TPTP Problem File: SLH0089^1.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 : Finite_Fields/0004_Finite_Fields_Factorization_Ext/prob_00250_008133__18112424_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1444 ( 388 unt; 164 typ; 0 def)
% Number of atoms : 4782 (1145 equ; 0 cnn)
% Maximal formula atoms : 14 ( 3 avg)
% Number of connectives : 17196 ( 246 ~; 31 |; 246 &;14009 @)
% ( 0 <=>;2664 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 8 avg)
% Number of types : 21 ( 20 usr)
% Number of type conns : 445 ( 445 >; 0 *; 0 +; 0 <<)
% Number of symbols : 145 ( 144 usr; 13 con; 0-4 aty)
% Number of variables : 3740 ( 142 ^;3431 !; 167 ?;3740 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 13:20:58.683
%------------------------------------------------------------------------------
% Could-be-implicit typings (20)
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% Explicit typings (144)
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thf(sy_c_List_Olist_Olist__all2_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
list_a5961261016436360967_set_a: ( set_a > set_a > $o ) > list_set_a > list_set_a > $o ).
thf(sy_c_List_Olist_Olist__all2_001t__Set__Oset_Itf__a_J_001tf__a,type,
list_all2_set_a_a: ( set_a > a > $o ) > list_set_a > list_a > $o ).
thf(sy_c_List_Olist_Olist__all2_001tf__a_001t__Set__Oset_Itf__a_J,type,
list_all2_a_set_a: ( a > set_a > $o ) > list_a > list_set_a > $o ).
thf(sy_c_List_Olist_Olist__all2_001tf__a_001tf__a,type,
list_all2_a_a: ( a > a > $o ) > list_a > list_a > $o ).
thf(sy_c_List_Olist_Oset_001t__Set__Oset_Itf__a_J,type,
set_set_a2: list_set_a > set_set_a ).
thf(sy_c_List_Olist_Oset_001tf__a,type,
set_a2: list_a > set_a ).
thf(sy_c_List_Olist__update_001t__Set__Oset_Itf__a_J,type,
list_update_set_a: list_set_a > nat > set_a > list_set_a ).
thf(sy_c_List_Olist__update_001tf__a,type,
list_update_a: list_a > nat > a > list_a ).
thf(sy_c_List_On__lists_001tf__a,type,
n_lists_a: nat > list_a > list_list_a ).
thf(sy_c_List_Onth_001t__Set__Oset_Itf__a_J,type,
nth_set_a: list_set_a > nat > set_a ).
thf(sy_c_List_Onth_001tf__a,type,
nth_a: list_a > nat > a ).
thf(sy_c_Multiset_Omset_001t__Set__Oset_Itf__a_J,type,
mset_set_a: list_set_a > multiset_set_a ).
thf(sy_c_Multiset_Omset_001tf__a,type,
mset_a: list_a > multiset_a ).
thf(sy_c_Multiset_Omultiset_Ocount_001t__Set__Oset_Itf__a_J,type,
count_set_a: multiset_set_a > set_a > nat ).
thf(sy_c_Multiset_Omultiset_Ocount_001tf__a,type,
count_a: multiset_a > a > nat ).
thf(sy_c_Multiset_Orepeat__mset_001t__Set__Oset_Itf__a_J,type,
repeat_mset_set_a: nat > multiset_set_a > multiset_set_a ).
thf(sy_c_Multiset_Oreplicate__mset_001t__Set__Oset_Itf__a_J,type,
replicate_mset_set_a: nat > set_a > multiset_set_a ).
thf(sy_c_Multiset_Oreplicate__mset_001tf__a,type,
replicate_mset_a: nat > a > multiset_a ).
thf(sy_c_Multiset_Oset__mset_001t__Set__Oset_Itf__a_J,type,
set_mset_set_a: multiset_set_a > set_set_a ).
thf(sy_c_Multiset_Oset__mset_001tf__a,type,
set_mset_a: multiset_a > set_a ).
thf(sy_c_Multiset_Osubseteq__mset_001t__Set__Oset_Itf__a_J,type,
subseteq_mset_set_a: multiset_set_a > multiset_set_a > $o ).
thf(sy_c_Multiset_Osubseteq__mset_001tf__a,type,
subseteq_mset_a: multiset_a > multiset_a > $o ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
size_size_list_set_a: list_set_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
size_size_list_a: list_a > nat ).
thf(sy_c_Order_OLower_001tf__a_001t__Product____Type__Ounit,type,
lower_a_Product_unit: partia6638023223214267844t_unit > set_a > set_a ).
thf(sy_c_Order_Ogorder_Ogorder__ext_001tf__a_001t__Product____Type__Ounit,type,
gorder6800608520584131120t_unit: ( a > a > $o ) > product_unit > gorder3081419850850201676t_unit ).
thf(sy_c_Order_Ogreatest_001tf__a_001t__Product____Type__Ounit,type,
greate1756078582999224458t_unit: partia6638023223214267844t_unit > a > set_a > $o ).
thf(sy_c_Order_Olless_001tf__a_001t__Product____Type__Ounit,type,
lless_a_Product_unit: partia6638023223214267844t_unit > a > a > $o ).
thf(sy_c_Order_Oweak__partial__order_001tf__a_001t__Product____Type__Ounit,type,
weak_p4024844561679123766t_unit: partia6638023223214267844t_unit > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
bot_bot_set_set_a: set_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
ord_le5765082015083327056_set_a: multiset_set_a > multiset_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_less_set_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
ord_less_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__List__Olist_Itf__a_J_M_062_It__List__Olist_Itf__a_J_M_Eo_J_J,type,
ord_le5542992221119063950st_a_o: ( list_a > list_a > $o ) > ( list_a > list_a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__a_M_062_Itf__a_M_Eo_J_J,type,
ord_less_eq_a_a_o: ( a > a > $o ) > ( a > a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
ord_le7905258569527593284_set_a: multiset_set_a > multiset_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_le3724670747650509150_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Product__Type_OUnity,type,
product_Unity: product_unit ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_Itf__a_J,type,
insert_set_a: set_a > set_set_a > set_set_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_G,type,
g: partia7680350978787392319xt_a_b ).
thf(sy_v_a,type,
a2: a ).
thf(sy_v_d,type,
d: a ).
thf(sy_v_m____,type,
m: nat ).
% Relevant facts (1279)
thf(fact_0_associated__sym,axiom,
! [A: a,B: a] :
( ( associated_a_b @ g @ A @ B )
=> ( associated_a_b @ g @ B @ A ) ) ).
% associated_sym
thf(fact_1_assms_I2_J,axiom,
member_a @ d @ ( partia7484183841585581558xt_a_b @ g ) ).
% assms(2)
thf(fact_2_assms_I1_J,axiom,
member_a @ a2 @ ( partia7484183841585581558xt_a_b @ g ) ).
% assms(1)
thf(fact_3_assoc__subst,axiom,
! [A: a,B: a,F: a > a] :
( ( associated_a_b @ g @ A @ B )
=> ( ! [A2: a,B2: a] :
( ( ( member_a @ A2 @ ( partia7484183841585581558xt_a_b @ g ) )
& ( member_a @ B2 @ ( partia7484183841585581558xt_a_b @ g ) )
& ( associated_a_b @ g @ A2 @ B2 ) )
=> ( ( member_a @ ( F @ A2 ) @ ( partia7484183841585581558xt_a_b @ g ) )
& ( member_a @ ( F @ B2 ) @ ( partia7484183841585581558xt_a_b @ g ) )
& ( associated_a_b @ g @ ( F @ A2 ) @ ( F @ B2 ) ) ) )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( associated_a_b @ g @ ( F @ A ) @ ( F @ B ) ) ) ) ) ) ).
% assoc_subst
thf(fact_4_associated__trans,axiom,
! [A: a,B: a,C: a] :
( ( associated_a_b @ g @ A @ B )
=> ( ( associated_a_b @ g @ B @ C )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ C @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( associated_a_b @ g @ A @ C ) ) ) ) ) ).
% associated_trans
thf(fact_5_divides__trans,axiom,
! [A: a,B: a,C: a] :
( ( factor_a_b @ g @ A @ B )
=> ( ( factor_a_b @ g @ B @ C )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( factor_a_b @ g @ A @ C ) ) ) ) ).
% divides_trans
thf(fact_6_carrier__not__empty,axiom,
( ( partia7484183841585581558xt_a_b @ g )
!= bot_bot_set_a ) ).
% carrier_not_empty
thf(fact_7_factorial__monoid__axioms,axiom,
factorial_monoid_a_b @ g ).
% factorial_monoid_axioms
thf(fact_8_divides__cong__l,axiom,
! [X: a,X2: a,Y: a] :
( ( associated_a_b @ g @ X @ X2 )
=> ( ( factor_a_b @ g @ X2 @ Y )
=> ( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( factor_a_b @ g @ X @ Y ) ) ) ) ).
% divides_cong_l
thf(fact_9_divides__cong__r,axiom,
! [X: a,Y: a,Y2: a] :
( ( factor_a_b @ g @ X @ Y )
=> ( ( associated_a_b @ g @ Y @ Y2 )
=> ( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( factor_a_b @ g @ X @ Y2 ) ) ) ) ).
% divides_cong_r
thf(fact_10_factor__mset__sim,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( associated_a_b @ g @ A @ B )
= ( ( finite8971978181520403909et_a_b @ g @ A )
= ( finite8971978181520403909et_a_b @ g @ B ) ) ) ) ) ).
% factor_mset_sim
thf(fact_11_factor__mset__divides,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( factor_a_b @ g @ A @ B )
= ( subseteq_mset_set_a @ ( finite8971978181520403909et_a_b @ g @ A ) @ ( finite8971978181520403909et_a_b @ g @ B ) ) ) ) ) ).
% factor_mset_divides
thf(fact_12_lcmof__exists,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ? [C2: a] :
( ( member_a @ C2 @ ( partia7484183841585581558xt_a_b @ g ) )
& ( islcm_a_b @ g @ C2 @ A @ B ) ) ) ) ).
% lcmof_exists
thf(fact_13_assocs__assoc,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( eq_clo186784744570230005t_unit @ ( partia6383399360842404908t_unit @ ( partia7484183841585581558xt_a_b @ g ) @ ( eq_eq_5233546288009311946t_unit @ ( associated_a_b @ g ) @ ( gorder6800608520584131120t_unit @ ( factor_a_b @ g ) @ product_Unity ) ) ) @ ( insert_a @ B @ bot_bot_set_a ) ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( associated_a_b @ g @ A @ B ) ) ) ).
% assocs_assoc
thf(fact_14_assocs__eqD,axiom,
! [B: a,A: a] :
( ( ( eq_clo186784744570230005t_unit @ ( partia6383399360842404908t_unit @ ( partia7484183841585581558xt_a_b @ g ) @ ( eq_eq_5233546288009311946t_unit @ ( associated_a_b @ g ) @ ( gorder6800608520584131120t_unit @ ( factor_a_b @ g ) @ product_Unity ) ) ) @ ( insert_a @ B @ bot_bot_set_a ) )
= ( eq_clo186784744570230005t_unit @ ( partia6383399360842404908t_unit @ ( partia7484183841585581558xt_a_b @ g ) @ ( eq_eq_5233546288009311946t_unit @ ( associated_a_b @ g ) @ ( gorder6800608520584131120t_unit @ ( factor_a_b @ g ) @ product_Unity ) ) ) @ ( insert_a @ A @ bot_bot_set_a ) ) )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( associated_a_b @ g @ A @ B ) ) ) ) ).
% assocs_eqD
thf(fact_15_assocs__repr__independence,axiom,
! [Y: a,X: a] :
( ( member_a @ Y @ ( eq_clo186784744570230005t_unit @ ( partia6383399360842404908t_unit @ ( partia7484183841585581558xt_a_b @ g ) @ ( eq_eq_5233546288009311946t_unit @ ( associated_a_b @ g ) @ ( gorder6800608520584131120t_unit @ ( factor_a_b @ g ) @ product_Unity ) ) ) @ ( insert_a @ X @ bot_bot_set_a ) ) )
=> ( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( eq_clo186784744570230005t_unit @ ( partia6383399360842404908t_unit @ ( partia7484183841585581558xt_a_b @ g ) @ ( eq_eq_5233546288009311946t_unit @ ( associated_a_b @ g ) @ ( gorder6800608520584131120t_unit @ ( factor_a_b @ g ) @ product_Unity ) ) ) @ ( insert_a @ X @ bot_bot_set_a ) )
= ( eq_clo186784744570230005t_unit @ ( partia6383399360842404908t_unit @ ( partia7484183841585581558xt_a_b @ g ) @ ( eq_eq_5233546288009311946t_unit @ ( associated_a_b @ g ) @ ( gorder6800608520584131120t_unit @ ( factor_a_b @ g ) @ product_Unity ) ) ) @ ( insert_a @ Y @ bot_bot_set_a ) ) ) ) ) ).
% assocs_repr_independence
thf(fact_16_assocs__repr__independenceD,axiom,
! [X: a,Y: a] :
( ( ( eq_clo186784744570230005t_unit @ ( partia6383399360842404908t_unit @ ( partia7484183841585581558xt_a_b @ g ) @ ( eq_eq_5233546288009311946t_unit @ ( associated_a_b @ g ) @ ( gorder6800608520584131120t_unit @ ( factor_a_b @ g ) @ product_Unity ) ) ) @ ( insert_a @ X @ bot_bot_set_a ) )
= ( eq_clo186784744570230005t_unit @ ( partia6383399360842404908t_unit @ ( partia7484183841585581558xt_a_b @ g ) @ ( eq_eq_5233546288009311946t_unit @ ( associated_a_b @ g ) @ ( gorder6800608520584131120t_unit @ ( factor_a_b @ g ) @ product_Unity ) ) ) @ ( insert_a @ Y @ bot_bot_set_a ) ) )
=> ( ( member_a @ Y @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( member_a @ Y @ ( eq_clo186784744570230005t_unit @ ( partia6383399360842404908t_unit @ ( partia7484183841585581558xt_a_b @ g ) @ ( eq_eq_5233546288009311946t_unit @ ( associated_a_b @ g ) @ ( gorder6800608520584131120t_unit @ ( factor_a_b @ g ) @ product_Unity ) ) ) @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% assocs_repr_independenceD
thf(fact_17_assocs__self,axiom,
! [X: a] :
( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( member_a @ X @ ( eq_clo186784744570230005t_unit @ ( partia6383399360842404908t_unit @ ( partia7484183841585581558xt_a_b @ g ) @ ( eq_eq_5233546288009311946t_unit @ ( associated_a_b @ g ) @ ( gorder6800608520584131120t_unit @ ( factor_a_b @ g ) @ product_Unity ) ) ) @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% assocs_self
thf(fact_18_associated__refl,axiom,
! [A: a] :
( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( associated_a_b @ g @ A @ A ) ) ).
% associated_refl
thf(fact_19_divides__refl,axiom,
! [A: a] :
( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( factor_a_b @ g @ A @ A ) ) ).
% divides_refl
thf(fact_20_assms_I3_J,axiom,
irreducible_a_b @ g @ d ).
% assms(3)
thf(fact_21_divisor__chain__condition__monoid__axioms,axiom,
diviso5244014645414798900id_a_b @ g ).
% divisor_chain_condition_monoid_axioms
thf(fact_22_primeness__condition__monoid__axioms,axiom,
primen900146951955507079id_a_b @ g ).
% primeness_condition_monoid_axioms
thf(fact_23_division__weak__lattice,axiom,
weak_l7828739606774570307t_unit @ ( partia6383399360842404908t_unit @ ( partia7484183841585581558xt_a_b @ g ) @ ( eq_eq_5233546288009311946t_unit @ ( associated_a_b @ g ) @ ( gorder6800608520584131120t_unit @ ( factor_a_b @ g ) @ product_Unity ) ) ) ).
% division_weak_lattice
thf(fact_24_divides__fcount,axiom,
! [A: a,B: a] :
( ( factor_a_b @ g @ A @ B )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ord_less_eq_nat @ ( factorcount_a_b @ g @ A ) @ ( factorcount_a_b @ g @ B ) ) ) ) ) ).
% divides_fcount
thf(fact_25_associated__fcount,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( associated_a_b @ g @ A @ B )
=> ( ( factorcount_a_b @ g @ A )
= ( factorcount_a_b @ g @ B ) ) ) ) ) ).
% associated_fcount
thf(fact_26_division__weak__lower__semilattice,axiom,
weak_l5773807957815819224t_unit @ ( partia6383399360842404908t_unit @ ( partia7484183841585581558xt_a_b @ g ) @ ( eq_eq_5233546288009311946t_unit @ ( associated_a_b @ g ) @ ( gorder6800608520584131120t_unit @ ( factor_a_b @ g ) @ product_Unity ) ) ) ).
% division_weak_lower_semilattice
thf(fact_27_isgcd__divides__l,axiom,
! [A: a,B: a] :
( ( factor_a_b @ g @ A @ B )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( isgcd_a_b @ g @ A @ A @ B ) ) ) ) ).
% isgcd_divides_l
thf(fact_28_isgcd__divides__r,axiom,
! [B: a,A: a] :
( ( factor_a_b @ g @ B @ A )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( isgcd_a_b @ g @ B @ A @ B ) ) ) ) ).
% isgcd_divides_r
thf(fact_29_gcdof__cong__l,axiom,
! [A3: a,A: a,B: a,C: a] :
( ( associated_a_b @ g @ A3 @ A )
=> ( ( isgcd_a_b @ g @ A @ B @ C )
=> ( ( member_a @ A3 @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ C @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( isgcd_a_b @ g @ A3 @ B @ C ) ) ) ) ) ) ) ).
% gcdof_cong_l
thf(fact_30_prime__cong,axiom,
! [P: a,P2: a] :
( ( prime_a_b @ g @ P )
=> ( ( associated_a_b @ g @ P @ P2 )
=> ( ( member_a @ P @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ P2 @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( prime_a_b @ g @ P2 ) ) ) ) ) ).
% prime_cong
thf(fact_31_gcdof__exists,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ? [C2: a] :
( ( member_a @ C2 @ ( partia7484183841585581558xt_a_b @ g ) )
& ( isgcd_a_b @ g @ C2 @ A @ B ) ) ) ) ).
% gcdof_exists
thf(fact_32_singletonI,axiom,
! [A: set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) ).
% singletonI
thf(fact_33_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_34_local_OgcdI,axiom,
! [A: a,B: a,C: a] :
( ( factor_a_b @ g @ A @ B )
=> ( ( factor_a_b @ g @ A @ C )
=> ( ! [Y3: a] :
( ( member_a @ Y3 @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( factor_a_b @ g @ Y3 @ B )
=> ( ( factor_a_b @ g @ Y3 @ C )
=> ( factor_a_b @ g @ Y3 @ A ) ) ) )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ C @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( associated_a_b @ g @ A @ ( somegcd_a_b @ g @ B @ C ) ) ) ) ) ) ) ) ).
% local.gcdI
thf(fact_35_gorder_Ocases,axiom,
! [R: partia6638023223214267844t_unit] :
~ ! [Carrier: set_a,Eq: a > a > $o,Le: a > a > $o] :
( R
!= ( partia6383399360842404908t_unit @ Carrier @ ( eq_eq_5233546288009311946t_unit @ Eq @ ( gorder6800608520584131120t_unit @ Le @ product_Unity ) ) ) ) ).
% gorder.cases
thf(fact_36_gcd__exists,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( member_a @ ( somegcd_a_b @ g @ A @ B ) @ ( partia7484183841585581558xt_a_b @ g ) ) ) ) ).
% gcd_exists
thf(fact_37_gcd__closed,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( member_a @ ( somegcd_a_b @ g @ A @ B ) @ ( partia7484183841585581558xt_a_b @ g ) ) ) ) ).
% gcd_closed
thf(fact_38_prime__irreducible,axiom,
! [P: a] :
( ( prime_a_b @ g @ P )
=> ( irreducible_a_b @ g @ P ) ) ).
% prime_irreducible
thf(fact_39_empty__Collect__eq,axiom,
! [P3: set_a > $o] :
( ( bot_bot_set_set_a
= ( collect_set_a @ P3 ) )
= ( ! [X3: set_a] :
~ ( P3 @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_40_empty__Collect__eq,axiom,
! [P3: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P3 ) )
= ( ! [X3: a] :
~ ( P3 @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_41_Collect__empty__eq,axiom,
! [P3: set_a > $o] :
( ( ( collect_set_a @ P3 )
= bot_bot_set_set_a )
= ( ! [X3: set_a] :
~ ( P3 @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_42_Collect__empty__eq,axiom,
! [P3: a > $o] :
( ( ( collect_a @ P3 )
= bot_bot_set_a )
= ( ! [X3: a] :
~ ( P3 @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_43_all__not__in__conv,axiom,
! [A4: set_set_a] :
( ( ! [X3: set_a] :
~ ( member_set_a @ X3 @ A4 ) )
= ( A4 = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_44_all__not__in__conv,axiom,
! [A4: set_a] :
( ( ! [X3: a] :
~ ( member_a @ X3 @ A4 ) )
= ( A4 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_45_empty__iff,axiom,
! [C: set_a] :
~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_46_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_47_insert__absorb2,axiom,
! [X: set_a,A4: set_set_a] :
( ( insert_set_a @ X @ ( insert_set_a @ X @ A4 ) )
= ( insert_set_a @ X @ A4 ) ) ).
% insert_absorb2
thf(fact_48_insert__absorb2,axiom,
! [X: a,A4: set_a] :
( ( insert_a @ X @ ( insert_a @ X @ A4 ) )
= ( insert_a @ X @ A4 ) ) ).
% insert_absorb2
thf(fact_49_insert__iff,axiom,
! [A: a,B: a,A4: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A4 ) )
= ( ( A = B )
| ( member_a @ A @ A4 ) ) ) ).
% insert_iff
thf(fact_50_insert__iff,axiom,
! [A: set_a,B: set_a,A4: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B @ A4 ) )
= ( ( A = B )
| ( member_set_a @ A @ A4 ) ) ) ).
% insert_iff
thf(fact_51_insertCI,axiom,
! [A: a,B3: set_a,B: a] :
( ( ~ ( member_a @ A @ B3 )
=> ( A = B ) )
=> ( member_a @ A @ ( insert_a @ B @ B3 ) ) ) ).
% insertCI
thf(fact_52_insertCI,axiom,
! [A: set_a,B3: set_set_a,B: set_a] :
( ( ~ ( member_set_a @ A @ B3 )
=> ( A = B ) )
=> ( member_set_a @ A @ ( insert_set_a @ B @ B3 ) ) ) ).
% insertCI
thf(fact_53_mem__Collect__eq,axiom,
! [A: a,P3: a > $o] :
( ( member_a @ A @ ( collect_a @ P3 ) )
= ( P3 @ A ) ) ).
% mem_Collect_eq
thf(fact_54_mem__Collect__eq,axiom,
! [A: set_a,P3: set_a > $o] :
( ( member_set_a @ A @ ( collect_set_a @ P3 ) )
= ( P3 @ A ) ) ).
% mem_Collect_eq
thf(fact_55_Collect__mem__eq,axiom,
! [A4: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_56_Collect__mem__eq,axiom,
! [A4: set_set_a] :
( ( collect_set_a
@ ^ [X3: set_a] : ( member_set_a @ X3 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_57_Collect__cong,axiom,
! [P3: a > $o,Q: a > $o] :
( ! [X4: a] :
( ( P3 @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_a @ P3 )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_58_Collect__cong,axiom,
! [P3: set_a > $o,Q: set_a > $o] :
( ! [X4: set_a] :
( ( P3 @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_set_a @ P3 )
= ( collect_set_a @ Q ) ) ) ).
% Collect_cong
thf(fact_59_irreducible__cong,axiom,
! [A: a,A3: a] :
( ( irreducible_a_b @ g @ A )
=> ( ( associated_a_b @ g @ A @ A3 )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ A3 @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( irreducible_a_b @ g @ A3 ) ) ) ) ) ).
% irreducible_cong
thf(fact_60_gcd__cong__r,axiom,
! [Y: a,Y2: a,X: a] :
( ( associated_a_b @ g @ Y @ Y2 )
=> ( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ Y @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ Y2 @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( associated_a_b @ g @ ( somegcd_a_b @ g @ X @ Y ) @ ( somegcd_a_b @ g @ X @ Y2 ) ) ) ) ) ) ).
% gcd_cong_r
thf(fact_61_gcd__cong__l,axiom,
! [X: a,X2: a,Y: a] :
( ( associated_a_b @ g @ X @ X2 )
=> ( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ X2 @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ Y @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( associated_a_b @ g @ ( somegcd_a_b @ g @ X @ Y ) @ ( somegcd_a_b @ g @ X2 @ Y ) ) ) ) ) ) ).
% gcd_cong_l
thf(fact_62_gcd__assoc,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ C @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( associated_a_b @ g @ ( somegcd_a_b @ g @ ( somegcd_a_b @ g @ A @ B ) @ C ) @ ( somegcd_a_b @ g @ A @ ( somegcd_a_b @ g @ B @ C ) ) ) ) ) ) ).
% gcd_assoc
thf(fact_63_gcd__divides__r,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( factor_a_b @ g @ ( somegcd_a_b @ g @ A @ B ) @ B ) ) ) ).
% gcd_divides_r
thf(fact_64_gcd__divides__l,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( factor_a_b @ g @ ( somegcd_a_b @ g @ A @ B ) @ A ) ) ) ).
% gcd_divides_l
thf(fact_65_gcd__divides,axiom,
! [Z: a,X: a,Y: a] :
( ( factor_a_b @ g @ Z @ X )
=> ( ( factor_a_b @ g @ Z @ Y )
=> ( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ Y @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ Z @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( factor_a_b @ g @ Z @ ( somegcd_a_b @ g @ X @ Y ) ) ) ) ) ) ) ).
% gcd_divides
thf(fact_66_gorder_Oext__inject,axiom,
! [Le2: a > a > $o,More: product_unit,Le3: a > a > $o,More2: product_unit] :
( ( ( gorder6800608520584131120t_unit @ Le2 @ More )
= ( gorder6800608520584131120t_unit @ Le3 @ More2 ) )
= ( ( Le2 = Le3 )
& ( More = More2 ) ) ) ).
% gorder.ext_inject
thf(fact_67_irreducible__prime,axiom,
! [P: a] :
( ( irreducible_a_b @ g @ P )
=> ( ( member_a @ P @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( prime_a_b @ g @ P ) ) ) ).
% irreducible_prime
thf(fact_68_gcd__isgcd,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( isgcd_a_b @ g @ ( somegcd_a_b @ g @ A @ B ) @ A @ B ) ) ) ).
% gcd_isgcd
thf(fact_69_gcdI2,axiom,
! [A: a,B: a,C: a] :
( ( isgcd_a_b @ g @ A @ B @ C )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ C @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( associated_a_b @ g @ A @ ( somegcd_a_b @ g @ B @ C ) ) ) ) ) ) ).
% gcdI2
thf(fact_70_ex__in__conv,axiom,
! [A4: set_set_a] :
( ( ? [X3: set_a] : ( member_set_a @ X3 @ A4 ) )
= ( A4 != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_71_ex__in__conv,axiom,
! [A4: set_a] :
( ( ? [X3: a] : ( member_a @ X3 @ A4 ) )
= ( A4 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_72_equals0I,axiom,
! [A4: set_set_a] :
( ! [Y3: set_a] :
~ ( member_set_a @ Y3 @ A4 )
=> ( A4 = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_73_equals0I,axiom,
! [A4: set_a] :
( ! [Y3: a] :
~ ( member_a @ Y3 @ A4 )
=> ( A4 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_74_equals0D,axiom,
! [A4: set_set_a,A: set_a] :
( ( A4 = bot_bot_set_set_a )
=> ~ ( member_set_a @ A @ A4 ) ) ).
% equals0D
thf(fact_75_equals0D,axiom,
! [A4: set_a,A: a] :
( ( A4 = bot_bot_set_a )
=> ~ ( member_a @ A @ A4 ) ) ).
% equals0D
thf(fact_76_emptyE,axiom,
! [A: set_a] :
~ ( member_set_a @ A @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_77_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_78_mk__disjoint__insert,axiom,
! [A: a,A4: set_a] :
( ( member_a @ A @ A4 )
=> ? [B4: set_a] :
( ( A4
= ( insert_a @ A @ B4 ) )
& ~ ( member_a @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_79_mk__disjoint__insert,axiom,
! [A: set_a,A4: set_set_a] :
( ( member_set_a @ A @ A4 )
=> ? [B4: set_set_a] :
( ( A4
= ( insert_set_a @ A @ B4 ) )
& ~ ( member_set_a @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_80_insert__commute,axiom,
! [X: set_a,Y: set_a,A4: set_set_a] :
( ( insert_set_a @ X @ ( insert_set_a @ Y @ A4 ) )
= ( insert_set_a @ Y @ ( insert_set_a @ X @ A4 ) ) ) ).
% insert_commute
thf(fact_81_insert__commute,axiom,
! [X: a,Y: a,A4: set_a] :
( ( insert_a @ X @ ( insert_a @ Y @ A4 ) )
= ( insert_a @ Y @ ( insert_a @ X @ A4 ) ) ) ).
% insert_commute
thf(fact_82_insert__eq__iff,axiom,
! [A: a,A4: set_a,B: a,B3: set_a] :
( ~ ( member_a @ A @ A4 )
=> ( ~ ( member_a @ B @ B3 )
=> ( ( ( insert_a @ A @ A4 )
= ( insert_a @ B @ B3 ) )
= ( ( ( A = B )
=> ( A4 = B3 ) )
& ( ( A != B )
=> ? [C3: set_a] :
( ( A4
= ( insert_a @ B @ C3 ) )
& ~ ( member_a @ B @ C3 )
& ( B3
= ( insert_a @ A @ C3 ) )
& ~ ( member_a @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_83_insert__eq__iff,axiom,
! [A: set_a,A4: set_set_a,B: set_a,B3: set_set_a] :
( ~ ( member_set_a @ A @ A4 )
=> ( ~ ( member_set_a @ B @ B3 )
=> ( ( ( insert_set_a @ A @ A4 )
= ( insert_set_a @ B @ B3 ) )
= ( ( ( A = B )
=> ( A4 = B3 ) )
& ( ( A != B )
=> ? [C3: set_set_a] :
( ( A4
= ( insert_set_a @ B @ C3 ) )
& ~ ( member_set_a @ B @ C3 )
& ( B3
= ( insert_set_a @ A @ C3 ) )
& ~ ( member_set_a @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_84_insert__absorb,axiom,
! [A: a,A4: set_a] :
( ( member_a @ A @ A4 )
=> ( ( insert_a @ A @ A4 )
= A4 ) ) ).
% insert_absorb
thf(fact_85_insert__absorb,axiom,
! [A: set_a,A4: set_set_a] :
( ( member_set_a @ A @ A4 )
=> ( ( insert_set_a @ A @ A4 )
= A4 ) ) ).
% insert_absorb
thf(fact_86_insert__ident,axiom,
! [X: a,A4: set_a,B3: set_a] :
( ~ ( member_a @ X @ A4 )
=> ( ~ ( member_a @ X @ B3 )
=> ( ( ( insert_a @ X @ A4 )
= ( insert_a @ X @ B3 ) )
= ( A4 = B3 ) ) ) ) ).
% insert_ident
thf(fact_87_insert__ident,axiom,
! [X: set_a,A4: set_set_a,B3: set_set_a] :
( ~ ( member_set_a @ X @ A4 )
=> ( ~ ( member_set_a @ X @ B3 )
=> ( ( ( insert_set_a @ X @ A4 )
= ( insert_set_a @ X @ B3 ) )
= ( A4 = B3 ) ) ) ) ).
% insert_ident
thf(fact_88_Set_Oset__insert,axiom,
! [X: a,A4: set_a] :
( ( member_a @ X @ A4 )
=> ~ ! [B4: set_a] :
( ( A4
= ( insert_a @ X @ B4 ) )
=> ( member_a @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_89_Set_Oset__insert,axiom,
! [X: set_a,A4: set_set_a] :
( ( member_set_a @ X @ A4 )
=> ~ ! [B4: set_set_a] :
( ( A4
= ( insert_set_a @ X @ B4 ) )
=> ( member_set_a @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_90_insertI2,axiom,
! [A: a,B3: set_a,B: a] :
( ( member_a @ A @ B3 )
=> ( member_a @ A @ ( insert_a @ B @ B3 ) ) ) ).
% insertI2
thf(fact_91_insertI2,axiom,
! [A: set_a,B3: set_set_a,B: set_a] :
( ( member_set_a @ A @ B3 )
=> ( member_set_a @ A @ ( insert_set_a @ B @ B3 ) ) ) ).
% insertI2
thf(fact_92_insertI1,axiom,
! [A: a,B3: set_a] : ( member_a @ A @ ( insert_a @ A @ B3 ) ) ).
% insertI1
thf(fact_93_insertI1,axiom,
! [A: set_a,B3: set_set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ B3 ) ) ).
% insertI1
thf(fact_94_insertE,axiom,
! [A: a,B: a,A4: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A4 ) )
=> ( ( A != B )
=> ( member_a @ A @ A4 ) ) ) ).
% insertE
thf(fact_95_insertE,axiom,
! [A: set_a,B: set_a,A4: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B @ A4 ) )
=> ( ( A != B )
=> ( member_set_a @ A @ A4 ) ) ) ).
% insertE
thf(fact_96_factorial__monoid_Ofactor__mset__sim,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a] :
( ( factor2046533344642582127t_unit @ G )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( associ6879500422977059064t_unit @ G @ A @ B )
= ( ( finite2487704559579920083t_unit @ G @ A )
= ( finite2487704559579920083t_unit @ G @ B ) ) ) ) ) ) ).
% factorial_monoid.factor_mset_sim
thf(fact_97_factorial__monoid_Ofactor__mset__sim,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a] :
( ( factorial_monoid_a_b @ G )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( associated_a_b @ G @ A @ B )
= ( ( finite8971978181520403909et_a_b @ G @ A )
= ( finite8971978181520403909et_a_b @ G @ B ) ) ) ) ) ) ).
% factorial_monoid.factor_mset_sim
thf(fact_98_gorder_Oext__induct,axiom,
! [P3: gorder3081419850850201676t_unit > $o,R: gorder3081419850850201676t_unit] :
( ! [Le: a > a > $o,More3: product_unit] : ( P3 @ ( gorder6800608520584131120t_unit @ Le @ More3 ) )
=> ( P3 @ R ) ) ).
% gorder.ext_induct
thf(fact_99_factorial__monoid_Ofactor__mset__divides,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a] :
( ( factor2046533344642582127t_unit @ G )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( factor3040189038382604065t_unit @ G @ A @ B )
= ( subseteq_mset_set_a @ ( finite2487704559579920083t_unit @ G @ A ) @ ( finite2487704559579920083t_unit @ G @ B ) ) ) ) ) ) ).
% factorial_monoid.factor_mset_divides
thf(fact_100_factorial__monoid_Ofactor__mset__divides,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a] :
( ( factorial_monoid_a_b @ G )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( factor_a_b @ G @ A @ B )
= ( subseteq_mset_set_a @ ( finite8971978181520403909et_a_b @ G @ A ) @ ( finite8971978181520403909et_a_b @ G @ B ) ) ) ) ) ) ).
% factorial_monoid.factor_mset_divides
thf(fact_101_singleton__inject,axiom,
! [A: set_a,B: set_a] :
( ( ( insert_set_a @ A @ bot_bot_set_set_a )
= ( insert_set_a @ B @ bot_bot_set_set_a ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_102_singleton__inject,axiom,
! [A: a,B: a] :
( ( ( insert_a @ A @ bot_bot_set_a )
= ( insert_a @ B @ bot_bot_set_a ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_103_insert__not__empty,axiom,
! [A: set_a,A4: set_set_a] :
( ( insert_set_a @ A @ A4 )
!= bot_bot_set_set_a ) ).
% insert_not_empty
thf(fact_104_insert__not__empty,axiom,
! [A: a,A4: set_a] :
( ( insert_a @ A @ A4 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_105_doubleton__eq__iff,axiom,
! [A: set_a,B: set_a,C: set_a,D: set_a] :
( ( ( insert_set_a @ A @ ( insert_set_a @ B @ bot_bot_set_set_a ) )
= ( insert_set_a @ C @ ( insert_set_a @ D @ bot_bot_set_set_a ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_106_doubleton__eq__iff,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) )
= ( insert_a @ C @ ( insert_a @ D @ bot_bot_set_a ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_107_singleton__iff,axiom,
! [B: set_a,A: set_a] :
( ( member_set_a @ B @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_108_singleton__iff,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_109_singletonD,axiom,
! [B: set_a,A: set_a] :
( ( member_set_a @ B @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_110_singletonD,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_111_gorder_Ocases__scheme,axiom,
! [R: partia6638023223214267844t_unit] :
~ ! [Carrier: set_a,Eq: a > a > $o,Le: a > a > $o,More3: product_unit] :
( R
!= ( partia6383399360842404908t_unit @ Carrier @ ( eq_eq_5233546288009311946t_unit @ Eq @ ( gorder6800608520584131120t_unit @ Le @ More3 ) ) ) ) ).
% gorder.cases_scheme
thf(fact_112_factorial__monoid_Odivision__weak__lattice,axiom,
! [G: partia8223610829204095565t_unit] :
( ( factor2046533344642582127t_unit @ G )
=> ( weak_l7828739606774570307t_unit @ ( partia6383399360842404908t_unit @ ( partia6735698275553448452t_unit @ G ) @ ( eq_eq_5233546288009311946t_unit @ ( associ6879500422977059064t_unit @ G ) @ ( gorder6800608520584131120t_unit @ ( factor3040189038382604065t_unit @ G ) @ product_Unity ) ) ) ) ) ).
% factorial_monoid.division_weak_lattice
thf(fact_113_factorial__monoid_Odivision__weak__lattice,axiom,
! [G: partia7680350978787392319xt_a_b] :
( ( factorial_monoid_a_b @ G )
=> ( weak_l7828739606774570307t_unit @ ( partia6383399360842404908t_unit @ ( partia7484183841585581558xt_a_b @ G ) @ ( eq_eq_5233546288009311946t_unit @ ( associated_a_b @ G ) @ ( gorder6800608520584131120t_unit @ ( factor_a_b @ G ) @ product_Unity ) ) ) ) ) ).
% factorial_monoid.division_weak_lattice
thf(fact_114_factor__mset__set,axiom,
! [A: a,X: set_a] :
( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_set_a @ X @ ( set_mset_set_a @ ( finite8971978181520403909et_a_b @ g @ A ) ) )
=> ~ ! [Y3: a] :
( ( member_a @ Y3 @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( irreducible_a_b @ g @ Y3 )
=> ( ( eq_clo186784744570230005t_unit @ ( partia6383399360842404908t_unit @ ( partia7484183841585581558xt_a_b @ g ) @ ( eq_eq_5233546288009311946t_unit @ ( associated_a_b @ g ) @ ( gorder6800608520584131120t_unit @ ( factor_a_b @ g ) @ product_Unity ) ) ) @ ( insert_a @ Y3 @ bot_bot_set_a ) )
!= X ) ) ) ) ) ).
% factor_mset_set
thf(fact_115_factorial__monoid_Odivides__fcount,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a] :
( ( factor2046533344642582127t_unit @ G )
=> ( ( factor3040189038382604065t_unit @ G @ A @ B )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( ord_less_eq_nat @ ( factor4067924603488134956t_unit @ G @ A ) @ ( factor4067924603488134956t_unit @ G @ B ) ) ) ) ) ) ).
% factorial_monoid.divides_fcount
thf(fact_116_factorial__monoid_Odivides__fcount,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a] :
( ( factorial_monoid_a_b @ G )
=> ( ( factor_a_b @ G @ A @ B )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ord_less_eq_nat @ ( factorcount_a_b @ G @ A ) @ ( factorcount_a_b @ G @ B ) ) ) ) ) ) ).
% factorial_monoid.divides_fcount
thf(fact_117_primeness__condition__monoid_Oirreducible__prime,axiom,
! [G: partia8223610829204095565t_unit,A: a] :
( ( primen965786292471834261t_unit @ G )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( irredu4023057619401689684t_unit @ G @ A )
=> ( prime_a_Product_unit @ G @ A ) ) ) ) ).
% primeness_condition_monoid.irreducible_prime
thf(fact_118_primeness__condition__monoid_Oirreducible__prime,axiom,
! [G: partia7680350978787392319xt_a_b,A: a] :
( ( primen900146951955507079id_a_b @ G )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( irreducible_a_b @ G @ A )
=> ( prime_a_b @ G @ A ) ) ) ) ).
% primeness_condition_monoid.irreducible_prime
thf(fact_119_factorial__monoid_Oassociated__fcount,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a] :
( ( factor2046533344642582127t_unit @ G )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( associ6879500422977059064t_unit @ G @ A @ B )
=> ( ( factor4067924603488134956t_unit @ G @ A )
= ( factor4067924603488134956t_unit @ G @ B ) ) ) ) ) ) ).
% factorial_monoid.associated_fcount
thf(fact_120_factorial__monoid_Oassociated__fcount,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a] :
( ( factorial_monoid_a_b @ G )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( associated_a_b @ G @ A @ B )
=> ( ( factorcount_a_b @ G @ A )
= ( factorcount_a_b @ G @ B ) ) ) ) ) ) ).
% factorial_monoid.associated_fcount
thf(fact_121_factorial__monoid_Ofactors__irreducible__prime,axiom,
! [G: partia8223610829204095565t_unit,P: a] :
( ( factor2046533344642582127t_unit @ G )
=> ( ( irredu4023057619401689684t_unit @ G @ P )
=> ( ( member_a @ P @ ( partia6735698275553448452t_unit @ G ) )
=> ( prime_a_Product_unit @ G @ P ) ) ) ) ).
% factorial_monoid.factors_irreducible_prime
thf(fact_122_factorial__monoid_Ofactors__irreducible__prime,axiom,
! [G: partia7680350978787392319xt_a_b,P: a] :
( ( factorial_monoid_a_b @ G )
=> ( ( irreducible_a_b @ G @ P )
=> ( ( member_a @ P @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( prime_a_b @ G @ P ) ) ) ) ).
% factorial_monoid.factors_irreducible_prime
thf(fact_123_count__le__replicate__mset__subset__eq,axiom,
! [N: nat,M: multiset_a,X: a] :
( ( ord_less_eq_nat @ N @ ( count_a @ M @ X ) )
= ( subseteq_mset_a @ ( replicate_mset_a @ N @ X ) @ M ) ) ).
% count_le_replicate_mset_subset_eq
thf(fact_124_count__le__replicate__mset__subset__eq,axiom,
! [N: nat,M: multiset_set_a,X: set_a] :
( ( ord_less_eq_nat @ N @ ( count_set_a @ M @ X ) )
= ( subseteq_mset_set_a @ ( replicate_mset_set_a @ N @ X ) @ M ) ) ).
% count_le_replicate_mset_subset_eq
thf(fact_125_factorial__condition__one,axiom,
! [G: partia8223610829204095565t_unit] :
( ( ( diviso6259607970152342594t_unit @ G )
& ( primen965786292471834261t_unit @ G ) )
= ( factor2046533344642582127t_unit @ G ) ) ).
% factorial_condition_one
thf(fact_126_factorial__condition__one,axiom,
! [G: partia7680350978787392319xt_a_b] :
( ( ( diviso5244014645414798900id_a_b @ G )
& ( primen900146951955507079id_a_b @ G ) )
= ( factorial_monoid_a_b @ G ) ) ).
% factorial_condition_one
thf(fact_127_factorial__monoid_Ofactor__mset__set,axiom,
! [G: partia8223610829204095565t_unit,A: a,X: set_a] :
( ( factor2046533344642582127t_unit @ G )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_set_a @ X @ ( set_mset_set_a @ ( finite2487704559579920083t_unit @ G @ A ) ) )
=> ~ ! [Y3: a] :
( ( member_a @ Y3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( irredu4023057619401689684t_unit @ G @ Y3 )
=> ( ( eq_clo186784744570230005t_unit @ ( partia6383399360842404908t_unit @ ( partia6735698275553448452t_unit @ G ) @ ( eq_eq_5233546288009311946t_unit @ ( associ6879500422977059064t_unit @ G ) @ ( gorder6800608520584131120t_unit @ ( factor3040189038382604065t_unit @ G ) @ product_Unity ) ) ) @ ( insert_a @ Y3 @ bot_bot_set_a ) )
!= X ) ) ) ) ) ) ).
% factorial_monoid.factor_mset_set
thf(fact_128_factorial__monoid_Ofactor__mset__set,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,X: set_a] :
( ( factorial_monoid_a_b @ G )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_set_a @ X @ ( set_mset_set_a @ ( finite8971978181520403909et_a_b @ G @ A ) ) )
=> ~ ! [Y3: a] :
( ( member_a @ Y3 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( irreducible_a_b @ G @ Y3 )
=> ( ( eq_clo186784744570230005t_unit @ ( partia6383399360842404908t_unit @ ( partia7484183841585581558xt_a_b @ G ) @ ( eq_eq_5233546288009311946t_unit @ ( associated_a_b @ G ) @ ( gorder6800608520584131120t_unit @ ( factor_a_b @ G ) @ product_Unity ) ) ) @ ( insert_a @ Y3 @ bot_bot_set_a ) )
!= X ) ) ) ) ) ) ).
% factorial_monoid.factor_mset_set
thf(fact_129_division__equiv,axiom,
equiva8593399568135932475t_unit @ ( partia6383399360842404908t_unit @ ( partia7484183841585581558xt_a_b @ g ) @ ( eq_eq_5233546288009311946t_unit @ ( associated_a_b @ g ) @ ( gorder6800608520584131120t_unit @ ( factor_a_b @ g ) @ product_Unity ) ) ) ).
% division_equiv
thf(fact_130_factorial__monoid_Olcmof__exists,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a] :
( ( factor2046533344642582127t_unit @ G )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ? [C2: a] :
( ( member_a @ C2 @ ( partia6735698275553448452t_unit @ G ) )
& ( islcm_a_Product_unit @ G @ C2 @ A @ B ) ) ) ) ) ).
% factorial_monoid.lcmof_exists
thf(fact_131_factorial__monoid_Olcmof__exists,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a] :
( ( factorial_monoid_a_b @ G )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ? [C2: a] :
( ( member_a @ C2 @ ( partia7484183841585581558xt_a_b @ G ) )
& ( islcm_a_b @ G @ C2 @ A @ B ) ) ) ) ) ).
% factorial_monoid.lcmof_exists
thf(fact_132_subset__mset_Oorder__refl,axiom,
! [X: multiset_a] : ( subseteq_mset_a @ X @ X ) ).
% subset_mset.order_refl
thf(fact_133_subset__mset_Oorder__refl,axiom,
! [X: multiset_set_a] : ( subseteq_mset_set_a @ X @ X ) ).
% subset_mset.order_refl
thf(fact_134_subset__mset_Odual__order_Orefl,axiom,
! [A: multiset_a] : ( subseteq_mset_a @ A @ A ) ).
% subset_mset.dual_order.refl
thf(fact_135_subset__mset_Odual__order_Orefl,axiom,
! [A: multiset_set_a] : ( subseteq_mset_set_a @ A @ A ) ).
% subset_mset.dual_order.refl
thf(fact_136_mset__subset__eqD,axiom,
! [A4: multiset_a,B3: multiset_a,X: a] :
( ( subseteq_mset_a @ A4 @ B3 )
=> ( ( member_a @ X @ ( set_mset_a @ A4 ) )
=> ( member_a @ X @ ( set_mset_a @ B3 ) ) ) ) ).
% mset_subset_eqD
thf(fact_137_mset__subset__eqD,axiom,
! [A4: multiset_set_a,B3: multiset_set_a,X: set_a] :
( ( subseteq_mset_set_a @ A4 @ B3 )
=> ( ( member_set_a @ X @ ( set_mset_set_a @ A4 ) )
=> ( member_set_a @ X @ ( set_mset_set_a @ B3 ) ) ) ) ).
% mset_subset_eqD
thf(fact_138_subset__mset_Otrans,axiom,
! [A: multiset_a,B: multiset_a,C: multiset_a] :
( ( subseteq_mset_a @ A @ B )
=> ( ( subseteq_mset_a @ B @ C )
=> ( subseteq_mset_a @ A @ C ) ) ) ).
% subset_mset.trans
thf(fact_139_subset__mset_Otrans,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( subseteq_mset_set_a @ A @ B )
=> ( ( subseteq_mset_set_a @ B @ C )
=> ( subseteq_mset_set_a @ A @ C ) ) ) ).
% subset_mset.trans
thf(fact_140_subset__mset_Oeq__iff,axiom,
( ( ^ [Y4: multiset_a,Z2: multiset_a] : ( Y4 = Z2 ) )
= ( ^ [A5: multiset_a,B5: multiset_a] :
( ( subseteq_mset_a @ A5 @ B5 )
& ( subseteq_mset_a @ B5 @ A5 ) ) ) ) ).
% subset_mset.eq_iff
thf(fact_141_subset__mset_Oeq__iff,axiom,
( ( ^ [Y4: multiset_set_a,Z2: multiset_set_a] : ( Y4 = Z2 ) )
= ( ^ [A5: multiset_set_a,B5: multiset_set_a] :
( ( subseteq_mset_set_a @ A5 @ B5 )
& ( subseteq_mset_set_a @ B5 @ A5 ) ) ) ) ).
% subset_mset.eq_iff
thf(fact_142_subset__mset_Oantisym,axiom,
! [A: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ A @ B )
=> ( ( subseteq_mset_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_mset.antisym
thf(fact_143_subset__mset_Oantisym,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( subseteq_mset_set_a @ A @ B )
=> ( ( subseteq_mset_set_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_mset.antisym
thf(fact_144_subset__mset_Oeq__refl,axiom,
! [X: multiset_a,Y: multiset_a] :
( ( X = Y )
=> ( subseteq_mset_a @ X @ Y ) ) ).
% subset_mset.eq_refl
thf(fact_145_subset__mset_Oeq__refl,axiom,
! [X: multiset_set_a,Y: multiset_set_a] :
( ( X = Y )
=> ( subseteq_mset_set_a @ X @ Y ) ) ).
% subset_mset.eq_refl
thf(fact_146_subset__mset_Oorder__trans,axiom,
! [X: multiset_a,Y: multiset_a,Z: multiset_a] :
( ( subseteq_mset_a @ X @ Y )
=> ( ( subseteq_mset_a @ Y @ Z )
=> ( subseteq_mset_a @ X @ Z ) ) ) ).
% subset_mset.order_trans
thf(fact_147_subset__mset_Oorder__trans,axiom,
! [X: multiset_set_a,Y: multiset_set_a,Z: multiset_set_a] :
( ( subseteq_mset_set_a @ X @ Y )
=> ( ( subseteq_mset_set_a @ Y @ Z )
=> ( subseteq_mset_set_a @ X @ Z ) ) ) ).
% subset_mset.order_trans
thf(fact_148_subset__mset_Oantisym__conv,axiom,
! [Y: multiset_a,X: multiset_a] :
( ( subseteq_mset_a @ Y @ X )
=> ( ( subseteq_mset_a @ X @ Y )
= ( X = Y ) ) ) ).
% subset_mset.antisym_conv
thf(fact_149_subset__mset_Oantisym__conv,axiom,
! [Y: multiset_set_a,X: multiset_set_a] :
( ( subseteq_mset_set_a @ Y @ X )
=> ( ( subseteq_mset_set_a @ X @ Y )
= ( X = Y ) ) ) ).
% subset_mset.antisym_conv
thf(fact_150_subset__mset_Oorder__eq__iff,axiom,
( ( ^ [Y4: multiset_a,Z2: multiset_a] : ( Y4 = Z2 ) )
= ( ^ [X3: multiset_a,Y5: multiset_a] :
( ( subseteq_mset_a @ X3 @ Y5 )
& ( subseteq_mset_a @ Y5 @ X3 ) ) ) ) ).
% subset_mset.order_eq_iff
thf(fact_151_subset__mset_Oorder__eq__iff,axiom,
( ( ^ [Y4: multiset_set_a,Z2: multiset_set_a] : ( Y4 = Z2 ) )
= ( ^ [X3: multiset_set_a,Y5: multiset_set_a] :
( ( subseteq_mset_set_a @ X3 @ Y5 )
& ( subseteq_mset_set_a @ Y5 @ X3 ) ) ) ) ).
% subset_mset.order_eq_iff
thf(fact_152_subset__mset_Oorder__antisym,axiom,
! [X: multiset_a,Y: multiset_a] :
( ( subseteq_mset_a @ X @ Y )
=> ( ( subseteq_mset_a @ Y @ X )
=> ( X = Y ) ) ) ).
% subset_mset.order_antisym
thf(fact_153_subset__mset_Oorder__antisym,axiom,
! [X: multiset_set_a,Y: multiset_set_a] :
( ( subseteq_mset_set_a @ X @ Y )
=> ( ( subseteq_mset_set_a @ Y @ X )
=> ( X = Y ) ) ) ).
% subset_mset.order_antisym
thf(fact_154_subset__mset_Oord__eq__le__trans,axiom,
! [A: multiset_a,B: multiset_a,C: multiset_a] :
( ( A = B )
=> ( ( subseteq_mset_a @ B @ C )
=> ( subseteq_mset_a @ A @ C ) ) ) ).
% subset_mset.ord_eq_le_trans
thf(fact_155_subset__mset_Oord__eq__le__trans,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( A = B )
=> ( ( subseteq_mset_set_a @ B @ C )
=> ( subseteq_mset_set_a @ A @ C ) ) ) ).
% subset_mset.ord_eq_le_trans
thf(fact_156_subset__mset_Oord__le__eq__trans,axiom,
! [A: multiset_a,B: multiset_a,C: multiset_a] :
( ( subseteq_mset_a @ A @ B )
=> ( ( B = C )
=> ( subseteq_mset_a @ A @ C ) ) ) ).
% subset_mset.ord_le_eq_trans
thf(fact_157_subset__mset_Oord__le__eq__trans,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( subseteq_mset_set_a @ A @ B )
=> ( ( B = C )
=> ( subseteq_mset_set_a @ A @ C ) ) ) ).
% subset_mset.ord_le_eq_trans
thf(fact_158_subset__mset_Odual__order_Otrans,axiom,
! [B: multiset_a,A: multiset_a,C: multiset_a] :
( ( subseteq_mset_a @ B @ A )
=> ( ( subseteq_mset_a @ C @ B )
=> ( subseteq_mset_a @ C @ A ) ) ) ).
% subset_mset.dual_order.trans
thf(fact_159_subset__mset_Odual__order_Otrans,axiom,
! [B: multiset_set_a,A: multiset_set_a,C: multiset_set_a] :
( ( subseteq_mset_set_a @ B @ A )
=> ( ( subseteq_mset_set_a @ C @ B )
=> ( subseteq_mset_set_a @ C @ A ) ) ) ).
% subset_mset.dual_order.trans
thf(fact_160_subset__mset_Odual__order_Oeq__iff,axiom,
( ( ^ [Y4: multiset_a,Z2: multiset_a] : ( Y4 = Z2 ) )
= ( ^ [A5: multiset_a,B5: multiset_a] :
( ( subseteq_mset_a @ B5 @ A5 )
& ( subseteq_mset_a @ A5 @ B5 ) ) ) ) ).
% subset_mset.dual_order.eq_iff
thf(fact_161_subset__mset_Odual__order_Oeq__iff,axiom,
( ( ^ [Y4: multiset_set_a,Z2: multiset_set_a] : ( Y4 = Z2 ) )
= ( ^ [A5: multiset_set_a,B5: multiset_set_a] :
( ( subseteq_mset_set_a @ B5 @ A5 )
& ( subseteq_mset_set_a @ A5 @ B5 ) ) ) ) ).
% subset_mset.dual_order.eq_iff
thf(fact_162_subset__mset_Odual__order_Oantisym,axiom,
! [B: multiset_a,A: multiset_a] :
( ( subseteq_mset_a @ B @ A )
=> ( ( subseteq_mset_a @ A @ B )
=> ( A = B ) ) ) ).
