,
! [N: multiset_a,M: multiset_a,R2: set_Product_prod_a_a] :
( ( member340150864iset_a @ ( produc2037245207iset_a @ N @ M ) @ ( mult1_a @ R2 ) )
=> ~ ! [A3: a,M0: multiset_a] :
( ( M
= ( add_mset_a @ A3 @ M0 ) )
=> ! [K: multiset_a] :
( ( N
= ( plus_plus_multiset_a @ M0 @ K ) )
=> ~ ! [B6: a] :
( ( member_a @ B6 @ ( set_mset_a @ K ) )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ B6 @ A3 ) @ R2 ) ) ) ) ) ).
% mult1E
thf(fact_303_mult1I,axiom,
! [M: multis599418605od_a_a,A: product_prod_a_a,M02: multis599418605od_a_a,N: multis599418605od_a_a,K2: multis599418605od_a_a,R2: set_Pr1948701895od_a_a] :
( ( M
= ( add_ms1725977093od_a_a @ A @ M02 ) )
=> ( ( N
= ( plus_p404985124od_a_a @ M02 @ K2 ) )
=> ( ! [B3: product_prod_a_a] :
( ( member449909584od_a_a @ B3 @ ( set_ms1818337124od_a_a @ K2 ) )
=> ( member2057358096od_a_a @ ( produc1474507607od_a_a @ B3 @ A ) @ R2 ) )
=> ( member1349055120od_a_a @ ( produc1859144151od_a_a @ N @ M ) @ ( mult1_1719650714od_a_a @ R2 ) ) ) ) ) ).
% mult1I
thf(fact_304_mult1I,axiom,
! [M: multiset_a,A: a,M02: multiset_a,N: multiset_a,K2: multiset_a,R2: set_Product_prod_a_a] :
( ( M
= ( add_mset_a @ A @ M02 ) )
=> ( ( N
= ( plus_plus_multiset_a @ M02 @ K2 ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ ( set_mset_a @ K2 ) )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ B3 @ A ) @ R2 ) )
=> ( member340150864iset_a @ ( produc2037245207iset_a @ N @ M ) @ ( mult1_a @ R2 ) ) ) ) ) ).
% mult1I
thf(fact_305_less__add,axiom,
! [N: multis599418605od_a_a,A: product_prod_a_a,M02: multis599418605od_a_a,R2: set_Pr1948701895od_a_a] :
( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ N @ ( add_ms1725977093od_a_a @ A @ M02 ) ) @ ( mult1_1719650714od_a_a @ R2 ) )
=> ( ? [M2: multis599418605od_a_a] :
( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ M2 @ M02 ) @ ( mult1_1719650714od_a_a @ R2 ) )
& ( N
= ( add_ms1725977093od_a_a @ A @ M2 ) ) )
| ? [K: multis599418605od_a_a] :
( ! [B6: product_prod_a_a] :
( ( member449909584od_a_a @ B6 @ ( set_ms1818337124od_a_a @ K ) )
=> ( member2057358096od_a_a @ ( produc1474507607od_a_a @ B6 @ A ) @ R2 ) )
& ( N
= ( plus_p404985124od_a_a @ M02 @ K ) ) ) ) ) ).
% less_add
thf(fact_306_less__add,axiom,
! [N: multiset_a,A: a,M02: multiset_a,R2: set_Product_prod_a_a] :
( ( member340150864iset_a @ ( produc2037245207iset_a @ N @ ( add_mset_a @ A @ M02 ) ) @ ( mult1_a @ R2 ) )
=> ( ? [M2: multiset_a] :
( ( member340150864iset_a @ ( produc2037245207iset_a @ M2 @ M02 ) @ ( mult1_a @ R2 ) )
& ( N
= ( add_mset_a @ A @ M2 ) ) )
| ? [K: multiset_a] :
( ! [B6: a] :
( ( member_a @ B6 @ ( set_mset_a @ K ) )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ B6 @ A ) @ R2 ) )
& ( N
= ( plus_plus_multiset_a @ M02 @ K ) ) ) ) ) ).
% less_add
thf(fact_307_mset__add,axiom,
! [A: a,A5: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ A5 ) )
=> ~ ! [B7: multiset_a] :
( A5
!= ( add_mset_a @ A @ B7 ) ) ) ).
