TPTP Problem File: SLH0832^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% 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_00111_004690__13269874_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1544 ( 528 unt; 266 typ; 0 def)
% Number of atoms : 4367 (1276 equ; 0 cnn)
% Maximal formula atoms : 30 ( 3 avg)
% Number of connectives : 16061 ( 349 ~; 43 |; 257 &;13215 @)
% ( 0 <=>;2197 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 8 avg)
% Number of types : 36 ( 35 usr)
% Number of type conns : 582 ( 582 >; 0 *; 0 +; 0 <<)
% Number of symbols : 234 ( 231 usr; 34 con; 0-5 aty)
% Number of variables : 4603 ( 229 ^;4244 !; 130 ?;4603 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:28:40.682
%------------------------------------------------------------------------------
% Could-be-implicit typings (35)
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% Relevant facts (1277)
thf(fact_0_add_Ohyps_I1_J,axiom,
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% add.hyps(1)
thf(fact_1_add_Ohyps_I2_J,axiom,
member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ nasa @ nbsa ) @ ( multis2373687813370458242ltpw_a @ ( sup_su3048258781599657691od_a_a @ ns @ s2 ) ) ).
% add.hyps(2)
thf(fact_2_add_Ohyps_I3_J,axiom,
? [Nas2: multiset_a,Sas2: multiset_a] :
( ( nasa
= ( plus_plus_multiset_a @ Nas2 @ Sas2 ) )
& ? [Nbs2: multiset_a,Sbs2: multiset_a] :
( ( nbsa
= ( plus_plus_multiset_a @ Nbs2 @ Sbs2 ) )
& ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Nas2 @ Nbs2 ) @ ( multis2373687813370458242ltpw_a @ ns ) )
& ! [X: a] :
( ( member_a @ X @ ( set_mset_a @ Sbs2 ) )
=> ? [A: a] :
( ( member_a @ A @ ( set_mset_a @ Sas2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ X ) @ s ) ) ) ) ) ).
% add.hyps(3)
thf(fact_3__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062nas2_Asas2_Anbs2_Asbs2_O_Anas_A_061_Anas2_A_L_Asas2_A_092_060and_062_Anbs_A_061_Anbs2_A_L_Asbs2_A_092_060and_062_A_Inas2_M_Anbs2_J_A_092_060in_062_Amultpw_ANS_A_092_060and_062_A_I_092_060forall_062b_092_060in_062_Dsbs2_O_A_092_060exists_062a_O_Aa_A_092_060in_062_D_Asas2_A_092_060and_062_A_Ia_M_Ab_J_A_092_060in_062_AS_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Nas2: multiset_a,Sas2: multiset_a,Nbs2: multiset_a,Sbs2: multiset_a] :
~ ( ( nasa
= ( plus_plus_multiset_a @ Nas2 @ Sas2 ) )
& ( nbsa
= ( plus_plus_multiset_a @ Nbs2 @ Sbs2 ) )
& ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Nas2 @ Nbs2 ) @ ( multis2373687813370458242ltpw_a @ ns ) )
& ! [X: a] :
( ( member_a @ X @ ( set_mset_a @ Sbs2 ) )
=> ? [A: a] :
( ( member_a @ A @ ( set_mset_a @ Sas2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ X ) @ s ) ) ) ) ).
% \<open>\<And>thesis. (\<And>nas2 sas2 nbs2 sbs2. nas = nas2 + sas2 \<and> nbs = nbs2 + sbs2 \<and> (nas2, nbs2) \<in> multpw NS \<and> (\<forall>b\<in>#sbs2. \<exists>a. a \<in># sas2 \<and> (a, b) \<in> S) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_4_assms,axiom,
ord_le746702958409616551od_a_a @ s2 @ s ).
% assms
thf(fact_5_ns,axiom,
member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ nas @ nbs ) @ ( multis2373687813370458242ltpw_a @ ( sup_su3048258781599657691od_a_a @ ns @ s2 ) ) ).
% ns
thf(fact_6__C_K_C,axiom,
( ( nasa
= ( plus_plus_multiset_a @ nas2 @ sas2 ) )
& ( nbsa
= ( plus_plus_multiset_a @ nbs2 @ sbs2 ) )
& ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ nas2 @ nbs2 ) @ ( multis2373687813370458242ltpw_a @ ns ) )
& ! [X: a] :
( ( member_a @ X @ ( set_mset_a @ sbs2 ) )
=> ? [A: a] :
( ( member_a @ A @ ( set_mset_a @ sas2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ X ) @ s ) ) ) ) ).
% "*"
thf(fact_7_s,axiom,
! [B: a] :
( ( member_a @ B @ ( set_mset_a @ sbs ) )
=> ? [A: a] :
( ( member_a @ A @ ( set_mset_a @ sas ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ s ) ) ) ).
% s
thf(fact_8_as,axiom,
( as
= ( plus_plus_multiset_a @ nas @ sas ) ) ).
% as
thf(fact_9_bs,axiom,
( bs
= ( plus_plus_multiset_a @ nbs @ sbs ) ) ).
% bs
thf(fact_10_union__mset__add__mset__left,axiom,
! [A2: a,A3: multiset_a,B2: multiset_a] :
( ( plus_plus_multiset_a @ ( add_mset_a @ A2 @ A3 ) @ B2 )
= ( add_mset_a @ A2 @ ( plus_plus_multiset_a @ A3 @ B2 ) ) ) ).
% union_mset_add_mset_left
thf(fact_11_union__mset__add__mset__right,axiom,
! [A3: multiset_a,A2: a,B2: multiset_a] :
( ( plus_plus_multiset_a @ A3 @ ( add_mset_a @ A2 @ B2 ) )
= ( add_mset_a @ A2 @ ( plus_plus_multiset_a @ A3 @ B2 ) ) ) ).
% union_mset_add_mset_right
thf(fact_12__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062nas_Asas_Anbs_Asbs_O_A_092_060lbrakk_062as_A_061_Anas_A_L_Asas_059_Abs_A_061_Anbs_A_L_Asbs_059_A_Inas_M_Anbs_J_A_092_060in_062_Amultpw_A_INS_A_092_060union_062_AS_H_J_059_A_092_060And_062b_O_Ab_A_092_060in_062_D_Asbs_A_092_060Longrightarrow_062_A_092_060exists_062a_O_Aa_A_092_060in_062_D_Asas_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,
~ ! [Nas: multiset_a,Sas: multiset_a] :
( ( as
= ( plus_plus_multiset_a @ Nas @ Sas ) )
=> ! [Nbs: multiset_a,Sbs: multiset_a] :
( ( bs
= ( plus_plus_multiset_a @ Nbs @ Sbs ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Nas @ Nbs ) @ ( multis2373687813370458242ltpw_a @ ( sup_su3048258781599657691od_a_a @ ns @ s2 ) ) )
=> ~ ! [B3: a] :
( ( member_a @ B3 @ ( set_mset_a @ Sbs ) )
=> ? [A: a] :
( ( member_a @ A @ ( set_mset_a @ Sas ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B3 ) @ s ) ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>nas sas nbs sbs. \<lbrakk>as = nas + sas; bs = nbs + sbs; (nas, nbs) \<in> multpw (NS \<union> S'); \<And>b. b \<in># sbs \<Longrightarrow> \<exists>a. a \<in># sas \<and> (a, b) \<in> S\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_13_multpw_Oadd,axiom,
! [X2: multiset_a,Y: multiset_a,Ns: set_Pr79727621955416071iset_a,X3: multiset_multiset_a,Y2: multiset_multiset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X2 @ Y ) @ Ns )
=> ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X3 @ Y2 ) @ ( multis451435756853798856iset_a @ Ns ) )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ ( add_mset_multiset_a @ X2 @ X3 ) @ ( add_mset_multiset_a @ Y @ Y2 ) ) @ ( multis451435756853798856iset_a @ Ns ) ) ) ) ).
% multpw.add
thf(fact_14_multpw_Oadd,axiom,
! [X2: a,Y: a,Ns: set_Product_prod_a_a,X3: multiset_a,Y2: multiset_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ Ns )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ Y2 ) @ ( multis2373687813370458242ltpw_a @ Ns ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( add_mset_a @ X2 @ X3 ) @ ( add_mset_a @ Y @ Y2 ) ) @ ( multis2373687813370458242ltpw_a @ Ns ) ) ) ) ).
% multpw.add
thf(fact_15_multpw__split1L,axiom,
! [X3: multiset_multiset_a,Y: multiset_a,Y2: multiset_multiset_a,Ns: set_Pr79727621955416071iset_a] :
( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X3 @ ( add_mset_multiset_a @ Y @ Y2 ) ) @ ( multis451435756853798856iset_a @ Ns ) )
=> ~ ! [Z: multiset_a,Z2: multiset_multiset_a] :
( ( X3
= ( add_mset_multiset_a @ Z @ Z2 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Z @ Y ) @ Ns )
=> ~ ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ Z2 @ Y2 ) @ ( multis451435756853798856iset_a @ Ns ) ) ) ) ) ).
% multpw_split1L
thf(fact_16_multpw__split1L,axiom,
! [X3: multiset_a,Y: a,Y2: multiset_a,Ns: set_Product_prod_a_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ ( add_mset_a @ Y @ Y2 ) ) @ ( multis2373687813370458242ltpw_a @ Ns ) )
=> ~ ! [Z: a,Z2: multiset_a] :
( ( X3
= ( add_mset_a @ Z @ Z2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ Ns )
=> ~ ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Z2 @ Y2 ) @ ( multis2373687813370458242ltpw_a @ Ns ) ) ) ) ) ).
% multpw_split1L
thf(fact_17_multpw__split1R,axiom,
! [X2: multiset_a,X3: multiset_multiset_a,Y2: multiset_multiset_a,Ns: set_Pr79727621955416071iset_a] :
( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ ( add_mset_multiset_a @ X2 @ X3 ) @ Y2 ) @ ( multis451435756853798856iset_a @ Ns ) )
=> ~ ! [Z: multiset_a,Z2: multiset_multiset_a] :
( ( Y2
= ( add_mset_multiset_a @ Z @ Z2 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X2 @ Z ) @ Ns )
=> ~ ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X3 @ Z2 ) @ ( multis451435756853798856iset_a @ Ns ) ) ) ) ) ).
% multpw_split1R
thf(fact_18_multpw__split1R,axiom,
! [X2: a,X3: multiset_a,Y2: multiset_a,Ns: set_Product_prod_a_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( add_mset_a @ X2 @ X3 ) @ Y2 ) @ ( multis2373687813370458242ltpw_a @ Ns ) )
=> ~ ! [Z: a,Z2: multiset_a] :
( ( Y2
= ( add_mset_a @ Z @ Z2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Z ) @ Ns )
=> ~ ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ Z2 ) @ ( multis2373687813370458242ltpw_a @ Ns ) ) ) ) ) ).
% multpw_split1R
thf(fact_19_multpw__add,axiom,
! [X1: multiset_a,Y1: multiset_a,Ns: set_Product_prod_a_a,X22: multiset_a,Y22: multiset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X1 @ Y1 ) @ ( multis2373687813370458242ltpw_a @ Ns ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X22 @ Y22 ) @ ( multis2373687813370458242ltpw_a @ Ns ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( plus_plus_multiset_a @ X1 @ X22 ) @ ( plus_plus_multiset_a @ Y1 @ Y22 ) ) @ ( multis2373687813370458242ltpw_a @ Ns ) ) ) ) ).
% multpw_add
thf(fact_20_multi__self__add__other__not__self,axiom,
! [M: multiset_a,X2: a] :
( M
!= ( add_mset_a @ X2 @ M ) ) ).
% multi_self_add_other_not_self
thf(fact_21_add__mset__add__mset__same__iff,axiom,
! [A2: a,A3: multiset_a,B2: multiset_a] :
( ( ( add_mset_a @ A2 @ A3 )
= ( add_mset_a @ A2 @ B2 ) )
= ( A3 = B2 ) ) ).
% add_mset_add_mset_same_iff
thf(fact_22_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_23_set__mset__union,axiom,
! [M: multis501812127501805293od_a_a,N: multis501812127501805293od_a_a] :
( ( set_ms119794563918438244od_a_a @ ( plus_p6967097483979887908od_a_a @ M @ N ) )
= ( sup_su3048258781599657691od_a_a @ ( set_ms119794563918438244od_a_a @ M ) @ ( set_ms119794563918438244od_a_a @ N ) ) ) ).
% set_mset_union
thf(fact_24__092_060open_062_Ias_M_Abs_J_A_092_060in_062_Ans__mul__ext_A_INS_A_092_060union_062_AS_H_J_AS_092_060close_062,axiom,
member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ as @ bs ) @ ( multis4023584895248362547_ext_a @ ( sup_su3048258781599657691od_a_a @ ns @ s2 ) @ s ) ).
% \<open>(as, bs) \<in> ns_mul_ext (NS \<union> S') S\<close>
thf(fact_25_mono__multpw,axiom,
! [Ns: set_Pr79727621955416071iset_a,Ns2: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ Ns @ Ns2 )
=> ( ord_le2895628588359463079iset_a @ ( multis451435756853798856iset_a @ Ns ) @ ( multis451435756853798856iset_a @ Ns2 ) ) ) ).
% mono_multpw
thf(fact_26_mono__multpw,axiom,
! [Ns: set_Product_prod_a_a,Ns2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ Ns @ Ns2 )
=> ( ord_le5871882978999109543iset_a @ ( multis2373687813370458242ltpw_a @ Ns ) @ ( multis2373687813370458242ltpw_a @ Ns2 ) ) ) ).
% mono_multpw
thf(fact_27_multi__union__self__other__eq,axiom,
! [A3: multiset_a,X3: multiset_a,Y2: multiset_a] :
( ( ( plus_plus_multiset_a @ A3 @ X3 )
= ( plus_plus_multiset_a @ A3 @ Y2 ) )
=> ( X3 = Y2 ) ) ).
% multi_union_self_other_eq
thf(fact_28_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_29_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_30_union__commute,axiom,
( plus_plus_multiset_a
= ( ^ [M2: multiset_a,N2: multiset_a] : ( plus_plus_multiset_a @ N2 @ M2 ) ) ) ).
% union_commute
thf(fact_31_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_32_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_33_add__mset__commute,axiom,
! [X2: a,Y: a,M: multiset_a] :
( ( add_mset_a @ X2 @ ( add_mset_a @ Y @ M ) )
= ( add_mset_a @ Y @ ( add_mset_a @ X2 @ M ) ) ) ).
% add_mset_commute
thf(fact_34_add__eq__conv__ex,axiom,
! [A2: a,M: multiset_a,B: a,N: multiset_a] :
( ( ( add_mset_a @ A2 @ M )
= ( add_mset_a @ B @ N ) )
= ( ( ( M = N )
& ( A2 = B ) )
| ? [K2: multiset_a] :
( ( M
= ( add_mset_a @ B @ K2 ) )
& ( N
= ( add_mset_a @ A2 @ K2 ) ) ) ) ) ).
% add_eq_conv_ex
thf(fact_35_union__iff,axiom,
! [A2: produc6518373309651786023iset_a,A3: multis4781622204924204013iset_a,B2: multis4781622204924204013iset_a] :
( ( member5199237121806060112iset_a @ A2 @ ( set_ms2379723421352415844iset_a @ ( plus_p1053901119268525860iset_a @ A3 @ B2 ) ) )
= ( ( member5199237121806060112iset_a @ A2 @ ( set_ms2379723421352415844iset_a @ A3 ) )
| ( member5199237121806060112iset_a @ A2 @ ( set_ms2379723421352415844iset_a @ B2 ) ) ) ) ).
% union_iff
thf(fact_36_union__iff,axiom,
! [A2: product_prod_a_a,A3: multis501812127501805293od_a_a,B2: multis501812127501805293od_a_a] :
( ( member1426531477525435216od_a_a @ A2 @ ( set_ms119794563918438244od_a_a @ ( plus_p6967097483979887908od_a_a @ A3 @ B2 ) ) )
= ( ( member1426531477525435216od_a_a @ A2 @ ( set_ms119794563918438244od_a_a @ A3 ) )
| ( member1426531477525435216od_a_a @ A2 @ ( set_ms119794563918438244od_a_a @ B2 ) ) ) ) ).
% union_iff
thf(fact_37_union__iff,axiom,
! [A2: a,A3: multiset_a,B2: multiset_a] :
( ( member_a @ A2 @ ( set_mset_a @ ( plus_plus_multiset_a @ A3 @ B2 ) ) )
= ( ( member_a @ A2 @ ( set_mset_a @ A3 ) )
| ( member_a @ A2 @ ( set_mset_a @ B2 ) ) ) ) ).
% union_iff
thf(fact_38_union__single__eq__member,axiom,
! [X2: produc6518373309651786023iset_a,M: multis4781622204924204013iset_a,N: multis4781622204924204013iset_a] :
( ( ( add_ms5348986034529679365iset_a @ X2 @ M )
= N )
=> ( member5199237121806060112iset_a @ X2 @ ( set_ms2379723421352415844iset_a @ N ) ) ) ).
% union_single_eq_member
thf(fact_39_union__single__eq__member,axiom,
! [X2: product_prod_a_a,M: multis501812127501805293od_a_a,N: multis501812127501805293od_a_a] :
( ( ( add_ms8655138167283798533od_a_a @ X2 @ M )
= N )
=> ( member1426531477525435216od_a_a @ X2 @ ( set_ms119794563918438244od_a_a @ N ) ) ) ).
% union_single_eq_member
thf(fact_40_union__single__eq__member,axiom,
! [X2: a,M: multiset_a,N: multiset_a] :
( ( ( add_mset_a @ X2 @ M )
= N )
=> ( member_a @ X2 @ ( set_mset_a @ N ) ) ) ).
% union_single_eq_member
thf(fact_41_insert__noteq__member,axiom,
! [B: produc6518373309651786023iset_a,B2: multis4781622204924204013iset_a,C: produc6518373309651786023iset_a,C2: multis4781622204924204013iset_a] :
( ( ( add_ms5348986034529679365iset_a @ B @ B2 )
= ( add_ms5348986034529679365iset_a @ C @ C2 ) )
=> ( ( B != C )
=> ( member5199237121806060112iset_a @ C @ ( set_ms2379723421352415844iset_a @ B2 ) ) ) ) ).
% insert_noteq_member
thf(fact_42_insert__noteq__member,axiom,
! [B: product_prod_a_a,B2: multis501812127501805293od_a_a,C: product_prod_a_a,C2: multis501812127501805293od_a_a] :
( ( ( add_ms8655138167283798533od_a_a @ B @ B2 )
= ( add_ms8655138167283798533od_a_a @ C @ C2 ) )
=> ( ( B != C )
=> ( member1426531477525435216od_a_a @ C @ ( set_ms119794563918438244od_a_a @ B2 ) ) ) ) ).
% insert_noteq_member
thf(fact_43_insert__noteq__member,axiom,
! [B: a,B2: multiset_a,C: a,C2: multiset_a] :
( ( ( add_mset_a @ B @ B2 )
= ( add_mset_a @ C @ C2 ) )
=> ( ( B != C )
=> ( member_a @ C @ ( set_mset_a @ B2 ) ) ) ) ).
% insert_noteq_member
thf(fact_44_multi__member__split,axiom,
! [X2: produc6518373309651786023iset_a,M: multis4781622204924204013iset_a] :
( ( member5199237121806060112iset_a @ X2 @ ( set_ms2379723421352415844iset_a @ M ) )
=> ? [A4: multis4781622204924204013iset_a] :
( M
= ( add_ms5348986034529679365iset_a @ X2 @ A4 ) ) ) ).
% multi_member_split
thf(fact_45_multi__member__split,axiom,
! [X2: product_prod_a_a,M: multis501812127501805293od_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ ( set_ms119794563918438244od_a_a @ M ) )
=> ? [A4: multis501812127501805293od_a_a] :
( M
= ( add_ms8655138167283798533od_a_a @ X2 @ A4 ) ) ) ).
% multi_member_split
thf(fact_46_multi__member__split,axiom,
! [X2: a,M: multiset_a] :
( ( member_a @ X2 @ ( set_mset_a @ M ) )
=> ? [A4: multiset_a] :
( M
= ( add_mset_a @ X2 @ A4 ) ) ) ).
% multi_member_split
thf(fact_47_mset__add,axiom,
! [A2: produc6518373309651786023iset_a,A3: multis4781622204924204013iset_a] :
( ( member5199237121806060112iset_a @ A2 @ ( set_ms2379723421352415844iset_a @ A3 ) )
=> ~ ! [B4: multis4781622204924204013iset_a] :
( A3
!= ( add_ms5348986034529679365iset_a @ A2 @ B4 ) ) ) ).
% mset_add
thf(fact_48_mset__add,axiom,
! [A2: product_prod_a_a,A3: multis501812127501805293od_a_a] :
( ( member1426531477525435216od_a_a @ A2 @ ( set_ms119794563918438244od_a_a @ A3 ) )
=> ~ ! [B4: multis501812127501805293od_a_a] :
( A3
!= ( add_ms8655138167283798533od_a_a @ A2 @ B4 ) ) ) ).
% mset_add
thf(fact_49_mset__add,axiom,
! [A2: a,A3: multiset_a] :
( ( member_a @ A2 @ ( set_mset_a @ A3 ) )
=> ~ ! [B4: multiset_a] :
( A3
!= ( add_mset_a @ A2 @ B4 ) ) ) ).
% mset_add
thf(fact_50_multpw__splitR,axiom,
! [X1: multiset_a,X22: multiset_a,Y2: multiset_a,Ns: set_Product_prod_a_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( plus_plus_multiset_a @ X1 @ X22 ) @ Y2 ) @ ( multis2373687813370458242ltpw_a @ Ns ) )
=> ~ ! [Y12: multiset_a,Y23: multiset_a] :
( ( Y2
= ( plus_plus_multiset_a @ Y12 @ Y23 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X1 @ Y12 ) @ ( multis2373687813370458242ltpw_a @ Ns ) )
=> ~ ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X22 @ Y23 ) @ ( multis2373687813370458242ltpw_a @ Ns ) ) ) ) ) ).
% multpw_splitR
thf(fact_51_multpw__splitL,axiom,
! [X3: multiset_a,Y1: multiset_a,Y22: multiset_a,Ns: set_Product_prod_a_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ ( plus_plus_multiset_a @ Y1 @ Y22 ) ) @ ( multis2373687813370458242ltpw_a @ Ns ) )
=> ~ ! [X12: multiset_a,X23: multiset_a] :
( ( X3
= ( plus_plus_multiset_a @ X12 @ X23 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X12 @ Y1 ) @ ( multis2373687813370458242ltpw_a @ Ns ) )
=> ~ ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X23 @ Y22 ) @ ( multis2373687813370458242ltpw_a @ Ns ) ) ) ) ) ).
% multpw_splitL
thf(fact_52_Un__subset__iff,axiom,
! [A3: set_Product_prod_a_a,B2: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( sup_su3048258781599657691od_a_a @ A3 @ B2 ) @ C2 )
= ( ( ord_le746702958409616551od_a_a @ A3 @ C2 )
& ( ord_le746702958409616551od_a_a @ B2 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_53_Un__subset__iff,axiom,
! [A3: set_Pr79727621955416071iset_a,B2: set_Pr79727621955416071iset_a,C2: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ ( sup_su1925787886501348827iset_a @ A3 @ B2 ) @ C2 )
= ( ( ord_le5871882978999109543iset_a @ A3 @ C2 )
& ( ord_le5871882978999109543iset_a @ B2 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_54_le__sup__iff,axiom,
! [X2: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( sup_su3048258781599657691od_a_a @ X2 @ Y ) @ Z3 )
= ( ( ord_le746702958409616551od_a_a @ X2 @ Z3 )
& ( ord_le746702958409616551od_a_a @ Y @ Z3 ) ) ) ).
% le_sup_iff
thf(fact_55_le__sup__iff,axiom,
! [X2: set_Pr79727621955416071iset_a,Y: set_Pr79727621955416071iset_a,Z3: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ ( sup_su1925787886501348827iset_a @ X2 @ Y ) @ Z3 )
= ( ( ord_le5871882978999109543iset_a @ X2 @ Z3 )
& ( ord_le5871882978999109543iset_a @ Y @ Z3 ) ) ) ).
% le_sup_iff
thf(fact_56_le__sup__iff,axiom,
! [X2: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X2 @ Y ) @ Z3 )
= ( ( ord_less_eq_nat @ X2 @ Z3 )
& ( ord_less_eq_nat @ Y @ Z3 ) ) ) ).
% le_sup_iff
thf(fact_57_sup_Obounded__iff,axiom,
! [B: set_Product_prod_a_a,C: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( sup_su3048258781599657691od_a_a @ B @ C ) @ A2 )
= ( ( ord_le746702958409616551od_a_a @ B @ A2 )
& ( ord_le746702958409616551od_a_a @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_58_sup_Obounded__iff,axiom,
! [B: set_Pr79727621955416071iset_a,C: set_Pr79727621955416071iset_a,A2: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ ( sup_su1925787886501348827iset_a @ B @ C ) @ A2 )
= ( ( ord_le5871882978999109543iset_a @ B @ A2 )
& ( ord_le5871882978999109543iset_a @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_59_sup_Obounded__iff,axiom,
! [B: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A2 )
= ( ( ord_less_eq_nat @ B @ A2 )
& ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_60_add__le__cancel__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_cancel_right
thf(fact_61_add__le__cancel__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_cancel_left
thf(fact_62_UnCI,axiom,
! [C: produc6518373309651786023iset_a,B2: set_Pr79727621955416071iset_a,A3: set_Pr79727621955416071iset_a] :
( ( ~ ( member5199237121806060112iset_a @ C @ B2 )
=> ( member5199237121806060112iset_a @ C @ A3 ) )
=> ( member5199237121806060112iset_a @ C @ ( sup_su1925787886501348827iset_a @ A3 @ B2 ) ) ) ).
% UnCI
thf(fact_63_UnCI,axiom,
! [C: a,B2: set_a,A3: set_a] :
( ( ~ ( member_a @ C @ B2 )
=> ( member_a @ C @ A3 ) )
=> ( member_a @ C @ ( sup_sup_set_a @ A3 @ B2 ) ) ) ).
% UnCI
thf(fact_64_UnCI,axiom,
! [C: product_prod_a_a,B2: set_Product_prod_a_a,A3: set_Product_prod_a_a] :
( ( ~ ( member1426531477525435216od_a_a @ C @ B2 )
=> ( member1426531477525435216od_a_a @ C @ A3 ) )
=> ( member1426531477525435216od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A3 @ B2 ) ) ) ).
% UnCI
thf(fact_65_Un__iff,axiom,
! [C: produc6518373309651786023iset_a,A3: set_Pr79727621955416071iset_a,B2: set_Pr79727621955416071iset_a] :
( ( member5199237121806060112iset_a @ C @ ( sup_su1925787886501348827iset_a @ A3 @ B2 ) )
= ( ( member5199237121806060112iset_a @ C @ A3 )
| ( member5199237121806060112iset_a @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_66_Un__iff,axiom,
! [C: a,A3: set_a,B2: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A3 @ B2 ) )
= ( ( member_a @ C @ A3 )
| ( member_a @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_67_Un__iff,axiom,
! [C: product_prod_a_a,A3: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A3 @ B2 ) )
= ( ( member1426531477525435216od_a_a @ C @ A3 )
| ( member1426531477525435216od_a_a @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_68_sup_Oidem,axiom,
! [A2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_69_mem__Collect__eq,axiom,
! [A2: produc6518373309651786023iset_a,P: produc6518373309651786023iset_a > $o] :
( ( member5199237121806060112iset_a @ A2 @ ( collec6344740439120713362iset_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_70_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_71_mem__Collect__eq,axiom,
! [A2: product_prod_a_a,P: product_prod_a_a > $o] :
( ( member1426531477525435216od_a_a @ A2 @ ( collec3336397797384452498od_a_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_72_Collect__mem__eq,axiom,
! [A3: set_Pr79727621955416071iset_a] :
( ( collec6344740439120713362iset_a
@ ^ [X4: produc6518373309651786023iset_a] : ( member5199237121806060112iset_a @ X4 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_73_Collect__mem__eq,axiom,
! [A3: set_a] :
( ( collect_a
@ ^ [X4: a] : ( member_a @ X4 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_74_Collect__mem__eq,axiom,
! [A3: set_Product_prod_a_a] :
( ( collec3336397797384452498od_a_a
@ ^ [X4: product_prod_a_a] : ( member1426531477525435216od_a_a @ X4 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_75_sup__idem,axiom,
! [X2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_76_sup_Oleft__idem,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A2 @ ( sup_su3048258781599657691od_a_a @ A2 @ B ) )
= ( sup_su3048258781599657691od_a_a @ A2 @ B ) ) ).
% sup.left_idem
thf(fact_77_sup__left__idem,axiom,
! [X2: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X2 @ ( sup_su3048258781599657691od_a_a @ X2 @ Y ) )
= ( sup_su3048258781599657691od_a_a @ X2 @ Y ) ) ).
% sup_left_idem
thf(fact_78_sup_Oright__idem,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( sup_su3048258781599657691od_a_a @ A2 @ B ) @ B )
= ( sup_su3048258781599657691od_a_a @ A2 @ B ) ) ).
% sup.right_idem
thf(fact_79_add__left__cancel,axiom,
! [A2: multiset_a,B: multiset_a,C: multiset_a] :
( ( ( plus_plus_multiset_a @ A2 @ B )
= ( plus_plus_multiset_a @ A2 @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_80_add__left__cancel,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ A2 @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_81_add__right__cancel,axiom,
! [B: multiset_a,A2: multiset_a,C: multiset_a] :
( ( ( plus_plus_multiset_a @ B @ A2 )
= ( plus_plus_multiset_a @ C @ A2 ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_82_add__right__cancel,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= ( plus_plus_nat @ C @ A2 ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_83_subset__antisym,axiom,
! [A3: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A3 @ B2 )
=> ( ( ord_le746702958409616551od_a_a @ B2 @ A3 )
=> ( A3 = B2 ) ) ) ).
% subset_antisym
thf(fact_84_subset__antisym,axiom,
! [A3: set_Pr79727621955416071iset_a,B2: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ A3 @ B2 )
=> ( ( ord_le5871882978999109543iset_a @ B2 @ A3 )
=> ( A3 = B2 ) ) ) ).
% subset_antisym
thf(fact_85_subsetI,axiom,
! [A3: set_a,B2: set_a] :
( ! [X5: a] :
( ( member_a @ X5 @ A3 )
=> ( member_a @ X5 @ B2 ) )
=> ( ord_less_eq_set_a @ A3 @ B2 ) ) ).
% subsetI
thf(fact_86_subsetI,axiom,
! [A3: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ! [X5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ A3 )
=> ( member1426531477525435216od_a_a @ X5 @ B2 ) )
=> ( ord_le746702958409616551od_a_a @ A3 @ B2 ) ) ).
% subsetI
thf(fact_87_subsetI,axiom,
! [A3: set_Pr79727621955416071iset_a,B2: set_Pr79727621955416071iset_a] :
( ! [X5: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X5 @ A3 )
=> ( member5199237121806060112iset_a @ X5 @ B2 ) )
=> ( ord_le5871882978999109543iset_a @ A3 @ B2 ) ) ).
% subsetI
thf(fact_88__092_060open_062ns__mul__ext_ANS_AS_A_092_060subseteq_062_Ans__mul__ext_A_INS_A_092_060union_062_AS_H_J_AS_092_060close_062,axiom,
ord_le5871882978999109543iset_a @ ( multis4023584895248362547_ext_a @ ns @ s ) @ ( multis4023584895248362547_ext_a @ ( sup_su3048258781599657691od_a_a @ ns @ s2 ) @ s ) ).
% \<open>ns_mul_ext NS S \<subseteq> ns_mul_ext (NS \<union> S') S\<close>
thf(fact_89_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: multiset_a,B: multiset_a,C: multiset_a] :
( ( plus_plus_multiset_a @ ( plus_plus_multiset_a @ A2 @ B ) @ C )
= ( plus_plus_multiset_a @ A2 @ ( plus_plus_multiset_a @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_90_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_91_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K3: nat,L: nat] :
( ( ( I = J )
& ( K3 = L ) )
=> ( ( plus_plus_nat @ I @ K3 )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_92_group__cancel_Oadd1,axiom,
! [A3: multiset_a,K3: multiset_a,A2: multiset_a,B: multiset_a] :
( ( A3
= ( plus_plus_multiset_a @ K3 @ A2 ) )
=> ( ( plus_plus_multiset_a @ A3 @ B )
= ( plus_plus_multiset_a @ K3 @ ( plus_plus_multiset_a @ A2 @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_93_group__cancel_Oadd1,axiom,
! [A3: nat,K3: nat,A2: nat,B: nat] :
( ( A3
= ( plus_plus_nat @ K3 @ A2 ) )
=> ( ( plus_plus_nat @ A3 @ B )
= ( plus_plus_nat @ K3 @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_94_group__cancel_Oadd2,axiom,
! [B2: multiset_a,K3: multiset_a,B: multiset_a,A2: multiset_a] :
( ( B2
= ( plus_plus_multiset_a @ K3 @ B ) )
=> ( ( plus_plus_multiset_a @ A2 @ B2 )
= ( plus_plus_multiset_a @ K3 @ ( plus_plus_multiset_a @ A2 @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_95_group__cancel_Oadd2,axiom,
! [B2: nat,K3: nat,B: nat,A2: nat] :
( ( B2
= ( plus_plus_nat @ K3 @ B ) )
=> ( ( plus_plus_nat @ A2 @ B2 )
= ( plus_plus_nat @ K3 @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_96_add_Oassoc,axiom,
! [A2: multiset_a,B: multiset_a,C: multiset_a] :
( ( plus_plus_multiset_a @ ( plus_plus_multiset_a @ A2 @ B ) @ C )
= ( plus_plus_multiset_a @ A2 @ ( plus_plus_multiset_a @ B @ C ) ) ) ).
% add.assoc
thf(fact_97_add_Oassoc,axiom,
! [A2: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_98_add_Ocommute,axiom,
( plus_plus_multiset_a
= ( ^ [A5: multiset_a,B5: multiset_a] : ( plus_plus_multiset_a @ B5 @ A5 ) ) ) ).
% add.commute
thf(fact_99_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A5: nat,B5: nat] : ( plus_plus_nat @ B5 @ A5 ) ) ) ).
% add.commute
thf(fact_100_add_Oleft__commute,axiom,
! [B: multiset_a,A2: multiset_a,C: multiset_a] :
( ( plus_plus_multiset_a @ B @ ( plus_plus_multiset_a @ A2 @ C ) )
= ( plus_plus_multiset_a @ A2 @ ( plus_plus_multiset_a @ B @ C ) ) ) ).
% add.left_commute
thf(fact_101_add_Oleft__commute,axiom,
! [B: nat,A2: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A2 @ C ) )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_102_add__left__imp__eq,axiom,
! [A2: multiset_a,B: multiset_a,C: multiset_a] :
( ( ( plus_plus_multiset_a @ A2 @ B )
= ( plus_plus_multiset_a @ A2 @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_103_add__left__imp__eq,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ A2 @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_104_add__right__imp__eq,axiom,
! [B: multiset_a,A2: multiset_a,C: multiset_a] :
( ( ( plus_plus_multiset_a @ B @ A2 )
= ( plus_plus_multiset_a @ C @ A2 ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_105_add__right__imp__eq,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= ( plus_plus_nat @ C @ A2 ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_106_Collect__mono__iff,axiom,
! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ( ord_le746702958409616551od_a_a @ ( collec3336397797384452498od_a_a @ P ) @ ( collec3336397797384452498od_a_a @ Q ) )
= ( ! [X4: product_prod_a_a] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_107_Collect__mono__iff,axiom,
! [P: produc6518373309651786023iset_a > $o,Q: produc6518373309651786023iset_a > $o] :
( ( ord_le5871882978999109543iset_a @ ( collec6344740439120713362iset_a @ P ) @ ( collec6344740439120713362iset_a @ Q ) )
= ( ! [X4: produc6518373309651786023iset_a] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_108_set__eq__subset,axiom,
( ( ^ [Y3: set_Product_prod_a_a,Z4: set_Product_prod_a_a] : ( Y3 = Z4 ) )
= ( ^ [A6: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A6 @ B6 )
& ( ord_le746702958409616551od_a_a @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_109_set__eq__subset,axiom,
( ( ^ [Y3: set_Pr79727621955416071iset_a,Z4: set_Pr79727621955416071iset_a] : ( Y3 = Z4 ) )
= ( ^ [A6: set_Pr79727621955416071iset_a,B6: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ A6 @ B6 )
& ( ord_le5871882978999109543iset_a @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_110_subset__trans,axiom,
! [A3: set_Product_prod_a_a,B2: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A3 @ B2 )
=> ( ( ord_le746702958409616551od_a_a @ B2 @ C2 )
=> ( ord_le746702958409616551od_a_a @ A3 @ C2 ) ) ) ).
% subset_trans
thf(fact_111_subset__trans,axiom,
! [A3: set_Pr79727621955416071iset_a,B2: set_Pr79727621955416071iset_a,C2: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ A3 @ B2 )
=> ( ( ord_le5871882978999109543iset_a @ B2 @ C2 )
=> ( ord_le5871882978999109543iset_a @ A3 @ C2 ) ) ) ).
% subset_trans
thf(fact_112_Collect__mono,axiom,
! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ! [X5: product_prod_a_a] :
( ( P @ X5 )
=> ( Q @ X5 ) )
=> ( ord_le746702958409616551od_a_a @ ( collec3336397797384452498od_a_a @ P ) @ ( collec3336397797384452498od_a_a @ Q ) ) ) ).
% Collect_mono
thf(fact_113_Collect__mono,axiom,
! [P: produc6518373309651786023iset_a > $o,Q: produc6518373309651786023iset_a > $o] :
( ! [X5: produc6518373309651786023iset_a] :
( ( P @ X5 )
=> ( Q @ X5 ) )
=> ( ord_le5871882978999109543iset_a @ ( collec6344740439120713362iset_a @ P ) @ ( collec6344740439120713362iset_a @ Q ) ) ) ).
% Collect_mono
thf(fact_114_subset__refl,axiom,
! [A3: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ A3 @ A3 ) ).
% subset_refl
thf(fact_115_subset__refl,axiom,
! [A3: set_Pr79727621955416071iset_a] : ( ord_le5871882978999109543iset_a @ A3 @ A3 ) ).
% subset_refl
thf(fact_116_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
! [T: a] :
( ( member_a @ T @ A6 )
=> ( member_a @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_117_subset__iff,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A6: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
! [T: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ T @ A6 )
=> ( member1426531477525435216od_a_a @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_118_subset__iff,axiom,
( ord_le5871882978999109543iset_a
= ( ^ [A6: set_Pr79727621955416071iset_a,B6: set_Pr79727621955416071iset_a] :
! [T: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ T @ A6 )
=> ( member5199237121806060112iset_a @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_119_equalityD2,axiom,
! [A3: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( A3 = B2 )
=> ( ord_le746702958409616551od_a_a @ B2 @ A3 ) ) ).
% equalityD2
thf(fact_120_equalityD2,axiom,
! [A3: set_Pr79727621955416071iset_a,B2: set_Pr79727621955416071iset_a] :
( ( A3 = B2 )
=> ( ord_le5871882978999109543iset_a @ B2 @ A3 ) ) ).
% equalityD2
thf(fact_121_equalityD1,axiom,
! [A3: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( A3 = B2 )
=> ( ord_le746702958409616551od_a_a @ A3 @ B2 ) ) ).
% equalityD1
thf(fact_122_equalityD1,axiom,
! [A3: set_Pr79727621955416071iset_a,B2: set_Pr79727621955416071iset_a] :
( ( A3 = B2 )
=> ( ord_le5871882978999109543iset_a @ A3 @ B2 ) ) ).
% equalityD1
thf(fact_123_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
! [X4: a] :
( ( member_a @ X4 @ A6 )
=> ( member_a @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_124_subset__eq,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A6: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
! [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ A6 )
=> ( member1426531477525435216od_a_a @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_125_subset__eq,axiom,
( ord_le5871882978999109543iset_a
= ( ^ [A6: set_Pr79727621955416071iset_a,B6: set_Pr79727621955416071iset_a] :
! [X4: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X4 @ A6 )
=> ( member5199237121806060112iset_a @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_126_equalityE,axiom,
! [A3: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( A3 = B2 )
=> ~ ( ( ord_le746702958409616551od_a_a @ A3 @ B2 )
=> ~ ( ord_le746702958409616551od_a_a @ B2 @ A3 ) ) ) ).
% equalityE
thf(fact_127_equalityE,axiom,
! [A3: set_Pr79727621955416071iset_a,B2: set_Pr79727621955416071iset_a] :
( ( A3 = B2 )
=> ~ ( ( ord_le5871882978999109543iset_a @ A3 @ B2 )
=> ~ ( ord_le5871882978999109543iset_a @ B2 @ A3 ) ) ) ).
% equalityE
thf(fact_128_subsetD,axiom,
! [A3: set_a,B2: set_a,C: a] :
( ( ord_less_eq_set_a @ A3 @ B2 )
=> ( ( member_a @ C @ A3 )
=> ( member_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_129_subsetD,axiom,
! [A3: set_Product_prod_a_a,B2: set_Product_prod_a_a,C: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A3 @ B2 )
=> ( ( member1426531477525435216od_a_a @ C @ A3 )
=> ( member1426531477525435216od_a_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_130_subsetD,axiom,
! [A3: set_Pr79727621955416071iset_a,B2: set_Pr79727621955416071iset_a,C: produc6518373309651786023iset_a] :
( ( ord_le5871882978999109543iset_a @ A3 @ B2 )
=> ( ( member5199237121806060112iset_a @ C @ A3 )
=> ( member5199237121806060112iset_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_131_in__mono,axiom,
! [A3: set_a,B2: set_a,X2: a] :
( ( ord_less_eq_set_a @ A3 @ B2 )
=> ( ( member_a @ X2 @ A3 )
=> ( member_a @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_132_in__mono,axiom,
! [A3: set_Product_prod_a_a,B2: set_Product_prod_a_a,X2: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A3 @ B2 )
=> ( ( member1426531477525435216od_a_a @ X2 @ A3 )
=> ( member1426531477525435216od_a_a @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_133_in__mono,axiom,
! [A3: set_Pr79727621955416071iset_a,B2: set_Pr79727621955416071iset_a,X2: produc6518373309651786023iset_a] :
( ( ord_le5871882978999109543iset_a @ A3 @ B2 )
=> ( ( member5199237121806060112iset_a @ X2 @ A3 )
=> ( member5199237121806060112iset_a @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_134_sup__left__commute,axiom,
! [X2: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z3: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X2 @ ( sup_su3048258781599657691od_a_a @ Y @ Z3 ) )
= ( sup_su3048258781599657691od_a_a @ Y @ ( sup_su3048258781599657691od_a_a @ X2 @ Z3 ) ) ) ).
% sup_left_commute
thf(fact_135_sup_Oleft__commute,axiom,
! [B: set_Product_prod_a_a,A2: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ B @ ( sup_su3048258781599657691od_a_a @ A2 @ C ) )
= ( sup_su3048258781599657691od_a_a @ A2 @ ( sup_su3048258781599657691od_a_a @ B @ C ) ) ) ).
% sup.left_commute
thf(fact_136_sup__commute,axiom,
( sup_su3048258781599657691od_a_a
= ( ^ [X4: set_Product_prod_a_a,Y4: set_Product_prod_a_a] : ( sup_su3048258781599657691od_a_a @ Y4 @ X4 ) ) ) ).
% sup_commute
thf(fact_137_sup_Ocommute,axiom,
( sup_su3048258781599657691od_a_a
= ( ^ [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] : ( sup_su3048258781599657691od_a_a @ B5 @ A5 ) ) ) ).
% sup.commute
thf(fact_138_sup__assoc,axiom,
! [X2: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z3: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( sup_su3048258781599657691od_a_a @ X2 @ Y ) @ Z3 )
= ( sup_su3048258781599657691od_a_a @ X2 @ ( sup_su3048258781599657691od_a_a @ Y @ Z3 ) ) ) ).
% sup_assoc
thf(fact_139_sup_Oassoc,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( sup_su3048258781599657691od_a_a @ A2 @ B ) @ C )
= ( sup_su3048258781599657691od_a_a @ A2 @ ( sup_su3048258781599657691od_a_a @ B @ C ) ) ) ).
% sup.assoc
thf(fact_140_inf__sup__aci_I5_J,axiom,
( sup_su3048258781599657691od_a_a
= ( ^ [X4: set_Product_prod_a_a,Y4: set_Product_prod_a_a] : ( sup_su3048258781599657691od_a_a @ Y4 @ X4 ) ) ) ).
% inf_sup_aci(5)
thf(fact_141_inf__sup__aci_I6_J,axiom,
! [X2: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z3: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( sup_su3048258781599657691od_a_a @ X2 @ Y ) @ Z3 )
= ( sup_su3048258781599657691od_a_a @ X2 @ ( sup_su3048258781599657691od_a_a @ Y @ Z3 ) ) ) ).
% inf_sup_aci(6)
thf(fact_142_inf__sup__aci_I7_J,axiom,
! [X2: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z3: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X2 @ ( sup_su3048258781599657691od_a_a @ Y @ Z3 ) )
= ( sup_su3048258781599657691od_a_a @ Y @ ( sup_su3048258781599657691od_a_a @ X2 @ Z3 ) ) ) ).
% inf_sup_aci(7)
thf(fact_143_inf__sup__aci_I8_J,axiom,
! [X2: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X2 @ ( sup_su3048258781599657691od_a_a @ X2 @ Y ) )
= ( sup_su3048258781599657691od_a_a @ X2 @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_144_Un__left__commute,axiom,
! [A3: set_Product_prod_a_a,B2: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A3 @ ( sup_su3048258781599657691od_a_a @ B2 @ C2 ) )
= ( sup_su3048258781599657691od_a_a @ B2 @ ( sup_su3048258781599657691od_a_a @ A3 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_145_Un__left__absorb,axiom,
! [A3: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A3 @ ( sup_su3048258781599657691od_a_a @ A3 @ B2 ) )
= ( sup_su3048258781599657691od_a_a @ A3 @ B2 ) ) ).
% Un_left_absorb
thf(fact_146_Un__commute,axiom,
( sup_su3048258781599657691od_a_a
= ( ^ [A6: set_Product_prod_a_a,B6: set_Product_prod_a_a] : ( sup_su3048258781599657691od_a_a @ B6 @ A6 ) ) ) ).
% Un_commute
thf(fact_147_Un__absorb,axiom,
! [A3: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A3 @ A3 )
= A3 ) ).
% Un_absorb
thf(fact_148_Un__assoc,axiom,
! [A3: set_Product_prod_a_a,B2: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( sup_su3048258781599657691od_a_a @ A3 @ B2 ) @ C2 )
= ( sup_su3048258781599657691od_a_a @ A3 @ ( sup_su3048258781599657691od_a_a @ B2 @ C2 ) ) ) ).
% Un_assoc
thf(fact_149_ball__Un,axiom,
! [A3: set_Product_prod_a_a,B2: set_Product_prod_a_a,P: product_prod_a_a > $o] :
( ( ! [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ ( sup_su3048258781599657691od_a_a @ A3 @ B2 ) )
=> ( P @ X4 ) ) )
= ( ! [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ A3 )
=> ( P @ X4 ) )
& ! [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ B2 )
=> ( P @ X4 ) ) ) ) ).
% ball_Un
thf(fact_150_bex__Un,axiom,
! [A3: set_Product_prod_a_a,B2: set_Product_prod_a_a,P: product_prod_a_a > $o] :
( ( ? [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ ( sup_su3048258781599657691od_a_a @ A3 @ B2 ) )
& ( P @ X4 ) ) )
= ( ? [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ A3 )
& ( P @ X4 ) )
| ? [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ B2 )
& ( P @ X4 ) ) ) ) ).
% bex_Un
thf(fact_151_UnI2,axiom,
! [C: produc6518373309651786023iset_a,B2: set_Pr79727621955416071iset_a,A3: set_Pr79727621955416071iset_a] :
( ( member5199237121806060112iset_a @ C @ B2 )
=> ( member5199237121806060112iset_a @ C @ ( sup_su1925787886501348827iset_a @ A3 @ B2 ) ) ) ).
% UnI2
thf(fact_152_UnI2,axiom,
! [C: a,B2: set_a,A3: set_a] :
( ( member_a @ C @ B2 )
=> ( member_a @ C @ ( sup_sup_set_a @ A3 @ B2 ) ) ) ).
% UnI2
thf(fact_153_UnI2,axiom,
! [C: product_prod_a_a,B2: set_Product_prod_a_a,A3: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ B2 )
=> ( member1426531477525435216od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A3 @ B2 ) ) ) ).
% UnI2
thf(fact_154_UnI1,axiom,
! [C: produc6518373309651786023iset_a,A3: set_Pr79727621955416071iset_a,B2: set_Pr79727621955416071iset_a] :
( ( member5199237121806060112iset_a @ C @ A3 )
=> ( member5199237121806060112iset_a @ C @ ( sup_su1925787886501348827iset_a @ A3 @ B2 ) ) ) ).
% UnI1
thf(fact_155_UnI1,axiom,
! [C: a,A3: set_a,B2: set_a] :
( ( member_a @ C @ A3 )
=> ( member_a @ C @ ( sup_sup_set_a @ A3 @ B2 ) ) ) ).
% UnI1
thf(fact_156_UnI1,axiom,
! [C: product_prod_a_a,A3: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ A3 )
=> ( member1426531477525435216od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A3 @ B2 ) ) ) ).
% UnI1
thf(fact_157_UnE,axiom,
! [C: produc6518373309651786023iset_a,A3: set_Pr79727621955416071iset_a,B2: set_Pr79727621955416071iset_a] :
( ( member5199237121806060112iset_a @ C @ ( sup_su1925787886501348827iset_a @ A3 @ B2 ) )
=> ( ~ ( member5199237121806060112iset_a @ C @ A3 )
=> ( member5199237121806060112iset_a @ C @ B2 ) ) ) ).
% UnE
thf(fact_158_UnE,axiom,
! [C: a,A3: set_a,B2: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A3 @ B2 ) )
=> ( ~ ( member_a @ C @ A3 )
=> ( member_a @ C @ B2 ) ) ) ).
% UnE
thf(fact_159_UnE,axiom,
! [C: product_prod_a_a,A3: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A3 @ B2 ) )
=> ( ~ ( member1426531477525435216od_a_a @ C @ A3 )
=> ( member1426531477525435216od_a_a @ C @ B2 ) ) ) ).
% UnE
thf(fact_160_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K3: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K3 = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_161_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K3: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K3 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_162_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K3: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K3 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_163_add__mono,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_164_add__left__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_165_less__eqE,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ~ ! [C3: nat] :
( B
!= ( plus_plus_nat @ A2 @ C3 ) ) ) ).
% less_eqE
thf(fact_166_add__right__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_167_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
? [C4: nat] :
( B5
= ( plus_plus_nat @ A5 @ C4 ) ) ) ) ).
% le_iff_add
thf(fact_168_add__le__imp__le__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_imp_le_left
thf(fact_169_add__le__imp__le__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_imp_le_right
thf(fact_170_sup_OcoboundedI2,axiom,
! [C: set_Product_prod_a_a,B: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ C @ B )
=> ( ord_le746702958409616551od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A2 @ B ) ) ) ).
% sup.coboundedI2
thf(fact_171_sup_OcoboundedI2,axiom,
! [C: set_Pr79727621955416071iset_a,B: set_Pr79727621955416071iset_a,A2: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ C @ B )
=> ( ord_le5871882978999109543iset_a @ C @ ( sup_su1925787886501348827iset_a @ A2 @ B ) ) ) ).
% sup.coboundedI2
thf(fact_172_sup_OcoboundedI2,axiom,
! [C: nat,B: nat,A2: nat] :
( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% sup.coboundedI2
thf(fact_173_sup_OcoboundedI1,axiom,
! [C: set_Product_prod_a_a,A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ C @ A2 )
=> ( ord_le746702958409616551od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A2 @ B ) ) ) ).
% sup.coboundedI1
thf(fact_174_sup_OcoboundedI1,axiom,
! [C: set_Pr79727621955416071iset_a,A2: set_Pr79727621955416071iset_a,B: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ C @ A2 )
=> ( ord_le5871882978999109543iset_a @ C @ ( sup_su1925787886501348827iset_a @ A2 @ B ) ) ) ).
% sup.coboundedI1
thf(fact_175_sup_OcoboundedI1,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% sup.coboundedI1
thf(fact_176_sup_Oabsorb__iff2,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A5 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_177_sup_Oabsorb__iff2,axiom,
( ord_le5871882978999109543iset_a
= ( ^ [A5: set_Pr79727621955416071iset_a,B5: set_Pr79727621955416071iset_a] :
( ( sup_su1925787886501348827iset_a @ A5 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_178_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
( ( sup_sup_nat @ A5 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_179_sup_Oabsorb__iff1,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [B5: set_Product_prod_a_a,A5: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A5 @ B5 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_180_sup_Oabsorb__iff1,axiom,
( ord_le5871882978999109543iset_a
= ( ^ [B5: set_Pr79727621955416071iset_a,A5: set_Pr79727621955416071iset_a] :
( ( sup_su1925787886501348827iset_a @ A5 @ B5 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_181_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A5: nat] :
( ( sup_sup_nat @ A5 @ B5 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_182_sup_Ocobounded2,axiom,
! [B: set_Product_prod_a_a,A2: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ B @ ( sup_su3048258781599657691od_a_a @ A2 @ B ) ) ).
% sup.cobounded2
thf(fact_183_sup_Ocobounded2,axiom,
! [B: set_Pr79727621955416071iset_a,A2: set_Pr79727621955416071iset_a] : ( ord_le5871882978999109543iset_a @ B @ ( sup_su1925787886501348827iset_a @ A2 @ B ) ) ).
% sup.cobounded2
thf(fact_184_sup_Ocobounded2,axiom,
! [B: nat,A2: nat] : ( ord_less_eq_nat @ B @ ( sup_sup_nat @ A2 @ B ) ) ).
% sup.cobounded2
thf(fact_185_sup_Ocobounded1,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ A2 @ ( sup_su3048258781599657691od_a_a @ A2 @ B ) ) ).
% sup.cobounded1
thf(fact_186_sup_Ocobounded1,axiom,
! [A2: set_Pr79727621955416071iset_a,B: set_Pr79727621955416071iset_a] : ( ord_le5871882978999109543iset_a @ A2 @ ( sup_su1925787886501348827iset_a @ A2 @ B ) ) ).
% sup.cobounded1
thf(fact_187_sup_Ocobounded1,axiom,
! [A2: nat,B: nat] : ( ord_less_eq_nat @ A2 @ ( sup_sup_nat @ A2 @ B ) ) ).
% sup.cobounded1
thf(fact_188_sup_Oorder__iff,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [B5: set_Product_prod_a_a,A5: set_Product_prod_a_a] :
( A5
= ( sup_su3048258781599657691od_a_a @ A5 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_189_sup_Oorder__iff,axiom,
( ord_le5871882978999109543iset_a
= ( ^ [B5: set_Pr79727621955416071iset_a,A5: set_Pr79727621955416071iset_a] :
( A5
= ( sup_su1925787886501348827iset_a @ A5 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_190_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A5: nat] :
( A5
= ( sup_sup_nat @ A5 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_191_sup_OboundedI,axiom,
! [B: set_Product_prod_a_a,A2: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B @ A2 )
=> ( ( ord_le746702958409616551od_a_a @ C @ A2 )
=> ( ord_le746702958409616551od_a_a @ ( sup_su3048258781599657691od_a_a @ B @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_192_sup_OboundedI,axiom,
! [B: set_Pr79727621955416071iset_a,A2: set_Pr79727621955416071iset_a,C: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ B @ A2 )
=> ( ( ord_le5871882978999109543iset_a @ C @ A2 )
=> ( ord_le5871882978999109543iset_a @ ( sup_su1925787886501348827iset_a @ B @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_193_sup_OboundedI,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_194_sup_OboundedE,axiom,
! [B: set_Product_prod_a_a,C: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( sup_su3048258781599657691od_a_a @ B @ C ) @ A2 )
=> ~ ( ( ord_le746702958409616551od_a_a @ B @ A2 )
=> ~ ( ord_le746702958409616551od_a_a @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_195_sup_OboundedE,axiom,
! [B: set_Pr79727621955416071iset_a,C: set_Pr79727621955416071iset_a,A2: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ ( sup_su1925787886501348827iset_a @ B @ C ) @ A2 )
=> ~ ( ( ord_le5871882978999109543iset_a @ B @ A2 )
=> ~ ( ord_le5871882978999109543iset_a @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_196_sup_OboundedE,axiom,
! [B: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A2 )
=> ~ ( ( ord_less_eq_nat @ B @ A2 )
=> ~ ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_197_sup__absorb2,axiom,
! [X2: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X2 @ Y )
=> ( ( sup_su3048258781599657691od_a_a @ X2 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_198_sup__absorb2,axiom,
! [X2: set_Pr79727621955416071iset_a,Y: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ X2 @ Y )
=> ( ( sup_su1925787886501348827iset_a @ X2 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_199_sup__absorb2,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( sup_sup_nat @ X2 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_200_sup__absorb1,axiom,
! [Y: set_Product_prod_a_a,X2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ Y @ X2 )
=> ( ( sup_su3048258781599657691od_a_a @ X2 @ Y )
= X2 ) ) ).
% sup_absorb1
thf(fact_201_sup__absorb1,axiom,
! [Y: set_Pr79727621955416071iset_a,X2: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ Y @ X2 )
=> ( ( sup_su1925787886501348827iset_a @ X2 @ Y )
= X2 ) ) ).
% sup_absorb1
thf(fact_202_sup__absorb1,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ( ( sup_sup_nat @ X2 @ Y )
= X2 ) ) ).
% sup_absorb1
thf(fact_203_sup_Oabsorb2,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B )
=> ( ( sup_su3048258781599657691od_a_a @ A2 @ B )
= B ) ) ).
% sup.absorb2
thf(fact_204_sup_Oabsorb2,axiom,
! [A2: set_Pr79727621955416071iset_a,B: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ A2 @ B )
=> ( ( sup_su1925787886501348827iset_a @ A2 @ B )
= B ) ) ).
% sup.absorb2
thf(fact_205_sup_Oabsorb2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( sup_sup_nat @ A2 @ B )
= B ) ) ).
% sup.absorb2
thf(fact_206_sup_Oabsorb1,axiom,
! [B: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B @ A2 )
=> ( ( sup_su3048258781599657691od_a_a @ A2 @ B )
= A2 ) ) ).
% sup.absorb1
thf(fact_207_sup_Oabsorb1,axiom,
! [B: set_Pr79727621955416071iset_a,A2: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ B @ A2 )
=> ( ( sup_su1925787886501348827iset_a @ A2 @ B )
= A2 ) ) ).
% sup.absorb1
thf(fact_208_sup_Oabsorb1,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( sup_sup_nat @ A2 @ B )
= A2 ) ) ).
% sup.absorb1
thf(fact_209_sup__unique,axiom,
! [F: set_Product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_a,X2: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ! [X5: set_Product_prod_a_a,Y5: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ X5 @ ( F @ X5 @ Y5 ) )
=> ( ! [X5: set_Product_prod_a_a,Y5: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ Y5 @ ( F @ X5 @ Y5 ) )
=> ( ! [X5: set_Product_prod_a_a,Y5: set_Product_prod_a_a,Z: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ Y5 @ X5 )
=> ( ( ord_le746702958409616551od_a_a @ Z @ X5 )
=> ( ord_le746702958409616551od_a_a @ ( F @ Y5 @ Z ) @ X5 ) ) )
=> ( ( sup_su3048258781599657691od_a_a @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_210_sup__unique,axiom,
! [F: set_Pr79727621955416071iset_a > set_Pr79727621955416071iset_a > set_Pr79727621955416071iset_a,X2: set_Pr79727621955416071iset_a,Y: set_Pr79727621955416071iset_a] :
( ! [X5: set_Pr79727621955416071iset_a,Y5: set_Pr79727621955416071iset_a] : ( ord_le5871882978999109543iset_a @ X5 @ ( F @ X5 @ Y5 ) )
=> ( ! [X5: set_Pr79727621955416071iset_a,Y5: set_Pr79727621955416071iset_a] : ( ord_le5871882978999109543iset_a @ Y5 @ ( F @ X5 @ Y5 ) )
=> ( ! [X5: set_Pr79727621955416071iset_a,Y5: set_Pr79727621955416071iset_a,Z: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ Y5 @ X5 )
=> ( ( ord_le5871882978999109543iset_a @ Z @ X5 )
=> ( ord_le5871882978999109543iset_a @ ( F @ Y5 @ Z ) @ X5 ) ) )
=> ( ( sup_su1925787886501348827iset_a @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_211_sup__unique,axiom,
! [F: nat > nat > nat,X2: nat,Y: nat] :
( ! [X5: nat,Y5: nat] : ( ord_less_eq_nat @ X5 @ ( F @ X5 @ Y5 ) )
=> ( ! [X5: nat,Y5: nat] : ( ord_less_eq_nat @ Y5 @ ( F @ X5 @ Y5 ) )
=> ( ! [X5: nat,Y5: nat,Z: nat] :
( ( ord_less_eq_nat @ Y5 @ X5 )
=> ( ( ord_less_eq_nat @ Z @ X5 )
=> ( ord_less_eq_nat @ ( F @ Y5 @ Z ) @ X5 ) ) )
=> ( ( sup_sup_nat @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_212_sup_OorderI,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( A2
= ( sup_su3048258781599657691od_a_a @ A2 @ B ) )
=> ( ord_le746702958409616551od_a_a @ B @ A2 ) ) ).
% sup.orderI
thf(fact_213_sup_OorderI,axiom,
! [A2: set_Pr79727621955416071iset_a,B: set_Pr79727621955416071iset_a] :
( ( A2
= ( sup_su1925787886501348827iset_a @ A2 @ B ) )
=> ( ord_le5871882978999109543iset_a @ B @ A2 ) ) ).
% sup.orderI
thf(fact_214_sup_OorderI,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( sup_sup_nat @ A2 @ B ) )
=> ( ord_less_eq_nat @ B @ A2 ) ) ).
% sup.orderI
thf(fact_215_sup_OorderE,axiom,
! [B: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B @ A2 )
=> ( A2
= ( sup_su3048258781599657691od_a_a @ A2 @ B ) ) ) ).
% sup.orderE
thf(fact_216_sup_OorderE,axiom,
! [B: set_Pr79727621955416071iset_a,A2: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ B @ A2 )
=> ( A2
= ( sup_su1925787886501348827iset_a @ A2 @ B ) ) ) ).