% subset_mset.dual_order.antisym
thf(fact_163_subset__mset_Odual__order_Oantisym,axiom,
! [B: multiset_set_a,A: multiset_set_a] :
( ( subseteq_mset_set_a @ B @ A )
=> ( ( subseteq_mset_set_a @ A @ B )
=> ( A = B ) ) ) ).
% subset_mset.dual_order.antisym
thf(fact_164_count__inject,axiom,
! [X: multiset_a,Y: multiset_a] :
( ( ( count_a @ X )
= ( count_a @ Y ) )
= ( X = Y ) ) ).
% count_inject
thf(fact_165_count__inject,axiom,
! [X: multiset_set_a,Y: multiset_set_a] :
( ( ( count_set_a @ X )
= ( count_set_a @ Y ) )
= ( X = Y ) ) ).
% count_inject
thf(fact_166_multiset__eqI,axiom,
! [A4: multiset_a,B3: multiset_a] :
( ! [X4: a] :
( ( count_a @ A4 @ X4 )
= ( count_a @ B3 @ X4 ) )
=> ( A4 = B3 ) ) ).
% multiset_eqI
thf(fact_167_multiset__eqI,axiom,
! [A4: multiset_set_a,B3: multiset_set_a] :
( ! [X4: set_a] :
( ( count_set_a @ A4 @ X4 )
= ( count_set_a @ B3 @ X4 ) )
=> ( A4 = B3 ) ) ).
% multiset_eqI
thf(fact_168_multiset__eq__iff,axiom,
( ( ^ [Y4: multiset_a,Z2: multiset_a] : ( Y4 = Z2 ) )
= ( ^ [M2: multiset_a,N2: multiset_a] :
! [A5: a] :
( ( count_a @ M2 @ A5 )
= ( count_a @ N2 @ A5 ) ) ) ) ).
% multiset_eq_iff
thf(fact_169_multiset__eq__iff,axiom,
( ( ^ [Y4: multiset_set_a,Z2: multiset_set_a] : ( Y4 = Z2 ) )
= ( ^ [M2: multiset_set_a,N2: multiset_set_a] :
! [A5: set_a] :
( ( count_set_a @ M2 @ A5 )
= ( count_set_a @ N2 @ A5 ) ) ) ) ).
% multiset_eq_iff
thf(fact_170_associatedD,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a] :
( ( associ6879500422977059064t_unit @ G @ A @ B )
=> ( factor3040189038382604065t_unit @ G @ A @ B ) ) ).
% associatedD
thf(fact_171_associatedD,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a] :
( ( associated_a_b @ G @ A @ B )
=> ( factor_a_b @ G @ A @ B ) ) ).
% associatedD
thf(fact_172_associatedE,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a] :
( ( associ6879500422977059064t_unit @ G @ A @ B )
=> ~ ( ( factor3040189038382604065t_unit @ G @ A @ B )
=> ~ ( factor3040189038382604065t_unit @ G @ B @ A ) ) ) ).
% associatedE
thf(fact_173_associatedE,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a] :
( ( associated_a_b @ G @ A @ B )
=> ~ ( ( factor_a_b @ G @ A @ B )
=> ~ ( factor_a_b @ G @ B @ A ) ) ) ).
% associatedE
thf(fact_174_associated__def,axiom,
( associ6879500422977059064t_unit
= ( ^ [G2: partia8223610829204095565t_unit,A5: a,B5: a] :
( ( factor3040189038382604065t_unit @ G2 @ A5 @ B5 )
& ( factor3040189038382604065t_unit @ G2 @ B5 @ A5 ) ) ) ) ).
% associated_def
thf(fact_175_associated__def,axiom,
( associated_a_b
= ( ^ [G2: partia7680350978787392319xt_a_b,A5: a,B5: a] :
( ( factor_a_b @ G2 @ A5 @ B5 )
& ( factor_a_b @ G2 @ B5 @ A5 ) ) ) ) ).
% associated_def
thf(fact_176_divides__antisym,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a] :
( ( factor3040189038382604065t_unit @ G @ A @ B )
=> ( ( factor3040189038382604065t_unit @ G @ B @ A )
=> ( associ6879500422977059064t_unit @ G @ A @ B ) ) ) ).
% divides_antisym
thf(fact_177_divides__antisym,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a] :
( ( factor_a_b @ G @ A @ B )
=> ( ( factor_a_b @ G @ B @ A )
=> ( associated_a_b @ G @ A @ B ) ) ) ).
% divides_antisym
thf(fact_178_factorial__monoid_Oprimeness__condition,axiom,
! [G: partia8223610829204095565t_unit] :
( ( factor2046533344642582127t_unit @ G )
=> ( primen965786292471834261t_unit @ G ) ) ).
% factorial_monoid.primeness_condition
thf(fact_179_factorial__monoid_Oprimeness__condition,axiom,
! [G: partia7680350978787392319xt_a_b] :
( ( factorial_monoid_a_b @ G )
=> ( primen900146951955507079id_a_b @ G ) ) ).
% factorial_monoid.primeness_condition
thf(fact_180_mset__subset__eqI,axiom,
! [A4: multiset_a,B3: multiset_a] :
( ! [A2: a] : ( ord_less_eq_nat @ ( count_a @ A4 @ A2 ) @ ( count_a @ B3 @ A2 ) )
=> ( subseteq_mset_a @ A4 @ B3 ) ) ).
% mset_subset_eqI
thf(fact_181_mset__subset__eqI,axiom,
! [A4: multiset_set_a,B3: multiset_set_a] :
( ! [A2: set_a] : ( ord_less_eq_nat @ ( count_set_a @ A4 @ A2 ) @ ( count_set_a @ B3 @ A2 ) )
=> ( subseteq_mset_set_a @ A4 @ B3 ) ) ).
% mset_subset_eqI
thf(fact_182_subseteq__mset__def,axiom,
( subseteq_mset_a
= ( ^ [A6: multiset_a,B6: multiset_a] :
! [A5: a] : ( ord_less_eq_nat @ ( count_a @ A6 @ A5 ) @ ( count_a @ B6 @ A5 ) ) ) ) ).
% subseteq_mset_def
thf(fact_183_subseteq__mset__def,axiom,
( subseteq_mset_set_a
= ( ^ [A6: multiset_set_a,B6: multiset_set_a] :
! [A5: set_a] : ( ord_less_eq_nat @ ( count_set_a @ A6 @ A5 ) @ ( count_set_a @ B6 @ A5 ) ) ) ) ).
% subseteq_mset_def
thf(fact_184_mset__subset__eq__count,axiom,
! [A4: multiset_a,B3: multiset_a,A: a] :
( ( subseteq_mset_a @ A4 @ B3 )
=> ( ord_less_eq_nat @ ( count_a @ A4 @ A ) @ ( count_a @ B3 @ A ) ) ) ).
% mset_subset_eq_count
thf(fact_185_mset__subset__eq__count,axiom,
! [A4: multiset_set_a,B3: multiset_set_a,A: set_a] :
( ( subseteq_mset_set_a @ A4 @ B3 )
=> ( ord_less_eq_nat @ ( count_set_a @ A4 @ A ) @ ( count_set_a @ B3 @ A ) ) ) ).
% mset_subset_eq_count
thf(fact_186_msubseteq__replicate__msetE,axiom,
! [A4: multiset_a,N: nat,A: a] :
( ( subseteq_mset_a @ A4 @ ( replicate_mset_a @ N @ A ) )
=> ~ ! [M3: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( A4
!= ( replicate_mset_a @ M3 @ A ) ) ) ) ).
% msubseteq_replicate_msetE
thf(fact_187_msubseteq__replicate__msetE,axiom,
! [A4: multiset_set_a,N: nat,A: set_a] :
( ( subseteq_mset_set_a @ A4 @ ( replicate_mset_set_a @ N @ A ) )
=> ~ ! [M3: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( A4
!= ( replicate_mset_set_a @ M3 @ A ) ) ) ) ).
% msubseteq_replicate_msetE
thf(fact_188_isgcd__def,axiom,
( isgcd_a_Product_unit
= ( ^ [G2: partia8223610829204095565t_unit,X3: a,A5: a,B5: a] :
( ( factor3040189038382604065t_unit @ G2 @ X3 @ A5 )
& ( factor3040189038382604065t_unit @ G2 @ X3 @ B5 )
& ! [Y5: a] :
( ( member_a @ Y5 @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( ( factor3040189038382604065t_unit @ G2 @ Y5 @ A5 )
& ( factor3040189038382604065t_unit @ G2 @ Y5 @ B5 ) )
=> ( factor3040189038382604065t_unit @ G2 @ Y5 @ X3 ) ) ) ) ) ) ).
% isgcd_def
thf(fact_189_isgcd__def,axiom,
( isgcd_a_b
= ( ^ [G2: partia7680350978787392319xt_a_b,X3: a,A5: a,B5: a] :
( ( factor_a_b @ G2 @ X3 @ A5 )
& ( factor_a_b @ G2 @ X3 @ B5 )
& ! [Y5: a] :
( ( member_a @ Y5 @ ( partia7484183841585581558xt_a_b @ G2 ) )
=> ( ( ( factor_a_b @ G2 @ Y5 @ A5 )
& ( factor_a_b @ G2 @ Y5 @ B5 ) )
=> ( factor_a_b @ G2 @ Y5 @ X3 ) ) ) ) ) ) ).
% isgcd_def
thf(fact_190_factorial__monoid_Ogcdof__exists,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a] :
( ( factor2046533344642582127t_unit @ G )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ? [C2: a] :
( ( member_a @ C2 @ ( partia6735698275553448452t_unit @ G ) )
& ( isgcd_a_Product_unit @ G @ C2 @ A @ B ) ) ) ) ) ).
% factorial_monoid.gcdof_exists
thf(fact_191_factorial__monoid_Ogcdof__exists,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a] :
( ( factorial_monoid_a_b @ G )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ? [C2: a] :
( ( member_a @ C2 @ ( partia7484183841585581558xt_a_b @ G ) )
& ( isgcd_a_b @ G @ C2 @ A @ B ) ) ) ) ) ).
% factorial_monoid.gcdof_exists
thf(fact_192_islcm__def,axiom,
( islcm_a_Product_unit
= ( ^ [G2: partia8223610829204095565t_unit,X3: a,A5: a,B5: a] :
( ( factor3040189038382604065t_unit @ G2 @ A5 @ X3 )
& ( factor3040189038382604065t_unit @ G2 @ B5 @ X3 )
& ! [Y5: a] :
( ( member_a @ Y5 @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( ( factor3040189038382604065t_unit @ G2 @ A5 @ Y5 )
& ( factor3040189038382604065t_unit @ G2 @ B5 @ Y5 ) )
=> ( factor3040189038382604065t_unit @ G2 @ X3 @ Y5 ) ) ) ) ) ) ).
% islcm_def
thf(fact_193_islcm__def,axiom,
( islcm_a_b
= ( ^ [G2: partia7680350978787392319xt_a_b,X3: a,A5: a,B5: a] :
( ( factor_a_b @ G2 @ A5 @ X3 )
& ( factor_a_b @ G2 @ B5 @ X3 )
& ! [Y5: a] :
( ( member_a @ Y5 @ ( partia7484183841585581558xt_a_b @ G2 ) )
=> ( ( ( factor_a_b @ G2 @ A5 @ Y5 )
& ( factor_a_b @ G2 @ B5 @ Y5 ) )
=> ( factor_a_b @ G2 @ X3 @ Y5 ) ) ) ) ) ) ).
% islcm_def
thf(fact_194_prime__divides,axiom,
! [A: a,B: a,P: a] :
( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( prime_a_b @ g @ P )
=> ( ( factor_a_b @ g @ P @ ( mult_a_b @ g @ A @ B ) )
=> ( ( factor_a_b @ g @ P @ A )
| ( factor_a_b @ g @ P @ B ) ) ) ) ) ) ).
% prime_divides
thf(fact_195_gcd__condition__monoid_Odivision__weak__lower__semilattice,axiom,
! [G: partia8223610829204095565t_unit] :
( ( gcd_co701944698663231555t_unit @ G )
=> ( weak_l5773807957815819224t_unit @ ( partia6383399360842404908t_unit @ ( partia6735698275553448452t_unit @ G ) @ ( eq_eq_5233546288009311946t_unit @ ( associ6879500422977059064t_unit @ G ) @ ( gorder6800608520584131120t_unit @ ( factor3040189038382604065t_unit @ G ) @ product_Unity ) ) ) ) ) ).
% gcd_condition_monoid.division_weak_lower_semilattice
thf(fact_196_gcd__condition__monoid_Odivision__weak__lower__semilattice,axiom,
! [G: partia7680350978787392319xt_a_b] :
( ( gcd_co4024338386749376309id_a_b @ G )
=> ( weak_l5773807957815819224t_unit @ ( partia6383399360842404908t_unit @ ( partia7484183841585581558xt_a_b @ G ) @ ( eq_eq_5233546288009311946t_unit @ ( associated_a_b @ G ) @ ( gorder6800608520584131120t_unit @ ( factor_a_b @ G ) @ product_Unity ) ) ) ) ) ).
% gcd_condition_monoid.division_weak_lower_semilattice
thf(fact_197_gcd__mult,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ C @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( associated_a_b @ g @ ( mult_a_b @ g @ C @ ( somegcd_a_b @ g @ A @ B ) ) @ ( somegcd_a_b @ g @ ( mult_a_b @ g @ C @ A ) @ ( mult_a_b @ g @ C @ B ) ) ) ) ) ) ).
% gcd_mult
thf(fact_198_division__weak__partial__order,axiom,
weak_p4024844561679123766t_unit @ ( partia6383399360842404908t_unit @ ( partia7484183841585581558xt_a_b @ g ) @ ( eq_eq_5233546288009311946t_unit @ ( associated_a_b @ g ) @ ( gorder6800608520584131120t_unit @ ( factor_a_b @ g ) @ product_Unity ) ) ) ).
% division_weak_partial_order
thf(fact_199_mset__fmsetEx,axiom,
! [Cs: multiset_set_a,P3: a > $o] :
( ! [X5: set_a] :
( ( member_set_a @ X5 @ ( set_mset_set_a @ Cs ) )
=> ? [X6: a] :
( ( P3 @ X6 )
& ( X5
= ( eq_clo186784744570230005t_unit @ ( partia6383399360842404908t_unit @ ( partia7484183841585581558xt_a_b @ g ) @ ( eq_eq_5233546288009311946t_unit @ ( associated_a_b @ g ) @ ( gorder6800608520584131120t_unit @ ( factor_a_b @ g ) @ product_Unity ) ) ) @ ( insert_a @ X6 @ bot_bot_set_a ) ) ) ) )
=> ? [Cs2: list_a] :
( ! [X6: a] :
( ( member_a @ X6 @ ( set_a2 @ Cs2 ) )
=> ( P3 @ X6 ) )
& ( ( fmset_a_b @ g @ Cs2 )
= Cs ) ) ) ).
% mset_fmsetEx
thf(fact_200_gcd__condition__monoid__axioms,axiom,
gcd_co4024338386749376309id_a_b @ g ).
% gcd_condition_monoid_axioms
thf(fact_201_comm__monoid_Oassocs__eqD,axiom,
! [G: partia8223610829204095565t_unit,B: a,A: a] :
( ( comm_m7681468956318391052t_unit @ G )
=> ( ( ( eq_clo186784744570230005t_unit @ ( partia6383399360842404908t_unit @ ( partia6735698275553448452t_unit @ G ) @ ( eq_eq_5233546288009311946t_unit @ ( associ6879500422977059064t_unit @ G ) @ ( gorder6800608520584131120t_unit @ ( factor3040189038382604065t_unit @ G ) @ product_Unity ) ) ) @ ( insert_a @ B @ bot_bot_set_a ) )
= ( eq_clo186784744570230005t_unit @ ( partia6383399360842404908t_unit @ ( partia6735698275553448452t_unit @ G ) @ ( eq_eq_5233546288009311946t_unit @ ( associ6879500422977059064t_unit @ G ) @ ( gorder6800608520584131120t_unit @ ( factor3040189038382604065t_unit @ G ) @ product_Unity ) ) ) @ ( insert_a @ A @ bot_bot_set_a ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( associ6879500422977059064t_unit @ G @ A @ B ) ) ) ) ) ).
% comm_monoid.assocs_eqD
thf(fact_202_comm__monoid_Oassocs__eqD,axiom,
! [G: partia7680350978787392319xt_a_b,B: a,A: a] :
( ( comm_monoid_a_b @ G )
=> ( ( ( eq_clo186784744570230005t_unit @ ( partia6383399360842404908t_unit @ ( partia7484183841585581558xt_a_b @ G ) @ ( eq_eq_5233546288009311946t_unit @ ( associated_a_b @ G ) @ ( gorder6800608520584131120t_unit @ ( factor_a_b @ G ) @ product_Unity ) ) ) @ ( insert_a @ B @ bot_bot_set_a ) )
= ( eq_clo186784744570230005t_unit @ ( partia6383399360842404908t_unit @ ( partia7484183841585581558xt_a_b @ G ) @ ( eq_eq_5233546288009311946t_unit @ ( associated_a_b @ G ) @ ( gorder6800608520584131120t_unit @ ( factor_a_b @ G ) @ product_Unity ) ) ) @ ( insert_a @ A @ bot_bot_set_a ) ) )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( associated_a_b @ G @ A @ B ) ) ) ) ) ).
% comm_monoid.assocs_eqD
thf(fact_203_comm__monoid_Oassocs__assoc,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a] :
( ( comm_m7681468956318391052t_unit @ G )
=> ( ( member_a @ A @ ( eq_clo186784744570230005t_unit @ ( partia6383399360842404908t_unit @ ( partia6735698275553448452t_unit @ G ) @ ( eq_eq_5233546288009311946t_unit @ ( associ6879500422977059064t_unit @ G ) @ ( gorder6800608520584131120t_unit @ ( factor3040189038382604065t_unit @ G ) @ product_Unity ) ) ) @ ( insert_a @ B @ bot_bot_set_a ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( associ6879500422977059064t_unit @ G @ A @ B ) ) ) ) ).
% comm_monoid.assocs_assoc
thf(fact_204_comm__monoid_Oassocs__assoc,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a] :
( ( comm_monoid_a_b @ G )
=> ( ( member_a @ A @ ( eq_clo186784744570230005t_unit @ ( partia6383399360842404908t_unit @ ( partia7484183841585581558xt_a_b @ G ) @ ( eq_eq_5233546288009311946t_unit @ ( associated_a_b @ G ) @ ( gorder6800608520584131120t_unit @ ( factor_a_b @ G ) @ product_Unity ) ) ) @ ( insert_a @ B @ bot_bot_set_a ) ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( associated_a_b @ G @ A @ B ) ) ) ) ).
% comm_monoid.assocs_assoc
thf(fact_205_somegcd__meet,axiom,
! [A: a,G: partia8223610829204095565t_unit,B: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( somegc8962790057355718400t_unit @ G @ A @ B )
= ( meet_a_Product_unit @ ( partia6383399360842404908t_unit @ ( partia6735698275553448452t_unit @ G ) @ ( eq_eq_5233546288009311946t_unit @ ( associ6879500422977059064t_unit @ G ) @ ( gorder6800608520584131120t_unit @ ( factor3040189038382604065t_unit @ G ) @ product_Unity ) ) ) @ A @ B ) ) ) ) ).
% somegcd_meet
thf(fact_206_somegcd__meet,axiom,
! [A: a,G: partia7680350978787392319xt_a_b,B: a] :
( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( somegcd_a_b @ G @ A @ B )
= ( meet_a_Product_unit @ ( partia6383399360842404908t_unit @ ( partia7484183841585581558xt_a_b @ G ) @ ( eq_eq_5233546288009311946t_unit @ ( associated_a_b @ G ) @ ( gorder6800608520584131120t_unit @ ( factor_a_b @ G ) @ product_Unity ) ) ) @ A @ B ) ) ) ) ).
% somegcd_meet
thf(fact_207_monoid_Oassocs__self,axiom,
! [G: partia8223610829204095565t_unit,X: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( member_a @ X @ ( eq_clo186784744570230005t_unit @ ( partia6383399360842404908t_unit @ ( partia6735698275553448452t_unit @ G ) @ ( eq_eq_5233546288009311946t_unit @ ( associ6879500422977059064t_unit @ G ) @ ( gorder6800608520584131120t_unit @ ( factor3040189038382604065t_unit @ G ) @ product_Unity ) ) ) @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% monoid.assocs_self
thf(fact_208_monoid_Oassocs__self,axiom,
! [G: partia7680350978787392319xt_a_b,X: a] :
( ( monoid_a_b @ G )
=> ( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( member_a @ X @ ( eq_clo186784744570230005t_unit @ ( partia6383399360842404908t_unit @ ( partia7484183841585581558xt_a_b @ G ) @ ( eq_eq_5233546288009311946t_unit @ ( associated_a_b @ G ) @ ( gorder6800608520584131120t_unit @ ( factor_a_b @ G ) @ product_Unity ) ) ) @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% monoid.assocs_self
thf(fact_209_monoid__axioms,axiom,
monoid_a_b @ g ).
% monoid_axioms
thf(fact_210_comm__monoid__axioms,axiom,
comm_monoid_a_b @ g ).
% comm_monoid_axioms
thf(fact_211_monoid__comm__monoidI,axiom,
( ! [X4: a,Y3: a] :
( ( member_a @ X4 @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ Y3 @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( mult_a_b @ g @ X4 @ Y3 )
= ( mult_a_b @ g @ Y3 @ X4 ) ) ) )
=> ( comm_monoid_a_b @ g ) ) ).
% monoid_comm_monoidI
thf(fact_212_r__cancel,axiom,
! [A: a,C: a,B: a] :
( ( ( mult_a_b @ g @ A @ C )
= ( mult_a_b @ g @ B @ C ) )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ C @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( A = B ) ) ) ) ) ).
% r_cancel
thf(fact_213_m__lcomm,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ Y @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ Z @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( mult_a_b @ g @ X @ ( mult_a_b @ g @ Y @ Z ) )
= ( mult_a_b @ g @ Y @ ( mult_a_b @ g @ X @ Z ) ) ) ) ) ) ).
% m_lcomm
thf(fact_214_m__comm,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ Y @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( mult_a_b @ g @ X @ Y )
= ( mult_a_b @ g @ Y @ X ) ) ) ) ).
% m_comm
thf(fact_215_m__assoc,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ Y @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ Z @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( mult_a_b @ g @ ( mult_a_b @ g @ X @ Y ) @ Z )
= ( mult_a_b @ g @ X @ ( mult_a_b @ g @ Y @ Z ) ) ) ) ) ) ).
% m_assoc
thf(fact_216_l__cancel,axiom,
! [C: a,A: a,B: a] :
( ( ( mult_a_b @ g @ C @ A )
= ( mult_a_b @ g @ C @ B ) )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ C @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( A = B ) ) ) ) ) ).
% l_cancel
thf(fact_217_mult__cong__r,axiom,
! [B: a,B7: a,A: a] :
( ( associated_a_b @ g @ B @ B7 )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B7 @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( associated_a_b @ g @ ( mult_a_b @ g @ A @ B ) @ ( mult_a_b @ g @ A @ B7 ) ) ) ) ) ) ).
% mult_cong_r
thf(fact_218_mult__cong__l,axiom,
! [A: a,A3: a,B: a] :
( ( associated_a_b @ g @ A @ A3 )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ A3 @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( associated_a_b @ g @ ( mult_a_b @ g @ A @ B ) @ ( mult_a_b @ g @ A3 @ B ) ) ) ) ) ) ).
% mult_cong_l
thf(fact_219_assoc__r__cancel,axiom,
! [A: a,B: a,A3: a] :
( ( associated_a_b @ g @ ( mult_a_b @ g @ A @ B ) @ ( mult_a_b @ g @ A3 @ B ) )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ A3 @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( associated_a_b @ g @ A @ A3 ) ) ) ) ) ).
% assoc_r_cancel
thf(fact_220_assoc__l__cancel,axiom,
! [A: a,B: a,B7: a] :
( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B7 @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( associated_a_b @ g @ ( mult_a_b @ g @ A @ B ) @ ( mult_a_b @ g @ A @ B7 ) )
=> ( associated_a_b @ g @ B @ B7 ) ) ) ) ) ).
% assoc_l_cancel
thf(fact_221_divides__prod__r,axiom,
! [A: a,B: a,C: a] :
( ( factor_a_b @ g @ A @ B )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ C @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( factor_a_b @ g @ A @ ( mult_a_b @ g @ B @ C ) ) ) ) ) ).
% divides_prod_r
thf(fact_222_divides__prod__l,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ C @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( factor_a_b @ g @ A @ B )
=> ( factor_a_b @ g @ A @ ( mult_a_b @ g @ C @ B ) ) ) ) ) ) ).
% divides_prod_l
thf(fact_223_dividesI_H,axiom,
! [B: a,G: partia8223610829204095565t_unit,A: a,C: a] :
( ( B
= ( mult_a_Product_unit @ G @ A @ C ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ G ) )
=> ( factor3040189038382604065t_unit @ G @ A @ B ) ) ) ).
% dividesI'
thf(fact_224_dividesI_H,axiom,
! [B: a,G: partia7680350978787392319xt_a_b,A: a,C: a] :
( ( B
= ( mult_a_b @ G @ A @ C ) )
=> ( ( member_a @ C @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( factor_a_b @ G @ A @ B ) ) ) ).
% dividesI'
thf(fact_225_m__closed,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ Y @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( member_a @ ( mult_a_b @ g @ X @ Y ) @ ( partia7484183841585581558xt_a_b @ g ) ) ) ) ).
% m_closed
thf(fact_226_divides__mult__rI,axiom,
! [A: a,B: a,C: a] :
( ( factor_a_b @ g @ A @ B )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ C @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( factor_a_b @ g @ ( mult_a_b @ g @ A @ C ) @ ( mult_a_b @ g @ B @ C ) ) ) ) ) ) ).
% divides_mult_rI
thf(fact_227_divides__mult__r,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ C @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( factor_a_b @ g @ ( mult_a_b @ g @ A @ C ) @ ( mult_a_b @ g @ B @ C ) )
= ( factor_a_b @ g @ A @ B ) ) ) ) ) ).
% divides_mult_r
thf(fact_228_divides__mult__lI,axiom,
! [A: a,B: a,C: a] :
( ( factor_a_b @ g @ A @ B )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ C @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( factor_a_b @ g @ ( mult_a_b @ g @ C @ A ) @ ( mult_a_b @ g @ C @ B ) ) ) ) ) ).
% divides_mult_lI
thf(fact_229_divides__mult__l,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ C @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( factor_a_b @ g @ ( mult_a_b @ g @ C @ A ) @ ( mult_a_b @ g @ C @ B ) )
= ( factor_a_b @ g @ A @ B ) ) ) ) ) ).
% divides_mult_l
thf(fact_230_monoid_Odivides__mult__lI,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a,C: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( factor3040189038382604065t_unit @ G @ A @ B )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ G ) )
=> ( factor3040189038382604065t_unit @ G @ ( mult_a_Product_unit @ G @ C @ A ) @ ( mult_a_Product_unit @ G @ C @ B ) ) ) ) ) ) ).
% monoid.divides_mult_lI
thf(fact_231_monoid_Odivides__mult__lI,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a,C: a] :
( ( monoid_a_b @ G )
=> ( ( factor_a_b @ G @ A @ B )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( factor_a_b @ G @ ( mult_a_b @ G @ C @ A ) @ ( mult_a_b @ G @ C @ B ) ) ) ) ) ) ).
% monoid.divides_mult_lI
thf(fact_232_monoid_Odivides__prod__r,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a,C: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( factor3040189038382604065t_unit @ G @ A @ B )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ G ) )
=> ( factor3040189038382604065t_unit @ G @ A @ ( mult_a_Product_unit @ G @ B @ C ) ) ) ) ) ) ).
% monoid.divides_prod_r
thf(fact_233_monoid_Odivides__prod__r,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a,C: a] :
( ( monoid_a_b @ G )
=> ( ( factor_a_b @ G @ A @ B )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( factor_a_b @ G @ A @ ( mult_a_b @ G @ B @ C ) ) ) ) ) ) ).
% monoid.divides_prod_r
thf(fact_234_monoid_Omult__cong__r,axiom,
! [G: partia8223610829204095565t_unit,B: a,B7: a,A: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( associ6879500422977059064t_unit @ G @ B @ B7 )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B7 @ ( partia6735698275553448452t_unit @ G ) )
=> ( associ6879500422977059064t_unit @ G @ ( mult_a_Product_unit @ G @ A @ B ) @ ( mult_a_Product_unit @ G @ A @ B7 ) ) ) ) ) ) ) ).
% monoid.mult_cong_r
thf(fact_235_monoid_Omult__cong__r,axiom,
! [G: partia7680350978787392319xt_a_b,B: a,B7: a,A: a] :
( ( monoid_a_b @ G )
=> ( ( associated_a_b @ G @ B @ B7 )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B7 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( associated_a_b @ G @ ( mult_a_b @ G @ A @ B ) @ ( mult_a_b @ G @ A @ B7 ) ) ) ) ) ) ) ).
% monoid.mult_cong_r
thf(fact_236_comm__monoid_Odivides__mult__rI,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a,C: a] :
( ( comm_m7681468956318391052t_unit @ G )
=> ( ( factor3040189038382604065t_unit @ G @ A @ B )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ G ) )
=> ( factor3040189038382604065t_unit @ G @ ( mult_a_Product_unit @ G @ A @ C ) @ ( mult_a_Product_unit @ G @ B @ C ) ) ) ) ) ) ) ).
% comm_monoid.divides_mult_rI
thf(fact_237_comm__monoid_Odivides__mult__rI,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a,C: a] :
( ( comm_monoid_a_b @ G )
=> ( ( factor_a_b @ G @ A @ B )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( factor_a_b @ G @ ( mult_a_b @ G @ A @ C ) @ ( mult_a_b @ G @ B @ C ) ) ) ) ) ) ) ).
% comm_monoid.divides_mult_rI
thf(fact_238_comm__monoid_Odivides__prod__l,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a,C: a] :
( ( comm_m7681468956318391052t_unit @ G )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( factor3040189038382604065t_unit @ G @ A @ B )
=> ( factor3040189038382604065t_unit @ G @ A @ ( mult_a_Product_unit @ G @ C @ B ) ) ) ) ) ) ) ).
% comm_monoid.divides_prod_l
thf(fact_239_comm__monoid_Odivides__prod__l,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a,C: a] :
( ( comm_monoid_a_b @ G )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( factor_a_b @ G @ A @ B )
=> ( factor_a_b @ G @ A @ ( mult_a_b @ G @ C @ B ) ) ) ) ) ) ) ).
% comm_monoid.divides_prod_l
thf(fact_240_comm__monoid_Omult__cong__l,axiom,
! [G: partia8223610829204095565t_unit,A: a,A3: a,B: a] :
( ( comm_m7681468956318391052t_unit @ G )
=> ( ( associ6879500422977059064t_unit @ G @ A @ A3 )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( associ6879500422977059064t_unit @ G @ ( mult_a_Product_unit @ G @ A @ B ) @ ( mult_a_Product_unit @ G @ A3 @ B ) ) ) ) ) ) ) ).
% comm_monoid.mult_cong_l
thf(fact_241_comm__monoid_Omult__cong__l,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,A3: a,B: a] :
( ( comm_monoid_a_b @ G )
=> ( ( associated_a_b @ G @ A @ A3 )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ A3 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( associated_a_b @ G @ ( mult_a_b @ G @ A @ B ) @ ( mult_a_b @ G @ A3 @ B ) ) ) ) ) ) ) ).
% comm_monoid.mult_cong_l
thf(fact_242_monoid_Oassociated__sym,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( associ6879500422977059064t_unit @ G @ A @ B )
=> ( associ6879500422977059064t_unit @ G @ B @ A ) ) ) ).
% monoid.associated_sym
thf(fact_243_monoid_Oassociated__sym,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a] :
( ( monoid_a_b @ G )
=> ( ( associated_a_b @ G @ A @ B )
=> ( associated_a_b @ G @ B @ A ) ) ) ).
% monoid.associated_sym
thf(fact_244_weak__partial__order_Oaxioms_I1_J,axiom,
! [L: partia6638023223214267844t_unit] :
( ( weak_p4024844561679123766t_unit @ L )
=> ( equiva8593399568135932475t_unit @ L ) ) ).
% weak_partial_order.axioms(1)
thf(fact_245_gcd__condition__monoid_Ogcd__mult,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a,C: a] :
( ( gcd_co701944698663231555t_unit @ G )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ G ) )
=> ( associ6879500422977059064t_unit @ G @ ( mult_a_Product_unit @ G @ C @ ( somegc8962790057355718400t_unit @ G @ A @ B ) ) @ ( somegc8962790057355718400t_unit @ G @ ( mult_a_Product_unit @ G @ C @ A ) @ ( mult_a_Product_unit @ G @ C @ B ) ) ) ) ) ) ) ).
% gcd_condition_monoid.gcd_mult
thf(fact_246_gcd__condition__monoid_Ogcd__mult,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a,C: a] :
( ( gcd_co4024338386749376309id_a_b @ G )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( associated_a_b @ G @ ( mult_a_b @ G @ C @ ( somegcd_a_b @ G @ A @ B ) ) @ ( somegcd_a_b @ G @ ( mult_a_b @ G @ C @ A ) @ ( mult_a_b @ G @ C @ B ) ) ) ) ) ) ) ).
% gcd_condition_monoid.gcd_mult
thf(fact_247_factorial__monoid_Ogcd__condition,axiom,
! [G: partia8223610829204095565t_unit] :
( ( factor2046533344642582127t_unit @ G )
=> ( gcd_co701944698663231555t_unit @ G ) ) ).
% factorial_monoid.gcd_condition
thf(fact_248_factorial__monoid_Ogcd__condition,axiom,
! [G: partia7680350978787392319xt_a_b] :
( ( factorial_monoid_a_b @ G )
=> ( gcd_co4024338386749376309id_a_b @ G ) ) ).
% factorial_monoid.gcd_condition
thf(fact_249_monoid_Odivides__trans,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a,C: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( factor3040189038382604065t_unit @ G @ A @ B )
=> ( ( factor3040189038382604065t_unit @ G @ B @ C )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( factor3040189038382604065t_unit @ G @ A @ C ) ) ) ) ) ).
% monoid.divides_trans
thf(fact_250_monoid_Odivides__trans,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a,C: a] :
( ( monoid_a_b @ G )
=> ( ( factor_a_b @ G @ A @ B )
=> ( ( factor_a_b @ G @ B @ C )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( factor_a_b @ G @ A @ C ) ) ) ) ) ).
% monoid.divides_trans
thf(fact_251_monoid_Odivides__refl,axiom,
! [G: partia8223610829204095565t_unit,A: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( factor3040189038382604065t_unit @ G @ A @ A ) ) ) ).
% monoid.divides_refl
thf(fact_252_monoid_Odivides__refl,axiom,
! [G: partia7680350978787392319xt_a_b,A: a] :
( ( monoid_a_b @ G )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( factor_a_b @ G @ A @ A ) ) ) ).
% monoid.divides_refl
thf(fact_253_monoid_Oassociated__trans,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a,C: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( associ6879500422977059064t_unit @ G @ A @ B )
=> ( ( associ6879500422977059064t_unit @ G @ B @ C )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ G ) )
=> ( associ6879500422977059064t_unit @ G @ A @ C ) ) ) ) ) ) ).
% monoid.associated_trans
thf(fact_254_monoid_Oassociated__trans,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a,C: a] :
( ( monoid_a_b @ G )
=> ( ( associated_a_b @ G @ A @ B )
=> ( ( associated_a_b @ G @ B @ C )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( associated_a_b @ G @ A @ C ) ) ) ) ) ) ).
% monoid.associated_trans
thf(fact_255_monoid_Oassociated__refl,axiom,
! [G: partia8223610829204095565t_unit,A: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( associ6879500422977059064t_unit @ G @ A @ A ) ) ) ).
% monoid.associated_refl
thf(fact_256_monoid_Oassociated__refl,axiom,
! [G: partia7680350978787392319xt_a_b,A: a] :
( ( monoid_a_b @ G )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( associated_a_b @ G @ A @ A ) ) ) ).
% monoid.associated_refl
thf(fact_257_monoid_Oassoc__subst,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a,F: a > a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( associ6879500422977059064t_unit @ G @ A @ B )
=> ( ! [A2: a,B2: a] :
( ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ G ) )
& ( member_a @ B2 @ ( partia6735698275553448452t_unit @ G ) )
& ( associ6879500422977059064t_unit @ G @ A2 @ B2 ) )
=> ( ( member_a @ ( F @ A2 ) @ ( partia6735698275553448452t_unit @ G ) )
& ( member_a @ ( F @ B2 ) @ ( partia6735698275553448452t_unit @ G ) )
& ( associ6879500422977059064t_unit @ G @ ( F @ A2 ) @ ( F @ B2 ) ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( associ6879500422977059064t_unit @ G @ ( F @ A ) @ ( F @ B ) ) ) ) ) ) ) ).
% monoid.assoc_subst
thf(fact_258_monoid_Oassoc__subst,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a,F: a > a] :
( ( monoid_a_b @ G )
=> ( ( associated_a_b @ G @ A @ B )
=> ( ! [A2: a,B2: a] :
( ( ( member_a @ A2 @ ( partia7484183841585581558xt_a_b @ G ) )
& ( member_a @ B2 @ ( partia7484183841585581558xt_a_b @ G ) )
& ( associated_a_b @ G @ A2 @ B2 ) )
=> ( ( member_a @ ( F @ A2 ) @ ( partia7484183841585581558xt_a_b @ G ) )
& ( member_a @ ( F @ B2 ) @ ( partia7484183841585581558xt_a_b @ G ) )
& ( associated_a_b @ G @ ( F @ A2 ) @ ( F @ B2 ) ) ) )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( associated_a_b @ G @ ( F @ A ) @ ( F @ B ) ) ) ) ) ) ) ).
% monoid.assoc_subst
thf(fact_259_factor__def,axiom,
( factor3040189038382604065t_unit
= ( ^ [G2: partia8223610829204095565t_unit,A5: a,B5: a] :
? [X3: a] :
( ( member_a @ X3 @ ( partia6735698275553448452t_unit @ G2 ) )
& ( B5
= ( mult_a_Product_unit @ G2 @ A5 @ X3 ) ) ) ) ) ).
% factor_def
thf(fact_260_factor__def,axiom,
( factor_a_b
= ( ^ [G2: partia7680350978787392319xt_a_b,A5: a,B5: a] :
? [X3: a] :
( ( member_a @ X3 @ ( partia7484183841585581558xt_a_b @ G2 ) )
& ( B5
= ( mult_a_b @ G2 @ A5 @ X3 ) ) ) ) ) ).
% factor_def
thf(fact_261_dividesI,axiom,
! [C: a,G: partia8223610829204095565t_unit,B: a,A: a] :
( ( member_a @ C @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( B
= ( mult_a_Product_unit @ G @ A @ C ) )
=> ( factor3040189038382604065t_unit @ G @ A @ B ) ) ) ).
% dividesI
thf(fact_262_dividesI,axiom,
! [C: a,G: partia7680350978787392319xt_a_b,B: a,A: a] :
( ( member_a @ C @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( B
= ( mult_a_b @ G @ A @ C ) )
=> ( factor_a_b @ G @ A @ B ) ) ) ).
% dividesI
thf(fact_263_dividesE,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a] :
( ( factor3040189038382604065t_unit @ G @ A @ B )
=> ~ ! [C2: a] :
( ( B
= ( mult_a_Product_unit @ G @ A @ C2 ) )
=> ~ ( member_a @ C2 @ ( partia6735698275553448452t_unit @ G ) ) ) ) ).
% dividesE
thf(fact_264_dividesE,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a] :
( ( factor_a_b @ G @ A @ B )
=> ~ ! [C2: a] :
( ( B
= ( mult_a_b @ G @ A @ C2 ) )
=> ~ ( member_a @ C2 @ ( partia7484183841585581558xt_a_b @ G ) ) ) ) ).
% dividesE
thf(fact_265_dividesD,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a] :
( ( factor3040189038382604065t_unit @ G @ A @ B )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia6735698275553448452t_unit @ G ) )
& ( B
= ( mult_a_Product_unit @ G @ A @ X4 ) ) ) ) ).
% dividesD
thf(fact_266_dividesD,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a] :
( ( factor_a_b @ G @ A @ B )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia7484183841585581558xt_a_b @ G ) )
& ( B
= ( mult_a_b @ G @ A @ X4 ) ) ) ) ).
% dividesD
thf(fact_267_gcd__condition__monoid_Oprimeness__condition,axiom,
! [G: partia7680350978787392319xt_a_b] :
( ( gcd_co4024338386749376309id_a_b @ G )
=> ( primen900146951955507079id_a_b @ G ) ) ).
% gcd_condition_monoid.primeness_condition
thf(fact_268_monoid_Odivides__cong__r,axiom,
! [G: partia8223610829204095565t_unit,X: a,Y: a,Y2: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( factor3040189038382604065t_unit @ G @ X @ Y )
=> ( ( associ6879500422977059064t_unit @ G @ Y @ Y2 )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( factor3040189038382604065t_unit @ G @ X @ Y2 ) ) ) ) ) ).
% monoid.divides_cong_r
thf(fact_269_monoid_Odivides__cong__r,axiom,
! [G: partia7680350978787392319xt_a_b,X: a,Y: a,Y2: a] :
( ( monoid_a_b @ G )
=> ( ( factor_a_b @ G @ X @ Y )
=> ( ( associated_a_b @ G @ Y @ Y2 )
=> ( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( factor_a_b @ G @ X @ Y2 ) ) ) ) ) ).
% monoid.divides_cong_r
thf(fact_270_monoid_Odivides__cong__l,axiom,
! [G: partia8223610829204095565t_unit,X: a,X2: a,Y: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( associ6879500422977059064t_unit @ G @ X @ X2 )
=> ( ( factor3040189038382604065t_unit @ G @ X2 @ Y )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( factor3040189038382604065t_unit @ G @ X @ Y ) ) ) ) ) ).
% monoid.divides_cong_l
thf(fact_271_monoid_Odivides__cong__l,axiom,
! [G: partia7680350978787392319xt_a_b,X: a,X2: a,Y: a] :
( ( monoid_a_b @ G )
=> ( ( associated_a_b @ G @ X @ X2 )
=> ( ( factor_a_b @ G @ X2 @ Y )
=> ( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( factor_a_b @ G @ X @ Y ) ) ) ) ) ).
% monoid.divides_cong_l
thf(fact_272_gcd__condition__monoid_Ogcd__closed,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a] :
( ( gcd_co701944698663231555t_unit @ G )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( member_a @ ( somegc8962790057355718400t_unit @ G @ A @ B ) @ ( partia6735698275553448452t_unit @ G ) ) ) ) ) ).
% gcd_condition_monoid.gcd_closed
thf(fact_273_gcd__condition__monoid_Ogcd__closed,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a] :
( ( gcd_co4024338386749376309id_a_b @ G )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( member_a @ ( somegcd_a_b @ G @ A @ B ) @ ( partia7484183841585581558xt_a_b @ G ) ) ) ) ) ).
% gcd_condition_monoid.gcd_closed
thf(fact_274_gcd__condition__monoid_Ogcd__exists,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a] :
( ( gcd_co701944698663231555t_unit @ G )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( member_a @ ( somegc8962790057355718400t_unit @ G @ A @ B ) @ ( partia6735698275553448452t_unit @ G ) ) ) ) ) ).
% gcd_condition_monoid.gcd_exists
thf(fact_275_gcd__condition__monoid_Ogcd__exists,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a] :
( ( gcd_co4024338386749376309id_a_b @ G )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( member_a @ ( somegcd_a_b @ G @ A @ B ) @ ( partia7484183841585581558xt_a_b @ G ) ) ) ) ) ).
% gcd_condition_monoid.gcd_exists
thf(fact_276_gcd__condition__monoid_Ogcdof__exists,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a] :
( ( gcd_co701944698663231555t_unit @ G )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ? [C2: a] :
( ( member_a @ C2 @ ( partia6735698275553448452t_unit @ G ) )
& ( isgcd_a_Product_unit @ G @ C2 @ A @ B ) ) ) ) ) ).
% gcd_condition_monoid.gcdof_exists
thf(fact_277_gcd__condition__monoid_Ogcdof__exists,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a] :
( ( gcd_co4024338386749376309id_a_b @ G )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ? [C2: a] :
( ( member_a @ C2 @ ( partia7484183841585581558xt_a_b @ G ) )
& ( isgcd_a_b @ G @ C2 @ A @ B ) ) ) ) ) ).
% gcd_condition_monoid.gcdof_exists
thf(fact_278_monoid_Oisgcd__divides__r,axiom,
! [G: partia8223610829204095565t_unit,B: a,A: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( factor3040189038382604065t_unit @ G @ B @ A )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( isgcd_a_Product_unit @ G @ B @ A @ B ) ) ) ) ) ).
% monoid.isgcd_divides_r
thf(fact_279_monoid_Oisgcd__divides__r,axiom,
! [G: partia7680350978787392319xt_a_b,B: a,A: a] :
( ( monoid_a_b @ G )
=> ( ( factor_a_b @ G @ B @ A )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( isgcd_a_b @ G @ B @ A @ B ) ) ) ) ) ).
% monoid.isgcd_divides_r
thf(fact_280_monoid_Oisgcd__divides__l,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( factor3040189038382604065t_unit @ G @ A @ B )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( isgcd_a_Product_unit @ G @ A @ A @ B ) ) ) ) ) ).
% monoid.isgcd_divides_l
thf(fact_281_monoid_Oisgcd__divides__l,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a] :
( ( monoid_a_b @ G )
=> ( ( factor_a_b @ G @ A @ B )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( isgcd_a_b @ G @ A @ A @ B ) ) ) ) ) ).
% monoid.isgcd_divides_l
thf(fact_282_monoid_Odivision__weak__partial__order,axiom,
! [G: partia8223610829204095565t_unit] :
( ( monoid2746444814143937472t_unit @ G )
=> ( weak_p4024844561679123766t_unit @ ( partia6383399360842404908t_unit @ ( partia6735698275553448452t_unit @ G ) @ ( eq_eq_5233546288009311946t_unit @ ( associ6879500422977059064t_unit @ G ) @ ( gorder6800608520584131120t_unit @ ( factor3040189038382604065t_unit @ G ) @ product_Unity ) ) ) ) ) ).
% monoid.division_weak_partial_order
thf(fact_283_monoid_Odivision__weak__partial__order,axiom,
! [G: partia7680350978787392319xt_a_b] :
( ( monoid_a_b @ G )
=> ( weak_p4024844561679123766t_unit @ ( partia6383399360842404908t_unit @ ( partia7484183841585581558xt_a_b @ G ) @ ( eq_eq_5233546288009311946t_unit @ ( associated_a_b @ G ) @ ( gorder6800608520584131120t_unit @ ( factor_a_b @ G ) @ product_Unity ) ) ) ) ) ).
% monoid.division_weak_partial_order
thf(fact_284_factorial__condition__two,axiom,
! [G: partia8223610829204095565t_unit] :
( ( ( diviso6259607970152342594t_unit @ G )
& ( gcd_co701944698663231555t_unit @ G ) )
= ( factor2046533344642582127t_unit @ G ) ) ).
% factorial_condition_two
thf(fact_285_factorial__condition__two,axiom,
! [G: partia7680350978787392319xt_a_b] :
( ( ( diviso5244014645414798900id_a_b @ G )
& ( gcd_co4024338386749376309id_a_b @ G ) )
= ( factorial_monoid_a_b @ G ) ) ).
% factorial_condition_two
thf(fact_286_monoid_Omset__fmsetEx,axiom,
! [G: partia8223610829204095565t_unit,Cs: multiset_set_a,P3: a > $o] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ! [X5: set_a] :
( ( member_set_a @ X5 @ ( set_mset_set_a @ Cs ) )
=> ? [X6: a] :
( ( P3 @ X6 )
& ( X5
= ( eq_clo186784744570230005t_unit @ ( partia6383399360842404908t_unit @ ( partia6735698275553448452t_unit @ G ) @ ( eq_eq_5233546288009311946t_unit @ ( associ6879500422977059064t_unit @ G ) @ ( gorder6800608520584131120t_unit @ ( factor3040189038382604065t_unit @ G ) @ product_Unity ) ) ) @ ( insert_a @ X6 @ bot_bot_set_a ) ) ) ) )
=> ? [Cs2: list_a] :
( ! [X6: a] :
( ( member_a @ X6 @ ( set_a2 @ Cs2 ) )
=> ( P3 @ X6 ) )
& ( ( fmset_a_Product_unit @ G @ Cs2 )
= Cs ) ) ) ) ).
% monoid.mset_fmsetEx
thf(fact_287_monoid_Omset__fmsetEx,axiom,
! [G: partia7680350978787392319xt_a_b,Cs: multiset_set_a,P3: a > $o] :
( ( monoid_a_b @ G )
=> ( ! [X5: set_a] :
( ( member_set_a @ X5 @ ( set_mset_set_a @ Cs ) )
=> ? [X6: a] :
( ( P3 @ X6 )
& ( X5
= ( eq_clo186784744570230005t_unit @ ( partia6383399360842404908t_unit @ ( partia7484183841585581558xt_a_b @ G ) @ ( eq_eq_5233546288009311946t_unit @ ( associated_a_b @ G ) @ ( gorder6800608520584131120t_unit @ ( factor_a_b @ G ) @ product_Unity ) ) ) @ ( insert_a @ X6 @ bot_bot_set_a ) ) ) ) )
=> ? [Cs2: list_a] :
( ! [X6: a] :
( ( member_a @ X6 @ ( set_a2 @ Cs2 ) )
=> ( P3 @ X6 ) )
& ( ( fmset_a_b @ G @ Cs2 )
= Cs ) ) ) ) ).
% monoid.mset_fmsetEx
thf(fact_288_gcd__condition__monoid_Ogcd__divides__r,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a] :
( ( gcd_co701944698663231555t_unit @ G )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( factor3040189038382604065t_unit @ G @ ( somegc8962790057355718400t_unit @ G @ A @ B ) @ B ) ) ) ) ).
% gcd_condition_monoid.gcd_divides_r
thf(fact_289_gcd__condition__monoid_Ogcd__divides__r,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a] :
( ( gcd_co4024338386749376309id_a_b @ G )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( factor_a_b @ G @ ( somegcd_a_b @ G @ A @ B ) @ B ) ) ) ) ).
% gcd_condition_monoid.gcd_divides_r
thf(fact_290_gcd__condition__monoid_Ogcd__divides__l,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a] :
( ( gcd_co701944698663231555t_unit @ G )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( factor3040189038382604065t_unit @ G @ ( somegc8962790057355718400t_unit @ G @ A @ B ) @ A ) ) ) ) ).
% gcd_condition_monoid.gcd_divides_l
thf(fact_291_gcd__condition__monoid_Ogcd__divides__l,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a] :
( ( gcd_co4024338386749376309id_a_b @ G )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( factor_a_b @ G @ ( somegcd_a_b @ G @ A @ B ) @ A ) ) ) ) ).
% gcd_condition_monoid.gcd_divides_l
thf(fact_292_gcd__condition__monoid_Ogcd__divides,axiom,
! [G: partia8223610829204095565t_unit,Z: a,X: a,Y: a] :
( ( gcd_co701944698663231555t_unit @ G )
=> ( ( factor3040189038382604065t_unit @ G @ Z @ X )
=> ( ( factor3040189038382604065t_unit @ G @ Z @ Y )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ Z @ ( partia6735698275553448452t_unit @ G ) )
=> ( factor3040189038382604065t_unit @ G @ Z @ ( somegc8962790057355718400t_unit @ G @ X @ Y ) ) ) ) ) ) ) ) ).
% gcd_condition_monoid.gcd_divides
thf(fact_293_gcd__condition__monoid_Ogcd__divides,axiom,
! [G: partia7680350978787392319xt_a_b,Z: a,X: a,Y: a] :
( ( gcd_co4024338386749376309id_a_b @ G )
=> ( ( factor_a_b @ G @ Z @ X )
=> ( ( factor_a_b @ G @ Z @ Y )
=> ( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ Z @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( factor_a_b @ G @ Z @ ( somegcd_a_b @ G @ X @ Y ) ) ) ) ) ) ) ) ).
% gcd_condition_monoid.gcd_divides
thf(fact_294_gcd__condition__monoid_Ogcd__cong__r,axiom,
! [G: partia8223610829204095565t_unit,Y: a,Y2: a,X: a] :
( ( gcd_co701944698663231555t_unit @ G )
=> ( ( associ6879500422977059064t_unit @ G @ Y @ Y2 )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ Y2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( associ6879500422977059064t_unit @ G @ ( somegc8962790057355718400t_unit @ G @ X @ Y ) @ ( somegc8962790057355718400t_unit @ G @ X @ Y2 ) ) ) ) ) ) ) ).
% gcd_condition_monoid.gcd_cong_r
thf(fact_295_gcd__condition__monoid_Ogcd__cong__r,axiom,
! [G: partia7680350978787392319xt_a_b,Y: a,Y2: a,X: a] :
( ( gcd_co4024338386749376309id_a_b @ G )
=> ( ( associated_a_b @ G @ Y @ Y2 )
=> ( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ Y2 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( associated_a_b @ G @ ( somegcd_a_b @ G @ X @ Y ) @ ( somegcd_a_b @ G @ X @ Y2 ) ) ) ) ) ) ) ).
% gcd_condition_monoid.gcd_cong_r
thf(fact_296_gcd__condition__monoid_Ogcd__cong__l,axiom,
! [G: partia8223610829204095565t_unit,X: a,X2: a,Y: a] :
( ( gcd_co701944698663231555t_unit @ G )
=> ( ( associ6879500422977059064t_unit @ G @ X @ X2 )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ X2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ G ) )
=> ( associ6879500422977059064t_unit @ G @ ( somegc8962790057355718400t_unit @ G @ X @ Y ) @ ( somegc8962790057355718400t_unit @ G @ X2 @ Y ) ) ) ) ) ) ) ).
% gcd_condition_monoid.gcd_cong_l
thf(fact_297_gcd__condition__monoid_Ogcd__cong__l,axiom,
! [G: partia7680350978787392319xt_a_b,X: a,X2: a,Y: a] :
( ( gcd_co4024338386749376309id_a_b @ G )
=> ( ( associated_a_b @ G @ X @ X2 )
=> ( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ X2 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( associated_a_b @ G @ ( somegcd_a_b @ G @ X @ Y ) @ ( somegcd_a_b @ G @ X2 @ Y ) ) ) ) ) ) ) ).
% gcd_condition_monoid.gcd_cong_l
thf(fact_298_gcd__condition__monoid_Ogcd__assoc,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a,C: a] :
( ( gcd_co701944698663231555t_unit @ G )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ G ) )
=> ( associ6879500422977059064t_unit @ G @ ( somegc8962790057355718400t_unit @ G @ ( somegc8962790057355718400t_unit @ G @ A @ B ) @ C ) @ ( somegc8962790057355718400t_unit @ G @ A @ ( somegc8962790057355718400t_unit @ G @ B @ C ) ) ) ) ) ) ) ).
% gcd_condition_monoid.gcd_assoc
thf(fact_299_gcd__condition__monoid_Ogcd__assoc,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a,C: a] :
( ( gcd_co4024338386749376309id_a_b @ G )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( associated_a_b @ G @ ( somegcd_a_b @ G @ ( somegcd_a_b @ G @ A @ B ) @ C ) @ ( somegcd_a_b @ G @ A @ ( somegcd_a_b @ G @ B @ C ) ) ) ) ) ) ) ).
