TPTP Problem File: SLH0308^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Finite_Fields/0008_Card_Irreducible_Polynomials_Aux/prob_00911_033598__18424170_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1553 ( 367 unt; 276 typ; 0 def)
% Number of atoms : 4570 (1437 equ; 0 cnn)
% Maximal formula atoms : 17 ( 3 avg)
% Number of connectives : 18761 ( 339 ~; 76 |; 302 &;15704 @)
% ( 0 <=>;2340 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 8 avg)
% Number of types : 25 ( 24 usr)
% Number of type conns : 979 ( 979 >; 0 *; 0 +; 0 <<)
% Number of symbols : 255 ( 252 usr; 14 con; 0-4 aty)
% Number of variables : 3517 ( 344 ^;3110 !; 63 ?;3517 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 13:23:48.143
%------------------------------------------------------------------------------
% Could-be-implicit typings (24)
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% Explicit typings (252)
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add_li7652885771158616974t_unit: partia2670972154091845814t_unit > list_a > list_a > list_a ).
thf(sy_c_Ring_Oring_Ozero_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
zero_l347298301471573063t_unit: partia2956882679547061052t_unit > list_list_a ).
thf(sy_c_Ring_Oring_Ozero_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
zero_l4142658623432671053t_unit: partia2670972154091845814t_unit > list_a ).
thf(sy_c_Ring_Oring__hom_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_h5031276006722532742t_unit: partia2956882679547061052t_unit > partia2670972154091845814t_unit > set_li3422455791611400469list_a ).
thf(sy_c_Ring_Oring__hom__cring_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_h453377649743177125t_unit: partia2956882679547061052t_unit > partia2670972154091845814t_unit > ( list_list_a > list_a ) > $o ).
thf(sy_c_Ring_Oring__hom__cring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_h8282015026914974507t_unit: partia2670972154091845814t_unit > partia2670972154091845814t_unit > ( list_a > list_a ) > $o ).
thf(sy_c_Ring_Oring__hom__cring_001tf__a_001tf__b_001tf__a_001tf__b,type,
ring_h661254511236296859_b_a_b: partia2175431115845679010xt_a_b > partia2175431115845679010xt_a_b > ( a > a ) > $o ).
thf(sy_c_Ring__Characteristic_Ochar_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_c500279861223467766t_unit: partia2670972154091845814t_unit > nat ).
thf(sy_c_Ring__Characteristic_Ochar_001t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J_001t__Product____Type__Ounit,type,
ring_c8395554250859618576t_unit: partia4960592913263135132t_unit > nat ).
thf(sy_c_Ring__Characteristic_Ochar_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_c6053888738502451990t_unit: partia7496981018696276118t_unit > nat ).
thf(sy_c_Ring__Characteristic_Ochar_001tf__a_001tf__b,type,
ring_char_a_b: partia2175431115845679010xt_a_b > nat ).
thf(sy_c_Ring__Characteristic_Ofinite__field_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_f3414212785825285533t_unit: partia2670972154091845814t_unit > $o ).
thf(sy_c_Ring__Characteristic_Ofinite__field_001t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J_001t__Product____Type__Ounit,type,
ring_f3227788147421470775t_unit: partia4960592913263135132t_unit > $o ).
thf(sy_c_Ring__Characteristic_Ofinite__field_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_f902895163656626621t_unit: partia7496981018696276118t_unit > $o ).
thf(sy_c_Ring__Characteristic_Ofinite__field_001tf__a_001tf__b,type,
ring_f9002592259703987221ld_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring__Divisibility_Oeuclidean__domain_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_e6434146001954145682t_unit: partia2956882679547061052t_unit > ( list_list_a > nat ) > $o ).
thf(sy_c_Ring__Divisibility_Oeuclidean__domain_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_e7478897652244013592t_unit: partia2670972154091845814t_unit > ( list_a > nat ) > $o ).
thf(sy_c_Ring__Divisibility_Oeuclidean__domain_001tf__a_001tf__b,type,
ring_e8745995371659049232in_a_b: partia2175431115845679010xt_a_b > ( a > nat ) > $o ).
thf(sy_c_Ring__Divisibility_Ofactorial__domain_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_f796907574329358751t_unit: partia2670972154091845814t_unit > $o ).
thf(sy_c_Ring__Divisibility_Ofactorial__domain_001tf__a_001tf__b,type,
ring_f5272581269873410839in_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring__Divisibility_Onoetherian__domain_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_n4705423059119889713t_unit: partia2670972154091845814t_unit > $o ).
thf(sy_c_Ring__Divisibility_Onoetherian__domain_001tf__a_001tf__b,type,
ring_n4045954140777738665in_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring__Divisibility_Onoetherian__ring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_n5188127996776581661t_unit: partia2670972154091845814t_unit > $o ).
thf(sy_c_Ring__Divisibility_Onoetherian__ring_001tf__a_001tf__b,type,
ring_n3639167112692572309ng_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring__Divisibility_Oprincipal__domain_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_p715737262848045090t_unit: partia2956882679547061052t_unit > $o ).
thf(sy_c_Ring__Divisibility_Oprincipal__domain_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_p8098905331641078952t_unit: partia2670972154091845814t_unit > $o ).
thf(sy_c_Ring__Divisibility_Oprincipal__domain_001tf__a_001tf__b,type,
ring_p8803135361686045600in_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring__Divisibility_Oring__irreducible_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_r360171070648044744t_unit: partia2956882679547061052t_unit > list_list_a > $o ).
thf(sy_c_Ring__Divisibility_Oring__irreducible_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_r932985474545269838t_unit: partia2670972154091845814t_unit > list_a > $o ).
thf(sy_c_Ring__Divisibility_Oring__prime_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_r5437400583859147359t_unit: partia2956882679547061052t_unit > list_list_a > $o ).
thf(sy_c_Ring__Divisibility_Oring__prime_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_r6430282645014804837t_unit: partia2670972154091845814t_unit > list_a > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Complex__Ocomplex,type,
dvd_dvd_complex: complex > complex > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
dvd_dvd_int: int > int > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
dvd_dvd_nat: nat > nat > $o ).
thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Nat__Onat,type,
zero_n2687167440665602831ol_nat: $o > nat ).
thf(sy_c_Set_OCollect_001_062_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__List__Olist_Itf__a_J_J,type,
collec3877118569217867188list_a: ( ( list_list_a > list_a ) > $o ) > set_li3422455791611400469list_a ).
thf(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
collect_complex: ( complex > $o ) > set_complex ).
thf(sy_c_Set_OCollect_001t__Int__Oint,type,
collect_int: ( int > $o ) > set_int ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
collect_list_list_a: ( list_list_a > $o ) > set_list_list_a ).
thf(sy_c_Set_OCollect_001t__List__Olist_Itf__a_J,type,
collect_list_a: ( list_a > $o ) > set_list_a ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
collect_set_list_a: ( set_list_a > $o ) > set_set_list_a ).
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_Oimage_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
image_5446147394702115205list_a: ( list_list_a > list_list_a ) > set_list_list_a > set_list_list_a ).
thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
image_8260866953997875467list_a: ( list_a > list_list_a ) > set_list_a > set_list_list_a ).
thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
image_list_a_list_a: ( list_a > list_a ) > set_list_a > set_list_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
image_nat_set_list_a: ( nat > set_list_a ) > set_nat > set_set_list_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__List__Olist_Itf__a_J,type,
image_a_list_a: ( a > list_a ) > set_a > set_list_a ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(sy_c_Subrings_Osubdomain_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
subdom7821232466298058046t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_Subrings_Osubfield_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
subfie1779122896746047282t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_Subrings_Osubfield_001tf__a_001tf__b,type,
subfield_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Subrings_Osubring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
subrin6918843898125473962t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_UnivPoly_Obound_001t__List__Olist_Itf__a_J,type,
bound_list_a: list_a > nat > ( nat > list_a ) > $o ).
thf(sy_c_member_001_062_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__List__Olist_Itf__a_J_J,type,
member7168557129179038582list_a: ( list_list_a > list_a ) > set_li3422455791611400469list_a > $o ).
thf(sy_c_member_001t__Complex__Ocomplex,type,
member_complex: complex > set_complex > $o ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
member_list_list_a: list_list_a > set_list_list_a > $o ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
member_set_list_a: set_list_a > set_set_list_a > $o ).
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_R,type,
r: partia2175431115845679010xt_a_b ).
thf(sy_v_n,type,
n: nat ).
% Relevant facts (1269)
thf(fact_0_finite__field__axioms,axiom,
ring_f9002592259703987221ld_a_b @ r ).
% finite_field_axioms
thf(fact_1_factorial__domain__axioms,axiom,
ring_f5272581269873410839in_a_b @ r ).
% factorial_domain_axioms
thf(fact_2_noetherian__domain__axioms,axiom,
ring_n4045954140777738665in_a_b @ r ).
% noetherian_domain_axioms
thf(fact_3_principal__domain__axioms,axiom,
ring_p8803135361686045600in_a_b @ r ).
% principal_domain_axioms
thf(fact_4__092_060open_062order_AR_A_094_An_A_061_Adegree_A_Igauss__poly_AR_A_Iorder_AR_A_094_An_J_J_092_060close_062,axiom,
( ( power_power_nat @ ( order_a_ring_ext_a_b @ r ) @ n )
= ( minus_minus_nat @ ( size_size_list_a @ ( card_I2373409586816755191ly_a_b @ r @ ( power_power_nat @ ( order_a_ring_ext_a_b @ r ) @ n ) ) ) @ one_one_nat ) ) ).
% \<open>order R ^ n = degree (gauss_poly R (order R ^ n))\<close>
thf(fact_5_noetherian__ring__axioms,axiom,
ring_n3639167112692572309ng_a_b @ r ).
% noetherian_ring_axioms
thf(fact_6__092_060open_062_I_092_060Sum_062d_A_124_Ad_Advd_An_O_A_092_060Sum_062___092_060in_062_123f_O_Am__i__p_AR_Af_A_092_060and_062_Adegree_Af_A_061_Ad_125_O_Ad_J_A_061_A_I_092_060Sum_062d_A_124_Ad_Advd_An_O_Ad_A_K_Acard_A_123f_O_Am__i__p_AR_Af_A_092_060and_062_Adegree_Af_A_061_Ad_125_J_092_060close_062,axiom,
( ( groups3542108847815614940at_nat
@ ^ [D: nat] :
( groups5521247699297860762_a_nat
@ ^ [Uu: list_a] : D
@ ( collect_list_a
@ ^ [F: list_a] :
( ( monic_4919232885364369782ly_a_b @ r @ F )
& ( ( minus_minus_nat @ ( size_size_list_a @ F ) @ one_one_nat )
= D ) ) ) )
@ ( collect_nat
@ ^ [D: nat] : ( dvd_dvd_nat @ D @ n ) ) )
= ( groups3542108847815614940at_nat
@ ^ [D: nat] :
( times_times_nat @ D
@ ( finite_card_list_a
@ ( collect_list_a
@ ^ [F: list_a] :
( ( monic_4919232885364369782ly_a_b @ r @ F )
& ( ( minus_minus_nat @ ( size_size_list_a @ F ) @ one_one_nat )
= D ) ) ) ) )
@ ( collect_nat
@ ^ [D: nat] : ( dvd_dvd_nat @ D @ n ) ) ) ) ).
% \<open>(\<Sum>d | d dvd n. \<Sum>_\<in>{f. m_i_p R f \<and> degree f = d}. d) = (\<Sum>d | d dvd n. d * card {f. m_i_p R f \<and> degree f = d})\<close>
thf(fact_7_unit__prod,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ).
% unit_prod
thf(fact_8_unit__prod,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ).
% unit_prod
thf(fact_9_c,axiom,
( finite_finite_list_a
@ ( collect_list_a
@ ^ [F: list_a] :
( ( monic_4919232885364369782ly_a_b @ r @ F )
& ( dvd_dvd_nat @ ( minus_minus_nat @ ( size_size_list_a @ F ) @ one_one_nat ) @ n ) ) ) ) ).
% c
thf(fact_10__092_060open_062_I_092_060Sum_062d_A_124_Ad_Advd_An_O_Asum_Adegree_A_123f_O_Am__i__p_AR_Af_A_092_060and_062_Adegree_Af_A_061_Ad_125_J_A_061_A_I_092_060Sum_062d_A_124_Ad_Advd_An_O_A_092_060Sum_062___092_060in_062_123f_O_Am__i__p_AR_Af_A_092_060and_062_Adegree_Af_A_061_Ad_125_O_Ad_J_092_060close_062,axiom,
( ( groups3542108847815614940at_nat
@ ^ [D: nat] :
( groups5521247699297860762_a_nat
@ ^ [P: list_a] : ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
@ ( collect_list_a
@ ^ [F: list_a] :
( ( monic_4919232885364369782ly_a_b @ r @ F )
& ( ( minus_minus_nat @ ( size_size_list_a @ F ) @ one_one_nat )
= D ) ) ) )
@ ( collect_nat
@ ^ [D: nat] : ( dvd_dvd_nat @ D @ n ) ) )
= ( groups3542108847815614940at_nat
@ ^ [D: nat] :
( groups5521247699297860762_a_nat
@ ^ [Uu: list_a] : D
@ ( collect_list_a
@ ^ [F: list_a] :
( ( monic_4919232885364369782ly_a_b @ r @ F )
& ( ( minus_minus_nat @ ( size_size_list_a @ F ) @ one_one_nat )
= D ) ) ) )
@ ( collect_nat
@ ^ [D: nat] : ( dvd_dvd_nat @ D @ n ) ) ) ) ).
% \<open>(\<Sum>d | d dvd n. sum degree {f. m_i_p R f \<and> degree f = d}) = (\<Sum>d | d dvd n. \<Sum>_\<in>{f. m_i_p R f \<and> degree f = d}. d)\<close>
thf(fact_11_d,axiom,
ord_less_nat @ one_one_nat @ ( power_power_nat @ ( order_a_ring_ext_a_b @ r ) @ n ) ).
