TPTP Problem File: SLH0246^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Multiset_Ordering_NPC/0001_Multiset_Ordering_More/prob_00231_010452__13303868_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1548 ( 714 unt; 271 typ; 0 def)
% Number of atoms : 2914 (1748 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 10708 ( 332 ~; 42 |; 208 &;9267 @)
% ( 0 <=>; 859 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 5 avg)
% Number of types : 37 ( 36 usr)
% Number of type conns : 595 ( 595 >; 0 *; 0 +; 0 <<)
% Number of symbols : 236 ( 235 usr; 24 con; 0-4 aty)
% Number of variables : 3357 ( 87 ^;3183 !; 87 ?;3357 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:28:52.162
%------------------------------------------------------------------------------
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thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
image_8071159009363422661od_a_a: ( produc6518373309651786023iset_a > product_prod_a_a ) > set_Pr79727621955416071iset_a > set_Product_prod_a_a ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_001tf__a,type,
image_830901103612899740et_a_a: ( produc6518373309651786023iset_a > a ) > set_Pr79727621955416071iset_a > set_a ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J,type,
image_9107824508038213061iset_a: ( product_prod_a_a > produc6518373309651786023iset_a ) > set_Product_prod_a_a > set_Pr79727621955416071iset_a ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
image_4636654165204879301od_a_a: ( product_prod_a_a > product_prod_a_a ) > set_Product_prod_a_a > set_Product_prod_a_a ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001tf__a,type,
image_3437945252899457948_a_a_a: ( product_prod_a_a > a ) > set_Product_prod_a_a > set_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J,type,
image_6641517091992857518iset_a: ( a > produc6518373309651786023iset_a ) > set_a > set_Pr79727621955416071iset_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
image_7400625782589995694od_a_a: ( a > product_prod_a_a ) > set_a > set_Product_prod_a_a ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set_Oinsert_001t__Multiset__Omultiset_Itf__a_J,type,
insert_multiset_a: multiset_a > set_multiset_a > set_multiset_a ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J,type,
insert4880209295958132471iset_a: produc6518373309651786023iset_a > set_Pr79727621955416071iset_a > set_Pr79727621955416071iset_a ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
insert4534936382041156343od_a_a: product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_member_001t__Multiset__Omultiset_It__Multiset__Omultiset_Itf__a_J_J,type,
member7618379257985549619iset_a: multiset_multiset_a > set_mu2969860852453053802iset_a > $o ).
thf(sy_c_member_001t__Multiset__Omultiset_It__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_J,type,
member2597461214246177430iset_a: multis4781622204924204013iset_a > set_mu7670847959134527309iset_a > $o ).
thf(sy_c_member_001t__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
member7700929758516679318od_a_a: multis501812127501805293od_a_a > set_mu7436117841396106061od_a_a > $o ).
thf(sy_c_member_001t__Multiset__Omultiset_Itf__a_J,type,
member_multiset_a: multiset_a > set_multiset_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Multiset__Omultiset_It__Multiset__Omultiset_It__Multiset__Omultiset_Itf__a_J_J_J_Mt__Multiset__Omultiset_It__Multiset__Omultiset_It__Multiset__Omultiset_Itf__a_J_J_J_J,type,
member7045735419414020688iset_a: produc477645357109558055iset_a > set_Pr8147813163840139271iset_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Multiset__Omultiset_It__Multiset__Omultiset_It__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_J_J_Mt__Multiset__Omultiset_It__Multiset__Omultiset_It__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_J_J_J,type,
member4376497788722562064iset_a: produc4419857851285853159iset_a > set_Pr8047240249530146247iset_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Multiset__Omultiset_It__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_Mt__Multiset__Omultiset_It__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_J,type,
member231996757139097616od_a_a: produc7223976429359476199od_a_a > set_Pr8876319265091017159od_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Multiset__Omultiset_It__Multiset__Omultiset_Itf__a_J_J_Mt__Multiset__Omultiset_It__Multiset__Omultiset_Itf__a_J_J_J,type,
member1345422370561994576iset_a: produc4424810744984089383iset_a > set_Pr1903681468693068039iset_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Multiset__Omultiset_It__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_J_Mt__Multiset__Omultiset_It__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_J_J,type,
member8574442277963079824iset_a: produc2869653011250641767iset_a > set_Pr6964934382344443463iset_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
member1567053331504362128od_a_a: produc5665077874975597415od_a_a > set_Pr3746329090258575431od_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J,type,
member5199237121806060112iset_a: produc6518373309651786023iset_a > set_Pr79727621955416071iset_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_Mt__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_J,type,
member1562461016398818064iset_a: produc8554440688619196647iset_a > set_Pr3059988933816007879iset_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,
member6330455413206600464od_a_a: produc3498347346309940967od_a_a > set_Pr8600417178894128327od_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
member1426531477525435216od_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_E____,type,
e: multiset_a ).
thf(sy_v_I____,type,
i: multiset_a ).
thf(sy_v_J____,type,
j: multiset_a ).
thf(sy_v_K____,type,
k: multiset_a ).
thf(sy_v_M,type,
m: multiset_a ).
thf(sy_v_N,type,
n: multiset_a ).
thf(sy_v_S,type,
s: set_Product_prod_a_a ).
% Relevant facts (1276)
thf(fact_0_JK_I2_J,axiom,
( k
= ( minus_3765977307040488491iset_a @ n @ m ) ) ).
% JK(2)
thf(fact_1_JK_I1_J,axiom,
( j
= ( minus_3765977307040488491iset_a @ m @ n ) ) ).
% JK(1)
thf(fact_2__092_060open_062_IN_A_N_AM_M_AM_A_N_AN_J_A_092_060in_062_Amult_AS_092_060close_062,axiom,
member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( minus_3765977307040488491iset_a @ n @ m ) @ ( minus_3765977307040488491iset_a @ m @ n ) ) @ ( mult_a @ s ) ).
% \<open>(N - M, M - N) \<in> mult S\<close>
thf(fact_3_prod_Oinject,axiom,
! [X1: multiset_multiset_a,X2: multiset_multiset_a,Y1: multiset_multiset_a,Y2: multiset_multiset_a] :
( ( ( produc8381460346935720215iset_a @ X1 @ X2 )
= ( produc8381460346935720215iset_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_4_prod_Oinject,axiom,
! [X1: product_prod_a_a,X2: product_prod_a_a,Y1: product_prod_a_a,Y2: product_prod_a_a] :
( ( ( produc7886510207707329367od_a_a @ X1 @ X2 )
= ( produc7886510207707329367od_a_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_5_prod_Oinject,axiom,
! [X1: multis501812127501805293od_a_a,X2: multis501812127501805293od_a_a,Y1: multis501812127501805293od_a_a,Y2: multis501812127501805293od_a_a] :
( ( ( produc6961845650964105687od_a_a @ X1 @ X2 )
= ( produc6961845650964105687od_a_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_6_prod_Oinject,axiom,
! [X1: produc6518373309651786023iset_a,X2: produc6518373309651786023iset_a,Y1: produc6518373309651786023iset_a,Y2: produc6518373309651786023iset_a] :
( ( ( produc2591098542590733143iset_a @ X1 @ X2 )
= ( produc2591098542590733143iset_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_7_prod_Oinject,axiom,
! [X1: multis4781622204924204013iset_a,X2: multis4781622204924204013iset_a,Y1: multis4781622204924204013iset_a,Y2: multis4781622204924204013iset_a] :
( ( ( produc1739639915645390807iset_a @ X1 @ X2 )
= ( produc1739639915645390807iset_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_8_prod_Oinject,axiom,
! [X1: a,X2: a,Y1: a,Y2: a] :
( ( ( product_Pair_a_a @ X1 @ X2 )
= ( product_Pair_a_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_9_prod_Oinject,axiom,
! [X1: multiset_a,X2: multiset_a,Y1: multiset_a,Y2: multiset_a] :
( ( ( produc654756711066625303iset_a @ X1 @ X2 )
= ( produc654756711066625303iset_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_10_old_Oprod_Oinject,axiom,
! [A: multiset_multiset_a,B: multiset_multiset_a,A2: multiset_multiset_a,B2: multiset_multiset_a] :
( ( ( produc8381460346935720215iset_a @ A @ B )
= ( produc8381460346935720215iset_a @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_11_old_Oprod_Oinject,axiom,
! [A: product_prod_a_a,B: product_prod_a_a,A2: product_prod_a_a,B2: product_prod_a_a] :
( ( ( produc7886510207707329367od_a_a @ A @ B )
= ( produc7886510207707329367od_a_a @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_12_old_Oprod_Oinject,axiom,
! [A: multis501812127501805293od_a_a,B: multis501812127501805293od_a_a,A2: multis501812127501805293od_a_a,B2: multis501812127501805293od_a_a] :
( ( ( produc6961845650964105687od_a_a @ A @ B )
= ( produc6961845650964105687od_a_a @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_13_old_Oprod_Oinject,axiom,
! [A: produc6518373309651786023iset_a,B: produc6518373309651786023iset_a,A2: produc6518373309651786023iset_a,B2: produc6518373309651786023iset_a] :
( ( ( produc2591098542590733143iset_a @ A @ B )
= ( produc2591098542590733143iset_a @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_14_old_Oprod_Oinject,axiom,
! [A: multis4781622204924204013iset_a,B: multis4781622204924204013iset_a,A2: multis4781622204924204013iset_a,B2: multis4781622204924204013iset_a] :
( ( ( produc1739639915645390807iset_a @ A @ B )
= ( produc1739639915645390807iset_a @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_15_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_16_old_Oprod_Oinject,axiom,
! [A: multiset_a,B: multiset_a,A2: multiset_a,B2: multiset_a] :
( ( ( produc654756711066625303iset_a @ A @ B )
= ( produc654756711066625303iset_a @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_17__092_060open_062_092_060exists_062I_AJ_AK_O_AM_A_N_AN_A_061_AI_A_L_AJ_A_092_060and_062_AN_A_N_AM_A_061_AI_A_L_AK_A_092_060and_062_AJ_A_092_060noteq_062_A_123_D_125_A_092_060and_062_A_I_092_060forall_062k_092_060in_062_DK_O_A_092_060exists_062j_092_060in_062_DJ_O_A_Ik_M_Aj_J_A_092_060in_062_AS_J_092_060close_062,axiom,
? [I: multiset_a,J: multiset_a] :
( ( ( minus_3765977307040488491iset_a @ m @ n )
= ( plus_plus_multiset_a @ I @ J ) )
& ? [K: multiset_a] :
( ( ( minus_3765977307040488491iset_a @ n @ m )
= ( plus_plus_multiset_a @ I @ K ) )
& ( J != zero_zero_multiset_a )
& ! [X: a] :
( ( member_a @ X @ ( set_mset_a @ K ) )
=> ? [Xa: a] :
( ( member_a @ Xa @ ( set_mset_a @ J ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Xa ) @ s ) ) ) ) ) ).
% \<open>\<exists>I J K. M - N = I + J \<and> N - M = I + K \<and> J \<noteq> {#} \<and> (\<forall>k\<in>#K. \<exists>j\<in>#J. (k, j) \<in> S)\<close>
thf(fact_18__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062E_AJ_AK_O_A_092_060lbrakk_062M_A_N_AN_A_061_AE_A_L_AJ_A_092_060and_062_AN_A_N_AM_A_061_AE_A_L_AK_059_AJ_A_092_060noteq_062_A_123_D_125_A_092_060and_062_A_I_092_060forall_062k_092_060in_062_DK_O_A_092_060exists_062j_092_060in_062_DJ_O_A_Ik_M_Aj_J_A_092_060in_062_AS_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [E: multiset_a,J: multiset_a,K: multiset_a] :
( ( ( ( minus_3765977307040488491iset_a @ m @ n )
= ( plus_plus_multiset_a @ E @ J ) )
& ( ( minus_3765977307040488491iset_a @ n @ m )
= ( plus_plus_multiset_a @ E @ K ) ) )
=> ~ ( ( J != zero_zero_multiset_a )
& ! [X: a] :
( ( member_a @ X @ ( set_mset_a @ K ) )
=> ? [Xa: a] :
( ( member_a @ Xa @ ( set_mset_a @ J ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Xa ) @ s ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>E J K. \<lbrakk>M - N = E + J \<and> N - M = E + K; J \<noteq> {#} \<and> (\<forall>k\<in>#K. \<exists>j\<in>#J. (k, j) \<in> S)\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_19_in__diffD,axiom,
! [A: produc5665077874975597415od_a_a,M: multis3377851686571972653od_a_a,N: multis3377851686571972653od_a_a] :
( ( member1567053331504362128od_a_a @ A @ ( set_ms5376707576967697060od_a_a @ ( minus_6383096217462597652od_a_a @ M @ N ) ) )
=> ( member1567053331504362128od_a_a @ A @ ( set_ms5376707576967697060od_a_a @ M ) ) ) ).
% in_diffD
thf(fact_20_in__diffD,axiom,
! [A: produc4424810744984089383iset_a,M: multis1386120040344743661iset_a,N: multis1386120040344743661iset_a] :
( ( member1345422370561994576iset_a @ A @ ( set_ms2882773486807528292iset_a @ ( minus_2536987971823575508iset_a @ M @ N ) ) )
=> ( member1345422370561994576iset_a @ A @ ( set_ms2882773486807528292iset_a @ M ) ) ) ).
% in_diffD
thf(fact_21_in__diffD,axiom,
! [A: multiset_a,M: multiset_multiset_a,N: multiset_multiset_a] :
( ( member_multiset_a @ A @ ( set_mset_multiset_a @ ( minus_3395427628221709681iset_a @ M @ N ) ) )
=> ( member_multiset_a @ A @ ( set_mset_multiset_a @ M ) ) ) ).
% in_diffD
thf(fact_22_in__diffD,axiom,
! [A: multis501812127501805293od_a_a,M: multis5935151272263360051od_a_a,N: multis5935151272263360051od_a_a] :
( ( member7700929758516679318od_a_a @ A @ ( set_ms8253844250382261418od_a_a @ ( minus_385666609031338394od_a_a @ M @ N ) ) )
=> ( member7700929758516679318od_a_a @ A @ ( set_ms8253844250382261418od_a_a @ M ) ) ) ).
% in_diffD
thf(fact_23_in__diffD,axiom,
! [A: multis4781622204924204013iset_a,M: multis7221419791219231795iset_a,N: multis7221419791219231795iset_a] :
( ( member2597461214246177430iset_a @ A @ ( set_ms5327031259215843498iset_a @ ( minus_4490418833869548698iset_a @ M @ N ) ) )
=> ( member2597461214246177430iset_a @ A @ ( set_ms5327031259215843498iset_a @ M ) ) ) ).
% in_diffD
thf(fact_24_in__diffD,axiom,
! [A: product_prod_a_a,M: multis501812127501805293od_a_a,N: multis501812127501805293od_a_a] :
( ( member1426531477525435216od_a_a @ A @ ( set_ms119794563918438244od_a_a @ ( minus_7313152569058643924od_a_a @ M @ N ) ) )
=> ( member1426531477525435216od_a_a @ A @ ( set_ms119794563918438244od_a_a @ M ) ) ) ).
% in_diffD
thf(fact_25_in__diffD,axiom,
! [A: produc6518373309651786023iset_a,M: multis4781622204924204013iset_a,N: multis4781622204924204013iset_a] :
( ( member5199237121806060112iset_a @ A @ ( set_ms2379723421352415844iset_a @ ( minus_9211619285413683668iset_a @ M @ N ) ) )
=> ( member5199237121806060112iset_a @ A @ ( set_ms2379723421352415844iset_a @ M ) ) ) ).
% in_diffD
thf(fact_26_in__diffD,axiom,
! [A: a,M: multiset_a,N: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ M @ N ) ) )
=> ( member_a @ A @ ( set_mset_a @ M ) ) ) ).
% in_diffD
thf(fact_27_rel,axiom,
( ( j != zero_zero_multiset_a )
& ! [X: a] :
( ( member_a @ X @ ( set_mset_a @ k ) )
=> ? [Xa: a] :
( ( member_a @ Xa @ ( set_mset_a @ j ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Xa ) @ s ) ) ) ) ).
% rel
thf(fact_28_calculation,axiom,
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ n @ m ) @ ( mult_a @ s ) )
= ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( minus_3765977307040488491iset_a @ n @ m ) @ ( minus_3765977307040488491iset_a @ m @ n ) ) @ ( mult_a @ s ) ) ) ).
% calculation
thf(fact_29_M,axiom,
( m
= ( plus_plus_multiset_a @ ( minus_3765977307040488491iset_a @ m @ n ) @ i ) ) ).
% M
thf(fact_30_N,axiom,
( n
= ( plus_plus_multiset_a @ ( minus_3765977307040488491iset_a @ n @ m ) @ i ) ) ).
% N
thf(fact_31__C_K_C,axiom,
( ( ( minus_3765977307040488491iset_a @ m @ n )
= ( plus_plus_multiset_a @ e @ j ) )
& ( ( minus_3765977307040488491iset_a @ n @ m )
= ( plus_plus_multiset_a @ e @ k ) ) ) ).
% "*"
thf(fact_32__092_060open_062E_A_061_A_123_D_125_092_060close_062,axiom,
e = zero_zero_multiset_a ).
% \<open>E = {#}\<close>
thf(fact_33_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
! [X3: multiset_multiset_a,Y: multiset_multiset_a] :
( ( ( plus_p6738641960240532161iset_a @ X3 @ Y )
= zero_z7799948378220188993iset_a )
= ( ( X3 = zero_z7799948378220188993iset_a )
& ( Y = zero_z7799948378220188993iset_a ) ) ) ).
% subset_mset.add_eq_0_iff_both_eq_0
thf(fact_34_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
! [X3: multis501812127501805293od_a_a,Y: multis501812127501805293od_a_a] :
( ( ( plus_p6967097483979887908od_a_a @ X3 @ Y )
= zero_z3024502163831676836od_a_a )
= ( ( X3 = zero_z3024502163831676836od_a_a )
& ( Y = zero_z3024502163831676836od_a_a ) ) ) ).
% subset_mset.add_eq_0_iff_both_eq_0
thf(fact_35_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
! [X3: multis4781622204924204013iset_a,Y: multis4781622204924204013iset_a] :
( ( ( plus_p1053901119268525860iset_a @ X3 @ Y )
= zero_z4591712717340762532iset_a )
= ( ( X3 = zero_z4591712717340762532iset_a )
& ( Y = zero_z4591712717340762532iset_a ) ) ) ).
% subset_mset.add_eq_0_iff_both_eq_0
thf(fact_36_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
! [X3: multiset_a,Y: multiset_a] :
( ( ( plus_plus_multiset_a @ X3 @ Y )
= zero_zero_multiset_a )
= ( ( X3 = zero_zero_multiset_a )
& ( Y = zero_zero_multiset_a ) ) ) ).
% subset_mset.add_eq_0_iff_both_eq_0
thf(fact_37_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
! [X3: multiset_multiset_a,Y: multiset_multiset_a] :
( ( zero_z7799948378220188993iset_a
= ( plus_p6738641960240532161iset_a @ X3 @ Y ) )
= ( ( X3 = zero_z7799948378220188993iset_a )
& ( Y = zero_z7799948378220188993iset_a ) ) ) ).
% subset_mset.zero_eq_add_iff_both_eq_0
thf(fact_38_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
! [X3: multis501812127501805293od_a_a,Y: multis501812127501805293od_a_a] :
( ( zero_z3024502163831676836od_a_a
= ( plus_p6967097483979887908od_a_a @ X3 @ Y ) )
= ( ( X3 = zero_z3024502163831676836od_a_a )
& ( Y = zero_z3024502163831676836od_a_a ) ) ) ).
% subset_mset.zero_eq_add_iff_both_eq_0
thf(fact_39_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
! [X3: multis4781622204924204013iset_a,Y: multis4781622204924204013iset_a] :
( ( zero_z4591712717340762532iset_a
= ( plus_p1053901119268525860iset_a @ X3 @ Y ) )
= ( ( X3 = zero_z4591712717340762532iset_a )
& ( Y = zero_z4591712717340762532iset_a ) ) ) ).
% subset_mset.zero_eq_add_iff_both_eq_0
thf(fact_40_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
! [X3: multiset_a,Y: multiset_a] :
( ( zero_zero_multiset_a
= ( plus_plus_multiset_a @ X3 @ Y ) )
= ( ( X3 = zero_zero_multiset_a )
& ( Y = zero_zero_multiset_a ) ) ) ).
% subset_mset.zero_eq_add_iff_both_eq_0
thf(fact_41_empty__eq__union,axiom,
! [M: multiset_multiset_a,N: multiset_multiset_a] :
( ( zero_z7799948378220188993iset_a
= ( plus_p6738641960240532161iset_a @ M @ N ) )
= ( ( M = zero_z7799948378220188993iset_a )
& ( N = zero_z7799948378220188993iset_a ) ) ) ).
% empty_eq_union
thf(fact_42_empty__eq__union,axiom,
! [M: multis501812127501805293od_a_a,N: multis501812127501805293od_a_a] :
( ( zero_z3024502163831676836od_a_a
= ( plus_p6967097483979887908od_a_a @ M @ N ) )
= ( ( M = zero_z3024502163831676836od_a_a )
& ( N = zero_z3024502163831676836od_a_a ) ) ) ).
% empty_eq_union
thf(fact_43_empty__eq__union,axiom,
! [M: multis4781622204924204013iset_a,N: multis4781622204924204013iset_a] :
( ( zero_z4591712717340762532iset_a
= ( plus_p1053901119268525860iset_a @ M @ N ) )
= ( ( M = zero_z4591712717340762532iset_a )
& ( N = zero_z4591712717340762532iset_a ) ) ) ).
% empty_eq_union
thf(fact_44_empty__eq__union,axiom,
! [M: multiset_a,N: multiset_a] :
( ( zero_zero_multiset_a
= ( plus_plus_multiset_a @ M @ N ) )
= ( ( M = zero_zero_multiset_a )
& ( N = zero_zero_multiset_a ) ) ) ).
% empty_eq_union
thf(fact_45_union__eq__empty,axiom,
! [M: multiset_multiset_a,N: multiset_multiset_a] :
( ( ( plus_p6738641960240532161iset_a @ M @ N )
= zero_z7799948378220188993iset_a )
= ( ( M = zero_z7799948378220188993iset_a )
& ( N = zero_z7799948378220188993iset_a ) ) ) ).
% union_eq_empty
thf(fact_46_union__eq__empty,axiom,
! [M: multis501812127501805293od_a_a,N: multis501812127501805293od_a_a] :
( ( ( plus_p6967097483979887908od_a_a @ M @ N )
= zero_z3024502163831676836od_a_a )
= ( ( M = zero_z3024502163831676836od_a_a )
& ( N = zero_z3024502163831676836od_a_a ) ) ) ).
% union_eq_empty
thf(fact_47_union__eq__empty,axiom,
! [M: multis4781622204924204013iset_a,N: multis4781622204924204013iset_a] :
( ( ( plus_p1053901119268525860iset_a @ M @ N )
= zero_z4591712717340762532iset_a )
= ( ( M = zero_z4591712717340762532iset_a )
& ( N = zero_z4591712717340762532iset_a ) ) ) ).
% union_eq_empty
thf(fact_48_union__eq__empty,axiom,
! [M: multiset_a,N: multiset_a] :
( ( ( plus_plus_multiset_a @ M @ N )
= zero_zero_multiset_a )
= ( ( M = zero_zero_multiset_a )
& ( N = zero_zero_multiset_a ) ) ) ).
% union_eq_empty
thf(fact_49_diff__diff__add__mset,axiom,
! [M: multiset_multiset_a,N: multiset_multiset_a,P: multiset_multiset_a] :
( ( minus_3395427628221709681iset_a @ ( minus_3395427628221709681iset_a @ M @ N ) @ P )
= ( minus_3395427628221709681iset_a @ M @ ( plus_p6738641960240532161iset_a @ N @ P ) ) ) ).
% diff_diff_add_mset
thf(fact_50_diff__diff__add__mset,axiom,
! [M: multiset_a,N: multiset_a,P: multiset_a] :
( ( minus_3765977307040488491iset_a @ ( minus_3765977307040488491iset_a @ M @ N ) @ P )
= ( minus_3765977307040488491iset_a @ M @ ( plus_plus_multiset_a @ N @ P ) ) ) ).
% diff_diff_add_mset
thf(fact_51_diff__diff__add__mset,axiom,
! [M: multis4781622204924204013iset_a,N: multis4781622204924204013iset_a,P: multis4781622204924204013iset_a] :
( ( minus_9211619285413683668iset_a @ ( minus_9211619285413683668iset_a @ M @ N ) @ P )
= ( minus_9211619285413683668iset_a @ M @ ( plus_p1053901119268525860iset_a @ N @ P ) ) ) ).
% diff_diff_add_mset
thf(fact_52_diff__diff__add__mset,axiom,
! [M: multis501812127501805293od_a_a,N: multis501812127501805293od_a_a,P: multis501812127501805293od_a_a] :
( ( minus_7313152569058643924od_a_a @ ( minus_7313152569058643924od_a_a @ M @ N ) @ P )
= ( minus_7313152569058643924od_a_a @ M @ ( plus_p6967097483979887908od_a_a @ N @ P ) ) ) ).
% diff_diff_add_mset
thf(fact_53__092_060open_062_I_IN_A_N_AM_A_L_AI_M_AM_A_N_AN_A_L_AI_J_A_092_060in_062_Amult_AS_J_A_061_A_I_IN_A_N_AM_M_AM_A_N_AN_J_A_092_060in_062_Amult_AS_J_092_060close_062,axiom,
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( plus_plus_multiset_a @ ( minus_3765977307040488491iset_a @ n @ m ) @ i ) @ ( plus_plus_multiset_a @ ( minus_3765977307040488491iset_a @ m @ n ) @ i ) ) @ ( mult_a @ s ) )
= ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( minus_3765977307040488491iset_a @ n @ m ) @ ( minus_3765977307040488491iset_a @ m @ n ) ) @ ( mult_a @ s ) ) ) ).
% \<open>((N - M + I, M - N + I) \<in> mult S) = ((N - M, M - N) \<in> mult S)\<close>
thf(fact_54__092_060open_062_I_IN_M_AM_J_A_092_060in_062_Amult_AS_J_A_061_A_I_IN_A_N_AM_A_L_AI_M_AM_A_N_AN_A_L_AI_J_A_092_060in_062_Amult_AS_J_092_060close_062,axiom,
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ n @ m ) @ ( mult_a @ s ) )
= ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( plus_plus_multiset_a @ ( minus_3765977307040488491iset_a @ n @ m ) @ i ) @ ( plus_plus_multiset_a @ ( minus_3765977307040488491iset_a @ m @ n ) @ i ) ) @ ( mult_a @ s ) ) ) ).
% \<open>((N, M) \<in> mult S) = ((N - M + I, M - N + I) \<in> mult S)\<close>
thf(fact_55_empty__neutral_I2_J,axiom,
! [X3: multiset_multiset_a] :
( ( plus_p6738641960240532161iset_a @ X3 @ zero_z7799948378220188993iset_a )
= X3 ) ).
% empty_neutral(2)
thf(fact_56_empty__neutral_I2_J,axiom,
! [X3: multis501812127501805293od_a_a] :
( ( plus_p6967097483979887908od_a_a @ X3 @ zero_z3024502163831676836od_a_a )
= X3 ) ).
% empty_neutral(2)
thf(fact_57_empty__neutral_I2_J,axiom,
! [X3: multis4781622204924204013iset_a] :
( ( plus_p1053901119268525860iset_a @ X3 @ zero_z4591712717340762532iset_a )
= X3 ) ).
% empty_neutral(2)
thf(fact_58_empty__neutral_I2_J,axiom,
! [X3: multiset_a] :
( ( plus_plus_multiset_a @ X3 @ zero_zero_multiset_a )
= X3 ) ).
% empty_neutral(2)
thf(fact_59_empty__neutral_I1_J,axiom,
! [X3: multiset_multiset_a] :
( ( plus_p6738641960240532161iset_a @ zero_z7799948378220188993iset_a @ X3 )
= X3 ) ).
% empty_neutral(1)
thf(fact_60_empty__neutral_I1_J,axiom,
! [X3: multis501812127501805293od_a_a] :
( ( plus_p6967097483979887908od_a_a @ zero_z3024502163831676836od_a_a @ X3 )
= X3 ) ).
% empty_neutral(1)
thf(fact_61_empty__neutral_I1_J,axiom,
! [X3: multis4781622204924204013iset_a] :
( ( plus_p1053901119268525860iset_a @ zero_z4591712717340762532iset_a @ X3 )
= X3 ) ).
% empty_neutral(1)
thf(fact_62_empty__neutral_I1_J,axiom,
! [X3: multiset_a] :
( ( plus_plus_multiset_a @ zero_zero_multiset_a @ X3 )
= X3 ) ).
% empty_neutral(1)
thf(fact_63_union__assoc,axiom,
! [M: multis501812127501805293od_a_a,N: multis501812127501805293od_a_a,K2: multis501812127501805293od_a_a] :
( ( plus_p6967097483979887908od_a_a @ ( plus_p6967097483979887908od_a_a @ M @ N ) @ K2 )
= ( plus_p6967097483979887908od_a_a @ M @ ( plus_p6967097483979887908od_a_a @ N @ K2 ) ) ) ).
% union_assoc
thf(fact_64_union__assoc,axiom,
! [M: multis4781622204924204013iset_a,N: multis4781622204924204013iset_a,K2: multis4781622204924204013iset_a] :
( ( plus_p1053901119268525860iset_a @ ( plus_p1053901119268525860iset_a @ M @ N ) @ K2 )
= ( plus_p1053901119268525860iset_a @ M @ ( plus_p1053901119268525860iset_a @ N @ K2 ) ) ) ).
% union_assoc
thf(fact_65_union__assoc,axiom,
! [M: multiset_multiset_a,N: multiset_multiset_a,K2: multiset_multiset_a] :
( ( plus_p6738641960240532161iset_a @ ( plus_p6738641960240532161iset_a @ M @ N ) @ K2 )
= ( plus_p6738641960240532161iset_a @ M @ ( plus_p6738641960240532161iset_a @ N @ K2 ) ) ) ).
% union_assoc
thf(fact_66_union__assoc,axiom,
! [M: multiset_a,N: multiset_a,K2: multiset_a] :
( ( plus_plus_multiset_a @ ( plus_plus_multiset_a @ M @ N ) @ K2 )
= ( plus_plus_multiset_a @ M @ ( plus_plus_multiset_a @ N @ K2 ) ) ) ).
% union_assoc
thf(fact_67_union__lcomm,axiom,
! [M: multis501812127501805293od_a_a,N: multis501812127501805293od_a_a,K2: multis501812127501805293od_a_a] :
( ( plus_p6967097483979887908od_a_a @ M @ ( plus_p6967097483979887908od_a_a @ N @ K2 ) )
= ( plus_p6967097483979887908od_a_a @ N @ ( plus_p6967097483979887908od_a_a @ M @ K2 ) ) ) ).
% union_lcomm
thf(fact_68_union__lcomm,axiom,
! [M: multis4781622204924204013iset_a,N: multis4781622204924204013iset_a,K2: multis4781622204924204013iset_a] :
( ( plus_p1053901119268525860iset_a @ M @ ( plus_p1053901119268525860iset_a @ N @ K2 ) )
= ( plus_p1053901119268525860iset_a @ N @ ( plus_p1053901119268525860iset_a @ M @ K2 ) ) ) ).
% union_lcomm
thf(fact_69_union__lcomm,axiom,
! [M: multiset_multiset_a,N: multiset_multiset_a,K2: multiset_multiset_a] :
( ( plus_p6738641960240532161iset_a @ M @ ( plus_p6738641960240532161iset_a @ N @ K2 ) )
= ( plus_p6738641960240532161iset_a @ N @ ( plus_p6738641960240532161iset_a @ M @ K2 ) ) ) ).
% union_lcomm
thf(fact_70_union__lcomm,axiom,
! [M: multiset_a,N: multiset_a,K2: multiset_a] :
( ( plus_plus_multiset_a @ M @ ( plus_plus_multiset_a @ N @ K2 ) )
= ( plus_plus_multiset_a @ N @ ( plus_plus_multiset_a @ M @ K2 ) ) ) ).
% union_lcomm
thf(fact_71_union__commute,axiom,
( plus_p6967097483979887908od_a_a
= ( ^ [M2: multis501812127501805293od_a_a,N2: multis501812127501805293od_a_a] : ( plus_p6967097483979887908od_a_a @ N2 @ M2 ) ) ) ).
% union_commute
thf(fact_72_union__commute,axiom,
( plus_p1053901119268525860iset_a
= ( ^ [M2: multis4781622204924204013iset_a,N2: multis4781622204924204013iset_a] : ( plus_p1053901119268525860iset_a @ N2 @ M2 ) ) ) ).
% union_commute
thf(fact_73_union__commute,axiom,
( plus_p6738641960240532161iset_a
= ( ^ [M2: multiset_multiset_a,N2: multiset_multiset_a] : ( plus_p6738641960240532161iset_a @ N2 @ M2 ) ) ) ).
% union_commute
thf(fact_74_union__commute,axiom,
( plus_plus_multiset_a
= ( ^ [M2: multiset_a,N2: multiset_a] : ( plus_plus_multiset_a @ N2 @ M2 ) ) ) ).
% union_commute
thf(fact_75_union__left__cancel,axiom,
! [K2: multis501812127501805293od_a_a,M: multis501812127501805293od_a_a,N: multis501812127501805293od_a_a] :
( ( ( plus_p6967097483979887908od_a_a @ K2 @ M )
= ( plus_p6967097483979887908od_a_a @ K2 @ N ) )
= ( M = N ) ) ).
% union_left_cancel
thf(fact_76_union__left__cancel,axiom,
! [K2: multis4781622204924204013iset_a,M: multis4781622204924204013iset_a,N: multis4781622204924204013iset_a] :
( ( ( plus_p1053901119268525860iset_a @ K2 @ M )
= ( plus_p1053901119268525860iset_a @ K2 @ N ) )
= ( M = N ) ) ).
% union_left_cancel
thf(fact_77_union__left__cancel,axiom,
! [K2: multiset_multiset_a,M: multiset_multiset_a,N: multiset_multiset_a] :
( ( ( plus_p6738641960240532161iset_a @ K2 @ M )
= ( plus_p6738641960240532161iset_a @ K2 @ N ) )
= ( M = N ) ) ).
% union_left_cancel
thf(fact_78_union__left__cancel,axiom,
! [K2: multiset_a,M: multiset_a,N: multiset_a] :
( ( ( plus_plus_multiset_a @ K2 @ M )
= ( plus_plus_multiset_a @ K2 @ N ) )
= ( M = N ) ) ).
% union_left_cancel
thf(fact_79_union__right__cancel,axiom,
! [M: multis501812127501805293od_a_a,K2: multis501812127501805293od_a_a,N: multis501812127501805293od_a_a] :
( ( ( plus_p6967097483979887908od_a_a @ M @ K2 )
= ( plus_p6967097483979887908od_a_a @ N @ K2 ) )
= ( M = N ) ) ).
% union_right_cancel
thf(fact_80_union__right__cancel,axiom,
! [M: multis4781622204924204013iset_a,K2: multis4781622204924204013iset_a,N: multis4781622204924204013iset_a] :
( ( ( plus_p1053901119268525860iset_a @ M @ K2 )
= ( plus_p1053901119268525860iset_a @ N @ K2 ) )
= ( M = N ) ) ).
% union_right_cancel
thf(fact_81_union__right__cancel,axiom,
! [M: multiset_multiset_a,K2: multiset_multiset_a,N: multiset_multiset_a] :
( ( ( plus_p6738641960240532161iset_a @ M @ K2 )
= ( plus_p6738641960240532161iset_a @ N @ K2 ) )
= ( M = N ) ) ).
% union_right_cancel
thf(fact_82_union__right__cancel,axiom,
! [M: multiset_a,K2: multiset_a,N: multiset_a] :
( ( ( plus_plus_multiset_a @ M @ K2 )
= ( plus_plus_multiset_a @ N @ K2 ) )
= ( M = N ) ) ).
% union_right_cancel
thf(fact_83_multi__union__self__other__eq,axiom,
! [A3: multis501812127501805293od_a_a,X4: multis501812127501805293od_a_a,Y3: multis501812127501805293od_a_a] :
( ( ( plus_p6967097483979887908od_a_a @ A3 @ X4 )
= ( plus_p6967097483979887908od_a_a @ A3 @ Y3 ) )
=> ( X4 = Y3 ) ) ).
% multi_union_self_other_eq
thf(fact_84_multi__union__self__other__eq,axiom,
! [A3: multis4781622204924204013iset_a,X4: multis4781622204924204013iset_a,Y3: multis4781622204924204013iset_a] :
( ( ( plus_p1053901119268525860iset_a @ A3 @ X4 )
= ( plus_p1053901119268525860iset_a @ A3 @ Y3 ) )
=> ( X4 = Y3 ) ) ).
% multi_union_self_other_eq
thf(fact_85_multi__union__self__other__eq,axiom,
! [A3: multiset_multiset_a,X4: multiset_multiset_a,Y3: multiset_multiset_a] :
( ( ( plus_p6738641960240532161iset_a @ A3 @ X4 )
= ( plus_p6738641960240532161iset_a @ A3 @ Y3 ) )
=> ( X4 = Y3 ) ) ).
% multi_union_self_other_eq
thf(fact_86_multi__union__self__other__eq,axiom,
! [A3: multiset_a,X4: multiset_a,Y3: multiset_a] :
( ( ( plus_plus_multiset_a @ A3 @ X4 )
= ( plus_plus_multiset_a @ A3 @ Y3 ) )
=> ( X4 = Y3 ) ) ).
% multi_union_self_other_eq
thf(fact_87_one__step__implies__mult,axiom,
! [J2: multis8895438461125693264iset_a,K2: multis8895438461125693264iset_a,R: set_Pr1903681468693068039iset_a,I2: multis8895438461125693264iset_a] :
( ( J2 != zero_z6863637113064401927iset_a )
=> ( ! [X5: multiset_multiset_a] :
( ( member7618379257985549619iset_a @ X5 @ ( set_ms6500351741012541767iset_a @ K2 ) )
=> ? [Xa2: multiset_multiset_a] :
( ( member7618379257985549619iset_a @ Xa2 @ ( set_ms6500351741012541767iset_a @ J2 ) )
& ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X5 @ Xa2 ) @ R ) ) )
=> ( member7045735419414020688iset_a @ ( produc2480746958397356823iset_a @ ( plus_p6063455927172555143iset_a @ I2 @ K2 ) @ ( plus_p6063455927172555143iset_a @ I2 @ J2 ) ) @ ( mult_m3296751365763795364iset_a @ R ) ) ) ) ).
% one_step_implies_mult
thf(fact_88_one__step__implies__mult,axiom,
! [J2: multis5935151272263360051od_a_a,K2: multis5935151272263360051od_a_a,R: set_Pr3746329090258575431od_a_a,I2: multis5935151272263360051od_a_a] :
( ( J2 != zero_z8896958667708014954od_a_a )
=> ( ! [X5: multis501812127501805293od_a_a] :
( ( member7700929758516679318od_a_a @ X5 @ ( set_ms8253844250382261418od_a_a @ K2 ) )
=> ? [Xa2: multis501812127501805293od_a_a] :
( ( member7700929758516679318od_a_a @ Xa2 @ ( set_ms8253844250382261418od_a_a @ J2 ) )
& ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ X5 @ Xa2 ) @ R ) ) )
=> ( member231996757139097616od_a_a @ ( produc288868108895620695od_a_a @ ( plus_p2133152728643999466od_a_a @ I2 @ K2 ) @ ( plus_p2133152728643999466od_a_a @ I2 @ J2 ) ) @ ( mult_m8775393902331661447od_a_a @ R ) ) ) ) ).
% one_step_implies_mult
thf(fact_89_one__step__implies__mult,axiom,
! [J2: multis7221419791219231795iset_a,K2: multis7221419791219231795iset_a,R: set_Pr6964934382344443463iset_a,I2: multis7221419791219231795iset_a] :
( ( J2 != zero_z8662914267040441450iset_a )
=> ( ! [X5: multis4781622204924204013iset_a] :
( ( member2597461214246177430iset_a @ X5 @ ( set_ms5327031259215843498iset_a @ K2 ) )
=> ? [Xa2: multis4781622204924204013iset_a] :
( ( member2597461214246177430iset_a @ Xa2 @ ( set_ms5327031259215843498iset_a @ J2 ) )
& ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ X5 @ Xa2 ) @ R ) ) )
=> ( member4376497788722562064iset_a @ ( produc5887428786155168343iset_a @ ( plus_p1436237118943648234iset_a @ I2 @ K2 ) @ ( plus_p1436237118943648234iset_a @ I2 @ J2 ) ) @ ( mult_m4260427137521788807iset_a @ R ) ) ) ) ).
% one_step_implies_mult
thf(fact_90_one__step__implies__mult,axiom,
! [J2: multis4781622204924204013iset_a,K2: multis4781622204924204013iset_a,R: set_Pr3059988933816007879iset_a,I2: multis4781622204924204013iset_a] :
( ( J2 != zero_z4591712717340762532iset_a )
=> ( ! [X5: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X5 @ ( set_ms2379723421352415844iset_a @ K2 ) )
=> ? [Xa2: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Xa2 @ ( set_ms2379723421352415844iset_a @ J2 ) )
& ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X5 @ Xa2 ) @ R ) ) )
=> ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ ( plus_p1053901119268525860iset_a @ I2 @ K2 ) @ ( plus_p1053901119268525860iset_a @ I2 @ J2 ) ) @ ( mult_P6168284979550713281iset_a @ R ) ) ) ) ).
% one_step_implies_mult
thf(fact_91_one__step__implies__mult,axiom,
! [J2: multis501812127501805293od_a_a,K2: multis501812127501805293od_a_a,R: set_Pr8600417178894128327od_a_a,I2: multis501812127501805293od_a_a] :
( ( J2 != zero_z3024502163831676836od_a_a )
=> ( ! [X5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ ( set_ms119794563918438244od_a_a @ K2 ) )
=> ? [Xa2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Xa2 @ ( set_ms119794563918438244od_a_a @ J2 ) )
& ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X5 @ Xa2 ) @ R ) ) )
=> ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ ( plus_p6967097483979887908od_a_a @ I2 @ K2 ) @ ( plus_p6967097483979887908od_a_a @ I2 @ J2 ) ) @ ( mult_P575886704022854593od_a_a @ R ) ) ) ) ).
% one_step_implies_mult
thf(fact_92_one__step__implies__mult,axiom,
! [J2: multiset_multiset_a,K2: multiset_multiset_a,R: set_Pr79727621955416071iset_a,I2: multiset_multiset_a] :
( ( J2 != zero_z7799948378220188993iset_a )
=> ( ! [X5: multiset_a] :
( ( member_multiset_a @ X5 @ ( set_mset_multiset_a @ K2 ) )
=> ? [Xa2: multiset_a] :
( ( member_multiset_a @ Xa2 @ ( set_mset_multiset_a @ J2 ) )
& ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Xa2 ) @ R ) ) )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ ( plus_p6738641960240532161iset_a @ I2 @ K2 ) @ ( plus_p6738641960240532161iset_a @ I2 @ J2 ) ) @ ( mult_multiset_a @ R ) ) ) ) ).
% one_step_implies_mult
thf(fact_93_one__step__implies__mult,axiom,
! [J2: multiset_a,K2: multiset_a,R: set_Product_prod_a_a,I2: multiset_a] :
( ( J2 != zero_zero_multiset_a )
=> ( ! [X5: a] :
( ( member_a @ X5 @ ( set_mset_a @ K2 ) )
=> ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_mset_a @ J2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Xa2 ) @ R ) ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( plus_plus_multiset_a @ I2 @ K2 ) @ ( plus_plus_multiset_a @ I2 @ J2 ) ) @ ( mult_a @ R ) ) ) ) ).
% one_step_implies_mult
thf(fact_94_union__iff,axiom,
! [A: produc5665077874975597415od_a_a,A3: multis3377851686571972653od_a_a,B3: multis3377851686571972653od_a_a] :
( ( member1567053331504362128od_a_a @ A @ ( set_ms5376707576967697060od_a_a @ ( plus_p1121120397469056356od_a_a @ A3 @ B3 ) ) )
= ( ( member1567053331504362128od_a_a @ A @ ( set_ms5376707576967697060od_a_a @ A3 ) )
| ( member1567053331504362128od_a_a @ A @ ( set_ms5376707576967697060od_a_a @ B3 ) ) ) ) ).
% union_iff
thf(fact_95_union__iff,axiom,
! [A: produc4424810744984089383iset_a,A3: multis1386120040344743661iset_a,B3: multis1386120040344743661iset_a] :
( ( member1345422370561994576iset_a @ A @ ( set_ms2882773486807528292iset_a @ ( plus_p2529605555928385316iset_a @ A3 @ B3 ) ) )
= ( ( member1345422370561994576iset_a @ A @ ( set_ms2882773486807528292iset_a @ A3 ) )
| ( member1345422370561994576iset_a @ A @ ( set_ms2882773486807528292iset_a @ B3 ) ) ) ) ).
% union_iff
thf(fact_96_union__iff,axiom,
! [A: multis501812127501805293od_a_a,A3: multis5935151272263360051od_a_a,B3: multis5935151272263360051od_a_a] :
( ( member7700929758516679318od_a_a @ A @ ( set_ms8253844250382261418od_a_a @ ( plus_p2133152728643999466od_a_a @ A3 @ B3 ) ) )
= ( ( member7700929758516679318od_a_a @ A @ ( set_ms8253844250382261418od_a_a @ A3 ) )
| ( member7700929758516679318od_a_a @ A @ ( set_ms8253844250382261418od_a_a @ B3 ) ) ) ) ).
% union_iff
thf(fact_97_union__iff,axiom,
! [A: multis4781622204924204013iset_a,A3: multis7221419791219231795iset_a,B3: multis7221419791219231795iset_a] :
( ( member2597461214246177430iset_a @ A @ ( set_ms5327031259215843498iset_a @ ( plus_p1436237118943648234iset_a @ A3 @ B3 ) ) )
= ( ( member2597461214246177430iset_a @ A @ ( set_ms5327031259215843498iset_a @ A3 ) )
| ( member2597461214246177430iset_a @ A @ ( set_ms5327031259215843498iset_a @ B3 ) ) ) ) ).
% union_iff
thf(fact_98_union__iff,axiom,
! [A: multiset_a,A3: multiset_multiset_a,B3: multiset_multiset_a] :
( ( member_multiset_a @ A @ ( set_mset_multiset_a @ ( plus_p6738641960240532161iset_a @ A3 @ B3 ) ) )
= ( ( member_multiset_a @ A @ ( set_mset_multiset_a @ A3 ) )
| ( member_multiset_a @ A @ ( set_mset_multiset_a @ B3 ) ) ) ) ).
% union_iff
thf(fact_99_union__iff,axiom,
! [A: produc6518373309651786023iset_a,A3: multis4781622204924204013iset_a,B3: multis4781622204924204013iset_a] :
( ( member5199237121806060112iset_a @ A @ ( set_ms2379723421352415844iset_a @ ( plus_p1053901119268525860iset_a @ A3 @ B3 ) ) )
= ( ( member5199237121806060112iset_a @ A @ ( set_ms2379723421352415844iset_a @ A3 ) )
| ( member5199237121806060112iset_a @ A @ ( set_ms2379723421352415844iset_a @ B3 ) ) ) ) ).
% union_iff
thf(fact_100_union__iff,axiom,
! [A: product_prod_a_a,A3: multis501812127501805293od_a_a,B3: multis501812127501805293od_a_a] :
( ( member1426531477525435216od_a_a @ A @ ( set_ms119794563918438244od_a_a @ ( plus_p6967097483979887908od_a_a @ A3 @ B3 ) ) )
= ( ( member1426531477525435216od_a_a @ A @ ( set_ms119794563918438244od_a_a @ A3 ) )
| ( member1426531477525435216od_a_a @ A @ ( set_ms119794563918438244od_a_a @ B3 ) ) ) ) ).
% union_iff
thf(fact_101_union__iff,axiom,
! [A: a,A3: multiset_a,B3: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ ( plus_plus_multiset_a @ A3 @ B3 ) ) )
= ( ( member_a @ A @ ( set_mset_a @ A3 ) )
| ( member_a @ A @ ( set_mset_a @ B3 ) ) ) ) ).
% union_iff
thf(fact_102_diff__union__cancelR,axiom,
! [M: multiset_multiset_a,N: multiset_multiset_a] :
( ( minus_3395427628221709681iset_a @ ( plus_p6738641960240532161iset_a @ M @ N ) @ N )
= M ) ).
% diff_union_cancelR
thf(fact_103_diff__union__cancelR,axiom,
! [M: multiset_a,N: multiset_a] :
( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ M @ N ) @ N )
= M ) ).
% diff_union_cancelR
thf(fact_104_diff__union__cancelR,axiom,
! [M: multis4781622204924204013iset_a,N: multis4781622204924204013iset_a] :
( ( minus_9211619285413683668iset_a @ ( plus_p1053901119268525860iset_a @ M @ N ) @ N )
= M ) ).
% diff_union_cancelR
thf(fact_105_diff__union__cancelR,axiom,
! [M: multis501812127501805293od_a_a,N: multis501812127501805293od_a_a] :
( ( minus_7313152569058643924od_a_a @ ( plus_p6967097483979887908od_a_a @ M @ N ) @ N )
= M ) ).
% diff_union_cancelR
thf(fact_106_diff__union__cancelL,axiom,
! [N: multiset_multiset_a,M: multiset_multiset_a] :
( ( minus_3395427628221709681iset_a @ ( plus_p6738641960240532161iset_a @ N @ M ) @ N )
= M ) ).
% diff_union_cancelL
thf(fact_107_diff__union__cancelL,axiom,
! [N: multiset_a,M: multiset_a] :
( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ N @ M ) @ N )
= M ) ).
% diff_union_cancelL
thf(fact_108_diff__union__cancelL,axiom,
! [N: multis4781622204924204013iset_a,M: multis4781622204924204013iset_a] :
( ( minus_9211619285413683668iset_a @ ( plus_p1053901119268525860iset_a @ N @ M ) @ N )
= M ) ).
% diff_union_cancelL
thf(fact_109_diff__union__cancelL,axiom,
! [N: multis501812127501805293od_a_a,M: multis501812127501805293od_a_a] :
( ( minus_7313152569058643924od_a_a @ ( plus_p6967097483979887908od_a_a @ N @ M ) @ N )
= M ) ).
% diff_union_cancelL
thf(fact_110_Multiset_Odiff__add,axiom,
! [M: multiset_multiset_a,N: multiset_multiset_a,Q: multiset_multiset_a] :
( ( minus_3395427628221709681iset_a @ M @ ( plus_p6738641960240532161iset_a @ N @ Q ) )
= ( minus_3395427628221709681iset_a @ ( minus_3395427628221709681iset_a @ M @ N ) @ Q ) ) ).
% Multiset.diff_add
thf(fact_111_Multiset_Odiff__add,axiom,
! [M: multiset_a,N: multiset_a,Q: multiset_a] :
( ( minus_3765977307040488491iset_a @ M @ ( plus_plus_multiset_a @ N @ Q ) )
= ( minus_3765977307040488491iset_a @ ( minus_3765977307040488491iset_a @ M @ N ) @ Q ) ) ).
% Multiset.diff_add
thf(fact_112_Multiset_Odiff__add,axiom,
! [M: multis4781622204924204013iset_a,N: multis4781622204924204013iset_a,Q: multis4781622204924204013iset_a] :
( ( minus_9211619285413683668iset_a @ M @ ( plus_p1053901119268525860iset_a @ N @ Q ) )
= ( minus_9211619285413683668iset_a @ ( minus_9211619285413683668iset_a @ M @ N ) @ Q ) ) ).
% Multiset.diff_add
thf(fact_113_Multiset_Odiff__add,axiom,
! [M: multis501812127501805293od_a_a,N: multis501812127501805293od_a_a,Q: multis501812127501805293od_a_a] :
( ( minus_7313152569058643924od_a_a @ M @ ( plus_p6967097483979887908od_a_a @ N @ Q ) )
= ( minus_7313152569058643924od_a_a @ ( minus_7313152569058643924od_a_a @ M @ N ) @ Q ) ) ).
% Multiset.diff_add
thf(fact_114_multiset__nonemptyE,axiom,
! [A3: multis3377851686571972653od_a_a] :
( ( A3 != zero_z3721308388212328420od_a_a )
=> ~ ! [X5: produc5665077874975597415od_a_a] :
~ ( member1567053331504362128od_a_a @ X5 @ ( set_ms5376707576967697060od_a_a @ A3 ) ) ) ).
% multiset_nonemptyE
thf(fact_115_multiset__nonemptyE,axiom,
! [A3: multis1386120040344743661iset_a] :
( ( A3 != zero_z9101083372863164836iset_a )
=> ~ ! [X5: produc4424810744984089383iset_a] :
~ ( member1345422370561994576iset_a @ X5 @ ( set_ms2882773486807528292iset_a @ A3 ) ) ) ).
% multiset_nonemptyE
thf(fact_116_multiset__nonemptyE,axiom,
! [A3: multis5935151272263360051od_a_a] :
( ( A3 != zero_z8896958667708014954od_a_a )
=> ~ ! [X5: multis501812127501805293od_a_a] :
~ ( member7700929758516679318od_a_a @ X5 @ ( set_ms8253844250382261418od_a_a @ A3 ) ) ) ).
% multiset_nonemptyE
thf(fact_117_multiset__nonemptyE,axiom,
! [A3: multis7221419791219231795iset_a] :
( ( A3 != zero_z8662914267040441450iset_a )
=> ~ ! [X5: multis4781622204924204013iset_a] :
~ ( member2597461214246177430iset_a @ X5 @ ( set_ms5327031259215843498iset_a @ A3 ) ) ) ).
% multiset_nonemptyE
thf(fact_118_multiset__nonemptyE,axiom,
! [A3: multiset_multiset_a] :
( ( A3 != zero_z7799948378220188993iset_a )
=> ~ ! [X5: multiset_a] :
~ ( member_multiset_a @ X5 @ ( set_mset_multiset_a @ A3 ) ) ) ).
% multiset_nonemptyE
thf(fact_119_multiset__nonemptyE,axiom,
! [A3: multis4781622204924204013iset_a] :
( ( A3 != zero_z4591712717340762532iset_a )
=> ~ ! [X5: produc6518373309651786023iset_a] :
~ ( member5199237121806060112iset_a @ X5 @ ( set_ms2379723421352415844iset_a @ A3 ) ) ) ).
% multiset_nonemptyE
thf(fact_120_multiset__nonemptyE,axiom,
! [A3: multis501812127501805293od_a_a] :
( ( A3 != zero_z3024502163831676836od_a_a )
=> ~ ! [X5: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ X5 @ ( set_ms119794563918438244od_a_a @ A3 ) ) ) ).
% multiset_nonemptyE
thf(fact_121_multiset__nonemptyE,axiom,
! [A3: multiset_a] :
( ( A3 != zero_zero_multiset_a )
=> ~ ! [X5: a] :
~ ( member_a @ X5 @ ( set_mset_a @ A3 ) ) ) ).
% multiset_nonemptyE
thf(fact_122_Multiset_Odiff__cancel,axiom,
! [A3: multiset_multiset_a] :
( ( minus_3395427628221709681iset_a @ A3 @ A3 )
= zero_z7799948378220188993iset_a ) ).
% Multiset.diff_cancel
thf(fact_123_Multiset_Odiff__cancel,axiom,
! [A3: multiset_a] :
( ( minus_3765977307040488491iset_a @ A3 @ A3 )
= zero_zero_multiset_a ) ).
% Multiset.diff_cancel
thf(fact_124_Multiset_Odiff__cancel,axiom,
! [A3: multis4781622204924204013iset_a] :
( ( minus_9211619285413683668iset_a @ A3 @ A3 )
= zero_z4591712717340762532iset_a ) ).
% Multiset.diff_cancel
thf(fact_125_Multiset_Odiff__cancel,axiom,
! [A3: multis501812127501805293od_a_a] :
( ( minus_7313152569058643924od_a_a @ A3 @ A3 )
= zero_z3024502163831676836od_a_a ) ).
% Multiset.diff_cancel
thf(fact_126_diff__empty,axiom,
! [M: multiset_multiset_a] :
( ( ( minus_3395427628221709681iset_a @ M @ zero_z7799948378220188993iset_a )
= M )
& ( ( minus_3395427628221709681iset_a @ zero_z7799948378220188993iset_a @ M )
= zero_z7799948378220188993iset_a ) ) ).
% diff_empty
thf(fact_127_diff__empty,axiom,
! [M: multiset_a] :
( ( ( minus_3765977307040488491iset_a @ M @ zero_zero_multiset_a )
= M )
& ( ( minus_3765977307040488491iset_a @ zero_zero_multiset_a @ M )
= zero_zero_multiset_a ) ) ).
% diff_empty
thf(fact_128_diff__empty,axiom,
! [M: multis4781622204924204013iset_a] :
( ( ( minus_9211619285413683668iset_a @ M @ zero_z4591712717340762532iset_a )
= M )
& ( ( minus_9211619285413683668iset_a @ zero_z4591712717340762532iset_a @ M )
= zero_z4591712717340762532iset_a ) ) ).
% diff_empty
thf(fact_129_diff__empty,axiom,
! [M: multis501812127501805293od_a_a] :
( ( ( minus_7313152569058643924od_a_a @ M @ zero_z3024502163831676836od_a_a )
= M )
& ( ( minus_7313152569058643924od_a_a @ zero_z3024502163831676836od_a_a @ M )
= zero_z3024502163831676836od_a_a ) ) ).
% diff_empty
thf(fact_130_prod__induct3,axiom,
! [P: produc3498347346309940967od_a_a > $o,X3: produc3498347346309940967od_a_a] :
( ! [A4: product_prod_a_a,B4: a,C: a] : ( P @ ( produc7886510207707329367od_a_a @ A4 @ ( product_Pair_a_a @ B4 @ C ) ) )
=> ( P @ X3 ) ) ).
% prod_induct3
thf(fact_131_prod__induct3,axiom,
! [P: produc8554440688619196647iset_a > $o,X3: produc8554440688619196647iset_a] :
( ! [A4: produc6518373309651786023iset_a,B4: multiset_a,C: multiset_a] : ( P @ ( produc2591098542590733143iset_a @ A4 @ ( produc654756711066625303iset_a @ B4 @ C ) ) )
=> ( P @ X3 ) ) ).
% prod_induct3
thf(fact_132_mem__Collect__eq,axiom,
! [A: produc5665077874975597415od_a_a,P: produc5665077874975597415od_a_a > $o] :
( ( member1567053331504362128od_a_a @ A @ ( collec632816824236251858od_a_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_133_mem__Collect__eq,axiom,
! [A: produc4424810744984089383iset_a,P: produc4424810744984089383iset_a > $o] :
( ( member1345422370561994576iset_a @ A @ ( collec300885903864160658iset_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_134_mem__Collect__eq,axiom,
! [A: multis4781622204924204013iset_a,P: multis4781622204924204013iset_a > $o] :
( ( member2597461214246177430iset_a @ A @ ( collec1060546129649387992iset_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_135_mem__Collect__eq,axiom,
! [A: multis501812127501805293od_a_a,P: multis501812127501805293od_a_a > $o] :
( ( member7700929758516679318od_a_a @ A @ ( collec9001736562947189720od_a_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_136_mem__Collect__eq,axiom,
! [A: multiset_a,P: multiset_a > $o] :
( ( member_multiset_a @ A @ ( collect_multiset_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_137_mem__Collect__eq,axiom,
! [A: product_prod_a_a,P: product_prod_a_a > $o] :
( ( member1426531477525435216od_a_a @ A @ ( collec3336397797384452498od_a_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_138_mem__Collect__eq,axiom,
! [A: produc6518373309651786023iset_a,P: produc6518373309651786023iset_a > $o] :
( ( member5199237121806060112iset_a @ A @ ( collec6344740439120713362iset_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_139_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_140_Collect__mem__eq,axiom,
! [A3: set_Pr3746329090258575431od_a_a] :
( ( collec632816824236251858od_a_a
@ ^ [X6: produc5665077874975597415od_a_a] : ( member1567053331504362128od_a_a @ X6 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_141_Collect__mem__eq,axiom,
! [A3: set_Pr1903681468693068039iset_a] :
( ( collec300885903864160658iset_a
@ ^ [X6: produc4424810744984089383iset_a] : ( member1345422370561994576iset_a @ X6 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_142_Collect__mem__eq,axiom,
! [A3: set_mu7670847959134527309iset_a] :
( ( collec1060546129649387992iset_a
@ ^ [X6: multis4781622204924204013iset_a] : ( member2597461214246177430iset_a @ X6 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_143_Collect__mem__eq,axiom,
! [A3: set_mu7436117841396106061od_a_a] :
( ( collec9001736562947189720od_a_a
@ ^ [X6: multis501812127501805293od_a_a] : ( member7700929758516679318od_a_a @ X6 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_144_Collect__mem__eq,axiom,
! [A3: set_multiset_a] :
( ( collect_multiset_a
@ ^ [X6: multiset_a] : ( member_multiset_a @ X6 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_145_Collect__mem__eq,axiom,
! [A3: set_Product_prod_a_a] :
( ( collec3336397797384452498od_a_a
@ ^ [X6: product_prod_a_a] : ( member1426531477525435216od_a_a @ X6 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_146_Collect__mem__eq,axiom,
! [A3: set_Pr79727621955416071iset_a] :
( ( collec6344740439120713362iset_a
@ ^ [X6: produc6518373309651786023iset_a] : ( member5199237121806060112iset_a @ X6 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_147_Collect__mem__eq,axiom,
! [A3: set_a] :
( ( collect_a
@ ^ [X6: a] : ( member_a @ X6 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_148_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X5: a] :
( ( P @ X5 )
= ( Q @ X5 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_149_Collect__cong,axiom,
! [P: produc6518373309651786023iset_a > $o,Q: produc6518373309651786023iset_a > $o] :
( ! [X5: produc6518373309651786023iset_a] :
( ( P @ X5 )
= ( Q @ X5 ) )
=> ( ( collec6344740439120713362iset_a @ P )
= ( collec6344740439120713362iset_a @ Q ) ) ) ).
% Collect_cong
thf(fact_150_Collect__cong,axiom,
! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ! [X5: product_prod_a_a] :
( ( P @ X5 )
= ( Q @ X5 ) )
=> ( ( collec3336397797384452498od_a_a @ P )
= ( collec3336397797384452498od_a_a @ Q ) ) ) ).
% Collect_cong
thf(fact_151_prod__cases3,axiom,
! [Y: produc3498347346309940967od_a_a] :
~ ! [A4: product_prod_a_a,B4: a,C: a] :
( Y
!= ( produc7886510207707329367od_a_a @ A4 @ ( product_Pair_a_a @ B4 @ C ) ) ) ).
% prod_cases3
thf(fact_152_prod__cases3,axiom,
! [Y: produc8554440688619196647iset_a] :
~ ! [A4: produc6518373309651786023iset_a,B4: multiset_a,C: multiset_a] :
( Y
!= ( produc2591098542590733143iset_a @ A4 @ ( produc654756711066625303iset_a @ B4 @ C ) ) ) ).
% prod_cases3
thf(fact_153_Pair__inject,axiom,
! [A: multiset_multiset_a,B: multiset_multiset_a,A2: multiset_multiset_a,B2: multiset_multiset_a] :
( ( ( produc8381460346935720215iset_a @ A @ B )
= ( produc8381460346935720215iset_a @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_154_Pair__inject,axiom,
! [A: product_prod_a_a,B: product_prod_a_a,A2: product_prod_a_a,B2: product_prod_a_a] :
( ( ( produc7886510207707329367od_a_a @ A @ B )
= ( produc7886510207707329367od_a_a @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_155_Pair__inject,axiom,
! [A: multis501812127501805293od_a_a,B: multis501812127501805293od_a_a,A2: multis501812127501805293od_a_a,B2: multis501812127501805293od_a_a] :
( ( ( produc6961845650964105687od_a_a @ A @ B )
= ( produc6961845650964105687od_a_a @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_156_Pair__inject,axiom,
! [A: produc6518373309651786023iset_a,B: produc6518373309651786023iset_a,A2: produc6518373309651786023iset_a,B2: produc6518373309651786023iset_a] :
( ( ( produc2591098542590733143iset_a @ A @ B )
= ( produc2591098542590733143iset_a @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_157_Pair__inject,axiom,
! [A: multis4781622204924204013iset_a,B: multis4781622204924204013iset_a,A2: multis4781622204924204013iset_a,B2: multis4781622204924204013iset_a] :
( ( ( produc1739639915645390807iset_a @ A @ B )
= ( produc1739639915645390807iset_a @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_158_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_159_Pair__inject,axiom,
! [A: multiset_a,B: multiset_a,A2: multiset_a,B2: multiset_a] :
( ( ( produc654756711066625303iset_a @ A @ B )
= ( produc654756711066625303iset_a @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_160_prod__cases,axiom,
! [P: produc4424810744984089383iset_a > $o,P2: produc4424810744984089383iset_a] :
( ! [A4: multiset_multiset_a,B4: multiset_multiset_a] : ( P @ ( produc8381460346935720215iset_a @ A4 @ B4 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_161_prod__cases,axiom,
! [P: produc3498347346309940967od_a_a > $o,P2: produc3498347346309940967od_a_a] :
( ! [A4: product_prod_a_a,B4: product_prod_a_a] : ( P @ ( produc7886510207707329367od_a_a @ A4 @ B4 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_162_prod__cases,axiom,
! [P: produc5665077874975597415od_a_a > $o,P2: produc5665077874975597415od_a_a] :
( ! [A4: multis501812127501805293od_a_a,B4: multis501812127501805293od_a_a] : ( P @ ( produc6961845650964105687od_a_a @ A4 @ B4 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_163_prod__cases,axiom,
! [P: produc8554440688619196647iset_a > $o,P2: produc8554440688619196647iset_a] :
( ! [A4: produc6518373309651786023iset_a,B4: produc6518373309651786023iset_a] : ( P @ ( produc2591098542590733143iset_a @ A4 @ B4 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_164_prod__cases,axiom,
! [P: produc2869653011250641767iset_a > $o,P2: produc2869653011250641767iset_a] :
( ! [A4: multis4781622204924204013iset_a,B4: multis4781622204924204013iset_a] : ( P @ ( produc1739639915645390807iset_a @ A4 @ B4 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_165_prod__cases,axiom,
! [P: product_prod_a_a > $o,P2: product_prod_a_a] :
( ! [A4: a,B4: a] : ( P @ ( product_Pair_a_a @ A4 @ B4 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_166_prod__cases,axiom,
! [P: produc6518373309651786023iset_a > $o,P2: produc6518373309651786023iset_a] :
( ! [A4: multiset_a,B4: multiset_a] : ( P @ ( produc654756711066625303iset_a @ A4 @ B4 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_167_surj__pair,axiom,
! [P2: produc4424810744984089383iset_a] :
? [X5: multiset_multiset_a,Y4: multiset_multiset_a] :
( P2
= ( produc8381460346935720215iset_a @ X5 @ Y4 ) ) ).