% gcd_condition_monoid.gcd_assoc
thf(fact_300_gcd__condition__monoid_Ogcdof__cong__l,axiom,
! [G: partia8223610829204095565t_unit,A3: a,A: a,B: a,C: a] :
( ( gcd_co701944698663231555t_unit @ G )
=> ( ( associ6879500422977059064t_unit @ G @ A3 @ A )
=> ( ( isgcd_a_Product_unit @ G @ A @ B @ C )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ G ) )
=> ( isgcd_a_Product_unit @ G @ A3 @ B @ C ) ) ) ) ) ) ) ) ).
% gcd_condition_monoid.gcdof_cong_l
thf(fact_301_gcd__condition__monoid_Ogcdof__cong__l,axiom,
! [G: partia7680350978787392319xt_a_b,A3: a,A: a,B: a,C: a] :
( ( gcd_co4024338386749376309id_a_b @ G )
=> ( ( associated_a_b @ G @ A3 @ A )
=> ( ( isgcd_a_b @ G @ A @ B @ C )
=> ( ( member_a @ A3 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( isgcd_a_b @ G @ A3 @ B @ C ) ) ) ) ) ) ) ) ).
% gcd_condition_monoid.gcdof_cong_l
thf(fact_302_gcd__condition__monoid_Ogcd__isgcd,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a] :
( ( gcd_co701944698663231555t_unit @ G )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( isgcd_a_Product_unit @ G @ ( somegc8962790057355718400t_unit @ G @ A @ B ) @ A @ B ) ) ) ) ).
% gcd_condition_monoid.gcd_isgcd
thf(fact_303_gcd__condition__monoid_Ogcd__isgcd,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a] :
( ( gcd_co4024338386749376309id_a_b @ G )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( isgcd_a_b @ G @ ( somegcd_a_b @ G @ A @ B ) @ A @ B ) ) ) ) ).
% gcd_condition_monoid.gcd_isgcd
thf(fact_304_gcd__condition__monoid_OgcdI,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a,C: a] :
( ( gcd_co701944698663231555t_unit @ G )
=> ( ( factor3040189038382604065t_unit @ G @ A @ B )
=> ( ( factor3040189038382604065t_unit @ G @ A @ C )
=> ( ! [Y3: a] :
( ( member_a @ Y3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( factor3040189038382604065t_unit @ G @ Y3 @ B )
=> ( ( factor3040189038382604065t_unit @ G @ Y3 @ C )
=> ( factor3040189038382604065t_unit @ G @ Y3 @ A ) ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ G ) )
=> ( associ6879500422977059064t_unit @ G @ A @ ( somegc8962790057355718400t_unit @ G @ B @ C ) ) ) ) ) ) ) ) ) ).
% gcd_condition_monoid.gcdI
thf(fact_305_gcd__condition__monoid_OgcdI,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a,C: a] :
( ( gcd_co4024338386749376309id_a_b @ G )
=> ( ( factor_a_b @ G @ A @ B )
=> ( ( factor_a_b @ G @ A @ C )
=> ( ! [Y3: a] :
( ( member_a @ Y3 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( factor_a_b @ G @ Y3 @ B )
=> ( ( factor_a_b @ G @ Y3 @ C )
=> ( factor_a_b @ G @ Y3 @ A ) ) ) )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( associated_a_b @ G @ A @ ( somegcd_a_b @ G @ B @ C ) ) ) ) ) ) ) ) ) ).
% gcd_condition_monoid.gcdI
thf(fact_306_gcd__condition__monoid_OgcdI2,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a,C: a] :
( ( gcd_co701944698663231555t_unit @ G )
=> ( ( isgcd_a_Product_unit @ G @ A @ B @ C )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ G ) )
=> ( associ6879500422977059064t_unit @ G @ A @ ( somegc8962790057355718400t_unit @ G @ B @ C ) ) ) ) ) ) ) ).
% gcd_condition_monoid.gcdI2
thf(fact_307_gcd__condition__monoid_OgcdI2,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a,C: a] :
( ( gcd_co4024338386749376309id_a_b @ G )
=> ( ( isgcd_a_b @ G @ A @ B @ C )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( associated_a_b @ G @ A @ ( somegcd_a_b @ G @ B @ C ) ) ) ) ) ) ) ).
% gcd_condition_monoid.gcdI2
thf(fact_308_monoid_Odivision__equiv,axiom,
! [G: partia8223610829204095565t_unit] :
( ( monoid2746444814143937472t_unit @ G )
=> ( equiva8593399568135932475t_unit @ ( partia6383399360842404908t_unit @ ( partia6735698275553448452t_unit @ G ) @ ( eq_eq_5233546288009311946t_unit @ ( associ6879500422977059064t_unit @ G ) @ ( gorder6800608520584131120t_unit @ ( factor3040189038382604065t_unit @ G ) @ product_Unity ) ) ) ) ) ).
% monoid.division_equiv
thf(fact_309_monoid_Odivision__equiv,axiom,
! [G: partia7680350978787392319xt_a_b] :
( ( monoid_a_b @ G )
=> ( equiva8593399568135932475t_unit @ ( partia6383399360842404908t_unit @ ( partia7484183841585581558xt_a_b @ G ) @ ( eq_eq_5233546288009311946t_unit @ ( associated_a_b @ G ) @ ( gorder6800608520584131120t_unit @ ( factor_a_b @ G ) @ product_Unity ) ) ) ) ) ).
% monoid.division_equiv
thf(fact_310_monoid_Oassocs__repr__independenceD,axiom,
! [G: partia8223610829204095565t_unit,X: a,Y: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( ( eq_clo186784744570230005t_unit @ ( partia6383399360842404908t_unit @ ( partia6735698275553448452t_unit @ G ) @ ( eq_eq_5233546288009311946t_unit @ ( associ6879500422977059064t_unit @ G ) @ ( gorder6800608520584131120t_unit @ ( factor3040189038382604065t_unit @ G ) @ product_Unity ) ) ) @ ( insert_a @ X @ bot_bot_set_a ) )
= ( eq_clo186784744570230005t_unit @ ( partia6383399360842404908t_unit @ ( partia6735698275553448452t_unit @ G ) @ ( eq_eq_5233546288009311946t_unit @ ( associ6879500422977059064t_unit @ G ) @ ( gorder6800608520584131120t_unit @ ( factor3040189038382604065t_unit @ G ) @ product_Unity ) ) ) @ ( insert_a @ Y @ bot_bot_set_a ) ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ G ) )
=> ( member_a @ Y @ ( eq_clo186784744570230005t_unit @ ( partia6383399360842404908t_unit @ ( partia6735698275553448452t_unit @ G ) @ ( eq_eq_5233546288009311946t_unit @ ( associ6879500422977059064t_unit @ G ) @ ( gorder6800608520584131120t_unit @ ( factor3040189038382604065t_unit @ G ) @ product_Unity ) ) ) @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ) ).
% monoid.assocs_repr_independenceD
thf(fact_311_monoid_Oassocs__repr__independenceD,axiom,
! [G: partia7680350978787392319xt_a_b,X: a,Y: a] :
( ( monoid_a_b @ G )
=> ( ( ( eq_clo186784744570230005t_unit @ ( partia6383399360842404908t_unit @ ( partia7484183841585581558xt_a_b @ G ) @ ( eq_eq_5233546288009311946t_unit @ ( associated_a_b @ G ) @ ( gorder6800608520584131120t_unit @ ( factor_a_b @ G ) @ product_Unity ) ) ) @ ( insert_a @ X @ bot_bot_set_a ) )
= ( eq_clo186784744570230005t_unit @ ( partia6383399360842404908t_unit @ ( partia7484183841585581558xt_a_b @ G ) @ ( eq_eq_5233546288009311946t_unit @ ( associated_a_b @ G ) @ ( gorder6800608520584131120t_unit @ ( factor_a_b @ G ) @ product_Unity ) ) ) @ ( insert_a @ Y @ bot_bot_set_a ) ) )
=> ( ( member_a @ Y @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( member_a @ Y @ ( eq_clo186784744570230005t_unit @ ( partia6383399360842404908t_unit @ ( partia7484183841585581558xt_a_b @ G ) @ ( eq_eq_5233546288009311946t_unit @ ( associated_a_b @ G ) @ ( gorder6800608520584131120t_unit @ ( factor_a_b @ G ) @ product_Unity ) ) ) @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ) ).
% monoid.assocs_repr_independenceD
thf(fact_312_monoid_Oassocs__repr__independence,axiom,
! [G: partia8223610829204095565t_unit,Y: a,X: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( member_a @ Y @ ( eq_clo186784744570230005t_unit @ ( partia6383399360842404908t_unit @ ( partia6735698275553448452t_unit @ G ) @ ( eq_eq_5233546288009311946t_unit @ ( associ6879500422977059064t_unit @ G ) @ ( gorder6800608520584131120t_unit @ ( factor3040189038382604065t_unit @ G ) @ product_Unity ) ) ) @ ( insert_a @ X @ bot_bot_set_a ) ) )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( eq_clo186784744570230005t_unit @ ( partia6383399360842404908t_unit @ ( partia6735698275553448452t_unit @ G ) @ ( eq_eq_5233546288009311946t_unit @ ( associ6879500422977059064t_unit @ G ) @ ( gorder6800608520584131120t_unit @ ( factor3040189038382604065t_unit @ G ) @ product_Unity ) ) ) @ ( insert_a @ X @ bot_bot_set_a ) )
= ( eq_clo186784744570230005t_unit @ ( partia6383399360842404908t_unit @ ( partia6735698275553448452t_unit @ G ) @ ( eq_eq_5233546288009311946t_unit @ ( associ6879500422977059064t_unit @ G ) @ ( gorder6800608520584131120t_unit @ ( factor3040189038382604065t_unit @ G ) @ product_Unity ) ) ) @ ( insert_a @ Y @ bot_bot_set_a ) ) ) ) ) ) ).
% monoid.assocs_repr_independence
thf(fact_313_monoid_Oassocs__repr__independence,axiom,
! [G: partia7680350978787392319xt_a_b,Y: a,X: a] :
( ( monoid_a_b @ G )
=> ( ( member_a @ Y @ ( eq_clo186784744570230005t_unit @ ( partia6383399360842404908t_unit @ ( partia7484183841585581558xt_a_b @ G ) @ ( eq_eq_5233546288009311946t_unit @ ( associated_a_b @ G ) @ ( gorder6800608520584131120t_unit @ ( factor_a_b @ G ) @ product_Unity ) ) ) @ ( insert_a @ X @ bot_bot_set_a ) ) )
=> ( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( eq_clo186784744570230005t_unit @ ( partia6383399360842404908t_unit @ ( partia7484183841585581558xt_a_b @ G ) @ ( eq_eq_5233546288009311946t_unit @ ( associated_a_b @ G ) @ ( gorder6800608520584131120t_unit @ ( factor_a_b @ G ) @ product_Unity ) ) ) @ ( insert_a @ X @ bot_bot_set_a ) )
= ( eq_clo186784744570230005t_unit @ ( partia6383399360842404908t_unit @ ( partia7484183841585581558xt_a_b @ G ) @ ( eq_eq_5233546288009311946t_unit @ ( associated_a_b @ G ) @ ( gorder6800608520584131120t_unit @ ( factor_a_b @ G ) @ product_Unity ) ) ) @ ( insert_a @ Y @ bot_bot_set_a ) ) ) ) ) ) ).
% monoid.assocs_repr_independence
thf(fact_314_monoid__cancelI,axiom,
( ! [A2: a,B2: a,C2: a] :
( ( ( mult_a_b @ g @ C2 @ A2 )
= ( mult_a_b @ g @ C2 @ B2 ) )
=> ( ( member_a @ A2 @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B2 @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ C2 @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( A2 = B2 ) ) ) ) )
=> ( ! [A2: a,B2: a,C2: a] :
( ( ( mult_a_b @ g @ A2 @ C2 )
= ( mult_a_b @ g @ B2 @ C2 ) )
=> ( ( member_a @ A2 @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B2 @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ C2 @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( A2 = B2 ) ) ) ) )
=> ( monoid_cancel_a_b @ g ) ) ) ).
% monoid_cancelI
thf(fact_315_monoid_Omonoid__comm__monoidI,axiom,
! [G: partia8223610829204095565t_unit] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ! [X4: a,Y3: a] :
( ( member_a @ X4 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ Y3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( mult_a_Product_unit @ G @ X4 @ Y3 )
= ( mult_a_Product_unit @ G @ Y3 @ X4 ) ) ) )
=> ( comm_m7681468956318391052t_unit @ G ) ) ) ).
% monoid.monoid_comm_monoidI
thf(fact_316_monoid_Omonoid__comm__monoidI,axiom,
! [G: partia7680350978787392319xt_a_b] :
( ( monoid_a_b @ G )
=> ( ! [X4: a,Y3: a] :
( ( member_a @ X4 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ Y3 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( mult_a_b @ G @ X4 @ Y3 )
= ( mult_a_b @ G @ Y3 @ X4 ) ) ) )
=> ( comm_monoid_a_b @ G ) ) ) ).
% monoid.monoid_comm_monoidI
thf(fact_317_relprime__mult,axiom,
! [A: a,B: a,C: a] :
( ( associated_a_b @ g @ ( somegcd_a_b @ g @ A @ B ) @ ( one_a_b @ g ) )
=> ( ( associated_a_b @ g @ ( somegcd_a_b @ g @ A @ C ) @ ( one_a_b @ g ) )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ C @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( associated_a_b @ g @ ( somegcd_a_b @ g @ A @ ( mult_a_b @ g @ B @ C ) ) @ ( one_a_b @ g ) ) ) ) ) ) ) ).
% relprime_mult
thf(fact_318_factor__mset__mult,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( finite8971978181520403909et_a_b @ g @ ( mult_a_b @ g @ A @ B ) )
= ( plus_p2331992037799027419_set_a @ ( finite8971978181520403909et_a_b @ g @ A ) @ ( finite8971978181520403909et_a_b @ g @ B ) ) ) ) ) ).
% factor_mset_mult
thf(fact_319_prime__pow__divides__iff,axiom,
! [P: a,A: a,B: a,N: nat] :
( ( member_a @ P @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( prime_a_b @ g @ P )
=> ( ~ ( factor_a_b @ g @ P @ A )
=> ( ( factor_a_b @ g @ ( pow_a_b_nat @ g @ P @ N ) @ ( mult_a_b @ g @ A @ B ) )
= ( factor_a_b @ g @ ( pow_a_b_nat @ g @ P @ N ) @ B ) ) ) ) ) ) ) ).
% prime_pow_divides_iff
thf(fact_320_mset__wfactorsEx,axiom,
! [Cs: multiset_set_a] :
( ! [X5: set_a] :
( ( member_set_a @ X5 @ ( set_mset_set_a @ Cs ) )
=> ? [X6: a] :
( ( member_a @ X6 @ ( partia7484183841585581558xt_a_b @ g ) )
& ( irreducible_a_b @ g @ X6 )
& ( X5
= ( eq_clo186784744570230005t_unit @ ( partia6383399360842404908t_unit @ ( partia7484183841585581558xt_a_b @ g ) @ ( eq_eq_5233546288009311946t_unit @ ( associated_a_b @ g ) @ ( gorder6800608520584131120t_unit @ ( factor_a_b @ g ) @ product_Unity ) ) ) @ ( insert_a @ X6 @ bot_bot_set_a ) ) ) ) )
=> ? [C2: a,Cs2: list_a] :
( ( member_a @ C2 @ ( partia7484183841585581558xt_a_b @ g ) )
& ( ord_less_eq_set_a @ ( set_a2 @ Cs2 ) @ ( partia7484183841585581558xt_a_b @ g ) )
& ( wfactors_a_b @ g @ Cs2 @ C2 )
& ( ( fmset_a_b @ g @ Cs2 )
= Cs ) ) ) ).
% mset_wfactorsEx
thf(fact_321_irreducible__prod__rI,axiom,
! [A: a,B: a] :
( ( irreducible_a_b @ g @ A )
=> ( ( member_a @ B @ ( units_a_b @ g ) )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( irreducible_a_b @ g @ ( mult_a_b @ g @ A @ B ) ) ) ) ) ) ).
% irreducible_prod_rI
thf(fact_322_irreducible__prod__lI,axiom,
! [B: a,A: a] :
( ( irreducible_a_b @ g @ B )
=> ( ( member_a @ A @ ( units_a_b @ g ) )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( irreducible_a_b @ g @ ( mult_a_b @ g @ A @ B ) ) ) ) ) ) ).
% irreducible_prod_lI
thf(fact_323_irreducible__prodE,axiom,
! [A: a,B: a] :
( ( irreducible_a_b @ g @ ( mult_a_b @ g @ A @ B ) )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ( irreducible_a_b @ g @ A )
=> ~ ( member_a @ B @ ( units_a_b @ g ) ) )
=> ~ ( ( member_a @ A @ ( units_a_b @ g ) )
=> ~ ( irreducible_a_b @ g @ B ) ) ) ) ) ) ).
% irreducible_prodE
thf(fact_324_Units__closed,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_b @ g ) )
=> ( member_a @ X @ ( partia7484183841585581558xt_a_b @ g ) ) ) ).
% Units_closed
thf(fact_325_monoid__cancel__axioms,axiom,
monoid_cancel_a_b @ g ).
% monoid_cancel_axioms
thf(fact_326_Units__assoc,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( units_a_b @ g ) )
=> ( ( member_a @ B @ ( units_a_b @ g ) )
=> ( associated_a_b @ g @ A @ B ) ) ) ).
% Units_assoc
thf(fact_327_Units__pow__closed,axiom,
! [X: a,D: nat] :
( ( member_a @ X @ ( units_a_b @ g ) )
=> ( member_a @ ( pow_a_b_nat @ g @ X @ D ) @ ( units_a_b @ g ) ) ) ).
% Units_pow_closed
thf(fact_328_subsetI,axiom,
! [A4: set_set_a,B3: set_set_a] :
( ! [X4: set_a] :
( ( member_set_a @ X4 @ A4 )
=> ( member_set_a @ X4 @ B3 ) )
=> ( ord_le3724670747650509150_set_a @ A4 @ B3 ) ) ).
% subsetI
thf(fact_329_subsetI,axiom,
! [A4: set_a,B3: set_a] :
( ! [X4: a] :
( ( member_a @ X4 @ A4 )
=> ( member_a @ X4 @ B3 ) )
=> ( ord_less_eq_set_a @ A4 @ B3 ) ) ).
% subsetI
thf(fact_330_subset__antisym,axiom,
! [A4: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ B3 )
=> ( ( ord_le3724670747650509150_set_a @ B3 @ A4 )
=> ( A4 = B3 ) ) ) ).
% subset_antisym
thf(fact_331_subset__antisym,axiom,
! [A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A4 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ A4 )
=> ( A4 = B3 ) ) ) ).
% subset_antisym
thf(fact_332_prod__unit__l,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_b @ g @ A @ B ) @ ( units_a_b @ g ) )
=> ( ( member_a @ A @ ( units_a_b @ g ) )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( member_a @ B @ ( units_a_b @ g ) ) ) ) ) ) ).
% prod_unit_l
thf(fact_333_prod__unit__r,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_b @ g @ A @ B ) @ ( units_a_b @ g ) )
=> ( ( member_a @ B @ ( units_a_b @ g ) )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( member_a @ A @ ( units_a_b @ g ) ) ) ) ) ) ).
% prod_unit_r
thf(fact_334_unit__factor,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_b @ g @ A @ B ) @ ( units_a_b @ g ) )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( member_a @ A @ ( units_a_b @ g ) ) ) ) ) ).
% unit_factor
thf(fact_335_inv__unique,axiom,
! [Y: a,X: a,Y2: a] :
( ( ( mult_a_b @ g @ Y @ X )
= ( one_a_b @ g ) )
=> ( ( ( mult_a_b @ g @ X @ Y2 )
= ( one_a_b @ g ) )
=> ( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ Y @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ Y2 @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% inv_unique
thf(fact_336_one__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( mult_a_b @ g @ U @ X4 )
= X4 ) )
=> ( U
= ( one_a_b @ g ) ) ) ) ).
% one_unique
thf(fact_337_assoc__unit__l,axiom,
! [A: a,B: a] :
( ( associated_a_b @ g @ A @ B )
=> ( ( member_a @ B @ ( units_a_b @ g ) )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( member_a @ A @ ( units_a_b @ g ) ) ) ) ) ).
% assoc_unit_l
thf(fact_338_assoc__unit__r,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( units_a_b @ g ) )
=> ( ( associated_a_b @ g @ A @ B )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( member_a @ B @ ( units_a_b @ g ) ) ) ) ) ).
% assoc_unit_r
thf(fact_339_divides__unit,axiom,
! [A: a,U: a] :
( ( factor_a_b @ g @ A @ U )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ U @ ( units_a_b @ g ) )
=> ( member_a @ A @ ( units_a_b @ g ) ) ) ) ) ).
% divides_unit
thf(fact_340_unit__divides,axiom,
! [U: a,A: a] :
( ( member_a @ U @ ( units_a_b @ g ) )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( factor_a_b @ g @ U @ A ) ) ) ).
% unit_divides
thf(fact_341_group__commutes__pow,axiom,
! [X: a,Y: a,N: nat] :
( ( ( mult_a_b @ g @ X @ Y )
= ( mult_a_b @ g @ Y @ X ) )
=> ( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ Y @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( mult_a_b @ g @ ( pow_a_b_nat @ g @ X @ N ) @ Y )
= ( mult_a_b @ g @ Y @ ( pow_a_b_nat @ g @ X @ N ) ) ) ) ) ) ).
% group_commutes_pow
thf(fact_342_nat__pow__comm,axiom,
! [X: a,N: nat,M4: nat] :
( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( mult_a_b @ g @ ( pow_a_b_nat @ g @ X @ N ) @ ( pow_a_b_nat @ g @ X @ M4 ) )
= ( mult_a_b @ g @ ( pow_a_b_nat @ g @ X @ M4 ) @ ( pow_a_b_nat @ g @ X @ N ) ) ) ) ).
% nat_pow_comm
thf(fact_343_nat__pow__distrib,axiom,
! [X: a,Y: a,N: nat] :
( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ Y @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( pow_a_b_nat @ g @ ( mult_a_b @ g @ X @ Y ) @ N )
= ( mult_a_b @ g @ ( pow_a_b_nat @ g @ X @ N ) @ ( pow_a_b_nat @ g @ Y @ N ) ) ) ) ) ).
% nat_pow_distrib
thf(fact_344_pow__mult__distrib,axiom,
! [X: a,Y: a,N: nat] :
( ( ( mult_a_b @ g @ X @ Y )
= ( mult_a_b @ g @ Y @ X ) )
=> ( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ Y @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( pow_a_b_nat @ g @ ( mult_a_b @ g @ X @ Y ) @ N )
= ( mult_a_b @ g @ ( pow_a_b_nat @ g @ X @ N ) @ ( pow_a_b_nat @ g @ Y @ N ) ) ) ) ) ) ).
% pow_mult_distrib
thf(fact_345_Units__inv__comm,axiom,
! [X: a,Y: a] :
( ( ( mult_a_b @ g @ X @ Y )
= ( one_a_b @ g ) )
=> ( ( member_a @ X @ ( units_a_b @ g ) )
=> ( ( member_a @ Y @ ( units_a_b @ g ) )
=> ( ( mult_a_b @ g @ Y @ X )
= ( one_a_b @ g ) ) ) ) ) ).
% Units_inv_comm
thf(fact_346_local_OassociatedD2,axiom,
! [A: a,B: a] :
( ( associated_a_b @ g @ A @ B )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( units_a_b @ g ) )
& ( A
= ( mult_a_b @ g @ B @ X4 ) ) ) ) ) ) ).
% local.associatedD2
thf(fact_347_associatedE2,axiom,
! [A: a,B: a] :
( ( associated_a_b @ g @ A @ B )
=> ( ! [U2: a] :
( ( A
= ( mult_a_b @ g @ B @ U2 ) )
=> ~ ( member_a @ U2 @ ( units_a_b @ g ) ) )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ~ ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) ) ) ) ) ).
% associatedE2
thf(fact_348_associatedI2,axiom,
! [U: a,A: a,B: a] :
( ( member_a @ U @ ( units_a_b @ g ) )
=> ( ( A
= ( mult_a_b @ g @ B @ U ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( associated_a_b @ g @ A @ B ) ) ) ) ).
% associatedI2
thf(fact_349_associatedI2_H,axiom,
! [A: a,B: a,U: a] :
( ( A
= ( mult_a_b @ g @ B @ U ) )
=> ( ( member_a @ U @ ( units_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( associated_a_b @ g @ A @ B ) ) ) ) ).
% associatedI2'
thf(fact_350_associated__iff,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( associated_a_b @ g @ A @ B )
= ( ? [X3: a] :
( ( member_a @ X3 @ ( units_a_b @ g ) )
& ( A
= ( mult_a_b @ g @ B @ X3 ) ) ) ) ) ) ) ).
% associated_iff
thf(fact_351_Units__l__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_b @ g ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia7484183841585581558xt_a_b @ g ) )
& ( ( mult_a_b @ g @ X4 @ X )
= ( one_a_b @ g ) ) ) ) ).
% Units_l_inv_ex
thf(fact_352_Units__r__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_b @ g ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia7484183841585581558xt_a_b @ g ) )
& ( ( mult_a_b @ g @ X @ X4 )
= ( one_a_b @ g ) ) ) ) ).
% Units_r_inv_ex
thf(fact_353_Unit__eq__dividesone,axiom,
! [U: a] :
( ( member_a @ U @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ U @ ( units_a_b @ g ) )
= ( factor_a_b @ g @ U @ ( one_a_b @ g ) ) ) ) ).
% Unit_eq_dividesone
thf(fact_354_wfactors__cong__r,axiom,
! [Fs: list_a,A: a,A3: a] :
( ( wfactors_a_b @ g @ Fs @ A )
=> ( ( associated_a_b @ g @ A @ A3 )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ A3 @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( wfactors_a_b @ g @ Fs @ A3 ) ) ) ) ) ) ).
% wfactors_cong_r
thf(fact_355_wfactors__dividesI,axiom,
! [Fs: list_a,A: a,F: a] :
( ( wfactors_a_b @ g @ Fs @ A )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ F @ ( set_a2 @ Fs ) )
=> ( factor_a_b @ g @ F @ A ) ) ) ) ) ).
% wfactors_dividesI
thf(fact_356_assoc__as__fmset__eq,axiom,
! [As: list_a,A: a,Bs: list_a,B: a] :
( ( wfactors_a_b @ g @ As @ A )
=> ( ( wfactors_a_b @ g @ Bs @ B )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( associated_a_b @ g @ A @ B )
= ( ( fmset_a_b @ g @ As )
= ( fmset_a_b @ g @ Bs ) ) ) ) ) ) ) ) ) ).
% assoc_as_fmset_eq
thf(fact_357_subset__empty,axiom,
! [A4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ bot_bot_set_set_a )
= ( A4 = bot_bot_set_set_a ) ) ).
% subset_empty
thf(fact_358_subset__empty,axiom,
! [A4: set_a] :
( ( ord_less_eq_set_a @ A4 @ bot_bot_set_a )
= ( A4 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_359_empty__subsetI,axiom,
! [A4: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A4 ) ).
% empty_subsetI
thf(fact_360_empty__subsetI,axiom,
! [A4: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A4 ) ).
% empty_subsetI
thf(fact_361_insert__subset,axiom,
! [X: set_a,A4: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X @ A4 ) @ B3 )
= ( ( member_set_a @ X @ B3 )
& ( ord_le3724670747650509150_set_a @ A4 @ B3 ) ) ) ).
% insert_subset
thf(fact_362_insert__subset,axiom,
! [X: a,A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A4 ) @ B3 )
= ( ( member_a @ X @ B3 )
& ( ord_less_eq_set_a @ A4 @ B3 ) ) ) ).
% insert_subset
thf(fact_363_factor__mset__aux,axiom,
! [A: a] :
( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ? [As2: list_a] :
( ( ( finite8971978181520403909et_a_b @ g @ A )
= ( fmset_a_b @ g @ As2 ) )
& ( wfactors_a_b @ g @ As2 @ A )
& ( ord_less_eq_set_a @ ( set_a2 @ As2 ) @ ( partia7484183841585581558xt_a_b @ g ) ) ) ) ).
% factor_mset_aux
thf(fact_364_factor__mset__aux__1,axiom,
! [A: a,As: list_a] :
( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( wfactors_a_b @ g @ As @ A )
=> ( ( finite8971978181520403909et_a_b @ g @ A )
= ( fmset_a_b @ g @ As ) ) ) ) ) ).
% factor_mset_aux_1
thf(fact_365_subset__mset_Oadd__le__cancel__left,axiom,
! [C: multiset_a,A: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ ( plus_plus_multiset_a @ C @ A ) @ ( plus_plus_multiset_a @ C @ B ) )
= ( subseteq_mset_a @ A @ B ) ) ).
% subset_mset.add_le_cancel_left
thf(fact_366_subset__mset_Oadd__le__cancel__left,axiom,
! [C: multiset_set_a,A: multiset_set_a,B: multiset_set_a] :
( ( subseteq_mset_set_a @ ( plus_p2331992037799027419_set_a @ C @ A ) @ ( plus_p2331992037799027419_set_a @ C @ B ) )
= ( subseteq_mset_set_a @ A @ B ) ) ).
% subset_mset.add_le_cancel_left
thf(fact_367_subset__mset_Oadd__le__cancel__right,axiom,
! [A: multiset_a,C: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ ( plus_plus_multiset_a @ A @ C ) @ ( plus_plus_multiset_a @ B @ C ) )
= ( subseteq_mset_a @ A @ B ) ) ).
% subset_mset.add_le_cancel_right
thf(fact_368_subset__mset_Oadd__le__cancel__right,axiom,
! [A: multiset_set_a,C: multiset_set_a,B: multiset_set_a] :
( ( subseteq_mset_set_a @ ( plus_p2331992037799027419_set_a @ A @ C ) @ ( plus_p2331992037799027419_set_a @ B @ C ) )
= ( subseteq_mset_set_a @ A @ B ) ) ).
% subset_mset.add_le_cancel_right
thf(fact_369_mset__subset__eq__mono__add__left__cancel,axiom,
! [C4: multiset_a,A4: multiset_a,B3: multiset_a] :
( ( subseteq_mset_a @ ( plus_plus_multiset_a @ C4 @ A4 ) @ ( plus_plus_multiset_a @ C4 @ B3 ) )
= ( subseteq_mset_a @ A4 @ B3 ) ) ).
% mset_subset_eq_mono_add_left_cancel
thf(fact_370_mset__subset__eq__mono__add__left__cancel,axiom,
! [C4: multiset_set_a,A4: multiset_set_a,B3: multiset_set_a] :
( ( subseteq_mset_set_a @ ( plus_p2331992037799027419_set_a @ C4 @ A4 ) @ ( plus_p2331992037799027419_set_a @ C4 @ B3 ) )
= ( subseteq_mset_set_a @ A4 @ B3 ) ) ).
% mset_subset_eq_mono_add_left_cancel
thf(fact_371_mset__subset__eq__mono__add__right__cancel,axiom,
! [A4: multiset_a,C4: multiset_a,B3: multiset_a] :
( ( subseteq_mset_a @ ( plus_plus_multiset_a @ A4 @ C4 ) @ ( plus_plus_multiset_a @ B3 @ C4 ) )
= ( subseteq_mset_a @ A4 @ B3 ) ) ).
% mset_subset_eq_mono_add_right_cancel
thf(fact_372_mset__subset__eq__mono__add__right__cancel,axiom,
! [A4: multiset_set_a,C4: multiset_set_a,B3: multiset_set_a] :
( ( subseteq_mset_set_a @ ( plus_p2331992037799027419_set_a @ A4 @ C4 ) @ ( plus_p2331992037799027419_set_a @ B3 @ C4 ) )
= ( subseteq_mset_set_a @ A4 @ B3 ) ) ).
% mset_subset_eq_mono_add_right_cancel
thf(fact_373_divides__as__fmsubset,axiom,
! [As: list_a,A: a,Bs: list_a,B: a] :
( ( wfactors_a_b @ g @ As @ A )
=> ( ( wfactors_a_b @ g @ Bs @ B )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( factor_a_b @ g @ A @ B )
= ( subseteq_mset_set_a @ ( fmset_a_b @ g @ As ) @ ( fmset_a_b @ g @ Bs ) ) ) ) ) ) ) ) ) ).
% divides_as_fmsubset
thf(fact_374_divides__fmsubset,axiom,
! [A: a,B: a,As: list_a,Bs: list_a] :
( ( factor_a_b @ g @ A @ B )
=> ( ( wfactors_a_b @ g @ As @ A )
=> ( ( wfactors_a_b @ g @ Bs @ B )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( subseteq_mset_set_a @ ( fmset_a_b @ g @ As ) @ ( fmset_a_b @ g @ Bs ) ) ) ) ) ) ) ) ) ).
% divides_fmsubset
thf(fact_375_fmsubset__divides,axiom,
! [As: list_a,Bs: list_a,A: a,B: a] :
( ( subseteq_mset_set_a @ ( fmset_a_b @ g @ As ) @ ( fmset_a_b @ g @ Bs ) )
=> ( ( wfactors_a_b @ g @ As @ A )
=> ( ( wfactors_a_b @ g @ Bs @ B )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( factor_a_b @ g @ A @ B ) ) ) ) ) ) ) ) ).
% fmsubset_divides
thf(fact_376_mult__wfactors__fmset,axiom,
! [As: list_a,A: a,Bs: list_a,B: a,Cs3: list_a] :
( ( wfactors_a_b @ g @ As @ A )
=> ( ( wfactors_a_b @ g @ Bs @ B )
=> ( ( wfactors_a_b @ g @ Cs3 @ ( mult_a_b @ g @ A @ B ) )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Cs3 ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( fmset_a_b @ g @ Cs3 )
= ( plus_p2331992037799027419_set_a @ ( fmset_a_b @ g @ As ) @ ( fmset_a_b @ g @ Bs ) ) ) ) ) ) ) ) ) ) ) ).
% mult_wfactors_fmset
thf(fact_377_fmset__wfactors__mult,axiom,
! [Cs3: list_a,As: list_a,Bs: list_a,A: a,B: a,C: a] :
( ( ( fmset_a_b @ g @ Cs3 )
= ( plus_p2331992037799027419_set_a @ ( fmset_a_b @ g @ As ) @ ( fmset_a_b @ g @ Bs ) ) )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ C @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Cs3 ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( wfactors_a_b @ g @ As @ A )
=> ( ( wfactors_a_b @ g @ Bs @ B )
=> ( ( wfactors_a_b @ g @ Cs3 @ C )
=> ( associated_a_b @ g @ C @ ( mult_a_b @ g @ A @ B ) ) ) ) ) ) ) ) ) ) ) ) ).
% fmset_wfactors_mult
thf(fact_378_singleton__insert__inj__eq,axiom,
! [B: set_a,A: set_a,A4: set_set_a] :
( ( ( insert_set_a @ B @ bot_bot_set_set_a )
= ( insert_set_a @ A @ A4 ) )
= ( ( A = B )
& ( ord_le3724670747650509150_set_a @ A4 @ ( insert_set_a @ B @ bot_bot_set_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_379_singleton__insert__inj__eq,axiom,
! [B: a,A: a,A4: set_a] :
( ( ( insert_a @ B @ bot_bot_set_a )
= ( insert_a @ A @ A4 ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A4 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_380_singleton__insert__inj__eq_H,axiom,
! [A: set_a,A4: set_set_a,B: set_a] :
( ( ( insert_set_a @ A @ A4 )
= ( insert_set_a @ B @ bot_bot_set_set_a ) )
= ( ( A = B )
& ( ord_le3724670747650509150_set_a @ A4 @ ( insert_set_a @ B @ bot_bot_set_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_381_singleton__insert__inj__eq_H,axiom,
! [A: a,A4: set_a,B: a] :
( ( ( insert_a @ A @ A4 )
= ( insert_a @ B @ bot_bot_set_a ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A4 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_382_monoid_Oone__closed,axiom,
! [G: partia8223610829204095565t_unit] :
( ( monoid2746444814143937472t_unit @ G )
=> ( member_a @ ( one_a_Product_unit @ G ) @ ( partia6735698275553448452t_unit @ G ) ) ) ).
% monoid.one_closed
thf(fact_383_monoid_Oone__closed,axiom,
! [G: partia7680350978787392319xt_a_b] :
( ( monoid_a_b @ G )
=> ( member_a @ ( one_a_b @ G ) @ ( partia7484183841585581558xt_a_b @ G ) ) ) ).
% monoid.one_closed
thf(fact_384_one__closed,axiom,
member_a @ ( one_a_b @ g ) @ ( partia7484183841585581558xt_a_b @ g ) ).
% one_closed
thf(fact_385_Units__m__closed,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( units_a_b @ g ) )
=> ( ( member_a @ Y @ ( units_a_b @ g ) )
=> ( member_a @ ( mult_a_b @ g @ X @ Y ) @ ( units_a_b @ g ) ) ) ) ).
% Units_m_closed
thf(fact_386_nat__pow__closed,axiom,
! [X: a,N: nat] :
( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( member_a @ ( pow_a_b_nat @ g @ X @ N ) @ ( partia7484183841585581558xt_a_b @ g ) ) ) ).
% nat_pow_closed
thf(fact_387_Units__one__closed,axiom,
member_a @ ( one_a_b @ g ) @ ( units_a_b @ g ) ).
% Units_one_closed
thf(fact_388_nat__pow__one,axiom,
! [N: nat] :
( ( pow_a_b_nat @ g @ ( one_a_b @ g ) @ N )
= ( one_a_b @ g ) ) ).
% nat_pow_one
thf(fact_389_monoid_Ol__one,axiom,
! [G: partia8223610829204095565t_unit,X: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( mult_a_Product_unit @ G @ ( one_a_Product_unit @ G ) @ X )
= X ) ) ) ).
% monoid.l_one
thf(fact_390_monoid_Ol__one,axiom,
! [G: partia7680350978787392319xt_a_b,X: a] :
( ( monoid_a_b @ G )
=> ( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( mult_a_b @ G @ ( one_a_b @ G ) @ X )
= X ) ) ) ).
% monoid.l_one
thf(fact_391_monoid_Or__one,axiom,
! [G: partia8223610829204095565t_unit,X: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( mult_a_Product_unit @ G @ X @ ( one_a_Product_unit @ G ) )
= X ) ) ) ).
% monoid.r_one
thf(fact_392_monoid_Or__one,axiom,
! [G: partia7680350978787392319xt_a_b,X: a] :
( ( monoid_a_b @ G )
=> ( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( mult_a_b @ G @ X @ ( one_a_b @ G ) )
= X ) ) ) ).
% monoid.r_one
thf(fact_393_Units__l__cancel,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( units_a_b @ g ) )
=> ( ( member_a @ Y @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ Z @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ( mult_a_b @ g @ X @ Y )
= ( mult_a_b @ g @ X @ Z ) )
= ( Y = Z ) ) ) ) ) ).
% Units_l_cancel
thf(fact_394_l__one,axiom,
! [X: a] :
( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( mult_a_b @ g @ ( one_a_b @ g ) @ X )
= X ) ) ).
% l_one
thf(fact_395_r__one,axiom,
! [X: a] :
( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( mult_a_b @ g @ X @ ( one_a_b @ g ) )
= X ) ) ).
% r_one
thf(fact_396_wfactors__exist,axiom,
! [A: a] :
( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ? [Fs2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Fs2 ) @ ( partia7484183841585581558xt_a_b @ g ) )
& ( wfactors_a_b @ g @ Fs2 @ A ) ) ) ).
% wfactors_exist
thf(fact_397_wfactors__prod__exists,axiom,
! [As: list_a] :
( ! [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ As ) )
=> ( irreducible_a_b @ g @ X4 ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ? [A2: a] :
( ( member_a @ A2 @ ( partia7484183841585581558xt_a_b @ g ) )
& ( wfactors_a_b @ g @ As @ A2 ) ) ) ) ).
% wfactors_prod_exists
thf(fact_398_monoid_Onat__pow__one,axiom,
! [G: partia8223610829204095565t_unit,N: nat] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( pow_a_1875594501834816709it_nat @ G @ ( one_a_Product_unit @ G ) @ N )
= ( one_a_Product_unit @ G ) ) ) ).
% monoid.nat_pow_one
thf(fact_399_monoid_Onat__pow__one,axiom,
! [G: partia7680350978787392319xt_a_b,N: nat] :
( ( monoid_a_b @ G )
=> ( ( pow_a_b_nat @ G @ ( one_a_b @ G ) @ N )
= ( one_a_b @ G ) ) ) ).
% monoid.nat_pow_one
thf(fact_400_monoid_OUnits__one__closed,axiom,
! [G: partia8223610829204095565t_unit] :
( ( monoid2746444814143937472t_unit @ G )
=> ( member_a @ ( one_a_Product_unit @ G ) @ ( units_a_Product_unit @ G ) ) ) ).
% monoid.Units_one_closed
thf(fact_401_monoid_OUnits__one__closed,axiom,
! [G: partia7680350978787392319xt_a_b] :
( ( monoid_a_b @ G )
=> ( member_a @ ( one_a_b @ G ) @ ( units_a_b @ G ) ) ) ).
% monoid.Units_one_closed
thf(fact_402_in__mono,axiom,
! [A4: set_set_a,B3: set_set_a,X: set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ B3 )
=> ( ( member_set_a @ X @ A4 )
=> ( member_set_a @ X @ B3 ) ) ) ).
% in_mono
thf(fact_403_in__mono,axiom,
! [A4: set_a,B3: set_a,X: a] :
( ( ord_less_eq_set_a @ A4 @ B3 )
=> ( ( member_a @ X @ A4 )
=> ( member_a @ X @ B3 ) ) ) ).
% in_mono
thf(fact_404_subsetD,axiom,
! [A4: set_set_a,B3: set_set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ B3 )
=> ( ( member_set_a @ C @ A4 )
=> ( member_set_a @ C @ B3 ) ) ) ).
% subsetD
thf(fact_405_subsetD,axiom,
! [A4: set_a,B3: set_a,C: a] :
( ( ord_less_eq_set_a @ A4 @ B3 )
=> ( ( member_a @ C @ A4 )
=> ( member_a @ C @ B3 ) ) ) ).
% subsetD
thf(fact_406_equalityE,axiom,
! [A4: set_set_a,B3: set_set_a] :
( ( A4 = B3 )
=> ~ ( ( ord_le3724670747650509150_set_a @ A4 @ B3 )
=> ~ ( ord_le3724670747650509150_set_a @ B3 @ A4 ) ) ) ).
% equalityE
thf(fact_407_equalityE,axiom,
! [A4: set_a,B3: set_a] :
( ( A4 = B3 )
=> ~ ( ( ord_less_eq_set_a @ A4 @ B3 )
=> ~ ( ord_less_eq_set_a @ B3 @ A4 ) ) ) ).
% equalityE
thf(fact_408_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A6: set_set_a,B6: set_set_a] :
! [X3: set_a] :
( ( member_set_a @ X3 @ A6 )
=> ( member_set_a @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_409_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A6 )
=> ( member_a @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_410_equalityD1,axiom,
! [A4: set_set_a,B3: set_set_a] :
( ( A4 = B3 )
=> ( ord_le3724670747650509150_set_a @ A4 @ B3 ) ) ).
% equalityD1
thf(fact_411_equalityD1,axiom,
! [A4: set_a,B3: set_a] :
( ( A4 = B3 )
=> ( ord_less_eq_set_a @ A4 @ B3 ) ) ).
% equalityD1
thf(fact_412_equalityD2,axiom,
! [A4: set_set_a,B3: set_set_a] :
( ( A4 = B3 )
=> ( ord_le3724670747650509150_set_a @ B3 @ A4 ) ) ).
% equalityD2
thf(fact_413_equalityD2,axiom,
! [A4: set_a,B3: set_a] :
( ( A4 = B3 )
=> ( ord_less_eq_set_a @ B3 @ A4 ) ) ).
% equalityD2
thf(fact_414_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A6: set_set_a,B6: set_set_a] :
! [T: set_a] :
( ( member_set_a @ T @ A6 )
=> ( member_set_a @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_415_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_416_subset__refl,axiom,
! [A4: set_set_a] : ( ord_le3724670747650509150_set_a @ A4 @ A4 ) ).
% subset_refl
thf(fact_417_subset__refl,axiom,
! [A4: set_a] : ( ord_less_eq_set_a @ A4 @ A4 ) ).
% subset_refl
thf(fact_418_Collect__mono,axiom,
! [P3: set_a > $o,Q: set_a > $o] :
( ! [X4: set_a] :
( ( P3 @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P3 ) @ ( collect_set_a @ Q ) ) ) ).
% Collect_mono
thf(fact_419_Collect__mono,axiom,
! [P3: a > $o,Q: a > $o] :
( ! [X4: a] :
( ( P3 @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P3 ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_420_subset__trans,axiom,
! [A4: set_set_a,B3: set_set_a,C4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ B3 )
=> ( ( ord_le3724670747650509150_set_a @ B3 @ C4 )
=> ( ord_le3724670747650509150_set_a @ A4 @ C4 ) ) ) ).
% subset_trans
thf(fact_421_subset__trans,axiom,
! [A4: set_a,B3: set_a,C4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C4 )
=> ( ord_less_eq_set_a @ A4 @ C4 ) ) ) ).
% subset_trans
thf(fact_422_set__eq__subset,axiom,
( ( ^ [Y4: set_set_a,Z2: set_set_a] : ( Y4 = Z2 ) )
= ( ^ [A6: set_set_a,B6: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A6 @ B6 )
& ( ord_le3724670747650509150_set_a @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_423_set__eq__subset,axiom,
( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 ) )
= ( ^ [A6: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A6 @ B6 )
& ( ord_less_eq_set_a @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_424_Collect__mono__iff,axiom,
! [P3: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P3 ) @ ( collect_set_a @ Q ) )
= ( ! [X3: set_a] :
( ( P3 @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_425_Collect__mono__iff,axiom,
! [P3: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P3 ) @ ( collect_a @ Q ) )
= ( ! [X3: a] :
( ( P3 @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_426_monoid__cancel_Ois__monoid__cancel,axiom,
! [G: partia7680350978787392319xt_a_b] :
( ( monoid_cancel_a_b @ G )
=> ( monoid_cancel_a_b @ G ) ) ).
% monoid_cancel.is_monoid_cancel
thf(fact_427_union__assoc,axiom,
! [M: multiset_a,N3: multiset_a,K: multiset_a] :
( ( plus_plus_multiset_a @ ( plus_plus_multiset_a @ M @ N3 ) @ K )
= ( plus_plus_multiset_a @ M @ ( plus_plus_multiset_a @ N3 @ K ) ) ) ).
% union_assoc
thf(fact_428_union__assoc,axiom,
! [M: multiset_set_a,N3: multiset_set_a,K: multiset_set_a] :
( ( plus_p2331992037799027419_set_a @ ( plus_p2331992037799027419_set_a @ M @ N3 ) @ K )
= ( plus_p2331992037799027419_set_a @ M @ ( plus_p2331992037799027419_set_a @ N3 @ K ) ) ) ).
% union_assoc
thf(fact_429_union__lcomm,axiom,
! [M: multiset_a,N3: multiset_a,K: multiset_a] :
( ( plus_plus_multiset_a @ M @ ( plus_plus_multiset_a @ N3 @ K ) )
= ( plus_plus_multiset_a @ N3 @ ( plus_plus_multiset_a @ M @ K ) ) ) ).
% union_lcomm
thf(fact_430_union__lcomm,axiom,
! [M: multiset_set_a,N3: multiset_set_a,K: multiset_set_a] :
( ( plus_p2331992037799027419_set_a @ M @ ( plus_p2331992037799027419_set_a @ N3 @ K ) )
= ( plus_p2331992037799027419_set_a @ N3 @ ( plus_p2331992037799027419_set_a @ M @ K ) ) ) ).
% union_lcomm
thf(fact_431_union__commute,axiom,
( plus_plus_multiset_a
= ( ^ [M2: multiset_a,N2: multiset_a] : ( plus_plus_multiset_a @ N2 @ M2 ) ) ) ).
% union_commute
thf(fact_432_union__commute,axiom,
( plus_p2331992037799027419_set_a
= ( ^ [M2: multiset_set_a,N2: multiset_set_a] : ( plus_p2331992037799027419_set_a @ N2 @ M2 ) ) ) ).
% union_commute
thf(fact_433_union__left__cancel,axiom,
! [K: multiset_a,M: multiset_a,N3: multiset_a] :
( ( ( plus_plus_multiset_a @ K @ M )
= ( plus_plus_multiset_a @ K @ N3 ) )
= ( M = N3 ) ) ).
% union_left_cancel
thf(fact_434_union__left__cancel,axiom,
! [K: multiset_set_a,M: multiset_set_a,N3: multiset_set_a] :
( ( ( plus_p2331992037799027419_set_a @ K @ M )
= ( plus_p2331992037799027419_set_a @ K @ N3 ) )
= ( M = N3 ) ) ).
% union_left_cancel
thf(fact_435_union__right__cancel,axiom,
! [M: multiset_a,K: multiset_a,N3: multiset_a] :
( ( ( plus_plus_multiset_a @ M @ K )
= ( plus_plus_multiset_a @ N3 @ K ) )
= ( M = N3 ) ) ).
% union_right_cancel
thf(fact_436_union__right__cancel,axiom,
! [M: multiset_set_a,K: multiset_set_a,N3: multiset_set_a] :
( ( ( plus_p2331992037799027419_set_a @ M @ K )
= ( plus_p2331992037799027419_set_a @ N3 @ K ) )
= ( M = N3 ) ) ).
% union_right_cancel
thf(fact_437_multi__union__self__other__eq,axiom,
! [A4: multiset_a,X7: multiset_a,Y6: multiset_a] :
( ( ( plus_plus_multiset_a @ A4 @ X7 )
= ( plus_plus_multiset_a @ A4 @ Y6 ) )
=> ( X7 = Y6 ) ) ).
% multi_union_self_other_eq
thf(fact_438_multi__union__self__other__eq,axiom,
! [A4: multiset_set_a,X7: multiset_set_a,Y6: multiset_set_a] :
( ( ( plus_p2331992037799027419_set_a @ A4 @ X7 )
= ( plus_p2331992037799027419_set_a @ A4 @ Y6 ) )
=> ( X7 = Y6 ) ) ).
% multi_union_self_other_eq
thf(fact_439_monoid_OUnits__inv__comm,axiom,
! [G: partia8223610829204095565t_unit,X: a,Y: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( ( mult_a_Product_unit @ G @ X @ Y )
= ( one_a_Product_unit @ G ) )
=> ( ( member_a @ X @ ( units_a_Product_unit @ G ) )
=> ( ( member_a @ Y @ ( units_a_Product_unit @ G ) )
=> ( ( mult_a_Product_unit @ G @ Y @ X )
= ( one_a_Product_unit @ G ) ) ) ) ) ) ).
% monoid.Units_inv_comm
thf(fact_440_monoid_OUnits__inv__comm,axiom,
! [G: partia7680350978787392319xt_a_b,X: a,Y: a] :
( ( monoid_a_b @ G )
=> ( ( ( mult_a_b @ G @ X @ Y )
= ( one_a_b @ G ) )
=> ( ( member_a @ X @ ( units_a_b @ G ) )
=> ( ( member_a @ Y @ ( units_a_b @ G ) )
=> ( ( mult_a_b @ G @ Y @ X )
= ( one_a_b @ G ) ) ) ) ) ) ).
% monoid.Units_inv_comm
thf(fact_441_monoid_OUnits__l__inv__ex,axiom,
! [G: partia8223610829204095565t_unit,X: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( member_a @ X @ ( units_a_Product_unit @ G ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia6735698275553448452t_unit @ G ) )
& ( ( mult_a_Product_unit @ G @ X4 @ X )
= ( one_a_Product_unit @ G ) ) ) ) ) ).
% monoid.Units_l_inv_ex
thf(fact_442_monoid_OUnits__l__inv__ex,axiom,
! [G: partia7680350978787392319xt_a_b,X: a] :
( ( monoid_a_b @ G )
=> ( ( member_a @ X @ ( units_a_b @ G ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia7484183841585581558xt_a_b @ G ) )
& ( ( mult_a_b @ G @ X4 @ X )
= ( one_a_b @ G ) ) ) ) ) ).
% monoid.Units_l_inv_ex
thf(fact_443_monoid_OUnits__r__inv__ex,axiom,
! [G: partia8223610829204095565t_unit,X: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( member_a @ X @ ( units_a_Product_unit @ G ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia6735698275553448452t_unit @ G ) )
& ( ( mult_a_Product_unit @ G @ X @ X4 )
= ( one_a_Product_unit @ G ) ) ) ) ) ).
% monoid.Units_r_inv_ex
thf(fact_444_monoid_OUnits__r__inv__ex,axiom,
! [G: partia7680350978787392319xt_a_b,X: a] :
( ( monoid_a_b @ G )
=> ( ( member_a @ X @ ( units_a_b @ G ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia7484183841585581558xt_a_b @ G ) )
& ( ( mult_a_b @ G @ X @ X4 )
= ( one_a_b @ G ) ) ) ) ) ).
% monoid.Units_r_inv_ex
thf(fact_445_monoid__cancel_Oassoc__unit__l,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a] :
( ( monoid1999574367301118026t_unit @ G )
=> ( ( associ6879500422977059064t_unit @ G @ A @ B )
=> ( ( member_a @ B @ ( units_a_Product_unit @ G ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( member_a @ A @ ( units_a_Product_unit @ G ) ) ) ) ) ) ).
% monoid_cancel.assoc_unit_l
thf(fact_446_monoid__cancel_Oassoc__unit__l,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a] :
( ( monoid_cancel_a_b @ G )
=> ( ( associated_a_b @ G @ A @ B )
=> ( ( member_a @ B @ ( units_a_b @ G ) )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( member_a @ A @ ( units_a_b @ G ) ) ) ) ) ) ).
% monoid_cancel.assoc_unit_l
thf(fact_447_monoid__cancel_Oassoc__unit__r,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a] :
( ( monoid1999574367301118026t_unit @ G )
=> ( ( member_a @ A @ ( units_a_Product_unit @ G ) )
=> ( ( associ6879500422977059064t_unit @ G @ A @ B )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( member_a @ B @ ( units_a_Product_unit @ G ) ) ) ) ) ) ).
% monoid_cancel.assoc_unit_r
thf(fact_448_monoid__cancel_Oassoc__unit__r,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a] :
( ( monoid_cancel_a_b @ G )
=> ( ( member_a @ A @ ( units_a_b @ G ) )
=> ( ( associated_a_b @ G @ A @ B )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( member_a @ B @ ( units_a_b @ G ) ) ) ) ) ) ).
% monoid_cancel.assoc_unit_r
thf(fact_449_subset__mset_Oadd__mono,axiom,
! [A: multiset_a,B: multiset_a,C: multiset_a,D: multiset_a] :
( ( subseteq_mset_a @ A @ B )
=> ( ( subseteq_mset_a @ C @ D )
=> ( subseteq_mset_a @ ( plus_plus_multiset_a @ A @ C ) @ ( plus_plus_multiset_a @ B @ D ) ) ) ) ).
% subset_mset.add_mono
thf(fact_450_subset__mset_Oadd__mono,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a,D: multiset_set_a] :
( ( subseteq_mset_set_a @ A @ B )
=> ( ( subseteq_mset_set_a @ C @ D )
=> ( subseteq_mset_set_a @ ( plus_p2331992037799027419_set_a @ A @ C ) @ ( plus_p2331992037799027419_set_a @ B @ D ) ) ) ) ).
% subset_mset.add_mono
thf(fact_451_subset__mset_Oless__eqE,axiom,
! [A: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ A @ B )
=> ~ ! [C2: multiset_a] :
( B
!= ( plus_plus_multiset_a @ A @ C2 ) ) ) ).
% subset_mset.less_eqE
thf(fact_452_subset__mset_Oless__eqE,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( subseteq_mset_set_a @ A @ B )
=> ~ ! [C2: multiset_set_a] :
( B
!= ( plus_p2331992037799027419_set_a @ A @ C2 ) ) ) ).