% mset_add
thf(fact_308_mset__add,axiom,
! [A: product_prod_a_a,A5: multis599418605od_a_a] :
( ( member449909584od_a_a @ A @ ( set_ms1818337124od_a_a @ A5 ) )
=> ~ ! [B7: multis599418605od_a_a] :
( A5
!= ( add_ms1725977093od_a_a @ A @ B7 ) ) ) ).
% mset_add
thf(fact_309_multi__member__split,axiom,
! [X: a,M: multiset_a] :
( ( member_a @ X @ ( set_mset_a @ M ) )
=> ? [A8: multiset_a] :
( M
= ( add_mset_a @ X @ A8 ) ) ) ).
% multi_member_split
thf(fact_310_multi__member__split,axiom,
! [X: product_prod_a_a,M: multis599418605od_a_a] :
( ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ M ) )
=> ? [A8: multis599418605od_a_a] :
( M
= ( add_ms1725977093od_a_a @ X @ A8 ) ) ) ).
% multi_member_split
thf(fact_311_insert__noteq__member,axiom,
! [B: a,B4: multiset_a,C2: a,C3: multiset_a] :
( ( ( add_mset_a @ B @ B4 )
= ( add_mset_a @ C2 @ C3 ) )
=> ( ( B != C2 )
=> ( member_a @ C2 @ ( set_mset_a @ B4 ) ) ) ) ).
% insert_noteq_member
thf(fact_312_insert__noteq__member,axiom,
! [B: product_prod_a_a,B4: multis599418605od_a_a,C2: product_prod_a_a,C3: multis599418605od_a_a] :
( ( ( add_ms1725977093od_a_a @ B @ B4 )
= ( add_ms1725977093od_a_a @ C2 @ C3 ) )
=> ( ( B != C2 )
=> ( member449909584od_a_a @ C2 @ ( set_ms1818337124od_a_a @ B4 ) ) ) ) ).
% insert_noteq_member
thf(fact_313_union__single__eq__member,axiom,
! [X: a,M: multiset_a,N: multiset_a] :
( ( ( add_mset_a @ X @ M )
= N )
=> ( member_a @ X @ ( set_mset_a @ N ) ) ) ).
% union_single_eq_member
thf(fact_314_union__single__eq__member,axiom,
! [X: product_prod_a_a,M: multis599418605od_a_a,N: multis599418605od_a_a] :
( ( ( add_ms1725977093od_a_a @ X @ M )
= N )
=> ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ N ) ) ) ).
% union_single_eq_member
thf(fact_315_union__iff,axiom,
! [A: a,A5: multiset_a,B4: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ ( plus_plus_multiset_a @ A5 @ B4 ) ) )
= ( ( member_a @ A @ ( set_mset_a @ A5 ) )
| ( member_a @ A @ ( set_mset_a @ B4 ) ) ) ) ).
% union_iff
thf(fact_316_union__iff,axiom,
! [A: product_prod_a_a,A5: multis599418605od_a_a,B4: multis599418605od_a_a] :
( ( member449909584od_a_a @ A @ ( set_ms1818337124od_a_a @ ( plus_p404985124od_a_a @ A5 @ B4 ) ) )
= ( ( member449909584od_a_a @ A @ ( set_ms1818337124od_a_a @ A5 ) )
| ( member449909584od_a_a @ A @ ( set_ms1818337124od_a_a @ B4 ) ) ) ) ).
% union_iff
thf(fact_317_mult__implies__one__step,axiom,
! [R2: set_Product_prod_a_a,M: multiset_a,N: multiset_a] :
( ( trans_a @ R2 )
=> ( ( member340150864iset_a @ ( produc2037245207iset_a @ M @ N ) @ ( mult_a @ R2 ) )
=> ? [I: multiset_a,J: multiset_a] :
( ( N
= ( plus_plus_multiset_a @ I @ J ) )
& ? [K: multiset_a] :
( ( M
= ( plus_plus_multiset_a @ I @ K ) )
& ( J != zero_zero_multiset_a )
& ! [X2: a] :
( ( member_a @ X2 @ ( set_mset_a @ K ) )
=> ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_mset_a @ J ) )
& ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Xa2 ) @ R2 ) ) ) ) ) ) ) ).