% surj_pair
thf(fact_168_surj__pair,axiom,
! [P2: produc3498347346309940967od_a_a] :
? [X5: product_prod_a_a,Y4: product_prod_a_a] :
( P2
= ( produc7886510207707329367od_a_a @ X5 @ Y4 ) ) ).
% surj_pair
thf(fact_169_surj__pair,axiom,
! [P2: produc5665077874975597415od_a_a] :
? [X5: multis501812127501805293od_a_a,Y4: multis501812127501805293od_a_a] :
( P2
= ( produc6961845650964105687od_a_a @ X5 @ Y4 ) ) ).
% surj_pair
thf(fact_170_surj__pair,axiom,
! [P2: produc8554440688619196647iset_a] :
? [X5: produc6518373309651786023iset_a,Y4: produc6518373309651786023iset_a] :
( P2
= ( produc2591098542590733143iset_a @ X5 @ Y4 ) ) ).
% surj_pair
thf(fact_171_surj__pair,axiom,
! [P2: produc2869653011250641767iset_a] :
? [X5: multis4781622204924204013iset_a,Y4: multis4781622204924204013iset_a] :
( P2
= ( produc1739639915645390807iset_a @ X5 @ Y4 ) ) ).
% surj_pair
thf(fact_172_surj__pair,axiom,
! [P2: product_prod_a_a] :
? [X5: a,Y4: a] :
( P2
= ( product_Pair_a_a @ X5 @ Y4 ) ) ).
% surj_pair
thf(fact_173_surj__pair,axiom,
! [P2: produc6518373309651786023iset_a] :
? [X5: multiset_a,Y4: multiset_a] :
( P2
= ( produc654756711066625303iset_a @ X5 @ Y4 ) ) ).
% surj_pair
thf(fact_174_old_Oprod_Oexhaust,axiom,
! [Y: produc4424810744984089383iset_a] :
~ ! [A4: multiset_multiset_a,B4: multiset_multiset_a] :
( Y
!= ( produc8381460346935720215iset_a @ A4 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_175_old_Oprod_Oexhaust,axiom,
! [Y: produc3498347346309940967od_a_a] :
~ ! [A4: product_prod_a_a,B4: product_prod_a_a] :
( Y
!= ( produc7886510207707329367od_a_a @ A4 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_176_old_Oprod_Oexhaust,axiom,
! [Y: produc5665077874975597415od_a_a] :
~ ! [A4: multis501812127501805293od_a_a,B4: multis501812127501805293od_a_a] :
( Y
!= ( produc6961845650964105687od_a_a @ A4 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_177_old_Oprod_Oexhaust,axiom,
! [Y: produc8554440688619196647iset_a] :
~ ! [A4: produc6518373309651786023iset_a,B4: produc6518373309651786023iset_a] :
( Y
!= ( produc2591098542590733143iset_a @ A4 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_178_old_Oprod_Oexhaust,axiom,
! [Y: produc2869653011250641767iset_a] :
~ ! [A4: multis4781622204924204013iset_a,B4: multis4781622204924204013iset_a] :
( Y
!= ( produc1739639915645390807iset_a @ A4 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_179_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_a_a] :
~ ! [A4: a,B4: a] :
( Y
!= ( product_Pair_a_a @ A4 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_180_old_Oprod_Oexhaust,axiom,
! [Y: produc6518373309651786023iset_a] :
~ ! [A4: multiset_a,B4: multiset_a] :
( Y
!= ( produc654756711066625303iset_a @ A4 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_181_Multiset_Odiff__right__commute,axiom,
! [M: multiset_a,N: multiset_a,Q: multiset_a] :
( ( minus_3765977307040488491iset_a @ ( minus_3765977307040488491iset_a @ M @ N ) @ Q )
= ( minus_3765977307040488491iset_a @ ( minus_3765977307040488491iset_a @ M @ Q ) @ N ) ) ).
% Multiset.diff_right_commute
thf(fact_182_Multiset_Odiff__right__commute,axiom,
! [M: multis4781622204924204013iset_a,N: multis4781622204924204013iset_a,Q: multis4781622204924204013iset_a] :
( ( minus_9211619285413683668iset_a @ ( minus_9211619285413683668iset_a @ M @ N ) @ Q )
= ( minus_9211619285413683668iset_a @ ( minus_9211619285413683668iset_a @ M @ Q ) @ N ) ) ).
% Multiset.diff_right_commute
thf(fact_183_Multiset_Odiff__right__commute,axiom,
! [M: multis501812127501805293od_a_a,N: multis501812127501805293od_a_a,Q: multis501812127501805293od_a_a] :
( ( minus_7313152569058643924od_a_a @ ( minus_7313152569058643924od_a_a @ M @ N ) @ Q )
= ( minus_7313152569058643924od_a_a @ ( minus_7313152569058643924od_a_a @ M @ Q ) @ N ) ) ).
% Multiset.diff_right_commute
thf(fact_184_diff__add__zero,axiom,
! [A: multiset_multiset_a,B: multiset_multiset_a] :
( ( minus_3395427628221709681iset_a @ A @ ( plus_p6738641960240532161iset_a @ A @ B ) )
= zero_z7799948378220188993iset_a ) ).
% diff_add_zero
thf(fact_185_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_186_diff__add__zero,axiom,
! [A: multiset_a,B: multiset_a] :
( ( minus_3765977307040488491iset_a @ A @ ( plus_plus_multiset_a @ A @ B ) )
= zero_zero_multiset_a ) ).
% diff_add_zero
thf(fact_187_diff__add__zero,axiom,
! [A: multis4781622204924204013iset_a,B: multis4781622204924204013iset_a] :
( ( minus_9211619285413683668iset_a @ A @ ( plus_p1053901119268525860iset_a @ A @ B ) )
= zero_z4591712717340762532iset_a ) ).
% diff_add_zero
thf(fact_188_diff__add__zero,axiom,
! [A: multis501812127501805293od_a_a,B: multis501812127501805293od_a_a] :
( ( minus_7313152569058643924od_a_a @ A @ ( plus_p6967097483979887908od_a_a @ A @ B ) )
= zero_z3024502163831676836od_a_a ) ).
% diff_add_zero
thf(fact_189_add__diff__cancel__right_H,axiom,
! [A: multiset_multiset_a,B: multiset_multiset_a] :
( ( minus_3395427628221709681iset_a @ ( plus_p6738641960240532161iset_a @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_190_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_191_add__diff__cancel__right_H,axiom,
! [A: multiset_a,B: multiset_a] :
( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_192_add__diff__cancel__right_H,axiom,
! [A: multis4781622204924204013iset_a,B: multis4781622204924204013iset_a] :
( ( minus_9211619285413683668iset_a @ ( plus_p1053901119268525860iset_a @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_193_add__diff__cancel__right_H,axiom,
! [A: multis501812127501805293od_a_a,B: multis501812127501805293od_a_a] :
( ( minus_7313152569058643924od_a_a @ ( plus_p6967097483979887908od_a_a @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_194_add__diff__cancel__right,axiom,
! [A: multiset_multiset_a,C2: multiset_multiset_a,B: multiset_multiset_a] :
( ( minus_3395427628221709681iset_a @ ( plus_p6738641960240532161iset_a @ A @ C2 ) @ ( plus_p6738641960240532161iset_a @ B @ C2 ) )
= ( minus_3395427628221709681iset_a @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_195_add__diff__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_196_add__diff__cancel__right,axiom,
! [A: multiset_a,C2: multiset_a,B: multiset_a] :
( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ A @ C2 ) @ ( plus_plus_multiset_a @ B @ C2 ) )
= ( minus_3765977307040488491iset_a @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_197_add__diff__cancel__right,axiom,
! [A: multis4781622204924204013iset_a,C2: multis4781622204924204013iset_a,B: multis4781622204924204013iset_a] :
( ( minus_9211619285413683668iset_a @ ( plus_p1053901119268525860iset_a @ A @ C2 ) @ ( plus_p1053901119268525860iset_a @ B @ C2 ) )
= ( minus_9211619285413683668iset_a @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_198_add__diff__cancel__right,axiom,
! [A: multis501812127501805293od_a_a,C2: multis501812127501805293od_a_a,B: multis501812127501805293od_a_a] :
( ( minus_7313152569058643924od_a_a @ ( plus_p6967097483979887908od_a_a @ A @ C2 ) @ ( plus_p6967097483979887908od_a_a @ B @ C2 ) )
= ( minus_7313152569058643924od_a_a @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_199_add__diff__cancel__left_H,axiom,
! [A: multiset_multiset_a,B: multiset_multiset_a] :
( ( minus_3395427628221709681iset_a @ ( plus_p6738641960240532161iset_a @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_200_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_201_add__diff__cancel__left_H,axiom,
! [A: multiset_a,B: multiset_a] :
( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_202_add__diff__cancel__left_H,axiom,
! [A: multis4781622204924204013iset_a,B: multis4781622204924204013iset_a] :
( ( minus_9211619285413683668iset_a @ ( plus_p1053901119268525860iset_a @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_203_add__diff__cancel__left_H,axiom,
! [A: multis501812127501805293od_a_a,B: multis501812127501805293od_a_a] :
( ( minus_7313152569058643924od_a_a @ ( plus_p6967097483979887908od_a_a @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_204_add__diff__cancel__left,axiom,
! [C2: multiset_multiset_a,A: multiset_multiset_a,B: multiset_multiset_a] :
( ( minus_3395427628221709681iset_a @ ( plus_p6738641960240532161iset_a @ C2 @ A ) @ ( plus_p6738641960240532161iset_a @ C2 @ B ) )
= ( minus_3395427628221709681iset_a @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_205_add__diff__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_206_add__diff__cancel__left,axiom,
! [C2: multiset_a,A: multiset_a,B: multiset_a] :
( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ C2 @ A ) @ ( plus_plus_multiset_a @ C2 @ B ) )
= ( minus_3765977307040488491iset_a @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_207_add__diff__cancel__left,axiom,
! [C2: multis4781622204924204013iset_a,A: multis4781622204924204013iset_a,B: multis4781622204924204013iset_a] :
( ( minus_9211619285413683668iset_a @ ( plus_p1053901119268525860iset_a @ C2 @ A ) @ ( plus_p1053901119268525860iset_a @ C2 @ B ) )
= ( minus_9211619285413683668iset_a @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_208_add__diff__cancel__left,axiom,
! [C2: multis501812127501805293od_a_a,A: multis501812127501805293od_a_a,B: multis501812127501805293od_a_a] :
( ( minus_7313152569058643924od_a_a @ ( plus_p6967097483979887908od_a_a @ C2 @ A ) @ ( plus_p6967097483979887908od_a_a @ C2 @ B ) )
= ( minus_7313152569058643924od_a_a @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_209_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: multiset_multiset_a] :
( ( minus_3395427628221709681iset_a @ A @ A )
= zero_z7799948378220188993iset_a ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_210_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_211_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: multiset_a] :
( ( minus_3765977307040488491iset_a @ A @ A )
= zero_zero_multiset_a ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_212_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: multis4781622204924204013iset_a] :
( ( minus_9211619285413683668iset_a @ A @ A )
= zero_z4591712717340762532iset_a ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_213_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: multis501812127501805293od_a_a] :
( ( minus_7313152569058643924od_a_a @ A @ A )
= zero_z3024502163831676836od_a_a ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_214_diff__zero,axiom,
! [A: multiset_multiset_a] :
( ( minus_3395427628221709681iset_a @ A @ zero_z7799948378220188993iset_a )
= A ) ).
% diff_zero
thf(fact_215_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_216_diff__zero,axiom,
! [A: multiset_a] :
( ( minus_3765977307040488491iset_a @ A @ zero_zero_multiset_a )
= A ) ).
% diff_zero
thf(fact_217_diff__zero,axiom,
! [A: multis4781622204924204013iset_a] :
( ( minus_9211619285413683668iset_a @ A @ zero_z4591712717340762532iset_a )
= A ) ).
% diff_zero
thf(fact_218_diff__zero,axiom,
! [A: multis501812127501805293od_a_a] :
( ( minus_7313152569058643924od_a_a @ A @ zero_z3024502163831676836od_a_a )
= A ) ).
% diff_zero
thf(fact_219_zero__diff,axiom,
! [A: multiset_multiset_a] :
( ( minus_3395427628221709681iset_a @ zero_z7799948378220188993iset_a @ A )
= zero_z7799948378220188993iset_a ) ).
% zero_diff
thf(fact_220_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_221_zero__diff,axiom,
! [A: multiset_a] :
( ( minus_3765977307040488491iset_a @ zero_zero_multiset_a @ A )
= zero_zero_multiset_a ) ).
% zero_diff
thf(fact_222_zero__diff,axiom,
! [A: multis4781622204924204013iset_a] :
( ( minus_9211619285413683668iset_a @ zero_z4591712717340762532iset_a @ A )
= zero_z4591712717340762532iset_a ) ).
% zero_diff
thf(fact_223_zero__diff,axiom,
! [A: multis501812127501805293od_a_a] :
( ( minus_7313152569058643924od_a_a @ zero_z3024502163831676836od_a_a @ A )
= zero_z3024502163831676836od_a_a ) ).
% zero_diff
thf(fact_224_add__right__cancel,axiom,
! [B: multis501812127501805293od_a_a,A: multis501812127501805293od_a_a,C2: multis501812127501805293od_a_a] :
( ( ( plus_p6967097483979887908od_a_a @ B @ A )
= ( plus_p6967097483979887908od_a_a @ C2 @ A ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_225_add__right__cancel,axiom,
! [B: multis4781622204924204013iset_a,A: multis4781622204924204013iset_a,C2: multis4781622204924204013iset_a] :
( ( ( plus_p1053901119268525860iset_a @ B @ A )
= ( plus_p1053901119268525860iset_a @ C2 @ A ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_226_add__right__cancel,axiom,
! [B: multiset_multiset_a,A: multiset_multiset_a,C2: multiset_multiset_a] :
( ( ( plus_p6738641960240532161iset_a @ B @ A )
= ( plus_p6738641960240532161iset_a @ C2 @ A ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_227_add__right__cancel,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C2 @ A ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_228_add__right__cancel,axiom,
! [B: multiset_a,A: multiset_a,C2: multiset_a] :
( ( ( plus_plus_multiset_a @ B @ A )
= ( plus_plus_multiset_a @ C2 @ A ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_229_add__left__cancel,axiom,
! [A: multis501812127501805293od_a_a,B: multis501812127501805293od_a_a,C2: multis501812127501805293od_a_a] :
( ( ( plus_p6967097483979887908od_a_a @ A @ B )
= ( plus_p6967097483979887908od_a_a @ A @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_230_add__left__cancel,axiom,
! [A: multis4781622204924204013iset_a,B: multis4781622204924204013iset_a,C2: multis4781622204924204013iset_a] :
( ( ( plus_p1053901119268525860iset_a @ A @ B )
= ( plus_p1053901119268525860iset_a @ A @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_231_add__left__cancel,axiom,
! [A: multiset_multiset_a,B: multiset_multiset_a,C2: multiset_multiset_a] :
( ( ( plus_p6738641960240532161iset_a @ A @ B )
= ( plus_p6738641960240532161iset_a @ A @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_232_add__left__cancel,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_233_add__left__cancel,axiom,
! [A: multiset_a,B: multiset_a,C2: multiset_a] :
( ( ( plus_plus_multiset_a @ A @ B )
= ( plus_plus_multiset_a @ A @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_234_add__0,axiom,
! [A: multiset_multiset_a] :
( ( plus_p6738641960240532161iset_a @ zero_z7799948378220188993iset_a @ A )
= A ) ).
% add_0
thf(fact_235_add__0,axiom,
! [A: multis501812127501805293od_a_a] :
( ( plus_p6967097483979887908od_a_a @ zero_z3024502163831676836od_a_a @ A )
= A ) ).
% add_0
thf(fact_236_add__0,axiom,
! [A: multis4781622204924204013iset_a] :
( ( plus_p1053901119268525860iset_a @ zero_z4591712717340762532iset_a @ A )
= A ) ).
% add_0
thf(fact_237_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_238_add__0,axiom,
! [A: multiset_a] :
( ( plus_plus_multiset_a @ zero_zero_multiset_a @ A )
= A ) ).
% add_0
thf(fact_239_zero__eq__add__iff__both__eq__0,axiom,
! [X3: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X3 @ Y ) )
= ( ( X3 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_240_add__eq__0__iff__both__eq__0,axiom,
! [X3: nat,Y: nat] :
( ( ( plus_plus_nat @ X3 @ Y )
= zero_zero_nat )
= ( ( X3 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_241_add__cancel__right__right,axiom,
! [A: multiset_multiset_a,B: multiset_multiset_a] :
( ( A
= ( plus_p6738641960240532161iset_a @ A @ B ) )
= ( B = zero_z7799948378220188993iset_a ) ) ).
% add_cancel_right_right
thf(fact_242_add__cancel__right__right,axiom,
! [A: multis501812127501805293od_a_a,B: multis501812127501805293od_a_a] :
( ( A
= ( plus_p6967097483979887908od_a_a @ A @ B ) )
= ( B = zero_z3024502163831676836od_a_a ) ) ).
% add_cancel_right_right
thf(fact_243_add__cancel__right__right,axiom,
! [A: multis4781622204924204013iset_a,B: multis4781622204924204013iset_a] :
( ( A
= ( plus_p1053901119268525860iset_a @ A @ B ) )
= ( B = zero_z4591712717340762532iset_a ) ) ).
% add_cancel_right_right
thf(fact_244_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_245_add__cancel__right__right,axiom,
! [A: multiset_a,B: multiset_a] :
( ( A
= ( plus_plus_multiset_a @ A @ B ) )
= ( B = zero_zero_multiset_a ) ) ).
% add_cancel_right_right
thf(fact_246_add__cancel__right__left,axiom,
! [A: multiset_multiset_a,B: multiset_multiset_a] :
( ( A
= ( plus_p6738641960240532161iset_a @ B @ A ) )
= ( B = zero_z7799948378220188993iset_a ) ) ).
% add_cancel_right_left
thf(fact_247_add__cancel__right__left,axiom,
! [A: multis501812127501805293od_a_a,B: multis501812127501805293od_a_a] :
( ( A
= ( plus_p6967097483979887908od_a_a @ B @ A ) )
= ( B = zero_z3024502163831676836od_a_a ) ) ).
% add_cancel_right_left
thf(fact_248_add__cancel__right__left,axiom,
! [A: multis4781622204924204013iset_a,B: multis4781622204924204013iset_a] :
( ( A
= ( plus_p1053901119268525860iset_a @ B @ A ) )
= ( B = zero_z4591712717340762532iset_a ) ) ).
% add_cancel_right_left
thf(fact_249_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_250_add__cancel__right__left,axiom,
! [A: multiset_a,B: multiset_a] :
( ( A
= ( plus_plus_multiset_a @ B @ A ) )
= ( B = zero_zero_multiset_a ) ) ).
% add_cancel_right_left
thf(fact_251_add__cancel__left__right,axiom,
! [A: multiset_multiset_a,B: multiset_multiset_a] :
( ( ( plus_p6738641960240532161iset_a @ A @ B )
= A )
= ( B = zero_z7799948378220188993iset_a ) ) ).
% add_cancel_left_right
thf(fact_252_add__cancel__left__right,axiom,
! [A: multis501812127501805293od_a_a,B: multis501812127501805293od_a_a] :
( ( ( plus_p6967097483979887908od_a_a @ A @ B )
= A )
= ( B = zero_z3024502163831676836od_a_a ) ) ).
% add_cancel_left_right
thf(fact_253_add__cancel__left__right,axiom,
! [A: multis4781622204924204013iset_a,B: multis4781622204924204013iset_a] :
( ( ( plus_p1053901119268525860iset_a @ A @ B )
= A )
= ( B = zero_z4591712717340762532iset_a ) ) ).
% add_cancel_left_right
thf(fact_254_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_255_add__cancel__left__right,axiom,
! [A: multiset_a,B: multiset_a] :
( ( ( plus_plus_multiset_a @ A @ B )
= A )
= ( B = zero_zero_multiset_a ) ) ).
% add_cancel_left_right
thf(fact_256_add__cancel__left__left,axiom,
! [B: multiset_multiset_a,A: multiset_multiset_a] :
( ( ( plus_p6738641960240532161iset_a @ B @ A )
= A )
= ( B = zero_z7799948378220188993iset_a ) ) ).
% add_cancel_left_left
thf(fact_257_add__cancel__left__left,axiom,
! [B: multis501812127501805293od_a_a,A: multis501812127501805293od_a_a] :
( ( ( plus_p6967097483979887908od_a_a @ B @ A )
= A )
= ( B = zero_z3024502163831676836od_a_a ) ) ).
% add_cancel_left_left
thf(fact_258_add__cancel__left__left,axiom,
! [B: multis4781622204924204013iset_a,A: multis4781622204924204013iset_a] :
( ( ( plus_p1053901119268525860iset_a @ B @ A )
= A )
= ( B = zero_z4591712717340762532iset_a ) ) ).
% add_cancel_left_left
thf(fact_259_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_260_add__cancel__left__left,axiom,
! [B: multiset_a,A: multiset_a] :
( ( ( plus_plus_multiset_a @ B @ A )
= A )
= ( B = zero_zero_multiset_a ) ) ).
% add_cancel_left_left
thf(fact_261_add_Oright__neutral,axiom,
! [A: multiset_multiset_a] :
( ( plus_p6738641960240532161iset_a @ A @ zero_z7799948378220188993iset_a )
= A ) ).
% add.right_neutral
thf(fact_262_add_Oright__neutral,axiom,
! [A: multis501812127501805293od_a_a] :
( ( plus_p6967097483979887908od_a_a @ A @ zero_z3024502163831676836od_a_a )
= A ) ).
% add.right_neutral
thf(fact_263_add_Oright__neutral,axiom,
! [A: multis4781622204924204013iset_a] :
( ( plus_p1053901119268525860iset_a @ A @ zero_z4591712717340762532iset_a )
= A ) ).
% add.right_neutral
thf(fact_264_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_265_add_Oright__neutral,axiom,
! [A: multiset_a] :
( ( plus_plus_multiset_a @ A @ zero_zero_multiset_a )
= A ) ).
% add.right_neutral
thf(fact_266_zero__reorient,axiom,
! [X3: multiset_multiset_a] :
( ( zero_z7799948378220188993iset_a = X3 )
= ( X3 = zero_z7799948378220188993iset_a ) ) ).
% zero_reorient
thf(fact_267_zero__reorient,axiom,
! [X3: multis501812127501805293od_a_a] :
( ( zero_z3024502163831676836od_a_a = X3 )
= ( X3 = zero_z3024502163831676836od_a_a ) ) ).
% zero_reorient
thf(fact_268_zero__reorient,axiom,
! [X3: multis4781622204924204013iset_a] :
( ( zero_z4591712717340762532iset_a = X3 )
= ( X3 = zero_z4591712717340762532iset_a ) ) ).
% zero_reorient
thf(fact_269_zero__reorient,axiom,
! [X3: nat] :
( ( zero_zero_nat = X3 )
= ( X3 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_270_zero__reorient,axiom,
! [X3: multiset_a] :
( ( zero_zero_multiset_a = X3 )
= ( X3 = zero_zero_multiset_a ) ) ).
% zero_reorient
thf(fact_271_add__right__imp__eq,axiom,
! [B: multis501812127501805293od_a_a,A: multis501812127501805293od_a_a,C2: multis501812127501805293od_a_a] :
( ( ( plus_p6967097483979887908od_a_a @ B @ A )
= ( plus_p6967097483979887908od_a_a @ C2 @ A ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_272_add__right__imp__eq,axiom,
! [B: multis4781622204924204013iset_a,A: multis4781622204924204013iset_a,C2: multis4781622204924204013iset_a] :
( ( ( plus_p1053901119268525860iset_a @ B @ A )
= ( plus_p1053901119268525860iset_a @ C2 @ A ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_273_add__right__imp__eq,axiom,
! [B: multiset_multiset_a,A: multiset_multiset_a,C2: multiset_multiset_a] :
( ( ( plus_p6738641960240532161iset_a @ B @ A )
= ( plus_p6738641960240532161iset_a @ C2 @ A ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_274_add__right__imp__eq,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C2 @ A ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_275_add__right__imp__eq,axiom,
! [B: multiset_a,A: multiset_a,C2: multiset_a] :
( ( ( plus_plus_multiset_a @ B @ A )
= ( plus_plus_multiset_a @ C2 @ A ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_276_add__left__imp__eq,axiom,
! [A: multis501812127501805293od_a_a,B: multis501812127501805293od_a_a,C2: multis501812127501805293od_a_a] :
( ( ( plus_p6967097483979887908od_a_a @ A @ B )
= ( plus_p6967097483979887908od_a_a @ A @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_277_add__left__imp__eq,axiom,
! [A: multis4781622204924204013iset_a,B: multis4781622204924204013iset_a,C2: multis4781622204924204013iset_a] :
( ( ( plus_p1053901119268525860iset_a @ A @ B )
= ( plus_p1053901119268525860iset_a @ A @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_278_add__left__imp__eq,axiom,
! [A: multiset_multiset_a,B: multiset_multiset_a,C2: multiset_multiset_a] :
( ( ( plus_p6738641960240532161iset_a @ A @ B )
= ( plus_p6738641960240532161iset_a @ A @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_279_add__left__imp__eq,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_280_add__left__imp__eq,axiom,
! [A: multiset_a,B: multiset_a,C2: multiset_a] :
( ( ( plus_plus_multiset_a @ A @ B )
= ( plus_plus_multiset_a @ A @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_281_add_Oleft__commute,axiom,
! [B: multis501812127501805293od_a_a,A: multis501812127501805293od_a_a,C2: multis501812127501805293od_a_a] :
( ( plus_p6967097483979887908od_a_a @ B @ ( plus_p6967097483979887908od_a_a @ A @ C2 ) )
= ( plus_p6967097483979887908od_a_a @ A @ ( plus_p6967097483979887908od_a_a @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_282_add_Oleft__commute,axiom,
! [B: multis4781622204924204013iset_a,A: multis4781622204924204013iset_a,C2: multis4781622204924204013iset_a] :
( ( plus_p1053901119268525860iset_a @ B @ ( plus_p1053901119268525860iset_a @ A @ C2 ) )
= ( plus_p1053901119268525860iset_a @ A @ ( plus_p1053901119268525860iset_a @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_283_add_Oleft__commute,axiom,
! [B: multiset_multiset_a,A: multiset_multiset_a,C2: multiset_multiset_a] :
( ( plus_p6738641960240532161iset_a @ B @ ( plus_p6738641960240532161iset_a @ A @ C2 ) )
= ( plus_p6738641960240532161iset_a @ A @ ( plus_p6738641960240532161iset_a @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_284_add_Oleft__commute,axiom,
! [B: nat,A: nat,C2: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C2 ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_285_add_Oleft__commute,axiom,
! [B: multiset_a,A: multiset_a,C2: multiset_a] :
( ( plus_plus_multiset_a @ B @ ( plus_plus_multiset_a @ A @ C2 ) )
= ( plus_plus_multiset_a @ A @ ( plus_plus_multiset_a @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_286_add_Ocommute,axiom,
( plus_p6967097483979887908od_a_a
= ( ^ [A5: multis501812127501805293od_a_a,B5: multis501812127501805293od_a_a] : ( plus_p6967097483979887908od_a_a @ B5 @ A5 ) ) ) ).
% add.commute
thf(fact_287_add_Ocommute,axiom,
( plus_p1053901119268525860iset_a
= ( ^ [A5: multis4781622204924204013iset_a,B5: multis4781622204924204013iset_a] : ( plus_p1053901119268525860iset_a @ B5 @ A5 ) ) ) ).
% add.commute
thf(fact_288_add_Ocommute,axiom,
( plus_p6738641960240532161iset_a
= ( ^ [A5: multiset_multiset_a,B5: multiset_multiset_a] : ( plus_p6738641960240532161iset_a @ B5 @ A5 ) ) ) ).
% add.commute
thf(fact_289_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A5: nat,B5: nat] : ( plus_plus_nat @ B5 @ A5 ) ) ) ).
% add.commute
thf(fact_290_add_Ocommute,axiom,
( plus_plus_multiset_a
= ( ^ [A5: multiset_a,B5: multiset_a] : ( plus_plus_multiset_a @ B5 @ A5 ) ) ) ).
% add.commute
thf(fact_291_add_Oassoc,axiom,
! [A: multis501812127501805293od_a_a,B: multis501812127501805293od_a_a,C2: multis501812127501805293od_a_a] :
( ( plus_p6967097483979887908od_a_a @ ( plus_p6967097483979887908od_a_a @ A @ B ) @ C2 )
= ( plus_p6967097483979887908od_a_a @ A @ ( plus_p6967097483979887908od_a_a @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_292_add_Oassoc,axiom,
! [A: multis4781622204924204013iset_a,B: multis4781622204924204013iset_a,C2: multis4781622204924204013iset_a] :
( ( plus_p1053901119268525860iset_a @ ( plus_p1053901119268525860iset_a @ A @ B ) @ C2 )
= ( plus_p1053901119268525860iset_a @ A @ ( plus_p1053901119268525860iset_a @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_293_add_Oassoc,axiom,
! [A: multiset_multiset_a,B: multiset_multiset_a,C2: multiset_multiset_a] :
( ( plus_p6738641960240532161iset_a @ ( plus_p6738641960240532161iset_a @ A @ B ) @ C2 )
= ( plus_p6738641960240532161iset_a @ A @ ( plus_p6738641960240532161iset_a @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_294_add_Oassoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_295_add_Oassoc,axiom,
! [A: multiset_a,B: multiset_a,C2: multiset_a] :
( ( plus_plus_multiset_a @ ( plus_plus_multiset_a @ A @ B ) @ C2 )
= ( plus_plus_multiset_a @ A @ ( plus_plus_multiset_a @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_296_group__cancel_Oadd2,axiom,
! [B3: multis501812127501805293od_a_a,K3: multis501812127501805293od_a_a,B: multis501812127501805293od_a_a,A: multis501812127501805293od_a_a] :
( ( B3
= ( plus_p6967097483979887908od_a_a @ K3 @ B ) )
=> ( ( plus_p6967097483979887908od_a_a @ A @ B3 )
= ( plus_p6967097483979887908od_a_a @ K3 @ ( plus_p6967097483979887908od_a_a @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_297_group__cancel_Oadd2,axiom,
! [B3: multis4781622204924204013iset_a,K3: multis4781622204924204013iset_a,B: multis4781622204924204013iset_a,A: multis4781622204924204013iset_a] :
( ( B3
= ( plus_p1053901119268525860iset_a @ K3 @ B ) )
=> ( ( plus_p1053901119268525860iset_a @ A @ B3 )
= ( plus_p1053901119268525860iset_a @ K3 @ ( plus_p1053901119268525860iset_a @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_298_group__cancel_Oadd2,axiom,
! [B3: multiset_multiset_a,K3: multiset_multiset_a,B: multiset_multiset_a,A: multiset_multiset_a] :
( ( B3
= ( plus_p6738641960240532161iset_a @ K3 @ B ) )
=> ( ( plus_p6738641960240532161iset_a @ A @ B3 )
= ( plus_p6738641960240532161iset_a @ K3 @ ( plus_p6738641960240532161iset_a @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_299_group__cancel_Oadd2,axiom,
! [B3: nat,K3: nat,B: nat,A: nat] :
( ( B3
= ( plus_plus_nat @ K3 @ B ) )
=> ( ( plus_plus_nat @ A @ B3 )
= ( plus_plus_nat @ K3 @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_300_group__cancel_Oadd2,axiom,
! [B3: multiset_a,K3: multiset_a,B: multiset_a,A: multiset_a] :
( ( B3
= ( plus_plus_multiset_a @ K3 @ B ) )
=> ( ( plus_plus_multiset_a @ A @ B3 )
= ( plus_plus_multiset_a @ K3 @ ( plus_plus_multiset_a @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_301_group__cancel_Oadd1,axiom,
! [A3: multis501812127501805293od_a_a,K3: multis501812127501805293od_a_a,A: multis501812127501805293od_a_a,B: multis501812127501805293od_a_a] :
( ( A3
= ( plus_p6967097483979887908od_a_a @ K3 @ A ) )
=> ( ( plus_p6967097483979887908od_a_a @ A3 @ B )
= ( plus_p6967097483979887908od_a_a @ K3 @ ( plus_p6967097483979887908od_a_a @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_302_group__cancel_Oadd1,axiom,
! [A3: multis4781622204924204013iset_a,K3: multis4781622204924204013iset_a,A: multis4781622204924204013iset_a,B: multis4781622204924204013iset_a] :
( ( A3
= ( plus_p1053901119268525860iset_a @ K3 @ A ) )
=> ( ( plus_p1053901119268525860iset_a @ A3 @ B )
= ( plus_p1053901119268525860iset_a @ K3 @ ( plus_p1053901119268525860iset_a @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_303_group__cancel_Oadd1,axiom,
! [A3: multiset_multiset_a,K3: multiset_multiset_a,A: multiset_multiset_a,B: multiset_multiset_a] :
( ( A3
= ( plus_p6738641960240532161iset_a @ K3 @ A ) )
=> ( ( plus_p6738641960240532161iset_a @ A3 @ B )
= ( plus_p6738641960240532161iset_a @ K3 @ ( plus_p6738641960240532161iset_a @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_304_group__cancel_Oadd1,axiom,
! [A3: nat,K3: nat,A: nat,B: nat] :
( ( A3
= ( plus_plus_nat @ K3 @ A ) )
=> ( ( plus_plus_nat @ A3 @ B )
= ( plus_plus_nat @ K3 @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_305_group__cancel_Oadd1,axiom,
! [A3: multiset_a,K3: multiset_a,A: multiset_a,B: multiset_a] :
( ( A3
= ( plus_plus_multiset_a @ K3 @ A ) )
=> ( ( plus_plus_multiset_a @ A3 @ B )
= ( plus_plus_multiset_a @ K3 @ ( plus_plus_multiset_a @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_306_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I3: nat,J3: nat,K3: nat,L: nat] :
( ( ( I3 = J3 )
& ( K3 = L ) )
=> ( ( plus_plus_nat @ I3 @ K3 )
= ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_307_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: multis501812127501805293od_a_a,B: multis501812127501805293od_a_a,C2: multis501812127501805293od_a_a] :
( ( plus_p6967097483979887908od_a_a @ ( plus_p6967097483979887908od_a_a @ A @ B ) @ C2 )
= ( plus_p6967097483979887908od_a_a @ A @ ( plus_p6967097483979887908od_a_a @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_308_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: multis4781622204924204013iset_a,B: multis4781622204924204013iset_a,C2: multis4781622204924204013iset_a] :
( ( plus_p1053901119268525860iset_a @ ( plus_p1053901119268525860iset_a @ A @ B ) @ C2 )
= ( plus_p1053901119268525860iset_a @ A @ ( plus_p1053901119268525860iset_a @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_309_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: multiset_multiset_a,B: multiset_multiset_a,C2: multiset_multiset_a] :
( ( plus_p6738641960240532161iset_a @ ( plus_p6738641960240532161iset_a @ A @ B ) @ C2 )
= ( plus_p6738641960240532161iset_a @ A @ ( plus_p6738641960240532161iset_a @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_310_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_311_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: multiset_a,B: multiset_a,C2: multiset_a] :
( ( plus_plus_multiset_a @ ( plus_plus_multiset_a @ A @ B ) @ C2 )
= ( plus_plus_multiset_a @ A @ ( plus_plus_multiset_a @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_312_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: nat,C2: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C2 ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C2 ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_313_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: multiset_a,C2: multiset_a,B: multiset_a] :
( ( minus_3765977307040488491iset_a @ ( minus_3765977307040488491iset_a @ A @ C2 ) @ B )
= ( minus_3765977307040488491iset_a @ ( minus_3765977307040488491iset_a @ A @ B ) @ C2 ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_314_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: multis4781622204924204013iset_a,C2: multis4781622204924204013iset_a,B: multis4781622204924204013iset_a] :
( ( minus_9211619285413683668iset_a @ ( minus_9211619285413683668iset_a @ A @ C2 ) @ B )
= ( minus_9211619285413683668iset_a @ ( minus_9211619285413683668iset_a @ A @ B ) @ C2 ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_315_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: multis501812127501805293od_a_a,C2: multis501812127501805293od_a_a,B: multis501812127501805293od_a_a] :
( ( minus_7313152569058643924od_a_a @ ( minus_7313152569058643924od_a_a @ A @ C2 ) @ B )
= ( minus_7313152569058643924od_a_a @ ( minus_7313152569058643924od_a_a @ A @ B ) @ C2 ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_316_add_Ocomm__neutral,axiom,
! [A: multiset_multiset_a] :
( ( plus_p6738641960240532161iset_a @ A @ zero_z7799948378220188993iset_a )
= A ) ).
% add.comm_neutral
thf(fact_317_add_Ocomm__neutral,axiom,
! [A: multis501812127501805293od_a_a] :
( ( plus_p6967097483979887908od_a_a @ A @ zero_z3024502163831676836od_a_a )
= A ) ).
% add.comm_neutral
thf(fact_318_add_Ocomm__neutral,axiom,
! [A: multis4781622204924204013iset_a] :
( ( plus_p1053901119268525860iset_a @ A @ zero_z4591712717340762532iset_a )
= A ) ).
% add.comm_neutral
thf(fact_319_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_320_add_Ocomm__neutral,axiom,
! [A: multiset_a] :
( ( plus_plus_multiset_a @ A @ zero_zero_multiset_a )
= A ) ).
% add.comm_neutral
thf(fact_321_comm__monoid__add__class_Oadd__0,axiom,
! [A: multiset_multiset_a] :
( ( plus_p6738641960240532161iset_a @ zero_z7799948378220188993iset_a @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_322_comm__monoid__add__class_Oadd__0,axiom,
! [A: multis501812127501805293od_a_a] :
( ( plus_p6967097483979887908od_a_a @ zero_z3024502163831676836od_a_a @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_323_comm__monoid__add__class_Oadd__0,axiom,
! [A: multis4781622204924204013iset_a] :
( ( plus_p1053901119268525860iset_a @ zero_z4591712717340762532iset_a @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_324_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_325_comm__monoid__add__class_Oadd__0,axiom,
! [A: multiset_a] :
( ( plus_plus_multiset_a @ zero_zero_multiset_a @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_326_add__implies__diff,axiom,
! [C2: multiset_multiset_a,B: multiset_multiset_a,A: multiset_multiset_a] :
( ( ( plus_p6738641960240532161iset_a @ C2 @ B )
= A )
=> ( C2
= ( minus_3395427628221709681iset_a @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_327_add__implies__diff,axiom,
! [C2: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C2 @ B )
= A )
=> ( C2
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_328_add__implies__diff,axiom,
! [C2: multiset_a,B: multiset_a,A: multiset_a] :
( ( ( plus_plus_multiset_a @ C2 @ B )
= A )
=> ( C2
= ( minus_3765977307040488491iset_a @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_329_add__implies__diff,axiom,
! [C2: multis4781622204924204013iset_a,B: multis4781622204924204013iset_a,A: multis4781622204924204013iset_a] :
( ( ( plus_p1053901119268525860iset_a @ C2 @ B )
= A )
=> ( C2
= ( minus_9211619285413683668iset_a @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_330_add__implies__diff,axiom,
! [C2: multis501812127501805293od_a_a,B: multis501812127501805293od_a_a,A: multis501812127501805293od_a_a] :
( ( ( plus_p6967097483979887908od_a_a @ C2 @ B )
= A )
=> ( C2
= ( minus_7313152569058643924od_a_a @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_331_diff__diff__eq,axiom,
! [A: multiset_multiset_a,B: multiset_multiset_a,C2: multiset_multiset_a] :
( ( minus_3395427628221709681iset_a @ ( minus_3395427628221709681iset_a @ A @ B ) @ C2 )
= ( minus_3395427628221709681iset_a @ A @ ( plus_p6738641960240532161iset_a @ B @ C2 ) ) ) ).
% diff_diff_eq
thf(fact_332_diff__diff__eq,axiom,
! [A: nat,B: nat,C2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C2 )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% diff_diff_eq
thf(fact_333_diff__diff__eq,axiom,
! [A: multiset_a,B: multiset_a,C2: multiset_a] :
( ( minus_3765977307040488491iset_a @ ( minus_3765977307040488491iset_a @ A @ B ) @ C2 )
= ( minus_3765977307040488491iset_a @ A @ ( plus_plus_multiset_a @ B @ C2 ) ) ) ).
% diff_diff_eq
thf(fact_334_diff__diff__eq,axiom,
! [A: multis4781622204924204013iset_a,B: multis4781622204924204013iset_a,C2: multis4781622204924204013iset_a] :
( ( minus_9211619285413683668iset_a @ ( minus_9211619285413683668iset_a @ A @ B ) @ C2 )
= ( minus_9211619285413683668iset_a @ A @ ( plus_p1053901119268525860iset_a @ B @ C2 ) ) ) ).
% diff_diff_eq
thf(fact_335_diff__diff__eq,axiom,
! [A: multis501812127501805293od_a_a,B: multis501812127501805293od_a_a,C2: multis501812127501805293od_a_a] :
( ( minus_7313152569058643924od_a_a @ ( minus_7313152569058643924od_a_a @ A @ B ) @ C2 )
= ( minus_7313152569058643924od_a_a @ A @ ( plus_p6967097483979887908od_a_a @ B @ C2 ) ) ) ).
% diff_diff_eq
thf(fact_336_I__def,axiom,
( i
= ( inter_mset_a @ m @ n ) ) ).
% I_def
thf(fact_337_subset__mset_Osum__mset__0__iff,axiom,
! [M: multis8895438461125693264iset_a] :
( ( ( comm_m7387914173477526523iset_a @ plus_p6738641960240532161iset_a @ zero_z7799948378220188993iset_a @ M )
= zero_z7799948378220188993iset_a )
= ( ! [X6: multiset_multiset_a] :
( ( member7618379257985549619iset_a @ X6 @ ( set_ms6500351741012541767iset_a @ M ) )
=> ( X6 = zero_z7799948378220188993iset_a ) ) ) ) ).
% subset_mset.sum_mset_0_iff
thf(fact_338_subset__mset_Osum__mset__0__iff,axiom,
! [M: multis5935151272263360051od_a_a] :
( ( ( comm_m7674388060029416798od_a_a @ plus_p6967097483979887908od_a_a @ zero_z3024502163831676836od_a_a @ M )
= zero_z3024502163831676836od_a_a )
= ( ! [X6: multis501812127501805293od_a_a] :
( ( member7700929758516679318od_a_a @ X6 @ ( set_ms8253844250382261418od_a_a @ M ) )
=> ( X6 = zero_z3024502163831676836od_a_a ) ) ) ) ).
% subset_mset.sum_mset_0_iff
thf(fact_339_subset__mset_Osum__mset__0__iff,axiom,
! [M: multis7221419791219231795iset_a] :
( ( ( comm_m7448400206062208350iset_a @ plus_p1053901119268525860iset_a @ zero_z4591712717340762532iset_a @ M )
= zero_z4591712717340762532iset_a )
= ( ! [X6: multis4781622204924204013iset_a] :
( ( member2597461214246177430iset_a @ X6 @ ( set_ms5327031259215843498iset_a @ M ) )
=> ( X6 = zero_z4591712717340762532iset_a ) ) ) ) ).
% subset_mset.sum_mset_0_iff
thf(fact_340_subset__mset_Osum__mset__0__iff,axiom,
! [M: multiset_multiset_a] :
( ( ( comm_m1977238983017651125iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ M )
= zero_zero_multiset_a )
= ( ! [X6: multiset_a] :
( ( member_multiset_a @ X6 @ ( set_mset_multiset_a @ M ) )
=> ( X6 = zero_zero_multiset_a ) ) ) ) ).
% subset_mset.sum_mset_0_iff
thf(fact_341_Multiset_Ois__empty__def,axiom,
( is_empty_multiset_a
= ( ^ [A6: multiset_multiset_a] : ( A6 = zero_z7799948378220188993iset_a ) ) ) ).
% Multiset.is_empty_def
thf(fact_342_Multiset_Ois__empty__def,axiom,
( is_emp1975280692358257496od_a_a
= ( ^ [A6: multis501812127501805293od_a_a] : ( A6 = zero_z3024502163831676836od_a_a ) ) ) ).
% Multiset.is_empty_def
thf(fact_343_Multiset_Ois__empty__def,axiom,
( is_emp916564191726758488iset_a
= ( ^ [A6: multis4781622204924204013iset_a] : ( A6 = zero_z4591712717340762532iset_a ) ) ) ).
% Multiset.is_empty_def
thf(fact_344_Multiset_Ois__empty__def,axiom,
( is_empty_a
= ( ^ [A6: multiset_a] : ( A6 = zero_zero_multiset_a ) ) ) ).
% Multiset.is_empty_def
thf(fact_345_verit__sum__simplify,axiom,
! [A: multiset_multiset_a] :
( ( plus_p6738641960240532161iset_a @ A @ zero_z7799948378220188993iset_a )
= A ) ).
% verit_sum_simplify
thf(fact_346_verit__sum__simplify,axiom,
! [A: multis501812127501805293od_a_a] :
( ( plus_p6967097483979887908od_a_a @ A @ zero_z3024502163831676836od_a_a )
= A ) ).
% verit_sum_simplify
thf(fact_347_verit__sum__simplify,axiom,
! [A: multis4781622204924204013iset_a] :
( ( plus_p1053901119268525860iset_a @ A @ zero_z4591712717340762532iset_a )
= A ) ).
% verit_sum_simplify
thf(fact_348_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_349_verit__sum__simplify,axiom,
! [A: multiset_a] :
( ( plus_plus_multiset_a @ A @ zero_zero_multiset_a )
= A ) ).
% verit_sum_simplify
thf(fact_350_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_351_insert__DiffM2,axiom,
! [X3: produc5665077874975597415od_a_a,M: multis3377851686571972653od_a_a] :
( ( member1567053331504362128od_a_a @ X3 @ ( set_ms5376707576967697060od_a_a @ M ) )
=> ( ( plus_p1121120397469056356od_a_a @ ( minus_6383096217462597652od_a_a @ M @ ( add_ms9136373847686211141od_a_a @ X3 @ zero_z3721308388212328420od_a_a ) ) @ ( add_ms9136373847686211141od_a_a @ X3 @ zero_z3721308388212328420od_a_a ) )
= M ) ) ).
% insert_DiffM2
thf(fact_352_insert__DiffM2,axiom,
! [X3: produc4424810744984089383iset_a,M: multis1386120040344743661iset_a] :
( ( member1345422370561994576iset_a @ X3 @ ( set_ms2882773486807528292iset_a @ M ) )
=> ( ( plus_p2529605555928385316iset_a @ ( minus_2536987971823575508iset_a @ M @ ( add_ms9106060982595931141iset_a @ X3 @ zero_z9101083372863164836iset_a ) ) @ ( add_ms9106060982595931141iset_a @ X3 @ zero_z9101083372863164836iset_a ) )
= M ) ) ).
% insert_DiffM2
thf(fact_353_insert__DiffM2,axiom,
! [X3: multis501812127501805293od_a_a,M: multis5935151272263360051od_a_a] :
( ( member7700929758516679318od_a_a @ X3 @ ( set_ms8253844250382261418od_a_a @ M ) )
=> ( ( plus_p2133152728643999466od_a_a @ ( minus_385666609031338394od_a_a @ M @ ( add_ms4156406267932395723od_a_a @ X3 @ zero_z8896958667708014954od_a_a ) ) @ ( add_ms4156406267932395723od_a_a @ X3 @ zero_z8896958667708014954od_a_a ) )
= M ) ) ).
% insert_DiffM2
thf(fact_354_insert__DiffM2,axiom,
! [X3: multis4781622204924204013iset_a,M: multis7221419791219231795iset_a] :
( ( member2597461214246177430iset_a @ X3 @ ( set_ms5327031259215843498iset_a @ M ) )
=> ( ( plus_p1436237118943648234iset_a @ ( minus_4490418833869548698iset_a @ M @ ( add_ms7610149848545062859iset_a @ X3 @ zero_z8662914267040441450iset_a ) ) @ ( add_ms7610149848545062859iset_a @ X3 @ zero_z8662914267040441450iset_a ) )
= M ) ) ).
% insert_DiffM2
thf(fact_355_insert__DiffM2,axiom,
! [X3: multiset_a,M: multiset_multiset_a] :
( ( member_multiset_a @ X3 @ ( set_mset_multiset_a @ M ) )
=> ( ( plus_p6738641960240532161iset_a @ ( minus_3395427628221709681iset_a @ M @ ( add_mset_multiset_a @ X3 @ zero_z7799948378220188993iset_a ) ) @ ( add_mset_multiset_a @ X3 @ zero_z7799948378220188993iset_a ) )
= M ) ) ).
% insert_DiffM2
thf(fact_356_insert__DiffM2,axiom,
! [X3: a,M: multiset_a] :
( ( member_a @ X3 @ ( set_mset_a @ M ) )
=> ( ( plus_plus_multiset_a @ ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ X3 @ zero_zero_multiset_a ) ) @ ( add_mset_a @ X3 @ zero_zero_multiset_a ) )
= M ) ) ).
% insert_DiffM2
thf(fact_357_insert__DiffM2,axiom,
! [X3: produc6518373309651786023iset_a,M: multis4781622204924204013iset_a] :
( ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ M ) )
=> ( ( plus_p1053901119268525860iset_a @ ( minus_9211619285413683668iset_a @ M @ ( add_ms5348986034529679365iset_a @ X3 @ zero_z4591712717340762532iset_a ) ) @ ( add_ms5348986034529679365iset_a @ X3 @ zero_z4591712717340762532iset_a ) )
= M ) ) ).
% insert_DiffM2
thf(fact_358_insert__DiffM2,axiom,
! [X3: product_prod_a_a,M: multis501812127501805293od_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ M ) )
=> ( ( plus_p6967097483979887908od_a_a @ ( minus_7313152569058643924od_a_a @ M @ ( add_ms8655138167283798533od_a_a @ X3 @ zero_z3024502163831676836od_a_a ) ) @ ( add_ms8655138167283798533od_a_a @ X3 @ zero_z3024502163831676836od_a_a ) )
= M ) ) ).
% insert_DiffM2
thf(fact_359_diff__union__single__conv,axiom,
! [A: produc5665077874975597415od_a_a,J2: multis3377851686571972653od_a_a,I2: multis3377851686571972653od_a_a] :
( ( member1567053331504362128od_a_a @ A @ ( set_ms5376707576967697060od_a_a @ J2 ) )
=> ( ( minus_6383096217462597652od_a_a @ ( plus_p1121120397469056356od_a_a @ I2 @ J2 ) @ ( add_ms9136373847686211141od_a_a @ A @ zero_z3721308388212328420od_a_a ) )
= ( plus_p1121120397469056356od_a_a @ I2 @ ( minus_6383096217462597652od_a_a @ J2 @ ( add_ms9136373847686211141od_a_a @ A @ zero_z3721308388212328420od_a_a ) ) ) ) ) ).
% diff_union_single_conv
thf(fact_360_diff__union__single__conv,axiom,
! [A: produc4424810744984089383iset_a,J2: multis1386120040344743661iset_a,I2: multis1386120040344743661iset_a] :
( ( member1345422370561994576iset_a @ A @ ( set_ms2882773486807528292iset_a @ J2 ) )
=> ( ( minus_2536987971823575508iset_a @ ( plus_p2529605555928385316iset_a @ I2 @ J2 ) @ ( add_ms9106060982595931141iset_a @ A @ zero_z9101083372863164836iset_a ) )
= ( plus_p2529605555928385316iset_a @ I2 @ ( minus_2536987971823575508iset_a @ J2 @ ( add_ms9106060982595931141iset_a @ A @ zero_z9101083372863164836iset_a ) ) ) ) ) ).
% diff_union_single_conv
thf(fact_361_diff__union__single__conv,axiom,
! [A: multis501812127501805293od_a_a,J2: multis5935151272263360051od_a_a,I2: multis5935151272263360051od_a_a] :
( ( member7700929758516679318od_a_a @ A @ ( set_ms8253844250382261418od_a_a @ J2 ) )
=> ( ( minus_385666609031338394od_a_a @ ( plus_p2133152728643999466od_a_a @ I2 @ J2 ) @ ( add_ms4156406267932395723od_a_a @ A @ zero_z8896958667708014954od_a_a ) )
= ( plus_p2133152728643999466od_a_a @ I2 @ ( minus_385666609031338394od_a_a @ J2 @ ( add_ms4156406267932395723od_a_a @ A @ zero_z8896958667708014954od_a_a ) ) ) ) ) ).
% diff_union_single_conv
thf(fact_362_diff__union__single__conv,axiom,
! [A: multis4781622204924204013iset_a,J2: multis7221419791219231795iset_a,I2: multis7221419791219231795iset_a] :
( ( member2597461214246177430iset_a @ A @ ( set_ms5327031259215843498iset_a @ J2 ) )
=> ( ( minus_4490418833869548698iset_a @ ( plus_p1436237118943648234iset_a @ I2 @ J2 ) @ ( add_ms7610149848545062859iset_a @ A @ zero_z8662914267040441450iset_a ) )
= ( plus_p1436237118943648234iset_a @ I2 @ ( minus_4490418833869548698iset_a @ J2 @ ( add_ms7610149848545062859iset_a @ A @ zero_z8662914267040441450iset_a ) ) ) ) ) ).
% diff_union_single_conv
thf(fact_363_diff__union__single__conv,axiom,
! [A: multiset_a,J2: multiset_multiset_a,I2: multiset_multiset_a] :
( ( member_multiset_a @ A @ ( set_mset_multiset_a @ J2 ) )
=> ( ( minus_3395427628221709681iset_a @ ( plus_p6738641960240532161iset_a @ I2 @ J2 ) @ ( add_mset_multiset_a @ A @ zero_z7799948378220188993iset_a ) )
= ( plus_p6738641960240532161iset_a @ I2 @ ( minus_3395427628221709681iset_a @ J2 @ ( add_mset_multiset_a @ A @ zero_z7799948378220188993iset_a ) ) ) ) ) ).
% diff_union_single_conv
thf(fact_364_diff__union__single__conv,axiom,
! [A: a,J2: multiset_a,I2: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ J2 ) )
=> ( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ I2 @ J2 ) @ ( add_mset_a @ A @ zero_zero_multiset_a ) )
= ( plus_plus_multiset_a @ I2 @ ( minus_3765977307040488491iset_a @ J2 @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) ) ) ).
% diff_union_single_conv
thf(fact_365_diff__union__single__conv,axiom,
! [A: produc6518373309651786023iset_a,J2: multis4781622204924204013iset_a,I2: multis4781622204924204013iset_a] :
( ( member5199237121806060112iset_a @ A @ ( set_ms2379723421352415844iset_a @ J2 ) )
=> ( ( minus_9211619285413683668iset_a @ ( plus_p1053901119268525860iset_a @ I2 @ J2 ) @ ( add_ms5348986034529679365iset_a @ A @ zero_z4591712717340762532iset_a ) )
= ( plus_p1053901119268525860iset_a @ I2 @ ( minus_9211619285413683668iset_a @ J2 @ ( add_ms5348986034529679365iset_a @ A @ zero_z4591712717340762532iset_a ) ) ) ) ) ).
% diff_union_single_conv
thf(fact_366_diff__union__single__conv,axiom,
! [A: product_prod_a_a,J2: multis501812127501805293od_a_a,I2: multis501812127501805293od_a_a] :
( ( member1426531477525435216od_a_a @ A @ ( set_ms119794563918438244od_a_a @ J2 ) )
=> ( ( minus_7313152569058643924od_a_a @ ( plus_p6967097483979887908od_a_a @ I2 @ J2 ) @ ( add_ms8655138167283798533od_a_a @ A @ zero_z3024502163831676836od_a_a ) )
= ( plus_p6967097483979887908od_a_a @ I2 @ ( minus_7313152569058643924od_a_a @ J2 @ ( add_ms8655138167283798533od_a_a @ A @ zero_z3024502163831676836od_a_a ) ) ) ) ) ).
% diff_union_single_conv
thf(fact_367_multi__self__add__other__not__self,axiom,
! [M: multis501812127501805293od_a_a,X3: product_prod_a_a] :
( M
!= ( add_ms8655138167283798533od_a_a @ X3 @ M ) ) ).
% multi_self_add_other_not_self
thf(fact_368_multi__self__add__other__not__self,axiom,
! [M: multis4781622204924204013iset_a,X3: produc6518373309651786023iset_a] :
( M
!= ( add_ms5348986034529679365iset_a @ X3 @ M ) ) ).
% multi_self_add_other_not_self
thf(fact_369_multi__self__add__other__not__self,axiom,
! [M: multiset_a,X3: a] :
( M
!= ( add_mset_a @ X3 @ M ) ) ).
% multi_self_add_other_not_self
thf(fact_370_multi__self__add__other__not__self,axiom,
! [M: multiset_multiset_a,X3: multiset_a] :
( M
!= ( add_mset_multiset_a @ X3 @ M ) ) ).
% multi_self_add_other_not_self
thf(fact_371_add__mset__add__mset__same__iff,axiom,
! [A: product_prod_a_a,A3: multis501812127501805293od_a_a,B3: multis501812127501805293od_a_a] :
( ( ( add_ms8655138167283798533od_a_a @ A @ A3 )
= ( add_ms8655138167283798533od_a_a @ A @ B3 ) )
= ( A3 = B3 ) ) ).
% add_mset_add_mset_same_iff
thf(fact_372_add__mset__add__mset__same__iff,axiom,
! [A: produc6518373309651786023iset_a,A3: multis4781622204924204013iset_a,B3: multis4781622204924204013iset_a] :
( ( ( add_ms5348986034529679365iset_a @ A @ A3 )
= ( add_ms5348986034529679365iset_a @ A @ B3 ) )
= ( A3 = B3 ) ) ).
% add_mset_add_mset_same_iff
thf(fact_373_add__mset__add__mset__same__iff,axiom,
! [A: a,A3: multiset_a,B3: multiset_a] :
( ( ( add_mset_a @ A @ A3 )
= ( add_mset_a @ A @ B3 ) )
= ( A3 = B3 ) ) ).
% add_mset_add_mset_same_iff
thf(fact_374_add__mset__add__mset__same__iff,axiom,
! [A: multiset_a,A3: multiset_multiset_a,B3: multiset_multiset_a] :
( ( ( add_mset_multiset_a @ A @ A3 )
= ( add_mset_multiset_a @ A @ B3 ) )
= ( A3 = B3 ) ) ).
% add_mset_add_mset_same_iff
thf(fact_375_subset__mset_Oinf__right__idem,axiom,
! [X3: multis501812127501805293od_a_a,Y: multis501812127501805293od_a_a] :
( ( inter_3651181809579396970od_a_a @ ( inter_3651181809579396970od_a_a @ X3 @ Y ) @ Y )
= ( inter_3651181809579396970od_a_a @ X3 @ Y ) ) ).
% subset_mset.inf_right_idem
thf(fact_376_subset__mset_Oinf__right__idem,axiom,
! [X3: multis4781622204924204013iset_a,Y: multis4781622204924204013iset_a] :
( ( inter_1588963877939557482iset_a @ ( inter_1588963877939557482iset_a @ X3 @ Y ) @ Y )
= ( inter_1588963877939557482iset_a @ X3 @ Y ) ) ).
% subset_mset.inf_right_idem
thf(fact_377_subset__mset_Oinf__right__idem,axiom,
! [X3: multiset_a,Y: multiset_a] :
( ( inter_mset_a @ ( inter_mset_a @ X3 @ Y ) @ Y )
= ( inter_mset_a @ X3 @ Y ) ) ).
% subset_mset.inf_right_idem
thf(fact_378_subset__mset_Oinf_Oright__idem,axiom,
! [A: multis501812127501805293od_a_a,B: multis501812127501805293od_a_a] :
( ( inter_3651181809579396970od_a_a @ ( inter_3651181809579396970od_a_a @ A @ B ) @ B )
= ( inter_3651181809579396970od_a_a @ A @ B ) ) ).
% subset_mset.inf.right_idem
thf(fact_379_subset__mset_Oinf_Oright__idem,axiom,
! [A: multis4781622204924204013iset_a,B: multis4781622204924204013iset_a] :
( ( inter_1588963877939557482iset_a @ ( inter_1588963877939557482iset_a @ A @ B ) @ B )
= ( inter_1588963877939557482iset_a @ A @ B ) ) ).
% subset_mset.inf.right_idem
thf(fact_380_subset__mset_Oinf_Oright__idem,axiom,
! [A: multiset_a,B: multiset_a] :
( ( inter_mset_a @ ( inter_mset_a @ A @ B ) @ B )
= ( inter_mset_a @ A @ B ) ) ).
% subset_mset.inf.right_idem
thf(fact_381_subset__mset_Oinf__left__idem,axiom,
! [X3: multis501812127501805293od_a_a,Y: multis501812127501805293od_a_a] :
( ( inter_3651181809579396970od_a_a @ X3 @ ( inter_3651181809579396970od_a_a @ X3 @ Y ) )
= ( inter_3651181809579396970od_a_a @ X3 @ Y ) ) ).
% subset_mset.inf_left_idem
thf(fact_382_subset__mset_Oinf__left__idem,axiom,
! [X3: multis4781622204924204013iset_a,Y: multis4781622204924204013iset_a] :
( ( inter_1588963877939557482iset_a @ X3 @ ( inter_1588963877939557482iset_a @ X3 @ Y ) )
= ( inter_1588963877939557482iset_a @ X3 @ Y ) ) ).
% subset_mset.inf_left_idem
thf(fact_383_subset__mset_Oinf__left__idem,axiom,
! [X3: multiset_a,Y: multiset_a] :
( ( inter_mset_a @ X3 @ ( inter_mset_a @ X3 @ Y ) )
= ( inter_mset_a @ X3 @ Y ) ) ).
% subset_mset.inf_left_idem
thf(fact_384_subset__mset_Oinf_Oleft__idem,axiom,
! [A: multis501812127501805293od_a_a,B: multis501812127501805293od_a_a] :
( ( inter_3651181809579396970od_a_a @ A @ ( inter_3651181809579396970od_a_a @ A @ B ) )
= ( inter_3651181809579396970od_a_a @ A @ B ) ) ).
% subset_mset.inf.left_idem
thf(fact_385_subset__mset_Oinf_Oleft__idem,axiom,
! [A: multis4781622204924204013iset_a,B: multis4781622204924204013iset_a] :
( ( inter_1588963877939557482iset_a @ A @ ( inter_1588963877939557482iset_a @ A @ B ) )
= ( inter_1588963877939557482iset_a @ A @ B ) ) ).
% subset_mset.inf.left_idem
thf(fact_386_subset__mset_Oinf_Oleft__idem,axiom,
! [A: multiset_a,B: multiset_a] :
( ( inter_mset_a @ A @ ( inter_mset_a @ A @ B ) )
= ( inter_mset_a @ A @ B ) ) ).
% subset_mset.inf.left_idem
thf(fact_387_subset__mset_Oinf__idem,axiom,
! [X3: multis501812127501805293od_a_a] :
( ( inter_3651181809579396970od_a_a @ X3 @ X3 )
= X3 ) ).
% subset_mset.inf_idem
thf(fact_388_subset__mset_Oinf__idem,axiom,
! [X3: multis4781622204924204013iset_a] :
( ( inter_1588963877939557482iset_a @ X3 @ X3 )
= X3 ) ).
% subset_mset.inf_idem
thf(fact_389_subset__mset_Oinf__idem,axiom,
! [X3: multiset_a] :
( ( inter_mset_a @ X3 @ X3 )
= X3 ) ).
% subset_mset.inf_idem
thf(fact_390_subset__mset_Oinf_Oidem,axiom,
! [A: multis501812127501805293od_a_a] :
( ( inter_3651181809579396970od_a_a @ A @ A )
= A ) ).
% subset_mset.inf.idem
thf(fact_391_subset__mset_Oinf_Oidem,axiom,
! [A: multis4781622204924204013iset_a] :
( ( inter_1588963877939557482iset_a @ A @ A )
= A ) ).
% subset_mset.inf.idem
thf(fact_392_subset__mset_Oinf_Oidem,axiom,
! [A: multiset_a] :
( ( inter_mset_a @ A @ A )
= A ) ).
% subset_mset.inf.idem
thf(fact_393_single__eq__single,axiom,
! [A: multiset_a,B: multiset_a] :
( ( ( add_mset_multiset_a @ A @ zero_z7799948378220188993iset_a )
= ( add_mset_multiset_a @ B @ zero_z7799948378220188993iset_a ) )
= ( A = B ) ) ).
% single_eq_single
thf(fact_394_single__eq__single,axiom,
! [A: product_prod_a_a,B: product_prod_a_a] :
( ( ( add_ms8655138167283798533od_a_a @ A @ zero_z3024502163831676836od_a_a )
= ( add_ms8655138167283798533od_a_a @ B @ zero_z3024502163831676836od_a_a ) )
= ( A = B ) ) ).
% single_eq_single
thf(fact_395_single__eq__single,axiom,
! [A: produc6518373309651786023iset_a,B: produc6518373309651786023iset_a] :
( ( ( add_ms5348986034529679365iset_a @ A @ zero_z4591712717340762532iset_a )
= ( add_ms5348986034529679365iset_a @ B @ zero_z4591712717340762532iset_a ) )
= ( A = B ) ) ).
% single_eq_single
thf(fact_396_single__eq__single,axiom,
! [A: a,B: a] :
( ( ( add_mset_a @ A @ zero_zero_multiset_a )
= ( add_mset_a @ B @ zero_zero_multiset_a ) )
= ( A = B ) ) ).
% single_eq_single
thf(fact_397_add__mset__eq__single,axiom,
! [B: multiset_a,M: multiset_multiset_a,A: multiset_a] :
( ( ( add_mset_multiset_a @ B @ M )
= ( add_mset_multiset_a @ A @ zero_z7799948378220188993iset_a ) )
= ( ( B = A )
& ( M = zero_z7799948378220188993iset_a ) ) ) ).
% add_mset_eq_single
thf(fact_398_add__mset__eq__single,axiom,
! [B: product_prod_a_a,M: multis501812127501805293od_a_a,A: product_prod_a_a] :
( ( ( add_ms8655138167283798533od_a_a @ B @ M )
= ( add_ms8655138167283798533od_a_a @ A @ zero_z3024502163831676836od_a_a ) )
= ( ( B = A )
& ( M = zero_z3024502163831676836od_a_a ) ) ) ).