% subset_mset.less_eqE
thf(fact_453_subset__mset_Ole__iff__add,axiom,
( subseteq_mset_a
= ( ^ [A5: multiset_a,B5: multiset_a] :
? [C5: multiset_a] :
( B5
= ( plus_plus_multiset_a @ A5 @ C5 ) ) ) ) ).
% subset_mset.le_iff_add
thf(fact_454_subset__mset_Ole__iff__add,axiom,
( subseteq_mset_set_a
= ( ^ [A5: multiset_set_a,B5: multiset_set_a] :
? [C5: multiset_set_a] :
( B5
= ( plus_p2331992037799027419_set_a @ A5 @ C5 ) ) ) ) ).
% subset_mset.le_iff_add
thf(fact_455_subset__mset_Oadd__left__mono,axiom,
! [A: multiset_a,B: multiset_a,C: multiset_a] :
( ( subseteq_mset_a @ A @ B )
=> ( subseteq_mset_a @ ( plus_plus_multiset_a @ C @ A ) @ ( plus_plus_multiset_a @ C @ B ) ) ) ).
% subset_mset.add_left_mono
thf(fact_456_subset__mset_Oadd__left__mono,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( subseteq_mset_set_a @ A @ B )
=> ( subseteq_mset_set_a @ ( plus_p2331992037799027419_set_a @ C @ A ) @ ( plus_p2331992037799027419_set_a @ C @ B ) ) ) ).
% subset_mset.add_left_mono
thf(fact_457_subset__mset_Oadd__right__mono,axiom,
! [A: multiset_a,B: multiset_a,C: multiset_a] :
( ( subseteq_mset_a @ A @ B )
=> ( subseteq_mset_a @ ( plus_plus_multiset_a @ A @ C ) @ ( plus_plus_multiset_a @ B @ C ) ) ) ).
% subset_mset.add_right_mono
thf(fact_458_subset__mset_Oadd__right__mono,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( subseteq_mset_set_a @ A @ B )
=> ( subseteq_mset_set_a @ ( plus_p2331992037799027419_set_a @ A @ C ) @ ( plus_p2331992037799027419_set_a @ B @ C ) ) ) ).
% subset_mset.add_right_mono
thf(fact_459_subset__mset_Oadd__le__imp__le__left,axiom,
! [C: multiset_a,A: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ ( plus_plus_multiset_a @ C @ A ) @ ( plus_plus_multiset_a @ C @ B ) )
=> ( subseteq_mset_a @ A @ B ) ) ).
% subset_mset.add_le_imp_le_left
thf(fact_460_subset__mset_Oadd__le__imp__le__left,axiom,
! [C: multiset_set_a,A: multiset_set_a,B: multiset_set_a] :
( ( subseteq_mset_set_a @ ( plus_p2331992037799027419_set_a @ C @ A ) @ ( plus_p2331992037799027419_set_a @ C @ B ) )
=> ( subseteq_mset_set_a @ A @ B ) ) ).
% subset_mset.add_le_imp_le_left
thf(fact_461_subset__mset_Oadd__le__imp__le__right,axiom,
! [A: multiset_a,C: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ ( plus_plus_multiset_a @ A @ C ) @ ( plus_plus_multiset_a @ B @ C ) )
=> ( subseteq_mset_a @ A @ B ) ) ).
% subset_mset.add_le_imp_le_right
thf(fact_462_subset__mset_Oadd__le__imp__le__right,axiom,
! [A: multiset_set_a,C: multiset_set_a,B: multiset_set_a] :
( ( subseteq_mset_set_a @ ( plus_p2331992037799027419_set_a @ A @ C ) @ ( plus_p2331992037799027419_set_a @ B @ C ) )
=> ( subseteq_mset_set_a @ A @ B ) ) ).
% subset_mset.add_le_imp_le_right
thf(fact_463_mset__subset__eq__add__left,axiom,
! [A4: multiset_a,B3: multiset_a] : ( subseteq_mset_a @ A4 @ ( plus_plus_multiset_a @ A4 @ B3 ) ) ).
% mset_subset_eq_add_left
thf(fact_464_mset__subset__eq__add__left,axiom,
! [A4: multiset_set_a,B3: multiset_set_a] : ( subseteq_mset_set_a @ A4 @ ( plus_p2331992037799027419_set_a @ A4 @ B3 ) ) ).
% mset_subset_eq_add_left
thf(fact_465_mset__subset__eq__mono__add,axiom,
! [A4: multiset_set_a,B3: multiset_set_a,C4: multiset_set_a,D2: multiset_set_a] :
( ( subseteq_mset_set_a @ A4 @ B3 )
=> ( ( subseteq_mset_set_a @ C4 @ D2 )
=> ( subseteq_mset_set_a @ ( plus_p2331992037799027419_set_a @ A4 @ C4 ) @ ( plus_p2331992037799027419_set_a @ B3 @ D2 ) ) ) ) ).
% mset_subset_eq_mono_add
thf(fact_466_mset__subset__eq__add__right,axiom,
! [B3: multiset_set_a,A4: multiset_set_a] : ( subseteq_mset_set_a @ B3 @ ( plus_p2331992037799027419_set_a @ A4 @ B3 ) ) ).
% mset_subset_eq_add_right
thf(fact_467_mset__subset__eq__exists__conv,axiom,
( subseteq_mset_set_a
= ( ^ [A6: multiset_set_a,B6: multiset_set_a] :
? [C3: multiset_set_a] :
( B6
= ( plus_p2331992037799027419_set_a @ A6 @ C3 ) ) ) ) ).
% mset_subset_eq_exists_conv
thf(fact_468_insert__mono,axiom,
! [C4: set_a,D2: set_a,A: a] :
( ( ord_less_eq_set_a @ C4 @ D2 )
=> ( ord_less_eq_set_a @ ( insert_a @ A @ C4 ) @ ( insert_a @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_469_subset__insert,axiom,
! [X: set_a,A4: set_set_a,B3: set_set_a] :
( ~ ( member_set_a @ X @ A4 )
=> ( ( ord_le3724670747650509150_set_a @ A4 @ ( insert_set_a @ X @ B3 ) )
= ( ord_le3724670747650509150_set_a @ A4 @ B3 ) ) ) ).
% subset_insert
thf(fact_470_subset__insert,axiom,
! [X: a,A4: set_a,B3: set_a] :
( ~ ( member_a @ X @ A4 )
=> ( ( ord_less_eq_set_a @ A4 @ ( insert_a @ X @ B3 ) )
= ( ord_less_eq_set_a @ A4 @ B3 ) ) ) ).
% subset_insert
thf(fact_471_subset__insertI,axiom,
! [B3: set_a,A: a] : ( ord_less_eq_set_a @ B3 @ ( insert_a @ A @ B3 ) ) ).
% subset_insertI
thf(fact_472_subset__insertI2,axiom,
! [A4: set_a,B3: set_a,B: a] :
( ( ord_less_eq_set_a @ A4 @ B3 )
=> ( ord_less_eq_set_a @ A4 @ ( insert_a @ B @ B3 ) ) ) ).
% subset_insertI2
thf(fact_473_union__iff,axiom,
! [A: a,A4: multiset_a,B3: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ ( plus_plus_multiset_a @ A4 @ B3 ) ) )
= ( ( member_a @ A @ ( set_mset_a @ A4 ) )
| ( member_a @ A @ ( set_mset_a @ B3 ) ) ) ) ).
% union_iff
thf(fact_474_union__iff,axiom,
! [A: set_a,A4: multiset_set_a,B3: multiset_set_a] :
( ( member_set_a @ A @ ( set_mset_set_a @ ( plus_p2331992037799027419_set_a @ A4 @ B3 ) ) )
= ( ( member_set_a @ A @ ( set_mset_set_a @ A4 ) )
| ( member_set_a @ A @ ( set_mset_set_a @ B3 ) ) ) ) ).
% union_iff
thf(fact_475_monoid_OUnits__closed,axiom,
! [G: partia7680350978787392319xt_a_b,X: a] :
( ( monoid_a_b @ G )
=> ( ( member_a @ X @ ( units_a_b @ G ) )
=> ( member_a @ X @ ( partia7484183841585581558xt_a_b @ G ) ) ) ) ).
% monoid.Units_closed
thf(fact_476_monoid_OUnits__m__closed,axiom,
! [G: partia7680350978787392319xt_a_b,X: a,Y: a] :
( ( monoid_a_b @ G )
=> ( ( member_a @ X @ ( units_a_b @ G ) )
=> ( ( member_a @ Y @ ( units_a_b @ G ) )
=> ( member_a @ ( mult_a_b @ G @ X @ Y ) @ ( units_a_b @ G ) ) ) ) ) ).
% monoid.Units_m_closed
thf(fact_477_monoid_Onat__pow__closed,axiom,
! [G: partia8223610829204095565t_unit,X: a,N: nat] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( member_a @ ( pow_a_1875594501834816709it_nat @ G @ X @ N ) @ ( partia6735698275553448452t_unit @ G ) ) ) ) ).
% monoid.nat_pow_closed
thf(fact_478_monoid_Onat__pow__closed,axiom,
! [G: partia7680350978787392319xt_a_b,X: a,N: nat] :
( ( monoid_a_b @ G )
=> ( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( member_a @ ( pow_a_b_nat @ G @ X @ N ) @ ( partia7484183841585581558xt_a_b @ G ) ) ) ) ).
% monoid.nat_pow_closed
thf(fact_479_comm__monoid_OUnit__eq__dividesone,axiom,
! [G: partia7680350978787392319xt_a_b,U: a] :
( ( comm_monoid_a_b @ G )
=> ( ( member_a @ U @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ U @ ( units_a_b @ G ) )
= ( factor_a_b @ G @ U @ ( one_a_b @ G ) ) ) ) ) ).
% comm_monoid.Unit_eq_dividesone
thf(fact_480_factorial__monoid_Owfactors__exist,axiom,
! [G: partia7680350978787392319xt_a_b,A: a] :
( ( factorial_monoid_a_b @ G )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ? [Fs2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Fs2 ) @ ( partia7484183841585581558xt_a_b @ G ) )
& ( wfactors_a_b @ G @ Fs2 @ A ) ) ) ) ).
% factorial_monoid.wfactors_exist
thf(fact_481_monoid__cancel_OassociatedD2,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a] :
( ( monoid_cancel_a_b @ G )
=> ( ( associated_a_b @ G @ A @ B )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( units_a_b @ G ) )
& ( A
= ( mult_a_b @ G @ B @ X4 ) ) ) ) ) ) ) ).
% monoid_cancel.associatedD2
thf(fact_482_monoid__cancel_OassociatedE2,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a] :
( ( monoid_cancel_a_b @ G )
=> ( ( associated_a_b @ G @ A @ B )
=> ( ! [U2: a] :
( ( A
= ( mult_a_b @ G @ B @ U2 ) )
=> ~ ( member_a @ U2 @ ( units_a_b @ G ) ) )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ~ ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) ) ) ) ) ) ).
% monoid_cancel.associatedE2
thf(fact_483_monoid__cancel_Oassociated__iff,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a] :
( ( monoid_cancel_a_b @ G )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( associated_a_b @ G @ A @ B )
= ( ? [X3: a] :
( ( member_a @ X3 @ ( units_a_b @ G ) )
& ( A
= ( mult_a_b @ G @ B @ X3 ) ) ) ) ) ) ) ) ).
% monoid_cancel.associated_iff
thf(fact_484_monoid__cancel_Oaxioms_I1_J,axiom,
! [G: partia7680350978787392319xt_a_b] :
( ( monoid_cancel_a_b @ G )
=> ( monoid_a_b @ G ) ) ).
% monoid_cancel.axioms(1)
thf(fact_485_factorial__monoid_Omult__wfactors__fmset,axiom,
! [G: partia7680350978787392319xt_a_b,As: list_a,A: a,Bs: list_a,B: a,Cs3: list_a] :
( ( factorial_monoid_a_b @ G )
=> ( ( wfactors_a_b @ G @ As @ A )
=> ( ( wfactors_a_b @ G @ Bs @ B )
=> ( ( wfactors_a_b @ G @ Cs3 @ ( mult_a_b @ G @ A @ B ) )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Cs3 ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( fmset_a_b @ G @ Cs3 )
= ( plus_p2331992037799027419_set_a @ ( fmset_a_b @ G @ As ) @ ( fmset_a_b @ G @ Bs ) ) ) ) ) ) ) ) ) ) ) ) ).
% factorial_monoid.mult_wfactors_fmset
thf(fact_486_divisor__chain__condition__monoid_Owfactors__exist,axiom,
! [G: partia7680350978787392319xt_a_b,A: a] :
( ( diviso5244014645414798900id_a_b @ G )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ? [As2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ As2 ) @ ( partia7484183841585581558xt_a_b @ G ) )
& ( wfactors_a_b @ G @ As2 @ A ) ) ) ) ).
% divisor_chain_condition_monoid.wfactors_exist
thf(fact_487_monoid_OUnits__l__cancel,axiom,
! [G: partia7680350978787392319xt_a_b,X: a,Y: a,Z: a] :
( ( monoid_a_b @ G )
=> ( ( member_a @ X @ ( units_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ Z @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ( mult_a_b @ G @ X @ Y )
= ( mult_a_b @ G @ X @ Z ) )
= ( Y = Z ) ) ) ) ) ) ).
% monoid.Units_l_cancel
thf(fact_488_monoid_Owfactors__cong__r,axiom,
! [G: partia7680350978787392319xt_a_b,Fs: list_a,A: a,A3: a] :
( ( monoid_a_b @ G )
=> ( ( wfactors_a_b @ G @ Fs @ A )
=> ( ( associated_a_b @ G @ A @ A3 )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ A3 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( wfactors_a_b @ G @ Fs @ A3 ) ) ) ) ) ) ) ).
% monoid.wfactors_cong_r
thf(fact_489_Group_Omonoid_Ointro,axiom,
! [G: partia7680350978787392319xt_a_b] :
( ! [X4: a,Y3: a] :
( ( member_a @ X4 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ Y3 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( member_a @ ( mult_a_b @ G @ X4 @ Y3 ) @ ( partia7484183841585581558xt_a_b @ G ) ) ) )
=> ( ! [X4: a,Y3: a,Z3: a] :
( ( member_a @ X4 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ Y3 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ Z3 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( mult_a_b @ G @ ( mult_a_b @ G @ X4 @ Y3 ) @ Z3 )
= ( mult_a_b @ G @ X4 @ ( mult_a_b @ G @ Y3 @ Z3 ) ) ) ) ) )
=> ( ( member_a @ ( one_a_b @ G ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( mult_a_b @ G @ ( one_a_b @ G ) @ X4 )
= X4 ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( mult_a_b @ G @ X4 @ ( one_a_b @ G ) )
= X4 ) )
=> ( monoid_a_b @ G ) ) ) ) ) ) ).
% Group.monoid.intro
thf(fact_490_monoid_Oinv__unique,axiom,
! [G: partia7680350978787392319xt_a_b,Y: a,X: a,Y2: a] :
( ( monoid_a_b @ G )
=> ( ( ( mult_a_b @ G @ Y @ X )
= ( one_a_b @ G ) )
=> ( ( ( mult_a_b @ G @ X @ Y2 )
= ( one_a_b @ G ) )
=> ( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ Y2 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( Y = Y2 ) ) ) ) ) ) ) ).
% monoid.inv_unique
thf(fact_491_monoid_Oone__unique,axiom,
! [G: partia7680350978787392319xt_a_b,U: a] :
( ( monoid_a_b @ G )
=> ( ( member_a @ U @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( mult_a_b @ G @ U @ X4 )
= X4 ) )
=> ( U
= ( one_a_b @ G ) ) ) ) ) ).
% monoid.one_unique
thf(fact_492_monoidI,axiom,
! [G: partia7680350978787392319xt_a_b] :
( ! [X4: a,Y3: a] :
( ( member_a @ X4 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ Y3 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( member_a @ ( mult_a_b @ G @ X4 @ Y3 ) @ ( partia7484183841585581558xt_a_b @ G ) ) ) )
=> ( ( member_a @ ( one_a_b @ G ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ! [X4: a,Y3: a,Z3: a] :
( ( member_a @ X4 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ Y3 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ Z3 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( mult_a_b @ G @ ( mult_a_b @ G @ X4 @ Y3 ) @ Z3 )
= ( mult_a_b @ G @ X4 @ ( mult_a_b @ G @ Y3 @ Z3 ) ) ) ) ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( mult_a_b @ G @ ( one_a_b @ G ) @ X4 )
= X4 ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( mult_a_b @ G @ X4 @ ( one_a_b @ G ) )
= X4 ) )
=> ( monoid_a_b @ G ) ) ) ) ) ) ).
% monoidI
thf(fact_493_Group_Omonoid__def,axiom,
( monoid_a_b
= ( ^ [G2: partia7680350978787392319xt_a_b] :
( ! [X3: a,Y5: a] :
( ( member_a @ X3 @ ( partia7484183841585581558xt_a_b @ G2 ) )
=> ( ( member_a @ Y5 @ ( partia7484183841585581558xt_a_b @ G2 ) )
=> ( member_a @ ( mult_a_b @ G2 @ X3 @ Y5 ) @ ( partia7484183841585581558xt_a_b @ G2 ) ) ) )
& ! [X3: a,Y5: a,Z4: a] :
( ( member_a @ X3 @ ( partia7484183841585581558xt_a_b @ G2 ) )
=> ( ( member_a @ Y5 @ ( partia7484183841585581558xt_a_b @ G2 ) )
=> ( ( member_a @ Z4 @ ( partia7484183841585581558xt_a_b @ G2 ) )
=> ( ( mult_a_b @ G2 @ ( mult_a_b @ G2 @ X3 @ Y5 ) @ Z4 )
= ( mult_a_b @ G2 @ X3 @ ( mult_a_b @ G2 @ Y5 @ Z4 ) ) ) ) ) )
& ( member_a @ ( one_a_b @ G2 ) @ ( partia7484183841585581558xt_a_b @ G2 ) )
& ! [X3: a] :
( ( member_a @ X3 @ ( partia7484183841585581558xt_a_b @ G2 ) )
=> ( ( mult_a_b @ G2 @ ( one_a_b @ G2 ) @ X3 )
= X3 ) )
& ! [X3: a] :
( ( member_a @ X3 @ ( partia7484183841585581558xt_a_b @ G2 ) )
=> ( ( mult_a_b @ G2 @ X3 @ ( one_a_b @ G2 ) )
= X3 ) ) ) ) ) ).
% Group.monoid_def
thf(fact_494_comm__monoid_Owfactors__dividesI,axiom,
! [G: partia7680350978787392319xt_a_b,Fs: list_a,A: a,F: a] :
( ( comm_monoid_a_b @ G )
=> ( ( wfactors_a_b @ G @ Fs @ A )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ F @ ( set_a2 @ Fs ) )
=> ( factor_a_b @ G @ F @ A ) ) ) ) ) ) ).
% comm_monoid.wfactors_dividesI
thf(fact_495_monoid_Onat__pow__comm,axiom,
! [G: partia8223610829204095565t_unit,X: a,N: nat,M4: nat] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( mult_a_Product_unit @ G @ ( pow_a_1875594501834816709it_nat @ G @ X @ N ) @ ( pow_a_1875594501834816709it_nat @ G @ X @ M4 ) )
= ( mult_a_Product_unit @ G @ ( pow_a_1875594501834816709it_nat @ G @ X @ M4 ) @ ( pow_a_1875594501834816709it_nat @ G @ X @ N ) ) ) ) ) ).
% monoid.nat_pow_comm
thf(fact_496_monoid_Onat__pow__comm,axiom,
! [G: partia7680350978787392319xt_a_b,X: a,N: nat,M4: nat] :
( ( monoid_a_b @ G )
=> ( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( mult_a_b @ G @ ( pow_a_b_nat @ G @ X @ N ) @ ( pow_a_b_nat @ G @ X @ M4 ) )
= ( mult_a_b @ G @ ( pow_a_b_nat @ G @ X @ M4 ) @ ( pow_a_b_nat @ G @ X @ N ) ) ) ) ) ).
% monoid.nat_pow_comm
thf(fact_497_monoid_Opow__mult__distrib,axiom,
! [G: partia8223610829204095565t_unit,X: a,Y: a,N: nat] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( ( mult_a_Product_unit @ G @ X @ Y )
= ( mult_a_Product_unit @ G @ Y @ X ) )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( pow_a_1875594501834816709it_nat @ G @ ( mult_a_Product_unit @ G @ X @ Y ) @ N )
= ( mult_a_Product_unit @ G @ ( pow_a_1875594501834816709it_nat @ G @ X @ N ) @ ( pow_a_1875594501834816709it_nat @ G @ Y @ N ) ) ) ) ) ) ) ).
% monoid.pow_mult_distrib
thf(fact_498_monoid_Opow__mult__distrib,axiom,
! [G: partia7680350978787392319xt_a_b,X: a,Y: a,N: nat] :
( ( monoid_a_b @ G )
=> ( ( ( mult_a_b @ G @ X @ Y )
= ( mult_a_b @ G @ Y @ X ) )
=> ( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( pow_a_b_nat @ G @ ( mult_a_b @ G @ X @ Y ) @ N )
= ( mult_a_b @ G @ ( pow_a_b_nat @ G @ X @ N ) @ ( pow_a_b_nat @ G @ Y @ N ) ) ) ) ) ) ) ).
% monoid.pow_mult_distrib
thf(fact_499_monoid_Ogroup__commutes__pow,axiom,
! [G: partia8223610829204095565t_unit,X: a,Y: a,N: nat] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( ( mult_a_Product_unit @ G @ X @ Y )
= ( mult_a_Product_unit @ G @ Y @ X ) )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( mult_a_Product_unit @ G @ ( pow_a_1875594501834816709it_nat @ G @ X @ N ) @ Y )
= ( mult_a_Product_unit @ G @ Y @ ( pow_a_1875594501834816709it_nat @ G @ X @ N ) ) ) ) ) ) ) ).
% monoid.group_commutes_pow
thf(fact_500_monoid_Ogroup__commutes__pow,axiom,
! [G: partia7680350978787392319xt_a_b,X: a,Y: a,N: nat] :
( ( monoid_a_b @ G )
=> ( ( ( mult_a_b @ G @ X @ Y )
= ( mult_a_b @ G @ Y @ X ) )
=> ( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( mult_a_b @ G @ ( pow_a_b_nat @ G @ X @ N ) @ Y )
= ( mult_a_b @ G @ Y @ ( pow_a_b_nat @ G @ X @ N ) ) ) ) ) ) ) ).
% monoid.group_commutes_pow
thf(fact_501_comm__monoidI,axiom,
! [G: partia7680350978787392319xt_a_b] :
( ! [X4: a,Y3: a] :
( ( member_a @ X4 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ Y3 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( member_a @ ( mult_a_b @ G @ X4 @ Y3 ) @ ( partia7484183841585581558xt_a_b @ G ) ) ) )
=> ( ( member_a @ ( one_a_b @ G ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ! [X4: a,Y3: a,Z3: a] :
( ( member_a @ X4 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ Y3 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ Z3 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( mult_a_b @ G @ ( mult_a_b @ G @ X4 @ Y3 ) @ Z3 )
= ( mult_a_b @ G @ X4 @ ( mult_a_b @ G @ Y3 @ Z3 ) ) ) ) ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( mult_a_b @ G @ ( one_a_b @ G ) @ X4 )
= X4 ) )
=> ( ! [X4: a,Y3: a] :
( ( member_a @ X4 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ Y3 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( mult_a_b @ G @ X4 @ Y3 )
= ( mult_a_b @ G @ Y3 @ X4 ) ) ) )
=> ( comm_monoid_a_b @ G ) ) ) ) ) ) ).
% comm_monoidI
thf(fact_502_monoid_Owfactors__prod__exists,axiom,
! [G: partia7680350978787392319xt_a_b,As: list_a] :
( ( monoid_a_b @ G )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ As ) )
=> ( irreducible_a_b @ G @ X4 ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ? [A2: a] :
( ( member_a @ A2 @ ( partia7484183841585581558xt_a_b @ G ) )
& ( wfactors_a_b @ G @ As @ A2 ) ) ) ) ) ).
% monoid.wfactors_prod_exists
thf(fact_503_comm__monoid_Onat__pow__distrib,axiom,
! [G: partia8223610829204095565t_unit,X: a,Y: a,N: nat] :
( ( comm_m7681468956318391052t_unit @ G )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( pow_a_1875594501834816709it_nat @ G @ ( mult_a_Product_unit @ G @ X @ Y ) @ N )
= ( mult_a_Product_unit @ G @ ( pow_a_1875594501834816709it_nat @ G @ X @ N ) @ ( pow_a_1875594501834816709it_nat @ G @ Y @ N ) ) ) ) ) ) ).
% comm_monoid.nat_pow_distrib
thf(fact_504_comm__monoid_Onat__pow__distrib,axiom,
! [G: partia7680350978787392319xt_a_b,X: a,Y: a,N: nat] :
( ( comm_monoid_a_b @ G )
=> ( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( pow_a_b_nat @ G @ ( mult_a_b @ G @ X @ Y ) @ N )
= ( mult_a_b @ G @ ( pow_a_b_nat @ G @ X @ N ) @ ( pow_a_b_nat @ G @ Y @ N ) ) ) ) ) ) ).
% comm_monoid.nat_pow_distrib
thf(fact_505_monoid_OUnits__assoc,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a] :
( ( monoid_a_b @ G )
=> ( ( member_a @ A @ ( units_a_b @ G ) )
=> ( ( member_a @ B @ ( units_a_b @ G ) )
=> ( associated_a_b @ G @ A @ B ) ) ) ) ).
% monoid.Units_assoc
thf(fact_506_subset__singletonD,axiom,
! [A4: set_a,X: a] :
( ( ord_less_eq_set_a @ A4 @ ( insert_a @ X @ bot_bot_set_a ) )
=> ( ( A4 = bot_bot_set_a )
| ( A4
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_507_subset__singleton__iff,axiom,
! [X7: set_a,A: a] :
( ( ord_less_eq_set_a @ X7 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( ( X7 = bot_bot_set_a )
| ( X7
= ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_508_set__mset__mono,axiom,
! [A4: multiset_set_a,B3: multiset_set_a] :
( ( subseteq_mset_set_a @ A4 @ B3 )
=> ( ord_le3724670747650509150_set_a @ ( set_mset_set_a @ A4 ) @ ( set_mset_set_a @ B3 ) ) ) ).
% set_mset_mono
thf(fact_509_set__mset__mono,axiom,
! [A4: multiset_a,B3: multiset_a] :
( ( subseteq_mset_a @ A4 @ B3 )
=> ( ord_less_eq_set_a @ ( set_mset_a @ A4 ) @ ( set_mset_a @ B3 ) ) ) ).
% set_mset_mono
thf(fact_510_factorial__monoid_Oassoc__as__fmset__eq,axiom,
! [G: partia7680350978787392319xt_a_b,As: list_a,A: a,Bs: list_a,B: a] :
( ( factorial_monoid_a_b @ G )
=> ( ( wfactors_a_b @ G @ As @ A )
=> ( ( wfactors_a_b @ G @ Bs @ B )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( associated_a_b @ G @ A @ B )
= ( ( fmset_a_b @ G @ As )
= ( fmset_a_b @ G @ Bs ) ) ) ) ) ) ) ) ) ) ).
% factorial_monoid.assoc_as_fmset_eq
thf(fact_511_factorial__monoid_Ofactor__mset__aux__1,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,As: list_a] :
( ( factorial_monoid_a_b @ G )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( wfactors_a_b @ G @ As @ A )
=> ( ( finite8971978181520403909et_a_b @ G @ A )
= ( fmset_a_b @ G @ As ) ) ) ) ) ) ).
% factorial_monoid.factor_mset_aux_1
thf(fact_512_factorial__monoid_Ofactor__mset__aux,axiom,
! [G: partia7680350978787392319xt_a_b,A: a] :
( ( factorial_monoid_a_b @ G )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ? [As2: list_a] :
( ( ( finite8971978181520403909et_a_b @ G @ A )
= ( fmset_a_b @ G @ As2 ) )
& ( wfactors_a_b @ G @ As2 @ A )
& ( ord_less_eq_set_a @ ( set_a2 @ As2 ) @ ( partia7484183841585581558xt_a_b @ G ) ) ) ) ) ).
% factorial_monoid.factor_mset_aux
thf(fact_513_monoid__cancel_Or__cancel,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,C: a,B: a] :
( ( monoid_cancel_a_b @ G )
=> ( ( ( mult_a_b @ G @ A @ C )
= ( mult_a_b @ G @ B @ C ) )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( A = B ) ) ) ) ) ) ).
% monoid_cancel.r_cancel
thf(fact_514_monoid__cancel_Ol__cancel,axiom,
! [G: partia7680350978787392319xt_a_b,C: a,A: a,B: a] :
( ( monoid_cancel_a_b @ G )
=> ( ( ( mult_a_b @ G @ C @ A )
= ( mult_a_b @ G @ C @ B ) )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( A = B ) ) ) ) ) ) ).
% monoid_cancel.l_cancel
thf(fact_515_monoid_Oprod__unit__l,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a] :
( ( monoid_a_b @ G )
=> ( ( member_a @ ( mult_a_b @ G @ A @ B ) @ ( units_a_b @ G ) )
=> ( ( member_a @ A @ ( units_a_b @ G ) )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( member_a @ B @ ( units_a_b @ G ) ) ) ) ) ) ) ).
% monoid.prod_unit_l
thf(fact_516_monoid_Oprod__unit__r,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a] :
( ( monoid_a_b @ G )
=> ( ( member_a @ ( mult_a_b @ G @ A @ B ) @ ( units_a_b @ G ) )
=> ( ( member_a @ B @ ( units_a_b @ G ) )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( member_a @ A @ ( units_a_b @ G ) ) ) ) ) ) ) ).
% monoid.prod_unit_r
thf(fact_517_comm__monoid_Ounit__factor,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a] :
( ( comm_monoid_a_b @ G )
=> ( ( member_a @ ( mult_a_b @ G @ A @ B ) @ ( units_a_b @ G ) )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( member_a @ A @ ( units_a_b @ G ) ) ) ) ) ) ).
% comm_monoid.unit_factor
thf(fact_518_monoid_Ounit__divides,axiom,
! [G: partia7680350978787392319xt_a_b,U: a,A: a] :
( ( monoid_a_b @ G )
=> ( ( member_a @ U @ ( units_a_b @ G ) )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( factor_a_b @ G @ U @ A ) ) ) ) ).
% monoid.unit_divides
thf(fact_519_comm__monoid_Odivides__unit,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,U: a] :
( ( comm_monoid_a_b @ G )
=> ( ( factor_a_b @ G @ A @ U )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ U @ ( units_a_b @ G ) )
=> ( member_a @ A @ ( units_a_b @ G ) ) ) ) ) ) ).
% comm_monoid.divides_unit
thf(fact_520_comm__monoid_OUnits__cong,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a] :
( ( comm_monoid_a_b @ G )
=> ( ( member_a @ A @ ( units_a_b @ G ) )
=> ( ( associated_a_b @ G @ A @ B )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( member_a @ B @ ( units_a_b @ G ) ) ) ) ) ) ).
% comm_monoid.Units_cong
thf(fact_521_factorial__monoid_Odivides__fmsubset,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a,As: list_a,Bs: list_a] :
( ( factorial_monoid_a_b @ G )
=> ( ( factor_a_b @ G @ A @ B )
=> ( ( wfactors_a_b @ G @ As @ A )
=> ( ( wfactors_a_b @ G @ Bs @ B )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( subseteq_mset_set_a @ ( fmset_a_b @ G @ As ) @ ( fmset_a_b @ G @ Bs ) ) ) ) ) ) ) ) ) ) ).
% factorial_monoid.divides_fmsubset
thf(fact_522_factorial__monoid_Odivides__as__fmsubset,axiom,
! [G: partia7680350978787392319xt_a_b,As: list_a,A: a,Bs: list_a,B: a] :
( ( factorial_monoid_a_b @ G )
=> ( ( wfactors_a_b @ G @ As @ A )
=> ( ( wfactors_a_b @ G @ Bs @ B )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( factor_a_b @ G @ A @ B )
= ( subseteq_mset_set_a @ ( fmset_a_b @ G @ As ) @ ( fmset_a_b @ G @ Bs ) ) ) ) ) ) ) ) ) ) ).
% factorial_monoid.divides_as_fmsubset
thf(fact_523_monoid_OassociatedI2_H,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a,U: a] :
( ( monoid_a_b @ G )
=> ( ( A
= ( mult_a_b @ G @ B @ U ) )
=> ( ( member_a @ U @ ( units_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( associated_a_b @ G @ A @ B ) ) ) ) ) ).
% monoid.associatedI2'
thf(fact_524_monoid_OassociatedI2,axiom,
! [G: partia7680350978787392319xt_a_b,U: a,A: a,B: a] :
( ( monoid_a_b @ G )
=> ( ( member_a @ U @ ( units_a_b @ G ) )
=> ( ( A
= ( mult_a_b @ G @ B @ U ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( associated_a_b @ G @ A @ B ) ) ) ) ) ).
% monoid.associatedI2
thf(fact_525_monoid_Oirreducible__prod__rI,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a] :
( ( monoid_a_b @ G )
=> ( ( irreducible_a_b @ G @ A )
=> ( ( member_a @ B @ ( units_a_b @ G ) )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( irreducible_a_b @ G @ ( mult_a_b @ G @ A @ B ) ) ) ) ) ) ) ).
% monoid.irreducible_prod_rI
thf(fact_526_comm__monoid_Oirreducible__prod__lI,axiom,
! [G: partia7680350978787392319xt_a_b,B: a,A: a] :
( ( comm_monoid_a_b @ G )
=> ( ( irreducible_a_b @ G @ B )
=> ( ( member_a @ A @ ( units_a_b @ G ) )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( irreducible_a_b @ G @ ( mult_a_b @ G @ A @ B ) ) ) ) ) ) ) ).
% comm_monoid.irreducible_prod_lI
thf(fact_527_divides__irreducible__condition,axiom,
! [G: partia7680350978787392319xt_a_b,R: a,A: a] :
( ( irreducible_a_b @ G @ R )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( factor_a_b @ G @ A @ R )
=> ( ( member_a @ A @ ( units_a_b @ G ) )
| ( associated_a_b @ G @ A @ R ) ) ) ) ) ).
% divides_irreducible_condition
thf(fact_528_primeE,axiom,
! [G: partia7680350978787392319xt_a_b,P: a] :
( ( prime_a_b @ G @ P )
=> ~ ( ~ ( member_a @ P @ ( units_a_b @ G ) )
=> ~ ! [X6: a] :
( ( member_a @ X6 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ! [Xa: a] :
( ( member_a @ Xa @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( factor_a_b @ G @ P @ ( mult_a_b @ G @ X6 @ Xa ) )
=> ( ( factor_a_b @ G @ P @ X6 )
| ( factor_a_b @ G @ P @ Xa ) ) ) ) ) ) ) ).
% primeE
thf(fact_529_primeI,axiom,
! [P: a,G: partia7680350978787392319xt_a_b] :
( ~ ( member_a @ P @ ( units_a_b @ G ) )
=> ( ! [A2: a,B2: a] :
( ( member_a @ A2 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B2 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( factor_a_b @ G @ P @ ( mult_a_b @ G @ A2 @ B2 ) )
=> ( ( factor_a_b @ G @ P @ A2 )
| ( factor_a_b @ G @ P @ B2 ) ) ) ) )
=> ( prime_a_b @ G @ P ) ) ) ).
% primeI
thf(fact_530_prime__def,axiom,
( prime_a_b
= ( ^ [G2: partia7680350978787392319xt_a_b,P4: a] :
( ~ ( member_a @ P4 @ ( units_a_b @ G2 ) )
& ! [X3: a] :
( ( member_a @ X3 @ ( partia7484183841585581558xt_a_b @ G2 ) )
=> ! [Y5: a] :
( ( member_a @ Y5 @ ( partia7484183841585581558xt_a_b @ G2 ) )
=> ( ( factor_a_b @ G2 @ P4 @ ( mult_a_b @ G2 @ X3 @ Y5 ) )
=> ( ( factor_a_b @ G2 @ P4 @ X3 )
| ( factor_a_b @ G2 @ P4 @ Y5 ) ) ) ) ) ) ) ) ).
% prime_def
thf(fact_531_monoid__cancel_Odivides__mult__l,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a,C: a] :
( ( monoid_cancel_a_b @ G )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( factor_a_b @ G @ ( mult_a_b @ G @ C @ A ) @ ( mult_a_b @ G @ C @ B ) )
= ( factor_a_b @ G @ A @ B ) ) ) ) ) ) ).
% monoid_cancel.divides_mult_l
thf(fact_532_monoid__cancel_Oassoc__l__cancel,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a,B7: a] :
( ( monoid_cancel_a_b @ G )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B7 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( associated_a_b @ G @ ( mult_a_b @ G @ A @ B ) @ ( mult_a_b @ G @ A @ B7 ) )
=> ( associated_a_b @ G @ B @ B7 ) ) ) ) ) ) ).
% monoid_cancel.assoc_l_cancel
thf(fact_533_monoid_Omonoid__cancelI,axiom,
! [G: partia7680350978787392319xt_a_b] :
( ( monoid_a_b @ G )
=> ( ! [A2: a,B2: a,C2: a] :
( ( ( mult_a_b @ G @ C2 @ A2 )
= ( mult_a_b @ G @ C2 @ B2 ) )
=> ( ( member_a @ A2 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B2 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ C2 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( A2 = B2 ) ) ) ) )
=> ( ! [A2: a,B2: a,C2: a] :
( ( ( mult_a_b @ G @ A2 @ C2 )
= ( mult_a_b @ G @ B2 @ C2 ) )
=> ( ( member_a @ A2 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B2 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ C2 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( A2 = B2 ) ) ) ) )
=> ( monoid_cancel_a_b @ G ) ) ) ) ).
% monoid.monoid_cancelI
thf(fact_534_monoid__cancel_Oirreducible__cong,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,A3: a] :
( ( monoid_cancel_a_b @ G )
=> ( ( irreducible_a_b @ G @ A )
=> ( ( associated_a_b @ G @ A @ A3 )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ A3 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( irreducible_a_b @ G @ A3 ) ) ) ) ) ) ).
% monoid_cancel.irreducible_cong
thf(fact_535_monoid__cancel_Oprime__cong,axiom,
! [G: partia7680350978787392319xt_a_b,P: a,P2: a] :
( ( monoid_cancel_a_b @ G )
=> ( ( prime_a_b @ G @ P )
=> ( ( associated_a_b @ G @ P @ P2 )
=> ( ( member_a @ P @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ P2 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( prime_a_b @ G @ P2 ) ) ) ) ) ) ).
% monoid_cancel.prime_cong
thf(fact_536_factorial__monoid_Ofactor__mset__mult,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a] :
( ( factorial_monoid_a_b @ G )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( finite8971978181520403909et_a_b @ G @ ( mult_a_b @ G @ A @ B ) )
= ( plus_p2331992037799027419_set_a @ ( finite8971978181520403909et_a_b @ G @ A ) @ ( finite8971978181520403909et_a_b @ G @ B ) ) ) ) ) ) ).
% factorial_monoid.factor_mset_mult
thf(fact_537_gcd__condition__monoid_Orelprime__mult,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a,C: a] :
( ( gcd_co4024338386749376309id_a_b @ G )
=> ( ( associated_a_b @ G @ ( somegcd_a_b @ G @ A @ B ) @ ( one_a_b @ G ) )
=> ( ( associated_a_b @ G @ ( somegcd_a_b @ G @ A @ C ) @ ( one_a_b @ G ) )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( associated_a_b @ G @ ( somegcd_a_b @ G @ A @ ( mult_a_b @ G @ B @ C ) ) @ ( one_a_b @ G ) ) ) ) ) ) ) ) ).
% gcd_condition_monoid.relprime_mult
thf(fact_538_Group_Ocomm__monoid_Oaxioms_I1_J,axiom,
! [G: partia7680350978787392319xt_a_b] :
( ( comm_monoid_a_b @ G )
=> ( monoid_a_b @ G ) ) ).
% Group.comm_monoid.axioms(1)
thf(fact_539_monoid_Omset__wfactorsEx,axiom,
! [G: partia7680350978787392319xt_a_b,Cs: multiset_set_a] :
( ( monoid_a_b @ G )
=> ( ! [X5: set_a] :
( ( member_set_a @ X5 @ ( set_mset_set_a @ Cs ) )
=> ? [X6: a] :
( ( member_a @ X6 @ ( partia7484183841585581558xt_a_b @ G ) )
& ( irreducible_a_b @ G @ X6 )
& ( X5
= ( eq_clo186784744570230005t_unit @ ( partia6383399360842404908t_unit @ ( partia7484183841585581558xt_a_b @ G ) @ ( eq_eq_5233546288009311946t_unit @ ( associated_a_b @ G ) @ ( gorder6800608520584131120t_unit @ ( factor_a_b @ G ) @ product_Unity ) ) ) @ ( insert_a @ X6 @ bot_bot_set_a ) ) ) ) )
=> ? [C2: a,Cs2: list_a] :
( ( member_a @ C2 @ ( partia7484183841585581558xt_a_b @ G ) )
& ( ord_less_eq_set_a @ ( set_a2 @ Cs2 ) @ ( partia7484183841585581558xt_a_b @ G ) )
& ( wfactors_a_b @ G @ Cs2 @ C2 )
& ( ( fmset_a_b @ G @ Cs2 )
= Cs ) ) ) ) ).
% monoid.mset_wfactorsEx
thf(fact_540_monoid_Ocarrier__not__empty,axiom,
! [G: partia7680350978787392319xt_a_b] :
( ( monoid_a_b @ G )
=> ( ( partia7484183841585581558xt_a_b @ G )
!= bot_bot_set_a ) ) ).
% monoid.carrier_not_empty
thf(fact_541_monoid_Om__assoc,axiom,
! [G: partia7680350978787392319xt_a_b,X: a,Y: a,Z: a] :
( ( monoid_a_b @ G )
=> ( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ Z @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( mult_a_b @ G @ ( mult_a_b @ G @ X @ Y ) @ Z )
= ( mult_a_b @ G @ X @ ( mult_a_b @ G @ Y @ Z ) ) ) ) ) ) ) ).
% monoid.m_assoc
thf(fact_542_monoid_Om__closed,axiom,
! [G: partia7680350978787392319xt_a_b,X: a,Y: a] :
( ( monoid_a_b @ G )
=> ( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( member_a @ ( mult_a_b @ G @ X @ Y ) @ ( partia7484183841585581558xt_a_b @ G ) ) ) ) ) ).
% monoid.m_closed
thf(fact_543_comm__monoid_Om__ac_I1_J,axiom,
! [G: partia7680350978787392319xt_a_b,X: a,Y: a,Z: a] :
( ( comm_monoid_a_b @ G )
=> ( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ Z @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( mult_a_b @ G @ ( mult_a_b @ G @ X @ Y ) @ Z )
= ( mult_a_b @ G @ X @ ( mult_a_b @ G @ Y @ Z ) ) ) ) ) ) ) ).
% comm_monoid.m_ac(1)
thf(fact_544_comm__monoid_Om__comm,axiom,
! [G: partia7680350978787392319xt_a_b,X: a,Y: a] :
( ( comm_monoid_a_b @ G )
=> ( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( mult_a_b @ G @ X @ Y )
= ( mult_a_b @ G @ Y @ X ) ) ) ) ) ).
% comm_monoid.m_comm
thf(fact_545_comm__monoid_Om__lcomm,axiom,
! [G: partia7680350978787392319xt_a_b,X: a,Y: a,Z: a] :
( ( comm_monoid_a_b @ G )
=> ( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ Z @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( mult_a_b @ G @ X @ ( mult_a_b @ G @ Y @ Z ) )
= ( mult_a_b @ G @ Y @ ( mult_a_b @ G @ X @ Z ) ) ) ) ) ) ) ).
% comm_monoid.m_lcomm
thf(fact_546_Units__are__ones,axiom,
set_eq3287599968972363709t_unit @ ( partia6383399360842404908t_unit @ ( partia7484183841585581558xt_a_b @ g ) @ ( eq_eq_5233546288009311946t_unit @ ( associated_a_b @ g ) @ ( gorder6800608520584131120t_unit @ ( factor_a_b @ g ) @ product_Unity ) ) ) @ ( units_a_b @ g ) @ ( insert_a @ ( one_a_b @ g ) @ bot_bot_set_a ) ).
% Units_are_ones
thf(fact_547_Units__Lower,axiom,
( ( units_a_b @ g )
= ( lower_a_Product_unit @ ( partia6383399360842404908t_unit @ ( partia7484183841585581558xt_a_b @ g ) @ ( eq_eq_5233546288009311946t_unit @ ( associated_a_b @ g ) @ ( gorder6800608520584131120t_unit @ ( factor_a_b @ g ) @ product_Unity ) ) ) @ ( partia7484183841585581558xt_a_b @ g ) ) ) ).
% Units_Lower
thf(fact_548_factorial__monoidI,axiom,
( ! [A2: a] :
( ( member_a @ A2 @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ~ ( member_a @ A2 @ ( units_a_b @ g ) )
=> ? [Fs3: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Fs3 ) @ ( partia7484183841585581558xt_a_b @ g ) )
& ( wfactors_a_b @ g @ Fs3 @ A2 ) ) ) )
=> ( ! [A2: a,Fs2: list_a,Fs4: list_a] :
( ( member_a @ A2 @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs2 ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs4 ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( wfactors_a_b @ g @ Fs2 @ A2 )
=> ( ( wfactors_a_b @ g @ Fs4 @ A2 )
=> ( essent1248755695173879485al_a_b @ g @ Fs2 @ Fs4 ) ) ) ) ) )
=> ( factorial_monoid_a_b @ g ) ) ) ).
% factorial_monoidI
thf(fact_549_mult__factors__fmset,axiom,
! [As: list_a,A: a,Bs: list_a,B: a,Cs3: list_a] :
( ( factors_a_b @ g @ As @ A )
=> ( ( factors_a_b @ g @ Bs @ B )
=> ( ( factors_a_b @ g @ Cs3 @ ( mult_a_b @ g @ A @ B ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Cs3 ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( fmset_a_b @ g @ Cs3 )
= ( plus_p2331992037799027419_set_a @ ( fmset_a_b @ g @ As ) @ ( fmset_a_b @ g @ Bs ) ) ) ) ) ) ) ) ) ).
% mult_factors_fmset
thf(fact_550_multlist__dividesI,axiom,
! [F: a,Fs: list_a] :
( ( member_a @ F @ ( set_a2 @ Fs ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( factor_a_b @ g @ F @ ( foldr_a_a @ ( mult_a_b @ g ) @ Fs @ ( one_a_b @ g ) ) ) ) ) ).
% multlist_dividesI
thf(fact_551_wfactors__mult__single,axiom,
! [A: a,Fb: list_a,B: a] :
( ( irreducible_a_b @ g @ A )
=> ( ( wfactors_a_b @ g @ Fb @ B )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fb ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( wfactors_a_b @ g @ ( cons_a @ A @ Fb ) @ ( mult_a_b @ g @ A @ B ) ) ) ) ) ) ) ).
% wfactors_mult_single
thf(fact_552_fmset__properfactor,axiom,
! [As: list_a,Bs: list_a,A: a,B: a] :
( ( subseteq_mset_set_a @ ( fmset_a_b @ g @ As ) @ ( fmset_a_b @ g @ Bs ) )
=> ( ( ( fmset_a_b @ g @ As )
!= ( fmset_a_b @ g @ Bs ) )
=> ( ( wfactors_a_b @ g @ As @ A )
=> ( ( wfactors_a_b @ g @ Bs @ B )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( properfactor_a_b @ g @ A @ B ) ) ) ) ) ) ) ) ) ).
% fmset_properfactor
thf(fact_553_properfactor__fmset,axiom,
! [A: a,B: a,As: list_a,Bs: list_a] :
( ( properfactor_a_b @ g @ A @ B )
=> ( ( wfactors_a_b @ g @ As @ A )
=> ( ( wfactors_a_b @ g @ Bs @ B )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( subseteq_mset_set_a @ ( fmset_a_b @ g @ As ) @ ( fmset_a_b @ g @ Bs ) ) ) ) ) ) ) ) ) ).
% properfactor_fmset
thf(fact_554_properfactor__divides,axiom,
! [A: a,B: a] :
( ( properfactor_a_b @ g @ A @ B )
=> ( factor_a_b @ g @ A @ B ) ) ).
% properfactor_divides
thf(fact_555_properfactor__prod__r,axiom,
! [A: a,B: a,C: a] :
( ( properfactor_a_b @ g @ A @ B )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ C @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( properfactor_a_b @ g @ A @ ( mult_a_b @ g @ B @ C ) ) ) ) ) ) ).
% properfactor_prod_r
thf(fact_556_properfactor__prod__l,axiom,
! [A: a,B: a,C: a] :
( ( properfactor_a_b @ g @ A @ B )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ C @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( properfactor_a_b @ g @ A @ ( mult_a_b @ g @ C @ B ) ) ) ) ) ) ).
% properfactor_prod_l
thf(fact_557_properfactor__cong__r,axiom,
! [X: a,Y: a,Y2: a] :
( ( properfactor_a_b @ g @ X @ Y )
=> ( ( associated_a_b @ g @ Y @ Y2 )
=> ( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ Y @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ Y2 @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( properfactor_a_b @ g @ X @ Y2 ) ) ) ) ) ) ).
% properfactor_cong_r
thf(fact_558_properfactor__cong__l,axiom,
! [X2: a,X: a,Y: a] :
( ( associated_a_b @ g @ X2 @ X )
=> ( ( properfactor_a_b @ g @ X @ Y )
=> ( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ X2 @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ Y @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( properfactor_a_b @ g @ X2 @ Y ) ) ) ) ) ) ).
% properfactor_cong_l
thf(fact_559_properfactor__trans2,axiom,
! [A: a,B: a,C: a] :
( ( properfactor_a_b @ g @ A @ B )
=> ( ( factor_a_b @ g @ B @ C )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( properfactor_a_b @ g @ A @ C ) ) ) ) ) ).
% properfactor_trans2
thf(fact_560_properfactor__trans1,axiom,
! [A: a,B: a,C: a] :
( ( factor_a_b @ g @ A @ B )
=> ( ( properfactor_a_b @ g @ B @ C )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ C @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( properfactor_a_b @ g @ A @ C ) ) ) ) ) ).
% properfactor_trans1
thf(fact_561_properfactor__unitE,axiom,
! [U: a,A: a] :
( ( member_a @ U @ ( units_a_b @ g ) )
=> ( ( properfactor_a_b @ g @ A @ U )
=> ~ ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) ) ) ) ).
% properfactor_unitE
thf(fact_562_factors__closed,axiom,
! [Fs: list_a,A: a] :
( ( factors_a_b @ g @ Fs @ A )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) ) ) ) ).
% factors_closed
thf(fact_563_ee__trans,axiom,
! [As: list_a,Bs: list_a,Cs3: list_a] :
( ( essent1248755695173879485al_a_b @ g @ As @ Bs )
=> ( ( essent1248755695173879485al_a_b @ g @ Bs @ Cs3 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Cs3 ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( essent1248755695173879485al_a_b @ g @ As @ Cs3 ) ) ) ) ) ) ).
% ee_trans
thf(fact_564_ee__sym,axiom,
! [As: list_a,Bs: list_a] :
( ( essent1248755695173879485al_a_b @ g @ As @ Bs )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( essent1248755695173879485al_a_b @ g @ Bs @ As ) ) ) ) ).
% ee_sym
thf(fact_565_properfactorI3,axiom,
! [P: a,A: a,B: a] :
( ( P
= ( mult_a_b @ g @ A @ B ) )
=> ( ~ ( member_a @ B @ ( units_a_b @ g ) )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( properfactor_a_b @ g @ A @ P ) ) ) ) ) ).
% properfactorI3
thf(fact_566_factors__dividesI,axiom,
! [Fs: list_a,A: a,F: a] :
( ( factors_a_b @ g @ Fs @ A )
=> ( ( member_a @ F @ ( set_a2 @ Fs ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( factor_a_b @ g @ F @ A ) ) ) ) ).
% factors_dividesI
thf(fact_567_factors__exist,axiom,
! [A: a] :
( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ~ ( member_a @ A @ ( units_a_b @ g ) )
=> ? [Fs2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Fs2 ) @ ( partia7484183841585581558xt_a_b @ g ) )
& ( factors_a_b @ g @ Fs2 @ A ) ) ) ) ).
% factors_exist
thf(fact_568_factors__wfactors,axiom,
! [As: list_a,A: a] :
( ( factors_a_b @ g @ As @ A )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( wfactors_a_b @ g @ As @ A ) ) ) ).
% factors_wfactors
thf(fact_569_wfactors__unique,axiom,
! [Fs: list_a,A: a,Fs5: list_a] :
( ( wfactors_a_b @ g @ Fs @ A )
=> ( ( wfactors_a_b @ g @ Fs5 @ A )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs5 ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( essent1248755695173879485al_a_b @ g @ Fs @ Fs5 ) ) ) ) ) ) ).
% wfactors_unique
thf(fact_570_wfactors__ee__cong__l,axiom,
! [As: list_a,Bs: list_a,B: a] :
( ( essent1248755695173879485al_a_b @ g @ As @ Bs )
=> ( ( wfactors_a_b @ g @ Bs @ B )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( wfactors_a_b @ g @ As @ B ) ) ) ) ) ) ).
% wfactors_ee_cong_l
thf(fact_571_fmset__ee,axiom,
! [As: list_a,Bs: list_a] :
( ( ( fmset_a_b @ g @ As )
= ( fmset_a_b @ g @ Bs ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( essent1248755695173879485al_a_b @ g @ As @ Bs ) ) ) ) ).
% fmset_ee
thf(fact_572_ee__is__fmset,axiom,
! [As: list_a,Bs: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( essent1248755695173879485al_a_b @ g @ As @ Bs )
= ( ( fmset_a_b @ g @ As )
= ( fmset_a_b @ g @ Bs ) ) ) ) ) ).
% ee_is_fmset
thf(fact_573_ee__fmset,axiom,
! [As: list_a,Bs: list_a] :
( ( essent1248755695173879485al_a_b @ g @ As @ Bs )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( fmset_a_b @ g @ As )
= ( fmset_a_b @ g @ Bs ) ) ) ) ) ).
% ee_fmset
thf(fact_574_wfactors__factors,axiom,
! [As: list_a,A: a] :
( ( wfactors_a_b @ g @ As @ A )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ? [A7: a] :
( ( factors_a_b @ g @ As @ A7 )
& ( associated_a_b @ g @ A7 @ A ) ) ) ) ).
% wfactors_factors
thf(fact_575_ee__wfactorsI,axiom,
! [A: a,B: a,As: list_a,Bs: list_a] :
( ( associated_a_b @ g @ A @ B )
=> ( ( wfactors_a_b @ g @ As @ A )
=> ( ( wfactors_a_b @ g @ Bs @ B )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( essent1248755695173879485al_a_b @ g @ As @ Bs ) ) ) ) ) ) ) ) ).
% ee_wfactorsI
thf(fact_576_ee__wfactorsD,axiom,
! [As: list_a,Bs: list_a,A: a,B: a] :
( ( essent1248755695173879485al_a_b @ g @ As @ Bs )
=> ( ( wfactors_a_b @ g @ As @ A )
=> ( ( wfactors_a_b @ g @ Bs @ B )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( associated_a_b @ g @ A @ B ) ) ) ) ) ) ) ) ).
% ee_wfactorsD
thf(fact_577_ee__wfactors,axiom,
! [As: list_a,A: a,Bs: list_a,B: a] :
( ( wfactors_a_b @ g @ As @ A )
=> ( ( wfactors_a_b @ g @ Bs @ B )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( associated_a_b @ g @ A @ B )
= ( essent1248755695173879485al_a_b @ g @ As @ Bs ) ) ) ) ) ) ) ) ).