% mult_implies_one_step
thf(fact_318_one__step__implies__mult,axiom,
! [J2: multiset_a,K2: multiset_a,R2: set_Product_prod_a_a,I2: multiset_a] :
( ( J2 != zero_zero_multiset_a )
=> ( ! [X3: a] :
( ( member_a @ X3 @ ( set_mset_a @ K2 ) )
=> ? [Xa: a] :
( ( member_a @ Xa @ ( set_mset_a @ J2 ) )
& ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Xa ) @ R2 ) ) )
=> ( member340150864iset_a @ ( produc2037245207iset_a @ ( plus_plus_multiset_a @ I2 @ K2 ) @ ( plus_plus_multiset_a @ I2 @ J2 ) ) @ ( mult_a @ R2 ) ) ) ) ).
% one_step_implies_mult
thf(fact_319_multiset__nonemptyE,axiom,
! [A5: multiset_a] :
( ( A5 != zero_zero_multiset_a )
=> ~ ! [X3: a] :
~ ( member_a @ X3 @ ( set_mset_a @ A5 ) ) ) ).
% multiset_nonemptyE
thf(fact_320_multiset__nonemptyE,axiom,
! [A5: multis599418605od_a_a] :
( ( A5 != zero_z160849828od_a_a )
=> ~ ! [X3: product_prod_a_a] :
~ ( member449909584od_a_a @ X3 @ ( set_ms1818337124od_a_a @ A5 ) ) ) ).
% multiset_nonemptyE
thf(fact_321_multi__member__last,axiom,
! [X: a] : ( member_a @ X @ ( set_mset_a @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) ) ).
% multi_member_last
thf(fact_322_multi__member__last,axiom,
! [X: product_prod_a_a] : ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ ( add_ms1725977093od_a_a @ X @ zero_z160849828od_a_a ) ) ) ).
% multi_member_last
thf(fact_323_multi__member__skip,axiom,
! [X: a,XS: multiset_a,Y: a] :
( ( member_a @ X @ ( set_mset_a @ XS ) )
=> ( member_a @ X @ ( set_mset_a @ ( plus_plus_multiset_a @ ( add_mset_a @ Y @ zero_zero_multiset_a ) @ XS ) ) ) ) ).
% multi_member_skip
thf(fact_324_multi__member__skip,axiom,
! [X: product_prod_a_a,XS: multis599418605od_a_a,Y: product_prod_a_a] :
( ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ XS ) )
=> ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ ( plus_p404985124od_a_a @ ( add_ms1725977093od_a_a @ Y @ zero_z160849828od_a_a ) @ XS ) ) ) ) ).
% multi_member_skip
thf(fact_325_multi__member__this,axiom,
! [X: a,XS: multiset_a] : ( member_a @ X @ ( set_mset_a @ ( plus_plus_multiset_a @ ( add_mset_a @ X @ zero_zero_multiset_a ) @ XS ) ) ) ).
% multi_member_this
thf(fact_326_multi__member__this,axiom,
! [X: product_prod_a_a,XS: multis599418605od_a_a] : ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ ( plus_p404985124od_a_a @ ( add_ms1725977093od_a_a @ X @ zero_z160849828od_a_a ) @ XS ) ) ) ).
% multi_member_this
thf(fact_327_in__mset__fold__plus__iff,axiom,
! [X: a,M: multiset_a,NN: multiset_multiset_a] :
( ( member_a @ X @ ( set_mset_a @ ( fold_m382157835iset_a @ plus_plus_multiset_a @ M @ NN ) ) )
= ( ( member_a @ X @ ( set_mset_a @ M ) )
| ? [N2: multiset_a] :
( ( member_multiset_a @ N2 @ ( set_mset_multiset_a @ NN ) )
& ( member_a @ X @ ( set_mset_a @ N2 ) ) ) ) ) ).
% in_mset_fold_plus_iff
thf(fact_328_in__mset__fold__plus__iff,axiom,
! [X: product_prod_a_a,M: multis599418605od_a_a,NN: multis1735859763od_a_a] :
( ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ ( fold_m773816523od_a_a @ plus_p404985124od_a_a @ M @ NN ) ) )
= ( ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ M ) )
| ? [N2: multis599418605od_a_a] :
( ( member1755255446od_a_a @ N2 @ ( set_ms1869569194od_a_a @ NN ) )
& ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ N2 ) ) ) ) ) ).
% in_mset_fold_plus_iff
thf(fact_329_insert__DiffM2,axiom,
! [X: a,M: multiset_a] :
( ( member_a @ X @ ( set_mset_a @ M ) )
=> ( ( plus_plus_multiset_a @ ( minus_1649712171iset_a @ M @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) @ ( add_mset_a @ X @ zero_zero_multiset_a ) )
= M ) ) ).