% add_mset_eq_single
thf(fact_399_add__mset__eq__single,axiom,
! [B: produc6518373309651786023iset_a,M: multis4781622204924204013iset_a,A: produc6518373309651786023iset_a] :
( ( ( add_ms5348986034529679365iset_a @ B @ M )
= ( add_ms5348986034529679365iset_a @ A @ zero_z4591712717340762532iset_a ) )
= ( ( B = A )
& ( M = zero_z4591712717340762532iset_a ) ) ) ).
% add_mset_eq_single
thf(fact_400_add__mset__eq__single,axiom,
! [B: a,M: multiset_a,A: a] :
( ( ( add_mset_a @ B @ M )
= ( add_mset_a @ A @ zero_zero_multiset_a ) )
= ( ( B = A )
& ( M = zero_zero_multiset_a ) ) ) ).
% add_mset_eq_single
thf(fact_401_single__eq__add__mset,axiom,
! [A: multiset_a,B: multiset_a,M: multiset_multiset_a] :
( ( ( add_mset_multiset_a @ A @ zero_z7799948378220188993iset_a )
= ( add_mset_multiset_a @ B @ M ) )
= ( ( B = A )
& ( M = zero_z7799948378220188993iset_a ) ) ) ).
% single_eq_add_mset
thf(fact_402_single__eq__add__mset,axiom,
! [A: product_prod_a_a,B: product_prod_a_a,M: multis501812127501805293od_a_a] :
( ( ( add_ms8655138167283798533od_a_a @ A @ zero_z3024502163831676836od_a_a )
= ( add_ms8655138167283798533od_a_a @ B @ M ) )
= ( ( B = A )
& ( M = zero_z3024502163831676836od_a_a ) ) ) ).
% single_eq_add_mset
thf(fact_403_single__eq__add__mset,axiom,
! [A: produc6518373309651786023iset_a,B: produc6518373309651786023iset_a,M: multis4781622204924204013iset_a] :
( ( ( add_ms5348986034529679365iset_a @ A @ zero_z4591712717340762532iset_a )
= ( add_ms5348986034529679365iset_a @ B @ M ) )
= ( ( B = A )
& ( M = zero_z4591712717340762532iset_a ) ) ) ).
% single_eq_add_mset
thf(fact_404_single__eq__add__mset,axiom,
! [A: a,B: a,M: multiset_a] :
( ( ( add_mset_a @ A @ zero_zero_multiset_a )
= ( add_mset_a @ B @ M ) )
= ( ( B = A )
& ( M = zero_zero_multiset_a ) ) ) ).
% single_eq_add_mset
thf(fact_405_add__mset__eq__singleton__iff,axiom,
! [X3: multiset_a,M: multiset_multiset_a,Y: multiset_a] :
( ( ( add_mset_multiset_a @ X3 @ M )
= ( add_mset_multiset_a @ Y @ zero_z7799948378220188993iset_a ) )
= ( ( M = zero_z7799948378220188993iset_a )
& ( X3 = Y ) ) ) ).
% add_mset_eq_singleton_iff
thf(fact_406_add__mset__eq__singleton__iff,axiom,
! [X3: product_prod_a_a,M: multis501812127501805293od_a_a,Y: product_prod_a_a] :
( ( ( add_ms8655138167283798533od_a_a @ X3 @ M )
= ( add_ms8655138167283798533od_a_a @ Y @ zero_z3024502163831676836od_a_a ) )
= ( ( M = zero_z3024502163831676836od_a_a )
& ( X3 = Y ) ) ) ).
% add_mset_eq_singleton_iff
thf(fact_407_add__mset__eq__singleton__iff,axiom,
! [X3: produc6518373309651786023iset_a,M: multis4781622204924204013iset_a,Y: produc6518373309651786023iset_a] :
( ( ( add_ms5348986034529679365iset_a @ X3 @ M )
= ( add_ms5348986034529679365iset_a @ Y @ zero_z4591712717340762532iset_a ) )
= ( ( M = zero_z4591712717340762532iset_a )
& ( X3 = Y ) ) ) ).
% add_mset_eq_singleton_iff
thf(fact_408_add__mset__eq__singleton__iff,axiom,
! [X3: a,M: multiset_a,Y: a] :
( ( ( add_mset_a @ X3 @ M )
= ( add_mset_a @ Y @ zero_zero_multiset_a ) )
= ( ( M = zero_zero_multiset_a )
& ( X3 = Y ) ) ) ).
% add_mset_eq_singleton_iff
thf(fact_409_union__mset__add__mset__left,axiom,
! [A: product_prod_a_a,A3: multis501812127501805293od_a_a,B3: multis501812127501805293od_a_a] :
( ( plus_p6967097483979887908od_a_a @ ( add_ms8655138167283798533od_a_a @ A @ A3 ) @ B3 )
= ( add_ms8655138167283798533od_a_a @ A @ ( plus_p6967097483979887908od_a_a @ A3 @ B3 ) ) ) ).
% union_mset_add_mset_left
thf(fact_410_union__mset__add__mset__left,axiom,
! [A: produc6518373309651786023iset_a,A3: multis4781622204924204013iset_a,B3: multis4781622204924204013iset_a] :
( ( plus_p1053901119268525860iset_a @ ( add_ms5348986034529679365iset_a @ A @ A3 ) @ B3 )
= ( add_ms5348986034529679365iset_a @ A @ ( plus_p1053901119268525860iset_a @ A3 @ B3 ) ) ) ).
% union_mset_add_mset_left
thf(fact_411_union__mset__add__mset__left,axiom,
! [A: multiset_a,A3: multiset_multiset_a,B3: multiset_multiset_a] :
( ( plus_p6738641960240532161iset_a @ ( add_mset_multiset_a @ A @ A3 ) @ B3 )
= ( add_mset_multiset_a @ A @ ( plus_p6738641960240532161iset_a @ A3 @ B3 ) ) ) ).
% union_mset_add_mset_left
thf(fact_412_union__mset__add__mset__left,axiom,
! [A: a,A3: multiset_a,B3: multiset_a] :
( ( plus_plus_multiset_a @ ( add_mset_a @ A @ A3 ) @ B3 )
= ( add_mset_a @ A @ ( plus_plus_multiset_a @ A3 @ B3 ) ) ) ).
% union_mset_add_mset_left
thf(fact_413_union__mset__add__mset__right,axiom,
! [A3: multis501812127501805293od_a_a,A: product_prod_a_a,B3: multis501812127501805293od_a_a] :
( ( plus_p6967097483979887908od_a_a @ A3 @ ( add_ms8655138167283798533od_a_a @ A @ B3 ) )
= ( add_ms8655138167283798533od_a_a @ A @ ( plus_p6967097483979887908od_a_a @ A3 @ B3 ) ) ) ).
% union_mset_add_mset_right
thf(fact_414_union__mset__add__mset__right,axiom,
! [A3: multis4781622204924204013iset_a,A: produc6518373309651786023iset_a,B3: multis4781622204924204013iset_a] :
( ( plus_p1053901119268525860iset_a @ A3 @ ( add_ms5348986034529679365iset_a @ A @ B3 ) )
= ( add_ms5348986034529679365iset_a @ A @ ( plus_p1053901119268525860iset_a @ A3 @ B3 ) ) ) ).
% union_mset_add_mset_right
thf(fact_415_union__mset__add__mset__right,axiom,
! [A3: multiset_multiset_a,A: multiset_a,B3: multiset_multiset_a] :
( ( plus_p6738641960240532161iset_a @ A3 @ ( add_mset_multiset_a @ A @ B3 ) )
= ( add_mset_multiset_a @ A @ ( plus_p6738641960240532161iset_a @ A3 @ B3 ) ) ) ).
% union_mset_add_mset_right
thf(fact_416_union__mset__add__mset__right,axiom,
! [A3: multiset_a,A: a,B3: multiset_a] :
( ( plus_plus_multiset_a @ A3 @ ( add_mset_a @ A @ B3 ) )
= ( add_mset_a @ A @ ( plus_plus_multiset_a @ A3 @ B3 ) ) ) ).
% union_mset_add_mset_right
thf(fact_417_subset__mset_Oinf__bot__left,axiom,
! [X3: multiset_multiset_a] :
( ( inter_3508108863637392903iset_a @ zero_z7799948378220188993iset_a @ X3 )
= zero_z7799948378220188993iset_a ) ).
% subset_mset.inf_bot_left
thf(fact_418_subset__mset_Oinf__bot__left,axiom,
! [X3: multis501812127501805293od_a_a] :
( ( inter_3651181809579396970od_a_a @ zero_z3024502163831676836od_a_a @ X3 )
= zero_z3024502163831676836od_a_a ) ).
% subset_mset.inf_bot_left
thf(fact_419_subset__mset_Oinf__bot__left,axiom,
! [X3: multis4781622204924204013iset_a] :
( ( inter_1588963877939557482iset_a @ zero_z4591712717340762532iset_a @ X3 )
= zero_z4591712717340762532iset_a ) ).
% subset_mset.inf_bot_left
thf(fact_420_subset__mset_Oinf__bot__left,axiom,
! [X3: multiset_a] :
( ( inter_mset_a @ zero_zero_multiset_a @ X3 )
= zero_zero_multiset_a ) ).
% subset_mset.inf_bot_left
thf(fact_421_subset__mset_Oinf__bot__right,axiom,
! [X3: multiset_multiset_a] :
( ( inter_3508108863637392903iset_a @ X3 @ zero_z7799948378220188993iset_a )
= zero_z7799948378220188993iset_a ) ).
% subset_mset.inf_bot_right
thf(fact_422_subset__mset_Oinf__bot__right,axiom,
! [X3: multis501812127501805293od_a_a] :
( ( inter_3651181809579396970od_a_a @ X3 @ zero_z3024502163831676836od_a_a )
= zero_z3024502163831676836od_a_a ) ).
% subset_mset.inf_bot_right
thf(fact_423_subset__mset_Oinf__bot__right,axiom,
! [X3: multis4781622204924204013iset_a] :
( ( inter_1588963877939557482iset_a @ X3 @ zero_z4591712717340762532iset_a )
= zero_z4591712717340762532iset_a ) ).
% subset_mset.inf_bot_right
thf(fact_424_subset__mset_Oinf__bot__right,axiom,
! [X3: multiset_a] :
( ( inter_mset_a @ X3 @ zero_zero_multiset_a )
= zero_zero_multiset_a ) ).
% subset_mset.inf_bot_right
thf(fact_425_add__mset__inter__add__mset,axiom,
! [A: multiset_a,A3: multiset_multiset_a,B3: multiset_multiset_a] :
( ( inter_3508108863637392903iset_a @ ( add_mset_multiset_a @ A @ A3 ) @ ( add_mset_multiset_a @ A @ B3 ) )
= ( add_mset_multiset_a @ A @ ( inter_3508108863637392903iset_a @ A3 @ B3 ) ) ) ).
% add_mset_inter_add_mset
thf(fact_426_add__mset__inter__add__mset,axiom,
! [A: product_prod_a_a,A3: multis501812127501805293od_a_a,B3: multis501812127501805293od_a_a] :
( ( inter_3651181809579396970od_a_a @ ( add_ms8655138167283798533od_a_a @ A @ A3 ) @ ( add_ms8655138167283798533od_a_a @ A @ B3 ) )
= ( add_ms8655138167283798533od_a_a @ A @ ( inter_3651181809579396970od_a_a @ A3 @ B3 ) ) ) ).
% add_mset_inter_add_mset
thf(fact_427_add__mset__inter__add__mset,axiom,
! [A: produc6518373309651786023iset_a,A3: multis4781622204924204013iset_a,B3: multis4781622204924204013iset_a] :
( ( inter_1588963877939557482iset_a @ ( add_ms5348986034529679365iset_a @ A @ A3 ) @ ( add_ms5348986034529679365iset_a @ A @ B3 ) )
= ( add_ms5348986034529679365iset_a @ A @ ( inter_1588963877939557482iset_a @ A3 @ B3 ) ) ) ).
% add_mset_inter_add_mset
thf(fact_428_add__mset__inter__add__mset,axiom,
! [A: a,A3: multiset_a,B3: multiset_a] :
( ( inter_mset_a @ ( add_mset_a @ A @ A3 ) @ ( add_mset_a @ A @ B3 ) )
= ( add_mset_a @ A @ ( inter_mset_a @ A3 @ B3 ) ) ) ).
% add_mset_inter_add_mset
thf(fact_429_diff__intersect__right__idem,axiom,
! [M: multiset_a,N: multiset_a] :
( ( minus_3765977307040488491iset_a @ M @ ( inter_mset_a @ N @ M ) )
= ( minus_3765977307040488491iset_a @ M @ N ) ) ).
% diff_intersect_right_idem
thf(fact_430_diff__intersect__right__idem,axiom,
! [M: multis4781622204924204013iset_a,N: multis4781622204924204013iset_a] :
( ( minus_9211619285413683668iset_a @ M @ ( inter_1588963877939557482iset_a @ N @ M ) )
= ( minus_9211619285413683668iset_a @ M @ N ) ) ).
% diff_intersect_right_idem
thf(fact_431_diff__intersect__right__idem,axiom,
! [M: multis501812127501805293od_a_a,N: multis501812127501805293od_a_a] :
( ( minus_7313152569058643924od_a_a @ M @ ( inter_3651181809579396970od_a_a @ N @ M ) )
= ( minus_7313152569058643924od_a_a @ M @ N ) ) ).
% diff_intersect_right_idem
thf(fact_432_diff__intersect__left__idem,axiom,
! [M: multiset_a,N: multiset_a] :
( ( minus_3765977307040488491iset_a @ M @ ( inter_mset_a @ M @ N ) )
= ( minus_3765977307040488491iset_a @ M @ N ) ) ).
% diff_intersect_left_idem
thf(fact_433_diff__intersect__left__idem,axiom,
! [M: multis4781622204924204013iset_a,N: multis4781622204924204013iset_a] :
( ( minus_9211619285413683668iset_a @ M @ ( inter_1588963877939557482iset_a @ M @ N ) )
= ( minus_9211619285413683668iset_a @ M @ N ) ) ).
% diff_intersect_left_idem
thf(fact_434_diff__intersect__left__idem,axiom,
! [M: multis501812127501805293od_a_a,N: multis501812127501805293od_a_a] :
( ( minus_7313152569058643924od_a_a @ M @ ( inter_3651181809579396970od_a_a @ M @ N ) )
= ( minus_7313152569058643924od_a_a @ M @ N ) ) ).
% diff_intersect_left_idem
thf(fact_435_multiset__inter__single,axiom,
! [A: multiset_a,B: multiset_a] :
( ( A != B )
=> ( ( inter_3508108863637392903iset_a @ ( add_mset_multiset_a @ A @ zero_z7799948378220188993iset_a ) @ ( add_mset_multiset_a @ B @ zero_z7799948378220188993iset_a ) )
= zero_z7799948378220188993iset_a ) ) ).
% multiset_inter_single
thf(fact_436_multiset__inter__single,axiom,
! [A: product_prod_a_a,B: product_prod_a_a] :
( ( A != B )
=> ( ( inter_3651181809579396970od_a_a @ ( add_ms8655138167283798533od_a_a @ A @ zero_z3024502163831676836od_a_a ) @ ( add_ms8655138167283798533od_a_a @ B @ zero_z3024502163831676836od_a_a ) )
= zero_z3024502163831676836od_a_a ) ) ).
% multiset_inter_single
thf(fact_437_multiset__inter__single,axiom,
! [A: produc6518373309651786023iset_a,B: produc6518373309651786023iset_a] :
( ( A != B )
=> ( ( inter_1588963877939557482iset_a @ ( add_ms5348986034529679365iset_a @ A @ zero_z4591712717340762532iset_a ) @ ( add_ms5348986034529679365iset_a @ B @ zero_z4591712717340762532iset_a ) )
= zero_z4591712717340762532iset_a ) ) ).
% multiset_inter_single
thf(fact_438_multiset__inter__single,axiom,
! [A: a,B: a] :
( ( A != B )
=> ( ( inter_mset_a @ ( add_mset_a @ A @ zero_zero_multiset_a ) @ ( add_mset_a @ B @ zero_zero_multiset_a ) )
= zero_zero_multiset_a ) ) ).
% multiset_inter_single
thf(fact_439_add__mset__remove__trivial,axiom,
! [X3: multiset_a,M: multiset_multiset_a] :
( ( minus_3395427628221709681iset_a @ ( add_mset_multiset_a @ X3 @ M ) @ ( add_mset_multiset_a @ X3 @ zero_z7799948378220188993iset_a ) )
= M ) ).
% add_mset_remove_trivial
thf(fact_440_add__mset__remove__trivial,axiom,
! [X3: a,M: multiset_a] :
( ( minus_3765977307040488491iset_a @ ( add_mset_a @ X3 @ M ) @ ( add_mset_a @ X3 @ zero_zero_multiset_a ) )
= M ) ).
% add_mset_remove_trivial
thf(fact_441_add__mset__remove__trivial,axiom,
! [X3: produc6518373309651786023iset_a,M: multis4781622204924204013iset_a] :
( ( minus_9211619285413683668iset_a @ ( add_ms5348986034529679365iset_a @ X3 @ M ) @ ( add_ms5348986034529679365iset_a @ X3 @ zero_z4591712717340762532iset_a ) )
= M ) ).
% add_mset_remove_trivial
thf(fact_442_add__mset__remove__trivial,axiom,
! [X3: product_prod_a_a,M: multis501812127501805293od_a_a] :
( ( minus_7313152569058643924od_a_a @ ( add_ms8655138167283798533od_a_a @ X3 @ M ) @ ( add_ms8655138167283798533od_a_a @ X3 @ zero_z3024502163831676836od_a_a ) )
= M ) ).
% add_mset_remove_trivial
thf(fact_443_diff__add__mset__swap,axiom,
! [B: produc5665077874975597415od_a_a,A3: multis3377851686571972653od_a_a,M: multis3377851686571972653od_a_a] :
( ~ ( member1567053331504362128od_a_a @ B @ ( set_ms5376707576967697060od_a_a @ A3 ) )
=> ( ( minus_6383096217462597652od_a_a @ ( add_ms9136373847686211141od_a_a @ B @ M ) @ A3 )
= ( add_ms9136373847686211141od_a_a @ B @ ( minus_6383096217462597652od_a_a @ M @ A3 ) ) ) ) ).
% diff_add_mset_swap
thf(fact_444_diff__add__mset__swap,axiom,
! [B: produc4424810744984089383iset_a,A3: multis1386120040344743661iset_a,M: multis1386120040344743661iset_a] :
( ~ ( member1345422370561994576iset_a @ B @ ( set_ms2882773486807528292iset_a @ A3 ) )
=> ( ( minus_2536987971823575508iset_a @ ( add_ms9106060982595931141iset_a @ B @ M ) @ A3 )
= ( add_ms9106060982595931141iset_a @ B @ ( minus_2536987971823575508iset_a @ M @ A3 ) ) ) ) ).
% diff_add_mset_swap
thf(fact_445_diff__add__mset__swap,axiom,
! [B: multiset_a,A3: multiset_multiset_a,M: multiset_multiset_a] :
( ~ ( member_multiset_a @ B @ ( set_mset_multiset_a @ A3 ) )
=> ( ( minus_3395427628221709681iset_a @ ( add_mset_multiset_a @ B @ M ) @ A3 )
= ( add_mset_multiset_a @ B @ ( minus_3395427628221709681iset_a @ M @ A3 ) ) ) ) ).
% diff_add_mset_swap
thf(fact_446_diff__add__mset__swap,axiom,
! [B: multis501812127501805293od_a_a,A3: multis5935151272263360051od_a_a,M: multis5935151272263360051od_a_a] :
( ~ ( member7700929758516679318od_a_a @ B @ ( set_ms8253844250382261418od_a_a @ A3 ) )
=> ( ( minus_385666609031338394od_a_a @ ( add_ms4156406267932395723od_a_a @ B @ M ) @ A3 )
= ( add_ms4156406267932395723od_a_a @ B @ ( minus_385666609031338394od_a_a @ M @ A3 ) ) ) ) ).
% diff_add_mset_swap
thf(fact_447_diff__add__mset__swap,axiom,
! [B: multis4781622204924204013iset_a,A3: multis7221419791219231795iset_a,M: multis7221419791219231795iset_a] :
( ~ ( member2597461214246177430iset_a @ B @ ( set_ms5327031259215843498iset_a @ A3 ) )
=> ( ( minus_4490418833869548698iset_a @ ( add_ms7610149848545062859iset_a @ B @ M ) @ A3 )
= ( add_ms7610149848545062859iset_a @ B @ ( minus_4490418833869548698iset_a @ M @ A3 ) ) ) ) ).
% diff_add_mset_swap
thf(fact_448_diff__add__mset__swap,axiom,
! [B: a,A3: multiset_a,M: multiset_a] :
( ~ ( member_a @ B @ ( set_mset_a @ A3 ) )
=> ( ( minus_3765977307040488491iset_a @ ( add_mset_a @ B @ M ) @ A3 )
= ( add_mset_a @ B @ ( minus_3765977307040488491iset_a @ M @ A3 ) ) ) ) ).
% diff_add_mset_swap
thf(fact_449_diff__add__mset__swap,axiom,
! [B: produc6518373309651786023iset_a,A3: multis4781622204924204013iset_a,M: multis4781622204924204013iset_a] :
( ~ ( member5199237121806060112iset_a @ B @ ( set_ms2379723421352415844iset_a @ A3 ) )
=> ( ( minus_9211619285413683668iset_a @ ( add_ms5348986034529679365iset_a @ B @ M ) @ A3 )
= ( add_ms5348986034529679365iset_a @ B @ ( minus_9211619285413683668iset_a @ M @ A3 ) ) ) ) ).
% diff_add_mset_swap
thf(fact_450_diff__add__mset__swap,axiom,
! [B: product_prod_a_a,A3: multis501812127501805293od_a_a,M: multis501812127501805293od_a_a] :
( ~ ( member1426531477525435216od_a_a @ B @ ( set_ms119794563918438244od_a_a @ A3 ) )
=> ( ( minus_7313152569058643924od_a_a @ ( add_ms8655138167283798533od_a_a @ B @ M ) @ A3 )
= ( add_ms8655138167283798533od_a_a @ B @ ( minus_7313152569058643924od_a_a @ M @ A3 ) ) ) ) ).
% diff_add_mset_swap
thf(fact_451_add__mset__disjoint_I2_J,axiom,
! [A: produc5665077874975597415od_a_a,A3: multis3377851686571972653od_a_a,B3: multis3377851686571972653od_a_a] :
( ( zero_z3721308388212328420od_a_a
= ( inter_2707069342228023466od_a_a @ ( add_ms9136373847686211141od_a_a @ A @ A3 ) @ B3 ) )
= ( ~ ( member1567053331504362128od_a_a @ A @ ( set_ms5376707576967697060od_a_a @ B3 ) )
& ( zero_z3721308388212328420od_a_a
= ( inter_2707069342228023466od_a_a @ A3 @ B3 ) ) ) ) ).
% add_mset_disjoint(2)
thf(fact_452_add__mset__disjoint_I2_J,axiom,
! [A: produc4424810744984089383iset_a,A3: multis1386120040344743661iset_a,B3: multis1386120040344743661iset_a] :
( ( zero_z9101083372863164836iset_a
= ( inter_7955219499984291690iset_a @ ( add_ms9106060982595931141iset_a @ A @ A3 ) @ B3 ) )
= ( ~ ( member1345422370561994576iset_a @ A @ ( set_ms2882773486807528292iset_a @ B3 ) )
& ( zero_z9101083372863164836iset_a
= ( inter_7955219499984291690iset_a @ A3 @ B3 ) ) ) ) ).
% add_mset_disjoint(2)
thf(fact_453_add__mset__disjoint_I2_J,axiom,
! [A: multis501812127501805293od_a_a,A3: multis5935151272263360051od_a_a,B3: multis5935151272263360051od_a_a] :
( ( zero_z8896958667708014954od_a_a
= ( inter_1480659854676493744od_a_a @ ( add_ms4156406267932395723od_a_a @ A @ A3 ) @ B3 ) )
= ( ~ ( member7700929758516679318od_a_a @ A @ ( set_ms8253844250382261418od_a_a @ B3 ) )
& ( zero_z8896958667708014954od_a_a
= ( inter_1480659854676493744od_a_a @ A3 @ B3 ) ) ) ) ).
% add_mset_disjoint(2)
thf(fact_454_add__mset__disjoint_I2_J,axiom,
! [A: multis4781622204924204013iset_a,A3: multis7221419791219231795iset_a,B3: multis7221419791219231795iset_a] :
( ( zero_z8662914267040441450iset_a
= ( inter_8933094105380514736iset_a @ ( add_ms7610149848545062859iset_a @ A @ A3 ) @ B3 ) )
= ( ~ ( member2597461214246177430iset_a @ A @ ( set_ms5327031259215843498iset_a @ B3 ) )
& ( zero_z8662914267040441450iset_a
= ( inter_8933094105380514736iset_a @ A3 @ B3 ) ) ) ) ).
% add_mset_disjoint(2)
thf(fact_455_add__mset__disjoint_I2_J,axiom,
! [A: multiset_a,A3: multiset_multiset_a,B3: multiset_multiset_a] :
( ( zero_z7799948378220188993iset_a
= ( inter_3508108863637392903iset_a @ ( add_mset_multiset_a @ A @ A3 ) @ B3 ) )
= ( ~ ( member_multiset_a @ A @ ( set_mset_multiset_a @ B3 ) )
& ( zero_z7799948378220188993iset_a
= ( inter_3508108863637392903iset_a @ A3 @ B3 ) ) ) ) ).
% add_mset_disjoint(2)
thf(fact_456_add__mset__disjoint_I2_J,axiom,
! [A: produc6518373309651786023iset_a,A3: multis4781622204924204013iset_a,B3: multis4781622204924204013iset_a] :
( ( zero_z4591712717340762532iset_a
= ( inter_1588963877939557482iset_a @ ( add_ms5348986034529679365iset_a @ A @ A3 ) @ B3 ) )
= ( ~ ( member5199237121806060112iset_a @ A @ ( set_ms2379723421352415844iset_a @ B3 ) )
& ( zero_z4591712717340762532iset_a
= ( inter_1588963877939557482iset_a @ A3 @ B3 ) ) ) ) ).
% add_mset_disjoint(2)
thf(fact_457_add__mset__disjoint_I2_J,axiom,
! [A: product_prod_a_a,A3: multis501812127501805293od_a_a,B3: multis501812127501805293od_a_a] :
( ( zero_z3024502163831676836od_a_a
= ( inter_3651181809579396970od_a_a @ ( add_ms8655138167283798533od_a_a @ A @ A3 ) @ B3 ) )
= ( ~ ( member1426531477525435216od_a_a @ A @ ( set_ms119794563918438244od_a_a @ B3 ) )
& ( zero_z3024502163831676836od_a_a
= ( inter_3651181809579396970od_a_a @ A3 @ B3 ) ) ) ) ).
% add_mset_disjoint(2)
thf(fact_458_add__mset__disjoint_I2_J,axiom,
! [A: a,A3: multiset_a,B3: multiset_a] :
( ( zero_zero_multiset_a
= ( inter_mset_a @ ( add_mset_a @ A @ A3 ) @ B3 ) )
= ( ~ ( member_a @ A @ ( set_mset_a @ B3 ) )
& ( zero_zero_multiset_a
= ( inter_mset_a @ A3 @ B3 ) ) ) ) ).
% add_mset_disjoint(2)
thf(fact_459_add__mset__disjoint_I1_J,axiom,
! [A: produc5665077874975597415od_a_a,A3: multis3377851686571972653od_a_a,B3: multis3377851686571972653od_a_a] :
( ( ( inter_2707069342228023466od_a_a @ ( add_ms9136373847686211141od_a_a @ A @ A3 ) @ B3 )
= zero_z3721308388212328420od_a_a )
= ( ~ ( member1567053331504362128od_a_a @ A @ ( set_ms5376707576967697060od_a_a @ B3 ) )
& ( ( inter_2707069342228023466od_a_a @ A3 @ B3 )
= zero_z3721308388212328420od_a_a ) ) ) ).
% add_mset_disjoint(1)
thf(fact_460_add__mset__disjoint_I1_J,axiom,
! [A: produc4424810744984089383iset_a,A3: multis1386120040344743661iset_a,B3: multis1386120040344743661iset_a] :
( ( ( inter_7955219499984291690iset_a @ ( add_ms9106060982595931141iset_a @ A @ A3 ) @ B3 )
= zero_z9101083372863164836iset_a )
= ( ~ ( member1345422370561994576iset_a @ A @ ( set_ms2882773486807528292iset_a @ B3 ) )
& ( ( inter_7955219499984291690iset_a @ A3 @ B3 )
= zero_z9101083372863164836iset_a ) ) ) ).
% add_mset_disjoint(1)
thf(fact_461_add__mset__disjoint_I1_J,axiom,
! [A: multis501812127501805293od_a_a,A3: multis5935151272263360051od_a_a,B3: multis5935151272263360051od_a_a] :
( ( ( inter_1480659854676493744od_a_a @ ( add_ms4156406267932395723od_a_a @ A @ A3 ) @ B3 )
= zero_z8896958667708014954od_a_a )
= ( ~ ( member7700929758516679318od_a_a @ A @ ( set_ms8253844250382261418od_a_a @ B3 ) )
& ( ( inter_1480659854676493744od_a_a @ A3 @ B3 )
= zero_z8896958667708014954od_a_a ) ) ) ).
% add_mset_disjoint(1)
thf(fact_462_add__mset__disjoint_I1_J,axiom,
! [A: multis4781622204924204013iset_a,A3: multis7221419791219231795iset_a,B3: multis7221419791219231795iset_a] :
( ( ( inter_8933094105380514736iset_a @ ( add_ms7610149848545062859iset_a @ A @ A3 ) @ B3 )
= zero_z8662914267040441450iset_a )
= ( ~ ( member2597461214246177430iset_a @ A @ ( set_ms5327031259215843498iset_a @ B3 ) )
& ( ( inter_8933094105380514736iset_a @ A3 @ B3 )
= zero_z8662914267040441450iset_a ) ) ) ).
% add_mset_disjoint(1)
thf(fact_463_add__mset__disjoint_I1_J,axiom,
! [A: multiset_a,A3: multiset_multiset_a,B3: multiset_multiset_a] :
( ( ( inter_3508108863637392903iset_a @ ( add_mset_multiset_a @ A @ A3 ) @ B3 )
= zero_z7799948378220188993iset_a )
= ( ~ ( member_multiset_a @ A @ ( set_mset_multiset_a @ B3 ) )
& ( ( inter_3508108863637392903iset_a @ A3 @ B3 )
= zero_z7799948378220188993iset_a ) ) ) ).
% add_mset_disjoint(1)
thf(fact_464_add__mset__disjoint_I1_J,axiom,
! [A: produc6518373309651786023iset_a,A3: multis4781622204924204013iset_a,B3: multis4781622204924204013iset_a] :
( ( ( inter_1588963877939557482iset_a @ ( add_ms5348986034529679365iset_a @ A @ A3 ) @ B3 )
= zero_z4591712717340762532iset_a )
= ( ~ ( member5199237121806060112iset_a @ A @ ( set_ms2379723421352415844iset_a @ B3 ) )
& ( ( inter_1588963877939557482iset_a @ A3 @ B3 )
= zero_z4591712717340762532iset_a ) ) ) ).
% add_mset_disjoint(1)
thf(fact_465_add__mset__disjoint_I1_J,axiom,
! [A: product_prod_a_a,A3: multis501812127501805293od_a_a,B3: multis501812127501805293od_a_a] :
( ( ( inter_3651181809579396970od_a_a @ ( add_ms8655138167283798533od_a_a @ A @ A3 ) @ B3 )
= zero_z3024502163831676836od_a_a )
= ( ~ ( member1426531477525435216od_a_a @ A @ ( set_ms119794563918438244od_a_a @ B3 ) )
& ( ( inter_3651181809579396970od_a_a @ A3 @ B3 )
= zero_z3024502163831676836od_a_a ) ) ) ).
% add_mset_disjoint(1)
thf(fact_466_add__mset__disjoint_I1_J,axiom,
! [A: a,A3: multiset_a,B3: multiset_a] :
( ( ( inter_mset_a @ ( add_mset_a @ A @ A3 ) @ B3 )
= zero_zero_multiset_a )
= ( ~ ( member_a @ A @ ( set_mset_a @ B3 ) )
& ( ( inter_mset_a @ A3 @ B3 )
= zero_zero_multiset_a ) ) ) ).
% add_mset_disjoint(1)
thf(fact_467_disjoint__add__mset_I2_J,axiom,
! [A3: multis3377851686571972653od_a_a,B: produc5665077874975597415od_a_a,B3: multis3377851686571972653od_a_a] :
( ( zero_z3721308388212328420od_a_a
= ( inter_2707069342228023466od_a_a @ A3 @ ( add_ms9136373847686211141od_a_a @ B @ B3 ) ) )
= ( ~ ( member1567053331504362128od_a_a @ B @ ( set_ms5376707576967697060od_a_a @ A3 ) )
& ( zero_z3721308388212328420od_a_a
= ( inter_2707069342228023466od_a_a @ A3 @ B3 ) ) ) ) ).
% disjoint_add_mset(2)
thf(fact_468_disjoint__add__mset_I2_J,axiom,
! [A3: multis1386120040344743661iset_a,B: produc4424810744984089383iset_a,B3: multis1386120040344743661iset_a] :
( ( zero_z9101083372863164836iset_a
= ( inter_7955219499984291690iset_a @ A3 @ ( add_ms9106060982595931141iset_a @ B @ B3 ) ) )
= ( ~ ( member1345422370561994576iset_a @ B @ ( set_ms2882773486807528292iset_a @ A3 ) )
& ( zero_z9101083372863164836iset_a
= ( inter_7955219499984291690iset_a @ A3 @ B3 ) ) ) ) ).
% disjoint_add_mset(2)
thf(fact_469_disjoint__add__mset_I2_J,axiom,
! [A3: multis5935151272263360051od_a_a,B: multis501812127501805293od_a_a,B3: multis5935151272263360051od_a_a] :
( ( zero_z8896958667708014954od_a_a
= ( inter_1480659854676493744od_a_a @ A3 @ ( add_ms4156406267932395723od_a_a @ B @ B3 ) ) )
= ( ~ ( member7700929758516679318od_a_a @ B @ ( set_ms8253844250382261418od_a_a @ A3 ) )
& ( zero_z8896958667708014954od_a_a
= ( inter_1480659854676493744od_a_a @ A3 @ B3 ) ) ) ) ).
% disjoint_add_mset(2)
thf(fact_470_disjoint__add__mset_I2_J,axiom,
! [A3: multis7221419791219231795iset_a,B: multis4781622204924204013iset_a,B3: multis7221419791219231795iset_a] :
( ( zero_z8662914267040441450iset_a
= ( inter_8933094105380514736iset_a @ A3 @ ( add_ms7610149848545062859iset_a @ B @ B3 ) ) )
= ( ~ ( member2597461214246177430iset_a @ B @ ( set_ms5327031259215843498iset_a @ A3 ) )
& ( zero_z8662914267040441450iset_a
= ( inter_8933094105380514736iset_a @ A3 @ B3 ) ) ) ) ).
% disjoint_add_mset(2)
thf(fact_471_disjoint__add__mset_I2_J,axiom,
! [A3: multiset_multiset_a,B: multiset_a,B3: multiset_multiset_a] :
( ( zero_z7799948378220188993iset_a
= ( inter_3508108863637392903iset_a @ A3 @ ( add_mset_multiset_a @ B @ B3 ) ) )
= ( ~ ( member_multiset_a @ B @ ( set_mset_multiset_a @ A3 ) )
& ( zero_z7799948378220188993iset_a
= ( inter_3508108863637392903iset_a @ A3 @ B3 ) ) ) ) ).
% disjoint_add_mset(2)
thf(fact_472_disjoint__add__mset_I2_J,axiom,
! [A3: multis4781622204924204013iset_a,B: produc6518373309651786023iset_a,B3: multis4781622204924204013iset_a] :
( ( zero_z4591712717340762532iset_a
= ( inter_1588963877939557482iset_a @ A3 @ ( add_ms5348986034529679365iset_a @ B @ B3 ) ) )
= ( ~ ( member5199237121806060112iset_a @ B @ ( set_ms2379723421352415844iset_a @ A3 ) )
& ( zero_z4591712717340762532iset_a
= ( inter_1588963877939557482iset_a @ A3 @ B3 ) ) ) ) ).
% disjoint_add_mset(2)
thf(fact_473_disjoint__add__mset_I2_J,axiom,
! [A3: multis501812127501805293od_a_a,B: product_prod_a_a,B3: multis501812127501805293od_a_a] :
( ( zero_z3024502163831676836od_a_a
= ( inter_3651181809579396970od_a_a @ A3 @ ( add_ms8655138167283798533od_a_a @ B @ B3 ) ) )
= ( ~ ( member1426531477525435216od_a_a @ B @ ( set_ms119794563918438244od_a_a @ A3 ) )
& ( zero_z3024502163831676836od_a_a
= ( inter_3651181809579396970od_a_a @ A3 @ B3 ) ) ) ) ).
% disjoint_add_mset(2)
thf(fact_474_disjoint__add__mset_I2_J,axiom,
! [A3: multiset_a,B: a,B3: multiset_a] :
( ( zero_zero_multiset_a
= ( inter_mset_a @ A3 @ ( add_mset_a @ B @ B3 ) ) )
= ( ~ ( member_a @ B @ ( set_mset_a @ A3 ) )
& ( zero_zero_multiset_a
= ( inter_mset_a @ A3 @ B3 ) ) ) ) ).
% disjoint_add_mset(2)
thf(fact_475_disjoint__add__mset_I1_J,axiom,
! [B3: multis3377851686571972653od_a_a,A: produc5665077874975597415od_a_a,A3: multis3377851686571972653od_a_a] :
( ( ( inter_2707069342228023466od_a_a @ B3 @ ( add_ms9136373847686211141od_a_a @ A @ A3 ) )
= zero_z3721308388212328420od_a_a )
= ( ~ ( member1567053331504362128od_a_a @ A @ ( set_ms5376707576967697060od_a_a @ B3 ) )
& ( ( inter_2707069342228023466od_a_a @ B3 @ A3 )
= zero_z3721308388212328420od_a_a ) ) ) ).
% disjoint_add_mset(1)
thf(fact_476_disjoint__add__mset_I1_J,axiom,
! [B3: multis1386120040344743661iset_a,A: produc4424810744984089383iset_a,A3: multis1386120040344743661iset_a] :
( ( ( inter_7955219499984291690iset_a @ B3 @ ( add_ms9106060982595931141iset_a @ A @ A3 ) )
= zero_z9101083372863164836iset_a )
= ( ~ ( member1345422370561994576iset_a @ A @ ( set_ms2882773486807528292iset_a @ B3 ) )
& ( ( inter_7955219499984291690iset_a @ B3 @ A3 )
= zero_z9101083372863164836iset_a ) ) ) ).
% disjoint_add_mset(1)
thf(fact_477_disjoint__add__mset_I1_J,axiom,
! [B3: multis5935151272263360051od_a_a,A: multis501812127501805293od_a_a,A3: multis5935151272263360051od_a_a] :
( ( ( inter_1480659854676493744od_a_a @ B3 @ ( add_ms4156406267932395723od_a_a @ A @ A3 ) )
= zero_z8896958667708014954od_a_a )
= ( ~ ( member7700929758516679318od_a_a @ A @ ( set_ms8253844250382261418od_a_a @ B3 ) )
& ( ( inter_1480659854676493744od_a_a @ B3 @ A3 )
= zero_z8896958667708014954od_a_a ) ) ) ).
% disjoint_add_mset(1)
thf(fact_478_disjoint__add__mset_I1_J,axiom,
! [B3: multis7221419791219231795iset_a,A: multis4781622204924204013iset_a,A3: multis7221419791219231795iset_a] :
( ( ( inter_8933094105380514736iset_a @ B3 @ ( add_ms7610149848545062859iset_a @ A @ A3 ) )
= zero_z8662914267040441450iset_a )
= ( ~ ( member2597461214246177430iset_a @ A @ ( set_ms5327031259215843498iset_a @ B3 ) )
& ( ( inter_8933094105380514736iset_a @ B3 @ A3 )
= zero_z8662914267040441450iset_a ) ) ) ).
% disjoint_add_mset(1)
thf(fact_479_disjoint__add__mset_I1_J,axiom,
! [B3: multiset_multiset_a,A: multiset_a,A3: multiset_multiset_a] :
( ( ( inter_3508108863637392903iset_a @ B3 @ ( add_mset_multiset_a @ A @ A3 ) )
= zero_z7799948378220188993iset_a )
= ( ~ ( member_multiset_a @ A @ ( set_mset_multiset_a @ B3 ) )
& ( ( inter_3508108863637392903iset_a @ B3 @ A3 )
= zero_z7799948378220188993iset_a ) ) ) ).
% disjoint_add_mset(1)
thf(fact_480_disjoint__add__mset_I1_J,axiom,
! [B3: multis4781622204924204013iset_a,A: produc6518373309651786023iset_a,A3: multis4781622204924204013iset_a] :
( ( ( inter_1588963877939557482iset_a @ B3 @ ( add_ms5348986034529679365iset_a @ A @ A3 ) )
= zero_z4591712717340762532iset_a )
= ( ~ ( member5199237121806060112iset_a @ A @ ( set_ms2379723421352415844iset_a @ B3 ) )
& ( ( inter_1588963877939557482iset_a @ B3 @ A3 )
= zero_z4591712717340762532iset_a ) ) ) ).
% disjoint_add_mset(1)
thf(fact_481_disjoint__add__mset_I1_J,axiom,
! [B3: multis501812127501805293od_a_a,A: product_prod_a_a,A3: multis501812127501805293od_a_a] :
( ( ( inter_3651181809579396970od_a_a @ B3 @ ( add_ms8655138167283798533od_a_a @ A @ A3 ) )
= zero_z3024502163831676836od_a_a )
= ( ~ ( member1426531477525435216od_a_a @ A @ ( set_ms119794563918438244od_a_a @ B3 ) )
& ( ( inter_3651181809579396970od_a_a @ B3 @ A3 )
= zero_z3024502163831676836od_a_a ) ) ) ).
% disjoint_add_mset(1)
thf(fact_482_disjoint__add__mset_I1_J,axiom,
! [B3: multiset_a,A: a,A3: multiset_a] :
( ( ( inter_mset_a @ B3 @ ( add_mset_a @ A @ A3 ) )
= zero_zero_multiset_a )
= ( ~ ( member_a @ A @ ( set_mset_a @ B3 ) )
& ( ( inter_mset_a @ B3 @ A3 )
= zero_zero_multiset_a ) ) ) ).
% disjoint_add_mset(1)
thf(fact_483_diff__union__swap2,axiom,
! [Y: produc5665077874975597415od_a_a,M: multis3377851686571972653od_a_a,X3: produc5665077874975597415od_a_a] :
( ( member1567053331504362128od_a_a @ Y @ ( set_ms5376707576967697060od_a_a @ M ) )
=> ( ( minus_6383096217462597652od_a_a @ ( add_ms9136373847686211141od_a_a @ X3 @ M ) @ ( add_ms9136373847686211141od_a_a @ Y @ zero_z3721308388212328420od_a_a ) )
= ( add_ms9136373847686211141od_a_a @ X3 @ ( minus_6383096217462597652od_a_a @ M @ ( add_ms9136373847686211141od_a_a @ Y @ zero_z3721308388212328420od_a_a ) ) ) ) ) ).
% diff_union_swap2
thf(fact_484_diff__union__swap2,axiom,
! [Y: produc4424810744984089383iset_a,M: multis1386120040344743661iset_a,X3: produc4424810744984089383iset_a] :
( ( member1345422370561994576iset_a @ Y @ ( set_ms2882773486807528292iset_a @ M ) )
=> ( ( minus_2536987971823575508iset_a @ ( add_ms9106060982595931141iset_a @ X3 @ M ) @ ( add_ms9106060982595931141iset_a @ Y @ zero_z9101083372863164836iset_a ) )
= ( add_ms9106060982595931141iset_a @ X3 @ ( minus_2536987971823575508iset_a @ M @ ( add_ms9106060982595931141iset_a @ Y @ zero_z9101083372863164836iset_a ) ) ) ) ) ).
% diff_union_swap2
thf(fact_485_diff__union__swap2,axiom,
! [Y: multis501812127501805293od_a_a,M: multis5935151272263360051od_a_a,X3: multis501812127501805293od_a_a] :
( ( member7700929758516679318od_a_a @ Y @ ( set_ms8253844250382261418od_a_a @ M ) )
=> ( ( minus_385666609031338394od_a_a @ ( add_ms4156406267932395723od_a_a @ X3 @ M ) @ ( add_ms4156406267932395723od_a_a @ Y @ zero_z8896958667708014954od_a_a ) )
= ( add_ms4156406267932395723od_a_a @ X3 @ ( minus_385666609031338394od_a_a @ M @ ( add_ms4156406267932395723od_a_a @ Y @ zero_z8896958667708014954od_a_a ) ) ) ) ) ).
% diff_union_swap2
thf(fact_486_diff__union__swap2,axiom,
! [Y: multis4781622204924204013iset_a,M: multis7221419791219231795iset_a,X3: multis4781622204924204013iset_a] :
( ( member2597461214246177430iset_a @ Y @ ( set_ms5327031259215843498iset_a @ M ) )
=> ( ( minus_4490418833869548698iset_a @ ( add_ms7610149848545062859iset_a @ X3 @ M ) @ ( add_ms7610149848545062859iset_a @ Y @ zero_z8662914267040441450iset_a ) )
= ( add_ms7610149848545062859iset_a @ X3 @ ( minus_4490418833869548698iset_a @ M @ ( add_ms7610149848545062859iset_a @ Y @ zero_z8662914267040441450iset_a ) ) ) ) ) ).
% diff_union_swap2
thf(fact_487_diff__union__swap2,axiom,
! [Y: multiset_a,M: multiset_multiset_a,X3: multiset_a] :
( ( member_multiset_a @ Y @ ( set_mset_multiset_a @ M ) )
=> ( ( minus_3395427628221709681iset_a @ ( add_mset_multiset_a @ X3 @ M ) @ ( add_mset_multiset_a @ Y @ zero_z7799948378220188993iset_a ) )
= ( add_mset_multiset_a @ X3 @ ( minus_3395427628221709681iset_a @ M @ ( add_mset_multiset_a @ Y @ zero_z7799948378220188993iset_a ) ) ) ) ) ).
% diff_union_swap2
thf(fact_488_diff__union__swap2,axiom,
! [Y: a,M: multiset_a,X3: a] :
( ( member_a @ Y @ ( set_mset_a @ M ) )
=> ( ( minus_3765977307040488491iset_a @ ( add_mset_a @ X3 @ M ) @ ( add_mset_a @ Y @ zero_zero_multiset_a ) )
= ( add_mset_a @ X3 @ ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ Y @ zero_zero_multiset_a ) ) ) ) ) ).
% diff_union_swap2
thf(fact_489_diff__union__swap2,axiom,
! [Y: produc6518373309651786023iset_a,M: multis4781622204924204013iset_a,X3: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Y @ ( set_ms2379723421352415844iset_a @ M ) )
=> ( ( minus_9211619285413683668iset_a @ ( add_ms5348986034529679365iset_a @ X3 @ M ) @ ( add_ms5348986034529679365iset_a @ Y @ zero_z4591712717340762532iset_a ) )
= ( add_ms5348986034529679365iset_a @ X3 @ ( minus_9211619285413683668iset_a @ M @ ( add_ms5348986034529679365iset_a @ Y @ zero_z4591712717340762532iset_a ) ) ) ) ) ).
% diff_union_swap2
thf(fact_490_diff__union__swap2,axiom,
! [Y: product_prod_a_a,M: multis501812127501805293od_a_a,X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y @ ( set_ms119794563918438244od_a_a @ M ) )
=> ( ( minus_7313152569058643924od_a_a @ ( add_ms8655138167283798533od_a_a @ X3 @ M ) @ ( add_ms8655138167283798533od_a_a @ Y @ zero_z3024502163831676836od_a_a ) )
= ( add_ms8655138167283798533od_a_a @ X3 @ ( minus_7313152569058643924od_a_a @ M @ ( add_ms8655138167283798533od_a_a @ Y @ zero_z3024502163831676836od_a_a ) ) ) ) ) ).
% diff_union_swap2
thf(fact_491_insert__DiffM,axiom,
! [X3: produc5665077874975597415od_a_a,M: multis3377851686571972653od_a_a] :
( ( member1567053331504362128od_a_a @ X3 @ ( set_ms5376707576967697060od_a_a @ M ) )
=> ( ( add_ms9136373847686211141od_a_a @ X3 @ ( minus_6383096217462597652od_a_a @ M @ ( add_ms9136373847686211141od_a_a @ X3 @ zero_z3721308388212328420od_a_a ) ) )
= M ) ) ).
% insert_DiffM
thf(fact_492_insert__DiffM,axiom,
! [X3: produc4424810744984089383iset_a,M: multis1386120040344743661iset_a] :
( ( member1345422370561994576iset_a @ X3 @ ( set_ms2882773486807528292iset_a @ M ) )
=> ( ( add_ms9106060982595931141iset_a @ X3 @ ( minus_2536987971823575508iset_a @ M @ ( add_ms9106060982595931141iset_a @ X3 @ zero_z9101083372863164836iset_a ) ) )
= M ) ) ).
% insert_DiffM
thf(fact_493_insert__DiffM,axiom,
! [X3: multis501812127501805293od_a_a,M: multis5935151272263360051od_a_a] :
( ( member7700929758516679318od_a_a @ X3 @ ( set_ms8253844250382261418od_a_a @ M ) )
=> ( ( add_ms4156406267932395723od_a_a @ X3 @ ( minus_385666609031338394od_a_a @ M @ ( add_ms4156406267932395723od_a_a @ X3 @ zero_z8896958667708014954od_a_a ) ) )
= M ) ) ).
% insert_DiffM
thf(fact_494_insert__DiffM,axiom,
! [X3: multis4781622204924204013iset_a,M: multis7221419791219231795iset_a] :
( ( member2597461214246177430iset_a @ X3 @ ( set_ms5327031259215843498iset_a @ M ) )
=> ( ( add_ms7610149848545062859iset_a @ X3 @ ( minus_4490418833869548698iset_a @ M @ ( add_ms7610149848545062859iset_a @ X3 @ zero_z8662914267040441450iset_a ) ) )
= M ) ) ).
% insert_DiffM
thf(fact_495_insert__DiffM,axiom,
! [X3: multiset_a,M: multiset_multiset_a] :
( ( member_multiset_a @ X3 @ ( set_mset_multiset_a @ M ) )
=> ( ( add_mset_multiset_a @ X3 @ ( minus_3395427628221709681iset_a @ M @ ( add_mset_multiset_a @ X3 @ zero_z7799948378220188993iset_a ) ) )
= M ) ) ).
% insert_DiffM
thf(fact_496_insert__DiffM,axiom,
! [X3: a,M: multiset_a] :
( ( member_a @ X3 @ ( set_mset_a @ M ) )
=> ( ( add_mset_a @ X3 @ ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ X3 @ zero_zero_multiset_a ) ) )
= M ) ) ).
% insert_DiffM
thf(fact_497_insert__DiffM,axiom,
! [X3: produc6518373309651786023iset_a,M: multis4781622204924204013iset_a] :
( ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ M ) )
=> ( ( add_ms5348986034529679365iset_a @ X3 @ ( minus_9211619285413683668iset_a @ M @ ( add_ms5348986034529679365iset_a @ X3 @ zero_z4591712717340762532iset_a ) ) )
= M ) ) ).
% insert_DiffM
thf(fact_498_insert__DiffM,axiom,
! [X3: product_prod_a_a,M: multis501812127501805293od_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ M ) )
=> ( ( add_ms8655138167283798533od_a_a @ X3 @ ( minus_7313152569058643924od_a_a @ M @ ( add_ms8655138167283798533od_a_a @ X3 @ zero_z3024502163831676836od_a_a ) ) )
= M ) ) ).
% insert_DiffM
thf(fact_499_inter__add__right1,axiom,
! [X3: produc5665077874975597415od_a_a,N: multis3377851686571972653od_a_a,M: multis3377851686571972653od_a_a] :
( ~ ( member1567053331504362128od_a_a @ X3 @ ( set_ms5376707576967697060od_a_a @ N ) )
=> ( ( inter_2707069342228023466od_a_a @ N @ ( add_ms9136373847686211141od_a_a @ X3 @ M ) )
= ( inter_2707069342228023466od_a_a @ N @ M ) ) ) ).
% inter_add_right1
thf(fact_500_inter__add__right1,axiom,
! [X3: produc4424810744984089383iset_a,N: multis1386120040344743661iset_a,M: multis1386120040344743661iset_a] :
( ~ ( member1345422370561994576iset_a @ X3 @ ( set_ms2882773486807528292iset_a @ N ) )
=> ( ( inter_7955219499984291690iset_a @ N @ ( add_ms9106060982595931141iset_a @ X3 @ M ) )
= ( inter_7955219499984291690iset_a @ N @ M ) ) ) ).
% inter_add_right1
thf(fact_501_inter__add__right1,axiom,
! [X3: multiset_a,N: multiset_multiset_a,M: multiset_multiset_a] :
( ~ ( member_multiset_a @ X3 @ ( set_mset_multiset_a @ N ) )
=> ( ( inter_3508108863637392903iset_a @ N @ ( add_mset_multiset_a @ X3 @ M ) )
= ( inter_3508108863637392903iset_a @ N @ M ) ) ) ).
% inter_add_right1
thf(fact_502_inter__add__right1,axiom,
! [X3: multis501812127501805293od_a_a,N: multis5935151272263360051od_a_a,M: multis5935151272263360051od_a_a] :
( ~ ( member7700929758516679318od_a_a @ X3 @ ( set_ms8253844250382261418od_a_a @ N ) )
=> ( ( inter_1480659854676493744od_a_a @ N @ ( add_ms4156406267932395723od_a_a @ X3 @ M ) )
= ( inter_1480659854676493744od_a_a @ N @ M ) ) ) ).
% inter_add_right1
thf(fact_503_inter__add__right1,axiom,
! [X3: multis4781622204924204013iset_a,N: multis7221419791219231795iset_a,M: multis7221419791219231795iset_a] :
( ~ ( member2597461214246177430iset_a @ X3 @ ( set_ms5327031259215843498iset_a @ N ) )
=> ( ( inter_8933094105380514736iset_a @ N @ ( add_ms7610149848545062859iset_a @ X3 @ M ) )
= ( inter_8933094105380514736iset_a @ N @ M ) ) ) ).
% inter_add_right1
thf(fact_504_inter__add__right1,axiom,
! [X3: produc6518373309651786023iset_a,N: multis4781622204924204013iset_a,M: multis4781622204924204013iset_a] :
( ~ ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ N ) )
=> ( ( inter_1588963877939557482iset_a @ N @ ( add_ms5348986034529679365iset_a @ X3 @ M ) )
= ( inter_1588963877939557482iset_a @ N @ M ) ) ) ).
% inter_add_right1
thf(fact_505_inter__add__right1,axiom,
! [X3: product_prod_a_a,N: multis501812127501805293od_a_a,M: multis501812127501805293od_a_a] :
( ~ ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ N ) )
=> ( ( inter_3651181809579396970od_a_a @ N @ ( add_ms8655138167283798533od_a_a @ X3 @ M ) )
= ( inter_3651181809579396970od_a_a @ N @ M ) ) ) ).
% inter_add_right1
thf(fact_506_inter__add__right1,axiom,
! [X3: a,N: multiset_a,M: multiset_a] :
( ~ ( member_a @ X3 @ ( set_mset_a @ N ) )
=> ( ( inter_mset_a @ N @ ( add_mset_a @ X3 @ M ) )
= ( inter_mset_a @ N @ M ) ) ) ).
% inter_add_right1
thf(fact_507_inter__add__left1,axiom,
! [X3: produc5665077874975597415od_a_a,N: multis3377851686571972653od_a_a,M: multis3377851686571972653od_a_a] :
( ~ ( member1567053331504362128od_a_a @ X3 @ ( set_ms5376707576967697060od_a_a @ N ) )
=> ( ( inter_2707069342228023466od_a_a @ ( add_ms9136373847686211141od_a_a @ X3 @ M ) @ N )
= ( inter_2707069342228023466od_a_a @ M @ N ) ) ) ).
% inter_add_left1
thf(fact_508_inter__add__left1,axiom,
! [X3: produc4424810744984089383iset_a,N: multis1386120040344743661iset_a,M: multis1386120040344743661iset_a] :
( ~ ( member1345422370561994576iset_a @ X3 @ ( set_ms2882773486807528292iset_a @ N ) )
=> ( ( inter_7955219499984291690iset_a @ ( add_ms9106060982595931141iset_a @ X3 @ M ) @ N )
= ( inter_7955219499984291690iset_a @ M @ N ) ) ) ).
% inter_add_left1
thf(fact_509_inter__add__left1,axiom,
! [X3: multiset_a,N: multiset_multiset_a,M: multiset_multiset_a] :
( ~ ( member_multiset_a @ X3 @ ( set_mset_multiset_a @ N ) )
=> ( ( inter_3508108863637392903iset_a @ ( add_mset_multiset_a @ X3 @ M ) @ N )
= ( inter_3508108863637392903iset_a @ M @ N ) ) ) ).
% inter_add_left1
thf(fact_510_inter__add__left1,axiom,
! [X3: multis501812127501805293od_a_a,N: multis5935151272263360051od_a_a,M: multis5935151272263360051od_a_a] :
( ~ ( member7700929758516679318od_a_a @ X3 @ ( set_ms8253844250382261418od_a_a @ N ) )
=> ( ( inter_1480659854676493744od_a_a @ ( add_ms4156406267932395723od_a_a @ X3 @ M ) @ N )
= ( inter_1480659854676493744od_a_a @ M @ N ) ) ) ).
% inter_add_left1
thf(fact_511_inter__add__left1,axiom,
! [X3: multis4781622204924204013iset_a,N: multis7221419791219231795iset_a,M: multis7221419791219231795iset_a] :
( ~ ( member2597461214246177430iset_a @ X3 @ ( set_ms5327031259215843498iset_a @ N ) )
=> ( ( inter_8933094105380514736iset_a @ ( add_ms7610149848545062859iset_a @ X3 @ M ) @ N )
= ( inter_8933094105380514736iset_a @ M @ N ) ) ) ).
% inter_add_left1
thf(fact_512_inter__add__left1,axiom,
! [X3: produc6518373309651786023iset_a,N: multis4781622204924204013iset_a,M: multis4781622204924204013iset_a] :
( ~ ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ N ) )
=> ( ( inter_1588963877939557482iset_a @ ( add_ms5348986034529679365iset_a @ X3 @ M ) @ N )
= ( inter_1588963877939557482iset_a @ M @ N ) ) ) ).
% inter_add_left1
thf(fact_513_inter__add__left1,axiom,
! [X3: product_prod_a_a,N: multis501812127501805293od_a_a,M: multis501812127501805293od_a_a] :
( ~ ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ N ) )
=> ( ( inter_3651181809579396970od_a_a @ ( add_ms8655138167283798533od_a_a @ X3 @ M ) @ N )
= ( inter_3651181809579396970od_a_a @ M @ N ) ) ) ).
% inter_add_left1
thf(fact_514_inter__add__left1,axiom,
! [X3: a,N: multiset_a,M: multiset_a] :
( ~ ( member_a @ X3 @ ( set_mset_a @ N ) )
=> ( ( inter_mset_a @ ( add_mset_a @ X3 @ M ) @ N )
= ( inter_mset_a @ M @ N ) ) ) ).
% inter_add_left1
thf(fact_515_multiset__inter__left__commute,axiom,
! [A3: multis501812127501805293od_a_a,B3: multis501812127501805293od_a_a,C3: multis501812127501805293od_a_a] :
( ( inter_3651181809579396970od_a_a @ A3 @ ( inter_3651181809579396970od_a_a @ B3 @ C3 ) )
= ( inter_3651181809579396970od_a_a @ B3 @ ( inter_3651181809579396970od_a_a @ A3 @ C3 ) ) ) ).
% multiset_inter_left_commute
thf(fact_516_multiset__inter__left__commute,axiom,
! [A3: multis4781622204924204013iset_a,B3: multis4781622204924204013iset_a,C3: multis4781622204924204013iset_a] :
( ( inter_1588963877939557482iset_a @ A3 @ ( inter_1588963877939557482iset_a @ B3 @ C3 ) )
= ( inter_1588963877939557482iset_a @ B3 @ ( inter_1588963877939557482iset_a @ A3 @ C3 ) ) ) ).
% multiset_inter_left_commute
thf(fact_517_multiset__inter__left__commute,axiom,
! [A3: multiset_a,B3: multiset_a,C3: multiset_a] :
( ( inter_mset_a @ A3 @ ( inter_mset_a @ B3 @ C3 ) )
= ( inter_mset_a @ B3 @ ( inter_mset_a @ A3 @ C3 ) ) ) ).
% multiset_inter_left_commute
thf(fact_518_multiset__inter__commute,axiom,
( inter_3651181809579396970od_a_a
= ( ^ [A6: multis501812127501805293od_a_a,B6: multis501812127501805293od_a_a] : ( inter_3651181809579396970od_a_a @ B6 @ A6 ) ) ) ).
% multiset_inter_commute
thf(fact_519_multiset__inter__commute,axiom,
( inter_1588963877939557482iset_a
= ( ^ [A6: multis4781622204924204013iset_a,B6: multis4781622204924204013iset_a] : ( inter_1588963877939557482iset_a @ B6 @ A6 ) ) ) ).
% multiset_inter_commute
thf(fact_520_multiset__inter__commute,axiom,
( inter_mset_a
= ( ^ [A6: multiset_a,B6: multiset_a] : ( inter_mset_a @ B6 @ A6 ) ) ) ).
% multiset_inter_commute
thf(fact_521_multiset__inter__assoc,axiom,
! [A3: multis501812127501805293od_a_a,B3: multis501812127501805293od_a_a,C3: multis501812127501805293od_a_a] :
( ( inter_3651181809579396970od_a_a @ A3 @ ( inter_3651181809579396970od_a_a @ B3 @ C3 ) )
= ( inter_3651181809579396970od_a_a @ ( inter_3651181809579396970od_a_a @ A3 @ B3 ) @ C3 ) ) ).
% multiset_inter_assoc
thf(fact_522_multiset__inter__assoc,axiom,
! [A3: multis4781622204924204013iset_a,B3: multis4781622204924204013iset_a,C3: multis4781622204924204013iset_a] :
( ( inter_1588963877939557482iset_a @ A3 @ ( inter_1588963877939557482iset_a @ B3 @ C3 ) )
= ( inter_1588963877939557482iset_a @ ( inter_1588963877939557482iset_a @ A3 @ B3 ) @ C3 ) ) ).
% multiset_inter_assoc
thf(fact_523_multiset__inter__assoc,axiom,
! [A3: multiset_a,B3: multiset_a,C3: multiset_a] :
( ( inter_mset_a @ A3 @ ( inter_mset_a @ B3 @ C3 ) )
= ( inter_mset_a @ ( inter_mset_a @ A3 @ B3 ) @ C3 ) ) ).
% multiset_inter_assoc
thf(fact_524_add__mset__commute,axiom,
! [X3: product_prod_a_a,Y: product_prod_a_a,M: multis501812127501805293od_a_a] :
( ( add_ms8655138167283798533od_a_a @ X3 @ ( add_ms8655138167283798533od_a_a @ Y @ M ) )
= ( add_ms8655138167283798533od_a_a @ Y @ ( add_ms8655138167283798533od_a_a @ X3 @ M ) ) ) ).
% add_mset_commute
thf(fact_525_add__mset__commute,axiom,
! [X3: produc6518373309651786023iset_a,Y: produc6518373309651786023iset_a,M: multis4781622204924204013iset_a] :
( ( add_ms5348986034529679365iset_a @ X3 @ ( add_ms5348986034529679365iset_a @ Y @ M ) )
= ( add_ms5348986034529679365iset_a @ Y @ ( add_ms5348986034529679365iset_a @ X3 @ M ) ) ) ).
% add_mset_commute
thf(fact_526_add__mset__commute,axiom,
! [X3: a,Y: a,M: multiset_a] :
( ( add_mset_a @ X3 @ ( add_mset_a @ Y @ M ) )
= ( add_mset_a @ Y @ ( add_mset_a @ X3 @ M ) ) ) ).
% add_mset_commute
thf(fact_527_add__mset__commute,axiom,
! [X3: multiset_a,Y: multiset_a,M: multiset_multiset_a] :
( ( add_mset_multiset_a @ X3 @ ( add_mset_multiset_a @ Y @ M ) )
= ( add_mset_multiset_a @ Y @ ( add_mset_multiset_a @ X3 @ M ) ) ) ).
% add_mset_commute
thf(fact_528_add__eq__conv__ex,axiom,
! [A: product_prod_a_a,M: multis501812127501805293od_a_a,B: product_prod_a_a,N: multis501812127501805293od_a_a] :
( ( ( add_ms8655138167283798533od_a_a @ A @ M )
= ( add_ms8655138167283798533od_a_a @ B @ N ) )
= ( ( ( M = N )
& ( A = B ) )
| ? [K4: multis501812127501805293od_a_a] :
( ( M
= ( add_ms8655138167283798533od_a_a @ B @ K4 ) )
& ( N
= ( add_ms8655138167283798533od_a_a @ A @ K4 ) ) ) ) ) ).
% add_eq_conv_ex
thf(fact_529_add__eq__conv__ex,axiom,
! [A: produc6518373309651786023iset_a,M: multis4781622204924204013iset_a,B: produc6518373309651786023iset_a,N: multis4781622204924204013iset_a] :
( ( ( add_ms5348986034529679365iset_a @ A @ M )
= ( add_ms5348986034529679365iset_a @ B @ N ) )
= ( ( ( M = N )
& ( A = B ) )
| ? [K4: multis4781622204924204013iset_a] :
( ( M
= ( add_ms5348986034529679365iset_a @ B @ K4 ) )
& ( N
= ( add_ms5348986034529679365iset_a @ A @ K4 ) ) ) ) ) ).
% add_eq_conv_ex
thf(fact_530_add__eq__conv__ex,axiom,
! [A: a,M: multiset_a,B: a,N: multiset_a] :
( ( ( add_mset_a @ A @ M )
= ( add_mset_a @ B @ N ) )
= ( ( ( M = N )
& ( A = B ) )
| ? [K4: multiset_a] :
( ( M
= ( add_mset_a @ B @ K4 ) )
& ( N
= ( add_mset_a @ A @ K4 ) ) ) ) ) ).