% ee_wfactors
thf(fact_578_properfactor__fmset__ne,axiom,
! [A: a,B: a,As: list_a,Bs: list_a] :
( ( properfactor_a_b @ g @ A @ B )
=> ( ( wfactors_a_b @ g @ As @ A )
=> ( ( wfactors_a_b @ g @ Bs @ B )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( fmset_a_b @ g @ As )
!= ( fmset_a_b @ g @ Bs ) ) ) ) ) ) ) ) ) ).
% properfactor_fmset_ne
thf(fact_579_factors__mult__single,axiom,
! [A: a,Fb: list_a,B: a] :
( ( irreducible_a_b @ g @ A )
=> ( ( factors_a_b @ g @ Fb @ B )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( factors_a_b @ g @ ( cons_a @ A @ Fb ) @ ( mult_a_b @ g @ A @ B ) ) ) ) ) ).
% factors_mult_single
thf(fact_580_ee__factorsD,axiom,
! [As: list_a,Bs: list_a,A: a,B: a] :
( ( essent1248755695173879485al_a_b @ g @ As @ Bs )
=> ( ( factors_a_b @ g @ As @ A )
=> ( ( factors_a_b @ g @ Bs @ B )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( associated_a_b @ g @ A @ B ) ) ) ) ) ) ).
% ee_factorsD
thf(fact_581_factors__unique,axiom,
! [Fs: list_a,A: a,Fs5: list_a] :
( ( factors_a_b @ g @ Fs @ A )
=> ( ( factors_a_b @ g @ Fs5 @ A )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ~ ( member_a @ A @ ( units_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs5 ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( essent1248755695173879485al_a_b @ g @ Fs @ Fs5 ) ) ) ) ) ) ) ).
% factors_unique
thf(fact_582_factorsI,axiom,
! [Fs: list_a,A: a] :
( ! [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Fs ) )
=> ( irreducible_a_b @ g @ X4 ) )
=> ( ( ( foldr_a_a @ ( mult_a_b @ g ) @ Fs @ ( one_a_b @ g ) )
= A )
=> ( factors_a_b @ g @ Fs @ A ) ) ) ).
% factorsI
thf(fact_583_ee__factorsI,axiom,
! [A: a,B: a,As: list_a,Bs: list_a] :
( ( associated_a_b @ g @ A @ B )
=> ( ( factors_a_b @ g @ As @ A )
=> ( ~ ( member_a @ A @ ( units_a_b @ g ) )
=> ( ( factors_a_b @ g @ Bs @ B )
=> ( ~ ( member_a @ B @ ( units_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( essent1248755695173879485al_a_b @ g @ As @ Bs ) ) ) ) ) ) ) ) ).
% ee_factorsI
thf(fact_584_multlist__ee__cong,axiom,
! [Fs: list_a,Fs5: list_a] :
( ( essent1248755695173879485al_a_b @ g @ Fs @ Fs5 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs5 ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( associated_a_b @ g @ ( foldr_a_a @ ( mult_a_b @ g ) @ Fs @ ( one_a_b @ g ) ) @ ( foldr_a_a @ ( mult_a_b @ g ) @ Fs5 @ ( one_a_b @ g ) ) ) ) ) ) ).
% multlist_ee_cong
thf(fact_585_properfactor__mult__rI,axiom,
! [A: a,B: a,C: a] :
( ( properfactor_a_b @ g @ A @ B )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ C @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( properfactor_a_b @ g @ ( mult_a_b @ g @ A @ C ) @ ( mult_a_b @ g @ B @ C ) ) ) ) ) ).
% properfactor_mult_rI
thf(fact_586_properfactor__mult__r,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ C @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( properfactor_a_b @ g @ ( mult_a_b @ g @ A @ C ) @ ( mult_a_b @ g @ B @ C ) )
= ( properfactor_a_b @ g @ A @ B ) ) ) ) ) ).
% properfactor_mult_r
thf(fact_587_properfactor__mult__lI,axiom,
! [A: a,B: a,C: a] :
( ( properfactor_a_b @ g @ A @ B )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ C @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( properfactor_a_b @ g @ ( mult_a_b @ g @ C @ A ) @ ( mult_a_b @ g @ C @ B ) ) ) ) ) ).
% properfactor_mult_lI
thf(fact_588_properfactor__mult__l,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ C @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( properfactor_a_b @ g @ ( mult_a_b @ g @ C @ A ) @ ( mult_a_b @ g @ C @ B ) )
= ( properfactor_a_b @ g @ A @ B ) ) ) ) ) ).
% properfactor_mult_l
thf(fact_589_ee__refl,axiom,
! [As: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( essent1248755695173879485al_a_b @ g @ As @ As ) ) ).
% ee_refl
thf(fact_590_multlist__closed,axiom,
! [Fs: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( member_a @ ( foldr_a_a @ ( mult_a_b @ g ) @ Fs @ ( one_a_b @ g ) ) @ ( partia7484183841585581558xt_a_b @ g ) ) ) ).
% multlist_closed
thf(fact_591_Lower__antimono,axiom,
! [A4: set_a,B3: set_a,L: partia6638023223214267844t_unit] :
( ( ord_less_eq_set_a @ A4 @ B3 )
=> ( ord_less_eq_set_a @ ( lower_a_Product_unit @ L @ B3 ) @ ( lower_a_Product_unit @ L @ A4 ) ) ) ).
% Lower_antimono
thf(fact_592_properfactor__def,axiom,
( properfactor_a_b
= ( ^ [G2: partia7680350978787392319xt_a_b,A5: a,B5: a] :
( ( factor_a_b @ G2 @ A5 @ B5 )
& ~ ( factor_a_b @ G2 @ B5 @ A5 ) ) ) ) ).
% properfactor_def
thf(fact_593_properfactorI,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a] :
( ( factor_a_b @ G @ A @ B )
=> ( ~ ( factor_a_b @ G @ B @ A )
=> ( properfactor_a_b @ G @ A @ B ) ) ) ).
% properfactorI
thf(fact_594_properfactorE,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a] :
( ( properfactor_a_b @ G @ A @ B )
=> ~ ( ( factor_a_b @ G @ A @ B )
=> ( factor_a_b @ G @ B @ A ) ) ) ).
% properfactorE
thf(fact_595_monoid_Ofactors__mult__single,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,Fb: list_a,B: a] :
( ( monoid_a_b @ G )
=> ( ( irreducible_a_b @ G @ A )
=> ( ( factors_a_b @ G @ Fb @ B )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( factors_a_b @ G @ ( cons_a @ A @ Fb ) @ ( mult_a_b @ G @ A @ B ) ) ) ) ) ) ).
% monoid.factors_mult_single
thf(fact_596_factorsE,axiom,
! [G: partia7680350978787392319xt_a_b,Fs: list_a,A: a] :
( ( factors_a_b @ G @ Fs @ A )
=> ~ ( ! [X6: a] :
( ( member_a @ X6 @ ( set_a2 @ Fs ) )
=> ( irreducible_a_b @ G @ X6 ) )
=> ( ( foldr_a_a @ ( mult_a_b @ G ) @ Fs @ ( one_a_b @ G ) )
!= A ) ) ) ).
% factorsE
thf(fact_597_factors__def,axiom,
( factors_a_b
= ( ^ [G2: partia7680350978787392319xt_a_b,Fs6: list_a,A5: a] :
( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Fs6 ) )
=> ( irreducible_a_b @ G2 @ X3 ) )
& ( ( foldr_a_a @ ( mult_a_b @ G2 ) @ Fs6 @ ( one_a_b @ G2 ) )
= A5 ) ) ) ) ).
% factors_def
thf(fact_598_properfactorI2,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a] :
( ( factor_a_b @ G @ A @ B )
=> ( ~ ( associated_a_b @ G @ A @ B )
=> ( properfactor_a_b @ G @ A @ B ) ) ) ).
% properfactorI2
thf(fact_599_properfactorE2,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a] :
( ( properfactor_a_b @ G @ A @ B )
=> ~ ( ( factor_a_b @ G @ A @ B )
=> ( associated_a_b @ G @ A @ B ) ) ) ).
% properfactorE2
thf(fact_600_monoid_Oproperfactor__divides,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a] :
( ( monoid_a_b @ G )
=> ( ( properfactor_a_b @ G @ A @ B )
=> ( factor_a_b @ G @ A @ B ) ) ) ).
% monoid.properfactor_divides
thf(fact_601_monoid_OfactorsI,axiom,
! [G: partia7680350978787392319xt_a_b,Fs: list_a,A: a] :
( ( monoid_a_b @ G )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Fs ) )
=> ( irreducible_a_b @ G @ X4 ) )
=> ( ( ( foldr_a_a @ ( mult_a_b @ G ) @ Fs @ ( one_a_b @ G ) )
= A )
=> ( factors_a_b @ G @ Fs @ A ) ) ) ) ).
% monoid.factorsI
thf(fact_602_factorial__monoid_Ofactors__unique,axiom,
! [G: partia7680350978787392319xt_a_b,Fs: list_a,A: a,Fs5: list_a] :
( ( factorial_monoid_a_b @ G )
=> ( ( factors_a_b @ G @ Fs @ A )
=> ( ( factors_a_b @ G @ Fs5 @ A )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ~ ( member_a @ A @ ( units_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs5 ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( essent1248755695173879485al_a_b @ G @ Fs @ Fs5 ) ) ) ) ) ) ) ) ).
% factorial_monoid.factors_unique
thf(fact_603_monoid_Oproperfactor__prod__r,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a,C: a] :
( ( monoid_a_b @ G )
=> ( ( properfactor_a_b @ G @ A @ B )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( properfactor_a_b @ G @ A @ ( mult_a_b @ G @ B @ C ) ) ) ) ) ) ) ).
% monoid.properfactor_prod_r
thf(fact_604_comm__monoid_Oproperfactor__prod__l,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a,C: a] :
( ( comm_monoid_a_b @ G )
=> ( ( properfactor_a_b @ G @ A @ B )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( properfactor_a_b @ G @ A @ ( mult_a_b @ G @ C @ B ) ) ) ) ) ) ) ).
% comm_monoid.properfactor_prod_l
thf(fact_605_monoid_Oproperfactor__unitE,axiom,
! [G: partia7680350978787392319xt_a_b,U: a,A: a] :
( ( monoid_a_b @ G )
=> ( ( member_a @ U @ ( units_a_b @ G ) )
=> ( ( properfactor_a_b @ G @ A @ U )
=> ~ ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) ) ) ) ) ).
% monoid.properfactor_unitE
thf(fact_606_monoid_Oproperfactor__trans2,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a,C: a] :
( ( monoid_a_b @ G )
=> ( ( properfactor_a_b @ G @ A @ B )
=> ( ( factor_a_b @ G @ B @ C )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( properfactor_a_b @ G @ A @ C ) ) ) ) ) ) ).
% monoid.properfactor_trans2
thf(fact_607_monoid_Oproperfactor__trans1,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a,C: a] :
( ( monoid_a_b @ G )
=> ( ( factor_a_b @ G @ A @ B )
=> ( ( properfactor_a_b @ G @ B @ C )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( properfactor_a_b @ G @ A @ C ) ) ) ) ) ) ).
% monoid.properfactor_trans1
thf(fact_608_monoid_Oproperfactor__cong__r,axiom,
! [G: partia7680350978787392319xt_a_b,X: a,Y: a,Y2: a] :
( ( monoid_a_b @ G )
=> ( ( properfactor_a_b @ G @ X @ Y )
=> ( ( associated_a_b @ G @ Y @ Y2 )
=> ( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ Y2 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( properfactor_a_b @ G @ X @ Y2 ) ) ) ) ) ) ) ).
% monoid.properfactor_cong_r
thf(fact_609_monoid_Oproperfactor__cong__l,axiom,
! [G: partia7680350978787392319xt_a_b,X2: a,X: a,Y: a] :
( ( monoid_a_b @ G )
=> ( ( associated_a_b @ G @ X2 @ X )
=> ( ( properfactor_a_b @ G @ X @ Y )
=> ( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ X2 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( properfactor_a_b @ G @ X2 @ Y ) ) ) ) ) ) ) ).
% monoid.properfactor_cong_l
thf(fact_610_irreducibleD,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a] :
( ( irreducible_a_b @ G @ A )
=> ( ( properfactor_a_b @ G @ B @ A )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( member_a @ B @ ( units_a_b @ G ) ) ) ) ) ).
% irreducibleD
thf(fact_611_irreducibleE,axiom,
! [G: partia7680350978787392319xt_a_b,A: a] :
( ( irreducible_a_b @ G @ A )
=> ~ ( ~ ( member_a @ A @ ( units_a_b @ G ) )
=> ~ ! [B8: a] :
( ( ( member_a @ B8 @ ( partia7484183841585581558xt_a_b @ G ) )
& ( properfactor_a_b @ G @ B8 @ A ) )
=> ( member_a @ B8 @ ( units_a_b @ G ) ) ) ) ) ).
% irreducibleE
thf(fact_612_irreducibleI,axiom,
! [A: a,G: partia7680350978787392319xt_a_b] :
( ~ ( member_a @ A @ ( units_a_b @ G ) )
=> ( ! [B2: a] :
( ( member_a @ B2 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( properfactor_a_b @ G @ B2 @ A )
=> ( member_a @ B2 @ ( units_a_b @ G ) ) ) )
=> ( irreducible_a_b @ G @ A ) ) ) ).
% irreducibleI
thf(fact_613_irreducible__def,axiom,
( irreducible_a_b
= ( ^ [G2: partia7680350978787392319xt_a_b,A5: a] :
( ~ ( member_a @ A5 @ ( units_a_b @ G2 ) )
& ! [X3: a] :
( ( member_a @ X3 @ ( partia7484183841585581558xt_a_b @ G2 ) )
=> ( ( properfactor_a_b @ G2 @ X3 @ A5 )
=> ( member_a @ X3 @ ( units_a_b @ G2 ) ) ) ) ) ) ) ).
% irreducible_def
thf(fact_614_monoid__cancel_Oproperfactor__mult__lI,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a,C: a] :
( ( monoid_cancel_a_b @ G )
=> ( ( properfactor_a_b @ G @ A @ B )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( properfactor_a_b @ G @ ( mult_a_b @ G @ C @ A ) @ ( mult_a_b @ G @ C @ B ) ) ) ) ) ) ).
% monoid_cancel.properfactor_mult_lI
thf(fact_615_monoid__cancel_Oproperfactor__mult__l,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a,C: a] :
( ( monoid_cancel_a_b @ G )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( properfactor_a_b @ G @ ( mult_a_b @ G @ C @ A ) @ ( mult_a_b @ G @ C @ B ) )
= ( properfactor_a_b @ G @ A @ B ) ) ) ) ) ) ).
% monoid_cancel.properfactor_mult_l
thf(fact_616_factorial__monoid_Oee__factorsI,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a,As: list_a,Bs: list_a] :
( ( factorial_monoid_a_b @ G )
=> ( ( associated_a_b @ G @ A @ B )
=> ( ( factors_a_b @ G @ As @ A )
=> ( ~ ( member_a @ A @ ( units_a_b @ G ) )
=> ( ( factors_a_b @ G @ Bs @ B )
=> ( ~ ( member_a @ B @ ( units_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( essent1248755695173879485al_a_b @ G @ As @ Bs ) ) ) ) ) ) ) ) ) ).
% factorial_monoid.ee_factorsI
thf(fact_617_monoid_Ofactors__closed,axiom,
! [G: partia7680350978787392319xt_a_b,Fs: list_a,A: a] :
( ( monoid_a_b @ G )
=> ( ( factors_a_b @ G @ Fs @ A )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) ) ) ) ) ).
% monoid.factors_closed
thf(fact_618_monoid_Oee__trans,axiom,
! [G: partia7680350978787392319xt_a_b,As: list_a,Bs: list_a,Cs3: list_a] :
( ( monoid_a_b @ G )
=> ( ( essent1248755695173879485al_a_b @ G @ As @ Bs )
=> ( ( essent1248755695173879485al_a_b @ G @ Bs @ Cs3 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Cs3 ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( essent1248755695173879485al_a_b @ G @ As @ Cs3 ) ) ) ) ) ) ) ).
% monoid.ee_trans
thf(fact_619_monoid_Oee__refl,axiom,
! [G: partia7680350978787392319xt_a_b,As: list_a] :
( ( monoid_a_b @ G )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( essent1248755695173879485al_a_b @ G @ As @ As ) ) ) ).
% monoid.ee_refl
thf(fact_620_monoid_Oee__sym,axiom,
! [G: partia7680350978787392319xt_a_b,As: list_a,Bs: list_a] :
( ( monoid_a_b @ G )
=> ( ( essent1248755695173879485al_a_b @ G @ As @ Bs )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( essent1248755695173879485al_a_b @ G @ Bs @ As ) ) ) ) ) ).
% monoid.ee_sym
thf(fact_621_comm__monoid_Ofactors__dividesI,axiom,
! [G: partia7680350978787392319xt_a_b,Fs: list_a,A: a,F: a] :
( ( comm_monoid_a_b @ G )
=> ( ( factors_a_b @ G @ Fs @ A )
=> ( ( member_a @ F @ ( set_a2 @ Fs ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( factor_a_b @ G @ F @ A ) ) ) ) ) ).
% comm_monoid.factors_dividesI
thf(fact_622_monoid_Ofactors__wfactors,axiom,
! [G: partia7680350978787392319xt_a_b,As: list_a,A: a] :
( ( monoid_a_b @ G )
=> ( ( factors_a_b @ G @ As @ A )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( wfactors_a_b @ G @ As @ A ) ) ) ) ).
% monoid.factors_wfactors
thf(fact_623_factorial__monoid_Ofactors__exist,axiom,
! [G: partia7680350978787392319xt_a_b,A: a] :
( ( factorial_monoid_a_b @ G )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ~ ( member_a @ A @ ( units_a_b @ G ) )
=> ? [Fs2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Fs2 ) @ ( partia7484183841585581558xt_a_b @ G ) )
& ( factors_a_b @ G @ Fs2 @ A ) ) ) ) ) ).
% factorial_monoid.factors_exist
thf(fact_624_monoid_Omultlist__closed,axiom,
! [G: partia7680350978787392319xt_a_b,Fs: list_a] :
( ( monoid_a_b @ G )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( member_a @ ( foldr_a_a @ ( mult_a_b @ G ) @ Fs @ ( one_a_b @ G ) ) @ ( partia7484183841585581558xt_a_b @ G ) ) ) ) ).
% monoid.multlist_closed
thf(fact_625_factorial__monoid_Owfactors__unique,axiom,
! [G: partia7680350978787392319xt_a_b,Fs: list_a,A: a,Fs5: list_a] :
( ( factorial_monoid_a_b @ G )
=> ( ( wfactors_a_b @ G @ Fs @ A )
=> ( ( wfactors_a_b @ G @ Fs5 @ A )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs5 ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( essent1248755695173879485al_a_b @ G @ Fs @ Fs5 ) ) ) ) ) ) ) ).
% factorial_monoid.wfactors_unique
thf(fact_626_wfactorsE,axiom,
! [G: partia7680350978787392319xt_a_b,Fs: list_a,A: a] :
( ( wfactors_a_b @ G @ Fs @ A )
=> ~ ( ! [X6: a] :
( ( member_a @ X6 @ ( set_a2 @ Fs ) )
=> ( irreducible_a_b @ G @ X6 ) )
=> ~ ( associated_a_b @ G @ ( foldr_a_a @ ( mult_a_b @ G ) @ Fs @ ( one_a_b @ G ) ) @ A ) ) ) ).
% wfactorsE
thf(fact_627_wfactorsI,axiom,
! [Fs: list_a,G: partia7680350978787392319xt_a_b,A: a] :
( ! [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Fs ) )
=> ( irreducible_a_b @ G @ X4 ) )
=> ( ( associated_a_b @ G @ ( foldr_a_a @ ( mult_a_b @ G ) @ Fs @ ( one_a_b @ G ) ) @ A )
=> ( wfactors_a_b @ G @ Fs @ A ) ) ) ).
% wfactorsI
thf(fact_628_wfactors__def,axiom,
( wfactors_a_b
= ( ^ [G2: partia7680350978787392319xt_a_b,Fs6: list_a,A5: a] :
( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Fs6 ) )
=> ( irreducible_a_b @ G2 @ X3 ) )
& ( associated_a_b @ G2 @ ( foldr_a_a @ ( mult_a_b @ G2 ) @ Fs6 @ ( one_a_b @ G2 ) ) @ A5 ) ) ) ) ).
% wfactors_def
thf(fact_629_primeness__condition__monoid_Owfactors__unique,axiom,
! [G: partia7680350978787392319xt_a_b,As: list_a,A: a,As3: list_a] :
( ( primen900146951955507079id_a_b @ G )
=> ( ( wfactors_a_b @ G @ As @ A )
=> ( ( wfactors_a_b @ G @ As3 @ A )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As3 ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( essent1248755695173879485al_a_b @ G @ As @ As3 ) ) ) ) ) ) ) ).
% primeness_condition_monoid.wfactors_unique
thf(fact_630_factorial__monoid_Oproperfactor__fmset__ne,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a,As: list_a,Bs: list_a] :
( ( factorial_monoid_a_b @ G )
=> ( ( properfactor_a_b @ G @ A @ B )
=> ( ( wfactors_a_b @ G @ As @ A )
=> ( ( wfactors_a_b @ G @ Bs @ B )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( fmset_a_b @ G @ As )
!= ( fmset_a_b @ G @ Bs ) ) ) ) ) ) ) ) ) ) ).
% factorial_monoid.properfactor_fmset_ne
thf(fact_631_monoid__cancel_Owfactors__mult__single,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,Fb: list_a,B: a] :
( ( monoid_cancel_a_b @ G )
=> ( ( irreducible_a_b @ G @ A )
=> ( ( wfactors_a_b @ G @ Fb @ B )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fb ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( wfactors_a_b @ G @ ( cons_a @ A @ Fb ) @ ( mult_a_b @ G @ A @ B ) ) ) ) ) ) ) ) ).
% monoid_cancel.wfactors_mult_single
thf(fact_632_monoid_Owfactors__factors,axiom,
! [G: partia7680350978787392319xt_a_b,As: list_a,A: a] :
( ( monoid_a_b @ G )
=> ( ( wfactors_a_b @ G @ As @ A )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ? [A7: a] :
( ( factors_a_b @ G @ As @ A7 )
& ( associated_a_b @ G @ A7 @ A ) ) ) ) ) ).
% monoid.wfactors_factors
thf(fact_633_comm__monoid_Omultlist__dividesI,axiom,
! [G: partia7680350978787392319xt_a_b,F: a,Fs: list_a] :
( ( comm_monoid_a_b @ G )
=> ( ( member_a @ F @ ( set_a2 @ Fs ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( factor_a_b @ G @ F @ ( foldr_a_a @ ( mult_a_b @ G ) @ Fs @ ( one_a_b @ G ) ) ) ) ) ) ).
% comm_monoid.multlist_dividesI
thf(fact_634_factorial__monoid_Oee__wfactors,axiom,
! [G: partia7680350978787392319xt_a_b,As: list_a,A: a,Bs: list_a,B: a] :
( ( factorial_monoid_a_b @ G )
=> ( ( wfactors_a_b @ G @ As @ A )
=> ( ( wfactors_a_b @ G @ Bs @ B )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( associated_a_b @ G @ A @ B )
= ( essent1248755695173879485al_a_b @ G @ As @ Bs ) ) ) ) ) ) ) ) ) ).
% factorial_monoid.ee_wfactors
thf(fact_635_factorial__monoid_Oee__wfactorsI,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a,As: list_a,Bs: list_a] :
( ( factorial_monoid_a_b @ G )
=> ( ( associated_a_b @ G @ A @ B )
=> ( ( wfactors_a_b @ G @ As @ A )
=> ( ( wfactors_a_b @ G @ Bs @ B )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( essent1248755695173879485al_a_b @ G @ As @ Bs ) ) ) ) ) ) ) ) ) ).
% factorial_monoid.ee_wfactorsI
thf(fact_636_factorial__monoid_Ofmset__properfactor,axiom,
! [G: partia7680350978787392319xt_a_b,As: list_a,Bs: list_a,A: a,B: a] :
( ( factorial_monoid_a_b @ G )
=> ( ( subseteq_mset_set_a @ ( fmset_a_b @ G @ As ) @ ( fmset_a_b @ G @ Bs ) )
=> ( ( ( fmset_a_b @ G @ As )
!= ( fmset_a_b @ G @ Bs ) )
=> ( ( wfactors_a_b @ G @ As @ A )
=> ( ( wfactors_a_b @ G @ Bs @ B )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( properfactor_a_b @ G @ A @ B ) ) ) ) ) ) ) ) ) ) ).
% factorial_monoid.fmset_properfactor
thf(fact_637_factorial__monoid_Oproperfactor__fmset,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a,As: list_a,Bs: list_a] :
( ( factorial_monoid_a_b @ G )
=> ( ( properfactor_a_b @ G @ A @ B )
=> ( ( wfactors_a_b @ G @ As @ A )
=> ( ( wfactors_a_b @ G @ Bs @ B )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( subseteq_mset_set_a @ ( fmset_a_b @ G @ As ) @ ( fmset_a_b @ G @ Bs ) ) ) ) ) ) ) ) ) ) ).
% factorial_monoid.properfactor_fmset
thf(fact_638_factorial__monoid_Omult__factors__fmset,axiom,
! [G: partia7680350978787392319xt_a_b,As: list_a,A: a,Bs: list_a,B: a,Cs3: list_a] :
( ( factorial_monoid_a_b @ G )
=> ( ( factors_a_b @ G @ As @ A )
=> ( ( factors_a_b @ G @ Bs @ B )
=> ( ( factors_a_b @ G @ Cs3 @ ( mult_a_b @ G @ A @ B ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Cs3 ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( fmset_a_b @ G @ Cs3 )
= ( plus_p2331992037799027419_set_a @ ( fmset_a_b @ G @ As ) @ ( fmset_a_b @ G @ Bs ) ) ) ) ) ) ) ) ) ) ).
% factorial_monoid.mult_factors_fmset
thf(fact_639_comm__monoid_OUnits__Lower,axiom,
! [G: partia7680350978787392319xt_a_b] :
( ( comm_monoid_a_b @ G )
=> ( ( units_a_b @ G )
= ( lower_a_Product_unit @ ( partia6383399360842404908t_unit @ ( partia7484183841585581558xt_a_b @ G ) @ ( eq_eq_5233546288009311946t_unit @ ( associated_a_b @ G ) @ ( gorder6800608520584131120t_unit @ ( factor_a_b @ G ) @ product_Unity ) ) ) @ ( partia7484183841585581558xt_a_b @ G ) ) ) ) ).
% comm_monoid.Units_Lower
thf(fact_640_monoid_OUnits__are__ones,axiom,
! [G: partia7680350978787392319xt_a_b] :
( ( monoid_a_b @ G )
=> ( set_eq3287599968972363709t_unit @ ( partia6383399360842404908t_unit @ ( partia7484183841585581558xt_a_b @ G ) @ ( eq_eq_5233546288009311946t_unit @ ( associated_a_b @ G ) @ ( gorder6800608520584131120t_unit @ ( factor_a_b @ G ) @ product_Unity ) ) ) @ ( units_a_b @ G ) @ ( insert_a @ ( one_a_b @ G ) @ bot_bot_set_a ) ) ) ).
% monoid.Units_are_ones
thf(fact_641_units__of__pow,axiom,
! [X: a,N: nat] :
( ( member_a @ X @ ( units_a_b @ g ) )
=> ( ( pow_a_1875594501834816709it_nat @ ( units_of_a_b @ g ) @ X @ N )
= ( pow_a_b_nat @ g @ X @ N ) ) ) ).
% units_of_pow
thf(fact_642_list_Osimps_I15_J,axiom,
! [X21: a,X22: list_a] :
( ( set_a2 @ ( cons_a @ X21 @ X22 ) )
= ( insert_a @ X21 @ ( set_a2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_643_unit__wfactors__empty,axiom,
! [A: a,Fs: list_a] :
( ( member_a @ A @ ( units_a_b @ g ) )
=> ( ( wfactors_a_b @ g @ Fs @ A )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( Fs = nil_a ) ) ) ) ).
% unit_wfactors_empty
thf(fact_644_multlist__listassoc__cong,axiom,
! [Fs: list_a,Fs5: list_a] :
( ( list_all2_a_a @ ( associated_a_b @ g ) @ Fs @ Fs5 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs5 ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( associated_a_b @ g @ ( foldr_a_a @ ( mult_a_b @ g ) @ Fs @ ( one_a_b @ g ) ) @ ( foldr_a_a @ ( mult_a_b @ g ) @ Fs5 @ ( one_a_b @ g ) ) ) ) ) ) ).
% multlist_listassoc_cong
thf(fact_645_factorcount__unique,axiom,
! [As: list_a,A: a] :
( ( wfactors_a_b @ g @ As @ A )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( factorcount_a_b @ g @ A )
= ( size_size_list_a @ As ) ) ) ) ) ).
% factorcount_unique
thf(fact_646_ee__length,axiom,
! [As: list_a,Bs: list_a] :
( ( essent1248755695173879485al_a_b @ g @ As @ Bs )
=> ( ( size_size_list_a @ As )
= ( size_size_list_a @ Bs ) ) ) ).
% ee_length
thf(fact_647_list_Oinject,axiom,
! [X21: a,X22: list_a,Y21: a,Y22: list_a] :
( ( ( cons_a @ X21 @ X22 )
= ( cons_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_648_unit__wfactors,axiom,
! [A: a] :
( ( member_a @ A @ ( units_a_b @ g ) )
=> ( wfactors_a_b @ g @ nil_a @ A ) ) ).
% unit_wfactors
thf(fact_649_listassoc__sym,axiom,
! [As: list_a,Bs: list_a] :
( ( list_all2_a_a @ ( associated_a_b @ g ) @ As @ Bs )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( list_all2_a_a @ ( associated_a_b @ g ) @ Bs @ As ) ) ) ) ).
% listassoc_sym
thf(fact_650_listassoc__trans,axiom,
! [As: list_a,Bs: list_a,Cs3: list_a] :
( ( list_all2_a_a @ ( associated_a_b @ g ) @ As @ Bs )
=> ( ( list_all2_a_a @ ( associated_a_b @ g ) @ Bs @ Cs3 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Cs3 ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( list_all2_a_a @ ( associated_a_b @ g ) @ As @ Cs3 ) ) ) ) ) ) ).
% listassoc_trans
thf(fact_651_factorcount__exists,axiom,
! [A: a] :
( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ? [C2: nat] :
! [As4: list_a] :
( ( ( ord_less_eq_set_a @ ( set_a2 @ As4 ) @ ( partia7484183841585581558xt_a_b @ g ) )
& ( wfactors_a_b @ g @ As4 @ A ) )
=> ( C2
= ( size_size_list_a @ As4 ) ) ) ) ).
% factorcount_exists
thf(fact_652_listassoc__wfactorsD,axiom,
! [As: list_a,Bs: list_a,A: a,B: a] :
( ( list_all2_a_a @ ( associated_a_b @ g ) @ As @ Bs )
=> ( ( wfactors_a_b @ g @ As @ A )
=> ( ( wfactors_a_b @ g @ Bs @ B )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( associated_a_b @ g @ A @ B ) ) ) ) ) ) ) ) ).
% listassoc_wfactorsD
thf(fact_653_wfactors__listassoc__cong__l,axiom,
! [Fs: list_a,A: a,Fs5: list_a] :
( ( wfactors_a_b @ g @ Fs @ A )
=> ( ( list_all2_a_a @ ( associated_a_b @ g ) @ Fs @ Fs5 )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs5 ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( wfactors_a_b @ g @ Fs5 @ A ) ) ) ) ) ) ).
% wfactors_listassoc_cong_l
thf(fact_654_fmset__listassoc__cong,axiom,
! [As: list_a,Bs: list_a] :
( ( list_all2_a_a @ ( associated_a_b @ g ) @ As @ Bs )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( fmset_a_b @ g @ As )
= ( fmset_a_b @ g @ Bs ) ) ) ) ) ).
% fmset_listassoc_cong
thf(fact_655_irrlist__listassoc__cong,axiom,
! [As: list_a,Bs: list_a] :
( ! [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ As ) )
=> ( irreducible_a_b @ g @ X4 ) )
=> ( ( list_all2_a_a @ ( associated_a_b @ g ) @ As @ Bs )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ! [X6: a] :
( ( member_a @ X6 @ ( set_a2 @ Bs ) )
=> ( irreducible_a_b @ g @ X6 ) ) ) ) ) ) ).
% irrlist_listassoc_cong
thf(fact_656_units__of__units,axiom,
! [G: partia7680350978787392319xt_a_b] :
( ( units_a_Product_unit @ ( units_of_a_b @ G ) )
= ( units_a_b @ G ) ) ).
% units_of_units
thf(fact_657_set__empty,axiom,
! [Xs: list_a] :
( ( ( set_a2 @ Xs )
= bot_bot_set_a )
= ( Xs = nil_a ) ) ).
% set_empty
thf(fact_658_set__empty2,axiom,
! [Xs: list_a] :
( ( bot_bot_set_a
= ( set_a2 @ Xs ) )
= ( Xs = nil_a ) ) ).
% set_empty2
thf(fact_659_listassoc__refl,axiom,
! [As: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( list_all2_a_a @ ( associated_a_b @ g ) @ As @ As ) ) ).
% listassoc_refl
thf(fact_660_list_Odistinct_I1_J,axiom,
! [X21: a,X22: list_a] :
( nil_a
!= ( cons_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_661_list_Orel__inject_I2_J,axiom,
! [R2: a > a > $o,X21: a,X22: list_a,Y21: a,Y22: list_a] :
( ( list_all2_a_a @ R2 @ ( cons_a @ X21 @ X22 ) @ ( cons_a @ Y21 @ Y22 ) )
= ( ( R2 @ X21 @ Y21 )
& ( list_all2_a_a @ R2 @ X22 @ Y22 ) ) ) ).
% list.rel_inject(2)
thf(fact_662_list_Orel__intros_I2_J,axiom,
! [R2: a > a > $o,X21: a,Y21: a,X22: list_a,Y22: list_a] :
( ( R2 @ X21 @ Y21 )
=> ( ( list_all2_a_a @ R2 @ X22 @ Y22 )
=> ( list_all2_a_a @ R2 @ ( cons_a @ X21 @ X22 ) @ ( cons_a @ Y21 @ Y22 ) ) ) ) ).
% list.rel_intros(2)
thf(fact_663_list_Orel__distinct_I2_J,axiom,
! [R2: a > a > $o,Y21: a,Y22: list_a] :
~ ( list_all2_a_a @ R2 @ ( cons_a @ Y21 @ Y22 ) @ nil_a ) ).
% list.rel_distinct(2)
thf(fact_664_list_Orel__distinct_I1_J,axiom,
! [R2: a > a > $o,Y21: a,Y22: list_a] :
~ ( list_all2_a_a @ R2 @ nil_a @ ( cons_a @ Y21 @ Y22 ) ) ).
% list.rel_distinct(1)
thf(fact_665_list_OdiscI,axiom,
! [List: list_a,X21: a,X22: list_a] :
( ( List
= ( cons_a @ X21 @ X22 ) )
=> ( List != nil_a ) ) ).
% list.discI
thf(fact_666_list_Oexhaust,axiom,
! [Y: list_a] :
( ( Y != nil_a )
=> ~ ! [X212: a,X222: list_a] :
( Y
!= ( cons_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_667_list_Orel__cases,axiom,
! [R2: a > a > $o,A: list_a,B: list_a] :
( ( list_all2_a_a @ R2 @ A @ B )
=> ( ( ( A = nil_a )
=> ( B != nil_a ) )
=> ~ ! [X1: a,X23: list_a] :
( ( A
= ( cons_a @ X1 @ X23 ) )
=> ! [Y1: a,Y23: list_a] :
( ( B
= ( cons_a @ Y1 @ Y23 ) )
=> ( ( R2 @ X1 @ Y1 )
=> ~ ( list_all2_a_a @ R2 @ X23 @ Y23 ) ) ) ) ) ) ).
% list.rel_cases
thf(fact_668_list_Orel__induct,axiom,
! [R2: a > a > $o,X: list_a,Y: list_a,Q: list_a > list_a > $o] :
( ( list_all2_a_a @ R2 @ X @ Y )
=> ( ( Q @ nil_a @ nil_a )
=> ( ! [A21: a,A22: list_a,B21: a,B22: list_a] :
( ( R2 @ A21 @ B21 )
=> ( ( Q @ A22 @ B22 )
=> ( Q @ ( cons_a @ A21 @ A22 ) @ ( cons_a @ B21 @ B22 ) ) ) )
=> ( Q @ X @ Y ) ) ) ) ).
% list.rel_induct
thf(fact_669_transpose_Ocases,axiom,
! [X: list_list_a] :
( ( X != nil_list_a )
=> ( ! [Xss: list_list_a] :
( X
!= ( cons_list_a @ nil_a @ Xss ) )
=> ~ ! [X4: a,Xs2: list_a,Xss: list_list_a] :
( X
!= ( cons_list_a @ ( cons_a @ X4 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_670_remdups__adj_Ocases,axiom,
! [X: list_a] :
( ( X != nil_a )
=> ( ! [X4: a] :
( X
!= ( cons_a @ X4 @ nil_a ) )
=> ~ ! [X4: a,Y3: a,Xs2: list_a] :
( X
!= ( cons_a @ X4 @ ( cons_a @ Y3 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_671_list__induct2,axiom,
! [Xs: list_a,Ys: list_a,P3: list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P3 @ nil_a @ nil_a )
=> ( ! [X4: a,Xs2: list_a,Y3: a,Ys2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons_a @ X4 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_672_list__induct3,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,P3: list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P3 @ nil_a @ nil_a @ nil_a )
=> ( ! [X4: a,Xs2: list_a,Y3: a,Ys2: list_a,Z3: a,Zs2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P3 @ Xs2 @ Ys2 @ Zs2 )
=> ( P3 @ ( cons_a @ X4 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) @ ( cons_a @ Z3 @ Zs2 ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_673_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,Ws: list_a,P3: list_a > list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P3 @ nil_a @ nil_a @ nil_a @ nil_a )
=> ( ! [X4: a,Xs2: list_a,Y3: a,Ys2: list_a,Z3: a,Zs2: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P3 @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P3 @ ( cons_a @ X4 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) @ ( cons_a @ Z3 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_674_neq__Nil__conv,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
= ( ? [Y5: a,Ys3: list_a] :
( Xs
= ( cons_a @ Y5 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_675_list__induct2_H,axiom,
! [P3: list_a > list_a > $o,Xs: list_a,Ys: list_a] :
( ( P3 @ nil_a @ nil_a )
=> ( ! [X4: a,Xs2: list_a] : ( P3 @ ( cons_a @ X4 @ Xs2 ) @ nil_a )
=> ( ! [Y3: a,Ys2: list_a] : ( P3 @ nil_a @ ( cons_a @ Y3 @ Ys2 ) )
=> ( ! [X4: a,Xs2: list_a,Y3: a,Ys2: list_a] :
( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons_a @ X4 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_676_list__all2__Cons,axiom,
! [P3: a > a > $o,X: a,Xs: list_a,Y: a,Ys: list_a] :
( ( list_all2_a_a @ P3 @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y @ Ys ) )
= ( ( P3 @ X @ Y )
& ( list_all2_a_a @ P3 @ Xs @ Ys ) ) ) ).
% list_all2_Cons
thf(fact_677_list__all2__Cons1,axiom,
! [P3: a > a > $o,X: a,Xs: list_a,Ys: list_a] :
( ( list_all2_a_a @ P3 @ ( cons_a @ X @ Xs ) @ Ys )
= ( ? [Z4: a,Zs3: list_a] :
( ( Ys
= ( cons_a @ Z4 @ Zs3 ) )
& ( P3 @ X @ Z4 )
& ( list_all2_a_a @ P3 @ Xs @ Zs3 ) ) ) ) ).
% list_all2_Cons1
thf(fact_678_list__all2__Cons2,axiom,
! [P3: a > a > $o,Xs: list_a,Y: a,Ys: list_a] :
( ( list_all2_a_a @ P3 @ Xs @ ( cons_a @ Y @ Ys ) )
= ( ? [Z4: a,Zs3: list_a] :
( ( Xs
= ( cons_a @ Z4 @ Zs3 ) )
& ( P3 @ Z4 @ Y )
& ( list_all2_a_a @ P3 @ Zs3 @ Ys ) ) ) ) ).
% list_all2_Cons2
thf(fact_679_list__all2__induct,axiom,
! [P3: a > a > $o,Xs: list_a,Ys: list_a,R2: list_a > list_a > $o] :
( ( list_all2_a_a @ P3 @ Xs @ Ys )
=> ( ( R2 @ nil_a @ nil_a )
=> ( ! [X4: a,Xs2: list_a,Y3: a,Ys2: list_a] :
( ( P3 @ X4 @ Y3 )
=> ( ( list_all2_a_a @ P3 @ Xs2 @ Ys2 )
=> ( ( R2 @ Xs2 @ Ys2 )
=> ( R2 @ ( cons_a @ X4 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) ) ) ) )
=> ( R2 @ Xs @ Ys ) ) ) ) ).
% list_all2_induct
thf(fact_680_list__nonempty__induct,axiom,
! [Xs: list_a,P3: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X4: a] : ( P3 @ ( cons_a @ X4 @ nil_a ) )
=> ( ! [X4: a,Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( P3 @ Xs2 )
=> ( P3 @ ( cons_a @ X4 @ Xs2 ) ) ) )
=> ( P3 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_681_list__all2__same,axiom,
! [P3: a > a > $o,Xs: list_a] :
( ( list_all2_a_a @ P3 @ Xs @ Xs )
= ( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( P3 @ X3 @ X3 ) ) ) ) ).
% list_all2_same
thf(fact_682_list_Orel__refl__strong,axiom,
! [X: list_set_a,Ra: set_a > set_a > $o] :
( ! [Z3: set_a] :
( ( member_set_a @ Z3 @ ( set_set_a2 @ X ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( list_a5961261016436360967_set_a @ Ra @ X @ X ) ) ).
% list.rel_refl_strong
thf(fact_683_list_Orel__refl__strong,axiom,
! [X: list_a,Ra: a > a > $o] :
( ! [Z3: a] :
( ( member_a @ Z3 @ ( set_a2 @ X ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( list_all2_a_a @ Ra @ X @ X ) ) ).
% list.rel_refl_strong
thf(fact_684_list_Orel__mono__strong,axiom,
! [R2: set_a > set_a > $o,X: list_set_a,Y: list_set_a,Ra: set_a > set_a > $o] :
( ( list_a5961261016436360967_set_a @ R2 @ X @ Y )
=> ( ! [Z3: set_a,Yb: set_a] :
( ( member_set_a @ Z3 @ ( set_set_a2 @ X ) )
=> ( ( member_set_a @ Yb @ ( set_set_a2 @ Y ) )
=> ( ( R2 @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( list_a5961261016436360967_set_a @ Ra @ X @ Y ) ) ) ).
% list.rel_mono_strong
thf(fact_685_list_Orel__mono__strong,axiom,
! [R2: set_a > a > $o,X: list_set_a,Y: list_a,Ra: set_a > a > $o] :
( ( list_all2_set_a_a @ R2 @ X @ Y )
=> ( ! [Z3: set_a,Yb: a] :
( ( member_set_a @ Z3 @ ( set_set_a2 @ X ) )
=> ( ( member_a @ Yb @ ( set_a2 @ Y ) )
=> ( ( R2 @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( list_all2_set_a_a @ Ra @ X @ Y ) ) ) ).
% list.rel_mono_strong
thf(fact_686_list_Orel__mono__strong,axiom,
! [R2: a > set_a > $o,X: list_a,Y: list_set_a,Ra: a > set_a > $o] :
( ( list_all2_a_set_a @ R2 @ X @ Y )
=> ( ! [Z3: a,Yb: set_a] :
( ( member_a @ Z3 @ ( set_a2 @ X ) )
=> ( ( member_set_a @ Yb @ ( set_set_a2 @ Y ) )
=> ( ( R2 @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( list_all2_a_set_a @ Ra @ X @ Y ) ) ) ).
% list.rel_mono_strong
thf(fact_687_list_Orel__mono__strong,axiom,
! [R2: a > a > $o,X: list_a,Y: list_a,Ra: a > a > $o] :
( ( list_all2_a_a @ R2 @ X @ Y )
=> ( ! [Z3: a,Yb: a] :
( ( member_a @ Z3 @ ( set_a2 @ X ) )
=> ( ( member_a @ Yb @ ( set_a2 @ Y ) )
=> ( ( R2 @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( list_all2_a_a @ Ra @ X @ Y ) ) ) ).
% list.rel_mono_strong
thf(fact_688_list_Orel__cong,axiom,
! [X: list_set_a,Ya: list_set_a,Y: list_set_a,Xa2: list_set_a,R2: set_a > set_a > $o,Ra: set_a > set_a > $o] :
( ( X = Ya )
=> ( ( Y = Xa2 )
=> ( ! [Z3: set_a,Yb: set_a] :
( ( member_set_a @ Z3 @ ( set_set_a2 @ Ya ) )
=> ( ( member_set_a @ Yb @ ( set_set_a2 @ Xa2 ) )
=> ( ( R2 @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( list_a5961261016436360967_set_a @ R2 @ X @ Y )
= ( list_a5961261016436360967_set_a @ Ra @ Ya @ Xa2 ) ) ) ) ) ).
% list.rel_cong
thf(fact_689_list_Orel__cong,axiom,
! [X: list_set_a,Ya: list_set_a,Y: list_a,Xa2: list_a,R2: set_a > a > $o,Ra: set_a > a > $o] :
( ( X = Ya )
=> ( ( Y = Xa2 )
=> ( ! [Z3: set_a,Yb: a] :
( ( member_set_a @ Z3 @ ( set_set_a2 @ Ya ) )
=> ( ( member_a @ Yb @ ( set_a2 @ Xa2 ) )
=> ( ( R2 @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( list_all2_set_a_a @ R2 @ X @ Y )
= ( list_all2_set_a_a @ Ra @ Ya @ Xa2 ) ) ) ) ) ).
% list.rel_cong
thf(fact_690_list_Orel__cong,axiom,
! [X: list_a,Ya: list_a,Y: list_set_a,Xa2: list_set_a,R2: a > set_a > $o,Ra: a > set_a > $o] :
( ( X = Ya )
=> ( ( Y = Xa2 )
=> ( ! [Z3: a,Yb: set_a] :
( ( member_a @ Z3 @ ( set_a2 @ Ya ) )
=> ( ( member_set_a @ Yb @ ( set_set_a2 @ Xa2 ) )
=> ( ( R2 @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( list_all2_a_set_a @ R2 @ X @ Y )
= ( list_all2_a_set_a @ Ra @ Ya @ Xa2 ) ) ) ) ) ).
% list.rel_cong
thf(fact_691_list_Orel__cong,axiom,
! [X: list_a,Ya: list_a,Y: list_a,Xa2: list_a,R2: a > a > $o,Ra: a > a > $o] :
( ( X = Ya )
=> ( ( Y = Xa2 )
=> ( ! [Z3: a,Yb: a] :
( ( member_a @ Z3 @ ( set_a2 @ Ya ) )
=> ( ( member_a @ Yb @ ( set_a2 @ Xa2 ) )
=> ( ( R2 @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( list_all2_a_a @ R2 @ X @ Y )
= ( list_all2_a_a @ Ra @ Ya @ Xa2 ) ) ) ) ) ).
% list.rel_cong
thf(fact_692_impossible__Cons,axiom,
! [Xs: list_a,Ys: list_a,X: a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) )
=> ( Xs
!= ( cons_a @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_693_empty__set,axiom,
( bot_bot_set_a
= ( set_a2 @ nil_a ) ) ).
% empty_set
thf(fact_694_units__of__mult,axiom,
! [G: partia7680350978787392319xt_a_b] :
( ( mult_a_Product_unit @ ( units_of_a_b @ G ) )
= ( mult_a_b @ G ) ) ).
% units_of_mult
thf(fact_695_units__of__one,axiom,
! [G: partia7680350978787392319xt_a_b] :
( ( one_a_Product_unit @ ( units_of_a_b @ G ) )
= ( one_a_b @ G ) ) ).
% units_of_one
thf(fact_696_units__of__carrier,axiom,
! [G: partia7680350978787392319xt_a_b] :
( ( partia6735698275553448452t_unit @ ( units_of_a_b @ G ) )
= ( units_a_b @ G ) ) ).
% units_of_carrier
thf(fact_697_monoid_Oee__length,axiom,
! [G: partia7680350978787392319xt_a_b,As: list_a,Bs: list_a] :
( ( monoid_a_b @ G )
=> ( ( essent1248755695173879485al_a_b @ G @ As @ Bs )
=> ( ( size_size_list_a @ As )
= ( size_size_list_a @ Bs ) ) ) ) ).
% monoid.ee_length
thf(fact_698_not__Cons__self2,axiom,
! [X: a,Xs: list_a] :
( ( cons_a @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_699_monoid_Ounit__wfactors,axiom,
! [G: partia7680350978787392319xt_a_b,A: a] :
( ( monoid_a_b @ G )
=> ( ( member_a @ A @ ( units_a_b @ G ) )
=> ( wfactors_a_b @ G @ nil_a @ A ) ) ) ).
% monoid.unit_wfactors
thf(fact_700_monoid_Olistassoc__sym,axiom,
! [G: partia7680350978787392319xt_a_b,As: list_a,Bs: list_a] :
( ( monoid_a_b @ G )
=> ( ( list_all2_a_a @ ( associated_a_b @ G ) @ As @ Bs )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( list_all2_a_a @ ( associated_a_b @ G ) @ Bs @ As ) ) ) ) ) ).
% monoid.listassoc_sym
thf(fact_701_monoid_Olistassoc__refl,axiom,
! [G: partia7680350978787392319xt_a_b,As: list_a] :
( ( monoid_a_b @ G )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( list_all2_a_a @ ( associated_a_b @ G ) @ As @ As ) ) ) ).
% monoid.listassoc_refl
thf(fact_702_monoid_Olistassoc__trans,axiom,
! [G: partia7680350978787392319xt_a_b,As: list_a,Bs: list_a,Cs3: list_a] :
( ( monoid_a_b @ G )
=> ( ( list_all2_a_a @ ( associated_a_b @ G ) @ As @ Bs )
=> ( ( list_all2_a_a @ ( associated_a_b @ G ) @ Bs @ Cs3 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Cs3 ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( list_all2_a_a @ ( associated_a_b @ G ) @ As @ Cs3 ) ) ) ) ) ) ) ).
% monoid.listassoc_trans
thf(fact_703_factorial__monoid_Ofactorcount__exists,axiom,
! [G: partia7680350978787392319xt_a_b,A: a] :
( ( factorial_monoid_a_b @ G )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ? [C2: nat] :
! [As4: list_a] :
( ( ( ord_less_eq_set_a @ ( set_a2 @ As4 ) @ ( partia7484183841585581558xt_a_b @ G ) )
& ( wfactors_a_b @ G @ As4 @ A ) )
=> ( C2
= ( size_size_list_a @ As4 ) ) ) ) ) ).
% factorial_monoid.factorcount_exists
thf(fact_704_monoid__cancel_Oirrlist__listassoc__cong,axiom,
! [G: partia7680350978787392319xt_a_b,As: list_a,Bs: list_a] :
( ( monoid_cancel_a_b @ G )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ As ) )
=> ( irreducible_a_b @ G @ X4 ) )
=> ( ( list_all2_a_a @ ( associated_a_b @ G ) @ As @ Bs )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ! [X6: a] :
( ( member_a @ X6 @ ( set_a2 @ Bs ) )
=> ( irreducible_a_b @ G @ X6 ) ) ) ) ) ) ) ).
% monoid_cancel.irrlist_listassoc_cong
thf(fact_705_factorial__monoid_Ofactorcount__unique,axiom,
! [G: partia7680350978787392319xt_a_b,As: list_a,A: a] :
( ( factorial_monoid_a_b @ G )
=> ( ( wfactors_a_b @ G @ As @ A )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( factorcount_a_b @ G @ A )
= ( size_size_list_a @ As ) ) ) ) ) ) ).
% factorial_monoid.factorcount_unique
thf(fact_706_subset__code_I1_J,axiom,
! [Xs: list_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ Xs ) @ B3 )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ ( set_set_a2 @ Xs ) )
=> ( member_set_a @ X3 @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_707_subset__code_I1_J,axiom,
! [Xs: list_a,B3: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ B3 )
= ( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( member_a @ X3 @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_708_set__ConsD,axiom,
! [Y: set_a,X: set_a,Xs: list_set_a] :
( ( member_set_a @ Y @ ( set_set_a2 @ ( cons_set_a @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_set_a @ Y @ ( set_set_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_709_set__ConsD,axiom,
! [Y: a,X: a,Xs: list_a] :
( ( member_a @ Y @ ( set_a2 @ ( cons_a @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_a @ Y @ ( set_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_710_list_Oset__cases,axiom,
! [E: set_a,A: list_set_a] :
( ( member_set_a @ E @ ( set_set_a2 @ A ) )
=> ( ! [Z22: list_set_a] :
( A
!= ( cons_set_a @ E @ Z22 ) )
=> ~ ! [Z1: set_a,Z22: list_set_a] :
( ( A
= ( cons_set_a @ Z1 @ Z22 ) )
=> ~ ( member_set_a @ E @ ( set_set_a2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_711_list_Oset__cases,axiom,
! [E: a,A: list_a] :
( ( member_a @ E @ ( set_a2 @ A ) )
=> ( ! [Z22: list_a] :
( A
!= ( cons_a @ E @ Z22 ) )
=> ~ ! [Z1: a,Z22: list_a] :
( ( A
= ( cons_a @ Z1 @ Z22 ) )
=> ~ ( member_a @ E @ ( set_a2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_712_list_Oset__intros_I1_J,axiom,
! [X21: set_a,X22: list_set_a] : ( member_set_a @ X21 @ ( set_set_a2 @ ( cons_set_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_713_list_Oset__intros_I1_J,axiom,
! [X21: a,X22: list_a] : ( member_a @ X21 @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_714_list_Oset__intros_I2_J,axiom,
! [Y: set_a,X22: list_set_a,X21: set_a] :
( ( member_set_a @ Y @ ( set_set_a2 @ X22 ) )
=> ( member_set_a @ Y @ ( set_set_a2 @ ( cons_set_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_715_list_Oset__intros_I2_J,axiom,
! [Y: a,X22: list_a,X21: a] :
( ( member_a @ Y @ ( set_a2 @ X22 ) )
=> ( member_a @ Y @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_716_foldr__cong,axiom,
! [A: a,B: a,L2: list_a,K2: list_a,F: a > a > a,G3: a > a > a] :
( ( A = B )
=> ( ( L2 = K2 )
=> ( ! [A2: a,X4: a] :
( ( member_a @ X4 @ ( set_a2 @ L2 ) )
=> ( ( F @ X4 @ A2 )
= ( G3 @ X4 @ A2 ) ) )
=> ( ( foldr_a_a @ F @ L2 @ A )
= ( foldr_a_a @ G3 @ K2 @ B ) ) ) ) ) ).
% foldr_cong
thf(fact_717_set__subset__Cons,axiom,
! [Xs: list_a,X: a] : ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ ( cons_a @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_718_monoid_Ounits__of__pow,axiom,
! [G: partia8223610829204095565t_unit,X: a,N: nat] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( member_a @ X @ ( units_a_Product_unit @ G ) )
=> ( ( pow_a_1875594501834816709it_nat @ ( units_7501539392726747778t_unit @ G ) @ X @ N )
= ( pow_a_1875594501834816709it_nat @ G @ X @ N ) ) ) ) ).
% monoid.units_of_pow
thf(fact_719_monoid_Ounits__of__pow,axiom,
! [G: partia7680350978787392319xt_a_b,X: a,N: nat] :
( ( monoid_a_b @ G )
=> ( ( member_a @ X @ ( units_a_b @ G ) )
=> ( ( pow_a_1875594501834816709it_nat @ ( units_of_a_b @ G ) @ X @ N )
= ( pow_a_b_nat @ G @ X @ N ) ) ) ) ).