% d
thf(fact_12_nat__dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd_nat @ M @ one_one_nat )
= ( M = one_one_nat ) ) ).
% nat_dvd_1_iff_1
thf(fact_13_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_14_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_15_power__one__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_16_power__one__right,axiom,
! [A: int] :
( ( power_power_int @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_17_power__one__right,axiom,
! [A: complex] :
( ( power_power_complex @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_18_finite__field__min__order,axiom,
ord_less_nat @ one_one_nat @ ( order_a_ring_ext_a_b @ r ) ).
% finite_field_min_order
thf(fact_19_gauss__poly__degree,axiom,
! [N: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( minus_minus_nat @ ( size_size_list_a @ ( card_I2373409586816755191ly_a_b @ r @ N ) ) @ one_one_nat )
= N ) ) ).
% gauss_poly_degree
thf(fact_20_b,axiom,
! [K: nat] :
( finite_finite_list_a
@ ( collect_list_a
@ ^ [F: list_a] :
( ( monic_4919232885364369782ly_a_b @ r @ F )
& ( ( minus_minus_nat @ ( size_size_list_a @ F ) @ one_one_nat )
= K ) ) ) ) ).
% b
thf(fact_21_assms,axiom,
ord_less_nat @ zero_zero_nat @ n ).
% assms
thf(fact_22_power__one,axiom,
! [N: nat] :
( ( power_power_nat @ one_one_nat @ N )
= one_one_nat ) ).
% power_one
thf(fact_23_power__one,axiom,
! [N: nat] :
( ( power_power_int @ one_one_int @ N )
= one_one_int ) ).
% power_one
thf(fact_24_power__one,axiom,
! [N: nat] :
( ( power_power_complex @ one_one_complex @ N )
= one_one_complex ) ).
% power_one
thf(fact_25_power__inject__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ( power_power_nat @ A @ M )
= ( power_power_nat @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_26_power__inject__exp,axiom,
! [A: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ( power_power_int @ A @ M )
= ( power_power_int @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_27_power__strict__increasing__iff,axiom,
! [B: nat,X: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_28_power__strict__increasing__iff,axiom,
! [B: int,X: nat,Y: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_29_linorder__neqE__linordered__idom,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_30_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_31_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_32_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_33_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_34_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_35_nat__less__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( P2 @ M2 ) )
=> ( P2 @ N2 ) )
=> ( P2 @ N ) ) ).
% nat_less_induct
thf(fact_36_infinite__descent,axiom,
! [P2: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P2 @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P2 @ M2 ) ) )
=> ( P2 @ N ) ) ).
% infinite_descent
thf(fact_37_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_38_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_39_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_40_power__strict__increasing,axiom,
! [N: nat,N3: nat,A: nat] :
( ( ord_less_nat @ N @ N3 )
=> ( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N3 ) ) ) ) ).
% power_strict_increasing
thf(fact_41_power__strict__increasing,axiom,
! [N: nat,N3: nat,A: int] :
( ( ord_less_nat @ N @ N3 )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N3 ) ) ) ) ).
% power_strict_increasing
thf(fact_42_power__less__imp__less__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_43_power__less__imp__less__exp,axiom,
! [A: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_44_less__1__mult,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ M )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_45_less__1__mult,axiom,
! [M: int,N: int] :
( ( ord_less_int @ one_one_int @ M )
=> ( ( ord_less_int @ one_one_int @ N )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_46_dvd__minus__self,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) )
= ( ( ord_less_nat @ N @ M )
| ( dvd_dvd_nat @ M @ N ) ) ) ).
% dvd_minus_self
thf(fact_47_power__less__power__Suc,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_48_power__less__power__Suc,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_49_power__gt1__lemma,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_50_power__gt1__lemma,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_51_size__neq__size__imp__neq,axiom,
! [X: list_a,Y: list_a] :
( ( ( size_size_list_a @ X )
!= ( size_size_list_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_52_size__neq__size__imp__neq,axiom,
! [X: multiset_a,Y: multiset_a] :
( ( ( size_size_multiset_a @ X )
!= ( size_size_multiset_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_53_size__neq__size__imp__neq,axiom,
! [X: multiset_list_a,Y: multiset_list_a] :
( ( ( size_s2335926164413107382list_a @ X )
!= ( size_s2335926164413107382list_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_54_size__neq__size__imp__neq,axiom,
! [X: list_list_a,Y: list_list_a] :
( ( ( size_s349497388124573686list_a @ X )
!= ( size_s349497388124573686list_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_55_dvd__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ B @ C )
=> ( dvd_dvd_nat @ A @ C ) ) ) ).
% dvd_trans
thf(fact_56_dvd__trans,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ B @ C )
=> ( dvd_dvd_int @ A @ C ) ) ) ).
% dvd_trans
thf(fact_57_dvd__refl,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).
% dvd_refl
thf(fact_58_dvd__refl,axiom,
! [A: int] : ( dvd_dvd_int @ A @ A ) ).
% dvd_refl
thf(fact_59_mem__Collect__eq,axiom,
! [A: set_list_a,P2: set_list_a > $o] :
( ( member_set_list_a @ A @ ( collect_set_list_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_60_mem__Collect__eq,axiom,
! [A: set_a,P2: set_a > $o] :
( ( member_set_a @ A @ ( collect_set_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_61_mem__Collect__eq,axiom,
! [A: list_list_a > list_a,P2: ( list_list_a > list_a ) > $o] :
( ( member7168557129179038582list_a @ A @ ( collec3877118569217867188list_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_62_mem__Collect__eq,axiom,
! [A: list_list_a,P2: list_list_a > $o] :
( ( member_list_list_a @ A @ ( collect_list_list_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_63_mem__Collect__eq,axiom,
! [A: list_a,P2: list_a > $o] :
( ( member_list_a @ A @ ( collect_list_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_64_mem__Collect__eq,axiom,
! [A: nat,P2: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_65_mem__Collect__eq,axiom,
! [A: int,P2: int > $o] :
( ( member_int @ A @ ( collect_int @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_66_mem__Collect__eq,axiom,
! [A: a,P2: a > $o] :
( ( member_a @ A @ ( collect_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_67_Collect__mem__eq,axiom,
! [A2: set_set_list_a] :
( ( collect_set_list_a
@ ^ [X2: set_list_a] : ( member_set_list_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_68_Collect__mem__eq,axiom,
! [A2: set_set_a] :
( ( collect_set_a
@ ^ [X2: set_a] : ( member_set_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_69_Collect__mem__eq,axiom,
! [A2: set_li3422455791611400469list_a] :
( ( collec3877118569217867188list_a
@ ^ [X2: list_list_a > list_a] : ( member7168557129179038582list_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_70_Collect__mem__eq,axiom,
! [A2: set_list_list_a] :
( ( collect_list_list_a
@ ^ [X2: list_list_a] : ( member_list_list_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_71_Collect__mem__eq,axiom,
! [A2: set_list_a] :
( ( collect_list_a
@ ^ [X2: list_a] : ( member_list_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_72_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_73_Collect__mem__eq,axiom,
! [A2: set_int] :
( ( collect_int
@ ^ [X2: int] : ( member_int @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_74_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_75_Collect__cong,axiom,
! [P2: list_list_a > $o,Q: list_list_a > $o] :
( ! [X3: list_list_a] :
( ( P2 @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_list_list_a @ P2 )
= ( collect_list_list_a @ Q ) ) ) ).
% Collect_cong
thf(fact_76_Collect__cong,axiom,
! [P2: list_a > $o,Q: list_a > $o] :
( ! [X3: list_a] :
( ( P2 @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_list_a @ P2 )
= ( collect_list_a @ Q ) ) ) ).
% Collect_cong
thf(fact_77_Collect__cong,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P2 @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_nat @ P2 )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_78_Collect__cong,axiom,
! [P2: int > $o,Q: int > $o] :
( ! [X3: int] :
( ( P2 @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_int @ P2 )
= ( collect_int @ Q ) ) ) ).
% Collect_cong
thf(fact_79_Collect__cong,axiom,
! [P2: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( P2 @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_a @ P2 )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_80_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_81_dvd__antisym,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ N )
=> ( ( dvd_dvd_nat @ N @ M )
=> ( M = N ) ) ) ).
% dvd_antisym
thf(fact_82_right__diff__distrib_H,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_83_right__diff__distrib_H,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_84_left__diff__distrib_H,axiom,
! [B: nat,C: nat,A: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
= ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_85_left__diff__distrib_H,axiom,
! [B: int,C: int,A: int] :
( ( times_times_int @ ( minus_minus_int @ B @ C ) @ A )
= ( minus_minus_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_86_right__diff__distrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% right_diff_distrib
thf(fact_87_left__diff__distrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_88_power__commuting__commutes,axiom,
! [X: complex,Y: complex,N: nat] :
( ( ( times_times_complex @ X @ Y )
= ( times_times_complex @ Y @ X ) )
=> ( ( times_times_complex @ ( power_power_complex @ X @ N ) @ Y )
= ( times_times_complex @ Y @ ( power_power_complex @ X @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_89_power__commuting__commutes,axiom,
! [X: nat,Y: nat,N: nat] :
( ( ( times_times_nat @ X @ Y )
= ( times_times_nat @ Y @ X ) )
=> ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ Y )
= ( times_times_nat @ Y @ ( power_power_nat @ X @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_90_power__commuting__commutes,axiom,
! [X: int,Y: int,N: nat] :
( ( ( times_times_int @ X @ Y )
= ( times_times_int @ Y @ X ) )
=> ( ( times_times_int @ ( power_power_int @ X @ N ) @ Y )
= ( times_times_int @ Y @ ( power_power_int @ X @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_91_power__mult__distrib,axiom,
! [A: complex,B: complex,N: nat] :
( ( power_power_complex @ ( times_times_complex @ A @ B ) @ N )
= ( times_times_complex @ ( power_power_complex @ A @ N ) @ ( power_power_complex @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_92_power__mult__distrib,axiom,
! [A: nat,B: nat,N: nat] :
( ( power_power_nat @ ( times_times_nat @ A @ B ) @ N )
= ( times_times_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_93_power__mult__distrib,axiom,
! [A: int,B: int,N: nat] :
( ( power_power_int @ ( times_times_int @ A @ B ) @ N )
= ( times_times_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_94_power__commutes,axiom,
! [A: complex,N: nat] :
( ( times_times_complex @ ( power_power_complex @ A @ N ) @ A )
= ( times_times_complex @ A @ ( power_power_complex @ A @ N ) ) ) ).
% power_commutes
thf(fact_95_power__commutes,axiom,
! [A: nat,N: nat] :
( ( times_times_nat @ ( power_power_nat @ A @ N ) @ A )
= ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% power_commutes
thf(fact_96_power__commutes,axiom,
! [A: int,N: nat] :
( ( times_times_int @ ( power_power_int @ A @ N ) @ A )
= ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% power_commutes
thf(fact_97_dvd__unit__imp__unit,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ A @ one_one_nat ) ) ) ).
% dvd_unit_imp_unit
thf(fact_98_dvd__unit__imp__unit,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ A @ one_one_int ) ) ) ).
% dvd_unit_imp_unit
thf(fact_99_unit__imp__dvd,axiom,
! [B: nat,A: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ B @ A ) ) ).
% unit_imp_dvd
thf(fact_100_unit__imp__dvd,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ B @ A ) ) ).
% unit_imp_dvd
thf(fact_101_one__dvd,axiom,
! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).
% one_dvd
thf(fact_102_one__dvd,axiom,
! [A: int] : ( dvd_dvd_int @ one_one_int @ A ) ).
% one_dvd
thf(fact_103_one__dvd,axiom,
! [A: complex] : ( dvd_dvd_complex @ one_one_complex @ A ) ).
% one_dvd
thf(fact_104_dvd__triv__right,axiom,
! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ A ) ) ).
% dvd_triv_right
thf(fact_105_dvd__triv__right,axiom,
! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ B @ A ) ) ).
% dvd_triv_right
thf(fact_106_dvd__mult__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
=> ( dvd_dvd_nat @ B @ C ) ) ).
% dvd_mult_right
thf(fact_107_dvd__mult__right,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
=> ( dvd_dvd_int @ B @ C ) ) ).
% dvd_mult_right
thf(fact_108_mult__dvd__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ C @ D2 )
=> ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ).
% mult_dvd_mono
thf(fact_109_mult__dvd__mono,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ C @ D2 )
=> ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ).
% mult_dvd_mono
thf(fact_110_dvd__triv__left,axiom,
! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ A @ B ) ) ).
% dvd_triv_left
thf(fact_111_dvd__triv__left,axiom,
! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ A @ B ) ) ).
% dvd_triv_left
thf(fact_112_dvd__mult__left,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
=> ( dvd_dvd_nat @ A @ C ) ) ).
% dvd_mult_left
thf(fact_113_dvd__mult__left,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
=> ( dvd_dvd_int @ A @ C ) ) ).
% dvd_mult_left
thf(fact_114_dvd__mult2,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% dvd_mult2
thf(fact_115_dvd__mult2,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ B )
=> ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% dvd_mult2
thf(fact_116_dvd__mult,axiom,
! [A: nat,C: nat,B: nat] :
( ( dvd_dvd_nat @ A @ C )
=> ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% dvd_mult
thf(fact_117_dvd__mult,axiom,
! [A: int,C: int,B: int] :
( ( dvd_dvd_int @ A @ C )
=> ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% dvd_mult
thf(fact_118_dvd__def,axiom,
( dvd_dvd_nat
= ( ^ [B2: nat,A3: nat] :
? [K2: nat] :
( A3
= ( times_times_nat @ B2 @ K2 ) ) ) ) ).
% dvd_def
thf(fact_119_dvd__def,axiom,
( dvd_dvd_int
= ( ^ [B2: int,A3: int] :
? [K2: int] :
( A3
= ( times_times_int @ B2 @ K2 ) ) ) ) ).