% add_eq_conv_ex
thf(fact_531_add__eq__conv__ex,axiom,
! [A: multiset_a,M: multiset_multiset_a,B: multiset_a,N: multiset_multiset_a] :
( ( ( add_mset_multiset_a @ A @ M )
= ( add_mset_multiset_a @ B @ N ) )
= ( ( ( M = N )
& ( A = B ) )
| ? [K4: multiset_multiset_a] :
( ( M
= ( add_mset_multiset_a @ B @ K4 ) )
& ( N
= ( add_mset_multiset_a @ A @ K4 ) ) ) ) ) ).
% add_eq_conv_ex
thf(fact_532_subset__mset_Oinf__left__commute,axiom,
! [X3: multis501812127501805293od_a_a,Y: multis501812127501805293od_a_a,Z: multis501812127501805293od_a_a] :
( ( inter_3651181809579396970od_a_a @ X3 @ ( inter_3651181809579396970od_a_a @ Y @ Z ) )
= ( inter_3651181809579396970od_a_a @ Y @ ( inter_3651181809579396970od_a_a @ X3 @ Z ) ) ) ).
% subset_mset.inf_left_commute
thf(fact_533_subset__mset_Oinf__left__commute,axiom,
! [X3: multis4781622204924204013iset_a,Y: multis4781622204924204013iset_a,Z: multis4781622204924204013iset_a] :
( ( inter_1588963877939557482iset_a @ X3 @ ( inter_1588963877939557482iset_a @ Y @ Z ) )
= ( inter_1588963877939557482iset_a @ Y @ ( inter_1588963877939557482iset_a @ X3 @ Z ) ) ) ).
% subset_mset.inf_left_commute
thf(fact_534_subset__mset_Oinf__left__commute,axiom,
! [X3: multiset_a,Y: multiset_a,Z: multiset_a] :
( ( inter_mset_a @ X3 @ ( inter_mset_a @ Y @ Z ) )
= ( inter_mset_a @ Y @ ( inter_mset_a @ X3 @ Z ) ) ) ).
% subset_mset.inf_left_commute
thf(fact_535_subset__mset_Oinf_Oleft__commute,axiom,
! [B: multis501812127501805293od_a_a,A: multis501812127501805293od_a_a,C2: multis501812127501805293od_a_a] :
( ( inter_3651181809579396970od_a_a @ B @ ( inter_3651181809579396970od_a_a @ A @ C2 ) )
= ( inter_3651181809579396970od_a_a @ A @ ( inter_3651181809579396970od_a_a @ B @ C2 ) ) ) ).
% subset_mset.inf.left_commute
thf(fact_536_subset__mset_Oinf_Oleft__commute,axiom,
! [B: multis4781622204924204013iset_a,A: multis4781622204924204013iset_a,C2: multis4781622204924204013iset_a] :
( ( inter_1588963877939557482iset_a @ B @ ( inter_1588963877939557482iset_a @ A @ C2 ) )
= ( inter_1588963877939557482iset_a @ A @ ( inter_1588963877939557482iset_a @ B @ C2 ) ) ) ).
% subset_mset.inf.left_commute
thf(fact_537_subset__mset_Oinf_Oleft__commute,axiom,
! [B: multiset_a,A: multiset_a,C2: multiset_a] :
( ( inter_mset_a @ B @ ( inter_mset_a @ A @ C2 ) )
= ( inter_mset_a @ A @ ( inter_mset_a @ B @ C2 ) ) ) ).
% subset_mset.inf.left_commute
thf(fact_538_subset__mset_Oinf__commute,axiom,
( inter_3651181809579396970od_a_a
= ( ^ [X6: multis501812127501805293od_a_a,Y5: multis501812127501805293od_a_a] : ( inter_3651181809579396970od_a_a @ Y5 @ X6 ) ) ) ).
% subset_mset.inf_commute
thf(fact_539_subset__mset_Oinf__commute,axiom,
( inter_1588963877939557482iset_a
= ( ^ [X6: multis4781622204924204013iset_a,Y5: multis4781622204924204013iset_a] : ( inter_1588963877939557482iset_a @ Y5 @ X6 ) ) ) ).
% subset_mset.inf_commute
thf(fact_540_subset__mset_Oinf__commute,axiom,
( inter_mset_a
= ( ^ [X6: multiset_a,Y5: multiset_a] : ( inter_mset_a @ Y5 @ X6 ) ) ) ).
% subset_mset.inf_commute
thf(fact_541_subset__mset_Oinf_Ocommute,axiom,
( inter_3651181809579396970od_a_a
= ( ^ [A5: multis501812127501805293od_a_a,B5: multis501812127501805293od_a_a] : ( inter_3651181809579396970od_a_a @ B5 @ A5 ) ) ) ).
% subset_mset.inf.commute
thf(fact_542_subset__mset_Oinf_Ocommute,axiom,
( inter_1588963877939557482iset_a
= ( ^ [A5: multis4781622204924204013iset_a,B5: multis4781622204924204013iset_a] : ( inter_1588963877939557482iset_a @ B5 @ A5 ) ) ) ).
% subset_mset.inf.commute
thf(fact_543_subset__mset_Oinf_Ocommute,axiom,
( inter_mset_a
= ( ^ [A5: multiset_a,B5: multiset_a] : ( inter_mset_a @ B5 @ A5 ) ) ) ).
% subset_mset.inf.commute
thf(fact_544_subset__mset_Oinf__assoc,axiom,
! [X3: multis501812127501805293od_a_a,Y: multis501812127501805293od_a_a,Z: multis501812127501805293od_a_a] :
( ( inter_3651181809579396970od_a_a @ ( inter_3651181809579396970od_a_a @ X3 @ Y ) @ Z )
= ( inter_3651181809579396970od_a_a @ X3 @ ( inter_3651181809579396970od_a_a @ Y @ Z ) ) ) ).
% subset_mset.inf_assoc
thf(fact_545_subset__mset_Oinf__assoc,axiom,
! [X3: multis4781622204924204013iset_a,Y: multis4781622204924204013iset_a,Z: multis4781622204924204013iset_a] :
( ( inter_1588963877939557482iset_a @ ( inter_1588963877939557482iset_a @ X3 @ Y ) @ Z )
= ( inter_1588963877939557482iset_a @ X3 @ ( inter_1588963877939557482iset_a @ Y @ Z ) ) ) ).
% subset_mset.inf_assoc
thf(fact_546_subset__mset_Oinf__assoc,axiom,
! [X3: multiset_a,Y: multiset_a,Z: multiset_a] :
( ( inter_mset_a @ ( inter_mset_a @ X3 @ Y ) @ Z )
= ( inter_mset_a @ X3 @ ( inter_mset_a @ Y @ Z ) ) ) ).
% subset_mset.inf_assoc
thf(fact_547_subset__mset_Oinf_Oassoc,axiom,
! [A: multis501812127501805293od_a_a,B: multis501812127501805293od_a_a,C2: multis501812127501805293od_a_a] :
( ( inter_3651181809579396970od_a_a @ ( inter_3651181809579396970od_a_a @ A @ B ) @ C2 )
= ( inter_3651181809579396970od_a_a @ A @ ( inter_3651181809579396970od_a_a @ B @ C2 ) ) ) ).
% subset_mset.inf.assoc
thf(fact_548_subset__mset_Oinf_Oassoc,axiom,
! [A: multis4781622204924204013iset_a,B: multis4781622204924204013iset_a,C2: multis4781622204924204013iset_a] :
( ( inter_1588963877939557482iset_a @ ( inter_1588963877939557482iset_a @ A @ B ) @ C2 )
= ( inter_1588963877939557482iset_a @ A @ ( inter_1588963877939557482iset_a @ B @ C2 ) ) ) ).
% subset_mset.inf.assoc
thf(fact_549_subset__mset_Oinf_Oassoc,axiom,
! [A: multiset_a,B: multiset_a,C2: multiset_a] :
( ( inter_mset_a @ ( inter_mset_a @ A @ B ) @ C2 )
= ( inter_mset_a @ A @ ( inter_mset_a @ B @ C2 ) ) ) ).
% subset_mset.inf.assoc
thf(fact_550_inter__union__distrib__left,axiom,
! [A3: multis501812127501805293od_a_a,B3: multis501812127501805293od_a_a,C3: multis501812127501805293od_a_a] :
( ( plus_p6967097483979887908od_a_a @ ( inter_3651181809579396970od_a_a @ A3 @ B3 ) @ C3 )
= ( inter_3651181809579396970od_a_a @ ( plus_p6967097483979887908od_a_a @ A3 @ C3 ) @ ( plus_p6967097483979887908od_a_a @ B3 @ C3 ) ) ) ).
% inter_union_distrib_left
thf(fact_551_inter__union__distrib__left,axiom,
! [A3: multis4781622204924204013iset_a,B3: multis4781622204924204013iset_a,C3: multis4781622204924204013iset_a] :
( ( plus_p1053901119268525860iset_a @ ( inter_1588963877939557482iset_a @ A3 @ B3 ) @ C3 )
= ( inter_1588963877939557482iset_a @ ( plus_p1053901119268525860iset_a @ A3 @ C3 ) @ ( plus_p1053901119268525860iset_a @ B3 @ C3 ) ) ) ).
% inter_union_distrib_left
thf(fact_552_inter__union__distrib__left,axiom,
! [A3: multiset_multiset_a,B3: multiset_multiset_a,C3: multiset_multiset_a] :
( ( plus_p6738641960240532161iset_a @ ( inter_3508108863637392903iset_a @ A3 @ B3 ) @ C3 )
= ( inter_3508108863637392903iset_a @ ( plus_p6738641960240532161iset_a @ A3 @ C3 ) @ ( plus_p6738641960240532161iset_a @ B3 @ C3 ) ) ) ).
% inter_union_distrib_left
thf(fact_553_inter__union__distrib__left,axiom,
! [A3: multiset_a,B3: multiset_a,C3: multiset_a] :
( ( plus_plus_multiset_a @ ( inter_mset_a @ A3 @ B3 ) @ C3 )
= ( inter_mset_a @ ( plus_plus_multiset_a @ A3 @ C3 ) @ ( plus_plus_multiset_a @ B3 @ C3 ) ) ) ).
% inter_union_distrib_left
thf(fact_554_inter__union__distrib__right,axiom,
! [C3: multis501812127501805293od_a_a,A3: multis501812127501805293od_a_a,B3: multis501812127501805293od_a_a] :
( ( plus_p6967097483979887908od_a_a @ C3 @ ( inter_3651181809579396970od_a_a @ A3 @ B3 ) )
= ( inter_3651181809579396970od_a_a @ ( plus_p6967097483979887908od_a_a @ C3 @ A3 ) @ ( plus_p6967097483979887908od_a_a @ C3 @ B3 ) ) ) ).
% inter_union_distrib_right
thf(fact_555_inter__union__distrib__right,axiom,
! [C3: multis4781622204924204013iset_a,A3: multis4781622204924204013iset_a,B3: multis4781622204924204013iset_a] :
( ( plus_p1053901119268525860iset_a @ C3 @ ( inter_1588963877939557482iset_a @ A3 @ B3 ) )
= ( inter_1588963877939557482iset_a @ ( plus_p1053901119268525860iset_a @ C3 @ A3 ) @ ( plus_p1053901119268525860iset_a @ C3 @ B3 ) ) ) ).
% inter_union_distrib_right
thf(fact_556_inter__union__distrib__right,axiom,
! [C3: multiset_multiset_a,A3: multiset_multiset_a,B3: multiset_multiset_a] :
( ( plus_p6738641960240532161iset_a @ C3 @ ( inter_3508108863637392903iset_a @ A3 @ B3 ) )
= ( inter_3508108863637392903iset_a @ ( plus_p6738641960240532161iset_a @ C3 @ A3 ) @ ( plus_p6738641960240532161iset_a @ C3 @ B3 ) ) ) ).
% inter_union_distrib_right
thf(fact_557_inter__union__distrib__right,axiom,
! [C3: multiset_a,A3: multiset_a,B3: multiset_a] :
( ( plus_plus_multiset_a @ C3 @ ( inter_mset_a @ A3 @ B3 ) )
= ( inter_mset_a @ ( plus_plus_multiset_a @ C3 @ A3 ) @ ( plus_plus_multiset_a @ C3 @ B3 ) ) ) ).
% inter_union_distrib_right
thf(fact_558_inter__iff,axiom,
! [A: produc5665077874975597415od_a_a,A3: multis3377851686571972653od_a_a,B3: multis3377851686571972653od_a_a] :
( ( member1567053331504362128od_a_a @ A @ ( set_ms5376707576967697060od_a_a @ ( inter_2707069342228023466od_a_a @ A3 @ B3 ) ) )
= ( ( member1567053331504362128od_a_a @ A @ ( set_ms5376707576967697060od_a_a @ A3 ) )
& ( member1567053331504362128od_a_a @ A @ ( set_ms5376707576967697060od_a_a @ B3 ) ) ) ) ).
% inter_iff
thf(fact_559_inter__iff,axiom,
! [A: produc4424810744984089383iset_a,A3: multis1386120040344743661iset_a,B3: multis1386120040344743661iset_a] :
( ( member1345422370561994576iset_a @ A @ ( set_ms2882773486807528292iset_a @ ( inter_7955219499984291690iset_a @ A3 @ B3 ) ) )
= ( ( member1345422370561994576iset_a @ A @ ( set_ms2882773486807528292iset_a @ A3 ) )
& ( member1345422370561994576iset_a @ A @ ( set_ms2882773486807528292iset_a @ B3 ) ) ) ) ).
% inter_iff
thf(fact_560_inter__iff,axiom,
! [A: multiset_a,A3: multiset_multiset_a,B3: multiset_multiset_a] :
( ( member_multiset_a @ A @ ( set_mset_multiset_a @ ( inter_3508108863637392903iset_a @ A3 @ B3 ) ) )
= ( ( member_multiset_a @ A @ ( set_mset_multiset_a @ A3 ) )
& ( member_multiset_a @ A @ ( set_mset_multiset_a @ B3 ) ) ) ) ).
% inter_iff
thf(fact_561_inter__iff,axiom,
! [A: multis501812127501805293od_a_a,A3: multis5935151272263360051od_a_a,B3: multis5935151272263360051od_a_a] :
( ( member7700929758516679318od_a_a @ A @ ( set_ms8253844250382261418od_a_a @ ( inter_1480659854676493744od_a_a @ A3 @ B3 ) ) )
= ( ( member7700929758516679318od_a_a @ A @ ( set_ms8253844250382261418od_a_a @ A3 ) )
& ( member7700929758516679318od_a_a @ A @ ( set_ms8253844250382261418od_a_a @ B3 ) ) ) ) ).
% inter_iff
thf(fact_562_inter__iff,axiom,
! [A: multis4781622204924204013iset_a,A3: multis7221419791219231795iset_a,B3: multis7221419791219231795iset_a] :
( ( member2597461214246177430iset_a @ A @ ( set_ms5327031259215843498iset_a @ ( inter_8933094105380514736iset_a @ A3 @ B3 ) ) )
= ( ( member2597461214246177430iset_a @ A @ ( set_ms5327031259215843498iset_a @ A3 ) )
& ( member2597461214246177430iset_a @ A @ ( set_ms5327031259215843498iset_a @ B3 ) ) ) ) ).
% inter_iff
thf(fact_563_inter__iff,axiom,
! [A: produc6518373309651786023iset_a,A3: multis4781622204924204013iset_a,B3: multis4781622204924204013iset_a] :
( ( member5199237121806060112iset_a @ A @ ( set_ms2379723421352415844iset_a @ ( inter_1588963877939557482iset_a @ A3 @ B3 ) ) )
= ( ( member5199237121806060112iset_a @ A @ ( set_ms2379723421352415844iset_a @ A3 ) )
& ( member5199237121806060112iset_a @ A @ ( set_ms2379723421352415844iset_a @ B3 ) ) ) ) ).
% inter_iff
thf(fact_564_inter__iff,axiom,
! [A: product_prod_a_a,A3: multis501812127501805293od_a_a,B3: multis501812127501805293od_a_a] :
( ( member1426531477525435216od_a_a @ A @ ( set_ms119794563918438244od_a_a @ ( inter_3651181809579396970od_a_a @ A3 @ B3 ) ) )
= ( ( member1426531477525435216od_a_a @ A @ ( set_ms119794563918438244od_a_a @ A3 ) )
& ( member1426531477525435216od_a_a @ A @ ( set_ms119794563918438244od_a_a @ B3 ) ) ) ) ).
% inter_iff
thf(fact_565_inter__iff,axiom,
! [A: a,A3: multiset_a,B3: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ ( inter_mset_a @ A3 @ B3 ) ) )
= ( ( member_a @ A @ ( set_mset_a @ A3 ) )
& ( member_a @ A @ ( set_mset_a @ B3 ) ) ) ) ).
% inter_iff
thf(fact_566_multiset__cases,axiom,
! [M: multiset_multiset_a] :
( ( M != zero_z7799948378220188993iset_a )
=> ~ ! [X5: multiset_a,N3: multiset_multiset_a] :
( M
!= ( add_mset_multiset_a @ X5 @ N3 ) ) ) ).
% multiset_cases
thf(fact_567_multiset__cases,axiom,
! [M: multis501812127501805293od_a_a] :
( ( M != zero_z3024502163831676836od_a_a )
=> ~ ! [X5: product_prod_a_a,N3: multis501812127501805293od_a_a] :
( M
!= ( add_ms8655138167283798533od_a_a @ X5 @ N3 ) ) ) ).
% multiset_cases
thf(fact_568_multiset__cases,axiom,
! [M: multis4781622204924204013iset_a] :
( ( M != zero_z4591712717340762532iset_a )
=> ~ ! [X5: produc6518373309651786023iset_a,N3: multis4781622204924204013iset_a] :
( M
!= ( add_ms5348986034529679365iset_a @ X5 @ N3 ) ) ) ).
% multiset_cases
thf(fact_569_multiset__cases,axiom,
! [M: multiset_a] :
( ( M != zero_zero_multiset_a )
=> ~ ! [X5: a,N3: multiset_a] :
( M
!= ( add_mset_a @ X5 @ N3 ) ) ) ).
% multiset_cases
thf(fact_570_multiset__induct,axiom,
! [P: multiset_multiset_a > $o,M: multiset_multiset_a] :
( ( P @ zero_z7799948378220188993iset_a )
=> ( ! [X5: multiset_a,M3: multiset_multiset_a] :
( ( P @ M3 )
=> ( P @ ( add_mset_multiset_a @ X5 @ M3 ) ) )
=> ( P @ M ) ) ) ).
% multiset_induct
thf(fact_571_multiset__induct,axiom,
! [P: multis501812127501805293od_a_a > $o,M: multis501812127501805293od_a_a] :
( ( P @ zero_z3024502163831676836od_a_a )
=> ( ! [X5: product_prod_a_a,M3: multis501812127501805293od_a_a] :
( ( P @ M3 )
=> ( P @ ( add_ms8655138167283798533od_a_a @ X5 @ M3 ) ) )
=> ( P @ M ) ) ) ).
% multiset_induct
thf(fact_572_multiset__induct,axiom,
! [P: multis4781622204924204013iset_a > $o,M: multis4781622204924204013iset_a] :
( ( P @ zero_z4591712717340762532iset_a )
=> ( ! [X5: produc6518373309651786023iset_a,M3: multis4781622204924204013iset_a] :
( ( P @ M3 )
=> ( P @ ( add_ms5348986034529679365iset_a @ X5 @ M3 ) ) )
=> ( P @ M ) ) ) ).
% multiset_induct
thf(fact_573_multiset__induct,axiom,
! [P: multiset_a > $o,M: multiset_a] :
( ( P @ zero_zero_multiset_a )
=> ( ! [X5: a,M3: multiset_a] :
( ( P @ M3 )
=> ( P @ ( add_mset_a @ X5 @ M3 ) ) )
=> ( P @ M ) ) ) ).
% multiset_induct
thf(fact_574_multiset__induct2,axiom,
! [P: multiset_a > multiset_a > $o,M: multiset_a,N: multiset_a] :
( ( P @ zero_zero_multiset_a @ zero_zero_multiset_a )
=> ( ! [A4: a,M3: multiset_a,N3: multiset_a] :
( ( P @ M3 @ N3 )
=> ( P @ ( add_mset_a @ A4 @ M3 ) @ N3 ) )
=> ( ! [A4: a,M3: multiset_a,N3: multiset_a] :
( ( P @ M3 @ N3 )
=> ( P @ M3 @ ( add_mset_a @ A4 @ N3 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% multiset_induct2
thf(fact_575_multiset__induct2,axiom,
! [P: multiset_a > multiset_multiset_a > $o,M: multiset_a,N: multiset_multiset_a] :
( ( P @ zero_zero_multiset_a @ zero_z7799948378220188993iset_a )
=> ( ! [A4: a,M3: multiset_a,N3: multiset_multiset_a] :
( ( P @ M3 @ N3 )
=> ( P @ ( add_mset_a @ A4 @ M3 ) @ N3 ) )
=> ( ! [A4: multiset_a,M3: multiset_a,N3: multiset_multiset_a] :
( ( P @ M3 @ N3 )
=> ( P @ M3 @ ( add_mset_multiset_a @ A4 @ N3 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% multiset_induct2
thf(fact_576_multiset__induct2,axiom,
! [P: multiset_multiset_a > multiset_a > $o,M: multiset_multiset_a,N: multiset_a] :
( ( P @ zero_z7799948378220188993iset_a @ zero_zero_multiset_a )
=> ( ! [A4: multiset_a,M3: multiset_multiset_a,N3: multiset_a] :
( ( P @ M3 @ N3 )
=> ( P @ ( add_mset_multiset_a @ A4 @ M3 ) @ N3 ) )
=> ( ! [A4: a,M3: multiset_multiset_a,N3: multiset_a] :
( ( P @ M3 @ N3 )
=> ( P @ M3 @ ( add_mset_a @ A4 @ N3 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% multiset_induct2
thf(fact_577_multiset__induct2,axiom,
! [P: multiset_a > multis501812127501805293od_a_a > $o,M: multiset_a,N: multis501812127501805293od_a_a] :
( ( P @ zero_zero_multiset_a @ zero_z3024502163831676836od_a_a )
=> ( ! [A4: a,M3: multiset_a,N3: multis501812127501805293od_a_a] :
( ( P @ M3 @ N3 )
=> ( P @ ( add_mset_a @ A4 @ M3 ) @ N3 ) )
=> ( ! [A4: product_prod_a_a,M3: multiset_a,N3: multis501812127501805293od_a_a] :
( ( P @ M3 @ N3 )
=> ( P @ M3 @ ( add_ms8655138167283798533od_a_a @ A4 @ N3 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% multiset_induct2
thf(fact_578_multiset__induct2,axiom,
! [P: multiset_multiset_a > multiset_multiset_a > $o,M: multiset_multiset_a,N: multiset_multiset_a] :
( ( P @ zero_z7799948378220188993iset_a @ zero_z7799948378220188993iset_a )
=> ( ! [A4: multiset_a,M3: multiset_multiset_a,N3: multiset_multiset_a] :
( ( P @ M3 @ N3 )
=> ( P @ ( add_mset_multiset_a @ A4 @ M3 ) @ N3 ) )
=> ( ! [A4: multiset_a,M3: multiset_multiset_a,N3: multiset_multiset_a] :
( ( P @ M3 @ N3 )
=> ( P @ M3 @ ( add_mset_multiset_a @ A4 @ N3 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% multiset_induct2
thf(fact_579_multiset__induct2,axiom,
! [P: multis501812127501805293od_a_a > multiset_a > $o,M: multis501812127501805293od_a_a,N: multiset_a] :
( ( P @ zero_z3024502163831676836od_a_a @ zero_zero_multiset_a )
=> ( ! [A4: product_prod_a_a,M3: multis501812127501805293od_a_a,N3: multiset_a] :
( ( P @ M3 @ N3 )
=> ( P @ ( add_ms8655138167283798533od_a_a @ A4 @ M3 ) @ N3 ) )
=> ( ! [A4: a,M3: multis501812127501805293od_a_a,N3: multiset_a] :
( ( P @ M3 @ N3 )
=> ( P @ M3 @ ( add_mset_a @ A4 @ N3 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% multiset_induct2
thf(fact_580_multiset__induct2,axiom,
! [P: multiset_multiset_a > multis501812127501805293od_a_a > $o,M: multiset_multiset_a,N: multis501812127501805293od_a_a] :
( ( P @ zero_z7799948378220188993iset_a @ zero_z3024502163831676836od_a_a )
=> ( ! [A4: multiset_a,M3: multiset_multiset_a,N3: multis501812127501805293od_a_a] :
( ( P @ M3 @ N3 )
=> ( P @ ( add_mset_multiset_a @ A4 @ M3 ) @ N3 ) )
=> ( ! [A4: product_prod_a_a,M3: multiset_multiset_a,N3: multis501812127501805293od_a_a] :
( ( P @ M3 @ N3 )
=> ( P @ M3 @ ( add_ms8655138167283798533od_a_a @ A4 @ N3 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% multiset_induct2
thf(fact_581_multiset__induct2,axiom,
! [P: multis501812127501805293od_a_a > multiset_multiset_a > $o,M: multis501812127501805293od_a_a,N: multiset_multiset_a] :
( ( P @ zero_z3024502163831676836od_a_a @ zero_z7799948378220188993iset_a )
=> ( ! [A4: product_prod_a_a,M3: multis501812127501805293od_a_a,N3: multiset_multiset_a] :
( ( P @ M3 @ N3 )
=> ( P @ ( add_ms8655138167283798533od_a_a @ A4 @ M3 ) @ N3 ) )
=> ( ! [A4: multiset_a,M3: multis501812127501805293od_a_a,N3: multiset_multiset_a] :
( ( P @ M3 @ N3 )
=> ( P @ M3 @ ( add_mset_multiset_a @ A4 @ N3 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% multiset_induct2
thf(fact_582_multiset__induct2,axiom,
! [P: multiset_a > multis4781622204924204013iset_a > $o,M: multiset_a,N: multis4781622204924204013iset_a] :
( ( P @ zero_zero_multiset_a @ zero_z4591712717340762532iset_a )
=> ( ! [A4: a,M3: multiset_a,N3: multis4781622204924204013iset_a] :
( ( P @ M3 @ N3 )
=> ( P @ ( add_mset_a @ A4 @ M3 ) @ N3 ) )
=> ( ! [A4: produc6518373309651786023iset_a,M3: multiset_a,N3: multis4781622204924204013iset_a] :
( ( P @ M3 @ N3 )
=> ( P @ M3 @ ( add_ms5348986034529679365iset_a @ A4 @ N3 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% multiset_induct2
thf(fact_583_multiset__induct2,axiom,
! [P: multis501812127501805293od_a_a > multis501812127501805293od_a_a > $o,M: multis501812127501805293od_a_a,N: multis501812127501805293od_a_a] :
( ( P @ zero_z3024502163831676836od_a_a @ zero_z3024502163831676836od_a_a )
=> ( ! [A4: product_prod_a_a,M3: multis501812127501805293od_a_a,N3: multis501812127501805293od_a_a] :
( ( P @ M3 @ N3 )
=> ( P @ ( add_ms8655138167283798533od_a_a @ A4 @ M3 ) @ N3 ) )
=> ( ! [A4: product_prod_a_a,M3: multis501812127501805293od_a_a,N3: multis501812127501805293od_a_a] :
( ( P @ M3 @ N3 )
=> ( P @ M3 @ ( add_ms8655138167283798533od_a_a @ A4 @ N3 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% multiset_induct2
thf(fact_584_empty__not__add__mset,axiom,
! [A: multiset_a,A3: multiset_multiset_a] :
( zero_z7799948378220188993iset_a
!= ( add_mset_multiset_a @ A @ A3 ) ) ).
% empty_not_add_mset
thf(fact_585_empty__not__add__mset,axiom,
! [A: product_prod_a_a,A3: multis501812127501805293od_a_a] :
( zero_z3024502163831676836od_a_a
!= ( add_ms8655138167283798533od_a_a @ A @ A3 ) ) ).
% empty_not_add_mset
thf(fact_586_empty__not__add__mset,axiom,
! [A: produc6518373309651786023iset_a,A3: multis4781622204924204013iset_a] :
( zero_z4591712717340762532iset_a
!= ( add_ms5348986034529679365iset_a @ A @ A3 ) ) ).
% empty_not_add_mset
thf(fact_587_empty__not__add__mset,axiom,
! [A: a,A3: multiset_a] :
( zero_zero_multiset_a
!= ( add_mset_a @ A @ A3 ) ) ).
% empty_not_add_mset
thf(fact_588_multi__nonempty__split,axiom,
! [M: multiset_multiset_a] :
( ( M != zero_z7799948378220188993iset_a )
=> ? [A7: multiset_multiset_a,A4: multiset_a] :
( M
= ( add_mset_multiset_a @ A4 @ A7 ) ) ) ).
% multi_nonempty_split
thf(fact_589_multi__nonempty__split,axiom,
! [M: multis501812127501805293od_a_a] :
( ( M != zero_z3024502163831676836od_a_a )
=> ? [A7: multis501812127501805293od_a_a,A4: product_prod_a_a] :
( M
= ( add_ms8655138167283798533od_a_a @ A4 @ A7 ) ) ) ).
% multi_nonempty_split
thf(fact_590_multi__nonempty__split,axiom,
! [M: multis4781622204924204013iset_a] :
( ( M != zero_z4591712717340762532iset_a )
=> ? [A7: multis4781622204924204013iset_a,A4: produc6518373309651786023iset_a] :
( M
= ( add_ms5348986034529679365iset_a @ A4 @ A7 ) ) ) ).
% multi_nonempty_split
thf(fact_591_multi__nonempty__split,axiom,
! [M: multiset_a] :
( ( M != zero_zero_multiset_a )
=> ? [A7: multiset_a,A4: a] :
( M
= ( add_mset_a @ A4 @ A7 ) ) ) ).
% multi_nonempty_split
thf(fact_592_union__single__eq__member,axiom,
! [X3: produc5665077874975597415od_a_a,M: multis3377851686571972653od_a_a,N: multis3377851686571972653od_a_a] :
( ( ( add_ms9136373847686211141od_a_a @ X3 @ M )
= N )
=> ( member1567053331504362128od_a_a @ X3 @ ( set_ms5376707576967697060od_a_a @ N ) ) ) ).
% union_single_eq_member
thf(fact_593_union__single__eq__member,axiom,
! [X3: produc4424810744984089383iset_a,M: multis1386120040344743661iset_a,N: multis1386120040344743661iset_a] :
( ( ( add_ms9106060982595931141iset_a @ X3 @ M )
= N )
=> ( member1345422370561994576iset_a @ X3 @ ( set_ms2882773486807528292iset_a @ N ) ) ) ).
% union_single_eq_member
thf(fact_594_union__single__eq__member,axiom,
! [X3: multiset_a,M: multiset_multiset_a,N: multiset_multiset_a] :
( ( ( add_mset_multiset_a @ X3 @ M )
= N )
=> ( member_multiset_a @ X3 @ ( set_mset_multiset_a @ N ) ) ) ).
% union_single_eq_member
thf(fact_595_union__single__eq__member,axiom,
! [X3: multis501812127501805293od_a_a,M: multis5935151272263360051od_a_a,N: multis5935151272263360051od_a_a] :
( ( ( add_ms4156406267932395723od_a_a @ X3 @ M )
= N )
=> ( member7700929758516679318od_a_a @ X3 @ ( set_ms8253844250382261418od_a_a @ N ) ) ) ).
% union_single_eq_member
thf(fact_596_union__single__eq__member,axiom,
! [X3: multis4781622204924204013iset_a,M: multis7221419791219231795iset_a,N: multis7221419791219231795iset_a] :
( ( ( add_ms7610149848545062859iset_a @ X3 @ M )
= N )
=> ( member2597461214246177430iset_a @ X3 @ ( set_ms5327031259215843498iset_a @ N ) ) ) ).
% union_single_eq_member
thf(fact_597_union__single__eq__member,axiom,
! [X3: a,M: multiset_a,N: multiset_a] :
( ( ( add_mset_a @ X3 @ M )
= N )
=> ( member_a @ X3 @ ( set_mset_a @ N ) ) ) ).
% union_single_eq_member
thf(fact_598_union__single__eq__member,axiom,
! [X3: produc6518373309651786023iset_a,M: multis4781622204924204013iset_a,N: multis4781622204924204013iset_a] :
( ( ( add_ms5348986034529679365iset_a @ X3 @ M )
= N )
=> ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ N ) ) ) ).
% union_single_eq_member
thf(fact_599_union__single__eq__member,axiom,
! [X3: product_prod_a_a,M: multis501812127501805293od_a_a,N: multis501812127501805293od_a_a] :
( ( ( add_ms8655138167283798533od_a_a @ X3 @ M )
= N )
=> ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ N ) ) ) ).
% union_single_eq_member
thf(fact_600_insert__noteq__member,axiom,
! [B: produc5665077874975597415od_a_a,B3: multis3377851686571972653od_a_a,C2: produc5665077874975597415od_a_a,C3: multis3377851686571972653od_a_a] :
( ( ( add_ms9136373847686211141od_a_a @ B @ B3 )
= ( add_ms9136373847686211141od_a_a @ C2 @ C3 ) )
=> ( ( B != C2 )
=> ( member1567053331504362128od_a_a @ C2 @ ( set_ms5376707576967697060od_a_a @ B3 ) ) ) ) ).
% insert_noteq_member
thf(fact_601_insert__noteq__member,axiom,
! [B: produc4424810744984089383iset_a,B3: multis1386120040344743661iset_a,C2: produc4424810744984089383iset_a,C3: multis1386120040344743661iset_a] :
( ( ( add_ms9106060982595931141iset_a @ B @ B3 )
= ( add_ms9106060982595931141iset_a @ C2 @ C3 ) )
=> ( ( B != C2 )
=> ( member1345422370561994576iset_a @ C2 @ ( set_ms2882773486807528292iset_a @ B3 ) ) ) ) ).
% insert_noteq_member
thf(fact_602_insert__noteq__member,axiom,
! [B: multiset_a,B3: multiset_multiset_a,C2: multiset_a,C3: multiset_multiset_a] :
( ( ( add_mset_multiset_a @ B @ B3 )
= ( add_mset_multiset_a @ C2 @ C3 ) )
=> ( ( B != C2 )
=> ( member_multiset_a @ C2 @ ( set_mset_multiset_a @ B3 ) ) ) ) ).
% insert_noteq_member
thf(fact_603_insert__noteq__member,axiom,
! [B: multis501812127501805293od_a_a,B3: multis5935151272263360051od_a_a,C2: multis501812127501805293od_a_a,C3: multis5935151272263360051od_a_a] :
( ( ( add_ms4156406267932395723od_a_a @ B @ B3 )
= ( add_ms4156406267932395723od_a_a @ C2 @ C3 ) )
=> ( ( B != C2 )
=> ( member7700929758516679318od_a_a @ C2 @ ( set_ms8253844250382261418od_a_a @ B3 ) ) ) ) ).
% insert_noteq_member
thf(fact_604_insert__noteq__member,axiom,
! [B: multis4781622204924204013iset_a,B3: multis7221419791219231795iset_a,C2: multis4781622204924204013iset_a,C3: multis7221419791219231795iset_a] :
( ( ( add_ms7610149848545062859iset_a @ B @ B3 )
= ( add_ms7610149848545062859iset_a @ C2 @ C3 ) )
=> ( ( B != C2 )
=> ( member2597461214246177430iset_a @ C2 @ ( set_ms5327031259215843498iset_a @ B3 ) ) ) ) ).
% insert_noteq_member
thf(fact_605_insert__noteq__member,axiom,
! [B: a,B3: multiset_a,C2: a,C3: multiset_a] :
( ( ( add_mset_a @ B @ B3 )
= ( add_mset_a @ C2 @ C3 ) )
=> ( ( B != C2 )
=> ( member_a @ C2 @ ( set_mset_a @ B3 ) ) ) ) ).
% insert_noteq_member
thf(fact_606_insert__noteq__member,axiom,
! [B: produc6518373309651786023iset_a,B3: multis4781622204924204013iset_a,C2: produc6518373309651786023iset_a,C3: multis4781622204924204013iset_a] :
( ( ( add_ms5348986034529679365iset_a @ B @ B3 )
= ( add_ms5348986034529679365iset_a @ C2 @ C3 ) )
=> ( ( B != C2 )
=> ( member5199237121806060112iset_a @ C2 @ ( set_ms2379723421352415844iset_a @ B3 ) ) ) ) ).
% insert_noteq_member
thf(fact_607_insert__noteq__member,axiom,
! [B: product_prod_a_a,B3: multis501812127501805293od_a_a,C2: product_prod_a_a,C3: multis501812127501805293od_a_a] :
( ( ( add_ms8655138167283798533od_a_a @ B @ B3 )
= ( add_ms8655138167283798533od_a_a @ C2 @ C3 ) )
=> ( ( B != C2 )
=> ( member1426531477525435216od_a_a @ C2 @ ( set_ms119794563918438244od_a_a @ B3 ) ) ) ) ).
% insert_noteq_member
thf(fact_608_multi__member__split,axiom,
! [X3: produc5665077874975597415od_a_a,M: multis3377851686571972653od_a_a] :
( ( member1567053331504362128od_a_a @ X3 @ ( set_ms5376707576967697060od_a_a @ M ) )
=> ? [A7: multis3377851686571972653od_a_a] :
( M
= ( add_ms9136373847686211141od_a_a @ X3 @ A7 ) ) ) ).
% multi_member_split
thf(fact_609_multi__member__split,axiom,
! [X3: produc4424810744984089383iset_a,M: multis1386120040344743661iset_a] :
( ( member1345422370561994576iset_a @ X3 @ ( set_ms2882773486807528292iset_a @ M ) )
=> ? [A7: multis1386120040344743661iset_a] :
( M
= ( add_ms9106060982595931141iset_a @ X3 @ A7 ) ) ) ).
% multi_member_split
thf(fact_610_multi__member__split,axiom,
! [X3: multiset_a,M: multiset_multiset_a] :
( ( member_multiset_a @ X3 @ ( set_mset_multiset_a @ M ) )
=> ? [A7: multiset_multiset_a] :
( M
= ( add_mset_multiset_a @ X3 @ A7 ) ) ) ).
% multi_member_split
thf(fact_611_multi__member__split,axiom,
! [X3: multis501812127501805293od_a_a,M: multis5935151272263360051od_a_a] :
( ( member7700929758516679318od_a_a @ X3 @ ( set_ms8253844250382261418od_a_a @ M ) )
=> ? [A7: multis5935151272263360051od_a_a] :
( M
= ( add_ms4156406267932395723od_a_a @ X3 @ A7 ) ) ) ).
% multi_member_split
thf(fact_612_multi__member__split,axiom,
! [X3: multis4781622204924204013iset_a,M: multis7221419791219231795iset_a] :
( ( member2597461214246177430iset_a @ X3 @ ( set_ms5327031259215843498iset_a @ M ) )
=> ? [A7: multis7221419791219231795iset_a] :
( M
= ( add_ms7610149848545062859iset_a @ X3 @ A7 ) ) ) ).
% multi_member_split
thf(fact_613_multi__member__split,axiom,
! [X3: a,M: multiset_a] :
( ( member_a @ X3 @ ( set_mset_a @ M ) )
=> ? [A7: multiset_a] :
( M
= ( add_mset_a @ X3 @ A7 ) ) ) ).
% multi_member_split
thf(fact_614_multi__member__split,axiom,
! [X3: produc6518373309651786023iset_a,M: multis4781622204924204013iset_a] :
( ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ M ) )
=> ? [A7: multis4781622204924204013iset_a] :
( M
= ( add_ms5348986034529679365iset_a @ X3 @ A7 ) ) ) ).
% multi_member_split
thf(fact_615_multi__member__split,axiom,
! [X3: product_prod_a_a,M: multis501812127501805293od_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ M ) )
=> ? [A7: multis501812127501805293od_a_a] :
( M
= ( add_ms8655138167283798533od_a_a @ X3 @ A7 ) ) ) ).
% multi_member_split
thf(fact_616_mset__add,axiom,
! [A: produc5665077874975597415od_a_a,A3: multis3377851686571972653od_a_a] :
( ( member1567053331504362128od_a_a @ A @ ( set_ms5376707576967697060od_a_a @ A3 ) )
=> ~ ! [B7: multis3377851686571972653od_a_a] :
( A3
!= ( add_ms9136373847686211141od_a_a @ A @ B7 ) ) ) ).
% mset_add
thf(fact_617_mset__add,axiom,
! [A: produc4424810744984089383iset_a,A3: multis1386120040344743661iset_a] :
( ( member1345422370561994576iset_a @ A @ ( set_ms2882773486807528292iset_a @ A3 ) )
=> ~ ! [B7: multis1386120040344743661iset_a] :
( A3
!= ( add_ms9106060982595931141iset_a @ A @ B7 ) ) ) ).
% mset_add
thf(fact_618_mset__add,axiom,
! [A: multiset_a,A3: multiset_multiset_a] :
( ( member_multiset_a @ A @ ( set_mset_multiset_a @ A3 ) )
=> ~ ! [B7: multiset_multiset_a] :
( A3
!= ( add_mset_multiset_a @ A @ B7 ) ) ) ).
% mset_add
thf(fact_619_mset__add,axiom,
! [A: multis501812127501805293od_a_a,A3: multis5935151272263360051od_a_a] :
( ( member7700929758516679318od_a_a @ A @ ( set_ms8253844250382261418od_a_a @ A3 ) )
=> ~ ! [B7: multis5935151272263360051od_a_a] :
( A3
!= ( add_ms4156406267932395723od_a_a @ A @ B7 ) ) ) ).
% mset_add
thf(fact_620_mset__add,axiom,
! [A: multis4781622204924204013iset_a,A3: multis7221419791219231795iset_a] :
( ( member2597461214246177430iset_a @ A @ ( set_ms5327031259215843498iset_a @ A3 ) )
=> ~ ! [B7: multis7221419791219231795iset_a] :
( A3
!= ( add_ms7610149848545062859iset_a @ A @ B7 ) ) ) ).
% mset_add
thf(fact_621_mset__add,axiom,
! [A: a,A3: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ A3 ) )
=> ~ ! [B7: multiset_a] :
( A3
!= ( add_mset_a @ A @ B7 ) ) ) ).
% mset_add
thf(fact_622_mset__add,axiom,
! [A: produc6518373309651786023iset_a,A3: multis4781622204924204013iset_a] :
( ( member5199237121806060112iset_a @ A @ ( set_ms2379723421352415844iset_a @ A3 ) )
=> ~ ! [B7: multis4781622204924204013iset_a] :
( A3
!= ( add_ms5348986034529679365iset_a @ A @ B7 ) ) ) ).
% mset_add
thf(fact_623_mset__add,axiom,
! [A: product_prod_a_a,A3: multis501812127501805293od_a_a] :
( ( member1426531477525435216od_a_a @ A @ ( set_ms119794563918438244od_a_a @ A3 ) )
=> ~ ! [B7: multis501812127501805293od_a_a] :
( A3
!= ( add_ms8655138167283798533od_a_a @ A @ B7 ) ) ) ).
% mset_add
thf(fact_624_inter__mset__def,axiom,
( inter_mset_a
= ( ^ [A6: multiset_a,B6: multiset_a] : ( minus_3765977307040488491iset_a @ A6 @ ( minus_3765977307040488491iset_a @ A6 @ B6 ) ) ) ) ).
% inter_mset_def
thf(fact_625_inter__mset__def,axiom,
( inter_1588963877939557482iset_a
= ( ^ [A6: multis4781622204924204013iset_a,B6: multis4781622204924204013iset_a] : ( minus_9211619285413683668iset_a @ A6 @ ( minus_9211619285413683668iset_a @ A6 @ B6 ) ) ) ) ).
% inter_mset_def
thf(fact_626_inter__mset__def,axiom,
( inter_3651181809579396970od_a_a
= ( ^ [A6: multis501812127501805293od_a_a,B6: multis501812127501805293od_a_a] : ( minus_7313152569058643924od_a_a @ A6 @ ( minus_7313152569058643924od_a_a @ A6 @ B6 ) ) ) ) ).
% inter_mset_def
thf(fact_627_comm__monoid__add_Osum__mset_Ocong,axiom,
comm_m1977238983017651125iset_a = comm_m1977238983017651125iset_a ).
% comm_monoid_add.sum_mset.cong
thf(fact_628_add__mset__diff__bothsides,axiom,
! [A: multiset_a,M: multiset_multiset_a,A3: multiset_multiset_a] :
( ( minus_3395427628221709681iset_a @ ( add_mset_multiset_a @ A @ M ) @ ( add_mset_multiset_a @ A @ A3 ) )
= ( minus_3395427628221709681iset_a @ M @ A3 ) ) ).
% add_mset_diff_bothsides
thf(fact_629_add__mset__diff__bothsides,axiom,
! [A: a,M: multiset_a,A3: multiset_a] :
( ( minus_3765977307040488491iset_a @ ( add_mset_a @ A @ M ) @ ( add_mset_a @ A @ A3 ) )
= ( minus_3765977307040488491iset_a @ M @ A3 ) ) ).
% add_mset_diff_bothsides
thf(fact_630_add__mset__diff__bothsides,axiom,
! [A: produc6518373309651786023iset_a,M: multis4781622204924204013iset_a,A3: multis4781622204924204013iset_a] :
( ( minus_9211619285413683668iset_a @ ( add_ms5348986034529679365iset_a @ A @ M ) @ ( add_ms5348986034529679365iset_a @ A @ A3 ) )
= ( minus_9211619285413683668iset_a @ M @ A3 ) ) ).
% add_mset_diff_bothsides
thf(fact_631_add__mset__diff__bothsides,axiom,
! [A: product_prod_a_a,M: multis501812127501805293od_a_a,A3: multis501812127501805293od_a_a] :
( ( minus_7313152569058643924od_a_a @ ( add_ms8655138167283798533od_a_a @ A @ M ) @ ( add_ms8655138167283798533od_a_a @ A @ A3 ) )
= ( minus_7313152569058643924od_a_a @ M @ A3 ) ) ).
% add_mset_diff_bothsides
thf(fact_632_inter__add__right2,axiom,
! [X3: produc5665077874975597415od_a_a,N: multis3377851686571972653od_a_a,M: multis3377851686571972653od_a_a] :
( ( member1567053331504362128od_a_a @ X3 @ ( set_ms5376707576967697060od_a_a @ N ) )
=> ( ( inter_2707069342228023466od_a_a @ N @ ( add_ms9136373847686211141od_a_a @ X3 @ M ) )
= ( add_ms9136373847686211141od_a_a @ X3 @ ( inter_2707069342228023466od_a_a @ ( minus_6383096217462597652od_a_a @ N @ ( add_ms9136373847686211141od_a_a @ X3 @ zero_z3721308388212328420od_a_a ) ) @ M ) ) ) ) ).
% inter_add_right2
thf(fact_633_inter__add__right2,axiom,
! [X3: produc4424810744984089383iset_a,N: multis1386120040344743661iset_a,M: multis1386120040344743661iset_a] :
( ( member1345422370561994576iset_a @ X3 @ ( set_ms2882773486807528292iset_a @ N ) )
=> ( ( inter_7955219499984291690iset_a @ N @ ( add_ms9106060982595931141iset_a @ X3 @ M ) )
= ( add_ms9106060982595931141iset_a @ X3 @ ( inter_7955219499984291690iset_a @ ( minus_2536987971823575508iset_a @ N @ ( add_ms9106060982595931141iset_a @ X3 @ zero_z9101083372863164836iset_a ) ) @ M ) ) ) ) ).
% inter_add_right2
thf(fact_634_inter__add__right2,axiom,
! [X3: multis501812127501805293od_a_a,N: multis5935151272263360051od_a_a,M: multis5935151272263360051od_a_a] :
( ( member7700929758516679318od_a_a @ X3 @ ( set_ms8253844250382261418od_a_a @ N ) )
=> ( ( inter_1480659854676493744od_a_a @ N @ ( add_ms4156406267932395723od_a_a @ X3 @ M ) )
= ( add_ms4156406267932395723od_a_a @ X3 @ ( inter_1480659854676493744od_a_a @ ( minus_385666609031338394od_a_a @ N @ ( add_ms4156406267932395723od_a_a @ X3 @ zero_z8896958667708014954od_a_a ) ) @ M ) ) ) ) ).
% inter_add_right2
thf(fact_635_inter__add__right2,axiom,
! [X3: multis4781622204924204013iset_a,N: multis7221419791219231795iset_a,M: multis7221419791219231795iset_a] :
( ( member2597461214246177430iset_a @ X3 @ ( set_ms5327031259215843498iset_a @ N ) )
=> ( ( inter_8933094105380514736iset_a @ N @ ( add_ms7610149848545062859iset_a @ X3 @ M ) )
= ( add_ms7610149848545062859iset_a @ X3 @ ( inter_8933094105380514736iset_a @ ( minus_4490418833869548698iset_a @ N @ ( add_ms7610149848545062859iset_a @ X3 @ zero_z8662914267040441450iset_a ) ) @ M ) ) ) ) ).
% inter_add_right2
thf(fact_636_inter__add__right2,axiom,
! [X3: multiset_a,N: multiset_multiset_a,M: multiset_multiset_a] :
( ( member_multiset_a @ X3 @ ( set_mset_multiset_a @ N ) )
=> ( ( inter_3508108863637392903iset_a @ N @ ( add_mset_multiset_a @ X3 @ M ) )
= ( add_mset_multiset_a @ X3 @ ( inter_3508108863637392903iset_a @ ( minus_3395427628221709681iset_a @ N @ ( add_mset_multiset_a @ X3 @ zero_z7799948378220188993iset_a ) ) @ M ) ) ) ) ).
% inter_add_right2
thf(fact_637_inter__add__right2,axiom,
! [X3: a,N: multiset_a,M: multiset_a] :
( ( member_a @ X3 @ ( set_mset_a @ N ) )
=> ( ( inter_mset_a @ N @ ( add_mset_a @ X3 @ M ) )
= ( add_mset_a @ X3 @ ( inter_mset_a @ ( minus_3765977307040488491iset_a @ N @ ( add_mset_a @ X3 @ zero_zero_multiset_a ) ) @ M ) ) ) ) ).
% inter_add_right2
thf(fact_638_inter__add__right2,axiom,
! [X3: produc6518373309651786023iset_a,N: multis4781622204924204013iset_a,M: multis4781622204924204013iset_a] :
( ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ N ) )
=> ( ( inter_1588963877939557482iset_a @ N @ ( add_ms5348986034529679365iset_a @ X3 @ M ) )
= ( add_ms5348986034529679365iset_a @ X3 @ ( inter_1588963877939557482iset_a @ ( minus_9211619285413683668iset_a @ N @ ( add_ms5348986034529679365iset_a @ X3 @ zero_z4591712717340762532iset_a ) ) @ M ) ) ) ) ).
% inter_add_right2
thf(fact_639_inter__add__right2,axiom,
! [X3: product_prod_a_a,N: multis501812127501805293od_a_a,M: multis501812127501805293od_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ N ) )
=> ( ( inter_3651181809579396970od_a_a @ N @ ( add_ms8655138167283798533od_a_a @ X3 @ M ) )
= ( add_ms8655138167283798533od_a_a @ X3 @ ( inter_3651181809579396970od_a_a @ ( minus_7313152569058643924od_a_a @ N @ ( add_ms8655138167283798533od_a_a @ X3 @ zero_z3024502163831676836od_a_a ) ) @ M ) ) ) ) ).
% inter_add_right2
thf(fact_640_inter__add__left2,axiom,
! [X3: produc5665077874975597415od_a_a,N: multis3377851686571972653od_a_a,M: multis3377851686571972653od_a_a] :
( ( member1567053331504362128od_a_a @ X3 @ ( set_ms5376707576967697060od_a_a @ N ) )
=> ( ( inter_2707069342228023466od_a_a @ ( add_ms9136373847686211141od_a_a @ X3 @ M ) @ N )
= ( add_ms9136373847686211141od_a_a @ X3 @ ( inter_2707069342228023466od_a_a @ M @ ( minus_6383096217462597652od_a_a @ N @ ( add_ms9136373847686211141od_a_a @ X3 @ zero_z3721308388212328420od_a_a ) ) ) ) ) ) ).
% inter_add_left2
thf(fact_641_inter__add__left2,axiom,
! [X3: produc4424810744984089383iset_a,N: multis1386120040344743661iset_a,M: multis1386120040344743661iset_a] :
( ( member1345422370561994576iset_a @ X3 @ ( set_ms2882773486807528292iset_a @ N ) )
=> ( ( inter_7955219499984291690iset_a @ ( add_ms9106060982595931141iset_a @ X3 @ M ) @ N )
= ( add_ms9106060982595931141iset_a @ X3 @ ( inter_7955219499984291690iset_a @ M @ ( minus_2536987971823575508iset_a @ N @ ( add_ms9106060982595931141iset_a @ X3 @ zero_z9101083372863164836iset_a ) ) ) ) ) ) ).
% inter_add_left2
thf(fact_642_inter__add__left2,axiom,
! [X3: multis501812127501805293od_a_a,N: multis5935151272263360051od_a_a,M: multis5935151272263360051od_a_a] :
( ( member7700929758516679318od_a_a @ X3 @ ( set_ms8253844250382261418od_a_a @ N ) )
=> ( ( inter_1480659854676493744od_a_a @ ( add_ms4156406267932395723od_a_a @ X3 @ M ) @ N )
= ( add_ms4156406267932395723od_a_a @ X3 @ ( inter_1480659854676493744od_a_a @ M @ ( minus_385666609031338394od_a_a @ N @ ( add_ms4156406267932395723od_a_a @ X3 @ zero_z8896958667708014954od_a_a ) ) ) ) ) ) ).
% inter_add_left2
thf(fact_643_inter__add__left2,axiom,
! [X3: multis4781622204924204013iset_a,N: multis7221419791219231795iset_a,M: multis7221419791219231795iset_a] :
( ( member2597461214246177430iset_a @ X3 @ ( set_ms5327031259215843498iset_a @ N ) )
=> ( ( inter_8933094105380514736iset_a @ ( add_ms7610149848545062859iset_a @ X3 @ M ) @ N )
= ( add_ms7610149848545062859iset_a @ X3 @ ( inter_8933094105380514736iset_a @ M @ ( minus_4490418833869548698iset_a @ N @ ( add_ms7610149848545062859iset_a @ X3 @ zero_z8662914267040441450iset_a ) ) ) ) ) ) ).
% inter_add_left2
thf(fact_644_inter__add__left2,axiom,
! [X3: multiset_a,N: multiset_multiset_a,M: multiset_multiset_a] :
( ( member_multiset_a @ X3 @ ( set_mset_multiset_a @ N ) )
=> ( ( inter_3508108863637392903iset_a @ ( add_mset_multiset_a @ X3 @ M ) @ N )
= ( add_mset_multiset_a @ X3 @ ( inter_3508108863637392903iset_a @ M @ ( minus_3395427628221709681iset_a @ N @ ( add_mset_multiset_a @ X3 @ zero_z7799948378220188993iset_a ) ) ) ) ) ) ).
% inter_add_left2
thf(fact_645_inter__add__left2,axiom,
! [X3: a,N: multiset_a,M: multiset_a] :
( ( member_a @ X3 @ ( set_mset_a @ N ) )
=> ( ( inter_mset_a @ ( add_mset_a @ X3 @ M ) @ N )
= ( add_mset_a @ X3 @ ( inter_mset_a @ M @ ( minus_3765977307040488491iset_a @ N @ ( add_mset_a @ X3 @ zero_zero_multiset_a ) ) ) ) ) ) ).
% inter_add_left2
thf(fact_646_inter__add__left2,axiom,
! [X3: produc6518373309651786023iset_a,N: multis4781622204924204013iset_a,M: multis4781622204924204013iset_a] :
( ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ N ) )
=> ( ( inter_1588963877939557482iset_a @ ( add_ms5348986034529679365iset_a @ X3 @ M ) @ N )
= ( add_ms5348986034529679365iset_a @ X3 @ ( inter_1588963877939557482iset_a @ M @ ( minus_9211619285413683668iset_a @ N @ ( add_ms5348986034529679365iset_a @ X3 @ zero_z4591712717340762532iset_a ) ) ) ) ) ) ).
% inter_add_left2
thf(fact_647_inter__add__left2,axiom,
! [X3: product_prod_a_a,N: multis501812127501805293od_a_a,M: multis501812127501805293od_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ N ) )
=> ( ( inter_3651181809579396970od_a_a @ ( add_ms8655138167283798533od_a_a @ X3 @ M ) @ N )
= ( add_ms8655138167283798533od_a_a @ X3 @ ( inter_3651181809579396970od_a_a @ M @ ( minus_7313152569058643924od_a_a @ N @ ( add_ms8655138167283798533od_a_a @ X3 @ zero_z3024502163831676836od_a_a ) ) ) ) ) ) ).
% inter_add_left2
thf(fact_648_inter__mset__empty__distrib__left,axiom,
! [A3: multiset_multiset_a,B3: multiset_multiset_a,C3: multiset_multiset_a] :
( ( ( inter_3508108863637392903iset_a @ ( plus_p6738641960240532161iset_a @ A3 @ B3 ) @ C3 )
= zero_z7799948378220188993iset_a )
= ( ( ( inter_3508108863637392903iset_a @ A3 @ C3 )
= zero_z7799948378220188993iset_a )
& ( ( inter_3508108863637392903iset_a @ B3 @ C3 )
= zero_z7799948378220188993iset_a ) ) ) ).
% inter_mset_empty_distrib_left
thf(fact_649_inter__mset__empty__distrib__left,axiom,
! [A3: multis501812127501805293od_a_a,B3: multis501812127501805293od_a_a,C3: multis501812127501805293od_a_a] :
( ( ( inter_3651181809579396970od_a_a @ ( plus_p6967097483979887908od_a_a @ A3 @ B3 ) @ C3 )
= zero_z3024502163831676836od_a_a )
= ( ( ( inter_3651181809579396970od_a_a @ A3 @ C3 )
= zero_z3024502163831676836od_a_a )
& ( ( inter_3651181809579396970od_a_a @ B3 @ C3 )
= zero_z3024502163831676836od_a_a ) ) ) ).
% inter_mset_empty_distrib_left
thf(fact_650_inter__mset__empty__distrib__left,axiom,
! [A3: multis4781622204924204013iset_a,B3: multis4781622204924204013iset_a,C3: multis4781622204924204013iset_a] :
( ( ( inter_1588963877939557482iset_a @ ( plus_p1053901119268525860iset_a @ A3 @ B3 ) @ C3 )
= zero_z4591712717340762532iset_a )
= ( ( ( inter_1588963877939557482iset_a @ A3 @ C3 )
= zero_z4591712717340762532iset_a )
& ( ( inter_1588963877939557482iset_a @ B3 @ C3 )
= zero_z4591712717340762532iset_a ) ) ) ).
% inter_mset_empty_distrib_left
thf(fact_651_inter__mset__empty__distrib__left,axiom,
! [A3: multiset_a,B3: multiset_a,C3: multiset_a] :
( ( ( inter_mset_a @ ( plus_plus_multiset_a @ A3 @ B3 ) @ C3 )
= zero_zero_multiset_a )
= ( ( ( inter_mset_a @ A3 @ C3 )
= zero_zero_multiset_a )
& ( ( inter_mset_a @ B3 @ C3 )
= zero_zero_multiset_a ) ) ) ).
% inter_mset_empty_distrib_left
thf(fact_652_inter__mset__empty__distrib__right,axiom,
! [A3: multiset_multiset_a,B3: multiset_multiset_a,C3: multiset_multiset_a] :
( ( ( inter_3508108863637392903iset_a @ A3 @ ( plus_p6738641960240532161iset_a @ B3 @ C3 ) )
= zero_z7799948378220188993iset_a )
= ( ( ( inter_3508108863637392903iset_a @ A3 @ B3 )
= zero_z7799948378220188993iset_a )
& ( ( inter_3508108863637392903iset_a @ A3 @ C3 )
= zero_z7799948378220188993iset_a ) ) ) ).
% inter_mset_empty_distrib_right
thf(fact_653_inter__mset__empty__distrib__right,axiom,
! [A3: multis501812127501805293od_a_a,B3: multis501812127501805293od_a_a,C3: multis501812127501805293od_a_a] :
( ( ( inter_3651181809579396970od_a_a @ A3 @ ( plus_p6967097483979887908od_a_a @ B3 @ C3 ) )
= zero_z3024502163831676836od_a_a )
= ( ( ( inter_3651181809579396970od_a_a @ A3 @ B3 )
= zero_z3024502163831676836od_a_a )
& ( ( inter_3651181809579396970od_a_a @ A3 @ C3 )
= zero_z3024502163831676836od_a_a ) ) ) ).
% inter_mset_empty_distrib_right
thf(fact_654_inter__mset__empty__distrib__right,axiom,
! [A3: multis4781622204924204013iset_a,B3: multis4781622204924204013iset_a,C3: multis4781622204924204013iset_a] :
( ( ( inter_1588963877939557482iset_a @ A3 @ ( plus_p1053901119268525860iset_a @ B3 @ C3 ) )
= zero_z4591712717340762532iset_a )
= ( ( ( inter_1588963877939557482iset_a @ A3 @ B3 )
= zero_z4591712717340762532iset_a )
& ( ( inter_1588963877939557482iset_a @ A3 @ C3 )
= zero_z4591712717340762532iset_a ) ) ) ).
% inter_mset_empty_distrib_right
thf(fact_655_inter__mset__empty__distrib__right,axiom,
! [A3: multiset_a,B3: multiset_a,C3: multiset_a] :
( ( ( inter_mset_a @ A3 @ ( plus_plus_multiset_a @ B3 @ C3 ) )
= zero_zero_multiset_a )
= ( ( ( inter_mset_a @ A3 @ B3 )
= zero_zero_multiset_a )
& ( ( inter_mset_a @ A3 @ C3 )
= zero_zero_multiset_a ) ) ) ).
% inter_mset_empty_distrib_right
thf(fact_656_disjunct__not__in,axiom,
! [A3: multis3377851686571972653od_a_a,B3: multis3377851686571972653od_a_a] :
( ( ( inter_2707069342228023466od_a_a @ A3 @ B3 )
= zero_z3721308388212328420od_a_a )
= ( ! [A5: produc5665077874975597415od_a_a] :
( ~ ( member1567053331504362128od_a_a @ A5 @ ( set_ms5376707576967697060od_a_a @ A3 ) )
| ~ ( member1567053331504362128od_a_a @ A5 @ ( set_ms5376707576967697060od_a_a @ B3 ) ) ) ) ) ).
% disjunct_not_in
thf(fact_657_disjunct__not__in,axiom,
! [A3: multis1386120040344743661iset_a,B3: multis1386120040344743661iset_a] :
( ( ( inter_7955219499984291690iset_a @ A3 @ B3 )
= zero_z9101083372863164836iset_a )
= ( ! [A5: produc4424810744984089383iset_a] :
( ~ ( member1345422370561994576iset_a @ A5 @ ( set_ms2882773486807528292iset_a @ A3 ) )
| ~ ( member1345422370561994576iset_a @ A5 @ ( set_ms2882773486807528292iset_a @ B3 ) ) ) ) ) ).
% disjunct_not_in
thf(fact_658_disjunct__not__in,axiom,
! [A3: multis5935151272263360051od_a_a,B3: multis5935151272263360051od_a_a] :
( ( ( inter_1480659854676493744od_a_a @ A3 @ B3 )
= zero_z8896958667708014954od_a_a )
= ( ! [A5: multis501812127501805293od_a_a] :
( ~ ( member7700929758516679318od_a_a @ A5 @ ( set_ms8253844250382261418od_a_a @ A3 ) )
| ~ ( member7700929758516679318od_a_a @ A5 @ ( set_ms8253844250382261418od_a_a @ B3 ) ) ) ) ) ).
% disjunct_not_in
thf(fact_659_disjunct__not__in,axiom,
! [A3: multis7221419791219231795iset_a,B3: multis7221419791219231795iset_a] :
( ( ( inter_8933094105380514736iset_a @ A3 @ B3 )
= zero_z8662914267040441450iset_a )
= ( ! [A5: multis4781622204924204013iset_a] :
( ~ ( member2597461214246177430iset_a @ A5 @ ( set_ms5327031259215843498iset_a @ A3 ) )
| ~ ( member2597461214246177430iset_a @ A5 @ ( set_ms5327031259215843498iset_a @ B3 ) ) ) ) ) ).
% disjunct_not_in
thf(fact_660_disjunct__not__in,axiom,
! [A3: multiset_multiset_a,B3: multiset_multiset_a] :
( ( ( inter_3508108863637392903iset_a @ A3 @ B3 )
= zero_z7799948378220188993iset_a )
= ( ! [A5: multiset_a] :
( ~ ( member_multiset_a @ A5 @ ( set_mset_multiset_a @ A3 ) )
| ~ ( member_multiset_a @ A5 @ ( set_mset_multiset_a @ B3 ) ) ) ) ) ).
% disjunct_not_in
thf(fact_661_disjunct__not__in,axiom,
! [A3: multis4781622204924204013iset_a,B3: multis4781622204924204013iset_a] :
( ( ( inter_1588963877939557482iset_a @ A3 @ B3 )
= zero_z4591712717340762532iset_a )
= ( ! [A5: produc6518373309651786023iset_a] :
( ~ ( member5199237121806060112iset_a @ A5 @ ( set_ms2379723421352415844iset_a @ A3 ) )
| ~ ( member5199237121806060112iset_a @ A5 @ ( set_ms2379723421352415844iset_a @ B3 ) ) ) ) ) ).
% disjunct_not_in
thf(fact_662_disjunct__not__in,axiom,
! [A3: multis501812127501805293od_a_a,B3: multis501812127501805293od_a_a] :
( ( ( inter_3651181809579396970od_a_a @ A3 @ B3 )
= zero_z3024502163831676836od_a_a )
= ( ! [A5: product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ A5 @ ( set_ms119794563918438244od_a_a @ A3 ) )
| ~ ( member1426531477525435216od_a_a @ A5 @ ( set_ms119794563918438244od_a_a @ B3 ) ) ) ) ) ).
% disjunct_not_in
thf(fact_663_disjunct__not__in,axiom,
! [A3: multiset_a,B3: multiset_a] :
( ( ( inter_mset_a @ A3 @ B3 )
= zero_zero_multiset_a )
= ( ! [A5: a] :
( ~ ( member_a @ A5 @ ( set_mset_a @ A3 ) )
| ~ ( member_a @ A5 @ ( set_mset_a @ B3 ) ) ) ) ) ).
% disjunct_not_in
thf(fact_664_single__is__union,axiom,
! [A: multiset_a,M: multiset_multiset_a,N: multiset_multiset_a] :
( ( ( add_mset_multiset_a @ A @ zero_z7799948378220188993iset_a )
= ( plus_p6738641960240532161iset_a @ M @ N ) )
= ( ( ( ( add_mset_multiset_a @ A @ zero_z7799948378220188993iset_a )
= M )
& ( N = zero_z7799948378220188993iset_a ) )
| ( ( M = zero_z7799948378220188993iset_a )
& ( ( add_mset_multiset_a @ A @ zero_z7799948378220188993iset_a )
= N ) ) ) ) ).