% sup.orderE
thf(fact_217_sup_OorderE,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( A2
= ( sup_sup_nat @ A2 @ B ) ) ) ).
% sup.orderE
thf(fact_218_le__iff__sup,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [X4: set_Product_prod_a_a,Y4: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X4 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_219_le__iff__sup,axiom,
( ord_le5871882978999109543iset_a
= ( ^ [X4: set_Pr79727621955416071iset_a,Y4: set_Pr79727621955416071iset_a] :
( ( sup_su1925787886501348827iset_a @ X4 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_220_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y4: nat] :
( ( sup_sup_nat @ X4 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_221_sup__least,axiom,
! [Y: set_Product_prod_a_a,X2: set_Product_prod_a_a,Z3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ Y @ X2 )
=> ( ( ord_le746702958409616551od_a_a @ Z3 @ X2 )
=> ( ord_le746702958409616551od_a_a @ ( sup_su3048258781599657691od_a_a @ Y @ Z3 ) @ X2 ) ) ) ).
% sup_least
thf(fact_222_sup__least,axiom,
! [Y: set_Pr79727621955416071iset_a,X2: set_Pr79727621955416071iset_a,Z3: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ Y @ X2 )
=> ( ( ord_le5871882978999109543iset_a @ Z3 @ X2 )
=> ( ord_le5871882978999109543iset_a @ ( sup_su1925787886501348827iset_a @ Y @ Z3 ) @ X2 ) ) ) ).
% sup_least
thf(fact_223_sup__least,axiom,
! [Y: nat,X2: nat,Z3: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ( ( ord_less_eq_nat @ Z3 @ X2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z3 ) @ X2 ) ) ) ).
% sup_least
thf(fact_224_sup__mono,axiom,
! [A2: set_Product_prod_a_a,C: set_Product_prod_a_a,B: set_Product_prod_a_a,D: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ C )
=> ( ( ord_le746702958409616551od_a_a @ B @ D )
=> ( ord_le746702958409616551od_a_a @ ( sup_su3048258781599657691od_a_a @ A2 @ B ) @ ( sup_su3048258781599657691od_a_a @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_225_sup__mono,axiom,
! [A2: set_Pr79727621955416071iset_a,C: set_Pr79727621955416071iset_a,B: set_Pr79727621955416071iset_a,D: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ A2 @ C )
=> ( ( ord_le5871882978999109543iset_a @ B @ D )
=> ( ord_le5871882978999109543iset_a @ ( sup_su1925787886501348827iset_a @ A2 @ B ) @ ( sup_su1925787886501348827iset_a @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_226_sup__mono,axiom,
! [A2: nat,C: nat,B: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B @ D )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B ) @ ( sup_sup_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_227_sup_Omono,axiom,
! [C: set_Product_prod_a_a,A2: set_Product_prod_a_a,D: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ C @ A2 )
=> ( ( ord_le746702958409616551od_a_a @ D @ B )
=> ( ord_le746702958409616551od_a_a @ ( sup_su3048258781599657691od_a_a @ C @ D ) @ ( sup_su3048258781599657691od_a_a @ A2 @ B ) ) ) ) ).
% sup.mono
thf(fact_228_sup_Omono,axiom,
! [C: set_Pr79727621955416071iset_a,A2: set_Pr79727621955416071iset_a,D: set_Pr79727621955416071iset_a,B: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ C @ A2 )
=> ( ( ord_le5871882978999109543iset_a @ D @ B )
=> ( ord_le5871882978999109543iset_a @ ( sup_su1925787886501348827iset_a @ C @ D ) @ ( sup_su1925787886501348827iset_a @ A2 @ B ) ) ) ) ).
% sup.mono
thf(fact_229_sup_Omono,axiom,
! [C: nat,A2: nat,D: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ( ord_less_eq_nat @ D @ B )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D ) @ ( sup_sup_nat @ A2 @ B ) ) ) ) ).
% sup.mono
thf(fact_230_le__supI2,axiom,
! [X2: set_Product_prod_a_a,B: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X2 @ B )
=> ( ord_le746702958409616551od_a_a @ X2 @ ( sup_su3048258781599657691od_a_a @ A2 @ B ) ) ) ).
% le_supI2
thf(fact_231_le__supI2,axiom,
! [X2: set_Pr79727621955416071iset_a,B: set_Pr79727621955416071iset_a,A2: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ X2 @ B )
=> ( ord_le5871882978999109543iset_a @ X2 @ ( sup_su1925787886501348827iset_a @ A2 @ B ) ) ) ).
% le_supI2
thf(fact_232_le__supI2,axiom,
! [X2: nat,B: nat,A2: nat] :
( ( ord_less_eq_nat @ X2 @ B )
=> ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% le_supI2
thf(fact_233_le__supI1,axiom,
! [X2: set_Product_prod_a_a,A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X2 @ A2 )
=> ( ord_le746702958409616551od_a_a @ X2 @ ( sup_su3048258781599657691od_a_a @ A2 @ B ) ) ) ).
% le_supI1
thf(fact_234_le__supI1,axiom,
! [X2: set_Pr79727621955416071iset_a,A2: set_Pr79727621955416071iset_a,B: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ X2 @ A2 )
=> ( ord_le5871882978999109543iset_a @ X2 @ ( sup_su1925787886501348827iset_a @ A2 @ B ) ) ) ).
% le_supI1
thf(fact_235_le__supI1,axiom,
! [X2: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ X2 @ A2 )
=> ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% le_supI1
thf(fact_236_sup__ge2,axiom,
! [Y: set_Product_prod_a_a,X2: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ Y @ ( sup_su3048258781599657691od_a_a @ X2 @ Y ) ) ).
% sup_ge2
thf(fact_237_sup__ge2,axiom,
! [Y: set_Pr79727621955416071iset_a,X2: set_Pr79727621955416071iset_a] : ( ord_le5871882978999109543iset_a @ Y @ ( sup_su1925787886501348827iset_a @ X2 @ Y ) ) ).
% sup_ge2
thf(fact_238_sup__ge2,axiom,
! [Y: nat,X2: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X2 @ Y ) ) ).
% sup_ge2
thf(fact_239_sup__ge1,axiom,
! [X2: set_Product_prod_a_a,Y: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ X2 @ ( sup_su3048258781599657691od_a_a @ X2 @ Y ) ) ).
% sup_ge1
thf(fact_240_sup__ge1,axiom,
! [X2: set_Pr79727621955416071iset_a,Y: set_Pr79727621955416071iset_a] : ( ord_le5871882978999109543iset_a @ X2 @ ( sup_su1925787886501348827iset_a @ X2 @ Y ) ) ).
% sup_ge1
thf(fact_241_sup__ge1,axiom,
! [X2: nat,Y: nat] : ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ X2 @ Y ) ) ).
% sup_ge1
thf(fact_242_le__supI,axiom,
! [A2: set_Product_prod_a_a,X2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ X2 )
=> ( ( ord_le746702958409616551od_a_a @ B @ X2 )
=> ( ord_le746702958409616551od_a_a @ ( sup_su3048258781599657691od_a_a @ A2 @ B ) @ X2 ) ) ) ).
% le_supI
thf(fact_243_le__supI,axiom,
! [A2: set_Pr79727621955416071iset_a,X2: set_Pr79727621955416071iset_a,B: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ A2 @ X2 )
=> ( ( ord_le5871882978999109543iset_a @ B @ X2 )
=> ( ord_le5871882978999109543iset_a @ ( sup_su1925787886501348827iset_a @ A2 @ B ) @ X2 ) ) ) ).
% le_supI
thf(fact_244_le__supI,axiom,
! [A2: nat,X2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ X2 )
=> ( ( ord_less_eq_nat @ B @ X2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B ) @ X2 ) ) ) ).
% le_supI
thf(fact_245_le__supE,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a,X2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( sup_su3048258781599657691od_a_a @ A2 @ B ) @ X2 )
=> ~ ( ( ord_le746702958409616551od_a_a @ A2 @ X2 )
=> ~ ( ord_le746702958409616551od_a_a @ B @ X2 ) ) ) ).
% le_supE
thf(fact_246_le__supE,axiom,
! [A2: set_Pr79727621955416071iset_a,B: set_Pr79727621955416071iset_a,X2: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ ( sup_su1925787886501348827iset_a @ A2 @ B ) @ X2 )
=> ~ ( ( ord_le5871882978999109543iset_a @ A2 @ X2 )
=> ~ ( ord_le5871882978999109543iset_a @ B @ X2 ) ) ) ).
% le_supE
thf(fact_247_le__supE,axiom,
! [A2: nat,B: nat,X2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B ) @ X2 )
=> ~ ( ( ord_less_eq_nat @ A2 @ X2 )
=> ~ ( ord_less_eq_nat @ B @ X2 ) ) ) ).
% le_supE
thf(fact_248_inf__sup__ord_I3_J,axiom,
! [X2: set_Product_prod_a_a,Y: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ X2 @ ( sup_su3048258781599657691od_a_a @ X2 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_249_inf__sup__ord_I3_J,axiom,
! [X2: set_Pr79727621955416071iset_a,Y: set_Pr79727621955416071iset_a] : ( ord_le5871882978999109543iset_a @ X2 @ ( sup_su1925787886501348827iset_a @ X2 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_250_inf__sup__ord_I3_J,axiom,
! [X2: nat,Y: nat] : ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ X2 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_251_inf__sup__ord_I4_J,axiom,
! [Y: set_Product_prod_a_a,X2: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ Y @ ( sup_su3048258781599657691od_a_a @ X2 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_252_inf__sup__ord_I4_J,axiom,
! [Y: set_Pr79727621955416071iset_a,X2: set_Pr79727621955416071iset_a] : ( ord_le5871882978999109543iset_a @ Y @ ( sup_su1925787886501348827iset_a @ X2 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_253_inf__sup__ord_I4_J,axiom,
! [Y: nat,X2: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X2 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_254_subset__Un__eq,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A6: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A6 @ B6 )
= B6 ) ) ) ).
% subset_Un_eq
thf(fact_255_subset__Un__eq,axiom,
( ord_le5871882978999109543iset_a
= ( ^ [A6: set_Pr79727621955416071iset_a,B6: set_Pr79727621955416071iset_a] :
( ( sup_su1925787886501348827iset_a @ A6 @ B6 )
= B6 ) ) ) ).
% subset_Un_eq
thf(fact_256_subset__UnE,axiom,
! [C2: set_Product_prod_a_a,A3: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ C2 @ ( sup_su3048258781599657691od_a_a @ A3 @ B2 ) )
=> ~ ! [A7: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A7 @ A3 )
=> ! [B7: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B7 @ B2 )
=> ( C2
!= ( sup_su3048258781599657691od_a_a @ A7 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_257_subset__UnE,axiom,
! [C2: set_Pr79727621955416071iset_a,A3: set_Pr79727621955416071iset_a,B2: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ C2 @ ( sup_su1925787886501348827iset_a @ A3 @ B2 ) )
=> ~ ! [A7: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ A7 @ A3 )
=> ! [B7: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ B7 @ B2 )
=> ( C2
!= ( sup_su1925787886501348827iset_a @ A7 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_258_Un__absorb2,axiom,
! [B2: set_Product_prod_a_a,A3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B2 @ A3 )
=> ( ( sup_su3048258781599657691od_a_a @ A3 @ B2 )
= A3 ) ) ).
% Un_absorb2
thf(fact_259_Un__absorb2,axiom,
! [B2: set_Pr79727621955416071iset_a,A3: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ B2 @ A3 )
=> ( ( sup_su1925787886501348827iset_a @ A3 @ B2 )
= A3 ) ) ).
% Un_absorb2
thf(fact_260_Un__absorb1,axiom,
! [A3: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A3 @ B2 )
=> ( ( sup_su3048258781599657691od_a_a @ A3 @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_261_Un__absorb1,axiom,
! [A3: set_Pr79727621955416071iset_a,B2: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ A3 @ B2 )
=> ( ( sup_su1925787886501348827iset_a @ A3 @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_262_Un__upper2,axiom,
! [B2: set_Product_prod_a_a,A3: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ B2 @ ( sup_su3048258781599657691od_a_a @ A3 @ B2 ) ) ).
% Un_upper2
thf(fact_263_Un__upper2,axiom,
! [B2: set_Pr79727621955416071iset_a,A3: set_Pr79727621955416071iset_a] : ( ord_le5871882978999109543iset_a @ B2 @ ( sup_su1925787886501348827iset_a @ A3 @ B2 ) ) ).
% Un_upper2
thf(fact_264_Un__upper1,axiom,
! [A3: set_Product_prod_a_a,B2: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ A3 @ ( sup_su3048258781599657691od_a_a @ A3 @ B2 ) ) ).
% Un_upper1
thf(fact_265_Un__upper1,axiom,
! [A3: set_Pr79727621955416071iset_a,B2: set_Pr79727621955416071iset_a] : ( ord_le5871882978999109543iset_a @ A3 @ ( sup_su1925787886501348827iset_a @ A3 @ B2 ) ) ).
% Un_upper1
thf(fact_266_Un__least,axiom,
! [A3: set_Product_prod_a_a,C2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A3 @ C2 )
=> ( ( ord_le746702958409616551od_a_a @ B2 @ C2 )
=> ( ord_le746702958409616551od_a_a @ ( sup_su3048258781599657691od_a_a @ A3 @ B2 ) @ C2 ) ) ) ).
% Un_least
thf(fact_267_Un__least,axiom,
! [A3: set_Pr79727621955416071iset_a,C2: set_Pr79727621955416071iset_a,B2: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ A3 @ C2 )
=> ( ( ord_le5871882978999109543iset_a @ B2 @ C2 )
=> ( ord_le5871882978999109543iset_a @ ( sup_su1925787886501348827iset_a @ A3 @ B2 ) @ C2 ) ) ) ).
% Un_least
thf(fact_268_Un__mono,axiom,
! [A3: set_Product_prod_a_a,C2: set_Product_prod_a_a,B2: set_Product_prod_a_a,D2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A3 @ C2 )
=> ( ( ord_le746702958409616551od_a_a @ B2 @ D2 )
=> ( ord_le746702958409616551od_a_a @ ( sup_su3048258781599657691od_a_a @ A3 @ B2 ) @ ( sup_su3048258781599657691od_a_a @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_269_Un__mono,axiom,
! [A3: set_Pr79727621955416071iset_a,C2: set_Pr79727621955416071iset_a,B2: set_Pr79727621955416071iset_a,D2: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ A3 @ C2 )
=> ( ( ord_le5871882978999109543iset_a @ B2 @ D2 )
=> ( ord_le5871882978999109543iset_a @ ( sup_su1925787886501348827iset_a @ A3 @ B2 ) @ ( sup_su1925787886501348827iset_a @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_270_ns__mul__extE,axiom,
! [A3: multis4781622204924204013iset_a,B2: multis4781622204924204013iset_a,Ns: set_Pr3059988933816007879iset_a,S: set_Pr3059988933816007879iset_a] :
( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ A3 @ B2 ) @ ( multis2244641948333282396iset_a @ Ns @ S ) )
=> ~ ! [A1: multis4781622204924204013iset_a,A22: multis4781622204924204013iset_a] :
( ( A3
= ( plus_p1053901119268525860iset_a @ A1 @ A22 ) )
=> ! [B1: multis4781622204924204013iset_a,B22: multis4781622204924204013iset_a] :
( ( B2
= ( plus_p1053901119268525860iset_a @ B1 @ B22 ) )
=> ( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ A1 @ B1 ) @ ( multis3430030276581740203iset_a @ Ns ) )
=> ~ ! [B3: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ B3 @ ( set_ms2379723421352415844iset_a @ B22 ) )
=> ? [A: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ A @ ( set_ms2379723421352415844iset_a @ A22 ) )
& ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ A @ B3 ) @ S ) ) ) ) ) ) ) ).
% ns_mul_extE
thf(fact_271_ns__mul__extE,axiom,
! [A3: multis501812127501805293od_a_a,B2: multis501812127501805293od_a_a,Ns: set_Pr8600417178894128327od_a_a,S: set_Pr8600417178894128327od_a_a] :
( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ A3 @ B2 ) @ ( multis1178328011278088540od_a_a @ Ns @ S ) )
=> ~ ! [A1: multis501812127501805293od_a_a,A22: multis501812127501805293od_a_a] :
( ( A3
= ( plus_p6967097483979887908od_a_a @ A1 @ A22 ) )
=> ! [B1: multis501812127501805293od_a_a,B22: multis501812127501805293od_a_a] :
( ( B2
= ( plus_p6967097483979887908od_a_a @ B1 @ B22 ) )
=> ( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ A1 @ B1 ) @ ( multis1821517007024690859od_a_a @ Ns ) )
=> ~ ! [B3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ B3 @ ( 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 @ B3 ) @ S ) ) ) ) ) ) ) ).
% ns_mul_extE
thf(fact_272_ns__mul__extE,axiom,
! [A3: multiset_multiset_a,B2: multiset_multiset_a,Ns: set_Pr79727621955416071iset_a,S: set_Pr79727621955416071iset_a] :
( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ A3 @ B2 ) @ ( multis8112516946959054073iset_a @ Ns @ S ) )
=> ~ ! [A1: multiset_multiset_a,A22: multiset_multiset_a] :
( ( A3
= ( plus_p6738641960240532161iset_a @ A1 @ A22 ) )
=> ! [B1: multiset_multiset_a,B22: multiset_multiset_a] :
( ( B2
= ( plus_p6738641960240532161iset_a @ B1 @ B22 ) )
=> ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ A1 @ B1 ) @ ( multis451435756853798856iset_a @ Ns ) )
=> ~ ! [B3: multiset_a] :
( ( member_multiset_a @ B3 @ ( set_mset_multiset_a @ B22 ) )
=> ? [A: multiset_a] :
( ( member_multiset_a @ A @ ( set_mset_multiset_a @ A22 ) )
& ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A @ B3 ) @ S ) ) ) ) ) ) ) ).
% ns_mul_extE
thf(fact_273_ns__mul__extE,axiom,
! [A3: multiset_a,B2: multiset_a,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A3 @ B2 ) @ ( multis4023584895248362547_ext_a @ Ns @ S ) )
=> ~ ! [A1: multiset_a,A22: multiset_a] :
( ( A3
= ( plus_plus_multiset_a @ A1 @ A22 ) )
=> ! [B1: multiset_a,B22: multiset_a] :
( ( B2
= ( plus_plus_multiset_a @ B1 @ B22 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A1 @ B1 ) @ ( multis2373687813370458242ltpw_a @ Ns ) )
=> ~ ! [B3: a] :
( ( member_a @ B3 @ ( set_mset_a @ B22 ) )
=> ? [A: a] :
( ( member_a @ A @ ( set_mset_a @ A22 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B3 ) @ S ) ) ) ) ) ) ) ).
% ns_mul_extE
thf(fact_274_ns__mul__extI,axiom,
! [A3: multis4781622204924204013iset_a,A12: multis4781622204924204013iset_a,A23: multis4781622204924204013iset_a,B2: multis4781622204924204013iset_a,B12: multis4781622204924204013iset_a,B23: multis4781622204924204013iset_a,Ns: set_Pr3059988933816007879iset_a,S: set_Pr3059988933816007879iset_a] :
( ( A3
= ( plus_p1053901119268525860iset_a @ A12 @ A23 ) )
=> ( ( B2
= ( plus_p1053901119268525860iset_a @ B12 @ B23 ) )
=> ( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ A12 @ B12 ) @ ( multis3430030276581740203iset_a @ Ns ) )
=> ( ! [B8: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ B8 @ ( set_ms2379723421352415844iset_a @ B23 ) )
=> ? [A8: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ A8 @ ( set_ms2379723421352415844iset_a @ A23 ) )
& ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ A8 @ B8 ) @ S ) ) )
=> ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ A3 @ B2 ) @ ( multis2244641948333282396iset_a @ Ns @ S ) ) ) ) ) ) ).
% ns_mul_extI
thf(fact_275_ns__mul__extI,axiom,
! [A3: multis501812127501805293od_a_a,A12: multis501812127501805293od_a_a,A23: multis501812127501805293od_a_a,B2: multis501812127501805293od_a_a,B12: multis501812127501805293od_a_a,B23: multis501812127501805293od_a_a,Ns: set_Pr8600417178894128327od_a_a,S: set_Pr8600417178894128327od_a_a] :
( ( A3
= ( plus_p6967097483979887908od_a_a @ A12 @ A23 ) )
=> ( ( B2
= ( plus_p6967097483979887908od_a_a @ B12 @ B23 ) )
=> ( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ A12 @ B12 ) @ ( multis1821517007024690859od_a_a @ Ns ) )
=> ( ! [B8: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ B8 @ ( set_ms119794563918438244od_a_a @ B23 ) )
=> ? [A8: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A8 @ ( set_ms119794563918438244od_a_a @ A23 ) )
& ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A8 @ B8 ) @ S ) ) )
=> ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ A3 @ B2 ) @ ( multis1178328011278088540od_a_a @ Ns @ S ) ) ) ) ) ) ).
% ns_mul_extI
thf(fact_276_ns__mul__extI,axiom,
! [A3: multiset_multiset_a,A12: multiset_multiset_a,A23: multiset_multiset_a,B2: multiset_multiset_a,B12: multiset_multiset_a,B23: multiset_multiset_a,Ns: set_Pr79727621955416071iset_a,S: set_Pr79727621955416071iset_a] :
( ( A3
= ( plus_p6738641960240532161iset_a @ A12 @ A23 ) )
=> ( ( B2
= ( plus_p6738641960240532161iset_a @ B12 @ B23 ) )
=> ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ A12 @ B12 ) @ ( multis451435756853798856iset_a @ Ns ) )
=> ( ! [B8: multiset_a] :
( ( member_multiset_a @ B8 @ ( set_mset_multiset_a @ B23 ) )
=> ? [A8: multiset_a] :
( ( member_multiset_a @ A8 @ ( set_mset_multiset_a @ A23 ) )
& ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A8 @ B8 ) @ S ) ) )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ A3 @ B2 ) @ ( multis8112516946959054073iset_a @ Ns @ S ) ) ) ) ) ) ).
% ns_mul_extI
thf(fact_277_ns__mul__extI,axiom,
! [A3: multiset_a,A12: multiset_a,A23: multiset_a,B2: multiset_a,B12: multiset_a,B23: multiset_a,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( A3
= ( plus_plus_multiset_a @ A12 @ A23 ) )
=> ( ( B2
= ( plus_plus_multiset_a @ B12 @ B23 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A12 @ B12 ) @ ( multis2373687813370458242ltpw_a @ Ns ) )
=> ( ! [B8: a] :
( ( member_a @ B8 @ ( set_mset_a @ B23 ) )
=> ? [A8: a] :
( ( member_a @ A8 @ ( set_mset_a @ A23 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A8 @ B8 ) @ S ) ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A3 @ B2 ) @ ( multis4023584895248362547_ext_a @ Ns @ S ) ) ) ) ) ) ).
% ns_mul_extI
thf(fact_278_ns__mul__ext__mono,axiom,
! [Ns: set_Pr79727621955416071iset_a,Ns2: set_Pr79727621955416071iset_a,S: set_Pr79727621955416071iset_a,S2: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ Ns @ Ns2 )
=> ( ( ord_le5871882978999109543iset_a @ S @ S2 )
=> ( ord_le2895628588359463079iset_a @ ( multis8112516946959054073iset_a @ Ns @ S ) @ ( multis8112516946959054073iset_a @ Ns2 @ S2 ) ) ) ) ).
% ns_mul_ext_mono
thf(fact_279_ns__mul__ext__mono,axiom,
! [Ns: set_Product_prod_a_a,Ns2: set_Product_prod_a_a,S: set_Product_prod_a_a,S2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ Ns @ Ns2 )
=> ( ( ord_le746702958409616551od_a_a @ S @ S2 )
=> ( ord_le5871882978999109543iset_a @ ( multis4023584895248362547_ext_a @ Ns @ S ) @ ( multis4023584895248362547_ext_a @ Ns2 @ S2 ) ) ) ) ).
% ns_mul_ext_mono
thf(fact_280_ns__ns__mul__ext__union__compat,axiom,
! [A3: multiset_a,B2: multiset_a,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a,C2: multiset_a,D2: multiset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A3 @ B2 ) @ ( multis4023584895248362547_ext_a @ Ns @ S ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ C2 @ D2 ) @ ( multis4023584895248362547_ext_a @ Ns @ S ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( plus_plus_multiset_a @ A3 @ C2 ) @ ( plus_plus_multiset_a @ B2 @ D2 ) ) @ ( multis4023584895248362547_ext_a @ Ns @ S ) ) ) ) ).
% ns_ns_mul_ext_union_compat
thf(fact_281_prod_Oinject,axiom,
! [X13: multiset_a,X24: multiset_a,Y13: multiset_a,Y24: multiset_a] :
( ( ( produc654756711066625303iset_a @ X13 @ X24 )
= ( produc654756711066625303iset_a @ Y13 @ Y24 ) )
= ( ( X13 = Y13 )
& ( X24 = Y24 ) ) ) ).
% prod.inject
thf(fact_282_prod_Oinject,axiom,
! [X13: a,X24: a,Y13: a,Y24: a] :
( ( ( product_Pair_a_a @ X13 @ X24 )
= ( product_Pair_a_a @ Y13 @ Y24 ) )
= ( ( X13 = Y13 )
& ( X24 = Y24 ) ) ) ).
% prod.inject
thf(fact_283_old_Oprod_Oinject,axiom,
! [A2: multiset_a,B: multiset_a,A9: multiset_a,B9: multiset_a] :
( ( ( produc654756711066625303iset_a @ A2 @ B )
= ( produc654756711066625303iset_a @ A9 @ B9 ) )
= ( ( A2 = A9 )
& ( B = B9 ) ) ) ).
% old.prod.inject
thf(fact_284_old_Oprod_Oinject,axiom,
! [A2: a,B: a,A9: a,B9: a] :
( ( ( product_Pair_a_a @ A2 @ B )
= ( product_Pair_a_a @ A9 @ B9 ) )
= ( ( A2 = A9 )
& ( B = B9 ) ) ) ).
% old.prod.inject
thf(fact_285_dual__order_Orefl,axiom,
! [A2: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_286_dual__order_Orefl,axiom,
! [A2: set_Pr79727621955416071iset_a] : ( ord_le5871882978999109543iset_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_287_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_288_order__refl,axiom,
! [X2: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ X2 @ X2 ) ).
% order_refl
thf(fact_289_order__refl,axiom,
! [X2: set_Pr79727621955416071iset_a] : ( ord_le5871882978999109543iset_a @ X2 @ X2 ) ).
% order_refl
thf(fact_290_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_291_mult1E,axiom,
! [N: multis4781622204924204013iset_a,M: multis4781622204924204013iset_a,R: set_Pr3059988933816007879iset_a] :
( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ N @ M ) @ ( mult1_5927536977011277466iset_a @ R ) )
=> ~ ! [A: produc6518373309651786023iset_a,M0: multis4781622204924204013iset_a] :
( ( M
= ( add_ms5348986034529679365iset_a @ A @ M0 ) )
=> ! [K4: multis4781622204924204013iset_a] :
( ( N
= ( plus_p1053901119268525860iset_a @ M0 @ K4 ) )
=> ~ ! [B3: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ B3 @ ( set_ms2379723421352415844iset_a @ K4 ) )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ B3 @ A ) @ R ) ) ) ) ) ).
% mult1E
thf(fact_292_mult1E,axiom,
! [N: multis501812127501805293od_a_a,M: multis501812127501805293od_a_a,R: set_Pr8600417178894128327od_a_a] :
( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ N @ M ) @ ( mult1_1845928942838466970od_a_a @ R ) )
=> ~ ! [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 ) )
=> ~ ! [B3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ B3 @ ( set_ms119794563918438244od_a_a @ K4 ) )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ B3 @ A ) @ R ) ) ) ) ) ).
% mult1E
thf(fact_293_mult1E,axiom,
! [N: multiset_multiset_a,M: multiset_multiset_a,R: set_Pr79727621955416071iset_a] :
( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ N @ M ) @ ( mult1_multiset_a @ R ) )
=> ~ ! [A: multiset_a,M0: multiset_multiset_a] :
( ( M
= ( add_mset_multiset_a @ A @ M0 ) )
=> ! [K4: multiset_multiset_a] :
( ( N
= ( plus_p6738641960240532161iset_a @ M0 @ K4 ) )
=> ~ ! [B3: multiset_a] :
( ( member_multiset_a @ B3 @ ( set_mset_multiset_a @ K4 ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ B3 @ A ) @ R ) ) ) ) ) ).
% mult1E
thf(fact_294_mult1E,axiom,
! [N: multiset_a,M: multiset_a,R: set_Product_prod_a_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ N @ M ) @ ( mult1_a @ R ) )
=> ~ ! [A: a,M0: multiset_a] :
( ( M
= ( add_mset_a @ A @ M0 ) )
=> ! [K4: multiset_a] :
( ( N
= ( plus_plus_multiset_a @ M0 @ K4 ) )
=> ~ ! [B3: a] :
( ( member_a @ B3 @ ( set_mset_a @ K4 ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B3 @ A ) @ R ) ) ) ) ) ).
% mult1E
thf(fact_295_mult1I,axiom,
! [M: multis4781622204924204013iset_a,A2: produc6518373309651786023iset_a,M02: multis4781622204924204013iset_a,N: multis4781622204924204013iset_a,K: multis4781622204924204013iset_a,R: set_Pr3059988933816007879iset_a] :
( ( M
= ( add_ms5348986034529679365iset_a @ A2 @ M02 ) )
=> ( ( N
= ( plus_p1053901119268525860iset_a @ M02 @ K ) )
=> ( ! [B8: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ B8 @ ( set_ms2379723421352415844iset_a @ K ) )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ B8 @ A2 ) @ R ) )
=> ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ N @ M ) @ ( mult1_5927536977011277466iset_a @ R ) ) ) ) ) ).
% mult1I
thf(fact_296_mult1I,axiom,
! [M: multis501812127501805293od_a_a,A2: product_prod_a_a,M02: multis501812127501805293od_a_a,N: multis501812127501805293od_a_a,K: multis501812127501805293od_a_a,R: set_Pr8600417178894128327od_a_a] :
( ( M
= ( add_ms8655138167283798533od_a_a @ A2 @ M02 ) )
=> ( ( N
= ( plus_p6967097483979887908od_a_a @ M02 @ K ) )
=> ( ! [B8: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ B8 @ ( set_ms119794563918438244od_a_a @ K ) )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ B8 @ A2 ) @ R ) )
=> ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ N @ M ) @ ( mult1_1845928942838466970od_a_a @ R ) ) ) ) ) ).
% mult1I
thf(fact_297_mult1I,axiom,
! [M: multiset_multiset_a,A2: multiset_a,M02: multiset_multiset_a,N: multiset_multiset_a,K: multiset_multiset_a,R: set_Pr79727621955416071iset_a] :
( ( M
= ( add_mset_multiset_a @ A2 @ M02 ) )
=> ( ( N
= ( plus_p6738641960240532161iset_a @ M02 @ K ) )
=> ( ! [B8: multiset_a] :
( ( member_multiset_a @ B8 @ ( set_mset_multiset_a @ K ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ B8 @ A2 ) @ R ) )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ N @ M ) @ ( mult1_multiset_a @ R ) ) ) ) ) ).
% mult1I
thf(fact_298_mult1I,axiom,
! [M: multiset_a,A2: a,M02: multiset_a,N: multiset_a,K: multiset_a,R: set_Product_prod_a_a] :
( ( M
= ( add_mset_a @ A2 @ M02 ) )
=> ( ( N
= ( plus_plus_multiset_a @ M02 @ K ) )
=> ( ! [B8: a] :
( ( member_a @ B8 @ ( set_mset_a @ K ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B8 @ A2 ) @ R ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ N @ M ) @ ( mult1_a @ R ) ) ) ) ) ).
% mult1I
thf(fact_299_less__add,axiom,
! [N: multis4781622204924204013iset_a,A2: produc6518373309651786023iset_a,M02: multis4781622204924204013iset_a,R: set_Pr3059988933816007879iset_a] :
( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ N @ ( add_ms5348986034529679365iset_a @ A2 @ M02 ) ) @ ( mult1_5927536977011277466iset_a @ R ) )
=> ( ? [M3: multis4781622204924204013iset_a] :
( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ M3 @ M02 ) @ ( mult1_5927536977011277466iset_a @ R ) )
& ( N
= ( add_ms5348986034529679365iset_a @ A2 @ M3 ) ) )
| ? [K4: multis4781622204924204013iset_a] :
( ! [B3: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ B3 @ ( set_ms2379723421352415844iset_a @ K4 ) )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ B3 @ A2 ) @ R ) )
& ( N
= ( plus_p1053901119268525860iset_a @ M02 @ K4 ) ) ) ) ) ).
% less_add
thf(fact_300_less__add,axiom,
! [N: multis501812127501805293od_a_a,A2: product_prod_a_a,M02: multis501812127501805293od_a_a,R: set_Pr8600417178894128327od_a_a] :
( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ N @ ( add_ms8655138167283798533od_a_a @ A2 @ M02 ) ) @ ( mult1_1845928942838466970od_a_a @ R ) )
=> ( ? [M3: multis501812127501805293od_a_a] :
( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ M3 @ M02 ) @ ( mult1_1845928942838466970od_a_a @ R ) )
& ( N
= ( add_ms8655138167283798533od_a_a @ A2 @ M3 ) ) )
| ? [K4: multis501812127501805293od_a_a] :
( ! [B3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ B3 @ ( set_ms119794563918438244od_a_a @ K4 ) )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ B3 @ A2 ) @ R ) )
& ( N
= ( plus_p6967097483979887908od_a_a @ M02 @ K4 ) ) ) ) ) ).
% less_add
thf(fact_301_less__add,axiom,
! [N: multiset_multiset_a,A2: multiset_a,M02: multiset_multiset_a,R: set_Pr79727621955416071iset_a] :
( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ N @ ( add_mset_multiset_a @ A2 @ M02 ) ) @ ( mult1_multiset_a @ R ) )
=> ( ? [M3: multiset_multiset_a] :
( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ M3 @ M02 ) @ ( mult1_multiset_a @ R ) )
& ( N
= ( add_mset_multiset_a @ A2 @ M3 ) ) )
| ? [K4: multiset_multiset_a] :
( ! [B3: multiset_a] :
( ( member_multiset_a @ B3 @ ( set_mset_multiset_a @ K4 ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ B3 @ A2 ) @ R ) )
& ( N
= ( plus_p6738641960240532161iset_a @ M02 @ K4 ) ) ) ) ) ).
% less_add
thf(fact_302_less__add,axiom,
! [N: multiset_a,A2: a,M02: multiset_a,R: set_Product_prod_a_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ N @ ( add_mset_a @ A2 @ M02 ) ) @ ( mult1_a @ R ) )
=> ( ? [M3: multiset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ M3 @ M02 ) @ ( mult1_a @ R ) )
& ( N
= ( add_mset_a @ A2 @ M3 ) ) )
| ? [K4: multiset_a] :
( ! [B3: a] :
( ( member_a @ B3 @ ( set_mset_a @ K4 ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B3 @ A2 ) @ R ) )
& ( N
= ( plus_plus_multiset_a @ M02 @ K4 ) ) ) ) ) ).
% less_add
thf(fact_303_nle__le,axiom,
! [A2: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B ) )
= ( ( ord_less_eq_nat @ B @ A2 )
& ( B != A2 ) ) ) ).
% nle_le
thf(fact_304_le__cases3,axiom,
! [X2: nat,Y: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_305_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_Product_prod_a_a,Z4: set_Product_prod_a_a] : ( Y3 = Z4 ) )
= ( ^ [X4: set_Product_prod_a_a,Y4: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X4 @ Y4 )
& ( ord_le746702958409616551od_a_a @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_306_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_Pr79727621955416071iset_a,Z4: set_Pr79727621955416071iset_a] : ( Y3 = Z4 ) )
= ( ^ [X4: set_Pr79727621955416071iset_a,Y4: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ X4 @ Y4 )
& ( ord_le5871882978999109543iset_a @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_307_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z4: nat] : ( Y3 = Z4 ) )
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_308_ord__eq__le__trans,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( A2 = B )
=> ( ( ord_le746702958409616551od_a_a @ B @ C )
=> ( ord_le746702958409616551od_a_a @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_309_ord__eq__le__trans,axiom,
! [A2: set_Pr79727621955416071iset_a,B: set_Pr79727621955416071iset_a,C: set_Pr79727621955416071iset_a] :
( ( A2 = B )
=> ( ( ord_le5871882978999109543iset_a @ B @ C )
=> ( ord_le5871882978999109543iset_a @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_310_ord__eq__le__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( A2 = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_311_ord__le__eq__trans,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B )
=> ( ( B = C )
=> ( ord_le746702958409616551od_a_a @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_312_ord__le__eq__trans,axiom,
! [A2: set_Pr79727621955416071iset_a,B: set_Pr79727621955416071iset_a,C: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ A2 @ B )
=> ( ( B = C )
=> ( ord_le5871882978999109543iset_a @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_313_ord__le__eq__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_314_order__antisym,axiom,
! [X2: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X2 @ Y )
=> ( ( ord_le746702958409616551od_a_a @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_315_order__antisym,axiom,
! [X2: set_Pr79727621955416071iset_a,Y: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ X2 @ Y )
=> ( ( ord_le5871882978999109543iset_a @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_316_order__antisym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_317_order_Otrans,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B )
=> ( ( ord_le746702958409616551od_a_a @ B @ C )
=> ( ord_le746702958409616551od_a_a @ A2 @ C ) ) ) ).
% order.trans
thf(fact_318_order_Otrans,axiom,
! [A2: set_Pr79727621955416071iset_a,B: set_Pr79727621955416071iset_a,C: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ A2 @ B )
=> ( ( ord_le5871882978999109543iset_a @ B @ C )
=> ( ord_le5871882978999109543iset_a @ A2 @ C ) ) ) ).
% order.trans
thf(fact_319_order_Otrans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_320_order__trans,axiom,
! [X2: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X2 @ Y )
=> ( ( ord_le746702958409616551od_a_a @ Y @ Z3 )
=> ( ord_le746702958409616551od_a_a @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_321_order__trans,axiom,
! [X2: set_Pr79727621955416071iset_a,Y: set_Pr79727621955416071iset_a,Z3: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ X2 @ Y )
=> ( ( ord_le5871882978999109543iset_a @ Y @ Z3 )
=> ( ord_le5871882978999109543iset_a @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_322_order__trans,axiom,
! [X2: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_eq_nat @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_323_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B: nat] :
( ! [A: nat,B8: nat] :
( ( ord_less_eq_nat @ A @ B8 )
=> ( P @ A @ B8 ) )
=> ( ! [A: nat,B8: nat] :
( ( P @ B8 @ A )
=> ( P @ A @ B8 ) )
=> ( P @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_324_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_Product_prod_a_a,Z4: set_Product_prod_a_a] : ( Y3 = Z4 ) )
= ( ^ [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B5 @ A5 )
& ( ord_le746702958409616551od_a_a @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_325_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_Pr79727621955416071iset_a,Z4: set_Pr79727621955416071iset_a] : ( Y3 = Z4 ) )
= ( ^ [A5: set_Pr79727621955416071iset_a,B5: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ B5 @ A5 )
& ( ord_le5871882978999109543iset_a @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_326_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z4: nat] : ( Y3 = Z4 ) )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ( ord_less_eq_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_327_dual__order_Oantisym,axiom,
! [B: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B @ A2 )
=> ( ( ord_le746702958409616551od_a_a @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_328_dual__order_Oantisym,axiom,
! [B: set_Pr79727621955416071iset_a,A2: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ B @ A2 )
=> ( ( ord_le5871882978999109543iset_a @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_329_dual__order_Oantisym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_330_dual__order_Otrans,axiom,
! [B: set_Product_prod_a_a,A2: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B @ A2 )
=> ( ( ord_le746702958409616551od_a_a @ C @ B )
=> ( ord_le746702958409616551od_a_a @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_331_dual__order_Otrans,axiom,
! [B: set_Pr79727621955416071iset_a,A2: set_Pr79727621955416071iset_a,C: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ B @ A2 )
=> ( ( ord_le5871882978999109543iset_a @ C @ B )
=> ( ord_le5871882978999109543iset_a @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_332_dual__order_Otrans,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_333_antisym,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B )
=> ( ( ord_le746702958409616551od_a_a @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_334_antisym,axiom,
! [A2: set_Pr79727621955416071iset_a,B: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ A2 @ B )
=> ( ( ord_le5871882978999109543iset_a @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_335_antisym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_336_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_Product_prod_a_a,Z4: set_Product_prod_a_a] : ( Y3 = Z4 ) )
= ( ^ [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A5 @ B5 )
& ( ord_le746702958409616551od_a_a @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_337_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_Pr79727621955416071iset_a,Z4: set_Pr79727621955416071iset_a] : ( Y3 = Z4 ) )
= ( ^ [A5: set_Pr79727621955416071iset_a,B5: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ A5 @ B5 )
& ( ord_le5871882978999109543iset_a @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_338_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z4: nat] : ( Y3 = Z4 ) )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ( ord_less_eq_nat @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_339_order__subst1,axiom,
! [A2: set_Product_prod_a_a,F: set_Product_prod_a_a > set_Product_prod_a_a,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ ( F @ B ) )
=> ( ( ord_le746702958409616551od_a_a @ B @ C )
=> ( ! [X5: set_Product_prod_a_a,Y5: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X5 @ Y5 )
=> ( ord_le746702958409616551od_a_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le746702958409616551od_a_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_340_order__subst1,axiom,
! [A2: set_Product_prod_a_a,F: set_Pr79727621955416071iset_a > set_Product_prod_a_a,B: set_Pr79727621955416071iset_a,C: set_Pr79727621955416071iset_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ ( F @ B ) )
=> ( ( ord_le5871882978999109543iset_a @ B @ C )
=> ( ! [X5: set_Pr79727621955416071iset_a,Y5: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ X5 @ Y5 )
=> ( ord_le746702958409616551od_a_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le746702958409616551od_a_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_341_order__subst1,axiom,
! [A2: set_Product_prod_a_a,F: nat > set_Product_prod_a_a,B: nat,C: nat] :
( ( ord_le746702958409616551od_a_a @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ord_le746702958409616551od_a_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le746702958409616551od_a_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_342_order__subst1,axiom,
! [A2: set_Pr79727621955416071iset_a,F: set_Product_prod_a_a > set_Pr79727621955416071iset_a,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le5871882978999109543iset_a @ A2 @ ( F @ B ) )
=> ( ( ord_le746702958409616551od_a_a @ B @ C )
=> ( ! [X5: set_Product_prod_a_a,Y5: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X5 @ Y5 )
=> ( ord_le5871882978999109543iset_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le5871882978999109543iset_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_343_order__subst1,axiom,
! [A2: set_Pr79727621955416071iset_a,F: set_Pr79727621955416071iset_a > set_Pr79727621955416071iset_a,B: set_Pr79727621955416071iset_a,C: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ A2 @ ( F @ B ) )
=> ( ( ord_le5871882978999109543iset_a @ B @ C )
=> ( ! [X5: set_Pr79727621955416071iset_a,Y5: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ X5 @ Y5 )
=> ( ord_le5871882978999109543iset_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le5871882978999109543iset_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_344_order__subst1,axiom,
! [A2: set_Pr79727621955416071iset_a,F: nat > set_Pr79727621955416071iset_a,B: nat,C: nat] :
( ( ord_le5871882978999109543iset_a @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ord_le5871882978999109543iset_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le5871882978999109543iset_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_345_order__subst1,axiom,
! [A2: nat,F: set_Product_prod_a_a > nat,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_le746702958409616551od_a_a @ B @ C )
=> ( ! [X5: set_Product_prod_a_a,Y5: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X5 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_346_order__subst1,axiom,
! [A2: nat,F: set_Pr79727621955416071iset_a > nat,B: set_Pr79727621955416071iset_a,C: set_Pr79727621955416071iset_a] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_le5871882978999109543iset_a @ B @ C )
=> ( ! [X5: set_Pr79727621955416071iset_a,Y5: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ X5 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_347_order__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_348_order__subst2,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a,F: set_Product_prod_a_a > set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B )
=> ( ( ord_le746702958409616551od_a_a @ ( F @ B ) @ C )
=> ( ! [X5: set_Product_prod_a_a,Y5: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X5 @ Y5 )
=> ( ord_le746702958409616551od_a_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le746702958409616551od_a_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_349_order__subst2,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a,F: set_Product_prod_a_a > set_Pr79727621955416071iset_a,C: set_Pr79727621955416071iset_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B )
=> ( ( ord_le5871882978999109543iset_a @ ( F @ B ) @ C )
=> ( ! [X5: set_Product_prod_a_a,Y5: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X5 @ Y5 )
=> ( ord_le5871882978999109543iset_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le5871882978999109543iset_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_350_order__subst2,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a,F: set_Product_prod_a_a > nat,C: nat] :
( ( ord_le746702958409616551od_a_a @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X5: set_Product_prod_a_a,Y5: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X5 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_351_order__subst2,axiom,
! [A2: set_Pr79727621955416071iset_a,B: set_Pr79727621955416071iset_a,F: set_Pr79727621955416071iset_a > set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le5871882978999109543iset_a @ A2 @ B )
=> ( ( ord_le746702958409616551od_a_a @ ( F @ B ) @ C )
=> ( ! [X5: set_Pr79727621955416071iset_a,Y5: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ X5 @ Y5 )
=> ( ord_le746702958409616551od_a_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le746702958409616551od_a_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_352_order__subst2,axiom,
! [A2: set_Pr79727621955416071iset_a,B: set_Pr79727621955416071iset_a,F: set_Pr79727621955416071iset_a > set_Pr79727621955416071iset_a,C: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ A2 @ B )
=> ( ( ord_le5871882978999109543iset_a @ ( F @ B ) @ C )
=> ( ! [X5: set_Pr79727621955416071iset_a,Y5: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ X5 @ Y5 )
=> ( ord_le5871882978999109543iset_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le5871882978999109543iset_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_353_order__subst2,axiom,
! [A2: set_Pr79727621955416071iset_a,B: set_Pr79727621955416071iset_a,F: set_Pr79727621955416071iset_a > nat,C: nat] :
( ( ord_le5871882978999109543iset_a @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X5: set_Pr79727621955416071iset_a,Y5: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ X5 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_354_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_le746702958409616551od_a_a @ ( F @ B ) @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ord_le746702958409616551od_a_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le746702958409616551od_a_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_355_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > set_Pr79727621955416071iset_a,C: set_Pr79727621955416071iset_a] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_le5871882978999109543iset_a @ ( F @ B ) @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ord_le5871882978999109543iset_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le5871882978999109543iset_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_356_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_357_order__eq__refl,axiom,
! [X2: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( X2 = Y )
=> ( ord_le746702958409616551od_a_a @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_358_order__eq__refl,axiom,
! [X2: set_Pr79727621955416071iset_a,Y: set_Pr79727621955416071iset_a] :
( ( X2 = Y )
=> ( ord_le5871882978999109543iset_a @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_359_order__eq__refl,axiom,
! [X2: nat,Y: nat] :
( ( X2 = Y )
=> ( ord_less_eq_nat @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_360_linorder__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
| ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_linear
thf(fact_361_ord__eq__le__subst,axiom,
! [A2: set_Product_prod_a_a,F: set_Product_prod_a_a > set_Product_prod_a_a,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le746702958409616551od_a_a @ B @ C )
=> ( ! [X5: set_Product_prod_a_a,Y5: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X5 @ Y5 )
=> ( ord_le746702958409616551od_a_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le746702958409616551od_a_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_362_ord__eq__le__subst,axiom,
! [A2: set_Pr79727621955416071iset_a,F: set_Product_prod_a_a > set_Pr79727621955416071iset_a,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le746702958409616551od_a_a @ B @ C )
=> ( ! [X5: set_Product_prod_a_a,Y5: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X5 @ Y5 )
=> ( ord_le5871882978999109543iset_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le5871882978999109543iset_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_363_ord__eq__le__subst,axiom,
! [A2: nat,F: set_Product_prod_a_a > nat,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le746702958409616551od_a_a @ B @ C )
=> ( ! [X5: set_Product_prod_a_a,Y5: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X5 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_364_ord__eq__le__subst,axiom,
! [A2: set_Product_prod_a_a,F: set_Pr79727621955416071iset_a > set_Product_prod_a_a,B: set_Pr79727621955416071iset_a,C: set_Pr79727621955416071iset_a] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le5871882978999109543iset_a @ B @ C )
=> ( ! [X5: set_Pr79727621955416071iset_a,Y5: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ X5 @ Y5 )
=> ( ord_le746702958409616551od_a_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le746702958409616551od_a_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_365_ord__eq__le__subst,axiom,
! [A2: set_Pr79727621955416071iset_a,F: set_Pr79727621955416071iset_a > set_Pr79727621955416071iset_a,B: set_Pr79727621955416071iset_a,C: set_Pr79727621955416071iset_a] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le5871882978999109543iset_a @ B @ C )
=> ( ! [X5: set_Pr79727621955416071iset_a,Y5: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ X5 @ Y5 )
=> ( ord_le5871882978999109543iset_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le5871882978999109543iset_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_366_ord__eq__le__subst,axiom,
! [A2: nat,F: set_Pr79727621955416071iset_a > nat,B: set_Pr79727621955416071iset_a,C: set_Pr79727621955416071iset_a] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le5871882978999109543iset_a @ B @ C )
=> ( ! [X5: set_Pr79727621955416071iset_a,Y5: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ X5 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_367_ord__eq__le__subst,axiom,
! [A2: set_Product_prod_a_a,F: nat > set_Product_prod_a_a,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ord_le746702958409616551od_a_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le746702958409616551od_a_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_368_ord__eq__le__subst,axiom,
! [A2: set_Pr79727621955416071iset_a,F: nat > set_Pr79727621955416071iset_a,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ord_le5871882978999109543iset_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le5871882978999109543iset_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_369_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_370_ord__le__eq__subst,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a,F: set_Product_prod_a_a > set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X5: set_Product_prod_a_a,Y5: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X5 @ Y5 )
=> ( ord_le746702958409616551od_a_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le746702958409616551od_a_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_371_ord__le__eq__subst,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a,F: set_Product_prod_a_a > set_Pr79727621955416071iset_a,C: set_Pr79727621955416071iset_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X5: set_Product_prod_a_a,Y5: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X5 @ Y5 )
=> ( ord_le5871882978999109543iset_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le5871882978999109543iset_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_372_ord__le__eq__subst,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a,F: set_Product_prod_a_a > nat,C: nat] :
( ( ord_le746702958409616551od_a_a @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X5: set_Product_prod_a_a,Y5: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X5 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_373_ord__le__eq__subst,axiom,
! [A2: set_Pr79727621955416071iset_a,B: set_Pr79727621955416071iset_a,F: set_Pr79727621955416071iset_a > set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le5871882978999109543iset_a @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X5: set_Pr79727621955416071iset_a,Y5: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ X5 @ Y5 )
=> ( ord_le746702958409616551od_a_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le746702958409616551od_a_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_374_ord__le__eq__subst,axiom,
! [A2: set_Pr79727621955416071iset_a,B: set_Pr79727621955416071iset_a,F: set_Pr79727621955416071iset_a > set_Pr79727621955416071iset_a,C: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X5: set_Pr79727621955416071iset_a,Y5: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ X5 @ Y5 )
=> ( ord_le5871882978999109543iset_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le5871882978999109543iset_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_375_ord__le__eq__subst,axiom,
! [A2: set_Pr79727621955416071iset_a,B: set_Pr79727621955416071iset_a,F: set_Pr79727621955416071iset_a > nat,C: nat] :
( ( ord_le5871882978999109543iset_a @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X5: set_Pr79727621955416071iset_a,Y5: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ X5 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_376_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ord_le746702958409616551od_a_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le746702958409616551od_a_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_377_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > set_Pr79727621955416071iset_a,C: set_Pr79727621955416071iset_a] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ord_le5871882978999109543iset_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le5871882978999109543iset_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_378_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_379_linorder__le__cases,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_le_cases
thf(fact_380_order__antisym__conv,axiom,
! [Y: set_Product_prod_a_a,X2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ Y @ X2 )
=> ( ( ord_le746702958409616551od_a_a @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_381_order__antisym__conv,axiom,
! [Y: set_Pr79727621955416071iset_a,X2: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ Y @ X2 )
=> ( ( ord_le5871882978999109543iset_a @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_382_order__antisym__conv,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_383_Pair__inject,axiom,
! [A2: multiset_a,B: multiset_a,A9: multiset_a,B9: multiset_a] :
( ( ( produc654756711066625303iset_a @ A2 @ B )
= ( produc654756711066625303iset_a @ A9 @ B9 ) )
=> ~ ( ( A2 = A9 )
=> ( B != B9 ) ) ) ).
% Pair_inject
thf(fact_384_Pair__inject,axiom,
! [A2: a,B: a,A9: a,B9: a] :
( ( ( product_Pair_a_a @ A2 @ B )
= ( product_Pair_a_a @ A9 @ B9 ) )
=> ~ ( ( A2 = A9 )
=> ( B != B9 ) ) ) ).
% Pair_inject
thf(fact_385_prod__cases,axiom,
! [P: produc6518373309651786023iset_a > $o,P2: produc6518373309651786023iset_a] :
( ! [A: multiset_a,B8: multiset_a] : ( P @ ( produc654756711066625303iset_a @ A @ B8 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_386_prod__cases,axiom,
! [P: product_prod_a_a > $o,P2: product_prod_a_a] :
( ! [A: a,B8: a] : ( P @ ( product_Pair_a_a @ A @ B8 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_387_surj__pair,axiom,
! [P2: produc6518373309651786023iset_a] :
? [X5: multiset_a,Y5: multiset_a] :
( P2
= ( produc654756711066625303iset_a @ X5 @ Y5 ) ) ).
% surj_pair
thf(fact_388_surj__pair,axiom,
! [P2: product_prod_a_a] :
? [X5: a,Y5: a] :
( P2
= ( product_Pair_a_a @ X5 @ Y5 ) ) ).
% surj_pair
thf(fact_389_old_Oprod_Oexhaust,axiom,
! [Y: produc6518373309651786023iset_a] :
~ ! [A: multiset_a,B8: multiset_a] :
( Y
!= ( produc654756711066625303iset_a @ A @ B8 ) ) ).
% old.prod.exhaust
thf(fact_390_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_a_a] :
~ ! [A: a,B8: a] :
( Y
!= ( product_Pair_a_a @ A @ B8 ) ) ).
% old.prod.exhaust
thf(fact_391_mult1__union,axiom,
! [B2: multiset_a,D2: multiset_a,R: set_Product_prod_a_a,C2: multiset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ B2 @ D2 ) @ ( mult1_a @ R ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( plus_plus_multiset_a @ C2 @ B2 ) @ ( plus_plus_multiset_a @ C2 @ D2 ) ) @ ( mult1_a @ R ) ) ) ).
% mult1_union
thf(fact_392_mono__mult1,axiom,
! [R: set_Product_prod_a_a,R2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ R @ R2 )
=> ( ord_le5871882978999109543iset_a @ ( mult1_a @ R ) @ ( mult1_a @ R2 ) ) ) ).
% mono_mult1
thf(fact_393_mono__mult1,axiom,
! [R: set_Pr79727621955416071iset_a,R2: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ R @ R2 )
=> ( ord_le2895628588359463079iset_a @ ( mult1_multiset_a @ R ) @ ( mult1_multiset_a @ R2 ) ) ) ).
% mono_mult1
thf(fact_394_subrelI,axiom,
! [R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ! [X5: a,Y5: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y5 ) @ R )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y5 ) @ S ) )
=> ( ord_le746702958409616551od_a_a @ R @ S ) ) ).
% subrelI
thf(fact_395_subrelI,axiom,
! [R: set_Pr79727621955416071iset_a,S: set_Pr79727621955416071iset_a] :
( ! [X5: multiset_a,Y5: multiset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Y5 ) @ R )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Y5 ) @ S ) )
=> ( ord_le5871882978999109543iset_a @ R @ S ) ) ).
% subrelI
thf(fact_396_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_397_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_398_mult2__alt__nsI,axiom,
! [X3: multis4781622204924204013iset_a,X1: multis4781622204924204013iset_a,X22: multis4781622204924204013iset_a,Y2: multis4781622204924204013iset_a,Y1: multis4781622204924204013iset_a,Y22: multis4781622204924204013iset_a,Ns: set_Pr3059988933816007879iset_a,S: set_Pr3059988933816007879iset_a] :
( ( X3
= ( plus_p1053901119268525860iset_a @ X1 @ X22 ) )
=> ( ( Y2
= ( plus_p1053901119268525860iset_a @ Y1 @ Y22 ) )
=> ( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ X1 @ Y1 ) @ ( multis3430030276581740203iset_a @ Ns ) )
=> ( ! [X5: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X5 @ ( set_ms2379723421352415844iset_a @ X22 ) )
=> ? [Y6: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Y6 @ ( set_ms2379723421352415844iset_a @ Y22 ) )
& ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X5 @ Y6 ) @ S ) ) )
=> ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ X3 @ Y2 ) @ ( multis1572145089415997755iset_a @ $true @ Ns @ S ) ) ) ) ) ) ).
% mult2_alt_nsI
thf(fact_399_mult2__alt__nsI,axiom,
! [X3: multis501812127501805293od_a_a,X1: multis501812127501805293od_a_a,X22: multis501812127501805293od_a_a,Y2: multis501812127501805293od_a_a,Y1: multis501812127501805293od_a_a,Y22: multis501812127501805293od_a_a,Ns: set_Pr8600417178894128327od_a_a,S: set_Pr8600417178894128327od_a_a] :
( ( X3
= ( plus_p6967097483979887908od_a_a @ X1 @ X22 ) )
=> ( ( Y2
= ( plus_p6967097483979887908od_a_a @ Y1 @ Y22 ) )
=> ( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ X1 @ Y1 ) @ ( multis1821517007024690859od_a_a @ Ns ) )
=> ( ! [X5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ ( set_ms119794563918438244od_a_a @ X22 ) )
=> ? [Y6: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y6 @ ( set_ms119794563918438244od_a_a @ Y22 ) )
& ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X5 @ Y6 ) @ S ) ) )
=> ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ X3 @ Y2 ) @ ( multis4249482161328750907od_a_a @ $true @ Ns @ S ) ) ) ) ) ) ).
% mult2_alt_nsI
thf(fact_400_mult2__alt__nsI,axiom,
! [X3: multiset_multiset_a,X1: multiset_multiset_a,X22: multiset_multiset_a,Y2: multiset_multiset_a,Y1: multiset_multiset_a,Y22: multiset_multiset_a,Ns: set_Pr79727621955416071iset_a,S: set_Pr79727621955416071iset_a] :
( ( X3
= ( plus_p6738641960240532161iset_a @ X1 @ X22 ) )
=> ( ( Y2
= ( plus_p6738641960240532161iset_a @ Y1 @ Y22 ) )
=> ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X1 @ Y1 ) @ ( multis451435756853798856iset_a @ Ns ) )
=> ( ! [X5: multiset_a] :
( ( member_multiset_a @ X5 @ ( set_mset_multiset_a @ X22 ) )
=> ? [Y6: multiset_a] :
( ( member_multiset_a @ Y6 @ ( set_mset_multiset_a @ Y22 ) )
& ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Y6 ) @ S ) ) )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X3 @ Y2 ) @ ( multis2006310815304088152iset_a @ $true @ Ns @ S ) ) ) ) ) ) ).
% mult2_alt_nsI
thf(fact_401_mult2__alt__nsI,axiom,
! [X3: multiset_a,X1: multiset_a,X22: multiset_a,Y2: multiset_a,Y1: multiset_a,Y22: multiset_a,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( X3
= ( plus_plus_multiset_a @ X1 @ X22 ) )
=> ( ( Y2
= ( plus_plus_multiset_a @ Y1 @ Y22 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X1 @ Y1 ) @ ( multis2373687813370458242ltpw_a @ Ns ) )
=> ( ! [X5: a] :
( ( member_a @ X5 @ ( set_mset_a @ X22 ) )
=> ? [Y6: a] :
( ( member_a @ Y6 @ ( set_mset_a @ Y22 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y6 ) @ S ) ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ Y2 ) @ ( multis1686031732355738386_alt_a @ $true @ Ns @ S ) ) ) ) ) ) ).
% mult2_alt_nsI
thf(fact_402_mult2__alt__nsE,axiom,
! [X3: multis4781622204924204013iset_a,Y2: multis4781622204924204013iset_a,Ns: set_Pr3059988933816007879iset_a,S: set_Pr3059988933816007879iset_a] :
( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ X3 @ Y2 ) @ ( multis1572145089415997755iset_a @ $true @ Ns @ S ) )
=> ~ ! [X12: multis4781622204924204013iset_a,X23: multis4781622204924204013iset_a] :
( ( X3
= ( plus_p1053901119268525860iset_a @ X12 @ X23 ) )
=> ! [Y12: multis4781622204924204013iset_a,Y23: multis4781622204924204013iset_a] :
( ( Y2
= ( plus_p1053901119268525860iset_a @ Y12 @ Y23 ) )
=> ( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ X12 @ Y12 ) @ ( multis3430030276581740203iset_a @ Ns ) )
=> ~ ! [X: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X @ ( set_ms2379723421352415844iset_a @ X23 ) )
=> ? [Y5: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Y5 @ ( set_ms2379723421352415844iset_a @ Y23 ) )
& ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X @ Y5 ) @ S ) ) ) ) ) ) ) ).
% mult2_alt_nsE
thf(fact_403_mult2__alt__nsE,axiom,
! [X3: multis501812127501805293od_a_a,Y2: multis501812127501805293od_a_a,Ns: set_Pr8600417178894128327od_a_a,S: set_Pr8600417178894128327od_a_a] :
( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ X3 @ Y2 ) @ ( multis4249482161328750907od_a_a @ $true @ Ns @ S ) )
=> ~ ! [X12: multis501812127501805293od_a_a,X23: multis501812127501805293od_a_a] :
( ( X3
= ( plus_p6967097483979887908od_a_a @ X12 @ X23 ) )
=> ! [Y12: multis501812127501805293od_a_a,Y23: multis501812127501805293od_a_a] :
( ( Y2
= ( plus_p6967097483979887908od_a_a @ Y12 @ Y23 ) )
=> ( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ X12 @ Y12 ) @ ( multis1821517007024690859od_a_a @ Ns ) )
=> ~ ! [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_ms119794563918438244od_a_a @ X23 ) )
=> ? [Y5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y5 @ ( set_ms119794563918438244od_a_a @ Y23 ) )
& ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Y5 ) @ S ) ) ) ) ) ) ) ).