% dvd_def
thf(fact_120_dvdI,axiom,
! [A: nat,B: nat,K: nat] :
( ( A
= ( times_times_nat @ B @ K ) )
=> ( dvd_dvd_nat @ B @ A ) ) ).
% dvdI
thf(fact_121_dvdI,axiom,
! [A: int,B: int,K: int] :
( ( A
= ( times_times_int @ B @ K ) )
=> ( dvd_dvd_int @ B @ A ) ) ).
% dvdI
thf(fact_122_dvdE,axiom,
! [B: nat,A: nat] :
( ( dvd_dvd_nat @ B @ A )
=> ~ ! [K3: nat] :
( A
!= ( times_times_nat @ B @ K3 ) ) ) ).
% dvdE
thf(fact_123_dvdE,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ A )
=> ~ ! [K3: int] :
( A
!= ( times_times_int @ B @ K3 ) ) ) ).
% dvdE
thf(fact_124_dvd__diff,axiom,
! [X: int,Y: int,Z: int] :
( ( dvd_dvd_int @ X @ Y )
=> ( ( dvd_dvd_int @ X @ Z )
=> ( dvd_dvd_int @ X @ ( minus_minus_int @ Y @ Z ) ) ) ) ).
% dvd_diff
thf(fact_125_dvd__power__same,axiom,
! [X: nat,Y: nat,N: nat] :
( ( dvd_dvd_nat @ X @ Y )
=> ( dvd_dvd_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ N ) ) ) ).
% dvd_power_same
thf(fact_126_dvd__power__same,axiom,
! [X: int,Y: int,N: nat] :
( ( dvd_dvd_int @ X @ Y )
=> ( dvd_dvd_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ N ) ) ) ).
% dvd_power_same
thf(fact_127_dvd__power__same,axiom,
! [X: complex,Y: complex,N: nat] :
( ( dvd_dvd_complex @ X @ Y )
=> ( dvd_dvd_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y @ N ) ) ) ).
% dvd_power_same
thf(fact_128_power__mult,axiom,
! [A: nat,M: nat,N: nat] :
( ( power_power_nat @ A @ ( times_times_nat @ M @ N ) )
= ( power_power_nat @ ( power_power_nat @ A @ M ) @ N ) ) ).
% power_mult
thf(fact_129_power__mult,axiom,
! [A: int,M: nat,N: nat] :
( ( power_power_int @ A @ ( times_times_nat @ M @ N ) )
= ( power_power_int @ ( power_power_int @ A @ M ) @ N ) ) ).
% power_mult
thf(fact_130_power__mult,axiom,
! [A: complex,M: nat,N: nat] :
( ( power_power_complex @ A @ ( times_times_nat @ M @ N ) )
= ( power_power_complex @ ( power_power_complex @ A @ M ) @ N ) ) ).
% power_mult
thf(fact_131_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_132_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_133_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_134_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_135_dvd__diff__nat,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K @ M )
=> ( ( dvd_dvd_nat @ K @ N )
=> ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% dvd_diff_nat
thf(fact_136_lambda__one,axiom,
( ( ^ [X2: complex] : X2 )
= ( times_times_complex @ one_one_complex ) ) ).
% lambda_one
thf(fact_137_lambda__one,axiom,
( ( ^ [X2: nat] : X2 )
= ( times_times_nat @ one_one_nat ) ) ).
% lambda_one
thf(fact_138_lambda__one,axiom,
( ( ^ [X2: int] : X2 )
= ( times_times_int @ one_one_int ) ) ).
% lambda_one
thf(fact_139_left__right__inverse__power,axiom,
! [X: complex,Y: complex,N: nat] :
( ( ( times_times_complex @ X @ Y )
= one_one_complex )
=> ( ( times_times_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y @ N ) )
= one_one_complex ) ) ).
% left_right_inverse_power
thf(fact_140_left__right__inverse__power,axiom,
! [X: nat,Y: nat,N: nat] :
( ( ( times_times_nat @ X @ Y )
= one_one_nat )
=> ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ N ) )
= one_one_nat ) ) ).
% left_right_inverse_power
thf(fact_141_left__right__inverse__power,axiom,
! [X: int,Y: int,N: nat] :
( ( ( times_times_int @ X @ Y )
= one_one_int )
=> ( ( times_times_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ N ) )
= one_one_int ) ) ).
% left_right_inverse_power
thf(fact_142_unit__mult__right__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( ( times_times_nat @ B @ A )
= ( times_times_nat @ C @ A ) )
= ( B = C ) ) ) ).
% unit_mult_right_cancel
thf(fact_143_unit__mult__right__cancel,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( ( times_times_int @ B @ A )
= ( times_times_int @ C @ A ) )
= ( B = C ) ) ) ).
% unit_mult_right_cancel
thf(fact_144_unit__mult__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( ( times_times_nat @ A @ B )
= ( times_times_nat @ A @ C ) )
= ( B = C ) ) ) ).
% unit_mult_left_cancel
thf(fact_145_unit__mult__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( ( times_times_int @ A @ B )
= ( times_times_int @ A @ C ) )
= ( B = C ) ) ) ).
% unit_mult_left_cancel
thf(fact_146_mult__unit__dvd__iff_H,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
= ( dvd_dvd_nat @ B @ C ) ) ) ).
% mult_unit_dvd_iff'
thf(fact_147_mult__unit__dvd__iff_H,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
= ( dvd_dvd_int @ B @ C ) ) ) ).
% mult_unit_dvd_iff'
thf(fact_148_dvd__mult__unit__iff_H,axiom,
! [B: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% dvd_mult_unit_iff'
thf(fact_149_dvd__mult__unit__iff_H,axiom,
! [B: int,A: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% dvd_mult_unit_iff'
thf(fact_150_mult__unit__dvd__iff,axiom,
! [B: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% mult_unit_dvd_iff
thf(fact_151_mult__unit__dvd__iff,axiom,
! [B: int,A: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% mult_unit_dvd_iff
thf(fact_152_dvd__mult__unit__iff,axiom,
! [B: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% dvd_mult_unit_iff
thf(fact_153_dvd__mult__unit__iff,axiom,
! [B: int,A: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% dvd_mult_unit_iff
thf(fact_154_is__unit__mult__iff,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat )
= ( ( dvd_dvd_nat @ A @ one_one_nat )
& ( dvd_dvd_nat @ B @ one_one_nat ) ) ) ).
% is_unit_mult_iff
thf(fact_155_is__unit__mult__iff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int )
= ( ( dvd_dvd_int @ A @ one_one_int )
& ( dvd_dvd_int @ B @ one_one_int ) ) ) ).
% is_unit_mult_iff
thf(fact_156_gauss__poly__splitted,axiom,
polyno8329700637149614481ed_a_b @ r @ ( card_I2373409586816755191ly_a_b @ r @ ( order_a_ring_ext_a_b @ r ) ) ).
% gauss_poly_splitted
thf(fact_157__092_060open_062sum_H_Adegree_A_123f_O_Am__i__p_AR_Af_A_092_060and_062_Adegree_Af_Advd_An_125_A_061_Asum_Adegree_A_123f_O_Am__i__p_AR_Af_A_092_060and_062_Adegree_Af_Advd_An_125_092_060close_062,axiom,
( ( groups1083796956622547633_a_nat
@ ^ [D: list_a] : ( minus_minus_nat @ ( size_size_list_a @ D ) @ one_one_nat )
@ ( collect_list_a
@ ^ [F: list_a] :
( ( monic_4919232885364369782ly_a_b @ r @ F )
& ( dvd_dvd_nat @ ( minus_minus_nat @ ( size_size_list_a @ F ) @ one_one_nat ) @ n ) ) ) )
= ( groups5521247699297860762_a_nat
@ ^ [D: list_a] : ( minus_minus_nat @ ( size_size_list_a @ D ) @ one_one_nat )
@ ( collect_list_a
@ ^ [F: list_a] :
( ( monic_4919232885364369782ly_a_b @ r @ F )
& ( dvd_dvd_nat @ ( minus_minus_nat @ ( size_size_list_a @ F ) @ one_one_nat ) @ n ) ) ) ) ) ).
% \<open>sum' degree {f. m_i_p R f \<and> degree f dvd n} = sum degree {f. m_i_p R f \<and> degree f dvd n}\<close>
thf(fact_158_mult__1,axiom,
! [A: complex] :
( ( times_times_complex @ one_one_complex @ A )
= A ) ).
% mult_1
thf(fact_159_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_160_mult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% mult_1
thf(fact_161_mult_Oright__neutral,axiom,
! [A: complex] :
( ( times_times_complex @ A @ one_one_complex )
= A ) ).
% mult.right_neutral
thf(fact_162_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_163_mult_Oright__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.right_neutral
thf(fact_164_monic__poly__min__degree,axiom,
! [F2: list_a] :
( ( monic_4919232885364369782ly_a_b @ r @ F2 )
=> ( ord_less_eq_nat @ one_one_nat @ ( minus_minus_nat @ ( size_size_list_a @ F2 ) @ one_one_nat ) ) ) ).
% monic_poly_min_degree
thf(fact_165_sum__multicount,axiom,
! [S2: set_int,T2: set_nat,R: int > nat > $o,K: nat] :
( ( finite_finite_int @ S2 )
=> ( ( finite_finite_nat @ T2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ T2 )
=> ( ( finite_card_int
@ ( collect_int
@ ^ [I2: int] :
( ( member_int @ I2 @ S2 )
& ( R @ I2 @ X3 ) ) ) )
= K ) )
=> ( ( groups4541462559716669496nt_nat
@ ^ [I2: int] :
( finite_card_nat
@ ( collect_nat
@ ^ [J2: nat] :
( ( member_nat @ J2 @ T2 )
& ( R @ I2 @ J2 ) ) ) )
@ S2 )
= ( times_times_nat @ K @ ( finite_card_nat @ T2 ) ) ) ) ) ) ).
% sum_multicount
thf(fact_166_sum__multicount,axiom,
! [S2: set_int,T2: set_int,R: int > int > $o,K: nat] :
( ( finite_finite_int @ S2 )
=> ( ( finite_finite_int @ T2 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ T2 )
=> ( ( finite_card_int
@ ( collect_int
@ ^ [I2: int] :
( ( member_int @ I2 @ S2 )
& ( R @ I2 @ X3 ) ) ) )
= K ) )
=> ( ( groups4541462559716669496nt_nat
@ ^ [I2: int] :
( finite_card_int
@ ( collect_int
@ ^ [J2: int] :
( ( member_int @ J2 @ T2 )
& ( R @ I2 @ J2 ) ) ) )
@ S2 )
= ( times_times_nat @ K @ ( finite_card_int @ T2 ) ) ) ) ) ) ).
% sum_multicount
thf(fact_167_sum__multicount,axiom,
! [S2: set_int,T2: set_a,R: int > a > $o,K: nat] :
( ( finite_finite_int @ S2 )
=> ( ( finite_finite_a @ T2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ T2 )
=> ( ( finite_card_int
@ ( collect_int
@ ^ [I2: int] :
( ( member_int @ I2 @ S2 )
& ( R @ I2 @ X3 ) ) ) )
= K ) )
=> ( ( groups4541462559716669496nt_nat
@ ^ [I2: int] :
( finite_card_a
@ ( collect_a
@ ^ [J2: a] :
( ( member_a @ J2 @ T2 )
& ( R @ I2 @ J2 ) ) ) )
@ S2 )
= ( times_times_nat @ K @ ( finite_card_a @ T2 ) ) ) ) ) ) ).
% sum_multicount
thf(fact_168_sum__multicount,axiom,
! [S2: set_a,T2: set_nat,R: a > nat > $o,K: nat] :
( ( finite_finite_a @ S2 )
=> ( ( finite_finite_nat @ T2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ T2 )
=> ( ( finite_card_a
@ ( collect_a
@ ^ [I2: a] :
( ( member_a @ I2 @ S2 )
& ( R @ I2 @ X3 ) ) ) )
= K ) )
=> ( ( groups6334556678337121940_a_nat
@ ^ [I2: a] :
( finite_card_nat
@ ( collect_nat
@ ^ [J2: nat] :
( ( member_nat @ J2 @ T2 )
& ( R @ I2 @ J2 ) ) ) )
@ S2 )
= ( times_times_nat @ K @ ( finite_card_nat @ T2 ) ) ) ) ) ) ).
% sum_multicount
thf(fact_169_sum__multicount,axiom,
! [S2: set_a,T2: set_int,R: a > int > $o,K: nat] :
( ( finite_finite_a @ S2 )
=> ( ( finite_finite_int @ T2 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ T2 )
=> ( ( finite_card_a
@ ( collect_a
@ ^ [I2: a] :
( ( member_a @ I2 @ S2 )
& ( R @ I2 @ X3 ) ) ) )
= K ) )
=> ( ( groups6334556678337121940_a_nat
@ ^ [I2: a] :
( finite_card_int
@ ( collect_int
@ ^ [J2: int] :
( ( member_int @ J2 @ T2 )
& ( R @ I2 @ J2 ) ) ) )
@ S2 )
= ( times_times_nat @ K @ ( finite_card_int @ T2 ) ) ) ) ) ) ).
% sum_multicount
thf(fact_170_sum__multicount,axiom,
! [S2: set_a,T2: set_a,R: a > a > $o,K: nat] :
( ( finite_finite_a @ S2 )
=> ( ( finite_finite_a @ T2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ T2 )
=> ( ( finite_card_a
@ ( collect_a
@ ^ [I2: a] :
( ( member_a @ I2 @ S2 )
& ( R @ I2 @ X3 ) ) ) )
= K ) )
=> ( ( groups6334556678337121940_a_nat
@ ^ [I2: a] :
( finite_card_a
@ ( collect_a
@ ^ [J2: a] :
( ( member_a @ J2 @ T2 )
& ( R @ I2 @ J2 ) ) ) )
@ S2 )
= ( times_times_nat @ K @ ( finite_card_a @ T2 ) ) ) ) ) ) ).