% insert_DiffM2
thf(fact_330_insert__DiffM2,axiom,
! [X: product_prod_a_a,M: multis599418605od_a_a] :
( ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ M ) )
=> ( ( plus_p404985124od_a_a @ ( minus_1257885652od_a_a @ M @ ( add_ms1725977093od_a_a @ X @ zero_z160849828od_a_a ) ) @ ( add_ms1725977093od_a_a @ X @ zero_z160849828od_a_a ) )
= M ) ) ).
% insert_DiffM2
thf(fact_331_diff__add__mset__swap,axiom,
! [B: a,A5: multiset_a,M: multiset_a] :
( ~ ( member_a @ B @ ( set_mset_a @ A5 ) )
=> ( ( minus_1649712171iset_a @ ( add_mset_a @ B @ M ) @ A5 )
= ( add_mset_a @ B @ ( minus_1649712171iset_a @ M @ A5 ) ) ) ) ).
% diff_add_mset_swap
thf(fact_332_diff__add__mset__swap,axiom,
! [B: product_prod_a_a,A5: multis599418605od_a_a,M: multis599418605od_a_a] :
( ~ ( member449909584od_a_a @ B @ ( set_ms1818337124od_a_a @ A5 ) )
=> ( ( minus_1257885652od_a_a @ ( add_ms1725977093od_a_a @ B @ M ) @ A5 )
= ( add_ms1725977093od_a_a @ B @ ( minus_1257885652od_a_a @ M @ A5 ) ) ) ) ).
% diff_add_mset_swap
thf(fact_333_insert__DiffM,axiom,
! [X: a,M: multiset_a] :
( ( member_a @ X @ ( set_mset_a @ M ) )
=> ( ( add_mset_a @ X @ ( minus_1649712171iset_a @ M @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) )
= M ) ) ).
% insert_DiffM
thf(fact_334_insert__DiffM,axiom,
! [X: product_prod_a_a,M: multis599418605od_a_a] :
( ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ M ) )
=> ( ( add_ms1725977093od_a_a @ X @ ( minus_1257885652od_a_a @ M @ ( add_ms1725977093od_a_a @ X @ zero_z160849828od_a_a ) ) )
= M ) ) ).
% insert_DiffM
thf(fact_335_diff__union__swap2,axiom,
! [Y: a,M: multiset_a,X: a] :
( ( member_a @ Y @ ( set_mset_a @ M ) )
=> ( ( minus_1649712171iset_a @ ( add_mset_a @ X @ M ) @ ( add_mset_a @ Y @ zero_zero_multiset_a ) )
= ( add_mset_a @ X @ ( minus_1649712171iset_a @ M @ ( add_mset_a @ Y @ zero_zero_multiset_a ) ) ) ) ) ).
% diff_union_swap2
thf(fact_336_diff__union__swap2,axiom,
! [Y: product_prod_a_a,M: multis599418605od_a_a,X: product_prod_a_a] :
( ( member449909584od_a_a @ Y @ ( set_ms1818337124od_a_a @ M ) )
=> ( ( minus_1257885652od_a_a @ ( add_ms1725977093od_a_a @ X @ M ) @ ( add_ms1725977093od_a_a @ Y @ zero_z160849828od_a_a ) )
= ( add_ms1725977093od_a_a @ X @ ( minus_1257885652od_a_a @ M @ ( add_ms1725977093od_a_a @ Y @ zero_z160849828od_a_a ) ) ) ) ) ).
% diff_union_swap2
thf(fact_337_in__diffD,axiom,
! [A: a,M: multiset_a,N: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ ( minus_1649712171iset_a @ M @ N ) ) )
=> ( member_a @ A @ ( set_mset_a @ M ) ) ) ).
% in_diffD
thf(fact_338_in__diffD,axiom,
! [A: product_prod_a_a,M: multis599418605od_a_a,N: multis599418605od_a_a] :
( ( member449909584od_a_a @ A @ ( set_ms1818337124od_a_a @ ( minus_1257885652od_a_a @ M @ N ) ) )
=> ( member449909584od_a_a @ A @ ( set_ms1818337124od_a_a @ M ) ) ) ).
