TPTP Problem File: SLH0338^1.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_00054_002245__13186458_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1684 ( 508 unt; 412 typ; 0 def)
% Number of atoms : 4245 (1679 equ; 0 cnn)
% Maximal formula atoms : 30 ( 3 avg)
% Number of connectives : 16959 ( 529 ~; 62 |; 303 &;13907 @)
% ( 0 <=>;2158 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 8 avg)
% Number of types : 91 ( 90 usr)
% Number of type conns : 692 ( 692 >; 0 *; 0 +; 0 <<)
% Number of symbols : 325 ( 322 usr; 31 con; 0-3 aty)
% Number of variables : 4685 ( 69 ^;4352 !; 264 ?;4685 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:25:20.726
%------------------------------------------------------------------------------
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thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member1096249278104964872at_nat: produc1932156733058919263at_nat > set_Pr7682762132356531903at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
member7226740684066999833at_nat: produc8199716216217303280at_nat > set_Pr9093778441882193744at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
member6987746275253522745st_nat: produc4155638208356863248st_nat > set_Pr5771542735269860976st_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Multiset__Omultiset_Itf__a_J_J_Mt__List__Olist_It__Multiset__Omultiset_Itf__a_J_J_J,type,
member1640862112313989456iset_a: produc924913089875786023iset_a > set_Pr2598276052810994439iset_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
member8191768239178080336list_a: produc9164743771328383783list_a > set_Pr4048851178543822343list_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Multiset__Omultiset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Multiset__Omultiset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
member3028814343351594408at_nat: produc5900369702073646847at_nat > set_Pr2387333698486958431at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Multiset__Omultiset_It__List__Olist_Itf__a_J_J_Mt__Multiset__Omultiset_It__List__Olist_Itf__a_J_J_J,type,
member8467872774006757712list_a: produc7751923751568554279list_a > set_Pr5870458363815695111list_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__Nat__Onat_J_Mt__Multiset__Omultiset_It__Nat__Onat_J_J,type,
member2326506860204807626et_nat: produc7112813594591200289et_nat > set_Pr7086239977703175297et_nat > $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__Nat__Onat_Mt__Nat__Onat_J,type,
member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $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_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
member6916131784381031696od_a_a: produc1790784290997491175od_a_a > set_Pr303625483582100167od_a_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_Mtf__a_J,type,
member2419060323508899175et_a_a: produc1703111301070695742et_a_a > set_Pr2303359794933361566et_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_J,type,
member7748342945563660560iset_a: produc2622995452180120039iset_a > set_Pr9112822110687641287iset_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_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J,type,
member2829916517802586983_a_a_a: produc3802892049952890430_a_a_a > set_Pr8876520727511657886_a_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_J,type,
member4303746980753173241iset_a: produc3587797958314969808iset_a > set_Pr7128563942666297392iset_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
member3071122053849602553od_a_a: produc4044097585999906000od_a_a > set_Pr5530083903271594800od_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_A1____,type,
a1: multiset_a ).
thf(sy_v_A2____,type,
a2: multiset_a ).
thf(sy_v_B1____,type,
b1: multiset_a ).
thf(sy_v_B2____,type,
b2: multiset_a ).
thf(sy_v_NS,type,
ns: set_Product_prod_a_a ).
thf(sy_v_S,type,
s: set_Product_prod_a_a ).
thf(sy_v_xs,type,
xs: multiset_a ).
thf(sy_v_xs1____,type,
xs1: list_a ).
thf(sy_v_xs2____,type,
xs2: list_a ).
thf(sy_v_ys,type,
ys: multiset_a ).
thf(sy_v_ys1____,type,
ys1: list_a ).
thf(sy_v_ys2____,type,
ys2: list_a ).
% Relevant facts (1268)
thf(fact_0__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062xs2_O_AA2_A_061_Amset_Axs2_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Xs2: list_a] :
( a2
!= ( mset_a @ Xs2 ) ) ).
% \<open>\<And>thesis. (\<And>xs2. A2 = mset xs2 \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_1__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062ys2_O_AB2_A_061_Amset_Ays2_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Ys2: list_a] :
( b2
!= ( mset_a @ Ys2 ) ) ).
% \<open>\<And>thesis. (\<And>ys2. B2 = mset ys2 \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_2_A2,axiom,
( a2
= ( mset_a @ xs2 ) ) ).
% A2
thf(fact_3_B2,axiom,
( b2
= ( mset_a @ ys2 ) ) ).
% B2
thf(fact_4_S,axiom,
! [B: a] :
( ( member_a @ B @ ( set_mset_a @ b2 ) )
=> ? [A: a] :
( ( member_a @ A @ ( set_mset_a @ a2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ s ) ) ) ).
% S
thf(fact_5__092_060open_062xs_A_061_Amset_Axs1_A_L_Amset_Axs2_092_060close_062,axiom,
( xs
= ( plus_plus_multiset_a @ ( mset_a @ xs1 ) @ ( mset_a @ xs2 ) ) ) ).
% \<open>xs = mset xs1 + mset xs2\<close>
thf(fact_6__092_060open_062ys_A_061_Amset_Ays1_A_L_Amset_Ays2_092_060close_062,axiom,
( ys
= ( plus_plus_multiset_a @ ( mset_a @ ys1 ) @ ( mset_a @ ys2 ) ) ) ).
% \<open>ys = mset ys1 + mset ys2\<close>
thf(fact_7_prod_Oinject,axiom,
! [X1: list_nat_nat,X2: list_nat,Y1: list_nat_nat,Y2: list_nat] :
( ( ( produc7978589510830832328st_nat @ X1 @ X2 )
= ( produc7978589510830832328st_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_8_prod_Oinject,axiom,
! [X1: list_a,X2: list_a,Y1: list_a,Y2: list_a] :
( ( ( produc6837034575241423639list_a @ X1 @ X2 )
= ( produc6837034575241423639list_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_9_prod_Oinject,axiom,
! [X1: nat,X2: a,Y1: nat,Y2: a] :
( ( ( product_Pair_nat_a @ X1 @ X2 )
= ( product_Pair_nat_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_10_prod_Oinject,axiom,
! [X1: a,X2: produc6518373309651786023iset_a,Y1: a,Y2: produc6518373309651786023iset_a] :
( ( ( produc1768535192049815360iset_a @ X1 @ X2 )
= ( produc1768535192049815360iset_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_11_prod_Oinject,axiom,
! [X1: a,X2: product_prod_a_a,Y1: a,Y2: product_prod_a_a] :
( ( ( produc431845341423274048od_a_a @ X1 @ X2 )
= ( produc431845341423274048od_a_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_12_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_13_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_14_prod_Oinject,axiom,
! [X1: nat > nat,X2: nat,Y1: nat > nat,Y2: nat] :
( ( ( produc72220940542539688at_nat @ X1 @ X2 )
= ( produc72220940542539688at_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_15_old_Oprod_Oinject,axiom,
! [A2: list_nat_nat,B: list_nat,A3: list_nat_nat,B2: list_nat] :
( ( ( produc7978589510830832328st_nat @ A2 @ B )
= ( produc7978589510830832328st_nat @ A3 @ B2 ) )
= ( ( A2 = A3 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_16_old_Oprod_Oinject,axiom,
! [A2: list_a,B: list_a,A3: list_a,B2: list_a] :
( ( ( produc6837034575241423639list_a @ A2 @ B )
= ( produc6837034575241423639list_a @ A3 @ B2 ) )
= ( ( A2 = A3 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_17_old_Oprod_Oinject,axiom,
! [A2: nat,B: a,A3: nat,B2: a] :
( ( ( product_Pair_nat_a @ A2 @ B )
= ( product_Pair_nat_a @ A3 @ B2 ) )
= ( ( A2 = A3 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_18_old_Oprod_Oinject,axiom,
! [A2: a,B: produc6518373309651786023iset_a,A3: a,B2: produc6518373309651786023iset_a] :
( ( ( produc1768535192049815360iset_a @ A2 @ B )
= ( produc1768535192049815360iset_a @ A3 @ B2 ) )
= ( ( A2 = A3 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_19_old_Oprod_Oinject,axiom,
! [A2: a,B: product_prod_a_a,A3: a,B2: product_prod_a_a] :
( ( ( produc431845341423274048od_a_a @ A2 @ B )
= ( produc431845341423274048od_a_a @ A3 @ B2 ) )
= ( ( A2 = A3 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_20_old_Oprod_Oinject,axiom,
! [A2: a,B: a,A3: a,B2: a] :
( ( ( product_Pair_a_a @ A2 @ B )
= ( product_Pair_a_a @ A3 @ B2 ) )
= ( ( A2 = A3 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_21_old_Oprod_Oinject,axiom,
! [A2: multiset_a,B: multiset_a,A3: multiset_a,B2: multiset_a] :
( ( ( produc654756711066625303iset_a @ A2 @ B )
= ( produc654756711066625303iset_a @ A3 @ B2 ) )
= ( ( A2 = A3 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_22_old_Oprod_Oinject,axiom,
! [A2: nat > nat,B: nat,A3: nat > nat,B2: nat] :
( ( ( produc72220940542539688at_nat @ A2 @ B )
= ( produc72220940542539688at_nat @ A3 @ B2 ) )
= ( ( A2 = A3 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_23_old_Oprod_Oexhaust,axiom,
! [Y: produc4155638208356863248st_nat] :
~ ! [A: list_nat_nat,B3: list_nat] :
( Y
!= ( produc7978589510830832328st_nat @ A @ B3 ) ) ).
% old.prod.exhaust
thf(fact_24_old_Oprod_Oexhaust,axiom,
! [Y: produc9164743771328383783list_a] :
~ ! [A: list_a,B3: list_a] :
( Y
!= ( produc6837034575241423639list_a @ A @ B3 ) ) ).
% old.prod.exhaust
thf(fact_25_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_nat_a] :
~ ! [A: nat,B3: a] :
( Y
!= ( product_Pair_nat_a @ A @ B3 ) ) ).
% old.prod.exhaust
thf(fact_26_old_Oprod_Oexhaust,axiom,
! [Y: produc3587797958314969808iset_a] :
~ ! [A: a,B3: produc6518373309651786023iset_a] :
( Y
!= ( produc1768535192049815360iset_a @ A @ B3 ) ) ).
% old.prod.exhaust
thf(fact_27_old_Oprod_Oexhaust,axiom,
! [Y: produc4044097585999906000od_a_a] :
~ ! [A: a,B3: product_prod_a_a] :
( Y
!= ( produc431845341423274048od_a_a @ A @ B3 ) ) ).
% old.prod.exhaust
thf(fact_28_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_a_a] :
~ ! [A: a,B3: a] :
( Y
!= ( product_Pair_a_a @ A @ B3 ) ) ).
% old.prod.exhaust
thf(fact_29_old_Oprod_Oexhaust,axiom,
! [Y: produc6518373309651786023iset_a] :
~ ! [A: multiset_a,B3: multiset_a] :
( Y
!= ( produc654756711066625303iset_a @ A @ B3 ) ) ).
% old.prod.exhaust
thf(fact_30_old_Oprod_Oexhaust,axiom,
! [Y: produc8199716216217303280at_nat] :
~ ! [A: nat > nat,B3: nat] :
( Y
!= ( produc72220940542539688at_nat @ A @ B3 ) ) ).
% old.prod.exhaust
thf(fact_31_surj__pair,axiom,
! [P: produc4155638208356863248st_nat] :
? [X: list_nat_nat,Y3: list_nat] :
( P
= ( produc7978589510830832328st_nat @ X @ Y3 ) ) ).
% surj_pair
thf(fact_32_surj__pair,axiom,
! [P: produc9164743771328383783list_a] :
? [X: list_a,Y3: list_a] :
( P
= ( produc6837034575241423639list_a @ X @ Y3 ) ) ).
% surj_pair
thf(fact_33_surj__pair,axiom,
! [P: product_prod_nat_a] :
? [X: nat,Y3: a] :
( P
= ( product_Pair_nat_a @ X @ Y3 ) ) ).
% surj_pair
thf(fact_34_surj__pair,axiom,
! [P: produc3587797958314969808iset_a] :
? [X: a,Y3: produc6518373309651786023iset_a] :
( P
= ( produc1768535192049815360iset_a @ X @ Y3 ) ) ).
% surj_pair
thf(fact_35_surj__pair,axiom,
! [P: produc4044097585999906000od_a_a] :
? [X: a,Y3: product_prod_a_a] :
( P
= ( produc431845341423274048od_a_a @ X @ Y3 ) ) ).
% surj_pair
thf(fact_36_surj__pair,axiom,
! [P: product_prod_a_a] :
? [X: a,Y3: a] :
( P
= ( product_Pair_a_a @ X @ Y3 ) ) ).
% surj_pair
thf(fact_37_surj__pair,axiom,
! [P: produc6518373309651786023iset_a] :
? [X: multiset_a,Y3: multiset_a] :
( P
= ( produc654756711066625303iset_a @ X @ Y3 ) ) ).
% surj_pair
thf(fact_38_surj__pair,axiom,
! [P: produc8199716216217303280at_nat] :
? [X: nat > nat,Y3: nat] :
( P
= ( produc72220940542539688at_nat @ X @ Y3 ) ) ).
% surj_pair
thf(fact_39_prod__cases,axiom,
! [P2: produc4155638208356863248st_nat > $o,P: produc4155638208356863248st_nat] :
( ! [A: list_nat_nat,B3: list_nat] : ( P2 @ ( produc7978589510830832328st_nat @ A @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_40_prod__cases,axiom,
! [P2: produc9164743771328383783list_a > $o,P: produc9164743771328383783list_a] :
( ! [A: list_a,B3: list_a] : ( P2 @ ( produc6837034575241423639list_a @ A @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_41_prod__cases,axiom,
! [P2: product_prod_nat_a > $o,P: product_prod_nat_a] :
( ! [A: nat,B3: a] : ( P2 @ ( product_Pair_nat_a @ A @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_42_prod__cases,axiom,
! [P2: produc3587797958314969808iset_a > $o,P: produc3587797958314969808iset_a] :
( ! [A: a,B3: produc6518373309651786023iset_a] : ( P2 @ ( produc1768535192049815360iset_a @ A @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_43_prod__cases,axiom,
! [P2: produc4044097585999906000od_a_a > $o,P: produc4044097585999906000od_a_a] :
( ! [A: a,B3: product_prod_a_a] : ( P2 @ ( produc431845341423274048od_a_a @ A @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_44_prod__cases,axiom,
! [P2: product_prod_a_a > $o,P: product_prod_a_a] :
( ! [A: a,B3: a] : ( P2 @ ( product_Pair_a_a @ A @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_45_prod__cases,axiom,
! [P2: produc6518373309651786023iset_a > $o,P: produc6518373309651786023iset_a] :
( ! [A: multiset_a,B3: multiset_a] : ( P2 @ ( produc654756711066625303iset_a @ A @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_46_prod__cases,axiom,
! [P2: produc8199716216217303280at_nat > $o,P: produc8199716216217303280at_nat] :
( ! [A: nat > nat,B3: nat] : ( P2 @ ( produc72220940542539688at_nat @ A @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_47_Pair__inject,axiom,
! [A2: list_nat_nat,B: list_nat,A3: list_nat_nat,B2: list_nat] :
( ( ( produc7978589510830832328st_nat @ A2 @ B )
= ( produc7978589510830832328st_nat @ A3 @ B2 ) )
=> ~ ( ( A2 = A3 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_48_Pair__inject,axiom,
! [A2: list_a,B: list_a,A3: list_a,B2: list_a] :
( ( ( produc6837034575241423639list_a @ A2 @ B )
= ( produc6837034575241423639list_a @ A3 @ B2 ) )
=> ~ ( ( A2 = A3 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_49_Pair__inject,axiom,
! [A2: nat,B: a,A3: nat,B2: a] :
( ( ( product_Pair_nat_a @ A2 @ B )
= ( product_Pair_nat_a @ A3 @ B2 ) )
=> ~ ( ( A2 = A3 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_50_Pair__inject,axiom,
! [A2: a,B: produc6518373309651786023iset_a,A3: a,B2: produc6518373309651786023iset_a] :
( ( ( produc1768535192049815360iset_a @ A2 @ B )
= ( produc1768535192049815360iset_a @ A3 @ B2 ) )
=> ~ ( ( A2 = A3 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_51_Pair__inject,axiom,
! [A2: a,B: product_prod_a_a,A3: a,B2: product_prod_a_a] :
( ( ( produc431845341423274048od_a_a @ A2 @ B )
= ( produc431845341423274048od_a_a @ A3 @ B2 ) )
=> ~ ( ( A2 = A3 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_52_Pair__inject,axiom,
! [A2: a,B: a,A3: a,B2: a] :
( ( ( product_Pair_a_a @ A2 @ B )
= ( product_Pair_a_a @ A3 @ B2 ) )
=> ~ ( ( A2 = A3 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_53_Pair__inject,axiom,
! [A2: multiset_a,B: multiset_a,A3: multiset_a,B2: multiset_a] :
( ( ( produc654756711066625303iset_a @ A2 @ B )
= ( produc654756711066625303iset_a @ A3 @ B2 ) )
=> ~ ( ( A2 = A3 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_54_Pair__inject,axiom,
! [A2: nat > nat,B: nat,A3: nat > nat,B2: nat] :
( ( ( produc72220940542539688at_nat @ A2 @ B )
= ( produc72220940542539688at_nat @ A3 @ B2 ) )
=> ~ ( ( A2 = A3 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_55_prod__cases3,axiom,
! [Y: produc3587797958314969808iset_a] :
~ ! [A: a,B3: multiset_a,C: multiset_a] :
( Y
!= ( produc1768535192049815360iset_a @ A @ ( produc654756711066625303iset_a @ B3 @ C ) ) ) ).
% prod_cases3
thf(fact_56_prod__cases3,axiom,
! [Y: produc4044097585999906000od_a_a] :
~ ! [A: a,B3: a,C: a] :
( Y
!= ( produc431845341423274048od_a_a @ A @ ( product_Pair_a_a @ B3 @ C ) ) ) ).
% prod_cases3
thf(fact_57__C_K_K_C_I3_J,axiom,
( b1
= ( mset_a @ ys1 ) ) ).
% "**"(3)
thf(fact_58__C_K_K_C_I2_J,axiom,
( a1
= ( mset_a @ xs1 ) ) ).
% "**"(2)
thf(fact_59__C_K_K_C_I1_J,axiom,
( ( size_size_list_a @ xs1 )
= ( size_size_list_a @ ys1 ) ) ).
% "**"(1)
thf(fact_60__C_K_C_I2_J,axiom,
( ys
= ( plus_plus_multiset_a @ b1 @ b2 ) ) ).
% "*"(2)
thf(fact_61__C_K_C_I1_J,axiom,
( xs
= ( plus_plus_multiset_a @ a1 @ a2 ) ) ).
% "*"(1)
thf(fact_62_prod__induct3,axiom,
! [P2: produc3587797958314969808iset_a > $o,X3: produc3587797958314969808iset_a] :
( ! [A: a,B3: multiset_a,C: multiset_a] : ( P2 @ ( produc1768535192049815360iset_a @ A @ ( produc654756711066625303iset_a @ B3 @ C ) ) )
=> ( P2 @ X3 ) ) ).
% prod_induct3
thf(fact_63_prod__induct3,axiom,
! [P2: produc4044097585999906000od_a_a > $o,X3: produc4044097585999906000od_a_a] :
( ! [A: a,B3: a,C: a] : ( P2 @ ( produc431845341423274048od_a_a @ A @ ( product_Pair_a_a @ B3 @ C ) ) )
=> ( P2 @ X3 ) ) ).
% prod_induct3
thf(fact_64_set__mset__mset,axiom,
! [Xs: list_list_a] :
( ( set_mset_list_a @ ( mset_list_a @ Xs ) )
= ( set_list_a2 @ Xs ) ) ).
% set_mset_mset
thf(fact_65_set__mset__mset,axiom,
! [Xs: list_nat_nat] :
( ( set_mset_nat_nat @ ( mset_nat_nat @ Xs ) )
= ( set_nat_nat2 @ Xs ) ) ).
% set_mset_mset
thf(fact_66_set__mset__mset,axiom,
! [Xs: list_P6988024003011616557iset_a] :
( ( set_ms2379723421352415844iset_a @ ( mset_P2771936761501114300iset_a @ Xs ) )
= ( set_Pr5550698219922361532iset_a @ Xs ) ) ).
% set_mset_mset
thf(fact_67_set__mset__mset,axiom,
! [Xs: list_P5366131564312172150at_nat] :
( ( set_ms7357618667746797101at_nat @ ( mset_P2828777261068555141at_nat @ Xs ) )
= ( set_Pr1941165898809491589at_nat @ Xs ) ) ).
% set_mset_mset
thf(fact_68_set__mset__mset,axiom,
! [Xs: list_P1396940483166286381od_a_a] :
( ( set_ms119794563918438244od_a_a @ ( mset_P3709024939784119484od_a_a @ Xs ) )
= ( set_Product_prod_a_a2 @ Xs ) ) ).
% set_mset_mset
thf(fact_69_set__mset__mset,axiom,
! [Xs: list_m1501550643479607155iset_a] :
( ( set_ms5327031259215843498iset_a @ ( mset_m5849828674940346882iset_a @ Xs ) )
= ( set_mu7857395473942892290iset_a @ Xs ) ) ).
% set_mset_mset
thf(fact_70_set__mset__mset,axiom,
! [Xs: list_m8731804080677374323od_a_a] :
( ( set_ms8253844250382261418od_a_a @ ( mset_m7731653587048867330od_a_a @ Xs ) )
= ( set_mu2092502360340255490od_a_a @ Xs ) ) ).
% set_mset_mset
thf(fact_71_set__mset__mset,axiom,
! [Xs: list_multiset_a] :
( ( set_mset_multiset_a @ ( mset_multiset_a @ Xs ) )
= ( set_multiset_a2 @ Xs ) ) ).
% set_mset_mset
thf(fact_72_set__mset__mset,axiom,
! [Xs: list_nat] :
( ( set_mset_nat @ ( mset_nat @ Xs ) )
= ( set_nat2 @ Xs ) ) ).
% set_mset_mset
thf(fact_73_set__mset__mset,axiom,
! [Xs: list_a] :
( ( set_mset_a @ ( mset_a @ Xs ) )
= ( set_a2 @ Xs ) ) ).
% set_mset_mset
thf(fact_74_assms,axiom,
member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ xs @ ys ) @ ( multis4023584895248362547_ext_a @ ns @ s ) ).
% assms
thf(fact_75__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062A1_AA2_AB1_AB2_O_A_092_060lbrakk_062xs_A_061_AA1_A_L_AA2_059_Ays_A_061_AB1_A_L_AB2_059_A_IA1_M_AB1_J_A_092_060in_062_Amultpw_ANS_059_A_092_060And_062b_O_Ab_A_092_060in_062_D_AB2_A_092_060Longrightarrow_062_A_092_060exists_062a_O_Aa_A_092_060in_062_D_AA2_A_092_060and_062_A_Ia_M_Ab_J_A_092_060in_062_AS_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [A1: multiset_a,A22: multiset_a] :
( ( xs
= ( plus_plus_multiset_a @ A1 @ A22 ) )
=> ! [B1: multiset_a,B22: multiset_a] :
( ( ys
= ( plus_plus_multiset_a @ B1 @ B22 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A1 @ B1 ) @ ( multis2373687813370458242ltpw_a @ ns ) )
=> ~ ! [B4: a] :
( ( member_a @ B4 @ ( set_mset_a @ B22 ) )
=> ? [A: a] :
( ( member_a @ A @ ( set_mset_a @ A22 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B4 ) @ s ) ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>A1 A2 B1 B2. \<lbrakk>xs = A1 + A2; ys = B1 + B2; (A1, B1) \<in> multpw NS; \<And>b. b \<in># B2 \<Longrightarrow> \<exists>a. a \<in># A2 \<and> (a, b) \<in> S\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_76_in__multiset__in__set,axiom,
! [X3: list_a,Xs: list_list_a] :
( ( member_list_a @ X3 @ ( set_mset_list_a @ ( mset_list_a @ Xs ) ) )
= ( member_list_a @ X3 @ ( set_list_a2 @ Xs ) ) ) ).
% in_multiset_in_set
thf(fact_77_in__multiset__in__set,axiom,
! [X3: nat > nat,Xs: list_nat_nat] :
( ( member_nat_nat @ X3 @ ( set_mset_nat_nat @ ( mset_nat_nat @ Xs ) ) )
= ( member_nat_nat @ X3 @ ( set_nat_nat2 @ Xs ) ) ) ).
% in_multiset_in_set
thf(fact_78_in__multiset__in__set,axiom,
! [X3: produc8199716216217303280at_nat,Xs: list_P5366131564312172150at_nat] :
( ( member7226740684066999833at_nat @ X3 @ ( set_ms7357618667746797101at_nat @ ( mset_P2828777261068555141at_nat @ Xs ) ) )
= ( member7226740684066999833at_nat @ X3 @ ( set_Pr1941165898809491589at_nat @ Xs ) ) ) ).
% in_multiset_in_set
thf(fact_79_in__multiset__in__set,axiom,
! [X3: multis4781622204924204013iset_a,Xs: list_m1501550643479607155iset_a] :
( ( member2597461214246177430iset_a @ X3 @ ( set_ms5327031259215843498iset_a @ ( mset_m5849828674940346882iset_a @ Xs ) ) )
= ( member2597461214246177430iset_a @ X3 @ ( set_mu7857395473942892290iset_a @ Xs ) ) ) ).
% in_multiset_in_set
thf(fact_80_in__multiset__in__set,axiom,
! [X3: multis501812127501805293od_a_a,Xs: list_m8731804080677374323od_a_a] :
( ( member7700929758516679318od_a_a @ X3 @ ( set_ms8253844250382261418od_a_a @ ( mset_m7731653587048867330od_a_a @ Xs ) ) )
= ( member7700929758516679318od_a_a @ X3 @ ( set_mu2092502360340255490od_a_a @ Xs ) ) ) ).
% in_multiset_in_set
thf(fact_81_in__multiset__in__set,axiom,
! [X3: multiset_a,Xs: list_multiset_a] :
( ( member_multiset_a @ X3 @ ( set_mset_multiset_a @ ( mset_multiset_a @ Xs ) ) )
= ( member_multiset_a @ X3 @ ( set_multiset_a2 @ Xs ) ) ) ).
% in_multiset_in_set
thf(fact_82_in__multiset__in__set,axiom,
! [X3: nat,Xs: list_nat] :
( ( member_nat @ X3 @ ( set_mset_nat @ ( mset_nat @ Xs ) ) )
= ( member_nat @ X3 @ ( set_nat2 @ Xs ) ) ) ).
% in_multiset_in_set
thf(fact_83_in__multiset__in__set,axiom,
! [X3: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ ( mset_P3709024939784119484od_a_a @ Xs ) ) )
= ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Xs ) ) ) ).
% in_multiset_in_set
thf(fact_84_in__multiset__in__set,axiom,
! [X3: produc6518373309651786023iset_a,Xs: list_P6988024003011616557iset_a] :
( ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ ( mset_P2771936761501114300iset_a @ Xs ) ) )
= ( member5199237121806060112iset_a @ X3 @ ( set_Pr5550698219922361532iset_a @ Xs ) ) ) ).
% in_multiset_in_set
thf(fact_85_in__multiset__in__set,axiom,
! [X3: a,Xs: list_a] :
( ( member_a @ X3 @ ( set_mset_a @ ( mset_a @ Xs ) ) )
= ( member_a @ X3 @ ( set_a2 @ Xs ) ) ) ).
% in_multiset_in_set
thf(fact_86_add__left__cancel,axiom,
! [A2: multis4781622204924204013iset_a,B: multis4781622204924204013iset_a,C2: multis4781622204924204013iset_a] :
( ( ( plus_p1053901119268525860iset_a @ A2 @ B )
= ( plus_p1053901119268525860iset_a @ A2 @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_87_add__left__cancel,axiom,
! [A2: multis501812127501805293od_a_a,B: multis501812127501805293od_a_a,C2: multis501812127501805293od_a_a] :
( ( ( plus_p6967097483979887908od_a_a @ A2 @ B )
= ( plus_p6967097483979887908od_a_a @ A2 @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_88_add__left__cancel,axiom,
! [A2: multiset_multiset_a,B: multiset_multiset_a,C2: multiset_multiset_a] :
( ( ( plus_p6738641960240532161iset_a @ A2 @ B )
= ( plus_p6738641960240532161iset_a @ A2 @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_89_add__left__cancel,axiom,
! [A2: multiset_nat,B: multiset_nat,C2: multiset_nat] :
( ( ( plus_p6334493942879108393et_nat @ A2 @ B )
= ( plus_p6334493942879108393et_nat @ A2 @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_90_add__left__cancel,axiom,
! [A2: multiset_a,B: multiset_a,C2: multiset_a] :
( ( ( plus_plus_multiset_a @ A2 @ B )
= ( plus_plus_multiset_a @ A2 @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_91_add__left__cancel,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ A2 @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_92_add__right__cancel,axiom,
! [B: multis4781622204924204013iset_a,A2: multis4781622204924204013iset_a,C2: multis4781622204924204013iset_a] :
( ( ( plus_p1053901119268525860iset_a @ B @ A2 )
= ( plus_p1053901119268525860iset_a @ C2 @ A2 ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_93_add__right__cancel,axiom,
! [B: multis501812127501805293od_a_a,A2: multis501812127501805293od_a_a,C2: multis501812127501805293od_a_a] :
( ( ( plus_p6967097483979887908od_a_a @ B @ A2 )
= ( plus_p6967097483979887908od_a_a @ C2 @ A2 ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_94_add__right__cancel,axiom,
! [B: multiset_multiset_a,A2: multiset_multiset_a,C2: multiset_multiset_a] :
( ( ( plus_p6738641960240532161iset_a @ B @ A2 )
= ( plus_p6738641960240532161iset_a @ C2 @ A2 ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_95_add__right__cancel,axiom,
! [B: multiset_nat,A2: multiset_nat,C2: multiset_nat] :
( ( ( plus_p6334493942879108393et_nat @ B @ A2 )
= ( plus_p6334493942879108393et_nat @ C2 @ A2 ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_96_add__right__cancel,axiom,
! [B: multiset_a,A2: multiset_a,C2: multiset_a] :
( ( ( plus_plus_multiset_a @ B @ A2 )
= ( plus_plus_multiset_a @ C2 @ A2 ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_97_add__right__cancel,axiom,
! [B: nat,A2: nat,C2: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= ( plus_plus_nat @ C2 @ A2 ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_98_mset__eq__setD,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( ( mset_list_a @ Xs )
= ( mset_list_a @ Ys ) )
=> ( ( set_list_a2 @ Xs )
= ( set_list_a2 @ Ys ) ) ) ).
% mset_eq_setD
thf(fact_99_mset__eq__setD,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( mset_nat @ Xs )
= ( mset_nat @ Ys ) )
=> ( ( set_nat2 @ Xs )
= ( set_nat2 @ Ys ) ) ) ).
% mset_eq_setD
thf(fact_100_mset__eq__setD,axiom,
! [Xs: list_nat_nat,Ys: list_nat_nat] :
( ( ( mset_nat_nat @ Xs )
= ( mset_nat_nat @ Ys ) )
=> ( ( set_nat_nat2 @ Xs )
= ( set_nat_nat2 @ Ys ) ) ) ).
% mset_eq_setD
thf(fact_101_mset__eq__setD,axiom,
! [Xs: list_P6988024003011616557iset_a,Ys: list_P6988024003011616557iset_a] :
( ( ( mset_P2771936761501114300iset_a @ Xs )
= ( mset_P2771936761501114300iset_a @ Ys ) )
=> ( ( set_Pr5550698219922361532iset_a @ Xs )
= ( set_Pr5550698219922361532iset_a @ Ys ) ) ) ).
% mset_eq_setD
thf(fact_102_mset__eq__setD,axiom,
! [Xs: list_P1396940483166286381od_a_a,Ys: list_P1396940483166286381od_a_a] :
( ( ( mset_P3709024939784119484od_a_a @ Xs )
= ( mset_P3709024939784119484od_a_a @ Ys ) )
=> ( ( set_Product_prod_a_a2 @ Xs )
= ( set_Product_prod_a_a2 @ Ys ) ) ) ).
% mset_eq_setD
thf(fact_103_mset__eq__setD,axiom,
! [Xs: list_multiset_a,Ys: list_multiset_a] :
( ( ( mset_multiset_a @ Xs )
= ( mset_multiset_a @ Ys ) )
=> ( ( set_multiset_a2 @ Xs )
= ( set_multiset_a2 @ Ys ) ) ) ).
% mset_eq_setD
thf(fact_104_mset__eq__setD,axiom,
! [Xs: list_P5366131564312172150at_nat,Ys: list_P5366131564312172150at_nat] :
( ( ( mset_P2828777261068555141at_nat @ Xs )
= ( mset_P2828777261068555141at_nat @ Ys ) )
=> ( ( set_Pr1941165898809491589at_nat @ Xs )
= ( set_Pr1941165898809491589at_nat @ Ys ) ) ) ).
% mset_eq_setD
thf(fact_105_mset__eq__setD,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( mset_a @ Xs )
= ( mset_a @ Ys ) )
=> ( ( set_a2 @ Xs )
= ( set_a2 @ Ys ) ) ) ).
% mset_eq_setD
thf(fact_106_union__iff,axiom,
! [A2: list_a,A4: multiset_list_a,B5: multiset_list_a] :
( ( member_list_a @ A2 @ ( set_mset_list_a @ ( plus_p690419498615200257list_a @ A4 @ B5 ) ) )
= ( ( member_list_a @ A2 @ ( set_mset_list_a @ A4 ) )
| ( member_list_a @ A2 @ ( set_mset_list_a @ B5 ) ) ) ) ).
% union_iff
thf(fact_107_union__iff,axiom,
! [A2: nat > nat,A4: multiset_nat_nat,B5: multiset_nat_nat] :
( ( member_nat_nat @ A2 @ ( set_mset_nat_nat @ ( plus_p6288751606316705304at_nat @ A4 @ B5 ) ) )
= ( ( member_nat_nat @ A2 @ ( set_mset_nat_nat @ A4 ) )
| ( member_nat_nat @ A2 @ ( set_mset_nat_nat @ B5 ) ) ) ) ).
% union_iff
thf(fact_108_union__iff,axiom,
! [A2: multis4781622204924204013iset_a,A4: multis7221419791219231795iset_a,B5: multis7221419791219231795iset_a] :
( ( member2597461214246177430iset_a @ A2 @ ( set_ms5327031259215843498iset_a @ ( plus_p1436237118943648234iset_a @ A4 @ B5 ) ) )
= ( ( member2597461214246177430iset_a @ A2 @ ( set_ms5327031259215843498iset_a @ A4 ) )
| ( member2597461214246177430iset_a @ A2 @ ( set_ms5327031259215843498iset_a @ B5 ) ) ) ) ).
% union_iff
thf(fact_109_union__iff,axiom,
! [A2: multis501812127501805293od_a_a,A4: multis5935151272263360051od_a_a,B5: multis5935151272263360051od_a_a] :
( ( member7700929758516679318od_a_a @ A2 @ ( set_ms8253844250382261418od_a_a @ ( plus_p2133152728643999466od_a_a @ A4 @ B5 ) ) )
= ( ( member7700929758516679318od_a_a @ A2 @ ( set_ms8253844250382261418od_a_a @ A4 ) )
| ( member7700929758516679318od_a_a @ A2 @ ( set_ms8253844250382261418od_a_a @ B5 ) ) ) ) ).
% union_iff
thf(fact_110_union__iff,axiom,
! [A2: multiset_a,A4: multiset_multiset_a,B5: multiset_multiset_a] :
( ( member_multiset_a @ A2 @ ( set_mset_multiset_a @ ( plus_p6738641960240532161iset_a @ A4 @ B5 ) ) )
= ( ( member_multiset_a @ A2 @ ( set_mset_multiset_a @ A4 ) )
| ( member_multiset_a @ A2 @ ( set_mset_multiset_a @ B5 ) ) ) ) ).
% union_iff
thf(fact_111_union__iff,axiom,
! [A2: nat,A4: multiset_nat,B5: multiset_nat] :
( ( member_nat @ A2 @ ( set_mset_nat @ ( plus_p6334493942879108393et_nat @ A4 @ B5 ) ) )
= ( ( member_nat @ A2 @ ( set_mset_nat @ A4 ) )
| ( member_nat @ A2 @ ( set_mset_nat @ B5 ) ) ) ) ).
% union_iff
thf(fact_112_union__iff,axiom,
! [A2: product_prod_a_a,A4: multis501812127501805293od_a_a,B5: multis501812127501805293od_a_a] :
( ( member1426531477525435216od_a_a @ A2 @ ( set_ms119794563918438244od_a_a @ ( plus_p6967097483979887908od_a_a @ A4 @ B5 ) ) )
= ( ( member1426531477525435216od_a_a @ A2 @ ( set_ms119794563918438244od_a_a @ A4 ) )
| ( member1426531477525435216od_a_a @ A2 @ ( set_ms119794563918438244od_a_a @ B5 ) ) ) ) ).
% union_iff
thf(fact_113_union__iff,axiom,
! [A2: produc6518373309651786023iset_a,A4: multis4781622204924204013iset_a,B5: multis4781622204924204013iset_a] :
( ( member5199237121806060112iset_a @ A2 @ ( set_ms2379723421352415844iset_a @ ( plus_p1053901119268525860iset_a @ A4 @ B5 ) ) )
= ( ( member5199237121806060112iset_a @ A2 @ ( set_ms2379723421352415844iset_a @ A4 ) )
| ( member5199237121806060112iset_a @ A2 @ ( set_ms2379723421352415844iset_a @ B5 ) ) ) ) ).
% union_iff
thf(fact_114_union__iff,axiom,
! [A2: a,A4: multiset_a,B5: multiset_a] :
( ( member_a @ A2 @ ( set_mset_a @ ( plus_plus_multiset_a @ A4 @ B5 ) ) )
= ( ( member_a @ A2 @ ( set_mset_a @ A4 ) )
| ( member_a @ A2 @ ( set_mset_a @ B5 ) ) ) ) ).
% union_iff
thf(fact_115_ex__mset,axiom,
! [X4: multis4781622204924204013iset_a] :
? [Xs3: list_P6988024003011616557iset_a] :
( ( mset_P2771936761501114300iset_a @ Xs3 )
= X4 ) ).
% ex_mset
thf(fact_116_ex__mset,axiom,
! [X4: multis501812127501805293od_a_a] :
? [Xs3: list_P1396940483166286381od_a_a] :
( ( mset_P3709024939784119484od_a_a @ Xs3 )
= X4 ) ).
% ex_mset
thf(fact_117_ex__mset,axiom,
! [X4: multiset_multiset_a] :
? [Xs3: list_multiset_a] :
( ( mset_multiset_a @ Xs3 )
= X4 ) ).
% ex_mset
thf(fact_118_ex__mset,axiom,
! [X4: multis4792816872523369270at_nat] :
? [Xs3: list_P5366131564312172150at_nat] :
( ( mset_P2828777261068555141at_nat @ Xs3 )
= X4 ) ).
% ex_mset
thf(fact_119_ex__mset,axiom,
! [X4: multiset_a] :
? [Xs3: list_a] :
( ( mset_a @ Xs3 )
= X4 ) ).
% ex_mset
thf(fact_120_union__assoc,axiom,
! [M: multis4781622204924204013iset_a,N: multis4781622204924204013iset_a,K: multis4781622204924204013iset_a] :
( ( plus_p1053901119268525860iset_a @ ( plus_p1053901119268525860iset_a @ M @ N ) @ K )
= ( plus_p1053901119268525860iset_a @ M @ ( plus_p1053901119268525860iset_a @ N @ K ) ) ) ).
% union_assoc
thf(fact_121_union__assoc,axiom,
! [M: multis501812127501805293od_a_a,N: multis501812127501805293od_a_a,K: multis501812127501805293od_a_a] :
( ( plus_p6967097483979887908od_a_a @ ( plus_p6967097483979887908od_a_a @ M @ N ) @ K )
= ( plus_p6967097483979887908od_a_a @ M @ ( plus_p6967097483979887908od_a_a @ N @ K ) ) ) ).
% union_assoc
thf(fact_122_union__assoc,axiom,
! [M: multiset_multiset_a,N: multiset_multiset_a,K: multiset_multiset_a] :
( ( plus_p6738641960240532161iset_a @ ( plus_p6738641960240532161iset_a @ M @ N ) @ K )
= ( plus_p6738641960240532161iset_a @ M @ ( plus_p6738641960240532161iset_a @ N @ K ) ) ) ).
% union_assoc
thf(fact_123_union__assoc,axiom,
! [M: multiset_nat,N: multiset_nat,K: multiset_nat] :
( ( plus_p6334493942879108393et_nat @ ( plus_p6334493942879108393et_nat @ M @ N ) @ K )
= ( plus_p6334493942879108393et_nat @ M @ ( plus_p6334493942879108393et_nat @ N @ K ) ) ) ).
% union_assoc
thf(fact_124_union__assoc,axiom,
! [M: multiset_a,N: multiset_a,K: multiset_a] :
( ( plus_plus_multiset_a @ ( plus_plus_multiset_a @ M @ N ) @ K )
= ( plus_plus_multiset_a @ M @ ( plus_plus_multiset_a @ N @ K ) ) ) ).
% union_assoc
thf(fact_125_mset__eq__length,axiom,
! [Xs: list_P6988024003011616557iset_a,Ys: list_P6988024003011616557iset_a] :
( ( ( mset_P2771936761501114300iset_a @ Xs )
= ( mset_P2771936761501114300iset_a @ Ys ) )
=> ( ( size_s7238940367606365593iset_a @ Xs )
= ( size_s7238940367606365593iset_a @ Ys ) ) ) ).
% mset_eq_length
thf(fact_126_mset__eq__length,axiom,
! [Xs: list_P5366131564312172150at_nat,Ys: list_P5366131564312172150at_nat] :
( ( ( mset_P2828777261068555141at_nat @ Xs )
= ( mset_P2828777261068555141at_nat @ Ys ) )
=> ( ( size_s765634767183340770at_nat @ Xs )
= ( size_s765634767183340770at_nat @ Ys ) ) ) ).
% mset_eq_length
thf(fact_127_mset__eq__length,axiom,
! [Xs: list_P1396940483166286381od_a_a,Ys: list_P1396940483166286381od_a_a] :
( ( ( mset_P3709024939784119484od_a_a @ Xs )
= ( mset_P3709024939784119484od_a_a @ Ys ) )
=> ( ( size_s3885678630836030617od_a_a @ Xs )
= ( size_s3885678630836030617od_a_a @ Ys ) ) ) ).
% mset_eq_length
thf(fact_128_mset__eq__length,axiom,
! [Xs: list_multiset_a,Ys: list_multiset_a] :
( ( ( mset_multiset_a @ Xs )
= ( mset_multiset_a @ Ys ) )
=> ( ( size_s3003218658903709366iset_a @ Xs )
= ( size_s3003218658903709366iset_a @ Ys ) ) ) ).
% mset_eq_length
thf(fact_129_mset__eq__length,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( ( mset_list_a @ Xs )
= ( mset_list_a @ Ys ) )
=> ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) ) ) ).
% mset_eq_length
thf(fact_130_mset__eq__length,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( mset_nat @ Xs )
= ( mset_nat @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% mset_eq_length
thf(fact_131_mset__eq__length,axiom,
! [Xs: list_nat_nat,Ys: list_nat_nat] :
( ( ( mset_nat_nat @ Xs )
= ( mset_nat_nat @ Ys ) )
=> ( ( size_s8208510060688613859at_nat @ Xs )
= ( size_s8208510060688613859at_nat @ Ys ) ) ) ).
% mset_eq_length
thf(fact_132_mset__eq__length,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( mset_a @ Xs )
= ( mset_a @ Ys ) )
=> ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) ) ) ).
% mset_eq_length
thf(fact_133_add__right__imp__eq,axiom,
! [B: multis4781622204924204013iset_a,A2: multis4781622204924204013iset_a,C2: multis4781622204924204013iset_a] :
( ( ( plus_p1053901119268525860iset_a @ B @ A2 )
= ( plus_p1053901119268525860iset_a @ C2 @ A2 ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_134_add__right__imp__eq,axiom,
! [B: multis501812127501805293od_a_a,A2: multis501812127501805293od_a_a,C2: multis501812127501805293od_a_a] :
( ( ( plus_p6967097483979887908od_a_a @ B @ A2 )
= ( plus_p6967097483979887908od_a_a @ C2 @ A2 ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_135_add__right__imp__eq,axiom,
! [B: multiset_multiset_a,A2: multiset_multiset_a,C2: multiset_multiset_a] :
( ( ( plus_p6738641960240532161iset_a @ B @ A2 )
= ( plus_p6738641960240532161iset_a @ C2 @ A2 ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_136_add__right__imp__eq,axiom,
! [B: multiset_nat,A2: multiset_nat,C2: multiset_nat] :
( ( ( plus_p6334493942879108393et_nat @ B @ A2 )
= ( plus_p6334493942879108393et_nat @ C2 @ A2 ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_137_add__right__imp__eq,axiom,
! [B: multiset_a,A2: multiset_a,C2: multiset_a] :
( ( ( plus_plus_multiset_a @ B @ A2 )
= ( plus_plus_multiset_a @ C2 @ A2 ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_138_add__right__imp__eq,axiom,
! [B: nat,A2: nat,C2: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= ( plus_plus_nat @ C2 @ A2 ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_139_add__left__imp__eq,axiom,
! [A2: multis4781622204924204013iset_a,B: multis4781622204924204013iset_a,C2: multis4781622204924204013iset_a] :
( ( ( plus_p1053901119268525860iset_a @ A2 @ B )
= ( plus_p1053901119268525860iset_a @ A2 @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_140_add__left__imp__eq,axiom,
! [A2: multis501812127501805293od_a_a,B: multis501812127501805293od_a_a,C2: multis501812127501805293od_a_a] :
( ( ( plus_p6967097483979887908od_a_a @ A2 @ B )
= ( plus_p6967097483979887908od_a_a @ A2 @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_141_add__left__imp__eq,axiom,
! [A2: multiset_multiset_a,B: multiset_multiset_a,C2: multiset_multiset_a] :
( ( ( plus_p6738641960240532161iset_a @ A2 @ B )
= ( plus_p6738641960240532161iset_a @ A2 @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_142_add__left__imp__eq,axiom,
! [A2: multiset_nat,B: multiset_nat,C2: multiset_nat] :
( ( ( plus_p6334493942879108393et_nat @ A2 @ B )
= ( plus_p6334493942879108393et_nat @ A2 @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_143_add__left__imp__eq,axiom,
! [A2: multiset_a,B: multiset_a,C2: multiset_a] :
( ( ( plus_plus_multiset_a @ A2 @ B )
= ( plus_plus_multiset_a @ A2 @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_144_add__left__imp__eq,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ A2 @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_145_add_Oleft__commute,axiom,
! [B: multis4781622204924204013iset_a,A2: multis4781622204924204013iset_a,C2: multis4781622204924204013iset_a] :
( ( plus_p1053901119268525860iset_a @ B @ ( plus_p1053901119268525860iset_a @ A2 @ C2 ) )
= ( plus_p1053901119268525860iset_a @ A2 @ ( plus_p1053901119268525860iset_a @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_146_add_Oleft__commute,axiom,
! [B: multis501812127501805293od_a_a,A2: multis501812127501805293od_a_a,C2: multis501812127501805293od_a_a] :
( ( plus_p6967097483979887908od_a_a @ B @ ( plus_p6967097483979887908od_a_a @ A2 @ C2 ) )
= ( plus_p6967097483979887908od_a_a @ A2 @ ( plus_p6967097483979887908od_a_a @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_147_add_Oleft__commute,axiom,
! [B: multiset_multiset_a,A2: multiset_multiset_a,C2: multiset_multiset_a] :
( ( plus_p6738641960240532161iset_a @ B @ ( plus_p6738641960240532161iset_a @ A2 @ C2 ) )
= ( plus_p6738641960240532161iset_a @ A2 @ ( plus_p6738641960240532161iset_a @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_148_add_Oleft__commute,axiom,
! [B: multiset_nat,A2: multiset_nat,C2: multiset_nat] :
( ( plus_p6334493942879108393et_nat @ B @ ( plus_p6334493942879108393et_nat @ A2 @ C2 ) )
= ( plus_p6334493942879108393et_nat @ A2 @ ( plus_p6334493942879108393et_nat @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_149_add_Oleft__commute,axiom,
! [B: multiset_a,A2: multiset_a,C2: multiset_a] :
( ( plus_plus_multiset_a @ B @ ( plus_plus_multiset_a @ A2 @ C2 ) )
= ( plus_plus_multiset_a @ A2 @ ( plus_plus_multiset_a @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_150_add_Oleft__commute,axiom,
! [B: nat,A2: nat,C2: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A2 @ C2 ) )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_151_mem__Collect__eq,axiom,
! [A2: multis501812127501805293od_a_a,P2: multis501812127501805293od_a_a > $o] :
( ( member7700929758516679318od_a_a @ A2 @ ( collec9001736562947189720od_a_a @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_152_mem__Collect__eq,axiom,
! [A2: multiset_a,P2: multiset_a > $o] :
( ( member_multiset_a @ A2 @ ( collect_multiset_a @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_153_mem__Collect__eq,axiom,
! [A2: list_a,P2: list_a > $o] :
( ( member_list_a @ A2 @ ( collect_list_a @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_154_mem__Collect__eq,axiom,
! [A2: nat,P2: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_155_mem__Collect__eq,axiom,
! [A2: nat > nat,P2: ( nat > nat ) > $o] :
( ( member_nat_nat @ A2 @ ( collect_nat_nat @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_156_mem__Collect__eq,axiom,
! [A2: product_prod_a_a,P2: product_prod_a_a > $o] :
( ( member1426531477525435216od_a_a @ A2 @ ( collec3336397797384452498od_a_a @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_157_mem__Collect__eq,axiom,
! [A2: a,P2: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_158_mem__Collect__eq,axiom,
! [A2: produc6518373309651786023iset_a,P2: produc6518373309651786023iset_a > $o] :
( ( member5199237121806060112iset_a @ A2 @ ( collec6344740439120713362iset_a @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_159_Collect__mem__eq,axiom,
! [A4: set_mu7436117841396106061od_a_a] :
( ( collec9001736562947189720od_a_a
@ ^ [X5: multis501812127501805293od_a_a] : ( member7700929758516679318od_a_a @ X5 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_160_Collect__mem__eq,axiom,
! [A4: set_multiset_a] :
( ( collect_multiset_a
@ ^ [X5: multiset_a] : ( member_multiset_a @ X5 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_161_Collect__mem__eq,axiom,
! [A4: set_list_a] :
( ( collect_list_a
@ ^ [X5: list_a] : ( member_list_a @ X5 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_162_Collect__mem__eq,axiom,
! [A4: set_nat] :
( ( collect_nat
@ ^ [X5: nat] : ( member_nat @ X5 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_163_Collect__mem__eq,axiom,
! [A4: set_nat_nat] :
( ( collect_nat_nat
@ ^ [X5: nat > nat] : ( member_nat_nat @ X5 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_164_Collect__mem__eq,axiom,
! [A4: set_Product_prod_a_a] :
( ( collec3336397797384452498od_a_a
@ ^ [X5: product_prod_a_a] : ( member1426531477525435216od_a_a @ X5 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_165_Collect__mem__eq,axiom,
! [A4: set_a] :
( ( collect_a
@ ^ [X5: a] : ( member_a @ X5 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_166_Collect__mem__eq,axiom,
! [A4: set_Pr79727621955416071iset_a] :
( ( collec6344740439120713362iset_a
@ ^ [X5: produc6518373309651786023iset_a] : ( member5199237121806060112iset_a @ X5 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_167_Collect__cong,axiom,
! [P2: produc6518373309651786023iset_a > $o,Q: produc6518373309651786023iset_a > $o] :
( ! [X: produc6518373309651786023iset_a] :
( ( P2 @ X )
= ( Q @ X ) )
=> ( ( collec6344740439120713362iset_a @ P2 )
= ( collec6344740439120713362iset_a @ Q ) ) ) ).
% Collect_cong
thf(fact_168_Collect__cong,axiom,
! [P2: a > $o,Q: a > $o] :
( ! [X: a] :
( ( P2 @ X )
= ( Q @ X ) )
=> ( ( collect_a @ P2 )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_169_Collect__cong,axiom,
! [P2: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ! [X: product_prod_a_a] :
( ( P2 @ X )
= ( Q @ X ) )
=> ( ( collec3336397797384452498od_a_a @ P2 )
= ( collec3336397797384452498od_a_a @ Q ) ) ) ).
% Collect_cong
thf(fact_170_add_Ocommute,axiom,
( plus_p1053901119268525860iset_a
= ( ^ [A5: multis4781622204924204013iset_a,B6: multis4781622204924204013iset_a] : ( plus_p1053901119268525860iset_a @ B6 @ A5 ) ) ) ).
% add.commute
thf(fact_171_add_Ocommute,axiom,
( plus_p6967097483979887908od_a_a
= ( ^ [A5: multis501812127501805293od_a_a,B6: multis501812127501805293od_a_a] : ( plus_p6967097483979887908od_a_a @ B6 @ A5 ) ) ) ).
% add.commute
thf(fact_172_add_Ocommute,axiom,
( plus_p6738641960240532161iset_a
= ( ^ [A5: multiset_multiset_a,B6: multiset_multiset_a] : ( plus_p6738641960240532161iset_a @ B6 @ A5 ) ) ) ).
% add.commute
thf(fact_173_add_Ocommute,axiom,
( plus_p6334493942879108393et_nat
= ( ^ [A5: multiset_nat,B6: multiset_nat] : ( plus_p6334493942879108393et_nat @ B6 @ A5 ) ) ) ).
% add.commute
thf(fact_174_add_Ocommute,axiom,
( plus_plus_multiset_a
= ( ^ [A5: multiset_a,B6: multiset_a] : ( plus_plus_multiset_a @ B6 @ A5 ) ) ) ).
% add.commute
thf(fact_175_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A5: nat,B6: nat] : ( plus_plus_nat @ B6 @ A5 ) ) ) ).
% add.commute
thf(fact_176_add_Oassoc,axiom,
! [A2: multis4781622204924204013iset_a,B: multis4781622204924204013iset_a,C2: multis4781622204924204013iset_a] :
( ( plus_p1053901119268525860iset_a @ ( plus_p1053901119268525860iset_a @ A2 @ B ) @ C2 )
= ( plus_p1053901119268525860iset_a @ A2 @ ( plus_p1053901119268525860iset_a @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_177_add_Oassoc,axiom,
! [A2: multis501812127501805293od_a_a,B: multis501812127501805293od_a_a,C2: multis501812127501805293od_a_a] :
( ( plus_p6967097483979887908od_a_a @ ( plus_p6967097483979887908od_a_a @ A2 @ B ) @ C2 )
= ( plus_p6967097483979887908od_a_a @ A2 @ ( plus_p6967097483979887908od_a_a @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_178_add_Oassoc,axiom,
! [A2: multiset_multiset_a,B: multiset_multiset_a,C2: multiset_multiset_a] :
( ( plus_p6738641960240532161iset_a @ ( plus_p6738641960240532161iset_a @ A2 @ B ) @ C2 )
= ( plus_p6738641960240532161iset_a @ A2 @ ( plus_p6738641960240532161iset_a @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_179_add_Oassoc,axiom,
! [A2: multiset_nat,B: multiset_nat,C2: multiset_nat] :
( ( plus_p6334493942879108393et_nat @ ( plus_p6334493942879108393et_nat @ A2 @ B ) @ C2 )
= ( plus_p6334493942879108393et_nat @ A2 @ ( plus_p6334493942879108393et_nat @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_180_add_Oassoc,axiom,
! [A2: multiset_a,B: multiset_a,C2: multiset_a] :
( ( plus_plus_multiset_a @ ( plus_plus_multiset_a @ A2 @ B ) @ C2 )
= ( plus_plus_multiset_a @ A2 @ ( plus_plus_multiset_a @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_181_add_Oassoc,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B ) @ C2 )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_182_group__cancel_Oadd2,axiom,
! [B5: multis4781622204924204013iset_a,K2: multis4781622204924204013iset_a,B: multis4781622204924204013iset_a,A2: multis4781622204924204013iset_a] :
( ( B5
= ( plus_p1053901119268525860iset_a @ K2 @ B ) )
=> ( ( plus_p1053901119268525860iset_a @ A2 @ B5 )
= ( plus_p1053901119268525860iset_a @ K2 @ ( plus_p1053901119268525860iset_a @ A2 @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_183_group__cancel_Oadd2,axiom,
! [B5: multis501812127501805293od_a_a,K2: multis501812127501805293od_a_a,B: multis501812127501805293od_a_a,A2: multis501812127501805293od_a_a] :
( ( B5
= ( plus_p6967097483979887908od_a_a @ K2 @ B ) )
=> ( ( plus_p6967097483979887908od_a_a @ A2 @ B5 )
= ( plus_p6967097483979887908od_a_a @ K2 @ ( plus_p6967097483979887908od_a_a @ A2 @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_184_group__cancel_Oadd2,axiom,
! [B5: multiset_multiset_a,K2: multiset_multiset_a,B: multiset_multiset_a,A2: multiset_multiset_a] :
( ( B5
= ( plus_p6738641960240532161iset_a @ K2 @ B ) )
=> ( ( plus_p6738641960240532161iset_a @ A2 @ B5 )
= ( plus_p6738641960240532161iset_a @ K2 @ ( plus_p6738641960240532161iset_a @ A2 @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_185_group__cancel_Oadd2,axiom,
! [B5: multiset_nat,K2: multiset_nat,B: multiset_nat,A2: multiset_nat] :
( ( B5
= ( plus_p6334493942879108393et_nat @ K2 @ B ) )
=> ( ( plus_p6334493942879108393et_nat @ A2 @ B5 )
= ( plus_p6334493942879108393et_nat @ K2 @ ( plus_p6334493942879108393et_nat @ A2 @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_186_group__cancel_Oadd2,axiom,
! [B5: multiset_a,K2: multiset_a,B: multiset_a,A2: multiset_a] :
( ( B5
= ( plus_plus_multiset_a @ K2 @ B ) )
=> ( ( plus_plus_multiset_a @ A2 @ B5 )
= ( plus_plus_multiset_a @ K2 @ ( plus_plus_multiset_a @ A2 @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_187_group__cancel_Oadd2,axiom,
! [B5: nat,K2: nat,B: nat,A2: nat] :
( ( B5
= ( plus_plus_nat @ K2 @ B ) )
=> ( ( plus_plus_nat @ A2 @ B5 )
= ( plus_plus_nat @ K2 @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_188_group__cancel_Oadd1,axiom,
! [A4: multis4781622204924204013iset_a,K2: multis4781622204924204013iset_a,A2: multis4781622204924204013iset_a,B: multis4781622204924204013iset_a] :
( ( A4
= ( plus_p1053901119268525860iset_a @ K2 @ A2 ) )
=> ( ( plus_p1053901119268525860iset_a @ A4 @ B )
= ( plus_p1053901119268525860iset_a @ K2 @ ( plus_p1053901119268525860iset_a @ A2 @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_189_group__cancel_Oadd1,axiom,
! [A4: multis501812127501805293od_a_a,K2: multis501812127501805293od_a_a,A2: multis501812127501805293od_a_a,B: multis501812127501805293od_a_a] :
( ( A4
= ( plus_p6967097483979887908od_a_a @ K2 @ A2 ) )
=> ( ( plus_p6967097483979887908od_a_a @ A4 @ B )
= ( plus_p6967097483979887908od_a_a @ K2 @ ( plus_p6967097483979887908od_a_a @ A2 @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_190_group__cancel_Oadd1,axiom,
! [A4: multiset_multiset_a,K2: multiset_multiset_a,A2: multiset_multiset_a,B: multiset_multiset_a] :
( ( A4
= ( plus_p6738641960240532161iset_a @ K2 @ A2 ) )
=> ( ( plus_p6738641960240532161iset_a @ A4 @ B )
= ( plus_p6738641960240532161iset_a @ K2 @ ( plus_p6738641960240532161iset_a @ A2 @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_191_group__cancel_Oadd1,axiom,
! [A4: multiset_nat,K2: multiset_nat,A2: multiset_nat,B: multiset_nat] :
( ( A4
= ( plus_p6334493942879108393et_nat @ K2 @ A2 ) )
=> ( ( plus_p6334493942879108393et_nat @ A4 @ B )
= ( plus_p6334493942879108393et_nat @ K2 @ ( plus_p6334493942879108393et_nat @ A2 @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_192_group__cancel_Oadd1,axiom,
! [A4: multiset_a,K2: multiset_a,A2: multiset_a,B: multiset_a] :
( ( A4
= ( plus_plus_multiset_a @ K2 @ A2 ) )
=> ( ( plus_plus_multiset_a @ A4 @ B )
= ( plus_plus_multiset_a @ K2 @ ( plus_plus_multiset_a @ A2 @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_193_group__cancel_Oadd1,axiom,
! [A4: nat,K2: nat,A2: nat,B: nat] :
( ( A4
= ( plus_plus_nat @ K2 @ A2 ) )
=> ( ( plus_plus_nat @ A4 @ B )
= ( plus_plus_nat @ K2 @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_194_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: multiset_nat,J: multiset_nat,K2: multiset_nat,L: multiset_nat] :
( ( ( I = J )
& ( K2 = L ) )
=> ( ( plus_p6334493942879108393et_nat @ I @ K2 )
= ( plus_p6334493942879108393et_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_195_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( I = J )
& ( K2 = L ) )
=> ( ( plus_plus_nat @ I @ K2 )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_196_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: multis4781622204924204013iset_a,B: multis4781622204924204013iset_a,C2: multis4781622204924204013iset_a] :
( ( plus_p1053901119268525860iset_a @ ( plus_p1053901119268525860iset_a @ A2 @ B ) @ C2 )
= ( plus_p1053901119268525860iset_a @ A2 @ ( plus_p1053901119268525860iset_a @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_197_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: multis501812127501805293od_a_a,B: multis501812127501805293od_a_a,C2: multis501812127501805293od_a_a] :
( ( plus_p6967097483979887908od_a_a @ ( plus_p6967097483979887908od_a_a @ A2 @ B ) @ C2 )
= ( plus_p6967097483979887908od_a_a @ A2 @ ( plus_p6967097483979887908od_a_a @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_198_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: multiset_multiset_a,B: multiset_multiset_a,C2: multiset_multiset_a] :
( ( plus_p6738641960240532161iset_a @ ( plus_p6738641960240532161iset_a @ A2 @ B ) @ C2 )
= ( plus_p6738641960240532161iset_a @ A2 @ ( plus_p6738641960240532161iset_a @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_199_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: multiset_nat,B: multiset_nat,C2: multiset_nat] :
( ( plus_p6334493942879108393et_nat @ ( plus_p6334493942879108393et_nat @ A2 @ B ) @ C2 )
= ( plus_p6334493942879108393et_nat @ A2 @ ( plus_p6334493942879108393et_nat @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_200_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: multiset_a,B: multiset_a,C2: multiset_a] :
( ( plus_plus_multiset_a @ ( plus_plus_multiset_a @ A2 @ B ) @ C2 )
= ( plus_plus_multiset_a @ A2 @ ( plus_plus_multiset_a @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_201_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B ) @ C2 )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_202_multi__union__self__other__eq,axiom,
! [A4: multis4781622204924204013iset_a,X4: multis4781622204924204013iset_a,Y4: multis4781622204924204013iset_a] :
( ( ( plus_p1053901119268525860iset_a @ A4 @ X4 )
= ( plus_p1053901119268525860iset_a @ A4 @ Y4 ) )
=> ( X4 = Y4 ) ) ).
% multi_union_self_other_eq
thf(fact_203_multi__union__self__other__eq,axiom,
! [A4: multis501812127501805293od_a_a,X4: multis501812127501805293od_a_a,Y4: multis501812127501805293od_a_a] :
( ( ( plus_p6967097483979887908od_a_a @ A4 @ X4 )
= ( plus_p6967097483979887908od_a_a @ A4 @ Y4 ) )
=> ( X4 = Y4 ) ) ).
% multi_union_self_other_eq
thf(fact_204_multi__union__self__other__eq,axiom,
! [A4: multiset_multiset_a,X4: multiset_multiset_a,Y4: multiset_multiset_a] :
( ( ( plus_p6738641960240532161iset_a @ A4 @ X4 )
= ( plus_p6738641960240532161iset_a @ A4 @ Y4 ) )
=> ( X4 = Y4 ) ) ).