% mult2_alt_nsE
thf(fact_404_mult2__alt__nsE,axiom,
! [X3: multiset_multiset_a,Y2: multiset_multiset_a,Ns: set_Pr79727621955416071iset_a,S: set_Pr79727621955416071iset_a] :
( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X3 @ Y2 ) @ ( multis2006310815304088152iset_a @ $true @ Ns @ S ) )
=> ~ ! [X12: multiset_multiset_a,X23: multiset_multiset_a] :
( ( X3
= ( plus_p6738641960240532161iset_a @ X12 @ X23 ) )
=> ! [Y12: multiset_multiset_a,Y23: multiset_multiset_a] :
( ( Y2
= ( plus_p6738641960240532161iset_a @ Y12 @ Y23 ) )
=> ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X12 @ Y12 ) @ ( multis451435756853798856iset_a @ Ns ) )
=> ~ ! [X: multiset_a] :
( ( member_multiset_a @ X @ ( set_mset_multiset_a @ X23 ) )
=> ? [Y5: multiset_a] :
( ( member_multiset_a @ Y5 @ ( set_mset_multiset_a @ Y23 ) )
& ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ Y5 ) @ S ) ) ) ) ) ) ) ).
% mult2_alt_nsE
thf(fact_405_mult2__alt__nsE,axiom,
! [X3: multiset_a,Y2: multiset_a,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ Y2 ) @ ( multis1686031732355738386_alt_a @ $true @ Ns @ S ) )
=> ~ ! [X12: multiset_a,X23: multiset_a] :
( ( X3
= ( plus_plus_multiset_a @ X12 @ X23 ) )
=> ! [Y12: multiset_a,Y23: multiset_a] :
( ( Y2
= ( plus_plus_multiset_a @ Y12 @ Y23 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X12 @ Y12 ) @ ( multis2373687813370458242ltpw_a @ Ns ) )
=> ~ ! [X: a] :
( ( member_a @ X @ ( set_mset_a @ X23 ) )
=> ? [Y5: a] :
( ( member_a @ Y5 @ ( set_mset_a @ Y23 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y5 ) @ S ) ) ) ) ) ) ) ).
% mult2_alt_nsE
thf(fact_406_ns__mul__ext__union__compat,axiom,
! [A3: multiset_a,B2: multiset_a,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a,C2: multiset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A3 @ B2 ) @ ( multis4023584895248362547_ext_a @ Ns @ S ) )
=> ( ( locally_refl_a @ Ns @ C2 )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( plus_plus_multiset_a @ A3 @ C2 ) @ ( plus_plus_multiset_a @ B2 @ C2 ) ) @ ( multis4023584895248362547_ext_a @ Ns @ S ) ) ) ) ).
% ns_mul_ext_union_compat
thf(fact_407_ns__mul__ext__IdI,axiom,
! [M: multiset_multiset_a,X3: multiset_multiset_a,Z5: multiset_multiset_a,N: multiset_multiset_a,Y2: multiset_multiset_a,R3: set_Pr79727621955416071iset_a] :
( ( M
= ( plus_p6738641960240532161iset_a @ X3 @ Z5 ) )
=> ( ( N
= ( plus_p6738641960240532161iset_a @ Y2 @ Z5 ) )
=> ( ! [X5: multiset_a] :
( ( member_multiset_a @ X5 @ ( set_mset_multiset_a @ Y2 ) )
=> ? [Xa: multiset_a] :
( ( member_multiset_a @ Xa @ ( set_mset_multiset_a @ X3 ) )
& ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Xa @ X5 ) @ R3 ) ) )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ M @ N ) @ ( multis8112516946959054073iset_a @ id_multiset_a @ R3 ) ) ) ) ) ).
% ns_mul_ext_IdI
thf(fact_408_ns__mul__ext__IdI,axiom,
! [M: multiset_a,X3: multiset_a,Z5: multiset_a,N: multiset_a,Y2: multiset_a,R3: set_Product_prod_a_a] :
( ( M
= ( plus_plus_multiset_a @ X3 @ Z5 ) )
=> ( ( N
= ( plus_plus_multiset_a @ Y2 @ Z5 ) )
=> ( ! [X5: a] :
( ( member_a @ X5 @ ( set_mset_a @ Y2 ) )
=> ? [Xa: a] :
( ( member_a @ Xa @ ( set_mset_a @ X3 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa @ X5 ) @ R3 ) ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ M @ N ) @ ( multis4023584895248362547_ext_a @ id_a @ R3 ) ) ) ) ) ).
% ns_mul_ext_IdI
thf(fact_409_ns__mul__ext__IdE,axiom,
! [M: multiset_multiset_a,N: multiset_multiset_a,R3: set_Pr79727621955416071iset_a] :
( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ M @ N ) @ ( multis8112516946959054073iset_a @ id_multiset_a @ R3 ) )
=> ~ ! [X6: multiset_multiset_a,Y7: multiset_multiset_a,Z2: multiset_multiset_a] :
( ( M
= ( plus_p6738641960240532161iset_a @ X6 @ Z2 ) )
=> ( ( N
= ( plus_p6738641960240532161iset_a @ Y7 @ Z2 ) )
=> ~ ! [X: multiset_a] :
( ( member_multiset_a @ X @ ( set_mset_multiset_a @ Y7 ) )
=> ? [Xa2: multiset_a] :
( ( member_multiset_a @ Xa2 @ ( set_mset_multiset_a @ X6 ) )
& ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Xa2 @ X ) @ R3 ) ) ) ) ) ) ).
% ns_mul_ext_IdE
thf(fact_410_ns__mul__ext__IdE,axiom,
! [M: multiset_a,N: multiset_a,R3: set_Product_prod_a_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ M @ N ) @ ( multis4023584895248362547_ext_a @ id_a @ R3 ) )
=> ~ ! [X6: multiset_a,Y7: multiset_a,Z2: multiset_a] :
( ( M
= ( plus_plus_multiset_a @ X6 @ Z2 ) )
=> ( ( N
= ( plus_plus_multiset_a @ Y7 @ Z2 ) )
=> ~ ! [X: a] :
( ( member_a @ X @ ( set_mset_a @ Y7 ) )
=> ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_mset_a @ X6 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa2 @ X ) @ R3 ) ) ) ) ) ) ).
% ns_mul_ext_IdE
thf(fact_411_le__zero__eq,axiom,
! [N3: nat] :
( ( ord_less_eq_nat @ N3 @ zero_zero_nat )
= ( N3 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_412_add_Oright__neutral,axiom,
! [A2: multiset_a] :
( ( plus_plus_multiset_a @ A2 @ zero_zero_multiset_a )
= A2 ) ).
% add.right_neutral
thf(fact_413_add_Oright__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.right_neutral
thf(fact_414_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_415_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_416_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_417_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_418_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_419_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_420_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_421_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_422_add__eq__0__iff__both__eq__0,axiom,
! [X2: nat,Y: nat] :
( ( ( plus_plus_nat @ X2 @ Y )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_423_zero__eq__add__iff__both__eq__0,axiom,
! [X2: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X2 @ Y ) )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_424_add__0,axiom,
! [A2: multiset_a] :
( ( plus_plus_multiset_a @ zero_zero_multiset_a @ A2 )
= A2 ) ).
% add_0
thf(fact_425_add__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% add_0
thf(fact_426_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
! [X2: multiset_a,Y: multiset_a] :
( ( ( plus_plus_multiset_a @ X2 @ Y )
= zero_zero_multiset_a )
= ( ( X2 = zero_zero_multiset_a )
& ( Y = zero_zero_multiset_a ) ) ) ).
% subset_mset.add_eq_0_iff_both_eq_0
thf(fact_427_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
! [X2: multiset_a,Y: multiset_a] :
( ( zero_zero_multiset_a
= ( plus_plus_multiset_a @ X2 @ Y ) )
= ( ( X2 = zero_zero_multiset_a )
& ( Y = zero_zero_multiset_a ) ) ) ).
% subset_mset.zero_eq_add_iff_both_eq_0
thf(fact_428_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_429_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_430_single__eq__single,axiom,
! [A2: a,B: a] :
( ( ( add_mset_a @ A2 @ zero_zero_multiset_a )
= ( add_mset_a @ B @ zero_zero_multiset_a ) )
= ( A2 = B ) ) ).
% single_eq_single
thf(fact_431_add__mset__eq__single,axiom,
! [B: a,M: multiset_a,A2: a] :
( ( ( add_mset_a @ B @ M )
= ( add_mset_a @ A2 @ zero_zero_multiset_a ) )
= ( ( B = A2 )
& ( M = zero_zero_multiset_a ) ) ) ).
% add_mset_eq_single
thf(fact_432_single__eq__add__mset,axiom,
! [A2: a,B: a,M: multiset_a] :
( ( ( add_mset_a @ A2 @ zero_zero_multiset_a )
= ( add_mset_a @ B @ M ) )
= ( ( B = A2 )
& ( M = zero_zero_multiset_a ) ) ) ).
% single_eq_add_mset
thf(fact_433_add__mset__eq__singleton__iff,axiom,
! [X2: a,M: multiset_a,Y: a] :
( ( ( add_mset_a @ X2 @ M )
= ( add_mset_a @ Y @ zero_zero_multiset_a ) )
= ( ( M = zero_zero_multiset_a )
& ( X2 = Y ) ) ) ).
% add_mset_eq_singleton_iff
thf(fact_434_IdI,axiom,
! [A2: multiset_a] : ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A2 @ A2 ) @ id_multiset_a ) ).
% IdI
thf(fact_435_IdI,axiom,
! [A2: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ A2 ) @ id_a ) ).
% IdI
thf(fact_436_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_437_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_438_multpw__Id__Id,axiom,
( ( multis2373687813370458242ltpw_a @ id_a )
= id_multiset_a ) ).
% multpw_Id_Id
thf(fact_439_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_440_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_441_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_442_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_443_multpw__emptyL,axiom,
! [X3: multiset_a,Ns: set_Product_prod_a_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ zero_zero_multiset_a @ X3 ) @ ( multis2373687813370458242ltpw_a @ Ns ) )
= ( X3 = zero_zero_multiset_a ) ) ).
% multpw_emptyL
thf(fact_444_multpw__emptyR,axiom,
! [X3: multiset_a,Ns: set_Product_prod_a_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ zero_zero_multiset_a ) @ ( multis2373687813370458242ltpw_a @ Ns ) )
= ( X3 = zero_zero_multiset_a ) ) ).
% multpw_emptyR
thf(fact_445_IdE,axiom,
! [P2: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ P2 @ id_multiset_a )
=> ~ ! [X5: multiset_a] :
( P2
!= ( produc654756711066625303iset_a @ X5 @ X5 ) ) ) ).
% IdE
thf(fact_446_IdE,axiom,
! [P2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ P2 @ id_a )
=> ~ ! [X5: a] :
( P2
!= ( product_Pair_a_a @ X5 @ X5 ) ) ) ).
% IdE
thf(fact_447_mult2__alt__emptyL,axiom,
! [Y2: multiset_a,B: $o,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ zero_zero_multiset_a @ Y2 ) @ ( multis1686031732355738386_alt_a @ B @ Ns @ S ) )
= ( B
| ( Y2 != zero_zero_multiset_a ) ) ) ).
% mult2_alt_emptyL
thf(fact_448_mult2__alt__emptyR,axiom,
! [X3: multiset_a,B: $o,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ zero_zero_multiset_a ) @ ( multis1686031732355738386_alt_a @ B @ Ns @ S ) )
= ( B
& ( X3 = zero_zero_multiset_a ) ) ) ).
% mult2_alt_emptyR
thf(fact_449_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_450_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_451_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_452_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_453_add_Ocomm__neutral,axiom,
! [A2: multiset_a] :
( ( plus_plus_multiset_a @ A2 @ zero_zero_multiset_a )
= A2 ) ).
% add.comm_neutral
thf(fact_454_add_Ocomm__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.comm_neutral
thf(fact_455_mult2__alt__s__implies__mult2__alt__ns,axiom,
! [X3: multiset_a,Y2: multiset_a,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ Y2 ) @ ( multis1686031732355738386_alt_a @ $false @ Ns @ S ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ Y2 ) @ ( multis1686031732355738386_alt_a @ $true @ Ns @ S ) ) ) ).
% mult2_alt_s_implies_mult2_alt_ns
thf(fact_456_empty__neutral_I2_J,axiom,
! [X2: multiset_a] :
( ( plus_plus_multiset_a @ X2 @ zero_zero_multiset_a )
= X2 ) ).
% empty_neutral(2)
thf(fact_457_empty__neutral_I1_J,axiom,
! [X2: multiset_a] :
( ( plus_plus_multiset_a @ zero_zero_multiset_a @ X2 )
= X2 ) ).
% empty_neutral(1)
thf(fact_458_multiset__nonemptyE,axiom,
! [A3: multis4781622204924204013iset_a] :
( ( A3 != zero_z4591712717340762532iset_a )
=> ~ ! [X5: produc6518373309651786023iset_a] :
~ ( member5199237121806060112iset_a @ X5 @ ( set_ms2379723421352415844iset_a @ A3 ) ) ) ).
% multiset_nonemptyE
thf(fact_459_multiset__nonemptyE,axiom,
! [A3: multis501812127501805293od_a_a] :
( ( A3 != zero_z3024502163831676836od_a_a )
=> ~ ! [X5: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ X5 @ ( set_ms119794563918438244od_a_a @ A3 ) ) ) ).
% multiset_nonemptyE
thf(fact_460_multiset__nonemptyE,axiom,
! [A3: multiset_a] :
( ( A3 != zero_zero_multiset_a )
=> ~ ! [X5: a] :
~ ( member_a @ X5 @ ( set_mset_a @ A3 ) ) ) ).
% multiset_nonemptyE
thf(fact_461_multiset__cases,axiom,
! [M: multiset_a] :
( ( M != zero_zero_multiset_a )
=> ~ ! [X5: a,N4: multiset_a] :
( M
!= ( add_mset_a @ X5 @ N4 ) ) ) ).
% multiset_cases
thf(fact_462_multiset__induct,axiom,
! [P: multiset_a > $o,M: multiset_a] :
( ( P @ zero_zero_multiset_a )
=> ( ! [X5: a,M3: multiset_a] :
( ( P @ M3 )
=> ( P @ ( add_mset_a @ X5 @ M3 ) ) )
=> ( P @ M ) ) ) ).
% multiset_induct
thf(fact_463_multiset__induct2,axiom,
! [P: multiset_a > multiset_a > $o,M: multiset_a,N: multiset_a] :
( ( P @ zero_zero_multiset_a @ zero_zero_multiset_a )
=> ( ! [A: a,M3: multiset_a,N4: multiset_a] :
( ( P @ M3 @ N4 )
=> ( P @ ( add_mset_a @ A @ M3 ) @ N4 ) )
=> ( ! [A: a,M3: multiset_a,N4: multiset_a] :
( ( P @ M3 @ N4 )
=> ( P @ M3 @ ( add_mset_a @ A @ N4 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% multiset_induct2
thf(fact_464_empty__not__add__mset,axiom,
! [A2: a,A3: multiset_a] :
( zero_zero_multiset_a
!= ( add_mset_a @ A2 @ A3 ) ) ).
% empty_not_add_mset
thf(fact_465_multi__nonempty__split,axiom,
! [M: multiset_a] :
( ( M != zero_zero_multiset_a )
=> ? [A4: multiset_a,A: a] :
( M
= ( add_mset_a @ A @ A4 ) ) ) ).
% multi_nonempty_split
thf(fact_466_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_467_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_468_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_469_add__nonpos__eq__0__iff,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X2 @ Y )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_470_add__nonneg__eq__0__iff,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X2 @ Y )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_471_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_472_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_473_add__increasing2,axiom,
! [C: nat,B: nat,A2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A2 )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_increasing2
thf(fact_474_add__decreasing2,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_475_add__increasing,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_increasing
thf(fact_476_add__decreasing,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_477_mult2__alt__add,axiom,
! [X1: multiset_a,Y1: multiset_a,B13: $o,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a,X22: multiset_a,Y22: multiset_a,B24: $o] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X1 @ Y1 ) @ ( multis1686031732355738386_alt_a @ B13 @ Ns @ S ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X22 @ Y22 ) @ ( multis1686031732355738386_alt_a @ B24 @ Ns @ S ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( plus_plus_multiset_a @ X1 @ X22 ) @ ( plus_plus_multiset_a @ Y1 @ Y22 ) )
@ ( multis1686031732355738386_alt_a
@ ( B13
& B24 )
@ Ns
@ S ) ) ) ) ).
% mult2_alt_add
thf(fact_478_mult2__alt__s__s__add,axiom,
! [X1: multiset_a,Y1: multiset_a,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a,X22: multiset_a,Y22: multiset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X1 @ Y1 ) @ ( multis1686031732355738386_alt_a @ $false @ Ns @ S ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X22 @ Y22 ) @ ( multis1686031732355738386_alt_a @ $false @ Ns @ S ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( plus_plus_multiset_a @ X1 @ X22 ) @ ( plus_plus_multiset_a @ Y1 @ Y22 ) ) @ ( multis1686031732355738386_alt_a @ $false @ Ns @ S ) ) ) ) ).
% mult2_alt_s_s_add
thf(fact_479_mult2__alt__ns__s__add,axiom,
! [X1: multiset_a,Y1: multiset_a,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a,X22: multiset_a,Y22: multiset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X1 @ Y1 ) @ ( multis1686031732355738386_alt_a @ $true @ Ns @ S ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X22 @ Y22 ) @ ( multis1686031732355738386_alt_a @ $false @ Ns @ S ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( plus_plus_multiset_a @ X1 @ X22 ) @ ( plus_plus_multiset_a @ Y1 @ Y22 ) ) @ ( multis1686031732355738386_alt_a @ $false @ Ns @ S ) ) ) ) ).
% mult2_alt_ns_s_add
thf(fact_480_mult2__alt__s__ns__add,axiom,
! [X1: multiset_a,Y1: multiset_a,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a,X22: multiset_a,Y22: multiset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X1 @ Y1 ) @ ( multis1686031732355738386_alt_a @ $false @ Ns @ S ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X22 @ Y22 ) @ ( multis1686031732355738386_alt_a @ $true @ Ns @ S ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( plus_plus_multiset_a @ X1 @ X22 ) @ ( plus_plus_multiset_a @ Y1 @ Y22 ) ) @ ( multis1686031732355738386_alt_a @ $false @ Ns @ S ) ) ) ) ).
% mult2_alt_s_ns_add
thf(fact_481_mult2__alt__ns__ns__add,axiom,
! [X1: multiset_a,Y1: multiset_a,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a,X22: multiset_a,Y22: multiset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X1 @ Y1 ) @ ( multis1686031732355738386_alt_a @ $true @ Ns @ S ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X22 @ Y22 ) @ ( multis1686031732355738386_alt_a @ $true @ Ns @ S ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( plus_plus_multiset_a @ X1 @ X22 ) @ ( plus_plus_multiset_a @ Y1 @ Y22 ) ) @ ( multis1686031732355738386_alt_a @ $true @ Ns @ S ) ) ) ) ).
% mult2_alt_ns_ns_add
thf(fact_482_multpw__implies__mult2__alt__ns,axiom,
! [X3: multiset_a,Y2: multiset_a,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ Y2 ) @ ( multis2373687813370458242ltpw_a @ Ns ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ Y2 ) @ ( multis1686031732355738386_alt_a @ $true @ Ns @ S ) ) ) ).
% multpw_implies_mult2_alt_ns
thf(fact_483_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_484_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_485_add__mset__add__single,axiom,
( add_mset_a
= ( ^ [A5: a,A6: multiset_a] : ( plus_plus_multiset_a @ A6 @ ( add_mset_a @ A5 @ zero_zero_multiset_a ) ) ) ) ).
% add_mset_add_single
thf(fact_486_multi__member__last,axiom,
! [X2: produc6518373309651786023iset_a] : ( member5199237121806060112iset_a @ X2 @ ( set_ms2379723421352415844iset_a @ ( add_ms5348986034529679365iset_a @ X2 @ zero_z4591712717340762532iset_a ) ) ) ).
% multi_member_last
thf(fact_487_multi__member__last,axiom,
! [X2: product_prod_a_a] : ( member1426531477525435216od_a_a @ X2 @ ( set_ms119794563918438244od_a_a @ ( add_ms8655138167283798533od_a_a @ X2 @ zero_z3024502163831676836od_a_a ) ) ) ).
% multi_member_last
thf(fact_488_multi__member__last,axiom,
! [X2: a] : ( member_a @ X2 @ ( set_mset_a @ ( add_mset_a @ X2 @ zero_zero_multiset_a ) ) ) ).
% multi_member_last
thf(fact_489_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_490_ns__mul__ext__bottom,axiom,
! [A3: multiset_a,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a] : ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A3 @ zero_zero_multiset_a ) @ ( multis4023584895248362547_ext_a @ Ns @ S ) ) ).
% ns_mul_ext_bottom
thf(fact_491_ns__mul__ext__bottom2,axiom,
! [A3: multiset_a,B2: multiset_a,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A3 @ B2 ) @ ( multis4023584895248362547_ext_a @ Ns @ S ) )
=> ( ( B2 != zero_zero_multiset_a )
=> ( A3 != zero_zero_multiset_a ) ) ) ).
% ns_mul_ext_bottom2
thf(fact_492_ns__mul__ext__bottom__uniqueness,axiom,
! [A3: multiset_a,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ zero_zero_multiset_a @ A3 ) @ ( multis4023584895248362547_ext_a @ Ns @ S ) )
=> ( A3 = zero_zero_multiset_a ) ) ).
% ns_mul_ext_bottom_uniqueness
thf(fact_493_mono__mult2__alt,axiom,
! [Ns: set_Product_prod_a_a,Ns2: set_Product_prod_a_a,S: set_Product_prod_a_a,S2: set_Product_prod_a_a,B: $o] :
( ( ord_le746702958409616551od_a_a @ Ns @ Ns2 )
=> ( ( ord_le746702958409616551od_a_a @ S @ S2 )
=> ( ord_le5871882978999109543iset_a @ ( multis1686031732355738386_alt_a @ B @ Ns @ S ) @ ( multis1686031732355738386_alt_a @ B @ Ns2 @ S2 ) ) ) ) ).
% mono_mult2_alt
thf(fact_494_mono__mult2__alt,axiom,
! [Ns: set_Pr79727621955416071iset_a,Ns2: set_Pr79727621955416071iset_a,S: set_Pr79727621955416071iset_a,S2: set_Pr79727621955416071iset_a,B: $o] :
( ( ord_le5871882978999109543iset_a @ Ns @ Ns2 )
=> ( ( ord_le5871882978999109543iset_a @ S @ S2 )
=> ( ord_le2895628588359463079iset_a @ ( multis2006310815304088152iset_a @ B @ Ns @ S ) @ ( multis2006310815304088152iset_a @ B @ Ns2 @ S2 ) ) ) ) ).
% mono_mult2_alt
thf(fact_495_mult2__altE,axiom,
! [X3: multis4781622204924204013iset_a,Y2: multis4781622204924204013iset_a,B: $o,Ns: set_Pr3059988933816007879iset_a,S: set_Pr3059988933816007879iset_a] :
( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ X3 @ Y2 ) @ ( multis1572145089415997755iset_a @ B @ Ns @ S ) )
=> ~ ! [X12: multis4781622204924204013iset_a,X23: multis4781622204924204013iset_a] :
( ( X3
= ( plus_p1053901119268525860iset_a @ X12 @ X23 ) )
=> ! [Y12: multis4781622204924204013iset_a,Y23: multis4781622204924204013iset_a] :
( ( Y2
= ( plus_p1053901119268525860iset_a @ Y12 @ Y23 ) )
=> ( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ X12 @ Y12 ) @ ( multis3430030276581740203iset_a @ Ns ) )
=> ( ( B
| ( Y23 != zero_z4591712717340762532iset_a ) )
=> ~ ! [X: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X @ ( set_ms2379723421352415844iset_a @ X23 ) )
=> ? [Y5: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Y5 @ ( set_ms2379723421352415844iset_a @ Y23 ) )
& ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X @ Y5 ) @ S ) ) ) ) ) ) ) ) ).
% mult2_altE
thf(fact_496_mult2__altE,axiom,
! [X3: multis501812127501805293od_a_a,Y2: multis501812127501805293od_a_a,B: $o,Ns: set_Pr8600417178894128327od_a_a,S: set_Pr8600417178894128327od_a_a] :
( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ X3 @ Y2 ) @ ( multis4249482161328750907od_a_a @ B @ Ns @ S ) )
=> ~ ! [X12: multis501812127501805293od_a_a,X23: multis501812127501805293od_a_a] :
( ( X3
= ( plus_p6967097483979887908od_a_a @ X12 @ X23 ) )
=> ! [Y12: multis501812127501805293od_a_a,Y23: multis501812127501805293od_a_a] :
( ( Y2
= ( plus_p6967097483979887908od_a_a @ Y12 @ Y23 ) )
=> ( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ X12 @ Y12 ) @ ( multis1821517007024690859od_a_a @ Ns ) )
=> ( ( B
| ( Y23 != zero_z3024502163831676836od_a_a ) )
=> ~ ! [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_ms119794563918438244od_a_a @ X23 ) )
=> ? [Y5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y5 @ ( set_ms119794563918438244od_a_a @ Y23 ) )
& ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Y5 ) @ S ) ) ) ) ) ) ) ) ).
% mult2_altE
thf(fact_497_mult2__altE,axiom,
! [X3: multiset_multiset_a,Y2: multiset_multiset_a,B: $o,Ns: set_Pr79727621955416071iset_a,S: set_Pr79727621955416071iset_a] :
( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X3 @ Y2 ) @ ( multis2006310815304088152iset_a @ B @ Ns @ S ) )
=> ~ ! [X12: multiset_multiset_a,X23: multiset_multiset_a] :
( ( X3
= ( plus_p6738641960240532161iset_a @ X12 @ X23 ) )
=> ! [Y12: multiset_multiset_a,Y23: multiset_multiset_a] :
( ( Y2
= ( plus_p6738641960240532161iset_a @ Y12 @ Y23 ) )
=> ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X12 @ Y12 ) @ ( multis451435756853798856iset_a @ Ns ) )
=> ( ( B
| ( Y23 != zero_z7799948378220188993iset_a ) )
=> ~ ! [X: multiset_a] :
( ( member_multiset_a @ X @ ( set_mset_multiset_a @ X23 ) )
=> ? [Y5: multiset_a] :
( ( member_multiset_a @ Y5 @ ( set_mset_multiset_a @ Y23 ) )
& ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ Y5 ) @ S ) ) ) ) ) ) ) ) ).
% mult2_altE
thf(fact_498_mult2__altE,axiom,
! [X3: multiset_a,Y2: multiset_a,B: $o,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ Y2 ) @ ( multis1686031732355738386_alt_a @ B @ Ns @ S ) )
=> ~ ! [X12: multiset_a,X23: multiset_a] :
( ( X3
= ( plus_plus_multiset_a @ X12 @ X23 ) )
=> ! [Y12: multiset_a,Y23: multiset_a] :
( ( Y2
= ( plus_plus_multiset_a @ Y12 @ Y23 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X12 @ Y12 ) @ ( multis2373687813370458242ltpw_a @ Ns ) )
=> ( ( B
| ( Y23 != zero_zero_multiset_a ) )
=> ~ ! [X: a] :
( ( member_a @ X @ ( set_mset_a @ X23 ) )
=> ? [Y5: a] :
( ( member_a @ Y5 @ ( set_mset_a @ Y23 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y5 ) @ S ) ) ) ) ) ) ) ) ).
% mult2_altE
thf(fact_499_mult2__altI,axiom,
! [X3: multis4781622204924204013iset_a,X1: multis4781622204924204013iset_a,X22: multis4781622204924204013iset_a,Y2: multis4781622204924204013iset_a,Y1: multis4781622204924204013iset_a,Y22: multis4781622204924204013iset_a,Ns: set_Pr3059988933816007879iset_a,B: $o,S: set_Pr3059988933816007879iset_a] :
( ( X3
= ( plus_p1053901119268525860iset_a @ X1 @ X22 ) )
=> ( ( Y2
= ( plus_p1053901119268525860iset_a @ Y1 @ Y22 ) )
=> ( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ X1 @ Y1 ) @ ( multis3430030276581740203iset_a @ Ns ) )
=> ( ( B
| ( Y22 != zero_z4591712717340762532iset_a ) )
=> ( ! [X5: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X5 @ ( set_ms2379723421352415844iset_a @ X22 ) )
=> ? [Y6: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Y6 @ ( set_ms2379723421352415844iset_a @ Y22 ) )
& ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X5 @ Y6 ) @ S ) ) )
=> ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ X3 @ Y2 ) @ ( multis1572145089415997755iset_a @ B @ Ns @ S ) ) ) ) ) ) ) ).
% mult2_altI
thf(fact_500_mult2__altI,axiom,
! [X3: multis501812127501805293od_a_a,X1: multis501812127501805293od_a_a,X22: multis501812127501805293od_a_a,Y2: multis501812127501805293od_a_a,Y1: multis501812127501805293od_a_a,Y22: multis501812127501805293od_a_a,Ns: set_Pr8600417178894128327od_a_a,B: $o,S: set_Pr8600417178894128327od_a_a] :
( ( X3
= ( plus_p6967097483979887908od_a_a @ X1 @ X22 ) )
=> ( ( Y2
= ( plus_p6967097483979887908od_a_a @ Y1 @ Y22 ) )
=> ( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ X1 @ Y1 ) @ ( multis1821517007024690859od_a_a @ Ns ) )
=> ( ( B
| ( Y22 != zero_z3024502163831676836od_a_a ) )
=> ( ! [X5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ ( set_ms119794563918438244od_a_a @ X22 ) )
=> ? [Y6: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y6 @ ( set_ms119794563918438244od_a_a @ Y22 ) )
& ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X5 @ Y6 ) @ S ) ) )
=> ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ X3 @ Y2 ) @ ( multis4249482161328750907od_a_a @ B @ Ns @ S ) ) ) ) ) ) ) ).
% mult2_altI
thf(fact_501_mult2__altI,axiom,
! [X3: multiset_multiset_a,X1: multiset_multiset_a,X22: multiset_multiset_a,Y2: multiset_multiset_a,Y1: multiset_multiset_a,Y22: multiset_multiset_a,Ns: set_Pr79727621955416071iset_a,B: $o,S: set_Pr79727621955416071iset_a] :
( ( X3
= ( plus_p6738641960240532161iset_a @ X1 @ X22 ) )
=> ( ( Y2
= ( plus_p6738641960240532161iset_a @ Y1 @ Y22 ) )
=> ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X1 @ Y1 ) @ ( multis451435756853798856iset_a @ Ns ) )
=> ( ( B
| ( Y22 != zero_z7799948378220188993iset_a ) )
=> ( ! [X5: multiset_a] :
( ( member_multiset_a @ X5 @ ( set_mset_multiset_a @ X22 ) )
=> ? [Y6: multiset_a] :
( ( member_multiset_a @ Y6 @ ( set_mset_multiset_a @ Y22 ) )
& ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Y6 ) @ S ) ) )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X3 @ Y2 ) @ ( multis2006310815304088152iset_a @ B @ Ns @ S ) ) ) ) ) ) ) ).
% mult2_altI
thf(fact_502_mult2__altI,axiom,
! [X3: multiset_a,X1: multiset_a,X22: multiset_a,Y2: multiset_a,Y1: multiset_a,Y22: multiset_a,Ns: set_Product_prod_a_a,B: $o,S: set_Product_prod_a_a] :
( ( X3
= ( plus_plus_multiset_a @ X1 @ X22 ) )
=> ( ( Y2
= ( plus_plus_multiset_a @ Y1 @ Y22 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X1 @ Y1 ) @ ( multis2373687813370458242ltpw_a @ Ns ) )
=> ( ( B
| ( Y22 != zero_zero_multiset_a ) )
=> ( ! [X5: a] :
( ( member_a @ X5 @ ( set_mset_a @ X22 ) )
=> ? [Y6: a] :
( ( member_a @ Y6 @ ( set_mset_a @ Y22 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y6 ) @ S ) ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ Y2 ) @ ( multis1686031732355738386_alt_a @ B @ Ns @ S ) ) ) ) ) ) ) ).
% mult2_altI
thf(fact_503_mult2__alt__sE,axiom,
! [X3: multis4781622204924204013iset_a,Y2: multis4781622204924204013iset_a,Ns: set_Pr3059988933816007879iset_a,S: set_Pr3059988933816007879iset_a] :
( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ X3 @ Y2 ) @ ( multis1572145089415997755iset_a @ $false @ Ns @ S ) )
=> ~ ! [X12: multis4781622204924204013iset_a,X23: multis4781622204924204013iset_a] :
( ( X3
= ( plus_p1053901119268525860iset_a @ X12 @ X23 ) )
=> ! [Y12: multis4781622204924204013iset_a,Y23: multis4781622204924204013iset_a] :
( ( Y2
= ( plus_p1053901119268525860iset_a @ Y12 @ Y23 ) )
=> ( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ X12 @ Y12 ) @ ( multis3430030276581740203iset_a @ Ns ) )
=> ( ( Y23 != zero_z4591712717340762532iset_a )
=> ~ ! [X: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X @ ( set_ms2379723421352415844iset_a @ X23 ) )
=> ? [Y5: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Y5 @ ( set_ms2379723421352415844iset_a @ Y23 ) )
& ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X @ Y5 ) @ S ) ) ) ) ) ) ) ) ).
% mult2_alt_sE
thf(fact_504_mult2__alt__sE,axiom,
! [X3: multis501812127501805293od_a_a,Y2: multis501812127501805293od_a_a,Ns: set_Pr8600417178894128327od_a_a,S: set_Pr8600417178894128327od_a_a] :
( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ X3 @ Y2 ) @ ( multis4249482161328750907od_a_a @ $false @ Ns @ S ) )
=> ~ ! [X12: multis501812127501805293od_a_a,X23: multis501812127501805293od_a_a] :
( ( X3
= ( plus_p6967097483979887908od_a_a @ X12 @ X23 ) )
=> ! [Y12: multis501812127501805293od_a_a,Y23: multis501812127501805293od_a_a] :
( ( Y2
= ( plus_p6967097483979887908od_a_a @ Y12 @ Y23 ) )
=> ( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ X12 @ Y12 ) @ ( multis1821517007024690859od_a_a @ Ns ) )
=> ( ( Y23 != zero_z3024502163831676836od_a_a )
=> ~ ! [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_ms119794563918438244od_a_a @ X23 ) )
=> ? [Y5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y5 @ ( set_ms119794563918438244od_a_a @ Y23 ) )
& ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Y5 ) @ S ) ) ) ) ) ) ) ) ).
% mult2_alt_sE
thf(fact_505_mult2__alt__sE,axiom,
! [X3: multiset_multiset_a,Y2: multiset_multiset_a,Ns: set_Pr79727621955416071iset_a,S: set_Pr79727621955416071iset_a] :
( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X3 @ Y2 ) @ ( multis2006310815304088152iset_a @ $false @ Ns @ S ) )
=> ~ ! [X12: multiset_multiset_a,X23: multiset_multiset_a] :
( ( X3
= ( plus_p6738641960240532161iset_a @ X12 @ X23 ) )
=> ! [Y12: multiset_multiset_a,Y23: multiset_multiset_a] :
( ( Y2
= ( plus_p6738641960240532161iset_a @ Y12 @ Y23 ) )
=> ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X12 @ Y12 ) @ ( multis451435756853798856iset_a @ Ns ) )
=> ( ( Y23 != zero_z7799948378220188993iset_a )
=> ~ ! [X: multiset_a] :
( ( member_multiset_a @ X @ ( set_mset_multiset_a @ X23 ) )
=> ? [Y5: multiset_a] :
( ( member_multiset_a @ Y5 @ ( set_mset_multiset_a @ Y23 ) )
& ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ Y5 ) @ S ) ) ) ) ) ) ) ) ).
% mult2_alt_sE
thf(fact_506_mult2__alt__sE,axiom,
! [X3: multiset_a,Y2: multiset_a,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ Y2 ) @ ( multis1686031732355738386_alt_a @ $false @ Ns @ S ) )
=> ~ ! [X12: multiset_a,X23: multiset_a] :
( ( X3
= ( plus_plus_multiset_a @ X12 @ X23 ) )
=> ! [Y12: multiset_a,Y23: multiset_a] :
( ( Y2
= ( plus_plus_multiset_a @ Y12 @ Y23 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X12 @ Y12 ) @ ( multis2373687813370458242ltpw_a @ Ns ) )
=> ( ( Y23 != zero_zero_multiset_a )
=> ~ ! [X: a] :
( ( member_a @ X @ ( set_mset_a @ X23 ) )
=> ? [Y5: a] :
( ( member_a @ Y5 @ ( set_mset_a @ Y23 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y5 ) @ S ) ) ) ) ) ) ) ) ).
% mult2_alt_sE
thf(fact_507_mult2__alt__sI,axiom,
! [X3: multis4781622204924204013iset_a,X1: multis4781622204924204013iset_a,X22: multis4781622204924204013iset_a,Y2: multis4781622204924204013iset_a,Y1: multis4781622204924204013iset_a,Y22: multis4781622204924204013iset_a,Ns: set_Pr3059988933816007879iset_a,S: set_Pr3059988933816007879iset_a] :
( ( X3
= ( plus_p1053901119268525860iset_a @ X1 @ X22 ) )
=> ( ( Y2
= ( plus_p1053901119268525860iset_a @ Y1 @ Y22 ) )
=> ( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ X1 @ Y1 ) @ ( multis3430030276581740203iset_a @ Ns ) )
=> ( ( Y22 != zero_z4591712717340762532iset_a )
=> ( ! [X5: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X5 @ ( set_ms2379723421352415844iset_a @ X22 ) )
=> ? [Y6: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Y6 @ ( set_ms2379723421352415844iset_a @ Y22 ) )
& ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X5 @ Y6 ) @ S ) ) )
=> ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ X3 @ Y2 ) @ ( multis1572145089415997755iset_a @ $false @ Ns @ S ) ) ) ) ) ) ) ).
% mult2_alt_sI
thf(fact_508_mult2__alt__sI,axiom,
! [X3: multis501812127501805293od_a_a,X1: multis501812127501805293od_a_a,X22: multis501812127501805293od_a_a,Y2: multis501812127501805293od_a_a,Y1: multis501812127501805293od_a_a,Y22: multis501812127501805293od_a_a,Ns: set_Pr8600417178894128327od_a_a,S: set_Pr8600417178894128327od_a_a] :
( ( X3
= ( plus_p6967097483979887908od_a_a @ X1 @ X22 ) )
=> ( ( Y2
= ( plus_p6967097483979887908od_a_a @ Y1 @ Y22 ) )
=> ( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ X1 @ Y1 ) @ ( multis1821517007024690859od_a_a @ Ns ) )
=> ( ( Y22 != zero_z3024502163831676836od_a_a )
=> ( ! [X5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ ( set_ms119794563918438244od_a_a @ X22 ) )
=> ? [Y6: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y6 @ ( set_ms119794563918438244od_a_a @ Y22 ) )
& ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X5 @ Y6 ) @ S ) ) )
=> ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ X3 @ Y2 ) @ ( multis4249482161328750907od_a_a @ $false @ Ns @ S ) ) ) ) ) ) ) ).
% mult2_alt_sI
thf(fact_509_mult2__alt__sI,axiom,
! [X3: multiset_multiset_a,X1: multiset_multiset_a,X22: multiset_multiset_a,Y2: multiset_multiset_a,Y1: multiset_multiset_a,Y22: multiset_multiset_a,Ns: set_Pr79727621955416071iset_a,S: set_Pr79727621955416071iset_a] :
( ( X3
= ( plus_p6738641960240532161iset_a @ X1 @ X22 ) )
=> ( ( Y2
= ( plus_p6738641960240532161iset_a @ Y1 @ Y22 ) )
=> ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X1 @ Y1 ) @ ( multis451435756853798856iset_a @ Ns ) )
=> ( ( Y22 != zero_z7799948378220188993iset_a )
=> ( ! [X5: multiset_a] :
( ( member_multiset_a @ X5 @ ( set_mset_multiset_a @ X22 ) )
=> ? [Y6: multiset_a] :
( ( member_multiset_a @ Y6 @ ( set_mset_multiset_a @ Y22 ) )
& ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Y6 ) @ S ) ) )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X3 @ Y2 ) @ ( multis2006310815304088152iset_a @ $false @ Ns @ S ) ) ) ) ) ) ) ).
% mult2_alt_sI
thf(fact_510_mult2__alt__sI,axiom,
! [X3: multiset_a,X1: multiset_a,X22: multiset_a,Y2: multiset_a,Y1: multiset_a,Y22: multiset_a,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( X3
= ( plus_plus_multiset_a @ X1 @ X22 ) )
=> ( ( Y2
= ( plus_plus_multiset_a @ Y1 @ Y22 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X1 @ Y1 ) @ ( multis2373687813370458242ltpw_a @ Ns ) )
=> ( ( Y22 != zero_zero_multiset_a )
=> ( ! [X5: a] :
( ( member_a @ X5 @ ( set_mset_a @ X22 ) )
=> ? [Y6: a] :
( ( member_a @ Y6 @ ( set_mset_a @ Y22 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y6 ) @ S ) ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ Y2 ) @ ( multis1686031732355738386_alt_a @ $false @ Ns @ S ) ) ) ) ) ) ) ).
% mult2_alt_sI
thf(fact_511_not__less__empty,axiom,
! [M: multiset_a,R: set_Product_prod_a_a] :
~ ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ M @ zero_zero_multiset_a ) @ ( mult1_a @ R ) ) ).
% not_less_empty
thf(fact_512_trans__mult2__alt__local,axiom,
! [X3: multis4781622204924204013iset_a,Y2: multis4781622204924204013iset_a,Z5: multis4781622204924204013iset_a,S: set_Pr3059988933816007879iset_a,Ns: set_Pr3059988933816007879iset_a,B13: $o,B24: $o] :
( ! [X5: produc6518373309651786023iset_a,Y5: produc6518373309651786023iset_a,Z: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X5 @ ( set_ms2379723421352415844iset_a @ X3 ) )
=> ( ( member5199237121806060112iset_a @ Y5 @ ( set_ms2379723421352415844iset_a @ Y2 ) )
=> ( ( member5199237121806060112iset_a @ Z @ ( set_ms2379723421352415844iset_a @ Z5 ) )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X5 @ Y5 ) @ S )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ Y5 @ Z ) @ S )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: produc6518373309651786023iset_a,Y5: produc6518373309651786023iset_a,Z: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X5 @ ( set_ms2379723421352415844iset_a @ X3 ) )
=> ( ( member5199237121806060112iset_a @ Y5 @ ( set_ms2379723421352415844iset_a @ Y2 ) )
=> ( ( member5199237121806060112iset_a @ Z @ ( set_ms2379723421352415844iset_a @ Z5 ) )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X5 @ Y5 ) @ Ns )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ Y5 @ Z ) @ S )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: produc6518373309651786023iset_a,Y5: produc6518373309651786023iset_a,Z: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X5 @ ( set_ms2379723421352415844iset_a @ X3 ) )
=> ( ( member5199237121806060112iset_a @ Y5 @ ( set_ms2379723421352415844iset_a @ Y2 ) )
=> ( ( member5199237121806060112iset_a @ Z @ ( set_ms2379723421352415844iset_a @ Z5 ) )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X5 @ Y5 ) @ S )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ Y5 @ Z ) @ Ns )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: produc6518373309651786023iset_a,Y5: produc6518373309651786023iset_a,Z: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X5 @ ( set_ms2379723421352415844iset_a @ X3 ) )
=> ( ( member5199237121806060112iset_a @ Y5 @ ( set_ms2379723421352415844iset_a @ Y2 ) )
=> ( ( member5199237121806060112iset_a @ Z @ ( set_ms2379723421352415844iset_a @ Z5 ) )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X5 @ Y5 ) @ Ns )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ Y5 @ Z ) @ Ns )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X5 @ Z ) @ Ns ) ) ) ) ) )
=> ( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ X3 @ Y2 ) @ ( multis1572145089415997755iset_a @ B13 @ Ns @ S ) )
=> ( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ Y2 @ Z5 ) @ ( multis1572145089415997755iset_a @ B24 @ Ns @ S ) )
=> ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ X3 @ Z5 )
@ ( multis1572145089415997755iset_a
@ ( B13
& B24 )
@ Ns
@ S ) ) ) ) ) ) ) ) ).
% trans_mult2_alt_local
thf(fact_513_trans__mult2__alt__local,axiom,
! [X3: multis501812127501805293od_a_a,Y2: multis501812127501805293od_a_a,Z5: multis501812127501805293od_a_a,S: set_Pr8600417178894128327od_a_a,Ns: set_Pr8600417178894128327od_a_a,B13: $o,B24: $o] :
( ! [X5: product_prod_a_a,Y5: product_prod_a_a,Z: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ ( set_ms119794563918438244od_a_a @ X3 ) )
=> ( ( member1426531477525435216od_a_a @ Y5 @ ( set_ms119794563918438244od_a_a @ Y2 ) )
=> ( ( member1426531477525435216od_a_a @ Z @ ( set_ms119794563918438244od_a_a @ Z5 ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X5 @ Y5 ) @ S )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y5 @ Z ) @ S )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: product_prod_a_a,Y5: product_prod_a_a,Z: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ ( set_ms119794563918438244od_a_a @ X3 ) )
=> ( ( member1426531477525435216od_a_a @ Y5 @ ( set_ms119794563918438244od_a_a @ Y2 ) )
=> ( ( member1426531477525435216od_a_a @ Z @ ( set_ms119794563918438244od_a_a @ Z5 ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X5 @ Y5 ) @ Ns )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y5 @ Z ) @ S )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: product_prod_a_a,Y5: product_prod_a_a,Z: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ ( set_ms119794563918438244od_a_a @ X3 ) )
=> ( ( member1426531477525435216od_a_a @ Y5 @ ( set_ms119794563918438244od_a_a @ Y2 ) )
=> ( ( member1426531477525435216od_a_a @ Z @ ( set_ms119794563918438244od_a_a @ Z5 ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X5 @ Y5 ) @ S )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y5 @ Z ) @ Ns )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: product_prod_a_a,Y5: product_prod_a_a,Z: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ ( set_ms119794563918438244od_a_a @ X3 ) )
=> ( ( member1426531477525435216od_a_a @ Y5 @ ( set_ms119794563918438244od_a_a @ Y2 ) )
=> ( ( member1426531477525435216od_a_a @ Z @ ( set_ms119794563918438244od_a_a @ Z5 ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X5 @ Y5 ) @ Ns )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y5 @ Z ) @ Ns )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X5 @ Z ) @ Ns ) ) ) ) ) )
=> ( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ X3 @ Y2 ) @ ( multis4249482161328750907od_a_a @ B13 @ Ns @ S ) )
=> ( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ Y2 @ Z5 ) @ ( multis4249482161328750907od_a_a @ B24 @ Ns @ S ) )
=> ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ X3 @ Z5 )
@ ( multis4249482161328750907od_a_a
@ ( B13
& B24 )
@ Ns
@ S ) ) ) ) ) ) ) ) ).
% trans_mult2_alt_local
thf(fact_514_trans__mult2__alt__local,axiom,
! [X3: multiset_multiset_a,Y2: multiset_multiset_a,Z5: multiset_multiset_a,S: set_Pr79727621955416071iset_a,Ns: set_Pr79727621955416071iset_a,B13: $o,B24: $o] :
( ! [X5: multiset_a,Y5: multiset_a,Z: multiset_a] :
( ( member_multiset_a @ X5 @ ( set_mset_multiset_a @ X3 ) )
=> ( ( member_multiset_a @ Y5 @ ( set_mset_multiset_a @ Y2 ) )
=> ( ( member_multiset_a @ Z @ ( set_mset_multiset_a @ Z5 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Y5 ) @ S )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Y5 @ Z ) @ S )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: multiset_a,Y5: multiset_a,Z: multiset_a] :
( ( member_multiset_a @ X5 @ ( set_mset_multiset_a @ X3 ) )
=> ( ( member_multiset_a @ Y5 @ ( set_mset_multiset_a @ Y2 ) )
=> ( ( member_multiset_a @ Z @ ( set_mset_multiset_a @ Z5 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Y5 ) @ Ns )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Y5 @ Z ) @ S )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: multiset_a,Y5: multiset_a,Z: multiset_a] :
( ( member_multiset_a @ X5 @ ( set_mset_multiset_a @ X3 ) )
=> ( ( member_multiset_a @ Y5 @ ( set_mset_multiset_a @ Y2 ) )
=> ( ( member_multiset_a @ Z @ ( set_mset_multiset_a @ Z5 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Y5 ) @ S )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Y5 @ Z ) @ Ns )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: multiset_a,Y5: multiset_a,Z: multiset_a] :
( ( member_multiset_a @ X5 @ ( set_mset_multiset_a @ X3 ) )
=> ( ( member_multiset_a @ Y5 @ ( set_mset_multiset_a @ Y2 ) )
=> ( ( member_multiset_a @ Z @ ( set_mset_multiset_a @ Z5 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Y5 ) @ Ns )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Y5 @ Z ) @ Ns )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Z ) @ Ns ) ) ) ) ) )
=> ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X3 @ Y2 ) @ ( multis2006310815304088152iset_a @ B13 @ Ns @ S ) )
=> ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ Y2 @ Z5 ) @ ( multis2006310815304088152iset_a @ B24 @ Ns @ S ) )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X3 @ Z5 )
@ ( multis2006310815304088152iset_a
@ ( B13
& B24 )
@ Ns
@ S ) ) ) ) ) ) ) ) ).
% trans_mult2_alt_local
thf(fact_515_trans__mult2__alt__local,axiom,
! [X3: multiset_a,Y2: multiset_a,Z5: multiset_a,S: set_Product_prod_a_a,Ns: set_Product_prod_a_a,B13: $o,B24: $o] :
( ! [X5: a,Y5: a,Z: a] :
( ( member_a @ X5 @ ( set_mset_a @ X3 ) )
=> ( ( member_a @ Y5 @ ( set_mset_a @ Y2 ) )
=> ( ( member_a @ Z @ ( set_mset_a @ Z5 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y5 ) @ S )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y5 @ Z ) @ S )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: a,Y5: a,Z: a] :
( ( member_a @ X5 @ ( set_mset_a @ X3 ) )
=> ( ( member_a @ Y5 @ ( set_mset_a @ Y2 ) )
=> ( ( member_a @ Z @ ( set_mset_a @ Z5 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y5 ) @ Ns )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y5 @ Z ) @ S )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: a,Y5: a,Z: a] :
( ( member_a @ X5 @ ( set_mset_a @ X3 ) )
=> ( ( member_a @ Y5 @ ( set_mset_a @ Y2 ) )
=> ( ( member_a @ Z @ ( set_mset_a @ Z5 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y5 ) @ S )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y5 @ Z ) @ Ns )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: a,Y5: a,Z: a] :
( ( member_a @ X5 @ ( set_mset_a @ X3 ) )
=> ( ( member_a @ Y5 @ ( set_mset_a @ Y2 ) )
=> ( ( member_a @ Z @ ( set_mset_a @ Z5 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y5 ) @ Ns )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y5 @ Z ) @ Ns )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Z ) @ Ns ) ) ) ) ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ Y2 ) @ ( multis1686031732355738386_alt_a @ B13 @ Ns @ S ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Y2 @ Z5 ) @ ( multis1686031732355738386_alt_a @ B24 @ Ns @ S ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ Z5 )
@ ( multis1686031732355738386_alt_a
@ ( B13
& B24 )
@ Ns
@ S ) ) ) ) ) ) ) ) ).
% trans_mult2_alt_local
thf(fact_516_trans__mult2__alt__s__s__local,axiom,
! [X3: multis4781622204924204013iset_a,Y2: multis4781622204924204013iset_a,Z5: multis4781622204924204013iset_a,S: set_Pr3059988933816007879iset_a,Ns: set_Pr3059988933816007879iset_a] :
( ! [X5: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X5 @ ( set_ms2379723421352415844iset_a @ X3 ) )
=> ! [Y5: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Y5 @ ( set_ms2379723421352415844iset_a @ Y2 ) )
=> ! [Z: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Z @ ( set_ms2379723421352415844iset_a @ Z5 ) )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X5 @ Y5 ) @ S )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ Y5 @ Z ) @ S )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X5 @ ( set_ms2379723421352415844iset_a @ X3 ) )
=> ! [Y5: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Y5 @ ( set_ms2379723421352415844iset_a @ Y2 ) )
=> ! [Z: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Z @ ( set_ms2379723421352415844iset_a @ Z5 ) )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X5 @ Y5 ) @ Ns )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ Y5 @ Z ) @ S )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X5 @ ( set_ms2379723421352415844iset_a @ X3 ) )
=> ! [Y5: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Y5 @ ( set_ms2379723421352415844iset_a @ Y2 ) )
=> ! [Z: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Z @ ( set_ms2379723421352415844iset_a @ Z5 ) )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X5 @ Y5 ) @ S )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ Y5 @ Z ) @ Ns )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X5 @ ( set_ms2379723421352415844iset_a @ X3 ) )
=> ! [Y5: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Y5 @ ( set_ms2379723421352415844iset_a @ Y2 ) )
=> ! [Z: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Z @ ( set_ms2379723421352415844iset_a @ Z5 ) )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X5 @ Y5 ) @ Ns )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ Y5 @ Z ) @ Ns )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X5 @ Z ) @ Ns ) ) ) ) ) )
=> ( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ X3 @ Y2 ) @ ( multis1572145089415997755iset_a @ $false @ Ns @ S ) )
=> ( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ Y2 @ Z5 ) @ ( multis1572145089415997755iset_a @ $false @ Ns @ S ) )
=> ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ X3 @ Z5 ) @ ( multis1572145089415997755iset_a @ $false @ Ns @ S ) ) ) ) ) ) ) ) ).
% trans_mult2_alt_s_s_local
thf(fact_517_trans__mult2__alt__s__s__local,axiom,
! [X3: multis501812127501805293od_a_a,Y2: multis501812127501805293od_a_a,Z5: multis501812127501805293od_a_a,S: set_Pr8600417178894128327od_a_a,Ns: set_Pr8600417178894128327od_a_a] :
( ! [X5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ ( set_ms119794563918438244od_a_a @ X3 ) )
=> ! [Y5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y5 @ ( set_ms119794563918438244od_a_a @ Y2 ) )
=> ! [Z: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Z @ ( set_ms119794563918438244od_a_a @ Z5 ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X5 @ Y5 ) @ S )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y5 @ Z ) @ S )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ ( set_ms119794563918438244od_a_a @ X3 ) )
=> ! [Y5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y5 @ ( set_ms119794563918438244od_a_a @ Y2 ) )
=> ! [Z: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Z @ ( set_ms119794563918438244od_a_a @ Z5 ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X5 @ Y5 ) @ Ns )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y5 @ Z ) @ S )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ ( set_ms119794563918438244od_a_a @ X3 ) )
=> ! [Y5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y5 @ ( set_ms119794563918438244od_a_a @ Y2 ) )
=> ! [Z: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Z @ ( set_ms119794563918438244od_a_a @ Z5 ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X5 @ Y5 ) @ S )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y5 @ Z ) @ Ns )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ ( set_ms119794563918438244od_a_a @ X3 ) )
=> ! [Y5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y5 @ ( set_ms119794563918438244od_a_a @ Y2 ) )
=> ! [Z: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Z @ ( set_ms119794563918438244od_a_a @ Z5 ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X5 @ Y5 ) @ Ns )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y5 @ Z ) @ Ns )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X5 @ Z ) @ Ns ) ) ) ) ) )
=> ( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ X3 @ Y2 ) @ ( multis4249482161328750907od_a_a @ $false @ Ns @ S ) )
=> ( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ Y2 @ Z5 ) @ ( multis4249482161328750907od_a_a @ $false @ Ns @ S ) )
=> ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ X3 @ Z5 ) @ ( multis4249482161328750907od_a_a @ $false @ Ns @ S ) ) ) ) ) ) ) ) ).
% trans_mult2_alt_s_s_local
thf(fact_518_trans__mult2__alt__s__s__local,axiom,
! [X3: multiset_multiset_a,Y2: multiset_multiset_a,Z5: multiset_multiset_a,S: set_Pr79727621955416071iset_a,Ns: set_Pr79727621955416071iset_a] :
( ! [X5: multiset_a] :
( ( member_multiset_a @ X5 @ ( set_mset_multiset_a @ X3 ) )
=> ! [Y5: multiset_a] :
( ( member_multiset_a @ Y5 @ ( set_mset_multiset_a @ Y2 ) )
=> ! [Z: multiset_a] :
( ( member_multiset_a @ Z @ ( set_mset_multiset_a @ Z5 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Y5 ) @ S )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Y5 @ Z ) @ S )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: multiset_a] :
( ( member_multiset_a @ X5 @ ( set_mset_multiset_a @ X3 ) )
=> ! [Y5: multiset_a] :
( ( member_multiset_a @ Y5 @ ( set_mset_multiset_a @ Y2 ) )
=> ! [Z: multiset_a] :
( ( member_multiset_a @ Z @ ( set_mset_multiset_a @ Z5 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Y5 ) @ Ns )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Y5 @ Z ) @ S )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: multiset_a] :
( ( member_multiset_a @ X5 @ ( set_mset_multiset_a @ X3 ) )
=> ! [Y5: multiset_a] :
( ( member_multiset_a @ Y5 @ ( set_mset_multiset_a @ Y2 ) )
=> ! [Z: multiset_a] :
( ( member_multiset_a @ Z @ ( set_mset_multiset_a @ Z5 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Y5 ) @ S )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Y5 @ Z ) @ Ns )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: multiset_a] :
( ( member_multiset_a @ X5 @ ( set_mset_multiset_a @ X3 ) )
=> ! [Y5: multiset_a] :
( ( member_multiset_a @ Y5 @ ( set_mset_multiset_a @ Y2 ) )
=> ! [Z: multiset_a] :
( ( member_multiset_a @ Z @ ( set_mset_multiset_a @ Z5 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Y5 ) @ Ns )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Y5 @ Z ) @ Ns )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Z ) @ Ns ) ) ) ) ) )
=> ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X3 @ Y2 ) @ ( multis2006310815304088152iset_a @ $false @ Ns @ S ) )
=> ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ Y2 @ Z5 ) @ ( multis2006310815304088152iset_a @ $false @ Ns @ S ) )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X3 @ Z5 ) @ ( multis2006310815304088152iset_a @ $false @ Ns @ S ) ) ) ) ) ) ) ) ).
% trans_mult2_alt_s_s_local
thf(fact_519_trans__mult2__alt__s__s__local,axiom,
! [X3: multiset_a,Y2: multiset_a,Z5: multiset_a,S: set_Product_prod_a_a,Ns: set_Product_prod_a_a] :
( ! [X5: a] :
( ( member_a @ X5 @ ( set_mset_a @ X3 ) )
=> ! [Y5: a] :
( ( member_a @ Y5 @ ( set_mset_a @ Y2 ) )
=> ! [Z: a] :
( ( member_a @ Z @ ( set_mset_a @ Z5 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y5 ) @ S )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y5 @ Z ) @ S )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: a] :
( ( member_a @ X5 @ ( set_mset_a @ X3 ) )
=> ! [Y5: a] :
( ( member_a @ Y5 @ ( set_mset_a @ Y2 ) )
=> ! [Z: a] :
( ( member_a @ Z @ ( set_mset_a @ Z5 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y5 ) @ Ns )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y5 @ Z ) @ S )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: a] :
( ( member_a @ X5 @ ( set_mset_a @ X3 ) )
=> ! [Y5: a] :
( ( member_a @ Y5 @ ( set_mset_a @ Y2 ) )
=> ! [Z: a] :
( ( member_a @ Z @ ( set_mset_a @ Z5 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y5 ) @ S )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y5 @ Z ) @ Ns )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: a] :
( ( member_a @ X5 @ ( set_mset_a @ X3 ) )
=> ! [Y5: a] :
( ( member_a @ Y5 @ ( set_mset_a @ Y2 ) )
=> ! [Z: a] :
( ( member_a @ Z @ ( set_mset_a @ Z5 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y5 ) @ Ns )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y5 @ Z ) @ Ns )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Z ) @ Ns ) ) ) ) ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ Y2 ) @ ( multis1686031732355738386_alt_a @ $false @ Ns @ S ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Y2 @ Z5 ) @ ( multis1686031732355738386_alt_a @ $false @ Ns @ S ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ Z5 ) @ ( multis1686031732355738386_alt_a @ $false @ Ns @ S ) ) ) ) ) ) ) ) ).
% trans_mult2_alt_s_s_local
thf(fact_520_trans__mult2__alt__ns__s__local,axiom,
! [X3: multis4781622204924204013iset_a,Y2: multis4781622204924204013iset_a,Z5: multis4781622204924204013iset_a,S: set_Pr3059988933816007879iset_a,Ns: set_Pr3059988933816007879iset_a] :
( ! [X5: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X5 @ ( set_ms2379723421352415844iset_a @ X3 ) )
=> ! [Y5: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Y5 @ ( set_ms2379723421352415844iset_a @ Y2 ) )
=> ! [Z: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Z @ ( set_ms2379723421352415844iset_a @ Z5 ) )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X5 @ Y5 ) @ S )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ Y5 @ Z ) @ S )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X5 @ ( set_ms2379723421352415844iset_a @ X3 ) )
=> ! [Y5: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Y5 @ ( set_ms2379723421352415844iset_a @ Y2 ) )
=> ! [Z: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Z @ ( set_ms2379723421352415844iset_a @ Z5 ) )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X5 @ Y5 ) @ Ns )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ Y5 @ Z ) @ S )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X5 @ ( set_ms2379723421352415844iset_a @ X3 ) )
=> ! [Y5: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Y5 @ ( set_ms2379723421352415844iset_a @ Y2 ) )
=> ! [Z: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Z @ ( set_ms2379723421352415844iset_a @ Z5 ) )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X5 @ Y5 ) @ S )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ Y5 @ Z ) @ Ns )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X5 @ ( set_ms2379723421352415844iset_a @ X3 ) )
=> ! [Y5: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Y5 @ ( set_ms2379723421352415844iset_a @ Y2 ) )
=> ! [Z: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Z @ ( set_ms2379723421352415844iset_a @ Z5 ) )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X5 @ Y5 ) @ Ns )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ Y5 @ Z ) @ Ns )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X5 @ Z ) @ Ns ) ) ) ) ) )
=> ( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ X3 @ Y2 ) @ ( multis1572145089415997755iset_a @ $true @ Ns @ S ) )
=> ( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ Y2 @ Z5 ) @ ( multis1572145089415997755iset_a @ $false @ Ns @ S ) )
=> ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ X3 @ Z5 ) @ ( multis1572145089415997755iset_a @ $false @ Ns @ S ) ) ) ) ) ) ) ) ).