% sum_multicount
thf(fact_171_sum__multicount,axiom,
! [S2: set_nat,T2: set_nat,R: nat > nat > $o,K: nat] :
( ( finite_finite_nat @ S2 )
=> ( ( finite_finite_nat @ T2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ T2 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [I2: nat] :
( ( member_nat @ I2 @ S2 )
& ( R @ I2 @ X3 ) ) ) )
= K ) )
=> ( ( groups3542108847815614940at_nat
@ ^ [I2: nat] :
( finite_card_nat
@ ( collect_nat
@ ^ [J2: nat] :
( ( member_nat @ J2 @ T2 )
& ( R @ I2 @ J2 ) ) ) )
@ S2 )
= ( times_times_nat @ K @ ( finite_card_nat @ T2 ) ) ) ) ) ) ).
% sum_multicount
thf(fact_172_sum__multicount,axiom,
! [S2: set_nat,T2: set_int,R: nat > int > $o,K: nat] :
( ( finite_finite_nat @ S2 )
=> ( ( finite_finite_int @ T2 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ T2 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [I2: nat] :
( ( member_nat @ I2 @ S2 )
& ( R @ I2 @ X3 ) ) ) )
= K ) )
=> ( ( groups3542108847815614940at_nat
@ ^ [I2: nat] :
( finite_card_int
@ ( collect_int
@ ^ [J2: int] :
( ( member_int @ J2 @ T2 )
& ( R @ I2 @ J2 ) ) ) )
@ S2 )
= ( times_times_nat @ K @ ( finite_card_int @ T2 ) ) ) ) ) ) ).
% sum_multicount
thf(fact_173_sum__multicount,axiom,
! [S2: set_nat,T2: set_a,R: nat > a > $o,K: nat] :
( ( finite_finite_nat @ S2 )
=> ( ( finite_finite_a @ T2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ T2 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [I2: nat] :
( ( member_nat @ I2 @ S2 )
& ( R @ I2 @ X3 ) ) ) )
= K ) )
=> ( ( groups3542108847815614940at_nat
@ ^ [I2: nat] :
( finite_card_a
@ ( collect_a
@ ^ [J2: a] :
( ( member_a @ J2 @ T2 )
& ( R @ I2 @ J2 ) ) ) )
@ S2 )
= ( times_times_nat @ K @ ( finite_card_a @ T2 ) ) ) ) ) ) ).
% sum_multicount
thf(fact_174_sum__multicount,axiom,
! [S2: set_set_a,T2: set_nat,R: set_a > nat > $o,K: nat] :
( ( finite_finite_set_a @ S2 )
=> ( ( finite_finite_nat @ T2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ T2 )
=> ( ( finite_card_set_a
@ ( collect_set_a
@ ^ [I2: set_a] :
( ( member_set_a @ I2 @ S2 )
& ( R @ I2 @ X3 ) ) ) )
= K ) )
=> ( ( groups6141743369313575924_a_nat
@ ^ [I2: set_a] :
( finite_card_nat
@ ( collect_nat
@ ^ [J2: nat] :
( ( member_nat @ J2 @ T2 )
& ( R @ I2 @ J2 ) ) ) )
@ S2 )
= ( times_times_nat @ K @ ( finite_card_nat @ T2 ) ) ) ) ) ) ).
% sum_multicount
thf(fact_175_finite__field_Ofinite__field__min__order,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_f3414212785825285533t_unit @ R )
=> ( ord_less_nat @ one_one_nat @ ( order_3240872107759947550t_unit @ R ) ) ) ).
% finite_field.finite_field_min_order
thf(fact_176_finite__field_Ofinite__field__min__order,axiom,
! [R: partia4960592913263135132t_unit] :
( ( ring_f3227788147421470775t_unit @ R )
=> ( ord_less_nat @ one_one_nat @ ( order_78496787366231454t_unit @ R ) ) ) ).
% finite_field.finite_field_min_order
thf(fact_177_finite__field_Ofinite__field__min__order,axiom,
! [R: partia7496981018696276118t_unit] :
( ( ring_f902895163656626621t_unit @ R )
=> ( ord_less_nat @ one_one_nat @ ( order_1351569949434154782t_unit @ R ) ) ) ).
% finite_field.finite_field_min_order
thf(fact_178_finite__field_Ofinite__field__min__order,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_f9002592259703987221ld_a_b @ R )
=> ( ord_less_nat @ one_one_nat @ ( order_a_ring_ext_a_b @ R ) ) ) ).
% finite_field.finite_field_min_order
thf(fact_179_gauss__poly__monic,axiom,
! [N: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( monic_3145109188698636716ly_a_b @ r @ ( card_I2373409586816755191ly_a_b @ r @ N ) ) ) ).
% gauss_poly_monic
thf(fact_180_geom__nat,axiom,
! [X: complex,Q2: nat] :
( ( times_times_complex @ ( minus_minus_complex @ X @ one_one_complex ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ Q2 ) ) )
= ( minus_minus_complex @ ( power_power_complex @ X @ Q2 ) @ one_one_complex ) ) ).
% geom_nat
thf(fact_181_geom__nat,axiom,
! [X: int,Q2: nat] :
( ( times_times_int @ ( minus_minus_int @ X @ one_one_int ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_lessThan_nat @ Q2 ) ) )
= ( minus_minus_int @ ( power_power_int @ X @ Q2 ) @ one_one_int ) ) ).
% geom_nat
thf(fact_182_card__eq__sum,axiom,
( finite7490188192133390220list_a
= ( groups5993734322560061562_a_nat
@ ^ [X2: set_list_a] : one_one_nat ) ) ).
% card_eq_sum
thf(fact_183_card__eq__sum,axiom,
( finite_card_set_a
= ( groups6141743369313575924_a_nat
@ ^ [X2: set_a] : one_one_nat ) ) ).
% card_eq_sum
thf(fact_184_card__eq__sum,axiom,
( finite_card_a
= ( groups6334556678337121940_a_nat
@ ^ [X2: a] : one_one_nat ) ) ).
% card_eq_sum
thf(fact_185_card__eq__sum,axiom,
( finite_card_nat
= ( groups3542108847815614940at_nat
@ ^ [X2: nat] : one_one_nat ) ) ).
% card_eq_sum
thf(fact_186_card__eq__sum,axiom,
( finite_card_list_a
= ( groups5521247699297860762_a_nat
@ ^ [X2: list_a] : one_one_nat ) ) ).
% card_eq_sum
thf(fact_187_sum__multicount__gen,axiom,
! [S: set_int,T: set_int,R: int > int > $o,K: int > nat] :
( ( finite_finite_int @ S )
=> ( ( finite_finite_int @ T )
=> ( ! [X3: int] :
( ( member_int @ X3 @ T )
=> ( ( finite_card_int
@ ( collect_int
@ ^ [I2: int] :
( ( member_int @ I2 @ S )
& ( R @ I2 @ X3 ) ) ) )
= ( K @ X3 ) ) )
=> ( ( groups4541462559716669496nt_nat
@ ^ [I2: int] :
( finite_card_int
@ ( collect_int
@ ^ [J2: int] :
( ( member_int @ J2 @ T )
& ( R @ I2 @ J2 ) ) ) )
@ S )
= ( groups4541462559716669496nt_nat @ K @ T ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_188_sum__multicount__gen,axiom,
! [S: set_int,T: set_a,R: int > a > $o,K: a > nat] :
( ( finite_finite_int @ S )
=> ( ( finite_finite_a @ T )
=> ( ! [X3: a] :
( ( member_a @ X3 @ T )
=> ( ( finite_card_int
@ ( collect_int
@ ^ [I2: int] :
( ( member_int @ I2 @ S )
& ( R @ I2 @ X3 ) ) ) )
= ( K @ X3 ) ) )
=> ( ( groups4541462559716669496nt_nat
@ ^ [I2: int] :
( finite_card_a
@ ( collect_a
@ ^ [J2: a] :
( ( member_a @ J2 @ T )
& ( R @ I2 @ J2 ) ) ) )
@ S )
= ( groups6334556678337121940_a_nat @ K @ T ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_189_sum__multicount__gen,axiom,
! [S: set_a,T: set_int,R: a > int > $o,K: int > nat] :
( ( finite_finite_a @ S )
=> ( ( finite_finite_int @ T )
=> ( ! [X3: int] :
( ( member_int @ X3 @ T )
=> ( ( finite_card_a
@ ( collect_a
@ ^ [I2: a] :
( ( member_a @ I2 @ S )
& ( R @ I2 @ X3 ) ) ) )
= ( K @ X3 ) ) )
=> ( ( groups6334556678337121940_a_nat
@ ^ [I2: a] :
( finite_card_int
@ ( collect_int
@ ^ [J2: int] :
( ( member_int @ J2 @ T )
& ( R @ I2 @ J2 ) ) ) )
@ S )
= ( groups4541462559716669496nt_nat @ K @ T ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_190_sum__multicount__gen,axiom,
! [S: set_a,T: set_a,R: a > a > $o,K: a > nat] :
( ( finite_finite_a @ S )
=> ( ( finite_finite_a @ T )
=> ( ! [X3: a] :
( ( member_a @ X3 @ T )
=> ( ( finite_card_a
@ ( collect_a
@ ^ [I2: a] :
( ( member_a @ I2 @ S )
& ( R @ I2 @ X3 ) ) ) )
= ( K @ X3 ) ) )
=> ( ( groups6334556678337121940_a_nat
@ ^ [I2: a] :
( finite_card_a
@ ( collect_a
@ ^ [J2: a] :
( ( member_a @ J2 @ T )
& ( R @ I2 @ J2 ) ) ) )
@ S )
= ( groups6334556678337121940_a_nat @ K @ T ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_191_sum__multicount__gen,axiom,
! [S: set_int,T: set_nat,R: int > nat > $o,K: nat > nat] :
( ( finite_finite_int @ S )
=> ( ( finite_finite_nat @ T )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ T )
=> ( ( finite_card_int
@ ( collect_int
@ ^ [I2: int] :
( ( member_int @ I2 @ S )
& ( R @ I2 @ X3 ) ) ) )
= ( K @ X3 ) ) )
=> ( ( groups4541462559716669496nt_nat
@ ^ [I2: int] :
( finite_card_nat
@ ( collect_nat
@ ^ [J2: nat] :
( ( member_nat @ J2 @ T )
& ( R @ I2 @ J2 ) ) ) )
@ S )
= ( groups3542108847815614940at_nat @ K @ T ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_192_sum__multicount__gen,axiom,
! [S: set_a,T: set_nat,R: a > nat > $o,K: nat > nat] :
( ( finite_finite_a @ S )
=> ( ( finite_finite_nat @ T )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ T )
=> ( ( finite_card_a
@ ( collect_a
@ ^ [I2: a] :
( ( member_a @ I2 @ S )
& ( R @ I2 @ X3 ) ) ) )
= ( K @ X3 ) ) )
=> ( ( groups6334556678337121940_a_nat
@ ^ [I2: a] :
( finite_card_nat
@ ( collect_nat
@ ^ [J2: nat] :
( ( member_nat @ J2 @ T )
& ( R @ I2 @ J2 ) ) ) )
@ S )
= ( groups3542108847815614940at_nat @ K @ T ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_193_sum__multicount__gen,axiom,
! [S: set_nat,T: set_int,R: nat > int > $o,K: int > nat] :
( ( finite_finite_nat @ S )
=> ( ( finite_finite_int @ T )
=> ( ! [X3: int] :
( ( member_int @ X3 @ T )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [I2: nat] :
( ( member_nat @ I2 @ S )
& ( R @ I2 @ X3 ) ) ) )
= ( K @ X3 ) ) )
=> ( ( groups3542108847815614940at_nat
@ ^ [I2: nat] :
( finite_card_int
@ ( collect_int
@ ^ [J2: int] :
( ( member_int @ J2 @ T )
& ( R @ I2 @ J2 ) ) ) )
@ S )
= ( groups4541462559716669496nt_nat @ K @ T ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_194_sum__multicount__gen,axiom,
! [S: set_nat,T: set_a,R: nat > a > $o,K: a > nat] :
( ( finite_finite_nat @ S )
=> ( ( finite_finite_a @ T )
=> ( ! [X3: a] :
( ( member_a @ X3 @ T )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [I2: nat] :
( ( member_nat @ I2 @ S )
& ( R @ I2 @ X3 ) ) ) )
= ( K @ X3 ) ) )
=> ( ( groups3542108847815614940at_nat
@ ^ [I2: nat] :
( finite_card_a
@ ( collect_a
@ ^ [J2: a] :
( ( member_a @ J2 @ T )
& ( R @ I2 @ J2 ) ) ) )
@ S )
= ( groups6334556678337121940_a_nat @ K @ T ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_195_sum__multicount__gen,axiom,
! [S: set_nat,T: set_nat,R: nat > nat > $o,K: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( finite_finite_nat @ T )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ T )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [I2: nat] :
( ( member_nat @ I2 @ S )
& ( R @ I2 @ X3 ) ) ) )
= ( K @ X3 ) ) )
=> ( ( groups3542108847815614940at_nat
@ ^ [I2: nat] :
( finite_card_nat
@ ( collect_nat
@ ^ [J2: nat] :
( ( member_nat @ J2 @ T )
& ( R @ I2 @ J2 ) ) ) )
@ S )
= ( groups3542108847815614940at_nat @ K @ T ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_196_sum__multicount__gen,axiom,
! [S: set_set_a,T: set_int,R: set_a > int > $o,K: int > nat] :
( ( finite_finite_set_a @ S )
=> ( ( finite_finite_int @ T )
=> ( ! [X3: int] :
( ( member_int @ X3 @ T )
=> ( ( finite_card_set_a
@ ( collect_set_a
@ ^ [I2: set_a] :
( ( member_set_a @ I2 @ S )
& ( R @ I2 @ X3 ) ) ) )
= ( K @ X3 ) ) )
=> ( ( groups6141743369313575924_a_nat
@ ^ [I2: set_a] :
( finite_card_int
@ ( collect_int
@ ^ [J2: int] :
( ( member_int @ J2 @ T )
& ( R @ I2 @ J2 ) ) ) )
@ S )
= ( groups4541462559716669496nt_nat @ K @ T ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_197_finite__Collect__conjI,axiom,
! [P2: list_list_a > $o,Q: list_list_a > $o] :
( ( ( finite1660835950917165235list_a @ ( collect_list_list_a @ P2 ) )
| ( finite1660835950917165235list_a @ ( collect_list_list_a @ Q ) ) )
=> ( finite1660835950917165235list_a
@ ( collect_list_list_a
@ ^ [X2: list_list_a] :
( ( P2 @ X2 )
& ( Q @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_198_finite__Collect__conjI,axiom,
! [P2: list_a > $o,Q: list_a > $o] :
( ( ( finite_finite_list_a @ ( collect_list_a @ P2 ) )
| ( finite_finite_list_a @ ( collect_list_a @ Q ) ) )
=> ( finite_finite_list_a
@ ( collect_list_a
@ ^ [X2: list_a] :
( ( P2 @ X2 )
& ( Q @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_199_finite__Collect__conjI,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
| ( finite_finite_nat @ ( collect_nat @ Q ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( P2 @ X2 )
& ( Q @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_200_finite__Collect__conjI,axiom,
! [P2: int > $o,Q: int > $o] :
( ( ( finite_finite_int @ ( collect_int @ P2 ) )
| ( finite_finite_int @ ( collect_int @ Q ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [X2: int] :
( ( P2 @ X2 )
& ( Q @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_201_finite__Collect__conjI,axiom,
! [P2: a > $o,Q: a > $o] :
( ( ( finite_finite_a @ ( collect_a @ P2 ) )
| ( finite_finite_a @ ( collect_a @ Q ) ) )
=> ( finite_finite_a
@ ( collect_a
@ ^ [X2: a] :
( ( P2 @ X2 )
& ( Q @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_202_gauss__poly__div__gauss__poly__iff__2,axiom,
! [L: nat,A: int,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ L )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ( dvd_dvd_int @ ( minus_minus_int @ ( power_power_int @ A @ L ) @ one_one_int ) @ ( minus_minus_int @ ( power_power_int @ A @ M ) @ one_one_int ) )
= ( dvd_dvd_nat @ L @ M ) ) ) ) ).