% multi_union_self_other_eq
thf(fact_205_multi__union__self__other__eq,axiom,
! [A4: multiset_nat,X4: multiset_nat,Y4: multiset_nat] :
( ( ( plus_p6334493942879108393et_nat @ A4 @ X4 )
= ( plus_p6334493942879108393et_nat @ A4 @ Y4 ) )
=> ( X4 = Y4 ) ) ).
% multi_union_self_other_eq
thf(fact_206_multi__union__self__other__eq,axiom,
! [A4: multiset_a,X4: multiset_a,Y4: multiset_a] :
( ( ( plus_plus_multiset_a @ A4 @ X4 )
= ( plus_plus_multiset_a @ A4 @ Y4 ) )
=> ( X4 = Y4 ) ) ).
% multi_union_self_other_eq
thf(fact_207_union__right__cancel,axiom,
! [M: multis4781622204924204013iset_a,K: multis4781622204924204013iset_a,N: multis4781622204924204013iset_a] :
( ( ( plus_p1053901119268525860iset_a @ M @ K )
= ( plus_p1053901119268525860iset_a @ N @ K ) )
= ( M = N ) ) ).
% union_right_cancel
thf(fact_208_union__right__cancel,axiom,
! [M: multis501812127501805293od_a_a,K: multis501812127501805293od_a_a,N: multis501812127501805293od_a_a] :
( ( ( plus_p6967097483979887908od_a_a @ M @ K )
= ( plus_p6967097483979887908od_a_a @ N @ K ) )
= ( M = N ) ) ).
% union_right_cancel
thf(fact_209_union__right__cancel,axiom,
! [M: multiset_multiset_a,K: multiset_multiset_a,N: multiset_multiset_a] :
( ( ( plus_p6738641960240532161iset_a @ M @ K )
= ( plus_p6738641960240532161iset_a @ N @ K ) )
= ( M = N ) ) ).
% union_right_cancel
thf(fact_210_union__right__cancel,axiom,
! [M: multiset_nat,K: multiset_nat,N: multiset_nat] :
( ( ( plus_p6334493942879108393et_nat @ M @ K )
= ( plus_p6334493942879108393et_nat @ N @ K ) )
= ( M = N ) ) ).
% union_right_cancel
thf(fact_211_union__right__cancel,axiom,
! [M: multiset_a,K: multiset_a,N: multiset_a] :
( ( ( plus_plus_multiset_a @ M @ K )
= ( plus_plus_multiset_a @ N @ K ) )
= ( M = N ) ) ).
% union_right_cancel
thf(fact_212_union__left__cancel,axiom,
! [K: multis4781622204924204013iset_a,M: multis4781622204924204013iset_a,N: multis4781622204924204013iset_a] :
( ( ( plus_p1053901119268525860iset_a @ K @ M )
= ( plus_p1053901119268525860iset_a @ K @ N ) )
= ( M = N ) ) ).
% union_left_cancel
thf(fact_213_union__left__cancel,axiom,
! [K: multis501812127501805293od_a_a,M: multis501812127501805293od_a_a,N: multis501812127501805293od_a_a] :
( ( ( plus_p6967097483979887908od_a_a @ K @ M )
= ( plus_p6967097483979887908od_a_a @ K @ N ) )
= ( M = N ) ) ).
% union_left_cancel
thf(fact_214_union__left__cancel,axiom,
! [K: multiset_multiset_a,M: multiset_multiset_a,N: multiset_multiset_a] :
( ( ( plus_p6738641960240532161iset_a @ K @ M )
= ( plus_p6738641960240532161iset_a @ K @ N ) )
= ( M = N ) ) ).
% union_left_cancel
thf(fact_215_union__left__cancel,axiom,
! [K: multiset_nat,M: multiset_nat,N: multiset_nat] :
( ( ( plus_p6334493942879108393et_nat @ K @ M )
= ( plus_p6334493942879108393et_nat @ K @ N ) )
= ( M = N ) ) ).
% union_left_cancel
thf(fact_216_union__left__cancel,axiom,
! [K: multiset_a,M: multiset_a,N: multiset_a] :
( ( ( plus_plus_multiset_a @ K @ M )
= ( plus_plus_multiset_a @ K @ N ) )
= ( M = N ) ) ).
% union_left_cancel
thf(fact_217_union__commute,axiom,
( plus_p1053901119268525860iset_a
= ( ^ [M2: multis4781622204924204013iset_a,N2: multis4781622204924204013iset_a] : ( plus_p1053901119268525860iset_a @ N2 @ M2 ) ) ) ).
% union_commute
thf(fact_218_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_219_union__commute,axiom,
( plus_p6738641960240532161iset_a
= ( ^ [M2: multiset_multiset_a,N2: multiset_multiset_a] : ( plus_p6738641960240532161iset_a @ N2 @ M2 ) ) ) ).
% union_commute
thf(fact_220_union__commute,axiom,
( plus_p6334493942879108393et_nat
= ( ^ [M2: multiset_nat,N2: multiset_nat] : ( plus_p6334493942879108393et_nat @ N2 @ M2 ) ) ) ).
% union_commute
thf(fact_221_union__commute,axiom,
( plus_plus_multiset_a
= ( ^ [M2: multiset_a,N2: multiset_a] : ( plus_plus_multiset_a @ N2 @ M2 ) ) ) ).
% union_commute
thf(fact_222_union__lcomm,axiom,
! [M: multis4781622204924204013iset_a,N: multis4781622204924204013iset_a,K: multis4781622204924204013iset_a] :
( ( plus_p1053901119268525860iset_a @ M @ ( plus_p1053901119268525860iset_a @ N @ K ) )
= ( plus_p1053901119268525860iset_a @ N @ ( plus_p1053901119268525860iset_a @ M @ K ) ) ) ).
% union_lcomm
thf(fact_223_union__lcomm,axiom,
! [M: multis501812127501805293od_a_a,N: multis501812127501805293od_a_a,K: multis501812127501805293od_a_a] :
( ( plus_p6967097483979887908od_a_a @ M @ ( plus_p6967097483979887908od_a_a @ N @ K ) )
= ( plus_p6967097483979887908od_a_a @ N @ ( plus_p6967097483979887908od_a_a @ M @ K ) ) ) ).
% union_lcomm
thf(fact_224_union__lcomm,axiom,
! [M: multiset_multiset_a,N: multiset_multiset_a,K: multiset_multiset_a] :
( ( plus_p6738641960240532161iset_a @ M @ ( plus_p6738641960240532161iset_a @ N @ K ) )
= ( plus_p6738641960240532161iset_a @ N @ ( plus_p6738641960240532161iset_a @ M @ K ) ) ) ).
% union_lcomm
thf(fact_225_union__lcomm,axiom,
! [M: multiset_nat,N: multiset_nat,K: multiset_nat] :
( ( plus_p6334493942879108393et_nat @ M @ ( plus_p6334493942879108393et_nat @ N @ K ) )
= ( plus_p6334493942879108393et_nat @ N @ ( plus_p6334493942879108393et_nat @ M @ K ) ) ) ).
% union_lcomm
thf(fact_226_union__lcomm,axiom,
! [M: multiset_a,N: multiset_a,K: multiset_a] :
( ( plus_plus_multiset_a @ M @ ( plus_plus_multiset_a @ N @ K ) )
= ( plus_plus_multiset_a @ N @ ( plus_plus_multiset_a @ M @ K ) ) ) ).
% union_lcomm
thf(fact_227_ns__mul__extI,axiom,
! [A4: multiset_nat_nat,A12: multiset_nat_nat,A23: multiset_nat_nat,B5: multiset_nat_nat,B12: multiset_nat_nat,B23: multiset_nat_nat,Ns: set_Pr7682762132356531903at_nat,S: set_Pr7682762132356531903at_nat] :
( ( A4
= ( plus_p6288751606316705304at_nat @ A12 @ A23 ) )
=> ( ( B5
= ( plus_p6288751606316705304at_nat @ B12 @ B23 ) )
=> ( ( member3028814343351594408at_nat @ ( produc7326294017233846199at_nat @ A12 @ B12 ) @ ( multis3696764012599529659at_nat @ Ns ) )
=> ( ! [B3: nat > nat] :
( ( member_nat_nat @ B3 @ ( set_mset_nat_nat @ B23 ) )
=> ? [A6: nat > nat] :
( ( member_nat_nat @ A6 @ ( set_mset_nat_nat @ A23 ) )
& ( member1096249278104964872at_nat @ ( produc5770335208449155351at_nat @ A6 @ B3 ) @ S ) ) )
=> ( member3028814343351594408at_nat @ ( produc7326294017233846199at_nat @ A4 @ B5 ) @ ( multis2832032611068704842at_nat @ Ns @ S ) ) ) ) ) ) ).
% ns_mul_extI
thf(fact_228_ns__mul__extI,axiom,
! [A4: multis7221419791219231795iset_a,A12: multis7221419791219231795iset_a,A23: multis7221419791219231795iset_a,B5: multis7221419791219231795iset_a,B12: multis7221419791219231795iset_a,B23: multis7221419791219231795iset_a,Ns: set_Pr6964934382344443463iset_a,S: set_Pr6964934382344443463iset_a] :
( ( A4
= ( plus_p1436237118943648234iset_a @ A12 @ A23 ) )
=> ( ( B5
= ( plus_p1436237118943648234iset_a @ B12 @ B23 ) )
=> ( ( member4376497788722562064iset_a @ ( produc5887428786155168343iset_a @ A12 @ B12 ) @ ( multis7851155180588674929iset_a @ Ns ) )
=> ( ! [B3: multis4781622204924204013iset_a] :
( ( member2597461214246177430iset_a @ B3 @ ( set_ms5327031259215843498iset_a @ B23 ) )
=> ? [A6: multis4781622204924204013iset_a] :
( ( member2597461214246177430iset_a @ A6 @ ( set_ms5327031259215843498iset_a @ A23 ) )
& ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ A6 @ B3 ) @ S ) ) )
=> ( member4376497788722562064iset_a @ ( produc5887428786155168343iset_a @ A4 @ B5 ) @ ( multis1613109581574850210iset_a @ Ns @ S ) ) ) ) ) ) ).
% ns_mul_extI
thf(fact_229_ns__mul__extI,axiom,
! [A4: multis5935151272263360051od_a_a,A12: multis5935151272263360051od_a_a,A23: multis5935151272263360051od_a_a,B5: multis5935151272263360051od_a_a,B12: multis5935151272263360051od_a_a,B23: multis5935151272263360051od_a_a,Ns: set_Pr3746329090258575431od_a_a,S: set_Pr3746329090258575431od_a_a] :
( ( A4
= ( plus_p2133152728643999466od_a_a @ A12 @ A23 ) )
=> ( ( B5
= ( plus_p2133152728643999466od_a_a @ B12 @ B23 ) )
=> ( ( member231996757139097616od_a_a @ ( produc288868108895620695od_a_a @ A12 @ B12 ) @ ( multis9199697240695551601od_a_a @ Ns ) )
=> ( ! [B3: multis501812127501805293od_a_a] :
( ( member7700929758516679318od_a_a @ B3 @ ( set_ms8253844250382261418od_a_a @ B23 ) )
=> ? [A6: multis501812127501805293od_a_a] :
( ( member7700929758516679318od_a_a @ A6 @ ( set_ms8253844250382261418od_a_a @ A23 ) )
& ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ A6 @ B3 ) @ S ) ) )
=> ( member231996757139097616od_a_a @ ( produc288868108895620695od_a_a @ A4 @ B5 ) @ ( multis3861150492729141922od_a_a @ Ns @ S ) ) ) ) ) ) ).
% ns_mul_extI
thf(fact_230_ns__mul__extI,axiom,
! [A4: multiset_list_a,A12: multiset_list_a,A23: multiset_list_a,B5: multiset_list_a,B12: multiset_list_a,B23: multiset_list_a,Ns: set_Pr4048851178543822343list_a,S: set_Pr4048851178543822343list_a] :
( ( A4
= ( plus_p690419498615200257list_a @ A12 @ A23 ) )
=> ( ( B5
= ( plus_p690419498615200257list_a @ B12 @ B23 ) )
=> ( ( member8467872774006757712list_a @ ( produc1470943644513211159list_a @ A12 @ B12 ) @ ( multis4869478336504412936list_a @ Ns ) )
=> ( ! [B3: list_a] :
( ( member_list_a @ B3 @ ( set_mset_list_a @ B23 ) )
=> ? [A6: list_a] :
( ( member_list_a @ A6 @ ( set_mset_list_a @ A23 ) )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ A6 @ B3 ) @ S ) ) )
=> ( member8467872774006757712list_a @ ( produc1470943644513211159list_a @ A4 @ B5 ) @ ( multis7921000248416496185list_a @ Ns @ S ) ) ) ) ) ) ).
% ns_mul_extI
thf(fact_231_ns__mul__extI,axiom,
! [A4: multiset_nat,A12: multiset_nat,A23: multiset_nat,B5: multiset_nat,B12: multiset_nat,B23: multiset_nat,Ns: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( A4
= ( plus_p6334493942879108393et_nat @ A12 @ A23 ) )
=> ( ( B5
= ( plus_p6334493942879108393et_nat @ B12 @ B23 ) )
=> ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ A12 @ B12 ) @ ( multis5152147736693828940pw_nat @ Ns ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( set_mset_nat @ B23 ) )
=> ? [A6: nat] :
( ( member_nat @ A6 @ ( set_mset_nat @ A23 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B3 ) @ S ) ) )
=> ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ A4 @ B5 ) @ ( multis8831328596575508315xt_nat @ Ns @ S ) ) ) ) ) ) ).
% ns_mul_extI
thf(fact_232_ns__mul__extI,axiom,
! [A4: multis501812127501805293od_a_a,A12: multis501812127501805293od_a_a,A23: multis501812127501805293od_a_a,B5: multis501812127501805293od_a_a,B12: multis501812127501805293od_a_a,B23: multis501812127501805293od_a_a,Ns: set_Pr8600417178894128327od_a_a,S: set_Pr8600417178894128327od_a_a] :
( ( A4
= ( plus_p6967097483979887908od_a_a @ A12 @ A23 ) )
=> ( ( B5
= ( plus_p6967097483979887908od_a_a @ B12 @ B23 ) )
=> ( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ A12 @ B12 ) @ ( multis1821517007024690859od_a_a @ Ns ) )
=> ( ! [B3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ B3 @ ( set_ms119794563918438244od_a_a @ B23 ) )
=> ? [A6: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A6 @ ( set_ms119794563918438244od_a_a @ A23 ) )
& ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A6 @ B3 ) @ S ) ) )
=> ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ A4 @ B5 ) @ ( multis1178328011278088540od_a_a @ Ns @ S ) ) ) ) ) ) ).
% ns_mul_extI
thf(fact_233_ns__mul__extI,axiom,
! [A4: multis4781622204924204013iset_a,A12: multis4781622204924204013iset_a,A23: multis4781622204924204013iset_a,B5: multis4781622204924204013iset_a,B12: multis4781622204924204013iset_a,B23: multis4781622204924204013iset_a,Ns: set_Pr3059988933816007879iset_a,S: set_Pr3059988933816007879iset_a] :
( ( A4
= ( plus_p1053901119268525860iset_a @ A12 @ A23 ) )
=> ( ( B5
= ( plus_p1053901119268525860iset_a @ B12 @ B23 ) )
=> ( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ A12 @ B12 ) @ ( multis3430030276581740203iset_a @ Ns ) )
=> ( ! [B3: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ B3 @ ( set_ms2379723421352415844iset_a @ B23 ) )
=> ? [A6: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ A6 @ ( set_ms2379723421352415844iset_a @ A23 ) )
& ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ A6 @ B3 ) @ S ) ) )
=> ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ A4 @ B5 ) @ ( multis2244641948333282396iset_a @ Ns @ S ) ) ) ) ) ) ).
% ns_mul_extI
thf(fact_234_ns__mul__extI,axiom,
! [A4: multiset_multiset_a,A12: multiset_multiset_a,A23: multiset_multiset_a,B5: multiset_multiset_a,B12: multiset_multiset_a,B23: multiset_multiset_a,Ns: set_Pr79727621955416071iset_a,S: set_Pr79727621955416071iset_a] :
( ( A4
= ( plus_p6738641960240532161iset_a @ A12 @ A23 ) )
=> ( ( B5
= ( plus_p6738641960240532161iset_a @ B12 @ B23 ) )
=> ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ A12 @ B12 ) @ ( multis451435756853798856iset_a @ Ns ) )
=> ( ! [B3: multiset_a] :
( ( member_multiset_a @ B3 @ ( set_mset_multiset_a @ B23 ) )
=> ? [A6: multiset_a] :
( ( member_multiset_a @ A6 @ ( set_mset_multiset_a @ A23 ) )
& ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A6 @ B3 ) @ S ) ) )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ A4 @ B5 ) @ ( multis8112516946959054073iset_a @ Ns @ S ) ) ) ) ) ) ).
% ns_mul_extI
thf(fact_235_ns__mul__extI,axiom,
! [A4: multiset_a,A12: multiset_a,A23: multiset_a,B5: multiset_a,B12: multiset_a,B23: multiset_a,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( A4
= ( plus_plus_multiset_a @ A12 @ A23 ) )
=> ( ( B5
= ( plus_plus_multiset_a @ B12 @ B23 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A12 @ B12 ) @ ( multis2373687813370458242ltpw_a @ Ns ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ ( set_mset_a @ B23 ) )
=> ? [A6: a] :
( ( member_a @ A6 @ ( set_mset_a @ A23 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A6 @ B3 ) @ S ) ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A4 @ B5 ) @ ( multis4023584895248362547_ext_a @ Ns @ S ) ) ) ) ) ) ).
% ns_mul_extI
thf(fact_236_ns__mul__extE,axiom,
! [A4: multiset_nat_nat,B5: multiset_nat_nat,Ns: set_Pr7682762132356531903at_nat,S: set_Pr7682762132356531903at_nat] :
( ( member3028814343351594408at_nat @ ( produc7326294017233846199at_nat @ A4 @ B5 ) @ ( multis2832032611068704842at_nat @ Ns @ S ) )
=> ~ ! [A1: multiset_nat_nat,A22: multiset_nat_nat] :
( ( A4
= ( plus_p6288751606316705304at_nat @ A1 @ A22 ) )
=> ! [B1: multiset_nat_nat,B22: multiset_nat_nat] :
( ( B5
= ( plus_p6288751606316705304at_nat @ B1 @ B22 ) )
=> ( ( member3028814343351594408at_nat @ ( produc7326294017233846199at_nat @ A1 @ B1 ) @ ( multis3696764012599529659at_nat @ Ns ) )
=> ~ ! [B4: nat > nat] :
( ( member_nat_nat @ B4 @ ( set_mset_nat_nat @ B22 ) )
=> ? [A: nat > nat] :
( ( member_nat_nat @ A @ ( set_mset_nat_nat @ A22 ) )
& ( member1096249278104964872at_nat @ ( produc5770335208449155351at_nat @ A @ B4 ) @ S ) ) ) ) ) ) ) ).
% ns_mul_extE
thf(fact_237_ns__mul__extE,axiom,
! [A4: multis7221419791219231795iset_a,B5: multis7221419791219231795iset_a,Ns: set_Pr6964934382344443463iset_a,S: set_Pr6964934382344443463iset_a] :
( ( member4376497788722562064iset_a @ ( produc5887428786155168343iset_a @ A4 @ B5 ) @ ( multis1613109581574850210iset_a @ Ns @ S ) )
=> ~ ! [A1: multis7221419791219231795iset_a,A22: multis7221419791219231795iset_a] :
( ( A4
= ( plus_p1436237118943648234iset_a @ A1 @ A22 ) )
=> ! [B1: multis7221419791219231795iset_a,B22: multis7221419791219231795iset_a] :
( ( B5
= ( plus_p1436237118943648234iset_a @ B1 @ B22 ) )
=> ( ( member4376497788722562064iset_a @ ( produc5887428786155168343iset_a @ A1 @ B1 ) @ ( multis7851155180588674929iset_a @ Ns ) )
=> ~ ! [B4: multis4781622204924204013iset_a] :
( ( member2597461214246177430iset_a @ B4 @ ( set_ms5327031259215843498iset_a @ B22 ) )
=> ? [A: multis4781622204924204013iset_a] :
( ( member2597461214246177430iset_a @ A @ ( set_ms5327031259215843498iset_a @ A22 ) )
& ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ A @ B4 ) @ S ) ) ) ) ) ) ) ).
% ns_mul_extE
thf(fact_238_ns__mul__extE,axiom,
! [A4: multis5935151272263360051od_a_a,B5: multis5935151272263360051od_a_a,Ns: set_Pr3746329090258575431od_a_a,S: set_Pr3746329090258575431od_a_a] :
( ( member231996757139097616od_a_a @ ( produc288868108895620695od_a_a @ A4 @ B5 ) @ ( multis3861150492729141922od_a_a @ Ns @ S ) )
=> ~ ! [A1: multis5935151272263360051od_a_a,A22: multis5935151272263360051od_a_a] :
( ( A4
= ( plus_p2133152728643999466od_a_a @ A1 @ A22 ) )
=> ! [B1: multis5935151272263360051od_a_a,B22: multis5935151272263360051od_a_a] :
( ( B5
= ( plus_p2133152728643999466od_a_a @ B1 @ B22 ) )
=> ( ( member231996757139097616od_a_a @ ( produc288868108895620695od_a_a @ A1 @ B1 ) @ ( multis9199697240695551601od_a_a @ Ns ) )
=> ~ ! [B4: multis501812127501805293od_a_a] :
( ( member7700929758516679318od_a_a @ B4 @ ( set_ms8253844250382261418od_a_a @ B22 ) )
=> ? [A: multis501812127501805293od_a_a] :
( ( member7700929758516679318od_a_a @ A @ ( set_ms8253844250382261418od_a_a @ A22 ) )
& ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ A @ B4 ) @ S ) ) ) ) ) ) ) ).
% ns_mul_extE
thf(fact_239_ns__mul__extE,axiom,
! [A4: multiset_list_a,B5: multiset_list_a,Ns: set_Pr4048851178543822343list_a,S: set_Pr4048851178543822343list_a] :
( ( member8467872774006757712list_a @ ( produc1470943644513211159list_a @ A4 @ B5 ) @ ( multis7921000248416496185list_a @ Ns @ S ) )
=> ~ ! [A1: multiset_list_a,A22: multiset_list_a] :
( ( A4
= ( plus_p690419498615200257list_a @ A1 @ A22 ) )
=> ! [B1: multiset_list_a,B22: multiset_list_a] :
( ( B5
= ( plus_p690419498615200257list_a @ B1 @ B22 ) )
=> ( ( member8467872774006757712list_a @ ( produc1470943644513211159list_a @ A1 @ B1 ) @ ( multis4869478336504412936list_a @ Ns ) )
=> ~ ! [B4: list_a] :
( ( member_list_a @ B4 @ ( set_mset_list_a @ B22 ) )
=> ? [A: list_a] :
( ( member_list_a @ A @ ( set_mset_list_a @ A22 ) )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ A @ B4 ) @ S ) ) ) ) ) ) ) ).
% ns_mul_extE
thf(fact_240_ns__mul__extE,axiom,
! [A4: multiset_nat,B5: multiset_nat,Ns: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ A4 @ B5 ) @ ( multis8831328596575508315xt_nat @ Ns @ S ) )
=> ~ ! [A1: multiset_nat,A22: multiset_nat] :
( ( A4
= ( plus_p6334493942879108393et_nat @ A1 @ A22 ) )
=> ! [B1: multiset_nat,B22: multiset_nat] :
( ( B5
= ( plus_p6334493942879108393et_nat @ B1 @ B22 ) )
=> ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ A1 @ B1 ) @ ( multis5152147736693828940pw_nat @ Ns ) )
=> ~ ! [B4: nat] :
( ( member_nat @ B4 @ ( set_mset_nat @ B22 ) )
=> ? [A: nat] :
( ( member_nat @ A @ ( set_mset_nat @ A22 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B4 ) @ S ) ) ) ) ) ) ) ).
% ns_mul_extE
thf(fact_241_ns__mul__extE,axiom,
! [A4: multis501812127501805293od_a_a,B5: multis501812127501805293od_a_a,Ns: set_Pr8600417178894128327od_a_a,S: set_Pr8600417178894128327od_a_a] :
( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ A4 @ B5 ) @ ( multis1178328011278088540od_a_a @ Ns @ S ) )
=> ~ ! [A1: multis501812127501805293od_a_a,A22: multis501812127501805293od_a_a] :
( ( A4
= ( plus_p6967097483979887908od_a_a @ A1 @ A22 ) )
=> ! [B1: multis501812127501805293od_a_a,B22: multis501812127501805293od_a_a] :
( ( B5
= ( plus_p6967097483979887908od_a_a @ B1 @ B22 ) )
=> ( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ A1 @ B1 ) @ ( multis1821517007024690859od_a_a @ Ns ) )
=> ~ ! [B4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ B4 @ ( set_ms119794563918438244od_a_a @ B22 ) )
=> ? [A: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A @ ( set_ms119794563918438244od_a_a @ A22 ) )
& ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A @ B4 ) @ S ) ) ) ) ) ) ) ).
% ns_mul_extE
thf(fact_242_ns__mul__extE,axiom,
! [A4: multis4781622204924204013iset_a,B5: multis4781622204924204013iset_a,Ns: set_Pr3059988933816007879iset_a,S: set_Pr3059988933816007879iset_a] :
( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ A4 @ B5 ) @ ( multis2244641948333282396iset_a @ Ns @ S ) )
=> ~ ! [A1: multis4781622204924204013iset_a,A22: multis4781622204924204013iset_a] :
( ( A4
= ( plus_p1053901119268525860iset_a @ A1 @ A22 ) )
=> ! [B1: multis4781622204924204013iset_a,B22: multis4781622204924204013iset_a] :
( ( B5
= ( plus_p1053901119268525860iset_a @ B1 @ B22 ) )
=> ( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ A1 @ B1 ) @ ( multis3430030276581740203iset_a @ Ns ) )
=> ~ ! [B4: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ B4 @ ( set_ms2379723421352415844iset_a @ B22 ) )
=> ? [A: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ A @ ( set_ms2379723421352415844iset_a @ A22 ) )
& ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ A @ B4 ) @ S ) ) ) ) ) ) ) ).
% ns_mul_extE
thf(fact_243_ns__mul__extE,axiom,
! [A4: multiset_multiset_a,B5: multiset_multiset_a,Ns: set_Pr79727621955416071iset_a,S: set_Pr79727621955416071iset_a] :
( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ A4 @ B5 ) @ ( multis8112516946959054073iset_a @ Ns @ S ) )
=> ~ ! [A1: multiset_multiset_a,A22: multiset_multiset_a] :
( ( A4
= ( plus_p6738641960240532161iset_a @ A1 @ A22 ) )
=> ! [B1: multiset_multiset_a,B22: multiset_multiset_a] :
( ( B5
= ( plus_p6738641960240532161iset_a @ B1 @ B22 ) )
=> ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ A1 @ B1 ) @ ( multis451435756853798856iset_a @ Ns ) )
=> ~ ! [B4: multiset_a] :
( ( member_multiset_a @ B4 @ ( set_mset_multiset_a @ B22 ) )
=> ? [A: multiset_a] :
( ( member_multiset_a @ A @ ( set_mset_multiset_a @ A22 ) )
& ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A @ B4 ) @ S ) ) ) ) ) ) ) ).
% ns_mul_extE
thf(fact_244_ns__mul__extE,axiom,
! [A4: multiset_a,B5: multiset_a,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A4 @ B5 ) @ ( multis4023584895248362547_ext_a @ Ns @ S ) )
=> ~ ! [A1: multiset_a,A22: multiset_a] :
( ( A4
= ( plus_plus_multiset_a @ A1 @ A22 ) )
=> ! [B1: multiset_a,B22: multiset_a] :
( ( B5
= ( plus_plus_multiset_a @ B1 @ B22 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A1 @ B1 ) @ ( multis2373687813370458242ltpw_a @ Ns ) )
=> ~ ! [B4: a] :
( ( member_a @ B4 @ ( set_mset_a @ B22 ) )
=> ? [A: a] :
( ( member_a @ A @ ( set_mset_a @ A22 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B4 ) @ S ) ) ) ) ) ) ) ).
% ns_mul_extE
thf(fact_245_ns__ns__mul__ext__union__compat,axiom,
! [A4: multiset_nat,B5: multiset_nat,Ns: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat,C3: multiset_nat,D: multiset_nat] :
( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ A4 @ B5 ) @ ( multis8831328596575508315xt_nat @ Ns @ S ) )
=> ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ C3 @ D ) @ ( multis8831328596575508315xt_nat @ Ns @ S ) )
=> ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( plus_p6334493942879108393et_nat @ A4 @ C3 ) @ ( plus_p6334493942879108393et_nat @ B5 @ D ) ) @ ( multis8831328596575508315xt_nat @ Ns @ S ) ) ) ) ).
% ns_ns_mul_ext_union_compat
thf(fact_246_ns__ns__mul__ext__union__compat,axiom,
! [A4: multiset_multiset_a,B5: multiset_multiset_a,Ns: set_Pr79727621955416071iset_a,S: set_Pr79727621955416071iset_a,C3: multiset_multiset_a,D: multiset_multiset_a] :
( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ A4 @ B5 ) @ ( multis8112516946959054073iset_a @ Ns @ S ) )
=> ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ C3 @ D ) @ ( multis8112516946959054073iset_a @ Ns @ S ) )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ ( plus_p6738641960240532161iset_a @ A4 @ C3 ) @ ( plus_p6738641960240532161iset_a @ B5 @ D ) ) @ ( multis8112516946959054073iset_a @ Ns @ S ) ) ) ) ).
% ns_ns_mul_ext_union_compat
thf(fact_247_ns__ns__mul__ext__union__compat,axiom,
! [A4: multis4781622204924204013iset_a,B5: multis4781622204924204013iset_a,Ns: set_Pr3059988933816007879iset_a,S: set_Pr3059988933816007879iset_a,C3: multis4781622204924204013iset_a,D: multis4781622204924204013iset_a] :
( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ A4 @ B5 ) @ ( multis2244641948333282396iset_a @ Ns @ S ) )
=> ( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ C3 @ D ) @ ( multis2244641948333282396iset_a @ Ns @ S ) )
=> ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ ( plus_p1053901119268525860iset_a @ A4 @ C3 ) @ ( plus_p1053901119268525860iset_a @ B5 @ D ) ) @ ( multis2244641948333282396iset_a @ Ns @ S ) ) ) ) ).
% ns_ns_mul_ext_union_compat
thf(fact_248_ns__ns__mul__ext__union__compat,axiom,
! [A4: multis501812127501805293od_a_a,B5: multis501812127501805293od_a_a,Ns: set_Pr8600417178894128327od_a_a,S: set_Pr8600417178894128327od_a_a,C3: multis501812127501805293od_a_a,D: multis501812127501805293od_a_a] :
( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ A4 @ B5 ) @ ( multis1178328011278088540od_a_a @ Ns @ S ) )
=> ( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ C3 @ D ) @ ( multis1178328011278088540od_a_a @ Ns @ S ) )
=> ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ ( plus_p6967097483979887908od_a_a @ A4 @ C3 ) @ ( plus_p6967097483979887908od_a_a @ B5 @ D ) ) @ ( multis1178328011278088540od_a_a @ Ns @ S ) ) ) ) ).
% ns_ns_mul_ext_union_compat
thf(fact_249_ns__ns__mul__ext__union__compat,axiom,
! [A4: multiset_a,B5: multiset_a,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a,C3: multiset_a,D: multiset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A4 @ B5 ) @ ( multis4023584895248362547_ext_a @ Ns @ S ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ C3 @ D ) @ ( multis4023584895248362547_ext_a @ Ns @ S ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( plus_plus_multiset_a @ A4 @ C3 ) @ ( plus_plus_multiset_a @ B5 @ D ) ) @ ( multis4023584895248362547_ext_a @ Ns @ S ) ) ) ) ).
% ns_ns_mul_ext_union_compat
thf(fact_250_multpw__splitR,axiom,
! [X12: multiset_nat,X22: multiset_nat,Y4: multiset_nat,Ns: set_Pr1261947904930325089at_nat] :
( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( plus_p6334493942879108393et_nat @ X12 @ X22 ) @ Y4 ) @ ( multis5152147736693828940pw_nat @ Ns ) )
=> ~ ! [Y12: multiset_nat,Y22: multiset_nat] :
( ( Y4
= ( plus_p6334493942879108393et_nat @ Y12 @ Y22 ) )
=> ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ X12 @ Y12 ) @ ( multis5152147736693828940pw_nat @ Ns ) )
=> ~ ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ X22 @ Y22 ) @ ( multis5152147736693828940pw_nat @ Ns ) ) ) ) ) ).
% multpw_splitR
thf(fact_251_multpw__splitR,axiom,
! [X12: multiset_multiset_a,X22: multiset_multiset_a,Y4: multiset_multiset_a,Ns: set_Pr79727621955416071iset_a] :
( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ ( plus_p6738641960240532161iset_a @ X12 @ X22 ) @ Y4 ) @ ( multis451435756853798856iset_a @ Ns ) )
=> ~ ! [Y12: multiset_multiset_a,Y22: multiset_multiset_a] :
( ( Y4
= ( plus_p6738641960240532161iset_a @ Y12 @ Y22 ) )
=> ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X12 @ Y12 ) @ ( multis451435756853798856iset_a @ Ns ) )
=> ~ ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X22 @ Y22 ) @ ( multis451435756853798856iset_a @ Ns ) ) ) ) ) ).
% multpw_splitR
thf(fact_252_multpw__splitR,axiom,
! [X12: multis4781622204924204013iset_a,X22: multis4781622204924204013iset_a,Y4: multis4781622204924204013iset_a,Ns: set_Pr3059988933816007879iset_a] :
( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ ( plus_p1053901119268525860iset_a @ X12 @ X22 ) @ Y4 ) @ ( multis3430030276581740203iset_a @ Ns ) )
=> ~ ! [Y12: multis4781622204924204013iset_a,Y22: multis4781622204924204013iset_a] :
( ( Y4
= ( plus_p1053901119268525860iset_a @ Y12 @ Y22 ) )
=> ( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ X12 @ Y12 ) @ ( multis3430030276581740203iset_a @ Ns ) )
=> ~ ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ X22 @ Y22 ) @ ( multis3430030276581740203iset_a @ Ns ) ) ) ) ) ).
% multpw_splitR
thf(fact_253_multpw__splitR,axiom,
! [X12: multis501812127501805293od_a_a,X22: multis501812127501805293od_a_a,Y4: multis501812127501805293od_a_a,Ns: set_Pr8600417178894128327od_a_a] :
( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ ( plus_p6967097483979887908od_a_a @ X12 @ X22 ) @ Y4 ) @ ( multis1821517007024690859od_a_a @ Ns ) )
=> ~ ! [Y12: multis501812127501805293od_a_a,Y22: multis501812127501805293od_a_a] :
( ( Y4
= ( plus_p6967097483979887908od_a_a @ Y12 @ Y22 ) )
=> ( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ X12 @ Y12 ) @ ( multis1821517007024690859od_a_a @ Ns ) )
=> ~ ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ X22 @ Y22 ) @ ( multis1821517007024690859od_a_a @ Ns ) ) ) ) ) ).
% multpw_splitR
thf(fact_254_multpw__splitR,axiom,
! [X12: multiset_a,X22: multiset_a,Y4: multiset_a,Ns: set_Product_prod_a_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( plus_plus_multiset_a @ X12 @ X22 ) @ Y4 ) @ ( multis2373687813370458242ltpw_a @ Ns ) )
=> ~ ! [Y12: multiset_a,Y22: multiset_a] :
( ( Y4
= ( plus_plus_multiset_a @ Y12 @ Y22 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X12 @ Y12 ) @ ( multis2373687813370458242ltpw_a @ Ns ) )
=> ~ ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X22 @ Y22 ) @ ( multis2373687813370458242ltpw_a @ Ns ) ) ) ) ) ).
% multpw_splitR
thf(fact_255_multpw__splitL,axiom,
! [X4: multiset_nat,Y13: multiset_nat,Y23: multiset_nat,Ns: set_Pr1261947904930325089at_nat] :
( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ X4 @ ( plus_p6334493942879108393et_nat @ Y13 @ Y23 ) ) @ ( multis5152147736693828940pw_nat @ Ns ) )
=> ~ ! [X13: multiset_nat,X23: multiset_nat] :
( ( X4
= ( plus_p6334493942879108393et_nat @ X13 @ X23 ) )
=> ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ X13 @ Y13 ) @ ( multis5152147736693828940pw_nat @ Ns ) )
=> ~ ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ X23 @ Y23 ) @ ( multis5152147736693828940pw_nat @ Ns ) ) ) ) ) ).
% multpw_splitL
thf(fact_256_multpw__splitL,axiom,
! [X4: multiset_multiset_a,Y13: multiset_multiset_a,Y23: multiset_multiset_a,Ns: set_Pr79727621955416071iset_a] :
( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X4 @ ( plus_p6738641960240532161iset_a @ Y13 @ Y23 ) ) @ ( multis451435756853798856iset_a @ Ns ) )
=> ~ ! [X13: multiset_multiset_a,X23: multiset_multiset_a] :
( ( X4
= ( plus_p6738641960240532161iset_a @ X13 @ X23 ) )
=> ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X13 @ Y13 ) @ ( multis451435756853798856iset_a @ Ns ) )
=> ~ ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X23 @ Y23 ) @ ( multis451435756853798856iset_a @ Ns ) ) ) ) ) ).
% multpw_splitL
thf(fact_257_multpw__splitL,axiom,
! [X4: multis4781622204924204013iset_a,Y13: multis4781622204924204013iset_a,Y23: multis4781622204924204013iset_a,Ns: set_Pr3059988933816007879iset_a] :
( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ X4 @ ( plus_p1053901119268525860iset_a @ Y13 @ Y23 ) ) @ ( multis3430030276581740203iset_a @ Ns ) )
=> ~ ! [X13: multis4781622204924204013iset_a,X23: multis4781622204924204013iset_a] :
( ( X4
= ( plus_p1053901119268525860iset_a @ X13 @ X23 ) )
=> ( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ X13 @ Y13 ) @ ( multis3430030276581740203iset_a @ Ns ) )
=> ~ ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ X23 @ Y23 ) @ ( multis3430030276581740203iset_a @ Ns ) ) ) ) ) ).
% multpw_splitL
thf(fact_258_multpw__splitL,axiom,
! [X4: multis501812127501805293od_a_a,Y13: multis501812127501805293od_a_a,Y23: multis501812127501805293od_a_a,Ns: set_Pr8600417178894128327od_a_a] :
( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ X4 @ ( plus_p6967097483979887908od_a_a @ Y13 @ Y23 ) ) @ ( multis1821517007024690859od_a_a @ Ns ) )
=> ~ ! [X13: multis501812127501805293od_a_a,X23: multis501812127501805293od_a_a] :
( ( X4
= ( plus_p6967097483979887908od_a_a @ X13 @ X23 ) )
=> ( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ X13 @ Y13 ) @ ( multis1821517007024690859od_a_a @ Ns ) )
=> ~ ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ X23 @ Y23 ) @ ( multis1821517007024690859od_a_a @ Ns ) ) ) ) ) ).
% multpw_splitL
thf(fact_259_multpw__splitL,axiom,
! [X4: multiset_a,Y13: multiset_a,Y23: multiset_a,Ns: set_Product_prod_a_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X4 @ ( plus_plus_multiset_a @ Y13 @ Y23 ) ) @ ( multis2373687813370458242ltpw_a @ Ns ) )
=> ~ ! [X13: multiset_a,X23: multiset_a] :
( ( X4
= ( plus_plus_multiset_a @ X13 @ X23 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X13 @ Y13 ) @ ( multis2373687813370458242ltpw_a @ Ns ) )
=> ~ ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X23 @ Y23 ) @ ( multis2373687813370458242ltpw_a @ Ns ) ) ) ) ) ).
% multpw_splitL
thf(fact_260_multpw__add,axiom,
! [X12: multiset_multiset_a,Y13: multiset_multiset_a,Ns: set_Pr79727621955416071iset_a,X22: multiset_multiset_a,Y23: multiset_multiset_a] :
( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X12 @ Y13 ) @ ( multis451435756853798856iset_a @ Ns ) )
=> ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X22 @ Y23 ) @ ( multis451435756853798856iset_a @ Ns ) )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ ( plus_p6738641960240532161iset_a @ X12 @ X22 ) @ ( plus_p6738641960240532161iset_a @ Y13 @ Y23 ) ) @ ( multis451435756853798856iset_a @ Ns ) ) ) ) ).
% multpw_add
thf(fact_261_multpw__add,axiom,
! [X12: multis4781622204924204013iset_a,Y13: multis4781622204924204013iset_a,Ns: set_Pr3059988933816007879iset_a,X22: multis4781622204924204013iset_a,Y23: multis4781622204924204013iset_a] :
( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ X12 @ Y13 ) @ ( multis3430030276581740203iset_a @ Ns ) )
=> ( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ X22 @ Y23 ) @ ( multis3430030276581740203iset_a @ Ns ) )
=> ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ ( plus_p1053901119268525860iset_a @ X12 @ X22 ) @ ( plus_p1053901119268525860iset_a @ Y13 @ Y23 ) ) @ ( multis3430030276581740203iset_a @ Ns ) ) ) ) ).
% multpw_add
thf(fact_262_multpw__add,axiom,
! [X12: multis501812127501805293od_a_a,Y13: multis501812127501805293od_a_a,Ns: set_Pr8600417178894128327od_a_a,X22: multis501812127501805293od_a_a,Y23: multis501812127501805293od_a_a] :
( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ X12 @ Y13 ) @ ( multis1821517007024690859od_a_a @ Ns ) )
=> ( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ X22 @ Y23 ) @ ( multis1821517007024690859od_a_a @ Ns ) )
=> ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ ( plus_p6967097483979887908od_a_a @ X12 @ X22 ) @ ( plus_p6967097483979887908od_a_a @ Y13 @ Y23 ) ) @ ( multis1821517007024690859od_a_a @ Ns ) ) ) ) ).
% multpw_add
thf(fact_263_multpw__add,axiom,
! [X12: multiset_a,Y13: multiset_a,Ns: set_Product_prod_a_a,X22: multiset_a,Y23: multiset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X12 @ Y13 ) @ ( multis2373687813370458242ltpw_a @ Ns ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X22 @ Y23 ) @ ( multis2373687813370458242ltpw_a @ Ns ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( plus_plus_multiset_a @ X12 @ X22 ) @ ( plus_plus_multiset_a @ Y13 @ Y23 ) ) @ ( multis2373687813370458242ltpw_a @ Ns ) ) ) ) ).
% multpw_add
thf(fact_264__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062xs_Ays_O_A_092_060lbrakk_062length_Axs_A_061_Alength_Ays_059_AA1_A_061_Amset_Axs_059_AB1_A_061_Amset_Ays_059_A_092_060forall_062i_060length_Ays_O_A_Ixs_A_B_Ai_M_Ays_A_B_Ai_J_A_092_060in_062_ANS_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Xs3: list_a,Ys3: list_a] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( a1
= ( mset_a @ Xs3 ) )
=> ( ( b1
= ( mset_a @ Ys3 ) )
=> ~ ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Ys3 ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs3 @ I2 ) @ ( nth_a @ Ys3 @ I2 ) ) @ ns ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>xs ys. \<lbrakk>length xs = length ys; A1 = mset xs; B1 = mset ys; \<forall>i<length ys. (xs ! i, ys ! i) \<in> NS\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_265__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062xs1_Ays1_O_A_092_060lbrakk_062length_Axs1_A_061_Alength_Ays1_059_AA1_A_061_Amset_Axs1_059_AB1_A_061_Amset_Ays1_059_A_092_060And_062i_O_Ai_A_060_Alength_Ays1_A_092_060Longrightarrow_062_A_Ixs1_A_B_Ai_M_Ays1_A_B_Ai_J_A_092_060in_062_ANS_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Xs1: list_a,Ys1: list_a] :
( ( ( size_size_list_a @ Xs1 )
= ( size_size_list_a @ Ys1 ) )
=> ( ( a1
= ( mset_a @ Xs1 ) )
=> ( ( b1
= ( mset_a @ Ys1 ) )
=> ~ ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Ys1 ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs1 @ I2 ) @ ( nth_a @ Ys1 @ I2 ) ) @ ns ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>xs1 ys1. \<lbrakk>length xs1 = length ys1; A1 = mset xs1; B1 = mset ys1; \<And>i. i < length ys1 \<Longrightarrow> (xs1 ! i, ys1 ! i) \<in> NS\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_266__092_060open_062_092_060forall_062i_060length_Ays1_O_A_Ixs1_A_B_Ai_M_Ays1_A_B_Ai_J_A_092_060in_062_ANS_092_060close_062,axiom,
! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ ys1 ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ xs1 @ I2 ) @ ( nth_a @ ys1 @ I2 ) ) @ ns ) ) ).
% \<open>\<forall>i<length ys1. (xs1 ! i, ys1 ! i) \<in> NS\<close>
thf(fact_267_NS,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ ys1 ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ xs1 @ I ) @ ( nth_a @ ys1 @ I ) ) @ ns ) ) ).
% NS
thf(fact_268_add__less__cancel__left,axiom,
! [C2: nat,A2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A2 ) @ ( plus_plus_nat @ C2 @ B ) )
= ( ord_less_nat @ A2 @ B ) ) ).
% add_less_cancel_left
thf(fact_269_add__less__cancel__right,axiom,
! [A2: nat,C2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( ord_less_nat @ A2 @ B ) ) ).
% add_less_cancel_right
thf(fact_270_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_271_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_272_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K2 = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_273_add__strict__mono,axiom,
! [A2: nat,B: nat,C2: nat,D2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ C2 @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_274_add__strict__left__mono,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C2 @ A2 ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).
% add_strict_left_mono
thf(fact_275_add__strict__right__mono,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_276_add__less__imp__less__left,axiom,
! [C2: nat,A2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A2 ) @ ( plus_plus_nat @ C2 @ B ) )
=> ( ord_less_nat @ A2 @ B ) ) ).
% add_less_imp_less_left
thf(fact_277_add__less__imp__less__right,axiom,
! [A2: nat,C2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
=> ( ord_less_nat @ A2 @ B ) ) ).
% add_less_imp_less_right
thf(fact_278_nth__mem__mset,axiom,
! [I: nat,Ls: list_P1396940483166286381od_a_a] :
( ( ord_less_nat @ I @ ( size_s3885678630836030617od_a_a @ Ls ) )
=> ( member1426531477525435216od_a_a @ ( nth_Product_prod_a_a @ Ls @ I ) @ ( set_ms119794563918438244od_a_a @ ( mset_P3709024939784119484od_a_a @ Ls ) ) ) ) ).
% nth_mem_mset
thf(fact_279_nth__mem__mset,axiom,
! [I: nat,Ls: list_P6988024003011616557iset_a] :
( ( ord_less_nat @ I @ ( size_s7238940367606365593iset_a @ Ls ) )
=> ( member5199237121806060112iset_a @ ( nth_Pr1079437516805851566iset_a @ Ls @ I ) @ ( set_ms2379723421352415844iset_a @ ( mset_P2771936761501114300iset_a @ Ls ) ) ) ) ).
% nth_mem_mset
thf(fact_280_nth__mem__mset,axiom,
! [I: nat,Ls: list_a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Ls ) )
=> ( member_a @ ( nth_a @ Ls @ I ) @ ( set_mset_a @ ( mset_a @ Ls ) ) ) ) ).
% nth_mem_mset
thf(fact_281_multpw__listI,axiom,
! [Xs: list_multiset_a,Ys: list_multiset_a,X4: multiset_multiset_a,Y4: multiset_multiset_a,Ns: set_Pr79727621955416071iset_a] :
( ( ( size_s3003218658903709366iset_a @ Xs )
= ( size_s3003218658903709366iset_a @ Ys ) )
=> ( ( X4
= ( mset_multiset_a @ Xs ) )
=> ( ( Y4
= ( mset_multiset_a @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3003218658903709366iset_a @ Ys ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( nth_multiset_a @ Xs @ I3 ) @ ( nth_multiset_a @ Ys @ I3 ) ) @ Ns ) )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X4 @ Y4 ) @ ( multis451435756853798856iset_a @ Ns ) ) ) ) ) ) ).
% multpw_listI
thf(fact_282_multpw__listI,axiom,
! [Xs: list_a,Ys: list_a,X4: multiset_a,Y4: multiset_a,Ns: set_Product_prod_a_a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( X4
= ( mset_a @ Xs ) )
=> ( ( Y4
= ( mset_a @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Ys ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs @ I3 ) @ ( nth_a @ Ys @ I3 ) ) @ Ns ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X4 @ Y4 ) @ ( multis2373687813370458242ltpw_a @ Ns ) ) ) ) ) ) ).
% multpw_listI
thf(fact_283_multpw__listE,axiom,
! [X4: multiset_multiset_a,Y4: multiset_multiset_a,Ns: set_Pr79727621955416071iset_a] :
( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X4 @ Y4 ) @ ( multis451435756853798856iset_a @ Ns ) )
=> ~ ! [Xs3: list_multiset_a,Ys3: list_multiset_a] :
( ( ( size_s3003218658903709366iset_a @ Xs3 )
= ( size_s3003218658903709366iset_a @ Ys3 ) )
=> ( ( X4
= ( mset_multiset_a @ Xs3 ) )
=> ( ( Y4
= ( mset_multiset_a @ Ys3 ) )
=> ~ ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3003218658903709366iset_a @ Ys3 ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( nth_multiset_a @ Xs3 @ I2 ) @ ( nth_multiset_a @ Ys3 @ I2 ) ) @ Ns ) ) ) ) ) ) ).
% multpw_listE
thf(fact_284_multpw__listE,axiom,
! [X4: multiset_a,Y4: multiset_a,Ns: set_Product_prod_a_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X4 @ Y4 ) @ ( multis2373687813370458242ltpw_a @ Ns ) )
=> ~ ! [Xs3: list_a,Ys3: list_a] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( X4
= ( mset_a @ Xs3 ) )
=> ( ( Y4
= ( mset_a @ Ys3 ) )
=> ~ ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Ys3 ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs3 @ I2 ) @ ( nth_a @ Ys3 @ I2 ) ) @ Ns ) ) ) ) ) ) ).
% multpw_listE
thf(fact_285_all__ns__ns__mul__ext,axiom,
! [As: list_multiset_a,Bs: list_multiset_a,Ns: set_Pr79727621955416071iset_a,S: set_Pr79727621955416071iset_a] :
( ( ( size_s3003218658903709366iset_a @ As )
= ( size_s3003218658903709366iset_a @ Bs ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3003218658903709366iset_a @ Bs ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( nth_multiset_a @ As @ I3 ) @ ( nth_multiset_a @ Bs @ I3 ) ) @ Ns ) )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ ( mset_multiset_a @ As ) @ ( mset_multiset_a @ Bs ) ) @ ( multis8112516946959054073iset_a @ Ns @ S ) ) ) ) ).
% all_ns_ns_mul_ext
thf(fact_286_all__ns__ns__mul__ext,axiom,
! [As: list_a,Bs: list_a,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( ( size_size_list_a @ As )
= ( size_size_list_a @ Bs ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Bs ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ As @ I3 ) @ ( nth_a @ Bs @ I3 ) ) @ Ns ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( mset_a @ As ) @ ( mset_a @ Bs ) ) @ ( multis4023584895248362547_ext_a @ Ns @ S ) ) ) ) ).
% all_ns_ns_mul_ext
thf(fact_287_ns__mul__extI__old,axiom,
! [A4: multis501812127501805293od_a_a,Xs: list_P1396940483166286381od_a_a,A23: multis501812127501805293od_a_a,B5: multis501812127501805293od_a_a,Ys: list_P1396940483166286381od_a_a,B23: multis501812127501805293od_a_a,Ns: set_Pr8600417178894128327od_a_a,S: set_Pr8600417178894128327od_a_a] :
( ( A4
= ( plus_p6967097483979887908od_a_a @ ( mset_P3709024939784119484od_a_a @ Xs ) @ A23 ) )
=> ( ( B5
= ( plus_p6967097483979887908od_a_a @ ( mset_P3709024939784119484od_a_a @ Ys ) @ B23 ) )
=> ( ( ( size_s3885678630836030617od_a_a @ Xs )
= ( size_s3885678630836030617od_a_a @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3885678630836030617od_a_a @ Ys ) )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ ( nth_Product_prod_a_a @ Xs @ I3 ) @ ( nth_Product_prod_a_a @ Ys @ I3 ) ) @ Ns ) )
=> ( ! [B3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ B3 @ ( set_ms119794563918438244od_a_a @ B23 ) )
=> ? [A6: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A6 @ ( set_ms119794563918438244od_a_a @ A23 ) )
& ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A6 @ B3 ) @ S ) ) )
=> ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ A4 @ B5 ) @ ( multis1178328011278088540od_a_a @ Ns @ S ) ) ) ) ) ) ) ).
% ns_mul_extI_old
thf(fact_288_ns__mul__extI__old,axiom,
! [A4: multis4781622204924204013iset_a,Xs: list_P6988024003011616557iset_a,A23: multis4781622204924204013iset_a,B5: multis4781622204924204013iset_a,Ys: list_P6988024003011616557iset_a,B23: multis4781622204924204013iset_a,Ns: set_Pr3059988933816007879iset_a,S: set_Pr3059988933816007879iset_a] :
( ( A4
= ( plus_p1053901119268525860iset_a @ ( mset_P2771936761501114300iset_a @ Xs ) @ A23 ) )
=> ( ( B5
= ( plus_p1053901119268525860iset_a @ ( mset_P2771936761501114300iset_a @ Ys ) @ B23 ) )
=> ( ( ( size_s7238940367606365593iset_a @ Xs )
= ( size_s7238940367606365593iset_a @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s7238940367606365593iset_a @ Ys ) )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ ( nth_Pr1079437516805851566iset_a @ Xs @ I3 ) @ ( nth_Pr1079437516805851566iset_a @ Ys @ I3 ) ) @ Ns ) )
=> ( ! [B3: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ B3 @ ( set_ms2379723421352415844iset_a @ B23 ) )
=> ? [A6: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ A6 @ ( set_ms2379723421352415844iset_a @ A23 ) )
& ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ A6 @ B3 ) @ S ) ) )
=> ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ A4 @ B5 ) @ ( multis2244641948333282396iset_a @ Ns @ S ) ) ) ) ) ) ) ).
% ns_mul_extI_old
thf(fact_289_ns__mul__extI__old,axiom,
! [A4: multiset_multiset_a,Xs: list_multiset_a,A23: multiset_multiset_a,B5: multiset_multiset_a,Ys: list_multiset_a,B23: multiset_multiset_a,Ns: set_Pr79727621955416071iset_a,S: set_Pr79727621955416071iset_a] :
( ( A4
= ( plus_p6738641960240532161iset_a @ ( mset_multiset_a @ Xs ) @ A23 ) )
=> ( ( B5
= ( plus_p6738641960240532161iset_a @ ( mset_multiset_a @ Ys ) @ B23 ) )
=> ( ( ( size_s3003218658903709366iset_a @ Xs )
= ( size_s3003218658903709366iset_a @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3003218658903709366iset_a @ Ys ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( nth_multiset_a @ Xs @ I3 ) @ ( nth_multiset_a @ Ys @ I3 ) ) @ Ns ) )
=> ( ! [B3: multiset_a] :
( ( member_multiset_a @ B3 @ ( set_mset_multiset_a @ B23 ) )
=> ? [A6: multiset_a] :
( ( member_multiset_a @ A6 @ ( set_mset_multiset_a @ A23 ) )
& ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A6 @ B3 ) @ S ) ) )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ A4 @ B5 ) @ ( multis8112516946959054073iset_a @ Ns @ S ) ) ) ) ) ) ) ).
% ns_mul_extI_old
thf(fact_290_ns__mul__extI__old,axiom,
! [A4: multiset_a,Xs: list_a,A23: multiset_a,B5: multiset_a,Ys: list_a,B23: multiset_a,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( A4
= ( plus_plus_multiset_a @ ( mset_a @ Xs ) @ A23 ) )
=> ( ( B5
= ( plus_plus_multiset_a @ ( mset_a @ Ys ) @ B23 ) )
=> ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Ys ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs @ I3 ) @ ( nth_a @ Ys @ I3 ) ) @ Ns ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ ( set_mset_a @ B23 ) )
=> ? [A6: a] :
( ( member_a @ A6 @ ( set_mset_a @ A23 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A6 @ B3 ) @ S ) ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A4 @ B5 ) @ ( multis4023584895248362547_ext_a @ Ns @ S ) ) ) ) ) ) ) ).
% ns_mul_extI_old
thf(fact_291_ns__mul__ext__intro,axiom,
! [Xs: multis501812127501805293od_a_a,Xs12: list_P1396940483166286381od_a_a,Xs22: list_P1396940483166286381od_a_a,Ys: multis501812127501805293od_a_a,Ys12: list_P1396940483166286381od_a_a,Ys22: list_P1396940483166286381od_a_a,NS: set_Pr8600417178894128327od_a_a,S2: set_Pr8600417178894128327od_a_a] :
( ( Xs
= ( plus_p6967097483979887908od_a_a @ ( mset_P3709024939784119484od_a_a @ Xs12 ) @ ( mset_P3709024939784119484od_a_a @ Xs22 ) ) )
=> ( ( Ys
= ( plus_p6967097483979887908od_a_a @ ( mset_P3709024939784119484od_a_a @ Ys12 ) @ ( mset_P3709024939784119484od_a_a @ Ys22 ) ) )
=> ( ( ( size_s3885678630836030617od_a_a @ Xs12 )
= ( size_s3885678630836030617od_a_a @ Ys12 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3885678630836030617od_a_a @ Ys12 ) )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ ( nth_Product_prod_a_a @ Xs12 @ I3 ) @ ( nth_Product_prod_a_a @ Ys12 @ I3 ) ) @ NS ) )
=> ( ! [Y3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y3 @ ( set_Product_prod_a_a2 @ Ys22 ) )
=> ? [X6: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X6 @ ( set_Product_prod_a_a2 @ Xs22 ) )
& ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X6 @ Y3 ) @ S2 ) ) )
=> ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ Xs @ Ys ) @ ( multis1178328011278088540od_a_a @ NS @ S2 ) ) ) ) ) ) ) ).
% ns_mul_ext_intro
thf(fact_292_ns__mul__ext__intro,axiom,
! [Xs: multis4781622204924204013iset_a,Xs12: list_P6988024003011616557iset_a,Xs22: list_P6988024003011616557iset_a,Ys: multis4781622204924204013iset_a,Ys12: list_P6988024003011616557iset_a,Ys22: list_P6988024003011616557iset_a,NS: set_Pr3059988933816007879iset_a,S2: set_Pr3059988933816007879iset_a] :
( ( Xs
= ( plus_p1053901119268525860iset_a @ ( mset_P2771936761501114300iset_a @ Xs12 ) @ ( mset_P2771936761501114300iset_a @ Xs22 ) ) )
=> ( ( Ys
= ( plus_p1053901119268525860iset_a @ ( mset_P2771936761501114300iset_a @ Ys12 ) @ ( mset_P2771936761501114300iset_a @ Ys22 ) ) )
=> ( ( ( size_s7238940367606365593iset_a @ Xs12 )
= ( size_s7238940367606365593iset_a @ Ys12 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s7238940367606365593iset_a @ Ys12 ) )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ ( nth_Pr1079437516805851566iset_a @ Xs12 @ I3 ) @ ( nth_Pr1079437516805851566iset_a @ Ys12 @ I3 ) ) @ NS ) )
=> ( ! [Y3: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Y3 @ ( set_Pr5550698219922361532iset_a @ Ys22 ) )
=> ? [X6: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X6 @ ( set_Pr5550698219922361532iset_a @ Xs22 ) )
& ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X6 @ Y3 ) @ S2 ) ) )
=> ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ Xs @ Ys ) @ ( multis2244641948333282396iset_a @ NS @ S2 ) ) ) ) ) ) ) ).
% ns_mul_ext_intro
thf(fact_293_ns__mul__ext__intro,axiom,
! [Xs: multiset_multiset_a,Xs12: list_multiset_a,Xs22: list_multiset_a,Ys: multiset_multiset_a,Ys12: list_multiset_a,Ys22: list_multiset_a,NS: set_Pr79727621955416071iset_a,S2: set_Pr79727621955416071iset_a] :
( ( Xs
= ( plus_p6738641960240532161iset_a @ ( mset_multiset_a @ Xs12 ) @ ( mset_multiset_a @ Xs22 ) ) )
=> ( ( Ys
= ( plus_p6738641960240532161iset_a @ ( mset_multiset_a @ Ys12 ) @ ( mset_multiset_a @ Ys22 ) ) )
=> ( ( ( size_s3003218658903709366iset_a @ Xs12 )
= ( size_s3003218658903709366iset_a @ Ys12 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3003218658903709366iset_a @ Ys12 ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( nth_multiset_a @ Xs12 @ I3 ) @ ( nth_multiset_a @ Ys12 @ I3 ) ) @ NS ) )
=> ( ! [Y3: multiset_a] :
( ( member_multiset_a @ Y3 @ ( set_multiset_a2 @ Ys22 ) )
=> ? [X6: multiset_a] :
( ( member_multiset_a @ X6 @ ( set_multiset_a2 @ Xs22 ) )
& ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X6 @ Y3 ) @ S2 ) ) )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ Xs @ Ys ) @ ( multis8112516946959054073iset_a @ NS @ S2 ) ) ) ) ) ) ) ).
% ns_mul_ext_intro
thf(fact_294_ns__mul__ext__intro,axiom,
! [Xs: multiset_a,Xs12: list_a,Xs22: list_a,Ys: multiset_a,Ys12: list_a,Ys22: list_a,NS: set_Product_prod_a_a,S2: set_Product_prod_a_a] :
( ( Xs
= ( plus_plus_multiset_a @ ( mset_a @ Xs12 ) @ ( mset_a @ Xs22 ) ) )
=> ( ( Ys
= ( plus_plus_multiset_a @ ( mset_a @ Ys12 ) @ ( mset_a @ Ys22 ) ) )
=> ( ( ( size_size_list_a @ Xs12 )
= ( size_size_list_a @ Ys12 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Ys12 ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs12 @ I3 ) @ ( nth_a @ Ys12 @ I3 ) ) @ NS ) )
=> ( ! [Y3: a] :
( ( member_a @ Y3 @ ( set_a2 @ Ys22 ) )
=> ? [X6: a] :
( ( member_a @ X6 @ ( set_a2 @ Xs22 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X6 @ Y3 ) @ S2 ) ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Xs @ Ys ) @ ( multis4023584895248362547_ext_a @ NS @ S2 ) ) ) ) ) ) ) ).
% ns_mul_ext_intro
thf(fact_295_nth__mem,axiom,
! [N3: nat,Xs: list_P1396940483166286381od_a_a] :
( ( ord_less_nat @ N3 @ ( size_s3885678630836030617od_a_a @ Xs ) )
=> ( member1426531477525435216od_a_a @ ( nth_Product_prod_a_a @ Xs @ N3 ) @ ( set_Product_prod_a_a2 @ Xs ) ) ) ).
% nth_mem
thf(fact_296_nth__mem,axiom,
! [N3: nat,Xs: list_P6988024003011616557iset_a] :
( ( ord_less_nat @ N3 @ ( size_s7238940367606365593iset_a @ Xs ) )
=> ( member5199237121806060112iset_a @ ( nth_Pr1079437516805851566iset_a @ Xs @ N3 ) @ ( set_Pr5550698219922361532iset_a @ Xs ) ) ) ).
% nth_mem
thf(fact_297_nth__mem,axiom,
! [N3: nat,Xs: list_a] :
( ( ord_less_nat @ N3 @ ( size_size_list_a @ Xs ) )
=> ( member_a @ ( nth_a @ Xs @ N3 ) @ ( set_a2 @ Xs ) ) ) ).
% nth_mem
thf(fact_298_list__ball__nth,axiom,
! [N3: nat,Xs: list_a,P2: a > $o] :
( ( ord_less_nat @ N3 @ ( size_size_list_a @ Xs ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( P2 @ X ) )
=> ( P2 @ ( nth_a @ Xs @ N3 ) ) ) ) ).
% list_ball_nth
thf(fact_299_in__set__conv__nth,axiom,
! [X3: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Xs ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s3885678630836030617od_a_a @ Xs ) )
& ( ( nth_Product_prod_a_a @ Xs @ I4 )
= X3 ) ) ) ) ).
% in_set_conv_nth
thf(fact_300_in__set__conv__nth,axiom,
! [X3: produc6518373309651786023iset_a,Xs: list_P6988024003011616557iset_a] :
( ( member5199237121806060112iset_a @ X3 @ ( set_Pr5550698219922361532iset_a @ Xs ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s7238940367606365593iset_a @ Xs ) )
& ( ( nth_Pr1079437516805851566iset_a @ Xs @ I4 )
= X3 ) ) ) ) ).