% in_diffD
thf(fact_339_diff__single__trivial,axiom,
! [X: a,M: multiset_a] :
( ~ ( member_a @ X @ ( set_mset_a @ M ) )
=> ( ( minus_1649712171iset_a @ M @ ( add_mset_a @ X @ zero_zero_multiset_a ) )
= M ) ) ).
% diff_single_trivial
thf(fact_340_diff__single__trivial,axiom,
! [X: product_prod_a_a,M: multis599418605od_a_a] :
( ~ ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ M ) )
=> ( ( minus_1257885652od_a_a @ M @ ( add_ms1725977093od_a_a @ X @ zero_z160849828od_a_a ) )
= M ) ) ).
% diff_single_trivial
thf(fact_341_diff__single__eq__union,axiom,
! [X: a,M: multiset_a,N: multiset_a] :
( ( member_a @ X @ ( set_mset_a @ M ) )
=> ( ( ( minus_1649712171iset_a @ M @ ( add_mset_a @ X @ zero_zero_multiset_a ) )
= N )
= ( M
= ( add_mset_a @ X @ N ) ) ) ) ).
% diff_single_eq_union
thf(fact_342_diff__single__eq__union,axiom,
! [X: product_prod_a_a,M: multis599418605od_a_a,N: multis599418605od_a_a] :
( ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ M ) )
=> ( ( ( minus_1257885652od_a_a @ M @ ( add_ms1725977093od_a_a @ X @ zero_z160849828od_a_a ) )
= N )
= ( M
= ( add_ms1725977093od_a_a @ X @ N ) ) ) ) ).
% diff_single_eq_union
thf(fact_343_multi__drop__mem__not__eq,axiom,
! [C2: a,B4: multiset_a] :
( ( member_a @ C2 @ ( set_mset_a @ B4 ) )
=> ( ( minus_1649712171iset_a @ B4 @ ( add_mset_a @ C2 @ zero_zero_multiset_a ) )
!= B4 ) ) ).
% multi_drop_mem_not_eq
thf(fact_344_multi__drop__mem__not__eq,axiom,
! [C2: product_prod_a_a,B4: multis599418605od_a_a] :
( ( member449909584od_a_a @ C2 @ ( set_ms1818337124od_a_a @ B4 ) )
=> ( ( minus_1257885652od_a_a @ B4 @ ( add_ms1725977093od_a_a @ C2 @ zero_z160849828od_a_a ) )
!= B4 ) ) ).
% multi_drop_mem_not_eq
thf(fact_345_add__mset__remove__trivial__If,axiom,
! [A: a,N: multiset_a] :
( ( ( member_a @ A @ ( set_mset_a @ N ) )
=> ( ( add_mset_a @ A @ ( minus_1649712171iset_a @ N @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) )
= N ) )
& ( ~ ( member_a @ A @ ( set_mset_a @ N ) )
=> ( ( add_mset_a @ A @ ( minus_1649712171iset_a @ N @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) )
= ( add_mset_a @ A @ N ) ) ) ) ).
% add_mset_remove_trivial_If
thf(fact_346_add__mset__remove__trivial__If,axiom,
! [A: product_prod_a_a,N: multis599418605od_a_a] :
( ( ( member449909584od_a_a @ A @ ( set_ms1818337124od_a_a @ N ) )
=> ( ( add_ms1725977093od_a_a @ A @ ( minus_1257885652od_a_a @ N @ ( add_ms1725977093od_a_a @ A @ zero_z160849828od_a_a ) ) )
= N ) )
& ( ~ ( member449909584od_a_a @ A @ ( set_ms1818337124od_a_a @ N ) )
=> ( ( add_ms1725977093od_a_a @ A @ ( minus_1257885652od_a_a @ N @ ( add_ms1725977093od_a_a @ A @ zero_z160849828od_a_a ) ) )
= ( add_ms1725977093od_a_a @ A @ N ) ) ) ) ).
% add_mset_remove_trivial_If
thf(fact_347_add__mset__remove__trivial__eq,axiom,
! [N: multiset_a,A: a] :
( ( N
= ( add_mset_a @ A @ ( minus_1649712171iset_a @ N @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) )
= ( member_a @ A @ ( set_mset_a @ N ) ) ) ).
% add_mset_remove_trivial_eq
thf(fact_348_add__mset__remove__trivial__eq,axiom,
! [N: multis599418605od_a_a,A: product_prod_a_a] :
( ( N
= ( add_ms1725977093od_a_a @ A @ ( minus_1257885652od_a_a @ N @ ( add_ms1725977093od_a_a @ A @ zero_z160849828od_a_a ) ) ) )
= ( member449909584od_a_a @ A @ ( set_ms1818337124od_a_a @ N ) ) ) ).