% trans_mult2_alt_ns_s_local
thf(fact_521_trans__mult2__alt__ns__s__local,axiom,
! [X3: multis501812127501805293od_a_a,Y2: multis501812127501805293od_a_a,Z5: multis501812127501805293od_a_a,S: set_Pr8600417178894128327od_a_a,Ns: set_Pr8600417178894128327od_a_a] :
( ! [X5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ ( set_ms119794563918438244od_a_a @ X3 ) )
=> ! [Y5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y5 @ ( set_ms119794563918438244od_a_a @ Y2 ) )
=> ! [Z: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Z @ ( set_ms119794563918438244od_a_a @ Z5 ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X5 @ Y5 ) @ S )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y5 @ Z ) @ S )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ ( set_ms119794563918438244od_a_a @ X3 ) )
=> ! [Y5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y5 @ ( set_ms119794563918438244od_a_a @ Y2 ) )
=> ! [Z: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Z @ ( set_ms119794563918438244od_a_a @ Z5 ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X5 @ Y5 ) @ Ns )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y5 @ Z ) @ S )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ ( set_ms119794563918438244od_a_a @ X3 ) )
=> ! [Y5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y5 @ ( set_ms119794563918438244od_a_a @ Y2 ) )
=> ! [Z: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Z @ ( set_ms119794563918438244od_a_a @ Z5 ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X5 @ Y5 ) @ S )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y5 @ Z ) @ Ns )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ ( set_ms119794563918438244od_a_a @ X3 ) )
=> ! [Y5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y5 @ ( set_ms119794563918438244od_a_a @ Y2 ) )
=> ! [Z: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Z @ ( set_ms119794563918438244od_a_a @ Z5 ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X5 @ Y5 ) @ Ns )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y5 @ Z ) @ Ns )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X5 @ Z ) @ Ns ) ) ) ) ) )
=> ( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ X3 @ Y2 ) @ ( multis4249482161328750907od_a_a @ $true @ Ns @ S ) )
=> ( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ Y2 @ Z5 ) @ ( multis4249482161328750907od_a_a @ $false @ Ns @ S ) )
=> ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ X3 @ Z5 ) @ ( multis4249482161328750907od_a_a @ $false @ Ns @ S ) ) ) ) ) ) ) ) ).
% trans_mult2_alt_ns_s_local
thf(fact_522_trans__mult2__alt__ns__s__local,axiom,
! [X3: multiset_multiset_a,Y2: multiset_multiset_a,Z5: multiset_multiset_a,S: set_Pr79727621955416071iset_a,Ns: set_Pr79727621955416071iset_a] :
( ! [X5: multiset_a] :
( ( member_multiset_a @ X5 @ ( set_mset_multiset_a @ X3 ) )
=> ! [Y5: multiset_a] :
( ( member_multiset_a @ Y5 @ ( set_mset_multiset_a @ Y2 ) )
=> ! [Z: multiset_a] :
( ( member_multiset_a @ Z @ ( set_mset_multiset_a @ Z5 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Y5 ) @ S )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Y5 @ Z ) @ S )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: multiset_a] :
( ( member_multiset_a @ X5 @ ( set_mset_multiset_a @ X3 ) )
=> ! [Y5: multiset_a] :
( ( member_multiset_a @ Y5 @ ( set_mset_multiset_a @ Y2 ) )
=> ! [Z: multiset_a] :
( ( member_multiset_a @ Z @ ( set_mset_multiset_a @ Z5 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Y5 ) @ Ns )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Y5 @ Z ) @ S )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: multiset_a] :
( ( member_multiset_a @ X5 @ ( set_mset_multiset_a @ X3 ) )
=> ! [Y5: multiset_a] :
( ( member_multiset_a @ Y5 @ ( set_mset_multiset_a @ Y2 ) )
=> ! [Z: multiset_a] :
( ( member_multiset_a @ Z @ ( set_mset_multiset_a @ Z5 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Y5 ) @ S )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Y5 @ Z ) @ Ns )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: multiset_a] :
( ( member_multiset_a @ X5 @ ( set_mset_multiset_a @ X3 ) )
=> ! [Y5: multiset_a] :
( ( member_multiset_a @ Y5 @ ( set_mset_multiset_a @ Y2 ) )
=> ! [Z: multiset_a] :
( ( member_multiset_a @ Z @ ( set_mset_multiset_a @ Z5 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Y5 ) @ Ns )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Y5 @ Z ) @ Ns )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Z ) @ Ns ) ) ) ) ) )
=> ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X3 @ Y2 ) @ ( multis2006310815304088152iset_a @ $true @ Ns @ S ) )
=> ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ Y2 @ Z5 ) @ ( multis2006310815304088152iset_a @ $false @ Ns @ S ) )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X3 @ Z5 ) @ ( multis2006310815304088152iset_a @ $false @ Ns @ S ) ) ) ) ) ) ) ) ).
% trans_mult2_alt_ns_s_local
thf(fact_523_trans__mult2__alt__ns__s__local,axiom,
! [X3: multiset_a,Y2: multiset_a,Z5: multiset_a,S: set_Product_prod_a_a,Ns: set_Product_prod_a_a] :
( ! [X5: a] :
( ( member_a @ X5 @ ( set_mset_a @ X3 ) )
=> ! [Y5: a] :
( ( member_a @ Y5 @ ( set_mset_a @ Y2 ) )
=> ! [Z: a] :
( ( member_a @ Z @ ( set_mset_a @ Z5 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y5 ) @ S )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y5 @ Z ) @ S )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: a] :
( ( member_a @ X5 @ ( set_mset_a @ X3 ) )
=> ! [Y5: a] :
( ( member_a @ Y5 @ ( set_mset_a @ Y2 ) )
=> ! [Z: a] :
( ( member_a @ Z @ ( set_mset_a @ Z5 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y5 ) @ Ns )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y5 @ Z ) @ S )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: a] :
( ( member_a @ X5 @ ( set_mset_a @ X3 ) )
=> ! [Y5: a] :
( ( member_a @ Y5 @ ( set_mset_a @ Y2 ) )
=> ! [Z: a] :
( ( member_a @ Z @ ( set_mset_a @ Z5 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y5 ) @ S )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y5 @ Z ) @ Ns )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: a] :
( ( member_a @ X5 @ ( set_mset_a @ X3 ) )
=> ! [Y5: a] :
( ( member_a @ Y5 @ ( set_mset_a @ Y2 ) )
=> ! [Z: a] :
( ( member_a @ Z @ ( set_mset_a @ Z5 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y5 ) @ Ns )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y5 @ Z ) @ Ns )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Z ) @ Ns ) ) ) ) ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ Y2 ) @ ( multis1686031732355738386_alt_a @ $true @ Ns @ S ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Y2 @ Z5 ) @ ( multis1686031732355738386_alt_a @ $false @ Ns @ S ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ Z5 ) @ ( multis1686031732355738386_alt_a @ $false @ Ns @ S ) ) ) ) ) ) ) ) ).
% trans_mult2_alt_ns_s_local
thf(fact_524_trans__mult2__alt__s__ns__local,axiom,
! [X3: multis4781622204924204013iset_a,Y2: multis4781622204924204013iset_a,Z5: multis4781622204924204013iset_a,S: set_Pr3059988933816007879iset_a,Ns: set_Pr3059988933816007879iset_a] :
( ! [X5: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X5 @ ( set_ms2379723421352415844iset_a @ X3 ) )
=> ! [Y5: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Y5 @ ( set_ms2379723421352415844iset_a @ Y2 ) )
=> ! [Z: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Z @ ( set_ms2379723421352415844iset_a @ Z5 ) )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X5 @ Y5 ) @ S )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ Y5 @ Z ) @ S )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X5 @ ( set_ms2379723421352415844iset_a @ X3 ) )
=> ! [Y5: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Y5 @ ( set_ms2379723421352415844iset_a @ Y2 ) )
=> ! [Z: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Z @ ( set_ms2379723421352415844iset_a @ Z5 ) )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X5 @ Y5 ) @ Ns )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ Y5 @ Z ) @ S )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X5 @ ( set_ms2379723421352415844iset_a @ X3 ) )
=> ! [Y5: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Y5 @ ( set_ms2379723421352415844iset_a @ Y2 ) )
=> ! [Z: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Z @ ( set_ms2379723421352415844iset_a @ Z5 ) )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X5 @ Y5 ) @ S )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ Y5 @ Z ) @ Ns )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X5 @ ( set_ms2379723421352415844iset_a @ X3 ) )
=> ! [Y5: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Y5 @ ( set_ms2379723421352415844iset_a @ Y2 ) )
=> ! [Z: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Z @ ( set_ms2379723421352415844iset_a @ Z5 ) )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X5 @ Y5 ) @ Ns )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ Y5 @ Z ) @ Ns )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X5 @ Z ) @ Ns ) ) ) ) ) )
=> ( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ X3 @ Y2 ) @ ( multis1572145089415997755iset_a @ $false @ Ns @ S ) )
=> ( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ Y2 @ Z5 ) @ ( multis1572145089415997755iset_a @ $true @ Ns @ S ) )
=> ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ X3 @ Z5 ) @ ( multis1572145089415997755iset_a @ $false @ Ns @ S ) ) ) ) ) ) ) ) ).
% trans_mult2_alt_s_ns_local
thf(fact_525_trans__mult2__alt__s__ns__local,axiom,
! [X3: multis501812127501805293od_a_a,Y2: multis501812127501805293od_a_a,Z5: multis501812127501805293od_a_a,S: set_Pr8600417178894128327od_a_a,Ns: set_Pr8600417178894128327od_a_a] :
( ! [X5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ ( set_ms119794563918438244od_a_a @ X3 ) )
=> ! [Y5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y5 @ ( set_ms119794563918438244od_a_a @ Y2 ) )
=> ! [Z: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Z @ ( set_ms119794563918438244od_a_a @ Z5 ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X5 @ Y5 ) @ S )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y5 @ Z ) @ S )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ ( set_ms119794563918438244od_a_a @ X3 ) )
=> ! [Y5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y5 @ ( set_ms119794563918438244od_a_a @ Y2 ) )
=> ! [Z: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Z @ ( set_ms119794563918438244od_a_a @ Z5 ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X5 @ Y5 ) @ Ns )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y5 @ Z ) @ S )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ ( set_ms119794563918438244od_a_a @ X3 ) )
=> ! [Y5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y5 @ ( set_ms119794563918438244od_a_a @ Y2 ) )
=> ! [Z: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Z @ ( set_ms119794563918438244od_a_a @ Z5 ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X5 @ Y5 ) @ S )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y5 @ Z ) @ Ns )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ ( set_ms119794563918438244od_a_a @ X3 ) )
=> ! [Y5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y5 @ ( set_ms119794563918438244od_a_a @ Y2 ) )
=> ! [Z: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Z @ ( set_ms119794563918438244od_a_a @ Z5 ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X5 @ Y5 ) @ Ns )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y5 @ Z ) @ Ns )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X5 @ Z ) @ Ns ) ) ) ) ) )
=> ( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ X3 @ Y2 ) @ ( multis4249482161328750907od_a_a @ $false @ Ns @ S ) )
=> ( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ Y2 @ Z5 ) @ ( multis4249482161328750907od_a_a @ $true @ Ns @ S ) )
=> ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ X3 @ Z5 ) @ ( multis4249482161328750907od_a_a @ $false @ Ns @ S ) ) ) ) ) ) ) ) ).
% trans_mult2_alt_s_ns_local
thf(fact_526_trans__mult2__alt__s__ns__local,axiom,
! [X3: multiset_multiset_a,Y2: multiset_multiset_a,Z5: multiset_multiset_a,S: set_Pr79727621955416071iset_a,Ns: set_Pr79727621955416071iset_a] :
( ! [X5: multiset_a] :
( ( member_multiset_a @ X5 @ ( set_mset_multiset_a @ X3 ) )
=> ! [Y5: multiset_a] :
( ( member_multiset_a @ Y5 @ ( set_mset_multiset_a @ Y2 ) )
=> ! [Z: multiset_a] :
( ( member_multiset_a @ Z @ ( set_mset_multiset_a @ Z5 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Y5 ) @ S )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Y5 @ Z ) @ S )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: multiset_a] :
( ( member_multiset_a @ X5 @ ( set_mset_multiset_a @ X3 ) )
=> ! [Y5: multiset_a] :
( ( member_multiset_a @ Y5 @ ( set_mset_multiset_a @ Y2 ) )
=> ! [Z: multiset_a] :
( ( member_multiset_a @ Z @ ( set_mset_multiset_a @ Z5 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Y5 ) @ Ns )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Y5 @ Z ) @ S )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: multiset_a] :
( ( member_multiset_a @ X5 @ ( set_mset_multiset_a @ X3 ) )
=> ! [Y5: multiset_a] :
( ( member_multiset_a @ Y5 @ ( set_mset_multiset_a @ Y2 ) )
=> ! [Z: multiset_a] :
( ( member_multiset_a @ Z @ ( set_mset_multiset_a @ Z5 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Y5 ) @ S )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Y5 @ Z ) @ Ns )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: multiset_a] :
( ( member_multiset_a @ X5 @ ( set_mset_multiset_a @ X3 ) )
=> ! [Y5: multiset_a] :
( ( member_multiset_a @ Y5 @ ( set_mset_multiset_a @ Y2 ) )
=> ! [Z: multiset_a] :
( ( member_multiset_a @ Z @ ( set_mset_multiset_a @ Z5 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Y5 ) @ Ns )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Y5 @ Z ) @ Ns )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Z ) @ Ns ) ) ) ) ) )
=> ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X3 @ Y2 ) @ ( multis2006310815304088152iset_a @ $false @ Ns @ S ) )
=> ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ Y2 @ Z5 ) @ ( multis2006310815304088152iset_a @ $true @ Ns @ S ) )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X3 @ Z5 ) @ ( multis2006310815304088152iset_a @ $false @ Ns @ S ) ) ) ) ) ) ) ) ).
% trans_mult2_alt_s_ns_local
thf(fact_527_trans__mult2__alt__s__ns__local,axiom,
! [X3: multiset_a,Y2: multiset_a,Z5: multiset_a,S: set_Product_prod_a_a,Ns: set_Product_prod_a_a] :
( ! [X5: a] :
( ( member_a @ X5 @ ( set_mset_a @ X3 ) )
=> ! [Y5: a] :
( ( member_a @ Y5 @ ( set_mset_a @ Y2 ) )
=> ! [Z: a] :
( ( member_a @ Z @ ( set_mset_a @ Z5 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y5 ) @ S )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y5 @ Z ) @ S )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: a] :
( ( member_a @ X5 @ ( set_mset_a @ X3 ) )
=> ! [Y5: a] :
( ( member_a @ Y5 @ ( set_mset_a @ Y2 ) )
=> ! [Z: a] :
( ( member_a @ Z @ ( set_mset_a @ Z5 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y5 ) @ Ns )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y5 @ Z ) @ S )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: a] :
( ( member_a @ X5 @ ( set_mset_a @ X3 ) )
=> ! [Y5: a] :
( ( member_a @ Y5 @ ( set_mset_a @ Y2 ) )
=> ! [Z: a] :
( ( member_a @ Z @ ( set_mset_a @ Z5 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y5 ) @ S )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y5 @ Z ) @ Ns )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: a] :
( ( member_a @ X5 @ ( set_mset_a @ X3 ) )
=> ! [Y5: a] :
( ( member_a @ Y5 @ ( set_mset_a @ Y2 ) )
=> ! [Z: a] :
( ( member_a @ Z @ ( set_mset_a @ Z5 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y5 ) @ Ns )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y5 @ Z ) @ Ns )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Z ) @ Ns ) ) ) ) ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ Y2 ) @ ( multis1686031732355738386_alt_a @ $false @ Ns @ S ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Y2 @ Z5 ) @ ( multis1686031732355738386_alt_a @ $true @ Ns @ S ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ Z5 ) @ ( multis1686031732355738386_alt_a @ $false @ Ns @ S ) ) ) ) ) ) ) ) ).
% trans_mult2_alt_s_ns_local
thf(fact_528_trans__mult2__alt__ns__ns__local,axiom,
! [X3: multis4781622204924204013iset_a,Y2: multis4781622204924204013iset_a,Z5: multis4781622204924204013iset_a,S: set_Pr3059988933816007879iset_a,Ns: set_Pr3059988933816007879iset_a] :
( ! [X5: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X5 @ ( set_ms2379723421352415844iset_a @ X3 ) )
=> ! [Y5: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Y5 @ ( set_ms2379723421352415844iset_a @ Y2 ) )
=> ! [Z: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Z @ ( set_ms2379723421352415844iset_a @ Z5 ) )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X5 @ Y5 ) @ S )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ Y5 @ Z ) @ S )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X5 @ ( set_ms2379723421352415844iset_a @ X3 ) )
=> ! [Y5: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Y5 @ ( set_ms2379723421352415844iset_a @ Y2 ) )
=> ! [Z: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Z @ ( set_ms2379723421352415844iset_a @ Z5 ) )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X5 @ Y5 ) @ Ns )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ Y5 @ Z ) @ S )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X5 @ ( set_ms2379723421352415844iset_a @ X3 ) )
=> ! [Y5: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Y5 @ ( set_ms2379723421352415844iset_a @ Y2 ) )
=> ! [Z: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Z @ ( set_ms2379723421352415844iset_a @ Z5 ) )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X5 @ Y5 ) @ S )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ Y5 @ Z ) @ Ns )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X5 @ ( set_ms2379723421352415844iset_a @ X3 ) )
=> ! [Y5: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Y5 @ ( set_ms2379723421352415844iset_a @ Y2 ) )
=> ! [Z: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Z @ ( set_ms2379723421352415844iset_a @ Z5 ) )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X5 @ Y5 ) @ Ns )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ Y5 @ Z ) @ Ns )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X5 @ Z ) @ Ns ) ) ) ) ) )
=> ( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ X3 @ Y2 ) @ ( multis1572145089415997755iset_a @ $true @ Ns @ S ) )
=> ( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ Y2 @ Z5 ) @ ( multis1572145089415997755iset_a @ $true @ Ns @ S ) )
=> ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ X3 @ Z5 ) @ ( multis1572145089415997755iset_a @ $true @ Ns @ S ) ) ) ) ) ) ) ) ).
% trans_mult2_alt_ns_ns_local
thf(fact_529_trans__mult2__alt__ns__ns__local,axiom,
! [X3: multis501812127501805293od_a_a,Y2: multis501812127501805293od_a_a,Z5: multis501812127501805293od_a_a,S: set_Pr8600417178894128327od_a_a,Ns: set_Pr8600417178894128327od_a_a] :
( ! [X5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ ( set_ms119794563918438244od_a_a @ X3 ) )
=> ! [Y5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y5 @ ( set_ms119794563918438244od_a_a @ Y2 ) )
=> ! [Z: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Z @ ( set_ms119794563918438244od_a_a @ Z5 ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X5 @ Y5 ) @ S )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y5 @ Z ) @ S )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ ( set_ms119794563918438244od_a_a @ X3 ) )
=> ! [Y5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y5 @ ( set_ms119794563918438244od_a_a @ Y2 ) )
=> ! [Z: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Z @ ( set_ms119794563918438244od_a_a @ Z5 ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X5 @ Y5 ) @ Ns )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y5 @ Z ) @ S )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ ( set_ms119794563918438244od_a_a @ X3 ) )
=> ! [Y5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y5 @ ( set_ms119794563918438244od_a_a @ Y2 ) )
=> ! [Z: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Z @ ( set_ms119794563918438244od_a_a @ Z5 ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X5 @ Y5 ) @ S )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y5 @ Z ) @ Ns )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ ( set_ms119794563918438244od_a_a @ X3 ) )
=> ! [Y5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y5 @ ( set_ms119794563918438244od_a_a @ Y2 ) )
=> ! [Z: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Z @ ( set_ms119794563918438244od_a_a @ Z5 ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X5 @ Y5 ) @ Ns )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y5 @ Z ) @ Ns )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X5 @ Z ) @ Ns ) ) ) ) ) )
=> ( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ X3 @ Y2 ) @ ( multis4249482161328750907od_a_a @ $true @ Ns @ S ) )
=> ( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ Y2 @ Z5 ) @ ( multis4249482161328750907od_a_a @ $true @ Ns @ S ) )
=> ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ X3 @ Z5 ) @ ( multis4249482161328750907od_a_a @ $true @ Ns @ S ) ) ) ) ) ) ) ) ).
% trans_mult2_alt_ns_ns_local
thf(fact_530_trans__mult2__alt__ns__ns__local,axiom,
! [X3: multiset_multiset_a,Y2: multiset_multiset_a,Z5: multiset_multiset_a,S: set_Pr79727621955416071iset_a,Ns: set_Pr79727621955416071iset_a] :
( ! [X5: multiset_a] :
( ( member_multiset_a @ X5 @ ( set_mset_multiset_a @ X3 ) )
=> ! [Y5: multiset_a] :
( ( member_multiset_a @ Y5 @ ( set_mset_multiset_a @ Y2 ) )
=> ! [Z: multiset_a] :
( ( member_multiset_a @ Z @ ( set_mset_multiset_a @ Z5 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Y5 ) @ S )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Y5 @ Z ) @ S )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: multiset_a] :
( ( member_multiset_a @ X5 @ ( set_mset_multiset_a @ X3 ) )
=> ! [Y5: multiset_a] :
( ( member_multiset_a @ Y5 @ ( set_mset_multiset_a @ Y2 ) )
=> ! [Z: multiset_a] :
( ( member_multiset_a @ Z @ ( set_mset_multiset_a @ Z5 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Y5 ) @ Ns )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Y5 @ Z ) @ S )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: multiset_a] :
( ( member_multiset_a @ X5 @ ( set_mset_multiset_a @ X3 ) )
=> ! [Y5: multiset_a] :
( ( member_multiset_a @ Y5 @ ( set_mset_multiset_a @ Y2 ) )
=> ! [Z: multiset_a] :
( ( member_multiset_a @ Z @ ( set_mset_multiset_a @ Z5 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Y5 ) @ S )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Y5 @ Z ) @ Ns )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: multiset_a] :
( ( member_multiset_a @ X5 @ ( set_mset_multiset_a @ X3 ) )
=> ! [Y5: multiset_a] :
( ( member_multiset_a @ Y5 @ ( set_mset_multiset_a @ Y2 ) )
=> ! [Z: multiset_a] :
( ( member_multiset_a @ Z @ ( set_mset_multiset_a @ Z5 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Y5 ) @ Ns )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Y5 @ Z ) @ Ns )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Z ) @ Ns ) ) ) ) ) )
=> ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X3 @ Y2 ) @ ( multis2006310815304088152iset_a @ $true @ Ns @ S ) )
=> ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ Y2 @ Z5 ) @ ( multis2006310815304088152iset_a @ $true @ Ns @ S ) )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X3 @ Z5 ) @ ( multis2006310815304088152iset_a @ $true @ Ns @ S ) ) ) ) ) ) ) ) ).
% trans_mult2_alt_ns_ns_local
thf(fact_531_trans__mult2__alt__ns__ns__local,axiom,
! [X3: multiset_a,Y2: multiset_a,Z5: multiset_a,S: set_Product_prod_a_a,Ns: set_Product_prod_a_a] :
( ! [X5: a] :
( ( member_a @ X5 @ ( set_mset_a @ X3 ) )
=> ! [Y5: a] :
( ( member_a @ Y5 @ ( set_mset_a @ Y2 ) )
=> ! [Z: a] :
( ( member_a @ Z @ ( set_mset_a @ Z5 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y5 ) @ S )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y5 @ Z ) @ S )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: a] :
( ( member_a @ X5 @ ( set_mset_a @ X3 ) )
=> ! [Y5: a] :
( ( member_a @ Y5 @ ( set_mset_a @ Y2 ) )
=> ! [Z: a] :
( ( member_a @ Z @ ( set_mset_a @ Z5 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y5 ) @ Ns )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y5 @ Z ) @ S )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: a] :
( ( member_a @ X5 @ ( set_mset_a @ X3 ) )
=> ! [Y5: a] :
( ( member_a @ Y5 @ ( set_mset_a @ Y2 ) )
=> ! [Z: a] :
( ( member_a @ Z @ ( set_mset_a @ Z5 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y5 ) @ S )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y5 @ Z ) @ Ns )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Z ) @ S ) ) ) ) ) )
=> ( ! [X5: a] :
( ( member_a @ X5 @ ( set_mset_a @ X3 ) )
=> ! [Y5: a] :
( ( member_a @ Y5 @ ( set_mset_a @ Y2 ) )
=> ! [Z: a] :
( ( member_a @ Z @ ( set_mset_a @ Z5 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y5 ) @ Ns )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y5 @ Z ) @ Ns )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Z ) @ Ns ) ) ) ) ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ Y2 ) @ ( multis1686031732355738386_alt_a @ $true @ Ns @ S ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Y2 @ Z5 ) @ ( multis1686031732355738386_alt_a @ $true @ Ns @ S ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ Z5 ) @ ( multis1686031732355738386_alt_a @ $true @ Ns @ S ) ) ) ) ) ) ) ) ).
% trans_mult2_alt_ns_ns_local
thf(fact_532_multiset__induct__max,axiom,
! [P: multiset_nat > $o,M: multiset_nat] :
( ( P @ zero_z7348594199698428585et_nat )
=> ( ! [X5: nat,M3: multiset_nat] :
( ( P @ M3 )
=> ( ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_mset_nat @ M3 ) )
=> ( ord_less_eq_nat @ Xa @ X5 ) )
=> ( P @ ( add_mset_nat @ X5 @ M3 ) ) ) )
=> ( P @ M ) ) ) ).
% multiset_induct_max
thf(fact_533_multiset__induct__min,axiom,
! [P: multiset_nat > $o,M: multiset_nat] :
( ( P @ zero_z7348594199698428585et_nat )
=> ( ! [X5: nat,M3: multiset_nat] :
( ( P @ M3 )
=> ( ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_mset_nat @ M3 ) )
=> ( ord_less_eq_nat @ X5 @ Xa ) )
=> ( P @ ( add_mset_nat @ X5 @ M3 ) ) ) )
=> ( P @ M ) ) ) ).
% multiset_induct_min
thf(fact_534_multi__member__skip,axiom,
! [X2: produc6518373309651786023iset_a,XS: multis4781622204924204013iset_a,Y: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X2 @ ( set_ms2379723421352415844iset_a @ XS ) )
=> ( member5199237121806060112iset_a @ X2 @ ( set_ms2379723421352415844iset_a @ ( plus_p1053901119268525860iset_a @ ( add_ms5348986034529679365iset_a @ Y @ zero_z4591712717340762532iset_a ) @ XS ) ) ) ) ).
% multi_member_skip
thf(fact_535_multi__member__skip,axiom,
! [X2: product_prod_a_a,XS: multis501812127501805293od_a_a,Y: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ ( set_ms119794563918438244od_a_a @ XS ) )
=> ( member1426531477525435216od_a_a @ X2 @ ( set_ms119794563918438244od_a_a @ ( plus_p6967097483979887908od_a_a @ ( add_ms8655138167283798533od_a_a @ Y @ zero_z3024502163831676836od_a_a ) @ XS ) ) ) ) ).
% multi_member_skip
thf(fact_536_multi__member__skip,axiom,
! [X2: a,XS: multiset_a,Y: a] :
( ( member_a @ X2 @ ( set_mset_a @ XS ) )
=> ( member_a @ X2 @ ( set_mset_a @ ( plus_plus_multiset_a @ ( add_mset_a @ Y @ zero_zero_multiset_a ) @ XS ) ) ) ) ).
% multi_member_skip
thf(fact_537_multi__member__this,axiom,
! [X2: produc6518373309651786023iset_a,XS: multis4781622204924204013iset_a] : ( member5199237121806060112iset_a @ X2 @ ( set_ms2379723421352415844iset_a @ ( plus_p1053901119268525860iset_a @ ( add_ms5348986034529679365iset_a @ X2 @ zero_z4591712717340762532iset_a ) @ XS ) ) ) ).
% multi_member_this
thf(fact_538_multi__member__this,axiom,
! [X2: product_prod_a_a,XS: multis501812127501805293od_a_a] : ( member1426531477525435216od_a_a @ X2 @ ( set_ms119794563918438244od_a_a @ ( plus_p6967097483979887908od_a_a @ ( add_ms8655138167283798533od_a_a @ X2 @ zero_z3024502163831676836od_a_a ) @ XS ) ) ) ).
% multi_member_this
thf(fact_539_multi__member__this,axiom,
! [X2: a,XS: multiset_a] : ( member_a @ X2 @ ( set_mset_a @ ( plus_plus_multiset_a @ ( add_mset_a @ X2 @ zero_zero_multiset_a ) @ XS ) ) ) ).
% multi_member_this
thf(fact_540_multpw__refl_H,axiom,
! [Ns: set_Product_prod_a_a,A3: multiset_a] :
( ( locally_refl_a @ Ns @ A3 )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A3 @ A3 ) @ ( multis2373687813370458242ltpw_a @ Ns ) ) ) ).
% multpw_refl'
thf(fact_541_ns__mul__ext__refl__local,axiom,
! [Ns: set_Product_prod_a_a,A3: multiset_a,S: set_Product_prod_a_a] :
( ( locally_refl_a @ Ns @ A3 )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A3 @ A3 ) @ ( multis4023584895248362547_ext_a @ Ns @ S ) ) ) ).
% ns_mul_ext_refl_local
thf(fact_542_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 ) )
=> ~ ! [X5: multiset_a,Y5: multiset_a,X6: multiset_multiset_a] :
( ( A13
= ( add_mset_multiset_a @ X5 @ X6 ) )
=> ! [Y7: multiset_multiset_a] :
( ( A24
= ( add_mset_multiset_a @ Y5 @ Y7 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Y5 ) @ Ns )
=> ~ ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X6 @ Y7 ) @ ( multis451435756853798856iset_a @ Ns ) ) ) ) ) ) ) ).
% multpw.cases
thf(fact_543_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 ) )
=> ~ ! [X5: a,Y5: a,X6: multiset_a] :
( ( A13
= ( add_mset_a @ X5 @ X6 ) )
=> ! [Y7: multiset_a] :
( ( A24
= ( add_mset_a @ Y5 @ Y7 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y5 ) @ Ns )
=> ~ ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X6 @ Y7 ) @ ( multis2373687813370458242ltpw_a @ Ns ) ) ) ) ) ) ) ).
% multpw.cases
thf(fact_544_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 ) )
| ? [X4: multiset_a,Y4: multiset_a,X7: multiset_multiset_a,Y8: multiset_multiset_a] :
( ( A13
= ( add_mset_multiset_a @ X4 @ X7 ) )
& ( A24
= ( add_mset_multiset_a @ Y4 @ Y8 ) )
& ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X4 @ Y4 ) @ Ns )
& ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X7 @ Y8 ) @ ( multis451435756853798856iset_a @ Ns ) ) ) ) ) ).
% multpw.simps
thf(fact_545_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 ) )
| ? [X4: a,Y4: a,X7: multiset_a,Y8: multiset_a] :
( ( A13
= ( add_mset_a @ X4 @ X7 ) )
& ( A24
= ( add_mset_a @ Y4 @ Y8 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ Ns )
& ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X7 @ Y8 ) @ ( multis2373687813370458242ltpw_a @ Ns ) ) ) ) ) ).
% multpw.simps
thf(fact_546_multpw__single,axiom,
! [X2: multiset_a,Y: multiset_a,Ns: set_Pr79727621955416071iset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X2 @ Y ) @ Ns )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ ( add_mset_multiset_a @ X2 @ zero_z7799948378220188993iset_a ) @ ( add_mset_multiset_a @ Y @ zero_z7799948378220188993iset_a ) ) @ ( multis451435756853798856iset_a @ Ns ) ) ) ).
% multpw_single
thf(fact_547_multpw__single,axiom,
! [X2: a,Y: a,Ns: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ Ns )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( add_mset_a @ X2 @ zero_zero_multiset_a ) @ ( add_mset_a @ Y @ zero_zero_multiset_a ) ) @ ( multis2373687813370458242ltpw_a @ Ns ) ) ) ).
% multpw_single
thf(fact_548_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_549_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_550_locally__refl__def,axiom,
( locall5617794901334676992iset_a
= ( ^ [R4: set_Pr3059988933816007879iset_a,A6: multis4781622204924204013iset_a] :
! [A5: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ A5 @ ( set_ms2379723421352415844iset_a @ A6 ) )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ A5 @ A5 ) @ R4 ) ) ) ) ).
% locally_refl_def
thf(fact_551_locally__refl__def,axiom,
( locall7179175945658863360od_a_a
= ( ^ [R4: set_Pr8600417178894128327od_a_a,A6: multis501812127501805293od_a_a] :
! [A5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A5 @ ( set_ms119794563918438244od_a_a @ A6 ) )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A5 @ A5 ) @ R4 ) ) ) ) ).
% locally_refl_def
thf(fact_552_locally__refl__def,axiom,
( locall5412118405798798493iset_a
= ( ^ [R4: set_Pr79727621955416071iset_a,A6: multiset_multiset_a] :
! [A5: multiset_a] :
( ( member_multiset_a @ A5 @ ( set_mset_multiset_a @ A6 ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A5 @ A5 ) @ R4 ) ) ) ) ).
% locally_refl_def
thf(fact_553_locally__refl__def,axiom,
( locally_refl_a
= ( ^ [R4: set_Product_prod_a_a,A6: multiset_a] :
! [A5: a] :
( ( member_a @ A5 @ ( set_mset_a @ A6 ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A5 @ A5 ) @ R4 ) ) ) ) ).
% locally_refl_def
thf(fact_554_lr__trans__l,axiom,
! [R3: set_Product_prod_a_a,A3: multiset_a,B2: multiset_a] :
( ( locally_refl_a @ R3 @ ( plus_plus_multiset_a @ A3 @ B2 ) )
=> ( locally_refl_a @ R3 @ A3 ) ) ).
% lr_trans_l
thf(fact_555_lr__trans__r,axiom,
! [R3: set_Product_prod_a_a,A3: multiset_a,B2: multiset_a] :
( ( locally_refl_a @ R3 @ ( plus_plus_multiset_a @ A3 @ B2 ) )
=> ( locally_refl_a @ R3 @ B2 ) ) ).
% lr_trans_r
thf(fact_556_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 )
= ( ! [X4: multiset_a] :
( ( member_multiset_a @ X4 @ ( set_mset_multiset_a @ M ) )
=> ( X4 = zero_zero_multiset_a ) ) ) ) ).
% subset_mset.sum_mset_0_iff
thf(fact_557_IdD,axiom,
! [A2: multiset_a,B: multiset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A2 @ B ) @ id_multiset_a )
=> ( A2 = B ) ) ).
% IdD
thf(fact_558_IdD,axiom,
! [A2: a,B: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B ) @ id_a )
=> ( A2 = B ) ) ).
% IdD
thf(fact_559_ns__s__mul__ext__union__multiset__l,axiom,
! [A3: multis4781622204924204013iset_a,B2: multis4781622204924204013iset_a,Ns: set_Pr3059988933816007879iset_a,S: set_Pr3059988933816007879iset_a,C2: multis4781622204924204013iset_a,D2: multis4781622204924204013iset_a] :
( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ A3 @ B2 ) @ ( multis2244641948333282396iset_a @ Ns @ S ) )
=> ( ( C2 != zero_z4591712717340762532iset_a )
=> ( ! [D3: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ D3 @ ( set_ms2379723421352415844iset_a @ D2 ) )
=> ? [C5: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ C5 @ ( set_ms2379723421352415844iset_a @ C2 ) )
& ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ C5 @ D3 ) @ S ) ) )
=> ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ ( plus_p1053901119268525860iset_a @ A3 @ C2 ) @ ( plus_p1053901119268525860iset_a @ B2 @ D2 ) ) @ ( multis6482401367445528638iset_a @ Ns @ S ) ) ) ) ) ).
% ns_s_mul_ext_union_multiset_l
thf(fact_560_ns__s__mul__ext__union__multiset__l,axiom,
! [A3: multis501812127501805293od_a_a,B2: multis501812127501805293od_a_a,Ns: set_Pr8600417178894128327od_a_a,S: set_Pr8600417178894128327od_a_a,C2: multis501812127501805293od_a_a,D2: multis501812127501805293od_a_a] :
( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ A3 @ B2 ) @ ( multis1178328011278088540od_a_a @ Ns @ S ) )
=> ( ( C2 != zero_z3024502163831676836od_a_a )
=> ( ! [D3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ D3 @ ( set_ms119794563918438244od_a_a @ D2 ) )
=> ? [C5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C5 @ ( set_ms119794563918438244od_a_a @ C2 ) )
& ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ C5 @ D3 ) @ S ) ) )
=> ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ ( plus_p6967097483979887908od_a_a @ A3 @ C2 ) @ ( plus_p6967097483979887908od_a_a @ B2 @ D2 ) ) @ ( multis6523615221270609726od_a_a @ Ns @ S ) ) ) ) ) ).
% ns_s_mul_ext_union_multiset_l
thf(fact_561_ns__s__mul__ext__union__multiset__l,axiom,
! [A3: multiset_multiset_a,B2: multiset_multiset_a,Ns: set_Pr79727621955416071iset_a,S: set_Pr79727621955416071iset_a,C2: multiset_multiset_a,D2: multiset_multiset_a] :
( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ A3 @ B2 ) @ ( multis8112516946959054073iset_a @ Ns @ S ) )
=> ( ( C2 != zero_z7799948378220188993iset_a )
=> ( ! [D3: multiset_a] :
( ( member_multiset_a @ D3 @ ( set_mset_multiset_a @ D2 ) )
=> ? [C5: multiset_a] :
( ( member_multiset_a @ C5 @ ( set_mset_multiset_a @ C2 ) )
& ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ C5 @ D3 ) @ S ) ) )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ ( plus_p6738641960240532161iset_a @ A3 @ C2 ) @ ( plus_p6738641960240532161iset_a @ B2 @ D2 ) ) @ ( multis7166037729339985883iset_a @ Ns @ S ) ) ) ) ) ).
% ns_s_mul_ext_union_multiset_l
thf(fact_562_ns__s__mul__ext__union__multiset__l,axiom,
! [A3: multiset_a,B2: multiset_a,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a,C2: multiset_a,D2: multiset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A3 @ B2 ) @ ( multis4023584895248362547_ext_a @ Ns @ S ) )
=> ( ( C2 != zero_zero_multiset_a )
=> ( ! [D3: a] :
( ( member_a @ D3 @ ( set_mset_a @ D2 ) )
=> ? [C5: a] :
( ( member_a @ C5 @ ( set_mset_a @ C2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ C5 @ D3 ) @ S ) ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( plus_plus_multiset_a @ A3 @ C2 ) @ ( plus_plus_multiset_a @ B2 @ D2 ) ) @ ( multiset_s_mul_ext_a @ Ns @ S ) ) ) ) ) ).
% ns_s_mul_ext_union_multiset_l
thf(fact_563_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_564_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_565_s__mul__ext__bottom,axiom,
! [A3: multiset_a,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( A3 != zero_zero_multiset_a )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A3 @ zero_zero_multiset_a ) @ ( multiset_s_mul_ext_a @ Ns @ S ) ) ) ).
% s_mul_ext_bottom
thf(fact_566_s__mul__ext__bottom__strict,axiom,
! [A3: multiset_a,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a] :
~ ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ zero_zero_multiset_a @ A3 ) @ ( multiset_s_mul_ext_a @ Ns @ S ) ) ).
% s_mul_ext_bottom_strict
thf(fact_567_s__mul__ext__extend__left,axiom,
! [B2: multiset_a,C2: multiset_a,W: set_Product_prod_a_a,S3: set_Product_prod_a_a,A3: multiset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ B2 @ C2 ) @ ( multiset_s_mul_ext_a @ W @ S3 ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( plus_plus_multiset_a @ A3 @ B2 ) @ C2 ) @ ( multiset_s_mul_ext_a @ W @ S3 ) ) ) ).
% s_mul_ext_extend_left
thf(fact_568_s__ns__mul__ext,axiom,
! [A3: multiset_a,B2: multiset_a,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A3 @ B2 ) @ ( multiset_s_mul_ext_a @ Ns @ S ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A3 @ B2 ) @ ( multis4023584895248362547_ext_a @ Ns @ S ) ) ) ).
% s_ns_mul_ext
thf(fact_569_ns__mul__ext__Id__eq,axiom,
! [R3: set_Product_prod_a_a] :
( ( multis4023584895248362547_ext_a @ id_a @ R3 )
= ( sup_su1925787886501348827iset_a @ ( multiset_s_mul_ext_a @ id_a @ R3 ) @ id_multiset_a ) ) ).
% ns_mul_ext_Id_eq
thf(fact_570_s__mul__ext__mono,axiom,
! [Ns: set_Product_prod_a_a,Ns2: set_Product_prod_a_a,S: set_Product_prod_a_a,S2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ Ns @ Ns2 )
=> ( ( ord_le746702958409616551od_a_a @ S @ S2 )
=> ( ord_le5871882978999109543iset_a @ ( multiset_s_mul_ext_a @ Ns @ S ) @ ( multiset_s_mul_ext_a @ Ns2 @ S2 ) ) ) ) ).
% s_mul_ext_mono
thf(fact_571_s__mul__ext__mono,axiom,
! [Ns: set_Pr79727621955416071iset_a,Ns2: set_Pr79727621955416071iset_a,S: set_Pr79727621955416071iset_a,S2: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ Ns @ Ns2 )
=> ( ( ord_le5871882978999109543iset_a @ S @ S2 )
=> ( ord_le2895628588359463079iset_a @ ( multis7166037729339985883iset_a @ Ns @ S ) @ ( multis7166037729339985883iset_a @ Ns2 @ S2 ) ) ) ) ).
% s_mul_ext_mono
thf(fact_572_s__ns__mul__ext__union__compat,axiom,
! [A3: multiset_a,B2: multiset_a,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a,C2: multiset_a,D2: multiset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A3 @ B2 ) @ ( multiset_s_mul_ext_a @ Ns @ S ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ C2 @ D2 ) @ ( multis4023584895248362547_ext_a @ Ns @ S ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( plus_plus_multiset_a @ A3 @ C2 ) @ ( plus_plus_multiset_a @ B2 @ D2 ) ) @ ( multiset_s_mul_ext_a @ Ns @ S ) ) ) ) ).
% s_ns_mul_ext_union_compat
thf(fact_573_s__mul__ext__union__compat,axiom,
! [A3: multiset_a,B2: multiset_a,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a,C2: multiset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A3 @ B2 ) @ ( multiset_s_mul_ext_a @ Ns @ S ) )
=> ( ( locally_refl_a @ Ns @ C2 )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( plus_plus_multiset_a @ A3 @ C2 ) @ ( plus_plus_multiset_a @ B2 @ C2 ) ) @ ( multiset_s_mul_ext_a @ Ns @ S ) ) ) ) ).
% s_mul_ext_union_compat
thf(fact_574_all__s__s__mul__ext,axiom,
! [A3: multis4781622204924204013iset_a,B2: multis4781622204924204013iset_a,S: set_Pr3059988933816007879iset_a,Ns: set_Pr3059988933816007879iset_a] :
( ( A3 != zero_z4591712717340762532iset_a )
=> ( ! [B8: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ B8 @ ( set_ms2379723421352415844iset_a @ B2 ) )
=> ? [A8: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ A8 @ ( set_ms2379723421352415844iset_a @ A3 ) )
& ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ A8 @ B8 ) @ S ) ) )
=> ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ A3 @ B2 ) @ ( multis6482401367445528638iset_a @ Ns @ S ) ) ) ) ).
% all_s_s_mul_ext
thf(fact_575_all__s__s__mul__ext,axiom,
! [A3: multis501812127501805293od_a_a,B2: multis501812127501805293od_a_a,S: set_Pr8600417178894128327od_a_a,Ns: set_Pr8600417178894128327od_a_a] :
( ( A3 != zero_z3024502163831676836od_a_a )
=> ( ! [B8: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ B8 @ ( set_ms119794563918438244od_a_a @ B2 ) )
=> ? [A8: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A8 @ ( set_ms119794563918438244od_a_a @ A3 ) )
& ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A8 @ B8 ) @ S ) ) )
=> ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ A3 @ B2 ) @ ( multis6523615221270609726od_a_a @ Ns @ S ) ) ) ) ).
% all_s_s_mul_ext
thf(fact_576_all__s__s__mul__ext,axiom,
! [A3: multiset_multiset_a,B2: multiset_multiset_a,S: set_Pr79727621955416071iset_a,Ns: set_Pr79727621955416071iset_a] :
( ( A3 != zero_z7799948378220188993iset_a )
=> ( ! [B8: multiset_a] :
( ( member_multiset_a @ B8 @ ( set_mset_multiset_a @ B2 ) )
=> ? [A8: multiset_a] :
( ( member_multiset_a @ A8 @ ( set_mset_multiset_a @ A3 ) )
& ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A8 @ B8 ) @ S ) ) )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ A3 @ B2 ) @ ( multis7166037729339985883iset_a @ Ns @ S ) ) ) ) ).
% all_s_s_mul_ext
thf(fact_577_all__s__s__mul__ext,axiom,
! [A3: multiset_a,B2: multiset_a,S: set_Product_prod_a_a,Ns: set_Product_prod_a_a] :
( ( A3 != zero_zero_multiset_a )
=> ( ! [B8: a] :
( ( member_a @ B8 @ ( set_mset_a @ B2 ) )
=> ? [A8: a] :
( ( member_a @ A8 @ ( set_mset_a @ A3 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A8 @ B8 ) @ S ) ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A3 @ B2 ) @ ( multiset_s_mul_ext_a @ Ns @ S ) ) ) ) ).
% all_s_s_mul_ext
thf(fact_578_s__mul__ext__ne__extend__left,axiom,
! [A3: multiset_a,B2: multiset_a,C2: multiset_a,W: set_Product_prod_a_a,S3: set_Product_prod_a_a] :
( ( A3 != zero_zero_multiset_a )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ B2 @ C2 ) @ ( multis4023584895248362547_ext_a @ W @ S3 ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( plus_plus_multiset_a @ A3 @ B2 ) @ C2 ) @ ( multiset_s_mul_ext_a @ W @ S3 ) ) ) ) ).
% s_mul_ext_ne_extend_left
thf(fact_579_s__mul__ext__self__extend__left,axiom,
! [A3: multiset_a,W: set_Product_prod_a_a,B2: multiset_a,S3: set_Product_prod_a_a] :
( ( A3 != zero_zero_multiset_a )
=> ( ( locally_refl_a @ W @ B2 )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( plus_plus_multiset_a @ A3 @ B2 ) @ B2 ) @ ( multiset_s_mul_ext_a @ W @ S3 ) ) ) ) ).
% s_mul_ext_self_extend_left
thf(fact_580_s__mul__ext__IdI,axiom,
! [X3: multiset_multiset_a,M: multiset_multiset_a,Z5: multiset_multiset_a,N: multiset_multiset_a,Y2: multiset_multiset_a,R3: set_Pr79727621955416071iset_a] :
( ( X3 != zero_z7799948378220188993iset_a )
=> ( ( M
= ( plus_p6738641960240532161iset_a @ X3 @ Z5 ) )
=> ( ( N
= ( plus_p6738641960240532161iset_a @ Y2 @ Z5 ) )
=> ( ! [X5: multiset_a] :
( ( member_multiset_a @ X5 @ ( set_mset_multiset_a @ Y2 ) )
=> ? [Xa: multiset_a] :
( ( member_multiset_a @ Xa @ ( set_mset_multiset_a @ X3 ) )
& ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Xa @ X5 ) @ R3 ) ) )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ M @ N ) @ ( multis7166037729339985883iset_a @ id_multiset_a @ R3 ) ) ) ) ) ) ).
% s_mul_ext_IdI
thf(fact_581_s__mul__ext__IdI,axiom,
! [X3: multiset_a,M: multiset_a,Z5: multiset_a,N: multiset_a,Y2: multiset_a,R3: set_Product_prod_a_a] :
( ( X3 != zero_zero_multiset_a )
=> ( ( M
= ( plus_plus_multiset_a @ X3 @ Z5 ) )
=> ( ( N
= ( plus_plus_multiset_a @ Y2 @ Z5 ) )
=> ( ! [X5: a] :
( ( member_a @ X5 @ ( set_mset_a @ Y2 ) )
=> ? [Xa: a] :
( ( member_a @ Xa @ ( set_mset_a @ X3 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa @ X5 ) @ R3 ) ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ M @ N ) @ ( multiset_s_mul_ext_a @ id_a @ R3 ) ) ) ) ) ) ).
% s_mul_ext_IdI
thf(fact_582_s__mul__ext__IdE,axiom,
! [M: multiset_multiset_a,N: multiset_multiset_a,R3: set_Pr79727621955416071iset_a] :
( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ M @ N ) @ ( multis7166037729339985883iset_a @ id_multiset_a @ R3 ) )
=> ~ ! [X6: multiset_multiset_a] :
( ( X6 != zero_z7799948378220188993iset_a )
=> ! [Y7: multiset_multiset_a,Z2: multiset_multiset_a] :
( ( M
= ( plus_p6738641960240532161iset_a @ X6 @ Z2 ) )
=> ( ( N
= ( plus_p6738641960240532161iset_a @ Y7 @ Z2 ) )
=> ~ ! [X: multiset_a] :
( ( member_multiset_a @ X @ ( set_mset_multiset_a @ Y7 ) )
=> ? [Xa2: multiset_a] :
( ( member_multiset_a @ Xa2 @ ( set_mset_multiset_a @ X6 ) )
& ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Xa2 @ X ) @ R3 ) ) ) ) ) ) ) ).
% s_mul_ext_IdE
thf(fact_583_s__mul__ext__IdE,axiom,
! [M: multiset_a,N: multiset_a,R3: set_Product_prod_a_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ M @ N ) @ ( multiset_s_mul_ext_a @ id_a @ R3 ) )
=> ~ ! [X6: multiset_a] :
( ( X6 != zero_zero_multiset_a )
=> ! [Y7: multiset_a,Z2: multiset_a] :
( ( M
= ( plus_plus_multiset_a @ X6 @ Z2 ) )
=> ( ( N
= ( plus_plus_multiset_a @ Y7 @ Z2 ) )
=> ~ ! [X: a] :
( ( member_a @ X @ ( set_mset_a @ Y7 ) )
=> ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_mset_a @ X6 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa2 @ X ) @ R3 ) ) ) ) ) ) ) ).
% s_mul_ext_IdE
thf(fact_584_s__mul__extI,axiom,
! [A3: multis4781622204924204013iset_a,A12: multis4781622204924204013iset_a,A23: multis4781622204924204013iset_a,B2: multis4781622204924204013iset_a,B12: multis4781622204924204013iset_a,B23: multis4781622204924204013iset_a,Ns: set_Pr3059988933816007879iset_a,S: set_Pr3059988933816007879iset_a] :
( ( A3
= ( plus_p1053901119268525860iset_a @ A12 @ A23 ) )
=> ( ( B2
= ( plus_p1053901119268525860iset_a @ B12 @ B23 ) )
=> ( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ A12 @ B12 ) @ ( multis3430030276581740203iset_a @ Ns ) )
=> ( ( A23 != zero_z4591712717340762532iset_a )
=> ( ! [B8: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ B8 @ ( set_ms2379723421352415844iset_a @ B23 ) )
=> ? [A8: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ A8 @ ( set_ms2379723421352415844iset_a @ A23 ) )
& ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ A8 @ B8 ) @ S ) ) )
=> ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ A3 @ B2 ) @ ( multis6482401367445528638iset_a @ Ns @ S ) ) ) ) ) ) ) ).
% s_mul_extI
thf(fact_585_s__mul__extI,axiom,
! [A3: multis501812127501805293od_a_a,A12: multis501812127501805293od_a_a,A23: multis501812127501805293od_a_a,B2: multis501812127501805293od_a_a,B12: multis501812127501805293od_a_a,B23: multis501812127501805293od_a_a,Ns: set_Pr8600417178894128327od_a_a,S: set_Pr8600417178894128327od_a_a] :
( ( A3
= ( plus_p6967097483979887908od_a_a @ A12 @ A23 ) )
=> ( ( B2
= ( plus_p6967097483979887908od_a_a @ B12 @ B23 ) )
=> ( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ A12 @ B12 ) @ ( multis1821517007024690859od_a_a @ Ns ) )
=> ( ( A23 != zero_z3024502163831676836od_a_a )
=> ( ! [B8: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ B8 @ ( set_ms119794563918438244od_a_a @ B23 ) )
=> ? [A8: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A8 @ ( set_ms119794563918438244od_a_a @ A23 ) )
& ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A8 @ B8 ) @ S ) ) )
=> ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ A3 @ B2 ) @ ( multis6523615221270609726od_a_a @ Ns @ S ) ) ) ) ) ) ) ).
% s_mul_extI
thf(fact_586_s__mul__extI,axiom,
! [A3: multiset_multiset_a,A12: multiset_multiset_a,A23: multiset_multiset_a,B2: multiset_multiset_a,B12: multiset_multiset_a,B23: multiset_multiset_a,Ns: set_Pr79727621955416071iset_a,S: set_Pr79727621955416071iset_a] :
( ( A3
= ( plus_p6738641960240532161iset_a @ A12 @ A23 ) )
=> ( ( B2
= ( plus_p6738641960240532161iset_a @ B12 @ B23 ) )
=> ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ A12 @ B12 ) @ ( multis451435756853798856iset_a @ Ns ) )
=> ( ( A23 != zero_z7799948378220188993iset_a )
=> ( ! [B8: multiset_a] :
( ( member_multiset_a @ B8 @ ( set_mset_multiset_a @ B23 ) )
=> ? [A8: multiset_a] :
( ( member_multiset_a @ A8 @ ( set_mset_multiset_a @ A23 ) )
& ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A8 @ B8 ) @ S ) ) )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ A3 @ B2 ) @ ( multis7166037729339985883iset_a @ Ns @ S ) ) ) ) ) ) ) ).
% s_mul_extI
thf(fact_587_s__mul__extI,axiom,
! [A3: multiset_a,A12: multiset_a,A23: multiset_a,B2: multiset_a,B12: multiset_a,B23: multiset_a,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( A3
= ( plus_plus_multiset_a @ A12 @ A23 ) )
=> ( ( B2
= ( plus_plus_multiset_a @ B12 @ B23 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A12 @ B12 ) @ ( multis2373687813370458242ltpw_a @ Ns ) )
=> ( ( A23 != zero_zero_multiset_a )
=> ( ! [B8: a] :
( ( member_a @ B8 @ ( set_mset_a @ B23 ) )
=> ? [A8: a] :
( ( member_a @ A8 @ ( set_mset_a @ A23 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A8 @ B8 ) @ S ) ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A3 @ B2 ) @ ( multiset_s_mul_ext_a @ Ns @ S ) ) ) ) ) ) ) ).
% s_mul_extI
thf(fact_588_s__mul__extE,axiom,
! [A3: multis4781622204924204013iset_a,B2: multis4781622204924204013iset_a,Ns: set_Pr3059988933816007879iset_a,S: set_Pr3059988933816007879iset_a] :
( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ A3 @ B2 ) @ ( multis6482401367445528638iset_a @ Ns @ S ) )
=> ~ ! [A1: multis4781622204924204013iset_a,A22: multis4781622204924204013iset_a] :
( ( A3
= ( plus_p1053901119268525860iset_a @ A1 @ A22 ) )
=> ! [B1: multis4781622204924204013iset_a,B22: multis4781622204924204013iset_a] :
( ( B2
= ( plus_p1053901119268525860iset_a @ B1 @ B22 ) )
=> ( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ A1 @ B1 ) @ ( multis3430030276581740203iset_a @ Ns ) )
=> ( ( A22 != zero_z4591712717340762532iset_a )
=> ~ ! [B3: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ B3 @ ( set_ms2379723421352415844iset_a @ B22 ) )
=> ? [A: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ A @ ( set_ms2379723421352415844iset_a @ A22 ) )
& ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ A @ B3 ) @ S ) ) ) ) ) ) ) ) ).
% s_mul_extE
thf(fact_589_s__mul__extE,axiom,
! [A3: multis501812127501805293od_a_a,B2: multis501812127501805293od_a_a,Ns: set_Pr8600417178894128327od_a_a,S: set_Pr8600417178894128327od_a_a] :
( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ A3 @ B2 ) @ ( multis6523615221270609726od_a_a @ Ns @ S ) )
=> ~ ! [A1: multis501812127501805293od_a_a,A22: multis501812127501805293od_a_a] :
( ( A3
= ( plus_p6967097483979887908od_a_a @ A1 @ A22 ) )
=> ! [B1: multis501812127501805293od_a_a,B22: multis501812127501805293od_a_a] :
( ( B2
= ( plus_p6967097483979887908od_a_a @ B1 @ B22 ) )
=> ( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ A1 @ B1 ) @ ( multis1821517007024690859od_a_a @ Ns ) )
=> ( ( A22 != zero_z3024502163831676836od_a_a )
=> ~ ! [B3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ B3 @ ( 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 @ B3 ) @ S ) ) ) ) ) ) ) ) ).
% s_mul_extE
thf(fact_590_s__mul__extE,axiom,
! [A3: multiset_multiset_a,B2: multiset_multiset_a,Ns: set_Pr79727621955416071iset_a,S: set_Pr79727621955416071iset_a] :
( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ A3 @ B2 ) @ ( multis7166037729339985883iset_a @ Ns @ S ) )
=> ~ ! [A1: multiset_multiset_a,A22: multiset_multiset_a] :
( ( A3
= ( plus_p6738641960240532161iset_a @ A1 @ A22 ) )
=> ! [B1: multiset_multiset_a,B22: multiset_multiset_a] :
( ( B2
= ( plus_p6738641960240532161iset_a @ B1 @ B22 ) )
=> ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ A1 @ B1 ) @ ( multis451435756853798856iset_a @ Ns ) )
=> ( ( A22 != zero_z7799948378220188993iset_a )
=> ~ ! [B3: multiset_a] :
( ( member_multiset_a @ B3 @ ( set_mset_multiset_a @ B22 ) )
=> ? [A: multiset_a] :
( ( member_multiset_a @ A @ ( set_mset_multiset_a @ A22 ) )
& ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A @ B3 ) @ S ) ) ) ) ) ) ) ) ).
% s_mul_extE
thf(fact_591_s__mul__extE,axiom,
! [A3: multiset_a,B2: multiset_a,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A3 @ B2 ) @ ( multiset_s_mul_ext_a @ Ns @ S ) )
=> ~ ! [A1: multiset_a,A22: multiset_a] :
( ( A3
= ( plus_plus_multiset_a @ A1 @ A22 ) )
=> ! [B1: multiset_a,B22: multiset_a] :
( ( B2
= ( plus_plus_multiset_a @ B1 @ B22 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A1 @ B1 ) @ ( multis2373687813370458242ltpw_a @ Ns ) )
=> ( ( A22 != zero_zero_multiset_a )
=> ~ ! [B3: a] :
( ( member_a @ B3 @ ( set_mset_a @ B22 ) )
=> ? [A: a] :
( ( member_a @ A @ ( set_mset_a @ A22 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B3 ) @ S ) ) ) ) ) ) ) ) ).
% s_mul_extE
thf(fact_592_in__mset__fold__plus__iff,axiom,
! [X2: produc6518373309651786023iset_a,M: multis4781622204924204013iset_a,NN: multis7221419791219231795iset_a] :
( ( member5199237121806060112iset_a @ X2 @ ( set_ms2379723421352415844iset_a @ ( fold_m7087023614847598283iset_a @ plus_p1053901119268525860iset_a @ M @ NN ) ) )
= ( ( member5199237121806060112iset_a @ X2 @ ( set_ms2379723421352415844iset_a @ M ) )
| ? [N2: multis4781622204924204013iset_a] :
( ( member2597461214246177430iset_a @ N2 @ ( set_ms5327031259215843498iset_a @ NN ) )
& ( member5199237121806060112iset_a @ X2 @ ( set_ms2379723421352415844iset_a @ N2 ) ) ) ) ) ).
% in_mset_fold_plus_iff
thf(fact_593_in__mset__fold__plus__iff,axiom,
! [X2: product_prod_a_a,M: multis501812127501805293od_a_a,NN: multis5935151272263360051od_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ ( set_ms119794563918438244od_a_a @ ( fold_m3854860582760972491od_a_a @ plus_p6967097483979887908od_a_a @ M @ NN ) ) )
= ( ( member1426531477525435216od_a_a @ X2 @ ( set_ms119794563918438244od_a_a @ M ) )
| ? [N2: multis501812127501805293od_a_a] :
( ( member7700929758516679318od_a_a @ N2 @ ( set_ms8253844250382261418od_a_a @ NN ) )
& ( member1426531477525435216od_a_a @ X2 @ ( set_ms119794563918438244od_a_a @ N2 ) ) ) ) ) ).
% in_mset_fold_plus_iff
thf(fact_594_in__mset__fold__plus__iff,axiom,
! [X2: a,M: multiset_a,NN: multiset_multiset_a] :
( ( member_a @ X2 @ ( set_mset_a @ ( fold_m6601649825673331723iset_a @ plus_plus_multiset_a @ M @ NN ) ) )
= ( ( member_a @ X2 @ ( set_mset_a @ M ) )
| ? [N2: multiset_a] :
( ( member_multiset_a @ N2 @ ( set_mset_multiset_a @ NN ) )
& ( member_a @ X2 @ ( set_mset_a @ N2 ) ) ) ) ) ).
% in_mset_fold_plus_iff
thf(fact_595_add__0__iff,axiom,
! [B: nat,A2: nat] :
( ( B
= ( plus_plus_nat @ B @ A2 ) )
= ( A2 = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_596_verit__sum__simplify,axiom,
! [A2: multiset_a] :
( ( plus_plus_multiset_a @ A2 @ zero_zero_multiset_a )
= A2 ) ).
% verit_sum_simplify
thf(fact_597_verit__sum__simplify,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% verit_sum_simplify
thf(fact_598_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_599_one__step__implies__mult,axiom,
! [J2: multiset_multiset_a,K: multiset_multiset_a,R: set_Pr79727621955416071iset_a,I2: multiset_multiset_a] :
( ( J2 != zero_z7799948378220188993iset_a )
=> ( ! [X5: multiset_a] :
( ( member_multiset_a @ X5 @ ( set_mset_multiset_a @ K ) )
=> ? [Xa: multiset_a] :
( ( member_multiset_a @ Xa @ ( set_mset_multiset_a @ J2 ) )
& ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Xa ) @ R ) ) )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ ( plus_p6738641960240532161iset_a @ I2 @ K ) @ ( plus_p6738641960240532161iset_a @ I2 @ J2 ) ) @ ( mult_multiset_a @ R ) ) ) ) ).