% in_set_conv_nth
thf(fact_301_in__set__conv__nth,axiom,
! [X3: a,Xs: list_a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs ) )
& ( ( nth_a @ Xs @ I4 )
= X3 ) ) ) ) ).
% in_set_conv_nth
thf(fact_302_all__nth__imp__all__set,axiom,
! [Xs: list_P1396940483166286381od_a_a,P2: product_prod_a_a > $o,X3: product_prod_a_a] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3885678630836030617od_a_a @ Xs ) )
=> ( P2 @ ( nth_Product_prod_a_a @ Xs @ I3 ) ) )
=> ( ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Xs ) )
=> ( P2 @ X3 ) ) ) ).
% all_nth_imp_all_set
thf(fact_303_all__nth__imp__all__set,axiom,
! [Xs: list_P6988024003011616557iset_a,P2: produc6518373309651786023iset_a > $o,X3: produc6518373309651786023iset_a] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s7238940367606365593iset_a @ Xs ) )
=> ( P2 @ ( nth_Pr1079437516805851566iset_a @ Xs @ I3 ) ) )
=> ( ( member5199237121806060112iset_a @ X3 @ ( set_Pr5550698219922361532iset_a @ Xs ) )
=> ( P2 @ X3 ) ) ) ).
% all_nth_imp_all_set
thf(fact_304_all__nth__imp__all__set,axiom,
! [Xs: list_a,P2: a > $o,X3: a] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
=> ( P2 @ ( nth_a @ Xs @ I3 ) ) )
=> ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( P2 @ X3 ) ) ) ).
% all_nth_imp_all_set
thf(fact_305_all__set__conv__all__nth,axiom,
! [Xs: list_a,P2: a > $o] :
( ( ! [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs ) )
=> ( P2 @ X5 ) ) )
= ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs ) )
=> ( P2 @ ( nth_a @ Xs @ I4 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_306_list__eq__iff__nth__eq,axiom,
( ( ^ [Y5: list_a,Z: list_a] : ( Y5 = Z ) )
= ( ^ [Xs4: list_a,Ys4: list_a] :
( ( ( size_size_list_a @ Xs4 )
= ( size_size_list_a @ Ys4 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs4 ) )
=> ( ( nth_a @ Xs4 @ I4 )
= ( nth_a @ Ys4 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_307_Skolem__list__nth,axiom,
! [K2: nat,P2: nat > a > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ? [X7: a] : ( P2 @ I4 @ X7 ) ) )
= ( ? [Xs4: list_a] :
( ( ( size_size_list_a @ Xs4 )
= K2 )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ( P2 @ I4 @ ( nth_a @ Xs4 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_308_nth__equalityI,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ Xs @ I3 )
= ( nth_a @ Ys @ I3 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_309_nat__add__left__cancel__less,axiom,
! [K2: nat,M3: nat,N3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K2 @ M3 ) @ ( plus_plus_nat @ K2 @ N3 ) )
= ( ord_less_nat @ M3 @ N3 ) ) ).
% nat_add_left_cancel_less
thf(fact_310_length__induct,axiom,
! [P2: list_a > $o,Xs: list_a] :
( ! [Xs3: list_a] :
( ! [Ys5: list_a] :
( ( ord_less_nat @ ( size_size_list_a @ Ys5 ) @ ( size_size_list_a @ Xs3 ) )
=> ( P2 @ Ys5 ) )
=> ( P2 @ Xs3 ) )
=> ( P2 @ Xs ) ) ).
% length_induct
thf(fact_311_linorder__neqE__nat,axiom,
! [X3: nat,Y: nat] :
( ( X3 != Y )
=> ( ~ ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ Y @ X3 ) ) ) ).
% linorder_neqE_nat
thf(fact_312_infinite__descent,axiom,
! [P2: nat > $o,N3: nat] :
( ! [N4: nat] :
( ~ ( P2 @ N4 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N4 )
& ~ ( P2 @ M4 ) ) )
=> ( P2 @ N3 ) ) ).
% infinite_descent
thf(fact_313_nat__less__induct,axiom,
! [P2: nat > $o,N3: nat] :
( ! [N4: nat] :
( ! [M4: nat] :
( ( ord_less_nat @ M4 @ N4 )
=> ( P2 @ M4 ) )
=> ( P2 @ N4 ) )
=> ( P2 @ N3 ) ) ).
% nat_less_induct
thf(fact_314_less__irrefl__nat,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ N3 ) ).
% less_irrefl_nat
thf(fact_315_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_316_less__not__refl2,axiom,
! [N3: nat,M3: nat] :
( ( ord_less_nat @ N3 @ M3 )
=> ( M3 != N3 ) ) ).
% less_not_refl2
thf(fact_317_less__not__refl,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ N3 ) ).
% less_not_refl
thf(fact_318_nat__neq__iff,axiom,
! [M3: nat,N3: nat] :
( ( M3 != N3 )
= ( ( ord_less_nat @ M3 @ N3 )
| ( ord_less_nat @ N3 @ M3 ) ) ) ).
% nat_neq_iff
thf(fact_319_less__add__eq__less,axiom,
! [K2: nat,L: nat,M3: nat,N3: nat] :
( ( ord_less_nat @ K2 @ L )
=> ( ( ( plus_plus_nat @ M3 @ L )
= ( plus_plus_nat @ K2 @ N3 ) )
=> ( ord_less_nat @ M3 @ N3 ) ) ) ).
% less_add_eq_less
thf(fact_320_trans__less__add2,axiom,
! [I: nat,J: nat,M3: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M3 @ J ) ) ) ).
% trans_less_add2
thf(fact_321_trans__less__add1,axiom,
! [I: nat,J: nat,M3: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M3 ) ) ) ).
% trans_less_add1
thf(fact_322_add__less__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% add_less_mono1
thf(fact_323_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_324_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_325_add__less__mono,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K2 @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_326_add__lessD1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K2 )
=> ( ord_less_nat @ I @ K2 ) ) ).
% add_lessD1
thf(fact_327_size__neq__size__imp__neq,axiom,
! [X3: list_a,Y: list_a] :
( ( ( size_size_list_a @ X3 )
!= ( size_size_list_a @ Y ) )
=> ( X3 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_328_size__neq__size__imp__neq,axiom,
! [X3: char,Y: char] :
( ( ( size_size_char @ X3 )
!= ( size_size_char @ Y ) )
=> ( X3 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_329_neq__if__length__neq,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( size_size_list_a @ Xs )
!= ( size_size_list_a @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_330_Ex__list__of__length,axiom,
! [N3: nat] :
? [Xs3: list_a] :
( ( size_size_list_a @ Xs3 )
= N3 ) ).
% Ex_list_of_length
thf(fact_331_nth__enumerate__eq,axiom,
! [M3: nat,Xs: list_a,N3: nat] :
( ( ord_less_nat @ M3 @ ( size_size_list_a @ Xs ) )
=> ( ( nth_Pr2476257081389315164_nat_a @ ( enumerate_a @ N3 @ Xs ) @ M3 )
= ( product_Pair_nat_a @ ( plus_plus_nat @ N3 @ M3 ) @ ( nth_a @ Xs @ M3 ) ) ) ) ).
% nth_enumerate_eq
thf(fact_332_mult2__alt__nsI,axiom,
! [X4: multis501812127501805293od_a_a,X12: multis501812127501805293od_a_a,X22: multis501812127501805293od_a_a,Y4: multis501812127501805293od_a_a,Y13: multis501812127501805293od_a_a,Y23: multis501812127501805293od_a_a,Ns: set_Pr8600417178894128327od_a_a,S: set_Pr8600417178894128327od_a_a] :
( ( X4
= ( plus_p6967097483979887908od_a_a @ X12 @ X22 ) )
=> ( ( Y4
= ( plus_p6967097483979887908od_a_a @ Y13 @ Y23 ) )
=> ( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ X12 @ Y13 ) @ ( multis1821517007024690859od_a_a @ Ns ) )
=> ( ! [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_ms119794563918438244od_a_a @ X22 ) )
=> ? [Y6: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y6 @ ( set_ms119794563918438244od_a_a @ Y23 ) )
& ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Y6 ) @ S ) ) )
=> ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ X4 @ Y4 ) @ ( multis4249482161328750907od_a_a @ $true @ Ns @ S ) ) ) ) ) ) ).
% mult2_alt_nsI
thf(fact_333_mult2__alt__nsI,axiom,
! [X4: multis4781622204924204013iset_a,X12: multis4781622204924204013iset_a,X22: multis4781622204924204013iset_a,Y4: multis4781622204924204013iset_a,Y13: multis4781622204924204013iset_a,Y23: multis4781622204924204013iset_a,Ns: set_Pr3059988933816007879iset_a,S: set_Pr3059988933816007879iset_a] :
( ( X4
= ( plus_p1053901119268525860iset_a @ X12 @ X22 ) )
=> ( ( Y4
= ( plus_p1053901119268525860iset_a @ Y13 @ Y23 ) )
=> ( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ X12 @ Y13 ) @ ( multis3430030276581740203iset_a @ Ns ) )
=> ( ! [X: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X @ ( set_ms2379723421352415844iset_a @ X22 ) )
=> ? [Y6: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Y6 @ ( set_ms2379723421352415844iset_a @ Y23 ) )
& ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X @ Y6 ) @ S ) ) )
=> ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ X4 @ Y4 ) @ ( multis1572145089415997755iset_a @ $true @ Ns @ S ) ) ) ) ) ) ).
% mult2_alt_nsI
thf(fact_334_mult2__alt__nsI,axiom,
! [X4: multiset_multiset_a,X12: multiset_multiset_a,X22: multiset_multiset_a,Y4: multiset_multiset_a,Y13: multiset_multiset_a,Y23: multiset_multiset_a,Ns: set_Pr79727621955416071iset_a,S: set_Pr79727621955416071iset_a] :
( ( X4
= ( plus_p6738641960240532161iset_a @ X12 @ X22 ) )
=> ( ( Y4
= ( plus_p6738641960240532161iset_a @ Y13 @ Y23 ) )
=> ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X12 @ Y13 ) @ ( multis451435756853798856iset_a @ Ns ) )
=> ( ! [X: multiset_a] :
( ( member_multiset_a @ X @ ( set_mset_multiset_a @ X22 ) )
=> ? [Y6: multiset_a] :
( ( member_multiset_a @ Y6 @ ( set_mset_multiset_a @ Y23 ) )
& ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ Y6 ) @ S ) ) )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X4 @ Y4 ) @ ( multis2006310815304088152iset_a @ $true @ Ns @ S ) ) ) ) ) ) ).
% mult2_alt_nsI
thf(fact_335_mult2__alt__nsI,axiom,
! [X4: multiset_a,X12: multiset_a,X22: multiset_a,Y4: multiset_a,Y13: multiset_a,Y23: multiset_a,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( X4
= ( plus_plus_multiset_a @ X12 @ X22 ) )
=> ( ( Y4
= ( plus_plus_multiset_a @ Y13 @ Y23 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X12 @ Y13 ) @ ( multis2373687813370458242ltpw_a @ Ns ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( set_mset_a @ X22 ) )
=> ? [Y6: a] :
( ( member_a @ Y6 @ ( set_mset_a @ Y23 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y6 ) @ S ) ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X4 @ Y4 ) @ ( multis1686031732355738386_alt_a @ $true @ Ns @ S ) ) ) ) ) ) ).
% mult2_alt_nsI
thf(fact_336_mult2__alt__nsE,axiom,
! [X4: multis501812127501805293od_a_a,Y4: multis501812127501805293od_a_a,Ns: set_Pr8600417178894128327od_a_a,S: set_Pr8600417178894128327od_a_a] :
( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ X4 @ Y4 ) @ ( multis4249482161328750907od_a_a @ $true @ Ns @ S ) )
=> ~ ! [X13: multis501812127501805293od_a_a,X23: multis501812127501805293od_a_a] :
( ( X4
= ( plus_p6967097483979887908od_a_a @ X13 @ X23 ) )
=> ! [Y12: multis501812127501805293od_a_a,Y22: multis501812127501805293od_a_a] :
( ( Y4
= ( plus_p6967097483979887908od_a_a @ Y12 @ Y22 ) )
=> ( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ X13 @ Y12 ) @ ( multis1821517007024690859od_a_a @ Ns ) )
=> ~ ! [X6: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X6 @ ( set_ms119794563918438244od_a_a @ X23 ) )
=> ? [Y3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y3 @ ( set_ms119794563918438244od_a_a @ Y22 ) )
& ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X6 @ Y3 ) @ S ) ) ) ) ) ) ) ).
% mult2_alt_nsE
thf(fact_337_mult2__alt__nsE,axiom,
! [X4: multis4781622204924204013iset_a,Y4: multis4781622204924204013iset_a,Ns: set_Pr3059988933816007879iset_a,S: set_Pr3059988933816007879iset_a] :
( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ X4 @ Y4 ) @ ( multis1572145089415997755iset_a @ $true @ Ns @ S ) )
=> ~ ! [X13: multis4781622204924204013iset_a,X23: multis4781622204924204013iset_a] :
( ( X4
= ( plus_p1053901119268525860iset_a @ X13 @ X23 ) )
=> ! [Y12: multis4781622204924204013iset_a,Y22: multis4781622204924204013iset_a] :
( ( Y4
= ( plus_p1053901119268525860iset_a @ Y12 @ Y22 ) )
=> ( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ X13 @ Y12 ) @ ( multis3430030276581740203iset_a @ Ns ) )
=> ~ ! [X6: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X6 @ ( set_ms2379723421352415844iset_a @ X23 ) )
=> ? [Y3: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Y3 @ ( set_ms2379723421352415844iset_a @ Y22 ) )
& ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X6 @ Y3 ) @ S ) ) ) ) ) ) ) ).
% mult2_alt_nsE
thf(fact_338_mult2__alt__nsE,axiom,
! [X4: multiset_multiset_a,Y4: multiset_multiset_a,Ns: set_Pr79727621955416071iset_a,S: set_Pr79727621955416071iset_a] :
( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X4 @ Y4 ) @ ( multis2006310815304088152iset_a @ $true @ Ns @ S ) )
=> ~ ! [X13: multiset_multiset_a,X23: multiset_multiset_a] :
( ( X4
= ( plus_p6738641960240532161iset_a @ X13 @ X23 ) )
=> ! [Y12: multiset_multiset_a,Y22: multiset_multiset_a] :
( ( Y4
= ( plus_p6738641960240532161iset_a @ Y12 @ Y22 ) )
=> ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X13 @ Y12 ) @ ( multis451435756853798856iset_a @ Ns ) )
=> ~ ! [X6: multiset_a] :
( ( member_multiset_a @ X6 @ ( set_mset_multiset_a @ X23 ) )
=> ? [Y3: multiset_a] :
( ( member_multiset_a @ Y3 @ ( set_mset_multiset_a @ Y22 ) )
& ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X6 @ Y3 ) @ S ) ) ) ) ) ) ) ).
% mult2_alt_nsE
thf(fact_339_mult2__alt__nsE,axiom,
! [X4: multiset_a,Y4: multiset_a,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X4 @ Y4 ) @ ( multis1686031732355738386_alt_a @ $true @ Ns @ S ) )
=> ~ ! [X13: multiset_a,X23: multiset_a] :
( ( X4
= ( plus_plus_multiset_a @ X13 @ X23 ) )
=> ! [Y12: multiset_a,Y22: multiset_a] :
( ( Y4
= ( plus_plus_multiset_a @ Y12 @ Y22 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X13 @ Y12 ) @ ( multis2373687813370458242ltpw_a @ Ns ) )
=> ~ ! [X6: a] :
( ( member_a @ X6 @ ( set_mset_a @ X23 ) )
=> ? [Y3: a] :
( ( member_a @ Y3 @ ( set_mset_a @ Y22 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X6 @ Y3 ) @ S ) ) ) ) ) ) ) ).
% mult2_alt_nsE
thf(fact_340_gen__length__def,axiom,
( gen_length_a
= ( ^ [N5: nat,Xs4: list_a] : ( plus_plus_nat @ N5 @ ( size_size_list_a @ Xs4 ) ) ) ) ).
% gen_length_def
thf(fact_341_ns__mul__ext__union__compat,axiom,
! [A4: multiset_a,B5: multiset_a,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a,C3: multiset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A4 @ B5 ) @ ( multis4023584895248362547_ext_a @ Ns @ S ) )
=> ( ( locally_refl_a @ Ns @ C3 )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( plus_plus_multiset_a @ A4 @ C3 ) @ ( plus_plus_multiset_a @ B5 @ C3 ) ) @ ( multis4023584895248362547_ext_a @ Ns @ S ) ) ) ) ).
% ns_mul_ext_union_compat
thf(fact_342_ns__mul__ext__IdI,axiom,
! [M: multiset_multiset_a,X4: multiset_multiset_a,Z2: multiset_multiset_a,N: multiset_multiset_a,Y4: multiset_multiset_a,R: set_Pr79727621955416071iset_a] :
( ( M
= ( plus_p6738641960240532161iset_a @ X4 @ Z2 ) )
=> ( ( N
= ( plus_p6738641960240532161iset_a @ Y4 @ Z2 ) )
=> ( ! [X: multiset_a] :
( ( member_multiset_a @ X @ ( set_mset_multiset_a @ Y4 ) )
=> ? [Xa: multiset_a] :
( ( member_multiset_a @ Xa @ ( set_mset_multiset_a @ X4 ) )
& ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Xa @ X ) @ R ) ) )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ M @ N ) @ ( multis8112516946959054073iset_a @ id_multiset_a @ R ) ) ) ) ) ).
% ns_mul_ext_IdI
thf(fact_343_ns__mul__ext__IdI,axiom,
! [M: multiset_a,X4: multiset_a,Z2: multiset_a,N: multiset_a,Y4: multiset_a,R: set_Product_prod_a_a] :
( ( M
= ( plus_plus_multiset_a @ X4 @ Z2 ) )
=> ( ( N
= ( plus_plus_multiset_a @ Y4 @ Z2 ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( set_mset_a @ Y4 ) )
=> ? [Xa: a] :
( ( member_a @ Xa @ ( set_mset_a @ X4 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa @ X ) @ R ) ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ M @ N ) @ ( multis4023584895248362547_ext_a @ id_a @ R ) ) ) ) ) ).
% ns_mul_ext_IdI
thf(fact_344_ns__mul__ext__IdE,axiom,
! [M: multiset_multiset_a,N: multiset_multiset_a,R: set_Pr79727621955416071iset_a] :
( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ M @ N ) @ ( multis8112516946959054073iset_a @ id_multiset_a @ R ) )
=> ~ ! [X8: multiset_multiset_a,Y7: multiset_multiset_a,Z3: multiset_multiset_a] :
( ( M
= ( plus_p6738641960240532161iset_a @ X8 @ Z3 ) )
=> ( ( N
= ( plus_p6738641960240532161iset_a @ Y7 @ Z3 ) )
=> ~ ! [X6: multiset_a] :
( ( member_multiset_a @ X6 @ ( set_mset_multiset_a @ Y7 ) )
=> ? [Xa2: multiset_a] :
( ( member_multiset_a @ Xa2 @ ( set_mset_multiset_a @ X8 ) )
& ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Xa2 @ X6 ) @ R ) ) ) ) ) ) ).
% ns_mul_ext_IdE
thf(fact_345_ns__mul__ext__IdE,axiom,
! [M: multiset_a,N: multiset_a,R: set_Product_prod_a_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ M @ N ) @ ( multis4023584895248362547_ext_a @ id_a @ R ) )
=> ~ ! [X8: multiset_a,Y7: multiset_a,Z3: multiset_a] :
( ( M
= ( plus_plus_multiset_a @ X8 @ Z3 ) )
=> ( ( N
= ( plus_plus_multiset_a @ Y7 @ Z3 ) )
=> ~ ! [X6: a] :
( ( member_a @ X6 @ ( set_mset_a @ Y7 ) )
=> ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_mset_a @ X8 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa2 @ X6 ) @ R ) ) ) ) ) ) ).
% ns_mul_ext_IdE
thf(fact_346_set__swap,axiom,
! [I: nat,Xs: list_a,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_a @ Xs ) )
=> ( ( set_a2 @ ( list_update_a @ ( list_update_a @ Xs @ I @ ( nth_a @ Xs @ J ) ) @ J @ ( nth_a @ Xs @ I ) ) )
= ( set_a2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_347_list__ex__length,axiom,
( list_ex_a
= ( ^ [P3: a > $o,Xs4: list_a] :
? [N5: nat] :
( ( ord_less_nat @ N5 @ ( size_size_list_a @ Xs4 ) )
& ( P3 @ ( nth_a @ Xs4 @ N5 ) ) ) ) ) ).
% list_ex_length
thf(fact_348_listrel__iff__nth,axiom,
! [Xs: list_multiset_a,Ys: list_multiset_a,R2: set_Pr79727621955416071iset_a] :
( ( member1640862112313989456iset_a @ ( produc1434922032272282391iset_a @ Xs @ Ys ) @ ( listre5045558437749241713iset_a @ R2 ) )
= ( ( ( size_s3003218658903709366iset_a @ Xs )
= ( size_s3003218658903709366iset_a @ Ys ) )
& ! [N5: nat] :
( ( ord_less_nat @ N5 @ ( size_s3003218658903709366iset_a @ Xs ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( nth_multiset_a @ Xs @ N5 ) @ ( nth_multiset_a @ Ys @ N5 ) ) @ R2 ) ) ) ) ).
% listrel_iff_nth
thf(fact_349_listrel__iff__nth,axiom,
! [Xs: list_nat_nat,Ys: list_nat,R2: set_Pr9093778441882193744at_nat] :
( ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ Xs @ Ys ) @ ( listrel_nat_nat_nat @ R2 ) )
= ( ( ( size_s8208510060688613859at_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
& ! [N5: nat] :
( ( ord_less_nat @ N5 @ ( size_s8208510060688613859at_nat @ Xs ) )
=> ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ ( nth_nat_nat @ Xs @ N5 ) @ ( nth_nat @ Ys @ N5 ) ) @ R2 ) ) ) ) ).
% listrel_iff_nth
thf(fact_350_listrel__iff__nth,axiom,
! [Xs: list_a,Ys: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel_a_a @ R2 ) )
= ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
& ! [N5: nat] :
( ( ord_less_nat @ N5 @ ( size_size_list_a @ Xs ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs @ N5 ) @ ( nth_a @ Ys @ N5 ) ) @ R2 ) ) ) ) ).
% listrel_iff_nth
thf(fact_351_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_352_size__mset,axiom,
! [Xs: list_a] :
( ( size_size_multiset_a @ ( mset_a @ Xs ) )
= ( size_size_list_a @ Xs ) ) ).
% size_mset
thf(fact_353_length__list__update,axiom,
! [Xs: list_a,I: nat,X3: a] :
( ( size_size_list_a @ ( list_update_a @ Xs @ I @ X3 ) )
= ( size_size_list_a @ Xs ) ) ).
% length_list_update
thf(fact_354_list__update__id,axiom,
! [Xs: list_a,I: nat] :
( ( list_update_a @ Xs @ I @ ( nth_a @ Xs @ I ) )
= Xs ) ).
% list_update_id
thf(fact_355_nth__list__update__neq,axiom,
! [I: nat,J: nat,Xs: list_a,X3: a] :
( ( I != J )
=> ( ( nth_a @ ( list_update_a @ Xs @ I @ X3 ) @ J )
= ( nth_a @ Xs @ J ) ) ) ).
% nth_list_update_neq
thf(fact_356_multpw__Id__Id,axiom,
( ( multis2373687813370458242ltpw_a @ id_a )
= id_multiset_a ) ).
% multpw_Id_Id
thf(fact_357_length__enumerate,axiom,
! [N3: nat,Xs: list_a] :
( ( size_s243904063682394823_nat_a @ ( enumerate_a @ N3 @ Xs ) )
= ( size_size_list_a @ Xs ) ) ).
% length_enumerate
thf(fact_358_nth__list__update__eq,axiom,
! [I: nat,Xs: list_a,X3: a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( list_update_a @ Xs @ I @ X3 ) @ I )
= X3 ) ) ).
% nth_list_update_eq
thf(fact_359_mult2__alt__s__implies__mult2__alt__ns,axiom,
! [X4: multiset_a,Y4: multiset_a,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X4 @ Y4 ) @ ( multis1686031732355738386_alt_a @ $false @ Ns @ S ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X4 @ Y4 ) @ ( multis1686031732355738386_alt_a @ $true @ Ns @ S ) ) ) ).
% mult2_alt_s_implies_mult2_alt_ns
thf(fact_360_listrel__eq__len,axiom,
! [Xs: list_a,Ys: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel_a_a @ R2 ) )
=> ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) ) ) ).
% listrel_eq_len
thf(fact_361_list__ex__cong,axiom,
! [Xs: list_P1396940483166286381od_a_a,Ys: list_P1396940483166286381od_a_a,F: product_prod_a_a > $o,G: product_prod_a_a > $o] :
( ( Xs = Ys )
=> ( ! [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Ys ) )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( list_e6552556518106193515od_a_a @ F @ Xs )
= ( list_e6552556518106193515od_a_a @ G @ Ys ) ) ) ) ).
% list_ex_cong
thf(fact_362_list__ex__cong,axiom,
! [Xs: list_P6988024003011616557iset_a,Ys: list_P6988024003011616557iset_a,F: produc6518373309651786023iset_a > $o,G: produc6518373309651786023iset_a > $o] :
( ( Xs = Ys )
=> ( ! [X: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X @ ( set_Pr5550698219922361532iset_a @ Ys ) )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( list_e2898061274115626603iset_a @ F @ Xs )
= ( list_e2898061274115626603iset_a @ G @ Ys ) ) ) ) ).
% list_ex_cong
thf(fact_363_list__ex__cong,axiom,
! [Xs: list_a,Ys: list_a,F: a > $o,G: a > $o] :
( ( Xs = Ys )
=> ( ! [X: a] :
( ( member_a @ X @ ( set_a2 @ Ys ) )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( list_ex_a @ F @ Xs )
= ( list_ex_a @ G @ Ys ) ) ) ) ).
% list_ex_cong
thf(fact_364_mult2__alt__add,axiom,
! [X12: multiset_a,Y13: multiset_a,B13: $o,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a,X22: multiset_a,Y23: multiset_a,B24: $o] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X12 @ Y13 ) @ ( multis1686031732355738386_alt_a @ B13 @ Ns @ S ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X22 @ Y23 ) @ ( multis1686031732355738386_alt_a @ B24 @ Ns @ S ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( plus_plus_multiset_a @ X12 @ X22 ) @ ( plus_plus_multiset_a @ Y13 @ Y23 ) )
@ ( multis1686031732355738386_alt_a
@ ( B13
& B24 )
@ Ns
@ S ) ) ) ) ).
% mult2_alt_add
thf(fact_365_mult2__alt__s__s__add,axiom,
! [X12: multiset_a,Y13: multiset_a,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a,X22: multiset_a,Y23: multiset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X12 @ Y13 ) @ ( multis1686031732355738386_alt_a @ $false @ Ns @ S ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X22 @ Y23 ) @ ( multis1686031732355738386_alt_a @ $false @ Ns @ S ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( plus_plus_multiset_a @ X12 @ X22 ) @ ( plus_plus_multiset_a @ Y13 @ Y23 ) ) @ ( multis1686031732355738386_alt_a @ $false @ Ns @ S ) ) ) ) ).
% mult2_alt_s_s_add
thf(fact_366_mult2__alt__ns__s__add,axiom,
! [X12: multiset_a,Y13: multiset_a,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a,X22: multiset_a,Y23: multiset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X12 @ Y13 ) @ ( multis1686031732355738386_alt_a @ $true @ Ns @ S ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X22 @ Y23 ) @ ( multis1686031732355738386_alt_a @ $false @ Ns @ S ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( plus_plus_multiset_a @ X12 @ X22 ) @ ( plus_plus_multiset_a @ Y13 @ Y23 ) ) @ ( multis1686031732355738386_alt_a @ $false @ Ns @ S ) ) ) ) ).
% mult2_alt_ns_s_add
thf(fact_367_mult2__alt__s__ns__add,axiom,
! [X12: multiset_a,Y13: multiset_a,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a,X22: multiset_a,Y23: multiset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X12 @ Y13 ) @ ( multis1686031732355738386_alt_a @ $false @ Ns @ S ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X22 @ Y23 ) @ ( multis1686031732355738386_alt_a @ $true @ Ns @ S ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( plus_plus_multiset_a @ X12 @ X22 ) @ ( plus_plus_multiset_a @ Y13 @ Y23 ) ) @ ( multis1686031732355738386_alt_a @ $false @ Ns @ S ) ) ) ) ).
% mult2_alt_s_ns_add
thf(fact_368_mult2__alt__ns__ns__add,axiom,
! [X12: multiset_a,Y13: multiset_a,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a,X22: multiset_a,Y23: multiset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X12 @ Y13 ) @ ( multis1686031732355738386_alt_a @ $true @ Ns @ S ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X22 @ Y23 ) @ ( multis1686031732355738386_alt_a @ $true @ Ns @ S ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( plus_plus_multiset_a @ X12 @ X22 ) @ ( plus_plus_multiset_a @ Y13 @ Y23 ) ) @ ( multis1686031732355738386_alt_a @ $true @ Ns @ S ) ) ) ) ).
% mult2_alt_ns_ns_add
thf(fact_369_multpw__implies__mult2__alt__ns,axiom,
! [X4: multiset_a,Y4: multiset_a,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X4 @ Y4 ) @ ( multis2373687813370458242ltpw_a @ Ns ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X4 @ Y4 ) @ ( multis1686031732355738386_alt_a @ $true @ Ns @ S ) ) ) ).
% multpw_implies_mult2_alt_ns
thf(fact_370_set__update__memI,axiom,
! [N3: nat,Xs: list_P1396940483166286381od_a_a,X3: product_prod_a_a] :
( ( ord_less_nat @ N3 @ ( size_s3885678630836030617od_a_a @ Xs ) )
=> ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ ( list_u4967751623001292181od_a_a @ Xs @ N3 @ X3 ) ) ) ) ).
% set_update_memI
thf(fact_371_set__update__memI,axiom,
! [N3: nat,Xs: list_P6988024003011616557iset_a,X3: produc6518373309651786023iset_a] :
( ( ord_less_nat @ N3 @ ( size_s7238940367606365593iset_a @ Xs ) )
=> ( member5199237121806060112iset_a @ X3 @ ( set_Pr5550698219922361532iset_a @ ( list_u2687166788783756181iset_a @ Xs @ N3 @ X3 ) ) ) ) ).
% set_update_memI
thf(fact_372_set__update__memI,axiom,
! [N3: nat,Xs: list_a,X3: a] :
( ( ord_less_nat @ N3 @ ( size_size_list_a @ Xs ) )
=> ( member_a @ X3 @ ( set_a2 @ ( list_update_a @ Xs @ N3 @ X3 ) ) ) ) ).
% set_update_memI
thf(fact_373_list__update__same__conv,axiom,
! [I: nat,Xs: list_a,X3: a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( ( list_update_a @ Xs @ I @ X3 )
= Xs )
= ( ( nth_a @ Xs @ I )
= X3 ) ) ) ).
% list_update_same_conv
thf(fact_374_nth__list__update,axiom,
! [I: nat,Xs: list_a,J: nat,X3: a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( ( I = J )
=> ( ( nth_a @ ( list_update_a @ Xs @ I @ X3 ) @ J )
= X3 ) )
& ( ( I != J )
=> ( ( nth_a @ ( list_update_a @ Xs @ I @ X3 ) @ J )
= ( nth_a @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_375_trans__mult2__alt__ns__ns__local,axiom,
! [X4: multis501812127501805293od_a_a,Y4: multis501812127501805293od_a_a,Z2: multis501812127501805293od_a_a,S: set_Pr8600417178894128327od_a_a,Ns: set_Pr8600417178894128327od_a_a] :
( ! [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_ms119794563918438244od_a_a @ X4 ) )
=> ! [Y3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y3 @ ( set_ms119794563918438244od_a_a @ Y4 ) )
=> ! [Z4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Z4 @ ( set_ms119794563918438244od_a_a @ Z2 ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Y3 ) @ S )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y3 @ Z4 ) @ S )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_ms119794563918438244od_a_a @ X4 ) )
=> ! [Y3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y3 @ ( set_ms119794563918438244od_a_a @ Y4 ) )
=> ! [Z4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Z4 @ ( set_ms119794563918438244od_a_a @ Z2 ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Y3 ) @ Ns )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y3 @ Z4 ) @ S )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_ms119794563918438244od_a_a @ X4 ) )
=> ! [Y3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y3 @ ( set_ms119794563918438244od_a_a @ Y4 ) )
=> ! [Z4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Z4 @ ( set_ms119794563918438244od_a_a @ Z2 ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Y3 ) @ S )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y3 @ Z4 ) @ Ns )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_ms119794563918438244od_a_a @ X4 ) )
=> ! [Y3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y3 @ ( set_ms119794563918438244od_a_a @ Y4 ) )
=> ! [Z4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Z4 @ ( set_ms119794563918438244od_a_a @ Z2 ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Y3 ) @ Ns )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y3 @ Z4 ) @ Ns )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Z4 ) @ Ns ) ) ) ) ) )
=> ( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ X4 @ Y4 ) @ ( multis4249482161328750907od_a_a @ $true @ Ns @ S ) )
=> ( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ Y4 @ Z2 ) @ ( multis4249482161328750907od_a_a @ $true @ Ns @ S ) )
=> ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ X4 @ Z2 ) @ ( multis4249482161328750907od_a_a @ $true @ Ns @ S ) ) ) ) ) ) ) ) ).
% trans_mult2_alt_ns_ns_local
thf(fact_376_trans__mult2__alt__ns__ns__local,axiom,
! [X4: multis4781622204924204013iset_a,Y4: multis4781622204924204013iset_a,Z2: multis4781622204924204013iset_a,S: set_Pr3059988933816007879iset_a,Ns: set_Pr3059988933816007879iset_a] :
( ! [X: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X @ ( set_ms2379723421352415844iset_a @ X4 ) )
=> ! [Y3: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Y3 @ ( set_ms2379723421352415844iset_a @ Y4 ) )
=> ! [Z4: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Z4 @ ( set_ms2379723421352415844iset_a @ Z2 ) )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X @ Y3 ) @ S )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ Y3 @ Z4 ) @ S )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X @ ( set_ms2379723421352415844iset_a @ X4 ) )
=> ! [Y3: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Y3 @ ( set_ms2379723421352415844iset_a @ Y4 ) )
=> ! [Z4: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Z4 @ ( set_ms2379723421352415844iset_a @ Z2 ) )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X @ Y3 ) @ Ns )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ Y3 @ Z4 ) @ S )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X @ ( set_ms2379723421352415844iset_a @ X4 ) )
=> ! [Y3: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Y3 @ ( set_ms2379723421352415844iset_a @ Y4 ) )
=> ! [Z4: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Z4 @ ( set_ms2379723421352415844iset_a @ Z2 ) )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X @ Y3 ) @ S )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ Y3 @ Z4 ) @ Ns )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X @ ( set_ms2379723421352415844iset_a @ X4 ) )
=> ! [Y3: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Y3 @ ( set_ms2379723421352415844iset_a @ Y4 ) )
=> ! [Z4: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Z4 @ ( set_ms2379723421352415844iset_a @ Z2 ) )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X @ Y3 ) @ Ns )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ Y3 @ Z4 ) @ Ns )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X @ Z4 ) @ Ns ) ) ) ) ) )
=> ( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ X4 @ Y4 ) @ ( multis1572145089415997755iset_a @ $true @ Ns @ S ) )
=> ( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ Y4 @ Z2 ) @ ( multis1572145089415997755iset_a @ $true @ Ns @ S ) )
=> ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ X4 @ Z2 ) @ ( multis1572145089415997755iset_a @ $true @ Ns @ S ) ) ) ) ) ) ) ) ).
% trans_mult2_alt_ns_ns_local
thf(fact_377_trans__mult2__alt__ns__ns__local,axiom,
! [X4: multiset_a,Y4: multiset_a,Z2: multiset_a,S: set_Product_prod_a_a,Ns: set_Product_prod_a_a] :
( ! [X: a] :
( ( member_a @ X @ ( set_mset_a @ X4 ) )
=> ! [Y3: a] :
( ( member_a @ Y3 @ ( set_mset_a @ Y4 ) )
=> ! [Z4: a] :
( ( member_a @ Z4 @ ( set_mset_a @ Z2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ S )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ Z4 ) @ S )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( set_mset_a @ X4 ) )
=> ! [Y3: a] :
( ( member_a @ Y3 @ ( set_mset_a @ Y4 ) )
=> ! [Z4: a] :
( ( member_a @ Z4 @ ( set_mset_a @ Z2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ Ns )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ Z4 ) @ S )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( set_mset_a @ X4 ) )
=> ! [Y3: a] :
( ( member_a @ Y3 @ ( set_mset_a @ Y4 ) )
=> ! [Z4: a] :
( ( member_a @ Z4 @ ( set_mset_a @ Z2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ S )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ Z4 ) @ Ns )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( set_mset_a @ X4 ) )
=> ! [Y3: a] :
( ( member_a @ Y3 @ ( set_mset_a @ Y4 ) )
=> ! [Z4: a] :
( ( member_a @ Z4 @ ( set_mset_a @ Z2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ Ns )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ Z4 ) @ Ns )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Z4 ) @ Ns ) ) ) ) ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X4 @ Y4 ) @ ( multis1686031732355738386_alt_a @ $true @ Ns @ S ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Y4 @ Z2 ) @ ( multis1686031732355738386_alt_a @ $true @ Ns @ S ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X4 @ Z2 ) @ ( multis1686031732355738386_alt_a @ $true @ Ns @ S ) ) ) ) ) ) ) ) ).
% trans_mult2_alt_ns_ns_local
thf(fact_378_trans__mult2__alt__ns__ns__local,axiom,
! [X4: multiset_multiset_a,Y4: multiset_multiset_a,Z2: multiset_multiset_a,S: set_Pr79727621955416071iset_a,Ns: set_Pr79727621955416071iset_a] :
( ! [X: multiset_a] :
( ( member_multiset_a @ X @ ( set_mset_multiset_a @ X4 ) )
=> ! [Y3: multiset_a] :
( ( member_multiset_a @ Y3 @ ( set_mset_multiset_a @ Y4 ) )
=> ! [Z4: multiset_a] :
( ( member_multiset_a @ Z4 @ ( set_mset_multiset_a @ Z2 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ Y3 ) @ S )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Y3 @ Z4 ) @ S )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: multiset_a] :
( ( member_multiset_a @ X @ ( set_mset_multiset_a @ X4 ) )
=> ! [Y3: multiset_a] :
( ( member_multiset_a @ Y3 @ ( set_mset_multiset_a @ Y4 ) )
=> ! [Z4: multiset_a] :
( ( member_multiset_a @ Z4 @ ( set_mset_multiset_a @ Z2 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ Y3 ) @ Ns )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Y3 @ Z4 ) @ S )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: multiset_a] :
( ( member_multiset_a @ X @ ( set_mset_multiset_a @ X4 ) )
=> ! [Y3: multiset_a] :
( ( member_multiset_a @ Y3 @ ( set_mset_multiset_a @ Y4 ) )
=> ! [Z4: multiset_a] :
( ( member_multiset_a @ Z4 @ ( set_mset_multiset_a @ Z2 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ Y3 ) @ S )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Y3 @ Z4 ) @ Ns )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: multiset_a] :
( ( member_multiset_a @ X @ ( set_mset_multiset_a @ X4 ) )
=> ! [Y3: multiset_a] :
( ( member_multiset_a @ Y3 @ ( set_mset_multiset_a @ Y4 ) )
=> ! [Z4: multiset_a] :
( ( member_multiset_a @ Z4 @ ( set_mset_multiset_a @ Z2 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ Y3 ) @ Ns )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Y3 @ Z4 ) @ Ns )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ Z4 ) @ Ns ) ) ) ) ) )
=> ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X4 @ Y4 ) @ ( multis2006310815304088152iset_a @ $true @ Ns @ S ) )
=> ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ Y4 @ Z2 ) @ ( multis2006310815304088152iset_a @ $true @ Ns @ S ) )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X4 @ Z2 ) @ ( multis2006310815304088152iset_a @ $true @ Ns @ S ) ) ) ) ) ) ) ) ).
% trans_mult2_alt_ns_ns_local
thf(fact_379_trans__mult2__alt__s__ns__local,axiom,
! [X4: multis501812127501805293od_a_a,Y4: multis501812127501805293od_a_a,Z2: multis501812127501805293od_a_a,S: set_Pr8600417178894128327od_a_a,Ns: set_Pr8600417178894128327od_a_a] :
( ! [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_ms119794563918438244od_a_a @ X4 ) )
=> ! [Y3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y3 @ ( set_ms119794563918438244od_a_a @ Y4 ) )
=> ! [Z4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Z4 @ ( set_ms119794563918438244od_a_a @ Z2 ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Y3 ) @ S )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y3 @ Z4 ) @ S )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_ms119794563918438244od_a_a @ X4 ) )
=> ! [Y3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y3 @ ( set_ms119794563918438244od_a_a @ Y4 ) )
=> ! [Z4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Z4 @ ( set_ms119794563918438244od_a_a @ Z2 ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Y3 ) @ Ns )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y3 @ Z4 ) @ S )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_ms119794563918438244od_a_a @ X4 ) )
=> ! [Y3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y3 @ ( set_ms119794563918438244od_a_a @ Y4 ) )
=> ! [Z4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Z4 @ ( set_ms119794563918438244od_a_a @ Z2 ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Y3 ) @ S )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y3 @ Z4 ) @ Ns )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_ms119794563918438244od_a_a @ X4 ) )
=> ! [Y3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y3 @ ( set_ms119794563918438244od_a_a @ Y4 ) )
=> ! [Z4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Z4 @ ( set_ms119794563918438244od_a_a @ Z2 ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Y3 ) @ Ns )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y3 @ Z4 ) @ Ns )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Z4 ) @ Ns ) ) ) ) ) )
=> ( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ X4 @ Y4 ) @ ( multis4249482161328750907od_a_a @ $false @ Ns @ S ) )
=> ( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ Y4 @ Z2 ) @ ( multis4249482161328750907od_a_a @ $true @ Ns @ S ) )
=> ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ X4 @ Z2 ) @ ( multis4249482161328750907od_a_a @ $false @ Ns @ S ) ) ) ) ) ) ) ) ).
% trans_mult2_alt_s_ns_local
thf(fact_380_trans__mult2__alt__s__ns__local,axiom,
! [X4: multis4781622204924204013iset_a,Y4: multis4781622204924204013iset_a,Z2: multis4781622204924204013iset_a,S: set_Pr3059988933816007879iset_a,Ns: set_Pr3059988933816007879iset_a] :
( ! [X: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X @ ( set_ms2379723421352415844iset_a @ X4 ) )
=> ! [Y3: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Y3 @ ( set_ms2379723421352415844iset_a @ Y4 ) )
=> ! [Z4: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Z4 @ ( set_ms2379723421352415844iset_a @ Z2 ) )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X @ Y3 ) @ S )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ Y3 @ Z4 ) @ S )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X @ ( set_ms2379723421352415844iset_a @ X4 ) )
=> ! [Y3: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Y3 @ ( set_ms2379723421352415844iset_a @ Y4 ) )
=> ! [Z4: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Z4 @ ( set_ms2379723421352415844iset_a @ Z2 ) )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X @ Y3 ) @ Ns )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ Y3 @ Z4 ) @ S )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X @ ( set_ms2379723421352415844iset_a @ X4 ) )
=> ! [Y3: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Y3 @ ( set_ms2379723421352415844iset_a @ Y4 ) )
=> ! [Z4: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Z4 @ ( set_ms2379723421352415844iset_a @ Z2 ) )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X @ Y3 ) @ S )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ Y3 @ Z4 ) @ Ns )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X @ ( set_ms2379723421352415844iset_a @ X4 ) )
=> ! [Y3: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Y3 @ ( set_ms2379723421352415844iset_a @ Y4 ) )
=> ! [Z4: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Z4 @ ( set_ms2379723421352415844iset_a @ Z2 ) )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X @ Y3 ) @ Ns )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ Y3 @ Z4 ) @ Ns )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X @ Z4 ) @ Ns ) ) ) ) ) )
=> ( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ X4 @ Y4 ) @ ( multis1572145089415997755iset_a @ $false @ Ns @ S ) )
=> ( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ Y4 @ Z2 ) @ ( multis1572145089415997755iset_a @ $true @ Ns @ S ) )
=> ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ X4 @ Z2 ) @ ( multis1572145089415997755iset_a @ $false @ Ns @ S ) ) ) ) ) ) ) ) ).
% trans_mult2_alt_s_ns_local
thf(fact_381_trans__mult2__alt__s__ns__local,axiom,
! [X4: multiset_a,Y4: multiset_a,Z2: multiset_a,S: set_Product_prod_a_a,Ns: set_Product_prod_a_a] :
( ! [X: a] :
( ( member_a @ X @ ( set_mset_a @ X4 ) )
=> ! [Y3: a] :
( ( member_a @ Y3 @ ( set_mset_a @ Y4 ) )
=> ! [Z4: a] :
( ( member_a @ Z4 @ ( set_mset_a @ Z2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ S )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ Z4 ) @ S )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( set_mset_a @ X4 ) )
=> ! [Y3: a] :
( ( member_a @ Y3 @ ( set_mset_a @ Y4 ) )
=> ! [Z4: a] :
( ( member_a @ Z4 @ ( set_mset_a @ Z2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ Ns )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ Z4 ) @ S )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( set_mset_a @ X4 ) )
=> ! [Y3: a] :
( ( member_a @ Y3 @ ( set_mset_a @ Y4 ) )
=> ! [Z4: a] :
( ( member_a @ Z4 @ ( set_mset_a @ Z2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ S )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ Z4 ) @ Ns )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( set_mset_a @ X4 ) )
=> ! [Y3: a] :
( ( member_a @ Y3 @ ( set_mset_a @ Y4 ) )
=> ! [Z4: a] :
( ( member_a @ Z4 @ ( set_mset_a @ Z2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ Ns )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ Z4 ) @ Ns )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Z4 ) @ Ns ) ) ) ) ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X4 @ Y4 ) @ ( multis1686031732355738386_alt_a @ $false @ Ns @ S ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Y4 @ Z2 ) @ ( multis1686031732355738386_alt_a @ $true @ Ns @ S ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X4 @ Z2 ) @ ( multis1686031732355738386_alt_a @ $false @ Ns @ S ) ) ) ) ) ) ) ) ).
% trans_mult2_alt_s_ns_local
thf(fact_382_trans__mult2__alt__s__ns__local,axiom,
! [X4: multiset_multiset_a,Y4: multiset_multiset_a,Z2: multiset_multiset_a,S: set_Pr79727621955416071iset_a,Ns: set_Pr79727621955416071iset_a] :
( ! [X: multiset_a] :
( ( member_multiset_a @ X @ ( set_mset_multiset_a @ X4 ) )
=> ! [Y3: multiset_a] :
( ( member_multiset_a @ Y3 @ ( set_mset_multiset_a @ Y4 ) )
=> ! [Z4: multiset_a] :
( ( member_multiset_a @ Z4 @ ( set_mset_multiset_a @ Z2 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ Y3 ) @ S )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Y3 @ Z4 ) @ S )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: multiset_a] :
( ( member_multiset_a @ X @ ( set_mset_multiset_a @ X4 ) )
=> ! [Y3: multiset_a] :
( ( member_multiset_a @ Y3 @ ( set_mset_multiset_a @ Y4 ) )
=> ! [Z4: multiset_a] :
( ( member_multiset_a @ Z4 @ ( set_mset_multiset_a @ Z2 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ Y3 ) @ Ns )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Y3 @ Z4 ) @ S )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: multiset_a] :
( ( member_multiset_a @ X @ ( set_mset_multiset_a @ X4 ) )
=> ! [Y3: multiset_a] :
( ( member_multiset_a @ Y3 @ ( set_mset_multiset_a @ Y4 ) )
=> ! [Z4: multiset_a] :
( ( member_multiset_a @ Z4 @ ( set_mset_multiset_a @ Z2 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ Y3 ) @ S )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Y3 @ Z4 ) @ Ns )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: multiset_a] :
( ( member_multiset_a @ X @ ( set_mset_multiset_a @ X4 ) )
=> ! [Y3: multiset_a] :
( ( member_multiset_a @ Y3 @ ( set_mset_multiset_a @ Y4 ) )
=> ! [Z4: multiset_a] :
( ( member_multiset_a @ Z4 @ ( set_mset_multiset_a @ Z2 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ Y3 ) @ Ns )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Y3 @ Z4 ) @ Ns )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ Z4 ) @ Ns ) ) ) ) ) )
=> ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X4 @ Y4 ) @ ( multis2006310815304088152iset_a @ $false @ Ns @ S ) )
=> ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ Y4 @ Z2 ) @ ( multis2006310815304088152iset_a @ $true @ Ns @ S ) )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X4 @ Z2 ) @ ( multis2006310815304088152iset_a @ $false @ Ns @ S ) ) ) ) ) ) ) ) ).
% trans_mult2_alt_s_ns_local
thf(fact_383_trans__mult2__alt__ns__s__local,axiom,
! [X4: multis501812127501805293od_a_a,Y4: multis501812127501805293od_a_a,Z2: multis501812127501805293od_a_a,S: set_Pr8600417178894128327od_a_a,Ns: set_Pr8600417178894128327od_a_a] :
( ! [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_ms119794563918438244od_a_a @ X4 ) )
=> ! [Y3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y3 @ ( set_ms119794563918438244od_a_a @ Y4 ) )
=> ! [Z4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Z4 @ ( set_ms119794563918438244od_a_a @ Z2 ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Y3 ) @ S )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y3 @ Z4 ) @ S )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_ms119794563918438244od_a_a @ X4 ) )
=> ! [Y3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y3 @ ( set_ms119794563918438244od_a_a @ Y4 ) )
=> ! [Z4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Z4 @ ( set_ms119794563918438244od_a_a @ Z2 ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Y3 ) @ Ns )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y3 @ Z4 ) @ S )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_ms119794563918438244od_a_a @ X4 ) )
=> ! [Y3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y3 @ ( set_ms119794563918438244od_a_a @ Y4 ) )
=> ! [Z4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Z4 @ ( set_ms119794563918438244od_a_a @ Z2 ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Y3 ) @ S )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y3 @ Z4 ) @ Ns )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_ms119794563918438244od_a_a @ X4 ) )
=> ! [Y3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y3 @ ( set_ms119794563918438244od_a_a @ Y4 ) )
=> ! [Z4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Z4 @ ( set_ms119794563918438244od_a_a @ Z2 ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Y3 ) @ Ns )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y3 @ Z4 ) @ Ns )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Z4 ) @ Ns ) ) ) ) ) )
=> ( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ X4 @ Y4 ) @ ( multis4249482161328750907od_a_a @ $true @ Ns @ S ) )
=> ( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ Y4 @ Z2 ) @ ( multis4249482161328750907od_a_a @ $false @ Ns @ S ) )
=> ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ X4 @ Z2 ) @ ( multis4249482161328750907od_a_a @ $false @ Ns @ S ) ) ) ) ) ) ) ) ).
% trans_mult2_alt_ns_s_local
thf(fact_384_trans__mult2__alt__ns__s__local,axiom,
! [X4: multis4781622204924204013iset_a,Y4: multis4781622204924204013iset_a,Z2: multis4781622204924204013iset_a,S: set_Pr3059988933816007879iset_a,Ns: set_Pr3059988933816007879iset_a] :
( ! [X: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X @ ( set_ms2379723421352415844iset_a @ X4 ) )
=> ! [Y3: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Y3 @ ( set_ms2379723421352415844iset_a @ Y4 ) )
=> ! [Z4: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Z4 @ ( set_ms2379723421352415844iset_a @ Z2 ) )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X @ Y3 ) @ S )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ Y3 @ Z4 ) @ S )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X @ ( set_ms2379723421352415844iset_a @ X4 ) )
=> ! [Y3: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Y3 @ ( set_ms2379723421352415844iset_a @ Y4 ) )
=> ! [Z4: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Z4 @ ( set_ms2379723421352415844iset_a @ Z2 ) )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X @ Y3 ) @ Ns )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ Y3 @ Z4 ) @ S )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X @ ( set_ms2379723421352415844iset_a @ X4 ) )
=> ! [Y3: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Y3 @ ( set_ms2379723421352415844iset_a @ Y4 ) )
=> ! [Z4: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Z4 @ ( set_ms2379723421352415844iset_a @ Z2 ) )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X @ Y3 ) @ S )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ Y3 @ Z4 ) @ Ns )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X @ ( set_ms2379723421352415844iset_a @ X4 ) )
=> ! [Y3: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Y3 @ ( set_ms2379723421352415844iset_a @ Y4 ) )
=> ! [Z4: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Z4 @ ( set_ms2379723421352415844iset_a @ Z2 ) )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X @ Y3 ) @ Ns )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ Y3 @ Z4 ) @ Ns )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X @ Z4 ) @ Ns ) ) ) ) ) )
=> ( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ X4 @ Y4 ) @ ( multis1572145089415997755iset_a @ $true @ Ns @ S ) )
=> ( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ Y4 @ Z2 ) @ ( multis1572145089415997755iset_a @ $false @ Ns @ S ) )
=> ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ X4 @ Z2 ) @ ( multis1572145089415997755iset_a @ $false @ Ns @ S ) ) ) ) ) ) ) ) ).
% trans_mult2_alt_ns_s_local
thf(fact_385_trans__mult2__alt__ns__s__local,axiom,
! [X4: multiset_a,Y4: multiset_a,Z2: multiset_a,S: set_Product_prod_a_a,Ns: set_Product_prod_a_a] :
( ! [X: a] :
( ( member_a @ X @ ( set_mset_a @ X4 ) )
=> ! [Y3: a] :
( ( member_a @ Y3 @ ( set_mset_a @ Y4 ) )
=> ! [Z4: a] :
( ( member_a @ Z4 @ ( set_mset_a @ Z2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ S )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ Z4 ) @ S )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( set_mset_a @ X4 ) )
=> ! [Y3: a] :
( ( member_a @ Y3 @ ( set_mset_a @ Y4 ) )
=> ! [Z4: a] :
( ( member_a @ Z4 @ ( set_mset_a @ Z2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ Ns )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ Z4 ) @ S )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( set_mset_a @ X4 ) )
=> ! [Y3: a] :
( ( member_a @ Y3 @ ( set_mset_a @ Y4 ) )
=> ! [Z4: a] :
( ( member_a @ Z4 @ ( set_mset_a @ Z2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ S )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ Z4 ) @ Ns )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( set_mset_a @ X4 ) )
=> ! [Y3: a] :
( ( member_a @ Y3 @ ( set_mset_a @ Y4 ) )
=> ! [Z4: a] :
( ( member_a @ Z4 @ ( set_mset_a @ Z2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ Ns )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ Z4 ) @ Ns )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Z4 ) @ Ns ) ) ) ) ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X4 @ Y4 ) @ ( multis1686031732355738386_alt_a @ $true @ Ns @ S ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Y4 @ Z2 ) @ ( multis1686031732355738386_alt_a @ $false @ Ns @ S ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X4 @ Z2 ) @ ( multis1686031732355738386_alt_a @ $false @ Ns @ S ) ) ) ) ) ) ) ) ).
% trans_mult2_alt_ns_s_local
thf(fact_386_trans__mult2__alt__ns__s__local,axiom,
! [X4: multiset_multiset_a,Y4: multiset_multiset_a,Z2: multiset_multiset_a,S: set_Pr79727621955416071iset_a,Ns: set_Pr79727621955416071iset_a] :
( ! [X: multiset_a] :
( ( member_multiset_a @ X @ ( set_mset_multiset_a @ X4 ) )
=> ! [Y3: multiset_a] :
( ( member_multiset_a @ Y3 @ ( set_mset_multiset_a @ Y4 ) )
=> ! [Z4: multiset_a] :
( ( member_multiset_a @ Z4 @ ( set_mset_multiset_a @ Z2 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ Y3 ) @ S )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Y3 @ Z4 ) @ S )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: multiset_a] :
( ( member_multiset_a @ X @ ( set_mset_multiset_a @ X4 ) )
=> ! [Y3: multiset_a] :
( ( member_multiset_a @ Y3 @ ( set_mset_multiset_a @ Y4 ) )
=> ! [Z4: multiset_a] :
( ( member_multiset_a @ Z4 @ ( set_mset_multiset_a @ Z2 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ Y3 ) @ Ns )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Y3 @ Z4 ) @ S )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: multiset_a] :
( ( member_multiset_a @ X @ ( set_mset_multiset_a @ X4 ) )
=> ! [Y3: multiset_a] :
( ( member_multiset_a @ Y3 @ ( set_mset_multiset_a @ Y4 ) )
=> ! [Z4: multiset_a] :
( ( member_multiset_a @ Z4 @ ( set_mset_multiset_a @ Z2 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ Y3 ) @ S )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Y3 @ Z4 ) @ Ns )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: multiset_a] :
( ( member_multiset_a @ X @ ( set_mset_multiset_a @ X4 ) )
=> ! [Y3: multiset_a] :
( ( member_multiset_a @ Y3 @ ( set_mset_multiset_a @ Y4 ) )
=> ! [Z4: multiset_a] :
( ( member_multiset_a @ Z4 @ ( set_mset_multiset_a @ Z2 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ Y3 ) @ Ns )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Y3 @ Z4 ) @ Ns )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ Z4 ) @ Ns ) ) ) ) ) )
=> ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X4 @ Y4 ) @ ( multis2006310815304088152iset_a @ $true @ Ns @ S ) )
=> ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ Y4 @ Z2 ) @ ( multis2006310815304088152iset_a @ $false @ Ns @ S ) )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X4 @ Z2 ) @ ( multis2006310815304088152iset_a @ $false @ Ns @ S ) ) ) ) ) ) ) ) ).
% trans_mult2_alt_ns_s_local
thf(fact_387_trans__mult2__alt__s__s__local,axiom,
! [X4: multis501812127501805293od_a_a,Y4: multis501812127501805293od_a_a,Z2: multis501812127501805293od_a_a,S: set_Pr8600417178894128327od_a_a,Ns: set_Pr8600417178894128327od_a_a] :
( ! [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_ms119794563918438244od_a_a @ X4 ) )
=> ! [Y3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y3 @ ( set_ms119794563918438244od_a_a @ Y4 ) )
=> ! [Z4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Z4 @ ( set_ms119794563918438244od_a_a @ Z2 ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Y3 ) @ S )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y3 @ Z4 ) @ S )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_ms119794563918438244od_a_a @ X4 ) )
=> ! [Y3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y3 @ ( set_ms119794563918438244od_a_a @ Y4 ) )
=> ! [Z4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Z4 @ ( set_ms119794563918438244od_a_a @ Z2 ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Y3 ) @ Ns )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y3 @ Z4 ) @ S )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_ms119794563918438244od_a_a @ X4 ) )
=> ! [Y3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y3 @ ( set_ms119794563918438244od_a_a @ Y4 ) )
=> ! [Z4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Z4 @ ( set_ms119794563918438244od_a_a @ Z2 ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Y3 ) @ S )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y3 @ Z4 ) @ Ns )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_ms119794563918438244od_a_a @ X4 ) )
=> ! [Y3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y3 @ ( set_ms119794563918438244od_a_a @ Y4 ) )
=> ! [Z4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Z4 @ ( set_ms119794563918438244od_a_a @ Z2 ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Y3 ) @ Ns )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y3 @ Z4 ) @ Ns )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Z4 ) @ Ns ) ) ) ) ) )
=> ( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ X4 @ Y4 ) @ ( multis4249482161328750907od_a_a @ $false @ Ns @ S ) )
=> ( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ Y4 @ Z2 ) @ ( multis4249482161328750907od_a_a @ $false @ Ns @ S ) )
=> ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ X4 @ Z2 ) @ ( multis4249482161328750907od_a_a @ $false @ Ns @ S ) ) ) ) ) ) ) ) ).
% trans_mult2_alt_s_s_local
thf(fact_388_trans__mult2__alt__s__s__local,axiom,
! [X4: multis4781622204924204013iset_a,Y4: multis4781622204924204013iset_a,Z2: multis4781622204924204013iset_a,S: set_Pr3059988933816007879iset_a,Ns: set_Pr3059988933816007879iset_a] :
( ! [X: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X @ ( set_ms2379723421352415844iset_a @ X4 ) )
=> ! [Y3: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Y3 @ ( set_ms2379723421352415844iset_a @ Y4 ) )
=> ! [Z4: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Z4 @ ( set_ms2379723421352415844iset_a @ Z2 ) )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X @ Y3 ) @ S )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ Y3 @ Z4 ) @ S )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X @ ( set_ms2379723421352415844iset_a @ X4 ) )
=> ! [Y3: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Y3 @ ( set_ms2379723421352415844iset_a @ Y4 ) )
=> ! [Z4: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Z4 @ ( set_ms2379723421352415844iset_a @ Z2 ) )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X @ Y3 ) @ Ns )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ Y3 @ Z4 ) @ S )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X @ ( set_ms2379723421352415844iset_a @ X4 ) )
=> ! [Y3: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Y3 @ ( set_ms2379723421352415844iset_a @ Y4 ) )
=> ! [Z4: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Z4 @ ( set_ms2379723421352415844iset_a @ Z2 ) )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X @ Y3 ) @ S )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ Y3 @ Z4 ) @ Ns )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X @ ( set_ms2379723421352415844iset_a @ X4 ) )
=> ! [Y3: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Y3 @ ( set_ms2379723421352415844iset_a @ Y4 ) )
=> ! [Z4: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Z4 @ ( set_ms2379723421352415844iset_a @ Z2 ) )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X @ Y3 ) @ Ns )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ Y3 @ Z4 ) @ Ns )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X @ Z4 ) @ Ns ) ) ) ) ) )
=> ( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ X4 @ Y4 ) @ ( multis1572145089415997755iset_a @ $false @ Ns @ S ) )
=> ( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ Y4 @ Z2 ) @ ( multis1572145089415997755iset_a @ $false @ Ns @ S ) )
=> ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ X4 @ Z2 ) @ ( multis1572145089415997755iset_a @ $false @ Ns @ S ) ) ) ) ) ) ) ) ).
% trans_mult2_alt_s_s_local
thf(fact_389_trans__mult2__alt__s__s__local,axiom,
! [X4: multiset_a,Y4: multiset_a,Z2: multiset_a,S: set_Product_prod_a_a,Ns: set_Product_prod_a_a] :
( ! [X: a] :
( ( member_a @ X @ ( set_mset_a @ X4 ) )
=> ! [Y3: a] :
( ( member_a @ Y3 @ ( set_mset_a @ Y4 ) )
=> ! [Z4: a] :
( ( member_a @ Z4 @ ( set_mset_a @ Z2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ S )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ Z4 ) @ S )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( set_mset_a @ X4 ) )
=> ! [Y3: a] :
( ( member_a @ Y3 @ ( set_mset_a @ Y4 ) )
=> ! [Z4: a] :
( ( member_a @ Z4 @ ( set_mset_a @ Z2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ Ns )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ Z4 ) @ S )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( set_mset_a @ X4 ) )
=> ! [Y3: a] :
( ( member_a @ Y3 @ ( set_mset_a @ Y4 ) )
=> ! [Z4: a] :
( ( member_a @ Z4 @ ( set_mset_a @ Z2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ S )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ Z4 ) @ Ns )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( set_mset_a @ X4 ) )
=> ! [Y3: a] :
( ( member_a @ Y3 @ ( set_mset_a @ Y4 ) )
=> ! [Z4: a] :
( ( member_a @ Z4 @ ( set_mset_a @ Z2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ Ns )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ Z4 ) @ Ns )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Z4 ) @ Ns ) ) ) ) ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X4 @ Y4 ) @ ( multis1686031732355738386_alt_a @ $false @ Ns @ S ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Y4 @ Z2 ) @ ( multis1686031732355738386_alt_a @ $false @ Ns @ S ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X4 @ Z2 ) @ ( multis1686031732355738386_alt_a @ $false @ Ns @ S ) ) ) ) ) ) ) ) ).