% monoid.units_of_pow
thf(fact_720_multlist__perm__cong,axiom,
! [As: list_a,Bs: list_a] :
( ( ( mset_a @ As )
= ( mset_a @ Bs ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( foldr_a_a @ ( mult_a_b @ g ) @ As @ ( one_a_b @ g ) )
= ( foldr_a_a @ ( mult_a_b @ g ) @ Bs @ ( one_a_b @ g ) ) ) ) ) ).
% multlist_perm_cong
thf(fact_721_factors__mult,axiom,
! [Fa: list_a,A: a,Fb: list_a,B: a] :
( ( factors_a_b @ g @ Fa @ A )
=> ( ( factors_a_b @ g @ Fb @ B )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fa ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fb ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( factors_a_b @ g @ ( append_a @ Fa @ Fb ) @ ( mult_a_b @ g @ A @ B ) ) ) ) ) ) ).
% factors_mult
thf(fact_722_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_723_fmset__perm__cong,axiom,
! [As: list_a,Bs: list_a] :
( ( ( mset_a @ As )
= ( mset_a @ Bs ) )
=> ( ( fmset_a_b @ g @ As )
= ( fmset_a_b @ g @ Bs ) ) ) ).
% fmset_perm_cong
thf(fact_724_add__left__cancel,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( ( plus_p2331992037799027419_set_a @ A @ B )
= ( plus_p2331992037799027419_set_a @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_725_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_726_add__right__cancel,axiom,
! [B: multiset_set_a,A: multiset_set_a,C: multiset_set_a] :
( ( ( plus_p2331992037799027419_set_a @ B @ A )
= ( plus_p2331992037799027419_set_a @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_727_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_728_nat__pow__mult,axiom,
! [X: a,N: nat,M4: nat] :
( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( mult_a_b @ g @ ( pow_a_b_nat @ g @ X @ N ) @ ( pow_a_b_nat @ g @ X @ M4 ) )
= ( pow_a_b_nat @ g @ X @ ( plus_plus_nat @ N @ M4 ) ) ) ) ).
% nat_pow_mult
thf(fact_729_irrlist__perm__cong,axiom,
! [As: list_a,Bs: list_a] :
( ( ( mset_a @ As )
= ( mset_a @ Bs ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ As ) )
=> ( irreducible_a_b @ g @ X4 ) )
=> ! [X6: a] :
( ( member_a @ X6 @ ( set_a2 @ Bs ) )
=> ( irreducible_a_b @ g @ X6 ) ) ) ) ).
% irrlist_perm_cong
thf(fact_730_perm__assoc__switch__r,axiom,
! [As: list_a,Bs: list_a,Cs3: list_a] :
( ( ( mset_a @ As )
= ( mset_a @ Bs ) )
=> ( ( list_all2_a_a @ ( associated_a_b @ g ) @ Bs @ Cs3 )
=> ? [Bs2: list_a] :
( ( list_all2_a_a @ ( associated_a_b @ g ) @ As @ Bs2 )
& ( ( mset_a @ Bs2 )
= ( mset_a @ Cs3 ) ) ) ) ) ).
% perm_assoc_switch_r
thf(fact_731_perm__assoc__switch,axiom,
! [As: list_a,Bs: list_a,Cs3: list_a] :
( ( list_all2_a_a @ ( associated_a_b @ g ) @ As @ Bs )
=> ( ( ( mset_a @ Bs )
= ( mset_a @ Cs3 ) )
=> ? [Bs2: list_a] :
( ( ( mset_a @ As )
= ( mset_a @ Bs2 ) )
& ( list_all2_a_a @ ( associated_a_b @ g ) @ Bs2 @ Cs3 ) ) ) ) ).
% perm_assoc_switch
thf(fact_732_perm__closed,axiom,
! [As: list_a,Bs: list_a] :
( ( ( mset_a @ As )
= ( mset_a @ Bs ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia7484183841585581558xt_a_b @ g ) ) ) ) ).
% perm_closed
thf(fact_733_essentially__equalI,axiom,
! [Fs1: list_a,Fs12: list_a,Fs22: list_a] :
( ( ( mset_a @ Fs1 )
= ( mset_a @ Fs12 ) )
=> ( ( list_all2_a_a @ ( associated_a_b @ g ) @ Fs12 @ Fs22 )
=> ( essent1248755695173879485al_a_b @ g @ Fs1 @ Fs22 ) ) ) ).
% essentially_equalI
thf(fact_734_essentially__equalE,axiom,
! [Fs1: list_a,Fs22: list_a] :
( ( essent1248755695173879485al_a_b @ g @ Fs1 @ Fs22 )
=> ~ ! [Fs13: list_a] :
( ( ( mset_a @ Fs1 )
= ( mset_a @ Fs13 ) )
=> ~ ( list_all2_a_a @ ( associated_a_b @ g ) @ Fs13 @ Fs22 ) ) ) ).
% essentially_equalE
thf(fact_735_wfactors__perm__cong__l,axiom,
! [Fs: list_a,A: a,Fs5: list_a] :
( ( wfactors_a_b @ g @ Fs @ A )
=> ( ( ( mset_a @ Fs )
= ( mset_a @ Fs5 ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( wfactors_a_b @ g @ Fs5 @ A ) ) ) ) ).
% wfactors_perm_cong_l
thf(fact_736_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_737_perm__wfactorsD,axiom,
! [As: list_a,Bs: list_a,A: a,B: a] :
( ( ( mset_a @ As )
= ( mset_a @ Bs ) )
=> ( ( wfactors_a_b @ g @ As @ A )
=> ( ( wfactors_a_b @ g @ Bs @ B )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( associated_a_b @ g @ A @ B ) ) ) ) ) ) ) ).
% perm_wfactorsD
thf(fact_738_length__append,axiom,
! [Xs: list_a,Ys: list_a] :
( ( size_size_list_a @ ( append_a @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) ) ) ).
% length_append
thf(fact_739_count__union,axiom,
! [M: multiset_set_a,N3: multiset_set_a,A: set_a] :
( ( count_set_a @ ( plus_p2331992037799027419_set_a @ M @ N3 ) @ A )
= ( plus_plus_nat @ ( count_set_a @ M @ A ) @ ( count_set_a @ N3 @ A ) ) ) ).
% count_union
thf(fact_740_foldr__append,axiom,
! [F: a > a > a,Xs: list_a,Ys: list_a,A: a] :
( ( foldr_a_a @ F @ ( append_a @ Xs @ Ys ) @ A )
= ( foldr_a_a @ F @ Xs @ ( foldr_a_a @ F @ Ys @ A ) ) ) ).
% foldr_append
thf(fact_741_append1__eq__conv,axiom,
! [Xs: list_a,X: a,Ys: list_a,Y: a] :
( ( ( append_a @ Xs @ ( cons_a @ X @ nil_a ) )
= ( append_a @ Ys @ ( cons_a @ Y @ nil_a ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_742_set__mset__mset,axiom,
! [Xs: list_a] :
( ( set_mset_a @ ( mset_a @ Xs ) )
= ( set_a2 @ Xs ) ) ).
% set_mset_mset
thf(fact_743_set__mset__mset,axiom,
! [Xs: list_set_a] :
( ( set_mset_set_a @ ( mset_set_a @ Xs ) )
= ( set_set_a2 @ Xs ) ) ).
% set_mset_mset
thf(fact_744_mset__append,axiom,
! [Xs: list_a,Ys: list_a] :
( ( mset_a @ ( append_a @ Xs @ Ys ) )
= ( plus_plus_multiset_a @ ( mset_a @ Xs ) @ ( mset_a @ Ys ) ) ) ).
% mset_append
thf(fact_745_mset__append,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( mset_set_a @ ( append_set_a @ Xs @ Ys ) )
= ( plus_p2331992037799027419_set_a @ ( mset_set_a @ Xs ) @ ( mset_set_a @ Ys ) ) ) ).
% mset_append
thf(fact_746_wfactors__mult,axiom,
! [As: list_a,A: a,Bs: list_a,B: a] :
( ( wfactors_a_b @ g @ As @ A )
=> ( ( wfactors_a_b @ g @ Bs @ B )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( wfactors_a_b @ g @ ( append_a @ As @ Bs ) @ ( mult_a_b @ g @ A @ B ) ) ) ) ) ) ) ) ).
% wfactors_mult
thf(fact_747_list_Orel__mono,axiom,
! [R2: a > a > $o,Ra: a > a > $o] :
( ( ord_less_eq_a_a_o @ R2 @ Ra )
=> ( ord_le5542992221119063950st_a_o @ ( list_all2_a_a @ R2 ) @ ( list_all2_a_a @ Ra ) ) ) ).
% list.rel_mono
thf(fact_748_append__Cons,axiom,
! [X: a,Xs: list_a,Ys: list_a] :
( ( append_a @ ( cons_a @ X @ Xs ) @ Ys )
= ( cons_a @ X @ ( append_a @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_749_Cons__eq__appendI,axiom,
! [X: a,Xs1: list_a,Ys: list_a,Xs: list_a,Zs: list_a] :
( ( ( cons_a @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_a @ Xs1 @ Zs ) )
=> ( ( cons_a @ X @ Xs )
= ( append_a @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_750_list__all2__reorder__left__invariance,axiom,
! [R2: a > a > $o,Xs: list_a,Ys: list_a,Xs3: list_a] :
( ( list_all2_a_a @ R2 @ Xs @ Ys )
=> ( ( ( mset_a @ Xs3 )
= ( mset_a @ Xs ) )
=> ? [Ys4: list_a] :
( ( list_all2_a_a @ R2 @ Xs3 @ Ys4 )
& ( ( mset_a @ Ys4 )
= ( mset_a @ Ys ) ) ) ) ) ).
% list_all2_reorder_left_invariance
thf(fact_751_ex__mset,axiom,
! [X7: multiset_a] :
? [Xs2: list_a] :
( ( mset_a @ Xs2 )
= X7 ) ).
% ex_mset
thf(fact_752_perm__setP,axiom,
! [As: list_a,Bs: list_a,P3: set_a > $o] :
( ( ( mset_a @ As )
= ( mset_a @ Bs ) )
=> ( ( P3 @ ( set_a2 @ As ) )
=> ( P3 @ ( set_a2 @ Bs ) ) ) ) ).
% perm_setP
thf(fact_753_in__multiset__in__set,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_mset_a @ ( mset_a @ Xs ) ) )
= ( member_a @ X @ ( set_a2 @ Xs ) ) ) ).
% in_multiset_in_set
thf(fact_754_in__multiset__in__set,axiom,
! [X: set_a,Xs: list_set_a] :
( ( member_set_a @ X @ ( set_mset_set_a @ ( mset_set_a @ Xs ) ) )
= ( member_set_a @ X @ ( set_set_a2 @ Xs ) ) ) ).
% in_multiset_in_set
thf(fact_755_split__list,axiom,
! [X: set_a,Xs: list_set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
=> ? [Ys2: list_set_a,Zs2: list_set_a] :
( Xs
= ( append_set_a @ Ys2 @ ( cons_set_a @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_756_split__list,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ? [Ys2: list_a,Zs2: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_757_split__list__last,axiom,
! [X: set_a,Xs: list_set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
=> ? [Ys2: list_set_a,Zs2: list_set_a] :
( ( Xs
= ( append_set_a @ Ys2 @ ( cons_set_a @ X @ Zs2 ) ) )
& ~ ( member_set_a @ X @ ( set_set_a2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_758_split__list__last,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ? [Ys2: list_a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ X @ Zs2 ) ) )
& ~ ( member_a @ X @ ( set_a2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_759_split__list__prop,axiom,
! [Xs: list_a,P3: a > $o] :
( ? [X6: a] :
( ( member_a @ X6 @ ( set_a2 @ Xs ) )
& ( P3 @ X6 ) )
=> ? [Ys2: list_a,X4: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X4 @ Zs2 ) ) )
& ( P3 @ X4 ) ) ) ).
% split_list_prop
thf(fact_760_split__list__first,axiom,
! [X: set_a,Xs: list_set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
=> ? [Ys2: list_set_a,Zs2: list_set_a] :
( ( Xs
= ( append_set_a @ Ys2 @ ( cons_set_a @ X @ Zs2 ) ) )
& ~ ( member_set_a @ X @ ( set_set_a2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_761_split__list__first,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ? [Ys2: list_a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ X @ Zs2 ) ) )
& ~ ( member_a @ X @ ( set_a2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_762_split__list__propE,axiom,
! [Xs: list_a,P3: a > $o] :
( ? [X6: a] :
( ( member_a @ X6 @ ( set_a2 @ Xs ) )
& ( P3 @ X6 ) )
=> ~ ! [Ys2: list_a,X4: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X4 @ Zs2 ) ) )
=> ~ ( P3 @ X4 ) ) ) ).
% split_list_propE
thf(fact_763_append__Cons__eq__iff,axiom,
! [X: set_a,Xs: list_set_a,Ys: list_set_a,Xs3: list_set_a,Ys5: list_set_a] :
( ~ ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
=> ( ~ ( member_set_a @ X @ ( set_set_a2 @ Ys ) )
=> ( ( ( append_set_a @ Xs @ ( cons_set_a @ X @ Ys ) )
= ( append_set_a @ Xs3 @ ( cons_set_a @ X @ Ys5 ) ) )
= ( ( Xs = Xs3 )
& ( Ys = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_764_append__Cons__eq__iff,axiom,
! [X: a,Xs: list_a,Ys: list_a,Xs3: list_a,Ys5: list_a] :
( ~ ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( ~ ( member_a @ X @ ( set_a2 @ Ys ) )
=> ( ( ( append_a @ Xs @ ( cons_a @ X @ Ys ) )
= ( append_a @ Xs3 @ ( cons_a @ X @ Ys5 ) ) )
= ( ( Xs = Xs3 )
& ( Ys = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_765_in__set__conv__decomp,axiom,
! [X: set_a,Xs: list_set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
= ( ? [Ys3: list_set_a,Zs3: list_set_a] :
( Xs
= ( append_set_a @ Ys3 @ ( cons_set_a @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_766_in__set__conv__decomp,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
= ( ? [Ys3: list_a,Zs3: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_767_split__list__last__prop,axiom,
! [Xs: list_a,P3: a > $o] :
( ? [X6: a] :
( ( member_a @ X6 @ ( set_a2 @ Xs ) )
& ( P3 @ X6 ) )
=> ? [Ys2: list_a,X4: a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ X4 @ Zs2 ) ) )
& ( P3 @ X4 )
& ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Zs2 ) )
=> ~ ( P3 @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_768_split__list__first__prop,axiom,
! [Xs: list_a,P3: a > $o] :
( ? [X6: a] :
( ( member_a @ X6 @ ( set_a2 @ Xs ) )
& ( P3 @ X6 ) )
=> ? [Ys2: list_a,X4: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X4 @ Zs2 ) ) )
& ( P3 @ X4 )
& ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Ys2 ) )
=> ~ ( P3 @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_769_split__list__last__propE,axiom,
! [Xs: list_a,P3: a > $o] :
( ? [X6: a] :
( ( member_a @ X6 @ ( set_a2 @ Xs ) )
& ( P3 @ X6 ) )
=> ~ ! [Ys2: list_a,X4: a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ X4 @ Zs2 ) ) )
=> ( ( P3 @ X4 )
=> ~ ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Zs2 ) )
=> ~ ( P3 @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_770_split__list__first__propE,axiom,
! [Xs: list_a,P3: a > $o] :
( ? [X6: a] :
( ( member_a @ X6 @ ( set_a2 @ Xs ) )
& ( P3 @ X6 ) )
=> ~ ! [Ys2: list_a,X4: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X4 @ Zs2 ) ) )
=> ( ( P3 @ X4 )
=> ~ ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Ys2 ) )
=> ~ ( P3 @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_771_in__set__conv__decomp__last,axiom,
! [X: set_a,Xs: list_set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
= ( ? [Ys3: list_set_a,Zs3: list_set_a] :
( ( Xs
= ( append_set_a @ Ys3 @ ( cons_set_a @ X @ Zs3 ) ) )
& ~ ( member_set_a @ X @ ( set_set_a2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_772_in__set__conv__decomp__last,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
= ( ? [Ys3: list_a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X @ Zs3 ) ) )
& ~ ( member_a @ X @ ( set_a2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_773_in__set__conv__decomp__first,axiom,
! [X: set_a,Xs: list_set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
= ( ? [Ys3: list_set_a,Zs3: list_set_a] :
( ( Xs
= ( append_set_a @ Ys3 @ ( cons_set_a @ X @ Zs3 ) ) )
& ~ ( member_set_a @ X @ ( set_set_a2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_774_in__set__conv__decomp__first,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
= ( ? [Ys3: list_a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X @ Zs3 ) ) )
& ~ ( member_a @ X @ ( set_a2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_775_split__list__last__prop__iff,axiom,
! [Xs: list_a,P3: a > $o] :
( ( ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
& ( P3 @ X3 ) ) )
= ( ? [Ys3: list_a,X3: a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X3 @ Zs3 ) ) )
& ( P3 @ X3 )
& ! [Y5: a] :
( ( member_a @ Y5 @ ( set_a2 @ Zs3 ) )
=> ~ ( P3 @ Y5 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_776_split__list__first__prop__iff,axiom,
! [Xs: list_a,P3: a > $o] :
( ( ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
& ( P3 @ X3 ) ) )
= ( ? [Ys3: list_a,X3: a] :
( ? [Zs3: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X3 @ Zs3 ) ) )
& ( P3 @ X3 )
& ! [Y5: a] :
( ( member_a @ Y5 @ ( set_a2 @ Ys3 ) )
=> ~ ( P3 @ Y5 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_777_rev__nonempty__induct,axiom,
! [Xs: list_a,P3: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X4: a] : ( P3 @ ( cons_a @ X4 @ nil_a ) )
=> ( ! [X4: a,Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( P3 @ Xs2 )
=> ( P3 @ ( append_a @ Xs2 @ ( cons_a @ X4 @ nil_a ) ) ) ) )
=> ( P3 @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_778_append__eq__Cons__conv,axiom,
! [Ys: list_a,Zs: list_a,X: a,Xs: list_a] :
( ( ( append_a @ Ys @ Zs )
= ( cons_a @ X @ Xs ) )
= ( ( ( Ys = nil_a )
& ( Zs
= ( cons_a @ X @ Xs ) ) )
| ? [Ys6: list_a] :
( ( Ys
= ( cons_a @ X @ Ys6 ) )
& ( ( append_a @ Ys6 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_779_Cons__eq__append__conv,axiom,
! [X: a,Xs: list_a,Ys: list_a,Zs: list_a] :
( ( ( cons_a @ X @ Xs )
= ( append_a @ Ys @ Zs ) )
= ( ( ( Ys = nil_a )
& ( ( cons_a @ X @ Xs )
= Zs ) )
| ? [Ys6: list_a] :
( ( ( cons_a @ X @ Ys6 )
= Ys )
& ( Xs
= ( append_a @ Ys6 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_780_rev__exhaust,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ~ ! [Ys2: list_a,Y3: a] :
( Xs
!= ( append_a @ Ys2 @ ( cons_a @ Y3 @ nil_a ) ) ) ) ).
% rev_exhaust
thf(fact_781_rev__induct,axiom,
! [P3: list_a > $o,Xs: list_a] :
( ( P3 @ nil_a )
=> ( ! [X4: a,Xs2: list_a] :
( ( P3 @ Xs2 )
=> ( P3 @ ( append_a @ Xs2 @ ( cons_a @ X4 @ nil_a ) ) ) )
=> ( P3 @ Xs ) ) ) ).
% rev_induct
thf(fact_782_monoid_Ofmset__perm__cong,axiom,
! [G: partia7680350978787392319xt_a_b,As: list_a,Bs: list_a] :
( ( monoid_a_b @ G )
=> ( ( ( mset_a @ As )
= ( mset_a @ Bs ) )
=> ( ( fmset_a_b @ G @ As )
= ( fmset_a_b @ G @ Bs ) ) ) ) ).
% monoid.fmset_perm_cong
thf(fact_783_plus__multiset_Orep__eq,axiom,
! [X: multiset_set_a,Xa2: multiset_set_a] :
( ( count_set_a @ ( plus_p2331992037799027419_set_a @ X @ Xa2 ) )
= ( ^ [A5: set_a] : ( plus_plus_nat @ ( count_set_a @ X @ A5 ) @ ( count_set_a @ Xa2 @ A5 ) ) ) ) ).
% plus_multiset.rep_eq
thf(fact_784_same__length__different,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != Ys )
=> ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ? [Pre: list_a,X4: a,Xs4: list_a,Y3: a,Ys4: list_a] :
( ( X4 != Y3 )
& ( Xs
= ( append_a @ Pre @ ( append_a @ ( cons_a @ X4 @ nil_a ) @ Xs4 ) ) )
& ( Ys
= ( append_a @ Pre @ ( append_a @ ( cons_a @ Y3 @ nil_a ) @ Ys4 ) ) ) ) ) ) ).
% same_length_different
thf(fact_785_monoid_Operm__assoc__switch,axiom,
! [G: partia7680350978787392319xt_a_b,As: list_a,Bs: list_a,Cs3: list_a] :
( ( monoid_a_b @ G )
=> ( ( list_all2_a_a @ ( associated_a_b @ G ) @ As @ Bs )
=> ( ( ( mset_a @ Bs )
= ( mset_a @ Cs3 ) )
=> ? [Bs2: list_a] :
( ( ( mset_a @ As )
= ( mset_a @ Bs2 ) )
& ( list_all2_a_a @ ( associated_a_b @ G ) @ Bs2 @ Cs3 ) ) ) ) ) ).
% monoid.perm_assoc_switch
thf(fact_786_monoid_Operm__assoc__switch__r,axiom,
! [G: partia7680350978787392319xt_a_b,As: list_a,Bs: list_a,Cs3: list_a] :
( ( monoid_a_b @ G )
=> ( ( ( mset_a @ As )
= ( mset_a @ Bs ) )
=> ( ( list_all2_a_a @ ( associated_a_b @ G ) @ Bs @ Cs3 )
=> ? [Bs2: list_a] :
( ( list_all2_a_a @ ( associated_a_b @ G ) @ As @ Bs2 )
& ( ( mset_a @ Bs2 )
= ( mset_a @ Cs3 ) ) ) ) ) ) ).
% monoid.perm_assoc_switch_r
thf(fact_787_monoid_Oirrlist__perm__cong,axiom,
! [G: partia7680350978787392319xt_a_b,As: list_a,Bs: list_a] :
( ( monoid_a_b @ G )
=> ( ( ( mset_a @ As )
= ( mset_a @ Bs ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ As ) )
=> ( irreducible_a_b @ G @ X4 ) )
=> ! [X6: a] :
( ( member_a @ X6 @ ( set_a2 @ Bs ) )
=> ( irreducible_a_b @ G @ X6 ) ) ) ) ) ).
% monoid.irrlist_perm_cong
thf(fact_788_essentially__equal__def,axiom,
( essent1248755695173879485al_a_b
= ( ^ [G2: partia7680350978787392319xt_a_b,Fs14: list_a,Fs23: list_a] :
? [Fs15: list_a] :
( ( ( mset_a @ Fs14 )
= ( mset_a @ Fs15 ) )
& ( list_all2_a_a @ ( associated_a_b @ G2 ) @ Fs15 @ Fs23 ) ) ) ) ).
% essentially_equal_def
thf(fact_789_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( plus_p2331992037799027419_set_a @ ( plus_p2331992037799027419_set_a @ A @ B ) @ C )
= ( plus_p2331992037799027419_set_a @ A @ ( plus_p2331992037799027419_set_a @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_790_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_791_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: multiset_set_a,J: multiset_set_a,K2: multiset_set_a,L2: multiset_set_a] :
( ( ( I = J )
& ( K2 = L2 ) )
=> ( ( plus_p2331992037799027419_set_a @ I @ K2 )
= ( plus_p2331992037799027419_set_a @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_792_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K2: nat,L2: nat] :
( ( ( I = J )
& ( K2 = L2 ) )
=> ( ( plus_plus_nat @ I @ K2 )
= ( plus_plus_nat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_793_group__cancel_Oadd1,axiom,
! [A4: multiset_set_a,K2: multiset_set_a,A: multiset_set_a,B: multiset_set_a] :
( ( A4
= ( plus_p2331992037799027419_set_a @ K2 @ A ) )
=> ( ( plus_p2331992037799027419_set_a @ A4 @ B )
= ( plus_p2331992037799027419_set_a @ K2 @ ( plus_p2331992037799027419_set_a @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_794_group__cancel_Oadd1,axiom,
! [A4: nat,K2: nat,A: nat,B: nat] :
( ( A4
= ( plus_plus_nat @ K2 @ A ) )
=> ( ( plus_plus_nat @ A4 @ B )
= ( plus_plus_nat @ K2 @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_795_group__cancel_Oadd2,axiom,
! [B3: multiset_set_a,K2: multiset_set_a,B: multiset_set_a,A: multiset_set_a] :
( ( B3
= ( plus_p2331992037799027419_set_a @ K2 @ B ) )
=> ( ( plus_p2331992037799027419_set_a @ A @ B3 )
= ( plus_p2331992037799027419_set_a @ K2 @ ( plus_p2331992037799027419_set_a @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_796_group__cancel_Oadd2,axiom,
! [B3: nat,K2: nat,B: nat,A: nat] :
( ( B3
= ( plus_plus_nat @ K2 @ B ) )
=> ( ( plus_plus_nat @ A @ B3 )
= ( plus_plus_nat @ K2 @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_797_add_Oassoc,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( plus_p2331992037799027419_set_a @ ( plus_p2331992037799027419_set_a @ A @ B ) @ C )
= ( plus_p2331992037799027419_set_a @ A @ ( plus_p2331992037799027419_set_a @ B @ C ) ) ) ).
% add.assoc
thf(fact_798_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_799_add_Ocommute,axiom,
( plus_p2331992037799027419_set_a
= ( ^ [A5: multiset_set_a,B5: multiset_set_a] : ( plus_p2331992037799027419_set_a @ B5 @ A5 ) ) ) ).
% add.commute
thf(fact_800_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A5: nat,B5: nat] : ( plus_plus_nat @ B5 @ A5 ) ) ) ).
% add.commute
thf(fact_801_add_Oleft__commute,axiom,
! [B: multiset_set_a,A: multiset_set_a,C: multiset_set_a] :
( ( plus_p2331992037799027419_set_a @ B @ ( plus_p2331992037799027419_set_a @ A @ C ) )
= ( plus_p2331992037799027419_set_a @ A @ ( plus_p2331992037799027419_set_a @ B @ C ) ) ) ).
% add.left_commute
thf(fact_802_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_803_add__left__imp__eq,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( ( plus_p2331992037799027419_set_a @ A @ B )
= ( plus_p2331992037799027419_set_a @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_804_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_805_add__right__imp__eq,axiom,
! [B: multiset_set_a,A: multiset_set_a,C: multiset_set_a] :
( ( ( plus_p2331992037799027419_set_a @ B @ A )
= ( plus_p2331992037799027419_set_a @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_806_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_807_monoid_Operm__closed,axiom,
! [G: partia7680350978787392319xt_a_b,As: list_a,Bs: list_a] :
( ( monoid_a_b @ G )
=> ( ( ( mset_a @ As )
= ( mset_a @ Bs ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia7484183841585581558xt_a_b @ G ) ) ) ) ) ).
% monoid.perm_closed
thf(fact_808_monoid_Oessentially__equalE,axiom,
! [G: partia7680350978787392319xt_a_b,Fs1: list_a,Fs22: list_a] :
( ( monoid_a_b @ G )
=> ( ( essent1248755695173879485al_a_b @ G @ Fs1 @ Fs22 )
=> ~ ! [Fs13: list_a] :
( ( ( mset_a @ Fs1 )
= ( mset_a @ Fs13 ) )
=> ~ ( list_all2_a_a @ ( associated_a_b @ G ) @ Fs13 @ Fs22 ) ) ) ) ).
% monoid.essentially_equalE
thf(fact_809_monoid_Oessentially__equalI,axiom,
! [G: partia7680350978787392319xt_a_b,Fs1: list_a,Fs12: list_a,Fs22: list_a] :
( ( monoid_a_b @ G )
=> ( ( ( mset_a @ Fs1 )
= ( mset_a @ Fs12 ) )
=> ( ( list_all2_a_a @ ( associated_a_b @ G ) @ Fs12 @ Fs22 )
=> ( essent1248755695173879485al_a_b @ G @ Fs1 @ Fs22 ) ) ) ) ).
% monoid.essentially_equalI
thf(fact_810_monoid_Onat__pow__mult,axiom,
! [G: partia8223610829204095565t_unit,X: a,N: nat,M4: nat] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( mult_a_Product_unit @ G @ ( pow_a_1875594501834816709it_nat @ G @ X @ N ) @ ( pow_a_1875594501834816709it_nat @ G @ X @ M4 ) )
= ( pow_a_1875594501834816709it_nat @ G @ X @ ( plus_plus_nat @ N @ M4 ) ) ) ) ) ).
% monoid.nat_pow_mult
thf(fact_811_monoid_Onat__pow__mult,axiom,
! [G: partia7680350978787392319xt_a_b,X: a,N: nat,M4: nat] :
( ( monoid_a_b @ G )
=> ( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( mult_a_b @ G @ ( pow_a_b_nat @ G @ X @ N ) @ ( pow_a_b_nat @ G @ X @ M4 ) )
= ( pow_a_b_nat @ G @ X @ ( plus_plus_nat @ N @ M4 ) ) ) ) ) ).
% monoid.nat_pow_mult
thf(fact_812_comm__monoid_Owfactors__perm__cong__l,axiom,
! [G: partia7680350978787392319xt_a_b,Fs: list_a,A: a,Fs5: list_a] :
( ( comm_monoid_a_b @ G )
=> ( ( wfactors_a_b @ G @ Fs @ A )
=> ( ( ( mset_a @ Fs )
= ( mset_a @ Fs5 ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( wfactors_a_b @ G @ Fs5 @ A ) ) ) ) ) ).
% comm_monoid.wfactors_perm_cong_l
thf(fact_813_comm__monoid_Operm__wfactorsD,axiom,
! [G: partia7680350978787392319xt_a_b,As: list_a,Bs: list_a,A: a,B: a] :
( ( comm_monoid_a_b @ G )
=> ( ( ( mset_a @ As )
= ( mset_a @ Bs ) )
=> ( ( wfactors_a_b @ G @ As @ A )
=> ( ( wfactors_a_b @ G @ Bs @ B )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( associated_a_b @ G @ A @ B ) ) ) ) ) ) ) ) ).
% comm_monoid.perm_wfactorsD
thf(fact_814_monoid_Ofactors__mult,axiom,
! [G: partia7680350978787392319xt_a_b,Fa: list_a,A: a,Fb: list_a,B: a] :
( ( monoid_a_b @ G )
=> ( ( factors_a_b @ G @ Fa @ A )
=> ( ( factors_a_b @ G @ Fb @ B )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fa ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fb ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( factors_a_b @ G @ ( append_a @ Fa @ Fb ) @ ( mult_a_b @ G @ A @ B ) ) ) ) ) ) ) ).
% monoid.factors_mult
thf(fact_815_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: multiset_set_a,J: multiset_set_a,K2: multiset_set_a,L2: multiset_set_a] :
( ( ( ord_le7905258569527593284_set_a @ I @ J )
& ( K2 = L2 ) )
=> ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ I @ K2 ) @ ( plus_p2331992037799027419_set_a @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_816_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K2: nat,L2: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K2 = L2 ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_817_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: multiset_set_a,J: multiset_set_a,K2: multiset_set_a,L2: multiset_set_a] :
( ( ( I = J )
& ( ord_le7905258569527593284_set_a @ K2 @ L2 ) )
=> ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ I @ K2 ) @ ( plus_p2331992037799027419_set_a @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_818_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K2: nat,L2: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K2 @ L2 ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_819_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: multiset_set_a,J: multiset_set_a,K2: multiset_set_a,L2: multiset_set_a] :
( ( ( ord_le7905258569527593284_set_a @ I @ J )
& ( ord_le7905258569527593284_set_a @ K2 @ L2 ) )
=> ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ I @ K2 ) @ ( plus_p2331992037799027419_set_a @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_820_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K2: nat,L2: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K2 @ L2 ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_821_add__mono,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a,D: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ A @ B )
=> ( ( ord_le7905258569527593284_set_a @ C @ D )
=> ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ A @ C ) @ ( plus_p2331992037799027419_set_a @ B @ D ) ) ) ) ).
% add_mono
thf(fact_822_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_823_add__left__mono,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ A @ B )
=> ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ C @ A ) @ ( plus_p2331992037799027419_set_a @ C @ B ) ) ) ).
% add_left_mono
thf(fact_824_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_825_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C2: nat] :
( B
!= ( plus_plus_nat @ A @ C2 ) ) ) ).
% less_eqE
thf(fact_826_add__right__mono,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ A @ B )
=> ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ A @ C ) @ ( plus_p2331992037799027419_set_a @ B @ C ) ) ) ).
% add_right_mono
thf(fact_827_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_828_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
? [C5: nat] :
( B5
= ( plus_plus_nat @ A5 @ C5 ) ) ) ) ).
% le_iff_add
thf(fact_829_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_830_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_831_comm__monoid_Omultlist__perm__cong,axiom,
! [G: partia7680350978787392319xt_a_b,As: list_a,Bs: list_a] :
( ( comm_monoid_a_b @ G )
=> ( ( ( mset_a @ As )
= ( mset_a @ Bs ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( foldr_a_a @ ( mult_a_b @ G ) @ As @ ( one_a_b @ G ) )
= ( foldr_a_a @ ( mult_a_b @ G ) @ Bs @ ( one_a_b @ G ) ) ) ) ) ) ).
% comm_monoid.multlist_perm_cong
thf(fact_832_perm__sing__eq2,axiom,
! [Y: a,Ys: list_a] :
( ( ( mset_a @ ( cons_a @ Y @ nil_a ) )
= ( mset_a @ Ys ) )
= ( Ys
= ( cons_a @ Y @ nil_a ) ) ) ).
% perm_sing_eq2
thf(fact_833_perm__sing__eq,axiom,
! [Ys: list_a,Y: a] :
( ( ( mset_a @ Ys )
= ( mset_a @ ( cons_a @ Y @ nil_a ) ) )
= ( Ys
= ( cons_a @ Y @ nil_a ) ) ) ).
% perm_sing_eq
thf(fact_834_cons__perm__eq,axiom,
! [Z: a,Xs: list_a,Ys: list_a] :
( ( ( mset_a @ ( cons_a @ Z @ Xs ) )
= ( mset_a @ ( cons_a @ Z @ Ys ) ) )
= ( ( mset_a @ Xs )
= ( mset_a @ Ys ) ) ) ).
% cons_perm_eq
thf(fact_835_factor__mset__pow,axiom,
! [A: a,N: nat] :
( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( finite8971978181520403909et_a_b @ g @ ( pow_a_b_nat @ g @ A @ N ) )
= ( repeat_mset_set_a @ N @ ( finite8971978181520403909et_a_b @ g @ A ) ) ) ) ).
% factor_mset_pow
thf(fact_836_repeat__mset__distrib2,axiom,
! [N: nat,A4: multiset_set_a,B3: multiset_set_a] :
( ( repeat_mset_set_a @ N @ ( plus_p2331992037799027419_set_a @ A4 @ B3 ) )
= ( plus_p2331992037799027419_set_a @ ( repeat_mset_set_a @ N @ A4 ) @ ( repeat_mset_set_a @ N @ B3 ) ) ) ).
% repeat_mset_distrib2
thf(fact_837_left__add__mult__distrib__mset,axiom,
! [I: nat,U: multiset_set_a,J: nat,K2: multiset_set_a] :
( ( plus_p2331992037799027419_set_a @ ( repeat_mset_set_a @ I @ U ) @ ( plus_p2331992037799027419_set_a @ ( repeat_mset_set_a @ J @ U ) @ K2 ) )
= ( plus_p2331992037799027419_set_a @ ( repeat_mset_set_a @ ( plus_plus_nat @ I @ J ) @ U ) @ K2 ) ) ).
% left_add_mult_distrib_mset
thf(fact_838_repeat__mset__distrib,axiom,
! [M4: nat,N: nat,A4: multiset_set_a] :
( ( repeat_mset_set_a @ ( plus_plus_nat @ M4 @ N ) @ A4 )
= ( plus_p2331992037799027419_set_a @ ( repeat_mset_set_a @ M4 @ A4 ) @ ( repeat_mset_set_a @ N @ A4 ) ) ) ).
% repeat_mset_distrib
thf(fact_839_factorial__monoid_Ofactor__mset__pow,axiom,
! [G: partia8223610829204095565t_unit,A: a,N: nat] :
( ( factor2046533344642582127t_unit @ G )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( finite2487704559579920083t_unit @ G @ ( pow_a_1875594501834816709it_nat @ G @ A @ N ) )
= ( repeat_mset_set_a @ N @ ( finite2487704559579920083t_unit @ G @ A ) ) ) ) ) ).
% factorial_monoid.factor_mset_pow
thf(fact_840_factorial__monoid_Ofactor__mset__pow,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,N: nat] :
( ( factorial_monoid_a_b @ G )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( finite8971978181520403909et_a_b @ G @ ( pow_a_b_nat @ G @ A @ N ) )
= ( repeat_mset_set_a @ N @ ( finite8971978181520403909et_a_b @ G @ A ) ) ) ) ) ).
% factorial_monoid.factor_mset_pow
thf(fact_841_cons__perm__imp__perm,axiom,
! [Z: a,Xs: list_a,Ys: list_a] :
( ( ( mset_a @ ( cons_a @ Z @ Xs ) )
= ( mset_a @ ( cons_a @ Z @ Ys ) ) )
=> ( ( mset_a @ Xs )
= ( mset_a @ Ys ) ) ) ).
% cons_perm_imp_perm
thf(fact_842_perm__set__eq,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( mset_a @ Xs )
= ( mset_a @ Ys ) )
=> ( ( set_a2 @ Xs )
= ( set_a2 @ Ys ) ) ) ).
% perm_set_eq
thf(fact_843_perm__sing__imp,axiom,
! [Ys: list_a,Xs: list_a,Y: a] :
( ( ( mset_a @ Ys )
= ( mset_a @ Xs ) )
=> ( ( Xs
= ( cons_a @ Y @ nil_a ) )
=> ( Ys
= ( cons_a @ Y @ nil_a ) ) ) ) ).
% perm_sing_imp
thf(fact_844_perm__append__Cons,axiom,
! [A: a,Xs: list_a,Ys: list_a] :
( ( mset_a @ ( cons_a @ A @ ( append_a @ Xs @ Ys ) ) )
= ( mset_a @ ( append_a @ Xs @ ( cons_a @ A @ Ys ) ) ) ) ).
% perm_append_Cons
thf(fact_845_mset__le__perm__append,axiom,
! [Xs: list_a,Ys: list_a] :
( ( subseteq_mset_a @ ( mset_a @ Xs ) @ ( mset_a @ Ys ) )
= ( ? [Zs3: list_a] :
( ( mset_a @ ( append_a @ Xs @ Zs3 ) )
= ( mset_a @ Ys ) ) ) ) ).
% mset_le_perm_append
thf(fact_846_mset__le__perm__append,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( subseteq_mset_set_a @ ( mset_set_a @ Xs ) @ ( mset_set_a @ Ys ) )
= ( ? [Zs3: list_set_a] :
( ( mset_set_a @ ( append_set_a @ Xs @ Zs3 ) )
= ( mset_set_a @ Ys ) ) ) ) ).
% mset_le_perm_append
thf(fact_847_perm__append__single,axiom,
! [A: a,Xs: list_a] :
( ( mset_a @ ( cons_a @ A @ Xs ) )
= ( mset_a @ ( append_a @ Xs @ ( cons_a @ A @ nil_a ) ) ) ) ).
% perm_append_single
thf(fact_848_nat__add__left__cancel__le,axiom,
! [K2: nat,M4: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K2 @ M4 ) @ ( plus_plus_nat @ K2 @ N ) )
= ( ord_less_eq_nat @ M4 @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_849_properfactor__fcount,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( properfactor_a_b @ g @ A @ B )
=> ( ord_less_nat @ ( factorcount_a_b @ g @ A ) @ ( factorcount_a_b @ g @ B ) ) ) ) ) ).
% properfactor_fcount
thf(fact_850_factorial__monoid__axioms__def,axiom,
( factor4787786757223245700ms_a_b
= ( ^ [G2: partia7680350978787392319xt_a_b] :
( ! [A5: a] :
( ( member_a @ A5 @ ( partia7484183841585581558xt_a_b @ G2 ) )
=> ( ~ ( member_a @ A5 @ ( units_a_b @ G2 ) )
=> ? [Fs6: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Fs6 ) @ ( partia7484183841585581558xt_a_b @ G2 ) )
& ( factors_a_b @ G2 @ Fs6 @ A5 ) ) ) )
& ! [Fs6: list_a,A5: a,Fs7: list_a] :
( ( factors_a_b @ G2 @ Fs6 @ A5 )
=> ( ( factors_a_b @ G2 @ Fs7 @ A5 )
=> ( ( member_a @ A5 @ ( partia7484183841585581558xt_a_b @ G2 ) )
=> ( ~ ( member_a @ A5 @ ( units_a_b @ G2 ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs6 ) @ ( partia7484183841585581558xt_a_b @ G2 ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs7 ) @ ( partia7484183841585581558xt_a_b @ G2 ) )
=> ( essent1248755695173879485al_a_b @ G2 @ Fs6 @ Fs7 ) ) ) ) ) ) ) ) ) ) ).
% factorial_monoid_axioms_def
thf(fact_851_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_852_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_853_nat__add__left__cancel__less,axiom,
! [K2: nat,M4: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K2 @ M4 ) @ ( plus_plus_nat @ K2 @ N ) )
= ( ord_less_nat @ M4 @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_854_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_855_le__neq__implies__less,axiom,
! [M4: nat,N: nat] :
( ( ord_less_eq_nat @ M4 @ N )
=> ( ( M4 != N )
=> ( ord_less_nat @ M4 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_856_less__or__eq__imp__le,axiom,
! [M4: nat,N: nat] :
( ( ( ord_less_nat @ M4 @ N )
| ( M4 = N ) )
=> ( ord_less_eq_nat @ M4 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_857_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N4: nat] :
( ( ord_less_nat @ M5 @ N4 )
| ( M5 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_858_less__imp__le__nat,axiom,
! [M4: nat,N: nat] :
( ( ord_less_nat @ M4 @ N )
=> ( ord_less_eq_nat @ M4 @ N ) ) ).
% less_imp_le_nat
thf(fact_859_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M5: nat,N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
& ( M5 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_860_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K2: nat,L2: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K2 @ L2 ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_861_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K2: nat,L2: nat] :
( ( ( I = J )
& ( ord_less_nat @ K2 @ L2 ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_862_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K2: nat,L2: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K2 = L2 ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_863_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_864_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_865_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_866_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_867_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_868_add__lessD1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K2 )
=> ( ord_less_nat @ I @ K2 ) ) ).
% add_lessD1
thf(fact_869_add__less__mono,axiom,
! [I: nat,J: nat,K2: nat,L2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K2 @ L2 )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L2 ) ) ) ) ).
% add_less_mono
thf(fact_870_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_871_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_872_add__less__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% add_less_mono1
thf(fact_873_trans__less__add1,axiom,
! [I: nat,J: nat,M4: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M4 ) ) ) ).
% trans_less_add1
thf(fact_874_trans__less__add2,axiom,
! [I: nat,J: nat,M4: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M4 @ J ) ) ) ).
% trans_less_add2
thf(fact_875_less__add__eq__less,axiom,
! [K2: nat,L2: nat,M4: nat,N: nat] :
( ( ord_less_nat @ K2 @ L2 )
=> ( ( ( plus_plus_nat @ M4 @ L2 )
= ( plus_plus_nat @ K2 @ N ) )
=> ( ord_less_nat @ M4 @ N ) ) ) ).
% less_add_eq_less
thf(fact_876_mono__nat__linear__lb,axiom,
! [F: nat > nat,M4: nat,K2: nat] :
( ! [M3: nat,N5: nat] :
( ( ord_less_nat @ M3 @ N5 )
=> ( ord_less_nat @ ( F @ M3 ) @ ( F @ N5 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M4 ) @ K2 ) @ ( F @ ( plus_plus_nat @ M4 @ K2 ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_877_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_878_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_879_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K2: nat,L2: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K2 @ L2 ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_880_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K2: nat,L2: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K2 @ L2 ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_881_factorial__monoid_Oaxioms_I2_J,axiom,
! [G: partia7680350978787392319xt_a_b] :
( ( factorial_monoid_a_b @ G )
=> ( factor4787786757223245700ms_a_b @ G ) ) ).
% factorial_monoid.axioms(2)
thf(fact_882_Nat_Oex__has__greatest__nat,axiom,
! [P3: nat > $o,K2: nat,B: nat] :
( ( P3 @ K2 )
=> ( ! [Y3: nat] :
( ( P3 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X4: nat] :
( ( P3 @ X4 )
& ! [Y7: nat] :
( ( P3 @ Y7 )
=> ( ord_less_eq_nat @ Y7 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_883_nat__le__linear,axiom,
! [M4: nat,N: nat] :
( ( ord_less_eq_nat @ M4 @ N )
| ( ord_less_eq_nat @ N @ M4 ) ) ).
% nat_le_linear
thf(fact_884_le__antisym,axiom,
! [M4: nat,N: nat] :
( ( ord_less_eq_nat @ M4 @ N )
=> ( ( ord_less_eq_nat @ N @ M4 )
=> ( M4 = N ) ) ) ).
% le_antisym
thf(fact_885_eq__imp__le,axiom,
! [M4: nat,N: nat] :
( ( M4 = N )
=> ( ord_less_eq_nat @ M4 @ N ) ) ).
% eq_imp_le
thf(fact_886_le__trans,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K2 )
=> ( ord_less_eq_nat @ I @ K2 ) ) ) ).
% le_trans
thf(fact_887_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_888_factorial__monoid_Oproperfactor__fcount,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a] :
( ( factorial_monoid_a_b @ G )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( properfactor_a_b @ G @ A @ B )
=> ( ord_less_nat @ ( factorcount_a_b @ G @ A ) @ ( factorcount_a_b @ G @ B ) ) ) ) ) ) ).
% factorial_monoid.properfactor_fcount
thf(fact_889_add__leE,axiom,
! [M4: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M4 @ K2 ) @ N )
=> ~ ( ( ord_less_eq_nat @ M4 @ N )
=> ~ ( ord_less_eq_nat @ K2 @ N ) ) ) ).
% add_leE
thf(fact_890_le__add1,axiom,
! [N: nat,M4: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M4 ) ) ).
% le_add1
thf(fact_891_le__add2,axiom,
! [N: nat,M4: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M4 @ N ) ) ).
% le_add2
thf(fact_892_add__leD1,axiom,
! [M4: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M4 @ K2 ) @ N )
=> ( ord_less_eq_nat @ M4 @ N ) ) ).
% add_leD1
thf(fact_893_add__leD2,axiom,
! [M4: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M4 @ K2 ) @ N )
=> ( ord_less_eq_nat @ K2 @ N ) ) ).
% add_leD2
thf(fact_894_le__Suc__ex,axiom,
! [K2: nat,L2: nat] :
( ( ord_less_eq_nat @ K2 @ L2 )
=> ? [N5: nat] :
( L2
= ( plus_plus_nat @ K2 @ N5 ) ) ) ).
% le_Suc_ex
thf(fact_895_add__le__mono,axiom,
! [I: nat,J: nat,K2: nat,L2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K2 @ L2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L2 ) ) ) ) ).
% add_le_mono
thf(fact_896_add__le__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% add_le_mono1
thf(fact_897_trans__le__add1,axiom,
! [I: nat,J: nat,M4: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M4 ) ) ) ).
% trans_le_add1
thf(fact_898_trans__le__add2,axiom,
! [I: nat,J: nat,M4: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M4 @ J ) ) ) ).
% trans_le_add2
thf(fact_899_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N4: nat] :
? [K3: nat] :
( N4
= ( plus_plus_nat @ M5 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_900_factorial__monoid__axioms_Ointro,axiom,
! [G: partia7680350978787392319xt_a_b] :
( ! [A2: a] :
( ( member_a @ A2 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ~ ( member_a @ A2 @ ( units_a_b @ G ) )
=> ? [Fs3: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Fs3 ) @ ( partia7484183841585581558xt_a_b @ G ) )
& ( factors_a_b @ G @ Fs3 @ A2 ) ) ) )
=> ( ! [Fs2: list_a,A2: a,Fs4: list_a] :
( ( factors_a_b @ G @ Fs2 @ A2 )
=> ( ( factors_a_b @ G @ Fs4 @ A2 )
=> ( ( member_a @ A2 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ~ ( member_a @ A2 @ ( units_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs2 ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs4 ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( essent1248755695173879485al_a_b @ G @ Fs2 @ Fs4 ) ) ) ) ) ) )
=> ( factor4787786757223245700ms_a_b @ G ) ) ) ).
% factorial_monoid_axioms.intro
thf(fact_901_multlist__prime__pos,axiom,
! [A: a,As: list_a] :
( ( prime_a_b @ g @ A )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( factor_a_b @ g @ A @ ( foldr_a_a @ ( mult_a_b @ g ) @ As @ ( one_a_b @ g ) ) )
=> ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ As ) )
& ( factor_a_b @ g @ A @ ( nth_a @ As @ I2 ) ) ) ) ) ) ) ).
% multlist_prime_pos
thf(fact_902_nunit__factors,axiom,
! [A: a,As: list_a] :
( ~ ( member_a @ A @ ( units_a_b @ g ) )
=> ( ( factors_a_b @ g @ As @ A )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ As ) ) ) ) ).
% nunit_factors
thf(fact_903_factor__mset__unit,axiom,
( ( finite8971978181520403909et_a_b @ g @ ( one_a_b @ g ) )
= zero_z5079479921072680283_set_a ) ).
% factor_mset_unit
thf(fact_904_psubsetI,axiom,
! [A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A4 @ B3 )
=> ( ( A4 != B3 )
=> ( ord_less_set_a @ A4 @ B3 ) ) ) ).
% psubsetI
thf(fact_905_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_906_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_907_add__0,axiom,
! [A: multiset_set_a] :
( ( plus_p2331992037799027419_set_a @ zero_z5079479921072680283_set_a @ A )
= A ) ).
% add_0
thf(fact_908_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_909_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_910_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_911_add__cancel__right__right,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( A
= ( plus_p2331992037799027419_set_a @ A @ B ) )
= ( B = zero_z5079479921072680283_set_a ) ) ).
% add_cancel_right_right
thf(fact_912_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_913_add__cancel__right__left,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( A
= ( plus_p2331992037799027419_set_a @ B @ A ) )
= ( B = zero_z5079479921072680283_set_a ) ) ).
% add_cancel_right_left
thf(fact_914_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_915_add__cancel__left__right,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( ( plus_p2331992037799027419_set_a @ A @ B )
= A )
= ( B = zero_z5079479921072680283_set_a ) ) ).
% add_cancel_left_right
thf(fact_916_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_917_add__cancel__left__left,axiom,
! [B: multiset_set_a,A: multiset_set_a] :
( ( ( plus_p2331992037799027419_set_a @ B @ A )
= A )
= ( B = zero_z5079479921072680283_set_a ) ) ).
% add_cancel_left_left
thf(fact_918_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_919_add_Oright__neutral,axiom,
! [A: multiset_set_a] :
( ( plus_p2331992037799027419_set_a @ A @ zero_z5079479921072680283_set_a )
= A ) ).
% add.right_neutral
thf(fact_920_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_921_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_922_add__is__0,axiom,
! [M4: nat,N: nat] :
( ( ( plus_plus_nat @ M4 @ N )
= zero_zero_nat )
= ( ( M4 = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_923_Nat_Oadd__0__right,axiom,
! [M4: nat] :
( ( plus_plus_nat @ M4 @ zero_zero_nat )
= M4 ) ).
% Nat.add_0_right
thf(fact_924_subset__mset_Oextremum__unique,axiom,
! [A: multiset_set_a] :
( ( subseteq_mset_set_a @ A @ zero_z5079479921072680283_set_a )
= ( A = zero_z5079479921072680283_set_a ) ) ).
% subset_mset.extremum_unique
thf(fact_925_subset__mset_Ole__zero__eq,axiom,
! [N: multiset_set_a] :
( ( subseteq_mset_set_a @ N @ zero_z5079479921072680283_set_a )
= ( N = zero_z5079479921072680283_set_a ) ) ).
% subset_mset.le_zero_eq
thf(fact_926_union__eq__empty,axiom,
! [M: multiset_set_a,N3: multiset_set_a] :
( ( ( plus_p2331992037799027419_set_a @ M @ N3 )
= zero_z5079479921072680283_set_a )
= ( ( M = zero_z5079479921072680283_set_a )
& ( N3 = zero_z5079479921072680283_set_a ) ) ) ).
% union_eq_empty
thf(fact_927_empty__eq__union,axiom,
! [M: multiset_set_a,N3: multiset_set_a] :
( ( zero_z5079479921072680283_set_a
= ( plus_p2331992037799027419_set_a @ M @ N3 ) )
= ( ( M = zero_z5079479921072680283_set_a )
& ( N3 = zero_z5079479921072680283_set_a ) ) ) ).
% empty_eq_union
thf(fact_928_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
! [X: multiset_set_a,Y: multiset_set_a] :
( ( zero_z5079479921072680283_set_a
= ( plus_p2331992037799027419_set_a @ X @ Y ) )
= ( ( X = zero_z5079479921072680283_set_a )
& ( Y = zero_z5079479921072680283_set_a ) ) ) ).
% subset_mset.zero_eq_add_iff_both_eq_0
thf(fact_929_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
! [X: multiset_set_a,Y: multiset_set_a] :
( ( ( plus_p2331992037799027419_set_a @ X @ Y )
= zero_z5079479921072680283_set_a )
= ( ( X = zero_z5079479921072680283_set_a )
& ( Y = zero_z5079479921072680283_set_a ) ) ) ).
% subset_mset.add_eq_0_iff_both_eq_0
thf(fact_930_repeat__mset__empty,axiom,
! [N: nat] :
( ( repeat_mset_set_a @ N @ zero_z5079479921072680283_set_a )
= zero_z5079479921072680283_set_a ) ).
% repeat_mset_empty
thf(fact_931_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_932_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_933_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_934_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_935_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_936_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_937_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_938_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_939_add__gr__0,axiom,
! [M4: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M4 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M4 )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_940_nth__Cons__0,axiom,
! [X: a,Xs: list_a] :
( ( nth_a @ ( cons_a @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_941_Group_Onat__pow__0,axiom,
! [G: partia8223610829204095565t_unit,X: a] :
( ( pow_a_1875594501834816709it_nat @ G @ X @ zero_zero_nat )
= ( one_a_Product_unit @ G ) ) ).
% Group.nat_pow_0
thf(fact_942_Group_Onat__pow__0,axiom,
! [G: partia7680350978787392319xt_a_b,X: a] :
( ( pow_a_b_nat @ G @ X @ zero_zero_nat )
= ( one_a_b @ G ) ) ).
% Group.nat_pow_0
thf(fact_943_set__mset__empty,axiom,
( ( set_mset_set_a @ zero_z5079479921072680283_set_a )
= bot_bot_set_set_a ) ).
% set_mset_empty
thf(fact_944_set__mset__empty,axiom,
( ( set_mset_a @ zero_zero_multiset_a )
= bot_bot_set_a ) ).
% set_mset_empty
thf(fact_945_set__mset__eq__empty__iff,axiom,
! [M: multiset_set_a] :
( ( ( set_mset_set_a @ M )
= bot_bot_set_set_a )
= ( M = zero_z5079479921072680283_set_a ) ) ).