% one_step_implies_mult
thf(fact_600_one__step__implies__mult,axiom,
! [J2: multiset_a,K: multiset_a,R: set_Product_prod_a_a,I2: multiset_a] :
( ( J2 != zero_zero_multiset_a )
=> ( ! [X5: a] :
( ( member_a @ X5 @ ( set_mset_a @ K ) )
=> ? [Xa: a] :
( ( member_a @ Xa @ ( set_mset_a @ J2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Xa ) @ R ) ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( plus_plus_multiset_a @ I2 @ K ) @ ( plus_plus_multiset_a @ I2 @ J2 ) ) @ ( mult_a @ R ) ) ) ) ).
% one_step_implies_mult
thf(fact_601_Multiset_Omono__mult,axiom,
! [R: set_Product_prod_a_a,R2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ R @ R2 )
=> ( ord_le5871882978999109543iset_a @ ( mult_a @ R ) @ ( mult_a @ R2 ) ) ) ).
% Multiset.mono_mult
thf(fact_602_Multiset_Omono__mult,axiom,
! [R: set_Pr79727621955416071iset_a,R2: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ R @ R2 )
=> ( ord_le2895628588359463079iset_a @ ( mult_multiset_a @ R ) @ ( mult_multiset_a @ R2 ) ) ) ).
% Multiset.mono_mult
thf(fact_603_union__fold__mset__add__mset,axiom,
( plus_plus_multiset_a
= ( fold_m7320414754419674833iset_a @ add_mset_a ) ) ).
% union_fold_mset_add_mset
thf(fact_604_verit__la__disequality,axiom,
! [A2: nat,B: nat] :
( ( A2 = B )
| ~ ( ord_less_eq_nat @ A2 @ B )
| ~ ( ord_less_eq_nat @ B @ A2 ) ) ).
% verit_la_disequality
thf(fact_605_verit__comp__simplify1_I2_J,axiom,
! [A2: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_606_verit__comp__simplify1_I2_J,axiom,
! [A2: set_Pr79727621955416071iset_a] : ( ord_le5871882978999109543iset_a @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_607_verit__comp__simplify1_I2_J,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_608_locally__sym__def,axiom,
( locall7196068196170061100iset_a
= ( ^ [R4: set_Pr3059988933816007879iset_a,A6: multis4781622204924204013iset_a] :
! [T: produc6518373309651786023iset_a,U: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ T @ ( set_ms2379723421352415844iset_a @ A6 ) )
=> ( ( member5199237121806060112iset_a @ U @ ( set_ms2379723421352415844iset_a @ A6 ) )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ T @ U ) @ R4 )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ U @ T ) @ R4 ) ) ) ) ) ) ).
% locally_sym_def
thf(fact_609_locally__sym__def,axiom,
( locall2390554470099405356od_a_a
= ( ^ [R4: set_Pr8600417178894128327od_a_a,A6: multis501812127501805293od_a_a] :
! [T: product_prod_a_a,U: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ T @ ( set_ms119794563918438244od_a_a @ A6 ) )
=> ( ( member1426531477525435216od_a_a @ U @ ( set_ms119794563918438244od_a_a @ A6 ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ T @ U ) @ R4 )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ U @ T ) @ R4 ) ) ) ) ) ) ).
% locally_sym_def
thf(fact_610_locally__sym__def,axiom,
( locall8646784519055707593iset_a
= ( ^ [R4: set_Pr79727621955416071iset_a,A6: multiset_multiset_a] :
! [T: multiset_a,U: multiset_a] :
( ( member_multiset_a @ T @ ( set_mset_multiset_a @ A6 ) )
=> ( ( member_multiset_a @ U @ ( set_mset_multiset_a @ A6 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ T @ U ) @ R4 )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ U @ T ) @ R4 ) ) ) ) ) ) ).
% locally_sym_def
thf(fact_611_locally__sym__def,axiom,
( locally_sym_a
= ( ^ [R4: set_Product_prod_a_a,A6: multiset_a] :
! [T: a,U: a] :
( ( member_a @ T @ ( set_mset_a @ A6 ) )
=> ( ( member_a @ U @ ( set_mset_a @ A6 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ T @ U ) @ R4 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ T ) @ R4 ) ) ) ) ) ) ).
% locally_sym_def
thf(fact_612_locally__antisym__def,axiom,
( locall5667257730485359854iset_a
= ( ^ [R4: set_Pr3059988933816007879iset_a,A6: multis4781622204924204013iset_a] :
! [T: produc6518373309651786023iset_a,U: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ T @ ( set_ms2379723421352415844iset_a @ A6 ) )
=> ( ( member5199237121806060112iset_a @ U @ ( set_ms2379723421352415844iset_a @ A6 ) )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ T @ U ) @ R4 )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ U @ T ) @ R4 )
=> ( T = U ) ) ) ) ) ) ) ).
% locally_antisym_def
thf(fact_613_locally__antisym__def,axiom,
( locall7522220404423730158od_a_a
= ( ^ [R4: set_Pr8600417178894128327od_a_a,A6: multis501812127501805293od_a_a] :
! [T: product_prod_a_a,U: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ T @ ( set_ms119794563918438244od_a_a @ A6 ) )
=> ( ( member1426531477525435216od_a_a @ U @ ( set_ms119794563918438244od_a_a @ A6 ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ T @ U ) @ R4 )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ U @ T ) @ R4 )
=> ( T = U ) ) ) ) ) ) ) ).
% locally_antisym_def
thf(fact_614_locally__antisym__def,axiom,
( locall4507398398213139595iset_a
= ( ^ [R4: set_Pr79727621955416071iset_a,A6: multiset_multiset_a] :
! [T: multiset_a,U: multiset_a] :
( ( member_multiset_a @ T @ ( set_mset_multiset_a @ A6 ) )
=> ( ( member_multiset_a @ U @ ( set_mset_multiset_a @ A6 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ T @ U ) @ R4 )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ U @ T ) @ R4 )
=> ( T = U ) ) ) ) ) ) ) ).
% locally_antisym_def
thf(fact_615_locally__antisym__def,axiom,
( locally_antisym_a
= ( ^ [R4: set_Product_prod_a_a,A6: multiset_a] :
! [T: a,U: a] :
( ( member_a @ T @ ( set_mset_a @ A6 ) )
=> ( ( member_a @ U @ ( set_mset_a @ A6 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ T @ U ) @ R4 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ T ) @ R4 )
=> ( T = U ) ) ) ) ) ) ) ).
% locally_antisym_def
thf(fact_616_locally__trans__multpw,axiom,
! [Ns: set_Product_prod_a_a,S3: multiset_a,T2: multiset_a,U2: multiset_a] :
( ( locally_trans_a @ Ns @ S3 @ T2 @ U2 )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ S3 @ T2 ) @ ( multis2373687813370458242ltpw_a @ Ns ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ T2 @ U2 ) @ ( multis2373687813370458242ltpw_a @ Ns ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ S3 @ U2 ) @ ( multis2373687813370458242ltpw_a @ Ns ) ) ) ) ) ).
% locally_trans_multpw
thf(fact_617_locally__included__def,axiom,
( locall1765895018968356541iset_a
= ( ^ [R1: set_Pr3059988933816007879iset_a,R22: set_Pr3059988933816007879iset_a,A6: multis4781622204924204013iset_a,B6: multis4781622204924204013iset_a] :
! [T: produc6518373309651786023iset_a,U: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ T @ ( set_ms2379723421352415844iset_a @ A6 ) )
=> ( ( member5199237121806060112iset_a @ U @ ( set_ms2379723421352415844iset_a @ B6 ) )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ T @ U ) @ R1 )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ T @ U ) @ R22 ) ) ) ) ) ) ).
% locally_included_def
thf(fact_618_locally__included__def,axiom,
( locall6047871833758861501od_a_a
= ( ^ [R1: set_Pr8600417178894128327od_a_a,R22: set_Pr8600417178894128327od_a_a,A6: multis501812127501805293od_a_a,B6: multis501812127501805293od_a_a] :
! [T: product_prod_a_a,U: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ T @ ( set_ms119794563918438244od_a_a @ A6 ) )
=> ( ( member1426531477525435216od_a_a @ U @ ( set_ms119794563918438244od_a_a @ B6 ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ T @ U ) @ R1 )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ T @ U ) @ R22 ) ) ) ) ) ) ).
% locally_included_def
thf(fact_619_locally__included__def,axiom,
( locall9031913696783017690iset_a
= ( ^ [R1: set_Pr79727621955416071iset_a,R22: set_Pr79727621955416071iset_a,A6: multiset_multiset_a,B6: multiset_multiset_a] :
! [T: multiset_a,U: multiset_a] :
( ( member_multiset_a @ T @ ( set_mset_multiset_a @ A6 ) )
=> ( ( member_multiset_a @ U @ ( set_mset_multiset_a @ B6 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ T @ U ) @ R1 )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ T @ U ) @ R22 ) ) ) ) ) ) ).
% locally_included_def
thf(fact_620_locally__included__def,axiom,
( locally_included_a
= ( ^ [R1: set_Product_prod_a_a,R22: set_Product_prod_a_a,A6: multiset_a,B6: multiset_a] :
! [T: a,U: a] :
( ( member_a @ T @ ( set_mset_a @ A6 ) )
=> ( ( member_a @ U @ ( set_mset_a @ B6 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ T @ U ) @ R1 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ T @ U ) @ R22 ) ) ) ) ) ) ).
% locally_included_def
thf(fact_621_comp__fun__commute__plus__mset,axiom,
finite6687201363011020241iset_a @ plus_plus_multiset_a ).
% comp_fun_commute_plus_mset
thf(fact_622_lt__trans__r,axiom,
! [R3: set_Product_prod_a_a,A3: multiset_a,B2: multiset_a,C2: multiset_a,D2: multiset_a,E: multiset_a,F2: multiset_a] :
( ( locally_trans_a @ R3 @ ( plus_plus_multiset_a @ A3 @ B2 ) @ ( plus_plus_multiset_a @ C2 @ D2 ) @ ( plus_plus_multiset_a @ E @ F2 ) )
=> ( locally_trans_a @ R3 @ B2 @ D2 @ F2 ) ) ).
% lt_trans_r
thf(fact_623_lt__trans__l,axiom,
! [R3: set_Product_prod_a_a,A3: multiset_a,B2: multiset_a,C2: multiset_a,D2: multiset_a,E: multiset_a,F2: multiset_a] :
( ( locally_trans_a @ R3 @ ( plus_plus_multiset_a @ A3 @ B2 ) @ ( plus_plus_multiset_a @ C2 @ D2 ) @ ( plus_plus_multiset_a @ E @ F2 ) )
=> ( locally_trans_a @ R3 @ A3 @ C2 @ E ) ) ).
% lt_trans_l
thf(fact_624_lin__trans__r,axiom,
! [R12: set_Product_prod_a_a,R23: set_Product_prod_a_a,A3: multiset_a,B2: multiset_a,C2: multiset_a,D2: multiset_a] :
( ( locally_included_a @ R12 @ R23 @ ( plus_plus_multiset_a @ A3 @ B2 ) @ ( plus_plus_multiset_a @ C2 @ D2 ) )
=> ( locally_included_a @ R12 @ R23 @ B2 @ D2 ) ) ).
% lin_trans_r
thf(fact_625_lin__trans__l,axiom,
! [R12: set_Product_prod_a_a,R23: set_Product_prod_a_a,A3: multiset_a,B2: multiset_a,C2: multiset_a,D2: multiset_a] :
( ( locally_included_a @ R12 @ R23 @ ( plus_plus_multiset_a @ A3 @ B2 ) @ ( plus_plus_multiset_a @ C2 @ D2 ) )
=> ( locally_included_a @ R12 @ R23 @ A3 @ C2 ) ) ).
% lin_trans_l
thf(fact_626_in__lin,axiom,
! [R12: set_Product_prod_a_a,R23: set_Product_prod_a_a,A3: multiset_a,B2: multiset_a] :
( ( ord_le746702958409616551od_a_a @ R12 @ R23 )
=> ( locally_included_a @ R12 @ R23 @ A3 @ B2 ) ) ).
% in_lin
thf(fact_627_in__lin,axiom,
! [R12: set_Pr79727621955416071iset_a,R23: set_Pr79727621955416071iset_a,A3: multiset_multiset_a,B2: multiset_multiset_a] :
( ( ord_le5871882978999109543iset_a @ R12 @ R23 )
=> ( locall9031913696783017690iset_a @ R12 @ R23 @ A3 @ B2 ) ) ).
% in_lin
thf(fact_628_las__trans__r,axiom,
! [R3: set_Product_prod_a_a,A3: multiset_a,B2: multiset_a] :
( ( locally_antisym_a @ R3 @ ( plus_plus_multiset_a @ A3 @ B2 ) )
=> ( locally_antisym_a @ R3 @ B2 ) ) ).
% las_trans_r
thf(fact_629_las__trans__l,axiom,
! [R3: set_Product_prod_a_a,A3: multiset_a,B2: multiset_a] :
( ( locally_antisym_a @ R3 @ ( plus_plus_multiset_a @ A3 @ B2 ) )
=> ( locally_antisym_a @ R3 @ A3 ) ) ).
% las_trans_l
thf(fact_630_ls__trans__r,axiom,
! [R3: set_Product_prod_a_a,A3: multiset_a,B2: multiset_a] :
( ( locally_sym_a @ R3 @ ( plus_plus_multiset_a @ A3 @ B2 ) )
=> ( locally_sym_a @ R3 @ B2 ) ) ).
% ls_trans_r
thf(fact_631_ls__trans__l,axiom,
! [R3: set_Product_prod_a_a,A3: multiset_a,B2: multiset_a] :
( ( locally_sym_a @ R3 @ ( plus_plus_multiset_a @ A3 @ B2 ) )
=> ( locally_sym_a @ R3 @ A3 ) ) ).
% ls_trans_l
thf(fact_632_locally__trans__def,axiom,
( locall8070124297340116779iset_a
= ( ^ [R4: set_Pr3059988933816007879iset_a,A6: multis4781622204924204013iset_a,B6: multis4781622204924204013iset_a,C6: multis4781622204924204013iset_a] :
! [T: produc6518373309651786023iset_a,U: produc6518373309651786023iset_a,V: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ T @ ( set_ms2379723421352415844iset_a @ A6 ) )
=> ( ( member5199237121806060112iset_a @ U @ ( set_ms2379723421352415844iset_a @ B6 ) )
=> ( ( member5199237121806060112iset_a @ V @ ( set_ms2379723421352415844iset_a @ C6 ) )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ T @ U ) @ R4 )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ U @ V ) @ R4 )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ T @ V ) @ R4 ) ) ) ) ) ) ) ) ).
% locally_trans_def
thf(fact_633_locally__trans__def,axiom,
( locall3823767776488073003od_a_a
= ( ^ [R4: set_Pr8600417178894128327od_a_a,A6: multis501812127501805293od_a_a,B6: multis501812127501805293od_a_a,C6: multis501812127501805293od_a_a] :
! [T: product_prod_a_a,U: product_prod_a_a,V: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ T @ ( set_ms119794563918438244od_a_a @ A6 ) )
=> ( ( member1426531477525435216od_a_a @ U @ ( set_ms119794563918438244od_a_a @ B6 ) )
=> ( ( member1426531477525435216od_a_a @ V @ ( set_ms119794563918438244od_a_a @ C6 ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ T @ U ) @ R4 )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ U @ V ) @ R4 )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ T @ V ) @ R4 ) ) ) ) ) ) ) ) ).
% locally_trans_def
thf(fact_634_locally__trans__def,axiom,
( locall205826944635074632iset_a
= ( ^ [R4: set_Pr79727621955416071iset_a,A6: multiset_multiset_a,B6: multiset_multiset_a,C6: multiset_multiset_a] :
! [T: multiset_a,U: multiset_a,V: multiset_a] :
( ( member_multiset_a @ T @ ( set_mset_multiset_a @ A6 ) )
=> ( ( member_multiset_a @ U @ ( set_mset_multiset_a @ B6 ) )
=> ( ( member_multiset_a @ V @ ( set_mset_multiset_a @ C6 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ T @ U ) @ R4 )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ U @ V ) @ R4 )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ T @ V ) @ R4 ) ) ) ) ) ) ) ) ).
% locally_trans_def
thf(fact_635_locally__trans__def,axiom,
( locally_trans_a
= ( ^ [R4: set_Product_prod_a_a,A6: multiset_a,B6: multiset_a,C6: multiset_a] :
! [T: a,U: a,V: a] :
( ( member_a @ T @ ( set_mset_a @ A6 ) )
=> ( ( member_a @ U @ ( set_mset_a @ B6 ) )
=> ( ( member_a @ V @ ( set_mset_a @ C6 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ T @ U ) @ R4 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ V ) @ R4 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ T @ V ) @ R4 ) ) ) ) ) ) ) ) ).
% locally_trans_def
thf(fact_636_locally__compatible__r__def,axiom,
( locall7572239673656552821iset_a
= ( ^ [R1: set_Pr3059988933816007879iset_a,R22: set_Pr3059988933816007879iset_a,A6: multis4781622204924204013iset_a,B6: multis4781622204924204013iset_a,C6: multis4781622204924204013iset_a] :
! [T: produc6518373309651786023iset_a,U: produc6518373309651786023iset_a,V: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ T @ ( set_ms2379723421352415844iset_a @ A6 ) )
=> ( ( member5199237121806060112iset_a @ U @ ( set_ms2379723421352415844iset_a @ B6 ) )
=> ( ( member5199237121806060112iset_a @ V @ ( set_ms2379723421352415844iset_a @ C6 ) )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ T @ U ) @ R22 )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ U @ V ) @ R1 )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ T @ V ) @ R22 ) ) ) ) ) ) ) ) ).
% locally_compatible_r_def
thf(fact_637_locally__compatible__r__def,axiom,
( locall4267964216904087413od_a_a
= ( ^ [R1: set_Pr8600417178894128327od_a_a,R22: set_Pr8600417178894128327od_a_a,A6: multis501812127501805293od_a_a,B6: multis501812127501805293od_a_a,C6: multis501812127501805293od_a_a] :
! [T: product_prod_a_a,U: product_prod_a_a,V: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ T @ ( set_ms119794563918438244od_a_a @ A6 ) )
=> ( ( member1426531477525435216od_a_a @ U @ ( set_ms119794563918438244od_a_a @ B6 ) )
=> ( ( member1426531477525435216od_a_a @ V @ ( set_ms119794563918438244od_a_a @ C6 ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ T @ U ) @ R22 )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ U @ V ) @ R1 )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ T @ V ) @ R22 ) ) ) ) ) ) ) ) ).
% locally_compatible_r_def
thf(fact_638_locally__compatible__r__def,axiom,
( locall2684350217271501202iset_a
= ( ^ [R1: set_Pr79727621955416071iset_a,R22: set_Pr79727621955416071iset_a,A6: multiset_multiset_a,B6: multiset_multiset_a,C6: multiset_multiset_a] :
! [T: multiset_a,U: multiset_a,V: multiset_a] :
( ( member_multiset_a @ T @ ( set_mset_multiset_a @ A6 ) )
=> ( ( member_multiset_a @ U @ ( set_mset_multiset_a @ B6 ) )
=> ( ( member_multiset_a @ V @ ( set_mset_multiset_a @ C6 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ T @ U ) @ R22 )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ U @ V ) @ R1 )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ T @ V ) @ R22 ) ) ) ) ) ) ) ) ).
% locally_compatible_r_def
thf(fact_639_locally__compatible__r__def,axiom,
( locall5496200431401003852le_r_a
= ( ^ [R1: set_Product_prod_a_a,R22: set_Product_prod_a_a,A6: multiset_a,B6: multiset_a,C6: multiset_a] :
! [T: a,U: a,V: a] :
( ( member_a @ T @ ( set_mset_a @ A6 ) )
=> ( ( member_a @ U @ ( set_mset_a @ B6 ) )
=> ( ( member_a @ V @ ( set_mset_a @ C6 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ T @ U ) @ R22 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ V ) @ R1 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ T @ V ) @ R22 ) ) ) ) ) ) ) ) ).
% locally_compatible_r_def
thf(fact_640_locally__compatible__l__def,axiom,
( locall8867585307893472251iset_a
= ( ^ [R1: set_Pr3059988933816007879iset_a,R22: set_Pr3059988933816007879iset_a,A6: multis4781622204924204013iset_a,B6: multis4781622204924204013iset_a,C6: multis4781622204924204013iset_a] :
! [T: produc6518373309651786023iset_a,U: produc6518373309651786023iset_a,V: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ T @ ( set_ms2379723421352415844iset_a @ A6 ) )
=> ( ( member5199237121806060112iset_a @ U @ ( set_ms2379723421352415844iset_a @ B6 ) )
=> ( ( member5199237121806060112iset_a @ V @ ( set_ms2379723421352415844iset_a @ C6 ) )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ T @ U ) @ R1 )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ U @ V ) @ R22 )
=> ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ T @ V ) @ R22 ) ) ) ) ) ) ) ) ).
% locally_compatible_l_def
thf(fact_641_locally__compatible__l__def,axiom,
( locall8104606018894984187od_a_a
= ( ^ [R1: set_Pr8600417178894128327od_a_a,R22: set_Pr8600417178894128327od_a_a,A6: multis501812127501805293od_a_a,B6: multis501812127501805293od_a_a,C6: multis501812127501805293od_a_a] :
! [T: product_prod_a_a,U: product_prod_a_a,V: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ T @ ( set_ms119794563918438244od_a_a @ A6 ) )
=> ( ( member1426531477525435216od_a_a @ U @ ( set_ms119794563918438244od_a_a @ B6 ) )
=> ( ( member1426531477525435216od_a_a @ V @ ( set_ms119794563918438244od_a_a @ C6 ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ T @ U ) @ R1 )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ U @ V ) @ R22 )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ T @ V ) @ R22 ) ) ) ) ) ) ) ) ).
% locally_compatible_l_def
thf(fact_642_locally__compatible__l__def,axiom,
( locall3852272748305501464iset_a
= ( ^ [R1: set_Pr79727621955416071iset_a,R22: set_Pr79727621955416071iset_a,A6: multiset_multiset_a,B6: multiset_multiset_a,C6: multiset_multiset_a] :
! [T: multiset_a,U: multiset_a,V: multiset_a] :
( ( member_multiset_a @ T @ ( set_mset_multiset_a @ A6 ) )
=> ( ( member_multiset_a @ U @ ( set_mset_multiset_a @ B6 ) )
=> ( ( member_multiset_a @ V @ ( set_mset_multiset_a @ C6 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ T @ U ) @ R1 )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ U @ V ) @ R22 )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ T @ V ) @ R22 ) ) ) ) ) ) ) ) ).
% locally_compatible_l_def
thf(fact_643_locally__compatible__l__def,axiom,
( locall3774189849527508434le_l_a
= ( ^ [R1: set_Product_prod_a_a,R22: set_Product_prod_a_a,A6: multiset_a,B6: multiset_a,C6: multiset_a] :
! [T: a,U: a,V: a] :
( ( member_a @ T @ ( set_mset_a @ A6 ) )
=> ( ( member_a @ U @ ( set_mset_a @ B6 ) )
=> ( ( member_a @ V @ ( set_mset_a @ C6 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ T @ U ) @ R1 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ V ) @ R22 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ T @ V ) @ R22 ) ) ) ) ) ) ) ) ).
% locally_compatible_l_def
thf(fact_644_diff__union__single__conv,axiom,
! [A2: produc6518373309651786023iset_a,J2: multis4781622204924204013iset_a,I2: multis4781622204924204013iset_a] :
( ( member5199237121806060112iset_a @ A2 @ ( set_ms2379723421352415844iset_a @ J2 ) )
=> ( ( minus_9211619285413683668iset_a @ ( plus_p1053901119268525860iset_a @ I2 @ J2 ) @ ( add_ms5348986034529679365iset_a @ A2 @ zero_z4591712717340762532iset_a ) )
= ( plus_p1053901119268525860iset_a @ I2 @ ( minus_9211619285413683668iset_a @ J2 @ ( add_ms5348986034529679365iset_a @ A2 @ zero_z4591712717340762532iset_a ) ) ) ) ) ).
% diff_union_single_conv
thf(fact_645_diff__union__single__conv,axiom,
! [A2: product_prod_a_a,J2: multis501812127501805293od_a_a,I2: multis501812127501805293od_a_a] :
( ( member1426531477525435216od_a_a @ A2 @ ( set_ms119794563918438244od_a_a @ J2 ) )
=> ( ( minus_7313152569058643924od_a_a @ ( plus_p6967097483979887908od_a_a @ I2 @ J2 ) @ ( add_ms8655138167283798533od_a_a @ A2 @ zero_z3024502163831676836od_a_a ) )
= ( plus_p6967097483979887908od_a_a @ I2 @ ( minus_7313152569058643924od_a_a @ J2 @ ( add_ms8655138167283798533od_a_a @ A2 @ zero_z3024502163831676836od_a_a ) ) ) ) ) ).
% diff_union_single_conv
thf(fact_646_diff__union__single__conv,axiom,
! [A2: a,J2: multiset_a,I2: multiset_a] :
( ( member_a @ A2 @ ( set_mset_a @ J2 ) )
=> ( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ I2 @ J2 ) @ ( add_mset_a @ A2 @ zero_zero_multiset_a ) )
= ( plus_plus_multiset_a @ I2 @ ( minus_3765977307040488491iset_a @ J2 @ ( add_mset_a @ A2 @ zero_zero_multiset_a ) ) ) ) ) ).
% diff_union_single_conv
thf(fact_647_insert__DiffM2,axiom,
! [X2: produc6518373309651786023iset_a,M: multis4781622204924204013iset_a] :
( ( member5199237121806060112iset_a @ X2 @ ( set_ms2379723421352415844iset_a @ M ) )
=> ( ( plus_p1053901119268525860iset_a @ ( minus_9211619285413683668iset_a @ M @ ( add_ms5348986034529679365iset_a @ X2 @ zero_z4591712717340762532iset_a ) ) @ ( add_ms5348986034529679365iset_a @ X2 @ zero_z4591712717340762532iset_a ) )
= M ) ) ).
% insert_DiffM2
thf(fact_648_insert__DiffM2,axiom,
! [X2: product_prod_a_a,M: multis501812127501805293od_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ ( set_ms119794563918438244od_a_a @ M ) )
=> ( ( plus_p6967097483979887908od_a_a @ ( minus_7313152569058643924od_a_a @ M @ ( add_ms8655138167283798533od_a_a @ X2 @ zero_z3024502163831676836od_a_a ) ) @ ( add_ms8655138167283798533od_a_a @ X2 @ zero_z3024502163831676836od_a_a ) )
= M ) ) ).
% insert_DiffM2
thf(fact_649_insert__DiffM2,axiom,
! [X2: a,M: multiset_a] :
( ( member_a @ X2 @ ( set_mset_a @ M ) )
=> ( ( plus_plus_multiset_a @ ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ X2 @ zero_zero_multiset_a ) ) @ ( add_mset_a @ X2 @ zero_zero_multiset_a ) )
= M ) ) ).
% insert_DiffM2
thf(fact_650_divides__aux__eq,axiom,
! [Q2: nat,R: nat] :
( ( unique5332122412489317741ux_nat @ ( product_Pair_nat_nat @ Q2 @ R ) )
= ( R = zero_zero_nat ) ) ).
% divides_aux_eq
thf(fact_651_mono__mult2__ns,axiom,
! [Ns: set_Product_prod_a_a,Ns2: set_Product_prod_a_a,S: set_Product_prod_a_a,S2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ Ns @ Ns2 )
=> ( ( ord_le746702958409616551od_a_a @ S @ S2 )
=> ( ord_le5871882978999109543iset_a @ ( multis89121590392594451782_ns_a @ Ns @ S ) @ ( multis89121590392594451782_ns_a @ Ns2 @ S2 ) ) ) ) ).
% mono_mult2_ns
thf(fact_652_mono__mult2__ns,axiom,
! [Ns: set_Pr79727621955416071iset_a,Ns2: set_Pr79727621955416071iset_a,S: set_Pr79727621955416071iset_a,S2: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ Ns @ Ns2 )
=> ( ( ord_le5871882978999109543iset_a @ S @ S2 )
=> ( ord_le2895628588359463079iset_a @ ( multis7342709121975751424iset_a @ Ns @ S ) @ ( multis7342709121975751424iset_a @ Ns2 @ S2 ) ) ) ) ).
% mono_mult2_ns
thf(fact_653_zero__diff,axiom,
! [A2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% zero_diff
thf(fact_654_diff__zero,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% diff_zero
thf(fact_655_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_656_add__diff__cancel__left,axiom,
! [C: multiset_a,A2: multiset_a,B: multiset_a] :
( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ C @ A2 ) @ ( plus_plus_multiset_a @ C @ B ) )
= ( minus_3765977307040488491iset_a @ A2 @ B ) ) ).
% add_diff_cancel_left
thf(fact_657_add__diff__cancel__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A2 @ B ) ) ).
% add_diff_cancel_left
thf(fact_658_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_659_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_660_add__diff__cancel__right,axiom,
! [A2: multiset_a,C: multiset_a,B: multiset_a] :
( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ A2 @ C ) @ ( plus_plus_multiset_a @ B @ C ) )
= ( minus_3765977307040488491iset_a @ A2 @ B ) ) ).
% add_diff_cancel_right
thf(fact_661_add__diff__cancel__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A2 @ B ) ) ).
% add_diff_cancel_right
thf(fact_662_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_663_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_664_diff__diff__add__mset,axiom,
! [M: multiset_a,N: multiset_a,P: multiset_a] :
( ( minus_3765977307040488491iset_a @ ( minus_3765977307040488491iset_a @ M @ N ) @ P )
= ( minus_3765977307040488491iset_a @ M @ ( plus_plus_multiset_a @ N @ P ) ) ) ).
% diff_diff_add_mset
thf(fact_665_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_666_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_667_add__mset__remove__trivial,axiom,
! [X2: a,M: multiset_a] :
( ( minus_3765977307040488491iset_a @ ( add_mset_a @ X2 @ M ) @ ( add_mset_a @ X2 @ zero_zero_multiset_a ) )
= M ) ).
% add_mset_remove_trivial
thf(fact_668_diff__add__mset__swap,axiom,
! [B: produc6518373309651786023iset_a,A3: multis4781622204924204013iset_a,M: multis4781622204924204013iset_a] :
( ~ ( member5199237121806060112iset_a @ B @ ( set_ms2379723421352415844iset_a @ A3 ) )
=> ( ( minus_9211619285413683668iset_a @ ( add_ms5348986034529679365iset_a @ B @ M ) @ A3 )
= ( add_ms5348986034529679365iset_a @ B @ ( minus_9211619285413683668iset_a @ M @ A3 ) ) ) ) ).
% diff_add_mset_swap
thf(fact_669_diff__add__mset__swap,axiom,
! [B: product_prod_a_a,A3: multis501812127501805293od_a_a,M: multis501812127501805293od_a_a] :
( ~ ( member1426531477525435216od_a_a @ B @ ( set_ms119794563918438244od_a_a @ A3 ) )
=> ( ( minus_7313152569058643924od_a_a @ ( add_ms8655138167283798533od_a_a @ B @ M ) @ A3 )
= ( add_ms8655138167283798533od_a_a @ B @ ( minus_7313152569058643924od_a_a @ M @ A3 ) ) ) ) ).
% diff_add_mset_swap
thf(fact_670_diff__add__mset__swap,axiom,
! [B: a,A3: multiset_a,M: multiset_a] :
( ~ ( member_a @ B @ ( set_mset_a @ A3 ) )
=> ( ( minus_3765977307040488491iset_a @ ( add_mset_a @ B @ M ) @ A3 )
= ( add_mset_a @ B @ ( minus_3765977307040488491iset_a @ M @ A3 ) ) ) ) ).
% diff_add_mset_swap
thf(fact_671_insert__DiffM,axiom,
! [X2: produc6518373309651786023iset_a,M: multis4781622204924204013iset_a] :
( ( member5199237121806060112iset_a @ X2 @ ( set_ms2379723421352415844iset_a @ M ) )
=> ( ( add_ms5348986034529679365iset_a @ X2 @ ( minus_9211619285413683668iset_a @ M @ ( add_ms5348986034529679365iset_a @ X2 @ zero_z4591712717340762532iset_a ) ) )
= M ) ) ).
% insert_DiffM
thf(fact_672_insert__DiffM,axiom,
! [X2: product_prod_a_a,M: multis501812127501805293od_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ ( set_ms119794563918438244od_a_a @ M ) )
=> ( ( add_ms8655138167283798533od_a_a @ X2 @ ( minus_7313152569058643924od_a_a @ M @ ( add_ms8655138167283798533od_a_a @ X2 @ zero_z3024502163831676836od_a_a ) ) )
= M ) ) ).
% insert_DiffM
thf(fact_673_insert__DiffM,axiom,
! [X2: a,M: multiset_a] :
( ( member_a @ X2 @ ( set_mset_a @ M ) )
=> ( ( add_mset_a @ X2 @ ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ X2 @ zero_zero_multiset_a ) ) )
= M ) ) ).
% insert_DiffM
thf(fact_674_diff__union__swap2,axiom,
! [Y: produc6518373309651786023iset_a,M: multis4781622204924204013iset_a,X2: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Y @ ( set_ms2379723421352415844iset_a @ M ) )
=> ( ( minus_9211619285413683668iset_a @ ( add_ms5348986034529679365iset_a @ X2 @ M ) @ ( add_ms5348986034529679365iset_a @ Y @ zero_z4591712717340762532iset_a ) )
= ( add_ms5348986034529679365iset_a @ X2 @ ( minus_9211619285413683668iset_a @ M @ ( add_ms5348986034529679365iset_a @ Y @ zero_z4591712717340762532iset_a ) ) ) ) ) ).
% diff_union_swap2
thf(fact_675_diff__union__swap2,axiom,
! [Y: product_prod_a_a,M: multis501812127501805293od_a_a,X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y @ ( set_ms119794563918438244od_a_a @ M ) )
=> ( ( minus_7313152569058643924od_a_a @ ( add_ms8655138167283798533od_a_a @ X2 @ M ) @ ( add_ms8655138167283798533od_a_a @ Y @ zero_z3024502163831676836od_a_a ) )
= ( add_ms8655138167283798533od_a_a @ X2 @ ( minus_7313152569058643924od_a_a @ M @ ( add_ms8655138167283798533od_a_a @ Y @ zero_z3024502163831676836od_a_a ) ) ) ) ) ).
% diff_union_swap2
thf(fact_676_diff__union__swap2,axiom,
! [Y: a,M: multiset_a,X2: a] :
( ( member_a @ Y @ ( set_mset_a @ M ) )
=> ( ( minus_3765977307040488491iset_a @ ( add_mset_a @ X2 @ M ) @ ( add_mset_a @ Y @ zero_zero_multiset_a ) )
= ( add_mset_a @ X2 @ ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ Y @ zero_zero_multiset_a ) ) ) ) ) ).
% diff_union_swap2
thf(fact_677_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A2: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_678_add__implies__diff,axiom,
! [C: multiset_a,B: multiset_a,A2: multiset_a] :
( ( ( plus_plus_multiset_a @ C @ B )
= A2 )
=> ( C
= ( minus_3765977307040488491iset_a @ A2 @ B ) ) ) ).
% add_implies_diff
thf(fact_679_add__implies__diff,axiom,
! [C: nat,B: nat,A2: nat] :
( ( ( plus_plus_nat @ C @ B )
= A2 )
=> ( C
= ( minus_minus_nat @ A2 @ B ) ) ) ).
% add_implies_diff
thf(fact_680_diff__diff__eq,axiom,
! [A2: multiset_a,B: multiset_a,C: multiset_a] :
( ( minus_3765977307040488491iset_a @ ( minus_3765977307040488491iset_a @ A2 @ B ) @ C )
= ( minus_3765977307040488491iset_a @ A2 @ ( plus_plus_multiset_a @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_681_diff__diff__eq,axiom,
! [A2: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B ) @ C )
= ( minus_minus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_682_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_683_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_684_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_685_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_686_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_687_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_688_add__mset__diff__bothsides,axiom,
! [A2: a,M: multiset_a,A3: multiset_a] :
( ( minus_3765977307040488491iset_a @ ( add_mset_a @ A2 @ M ) @ ( add_mset_a @ A2 @ A3 ) )
= ( minus_3765977307040488491iset_a @ M @ A3 ) ) ).
% add_mset_diff_bothsides
thf(fact_689_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( minus_minus_nat @ B @ A2 )
= C )
= ( B
= ( plus_plus_nat @ C @ A2 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_690_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_691_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_692_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A2 )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A2 ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_693_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A2 ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_694_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A2 )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_695_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_696_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_697_le__add__diff,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A2 ) ) ) ).
% le_add_diff
thf(fact_698_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_699_diff__union__swap,axiom,
! [A2: a,B: a,M: multiset_a] :
( ( A2 != B )
=> ( ( add_mset_a @ B @ ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ A2 @ zero_zero_multiset_a ) ) )
= ( minus_3765977307040488491iset_a @ ( add_mset_a @ B @ M ) @ ( add_mset_a @ A2 @ zero_zero_multiset_a ) ) ) ) ).
% diff_union_swap
thf(fact_700_add__eq__conv__diff,axiom,
! [A2: a,M: multiset_a,B: a,N: multiset_a] :
( ( ( add_mset_a @ A2 @ M )
= ( add_mset_a @ B @ N ) )
= ( ( ( M = N )
& ( A2 = B ) )
| ( ( M
= ( add_mset_a @ B @ ( minus_3765977307040488491iset_a @ N @ ( add_mset_a @ A2 @ zero_zero_multiset_a ) ) ) )
& ( N
= ( add_mset_a @ A2 @ ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ B @ zero_zero_multiset_a ) ) ) ) ) ) ) ).
% add_eq_conv_diff
thf(fact_701_union__single__eq__diff,axiom,
! [X2: a,M: multiset_a,N: multiset_a] :
( ( ( add_mset_a @ X2 @ M )
= N )
=> ( M
= ( minus_3765977307040488491iset_a @ N @ ( add_mset_a @ X2 @ zero_zero_multiset_a ) ) ) ) ).
% union_single_eq_diff
thf(fact_702_lcl__trans__l,axiom,
! [R12: set_Product_prod_a_a,R23: set_Product_prod_a_a,A3: multiset_a,B2: multiset_a,C2: multiset_a,D2: multiset_a,E: multiset_a,F2: multiset_a] :
( ( locall3774189849527508434le_l_a @ R12 @ R23 @ ( plus_plus_multiset_a @ A3 @ B2 ) @ ( plus_plus_multiset_a @ C2 @ D2 ) @ ( plus_plus_multiset_a @ E @ F2 ) )
=> ( locall3774189849527508434le_l_a @ R12 @ R23 @ A3 @ C2 @ E ) ) ).
% lcl_trans_l
thf(fact_703_lcl__trans__r,axiom,
! [R12: set_Product_prod_a_a,R23: set_Product_prod_a_a,A3: multiset_a,B2: multiset_a,C2: multiset_a,D2: multiset_a,E: multiset_a,F2: multiset_a] :
( ( locall3774189849527508434le_l_a @ R12 @ R23 @ ( plus_plus_multiset_a @ A3 @ B2 ) @ ( plus_plus_multiset_a @ C2 @ D2 ) @ ( plus_plus_multiset_a @ E @ F2 ) )
=> ( locall3774189849527508434le_l_a @ R12 @ R23 @ B2 @ D2 @ F2 ) ) ).
% lcl_trans_r
thf(fact_704_lcr__trans__l,axiom,
! [R12: set_Product_prod_a_a,R23: set_Product_prod_a_a,A3: multiset_a,B2: multiset_a,C2: multiset_a,D2: multiset_a,E: multiset_a,F2: multiset_a] :
( ( locall5496200431401003852le_r_a @ R12 @ R23 @ ( plus_plus_multiset_a @ A3 @ B2 ) @ ( plus_plus_multiset_a @ C2 @ D2 ) @ ( plus_plus_multiset_a @ E @ F2 ) )
=> ( locall5496200431401003852le_r_a @ R12 @ R23 @ A3 @ C2 @ E ) ) ).
% lcr_trans_l
thf(fact_705_lcr__trans__r,axiom,
! [R12: set_Product_prod_a_a,R23: set_Product_prod_a_a,A3: multiset_a,B2: multiset_a,C2: multiset_a,D2: multiset_a,E: multiset_a,F2: multiset_a] :
( ( locall5496200431401003852le_r_a @ R12 @ R23 @ ( plus_plus_multiset_a @ A3 @ B2 ) @ ( plus_plus_multiset_a @ C2 @ D2 ) @ ( plus_plus_multiset_a @ E @ F2 ) )
=> ( locall5496200431401003852le_r_a @ R12 @ R23 @ B2 @ D2 @ F2 ) ) ).
% lcr_trans_r
thf(fact_706_diff__single__trivial,axiom,
! [X2: produc6518373309651786023iset_a,M: multis4781622204924204013iset_a] :
( ~ ( member5199237121806060112iset_a @ X2 @ ( set_ms2379723421352415844iset_a @ M ) )
=> ( ( minus_9211619285413683668iset_a @ M @ ( add_ms5348986034529679365iset_a @ X2 @ zero_z4591712717340762532iset_a ) )
= M ) ) ).
% diff_single_trivial
thf(fact_707_diff__single__trivial,axiom,
! [X2: product_prod_a_a,M: multis501812127501805293od_a_a] :
( ~ ( member1426531477525435216od_a_a @ X2 @ ( set_ms119794563918438244od_a_a @ M ) )
=> ( ( minus_7313152569058643924od_a_a @ M @ ( add_ms8655138167283798533od_a_a @ X2 @ zero_z3024502163831676836od_a_a ) )
= M ) ) ).
% diff_single_trivial
thf(fact_708_diff__single__trivial,axiom,
! [X2: a,M: multiset_a] :
( ~ ( member_a @ X2 @ ( set_mset_a @ M ) )
=> ( ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ X2 @ zero_zero_multiset_a ) )
= M ) ) ).
% diff_single_trivial
thf(fact_709_diff__single__eq__union,axiom,
! [X2: produc6518373309651786023iset_a,M: multis4781622204924204013iset_a,N: multis4781622204924204013iset_a] :
( ( member5199237121806060112iset_a @ X2 @ ( set_ms2379723421352415844iset_a @ M ) )
=> ( ( ( minus_9211619285413683668iset_a @ M @ ( add_ms5348986034529679365iset_a @ X2 @ zero_z4591712717340762532iset_a ) )
= N )
= ( M
= ( add_ms5348986034529679365iset_a @ X2 @ N ) ) ) ) ).
% diff_single_eq_union
thf(fact_710_diff__single__eq__union,axiom,
! [X2: product_prod_a_a,M: multis501812127501805293od_a_a,N: multis501812127501805293od_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ ( set_ms119794563918438244od_a_a @ M ) )
=> ( ( ( minus_7313152569058643924od_a_a @ M @ ( add_ms8655138167283798533od_a_a @ X2 @ zero_z3024502163831676836od_a_a ) )
= N )
= ( M
= ( add_ms8655138167283798533od_a_a @ X2 @ N ) ) ) ) ).
% diff_single_eq_union
thf(fact_711_diff__single__eq__union,axiom,
! [X2: a,M: multiset_a,N: multiset_a] :
( ( member_a @ X2 @ ( set_mset_a @ M ) )
=> ( ( ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ X2 @ zero_zero_multiset_a ) )
= N )
= ( M
= ( add_mset_a @ X2 @ N ) ) ) ) ).
% diff_single_eq_union
thf(fact_712_multi__drop__mem__not__eq,axiom,
! [C: produc6518373309651786023iset_a,B2: multis4781622204924204013iset_a] :
( ( member5199237121806060112iset_a @ C @ ( set_ms2379723421352415844iset_a @ B2 ) )
=> ( ( minus_9211619285413683668iset_a @ B2 @ ( add_ms5348986034529679365iset_a @ C @ zero_z4591712717340762532iset_a ) )
!= B2 ) ) ).
% multi_drop_mem_not_eq
thf(fact_713_multi__drop__mem__not__eq,axiom,
! [C: product_prod_a_a,B2: multis501812127501805293od_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( set_ms119794563918438244od_a_a @ B2 ) )
=> ( ( minus_7313152569058643924od_a_a @ B2 @ ( add_ms8655138167283798533od_a_a @ C @ zero_z3024502163831676836od_a_a ) )
!= B2 ) ) ).
% multi_drop_mem_not_eq
thf(fact_714_multi__drop__mem__not__eq,axiom,
! [C: a,B2: multiset_a] :
( ( member_a @ C @ ( set_mset_a @ B2 ) )
=> ( ( minus_3765977307040488491iset_a @ B2 @ ( add_mset_a @ C @ zero_zero_multiset_a ) )
!= B2 ) ) ).
% multi_drop_mem_not_eq
thf(fact_715_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_716_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_717_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_718_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_719_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_720_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_721_multiset__add__sub__el__shuffle,axiom,
! [C: produc6518373309651786023iset_a,B2: multis4781622204924204013iset_a,B: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ C @ ( set_ms2379723421352415844iset_a @ B2 ) )
=> ( ( B != C )
=> ( ( add_ms5348986034529679365iset_a @ B @ ( minus_9211619285413683668iset_a @ B2 @ ( add_ms5348986034529679365iset_a @ C @ zero_z4591712717340762532iset_a ) ) )
= ( minus_9211619285413683668iset_a @ ( add_ms5348986034529679365iset_a @ B @ B2 ) @ ( add_ms5348986034529679365iset_a @ C @ zero_z4591712717340762532iset_a ) ) ) ) ) ).
% multiset_add_sub_el_shuffle
thf(fact_722_multiset__add__sub__el__shuffle,axiom,
! [C: product_prod_a_a,B2: multis501812127501805293od_a_a,B: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( set_ms119794563918438244od_a_a @ B2 ) )
=> ( ( B != C )
=> ( ( add_ms8655138167283798533od_a_a @ B @ ( minus_7313152569058643924od_a_a @ B2 @ ( add_ms8655138167283798533od_a_a @ C @ zero_z3024502163831676836od_a_a ) ) )
= ( minus_7313152569058643924od_a_a @ ( add_ms8655138167283798533od_a_a @ B @ B2 ) @ ( add_ms8655138167283798533od_a_a @ C @ zero_z3024502163831676836od_a_a ) ) ) ) ) ).
% multiset_add_sub_el_shuffle
thf(fact_723_multiset__add__sub__el__shuffle,axiom,
! [C: a,B2: multiset_a,B: a] :
( ( member_a @ C @ ( set_mset_a @ B2 ) )
=> ( ( B != C )
=> ( ( add_mset_a @ B @ ( minus_3765977307040488491iset_a @ B2 @ ( add_mset_a @ C @ zero_zero_multiset_a ) ) )
= ( minus_3765977307040488491iset_a @ ( add_mset_a @ B @ B2 ) @ ( add_mset_a @ C @ zero_zero_multiset_a ) ) ) ) ) ).
% multiset_add_sub_el_shuffle
thf(fact_724_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_725_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_726_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_727_multpw__implies__mult2__ns,axiom,
! [X3: multiset_a,Y2: multiset_a,Ns: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ Y2 ) @ ( multis2373687813370458242ltpw_a @ Ns ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ Y2 ) @ ( multis89121590392594451782_ns_a @ Ns @ S ) ) ) ).
% multpw_implies_mult2_ns
thf(fact_728_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_729_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_730_add__le__add__imp__diff__le,axiom,
! [I: nat,K3: nat,N3: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K3 ) @ N3 )
=> ( ( ord_less_eq_nat @ N3 @ ( plus_plus_nat @ J @ K3 ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K3 ) @ N3 )
=> ( ( ord_less_eq_nat @ N3 @ ( plus_plus_nat @ J @ K3 ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N3 @ K3 ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_731_add__le__imp__le__diff,axiom,
! [I: nat,K3: nat,N3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K3 ) @ N3 )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N3 @ K3 ) ) ) ).
% add_le_imp_le_diff
thf(fact_732_mult2__ns__def,axiom,
( multis89121590392594451782_ns_a
= ( ^ [Ns3: set_Product_prod_a_a,S4: set_Product_prod_a_a] : ( relcom5893175440217691907iset_a @ ( multis2373687813370458242ltpw_a @ Ns3 ) @ ( sup_su1925787886501348827iset_a @ ( mult_a @ S4 ) @ id_multiset_a ) ) ) ) ).
% mult2_ns_def
thf(fact_733_Un__Diff__cancel,axiom,
! [A3: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A3 @ ( minus_6817036919807184750od_a_a @ B2 @ A3 ) )
= ( sup_su3048258781599657691od_a_a @ A3 @ B2 ) ) ).
% Un_Diff_cancel
thf(fact_734_Un__Diff__cancel2,axiom,
! [B2: set_Product_prod_a_a,A3: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( minus_6817036919807184750od_a_a @ B2 @ A3 ) @ A3 )
= ( sup_su3048258781599657691od_a_a @ B2 @ A3 ) ) ).
% Un_Diff_cancel2
thf(fact_735_relcomp__distrib,axiom,
! [R3: set_Product_prod_a_a,S3: set_Product_prod_a_a,T2: set_Product_prod_a_a] :
( ( relcomp_a_a_a @ R3 @ ( sup_su3048258781599657691od_a_a @ S3 @ T2 ) )
= ( sup_su3048258781599657691od_a_a @ ( relcomp_a_a_a @ R3 @ S3 ) @ ( relcomp_a_a_a @ R3 @ T2 ) ) ) ).
% relcomp_distrib
thf(fact_736_relcomp__distrib2,axiom,
! [S3: set_Product_prod_a_a,T2: set_Product_prod_a_a,R3: set_Product_prod_a_a] :
( ( relcomp_a_a_a @ ( sup_su3048258781599657691od_a_a @ S3 @ T2 ) @ R3 )
= ( sup_su3048258781599657691od_a_a @ ( relcomp_a_a_a @ S3 @ R3 ) @ ( relcomp_a_a_a @ T2 @ R3 ) ) ) ).
% relcomp_distrib2
thf(fact_737_relcomp_Ocases,axiom,
! [A13: multiset_a,A24: multiset_a,R: set_Pr79727621955416071iset_a,S: set_Pr79727621955416071iset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A13 @ A24 ) @ ( relcom5893175440217691907iset_a @ R @ S ) )
=> ~ ! [B8: multiset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A13 @ B8 ) @ R )
=> ~ ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ B8 @ A24 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_738_relcomp_Ocases,axiom,
! [A13: a,A24: a,R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A13 @ A24 ) @ ( relcomp_a_a_a @ R @ S ) )
=> ~ ! [B8: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A13 @ B8 ) @ R )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B8 @ A24 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_739_relcomp_Osimps,axiom,
! [A13: multiset_a,A24: multiset_a,R: set_Pr79727621955416071iset_a,S: set_Pr79727621955416071iset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A13 @ A24 ) @ ( relcom5893175440217691907iset_a @ R @ S ) )
= ( ? [A5: multiset_a,B5: multiset_a,C4: multiset_a] :
( ( A13 = A5 )
& ( A24 = C4 )
& ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A5 @ B5 ) @ R )
& ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ B5 @ C4 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_740_relcomp_Osimps,axiom,
! [A13: a,A24: a,R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A13 @ A24 ) @ ( relcomp_a_a_a @ R @ S ) )
= ( ? [A5: a,B5: a,C4: a] :
( ( A13 = A5 )
& ( A24 = C4 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A5 @ B5 ) @ R )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B5 @ C4 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_741_relcomp_OrelcompI,axiom,
! [A2: multiset_a,B: multiset_a,R: set_Pr79727621955416071iset_a,C: multiset_a,S: set_Pr79727621955416071iset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A2 @ B ) @ R )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ B @ C ) @ S )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A2 @ C ) @ ( relcom5893175440217691907iset_a @ R @ S ) ) ) ) ).
% relcomp.relcompI
thf(fact_742_relcomp_OrelcompI,axiom,
! [A2: a,B: a,R: set_Product_prod_a_a,C: a,S: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B ) @ R )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ C ) @ S )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ C ) @ ( relcomp_a_a_a @ R @ S ) ) ) ) ).
% relcomp.relcompI
thf(fact_743_relcompE,axiom,
! [Xz: produc6518373309651786023iset_a,R: set_Pr79727621955416071iset_a,S: set_Pr79727621955416071iset_a] :
( ( member5199237121806060112iset_a @ Xz @ ( relcom5893175440217691907iset_a @ R @ S ) )
=> ~ ! [X5: multiset_a,Y5: multiset_a,Z: multiset_a] :
( ( Xz
= ( produc654756711066625303iset_a @ X5 @ Z ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Y5 ) @ R )
=> ~ ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Y5 @ Z ) @ S ) ) ) ) ).
% relcompE
thf(fact_744_relcompE,axiom,
! [Xz: product_prod_a_a,R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Xz @ ( relcomp_a_a_a @ R @ S ) )
=> ~ ! [X5: a,Y5: a,Z: a] :
( ( Xz
= ( product_Pair_a_a @ X5 @ Z ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y5 ) @ R )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y5 @ Z ) @ S ) ) ) ) ).
% relcompE
thf(fact_745_relcompEpair,axiom,
! [A2: multiset_a,C: multiset_a,R: set_Pr79727621955416071iset_a,S: set_Pr79727621955416071iset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A2 @ C ) @ ( relcom5893175440217691907iset_a @ R @ S ) )
=> ~ ! [B8: multiset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A2 @ B8 ) @ R )
=> ~ ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ B8 @ C ) @ S ) ) ) ).
% relcompEpair
thf(fact_746_relcompEpair,axiom,
! [A2: a,C: a,R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ C ) @ ( relcomp_a_a_a @ R @ S ) )
=> ~ ! [B8: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B8 ) @ R )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B8 @ C ) @ S ) ) ) ).
% relcompEpair
thf(fact_747_Diff__mono,axiom,
! [A3: set_Product_prod_a_a,C2: set_Product_prod_a_a,D2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A3 @ C2 )
=> ( ( ord_le746702958409616551od_a_a @ D2 @ B2 )
=> ( ord_le746702958409616551od_a_a @ ( minus_6817036919807184750od_a_a @ A3 @ B2 ) @ ( minus_6817036919807184750od_a_a @ C2 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_748_Diff__mono,axiom,
! [A3: set_Pr79727621955416071iset_a,C2: set_Pr79727621955416071iset_a,D2: set_Pr79727621955416071iset_a,B2: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ A3 @ C2 )
=> ( ( ord_le5871882978999109543iset_a @ D2 @ B2 )
=> ( ord_le5871882978999109543iset_a @ ( minus_2461365523287052142iset_a @ A3 @ B2 ) @ ( minus_2461365523287052142iset_a @ C2 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_749_Diff__subset,axiom,
! [A3: set_Product_prod_a_a,B2: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( minus_6817036919807184750od_a_a @ A3 @ B2 ) @ A3 ) ).
% Diff_subset
thf(fact_750_Diff__subset,axiom,
! [A3: set_Pr79727621955416071iset_a,B2: set_Pr79727621955416071iset_a] : ( ord_le5871882978999109543iset_a @ ( minus_2461365523287052142iset_a @ A3 @ B2 ) @ A3 ) ).
% Diff_subset
thf(fact_751_double__diff,axiom,
! [A3: set_Product_prod_a_a,B2: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A3 @ B2 )
=> ( ( ord_le746702958409616551od_a_a @ B2 @ C2 )
=> ( ( minus_6817036919807184750od_a_a @ B2 @ ( minus_6817036919807184750od_a_a @ C2 @ A3 ) )
= A3 ) ) ) ).
% double_diff
thf(fact_752_double__diff,axiom,
! [A3: set_Pr79727621955416071iset_a,B2: set_Pr79727621955416071iset_a,C2: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ A3 @ B2 )
=> ( ( ord_le5871882978999109543iset_a @ B2 @ C2 )
=> ( ( minus_2461365523287052142iset_a @ B2 @ ( minus_2461365523287052142iset_a @ C2 @ A3 ) )
= A3 ) ) ) ).
% double_diff
thf(fact_753_relcomp__mono,axiom,
! [R2: set_Product_prod_a_a,R: set_Product_prod_a_a,S2: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ R2 @ R )
=> ( ( ord_le746702958409616551od_a_a @ S2 @ S )
=> ( ord_le746702958409616551od_a_a @ ( relcomp_a_a_a @ R2 @ S2 ) @ ( relcomp_a_a_a @ R @ S ) ) ) ) ).
% relcomp_mono
thf(fact_754_relcomp__mono,axiom,
! [R2: set_Pr79727621955416071iset_a,R: set_Pr79727621955416071iset_a,S2: set_Pr79727621955416071iset_a,S: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ R2 @ R )
=> ( ( ord_le5871882978999109543iset_a @ S2 @ S )
=> ( ord_le5871882978999109543iset_a @ ( relcom5893175440217691907iset_a @ R2 @ S2 ) @ ( relcom5893175440217691907iset_a @ R @ S ) ) ) ) ).
% relcomp_mono
thf(fact_755_Un__Diff,axiom,
! [A3: set_Product_prod_a_a,B2: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( minus_6817036919807184750od_a_a @ ( sup_su3048258781599657691od_a_a @ A3 @ B2 ) @ C2 )
= ( sup_su3048258781599657691od_a_a @ ( minus_6817036919807184750od_a_a @ A3 @ C2 ) @ ( minus_6817036919807184750od_a_a @ B2 @ C2 ) ) ) ).
% Un_Diff
thf(fact_756_Diff__partition,axiom,
! [A3: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A3 @ B2 )
=> ( ( sup_su3048258781599657691od_a_a @ A3 @ ( minus_6817036919807184750od_a_a @ B2 @ A3 ) )
= B2 ) ) ).
% Diff_partition
thf(fact_757_Diff__partition,axiom,
! [A3: set_Pr79727621955416071iset_a,B2: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ A3 @ B2 )
=> ( ( sup_su1925787886501348827iset_a @ A3 @ ( minus_2461365523287052142iset_a @ B2 @ A3 ) )
= B2 ) ) ).
% Diff_partition
thf(fact_758_Diff__subset__conv,axiom,
! [A3: set_Product_prod_a_a,B2: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( minus_6817036919807184750od_a_a @ A3 @ B2 ) @ C2 )
= ( ord_le746702958409616551od_a_a @ A3 @ ( sup_su3048258781599657691od_a_a @ B2 @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_759_Diff__subset__conv,axiom,
! [A3: set_Pr79727621955416071iset_a,B2: set_Pr79727621955416071iset_a,C2: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ ( minus_2461365523287052142iset_a @ A3 @ B2 ) @ C2 )
= ( ord_le5871882978999109543iset_a @ A3 @ ( sup_su1925787886501348827iset_a @ B2 @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_760_O__mono1,axiom,
! [R3: set_Product_prod_a_a,R5: set_Product_prod_a_a,S3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ R3 @ R5 )
=> ( ord_le746702958409616551od_a_a @ ( relcomp_a_a_a @ S3 @ R3 ) @ ( relcomp_a_a_a @ S3 @ R5 ) ) ) ).
% O_mono1
thf(fact_761_O__mono1,axiom,
! [R3: set_Pr79727621955416071iset_a,R5: set_Pr79727621955416071iset_a,S3: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ R3 @ R5 )
=> ( ord_le5871882978999109543iset_a @ ( relcom5893175440217691907iset_a @ S3 @ R3 ) @ ( relcom5893175440217691907iset_a @ S3 @ R5 ) ) ) ).
% O_mono1
thf(fact_762_O__mono2,axiom,
! [R3: set_Product_prod_a_a,R5: set_Product_prod_a_a,T2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ R3 @ R5 )
=> ( ord_le746702958409616551od_a_a @ ( relcomp_a_a_a @ R3 @ T2 ) @ ( relcomp_a_a_a @ R5 @ T2 ) ) ) ).
% O_mono2
thf(fact_763_O__mono2,axiom,
! [R3: set_Pr79727621955416071iset_a,R5: set_Pr79727621955416071iset_a,T2: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ R3 @ R5 )
=> ( ord_le5871882978999109543iset_a @ ( relcom5893175440217691907iset_a @ R3 @ T2 ) @ ( relcom5893175440217691907iset_a @ R5 @ T2 ) ) ) ).
% O_mono2
thf(fact_764_relcomp3__I,axiom,
! [T3: multiset_a,U3: multiset_a,A3: set_Pr79727621955416071iset_a,S: multiset_a,B2: set_Pr79727621955416071iset_a,V2: multiset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ T3 @ U3 ) @ A3 )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ S @ T3 ) @ B2 )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ U3 @ V2 ) @ B2 )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ S @ V2 ) @ ( relcom5893175440217691907iset_a @ B2 @ ( relcom5893175440217691907iset_a @ A3 @ B2 ) ) ) ) ) ) ).
% relcomp3_I
thf(fact_765_relcomp3__I,axiom,
! [T3: a,U3: a,A3: set_Product_prod_a_a,S: a,B2: set_Product_prod_a_a,V2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ T3 @ U3 ) @ A3 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ S @ T3 ) @ B2 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U3 @ V2 ) @ B2 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ S @ V2 ) @ ( relcomp_a_a_a @ B2 @ ( relcomp_a_a_a @ A3 @ B2 ) ) ) ) ) ) ).
% relcomp3_I
thf(fact_766_sum__mset_Oremove,axiom,
! [X2: multiset_a,A3: multiset_multiset_a] :
( ( member_multiset_a @ X2 @ ( set_mset_multiset_a @ A3 ) )
=> ( ( comm_m4595494212910443721iset_a @ A3 )
= ( plus_plus_multiset_a @ X2 @ ( comm_m4595494212910443721iset_a @ ( minus_3395427628221709681iset_a @ A3 @ ( add_mset_multiset_a @ X2 @ zero_z7799948378220188993iset_a ) ) ) ) ) ) ).
% sum_mset.remove
thf(fact_767_sum__mset_Oremove,axiom,
! [X2: nat,A3: multiset_nat] :
( ( member_nat @ X2 @ ( set_mset_nat @ A3 ) )
=> ( ( comm_m762188921832702859et_nat @ A3 )
= ( plus_plus_nat @ X2 @ ( comm_m762188921832702859et_nat @ ( minus_8522176038001411705et_nat @ A3 @ ( add_mset_nat @ X2 @ zero_z7348594199698428585et_nat ) ) ) ) ) ) ).
% sum_mset.remove
thf(fact_768_mult2__ns__conv,axiom,
( multis89121590392594451782_ns_a
= ( ^ [Ns3: set_Product_prod_a_a,S4: set_Product_prod_a_a] : ( sup_su1925787886501348827iset_a @ ( multis7561474421521238792t2_s_a @ Ns3 @ S4 ) @ ( multis2373687813370458242ltpw_a @ Ns3 ) ) ) ) ).