% trans_mult2_alt_s_s_local
thf(fact_390_trans__mult2__alt__s__s__local,axiom,
! [X4: multiset_multiset_a,Y4: multiset_multiset_a,Z2: multiset_multiset_a,S: set_Pr79727621955416071iset_a,Ns: set_Pr79727621955416071iset_a] :
( ! [X: multiset_a] :
( ( member_multiset_a @ X @ ( set_mset_multiset_a @ X4 ) )
=> ! [Y3: multiset_a] :
( ( member_multiset_a @ Y3 @ ( set_mset_multiset_a @ Y4 ) )
=> ! [Z4: multiset_a] :
( ( member_multiset_a @ Z4 @ ( set_mset_multiset_a @ Z2 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ Y3 ) @ S )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Y3 @ Z4 ) @ S )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: multiset_a] :
( ( member_multiset_a @ X @ ( set_mset_multiset_a @ X4 ) )
=> ! [Y3: multiset_a] :
( ( member_multiset_a @ Y3 @ ( set_mset_multiset_a @ Y4 ) )
=> ! [Z4: multiset_a] :
( ( member_multiset_a @ Z4 @ ( set_mset_multiset_a @ Z2 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ Y3 ) @ Ns )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Y3 @ Z4 ) @ S )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: multiset_a] :
( ( member_multiset_a @ X @ ( set_mset_multiset_a @ X4 ) )
=> ! [Y3: multiset_a] :
( ( member_multiset_a @ Y3 @ ( set_mset_multiset_a @ Y4 ) )
=> ! [Z4: multiset_a] :
( ( member_multiset_a @ Z4 @ ( set_mset_multiset_a @ Z2 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ Y3 ) @ S )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Y3 @ Z4 ) @ Ns )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: multiset_a] :
( ( member_multiset_a @ X @ ( set_mset_multiset_a @ X4 ) )
=> ! [Y3: multiset_a] :
( ( member_multiset_a @ Y3 @ ( set_mset_multiset_a @ Y4 ) )
=> ! [Z4: multiset_a] :
( ( member_multiset_a @ Z4 @ ( set_mset_multiset_a @ Z2 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ Y3 ) @ Ns )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Y3 @ Z4 ) @ Ns )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ Z4 ) @ Ns ) ) ) ) ) )
=> ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X4 @ Y4 ) @ ( multis2006310815304088152iset_a @ $false @ Ns @ S ) )
=> ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ Y4 @ Z2 ) @ ( multis2006310815304088152iset_a @ $false @ Ns @ S ) )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X4 @ Z2 ) @ ( multis2006310815304088152iset_a @ $false @ Ns @ S ) ) ) ) ) ) ) ) ).
% trans_mult2_alt_s_s_local
thf(fact_391_trans__mult2__alt__local,axiom,
! [X4: multis501812127501805293od_a_a,Y4: multis501812127501805293od_a_a,Z2: multis501812127501805293od_a_a,S: set_Pr8600417178894128327od_a_a,Ns: set_Pr8600417178894128327od_a_a,B13: $o,B24: $o] :
( ! [X: product_prod_a_a,Y3: product_prod_a_a,Z4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_ms119794563918438244od_a_a @ X4 ) )
=> ( ( member1426531477525435216od_a_a @ Y3 @ ( set_ms119794563918438244od_a_a @ Y4 ) )
=> ( ( member1426531477525435216od_a_a @ Z4 @ ( set_ms119794563918438244od_a_a @ Z2 ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Y3 ) @ S )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y3 @ Z4 ) @ S )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: product_prod_a_a,Y3: product_prod_a_a,Z4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_ms119794563918438244od_a_a @ X4 ) )
=> ( ( member1426531477525435216od_a_a @ Y3 @ ( set_ms119794563918438244od_a_a @ Y4 ) )
=> ( ( member1426531477525435216od_a_a @ Z4 @ ( set_ms119794563918438244od_a_a @ Z2 ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Y3 ) @ Ns )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y3 @ Z4 ) @ S )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: product_prod_a_a,Y3: product_prod_a_a,Z4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_ms119794563918438244od_a_a @ X4 ) )
=> ( ( member1426531477525435216od_a_a @ Y3 @ ( set_ms119794563918438244od_a_a @ Y4 ) )
=> ( ( member1426531477525435216od_a_a @ Z4 @ ( set_ms119794563918438244od_a_a @ Z2 ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Y3 ) @ S )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y3 @ Z4 ) @ Ns )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: product_prod_a_a,Y3: product_prod_a_a,Z4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_ms119794563918438244od_a_a @ X4 ) )
=> ( ( member1426531477525435216od_a_a @ Y3 @ ( set_ms119794563918438244od_a_a @ Y4 ) )
=> ( ( member1426531477525435216od_a_a @ Z4 @ ( set_ms119794563918438244od_a_a @ Z2 ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Y3 ) @ Ns )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y3 @ Z4 ) @ Ns )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Z4 ) @ Ns ) ) ) ) ) )
=> ( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ X4 @ Y4 ) @ ( multis4249482161328750907od_a_a @ B13 @ Ns @ S ) )
=> ( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ Y4 @ Z2 ) @ ( multis4249482161328750907od_a_a @ B24 @ Ns @ S ) )
=> ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ X4 @ Z2 )
@ ( multis4249482161328750907od_a_a
@ ( B13
& B24 )
@ Ns
@ S ) ) ) ) ) ) ) ) ).
% trans_mult2_alt_local
thf(fact_392_trans__mult2__alt__local,axiom,
! [X4: multis4781622204924204013iset_a,Y4: multis4781622204924204013iset_a,Z2: multis4781622204924204013iset_a,S: set_Pr3059988933816007879iset_a,Ns: set_Pr3059988933816007879iset_a,B13: $o,B24: $o] :
( ! [X: produc6518373309651786023iset_a,Y3: produc6518373309651786023iset_a,Z4: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X @ ( set_ms2379723421352415844iset_a @ X4 ) )
=> ( ( member5199237121806060112iset_a @ Y3 @ ( set_ms2379723421352415844iset_a @ Y4 ) )
=> ( ( member5199237121806060112iset_a @ Z4 @ ( set_ms2379723421352415844iset_a @ Z2 ) )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X @ Y3 ) @ S )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ Y3 @ Z4 ) @ S )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: produc6518373309651786023iset_a,Y3: produc6518373309651786023iset_a,Z4: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X @ ( set_ms2379723421352415844iset_a @ X4 ) )
=> ( ( member5199237121806060112iset_a @ Y3 @ ( set_ms2379723421352415844iset_a @ Y4 ) )
=> ( ( member5199237121806060112iset_a @ Z4 @ ( set_ms2379723421352415844iset_a @ Z2 ) )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X @ Y3 ) @ Ns )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ Y3 @ Z4 ) @ S )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: produc6518373309651786023iset_a,Y3: produc6518373309651786023iset_a,Z4: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X @ ( set_ms2379723421352415844iset_a @ X4 ) )
=> ( ( member5199237121806060112iset_a @ Y3 @ ( set_ms2379723421352415844iset_a @ Y4 ) )
=> ( ( member5199237121806060112iset_a @ Z4 @ ( set_ms2379723421352415844iset_a @ Z2 ) )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X @ Y3 ) @ S )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ Y3 @ Z4 ) @ Ns )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: produc6518373309651786023iset_a,Y3: produc6518373309651786023iset_a,Z4: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X @ ( set_ms2379723421352415844iset_a @ X4 ) )
=> ( ( member5199237121806060112iset_a @ Y3 @ ( set_ms2379723421352415844iset_a @ Y4 ) )
=> ( ( member5199237121806060112iset_a @ Z4 @ ( set_ms2379723421352415844iset_a @ Z2 ) )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X @ Y3 ) @ Ns )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ Y3 @ Z4 ) @ Ns )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X @ Z4 ) @ Ns ) ) ) ) ) )
=> ( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ X4 @ Y4 ) @ ( multis1572145089415997755iset_a @ B13 @ Ns @ S ) )
=> ( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ Y4 @ Z2 ) @ ( multis1572145089415997755iset_a @ B24 @ Ns @ S ) )
=> ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ X4 @ Z2 )
@ ( multis1572145089415997755iset_a
@ ( B13
& B24 )
@ Ns
@ S ) ) ) ) ) ) ) ) ).
% trans_mult2_alt_local
thf(fact_393_trans__mult2__alt__local,axiom,
! [X4: multiset_a,Y4: multiset_a,Z2: multiset_a,S: set_Product_prod_a_a,Ns: set_Product_prod_a_a,B13: $o,B24: $o] :
( ! [X: a,Y3: a,Z4: a] :
( ( member_a @ X @ ( set_mset_a @ X4 ) )
=> ( ( member_a @ Y3 @ ( set_mset_a @ Y4 ) )
=> ( ( member_a @ Z4 @ ( set_mset_a @ Z2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ S )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ Z4 ) @ S )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: a,Y3: a,Z4: a] :
( ( member_a @ X @ ( set_mset_a @ X4 ) )
=> ( ( member_a @ Y3 @ ( set_mset_a @ Y4 ) )
=> ( ( member_a @ Z4 @ ( set_mset_a @ Z2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ Ns )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ Z4 ) @ S )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: a,Y3: a,Z4: a] :
( ( member_a @ X @ ( set_mset_a @ X4 ) )
=> ( ( member_a @ Y3 @ ( set_mset_a @ Y4 ) )
=> ( ( member_a @ Z4 @ ( set_mset_a @ Z2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ S )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ Z4 ) @ Ns )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: a,Y3: a,Z4: a] :
( ( member_a @ X @ ( set_mset_a @ X4 ) )
=> ( ( member_a @ Y3 @ ( set_mset_a @ Y4 ) )
=> ( ( member_a @ Z4 @ ( set_mset_a @ Z2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ Ns )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ Z4 ) @ Ns )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Z4 ) @ Ns ) ) ) ) ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X4 @ Y4 ) @ ( multis1686031732355738386_alt_a @ B13 @ Ns @ S ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Y4 @ Z2 ) @ ( multis1686031732355738386_alt_a @ B24 @ Ns @ S ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X4 @ Z2 )
@ ( multis1686031732355738386_alt_a
@ ( B13
& B24 )
@ Ns
@ S ) ) ) ) ) ) ) ) ).
% trans_mult2_alt_local
thf(fact_394_trans__mult2__alt__local,axiom,
! [X4: multiset_multiset_a,Y4: multiset_multiset_a,Z2: multiset_multiset_a,S: set_Pr79727621955416071iset_a,Ns: set_Pr79727621955416071iset_a,B13: $o,B24: $o] :
( ! [X: multiset_a,Y3: multiset_a,Z4: multiset_a] :
( ( member_multiset_a @ X @ ( set_mset_multiset_a @ X4 ) )
=> ( ( member_multiset_a @ Y3 @ ( set_mset_multiset_a @ Y4 ) )
=> ( ( member_multiset_a @ Z4 @ ( set_mset_multiset_a @ Z2 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ Y3 ) @ S )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Y3 @ Z4 ) @ S )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: multiset_a,Y3: multiset_a,Z4: multiset_a] :
( ( member_multiset_a @ X @ ( set_mset_multiset_a @ X4 ) )
=> ( ( member_multiset_a @ Y3 @ ( set_mset_multiset_a @ Y4 ) )
=> ( ( member_multiset_a @ Z4 @ ( set_mset_multiset_a @ Z2 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ Y3 ) @ Ns )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Y3 @ Z4 ) @ S )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: multiset_a,Y3: multiset_a,Z4: multiset_a] :
( ( member_multiset_a @ X @ ( set_mset_multiset_a @ X4 ) )
=> ( ( member_multiset_a @ Y3 @ ( set_mset_multiset_a @ Y4 ) )
=> ( ( member_multiset_a @ Z4 @ ( set_mset_multiset_a @ Z2 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ Y3 ) @ S )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Y3 @ Z4 ) @ Ns )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ Z4 ) @ S ) ) ) ) ) )
=> ( ! [X: multiset_a,Y3: multiset_a,Z4: multiset_a] :
( ( member_multiset_a @ X @ ( set_mset_multiset_a @ X4 ) )
=> ( ( member_multiset_a @ Y3 @ ( set_mset_multiset_a @ Y4 ) )
=> ( ( member_multiset_a @ Z4 @ ( set_mset_multiset_a @ Z2 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ Y3 ) @ Ns )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Y3 @ Z4 ) @ Ns )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ Z4 ) @ Ns ) ) ) ) ) )
=> ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X4 @ Y4 ) @ ( multis2006310815304088152iset_a @ B13 @ Ns @ S ) )
=> ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ Y4 @ Z2 ) @ ( multis2006310815304088152iset_a @ B24 @ Ns @ S ) )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X4 @ Z2 )
@ ( multis2006310815304088152iset_a
@ ( B13
& B24 )
@ Ns
@ S ) ) ) ) ) ) ) ) ).
% trans_mult2_alt_local
thf(fact_395_multpw__refl_H,axiom,
! [Ns: set_Product_prod_a_a,A4: multiset_a] :
( ( locally_refl_a @ Ns @ A4 )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A4 @ A4 ) @ ( multis2373687813370458242ltpw_a @ Ns ) ) ) ).
% multpw_refl'
thf(fact_396_ns__mul__ext__refl__local,axiom,
! [Ns: set_Product_prod_a_a,A4: multiset_a,S: set_Product_prod_a_a] :
( ( locally_refl_a @ Ns @ A4 )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A4 @ A4 ) @ ( multis4023584895248362547_ext_a @ Ns @ S ) ) ) ).
% ns_mul_ext_refl_local
thf(fact_397_mset__swap,axiom,
! [I: nat,Ls: list_a,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Ls ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_a @ Ls ) )
=> ( ( mset_a @ ( list_update_a @ ( list_update_a @ Ls @ J @ ( nth_a @ Ls @ I ) ) @ I @ ( nth_a @ Ls @ J ) ) )
= ( mset_a @ Ls ) ) ) ) ).
% mset_swap
thf(fact_398_pair__in__Id__conv,axiom,
! [A2: a,B: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B ) @ id_a )
= ( A2 = B ) ) ).
% pair_in_Id_conv
thf(fact_399_pair__in__Id__conv,axiom,
! [A2: multiset_a,B: multiset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A2 @ B ) @ id_multiset_a )
= ( A2 = B ) ) ).
% pair_in_Id_conv
thf(fact_400_IdI,axiom,
! [A2: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ A2 ) @ id_a ) ).
% IdI
thf(fact_401_IdI,axiom,
! [A2: multiset_a] : ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A2 @ A2 ) @ id_multiset_a ) ).
% IdI
thf(fact_402_locally__refl__def,axiom,
( locall7179175945658863360od_a_a
= ( ^ [R3: set_Pr8600417178894128327od_a_a,A7: multis501812127501805293od_a_a] :
! [A5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A5 @ ( set_ms119794563918438244od_a_a @ A7 ) )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A5 @ A5 ) @ R3 ) ) ) ) ).
% locally_refl_def
thf(fact_403_locally__refl__def,axiom,
( locall5617794901334676992iset_a
= ( ^ [R3: set_Pr3059988933816007879iset_a,A7: multis4781622204924204013iset_a] :
! [A5: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ A5 @ ( set_ms2379723421352415844iset_a @ A7 ) )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ A5 @ A5 ) @ R3 ) ) ) ) ).
% locally_refl_def
thf(fact_404_locally__refl__def,axiom,
( locally_refl_a
= ( ^ [R3: set_Product_prod_a_a,A7: multiset_a] :
! [A5: a] :
( ( member_a @ A5 @ ( set_mset_a @ A7 ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A5 @ A5 ) @ R3 ) ) ) ) ).
% locally_refl_def
thf(fact_405_locally__refl__def,axiom,
( locall5412118405798798493iset_a
= ( ^ [R3: set_Pr79727621955416071iset_a,A7: multiset_multiset_a] :
! [A5: multiset_a] :
( ( member_multiset_a @ A5 @ ( set_mset_multiset_a @ A7 ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A5 @ A5 ) @ R3 ) ) ) ) ).
% locally_refl_def
thf(fact_406_lr__trans__l,axiom,
! [R: set_Product_prod_a_a,A4: multiset_a,B5: multiset_a] :
( ( locally_refl_a @ R @ ( plus_plus_multiset_a @ A4 @ B5 ) )
=> ( locally_refl_a @ R @ A4 ) ) ).
% lr_trans_l
thf(fact_407_lr__trans__r,axiom,
! [R: set_Product_prod_a_a,A4: multiset_a,B5: multiset_a] :
( ( locally_refl_a @ R @ ( plus_plus_multiset_a @ A4 @ B5 ) )
=> ( locally_refl_a @ R @ B5 ) ) ).
% lr_trans_r
thf(fact_408_IdD,axiom,
! [A2: a,B: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B ) @ id_a )
=> ( A2 = B ) ) ).
% IdD
thf(fact_409_IdD,axiom,
! [A2: multiset_a,B: multiset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A2 @ B ) @ id_multiset_a )
=> ( A2 = B ) ) ).
% IdD
thf(fact_410_IdE,axiom,
! [P: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ P @ id_a )
=> ~ ! [X: a] :
( P
!= ( product_Pair_a_a @ X @ X ) ) ) ).
% IdE
thf(fact_411_IdE,axiom,
! [P: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ P @ id_multiset_a )
=> ~ ! [X: multiset_a] :
( P
!= ( produc654756711066625303iset_a @ X @ X ) ) ) ).
% IdE
thf(fact_412_mult2__altE,axiom,
! [X4: multis501812127501805293od_a_a,Y4: multis501812127501805293od_a_a,B: $o,Ns: set_Pr8600417178894128327od_a_a,S: set_Pr8600417178894128327od_a_a] :
( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ X4 @ Y4 ) @ ( multis4249482161328750907od_a_a @ B @ Ns @ S ) )
=> ~ ! [X13: multis501812127501805293od_a_a,X23: multis501812127501805293od_a_a] :
( ( X4
= ( plus_p6967097483979887908od_a_a @ X13 @ X23 ) )
=> ! [Y12: multis501812127501805293od_a_a,Y22: multis501812127501805293od_a_a] :
( ( Y4
= ( plus_p6967097483979887908od_a_a @ Y12 @ Y22 ) )
=> ( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ X13 @ Y12 ) @ ( multis1821517007024690859od_a_a @ Ns ) )
=> ( ( B
| ( Y22 != zero_z3024502163831676836od_a_a ) )
=> ~ ! [X6: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X6 @ ( set_ms119794563918438244od_a_a @ X23 ) )
=> ? [Y3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y3 @ ( set_ms119794563918438244od_a_a @ Y22 ) )
& ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X6 @ Y3 ) @ S ) ) ) ) ) ) ) ) ).
% mult2_altE
thf(fact_413_mult2__altE,axiom,
! [X4: multis4781622204924204013iset_a,Y4: multis4781622204924204013iset_a,B: $o,Ns: set_Pr3059988933816007879iset_a,S: set_Pr3059988933816007879iset_a] :
( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ X4 @ Y4 ) @ ( multis1572145089415997755iset_a @ B @ Ns @ S ) )
=> ~ ! [X13: multis4781622204924204013iset_a,X23: multis4781622204924204013iset_a] :
( ( X4
= ( plus_p1053901119268525860iset_a @ X13 @ X23 ) )
=> ! [Y12: multis4781622204924204013iset_a,Y22: multis4781622204924204013iset_a] :
( ( Y4
= ( plus_p1053901119268525860iset_a @ Y12 @ Y22 ) )
=> ( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ X13 @ Y12 ) @ ( multis3430030276581740203iset_a @ Ns ) )
=> ( ( B
| ( Y22 != zero_z4591712717340762532iset_a ) )
=> ~ ! [X6: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X6 @ ( set_ms2379723421352415844iset_a @ X23 ) )
=> ? [Y3: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Y3 @ ( set_ms2379723421352415844iset_a @ Y22 ) )
& ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X6 @ Y3 ) @ S ) ) ) ) ) ) ) ) ).
% mult2_altE
thf(fact_414_mult2__altE,axiom,
! [X4: multiset_multiset_a,Y4: multiset_multiset_a,B: $o,Ns: set_Pr79727621955416071iset_a,S: set_Pr79727621955416071iset_a] :
( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X4 @ Y4 ) @ ( multis2006310815304088152iset_a @ B @ Ns @ S ) )
=> ~ ! [X13: multiset_multiset_a,X23: multiset_multiset_a] :
( ( X4
= ( plus_p6738641960240532161iset_a @ X13 @ X23 ) )
=> ! [Y12: multiset_multiset_a,Y22: multiset_multiset_a] :
( ( Y4
= ( plus_p6738641960240532161iset_a @ Y12 @ Y22 ) )
=> ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X13 @ Y12 ) @ ( multis451435756853798856iset_a @ Ns ) )
=> ( ( B
| ( Y22 != zero_z7799948378220188993iset_a ) )
=> ~ ! [X6: multiset_a] :
( ( member_multiset_a @ X6 @ ( set_mset_multiset_a @ X23 ) )
=> ? [Y3: multiset_a] :
( ( member_multiset_a @ Y3 @ ( set_mset_multiset_a @ Y22 ) )
& ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X6 @ Y3 ) @ S ) ) ) ) ) ) ) ) ).
% mult2_altE
thf(fact_415_mult2__altE,axiom,
! [X4: multiset_a,Y4: multiset_a,B: $o,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X4 @ Y4 ) @ ( multis1686031732355738386_alt_a @ B @ Ns @ S ) )
=> ~ ! [X13: multiset_a,X23: multiset_a] :
( ( X4
= ( plus_plus_multiset_a @ X13 @ X23 ) )
=> ! [Y12: multiset_a,Y22: multiset_a] :
( ( Y4
= ( plus_plus_multiset_a @ Y12 @ Y22 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X13 @ Y12 ) @ ( multis2373687813370458242ltpw_a @ Ns ) )
=> ( ( B
| ( Y22 != zero_zero_multiset_a ) )
=> ~ ! [X6: a] :
( ( member_a @ X6 @ ( set_mset_a @ X23 ) )
=> ? [Y3: a] :
( ( member_a @ Y3 @ ( set_mset_a @ Y22 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X6 @ Y3 ) @ S ) ) ) ) ) ) ) ) ).
% mult2_altE
thf(fact_416_mult2__altI,axiom,
! [X4: multis501812127501805293od_a_a,X12: multis501812127501805293od_a_a,X22: multis501812127501805293od_a_a,Y4: multis501812127501805293od_a_a,Y13: multis501812127501805293od_a_a,Y23: multis501812127501805293od_a_a,Ns: set_Pr8600417178894128327od_a_a,B: $o,S: set_Pr8600417178894128327od_a_a] :
( ( X4
= ( plus_p6967097483979887908od_a_a @ X12 @ X22 ) )
=> ( ( Y4
= ( plus_p6967097483979887908od_a_a @ Y13 @ Y23 ) )
=> ( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ X12 @ Y13 ) @ ( multis1821517007024690859od_a_a @ Ns ) )
=> ( ( B
| ( Y23 != zero_z3024502163831676836od_a_a ) )
=> ( ! [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_ms119794563918438244od_a_a @ X22 ) )
=> ? [Y6: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y6 @ ( set_ms119794563918438244od_a_a @ Y23 ) )
& ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Y6 ) @ S ) ) )
=> ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ X4 @ Y4 ) @ ( multis4249482161328750907od_a_a @ B @ Ns @ S ) ) ) ) ) ) ) ).
% mult2_altI
thf(fact_417_mult2__altI,axiom,
! [X4: multis4781622204924204013iset_a,X12: multis4781622204924204013iset_a,X22: multis4781622204924204013iset_a,Y4: multis4781622204924204013iset_a,Y13: multis4781622204924204013iset_a,Y23: multis4781622204924204013iset_a,Ns: set_Pr3059988933816007879iset_a,B: $o,S: set_Pr3059988933816007879iset_a] :
( ( X4
= ( plus_p1053901119268525860iset_a @ X12 @ X22 ) )
=> ( ( Y4
= ( plus_p1053901119268525860iset_a @ Y13 @ Y23 ) )
=> ( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ X12 @ Y13 ) @ ( multis3430030276581740203iset_a @ Ns ) )
=> ( ( B
| ( Y23 != zero_z4591712717340762532iset_a ) )
=> ( ! [X: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X @ ( set_ms2379723421352415844iset_a @ X22 ) )
=> ? [Y6: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Y6 @ ( set_ms2379723421352415844iset_a @ Y23 ) )
& ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X @ Y6 ) @ S ) ) )
=> ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ X4 @ Y4 ) @ ( multis1572145089415997755iset_a @ B @ Ns @ S ) ) ) ) ) ) ) ).
% mult2_altI
thf(fact_418_mult2__altI,axiom,
! [X4: multiset_multiset_a,X12: multiset_multiset_a,X22: multiset_multiset_a,Y4: multiset_multiset_a,Y13: multiset_multiset_a,Y23: multiset_multiset_a,Ns: set_Pr79727621955416071iset_a,B: $o,S: set_Pr79727621955416071iset_a] :
( ( X4
= ( plus_p6738641960240532161iset_a @ X12 @ X22 ) )
=> ( ( Y4
= ( plus_p6738641960240532161iset_a @ Y13 @ Y23 ) )
=> ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X12 @ Y13 ) @ ( multis451435756853798856iset_a @ Ns ) )
=> ( ( B
| ( Y23 != zero_z7799948378220188993iset_a ) )
=> ( ! [X: multiset_a] :
( ( member_multiset_a @ X @ ( set_mset_multiset_a @ X22 ) )
=> ? [Y6: multiset_a] :
( ( member_multiset_a @ Y6 @ ( set_mset_multiset_a @ Y23 ) )
& ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ Y6 ) @ S ) ) )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X4 @ Y4 ) @ ( multis2006310815304088152iset_a @ B @ Ns @ S ) ) ) ) ) ) ) ).
% mult2_altI
thf(fact_419_mult2__altI,axiom,
! [X4: multiset_a,X12: multiset_a,X22: multiset_a,Y4: multiset_a,Y13: multiset_a,Y23: multiset_a,Ns: set_Product_prod_a_a,B: $o,S: set_Product_prod_a_a] :
( ( X4
= ( plus_plus_multiset_a @ X12 @ X22 ) )
=> ( ( Y4
= ( plus_plus_multiset_a @ Y13 @ Y23 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X12 @ Y13 ) @ ( multis2373687813370458242ltpw_a @ Ns ) )
=> ( ( B
| ( Y23 != zero_zero_multiset_a ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( set_mset_a @ X22 ) )
=> ? [Y6: a] :
( ( member_a @ Y6 @ ( set_mset_a @ Y23 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y6 ) @ S ) ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X4 @ Y4 ) @ ( multis1686031732355738386_alt_a @ B @ Ns @ S ) ) ) ) ) ) ) ).
% mult2_altI
thf(fact_420_not__gr__zero,axiom,
! [N3: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
= ( N3 = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_421_add__0,axiom,
! [A2: multiset_a] :
( ( plus_plus_multiset_a @ zero_zero_multiset_a @ A2 )
= A2 ) ).
% add_0
thf(fact_422_add__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% add_0
thf(fact_423_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_424_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_425_add__cancel__right__right,axiom,
! [A2: multiset_a,B: multiset_a] :
( ( A2
= ( plus_plus_multiset_a @ A2 @ B ) )
= ( B = zero_zero_multiset_a ) ) ).
% add_cancel_right_right
thf(fact_426_add__cancel__right__right,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ A2 @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_427_add__cancel__right__left,axiom,
! [A2: multiset_a,B: multiset_a] :
( ( A2
= ( plus_plus_multiset_a @ B @ A2 ) )
= ( B = zero_zero_multiset_a ) ) ).
% add_cancel_right_left
thf(fact_428_add__cancel__right__left,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ B @ A2 ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_429_add__cancel__left__right,axiom,
! [A2: multiset_a,B: multiset_a] :
( ( ( plus_plus_multiset_a @ A2 @ B )
= A2 )
= ( B = zero_zero_multiset_a ) ) ).
% add_cancel_left_right
thf(fact_430_add__cancel__left__right,axiom,
! [A2: nat,B: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= A2 )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_431_add__cancel__left__left,axiom,
! [B: multiset_a,A2: multiset_a] :
( ( ( plus_plus_multiset_a @ B @ A2 )
= A2 )
= ( B = zero_zero_multiset_a ) ) ).
% add_cancel_left_left
thf(fact_432_add__cancel__left__left,axiom,
! [B: nat,A2: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= A2 )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_433_add_Oright__neutral,axiom,
! [A2: multiset_a] :
( ( plus_plus_multiset_a @ A2 @ zero_zero_multiset_a )
= A2 ) ).
% add.right_neutral
thf(fact_434_add_Oright__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.right_neutral
thf(fact_435_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_436_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_437_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_438_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_439_add__less__same__cancel1,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A2 ) @ B )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_440_add__less__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_441_less__add__same__cancel1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_442_less__add__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ B @ A2 ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_443_multpw__emptyL,axiom,
! [X4: multiset_a,Ns: set_Product_prod_a_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ zero_zero_multiset_a @ X4 ) @ ( multis2373687813370458242ltpw_a @ Ns ) )
= ( X4 = zero_zero_multiset_a ) ) ).
% multpw_emptyL
thf(fact_444_multpw__emptyR,axiom,
! [X4: multiset_a,Ns: set_Product_prod_a_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X4 @ zero_zero_multiset_a ) @ ( multis2373687813370458242ltpw_a @ Ns ) )
= ( X4 = zero_zero_multiset_a ) ) ).
% multpw_emptyR
thf(fact_445_zero__reorient,axiom,
! [X3: nat] :
( ( zero_zero_nat = X3 )
= ( X3 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_446_gr__zeroI,axiom,
! [N3: nat] :
( ( N3 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).
% gr_zeroI
thf(fact_447_not__less__zero,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ zero_zero_nat ) ).
% not_less_zero
thf(fact_448_gr__implies__not__zero,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( N3 != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_449_zero__less__iff__neq__zero,axiom,
! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
= ( N3 != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_450_add_Ocomm__neutral,axiom,
! [A2: multiset_a] :
( ( plus_plus_multiset_a @ A2 @ zero_zero_multiset_a )
= A2 ) ).
% add.comm_neutral
thf(fact_451_add_Ocomm__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.comm_neutral
thf(fact_452_comm__monoid__add__class_Oadd__0,axiom,
! [A2: multiset_a] :
( ( plus_plus_multiset_a @ zero_zero_multiset_a @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_453_comm__monoid__add__class_Oadd__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_454_empty__neutral_I1_J,axiom,
! [X3: multiset_a] :
( ( plus_plus_multiset_a @ zero_zero_multiset_a @ X3 )
= X3 ) ).
% empty_neutral(1)
thf(fact_455_empty__neutral_I2_J,axiom,
! [X3: multiset_a] :
( ( plus_plus_multiset_a @ X3 @ zero_zero_multiset_a )
= X3 ) ).
% empty_neutral(2)
thf(fact_456_multiset__nonemptyE,axiom,
! [A4: multis501812127501805293od_a_a] :
( ( A4 != zero_z3024502163831676836od_a_a )
=> ~ ! [X: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ X @ ( set_ms119794563918438244od_a_a @ A4 ) ) ) ).
% multiset_nonemptyE
thf(fact_457_multiset__nonemptyE,axiom,
! [A4: multis4781622204924204013iset_a] :
( ( A4 != zero_z4591712717340762532iset_a )
=> ~ ! [X: produc6518373309651786023iset_a] :
~ ( member5199237121806060112iset_a @ X @ ( set_ms2379723421352415844iset_a @ A4 ) ) ) ).
% multiset_nonemptyE
thf(fact_458_multiset__nonemptyE,axiom,
! [A4: multiset_a] :
( ( A4 != zero_zero_multiset_a )
=> ~ ! [X: a] :
~ ( member_a @ X @ ( set_mset_a @ A4 ) ) ) ).
% multiset_nonemptyE
thf(fact_459_add__neg__neg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_460_add__pos__pos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% add_pos_pos
thf(fact_461_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ! [C: nat] :
( ( B
= ( plus_plus_nat @ A2 @ C ) )
=> ( C = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_462_pos__add__strict,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A2 @ C2 ) ) ) ) ).
% pos_add_strict
thf(fact_463_multpw_Oempty,axiom,
! [Ns: set_Product_prod_a_a] : ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ zero_zero_multiset_a @ zero_zero_multiset_a ) @ ( multis2373687813370458242ltpw_a @ Ns ) ) ).
% multpw.empty
thf(fact_464_mult2__alt__emptyR,axiom,
! [X4: multiset_a,B: $o,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X4 @ zero_zero_multiset_a ) @ ( multis1686031732355738386_alt_a @ B @ Ns @ S ) )
= ( B
& ( X4 = zero_zero_multiset_a ) ) ) ).
% mult2_alt_emptyR
thf(fact_465_mult2__alt__emptyL,axiom,
! [Y4: multiset_a,B: $o,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ zero_zero_multiset_a @ Y4 ) @ ( multis1686031732355738386_alt_a @ B @ Ns @ S ) )
= ( B
| ( Y4 != zero_zero_multiset_a ) ) ) ).
% mult2_alt_emptyL
thf(fact_466_ns__mul__ext__bottom,axiom,
! [A4: multiset_a,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a] : ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A4 @ zero_zero_multiset_a ) @ ( multis4023584895248362547_ext_a @ Ns @ S ) ) ).
% ns_mul_ext_bottom
thf(fact_467_ns__mul__ext__bottom2,axiom,
! [A4: multiset_a,B5: multiset_a,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A4 @ B5 ) @ ( multis4023584895248362547_ext_a @ Ns @ S ) )
=> ( ( B5 != zero_zero_multiset_a )
=> ( A4 != zero_zero_multiset_a ) ) ) ).
% ns_mul_ext_bottom2
thf(fact_468_ns__mul__ext__bottom__uniqueness,axiom,
! [A4: multiset_a,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ zero_zero_multiset_a @ A4 ) @ ( multis4023584895248362547_ext_a @ Ns @ S ) )
=> ( A4 = zero_zero_multiset_a ) ) ).
% ns_mul_ext_bottom_uniqueness
thf(fact_469_mult2__alt__sI,axiom,
! [X4: multis501812127501805293od_a_a,X12: multis501812127501805293od_a_a,X22: multis501812127501805293od_a_a,Y4: multis501812127501805293od_a_a,Y13: multis501812127501805293od_a_a,Y23: multis501812127501805293od_a_a,Ns: set_Pr8600417178894128327od_a_a,S: set_Pr8600417178894128327od_a_a] :
( ( X4
= ( plus_p6967097483979887908od_a_a @ X12 @ X22 ) )
=> ( ( Y4
= ( plus_p6967097483979887908od_a_a @ Y13 @ Y23 ) )
=> ( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ X12 @ Y13 ) @ ( multis1821517007024690859od_a_a @ Ns ) )
=> ( ( Y23 != zero_z3024502163831676836od_a_a )
=> ( ! [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_ms119794563918438244od_a_a @ X22 ) )
=> ? [Y6: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y6 @ ( set_ms119794563918438244od_a_a @ Y23 ) )
& ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Y6 ) @ S ) ) )
=> ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ X4 @ Y4 ) @ ( multis4249482161328750907od_a_a @ $false @ Ns @ S ) ) ) ) ) ) ) ).
% mult2_alt_sI
thf(fact_470_mult2__alt__sI,axiom,
! [X4: multis4781622204924204013iset_a,X12: multis4781622204924204013iset_a,X22: multis4781622204924204013iset_a,Y4: multis4781622204924204013iset_a,Y13: multis4781622204924204013iset_a,Y23: multis4781622204924204013iset_a,Ns: set_Pr3059988933816007879iset_a,S: set_Pr3059988933816007879iset_a] :
( ( X4
= ( plus_p1053901119268525860iset_a @ X12 @ X22 ) )
=> ( ( Y4
= ( plus_p1053901119268525860iset_a @ Y13 @ Y23 ) )
=> ( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ X12 @ Y13 ) @ ( multis3430030276581740203iset_a @ Ns ) )
=> ( ( Y23 != zero_z4591712717340762532iset_a )
=> ( ! [X: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X @ ( set_ms2379723421352415844iset_a @ X22 ) )
=> ? [Y6: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Y6 @ ( set_ms2379723421352415844iset_a @ Y23 ) )
& ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X @ Y6 ) @ S ) ) )
=> ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ X4 @ Y4 ) @ ( multis1572145089415997755iset_a @ $false @ Ns @ S ) ) ) ) ) ) ) ).
% mult2_alt_sI
thf(fact_471_mult2__alt__sI,axiom,
! [X4: multiset_multiset_a,X12: multiset_multiset_a,X22: multiset_multiset_a,Y4: multiset_multiset_a,Y13: multiset_multiset_a,Y23: multiset_multiset_a,Ns: set_Pr79727621955416071iset_a,S: set_Pr79727621955416071iset_a] :
( ( X4
= ( plus_p6738641960240532161iset_a @ X12 @ X22 ) )
=> ( ( Y4
= ( plus_p6738641960240532161iset_a @ Y13 @ Y23 ) )
=> ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X12 @ Y13 ) @ ( multis451435756853798856iset_a @ Ns ) )
=> ( ( Y23 != zero_z7799948378220188993iset_a )
=> ( ! [X: multiset_a] :
( ( member_multiset_a @ X @ ( set_mset_multiset_a @ X22 ) )
=> ? [Y6: multiset_a] :
( ( member_multiset_a @ Y6 @ ( set_mset_multiset_a @ Y23 ) )
& ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ Y6 ) @ S ) ) )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X4 @ Y4 ) @ ( multis2006310815304088152iset_a @ $false @ Ns @ S ) ) ) ) ) ) ) ).
% mult2_alt_sI
thf(fact_472_mult2__alt__sI,axiom,
! [X4: multiset_a,X12: multiset_a,X22: multiset_a,Y4: multiset_a,Y13: multiset_a,Y23: multiset_a,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( X4
= ( plus_plus_multiset_a @ X12 @ X22 ) )
=> ( ( Y4
= ( plus_plus_multiset_a @ Y13 @ Y23 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X12 @ Y13 ) @ ( multis2373687813370458242ltpw_a @ Ns ) )
=> ( ( Y23 != zero_zero_multiset_a )
=> ( ! [X: a] :
( ( member_a @ X @ ( set_mset_a @ X22 ) )
=> ? [Y6: a] :
( ( member_a @ Y6 @ ( set_mset_a @ Y23 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y6 ) @ S ) ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X4 @ Y4 ) @ ( multis1686031732355738386_alt_a @ $false @ Ns @ S ) ) ) ) ) ) ) ).
% mult2_alt_sI
thf(fact_473_mult2__alt__sE,axiom,
! [X4: multis501812127501805293od_a_a,Y4: multis501812127501805293od_a_a,Ns: set_Pr8600417178894128327od_a_a,S: set_Pr8600417178894128327od_a_a] :
( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ X4 @ Y4 ) @ ( multis4249482161328750907od_a_a @ $false @ Ns @ S ) )
=> ~ ! [X13: multis501812127501805293od_a_a,X23: multis501812127501805293od_a_a] :
( ( X4
= ( plus_p6967097483979887908od_a_a @ X13 @ X23 ) )
=> ! [Y12: multis501812127501805293od_a_a,Y22: multis501812127501805293od_a_a] :
( ( Y4
= ( plus_p6967097483979887908od_a_a @ Y12 @ Y22 ) )
=> ( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ X13 @ Y12 ) @ ( multis1821517007024690859od_a_a @ Ns ) )
=> ( ( Y22 != zero_z3024502163831676836od_a_a )
=> ~ ! [X6: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X6 @ ( set_ms119794563918438244od_a_a @ X23 ) )
=> ? [Y3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y3 @ ( set_ms119794563918438244od_a_a @ Y22 ) )
& ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X6 @ Y3 ) @ S ) ) ) ) ) ) ) ) ).
% mult2_alt_sE
thf(fact_474_mult2__alt__sE,axiom,
! [X4: multis4781622204924204013iset_a,Y4: multis4781622204924204013iset_a,Ns: set_Pr3059988933816007879iset_a,S: set_Pr3059988933816007879iset_a] :
( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ X4 @ Y4 ) @ ( multis1572145089415997755iset_a @ $false @ Ns @ S ) )
=> ~ ! [X13: multis4781622204924204013iset_a,X23: multis4781622204924204013iset_a] :
( ( X4
= ( plus_p1053901119268525860iset_a @ X13 @ X23 ) )
=> ! [Y12: multis4781622204924204013iset_a,Y22: multis4781622204924204013iset_a] :
( ( Y4
= ( plus_p1053901119268525860iset_a @ Y12 @ Y22 ) )
=> ( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ X13 @ Y12 ) @ ( multis3430030276581740203iset_a @ Ns ) )
=> ( ( Y22 != zero_z4591712717340762532iset_a )
=> ~ ! [X6: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X6 @ ( set_ms2379723421352415844iset_a @ X23 ) )
=> ? [Y3: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Y3 @ ( set_ms2379723421352415844iset_a @ Y22 ) )
& ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X6 @ Y3 ) @ S ) ) ) ) ) ) ) ) ).
% mult2_alt_sE
thf(fact_475_mult2__alt__sE,axiom,
! [X4: multiset_multiset_a,Y4: multiset_multiset_a,Ns: set_Pr79727621955416071iset_a,S: set_Pr79727621955416071iset_a] :
( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X4 @ Y4 ) @ ( multis2006310815304088152iset_a @ $false @ Ns @ S ) )
=> ~ ! [X13: multiset_multiset_a,X23: multiset_multiset_a] :
( ( X4
= ( plus_p6738641960240532161iset_a @ X13 @ X23 ) )
=> ! [Y12: multiset_multiset_a,Y22: multiset_multiset_a] :
( ( Y4
= ( plus_p6738641960240532161iset_a @ Y12 @ Y22 ) )
=> ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X13 @ Y12 ) @ ( multis451435756853798856iset_a @ Ns ) )
=> ( ( Y22 != zero_z7799948378220188993iset_a )
=> ~ ! [X6: multiset_a] :
( ( member_multiset_a @ X6 @ ( set_mset_multiset_a @ X23 ) )
=> ? [Y3: multiset_a] :
( ( member_multiset_a @ Y3 @ ( set_mset_multiset_a @ Y22 ) )
& ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X6 @ Y3 ) @ S ) ) ) ) ) ) ) ) ).
% mult2_alt_sE
thf(fact_476_mult2__alt__sE,axiom,
! [X4: multiset_a,Y4: multiset_a,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X4 @ Y4 ) @ ( multis1686031732355738386_alt_a @ $false @ Ns @ S ) )
=> ~ ! [X13: multiset_a,X23: multiset_a] :
( ( X4
= ( plus_plus_multiset_a @ X13 @ X23 ) )
=> ! [Y12: multiset_a,Y22: multiset_a] :
( ( Y4
= ( plus_plus_multiset_a @ Y12 @ Y22 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X13 @ Y12 ) @ ( multis2373687813370458242ltpw_a @ Ns ) )
=> ( ( Y22 != zero_zero_multiset_a )
=> ~ ! [X6: a] :
( ( member_a @ X6 @ ( set_mset_a @ X23 ) )
=> ? [Y3: a] :
( ( member_a @ Y3 @ ( set_mset_a @ Y22 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X6 @ Y3 ) @ S ) ) ) ) ) ) ) ) ).
% mult2_alt_sE
thf(fact_477_s__mul__extI__old,axiom,
! [A4: multis501812127501805293od_a_a,Xs: list_P1396940483166286381od_a_a,A23: multis501812127501805293od_a_a,B5: multis501812127501805293od_a_a,Ys: list_P1396940483166286381od_a_a,B23: multis501812127501805293od_a_a,Ns: set_Pr8600417178894128327od_a_a,S: set_Pr8600417178894128327od_a_a] :
( ( A4
= ( plus_p6967097483979887908od_a_a @ ( mset_P3709024939784119484od_a_a @ Xs ) @ A23 ) )
=> ( ( B5
= ( plus_p6967097483979887908od_a_a @ ( mset_P3709024939784119484od_a_a @ Ys ) @ B23 ) )
=> ( ( ( size_s3885678630836030617od_a_a @ Xs )
= ( size_s3885678630836030617od_a_a @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3885678630836030617od_a_a @ Ys ) )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ ( nth_Product_prod_a_a @ Xs @ I3 ) @ ( nth_Product_prod_a_a @ Ys @ I3 ) ) @ Ns ) )
=> ( ( A23 != zero_z3024502163831676836od_a_a )
=> ( ! [B3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ B3 @ ( set_ms119794563918438244od_a_a @ B23 ) )
=> ? [A6: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A6 @ ( set_ms119794563918438244od_a_a @ A23 ) )
& ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A6 @ B3 ) @ S ) ) )
=> ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ A4 @ B5 ) @ ( multis6523615221270609726od_a_a @ Ns @ S ) ) ) ) ) ) ) ) ).
% s_mul_extI_old
thf(fact_478_s__mul__extI__old,axiom,
! [A4: multis4781622204924204013iset_a,Xs: list_P6988024003011616557iset_a,A23: multis4781622204924204013iset_a,B5: multis4781622204924204013iset_a,Ys: list_P6988024003011616557iset_a,B23: multis4781622204924204013iset_a,Ns: set_Pr3059988933816007879iset_a,S: set_Pr3059988933816007879iset_a] :
( ( A4
= ( plus_p1053901119268525860iset_a @ ( mset_P2771936761501114300iset_a @ Xs ) @ A23 ) )
=> ( ( B5
= ( plus_p1053901119268525860iset_a @ ( mset_P2771936761501114300iset_a @ Ys ) @ B23 ) )
=> ( ( ( size_s7238940367606365593iset_a @ Xs )
= ( size_s7238940367606365593iset_a @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s7238940367606365593iset_a @ Ys ) )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ ( nth_Pr1079437516805851566iset_a @ Xs @ I3 ) @ ( nth_Pr1079437516805851566iset_a @ Ys @ I3 ) ) @ Ns ) )
=> ( ( A23 != zero_z4591712717340762532iset_a )
=> ( ! [B3: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ B3 @ ( set_ms2379723421352415844iset_a @ B23 ) )
=> ? [A6: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ A6 @ ( set_ms2379723421352415844iset_a @ A23 ) )
& ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ A6 @ B3 ) @ S ) ) )
=> ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ A4 @ B5 ) @ ( multis6482401367445528638iset_a @ Ns @ S ) ) ) ) ) ) ) ) ).
% s_mul_extI_old
thf(fact_479_s__mul__extI__old,axiom,
! [A4: multiset_multiset_a,Xs: list_multiset_a,A23: multiset_multiset_a,B5: multiset_multiset_a,Ys: list_multiset_a,B23: multiset_multiset_a,Ns: set_Pr79727621955416071iset_a,S: set_Pr79727621955416071iset_a] :
( ( A4
= ( plus_p6738641960240532161iset_a @ ( mset_multiset_a @ Xs ) @ A23 ) )
=> ( ( B5
= ( plus_p6738641960240532161iset_a @ ( mset_multiset_a @ Ys ) @ B23 ) )
=> ( ( ( size_s3003218658903709366iset_a @ Xs )
= ( size_s3003218658903709366iset_a @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3003218658903709366iset_a @ Ys ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( nth_multiset_a @ Xs @ I3 ) @ ( nth_multiset_a @ Ys @ I3 ) ) @ Ns ) )
=> ( ( A23 != zero_z7799948378220188993iset_a )
=> ( ! [B3: multiset_a] :
( ( member_multiset_a @ B3 @ ( set_mset_multiset_a @ B23 ) )
=> ? [A6: multiset_a] :
( ( member_multiset_a @ A6 @ ( set_mset_multiset_a @ A23 ) )
& ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A6 @ B3 ) @ S ) ) )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ A4 @ B5 ) @ ( multis7166037729339985883iset_a @ Ns @ S ) ) ) ) ) ) ) ) ).
% s_mul_extI_old
thf(fact_480_s__mul__extI__old,axiom,
! [A4: multiset_a,Xs: list_a,A23: multiset_a,B5: multiset_a,Ys: list_a,B23: multiset_a,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( A4
= ( plus_plus_multiset_a @ ( mset_a @ Xs ) @ A23 ) )
=> ( ( B5
= ( plus_plus_multiset_a @ ( mset_a @ Ys ) @ B23 ) )
=> ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Ys ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs @ I3 ) @ ( nth_a @ Ys @ I3 ) ) @ Ns ) )
=> ( ( A23 != zero_zero_multiset_a )
=> ( ! [B3: a] :
( ( member_a @ B3 @ ( set_mset_a @ B23 ) )
=> ? [A6: a] :
( ( member_a @ A6 @ ( set_mset_a @ A23 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A6 @ B3 ) @ S ) ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A4 @ B5 ) @ ( multiset_s_mul_ext_a @ Ns @ S ) ) ) ) ) ) ) ) ).
% s_mul_extI_old
thf(fact_481_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 )
= ( ! [X5: multiset_a] :
( ( member_multiset_a @ X5 @ ( set_mset_multiset_a @ M ) )
=> ( X5 = zero_zero_multiset_a ) ) ) ) ).
% subset_mset.sum_mset_0_iff
thf(fact_482_listrel1__iff__update,axiom,
! [Xs: list_multiset_a,Ys: list_multiset_a,R2: set_Pr79727621955416071iset_a] :
( ( member1640862112313989456iset_a @ ( produc1434922032272282391iset_a @ Xs @ Ys ) @ ( listrel1_multiset_a @ R2 ) )
= ( ? [Y8: multiset_a,N5: nat] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( nth_multiset_a @ Xs @ N5 ) @ Y8 ) @ R2 )
& ( ord_less_nat @ N5 @ ( size_s3003218658903709366iset_a @ Xs ) )
& ( Ys
= ( list_u788720842672076722iset_a @ Xs @ N5 @ Y8 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_483_listrel1__iff__update,axiom,
! [Xs: list_a,Ys: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R2 ) )
= ( ? [Y8: a,N5: nat] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs @ N5 ) @ Y8 ) @ R2 )
& ( ord_less_nat @ N5 @ ( size_size_list_a @ Xs ) )
& ( Ys
= ( list_update_a @ Xs @ N5 @ Y8 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_484_ns__s__mul__ext__union__multiset__l,axiom,
! [A4: multis501812127501805293od_a_a,B5: multis501812127501805293od_a_a,Ns: set_Pr8600417178894128327od_a_a,S: set_Pr8600417178894128327od_a_a,C3: multis501812127501805293od_a_a,D: multis501812127501805293od_a_a] :
( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ A4 @ B5 ) @ ( multis1178328011278088540od_a_a @ Ns @ S ) )
=> ( ( C3 != zero_z3024502163831676836od_a_a )
=> ( ! [D3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ D3 @ ( set_ms119794563918438244od_a_a @ D ) )
=> ? [C4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C4 @ ( set_ms119794563918438244od_a_a @ C3 ) )
& ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ C4 @ D3 ) @ S ) ) )
=> ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ ( plus_p6967097483979887908od_a_a @ A4 @ C3 ) @ ( plus_p6967097483979887908od_a_a @ B5 @ D ) ) @ ( multis6523615221270609726od_a_a @ Ns @ S ) ) ) ) ) ).
% ns_s_mul_ext_union_multiset_l
thf(fact_485_ns__s__mul__ext__union__multiset__l,axiom,
! [A4: multis4781622204924204013iset_a,B5: multis4781622204924204013iset_a,Ns: set_Pr3059988933816007879iset_a,S: set_Pr3059988933816007879iset_a,C3: multis4781622204924204013iset_a,D: multis4781622204924204013iset_a] :
( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ A4 @ B5 ) @ ( multis2244641948333282396iset_a @ Ns @ S ) )
=> ( ( C3 != zero_z4591712717340762532iset_a )
=> ( ! [D3: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ D3 @ ( set_ms2379723421352415844iset_a @ D ) )
=> ? [C4: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ C4 @ ( set_ms2379723421352415844iset_a @ C3 ) )
& ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ C4 @ D3 ) @ S ) ) )
=> ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ ( plus_p1053901119268525860iset_a @ A4 @ C3 ) @ ( plus_p1053901119268525860iset_a @ B5 @ D ) ) @ ( multis6482401367445528638iset_a @ Ns @ S ) ) ) ) ) ).
% ns_s_mul_ext_union_multiset_l
thf(fact_486_ns__s__mul__ext__union__multiset__l,axiom,
! [A4: multiset_multiset_a,B5: multiset_multiset_a,Ns: set_Pr79727621955416071iset_a,S: set_Pr79727621955416071iset_a,C3: multiset_multiset_a,D: multiset_multiset_a] :
( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ A4 @ B5 ) @ ( multis8112516946959054073iset_a @ Ns @ S ) )
=> ( ( C3 != zero_z7799948378220188993iset_a )
=> ( ! [D3: multiset_a] :
( ( member_multiset_a @ D3 @ ( set_mset_multiset_a @ D ) )
=> ? [C4: multiset_a] :
( ( member_multiset_a @ C4 @ ( set_mset_multiset_a @ C3 ) )
& ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ C4 @ D3 ) @ S ) ) )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ ( plus_p6738641960240532161iset_a @ A4 @ C3 ) @ ( plus_p6738641960240532161iset_a @ B5 @ D ) ) @ ( multis7166037729339985883iset_a @ Ns @ S ) ) ) ) ) ).
% ns_s_mul_ext_union_multiset_l
thf(fact_487_ns__s__mul__ext__union__multiset__l,axiom,
! [A4: multiset_a,B5: multiset_a,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a,C3: multiset_a,D: multiset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A4 @ B5 ) @ ( multis4023584895248362547_ext_a @ Ns @ S ) )
=> ( ( C3 != zero_zero_multiset_a )
=> ( ! [D3: a] :
( ( member_a @ D3 @ ( set_mset_a @ D ) )
=> ? [C4: a] :
( ( member_a @ C4 @ ( set_mset_a @ C3 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ C4 @ D3 ) @ S ) ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( plus_plus_multiset_a @ A4 @ C3 ) @ ( plus_plus_multiset_a @ B5 @ D ) ) @ ( multiset_s_mul_ext_a @ Ns @ S ) ) ) ) ) ).
% ns_s_mul_ext_union_multiset_l
thf(fact_488_s__mul__extI,axiom,
! [A4: multis501812127501805293od_a_a,A12: multis501812127501805293od_a_a,A23: multis501812127501805293od_a_a,B5: multis501812127501805293od_a_a,B12: multis501812127501805293od_a_a,B23: multis501812127501805293od_a_a,Ns: set_Pr8600417178894128327od_a_a,S: set_Pr8600417178894128327od_a_a] :
( ( A4
= ( plus_p6967097483979887908od_a_a @ A12 @ A23 ) )
=> ( ( B5
= ( plus_p6967097483979887908od_a_a @ B12 @ B23 ) )
=> ( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ A12 @ B12 ) @ ( multis1821517007024690859od_a_a @ Ns ) )
=> ( ( A23 != zero_z3024502163831676836od_a_a )
=> ( ! [B3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ B3 @ ( set_ms119794563918438244od_a_a @ B23 ) )
=> ? [A6: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A6 @ ( set_ms119794563918438244od_a_a @ A23 ) )
& ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A6 @ B3 ) @ S ) ) )
=> ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ A4 @ B5 ) @ ( multis6523615221270609726od_a_a @ Ns @ S ) ) ) ) ) ) ) ).
% s_mul_extI
thf(fact_489_s__mul__extI,axiom,
! [A4: multis4781622204924204013iset_a,A12: multis4781622204924204013iset_a,A23: multis4781622204924204013iset_a,B5: multis4781622204924204013iset_a,B12: multis4781622204924204013iset_a,B23: multis4781622204924204013iset_a,Ns: set_Pr3059988933816007879iset_a,S: set_Pr3059988933816007879iset_a] :
( ( A4
= ( plus_p1053901119268525860iset_a @ A12 @ A23 ) )
=> ( ( B5
= ( plus_p1053901119268525860iset_a @ B12 @ B23 ) )
=> ( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ A12 @ B12 ) @ ( multis3430030276581740203iset_a @ Ns ) )
=> ( ( A23 != zero_z4591712717340762532iset_a )
=> ( ! [B3: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ B3 @ ( set_ms2379723421352415844iset_a @ B23 ) )
=> ? [A6: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ A6 @ ( set_ms2379723421352415844iset_a @ A23 ) )
& ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ A6 @ B3 ) @ S ) ) )
=> ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ A4 @ B5 ) @ ( multis6482401367445528638iset_a @ Ns @ S ) ) ) ) ) ) ) ).
% s_mul_extI
thf(fact_490_s__mul__extI,axiom,
! [A4: multiset_multiset_a,A12: multiset_multiset_a,A23: multiset_multiset_a,B5: multiset_multiset_a,B12: multiset_multiset_a,B23: multiset_multiset_a,Ns: set_Pr79727621955416071iset_a,S: set_Pr79727621955416071iset_a] :
( ( A4
= ( plus_p6738641960240532161iset_a @ A12 @ A23 ) )
=> ( ( B5
= ( plus_p6738641960240532161iset_a @ B12 @ B23 ) )
=> ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ A12 @ B12 ) @ ( multis451435756853798856iset_a @ Ns ) )
=> ( ( A23 != zero_z7799948378220188993iset_a )
=> ( ! [B3: multiset_a] :
( ( member_multiset_a @ B3 @ ( set_mset_multiset_a @ B23 ) )
=> ? [A6: multiset_a] :
( ( member_multiset_a @ A6 @ ( set_mset_multiset_a @ A23 ) )
& ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A6 @ B3 ) @ S ) ) )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ A4 @ B5 ) @ ( multis7166037729339985883iset_a @ Ns @ S ) ) ) ) ) ) ) ).
% s_mul_extI
thf(fact_491_s__mul__extI,axiom,
! [A4: multiset_a,A12: multiset_a,A23: multiset_a,B5: multiset_a,B12: multiset_a,B23: multiset_a,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( A4
= ( plus_plus_multiset_a @ A12 @ A23 ) )
=> ( ( B5
= ( plus_plus_multiset_a @ B12 @ B23 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A12 @ B12 ) @ ( multis2373687813370458242ltpw_a @ Ns ) )
=> ( ( A23 != zero_zero_multiset_a )
=> ( ! [B3: a] :
( ( member_a @ B3 @ ( set_mset_a @ B23 ) )
=> ? [A6: a] :
( ( member_a @ A6 @ ( set_mset_a @ A23 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A6 @ B3 ) @ S ) ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A4 @ B5 ) @ ( multiset_s_mul_ext_a @ Ns @ S ) ) ) ) ) ) ) ).
% s_mul_extI
thf(fact_492_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_493_neq0__conv,axiom,
! [N3: nat] :
( ( N3 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).
% neq0_conv
thf(fact_494_less__nat__zero__code,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_495_add__is__0,axiom,
! [M3: nat,N3: nat] :
( ( ( plus_plus_nat @ M3 @ N3 )
= zero_zero_nat )
= ( ( M3 = zero_zero_nat )
& ( N3 = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_496_Nat_Oadd__0__right,axiom,
! [M3: nat] :
( ( plus_plus_nat @ M3 @ zero_zero_nat )
= M3 ) ).
% Nat.add_0_right
thf(fact_497_add__gr__0,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M3 @ N3 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M3 )
| ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% add_gr_0
thf(fact_498_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_499_gr0I,axiom,
! [N3: nat] :
( ( N3 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).
% gr0I
thf(fact_500_not__gr0,axiom,
! [N3: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
= ( N3 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_501_not__less0,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ zero_zero_nat ) ).
% not_less0
thf(fact_502_less__zeroE,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_503_gr__implies__not0,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( N3 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_504_infinite__descent0,axiom,
! [P2: nat > $o,N3: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ~ ( P2 @ N4 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N4 )
& ~ ( P2 @ M4 ) ) ) )
=> ( P2 @ N3 ) ) ) ).
% infinite_descent0
thf(fact_505_plus__nat_Oadd__0,axiom,
! [N3: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N3 )
= N3 ) ).
% plus_nat.add_0
thf(fact_506_add__eq__self__zero,axiom,
! [M3: nat,N3: nat] :
( ( ( plus_plus_nat @ M3 @ N3 )
= M3 )
=> ( N3 = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_507_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_508_s__mul__ext__bottom__strict,axiom,
! [A4: multiset_a,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a] :
~ ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ zero_zero_multiset_a @ A4 ) @ ( multiset_s_mul_ext_a @ Ns @ S ) ) ).
% s_mul_ext_bottom_strict
thf(fact_509_s__mul__ext__bottom,axiom,
! [A4: multiset_a,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( A4 != zero_zero_multiset_a )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A4 @ zero_zero_multiset_a ) @ ( multiset_s_mul_ext_a @ Ns @ S ) ) ) ).
% s_mul_ext_bottom
thf(fact_510_s__mul__ext__extend__left,axiom,
! [B5: multiset_a,C3: multiset_a,W: set_Product_prod_a_a,S2: set_Product_prod_a_a,A4: multiset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ B5 @ C3 ) @ ( multiset_s_mul_ext_a @ W @ S2 ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( plus_plus_multiset_a @ A4 @ B5 ) @ C3 ) @ ( multiset_s_mul_ext_a @ W @ S2 ) ) ) ).
% s_mul_ext_extend_left
thf(fact_511_listrel1__eq__len,axiom,
! [Xs: list_a,Ys: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R2 ) )
=> ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) ) ) ).
% listrel1_eq_len
thf(fact_512_s__ns__mul__ext,axiom,
! [A4: multiset_a,B5: multiset_a,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A4 @ B5 ) @ ( multiset_s_mul_ext_a @ Ns @ S ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A4 @ B5 ) @ ( multis4023584895248362547_ext_a @ Ns @ S ) ) ) ).
% s_ns_mul_ext
thf(fact_513_length__code,axiom,
( size_size_list_a
= ( gen_length_a @ zero_zero_nat ) ) ).
% length_code
thf(fact_514_length__pos__if__in__set,axiom,
! [X3: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s3885678630836030617od_a_a @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_515_length__pos__if__in__set,axiom,
! [X3: produc6518373309651786023iset_a,Xs: list_P6988024003011616557iset_a] :
( ( member5199237121806060112iset_a @ X3 @ ( set_Pr5550698219922361532iset_a @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s7238940367606365593iset_a @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_516_length__pos__if__in__set,axiom,
! [X3: a,Xs: list_a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_517_s__ns__mul__ext__union__compat,axiom,
! [A4: multiset_a,B5: multiset_a,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a,C3: multiset_a,D: multiset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A4 @ B5 ) @ ( multiset_s_mul_ext_a @ Ns @ S ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ C3 @ D ) @ ( multis4023584895248362547_ext_a @ Ns @ S ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( plus_plus_multiset_a @ A4 @ C3 ) @ ( plus_plus_multiset_a @ B5 @ D ) ) @ ( multiset_s_mul_ext_a @ Ns @ S ) ) ) ) ).
% s_ns_mul_ext_union_compat
thf(fact_518_s__mul__ext__union__compat,axiom,
! [A4: multiset_a,B5: multiset_a,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a,C3: multiset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A4 @ B5 ) @ ( multiset_s_mul_ext_a @ Ns @ S ) )
=> ( ( locally_refl_a @ Ns @ C3 )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( plus_plus_multiset_a @ A4 @ C3 ) @ ( plus_plus_multiset_a @ B5 @ C3 ) ) @ ( multiset_s_mul_ext_a @ Ns @ S ) ) ) ) ).
% s_mul_ext_union_compat
thf(fact_519_all__s__s__mul__ext,axiom,
! [A4: multis501812127501805293od_a_a,B5: multis501812127501805293od_a_a,S: set_Pr8600417178894128327od_a_a,Ns: set_Pr8600417178894128327od_a_a] :
( ( A4 != zero_z3024502163831676836od_a_a )
=> ( ! [B3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ B3 @ ( set_ms119794563918438244od_a_a @ B5 ) )
=> ? [A6: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A6 @ ( set_ms119794563918438244od_a_a @ A4 ) )
& ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A6 @ B3 ) @ S ) ) )
=> ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ A4 @ B5 ) @ ( multis6523615221270609726od_a_a @ Ns @ S ) ) ) ) ).
% all_s_s_mul_ext
thf(fact_520_all__s__s__mul__ext,axiom,
! [A4: multis4781622204924204013iset_a,B5: multis4781622204924204013iset_a,S: set_Pr3059988933816007879iset_a,Ns: set_Pr3059988933816007879iset_a] :
( ( A4 != zero_z4591712717340762532iset_a )
=> ( ! [B3: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ B3 @ ( set_ms2379723421352415844iset_a @ B5 ) )
=> ? [A6: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ A6 @ ( set_ms2379723421352415844iset_a @ A4 ) )
& ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ A6 @ B3 ) @ S ) ) )
=> ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ A4 @ B5 ) @ ( multis6482401367445528638iset_a @ Ns @ S ) ) ) ) ).
% all_s_s_mul_ext
thf(fact_521_all__s__s__mul__ext,axiom,
! [A4: multiset_a,B5: multiset_a,S: set_Product_prod_a_a,Ns: set_Product_prod_a_a] :
( ( A4 != zero_zero_multiset_a )
=> ( ! [B3: a] :
( ( member_a @ B3 @ ( set_mset_a @ B5 ) )
=> ? [A6: a] :
( ( member_a @ A6 @ ( set_mset_a @ A4 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A6 @ B3 ) @ S ) ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A4 @ B5 ) @ ( multiset_s_mul_ext_a @ Ns @ S ) ) ) ) ).