% single_is_union
thf(fact_665_single__is__union,axiom,
! [A: product_prod_a_a,M: multis501812127501805293od_a_a,N: multis501812127501805293od_a_a] :
( ( ( add_ms8655138167283798533od_a_a @ A @ zero_z3024502163831676836od_a_a )
= ( plus_p6967097483979887908od_a_a @ M @ N ) )
= ( ( ( ( add_ms8655138167283798533od_a_a @ A @ zero_z3024502163831676836od_a_a )
= M )
& ( N = zero_z3024502163831676836od_a_a ) )
| ( ( M = zero_z3024502163831676836od_a_a )
& ( ( add_ms8655138167283798533od_a_a @ A @ zero_z3024502163831676836od_a_a )
= N ) ) ) ) ).
% single_is_union
thf(fact_666_single__is__union,axiom,
! [A: produc6518373309651786023iset_a,M: multis4781622204924204013iset_a,N: multis4781622204924204013iset_a] :
( ( ( add_ms5348986034529679365iset_a @ A @ zero_z4591712717340762532iset_a )
= ( plus_p1053901119268525860iset_a @ M @ N ) )
= ( ( ( ( add_ms5348986034529679365iset_a @ A @ zero_z4591712717340762532iset_a )
= M )
& ( N = zero_z4591712717340762532iset_a ) )
| ( ( M = zero_z4591712717340762532iset_a )
& ( ( add_ms5348986034529679365iset_a @ A @ zero_z4591712717340762532iset_a )
= N ) ) ) ) ).
% single_is_union
thf(fact_667_single__is__union,axiom,
! [A: a,M: multiset_a,N: multiset_a] :
( ( ( add_mset_a @ A @ zero_zero_multiset_a )
= ( plus_plus_multiset_a @ M @ N ) )
= ( ( ( ( add_mset_a @ A @ zero_zero_multiset_a )
= M )
& ( N = zero_zero_multiset_a ) )
| ( ( M = zero_zero_multiset_a )
& ( ( add_mset_a @ A @ zero_zero_multiset_a )
= N ) ) ) ) ).
% single_is_union
thf(fact_668_union__is__single,axiom,
! [M: multiset_multiset_a,N: multiset_multiset_a,A: multiset_a] :
( ( ( plus_p6738641960240532161iset_a @ M @ N )
= ( add_mset_multiset_a @ A @ zero_z7799948378220188993iset_a ) )
= ( ( ( M
= ( add_mset_multiset_a @ A @ zero_z7799948378220188993iset_a ) )
& ( N = zero_z7799948378220188993iset_a ) )
| ( ( M = zero_z7799948378220188993iset_a )
& ( N
= ( add_mset_multiset_a @ A @ zero_z7799948378220188993iset_a ) ) ) ) ) ).
% union_is_single
thf(fact_669_union__is__single,axiom,
! [M: multis501812127501805293od_a_a,N: multis501812127501805293od_a_a,A: product_prod_a_a] :
( ( ( plus_p6967097483979887908od_a_a @ M @ N )
= ( add_ms8655138167283798533od_a_a @ A @ zero_z3024502163831676836od_a_a ) )
= ( ( ( M
= ( add_ms8655138167283798533od_a_a @ A @ zero_z3024502163831676836od_a_a ) )
& ( N = zero_z3024502163831676836od_a_a ) )
| ( ( M = zero_z3024502163831676836od_a_a )
& ( N
= ( add_ms8655138167283798533od_a_a @ A @ zero_z3024502163831676836od_a_a ) ) ) ) ) ).
% union_is_single
thf(fact_670_union__is__single,axiom,
! [M: multis4781622204924204013iset_a,N: multis4781622204924204013iset_a,A: produc6518373309651786023iset_a] :
( ( ( plus_p1053901119268525860iset_a @ M @ N )
= ( add_ms5348986034529679365iset_a @ A @ zero_z4591712717340762532iset_a ) )
= ( ( ( M
= ( add_ms5348986034529679365iset_a @ A @ zero_z4591712717340762532iset_a ) )
& ( N = zero_z4591712717340762532iset_a ) )
| ( ( M = zero_z4591712717340762532iset_a )
& ( N
= ( add_ms5348986034529679365iset_a @ A @ zero_z4591712717340762532iset_a ) ) ) ) ) ).
% union_is_single
thf(fact_671_union__is__single,axiom,
! [M: multiset_a,N: multiset_a,A: a] :
( ( ( plus_plus_multiset_a @ M @ N )
= ( add_mset_a @ A @ zero_zero_multiset_a ) )
= ( ( ( M
= ( add_mset_a @ A @ zero_zero_multiset_a ) )
& ( N = zero_zero_multiset_a ) )
| ( ( M = zero_zero_multiset_a )
& ( N
= ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) ) ) ).
% union_is_single
thf(fact_672_add__mset__add__single,axiom,
( add_mset_multiset_a
= ( ^ [A5: multiset_a,A6: multiset_multiset_a] : ( plus_p6738641960240532161iset_a @ A6 @ ( add_mset_multiset_a @ A5 @ zero_z7799948378220188993iset_a ) ) ) ) ).
% add_mset_add_single
thf(fact_673_add__mset__add__single,axiom,
( add_ms8655138167283798533od_a_a
= ( ^ [A5: product_prod_a_a,A6: multis501812127501805293od_a_a] : ( plus_p6967097483979887908od_a_a @ A6 @ ( add_ms8655138167283798533od_a_a @ A5 @ zero_z3024502163831676836od_a_a ) ) ) ) ).
% add_mset_add_single
thf(fact_674_add__mset__add__single,axiom,
( add_ms5348986034529679365iset_a
= ( ^ [A5: produc6518373309651786023iset_a,A6: multis4781622204924204013iset_a] : ( plus_p1053901119268525860iset_a @ A6 @ ( add_ms5348986034529679365iset_a @ A5 @ zero_z4591712717340762532iset_a ) ) ) ) ).
% add_mset_add_single
thf(fact_675_add__mset__add__single,axiom,
( add_mset_a
= ( ^ [A5: a,A6: multiset_a] : ( plus_plus_multiset_a @ A6 @ ( add_mset_a @ A5 @ zero_zero_multiset_a ) ) ) ) ).
% add_mset_add_single
thf(fact_676_multi__member__last,axiom,
! [X3: produc5665077874975597415od_a_a] : ( member1567053331504362128od_a_a @ X3 @ ( set_ms5376707576967697060od_a_a @ ( add_ms9136373847686211141od_a_a @ X3 @ zero_z3721308388212328420od_a_a ) ) ) ).
% multi_member_last
thf(fact_677_multi__member__last,axiom,
! [X3: produc4424810744984089383iset_a] : ( member1345422370561994576iset_a @ X3 @ ( set_ms2882773486807528292iset_a @ ( add_ms9106060982595931141iset_a @ X3 @ zero_z9101083372863164836iset_a ) ) ) ).
% multi_member_last
thf(fact_678_multi__member__last,axiom,
! [X3: multis501812127501805293od_a_a] : ( member7700929758516679318od_a_a @ X3 @ ( set_ms8253844250382261418od_a_a @ ( add_ms4156406267932395723od_a_a @ X3 @ zero_z8896958667708014954od_a_a ) ) ) ).
% multi_member_last
thf(fact_679_multi__member__last,axiom,
! [X3: multis4781622204924204013iset_a] : ( member2597461214246177430iset_a @ X3 @ ( set_ms5327031259215843498iset_a @ ( add_ms7610149848545062859iset_a @ X3 @ zero_z8662914267040441450iset_a ) ) ) ).
% multi_member_last
thf(fact_680_multi__member__last,axiom,
! [X3: multiset_a] : ( member_multiset_a @ X3 @ ( set_mset_multiset_a @ ( add_mset_multiset_a @ X3 @ zero_z7799948378220188993iset_a ) ) ) ).
% multi_member_last
thf(fact_681_multi__member__last,axiom,
! [X3: produc6518373309651786023iset_a] : ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ ( add_ms5348986034529679365iset_a @ X3 @ zero_z4591712717340762532iset_a ) ) ) ).
% multi_member_last
thf(fact_682_multi__member__last,axiom,
! [X3: product_prod_a_a] : ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ ( add_ms8655138167283798533od_a_a @ X3 @ zero_z3024502163831676836od_a_a ) ) ) ).
% multi_member_last
thf(fact_683_multi__member__last,axiom,
! [X3: a] : ( member_a @ X3 @ ( set_mset_a @ ( add_mset_a @ X3 @ zero_zero_multiset_a ) ) ) ).
% multi_member_last
thf(fact_684_diff__union__swap,axiom,
! [A: multiset_a,B: multiset_a,M: multiset_multiset_a] :
( ( A != B )
=> ( ( add_mset_multiset_a @ B @ ( minus_3395427628221709681iset_a @ M @ ( add_mset_multiset_a @ A @ zero_z7799948378220188993iset_a ) ) )
= ( minus_3395427628221709681iset_a @ ( add_mset_multiset_a @ B @ M ) @ ( add_mset_multiset_a @ A @ zero_z7799948378220188993iset_a ) ) ) ) ).
% diff_union_swap
thf(fact_685_diff__union__swap,axiom,
! [A: a,B: a,M: multiset_a] :
( ( A != B )
=> ( ( add_mset_a @ B @ ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) )
= ( minus_3765977307040488491iset_a @ ( add_mset_a @ B @ M ) @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) ) ).
% diff_union_swap
thf(fact_686_diff__union__swap,axiom,
! [A: produc6518373309651786023iset_a,B: produc6518373309651786023iset_a,M: multis4781622204924204013iset_a] :
( ( A != B )
=> ( ( add_ms5348986034529679365iset_a @ B @ ( minus_9211619285413683668iset_a @ M @ ( add_ms5348986034529679365iset_a @ A @ zero_z4591712717340762532iset_a ) ) )
= ( minus_9211619285413683668iset_a @ ( add_ms5348986034529679365iset_a @ B @ M ) @ ( add_ms5348986034529679365iset_a @ A @ zero_z4591712717340762532iset_a ) ) ) ) ).
% diff_union_swap
thf(fact_687_diff__union__swap,axiom,
! [A: product_prod_a_a,B: product_prod_a_a,M: multis501812127501805293od_a_a] :
( ( A != B )
=> ( ( add_ms8655138167283798533od_a_a @ B @ ( minus_7313152569058643924od_a_a @ M @ ( add_ms8655138167283798533od_a_a @ A @ zero_z3024502163831676836od_a_a ) ) )
= ( minus_7313152569058643924od_a_a @ ( add_ms8655138167283798533od_a_a @ B @ M ) @ ( add_ms8655138167283798533od_a_a @ A @ zero_z3024502163831676836od_a_a ) ) ) ) ).
% diff_union_swap
thf(fact_688_add__eq__conv__diff,axiom,
! [A: multiset_a,M: multiset_multiset_a,B: multiset_a,N: multiset_multiset_a] :
( ( ( add_mset_multiset_a @ A @ M )
= ( add_mset_multiset_a @ B @ N ) )
= ( ( ( M = N )
& ( A = B ) )
| ( ( M
= ( add_mset_multiset_a @ B @ ( minus_3395427628221709681iset_a @ N @ ( add_mset_multiset_a @ A @ zero_z7799948378220188993iset_a ) ) ) )
& ( N
= ( add_mset_multiset_a @ A @ ( minus_3395427628221709681iset_a @ M @ ( add_mset_multiset_a @ B @ zero_z7799948378220188993iset_a ) ) ) ) ) ) ) ).
% add_eq_conv_diff
thf(fact_689_add__eq__conv__diff,axiom,
! [A: a,M: multiset_a,B: a,N: multiset_a] :
( ( ( add_mset_a @ A @ M )
= ( add_mset_a @ B @ N ) )
= ( ( ( M = N )
& ( A = B ) )
| ( ( M
= ( add_mset_a @ B @ ( minus_3765977307040488491iset_a @ N @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) )
& ( N
= ( add_mset_a @ A @ ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ B @ zero_zero_multiset_a ) ) ) ) ) ) ) ).
% add_eq_conv_diff
thf(fact_690_add__eq__conv__diff,axiom,
! [A: produc6518373309651786023iset_a,M: multis4781622204924204013iset_a,B: produc6518373309651786023iset_a,N: multis4781622204924204013iset_a] :
( ( ( add_ms5348986034529679365iset_a @ A @ M )
= ( add_ms5348986034529679365iset_a @ B @ N ) )
= ( ( ( M = N )
& ( A = B ) )
| ( ( M
= ( add_ms5348986034529679365iset_a @ B @ ( minus_9211619285413683668iset_a @ N @ ( add_ms5348986034529679365iset_a @ A @ zero_z4591712717340762532iset_a ) ) ) )
& ( N
= ( add_ms5348986034529679365iset_a @ A @ ( minus_9211619285413683668iset_a @ M @ ( add_ms5348986034529679365iset_a @ B @ zero_z4591712717340762532iset_a ) ) ) ) ) ) ) ).
% add_eq_conv_diff
thf(fact_691_add__eq__conv__diff,axiom,
! [A: product_prod_a_a,M: multis501812127501805293od_a_a,B: product_prod_a_a,N: multis501812127501805293od_a_a] :
( ( ( add_ms8655138167283798533od_a_a @ A @ M )
= ( add_ms8655138167283798533od_a_a @ B @ N ) )
= ( ( ( M = N )
& ( A = B ) )
| ( ( M
= ( add_ms8655138167283798533od_a_a @ B @ ( minus_7313152569058643924od_a_a @ N @ ( add_ms8655138167283798533od_a_a @ A @ zero_z3024502163831676836od_a_a ) ) ) )
& ( N
= ( add_ms8655138167283798533od_a_a @ A @ ( minus_7313152569058643924od_a_a @ M @ ( add_ms8655138167283798533od_a_a @ B @ zero_z3024502163831676836od_a_a ) ) ) ) ) ) ) ).
% add_eq_conv_diff
thf(fact_692_union__single__eq__diff,axiom,
! [X3: multiset_a,M: multiset_multiset_a,N: multiset_multiset_a] :
( ( ( add_mset_multiset_a @ X3 @ M )
= N )
=> ( M
= ( minus_3395427628221709681iset_a @ N @ ( add_mset_multiset_a @ X3 @ zero_z7799948378220188993iset_a ) ) ) ) ).
% union_single_eq_diff
thf(fact_693_union__single__eq__diff,axiom,
! [X3: a,M: multiset_a,N: multiset_a] :
( ( ( add_mset_a @ X3 @ M )
= N )
=> ( M
= ( minus_3765977307040488491iset_a @ N @ ( add_mset_a @ X3 @ zero_zero_multiset_a ) ) ) ) ).
% union_single_eq_diff
thf(fact_694_union__single__eq__diff,axiom,
! [X3: produc6518373309651786023iset_a,M: multis4781622204924204013iset_a,N: multis4781622204924204013iset_a] :
( ( ( add_ms5348986034529679365iset_a @ X3 @ M )
= N )
=> ( M
= ( minus_9211619285413683668iset_a @ N @ ( add_ms5348986034529679365iset_a @ X3 @ zero_z4591712717340762532iset_a ) ) ) ) ).
% union_single_eq_diff
thf(fact_695_union__single__eq__diff,axiom,
! [X3: product_prod_a_a,M: multis501812127501805293od_a_a,N: multis501812127501805293od_a_a] :
( ( ( add_ms8655138167283798533od_a_a @ X3 @ M )
= N )
=> ( M
= ( minus_7313152569058643924od_a_a @ N @ ( add_ms8655138167283798533od_a_a @ X3 @ zero_z3024502163831676836od_a_a ) ) ) ) ).
% union_single_eq_diff
thf(fact_696_multi__member__skip,axiom,
! [X3: multis4781622204924204013iset_a,XS: multis7221419791219231795iset_a,Y: multis4781622204924204013iset_a] :
( ( member2597461214246177430iset_a @ X3 @ ( set_ms5327031259215843498iset_a @ XS ) )
=> ( member2597461214246177430iset_a @ X3 @ ( set_ms5327031259215843498iset_a @ ( plus_p1436237118943648234iset_a @ ( add_ms7610149848545062859iset_a @ Y @ zero_z8662914267040441450iset_a ) @ XS ) ) ) ) ).
% multi_member_skip
thf(fact_697_multi__member__skip,axiom,
! [X3: multiset_a,XS: multiset_multiset_a,Y: multiset_a] :
( ( member_multiset_a @ X3 @ ( set_mset_multiset_a @ XS ) )
=> ( member_multiset_a @ X3 @ ( set_mset_multiset_a @ ( plus_p6738641960240532161iset_a @ ( add_mset_multiset_a @ Y @ zero_z7799948378220188993iset_a ) @ XS ) ) ) ) ).
% multi_member_skip
thf(fact_698_multi__member__skip,axiom,
! [X3: produc6518373309651786023iset_a,XS: multis4781622204924204013iset_a,Y: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ XS ) )
=> ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ ( plus_p1053901119268525860iset_a @ ( add_ms5348986034529679365iset_a @ Y @ zero_z4591712717340762532iset_a ) @ XS ) ) ) ) ).
% multi_member_skip
thf(fact_699_multi__member__skip,axiom,
! [X3: product_prod_a_a,XS: multis501812127501805293od_a_a,Y: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ XS ) )
=> ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ ( plus_p6967097483979887908od_a_a @ ( add_ms8655138167283798533od_a_a @ Y @ zero_z3024502163831676836od_a_a ) @ XS ) ) ) ) ).
% multi_member_skip
thf(fact_700_multi__member__skip,axiom,
! [X3: a,XS: multiset_a,Y: a] :
( ( member_a @ X3 @ ( set_mset_a @ XS ) )
=> ( member_a @ X3 @ ( set_mset_a @ ( plus_plus_multiset_a @ ( add_mset_a @ Y @ zero_zero_multiset_a ) @ XS ) ) ) ) ).
% multi_member_skip
thf(fact_701_multi__member__this,axiom,
! [X3: produc6518373309651786023iset_a,XS: multis4781622204924204013iset_a] : ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ ( plus_p1053901119268525860iset_a @ ( add_ms5348986034529679365iset_a @ X3 @ zero_z4591712717340762532iset_a ) @ XS ) ) ) ).
% multi_member_this
thf(fact_702_multi__member__this,axiom,
! [X3: product_prod_a_a,XS: multis501812127501805293od_a_a] : ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ ( plus_p6967097483979887908od_a_a @ ( add_ms8655138167283798533od_a_a @ X3 @ zero_z3024502163831676836od_a_a ) @ XS ) ) ) ).
% multi_member_this
thf(fact_703_multi__member__this,axiom,
! [X3: a,XS: multiset_a] : ( member_a @ X3 @ ( set_mset_a @ ( plus_plus_multiset_a @ ( add_mset_a @ X3 @ zero_zero_multiset_a ) @ XS ) ) ) ).
% multi_member_this
thf(fact_704_multiset__union__diff__commute,axiom,
! [B3: multiset_a,C3: multiset_a,A3: multiset_a] :
( ( ( inter_mset_a @ B3 @ C3 )
= zero_zero_multiset_a )
=> ( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ A3 @ B3 ) @ C3 )
= ( plus_plus_multiset_a @ ( minus_3765977307040488491iset_a @ A3 @ C3 ) @ B3 ) ) ) ).
% multiset_union_diff_commute
thf(fact_705_multiset__union__diff__commute,axiom,
! [B3: multis4781622204924204013iset_a,C3: multis4781622204924204013iset_a,A3: multis4781622204924204013iset_a] :
( ( ( inter_1588963877939557482iset_a @ B3 @ C3 )
= zero_z4591712717340762532iset_a )
=> ( ( minus_9211619285413683668iset_a @ ( plus_p1053901119268525860iset_a @ A3 @ B3 ) @ C3 )
= ( plus_p1053901119268525860iset_a @ ( minus_9211619285413683668iset_a @ A3 @ C3 ) @ B3 ) ) ) ).
% multiset_union_diff_commute
thf(fact_706_multiset__union__diff__commute,axiom,
! [B3: multis501812127501805293od_a_a,C3: multis501812127501805293od_a_a,A3: multis501812127501805293od_a_a] :
( ( ( inter_3651181809579396970od_a_a @ B3 @ C3 )
= zero_z3024502163831676836od_a_a )
=> ( ( minus_7313152569058643924od_a_a @ ( plus_p6967097483979887908od_a_a @ A3 @ B3 ) @ C3 )
= ( plus_p6967097483979887908od_a_a @ ( minus_7313152569058643924od_a_a @ A3 @ C3 ) @ B3 ) ) ) ).
% multiset_union_diff_commute
thf(fact_707_disjunct__set__mset__diff,axiom,
! [M: multiset_a,N: multiset_a] :
( ( ( inter_mset_a @ M @ N )
= zero_zero_multiset_a )
=> ( ( set_mset_a @ ( minus_3765977307040488491iset_a @ M @ N ) )
= ( set_mset_a @ M ) ) ) ).
% disjunct_set_mset_diff
thf(fact_708_disjunct__set__mset__diff,axiom,
! [M: multis4781622204924204013iset_a,N: multis4781622204924204013iset_a] :
( ( ( inter_1588963877939557482iset_a @ M @ N )
= zero_z4591712717340762532iset_a )
=> ( ( set_ms2379723421352415844iset_a @ ( minus_9211619285413683668iset_a @ M @ N ) )
= ( set_ms2379723421352415844iset_a @ M ) ) ) ).
% disjunct_set_mset_diff
thf(fact_709_disjunct__set__mset__diff,axiom,
! [M: multis501812127501805293od_a_a,N: multis501812127501805293od_a_a] :
( ( ( inter_3651181809579396970od_a_a @ M @ N )
= zero_z3024502163831676836od_a_a )
=> ( ( set_ms119794563918438244od_a_a @ ( minus_7313152569058643924od_a_a @ M @ N ) )
= ( set_ms119794563918438244od_a_a @ M ) ) ) ).
% disjunct_set_mset_diff
thf(fact_710_more__than__one__mset__mset__diff,axiom,
! [A: a,M: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) )
=> ( ( set_mset_a @ ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) )
= ( set_mset_a @ M ) ) ) ).
% more_than_one_mset_mset_diff
thf(fact_711_more__than__one__mset__mset__diff,axiom,
! [A: produc6518373309651786023iset_a,M: multis4781622204924204013iset_a] :
( ( member5199237121806060112iset_a @ A @ ( set_ms2379723421352415844iset_a @ ( minus_9211619285413683668iset_a @ M @ ( add_ms5348986034529679365iset_a @ A @ zero_z4591712717340762532iset_a ) ) ) )
=> ( ( set_ms2379723421352415844iset_a @ ( minus_9211619285413683668iset_a @ M @ ( add_ms5348986034529679365iset_a @ A @ zero_z4591712717340762532iset_a ) ) )
= ( set_ms2379723421352415844iset_a @ M ) ) ) ).
% more_than_one_mset_mset_diff
thf(fact_712_more__than__one__mset__mset__diff,axiom,
! [A: product_prod_a_a,M: multis501812127501805293od_a_a] :
( ( member1426531477525435216od_a_a @ A @ ( set_ms119794563918438244od_a_a @ ( minus_7313152569058643924od_a_a @ M @ ( add_ms8655138167283798533od_a_a @ A @ zero_z3024502163831676836od_a_a ) ) ) )
=> ( ( set_ms119794563918438244od_a_a @ ( minus_7313152569058643924od_a_a @ M @ ( add_ms8655138167283798533od_a_a @ A @ zero_z3024502163831676836od_a_a ) ) )
= ( set_ms119794563918438244od_a_a @ M ) ) ) ).
% more_than_one_mset_mset_diff
thf(fact_713_multiset__add__sub__el__shuffle,axiom,
! [C2: a,B3: multiset_a,B: a] :
( ( member_a @ C2 @ ( set_mset_a @ B3 ) )
=> ( ( B != C2 )
=> ( ( add_mset_a @ B @ ( minus_3765977307040488491iset_a @ B3 @ ( add_mset_a @ C2 @ zero_zero_multiset_a ) ) )
= ( minus_3765977307040488491iset_a @ ( add_mset_a @ B @ B3 ) @ ( add_mset_a @ C2 @ zero_zero_multiset_a ) ) ) ) ) ).
% multiset_add_sub_el_shuffle
thf(fact_714_multiset__add__sub__el__shuffle,axiom,
! [C2: produc6518373309651786023iset_a,B3: multis4781622204924204013iset_a,B: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ C2 @ ( set_ms2379723421352415844iset_a @ B3 ) )
=> ( ( B != C2 )
=> ( ( add_ms5348986034529679365iset_a @ B @ ( minus_9211619285413683668iset_a @ B3 @ ( add_ms5348986034529679365iset_a @ C2 @ zero_z4591712717340762532iset_a ) ) )
= ( minus_9211619285413683668iset_a @ ( add_ms5348986034529679365iset_a @ B @ B3 ) @ ( add_ms5348986034529679365iset_a @ C2 @ zero_z4591712717340762532iset_a ) ) ) ) ) ).
% multiset_add_sub_el_shuffle
thf(fact_715_multiset__add__sub__el__shuffle,axiom,
! [C2: product_prod_a_a,B3: multis501812127501805293od_a_a,B: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C2 @ ( set_ms119794563918438244od_a_a @ B3 ) )
=> ( ( B != C2 )
=> ( ( add_ms8655138167283798533od_a_a @ B @ ( minus_7313152569058643924od_a_a @ B3 @ ( add_ms8655138167283798533od_a_a @ C2 @ zero_z3024502163831676836od_a_a ) ) )
= ( minus_7313152569058643924od_a_a @ ( add_ms8655138167283798533od_a_a @ B @ B3 ) @ ( add_ms8655138167283798533od_a_a @ C2 @ zero_z3024502163831676836od_a_a ) ) ) ) ) ).
% multiset_add_sub_el_shuffle
thf(fact_716_add__mset__remove__trivial__eq,axiom,
! [N: multiset_a,A: a] :
( ( N
= ( add_mset_a @ A @ ( minus_3765977307040488491iset_a @ N @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) )
= ( member_a @ A @ ( set_mset_a @ N ) ) ) ).
% add_mset_remove_trivial_eq
thf(fact_717_add__mset__remove__trivial__eq,axiom,
! [N: multis4781622204924204013iset_a,A: produc6518373309651786023iset_a] :
( ( N
= ( add_ms5348986034529679365iset_a @ A @ ( minus_9211619285413683668iset_a @ N @ ( add_ms5348986034529679365iset_a @ A @ zero_z4591712717340762532iset_a ) ) ) )
= ( member5199237121806060112iset_a @ A @ ( set_ms2379723421352415844iset_a @ N ) ) ) ).
% add_mset_remove_trivial_eq
thf(fact_718_add__mset__remove__trivial__eq,axiom,
! [N: multis501812127501805293od_a_a,A: product_prod_a_a] :
( ( N
= ( add_ms8655138167283798533od_a_a @ A @ ( minus_7313152569058643924od_a_a @ N @ ( add_ms8655138167283798533od_a_a @ A @ zero_z3024502163831676836od_a_a ) ) ) )
= ( member1426531477525435216od_a_a @ A @ ( set_ms119794563918438244od_a_a @ N ) ) ) ).
% add_mset_remove_trivial_eq
thf(fact_719_add__mset__remove__trivial__If,axiom,
! [A: a,N: multiset_a] :
( ( ( member_a @ A @ ( set_mset_a @ N ) )
=> ( ( add_mset_a @ A @ ( minus_3765977307040488491iset_a @ N @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) )
= N ) )
& ( ~ ( member_a @ A @ ( set_mset_a @ N ) )
=> ( ( add_mset_a @ A @ ( minus_3765977307040488491iset_a @ N @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) )
= ( add_mset_a @ A @ N ) ) ) ) ).
% add_mset_remove_trivial_If
thf(fact_720_add__mset__remove__trivial__If,axiom,
! [A: produc6518373309651786023iset_a,N: multis4781622204924204013iset_a] :
( ( ( member5199237121806060112iset_a @ A @ ( set_ms2379723421352415844iset_a @ N ) )
=> ( ( add_ms5348986034529679365iset_a @ A @ ( minus_9211619285413683668iset_a @ N @ ( add_ms5348986034529679365iset_a @ A @ zero_z4591712717340762532iset_a ) ) )
= N ) )
& ( ~ ( member5199237121806060112iset_a @ A @ ( set_ms2379723421352415844iset_a @ N ) )
=> ( ( add_ms5348986034529679365iset_a @ A @ ( minus_9211619285413683668iset_a @ N @ ( add_ms5348986034529679365iset_a @ A @ zero_z4591712717340762532iset_a ) ) )
= ( add_ms5348986034529679365iset_a @ A @ N ) ) ) ) ).
% add_mset_remove_trivial_If
thf(fact_721_add__mset__remove__trivial__If,axiom,
! [A: product_prod_a_a,N: multis501812127501805293od_a_a] :
( ( ( member1426531477525435216od_a_a @ A @ ( set_ms119794563918438244od_a_a @ N ) )
=> ( ( add_ms8655138167283798533od_a_a @ A @ ( minus_7313152569058643924od_a_a @ N @ ( add_ms8655138167283798533od_a_a @ A @ zero_z3024502163831676836od_a_a ) ) )
= N ) )
& ( ~ ( member1426531477525435216od_a_a @ A @ ( set_ms119794563918438244od_a_a @ N ) )
=> ( ( add_ms8655138167283798533od_a_a @ A @ ( minus_7313152569058643924od_a_a @ N @ ( add_ms8655138167283798533od_a_a @ A @ zero_z3024502163831676836od_a_a ) ) )
= ( add_ms8655138167283798533od_a_a @ A @ N ) ) ) ) ).
% add_mset_remove_trivial_If
thf(fact_722_multi__drop__mem__not__eq,axiom,
! [C2: a,B3: multiset_a] :
( ( member_a @ C2 @ ( set_mset_a @ B3 ) )
=> ( ( minus_3765977307040488491iset_a @ B3 @ ( add_mset_a @ C2 @ zero_zero_multiset_a ) )
!= B3 ) ) ).
% multi_drop_mem_not_eq
thf(fact_723_multi__drop__mem__not__eq,axiom,
! [C2: produc6518373309651786023iset_a,B3: multis4781622204924204013iset_a] :
( ( member5199237121806060112iset_a @ C2 @ ( set_ms2379723421352415844iset_a @ B3 ) )
=> ( ( minus_9211619285413683668iset_a @ B3 @ ( add_ms5348986034529679365iset_a @ C2 @ zero_z4591712717340762532iset_a ) )
!= B3 ) ) ).
% multi_drop_mem_not_eq
thf(fact_724_multi__drop__mem__not__eq,axiom,
! [C2: product_prod_a_a,B3: multis501812127501805293od_a_a] :
( ( member1426531477525435216od_a_a @ C2 @ ( set_ms119794563918438244od_a_a @ B3 ) )
=> ( ( minus_7313152569058643924od_a_a @ B3 @ ( add_ms8655138167283798533od_a_a @ C2 @ zero_z3024502163831676836od_a_a ) )
!= B3 ) ) ).
% multi_drop_mem_not_eq
thf(fact_725_diff__single__eq__union,axiom,
! [X3: a,M: multiset_a,N: multiset_a] :
( ( member_a @ X3 @ ( set_mset_a @ M ) )
=> ( ( ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ X3 @ zero_zero_multiset_a ) )
= N )
= ( M
= ( add_mset_a @ X3 @ N ) ) ) ) ).
% diff_single_eq_union
thf(fact_726_diff__single__eq__union,axiom,
! [X3: produc6518373309651786023iset_a,M: multis4781622204924204013iset_a,N: multis4781622204924204013iset_a] :
( ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ M ) )
=> ( ( ( minus_9211619285413683668iset_a @ M @ ( add_ms5348986034529679365iset_a @ X3 @ zero_z4591712717340762532iset_a ) )
= N )
= ( M
= ( add_ms5348986034529679365iset_a @ X3 @ N ) ) ) ) ).
% diff_single_eq_union
thf(fact_727_diff__single__eq__union,axiom,
! [X3: product_prod_a_a,M: multis501812127501805293od_a_a,N: multis501812127501805293od_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ M ) )
=> ( ( ( minus_7313152569058643924od_a_a @ M @ ( add_ms8655138167283798533od_a_a @ X3 @ zero_z3024502163831676836od_a_a ) )
= N )
= ( M
= ( add_ms8655138167283798533od_a_a @ X3 @ N ) ) ) ) ).
% diff_single_eq_union
thf(fact_728_diff__single__trivial,axiom,
! [X3: a,M: multiset_a] :
( ~ ( member_a @ X3 @ ( set_mset_a @ M ) )
=> ( ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ X3 @ zero_zero_multiset_a ) )
= M ) ) ).
% diff_single_trivial
thf(fact_729_diff__single__trivial,axiom,
! [X3: produc6518373309651786023iset_a,M: multis4781622204924204013iset_a] :
( ~ ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ M ) )
=> ( ( minus_9211619285413683668iset_a @ M @ ( add_ms5348986034529679365iset_a @ X3 @ zero_z4591712717340762532iset_a ) )
= M ) ) ).
% diff_single_trivial
thf(fact_730_diff__single__trivial,axiom,
! [X3: product_prod_a_a,M: multis501812127501805293od_a_a] :
( ~ ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ M ) )
=> ( ( minus_7313152569058643924od_a_a @ M @ ( add_ms8655138167283798533od_a_a @ X3 @ zero_z3024502163831676836od_a_a ) )
= M ) ) ).
% diff_single_trivial
thf(fact_731_less__add,axiom,
! [N: multis4781622204924204013iset_a,A: produc6518373309651786023iset_a,M0: multis4781622204924204013iset_a,R: set_Pr3059988933816007879iset_a] :
( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ N @ ( add_ms5348986034529679365iset_a @ A @ M0 ) ) @ ( mult1_5927536977011277466iset_a @ R ) )
=> ( ? [M3: multis4781622204924204013iset_a] :
( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ M3 @ M0 ) @ ( mult1_5927536977011277466iset_a @ R ) )
& ( N
= ( add_ms5348986034529679365iset_a @ A @ M3 ) ) )
| ? [K: multis4781622204924204013iset_a] :
( ! [B8: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ B8 @ ( set_ms2379723421352415844iset_a @ K ) )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ B8 @ A ) @ R ) )
& ( N
= ( plus_p1053901119268525860iset_a @ M0 @ K ) ) ) ) ) ).
% less_add
thf(fact_732_less__add,axiom,
! [N: multis501812127501805293od_a_a,A: product_prod_a_a,M0: multis501812127501805293od_a_a,R: set_Pr8600417178894128327od_a_a] :
( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ N @ ( add_ms8655138167283798533od_a_a @ A @ M0 ) ) @ ( mult1_1845928942838466970od_a_a @ R ) )
=> ( ? [M3: multis501812127501805293od_a_a] :
( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ M3 @ M0 ) @ ( mult1_1845928942838466970od_a_a @ R ) )
& ( N
= ( add_ms8655138167283798533od_a_a @ A @ M3 ) ) )
| ? [K: multis501812127501805293od_a_a] :
( ! [B8: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ B8 @ ( set_ms119794563918438244od_a_a @ K ) )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ B8 @ A ) @ R ) )
& ( N
= ( plus_p6967097483979887908od_a_a @ M0 @ K ) ) ) ) ) ).
% less_add
thf(fact_733_less__add,axiom,
! [N: multiset_multiset_a,A: multiset_a,M0: multiset_multiset_a,R: set_Pr79727621955416071iset_a] :
( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ N @ ( add_mset_multiset_a @ A @ M0 ) ) @ ( mult1_multiset_a @ R ) )
=> ( ? [M3: multiset_multiset_a] :
( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ M3 @ M0 ) @ ( mult1_multiset_a @ R ) )
& ( N
= ( add_mset_multiset_a @ A @ M3 ) ) )
| ? [K: multiset_multiset_a] :
( ! [B8: multiset_a] :
( ( member_multiset_a @ B8 @ ( set_mset_multiset_a @ K ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ B8 @ A ) @ R ) )
& ( N
= ( plus_p6738641960240532161iset_a @ M0 @ K ) ) ) ) ) ).
% less_add
thf(fact_734_less__add,axiom,
! [N: multiset_a,A: a,M0: multiset_a,R: set_Product_prod_a_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ N @ ( add_mset_a @ A @ M0 ) ) @ ( mult1_a @ R ) )
=> ( ? [M3: multiset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ M3 @ M0 ) @ ( mult1_a @ R ) )
& ( N
= ( add_mset_a @ A @ M3 ) ) )
| ? [K: multiset_a] :
( ! [B8: a] :
( ( member_a @ B8 @ ( set_mset_a @ K ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B8 @ A ) @ R ) )
& ( N
= ( plus_plus_multiset_a @ M0 @ K ) ) ) ) ) ).
% less_add
thf(fact_735_mult1I,axiom,
! [M: multis4781622204924204013iset_a,A: produc6518373309651786023iset_a,M0: multis4781622204924204013iset_a,N: multis4781622204924204013iset_a,K2: multis4781622204924204013iset_a,R: set_Pr3059988933816007879iset_a] :
( ( M
= ( add_ms5348986034529679365iset_a @ A @ M0 ) )
=> ( ( N
= ( plus_p1053901119268525860iset_a @ M0 @ K2 ) )
=> ( ! [B4: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ B4 @ ( set_ms2379723421352415844iset_a @ K2 ) )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ B4 @ A ) @ R ) )
=> ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ N @ M ) @ ( mult1_5927536977011277466iset_a @ R ) ) ) ) ) ).
% mult1I
thf(fact_736_mult1I,axiom,
! [M: multis501812127501805293od_a_a,A: product_prod_a_a,M0: multis501812127501805293od_a_a,N: multis501812127501805293od_a_a,K2: multis501812127501805293od_a_a,R: set_Pr8600417178894128327od_a_a] :
( ( M
= ( add_ms8655138167283798533od_a_a @ A @ M0 ) )
=> ( ( N
= ( plus_p6967097483979887908od_a_a @ M0 @ K2 ) )
=> ( ! [B4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ B4 @ ( set_ms119794563918438244od_a_a @ K2 ) )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ B4 @ A ) @ R ) )
=> ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ N @ M ) @ ( mult1_1845928942838466970od_a_a @ R ) ) ) ) ) ).
% mult1I
thf(fact_737_mult1I,axiom,
! [M: multiset_multiset_a,A: multiset_a,M0: multiset_multiset_a,N: multiset_multiset_a,K2: multiset_multiset_a,R: set_Pr79727621955416071iset_a] :
( ( M
= ( add_mset_multiset_a @ A @ M0 ) )
=> ( ( N
= ( plus_p6738641960240532161iset_a @ M0 @ K2 ) )
=> ( ! [B4: multiset_a] :
( ( member_multiset_a @ B4 @ ( set_mset_multiset_a @ K2 ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ B4 @ A ) @ R ) )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ N @ M ) @ ( mult1_multiset_a @ R ) ) ) ) ) ).
% mult1I
thf(fact_738_mult1I,axiom,
! [M: multiset_a,A: a,M0: multiset_a,N: multiset_a,K2: multiset_a,R: set_Product_prod_a_a] :
( ( M
= ( add_mset_a @ A @ M0 ) )
=> ( ( N
= ( plus_plus_multiset_a @ M0 @ K2 ) )
=> ( ! [B4: a] :
( ( member_a @ B4 @ ( set_mset_a @ K2 ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B4 @ A ) @ R ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ N @ M ) @ ( mult1_a @ R ) ) ) ) ) ).
% mult1I
thf(fact_739_mult1E,axiom,
! [N: multis4781622204924204013iset_a,M: multis4781622204924204013iset_a,R: set_Pr3059988933816007879iset_a] :
( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ N @ M ) @ ( mult1_5927536977011277466iset_a @ R ) )
=> ~ ! [A4: produc6518373309651786023iset_a,M02: multis4781622204924204013iset_a] :
( ( M
= ( add_ms5348986034529679365iset_a @ A4 @ M02 ) )
=> ! [K: multis4781622204924204013iset_a] :
( ( N
= ( plus_p1053901119268525860iset_a @ M02 @ K ) )
=> ~ ! [B8: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ B8 @ ( set_ms2379723421352415844iset_a @ K ) )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ B8 @ A4 ) @ R ) ) ) ) ) ).
% mult1E
thf(fact_740_mult1E,axiom,
! [N: multis501812127501805293od_a_a,M: multis501812127501805293od_a_a,R: set_Pr8600417178894128327od_a_a] :
( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ N @ M ) @ ( mult1_1845928942838466970od_a_a @ R ) )
=> ~ ! [A4: product_prod_a_a,M02: multis501812127501805293od_a_a] :
( ( M
= ( add_ms8655138167283798533od_a_a @ A4 @ M02 ) )
=> ! [K: multis501812127501805293od_a_a] :
( ( N
= ( plus_p6967097483979887908od_a_a @ M02 @ K ) )
=> ~ ! [B8: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ B8 @ ( set_ms119794563918438244od_a_a @ K ) )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ B8 @ A4 ) @ R ) ) ) ) ) ).
% mult1E
thf(fact_741_mult1E,axiom,
! [N: multiset_multiset_a,M: multiset_multiset_a,R: set_Pr79727621955416071iset_a] :
( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ N @ M ) @ ( mult1_multiset_a @ R ) )
=> ~ ! [A4: multiset_a,M02: multiset_multiset_a] :
( ( M
= ( add_mset_multiset_a @ A4 @ M02 ) )
=> ! [K: multiset_multiset_a] :
( ( N
= ( plus_p6738641960240532161iset_a @ M02 @ K ) )
=> ~ ! [B8: multiset_a] :
( ( member_multiset_a @ B8 @ ( set_mset_multiset_a @ K ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ B8 @ A4 ) @ R ) ) ) ) ) ).
% mult1E
thf(fact_742_mult1E,axiom,
! [N: multiset_a,M: multiset_a,R: set_Product_prod_a_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ N @ M ) @ ( mult1_a @ R ) )
=> ~ ! [A4: a,M02: multiset_a] :
( ( M
= ( add_mset_a @ A4 @ M02 ) )
=> ! [K: multiset_a] :
( ( N
= ( plus_plus_multiset_a @ M02 @ K ) )
=> ~ ! [B8: a] :
( ( member_a @ B8 @ ( set_mset_a @ K ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B8 @ A4 ) @ R ) ) ) ) ) ).
% mult1E
thf(fact_743_in__mset__fold__plus__iff,axiom,
! [X3: produc6518373309651786023iset_a,M: multis4781622204924204013iset_a,NN: multis7221419791219231795iset_a] :
( ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ ( fold_m7087023614847598283iset_a @ plus_p1053901119268525860iset_a @ M @ NN ) ) )
= ( ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ M ) )
| ? [N2: multis4781622204924204013iset_a] :
( ( member2597461214246177430iset_a @ N2 @ ( set_ms5327031259215843498iset_a @ NN ) )
& ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ N2 ) ) ) ) ) ).
% in_mset_fold_plus_iff
thf(fact_744_in__mset__fold__plus__iff,axiom,
! [X3: product_prod_a_a,M: multis501812127501805293od_a_a,NN: multis5935151272263360051od_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ ( fold_m3854860582760972491od_a_a @ plus_p6967097483979887908od_a_a @ M @ NN ) ) )
= ( ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ M ) )
| ? [N2: multis501812127501805293od_a_a] :
( ( member7700929758516679318od_a_a @ N2 @ ( set_ms8253844250382261418od_a_a @ NN ) )
& ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ N2 ) ) ) ) ) ).
% in_mset_fold_plus_iff
thf(fact_745_in__mset__fold__plus__iff,axiom,
! [X3: a,M: multiset_a,NN: multiset_multiset_a] :
( ( member_a @ X3 @ ( set_mset_a @ ( fold_m6601649825673331723iset_a @ plus_plus_multiset_a @ M @ NN ) ) )
= ( ( member_a @ X3 @ ( set_mset_a @ M ) )
| ? [N2: multiset_a] :
( ( member_multiset_a @ N2 @ ( set_mset_multiset_a @ NN ) )
& ( member_a @ X3 @ ( set_mset_a @ N2 ) ) ) ) ) ).
% in_mset_fold_plus_iff
thf(fact_746_add__mset__replicate__mset__safe,axiom,
! [M: multiset_a,A: a] :
( ( nO_MAT1665547570504182763iset_a @ zero_zero_multiset_a @ M )
=> ( ( add_mset_a @ A @ M )
= ( plus_plus_multiset_a @ ( add_mset_a @ A @ zero_zero_multiset_a ) @ M ) ) ) ).
% add_mset_replicate_mset_safe
thf(fact_747_sum__mset_Oremove,axiom,
! [X3: multiset_a,A3: multiset_multiset_a] :
( ( member_multiset_a @ X3 @ ( set_mset_multiset_a @ A3 ) )
=> ( ( comm_m4595494212910443721iset_a @ A3 )
= ( plus_plus_multiset_a @ X3 @ ( comm_m4595494212910443721iset_a @ ( minus_3395427628221709681iset_a @ A3 @ ( add_mset_multiset_a @ X3 @ zero_z7799948378220188993iset_a ) ) ) ) ) ) ).
% sum_mset.remove
thf(fact_748_insert__subset__eq__iff,axiom,
! [A: a,A3: multiset_a,B3: multiset_a] :
( ( subseteq_mset_a @ ( add_mset_a @ A @ A3 ) @ B3 )
= ( ( member_a @ A @ ( set_mset_a @ B3 ) )
& ( subseteq_mset_a @ A3 @ ( minus_3765977307040488491iset_a @ B3 @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) ) ) ).
% insert_subset_eq_iff
thf(fact_749_insert__subset__eq__iff,axiom,
! [A: produc6518373309651786023iset_a,A3: multis4781622204924204013iset_a,B3: multis4781622204924204013iset_a] :
( ( subset6292090195676340366iset_a @ ( add_ms5348986034529679365iset_a @ A @ A3 ) @ B3 )
= ( ( member5199237121806060112iset_a @ A @ ( set_ms2379723421352415844iset_a @ B3 ) )
& ( subset6292090195676340366iset_a @ A3 @ ( minus_9211619285413683668iset_a @ B3 @ ( add_ms5348986034529679365iset_a @ A @ zero_z4591712717340762532iset_a ) ) ) ) ) ).
% insert_subset_eq_iff
thf(fact_750_insert__subset__eq__iff,axiom,
! [A: product_prod_a_a,A3: multis501812127501805293od_a_a,B3: multis501812127501805293od_a_a] :
( ( subset4108007524207251342od_a_a @ ( add_ms8655138167283798533od_a_a @ A @ A3 ) @ B3 )
= ( ( member1426531477525435216od_a_a @ A @ ( set_ms119794563918438244od_a_a @ B3 ) )
& ( subset4108007524207251342od_a_a @ A3 @ ( minus_7313152569058643924od_a_a @ B3 @ ( add_ms8655138167283798533od_a_a @ A @ zero_z3024502163831676836od_a_a ) ) ) ) ) ).
% insert_subset_eq_iff
thf(fact_751_insert__union__subset__iff,axiom,
! [A: a,A3: multiset_a,B3: multiset_a] :
( ( subset_mset_a @ ( add_mset_a @ A @ A3 ) @ B3 )
= ( ( member_a @ A @ ( set_mset_a @ B3 ) )
& ( subset_mset_a @ A3 @ ( minus_3765977307040488491iset_a @ B3 @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) ) ) ).
% insert_union_subset_iff
thf(fact_752_insert__union__subset__iff,axiom,
! [A: produc6518373309651786023iset_a,A3: multis4781622204924204013iset_a,B3: multis4781622204924204013iset_a] :
( ( subset5376938936802007386iset_a @ ( add_ms5348986034529679365iset_a @ A @ A3 ) @ B3 )
= ( ( member5199237121806060112iset_a @ A @ ( set_ms2379723421352415844iset_a @ B3 ) )
& ( subset5376938936802007386iset_a @ A3 @ ( minus_9211619285413683668iset_a @ B3 @ ( add_ms5348986034529679365iset_a @ A @ zero_z4591712717340762532iset_a ) ) ) ) ) ).
% insert_union_subset_iff
thf(fact_753_insert__union__subset__iff,axiom,
! [A: product_prod_a_a,A3: multis501812127501805293od_a_a,B3: multis501812127501805293od_a_a] :
( ( subset8265789482553336922od_a_a @ ( add_ms8655138167283798533od_a_a @ A @ A3 ) @ B3 )
= ( ( member1426531477525435216od_a_a @ A @ ( set_ms119794563918438244od_a_a @ B3 ) )
& ( subset8265789482553336922od_a_a @ A3 @ ( minus_7313152569058643924od_a_a @ B3 @ ( add_ms8655138167283798533od_a_a @ A @ zero_z3024502163831676836od_a_a ) ) ) ) ) ).
% insert_union_subset_iff
thf(fact_754_mset__subset__diff__self,axiom,
! [C2: a,B3: multiset_a] :
( ( member_a @ C2 @ ( set_mset_a @ B3 ) )
=> ( subset_mset_a @ ( minus_3765977307040488491iset_a @ B3 @ ( add_mset_a @ C2 @ zero_zero_multiset_a ) ) @ B3 ) ) ).
% mset_subset_diff_self
thf(fact_755_mset__subset__diff__self,axiom,
! [C2: produc6518373309651786023iset_a,B3: multis4781622204924204013iset_a] :
( ( member5199237121806060112iset_a @ C2 @ ( set_ms2379723421352415844iset_a @ B3 ) )
=> ( subset5376938936802007386iset_a @ ( minus_9211619285413683668iset_a @ B3 @ ( add_ms5348986034529679365iset_a @ C2 @ zero_z4591712717340762532iset_a ) ) @ B3 ) ) ).
% mset_subset_diff_self
thf(fact_756_mset__subset__diff__self,axiom,
! [C2: product_prod_a_a,B3: multis501812127501805293od_a_a] :
( ( member1426531477525435216od_a_a @ C2 @ ( set_ms119794563918438244od_a_a @ B3 ) )
=> ( subset8265789482553336922od_a_a @ ( minus_7313152569058643924od_a_a @ B3 @ ( add_ms8655138167283798533od_a_a @ C2 @ zero_z3024502163831676836od_a_a ) ) @ B3 ) ) ).
% mset_subset_diff_self
thf(fact_757_subset__mset_Oextremum__unique,axiom,
! [A: multiset_a] :
( ( subseteq_mset_a @ A @ zero_zero_multiset_a )
= ( A = zero_zero_multiset_a ) ) ).
% subset_mset.extremum_unique
thf(fact_758_subset__mset_Ole__zero__eq,axiom,
! [N4: multiset_a] :
( ( subseteq_mset_a @ N4 @ zero_zero_multiset_a )
= ( N4 = zero_zero_multiset_a ) ) ).
% subset_mset.le_zero_eq
thf(fact_759_mset__subset__eq__mono__add__right__cancel,axiom,
! [A3: multiset_a,C3: multiset_a,B3: multiset_a] :
( ( subseteq_mset_a @ ( plus_plus_multiset_a @ A3 @ C3 ) @ ( plus_plus_multiset_a @ B3 @ C3 ) )
= ( subseteq_mset_a @ A3 @ B3 ) ) ).
% mset_subset_eq_mono_add_right_cancel
thf(fact_760_mset__subset__eq__mono__add__left__cancel,axiom,
! [C3: multiset_a,A3: multiset_a,B3: multiset_a] :
( ( subseteq_mset_a @ ( plus_plus_multiset_a @ C3 @ A3 ) @ ( plus_plus_multiset_a @ C3 @ B3 ) )
= ( subseteq_mset_a @ A3 @ B3 ) ) ).
% mset_subset_eq_mono_add_left_cancel
thf(fact_761_subset__mset_Oadd__le__cancel__right,axiom,
! [A: multiset_a,C2: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ ( plus_plus_multiset_a @ A @ C2 ) @ ( plus_plus_multiset_a @ B @ C2 ) )
= ( subseteq_mset_a @ A @ B ) ) ).
% subset_mset.add_le_cancel_right
thf(fact_762_subset__mset_Oadd__le__cancel__left,axiom,
! [C2: multiset_a,A: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ ( plus_plus_multiset_a @ C2 @ A ) @ ( plus_plus_multiset_a @ C2 @ B ) )
= ( subseteq_mset_a @ A @ B ) ) ).
% subset_mset.add_le_cancel_left
thf(fact_763_mset__subset__of__empty,axiom,
! [A3: multiset_a] :
~ ( subset_mset_a @ A3 @ zero_zero_multiset_a ) ).
% mset_subset_of_empty
thf(fact_764_subset__mset_Onot__gr__zero,axiom,
! [N4: multiset_a] :
( ( ~ ( subset_mset_a @ zero_zero_multiset_a @ N4 ) )
= ( N4 = zero_zero_multiset_a ) ) ).
% subset_mset.not_gr_zero
thf(fact_765_subset__mset_Oadd__less__cancel__right,axiom,
! [A: multiset_a,C2: multiset_a,B: multiset_a] :
( ( subset_mset_a @ ( plus_plus_multiset_a @ A @ C2 ) @ ( plus_plus_multiset_a @ B @ C2 ) )
= ( subset_mset_a @ A @ B ) ) ).
% subset_mset.add_less_cancel_right
thf(fact_766_subset__mset_Oadd__less__cancel__left,axiom,
! [C2: multiset_a,A: multiset_a,B: multiset_a] :
( ( subset_mset_a @ ( plus_plus_multiset_a @ C2 @ A ) @ ( plus_plus_multiset_a @ C2 @ B ) )
= ( subset_mset_a @ A @ B ) ) ).
% subset_mset.add_less_cancel_left
thf(fact_767_subset__mset_Ole__inf__iff,axiom,
! [X3: multiset_a,Y: multiset_a,Z: multiset_a] :
( ( subseteq_mset_a @ X3 @ ( inter_mset_a @ Y @ Z ) )
= ( ( subseteq_mset_a @ X3 @ Y )
& ( subseteq_mset_a @ X3 @ Z ) ) ) ).
% subset_mset.le_inf_iff
thf(fact_768_subset__mset_Oinf_Obounded__iff,axiom,
! [A: multiset_a,B: multiset_a,C2: multiset_a] :
( ( subseteq_mset_a @ A @ ( inter_mset_a @ B @ C2 ) )
= ( ( subseteq_mset_a @ A @ B )
& ( subseteq_mset_a @ A @ C2 ) ) ) ).
% subset_mset.inf.bounded_iff
thf(fact_769_Union__mset__empty__conv,axiom,
! [M: multiset_multiset_a] :
( ( ( comm_m4595494212910443721iset_a @ M )
= zero_zero_multiset_a )
= ( ! [X6: multiset_a] :
( ( member_multiset_a @ X6 @ ( set_mset_multiset_a @ M ) )
=> ( X6 = zero_zero_multiset_a ) ) ) ) ).
% Union_mset_empty_conv
thf(fact_770_in__Union__mset__iff,axiom,
! [X3: a,MM: multiset_multiset_a] :
( ( member_a @ X3 @ ( set_mset_a @ ( comm_m4595494212910443721iset_a @ MM ) ) )
= ( ? [M2: multiset_a] :
( ( member_multiset_a @ M2 @ ( set_mset_multiset_a @ MM ) )
& ( member_a @ X3 @ ( set_mset_a @ M2 ) ) ) ) ) ).
% in_Union_mset_iff
thf(fact_771_in__Union__mset__iff,axiom,
! [X3: produc6518373309651786023iset_a,MM: multis7221419791219231795iset_a] :
( ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ ( comm_m5130607462838410866iset_a @ MM ) ) )
= ( ? [M2: multis4781622204924204013iset_a] :
( ( member2597461214246177430iset_a @ M2 @ ( set_ms5327031259215843498iset_a @ MM ) )
& ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ M2 ) ) ) ) ) ).
% in_Union_mset_iff
thf(fact_772_in__Union__mset__iff,axiom,
! [X3: product_prod_a_a,MM: multis5935151272263360051od_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ ( comm_m9185394553243882098od_a_a @ MM ) ) )
= ( ? [M2: multis501812127501805293od_a_a] :
( ( member7700929758516679318od_a_a @ M2 @ ( set_ms8253844250382261418od_a_a @ MM ) )
& ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ M2 ) ) ) ) ) ).
% in_Union_mset_iff
thf(fact_773_subset__mset_Ole__add__same__cancel2,axiom,
! [A: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ A @ ( plus_plus_multiset_a @ B @ A ) )
= ( subseteq_mset_a @ zero_zero_multiset_a @ B ) ) ).
% subset_mset.le_add_same_cancel2
thf(fact_774_subset__mset_Ole__add__same__cancel1,axiom,
! [A: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ A @ ( plus_plus_multiset_a @ A @ B ) )
= ( subseteq_mset_a @ zero_zero_multiset_a @ B ) ) ).
% subset_mset.le_add_same_cancel1
thf(fact_775_subset__mset_Oadd__le__same__cancel2,axiom,
! [A: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ ( plus_plus_multiset_a @ A @ B ) @ B )
= ( subseteq_mset_a @ A @ zero_zero_multiset_a ) ) ).
% subset_mset.add_le_same_cancel2
thf(fact_776_subset__mset_Oadd__le__same__cancel1,axiom,
! [B: multiset_a,A: multiset_a] :
( ( subseteq_mset_a @ ( plus_plus_multiset_a @ B @ A ) @ B )
= ( subseteq_mset_a @ A @ zero_zero_multiset_a ) ) ).
% subset_mset.add_le_same_cancel1
thf(fact_777_add__mset__subseteq__single__iff,axiom,
! [A: a,M: multiset_a,B: a] :
( ( subseteq_mset_a @ ( add_mset_a @ A @ M ) @ ( add_mset_a @ B @ zero_zero_multiset_a ) )
= ( ( M = zero_zero_multiset_a )
& ( A = B ) ) ) ).
% add_mset_subseteq_single_iff
thf(fact_778_subset__mset_Oless__add__same__cancel2,axiom,
! [A: multiset_a,B: multiset_a] :
( ( subset_mset_a @ A @ ( plus_plus_multiset_a @ B @ A ) )
= ( subset_mset_a @ zero_zero_multiset_a @ B ) ) ).
% subset_mset.less_add_same_cancel2
thf(fact_779_subset__mset_Oless__add__same__cancel1,axiom,
! [A: multiset_a,B: multiset_a] :
( ( subset_mset_a @ A @ ( plus_plus_multiset_a @ A @ B ) )
= ( subset_mset_a @ zero_zero_multiset_a @ B ) ) ).
% subset_mset.less_add_same_cancel1
thf(fact_780_subset__mset_Oadd__less__same__cancel2,axiom,
! [A: multiset_a,B: multiset_a] :
( ( subset_mset_a @ ( plus_plus_multiset_a @ A @ B ) @ B )
= ( subset_mset_a @ A @ zero_zero_multiset_a ) ) ).
% subset_mset.add_less_same_cancel2
thf(fact_781_subset__mset_Oadd__less__same__cancel1,axiom,
! [B: multiset_a,A: multiset_a] :
( ( subset_mset_a @ ( plus_plus_multiset_a @ B @ A ) @ B )
= ( subset_mset_a @ A @ zero_zero_multiset_a ) ) ).
% subset_mset.add_less_same_cancel1
thf(fact_782_mset__subset__eq__multiset__union__diff__commute,axiom,
! [B3: multiset_a,A3: multiset_a,C3: multiset_a] :
( ( subseteq_mset_a @ B3 @ A3 )
=> ( ( plus_plus_multiset_a @ ( minus_3765977307040488491iset_a @ A3 @ B3 ) @ C3 )
= ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ A3 @ C3 ) @ B3 ) ) ) ).
% mset_subset_eq_multiset_union_diff_commute
thf(fact_783_mset__subset__eq__multiset__union__diff__commute,axiom,
! [B3: multis4781622204924204013iset_a,A3: multis4781622204924204013iset_a,C3: multis4781622204924204013iset_a] :
( ( subset6292090195676340366iset_a @ B3 @ A3 )
=> ( ( plus_p1053901119268525860iset_a @ ( minus_9211619285413683668iset_a @ A3 @ B3 ) @ C3 )
= ( minus_9211619285413683668iset_a @ ( plus_p1053901119268525860iset_a @ A3 @ C3 ) @ B3 ) ) ) ).
% mset_subset_eq_multiset_union_diff_commute
thf(fact_784_mset__subset__eq__multiset__union__diff__commute,axiom,
! [B3: multis501812127501805293od_a_a,A3: multis501812127501805293od_a_a,C3: multis501812127501805293od_a_a] :
( ( subset4108007524207251342od_a_a @ B3 @ A3 )
=> ( ( plus_p6967097483979887908od_a_a @ ( minus_7313152569058643924od_a_a @ A3 @ B3 ) @ C3 )
= ( minus_7313152569058643924od_a_a @ ( plus_p6967097483979887908od_a_a @ A3 @ C3 ) @ B3 ) ) ) ).
% mset_subset_eq_multiset_union_diff_commute
thf(fact_785_subset__mset_Oadd__diff__assoc2,axiom,
! [A: multiset_a,B: multiset_a,C2: multiset_a] :
( ( subseteq_mset_a @ A @ B )
=> ( ( plus_plus_multiset_a @ ( minus_3765977307040488491iset_a @ B @ A ) @ C2 )
= ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ B @ C2 ) @ A ) ) ) ).
% subset_mset.add_diff_assoc2
thf(fact_786_subset__mset_Oadd__diff__assoc2,axiom,
! [A: multis4781622204924204013iset_a,B: multis4781622204924204013iset_a,C2: multis4781622204924204013iset_a] :
( ( subset6292090195676340366iset_a @ A @ B )
=> ( ( plus_p1053901119268525860iset_a @ ( minus_9211619285413683668iset_a @ B @ A ) @ C2 )
= ( minus_9211619285413683668iset_a @ ( plus_p1053901119268525860iset_a @ B @ C2 ) @ A ) ) ) ).
% subset_mset.add_diff_assoc2
thf(fact_787_subset__mset_Oadd__diff__assoc2,axiom,
! [A: multis501812127501805293od_a_a,B: multis501812127501805293od_a_a,C2: multis501812127501805293od_a_a] :
( ( subset4108007524207251342od_a_a @ A @ B )
=> ( ( plus_p6967097483979887908od_a_a @ ( minus_7313152569058643924od_a_a @ B @ A ) @ C2 )
= ( minus_7313152569058643924od_a_a @ ( plus_p6967097483979887908od_a_a @ B @ C2 ) @ A ) ) ) ).
% subset_mset.add_diff_assoc2
thf(fact_788_subset__mset_Oadd__diff__assoc,axiom,
! [A: multiset_a,B: multiset_a,C2: multiset_a] :
( ( subseteq_mset_a @ A @ B )
=> ( ( plus_plus_multiset_a @ C2 @ ( minus_3765977307040488491iset_a @ B @ A ) )
= ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ C2 @ B ) @ A ) ) ) ).
% subset_mset.add_diff_assoc
thf(fact_789_subset__mset_Oadd__diff__assoc,axiom,
! [A: multis4781622204924204013iset_a,B: multis4781622204924204013iset_a,C2: multis4781622204924204013iset_a] :
( ( subset6292090195676340366iset_a @ A @ B )
=> ( ( plus_p1053901119268525860iset_a @ C2 @ ( minus_9211619285413683668iset_a @ B @ A ) )
= ( minus_9211619285413683668iset_a @ ( plus_p1053901119268525860iset_a @ C2 @ B ) @ A ) ) ) ).
% subset_mset.add_diff_assoc
thf(fact_790_subset__mset_Oadd__diff__assoc,axiom,
! [A: multis501812127501805293od_a_a,B: multis501812127501805293od_a_a,C2: multis501812127501805293od_a_a] :
( ( subset4108007524207251342od_a_a @ A @ B )
=> ( ( plus_p6967097483979887908od_a_a @ C2 @ ( minus_7313152569058643924od_a_a @ B @ A ) )
= ( minus_7313152569058643924od_a_a @ ( plus_p6967097483979887908od_a_a @ C2 @ B ) @ A ) ) ) ).
% subset_mset.add_diff_assoc
thf(fact_791_sum__mset_Oempty,axiom,
( ( comm_m4595494212910443721iset_a @ zero_z7799948378220188993iset_a )
= zero_zero_multiset_a ) ).
% sum_mset.empty
thf(fact_792_sum__mset_Ounion,axiom,
! [M: multiset_multiset_a,N: multiset_multiset_a] :
( ( comm_m4595494212910443721iset_a @ ( plus_p6738641960240532161iset_a @ M @ N ) )
= ( plus_plus_multiset_a @ ( comm_m4595494212910443721iset_a @ M ) @ ( comm_m4595494212910443721iset_a @ N ) ) ) ).
% sum_mset.union
thf(fact_793_sum__mset_Oadd__mset,axiom,
! [X3: multiset_a,N: multiset_multiset_a] :
( ( comm_m4595494212910443721iset_a @ ( add_mset_multiset_a @ X3 @ N ) )
= ( plus_plus_multiset_a @ X3 @ ( comm_m4595494212910443721iset_a @ N ) ) ) ).
% sum_mset.add_mset
thf(fact_794_single__subset__iff,axiom,
! [A: produc6518373309651786023iset_a,M: multis4781622204924204013iset_a] :
( ( subset6292090195676340366iset_a @ ( add_ms5348986034529679365iset_a @ A @ zero_z4591712717340762532iset_a ) @ M )
= ( member5199237121806060112iset_a @ A @ ( set_ms2379723421352415844iset_a @ M ) ) ) ).
% single_subset_iff
thf(fact_795_single__subset__iff,axiom,
! [A: product_prod_a_a,M: multis501812127501805293od_a_a] :
( ( subset4108007524207251342od_a_a @ ( add_ms8655138167283798533od_a_a @ A @ zero_z3024502163831676836od_a_a ) @ M )
= ( member1426531477525435216od_a_a @ A @ ( set_ms119794563918438244od_a_a @ M ) ) ) ).
% single_subset_iff
thf(fact_796_single__subset__iff,axiom,
! [A: a,M: multiset_a] :
( ( subseteq_mset_a @ ( add_mset_a @ A @ zero_zero_multiset_a ) @ M )
= ( member_a @ A @ ( set_mset_a @ M ) ) ) ).
% single_subset_iff
thf(fact_797_subset__mset_Oadd__le__less__mono,axiom,
! [A: multiset_a,B: multiset_a,C2: multiset_a,D: multiset_a] :
( ( subseteq_mset_a @ A @ B )
=> ( ( subset_mset_a @ C2 @ D )
=> ( subset_mset_a @ ( plus_plus_multiset_a @ A @ C2 ) @ ( plus_plus_multiset_a @ B @ D ) ) ) ) ).