% mult2_ns_conv
thf(fact_769_DiffI,axiom,
! [C: produc6518373309651786023iset_a,A3: set_Pr79727621955416071iset_a,B2: set_Pr79727621955416071iset_a] :
( ( member5199237121806060112iset_a @ C @ A3 )
=> ( ~ ( member5199237121806060112iset_a @ C @ B2 )
=> ( member5199237121806060112iset_a @ C @ ( minus_2461365523287052142iset_a @ A3 @ B2 ) ) ) ) ).
% DiffI
thf(fact_770_DiffI,axiom,
! [C: a,A3: set_a,B2: set_a] :
( ( member_a @ C @ A3 )
=> ( ~ ( member_a @ C @ B2 )
=> ( member_a @ C @ ( minus_minus_set_a @ A3 @ B2 ) ) ) ) ).
% DiffI
thf(fact_771_DiffI,axiom,
! [C: product_prod_a_a,A3: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ A3 )
=> ( ~ ( member1426531477525435216od_a_a @ C @ B2 )
=> ( member1426531477525435216od_a_a @ C @ ( minus_6817036919807184750od_a_a @ A3 @ B2 ) ) ) ) ).
% DiffI
thf(fact_772_Diff__iff,axiom,
! [C: produc6518373309651786023iset_a,A3: set_Pr79727621955416071iset_a,B2: set_Pr79727621955416071iset_a] :
( ( member5199237121806060112iset_a @ C @ ( minus_2461365523287052142iset_a @ A3 @ B2 ) )
= ( ( member5199237121806060112iset_a @ C @ A3 )
& ~ ( member5199237121806060112iset_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_773_Diff__iff,axiom,
! [C: a,A3: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A3 @ B2 ) )
= ( ( member_a @ C @ A3 )
& ~ ( member_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_774_Diff__iff,axiom,
! [C: product_prod_a_a,A3: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( minus_6817036919807184750od_a_a @ A3 @ B2 ) )
= ( ( member1426531477525435216od_a_a @ C @ A3 )
& ~ ( member1426531477525435216od_a_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_775_in__Union__mset__iff,axiom,
! [X2: produc6518373309651786023iset_a,MM: multis7221419791219231795iset_a] :
( ( member5199237121806060112iset_a @ X2 @ ( set_ms2379723421352415844iset_a @ ( comm_m5130607462838410866iset_a @ MM ) ) )
= ( ? [M2: multis4781622204924204013iset_a] :
( ( member2597461214246177430iset_a @ M2 @ ( set_ms5327031259215843498iset_a @ MM ) )
& ( member5199237121806060112iset_a @ X2 @ ( set_ms2379723421352415844iset_a @ M2 ) ) ) ) ) ).
% in_Union_mset_iff
thf(fact_776_in__Union__mset__iff,axiom,
! [X2: product_prod_a_a,MM: multis5935151272263360051od_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ ( 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 @ X2 @ ( set_ms119794563918438244od_a_a @ M2 ) ) ) ) ) ).
% in_Union_mset_iff
thf(fact_777_in__Union__mset__iff,axiom,
! [X2: a,MM: multiset_multiset_a] :
( ( member_a @ X2 @ ( set_mset_a @ ( comm_m4595494212910443721iset_a @ MM ) ) )
= ( ? [M2: multiset_a] :
( ( member_multiset_a @ M2 @ ( set_mset_multiset_a @ MM ) )
& ( member_a @ X2 @ ( set_mset_a @ M2 ) ) ) ) ) ).
% in_Union_mset_iff
thf(fact_778_sum__mset_Oempty,axiom,
( ( comm_m762188921832702859et_nat @ zero_z7348594199698428585et_nat )
= zero_zero_nat ) ).
% sum_mset.empty
thf(fact_779_sum__mset__0__iff,axiom,
! [M: multiset_nat] :
( ( ( comm_m762188921832702859et_nat @ M )
= zero_zero_nat )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_mset_nat @ M ) )
=> ( X4 = zero_zero_nat ) ) ) ) ).
% sum_mset_0_iff
thf(fact_780_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_781_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_782_sum__mset_Oadd__mset,axiom,
! [X2: multiset_a,N: multiset_multiset_a] :
( ( comm_m4595494212910443721iset_a @ ( add_mset_multiset_a @ X2 @ N ) )
= ( plus_plus_multiset_a @ X2 @ ( comm_m4595494212910443721iset_a @ N ) ) ) ).
% sum_mset.add_mset
thf(fact_783_sum__mset_Oadd__mset,axiom,
! [X2: nat,N: multiset_nat] :
( ( comm_m762188921832702859et_nat @ ( add_mset_nat @ X2 @ N ) )
= ( plus_plus_nat @ X2 @ ( comm_m762188921832702859et_nat @ N ) ) ) ).
% sum_mset.add_mset
thf(fact_784_DiffE,axiom,
! [C: produc6518373309651786023iset_a,A3: set_Pr79727621955416071iset_a,B2: set_Pr79727621955416071iset_a] :
( ( member5199237121806060112iset_a @ C @ ( minus_2461365523287052142iset_a @ A3 @ B2 ) )
=> ~ ( ( member5199237121806060112iset_a @ C @ A3 )
=> ( member5199237121806060112iset_a @ C @ B2 ) ) ) ).
% DiffE
thf(fact_785_DiffE,axiom,
! [C: a,A3: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A3 @ B2 ) )
=> ~ ( ( member_a @ C @ A3 )
=> ( member_a @ C @ B2 ) ) ) ).
% DiffE
thf(fact_786_DiffE,axiom,
! [C: product_prod_a_a,A3: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( minus_6817036919807184750od_a_a @ A3 @ B2 ) )
=> ~ ( ( member1426531477525435216od_a_a @ C @ A3 )
=> ( member1426531477525435216od_a_a @ C @ B2 ) ) ) ).
% DiffE
thf(fact_787_DiffD1,axiom,
! [C: produc6518373309651786023iset_a,A3: set_Pr79727621955416071iset_a,B2: set_Pr79727621955416071iset_a] :
( ( member5199237121806060112iset_a @ C @ ( minus_2461365523287052142iset_a @ A3 @ B2 ) )
=> ( member5199237121806060112iset_a @ C @ A3 ) ) ).
% DiffD1
thf(fact_788_DiffD1,axiom,
! [C: a,A3: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A3 @ B2 ) )
=> ( member_a @ C @ A3 ) ) ).
% DiffD1
thf(fact_789_DiffD1,axiom,
! [C: product_prod_a_a,A3: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( minus_6817036919807184750od_a_a @ A3 @ B2 ) )
=> ( member1426531477525435216od_a_a @ C @ A3 ) ) ).
% DiffD1
thf(fact_790_DiffD2,axiom,
! [C: produc6518373309651786023iset_a,A3: set_Pr79727621955416071iset_a,B2: set_Pr79727621955416071iset_a] :
( ( member5199237121806060112iset_a @ C @ ( minus_2461365523287052142iset_a @ A3 @ B2 ) )
=> ~ ( member5199237121806060112iset_a @ C @ B2 ) ) ).
% DiffD2
thf(fact_791_DiffD2,axiom,
! [C: a,A3: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A3 @ B2 ) )
=> ~ ( member_a @ C @ B2 ) ) ).
% DiffD2
thf(fact_792_DiffD2,axiom,
! [C: product_prod_a_a,A3: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( minus_6817036919807184750od_a_a @ A3 @ B2 ) )
=> ~ ( member1426531477525435216od_a_a @ C @ B2 ) ) ).
% DiffD2
thf(fact_793_sum__mset_Oneutral,axiom,
! [A3: multiset_nat] :
( ! [X5: nat] :
( ( member_nat @ X5 @ ( set_mset_nat @ A3 ) )
=> ( X5 = zero_zero_nat ) )
=> ( ( comm_m762188921832702859et_nat @ A3 )
= zero_zero_nat ) ) ).
% sum_mset.neutral
thf(fact_794_mono__mult2__s,axiom,
! [Ns: set_Product_prod_a_a,Ns2: set_Product_prod_a_a,S: set_Product_prod_a_a,S2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ Ns @ Ns2 )
=> ( ( ord_le746702958409616551od_a_a @ S @ S2 )
=> ( ord_le5871882978999109543iset_a @ ( multis7561474421521238792t2_s_a @ Ns @ S ) @ ( multis7561474421521238792t2_s_a @ Ns2 @ S2 ) ) ) ) ).
% mono_mult2_s
thf(fact_795_mono__mult2__s,axiom,
! [Ns: set_Pr79727621955416071iset_a,Ns2: set_Pr79727621955416071iset_a,S: set_Pr79727621955416071iset_a,S2: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ Ns @ Ns2 )
=> ( ( ord_le5871882978999109543iset_a @ S @ S2 )
=> ( ord_le2895628588359463079iset_a @ ( multis5679217632910424910iset_a @ Ns @ S ) @ ( multis5679217632910424910iset_a @ Ns2 @ S2 ) ) ) ) ).
% mono_mult2_s
thf(fact_796_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_797_sum__mset_Oeq__fold,axiom,
( comm_m762188921832702859et_nat
= ( fold_mset_nat_nat @ plus_plus_nat @ zero_zero_nat ) ) ).
% sum_mset.eq_fold
thf(fact_798_mult2__s__def,axiom,
( multis7561474421521238792t2_s_a
= ( ^ [Ns3: set_Product_prod_a_a,S4: set_Product_prod_a_a] : ( relcom5893175440217691907iset_a @ ( multis2373687813370458242ltpw_a @ Ns3 ) @ ( mult_a @ S4 ) ) ) ) ).
% mult2_s_def
thf(fact_799_compatible__l__def,axiom,
( compatible_l_a
= ( ^ [R1: set_Product_prod_a_a,R22: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( relcomp_a_a_a @ R1 @ R22 ) @ R22 ) ) ) ).
% compatible_l_def
thf(fact_800_compatible__l__def,axiom,
( compat8834835292239961078iset_a
= ( ^ [R1: set_Pr79727621955416071iset_a,R22: set_Pr79727621955416071iset_a] : ( ord_le5871882978999109543iset_a @ ( relcom5893175440217691907iset_a @ R1 @ R22 ) @ R22 ) ) ) ).
% compatible_l_def
thf(fact_801_compatible__r__def,axiom,
( compatible_r_a
= ( ^ [R1: set_Product_prod_a_a,R22: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( relcomp_a_a_a @ R22 @ R1 ) @ R22 ) ) ) ).
% compatible_r_def
thf(fact_802_compatible__r__def,axiom,
( compat7666912761205960816iset_a
= ( ^ [R1: set_Pr79727621955416071iset_a,R22: set_Pr79727621955416071iset_a] : ( ord_le5871882978999109543iset_a @ ( relcom5893175440217691907iset_a @ R22 @ R1 ) @ R22 ) ) ) ).
% compatible_r_def
thf(fact_803_insert__subset__eq__iff,axiom,
! [A2: produc6518373309651786023iset_a,A3: multis4781622204924204013iset_a,B2: multis4781622204924204013iset_a] :
( ( subset6292090195676340366iset_a @ ( add_ms5348986034529679365iset_a @ A2 @ A3 ) @ B2 )
= ( ( member5199237121806060112iset_a @ A2 @ ( set_ms2379723421352415844iset_a @ B2 ) )
& ( subset6292090195676340366iset_a @ A3 @ ( minus_9211619285413683668iset_a @ B2 @ ( add_ms5348986034529679365iset_a @ A2 @ zero_z4591712717340762532iset_a ) ) ) ) ) ).
% insert_subset_eq_iff
thf(fact_804_insert__subset__eq__iff,axiom,
! [A2: product_prod_a_a,A3: multis501812127501805293od_a_a,B2: multis501812127501805293od_a_a] :
( ( subset4108007524207251342od_a_a @ ( add_ms8655138167283798533od_a_a @ A2 @ A3 ) @ B2 )
= ( ( member1426531477525435216od_a_a @ A2 @ ( set_ms119794563918438244od_a_a @ B2 ) )
& ( subset4108007524207251342od_a_a @ A3 @ ( minus_7313152569058643924od_a_a @ B2 @ ( add_ms8655138167283798533od_a_a @ A2 @ zero_z3024502163831676836od_a_a ) ) ) ) ) ).
% insert_subset_eq_iff
thf(fact_805_insert__subset__eq__iff,axiom,
! [A2: a,A3: multiset_a,B2: multiset_a] :
( ( subseteq_mset_a @ ( add_mset_a @ A2 @ A3 ) @ B2 )
= ( ( member_a @ A2 @ ( set_mset_a @ B2 ) )
& ( subseteq_mset_a @ A3 @ ( minus_3765977307040488491iset_a @ B2 @ ( add_mset_a @ A2 @ zero_zero_multiset_a ) ) ) ) ) ).
% insert_subset_eq_iff
thf(fact_806_insert__union__subset__iff,axiom,
! [A2: produc6518373309651786023iset_a,A3: multis4781622204924204013iset_a,B2: multis4781622204924204013iset_a] :
( ( subset5376938936802007386iset_a @ ( add_ms5348986034529679365iset_a @ A2 @ A3 ) @ B2 )
= ( ( member5199237121806060112iset_a @ A2 @ ( set_ms2379723421352415844iset_a @ B2 ) )
& ( subset5376938936802007386iset_a @ A3 @ ( minus_9211619285413683668iset_a @ B2 @ ( add_ms5348986034529679365iset_a @ A2 @ zero_z4591712717340762532iset_a ) ) ) ) ) ).
% insert_union_subset_iff
thf(fact_807_insert__union__subset__iff,axiom,
! [A2: product_prod_a_a,A3: multis501812127501805293od_a_a,B2: multis501812127501805293od_a_a] :
( ( subset8265789482553336922od_a_a @ ( add_ms8655138167283798533od_a_a @ A2 @ A3 ) @ B2 )
= ( ( member1426531477525435216od_a_a @ A2 @ ( set_ms119794563918438244od_a_a @ B2 ) )
& ( subset8265789482553336922od_a_a @ A3 @ ( minus_7313152569058643924od_a_a @ B2 @ ( add_ms8655138167283798533od_a_a @ A2 @ zero_z3024502163831676836od_a_a ) ) ) ) ) ).
% insert_union_subset_iff
thf(fact_808_insert__union__subset__iff,axiom,
! [A2: a,A3: multiset_a,B2: multiset_a] :
( ( subset_mset_a @ ( add_mset_a @ A2 @ A3 ) @ B2 )
= ( ( member_a @ A2 @ ( set_mset_a @ B2 ) )
& ( subset_mset_a @ A3 @ ( minus_3765977307040488491iset_a @ B2 @ ( add_mset_a @ A2 @ zero_zero_multiset_a ) ) ) ) ) ).
% insert_union_subset_iff
thf(fact_809_mset__subset__eq__mono__add__right__cancel,axiom,
! [A3: multiset_a,C2: multiset_a,B2: multiset_a] :
( ( subseteq_mset_a @ ( plus_plus_multiset_a @ A3 @ C2 ) @ ( plus_plus_multiset_a @ B2 @ C2 ) )
= ( subseteq_mset_a @ A3 @ B2 ) ) ).
% mset_subset_eq_mono_add_right_cancel
thf(fact_810_mset__subset__eq__mono__add__left__cancel,axiom,
! [C2: multiset_a,A3: multiset_a,B2: multiset_a] :
( ( subseteq_mset_a @ ( plus_plus_multiset_a @ C2 @ A3 ) @ ( plus_plus_multiset_a @ C2 @ B2 ) )
= ( subseteq_mset_a @ A3 @ B2 ) ) ).
% mset_subset_eq_mono_add_left_cancel
thf(fact_811_subset__mset_Oadd__le__cancel__right,axiom,
! [A2: multiset_a,C: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ ( plus_plus_multiset_a @ A2 @ C ) @ ( plus_plus_multiset_a @ B @ C ) )
= ( subseteq_mset_a @ A2 @ B ) ) ).
% subset_mset.add_le_cancel_right
thf(fact_812_subset__mset_Oadd__le__cancel__left,axiom,
! [C: multiset_a,A2: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ ( plus_plus_multiset_a @ C @ A2 ) @ ( plus_plus_multiset_a @ C @ B ) )
= ( subseteq_mset_a @ A2 @ B ) ) ).
% subset_mset.add_le_cancel_left
thf(fact_813_subset__mset_Oadd__less__cancel__right,axiom,
! [A2: multiset_a,C: multiset_a,B: multiset_a] :
( ( subset_mset_a @ ( plus_plus_multiset_a @ A2 @ C ) @ ( plus_plus_multiset_a @ B @ C ) )
= ( subset_mset_a @ A2 @ B ) ) ).
% subset_mset.add_less_cancel_right
thf(fact_814_subset__mset_Oadd__less__cancel__left,axiom,
! [C: multiset_a,A2: multiset_a,B: multiset_a] :
( ( subset_mset_a @ ( plus_plus_multiset_a @ C @ A2 ) @ ( plus_plus_multiset_a @ C @ B ) )
= ( subset_mset_a @ A2 @ B ) ) ).
% subset_mset.add_less_cancel_left
thf(fact_815_mset__subset__add__mset,axiom,
! [X2: a,N: multiset_a,M: multiset_a] :
( ( subset_mset_a @ ( add_mset_a @ X2 @ N ) @ ( add_mset_a @ X2 @ M ) )
= ( subset_mset_a @ N @ M ) ) ).
% mset_subset_add_mset
thf(fact_816_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_817_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_818_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_819_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_820_add__mset__subseteq__single__iff,axiom,
! [A2: a,M: multiset_a,B: a] :
( ( subseteq_mset_a @ ( add_mset_a @ A2 @ M ) @ ( add_mset_a @ B @ zero_zero_multiset_a ) )
= ( ( M = zero_zero_multiset_a )
& ( A2 = B ) ) ) ).
% add_mset_subseteq_single_iff
thf(fact_821_subset__mset_Oless__add__same__cancel2,axiom,
! [A2: multiset_a,B: multiset_a] :
( ( subset_mset_a @ A2 @ ( plus_plus_multiset_a @ B @ A2 ) )
= ( subset_mset_a @ zero_zero_multiset_a @ B ) ) ).
% subset_mset.less_add_same_cancel2
thf(fact_822_subset__mset_Oless__add__same__cancel1,axiom,
! [A2: multiset_a,B: multiset_a] :
( ( subset_mset_a @ A2 @ ( plus_plus_multiset_a @ A2 @ B ) )
= ( subset_mset_a @ zero_zero_multiset_a @ B ) ) ).
% subset_mset.less_add_same_cancel1
thf(fact_823_subset__mset_Oadd__less__same__cancel2,axiom,
! [A2: multiset_a,B: multiset_a] :
( ( subset_mset_a @ ( plus_plus_multiset_a @ A2 @ B ) @ B )
= ( subset_mset_a @ A2 @ zero_zero_multiset_a ) ) ).
% subset_mset.add_less_same_cancel2
thf(fact_824_subset__mset_Oadd__less__same__cancel1,axiom,
! [B: multiset_a,A2: multiset_a] :
( ( subset_mset_a @ ( plus_plus_multiset_a @ B @ A2 ) @ B )
= ( subset_mset_a @ A2 @ zero_zero_multiset_a ) ) ).
% subset_mset.add_less_same_cancel1
thf(fact_825_mset__subset__eq__multiset__union__diff__commute,axiom,
! [B2: multiset_a,A3: multiset_a,C2: multiset_a] :
( ( subseteq_mset_a @ B2 @ A3 )
=> ( ( plus_plus_multiset_a @ ( minus_3765977307040488491iset_a @ A3 @ B2 ) @ C2 )
= ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ A3 @ C2 ) @ B2 ) ) ) ).
% mset_subset_eq_multiset_union_diff_commute
thf(fact_826_subset__mset_Oadd__diff__assoc2,axiom,
! [A2: multiset_a,B: multiset_a,C: multiset_a] :
( ( subseteq_mset_a @ A2 @ B )
=> ( ( plus_plus_multiset_a @ ( minus_3765977307040488491iset_a @ B @ A2 ) @ C )
= ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ B @ C ) @ A2 ) ) ) ).
% subset_mset.add_diff_assoc2
thf(fact_827_subset__mset_Oadd__diff__assoc,axiom,
! [A2: multiset_a,B: multiset_a,C: multiset_a] :
( ( subseteq_mset_a @ A2 @ B )
=> ( ( plus_plus_multiset_a @ C @ ( minus_3765977307040488491iset_a @ B @ A2 ) )
= ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ C @ B ) @ A2 ) ) ) ).
% subset_mset.add_diff_assoc
thf(fact_828_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_829_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_830_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_831_subset__mset_Oadd__strict__increasing2,axiom,
! [A2: multiset_a,B: multiset_a,C: multiset_a] :
( ( subseteq_mset_a @ zero_zero_multiset_a @ A2 )
=> ( ( subset_mset_a @ B @ C )
=> ( subset_mset_a @ B @ ( plus_plus_multiset_a @ A2 @ C ) ) ) ) ).
% subset_mset.add_strict_increasing2
thf(fact_832_subset__mset_Oadd__strict__increasing,axiom,
! [A2: multiset_a,B: multiset_a,C: multiset_a] :
( ( subset_mset_a @ zero_zero_multiset_a @ A2 )
=> ( ( subseteq_mset_a @ B @ C )
=> ( subset_mset_a @ B @ ( plus_plus_multiset_a @ A2 @ C ) ) ) ) ).
% subset_mset.add_strict_increasing
thf(fact_833_subset__mset_Oadd__pos__nonneg,axiom,
! [A2: multiset_a,B: multiset_a] :
( ( subset_mset_a @ zero_zero_multiset_a @ A2 )
=> ( ( subseteq_mset_a @ zero_zero_multiset_a @ B )
=> ( subset_mset_a @ zero_zero_multiset_a @ ( plus_plus_multiset_a @ A2 @ B ) ) ) ) ).
% subset_mset.add_pos_nonneg
thf(fact_834_subset__mset_Oadd__nonpos__neg,axiom,
! [A2: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ A2 @ zero_zero_multiset_a )
=> ( ( subset_mset_a @ B @ zero_zero_multiset_a )
=> ( subset_mset_a @ ( plus_plus_multiset_a @ A2 @ B ) @ zero_zero_multiset_a ) ) ) ).
% subset_mset.add_nonpos_neg
thf(fact_835_subset__mset_Oadd__nonneg__pos,axiom,
! [A2: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ zero_zero_multiset_a @ A2 )
=> ( ( subset_mset_a @ zero_zero_multiset_a @ B )
=> ( subset_mset_a @ zero_zero_multiset_a @ ( plus_plus_multiset_a @ A2 @ B ) ) ) ) ).
% subset_mset.add_nonneg_pos
thf(fact_836_subset__mset_Oadd__neg__nonpos,axiom,
! [A2: multiset_a,B: multiset_a] :
( ( subset_mset_a @ A2 @ zero_zero_multiset_a )
=> ( ( subseteq_mset_a @ B @ zero_zero_multiset_a )
=> ( subset_mset_a @ ( plus_plus_multiset_a @ A2 @ B ) @ zero_zero_multiset_a ) ) ) ).
% subset_mset.add_neg_nonpos
thf(fact_837_mset__subset__eq__insertD,axiom,
! [X2: produc6518373309651786023iset_a,A3: multis4781622204924204013iset_a,B2: multis4781622204924204013iset_a] :
( ( subset6292090195676340366iset_a @ ( add_ms5348986034529679365iset_a @ X2 @ A3 ) @ B2 )
=> ( ( member5199237121806060112iset_a @ X2 @ ( set_ms2379723421352415844iset_a @ B2 ) )
& ( subset5376938936802007386iset_a @ A3 @ B2 ) ) ) ).
% mset_subset_eq_insertD
thf(fact_838_mset__subset__eq__insertD,axiom,
! [X2: product_prod_a_a,A3: multis501812127501805293od_a_a,B2: multis501812127501805293od_a_a] :
( ( subset4108007524207251342od_a_a @ ( add_ms8655138167283798533od_a_a @ X2 @ A3 ) @ B2 )
=> ( ( member1426531477525435216od_a_a @ X2 @ ( set_ms119794563918438244od_a_a @ B2 ) )
& ( subset8265789482553336922od_a_a @ A3 @ B2 ) ) ) ).
% mset_subset_eq_insertD
thf(fact_839_mset__subset__eq__insertD,axiom,
! [X2: a,A3: multiset_a,B2: multiset_a] :
( ( subseteq_mset_a @ ( add_mset_a @ X2 @ A3 ) @ B2 )
=> ( ( member_a @ X2 @ ( set_mset_a @ B2 ) )
& ( subset_mset_a @ A3 @ B2 ) ) ) ).
% mset_subset_eq_insertD
thf(fact_840_subset__mset_Oadd__less__le__mono,axiom,
! [A2: multiset_a,B: multiset_a,C: multiset_a,D: multiset_a] :
( ( subset_mset_a @ A2 @ B )
=> ( ( subseteq_mset_a @ C @ D )
=> ( subset_mset_a @ ( plus_plus_multiset_a @ A2 @ C ) @ ( plus_plus_multiset_a @ B @ D ) ) ) ) ).
% subset_mset.add_less_le_mono
thf(fact_841_subset__mset_Oadd__le__less__mono,axiom,
! [A2: multiset_a,B: multiset_a,C: multiset_a,D: multiset_a] :
( ( subseteq_mset_a @ A2 @ B )
=> ( ( subset_mset_a @ C @ D )
=> ( subset_mset_a @ ( plus_plus_multiset_a @ A2 @ C ) @ ( plus_plus_multiset_a @ B @ D ) ) ) ) ).
% subset_mset.add_le_less_mono
thf(fact_842_mset__subset__eq__add__mset__cancel,axiom,
! [A2: a,A3: multiset_a,B2: multiset_a] :
( ( subseteq_mset_a @ ( add_mset_a @ A2 @ A3 ) @ ( add_mset_a @ A2 @ B2 ) )
= ( subseteq_mset_a @ A3 @ B2 ) ) ).
% mset_subset_eq_add_mset_cancel
thf(fact_843_mset__subset__eqD,axiom,
! [A3: multis4781622204924204013iset_a,B2: multis4781622204924204013iset_a,X2: produc6518373309651786023iset_a] :
( ( subset6292090195676340366iset_a @ A3 @ B2 )
=> ( ( member5199237121806060112iset_a @ X2 @ ( set_ms2379723421352415844iset_a @ A3 ) )
=> ( member5199237121806060112iset_a @ X2 @ ( set_ms2379723421352415844iset_a @ B2 ) ) ) ) ).
% mset_subset_eqD
thf(fact_844_mset__subset__eqD,axiom,
! [A3: multis501812127501805293od_a_a,B2: multis501812127501805293od_a_a,X2: product_prod_a_a] :
( ( subset4108007524207251342od_a_a @ A3 @ B2 )
=> ( ( member1426531477525435216od_a_a @ X2 @ ( set_ms119794563918438244od_a_a @ A3 ) )
=> ( member1426531477525435216od_a_a @ X2 @ ( set_ms119794563918438244od_a_a @ B2 ) ) ) ) ).
% mset_subset_eqD
thf(fact_845_mset__subset__eqD,axiom,
! [A3: multiset_a,B2: multiset_a,X2: a] :
( ( subseteq_mset_a @ A3 @ B2 )
=> ( ( member_a @ X2 @ ( set_mset_a @ A3 ) )
=> ( member_a @ X2 @ ( set_mset_a @ B2 ) ) ) ) ).
% mset_subset_eqD
thf(fact_846_subset__mset_Oadd__mono,axiom,
! [A2: multiset_a,B: multiset_a,C: multiset_a,D: multiset_a] :
( ( subseteq_mset_a @ A2 @ B )
=> ( ( subseteq_mset_a @ C @ D )
=> ( subseteq_mset_a @ ( plus_plus_multiset_a @ A2 @ C ) @ ( plus_plus_multiset_a @ B @ D ) ) ) ) ).
% subset_mset.add_mono
thf(fact_847_subset__mset_Oless__eqE,axiom,
! [A2: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ A2 @ B )
=> ~ ! [C3: multiset_a] :
( B
!= ( plus_plus_multiset_a @ A2 @ C3 ) ) ) ).
% subset_mset.less_eqE
thf(fact_848_subset__mset_Ole__iff__add,axiom,
( subseteq_mset_a
= ( ^ [A5: multiset_a,B5: multiset_a] :
? [C4: multiset_a] :
( B5
= ( plus_plus_multiset_a @ A5 @ C4 ) ) ) ) ).
% subset_mset.le_iff_add
thf(fact_849_subset__mset_Oadd__left__mono,axiom,
! [A2: multiset_a,B: multiset_a,C: multiset_a] :
( ( subseteq_mset_a @ A2 @ B )
=> ( subseteq_mset_a @ ( plus_plus_multiset_a @ C @ A2 ) @ ( plus_plus_multiset_a @ C @ B ) ) ) ).
% subset_mset.add_left_mono
thf(fact_850_subset__mset_Oadd__right__mono,axiom,
! [A2: multiset_a,B: multiset_a,C: multiset_a] :
( ( subseteq_mset_a @ A2 @ B )
=> ( subseteq_mset_a @ ( plus_plus_multiset_a @ A2 @ C ) @ ( plus_plus_multiset_a @ B @ C ) ) ) ).
% subset_mset.add_right_mono
thf(fact_851_subset__mset_Oadd__le__imp__le__left,axiom,
! [C: multiset_a,A2: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ ( plus_plus_multiset_a @ C @ A2 ) @ ( plus_plus_multiset_a @ C @ B ) )
=> ( subseteq_mset_a @ A2 @ B ) ) ).
% subset_mset.add_le_imp_le_left
thf(fact_852_subset__mset_Oadd__le__imp__le__right,axiom,
! [A2: multiset_a,C: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ ( plus_plus_multiset_a @ A2 @ C ) @ ( plus_plus_multiset_a @ B @ C ) )
=> ( subseteq_mset_a @ A2 @ B ) ) ).
% subset_mset.add_le_imp_le_right
thf(fact_853_mset__subset__eq__add__left,axiom,
! [A3: multiset_a,B2: multiset_a] : ( subseteq_mset_a @ A3 @ ( plus_plus_multiset_a @ A3 @ B2 ) ) ).
% mset_subset_eq_add_left
thf(fact_854_mset__subset__eq__mono__add,axiom,
! [A3: multiset_a,B2: multiset_a,C2: multiset_a,D2: multiset_a] :
( ( subseteq_mset_a @ A3 @ B2 )
=> ( ( subseteq_mset_a @ C2 @ D2 )
=> ( subseteq_mset_a @ ( plus_plus_multiset_a @ A3 @ C2 ) @ ( plus_plus_multiset_a @ B2 @ D2 ) ) ) ) ).
% mset_subset_eq_mono_add
thf(fact_855_mset__subset__eq__add__right,axiom,
! [B2: multiset_a,A3: multiset_a] : ( subseteq_mset_a @ B2 @ ( plus_plus_multiset_a @ A3 @ B2 ) ) ).
% mset_subset_eq_add_right
thf(fact_856_mset__subset__eq__exists__conv,axiom,
( subseteq_mset_a
= ( ^ [A6: multiset_a,B6: multiset_a] :
? [C6: multiset_a] :
( B6
= ( plus_plus_multiset_a @ A6 @ C6 ) ) ) ) ).
% mset_subset_eq_exists_conv
thf(fact_857_multi__psub__of__add__self,axiom,
! [A3: multiset_a,X2: a] : ( subset_mset_a @ A3 @ ( add_mset_a @ X2 @ A3 ) ) ).
% multi_psub_of_add_self
thf(fact_858_mset__subsetD,axiom,
! [A3: multis4781622204924204013iset_a,B2: multis4781622204924204013iset_a,X2: produc6518373309651786023iset_a] :
( ( subset5376938936802007386iset_a @ A3 @ B2 )
=> ( ( member5199237121806060112iset_a @ X2 @ ( set_ms2379723421352415844iset_a @ A3 ) )
=> ( member5199237121806060112iset_a @ X2 @ ( set_ms2379723421352415844iset_a @ B2 ) ) ) ) ).
% mset_subsetD
thf(fact_859_mset__subsetD,axiom,
! [A3: multis501812127501805293od_a_a,B2: multis501812127501805293od_a_a,X2: product_prod_a_a] :
( ( subset8265789482553336922od_a_a @ A3 @ B2 )
=> ( ( member1426531477525435216od_a_a @ X2 @ ( set_ms119794563918438244od_a_a @ A3 ) )
=> ( member1426531477525435216od_a_a @ X2 @ ( set_ms119794563918438244od_a_a @ B2 ) ) ) ) ).
% mset_subsetD
thf(fact_860_mset__subsetD,axiom,
! [A3: multiset_a,B2: multiset_a,X2: a] :
( ( subset_mset_a @ A3 @ B2 )
=> ( ( member_a @ X2 @ ( set_mset_a @ A3 ) )
=> ( member_a @ X2 @ ( set_mset_a @ B2 ) ) ) ) ).
% mset_subsetD
thf(fact_861_subset__mset_Oadd__strict__mono,axiom,
! [A2: multiset_a,B: multiset_a,C: multiset_a,D: multiset_a] :
( ( subset_mset_a @ A2 @ B )
=> ( ( subset_mset_a @ C @ D )
=> ( subset_mset_a @ ( plus_plus_multiset_a @ A2 @ C ) @ ( plus_plus_multiset_a @ B @ D ) ) ) ) ).
% subset_mset.add_strict_mono
thf(fact_862_subset__mset_Oadd__strict__left__mono,axiom,
! [A2: multiset_a,B: multiset_a,C: multiset_a] :
( ( subset_mset_a @ A2 @ B )
=> ( subset_mset_a @ ( plus_plus_multiset_a @ C @ A2 ) @ ( plus_plus_multiset_a @ C @ B ) ) ) ).
% subset_mset.add_strict_left_mono
thf(fact_863_subset__mset_Oadd__strict__right__mono,axiom,
! [A2: multiset_a,B: multiset_a,C: multiset_a] :
( ( subset_mset_a @ A2 @ B )
=> ( subset_mset_a @ ( plus_plus_multiset_a @ A2 @ C ) @ ( plus_plus_multiset_a @ B @ C ) ) ) ).
% subset_mset.add_strict_right_mono
thf(fact_864_subset__mset_Oadd__less__imp__less__left,axiom,
! [C: multiset_a,A2: multiset_a,B: multiset_a] :
( ( subset_mset_a @ ( plus_plus_multiset_a @ C @ A2 ) @ ( plus_plus_multiset_a @ C @ B ) )
=> ( subset_mset_a @ A2 @ B ) ) ).
% subset_mset.add_less_imp_less_left
thf(fact_865_subset__mset_Oadd__less__imp__less__right,axiom,
! [A2: multiset_a,C: multiset_a,B: multiset_a] :
( ( subset_mset_a @ ( plus_plus_multiset_a @ A2 @ C ) @ ( plus_plus_multiset_a @ B @ C ) )
=> ( subset_mset_a @ A2 @ B ) ) ).
% subset_mset.add_less_imp_less_right
thf(fact_866_subset__mset_Opos__add__strict,axiom,
! [A2: multiset_a,B: multiset_a,C: multiset_a] :
( ( subset_mset_a @ zero_zero_multiset_a @ A2 )
=> ( ( subset_mset_a @ B @ C )
=> ( subset_mset_a @ B @ ( plus_plus_multiset_a @ A2 @ C ) ) ) ) ).
% subset_mset.pos_add_strict
thf(fact_867_subset__mset_Oadd__pos__pos,axiom,
! [A2: multiset_a,B: multiset_a] :
( ( subset_mset_a @ zero_zero_multiset_a @ A2 )
=> ( ( subset_mset_a @ zero_zero_multiset_a @ B )
=> ( subset_mset_a @ zero_zero_multiset_a @ ( plus_plus_multiset_a @ A2 @ B ) ) ) ) ).
% subset_mset.add_pos_pos
thf(fact_868_subset__mset_Oadd__neg__neg,axiom,
! [A2: multiset_a,B: multiset_a] :
( ( subset_mset_a @ A2 @ zero_zero_multiset_a )
=> ( ( subset_mset_a @ B @ zero_zero_multiset_a )
=> ( subset_mset_a @ ( plus_plus_multiset_a @ A2 @ B ) @ zero_zero_multiset_a ) ) ) ).
% subset_mset.add_neg_neg
thf(fact_869_subset__mset_OlessE,axiom,
! [A2: multiset_a,B: multiset_a] :
( ( subset_mset_a @ A2 @ B )
=> ~ ! [C3: multiset_a] :
( ( B
= ( plus_plus_multiset_a @ A2 @ C3 ) )
=> ( C3 = zero_zero_multiset_a ) ) ) ).
% subset_mset.lessE
thf(fact_870_empty__subset__add__mset,axiom,
! [X2: a,M: multiset_a] : ( subset_mset_a @ zero_zero_multiset_a @ ( add_mset_a @ X2 @ M ) ) ).
% empty_subset_add_mset
thf(fact_871_mset__subset__insertD,axiom,
! [X2: produc6518373309651786023iset_a,A3: multis4781622204924204013iset_a,B2: multis4781622204924204013iset_a] :
( ( subset5376938936802007386iset_a @ ( add_ms5348986034529679365iset_a @ X2 @ A3 ) @ B2 )
=> ( ( member5199237121806060112iset_a @ X2 @ ( set_ms2379723421352415844iset_a @ B2 ) )
& ( subset5376938936802007386iset_a @ A3 @ B2 ) ) ) ).
% mset_subset_insertD
thf(fact_872_mset__subset__insertD,axiom,
! [X2: product_prod_a_a,A3: multis501812127501805293od_a_a,B2: multis501812127501805293od_a_a] :
( ( subset8265789482553336922od_a_a @ ( add_ms8655138167283798533od_a_a @ X2 @ A3 ) @ B2 )
=> ( ( member1426531477525435216od_a_a @ X2 @ ( set_ms119794563918438244od_a_a @ B2 ) )
& ( subset8265789482553336922od_a_a @ A3 @ B2 ) ) ) ).
% mset_subset_insertD
thf(fact_873_mset__subset__insertD,axiom,
! [X2: a,A3: multiset_a,B2: multiset_a] :
( ( subset_mset_a @ ( add_mset_a @ X2 @ A3 ) @ B2 )
=> ( ( member_a @ X2 @ ( set_mset_a @ B2 ) )
& ( subset_mset_a @ A3 @ B2 ) ) ) ).
% mset_subset_insertD
thf(fact_874_set__mset__mono,axiom,
! [A3: multiset_a,B2: multiset_a] :
( ( subseteq_mset_a @ A3 @ B2 )
=> ( ord_less_eq_set_a @ ( set_mset_a @ A3 ) @ ( set_mset_a @ B2 ) ) ) ).
% set_mset_mono
thf(fact_875_set__mset__mono,axiom,
! [A3: multis501812127501805293od_a_a,B2: multis501812127501805293od_a_a] :
( ( subset4108007524207251342od_a_a @ A3 @ B2 )
=> ( ord_le746702958409616551od_a_a @ ( set_ms119794563918438244od_a_a @ A3 ) @ ( set_ms119794563918438244od_a_a @ B2 ) ) ) ).
% set_mset_mono
thf(fact_876_set__mset__mono,axiom,
! [A3: multis4781622204924204013iset_a,B2: multis4781622204924204013iset_a] :
( ( subset6292090195676340366iset_a @ A3 @ B2 )
=> ( ord_le5871882978999109543iset_a @ ( set_ms2379723421352415844iset_a @ A3 ) @ ( set_ms2379723421352415844iset_a @ B2 ) ) ) ).
% set_mset_mono
thf(fact_877_subset__mset_Oadd__nonpos__eq__0__iff,axiom,
! [X2: multiset_a,Y: multiset_a] :
( ( subseteq_mset_a @ X2 @ zero_zero_multiset_a )
=> ( ( subseteq_mset_a @ Y @ zero_zero_multiset_a )
=> ( ( ( plus_plus_multiset_a @ X2 @ Y )
= zero_zero_multiset_a )
= ( ( X2 = zero_zero_multiset_a )
& ( Y = zero_zero_multiset_a ) ) ) ) ) ).
% subset_mset.add_nonpos_eq_0_iff
thf(fact_878_subset__mset_Oadd__nonneg__eq__0__iff,axiom,
! [X2: multiset_a,Y: multiset_a] :
( ( subseteq_mset_a @ zero_zero_multiset_a @ X2 )
=> ( ( subseteq_mset_a @ zero_zero_multiset_a @ Y )
=> ( ( ( plus_plus_multiset_a @ X2 @ Y )
= zero_zero_multiset_a )
= ( ( X2 = zero_zero_multiset_a )
& ( Y = zero_zero_multiset_a ) ) ) ) ) ).
% subset_mset.add_nonneg_eq_0_iff
thf(fact_879_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_880_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_881_subset__mset_Oadd__increasing2,axiom,
! [C: multiset_a,B: multiset_a,A2: multiset_a] :
( ( subseteq_mset_a @ zero_zero_multiset_a @ C )
=> ( ( subseteq_mset_a @ B @ A2 )
=> ( subseteq_mset_a @ B @ ( plus_plus_multiset_a @ A2 @ C ) ) ) ) ).
% subset_mset.add_increasing2
thf(fact_882_subset__mset_Oadd__decreasing2,axiom,
! [C: multiset_a,A2: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ C @ zero_zero_multiset_a )
=> ( ( subseteq_mset_a @ A2 @ B )
=> ( subseteq_mset_a @ ( plus_plus_multiset_a @ A2 @ C ) @ B ) ) ) ).
% subset_mset.add_decreasing2
thf(fact_883_subset__mset_Oadd__increasing,axiom,
! [A2: multiset_a,B: multiset_a,C: multiset_a] :
( ( subseteq_mset_a @ zero_zero_multiset_a @ A2 )
=> ( ( subseteq_mset_a @ B @ C )
=> ( subseteq_mset_a @ B @ ( plus_plus_multiset_a @ A2 @ C ) ) ) ) ).
% subset_mset.add_increasing
thf(fact_884_subset__mset_Oadd__decreasing,axiom,
! [A2: multiset_a,C: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ A2 @ zero_zero_multiset_a )
=> ( ( subseteq_mset_a @ C @ B )
=> ( subseteq_mset_a @ ( plus_plus_multiset_a @ A2 @ C ) @ B ) ) ) ).
% subset_mset.add_decreasing
thf(fact_885_multiset__diff__union__assoc,axiom,
! [C2: multiset_a,B2: multiset_a,A3: multiset_a] :
( ( subseteq_mset_a @ C2 @ B2 )
=> ( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ A3 @ B2 ) @ C2 )
= ( plus_plus_multiset_a @ A3 @ ( minus_3765977307040488491iset_a @ B2 @ C2 ) ) ) ) ).
% multiset_diff_union_assoc
thf(fact_886_subset__eq__diff__conv,axiom,
! [A3: multiset_a,C2: multiset_a,B2: multiset_a] :
( ( subseteq_mset_a @ ( minus_3765977307040488491iset_a @ A3 @ C2 ) @ B2 )
= ( subseteq_mset_a @ A3 @ ( plus_plus_multiset_a @ B2 @ C2 ) ) ) ).
% subset_eq_diff_conv
thf(fact_887_subset__mset_Ole__imp__diff__is__add,axiom,
! [A2: multiset_a,B: multiset_a,C: multiset_a] :
( ( subseteq_mset_a @ A2 @ B )
=> ( ( subseteq_mset_a @ A2 @ B )
=> ( ( ( minus_3765977307040488491iset_a @ B @ A2 )
= C )
= ( B
= ( plus_plus_multiset_a @ C @ A2 ) ) ) ) ) ).
% subset_mset.le_imp_diff_is_add
thf(fact_888_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_889_subset__mset_Odiff__diff__right,axiom,
! [A2: multiset_a,B: multiset_a,C: multiset_a] :
( ( subseteq_mset_a @ A2 @ B )
=> ( ( minus_3765977307040488491iset_a @ C @ ( minus_3765977307040488491iset_a @ B @ A2 ) )
= ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ C @ A2 ) @ B ) ) ) ).
% subset_mset.diff_diff_right
thf(fact_890_subset__mset_Odiff__add__assoc2,axiom,
! [A2: multiset_a,B: multiset_a,C: multiset_a] :
( ( subseteq_mset_a @ A2 @ B )
=> ( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ B @ C ) @ A2 )
= ( plus_plus_multiset_a @ ( minus_3765977307040488491iset_a @ B @ A2 ) @ C ) ) ) ).
% subset_mset.diff_add_assoc2
thf(fact_891_subset__mset_Odiff__add__assoc,axiom,
! [A2: multiset_a,B: multiset_a,C: multiset_a] :
( ( subseteq_mset_a @ A2 @ B )
=> ( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ C @ B ) @ A2 )
= ( plus_plus_multiset_a @ C @ ( minus_3765977307040488491iset_a @ B @ A2 ) ) ) ) ).
% subset_mset.diff_add_assoc
thf(fact_892_subset__mset_Ole__diff__conv2,axiom,
! [A2: multiset_a,B: multiset_a,C: multiset_a] :
( ( subseteq_mset_a @ A2 @ B )
=> ( ( subseteq_mset_a @ C @ ( minus_3765977307040488491iset_a @ B @ A2 ) )
= ( subseteq_mset_a @ ( plus_plus_multiset_a @ C @ A2 ) @ B ) ) ) ).
% subset_mset.le_diff_conv2
thf(fact_893_subset__mset_Ole__add__diff,axiom,
! [A2: multiset_a,B: multiset_a,C: multiset_a] :
( ( subseteq_mset_a @ A2 @ B )
=> ( subseteq_mset_a @ C @ ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ B @ C ) @ A2 ) ) ) ).
% subset_mset.le_add_diff
thf(fact_894_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_895_subset__implies__mult,axiom,
! [A3: multiset_a,B2: multiset_a,R: set_Product_prod_a_a] :
( ( subset_mset_a @ A3 @ B2 )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A3 @ B2 ) @ ( mult_a @ R ) ) ) ).
% subset_implies_mult
thf(fact_896_mset__subset__eq__single,axiom,
! [A2: produc6518373309651786023iset_a,B2: multis4781622204924204013iset_a] :
( ( member5199237121806060112iset_a @ A2 @ ( set_ms2379723421352415844iset_a @ B2 ) )
=> ( subset6292090195676340366iset_a @ ( add_ms5348986034529679365iset_a @ A2 @ zero_z4591712717340762532iset_a ) @ B2 ) ) ).
% mset_subset_eq_single
thf(fact_897_mset__subset__eq__single,axiom,
! [A2: product_prod_a_a,B2: multis501812127501805293od_a_a] :
( ( member1426531477525435216od_a_a @ A2 @ ( set_ms119794563918438244od_a_a @ B2 ) )
=> ( subset4108007524207251342od_a_a @ ( add_ms8655138167283798533od_a_a @ A2 @ zero_z3024502163831676836od_a_a ) @ B2 ) ) ).
% mset_subset_eq_single
thf(fact_898_mset__subset__eq__single,axiom,
! [A2: a,B2: multiset_a] :
( ( member_a @ A2 @ ( set_mset_a @ B2 ) )
=> ( subseteq_mset_a @ ( add_mset_a @ A2 @ zero_zero_multiset_a ) @ B2 ) ) ).
% mset_subset_eq_single
thf(fact_899_multi__subset__induct,axiom,
! [F2: multis4781622204924204013iset_a,A3: multis4781622204924204013iset_a,P: multis4781622204924204013iset_a > $o] :
( ( subset6292090195676340366iset_a @ F2 @ A3 )
=> ( ( P @ zero_z4591712717340762532iset_a )
=> ( ! [A: produc6518373309651786023iset_a,F3: multis4781622204924204013iset_a] :
( ( member5199237121806060112iset_a @ A @ ( set_ms2379723421352415844iset_a @ A3 ) )
=> ( ( P @ F3 )
=> ( P @ ( add_ms5348986034529679365iset_a @ A @ F3 ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% multi_subset_induct
thf(fact_900_multi__subset__induct,axiom,
! [F2: multis501812127501805293od_a_a,A3: multis501812127501805293od_a_a,P: multis501812127501805293od_a_a > $o] :
( ( subset4108007524207251342od_a_a @ F2 @ A3 )
=> ( ( P @ zero_z3024502163831676836od_a_a )
=> ( ! [A: product_prod_a_a,F3: multis501812127501805293od_a_a] :
( ( member1426531477525435216od_a_a @ A @ ( set_ms119794563918438244od_a_a @ A3 ) )
=> ( ( P @ F3 )
=> ( P @ ( add_ms8655138167283798533od_a_a @ A @ F3 ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% multi_subset_induct
thf(fact_901_multi__subset__induct,axiom,
! [F2: multiset_a,A3: multiset_a,P: multiset_a > $o] :
( ( subseteq_mset_a @ F2 @ A3 )
=> ( ( P @ zero_zero_multiset_a )
=> ( ! [A: a,F3: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ A3 ) )
=> ( ( P @ F3 )
=> ( P @ ( add_mset_a @ A @ F3 ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% multi_subset_induct
thf(fact_902_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_903_subset__mset__imp__s__mul__ext__Id,axiom,
! [A3: multiset_a,B2: multiset_a,R3: set_Product_prod_a_a] :
( ( subset_mset_a @ A3 @ B2 )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ B2 @ A3 ) @ ( multiset_s_mul_ext_a @ id_a @ R3 ) ) ) ).
% subset_mset_imp_s_mul_ext_Id
thf(fact_904_subseteq__mset__imp__ns__mul__ext__Id,axiom,
! [A3: multiset_a,B2: multiset_a,R3: set_Product_prod_a_a] :
( ( subseteq_mset_a @ A3 @ B2 )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ B2 @ A3 ) @ ( multis4023584895248362547_ext_a @ id_a @ R3 ) ) ) ).
% subseteq_mset_imp_ns_mul_ext_Id
thf(fact_905_mset__subset__diff__self,axiom,
! [C: produc6518373309651786023iset_a,B2: multis4781622204924204013iset_a] :
( ( member5199237121806060112iset_a @ C @ ( set_ms2379723421352415844iset_a @ B2 ) )
=> ( subset5376938936802007386iset_a @ ( minus_9211619285413683668iset_a @ B2 @ ( add_ms5348986034529679365iset_a @ C @ zero_z4591712717340762532iset_a ) ) @ B2 ) ) ).
% mset_subset_diff_self
thf(fact_906_mset__subset__diff__self,axiom,
! [C: product_prod_a_a,B2: multis501812127501805293od_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( set_ms119794563918438244od_a_a @ B2 ) )
=> ( subset8265789482553336922od_a_a @ ( minus_7313152569058643924od_a_a @ B2 @ ( add_ms8655138167283798533od_a_a @ C @ zero_z3024502163831676836od_a_a ) ) @ B2 ) ) ).
% mset_subset_diff_self
thf(fact_907_mset__subset__diff__self,axiom,
! [C: a,B2: multiset_a] :
( ( member_a @ C @ ( set_mset_a @ B2 ) )
=> ( subset_mset_a @ ( minus_3765977307040488491iset_a @ B2 @ ( add_mset_a @ C @ zero_zero_multiset_a ) ) @ B2 ) ) ).
% mset_subset_diff_self
thf(fact_908_subset__mset_Osum__mset__mono,axiom,
! [K: multis4781622204924204013iset_a,F: produc6518373309651786023iset_a > multiset_a,G: produc6518373309651786023iset_a > multiset_a] :
( ! [I3: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ I3 @ ( set_ms2379723421352415844iset_a @ K ) )
=> ( subseteq_mset_a @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( subseteq_mset_a @ ( comm_m1977238983017651125iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ ( image_5610718949088693016iset_a @ F @ K ) ) @ ( comm_m1977238983017651125iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ ( image_5610718949088693016iset_a @ G @ K ) ) ) ) ).
% subset_mset.sum_mset_mono
thf(fact_909_subset__mset_Osum__mset__mono,axiom,
! [K: multis501812127501805293od_a_a,F: product_prod_a_a > multiset_a,G: product_prod_a_a > multiset_a] :
( ! [I3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ I3 @ ( set_ms119794563918438244od_a_a @ K ) )
=> ( subseteq_mset_a @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( subseteq_mset_a @ ( comm_m1977238983017651125iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ ( image_119670572209922840iset_a @ F @ K ) ) @ ( comm_m1977238983017651125iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ ( image_119670572209922840iset_a @ G @ K ) ) ) ) ).
% subset_mset.sum_mset_mono
thf(fact_910_subset__mset_Osum__mset__mono,axiom,
! [K: multiset_a,F: a > multiset_a,G: a > multiset_a] :
( ! [I3: a] :
( ( member_a @ I3 @ ( set_mset_a @ K ) )
=> ( subseteq_mset_a @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( subseteq_mset_a @ ( comm_m1977238983017651125iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ ( image_2204096306494911105iset_a @ F @ K ) ) @ ( comm_m1977238983017651125iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ ( image_2204096306494911105iset_a @ G @ K ) ) ) ) ).
% subset_mset.sum_mset_mono
thf(fact_911_image__mset__union,axiom,
! [F: a > a,M: multiset_a,N: multiset_a] :
( ( image_mset_a_a @ F @ ( plus_plus_multiset_a @ M @ N ) )
= ( plus_plus_multiset_a @ ( image_mset_a_a @ F @ M ) @ ( image_mset_a_a @ F @ N ) ) ) ).
% image_mset_union
thf(fact_912_image__mset__add__mset,axiom,
! [F: a > a,A2: a,M: multiset_a] :
( ( image_mset_a_a @ F @ ( add_mset_a @ A2 @ M ) )
= ( add_mset_a @ ( F @ A2 ) @ ( image_mset_a_a @ F @ M ) ) ) ).
% image_mset_add_mset
thf(fact_913_sum__mset_Oinsert,axiom,
! [G: a > multiset_a,X2: a,A3: multiset_a] :
( ( comm_m4595494212910443721iset_a @ ( image_2204096306494911105iset_a @ G @ ( add_mset_a @ X2 @ A3 ) ) )
= ( plus_plus_multiset_a @ ( G @ X2 ) @ ( comm_m4595494212910443721iset_a @ ( image_2204096306494911105iset_a @ G @ A3 ) ) ) ) ).
% sum_mset.insert
thf(fact_914_sum__mset_Oinsert,axiom,
! [G: a > nat,X2: a,A3: multiset_a] :
( ( comm_m762188921832702859et_nat @ ( image_mset_a_nat @ G @ ( add_mset_a @ X2 @ A3 ) ) )
= ( plus_plus_nat @ ( G @ X2 ) @ ( comm_m762188921832702859et_nat @ ( image_mset_a_nat @ G @ A3 ) ) ) ) ).
% sum_mset.insert
thf(fact_915_image__mset__eq__plusD,axiom,
! [F: a > a,A3: multiset_a,B2: multiset_a,C2: multiset_a] :
( ( ( image_mset_a_a @ F @ A3 )
= ( plus_plus_multiset_a @ B2 @ C2 ) )
=> ? [B7: multiset_a,C7: multiset_a] :
( ( A3
= ( plus_plus_multiset_a @ B7 @ C7 ) )
& ( B2
= ( image_mset_a_a @ F @ B7 ) )
& ( C2
= ( image_mset_a_a @ F @ C7 ) ) ) ) ).
% image_mset_eq_plusD
thf(fact_916_msed__map__invR,axiom,
! [F: a > a,M: multiset_a,B: a,N: multiset_a] :
( ( ( image_mset_a_a @ F @ M )
= ( add_mset_a @ B @ N ) )
=> ? [M1: multiset_a,A: a] :
( ( M
= ( add_mset_a @ A @ M1 ) )
& ( ( F @ A )
= B )
& ( ( image_mset_a_a @ F @ M1 )
= N ) ) ) ).
% msed_map_invR
thf(fact_917_msed__map__invL,axiom,
! [F: a > a,A2: a,M: multiset_a,N: multiset_a] :
( ( ( image_mset_a_a @ F @ ( add_mset_a @ A2 @ M ) )
= N )
=> ? [N1: multiset_a] :
( ( N
= ( add_mset_a @ ( F @ A2 ) @ N1 ) )
& ( ( image_mset_a_a @ F @ M )
= N1 ) ) ) ).
% msed_map_invL
thf(fact_918_multiset_Omap__ident__strong,axiom,
! [T3: multis4781622204924204013iset_a,F: produc6518373309651786023iset_a > produc6518373309651786023iset_a] :
( ! [Z: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ Z @ ( set_ms2379723421352415844iset_a @ T3 ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( image_1581608793515940347iset_a @ F @ T3 )
= T3 ) ) ).
% multiset.map_ident_strong
thf(fact_919_multiset_Omap__ident__strong,axiom,
! [T3: multis501812127501805293od_a_a,F: product_prod_a_a > product_prod_a_a] :
( ! [Z: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Z @ ( set_ms119794563918438244od_a_a @ T3 ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( image_1827833358229211643od_a_a @ F @ T3 )
= T3 ) ) ).
% multiset.map_ident_strong
thf(fact_920_multiset_Omap__ident__strong,axiom,
! [T3: multiset_a,F: a > a] :
( ! [Z: a] :
( ( member_a @ Z @ ( set_mset_a @ T3 ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( image_mset_a_a @ F @ T3 )
= T3 ) ) ).
% multiset.map_ident_strong
thf(fact_921_image__mset__single,axiom,
! [F: a > a,X2: a] :
( ( image_mset_a_a @ F @ ( add_mset_a @ X2 @ zero_zero_multiset_a ) )
= ( add_mset_a @ ( F @ X2 ) @ zero_zero_multiset_a ) ) ).
% image_mset_single
thf(fact_922_sum__mset__mono,axiom,
! [K: multis4781622204924204013iset_a,F: produc6518373309651786023iset_a > nat,G: produc6518373309651786023iset_a > nat] :
( ! [I3: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ I3 @ ( set_ms2379723421352415844iset_a @ K ) )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_nat @ ( comm_m762188921832702859et_nat @ ( image_7071356937311112700_a_nat @ F @ K ) ) @ ( comm_m762188921832702859et_nat @ ( image_7071356937311112700_a_nat @ G @ K ) ) ) ) ).
% sum_mset_mono
thf(fact_923_sum__mset__mono,axiom,
! [K: multis501812127501805293od_a_a,F: product_prod_a_a > nat,G: product_prod_a_a > nat] :
( ! [I3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ I3 @ ( set_ms119794563918438244od_a_a @ K ) )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_nat @ ( comm_m762188921832702859et_nat @ ( image_2949645167302440444_a_nat @ F @ K ) ) @ ( comm_m762188921832702859et_nat @ ( image_2949645167302440444_a_nat @ G @ K ) ) ) ) ).
% sum_mset_mono
thf(fact_924_sum__mset__mono,axiom,
! [K: multiset_a,F: a > nat,G: a > nat] :
( ! [I3: a] :
( ( member_a @ I3 @ ( set_mset_a @ K ) )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_nat @ ( comm_m762188921832702859et_nat @ ( image_mset_a_nat @ F @ K ) ) @ ( comm_m762188921832702859et_nat @ ( image_mset_a_nat @ G @ K ) ) ) ) ).
% sum_mset_mono
thf(fact_925_subset__mset_Omax__add__distrib__left,axiom,
! [X2: multiset_a,Y: multiset_a,Z3: multiset_a] :
( ( plus_plus_multiset_a @ ( max_multiset_a @ subseteq_mset_a @ X2 @ Y ) @ Z3 )
= ( max_multiset_a @ subseteq_mset_a @ ( plus_plus_multiset_a @ X2 @ Z3 ) @ ( plus_plus_multiset_a @ Y @ Z3 ) ) ) ).
% subset_mset.max_add_distrib_left
thf(fact_926_subset__mset_Omin__add__distrib__left,axiom,
! [X2: multiset_a,Y: multiset_a,Z3: multiset_a] :
( ( plus_plus_multiset_a @ ( min_multiset_a @ subseteq_mset_a @ X2 @ Y ) @ Z3 )
= ( min_multiset_a @ subseteq_mset_a @ ( plus_plus_multiset_a @ X2 @ Z3 ) @ ( plus_plus_multiset_a @ Y @ Z3 ) ) ) ).
% subset_mset.min_add_distrib_left
thf(fact_927_subset__mset_Omax__add__distrib__right,axiom,
! [X2: multiset_a,Y: multiset_a,Z3: multiset_a] :
( ( plus_plus_multiset_a @ X2 @ ( max_multiset_a @ subseteq_mset_a @ Y @ Z3 ) )
= ( max_multiset_a @ subseteq_mset_a @ ( plus_plus_multiset_a @ X2 @ Y ) @ ( plus_plus_multiset_a @ X2 @ Z3 ) ) ) ).
% subset_mset.max_add_distrib_right
thf(fact_928_subset__mset_Omin__add__distrib__right,axiom,
! [X2: multiset_a,Y: multiset_a,Z3: multiset_a] :
( ( plus_plus_multiset_a @ X2 @ ( min_multiset_a @ subseteq_mset_a @ Y @ Z3 ) )
= ( min_multiset_a @ subseteq_mset_a @ ( plus_plus_multiset_a @ X2 @ Y ) @ ( plus_plus_multiset_a @ X2 @ Z3 ) ) ) ).
% subset_mset.min_add_distrib_right
thf(fact_929_multpw__map,axiom,
! [X3: multis4781622204924204013iset_a,Y2: multis4781622204924204013iset_a,Ns: set_Pr3059988933816007879iset_a,F: produc6518373309651786023iset_a > multiset_a,G: produc6518373309651786023iset_a > multiset_a,Ns2: set_Pr79727621955416071iset_a] :
( ! [X5: produc6518373309651786023iset_a,Y5: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X5 @ ( set_ms2379723421352415844iset_a @ X3 ) )
=> ( ( member5199237121806060112iset_a @ Y5 @ ( set_ms2379723421352415844iset_a @ Y2 ) )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X5 @ Y5 ) @ Ns )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( F @ X5 ) @ ( G @ Y5 ) ) @ Ns2 ) ) ) )
=> ( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ X3 @ Y2 ) @ ( multis3430030276581740203iset_a @ Ns ) )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ ( image_5610718949088693016iset_a @ F @ X3 ) @ ( image_5610718949088693016iset_a @ G @ Y2 ) ) @ ( multis451435756853798856iset_a @ Ns2 ) ) ) ) ).