% all_s_s_mul_ext
thf(fact_522_all__s__s__mul__ext,axiom,
! [A4: multiset_multiset_a,B5: multiset_multiset_a,S: set_Pr79727621955416071iset_a,Ns: set_Pr79727621955416071iset_a] :
( ( A4 != zero_z7799948378220188993iset_a )
=> ( ! [B3: multiset_a] :
( ( member_multiset_a @ B3 @ ( set_mset_multiset_a @ B5 ) )
=> ? [A6: multiset_a] :
( ( member_multiset_a @ A6 @ ( set_mset_multiset_a @ A4 ) )
& ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A6 @ B3 ) @ S ) ) )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ A4 @ B5 ) @ ( multis7166037729339985883iset_a @ Ns @ S ) ) ) ) ).
% all_s_s_mul_ext
thf(fact_523_s__mul__ext__ne__extend__left,axiom,
! [A4: multiset_a,B5: multiset_a,C3: multiset_a,W: set_Product_prod_a_a,S2: set_Product_prod_a_a] :
( ( A4 != zero_zero_multiset_a )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ B5 @ C3 ) @ ( multis4023584895248362547_ext_a @ W @ S2 ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( plus_plus_multiset_a @ A4 @ B5 ) @ C3 ) @ ( multiset_s_mul_ext_a @ W @ S2 ) ) ) ) ).
% s_mul_ext_ne_extend_left
thf(fact_524_s__mul__ext__self__extend__left,axiom,
! [A4: multiset_a,W: set_Product_prod_a_a,B5: multiset_a,S2: set_Product_prod_a_a] :
( ( A4 != zero_zero_multiset_a )
=> ( ( locally_refl_a @ W @ B5 )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( plus_plus_multiset_a @ A4 @ B5 ) @ B5 ) @ ( multiset_s_mul_ext_a @ W @ S2 ) ) ) ) ).
% s_mul_ext_self_extend_left
thf(fact_525_s__mul__ext__IdE,axiom,
! [M: multiset_multiset_a,N: multiset_multiset_a,R: set_Pr79727621955416071iset_a] :
( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ M @ N ) @ ( multis7166037729339985883iset_a @ id_multiset_a @ R ) )
=> ~ ! [X8: multiset_multiset_a] :
( ( X8 != zero_z7799948378220188993iset_a )
=> ! [Y7: multiset_multiset_a,Z3: multiset_multiset_a] :
( ( M
= ( plus_p6738641960240532161iset_a @ X8 @ Z3 ) )
=> ( ( N
= ( plus_p6738641960240532161iset_a @ Y7 @ Z3 ) )
=> ~ ! [X6: multiset_a] :
( ( member_multiset_a @ X6 @ ( set_mset_multiset_a @ Y7 ) )
=> ? [Xa2: multiset_a] :
( ( member_multiset_a @ Xa2 @ ( set_mset_multiset_a @ X8 ) )
& ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Xa2 @ X6 ) @ R ) ) ) ) ) ) ) ).
% s_mul_ext_IdE
thf(fact_526_s__mul__ext__IdE,axiom,
! [M: multiset_a,N: multiset_a,R: set_Product_prod_a_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ M @ N ) @ ( multiset_s_mul_ext_a @ id_a @ R ) )
=> ~ ! [X8: multiset_a] :
( ( X8 != zero_zero_multiset_a )
=> ! [Y7: multiset_a,Z3: multiset_a] :
( ( M
= ( plus_plus_multiset_a @ X8 @ Z3 ) )
=> ( ( N
= ( plus_plus_multiset_a @ Y7 @ Z3 ) )
=> ~ ! [X6: a] :
( ( member_a @ X6 @ ( set_mset_a @ Y7 ) )
=> ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_mset_a @ X8 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa2 @ X6 ) @ R ) ) ) ) ) ) ) ).
% s_mul_ext_IdE
thf(fact_527_s__mul__ext__IdI,axiom,
! [X4: multiset_multiset_a,M: multiset_multiset_a,Z2: multiset_multiset_a,N: multiset_multiset_a,Y4: multiset_multiset_a,R: set_Pr79727621955416071iset_a] :
( ( X4 != zero_z7799948378220188993iset_a )
=> ( ( M
= ( plus_p6738641960240532161iset_a @ X4 @ Z2 ) )
=> ( ( N
= ( plus_p6738641960240532161iset_a @ Y4 @ Z2 ) )
=> ( ! [X: multiset_a] :
( ( member_multiset_a @ X @ ( set_mset_multiset_a @ Y4 ) )
=> ? [Xa: multiset_a] :
( ( member_multiset_a @ Xa @ ( set_mset_multiset_a @ X4 ) )
& ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Xa @ X ) @ R ) ) )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ M @ N ) @ ( multis7166037729339985883iset_a @ id_multiset_a @ R ) ) ) ) ) ) ).
% s_mul_ext_IdI
thf(fact_528_s__mul__ext__IdI,axiom,
! [X4: multiset_a,M: multiset_a,Z2: multiset_a,N: multiset_a,Y4: multiset_a,R: set_Product_prod_a_a] :
( ( X4 != zero_zero_multiset_a )
=> ( ( M
= ( plus_plus_multiset_a @ X4 @ Z2 ) )
=> ( ( N
= ( plus_plus_multiset_a @ Y4 @ Z2 ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( set_mset_a @ Y4 ) )
=> ? [Xa: a] :
( ( member_a @ Xa @ ( set_mset_a @ X4 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa @ X ) @ R ) ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ M @ N ) @ ( multiset_s_mul_ext_a @ id_a @ R ) ) ) ) ) ) ).
% s_mul_ext_IdI
thf(fact_529_s__mul__extE,axiom,
! [A4: multis501812127501805293od_a_a,B5: multis501812127501805293od_a_a,Ns: set_Pr8600417178894128327od_a_a,S: set_Pr8600417178894128327od_a_a] :
( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ A4 @ B5 ) @ ( multis6523615221270609726od_a_a @ Ns @ S ) )
=> ~ ! [A1: multis501812127501805293od_a_a,A22: multis501812127501805293od_a_a] :
( ( A4
= ( plus_p6967097483979887908od_a_a @ A1 @ A22 ) )
=> ! [B1: multis501812127501805293od_a_a,B22: multis501812127501805293od_a_a] :
( ( B5
= ( plus_p6967097483979887908od_a_a @ B1 @ B22 ) )
=> ( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ A1 @ B1 ) @ ( multis1821517007024690859od_a_a @ Ns ) )
=> ( ( A22 != zero_z3024502163831676836od_a_a )
=> ~ ! [B4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ B4 @ ( set_ms119794563918438244od_a_a @ B22 ) )
=> ? [A: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A @ ( set_ms119794563918438244od_a_a @ A22 ) )
& ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A @ B4 ) @ S ) ) ) ) ) ) ) ) ).
% s_mul_extE
thf(fact_530_s__mul__extE,axiom,
! [A4: multis4781622204924204013iset_a,B5: multis4781622204924204013iset_a,Ns: set_Pr3059988933816007879iset_a,S: set_Pr3059988933816007879iset_a] :
( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ A4 @ B5 ) @ ( multis6482401367445528638iset_a @ Ns @ S ) )
=> ~ ! [A1: multis4781622204924204013iset_a,A22: multis4781622204924204013iset_a] :
( ( A4
= ( plus_p1053901119268525860iset_a @ A1 @ A22 ) )
=> ! [B1: multis4781622204924204013iset_a,B22: multis4781622204924204013iset_a] :
( ( B5
= ( plus_p1053901119268525860iset_a @ B1 @ B22 ) )
=> ( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ A1 @ B1 ) @ ( multis3430030276581740203iset_a @ Ns ) )
=> ( ( A22 != zero_z4591712717340762532iset_a )
=> ~ ! [B4: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ B4 @ ( set_ms2379723421352415844iset_a @ B22 ) )
=> ? [A: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ A @ ( set_ms2379723421352415844iset_a @ A22 ) )
& ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ A @ B4 ) @ S ) ) ) ) ) ) ) ) ).
% s_mul_extE
thf(fact_531_s__mul__extE,axiom,
! [A4: multiset_multiset_a,B5: multiset_multiset_a,Ns: set_Pr79727621955416071iset_a,S: set_Pr79727621955416071iset_a] :
( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ A4 @ B5 ) @ ( multis7166037729339985883iset_a @ Ns @ S ) )
=> ~ ! [A1: multiset_multiset_a,A22: multiset_multiset_a] :
( ( A4
= ( plus_p6738641960240532161iset_a @ A1 @ A22 ) )
=> ! [B1: multiset_multiset_a,B22: multiset_multiset_a] :
( ( B5
= ( plus_p6738641960240532161iset_a @ B1 @ B22 ) )
=> ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ A1 @ B1 ) @ ( multis451435756853798856iset_a @ Ns ) )
=> ( ( A22 != zero_z7799948378220188993iset_a )
=> ~ ! [B4: multiset_a] :
( ( member_multiset_a @ B4 @ ( set_mset_multiset_a @ B22 ) )
=> ? [A: multiset_a] :
( ( member_multiset_a @ A @ ( set_mset_multiset_a @ A22 ) )
& ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A @ B4 ) @ S ) ) ) ) ) ) ) ) ).
% s_mul_extE
thf(fact_532_s__mul__extE,axiom,
! [A4: multiset_a,B5: multiset_a,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A4 @ B5 ) @ ( multiset_s_mul_ext_a @ Ns @ S ) )
=> ~ ! [A1: multiset_a,A22: multiset_a] :
( ( A4
= ( plus_plus_multiset_a @ A1 @ A22 ) )
=> ! [B1: multiset_a,B22: multiset_a] :
( ( B5
= ( plus_plus_multiset_a @ B1 @ B22 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A1 @ B1 ) @ ( multis2373687813370458242ltpw_a @ Ns ) )
=> ( ( A22 != zero_zero_multiset_a )
=> ~ ! [B4: a] :
( ( member_a @ B4 @ ( set_mset_a @ B22 ) )
=> ? [A: a] :
( ( member_a @ A @ ( set_mset_a @ A22 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B4 ) @ S ) ) ) ) ) ) ) ) ).
% s_mul_extE
thf(fact_533_s__mul__ext__intro,axiom,
! [Xs: multis501812127501805293od_a_a,Xs12: list_P1396940483166286381od_a_a,Xs22: list_P1396940483166286381od_a_a,Ys: multis501812127501805293od_a_a,Ys12: list_P1396940483166286381od_a_a,Ys22: list_P1396940483166286381od_a_a,NS: set_Pr8600417178894128327od_a_a,S2: set_Pr8600417178894128327od_a_a] :
( ( Xs
= ( plus_p6967097483979887908od_a_a @ ( mset_P3709024939784119484od_a_a @ Xs12 ) @ ( mset_P3709024939784119484od_a_a @ Xs22 ) ) )
=> ( ( Ys
= ( plus_p6967097483979887908od_a_a @ ( mset_P3709024939784119484od_a_a @ Ys12 ) @ ( mset_P3709024939784119484od_a_a @ Ys22 ) ) )
=> ( ( ( size_s3885678630836030617od_a_a @ Xs12 )
= ( size_s3885678630836030617od_a_a @ Ys12 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3885678630836030617od_a_a @ Ys12 ) )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ ( nth_Product_prod_a_a @ Xs12 @ I3 ) @ ( nth_Product_prod_a_a @ Ys12 @ I3 ) ) @ NS ) )
=> ( ( Xs22 != nil_Product_prod_a_a )
=> ( ! [Y3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y3 @ ( set_Product_prod_a_a2 @ Ys22 ) )
=> ? [X6: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X6 @ ( set_Product_prod_a_a2 @ Xs22 ) )
& ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X6 @ Y3 ) @ S2 ) ) )
=> ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ Xs @ Ys ) @ ( multis6523615221270609726od_a_a @ NS @ S2 ) ) ) ) ) ) ) ) ).
% s_mul_ext_intro
thf(fact_534_s__mul__ext__intro,axiom,
! [Xs: multis4781622204924204013iset_a,Xs12: list_P6988024003011616557iset_a,Xs22: list_P6988024003011616557iset_a,Ys: multis4781622204924204013iset_a,Ys12: list_P6988024003011616557iset_a,Ys22: list_P6988024003011616557iset_a,NS: set_Pr3059988933816007879iset_a,S2: set_Pr3059988933816007879iset_a] :
( ( Xs
= ( plus_p1053901119268525860iset_a @ ( mset_P2771936761501114300iset_a @ Xs12 ) @ ( mset_P2771936761501114300iset_a @ Xs22 ) ) )
=> ( ( Ys
= ( plus_p1053901119268525860iset_a @ ( mset_P2771936761501114300iset_a @ Ys12 ) @ ( mset_P2771936761501114300iset_a @ Ys22 ) ) )
=> ( ( ( size_s7238940367606365593iset_a @ Xs12 )
= ( size_s7238940367606365593iset_a @ Ys12 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s7238940367606365593iset_a @ Ys12 ) )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ ( nth_Pr1079437516805851566iset_a @ Xs12 @ I3 ) @ ( nth_Pr1079437516805851566iset_a @ Ys12 @ I3 ) ) @ NS ) )
=> ( ( Xs22 != nil_Pr2722442113885406509iset_a )
=> ( ! [Y3: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Y3 @ ( set_Pr5550698219922361532iset_a @ Ys22 ) )
=> ? [X6: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X6 @ ( set_Pr5550698219922361532iset_a @ Xs22 ) )
& ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X6 @ Y3 ) @ S2 ) ) )
=> ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ Xs @ Ys ) @ ( multis6482401367445528638iset_a @ NS @ S2 ) ) ) ) ) ) ) ) ).
% s_mul_ext_intro
thf(fact_535_s__mul__ext__intro,axiom,
! [Xs: multiset_multiset_a,Xs12: list_multiset_a,Xs22: list_multiset_a,Ys: multiset_multiset_a,Ys12: list_multiset_a,Ys22: list_multiset_a,NS: set_Pr79727621955416071iset_a,S2: set_Pr79727621955416071iset_a] :
( ( Xs
= ( plus_p6738641960240532161iset_a @ ( mset_multiset_a @ Xs12 ) @ ( mset_multiset_a @ Xs22 ) ) )
=> ( ( Ys
= ( plus_p6738641960240532161iset_a @ ( mset_multiset_a @ Ys12 ) @ ( mset_multiset_a @ Ys22 ) ) )
=> ( ( ( size_s3003218658903709366iset_a @ Xs12 )
= ( size_s3003218658903709366iset_a @ Ys12 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3003218658903709366iset_a @ Ys12 ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( nth_multiset_a @ Xs12 @ I3 ) @ ( nth_multiset_a @ Ys12 @ I3 ) ) @ NS ) )
=> ( ( Xs22 != nil_multiset_a )
=> ( ! [Y3: multiset_a] :
( ( member_multiset_a @ Y3 @ ( set_multiset_a2 @ Ys22 ) )
=> ? [X6: multiset_a] :
( ( member_multiset_a @ X6 @ ( set_multiset_a2 @ Xs22 ) )
& ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X6 @ Y3 ) @ S2 ) ) )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ Xs @ Ys ) @ ( multis7166037729339985883iset_a @ NS @ S2 ) ) ) ) ) ) ) ) ).
% s_mul_ext_intro
thf(fact_536_s__mul__ext__intro,axiom,
! [Xs: multiset_a,Xs12: list_a,Xs22: list_a,Ys: multiset_a,Ys12: list_a,Ys22: list_a,NS: set_Product_prod_a_a,S2: set_Product_prod_a_a] :
( ( Xs
= ( plus_plus_multiset_a @ ( mset_a @ Xs12 ) @ ( mset_a @ Xs22 ) ) )
=> ( ( Ys
= ( plus_plus_multiset_a @ ( mset_a @ Ys12 ) @ ( mset_a @ Ys22 ) ) )
=> ( ( ( size_size_list_a @ Xs12 )
= ( size_size_list_a @ Ys12 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Ys12 ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs12 @ I3 ) @ ( nth_a @ Ys12 @ I3 ) ) @ NS ) )
=> ( ( Xs22 != nil_a )
=> ( ! [Y3: a] :
( ( member_a @ Y3 @ ( set_a2 @ Ys22 ) )
=> ? [X6: a] :
( ( member_a @ X6 @ ( set_a2 @ Xs22 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X6 @ Y3 ) @ S2 ) ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Xs @ Ys ) @ ( multiset_s_mul_ext_a @ NS @ S2 ) ) ) ) ) ) ) ) ).
% s_mul_ext_intro
thf(fact_537_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_538_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_539_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_540_add__0__iff,axiom,
! [B: nat,A2: nat] :
( ( B
= ( plus_plus_nat @ B @ A2 ) )
= ( A2 = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_541_verit__sum__simplify,axiom,
! [A2: multiset_a] :
( ( plus_plus_multiset_a @ A2 @ zero_zero_multiset_a )
= A2 ) ).
% verit_sum_simplify
thf(fact_542_verit__sum__simplify,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% verit_sum_simplify
thf(fact_543_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_544_nth__zip,axiom,
! [I: nat,Xs: list_multiset_a,Ys: list_multiset_a] :
( ( ord_less_nat @ I @ ( size_s3003218658903709366iset_a @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_s3003218658903709366iset_a @ Ys ) )
=> ( ( nth_Pr1079437516805851566iset_a @ ( zip_mu5258324667929623607iset_a @ Xs @ Ys ) @ I )
= ( produc654756711066625303iset_a @ ( nth_multiset_a @ Xs @ I ) @ ( nth_multiset_a @ Ys @ I ) ) ) ) ) ).
% nth_zip
thf(fact_545_nth__zip,axiom,
! [I: nat,Xs: list_nat_nat,Ys: list_nat] :
( ( ord_less_nat @ I @ ( size_s8208510060688613859at_nat @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
=> ( ( nth_Pr4952906083868132727at_nat @ ( zip_nat_nat_nat @ Xs @ Ys ) @ I )
= ( produc72220940542539688at_nat @ ( nth_nat_nat @ Xs @ I ) @ ( nth_nat @ Ys @ I ) ) ) ) ) ).
% nth_zip
thf(fact_546_nth__zip,axiom,
! [I: nat,Xs: list_a,Ys: list_a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_a @ Ys ) )
=> ( ( nth_Product_prod_a_a @ ( zip_a_a @ Xs @ Ys ) @ I )
= ( product_Pair_a_a @ ( nth_a @ Xs @ I ) @ ( nth_a @ Ys @ I ) ) ) ) ) ).
% nth_zip
thf(fact_547_one__step__implies__mult,axiom,
! [J2: multiset_multiset_a,K: multiset_multiset_a,R2: set_Pr79727621955416071iset_a,I5: multiset_multiset_a] :
( ( J2 != zero_z7799948378220188993iset_a )
=> ( ! [X: multiset_a] :
( ( member_multiset_a @ X @ ( set_mset_multiset_a @ K ) )
=> ? [Xa: multiset_a] :
( ( member_multiset_a @ Xa @ ( set_mset_multiset_a @ J2 ) )
& ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ Xa ) @ R2 ) ) )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ ( plus_p6738641960240532161iset_a @ I5 @ K ) @ ( plus_p6738641960240532161iset_a @ I5 @ J2 ) ) @ ( mult_multiset_a @ R2 ) ) ) ) ).
% one_step_implies_mult
thf(fact_548_one__step__implies__mult,axiom,
! [J2: multiset_a,K: multiset_a,R2: set_Product_prod_a_a,I5: multiset_a] :
( ( J2 != zero_zero_multiset_a )
=> ( ! [X: a] :
( ( member_a @ X @ ( set_mset_a @ K ) )
=> ? [Xa: a] :
( ( member_a @ Xa @ ( set_mset_a @ J2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Xa ) @ R2 ) ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( plus_plus_multiset_a @ I5 @ K ) @ ( plus_plus_multiset_a @ I5 @ J2 ) ) @ ( mult_a @ R2 ) ) ) ) ).
% one_step_implies_mult
thf(fact_549_length__0__conv,axiom,
! [Xs: list_a] :
( ( ( size_size_list_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_a ) ) ).
% length_0_conv
thf(fact_550_mset__zero__iff,axiom,
! [X3: list_a] :
( ( ( mset_a @ X3 )
= zero_zero_multiset_a )
= ( X3 = nil_a ) ) ).
% mset_zero_iff
thf(fact_551_mset__zero__iff__right,axiom,
! [X3: list_a] :
( ( zero_zero_multiset_a
= ( mset_a @ X3 ) )
= ( X3 = nil_a ) ) ).
% mset_zero_iff_right
thf(fact_552_length__greater__0__conv,axiom,
! [Xs: list_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs ) )
= ( Xs != nil_a ) ) ).
% length_greater_0_conv
thf(fact_553_list_Osize_I3_J,axiom,
( ( size_size_list_a @ nil_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_554_zip__same,axiom,
! [A2: product_prod_a_a,B: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A2 @ B ) @ ( set_Pr6493056779096584572od_a_a @ ( zip_Pr1329156589218441847od_a_a @ Xs @ Xs ) ) )
= ( ( member1426531477525435216od_a_a @ A2 @ ( set_Product_prod_a_a2 @ Xs ) )
& ( A2 = B ) ) ) ).
% zip_same
thf(fact_555_zip__same,axiom,
! [A2: produc6518373309651786023iset_a,B: produc6518373309651786023iset_a,Xs: list_P6988024003011616557iset_a] :
( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ A2 @ B ) @ ( set_Pr3130690497449335164iset_a @ ( zip_Pr8115149548399026807iset_a @ Xs @ Xs ) ) )
= ( ( member5199237121806060112iset_a @ A2 @ ( set_Pr5550698219922361532iset_a @ Xs ) )
& ( A2 = B ) ) ) ).
% zip_same
thf(fact_556_zip__same,axiom,
! [A2: a,B: a,Xs: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B ) @ ( set_Product_prod_a_a2 @ ( zip_a_a @ Xs @ Xs ) ) )
= ( ( member_a @ A2 @ ( set_a2 @ Xs ) )
& ( A2 = B ) ) ) ).
% zip_same
thf(fact_557_zip__same,axiom,
! [A2: multiset_a,B: multiset_a,Xs: list_multiset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A2 @ B ) @ ( set_Pr5550698219922361532iset_a @ ( zip_mu5258324667929623607iset_a @ Xs @ Xs ) ) )
= ( ( member_multiset_a @ A2 @ ( set_multiset_a2 @ Xs ) )
& ( A2 = B ) ) ) ).
% zip_same
thf(fact_558_in__set__zipE,axiom,
! [X3: a,Y: a,Xs: list_a,Ys: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ ( set_Product_prod_a_a2 @ ( zip_a_a @ Xs @ Ys ) ) )
=> ~ ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ~ ( member_a @ Y @ ( set_a2 @ Ys ) ) ) ) ).
% in_set_zipE
thf(fact_559_in__set__zipE,axiom,
! [X3: product_prod_a_a,Y: a,Xs: list_P1396940483166286381od_a_a,Ys: list_a] :
( ( member2829916517802586983_a_a_a @ ( produc5692536848587512110_a_a_a @ X3 @ Y ) @ ( set_Pr6767713769399854547_a_a_a @ ( zip_Pr9109025954503487566_a_a_a @ Xs @ Ys ) ) )
=> ~ ( ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Xs ) )
=> ~ ( member_a @ Y @ ( set_a2 @ Ys ) ) ) ) ).
% in_set_zipE
thf(fact_560_in__set__zipE,axiom,
! [X3: a,Y: product_prod_a_a,Xs: list_a,Ys: list_P1396940483166286381od_a_a] :
( ( member3071122053849602553od_a_a @ ( produc431845341423274048od_a_a @ X3 @ Y ) @ ( set_Pr7008919305446870117od_a_a @ ( zip_a_3848334447339249504od_a_a @ Xs @ Ys ) ) )
=> ~ ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ~ ( member1426531477525435216od_a_a @ Y @ ( set_Product_prod_a_a2 @ Ys ) ) ) ) ).
% in_set_zipE
thf(fact_561_in__set__zipE,axiom,
! [X3: multiset_a,Y: multiset_a,Xs: list_multiset_a,Ys: list_multiset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ Y ) @ ( set_Pr5550698219922361532iset_a @ ( zip_mu5258324667929623607iset_a @ Xs @ Ys ) ) )
=> ~ ( ( member_multiset_a @ X3 @ ( set_multiset_a2 @ Xs ) )
=> ~ ( member_multiset_a @ Y @ ( set_multiset_a2 @ Ys ) ) ) ) ).
% in_set_zipE
thf(fact_562_in__set__zipE,axiom,
! [X3: nat > nat,Y: nat,Xs: list_nat_nat,Ys: list_nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X3 @ Y ) @ ( set_Pr1941165898809491589at_nat @ ( zip_nat_nat_nat @ Xs @ Ys ) ) )
=> ~ ( ( member_nat_nat @ X3 @ ( set_nat_nat2 @ Xs ) )
=> ~ ( member_nat @ Y @ ( set_nat2 @ Ys ) ) ) ) ).
% in_set_zipE
thf(fact_563_in__set__zipE,axiom,
! [X3: product_prod_a_a,Y: product_prod_a_a,Xs: list_P1396940483166286381od_a_a,Ys: list_P1396940483166286381od_a_a] :
( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X3 @ Y ) @ ( set_Pr6493056779096584572od_a_a @ ( zip_Pr1329156589218441847od_a_a @ Xs @ Ys ) ) )
=> ~ ( ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Xs ) )
=> ~ ( member1426531477525435216od_a_a @ Y @ ( set_Product_prod_a_a2 @ Ys ) ) ) ) ).
% in_set_zipE
thf(fact_564_in__set__zipE,axiom,
! [X3: produc6518373309651786023iset_a,Y: a,Xs: list_P6988024003011616557iset_a,Ys: list_a] :
( ( member2419060323508899175et_a_a @ ( produc5181291240524633390et_a_a @ X3 @ Y ) @ ( set_Pr5466182931555501011et_a_a @ ( zip_Pr3459117997753158734et_a_a @ Xs @ Ys ) ) )
=> ~ ( ( member5199237121806060112iset_a @ X3 @ ( set_Pr5550698219922361532iset_a @ Xs ) )
=> ~ ( member_a @ Y @ ( set_a2 @ Ys ) ) ) ) ).
% in_set_zipE
thf(fact_565_in__set__zipE,axiom,
! [X3: a,Y: produc6518373309651786023iset_a,Xs: list_a,Ys: list_P6988024003011616557iset_a] :
( ( member4303746980753173241iset_a @ ( produc1768535192049815360iset_a @ X3 @ Y ) @ ( set_Pr7350869588799775077iset_a @ ( zip_a_46361949278340704iset_a @ Xs @ Ys ) ) )
=> ~ ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ~ ( member5199237121806060112iset_a @ Y @ ( set_Pr5550698219922361532iset_a @ Ys ) ) ) ) ).
% in_set_zipE
thf(fact_566_in__set__zipE,axiom,
! [X3: product_prod_a_a,Y: produc6518373309651786023iset_a,Xs: list_P1396940483166286381od_a_a,Ys: list_P6988024003011616557iset_a] :
( ( member7748342945563660560iset_a @ ( produc2236536367857282903iset_a @ X3 @ Y ) @ ( set_Pr5806744667188525948iset_a @ ( zip_Pr3794690320804382327iset_a @ Xs @ Ys ) ) )
=> ~ ( ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Xs ) )
=> ~ ( member5199237121806060112iset_a @ Y @ ( set_Pr5550698219922361532iset_a @ Ys ) ) ) ) ).
% in_set_zipE
thf(fact_567_in__set__zipE,axiom,
! [X3: produc6518373309651786023iset_a,Y: product_prod_a_a,Xs: list_P6988024003011616557iset_a,Ys: list_P1396940483166286381od_a_a] :
( ( member6916131784381031696od_a_a @ ( produc1199870869182492503od_a_a @ X3 @ Y ) @ ( set_Pr4974533506005897084od_a_a @ ( zip_Pr2758024822129591927od_a_a @ Xs @ Ys ) ) )
=> ~ ( ( member5199237121806060112iset_a @ X3 @ ( set_Pr5550698219922361532iset_a @ Xs ) )
=> ~ ( member1426531477525435216od_a_a @ Y @ ( set_Product_prod_a_a2 @ Ys ) ) ) ) ).
% in_set_zipE
thf(fact_568_set__zip__leftD,axiom,
! [X3: a,Y: a,Xs: list_a,Ys: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ ( set_Product_prod_a_a2 @ ( zip_a_a @ Xs @ Ys ) ) )
=> ( member_a @ X3 @ ( set_a2 @ Xs ) ) ) ).
% set_zip_leftD
thf(fact_569_set__zip__leftD,axiom,
! [X3: multiset_a,Y: multiset_a,Xs: list_multiset_a,Ys: list_multiset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ Y ) @ ( set_Pr5550698219922361532iset_a @ ( zip_mu5258324667929623607iset_a @ Xs @ Ys ) ) )
=> ( member_multiset_a @ X3 @ ( set_multiset_a2 @ Xs ) ) ) ).
% set_zip_leftD
thf(fact_570_set__zip__leftD,axiom,
! [X3: nat > nat,Y: nat,Xs: list_nat_nat,Ys: list_nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X3 @ Y ) @ ( set_Pr1941165898809491589at_nat @ ( zip_nat_nat_nat @ Xs @ Ys ) ) )
=> ( member_nat_nat @ X3 @ ( set_nat_nat2 @ Xs ) ) ) ).
% set_zip_leftD
thf(fact_571_set__zip__rightD,axiom,
! [X3: a,Y: a,Xs: list_a,Ys: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ ( set_Product_prod_a_a2 @ ( zip_a_a @ Xs @ Ys ) ) )
=> ( member_a @ Y @ ( set_a2 @ Ys ) ) ) ).
% set_zip_rightD
thf(fact_572_set__zip__rightD,axiom,
! [X3: multiset_a,Y: multiset_a,Xs: list_multiset_a,Ys: list_multiset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ Y ) @ ( set_Pr5550698219922361532iset_a @ ( zip_mu5258324667929623607iset_a @ Xs @ Ys ) ) )
=> ( member_multiset_a @ Y @ ( set_multiset_a2 @ Ys ) ) ) ).
% set_zip_rightD
thf(fact_573_set__zip__rightD,axiom,
! [X3: nat > nat,Y: nat,Xs: list_nat_nat,Ys: list_nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X3 @ Y ) @ ( set_Pr1941165898809491589at_nat @ ( zip_nat_nat_nat @ Xs @ Ys ) ) )
=> ( member_nat @ Y @ ( set_nat2 @ Ys ) ) ) ).
% set_zip_rightD
thf(fact_574_ex__mset__zip__left,axiom,
! [Xs: list_a,Ys: list_a,Xs5: list_a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( mset_a @ Xs5 )
= ( mset_a @ Xs ) )
=> ? [Ys6: list_a] :
( ( ( size_size_list_a @ Ys6 )
= ( size_size_list_a @ Xs5 ) )
& ( ( mset_P3709024939784119484od_a_a @ ( zip_a_a @ Xs5 @ Ys6 ) )
= ( mset_P3709024939784119484od_a_a @ ( zip_a_a @ Xs @ Ys ) ) ) ) ) ) ).
% ex_mset_zip_left
thf(fact_575_zip__update,axiom,
! [Xs: list_a,I: nat,X3: a,Ys: list_a,Y: a] :
( ( zip_a_a @ ( list_update_a @ Xs @ I @ X3 ) @ ( list_update_a @ Ys @ I @ Y ) )
= ( list_u4967751623001292181od_a_a @ ( zip_a_a @ Xs @ Ys ) @ I @ ( product_Pair_a_a @ X3 @ Y ) ) ) ).
% zip_update
thf(fact_576_zip__update,axiom,
! [Xs: list_multiset_a,I: nat,X3: multiset_a,Ys: list_multiset_a,Y: multiset_a] :
( ( zip_mu5258324667929623607iset_a @ ( list_u788720842672076722iset_a @ Xs @ I @ X3 ) @ ( list_u788720842672076722iset_a @ Ys @ I @ Y ) )
= ( list_u2687166788783756181iset_a @ ( zip_mu5258324667929623607iset_a @ Xs @ Ys ) @ I @ ( produc654756711066625303iset_a @ X3 @ Y ) ) ) ).
% zip_update
thf(fact_577_zip__update,axiom,
! [Xs: list_nat_nat,I: nat,X3: nat > nat,Ys: list_nat,Y: nat] :
( ( zip_nat_nat_nat @ ( list_update_nat_nat @ Xs @ I @ X3 ) @ ( list_update_nat @ Ys @ I @ Y ) )
= ( list_u5493879213508619102at_nat @ ( zip_nat_nat_nat @ Xs @ Ys ) @ I @ ( produc72220940542539688at_nat @ X3 @ Y ) ) ) ).
% zip_update
thf(fact_578_mset_Osimps_I1_J,axiom,
( ( mset_a @ nil_a )
= zero_zero_multiset_a ) ).
% mset.simps(1)
thf(fact_579_in__set__impl__in__set__zip1,axiom,
! [Xs: list_multiset_a,Ys: list_multiset_a,X3: multiset_a] :
( ( ( size_s3003218658903709366iset_a @ Xs )
= ( size_s3003218658903709366iset_a @ Ys ) )
=> ( ( member_multiset_a @ X3 @ ( set_multiset_a2 @ Xs ) )
=> ~ ! [Y3: multiset_a] :
~ ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ Y3 ) @ ( set_Pr5550698219922361532iset_a @ ( zip_mu5258324667929623607iset_a @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip1
thf(fact_580_in__set__impl__in__set__zip1,axiom,
! [Xs: list_nat_nat,Ys: list_nat,X3: nat > nat] :
( ( ( size_s8208510060688613859at_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( member_nat_nat @ X3 @ ( set_nat_nat2 @ Xs ) )
=> ~ ! [Y3: nat] :
~ ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X3 @ Y3 ) @ ( set_Pr1941165898809491589at_nat @ ( zip_nat_nat_nat @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip1
thf(fact_581_in__set__impl__in__set__zip1,axiom,
! [Xs: list_P1396940483166286381od_a_a,Ys: list_a,X3: product_prod_a_a] :
( ( ( size_s3885678630836030617od_a_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Xs ) )
=> ~ ! [Y3: a] :
~ ( member2829916517802586983_a_a_a @ ( produc5692536848587512110_a_a_a @ X3 @ Y3 ) @ ( set_Pr6767713769399854547_a_a_a @ ( zip_Pr9109025954503487566_a_a_a @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip1
thf(fact_582_in__set__impl__in__set__zip1,axiom,
! [Xs: list_P6988024003011616557iset_a,Ys: list_a,X3: produc6518373309651786023iset_a] :
( ( ( size_s7238940367606365593iset_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( member5199237121806060112iset_a @ X3 @ ( set_Pr5550698219922361532iset_a @ Xs ) )
=> ~ ! [Y3: a] :
~ ( member2419060323508899175et_a_a @ ( produc5181291240524633390et_a_a @ X3 @ Y3 ) @ ( set_Pr5466182931555501011et_a_a @ ( zip_Pr3459117997753158734et_a_a @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip1
thf(fact_583_in__set__impl__in__set__zip1,axiom,
! [Xs: list_a,Ys: list_a,X3: a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ~ ! [Y3: a] :
~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ ( set_Product_prod_a_a2 @ ( zip_a_a @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip1
thf(fact_584_in__set__impl__in__set__zip2,axiom,
! [Xs: list_multiset_a,Ys: list_multiset_a,Y: multiset_a] :
( ( ( size_s3003218658903709366iset_a @ Xs )
= ( size_s3003218658903709366iset_a @ Ys ) )
=> ( ( member_multiset_a @ Y @ ( set_multiset_a2 @ Ys ) )
=> ~ ! [X: multiset_a] :
~ ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ Y ) @ ( set_Pr5550698219922361532iset_a @ ( zip_mu5258324667929623607iset_a @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_585_in__set__impl__in__set__zip2,axiom,
! [Xs: list_nat_nat,Ys: list_nat,Y: nat] :
( ( ( size_s8208510060688613859at_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( member_nat @ Y @ ( set_nat2 @ Ys ) )
=> ~ ! [X: nat > nat] :
~ ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X @ Y ) @ ( set_Pr1941165898809491589at_nat @ ( zip_nat_nat_nat @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_586_in__set__impl__in__set__zip2,axiom,
! [Xs: list_a,Ys: list_P1396940483166286381od_a_a,Y: product_prod_a_a] :
( ( ( size_size_list_a @ Xs )
= ( size_s3885678630836030617od_a_a @ Ys ) )
=> ( ( member1426531477525435216od_a_a @ Y @ ( set_Product_prod_a_a2 @ Ys ) )
=> ~ ! [X: a] :
~ ( member3071122053849602553od_a_a @ ( produc431845341423274048od_a_a @ X @ Y ) @ ( set_Pr7008919305446870117od_a_a @ ( zip_a_3848334447339249504od_a_a @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_587_in__set__impl__in__set__zip2,axiom,
! [Xs: list_a,Ys: list_P6988024003011616557iset_a,Y: produc6518373309651786023iset_a] :
( ( ( size_size_list_a @ Xs )
= ( size_s7238940367606365593iset_a @ Ys ) )
=> ( ( member5199237121806060112iset_a @ Y @ ( set_Pr5550698219922361532iset_a @ Ys ) )
=> ~ ! [X: a] :
~ ( member4303746980753173241iset_a @ ( produc1768535192049815360iset_a @ X @ Y ) @ ( set_Pr7350869588799775077iset_a @ ( zip_a_46361949278340704iset_a @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_588_in__set__impl__in__set__zip2,axiom,
! [Xs: list_a,Ys: list_a,Y: a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( member_a @ Y @ ( set_a2 @ Ys ) )
=> ~ ! [X: a] :
~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( set_Product_prod_a_a2 @ ( zip_a_a @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_589_verit__comp__simplify1_I1_J,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_590_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C5: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_591_Euclid__induct,axiom,
! [P2: nat > nat > $o,A2: nat,B: nat] :
( ! [A: nat,B3: nat] :
( ( P2 @ A @ B3 )
= ( P2 @ B3 @ A ) )
=> ( ! [A: nat] : ( P2 @ A @ zero_zero_nat )
=> ( ! [A: nat,B3: nat] :
( ( P2 @ A @ B3 )
=> ( P2 @ A @ ( plus_plus_nat @ A @ B3 ) ) )
=> ( P2 @ A2 @ B ) ) ) ) ).
% Euclid_induct
thf(fact_592_list__order__extension_Oall__ns__imp__ns,axiom,
! [S_list: set_Pr79727621955416071iset_a > set_Pr79727621955416071iset_a > set_Pr2598276052810994439iset_a,Ns_list: set_Pr79727621955416071iset_a > set_Pr79727621955416071iset_a > set_Pr2598276052810994439iset_a,As: list_multiset_a,Bs: list_multiset_a,NS: set_Pr79727621955416071iset_a,S2: set_Pr79727621955416071iset_a] :
( ( list_l1107371739660948112iset_a @ S_list @ Ns_list )
=> ( ( ( size_s3003218658903709366iset_a @ As )
= ( size_s3003218658903709366iset_a @ Bs ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3003218658903709366iset_a @ Bs ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( nth_multiset_a @ As @ I3 ) @ ( nth_multiset_a @ Bs @ I3 ) ) @ NS ) )
=> ( member1640862112313989456iset_a @ ( produc1434922032272282391iset_a @ As @ Bs ) @ ( Ns_list @ S2 @ NS ) ) ) ) ) ).
% list_order_extension.all_ns_imp_ns
thf(fact_593_list__order__extension_Oall__ns__imp__ns,axiom,
! [S_list: set_Product_prod_a_a > set_Product_prod_a_a > set_Pr4048851178543822343list_a,Ns_list: set_Product_prod_a_a > set_Product_prod_a_a > set_Pr4048851178543822343list_a,As: list_a,Bs: list_a,NS: set_Product_prod_a_a,S2: set_Product_prod_a_a] :
( ( list_l5019108106899435338sion_a @ S_list @ Ns_list )
=> ( ( ( size_size_list_a @ As )
= ( size_size_list_a @ Bs ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Bs ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ As @ I3 ) @ ( nth_a @ Bs @ I3 ) ) @ NS ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ As @ Bs ) @ ( Ns_list @ S2 @ NS ) ) ) ) ) ).
% list_order_extension.all_ns_imp_ns
thf(fact_594_in__replicate__mset,axiom,
! [X3: product_prod_a_a,N3: nat,Y: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ ( replic5559486381873801085od_a_a @ N3 @ Y ) ) )
= ( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( X3 = Y ) ) ) ).
% in_replicate_mset
thf(fact_595_in__replicate__mset,axiom,
! [X3: produc6518373309651786023iset_a,N3: nat,Y: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ ( replic6672042396387243389iset_a @ N3 @ Y ) ) )
= ( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( X3 = Y ) ) ) ).
% in_replicate_mset
thf(fact_596_in__replicate__mset,axiom,
! [X3: a,N3: nat,Y: a] :
( ( member_a @ X3 @ ( set_mset_a @ ( replicate_mset_a @ N3 @ Y ) ) )
= ( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( X3 = Y ) ) ) ).
% in_replicate_mset
thf(fact_597_subset__mset_Osum__list__eq__0__iff,axiom,
! [Ns: list_multiset_a] :
( ( ( groups778166972783649551iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ Ns )
= zero_zero_multiset_a )
= ( ! [X5: multiset_a] :
( ( member_multiset_a @ X5 @ ( set_multiset_a2 @ Ns ) )
=> ( X5 = zero_zero_multiset_a ) ) ) ) ).
% subset_mset.sum_list_eq_0_iff
thf(fact_598_ns__mul__ext__Id__eq,axiom,
! [R: set_Product_prod_a_a] :
( ( multis4023584895248362547_ext_a @ id_a @ R )
= ( sup_su1925787886501348827iset_a @ ( multiset_s_mul_ext_a @ id_a @ R ) @ id_multiset_a ) ) ).
% ns_mul_ext_Id_eq
thf(fact_599_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_600_mult2__alt__ns__conv,axiom,
! [Ns: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( multis1686031732355738386_alt_a @ $true @ Ns @ S )
= ( sup_su1925787886501348827iset_a @ ( multis1686031732355738386_alt_a @ $false @ Ns @ S ) @ ( multis2373687813370458242ltpw_a @ Ns ) ) ) ).
% mult2_alt_ns_conv
thf(fact_601_set__union,axiom,
! [Xs: list_a,Ys: list_a] :
( ( set_a2 @ ( union_a @ Xs @ Ys ) )
= ( sup_sup_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys ) ) ) ).
% set_union
thf(fact_602_less__supI1,axiom,
! [X3: nat,A2: nat,B: nat] :
( ( ord_less_nat @ X3 @ A2 )
=> ( ord_less_nat @ X3 @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% less_supI1
thf(fact_603_less__supI2,axiom,
! [X3: nat,B: nat,A2: nat] :
( ( ord_less_nat @ X3 @ B )
=> ( ord_less_nat @ X3 @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% less_supI2
thf(fact_604_sup_Oabsorb3,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( sup_sup_nat @ A2 @ B )
= A2 ) ) ).
% sup.absorb3
thf(fact_605_sup_Oabsorb4,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( sup_sup_nat @ A2 @ B )
= B ) ) ).
% sup.absorb4
thf(fact_606_sup_Ostrict__boundedE,axiom,
! [B: nat,C2: nat,A2: nat] :
( ( ord_less_nat @ ( sup_sup_nat @ B @ C2 ) @ A2 )
=> ~ ( ( ord_less_nat @ B @ A2 )
=> ~ ( ord_less_nat @ C2 @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_607_sup_Ostrict__coboundedI2,axiom,
! [C2: nat,B: nat,A2: nat] :
( ( ord_less_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_608_sup_Ostrict__coboundedI1,axiom,
! [C2: nat,A2: nat,B: nat] :
( ( ord_less_nat @ C2 @ A2 )
=> ( ord_less_nat @ C2 @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_609_sup_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B6: nat,A5: nat] :
( ( A5
= ( sup_sup_nat @ A5 @ B6 ) )
& ( A5 != B6 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_610_partition__set,axiom,
! [P2: a > $o,Xs: list_a,Yes: list_a,No: list_a] :
( ( ( partition_a @ P2 @ Xs )
= ( produc6837034575241423639list_a @ Yes @ No ) )
=> ( ( sup_sup_set_a @ ( set_a2 @ Yes ) @ ( set_a2 @ No ) )
= ( set_a2 @ Xs ) ) ) ).
% partition_set
thf(fact_611_subset__mset_Osum__list__nonneg,axiom,
! [Xs: list_multiset_a] :
( ! [X: multiset_a] :
( ( member_multiset_a @ X @ ( set_multiset_a2 @ Xs ) )
=> ( subseteq_mset_a @ zero_zero_multiset_a @ X ) )
=> ( subseteq_mset_a @ zero_zero_multiset_a @ ( groups778166972783649551iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ Xs ) ) ) ).
% subset_mset.sum_list_nonneg
thf(fact_612_subset__mset_Osum__list__nonpos,axiom,
! [Xs: list_multiset_a] :
( ! [X: multiset_a] :
( ( member_multiset_a @ X @ ( set_multiset_a2 @ Xs ) )
=> ( subseteq_mset_a @ X @ zero_zero_multiset_a ) )
=> ( subseteq_mset_a @ ( groups778166972783649551iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ Xs ) @ zero_zero_multiset_a ) ) ).
% subset_mset.sum_list_nonpos
thf(fact_613_subset__mset_Omember__le__sum__list,axiom,
! [X3: multiset_a,Xs: list_multiset_a] :
( ( member_multiset_a @ X3 @ ( set_multiset_a2 @ Xs ) )
=> ( subseteq_mset_a @ X3 @ ( groups778166972783649551iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ Xs ) ) ) ).
% subset_mset.member_le_sum_list
thf(fact_614_subset__mset_Osum__list__nonneg__eq__0__iff,axiom,
! [Xs: list_multiset_a] :
( ! [X: multiset_a] :
( ( member_multiset_a @ X @ ( set_multiset_a2 @ Xs ) )
=> ( subseteq_mset_a @ zero_zero_multiset_a @ X ) )
=> ( ( ( groups778166972783649551iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ Xs )
= zero_zero_multiset_a )
= ( ! [X5: multiset_a] :
( ( member_multiset_a @ X5 @ ( set_multiset_a2 @ Xs ) )
=> ( X5 = zero_zero_multiset_a ) ) ) ) ) ).
% subset_mset.sum_list_nonneg_eq_0_iff
thf(fact_615_s__mul__ext__singleton,axiom,
! [A2: a,B: a,S: set_Product_prod_a_a,Ns: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B ) @ S )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( add_mset_a @ A2 @ zero_zero_multiset_a ) @ ( add_mset_a @ B @ zero_zero_multiset_a ) ) @ ( multiset_s_mul_ext_a @ Ns @ S ) ) ) ).
% s_mul_ext_singleton
thf(fact_616_s__mul__ext__singleton,axiom,
! [A2: multiset_a,B: multiset_a,S: set_Pr79727621955416071iset_a,Ns: set_Pr79727621955416071iset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A2 @ B ) @ S )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ ( add_mset_multiset_a @ A2 @ zero_z7799948378220188993iset_a ) @ ( add_mset_multiset_a @ B @ zero_z7799948378220188993iset_a ) ) @ ( multis7166037729339985883iset_a @ Ns @ S ) ) ) ).
% s_mul_ext_singleton
thf(fact_617_subset__mset_Oadd__le__cancel__left,axiom,
! [C2: multiset_a,A2: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ ( plus_plus_multiset_a @ C2 @ A2 ) @ ( plus_plus_multiset_a @ C2 @ B ) )
= ( subseteq_mset_a @ A2 @ B ) ) ).
% subset_mset.add_le_cancel_left
thf(fact_618_subset__mset_Oadd__le__cancel__right,axiom,
! [A2: multiset_a,C2: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ ( plus_plus_multiset_a @ A2 @ C2 ) @ ( plus_plus_multiset_a @ B @ C2 ) )
= ( subseteq_mset_a @ A2 @ B ) ) ).
% subset_mset.add_le_cancel_right
thf(fact_619_mset__subset__eq__mono__add__left__cancel,axiom,
! [C3: multiset_a,A4: multiset_a,B5: multiset_a] :
( ( subseteq_mset_a @ ( plus_plus_multiset_a @ C3 @ A4 ) @ ( plus_plus_multiset_a @ C3 @ B5 ) )
= ( subseteq_mset_a @ A4 @ B5 ) ) ).
% mset_subset_eq_mono_add_left_cancel
thf(fact_620_mset__subset__eq__mono__add__right__cancel,axiom,
! [A4: multiset_a,C3: multiset_a,B5: multiset_a] :
( ( subseteq_mset_a @ ( plus_plus_multiset_a @ A4 @ C3 ) @ ( plus_plus_multiset_a @ B5 @ C3 ) )
= ( subseteq_mset_a @ A4 @ B5 ) ) ).
% mset_subset_eq_mono_add_right_cancel
thf(fact_621_union__mset__add__mset__left,axiom,
! [A2: a,A4: multiset_a,B5: multiset_a] :
( ( plus_plus_multiset_a @ ( add_mset_a @ A2 @ A4 ) @ B5 )
= ( add_mset_a @ A2 @ ( plus_plus_multiset_a @ A4 @ B5 ) ) ) ).
% union_mset_add_mset_left
thf(fact_622_union__mset__add__mset__right,axiom,
! [A4: multiset_a,A2: a,B5: multiset_a] :
( ( plus_plus_multiset_a @ A4 @ ( add_mset_a @ A2 @ B5 ) )
= ( add_mset_a @ A2 @ ( plus_plus_multiset_a @ A4 @ B5 ) ) ) ).
% union_mset_add_mset_right
thf(fact_623_subset__mset_Ole__add__same__cancel2,axiom,
! [A2: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ A2 @ ( plus_plus_multiset_a @ B @ A2 ) )
= ( subseteq_mset_a @ zero_zero_multiset_a @ B ) ) ).
% subset_mset.le_add_same_cancel2
thf(fact_624_subset__mset_Ole__add__same__cancel1,axiom,
! [A2: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ A2 @ ( plus_plus_multiset_a @ A2 @ B ) )
= ( subseteq_mset_a @ zero_zero_multiset_a @ B ) ) ).
% subset_mset.le_add_same_cancel1
thf(fact_625_subset__mset_Oadd__le__same__cancel2,axiom,
! [A2: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ ( plus_plus_multiset_a @ A2 @ B ) @ B )
= ( subseteq_mset_a @ A2 @ zero_zero_multiset_a ) ) ).
% subset_mset.add_le_same_cancel2
thf(fact_626_subset__mset_Oadd__le__same__cancel1,axiom,
! [B: multiset_a,A2: multiset_a] :
( ( subseteq_mset_a @ ( plus_plus_multiset_a @ B @ A2 ) @ B )
= ( subseteq_mset_a @ A2 @ zero_zero_multiset_a ) ) ).
% subset_mset.add_le_same_cancel1
thf(fact_627_single__subset__iff,axiom,
! [A2: product_prod_a_a,M: multis501812127501805293od_a_a] :
( ( subset4108007524207251342od_a_a @ ( add_ms8655138167283798533od_a_a @ A2 @ zero_z3024502163831676836od_a_a ) @ M )
= ( member1426531477525435216od_a_a @ A2 @ ( set_ms119794563918438244od_a_a @ M ) ) ) ).
% single_subset_iff
thf(fact_628_single__subset__iff,axiom,
! [A2: produc6518373309651786023iset_a,M: multis4781622204924204013iset_a] :
( ( subset6292090195676340366iset_a @ ( add_ms5348986034529679365iset_a @ A2 @ zero_z4591712717340762532iset_a ) @ M )
= ( member5199237121806060112iset_a @ A2 @ ( set_ms2379723421352415844iset_a @ M ) ) ) ).
% single_subset_iff
thf(fact_629_single__subset__iff,axiom,
! [A2: a,M: multiset_a] :
( ( subseteq_mset_a @ ( add_mset_a @ A2 @ zero_zero_multiset_a ) @ M )
= ( member_a @ A2 @ ( set_mset_a @ M ) ) ) ).
% single_subset_iff
thf(fact_630_ns__mul__ext__singleton,axiom,
! [A2: multiset_a,B: multiset_a,Ns: set_Pr79727621955416071iset_a,S: set_Pr79727621955416071iset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A2 @ B ) @ Ns )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ ( add_mset_multiset_a @ A2 @ zero_z7799948378220188993iset_a ) @ ( add_mset_multiset_a @ B @ zero_z7799948378220188993iset_a ) ) @ ( multis8112516946959054073iset_a @ Ns @ S ) ) ) ).
% ns_mul_ext_singleton
thf(fact_631_ns__mul__ext__singleton,axiom,
! [A2: a,B: a,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B ) @ Ns )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( add_mset_a @ A2 @ zero_zero_multiset_a ) @ ( add_mset_a @ B @ zero_zero_multiset_a ) ) @ ( multis4023584895248362547_ext_a @ Ns @ S ) ) ) ).
% ns_mul_ext_singleton
thf(fact_632_mset__subset__eq__single,axiom,
! [A2: product_prod_a_a,B5: multis501812127501805293od_a_a] :
( ( member1426531477525435216od_a_a @ A2 @ ( set_ms119794563918438244od_a_a @ B5 ) )
=> ( subset4108007524207251342od_a_a @ ( add_ms8655138167283798533od_a_a @ A2 @ zero_z3024502163831676836od_a_a ) @ B5 ) ) ).
% mset_subset_eq_single
thf(fact_633_mset__subset__eq__single,axiom,
! [A2: produc6518373309651786023iset_a,B5: multis4781622204924204013iset_a] :
( ( member5199237121806060112iset_a @ A2 @ ( set_ms2379723421352415844iset_a @ B5 ) )
=> ( subset6292090195676340366iset_a @ ( add_ms5348986034529679365iset_a @ A2 @ zero_z4591712717340762532iset_a ) @ B5 ) ) ).
% mset_subset_eq_single
thf(fact_634_mset__subset__eq__single,axiom,
! [A2: a,B5: multiset_a] :
( ( member_a @ A2 @ ( set_mset_a @ B5 ) )
=> ( subseteq_mset_a @ ( add_mset_a @ A2 @ zero_zero_multiset_a ) @ B5 ) ) ).
% mset_subset_eq_single
thf(fact_635_multi__subset__induct,axiom,
! [F2: multis501812127501805293od_a_a,A4: multis501812127501805293od_a_a,P2: multis501812127501805293od_a_a > $o] :
( ( subset4108007524207251342od_a_a @ F2 @ A4 )
=> ( ( P2 @ zero_z3024502163831676836od_a_a )
=> ( ! [A: product_prod_a_a,F3: multis501812127501805293od_a_a] :
( ( member1426531477525435216od_a_a @ A @ ( set_ms119794563918438244od_a_a @ A4 ) )
=> ( ( P2 @ F3 )
=> ( P2 @ ( add_ms8655138167283798533od_a_a @ A @ F3 ) ) ) )
=> ( P2 @ F2 ) ) ) ) ).
% multi_subset_induct
thf(fact_636_multi__subset__induct,axiom,
! [F2: multis4781622204924204013iset_a,A4: multis4781622204924204013iset_a,P2: multis4781622204924204013iset_a > $o] :
( ( subset6292090195676340366iset_a @ F2 @ A4 )
=> ( ( P2 @ zero_z4591712717340762532iset_a )
=> ( ! [A: produc6518373309651786023iset_a,F3: multis4781622204924204013iset_a] :
( ( member5199237121806060112iset_a @ A @ ( set_ms2379723421352415844iset_a @ A4 ) )
=> ( ( P2 @ F3 )
=> ( P2 @ ( add_ms5348986034529679365iset_a @ A @ F3 ) ) ) )
=> ( P2 @ F2 ) ) ) ) ).
% multi_subset_induct
thf(fact_637_multi__subset__induct,axiom,
! [F2: multiset_a,A4: multiset_a,P2: multiset_a > $o] :
( ( subseteq_mset_a @ F2 @ A4 )
=> ( ( P2 @ zero_zero_multiset_a )
=> ( ! [A: a,F3: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ A4 ) )
=> ( ( P2 @ F3 )
=> ( P2 @ ( add_mset_a @ A @ F3 ) ) ) )
=> ( P2 @ F2 ) ) ) ) ).
% multi_subset_induct
thf(fact_638_mset__add,axiom,
! [A2: product_prod_a_a,A4: multis501812127501805293od_a_a] :
( ( member1426531477525435216od_a_a @ A2 @ ( set_ms119794563918438244od_a_a @ A4 ) )
=> ~ ! [B7: multis501812127501805293od_a_a] :
( A4
!= ( add_ms8655138167283798533od_a_a @ A2 @ B7 ) ) ) ).
% mset_add
thf(fact_639_mset__add,axiom,
! [A2: produc6518373309651786023iset_a,A4: multis4781622204924204013iset_a] :
( ( member5199237121806060112iset_a @ A2 @ ( set_ms2379723421352415844iset_a @ A4 ) )
=> ~ ! [B7: multis4781622204924204013iset_a] :
( A4
!= ( add_ms5348986034529679365iset_a @ A2 @ B7 ) ) ) ).
% mset_add
thf(fact_640_mset__add,axiom,
! [A2: a,A4: multiset_a] :
( ( member_a @ A2 @ ( set_mset_a @ A4 ) )
=> ~ ! [B7: multiset_a] :
( A4
!= ( add_mset_a @ A2 @ B7 ) ) ) ).
% mset_add
thf(fact_641_multi__member__split,axiom,
! [X3: product_prod_a_a,M: multis501812127501805293od_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ M ) )
=> ? [A8: multis501812127501805293od_a_a] :
( M
= ( add_ms8655138167283798533od_a_a @ X3 @ A8 ) ) ) ).
% multi_member_split
thf(fact_642_multi__member__split,axiom,
! [X3: produc6518373309651786023iset_a,M: multis4781622204924204013iset_a] :
( ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ M ) )
=> ? [A8: multis4781622204924204013iset_a] :
( M
= ( add_ms5348986034529679365iset_a @ X3 @ A8 ) ) ) ).
% multi_member_split
thf(fact_643_multi__member__split,axiom,
! [X3: a,M: multiset_a] :
( ( member_a @ X3 @ ( set_mset_a @ M ) )
=> ? [A8: multiset_a] :
( M
= ( add_mset_a @ X3 @ A8 ) ) ) ).
% multi_member_split
thf(fact_644_insert__noteq__member,axiom,
! [B: product_prod_a_a,B5: multis501812127501805293od_a_a,C2: product_prod_a_a,C3: multis501812127501805293od_a_a] :
( ( ( add_ms8655138167283798533od_a_a @ B @ B5 )
= ( add_ms8655138167283798533od_a_a @ C2 @ C3 ) )
=> ( ( B != C2 )
=> ( member1426531477525435216od_a_a @ C2 @ ( set_ms119794563918438244od_a_a @ B5 ) ) ) ) ).
% insert_noteq_member
thf(fact_645_insert__noteq__member,axiom,
! [B: produc6518373309651786023iset_a,B5: multis4781622204924204013iset_a,C2: produc6518373309651786023iset_a,C3: multis4781622204924204013iset_a] :
( ( ( add_ms5348986034529679365iset_a @ B @ B5 )
= ( add_ms5348986034529679365iset_a @ C2 @ C3 ) )
=> ( ( B != C2 )
=> ( member5199237121806060112iset_a @ C2 @ ( set_ms2379723421352415844iset_a @ B5 ) ) ) ) ).
% insert_noteq_member
thf(fact_646_insert__noteq__member,axiom,
! [B: a,B5: multiset_a,C2: a,C3: multiset_a] :
( ( ( add_mset_a @ B @ B5 )
= ( add_mset_a @ C2 @ C3 ) )
=> ( ( B != C2 )
=> ( member_a @ C2 @ ( set_mset_a @ B5 ) ) ) ) ).
% insert_noteq_member
thf(fact_647_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_648_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_649_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_650_subset__mset_Oadd__mono,axiom,
! [A2: multiset_a,B: multiset_a,C2: multiset_a,D2: multiset_a] :
( ( subseteq_mset_a @ A2 @ B )
=> ( ( subseteq_mset_a @ C2 @ D2 )
=> ( subseteq_mset_a @ ( plus_plus_multiset_a @ A2 @ C2 ) @ ( plus_plus_multiset_a @ B @ D2 ) ) ) ) ).
% subset_mset.add_mono
thf(fact_651_subset__mset_Oless__eqE,axiom,
! [A2: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ A2 @ B )
=> ~ ! [C: multiset_a] :
( B
!= ( plus_plus_multiset_a @ A2 @ C ) ) ) ).
% subset_mset.less_eqE
thf(fact_652_subset__mset_Ole__iff__add,axiom,
( subseteq_mset_a
= ( ^ [A5: multiset_a,B6: multiset_a] :
? [C5: multiset_a] :
( B6
= ( plus_plus_multiset_a @ A5 @ C5 ) ) ) ) ).
% subset_mset.le_iff_add
thf(fact_653_subset__mset_Oadd__left__mono,axiom,
! [A2: multiset_a,B: multiset_a,C2: multiset_a] :
( ( subseteq_mset_a @ A2 @ B )
=> ( subseteq_mset_a @ ( plus_plus_multiset_a @ C2 @ A2 ) @ ( plus_plus_multiset_a @ C2 @ B ) ) ) ).
% subset_mset.add_left_mono
thf(fact_654_subset__mset_Oadd__right__mono,axiom,
! [A2: multiset_a,B: multiset_a,C2: multiset_a] :
( ( subseteq_mset_a @ A2 @ B )
=> ( subseteq_mset_a @ ( plus_plus_multiset_a @ A2 @ C2 ) @ ( plus_plus_multiset_a @ B @ C2 ) ) ) ).
% subset_mset.add_right_mono
thf(fact_655_subset__mset_Oadd__le__imp__le__left,axiom,
! [C2: multiset_a,A2: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ ( plus_plus_multiset_a @ C2 @ A2 ) @ ( plus_plus_multiset_a @ C2 @ B ) )
=> ( subseteq_mset_a @ A2 @ B ) ) ).
% subset_mset.add_le_imp_le_left
thf(fact_656_subset__mset_Oadd__le__imp__le__right,axiom,
! [A2: multiset_a,C2: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ ( plus_plus_multiset_a @ A2 @ C2 ) @ ( plus_plus_multiset_a @ B @ C2 ) )
=> ( subseteq_mset_a @ A2 @ B ) ) ).
% subset_mset.add_le_imp_le_right
thf(fact_657_mset__subset__eq__add__left,axiom,
! [A4: multiset_a,B5: multiset_a] : ( subseteq_mset_a @ A4 @ ( plus_plus_multiset_a @ A4 @ B5 ) ) ).
% mset_subset_eq_add_left
thf(fact_658_mset__subset__eq__mono__add,axiom,
! [A4: multiset_a,B5: multiset_a,C3: multiset_a,D: multiset_a] :
( ( subseteq_mset_a @ A4 @ B5 )
=> ( ( subseteq_mset_a @ C3 @ D )
=> ( subseteq_mset_a @ ( plus_plus_multiset_a @ A4 @ C3 ) @ ( plus_plus_multiset_a @ B5 @ D ) ) ) ) ).
% mset_subset_eq_mono_add
thf(fact_659_mset__subset__eq__add__right,axiom,
! [B5: multiset_a,A4: multiset_a] : ( subseteq_mset_a @ B5 @ ( plus_plus_multiset_a @ A4 @ B5 ) ) ).
% mset_subset_eq_add_right
thf(fact_660_mset__subset__eq__exists__conv,axiom,
( subseteq_mset_a
= ( ^ [A7: multiset_a,B8: multiset_a] :
? [C6: multiset_a] :
( B8
= ( plus_plus_multiset_a @ A7 @ C6 ) ) ) ) ).
% mset_subset_eq_exists_conv
thf(fact_661_mset__subset__eqD,axiom,
! [A4: multis501812127501805293od_a_a,B5: multis501812127501805293od_a_a,X3: product_prod_a_a] :
( ( subset4108007524207251342od_a_a @ A4 @ B5 )
=> ( ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ A4 ) )
=> ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ B5 ) ) ) ) ).
% mset_subset_eqD
thf(fact_662_mset__subset__eqD,axiom,
! [A4: multis4781622204924204013iset_a,B5: multis4781622204924204013iset_a,X3: produc6518373309651786023iset_a] :
( ( subset6292090195676340366iset_a @ A4 @ B5 )
=> ( ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ A4 ) )
=> ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ B5 ) ) ) ) ).
% mset_subset_eqD
thf(fact_663_mset__subset__eqD,axiom,
! [A4: multiset_a,B5: multiset_a,X3: a] :
( ( subseteq_mset_a @ A4 @ B5 )
=> ( ( member_a @ X3 @ ( set_mset_a @ A4 ) )
=> ( member_a @ X3 @ ( set_mset_a @ B5 ) ) ) ) ).