% subset_mset.add_le_less_mono
thf(fact_798_subset__mset_Oadd__less__le__mono,axiom,
! [A: multiset_a,B: multiset_a,C2: multiset_a,D: multiset_a] :
( ( subset_mset_a @ A @ B )
=> ( ( subseteq_mset_a @ C2 @ D )
=> ( subset_mset_a @ ( plus_plus_multiset_a @ A @ C2 ) @ ( plus_plus_multiset_a @ B @ D ) ) ) ) ).
% subset_mset.add_less_le_mono
thf(fact_799_subset__mset_Oadd__strict__increasing2,axiom,
! [A: multiset_a,B: multiset_a,C2: multiset_a] :
( ( subseteq_mset_a @ zero_zero_multiset_a @ A )
=> ( ( subset_mset_a @ B @ C2 )
=> ( subset_mset_a @ B @ ( plus_plus_multiset_a @ A @ C2 ) ) ) ) ).
% subset_mset.add_strict_increasing2
thf(fact_800_subset__mset_Oadd__strict__increasing,axiom,
! [A: multiset_a,B: multiset_a,C2: multiset_a] :
( ( subset_mset_a @ zero_zero_multiset_a @ A )
=> ( ( subseteq_mset_a @ B @ C2 )
=> ( subset_mset_a @ B @ ( plus_plus_multiset_a @ A @ C2 ) ) ) ) ).
% subset_mset.add_strict_increasing
thf(fact_801_subset__mset_Oadd__pos__nonneg,axiom,
! [A: multiset_a,B: multiset_a] :
( ( subset_mset_a @ zero_zero_multiset_a @ A )
=> ( ( subseteq_mset_a @ zero_zero_multiset_a @ B )
=> ( subset_mset_a @ zero_zero_multiset_a @ ( plus_plus_multiset_a @ A @ B ) ) ) ) ).
% subset_mset.add_pos_nonneg
thf(fact_802_subset__mset_Oadd__nonpos__neg,axiom,
! [A: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ A @ zero_zero_multiset_a )
=> ( ( subset_mset_a @ B @ zero_zero_multiset_a )
=> ( subset_mset_a @ ( plus_plus_multiset_a @ A @ B ) @ zero_zero_multiset_a ) ) ) ).
% subset_mset.add_nonpos_neg
thf(fact_803_subset__mset_Oadd__nonneg__pos,axiom,
! [A: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ zero_zero_multiset_a @ A )
=> ( ( subset_mset_a @ zero_zero_multiset_a @ B )
=> ( subset_mset_a @ zero_zero_multiset_a @ ( plus_plus_multiset_a @ A @ B ) ) ) ) ).
% subset_mset.add_nonneg_pos
thf(fact_804_subset__mset_Oadd__neg__nonpos,axiom,
! [A: multiset_a,B: multiset_a] :
( ( subset_mset_a @ A @ zero_zero_multiset_a )
=> ( ( subseteq_mset_a @ B @ zero_zero_multiset_a )
=> ( subset_mset_a @ ( plus_plus_multiset_a @ A @ B ) @ zero_zero_multiset_a ) ) ) ).
% subset_mset.add_neg_nonpos
thf(fact_805_mset__subset__eq__insertD,axiom,
! [X3: a,A3: multiset_a,B3: multiset_a] :
( ( subseteq_mset_a @ ( add_mset_a @ X3 @ A3 ) @ B3 )
=> ( ( member_a @ X3 @ ( set_mset_a @ B3 ) )
& ( subset_mset_a @ A3 @ B3 ) ) ) ).
% mset_subset_eq_insertD
thf(fact_806_mset__subset__eq__insertD,axiom,
! [X3: produc6518373309651786023iset_a,A3: multis4781622204924204013iset_a,B3: multis4781622204924204013iset_a] :
( ( subset6292090195676340366iset_a @ ( add_ms5348986034529679365iset_a @ X3 @ A3 ) @ B3 )
=> ( ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ B3 ) )
& ( subset5376938936802007386iset_a @ A3 @ B3 ) ) ) ).
% mset_subset_eq_insertD
thf(fact_807_mset__subset__eq__insertD,axiom,
! [X3: product_prod_a_a,A3: multis501812127501805293od_a_a,B3: multis501812127501805293od_a_a] :
( ( subset4108007524207251342od_a_a @ ( add_ms8655138167283798533od_a_a @ X3 @ A3 ) @ B3 )
=> ( ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ B3 ) )
& ( subset8265789482553336922od_a_a @ A3 @ B3 ) ) ) ).
% mset_subset_eq_insertD
thf(fact_808_sum__mset__diff,axiom,
! [N: multiset_multiset_a,M: multiset_multiset_a] :
( ( subset6997503458918918187iset_a @ N @ M )
=> ( ( comm_m4595494212910443721iset_a @ ( minus_3395427628221709681iset_a @ M @ N ) )
= ( minus_3765977307040488491iset_a @ ( comm_m4595494212910443721iset_a @ M ) @ ( comm_m4595494212910443721iset_a @ N ) ) ) ) ).
% sum_mset_diff
thf(fact_809_sum__mset__diff,axiom,
! [N: multis7221419791219231795iset_a,M: multis7221419791219231795iset_a] :
( ( subset8727788146060275156iset_a @ N @ M )
=> ( ( comm_m5130607462838410866iset_a @ ( minus_4490418833869548698iset_a @ M @ N ) )
= ( minus_9211619285413683668iset_a @ ( comm_m5130607462838410866iset_a @ M ) @ ( comm_m5130607462838410866iset_a @ N ) ) ) ) ).
% sum_mset_diff
thf(fact_810_sum__mset__diff,axiom,
! [N: multis5935151272263360051od_a_a,M: multis5935151272263360051od_a_a] :
( ( subset6416153986489326548od_a_a @ N @ M )
=> ( ( comm_m9185394553243882098od_a_a @ ( minus_385666609031338394od_a_a @ M @ N ) )
= ( minus_7313152569058643924od_a_a @ ( comm_m9185394553243882098od_a_a @ M ) @ ( comm_m9185394553243882098od_a_a @ N ) ) ) ) ).
% sum_mset_diff
thf(fact_811_subset__mset_Ozero__less__iff__neq__zero,axiom,
! [N4: multiset_a] :
( ( subset_mset_a @ zero_zero_multiset_a @ N4 )
= ( N4 != zero_zero_multiset_a ) ) ).
% subset_mset.zero_less_iff_neq_zero
thf(fact_812_subset__mset_Ogr__implies__not__zero,axiom,
! [M4: multiset_a,N4: multiset_a] :
( ( subset_mset_a @ M4 @ N4 )
=> ( N4 != zero_zero_multiset_a ) ) ).
% subset_mset.gr_implies_not_zero
thf(fact_813_subset__mset_Onot__eq__extremum,axiom,
! [A: multiset_a] :
( ( A != zero_zero_multiset_a )
= ( subset_mset_a @ zero_zero_multiset_a @ A ) ) ).
% subset_mset.not_eq_extremum
thf(fact_814_subset__mset_Oextremum__strict,axiom,
! [A: multiset_a] :
~ ( subset_mset_a @ A @ zero_zero_multiset_a ) ).
% subset_mset.extremum_strict
thf(fact_815_subset__mset_Onot__less__zero,axiom,
! [N4: multiset_a] :
~ ( subset_mset_a @ N4 @ zero_zero_multiset_a ) ).
% subset_mset.not_less_zero
thf(fact_816_subset__mset_Ogr__zeroI,axiom,
! [N4: multiset_a] :
( ( N4 != zero_zero_multiset_a )
=> ( subset_mset_a @ zero_zero_multiset_a @ N4 ) ) ).
% subset_mset.gr_zeroI
thf(fact_817_subset__mset_Oadd__less__imp__less__right,axiom,
! [A: multiset_a,C2: multiset_a,B: multiset_a] :
( ( subset_mset_a @ ( plus_plus_multiset_a @ A @ C2 ) @ ( plus_plus_multiset_a @ B @ C2 ) )
=> ( subset_mset_a @ A @ B ) ) ).
% subset_mset.add_less_imp_less_right
thf(fact_818_subset__mset_Oadd__less__imp__less__left,axiom,
! [C2: multiset_a,A: multiset_a,B: multiset_a] :
( ( subset_mset_a @ ( plus_plus_multiset_a @ C2 @ A ) @ ( plus_plus_multiset_a @ C2 @ B ) )
=> ( subset_mset_a @ A @ B ) ) ).
% subset_mset.add_less_imp_less_left
thf(fact_819_subset__mset_Oadd__strict__right__mono,axiom,
! [A: multiset_a,B: multiset_a,C2: multiset_a] :
( ( subset_mset_a @ A @ B )
=> ( subset_mset_a @ ( plus_plus_multiset_a @ A @ C2 ) @ ( plus_plus_multiset_a @ B @ C2 ) ) ) ).
% subset_mset.add_strict_right_mono
thf(fact_820_subset__mset_Oadd__strict__left__mono,axiom,
! [A: multiset_a,B: multiset_a,C2: multiset_a] :
( ( subset_mset_a @ A @ B )
=> ( subset_mset_a @ ( plus_plus_multiset_a @ C2 @ A ) @ ( plus_plus_multiset_a @ C2 @ B ) ) ) ).
% subset_mset.add_strict_left_mono
thf(fact_821_subset__mset_Oadd__strict__mono,axiom,
! [A: multiset_a,B: multiset_a,C2: multiset_a,D: multiset_a] :
( ( subset_mset_a @ A @ B )
=> ( ( subset_mset_a @ C2 @ D )
=> ( subset_mset_a @ ( plus_plus_multiset_a @ A @ C2 ) @ ( plus_plus_multiset_a @ B @ D ) ) ) ) ).
% subset_mset.add_strict_mono
thf(fact_822_mset__subsetD,axiom,
! [A3: multiset_a,B3: multiset_a,X3: a] :
( ( subset_mset_a @ A3 @ B3 )
=> ( ( member_a @ X3 @ ( set_mset_a @ A3 ) )
=> ( member_a @ X3 @ ( set_mset_a @ B3 ) ) ) ) ).
% mset_subsetD
thf(fact_823_mset__subsetD,axiom,
! [A3: multis4781622204924204013iset_a,B3: multis4781622204924204013iset_a,X3: produc6518373309651786023iset_a] :
( ( subset5376938936802007386iset_a @ A3 @ B3 )
=> ( ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ A3 ) )
=> ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ B3 ) ) ) ) ).
% mset_subsetD
thf(fact_824_mset__subsetD,axiom,
! [A3: multis501812127501805293od_a_a,B3: multis501812127501805293od_a_a,X3: product_prod_a_a] :
( ( subset8265789482553336922od_a_a @ A3 @ B3 )
=> ( ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ A3 ) )
=> ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ B3 ) ) ) ) ).
% mset_subsetD
thf(fact_825_subset__mset_Oless__infI1,axiom,
! [A: multiset_a,X3: multiset_a,B: multiset_a] :
( ( subset_mset_a @ A @ X3 )
=> ( subset_mset_a @ ( inter_mset_a @ A @ B ) @ X3 ) ) ).
% subset_mset.less_infI1
thf(fact_826_subset__mset_Oless__infI2,axiom,
! [B: multiset_a,X3: multiset_a,A: multiset_a] :
( ( subset_mset_a @ B @ X3 )
=> ( subset_mset_a @ ( inter_mset_a @ A @ B ) @ X3 ) ) ).
% subset_mset.less_infI2
thf(fact_827_subset__mset_Oinf_Oabsorb3,axiom,
! [A: multiset_a,B: multiset_a] :
( ( subset_mset_a @ A @ B )
=> ( ( inter_mset_a @ A @ B )
= A ) ) ).
% subset_mset.inf.absorb3
thf(fact_828_subset__mset_Oinf_Oabsorb4,axiom,
! [B: multiset_a,A: multiset_a] :
( ( subset_mset_a @ B @ A )
=> ( ( inter_mset_a @ A @ B )
= B ) ) ).
% subset_mset.inf.absorb4
thf(fact_829_subset__mset_Oinf_Ostrict__boundedE,axiom,
! [A: multiset_a,B: multiset_a,C2: multiset_a] :
( ( subset_mset_a @ A @ ( inter_mset_a @ B @ C2 ) )
=> ~ ( ( subset_mset_a @ A @ B )
=> ~ ( subset_mset_a @ A @ C2 ) ) ) ).
% subset_mset.inf.strict_boundedE
thf(fact_830_subset__mset_Oinf_Ostrict__order__iff,axiom,
( subset_mset_a
= ( ^ [A5: multiset_a,B5: multiset_a] :
( ( A5
= ( inter_mset_a @ A5 @ B5 ) )
& ( A5 != B5 ) ) ) ) ).
% subset_mset.inf.strict_order_iff
thf(fact_831_subset__mset_Oinf_Ostrict__coboundedI1,axiom,
! [A: multiset_a,C2: multiset_a,B: multiset_a] :
( ( subset_mset_a @ A @ C2 )
=> ( subset_mset_a @ ( inter_mset_a @ A @ B ) @ C2 ) ) ).
% subset_mset.inf.strict_coboundedI1
thf(fact_832_subset__mset_Oinf_Ostrict__coboundedI2,axiom,
! [B: multiset_a,C2: multiset_a,A: multiset_a] :
( ( subset_mset_a @ B @ C2 )
=> ( subset_mset_a @ ( inter_mset_a @ A @ B ) @ C2 ) ) ).
% subset_mset.inf.strict_coboundedI2
thf(fact_833_empty__le,axiom,
! [A3: multiset_a] : ( subseteq_mset_a @ zero_zero_multiset_a @ A3 ) ).
% empty_le
thf(fact_834_subset__mset_Oextremum__uniqueI,axiom,
! [A: multiset_a] :
( ( subseteq_mset_a @ A @ zero_zero_multiset_a )
=> ( A = zero_zero_multiset_a ) ) ).
% subset_mset.extremum_uniqueI
thf(fact_835_subset__mset_Obot__least,axiom,
! [A: multiset_a] : ( subseteq_mset_a @ zero_zero_multiset_a @ A ) ).
% subset_mset.bot_least
thf(fact_836_subset__mset_Ozero__le,axiom,
! [X3: multiset_a] : ( subseteq_mset_a @ zero_zero_multiset_a @ X3 ) ).
% subset_mset.zero_le
thf(fact_837_mset__subset__eq__exists__conv,axiom,
( subseteq_mset_a
= ( ^ [A6: multiset_a,B6: multiset_a] :
? [C4: multiset_a] :
( B6
= ( plus_plus_multiset_a @ A6 @ C4 ) ) ) ) ).
% mset_subset_eq_exists_conv
thf(fact_838_mset__subset__eq__add__right,axiom,
! [B3: multiset_a,A3: multiset_a] : ( subseteq_mset_a @ B3 @ ( plus_plus_multiset_a @ A3 @ B3 ) ) ).
% mset_subset_eq_add_right
thf(fact_839_mset__subset__eq__mono__add,axiom,
! [A3: multiset_a,B3: multiset_a,C3: multiset_a,D2: multiset_a] :
( ( subseteq_mset_a @ A3 @ B3 )
=> ( ( subseteq_mset_a @ C3 @ D2 )
=> ( subseteq_mset_a @ ( plus_plus_multiset_a @ A3 @ C3 ) @ ( plus_plus_multiset_a @ B3 @ D2 ) ) ) ) ).
% mset_subset_eq_mono_add
thf(fact_840_mset__subset__eq__add__left,axiom,
! [A3: multiset_a,B3: multiset_a] : ( subseteq_mset_a @ A3 @ ( plus_plus_multiset_a @ A3 @ B3 ) ) ).
% mset_subset_eq_add_left
thf(fact_841_subset__mset_Oadd__le__imp__le__right,axiom,
! [A: multiset_a,C2: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ ( plus_plus_multiset_a @ A @ C2 ) @ ( plus_plus_multiset_a @ B @ C2 ) )
=> ( subseteq_mset_a @ A @ B ) ) ).
% subset_mset.add_le_imp_le_right
thf(fact_842_subset__mset_Oadd__le__imp__le__left,axiom,
! [C2: multiset_a,A: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ ( plus_plus_multiset_a @ C2 @ A ) @ ( plus_plus_multiset_a @ C2 @ B ) )
=> ( subseteq_mset_a @ A @ B ) ) ).
% subset_mset.add_le_imp_le_left
thf(fact_843_subset__mset_Oadd__right__mono,axiom,
! [A: multiset_a,B: multiset_a,C2: multiset_a] :
( ( subseteq_mset_a @ A @ B )
=> ( subseteq_mset_a @ ( plus_plus_multiset_a @ A @ C2 ) @ ( plus_plus_multiset_a @ B @ C2 ) ) ) ).
% subset_mset.add_right_mono
thf(fact_844_subset__mset_Oadd__left__mono,axiom,
! [A: multiset_a,B: multiset_a,C2: multiset_a] :
( ( subseteq_mset_a @ A @ B )
=> ( subseteq_mset_a @ ( plus_plus_multiset_a @ C2 @ A ) @ ( plus_plus_multiset_a @ C2 @ B ) ) ) ).
% subset_mset.add_left_mono
thf(fact_845_subset__mset_Ole__iff__add,axiom,
( subseteq_mset_a
= ( ^ [A5: multiset_a,B5: multiset_a] :
? [C5: multiset_a] :
( B5
= ( plus_plus_multiset_a @ A5 @ C5 ) ) ) ) ).
% subset_mset.le_iff_add
thf(fact_846_subset__mset_Oless__eqE,axiom,
! [A: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ A @ B )
=> ~ ! [C: multiset_a] :
( B
!= ( plus_plus_multiset_a @ A @ C ) ) ) ).
% subset_mset.less_eqE
thf(fact_847_subset__mset_Oadd__mono,axiom,
! [A: multiset_a,B: multiset_a,C2: multiset_a,D: multiset_a] :
( ( subseteq_mset_a @ A @ B )
=> ( ( subseteq_mset_a @ C2 @ D )
=> ( subseteq_mset_a @ ( plus_plus_multiset_a @ A @ C2 ) @ ( plus_plus_multiset_a @ B @ D ) ) ) ) ).
% subset_mset.add_mono
thf(fact_848_mset__subset__eqD,axiom,
! [A3: multiset_a,B3: multiset_a,X3: a] :
( ( subseteq_mset_a @ A3 @ B3 )
=> ( ( member_a @ X3 @ ( set_mset_a @ A3 ) )
=> ( member_a @ X3 @ ( set_mset_a @ B3 ) ) ) ) ).
% mset_subset_eqD
thf(fact_849_mset__subset__eqD,axiom,
! [A3: multis4781622204924204013iset_a,B3: multis4781622204924204013iset_a,X3: produc6518373309651786023iset_a] :
( ( subset6292090195676340366iset_a @ A3 @ B3 )
=> ( ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ A3 ) )
=> ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ B3 ) ) ) ) ).
% mset_subset_eqD
thf(fact_850_mset__subset__eqD,axiom,
! [A3: multis501812127501805293od_a_a,B3: multis501812127501805293od_a_a,X3: product_prod_a_a] :
( ( subset4108007524207251342od_a_a @ A3 @ B3 )
=> ( ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ A3 ) )
=> ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ B3 ) ) ) ) ).
% mset_subset_eqD
thf(fact_851_sum__mset_Oeq__fold,axiom,
( comm_m4595494212910443721iset_a
= ( fold_m6601649825673331723iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a ) ) ).
% sum_mset.eq_fold
thf(fact_852_subset__mset_Oinf__le1,axiom,
! [X3: multiset_a,Y: multiset_a] : ( subseteq_mset_a @ ( inter_mset_a @ X3 @ Y ) @ X3 ) ).
% subset_mset.inf_le1
thf(fact_853_subset__mset_Oinf__le2,axiom,
! [X3: multiset_a,Y: multiset_a] : ( subseteq_mset_a @ ( inter_mset_a @ X3 @ Y ) @ Y ) ).
% subset_mset.inf_le2
thf(fact_854_subset__mset_Ole__infE,axiom,
! [X3: multiset_a,A: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ X3 @ ( inter_mset_a @ A @ B ) )
=> ~ ( ( subseteq_mset_a @ X3 @ A )
=> ~ ( subseteq_mset_a @ X3 @ B ) ) ) ).
% subset_mset.le_infE
thf(fact_855_subset__mset_Ole__infI,axiom,
! [X3: multiset_a,A: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ X3 @ A )
=> ( ( subseteq_mset_a @ X3 @ B )
=> ( subseteq_mset_a @ X3 @ ( inter_mset_a @ A @ B ) ) ) ) ).
% subset_mset.le_infI
thf(fact_856_subset__mset_Oinf__mono,axiom,
! [A: multiset_a,C2: multiset_a,B: multiset_a,D: multiset_a] :
( ( subseteq_mset_a @ A @ C2 )
=> ( ( subseteq_mset_a @ B @ D )
=> ( subseteq_mset_a @ ( inter_mset_a @ A @ B ) @ ( inter_mset_a @ C2 @ D ) ) ) ) ).
% subset_mset.inf_mono
thf(fact_857_subset__mset_Ole__infI1,axiom,
! [A: multiset_a,X3: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ A @ X3 )
=> ( subseteq_mset_a @ ( inter_mset_a @ A @ B ) @ X3 ) ) ).
% subset_mset.le_infI1
thf(fact_858_subset__mset_Ole__infI2,axiom,
! [B: multiset_a,X3: multiset_a,A: multiset_a] :
( ( subseteq_mset_a @ B @ X3 )
=> ( subseteq_mset_a @ ( inter_mset_a @ A @ B ) @ X3 ) ) ).
% subset_mset.le_infI2
thf(fact_859_subset__mset_Oinf_OorderE,axiom,
! [A: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ A @ B )
=> ( A
= ( inter_mset_a @ A @ B ) ) ) ).
% subset_mset.inf.orderE
thf(fact_860_subset__mset_Oinf_OorderI,axiom,
! [A: multiset_a,B: multiset_a] :
( ( A
= ( inter_mset_a @ A @ B ) )
=> ( subseteq_mset_a @ A @ B ) ) ).
% subset_mset.inf.orderI
thf(fact_861_subset__mset_Oinf__unique,axiom,
! [F: multiset_a > multiset_a > multiset_a,X3: multiset_a,Y: multiset_a] :
( ! [X5: multiset_a,Y4: multiset_a] : ( subseteq_mset_a @ ( F @ X5 @ Y4 ) @ X5 )
=> ( ! [X5: multiset_a,Y4: multiset_a] : ( subseteq_mset_a @ ( F @ X5 @ Y4 ) @ Y4 )
=> ( ! [X5: multiset_a,Y4: multiset_a,Z2: multiset_a] :
( ( subseteq_mset_a @ X5 @ Y4 )
=> ( ( subseteq_mset_a @ X5 @ Z2 )
=> ( subseteq_mset_a @ X5 @ ( F @ Y4 @ Z2 ) ) ) )
=> ( ( inter_mset_a @ X3 @ Y )
= ( F @ X3 @ Y ) ) ) ) ) ).
% subset_mset.inf_unique
thf(fact_862_subset__mset_Ole__iff__inf,axiom,
( subseteq_mset_a
= ( ^ [X6: multiset_a,Y5: multiset_a] :
( ( inter_mset_a @ X6 @ Y5 )
= X6 ) ) ) ).
% subset_mset.le_iff_inf
thf(fact_863_subset__mset_Oinf_Oabsorb1,axiom,
! [A: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ A @ B )
=> ( ( inter_mset_a @ A @ B )
= A ) ) ).
% subset_mset.inf.absorb1
thf(fact_864_subset__mset_Oinf_Oabsorb2,axiom,
! [B: multiset_a,A: multiset_a] :
( ( subseteq_mset_a @ B @ A )
=> ( ( inter_mset_a @ A @ B )
= B ) ) ).
% subset_mset.inf.absorb2
thf(fact_865_subset__mset_Oinf__absorb1,axiom,
! [X3: multiset_a,Y: multiset_a] :
( ( subseteq_mset_a @ X3 @ Y )
=> ( ( inter_mset_a @ X3 @ Y )
= X3 ) ) ).
% subset_mset.inf_absorb1
thf(fact_866_subset__mset_Oinf__absorb2,axiom,
! [Y: multiset_a,X3: multiset_a] :
( ( subseteq_mset_a @ Y @ X3 )
=> ( ( inter_mset_a @ X3 @ Y )
= Y ) ) ).
% subset_mset.inf_absorb2
thf(fact_867_subset__mset_Oinf_OboundedE,axiom,
! [A: multiset_a,B: multiset_a,C2: multiset_a] :
( ( subseteq_mset_a @ A @ ( inter_mset_a @ B @ C2 ) )
=> ~ ( ( subseteq_mset_a @ A @ B )
=> ~ ( subseteq_mset_a @ A @ C2 ) ) ) ).
% subset_mset.inf.boundedE
thf(fact_868_subset__mset_Oinf_OboundedI,axiom,
! [A: multiset_a,B: multiset_a,C2: multiset_a] :
( ( subseteq_mset_a @ A @ B )
=> ( ( subseteq_mset_a @ A @ C2 )
=> ( subseteq_mset_a @ A @ ( inter_mset_a @ B @ C2 ) ) ) ) ).
% subset_mset.inf.boundedI
thf(fact_869_subset__mset_Oinf__greatest,axiom,
! [X3: multiset_a,Y: multiset_a,Z: multiset_a] :
( ( subseteq_mset_a @ X3 @ Y )
=> ( ( subseteq_mset_a @ X3 @ Z )
=> ( subseteq_mset_a @ X3 @ ( inter_mset_a @ Y @ Z ) ) ) ) ).
% subset_mset.inf_greatest
thf(fact_870_subset__mset_Oinf_Oorder__iff,axiom,
( subseteq_mset_a
= ( ^ [A5: multiset_a,B5: multiset_a] :
( A5
= ( inter_mset_a @ A5 @ B5 ) ) ) ) ).
% subset_mset.inf.order_iff
thf(fact_871_subset__mset_Oinf_Ocobounded1,axiom,
! [A: multiset_a,B: multiset_a] : ( subseteq_mset_a @ ( inter_mset_a @ A @ B ) @ A ) ).
% subset_mset.inf.cobounded1
thf(fact_872_subset__mset_Oinf_Ocobounded2,axiom,
! [A: multiset_a,B: multiset_a] : ( subseteq_mset_a @ ( inter_mset_a @ A @ B ) @ B ) ).
% subset_mset.inf.cobounded2
thf(fact_873_subset__mset_Oinf_Oabsorb__iff1,axiom,
( subseteq_mset_a
= ( ^ [A5: multiset_a,B5: multiset_a] :
( ( inter_mset_a @ A5 @ B5 )
= A5 ) ) ) ).
% subset_mset.inf.absorb_iff1
thf(fact_874_subset__mset_Oinf_Oabsorb__iff2,axiom,
( subseteq_mset_a
= ( ^ [B5: multiset_a,A5: multiset_a] :
( ( inter_mset_a @ A5 @ B5 )
= B5 ) ) ) ).
% subset_mset.inf.absorb_iff2
thf(fact_875_subset__mset_Oinf_OcoboundedI1,axiom,
! [A: multiset_a,C2: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ A @ C2 )
=> ( subseteq_mset_a @ ( inter_mset_a @ A @ B ) @ C2 ) ) ).
% subset_mset.inf.coboundedI1
thf(fact_876_subset__mset_Oinf_OcoboundedI2,axiom,
! [B: multiset_a,C2: multiset_a,A: multiset_a] :
( ( subseteq_mset_a @ B @ C2 )
=> ( subseteq_mset_a @ ( inter_mset_a @ A @ B ) @ C2 ) ) ).
% subset_mset.inf.coboundedI2
thf(fact_877_diff__subset__eq__self,axiom,
! [M: multiset_a,N: multiset_a] : ( subseteq_mset_a @ ( minus_3765977307040488491iset_a @ M @ N ) @ M ) ).
% diff_subset_eq_self
thf(fact_878_diff__subset__eq__self,axiom,
! [M: multis4781622204924204013iset_a,N: multis4781622204924204013iset_a] : ( subset6292090195676340366iset_a @ ( minus_9211619285413683668iset_a @ M @ N ) @ M ) ).
% diff_subset_eq_self
thf(fact_879_diff__subset__eq__self,axiom,
! [M: multis501812127501805293od_a_a,N: multis501812127501805293od_a_a] : ( subset4108007524207251342od_a_a @ ( minus_7313152569058643924od_a_a @ M @ N ) @ M ) ).
% diff_subset_eq_self
thf(fact_880_subset__mset_Opos__add__strict,axiom,
! [A: multiset_a,B: multiset_a,C2: multiset_a] :
( ( subset_mset_a @ zero_zero_multiset_a @ A )
=> ( ( subset_mset_a @ B @ C2 )
=> ( subset_mset_a @ B @ ( plus_plus_multiset_a @ A @ C2 ) ) ) ) ).
% subset_mset.pos_add_strict
thf(fact_881_subset__mset_Oadd__pos__pos,axiom,
! [A: multiset_a,B: multiset_a] :
( ( subset_mset_a @ zero_zero_multiset_a @ A )
=> ( ( subset_mset_a @ zero_zero_multiset_a @ B )
=> ( subset_mset_a @ zero_zero_multiset_a @ ( plus_plus_multiset_a @ A @ B ) ) ) ) ).
% subset_mset.add_pos_pos
thf(fact_882_subset__mset_Oadd__neg__neg,axiom,
! [A: multiset_a,B: multiset_a] :
( ( subset_mset_a @ A @ zero_zero_multiset_a )
=> ( ( subset_mset_a @ B @ zero_zero_multiset_a )
=> ( subset_mset_a @ ( plus_plus_multiset_a @ A @ B ) @ zero_zero_multiset_a ) ) ) ).
% subset_mset.add_neg_neg
thf(fact_883_subset__mset_OlessE,axiom,
! [A: multiset_a,B: multiset_a] :
( ( subset_mset_a @ A @ B )
=> ~ ! [C: multiset_a] :
( ( B
= ( plus_plus_multiset_a @ A @ C ) )
=> ( C = zero_zero_multiset_a ) ) ) ).
% subset_mset.lessE
thf(fact_884_empty__subset__add__mset,axiom,
! [X3: a,M: multiset_a] : ( subset_mset_a @ zero_zero_multiset_a @ ( add_mset_a @ X3 @ M ) ) ).
% empty_subset_add_mset
thf(fact_885_mset__subset__insertD,axiom,
! [X3: a,A3: multiset_a,B3: multiset_a] :
( ( subset_mset_a @ ( add_mset_a @ X3 @ A3 ) @ B3 )
=> ( ( member_a @ X3 @ ( set_mset_a @ B3 ) )
& ( subset_mset_a @ A3 @ B3 ) ) ) ).
% mset_subset_insertD
thf(fact_886_mset__subset__insertD,axiom,
! [X3: produc6518373309651786023iset_a,A3: multis4781622204924204013iset_a,B3: multis4781622204924204013iset_a] :
( ( subset5376938936802007386iset_a @ ( add_ms5348986034529679365iset_a @ X3 @ A3 ) @ B3 )
=> ( ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ B3 ) )
& ( subset5376938936802007386iset_a @ A3 @ B3 ) ) ) ).
% mset_subset_insertD
thf(fact_887_mset__subset__insertD,axiom,
! [X3: product_prod_a_a,A3: multis501812127501805293od_a_a,B3: multis501812127501805293od_a_a] :
( ( subset8265789482553336922od_a_a @ ( add_ms8655138167283798533od_a_a @ X3 @ A3 ) @ B3 )
=> ( ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ B3 ) )
& ( subset8265789482553336922od_a_a @ A3 @ B3 ) ) ) ).
% mset_subset_insertD
thf(fact_888_subset__mset_Oadd__nonpos__eq__0__iff,axiom,
! [X3: multiset_a,Y: multiset_a] :
( ( subseteq_mset_a @ X3 @ zero_zero_multiset_a )
=> ( ( subseteq_mset_a @ Y @ zero_zero_multiset_a )
=> ( ( ( plus_plus_multiset_a @ X3 @ Y )
= zero_zero_multiset_a )
= ( ( X3 = zero_zero_multiset_a )
& ( Y = zero_zero_multiset_a ) ) ) ) ) ).
% subset_mset.add_nonpos_eq_0_iff
thf(fact_889_subset__mset_Oadd__nonneg__eq__0__iff,axiom,
! [X3: multiset_a,Y: multiset_a] :
( ( subseteq_mset_a @ zero_zero_multiset_a @ X3 )
=> ( ( subseteq_mset_a @ zero_zero_multiset_a @ Y )
=> ( ( ( plus_plus_multiset_a @ X3 @ Y )
= zero_zero_multiset_a )
= ( ( X3 = zero_zero_multiset_a )
& ( Y = zero_zero_multiset_a ) ) ) ) ) ).
% subset_mset.add_nonneg_eq_0_iff
thf(fact_890_subset__mset_Oadd__nonpos__nonpos,axiom,
! [A: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ A @ zero_zero_multiset_a )
=> ( ( subseteq_mset_a @ B @ zero_zero_multiset_a )
=> ( subseteq_mset_a @ ( plus_plus_multiset_a @ A @ B ) @ zero_zero_multiset_a ) ) ) ).
% subset_mset.add_nonpos_nonpos
thf(fact_891_subset__mset_Oadd__nonneg__nonneg,axiom,
! [A: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ zero_zero_multiset_a @ A )
=> ( ( subseteq_mset_a @ zero_zero_multiset_a @ B )
=> ( subseteq_mset_a @ zero_zero_multiset_a @ ( plus_plus_multiset_a @ A @ B ) ) ) ) ).
% subset_mset.add_nonneg_nonneg
thf(fact_892_subset__mset_Oadd__increasing2,axiom,
! [C2: multiset_a,B: multiset_a,A: multiset_a] :
( ( subseteq_mset_a @ zero_zero_multiset_a @ C2 )
=> ( ( subseteq_mset_a @ B @ A )
=> ( subseteq_mset_a @ B @ ( plus_plus_multiset_a @ A @ C2 ) ) ) ) ).
% subset_mset.add_increasing2
thf(fact_893_subset__mset_Oadd__decreasing2,axiom,
! [C2: multiset_a,A: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ C2 @ zero_zero_multiset_a )
=> ( ( subseteq_mset_a @ A @ B )
=> ( subseteq_mset_a @ ( plus_plus_multiset_a @ A @ C2 ) @ B ) ) ) ).
% subset_mset.add_decreasing2
thf(fact_894_subset__mset_Oadd__increasing,axiom,
! [A: multiset_a,B: multiset_a,C2: multiset_a] :
( ( subseteq_mset_a @ zero_zero_multiset_a @ A )
=> ( ( subseteq_mset_a @ B @ C2 )
=> ( subseteq_mset_a @ B @ ( plus_plus_multiset_a @ A @ C2 ) ) ) ) ).
% subset_mset.add_increasing
thf(fact_895_subset__mset_Oadd__decreasing,axiom,
! [A: multiset_a,C2: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ A @ zero_zero_multiset_a )
=> ( ( subseteq_mset_a @ C2 @ B )
=> ( subseteq_mset_a @ ( plus_plus_multiset_a @ A @ C2 ) @ B ) ) ) ).
% subset_mset.add_decreasing
thf(fact_896_inter__subset__eq__union,axiom,
! [A3: multiset_a,B3: multiset_a] : ( subseteq_mset_a @ ( inter_mset_a @ A3 @ B3 ) @ ( plus_plus_multiset_a @ A3 @ B3 ) ) ).
% inter_subset_eq_union
thf(fact_897_Diff__eq__empty__iff__mset,axiom,
! [A3: multiset_a,B3: multiset_a] :
( ( ( minus_3765977307040488491iset_a @ A3 @ B3 )
= zero_zero_multiset_a )
= ( subseteq_mset_a @ A3 @ B3 ) ) ).
% Diff_eq_empty_iff_mset
thf(fact_898_Diff__eq__empty__iff__mset,axiom,
! [A3: multis4781622204924204013iset_a,B3: multis4781622204924204013iset_a] :
( ( ( minus_9211619285413683668iset_a @ A3 @ B3 )
= zero_z4591712717340762532iset_a )
= ( subset6292090195676340366iset_a @ A3 @ B3 ) ) ).
% Diff_eq_empty_iff_mset
thf(fact_899_Diff__eq__empty__iff__mset,axiom,
! [A3: multis501812127501805293od_a_a,B3: multis501812127501805293od_a_a] :
( ( ( minus_7313152569058643924od_a_a @ A3 @ B3 )
= zero_z3024502163831676836od_a_a )
= ( subset4108007524207251342od_a_a @ A3 @ B3 ) ) ).
% Diff_eq_empty_iff_mset
thf(fact_900_multiset__diff__union__assoc,axiom,
! [C3: multiset_a,B3: multiset_a,A3: multiset_a] :
( ( subseteq_mset_a @ C3 @ B3 )
=> ( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ A3 @ B3 ) @ C3 )
= ( plus_plus_multiset_a @ A3 @ ( minus_3765977307040488491iset_a @ B3 @ C3 ) ) ) ) ).
% multiset_diff_union_assoc
thf(fact_901_multiset__diff__union__assoc,axiom,
! [C3: multis4781622204924204013iset_a,B3: multis4781622204924204013iset_a,A3: multis4781622204924204013iset_a] :
( ( subset6292090195676340366iset_a @ C3 @ B3 )
=> ( ( minus_9211619285413683668iset_a @ ( plus_p1053901119268525860iset_a @ A3 @ B3 ) @ C3 )
= ( plus_p1053901119268525860iset_a @ A3 @ ( minus_9211619285413683668iset_a @ B3 @ C3 ) ) ) ) ).
% multiset_diff_union_assoc
thf(fact_902_multiset__diff__union__assoc,axiom,
! [C3: multis501812127501805293od_a_a,B3: multis501812127501805293od_a_a,A3: multis501812127501805293od_a_a] :
( ( subset4108007524207251342od_a_a @ C3 @ B3 )
=> ( ( minus_7313152569058643924od_a_a @ ( plus_p6967097483979887908od_a_a @ A3 @ B3 ) @ C3 )
= ( plus_p6967097483979887908od_a_a @ A3 @ ( minus_7313152569058643924od_a_a @ B3 @ C3 ) ) ) ) ).
% multiset_diff_union_assoc
thf(fact_903_subset__eq__diff__conv,axiom,
! [A3: multiset_a,C3: multiset_a,B3: multiset_a] :
( ( subseteq_mset_a @ ( minus_3765977307040488491iset_a @ A3 @ C3 ) @ B3 )
= ( subseteq_mset_a @ A3 @ ( plus_plus_multiset_a @ B3 @ C3 ) ) ) ).
% subset_eq_diff_conv
thf(fact_904_subset__eq__diff__conv,axiom,
! [A3: multis4781622204924204013iset_a,C3: multis4781622204924204013iset_a,B3: multis4781622204924204013iset_a] :
( ( subset6292090195676340366iset_a @ ( minus_9211619285413683668iset_a @ A3 @ C3 ) @ B3 )
= ( subset6292090195676340366iset_a @ A3 @ ( plus_p1053901119268525860iset_a @ B3 @ C3 ) ) ) ).
% subset_eq_diff_conv
thf(fact_905_subset__eq__diff__conv,axiom,
! [A3: multis501812127501805293od_a_a,C3: multis501812127501805293od_a_a,B3: multis501812127501805293od_a_a] :
( ( subset4108007524207251342od_a_a @ ( minus_7313152569058643924od_a_a @ A3 @ C3 ) @ B3 )
= ( subset4108007524207251342od_a_a @ A3 @ ( plus_p6967097483979887908od_a_a @ B3 @ C3 ) ) ) ).
% subset_eq_diff_conv
thf(fact_906_subset__mset_Ole__imp__diff__is__add,axiom,
! [A: multiset_a,B: multiset_a,C2: multiset_a] :
( ( subseteq_mset_a @ A @ B )
=> ( ( subseteq_mset_a @ A @ B )
=> ( ( ( minus_3765977307040488491iset_a @ B @ A )
= C2 )
= ( B
= ( plus_plus_multiset_a @ C2 @ A ) ) ) ) ) ).
% subset_mset.le_imp_diff_is_add
thf(fact_907_subset__mset_Ole__imp__diff__is__add,axiom,
! [A: multis4781622204924204013iset_a,B: multis4781622204924204013iset_a,C2: multis4781622204924204013iset_a] :
( ( subset6292090195676340366iset_a @ A @ B )
=> ( ( subset6292090195676340366iset_a @ A @ B )
=> ( ( ( minus_9211619285413683668iset_a @ B @ A )
= C2 )
= ( B
= ( plus_p1053901119268525860iset_a @ C2 @ A ) ) ) ) ) ).
% subset_mset.le_imp_diff_is_add
thf(fact_908_subset__mset_Ole__imp__diff__is__add,axiom,
! [A: multis501812127501805293od_a_a,B: multis501812127501805293od_a_a,C2: multis501812127501805293od_a_a] :
( ( subset4108007524207251342od_a_a @ A @ B )
=> ( ( subset4108007524207251342od_a_a @ A @ B )
=> ( ( ( minus_7313152569058643924od_a_a @ B @ A )
= C2 )
= ( B
= ( plus_p6967097483979887908od_a_a @ C2 @ A ) ) ) ) ) ).
% subset_mset.le_imp_diff_is_add
thf(fact_909_subset__mset_Oadd__diff__inverse,axiom,
! [A: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ A @ B )
=> ( ( plus_plus_multiset_a @ A @ ( minus_3765977307040488491iset_a @ B @ A ) )
= B ) ) ).
% subset_mset.add_diff_inverse
thf(fact_910_subset__mset_Oadd__diff__inverse,axiom,
! [A: multis4781622204924204013iset_a,B: multis4781622204924204013iset_a] :
( ( subset6292090195676340366iset_a @ A @ B )
=> ( ( plus_p1053901119268525860iset_a @ A @ ( minus_9211619285413683668iset_a @ B @ A ) )
= B ) ) ).
% subset_mset.add_diff_inverse
thf(fact_911_subset__mset_Oadd__diff__inverse,axiom,
! [A: multis501812127501805293od_a_a,B: multis501812127501805293od_a_a] :
( ( subset4108007524207251342od_a_a @ A @ B )
=> ( ( plus_p6967097483979887908od_a_a @ A @ ( minus_7313152569058643924od_a_a @ B @ A ) )
= B ) ) ).
% subset_mset.add_diff_inverse
thf(fact_912_subset__mset_Odiff__diff__right,axiom,
! [A: multiset_a,B: multiset_a,C2: multiset_a] :
( ( subseteq_mset_a @ A @ B )
=> ( ( minus_3765977307040488491iset_a @ C2 @ ( minus_3765977307040488491iset_a @ B @ A ) )
= ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ C2 @ A ) @ B ) ) ) ).
% subset_mset.diff_diff_right
thf(fact_913_subset__mset_Odiff__diff__right,axiom,
! [A: multis4781622204924204013iset_a,B: multis4781622204924204013iset_a,C2: multis4781622204924204013iset_a] :
( ( subset6292090195676340366iset_a @ A @ B )
=> ( ( minus_9211619285413683668iset_a @ C2 @ ( minus_9211619285413683668iset_a @ B @ A ) )
= ( minus_9211619285413683668iset_a @ ( plus_p1053901119268525860iset_a @ C2 @ A ) @ B ) ) ) ).
% subset_mset.diff_diff_right
thf(fact_914_subset__mset_Odiff__diff__right,axiom,
! [A: multis501812127501805293od_a_a,B: multis501812127501805293od_a_a,C2: multis501812127501805293od_a_a] :
( ( subset4108007524207251342od_a_a @ A @ B )
=> ( ( minus_7313152569058643924od_a_a @ C2 @ ( minus_7313152569058643924od_a_a @ B @ A ) )
= ( minus_7313152569058643924od_a_a @ ( plus_p6967097483979887908od_a_a @ C2 @ A ) @ B ) ) ) ).
% subset_mset.diff_diff_right
thf(fact_915_subset__mset_Odiff__add__assoc2,axiom,
! [A: multiset_a,B: multiset_a,C2: multiset_a] :
( ( subseteq_mset_a @ A @ B )
=> ( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ B @ C2 ) @ A )
= ( plus_plus_multiset_a @ ( minus_3765977307040488491iset_a @ B @ A ) @ C2 ) ) ) ).
% subset_mset.diff_add_assoc2
thf(fact_916_subset__mset_Odiff__add__assoc2,axiom,
! [A: multis4781622204924204013iset_a,B: multis4781622204924204013iset_a,C2: multis4781622204924204013iset_a] :
( ( subset6292090195676340366iset_a @ A @ B )
=> ( ( minus_9211619285413683668iset_a @ ( plus_p1053901119268525860iset_a @ B @ C2 ) @ A )
= ( plus_p1053901119268525860iset_a @ ( minus_9211619285413683668iset_a @ B @ A ) @ C2 ) ) ) ).
% subset_mset.diff_add_assoc2
thf(fact_917_subset__mset_Odiff__add__assoc2,axiom,
! [A: multis501812127501805293od_a_a,B: multis501812127501805293od_a_a,C2: multis501812127501805293od_a_a] :
( ( subset4108007524207251342od_a_a @ A @ B )
=> ( ( minus_7313152569058643924od_a_a @ ( plus_p6967097483979887908od_a_a @ B @ C2 ) @ A )
= ( plus_p6967097483979887908od_a_a @ ( minus_7313152569058643924od_a_a @ B @ A ) @ C2 ) ) ) ).
% subset_mset.diff_add_assoc2
thf(fact_918_subset__mset_Odiff__add__assoc,axiom,
! [A: multiset_a,B: multiset_a,C2: multiset_a] :
( ( subseteq_mset_a @ A @ B )
=> ( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ C2 @ B ) @ A )
= ( plus_plus_multiset_a @ C2 @ ( minus_3765977307040488491iset_a @ B @ A ) ) ) ) ).
% subset_mset.diff_add_assoc
thf(fact_919_subset__mset_Odiff__add__assoc,axiom,
! [A: multis4781622204924204013iset_a,B: multis4781622204924204013iset_a,C2: multis4781622204924204013iset_a] :
( ( subset6292090195676340366iset_a @ A @ B )
=> ( ( minus_9211619285413683668iset_a @ ( plus_p1053901119268525860iset_a @ C2 @ B ) @ A )
= ( plus_p1053901119268525860iset_a @ C2 @ ( minus_9211619285413683668iset_a @ B @ A ) ) ) ) ).
% subset_mset.diff_add_assoc
thf(fact_920_subset__mset_Odiff__add__assoc,axiom,
! [A: multis501812127501805293od_a_a,B: multis501812127501805293od_a_a,C2: multis501812127501805293od_a_a] :
( ( subset4108007524207251342od_a_a @ A @ B )
=> ( ( minus_7313152569058643924od_a_a @ ( plus_p6967097483979887908od_a_a @ C2 @ B ) @ A )
= ( plus_p6967097483979887908od_a_a @ C2 @ ( minus_7313152569058643924od_a_a @ B @ A ) ) ) ) ).
% subset_mset.diff_add_assoc
thf(fact_921_subset__mset_Ole__diff__conv2,axiom,
! [A: multiset_a,B: multiset_a,C2: multiset_a] :
( ( subseteq_mset_a @ A @ B )
=> ( ( subseteq_mset_a @ C2 @ ( minus_3765977307040488491iset_a @ B @ A ) )
= ( subseteq_mset_a @ ( plus_plus_multiset_a @ C2 @ A ) @ B ) ) ) ).
% subset_mset.le_diff_conv2
thf(fact_922_subset__mset_Ole__diff__conv2,axiom,
! [A: multis4781622204924204013iset_a,B: multis4781622204924204013iset_a,C2: multis4781622204924204013iset_a] :
( ( subset6292090195676340366iset_a @ A @ B )
=> ( ( subset6292090195676340366iset_a @ C2 @ ( minus_9211619285413683668iset_a @ B @ A ) )
= ( subset6292090195676340366iset_a @ ( plus_p1053901119268525860iset_a @ C2 @ A ) @ B ) ) ) ).
% subset_mset.le_diff_conv2
thf(fact_923_subset__mset_Ole__diff__conv2,axiom,
! [A: multis501812127501805293od_a_a,B: multis501812127501805293od_a_a,C2: multis501812127501805293od_a_a] :
( ( subset4108007524207251342od_a_a @ A @ B )
=> ( ( subset4108007524207251342od_a_a @ C2 @ ( minus_7313152569058643924od_a_a @ B @ A ) )
= ( subset4108007524207251342od_a_a @ ( plus_p6967097483979887908od_a_a @ C2 @ A ) @ B ) ) ) ).
% subset_mset.le_diff_conv2
thf(fact_924_subset__mset_Ole__add__diff,axiom,
! [A: multiset_a,B: multiset_a,C2: multiset_a] :
( ( subseteq_mset_a @ A @ B )
=> ( subseteq_mset_a @ C2 @ ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ B @ C2 ) @ A ) ) ) ).
% subset_mset.le_add_diff
thf(fact_925_subset__mset_Ole__add__diff,axiom,
! [A: multis4781622204924204013iset_a,B: multis4781622204924204013iset_a,C2: multis4781622204924204013iset_a] :
( ( subset6292090195676340366iset_a @ A @ B )
=> ( subset6292090195676340366iset_a @ C2 @ ( minus_9211619285413683668iset_a @ ( plus_p1053901119268525860iset_a @ B @ C2 ) @ A ) ) ) ).
% subset_mset.le_add_diff
thf(fact_926_subset__mset_Ole__add__diff,axiom,
! [A: multis501812127501805293od_a_a,B: multis501812127501805293od_a_a,C2: multis501812127501805293od_a_a] :
( ( subset4108007524207251342od_a_a @ A @ B )
=> ( subset4108007524207251342od_a_a @ C2 @ ( minus_7313152569058643924od_a_a @ ( plus_p6967097483979887908od_a_a @ B @ C2 ) @ A ) ) ) ).
% subset_mset.le_add_diff
thf(fact_927_subset__mset_Odiff__add,axiom,
! [A: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ A @ B )
=> ( ( plus_plus_multiset_a @ ( minus_3765977307040488491iset_a @ B @ A ) @ A )
= B ) ) ).
% subset_mset.diff_add
thf(fact_928_subset__mset_Odiff__add,axiom,
! [A: multis4781622204924204013iset_a,B: multis4781622204924204013iset_a] :
( ( subset6292090195676340366iset_a @ A @ B )
=> ( ( plus_p1053901119268525860iset_a @ ( minus_9211619285413683668iset_a @ B @ A ) @ A )
= B ) ) ).
% subset_mset.diff_add
thf(fact_929_subset__mset_Odiff__add,axiom,
! [A: multis501812127501805293od_a_a,B: multis501812127501805293od_a_a] :
( ( subset4108007524207251342od_a_a @ A @ B )
=> ( ( plus_p6967097483979887908od_a_a @ ( minus_7313152569058643924od_a_a @ B @ A ) @ A )
= B ) ) ).
% subset_mset.diff_add
thf(fact_930_subset__implies__mult,axiom,
! [A3: multiset_a,B3: multiset_a,R: set_Product_prod_a_a] :
( ( subset_mset_a @ A3 @ B3 )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A3 @ B3 ) @ ( mult_a @ R ) ) ) ).
% subset_implies_mult
thf(fact_931_sum__mset_Oneutral,axiom,
! [A3: multiset_multiset_a] :
( ! [X5: multiset_a] :
( ( member_multiset_a @ X5 @ ( set_mset_multiset_a @ A3 ) )
=> ( X5 = zero_zero_multiset_a ) )
=> ( ( comm_m4595494212910443721iset_a @ A3 )
= zero_zero_multiset_a ) ) ).
% sum_mset.neutral
thf(fact_932_union__fold__mset__add__mset,axiom,
( plus_plus_multiset_a
= ( fold_m7320414754419674833iset_a @ add_mset_a ) ) ).
% union_fold_mset_add_mset
thf(fact_933_mset__subset__eq__single,axiom,
! [A: produc6518373309651786023iset_a,B3: multis4781622204924204013iset_a] :
( ( member5199237121806060112iset_a @ A @ ( set_ms2379723421352415844iset_a @ B3 ) )
=> ( subset6292090195676340366iset_a @ ( add_ms5348986034529679365iset_a @ A @ zero_z4591712717340762532iset_a ) @ B3 ) ) ).
% mset_subset_eq_single
thf(fact_934_mset__subset__eq__single,axiom,
! [A: product_prod_a_a,B3: multis501812127501805293od_a_a] :
( ( member1426531477525435216od_a_a @ A @ ( set_ms119794563918438244od_a_a @ B3 ) )
=> ( subset4108007524207251342od_a_a @ ( add_ms8655138167283798533od_a_a @ A @ zero_z3024502163831676836od_a_a ) @ B3 ) ) ).
% mset_subset_eq_single
thf(fact_935_mset__subset__eq__single,axiom,
! [A: a,B3: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ B3 ) )
=> ( subseteq_mset_a @ ( add_mset_a @ A @ zero_zero_multiset_a ) @ B3 ) ) ).
% mset_subset_eq_single
thf(fact_936_multi__subset__induct,axiom,
! [F2: multis4781622204924204013iset_a,A3: multis4781622204924204013iset_a,P: multis4781622204924204013iset_a > $o] :
( ( subset6292090195676340366iset_a @ F2 @ A3 )
=> ( ( P @ zero_z4591712717340762532iset_a )
=> ( ! [A4: produc6518373309651786023iset_a,F3: multis4781622204924204013iset_a] :
( ( member5199237121806060112iset_a @ A4 @ ( set_ms2379723421352415844iset_a @ A3 ) )
=> ( ( P @ F3 )
=> ( P @ ( add_ms5348986034529679365iset_a @ A4 @ F3 ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% multi_subset_induct
thf(fact_937_multi__subset__induct,axiom,
! [F2: multis501812127501805293od_a_a,A3: multis501812127501805293od_a_a,P: multis501812127501805293od_a_a > $o] :
( ( subset4108007524207251342od_a_a @ F2 @ A3 )
=> ( ( P @ zero_z3024502163831676836od_a_a )
=> ( ! [A4: product_prod_a_a,F3: multis501812127501805293od_a_a] :
( ( member1426531477525435216od_a_a @ A4 @ ( set_ms119794563918438244od_a_a @ A3 ) )
=> ( ( P @ F3 )
=> ( P @ ( add_ms8655138167283798533od_a_a @ A4 @ F3 ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% multi_subset_induct
thf(fact_938_multi__subset__induct,axiom,
! [F2: multiset_a,A3: multiset_a,P: multiset_a > $o] :
( ( subseteq_mset_a @ F2 @ A3 )
=> ( ( P @ zero_zero_multiset_a )
=> ( ! [A4: a,F3: multiset_a] :
( ( member_a @ A4 @ ( set_mset_a @ A3 ) )
=> ( ( P @ F3 )
=> ( P @ ( add_mset_a @ A4 @ F3 ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% multi_subset_induct
thf(fact_939_not__less__empty,axiom,
! [M: multiset_a,R: set_Product_prod_a_a] :
~ ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ M @ zero_zero_multiset_a ) @ ( mult1_a @ R ) ) ).
% not_less_empty
thf(fact_940_mult1__union,axiom,
! [B3: multiset_a,D2: multiset_a,R: set_Product_prod_a_a,C3: multiset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ B3 @ D2 ) @ ( mult1_a @ R ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( plus_plus_multiset_a @ C3 @ B3 ) @ ( plus_plus_multiset_a @ C3 @ D2 ) ) @ ( mult1_a @ R ) ) ) ).
% mult1_union
thf(fact_941_subset__mset_Osum__mset__mono,axiom,
! [K2: multiset_a,F: a > multiset_a,G: a > multiset_a] :
( ! [I4: a] :
( ( member_a @ I4 @ ( set_mset_a @ K2 ) )
=> ( subseteq_mset_a @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( subseteq_mset_a @ ( comm_m1977238983017651125iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ ( image_2204096306494911105iset_a @ F @ K2 ) ) @ ( comm_m1977238983017651125iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ ( image_2204096306494911105iset_a @ G @ K2 ) ) ) ) ).
% subset_mset.sum_mset_mono
thf(fact_942_subset__mset_Osum__mset__mono,axiom,
! [K2: multis4781622204924204013iset_a,F: produc6518373309651786023iset_a > multiset_a,G: produc6518373309651786023iset_a > multiset_a] :
( ! [I4: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ I4 @ ( set_ms2379723421352415844iset_a @ K2 ) )
=> ( subseteq_mset_a @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( subseteq_mset_a @ ( comm_m1977238983017651125iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ ( image_5610718949088693016iset_a @ F @ K2 ) ) @ ( comm_m1977238983017651125iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ ( image_5610718949088693016iset_a @ G @ K2 ) ) ) ) ).
% subset_mset.sum_mset_mono
thf(fact_943_subset__mset_Osum__mset__mono,axiom,
! [K2: multis501812127501805293od_a_a,F: product_prod_a_a > multiset_a,G: product_prod_a_a > multiset_a] :
( ! [I4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ I4 @ ( set_ms119794563918438244od_a_a @ K2 ) )
=> ( subseteq_mset_a @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( subseteq_mset_a @ ( comm_m1977238983017651125iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ ( image_119670572209922840iset_a @ F @ K2 ) ) @ ( comm_m1977238983017651125iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ ( image_119670572209922840iset_a @ G @ K2 ) ) ) ) ).
% subset_mset.sum_mset_mono
thf(fact_944_subset__mset_Omin__bot2,axiom,
! [X3: multiset_a] :
( ( min_multiset_a @ subseteq_mset_a @ X3 @ zero_zero_multiset_a )
= zero_zero_multiset_a ) ).
% subset_mset.min_bot2
thf(fact_945_subset__mset_Omax__bot2,axiom,
! [X3: multiset_a] :
( ( max_multiset_a @ subseteq_mset_a @ X3 @ zero_zero_multiset_a )
= X3 ) ).
% subset_mset.max_bot2
thf(fact_946_subset__mset_Omin__bot,axiom,
! [X3: multiset_a] :
( ( min_multiset_a @ subseteq_mset_a @ zero_zero_multiset_a @ X3 )
= zero_zero_multiset_a ) ).
% subset_mset.min_bot
thf(fact_947_subset__mset_Omax__bot,axiom,
! [X3: multiset_a] :
( ( max_multiset_a @ subseteq_mset_a @ zero_zero_multiset_a @ X3 )
= X3 ) ).
% subset_mset.max_bot
thf(fact_948_image__mset__empty,axiom,
! [F: a > a] :
( ( image_mset_a_a @ F @ zero_zero_multiset_a )
= zero_zero_multiset_a ) ).
% image_mset_empty
thf(fact_949_image__mset__is__empty__iff,axiom,
! [F: a > a,M: multiset_a] :
( ( ( image_mset_a_a @ F @ M )
= zero_zero_multiset_a )
= ( M = zero_zero_multiset_a ) ) ).
% image_mset_is_empty_iff
thf(fact_950_image__mset__union,axiom,
! [F: a > a,M: multiset_a,N: multiset_a] :
( ( image_mset_a_a @ F @ ( plus_plus_multiset_a @ M @ N ) )
= ( plus_plus_multiset_a @ ( image_mset_a_a @ F @ M ) @ ( image_mset_a_a @ F @ N ) ) ) ).
% image_mset_union
thf(fact_951_image__mset__eq__plusD,axiom,
! [F: a > a,A3: multiset_a,B3: multiset_a,C3: multiset_a] :
( ( ( image_mset_a_a @ F @ A3 )
= ( plus_plus_multiset_a @ B3 @ C3 ) )
=> ? [B9: multiset_a,C6: multiset_a] :
( ( A3
= ( plus_plus_multiset_a @ B9 @ C6 ) )
& ( B3
= ( image_mset_a_a @ F @ B9 ) )
& ( C3
= ( image_mset_a_a @ F @ C6 ) ) ) ) ).
% image_mset_eq_plusD
thf(fact_952_multiset_Omap__ident__strong,axiom,
! [T: multiset_a,F: a > a] :
( ! [Z2: a] :
( ( member_a @ Z2 @ ( set_mset_a @ T ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( image_mset_a_a @ F @ T )
= T ) ) ).
% multiset.map_ident_strong
thf(fact_953_multiset_Omap__ident__strong,axiom,
! [T: multis4781622204924204013iset_a,F: produc6518373309651786023iset_a > produc6518373309651786023iset_a] :
( ! [Z2: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Z2 @ ( set_ms2379723421352415844iset_a @ T ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( image_1581608793515940347iset_a @ F @ T )
= T ) ) ).
% multiset.map_ident_strong
thf(fact_954_multiset_Omap__ident__strong,axiom,
! [T: multis501812127501805293od_a_a,F: product_prod_a_a > product_prod_a_a] :
( ! [Z2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Z2 @ ( set_ms119794563918438244od_a_a @ T ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( image_1827833358229211643od_a_a @ F @ T )
= T ) ) ).
% multiset.map_ident_strong
thf(fact_955_comp__fun__commute__plus__mset,axiom,
finite6687201363011020241iset_a @ plus_plus_multiset_a ).
% comp_fun_commute_plus_mset
thf(fact_956_image__mset__single,axiom,
! [F: a > a,X3: a] :
( ( image_mset_a_a @ F @ ( add_mset_a @ X3 @ zero_zero_multiset_a ) )
= ( add_mset_a @ ( F @ X3 ) @ zero_zero_multiset_a ) ) ).
% image_mset_single
thf(fact_957_image__mset__Diff,axiom,
! [B3: multiset_a,A3: multiset_a,F: a > a] :
( ( subseteq_mset_a @ B3 @ A3 )
=> ( ( image_mset_a_a @ F @ ( minus_3765977307040488491iset_a @ A3 @ B3 ) )
= ( minus_3765977307040488491iset_a @ ( image_mset_a_a @ F @ A3 ) @ ( image_mset_a_a @ F @ B3 ) ) ) ) ).
% image_mset_Diff
thf(fact_958_image__mset__Diff,axiom,
! [B3: multiset_a,A3: multiset_a,F: a > produc6518373309651786023iset_a] :
( ( subseteq_mset_a @ B3 @ A3 )
=> ( ( image_8645235558099244516iset_a @ F @ ( minus_3765977307040488491iset_a @ A3 @ B3 ) )
= ( minus_9211619285413683668iset_a @ ( image_8645235558099244516iset_a @ F @ A3 ) @ ( image_8645235558099244516iset_a @ F @ B3 ) ) ) ) ).
% image_mset_Diff
thf(fact_959_image__mset__Diff,axiom,
! [B3: multiset_a,A3: multiset_a,F: a > product_prod_a_a] :
( ( subseteq_mset_a @ B3 @ A3 )
=> ( ( image_7325134204146066148od_a_a @ F @ ( minus_3765977307040488491iset_a @ A3 @ B3 ) )
= ( minus_7313152569058643924od_a_a @ ( image_7325134204146066148od_a_a @ F @ A3 ) @ ( image_7325134204146066148od_a_a @ F @ B3 ) ) ) ) ).
% image_mset_Diff
thf(fact_960_image__mset__Diff,axiom,
! [B3: multis4781622204924204013iset_a,A3: multis4781622204924204013iset_a,F: produc6518373309651786023iset_a > a] :
( ( subset6292090195676340366iset_a @ B3 @ A3 )
=> ( ( image_2834619569719286738et_a_a @ F @ ( minus_9211619285413683668iset_a @ A3 @ B3 ) )
= ( minus_3765977307040488491iset_a @ ( image_2834619569719286738et_a_a @ F @ A3 ) @ ( image_2834619569719286738et_a_a @ F @ B3 ) ) ) ) ).
% image_mset_Diff
thf(fact_961_image__mset__Diff,axiom,
! [B3: multis4781622204924204013iset_a,A3: multis4781622204924204013iset_a,F: produc6518373309651786023iset_a > produc6518373309651786023iset_a] :
( ( subset6292090195676340366iset_a @ B3 @ A3 )
=> ( ( image_1581608793515940347iset_a @ F @ ( minus_9211619285413683668iset_a @ A3 @ B3 ) )
= ( minus_9211619285413683668iset_a @ ( image_1581608793515940347iset_a @ F @ A3 ) @ ( image_1581608793515940347iset_a @ F @ B3 ) ) ) ) ).
% image_mset_Diff
thf(fact_962_image__mset__Diff,axiom,
! [B3: multis4781622204924204013iset_a,A3: multis4781622204924204013iset_a,F: produc6518373309651786023iset_a > product_prod_a_a] :
( ( subset6292090195676340366iset_a @ B3 @ A3 )
=> ( ( image_507504510031147515od_a_a @ F @ ( minus_9211619285413683668iset_a @ A3 @ B3 ) )
= ( minus_7313152569058643924od_a_a @ ( image_507504510031147515od_a_a @ F @ A3 ) @ ( image_507504510031147515od_a_a @ F @ B3 ) ) ) ) ).
% image_mset_Diff
thf(fact_963_image__mset__Diff,axiom,
! [B3: multis501812127501805293od_a_a,A3: multis501812127501805293od_a_a,F: product_prod_a_a > a] :
( ( subset4108007524207251342od_a_a @ B3 @ A3 )
=> ( ( image_3362453674455528402_a_a_a @ F @ ( minus_7313152569058643924od_a_a @ A3 @ B3 ) )
= ( minus_3765977307040488491iset_a @ ( image_3362453674455528402_a_a_a @ F @ A3 ) @ ( image_3362453674455528402_a_a_a @ F @ B3 ) ) ) ) ).
% image_mset_Diff
thf(fact_964_image__mset__Diff,axiom,
! [B3: multis501812127501805293od_a_a,A3: multis501812127501805293od_a_a,F: product_prod_a_a > produc6518373309651786023iset_a] :
( ( subset4108007524207251342od_a_a @ B3 @ A3 )
=> ( ( image_1544170008705937915iset_a @ F @ ( minus_7313152569058643924od_a_a @ A3 @ B3 ) )
= ( minus_9211619285413683668iset_a @ ( image_1544170008705937915iset_a @ F @ A3 ) @ ( image_1544170008705937915iset_a @ F @ B3 ) ) ) ) ).
% image_mset_Diff
thf(fact_965_image__mset__Diff,axiom,
! [B3: multis501812127501805293od_a_a,A3: multis501812127501805293od_a_a,F: product_prod_a_a > product_prod_a_a] :
( ( subset4108007524207251342od_a_a @ B3 @ A3 )
=> ( ( image_1827833358229211643od_a_a @ F @ ( minus_7313152569058643924od_a_a @ A3 @ B3 ) )
= ( minus_7313152569058643924od_a_a @ ( image_1827833358229211643od_a_a @ F @ A3 ) @ ( image_1827833358229211643od_a_a @ F @ B3 ) ) ) ) ).
% image_mset_Diff
thf(fact_966_subset__mset_Omax__add__distrib__left,axiom,
! [X3: multiset_a,Y: multiset_a,Z: multiset_a] :
( ( plus_plus_multiset_a @ ( max_multiset_a @ subseteq_mset_a @ X3 @ Y ) @ Z )
= ( max_multiset_a @ subseteq_mset_a @ ( plus_plus_multiset_a @ X3 @ Z ) @ ( plus_plus_multiset_a @ Y @ Z ) ) ) ).