% gauss_poly_div_gauss_poly_iff_2
thf(fact_203_a,axiom,
( finite_finite_nat
@ ( collect_nat
@ ^ [D: nat] : ( dvd_dvd_nat @ D @ n ) ) ) ).
% a
thf(fact_204_finite__Diff2,axiom,
! [B3: set_list_list_a,A2: set_list_list_a] :
( ( finite1660835950917165235list_a @ B3 )
=> ( ( finite1660835950917165235list_a @ ( minus_5335179877275218001list_a @ A2 @ B3 ) )
= ( finite1660835950917165235list_a @ A2 ) ) ) ).
% finite_Diff2
thf(fact_205_finite__Diff2,axiom,
! [B3: set_list_a,A2: set_list_a] :
( ( finite_finite_list_a @ B3 )
=> ( ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A2 @ B3 ) )
= ( finite_finite_list_a @ A2 ) ) ) ).
% finite_Diff2
thf(fact_206_finite__Diff2,axiom,
! [B3: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B3 )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B3 ) )
= ( finite_finite_nat @ A2 ) ) ) ).
% finite_Diff2
thf(fact_207_finite__Diff2,axiom,
! [B3: set_int,A2: set_int] :
( ( finite_finite_int @ B3 )
=> ( ( finite_finite_int @ ( minus_minus_set_int @ A2 @ B3 ) )
= ( finite_finite_int @ A2 ) ) ) ).
% finite_Diff2
thf(fact_208_finite__Diff2,axiom,
! [B3: set_a,A2: set_a] :
( ( finite_finite_a @ B3 )
=> ( ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B3 ) )
= ( finite_finite_a @ A2 ) ) ) ).
% finite_Diff2
thf(fact_209_finite__Diff,axiom,
! [A2: set_list_list_a,B3: set_list_list_a] :
( ( finite1660835950917165235list_a @ A2 )
=> ( finite1660835950917165235list_a @ ( minus_5335179877275218001list_a @ A2 @ B3 ) ) ) ).
% finite_Diff
thf(fact_210_finite__Diff,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( finite_finite_list_a @ A2 )
=> ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A2 @ B3 ) ) ) ).
% finite_Diff
thf(fact_211_finite__Diff,axiom,
! [A2: set_nat,B3: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B3 ) ) ) ).
% finite_Diff
thf(fact_212_finite__Diff,axiom,
! [A2: set_int,B3: set_int] :
( ( finite_finite_int @ A2 )
=> ( finite_finite_int @ ( minus_minus_set_int @ A2 @ B3 ) ) ) ).
% finite_Diff
thf(fact_213_finite__Diff,axiom,
! [A2: set_a,B3: set_a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B3 ) ) ) ).
% finite_Diff
thf(fact_214_finite__Collect__disjI,axiom,
! [P2: list_list_a > $o,Q: list_list_a > $o] :
( ( finite1660835950917165235list_a
@ ( collect_list_list_a
@ ^ [X2: list_list_a] :
( ( P2 @ X2 )
| ( Q @ X2 ) ) ) )
= ( ( finite1660835950917165235list_a @ ( collect_list_list_a @ P2 ) )
& ( finite1660835950917165235list_a @ ( collect_list_list_a @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_215_finite__Collect__disjI,axiom,
! [P2: list_a > $o,Q: list_a > $o] :
( ( finite_finite_list_a
@ ( collect_list_a
@ ^ [X2: list_a] :
( ( P2 @ X2 )
| ( Q @ X2 ) ) ) )
= ( ( finite_finite_list_a @ ( collect_list_a @ P2 ) )
& ( finite_finite_list_a @ ( collect_list_a @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_216_finite__Collect__disjI,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( P2 @ X2 )
| ( Q @ X2 ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
& ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_217_finite__Collect__disjI,axiom,
! [P2: int > $o,Q: int > $o] :
( ( finite_finite_int
@ ( collect_int
@ ^ [X2: int] :
( ( P2 @ X2 )
| ( Q @ X2 ) ) ) )
= ( ( finite_finite_int @ ( collect_int @ P2 ) )
& ( finite_finite_int @ ( collect_int @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_218_finite__Collect__disjI,axiom,
! [P2: a > $o,Q: a > $o] :
( ( finite_finite_a
@ ( collect_a
@ ^ [X2: a] :
( ( P2 @ X2 )
| ( Q @ X2 ) ) ) )
= ( ( finite_finite_a @ ( collect_a @ P2 ) )
& ( finite_finite_a @ ( collect_a @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_219_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I2: nat] : ( ord_less_nat @ I2 @ N ) ) )
= N ) ).
% card_Collect_less_nat
thf(fact_220_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_nat @ N4 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_221_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_eq_nat @ N4 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_222_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_223_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_224_mult__zero__left,axiom,
! [A: complex] :
( ( times_times_complex @ zero_zero_complex @ A )
= zero_zero_complex ) ).
% mult_zero_left
thf(fact_225_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_226_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_227_mult__zero__right,axiom,
! [A: complex] :
( ( times_times_complex @ A @ zero_zero_complex )
= zero_zero_complex ) ).
% mult_zero_right
thf(fact_228_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_229_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_230_mult__eq__0__iff,axiom,
! [A: complex,B: complex] :
( ( ( times_times_complex @ A @ B )
= zero_zero_complex )
= ( ( A = zero_zero_complex )
| ( B = zero_zero_complex ) ) ) ).
% mult_eq_0_iff
thf(fact_231_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_232_mult__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
= ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_233_mult__cancel__left,axiom,
! [C: complex,A: complex,B: complex] :
( ( ( times_times_complex @ C @ A )
= ( times_times_complex @ C @ B ) )
= ( ( C = zero_zero_complex )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_234_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_235_mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_236_mult__cancel__right,axiom,
! [A: complex,C: complex,B: complex] :
( ( ( times_times_complex @ A @ C )
= ( times_times_complex @ B @ C ) )
= ( ( C = zero_zero_complex )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_237_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_238_mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_239_diff__self,axiom,
! [A: complex] :
( ( minus_minus_complex @ A @ A )
= zero_zero_complex ) ).
% diff_self
thf(fact_240_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_241_diff__0__right,axiom,
! [A: complex] :
( ( minus_minus_complex @ A @ zero_zero_complex )
= A ) ).
% diff_0_right
thf(fact_242_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_243_zero__diff,axiom,
! [A: multiset_a] :
( ( minus_3765977307040488491iset_a @ zero_zero_multiset_a @ A )
= zero_zero_multiset_a ) ).
% zero_diff
thf(fact_244_zero__diff,axiom,
! [A: multiset_list_a] :
( ( minus_7431248565939055793list_a @ zero_z4454100511807792257list_a @ A )
= zero_z4454100511807792257list_a ) ).
% zero_diff
thf(fact_245_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_246_diff__zero,axiom,
! [A: complex] :
( ( minus_minus_complex @ A @ zero_zero_complex )
= A ) ).
% diff_zero
thf(fact_247_diff__zero,axiom,
! [A: multiset_a] :
( ( minus_3765977307040488491iset_a @ A @ zero_zero_multiset_a )
= A ) ).
% diff_zero
thf(fact_248_diff__zero,axiom,
! [A: multiset_list_a] :
( ( minus_7431248565939055793list_a @ A @ zero_z4454100511807792257list_a )
= A ) ).
% diff_zero
thf(fact_249_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_250_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_251_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: complex] :
( ( minus_minus_complex @ A @ A )
= zero_zero_complex ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_252_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: multiset_a] :
( ( minus_3765977307040488491iset_a @ A @ A )
= zero_zero_multiset_a ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_253_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: multiset_list_a] :
( ( minus_7431248565939055793list_a @ A @ A )
= zero_z4454100511807792257list_a ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_254_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_255_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_256_dvd__0__right,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% dvd_0_right
thf(fact_257_dvd__0__right,axiom,
! [A: int] : ( dvd_dvd_int @ A @ zero_zero_int ) ).
% dvd_0_right
thf(fact_258_dvd__0__right,axiom,
! [A: complex] : ( dvd_dvd_complex @ A @ zero_zero_complex ) ).
% dvd_0_right
thf(fact_259_dvd__0__left__iff,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
= ( A = zero_zero_nat ) ) ).
% dvd_0_left_iff
thf(fact_260_dvd__0__left__iff,axiom,
! [A: int] :
( ( dvd_dvd_int @ zero_zero_int @ A )
= ( A = zero_zero_int ) ) ).
% dvd_0_left_iff
thf(fact_261_dvd__0__left__iff,axiom,
! [A: complex] :
( ( dvd_dvd_complex @ zero_zero_complex @ A )
= ( A = zero_zero_complex ) ) ).
% dvd_0_left_iff
thf(fact_262_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_263_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_264_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_265_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_266_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_267_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_268_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_269_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_270_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_271_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_272_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_273_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_274_sum_Oneutral__const,axiom,
! [A2: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [Uu: nat] : zero_zero_nat
@ A2 )
= zero_zero_nat ) ).
% sum.neutral_const
thf(fact_275_sum_Oneutral__const,axiom,
! [A2: set_list_a] :
( ( groups5521247699297860762_a_nat
@ ^ [Uu: list_a] : zero_zero_nat
@ A2 )
= zero_zero_nat ) ).
% sum.neutral_const
thf(fact_276_sum_Oneutral__const,axiom,
! [A2: set_complex] :
( ( groups7754918857620584856omplex
@ ^ [Uu: complex] : zero_zero_complex
@ A2 )
= zero_zero_complex ) ).
% sum.neutral_const
thf(fact_277_diff__ge__0__iff__ge,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_278_diff__gt__0__iff__gt,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_int @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_279_mult__cancel__left1,axiom,
! [C: complex,B: complex] :
( ( C
= ( times_times_complex @ C @ B ) )
= ( ( C = zero_zero_complex )
| ( B = one_one_complex ) ) ) ).
% mult_cancel_left1
thf(fact_280_mult__cancel__left1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_281_mult__cancel__left2,axiom,
! [C: complex,A: complex] :
( ( ( times_times_complex @ C @ A )
= C )
= ( ( C = zero_zero_complex )
| ( A = one_one_complex ) ) ) ).
% mult_cancel_left2
thf(fact_282_mult__cancel__left2,axiom,
! [C: int,A: int] :
( ( ( times_times_int @ C @ A )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_283_mult__cancel__right1,axiom,
! [C: complex,B: complex] :
( ( C
= ( times_times_complex @ B @ C ) )
= ( ( C = zero_zero_complex )
| ( B = one_one_complex ) ) ) ).
% mult_cancel_right1
thf(fact_284_mult__cancel__right1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_285_mult__cancel__right2,axiom,
! [A: complex,C: complex] :
( ( ( times_times_complex @ A @ C )
= C )
= ( ( C = zero_zero_complex )
| ( A = one_one_complex ) ) ) ).
% mult_cancel_right2
thf(fact_286_mult__cancel__right2,axiom,
! [A: int,C: int] :
( ( ( times_times_int @ A @ C )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_287_dvd__mult__cancel__left,axiom,
! [C: complex,A: complex,B: complex] :
( ( dvd_dvd_complex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
= ( ( C = zero_zero_complex )
| ( dvd_dvd_complex @ A @ B ) ) ) ).
% dvd_mult_cancel_left
thf(fact_288_dvd__mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( dvd_dvd_int @ A @ B ) ) ) ).
% dvd_mult_cancel_left
thf(fact_289_dvd__mult__cancel__right,axiom,
! [A: complex,C: complex,B: complex] :
( ( dvd_dvd_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) )
= ( ( C = zero_zero_complex )
| ( dvd_dvd_complex @ A @ B ) ) ) ).