% set_mset_eq_empty_iff
thf(fact_946_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_947_mset__zero__iff__right,axiom,
! [X: list_a] :
( ( zero_zero_multiset_a
= ( mset_a @ X ) )
= ( X = nil_a ) ) ).
% mset_zero_iff_right
thf(fact_948_mset__zero__iff__right,axiom,
! [X: list_set_a] :
( ( zero_z5079479921072680283_set_a
= ( mset_set_a @ X ) )
= ( X = nil_set_a ) ) ).
% mset_zero_iff_right
thf(fact_949_mset__zero__iff,axiom,
! [X: list_a] :
( ( ( mset_a @ X )
= zero_zero_multiset_a )
= ( X = nil_a ) ) ).
% mset_zero_iff
thf(fact_950_mset__zero__iff,axiom,
! [X: list_set_a] :
( ( ( mset_set_a @ X )
= zero_z5079479921072680283_set_a )
= ( X = nil_set_a ) ) ).
% mset_zero_iff
thf(fact_951_subset__mset_Ole__add__same__cancel2,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( subseteq_mset_set_a @ A @ ( plus_p2331992037799027419_set_a @ B @ A ) )
= ( subseteq_mset_set_a @ zero_z5079479921072680283_set_a @ B ) ) ).
% subset_mset.le_add_same_cancel2
thf(fact_952_subset__mset_Ole__add__same__cancel1,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( subseteq_mset_set_a @ A @ ( plus_p2331992037799027419_set_a @ A @ B ) )
= ( subseteq_mset_set_a @ zero_z5079479921072680283_set_a @ B ) ) ).
% subset_mset.le_add_same_cancel1
thf(fact_953_subset__mset_Oadd__le__same__cancel2,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( subseteq_mset_set_a @ ( plus_p2331992037799027419_set_a @ A @ B ) @ B )
= ( subseteq_mset_set_a @ A @ zero_z5079479921072680283_set_a ) ) ).
% subset_mset.add_le_same_cancel2
thf(fact_954_subset__mset_Oadd__le__same__cancel1,axiom,
! [B: multiset_set_a,A: multiset_set_a] :
( ( subseteq_mset_set_a @ ( plus_p2331992037799027419_set_a @ B @ A ) @ B )
= ( subseteq_mset_set_a @ A @ zero_z5079479921072680283_set_a ) ) ).
% subset_mset.add_le_same_cancel1
thf(fact_955_count__empty,axiom,
! [A: set_a] :
( ( count_set_a @ zero_z5079479921072680283_set_a @ A )
= zero_zero_nat ) ).
% count_empty
thf(fact_956_repeat__mset__0,axiom,
! [M: multiset_set_a] :
( ( repeat_mset_set_a @ zero_zero_nat @ M )
= zero_z5079479921072680283_set_a ) ).
% repeat_mset_0
thf(fact_957_replicate__mset__eq__empty__iff,axiom,
! [N: nat,A: set_a] :
( ( ( replicate_mset_set_a @ N @ A )
= zero_z5079479921072680283_set_a )
= ( N = zero_zero_nat ) ) ).
% replicate_mset_eq_empty_iff
thf(fact_958_replicate__mset__0,axiom,
! [X: set_a] :
( ( replicate_mset_set_a @ zero_zero_nat @ X )
= zero_z5079479921072680283_set_a ) ).
% replicate_mset_0
thf(fact_959_count__replicate__mset,axiom,
! [Y: set_a,X: set_a,N: nat] :
( ( ( Y = X )
=> ( ( count_set_a @ ( replicate_mset_set_a @ N @ X ) @ Y )
= N ) )
& ( ( Y != X )
=> ( ( count_set_a @ ( replicate_mset_set_a @ N @ X ) @ Y )
= zero_zero_nat ) ) ) ).
% count_replicate_mset
thf(fact_960_nth__append__length,axiom,
! [Xs: list_a,X: a,Ys: list_a] :
( ( nth_a @ ( append_a @ Xs @ ( cons_a @ X @ Ys ) ) @ ( size_size_list_a @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_961_nth__append__length__plus,axiom,
! [Xs: list_a,Ys: list_a,N: nat] :
( ( nth_a @ ( append_a @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_size_list_a @ Xs ) @ N ) )
= ( nth_a @ Ys @ N ) ) ).
% nth_append_length_plus
thf(fact_962_count__mset__0__iff,axiom,
! [Xs: list_a,X: a] :
( ( ( count_a @ ( mset_a @ Xs ) @ X )
= zero_zero_nat )
= ( ~ ( member_a @ X @ ( set_a2 @ Xs ) ) ) ) ).
% count_mset_0_iff
thf(fact_963_count__mset__0__iff,axiom,
! [Xs: list_set_a,X: set_a] :
( ( ( count_set_a @ ( mset_set_a @ Xs ) @ X )
= zero_zero_nat )
= ( ~ ( member_set_a @ X @ ( set_set_a2 @ Xs ) ) ) ) ).
% count_mset_0_iff
thf(fact_964_count__greater__zero__iff,axiom,
! [M: multiset_a,X: a] :
( ( ord_less_nat @ zero_zero_nat @ ( count_a @ M @ X ) )
= ( member_a @ X @ ( set_mset_a @ M ) ) ) ).
% count_greater_zero_iff
thf(fact_965_count__greater__zero__iff,axiom,
! [M: multiset_set_a,X: set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( count_set_a @ M @ X ) )
= ( member_set_a @ X @ ( set_mset_set_a @ M ) ) ) ).
% count_greater_zero_iff
thf(fact_966_in__replicate__mset,axiom,
! [X: a,N: nat,Y: a] :
( ( member_a @ X @ ( set_mset_a @ ( replicate_mset_a @ N @ Y ) ) )
= ( ( ord_less_nat @ zero_zero_nat @ N )
& ( X = Y ) ) ) ).
% in_replicate_mset
thf(fact_967_in__replicate__mset,axiom,
! [X: set_a,N: nat,Y: set_a] :
( ( member_set_a @ X @ ( set_mset_set_a @ ( replicate_mset_set_a @ N @ Y ) ) )
= ( ( ord_less_nat @ zero_zero_nat @ N )
& ( X = Y ) ) ) ).
% in_replicate_mset
thf(fact_968_local_Onat__pow__0,axiom,
! [X: a] :
( ( pow_a_b_nat @ g @ X @ zero_zero_nat )
= ( one_a_b @ g ) ) ).
% local.nat_pow_0
thf(fact_969_set__mset__replicate__mset__subset,axiom,
! [N: nat,X: set_a] :
( ( ( N = zero_zero_nat )
=> ( ( set_mset_set_a @ ( replicate_mset_set_a @ N @ X ) )
= bot_bot_set_set_a ) )
& ( ( N != zero_zero_nat )
=> ( ( set_mset_set_a @ ( replicate_mset_set_a @ N @ X ) )
= ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) ) ).
% set_mset_replicate_mset_subset
thf(fact_970_set__mset__replicate__mset__subset,axiom,
! [N: nat,X: a] :
( ( ( N = zero_zero_nat )
=> ( ( set_mset_a @ ( replicate_mset_a @ N @ X ) )
= bot_bot_set_a ) )
& ( ( N != zero_zero_nat )
=> ( ( set_mset_a @ ( replicate_mset_a @ N @ X ) )
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% set_mset_replicate_mset_subset
thf(fact_971_psubsetE,axiom,
! [A4: set_a,B3: set_a] :
( ( ord_less_set_a @ A4 @ B3 )
=> ~ ( ( ord_less_eq_set_a @ A4 @ B3 )
=> ( ord_less_eq_set_a @ B3 @ A4 ) ) ) ).
% psubsetE
thf(fact_972_psubset__eq,axiom,
( ord_less_set_a
= ( ^ [A6: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A6 @ B6 )
& ( A6 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_973_psubset__imp__subset,axiom,
! [A4: set_a,B3: set_a] :
( ( ord_less_set_a @ A4 @ B3 )
=> ( ord_less_eq_set_a @ A4 @ B3 ) ) ).
% psubset_imp_subset
thf(fact_974_psubset__subset__trans,axiom,
! [A4: set_a,B3: set_a,C4: set_a] :
( ( ord_less_set_a @ A4 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C4 )
=> ( ord_less_set_a @ A4 @ C4 ) ) ) ).
% psubset_subset_trans
thf(fact_975_subset__not__subset__eq,axiom,
( ord_less_set_a
= ( ^ [A6: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A6 @ B6 )
& ~ ( ord_less_eq_set_a @ B6 @ A6 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_976_subset__psubset__trans,axiom,
! [A4: set_a,B3: set_a,C4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B3 )
=> ( ( ord_less_set_a @ B3 @ C4 )
=> ( ord_less_set_a @ A4 @ C4 ) ) ) ).
% subset_psubset_trans
thf(fact_977_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
( ( ord_less_set_a @ A6 @ B6 )
| ( A6 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_978_union__le__mono1,axiom,
! [B3: multiset_set_a,D2: multiset_set_a,C4: multiset_set_a] :
( ( ord_le5765082015083327056_set_a @ B3 @ D2 )
=> ( ord_le5765082015083327056_set_a @ ( plus_p2331992037799027419_set_a @ B3 @ C4 ) @ ( plus_p2331992037799027419_set_a @ D2 @ C4 ) ) ) ).
% union_le_mono1
thf(fact_979_union__le__mono2,axiom,
! [B3: multiset_set_a,D2: multiset_set_a,C4: multiset_set_a] :
( ( ord_le5765082015083327056_set_a @ B3 @ D2 )
=> ( ord_le5765082015083327056_set_a @ ( plus_p2331992037799027419_set_a @ C4 @ B3 ) @ ( plus_p2331992037799027419_set_a @ C4 @ D2 ) ) ) ).
% union_le_mono2
thf(fact_980_union__less__mono,axiom,
! [A4: multiset_set_a,C4: multiset_set_a,B3: multiset_set_a,D2: multiset_set_a] :
( ( ord_le5765082015083327056_set_a @ A4 @ C4 )
=> ( ( ord_le5765082015083327056_set_a @ B3 @ D2 )
=> ( ord_le5765082015083327056_set_a @ ( plus_p2331992037799027419_set_a @ A4 @ B3 ) @ ( plus_p2331992037799027419_set_a @ C4 @ D2 ) ) ) ) ).
% union_less_mono
thf(fact_981_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_982_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_983_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_984_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_985_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_986_add__eq__self__zero,axiom,
! [M4: nat,N: nat] :
( ( ( plus_plus_nat @ M4 @ N )
= M4 )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_987_repeat__mset__cancel1,axiom,
! [A: nat,A4: multiset_set_a,B3: multiset_set_a] :
( ( ( repeat_mset_set_a @ A @ A4 )
= ( repeat_mset_set_a @ A @ B3 ) )
= ( ( A4 = B3 )
| ( A = zero_zero_nat ) ) ) ).
% repeat_mset_cancel1
thf(fact_988_repeat__mset__cancel2,axiom,
! [A: nat,A4: multiset_set_a,B: nat] :
( ( ( repeat_mset_set_a @ A @ A4 )
= ( repeat_mset_set_a @ B @ A4 ) )
= ( ( A = B )
| ( A4 = zero_z5079479921072680283_set_a ) ) ) ).
% repeat_mset_cancel2
thf(fact_989_repeat__mset__eq__empty__iff,axiom,
! [N: nat,A4: multiset_set_a] :
( ( ( repeat_mset_set_a @ N @ A4 )
= zero_z5079479921072680283_set_a )
= ( ( N = zero_zero_nat )
| ( A4 = zero_z5079479921072680283_set_a ) ) ) ).
% repeat_mset_eq_empty_iff
thf(fact_990_not__psubset__empty,axiom,
! [A4: set_a] :
~ ( ord_less_set_a @ A4 @ bot_bot_set_a ) ).
% not_psubset_empty
thf(fact_991_replicate__mset__eq__iff,axiom,
! [M4: nat,A: set_a,N: nat,B: set_a] :
( ( ( replicate_mset_set_a @ M4 @ A )
= ( replicate_mset_set_a @ N @ B ) )
= ( ( ( M4 = zero_zero_nat )
& ( N = zero_zero_nat ) )
| ( ( M4 = N )
& ( A = B ) ) ) ) ).
% replicate_mset_eq_iff
thf(fact_992_zero__multiset_Orep__eq,axiom,
( ( count_set_a @ zero_z5079479921072680283_set_a )
= ( ^ [A5: set_a] : zero_zero_nat ) ) ).
% zero_multiset.rep_eq
thf(fact_993_multiset__nonemptyE,axiom,
! [A4: multiset_a] :
( ( A4 != zero_zero_multiset_a )
=> ~ ! [X4: a] :
~ ( member_a @ X4 @ ( set_mset_a @ A4 ) ) ) ).
% multiset_nonemptyE
thf(fact_994_multiset__nonemptyE,axiom,
! [A4: multiset_set_a] :
( ( A4 != zero_z5079479921072680283_set_a )
=> ~ ! [X4: set_a] :
~ ( member_set_a @ X4 @ ( set_mset_set_a @ A4 ) ) ) ).
% multiset_nonemptyE
thf(fact_995_empty__le,axiom,
! [A4: multiset_set_a] : ( subseteq_mset_set_a @ zero_z5079479921072680283_set_a @ A4 ) ).
% empty_le
thf(fact_996_subset__mset_Oextremum__uniqueI,axiom,
! [A: multiset_set_a] :
( ( subseteq_mset_set_a @ A @ zero_z5079479921072680283_set_a )
=> ( A = zero_z5079479921072680283_set_a ) ) ).
% subset_mset.extremum_uniqueI
thf(fact_997_subset__mset_Obot__least,axiom,
! [A: multiset_set_a] : ( subseteq_mset_set_a @ zero_z5079479921072680283_set_a @ A ) ).
% subset_mset.bot_least
thf(fact_998_subset__mset_Ozero__le,axiom,
! [X: multiset_set_a] : ( subseteq_mset_set_a @ zero_z5079479921072680283_set_a @ X ) ).
% subset_mset.zero_le
thf(fact_999_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_1000_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_1001_add_Ocomm__neutral,axiom,
! [A: multiset_set_a] :
( ( plus_p2331992037799027419_set_a @ A @ zero_z5079479921072680283_set_a )
= A ) ).
% add.comm_neutral
thf(fact_1002_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_1003_comm__monoid__add__class_Oadd__0,axiom,
! [A: multiset_set_a] :
( ( plus_p2331992037799027419_set_a @ zero_z5079479921072680283_set_a @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_1004_empty__neutral_I1_J,axiom,
! [X: multiset_set_a] :
( ( plus_p2331992037799027419_set_a @ zero_z5079479921072680283_set_a @ X )
= X ) ).
% empty_neutral(1)
thf(fact_1005_empty__neutral_I2_J,axiom,
! [X: multiset_set_a] :
( ( plus_p2331992037799027419_set_a @ X @ zero_z5079479921072680283_set_a )
= X ) ).
% empty_neutral(2)
thf(fact_1006_nth__equal__first__eq,axiom,
! [X: set_a,Xs: list_set_a,N: nat] :
( ~ ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_size_list_set_a @ Xs ) )
=> ( ( ( nth_set_a @ ( cons_set_a @ X @ Xs ) @ N )
= X )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_1007_nth__equal__first__eq,axiom,
! [X: a,Xs: list_a,N: nat] :
( ~ ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( ( nth_a @ ( cons_a @ X @ Xs ) @ N )
= X )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_1008_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1009_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1010_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_1011_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1012_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_1013_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_1014_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_1015_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_1016_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_1017_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C2: nat] :
( ( B
= ( plus_plus_nat @ A @ C2 ) )
=> ( C2 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_1018_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_1019_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_1020_ex__least__nat__le,axiom,
! [P3: nat > $o,N: nat] :
( ( P3 @ N )
=> ( ~ ( P3 @ zero_zero_nat )
=> ? [K4: nat] :
( ( ord_less_eq_nat @ K4 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K4 )
=> ~ ( P3 @ I3 ) )
& ( P3 @ K4 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1021_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K4: nat] :
( ( ord_less_nat @ zero_zero_nat @ K4 )
& ( ( plus_plus_nat @ I @ K4 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1022_mset_Osimps_I1_J,axiom,
( ( mset_a @ nil_a )
= zero_zero_multiset_a ) ).
% mset.simps(1)
thf(fact_1023_mset_Osimps_I1_J,axiom,
( ( mset_set_a @ nil_set_a )
= zero_z5079479921072680283_set_a ) ).
% mset.simps(1)
thf(fact_1024_subset__mset_Oadd__decreasing,axiom,
! [A: multiset_set_a,C: multiset_set_a,B: multiset_set_a] :
( ( subseteq_mset_set_a @ A @ zero_z5079479921072680283_set_a )
=> ( ( subseteq_mset_set_a @ C @ B )
=> ( subseteq_mset_set_a @ ( plus_p2331992037799027419_set_a @ A @ C ) @ B ) ) ) ).
% subset_mset.add_decreasing
thf(fact_1025_subset__mset_Oadd__increasing,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( subseteq_mset_set_a @ zero_z5079479921072680283_set_a @ A )
=> ( ( subseteq_mset_set_a @ B @ C )
=> ( subseteq_mset_set_a @ B @ ( plus_p2331992037799027419_set_a @ A @ C ) ) ) ) ).
% subset_mset.add_increasing
thf(fact_1026_subset__mset_Oadd__decreasing2,axiom,
! [C: multiset_set_a,A: multiset_set_a,B: multiset_set_a] :
( ( subseteq_mset_set_a @ C @ zero_z5079479921072680283_set_a )
=> ( ( subseteq_mset_set_a @ A @ B )
=> ( subseteq_mset_set_a @ ( plus_p2331992037799027419_set_a @ A @ C ) @ B ) ) ) ).
% subset_mset.add_decreasing2
thf(fact_1027_subset__mset_Oadd__increasing2,axiom,
! [C: multiset_set_a,B: multiset_set_a,A: multiset_set_a] :
( ( subseteq_mset_set_a @ zero_z5079479921072680283_set_a @ C )
=> ( ( subseteq_mset_set_a @ B @ A )
=> ( subseteq_mset_set_a @ B @ ( plus_p2331992037799027419_set_a @ A @ C ) ) ) ) ).
% subset_mset.add_increasing2
thf(fact_1028_subset__mset_Oadd__nonneg__nonneg,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( subseteq_mset_set_a @ zero_z5079479921072680283_set_a @ A )
=> ( ( subseteq_mset_set_a @ zero_z5079479921072680283_set_a @ B )
=> ( subseteq_mset_set_a @ zero_z5079479921072680283_set_a @ ( plus_p2331992037799027419_set_a @ A @ B ) ) ) ) ).
% subset_mset.add_nonneg_nonneg
thf(fact_1029_subset__mset_Oadd__nonpos__nonpos,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( subseteq_mset_set_a @ A @ zero_z5079479921072680283_set_a )
=> ( ( subseteq_mset_set_a @ B @ zero_z5079479921072680283_set_a )
=> ( subseteq_mset_set_a @ ( plus_p2331992037799027419_set_a @ A @ B ) @ zero_z5079479921072680283_set_a ) ) ) ).
% subset_mset.add_nonpos_nonpos
thf(fact_1030_subset__mset_Oadd__nonneg__eq__0__iff,axiom,
! [X: multiset_set_a,Y: multiset_set_a] :
( ( subseteq_mset_set_a @ zero_z5079479921072680283_set_a @ X )
=> ( ( subseteq_mset_set_a @ zero_z5079479921072680283_set_a @ Y )
=> ( ( ( plus_p2331992037799027419_set_a @ X @ Y )
= zero_z5079479921072680283_set_a )
= ( ( X = zero_z5079479921072680283_set_a )
& ( Y = zero_z5079479921072680283_set_a ) ) ) ) ) ).
% subset_mset.add_nonneg_eq_0_iff
thf(fact_1031_subset__mset_Oadd__nonpos__eq__0__iff,axiom,
! [X: multiset_set_a,Y: multiset_set_a] :
( ( subseteq_mset_set_a @ X @ zero_z5079479921072680283_set_a )
=> ( ( subseteq_mset_set_a @ Y @ zero_z5079479921072680283_set_a )
=> ( ( ( plus_p2331992037799027419_set_a @ X @ Y )
= zero_z5079479921072680283_set_a )
= ( ( X = zero_z5079479921072680283_set_a )
& ( Y = zero_z5079479921072680283_set_a ) ) ) ) ) ).
% subset_mset.add_nonpos_eq_0_iff
thf(fact_1032_count__eq__zero__iff,axiom,
! [M: multiset_a,X: a] :
( ( ( count_a @ M @ X )
= zero_zero_nat )
= ( ~ ( member_a @ X @ ( set_mset_a @ M ) ) ) ) ).
% count_eq_zero_iff
thf(fact_1033_count__eq__zero__iff,axiom,
! [M: multiset_set_a,X: set_a] :
( ( ( count_set_a @ M @ X )
= zero_zero_nat )
= ( ~ ( member_set_a @ X @ ( set_mset_set_a @ M ) ) ) ) ).
% count_eq_zero_iff
thf(fact_1034_count__inI,axiom,
! [M: multiset_a,X: a] :
( ( ( count_a @ M @ X )
!= zero_zero_nat )
=> ( member_a @ X @ ( set_mset_a @ M ) ) ) ).
% count_inI
thf(fact_1035_count__inI,axiom,
! [M: multiset_set_a,X: set_a] :
( ( ( count_set_a @ M @ X )
!= zero_zero_nat )
=> ( member_set_a @ X @ ( set_mset_set_a @ M ) ) ) ).
% count_inI
thf(fact_1036_all__set__conv__all__nth,axiom,
! [Xs: list_a,P3: a > $o] :
( ( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( P3 @ X3 ) ) )
= ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs ) )
=> ( P3 @ ( nth_a @ Xs @ I4 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_1037_all__nth__imp__all__set,axiom,
! [Xs: list_set_a,P3: set_a > $o,X: set_a] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_set_a @ Xs ) )
=> ( P3 @ ( nth_set_a @ Xs @ I2 ) ) )
=> ( ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
=> ( P3 @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_1038_all__nth__imp__all__set,axiom,
! [Xs: list_a,P3: a > $o,X: a] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
=> ( P3 @ ( nth_a @ Xs @ I2 ) ) )
=> ( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( P3 @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_1039_in__set__conv__nth,axiom,
! [X: set_a,Xs: list_set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_set_a @ Xs ) )
& ( ( nth_set_a @ Xs @ I4 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_1040_in__set__conv__nth,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs ) )
& ( ( nth_a @ Xs @ I4 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_1041_list__ball__nth,axiom,
! [N: nat,Xs: list_a,P3: a > $o] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
=> ( P3 @ X4 ) )
=> ( P3 @ ( nth_a @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_1042_nth__mem,axiom,
! [N: nat,Xs: list_set_a] :
( ( ord_less_nat @ N @ ( size_size_list_set_a @ Xs ) )
=> ( member_set_a @ ( nth_set_a @ Xs @ N ) @ ( set_set_a2 @ Xs ) ) ) ).
% nth_mem
thf(fact_1043_nth__mem,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( member_a @ ( nth_a @ Xs @ N ) @ ( set_a2 @ Xs ) ) ) ).
% nth_mem
thf(fact_1044_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_1045_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_1046_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_1047_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_1048_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_1049_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_1050_length__pos__if__in__set,axiom,
! [X: set_a,Xs: list_set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_set_a @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_1051_length__pos__if__in__set,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_1052_monoid_Onat__pow__0,axiom,
! [G: partia8223610829204095565t_unit,X: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( pow_a_1875594501834816709it_nat @ G @ X @ zero_zero_nat )
= ( one_a_Product_unit @ G ) ) ) ).
% monoid.nat_pow_0
thf(fact_1053_monoid_Onat__pow__0,axiom,
! [G: partia7680350978787392319xt_a_b,X: a] :
( ( monoid_a_b @ G )
=> ( ( pow_a_b_nat @ G @ X @ zero_zero_nat )
= ( one_a_b @ G ) ) ) ).
% monoid.nat_pow_0
thf(fact_1054_replicate__mset__msubseteq__iff,axiom,
! [M4: nat,A: set_a,N: nat,B: set_a] :
( ( subseteq_mset_set_a @ ( replicate_mset_set_a @ M4 @ A ) @ ( replicate_mset_set_a @ N @ B ) )
= ( ( M4 = zero_zero_nat )
| ( ( A = B )
& ( ord_less_eq_nat @ M4 @ N ) ) ) ) ).
% replicate_mset_msubseteq_iff
thf(fact_1055_factorial__monoid_Ofactor__mset__unit,axiom,
! [G: partia7680350978787392319xt_a_b] :
( ( factorial_monoid_a_b @ G )
=> ( ( finite8971978181520403909et_a_b @ G @ ( one_a_b @ G ) )
= zero_z5079479921072680283_set_a ) ) ).
% factorial_monoid.factor_mset_unit
thf(fact_1056_nth__mem__mset,axiom,
! [I: nat,Ls: list_set_a] :
( ( ord_less_nat @ I @ ( size_size_list_set_a @ Ls ) )
=> ( member_set_a @ ( nth_set_a @ Ls @ I ) @ ( set_mset_set_a @ ( mset_set_a @ Ls ) ) ) ) ).
% nth_mem_mset
thf(fact_1057_nth__mem__mset,axiom,
! [I: nat,Ls: list_a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Ls ) )
=> ( member_a @ ( nth_a @ Ls @ I ) @ ( set_mset_a @ ( mset_a @ Ls ) ) ) ) ).
% nth_mem_mset
thf(fact_1058_count__mset__gt__0,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( count_a @ ( mset_a @ Xs ) @ X ) ) ) ).
% count_mset_gt_0
thf(fact_1059_count__mset__gt__0,axiom,
! [X: set_a,Xs: list_set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( count_set_a @ ( mset_set_a @ Xs ) @ X ) ) ) ).
% count_mset_gt_0
thf(fact_1060_monoid_Onunit__factors,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,As: list_a] :
( ( monoid_a_b @ G )
=> ( ~ ( member_a @ A @ ( units_a_b @ G ) )
=> ( ( factors_a_b @ G @ As @ A )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ As ) ) ) ) ) ).
% monoid.nunit_factors
thf(fact_1061_unitfactor__ee,axiom,
! [U: a,As: list_a] :
( ( member_a @ U @ ( units_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( essent1248755695173879485al_a_b @ g @ ( list_update_a @ As @ zero_zero_nat @ ( mult_a_b @ g @ ( nth_a @ As @ zero_zero_nat ) @ U ) ) @ As ) ) ) ).
% unitfactor_ee
thf(fact_1062_factors__cong__unit,axiom,
! [U: a,A: a,As: list_a] :
( ( member_a @ U @ ( units_a_b @ g ) )
=> ( ~ ( member_a @ A @ ( units_a_b @ g ) )
=> ( ( factors_a_b @ g @ As @ A )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( factors_a_b @ g @ ( list_update_a @ As @ zero_zero_nat @ ( mult_a_b @ g @ ( nth_a @ As @ zero_zero_nat ) @ U ) ) @ ( mult_a_b @ g @ A @ U ) ) ) ) ) ) ).
% factors_cong_unit
thf(fact_1063_list__update__beyond,axiom,
! [Xs: list_a,I: nat,X: a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ I )
=> ( ( list_update_a @ Xs @ I @ X )
= Xs ) ) ).
% list_update_beyond
thf(fact_1064_list__update__length,axiom,
! [Xs: list_a,X: a,Ys: list_a,Y: a] :
( ( list_update_a @ ( append_a @ Xs @ ( cons_a @ X @ Ys ) ) @ ( size_size_list_a @ Xs ) @ Y )
= ( append_a @ Xs @ ( cons_a @ Y @ Ys ) ) ) ).
% list_update_length
thf(fact_1065_set__swap,axiom,
! [I: nat,Xs: list_a,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_a @ Xs ) )
=> ( ( set_a2 @ ( list_update_a @ ( list_update_a @ Xs @ I @ ( nth_a @ Xs @ J ) ) @ J @ ( nth_a @ Xs @ I ) ) )
= ( set_a2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_1066_psubsetD,axiom,
! [A4: set_a,B3: set_a,C: a] :
( ( ord_less_set_a @ A4 @ B3 )
=> ( ( member_a @ C @ A4 )
=> ( member_a @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_1067_psubsetD,axiom,
! [A4: set_set_a,B3: set_set_a,C: set_a] :
( ( ord_less_set_set_a @ A4 @ B3 )
=> ( ( member_set_a @ C @ A4 )
=> ( member_set_a @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_1068_list__update__code_I2_J,axiom,
! [X: a,Xs: list_a,Y: a] :
( ( list_update_a @ ( cons_a @ X @ Xs ) @ zero_zero_nat @ Y )
= ( cons_a @ Y @ Xs ) ) ).
% list_update_code(2)
thf(fact_1069_set__update__subsetI,axiom,
! [Xs: list_set_a,A4: set_set_a,X: set_a,I: nat] :
( ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ Xs ) @ A4 )
=> ( ( member_set_a @ X @ A4 )
=> ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ ( list_update_set_a @ Xs @ I @ X ) ) @ A4 ) ) ) ).
% set_update_subsetI
thf(fact_1070_set__update__subsetI,axiom,
! [Xs: list_a,A4: set_a,X: a,I: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ A4 )
=> ( ( member_a @ X @ A4 )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( list_update_a @ Xs @ I @ X ) ) @ A4 ) ) ) ).
% set_update_subsetI
thf(fact_1071_set__update__subset__insert,axiom,
! [Xs: list_a,I: nat,X: a] : ( ord_less_eq_set_a @ ( set_a2 @ ( list_update_a @ Xs @ I @ X ) ) @ ( insert_a @ X @ ( set_a2 @ Xs ) ) ) ).
% set_update_subset_insert
thf(fact_1072_set__update__memI,axiom,
! [N: nat,Xs: list_set_a,X: set_a] :
( ( ord_less_nat @ N @ ( size_size_list_set_a @ Xs ) )
=> ( member_set_a @ X @ ( set_set_a2 @ ( list_update_set_a @ Xs @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_1073_set__update__memI,axiom,
! [N: nat,Xs: list_a,X: a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( member_a @ X @ ( set_a2 @ ( list_update_a @ Xs @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_1074_mset__swap,axiom,
! [I: nat,Ls: list_a,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Ls ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_a @ Ls ) )
=> ( ( mset_a @ ( list_update_a @ ( list_update_a @ Ls @ J @ ( nth_a @ Ls @ I ) ) @ I @ ( nth_a @ Ls @ J ) ) )
= ( mset_a @ Ls ) ) ) ) ).
% mset_swap
thf(fact_1075_nat__pow__Suc2,axiom,
! [X: a,N: nat] :
( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( pow_a_b_nat @ g @ X @ ( suc @ N ) )
= ( mult_a_b @ g @ X @ ( pow_a_b_nat @ g @ X @ N ) ) ) ) ).
% nat_pow_Suc2
thf(fact_1076_Suc__le__mono,axiom,
! [N: nat,M4: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M4 ) )
= ( ord_less_eq_nat @ N @ M4 ) ) ).
% Suc_le_mono
thf(fact_1077_add__Suc__right,axiom,
! [M4: nat,N: nat] :
( ( plus_plus_nat @ M4 @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M4 @ N ) ) ) ).
% add_Suc_right
thf(fact_1078_Group_Onat__pow__Suc,axiom,
! [G: partia8223610829204095565t_unit,X: a,N: nat] :
( ( pow_a_1875594501834816709it_nat @ G @ X @ ( suc @ N ) )
= ( mult_a_Product_unit @ G @ ( pow_a_1875594501834816709it_nat @ G @ X @ N ) @ X ) ) ).
% Group.nat_pow_Suc
thf(fact_1079_Group_Onat__pow__Suc,axiom,
! [G: partia7680350978787392319xt_a_b,X: a,N: nat] :
( ( pow_a_b_nat @ G @ X @ ( suc @ N ) )
= ( mult_a_b @ G @ ( pow_a_b_nat @ G @ X @ N ) @ X ) ) ).
% Group.nat_pow_Suc
thf(fact_1080_nth__Cons__Suc,axiom,
! [X: a,Xs: list_a,N: nat] :
( ( nth_a @ ( cons_a @ X @ Xs ) @ ( suc @ N ) )
= ( nth_a @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_1081_repeat__mset__Suc,axiom,
! [N: nat,M: multiset_set_a] :
( ( repeat_mset_set_a @ ( suc @ N ) @ M )
= ( plus_p2331992037799027419_set_a @ M @ ( repeat_mset_set_a @ N @ M ) ) ) ).
% repeat_mset_Suc
thf(fact_1082_local_Onat__pow__Suc,axiom,
! [X: a,N: nat] :
( ( pow_a_b_nat @ g @ X @ ( suc @ N ) )
= ( mult_a_b @ g @ ( pow_a_b_nat @ g @ X @ N ) @ X ) ) ).
% local.nat_pow_Suc
thf(fact_1083_count__greater__eq__Suc__zero__iff,axiom,
! [M: multiset_a,X: a] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( count_a @ M @ X ) )
= ( member_a @ X @ ( set_mset_a @ M ) ) ) ).
% count_greater_eq_Suc_zero_iff
thf(fact_1084_count__greater__eq__Suc__zero__iff,axiom,
! [M: multiset_set_a,X: set_a] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( count_set_a @ M @ X ) )
= ( member_set_a @ X @ ( set_mset_set_a @ M ) ) ) ).
% count_greater_eq_Suc_zero_iff
thf(fact_1085_list__update__code_I3_J,axiom,
! [X: a,Xs: list_a,I: nat,Y: a] :
( ( list_update_a @ ( cons_a @ X @ Xs ) @ ( suc @ I ) @ Y )
= ( cons_a @ X @ ( list_update_a @ Xs @ I @ Y ) ) ) ).
% list_update_code(3)
thf(fact_1086_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N6: nat] :
( ! [N5: nat] : ( ord_less_eq_nat @ ( F @ N5 ) @ ( F @ ( suc @ N5 ) ) )
=> ( ( ord_less_eq_nat @ N @ N6 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N6 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1087_lift__Suc__mono__le,axiom,
! [F: nat > set_a,N: nat,N6: nat] :
( ! [N5: nat] : ( ord_less_eq_set_a @ ( F @ N5 ) @ ( F @ ( suc @ N5 ) ) )
=> ( ( ord_less_eq_nat @ N @ N6 )
=> ( ord_less_eq_set_a @ ( F @ N ) @ ( F @ N6 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1088_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N6: nat] :
( ! [N5: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N5 ) ) @ ( F @ N5 ) )
=> ( ( ord_less_eq_nat @ N @ N6 )
=> ( ord_less_eq_nat @ ( F @ N6 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1089_lift__Suc__antimono__le,axiom,
! [F: nat > set_a,N: nat,N6: nat] :
( ! [N5: nat] : ( ord_less_eq_set_a @ ( F @ ( suc @ N5 ) ) @ ( F @ N5 ) )
=> ( ( ord_less_eq_nat @ N @ N6 )
=> ( ord_less_eq_set_a @ ( F @ N6 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1090_length__Suc__conv,axiom,
! [Xs: list_a,N: nat] :
( ( ( size_size_list_a @ Xs )
= ( suc @ N ) )
= ( ? [Y5: a,Ys3: list_a] :
( ( Xs
= ( cons_a @ Y5 @ Ys3 ) )
& ( ( size_size_list_a @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_1091_Suc__length__conv,axiom,
! [N: nat,Xs: list_a] :
( ( ( suc @ N )
= ( size_size_list_a @ Xs ) )
= ( ? [Y5: a,Ys3: list_a] :
( ( Xs
= ( cons_a @ Y5 @ Ys3 ) )
& ( ( size_size_list_a @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_1092_Suc__leI,axiom,
! [M4: nat,N: nat] :
( ( ord_less_nat @ M4 @ N )
=> ( ord_less_eq_nat @ ( suc @ M4 ) @ N ) ) ).
% Suc_leI
thf(fact_1093_Suc__le__eq,axiom,
! [M4: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N )
= ( ord_less_nat @ M4 @ N ) ) ).
% Suc_le_eq
thf(fact_1094_dec__induct,axiom,
! [I: nat,J: nat,P3: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P3 @ I )
=> ( ! [N5: nat] :
( ( ord_less_eq_nat @ I @ N5 )
=> ( ( ord_less_nat @ N5 @ J )
=> ( ( P3 @ N5 )
=> ( P3 @ ( suc @ N5 ) ) ) ) )
=> ( P3 @ J ) ) ) ) ).
% dec_induct
thf(fact_1095_inc__induct,axiom,
! [I: nat,J: nat,P3: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P3 @ J )
=> ( ! [N5: nat] :
( ( ord_less_eq_nat @ I @ N5 )
=> ( ( ord_less_nat @ N5 @ J )
=> ( ( P3 @ ( suc @ N5 ) )
=> ( P3 @ N5 ) ) ) )
=> ( P3 @ I ) ) ) ) ).
% inc_induct
thf(fact_1096_Suc__le__lessD,axiom,
! [M4: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N )
=> ( ord_less_nat @ M4 @ N ) ) ).
% Suc_le_lessD
thf(fact_1097_le__less__Suc__eq,axiom,
! [M4: nat,N: nat] :
( ( ord_less_eq_nat @ M4 @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M4 ) )
= ( N = M4 ) ) ) ).
% le_less_Suc_eq
thf(fact_1098_less__Suc__eq__le,axiom,
! [M4: nat,N: nat] :
( ( ord_less_nat @ M4 @ ( suc @ N ) )
= ( ord_less_eq_nat @ M4 @ N ) ) ).
% less_Suc_eq_le
thf(fact_1099_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1100_le__imp__less__Suc,axiom,
! [M4: nat,N: nat] :
( ( ord_less_eq_nat @ M4 @ N )
=> ( ord_less_nat @ M4 @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_1101_one__is__add,axiom,
! [M4: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M4 @ N ) )
= ( ( ( M4
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M4 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1102_add__is__1,axiom,
! [M4: nat,N: nat] :
( ( ( plus_plus_nat @ M4 @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M4
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M4 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1103_less__natE,axiom,
! [M4: nat,N: nat] :
( ( ord_less_nat @ M4 @ N )
=> ~ ! [Q2: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M4 @ Q2 ) ) ) ) ).
% less_natE
thf(fact_1104_less__add__Suc1,axiom,
! [I: nat,M4: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M4 ) ) ) ).
% less_add_Suc1
thf(fact_1105_less__add__Suc2,axiom,
! [I: nat,M4: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M4 @ I ) ) ) ).
% less_add_Suc2
thf(fact_1106_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M5: nat,N4: nat] :
? [K3: nat] :
( N4
= ( suc @ ( plus_plus_nat @ M5 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1107_less__imp__Suc__add,axiom,
! [M4: nat,N: nat] :
( ( ord_less_nat @ M4 @ N )
=> ? [K4: nat] :
( N
= ( suc @ ( plus_plus_nat @ M4 @ K4 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1108_subset__mset_Olift__Suc__antimono__le,axiom,
! [F: nat > multiset_set_a,N: nat,N6: nat] :
( ! [N5: nat] : ( subseteq_mset_set_a @ ( F @ ( suc @ N5 ) ) @ ( F @ N5 ) )
=> ( ( ord_less_eq_nat @ N @ N6 )
=> ( subseteq_mset_set_a @ ( F @ N6 ) @ ( F @ N ) ) ) ) ).
% subset_mset.lift_Suc_antimono_le
thf(fact_1109_subset__mset_Olift__Suc__mono__le,axiom,
! [F: nat > multiset_set_a,N: nat,N6: nat] :
( ! [N5: nat] : ( subseteq_mset_set_a @ ( F @ N5 ) @ ( F @ ( suc @ N5 ) ) )
=> ( ( ord_less_eq_nat @ N @ N6 )
=> ( subseteq_mset_set_a @ ( F @ N ) @ ( F @ N6 ) ) ) ) ).
% subset_mset.lift_Suc_mono_le
thf(fact_1110_in__countE,axiom,
! [X: a,M: multiset_a] :
( ( member_a @ X @ ( set_mset_a @ M ) )
=> ~ ! [N5: nat] :
( ( count_a @ M @ X )
!= ( suc @ N5 ) ) ) ).
% in_countE
thf(fact_1111_in__countE,axiom,
! [X: set_a,M: multiset_set_a] :
( ( member_set_a @ X @ ( set_mset_set_a @ M ) )
=> ~ ! [N5: nat] :
( ( count_set_a @ M @ X )
!= ( suc @ N5 ) ) ) ).
% in_countE
thf(fact_1112_nat__arith_Osuc1,axiom,
! [A4: nat,K2: nat,A: nat] :
( ( A4
= ( plus_plus_nat @ K2 @ A ) )
=> ( ( suc @ A4 )
= ( plus_plus_nat @ K2 @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_1113_add__Suc,axiom,
! [M4: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M4 ) @ N )
= ( suc @ ( plus_plus_nat @ M4 @ N ) ) ) ).
% add_Suc
thf(fact_1114_add__Suc__shift,axiom,
! [M4: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M4 ) @ N )
= ( plus_plus_nat @ M4 @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_1115_Suc__leD,axiom,
! [M4: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N )
=> ( ord_less_eq_nat @ M4 @ N ) ) ).
% Suc_leD
thf(fact_1116_le__SucE,axiom,
! [M4: nat,N: nat] :
( ( ord_less_eq_nat @ M4 @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M4 @ N )
=> ( M4
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_1117_le__SucI,axiom,
! [M4: nat,N: nat] :
( ( ord_less_eq_nat @ M4 @ N )
=> ( ord_less_eq_nat @ M4 @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1118_Suc__le__D,axiom,
! [N: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
=> ? [M3: nat] :
( M6
= ( suc @ M3 ) ) ) ).
% Suc_le_D
thf(fact_1119_le__Suc__eq,axiom,
! [M4: nat,N: nat] :
( ( ord_less_eq_nat @ M4 @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M4 @ N )
| ( M4
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_1120_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1121_not__less__eq__eq,axiom,
! [M4: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M4 @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M4 ) ) ).
% not_less_eq_eq
thf(fact_1122_full__nat__induct,axiom,
! [P3: nat > $o,N: nat] :
( ! [N5: nat] :
( ! [M7: nat] :
( ( ord_less_eq_nat @ ( suc @ M7 ) @ N5 )
=> ( P3 @ M7 ) )
=> ( P3 @ N5 ) )
=> ( P3 @ N ) ) ).
% full_nat_induct
thf(fact_1123_nat__induct__at__least,axiom,
! [M4: nat,N: nat,P3: nat > $o] :
( ( ord_less_eq_nat @ M4 @ N )
=> ( ( P3 @ M4 )
=> ( ! [N5: nat] :
( ( ord_less_eq_nat @ M4 @ N5 )
=> ( ( P3 @ N5 )
=> ( P3 @ ( suc @ N5 ) ) ) )
=> ( P3 @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_1124_transitive__stepwise__le,axiom,
! [M4: nat,N: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M4 @ N )
=> ( ! [X4: nat] : ( R2 @ X4 @ X4 )
=> ( ! [X4: nat,Y3: nat,Z3: nat] :
( ( R2 @ X4 @ Y3 )
=> ( ( R2 @ Y3 @ Z3 )
=> ( R2 @ X4 @ Z3 ) ) )
=> ( ! [N5: nat] : ( R2 @ N5 @ ( suc @ N5 ) )
=> ( R2 @ M4 @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1125_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_a @ Xs ) )
= ( ? [X3: a,Ys3: list_a] :
( ( Xs
= ( cons_a @ X3 @ Ys3 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_a @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_1126_ex__least__nat__less,axiom,
! [P3: nat > $o,N: nat] :
( ( P3 @ N )
=> ( ~ ( P3 @ zero_zero_nat )
=> ? [K4: nat] :
( ( ord_less_nat @ K4 @ N )
& ! [I3: nat] :
( ( ord_less_eq_nat @ I3 @ K4 )
=> ~ ( P3 @ I3 ) )
& ( P3 @ ( suc @ K4 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1127_monoid_Onat__pow__Suc,axiom,
! [G: partia8223610829204095565t_unit,X: a,N: nat] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( pow_a_1875594501834816709it_nat @ G @ X @ ( suc @ N ) )
= ( mult_a_Product_unit @ G @ ( pow_a_1875594501834816709it_nat @ G @ X @ N ) @ X ) ) ) ).
% monoid.nat_pow_Suc
thf(fact_1128_monoid_Onat__pow__Suc,axiom,
! [G: partia7680350978787392319xt_a_b,X: a,N: nat] :
( ( monoid_a_b @ G )
=> ( ( pow_a_b_nat @ G @ X @ ( suc @ N ) )
= ( mult_a_b @ G @ ( pow_a_b_nat @ G @ X @ N ) @ X ) ) ) ).
% monoid.nat_pow_Suc
thf(fact_1129_list_Osize_I4_J,axiom,
! [X21: a,X22: list_a] :
( ( size_size_list_a @ ( cons_a @ X21 @ X22 ) )
= ( plus_plus_nat @ ( size_size_list_a @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_1130_length__Suc__conv__rev,axiom,
! [Xs: list_a,N: nat] :
( ( ( size_size_list_a @ Xs )
= ( suc @ N ) )
= ( ? [Y5: a,Ys3: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ Y5 @ nil_a ) ) )
& ( ( size_size_list_a @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_1131_monoid_Onat__pow__Suc2,axiom,
! [G: partia8223610829204095565t_unit,X: a,N: nat] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( pow_a_1875594501834816709it_nat @ G @ X @ ( suc @ N ) )
= ( mult_a_Product_unit @ G @ X @ ( pow_a_1875594501834816709it_nat @ G @ X @ N ) ) ) ) ) ).
% monoid.nat_pow_Suc2
thf(fact_1132_monoid_Onat__pow__Suc2,axiom,
! [G: partia7680350978787392319xt_a_b,X: a,N: nat] :
( ( monoid_a_b @ G )
=> ( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( pow_a_b_nat @ G @ X @ ( suc @ N ) )
= ( mult_a_b @ G @ X @ ( pow_a_b_nat @ G @ X @ N ) ) ) ) ) ).
% monoid.nat_pow_Suc2
thf(fact_1133_length__append__singleton,axiom,
! [Xs: list_a,X: a] :
( ( size_size_list_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) )
= ( suc @ ( size_size_list_a @ Xs ) ) ) ).
% length_append_singleton
thf(fact_1134_gcdof__greatestLower,axiom,
! [A: a,G: partia7680350978787392319xt_a_b,B: a,X: a] :
( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ G ) )
& ( isgcd_a_b @ G @ X @ A @ B ) )
= ( greate1756078582999224458t_unit @ ( partia6383399360842404908t_unit @ ( partia7484183841585581558xt_a_b @ G ) @ ( eq_eq_5233546288009311946t_unit @ ( associated_a_b @ G ) @ ( gorder6800608520584131120t_unit @ ( factor_a_b @ G ) @ product_Unity ) ) ) @ X @ ( lower_a_Product_unit @ ( partia6383399360842404908t_unit @ ( partia7484183841585581558xt_a_b @ G ) @ ( eq_eq_5233546288009311946t_unit @ ( associated_a_b @ G ) @ ( gorder6800608520584131120t_unit @ ( factor_a_b @ G ) @ product_Unity ) ) ) @ ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ) ) ) ).
% gcdof_greatestLower
thf(fact_1135_length__Cons,axiom,
! [X: a,Xs: list_a] :
( ( size_size_list_a @ ( cons_a @ X @ Xs ) )
= ( suc @ ( size_size_list_a @ Xs ) ) ) ).
% length_Cons
thf(fact_1136_properfactor__lless,axiom,
( properfactor_a_b
= ( ^ [G2: partia7680350978787392319xt_a_b] : ( lless_a_Product_unit @ ( partia6383399360842404908t_unit @ ( partia7484183841585581558xt_a_b @ G2 ) @ ( eq_eq_5233546288009311946t_unit @ ( associated_a_b @ G2 ) @ ( gorder6800608520584131120t_unit @ ( factor_a_b @ G2 ) @ product_Unity ) ) ) ) ) ) ).
% properfactor_lless
thf(fact_1137_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_a @ N @ nil_a )
= ( cons_list_a @ nil_a @ nil_list_a ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_a @ N @ nil_a )
= nil_list_a ) ) ) ).
% n_lists_Nil
thf(fact_1138_n__lists_Osimps_I1_J,axiom,
! [Xs: list_a] :
( ( n_lists_a @ zero_zero_nat @ Xs )
= ( cons_list_a @ nil_a @ nil_list_a ) ) ).
% n_lists.simps(1)
thf(fact_1139_equivalence_Oclosure__idem__weak,axiom,
! [S: partia6638023223214267844t_unit,A4: set_a] :
( ( equiva8593399568135932475t_unit @ S )
=> ( set_eq3287599968972363709t_unit @ S @ ( eq_clo186784744570230005t_unit @ S @ ( eq_clo186784744570230005t_unit @ S @ A4 ) ) @ ( eq_clo186784744570230005t_unit @ S @ A4 ) ) ) ).
% equivalence.closure_idem_weak
thf(fact_1140_comm__monoid__cancel_Omultlist__prime__pos,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,As: list_a] :
( ( comm_m4873603268227677576el_a_b @ G )
=> ( ( prime_a_b @ G @ A )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( factor_a_b @ G @ A @ ( foldr_a_a @ ( mult_a_b @ G ) @ As @ ( one_a_b @ G ) ) )
=> ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ As ) )
& ( factor_a_b @ G @ A @ ( nth_a @ As @ I2 ) ) ) ) ) ) ) ) ).
% comm_monoid_cancel.multlist_prime_pos
thf(fact_1141_is__comm__monoid__cancel,axiom,
comm_m4873603268227677576el_a_b @ g ).
% is_comm_monoid_cancel
thf(fact_1142_partial__object_Oext__inject,axiom,
! [Carrier2: set_a,More: eq_eq_8366045168579070496t_unit,Carrier3: set_a,More2: eq_eq_8366045168579070496t_unit] :
( ( ( partia6383399360842404908t_unit @ Carrier2 @ More )
= ( partia6383399360842404908t_unit @ Carrier3 @ More2 ) )
= ( ( Carrier2 = Carrier3 )
& ( More = More2 ) ) ) ).
% partial_object.ext_inject
thf(fact_1143_eq__object_Oext__inject,axiom,
! [Eq2: a > a > $o,More: gorder3081419850850201676t_unit,Eq3: a > a > $o,More2: gorder3081419850850201676t_unit] :
( ( ( eq_eq_5233546288009311946t_unit @ Eq2 @ More )
= ( eq_eq_5233546288009311946t_unit @ Eq3 @ More2 ) )
= ( ( Eq2 = Eq3 )
& ( More = More2 ) ) ) ).
% eq_object.ext_inject
thf(fact_1144_comm__monoid__cancel_Ointro,axiom,
! [G: partia7680350978787392319xt_a_b] :
( ( monoid_cancel_a_b @ G )
=> ( ( comm_monoid_a_b @ G )
=> ( comm_m4873603268227677576el_a_b @ G ) ) ) ).
% comm_monoid_cancel.intro
thf(fact_1145_comm__monoid__cancel__def,axiom,
( comm_m4873603268227677576el_a_b
= ( ^ [G2: partia7680350978787392319xt_a_b] :
( ( monoid_cancel_a_b @ G2 )
& ( comm_monoid_a_b @ G2 ) ) ) ) ).
% comm_monoid_cancel_def
thf(fact_1146_comm__monoid__cancel_Oprime__irreducible,axiom,
! [G: partia7680350978787392319xt_a_b,P: a] :
( ( comm_m4873603268227677576el_a_b @ G )
=> ( ( prime_a_b @ G @ P )
=> ( irreducible_a_b @ G @ P ) ) ) ).
% comm_monoid_cancel.prime_irreducible
thf(fact_1147_primeness__condition__monoid_Oaxioms_I1_J,axiom,
! [G: partia7680350978787392319xt_a_b] :
( ( primen900146951955507079id_a_b @ G )
=> ( comm_m4873603268227677576el_a_b @ G ) ) ).
% primeness_condition_monoid.axioms(1)
thf(fact_1148_divisor__chain__condition__monoid_Oaxioms_I1_J,axiom,
! [G: partia7680350978787392319xt_a_b] :
( ( diviso5244014645414798900id_a_b @ G )
=> ( comm_m4873603268227677576el_a_b @ G ) ) ).
% divisor_chain_condition_monoid.axioms(1)
thf(fact_1149_comm__monoid__cancel_Ois__comm__monoid__cancel,axiom,
! [G: partia7680350978787392319xt_a_b] :
( ( comm_m4873603268227677576el_a_b @ G )
=> ( comm_m4873603268227677576el_a_b @ G ) ) ).
% comm_monoid_cancel.is_comm_monoid_cancel
thf(fact_1150_gcd__condition__monoid_Oaxioms_I1_J,axiom,
! [G: partia7680350978787392319xt_a_b] :
( ( gcd_co4024338386749376309id_a_b @ G )
=> ( comm_m4873603268227677576el_a_b @ G ) ) ).
% gcd_condition_monoid.axioms(1)
thf(fact_1151_comm__monoid__cancel_Oaxioms_I1_J,axiom,
! [G: partia7680350978787392319xt_a_b] :
( ( comm_m4873603268227677576el_a_b @ G )
=> ( monoid_cancel_a_b @ G ) ) ).
% comm_monoid_cancel.axioms(1)
thf(fact_1152_factorial__monoid_Oaxioms_I1_J,axiom,
! [G: partia7680350978787392319xt_a_b] :
( ( factorial_monoid_a_b @ G )
=> ( comm_m4873603268227677576el_a_b @ G ) ) ).
% factorial_monoid.axioms(1)
thf(fact_1153_comm__monoid__cancel_Oaxioms_I2_J,axiom,
! [G: partia7680350978787392319xt_a_b] :
( ( comm_m4873603268227677576el_a_b @ G )
=> ( comm_monoid_a_b @ G ) ) ).
% comm_monoid_cancel.axioms(2)
thf(fact_1154_factorial__monoid__def,axiom,
( factorial_monoid_a_b
= ( ^ [G2: partia7680350978787392319xt_a_b] :
( ( comm_m4873603268227677576el_a_b @ G2 )
& ( factor4787786757223245700ms_a_b @ G2 ) ) ) ) ).
% factorial_monoid_def
thf(fact_1155_factorial__monoid_Ointro,axiom,
! [G: partia7680350978787392319xt_a_b] :
( ( comm_m4873603268227677576el_a_b @ G )
=> ( ( factor4787786757223245700ms_a_b @ G )
=> ( factorial_monoid_a_b @ G ) ) ) ).
% factorial_monoid.intro
thf(fact_1156_partial__object_Ocases__scheme,axiom,
! [R: partia6638023223214267844t_unit] :
~ ! [Carrier: set_a,More3: eq_eq_8366045168579070496t_unit] :
( R
!= ( partia6383399360842404908t_unit @ Carrier @ More3 ) ) ).
% partial_object.cases_scheme
thf(fact_1157_comm__monoid__cancel_Odivides__mult__r,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a,C: a] :
( ( comm_m4873603268227677576el_a_b @ G )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( factor_a_b @ G @ ( mult_a_b @ G @ A @ C ) @ ( mult_a_b @ G @ B @ C ) )
= ( factor_a_b @ G @ A @ B ) ) ) ) ) ) ).
% comm_monoid_cancel.divides_mult_r
thf(fact_1158_comm__monoid__cancel_Oassoc__r__cancel,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a,A3: a] :
( ( comm_m4873603268227677576el_a_b @ G )
=> ( ( associated_a_b @ G @ ( mult_a_b @ G @ A @ B ) @ ( mult_a_b @ G @ A3 @ B ) )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ A3 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( associated_a_b @ G @ A @ A3 ) ) ) ) ) ) ).
% comm_monoid_cancel.assoc_r_cancel
thf(fact_1159_comm__monoid__cancelI,axiom,
! [G: partia7680350978787392319xt_a_b] :
( ( comm_monoid_a_b @ G )
=> ( ! [A2: a,B2: a,C2: a] :
( ( ( mult_a_b @ G @ A2 @ C2 )
= ( mult_a_b @ G @ B2 @ C2 ) )
=> ( ( member_a @ A2 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B2 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ C2 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( A2 = B2 ) ) ) ) )
=> ( comm_m4873603268227677576el_a_b @ G ) ) ) ).