% mset_subset_eqD
thf(fact_664_single__is__union,axiom,
! [A2: a,M: multiset_a,N: multiset_a] :
( ( ( add_mset_a @ A2 @ zero_zero_multiset_a )
= ( plus_plus_multiset_a @ M @ N ) )
= ( ( ( ( add_mset_a @ A2 @ zero_zero_multiset_a )
= M )
& ( N = zero_zero_multiset_a ) )
| ( ( M = zero_zero_multiset_a )
& ( ( add_mset_a @ A2 @ zero_zero_multiset_a )
= N ) ) ) ) ).
% single_is_union
thf(fact_665_union__is__single,axiom,
! [M: multiset_a,N: multiset_a,A2: a] :
( ( ( plus_plus_multiset_a @ M @ N )
= ( add_mset_a @ A2 @ zero_zero_multiset_a ) )
= ( ( ( M
= ( add_mset_a @ A2 @ zero_zero_multiset_a ) )
& ( N = zero_zero_multiset_a ) )
| ( ( M = zero_zero_multiset_a )
& ( N
= ( add_mset_a @ A2 @ zero_zero_multiset_a ) ) ) ) ) ).
% union_is_single
thf(fact_666_add__mset__add__single,axiom,
( add_mset_a
= ( ^ [A5: a,A7: multiset_a] : ( plus_plus_multiset_a @ A7 @ ( add_mset_a @ A5 @ zero_zero_multiset_a ) ) ) ) ).
% add_mset_add_single
thf(fact_667_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_668_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_669_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_670_subset__mset_Oadd__decreasing,axiom,
! [A2: multiset_a,C2: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ A2 @ zero_zero_multiset_a )
=> ( ( subseteq_mset_a @ C2 @ B )
=> ( subseteq_mset_a @ ( plus_plus_multiset_a @ A2 @ C2 ) @ B ) ) ) ).
% subset_mset.add_decreasing
thf(fact_671_subset__mset_Oadd__increasing,axiom,
! [A2: multiset_a,B: multiset_a,C2: multiset_a] :
( ( subseteq_mset_a @ zero_zero_multiset_a @ A2 )
=> ( ( subseteq_mset_a @ B @ C2 )
=> ( subseteq_mset_a @ B @ ( plus_plus_multiset_a @ A2 @ C2 ) ) ) ) ).
% subset_mset.add_increasing
thf(fact_672_subset__mset_Oadd__decreasing2,axiom,
! [C2: multiset_a,A2: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ C2 @ zero_zero_multiset_a )
=> ( ( subseteq_mset_a @ A2 @ B )
=> ( subseteq_mset_a @ ( plus_plus_multiset_a @ A2 @ C2 ) @ B ) ) ) ).
% subset_mset.add_decreasing2
thf(fact_673_subset__mset_Oadd__increasing2,axiom,
! [C2: multiset_a,B: multiset_a,A2: multiset_a] :
( ( subseteq_mset_a @ zero_zero_multiset_a @ C2 )
=> ( ( subseteq_mset_a @ B @ A2 )
=> ( subseteq_mset_a @ B @ ( plus_plus_multiset_a @ A2 @ C2 ) ) ) ) ).
% subset_mset.add_increasing2
thf(fact_674_subset__mset_Oadd__nonneg__nonneg,axiom,
! [A2: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ zero_zero_multiset_a @ A2 )
=> ( ( subseteq_mset_a @ zero_zero_multiset_a @ B )
=> ( subseteq_mset_a @ zero_zero_multiset_a @ ( plus_plus_multiset_a @ A2 @ B ) ) ) ) ).
% subset_mset.add_nonneg_nonneg
thf(fact_675_subset__mset_Oadd__nonpos__nonpos,axiom,
! [A2: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ A2 @ zero_zero_multiset_a )
=> ( ( subseteq_mset_a @ B @ zero_zero_multiset_a )
=> ( subseteq_mset_a @ ( plus_plus_multiset_a @ A2 @ B ) @ zero_zero_multiset_a ) ) ) ).
% subset_mset.add_nonpos_nonpos
thf(fact_676_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_677_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_678_union__fold__mset__add__mset,axiom,
( plus_plus_multiset_a
= ( fold_m7320414754419674833iset_a @ add_mset_a ) ) ).
% union_fold_mset_add_mset
thf(fact_679_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_680_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_681_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_682_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_683_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_684_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_685_multpw__split1R,axiom,
! [X3: multiset_a,X4: multiset_multiset_a,Y4: multiset_multiset_a,Ns: set_Pr79727621955416071iset_a] :
( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ ( add_mset_multiset_a @ X3 @ X4 ) @ Y4 ) @ ( multis451435756853798856iset_a @ Ns ) )
=> ~ ! [Z4: multiset_a,Z3: multiset_multiset_a] :
( ( Y4
= ( add_mset_multiset_a @ Z4 @ Z3 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ Z4 ) @ Ns )
=> ~ ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X4 @ Z3 ) @ ( multis451435756853798856iset_a @ Ns ) ) ) ) ) ).
% multpw_split1R
thf(fact_686_multpw__split1R,axiom,
! [X3: a,X4: multiset_a,Y4: multiset_a,Ns: set_Product_prod_a_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( add_mset_a @ X3 @ X4 ) @ Y4 ) @ ( multis2373687813370458242ltpw_a @ Ns ) )
=> ~ ! [Z4: a,Z3: multiset_a] :
( ( Y4
= ( add_mset_a @ Z4 @ Z3 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Z4 ) @ Ns )
=> ~ ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X4 @ Z3 ) @ ( multis2373687813370458242ltpw_a @ Ns ) ) ) ) ) ).
% multpw_split1R
thf(fact_687_multpw__split1L,axiom,
! [X4: multiset_multiset_a,Y: multiset_a,Y4: multiset_multiset_a,Ns: set_Pr79727621955416071iset_a] :
( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X4 @ ( add_mset_multiset_a @ Y @ Y4 ) ) @ ( multis451435756853798856iset_a @ Ns ) )
=> ~ ! [Z4: multiset_a,Z3: multiset_multiset_a] :
( ( X4
= ( add_mset_multiset_a @ Z4 @ Z3 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Z4 @ Y ) @ Ns )
=> ~ ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ Z3 @ Y4 ) @ ( multis451435756853798856iset_a @ Ns ) ) ) ) ) ).
% multpw_split1L
thf(fact_688_multpw__split1L,axiom,
! [X4: multiset_a,Y: a,Y4: multiset_a,Ns: set_Product_prod_a_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X4 @ ( add_mset_a @ Y @ Y4 ) ) @ ( multis2373687813370458242ltpw_a @ Ns ) )
=> ~ ! [Z4: a,Z3: multiset_a] :
( ( X4
= ( add_mset_a @ Z4 @ Z3 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Z4 @ Y ) @ Ns )
=> ~ ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Z3 @ Y4 ) @ ( multis2373687813370458242ltpw_a @ Ns ) ) ) ) ) ).
% multpw_split1L
thf(fact_689_multpw_Oadd,axiom,
! [X3: multiset_a,Y: multiset_a,Ns: set_Pr79727621955416071iset_a,X4: multiset_multiset_a,Y4: multiset_multiset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ Y ) @ Ns )
=> ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X4 @ Y4 ) @ ( multis451435756853798856iset_a @ Ns ) )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ ( add_mset_multiset_a @ X3 @ X4 ) @ ( add_mset_multiset_a @ Y @ Y4 ) ) @ ( multis451435756853798856iset_a @ Ns ) ) ) ) ).
% multpw.add
thf(fact_690_multpw_Oadd,axiom,
! [X3: a,Y: a,Ns: set_Product_prod_a_a,X4: multiset_a,Y4: multiset_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ Ns )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X4 @ Y4 ) @ ( multis2373687813370458242ltpw_a @ Ns ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( add_mset_a @ X3 @ X4 ) @ ( add_mset_a @ Y @ Y4 ) ) @ ( multis2373687813370458242ltpw_a @ Ns ) ) ) ) ).
% multpw.add
thf(fact_691_subseteq__mset__imp__ns__mul__ext__Id,axiom,
! [A4: multiset_a,B5: multiset_a,R: set_Product_prod_a_a] :
( ( subseteq_mset_a @ A4 @ B5 )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ B5 @ A4 ) @ ( multis4023584895248362547_ext_a @ id_a @ R ) ) ) ).
% subseteq_mset_imp_ns_mul_ext_Id
thf(fact_692_partition__P,axiom,
! [P2: a > $o,Xs: list_a,Yes: list_a,No: list_a] :
( ( ( partition_a @ P2 @ Xs )
= ( produc6837034575241423639list_a @ Yes @ No ) )
=> ( ! [X6: a] :
( ( member_a @ X6 @ ( set_a2 @ Yes ) )
=> ( P2 @ X6 ) )
& ! [X6: a] :
( ( member_a @ X6 @ ( set_a2 @ No ) )
=> ~ ( P2 @ X6 ) ) ) ) ).
% partition_P
thf(fact_693_multpw__single,axiom,
! [X3: multiset_a,Y: multiset_a,Ns: set_Pr79727621955416071iset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ Y ) @ Ns )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ ( add_mset_multiset_a @ X3 @ zero_z7799948378220188993iset_a ) @ ( add_mset_multiset_a @ Y @ zero_z7799948378220188993iset_a ) ) @ ( multis451435756853798856iset_a @ Ns ) ) ) ).
% multpw_single
thf(fact_694_multpw__single,axiom,
! [X3: a,Y: a,Ns: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ Ns )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( add_mset_a @ X3 @ zero_zero_multiset_a ) @ ( add_mset_a @ Y @ zero_zero_multiset_a ) ) @ ( multis2373687813370458242ltpw_a @ Ns ) ) ) ).
% multpw_single
thf(fact_695_multpw_Osimps,axiom,
! [A13: multiset_multiset_a,A24: multiset_multiset_a,Ns: set_Pr79727621955416071iset_a] :
( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ A13 @ A24 ) @ ( multis451435756853798856iset_a @ Ns ) )
= ( ( ( A13 = zero_z7799948378220188993iset_a )
& ( A24 = zero_z7799948378220188993iset_a ) )
| ? [X5: multiset_a,Y8: multiset_a,X7: multiset_multiset_a,Y9: multiset_multiset_a] :
( ( A13
= ( add_mset_multiset_a @ X5 @ X7 ) )
& ( A24
= ( add_mset_multiset_a @ Y8 @ Y9 ) )
& ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Y8 ) @ Ns )
& ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X7 @ Y9 ) @ ( multis451435756853798856iset_a @ Ns ) ) ) ) ) ).
% multpw.simps
thf(fact_696_multpw_Osimps,axiom,
! [A13: multiset_a,A24: multiset_a,Ns: set_Product_prod_a_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A13 @ A24 ) @ ( multis2373687813370458242ltpw_a @ Ns ) )
= ( ( ( A13 = zero_zero_multiset_a )
& ( A24 = zero_zero_multiset_a ) )
| ? [X5: a,Y8: a,X7: multiset_a,Y9: multiset_a] :
( ( A13
= ( add_mset_a @ X5 @ X7 ) )
& ( A24
= ( add_mset_a @ Y8 @ Y9 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y8 ) @ Ns )
& ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X7 @ Y9 ) @ ( multis2373687813370458242ltpw_a @ Ns ) ) ) ) ) ).
% multpw.simps
thf(fact_697_multpw_Ocases,axiom,
! [A13: multiset_multiset_a,A24: multiset_multiset_a,Ns: set_Pr79727621955416071iset_a] :
( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ A13 @ A24 ) @ ( multis451435756853798856iset_a @ Ns ) )
=> ( ( ( A13 = zero_z7799948378220188993iset_a )
=> ( A24 != zero_z7799948378220188993iset_a ) )
=> ~ ! [X: multiset_a,Y3: multiset_a,X8: multiset_multiset_a] :
( ( A13
= ( add_mset_multiset_a @ X @ X8 ) )
=> ! [Y7: multiset_multiset_a] :
( ( A24
= ( add_mset_multiset_a @ Y3 @ Y7 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ Y3 ) @ Ns )
=> ~ ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X8 @ Y7 ) @ ( multis451435756853798856iset_a @ Ns ) ) ) ) ) ) ) ).
% multpw.cases
thf(fact_698_multpw_Ocases,axiom,
! [A13: multiset_a,A24: multiset_a,Ns: set_Product_prod_a_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A13 @ A24 ) @ ( multis2373687813370458242ltpw_a @ Ns ) )
=> ( ( ( A13 = zero_zero_multiset_a )
=> ( A24 != zero_zero_multiset_a ) )
=> ~ ! [X: a,Y3: a,X8: multiset_a] :
( ( A13
= ( add_mset_a @ X @ X8 ) )
=> ! [Y7: multiset_a] :
( ( A24
= ( add_mset_a @ Y3 @ Y7 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ Ns )
=> ~ ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X8 @ Y7 ) @ ( multis2373687813370458242ltpw_a @ Ns ) ) ) ) ) ) ) ).
% multpw.cases
thf(fact_699_mult2__alt__s__single,axiom,
! [A2: a,B: a,S: set_Product_prod_a_a,Ns: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B ) @ S )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( add_mset_a @ A2 @ zero_zero_multiset_a ) @ ( add_mset_a @ B @ zero_zero_multiset_a ) ) @ ( multis1686031732355738386_alt_a @ $false @ Ns @ S ) ) ) ).
% mult2_alt_s_single
thf(fact_700_mult2__alt__s__single,axiom,
! [A2: multiset_a,B: multiset_a,S: set_Pr79727621955416071iset_a,Ns: set_Pr79727621955416071iset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A2 @ B ) @ S )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ ( add_mset_multiset_a @ A2 @ zero_z7799948378220188993iset_a ) @ ( add_mset_multiset_a @ B @ zero_z7799948378220188993iset_a ) ) @ ( multis2006310815304088152iset_a @ $false @ Ns @ S ) ) ) ).
% mult2_alt_s_single
thf(fact_701_ns__mul__ext__singleton2,axiom,
! [A2: multiset_a,B: multiset_a,S: set_Pr79727621955416071iset_a,Ns: set_Pr79727621955416071iset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A2 @ B ) @ S )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ ( add_mset_multiset_a @ A2 @ zero_z7799948378220188993iset_a ) @ ( add_mset_multiset_a @ B @ zero_z7799948378220188993iset_a ) ) @ ( multis8112516946959054073iset_a @ Ns @ S ) ) ) ).
% ns_mul_ext_singleton2
thf(fact_702_ns__mul__ext__singleton2,axiom,
! [A2: a,B: a,S: set_Product_prod_a_a,Ns: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B ) @ S )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( add_mset_a @ A2 @ zero_zero_multiset_a ) @ ( add_mset_a @ B @ zero_zero_multiset_a ) ) @ ( multis4023584895248362547_ext_a @ Ns @ S ) ) ) ).
% ns_mul_ext_singleton2
thf(fact_703_subset__mset_Oelem__le__sum__list,axiom,
! [K2: nat,Ns: list_multiset_a] :
( ( ord_less_nat @ K2 @ ( size_s3003218658903709366iset_a @ Ns ) )
=> ( subseteq_mset_a @ ( nth_multiset_a @ Ns @ K2 ) @ ( groups778166972783649551iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ Ns ) ) ) ).
% subset_mset.elem_le_sum_list
thf(fact_704_mult1E,axiom,
! [N: multis501812127501805293od_a_a,M: multis501812127501805293od_a_a,R2: set_Pr8600417178894128327od_a_a] :
( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ N @ M ) @ ( mult1_1845928942838466970od_a_a @ R2 ) )
=> ~ ! [A: product_prod_a_a,M0: multis501812127501805293od_a_a] :
( ( M
= ( add_ms8655138167283798533od_a_a @ A @ M0 ) )
=> ! [K4: multis501812127501805293od_a_a] :
( ( N
= ( plus_p6967097483979887908od_a_a @ M0 @ K4 ) )
=> ~ ! [B4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ B4 @ ( set_ms119794563918438244od_a_a @ K4 ) )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ B4 @ A ) @ R2 ) ) ) ) ) ).
% mult1E
thf(fact_705_mult1E,axiom,
! [N: multis4781622204924204013iset_a,M: multis4781622204924204013iset_a,R2: set_Pr3059988933816007879iset_a] :
( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ N @ M ) @ ( mult1_5927536977011277466iset_a @ R2 ) )
=> ~ ! [A: produc6518373309651786023iset_a,M0: multis4781622204924204013iset_a] :
( ( M
= ( add_ms5348986034529679365iset_a @ A @ M0 ) )
=> ! [K4: multis4781622204924204013iset_a] :
( ( N
= ( plus_p1053901119268525860iset_a @ M0 @ K4 ) )
=> ~ ! [B4: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ B4 @ ( set_ms2379723421352415844iset_a @ K4 ) )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ B4 @ A ) @ R2 ) ) ) ) ) ).
% mult1E
thf(fact_706_mult1E,axiom,
! [N: multiset_multiset_a,M: multiset_multiset_a,R2: set_Pr79727621955416071iset_a] :
( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ N @ M ) @ ( mult1_multiset_a @ R2 ) )
=> ~ ! [A: multiset_a,M0: multiset_multiset_a] :
( ( M
= ( add_mset_multiset_a @ A @ M0 ) )
=> ! [K4: multiset_multiset_a] :
( ( N
= ( plus_p6738641960240532161iset_a @ M0 @ K4 ) )
=> ~ ! [B4: multiset_a] :
( ( member_multiset_a @ B4 @ ( set_mset_multiset_a @ K4 ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ B4 @ A ) @ R2 ) ) ) ) ) ).
% mult1E
thf(fact_707_mult1E,axiom,
! [N: multiset_a,M: multiset_a,R2: set_Product_prod_a_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ N @ M ) @ ( mult1_a @ R2 ) )
=> ~ ! [A: a,M0: multiset_a] :
( ( M
= ( add_mset_a @ A @ M0 ) )
=> ! [K4: multiset_a] :
( ( N
= ( plus_plus_multiset_a @ M0 @ K4 ) )
=> ~ ! [B4: a] :
( ( member_a @ B4 @ ( set_mset_a @ K4 ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B4 @ A ) @ R2 ) ) ) ) ) ).
% mult1E
thf(fact_708_mult1I,axiom,
! [M: multis501812127501805293od_a_a,A2: product_prod_a_a,M02: multis501812127501805293od_a_a,N: multis501812127501805293od_a_a,K: multis501812127501805293od_a_a,R2: set_Pr8600417178894128327od_a_a] :
( ( M
= ( add_ms8655138167283798533od_a_a @ A2 @ M02 ) )
=> ( ( N
= ( plus_p6967097483979887908od_a_a @ M02 @ K ) )
=> ( ! [B3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ B3 @ ( set_ms119794563918438244od_a_a @ K ) )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ B3 @ A2 ) @ R2 ) )
=> ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ N @ M ) @ ( mult1_1845928942838466970od_a_a @ R2 ) ) ) ) ) ).
% mult1I
thf(fact_709_mult1I,axiom,
! [M: multis4781622204924204013iset_a,A2: produc6518373309651786023iset_a,M02: multis4781622204924204013iset_a,N: multis4781622204924204013iset_a,K: multis4781622204924204013iset_a,R2: set_Pr3059988933816007879iset_a] :
( ( M
= ( add_ms5348986034529679365iset_a @ A2 @ M02 ) )
=> ( ( N
= ( plus_p1053901119268525860iset_a @ M02 @ K ) )
=> ( ! [B3: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ B3 @ ( set_ms2379723421352415844iset_a @ K ) )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ B3 @ A2 ) @ R2 ) )
=> ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ N @ M ) @ ( mult1_5927536977011277466iset_a @ R2 ) ) ) ) ) ).
% mult1I
thf(fact_710_mult1I,axiom,
! [M: multiset_multiset_a,A2: multiset_a,M02: multiset_multiset_a,N: multiset_multiset_a,K: multiset_multiset_a,R2: set_Pr79727621955416071iset_a] :
( ( M
= ( add_mset_multiset_a @ A2 @ M02 ) )
=> ( ( N
= ( plus_p6738641960240532161iset_a @ M02 @ K ) )
=> ( ! [B3: multiset_a] :
( ( member_multiset_a @ B3 @ ( set_mset_multiset_a @ K ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ B3 @ A2 ) @ R2 ) )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ N @ M ) @ ( mult1_multiset_a @ R2 ) ) ) ) ) ).
% mult1I
thf(fact_711_mult1I,axiom,
! [M: multiset_a,A2: a,M02: multiset_a,N: multiset_a,K: multiset_a,R2: set_Product_prod_a_a] :
( ( M
= ( add_mset_a @ A2 @ M02 ) )
=> ( ( N
= ( plus_plus_multiset_a @ M02 @ K ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ ( set_mset_a @ K ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B3 @ A2 ) @ R2 ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ N @ M ) @ ( mult1_a @ R2 ) ) ) ) ) ).
% mult1I
thf(fact_712_less__add,axiom,
! [N: multis501812127501805293od_a_a,A2: product_prod_a_a,M02: multis501812127501805293od_a_a,R2: set_Pr8600417178894128327od_a_a] :
( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ N @ ( add_ms8655138167283798533od_a_a @ A2 @ M02 ) ) @ ( mult1_1845928942838466970od_a_a @ R2 ) )
=> ( ? [M5: multis501812127501805293od_a_a] :
( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ M5 @ M02 ) @ ( mult1_1845928942838466970od_a_a @ R2 ) )
& ( N
= ( add_ms8655138167283798533od_a_a @ A2 @ M5 ) ) )
| ? [K4: multis501812127501805293od_a_a] :
( ! [B4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ B4 @ ( set_ms119794563918438244od_a_a @ K4 ) )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ B4 @ A2 ) @ R2 ) )
& ( N
= ( plus_p6967097483979887908od_a_a @ M02 @ K4 ) ) ) ) ) ).
% less_add
thf(fact_713_less__add,axiom,
! [N: multis4781622204924204013iset_a,A2: produc6518373309651786023iset_a,M02: multis4781622204924204013iset_a,R2: set_Pr3059988933816007879iset_a] :
( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ N @ ( add_ms5348986034529679365iset_a @ A2 @ M02 ) ) @ ( mult1_5927536977011277466iset_a @ R2 ) )
=> ( ? [M5: multis4781622204924204013iset_a] :
( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ M5 @ M02 ) @ ( mult1_5927536977011277466iset_a @ R2 ) )
& ( N
= ( add_ms5348986034529679365iset_a @ A2 @ M5 ) ) )
| ? [K4: multis4781622204924204013iset_a] :
( ! [B4: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ B4 @ ( set_ms2379723421352415844iset_a @ K4 ) )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ B4 @ A2 ) @ R2 ) )
& ( N
= ( plus_p1053901119268525860iset_a @ M02 @ K4 ) ) ) ) ) ).
% less_add
thf(fact_714_less__add,axiom,
! [N: multiset_multiset_a,A2: multiset_a,M02: multiset_multiset_a,R2: set_Pr79727621955416071iset_a] :
( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ N @ ( add_mset_multiset_a @ A2 @ M02 ) ) @ ( mult1_multiset_a @ R2 ) )
=> ( ? [M5: multiset_multiset_a] :
( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ M5 @ M02 ) @ ( mult1_multiset_a @ R2 ) )
& ( N
= ( add_mset_multiset_a @ A2 @ M5 ) ) )
| ? [K4: multiset_multiset_a] :
( ! [B4: multiset_a] :
( ( member_multiset_a @ B4 @ ( set_mset_multiset_a @ K4 ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ B4 @ A2 ) @ R2 ) )
& ( N
= ( plus_p6738641960240532161iset_a @ M02 @ K4 ) ) ) ) ) ).
% less_add
thf(fact_715_less__add,axiom,
! [N: multiset_a,A2: a,M02: multiset_a,R2: set_Product_prod_a_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ N @ ( add_mset_a @ A2 @ M02 ) ) @ ( mult1_a @ R2 ) )
=> ( ? [M5: multiset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ M5 @ M02 ) @ ( mult1_a @ R2 ) )
& ( N
= ( add_mset_a @ A2 @ M5 ) ) )
| ? [K4: multiset_a] :
( ! [B4: a] :
( ( member_a @ B4 @ ( set_mset_a @ K4 ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B4 @ A2 ) @ R2 ) )
& ( N
= ( plus_plus_multiset_a @ M02 @ K4 ) ) ) ) ) ).
% less_add
thf(fact_716_mset__update,axiom,
! [I: nat,Ls: list_a,V: a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Ls ) )
=> ( ( mset_a @ ( list_update_a @ Ls @ I @ V ) )
= ( add_mset_a @ V @ ( minus_3765977307040488491iset_a @ ( mset_a @ Ls ) @ ( add_mset_a @ ( nth_a @ Ls @ I ) @ zero_zero_multiset_a ) ) ) ) ) ).
% mset_update
thf(fact_717_zero__diff,axiom,
! [A2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% zero_diff
thf(fact_718_diff__zero,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% diff_zero
thf(fact_719_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ A2 )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_720_add__diff__cancel__right_H,axiom,
! [A2: multiset_a,B: multiset_a] :
( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ A2 @ B ) @ B )
= A2 ) ).
% add_diff_cancel_right'
thf(fact_721_add__diff__cancel__right_H,axiom,
! [A2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
= A2 ) ).
% add_diff_cancel_right'
thf(fact_722_add__diff__cancel__right,axiom,
! [A2: multiset_a,C2: multiset_a,B: multiset_a] :
( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ A2 @ C2 ) @ ( plus_plus_multiset_a @ B @ C2 ) )
= ( minus_3765977307040488491iset_a @ A2 @ B ) ) ).
% add_diff_cancel_right
thf(fact_723_add__diff__cancel__right,axiom,
! [A2: nat,C2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( minus_minus_nat @ A2 @ B ) ) ).
% add_diff_cancel_right
thf(fact_724_add__diff__cancel__left_H,axiom,
! [A2: multiset_a,B: multiset_a] :
( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ A2 @ B ) @ A2 )
= B ) ).
% add_diff_cancel_left'
thf(fact_725_add__diff__cancel__left_H,axiom,
! [A2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B ) @ A2 )
= B ) ).
% add_diff_cancel_left'
thf(fact_726_add__diff__cancel__left,axiom,
! [C2: multiset_a,A2: multiset_a,B: multiset_a] :
( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ C2 @ A2 ) @ ( plus_plus_multiset_a @ C2 @ B ) )
= ( minus_3765977307040488491iset_a @ A2 @ B ) ) ).
% add_diff_cancel_left
thf(fact_727_add__diff__cancel__left,axiom,
! [C2: nat,A2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A2 ) @ ( plus_plus_nat @ C2 @ B ) )
= ( minus_minus_nat @ A2 @ B ) ) ).
% add_diff_cancel_left
thf(fact_728_diff__diff__add__mset,axiom,
! [M: multiset_a,N: multiset_a,P2: multiset_a] :
( ( minus_3765977307040488491iset_a @ ( minus_3765977307040488491iset_a @ M @ N ) @ P2 )
= ( minus_3765977307040488491iset_a @ M @ ( plus_plus_multiset_a @ N @ P2 ) ) ) ).
% diff_diff_add_mset
thf(fact_729_diff__add__zero,axiom,
! [A2: multiset_a,B: multiset_a] :
( ( minus_3765977307040488491iset_a @ A2 @ ( plus_plus_multiset_a @ A2 @ B ) )
= zero_zero_multiset_a ) ).
% diff_add_zero
thf(fact_730_diff__add__zero,axiom,
! [A2: nat,B: nat] :
( ( minus_minus_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_731_nth__Cons__0,axiom,
! [X3: a,Xs: list_a] :
( ( nth_a @ ( cons_a @ X3 @ Xs ) @ zero_zero_nat )
= X3 ) ).
% nth_Cons_0
thf(fact_732_zip__Cons__Cons,axiom,
! [X3: a,Xs: list_a,Y: a,Ys: list_a] :
( ( zip_a_a @ ( cons_a @ X3 @ Xs ) @ ( cons_a @ Y @ Ys ) )
= ( cons_P7316939126706565853od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ ( zip_a_a @ Xs @ Ys ) ) ) ).
% zip_Cons_Cons
thf(fact_733_zip__Cons__Cons,axiom,
! [X3: multiset_a,Xs: list_multiset_a,Y: multiset_a,Ys: list_multiset_a] :
( ( zip_mu5258324667929623607iset_a @ ( cons_multiset_a @ X3 @ Xs ) @ ( cons_multiset_a @ Y @ Ys ) )
= ( cons_P7086145107730948317iset_a @ ( produc654756711066625303iset_a @ X3 @ Y ) @ ( zip_mu5258324667929623607iset_a @ Xs @ Ys ) ) ) ).
% zip_Cons_Cons
thf(fact_734_zip__Cons__Cons,axiom,
! [X3: nat > nat,Xs: list_nat_nat,Y: nat,Ys: list_nat] :
( ( zip_nat_nat_nat @ ( cons_nat_nat @ X3 @ Xs ) @ ( cons_nat @ Y @ Ys ) )
= ( cons_P4219629788700907686at_nat @ ( produc72220940542539688at_nat @ X3 @ Y ) @ ( zip_nat_nat_nat @ Xs @ Ys ) ) ) ).
% zip_Cons_Cons
thf(fact_735_mset__subset__eq__multiset__union__diff__commute,axiom,
! [B5: multiset_a,A4: multiset_a,C3: multiset_a] :
( ( subseteq_mset_a @ B5 @ A4 )
=> ( ( plus_plus_multiset_a @ ( minus_3765977307040488491iset_a @ A4 @ B5 ) @ C3 )
= ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ A4 @ C3 ) @ B5 ) ) ) ).
% mset_subset_eq_multiset_union_diff_commute
thf(fact_736_subset__mset_Oadd__diff__assoc2,axiom,
! [A2: multiset_a,B: multiset_a,C2: multiset_a] :
( ( subseteq_mset_a @ A2 @ B )
=> ( ( plus_plus_multiset_a @ ( minus_3765977307040488491iset_a @ B @ A2 ) @ C2 )
= ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ B @ C2 ) @ A2 ) ) ) ).
% subset_mset.add_diff_assoc2
thf(fact_737_subset__mset_Oadd__diff__assoc,axiom,
! [A2: multiset_a,B: multiset_a,C2: multiset_a] :
( ( subseteq_mset_a @ A2 @ B )
=> ( ( plus_plus_multiset_a @ C2 @ ( minus_3765977307040488491iset_a @ B @ A2 ) )
= ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ C2 @ B ) @ A2 ) ) ) ).
% subset_mset.add_diff_assoc
thf(fact_738_diff__add__mset__swap,axiom,
! [B: product_prod_a_a,A4: multis501812127501805293od_a_a,M: multis501812127501805293od_a_a] :
( ~ ( member1426531477525435216od_a_a @ B @ ( set_ms119794563918438244od_a_a @ A4 ) )
=> ( ( minus_7313152569058643924od_a_a @ ( add_ms8655138167283798533od_a_a @ B @ M ) @ A4 )
= ( add_ms8655138167283798533od_a_a @ B @ ( minus_7313152569058643924od_a_a @ M @ A4 ) ) ) ) ).
% diff_add_mset_swap
thf(fact_739_diff__add__mset__swap,axiom,
! [B: produc6518373309651786023iset_a,A4: multis4781622204924204013iset_a,M: multis4781622204924204013iset_a] :
( ~ ( member5199237121806060112iset_a @ B @ ( set_ms2379723421352415844iset_a @ A4 ) )
=> ( ( minus_9211619285413683668iset_a @ ( add_ms5348986034529679365iset_a @ B @ M ) @ A4 )
= ( add_ms5348986034529679365iset_a @ B @ ( minus_9211619285413683668iset_a @ M @ A4 ) ) ) ) ).
% diff_add_mset_swap
thf(fact_740_diff__add__mset__swap,axiom,
! [B: a,A4: multiset_a,M: multiset_a] :
( ~ ( member_a @ B @ ( set_mset_a @ A4 ) )
=> ( ( minus_3765977307040488491iset_a @ ( add_mset_a @ B @ M ) @ A4 )
= ( add_mset_a @ B @ ( minus_3765977307040488491iset_a @ M @ A4 ) ) ) ) ).
% diff_add_mset_swap
thf(fact_741_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_742_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_743_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_744_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_745_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_746_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_747_Cons__listrel1__Cons,axiom,
! [X3: a,Xs: list_a,Y: a,Ys: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X3 @ Xs ) @ ( cons_a @ Y @ Ys ) ) @ ( listrel1_a @ R2 ) )
= ( ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ R2 )
& ( Xs = Ys ) )
| ( ( X3 = Y )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R2 ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_748_Cons__listrel1__Cons,axiom,
! [X3: multiset_a,Xs: list_multiset_a,Y: multiset_a,Ys: list_multiset_a,R2: set_Pr79727621955416071iset_a] :
( ( member1640862112313989456iset_a @ ( produc1434922032272282391iset_a @ ( cons_multiset_a @ X3 @ Xs ) @ ( cons_multiset_a @ Y @ Ys ) ) @ ( listrel1_multiset_a @ R2 ) )
= ( ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ Y ) @ R2 )
& ( Xs = Ys ) )
| ( ( X3 = Y )
& ( member1640862112313989456iset_a @ ( produc1434922032272282391iset_a @ Xs @ Ys ) @ ( listrel1_multiset_a @ R2 ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_749_mset__single__iff__right,axiom,
! [X3: a,Xs: list_a] :
( ( ( add_mset_a @ X3 @ zero_zero_multiset_a )
= ( mset_a @ Xs ) )
= ( Xs
= ( cons_a @ X3 @ nil_a ) ) ) ).
% mset_single_iff_right
thf(fact_750_mset__single__iff,axiom,
! [Xs: list_a,X3: a] :
( ( ( mset_a @ Xs )
= ( add_mset_a @ X3 @ zero_zero_multiset_a ) )
= ( Xs
= ( cons_a @ X3 @ nil_a ) ) ) ).
% mset_single_iff
thf(fact_751_zip__eq__ConsE,axiom,
! [Xs: list_a,Ys: list_a,Xy: product_prod_a_a,Xys: list_P1396940483166286381od_a_a] :
( ( ( zip_a_a @ Xs @ Ys )
= ( cons_P7316939126706565853od_a_a @ Xy @ Xys ) )
=> ~ ! [X: a,Xs6: list_a] :
( ( Xs
= ( cons_a @ X @ Xs6 ) )
=> ! [Y3: a,Ys6: list_a] :
( ( Ys
= ( cons_a @ Y3 @ Ys6 ) )
=> ( ( Xy
= ( product_Pair_a_a @ X @ Y3 ) )
=> ( Xys
!= ( zip_a_a @ Xs6 @ Ys6 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_752_zip__eq__ConsE,axiom,
! [Xs: list_multiset_a,Ys: list_multiset_a,Xy: produc6518373309651786023iset_a,Xys: list_P6988024003011616557iset_a] :
( ( ( zip_mu5258324667929623607iset_a @ Xs @ Ys )
= ( cons_P7086145107730948317iset_a @ Xy @ Xys ) )
=> ~ ! [X: multiset_a,Xs6: list_multiset_a] :
( ( Xs
= ( cons_multiset_a @ X @ Xs6 ) )
=> ! [Y3: multiset_a,Ys6: list_multiset_a] :
( ( Ys
= ( cons_multiset_a @ Y3 @ Ys6 ) )
=> ( ( Xy
= ( produc654756711066625303iset_a @ X @ Y3 ) )
=> ( Xys
!= ( zip_mu5258324667929623607iset_a @ Xs6 @ Ys6 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_753_zip__eq__ConsE,axiom,
! [Xs: list_nat_nat,Ys: list_nat,Xy: produc8199716216217303280at_nat,Xys: list_P5366131564312172150at_nat] :
( ( ( zip_nat_nat_nat @ Xs @ Ys )
= ( cons_P4219629788700907686at_nat @ Xy @ Xys ) )
=> ~ ! [X: nat > nat,Xs6: list_nat_nat] :
( ( Xs
= ( cons_nat_nat @ X @ Xs6 ) )
=> ! [Y3: nat,Ys6: list_nat] :
( ( Ys
= ( cons_nat @ Y3 @ Ys6 ) )
=> ( ( Xy
= ( produc72220940542539688at_nat @ X @ Y3 ) )
=> ( Xys
!= ( zip_nat_nat_nat @ Xs6 @ Ys6 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_754_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A2: nat,C2: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ C2 ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B ) @ C2 ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_755_diff__diff__eq,axiom,
! [A2: multiset_a,B: multiset_a,C2: multiset_a] :
( ( minus_3765977307040488491iset_a @ ( minus_3765977307040488491iset_a @ A2 @ B ) @ C2 )
= ( minus_3765977307040488491iset_a @ A2 @ ( plus_plus_multiset_a @ B @ C2 ) ) ) ).
% diff_diff_eq
thf(fact_756_diff__diff__eq,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B ) @ C2 )
= ( minus_minus_nat @ A2 @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% diff_diff_eq
thf(fact_757_add__implies__diff,axiom,
! [C2: multiset_a,B: multiset_a,A2: multiset_a] :
( ( ( plus_plus_multiset_a @ C2 @ B )
= A2 )
=> ( C2
= ( minus_3765977307040488491iset_a @ A2 @ B ) ) ) ).
% add_implies_diff
thf(fact_758_add__implies__diff,axiom,
! [C2: nat,B: nat,A2: nat] :
( ( ( plus_plus_nat @ C2 @ B )
= A2 )
=> ( C2
= ( minus_minus_nat @ A2 @ B ) ) ) ).
% add_implies_diff
thf(fact_759_list_Oset__intros_I2_J,axiom,
! [Y: product_prod_a_a,X222: list_P1396940483166286381od_a_a,X21: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y @ ( set_Product_prod_a_a2 @ X222 ) )
=> ( member1426531477525435216od_a_a @ Y @ ( set_Product_prod_a_a2 @ ( cons_P7316939126706565853od_a_a @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_760_list_Oset__intros_I2_J,axiom,
! [Y: produc6518373309651786023iset_a,X222: list_P6988024003011616557iset_a,X21: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Y @ ( set_Pr5550698219922361532iset_a @ X222 ) )
=> ( member5199237121806060112iset_a @ Y @ ( set_Pr5550698219922361532iset_a @ ( cons_P7086145107730948317iset_a @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_761_list_Oset__intros_I2_J,axiom,
! [Y: a,X222: list_a,X21: a] :
( ( member_a @ Y @ ( set_a2 @ X222 ) )
=> ( member_a @ Y @ ( set_a2 @ ( cons_a @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_762_list_Oset__intros_I1_J,axiom,
! [X21: product_prod_a_a,X222: list_P1396940483166286381od_a_a] : ( member1426531477525435216od_a_a @ X21 @ ( set_Product_prod_a_a2 @ ( cons_P7316939126706565853od_a_a @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_763_list_Oset__intros_I1_J,axiom,
! [X21: produc6518373309651786023iset_a,X222: list_P6988024003011616557iset_a] : ( member5199237121806060112iset_a @ X21 @ ( set_Pr5550698219922361532iset_a @ ( cons_P7086145107730948317iset_a @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_764_list_Oset__intros_I1_J,axiom,
! [X21: a,X222: list_a] : ( member_a @ X21 @ ( set_a2 @ ( cons_a @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_765_list_Oset__cases,axiom,
! [E: product_prod_a_a,A2: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ E @ ( set_Product_prod_a_a2 @ A2 ) )
=> ( ! [Z22: list_P1396940483166286381od_a_a] :
( A2
!= ( cons_P7316939126706565853od_a_a @ E @ Z22 ) )
=> ~ ! [Z1: product_prod_a_a,Z22: list_P1396940483166286381od_a_a] :
( ( A2
= ( cons_P7316939126706565853od_a_a @ Z1 @ Z22 ) )
=> ~ ( member1426531477525435216od_a_a @ E @ ( set_Product_prod_a_a2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_766_list_Oset__cases,axiom,
! [E: produc6518373309651786023iset_a,A2: list_P6988024003011616557iset_a] :
( ( member5199237121806060112iset_a @ E @ ( set_Pr5550698219922361532iset_a @ A2 ) )
=> ( ! [Z22: list_P6988024003011616557iset_a] :
( A2
!= ( cons_P7086145107730948317iset_a @ E @ Z22 ) )
=> ~ ! [Z1: produc6518373309651786023iset_a,Z22: list_P6988024003011616557iset_a] :
( ( A2
= ( cons_P7086145107730948317iset_a @ Z1 @ Z22 ) )
=> ~ ( member5199237121806060112iset_a @ E @ ( set_Pr5550698219922361532iset_a @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_767_list_Oset__cases,axiom,
! [E: a,A2: list_a] :
( ( member_a @ E @ ( set_a2 @ A2 ) )
=> ( ! [Z22: list_a] :
( A2
!= ( cons_a @ E @ Z22 ) )
=> ~ ! [Z1: a,Z22: list_a] :
( ( A2
= ( cons_a @ Z1 @ Z22 ) )
=> ~ ( member_a @ E @ ( set_a2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_768_set__ConsD,axiom,
! [Y: product_prod_a_a,X3: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ Y @ ( set_Product_prod_a_a2 @ ( cons_P7316939126706565853od_a_a @ X3 @ Xs ) ) )
=> ( ( Y = X3 )
| ( member1426531477525435216od_a_a @ Y @ ( set_Product_prod_a_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_769_set__ConsD,axiom,
! [Y: produc6518373309651786023iset_a,X3: produc6518373309651786023iset_a,Xs: list_P6988024003011616557iset_a] :
( ( member5199237121806060112iset_a @ Y @ ( set_Pr5550698219922361532iset_a @ ( cons_P7086145107730948317iset_a @ X3 @ Xs ) ) )
=> ( ( Y = X3 )
| ( member5199237121806060112iset_a @ Y @ ( set_Pr5550698219922361532iset_a @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_770_set__ConsD,axiom,
! [Y: a,X3: a,Xs: list_a] :
( ( member_a @ Y @ ( set_a2 @ ( cons_a @ X3 @ Xs ) ) )
=> ( ( Y = X3 )
| ( member_a @ Y @ ( set_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_771_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_772_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_773_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_774_in__diffD,axiom,
! [A2: product_prod_a_a,M: multis501812127501805293od_a_a,N: multis501812127501805293od_a_a] :
( ( member1426531477525435216od_a_a @ A2 @ ( set_ms119794563918438244od_a_a @ ( minus_7313152569058643924od_a_a @ M @ N ) ) )
=> ( member1426531477525435216od_a_a @ A2 @ ( set_ms119794563918438244od_a_a @ M ) ) ) ).
% in_diffD
thf(fact_775_in__diffD,axiom,
! [A2: produc6518373309651786023iset_a,M: multis4781622204924204013iset_a,N: multis4781622204924204013iset_a] :
( ( member5199237121806060112iset_a @ A2 @ ( set_ms2379723421352415844iset_a @ ( minus_9211619285413683668iset_a @ M @ N ) ) )
=> ( member5199237121806060112iset_a @ A2 @ ( set_ms2379723421352415844iset_a @ M ) ) ) ).
% in_diffD
thf(fact_776_in__diffD,axiom,
! [A2: a,M: multiset_a,N: multiset_a] :
( ( member_a @ A2 @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ M @ N ) ) )
=> ( member_a @ A2 @ ( set_mset_a @ M ) ) ) ).
% in_diffD
thf(fact_777_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A2: nat,B: nat] :
( ~ ( ord_less_nat @ A2 @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A2 @ B ) )
= A2 ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_778_multiset__diff__union__assoc,axiom,
! [C3: multiset_a,B5: multiset_a,A4: multiset_a] :
( ( subseteq_mset_a @ C3 @ B5 )
=> ( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ A4 @ B5 ) @ C3 )
= ( plus_plus_multiset_a @ A4 @ ( minus_3765977307040488491iset_a @ B5 @ C3 ) ) ) ) ).
% multiset_diff_union_assoc
thf(fact_779_subset__eq__diff__conv,axiom,
! [A4: multiset_a,C3: multiset_a,B5: multiset_a] :
( ( subseteq_mset_a @ ( minus_3765977307040488491iset_a @ A4 @ C3 ) @ B5 )
= ( subseteq_mset_a @ A4 @ ( plus_plus_multiset_a @ B5 @ C3 ) ) ) ).
% subset_eq_diff_conv
thf(fact_780_subset__mset_Ole__imp__diff__is__add,axiom,
! [A2: multiset_a,B: multiset_a,C2: multiset_a] :
( ( subseteq_mset_a @ A2 @ B )
=> ( ( subseteq_mset_a @ A2 @ B )
=> ( ( ( minus_3765977307040488491iset_a @ B @ A2 )
= C2 )
= ( B
= ( plus_plus_multiset_a @ C2 @ A2 ) ) ) ) ) ).
% subset_mset.le_imp_diff_is_add
thf(fact_781_subset__mset_Oadd__diff__inverse,axiom,
! [A2: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ A2 @ B )
=> ( ( plus_plus_multiset_a @ A2 @ ( minus_3765977307040488491iset_a @ B @ A2 ) )
= B ) ) ).
% subset_mset.add_diff_inverse
thf(fact_782_subset__mset_Odiff__diff__right,axiom,
! [A2: multiset_a,B: multiset_a,C2: multiset_a] :
( ( subseteq_mset_a @ A2 @ B )
=> ( ( minus_3765977307040488491iset_a @ C2 @ ( minus_3765977307040488491iset_a @ B @ A2 ) )
= ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ C2 @ A2 ) @ B ) ) ) ).
% subset_mset.diff_diff_right
thf(fact_783_subset__mset_Odiff__add__assoc2,axiom,
! [A2: multiset_a,B: multiset_a,C2: multiset_a] :
( ( subseteq_mset_a @ A2 @ B )
=> ( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ B @ C2 ) @ A2 )
= ( plus_plus_multiset_a @ ( minus_3765977307040488491iset_a @ B @ A2 ) @ C2 ) ) ) ).
% subset_mset.diff_add_assoc2
thf(fact_784_subset__mset_Odiff__add__assoc,axiom,
! [A2: multiset_a,B: multiset_a,C2: multiset_a] :
( ( subseteq_mset_a @ A2 @ B )
=> ( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ C2 @ B ) @ A2 )
= ( plus_plus_multiset_a @ C2 @ ( minus_3765977307040488491iset_a @ B @ A2 ) ) ) ) ).
% subset_mset.diff_add_assoc
thf(fact_785_subset__mset_Ole__diff__conv2,axiom,
! [A2: multiset_a,B: multiset_a,C2: multiset_a] :
( ( subseteq_mset_a @ A2 @ B )
=> ( ( subseteq_mset_a @ C2 @ ( minus_3765977307040488491iset_a @ B @ A2 ) )
= ( subseteq_mset_a @ ( plus_plus_multiset_a @ C2 @ A2 ) @ B ) ) ) ).
% subset_mset.le_diff_conv2
thf(fact_786_subset__mset_Ole__add__diff,axiom,
! [A2: multiset_a,B: multiset_a,C2: multiset_a] :
( ( subseteq_mset_a @ A2 @ B )
=> ( subseteq_mset_a @ C2 @ ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ B @ C2 ) @ A2 ) ) ) ).
% subset_mset.le_add_diff
thf(fact_787_subset__mset_Odiff__add,axiom,
! [A2: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ A2 @ B )
=> ( ( plus_plus_multiset_a @ ( minus_3765977307040488491iset_a @ B @ A2 ) @ A2 )
= B ) ) ).
% subset_mset.diff_add
thf(fact_788_list__induct2,axiom,
! [Xs: list_a,Ys: list_a,P2: list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P2 @ nil_a @ nil_a )
=> ( ! [X: a,Xs3: list_a,Y3: a,Ys3: list_a] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( P2 @ Xs3 @ Ys3 )
=> ( P2 @ ( cons_a @ X @ Xs3 ) @ ( cons_a @ Y3 @ Ys3 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_789_list__induct3,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,P2: list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P2 @ nil_a @ nil_a @ nil_a )
=> ( ! [X: a,Xs3: list_a,Y3: a,Ys3: list_a,Z4: a,Zs2: list_a] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P2 @ Xs3 @ Ys3 @ Zs2 )
=> ( P2 @ ( cons_a @ X @ Xs3 ) @ ( cons_a @ Y3 @ Ys3 ) @ ( cons_a @ Z4 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_790_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,Ws: list_a,P2: list_a > list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P2 @ nil_a @ nil_a @ nil_a @ nil_a )
=> ( ! [X: a,Xs3: list_a,Y3: a,Ys3: list_a,Z4: a,Zs2: list_a,W2: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P2 @ Xs3 @ Ys3 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_a @ X @ Xs3 ) @ ( cons_a @ Y3 @ Ys3 ) @ ( cons_a @ Z4 @ Zs2 ) @ ( cons_a @ W2 @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_791_mset_Osimps_I2_J,axiom,
! [A2: a,X3: list_a] :
( ( mset_a @ ( cons_a @ A2 @ X3 ) )
= ( add_mset_a @ A2 @ ( mset_a @ X3 ) ) ) ).
% mset.simps(2)
thf(fact_792_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_793_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_794_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_795_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_796_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_797_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_798_multi__drop__mem__not__eq,axiom,
! [C2: product_prod_a_a,B5: multis501812127501805293od_a_a] :
( ( member1426531477525435216od_a_a @ C2 @ ( set_ms119794563918438244od_a_a @ B5 ) )
=> ( ( minus_7313152569058643924od_a_a @ B5 @ ( add_ms8655138167283798533od_a_a @ C2 @ zero_z3024502163831676836od_a_a ) )
!= B5 ) ) ).
% multi_drop_mem_not_eq
thf(fact_799_multi__drop__mem__not__eq,axiom,
! [C2: produc6518373309651786023iset_a,B5: multis4781622204924204013iset_a] :
( ( member5199237121806060112iset_a @ C2 @ ( set_ms2379723421352415844iset_a @ B5 ) )
=> ( ( minus_9211619285413683668iset_a @ B5 @ ( add_ms5348986034529679365iset_a @ C2 @ zero_z4591712717340762532iset_a ) )
!= B5 ) ) ).
% multi_drop_mem_not_eq
thf(fact_800_multi__drop__mem__not__eq,axiom,
! [C2: a,B5: multiset_a] :
( ( member_a @ C2 @ ( set_mset_a @ B5 ) )
=> ( ( minus_3765977307040488491iset_a @ B5 @ ( add_mset_a @ C2 @ zero_zero_multiset_a ) )
!= B5 ) ) ).
% multi_drop_mem_not_eq
thf(fact_801_add__mset__remove__trivial__If,axiom,
! [A2: product_prod_a_a,N: multis501812127501805293od_a_a] :
( ( ( member1426531477525435216od_a_a @ A2 @ ( set_ms119794563918438244od_a_a @ N ) )
=> ( ( add_ms8655138167283798533od_a_a @ A2 @ ( minus_7313152569058643924od_a_a @ N @ ( add_ms8655138167283798533od_a_a @ A2 @ zero_z3024502163831676836od_a_a ) ) )
= N ) )
& ( ~ ( member1426531477525435216od_a_a @ A2 @ ( set_ms119794563918438244od_a_a @ N ) )
=> ( ( add_ms8655138167283798533od_a_a @ A2 @ ( minus_7313152569058643924od_a_a @ N @ ( add_ms8655138167283798533od_a_a @ A2 @ zero_z3024502163831676836od_a_a ) ) )
= ( add_ms8655138167283798533od_a_a @ A2 @ N ) ) ) ) ).
% add_mset_remove_trivial_If
thf(fact_802_add__mset__remove__trivial__If,axiom,
! [A2: produc6518373309651786023iset_a,N: multis4781622204924204013iset_a] :
( ( ( member5199237121806060112iset_a @ A2 @ ( set_ms2379723421352415844iset_a @ N ) )
=> ( ( add_ms5348986034529679365iset_a @ A2 @ ( minus_9211619285413683668iset_a @ N @ ( add_ms5348986034529679365iset_a @ A2 @ zero_z4591712717340762532iset_a ) ) )
= N ) )
& ( ~ ( member5199237121806060112iset_a @ A2 @ ( set_ms2379723421352415844iset_a @ N ) )
=> ( ( add_ms5348986034529679365iset_a @ A2 @ ( minus_9211619285413683668iset_a @ N @ ( add_ms5348986034529679365iset_a @ A2 @ zero_z4591712717340762532iset_a ) ) )
= ( add_ms5348986034529679365iset_a @ A2 @ N ) ) ) ) ).
% add_mset_remove_trivial_If
thf(fact_803_add__mset__remove__trivial__If,axiom,
! [A2: a,N: multiset_a] :
( ( ( member_a @ A2 @ ( set_mset_a @ N ) )
=> ( ( add_mset_a @ A2 @ ( minus_3765977307040488491iset_a @ N @ ( add_mset_a @ A2 @ zero_zero_multiset_a ) ) )
= N ) )
& ( ~ ( member_a @ A2 @ ( set_mset_a @ N ) )
=> ( ( add_mset_a @ A2 @ ( minus_3765977307040488491iset_a @ N @ ( add_mset_a @ A2 @ zero_zero_multiset_a ) ) )
= ( add_mset_a @ A2 @ N ) ) ) ) ).
% add_mset_remove_trivial_If
thf(fact_804_add__mset__remove__trivial__eq,axiom,
! [N: multis501812127501805293od_a_a,A2: product_prod_a_a] :
( ( N
= ( add_ms8655138167283798533od_a_a @ A2 @ ( minus_7313152569058643924od_a_a @ N @ ( add_ms8655138167283798533od_a_a @ A2 @ zero_z3024502163831676836od_a_a ) ) ) )
= ( member1426531477525435216od_a_a @ A2 @ ( set_ms119794563918438244od_a_a @ N ) ) ) ).
% add_mset_remove_trivial_eq
thf(fact_805_add__mset__remove__trivial__eq,axiom,
! [N: multis4781622204924204013iset_a,A2: produc6518373309651786023iset_a] :
( ( N
= ( add_ms5348986034529679365iset_a @ A2 @ ( minus_9211619285413683668iset_a @ N @ ( add_ms5348986034529679365iset_a @ A2 @ zero_z4591712717340762532iset_a ) ) ) )
= ( member5199237121806060112iset_a @ A2 @ ( set_ms2379723421352415844iset_a @ N ) ) ) ).
% add_mset_remove_trivial_eq
thf(fact_806_add__mset__remove__trivial__eq,axiom,
! [N: multiset_a,A2: a] :
( ( N
= ( add_mset_a @ A2 @ ( minus_3765977307040488491iset_a @ N @ ( add_mset_a @ A2 @ zero_zero_multiset_a ) ) ) )
= ( member_a @ A2 @ ( set_mset_a @ N ) ) ) ).
% add_mset_remove_trivial_eq
thf(fact_807_multiset__add__sub__el__shuffle,axiom,
! [C2: product_prod_a_a,B5: multis501812127501805293od_a_a,B: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C2 @ ( set_ms119794563918438244od_a_a @ B5 ) )
=> ( ( B != C2 )
=> ( ( add_ms8655138167283798533od_a_a @ B @ ( minus_7313152569058643924od_a_a @ B5 @ ( add_ms8655138167283798533od_a_a @ C2 @ zero_z3024502163831676836od_a_a ) ) )
= ( minus_7313152569058643924od_a_a @ ( add_ms8655138167283798533od_a_a @ B @ B5 ) @ ( add_ms8655138167283798533od_a_a @ C2 @ zero_z3024502163831676836od_a_a ) ) ) ) ) ).
% multiset_add_sub_el_shuffle
thf(fact_808_multiset__add__sub__el__shuffle,axiom,
! [C2: produc6518373309651786023iset_a,B5: multis4781622204924204013iset_a,B: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ C2 @ ( set_ms2379723421352415844iset_a @ B5 ) )
=> ( ( B != C2 )
=> ( ( add_ms5348986034529679365iset_a @ B @ ( minus_9211619285413683668iset_a @ B5 @ ( add_ms5348986034529679365iset_a @ C2 @ zero_z4591712717340762532iset_a ) ) )
= ( minus_9211619285413683668iset_a @ ( add_ms5348986034529679365iset_a @ B @ B5 ) @ ( add_ms5348986034529679365iset_a @ C2 @ zero_z4591712717340762532iset_a ) ) ) ) ) ).
% multiset_add_sub_el_shuffle
thf(fact_809_multiset__add__sub__el__shuffle,axiom,
! [C2: a,B5: multiset_a,B: a] :
( ( member_a @ C2 @ ( set_mset_a @ B5 ) )
=> ( ( B != C2 )
=> ( ( add_mset_a @ B @ ( minus_3765977307040488491iset_a @ B5 @ ( add_mset_a @ C2 @ zero_zero_multiset_a ) ) )
= ( minus_3765977307040488491iset_a @ ( add_mset_a @ B @ B5 ) @ ( add_mset_a @ C2 @ zero_zero_multiset_a ) ) ) ) ) ).
% multiset_add_sub_el_shuffle
thf(fact_810_more__than__one__mset__mset__diff,axiom,
! [A2: product_prod_a_a,M: multis501812127501805293od_a_a] :
( ( member1426531477525435216od_a_a @ A2 @ ( set_ms119794563918438244od_a_a @ ( minus_7313152569058643924od_a_a @ M @ ( add_ms8655138167283798533od_a_a @ A2 @ zero_z3024502163831676836od_a_a ) ) ) )
=> ( ( set_ms119794563918438244od_a_a @ ( minus_7313152569058643924od_a_a @ M @ ( add_ms8655138167283798533od_a_a @ A2 @ zero_z3024502163831676836od_a_a ) ) )
= ( set_ms119794563918438244od_a_a @ M ) ) ) ).
% more_than_one_mset_mset_diff
thf(fact_811_more__than__one__mset__mset__diff,axiom,
! [A2: produc6518373309651786023iset_a,M: multis4781622204924204013iset_a] :
( ( member5199237121806060112iset_a @ A2 @ ( set_ms2379723421352415844iset_a @ ( minus_9211619285413683668iset_a @ M @ ( add_ms5348986034529679365iset_a @ A2 @ zero_z4591712717340762532iset_a ) ) ) )
=> ( ( set_ms2379723421352415844iset_a @ ( minus_9211619285413683668iset_a @ M @ ( add_ms5348986034529679365iset_a @ A2 @ zero_z4591712717340762532iset_a ) ) )
= ( set_ms2379723421352415844iset_a @ M ) ) ) ).
% more_than_one_mset_mset_diff
thf(fact_812_more__than__one__mset__mset__diff,axiom,
! [A2: a,M: multiset_a] :
( ( member_a @ A2 @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ A2 @ zero_zero_multiset_a ) ) ) )
=> ( ( set_mset_a @ ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ A2 @ zero_zero_multiset_a ) ) )
= ( set_mset_a @ M ) ) ) ).
% more_than_one_mset_mset_diff
thf(fact_813_Cons__listrel1E2,axiom,
! [Xs: list_a,Y: a,Ys: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ ( cons_a @ Y @ Ys ) ) @ ( listrel1_a @ R2 ) )
=> ( ! [X: a] :
( ( Xs
= ( cons_a @ X @ Ys ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 ) )
=> ~ ! [Zs2: list_a] :
( ( Xs
= ( cons_a @ Y @ Zs2 ) )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Zs2 @ Ys ) @ ( listrel1_a @ R2 ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_814_Cons__listrel1E2,axiom,
! [Xs: list_multiset_a,Y: multiset_a,Ys: list_multiset_a,R2: set_Pr79727621955416071iset_a] :
( ( member1640862112313989456iset_a @ ( produc1434922032272282391iset_a @ Xs @ ( cons_multiset_a @ Y @ Ys ) ) @ ( listrel1_multiset_a @ R2 ) )
=> ( ! [X: multiset_a] :
( ( Xs
= ( cons_multiset_a @ X @ Ys ) )
=> ~ ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ Y ) @ R2 ) )
=> ~ ! [Zs2: list_multiset_a] :
( ( Xs
= ( cons_multiset_a @ Y @ Zs2 ) )
=> ~ ( member1640862112313989456iset_a @ ( produc1434922032272282391iset_a @ Zs2 @ Ys ) @ ( listrel1_multiset_a @ R2 ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_815_Cons__listrel1E1,axiom,
! [X3: a,Xs: list_a,Ys: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X3 @ Xs ) @ Ys ) @ ( listrel1_a @ R2 ) )
=> ( ! [Y3: a] :
( ( Ys
= ( cons_a @ Y3 @ Xs ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R2 ) )
=> ~ ! [Zs2: list_a] :
( ( Ys
= ( cons_a @ X3 @ Zs2 ) )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Zs2 ) @ ( listrel1_a @ R2 ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_816_Cons__listrel1E1,axiom,
! [X3: multiset_a,Xs: list_multiset_a,Ys: list_multiset_a,R2: set_Pr79727621955416071iset_a] :
( ( member1640862112313989456iset_a @ ( produc1434922032272282391iset_a @ ( cons_multiset_a @ X3 @ Xs ) @ Ys ) @ ( listrel1_multiset_a @ R2 ) )
=> ( ! [Y3: multiset_a] :
( ( Ys
= ( cons_multiset_a @ Y3 @ Xs ) )
=> ~ ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ Y3 ) @ R2 ) )
=> ~ ! [Zs2: list_multiset_a] :
( ( Ys
= ( cons_multiset_a @ X3 @ Zs2 ) )
=> ~ ( member1640862112313989456iset_a @ ( produc1434922032272282391iset_a @ Xs @ Zs2 ) @ ( listrel1_multiset_a @ R2 ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_817_listrel1I1,axiom,
! [X3: a,Y: a,R2: set_Product_prod_a_a,Xs: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ R2 )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X3 @ Xs ) @ ( cons_a @ Y @ Xs ) ) @ ( listrel1_a @ R2 ) ) ) ).
% listrel1I1
thf(fact_818_listrel1I1,axiom,
! [X3: multiset_a,Y: multiset_a,R2: set_Pr79727621955416071iset_a,Xs: list_multiset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ Y ) @ R2 )
=> ( member1640862112313989456iset_a @ ( produc1434922032272282391iset_a @ ( cons_multiset_a @ X3 @ Xs ) @ ( cons_multiset_a @ Y @ Xs ) ) @ ( listrel1_multiset_a @ R2 ) ) ) ).
% listrel1I1
thf(fact_819_listrel__Cons2,axiom,
! [Xs: list_a,Y: a,Ys: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ ( cons_a @ Y @ Ys ) ) @ ( listrel_a_a @ R2 ) )
=> ~ ! [X: a,Xs3: list_a] :
( ( Xs
= ( cons_a @ X @ Xs3 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs3 @ Ys ) @ ( listrel_a_a @ R2 ) ) ) ) ) ).
% listrel_Cons2
thf(fact_820_listrel__Cons2,axiom,
! [Xs: list_multiset_a,Y: multiset_a,Ys: list_multiset_a,R2: set_Pr79727621955416071iset_a] :
( ( member1640862112313989456iset_a @ ( produc1434922032272282391iset_a @ Xs @ ( cons_multiset_a @ Y @ Ys ) ) @ ( listre5045558437749241713iset_a @ R2 ) )
=> ~ ! [X: multiset_a,Xs3: list_multiset_a] :
( ( Xs
= ( cons_multiset_a @ X @ Xs3 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ Y ) @ R2 )
=> ~ ( member1640862112313989456iset_a @ ( produc1434922032272282391iset_a @ Xs3 @ Ys ) @ ( listre5045558437749241713iset_a @ R2 ) ) ) ) ) ).
% listrel_Cons2
thf(fact_821_listrel__Cons2,axiom,
! [Xs: list_nat_nat,Y: nat,Ys: list_nat,R2: set_Pr9093778441882193744at_nat] :
( ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ Xs @ ( cons_nat @ Y @ Ys ) ) @ ( listrel_nat_nat_nat @ R2 ) )
=> ~ ! [X: nat > nat,Xs3: list_nat_nat] :
( ( Xs
= ( cons_nat_nat @ X @ Xs3 ) )
=> ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X @ Y ) @ R2 )
=> ~ ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ Xs3 @ Ys ) @ ( listrel_nat_nat_nat @ R2 ) ) ) ) ) ).
% listrel_Cons2
thf(fact_822_listrel__Cons1,axiom,
! [Y: a,Ys: list_a,Xs: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ Y @ Ys ) @ Xs ) @ ( listrel_a_a @ R2 ) )
=> ~ ! [Y3: a,Ys3: list_a] :
( ( Xs
= ( cons_a @ Y3 @ Ys3 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ Y3 ) @ R2 )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ys @ Ys3 ) @ ( listrel_a_a @ R2 ) ) ) ) ) ).
% listrel_Cons1
thf(fact_823_listrel__Cons1,axiom,
! [Y: multiset_a,Ys: list_multiset_a,Xs: list_multiset_a,R2: set_Pr79727621955416071iset_a] :
( ( member1640862112313989456iset_a @ ( produc1434922032272282391iset_a @ ( cons_multiset_a @ Y @ Ys ) @ Xs ) @ ( listre5045558437749241713iset_a @ R2 ) )
=> ~ ! [Y3: multiset_a,Ys3: list_multiset_a] :
( ( Xs
= ( cons_multiset_a @ Y3 @ Ys3 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Y @ Y3 ) @ R2 )
=> ~ ( member1640862112313989456iset_a @ ( produc1434922032272282391iset_a @ Ys @ Ys3 ) @ ( listre5045558437749241713iset_a @ R2 ) ) ) ) ) ).