% subset_mset.max_add_distrib_left
thf(fact_967_subset__mset_Omin__add__distrib__left,axiom,
! [X3: multiset_a,Y: multiset_a,Z: multiset_a] :
( ( plus_plus_multiset_a @ ( min_multiset_a @ subseteq_mset_a @ X3 @ Y ) @ Z )
= ( min_multiset_a @ subseteq_mset_a @ ( plus_plus_multiset_a @ X3 @ Z ) @ ( plus_plus_multiset_a @ Y @ Z ) ) ) ).
% subset_mset.min_add_distrib_left
thf(fact_968_subset__mset_Omax__add__distrib__right,axiom,
! [X3: multiset_a,Y: multiset_a,Z: multiset_a] :
( ( plus_plus_multiset_a @ X3 @ ( max_multiset_a @ subseteq_mset_a @ Y @ Z ) )
= ( max_multiset_a @ subseteq_mset_a @ ( plus_plus_multiset_a @ X3 @ Y ) @ ( plus_plus_multiset_a @ X3 @ Z ) ) ) ).
% subset_mset.max_add_distrib_right
thf(fact_969_subset__mset_Omin__add__distrib__right,axiom,
! [X3: multiset_a,Y: multiset_a,Z: multiset_a] :
( ( plus_plus_multiset_a @ X3 @ ( min_multiset_a @ subseteq_mset_a @ Y @ Z ) )
= ( min_multiset_a @ subseteq_mset_a @ ( plus_plus_multiset_a @ X3 @ Y ) @ ( plus_plus_multiset_a @ X3 @ Z ) ) ) ).
% subset_mset.min_add_distrib_right
thf(fact_970_sum__mset_Ounion__disjoint,axiom,
! [A3: multiset_a,B3: multiset_a,G: a > multiset_a] :
( ( ( inter_mset_a @ A3 @ B3 )
= zero_zero_multiset_a )
=> ( ( comm_m4595494212910443721iset_a @ ( image_2204096306494911105iset_a @ G @ ( union_mset_a @ A3 @ B3 ) ) )
= ( plus_plus_multiset_a @ ( comm_m4595494212910443721iset_a @ ( image_2204096306494911105iset_a @ G @ A3 ) ) @ ( comm_m4595494212910443721iset_a @ ( image_2204096306494911105iset_a @ G @ B3 ) ) ) ) ) ).
% sum_mset.union_disjoint
thf(fact_971_subset__mset_Osup__eq__bot__iff,axiom,
! [X3: multiset_a,Y: multiset_a] :
( ( ( union_mset_a @ X3 @ Y )
= zero_zero_multiset_a )
= ( ( X3 = zero_zero_multiset_a )
& ( Y = zero_zero_multiset_a ) ) ) ).
% subset_mset.sup_eq_bot_iff
thf(fact_972_subset__mset_Obot__eq__sup__iff,axiom,
! [X3: multiset_a,Y: multiset_a] :
( ( zero_zero_multiset_a
= ( union_mset_a @ X3 @ Y ) )
= ( ( X3 = zero_zero_multiset_a )
& ( Y = zero_zero_multiset_a ) ) ) ).
% subset_mset.bot_eq_sup_iff
thf(fact_973_subset__mset_Osup__bot__right,axiom,
! [X3: multiset_a] :
( ( union_mset_a @ X3 @ zero_zero_multiset_a )
= X3 ) ).
% subset_mset.sup_bot_right
thf(fact_974_subset__mset_Oright__neutral,axiom,
! [A: multiset_a] :
( ( union_mset_a @ A @ zero_zero_multiset_a )
= A ) ).
% subset_mset.right_neutral
thf(fact_975_subset__mset_Osup__bot__left,axiom,
! [X3: multiset_a] :
( ( union_mset_a @ zero_zero_multiset_a @ X3 )
= X3 ) ).
% subset_mset.sup_bot_left
thf(fact_976_subset__mset_Oneutr__eq__iff,axiom,
! [A: multiset_a,B: multiset_a] :
( ( zero_zero_multiset_a
= ( union_mset_a @ A @ B ) )
= ( ( A = zero_zero_multiset_a )
& ( B = zero_zero_multiset_a ) ) ) ).
% subset_mset.neutr_eq_iff
thf(fact_977_subset__mset_Oleft__neutral,axiom,
! [A: multiset_a] :
( ( union_mset_a @ zero_zero_multiset_a @ A )
= A ) ).
% subset_mset.left_neutral
thf(fact_978_subset__mset_Oeq__neutr__iff,axiom,
! [A: multiset_a,B: multiset_a] :
( ( ( union_mset_a @ A @ B )
= zero_zero_multiset_a )
= ( ( A = zero_zero_multiset_a )
& ( B = zero_zero_multiset_a ) ) ) ).
% subset_mset.eq_neutr_iff
thf(fact_979_subset__mset_Oinf__sup__absorb,axiom,
! [X3: multiset_a,Y: multiset_a] :
( ( inter_mset_a @ X3 @ ( union_mset_a @ X3 @ Y ) )
= X3 ) ).
% subset_mset.inf_sup_absorb
thf(fact_980_subset__mset_Osup__inf__absorb,axiom,
! [X3: multiset_a,Y: multiset_a] :
( ( union_mset_a @ X3 @ ( inter_mset_a @ X3 @ Y ) )
= X3 ) ).
% subset_mset.sup_inf_absorb
thf(fact_981_sup__union__right1,axiom,
! [X3: a,N: multiset_a,M: multiset_a] :
( ~ ( member_a @ X3 @ ( set_mset_a @ N ) )
=> ( ( union_mset_a @ N @ ( add_mset_a @ X3 @ M ) )
= ( add_mset_a @ X3 @ ( union_mset_a @ N @ M ) ) ) ) ).
% sup_union_right1
thf(fact_982_sup__union__right1,axiom,
! [X3: produc6518373309651786023iset_a,N: multis4781622204924204013iset_a,M: multis4781622204924204013iset_a] :
( ~ ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ N ) )
=> ( ( union_4549860014028270007iset_a @ N @ ( add_ms5348986034529679365iset_a @ X3 @ M ) )
= ( add_ms5348986034529679365iset_a @ X3 @ ( union_4549860014028270007iset_a @ N @ M ) ) ) ) ).
% sup_union_right1
thf(fact_983_sup__union__right1,axiom,
! [X3: product_prod_a_a,N: multis501812127501805293od_a_a,M: multis501812127501805293od_a_a] :
( ~ ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ N ) )
=> ( ( union_8952997168100052407od_a_a @ N @ ( add_ms8655138167283798533od_a_a @ X3 @ M ) )
= ( add_ms8655138167283798533od_a_a @ X3 @ ( union_8952997168100052407od_a_a @ N @ M ) ) ) ) ).
% sup_union_right1
thf(fact_984_sup__union__left1,axiom,
! [X3: a,N: multiset_a,M: multiset_a] :
( ~ ( member_a @ X3 @ ( set_mset_a @ N ) )
=> ( ( union_mset_a @ ( add_mset_a @ X3 @ M ) @ N )
= ( add_mset_a @ X3 @ ( union_mset_a @ M @ N ) ) ) ) ).
% sup_union_left1
thf(fact_985_sup__union__left1,axiom,
! [X3: produc6518373309651786023iset_a,N: multis4781622204924204013iset_a,M: multis4781622204924204013iset_a] :
( ~ ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ N ) )
=> ( ( union_4549860014028270007iset_a @ ( add_ms5348986034529679365iset_a @ X3 @ M ) @ N )
= ( add_ms5348986034529679365iset_a @ X3 @ ( union_4549860014028270007iset_a @ M @ N ) ) ) ) ).
% sup_union_left1
thf(fact_986_sup__union__left1,axiom,
! [X3: product_prod_a_a,N: multis501812127501805293od_a_a,M: multis501812127501805293od_a_a] :
( ~ ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ N ) )
=> ( ( union_8952997168100052407od_a_a @ ( add_ms8655138167283798533od_a_a @ X3 @ M ) @ N )
= ( add_ms8655138167283798533od_a_a @ X3 @ ( union_8952997168100052407od_a_a @ M @ N ) ) ) ) ).
% sup_union_left1
thf(fact_987_sup__union__distrib__left,axiom,
! [A3: multiset_a,B3: multiset_a,C3: multiset_a] :
( ( plus_plus_multiset_a @ ( union_mset_a @ A3 @ B3 ) @ C3 )
= ( union_mset_a @ ( plus_plus_multiset_a @ A3 @ C3 ) @ ( plus_plus_multiset_a @ B3 @ C3 ) ) ) ).
% sup_union_distrib_left
thf(fact_988_union__sup__distrib__right,axiom,
! [C3: multiset_a,A3: multiset_a,B3: multiset_a] :
( ( plus_plus_multiset_a @ C3 @ ( union_mset_a @ A3 @ B3 ) )
= ( union_mset_a @ ( plus_plus_multiset_a @ C3 @ A3 ) @ ( plus_plus_multiset_a @ C3 @ B3 ) ) ) ).
% union_sup_distrib_right
thf(fact_989_subset__mset_Osup__inf__distrib2,axiom,
! [Y: multiset_a,Z: multiset_a,X3: multiset_a] :
( ( union_mset_a @ ( inter_mset_a @ Y @ Z ) @ X3 )
= ( inter_mset_a @ ( union_mset_a @ Y @ X3 ) @ ( union_mset_a @ Z @ X3 ) ) ) ).
% subset_mset.sup_inf_distrib2
thf(fact_990_subset__mset_Osup__inf__distrib1,axiom,
! [X3: multiset_a,Y: multiset_a,Z: multiset_a] :
( ( union_mset_a @ X3 @ ( inter_mset_a @ Y @ Z ) )
= ( inter_mset_a @ ( union_mset_a @ X3 @ Y ) @ ( union_mset_a @ X3 @ Z ) ) ) ).
% subset_mset.sup_inf_distrib1
thf(fact_991_subset__mset_Oinf__sup__distrib2,axiom,
! [Y: multiset_a,Z: multiset_a,X3: multiset_a] :
( ( inter_mset_a @ ( union_mset_a @ Y @ Z ) @ X3 )
= ( union_mset_a @ ( inter_mset_a @ Y @ X3 ) @ ( inter_mset_a @ Z @ X3 ) ) ) ).
% subset_mset.inf_sup_distrib2
thf(fact_992_subset__mset_Oinf__sup__distrib1,axiom,
! [X3: multiset_a,Y: multiset_a,Z: multiset_a] :
( ( inter_mset_a @ X3 @ ( union_mset_a @ Y @ Z ) )
= ( union_mset_a @ ( inter_mset_a @ X3 @ Y ) @ ( inter_mset_a @ X3 @ Z ) ) ) ).
% subset_mset.inf_sup_distrib1
thf(fact_993_subset__mset_Odistrib__imp2,axiom,
! [X3: multiset_a,Y: multiset_a,Z: multiset_a] :
( ! [X5: multiset_a,Y4: multiset_a,Z2: multiset_a] :
( ( union_mset_a @ X5 @ ( inter_mset_a @ Y4 @ Z2 ) )
= ( inter_mset_a @ ( union_mset_a @ X5 @ Y4 ) @ ( union_mset_a @ X5 @ Z2 ) ) )
=> ( ( inter_mset_a @ X3 @ ( union_mset_a @ Y @ Z ) )
= ( union_mset_a @ ( inter_mset_a @ X3 @ Y ) @ ( inter_mset_a @ X3 @ Z ) ) ) ) ).
% subset_mset.distrib_imp2
thf(fact_994_subset__mset_Odistrib__imp1,axiom,
! [X3: multiset_a,Y: multiset_a,Z: multiset_a] :
( ! [X5: multiset_a,Y4: multiset_a,Z2: multiset_a] :
( ( inter_mset_a @ X5 @ ( union_mset_a @ Y4 @ Z2 ) )
= ( union_mset_a @ ( inter_mset_a @ X5 @ Y4 ) @ ( inter_mset_a @ X5 @ Z2 ) ) )
=> ( ( union_mset_a @ X3 @ ( inter_mset_a @ Y @ Z ) )
= ( inter_mset_a @ ( union_mset_a @ X3 @ Y ) @ ( union_mset_a @ X3 @ Z ) ) ) ) ).
% subset_mset.distrib_imp1
thf(fact_995_subset__mset_Odistrib__sup__le,axiom,
! [X3: multiset_a,Y: multiset_a,Z: multiset_a] : ( subseteq_mset_a @ ( union_mset_a @ X3 @ ( inter_mset_a @ Y @ Z ) ) @ ( inter_mset_a @ ( union_mset_a @ X3 @ Y ) @ ( union_mset_a @ X3 @ Z ) ) ) ).
% subset_mset.distrib_sup_le
thf(fact_996_subset__mset_Odistrib__inf__le,axiom,
! [X3: multiset_a,Y: multiset_a,Z: multiset_a] : ( subseteq_mset_a @ ( union_mset_a @ ( inter_mset_a @ X3 @ Y ) @ ( inter_mset_a @ X3 @ Z ) ) @ ( inter_mset_a @ X3 @ ( union_mset_a @ Y @ Z ) ) ) ).
% subset_mset.distrib_inf_le
thf(fact_997_union__mset__def,axiom,
( union_mset_a
= ( ^ [A6: multiset_a,B6: multiset_a] : ( plus_plus_multiset_a @ A6 @ ( minus_3765977307040488491iset_a @ B6 @ A6 ) ) ) ) ).
% union_mset_def
thf(fact_998_union__mset__def,axiom,
( union_4549860014028270007iset_a
= ( ^ [A6: multis4781622204924204013iset_a,B6: multis4781622204924204013iset_a] : ( plus_p1053901119268525860iset_a @ A6 @ ( minus_9211619285413683668iset_a @ B6 @ A6 ) ) ) ) ).
% union_mset_def
thf(fact_999_union__mset__def,axiom,
( union_8952997168100052407od_a_a
= ( ^ [A6: multis501812127501805293od_a_a,B6: multis501812127501805293od_a_a] : ( plus_p6967097483979887908od_a_a @ A6 @ ( minus_7313152569058643924od_a_a @ B6 @ A6 ) ) ) ) ).
% union_mset_def
thf(fact_1000_union__diff__sup__eq__inter,axiom,
! [A3: multiset_a,B3: multiset_a] :
( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ A3 @ B3 ) @ ( union_mset_a @ A3 @ B3 ) )
= ( inter_mset_a @ A3 @ B3 ) ) ).
% union_diff_sup_eq_inter
thf(fact_1001_union__diff__sup__eq__inter,axiom,
! [A3: multis4781622204924204013iset_a,B3: multis4781622204924204013iset_a] :
( ( minus_9211619285413683668iset_a @ ( plus_p1053901119268525860iset_a @ A3 @ B3 ) @ ( union_4549860014028270007iset_a @ A3 @ B3 ) )
= ( inter_1588963877939557482iset_a @ A3 @ B3 ) ) ).
% union_diff_sup_eq_inter
thf(fact_1002_union__diff__sup__eq__inter,axiom,
! [A3: multis501812127501805293od_a_a,B3: multis501812127501805293od_a_a] :
( ( minus_7313152569058643924od_a_a @ ( plus_p6967097483979887908od_a_a @ A3 @ B3 ) @ ( union_8952997168100052407od_a_a @ A3 @ B3 ) )
= ( inter_3651181809579396970od_a_a @ A3 @ B3 ) ) ).
% union_diff_sup_eq_inter
thf(fact_1003_union__diff__inter__eq__sup,axiom,
! [A3: multiset_a,B3: multiset_a] :
( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ A3 @ B3 ) @ ( inter_mset_a @ A3 @ B3 ) )
= ( union_mset_a @ A3 @ B3 ) ) ).
% union_diff_inter_eq_sup
thf(fact_1004_union__diff__inter__eq__sup,axiom,
! [A3: multis4781622204924204013iset_a,B3: multis4781622204924204013iset_a] :
( ( minus_9211619285413683668iset_a @ ( plus_p1053901119268525860iset_a @ A3 @ B3 ) @ ( inter_1588963877939557482iset_a @ A3 @ B3 ) )
= ( union_4549860014028270007iset_a @ A3 @ B3 ) ) ).
% union_diff_inter_eq_sup
thf(fact_1005_union__diff__inter__eq__sup,axiom,
! [A3: multis501812127501805293od_a_a,B3: multis501812127501805293od_a_a] :
( ( minus_7313152569058643924od_a_a @ ( plus_p6967097483979887908od_a_a @ A3 @ B3 ) @ ( inter_3651181809579396970od_a_a @ A3 @ B3 ) )
= ( union_8952997168100052407od_a_a @ A3 @ B3 ) ) ).
% union_diff_inter_eq_sup
thf(fact_1006_sup__union__left2,axiom,
! [X3: a,N: multiset_a,M: multiset_a] :
( ( member_a @ X3 @ ( set_mset_a @ N ) )
=> ( ( union_mset_a @ ( add_mset_a @ X3 @ M ) @ N )
= ( add_mset_a @ X3 @ ( union_mset_a @ M @ ( minus_3765977307040488491iset_a @ N @ ( add_mset_a @ X3 @ zero_zero_multiset_a ) ) ) ) ) ) ).
% sup_union_left2
thf(fact_1007_sup__union__left2,axiom,
! [X3: produc6518373309651786023iset_a,N: multis4781622204924204013iset_a,M: multis4781622204924204013iset_a] :
( ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ N ) )
=> ( ( union_4549860014028270007iset_a @ ( add_ms5348986034529679365iset_a @ X3 @ M ) @ N )
= ( add_ms5348986034529679365iset_a @ X3 @ ( union_4549860014028270007iset_a @ M @ ( minus_9211619285413683668iset_a @ N @ ( add_ms5348986034529679365iset_a @ X3 @ zero_z4591712717340762532iset_a ) ) ) ) ) ) ).
% sup_union_left2
thf(fact_1008_sup__union__left2,axiom,
! [X3: product_prod_a_a,N: multis501812127501805293od_a_a,M: multis501812127501805293od_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ N ) )
=> ( ( union_8952997168100052407od_a_a @ ( add_ms8655138167283798533od_a_a @ X3 @ M ) @ N )
= ( add_ms8655138167283798533od_a_a @ X3 @ ( union_8952997168100052407od_a_a @ M @ ( minus_7313152569058643924od_a_a @ N @ ( add_ms8655138167283798533od_a_a @ X3 @ zero_z3024502163831676836od_a_a ) ) ) ) ) ) ).
% sup_union_left2
thf(fact_1009_sup__union__right2,axiom,
! [X3: a,N: multiset_a,M: multiset_a] :
( ( member_a @ X3 @ ( set_mset_a @ N ) )
=> ( ( union_mset_a @ N @ ( add_mset_a @ X3 @ M ) )
= ( add_mset_a @ X3 @ ( union_mset_a @ ( minus_3765977307040488491iset_a @ N @ ( add_mset_a @ X3 @ zero_zero_multiset_a ) ) @ M ) ) ) ) ).
% sup_union_right2
thf(fact_1010_sup__union__right2,axiom,
! [X3: produc6518373309651786023iset_a,N: multis4781622204924204013iset_a,M: multis4781622204924204013iset_a] :
( ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ N ) )
=> ( ( union_4549860014028270007iset_a @ N @ ( add_ms5348986034529679365iset_a @ X3 @ M ) )
= ( add_ms5348986034529679365iset_a @ X3 @ ( union_4549860014028270007iset_a @ ( minus_9211619285413683668iset_a @ N @ ( add_ms5348986034529679365iset_a @ X3 @ zero_z4591712717340762532iset_a ) ) @ M ) ) ) ) ).
% sup_union_right2
thf(fact_1011_sup__union__right2,axiom,
! [X3: product_prod_a_a,N: multis501812127501805293od_a_a,M: multis501812127501805293od_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ N ) )
=> ( ( union_8952997168100052407od_a_a @ N @ ( add_ms8655138167283798533od_a_a @ X3 @ M ) )
= ( add_ms8655138167283798533od_a_a @ X3 @ ( union_8952997168100052407od_a_a @ ( minus_7313152569058643924od_a_a @ N @ ( add_ms8655138167283798533od_a_a @ X3 @ zero_z3024502163831676836od_a_a ) ) @ M ) ) ) ) ).
% sup_union_right2
thf(fact_1012_subset__mset_OInf__fin_Osemilattice__order__set__axioms,axiom,
lattic1827867734330664222iset_a @ inter_mset_a @ subseteq_mset_a @ subset_mset_a ).
% subset_mset.Inf_fin.semilattice_order_set_axioms
thf(fact_1013_subset__mset_Osum__nonpos,axiom,
! [A3: set_Product_prod_a_a,F: product_prod_a_a > multiset_a] :
( ! [X5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ A3 )
=> ( subseteq_mset_a @ ( F @ X5 ) @ zero_zero_multiset_a ) )
=> ( subseteq_mset_a @ ( groups1455199411266135047od_a_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ F @ A3 ) @ zero_zero_multiset_a ) ) ).
% subset_mset.sum_nonpos
thf(fact_1014_subset__mset_Osum__nonpos,axiom,
! [A3: set_Pr79727621955416071iset_a,F: produc6518373309651786023iset_a > multiset_a] :
( ! [X5: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X5 @ A3 )
=> ( subseteq_mset_a @ ( F @ X5 ) @ zero_zero_multiset_a ) )
=> ( subseteq_mset_a @ ( groups6473759573426450439iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ F @ A3 ) @ zero_zero_multiset_a ) ) ).
% subset_mset.sum_nonpos
thf(fact_1015_subset__mset_Osum__nonpos,axiom,
! [A3: set_a,F: a > multiset_a] :
( ! [X5: a] :
( ( member_a @ X5 @ A3 )
=> ( subseteq_mset_a @ ( F @ X5 ) @ zero_zero_multiset_a ) )
=> ( subseteq_mset_a @ ( groups409869659785034206et_a_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ F @ A3 ) @ zero_zero_multiset_a ) ) ).
% subset_mset.sum_nonpos
thf(fact_1016_subset__mset_Osum__nonneg,axiom,
! [A3: set_Product_prod_a_a,F: product_prod_a_a > multiset_a] :
( ! [X5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ A3 )
=> ( subseteq_mset_a @ zero_zero_multiset_a @ ( F @ X5 ) ) )
=> ( subseteq_mset_a @ zero_zero_multiset_a @ ( groups1455199411266135047od_a_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ F @ A3 ) ) ) ).
% subset_mset.sum_nonneg
thf(fact_1017_subset__mset_Osum__nonneg,axiom,
! [A3: set_Pr79727621955416071iset_a,F: produc6518373309651786023iset_a > multiset_a] :
( ! [X5: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X5 @ A3 )
=> ( subseteq_mset_a @ zero_zero_multiset_a @ ( F @ X5 ) ) )
=> ( subseteq_mset_a @ zero_zero_multiset_a @ ( groups6473759573426450439iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ F @ A3 ) ) ) ).
% subset_mset.sum_nonneg
thf(fact_1018_subset__mset_Osum__nonneg,axiom,
! [A3: set_a,F: a > multiset_a] :
( ! [X5: a] :
( ( member_a @ X5 @ A3 )
=> ( subseteq_mset_a @ zero_zero_multiset_a @ ( F @ X5 ) ) )
=> ( subseteq_mset_a @ zero_zero_multiset_a @ ( groups409869659785034206et_a_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ F @ A3 ) ) ) ).
% subset_mset.sum_nonneg
thf(fact_1019_subset__mset_Osum__pos2,axiom,
! [I2: set_Product_prod_a_a,I3: product_prod_a_a,F: product_prod_a_a > multiset_a] :
( ( finite6544458595007987280od_a_a @ I2 )
=> ( ( member1426531477525435216od_a_a @ I3 @ I2 )
=> ( ( subset_mset_a @ zero_zero_multiset_a @ ( F @ I3 ) )
=> ( ! [I4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ I4 @ I2 )
=> ( subseteq_mset_a @ zero_zero_multiset_a @ ( F @ I4 ) ) )
=> ( subset_mset_a @ zero_zero_multiset_a @ ( groups1455199411266135047od_a_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ F @ I2 ) ) ) ) ) ) ).
% subset_mset.sum_pos2
thf(fact_1020_subset__mset_Osum__pos2,axiom,
! [I2: set_Pr79727621955416071iset_a,I3: produc6518373309651786023iset_a,F: produc6518373309651786023iset_a > multiset_a] :
( ( finite1099724235734660432iset_a @ I2 )
=> ( ( member5199237121806060112iset_a @ I3 @ I2 )
=> ( ( subset_mset_a @ zero_zero_multiset_a @ ( F @ I3 ) )
=> ( ! [I4: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ I4 @ I2 )
=> ( subseteq_mset_a @ zero_zero_multiset_a @ ( F @ I4 ) ) )
=> ( subset_mset_a @ zero_zero_multiset_a @ ( groups6473759573426450439iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ F @ I2 ) ) ) ) ) ) ).
% subset_mset.sum_pos2
thf(fact_1021_subset__mset_Osum__pos2,axiom,
! [I2: set_a,I3: a,F: a > multiset_a] :
( ( finite_finite_a @ I2 )
=> ( ( member_a @ I3 @ I2 )
=> ( ( subset_mset_a @ zero_zero_multiset_a @ ( F @ I3 ) )
=> ( ! [I4: a] :
( ( member_a @ I4 @ I2 )
=> ( subseteq_mset_a @ zero_zero_multiset_a @ ( F @ I4 ) ) )
=> ( subset_mset_a @ zero_zero_multiset_a @ ( groups409869659785034206et_a_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ F @ I2 ) ) ) ) ) ) ).
% subset_mset.sum_pos2
thf(fact_1022_finite__set__mset,axiom,
! [M: multiset_a] : ( finite_finite_a @ ( set_mset_a @ M ) ) ).
% finite_set_mset
thf(fact_1023_finite__set__mset,axiom,
! [M: multis4781622204924204013iset_a] : ( finite1099724235734660432iset_a @ ( set_ms2379723421352415844iset_a @ M ) ) ).
% finite_set_mset
thf(fact_1024_finite__set__mset,axiom,
! [M: multis501812127501805293od_a_a] : ( finite6544458595007987280od_a_a @ ( set_ms119794563918438244od_a_a @ M ) ) ).
% finite_set_mset
thf(fact_1025_set__mset__empty,axiom,
( ( set_ms2379723421352415844iset_a @ zero_z4591712717340762532iset_a )
= bot_bo5898781664966027891iset_a ) ).
% set_mset_empty
thf(fact_1026_set__mset__empty,axiom,
( ( set_ms119794563918438244od_a_a @ zero_z3024502163831676836od_a_a )
= bot_bo3357376287454694259od_a_a ) ).
% set_mset_empty
thf(fact_1027_set__mset__empty,axiom,
( ( set_mset_a @ zero_zero_multiset_a )
= bot_bot_set_a ) ).
% set_mset_empty
thf(fact_1028_set__mset__eq__empty__iff,axiom,
! [M: multis4781622204924204013iset_a] :
( ( ( set_ms2379723421352415844iset_a @ M )
= bot_bo5898781664966027891iset_a )
= ( M = zero_z4591712717340762532iset_a ) ) ).
% set_mset_eq_empty_iff
thf(fact_1029_set__mset__eq__empty__iff,axiom,
! [M: multis501812127501805293od_a_a] :
( ( ( set_ms119794563918438244od_a_a @ M )
= bot_bo3357376287454694259od_a_a )
= ( M = zero_z3024502163831676836od_a_a ) ) ).
% set_mset_eq_empty_iff
thf(fact_1030_set__mset__eq__empty__iff,axiom,
! [M: multiset_a] :
( ( ( set_mset_a @ M )
= bot_bot_set_a )
= ( M = zero_zero_multiset_a ) ) ).
% set_mset_eq_empty_iff
thf(fact_1031_subset__mset_Osum__pos,axiom,
! [I2: set_Product_prod_a_a,F: product_prod_a_a > multiset_a] :
( ( finite6544458595007987280od_a_a @ I2 )
=> ( ( I2 != bot_bo3357376287454694259od_a_a )
=> ( ! [I4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ I4 @ I2 )
=> ( subset_mset_a @ zero_zero_multiset_a @ ( F @ I4 ) ) )
=> ( subset_mset_a @ zero_zero_multiset_a @ ( groups1455199411266135047od_a_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ F @ I2 ) ) ) ) ) ).
% subset_mset.sum_pos
thf(fact_1032_subset__mset_Osum__pos,axiom,
! [I2: set_Pr79727621955416071iset_a,F: produc6518373309651786023iset_a > multiset_a] :
( ( finite1099724235734660432iset_a @ I2 )
=> ( ( I2 != bot_bo5898781664966027891iset_a )
=> ( ! [I4: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ I4 @ I2 )
=> ( subset_mset_a @ zero_zero_multiset_a @ ( F @ I4 ) ) )
=> ( subset_mset_a @ zero_zero_multiset_a @ ( groups6473759573426450439iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ F @ I2 ) ) ) ) ) ).
% subset_mset.sum_pos
thf(fact_1033_subset__mset_Osum__pos,axiom,
! [I2: set_a,F: a > multiset_a] :
( ( finite_finite_a @ I2 )
=> ( ( I2 != bot_bot_set_a )
=> ( ! [I4: a] :
( ( member_a @ I4 @ I2 )
=> ( subset_mset_a @ zero_zero_multiset_a @ ( F @ I4 ) ) )
=> ( subset_mset_a @ zero_zero_multiset_a @ ( groups409869659785034206et_a_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ F @ I2 ) ) ) ) ) ).
% subset_mset.sum_pos
thf(fact_1034_subset__mset_Osum__strict__mono,axiom,
! [A3: set_Product_prod_a_a,F: product_prod_a_a > multiset_a,G: product_prod_a_a > multiset_a] :
( ( finite6544458595007987280od_a_a @ A3 )
=> ( ( A3 != bot_bo3357376287454694259od_a_a )
=> ( ! [X5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ A3 )
=> ( subset_mset_a @ ( F @ X5 ) @ ( G @ X5 ) ) )
=> ( subset_mset_a @ ( groups1455199411266135047od_a_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ F @ A3 ) @ ( groups1455199411266135047od_a_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ G @ A3 ) ) ) ) ) ).
% subset_mset.sum_strict_mono
thf(fact_1035_subset__mset_Osum__strict__mono,axiom,
! [A3: set_Pr79727621955416071iset_a,F: produc6518373309651786023iset_a > multiset_a,G: produc6518373309651786023iset_a > multiset_a] :
( ( finite1099724235734660432iset_a @ A3 )
=> ( ( A3 != bot_bo5898781664966027891iset_a )
=> ( ! [X5: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X5 @ A3 )
=> ( subset_mset_a @ ( F @ X5 ) @ ( G @ X5 ) ) )
=> ( subset_mset_a @ ( groups6473759573426450439iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ F @ A3 ) @ ( groups6473759573426450439iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ G @ A3 ) ) ) ) ) ).
% subset_mset.sum_strict_mono
thf(fact_1036_subset__mset_Osum__strict__mono,axiom,
! [A3: set_a,F: a > multiset_a,G: a > multiset_a] :
( ( finite_finite_a @ A3 )
=> ( ( A3 != bot_bot_set_a )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A3 )
=> ( subset_mset_a @ ( F @ X5 ) @ ( G @ X5 ) ) )
=> ( subset_mset_a @ ( groups409869659785034206et_a_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ F @ A3 ) @ ( groups409869659785034206et_a_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ G @ A3 ) ) ) ) ) ).
% subset_mset.sum_strict_mono
thf(fact_1037_subset__mset_Osum__nonneg__eq__0__iff,axiom,
! [A3: set_Product_prod_a_a,F: product_prod_a_a > multiset_a] :
( ( finite6544458595007987280od_a_a @ A3 )
=> ( ! [X5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ A3 )
=> ( subseteq_mset_a @ zero_zero_multiset_a @ ( F @ X5 ) ) )
=> ( ( ( groups1455199411266135047od_a_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ F @ A3 )
= zero_zero_multiset_a )
= ( ! [X6: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X6 @ A3 )
=> ( ( F @ X6 )
= zero_zero_multiset_a ) ) ) ) ) ) ).
% subset_mset.sum_nonneg_eq_0_iff
thf(fact_1038_subset__mset_Osum__nonneg__eq__0__iff,axiom,
! [A3: set_Pr79727621955416071iset_a,F: produc6518373309651786023iset_a > multiset_a] :
( ( finite1099724235734660432iset_a @ A3 )
=> ( ! [X5: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X5 @ A3 )
=> ( subseteq_mset_a @ zero_zero_multiset_a @ ( F @ X5 ) ) )
=> ( ( ( groups6473759573426450439iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ F @ A3 )
= zero_zero_multiset_a )
= ( ! [X6: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X6 @ A3 )
=> ( ( F @ X6 )
= zero_zero_multiset_a ) ) ) ) ) ) ).
% subset_mset.sum_nonneg_eq_0_iff
thf(fact_1039_subset__mset_Osum__nonneg__eq__0__iff,axiom,
! [A3: set_a,F: a > multiset_a] :
( ( finite_finite_a @ A3 )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A3 )
=> ( subseteq_mset_a @ zero_zero_multiset_a @ ( F @ X5 ) ) )
=> ( ( ( groups409869659785034206et_a_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ F @ A3 )
= zero_zero_multiset_a )
= ( ! [X6: a] :
( ( member_a @ X6 @ A3 )
=> ( ( F @ X6 )
= zero_zero_multiset_a ) ) ) ) ) ) ).
% subset_mset.sum_nonneg_eq_0_iff
thf(fact_1040_subset__mset_Osum__mono2,axiom,
! [B3: set_Product_prod_a_a,A3: set_Product_prod_a_a,F: product_prod_a_a > multiset_a] :
( ( finite6544458595007987280od_a_a @ B3 )
=> ( ( ord_le746702958409616551od_a_a @ A3 @ B3 )
=> ( ! [B4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ B4 @ ( minus_6817036919807184750od_a_a @ B3 @ A3 ) )
=> ( subseteq_mset_a @ zero_zero_multiset_a @ ( F @ B4 ) ) )
=> ( subseteq_mset_a @ ( groups1455199411266135047od_a_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ F @ A3 ) @ ( groups1455199411266135047od_a_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ F @ B3 ) ) ) ) ) ).
% subset_mset.sum_mono2
thf(fact_1041_subset__mset_Osum__mono2,axiom,
! [B3: set_Pr79727621955416071iset_a,A3: set_Pr79727621955416071iset_a,F: produc6518373309651786023iset_a > multiset_a] :
( ( finite1099724235734660432iset_a @ B3 )
=> ( ( ord_le5871882978999109543iset_a @ A3 @ B3 )
=> ( ! [B4: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ B4 @ ( minus_2461365523287052142iset_a @ B3 @ A3 ) )
=> ( subseteq_mset_a @ zero_zero_multiset_a @ ( F @ B4 ) ) )
=> ( subseteq_mset_a @ ( groups6473759573426450439iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ F @ A3 ) @ ( groups6473759573426450439iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ F @ B3 ) ) ) ) ) ).
% subset_mset.sum_mono2
thf(fact_1042_subset__mset_Osum__mono2,axiom,
! [B3: set_a,A3: set_a,F: a > multiset_a] :
( ( finite_finite_a @ B3 )
=> ( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ! [B4: a] :
( ( member_a @ B4 @ ( minus_minus_set_a @ B3 @ A3 ) )
=> ( subseteq_mset_a @ zero_zero_multiset_a @ ( F @ B4 ) ) )
=> ( subseteq_mset_a @ ( groups409869659785034206et_a_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ F @ A3 ) @ ( groups409869659785034206et_a_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ F @ B3 ) ) ) ) ) ).
% subset_mset.sum_mono2
thf(fact_1043_at__most__one__mset__mset__diff,axiom,
! [A: a,M: multiset_a] :
( ~ ( member_a @ A @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) )
=> ( ( set_mset_a @ ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) )
= ( minus_minus_set_a @ ( set_mset_a @ M ) @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% at_most_one_mset_mset_diff
thf(fact_1044_at__most__one__mset__mset__diff,axiom,
! [A: produc6518373309651786023iset_a,M: multis4781622204924204013iset_a] :
( ~ ( member5199237121806060112iset_a @ A @ ( set_ms2379723421352415844iset_a @ ( minus_9211619285413683668iset_a @ M @ ( add_ms5348986034529679365iset_a @ A @ zero_z4591712717340762532iset_a ) ) ) )
=> ( ( set_ms2379723421352415844iset_a @ ( minus_9211619285413683668iset_a @ M @ ( add_ms5348986034529679365iset_a @ A @ zero_z4591712717340762532iset_a ) ) )
= ( minus_2461365523287052142iset_a @ ( set_ms2379723421352415844iset_a @ M ) @ ( insert4880209295958132471iset_a @ A @ bot_bo5898781664966027891iset_a ) ) ) ) ).
% at_most_one_mset_mset_diff
thf(fact_1045_at__most__one__mset__mset__diff,axiom,
! [A: product_prod_a_a,M: multis501812127501805293od_a_a] :
( ~ ( member1426531477525435216od_a_a @ A @ ( set_ms119794563918438244od_a_a @ ( minus_7313152569058643924od_a_a @ M @ ( add_ms8655138167283798533od_a_a @ A @ zero_z3024502163831676836od_a_a ) ) ) )
=> ( ( set_ms119794563918438244od_a_a @ ( minus_7313152569058643924od_a_a @ M @ ( add_ms8655138167283798533od_a_a @ A @ zero_z3024502163831676836od_a_a ) ) )
= ( minus_6817036919807184750od_a_a @ ( set_ms119794563918438244od_a_a @ M ) @ ( insert4534936382041156343od_a_a @ A @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% at_most_one_mset_mset_diff
thf(fact_1046_set__mset__add__mset__insert,axiom,
! [A: a,A3: multiset_a] :
( ( set_mset_a @ ( add_mset_a @ A @ A3 ) )
= ( insert_a @ A @ ( set_mset_a @ A3 ) ) ) ).
% set_mset_add_mset_insert
thf(fact_1047_set__mset__add__mset__insert,axiom,
! [A: produc6518373309651786023iset_a,A3: multis4781622204924204013iset_a] :
( ( set_ms2379723421352415844iset_a @ ( add_ms5348986034529679365iset_a @ A @ A3 ) )
= ( insert4880209295958132471iset_a @ A @ ( set_ms2379723421352415844iset_a @ A3 ) ) ) ).
% set_mset_add_mset_insert
thf(fact_1048_set__mset__add__mset__insert,axiom,
! [A: product_prod_a_a,A3: multis501812127501805293od_a_a] :
( ( set_ms119794563918438244od_a_a @ ( add_ms8655138167283798533od_a_a @ A @ A3 ) )
= ( insert4534936382041156343od_a_a @ A @ ( set_ms119794563918438244od_a_a @ A3 ) ) ) ).
% set_mset_add_mset_insert
thf(fact_1049_Multiset_Omono__mult,axiom,
! [R: set_Product_prod_a_a,R2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ R @ R2 )
=> ( ord_le5871882978999109543iset_a @ ( mult_a @ R ) @ ( mult_a @ R2 ) ) ) ).
% Multiset.mono_mult
thf(fact_1050_set__mset__mono,axiom,
! [A3: multiset_a,B3: multiset_a] :
( ( subseteq_mset_a @ A3 @ B3 )
=> ( ord_less_eq_set_a @ ( set_mset_a @ A3 ) @ ( set_mset_a @ B3 ) ) ) ).
% set_mset_mono
thf(fact_1051_set__mset__mono,axiom,
! [A3: multis4781622204924204013iset_a,B3: multis4781622204924204013iset_a] :
( ( subset6292090195676340366iset_a @ A3 @ B3 )
=> ( ord_le5871882978999109543iset_a @ ( set_ms2379723421352415844iset_a @ A3 ) @ ( set_ms2379723421352415844iset_a @ B3 ) ) ) ).
% set_mset_mono
thf(fact_1052_set__mset__mono,axiom,
! [A3: multis501812127501805293od_a_a,B3: multis501812127501805293od_a_a] :
( ( subset4108007524207251342od_a_a @ A3 @ B3 )
=> ( ord_le746702958409616551od_a_a @ ( set_ms119794563918438244od_a_a @ A3 ) @ ( set_ms119794563918438244od_a_a @ B3 ) ) ) ).
% set_mset_mono
thf(fact_1053_set__mset__single,axiom,
! [B: produc6518373309651786023iset_a] :
( ( set_ms2379723421352415844iset_a @ ( add_ms5348986034529679365iset_a @ B @ zero_z4591712717340762532iset_a ) )
= ( insert4880209295958132471iset_a @ B @ bot_bo5898781664966027891iset_a ) ) ).
% set_mset_single
thf(fact_1054_set__mset__single,axiom,
! [B: product_prod_a_a] :
( ( set_ms119794563918438244od_a_a @ ( add_ms8655138167283798533od_a_a @ B @ zero_z3024502163831676836od_a_a ) )
= ( insert4534936382041156343od_a_a @ B @ bot_bo3357376287454694259od_a_a ) ) ).
% set_mset_single
thf(fact_1055_set__mset__single,axiom,
! [B: a] :
( ( set_mset_a @ ( add_mset_a @ B @ zero_zero_multiset_a ) )
= ( insert_a @ B @ bot_bot_set_a ) ) ).
% set_mset_single
thf(fact_1056_subrelI,axiom,
! [R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ! [X5: a,Y4: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y4 ) @ R )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y4 ) @ S ) )
=> ( ord_le746702958409616551od_a_a @ R @ S ) ) ).
% subrelI
thf(fact_1057_subrelI,axiom,
! [R: set_Pr79727621955416071iset_a,S: set_Pr79727621955416071iset_a] :
( ! [X5: multiset_a,Y4: multiset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Y4 ) @ R )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Y4 ) @ S ) )
=> ( ord_le5871882978999109543iset_a @ R @ S ) ) ).
% subrelI
thf(fact_1058_subset__mset_Oinf__Sup__absorb,axiom,
! [A3: set_multiset_a,A: multiset_a] :
( ( finite2463020702752857069iset_a @ A3 )
=> ( ( member_multiset_a @ A @ A3 )
=> ( ( inter_mset_a @ A @ ( lattic6457931440061790233iset_a @ union_mset_a @ A3 ) )
= A ) ) ) ).
% subset_mset.inf_Sup_absorb
thf(fact_1059_subset__mset_OInf__fin__le__Sup__fin,axiom,
! [A3: set_multiset_a] :
( ( finite2463020702752857069iset_a @ A3 )
=> ( ( A3 != bot_bo6997605411617904272iset_a )
=> ( subseteq_mset_a @ ( lattic1863195236566259673iset_a @ inter_mset_a @ A3 ) @ ( lattic6457931440061790233iset_a @ union_mset_a @ A3 ) ) ) ) ).
% subset_mset.Inf_fin_le_Sup_fin
thf(fact_1060_subset__mset_OInf__fin_Oremove,axiom,
! [A3: set_multiset_a,X3: multiset_a] :
( ( finite2463020702752857069iset_a @ A3 )
=> ( ( member_multiset_a @ X3 @ A3 )
=> ( ( ( ( minus_3527600028014272011iset_a @ A3 @ ( insert_multiset_a @ X3 @ bot_bo6997605411617904272iset_a ) )
= bot_bo6997605411617904272iset_a )
=> ( ( lattic1863195236566259673iset_a @ inter_mset_a @ A3 )
= X3 ) )
& ( ( ( minus_3527600028014272011iset_a @ A3 @ ( insert_multiset_a @ X3 @ bot_bo6997605411617904272iset_a ) )
!= bot_bo6997605411617904272iset_a )
=> ( ( lattic1863195236566259673iset_a @ inter_mset_a @ A3 )
= ( inter_mset_a @ X3 @ ( lattic1863195236566259673iset_a @ inter_mset_a @ ( minus_3527600028014272011iset_a @ A3 @ ( insert_multiset_a @ X3 @ bot_bo6997605411617904272iset_a ) ) ) ) ) ) ) ) ) ).
% subset_mset.Inf_fin.remove
thf(fact_1061_set__mset__inter,axiom,
! [A3: multis4781622204924204013iset_a,B3: multis4781622204924204013iset_a] :
( ( set_ms2379723421352415844iset_a @ ( inter_1588963877939557482iset_a @ A3 @ B3 ) )
= ( inf_in7847659125592275445iset_a @ ( set_ms2379723421352415844iset_a @ A3 ) @ ( set_ms2379723421352415844iset_a @ B3 ) ) ) ).
% set_mset_inter
thf(fact_1062_set__mset__inter,axiom,
! [A3: multis501812127501805293od_a_a,B3: multis501812127501805293od_a_a] :
( ( set_ms119794563918438244od_a_a @ ( inter_3651181809579396970od_a_a @ A3 @ B3 ) )
= ( inf_in8905007599844390133od_a_a @ ( set_ms119794563918438244od_a_a @ A3 ) @ ( set_ms119794563918438244od_a_a @ B3 ) ) ) ).
% set_mset_inter
thf(fact_1063_set__mset__inter,axiom,
! [A3: multiset_a,B3: multiset_a] :
( ( set_mset_a @ ( inter_mset_a @ A3 @ B3 ) )
= ( inf_inf_set_a @ ( set_mset_a @ A3 ) @ ( set_mset_a @ B3 ) ) ) ).
% set_mset_inter
thf(fact_1064_subset__mset_Osup__Inf__absorb,axiom,
! [A3: set_multiset_a,A: multiset_a] :
( ( finite2463020702752857069iset_a @ A3 )
=> ( ( member_multiset_a @ A @ A3 )
=> ( ( union_mset_a @ ( lattic1863195236566259673iset_a @ inter_mset_a @ A3 ) @ A )
= A ) ) ) ).
% subset_mset.sup_Inf_absorb
thf(fact_1065_subset__mset_OInf__fin_Osingleton,axiom,
! [X3: multiset_a] :
( ( lattic1863195236566259673iset_a @ inter_mset_a @ ( insert_multiset_a @ X3 @ bot_bo6997605411617904272iset_a ) )
= X3 ) ).
% subset_mset.Inf_fin.singleton
thf(fact_1066_subset__mset_OInf__fin_Oinsert,axiom,
! [A3: set_multiset_a,X3: multiset_a] :
( ( finite2463020702752857069iset_a @ A3 )
=> ( ( A3 != bot_bo6997605411617904272iset_a )
=> ( ( lattic1863195236566259673iset_a @ inter_mset_a @ ( insert_multiset_a @ X3 @ A3 ) )
= ( inter_mset_a @ X3 @ ( lattic1863195236566259673iset_a @ inter_mset_a @ A3 ) ) ) ) ) ).
% subset_mset.Inf_fin.insert
thf(fact_1067_subset__mset_OInf__fin_Oin__idem,axiom,
! [A3: set_multiset_a,X3: multiset_a] :
( ( finite2463020702752857069iset_a @ A3 )
=> ( ( member_multiset_a @ X3 @ A3 )
=> ( ( inter_mset_a @ X3 @ ( lattic1863195236566259673iset_a @ inter_mset_a @ A3 ) )
= ( lattic1863195236566259673iset_a @ inter_mset_a @ A3 ) ) ) ) ).
% subset_mset.Inf_fin.in_idem
thf(fact_1068_subset__mset_OInf__fin_OcoboundedI,axiom,
! [A3: set_multiset_a,A: multiset_a] :
( ( finite2463020702752857069iset_a @ A3 )
=> ( ( member_multiset_a @ A @ A3 )
=> ( subseteq_mset_a @ ( lattic1863195236566259673iset_a @ inter_mset_a @ A3 ) @ A ) ) ) ).
% subset_mset.Inf_fin.coboundedI
thf(fact_1069_subset__mset_OInf__fin_OboundedE,axiom,
! [A3: set_multiset_a,X3: multiset_a] :
( ( finite2463020702752857069iset_a @ A3 )
=> ( ( A3 != bot_bo6997605411617904272iset_a )
=> ( ( subseteq_mset_a @ X3 @ ( lattic1863195236566259673iset_a @ inter_mset_a @ A3 ) )
=> ! [A8: multiset_a] :
( ( member_multiset_a @ A8 @ A3 )
=> ( subseteq_mset_a @ X3 @ A8 ) ) ) ) ) ).
% subset_mset.Inf_fin.boundedE
thf(fact_1070_subset__mset_OInf__fin_OboundedI,axiom,
! [A3: set_multiset_a,X3: multiset_a] :
( ( finite2463020702752857069iset_a @ A3 )
=> ( ( A3 != bot_bo6997605411617904272iset_a )
=> ( ! [A4: multiset_a] :
( ( member_multiset_a @ A4 @ A3 )
=> ( subseteq_mset_a @ X3 @ A4 ) )
=> ( subseteq_mset_a @ X3 @ ( lattic1863195236566259673iset_a @ inter_mset_a @ A3 ) ) ) ) ) ).
% subset_mset.Inf_fin.boundedI
thf(fact_1071_subset__mset_OInf__fin_Obounded__iff,axiom,
! [A3: set_multiset_a,X3: multiset_a] :
( ( finite2463020702752857069iset_a @ A3 )
=> ( ( A3 != bot_bo6997605411617904272iset_a )
=> ( ( subseteq_mset_a @ X3 @ ( lattic1863195236566259673iset_a @ inter_mset_a @ A3 ) )
= ( ! [X6: multiset_a] :
( ( member_multiset_a @ X6 @ A3 )
=> ( subseteq_mset_a @ X3 @ X6 ) ) ) ) ) ) ).
% subset_mset.Inf_fin.bounded_iff
thf(fact_1072_subset__mset_OInf__fin_Oinsert__not__elem,axiom,
! [A3: set_multiset_a,X3: multiset_a] :
( ( finite2463020702752857069iset_a @ A3 )
=> ( ~ ( member_multiset_a @ X3 @ A3 )
=> ( ( A3 != bot_bo6997605411617904272iset_a )
=> ( ( lattic1863195236566259673iset_a @ inter_mset_a @ ( insert_multiset_a @ X3 @ A3 ) )
= ( inter_mset_a @ X3 @ ( lattic1863195236566259673iset_a @ inter_mset_a @ A3 ) ) ) ) ) ) ).
% subset_mset.Inf_fin.insert_not_elem
thf(fact_1073_subset__mset_OInf__fin_Oclosed,axiom,
! [A3: set_multiset_a] :
( ( finite2463020702752857069iset_a @ A3 )
=> ( ( A3 != bot_bo6997605411617904272iset_a )
=> ( ! [X5: multiset_a,Y4: multiset_a] : ( member_multiset_a @ ( inter_mset_a @ X5 @ Y4 ) @ ( insert_multiset_a @ X5 @ ( insert_multiset_a @ Y4 @ bot_bo6997605411617904272iset_a ) ) )
=> ( member_multiset_a @ ( lattic1863195236566259673iset_a @ inter_mset_a @ A3 ) @ A3 ) ) ) ) ).
% subset_mset.Inf_fin.closed
thf(fact_1074_subset__mset_OInf__fin_Osubset,axiom,
! [A3: set_multiset_a,B3: set_multiset_a] :
( ( finite2463020702752857069iset_a @ A3 )
=> ( ( B3 != bot_bo6997605411617904272iset_a )
=> ( ( ord_le1502830050749340100iset_a @ B3 @ A3 )
=> ( ( inter_mset_a @ ( lattic1863195236566259673iset_a @ inter_mset_a @ B3 ) @ ( lattic1863195236566259673iset_a @ inter_mset_a @ A3 ) )
= ( lattic1863195236566259673iset_a @ inter_mset_a @ A3 ) ) ) ) ) ).
% subset_mset.Inf_fin.subset
thf(fact_1075_subset__mset_OInf__fin_Osubset__imp,axiom,
! [A3: set_multiset_a,B3: set_multiset_a] :
( ( ord_le1502830050749340100iset_a @ A3 @ B3 )
=> ( ( A3 != bot_bo6997605411617904272iset_a )
=> ( ( finite2463020702752857069iset_a @ B3 )
=> ( subseteq_mset_a @ ( lattic1863195236566259673iset_a @ inter_mset_a @ B3 ) @ ( lattic1863195236566259673iset_a @ inter_mset_a @ A3 ) ) ) ) ) ).
% subset_mset.Inf_fin.subset_imp
thf(fact_1076_subset__mset_OInf__fin_Oinsert__remove,axiom,
! [A3: set_multiset_a,X3: multiset_a] :
( ( finite2463020702752857069iset_a @ A3 )
=> ( ( ( ( minus_3527600028014272011iset_a @ A3 @ ( insert_multiset_a @ X3 @ bot_bo6997605411617904272iset_a ) )
= bot_bo6997605411617904272iset_a )
=> ( ( lattic1863195236566259673iset_a @ inter_mset_a @ ( insert_multiset_a @ X3 @ A3 ) )
= X3 ) )
& ( ( ( minus_3527600028014272011iset_a @ A3 @ ( insert_multiset_a @ X3 @ bot_bo6997605411617904272iset_a ) )
!= bot_bo6997605411617904272iset_a )
=> ( ( lattic1863195236566259673iset_a @ inter_mset_a @ ( insert_multiset_a @ X3 @ A3 ) )
= ( inter_mset_a @ X3 @ ( lattic1863195236566259673iset_a @ inter_mset_a @ ( minus_3527600028014272011iset_a @ A3 @ ( insert_multiset_a @ X3 @ bot_bo6997605411617904272iset_a ) ) ) ) ) ) ) ) ).
% subset_mset.Inf_fin.insert_remove
thf(fact_1077_mset__set_Oremove,axiom,
! [A3: set_Product_prod_a_a,X3: product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ A3 )
=> ( ( member1426531477525435216od_a_a @ X3 @ A3 )
=> ( ( mset_s1230098591470296142od_a_a @ A3 )
= ( add_ms8655138167283798533od_a_a @ X3 @ ( mset_s1230098591470296142od_a_a @ ( minus_6817036919807184750od_a_a @ A3 @ ( insert4534936382041156343od_a_a @ X3 @ bot_bo3357376287454694259od_a_a ) ) ) ) ) ) ) ).
% mset_set.remove
thf(fact_1078_mset__set_Oremove,axiom,
! [A3: set_Pr79727621955416071iset_a,X3: produc6518373309651786023iset_a] :
( ( finite1099724235734660432iset_a @ A3 )
=> ( ( member5199237121806060112iset_a @ X3 @ A3 )
=> ( ( mset_s4664099280310381902iset_a @ A3 )
= ( add_ms5348986034529679365iset_a @ X3 @ ( mset_s4664099280310381902iset_a @ ( minus_2461365523287052142iset_a @ A3 @ ( insert4880209295958132471iset_a @ X3 @ bot_bo5898781664966027891iset_a ) ) ) ) ) ) ) ).
% mset_set.remove
thf(fact_1079_mset__set_Oremove,axiom,
! [A3: set_a,X3: a] :
( ( finite_finite_a @ A3 )
=> ( ( member_a @ X3 @ A3 )
=> ( ( mset_set_a @ A3 )
= ( add_mset_a @ X3 @ ( mset_set_a @ ( minus_minus_set_a @ A3 @ ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ) ) ) ).
% mset_set.remove
thf(fact_1080_set__mset__replicate__mset__subset,axiom,
! [N4: nat,X3: a] :
( ( ( N4 = zero_zero_nat )
=> ( ( set_mset_a @ ( replicate_mset_a @ N4 @ X3 ) )
= bot_bot_set_a ) )
& ( ( N4 != zero_zero_nat )
=> ( ( set_mset_a @ ( replicate_mset_a @ N4 @ X3 ) )
= ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ).
% set_mset_replicate_mset_subset
thf(fact_1081_set__mset__replicate__mset__subset,axiom,
! [N4: nat,X3: produc6518373309651786023iset_a] :
( ( ( N4 = zero_zero_nat )
=> ( ( set_ms2379723421352415844iset_a @ ( replic6672042396387243389iset_a @ N4 @ X3 ) )
= bot_bo5898781664966027891iset_a ) )
& ( ( N4 != zero_zero_nat )
=> ( ( set_ms2379723421352415844iset_a @ ( replic6672042396387243389iset_a @ N4 @ X3 ) )
= ( insert4880209295958132471iset_a @ X3 @ bot_bo5898781664966027891iset_a ) ) ) ) ).
% set_mset_replicate_mset_subset
thf(fact_1082_set__mset__replicate__mset__subset,axiom,
! [N4: nat,X3: product_prod_a_a] :
( ( ( N4 = zero_zero_nat )
=> ( ( set_ms119794563918438244od_a_a @ ( replic5559486381873801085od_a_a @ N4 @ X3 ) )
= bot_bo3357376287454694259od_a_a ) )
& ( ( N4 != zero_zero_nat )
=> ( ( set_ms119794563918438244od_a_a @ ( replic5559486381873801085od_a_a @ N4 @ X3 ) )
= ( insert4534936382041156343od_a_a @ X3 @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% set_mset_replicate_mset_subset
thf(fact_1083_mset__set_Oempty,axiom,
( ( mset_set_a @ bot_bot_set_a )
= zero_zero_multiset_a ) ).
% mset_set.empty
thf(fact_1084_mset__set_Oinfinite,axiom,
! [A3: set_a] :
( ~ ( finite_finite_a @ A3 )
=> ( ( mset_set_a @ A3 )
= zero_zero_multiset_a ) ) ).
% mset_set.infinite
thf(fact_1085_elem__mset__set,axiom,
! [A3: set_a,X3: a] :
( ( finite_finite_a @ A3 )
=> ( ( member_a @ X3 @ ( set_mset_a @ ( mset_set_a @ A3 ) ) )
= ( member_a @ X3 @ A3 ) ) ) ).
% elem_mset_set
thf(fact_1086_elem__mset__set,axiom,
! [A3: set_Pr79727621955416071iset_a,X3: produc6518373309651786023iset_a] :
( ( finite1099724235734660432iset_a @ A3 )
=> ( ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ ( mset_s4664099280310381902iset_a @ A3 ) ) )
= ( member5199237121806060112iset_a @ X3 @ A3 ) ) ) ).
% elem_mset_set
thf(fact_1087_elem__mset__set,axiom,
! [A3: set_Product_prod_a_a,X3: product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ A3 )
=> ( ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ ( mset_s1230098591470296142od_a_a @ A3 ) ) )
= ( member1426531477525435216od_a_a @ X3 @ A3 ) ) ) ).
% elem_mset_set
thf(fact_1088_finite__set__mset__mset__set,axiom,
! [A3: set_a] :
( ( finite_finite_a @ A3 )
=> ( ( set_mset_a @ ( mset_set_a @ A3 ) )
= A3 ) ) ).
% finite_set_mset_mset_set
thf(fact_1089_finite__set__mset__mset__set,axiom,
! [A3: set_Pr79727621955416071iset_a] :
( ( finite1099724235734660432iset_a @ A3 )
=> ( ( set_ms2379723421352415844iset_a @ ( mset_s4664099280310381902iset_a @ A3 ) )
= A3 ) ) ).
% finite_set_mset_mset_set
thf(fact_1090_finite__set__mset__mset__set,axiom,
! [A3: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ A3 )
=> ( ( set_ms119794563918438244od_a_a @ ( mset_s1230098591470296142od_a_a @ A3 ) )
= A3 ) ) ).
% finite_set_mset_mset_set
thf(fact_1091_replicate__mset__0,axiom,
! [X3: a] :
( ( replicate_mset_a @ zero_zero_nat @ X3 )
= zero_zero_multiset_a ) ).
% replicate_mset_0
thf(fact_1092_replicate__mset__eq__empty__iff,axiom,
! [N4: nat,A: a] :
( ( ( replicate_mset_a @ N4 @ A )
= zero_zero_multiset_a )
= ( N4 = zero_zero_nat ) ) ).
% replicate_mset_eq_empty_iff
thf(fact_1093_mset__set_Oinsert,axiom,
! [A3: set_Product_prod_a_a,X3: product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ A3 )
=> ( ~ ( member1426531477525435216od_a_a @ X3 @ A3 )
=> ( ( mset_s1230098591470296142od_a_a @ ( insert4534936382041156343od_a_a @ X3 @ A3 ) )
= ( add_ms8655138167283798533od_a_a @ X3 @ ( mset_s1230098591470296142od_a_a @ A3 ) ) ) ) ) ).
% mset_set.insert
thf(fact_1094_mset__set_Oinsert,axiom,
! [A3: set_Pr79727621955416071iset_a,X3: produc6518373309651786023iset_a] :
( ( finite1099724235734660432iset_a @ A3 )
=> ( ~ ( member5199237121806060112iset_a @ X3 @ A3 )
=> ( ( mset_s4664099280310381902iset_a @ ( insert4880209295958132471iset_a @ X3 @ A3 ) )
= ( add_ms5348986034529679365iset_a @ X3 @ ( mset_s4664099280310381902iset_a @ A3 ) ) ) ) ) ).
% mset_set.insert
thf(fact_1095_mset__set_Oinsert,axiom,
! [A3: set_a,X3: a] :
( ( finite_finite_a @ A3 )
=> ( ~ ( member_a @ X3 @ A3 )
=> ( ( mset_set_a @ ( insert_a @ X3 @ A3 ) )
= ( add_mset_a @ X3 @ ( mset_set_a @ A3 ) ) ) ) ) ).
% mset_set.insert
thf(fact_1096_mset__set__set__mset__msubset,axiom,
! [A3: multiset_a] : ( subseteq_mset_a @ ( mset_set_a @ ( set_mset_a @ A3 ) ) @ A3 ) ).
% mset_set_set_mset_msubset
thf(fact_1097_mset__set__set__mset__msubset,axiom,
! [A3: multis4781622204924204013iset_a] : ( subset6292090195676340366iset_a @ ( mset_s4664099280310381902iset_a @ ( set_ms2379723421352415844iset_a @ A3 ) ) @ A3 ) ).
% mset_set_set_mset_msubset
thf(fact_1098_mset__set__set__mset__msubset,axiom,
! [A3: multis501812127501805293od_a_a] : ( subset4108007524207251342od_a_a @ ( mset_s1230098591470296142od_a_a @ ( set_ms119794563918438244od_a_a @ A3 ) ) @ A3 ) ).
% mset_set_set_mset_msubset
thf(fact_1099_mset__set__empty__iff,axiom,
! [A3: set_a] :
( ( ( mset_set_a @ A3 )
= zero_zero_multiset_a )
= ( ( A3 = bot_bot_set_a )
| ~ ( finite_finite_a @ A3 ) ) ) ).
% mset_set_empty_iff
thf(fact_1100_infinite__set__mset__mset__set,axiom,
! [A3: set_a] :
( ~ ( finite_finite_a @ A3 )
=> ( ( set_mset_a @ ( mset_set_a @ A3 ) )
= bot_bot_set_a ) ) ).
% infinite_set_mset_mset_set
thf(fact_1101_infinite__set__mset__mset__set,axiom,
! [A3: set_Pr79727621955416071iset_a] :
( ~ ( finite1099724235734660432iset_a @ A3 )
=> ( ( set_ms2379723421352415844iset_a @ ( mset_s4664099280310381902iset_a @ A3 ) )
= bot_bo5898781664966027891iset_a ) ) ).
% infinite_set_mset_mset_set
thf(fact_1102_infinite__set__mset__mset__set,axiom,
! [A3: set_Product_prod_a_a] :
( ~ ( finite6544458595007987280od_a_a @ A3 )
=> ( ( set_ms119794563918438244od_a_a @ ( mset_s1230098591470296142od_a_a @ A3 ) )
= bot_bo3357376287454694259od_a_a ) ) ).
% infinite_set_mset_mset_set
thf(fact_1103_mset__set__Diff,axiom,
! [A3: set_a,B3: set_a] :
( ( finite_finite_a @ A3 )
=> ( ( ord_less_eq_set_a @ B3 @ A3 )
=> ( ( mset_set_a @ ( minus_minus_set_a @ A3 @ B3 ) )
= ( minus_3765977307040488491iset_a @ ( mset_set_a @ A3 ) @ ( mset_set_a @ B3 ) ) ) ) ) ).
% mset_set_Diff
thf(fact_1104_mset__set__Diff,axiom,
! [A3: set_Pr79727621955416071iset_a,B3: set_Pr79727621955416071iset_a] :
( ( finite1099724235734660432iset_a @ A3 )
=> ( ( ord_le5871882978999109543iset_a @ B3 @ A3 )
=> ( ( mset_s4664099280310381902iset_a @ ( minus_2461365523287052142iset_a @ A3 @ B3 ) )
= ( minus_9211619285413683668iset_a @ ( mset_s4664099280310381902iset_a @ A3 ) @ ( mset_s4664099280310381902iset_a @ B3 ) ) ) ) ) ).
% mset_set_Diff
thf(fact_1105_mset__set__Diff,axiom,
! [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ A3 )
=> ( ( ord_le746702958409616551od_a_a @ B3 @ A3 )
=> ( ( mset_s1230098591470296142od_a_a @ ( minus_6817036919807184750od_a_a @ A3 @ B3 ) )
= ( minus_7313152569058643924od_a_a @ ( mset_s1230098591470296142od_a_a @ A3 ) @ ( mset_s1230098591470296142od_a_a @ B3 ) ) ) ) ) ).
% mset_set_Diff
thf(fact_1106_mset__set__Union,axiom,
! [A3: set_a,B3: set_a] :
( ( finite_finite_a @ A3 )
=> ( ( finite_finite_a @ B3 )
=> ( ( ( inf_inf_set_a @ A3 @ B3 )
= bot_bot_set_a )
=> ( ( mset_set_a @ ( sup_sup_set_a @ A3 @ B3 ) )
= ( plus_plus_multiset_a @ ( mset_set_a @ A3 ) @ ( mset_set_a @ B3 ) ) ) ) ) ) ).
% mset_set_Union
thf(fact_1107_mset__set__def,axiom,
( mset_set_a
= ( finite4234324634300123651iset_a @ add_mset_a @ zero_zero_multiset_a ) ) ).
% mset_set_def
thf(fact_1108_repeat__mset__distrib__add__mset,axiom,
! [N4: nat,A: a,A3: multiset_a] :
( ( repeat_mset_a @ N4 @ ( add_mset_a @ A @ A3 ) )
= ( plus_plus_multiset_a @ ( replicate_mset_a @ N4 @ A ) @ ( repeat_mset_a @ N4 @ A3 ) ) ) ).
% repeat_mset_distrib_add_mset
thf(fact_1109_repeat__mset__empty,axiom,
! [N4: nat] :
( ( repeat_mset_a @ N4 @ zero_zero_multiset_a )
= zero_zero_multiset_a ) ).
% repeat_mset_empty
thf(fact_1110_repeat__mset__distrib2,axiom,
! [N4: nat,A3: multiset_a,B3: multiset_a] :
( ( repeat_mset_a @ N4 @ ( plus_plus_multiset_a @ A3 @ B3 ) )
= ( plus_plus_multiset_a @ ( repeat_mset_a @ N4 @ A3 ) @ ( repeat_mset_a @ N4 @ B3 ) ) ) ).