% add_mset_remove_trivial_eq
thf(fact_349_multiset__add__sub__el__shuffle,axiom,
! [C2: a,B4: multiset_a,B: a] :
( ( member_a @ C2 @ ( set_mset_a @ B4 ) )
=> ( ( B != C2 )
=> ( ( add_mset_a @ B @ ( minus_1649712171iset_a @ B4 @ ( add_mset_a @ C2 @ zero_zero_multiset_a ) ) )
= ( minus_1649712171iset_a @ ( add_mset_a @ B @ B4 ) @ ( add_mset_a @ C2 @ zero_zero_multiset_a ) ) ) ) ) ).
% multiset_add_sub_el_shuffle
thf(fact_350_multiset__add__sub__el__shuffle,axiom,
! [C2: product_prod_a_a,B4: multis599418605od_a_a,B: product_prod_a_a] :
( ( member449909584od_a_a @ C2 @ ( set_ms1818337124od_a_a @ B4 ) )
=> ( ( B != C2 )
=> ( ( add_ms1725977093od_a_a @ B @ ( minus_1257885652od_a_a @ B4 @ ( add_ms1725977093od_a_a @ C2 @ zero_z160849828od_a_a ) ) )
= ( minus_1257885652od_a_a @ ( add_ms1725977093od_a_a @ B @ B4 ) @ ( add_ms1725977093od_a_a @ C2 @ zero_z160849828od_a_a ) ) ) ) ) ).
% multiset_add_sub_el_shuffle
thf(fact_351_more__than__one__mset__mset__diff,axiom,
! [A: a,M: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ ( minus_1649712171iset_a @ M @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) )
=> ( ( set_mset_a @ ( minus_1649712171iset_a @ M @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) )
= ( set_mset_a @ M ) ) ) ).
% more_than_one_mset_mset_diff
thf(fact_352_more__than__one__mset__mset__diff,axiom,
! [A: product_prod_a_a,M: multis599418605od_a_a] :
( ( member449909584od_a_a @ A @ ( set_ms1818337124od_a_a @ ( minus_1257885652od_a_a @ M @ ( add_ms1725977093od_a_a @ A @ zero_z160849828od_a_a ) ) ) )
=> ( ( set_ms1818337124od_a_a @ ( minus_1257885652od_a_a @ M @ ( add_ms1725977093od_a_a @ A @ zero_z160849828od_a_a ) ) )
= ( set_ms1818337124od_a_a @ M ) ) ) ).
% more_than_one_mset_mset_diff
thf(fact_353_diff__union__single__conv,axiom,
! [A: a,J2: multiset_a,I2: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ J2 ) )
=> ( ( minus_1649712171iset_a @ ( plus_plus_multiset_a @ I2 @ J2 ) @ ( add_mset_a @ A @ zero_zero_multiset_a ) )
= ( plus_plus_multiset_a @ I2 @ ( minus_1649712171iset_a @ J2 @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) ) ) ).
% diff_union_single_conv
thf(fact_354_diff__union__single__conv,axiom,
! [A: product_prod_a_a,J2: multis599418605od_a_a,I2: multis599418605od_a_a] :
( ( member449909584od_a_a @ A @ ( set_ms1818337124od_a_a @ J2 ) )
=> ( ( minus_1257885652od_a_a @ ( plus_p404985124od_a_a @ I2 @ J2 ) @ ( add_ms1725977093od_a_a @ A @ zero_z160849828od_a_a ) )
= ( plus_p404985124od_a_a @ I2 @ ( minus_1257885652od_a_a @ J2 @ ( add_ms1725977093od_a_a @ A @ zero_z160849828od_a_a ) ) ) ) ) ).
% diff_union_single_conv
% Conjectures (4)
thf(conj_0,hypothesis,
member_a @ x @ carrier ).
thf(conj_1,hypothesis,
member_a @ y @ carrier ).
thf(conj_2,hypothesis,
x != y ).
thf(conj_3,conjecture,
( ( member449909584od_a_a @ ( product_Pair_a_a @ x @ y ) @ relation )
| ( member449909584od_a_a @ ( product_Pair_a_a @ y @ x ) @ relation ) ) ).
%------------------------------------------------------------------------------