% listrel_Cons1
thf(fact_824_listrel__Cons1,axiom,
! [Y: nat > nat,Ys: list_nat_nat,Xs: list_nat,R2: set_Pr9093778441882193744at_nat] :
( ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ ( cons_nat_nat @ Y @ Ys ) @ Xs ) @ ( listrel_nat_nat_nat @ R2 ) )
=> ~ ! [Y3: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ Y3 @ Ys3 ) )
=> ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ Y @ Y3 ) @ R2 )
=> ~ ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ Ys @ Ys3 ) @ ( listrel_nat_nat_nat @ R2 ) ) ) ) ) ).
% listrel_Cons1
thf(fact_825_listrel_OCons,axiom,
! [X3: a,Y: a,R2: set_Product_prod_a_a,Xs: list_a,Ys: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ R2 )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel_a_a @ R2 ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X3 @ Xs ) @ ( cons_a @ Y @ Ys ) ) @ ( listrel_a_a @ R2 ) ) ) ) ).
% listrel.Cons
thf(fact_826_listrel_OCons,axiom,
! [X3: multiset_a,Y: multiset_a,R2: set_Pr79727621955416071iset_a,Xs: list_multiset_a,Ys: list_multiset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ Y ) @ R2 )
=> ( ( member1640862112313989456iset_a @ ( produc1434922032272282391iset_a @ Xs @ Ys ) @ ( listre5045558437749241713iset_a @ R2 ) )
=> ( member1640862112313989456iset_a @ ( produc1434922032272282391iset_a @ ( cons_multiset_a @ X3 @ Xs ) @ ( cons_multiset_a @ Y @ Ys ) ) @ ( listre5045558437749241713iset_a @ R2 ) ) ) ) ).
% listrel.Cons
thf(fact_827_listrel_OCons,axiom,
! [X3: nat > nat,Y: nat,R2: set_Pr9093778441882193744at_nat,Xs: list_nat_nat,Ys: list_nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X3 @ Y ) @ R2 )
=> ( ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ Xs @ Ys ) @ ( listrel_nat_nat_nat @ R2 ) )
=> ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ ( cons_nat_nat @ X3 @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( listrel_nat_nat_nat @ R2 ) ) ) ) ).
% listrel.Cons
thf(fact_828_not__less__empty,axiom,
! [M: multiset_a,R2: set_Product_prod_a_a] :
~ ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ M @ zero_zero_multiset_a ) @ ( mult1_a @ R2 ) ) ).
% not_less_empty
thf(fact_829_mult1__union,axiom,
! [B5: multiset_a,D: multiset_a,R2: set_Product_prod_a_a,C3: multiset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ B5 @ D ) @ ( mult1_a @ R2 ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( plus_plus_multiset_a @ C3 @ B5 ) @ ( plus_plus_multiset_a @ C3 @ D ) ) @ ( mult1_a @ R2 ) ) ) ).
% mult1_union
thf(fact_830_insert__subset__eq__iff,axiom,
! [A2: product_prod_a_a,A4: multis501812127501805293od_a_a,B5: multis501812127501805293od_a_a] :
( ( subset4108007524207251342od_a_a @ ( add_ms8655138167283798533od_a_a @ A2 @ A4 ) @ B5 )
= ( ( member1426531477525435216od_a_a @ A2 @ ( set_ms119794563918438244od_a_a @ B5 ) )
& ( subset4108007524207251342od_a_a @ A4 @ ( minus_7313152569058643924od_a_a @ B5 @ ( add_ms8655138167283798533od_a_a @ A2 @ zero_z3024502163831676836od_a_a ) ) ) ) ) ).
% insert_subset_eq_iff
thf(fact_831_insert__subset__eq__iff,axiom,
! [A2: produc6518373309651786023iset_a,A4: multis4781622204924204013iset_a,B5: multis4781622204924204013iset_a] :
( ( subset6292090195676340366iset_a @ ( add_ms5348986034529679365iset_a @ A2 @ A4 ) @ B5 )
= ( ( member5199237121806060112iset_a @ A2 @ ( set_ms2379723421352415844iset_a @ B5 ) )
& ( subset6292090195676340366iset_a @ A4 @ ( minus_9211619285413683668iset_a @ B5 @ ( add_ms5348986034529679365iset_a @ A2 @ zero_z4591712717340762532iset_a ) ) ) ) ) ).
% insert_subset_eq_iff
thf(fact_832_insert__subset__eq__iff,axiom,
! [A2: a,A4: multiset_a,B5: multiset_a] :
( ( subseteq_mset_a @ ( add_mset_a @ A2 @ A4 ) @ B5 )
= ( ( member_a @ A2 @ ( set_mset_a @ B5 ) )
& ( subseteq_mset_a @ A4 @ ( minus_3765977307040488491iset_a @ B5 @ ( add_mset_a @ A2 @ zero_zero_multiset_a ) ) ) ) ) ).
% insert_subset_eq_iff
thf(fact_833_diff__union__single__conv,axiom,
! [A2: product_prod_a_a,J2: multis501812127501805293od_a_a,I5: multis501812127501805293od_a_a] :
( ( member1426531477525435216od_a_a @ A2 @ ( set_ms119794563918438244od_a_a @ J2 ) )
=> ( ( minus_7313152569058643924od_a_a @ ( plus_p6967097483979887908od_a_a @ I5 @ J2 ) @ ( add_ms8655138167283798533od_a_a @ A2 @ zero_z3024502163831676836od_a_a ) )
= ( plus_p6967097483979887908od_a_a @ I5 @ ( minus_7313152569058643924od_a_a @ J2 @ ( add_ms8655138167283798533od_a_a @ A2 @ zero_z3024502163831676836od_a_a ) ) ) ) ) ).
% diff_union_single_conv
thf(fact_834_diff__union__single__conv,axiom,
! [A2: produc6518373309651786023iset_a,J2: multis4781622204924204013iset_a,I5: multis4781622204924204013iset_a] :
( ( member5199237121806060112iset_a @ A2 @ ( set_ms2379723421352415844iset_a @ J2 ) )
=> ( ( minus_9211619285413683668iset_a @ ( plus_p1053901119268525860iset_a @ I5 @ J2 ) @ ( add_ms5348986034529679365iset_a @ A2 @ zero_z4591712717340762532iset_a ) )
= ( plus_p1053901119268525860iset_a @ I5 @ ( minus_9211619285413683668iset_a @ J2 @ ( add_ms5348986034529679365iset_a @ A2 @ zero_z4591712717340762532iset_a ) ) ) ) ) ).
% diff_union_single_conv
thf(fact_835_diff__union__single__conv,axiom,
! [A2: a,J2: multiset_a,I5: multiset_a] :
( ( member_a @ A2 @ ( set_mset_a @ J2 ) )
=> ( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ I5 @ J2 ) @ ( add_mset_a @ A2 @ zero_zero_multiset_a ) )
= ( plus_plus_multiset_a @ I5 @ ( minus_3765977307040488491iset_a @ J2 @ ( add_mset_a @ A2 @ zero_zero_multiset_a ) ) ) ) ) ).
% diff_union_single_conv
thf(fact_836_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_837_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_838_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_839_listrel_Osimps,axiom,
! [A13: list_a,A24: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ A13 @ A24 ) @ ( listrel_a_a @ R2 ) )
= ( ( ( A13 = nil_a )
& ( A24 = nil_a ) )
| ? [X5: a,Y8: a,Xs4: list_a,Ys4: list_a] :
( ( A13
= ( cons_a @ X5 @ Xs4 ) )
& ( A24
= ( cons_a @ Y8 @ Ys4 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y8 ) @ R2 )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs4 @ Ys4 ) @ ( listrel_a_a @ R2 ) ) ) ) ) ).
% listrel.simps
thf(fact_840_listrel_Osimps,axiom,
! [A13: list_multiset_a,A24: list_multiset_a,R2: set_Pr79727621955416071iset_a] :
( ( member1640862112313989456iset_a @ ( produc1434922032272282391iset_a @ A13 @ A24 ) @ ( listre5045558437749241713iset_a @ R2 ) )
= ( ( ( A13 = nil_multiset_a )
& ( A24 = nil_multiset_a ) )
| ? [X5: multiset_a,Y8: multiset_a,Xs4: list_multiset_a,Ys4: list_multiset_a] :
( ( A13
= ( cons_multiset_a @ X5 @ Xs4 ) )
& ( A24
= ( cons_multiset_a @ Y8 @ Ys4 ) )
& ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Y8 ) @ R2 )
& ( member1640862112313989456iset_a @ ( produc1434922032272282391iset_a @ Xs4 @ Ys4 ) @ ( listre5045558437749241713iset_a @ R2 ) ) ) ) ) ).
% listrel.simps
thf(fact_841_listrel_Osimps,axiom,
! [A13: list_nat_nat,A24: list_nat,R2: set_Pr9093778441882193744at_nat] :
( ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ A13 @ A24 ) @ ( listrel_nat_nat_nat @ R2 ) )
= ( ( ( A13 = nil_nat_nat )
& ( A24 = nil_nat ) )
| ? [X5: nat > nat,Y8: nat,Xs4: list_nat_nat,Ys4: list_nat] :
( ( A13
= ( cons_nat_nat @ X5 @ Xs4 ) )
& ( A24
= ( cons_nat @ Y8 @ Ys4 ) )
& ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X5 @ Y8 ) @ R2 )
& ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ Xs4 @ Ys4 ) @ ( listrel_nat_nat_nat @ R2 ) ) ) ) ) ).
% listrel.simps
thf(fact_842_listrel_Ocases,axiom,
! [A13: list_a,A24: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ A13 @ A24 ) @ ( listrel_a_a @ R2 ) )
=> ( ( ( A13 = nil_a )
=> ( A24 != nil_a ) )
=> ~ ! [X: a,Y3: a,Xs3: list_a] :
( ( A13
= ( cons_a @ X @ Xs3 ) )
=> ! [Ys3: list_a] :
( ( A24
= ( cons_a @ Y3 @ Ys3 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ R2 )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs3 @ Ys3 ) @ ( listrel_a_a @ R2 ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_843_listrel_Ocases,axiom,
! [A13: list_multiset_a,A24: list_multiset_a,R2: set_Pr79727621955416071iset_a] :
( ( member1640862112313989456iset_a @ ( produc1434922032272282391iset_a @ A13 @ A24 ) @ ( listre5045558437749241713iset_a @ R2 ) )
=> ( ( ( A13 = nil_multiset_a )
=> ( A24 != nil_multiset_a ) )
=> ~ ! [X: multiset_a,Y3: multiset_a,Xs3: list_multiset_a] :
( ( A13
= ( cons_multiset_a @ X @ Xs3 ) )
=> ! [Ys3: list_multiset_a] :
( ( A24
= ( cons_multiset_a @ Y3 @ Ys3 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ Y3 ) @ R2 )
=> ~ ( member1640862112313989456iset_a @ ( produc1434922032272282391iset_a @ Xs3 @ Ys3 ) @ ( listre5045558437749241713iset_a @ R2 ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_844_listrel_Ocases,axiom,
! [A13: list_nat_nat,A24: list_nat,R2: set_Pr9093778441882193744at_nat] :
( ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ A13 @ A24 ) @ ( listrel_nat_nat_nat @ R2 ) )
=> ( ( ( A13 = nil_nat_nat )
=> ( A24 != nil_nat ) )
=> ~ ! [X: nat > nat,Y3: nat,Xs3: list_nat_nat] :
( ( A13
= ( cons_nat_nat @ X @ Xs3 ) )
=> ! [Ys3: list_nat] :
( ( A24
= ( cons_nat @ Y3 @ Ys3 ) )
=> ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X @ Y3 ) @ R2 )
=> ~ ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ Xs3 @ Ys3 ) @ ( listrel_nat_nat_nat @ R2 ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_845_subset__mset_Osum__list__update,axiom,
! [K2: nat,Xs: list_multiset_a,X3: multiset_a] :
( ( ord_less_nat @ K2 @ ( size_s3003218658903709366iset_a @ Xs ) )
=> ( ( groups778166972783649551iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ ( list_u788720842672076722iset_a @ Xs @ K2 @ X3 ) )
= ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ ( groups778166972783649551iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ Xs ) @ X3 ) @ ( nth_multiset_a @ Xs @ K2 ) ) ) ) ).
% subset_mset.sum_list_update
thf(fact_846_size__Diff2__less,axiom,
! [X3: product_prod_a_a,M: multis501812127501805293od_a_a,Y: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ M ) )
=> ( ( member1426531477525435216od_a_a @ Y @ ( set_ms119794563918438244od_a_a @ M ) )
=> ( ord_less_nat @ ( size_s781968976467208537od_a_a @ ( minus_7313152569058643924od_a_a @ ( minus_7313152569058643924od_a_a @ M @ ( add_ms8655138167283798533od_a_a @ X3 @ zero_z3024502163831676836od_a_a ) ) @ ( add_ms8655138167283798533od_a_a @ Y @ zero_z3024502163831676836od_a_a ) ) ) @ ( size_s781968976467208537od_a_a @ M ) ) ) ) ).
% size_Diff2_less
thf(fact_847_size__Diff2__less,axiom,
! [X3: produc6518373309651786023iset_a,M: multis4781622204924204013iset_a,Y: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ M ) )
=> ( ( member5199237121806060112iset_a @ Y @ ( set_ms2379723421352415844iset_a @ M ) )
=> ( ord_less_nat @ ( size_s2629572463880657497iset_a @ ( minus_9211619285413683668iset_a @ ( minus_9211619285413683668iset_a @ M @ ( add_ms5348986034529679365iset_a @ X3 @ zero_z4591712717340762532iset_a ) ) @ ( add_ms5348986034529679365iset_a @ Y @ zero_z4591712717340762532iset_a ) ) ) @ ( size_s2629572463880657497iset_a @ M ) ) ) ) ).
% size_Diff2_less
thf(fact_848_size__Diff2__less,axiom,
! [X3: a,M: multiset_a,Y: a] :
( ( member_a @ X3 @ ( set_mset_a @ M ) )
=> ( ( member_a @ Y @ ( set_mset_a @ M ) )
=> ( ord_less_nat @ ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ X3 @ zero_zero_multiset_a ) ) @ ( add_mset_a @ Y @ zero_zero_multiset_a ) ) ) @ ( size_size_multiset_a @ M ) ) ) ) ).
% size_Diff2_less
thf(fact_849_size__Diff1__less,axiom,
! [X3: product_prod_a_a,M: multis501812127501805293od_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ M ) )
=> ( ord_less_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_less
thf(fact_850_size__Diff1__less,axiom,
! [X3: produc6518373309651786023iset_a,M: multis4781622204924204013iset_a] :
( ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ M ) )
=> ( ord_less_nat @ ( size_s2629572463880657497iset_a @ ( minus_9211619285413683668iset_a @ M @ ( add_ms5348986034529679365iset_a @ X3 @ zero_z4591712717340762532iset_a ) ) ) @ ( size_s2629572463880657497iset_a @ M ) ) ) ).
% size_Diff1_less
thf(fact_851_size__Diff1__less,axiom,
! [X3: a,M: multiset_a] :
( ( member_a @ X3 @ ( set_mset_a @ M ) )
=> ( ord_less_nat @ ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ X3 @ zero_zero_multiset_a ) ) ) @ ( size_size_multiset_a @ M ) ) ) ).
% size_Diff1_less
thf(fact_852_the__elem__set,axiom,
! [X3: a] :
( ( the_elem_a @ ( set_a2 @ ( cons_a @ X3 @ nil_a ) ) )
= X3 ) ).
% the_elem_set
thf(fact_853_Cons__lenlex__iff,axiom,
! [M3: multiset_a,Ms: list_multiset_a,N3: multiset_a,Ns: list_multiset_a,R2: set_Pr79727621955416071iset_a] :
( ( member1640862112313989456iset_a @ ( produc1434922032272282391iset_a @ ( cons_multiset_a @ M3 @ Ms ) @ ( cons_multiset_a @ N3 @ Ns ) ) @ ( lenlex_multiset_a @ R2 ) )
= ( ( ord_less_nat @ ( size_s3003218658903709366iset_a @ Ms ) @ ( size_s3003218658903709366iset_a @ Ns ) )
| ( ( ( size_s3003218658903709366iset_a @ Ms )
= ( size_s3003218658903709366iset_a @ Ns ) )
& ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ M3 @ N3 ) @ R2 ) )
| ( ( M3 = N3 )
& ( member1640862112313989456iset_a @ ( produc1434922032272282391iset_a @ Ms @ Ns ) @ ( lenlex_multiset_a @ R2 ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_854_Cons__lenlex__iff,axiom,
! [M3: a,Ms: list_a,N3: a,Ns: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ M3 @ Ms ) @ ( cons_a @ N3 @ Ns ) ) @ ( lenlex_a @ R2 ) )
= ( ( ord_less_nat @ ( size_size_list_a @ Ms ) @ ( size_size_list_a @ Ns ) )
| ( ( ( size_size_list_a @ Ms )
= ( size_size_list_a @ Ns ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ M3 @ N3 ) @ R2 ) )
| ( ( M3 = N3 )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ms @ Ns ) @ ( lenlex_a @ R2 ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_855_Cons__in__lex,axiom,
! [X3: multiset_a,Xs: list_multiset_a,Y: multiset_a,Ys: list_multiset_a,R2: set_Pr79727621955416071iset_a] :
( ( member1640862112313989456iset_a @ ( produc1434922032272282391iset_a @ ( cons_multiset_a @ X3 @ Xs ) @ ( cons_multiset_a @ Y @ Ys ) ) @ ( lex_multiset_a @ R2 ) )
= ( ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ Y ) @ R2 )
& ( ( size_s3003218658903709366iset_a @ Xs )
= ( size_s3003218658903709366iset_a @ Ys ) ) )
| ( ( X3 = Y )
& ( member1640862112313989456iset_a @ ( produc1434922032272282391iset_a @ Xs @ Ys ) @ ( lex_multiset_a @ R2 ) ) ) ) ) ).
% Cons_in_lex
thf(fact_856_Cons__in__lex,axiom,
! [X3: a,Xs: list_a,Y: a,Ys: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X3 @ Xs ) @ ( cons_a @ Y @ Ys ) ) @ ( lex_a @ R2 ) )
= ( ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ R2 )
& ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) ) )
| ( ( X3 = Y )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( lex_a @ R2 ) ) ) ) ) ).
% Cons_in_lex
thf(fact_857_mset__remove1,axiom,
! [A2: a,Xs: list_a] :
( ( mset_a @ ( remove1_a @ A2 @ Xs ) )
= ( minus_3765977307040488491iset_a @ ( mset_a @ Xs ) @ ( add_mset_a @ A2 @ zero_zero_multiset_a ) ) ) ).
% mset_remove1
thf(fact_858_diff__self__eq__0,axiom,
! [M3: nat] :
( ( minus_minus_nat @ M3 @ M3 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_859_diff__0__eq__0,axiom,
! [N3: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N3 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_860_diff__diff__left,axiom,
! [I: nat,J: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K2 )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% diff_diff_left
thf(fact_861_zero__less__diff,axiom,
! [N3: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N3 @ M3 ) )
= ( ord_less_nat @ M3 @ N3 ) ) ).
% zero_less_diff
thf(fact_862_in__set__remove1,axiom,
! [A2: product_prod_a_a,B: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( A2 != B )
=> ( ( member1426531477525435216od_a_a @ A2 @ ( set_Product_prod_a_a2 @ ( remove3530114606240936131od_a_a @ B @ Xs ) ) )
= ( member1426531477525435216od_a_a @ A2 @ ( set_Product_prod_a_a2 @ Xs ) ) ) ) ).
% in_set_remove1
thf(fact_863_in__set__remove1,axiom,
! [A2: produc6518373309651786023iset_a,B: produc6518373309651786023iset_a,Xs: list_P6988024003011616557iset_a] :
( ( A2 != B )
=> ( ( member5199237121806060112iset_a @ A2 @ ( set_Pr5550698219922361532iset_a @ ( remove5063108440011681987iset_a @ B @ Xs ) ) )
= ( member5199237121806060112iset_a @ A2 @ ( set_Pr5550698219922361532iset_a @ Xs ) ) ) ) ).
% in_set_remove1
thf(fact_864_in__set__remove1,axiom,
! [A2: a,B: a,Xs: list_a] :
( ( A2 != B )
=> ( ( member_a @ A2 @ ( set_a2 @ ( remove1_a @ B @ Xs ) ) )
= ( member_a @ A2 @ ( set_a2 @ Xs ) ) ) ) ).
% in_set_remove1
thf(fact_865_minus__nat_Odiff__0,axiom,
! [M3: nat] :
( ( minus_minus_nat @ M3 @ zero_zero_nat )
= M3 ) ).
% minus_nat.diff_0
thf(fact_866_diffs0__imp__equal,axiom,
! [M3: nat,N3: nat] :
( ( ( minus_minus_nat @ M3 @ N3 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N3 @ M3 )
= zero_zero_nat )
=> ( M3 = N3 ) ) ) ).
% diffs0_imp_equal
thf(fact_867_less__imp__diff__less,axiom,
! [J: nat,K2: nat,N3: nat] :
( ( ord_less_nat @ J @ K2 )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N3 ) @ K2 ) ) ).
% less_imp_diff_less
thf(fact_868_diff__less__mono2,axiom,
! [M3: nat,N3: nat,L: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( ( ord_less_nat @ M3 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N3 ) @ ( minus_minus_nat @ L @ M3 ) ) ) ) ).
% diff_less_mono2
thf(fact_869_diff__add__inverse2,axiom,
! [M3: nat,N3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M3 @ N3 ) @ N3 )
= M3 ) ).
% diff_add_inverse2
thf(fact_870_diff__add__inverse,axiom,
! [N3: nat,M3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N3 @ M3 ) @ N3 )
= M3 ) ).
% diff_add_inverse
thf(fact_871_diff__cancel2,axiom,
! [M3: nat,K2: nat,N3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M3 @ K2 ) @ ( plus_plus_nat @ N3 @ K2 ) )
= ( minus_minus_nat @ M3 @ N3 ) ) ).
% diff_cancel2
thf(fact_872_Nat_Odiff__cancel,axiom,
! [K2: nat,M3: nat,N3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K2 @ M3 ) @ ( plus_plus_nat @ K2 @ N3 ) )
= ( minus_minus_nat @ M3 @ N3 ) ) ).
% Nat.diff_cancel
thf(fact_873_notin__set__remove1,axiom,
! [X3: product_prod_a_a,Xs: list_P1396940483166286381od_a_a,Y: product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Xs ) )
=> ~ ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ ( remove3530114606240936131od_a_a @ Y @ Xs ) ) ) ) ).
% notin_set_remove1
thf(fact_874_notin__set__remove1,axiom,
! [X3: produc6518373309651786023iset_a,Xs: list_P6988024003011616557iset_a,Y: produc6518373309651786023iset_a] :
( ~ ( member5199237121806060112iset_a @ X3 @ ( set_Pr5550698219922361532iset_a @ Xs ) )
=> ~ ( member5199237121806060112iset_a @ X3 @ ( set_Pr5550698219922361532iset_a @ ( remove5063108440011681987iset_a @ Y @ Xs ) ) ) ) ).
% notin_set_remove1
thf(fact_875_notin__set__remove1,axiom,
! [X3: a,Xs: list_a,Y: a] :
( ~ ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ~ ( member_a @ X3 @ ( set_a2 @ ( remove1_a @ Y @ Xs ) ) ) ) ).
% notin_set_remove1
thf(fact_876_remove1__idem,axiom,
! [X3: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ~ ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Xs ) )
=> ( ( remove3530114606240936131od_a_a @ X3 @ Xs )
= Xs ) ) ).
% remove1_idem
thf(fact_877_remove1__idem,axiom,
! [X3: produc6518373309651786023iset_a,Xs: list_P6988024003011616557iset_a] :
( ~ ( member5199237121806060112iset_a @ X3 @ ( set_Pr5550698219922361532iset_a @ Xs ) )
=> ( ( remove5063108440011681987iset_a @ X3 @ Xs )
= Xs ) ) ).
% remove1_idem
thf(fact_878_remove1__idem,axiom,
! [X3: a,Xs: list_a] :
( ~ ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( ( remove1_a @ X3 @ Xs )
= Xs ) ) ).
% remove1_idem
thf(fact_879_diff__less,axiom,
! [N3: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( ord_less_nat @ zero_zero_nat @ M3 )
=> ( ord_less_nat @ ( minus_minus_nat @ M3 @ N3 ) @ M3 ) ) ) ).
% diff_less
thf(fact_880_diff__add__0,axiom,
! [N3: nat,M3: nat] :
( ( minus_minus_nat @ N3 @ ( plus_plus_nat @ N3 @ M3 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_881_less__diff__conv,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ).
% less_diff_conv
thf(fact_882_add__diff__inverse__nat,axiom,
! [M3: nat,N3: nat] :
( ~ ( ord_less_nat @ M3 @ N3 )
=> ( ( plus_plus_nat @ N3 @ ( minus_minus_nat @ M3 @ N3 ) )
= M3 ) ) ).
% add_diff_inverse_nat
thf(fact_883_length__n__lists__elem,axiom,
! [Ys: list_a,N3: nat,Xs: list_a] :
( ( member_list_a @ Ys @ ( set_list_a2 @ ( n_lists_a @ N3 @ Xs ) ) )
=> ( ( size_size_list_a @ Ys )
= N3 ) ) ).
% length_n_lists_elem
thf(fact_884_lenlex__irreflexive,axiom,
! [R2: set_Product_prod_a_a,Xs: list_a] :
( ! [X: a] :
~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ X ) @ R2 )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Xs ) @ ( lenlex_a @ R2 ) ) ) ).
% lenlex_irreflexive
thf(fact_885_lenlex__irreflexive,axiom,
! [R2: set_Pr79727621955416071iset_a,Xs: list_multiset_a] :
( ! [X: multiset_a] :
~ ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ X ) @ R2 )
=> ~ ( member1640862112313989456iset_a @ ( produc1434922032272282391iset_a @ Xs @ Xs ) @ ( lenlex_multiset_a @ R2 ) ) ) ).
% lenlex_irreflexive
thf(fact_886_nat__diff__split__asm,axiom,
! [P2: nat > $o,A2: nat,B: nat] :
( ( P2 @ ( minus_minus_nat @ A2 @ B ) )
= ( ~ ( ( ( ord_less_nat @ A2 @ B )
& ~ ( P2 @ zero_zero_nat ) )
| ? [D4: nat] :
( ( A2
= ( plus_plus_nat @ B @ D4 ) )
& ~ ( P2 @ D4 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_887_nat__diff__split,axiom,
! [P2: nat > $o,A2: nat,B: nat] :
( ( P2 @ ( minus_minus_nat @ A2 @ B ) )
= ( ( ( ord_less_nat @ A2 @ B )
=> ( P2 @ zero_zero_nat ) )
& ! [D4: nat] :
( ( A2
= ( plus_plus_nat @ B @ D4 ) )
=> ( P2 @ D4 ) ) ) ) ).
% nat_diff_split
thf(fact_888_lex__take__index,axiom,
! [Xs: list_multiset_a,Ys: list_multiset_a,R2: set_Pr79727621955416071iset_a] :
( ( member1640862112313989456iset_a @ ( produc1434922032272282391iset_a @ Xs @ Ys ) @ ( lex_multiset_a @ R2 ) )
=> ~ ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3003218658903709366iset_a @ Xs ) )
=> ( ( ord_less_nat @ I3 @ ( size_s3003218658903709366iset_a @ Ys ) )
=> ( ( ( take_multiset_a @ I3 @ Xs )
= ( take_multiset_a @ I3 @ Ys ) )
=> ~ ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( nth_multiset_a @ Xs @ I3 ) @ ( nth_multiset_a @ Ys @ I3 ) ) @ R2 ) ) ) ) ) ).
% lex_take_index
thf(fact_889_lex__take__index,axiom,
! [Xs: list_a,Ys: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( lex_a @ R2 ) )
=> ~ ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
=> ( ( ord_less_nat @ I3 @ ( size_size_list_a @ Ys ) )
=> ( ( ( take_a @ I3 @ Xs )
= ( take_a @ I3 @ Ys ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs @ I3 ) @ ( nth_a @ Ys @ I3 ) ) @ R2 ) ) ) ) ) ).
% lex_take_index
thf(fact_890_in__measures_I2_J,axiom,
! [X3: a,Y: a,F: a > nat,Fs: list_a_nat] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ ( measures_a @ ( cons_a_nat @ F @ Fs ) ) )
= ( ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) )
| ( ( ( F @ X3 )
= ( F @ Y ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ ( measures_a @ Fs ) ) ) ) ) ).
% in_measures(2)
thf(fact_891_in__measures_I2_J,axiom,
! [X3: multiset_a,Y: multiset_a,F: multiset_a > nat,Fs: list_multiset_a_nat] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ Y ) @ ( measures_multiset_a @ ( cons_multiset_a_nat @ F @ Fs ) ) )
= ( ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) )
| ( ( ( F @ X3 )
= ( F @ Y ) )
& ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ Y ) @ ( measures_multiset_a @ Fs ) ) ) ) ) ).
% in_measures(2)
thf(fact_892_in__measures_I1_J,axiom,
! [X3: a,Y: a] :
~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ ( measures_a @ nil_a_nat ) ) ).
% in_measures(1)
thf(fact_893_in__measures_I1_J,axiom,
! [X3: multiset_a,Y: multiset_a] :
~ ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ Y ) @ ( measures_multiset_a @ nil_multiset_a_nat ) ) ).
% in_measures(1)
thf(fact_894_nth__take,axiom,
! [I: nat,N3: nat,Xs: list_a] :
( ( ord_less_nat @ I @ N3 )
=> ( ( nth_a @ ( take_a @ N3 @ Xs ) @ I )
= ( nth_a @ Xs @ I ) ) ) ).
% nth_take
thf(fact_895_diff__commute,axiom,
! [I: nat,J: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K2 )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K2 ) @ J ) ) ).
% diff_commute
thf(fact_896_in__set__takeD,axiom,
! [X3: product_prod_a_a,N3: nat,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ ( take_P1986783995523548949od_a_a @ N3 @ Xs ) ) )
=> ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Xs ) ) ) ).
% in_set_takeD
thf(fact_897_in__set__takeD,axiom,
! [X3: produc6518373309651786023iset_a,N3: nat,Xs: list_P6988024003011616557iset_a] :
( ( member5199237121806060112iset_a @ X3 @ ( set_Pr5550698219922361532iset_a @ ( take_P4419225098117143829iset_a @ N3 @ Xs ) ) )
=> ( member5199237121806060112iset_a @ X3 @ ( set_Pr5550698219922361532iset_a @ Xs ) ) ) ).
% in_set_takeD
thf(fact_898_in__set__takeD,axiom,
! [X3: a,N3: nat,Xs: list_a] :
( ( member_a @ X3 @ ( set_a2 @ ( take_a @ N3 @ Xs ) ) )
=> ( member_a @ X3 @ ( set_a2 @ Xs ) ) ) ).
% in_set_takeD
thf(fact_899_in__set__product__lists__length,axiom,
! [Xs: list_a,Xss: list_list_a] :
( ( member_list_a @ Xs @ ( set_list_a2 @ ( product_lists_a @ Xss ) ) )
=> ( ( size_size_list_a @ Xs )
= ( size_s349497388124573686list_a @ Xss ) ) ) ).
% in_set_product_lists_length
thf(fact_900_measures__less,axiom,
! [F: a > nat,X3: a,Y: a,Fs: list_a_nat] :
( ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ ( measures_a @ ( cons_a_nat @ F @ Fs ) ) ) ) ).
% measures_less
thf(fact_901_measures__less,axiom,
! [F: multiset_a > nat,X3: multiset_a,Y: multiset_a,Fs: list_multiset_a_nat] :
( ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ Y ) @ ( measures_multiset_a @ ( cons_multiset_a_nat @ F @ Fs ) ) ) ) ).
% measures_less
thf(fact_902_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
thf(fact_903_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_904_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_905_size__Diff__singleton__if,axiom,
! [X3: product_prod_a_a,A4: multis501812127501805293od_a_a] :
( ( ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ A4 ) )
=> ( ( size_s781968976467208537od_a_a @ ( minus_7313152569058643924od_a_a @ A4 @ ( add_ms8655138167283798533od_a_a @ X3 @ zero_z3024502163831676836od_a_a ) ) )
= ( minus_minus_nat @ ( size_s781968976467208537od_a_a @ A4 ) @ one_one_nat ) ) )
& ( ~ ( member1426531477525435216od_a_a @ X3 @ ( set_ms119794563918438244od_a_a @ A4 ) )
=> ( ( size_s781968976467208537od_a_a @ ( minus_7313152569058643924od_a_a @ A4 @ ( add_ms8655138167283798533od_a_a @ X3 @ zero_z3024502163831676836od_a_a ) ) )
= ( size_s781968976467208537od_a_a @ A4 ) ) ) ) ).
% size_Diff_singleton_if
thf(fact_906_size__Diff__singleton__if,axiom,
! [X3: produc6518373309651786023iset_a,A4: multis4781622204924204013iset_a] :
( ( ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ A4 ) )
=> ( ( size_s2629572463880657497iset_a @ ( minus_9211619285413683668iset_a @ A4 @ ( add_ms5348986034529679365iset_a @ X3 @ zero_z4591712717340762532iset_a ) ) )
= ( minus_minus_nat @ ( size_s2629572463880657497iset_a @ A4 ) @ one_one_nat ) ) )
& ( ~ ( member5199237121806060112iset_a @ X3 @ ( set_ms2379723421352415844iset_a @ A4 ) )
=> ( ( size_s2629572463880657497iset_a @ ( minus_9211619285413683668iset_a @ A4 @ ( add_ms5348986034529679365iset_a @ X3 @ zero_z4591712717340762532iset_a ) ) )
= ( size_s2629572463880657497iset_a @ A4 ) ) ) ) ).
% size_Diff_singleton_if
thf(fact_907_size__Diff__singleton__if,axiom,
! [X3: a,A4: multiset_a] :
( ( ( member_a @ X3 @ ( set_mset_a @ A4 ) )
=> ( ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ A4 @ ( add_mset_a @ X3 @ zero_zero_multiset_a ) ) )
= ( minus_minus_nat @ ( size_size_multiset_a @ A4 ) @ one_one_nat ) ) )
& ( ~ ( member_a @ X3 @ ( set_mset_a @ A4 ) )
=> ( ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ A4 @ ( add_mset_a @ X3 @ zero_zero_multiset_a ) ) )
= ( size_size_multiset_a @ A4 ) ) ) ) ).
% size_Diff_singleton_if
thf(fact_908_nth__Cons__pos,axiom,
! [N3: nat,X3: a,Xs: list_a] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( nth_a @ ( cons_a @ X3 @ Xs ) @ N3 )
= ( nth_a @ Xs @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_909_less__one,axiom,
! [N3: nat] :
( ( ord_less_nat @ N3 @ one_one_nat )
= ( N3 = zero_zero_nat ) ) ).
% less_one
thf(fact_910_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_911_one__reorient,axiom,
! [X3: nat] :
( ( one_one_nat = X3 )
= ( X3 = one_one_nat ) ) ).
% one_reorient
thf(fact_912_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_913_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_914_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_915_less__add__one,axiom,
! [A2: nat] : ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ one_one_nat ) ) ).
% less_add_one
thf(fact_916_add__mono1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_917_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_918_nth__Cons_H,axiom,
! [N3: nat,X3: a,Xs: list_a] :
( ( ( N3 = zero_zero_nat )
=> ( ( nth_a @ ( cons_a @ X3 @ Xs ) @ N3 )
= X3 ) )
& ( ( N3 != zero_zero_nat )
=> ( ( nth_a @ ( cons_a @ X3 @ Xs ) @ N3 )
= ( nth_a @ Xs @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_919_length__remove1,axiom,
! [X3: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Xs ) )
=> ( ( size_s3885678630836030617od_a_a @ ( remove3530114606240936131od_a_a @ X3 @ Xs ) )
= ( minus_minus_nat @ ( size_s3885678630836030617od_a_a @ Xs ) @ one_one_nat ) ) )
& ( ~ ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Xs ) )
=> ( ( size_s3885678630836030617od_a_a @ ( remove3530114606240936131od_a_a @ X3 @ Xs ) )
= ( size_s3885678630836030617od_a_a @ Xs ) ) ) ) ).
% length_remove1
thf(fact_920_length__remove1,axiom,
! [X3: produc6518373309651786023iset_a,Xs: list_P6988024003011616557iset_a] :
( ( ( member5199237121806060112iset_a @ X3 @ ( set_Pr5550698219922361532iset_a @ Xs ) )
=> ( ( size_s7238940367606365593iset_a @ ( remove5063108440011681987iset_a @ X3 @ Xs ) )
= ( minus_minus_nat @ ( size_s7238940367606365593iset_a @ Xs ) @ one_one_nat ) ) )
& ( ~ ( member5199237121806060112iset_a @ X3 @ ( set_Pr5550698219922361532iset_a @ Xs ) )
=> ( ( size_s7238940367606365593iset_a @ ( remove5063108440011681987iset_a @ X3 @ Xs ) )
= ( size_s7238940367606365593iset_a @ Xs ) ) ) ) ).
% length_remove1
thf(fact_921_length__remove1,axiom,
! [X3: a,Xs: list_a] :
( ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( ( size_size_list_a @ ( remove1_a @ X3 @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) ) )
& ( ~ ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( ( size_size_list_a @ ( remove1_a @ X3 @ Xs ) )
= ( size_size_list_a @ Xs ) ) ) ) ).
% length_remove1
thf(fact_922_nth__non__equal__first__eq,axiom,
! [X3: a,Y: a,Xs: list_a,N3: nat] :
( ( X3 != Y )
=> ( ( ( nth_a @ ( cons_a @ X3 @ Xs ) @ N3 )
= Y )
= ( ( ( nth_a @ Xs @ ( minus_minus_nat @ N3 @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_923_last__list__update,axiom,
! [Xs: list_a,K2: nat,X3: a] :
( ( Xs != nil_a )
=> ( ( ( K2
= ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) )
=> ( ( last_a @ ( list_update_a @ Xs @ K2 @ X3 ) )
= X3 ) )
& ( ( K2
!= ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) )
=> ( ( last_a @ ( list_update_a @ Xs @ K2 @ X3 ) )
= ( last_a @ Xs ) ) ) ) ) ).
% last_list_update
thf(fact_924_last__zip,axiom,
! [Xs: list_multiset_a,Ys: list_multiset_a] :
( ( Xs != nil_multiset_a )
=> ( ( Ys != nil_multiset_a )
=> ( ( ( size_s3003218658903709366iset_a @ Xs )
= ( size_s3003218658903709366iset_a @ Ys ) )
=> ( ( last_P8747557137360899270iset_a @ ( zip_mu5258324667929623607iset_a @ Xs @ Ys ) )
= ( produc654756711066625303iset_a @ ( last_multiset_a @ Xs ) @ ( last_multiset_a @ Ys ) ) ) ) ) ) ).
% last_zip
thf(fact_925_last__zip,axiom,
! [Xs: list_nat_nat,Ys: list_nat] :
( ( Xs != nil_nat_nat )
=> ( ( Ys != nil_nat )
=> ( ( ( size_s8208510060688613859at_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( last_P1651921108130080911at_nat @ ( zip_nat_nat_nat @ Xs @ Ys ) )
= ( produc72220940542539688at_nat @ ( last_nat_nat @ Xs ) @ ( last_nat @ Ys ) ) ) ) ) ) ).
% last_zip
thf(fact_926_last__zip,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != nil_a )
=> ( ( Ys != nil_a )
=> ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( last_P8790725268278465478od_a_a @ ( zip_a_a @ Xs @ Ys ) )
= ( product_Pair_a_a @ ( last_a @ Xs ) @ ( last_a @ Ys ) ) ) ) ) ) ).
% last_zip
thf(fact_927_last__in__set,axiom,
! [As: list_P1396940483166286381od_a_a] :
( ( As != nil_Product_prod_a_a )
=> ( member1426531477525435216od_a_a @ ( last_P8790725268278465478od_a_a @ As ) @ ( set_Product_prod_a_a2 @ As ) ) ) ).
% last_in_set
thf(fact_928_last__in__set,axiom,
! [As: list_P6988024003011616557iset_a] :
( ( As != nil_Pr2722442113885406509iset_a )
=> ( member5199237121806060112iset_a @ ( last_P8747557137360899270iset_a @ As ) @ ( set_Pr5550698219922361532iset_a @ As ) ) ) ).
% last_in_set
thf(fact_929_last__in__set,axiom,
! [As: list_a] :
( ( As != nil_a )
=> ( member_a @ ( last_a @ As ) @ ( set_a2 @ As ) ) ) ).
% last_in_set
thf(fact_930_last__conv__nth,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( last_a @ Xs )
= ( nth_a @ Xs @ ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) ) ) ) ).
% last_conv_nth
thf(fact_931_sum__mset_Oremove,axiom,
! [X3: multiset_a,A4: multiset_multiset_a] :
( ( member_multiset_a @ X3 @ ( set_mset_multiset_a @ A4 ) )
=> ( ( comm_m4595494212910443721iset_a @ A4 )
= ( plus_plus_multiset_a @ X3 @ ( comm_m4595494212910443721iset_a @ ( minus_3395427628221709681iset_a @ A4 @ ( add_mset_multiset_a @ X3 @ zero_z7799948378220188993iset_a ) ) ) ) ) ) ).
% sum_mset.remove
thf(fact_932_sum__mset_Oremove,axiom,
! [X3: nat,A4: multiset_nat] :
( ( member_nat @ X3 @ ( set_mset_nat @ A4 ) )
=> ( ( comm_m762188921832702859et_nat @ A4 )
= ( plus_plus_nat @ X3 @ ( comm_m762188921832702859et_nat @ ( minus_8522176038001411705et_nat @ A4 @ ( add_mset_nat @ X3 @ zero_z7348594199698428585et_nat ) ) ) ) ) ) ).
% sum_mset.remove
thf(fact_933_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_934_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_935_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_936_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_937_nat_Oinject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nat.inject
thf(fact_938_Suc__less__eq,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_nat @ ( suc @ M3 ) @ ( suc @ N3 ) )
= ( ord_less_nat @ M3 @ N3 ) ) ).
% Suc_less_eq
thf(fact_939_Suc__mono,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( ord_less_nat @ ( suc @ M3 ) @ ( suc @ N3 ) ) ) ).
% Suc_mono
thf(fact_940_lessI,axiom,
! [N3: nat] : ( ord_less_nat @ N3 @ ( suc @ N3 ) ) ).
% lessI
thf(fact_941_add__Suc__right,axiom,
! [M3: nat,N3: nat] :
( ( plus_plus_nat @ M3 @ ( suc @ N3 ) )
= ( suc @ ( plus_plus_nat @ M3 @ N3 ) ) ) ).
% add_Suc_right
thf(fact_942_diff__Suc__Suc,axiom,
! [M3: nat,N3: nat] :
( ( minus_minus_nat @ ( suc @ M3 ) @ ( suc @ N3 ) )
= ( minus_minus_nat @ M3 @ N3 ) ) ).
% diff_Suc_Suc
thf(fact_943_Suc__diff__diff,axiom,
! [M3: nat,N3: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M3 ) @ N3 ) @ ( suc @ K2 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M3 @ N3 ) @ K2 ) ) ).
% Suc_diff_diff
thf(fact_944_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_945_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_946_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_947_zero__less__Suc,axiom,
! [N3: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N3 ) ) ).
% zero_less_Suc
thf(fact_948_less__Suc0,axiom,
! [N3: nat] :
( ( ord_less_nat @ N3 @ ( suc @ zero_zero_nat ) )
= ( N3 = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_949_nth__Cons__Suc,axiom,
! [X3: a,Xs: list_a,N3: nat] :
( ( nth_a @ ( cons_a @ X3 @ Xs ) @ ( suc @ N3 ) )
= ( nth_a @ Xs @ N3 ) ) ).
% nth_Cons_Suc
thf(fact_950_diff__Suc__1,axiom,
! [N3: nat] :
( ( minus_minus_nat @ ( suc @ N3 ) @ one_one_nat )
= N3 ) ).
% diff_Suc_1
thf(fact_951_sum__mset_Oempty,axiom,
( ( comm_m762188921832702859et_nat @ zero_z7348594199698428585et_nat )
= zero_zero_nat ) ).
% sum_mset.empty
thf(fact_952_sum__mset__0__iff,axiom,
! [M: multiset_nat] :
( ( ( comm_m762188921832702859et_nat @ M )
= zero_zero_nat )
= ( ! [X5: nat] :
( ( member_nat @ X5 @ ( set_mset_nat @ M ) )
=> ( X5 = zero_zero_nat ) ) ) ) ).
% sum_mset_0_iff
thf(fact_953_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_954_sum__mset_Ounion,axiom,
! [M: multiset_nat,N: multiset_nat] :
( ( comm_m762188921832702859et_nat @ ( plus_p6334493942879108393et_nat @ M @ N ) )
= ( plus_plus_nat @ ( comm_m762188921832702859et_nat @ M ) @ ( comm_m762188921832702859et_nat @ N ) ) ) ).
% sum_mset.union
thf(fact_955_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_956_sum__mset_Oadd__mset,axiom,
! [X3: nat,N: multiset_nat] :
( ( comm_m762188921832702859et_nat @ ( add_mset_nat @ X3 @ N ) )
= ( plus_plus_nat @ X3 @ ( comm_m762188921832702859et_nat @ N ) ) ) ).
% sum_mset.add_mset
thf(fact_957_Suc__pred,axiom,
! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( suc @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) )
= N3 ) ) ).
% Suc_pred
thf(fact_958_Suc__diff__1,axiom,
! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( suc @ ( minus_minus_nat @ N3 @ one_one_nat ) )
= N3 ) ) ).
% Suc_diff_1
thf(fact_959_zero__induct__lemma,axiom,
! [P2: nat > $o,K2: nat,I: nat] :
( ( P2 @ K2 )
=> ( ! [N4: nat] :
( ( P2 @ ( suc @ N4 ) )
=> ( P2 @ N4 ) )
=> ( P2 @ ( minus_minus_nat @ K2 @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_960_nat_Odistinct_I1_J,axiom,
! [X2: nat] :
( zero_zero_nat
!= ( suc @ X2 ) ) ).
% nat.distinct(1)
thf(fact_961_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_962_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_963_nat_OdiscI,axiom,
! [Nat: nat,X2: nat] :
( ( Nat
= ( suc @ X2 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_964_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_965_nat__induct,axiom,
! [P2: nat > $o,N3: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( P2 @ N4 )
=> ( P2 @ ( suc @ N4 ) ) )
=> ( P2 @ N3 ) ) ) ).
% nat_induct
thf(fact_966_diff__induct,axiom,
! [P2: nat > nat > $o,M3: nat,N3: nat] :
( ! [X: nat] : ( P2 @ X @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P2 @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X: nat,Y3: nat] :
( ( P2 @ X @ Y3 )
=> ( P2 @ ( suc @ X ) @ ( suc @ Y3 ) ) )
=> ( P2 @ M3 @ N3 ) ) ) ) ).
% diff_induct
thf(fact_967_zero__induct,axiom,
! [P2: nat > $o,K2: nat] :
( ( P2 @ K2 )
=> ( ! [N4: nat] :
( ( P2 @ ( suc @ N4 ) )
=> ( P2 @ N4 ) )
=> ( P2 @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_968_Suc__neq__Zero,axiom,
! [M3: nat] :
( ( suc @ M3 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_969_Zero__neq__Suc,axiom,
! [M3: nat] :
( zero_zero_nat
!= ( suc @ M3 ) ) ).
% Zero_neq_Suc
thf(fact_970_Zero__not__Suc,axiom,
! [M3: nat] :
( zero_zero_nat
!= ( suc @ M3 ) ) ).
% Zero_not_Suc
thf(fact_971_not0__implies__Suc,axiom,
! [N3: nat] :
( ( N3 != zero_zero_nat )
=> ? [M6: nat] :
( N3
= ( suc @ M6 ) ) ) ).
% not0_implies_Suc
thf(fact_972_not__less__less__Suc__eq,axiom,
! [N3: nat,M3: nat] :
( ~ ( ord_less_nat @ N3 @ M3 )
=> ( ( ord_less_nat @ N3 @ ( suc @ M3 ) )
= ( N3 = M3 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_973_strict__inc__induct,axiom,
! [I: nat,J: nat,P2: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P2 @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( P2 @ ( suc @ I3 ) )
=> ( P2 @ I3 ) ) )
=> ( P2 @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_974_less__Suc__induct,axiom,
! [I: nat,J: nat,P2: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] : ( P2 @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J3: nat,K3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ K3 )
=> ( ( P2 @ I3 @ J3 )
=> ( ( P2 @ J3 @ K3 )
=> ( P2 @ I3 @ K3 ) ) ) ) )
=> ( P2 @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_975_less__trans__Suc,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K2 )
=> ( ord_less_nat @ ( suc @ I ) @ K2 ) ) ) ).
% less_trans_Suc
thf(fact_976_Suc__less__SucD,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_nat @ ( suc @ M3 ) @ ( suc @ N3 ) )
=> ( ord_less_nat @ M3 @ N3 ) ) ).
% Suc_less_SucD
thf(fact_977_less__antisym,axiom,
! [N3: nat,M3: nat] :
( ~ ( ord_less_nat @ N3 @ M3 )
=> ( ( ord_less_nat @ N3 @ ( suc @ M3 ) )
=> ( M3 = N3 ) ) ) ).
% less_antisym
thf(fact_978_Suc__less__eq2,axiom,
! [N3: nat,M3: nat] :
( ( ord_less_nat @ ( suc @ N3 ) @ M3 )
= ( ? [M7: nat] :
( ( M3
= ( suc @ M7 ) )
& ( ord_less_nat @ N3 @ M7 ) ) ) ) ).
% Suc_less_eq2
thf(fact_979_All__less__Suc,axiom,
! [N3: nat,P2: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N3 ) )
=> ( P2 @ I4 ) ) )
= ( ( P2 @ N3 )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N3 )
=> ( P2 @ I4 ) ) ) ) ).
% All_less_Suc
thf(fact_980_not__less__eq,axiom,
! [M3: nat,N3: nat] :
( ( ~ ( ord_less_nat @ M3 @ N3 ) )
= ( ord_less_nat @ N3 @ ( suc @ M3 ) ) ) ).
% not_less_eq
thf(fact_981_less__Suc__eq,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ ( suc @ N3 ) )
= ( ( ord_less_nat @ M3 @ N3 )
| ( M3 = N3 ) ) ) ).
% less_Suc_eq
thf(fact_982_Ex__less__Suc,axiom,
! [N3: nat,P2: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N3 ) )
& ( P2 @ I4 ) ) )
= ( ( P2 @ N3 )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N3 )
& ( P2 @ I4 ) ) ) ) ).
% Ex_less_Suc
thf(fact_983_less__SucI,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( ord_less_nat @ M3 @ ( suc @ N3 ) ) ) ).
% less_SucI
thf(fact_984_less__SucE,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ ( suc @ N3 ) )
=> ( ~ ( ord_less_nat @ M3 @ N3 )
=> ( M3 = N3 ) ) ) ).
% less_SucE
thf(fact_985_Suc__lessI,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( ( ( suc @ M3 )
!= N3 )
=> ( ord_less_nat @ ( suc @ M3 ) @ N3 ) ) ) ).
% Suc_lessI
thf(fact_986_Suc__lessE,axiom,
! [I: nat,K2: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K2 )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( K2
!= ( suc @ J3 ) ) ) ) ).
% Suc_lessE
thf(fact_987_Suc__lessD,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_nat @ ( suc @ M3 ) @ N3 )
=> ( ord_less_nat @ M3 @ N3 ) ) ).
% Suc_lessD
thf(fact_988_Nat_OlessE,axiom,
! [I: nat,K2: nat] :
( ( ord_less_nat @ I @ K2 )
=> ( ( K2
!= ( suc @ I ) )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( K2
!= ( suc @ J3 ) ) ) ) ) ).
% Nat.lessE
thf(fact_989_add__Suc__shift,axiom,
! [M3: nat,N3: nat] :
( ( plus_plus_nat @ ( suc @ M3 ) @ N3 )
= ( plus_plus_nat @ M3 @ ( suc @ N3 ) ) ) ).
% add_Suc_shift
thf(fact_990_add__Suc,axiom,
! [M3: nat,N3: nat] :
( ( plus_plus_nat @ ( suc @ M3 ) @ N3 )
= ( suc @ ( plus_plus_nat @ M3 @ N3 ) ) ) ).
% add_Suc
thf(fact_991_nat__arith_Osuc1,axiom,
! [A4: nat,K2: nat,A2: nat] :
( ( A4
= ( plus_plus_nat @ K2 @ A2 ) )
=> ( ( suc @ A4 )
= ( plus_plus_nat @ K2 @ ( suc @ A2 ) ) ) ) ).
% nat_arith.suc1
thf(fact_992_n__not__Suc__n,axiom,
! [N3: nat] :
( N3
!= ( suc @ N3 ) ) ).
% n_not_Suc_n
thf(fact_993_Suc__inject,axiom,
! [X3: nat,Y: nat] :
( ( ( suc @ X3 )
= ( suc @ Y ) )
=> ( X3 = Y ) ) ).
% Suc_inject
thf(fact_994_lift__Suc__mono__less,axiom,
! [F: nat > nat,N3: nat,N6: nat] :
( ! [N4: nat] : ( ord_less_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_nat @ N3 @ N6 )
=> ( ord_less_nat @ ( F @ N3 ) @ ( F @ N6 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_995_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N3: nat,M3: nat] :
( ! [N4: nat] : ( ord_less_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_nat @ ( F @ N3 ) @ ( F @ M3 ) )
= ( ord_less_nat @ N3 @ M3 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_996_Ex__less__Suc2,axiom,
! [N3: nat,P2: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N3 ) )
& ( P2 @ I4 ) ) )
= ( ( P2 @ zero_zero_nat )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N3 )
& ( P2 @ ( suc @ I4 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_997_gr0__conv__Suc,axiom,
! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
= ( ? [M8: nat] :
( N3
= ( suc @ M8 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_998_All__less__Suc2,axiom,
! [N3: nat,P2: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N3 ) )
=> ( P2 @ I4 ) ) )
= ( ( P2 @ zero_zero_nat )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N3 )
=> ( P2 @ ( suc @ I4 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_999_gr0__implies__Suc,axiom,
! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ? [M6: nat] :
( N3
= ( suc @ M6 ) ) ) ).
% gr0_implies_Suc
thf(fact_1000_less__Suc__eq__0__disj,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ ( suc @ N3 ) )
= ( ( M3 = zero_zero_nat )
| ? [J4: nat] :
( ( M3
= ( suc @ J4 ) )
& ( ord_less_nat @ J4 @ N3 ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1001_length__Cons,axiom,
! [X3: a,Xs: list_a] :
( ( size_size_list_a @ ( cons_a @ X3 @ Xs ) )
= ( suc @ ( size_size_list_a @ Xs ) ) ) ).
% length_Cons
thf(fact_1002_Suc__length__conv,axiom,
! [N3: nat,Xs: list_a] :
( ( ( suc @ N3 )
= ( size_size_list_a @ Xs ) )
= ( ? [Y8: a,Ys4: list_a] :
( ( Xs
= ( cons_a @ Y8 @ Ys4 ) )
& ( ( size_size_list_a @ Ys4 )
= N3 ) ) ) ) ).
% Suc_length_conv
thf(fact_1003_length__Suc__conv,axiom,
! [Xs: list_a,N3: nat] :
( ( ( size_size_list_a @ Xs )
= ( suc @ N3 ) )
= ( ? [Y8: a,Ys4: list_a] :
( ( Xs
= ( cons_a @ Y8 @ Ys4 ) )
& ( ( size_size_list_a @ Ys4 )
= N3 ) ) ) ) ).
% length_Suc_conv
thf(fact_1004_add__is__1,axiom,
! [M3: nat,N3: nat] :
( ( ( plus_plus_nat @ M3 @ N3 )
= ( suc @ zero_zero_nat ) )
= ( ( ( M3
= ( suc @ zero_zero_nat ) )
& ( N3 = zero_zero_nat ) )
| ( ( M3 = zero_zero_nat )
& ( N3
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1005_one__is__add,axiom,
! [M3: nat,N3: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M3 @ N3 ) )
= ( ( ( M3
= ( suc @ zero_zero_nat ) )
& ( N3 = zero_zero_nat ) )
| ( ( M3 = zero_zero_nat )
& ( N3
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1006_less__natE,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ~ ! [Q2: nat] :
( N3
!= ( suc @ ( plus_plus_nat @ M3 @ Q2 ) ) ) ) ).
% less_natE
thf(fact_1007_less__add__Suc1,axiom,
! [I: nat,M3: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M3 ) ) ) ).
% less_add_Suc1
thf(fact_1008_less__add__Suc2,axiom,
! [I: nat,M3: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M3 @ I ) ) ) ).
% less_add_Suc2
thf(fact_1009_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M8: nat,N5: nat] :
? [K5: nat] :
( N5
= ( suc @ ( plus_plus_nat @ M8 @ K5 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1010_less__imp__Suc__add,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ? [K3: nat] :
( N3
= ( suc @ ( plus_plus_nat @ M3 @ K3 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1011_diff__less__Suc,axiom,
! [M3: nat,N3: nat] : ( ord_less_nat @ ( minus_minus_nat @ M3 @ N3 ) @ ( suc @ M3 ) ) ).
% diff_less_Suc
thf(fact_1012_Suc__diff__Suc,axiom,
! [N3: nat,M3: nat] :
( ( ord_less_nat @ N3 @ M3 )
=> ( ( suc @ ( minus_minus_nat @ M3 @ ( suc @ N3 ) ) )
= ( minus_minus_nat @ M3 @ N3 ) ) ) ).
% Suc_diff_Suc
thf(fact_1013_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1014_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1015_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1016_Suc__eq__plus1,axiom,
( suc
= ( ^ [N5: nat] : ( plus_plus_nat @ N5 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1017_diff__Suc__eq__diff__pred,axiom,
! [M3: nat,N3: nat] :
( ( minus_minus_nat @ M3 @ ( suc @ N3 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N3 ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1018_sum__mset_Oneutral,axiom,
! [A4: multiset_nat] :
( ! [X: nat] :
( ( member_nat @ X @ ( set_mset_nat @ A4 ) )
=> ( X = zero_zero_nat ) )
=> ( ( comm_m762188921832702859et_nat @ A4 )
= zero_zero_nat ) ) ).
% sum_mset.neutral
thf(fact_1019_size__eq__Suc__imp__elem,axiom,
! [M: multis501812127501805293od_a_a,N3: nat] :
( ( ( size_s781968976467208537od_a_a @ M )
= ( suc @ N3 ) )
=> ? [A: product_prod_a_a] : ( member1426531477525435216od_a_a @ A @ ( set_ms119794563918438244od_a_a @ M ) ) ) ).
% size_eq_Suc_imp_elem
thf(fact_1020_size__eq__Suc__imp__elem,axiom,
! [M: multis4781622204924204013iset_a,N3: nat] :
( ( ( size_s2629572463880657497iset_a @ M )
= ( suc @ N3 ) )
=> ? [A: produc6518373309651786023iset_a] : ( member5199237121806060112iset_a @ A @ ( set_ms2379723421352415844iset_a @ M ) ) ) ).
% size_eq_Suc_imp_elem
thf(fact_1021_size__eq__Suc__imp__elem,axiom,
! [M: multiset_a,N3: nat] :
( ( ( size_size_multiset_a @ M )
= ( suc @ N3 ) )
=> ? [A: a] : ( member_a @ A @ ( set_mset_a @ M ) ) ) ).
% size_eq_Suc_imp_elem
thf(fact_1022_diff__Suc__less,axiom,
! [N3: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ord_less_nat @ ( minus_minus_nat @ N3 @ ( suc @ I ) ) @ N3 ) ) ).
% diff_Suc_less
thf(fact_1023_nat__induct__non__zero,axiom,
! [N3: nat,P2: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P2 @ one_one_nat )
=> ( ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ( P2 @ N4 )
=> ( P2 @ ( suc @ N4 ) ) ) )
=> ( P2 @ N3 ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1024_sum__mset__diff,axiom,
! [N: multiset_nat,M: multiset_nat] :
( ( subseteq_mset_nat @ N @ M )
=> ( ( comm_m762188921832702859et_nat @ ( minus_8522176038001411705et_nat @ M @ N ) )
= ( minus_minus_nat @ ( comm_m762188921832702859et_nat @ M ) @ ( comm_m762188921832702859et_nat @ N ) ) ) ) ).
% sum_mset_diff
thf(fact_1025_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_1026_sum__mset_Oeq__fold,axiom,
( comm_m762188921832702859et_nat
= ( fold_mset_nat_nat @ plus_plus_nat @ zero_zero_nat ) ) ).
% sum_mset.eq_fold
thf(fact_1027_list_Osize_I4_J,axiom,
! [X21: a,X222: list_a] :
( ( size_size_list_a @ ( cons_a @ X21 @ X222 ) )
= ( plus_plus_nat @ ( size_size_list_a @ X222 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_1028_Suc__diff__eq__diff__pred,axiom,
! [N3: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( minus_minus_nat @ ( suc @ M3 ) @ N3 )
= ( minus_minus_nat @ M3 @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1029_Suc__pred_H,axiom,
! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( N3
= ( suc @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_1030_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M8: nat,N5: nat] : ( if_nat @ ( M8 = zero_zero_nat ) @ N5 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M8 @ one_one_nat ) @ N5 ) ) ) ) ) ).
% add_eq_if
thf(fact_1031_size__mset__SucE,axiom,
! [A4: multiset_a,N3: nat] :
( ( ( size_size_multiset_a @ A4 )
= ( suc @ N3 ) )
=> ~ ! [A: a,B7: multiset_a] :
( ( A4
= ( plus_plus_multiset_a @ ( add_mset_a @ A @ zero_zero_multiset_a ) @ B7 ) )
=> ( ( size_size_multiset_a @ B7 )
!= N3 ) ) ) ).
% size_mset_SucE
thf(fact_1032_take__Suc__conv__app__nth,axiom,
! [I: nat,Xs: list_a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( take_a @ ( suc @ I ) @ Xs )
= ( append_a @ ( take_a @ I @ Xs ) @ ( cons_a @ ( nth_a @ Xs @ I ) @ nil_a ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_1033_append__eq__append__conv,axiom,
! [Xs: list_a,Ys: list_a,Us: list_a,Vs: list_a] :
( ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
| ( ( size_size_list_a @ Us )
= ( size_size_list_a @ Vs ) ) )
=> ( ( ( append_a @ Xs @ Us )
= ( append_a @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_1034_set__append,axiom,
! [Xs: list_a,Ys: list_a] :
( ( set_a2 @ ( append_a @ Xs @ Ys ) )
= ( sup_sup_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys ) ) ) ).
% set_append
thf(fact_1035_length__append,axiom,
! [Xs: list_a,Ys: list_a] :
( ( size_size_list_a @ ( append_a @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) ) ) ).
% length_append
thf(fact_1036_mset__append,axiom,
! [Xs: list_a,Ys: list_a] :
( ( mset_a @ ( append_a @ Xs @ Ys ) )
= ( plus_plus_multiset_a @ ( mset_a @ Xs ) @ ( mset_a @ Ys ) ) ) ).
% mset_append
thf(fact_1037_zip__append,axiom,
! [Xs: list_a,Us: list_a,Ys: list_a,Vs: list_a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Us ) )
=> ( ( zip_a_a @ ( append_a @ Xs @ Ys ) @ ( append_a @ Us @ Vs ) )
= ( append5335208819046833346od_a_a @ ( zip_a_a @ Xs @ Us ) @ ( zip_a_a @ Ys @ Vs ) ) ) ) ).
% zip_append
thf(fact_1038_nth__append__length,axiom,
! [Xs: list_a,X3: a,Ys: list_a] :
( ( nth_a @ ( append_a @ Xs @ ( cons_a @ X3 @ Ys ) ) @ ( size_size_list_a @ Xs ) )
= X3 ) ).
% nth_append_length
thf(fact_1039_nth__append__length__plus,axiom,
! [Xs: list_a,Ys: list_a,N3: nat] :
( ( nth_a @ ( append_a @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_size_list_a @ Xs ) @ N3 ) )
= ( nth_a @ Ys @ N3 ) ) ).