% multpw_map
thf(fact_930_multpw__map,axiom,
! [X3: multis501812127501805293od_a_a,Y2: multis501812127501805293od_a_a,Ns: set_Pr8600417178894128327od_a_a,F: product_prod_a_a > multiset_a,G: product_prod_a_a > multiset_a,Ns2: set_Pr79727621955416071iset_a] :
( ! [X5: product_prod_a_a,Y5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ ( set_ms119794563918438244od_a_a @ X3 ) )
=> ( ( member1426531477525435216od_a_a @ Y5 @ ( set_ms119794563918438244od_a_a @ Y2 ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X5 @ Y5 ) @ Ns )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( F @ X5 ) @ ( G @ Y5 ) ) @ Ns2 ) ) ) )
=> ( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ X3 @ Y2 ) @ ( multis1821517007024690859od_a_a @ Ns ) )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ ( image_119670572209922840iset_a @ F @ X3 ) @ ( image_119670572209922840iset_a @ G @ Y2 ) ) @ ( multis451435756853798856iset_a @ Ns2 ) ) ) ) ).
% multpw_map
thf(fact_931_multpw__map,axiom,
! [X3: multiset_multiset_a,Y2: multiset_multiset_a,Ns: set_Pr79727621955416071iset_a,F: multiset_a > multiset_a,G: multiset_a > multiset_a,Ns2: set_Pr79727621955416071iset_a] :
( ! [X5: multiset_a,Y5: multiset_a] :
( ( member_multiset_a @ X5 @ ( set_mset_multiset_a @ X3 ) )
=> ( ( member_multiset_a @ Y5 @ ( set_mset_multiset_a @ Y2 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Y5 ) @ Ns )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( F @ X5 ) @ ( G @ Y5 ) ) @ Ns2 ) ) ) )
=> ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X3 @ Y2 ) @ ( multis451435756853798856iset_a @ Ns ) )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ ( image_7899907538157629371iset_a @ F @ X3 ) @ ( image_7899907538157629371iset_a @ G @ Y2 ) ) @ ( multis451435756853798856iset_a @ Ns2 ) ) ) ) ).
% multpw_map
thf(fact_932_multpw__map,axiom,
! [X3: multis4781622204924204013iset_a,Y2: multis4781622204924204013iset_a,Ns: set_Pr3059988933816007879iset_a,F: produc6518373309651786023iset_a > a,G: produc6518373309651786023iset_a > a,Ns2: set_Product_prod_a_a] :
( ! [X5: produc6518373309651786023iset_a,Y5: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X5 @ ( set_ms2379723421352415844iset_a @ X3 ) )
=> ( ( member5199237121806060112iset_a @ Y5 @ ( set_ms2379723421352415844iset_a @ Y2 ) )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X5 @ Y5 ) @ Ns )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F @ X5 ) @ ( G @ Y5 ) ) @ Ns2 ) ) ) )
=> ( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ X3 @ Y2 ) @ ( multis3430030276581740203iset_a @ Ns ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( image_2834619569719286738et_a_a @ F @ X3 ) @ ( image_2834619569719286738et_a_a @ G @ Y2 ) ) @ ( multis2373687813370458242ltpw_a @ Ns2 ) ) ) ) ).
% multpw_map
thf(fact_933_multpw__map,axiom,
! [X3: multis501812127501805293od_a_a,Y2: multis501812127501805293od_a_a,Ns: set_Pr8600417178894128327od_a_a,F: product_prod_a_a > a,G: product_prod_a_a > a,Ns2: set_Product_prod_a_a] :
( ! [X5: product_prod_a_a,Y5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ ( set_ms119794563918438244od_a_a @ X3 ) )
=> ( ( member1426531477525435216od_a_a @ Y5 @ ( set_ms119794563918438244od_a_a @ Y2 ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X5 @ Y5 ) @ Ns )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F @ X5 ) @ ( G @ Y5 ) ) @ Ns2 ) ) ) )
=> ( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ X3 @ Y2 ) @ ( multis1821517007024690859od_a_a @ Ns ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( image_3362453674455528402_a_a_a @ F @ X3 ) @ ( image_3362453674455528402_a_a_a @ G @ Y2 ) ) @ ( multis2373687813370458242ltpw_a @ Ns2 ) ) ) ) ).
% multpw_map
thf(fact_934_multpw__map,axiom,
! [X3: multiset_multiset_a,Y2: multiset_multiset_a,Ns: set_Pr79727621955416071iset_a,F: multiset_a > a,G: multiset_a > a,Ns2: set_Product_prod_a_a] :
( ! [X5: multiset_a,Y5: multiset_a] :
( ( member_multiset_a @ X5 @ ( set_mset_multiset_a @ X3 ) )
=> ( ( member_multiset_a @ Y5 @ ( set_mset_multiset_a @ Y2 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Y5 ) @ Ns )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F @ X5 ) @ ( G @ Y5 ) ) @ Ns2 ) ) ) )
=> ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X3 @ Y2 ) @ ( multis451435756853798856iset_a @ Ns ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( image_4765768716773618677et_a_a @ F @ X3 ) @ ( image_4765768716773618677et_a_a @ G @ Y2 ) ) @ ( multis2373687813370458242ltpw_a @ Ns2 ) ) ) ) ).
% multpw_map
thf(fact_935_multpw__map,axiom,
! [X3: multiset_a,Y2: multiset_a,Ns: set_Product_prod_a_a,F: a > multiset_a,G: a > multiset_a,Ns2: set_Pr79727621955416071iset_a] :
( ! [X5: a,Y5: a] :
( ( member_a @ X5 @ ( set_mset_a @ X3 ) )
=> ( ( member_a @ Y5 @ ( set_mset_a @ Y2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y5 ) @ Ns )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( F @ X5 ) @ ( G @ Y5 ) ) @ Ns2 ) ) ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ Y2 ) @ ( multis2373687813370458242ltpw_a @ Ns ) )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ ( image_2204096306494911105iset_a @ F @ X3 ) @ ( image_2204096306494911105iset_a @ G @ Y2 ) ) @ ( multis451435756853798856iset_a @ Ns2 ) ) ) ) ).
% multpw_map
thf(fact_936_multpw__map,axiom,
! [X3: multiset_a,Y2: multiset_a,Ns: set_Product_prod_a_a,F: a > a,G: a > a,Ns2: set_Product_prod_a_a] :
( ! [X5: a,Y5: a] :
( ( member_a @ X5 @ ( set_mset_a @ X3 ) )
=> ( ( member_a @ Y5 @ ( set_mset_a @ Y2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y5 ) @ Ns )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F @ X5 ) @ ( G @ Y5 ) ) @ Ns2 ) ) ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ Y2 ) @ ( multis2373687813370458242ltpw_a @ Ns ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( image_mset_a_a @ F @ X3 ) @ ( image_mset_a_a @ G @ Y2 ) ) @ ( multis2373687813370458242ltpw_a @ Ns2 ) ) ) ) ).
% multpw_map
thf(fact_937_mult2__alt__map,axiom,
! [X3: multis4781622204924204013iset_a,Y2: multis4781622204924204013iset_a,Ns: set_Pr3059988933816007879iset_a,F: produc6518373309651786023iset_a > multiset_a,G: produc6518373309651786023iset_a > multiset_a,Ns2: set_Pr79727621955416071iset_a,S: set_Pr3059988933816007879iset_a,S2: set_Pr79727621955416071iset_a,B: $o] :
( ! [X5: produc6518373309651786023iset_a,Y5: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X5 @ ( set_ms2379723421352415844iset_a @ X3 ) )
=> ( ( member5199237121806060112iset_a @ Y5 @ ( set_ms2379723421352415844iset_a @ Y2 ) )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X5 @ Y5 ) @ Ns )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( F @ X5 ) @ ( G @ Y5 ) ) @ Ns2 ) ) ) )
=> ( ! [X5: produc6518373309651786023iset_a,Y5: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X5 @ ( set_ms2379723421352415844iset_a @ X3 ) )
=> ( ( member5199237121806060112iset_a @ Y5 @ ( set_ms2379723421352415844iset_a @ Y2 ) )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X5 @ Y5 ) @ S )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( F @ X5 ) @ ( G @ Y5 ) ) @ S2 ) ) ) )
=> ( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ X3 @ Y2 ) @ ( multis1572145089415997755iset_a @ B @ Ns @ S ) )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ ( image_5610718949088693016iset_a @ F @ X3 ) @ ( image_5610718949088693016iset_a @ G @ Y2 ) ) @ ( multis2006310815304088152iset_a @ B @ Ns2 @ S2 ) ) ) ) ) ).
% mult2_alt_map
thf(fact_938_mult2__alt__map,axiom,
! [X3: multis501812127501805293od_a_a,Y2: multis501812127501805293od_a_a,Ns: set_Pr8600417178894128327od_a_a,F: product_prod_a_a > multiset_a,G: product_prod_a_a > multiset_a,Ns2: set_Pr79727621955416071iset_a,S: set_Pr8600417178894128327od_a_a,S2: set_Pr79727621955416071iset_a,B: $o] :
( ! [X5: product_prod_a_a,Y5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ ( set_ms119794563918438244od_a_a @ X3 ) )
=> ( ( member1426531477525435216od_a_a @ Y5 @ ( set_ms119794563918438244od_a_a @ Y2 ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X5 @ Y5 ) @ Ns )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( F @ X5 ) @ ( G @ Y5 ) ) @ Ns2 ) ) ) )
=> ( ! [X5: product_prod_a_a,Y5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ ( set_ms119794563918438244od_a_a @ X3 ) )
=> ( ( member1426531477525435216od_a_a @ Y5 @ ( set_ms119794563918438244od_a_a @ Y2 ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X5 @ Y5 ) @ S )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( F @ X5 ) @ ( G @ Y5 ) ) @ S2 ) ) ) )
=> ( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ X3 @ Y2 ) @ ( multis4249482161328750907od_a_a @ B @ Ns @ S ) )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ ( image_119670572209922840iset_a @ F @ X3 ) @ ( image_119670572209922840iset_a @ G @ Y2 ) ) @ ( multis2006310815304088152iset_a @ B @ Ns2 @ S2 ) ) ) ) ) ).
% mult2_alt_map
thf(fact_939_mult2__alt__map,axiom,
! [X3: multis4781622204924204013iset_a,Y2: multis4781622204924204013iset_a,Ns: set_Pr3059988933816007879iset_a,F: produc6518373309651786023iset_a > a,G: produc6518373309651786023iset_a > a,Ns2: set_Product_prod_a_a,S: set_Pr3059988933816007879iset_a,S2: set_Product_prod_a_a,B: $o] :
( ! [X5: produc6518373309651786023iset_a,Y5: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X5 @ ( set_ms2379723421352415844iset_a @ X3 ) )
=> ( ( member5199237121806060112iset_a @ Y5 @ ( set_ms2379723421352415844iset_a @ Y2 ) )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X5 @ Y5 ) @ Ns )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F @ X5 ) @ ( G @ Y5 ) ) @ Ns2 ) ) ) )
=> ( ! [X5: produc6518373309651786023iset_a,Y5: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X5 @ ( set_ms2379723421352415844iset_a @ X3 ) )
=> ( ( member5199237121806060112iset_a @ Y5 @ ( set_ms2379723421352415844iset_a @ Y2 ) )
=> ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X5 @ Y5 ) @ S )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F @ X5 ) @ ( G @ Y5 ) ) @ S2 ) ) ) )
=> ( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ X3 @ Y2 ) @ ( multis1572145089415997755iset_a @ B @ Ns @ S ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( image_2834619569719286738et_a_a @ F @ X3 ) @ ( image_2834619569719286738et_a_a @ G @ Y2 ) ) @ ( multis1686031732355738386_alt_a @ B @ Ns2 @ S2 ) ) ) ) ) ).
% mult2_alt_map
thf(fact_940_mult2__alt__map,axiom,
! [X3: multis501812127501805293od_a_a,Y2: multis501812127501805293od_a_a,Ns: set_Pr8600417178894128327od_a_a,F: product_prod_a_a > a,G: product_prod_a_a > a,Ns2: set_Product_prod_a_a,S: set_Pr8600417178894128327od_a_a,S2: set_Product_prod_a_a,B: $o] :
( ! [X5: product_prod_a_a,Y5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ ( set_ms119794563918438244od_a_a @ X3 ) )
=> ( ( member1426531477525435216od_a_a @ Y5 @ ( set_ms119794563918438244od_a_a @ Y2 ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X5 @ Y5 ) @ Ns )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F @ X5 ) @ ( G @ Y5 ) ) @ Ns2 ) ) ) )
=> ( ! [X5: product_prod_a_a,Y5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ ( set_ms119794563918438244od_a_a @ X3 ) )
=> ( ( member1426531477525435216od_a_a @ Y5 @ ( set_ms119794563918438244od_a_a @ Y2 ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X5 @ Y5 ) @ S )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F @ X5 ) @ ( G @ Y5 ) ) @ S2 ) ) ) )
=> ( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ X3 @ Y2 ) @ ( multis4249482161328750907od_a_a @ B @ Ns @ S ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( image_3362453674455528402_a_a_a @ F @ X3 ) @ ( image_3362453674455528402_a_a_a @ G @ Y2 ) ) @ ( multis1686031732355738386_alt_a @ B @ Ns2 @ S2 ) ) ) ) ) ).
% mult2_alt_map
thf(fact_941_mult2__alt__map,axiom,
! [X3: multiset_multiset_a,Y2: multiset_multiset_a,Ns: set_Pr79727621955416071iset_a,F: multiset_a > multiset_a,G: multiset_a > multiset_a,Ns2: set_Pr79727621955416071iset_a,S: set_Pr79727621955416071iset_a,S2: set_Pr79727621955416071iset_a,B: $o] :
( ! [X5: multiset_a,Y5: multiset_a] :
( ( member_multiset_a @ X5 @ ( set_mset_multiset_a @ X3 ) )
=> ( ( member_multiset_a @ Y5 @ ( set_mset_multiset_a @ Y2 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Y5 ) @ Ns )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( F @ X5 ) @ ( G @ Y5 ) ) @ Ns2 ) ) ) )
=> ( ! [X5: multiset_a,Y5: multiset_a] :
( ( member_multiset_a @ X5 @ ( set_mset_multiset_a @ X3 ) )
=> ( ( member_multiset_a @ Y5 @ ( set_mset_multiset_a @ Y2 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Y5 ) @ S )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( F @ X5 ) @ ( G @ Y5 ) ) @ S2 ) ) ) )
=> ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X3 @ Y2 ) @ ( multis2006310815304088152iset_a @ B @ Ns @ S ) )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ ( image_7899907538157629371iset_a @ F @ X3 ) @ ( image_7899907538157629371iset_a @ G @ Y2 ) ) @ ( multis2006310815304088152iset_a @ B @ Ns2 @ S2 ) ) ) ) ) ).
% mult2_alt_map
thf(fact_942_mult2__alt__map,axiom,
! [X3: multiset_multiset_a,Y2: multiset_multiset_a,Ns: set_Pr79727621955416071iset_a,F: multiset_a > a,G: multiset_a > a,Ns2: set_Product_prod_a_a,S: set_Pr79727621955416071iset_a,S2: set_Product_prod_a_a,B: $o] :
( ! [X5: multiset_a,Y5: multiset_a] :
( ( member_multiset_a @ X5 @ ( set_mset_multiset_a @ X3 ) )
=> ( ( member_multiset_a @ Y5 @ ( set_mset_multiset_a @ Y2 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Y5 ) @ Ns )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F @ X5 ) @ ( G @ Y5 ) ) @ Ns2 ) ) ) )
=> ( ! [X5: multiset_a,Y5: multiset_a] :
( ( member_multiset_a @ X5 @ ( set_mset_multiset_a @ X3 ) )
=> ( ( member_multiset_a @ Y5 @ ( set_mset_multiset_a @ Y2 ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X5 @ Y5 ) @ S )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F @ X5 ) @ ( G @ Y5 ) ) @ S2 ) ) ) )
=> ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ X3 @ Y2 ) @ ( multis2006310815304088152iset_a @ B @ Ns @ S ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( image_4765768716773618677et_a_a @ F @ X3 ) @ ( image_4765768716773618677et_a_a @ G @ Y2 ) ) @ ( multis1686031732355738386_alt_a @ B @ Ns2 @ S2 ) ) ) ) ) ).
% mult2_alt_map
thf(fact_943_mult2__alt__map,axiom,
! [X3: multiset_a,Y2: multiset_a,Ns: set_Product_prod_a_a,F: a > multiset_a,G: a > multiset_a,Ns2: set_Pr79727621955416071iset_a,S: set_Product_prod_a_a,S2: set_Pr79727621955416071iset_a,B: $o] :
( ! [X5: a,Y5: a] :
( ( member_a @ X5 @ ( set_mset_a @ X3 ) )
=> ( ( member_a @ Y5 @ ( set_mset_a @ Y2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y5 ) @ Ns )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( F @ X5 ) @ ( G @ Y5 ) ) @ Ns2 ) ) ) )
=> ( ! [X5: a,Y5: a] :
( ( member_a @ X5 @ ( set_mset_a @ X3 ) )
=> ( ( member_a @ Y5 @ ( set_mset_a @ Y2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y5 ) @ S )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( F @ X5 ) @ ( G @ Y5 ) ) @ S2 ) ) ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ Y2 ) @ ( multis1686031732355738386_alt_a @ B @ Ns @ S ) )
=> ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ ( image_2204096306494911105iset_a @ F @ X3 ) @ ( image_2204096306494911105iset_a @ G @ Y2 ) ) @ ( multis2006310815304088152iset_a @ B @ Ns2 @ S2 ) ) ) ) ) ).
% mult2_alt_map
thf(fact_944_mult2__alt__map,axiom,
! [X3: multiset_a,Y2: multiset_a,Ns: set_Product_prod_a_a,F: a > a,G: a > a,Ns2: set_Product_prod_a_a,S: set_Product_prod_a_a,S2: set_Product_prod_a_a,B: $o] :
( ! [X5: a,Y5: a] :
( ( member_a @ X5 @ ( set_mset_a @ X3 ) )
=> ( ( member_a @ Y5 @ ( set_mset_a @ Y2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y5 ) @ Ns )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F @ X5 ) @ ( G @ Y5 ) ) @ Ns2 ) ) ) )
=> ( ! [X5: a,Y5: a] :
( ( member_a @ X5 @ ( set_mset_a @ X3 ) )
=> ( ( member_a @ Y5 @ ( set_mset_a @ Y2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y5 ) @ S )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F @ X5 ) @ ( G @ Y5 ) ) @ S2 ) ) ) )
=> ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ Y2 ) @ ( multis1686031732355738386_alt_a @ B @ Ns @ S ) )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( image_mset_a_a @ F @ X3 ) @ ( image_mset_a_a @ G @ Y2 ) ) @ ( multis1686031732355738386_alt_a @ B @ Ns2 @ S2 ) ) ) ) ) ).
% mult2_alt_map
thf(fact_945_empty__iff,axiom,
! [C: produc6518373309651786023iset_a] :
~ ( member5199237121806060112iset_a @ C @ bot_bo5898781664966027891iset_a ) ).
% empty_iff
thf(fact_946_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_947_empty__iff,axiom,
! [C: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ C @ bot_bo3357376287454694259od_a_a ) ).
% empty_iff
thf(fact_948_all__not__in__conv,axiom,
! [A3: set_Pr79727621955416071iset_a] :
( ( ! [X4: produc6518373309651786023iset_a] :
~ ( member5199237121806060112iset_a @ X4 @ A3 ) )
= ( A3 = bot_bo5898781664966027891iset_a ) ) ).
% all_not_in_conv
thf(fact_949_all__not__in__conv,axiom,
! [A3: set_a] :
( ( ! [X4: a] :
~ ( member_a @ X4 @ A3 ) )
= ( A3 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_950_all__not__in__conv,axiom,
! [A3: set_Product_prod_a_a] :
( ( ! [X4: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ X4 @ A3 ) )
= ( A3 = bot_bo3357376287454694259od_a_a ) ) ).
% all_not_in_conv
thf(fact_951_subset__empty,axiom,
! [A3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A3 @ bot_bo3357376287454694259od_a_a )
= ( A3 = bot_bo3357376287454694259od_a_a ) ) ).
% subset_empty
thf(fact_952_subset__empty,axiom,
! [A3: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ A3 @ bot_bo5898781664966027891iset_a )
= ( A3 = bot_bo5898781664966027891iset_a ) ) ).
% subset_empty
thf(fact_953_empty__subsetI,axiom,
! [A3: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ bot_bo3357376287454694259od_a_a @ A3 ) ).
% empty_subsetI
thf(fact_954_empty__subsetI,axiom,
! [A3: set_Pr79727621955416071iset_a] : ( ord_le5871882978999109543iset_a @ bot_bo5898781664966027891iset_a @ A3 ) ).
% empty_subsetI
thf(fact_955_sup__bot__left,axiom,
! [X2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ bot_bo3357376287454694259od_a_a @ X2 )
= X2 ) ).
% sup_bot_left
thf(fact_956_sup__bot__right,axiom,
! [X2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X2 @ bot_bo3357376287454694259od_a_a )
= X2 ) ).
% sup_bot_right
thf(fact_957_bot__eq__sup__iff,axiom,
! [X2: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( bot_bo3357376287454694259od_a_a
= ( sup_su3048258781599657691od_a_a @ X2 @ Y ) )
= ( ( X2 = bot_bo3357376287454694259od_a_a )
& ( Y = bot_bo3357376287454694259od_a_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_958_sup__eq__bot__iff,axiom,
! [X2: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( ( sup_su3048258781599657691od_a_a @ X2 @ Y )
= bot_bo3357376287454694259od_a_a )
= ( ( X2 = bot_bo3357376287454694259od_a_a )
& ( Y = bot_bo3357376287454694259od_a_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_959_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ( sup_su3048258781599657691od_a_a @ A2 @ B )
= bot_bo3357376287454694259od_a_a )
= ( ( A2 = bot_bo3357376287454694259od_a_a )
& ( B = bot_bo3357376287454694259od_a_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_960_sup__bot_Oleft__neutral,axiom,
! [A2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ bot_bo3357376287454694259od_a_a @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_961_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( bot_bo3357376287454694259od_a_a
= ( sup_su3048258781599657691od_a_a @ A2 @ B ) )
= ( ( A2 = bot_bo3357376287454694259od_a_a )
& ( B = bot_bo3357376287454694259od_a_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_962_sup__bot_Oright__neutral,axiom,
! [A2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A2 @ bot_bo3357376287454694259od_a_a )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_963_Un__empty,axiom,
! [A3: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ( sup_su3048258781599657691od_a_a @ A3 @ B2 )
= bot_bo3357376287454694259od_a_a )
= ( ( A3 = bot_bo3357376287454694259od_a_a )
& ( B2 = bot_bo3357376287454694259od_a_a ) ) ) ).
% Un_empty
thf(fact_964_Diff__eq__empty__iff,axiom,
! [A3: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ( minus_6817036919807184750od_a_a @ A3 @ B2 )
= bot_bo3357376287454694259od_a_a )
= ( ord_le746702958409616551od_a_a @ A3 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_965_Diff__eq__empty__iff,axiom,
! [A3: set_Pr79727621955416071iset_a,B2: set_Pr79727621955416071iset_a] :
( ( ( minus_2461365523287052142iset_a @ A3 @ B2 )
= bot_bo5898781664966027891iset_a )
= ( ord_le5871882978999109543iset_a @ A3 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_966_set__mset__empty,axiom,
( ( set_mset_a @ zero_zero_multiset_a )
= bot_bot_set_a ) ).
% set_mset_empty
thf(fact_967_set__mset__eq__empty__iff,axiom,
! [M: multiset_a] :
( ( ( set_mset_a @ M )
= bot_bot_set_a )
= ( M = zero_zero_multiset_a ) ) ).
% set_mset_eq_empty_iff
thf(fact_968_mset__set_Ocomp__fun__commute__on,axiom,
! [Y: a,X2: a] :
( ( comp_m5768739296785061537iset_a @ ( add_mset_a @ Y ) @ ( add_mset_a @ X2 ) )
= ( comp_m5768739296785061537iset_a @ ( add_mset_a @ X2 ) @ ( add_mset_a @ Y ) ) ) ).
% mset_set.comp_fun_commute_on
thf(fact_969_emptyE,axiom,
! [A2: produc6518373309651786023iset_a] :
~ ( member5199237121806060112iset_a @ A2 @ bot_bo5898781664966027891iset_a ) ).
% emptyE
thf(fact_970_emptyE,axiom,
! [A2: a] :
~ ( member_a @ A2 @ bot_bot_set_a ) ).
% emptyE
thf(fact_971_emptyE,axiom,
! [A2: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) ).
% emptyE
thf(fact_972_equals0D,axiom,
! [A3: set_Pr79727621955416071iset_a,A2: produc6518373309651786023iset_a] :
( ( A3 = bot_bo5898781664966027891iset_a )
=> ~ ( member5199237121806060112iset_a @ A2 @ A3 ) ) ).
% equals0D
thf(fact_973_equals0D,axiom,
! [A3: set_a,A2: a] :
( ( A3 = bot_bot_set_a )
=> ~ ( member_a @ A2 @ A3 ) ) ).
% equals0D
thf(fact_974_equals0D,axiom,
! [A3: set_Product_prod_a_a,A2: product_prod_a_a] :
( ( A3 = bot_bo3357376287454694259od_a_a )
=> ~ ( member1426531477525435216od_a_a @ A2 @ A3 ) ) ).
% equals0D
thf(fact_975_equals0I,axiom,
! [A3: set_Pr79727621955416071iset_a] :
( ! [Y5: produc6518373309651786023iset_a] :
~ ( member5199237121806060112iset_a @ Y5 @ A3 )
=> ( A3 = bot_bo5898781664966027891iset_a ) ) ).
% equals0I
thf(fact_976_equals0I,axiom,
! [A3: set_a] :
( ! [Y5: a] :
~ ( member_a @ Y5 @ A3 )
=> ( A3 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_977_equals0I,axiom,
! [A3: set_Product_prod_a_a] :
( ! [Y5: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ Y5 @ A3 )
=> ( A3 = bot_bo3357376287454694259od_a_a ) ) ).
% equals0I
thf(fact_978_ex__in__conv,axiom,
! [A3: set_Pr79727621955416071iset_a] :
( ( ? [X4: produc6518373309651786023iset_a] : ( member5199237121806060112iset_a @ X4 @ A3 ) )
= ( A3 != bot_bo5898781664966027891iset_a ) ) ).
% ex_in_conv
thf(fact_979_ex__in__conv,axiom,
! [A3: set_a] :
( ( ? [X4: a] : ( member_a @ X4 @ A3 ) )
= ( A3 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_980_ex__in__conv,axiom,
! [A3: set_Product_prod_a_a] :
( ( ? [X4: product_prod_a_a] : ( member1426531477525435216od_a_a @ X4 @ A3 ) )
= ( A3 != bot_bo3357376287454694259od_a_a ) ) ).
% ex_in_conv
thf(fact_981_bot_Oextremum,axiom,
! [A2: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ bot_bo3357376287454694259od_a_a @ A2 ) ).
% bot.extremum
thf(fact_982_bot_Oextremum,axiom,
! [A2: set_Pr79727621955416071iset_a] : ( ord_le5871882978999109543iset_a @ bot_bo5898781664966027891iset_a @ A2 ) ).
% bot.extremum
thf(fact_983_bot_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).
% bot.extremum
thf(fact_984_bot_Oextremum__unique,axiom,
! [A2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ bot_bo3357376287454694259od_a_a )
= ( A2 = bot_bo3357376287454694259od_a_a ) ) ).
% bot.extremum_unique
thf(fact_985_bot_Oextremum__unique,axiom,
! [A2: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ A2 @ bot_bo5898781664966027891iset_a )
= ( A2 = bot_bo5898781664966027891iset_a ) ) ).
% bot.extremum_unique
thf(fact_986_bot_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
= ( A2 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_987_bot_Oextremum__uniqueI,axiom,
! [A2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ bot_bo3357376287454694259od_a_a )
=> ( A2 = bot_bo3357376287454694259od_a_a ) ) ).
% bot.extremum_uniqueI
thf(fact_988_bot_Oextremum__uniqueI,axiom,
! [A2: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ A2 @ bot_bo5898781664966027891iset_a )
=> ( A2 = bot_bo5898781664966027891iset_a ) ) ).
% bot.extremum_uniqueI
thf(fact_989_bot_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
=> ( A2 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_990_Un__empty__right,axiom,
! [A3: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A3 @ bot_bo3357376287454694259od_a_a )
= A3 ) ).
% Un_empty_right
thf(fact_991_Un__empty__left,axiom,
! [B2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ bot_bo3357376287454694259od_a_a @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_992_diff__shunt__var,axiom,
! [X2: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( ( minus_6817036919807184750od_a_a @ X2 @ Y )
= bot_bo3357376287454694259od_a_a )
= ( ord_le746702958409616551od_a_a @ X2 @ Y ) ) ).
% diff_shunt_var
thf(fact_993_diff__shunt__var,axiom,
! [X2: set_Pr79727621955416071iset_a,Y: set_Pr79727621955416071iset_a] :
( ( ( minus_2461365523287052142iset_a @ X2 @ Y )
= bot_bo5898781664966027891iset_a )
= ( ord_le5871882978999109543iset_a @ X2 @ Y ) ) ).
% diff_shunt_var
thf(fact_994_boolean__algebra_Odisj__zero__right,axiom,
! [X2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X2 @ bot_bo3357376287454694259od_a_a )
= X2 ) ).
% boolean_algebra.disj_zero_right
thf(fact_995_bot__empty__eq,axiom,
( bot_bo6270452684981427242et_a_o
= ( ^ [X4: produc6518373309651786023iset_a] : ( member5199237121806060112iset_a @ X4 @ bot_bo5898781664966027891iset_a ) ) ) ).
% bot_empty_eq
thf(fact_996_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X4: a] : ( member_a @ X4 @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_997_bot__empty__eq,axiom,
( bot_bo4160289986317612842_a_a_o
= ( ^ [X4: product_prod_a_a] : ( member1426531477525435216od_a_a @ X4 @ bot_bo3357376287454694259od_a_a ) ) ) ).
% bot_empty_eq
thf(fact_998_boolean__algebra__cancel_Osup1,axiom,
! [A3: set_Product_prod_a_a,K3: set_Product_prod_a_a,A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( A3
= ( sup_su3048258781599657691od_a_a @ K3 @ A2 ) )
=> ( ( sup_su3048258781599657691od_a_a @ A3 @ B )
= ( sup_su3048258781599657691od_a_a @ K3 @ ( sup_su3048258781599657691od_a_a @ A2 @ B ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_999_boolean__algebra__cancel_Osup2,axiom,
! [B2: set_Product_prod_a_a,K3: set_Product_prod_a_a,B: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
( ( B2
= ( sup_su3048258781599657691od_a_a @ K3 @ B ) )
=> ( ( sup_su3048258781599657691od_a_a @ A2 @ B2 )
= ( sup_su3048258781599657691od_a_a @ K3 @ ( sup_su3048258781599657691od_a_a @ A2 @ B ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_1000_union__comp__emptyL,axiom,
! [A3: set_Product_prod_a_a,C2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ( relcomp_a_a_a @ A3 @ C2 )
= bot_bo3357376287454694259od_a_a )
=> ( ( ( relcomp_a_a_a @ B2 @ C2 )
= bot_bo3357376287454694259od_a_a )
=> ( ( relcomp_a_a_a @ ( sup_su3048258781599657691od_a_a @ A3 @ B2 ) @ C2 )
= bot_bo3357376287454694259od_a_a ) ) ) ).
% union_comp_emptyL
thf(fact_1001_union__comp__emptyR,axiom,
! [A3: set_Product_prod_a_a,B2: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( ( relcomp_a_a_a @ A3 @ B2 )
= bot_bo3357376287454694259od_a_a )
=> ( ( ( relcomp_a_a_a @ A3 @ C2 )
= bot_bo3357376287454694259od_a_a )
=> ( ( relcomp_a_a_a @ A3 @ ( sup_su3048258781599657691od_a_a @ B2 @ C2 ) )
= bot_bo3357376287454694259od_a_a ) ) ) ).
% union_comp_emptyR
thf(fact_1002_subset__emptyI,axiom,
! [A3: set_a] :
( ! [X5: a] :
~ ( member_a @ X5 @ A3 )
=> ( ord_less_eq_set_a @ A3 @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_1003_subset__emptyI,axiom,
! [A3: set_Product_prod_a_a] :
( ! [X5: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ X5 @ A3 )
=> ( ord_le746702958409616551od_a_a @ A3 @ bot_bo3357376287454694259od_a_a ) ) ).
% subset_emptyI
thf(fact_1004_subset__emptyI,axiom,
! [A3: set_Pr79727621955416071iset_a] :
( ! [X5: produc6518373309651786023iset_a] :
~ ( member5199237121806060112iset_a @ X5 @ A3 )
=> ( ord_le5871882978999109543iset_a @ A3 @ bot_bo5898781664966027891iset_a ) ) ).
% subset_emptyI
thf(fact_1005_ssubst__Pair__rhs,axiom,
! [R: multiset_a,S: multiset_a,R3: set_Pr79727621955416071iset_a,S2: multiset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ R @ S ) @ R3 )
=> ( ( S2 = S )
=> ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ R @ S2 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_1006_ssubst__Pair__rhs,axiom,
! [R: a,S: a,R3: set_Product_prod_a_a,S2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ R @ S ) @ R3 )
=> ( ( S2 = S )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ R @ S2 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_1007_less__by__empty,axiom,
! [A3: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( A3 = bot_bo3357376287454694259od_a_a )
=> ( ord_le746702958409616551od_a_a @ A3 @ B2 ) ) ).
% less_by_empty
thf(fact_1008_less__by__empty,axiom,
! [A3: set_Pr79727621955416071iset_a,B2: set_Pr79727621955416071iset_a] :
( ( A3 = bot_bo5898781664966027891iset_a )
=> ( ord_le5871882978999109543iset_a @ A3 @ B2 ) ) ).
% less_by_empty
thf(fact_1009_at__most__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 ) ) )
= ( minus_2461365523287052142iset_a @ ( set_ms2379723421352415844iset_a @ M ) @ ( insert4880209295958132471iset_a @ A2 @ bot_bo5898781664966027891iset_a ) ) ) ) ).
% at_most_one_mset_mset_diff
thf(fact_1010_at__most__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 ) ) )
= ( minus_6817036919807184750od_a_a @ ( set_ms119794563918438244od_a_a @ M ) @ ( insert4534936382041156343od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% at_most_one_mset_mset_diff
thf(fact_1011_at__most__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 ) ) )
= ( minus_minus_set_a @ ( set_mset_a @ M ) @ ( insert_a @ A2 @ bot_bot_set_a ) ) ) ) ).
% at_most_one_mset_mset_diff
thf(fact_1012_insert__iff,axiom,
! [A2: produc6518373309651786023iset_a,B: produc6518373309651786023iset_a,A3: set_Pr79727621955416071iset_a] :
( ( member5199237121806060112iset_a @ A2 @ ( insert4880209295958132471iset_a @ B @ A3 ) )
= ( ( A2 = B )
| ( member5199237121806060112iset_a @ A2 @ A3 ) ) ) ).
% insert_iff
thf(fact_1013_insert__iff,axiom,
! [A2: a,B: a,A3: set_a] :
( ( member_a @ A2 @ ( insert_a @ B @ A3 ) )
= ( ( A2 = B )
| ( member_a @ A2 @ A3 ) ) ) ).
% insert_iff
thf(fact_1014_insert__iff,axiom,
! [A2: product_prod_a_a,B: product_prod_a_a,A3: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ B @ A3 ) )
= ( ( A2 = B )
| ( member1426531477525435216od_a_a @ A2 @ A3 ) ) ) ).
% insert_iff
thf(fact_1015_insertCI,axiom,
! [A2: produc6518373309651786023iset_a,B2: set_Pr79727621955416071iset_a,B: produc6518373309651786023iset_a] :
( ( ~ ( member5199237121806060112iset_a @ A2 @ B2 )
=> ( A2 = B ) )
=> ( member5199237121806060112iset_a @ A2 @ ( insert4880209295958132471iset_a @ B @ B2 ) ) ) ).
% insertCI
thf(fact_1016_insertCI,axiom,
! [A2: a,B2: set_a,B: a] :
( ( ~ ( member_a @ A2 @ B2 )
=> ( A2 = B ) )
=> ( member_a @ A2 @ ( insert_a @ B @ B2 ) ) ) ).
% insertCI
thf(fact_1017_insertCI,axiom,
! [A2: product_prod_a_a,B2: set_Product_prod_a_a,B: product_prod_a_a] :
( ( ~ ( member1426531477525435216od_a_a @ A2 @ B2 )
=> ( A2 = B ) )
=> ( member1426531477525435216od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ B @ B2 ) ) ) ).
% insertCI
thf(fact_1018_singletonI,axiom,
! [A2: produc6518373309651786023iset_a] : ( member5199237121806060112iset_a @ A2 @ ( insert4880209295958132471iset_a @ A2 @ bot_bo5898781664966027891iset_a ) ) ).
% singletonI
thf(fact_1019_singletonI,axiom,
! [A2: a] : ( member_a @ A2 @ ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_1020_singletonI,axiom,
! [A2: product_prod_a_a] : ( member1426531477525435216od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) ) ).
% singletonI
thf(fact_1021_insert__subset,axiom,
! [X2: a,A3: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X2 @ A3 ) @ B2 )
= ( ( member_a @ X2 @ B2 )
& ( ord_less_eq_set_a @ A3 @ B2 ) ) ) ).
% insert_subset
thf(fact_1022_insert__subset,axiom,
! [X2: product_prod_a_a,A3: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( insert4534936382041156343od_a_a @ X2 @ A3 ) @ B2 )
= ( ( member1426531477525435216od_a_a @ X2 @ B2 )
& ( ord_le746702958409616551od_a_a @ A3 @ B2 ) ) ) ).
% insert_subset
thf(fact_1023_insert__subset,axiom,
! [X2: produc6518373309651786023iset_a,A3: set_Pr79727621955416071iset_a,B2: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ ( insert4880209295958132471iset_a @ X2 @ A3 ) @ B2 )
= ( ( member5199237121806060112iset_a @ X2 @ B2 )
& ( ord_le5871882978999109543iset_a @ A3 @ B2 ) ) ) ).
% insert_subset
thf(fact_1024_Un__insert__right,axiom,
! [A3: set_Product_prod_a_a,A2: product_prod_a_a,B2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A3 @ ( insert4534936382041156343od_a_a @ A2 @ B2 ) )
= ( insert4534936382041156343od_a_a @ A2 @ ( sup_su3048258781599657691od_a_a @ A3 @ B2 ) ) ) ).
% Un_insert_right
thf(fact_1025_Un__insert__left,axiom,
! [A2: product_prod_a_a,B2: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( insert4534936382041156343od_a_a @ A2 @ B2 ) @ C2 )
= ( insert4534936382041156343od_a_a @ A2 @ ( sup_su3048258781599657691od_a_a @ B2 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_1026_Diff__insert0,axiom,
! [X2: produc6518373309651786023iset_a,A3: set_Pr79727621955416071iset_a,B2: set_Pr79727621955416071iset_a] :
( ~ ( member5199237121806060112iset_a @ X2 @ A3 )
=> ( ( minus_2461365523287052142iset_a @ A3 @ ( insert4880209295958132471iset_a @ X2 @ B2 ) )
= ( minus_2461365523287052142iset_a @ A3 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_1027_Diff__insert0,axiom,
! [X2: a,A3: set_a,B2: set_a] :
( ~ ( member_a @ X2 @ A3 )
=> ( ( minus_minus_set_a @ A3 @ ( insert_a @ X2 @ B2 ) )
= ( minus_minus_set_a @ A3 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_1028_Diff__insert0,axiom,
! [X2: product_prod_a_a,A3: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ X2 @ A3 )
=> ( ( minus_6817036919807184750od_a_a @ A3 @ ( insert4534936382041156343od_a_a @ X2 @ B2 ) )
= ( minus_6817036919807184750od_a_a @ A3 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_1029_insert__Diff1,axiom,
! [X2: produc6518373309651786023iset_a,B2: set_Pr79727621955416071iset_a,A3: set_Pr79727621955416071iset_a] :
( ( member5199237121806060112iset_a @ X2 @ B2 )
=> ( ( minus_2461365523287052142iset_a @ ( insert4880209295958132471iset_a @ X2 @ A3 ) @ B2 )
= ( minus_2461365523287052142iset_a @ A3 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_1030_insert__Diff1,axiom,
! [X2: a,B2: set_a,A3: set_a] :
( ( member_a @ X2 @ B2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X2 @ A3 ) @ B2 )
= ( minus_minus_set_a @ A3 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_1031_insert__Diff1,axiom,
! [X2: product_prod_a_a,B2: set_Product_prod_a_a,A3: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ B2 )
=> ( ( minus_6817036919807184750od_a_a @ ( insert4534936382041156343od_a_a @ X2 @ A3 ) @ B2 )
= ( minus_6817036919807184750od_a_a @ A3 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_1032_singleton__insert__inj__eq_H,axiom,
! [A2: product_prod_a_a,A3: set_Product_prod_a_a,B: product_prod_a_a] :
( ( ( insert4534936382041156343od_a_a @ A2 @ A3 )
= ( insert4534936382041156343od_a_a @ B @ bot_bo3357376287454694259od_a_a ) )
= ( ( A2 = B )
& ( ord_le746702958409616551od_a_a @ A3 @ ( insert4534936382041156343od_a_a @ B @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_1033_singleton__insert__inj__eq_H,axiom,
! [A2: produc6518373309651786023iset_a,A3: set_Pr79727621955416071iset_a,B: produc6518373309651786023iset_a] :
( ( ( insert4880209295958132471iset_a @ A2 @ A3 )
= ( insert4880209295958132471iset_a @ B @ bot_bo5898781664966027891iset_a ) )
= ( ( A2 = B )
& ( ord_le5871882978999109543iset_a @ A3 @ ( insert4880209295958132471iset_a @ B @ bot_bo5898781664966027891iset_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_1034_singleton__insert__inj__eq,axiom,
! [B: product_prod_a_a,A2: product_prod_a_a,A3: set_Product_prod_a_a] :
( ( ( insert4534936382041156343od_a_a @ B @ bot_bo3357376287454694259od_a_a )
= ( insert4534936382041156343od_a_a @ A2 @ A3 ) )
= ( ( A2 = B )
& ( ord_le746702958409616551od_a_a @ A3 @ ( insert4534936382041156343od_a_a @ B @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_1035_singleton__insert__inj__eq,axiom,
! [B: produc6518373309651786023iset_a,A2: produc6518373309651786023iset_a,A3: set_Pr79727621955416071iset_a] :
( ( ( insert4880209295958132471iset_a @ B @ bot_bo5898781664966027891iset_a )
= ( insert4880209295958132471iset_a @ A2 @ A3 ) )
= ( ( A2 = B )
& ( ord_le5871882978999109543iset_a @ A3 @ ( insert4880209295958132471iset_a @ B @ bot_bo5898781664966027891iset_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_1036_set__mset__add__mset__insert,axiom,
! [A2: a,A3: multiset_a] :
( ( set_mset_a @ ( add_mset_a @ A2 @ A3 ) )
= ( insert_a @ A2 @ ( set_mset_a @ A3 ) ) ) ).
% set_mset_add_mset_insert
thf(fact_1037_singletonD,axiom,
! [B: produc6518373309651786023iset_a,A2: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ B @ ( insert4880209295958132471iset_a @ A2 @ bot_bo5898781664966027891iset_a ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_1038_singletonD,axiom,
! [B: a,A2: a] :
( ( member_a @ B @ ( insert_a @ A2 @ bot_bot_set_a ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_1039_singletonD,axiom,
! [B: product_prod_a_a,A2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ B @ ( insert4534936382041156343od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_1040_singleton__iff,axiom,
! [B: produc6518373309651786023iset_a,A2: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ B @ ( insert4880209295958132471iset_a @ A2 @ bot_bo5898781664966027891iset_a ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_1041_singleton__iff,axiom,
! [B: a,A2: a] :
( ( member_a @ B @ ( insert_a @ A2 @ bot_bot_set_a ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_1042_singleton__iff,axiom,
! [B: product_prod_a_a,A2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ B @ ( insert4534936382041156343od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_1043_subset__singleton__iff,axiom,
! [X3: set_Product_prod_a_a,A2: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X3 @ ( insert4534936382041156343od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) )
= ( ( X3 = bot_bo3357376287454694259od_a_a )
| ( X3
= ( insert4534936382041156343od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_1044_subset__singleton__iff,axiom,
! [X3: set_Pr79727621955416071iset_a,A2: produc6518373309651786023iset_a] :
( ( ord_le5871882978999109543iset_a @ X3 @ ( insert4880209295958132471iset_a @ A2 @ bot_bo5898781664966027891iset_a ) )
= ( ( X3 = bot_bo5898781664966027891iset_a )
| ( X3
= ( insert4880209295958132471iset_a @ A2 @ bot_bo5898781664966027891iset_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_1045_subset__singletonD,axiom,
! [A3: set_Product_prod_a_a,X2: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A3 @ ( insert4534936382041156343od_a_a @ X2 @ bot_bo3357376287454694259od_a_a ) )
=> ( ( A3 = bot_bo3357376287454694259od_a_a )
| ( A3
= ( insert4534936382041156343od_a_a @ X2 @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% subset_singletonD
thf(fact_1046_subset__singletonD,axiom,
! [A3: set_Pr79727621955416071iset_a,X2: produc6518373309651786023iset_a] :
( ( ord_le5871882978999109543iset_a @ A3 @ ( insert4880209295958132471iset_a @ X2 @ bot_bo5898781664966027891iset_a ) )
=> ( ( A3 = bot_bo5898781664966027891iset_a )
| ( A3
= ( insert4880209295958132471iset_a @ X2 @ bot_bo5898781664966027891iset_a ) ) ) ) ).
% subset_singletonD
thf(fact_1047_insert__is__Un,axiom,
( insert4534936382041156343od_a_a
= ( ^ [A5: product_prod_a_a] : ( sup_su3048258781599657691od_a_a @ ( insert4534936382041156343od_a_a @ A5 @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% insert_is_Un
thf(fact_1048_Un__singleton__iff,axiom,
! [A3: set_Product_prod_a_a,B2: set_Product_prod_a_a,X2: product_prod_a_a] :
( ( ( sup_su3048258781599657691od_a_a @ A3 @ B2 )
= ( insert4534936382041156343od_a_a @ X2 @ bot_bo3357376287454694259od_a_a ) )
= ( ( ( A3 = bot_bo3357376287454694259od_a_a )
& ( B2
= ( insert4534936382041156343od_a_a @ X2 @ bot_bo3357376287454694259od_a_a ) ) )
| ( ( A3
= ( insert4534936382041156343od_a_a @ X2 @ bot_bo3357376287454694259od_a_a ) )
& ( B2 = bot_bo3357376287454694259od_a_a ) )
| ( ( A3
= ( insert4534936382041156343od_a_a @ X2 @ bot_bo3357376287454694259od_a_a ) )
& ( B2
= ( insert4534936382041156343od_a_a @ X2 @ bot_bo3357376287454694259od_a_a ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_1049_singleton__Un__iff,axiom,
! [X2: product_prod_a_a,A3: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ( insert4534936382041156343od_a_a @ X2 @ bot_bo3357376287454694259od_a_a )
= ( sup_su3048258781599657691od_a_a @ A3 @ B2 ) )
= ( ( ( A3 = bot_bo3357376287454694259od_a_a )
& ( B2
= ( insert4534936382041156343od_a_a @ X2 @ bot_bo3357376287454694259od_a_a ) ) )
| ( ( A3
= ( insert4534936382041156343od_a_a @ X2 @ bot_bo3357376287454694259od_a_a ) )
& ( B2 = bot_bo3357376287454694259od_a_a ) )
| ( ( A3
= ( insert4534936382041156343od_a_a @ X2 @ bot_bo3357376287454694259od_a_a ) )
& ( B2
= ( insert4534936382041156343od_a_a @ X2 @ bot_bo3357376287454694259od_a_a ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_1050_insert__Diff,axiom,
! [A2: produc6518373309651786023iset_a,A3: set_Pr79727621955416071iset_a] :
( ( member5199237121806060112iset_a @ A2 @ A3 )
=> ( ( insert4880209295958132471iset_a @ A2 @ ( minus_2461365523287052142iset_a @ A3 @ ( insert4880209295958132471iset_a @ A2 @ bot_bo5898781664966027891iset_a ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_1051_insert__Diff,axiom,
! [A2: a,A3: set_a] :
( ( member_a @ A2 @ A3 )
=> ( ( insert_a @ A2 @ ( minus_minus_set_a @ A3 @ ( insert_a @ A2 @ bot_bot_set_a ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_1052_insert__Diff,axiom,
! [A2: product_prod_a_a,A3: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A2 @ A3 )
=> ( ( insert4534936382041156343od_a_a @ A2 @ ( minus_6817036919807184750od_a_a @ A3 @ ( insert4534936382041156343od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_1053_Diff__insert__absorb,axiom,
! [X2: produc6518373309651786023iset_a,A3: set_Pr79727621955416071iset_a] :
( ~ ( member5199237121806060112iset_a @ X2 @ A3 )
=> ( ( minus_2461365523287052142iset_a @ ( insert4880209295958132471iset_a @ X2 @ A3 ) @ ( insert4880209295958132471iset_a @ X2 @ bot_bo5898781664966027891iset_a ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_1054_Diff__insert__absorb,axiom,
! [X2: a,A3: set_a] :
( ~ ( member_a @ X2 @ A3 )
=> ( ( minus_minus_set_a @ ( insert_a @ X2 @ A3 ) @ ( insert_a @ X2 @ bot_bot_set_a ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_1055_Diff__insert__absorb,axiom,
! [X2: product_prod_a_a,A3: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ X2 @ A3 )
=> ( ( minus_6817036919807184750od_a_a @ ( insert4534936382041156343od_a_a @ X2 @ A3 ) @ ( insert4534936382041156343od_a_a @ X2 @ bot_bo3357376287454694259od_a_a ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_1056_insert__subsetI,axiom,
! [X2: a,A3: set_a,X3: set_a] :
( ( member_a @ X2 @ A3 )
=> ( ( ord_less_eq_set_a @ X3 @ A3 )
=> ( ord_less_eq_set_a @ ( insert_a @ X2 @ X3 ) @ A3 ) ) ) ).
% insert_subsetI
thf(fact_1057_insert__subsetI,axiom,
! [X2: product_prod_a_a,A3: set_Product_prod_a_a,X3: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ A3 )
=> ( ( ord_le746702958409616551od_a_a @ X3 @ A3 )
=> ( ord_le746702958409616551od_a_a @ ( insert4534936382041156343od_a_a @ X2 @ X3 ) @ A3 ) ) ) ).
% insert_subsetI
thf(fact_1058_insert__subsetI,axiom,
! [X2: produc6518373309651786023iset_a,A3: set_Pr79727621955416071iset_a,X3: set_Pr79727621955416071iset_a] :
( ( member5199237121806060112iset_a @ X2 @ A3 )
=> ( ( ord_le5871882978999109543iset_a @ X3 @ A3 )
=> ( ord_le5871882978999109543iset_a @ ( insert4880209295958132471iset_a @ X2 @ X3 ) @ A3 ) ) ) ).
% insert_subsetI
thf(fact_1059_insert__Diff__if,axiom,
! [X2: produc6518373309651786023iset_a,B2: set_Pr79727621955416071iset_a,A3: set_Pr79727621955416071iset_a] :
( ( ( member5199237121806060112iset_a @ X2 @ B2 )
=> ( ( minus_2461365523287052142iset_a @ ( insert4880209295958132471iset_a @ X2 @ A3 ) @ B2 )
= ( minus_2461365523287052142iset_a @ A3 @ B2 ) ) )
& ( ~ ( member5199237121806060112iset_a @ X2 @ B2 )
=> ( ( minus_2461365523287052142iset_a @ ( insert4880209295958132471iset_a @ X2 @ A3 ) @ B2 )
= ( insert4880209295958132471iset_a @ X2 @ ( minus_2461365523287052142iset_a @ A3 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1060_insert__Diff__if,axiom,
! [X2: a,B2: set_a,A3: set_a] :
( ( ( member_a @ X2 @ B2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X2 @ A3 ) @ B2 )
= ( minus_minus_set_a @ A3 @ B2 ) ) )
& ( ~ ( member_a @ X2 @ B2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X2 @ A3 ) @ B2 )
= ( insert_a @ X2 @ ( minus_minus_set_a @ A3 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1061_insert__Diff__if,axiom,
! [X2: product_prod_a_a,B2: set_Product_prod_a_a,A3: set_Product_prod_a_a] :
( ( ( member1426531477525435216od_a_a @ X2 @ B2 )
=> ( ( minus_6817036919807184750od_a_a @ ( insert4534936382041156343od_a_a @ X2 @ A3 ) @ B2 )
= ( minus_6817036919807184750od_a_a @ A3 @ B2 ) ) )
& ( ~ ( member1426531477525435216od_a_a @ X2 @ B2 )
=> ( ( minus_6817036919807184750od_a_a @ ( insert4534936382041156343od_a_a @ X2 @ A3 ) @ B2 )
= ( insert4534936382041156343od_a_a @ X2 @ ( minus_6817036919807184750od_a_a @ A3 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1062_mk__disjoint__insert,axiom,
! [A2: produc6518373309651786023iset_a,A3: set_Pr79727621955416071iset_a] :
( ( member5199237121806060112iset_a @ A2 @ A3 )
=> ? [B4: set_Pr79727621955416071iset_a] :
( ( A3
= ( insert4880209295958132471iset_a @ A2 @ B4 ) )
& ~ ( member5199237121806060112iset_a @ A2 @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_1063_mk__disjoint__insert,axiom,
! [A2: a,A3: set_a] :
( ( member_a @ A2 @ A3 )
=> ? [B4: set_a] :
( ( A3
= ( insert_a @ A2 @ B4 ) )
& ~ ( member_a @ A2 @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_1064_mk__disjoint__insert,axiom,
! [A2: product_prod_a_a,A3: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A2 @ A3 )
=> ? [B4: set_Product_prod_a_a] :
( ( A3
= ( insert4534936382041156343od_a_a @ A2 @ B4 ) )
& ~ ( member1426531477525435216od_a_a @ A2 @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_1065_insert__eq__iff,axiom,
! [A2: produc6518373309651786023iset_a,A3: set_Pr79727621955416071iset_a,B: produc6518373309651786023iset_a,B2: set_Pr79727621955416071iset_a] :
( ~ ( member5199237121806060112iset_a @ A2 @ A3 )
=> ( ~ ( member5199237121806060112iset_a @ B @ B2 )
=> ( ( ( insert4880209295958132471iset_a @ A2 @ A3 )
= ( insert4880209295958132471iset_a @ B @ B2 ) )
= ( ( ( A2 = B )
=> ( A3 = B2 ) )
& ( ( A2 != B )
=> ? [C6: set_Pr79727621955416071iset_a] :
( ( A3
= ( insert4880209295958132471iset_a @ B @ C6 ) )
& ~ ( member5199237121806060112iset_a @ B @ C6 )
& ( B2
= ( insert4880209295958132471iset_a @ A2 @ C6 ) )
& ~ ( member5199237121806060112iset_a @ A2 @ C6 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_1066_insert__eq__iff,axiom,
! [A2: a,A3: set_a,B: a,B2: set_a] :
( ~ ( member_a @ A2 @ A3 )
=> ( ~ ( member_a @ B @ B2 )
=> ( ( ( insert_a @ A2 @ A3 )
= ( insert_a @ B @ B2 ) )
= ( ( ( A2 = B )
=> ( A3 = B2 ) )
& ( ( A2 != B )
=> ? [C6: set_a] :
( ( A3
= ( insert_a @ B @ C6 ) )
& ~ ( member_a @ B @ C6 )
& ( B2
= ( insert_a @ A2 @ C6 ) )
& ~ ( member_a @ A2 @ C6 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_1067_insert__eq__iff,axiom,
! [A2: product_prod_a_a,A3: set_Product_prod_a_a,B: product_prod_a_a,B2: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ A2 @ A3 )
=> ( ~ ( member1426531477525435216od_a_a @ B @ B2 )
=> ( ( ( insert4534936382041156343od_a_a @ A2 @ A3 )
= ( insert4534936382041156343od_a_a @ B @ B2 ) )
= ( ( ( A2 = B )
=> ( A3 = B2 ) )
& ( ( A2 != B )
=> ? [C6: set_Product_prod_a_a] :
( ( A3
= ( insert4534936382041156343od_a_a @ B @ C6 ) )
& ~ ( member1426531477525435216od_a_a @ B @ C6 )
& ( B2
= ( insert4534936382041156343od_a_a @ A2 @ C6 ) )
& ~ ( member1426531477525435216od_a_a @ A2 @ C6 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_1068_insert__absorb,axiom,
! [A2: produc6518373309651786023iset_a,A3: set_Pr79727621955416071iset_a] :
( ( member5199237121806060112iset_a @ A2 @ A3 )
=> ( ( insert4880209295958132471iset_a @ A2 @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_1069_insert__absorb,axiom,
! [A2: a,A3: set_a] :
( ( member_a @ A2 @ A3 )
=> ( ( insert_a @ A2 @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_1070_insert__absorb,axiom,
! [A2: product_prod_a_a,A3: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A2 @ A3 )
=> ( ( insert4534936382041156343od_a_a @ A2 @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_1071_insert__ident,axiom,
! [X2: produc6518373309651786023iset_a,A3: set_Pr79727621955416071iset_a,B2: set_Pr79727621955416071iset_a] :
( ~ ( member5199237121806060112iset_a @ X2 @ A3 )
=> ( ~ ( member5199237121806060112iset_a @ X2 @ B2 )
=> ( ( ( insert4880209295958132471iset_a @ X2 @ A3 )
= ( insert4880209295958132471iset_a @ X2 @ B2 ) )
= ( A3 = B2 ) ) ) ) ).
% insert_ident
thf(fact_1072_insert__ident,axiom,
! [X2: a,A3: set_a,B2: set_a] :
( ~ ( member_a @ X2 @ A3 )
=> ( ~ ( member_a @ X2 @ B2 )
=> ( ( ( insert_a @ X2 @ A3 )
= ( insert_a @ X2 @ B2 ) )
= ( A3 = B2 ) ) ) ) ).
% insert_ident
thf(fact_1073_insert__ident,axiom,
! [X2: product_prod_a_a,A3: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ X2 @ A3 )
=> ( ~ ( member1426531477525435216od_a_a @ X2 @ B2 )
=> ( ( ( insert4534936382041156343od_a_a @ X2 @ A3 )
= ( insert4534936382041156343od_a_a @ X2 @ B2 ) )
= ( A3 = B2 ) ) ) ) ).
% insert_ident
thf(fact_1074_Set_Oset__insert,axiom,
! [X2: produc6518373309651786023iset_a,A3: set_Pr79727621955416071iset_a] :
( ( member5199237121806060112iset_a @ X2 @ A3 )
=> ~ ! [B4: set_Pr79727621955416071iset_a] :
( ( A3
= ( insert4880209295958132471iset_a @ X2 @ B4 ) )
=> ( member5199237121806060112iset_a @ X2 @ B4 ) ) ) ).
% Set.set_insert
thf(fact_1075_Set_Oset__insert,axiom,
! [X2: a,A3: set_a] :
( ( member_a @ X2 @ A3 )
=> ~ ! [B4: set_a] :
( ( A3
= ( insert_a @ X2 @ B4 ) )
=> ( member_a @ X2 @ B4 ) ) ) ).
% Set.set_insert
thf(fact_1076_Set_Oset__insert,axiom,
! [X2: product_prod_a_a,A3: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ A3 )
=> ~ ! [B4: set_Product_prod_a_a] :
( ( A3
= ( insert4534936382041156343od_a_a @ X2 @ B4 ) )
=> ( member1426531477525435216od_a_a @ X2 @ B4 ) ) ) ).
% Set.set_insert
thf(fact_1077_insertI2,axiom,
! [A2: produc6518373309651786023iset_a,B2: set_Pr79727621955416071iset_a,B: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ A2 @ B2 )
=> ( member5199237121806060112iset_a @ A2 @ ( insert4880209295958132471iset_a @ B @ B2 ) ) ) ).
% insertI2
thf(fact_1078_insertI2,axiom,
! [A2: a,B2: set_a,B: a] :
( ( member_a @ A2 @ B2 )
=> ( member_a @ A2 @ ( insert_a @ B @ B2 ) ) ) ).
% insertI2
thf(fact_1079_insertI2,axiom,
! [A2: product_prod_a_a,B2: set_Product_prod_a_a,B: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A2 @ B2 )
=> ( member1426531477525435216od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ B @ B2 ) ) ) ).
% insertI2
thf(fact_1080_insertI1,axiom,
! [A2: produc6518373309651786023iset_a,B2: set_Pr79727621955416071iset_a] : ( member5199237121806060112iset_a @ A2 @ ( insert4880209295958132471iset_a @ A2 @ B2 ) ) ).
% insertI1
thf(fact_1081_insertI1,axiom,
! [A2: a,B2: set_a] : ( member_a @ A2 @ ( insert_a @ A2 @ B2 ) ) ).
% insertI1
thf(fact_1082_insertI1,axiom,
! [A2: product_prod_a_a,B2: set_Product_prod_a_a] : ( member1426531477525435216od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ A2 @ B2 ) ) ).
% insertI1
thf(fact_1083_insertE,axiom,
! [A2: produc6518373309651786023iset_a,B: produc6518373309651786023iset_a,A3: set_Pr79727621955416071iset_a] :
( ( member5199237121806060112iset_a @ A2 @ ( insert4880209295958132471iset_a @ B @ A3 ) )
=> ( ( A2 != B )
=> ( member5199237121806060112iset_a @ A2 @ A3 ) ) ) ).
% insertE
thf(fact_1084_insertE,axiom,
! [A2: a,B: a,A3: set_a] :
( ( member_a @ A2 @ ( insert_a @ B @ A3 ) )
=> ( ( A2 != B )
=> ( member_a @ A2 @ A3 ) ) ) ).
% insertE
thf(fact_1085_insertE,axiom,
! [A2: product_prod_a_a,B: product_prod_a_a,A3: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ B @ A3 ) )
=> ( ( A2 != B )
=> ( member1426531477525435216od_a_a @ A2 @ A3 ) ) ) ).
% insertE
thf(fact_1086_subset__insertI2,axiom,
! [A3: set_Product_prod_a_a,B2: set_Product_prod_a_a,B: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A3 @ B2 )
=> ( ord_le746702958409616551od_a_a @ A3 @ ( insert4534936382041156343od_a_a @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_1087_subset__insertI2,axiom,
! [A3: set_Pr79727621955416071iset_a,B2: set_Pr79727621955416071iset_a,B: produc6518373309651786023iset_a] :
( ( ord_le5871882978999109543iset_a @ A3 @ B2 )
=> ( ord_le5871882978999109543iset_a @ A3 @ ( insert4880209295958132471iset_a @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_1088_subset__insertI,axiom,
! [B2: set_Product_prod_a_a,A2: product_prod_a_a] : ( ord_le746702958409616551od_a_a @ B2 @ ( insert4534936382041156343od_a_a @ A2 @ B2 ) ) ).