% dvd_mult_cancel_right
thf(fact_290_dvd__mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( dvd_dvd_int @ A @ B ) ) ) ).
% dvd_mult_cancel_right
thf(fact_291_dvd__times__left__cancel__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( A != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) )
= ( dvd_dvd_nat @ B @ C ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_292_dvd__times__left__cancel__iff,axiom,
! [A: int,B: int,C: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) )
= ( dvd_dvd_int @ B @ C ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_293_dvd__times__right__cancel__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( A != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) )
= ( dvd_dvd_nat @ B @ C ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_294_dvd__times__right__cancel__iff,axiom,
! [A: int,B: int,C: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) )
= ( dvd_dvd_int @ B @ C ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_295_sum__eq__0__iff,axiom,
! [F3: set_list_list_a,F2: list_list_a > nat] :
( ( finite1660835950917165235list_a @ F3 )
=> ( ( ( groups7548105480907152928_a_nat @ F2 @ F3 )
= zero_zero_nat )
= ( ! [X2: list_list_a] :
( ( member_list_list_a @ X2 @ F3 )
=> ( ( F2 @ X2 )
= zero_zero_nat ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_296_sum__eq__0__iff,axiom,
! [F3: set_int,F2: int > nat] :
( ( finite_finite_int @ F3 )
=> ( ( ( groups4541462559716669496nt_nat @ F2 @ F3 )
= zero_zero_nat )
= ( ! [X2: int] :
( ( member_int @ X2 @ F3 )
=> ( ( F2 @ X2 )
= zero_zero_nat ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_297_sum__eq__0__iff,axiom,
! [F3: set_a,F2: a > nat] :
( ( finite_finite_a @ F3 )
=> ( ( ( groups6334556678337121940_a_nat @ F2 @ F3 )
= zero_zero_nat )
= ( ! [X2: a] :
( ( member_a @ X2 @ F3 )
=> ( ( F2 @ X2 )
= zero_zero_nat ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_298_sum__eq__0__iff,axiom,
! [F3: set_nat,F2: nat > nat] :
( ( finite_finite_nat @ F3 )
=> ( ( ( groups3542108847815614940at_nat @ F2 @ F3 )
= zero_zero_nat )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ F3 )
=> ( ( F2 @ X2 )
= zero_zero_nat ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_299_sum__eq__0__iff,axiom,
! [F3: set_list_a,F2: list_a > nat] :
( ( finite_finite_list_a @ F3 )
=> ( ( ( groups5521247699297860762_a_nat @ F2 @ F3 )
= zero_zero_nat )
= ( ! [X2: list_a] :
( ( member_list_a @ X2 @ F3 )
=> ( ( F2 @ X2 )
= zero_zero_nat ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_300_sum_Oinfinite,axiom,
! [A2: set_int,G: int > nat] :
( ~ ( finite_finite_int @ A2 )
=> ( ( groups4541462559716669496nt_nat @ G @ A2 )
= zero_zero_nat ) ) ).
% sum.infinite
thf(fact_301_sum_Oinfinite,axiom,
! [A2: set_a,G: a > nat] :
( ~ ( finite_finite_a @ A2 )
=> ( ( groups6334556678337121940_a_nat @ G @ A2 )
= zero_zero_nat ) ) ).
% sum.infinite
thf(fact_302_sum_Oinfinite,axiom,
! [A2: set_nat,G: nat > int] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( groups3539618377306564664at_int @ G @ A2 )
= zero_zero_int ) ) ).
% sum.infinite
thf(fact_303_sum_Oinfinite,axiom,
! [A2: set_int,G: int > int] :
( ~ ( finite_finite_int @ A2 )
=> ( ( groups4538972089207619220nt_int @ G @ A2 )
= zero_zero_int ) ) ).
% sum.infinite
thf(fact_304_sum_Oinfinite,axiom,
! [A2: set_a,G: a > int] :
( ~ ( finite_finite_a @ A2 )
=> ( ( groups6332066207828071664_a_int @ G @ A2 )
= zero_zero_int ) ) ).
% sum.infinite
thf(fact_305_sum_Oinfinite,axiom,
! [A2: set_nat,G: nat > complex] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( groups2073611262835488442omplex @ G @ A2 )
= zero_zero_complex ) ) ).
% sum.infinite
thf(fact_306_sum_Oinfinite,axiom,
! [A2: set_int,G: int > complex] :
( ~ ( finite_finite_int @ A2 )
=> ( ( groups3049146728041665814omplex @ G @ A2 )
= zero_zero_complex ) ) ).
% sum.infinite
thf(fact_307_sum_Oinfinite,axiom,
! [A2: set_a,G: a > complex] :
( ~ ( finite_finite_a @ A2 )
=> ( ( groups8331919209915413362omplex @ G @ A2 )
= zero_zero_complex ) ) ).
% sum.infinite
thf(fact_308_sum_Oinfinite,axiom,
! [A2: set_nat,G: nat > nat] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( groups3542108847815614940at_nat @ G @ A2 )
= zero_zero_nat ) ) ).
% sum.infinite
thf(fact_309_sum_Oinfinite,axiom,
! [A2: set_complex,G: complex > complex] :
( ~ ( finite3207457112153483333omplex @ A2 )
=> ( ( groups7754918857620584856omplex @ G @ A2 )
= zero_zero_complex ) ) ).
% sum.infinite
thf(fact_310_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_311_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_312_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_313_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_314_card_Oinfinite,axiom,
! [A2: set_set_list_a] :
( ~ ( finite5282473924520328461list_a @ A2 )
=> ( ( finite7490188192133390220list_a @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_315_card_Oinfinite,axiom,
! [A2: set_set_a] :
( ~ ( finite_finite_set_a @ A2 )
=> ( ( finite_card_set_a @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_316_card_Oinfinite,axiom,
! [A2: set_list_list_a] :
( ~ ( finite1660835950917165235list_a @ A2 )
=> ( ( finite9134805042761151410list_a @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_317_card_Oinfinite,axiom,
! [A2: set_list_a] :
( ~ ( finite_finite_list_a @ A2 )
=> ( ( finite_card_list_a @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_318_card_Oinfinite,axiom,
! [A2: set_nat] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_card_nat @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_319_card_Oinfinite,axiom,
! [A2: set_int] :
( ~ ( finite_finite_int @ A2 )
=> ( ( finite_card_int @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_320_card_Oinfinite,axiom,
! [A2: set_a] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_card_a @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_321_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_322_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_323_nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_324_sum_Oeq__sum,axiom,
! [I3: set_nat,P3: nat > nat] :
( ( finite_finite_nat @ I3 )
=> ( ( groups1986416967739987077at_nat @ P3 @ I3 )
= ( groups3542108847815614940at_nat @ P3 @ I3 ) ) ) ).
% sum.eq_sum
thf(fact_325_sum_Oeq__sum,axiom,
! [I3: set_complex,P3: complex > complex] :
( ( finite3207457112153483333omplex @ I3 )
=> ( ( groups808145749697022017omplex @ P3 @ I3 )
= ( groups7754918857620584856omplex @ P3 @ I3 ) ) ) ).
% sum.eq_sum
thf(fact_326_sum_Oeq__sum,axiom,
! [I3: set_list_a,P3: list_a > nat] :
( ( finite_finite_list_a @ I3 )
=> ( ( groups1083796956622547633_a_nat @ P3 @ I3 )
= ( groups5521247699297860762_a_nat @ P3 @ I3 ) ) ) ).
% sum.eq_sum
thf(fact_327_sum_Odelta,axiom,
! [S2: set_int,A: int,B: int > nat] :
( ( finite_finite_int @ S2 )
=> ( ( ( member_int @ A @ S2 )
=> ( ( groups4541462559716669496nt_nat
@ ^ [K2: int] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_nat )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_int @ A @ S2 )
=> ( ( groups4541462559716669496nt_nat
@ ^ [K2: int] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_nat )
@ S2 )
= zero_zero_nat ) ) ) ) ).
% sum.delta
thf(fact_328_sum_Odelta,axiom,
! [S2: set_a,A: a,B: a > nat] :
( ( finite_finite_a @ S2 )
=> ( ( ( member_a @ A @ S2 )
=> ( ( groups6334556678337121940_a_nat
@ ^ [K2: a] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_nat )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_a @ A @ S2 )
=> ( ( groups6334556678337121940_a_nat
@ ^ [K2: a] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_nat )
@ S2 )
= zero_zero_nat ) ) ) ) ).
% sum.delta
thf(fact_329_sum_Odelta,axiom,
! [S2: set_nat,A: nat,B: nat > int] :
( ( finite_finite_nat @ S2 )
=> ( ( ( member_nat @ A @ S2 )
=> ( ( groups3539618377306564664at_int
@ ^ [K2: nat] : ( if_int @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_int )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S2 )
=> ( ( groups3539618377306564664at_int
@ ^ [K2: nat] : ( if_int @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_int )
@ S2 )
= zero_zero_int ) ) ) ) ).
% sum.delta
thf(fact_330_sum_Odelta,axiom,
! [S2: set_int,A: int,B: int > int] :
( ( finite_finite_int @ S2 )
=> ( ( ( member_int @ A @ S2 )
=> ( ( groups4538972089207619220nt_int
@ ^ [K2: int] : ( if_int @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_int )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_int @ A @ S2 )
=> ( ( groups4538972089207619220nt_int
@ ^ [K2: int] : ( if_int @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_int )
@ S2 )
= zero_zero_int ) ) ) ) ).
% sum.delta
thf(fact_331_sum_Odelta,axiom,
! [S2: set_a,A: a,B: a > int] :
( ( finite_finite_a @ S2 )
=> ( ( ( member_a @ A @ S2 )
=> ( ( groups6332066207828071664_a_int
@ ^ [K2: a] : ( if_int @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_int )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_a @ A @ S2 )
=> ( ( groups6332066207828071664_a_int
@ ^ [K2: a] : ( if_int @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_int )
@ S2 )
= zero_zero_int ) ) ) ) ).
% sum.delta
thf(fact_332_sum_Odelta,axiom,
! [S2: set_nat,A: nat,B: nat > complex] :
( ( finite_finite_nat @ S2 )
=> ( ( ( member_nat @ A @ S2 )
=> ( ( groups2073611262835488442omplex
@ ^ [K2: nat] : ( if_complex @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_complex )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S2 )
=> ( ( groups2073611262835488442omplex
@ ^ [K2: nat] : ( if_complex @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_complex )
@ S2 )
= zero_zero_complex ) ) ) ) ).
% sum.delta
thf(fact_333_sum_Odelta,axiom,
! [S2: set_int,A: int,B: int > complex] :
( ( finite_finite_int @ S2 )
=> ( ( ( member_int @ A @ S2 )
=> ( ( groups3049146728041665814omplex
@ ^ [K2: int] : ( if_complex @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_complex )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_int @ A @ S2 )
=> ( ( groups3049146728041665814omplex
@ ^ [K2: int] : ( if_complex @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_complex )
@ S2 )
= zero_zero_complex ) ) ) ) ).
% sum.delta
thf(fact_334_sum_Odelta,axiom,
! [S2: set_a,A: a,B: a > complex] :
( ( finite_finite_a @ S2 )
=> ( ( ( member_a @ A @ S2 )
=> ( ( groups8331919209915413362omplex
@ ^ [K2: a] : ( if_complex @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_complex )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_a @ A @ S2 )
=> ( ( groups8331919209915413362omplex
@ ^ [K2: a] : ( if_complex @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_complex )
@ S2 )
= zero_zero_complex ) ) ) ) ).
% sum.delta
thf(fact_335_sum_Odelta,axiom,
! [S2: set_nat,A: nat,B: nat > nat] :
( ( finite_finite_nat @ S2 )
=> ( ( ( member_nat @ A @ S2 )
=> ( ( groups3542108847815614940at_nat
@ ^ [K2: nat] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_nat )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S2 )
=> ( ( groups3542108847815614940at_nat
@ ^ [K2: nat] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_nat )
@ S2 )
= zero_zero_nat ) ) ) ) ).
% sum.delta
thf(fact_336_sum_Odelta,axiom,
! [S2: set_complex,A: complex,B: complex > complex] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( ( member_complex @ A @ S2 )
=> ( ( groups7754918857620584856omplex
@ ^ [K2: complex] : ( if_complex @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_complex )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_complex @ A @ S2 )
=> ( ( groups7754918857620584856omplex
@ ^ [K2: complex] : ( if_complex @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_complex )
@ S2 )
= zero_zero_complex ) ) ) ) ).
% sum.delta
thf(fact_337_sum_Odelta_H,axiom,
! [S2: set_int,A: int,B: int > nat] :
( ( finite_finite_int @ S2 )
=> ( ( ( member_int @ A @ S2 )
=> ( ( groups4541462559716669496nt_nat
@ ^ [K2: int] : ( if_nat @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_nat )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_int @ A @ S2 )
=> ( ( groups4541462559716669496nt_nat
@ ^ [K2: int] : ( if_nat @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_nat )
@ S2 )
= zero_zero_nat ) ) ) ) ).
% sum.delta'
thf(fact_338_sum_Odelta_H,axiom,
! [S2: set_a,A: a,B: a > nat] :
( ( finite_finite_a @ S2 )
=> ( ( ( member_a @ A @ S2 )
=> ( ( groups6334556678337121940_a_nat
@ ^ [K2: a] : ( if_nat @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_nat )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_a @ A @ S2 )
=> ( ( groups6334556678337121940_a_nat
@ ^ [K2: a] : ( if_nat @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_nat )
@ S2 )
= zero_zero_nat ) ) ) ) ).
% sum.delta'
thf(fact_339_sum_Odelta_H,axiom,
! [S2: set_nat,A: nat,B: nat > int] :
( ( finite_finite_nat @ S2 )
=> ( ( ( member_nat @ A @ S2 )
=> ( ( groups3539618377306564664at_int
@ ^ [K2: nat] : ( if_int @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_int )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S2 )
=> ( ( groups3539618377306564664at_int
@ ^ [K2: nat] : ( if_int @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_int )
@ S2 )
= zero_zero_int ) ) ) ) ).