% nth_append_length_plus
thf(fact_1040_take__append,axiom,
! [N3: nat,Xs: list_a,Ys: list_a] :
( ( take_a @ N3 @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( take_a @ N3 @ Xs ) @ ( take_a @ ( minus_minus_nat @ N3 @ ( size_size_list_a @ Xs ) ) @ Ys ) ) ) ).
% take_append
thf(fact_1041_list__update__length,axiom,
! [Xs: list_a,X3: a,Ys: list_a,Y: a] :
( ( list_update_a @ ( append_a @ Xs @ ( cons_a @ X3 @ Ys ) ) @ ( size_size_list_a @ Xs ) @ Y )
= ( append_a @ Xs @ ( cons_a @ Y @ Ys ) ) ) ).
% list_update_length
thf(fact_1042_enumerate__append__eq,axiom,
! [N3: nat,Xs: list_a,Ys: list_a] :
( ( enumerate_a @ N3 @ ( append_a @ Xs @ Ys ) )
= ( append1694031006427026248_nat_a @ ( enumerate_a @ N3 @ Xs ) @ ( enumerate_a @ ( plus_plus_nat @ N3 @ ( size_size_list_a @ Xs ) ) @ Ys ) ) ) ).
% enumerate_append_eq
thf(fact_1043_split__list,axiom,
! [X3: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Xs ) )
=> ? [Ys3: list_P1396940483166286381od_a_a,Zs2: list_P1396940483166286381od_a_a] :
( Xs
= ( append5335208819046833346od_a_a @ Ys3 @ ( cons_P7316939126706565853od_a_a @ X3 @ Zs2 ) ) ) ) ).
% split_list
thf(fact_1044_split__list,axiom,
! [X3: produc6518373309651786023iset_a,Xs: list_P6988024003011616557iset_a] :
( ( member5199237121806060112iset_a @ X3 @ ( set_Pr5550698219922361532iset_a @ Xs ) )
=> ? [Ys3: list_P6988024003011616557iset_a,Zs2: list_P6988024003011616557iset_a] :
( Xs
= ( append1975991668658899394iset_a @ Ys3 @ ( cons_P7086145107730948317iset_a @ X3 @ Zs2 ) ) ) ) ).
% split_list
thf(fact_1045_split__list,axiom,
! [X3: a,Xs: list_a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ? [Ys3: list_a,Zs2: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X3 @ Zs2 ) ) ) ) ).
% split_list
thf(fact_1046_split__list__last,axiom,
! [X3: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Xs ) )
=> ? [Ys3: list_P1396940483166286381od_a_a,Zs2: list_P1396940483166286381od_a_a] :
( ( Xs
= ( append5335208819046833346od_a_a @ Ys3 @ ( cons_P7316939126706565853od_a_a @ X3 @ Zs2 ) ) )
& ~ ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_1047_split__list__last,axiom,
! [X3: produc6518373309651786023iset_a,Xs: list_P6988024003011616557iset_a] :
( ( member5199237121806060112iset_a @ X3 @ ( set_Pr5550698219922361532iset_a @ Xs ) )
=> ? [Ys3: list_P6988024003011616557iset_a,Zs2: list_P6988024003011616557iset_a] :
( ( Xs
= ( append1975991668658899394iset_a @ Ys3 @ ( cons_P7086145107730948317iset_a @ X3 @ Zs2 ) ) )
& ~ ( member5199237121806060112iset_a @ X3 @ ( set_Pr5550698219922361532iset_a @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_1048_split__list__last,axiom,
! [X3: a,Xs: list_a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ? [Ys3: list_a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X3 @ Zs2 ) ) )
& ~ ( member_a @ X3 @ ( set_a2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_1049_split__list__prop,axiom,
! [Xs: list_a,P2: a > $o] :
( ? [X6: a] :
( ( member_a @ X6 @ ( set_a2 @ Xs ) )
& ( P2 @ X6 ) )
=> ? [Ys3: list_a,X: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X @ Zs2 ) ) )
& ( P2 @ X ) ) ) ).
% split_list_prop
thf(fact_1050_split__list__first,axiom,
! [X3: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Xs ) )
=> ? [Ys3: list_P1396940483166286381od_a_a,Zs2: list_P1396940483166286381od_a_a] :
( ( Xs
= ( append5335208819046833346od_a_a @ Ys3 @ ( cons_P7316939126706565853od_a_a @ X3 @ Zs2 ) ) )
& ~ ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_1051_split__list__first,axiom,
! [X3: produc6518373309651786023iset_a,Xs: list_P6988024003011616557iset_a] :
( ( member5199237121806060112iset_a @ X3 @ ( set_Pr5550698219922361532iset_a @ Xs ) )
=> ? [Ys3: list_P6988024003011616557iset_a,Zs2: list_P6988024003011616557iset_a] :
( ( Xs
= ( append1975991668658899394iset_a @ Ys3 @ ( cons_P7086145107730948317iset_a @ X3 @ Zs2 ) ) )
& ~ ( member5199237121806060112iset_a @ X3 @ ( set_Pr5550698219922361532iset_a @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_1052_split__list__first,axiom,
! [X3: a,Xs: list_a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ? [Ys3: list_a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X3 @ Zs2 ) ) )
& ~ ( member_a @ X3 @ ( set_a2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_1053_split__list__propE,axiom,
! [Xs: list_a,P2: a > $o] :
( ? [X6: a] :
( ( member_a @ X6 @ ( set_a2 @ Xs ) )
& ( P2 @ X6 ) )
=> ~ ! [Ys3: list_a,X: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X @ Zs2 ) ) )
=> ~ ( P2 @ X ) ) ) ).
% split_list_propE
thf(fact_1054_append__Cons__eq__iff,axiom,
! [X3: product_prod_a_a,Xs: list_P1396940483166286381od_a_a,Ys: list_P1396940483166286381od_a_a,Xs5: list_P1396940483166286381od_a_a,Ys7: list_P1396940483166286381od_a_a] :
( ~ ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Xs ) )
=> ( ~ ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Ys ) )
=> ( ( ( append5335208819046833346od_a_a @ Xs @ ( cons_P7316939126706565853od_a_a @ X3 @ Ys ) )
= ( append5335208819046833346od_a_a @ Xs5 @ ( cons_P7316939126706565853od_a_a @ X3 @ Ys7 ) ) )
= ( ( Xs = Xs5 )
& ( Ys = Ys7 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_1055_append__Cons__eq__iff,axiom,
! [X3: produc6518373309651786023iset_a,Xs: list_P6988024003011616557iset_a,Ys: list_P6988024003011616557iset_a,Xs5: list_P6988024003011616557iset_a,Ys7: list_P6988024003011616557iset_a] :
( ~ ( member5199237121806060112iset_a @ X3 @ ( set_Pr5550698219922361532iset_a @ Xs ) )
=> ( ~ ( member5199237121806060112iset_a @ X3 @ ( set_Pr5550698219922361532iset_a @ Ys ) )
=> ( ( ( append1975991668658899394iset_a @ Xs @ ( cons_P7086145107730948317iset_a @ X3 @ Ys ) )
= ( append1975991668658899394iset_a @ Xs5 @ ( cons_P7086145107730948317iset_a @ X3 @ Ys7 ) ) )
= ( ( Xs = Xs5 )
& ( Ys = Ys7 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_1056_append__Cons__eq__iff,axiom,
! [X3: a,Xs: list_a,Ys: list_a,Xs5: list_a,Ys7: list_a] :
( ~ ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( ~ ( member_a @ X3 @ ( set_a2 @ Ys ) )
=> ( ( ( append_a @ Xs @ ( cons_a @ X3 @ Ys ) )
= ( append_a @ Xs5 @ ( cons_a @ X3 @ Ys7 ) ) )
= ( ( Xs = Xs5 )
& ( Ys = Ys7 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_1057_in__set__conv__decomp,axiom,
! [X3: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Xs ) )
= ( ? [Ys4: list_P1396940483166286381od_a_a,Zs3: list_P1396940483166286381od_a_a] :
( Xs
= ( append5335208819046833346od_a_a @ Ys4 @ ( cons_P7316939126706565853od_a_a @ X3 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_1058_in__set__conv__decomp,axiom,
! [X3: produc6518373309651786023iset_a,Xs: list_P6988024003011616557iset_a] :
( ( member5199237121806060112iset_a @ X3 @ ( set_Pr5550698219922361532iset_a @ Xs ) )
= ( ? [Ys4: list_P6988024003011616557iset_a,Zs3: list_P6988024003011616557iset_a] :
( Xs
= ( append1975991668658899394iset_a @ Ys4 @ ( cons_P7086145107730948317iset_a @ X3 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_1059_in__set__conv__decomp,axiom,
! [X3: a,Xs: list_a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
= ( ? [Ys4: list_a,Zs3: list_a] :
( Xs
= ( append_a @ Ys4 @ ( cons_a @ X3 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_1060_split__list__last__prop,axiom,
! [Xs: list_a,P2: a > $o] :
( ? [X6: a] :
( ( member_a @ X6 @ ( set_a2 @ Xs ) )
& ( P2 @ X6 ) )
=> ? [Ys3: list_a,X: a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X @ Zs2 ) ) )
& ( P2 @ X )
& ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Zs2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_1061_split__list__first__prop,axiom,
! [Xs: list_a,P2: a > $o] :
( ? [X6: a] :
( ( member_a @ X6 @ ( set_a2 @ Xs ) )
& ( P2 @ X6 ) )
=> ? [Ys3: list_a,X: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X @ Zs2 ) ) )
& ( P2 @ X )
& ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Ys3 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_1062_split__list__last__propE,axiom,
! [Xs: list_a,P2: a > $o] :
( ? [X6: a] :
( ( member_a @ X6 @ ( set_a2 @ Xs ) )
& ( P2 @ X6 ) )
=> ~ ! [Ys3: list_a,X: a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X @ Zs2 ) ) )
=> ( ( P2 @ X )
=> ~ ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Zs2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_1063_split__list__first__propE,axiom,
! [Xs: list_a,P2: a > $o] :
( ? [X6: a] :
( ( member_a @ X6 @ ( set_a2 @ Xs ) )
& ( P2 @ X6 ) )
=> ~ ! [Ys3: list_a,X: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X @ Zs2 ) ) )
=> ( ( P2 @ X )
=> ~ ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Ys3 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_1064_in__set__conv__decomp__last,axiom,
! [X3: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Xs ) )
= ( ? [Ys4: list_P1396940483166286381od_a_a,Zs3: list_P1396940483166286381od_a_a] :
( ( Xs
= ( append5335208819046833346od_a_a @ Ys4 @ ( cons_P7316939126706565853od_a_a @ X3 @ Zs3 ) ) )
& ~ ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_1065_in__set__conv__decomp__last,axiom,
! [X3: produc6518373309651786023iset_a,Xs: list_P6988024003011616557iset_a] :
( ( member5199237121806060112iset_a @ X3 @ ( set_Pr5550698219922361532iset_a @ Xs ) )
= ( ? [Ys4: list_P6988024003011616557iset_a,Zs3: list_P6988024003011616557iset_a] :
( ( Xs
= ( append1975991668658899394iset_a @ Ys4 @ ( cons_P7086145107730948317iset_a @ X3 @ Zs3 ) ) )
& ~ ( member5199237121806060112iset_a @ X3 @ ( set_Pr5550698219922361532iset_a @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_1066_in__set__conv__decomp__last,axiom,
! [X3: a,Xs: list_a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
= ( ? [Ys4: list_a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys4 @ ( cons_a @ X3 @ Zs3 ) ) )
& ~ ( member_a @ X3 @ ( set_a2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_1067_in__set__conv__decomp__first,axiom,
! [X3: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Xs ) )
= ( ? [Ys4: list_P1396940483166286381od_a_a,Zs3: list_P1396940483166286381od_a_a] :
( ( Xs
= ( append5335208819046833346od_a_a @ Ys4 @ ( cons_P7316939126706565853od_a_a @ X3 @ Zs3 ) ) )
& ~ ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Ys4 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_1068_in__set__conv__decomp__first,axiom,
! [X3: produc6518373309651786023iset_a,Xs: list_P6988024003011616557iset_a] :
( ( member5199237121806060112iset_a @ X3 @ ( set_Pr5550698219922361532iset_a @ Xs ) )
= ( ? [Ys4: list_P6988024003011616557iset_a,Zs3: list_P6988024003011616557iset_a] :
( ( Xs
= ( append1975991668658899394iset_a @ Ys4 @ ( cons_P7086145107730948317iset_a @ X3 @ Zs3 ) ) )
& ~ ( member5199237121806060112iset_a @ X3 @ ( set_Pr5550698219922361532iset_a @ Ys4 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_1069_in__set__conv__decomp__first,axiom,
! [X3: a,Xs: list_a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
= ( ? [Ys4: list_a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys4 @ ( cons_a @ X3 @ Zs3 ) ) )
& ~ ( member_a @ X3 @ ( set_a2 @ Ys4 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_1070_split__list__last__prop__iff,axiom,
! [Xs: list_a,P2: a > $o] :
( ( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs ) )
& ( P2 @ X5 ) ) )
= ( ? [Ys4: list_a,X5: a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys4 @ ( cons_a @ X5 @ Zs3 ) ) )
& ( P2 @ X5 )
& ! [Y8: a] :
( ( member_a @ Y8 @ ( set_a2 @ Zs3 ) )
=> ~ ( P2 @ Y8 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_1071_split__list__first__prop__iff,axiom,
! [Xs: list_a,P2: a > $o] :
( ( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs ) )
& ( P2 @ X5 ) ) )
= ( ? [Ys4: list_a,X5: a] :
( ? [Zs3: list_a] :
( Xs
= ( append_a @ Ys4 @ ( cons_a @ X5 @ Zs3 ) ) )
& ( P2 @ X5 )
& ! [Y8: a] :
( ( member_a @ Y8 @ ( set_a2 @ Ys4 ) )
=> ~ ( P2 @ Y8 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_1072_remove1__append,axiom,
! [X3: product_prod_a_a,Xs: list_P1396940483166286381od_a_a,Ys: list_P1396940483166286381od_a_a] :
( ( ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Xs ) )
=> ( ( remove3530114606240936131od_a_a @ X3 @ ( append5335208819046833346od_a_a @ Xs @ Ys ) )
= ( append5335208819046833346od_a_a @ ( remove3530114606240936131od_a_a @ X3 @ Xs ) @ Ys ) ) )
& ( ~ ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Xs ) )
=> ( ( remove3530114606240936131od_a_a @ X3 @ ( append5335208819046833346od_a_a @ Xs @ Ys ) )
= ( append5335208819046833346od_a_a @ Xs @ ( remove3530114606240936131od_a_a @ X3 @ Ys ) ) ) ) ) ).
% remove1_append
thf(fact_1073_remove1__append,axiom,
! [X3: produc6518373309651786023iset_a,Xs: list_P6988024003011616557iset_a,Ys: list_P6988024003011616557iset_a] :
( ( ( member5199237121806060112iset_a @ X3 @ ( set_Pr5550698219922361532iset_a @ Xs ) )
=> ( ( remove5063108440011681987iset_a @ X3 @ ( append1975991668658899394iset_a @ Xs @ Ys ) )
= ( append1975991668658899394iset_a @ ( remove5063108440011681987iset_a @ X3 @ Xs ) @ Ys ) ) )
& ( ~ ( member5199237121806060112iset_a @ X3 @ ( set_Pr5550698219922361532iset_a @ Xs ) )
=> ( ( remove5063108440011681987iset_a @ X3 @ ( append1975991668658899394iset_a @ Xs @ Ys ) )
= ( append1975991668658899394iset_a @ Xs @ ( remove5063108440011681987iset_a @ X3 @ Ys ) ) ) ) ) ).
% remove1_append
thf(fact_1074_remove1__append,axiom,
! [X3: a,Xs: list_a,Ys: list_a] :
( ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( ( remove1_a @ X3 @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( remove1_a @ X3 @ Xs ) @ Ys ) ) )
& ( ~ ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( ( remove1_a @ X3 @ ( append_a @ Xs @ Ys ) )
= ( append_a @ Xs @ ( remove1_a @ X3 @ Ys ) ) ) ) ) ).
% remove1_append
thf(fact_1075_same__length__different,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != Ys )
=> ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ? [Pre: list_a,X: a,Xs6: list_a,Y3: a,Ys6: list_a] :
( ( X != Y3 )
& ( Xs
= ( append_a @ Pre @ ( append_a @ ( cons_a @ X @ nil_a ) @ Xs6 ) ) )
& ( Ys
= ( append_a @ Pre @ ( append_a @ ( cons_a @ Y3 @ nil_a ) @ Ys6 ) ) ) ) ) ) ).
% same_length_different
thf(fact_1076_list__update__append1,axiom,
! [I: nat,Xs: list_a,Ys: list_a,X3: a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( list_update_a @ ( append_a @ Xs @ Ys ) @ I @ X3 )
= ( append_a @ ( list_update_a @ Xs @ I @ X3 ) @ Ys ) ) ) ).
% list_update_append1
thf(fact_1077_remove1__split,axiom,
! [A2: product_prod_a_a,Xs: list_P1396940483166286381od_a_a,Ys: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ A2 @ ( set_Product_prod_a_a2 @ Xs ) )
=> ( ( ( remove3530114606240936131od_a_a @ A2 @ Xs )
= Ys )
= ( ? [Ls2: list_P1396940483166286381od_a_a,Rs: list_P1396940483166286381od_a_a] :
( ( Xs
= ( append5335208819046833346od_a_a @ Ls2 @ ( cons_P7316939126706565853od_a_a @ A2 @ Rs ) ) )
& ~ ( member1426531477525435216od_a_a @ A2 @ ( set_Product_prod_a_a2 @ Ls2 ) )
& ( Ys
= ( append5335208819046833346od_a_a @ Ls2 @ Rs ) ) ) ) ) ) ).
% remove1_split
thf(fact_1078_remove1__split,axiom,
! [A2: produc6518373309651786023iset_a,Xs: list_P6988024003011616557iset_a,Ys: list_P6988024003011616557iset_a] :
( ( member5199237121806060112iset_a @ A2 @ ( set_Pr5550698219922361532iset_a @ Xs ) )
=> ( ( ( remove5063108440011681987iset_a @ A2 @ Xs )
= Ys )
= ( ? [Ls2: list_P6988024003011616557iset_a,Rs: list_P6988024003011616557iset_a] :
( ( Xs
= ( append1975991668658899394iset_a @ Ls2 @ ( cons_P7086145107730948317iset_a @ A2 @ Rs ) ) )
& ~ ( member5199237121806060112iset_a @ A2 @ ( set_Pr5550698219922361532iset_a @ Ls2 ) )
& ( Ys
= ( append1975991668658899394iset_a @ Ls2 @ Rs ) ) ) ) ) ) ).
% remove1_split
thf(fact_1079_remove1__split,axiom,
! [A2: a,Xs: list_a,Ys: list_a] :
( ( member_a @ A2 @ ( set_a2 @ Xs ) )
=> ( ( ( remove1_a @ A2 @ Xs )
= Ys )
= ( ? [Ls2: list_a,Rs: list_a] :
( ( Xs
= ( append_a @ Ls2 @ ( cons_a @ A2 @ Rs ) ) )
& ~ ( member_a @ A2 @ ( set_a2 @ Ls2 ) )
& ( Ys
= ( append_a @ Ls2 @ Rs ) ) ) ) ) ) ).
% remove1_split
thf(fact_1080_lex__append__left__iff,axiom,
! [R2: set_Product_prod_a_a,Xs: list_a,Ys: list_a,Zs: list_a] :
( ! [X: a] :
~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ X ) @ R2 )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs @ Ys ) @ ( append_a @ Xs @ Zs ) ) @ ( lex_a @ R2 ) )
= ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ys @ Zs ) @ ( lex_a @ R2 ) ) ) ) ).
% lex_append_left_iff
thf(fact_1081_lex__append__left__iff,axiom,
! [R2: set_Pr79727621955416071iset_a,Xs: list_multiset_a,Ys: list_multiset_a,Zs: list_multiset_a] :
( ! [X: multiset_a] :
~ ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ X ) @ R2 )
=> ( ( member1640862112313989456iset_a @ ( produc1434922032272282391iset_a @ ( append_multiset_a @ Xs @ Ys ) @ ( append_multiset_a @ Xs @ Zs ) ) @ ( lex_multiset_a @ R2 ) )
= ( member1640862112313989456iset_a @ ( produc1434922032272282391iset_a @ Ys @ Zs ) @ ( lex_multiset_a @ R2 ) ) ) ) ).
% lex_append_left_iff
thf(fact_1082_lex__append__leftD,axiom,
! [R2: set_Product_prod_a_a,Xs: list_a,Ys: list_a,Zs: list_a] :
( ! [X: a] :
~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ X ) @ R2 )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs @ Ys ) @ ( append_a @ Xs @ Zs ) ) @ ( lex_a @ R2 ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ys @ Zs ) @ ( lex_a @ R2 ) ) ) ) ).
% lex_append_leftD
thf(fact_1083_lex__append__leftD,axiom,
! [R2: set_Pr79727621955416071iset_a,Xs: list_multiset_a,Ys: list_multiset_a,Zs: list_multiset_a] :
( ! [X: multiset_a] :
~ ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ X ) @ R2 )
=> ( ( member1640862112313989456iset_a @ ( produc1434922032272282391iset_a @ ( append_multiset_a @ Xs @ Ys ) @ ( append_multiset_a @ Xs @ Zs ) ) @ ( lex_multiset_a @ R2 ) )
=> ( member1640862112313989456iset_a @ ( produc1434922032272282391iset_a @ Ys @ Zs ) @ ( lex_multiset_a @ R2 ) ) ) ) ).
% lex_append_leftD
thf(fact_1084_lex__append__rightI,axiom,
! [Xs: list_a,Ys: list_a,R2: set_Product_prod_a_a,Vs: list_a,Us: list_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( lex_a @ R2 ) )
=> ( ( ( size_size_list_a @ Vs )
= ( size_size_list_a @ Us ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs @ Us ) @ ( append_a @ Ys @ Vs ) ) @ ( lex_a @ R2 ) ) ) ) ).
% lex_append_rightI
thf(fact_1085_lenlex__append1,axiom,
! [Us: list_a,Xs: list_a,R: set_Product_prod_a_a,Vs: list_a,Ys: list_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Us @ Xs ) @ ( lenlex_a @ R ) )
=> ( ( ( size_size_list_a @ Vs )
= ( size_size_list_a @ Ys ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Us @ Vs ) @ ( append_a @ Xs @ Ys ) ) @ ( lenlex_a @ R ) ) ) ) ).
% lenlex_append1
thf(fact_1086_wcount__union,axiom,
! [F: a > nat,M: multiset_a,N: multiset_a,A2: a] :
( ( wcount_a @ F @ ( plus_plus_multiset_a @ M @ N ) @ A2 )
= ( plus_plus_nat @ ( wcount_a @ F @ M @ A2 ) @ ( wcount_a @ F @ N @ A2 ) ) ) ).
% wcount_union
thf(fact_1087_length__Suc__conv__rev,axiom,
! [Xs: list_a,N3: nat] :
( ( ( size_size_list_a @ Xs )
= ( suc @ N3 ) )
= ( ? [Y8: a,Ys4: list_a] :
( ( Xs
= ( append_a @ Ys4 @ ( cons_a @ Y8 @ nil_a ) ) )
& ( ( size_size_list_a @ Ys4 )
= N3 ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_1088_length__append__singleton,axiom,
! [Xs: list_a,X3: a] :
( ( size_size_list_a @ ( append_a @ Xs @ ( cons_a @ X3 @ nil_a ) ) )
= ( suc @ ( size_size_list_a @ Xs ) ) ) ).
% length_append_singleton
thf(fact_1089_nth__append,axiom,
! [N3: nat,Xs: list_a,Ys: list_a] :
( ( ( ord_less_nat @ N3 @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( append_a @ Xs @ Ys ) @ N3 )
= ( nth_a @ Xs @ N3 ) ) )
& ( ~ ( ord_less_nat @ N3 @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( append_a @ Xs @ Ys ) @ N3 )
= ( nth_a @ Ys @ ( minus_minus_nat @ N3 @ ( size_size_list_a @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_1090_list__update__append,axiom,
! [N3: nat,Xs: list_a,Ys: list_a,X3: a] :
( ( ( ord_less_nat @ N3 @ ( size_size_list_a @ Xs ) )
=> ( ( list_update_a @ ( append_a @ Xs @ Ys ) @ N3 @ X3 )
= ( append_a @ ( list_update_a @ Xs @ N3 @ X3 ) @ Ys ) ) )
& ( ~ ( ord_less_nat @ N3 @ ( size_size_list_a @ Xs ) )
=> ( ( list_update_a @ ( append_a @ Xs @ Ys ) @ N3 @ X3 )
= ( append_a @ Xs @ ( list_update_a @ Ys @ ( minus_minus_nat @ N3 @ ( size_size_list_a @ Xs ) ) @ X3 ) ) ) ) ) ).
% list_update_append
thf(fact_1091_listrel1E,axiom,
! [Xs: list_a,Ys: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R2 ) )
=> ~ ! [X: a,Y3: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ R2 )
=> ! [Us2: list_a,Vs2: list_a] :
( ( Xs
= ( append_a @ Us2 @ ( cons_a @ X @ Vs2 ) ) )
=> ( Ys
!= ( append_a @ Us2 @ ( cons_a @ Y3 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_1092_listrel1E,axiom,
! [Xs: list_multiset_a,Ys: list_multiset_a,R2: set_Pr79727621955416071iset_a] :
( ( member1640862112313989456iset_a @ ( produc1434922032272282391iset_a @ Xs @ Ys ) @ ( listrel1_multiset_a @ R2 ) )
=> ~ ! [X: multiset_a,Y3: multiset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ Y3 ) @ R2 )
=> ! [Us2: list_multiset_a,Vs2: list_multiset_a] :
( ( Xs
= ( append_multiset_a @ Us2 @ ( cons_multiset_a @ X @ Vs2 ) ) )
=> ( Ys
!= ( append_multiset_a @ Us2 @ ( cons_multiset_a @ Y3 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_1093_listrel1I,axiom,
! [X3: a,Y: a,R2: set_Product_prod_a_a,Xs: list_a,Us: list_a,Vs: list_a,Ys: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ R2 )
=> ( ( Xs
= ( append_a @ Us @ ( cons_a @ X3 @ Vs ) ) )
=> ( ( Ys
= ( append_a @ Us @ ( cons_a @ Y @ Vs ) ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R2 ) ) ) ) ) ).
% listrel1I
thf(fact_1094_listrel1I,axiom,
! [X3: multiset_a,Y: multiset_a,R2: set_Pr79727621955416071iset_a,Xs: list_multiset_a,Us: list_multiset_a,Vs: list_multiset_a,Ys: list_multiset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ Y ) @ R2 )
=> ( ( Xs
= ( append_multiset_a @ Us @ ( cons_multiset_a @ X3 @ Vs ) ) )
=> ( ( Ys
= ( append_multiset_a @ Us @ ( cons_multiset_a @ Y @ Vs ) ) )
=> ( member1640862112313989456iset_a @ ( produc1434922032272282391iset_a @ Xs @ Ys ) @ ( listrel1_multiset_a @ R2 ) ) ) ) ) ).
% listrel1I
thf(fact_1095_snoc__listrel1__snoc__iff,axiom,
! [Xs: list_a,X3: a,Ys: list_a,Y: a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs @ ( cons_a @ X3 @ nil_a ) ) @ ( append_a @ Ys @ ( cons_a @ Y @ nil_a ) ) ) @ ( listrel1_a @ R2 ) )
= ( ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R2 ) )
& ( X3 = Y ) )
| ( ( Xs = Ys )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ R2 ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_1096_snoc__listrel1__snoc__iff,axiom,
! [Xs: list_multiset_a,X3: multiset_a,Ys: list_multiset_a,Y: multiset_a,R2: set_Pr79727621955416071iset_a] :
( ( member1640862112313989456iset_a @ ( produc1434922032272282391iset_a @ ( append_multiset_a @ Xs @ ( cons_multiset_a @ X3 @ nil_multiset_a ) ) @ ( append_multiset_a @ Ys @ ( cons_multiset_a @ Y @ nil_multiset_a ) ) ) @ ( listrel1_multiset_a @ R2 ) )
= ( ( ( member1640862112313989456iset_a @ ( produc1434922032272282391iset_a @ Xs @ Ys ) @ ( listrel1_multiset_a @ R2 ) )
& ( X3 = Y ) )
| ( ( Xs = Ys )
& ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ Y ) @ R2 ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_1097_inf__concat__simple_Ocases,axiom,
! [X3: produc8199716216217303280at_nat] :
( ! [F4: nat > nat] :
( X3
!= ( produc72220940542539688at_nat @ F4 @ zero_zero_nat ) )
=> ~ ! [F4: nat > nat,N4: nat] :
( X3
!= ( produc72220940542539688at_nat @ F4 @ ( suc @ N4 ) ) ) ) ).
% inf_concat_simple.cases
thf(fact_1098_upd__conv__take__nth__drop,axiom,
! [I: nat,Xs: list_a,A2: a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( list_update_a @ Xs @ I @ A2 )
= ( append_a @ ( take_a @ I @ Xs ) @ ( cons_a @ A2 @ ( drop_a @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_1099_id__take__nth__drop,axiom,
! [I: nat,Xs: list_a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( Xs
= ( append_a @ ( take_a @ I @ Xs ) @ ( cons_a @ ( nth_a @ Xs @ I ) @ ( drop_a @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_1100_length__drop,axiom,
! [N3: nat,Xs: list_a] :
( ( size_size_list_a @ ( drop_a @ N3 @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ N3 ) ) ).
% length_drop
thf(fact_1101_drop__append,axiom,
! [N3: nat,Xs: list_a,Ys: list_a] :
( ( drop_a @ N3 @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( drop_a @ N3 @ Xs ) @ ( drop_a @ ( minus_minus_nat @ N3 @ ( size_size_list_a @ Xs ) ) @ Ys ) ) ) ).
% drop_append
thf(fact_1102_last__drop,axiom,
! [N3: nat,Xs: list_a] :
( ( ord_less_nat @ N3 @ ( size_size_list_a @ Xs ) )
=> ( ( last_a @ ( drop_a @ N3 @ Xs ) )
= ( last_a @ Xs ) ) ) ).
% last_drop
thf(fact_1103_nth__via__drop,axiom,
! [N3: nat,Xs: list_a,Y: a,Ys: list_a] :
( ( ( drop_a @ N3 @ Xs )
= ( cons_a @ Y @ Ys ) )
=> ( ( nth_a @ Xs @ N3 )
= Y ) ) ).
% nth_via_drop
thf(fact_1104_in__set__dropD,axiom,
! [X3: product_prod_a_a,N3: nat,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ ( drop_P8456769997282094189od_a_a @ N3 @ Xs ) ) )
=> ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Xs ) ) ) ).
% in_set_dropD
thf(fact_1105_in__set__dropD,axiom,
! [X3: produc6518373309651786023iset_a,N3: nat,Xs: list_P6988024003011616557iset_a] :
( ( member5199237121806060112iset_a @ X3 @ ( set_Pr5550698219922361532iset_a @ ( drop_P694518542769080941iset_a @ N3 @ Xs ) ) )
=> ( member5199237121806060112iset_a @ X3 @ ( set_Pr5550698219922361532iset_a @ Xs ) ) ) ).
% in_set_dropD
thf(fact_1106_in__set__dropD,axiom,
! [X3: a,N3: nat,Xs: list_a] :
( ( member_a @ X3 @ ( set_a2 @ ( drop_a @ N3 @ Xs ) ) )
=> ( member_a @ X3 @ ( set_a2 @ Xs ) ) ) ).
% in_set_dropD
thf(fact_1107_append__eq__conv__conj,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a] :
( ( ( append_a @ Xs @ Ys )
= Zs )
= ( ( Xs
= ( take_a @ ( size_size_list_a @ Xs ) @ Zs ) )
& ( Ys
= ( drop_a @ ( size_size_list_a @ Xs ) @ Zs ) ) ) ) ).
% append_eq_conv_conj
thf(fact_1108_Cons__nth__drop__Suc,axiom,
! [I: nat,Xs: list_a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( cons_a @ ( nth_a @ Xs @ I ) @ ( drop_a @ ( suc @ I ) @ Xs ) )
= ( drop_a @ I @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_1109_mset__zip__take__Cons__drop__twice,axiom,
! [Xs: list_multiset_a,Ys: list_multiset_a,J: nat,X3: multiset_a,Y: multiset_a] :
( ( ( size_s3003218658903709366iset_a @ Xs )
= ( size_s3003218658903709366iset_a @ Ys ) )
=> ( ( ord_less_eq_nat @ J @ ( size_s3003218658903709366iset_a @ Xs ) )
=> ( ( mset_P2771936761501114300iset_a @ ( zip_mu5258324667929623607iset_a @ ( append_multiset_a @ ( take_multiset_a @ J @ Xs ) @ ( cons_multiset_a @ X3 @ ( drop_multiset_a @ J @ Xs ) ) ) @ ( append_multiset_a @ ( take_multiset_a @ J @ Ys ) @ ( cons_multiset_a @ Y @ ( drop_multiset_a @ J @ Ys ) ) ) ) )
= ( add_ms5348986034529679365iset_a @ ( produc654756711066625303iset_a @ X3 @ Y ) @ ( mset_P2771936761501114300iset_a @ ( zip_mu5258324667929623607iset_a @ Xs @ Ys ) ) ) ) ) ) ).
% mset_zip_take_Cons_drop_twice
thf(fact_1110_mset__zip__take__Cons__drop__twice,axiom,
! [Xs: list_nat_nat,Ys: list_nat,J: nat,X3: nat > nat,Y: nat] :
( ( ( size_s8208510060688613859at_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ord_less_eq_nat @ J @ ( size_s8208510060688613859at_nat @ Xs ) )
=> ( ( mset_P2828777261068555141at_nat @ ( zip_nat_nat_nat @ ( append_nat_nat @ ( take_nat_nat @ J @ Xs ) @ ( cons_nat_nat @ X3 @ ( drop_nat_nat @ J @ Xs ) ) ) @ ( append_nat @ ( take_nat @ J @ Ys ) @ ( cons_nat @ Y @ ( drop_nat @ J @ Ys ) ) ) ) )
= ( add_ms4680508776633798606at_nat @ ( produc72220940542539688at_nat @ X3 @ Y ) @ ( mset_P2828777261068555141at_nat @ ( zip_nat_nat_nat @ Xs @ Ys ) ) ) ) ) ) ).
% mset_zip_take_Cons_drop_twice
thf(fact_1111_mset__zip__take__Cons__drop__twice,axiom,
! [Xs: list_a,Ys: list_a,J: nat,X3: a,Y: a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ord_less_eq_nat @ J @ ( size_size_list_a @ Xs ) )
=> ( ( mset_P3709024939784119484od_a_a @ ( zip_a_a @ ( append_a @ ( take_a @ J @ Xs ) @ ( cons_a @ X3 @ ( drop_a @ J @ Xs ) ) ) @ ( append_a @ ( take_a @ J @ Ys ) @ ( cons_a @ Y @ ( drop_a @ J @ Ys ) ) ) ) )
= ( add_ms8655138167283798533od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ ( mset_P3709024939784119484od_a_a @ ( zip_a_a @ Xs @ Ys ) ) ) ) ) ) ).
% mset_zip_take_Cons_drop_twice
thf(fact_1112_take__hd__drop,axiom,
! [N3: nat,Xs: list_a] :
( ( ord_less_nat @ N3 @ ( size_size_list_a @ Xs ) )
=> ( ( append_a @ ( take_a @ N3 @ Xs ) @ ( cons_a @ ( hd_a @ ( drop_a @ N3 @ Xs ) ) @ nil_a ) )
= ( take_a @ ( suc @ N3 ) @ Xs ) ) ) ).
% take_hd_drop
thf(fact_1113_le__zero__eq,axiom,
! [N3: nat] :
( ( ord_less_eq_nat @ N3 @ zero_zero_nat )
= ( N3 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_1114_add__le__cancel__left,axiom,
! [C2: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A2 ) @ ( plus_plus_nat @ C2 @ B ) )
= ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_cancel_left
thf(fact_1115_add__le__cancel__right,axiom,
! [A2: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_cancel_right
thf(fact_1116_le0,axiom,
! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N3 ) ).
% le0
thf(fact_1117_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_1118_Suc__le__mono,axiom,
! [N3: nat,M3: nat] :
( ( ord_less_eq_nat @ ( suc @ N3 ) @ ( suc @ M3 ) )
= ( ord_less_eq_nat @ N3 @ M3 ) ) ).
% Suc_le_mono
thf(fact_1119_nat__add__left__cancel__le,axiom,
! [K2: nat,M3: nat,N3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K2 @ M3 ) @ ( plus_plus_nat @ K2 @ N3 ) )
= ( ord_less_eq_nat @ M3 @ N3 ) ) ).
% nat_add_left_cancel_le
thf(fact_1120_diff__diff__cancel,axiom,
! [I: nat,N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( minus_minus_nat @ N3 @ ( minus_minus_nat @ N3 @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1121_add__le__same__cancel1,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A2 ) @ B )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_1122_add__le__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_1123_le__add__same__cancel1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_1124_le__add__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ B @ A2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_1125_le__add__diff__inverse,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A2 @ B ) )
= A2 ) ) ).
% le_add_diff_inverse
thf(fact_1126_le__add__diff__inverse2,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A2 @ B ) @ B )
= A2 ) ) ).
% le_add_diff_inverse2
thf(fact_1127_diff__is__0__eq_H,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
=> ( ( minus_minus_nat @ M3 @ N3 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1128_diff__is__0__eq,axiom,
! [M3: nat,N3: nat] :
( ( ( minus_minus_nat @ M3 @ N3 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M3 @ N3 ) ) ).
% diff_is_0_eq
thf(fact_1129_Nat_Oadd__diff__assoc,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K2 ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1130_Nat_Oadd__diff__assoc2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K2 ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1131_Nat_Odiff__diff__right,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1132_take__all__iff,axiom,
! [N3: nat,Xs: list_a] :
( ( ( take_a @ N3 @ Xs )
= Xs )
= ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N3 ) ) ).
% take_all_iff
thf(fact_1133_take__all,axiom,
! [Xs: list_a,N3: nat] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N3 )
=> ( ( take_a @ N3 @ Xs )
= Xs ) ) ).
% take_all
thf(fact_1134_list__update__beyond,axiom,
! [Xs: list_a,I: nat,X3: a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ I )
=> ( ( list_update_a @ Xs @ I @ X3 )
= Xs ) ) ).
% list_update_beyond
thf(fact_1135_diff__Suc__diff__eq2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K2 ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K2 @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1136_diff__Suc__diff__eq1,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K2 ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K2 ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1137_drop__eq__Nil2,axiom,
! [N3: nat,Xs: list_a] :
( ( nil_a
= ( drop_a @ N3 @ Xs ) )
= ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N3 ) ) ).
% drop_eq_Nil2
thf(fact_1138_drop__eq__Nil,axiom,
! [N3: nat,Xs: list_a] :
( ( ( drop_a @ N3 @ Xs )
= nil_a )
= ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N3 ) ) ).
% drop_eq_Nil
thf(fact_1139_drop__all,axiom,
! [Xs: list_a,N3: nat] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N3 )
=> ( ( drop_a @ N3 @ Xs )
= nil_a ) ) ).
% drop_all
thf(fact_1140_nth__drop,axiom,
! [N3: nat,Xs: list_a,I: nat] :
( ( ord_less_eq_nat @ N3 @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( drop_a @ N3 @ Xs ) @ I )
= ( nth_a @ Xs @ ( plus_plus_nat @ N3 @ I ) ) ) ) ).
% nth_drop
thf(fact_1141_hd__zip,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != nil_a )
=> ( ( Ys != nil_a )
=> ( ( hd_Product_prod_a_a @ ( zip_a_a @ Xs @ Ys ) )
= ( product_Pair_a_a @ ( hd_a @ Xs ) @ ( hd_a @ Ys ) ) ) ) ) ).
% hd_zip
thf(fact_1142_hd__zip,axiom,
! [Xs: list_multiset_a,Ys: list_multiset_a] :
( ( Xs != nil_multiset_a )
=> ( ( Ys != nil_multiset_a )
=> ( ( hd_Pro9105252180280396818iset_a @ ( zip_mu5258324667929623607iset_a @ Xs @ Ys ) )
= ( produc654756711066625303iset_a @ ( hd_multiset_a @ Xs ) @ ( hd_multiset_a @ Ys ) ) ) ) ) ).
% hd_zip
thf(fact_1143_hd__zip,axiom,
! [Xs: list_nat_nat,Ys: list_nat] :
( ( Xs != nil_nat_nat )
=> ( ( Ys != nil_nat )
=> ( ( hd_Pro5468458281938048987at_nat @ ( zip_nat_nat_nat @ Xs @ Ys ) )
= ( produc72220940542539688at_nat @ ( hd_nat_nat @ Xs ) @ ( hd_nat @ Ys ) ) ) ) ) ).
% hd_zip
thf(fact_1144_transitive__stepwise__le,axiom,
! [M3: nat,N3: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M3 @ N3 )
=> ( ! [X: nat] : ( R @ X @ X )
=> ( ! [X: nat,Y3: nat,Z4: nat] :
( ( R @ X @ Y3 )
=> ( ( R @ Y3 @ Z4 )
=> ( R @ X @ Z4 ) ) )
=> ( ! [N4: nat] : ( R @ N4 @ ( suc @ N4 ) )
=> ( R @ M3 @ N3 ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1145_nat__induct__at__least,axiom,
! [M3: nat,N3: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ M3 @ N3 )
=> ( ( P2 @ M3 )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ M3 @ N4 )
=> ( ( P2 @ N4 )
=> ( P2 @ ( suc @ N4 ) ) ) )
=> ( P2 @ N3 ) ) ) ) ).
% nat_induct_at_least
thf(fact_1146_full__nat__induct,axiom,
! [P2: nat > $o,N3: nat] :
( ! [N4: nat] :
( ! [M4: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N4 )
=> ( P2 @ M4 ) )
=> ( P2 @ N4 ) )
=> ( P2 @ N3 ) ) ).
% full_nat_induct
thf(fact_1147_not__less__eq__eq,axiom,
! [M3: nat,N3: nat] :
( ( ~ ( ord_less_eq_nat @ M3 @ N3 ) )
= ( ord_less_eq_nat @ ( suc @ N3 ) @ M3 ) ) ).
% not_less_eq_eq
thf(fact_1148_Suc__n__not__le__n,axiom,
! [N3: nat] :
~ ( ord_less_eq_nat @ ( suc @ N3 ) @ N3 ) ).
% Suc_n_not_le_n
thf(fact_1149_le__Suc__eq,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ ( suc @ N3 ) )
= ( ( ord_less_eq_nat @ M3 @ N3 )
| ( M3
= ( suc @ N3 ) ) ) ) ).
% le_Suc_eq
thf(fact_1150_Suc__le__D,axiom,
! [N3: nat,M9: nat] :
( ( ord_less_eq_nat @ ( suc @ N3 ) @ M9 )
=> ? [M6: nat] :
( M9
= ( suc @ M6 ) ) ) ).
% Suc_le_D
thf(fact_1151_le__SucI,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
=> ( ord_less_eq_nat @ M3 @ ( suc @ N3 ) ) ) ).
% le_SucI
thf(fact_1152_le__SucE,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ ( suc @ N3 ) )
=> ( ~ ( ord_less_eq_nat @ M3 @ N3 )
=> ( M3
= ( suc @ N3 ) ) ) ) ).
% le_SucE
thf(fact_1153_Suc__leD,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N3 )
=> ( ord_less_eq_nat @ M3 @ N3 ) ) ).
% Suc_leD
thf(fact_1154_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N3: nat,N6: nat] :
( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
=> ( ( ord_less_eq_nat @ N3 @ N6 )
=> ( ord_less_eq_nat @ ( F @ N6 ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1155_lift__Suc__mono__le,axiom,
! [F: nat > nat,N3: nat,N6: nat] :
( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_eq_nat @ N3 @ N6 )
=> ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ N6 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1156_le__refl,axiom,
! [N3: nat] : ( ord_less_eq_nat @ N3 @ N3 ) ).
% le_refl
thf(fact_1157_le__trans,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K2 )
=> ( ord_less_eq_nat @ I @ K2 ) ) ) ).
% le_trans
thf(fact_1158_eq__imp__le,axiom,
! [M3: nat,N3: nat] :
( ( M3 = N3 )
=> ( ord_less_eq_nat @ M3 @ N3 ) ) ).
% eq_imp_le
thf(fact_1159_le__antisym,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
=> ( ( ord_less_eq_nat @ N3 @ M3 )
=> ( M3 = N3 ) ) ) ).
% le_antisym
thf(fact_1160_nat__le__linear,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
| ( ord_less_eq_nat @ N3 @ M3 ) ) ).
% nat_le_linear
thf(fact_1161_Nat_Oex__has__greatest__nat,axiom,
! [P2: nat > $o,K2: nat,B: nat] :
( ( P2 @ K2 )
=> ( ! [Y3: nat] :
( ( P2 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X: nat] :
( ( P2 @ X )
& ! [Y6: nat] :
( ( P2 @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1162_add__leE,axiom,
! [M3: nat,K2: nat,N3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K2 ) @ N3 )
=> ~ ( ( ord_less_eq_nat @ M3 @ N3 )
=> ~ ( ord_less_eq_nat @ K2 @ N3 ) ) ) ).
% add_leE
thf(fact_1163_le__add1,axiom,
! [N3: nat,M3: nat] : ( ord_less_eq_nat @ N3 @ ( plus_plus_nat @ N3 @ M3 ) ) ).
% le_add1
thf(fact_1164_le__add2,axiom,
! [N3: nat,M3: nat] : ( ord_less_eq_nat @ N3 @ ( plus_plus_nat @ M3 @ N3 ) ) ).
% le_add2
thf(fact_1165_add__leD1,axiom,
! [M3: nat,K2: nat,N3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K2 ) @ N3 )
=> ( ord_less_eq_nat @ M3 @ N3 ) ) ).
% add_leD1
thf(fact_1166_add__leD2,axiom,
! [M3: nat,K2: nat,N3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K2 ) @ N3 )
=> ( ord_less_eq_nat @ K2 @ N3 ) ) ).
% add_leD2
thf(fact_1167_le__Suc__ex,axiom,
! [K2: nat,L: nat] :
( ( ord_less_eq_nat @ K2 @ L )
=> ? [N4: nat] :
( L
= ( plus_plus_nat @ K2 @ N4 ) ) ) ).
% le_Suc_ex
thf(fact_1168_add__le__mono,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K2 @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1169_add__le__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% add_le_mono1
thf(fact_1170_trans__le__add1,axiom,
! [I: nat,J: nat,M3: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M3 ) ) ) ).
% trans_le_add1
thf(fact_1171_trans__le__add2,axiom,
! [I: nat,J: nat,M3: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M3 @ J ) ) ) ).
% trans_le_add2
thf(fact_1172_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M8: nat,N5: nat] :
? [K5: nat] :
( N5
= ( plus_plus_nat @ M8 @ K5 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1173_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M8: nat,N5: nat] :
( ( ord_less_eq_nat @ M8 @ N5 )
& ( M8 != N5 ) ) ) ) ).
% nat_less_le
thf(fact_1174_less__imp__le__nat,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( ord_less_eq_nat @ M3 @ N3 ) ) ).
% less_imp_le_nat
thf(fact_1175_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M8: nat,N5: nat] :
( ( ord_less_nat @ M8 @ N5 )
| ( M8 = N5 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1176_less__or__eq__imp__le,axiom,
! [M3: nat,N3: nat] :
( ( ( ord_less_nat @ M3 @ N3 )
| ( M3 = N3 ) )
=> ( ord_less_eq_nat @ M3 @ N3 ) ) ).
% less_or_eq_imp_le
thf(fact_1177_le__neq__implies__less,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
=> ( ( M3 != N3 )
=> ( ord_less_nat @ M3 @ N3 ) ) ) ).
% le_neq_implies_less
thf(fact_1178_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1179_le__0__eq,axiom,
! [N3: nat] :
( ( ord_less_eq_nat @ N3 @ zero_zero_nat )
= ( N3 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1180_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1181_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1182_less__eq__nat_Osimps_I1_J,axiom,
! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N3 ) ).
% less_eq_nat.simps(1)
thf(fact_1183_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K2 = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1184_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1185_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1186_add__mono,axiom,
! [A2: nat,B: nat,C2: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C2 @ D2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_1187_add__left__mono,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A2 ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).
% add_left_mono
thf(fact_1188_less__eqE,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ~ ! [C: nat] :
( B
!= ( plus_plus_nat @ A2 @ C ) ) ) ).
% less_eqE
thf(fact_1189_add__right__mono,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add_right_mono
thf(fact_1190_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B6: nat] :
? [C5: nat] :
( B6
= ( plus_plus_nat @ A5 @ C5 ) ) ) ) ).
% le_iff_add
thf(fact_1191_add__le__imp__le__left,axiom,
! [C2: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A2 ) @ ( plus_plus_nat @ C2 @ B ) )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_imp_le_left
thf(fact_1192_add__le__imp__le__right,axiom,
! [A2: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_imp_le_right
thf(fact_1193_verit__comp__simplify1_I3_J,axiom,
! [B2: nat,A3: nat] :
( ( ~ ( ord_less_eq_nat @ B2 @ A3 ) )
= ( ord_less_nat @ A3 @ B2 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1194_zero__le,axiom,
! [X3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X3 ) ).
% zero_le
thf(fact_1195_diff__le__mono2,axiom,
! [M3: nat,N3: nat,L: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N3 ) @ ( minus_minus_nat @ L @ M3 ) ) ) ).
% diff_le_mono2
thf(fact_1196_le__diff__iff_H,axiom,
! [A2: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ C2 )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A2 ) @ ( minus_minus_nat @ C2 @ B ) )
= ( ord_less_eq_nat @ B @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_1197_diff__le__self,axiom,
! [M3: nat,N3: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ N3 ) @ M3 ) ).
% diff_le_self
thf(fact_1198_diff__le__mono,axiom,
! [M3: nat,N3: nat,L: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ L ) @ ( minus_minus_nat @ N3 @ L ) ) ) ).
% diff_le_mono
thf(fact_1199_Nat_Odiff__diff__eq,axiom,
! [K2: nat,M3: nat,N3: nat] :
( ( ord_less_eq_nat @ K2 @ M3 )
=> ( ( ord_less_eq_nat @ K2 @ N3 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M3 @ K2 ) @ ( minus_minus_nat @ N3 @ K2 ) )
= ( minus_minus_nat @ M3 @ N3 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1200_le__diff__iff,axiom,
! [K2: nat,M3: nat,N3: nat] :
( ( ord_less_eq_nat @ K2 @ M3 )
=> ( ( ord_less_eq_nat @ K2 @ N3 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ K2 ) @ ( minus_minus_nat @ N3 @ K2 ) )
= ( ord_less_eq_nat @ M3 @ N3 ) ) ) ) ).
% le_diff_iff
thf(fact_1201_eq__diff__iff,axiom,
! [K2: nat,M3: nat,N3: nat] :
( ( ord_less_eq_nat @ K2 @ M3 )
=> ( ( ord_less_eq_nat @ K2 @ N3 )
=> ( ( ( minus_minus_nat @ M3 @ K2 )
= ( minus_minus_nat @ N3 @ K2 ) )
= ( M3 = N3 ) ) ) ) ).
% eq_diff_iff
thf(fact_1202_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K2 )
= ( J
= ( plus_plus_nat @ K2 @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1203_Nat_Odiff__add__assoc2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K2 )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K2 ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1204_Nat_Odiff__add__assoc,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K2 )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K2 ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1205_Nat_Ole__diff__conv2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1206_le__diff__conv,axiom,
! [J: nat,K2: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K2 ) ) ) ).
% le_diff_conv
thf(fact_1207_diff__less__mono,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C2 @ A2 )
=> ( ord_less_nat @ ( minus_minus_nat @ A2 @ C2 ) @ ( minus_minus_nat @ B @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_1208_less__diff__iff,axiom,
! [K2: nat,M3: nat,N3: nat] :
( ( ord_less_eq_nat @ K2 @ M3 )
=> ( ( ord_less_eq_nat @ K2 @ N3 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M3 @ K2 ) @ ( minus_minus_nat @ N3 @ K2 ) )
= ( ord_less_nat @ M3 @ N3 ) ) ) ) ).
% less_diff_iff
thf(fact_1209_Suc__diff__le,axiom,
! [N3: nat,M3: nat] :
( ( ord_less_eq_nat @ N3 @ M3 )
=> ( ( minus_minus_nat @ ( suc @ M3 ) @ N3 )
= ( suc @ ( minus_minus_nat @ M3 @ N3 ) ) ) ) ).
% Suc_diff_le
thf(fact_1210_mono__nat__linear__lb,axiom,
! [F: nat > nat,M3: nat,K2: nat] :
( ! [M6: nat,N4: nat] :
( ( ord_less_nat @ M6 @ N4 )
=> ( ord_less_nat @ ( F @ M6 ) @ ( F @ N4 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M3 ) @ K2 ) @ ( F @ ( plus_plus_nat @ M3 @ K2 ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1211_impossible__Cons,axiom,
! [Xs: list_a,Ys: list_a,X3: a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) )
=> ( Xs
!= ( cons_a @ X3 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_1212_le__imp__less__Suc,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
=> ( ord_less_nat @ M3 @ ( suc @ N3 ) ) ) ).
% le_imp_less_Suc
thf(fact_1213_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N5: nat] : ( ord_less_eq_nat @ ( suc @ N5 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1214_less__Suc__eq__le,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ ( suc @ N3 ) )
= ( ord_less_eq_nat @ M3 @ N3 ) ) ).
% less_Suc_eq_le
thf(fact_1215_le__less__Suc__eq,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
=> ( ( ord_less_nat @ N3 @ ( suc @ M3 ) )
= ( N3 = M3 ) ) ) ).
% le_less_Suc_eq
thf(fact_1216_Suc__le__lessD,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N3 )
=> ( ord_less_nat @ M3 @ N3 ) ) ).
% Suc_le_lessD
thf(fact_1217_inc__induct,axiom,
! [I: nat,J: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P2 @ J )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ I @ N4 )
=> ( ( ord_less_nat @ N4 @ J )
=> ( ( P2 @ ( suc @ N4 ) )
=> ( P2 @ N4 ) ) ) )
=> ( P2 @ I ) ) ) ) ).
% inc_induct
thf(fact_1218_dec__induct,axiom,
! [I: nat,J: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P2 @ I )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ I @ N4 )
=> ( ( ord_less_nat @ N4 @ J )
=> ( ( P2 @ N4 )
=> ( P2 @ ( suc @ N4 ) ) ) ) )
=> ( P2 @ J ) ) ) ) ).
% dec_induct
thf(fact_1219_Suc__le__eq,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N3 )
= ( ord_less_nat @ M3 @ N3 ) ) ).
% Suc_le_eq
thf(fact_1220_Suc__leI,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( ord_less_eq_nat @ ( suc @ M3 ) @ N3 ) ) ).
% Suc_leI
thf(fact_1221_ex__least__nat__le,axiom,
! [P2: nat > $o,N3: nat] :
( ( P2 @ N3 )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N3 )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K3 )
=> ~ ( P2 @ I2 ) )
& ( P2 @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1222_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( minus_minus_nat @ B @ A2 )
= C2 )
= ( B
= ( plus_plus_nat @ C2 @ A2 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1223_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( plus_plus_nat @ A2 @ ( minus_minus_nat @ B @ A2 ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1224_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( minus_minus_nat @ C2 @ ( minus_minus_nat @ B @ A2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A2 ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1225_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A2 )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A2 ) @ C2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1226_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A2 ) @ C2 )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1227_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ B ) @ A2 )
= ( plus_plus_nat @ C2 @ ( minus_minus_nat @ B @ A2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1228_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( plus_plus_nat @ C2 @ ( minus_minus_nat @ B @ A2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C2 @ B ) @ A2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1229_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C2 @ ( minus_minus_nat @ B @ A2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A2 ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1230_le__add__diff,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ C2 @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A2 ) ) ) ).
% le_add_diff
thf(fact_1231_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A2 ) @ A2 )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_1232_add__le__imp__le__diff,axiom,
! [I: nat,K2: nat,N3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ N3 )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N3 @ K2 ) ) ) ).
% add_le_imp_le_diff
thf(fact_1233_add__le__add__imp__diff__le,axiom,
! [I: nat,K2: nat,N3: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ N3 )
=> ( ( ord_less_eq_nat @ N3 @ ( plus_plus_nat @ J @ K2 ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ N3 )
=> ( ( ord_less_eq_nat @ N3 @ ( plus_plus_nat @ J @ K2 ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N3 @ K2 ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1234_add__less__le__mono,axiom,
! [A2: nat,B: nat,C2: nat,D2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C2 @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_1235_add__le__less__mono,axiom,
! [A2: nat,B: nat,C2: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ C2 @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_1236_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1237_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1238_add__decreasing,axiom,
! [A2: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C2 ) @ B ) ) ) ).
% add_decreasing
thf(fact_1239_add__increasing,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A2 @ C2 ) ) ) ) ).
% add_increasing
thf(fact_1240_add__decreasing2,axiom,
! [C2: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ C2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C2 ) @ B ) ) ) ).
% add_decreasing2
thf(fact_1241_add__increasing2,axiom,
! [C2: nat,B: nat,A2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ( ord_less_eq_nat @ B @ A2 )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A2 @ C2 ) ) ) ) ).
% add_increasing2
thf(fact_1242_add__nonneg__nonneg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1243_add__nonpos__nonpos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_1244_add__nonneg__eq__0__iff,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X3 @ Y )
= zero_zero_nat )
= ( ( X3 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1245_add__nonpos__eq__0__iff,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X3 @ Y )
= zero_zero_nat )
= ( ( X3 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1246_list_Oset__sel_I1_J,axiom,
! [A2: list_P1396940483166286381od_a_a] :
( ( A2 != nil_Product_prod_a_a )
=> ( member1426531477525435216od_a_a @ ( hd_Product_prod_a_a @ A2 ) @ ( set_Product_prod_a_a2 @ A2 ) ) ) ).
% list.set_sel(1)
thf(fact_1247_list_Oset__sel_I1_J,axiom,
! [A2: list_P6988024003011616557iset_a] :
( ( A2 != nil_Pr2722442113885406509iset_a )
=> ( member5199237121806060112iset_a @ ( hd_Pro9105252180280396818iset_a @ A2 ) @ ( set_Pr5550698219922361532iset_a @ A2 ) ) ) ).
% list.set_sel(1)
thf(fact_1248_list_Oset__sel_I1_J,axiom,
! [A2: list_a] :
( ( A2 != nil_a )
=> ( member_a @ ( hd_a @ A2 ) @ ( set_a2 @ A2 ) ) ) ).
% list.set_sel(1)
thf(fact_1249_hd__in__set,axiom,
! [Xs: list_P1396940483166286381od_a_a] :
( ( Xs != nil_Product_prod_a_a )
=> ( member1426531477525435216od_a_a @ ( hd_Product_prod_a_a @ Xs ) @ ( set_Product_prod_a_a2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_1250_hd__in__set,axiom,
! [Xs: list_P6988024003011616557iset_a] :
( ( Xs != nil_Pr2722442113885406509iset_a )
=> ( member5199237121806060112iset_a @ ( hd_Pro9105252180280396818iset_a @ Xs ) @ ( set_Pr5550698219922361532iset_a @ Xs ) ) ) ).
% hd_in_set
thf(fact_1251_hd__in__set,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( member_a @ ( hd_a @ Xs ) @ ( set_a2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_1252_hd__conv__nth,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( hd_a @ Xs )
= ( nth_a @ Xs @ zero_zero_nat ) ) ) ).
% hd_conv_nth
thf(fact_1253_add__strict__increasing2,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A2 @ C2 ) ) ) ) ).
% add_strict_increasing2
thf(fact_1254_add__strict__increasing,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A2 @ C2 ) ) ) ) ).
% add_strict_increasing
thf(fact_1255_add__pos__nonneg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_1256_add__nonpos__neg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_1257_add__nonneg__pos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_1258_add__neg__nonpos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_1259_ex__least__nat__less,axiom,
! [P2: nat > $o,N3: nat] :
( ( P2 @ N3 )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_nat @ K3 @ N3 )
& ! [I2: nat] :
( ( ord_less_eq_nat @ I2 @ K3 )
=> ~ ( P2 @ I2 ) )
& ( P2 @ ( suc @ K3 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1260_Suc__le__length__iff,axiom,
! [N3: nat,Xs: list_a] :
( ( ord_less_eq_nat @ ( suc @ N3 ) @ ( size_size_list_a @ Xs ) )
= ( ? [X5: a,Ys4: list_a] :
( ( Xs
= ( cons_a @ X5 @ Ys4 ) )
& ( ord_less_eq_nat @ N3 @ ( size_size_list_a @ Ys4 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_1261_less__diff__conv2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K2 ) ) ) ) ).
% less_diff_conv2
thf(fact_1262_multiset__induct__max,axiom,
! [P2: multiset_nat > $o,M: multiset_nat] :
( ( P2 @ zero_z7348594199698428585et_nat )
=> ( ! [X: nat,M5: multiset_nat] :
( ( P2 @ M5 )
=> ( ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_mset_nat @ M5 ) )
=> ( ord_less_eq_nat @ Xa @ X ) )
=> ( P2 @ ( add_mset_nat @ X @ M5 ) ) ) )
=> ( P2 @ M ) ) ) ).
% multiset_induct_max
thf(fact_1263_multiset__induct__min,axiom,
! [P2: multiset_nat > $o,M: multiset_nat] :
( ( P2 @ zero_z7348594199698428585et_nat )
=> ( ! [X: nat,M5: multiset_nat] :
( ( P2 @ M5 )
=> ( ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_mset_nat @ M5 ) )
=> ( ord_less_eq_nat @ X @ Xa ) )
=> ( P2 @ ( add_mset_nat @ X @ M5 ) ) ) )
=> ( P2 @ M ) ) ) ).
% multiset_induct_min
thf(fact_1264_lenlex__length,axiom,
! [Ms: list_a,Ns: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ms @ Ns ) @ ( lenlex_a @ R2 ) )
=> ( ord_less_eq_nat @ ( size_size_list_a @ Ms ) @ ( size_size_list_a @ Ns ) ) ) ).
% lenlex_length
thf(fact_1265_hd__drop__conv__nth,axiom,
! [N3: nat,Xs: list_a] :
( ( ord_less_nat @ N3 @ ( size_size_list_a @ Xs ) )
=> ( ( hd_a @ ( drop_a @ N3 @ Xs ) )
= ( nth_a @ Xs @ N3 ) ) ) ).
% hd_drop_conv_nth
thf(fact_1266_nth__take__lemma,axiom,
! [K2: nat,Xs: list_a,Ys: list_a] :
( ( ord_less_eq_nat @ K2 @ ( size_size_list_a @ Xs ) )
=> ( ( ord_less_eq_nat @ K2 @ ( size_size_list_a @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ( ( nth_a @ Xs @ I3 )
= ( nth_a @ Ys @ I3 ) ) )
=> ( ( take_a @ K2 @ Xs )
= ( take_a @ K2 @ Ys ) ) ) ) ) ).
% nth_take_lemma
thf(fact_1267_append__eq__append__conv__if,axiom,
! [Xs_1: list_a,Xs_2: list_a,Ys_1: list_a,Ys_2: list_a] :
( ( ( append_a @ Xs_1 @ Xs_2 )
= ( append_a @ Ys_1 @ Ys_2 ) )
= ( ( ( ord_less_eq_nat @ ( size_size_list_a @ Xs_1 ) @ ( size_size_list_a @ Ys_1 ) )
=> ( ( Xs_1
= ( take_a @ ( size_size_list_a @ Xs_1 ) @ Ys_1 ) )
& ( Xs_2
= ( append_a @ ( drop_a @ ( size_size_list_a @ Xs_1 ) @ Ys_1 ) @ Ys_2 ) ) ) )
& ( ~ ( ord_less_eq_nat @ ( size_size_list_a @ Xs_1 ) @ ( size_size_list_a @ Ys_1 ) )
=> ( ( ( take_a @ ( size_size_list_a @ Ys_1 ) @ Xs_1 )
= Ys_1 )
& ( ( append_a @ ( drop_a @ ( size_size_list_a @ Ys_1 ) @ Xs_1 ) @ Xs_2 )
= Ys_2 ) ) ) ) ) ).
% append_eq_append_conv_if
% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X3: nat,Y: nat] :
( ( if_nat @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X3: nat,Y: nat] :
( ( if_nat @ $true @ X3 @ Y )
= X3 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
! [X: a] :
( ( member_a @ X @ ( set_a2 @ ys2 ) )
=> ? [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ xs2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa @ X ) @ s ) ) ) ).
%------------------------------------------------------------------------------