% subset_insertI
thf(fact_1089_subset__insertI,axiom,
! [B2: set_Pr79727621955416071iset_a,A2: produc6518373309651786023iset_a] : ( ord_le5871882978999109543iset_a @ B2 @ ( insert4880209295958132471iset_a @ A2 @ B2 ) ) ).
% subset_insertI
thf(fact_1090_subset__insert,axiom,
! [X2: a,A3: set_a,B2: set_a] :
( ~ ( member_a @ X2 @ A3 )
=> ( ( ord_less_eq_set_a @ A3 @ ( insert_a @ X2 @ B2 ) )
= ( ord_less_eq_set_a @ A3 @ B2 ) ) ) ).
% subset_insert
thf(fact_1091_subset__insert,axiom,
! [X2: product_prod_a_a,A3: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ X2 @ A3 )
=> ( ( ord_le746702958409616551od_a_a @ A3 @ ( insert4534936382041156343od_a_a @ X2 @ B2 ) )
= ( ord_le746702958409616551od_a_a @ A3 @ B2 ) ) ) ).
% subset_insert
thf(fact_1092_subset__insert,axiom,
! [X2: produc6518373309651786023iset_a,A3: set_Pr79727621955416071iset_a,B2: set_Pr79727621955416071iset_a] :
( ~ ( member5199237121806060112iset_a @ X2 @ A3 )
=> ( ( ord_le5871882978999109543iset_a @ A3 @ ( insert4880209295958132471iset_a @ X2 @ B2 ) )
= ( ord_le5871882978999109543iset_a @ A3 @ B2 ) ) ) ).
% subset_insert
thf(fact_1093_insert__mono,axiom,
! [C2: set_Product_prod_a_a,D2: set_Product_prod_a_a,A2: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ C2 @ D2 )
=> ( ord_le746702958409616551od_a_a @ ( insert4534936382041156343od_a_a @ A2 @ C2 ) @ ( insert4534936382041156343od_a_a @ A2 @ D2 ) ) ) ).
% insert_mono
thf(fact_1094_insert__mono,axiom,
! [C2: set_Pr79727621955416071iset_a,D2: set_Pr79727621955416071iset_a,A2: produc6518373309651786023iset_a] :
( ( ord_le5871882978999109543iset_a @ C2 @ D2 )
=> ( ord_le5871882978999109543iset_a @ ( insert4880209295958132471iset_a @ A2 @ C2 ) @ ( insert4880209295958132471iset_a @ A2 @ D2 ) ) ) ).
% insert_mono
thf(fact_1095_subset__Diff__insert,axiom,
! [A3: set_a,B2: set_a,X2: a,C2: set_a] :
( ( ord_less_eq_set_a @ A3 @ ( minus_minus_set_a @ B2 @ ( insert_a @ X2 @ C2 ) ) )
= ( ( ord_less_eq_set_a @ A3 @ ( minus_minus_set_a @ B2 @ C2 ) )
& ~ ( member_a @ X2 @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_1096_subset__Diff__insert,axiom,
! [A3: set_Product_prod_a_a,B2: set_Product_prod_a_a,X2: product_prod_a_a,C2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A3 @ ( minus_6817036919807184750od_a_a @ B2 @ ( insert4534936382041156343od_a_a @ X2 @ C2 ) ) )
= ( ( ord_le746702958409616551od_a_a @ A3 @ ( minus_6817036919807184750od_a_a @ B2 @ C2 ) )
& ~ ( member1426531477525435216od_a_a @ X2 @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_1097_subset__Diff__insert,axiom,
! [A3: set_Pr79727621955416071iset_a,B2: set_Pr79727621955416071iset_a,X2: produc6518373309651786023iset_a,C2: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ A3 @ ( minus_2461365523287052142iset_a @ B2 @ ( insert4880209295958132471iset_a @ X2 @ C2 ) ) )
= ( ( ord_le5871882978999109543iset_a @ A3 @ ( minus_2461365523287052142iset_a @ B2 @ C2 ) )
& ~ ( member5199237121806060112iset_a @ X2 @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_1098_Diff__single__insert,axiom,
! [A3: set_Product_prod_a_a,X2: product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( minus_6817036919807184750od_a_a @ A3 @ ( insert4534936382041156343od_a_a @ X2 @ bot_bo3357376287454694259od_a_a ) ) @ B2 )
=> ( ord_le746702958409616551od_a_a @ A3 @ ( insert4534936382041156343od_a_a @ X2 @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_1099_Diff__single__insert,axiom,
! [A3: set_Pr79727621955416071iset_a,X2: produc6518373309651786023iset_a,B2: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ ( minus_2461365523287052142iset_a @ A3 @ ( insert4880209295958132471iset_a @ X2 @ bot_bo5898781664966027891iset_a ) ) @ B2 )
=> ( ord_le5871882978999109543iset_a @ A3 @ ( insert4880209295958132471iset_a @ X2 @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_1100_subset__insert__iff,axiom,
! [A3: set_a,X2: a,B2: set_a] :
( ( ord_less_eq_set_a @ A3 @ ( insert_a @ X2 @ B2 ) )
= ( ( ( member_a @ X2 @ A3 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A3 @ ( insert_a @ X2 @ bot_bot_set_a ) ) @ B2 ) )
& ( ~ ( member_a @ X2 @ A3 )
=> ( ord_less_eq_set_a @ A3 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_1101_subset__insert__iff,axiom,
! [A3: set_Product_prod_a_a,X2: product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A3 @ ( insert4534936382041156343od_a_a @ X2 @ B2 ) )
= ( ( ( member1426531477525435216od_a_a @ X2 @ A3 )
=> ( ord_le746702958409616551od_a_a @ ( minus_6817036919807184750od_a_a @ A3 @ ( insert4534936382041156343od_a_a @ X2 @ bot_bo3357376287454694259od_a_a ) ) @ B2 ) )
& ( ~ ( member1426531477525435216od_a_a @ X2 @ A3 )
=> ( ord_le746702958409616551od_a_a @ A3 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_1102_subset__insert__iff,axiom,
! [A3: set_Pr79727621955416071iset_a,X2: produc6518373309651786023iset_a,B2: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ A3 @ ( insert4880209295958132471iset_a @ X2 @ B2 ) )
= ( ( ( member5199237121806060112iset_a @ X2 @ A3 )
=> ( ord_le5871882978999109543iset_a @ ( minus_2461365523287052142iset_a @ A3 @ ( insert4880209295958132471iset_a @ X2 @ bot_bo5898781664966027891iset_a ) ) @ B2 ) )
& ( ~ ( member5199237121806060112iset_a @ X2 @ A3 )
=> ( ord_le5871882978999109543iset_a @ A3 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_1103_set__mset__single,axiom,
! [B: a] :
( ( set_mset_a @ ( add_mset_a @ B @ zero_zero_multiset_a ) )
= ( insert_a @ B @ bot_bot_set_a ) ) ).
% set_mset_single
thf(fact_1104_min__ext__compat,axiom,
! [R3: set_Product_prod_a_a,S3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( relcomp_a_a_a @ R3 @ S3 ) @ R3 )
=> ( ord_le8376522849517564071_set_a @ ( relcom2715300867007160733_set_a @ ( min_ext_a @ R3 ) @ ( sup_su5624169484884619995_set_a @ ( min_ext_a @ S3 ) @ ( insert5612341225869342455_set_a @ ( produc9088192753505129239_set_a @ bot_bot_set_a @ bot_bot_set_a ) @ bot_bo5799363139946352499_set_a ) ) ) @ ( min_ext_a @ R3 ) ) ) ).
% min_ext_compat
thf(fact_1105_min__ext__compat,axiom,
! [R3: set_Pr79727621955416071iset_a,S3: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ ( relcom5893175440217691907iset_a @ R3 @ S3 ) @ R3 )
=> ( ord_le1923108481298371495iset_a @ ( relcom9080871063811409123iset_a @ ( min_ext_multiset_a @ R3 ) @ ( sup_su3864225725560205787iset_a @ ( min_ext_multiset_a @ S3 ) @ ( insert8965112373804595959iset_a @ ( produc8266138482805055767iset_a @ bot_bo6997605411617904272iset_a @ bot_bo6997605411617904272iset_a ) @ bot_bo2681478344426247795iset_a ) ) ) @ ( min_ext_multiset_a @ R3 ) ) ) ).
% min_ext_compat
thf(fact_1106_max__ext__compat,axiom,
! [R3: set_Product_prod_a_a,S3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( relcomp_a_a_a @ R3 @ S3 ) @ R3 )
=> ( ord_le8376522849517564071_set_a @ ( relcom2715300867007160733_set_a @ ( max_ext_a @ R3 ) @ ( sup_su5624169484884619995_set_a @ ( max_ext_a @ S3 ) @ ( insert5612341225869342455_set_a @ ( produc9088192753505129239_set_a @ bot_bot_set_a @ bot_bot_set_a ) @ bot_bo5799363139946352499_set_a ) ) ) @ ( max_ext_a @ R3 ) ) ) ).
% max_ext_compat
thf(fact_1107_max__ext__compat,axiom,
! [R3: set_Pr79727621955416071iset_a,S3: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ ( relcom5893175440217691907iset_a @ R3 @ S3 ) @ R3 )
=> ( ord_le1923108481298371495iset_a @ ( relcom9080871063811409123iset_a @ ( max_ext_multiset_a @ R3 ) @ ( sup_su3864225725560205787iset_a @ ( max_ext_multiset_a @ S3 ) @ ( insert8965112373804595959iset_a @ ( produc8266138482805055767iset_a @ bot_bo6997605411617904272iset_a @ bot_bo6997605411617904272iset_a ) @ bot_bo2681478344426247795iset_a ) ) ) @ ( max_ext_multiset_a @ R3 ) ) ) ).
% max_ext_compat
thf(fact_1108_is__singletonI_H,axiom,
! [A3: set_Pr79727621955416071iset_a] :
( ( A3 != bot_bo5898781664966027891iset_a )
=> ( ! [X5: produc6518373309651786023iset_a,Y5: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X5 @ A3 )
=> ( ( member5199237121806060112iset_a @ Y5 @ A3 )
=> ( X5 = Y5 ) ) )
=> ( is_sin8627954445697999899iset_a @ A3 ) ) ) ).
% is_singletonI'
thf(fact_1109_is__singletonI_H,axiom,
! [A3: set_a] :
( ( A3 != bot_bot_set_a )
=> ( ! [X5: a,Y5: a] :
( ( member_a @ X5 @ A3 )
=> ( ( member_a @ Y5 @ A3 )
=> ( X5 = Y5 ) ) )
=> ( is_singleton_a @ A3 ) ) ) ).
% is_singletonI'
thf(fact_1110_is__singletonI_H,axiom,
! [A3: set_Product_prod_a_a] :
( ( A3 != bot_bo3357376287454694259od_a_a )
=> ( ! [X5: product_prod_a_a,Y5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ A3 )
=> ( ( member1426531477525435216od_a_a @ Y5 @ A3 )
=> ( X5 = Y5 ) ) )
=> ( is_sin3171834905898671131od_a_a @ A3 ) ) ) ).
% is_singletonI'
thf(fact_1111_max__ext__additive,axiom,
! [A3: set_Product_prod_a_a,B2: set_Product_prod_a_a,R3: set_Pr8600417178894128327od_a_a,C2: set_Product_prod_a_a,D2: set_Product_prod_a_a] :
( ( member4903802553882211088od_a_a @ ( produc3372137660326521687od_a_a @ A3 @ B2 ) @ ( max_ex7229726517752475106od_a_a @ R3 ) )
=> ( ( member4903802553882211088od_a_a @ ( produc3372137660326521687od_a_a @ C2 @ D2 ) @ ( max_ex7229726517752475106od_a_a @ R3 ) )
=> ( member4903802553882211088od_a_a @ ( produc3372137660326521687od_a_a @ ( sup_su3048258781599657691od_a_a @ A3 @ C2 ) @ ( sup_su3048258781599657691od_a_a @ B2 @ D2 ) ) @ ( max_ex7229726517752475106od_a_a @ R3 ) ) ) ) ).
% max_ext_additive
thf(fact_1112_Field__insert,axiom,
! [A2: multiset_a,B: multiset_a,R: set_Pr79727621955416071iset_a] :
( ( field_multiset_a @ ( insert4880209295958132471iset_a @ ( produc654756711066625303iset_a @ A2 @ B ) @ R ) )
= ( sup_su2684633059243589112iset_a @ ( insert_multiset_a @ A2 @ ( insert_multiset_a @ B @ bot_bo6997605411617904272iset_a ) ) @ ( field_multiset_a @ R ) ) ) ).
% Field_insert
thf(fact_1113_Field__insert,axiom,
! [A2: a,B: a,R: set_Product_prod_a_a] :
( ( field_a @ ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ A2 @ B ) @ R ) )
= ( sup_sup_set_a @ ( insert_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) @ ( field_a @ R ) ) ) ).
% Field_insert
thf(fact_1114_Field__insert,axiom,
! [A2: product_prod_a_a,B: product_prod_a_a,R: set_Pr8600417178894128327od_a_a] :
( ( field_1126092520709947252od_a_a @ ( insert8933621553020740791od_a_a @ ( produc7886510207707329367od_a_a @ A2 @ B ) @ R ) )
= ( sup_su3048258781599657691od_a_a @ ( insert4534936382041156343od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ B @ bot_bo3357376287454694259od_a_a ) ) @ ( field_1126092520709947252od_a_a @ R ) ) ) ).
% Field_insert
thf(fact_1115_Field__Un,axiom,
! [R: set_Pr8600417178894128327od_a_a,S: set_Pr8600417178894128327od_a_a] :
( ( field_1126092520709947252od_a_a @ ( sup_su8193676293155882651od_a_a @ R @ S ) )
= ( sup_su3048258781599657691od_a_a @ ( field_1126092520709947252od_a_a @ R ) @ ( field_1126092520709947252od_a_a @ S ) ) ) ).
% Field_Un
thf(fact_1116_Field__Un,axiom,
! [R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( field_a @ ( sup_su3048258781599657691od_a_a @ R @ S ) )
= ( sup_sup_set_a @ ( field_a @ R ) @ ( field_a @ S ) ) ) ).
% Field_Un
thf(fact_1117_mono__Field,axiom,
! [R: set_Pr8600417178894128327od_a_a,S: set_Pr8600417178894128327od_a_a] :
( ( ord_le3469131294019144807od_a_a @ R @ S )
=> ( ord_le746702958409616551od_a_a @ ( field_1126092520709947252od_a_a @ R ) @ ( field_1126092520709947252od_a_a @ S ) ) ) ).
% mono_Field
thf(fact_1118_mono__Field,axiom,
! [R: set_Pr3059988933816007879iset_a,S: set_Pr3059988933816007879iset_a] :
( ( ord_le2922206065357059175iset_a @ R @ S )
=> ( ord_le5871882978999109543iset_a @ ( field_2318967117295313524iset_a @ R ) @ ( field_2318967117295313524iset_a @ S ) ) ) ).
% mono_Field
thf(fact_1119_mono__Field,axiom,
! [R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ R @ S )
=> ( ord_less_eq_set_a @ ( field_a @ R ) @ ( field_a @ S ) ) ) ).
% mono_Field
thf(fact_1120_mono__Field,axiom,
! [R: set_Pr79727621955416071iset_a,S: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ R @ S )
=> ( ord_le1502830050749340100iset_a @ ( field_multiset_a @ R ) @ ( field_multiset_a @ S ) ) ) ).
% mono_Field
thf(fact_1121_FieldI1,axiom,
! [I: produc6518373309651786023iset_a,J: produc6518373309651786023iset_a,R3: set_Pr3059988933816007879iset_a] :
( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ I @ J ) @ R3 )
=> ( member5199237121806060112iset_a @ I @ ( field_2318967117295313524iset_a @ R3 ) ) ) ).
% FieldI1
thf(fact_1122_FieldI1,axiom,
! [I: product_prod_a_a,J: product_prod_a_a,R3: set_Pr8600417178894128327od_a_a] :
( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ I @ J ) @ R3 )
=> ( member1426531477525435216od_a_a @ I @ ( field_1126092520709947252od_a_a @ R3 ) ) ) ).
% FieldI1
thf(fact_1123_FieldI1,axiom,
! [I: multiset_a,J: multiset_a,R3: set_Pr79727621955416071iset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ I @ J ) @ R3 )
=> ( member_multiset_a @ I @ ( field_multiset_a @ R3 ) ) ) ).
% FieldI1
thf(fact_1124_FieldI1,axiom,
! [I: a,J: a,R3: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ I @ J ) @ R3 )
=> ( member_a @ I @ ( field_a @ R3 ) ) ) ).
% FieldI1
thf(fact_1125_FieldI2,axiom,
! [I: produc6518373309651786023iset_a,J: produc6518373309651786023iset_a,R3: set_Pr3059988933816007879iset_a] :
( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ I @ J ) @ R3 )
=> ( member5199237121806060112iset_a @ J @ ( field_2318967117295313524iset_a @ R3 ) ) ) ).
% FieldI2
thf(fact_1126_FieldI2,axiom,
! [I: product_prod_a_a,J: product_prod_a_a,R3: set_Pr8600417178894128327od_a_a] :
( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ I @ J ) @ R3 )
=> ( member1426531477525435216od_a_a @ J @ ( field_1126092520709947252od_a_a @ R3 ) ) ) ).
% FieldI2
thf(fact_1127_FieldI2,axiom,
! [I: multiset_a,J: multiset_a,R3: set_Pr79727621955416071iset_a] :
( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ I @ J ) @ R3 )
=> ( member_multiset_a @ J @ ( field_multiset_a @ R3 ) ) ) ).
% FieldI2
thf(fact_1128_FieldI2,axiom,
! [I: a,J: a,R3: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ I @ J ) @ R3 )
=> ( member_a @ J @ ( field_a @ R3 ) ) ) ).
% FieldI2
thf(fact_1129_member__remove,axiom,
! [X2: produc6518373309651786023iset_a,Y: produc6518373309651786023iset_a,A3: set_Pr79727621955416071iset_a] :
( ( member5199237121806060112iset_a @ X2 @ ( remove1331213385245781516iset_a @ Y @ A3 ) )
= ( ( member5199237121806060112iset_a @ X2 @ A3 )
& ( X2 != Y ) ) ) ).
% member_remove
thf(fact_1130_member__remove,axiom,
! [X2: a,Y: a,A3: set_a] :
( ( member_a @ X2 @ ( remove_a @ Y @ A3 ) )
= ( ( member_a @ X2 @ A3 )
& ( X2 != Y ) ) ) ).
% member_remove
thf(fact_1131_member__remove,axiom,
! [X2: product_prod_a_a,Y: product_prod_a_a,A3: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ ( remove8198300757409973004od_a_a @ Y @ A3 ) )
= ( ( member1426531477525435216od_a_a @ X2 @ A3 )
& ( X2 != Y ) ) ) ).
% member_remove
thf(fact_1132_set__mset__replicate__mset__subset,axiom,
! [N3: nat,X2: a] :
( ( ( N3 = zero_zero_nat )
=> ( ( set_mset_a @ ( replicate_mset_a @ N3 @ X2 ) )
= bot_bot_set_a ) )
& ( ( N3 != zero_zero_nat )
=> ( ( set_mset_a @ ( replicate_mset_a @ N3 @ X2 ) )
= ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ).
% set_mset_replicate_mset_subset
thf(fact_1133_Int__iff,axiom,
! [C: produc6518373309651786023iset_a,A3: set_Pr79727621955416071iset_a,B2: set_Pr79727621955416071iset_a] :
( ( member5199237121806060112iset_a @ C @ ( inf_in7847659125592275445iset_a @ A3 @ B2 ) )
= ( ( member5199237121806060112iset_a @ C @ A3 )
& ( member5199237121806060112iset_a @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_1134_Int__iff,axiom,
! [C: a,A3: set_a,B2: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A3 @ B2 ) )
= ( ( member_a @ C @ A3 )
& ( member_a @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_1135_Int__iff,axiom,
! [C: product_prod_a_a,A3: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A3 @ B2 ) )
= ( ( member1426531477525435216od_a_a @ C @ A3 )
& ( member1426531477525435216od_a_a @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_1136_IntI,axiom,
! [C: produc6518373309651786023iset_a,A3: set_Pr79727621955416071iset_a,B2: set_Pr79727621955416071iset_a] :
( ( member5199237121806060112iset_a @ C @ A3 )
=> ( ( member5199237121806060112iset_a @ C @ B2 )
=> ( member5199237121806060112iset_a @ C @ ( inf_in7847659125592275445iset_a @ A3 @ B2 ) ) ) ) ).
% IntI
thf(fact_1137_IntI,axiom,
! [C: a,A3: set_a,B2: set_a] :
( ( member_a @ C @ A3 )
=> ( ( member_a @ C @ B2 )
=> ( member_a @ C @ ( inf_inf_set_a @ A3 @ B2 ) ) ) ) ).
% IntI
thf(fact_1138_IntI,axiom,
! [C: product_prod_a_a,A3: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ A3 )
=> ( ( member1426531477525435216od_a_a @ C @ B2 )
=> ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A3 @ B2 ) ) ) ) ).
% IntI
thf(fact_1139_le__inf__iff,axiom,
! [X2: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X2 @ ( inf_in8905007599844390133od_a_a @ Y @ Z3 ) )
= ( ( ord_le746702958409616551od_a_a @ X2 @ Y )
& ( ord_le746702958409616551od_a_a @ X2 @ Z3 ) ) ) ).
% le_inf_iff
thf(fact_1140_le__inf__iff,axiom,
! [X2: set_Pr79727621955416071iset_a,Y: set_Pr79727621955416071iset_a,Z3: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ X2 @ ( inf_in7847659125592275445iset_a @ Y @ Z3 ) )
= ( ( ord_le5871882978999109543iset_a @ X2 @ Y )
& ( ord_le5871882978999109543iset_a @ X2 @ Z3 ) ) ) ).
% le_inf_iff
thf(fact_1141_le__inf__iff,axiom,
! [X2: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ Y @ Z3 ) )
= ( ( ord_less_eq_nat @ X2 @ Y )
& ( ord_less_eq_nat @ X2 @ Z3 ) ) ) ).
% le_inf_iff
thf(fact_1142_inf_Obounded__iff,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ ( inf_in8905007599844390133od_a_a @ B @ C ) )
= ( ( ord_le746702958409616551od_a_a @ A2 @ B )
& ( ord_le746702958409616551od_a_a @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_1143_inf_Obounded__iff,axiom,
! [A2: set_Pr79727621955416071iset_a,B: set_Pr79727621955416071iset_a,C: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ A2 @ ( inf_in7847659125592275445iset_a @ B @ C ) )
= ( ( ord_le5871882978999109543iset_a @ A2 @ B )
& ( ord_le5871882978999109543iset_a @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_1144_inf_Obounded__iff,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B @ C ) )
= ( ( ord_less_eq_nat @ A2 @ B )
& ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_1145_inf__sup__absorb,axiom,
! [X2: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ X2 @ ( sup_su3048258781599657691od_a_a @ X2 @ Y ) )
= X2 ) ).
% inf_sup_absorb
thf(fact_1146_sup__inf__absorb,axiom,
! [X2: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X2 @ ( inf_in8905007599844390133od_a_a @ X2 @ Y ) )
= X2 ) ).
% sup_inf_absorb
thf(fact_1147_Int__subset__iff,axiom,
! [C2: set_Product_prod_a_a,A3: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ C2 @ ( inf_in8905007599844390133od_a_a @ A3 @ B2 ) )
= ( ( ord_le746702958409616551od_a_a @ C2 @ A3 )
& ( ord_le746702958409616551od_a_a @ C2 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_1148_Int__subset__iff,axiom,
! [C2: set_Pr79727621955416071iset_a,A3: set_Pr79727621955416071iset_a,B2: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ C2 @ ( inf_in7847659125592275445iset_a @ A3 @ B2 ) )
= ( ( ord_le5871882978999109543iset_a @ C2 @ A3 )
& ( ord_le5871882978999109543iset_a @ C2 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_1149_Int__insert__left__if0,axiom,
! [A2: produc6518373309651786023iset_a,C2: set_Pr79727621955416071iset_a,B2: set_Pr79727621955416071iset_a] :
( ~ ( member5199237121806060112iset_a @ A2 @ C2 )
=> ( ( inf_in7847659125592275445iset_a @ ( insert4880209295958132471iset_a @ A2 @ B2 ) @ C2 )
= ( inf_in7847659125592275445iset_a @ B2 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_1150_Int__insert__left__if0,axiom,
! [A2: a,C2: set_a,B2: set_a] :
( ~ ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B2 ) @ C2 )
= ( inf_inf_set_a @ B2 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_1151_Int__insert__left__if0,axiom,
! [A2: product_prod_a_a,C2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ A2 @ C2 )
=> ( ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A2 @ B2 ) @ C2 )
= ( inf_in8905007599844390133od_a_a @ B2 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_1152_Int__insert__left__if1,axiom,
! [A2: produc6518373309651786023iset_a,C2: set_Pr79727621955416071iset_a,B2: set_Pr79727621955416071iset_a] :
( ( member5199237121806060112iset_a @ A2 @ C2 )
=> ( ( inf_in7847659125592275445iset_a @ ( insert4880209295958132471iset_a @ A2 @ B2 ) @ C2 )
= ( insert4880209295958132471iset_a @ A2 @ ( inf_in7847659125592275445iset_a @ B2 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1153_Int__insert__left__if1,axiom,
! [A2: a,C2: set_a,B2: set_a] :
( ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B2 ) @ C2 )
= ( insert_a @ A2 @ ( inf_inf_set_a @ B2 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1154_Int__insert__left__if1,axiom,
! [A2: product_prod_a_a,C2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A2 @ C2 )
=> ( ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A2 @ B2 ) @ C2 )
= ( insert4534936382041156343od_a_a @ A2 @ ( inf_in8905007599844390133od_a_a @ B2 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1155_Int__insert__right__if0,axiom,
! [A2: produc6518373309651786023iset_a,A3: set_Pr79727621955416071iset_a,B2: set_Pr79727621955416071iset_a] :
( ~ ( member5199237121806060112iset_a @ A2 @ A3 )
=> ( ( inf_in7847659125592275445iset_a @ A3 @ ( insert4880209295958132471iset_a @ A2 @ B2 ) )
= ( inf_in7847659125592275445iset_a @ A3 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_1156_Int__insert__right__if0,axiom,
! [A2: a,A3: set_a,B2: set_a] :
( ~ ( member_a @ A2 @ A3 )
=> ( ( inf_inf_set_a @ A3 @ ( insert_a @ A2 @ B2 ) )
= ( inf_inf_set_a @ A3 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_1157_Int__insert__right__if0,axiom,
! [A2: product_prod_a_a,A3: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ A2 @ A3 )
=> ( ( inf_in8905007599844390133od_a_a @ A3 @ ( insert4534936382041156343od_a_a @ A2 @ B2 ) )
= ( inf_in8905007599844390133od_a_a @ A3 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_1158_Int__insert__right__if1,axiom,
! [A2: produc6518373309651786023iset_a,A3: set_Pr79727621955416071iset_a,B2: set_Pr79727621955416071iset_a] :
( ( member5199237121806060112iset_a @ A2 @ A3 )
=> ( ( inf_in7847659125592275445iset_a @ A3 @ ( insert4880209295958132471iset_a @ A2 @ B2 ) )
= ( insert4880209295958132471iset_a @ A2 @ ( inf_in7847659125592275445iset_a @ A3 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1159_Int__insert__right__if1,axiom,
! [A2: a,A3: set_a,B2: set_a] :
( ( member_a @ A2 @ A3 )
=> ( ( inf_inf_set_a @ A3 @ ( insert_a @ A2 @ B2 ) )
= ( insert_a @ A2 @ ( inf_inf_set_a @ A3 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1160_Int__insert__right__if1,axiom,
! [A2: product_prod_a_a,A3: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A2 @ A3 )
=> ( ( inf_in8905007599844390133od_a_a @ A3 @ ( insert4534936382041156343od_a_a @ A2 @ B2 ) )
= ( insert4534936382041156343od_a_a @ A2 @ ( inf_in8905007599844390133od_a_a @ A3 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1161_Int__Un__eq_I4_J,axiom,
! [T2: set_Product_prod_a_a,S3: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ T2 @ ( inf_in8905007599844390133od_a_a @ S3 @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_1162_Int__Un__eq_I3_J,axiom,
! [S3: set_Product_prod_a_a,T2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ S3 @ ( inf_in8905007599844390133od_a_a @ S3 @ T2 ) )
= S3 ) ).
% Int_Un_eq(3)
thf(fact_1163_Int__Un__eq_I2_J,axiom,
! [S3: set_Product_prod_a_a,T2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( inf_in8905007599844390133od_a_a @ S3 @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_1164_Int__Un__eq_I1_J,axiom,
! [S3: set_Product_prod_a_a,T2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( inf_in8905007599844390133od_a_a @ S3 @ T2 ) @ S3 )
= S3 ) ).
% Int_Un_eq(1)
thf(fact_1165_Un__Int__eq_I4_J,axiom,
! [T2: set_Product_prod_a_a,S3: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ T2 @ ( sup_su3048258781599657691od_a_a @ S3 @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_1166_Un__Int__eq_I3_J,axiom,
! [S3: set_Product_prod_a_a,T2: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ S3 @ ( sup_su3048258781599657691od_a_a @ S3 @ T2 ) )
= S3 ) ).
% Un_Int_eq(3)
thf(fact_1167_Un__Int__eq_I2_J,axiom,
! [S3: set_Product_prod_a_a,T2: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ ( sup_su3048258781599657691od_a_a @ S3 @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_1168_Un__Int__eq_I1_J,axiom,
! [S3: set_Product_prod_a_a,T2: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ ( sup_su3048258781599657691od_a_a @ S3 @ T2 ) @ S3 )
= S3 ) ).
% Un_Int_eq(1)
thf(fact_1169_disjoint__insert_I2_J,axiom,
! [A3: set_Pr79727621955416071iset_a,B: produc6518373309651786023iset_a,B2: set_Pr79727621955416071iset_a] :
( ( bot_bo5898781664966027891iset_a
= ( inf_in7847659125592275445iset_a @ A3 @ ( insert4880209295958132471iset_a @ B @ B2 ) ) )
= ( ~ ( member5199237121806060112iset_a @ B @ A3 )
& ( bot_bo5898781664966027891iset_a
= ( inf_in7847659125592275445iset_a @ A3 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1170_disjoint__insert_I2_J,axiom,
! [A3: set_a,B: a,B2: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A3 @ ( insert_a @ B @ B2 ) ) )
= ( ~ ( member_a @ B @ A3 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A3 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1171_disjoint__insert_I2_J,axiom,
! [A3: set_Product_prod_a_a,B: product_prod_a_a,B2: set_Product_prod_a_a] :
( ( bot_bo3357376287454694259od_a_a
= ( inf_in8905007599844390133od_a_a @ A3 @ ( insert4534936382041156343od_a_a @ B @ B2 ) ) )
= ( ~ ( member1426531477525435216od_a_a @ B @ A3 )
& ( bot_bo3357376287454694259od_a_a
= ( inf_in8905007599844390133od_a_a @ A3 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1172_disjoint__insert_I1_J,axiom,
! [B2: set_Pr79727621955416071iset_a,A2: produc6518373309651786023iset_a,A3: set_Pr79727621955416071iset_a] :
( ( ( inf_in7847659125592275445iset_a @ B2 @ ( insert4880209295958132471iset_a @ A2 @ A3 ) )
= bot_bo5898781664966027891iset_a )
= ( ~ ( member5199237121806060112iset_a @ A2 @ B2 )
& ( ( inf_in7847659125592275445iset_a @ B2 @ A3 )
= bot_bo5898781664966027891iset_a ) ) ) ).
% disjoint_insert(1)
thf(fact_1173_disjoint__insert_I1_J,axiom,
! [B2: set_a,A2: a,A3: set_a] :
( ( ( inf_inf_set_a @ B2 @ ( insert_a @ A2 @ A3 ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A2 @ B2 )
& ( ( inf_inf_set_a @ B2 @ A3 )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_1174_disjoint__insert_I1_J,axiom,
! [B2: set_Product_prod_a_a,A2: product_prod_a_a,A3: set_Product_prod_a_a] :
( ( ( inf_in8905007599844390133od_a_a @ B2 @ ( insert4534936382041156343od_a_a @ A2 @ A3 ) )
= bot_bo3357376287454694259od_a_a )
= ( ~ ( member1426531477525435216od_a_a @ A2 @ B2 )
& ( ( inf_in8905007599844390133od_a_a @ B2 @ A3 )
= bot_bo3357376287454694259od_a_a ) ) ) ).
% disjoint_insert(1)
thf(fact_1175_insert__disjoint_I2_J,axiom,
! [A2: produc6518373309651786023iset_a,A3: set_Pr79727621955416071iset_a,B2: set_Pr79727621955416071iset_a] :
( ( bot_bo5898781664966027891iset_a
= ( inf_in7847659125592275445iset_a @ ( insert4880209295958132471iset_a @ A2 @ A3 ) @ B2 ) )
= ( ~ ( member5199237121806060112iset_a @ A2 @ B2 )
& ( bot_bo5898781664966027891iset_a
= ( inf_in7847659125592275445iset_a @ A3 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1176_insert__disjoint_I2_J,axiom,
! [A2: a,A3: set_a,B2: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A2 @ A3 ) @ B2 ) )
= ( ~ ( member_a @ A2 @ B2 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A3 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1177_insert__disjoint_I2_J,axiom,
! [A2: product_prod_a_a,A3: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( bot_bo3357376287454694259od_a_a
= ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A2 @ A3 ) @ B2 ) )
= ( ~ ( member1426531477525435216od_a_a @ A2 @ B2 )
& ( bot_bo3357376287454694259od_a_a
= ( inf_in8905007599844390133od_a_a @ A3 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1178_insert__disjoint_I1_J,axiom,
! [A2: produc6518373309651786023iset_a,A3: set_Pr79727621955416071iset_a,B2: set_Pr79727621955416071iset_a] :
( ( ( inf_in7847659125592275445iset_a @ ( insert4880209295958132471iset_a @ A2 @ A3 ) @ B2 )
= bot_bo5898781664966027891iset_a )
= ( ~ ( member5199237121806060112iset_a @ A2 @ B2 )
& ( ( inf_in7847659125592275445iset_a @ A3 @ B2 )
= bot_bo5898781664966027891iset_a ) ) ) ).
% insert_disjoint(1)
thf(fact_1179_insert__disjoint_I1_J,axiom,
! [A2: a,A3: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A2 @ A3 ) @ B2 )
= bot_bot_set_a )
= ( ~ ( member_a @ A2 @ B2 )
& ( ( inf_inf_set_a @ A3 @ B2 )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_1180_insert__disjoint_I1_J,axiom,
! [A2: product_prod_a_a,A3: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A2 @ A3 ) @ B2 )
= bot_bo3357376287454694259od_a_a )
= ( ~ ( member1426531477525435216od_a_a @ A2 @ B2 )
& ( ( inf_in8905007599844390133od_a_a @ A3 @ B2 )
= bot_bo3357376287454694259od_a_a ) ) ) ).
% insert_disjoint(1)
thf(fact_1181_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_Product_prod_a_a,Z3: set_Product_prod_a_a,X2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( inf_in8905007599844390133od_a_a @ Y @ Z3 ) @ X2 )
= ( inf_in8905007599844390133od_a_a @ ( sup_su3048258781599657691od_a_a @ Y @ X2 ) @ ( sup_su3048258781599657691od_a_a @ Z3 @ X2 ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_1182_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_Product_prod_a_a,Z3: set_Product_prod_a_a,X2: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ ( sup_su3048258781599657691od_a_a @ Y @ Z3 ) @ X2 )
= ( sup_su3048258781599657691od_a_a @ ( inf_in8905007599844390133od_a_a @ Y @ X2 ) @ ( inf_in8905007599844390133od_a_a @ Z3 @ X2 ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_1183_boolean__algebra_Odisj__conj__distrib,axiom,
! [X2: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z3: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X2 @ ( inf_in8905007599844390133od_a_a @ Y @ Z3 ) )
= ( inf_in8905007599844390133od_a_a @ ( sup_su3048258781599657691od_a_a @ X2 @ Y ) @ ( sup_su3048258781599657691od_a_a @ X2 @ Z3 ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_1184_boolean__algebra_Oconj__disj__distrib,axiom,
! [X2: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z3: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ X2 @ ( sup_su3048258781599657691od_a_a @ Y @ Z3 ) )
= ( sup_su3048258781599657691od_a_a @ ( inf_in8905007599844390133od_a_a @ X2 @ Y ) @ ( inf_in8905007599844390133od_a_a @ X2 @ Z3 ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_1185_Un__Int__assoc__eq,axiom,
! [A3: set_Product_prod_a_a,B2: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( ( sup_su3048258781599657691od_a_a @ ( inf_in8905007599844390133od_a_a @ A3 @ B2 ) @ C2 )
= ( inf_in8905007599844390133od_a_a @ A3 @ ( sup_su3048258781599657691od_a_a @ B2 @ C2 ) ) )
= ( ord_le746702958409616551od_a_a @ C2 @ A3 ) ) ).
% Un_Int_assoc_eq
thf(fact_1186_Un__Int__assoc__eq,axiom,
! [A3: set_Pr79727621955416071iset_a,B2: set_Pr79727621955416071iset_a,C2: set_Pr79727621955416071iset_a] :
( ( ( sup_su1925787886501348827iset_a @ ( inf_in7847659125592275445iset_a @ A3 @ B2 ) @ C2 )
= ( inf_in7847659125592275445iset_a @ A3 @ ( sup_su1925787886501348827iset_a @ B2 @ C2 ) ) )
= ( ord_le5871882978999109543iset_a @ C2 @ A3 ) ) ).
% Un_Int_assoc_eq
thf(fact_1187_Un__Diff__Int,axiom,
! [A3: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( minus_6817036919807184750od_a_a @ A3 @ B2 ) @ ( inf_in8905007599844390133od_a_a @ A3 @ B2 ) )
= A3 ) ).
% Un_Diff_Int
thf(fact_1188_Int__Diff__Un,axiom,
! [A3: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( inf_in8905007599844390133od_a_a @ A3 @ B2 ) @ ( minus_6817036919807184750od_a_a @ A3 @ B2 ) )
= A3 ) ).
% Int_Diff_Un
thf(fact_1189_Diff__Int,axiom,
! [A3: set_Product_prod_a_a,B2: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( minus_6817036919807184750od_a_a @ A3 @ ( inf_in8905007599844390133od_a_a @ B2 @ C2 ) )
= ( sup_su3048258781599657691od_a_a @ ( minus_6817036919807184750od_a_a @ A3 @ B2 ) @ ( minus_6817036919807184750od_a_a @ A3 @ C2 ) ) ) ).
% Diff_Int
thf(fact_1190_Diff__Un,axiom,
! [A3: set_Product_prod_a_a,B2: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( minus_6817036919807184750od_a_a @ A3 @ ( sup_su3048258781599657691od_a_a @ B2 @ C2 ) )
= ( inf_in8905007599844390133od_a_a @ ( minus_6817036919807184750od_a_a @ A3 @ B2 ) @ ( minus_6817036919807184750od_a_a @ A3 @ C2 ) ) ) ).
% Diff_Un
thf(fact_1191_distrib__inf__le,axiom,
! [X2: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z3: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( sup_su3048258781599657691od_a_a @ ( inf_in8905007599844390133od_a_a @ X2 @ Y ) @ ( inf_in8905007599844390133od_a_a @ X2 @ Z3 ) ) @ ( inf_in8905007599844390133od_a_a @ X2 @ ( sup_su3048258781599657691od_a_a @ Y @ Z3 ) ) ) ).
% distrib_inf_le
thf(fact_1192_distrib__inf__le,axiom,
! [X2: set_Pr79727621955416071iset_a,Y: set_Pr79727621955416071iset_a,Z3: set_Pr79727621955416071iset_a] : ( ord_le5871882978999109543iset_a @ ( sup_su1925787886501348827iset_a @ ( inf_in7847659125592275445iset_a @ X2 @ Y ) @ ( inf_in7847659125592275445iset_a @ X2 @ Z3 ) ) @ ( inf_in7847659125592275445iset_a @ X2 @ ( sup_su1925787886501348827iset_a @ Y @ Z3 ) ) ) ).
% distrib_inf_le
thf(fact_1193_distrib__inf__le,axiom,
! [X2: nat,Y: nat,Z3: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X2 @ Y ) @ ( inf_inf_nat @ X2 @ Z3 ) ) @ ( inf_inf_nat @ X2 @ ( sup_sup_nat @ Y @ Z3 ) ) ) ).
% distrib_inf_le
thf(fact_1194_distrib__sup__le,axiom,
! [X2: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z3: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( sup_su3048258781599657691od_a_a @ X2 @ ( inf_in8905007599844390133od_a_a @ Y @ Z3 ) ) @ ( inf_in8905007599844390133od_a_a @ ( sup_su3048258781599657691od_a_a @ X2 @ Y ) @ ( sup_su3048258781599657691od_a_a @ X2 @ Z3 ) ) ) ).
% distrib_sup_le
thf(fact_1195_distrib__sup__le,axiom,
! [X2: set_Pr79727621955416071iset_a,Y: set_Pr79727621955416071iset_a,Z3: set_Pr79727621955416071iset_a] : ( ord_le5871882978999109543iset_a @ ( sup_su1925787886501348827iset_a @ X2 @ ( inf_in7847659125592275445iset_a @ Y @ Z3 ) ) @ ( inf_in7847659125592275445iset_a @ ( sup_su1925787886501348827iset_a @ X2 @ Y ) @ ( sup_su1925787886501348827iset_a @ X2 @ Z3 ) ) ) ).
% distrib_sup_le
thf(fact_1196_distrib__sup__le,axiom,
! [X2: nat,Y: nat,Z3: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X2 @ ( inf_inf_nat @ Y @ Z3 ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X2 @ Y ) @ ( sup_sup_nat @ X2 @ Z3 ) ) ) ).
% distrib_sup_le
thf(fact_1197_Int__mono,axiom,
! [A3: set_Product_prod_a_a,C2: set_Product_prod_a_a,B2: set_Product_prod_a_a,D2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A3 @ C2 )
=> ( ( ord_le746702958409616551od_a_a @ B2 @ D2 )
=> ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A3 @ B2 ) @ ( inf_in8905007599844390133od_a_a @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_1198_Int__mono,axiom,
! [A3: set_Pr79727621955416071iset_a,C2: set_Pr79727621955416071iset_a,B2: set_Pr79727621955416071iset_a,D2: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ A3 @ C2 )
=> ( ( ord_le5871882978999109543iset_a @ B2 @ D2 )
=> ( ord_le5871882978999109543iset_a @ ( inf_in7847659125592275445iset_a @ A3 @ B2 ) @ ( inf_in7847659125592275445iset_a @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_1199_Int__lower1,axiom,
! [A3: set_Product_prod_a_a,B2: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A3 @ B2 ) @ A3 ) ).
% Int_lower1
thf(fact_1200_Int__lower1,axiom,
! [A3: set_Pr79727621955416071iset_a,B2: set_Pr79727621955416071iset_a] : ( ord_le5871882978999109543iset_a @ ( inf_in7847659125592275445iset_a @ A3 @ B2 ) @ A3 ) ).
% Int_lower1
thf(fact_1201_Int__lower2,axiom,
! [A3: set_Product_prod_a_a,B2: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A3 @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_1202_Int__lower2,axiom,
! [A3: set_Pr79727621955416071iset_a,B2: set_Pr79727621955416071iset_a] : ( ord_le5871882978999109543iset_a @ ( inf_in7847659125592275445iset_a @ A3 @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_1203_Int__absorb1,axiom,
! [B2: set_Product_prod_a_a,A3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B2 @ A3 )
=> ( ( inf_in8905007599844390133od_a_a @ A3 @ B2 )
= B2 ) ) ).
% Int_absorb1
thf(fact_1204_Int__absorb1,axiom,
! [B2: set_Pr79727621955416071iset_a,A3: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ B2 @ A3 )
=> ( ( inf_in7847659125592275445iset_a @ A3 @ B2 )
= B2 ) ) ).
% Int_absorb1
thf(fact_1205_Int__absorb2,axiom,
! [A3: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A3 @ B2 )
=> ( ( inf_in8905007599844390133od_a_a @ A3 @ B2 )
= A3 ) ) ).
% Int_absorb2
thf(fact_1206_Int__absorb2,axiom,
! [A3: set_Pr79727621955416071iset_a,B2: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ A3 @ B2 )
=> ( ( inf_in7847659125592275445iset_a @ A3 @ B2 )
= A3 ) ) ).
% Int_absorb2
thf(fact_1207_Int__greatest,axiom,
! [C2: set_Product_prod_a_a,A3: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ C2 @ A3 )
=> ( ( ord_le746702958409616551od_a_a @ C2 @ B2 )
=> ( ord_le746702958409616551od_a_a @ C2 @ ( inf_in8905007599844390133od_a_a @ A3 @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_1208_Int__greatest,axiom,
! [C2: set_Pr79727621955416071iset_a,A3: set_Pr79727621955416071iset_a,B2: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ C2 @ A3 )
=> ( ( ord_le5871882978999109543iset_a @ C2 @ B2 )
=> ( ord_le5871882978999109543iset_a @ C2 @ ( inf_in7847659125592275445iset_a @ A3 @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_1209_Int__Collect__mono,axiom,
! [A3: set_a,B2: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A3 @ B2 )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A3 )
=> ( ( P @ X5 )
=> ( Q @ X5 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B2 @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1210_Int__Collect__mono,axiom,
! [A3: set_Product_prod_a_a,B2: set_Product_prod_a_a,P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ( ord_le746702958409616551od_a_a @ A3 @ B2 )
=> ( ! [X5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ A3 )
=> ( ( P @ X5 )
=> ( Q @ X5 ) ) )
=> ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A3 @ ( collec3336397797384452498od_a_a @ P ) ) @ ( inf_in8905007599844390133od_a_a @ B2 @ ( collec3336397797384452498od_a_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1211_Int__Collect__mono,axiom,
! [A3: set_Pr79727621955416071iset_a,B2: set_Pr79727621955416071iset_a,P: produc6518373309651786023iset_a > $o,Q: produc6518373309651786023iset_a > $o] :
( ( ord_le5871882978999109543iset_a @ A3 @ B2 )
=> ( ! [X5: produc6518373309651786023iset_a] :
( ( member5199237121806060112iset_a @ X5 @ A3 )
=> ( ( P @ X5 )
=> ( Q @ X5 ) ) )
=> ( ord_le5871882978999109543iset_a @ ( inf_in7847659125592275445iset_a @ A3 @ ( collec6344740439120713362iset_a @ P ) ) @ ( inf_in7847659125592275445iset_a @ B2 @ ( collec6344740439120713362iset_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1212_inf__sup__ord_I2_J,axiom,
! [X2: set_Product_prod_a_a,Y: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ X2 @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_1213_inf__sup__ord_I2_J,axiom,
! [X2: set_Pr79727621955416071iset_a,Y: set_Pr79727621955416071iset_a] : ( ord_le5871882978999109543iset_a @ ( inf_in7847659125592275445iset_a @ X2 @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_1214_inf__sup__ord_I2_J,axiom,
! [X2: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_1215_inf__sup__ord_I1_J,axiom,
! [X2: set_Product_prod_a_a,Y: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ X2 @ Y ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_1216_inf__sup__ord_I1_J,axiom,
! [X2: set_Pr79727621955416071iset_a,Y: set_Pr79727621955416071iset_a] : ( ord_le5871882978999109543iset_a @ ( inf_in7847659125592275445iset_a @ X2 @ Y ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_1217_inf__sup__ord_I1_J,axiom,
! [X2: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_1218_inf__le1,axiom,
! [X2: set_Product_prod_a_a,Y: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ X2 @ Y ) @ X2 ) ).
% inf_le1
thf(fact_1219_inf__le1,axiom,
! [X2: set_Pr79727621955416071iset_a,Y: set_Pr79727621955416071iset_a] : ( ord_le5871882978999109543iset_a @ ( inf_in7847659125592275445iset_a @ X2 @ Y ) @ X2 ) ).
% inf_le1
thf(fact_1220_inf__le1,axiom,
! [X2: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y ) @ X2 ) ).
% inf_le1
thf(fact_1221_inf__le2,axiom,
! [X2: set_Product_prod_a_a,Y: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ X2 @ Y ) @ Y ) ).
% inf_le2
thf(fact_1222_inf__le2,axiom,
! [X2: set_Pr79727621955416071iset_a,Y: set_Pr79727621955416071iset_a] : ( ord_le5871882978999109543iset_a @ ( inf_in7847659125592275445iset_a @ X2 @ Y ) @ Y ) ).
% inf_le2
thf(fact_1223_inf__le2,axiom,
! [X2: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y ) @ Y ) ).
% inf_le2
thf(fact_1224_le__infE,axiom,
! [X2: set_Product_prod_a_a,A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X2 @ ( inf_in8905007599844390133od_a_a @ A2 @ B ) )
=> ~ ( ( ord_le746702958409616551od_a_a @ X2 @ A2 )
=> ~ ( ord_le746702958409616551od_a_a @ X2 @ B ) ) ) ).
% le_infE
thf(fact_1225_le__infE,axiom,
! [X2: set_Pr79727621955416071iset_a,A2: set_Pr79727621955416071iset_a,B: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ X2 @ ( inf_in7847659125592275445iset_a @ A2 @ B ) )
=> ~ ( ( ord_le5871882978999109543iset_a @ X2 @ A2 )
=> ~ ( ord_le5871882978999109543iset_a @ X2 @ B ) ) ) ).
% le_infE
thf(fact_1226_le__infE,axiom,
! [X2: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ A2 @ B ) )
=> ~ ( ( ord_less_eq_nat @ X2 @ A2 )
=> ~ ( ord_less_eq_nat @ X2 @ B ) ) ) ).
% le_infE
thf(fact_1227_le__infI,axiom,
! [X2: set_Product_prod_a_a,A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X2 @ A2 )
=> ( ( ord_le746702958409616551od_a_a @ X2 @ B )
=> ( ord_le746702958409616551od_a_a @ X2 @ ( inf_in8905007599844390133od_a_a @ A2 @ B ) ) ) ) ).
% le_infI
thf(fact_1228_le__infI,axiom,
! [X2: set_Pr79727621955416071iset_a,A2: set_Pr79727621955416071iset_a,B: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ X2 @ A2 )
=> ( ( ord_le5871882978999109543iset_a @ X2 @ B )
=> ( ord_le5871882978999109543iset_a @ X2 @ ( inf_in7847659125592275445iset_a @ A2 @ B ) ) ) ) ).
% le_infI
thf(fact_1229_le__infI,axiom,
! [X2: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ X2 @ A2 )
=> ( ( ord_less_eq_nat @ X2 @ B )
=> ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ A2 @ B ) ) ) ) ).
% le_infI
thf(fact_1230_inf__mono,axiom,
! [A2: set_Product_prod_a_a,C: set_Product_prod_a_a,B: set_Product_prod_a_a,D: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ C )
=> ( ( ord_le746702958409616551od_a_a @ B @ D )
=> ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A2 @ B ) @ ( inf_in8905007599844390133od_a_a @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_1231_inf__mono,axiom,
! [A2: set_Pr79727621955416071iset_a,C: set_Pr79727621955416071iset_a,B: set_Pr79727621955416071iset_a,D: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ A2 @ C )
=> ( ( ord_le5871882978999109543iset_a @ B @ D )
=> ( ord_le5871882978999109543iset_a @ ( inf_in7847659125592275445iset_a @ A2 @ B ) @ ( inf_in7847659125592275445iset_a @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_1232_inf__mono,axiom,
! [A2: nat,C: nat,B: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B @ D )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ ( inf_inf_nat @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_1233_le__infI1,axiom,
! [A2: set_Product_prod_a_a,X2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ X2 )
=> ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A2 @ B ) @ X2 ) ) ).
% le_infI1
thf(fact_1234_le__infI1,axiom,
! [A2: set_Pr79727621955416071iset_a,X2: set_Pr79727621955416071iset_a,B: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ A2 @ X2 )
=> ( ord_le5871882978999109543iset_a @ ( inf_in7847659125592275445iset_a @ A2 @ B ) @ X2 ) ) ).
% le_infI1
thf(fact_1235_le__infI1,axiom,
! [A2: nat,X2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ X2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ X2 ) ) ).
% le_infI1
thf(fact_1236_le__infI2,axiom,
! [B: set_Product_prod_a_a,X2: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B @ X2 )
=> ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A2 @ B ) @ X2 ) ) ).
% le_infI2
thf(fact_1237_le__infI2,axiom,
! [B: set_Pr79727621955416071iset_a,X2: set_Pr79727621955416071iset_a,A2: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ B @ X2 )
=> ( ord_le5871882978999109543iset_a @ ( inf_in7847659125592275445iset_a @ A2 @ B ) @ X2 ) ) ).
% le_infI2
thf(fact_1238_le__infI2,axiom,
! [B: nat,X2: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ X2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ X2 ) ) ).
% le_infI2
thf(fact_1239_inf_OorderE,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B )
=> ( A2
= ( inf_in8905007599844390133od_a_a @ A2 @ B ) ) ) ).
% inf.orderE
thf(fact_1240_inf_OorderE,axiom,
! [A2: set_Pr79727621955416071iset_a,B: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ A2 @ B )
=> ( A2
= ( inf_in7847659125592275445iset_a @ A2 @ B ) ) ) ).
% inf.orderE
thf(fact_1241_inf_OorderE,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( A2
= ( inf_inf_nat @ A2 @ B ) ) ) ).
% inf.orderE
thf(fact_1242_inf_OorderI,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( A2
= ( inf_in8905007599844390133od_a_a @ A2 @ B ) )
=> ( ord_le746702958409616551od_a_a @ A2 @ B ) ) ).
% inf.orderI
thf(fact_1243_inf_OorderI,axiom,
! [A2: set_Pr79727621955416071iset_a,B: set_Pr79727621955416071iset_a] :
( ( A2
= ( inf_in7847659125592275445iset_a @ A2 @ B ) )
=> ( ord_le5871882978999109543iset_a @ A2 @ B ) ) ).
% inf.orderI
thf(fact_1244_inf_OorderI,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( inf_inf_nat @ A2 @ B ) )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% inf.orderI
thf(fact_1245_inf__unique,axiom,
! [F: set_Product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_a,X2: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ! [X5: set_Product_prod_a_a,Y5: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( F @ X5 @ Y5 ) @ X5 )
=> ( ! [X5: set_Product_prod_a_a,Y5: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( F @ X5 @ Y5 ) @ Y5 )
=> ( ! [X5: set_Product_prod_a_a,Y5: set_Product_prod_a_a,Z: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X5 @ Y5 )
=> ( ( ord_le746702958409616551od_a_a @ X5 @ Z )
=> ( ord_le746702958409616551od_a_a @ X5 @ ( F @ Y5 @ Z ) ) ) )
=> ( ( inf_in8905007599844390133od_a_a @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_1246_inf__unique,axiom,
! [F: set_Pr79727621955416071iset_a > set_Pr79727621955416071iset_a > set_Pr79727621955416071iset_a,X2: set_Pr79727621955416071iset_a,Y: set_Pr79727621955416071iset_a] :
( ! [X5: set_Pr79727621955416071iset_a,Y5: set_Pr79727621955416071iset_a] : ( ord_le5871882978999109543iset_a @ ( F @ X5 @ Y5 ) @ X5 )
=> ( ! [X5: set_Pr79727621955416071iset_a,Y5: set_Pr79727621955416071iset_a] : ( ord_le5871882978999109543iset_a @ ( F @ X5 @ Y5 ) @ Y5 )
=> ( ! [X5: set_Pr79727621955416071iset_a,Y5: set_Pr79727621955416071iset_a,Z: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ X5 @ Y5 )
=> ( ( ord_le5871882978999109543iset_a @ X5 @ Z )
=> ( ord_le5871882978999109543iset_a @ X5 @ ( F @ Y5 @ Z ) ) ) )
=> ( ( inf_in7847659125592275445iset_a @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_1247_inf__unique,axiom,
! [F: nat > nat > nat,X2: nat,Y: nat] :
( ! [X5: nat,Y5: nat] : ( ord_less_eq_nat @ ( F @ X5 @ Y5 ) @ X5 )
=> ( ! [X5: nat,Y5: nat] : ( ord_less_eq_nat @ ( F @ X5 @ Y5 ) @ Y5 )
=> ( ! [X5: nat,Y5: nat,Z: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ( ord_less_eq_nat @ X5 @ Z )
=> ( ord_less_eq_nat @ X5 @ ( F @ Y5 @ Z ) ) ) )
=> ( ( inf_inf_nat @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_1248_le__iff__inf,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [X4: set_Product_prod_a_a,Y4: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ X4 @ Y4 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_1249_le__iff__inf,axiom,
( ord_le5871882978999109543iset_a
= ( ^ [X4: set_Pr79727621955416071iset_a,Y4: set_Pr79727621955416071iset_a] :
( ( inf_in7847659125592275445iset_a @ X4 @ Y4 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_1250_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y4: nat] :
( ( inf_inf_nat @ X4 @ Y4 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_1251_inf_Oabsorb1,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B )
=> ( ( inf_in8905007599844390133od_a_a @ A2 @ B )
= A2 ) ) ).
% inf.absorb1
thf(fact_1252_inf_Oabsorb1,axiom,
! [A2: set_Pr79727621955416071iset_a,B: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ A2 @ B )
=> ( ( inf_in7847659125592275445iset_a @ A2 @ B )
= A2 ) ) ).
% inf.absorb1
thf(fact_1253_inf_Oabsorb1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( inf_inf_nat @ A2 @ B )
= A2 ) ) ).
% inf.absorb1
thf(fact_1254_inf_Oabsorb2,axiom,
! [B: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B @ A2 )
=> ( ( inf_in8905007599844390133od_a_a @ A2 @ B )
= B ) ) ).
% inf.absorb2
thf(fact_1255_inf_Oabsorb2,axiom,
! [B: set_Pr79727621955416071iset_a,A2: set_Pr79727621955416071iset_a] :
( ( ord_le5871882978999109543iset_a @ B @ A2 )
=> ( ( inf_in7847659125592275445iset_a @ A2 @ B )
= B ) ) ).
% inf.absorb2
thf(fact_1256_inf_Oabsorb2,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( inf_inf_nat @ A2 @ B )
= B ) ) ).
% inf.absorb2
thf(fact_1257_inf__absorb1,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( inf_inf_nat @ X2 @ Y )
= X2 ) ) ).
% inf_absorb1
thf(fact_1258_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_1259_Nat_Oadd__0__right,axiom,
! [M4: nat] :
( ( plus_plus_nat @ M4 @ zero_zero_nat )
= M4 ) ).
% Nat.add_0_right
thf(fact_1260_add__is__0,axiom,
! [M4: nat,N3: nat] :
( ( ( plus_plus_nat @ M4 @ N3 )
= zero_zero_nat )
= ( ( M4 = zero_zero_nat )
& ( N3 = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1261_diff__self__eq__0,axiom,
! [M4: nat] :
( ( minus_minus_nat @ M4 @ M4 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1262_diff__0__eq__0,axiom,
! [N3: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N3 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1263_diff__is__0__eq,axiom,
! [M4: nat,N3: nat] :
( ( ( minus_minus_nat @ M4 @ N3 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M4 @ N3 ) ) ).
% diff_is_0_eq
thf(fact_1264_diff__is__0__eq_H,axiom,
! [M4: nat,N3: nat] :
( ( ord_less_eq_nat @ M4 @ N3 )
=> ( ( minus_minus_nat @ M4 @ N3 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1265_le0,axiom,
! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N3 ) ).
% le0
thf(fact_1266_add__eq__self__zero,axiom,
! [M4: nat,N3: nat] :
( ( ( plus_plus_nat @ M4 @ N3 )
= M4 )
=> ( N3 = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1267_plus__nat_Oadd__0,axiom,
! [N3: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N3 )
= N3 ) ).
% plus_nat.add_0
thf(fact_1268_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1269_diffs0__imp__equal,axiom,
! [M4: nat,N3: nat] :
( ( ( minus_minus_nat @ M4 @ N3 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N3 @ M4 )
= zero_zero_nat )
=> ( M4 = N3 ) ) ) ).
% diffs0_imp_equal
thf(fact_1270_minus__nat_Odiff__0,axiom,
! [M4: nat] :
( ( minus_minus_nat @ M4 @ zero_zero_nat )
= M4 ) ).
% minus_nat.diff_0
thf(fact_1271_diff__add__0,axiom,
! [N3: nat,M4: nat] :
( ( minus_minus_nat @ N3 @ ( plus_plus_nat @ N3 @ M4 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1272_le__0__eq,axiom,
! [N3: nat] :
( ( ord_less_eq_nat @ N3 @ zero_zero_nat )
= ( N3 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1273_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_1274_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_1275_less__eq__nat_Osimps_I1_J,axiom,
! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N3 ) ).
% less_eq_nat.simps(1)
thf(fact_1276_Euclid__induct,axiom,
! [P: nat > nat > $o,A2: nat,B: nat] :
( ! [A: nat,B8: nat] :
( ( P @ A @ B8 )
= ( P @ B8 @ A ) )
=> ( ! [A: nat] : ( P @ A @ zero_zero_nat )
=> ( ! [A: nat,B8: nat] :
( ( P @ A @ B8 )
=> ( P @ A @ ( plus_plus_nat @ A @ B8 ) ) )
=> ( P @ A2 @ B ) ) ) ) ).
% Euclid_induct
% Conjectures (1)
thf(conj_0,conjecture,
? [Nas22: multiset_a,Sas22: multiset_a] :
( ( ( add_mset_a @ a2 @ nasa )
= ( plus_plus_multiset_a @ Nas22 @ Sas22 ) )
& ? [Nbs22: multiset_a,Sbs22: multiset_a] :
( ( ( add_mset_a @ b @ nbsa )
= ( plus_plus_multiset_a @ Nbs22 @ Sbs22 ) )
& ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Nas22 @ Nbs22 ) @ ( multis2373687813370458242ltpw_a @ ns ) )
& ! [X5: a] :
( ( member_a @ X5 @ ( set_mset_a @ Sbs22 ) )
=> ? [A8: a] :
( ( member_a @ A8 @ ( set_mset_a @ Sas22 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A8 @ X5 ) @ s ) ) ) ) ) ).
%------------------------------------------------------------------------------