% sum.delta'
thf(fact_340_sum_Odelta_H,axiom,
! [S2: set_int,A: int,B: int > int] :
( ( finite_finite_int @ S2 )
=> ( ( ( member_int @ A @ S2 )
=> ( ( groups4538972089207619220nt_int
@ ^ [K2: int] : ( if_int @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_int )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_int @ A @ S2 )
=> ( ( groups4538972089207619220nt_int
@ ^ [K2: int] : ( if_int @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_int )
@ S2 )
= zero_zero_int ) ) ) ) ).
% sum.delta'
thf(fact_341_sum_Odelta_H,axiom,
! [S2: set_a,A: a,B: a > int] :
( ( finite_finite_a @ S2 )
=> ( ( ( member_a @ A @ S2 )
=> ( ( groups6332066207828071664_a_int
@ ^ [K2: a] : ( if_int @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_int )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_a @ A @ S2 )
=> ( ( groups6332066207828071664_a_int
@ ^ [K2: a] : ( if_int @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_int )
@ S2 )
= zero_zero_int ) ) ) ) ).
% sum.delta'
thf(fact_342_sum_Odelta_H,axiom,
! [S2: set_nat,A: nat,B: nat > complex] :
( ( finite_finite_nat @ S2 )
=> ( ( ( member_nat @ A @ S2 )
=> ( ( groups2073611262835488442omplex
@ ^ [K2: nat] : ( if_complex @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_complex )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S2 )
=> ( ( groups2073611262835488442omplex
@ ^ [K2: nat] : ( if_complex @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_complex )
@ S2 )
= zero_zero_complex ) ) ) ) ).
% sum.delta'
thf(fact_343_sum_Odelta_H,axiom,
! [S2: set_int,A: int,B: int > complex] :
( ( finite_finite_int @ S2 )
=> ( ( ( member_int @ A @ S2 )
=> ( ( groups3049146728041665814omplex
@ ^ [K2: int] : ( if_complex @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_complex )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_int @ A @ S2 )
=> ( ( groups3049146728041665814omplex
@ ^ [K2: int] : ( if_complex @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_complex )
@ S2 )
= zero_zero_complex ) ) ) ) ).
% sum.delta'
thf(fact_344_sum_Odelta_H,axiom,
! [S2: set_a,A: a,B: a > complex] :
( ( finite_finite_a @ S2 )
=> ( ( ( member_a @ A @ S2 )
=> ( ( groups8331919209915413362omplex
@ ^ [K2: a] : ( if_complex @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_complex )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_a @ A @ S2 )
=> ( ( groups8331919209915413362omplex
@ ^ [K2: a] : ( if_complex @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_complex )
@ S2 )
= zero_zero_complex ) ) ) ) ).
% sum.delta'
thf(fact_345_sum_Odelta_H,axiom,
! [S2: set_nat,A: nat,B: nat > nat] :
( ( finite_finite_nat @ S2 )
=> ( ( ( member_nat @ A @ S2 )
=> ( ( groups3542108847815614940at_nat
@ ^ [K2: nat] : ( if_nat @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_nat )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S2 )
=> ( ( groups3542108847815614940at_nat
@ ^ [K2: nat] : ( if_nat @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_nat )
@ S2 )
= zero_zero_nat ) ) ) ) ).
% sum.delta'
thf(fact_346_sum_Odelta_H,axiom,
! [S2: set_complex,A: complex,B: complex > complex] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( ( member_complex @ A @ S2 )
=> ( ( groups7754918857620584856omplex
@ ^ [K2: complex] : ( if_complex @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_complex )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_complex @ A @ S2 )
=> ( ( groups7754918857620584856omplex
@ ^ [K2: complex] : ( if_complex @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_complex )
@ S2 )
= zero_zero_complex ) ) ) ) ).
% sum.delta'
thf(fact_347_power__eq__0__iff,axiom,
! [A: nat,N: nat] :
( ( ( power_power_nat @ A @ N )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_348_power__eq__0__iff,axiom,
! [A: int,N: nat] :
( ( ( power_power_int @ A @ N )
= zero_zero_int )
= ( ( A = zero_zero_int )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_349_power__eq__0__iff,axiom,
! [A: complex,N: nat] :
( ( ( power_power_complex @ A @ N )
= zero_zero_complex )
= ( ( A = zero_zero_complex )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_350_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_351_power__strict__decreasing__iff,axiom,
! [B: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_352_power__strict__decreasing__iff,axiom,
! [B: int,M: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ one_one_int )
=> ( ( ord_less_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_353_power__increasing__iff,axiom,
! [B: nat,X: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_354_power__increasing__iff,axiom,
! [B: int,X: nat,Y: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_eq_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_355_power__mono__iff,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_356_power__mono__iff,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
= ( ord_less_eq_int @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_357_sum__zero__power,axiom,
! [A2: set_nat,C: nat > complex] :
( ( ( ( finite_finite_nat @ A2 )
& ( member_nat @ zero_zero_nat @ A2 ) )
=> ( ( groups2073611262835488442omplex
@ ^ [I2: nat] : ( times_times_complex @ ( C @ I2 ) @ ( power_power_complex @ zero_zero_complex @ I2 ) )
@ A2 )
= ( C @ zero_zero_nat ) ) )
& ( ~ ( ( finite_finite_nat @ A2 )
& ( member_nat @ zero_zero_nat @ A2 ) )
=> ( ( groups2073611262835488442omplex
@ ^ [I2: nat] : ( times_times_complex @ ( C @ I2 ) @ ( power_power_complex @ zero_zero_complex @ I2 ) )
@ A2 )
= zero_zero_complex ) ) ) ).
% sum_zero_power
thf(fact_358_power__decreasing__iff,axiom,
! [B: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_359_power__decreasing__iff,axiom,
! [B: int,M: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ one_one_int )
=> ( ( ord_less_eq_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_360_euclidean__domain__axioms,axiom,
( ring_e8745995371659049232in_a_b @ r
@ ^ [Uu: a] : zero_zero_nat ) ).
% euclidean_domain_axioms
thf(fact_361_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_362_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_363_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_364_sum__mono2,axiom,
! [B3: set_int,A2: set_int,F2: int > nat] :
( ( finite_finite_int @ B3 )
=> ( ( ord_less_eq_set_int @ A2 @ B3 )
=> ( ! [B4: int] :
( ( member_int @ B4 @ ( minus_minus_set_int @ B3 @ A2 ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ B4 ) ) )
=> ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F2 @ A2 ) @ ( groups4541462559716669496nt_nat @ F2 @ B3 ) ) ) ) ) ).
% sum_mono2
thf(fact_365_sum__mono2,axiom,
! [B3: set_nat,A2: set_nat,F2: nat > int] :
( ( finite_finite_nat @ B3 )
=> ( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ! [B4: nat] :
( ( member_nat @ B4 @ ( minus_minus_set_nat @ B3 @ A2 ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( F2 @ B4 ) ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F2 @ A2 ) @ ( groups3539618377306564664at_int @ F2 @ B3 ) ) ) ) ) ).
% sum_mono2
thf(fact_366_sum__mono2,axiom,
! [B3: set_int,A2: set_int,F2: int > int] :
( ( finite_finite_int @ B3 )
=> ( ( ord_less_eq_set_int @ A2 @ B3 )
=> ( ! [B4: int] :
( ( member_int @ B4 @ ( minus_minus_set_int @ B3 @ A2 ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( F2 @ B4 ) ) )
=> ( ord_less_eq_int @ ( groups4538972089207619220nt_int @ F2 @ A2 ) @ ( groups4538972089207619220nt_int @ F2 @ B3 ) ) ) ) ) ).
% sum_mono2
thf(fact_367_sum__mono2,axiom,
! [B3: set_a,A2: set_a,F2: a > nat] :
( ( finite_finite_a @ B3 )
=> ( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ! [B4: a] :
( ( member_a @ B4 @ ( minus_minus_set_a @ B3 @ A2 ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ B4 ) ) )
=> ( ord_less_eq_nat @ ( groups6334556678337121940_a_nat @ F2 @ A2 ) @ ( groups6334556678337121940_a_nat @ F2 @ B3 ) ) ) ) ) ).
% sum_mono2
thf(fact_368_sum__mono2,axiom,
! [B3: set_a,A2: set_a,F2: a > int] :
( ( finite_finite_a @ B3 )
=> ( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ! [B4: a] :
( ( member_a @ B4 @ ( minus_minus_set_a @ B3 @ A2 ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( F2 @ B4 ) ) )
=> ( ord_less_eq_int @ ( groups6332066207828071664_a_int @ F2 @ A2 ) @ ( groups6332066207828071664_a_int @ F2 @ B3 ) ) ) ) ) ).
% sum_mono2
thf(fact_369_sum__mono2,axiom,
! [B3: set_nat,A2: set_nat,F2: nat > nat] :
( ( finite_finite_nat @ B3 )
=> ( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ! [B4: nat] :
( ( member_nat @ B4 @ ( minus_minus_set_nat @ B3 @ A2 ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ B4 ) ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F2 @ A2 ) @ ( groups3542108847815614940at_nat @ F2 @ B3 ) ) ) ) ) ).
% sum_mono2
thf(fact_370_sum__mono2,axiom,
! [B3: set_set_a,A2: set_set_a,F2: set_a > nat] :
( ( finite_finite_set_a @ B3 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B3 )
=> ( ! [B4: set_a] :
( ( member_set_a @ B4 @ ( minus_5736297505244876581_set_a @ B3 @ A2 ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ B4 ) ) )
=> ( ord_less_eq_nat @ ( groups6141743369313575924_a_nat @ F2 @ A2 ) @ ( groups6141743369313575924_a_nat @ F2 @ B3 ) ) ) ) ) ).
% sum_mono2
thf(fact_371_sum__mono2,axiom,
! [B3: set_set_a,A2: set_set_a,F2: set_a > int] :
( ( finite_finite_set_a @ B3 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B3 )
=> ( ! [B4: set_a] :
( ( member_set_a @ B4 @ ( minus_5736297505244876581_set_a @ B3 @ A2 ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( F2 @ B4 ) ) )
=> ( ord_less_eq_int @ ( groups6139252898804525648_a_int @ F2 @ A2 ) @ ( groups6139252898804525648_a_int @ F2 @ B3 ) ) ) ) ) ).
% sum_mono2
thf(fact_372_sum__mono2,axiom,
! [B3: set_list_a,A2: set_list_a,F2: list_a > int] :
( ( finite_finite_list_a @ B3 )
=> ( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( ! [B4: list_a] :
( ( member_list_a @ B4 @ ( minus_646659088055828811list_a @ B3 @ A2 ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( F2 @ B4 ) ) )
=> ( ord_less_eq_int @ ( groups5518757228788810486_a_int @ F2 @ A2 ) @ ( groups5518757228788810486_a_int @ F2 @ B3 ) ) ) ) ) ).
% sum_mono2
thf(fact_373_sum__mono2,axiom,
! [B3: set_list_a,A2: set_list_a,F2: list_a > nat] :
( ( finite_finite_list_a @ B3 )
=> ( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( ! [B4: list_a] :
( ( member_list_a @ B4 @ ( minus_646659088055828811list_a @ B3 @ A2 ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ B4 ) ) )
=> ( ord_less_eq_nat @ ( groups5521247699297860762_a_nat @ F2 @ A2 ) @ ( groups5521247699297860762_a_nat @ F2 @ B3 ) ) ) ) ) ).
% sum_mono2
thf(fact_374_sum__nonneg,axiom,
! [A2: set_a,F2: a > nat] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ X3 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups6334556678337121940_a_nat @ F2 @ A2 ) ) ) ).
% sum_nonneg
thf(fact_375_sum__nonneg,axiom,
! [A2: set_a,F2: a > int] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F2 @ X3 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups6332066207828071664_a_int @ F2 @ A2 ) ) ) ).
% sum_nonneg
thf(fact_376_sum__nonneg,axiom,
! [A2: set_nat,F2: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ X3 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups3542108847815614940at_nat @ F2 @ A2 ) ) ) ).
% sum_nonneg
thf(fact_377_sum__nonneg,axiom,
! [A2: set_set_a,F2: set_a > nat] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ X3 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups6141743369313575924_a_nat @ F2 @ A2 ) ) ) ).
% sum_nonneg
thf(fact_378_sum__nonneg,axiom,
! [A2: set_list_a,F2: list_a > int] :
( ! [X3: list_a] :
( ( member_list_a @ X3 @ A2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F2 @ X3 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups5518757228788810486_a_int @ F2 @ A2 ) ) ) ).
% sum_nonneg
thf(fact_379_sum__nonneg,axiom,
! [A2: set_set_a,F2: set_a > int] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F2 @ X3 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups6139252898804525648_a_int @ F2 @ A2 ) ) ) ).
% sum_nonneg
thf(fact_380_sum__nonneg,axiom,
! [A2: set_list_a,F2: list_a > nat] :
( ! [X3: list_a] :
( ( member_list_a @ X3 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ X3 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups5521247699297860762_a_nat @ F2 @ A2 ) ) ) ).
% sum_nonneg
thf(fact_381_sum__nonneg,axiom,
! [A2: set_set_list_a,F2: set_list_a > nat] :
( ! [X3: set_list_a] :
( ( member_set_list_a @ X3 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ X3 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups5993734322560061562_a_nat @ F2 @ A2 ) ) ) ).
% sum_nonneg
thf(fact_382_sum__nonneg,axiom,
! [A2: set_set_list_a,F2: set_list_a > int] :
( ! [X3: set_list_a] :
( ( member_set_list_a @ X3 @ A2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F2 @ X3 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups5991243852051011286_a_int @ F2 @ A2 ) ) ) ).
% sum_nonneg
thf(fact_383_sum__nonneg,axiom,
! [A2: set_li3422455791611400469list_a,F2: ( list_list_a > list_a ) > nat] :
( ! [X3: list_list_a > list_a] :
( ( member7168557129179038582list_a @ X3 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ X3 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups5819086642190997227_a_nat @ F2 @ A2 ) ) ) ).