% comm_monoid_cancelI
thf(fact_1160_comm__monoid__cancel_Oproperfactor__mult__rI,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a,C: a] :
( ( comm_m4873603268227677576el_a_b @ G )
=> ( ( properfactor_a_b @ G @ A @ B )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( properfactor_a_b @ G @ ( mult_a_b @ G @ A @ C ) @ ( mult_a_b @ G @ B @ C ) ) ) ) ) ) ).
% comm_monoid_cancel.properfactor_mult_rI
thf(fact_1161_comm__monoid__cancel_Oproperfactor__mult__r,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a,C: a] :
( ( comm_m4873603268227677576el_a_b @ G )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( properfactor_a_b @ G @ ( mult_a_b @ G @ A @ C ) @ ( mult_a_b @ G @ B @ C ) )
= ( properfactor_a_b @ G @ A @ B ) ) ) ) ) ) ).
% comm_monoid_cancel.properfactor_mult_r
thf(fact_1162_eq__object_Oext__induct,axiom,
! [P3: eq_eq_8366045168579070496t_unit > $o,R: eq_eq_8366045168579070496t_unit] :
( ! [Eq: a > a > $o,More3: gorder3081419850850201676t_unit] : ( P3 @ ( eq_eq_5233546288009311946t_unit @ Eq @ More3 ) )
=> ( P3 @ R ) ) ).
% eq_object.ext_induct
thf(fact_1163_comm__monoid__cancel_OproperfactorI3,axiom,
! [G: partia7680350978787392319xt_a_b,P: a,A: a,B: a] :
( ( comm_m4873603268227677576el_a_b @ G )
=> ( ( P
= ( mult_a_b @ G @ A @ B ) )
=> ( ~ ( member_a @ B @ ( units_a_b @ G ) )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( properfactor_a_b @ G @ A @ P ) ) ) ) ) ) ).
% comm_monoid_cancel.properfactorI3
thf(fact_1164_comm__monoid__cancel_Oirreducible__prodE,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a] :
( ( comm_m4873603268227677576el_a_b @ G )
=> ( ( irreducible_a_b @ G @ ( mult_a_b @ G @ A @ B ) )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ( irreducible_a_b @ G @ A )
=> ~ ( member_a @ B @ ( units_a_b @ G ) ) )
=> ~ ( ( member_a @ A @ ( units_a_b @ G ) )
=> ~ ( irreducible_a_b @ G @ B ) ) ) ) ) ) ) ).
% comm_monoid_cancel.irreducible_prodE
thf(fact_1165_comm__monoid__cancel_Oprime__divides,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,B: a,P: a] :
( ( comm_m4873603268227677576el_a_b @ G )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( prime_a_b @ G @ P )
=> ( ( factor_a_b @ G @ P @ ( mult_a_b @ G @ A @ B ) )
=> ( ( factor_a_b @ G @ P @ A )
| ( factor_a_b @ G @ P @ B ) ) ) ) ) ) ) ).
% comm_monoid_cancel.prime_divides
thf(fact_1166_comm__monoid__cancel_Owfactors__ee__cong__l,axiom,
! [G: partia7680350978787392319xt_a_b,As: list_a,Bs: list_a,B: a] :
( ( comm_m4873603268227677576el_a_b @ G )
=> ( ( essent1248755695173879485al_a_b @ G @ As @ Bs )
=> ( ( wfactors_a_b @ G @ Bs @ B )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( wfactors_a_b @ G @ As @ B ) ) ) ) ) ) ) ).
% comm_monoid_cancel.wfactors_ee_cong_l
thf(fact_1167_comm__monoid__cancel_Oee__is__fmset,axiom,
! [G: partia7680350978787392319xt_a_b,As: list_a,Bs: list_a] :
( ( comm_m4873603268227677576el_a_b @ G )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( essent1248755695173879485al_a_b @ G @ As @ Bs )
= ( ( fmset_a_b @ G @ As )
= ( fmset_a_b @ G @ Bs ) ) ) ) ) ) ).
% comm_monoid_cancel.ee_is_fmset
thf(fact_1168_comm__monoid__cancel_Ofmset__ee,axiom,
! [G: partia7680350978787392319xt_a_b,As: list_a,Bs: list_a] :
( ( comm_m4873603268227677576el_a_b @ G )
=> ( ( ( fmset_a_b @ G @ As )
= ( fmset_a_b @ G @ Bs ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( essent1248755695173879485al_a_b @ G @ As @ Bs ) ) ) ) ) ).
% comm_monoid_cancel.fmset_ee
thf(fact_1169_comm__monoid__cancel_Oee__fmset,axiom,
! [G: partia7680350978787392319xt_a_b,As: list_a,Bs: list_a] :
( ( comm_m4873603268227677576el_a_b @ G )
=> ( ( essent1248755695173879485al_a_b @ G @ As @ Bs )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( fmset_a_b @ G @ As )
= ( fmset_a_b @ G @ Bs ) ) ) ) ) ) ).
% comm_monoid_cancel.ee_fmset
thf(fact_1170_comm__monoid__cancel_Oprime__pow__divides__iff,axiom,
! [G: partia8223610829204095565t_unit,P: a,A: a,B: a,N: nat] :
( ( comm_m7120069553834612630t_unit @ G )
=> ( ( member_a @ P @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( prime_a_Product_unit @ G @ P )
=> ( ~ ( factor3040189038382604065t_unit @ G @ P @ A )
=> ( ( factor3040189038382604065t_unit @ G @ ( pow_a_1875594501834816709it_nat @ G @ P @ N ) @ ( mult_a_Product_unit @ G @ A @ B ) )
= ( factor3040189038382604065t_unit @ G @ ( pow_a_1875594501834816709it_nat @ G @ P @ N ) @ B ) ) ) ) ) ) ) ) ).
% comm_monoid_cancel.prime_pow_divides_iff
thf(fact_1171_comm__monoid__cancel_Oprime__pow__divides__iff,axiom,
! [G: partia7680350978787392319xt_a_b,P: a,A: a,B: a,N: nat] :
( ( comm_m4873603268227677576el_a_b @ G )
=> ( ( member_a @ P @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( prime_a_b @ G @ P )
=> ( ~ ( factor_a_b @ G @ P @ A )
=> ( ( factor_a_b @ G @ ( pow_a_b_nat @ G @ P @ N ) @ ( mult_a_b @ G @ A @ B ) )
= ( factor_a_b @ G @ ( pow_a_b_nat @ G @ P @ N ) @ B ) ) ) ) ) ) ) ) ).
% comm_monoid_cancel.prime_pow_divides_iff
thf(fact_1172_comm__monoid__cancel_Owfactors__mult,axiom,
! [G: partia7680350978787392319xt_a_b,As: list_a,A: a,Bs: list_a,B: a] :
( ( comm_m4873603268227677576el_a_b @ G )
=> ( ( wfactors_a_b @ G @ As @ A )
=> ( ( wfactors_a_b @ G @ Bs @ B )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( wfactors_a_b @ G @ ( append_a @ As @ Bs ) @ ( mult_a_b @ G @ A @ B ) ) ) ) ) ) ) ) ) ).
% comm_monoid_cancel.wfactors_mult
thf(fact_1173_comm__monoid__cancel_Ounit__wfactors__empty,axiom,
! [G: partia7680350978787392319xt_a_b,A: a,Fs: list_a] :
( ( comm_m4873603268227677576el_a_b @ G )
=> ( ( member_a @ A @ ( units_a_b @ G ) )
=> ( ( wfactors_a_b @ G @ Fs @ A )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( Fs = nil_a ) ) ) ) ) ).
% comm_monoid_cancel.unit_wfactors_empty
thf(fact_1174_comm__monoid__cancel_Owfactors__listassoc__cong__l,axiom,
! [G: partia7680350978787392319xt_a_b,Fs: list_a,A: a,Fs5: list_a] :
( ( comm_m4873603268227677576el_a_b @ G )
=> ( ( wfactors_a_b @ G @ Fs @ A )
=> ( ( list_all2_a_a @ ( associated_a_b @ G ) @ Fs @ Fs5 )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs5 ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( wfactors_a_b @ G @ Fs5 @ A ) ) ) ) ) ) ) ).
% comm_monoid_cancel.wfactors_listassoc_cong_l
thf(fact_1175_comm__monoid__cancel_Olistassoc__wfactorsD,axiom,
! [G: partia7680350978787392319xt_a_b,As: list_a,Bs: list_a,A: a,B: a] :
( ( comm_m4873603268227677576el_a_b @ G )
=> ( ( list_all2_a_a @ ( associated_a_b @ G ) @ As @ Bs )
=> ( ( wfactors_a_b @ G @ As @ A )
=> ( ( wfactors_a_b @ G @ Bs @ B )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( associated_a_b @ G @ A @ B ) ) ) ) ) ) ) ) ) ).
% comm_monoid_cancel.listassoc_wfactorsD
thf(fact_1176_comm__monoid__cancel_Ofmset__listassoc__cong,axiom,
! [G: partia7680350978787392319xt_a_b,As: list_a,Bs: list_a] :
( ( comm_m4873603268227677576el_a_b @ G )
=> ( ( list_all2_a_a @ ( associated_a_b @ G ) @ As @ Bs )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( fmset_a_b @ G @ As )
= ( fmset_a_b @ G @ Bs ) ) ) ) ) ) ).
% comm_monoid_cancel.fmset_listassoc_cong
thf(fact_1177_partial__object_Oselect__convs_I1_J,axiom,
! [Carrier2: set_a,More: monoid_ext_a_b] :
( ( partia7484183841585581558xt_a_b @ ( partia6120360514654833473xt_a_b @ Carrier2 @ More ) )
= Carrier2 ) ).
% partial_object.select_convs(1)
thf(fact_1178_partial__object_Oselect__convs_I1_J,axiom,
! [Carrier2: set_a,More: eq_eq_8366045168579070496t_unit] :
( ( partia3294672023810056887t_unit @ ( partia6383399360842404908t_unit @ Carrier2 @ More ) )
= Carrier2 ) ).
% partial_object.select_convs(1)
thf(fact_1179_comm__monoid__cancel_Oee__wfactorsD,axiom,
! [G: partia7680350978787392319xt_a_b,As: list_a,Bs: list_a,A: a,B: a] :
( ( comm_m4873603268227677576el_a_b @ G )
=> ( ( essent1248755695173879485al_a_b @ G @ As @ Bs )
=> ( ( wfactors_a_b @ G @ As @ A )
=> ( ( wfactors_a_b @ G @ Bs @ B )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( associated_a_b @ G @ A @ B ) ) ) ) ) ) ) ) ) ).
% comm_monoid_cancel.ee_wfactorsD
thf(fact_1180_comm__monoid__cancel_Oee__factorsD,axiom,
! [G: partia7680350978787392319xt_a_b,As: list_a,Bs: list_a,A: a,B: a] :
( ( comm_m4873603268227677576el_a_b @ G )
=> ( ( essent1248755695173879485al_a_b @ G @ As @ Bs )
=> ( ( factors_a_b @ G @ As @ A )
=> ( ( factors_a_b @ G @ Bs @ B )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( associated_a_b @ G @ A @ B ) ) ) ) ) ) ) ).
% comm_monoid_cancel.ee_factorsD
thf(fact_1181_eq__object_Ocases__scheme,axiom,
! [R: partia6638023223214267844t_unit] :
~ ! [Carrier: set_a,Eq: a > a > $o,More3: gorder3081419850850201676t_unit] :
( R
!= ( partia6383399360842404908t_unit @ Carrier @ ( eq_eq_5233546288009311946t_unit @ Eq @ More3 ) ) ) ).
% eq_object.cases_scheme
thf(fact_1182_comm__monoid__cancel_Ofactorial__monoidI,axiom,
! [G: partia7680350978787392319xt_a_b] :
( ( comm_m4873603268227677576el_a_b @ G )
=> ( ! [A2: a] :
( ( member_a @ A2 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ~ ( member_a @ A2 @ ( units_a_b @ G ) )
=> ? [Fs3: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Fs3 ) @ ( partia7484183841585581558xt_a_b @ G ) )
& ( wfactors_a_b @ G @ Fs3 @ A2 ) ) ) )
=> ( ! [A2: a,Fs2: list_a,Fs4: list_a] :
( ( member_a @ A2 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs2 ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs4 ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( wfactors_a_b @ G @ Fs2 @ A2 )
=> ( ( wfactors_a_b @ G @ Fs4 @ A2 )
=> ( essent1248755695173879485al_a_b @ G @ Fs2 @ Fs4 ) ) ) ) ) )
=> ( factorial_monoid_a_b @ G ) ) ) ) ).
% comm_monoid_cancel.factorial_monoidI
thf(fact_1183_comm__monoid__cancel_Ofmsubset__divides,axiom,
! [G: partia7680350978787392319xt_a_b,As: list_a,Bs: list_a,A: a,B: a] :
( ( comm_m4873603268227677576el_a_b @ G )
=> ( ( subseteq_mset_set_a @ ( fmset_a_b @ G @ As ) @ ( fmset_a_b @ G @ Bs ) )
=> ( ( wfactors_a_b @ G @ As @ A )
=> ( ( wfactors_a_b @ G @ Bs @ B )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( factor_a_b @ G @ A @ B ) ) ) ) ) ) ) ) ) ).
% comm_monoid_cancel.fmsubset_divides
thf(fact_1184_equivalence_Oequal__set__eq__trans,axiom,
! [S: partia6638023223214267844t_unit,A4: set_a,B3: set_a,C4: set_a] :
( ( equiva8593399568135932475t_unit @ S )
=> ( ( A4 = B3 )
=> ( ( set_eq3287599968972363709t_unit @ S @ B3 @ C4 )
=> ( set_eq3287599968972363709t_unit @ S @ A4 @ C4 ) ) ) ) ).
% equivalence.equal_set_eq_trans
thf(fact_1185_equivalence_Oset__eq__equal__trans,axiom,
! [S: partia6638023223214267844t_unit,A4: set_a,B3: set_a,C4: set_a] :
( ( equiva8593399568135932475t_unit @ S )
=> ( ( set_eq3287599968972363709t_unit @ S @ A4 @ B3 )
=> ( ( B3 = C4 )
=> ( set_eq3287599968972363709t_unit @ S @ A4 @ C4 ) ) ) ) ).
% equivalence.set_eq_equal_trans
thf(fact_1186_comm__monoid__cancel_Omultlist__listassoc__cong,axiom,
! [G: partia7680350978787392319xt_a_b,Fs: list_a,Fs5: list_a] :
( ( comm_m4873603268227677576el_a_b @ G )
=> ( ( list_all2_a_a @ ( associated_a_b @ G ) @ Fs @ Fs5 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs5 ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( associated_a_b @ G @ ( foldr_a_a @ ( mult_a_b @ G ) @ Fs @ ( one_a_b @ G ) ) @ ( foldr_a_a @ ( mult_a_b @ G ) @ Fs5 @ ( one_a_b @ G ) ) ) ) ) ) ) ).
% comm_monoid_cancel.multlist_listassoc_cong
thf(fact_1187_comm__monoid__cancel_Omultlist__ee__cong,axiom,
! [G: partia7680350978787392319xt_a_b,Fs: list_a,Fs5: list_a] :
( ( comm_m4873603268227677576el_a_b @ G )
=> ( ( essent1248755695173879485al_a_b @ G @ Fs @ Fs5 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs5 ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( associated_a_b @ G @ ( foldr_a_a @ ( mult_a_b @ G ) @ Fs @ ( one_a_b @ G ) ) @ ( foldr_a_a @ ( mult_a_b @ G ) @ Fs5 @ ( one_a_b @ G ) ) ) ) ) ) ) ).
% comm_monoid_cancel.multlist_ee_cong
thf(fact_1188_comm__monoid__cancel_Ofmset__wfactors__mult,axiom,
! [G: partia7680350978787392319xt_a_b,Cs3: list_a,As: list_a,Bs: list_a,A: a,B: a,C: a] :
( ( comm_m4873603268227677576el_a_b @ G )
=> ( ( ( fmset_a_b @ G @ Cs3 )
= ( plus_p2331992037799027419_set_a @ ( fmset_a_b @ G @ As ) @ ( fmset_a_b @ G @ Bs ) ) )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Cs3 ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( wfactors_a_b @ G @ As @ A )
=> ( ( wfactors_a_b @ G @ Bs @ B )
=> ( ( wfactors_a_b @ G @ Cs3 @ C )
=> ( associated_a_b @ G @ C @ ( mult_a_b @ G @ A @ B ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% comm_monoid_cancel.fmset_wfactors_mult
thf(fact_1189_comm__monoid__cancel_Ofactors__cong__unit,axiom,
! [G: partia7680350978787392319xt_a_b,U: a,A: a,As: list_a] :
( ( comm_m4873603268227677576el_a_b @ G )
=> ( ( member_a @ U @ ( units_a_b @ G ) )
=> ( ~ ( member_a @ A @ ( units_a_b @ G ) )
=> ( ( factors_a_b @ G @ As @ A )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( factors_a_b @ G @ ( list_update_a @ As @ zero_zero_nat @ ( mult_a_b @ G @ ( nth_a @ As @ zero_zero_nat ) @ U ) ) @ ( mult_a_b @ G @ A @ U ) ) ) ) ) ) ) ).
% comm_monoid_cancel.factors_cong_unit
thf(fact_1190_comm__monoid__cancel_Ounitfactor__ee,axiom,
! [G: partia7680350978787392319xt_a_b,U: a,As: list_a] :
( ( comm_m4873603268227677576el_a_b @ G )
=> ( ( member_a @ U @ ( units_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( essent1248755695173879485al_a_b @ G @ ( list_update_a @ As @ zero_zero_nat @ ( mult_a_b @ G @ ( nth_a @ As @ zero_zero_nat ) @ U ) ) @ As ) ) ) ) ).
% comm_monoid_cancel.unitfactor_ee
thf(fact_1191_equivalence_Oclosure__inclusion,axiom,
! [S: partia6638023223214267844t_unit,A4: set_a,B3: set_a] :
( ( equiva8593399568135932475t_unit @ S )
=> ( ( ord_less_eq_set_a @ A4 @ B3 )
=> ( ord_less_eq_set_a @ ( eq_clo186784744570230005t_unit @ S @ A4 ) @ ( eq_clo186784744570230005t_unit @ S @ B3 ) ) ) ) ).
% equivalence.closure_inclusion
thf(fact_1192_group__l__invI,axiom,
( ! [X4: a] :
( ( member_a @ X4 @ ( partia7484183841585581558xt_a_b @ g ) )
=> ? [Xa: a] :
( ( member_a @ Xa @ ( partia7484183841585581558xt_a_b @ g ) )
& ( ( mult_a_b @ g @ Xa @ X4 )
= ( one_a_b @ g ) ) ) )
=> ( group_a_b @ g ) ) ).
% group_l_invI
thf(fact_1193_monoid_OUnits__pow__closed,axiom,
! [G: partia8223610829204095565t_unit,X: a,D: nat] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( member_a @ X @ ( units_a_Product_unit @ G ) )
=> ( member_a @ ( pow_a_1875594501834816709it_nat @ G @ X @ D ) @ ( units_a_Product_unit @ G ) ) ) ) ).
% monoid.Units_pow_closed
thf(fact_1194_monoid_OUnits__pow__closed,axiom,
! [G: partia7680350978787392319xt_a_b,X: a,D: nat] :
( ( monoid_a_b @ G )
=> ( ( member_a @ X @ ( units_a_b @ G ) )
=> ( member_a @ ( pow_a_b_nat @ G @ X @ D ) @ ( units_a_b @ G ) ) ) ) ).
% monoid.Units_pow_closed
thf(fact_1195_closure__of__closed,axiom,
! [S: partia6638023223214267844t_unit,A4: set_a] : ( ord_less_eq_set_a @ ( eq_clo186784744570230005t_unit @ S @ A4 ) @ ( partia3294672023810056887t_unit @ S ) ) ).
% closure_of_closed
thf(fact_1196_Lower__closed,axiom,
! [L: partia6638023223214267844t_unit,A4: set_a] : ( ord_less_eq_set_a @ ( lower_a_Product_unit @ L @ A4 ) @ ( partia3294672023810056887t_unit @ L ) ) ).
% Lower_closed
thf(fact_1197_Lower__empty,axiom,
! [L: partia6638023223214267844t_unit] :
( ( lower_a_Product_unit @ L @ bot_bot_set_a )
= ( partia3294672023810056887t_unit @ L ) ) ).
% Lower_empty
thf(fact_1198_equivalenceI,axiom,
! [E2: set_a,P3: a > a > $o] :
( ! [X4: a] :
( ( member_a @ X4 @ E2 )
=> ( P3 @ X4 @ X4 ) )
=> ( ! [X4: a,Y3: a] :
( ( member_a @ X4 @ E2 )
=> ( ( member_a @ Y3 @ E2 )
=> ( ( P3 @ X4 @ Y3 )
=> ( P3 @ Y3 @ X4 ) ) ) )
=> ( ! [X4: a,Y3: a,Z3: a] :
( ( member_a @ X4 @ E2 )
=> ( ( member_a @ Y3 @ E2 )
=> ( ( member_a @ Z3 @ E2 )
=> ( ( P3 @ X4 @ Y3 )
=> ( ( P3 @ Y3 @ Z3 )
=> ( P3 @ X4 @ Z3 ) ) ) ) ) )
=> ( equiva4991551782370822396t_unit @ ( partia295254675854492781t_unit @ E2 @ ( eq_eq_4790507112745384587t_unit @ P3 @ product_Unity ) ) ) ) ) ) ).
% equivalenceI
thf(fact_1199_equivalenceI,axiom,
! [E2: set_set_a,P3: set_a > set_a > $o] :
( ! [X4: set_a] :
( ( member_set_a @ X4 @ E2 )
=> ( P3 @ X4 @ X4 ) )
=> ( ! [X4: set_a,Y3: set_a] :
( ( member_set_a @ X4 @ E2 )
=> ( ( member_set_a @ Y3 @ E2 )
=> ( ( P3 @ X4 @ Y3 )
=> ( P3 @ Y3 @ X4 ) ) ) )
=> ( ! [X4: set_a,Y3: set_a,Z3: set_a] :
( ( member_set_a @ X4 @ E2 )
=> ( ( member_set_a @ Y3 @ E2 )
=> ( ( member_set_a @ Z3 @ E2 )
=> ( ( P3 @ X4 @ Y3 )
=> ( ( P3 @ Y3 @ Z3 )
=> ( P3 @ X4 @ Z3 ) ) ) ) ) )
=> ( equiva3674511908918812572t_unit @ ( partia519936911669663853t_unit @ E2 @ ( eq_eq_6085121125334127915t_unit @ P3 @ product_Unity ) ) ) ) ) ) ).
% equivalenceI
thf(fact_1200_group_Ois__group,axiom,
! [G: partia7680350978787392319xt_a_b] :
( ( group_a_b @ G )
=> ( group_a_b @ G ) ) ).
% group.is_group
thf(fact_1201_group_Ois__group,axiom,
! [G: partia8223610829204095565t_unit] :
( ( group_a_Product_unit @ G )
=> ( group_a_Product_unit @ G ) ) ).
% group.is_group
thf(fact_1202_group_Ois__monoid,axiom,
! [G: partia8223610829204095565t_unit] :
( ( group_a_Product_unit @ G )
=> ( monoid2746444814143937472t_unit @ G ) ) ).
% group.is_monoid
thf(fact_1203_group_Ois__monoid,axiom,
! [G: partia7680350978787392319xt_a_b] :
( ( group_a_b @ G )
=> ( monoid_a_b @ G ) ) ).
% group.is_monoid
thf(fact_1204_group_OUnits__eq,axiom,
! [G: partia8223610829204095565t_unit] :
( ( group_a_Product_unit @ G )
=> ( ( units_a_Product_unit @ G )
= ( partia6735698275553448452t_unit @ G ) ) ) ).
% group.Units_eq
thf(fact_1205_group_OUnits__eq,axiom,
! [G: partia7680350978787392319xt_a_b] :
( ( group_a_b @ G )
=> ( ( units_a_b @ G )
= ( partia7484183841585581558xt_a_b @ G ) ) ) ).
% group.Units_eq
thf(fact_1206_Group_Ogroup_Oright__cancel,axiom,
! [G: partia8223610829204095565t_unit,X: a,Y: a,Z: a] :
( ( group_a_Product_unit @ G )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ Z @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( ( mult_a_Product_unit @ G @ Y @ X )
= ( mult_a_Product_unit @ G @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ) ).
% Group.group.right_cancel
thf(fact_1207_Group_Ogroup_Oright__cancel,axiom,
! [G: partia7680350978787392319xt_a_b,X: a,Y: a,Z: a] :
( ( group_a_b @ G )
=> ( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ Z @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ( mult_a_b @ G @ Y @ X )
= ( mult_a_b @ G @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ) ).
% Group.group.right_cancel
thf(fact_1208_group_OUnits,axiom,
! [G: partia8223610829204095565t_unit] :
( ( group_a_Product_unit @ G )
=> ( ord_less_eq_set_a @ ( partia6735698275553448452t_unit @ G ) @ ( units_a_Product_unit @ G ) ) ) ).
% group.Units
thf(fact_1209_group_OUnits,axiom,
! [G: partia7680350978787392319xt_a_b] :
( ( group_a_b @ G )
=> ( ord_less_eq_set_a @ ( partia7484183841585581558xt_a_b @ G ) @ ( units_a_b @ G ) ) ) ).
% group.Units
thf(fact_1210_groupI,axiom,
! [G: partia8223610829204095565t_unit] :
( ! [X4: a] :
( ( member_a @ X4 @ ( partia6735698275553448452t_unit @ G ) )
=> ! [Y3: a] :
( ( member_a @ Y3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( member_a @ ( mult_a_Product_unit @ G @ X4 @ Y3 ) @ ( partia6735698275553448452t_unit @ G ) ) ) )
=> ( ( member_a @ ( one_a_Product_unit @ G ) @ ( partia6735698275553448452t_unit @ G ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia6735698275553448452t_unit @ G ) )
=> ! [Y3: a] :
( ( member_a @ Y3 @ ( partia6735698275553448452t_unit @ G ) )
=> ! [Z3: a] :
( ( member_a @ Z3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( mult_a_Product_unit @ G @ ( mult_a_Product_unit @ G @ X4 @ Y3 ) @ Z3 )
= ( mult_a_Product_unit @ G @ X4 @ ( mult_a_Product_unit @ G @ Y3 @ Z3 ) ) ) ) ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( mult_a_Product_unit @ G @ ( one_a_Product_unit @ G ) @ X4 )
= X4 ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia6735698275553448452t_unit @ G ) )
=> ? [Xa: a] :
( ( member_a @ Xa @ ( partia6735698275553448452t_unit @ G ) )
& ( ( mult_a_Product_unit @ G @ Xa @ X4 )
= ( one_a_Product_unit @ G ) ) ) )
=> ( group_a_Product_unit @ G ) ) ) ) ) ) ).
% groupI
thf(fact_1211_groupI,axiom,
! [G: partia7680350978787392319xt_a_b] :
( ! [X4: a] :
( ( member_a @ X4 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ! [Y3: a] :
( ( member_a @ Y3 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( member_a @ ( mult_a_b @ G @ X4 @ Y3 ) @ ( partia7484183841585581558xt_a_b @ G ) ) ) )
=> ( ( member_a @ ( one_a_b @ G ) @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ! [Y3: a] :
( ( member_a @ Y3 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ! [Z3: a] :
( ( member_a @ Z3 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( mult_a_b @ G @ ( mult_a_b @ G @ X4 @ Y3 ) @ Z3 )
= ( mult_a_b @ G @ X4 @ ( mult_a_b @ G @ Y3 @ Z3 ) ) ) ) ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( mult_a_b @ G @ ( one_a_b @ G ) @ X4 )
= X4 ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ? [Xa: a] :
( ( member_a @ Xa @ ( partia7484183841585581558xt_a_b @ G ) )
& ( ( mult_a_b @ G @ Xa @ X4 )
= ( one_a_b @ G ) ) ) )
=> ( group_a_b @ G ) ) ) ) ) ) ).
% groupI
thf(fact_1212_group_Or__cancel__one_H,axiom,
! [G: partia8223610829204095565t_unit,X: a,A: a] :
( ( group_a_Product_unit @ G )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( X
= ( mult_a_Product_unit @ G @ A @ X ) )
= ( A
= ( one_a_Product_unit @ G ) ) ) ) ) ) ).
% group.r_cancel_one'
thf(fact_1213_group_Or__cancel__one_H,axiom,
! [G: partia7680350978787392319xt_a_b,X: a,A: a] :
( ( group_a_b @ G )
=> ( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( X
= ( mult_a_b @ G @ A @ X ) )
= ( A
= ( one_a_b @ G ) ) ) ) ) ) ).
% group.r_cancel_one'
thf(fact_1214_group_Ol__cancel__one_H,axiom,
! [G: partia8223610829204095565t_unit,X: a,A: a] :
( ( group_a_Product_unit @ G )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( X
= ( mult_a_Product_unit @ G @ X @ A ) )
= ( A
= ( one_a_Product_unit @ G ) ) ) ) ) ) ).
% group.l_cancel_one'
thf(fact_1215_group_Ol__cancel__one_H,axiom,
! [G: partia7680350978787392319xt_a_b,X: a,A: a] :
( ( group_a_b @ G )
=> ( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( X
= ( mult_a_b @ G @ X @ A ) )
= ( A
= ( one_a_b @ G ) ) ) ) ) ) ).
% group.l_cancel_one'
thf(fact_1216_group_Or__cancel__one,axiom,
! [G: partia8223610829204095565t_unit,X: a,A: a] :
( ( group_a_Product_unit @ G )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( ( mult_a_Product_unit @ G @ A @ X )
= X )
= ( A
= ( one_a_Product_unit @ G ) ) ) ) ) ) ).
% group.r_cancel_one
thf(fact_1217_group_Or__cancel__one,axiom,
! [G: partia7680350978787392319xt_a_b,X: a,A: a] :
( ( group_a_b @ G )
=> ( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ( mult_a_b @ G @ A @ X )
= X )
= ( A
= ( one_a_b @ G ) ) ) ) ) ) ).
% group.r_cancel_one
thf(fact_1218_group_Ol__cancel__one,axiom,
! [G: partia8223610829204095565t_unit,X: a,A: a] :
( ( group_a_Product_unit @ G )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( ( mult_a_Product_unit @ G @ X @ A )
= X )
= ( A
= ( one_a_Product_unit @ G ) ) ) ) ) ) ).
% group.l_cancel_one
thf(fact_1219_group_Ol__cancel__one,axiom,
! [G: partia7680350978787392319xt_a_b,X: a,A: a] :
( ( group_a_b @ G )
=> ( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ A @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( ( mult_a_b @ G @ X @ A )
= X )
= ( A
= ( one_a_b @ G ) ) ) ) ) ) ).
% group.l_cancel_one
thf(fact_1220_group_Or__inv__ex,axiom,
! [G: partia8223610829204095565t_unit,X: a] :
( ( group_a_Product_unit @ G )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia6735698275553448452t_unit @ G ) )
& ( ( mult_a_Product_unit @ G @ X @ X4 )
= ( one_a_Product_unit @ G ) ) ) ) ) ).
% group.r_inv_ex
thf(fact_1221_group_Or__inv__ex,axiom,
! [G: partia7680350978787392319xt_a_b,X: a] :
( ( group_a_b @ G )
=> ( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ G ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia7484183841585581558xt_a_b @ G ) )
& ( ( mult_a_b @ G @ X @ X4 )
= ( one_a_b @ G ) ) ) ) ) ).
% group.r_inv_ex
thf(fact_1222_group_Ol__inv__ex,axiom,
! [G: partia8223610829204095565t_unit,X: a] :
( ( group_a_Product_unit @ G )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia6735698275553448452t_unit @ G ) )
& ( ( mult_a_Product_unit @ G @ X4 @ X )
= ( one_a_Product_unit @ G ) ) ) ) ) ).
% group.l_inv_ex
thf(fact_1223_group_Ol__inv__ex,axiom,
! [G: partia7680350978787392319xt_a_b,X: a] :
( ( group_a_b @ G )
=> ( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ G ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia7484183841585581558xt_a_b @ G ) )
& ( ( mult_a_b @ G @ X4 @ X )
= ( one_a_b @ G ) ) ) ) ) ).
% group.l_inv_ex
thf(fact_1224_group_Oinv__comm,axiom,
! [G: partia8223610829204095565t_unit,X: a,Y: a] :
( ( group_a_Product_unit @ G )
=> ( ( ( mult_a_Product_unit @ G @ X @ Y )
= ( one_a_Product_unit @ G ) )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( mult_a_Product_unit @ G @ Y @ X )
= ( one_a_Product_unit @ G ) ) ) ) ) ) ).
% group.inv_comm
thf(fact_1225_group_Oinv__comm,axiom,
! [G: partia7680350978787392319xt_a_b,X: a,Y: a] :
( ( group_a_b @ G )
=> ( ( ( mult_a_b @ G @ X @ Y )
= ( one_a_b @ G ) )
=> ( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( mult_a_b @ G @ Y @ X )
= ( one_a_b @ G ) ) ) ) ) ) ).
% group.inv_comm
thf(fact_1226_weak__partial__order_Olless__trans,axiom,
! [L: partia6638023223214267844t_unit,A: a,B: a,C: a] :
( ( weak_p4024844561679123766t_unit @ L )
=> ( ( lless_a_Product_unit @ L @ A @ B )
=> ( ( lless_a_Product_unit @ L @ B @ C )
=> ( ( member_a @ A @ ( partia3294672023810056887t_unit @ L ) )
=> ( ( member_a @ B @ ( partia3294672023810056887t_unit @ L ) )
=> ( ( member_a @ C @ ( partia3294672023810056887t_unit @ L ) )
=> ( lless_a_Product_unit @ L @ A @ C ) ) ) ) ) ) ) ).
% weak_partial_order.lless_trans
thf(fact_1227_weak__partial__order_Olless__antisym,axiom,
! [L: partia6638023223214267844t_unit,A: a,B: a] :
( ( weak_p4024844561679123766t_unit @ L )
=> ( ( member_a @ A @ ( partia3294672023810056887t_unit @ L ) )
=> ( ( member_a @ B @ ( partia3294672023810056887t_unit @ L ) )
=> ( ( lless_a_Product_unit @ L @ A @ B )
=> ~ ( lless_a_Product_unit @ L @ B @ A ) ) ) ) ) ).
% weak_partial_order.lless_antisym
thf(fact_1228_equivalence_Oclosure__idem__strong,axiom,
! [S: partia6638023223214267844t_unit,A4: set_a] :
( ( equiva8593399568135932475t_unit @ S )
=> ( ( ord_less_eq_set_a @ A4 @ ( partia3294672023810056887t_unit @ S ) )
=> ( ( eq_clo186784744570230005t_unit @ S @ ( eq_clo186784744570230005t_unit @ S @ A4 ) )
= ( eq_clo186784744570230005t_unit @ S @ A4 ) ) ) ) ).
% equivalence.closure_idem_strong
thf(fact_1229_equivalence_Oset__eq__sym,axiom,
! [S: partia6638023223214267844t_unit,A4: set_a,B3: set_a] :
( ( equiva8593399568135932475t_unit @ S )
=> ( ( ord_less_eq_set_a @ A4 @ ( partia3294672023810056887t_unit @ S ) )
=> ( ( ord_less_eq_set_a @ B3 @ ( partia3294672023810056887t_unit @ S ) )
=> ( ( set_eq3287599968972363709t_unit @ S @ A4 @ B3 )
=> ( set_eq3287599968972363709t_unit @ S @ B3 @ A4 ) ) ) ) ) ).
% equivalence.set_eq_sym
thf(fact_1230_equivalence_Oset__eq__trans,axiom,
! [S: partia6638023223214267844t_unit,A4: set_a,B3: set_a,C4: set_a] :
( ( equiva8593399568135932475t_unit @ S )
=> ( ( ord_less_eq_set_a @ A4 @ ( partia3294672023810056887t_unit @ S ) )
=> ( ( ord_less_eq_set_a @ B3 @ ( partia3294672023810056887t_unit @ S ) )
=> ( ( ord_less_eq_set_a @ C4 @ ( partia3294672023810056887t_unit @ S ) )
=> ( ( set_eq3287599968972363709t_unit @ S @ A4 @ B3 )
=> ( ( set_eq3287599968972363709t_unit @ S @ B3 @ C4 )
=> ( set_eq3287599968972363709t_unit @ S @ A4 @ C4 ) ) ) ) ) ) ) ).
% equivalence.set_eq_trans
thf(fact_1231_monoid_Ogroup__l__invI,axiom,
! [G: partia8223610829204095565t_unit] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia6735698275553448452t_unit @ G ) )
=> ? [Xa: a] :
( ( member_a @ Xa @ ( partia6735698275553448452t_unit @ G ) )
& ( ( mult_a_Product_unit @ G @ Xa @ X4 )
= ( one_a_Product_unit @ G ) ) ) )
=> ( group_a_Product_unit @ G ) ) ) ).
% monoid.group_l_invI
thf(fact_1232_monoid_Ogroup__l__invI,axiom,
! [G: partia7680350978787392319xt_a_b] :
( ( monoid_a_b @ G )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ? [Xa: a] :
( ( member_a @ Xa @ ( partia7484183841585581558xt_a_b @ G ) )
& ( ( mult_a_b @ G @ Xa @ X4 )
= ( one_a_b @ G ) ) ) )
=> ( group_a_b @ G ) ) ) ).
% monoid.group_l_invI
thf(fact_1233_equivalence_Oclasses__eq,axiom,
! [S: partia6638023223214267844t_unit,A4: set_a] :
( ( equiva8593399568135932475t_unit @ S )
=> ( ( ord_less_eq_set_a @ A4 @ ( partia3294672023810056887t_unit @ S ) )
=> ( set_eq3287599968972363709t_unit @ S @ A4 @ ( eq_clo186784744570230005t_unit @ S @ A4 ) ) ) ) ).
% equivalence.classes_eq
thf(fact_1234_weak__partial__order_OLower__cong,axiom,
! [L: partia6638023223214267844t_unit,A4: set_a,A8: set_a] :
( ( weak_p4024844561679123766t_unit @ L )
=> ( ( ord_less_eq_set_a @ A4 @ ( partia3294672023810056887t_unit @ L ) )
=> ( ( ord_less_eq_set_a @ A8 @ ( partia3294672023810056887t_unit @ L ) )
=> ( ( set_eq3287599968972363709t_unit @ L @ A4 @ A8 )
=> ( ( lower_a_Product_unit @ L @ A4 )
= ( lower_a_Product_unit @ L @ A8 ) ) ) ) ) ) ).
% weak_partial_order.Lower_cong
thf(fact_1235_weak__partial__order_Ogreatest__Lower__cong__r,axiom,
! [L: partia6638023223214267844t_unit,A4: set_a,A8: set_a,X: a] :
( ( weak_p4024844561679123766t_unit @ L )
=> ( ( ord_less_eq_set_a @ A4 @ ( partia3294672023810056887t_unit @ L ) )
=> ( ( ord_less_eq_set_a @ A8 @ ( partia3294672023810056887t_unit @ L ) )
=> ( ( set_eq3287599968972363709t_unit @ L @ A4 @ A8 )
=> ( ( greate1756078582999224458t_unit @ L @ X @ ( lower_a_Product_unit @ L @ A4 ) )
= ( greate1756078582999224458t_unit @ L @ X @ ( lower_a_Product_unit @ L @ A8 ) ) ) ) ) ) ) ).
% weak_partial_order.greatest_Lower_cong_r
thf(fact_1236_weak__lower__semilattice_OmeetI,axiom,
! [L: partia6638023223214267844t_unit,X: a,Y: a,P3: a > $o] :
( ( weak_l5773807957815819224t_unit @ L )
=> ( ! [I2: a] :
( ( greate1756078582999224458t_unit @ L @ I2 @ ( lower_a_Product_unit @ L @ ( insert_a @ X @ ( insert_a @ Y @ bot_bot_set_a ) ) ) )
=> ( P3 @ I2 ) )
=> ( ( member_a @ X @ ( partia3294672023810056887t_unit @ L ) )
=> ( ( member_a @ Y @ ( partia3294672023810056887t_unit @ L ) )
=> ( P3 @ ( meet_a_Product_unit @ L @ X @ Y ) ) ) ) ) ) ).
% weak_lower_semilattice.meetI
thf(fact_1237_weak__lower__semilattice_Oinf__of__two__exists,axiom,
! [L: partia6638023223214267844t_unit,X: a,Y: a] :
( ( weak_l5773807957815819224t_unit @ L )
=> ( ( member_a @ X @ ( partia3294672023810056887t_unit @ L ) )
=> ( ( member_a @ Y @ ( partia3294672023810056887t_unit @ L ) )
=> ? [S2: a] : ( greate1756078582999224458t_unit @ L @ S2 @ ( lower_a_Product_unit @ L @ ( insert_a @ X @ ( insert_a @ Y @ bot_bot_set_a ) ) ) ) ) ) ) ).
% weak_lower_semilattice.inf_of_two_exists
thf(fact_1238_units__group,axiom,
group_a_Product_unit @ ( units_of_a_b @ g ) ).
% units_group
thf(fact_1239_monoid_Ounits__group,axiom,
! [G: partia7680350978787392319xt_a_b] :
( ( monoid_a_b @ G )
=> ( group_a_Product_unit @ ( units_of_a_b @ G ) ) ) ).
% monoid.units_group
thf(fact_1240_weak__lower__semilattice_Oaxioms_I1_J,axiom,
! [L: partia6638023223214267844t_unit] :
( ( weak_l5773807957815819224t_unit @ L )
=> ( weak_p4024844561679123766t_unit @ L ) ) ).
% weak_lower_semilattice.axioms(1)
thf(fact_1241_weak__lower__semilattice_Omeet__closed,axiom,
! [L: partia6638023223214267844t_unit,X: a,Y: a] :
( ( weak_l5773807957815819224t_unit @ L )
=> ( ( member_a @ X @ ( partia3294672023810056887t_unit @ L ) )
=> ( ( member_a @ Y @ ( partia3294672023810056887t_unit @ L ) )
=> ( member_a @ ( meet_a_Product_unit @ L @ X @ Y ) @ ( partia3294672023810056887t_unit @ L ) ) ) ) ) ).
% weak_lower_semilattice.meet_closed
thf(fact_1242_weak__lattice_Oaxioms_I2_J,axiom,
! [L: partia6638023223214267844t_unit] :
( ( weak_l7828739606774570307t_unit @ L )
=> ( weak_l5773807957815819224t_unit @ L ) ) ).
% weak_lattice.axioms(2)
thf(fact_1243_weak__partial__order_Oinf__of__singletonI,axiom,
! [L: partia6638023223214267844t_unit,X: a] :
( ( weak_p4024844561679123766t_unit @ L )
=> ( ( member_a @ X @ ( partia3294672023810056887t_unit @ L ) )
=> ( greate1756078582999224458t_unit @ L @ X @ ( lower_a_Product_unit @ L @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% weak_partial_order.inf_of_singletonI
thf(fact_1244_weak__lower__semilattice__axioms__def,axiom,
( weak_l1711373400843467003t_unit
= ( ^ [L3: partia6638023223214267844t_unit] :
! [X3: a,Y5: a] :
( ( member_a @ X3 @ ( partia3294672023810056887t_unit @ L3 ) )
=> ( ( member_a @ Y5 @ ( partia3294672023810056887t_unit @ L3 ) )
=> ? [S3: a] : ( greate1756078582999224458t_unit @ L3 @ S3 @ ( lower_a_Product_unit @ L3 @ ( insert_a @ X3 @ ( insert_a @ Y5 @ bot_bot_set_a ) ) ) ) ) ) ) ) ).
% weak_lower_semilattice_axioms_def
thf(fact_1245_lower__semilattice__axioms__def,axiom,
( lower_30033100221001027t_unit
= ( ^ [L3: partia6638023223214267844t_unit] :
! [X3: a,Y5: a] :
( ( member_a @ X3 @ ( partia3294672023810056887t_unit @ L3 ) )
=> ( ( member_a @ Y5 @ ( partia3294672023810056887t_unit @ L3 ) )
=> ? [S3: a] : ( greate1756078582999224458t_unit @ L3 @ S3 @ ( lower_a_Product_unit @ L3 @ ( insert_a @ X3 @ ( insert_a @ Y5 @ bot_bot_set_a ) ) ) ) ) ) ) ) ).
% lower_semilattice_axioms_def
thf(fact_1246_weak__lower__semilattice_Oaxioms_I2_J,axiom,
! [L: partia6638023223214267844t_unit] :
( ( weak_l5773807957815819224t_unit @ L )
=> ( weak_l1711373400843467003t_unit @ L ) ) ).
% weak_lower_semilattice.axioms(2)
thf(fact_1247_weak__lower__semilattice__def,axiom,
( weak_l5773807957815819224t_unit
= ( ^ [L3: partia6638023223214267844t_unit] :
( ( weak_p4024844561679123766t_unit @ L3 )
& ( weak_l1711373400843467003t_unit @ L3 ) ) ) ) ).
% weak_lower_semilattice_def
thf(fact_1248_weak__lower__semilattice_Ointro,axiom,
! [L: partia6638023223214267844t_unit] :
( ( weak_p4024844561679123766t_unit @ L )
=> ( ( weak_l1711373400843467003t_unit @ L )
=> ( weak_l5773807957815819224t_unit @ L ) ) ) ).
% weak_lower_semilattice.intro
thf(fact_1249_lower__semilattice__axioms_Ointro,axiom,
! [L: partia6638023223214267844t_unit] :
( ! [X4: a,Y3: a] :
( ( member_a @ X4 @ ( partia3294672023810056887t_unit @ L ) )
=> ( ( member_a @ Y3 @ ( partia3294672023810056887t_unit @ L ) )
=> ? [S4: a] : ( greate1756078582999224458t_unit @ L @ S4 @ ( lower_a_Product_unit @ L @ ( insert_a @ X4 @ ( insert_a @ Y3 @ bot_bot_set_a ) ) ) ) ) )
=> ( lower_30033100221001027t_unit @ L ) ) ).
% lower_semilattice_axioms.intro
thf(fact_1250_weak__lower__semilattice__axioms_Ointro,axiom,
! [L: partia6638023223214267844t_unit] :
( ! [X4: a,Y3: a] :
( ( member_a @ X4 @ ( partia3294672023810056887t_unit @ L ) )
=> ( ( member_a @ Y3 @ ( partia3294672023810056887t_unit @ L ) )
=> ? [S4: a] : ( greate1756078582999224458t_unit @ L @ S4 @ ( lower_a_Product_unit @ L @ ( insert_a @ X4 @ ( insert_a @ Y3 @ bot_bot_set_a ) ) ) ) ) )
=> ( weak_l1711373400843467003t_unit @ L ) ) ).
% weak_lower_semilattice_axioms.intro
thf(fact_1251_units__comm__group,axiom,
comm_g1850867397131805039t_unit @ ( units_of_a_b @ g ) ).
% units_comm_group
thf(fact_1252_nat__pow__pow,axiom,
! [X: a,N: nat,M4: nat] :
( ( member_a @ X @ ( partia7484183841585581558xt_a_b @ g ) )
=> ( ( pow_a_b_nat @ g @ ( pow_a_b_nat @ g @ X @ N ) @ M4 )
= ( pow_a_b_nat @ g @ X @ ( times_times_nat @ N @ M4 ) ) ) ) ).
% nat_pow_pow
thf(fact_1253_repeat__mset__right,axiom,
! [A: nat,B: nat,A4: multiset_set_a] :
( ( repeat_mset_set_a @ A @ ( repeat_mset_set_a @ B @ A4 ) )
= ( repeat_mset_set_a @ ( times_times_nat @ A @ B ) @ A4 ) ) ).
% repeat_mset_right
thf(fact_1254_mult__Suc__right,axiom,
! [M4: nat,N: nat] :
( ( times_times_nat @ M4 @ ( suc @ N ) )
= ( plus_plus_nat @ M4 @ ( times_times_nat @ M4 @ N ) ) ) ).
% mult_Suc_right
thf(fact_1255_count__repeat__mset,axiom,
! [I: nat,A4: multiset_set_a,A: set_a] :
( ( count_set_a @ ( repeat_mset_set_a @ I @ A4 ) @ A )
= ( times_times_nat @ I @ ( count_set_a @ A4 @ A ) ) ) ).
% count_repeat_mset
thf(fact_1256_one__le__mult__iff,axiom,
! [M4: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M4 @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M4 )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_1257_mult__le__cancel2,axiom,
! [M4: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M4 @ K2 ) @ ( times_times_nat @ N @ K2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_eq_nat @ M4 @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1258_comm__group_Oaxioms_I2_J,axiom,
! [G: partia7680350978787392319xt_a_b] :
( ( comm_group_a_b @ G )
=> ( group_a_b @ G ) ) ).
% comm_group.axioms(2)
thf(fact_1259_comm__group_Oaxioms_I2_J,axiom,
! [G: partia8223610829204095565t_unit] :
( ( comm_g1850867397131805039t_unit @ G )
=> ( group_a_Product_unit @ G ) ) ).
% comm_group.axioms(2)
thf(fact_1260_comm__group__def,axiom,
( comm_g1850867397131805039t_unit
= ( ^ [G2: partia8223610829204095565t_unit] :
( ( comm_m7681468956318391052t_unit @ G2 )
& ( group_a_Product_unit @ G2 ) ) ) ) ).
% comm_group_def
thf(fact_1261_comm__group__def,axiom,
( comm_group_a_b
= ( ^ [G2: partia7680350978787392319xt_a_b] :
( ( comm_monoid_a_b @ G2 )
& ( group_a_b @ G2 ) ) ) ) ).
% comm_group_def
thf(fact_1262_comm__group_Ointro,axiom,
! [G: partia8223610829204095565t_unit] :
( ( comm_m7681468956318391052t_unit @ G )
=> ( ( group_a_Product_unit @ G )
=> ( comm_g1850867397131805039t_unit @ G ) ) ) ).
% comm_group.intro
thf(fact_1263_comm__group_Ointro,axiom,
! [G: partia7680350978787392319xt_a_b] :
( ( comm_monoid_a_b @ G )
=> ( ( group_a_b @ G )
=> ( comm_group_a_b @ G ) ) ) ).
% comm_group.intro
thf(fact_1264_group_Ogroup__comm__groupI,axiom,
! [G: partia8223610829204095565t_unit] :
( ( group_a_Product_unit @ G )
=> ( ! [X4: a,Y3: a] :
( ( member_a @ X4 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ Y3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( mult_a_Product_unit @ G @ X4 @ Y3 )
= ( mult_a_Product_unit @ G @ Y3 @ X4 ) ) ) )
=> ( comm_g1850867397131805039t_unit @ G ) ) ) ).
% group.group_comm_groupI
thf(fact_1265_group_Ogroup__comm__groupI,axiom,
! [G: partia7680350978787392319xt_a_b] :
( ( group_a_b @ G )
=> ( ! [X4: a,Y3: a] :
( ( member_a @ X4 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( member_a @ Y3 @ ( partia7484183841585581558xt_a_b @ G ) )
=> ( ( mult_a_b @ G @ X4 @ Y3 )
= ( mult_a_b @ G @ Y3 @ X4 ) ) ) )
=> ( comm_group_a_b @ G ) ) ) ).
% group.group_comm_groupI
thf(fact_1266_mult__le__mono2,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K2 @ I ) @ ( times_times_nat @ K2 @ J ) ) ) ).
% mult_le_mono2
thf(fact_1267_mult__le__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K2 ) @ ( times_times_nat @ J @ K2 ) ) ) ).
% mult_le_mono1
thf(fact_1268_mult__le__mono,axiom,
! [I: nat,J: nat,K2: nat,L2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K2 @ L2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K2 ) @ ( times_times_nat @ J @ L2 ) ) ) ) ).
% mult_le_mono
thf(fact_1269_le__square,axiom,
! [M4: nat] : ( ord_less_eq_nat @ M4 @ ( times_times_nat @ M4 @ M4 ) ) ).
% le_square
thf(fact_1270_le__cube,axiom,
! [M4: nat] : ( ord_less_eq_nat @ M4 @ ( times_times_nat @ M4 @ ( times_times_nat @ M4 @ M4 ) ) ) ).
% le_cube
thf(fact_1271_mult__Suc,axiom,
! [M4: nat,N: nat] :
( ( times_times_nat @ ( suc @ M4 ) @ N )
= ( plus_plus_nat @ N @ ( times_times_nat @ M4 @ N ) ) ) ).
% mult_Suc
thf(fact_1272_Suc__mult__le__cancel1,axiom,
! [K2: nat,M4: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K2 ) @ M4 ) @ ( times_times_nat @ ( suc @ K2 ) @ N ) )
= ( ord_less_eq_nat @ M4 @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_1273_comm__monoid_Ounits__comm__group,axiom,
! [G: partia7680350978787392319xt_a_b] :
( ( comm_monoid_a_b @ G )
=> ( comm_g1850867397131805039t_unit @ ( units_of_a_b @ G ) ) ) ).
% comm_monoid.units_comm_group
thf(fact_1274_comm__group_Oaxioms_I1_J,axiom,
! [G: partia8223610829204095565t_unit] :
( ( comm_g1850867397131805039t_unit @ G )
=> ( comm_m7681468956318391052t_unit @ G ) ) ).
% comm_group.axioms(1)
thf(fact_1275_comm__group_Oaxioms_I1_J,axiom,
! [G: partia7680350978787392319xt_a_b] :
( ( comm_group_a_b @ G )
=> ( comm_monoid_a_b @ G ) ) ).
% comm_group.axioms(1)
thf(fact_1276_add__mult__distrib2,axiom,
! [K2: nat,M4: nat,N: nat] :
( ( times_times_nat @ K2 @ ( plus_plus_nat @ M4 @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K2 @ M4 ) @ ( times_times_nat @ K2 @ N ) ) ) ).
% add_mult_distrib2
thf(fact_1277_add__mult__distrib,axiom,
! [M4: nat,N: nat,K2: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M4 @ N ) @ K2 )
= ( plus_plus_nat @ ( times_times_nat @ M4 @ K2 ) @ ( times_times_nat @ N @ K2 ) ) ) ).
% add_mult_distrib
thf(fact_1278_repeat__mset_Orep__eq,axiom,
! [X: nat,Xa2: multiset_set_a] :
( ( count_set_a @ ( repeat_mset_set_a @ X @ Xa2 ) )
= ( ^ [A5: set_a] : ( times_times_nat @ X @ ( count_set_a @ Xa2 @ A5 ) ) ) ) ).
% repeat_mset.rep_eq
% Conjectures (1)
thf(conj_0,conjecture,
( ( ord_less_eq_nat @ m @ ( count_set_a @ ( finite8971978181520403909et_a_b @ g @ a2 ) @ ( eq_clo186784744570230005t_unit @ ( partia6383399360842404908t_unit @ ( partia7484183841585581558xt_a_b @ g ) @ ( eq_eq_5233546288009311946t_unit @ ( associated_a_b @ g ) @ ( gorder6800608520584131120t_unit @ ( factor_a_b @ g ) @ product_Unity ) ) ) @ ( insert_a @ d @ bot_bot_set_a ) ) ) )
= ( subseteq_mset_set_a @ ( replicate_mset_set_a @ m @ ( eq_clo186784744570230005t_unit @ ( partia6383399360842404908t_unit @ ( partia7484183841585581558xt_a_b @ g ) @ ( eq_eq_5233546288009311946t_unit @ ( associated_a_b @ g ) @ ( gorder6800608520584131120t_unit @ ( factor_a_b @ g ) @ product_Unity ) ) ) @ ( insert_a @ d @ bot_bot_set_a ) ) ) @ ( finite8971978181520403909et_a_b @ g @ a2 ) ) ) ).
%------------------------------------------------------------------------------