% repeat_mset_distrib2
thf(fact_1111_set__mset__union,axiom,
! [M: multis4781622204924204013iset_a,N: multis4781622204924204013iset_a] :
( ( set_ms2379723421352415844iset_a @ ( plus_p1053901119268525860iset_a @ M @ N ) )
= ( sup_su1925787886501348827iset_a @ ( set_ms2379723421352415844iset_a @ M ) @ ( set_ms2379723421352415844iset_a @ N ) ) ) ).
% set_mset_union
thf(fact_1112_set__mset__union,axiom,
! [M: multis501812127501805293od_a_a,N: multis501812127501805293od_a_a] :
( ( set_ms119794563918438244od_a_a @ ( plus_p6967097483979887908od_a_a @ M @ N ) )
= ( sup_su3048258781599657691od_a_a @ ( set_ms119794563918438244od_a_a @ M ) @ ( set_ms119794563918438244od_a_a @ N ) ) ) ).
% set_mset_union
thf(fact_1113_set__mset__union,axiom,
! [M: multiset_a,N: multiset_a] :
( ( set_mset_a @ ( plus_plus_multiset_a @ M @ N ) )
= ( sup_sup_set_a @ ( set_mset_a @ M ) @ ( set_mset_a @ N ) ) ) ).
% set_mset_union
thf(fact_1114_set__mset__sup,axiom,
! [A3: multiset_a,B3: multiset_a] :
( ( set_mset_a @ ( union_mset_a @ A3 @ B3 ) )
= ( sup_sup_set_a @ ( set_mset_a @ A3 ) @ ( set_mset_a @ B3 ) ) ) ).
% set_mset_sup
thf(fact_1115_set__mset__sup,axiom,
! [A3: multis4781622204924204013iset_a,B3: multis4781622204924204013iset_a] :
( ( set_ms2379723421352415844iset_a @ ( union_4549860014028270007iset_a @ A3 @ B3 ) )
= ( sup_su1925787886501348827iset_a @ ( set_ms2379723421352415844iset_a @ A3 ) @ ( set_ms2379723421352415844iset_a @ B3 ) ) ) ).
% set_mset_sup
thf(fact_1116_set__mset__sup,axiom,
! [A3: multis501812127501805293od_a_a,B3: multis501812127501805293od_a_a] :
( ( set_ms119794563918438244od_a_a @ ( union_8952997168100052407od_a_a @ A3 @ B3 ) )
= ( sup_su3048258781599657691od_a_a @ ( set_ms119794563918438244od_a_a @ A3 ) @ ( set_ms119794563918438244od_a_a @ B3 ) ) ) ).
% set_mset_sup
thf(fact_1117_repeat__mset__0,axiom,
! [M: multiset_a] :
( ( repeat_mset_a @ zero_zero_nat @ M )
= zero_zero_multiset_a ) ).
% repeat_mset_0
thf(fact_1118_repeat__mset__replicate__mset,axiom,
! [N4: nat,A: a] :
( ( repeat_mset_a @ N4 @ ( add_mset_a @ A @ zero_zero_multiset_a ) )
= ( replicate_mset_a @ N4 @ A ) ) ).
% repeat_mset_replicate_mset
thf(fact_1119_repeat__mset__cancel2,axiom,
! [A: nat,A3: multiset_a,B: nat] :
( ( ( repeat_mset_a @ A @ A3 )
= ( repeat_mset_a @ B @ A3 ) )
= ( ( A = B )
| ( A3 = zero_zero_multiset_a ) ) ) ).
% repeat_mset_cancel2
thf(fact_1120_left__add__mult__distrib__mset,axiom,
! [I3: nat,U: multiset_a,J3: nat,K3: multiset_a] :
( ( plus_plus_multiset_a @ ( repeat_mset_a @ I3 @ U ) @ ( plus_plus_multiset_a @ ( repeat_mset_a @ J3 @ U ) @ K3 ) )
= ( plus_plus_multiset_a @ ( repeat_mset_a @ ( plus_plus_nat @ I3 @ J3 ) @ U ) @ K3 ) ) ).
% left_add_mult_distrib_mset
thf(fact_1121_repeat__mset__distrib,axiom,
! [M4: nat,N4: nat,A3: multiset_a] :
( ( repeat_mset_a @ ( plus_plus_nat @ M4 @ N4 ) @ A3 )
= ( plus_plus_multiset_a @ ( repeat_mset_a @ M4 @ A3 ) @ ( repeat_mset_a @ N4 @ A3 ) ) ) ).
% repeat_mset_distrib
thf(fact_1122_left__diff__repeat__mset__distrib_H,axiom,
! [I3: nat,J3: nat,U: multiset_a] :
( ( repeat_mset_a @ ( minus_minus_nat @ I3 @ J3 ) @ U )
= ( minus_3765977307040488491iset_a @ ( repeat_mset_a @ I3 @ U ) @ ( repeat_mset_a @ J3 @ U ) ) ) ).
% left_diff_repeat_mset_distrib'
thf(fact_1123_left__diff__repeat__mset__distrib_H,axiom,
! [I3: nat,J3: nat,U: multis4781622204924204013iset_a] :
( ( repeat1211446897355296257iset_a @ ( minus_minus_nat @ I3 @ J3 ) @ U )
= ( minus_9211619285413683668iset_a @ ( repeat1211446897355296257iset_a @ I3 @ U ) @ ( repeat1211446897355296257iset_a @ J3 @ U ) ) ) ).
% left_diff_repeat_mset_distrib'
thf(fact_1124_left__diff__repeat__mset__distrib_H,axiom,
! [I3: nat,J3: nat,U: multis501812127501805293od_a_a] :
( ( repeat4794893366960026625od_a_a @ ( minus_minus_nat @ I3 @ J3 ) @ U )
= ( minus_7313152569058643924od_a_a @ ( repeat4794893366960026625od_a_a @ I3 @ U ) @ ( repeat4794893366960026625od_a_a @ J3 @ U ) ) ) ).
% left_diff_repeat_mset_distrib'
thf(fact_1125_repeat__mset__eq__empty__iff,axiom,
! [N4: nat,A3: multiset_a] :
( ( ( repeat_mset_a @ N4 @ A3 )
= zero_zero_multiset_a )
= ( ( N4 = zero_zero_nat )
| ( A3 = zero_zero_multiset_a ) ) ) ).
% repeat_mset_eq_empty_iff
thf(fact_1126_mset__subseteq__add__iff1,axiom,
! [J3: nat,I3: nat,U: multiset_a,M4: multiset_a,N4: multiset_a] :
( ( ord_less_eq_nat @ J3 @ I3 )
=> ( ( subseteq_mset_a @ ( plus_plus_multiset_a @ ( repeat_mset_a @ I3 @ U ) @ M4 ) @ ( plus_plus_multiset_a @ ( repeat_mset_a @ J3 @ U ) @ N4 ) )
= ( subseteq_mset_a @ ( plus_plus_multiset_a @ ( repeat_mset_a @ ( minus_minus_nat @ I3 @ J3 ) @ U ) @ M4 ) @ N4 ) ) ) ).
% mset_subseteq_add_iff1
thf(fact_1127_mset__subseteq__add__iff2,axiom,
! [I3: nat,J3: nat,U: multiset_a,M4: multiset_a,N4: multiset_a] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ( subseteq_mset_a @ ( plus_plus_multiset_a @ ( repeat_mset_a @ I3 @ U ) @ M4 ) @ ( plus_plus_multiset_a @ ( repeat_mset_a @ J3 @ U ) @ N4 ) )
= ( subseteq_mset_a @ M4 @ ( plus_plus_multiset_a @ ( repeat_mset_a @ ( minus_minus_nat @ J3 @ I3 ) @ U ) @ N4 ) ) ) ) ).
% mset_subseteq_add_iff2
thf(fact_1128_mset__subset__add__iff1,axiom,
! [J3: nat,I3: nat,U: multiset_a,M4: multiset_a,N4: multiset_a] :
( ( ord_less_eq_nat @ J3 @ I3 )
=> ( ( subset_mset_a @ ( plus_plus_multiset_a @ ( repeat_mset_a @ I3 @ U ) @ M4 ) @ ( plus_plus_multiset_a @ ( repeat_mset_a @ J3 @ U ) @ N4 ) )
= ( subset_mset_a @ ( plus_plus_multiset_a @ ( repeat_mset_a @ ( minus_minus_nat @ I3 @ J3 ) @ U ) @ M4 ) @ N4 ) ) ) ).
% mset_subset_add_iff1
thf(fact_1129_mset__subset__add__iff2,axiom,
! [I3: nat,J3: nat,U: multiset_a,M4: multiset_a,N4: multiset_a] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ( subset_mset_a @ ( plus_plus_multiset_a @ ( repeat_mset_a @ I3 @ U ) @ M4 ) @ ( plus_plus_multiset_a @ ( repeat_mset_a @ J3 @ U ) @ N4 ) )
= ( subset_mset_a @ M4 @ ( plus_plus_multiset_a @ ( repeat_mset_a @ ( minus_minus_nat @ J3 @ I3 ) @ U ) @ N4 ) ) ) ) ).
% mset_subset_add_iff2
thf(fact_1130_subset__mset_OInf__fin_Ounion,axiom,
! [A3: set_multiset_a,B3: set_multiset_a] :
( ( finite2463020702752857069iset_a @ A3 )
=> ( ( A3 != bot_bo6997605411617904272iset_a )
=> ( ( finite2463020702752857069iset_a @ B3 )
=> ( ( B3 != bot_bo6997605411617904272iset_a )
=> ( ( lattic1863195236566259673iset_a @ inter_mset_a @ ( sup_su2684633059243589112iset_a @ A3 @ B3 ) )
= ( inter_mset_a @ ( lattic1863195236566259673iset_a @ inter_mset_a @ A3 ) @ ( lattic1863195236566259673iset_a @ inter_mset_a @ B3 ) ) ) ) ) ) ) ).
% subset_mset.Inf_fin.union
thf(fact_1131_Field__insert,axiom,
! [A: a,B: a,R: set_Product_prod_a_a] :
( ( field_a @ ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ A @ B ) @ R ) )
= ( sup_sup_set_a @ ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) ) @ ( field_a @ R ) ) ) ).
% Field_insert
thf(fact_1132_Field__insert,axiom,
! [A: multiset_a,B: multiset_a,R: set_Pr79727621955416071iset_a] :
( ( field_multiset_a @ ( insert4880209295958132471iset_a @ ( produc654756711066625303iset_a @ A @ B ) @ R ) )
= ( sup_su2684633059243589112iset_a @ ( insert_multiset_a @ A @ ( insert_multiset_a @ B @ bot_bo6997605411617904272iset_a ) ) @ ( field_multiset_a @ R ) ) ) ).
% Field_insert
thf(fact_1133_FieldI1,axiom,
! [I3: product_prod_a_a,J3: product_prod_a_a,R3: set_Pr8600417178894128327od_a_a] :
( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ I3 @ J3 ) @ R3 )
=> ( member1426531477525435216od_a_a @ I3 @ ( field_1126092520709947252od_a_a @ R3 ) ) ) ).
% FieldI1
thf(fact_1134_FieldI1,axiom,
! [I3: produc6518373309651786023iset_a,J3: produc6518373309651786023iset_a,R3: set_Pr3059988933816007879iset_a] :
( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ I3 @ J3 ) @ R3 )
=> ( member5199237121806060112iset_a @ I3 @ ( field_2318967117295313524iset_a @ R3 ) ) ) ).
% FieldI1
thf(fact_1135_FieldI1,axiom,
! [I3: a,J3: a,R3: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ I3 @ J3 ) @ R3 )
=> ( member_a @ I3 @ ( field_a @ R3 ) ) ) ).
% FieldI1
thf(fact_1136_FieldI1,axiom,
! [I3: multiset_a,J3: multiset_a,R3: set_Pr79727621955416071iset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ I3 @ J3 ) @ R3 )
=> ( member_multiset_a @ I3 @ ( field_multiset_a @ R3 ) ) ) ).
% FieldI1
thf(fact_1137_FieldI2,axiom,
! [I3: product_prod_a_a,J3: product_prod_a_a,R3: set_Pr8600417178894128327od_a_a] :
( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ I3 @ J3 ) @ R3 )
=> ( member1426531477525435216od_a_a @ J3 @ ( field_1126092520709947252od_a_a @ R3 ) ) ) ).
% FieldI2
thf(fact_1138_FieldI2,axiom,
! [I3: produc6518373309651786023iset_a,J3: produc6518373309651786023iset_a,R3: set_Pr3059988933816007879iset_a] :
( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ I3 @ J3 ) @ R3 )
=> ( member5199237121806060112iset_a @ J3 @ ( field_2318967117295313524iset_a @ R3 ) ) ) ).
% FieldI2
thf(fact_1139_FieldI2,axiom,
! [I3: a,J3: a,R3: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ I3 @ J3 ) @ R3 )
=> ( member_a @ J3 @ ( field_a @ R3 ) ) ) ).
% FieldI2
thf(fact_1140_FieldI2,axiom,
! [I3: multiset_a,J3: multiset_a,R3: set_Pr79727621955416071iset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ I3 @ J3 ) @ R3 )
=> ( member_multiset_a @ J3 @ ( field_multiset_a @ R3 ) ) ) ).
% FieldI2
thf(fact_1141_mset__set_Oeq__fold,axiom,
( mset_set_a
= ( finite3188011573590188174iset_a @ add_mset_a @ zero_zero_multiset_a ) ) ).
% mset_set.eq_fold
thf(fact_1142_subset__mset_OInf__fin_Oeq__fold,axiom,
! [A3: set_multiset_a,X3: multiset_a] :
( ( finite2463020702752857069iset_a @ A3 )
=> ( ( lattic1863195236566259673iset_a @ inter_mset_a @ ( insert_multiset_a @ X3 @ A3 ) )
= ( finite5558441502610807112iset_a @ inter_mset_a @ X3 @ A3 ) ) ) ).
% subset_mset.Inf_fin.eq_fold
thf(fact_1143_size__multiset__union,axiom,
! [F: a > nat,M: multiset_a,N: multiset_a] :
( ( size_multiset_a @ F @ ( plus_plus_multiset_a @ M @ N ) )
= ( plus_plus_nat @ ( size_multiset_a @ F @ M ) @ ( size_multiset_a @ F @ N ) ) ) ).
% size_multiset_union
thf(fact_1144_Range__insert,axiom,
! [A: a,B: a,R: set_Product_prod_a_a] :
( ( range_a_a @ ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ A @ B ) @ R ) )
= ( insert_a @ B @ ( range_a_a @ R ) ) ) ).
% Range_insert
thf(fact_1145_Range__insert,axiom,
! [A: multiset_a,B: multiset_a,R: set_Pr79727621955416071iset_a] :
( ( range_1237004097883317749iset_a @ ( insert4880209295958132471iset_a @ ( produc654756711066625303iset_a @ A @ B ) @ R ) )
= ( insert_multiset_a @ B @ ( range_1237004097883317749iset_a @ R ) ) ) ).
% Range_insert
thf(fact_1146_size__multiset__empty,axiom,
! [F: a > nat] :
( ( size_multiset_a @ F @ zero_zero_multiset_a )
= zero_zero_nat ) ).
% size_multiset_empty
thf(fact_1147_size__multiset__eq__0__iff__empty,axiom,
! [F: a > nat,M: multiset_a] :
( ( ( size_multiset_a @ F @ M )
= zero_zero_nat )
= ( M = zero_zero_multiset_a ) ) ).
% size_multiset_eq_0_iff_empty
thf(fact_1148_Range__iff,axiom,
! [A: a,R: set_Product_prod_a_a] :
( ( member_a @ A @ ( range_a_a @ R ) )
= ( ? [Y5: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y5 @ A ) @ R ) ) ) ).
% Range_iff
thf(fact_1149_Range__iff,axiom,
! [A: multiset_a,R: set_Pr79727621955416071iset_a] :
( ( member_multiset_a @ A @ ( range_1237004097883317749iset_a @ R ) )
= ( ? [Y5: multiset_a] : ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Y5 @ A ) @ R ) ) ) ).
% Range_iff
thf(fact_1150_RangeE,axiom,
! [B: a,R: set_Product_prod_a_a] :
( ( member_a @ B @ ( range_a_a @ R ) )
=> ~ ! [A4: a] :
~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A4 @ B ) @ R ) ) ).
% RangeE
thf(fact_1151_RangeE,axiom,
! [B: multiset_a,R: set_Pr79727621955416071iset_a] :
( ( member_multiset_a @ B @ ( range_1237004097883317749iset_a @ R ) )
=> ~ ! [A4: multiset_a] :
~ ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A4 @ B ) @ R ) ) ).
% RangeE
thf(fact_1152_Range_Ointros,axiom,
! [A: a,B: a,R: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R )
=> ( member_a @ B @ ( range_a_a @ R ) ) ) ).
% Range.intros
thf(fact_1153_Range_Ointros,axiom,
! [A: multiset_a,B: multiset_a,R: set_Pr79727621955416071iset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A @ B ) @ R )
=> ( member_multiset_a @ B @ ( range_1237004097883317749iset_a @ R ) ) ) ).
% Range.intros
thf(fact_1154_Range_Osimps,axiom,
! [A: a,R: set_Product_prod_a_a] :
( ( member_a @ A @ ( range_a_a @ R ) )
= ( ? [A5: a,B5: a] :
( ( A = B5 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A5 @ B5 ) @ R ) ) ) ) ).
% Range.simps
thf(fact_1155_Range_Osimps,axiom,
! [A: multiset_a,R: set_Pr79727621955416071iset_a] :
( ( member_multiset_a @ A @ ( range_1237004097883317749iset_a @ R ) )
= ( ? [A5: multiset_a,B5: multiset_a] :
( ( A = B5 )
& ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A5 @ B5 ) @ R ) ) ) ) ).
% Range.simps
thf(fact_1156_Range_Ocases,axiom,
! [A: a,R: set_Product_prod_a_a] :
( ( member_a @ A @ ( range_a_a @ R ) )
=> ~ ! [A4: a] :
~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A4 @ A ) @ R ) ) ).
% Range.cases
thf(fact_1157_Range_Ocases,axiom,
! [A: multiset_a,R: set_Pr79727621955416071iset_a] :
( ( member_multiset_a @ A @ ( range_1237004097883317749iset_a @ R ) )
=> ~ ! [A4: multiset_a] :
~ ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A4 @ A ) @ R ) ) ).
% Range.cases
thf(fact_1158_Domain__insert,axiom,
! [A: a,B: a,R: set_Product_prod_a_a] :
( ( domain_a_a @ ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ A @ B ) @ R ) )
= ( insert_a @ A @ ( domain_a_a @ R ) ) ) ).
% Domain_insert
thf(fact_1159_Domain__insert,axiom,
! [A: multiset_a,B: multiset_a,R: set_Pr79727621955416071iset_a] :
( ( domain2009159847835016844iset_a @ ( insert4880209295958132471iset_a @ ( produc654756711066625303iset_a @ A @ B ) @ R ) )
= ( insert_multiset_a @ A @ ( domain2009159847835016844iset_a @ R ) ) ) ).
% Domain_insert
thf(fact_1160_in__Sup__multisetD,axiom,
! [X3: a,A3: set_multiset_a] :
( ( member_a @ X3 @ ( set_mset_a @ ( comple1815993499545739569iset_a @ A3 ) ) )
=> ? [X5: multiset_a] :
( ( member_multiset_a @ X5 @ A3 )
& ( member_a @ X3 @ ( set_mset_a @ X5 ) ) ) ) ).
% in_Sup_multisetD
thf(fact_1161_in__Sup__multisetD,axiom,
! [X3: produc6518373309651786023iset_a,A3: set_mu7670847959134527309iset_a] :
( ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ ( comple6783140961160028890iset_a @ A3 ) ) )
=> ? [X5: multis4781622204924204013iset_a] :
( ( member2597461214246177430iset_a @ X5 @ A3 )
& ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ X5 ) ) ) ) ).
% in_Sup_multisetD
thf(fact_1162_in__Sup__multisetD,axiom,
! [X3: product_prod_a_a,A3: set_mu7436117841396106061od_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ ( comple2555304808976473818od_a_a @ A3 ) ) )
=> ? [X5: multis501812127501805293od_a_a] :
( ( member7700929758516679318od_a_a @ X5 @ A3 )
& ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ X5 ) ) ) ) ).
% in_Sup_multisetD
thf(fact_1163_Domain_Ocases,axiom,
! [A: a,R: set_Product_prod_a_a] :
( ( member_a @ A @ ( domain_a_a @ R ) )
=> ~ ! [B4: a] :
~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B4 ) @ R ) ) ).
% Domain.cases
thf(fact_1164_Domain_Ocases,axiom,
! [A: multiset_a,R: set_Pr79727621955416071iset_a] :
( ( member_multiset_a @ A @ ( domain2009159847835016844iset_a @ R ) )
=> ~ ! [B4: multiset_a] :
~ ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A @ B4 ) @ R ) ) ).
% Domain.cases
thf(fact_1165_Domain_Osimps,axiom,
! [A: a,R: set_Product_prod_a_a] :
( ( member_a @ A @ ( domain_a_a @ R ) )
= ( ? [A5: a,B5: a] :
( ( A = A5 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A5 @ B5 ) @ R ) ) ) ) ).
% Domain.simps
thf(fact_1166_Domain_Osimps,axiom,
! [A: multiset_a,R: set_Pr79727621955416071iset_a] :
( ( member_multiset_a @ A @ ( domain2009159847835016844iset_a @ R ) )
= ( ? [A5: multiset_a,B5: multiset_a] :
( ( A = A5 )
& ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A5 @ B5 ) @ R ) ) ) ) ).
% Domain.simps
thf(fact_1167_Domain_ODomainI,axiom,
! [A: a,B: a,R: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R )
=> ( member_a @ A @ ( domain_a_a @ R ) ) ) ).
% Domain.DomainI
thf(fact_1168_Domain_ODomainI,axiom,
! [A: multiset_a,B: multiset_a,R: set_Pr79727621955416071iset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A @ B ) @ R )
=> ( member_multiset_a @ A @ ( domain2009159847835016844iset_a @ R ) ) ) ).
% Domain.DomainI
thf(fact_1169_DomainE,axiom,
! [A: a,R: set_Product_prod_a_a] :
( ( member_a @ A @ ( domain_a_a @ R ) )
=> ~ ! [B4: a] :
~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B4 ) @ R ) ) ).
% DomainE
thf(fact_1170_DomainE,axiom,
! [A: multiset_a,R: set_Pr79727621955416071iset_a] :
( ( member_multiset_a @ A @ ( domain2009159847835016844iset_a @ R ) )
=> ~ ! [B4: multiset_a] :
~ ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A @ B4 ) @ R ) ) ).
% DomainE
thf(fact_1171_Domain__iff,axiom,
! [A: a,R: set_Product_prod_a_a] :
( ( member_a @ A @ ( domain_a_a @ R ) )
= ( ? [Y5: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ Y5 ) @ R ) ) ) ).
% Domain_iff
thf(fact_1172_Domain__iff,axiom,
! [A: multiset_a,R: set_Pr79727621955416071iset_a] :
( ( member_multiset_a @ A @ ( domain2009159847835016844iset_a @ R ) )
= ( ? [Y5: multiset_a] : ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A @ Y5 ) @ R ) ) ) ).
% Domain_iff
thf(fact_1173_Sup__multiset__empty,axiom,
( ( comple1815993499545739569iset_a @ bot_bo6997605411617904272iset_a )
= zero_zero_multiset_a ) ).
% Sup_multiset_empty
thf(fact_1174_Sup__multiset__unbounded,axiom,
! [A3: set_multiset_a] :
( ~ ( condit399414615940703728iset_a @ subseteq_mset_a @ A3 )
=> ( ( comple1815993499545739569iset_a @ A3 )
= zero_zero_multiset_a ) ) ).
% Sup_multiset_unbounded
thf(fact_1175_in__Sup__multiset__iff,axiom,
! [A3: set_multiset_a,X3: a] :
( ( condit399414615940703728iset_a @ subseteq_mset_a @ A3 )
=> ( ( member_a @ X3 @ ( set_mset_a @ ( comple1815993499545739569iset_a @ A3 ) ) )
= ( ? [X6: multiset_a] :
( ( member_multiset_a @ X6 @ A3 )
& ( member_a @ X3 @ ( set_mset_a @ X6 ) ) ) ) ) ) ).
% in_Sup_multiset_iff
thf(fact_1176_in__Sup__multiset__iff,axiom,
! [A3: set_mu7670847959134527309iset_a,X3: produc6518373309651786023iset_a] :
( ( condit1245709839486664089iset_a @ subset6292090195676340366iset_a @ A3 )
=> ( ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ ( comple6783140961160028890iset_a @ A3 ) ) )
= ( ? [X6: multis4781622204924204013iset_a] :
( ( member2597461214246177430iset_a @ X6 @ A3 )
& ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ X6 ) ) ) ) ) ) ).
% in_Sup_multiset_iff
thf(fact_1177_in__Sup__multiset__iff,axiom,
! [A3: set_mu7436117841396106061od_a_a,X3: product_prod_a_a] :
( ( condit4715417034122644633od_a_a @ subset4108007524207251342od_a_a @ A3 )
=> ( ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ ( comple2555304808976473818od_a_a @ A3 ) ) )
= ( ? [X6: multis501812127501805293od_a_a] :
( ( member7700929758516679318od_a_a @ X6 @ A3 )
& ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ X6 ) ) ) ) ) ) ).
% in_Sup_multiset_iff
thf(fact_1178_set__image__mset,axiom,
! [F: a > a,M: multiset_a] :
( ( set_mset_a @ ( image_mset_a_a @ F @ M ) )
= ( image_a_a @ F @ ( set_mset_a @ M ) ) ) ).
% set_image_mset
thf(fact_1179_set__image__mset,axiom,
! [F: produc6518373309651786023iset_a > a,M: multis4781622204924204013iset_a] :
( ( set_mset_a @ ( image_2834619569719286738et_a_a @ F @ M ) )
= ( image_830901103612899740et_a_a @ F @ ( set_ms2379723421352415844iset_a @ M ) ) ) ).
% set_image_mset
thf(fact_1180_set__image__mset,axiom,
! [F: product_prod_a_a > a,M: multis501812127501805293od_a_a] :
( ( set_mset_a @ ( image_3362453674455528402_a_a_a @ F @ M ) )
= ( image_3437945252899457948_a_a_a @ F @ ( set_ms119794563918438244od_a_a @ M ) ) ) ).
% set_image_mset
thf(fact_1181_set__image__mset,axiom,
! [F: a > produc6518373309651786023iset_a,M: multiset_a] :
( ( set_ms2379723421352415844iset_a @ ( image_8645235558099244516iset_a @ F @ M ) )
= ( image_6641517091992857518iset_a @ F @ ( set_mset_a @ M ) ) ) ).
% set_image_mset
thf(fact_1182_set__image__mset,axiom,
! [F: produc6518373309651786023iset_a > produc6518373309651786023iset_a,M: multis4781622204924204013iset_a] :
( ( set_ms2379723421352415844iset_a @ ( image_1581608793515940347iset_a @ F @ M ) )
= ( image_149051612788593605iset_a @ F @ ( set_ms2379723421352415844iset_a @ M ) ) ) ).
% set_image_mset
thf(fact_1183_set__image__mset,axiom,
! [F: product_prod_a_a > produc6518373309651786023iset_a,M: multis501812127501805293od_a_a] :
( ( set_ms2379723421352415844iset_a @ ( image_1544170008705937915iset_a @ F @ M ) )
= ( image_9107824508038213061iset_a @ F @ ( set_ms119794563918438244od_a_a @ M ) ) ) ).
% set_image_mset
thf(fact_1184_set__image__mset,axiom,
! [F: a > product_prod_a_a,M: multiset_a] :
( ( set_ms119794563918438244od_a_a @ ( image_7325134204146066148od_a_a @ F @ M ) )
= ( image_7400625782589995694od_a_a @ F @ ( set_mset_a @ M ) ) ) ).
% set_image_mset
thf(fact_1185_set__image__mset,axiom,
! [F: produc6518373309651786023iset_a > product_prod_a_a,M: multis4781622204924204013iset_a] :
( ( set_ms119794563918438244od_a_a @ ( image_507504510031147515od_a_a @ F @ M ) )
= ( image_8071159009363422661od_a_a @ F @ ( set_ms2379723421352415844iset_a @ M ) ) ) ).
% set_image_mset
thf(fact_1186_set__image__mset,axiom,
! [F: product_prod_a_a > product_prod_a_a,M: multis501812127501805293od_a_a] :
( ( set_ms119794563918438244od_a_a @ ( image_1827833358229211643od_a_a @ F @ M ) )
= ( image_4636654165204879301od_a_a @ F @ ( set_ms119794563918438244od_a_a @ M ) ) ) ).
% set_image_mset
thf(fact_1187_multiset_Oset__map,axiom,
! [F: a > a,V: multiset_a] :
( ( set_mset_a @ ( image_mset_a_a @ F @ V ) )
= ( image_a_a @ F @ ( set_mset_a @ V ) ) ) ).
% multiset.set_map
thf(fact_1188_multiset_Oset__map,axiom,
! [F: produc6518373309651786023iset_a > a,V: multis4781622204924204013iset_a] :
( ( set_mset_a @ ( image_2834619569719286738et_a_a @ F @ V ) )
= ( image_830901103612899740et_a_a @ F @ ( set_ms2379723421352415844iset_a @ V ) ) ) ).
% multiset.set_map
thf(fact_1189_multiset_Oset__map,axiom,
! [F: product_prod_a_a > a,V: multis501812127501805293od_a_a] :
( ( set_mset_a @ ( image_3362453674455528402_a_a_a @ F @ V ) )
= ( image_3437945252899457948_a_a_a @ F @ ( set_ms119794563918438244od_a_a @ V ) ) ) ).
% multiset.set_map
thf(fact_1190_multiset_Oset__map,axiom,
! [F: a > produc6518373309651786023iset_a,V: multiset_a] :
( ( set_ms2379723421352415844iset_a @ ( image_8645235558099244516iset_a @ F @ V ) )
= ( image_6641517091992857518iset_a @ F @ ( set_mset_a @ V ) ) ) ).
% multiset.set_map
thf(fact_1191_multiset_Oset__map,axiom,
! [F: produc6518373309651786023iset_a > produc6518373309651786023iset_a,V: multis4781622204924204013iset_a] :
( ( set_ms2379723421352415844iset_a @ ( image_1581608793515940347iset_a @ F @ V ) )
= ( image_149051612788593605iset_a @ F @ ( set_ms2379723421352415844iset_a @ V ) ) ) ).
% multiset.set_map
thf(fact_1192_multiset_Oset__map,axiom,
! [F: product_prod_a_a > produc6518373309651786023iset_a,V: multis501812127501805293od_a_a] :
( ( set_ms2379723421352415844iset_a @ ( image_1544170008705937915iset_a @ F @ V ) )
= ( image_9107824508038213061iset_a @ F @ ( set_ms119794563918438244od_a_a @ V ) ) ) ).
% multiset.set_map
thf(fact_1193_multiset_Oset__map,axiom,
! [F: a > product_prod_a_a,V: multiset_a] :
( ( set_ms119794563918438244od_a_a @ ( image_7325134204146066148od_a_a @ F @ V ) )
= ( image_7400625782589995694od_a_a @ F @ ( set_mset_a @ V ) ) ) ).
% multiset.set_map
thf(fact_1194_multiset_Oset__map,axiom,
! [F: produc6518373309651786023iset_a > product_prod_a_a,V: multis4781622204924204013iset_a] :
( ( set_ms119794563918438244od_a_a @ ( image_507504510031147515od_a_a @ F @ V ) )
= ( image_8071159009363422661od_a_a @ F @ ( set_ms2379723421352415844iset_a @ V ) ) ) ).
% multiset.set_map
thf(fact_1195_multiset_Oset__map,axiom,
! [F: product_prod_a_a > product_prod_a_a,V: multis501812127501805293od_a_a] :
( ( set_ms119794563918438244od_a_a @ ( image_1827833358229211643od_a_a @ F @ V ) )
= ( image_4636654165204879301od_a_a @ F @ ( set_ms119794563918438244od_a_a @ V ) ) ) ).
% multiset.set_map
thf(fact_1196_subset__mset_OInf__fin_Ohom__commute,axiom,
! [H: multiset_a > multiset_a,N: set_multiset_a] :
( ! [X5: multiset_a,Y4: multiset_a] :
( ( H @ ( inter_mset_a @ X5 @ Y4 ) )
= ( inter_mset_a @ ( H @ X5 ) @ ( H @ Y4 ) ) )
=> ( ( finite2463020702752857069iset_a @ N )
=> ( ( N != bot_bo6997605411617904272iset_a )
=> ( ( H @ ( lattic1863195236566259673iset_a @ inter_mset_a @ N ) )
= ( lattic1863195236566259673iset_a @ inter_mset_a @ ( image_1938778250527104901iset_a @ H @ N ) ) ) ) ) ) ).
% subset_mset.Inf_fin.hom_commute
thf(fact_1197_image__add__0,axiom,
! [S2: set_multiset_a] :
( ( image_1938778250527104901iset_a @ ( plus_plus_multiset_a @ zero_zero_multiset_a ) @ S2 )
= S2 ) ).
% image_add_0
thf(fact_1198_subset__mset_OcInf__eq__Inf__fin,axiom,
! [X4: set_multiset_a] :
( ( finite2463020702752857069iset_a @ X4 )
=> ( ( X4 != bot_bo6997605411617904272iset_a )
=> ( ( comple6734739994952872011iset_a @ X4 )
= ( lattic1863195236566259673iset_a @ inter_mset_a @ X4 ) ) ) ) ).
% subset_mset.cInf_eq_Inf_fin
thf(fact_1199_repeat__mset__Suc,axiom,
! [N4: nat,M: multiset_a] :
( ( repeat_mset_a @ ( suc @ N4 ) @ M )
= ( plus_plus_multiset_a @ M @ ( repeat_mset_a @ N4 @ M ) ) ) ).
% repeat_mset_Suc
thf(fact_1200_in__Inf__multiset__iff,axiom,
! [A3: set_multiset_a,X3: a] :
( ( A3 != bot_bo6997605411617904272iset_a )
=> ( ( member_a @ X3 @ ( set_mset_a @ ( comple6734739994952872011iset_a @ A3 ) ) )
= ( ! [X6: multiset_a] :
( ( member_multiset_a @ X6 @ A3 )
=> ( member_a @ X3 @ ( set_mset_a @ X6 ) ) ) ) ) ) ).
% in_Inf_multiset_iff
thf(fact_1201_in__Inf__multiset__iff,axiom,
! [A3: set_mu7670847959134527309iset_a,X3: produc6518373309651786023iset_a] :
( ( A3 != bot_bo4307782150212492217iset_a )
=> ( ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ ( comple8818024563194017268iset_a @ A3 ) ) )
= ( ! [X6: multis4781622204924204013iset_a] :
( ( member2597461214246177430iset_a @ X6 @ A3 )
=> ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ X6 ) ) ) ) ) ) ).
% in_Inf_multiset_iff
thf(fact_1202_in__Inf__multiset__iff,axiom,
! [A3: set_mu7436117841396106061od_a_a,X3: product_prod_a_a] :
( ( A3 != bot_bo4878009912801868729od_a_a )
=> ( ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ ( comple134339875127444468od_a_a @ A3 ) ) )
= ( ! [X6: multis501812127501805293od_a_a] :
( ( member7700929758516679318od_a_a @ X6 @ A3 )
=> ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ X6 ) ) ) ) ) ) ).
% in_Inf_multiset_iff
thf(fact_1203_in__Inf__multisetD,axiom,
! [X3: a,A3: set_multiset_a,X4: multiset_a] :
( ( member_a @ X3 @ ( set_mset_a @ ( comple6734739994952872011iset_a @ A3 ) ) )
=> ( ( member_multiset_a @ X4 @ A3 )
=> ( member_a @ X3 @ ( set_mset_a @ X4 ) ) ) ) ).
% in_Inf_multisetD
thf(fact_1204_in__Inf__multisetD,axiom,
! [X3: produc6518373309651786023iset_a,A3: set_mu7670847959134527309iset_a,X4: multis4781622204924204013iset_a] :
( ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ ( comple8818024563194017268iset_a @ A3 ) ) )
=> ( ( member2597461214246177430iset_a @ X4 @ A3 )
=> ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ X4 ) ) ) ) ).
% in_Inf_multisetD
thf(fact_1205_in__Inf__multisetD,axiom,
! [X3: product_prod_a_a,A3: set_mu7436117841396106061od_a_a,X4: multis501812127501805293od_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ ( comple134339875127444468od_a_a @ A3 ) ) )
=> ( ( member7700929758516679318od_a_a @ X4 @ A3 )
=> ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ X4 ) ) ) ) ).
% in_Inf_multisetD
thf(fact_1206_Inf__multiset__empty,axiom,
( ( comple6734739994952872011iset_a @ bot_bo6997605411617904272iset_a )
= zero_zero_multiset_a ) ).
% Inf_multiset_empty
thf(fact_1207_size__multiset__single,axiom,
! [F: a > nat,B: a] :
( ( size_multiset_a @ F @ ( add_mset_a @ B @ zero_zero_multiset_a ) )
= ( suc @ ( F @ B ) ) ) ).
% size_multiset_single
thf(fact_1208_Abstract__Rewriting_Ochain__mono,axiom,
! [R4: set_Product_prod_a_a,R3: set_Product_prod_a_a,Seq: nat > a] :
( ( ord_le746702958409616551od_a_a @ R4 @ R3 )
=> ( ! [I4: nat] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( Seq @ I4 ) @ ( Seq @ ( suc @ I4 ) ) ) @ R4 )
=> ! [I5: nat] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( Seq @ I5 ) @ ( Seq @ ( suc @ I5 ) ) ) @ R3 ) ) ) ).
% Abstract_Rewriting.chain_mono
thf(fact_1209_Abstract__Rewriting_Ochain__mono,axiom,
! [R4: set_Pr79727621955416071iset_a,R3: set_Pr79727621955416071iset_a,Seq: nat > multiset_a] :
( ( ord_le5871882978999109543iset_a @ R4 @ R3 )
=> ( ! [I4: nat] : ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( Seq @ I4 ) @ ( Seq @ ( suc @ I4 ) ) ) @ R4 )
=> ! [I5: nat] : ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( Seq @ I5 ) @ ( Seq @ ( suc @ I5 ) ) ) @ R3 ) ) ) ).
% Abstract_Rewriting.chain_mono
thf(fact_1210_wcount__union,axiom,
! [F: a > nat,M: multiset_a,N: multiset_a,A: a] :
( ( wcount_a @ F @ ( plus_plus_multiset_a @ M @ N ) @ A )
= ( plus_plus_nat @ ( wcount_a @ F @ M @ A ) @ ( wcount_a @ F @ N @ A ) ) ) ).
% wcount_union
thf(fact_1211_subset__mset_OcInf__insert,axiom,
! [X4: set_multiset_a,A: multiset_a] :
( ( X4 != bot_bo6997605411617904272iset_a )
=> ( ( condit6464861502438683612iset_a @ subseteq_mset_a @ X4 )
=> ( ( comple6734739994952872011iset_a @ ( insert_multiset_a @ A @ X4 ) )
= ( inter_mset_a @ A @ ( comple6734739994952872011iset_a @ X4 ) ) ) ) ) ).
% subset_mset.cInf_insert
thf(fact_1212_subset__mset_OcInf__insert__If,axiom,
! [X4: set_multiset_a,A: multiset_a] :
( ( condit6464861502438683612iset_a @ subseteq_mset_a @ X4 )
=> ( ( ( X4 = bot_bo6997605411617904272iset_a )
=> ( ( comple6734739994952872011iset_a @ ( insert_multiset_a @ A @ X4 ) )
= A ) )
& ( ( X4 != bot_bo6997605411617904272iset_a )
=> ( ( comple6734739994952872011iset_a @ ( insert_multiset_a @ A @ X4 ) )
= ( inter_mset_a @ A @ ( comple6734739994952872011iset_a @ X4 ) ) ) ) ) ) ).
% subset_mset.cInf_insert_If
thf(fact_1213_subset__mset_Oless__eq__cInf__inter,axiom,
! [A3: set_multiset_a,B3: set_multiset_a] :
( ( condit6464861502438683612iset_a @ subseteq_mset_a @ A3 )
=> ( ( condit6464861502438683612iset_a @ subseteq_mset_a @ B3 )
=> ( ( ( inf_in767754420712842514iset_a @ A3 @ B3 )
!= bot_bo6997605411617904272iset_a )
=> ( subseteq_mset_a @ ( inter_mset_a @ ( comple6734739994952872011iset_a @ A3 ) @ ( comple6734739994952872011iset_a @ B3 ) ) @ ( comple6734739994952872011iset_a @ ( inf_in767754420712842514iset_a @ A3 @ B3 ) ) ) ) ) ) ).
% subset_mset.less_eq_cInf_inter
thf(fact_1214_subset__mset_OcInf__union__distrib,axiom,
! [A3: set_multiset_a,B3: set_multiset_a] :
( ( A3 != bot_bo6997605411617904272iset_a )
=> ( ( condit6464861502438683612iset_a @ subseteq_mset_a @ A3 )
=> ( ( B3 != bot_bo6997605411617904272iset_a )
=> ( ( condit6464861502438683612iset_a @ subseteq_mset_a @ B3 )
=> ( ( comple6734739994952872011iset_a @ ( sup_su2684633059243589112iset_a @ A3 @ B3 ) )
= ( inter_mset_a @ ( comple6734739994952872011iset_a @ A3 ) @ ( comple6734739994952872011iset_a @ B3 ) ) ) ) ) ) ) ).
% subset_mset.cInf_union_distrib
thf(fact_1215_size__Suc__Diff1,axiom,
! [X3: a,M: multiset_a] :
( ( member_a @ X3 @ ( set_mset_a @ M ) )
=> ( ( suc @ ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ X3 @ zero_zero_multiset_a ) ) ) )
= ( size_size_multiset_a @ M ) ) ) ).
% size_Suc_Diff1
thf(fact_1216_size__Suc__Diff1,axiom,
! [X3: produc6518373309651786023iset_a,M: multis4781622204924204013iset_a] :
( ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ M ) )
=> ( ( suc @ ( size_s2629572463880657497iset_a @ ( minus_9211619285413683668iset_a @ M @ ( add_ms5348986034529679365iset_a @ X3 @ zero_z4591712717340762532iset_a ) ) ) )
= ( size_s2629572463880657497iset_a @ M ) ) ) ).
% size_Suc_Diff1
thf(fact_1217_size__Suc__Diff1,axiom,
! [X3: product_prod_a_a,M: multis501812127501805293od_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ M ) )
=> ( ( suc @ ( size_s781968976467208537od_a_a @ ( minus_7313152569058643924od_a_a @ M @ ( add_ms8655138167283798533od_a_a @ X3 @ zero_z3024502163831676836od_a_a ) ) ) )
= ( size_s781968976467208537od_a_a @ M ) ) ) ).
% size_Suc_Diff1
thf(fact_1218_count__greater__eq__Suc__zero__iff,axiom,
! [M: multiset_a,X3: a] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( count_a @ M @ X3 ) )
= ( member_a @ X3 @ ( set_mset_a @ M ) ) ) ).
% count_greater_eq_Suc_zero_iff
thf(fact_1219_count__greater__eq__Suc__zero__iff,axiom,
! [M: multis4781622204924204013iset_a,X3: produc6518373309651786023iset_a] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( count_5092915046799791205iset_a @ M @ X3 ) )
= ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ M ) ) ) ).
% count_greater_eq_Suc_zero_iff
thf(fact_1220_count__greater__eq__Suc__zero__iff,axiom,
! [M: multis501812127501805293od_a_a,X3: product_prod_a_a] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( count_6046493980532723301od_a_a @ M @ X3 ) )
= ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ M ) ) ) ).
% count_greater_eq_Suc_zero_iff
thf(fact_1221_count__empty,axiom,
! [A: a] :
( ( count_a @ zero_zero_multiset_a @ A )
= zero_zero_nat ) ).
% count_empty
thf(fact_1222_count__union,axiom,
! [M: multiset_a,N: multiset_a,A: a] :
( ( count_a @ ( plus_plus_multiset_a @ M @ N ) @ A )
= ( plus_plus_nat @ ( count_a @ M @ A ) @ ( count_a @ N @ A ) ) ) ).
% count_union
thf(fact_1223_size__empty,axiom,
( ( size_size_multiset_a @ zero_zero_multiset_a )
= zero_zero_nat ) ).
% size_empty
thf(fact_1224_size__eq__0__iff__empty,axiom,
! [M: multiset_a] :
( ( ( size_size_multiset_a @ M )
= zero_zero_nat )
= ( M = zero_zero_multiset_a ) ) ).
% size_eq_0_iff_empty
thf(fact_1225_size__union,axiom,
! [M: multiset_a,N: multiset_a] :
( ( size_size_multiset_a @ ( plus_plus_multiset_a @ M @ N ) )
= ( plus_plus_nat @ ( size_size_multiset_a @ M ) @ ( size_size_multiset_a @ N ) ) ) ).
% size_union
thf(fact_1226_count__diff,axiom,
! [M: multiset_a,N: multiset_a,A: a] :
( ( count_a @ ( minus_3765977307040488491iset_a @ M @ N ) @ A )
= ( minus_minus_nat @ ( count_a @ M @ A ) @ ( count_a @ N @ A ) ) ) ).
% count_diff
thf(fact_1227_count__diff,axiom,
! [M: multis4781622204924204013iset_a,N: multis4781622204924204013iset_a,A: produc6518373309651786023iset_a] :
( ( count_5092915046799791205iset_a @ ( minus_9211619285413683668iset_a @ M @ N ) @ A )
= ( minus_minus_nat @ ( count_5092915046799791205iset_a @ M @ A ) @ ( count_5092915046799791205iset_a @ N @ A ) ) ) ).
% count_diff
thf(fact_1228_count__diff,axiom,
! [M: multis501812127501805293od_a_a,N: multis501812127501805293od_a_a,A: product_prod_a_a] :
( ( count_6046493980532723301od_a_a @ ( minus_7313152569058643924od_a_a @ M @ N ) @ A )
= ( minus_minus_nat @ ( count_6046493980532723301od_a_a @ M @ A ) @ ( count_6046493980532723301od_a_a @ N @ A ) ) ) ).
% count_diff
thf(fact_1229_count__mset__set_I3_J,axiom,
! [X3: product_prod_a_a,A3: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ X3 @ A3 )
=> ( ( count_6046493980532723301od_a_a @ ( mset_s1230098591470296142od_a_a @ A3 ) @ X3 )
= zero_zero_nat ) ) ).
% count_mset_set(3)
thf(fact_1230_count__mset__set_I3_J,axiom,
! [X3: produc6518373309651786023iset_a,A3: set_Pr79727621955416071iset_a] :
( ~ ( member5199237121806060112iset_a @ X3 @ A3 )
=> ( ( count_5092915046799791205iset_a @ ( mset_s4664099280310381902iset_a @ A3 ) @ X3 )
= zero_zero_nat ) ) ).
% count_mset_set(3)
thf(fact_1231_count__mset__set_I3_J,axiom,
! [X3: a,A3: set_a] :
( ~ ( member_a @ X3 @ A3 )
=> ( ( count_a @ ( mset_set_a @ A3 ) @ X3 )
= zero_zero_nat ) ) ).
% count_mset_set(3)
thf(fact_1232_in__countE,axiom,
! [X3: a,M: multiset_a] :
( ( member_a @ X3 @ ( set_mset_a @ M ) )
=> ~ ! [N5: nat] :
( ( count_a @ M @ X3 )
!= ( suc @ N5 ) ) ) ).
% in_countE
thf(fact_1233_in__countE,axiom,
! [X3: produc6518373309651786023iset_a,M: multis4781622204924204013iset_a] :
( ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ M ) )
=> ~ ! [N5: nat] :
( ( count_5092915046799791205iset_a @ M @ X3 )
!= ( suc @ N5 ) ) ) ).
% in_countE
thf(fact_1234_in__countE,axiom,
! [X3: product_prod_a_a,M: multis501812127501805293od_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ M ) )
=> ~ ! [N5: nat] :
( ( count_6046493980532723301od_a_a @ M @ X3 )
!= ( suc @ N5 ) ) ) ).
% in_countE
thf(fact_1235_size__eq__Suc__imp__elem,axiom,
! [M: multiset_a,N4: nat] :
( ( ( size_size_multiset_a @ M )
= ( suc @ N4 ) )
=> ? [A4: a] : ( member_a @ A4 @ ( set_mset_a @ M ) ) ) ).
% size_eq_Suc_imp_elem
thf(fact_1236_size__eq__Suc__imp__elem,axiom,
! [M: multis4781622204924204013iset_a,N4: nat] :
( ( ( size_s2629572463880657497iset_a @ M )
= ( suc @ N4 ) )
=> ? [A4: produc6518373309651786023iset_a] : ( member5199237121806060112iset_a @ A4 @ ( set_ms2379723421352415844iset_a @ M ) ) ) ).
% size_eq_Suc_imp_elem
thf(fact_1237_size__eq__Suc__imp__elem,axiom,
! [M: multis501812127501805293od_a_a,N4: nat] :
( ( ( size_s781968976467208537od_a_a @ M )
= ( suc @ N4 ) )
=> ? [A4: product_prod_a_a] : ( member1426531477525435216od_a_a @ A4 @ ( set_ms119794563918438244od_a_a @ M ) ) ) ).
% size_eq_Suc_imp_elem
thf(fact_1238_count__in__diffI,axiom,
! [N: multiset_a,X3: a,M: multiset_a] :
( ! [N5: nat] :
( ( count_a @ N @ X3 )
!= ( plus_plus_nat @ N5 @ ( count_a @ M @ X3 ) ) )
=> ( member_a @ X3 @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ M @ N ) ) ) ) ).
% count_in_diffI
thf(fact_1239_count__in__diffI,axiom,
! [N: multis4781622204924204013iset_a,X3: produc6518373309651786023iset_a,M: multis4781622204924204013iset_a] :
( ! [N5: nat] :
( ( count_5092915046799791205iset_a @ N @ X3 )
!= ( plus_plus_nat @ N5 @ ( count_5092915046799791205iset_a @ M @ X3 ) ) )
=> ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ ( minus_9211619285413683668iset_a @ M @ N ) ) ) ) ).
% count_in_diffI
thf(fact_1240_count__in__diffI,axiom,
! [N: multis501812127501805293od_a_a,X3: product_prod_a_a,M: multis501812127501805293od_a_a] :
( ! [N5: nat] :
( ( count_6046493980532723301od_a_a @ N @ X3 )
!= ( plus_plus_nat @ N5 @ ( count_6046493980532723301od_a_a @ M @ X3 ) ) )
=> ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ ( minus_7313152569058643924od_a_a @ M @ N ) ) ) ) ).
% count_in_diffI
thf(fact_1241_minus__multiset_Orep__eq,axiom,
! [X3: multiset_a,Xa3: multiset_a] :
( ( count_a @ ( minus_3765977307040488491iset_a @ X3 @ Xa3 ) )
= ( ^ [A5: a] : ( minus_minus_nat @ ( count_a @ X3 @ A5 ) @ ( count_a @ Xa3 @ A5 ) ) ) ) ).
% minus_multiset.rep_eq
thf(fact_1242_minus__multiset_Orep__eq,axiom,
! [X3: multis4781622204924204013iset_a,Xa3: multis4781622204924204013iset_a] :
( ( count_5092915046799791205iset_a @ ( minus_9211619285413683668iset_a @ X3 @ Xa3 ) )
= ( ^ [A5: produc6518373309651786023iset_a] : ( minus_minus_nat @ ( count_5092915046799791205iset_a @ X3 @ A5 ) @ ( count_5092915046799791205iset_a @ Xa3 @ A5 ) ) ) ) ).
% minus_multiset.rep_eq
thf(fact_1243_minus__multiset_Orep__eq,axiom,
! [X3: multis501812127501805293od_a_a,Xa3: multis501812127501805293od_a_a] :
( ( count_6046493980532723301od_a_a @ ( minus_7313152569058643924od_a_a @ X3 @ Xa3 ) )
= ( ^ [A5: product_prod_a_a] : ( minus_minus_nat @ ( count_6046493980532723301od_a_a @ X3 @ A5 ) @ ( count_6046493980532723301od_a_a @ Xa3 @ A5 ) ) ) ) ).
% minus_multiset.rep_eq
thf(fact_1244_plus__multiset_Orep__eq,axiom,
! [X3: multiset_a,Xa3: multiset_a] :
( ( count_a @ ( plus_plus_multiset_a @ X3 @ Xa3 ) )
= ( ^ [A5: a] : ( plus_plus_nat @ ( count_a @ X3 @ A5 ) @ ( count_a @ Xa3 @ A5 ) ) ) ) ).
% plus_multiset.rep_eq
thf(fact_1245_zero__multiset_Orep__eq,axiom,
( ( count_a @ zero_zero_multiset_a )
= ( ^ [A5: a] : zero_zero_nat ) ) ).
% zero_multiset.rep_eq
thf(fact_1246_count__inI,axiom,
! [M: multiset_a,X3: a] :
( ( ( count_a @ M @ X3 )
!= zero_zero_nat )
=> ( member_a @ X3 @ ( set_mset_a @ M ) ) ) ).
% count_inI
thf(fact_1247_count__inI,axiom,
! [M: multis4781622204924204013iset_a,X3: produc6518373309651786023iset_a] :
( ( ( count_5092915046799791205iset_a @ M @ X3 )
!= zero_zero_nat )
=> ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ M ) ) ) ).
% count_inI
thf(fact_1248_count__inI,axiom,
! [M: multis501812127501805293od_a_a,X3: product_prod_a_a] :
( ( ( count_6046493980532723301od_a_a @ M @ X3 )
!= zero_zero_nat )
=> ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ M ) ) ) ).
% count_inI
thf(fact_1249_count__eq__zero__iff,axiom,
! [M: multiset_a,X3: a] :
( ( ( count_a @ M @ X3 )
= zero_zero_nat )
= ( ~ ( member_a @ X3 @ ( set_mset_a @ M ) ) ) ) ).
% count_eq_zero_iff
thf(fact_1250_count__eq__zero__iff,axiom,
! [M: multis4781622204924204013iset_a,X3: produc6518373309651786023iset_a] :
( ( ( count_5092915046799791205iset_a @ M @ X3 )
= zero_zero_nat )
= ( ~ ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ M ) ) ) ) ).
% count_eq_zero_iff
thf(fact_1251_count__eq__zero__iff,axiom,
! [M: multis501812127501805293od_a_a,X3: product_prod_a_a] :
( ( ( count_6046493980532723301od_a_a @ M @ X3 )
= zero_zero_nat )
= ( ~ ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ M ) ) ) ) ).
% count_eq_zero_iff
thf(fact_1252_size__Diff__submset,axiom,
! [M: multiset_a,M5: multiset_a] :
( ( subseteq_mset_a @ M @ M5 )
=> ( ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ M5 @ M ) )
= ( minus_minus_nat @ ( size_size_multiset_a @ M5 ) @ ( size_size_multiset_a @ M ) ) ) ) ).
% size_Diff_submset
thf(fact_1253_size__Diff__submset,axiom,
! [M: multis4781622204924204013iset_a,M5: multis4781622204924204013iset_a] :
( ( subset6292090195676340366iset_a @ M @ M5 )
=> ( ( size_s2629572463880657497iset_a @ ( minus_9211619285413683668iset_a @ M5 @ M ) )
= ( minus_minus_nat @ ( size_s2629572463880657497iset_a @ M5 ) @ ( size_s2629572463880657497iset_a @ M ) ) ) ) ).
% size_Diff_submset
thf(fact_1254_size__Diff__submset,axiom,
! [M: multis501812127501805293od_a_a,M5: multis501812127501805293od_a_a] :
( ( subset4108007524207251342od_a_a @ M @ M5 )
=> ( ( size_s781968976467208537od_a_a @ ( minus_7313152569058643924od_a_a @ M5 @ M ) )
= ( minus_minus_nat @ ( size_s781968976467208537od_a_a @ M5 ) @ ( size_s781968976467208537od_a_a @ M ) ) ) ) ).
% size_Diff_submset
thf(fact_1255_diff__size__le__size__Diff,axiom,
! [M: multiset_a,M5: multiset_a] : ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_multiset_a @ M ) @ ( size_size_multiset_a @ M5 ) ) @ ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ M @ M5 ) ) ) ).
% diff_size_le_size_Diff
thf(fact_1256_diff__size__le__size__Diff,axiom,
! [M: multis4781622204924204013iset_a,M5: multis4781622204924204013iset_a] : ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s2629572463880657497iset_a @ M ) @ ( size_s2629572463880657497iset_a @ M5 ) ) @ ( size_s2629572463880657497iset_a @ ( minus_9211619285413683668iset_a @ M @ M5 ) ) ) ).
% diff_size_le_size_Diff
thf(fact_1257_diff__size__le__size__Diff,axiom,
! [M: multis501812127501805293od_a_a,M5: multis501812127501805293od_a_a] : ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s781968976467208537od_a_a @ M ) @ ( size_s781968976467208537od_a_a @ M5 ) ) @ ( size_s781968976467208537od_a_a @ ( minus_7313152569058643924od_a_a @ M @ M5 ) ) ) ).
% diff_size_le_size_Diff
thf(fact_1258_size__Un__Int,axiom,
! [A3: multiset_a,B3: multiset_a] :
( ( plus_plus_nat @ ( size_size_multiset_a @ A3 ) @ ( size_size_multiset_a @ B3 ) )
= ( plus_plus_nat @ ( size_size_multiset_a @ ( union_mset_a @ A3 @ B3 ) ) @ ( size_size_multiset_a @ ( inter_mset_a @ A3 @ B3 ) ) ) ) ).
% size_Un_Int
thf(fact_1259_size__Diff__subset__Int,axiom,
! [M: multiset_a,M5: multiset_a] :
( ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ M @ M5 ) )
= ( minus_minus_nat @ ( size_size_multiset_a @ M ) @ ( size_size_multiset_a @ ( inter_mset_a @ M @ M5 ) ) ) ) ).
% size_Diff_subset_Int
thf(fact_1260_size__Diff__subset__Int,axiom,
! [M: multis4781622204924204013iset_a,M5: multis4781622204924204013iset_a] :
( ( size_s2629572463880657497iset_a @ ( minus_9211619285413683668iset_a @ M @ M5 ) )
= ( minus_minus_nat @ ( size_s2629572463880657497iset_a @ M ) @ ( size_s2629572463880657497iset_a @ ( inter_1588963877939557482iset_a @ M @ M5 ) ) ) ) ).
% size_Diff_subset_Int
thf(fact_1261_size__Diff__subset__Int,axiom,
! [M: multis501812127501805293od_a_a,M5: multis501812127501805293od_a_a] :
( ( size_s781968976467208537od_a_a @ ( minus_7313152569058643924od_a_a @ M @ M5 ) )
= ( minus_minus_nat @ ( size_s781968976467208537od_a_a @ M ) @ ( size_s781968976467208537od_a_a @ ( inter_3651181809579396970od_a_a @ M @ M5 ) ) ) ) ).
% size_Diff_subset_Int
thf(fact_1262_in__diff__countE,axiom,
! [X3: a,M: multiset_a,N: multiset_a] :
( ( member_a @ X3 @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ M @ N ) ) )
=> ~ ! [N5: nat] :
( ( count_a @ M @ X3 )
!= ( plus_plus_nat @ ( suc @ N5 ) @ ( count_a @ N @ X3 ) ) ) ) ).
% in_diff_countE
thf(fact_1263_in__diff__countE,axiom,
! [X3: produc6518373309651786023iset_a,M: multis4781622204924204013iset_a,N: multis4781622204924204013iset_a] :
( ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ ( minus_9211619285413683668iset_a @ M @ N ) ) )
=> ~ ! [N5: nat] :
( ( count_5092915046799791205iset_a @ M @ X3 )
!= ( plus_plus_nat @ ( suc @ N5 ) @ ( count_5092915046799791205iset_a @ N @ X3 ) ) ) ) ).
% in_diff_countE
thf(fact_1264_in__diff__countE,axiom,
! [X3: product_prod_a_a,M: multis501812127501805293od_a_a,N: multis501812127501805293od_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ ( minus_7313152569058643924od_a_a @ M @ N ) ) )
=> ~ ! [N5: nat] :
( ( count_6046493980532723301od_a_a @ M @ X3 )
!= ( plus_plus_nat @ ( suc @ N5 ) @ ( count_6046493980532723301od_a_a @ N @ X3 ) ) ) ) ).
% in_diff_countE
thf(fact_1265_size__mset__SucE,axiom,
! [A3: multiset_a,N4: nat] :
( ( ( size_size_multiset_a @ A3 )
= ( suc @ N4 ) )
=> ~ ! [A4: a,B7: multiset_a] :
( ( A3
= ( plus_plus_multiset_a @ ( add_mset_a @ A4 @ zero_zero_multiset_a ) @ B7 ) )
=> ( ( size_size_multiset_a @ B7 )
!= N4 ) ) ) ).
% size_mset_SucE
thf(fact_1266_size__Diff1__le,axiom,
! [M: multiset_a,X3: a] : ( ord_less_eq_nat @ ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ X3 @ zero_zero_multiset_a ) ) ) @ ( size_size_multiset_a @ M ) ) ).
% size_Diff1_le
thf(fact_1267_size__Diff1__le,axiom,
! [M: multis4781622204924204013iset_a,X3: produc6518373309651786023iset_a] : ( ord_less_eq_nat @ ( size_s2629572463880657497iset_a @ ( minus_9211619285413683668iset_a @ M @ ( add_ms5348986034529679365iset_a @ X3 @ zero_z4591712717340762532iset_a ) ) ) @ ( size_s2629572463880657497iset_a @ M ) ) ).
% size_Diff1_le
thf(fact_1268_size__Diff1__le,axiom,
! [M: multis501812127501805293od_a_a,X3: product_prod_a_a] : ( ord_less_eq_nat @ ( size_s781968976467208537od_a_a @ ( minus_7313152569058643924od_a_a @ M @ ( add_ms8655138167283798533od_a_a @ X3 @ zero_z3024502163831676836od_a_a ) ) ) @ ( size_s781968976467208537od_a_a @ M ) ) ).
% size_Diff1_le
thf(fact_1269_size__Un__disjoint,axiom,
! [A3: multiset_a,B3: multiset_a] :
( ( ( inter_mset_a @ A3 @ B3 )
= zero_zero_multiset_a )
=> ( ( size_size_multiset_a @ ( union_mset_a @ A3 @ B3 ) )
= ( plus_plus_nat @ ( size_size_multiset_a @ A3 ) @ ( size_size_multiset_a @ B3 ) ) ) ) ).
% size_Un_disjoint
thf(fact_1270_size__Diff__singleton__if,axiom,
! [X3: a,A3: multiset_a] :
( ( ( member_a @ X3 @ ( set_mset_a @ A3 ) )
=> ( ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ A3 @ ( add_mset_a @ X3 @ zero_zero_multiset_a ) ) )
= ( minus_minus_nat @ ( size_size_multiset_a @ A3 ) @ one_one_nat ) ) )
& ( ~ ( member_a @ X3 @ ( set_mset_a @ A3 ) )
=> ( ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ A3 @ ( add_mset_a @ X3 @ zero_zero_multiset_a ) ) )
= ( size_size_multiset_a @ A3 ) ) ) ) ).
% size_Diff_singleton_if
thf(fact_1271_size__Diff__singleton__if,axiom,
! [X3: produc6518373309651786023iset_a,A3: multis4781622204924204013iset_a] :
( ( ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ A3 ) )
=> ( ( size_s2629572463880657497iset_a @ ( minus_9211619285413683668iset_a @ A3 @ ( add_ms5348986034529679365iset_a @ X3 @ zero_z4591712717340762532iset_a ) ) )
= ( minus_minus_nat @ ( size_s2629572463880657497iset_a @ A3 ) @ one_one_nat ) ) )
& ( ~ ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ A3 ) )
=> ( ( size_s2629572463880657497iset_a @ ( minus_9211619285413683668iset_a @ A3 @ ( add_ms5348986034529679365iset_a @ X3 @ zero_z4591712717340762532iset_a ) ) )
= ( size_s2629572463880657497iset_a @ A3 ) ) ) ) ).
% size_Diff_singleton_if
thf(fact_1272_size__Diff__singleton__if,axiom,
! [X3: product_prod_a_a,A3: multis501812127501805293od_a_a] :
( ( ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ A3 ) )
=> ( ( size_s781968976467208537od_a_a @ ( minus_7313152569058643924od_a_a @ A3 @ ( add_ms8655138167283798533od_a_a @ X3 @ zero_z3024502163831676836od_a_a ) ) )
= ( minus_minus_nat @ ( size_s781968976467208537od_a_a @ A3 ) @ one_one_nat ) ) )
& ( ~ ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ A3 ) )
=> ( ( size_s781968976467208537od_a_a @ ( minus_7313152569058643924od_a_a @ A3 @ ( add_ms8655138167283798533od_a_a @ X3 @ zero_z3024502163831676836od_a_a ) ) )
= ( size_s781968976467208537od_a_a @ A3 ) ) ) ) ).
% size_Diff_singleton_if
thf(fact_1273_size__Diff__singleton,axiom,
! [X3: a,M: multiset_a] :
( ( member_a @ X3 @ ( set_mset_a @ M ) )
=> ( ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ X3 @ zero_zero_multiset_a ) ) )
= ( minus_minus_nat @ ( size_size_multiset_a @ M ) @ one_one_nat ) ) ) ).
% size_Diff_singleton
thf(fact_1274_size__Diff__singleton,axiom,
! [X3: produc6518373309651786023iset_a,M: multis4781622204924204013iset_a] :
( ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ M ) )
=> ( ( size_s2629572463880657497iset_a @ ( minus_9211619285413683668iset_a @ M @ ( add_ms5348986034529679365iset_a @ X3 @ zero_z4591712717340762532iset_a ) ) )
= ( minus_minus_nat @ ( size_s2629572463880657497iset_a @ M ) @ one_one_nat ) ) ) ).
% size_Diff_singleton
thf(fact_1275_size__Diff__singleton,axiom,
! [X3: product_prod_a_a,M: multis501812127501805293od_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ M ) )
=> ( ( size_s781968976467208537od_a_a @ ( minus_7313152569058643924od_a_a @ M @ ( add_ms8655138167283798533od_a_a @ X3 @ zero_z3024502163831676836od_a_a ) ) )
= ( minus_minus_nat @ ( size_s781968976467208537od_a_a @ M ) @ one_one_nat ) ) ) ).
% size_Diff_singleton
% Conjectures (1)
thf(conj_0,conjecture,
( ( m != n )
& ! [X5: a] :
( ( member_a @ X5 @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ n @ m ) ) )
=> ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ m @ n ) ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Xa2 ) @ s ) ) ) ) ).
%------------------------------------------------------------------------------