% sum_nonneg
thf(fact_384_sum__nonpos,axiom,
! [A2: set_a,F2: a > nat] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups6334556678337121940_a_nat @ F2 @ A2 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_385_sum__nonpos,axiom,
! [A2: set_a,F2: a > int] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ zero_zero_int ) )
=> ( ord_less_eq_int @ ( groups6332066207828071664_a_int @ F2 @ A2 ) @ zero_zero_int ) ) ).
% sum_nonpos
thf(fact_386_sum__nonpos,axiom,
! [A2: set_nat,F2: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F2 @ A2 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_387_sum__nonpos,axiom,
! [A2: set_set_a,F2: set_a > nat] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups6141743369313575924_a_nat @ F2 @ A2 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_388_sum__nonpos,axiom,
! [A2: set_list_a,F2: list_a > int] :
( ! [X3: list_a] :
( ( member_list_a @ X3 @ A2 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ zero_zero_int ) )
=> ( ord_less_eq_int @ ( groups5518757228788810486_a_int @ F2 @ A2 ) @ zero_zero_int ) ) ).
% sum_nonpos
thf(fact_389_sum__nonpos,axiom,
! [A2: set_set_a,F2: set_a > int] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ zero_zero_int ) )
=> ( ord_less_eq_int @ ( groups6139252898804525648_a_int @ F2 @ A2 ) @ zero_zero_int ) ) ).
% sum_nonpos
thf(fact_390_sum__nonpos,axiom,
! [A2: set_list_a,F2: list_a > nat] :
( ! [X3: list_a] :
( ( member_list_a @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups5521247699297860762_a_nat @ F2 @ A2 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_391_sum__nonpos,axiom,
! [A2: set_set_list_a,F2: set_list_a > nat] :
( ! [X3: set_list_a] :
( ( member_set_list_a @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups5993734322560061562_a_nat @ F2 @ A2 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_392_sum__nonpos,axiom,
! [A2: set_set_list_a,F2: set_list_a > int] :
( ! [X3: set_list_a] :
( ( member_set_list_a @ X3 @ A2 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ zero_zero_int ) )
=> ( ord_less_eq_int @ ( groups5991243852051011286_a_int @ F2 @ A2 ) @ zero_zero_int ) ) ).
% sum_nonpos
thf(fact_393_sum__nonpos,axiom,
! [A2: set_li3422455791611400469list_a,F2: ( list_list_a > list_a ) > nat] :
( ! [X3: list_list_a > list_a] :
( ( member7168557129179038582list_a @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups5819086642190997227_a_nat @ F2 @ A2 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_394_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_395_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_396_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_397_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_398_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_399_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_400_Nat_Oex__has__greatest__nat,axiom,
! [P2: nat > $o,K: nat,B: nat] :
( ( P2 @ K )
=> ( ! [Y2: nat] :
( ( P2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ? [X3: nat] :
( ( P2 @ X3 )
& ! [Y3: nat] :
( ( P2 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_401_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_402_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_403_zero__reorient,axiom,
! [X: complex] :
( ( zero_zero_complex = X )
= ( X = zero_zero_complex ) ) ).
% zero_reorient
thf(fact_404_zero__reorient,axiom,
! [X: multiset_a] :
( ( zero_zero_multiset_a = X )
= ( X = zero_zero_multiset_a ) ) ).
% zero_reorient
thf(fact_405_zero__reorient,axiom,
! [X: multiset_list_a] :
( ( zero_z4454100511807792257list_a = X )
= ( X = zero_z4454100511807792257list_a ) ) ).
% zero_reorient
thf(fact_406_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_407_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A3: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ A3 @ B2 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_408_sum_Onon__neutral_H,axiom,
! [G: nat > nat,I3: set_nat] :
( ( groups1986416967739987077at_nat @ G
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ I3 )
& ( ( G @ X2 )
!= zero_zero_nat ) ) ) )
= ( groups1986416967739987077at_nat @ G @ I3 ) ) ).
% sum.non_neutral'
thf(fact_409_sum_Onon__neutral_H,axiom,
! [G: int > nat,I3: set_int] :
( ( groups2985770679641041633nt_nat @ G
@ ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ I3 )
& ( ( G @ X2 )
!= zero_zero_nat ) ) ) )
= ( groups2985770679641041633nt_nat @ G @ I3 ) ) ).
% sum.non_neutral'
thf(fact_410_sum_Onon__neutral_H,axiom,
! [G: a > nat,I3: set_a] :
( ( groups7493686093766423595_a_nat @ G
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ I3 )
& ( ( G @ X2 )
!= zero_zero_nat ) ) ) )
= ( groups7493686093766423595_a_nat @ G @ I3 ) ) ).
% sum.non_neutral'
thf(fact_411_sum_Onon__neutral_H,axiom,
! [G: nat > int,I3: set_nat] :
( ( groups1983926497230936801at_int @ G
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ I3 )
& ( ( G @ X2 )
!= zero_zero_int ) ) ) )
= ( groups1983926497230936801at_int @ G @ I3 ) ) ).
% sum.non_neutral'
thf(fact_412_sum_Onon__neutral_H,axiom,
! [G: int > int,I3: set_int] :
( ( groups2983280209131991357nt_int @ G
@ ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ I3 )
& ( ( G @ X2 )
!= zero_zero_int ) ) ) )
= ( groups2983280209131991357nt_int @ G @ I3 ) ) ).
% sum.non_neutral'
thf(fact_413_sum_Onon__neutral_H,axiom,
! [G: a > int,I3: set_a] :
( ( groups7491195623257373319_a_int @ G
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ I3 )
& ( ( G @ X2 )
!= zero_zero_int ) ) ) )
= ( groups7491195623257373319_a_int @ G @ I3 ) ) ).
% sum.non_neutral'
thf(fact_414_sum_Onon__neutral_H,axiom,
! [G: nat > complex,I3: set_nat] :
( ( groups8515261248781899619omplex @ G
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ I3 )
& ( ( G @ X2 )
!= zero_zero_complex ) ) ) )
= ( groups8515261248781899619omplex @ G @ I3 ) ) ).
% sum.non_neutral'
thf(fact_415_sum_Onon__neutral_H,axiom,
! [G: int > complex,I3: set_int] :
( ( groups267424677133301183omplex @ G
@ ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ I3 )
& ( ( G @ X2 )
!= zero_zero_complex ) ) ) )
= ( groups267424677133301183omplex @ G @ I3 ) ) ).
% sum.non_neutral'
thf(fact_416_sum_Onon__neutral_H,axiom,
! [G: a > complex,I3: set_a] :
( ( groups3888493890042768137omplex @ G
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ I3 )
& ( ( G @ X2 )
!= zero_zero_complex ) ) ) )
= ( groups3888493890042768137omplex @ G @ I3 ) ) ).
% sum.non_neutral'
thf(fact_417_sum_Onon__neutral_H,axiom,
! [G: set_a > nat,I3: set_set_a] :
( ( groups3149302966386732427_a_nat @ G
@ ( collect_set_a
@ ^ [X2: set_a] :
( ( member_set_a @ X2 @ I3 )
& ( ( G @ X2 )
!= zero_zero_nat ) ) ) )
= ( groups3149302966386732427_a_nat @ G @ I3 ) ) ).
% sum.non_neutral'
thf(fact_418_sum_Ocong_H,axiom,
! [A2: set_list_a,B3: set_list_a,G: list_a > nat,H: list_a > nat] :
( ( A2 = B3 )
=> ( ! [X3: list_a] :
( ( member_list_a @ X3 @ B3 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups1083796956622547633_a_nat @ G @ A2 )
= ( groups1083796956622547633_a_nat @ H @ B3 ) ) ) ) ).
% sum.cong'
thf(fact_419_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_420_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_421_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_422_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_423_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_424_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_le_one
thf(fact_425_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_426_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_427_zero__le__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_428_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_429_mult__nonneg__nonpos2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_430_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_431_mult__nonpos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_432_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_433_mult__nonneg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_434_mult__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 @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_435_mult__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_436_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_437_split__mult__neg__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_438_mult__le__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_439_mult__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_440_mult__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_441_mult__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_442_mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_443_mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_444_mult__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_445_mult__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_446_split__mult__pos__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_447_zero__le__square,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% zero_le_square
thf(fact_448_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_449_mult__mono_H,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_450_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_451_mult__mono,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_452_sum__strict__mono2,axiom,
! [B3: set_int,A2: set_int,B: int,F2: int > nat] :
( ( finite_finite_int @ B3 )
=> ( ( ord_less_eq_set_int @ A2 @ B3 )
=> ( ( member_int @ B @ ( minus_minus_set_int @ B3 @ A2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F2 @ B ) )
=> ( ! [X3: int] :
( ( member_int @ X3 @ B3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ X3 ) ) )
=> ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F2 @ A2 ) @ ( groups4541462559716669496nt_nat @ F2 @ B3 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_453_sum__strict__mono2,axiom,
! [B3: set_nat,A2: set_nat,B: nat,F2: nat > int] :
( ( finite_finite_nat @ B3 )
=> ( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( member_nat @ B @ ( minus_minus_set_nat @ B3 @ A2 ) )
=> ( ( ord_less_int @ zero_zero_int @ ( F2 @ B ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B3 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F2 @ X3 ) ) )
=> ( ord_less_int @ ( groups3539618377306564664at_int @ F2 @ A2 ) @ ( groups3539618377306564664at_int @ F2 @ B3 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_454_sum__strict__mono2,axiom,
! [B3: set_int,A2: set_int,B: int,F2: int > int] :
( ( finite_finite_int @ B3 )
=> ( ( ord_less_eq_set_int @ A2 @ B3 )
=> ( ( member_int @ B @ ( minus_minus_set_int @ B3 @ A2 ) )
=> ( ( ord_less_int @ zero_zero_int @ ( F2 @ B ) )
=> ( ! [X3: int] :
( ( member_int @ X3 @ B3 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F2 @ X3 ) ) )
=> ( ord_less_int @ ( groups4538972089207619220nt_int @ F2 @ A2 ) @ ( groups4538972089207619220nt_int @ F2 @ B3 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_455_sum__strict__mono2,axiom,
! [B3: set_a,A2: set_a,B: a,F2: a > nat] :
( ( finite_finite_a @ B3 )
=> ( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( member_a @ B @ ( minus_minus_set_a @ B3 @ A2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F2 @ B ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ B3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ X3 ) ) )
=> ( ord_less_nat @ ( groups6334556678337121940_a_nat @ F2 @ A2 ) @ ( groups6334556678337121940_a_nat @ F2 @ B3 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_456_sum__strict__mono2,axiom,
! [B3: set_a,A2: set_a,B: a,F2: a > int] :
( ( finite_finite_a @ B3 )
=> ( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( member_a @ B @ ( minus_minus_set_a @ B3 @ A2 ) )
=> ( ( ord_less_int @ zero_zero_int @ ( F2 @ B ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ B3 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F2 @ X3 ) ) )
=> ( ord_less_int @ ( groups6332066207828071664_a_int @ F2 @ A2 ) @ ( groups6332066207828071664_a_int @ F2 @ B3 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_457_sum__strict__mono2,axiom,
! [B3: set_nat,A2: set_nat,B: nat,F2: nat > nat] :
( ( finite_finite_nat @ B3 )
=> ( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( member_nat @ B @ ( minus_minus_set_nat @ B3 @ A2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F2 @ B ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ X3 ) ) )
=> ( ord_less_nat @ ( groups3542108847815614940at_nat @ F2 @ A2 ) @ ( groups3542108847815614940at_nat @ F2 @ B3 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_458_sum__strict__mono2,axiom,
! [B3: set_set_a,A2: set_set_a,B: set_a,F2: set_a > nat] :
( ( finite_finite_set_a @ B3 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B3 )
=> ( ( member_set_a @ B @ ( minus_5736297505244876581_set_a @ B3 @ A2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F2 @ B ) )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ B3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ X3 ) ) )
=> ( ord_less_nat @ ( groups6141743369313575924_a_nat @ F2 @ A2 ) @ ( groups6141743369313575924_a_nat @ F2 @ B3 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_459_sum__strict__mono2,axiom,
! [B3: set_set_a,A2: set_set_a,B: set_a,F2: set_a > int] :
( ( finite_finite_set_a @ B3 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B3 )
=> ( ( member_set_a @ B @ ( minus_5736297505244876581_set_a @ B3 @ A2 ) )
=> ( ( ord_less_int @ zero_zero_int @ ( F2 @ B ) )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ B3 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F2 @ X3 ) ) )
=> ( ord_less_int @ ( groups6139252898804525648_a_int @ F2 @ A2 ) @ ( groups6139252898804525648_a_int @ F2 @ B3 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_460_sum__strict__mono2,axiom,
! [B3: set_list_a,A2: set_list_a,B: list_a,F2: list_a > int] :
( ( finite_finite_list_a @ B3 )
=> ( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( ( member_list_a @ B @ ( minus_646659088055828811list_a @ B3 @ A2 ) )
=> ( ( ord_less_int @ zero_zero_int @ ( F2 @ B ) )
=> ( ! [X3: list_a] :
( ( member_list_a @ X3 @ B3 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F2 @ X3 ) ) )
=> ( ord_less_int @ ( groups5518757228788810486_a_int @ F2 @ A2 ) @ ( groups5518757228788810486_a_int @ F2 @ B3 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_461_sum__strict__mono2,axiom,
! [B3: set_list_a,A2: set_list_a,B: list_a,F2: list_a > nat] :
( ( finite_finite_list_a @ B3 )
=> ( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( ( member_list_a @ B @ ( minus_646659088055828811list_a @ B3 @ A2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F2 @ B ) )
=> ( ! [X3: list_a] :
( ( member_list_a @ X3 @ B3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ X3 ) ) )
=> ( ord_less_nat @ ( groups5521247699297860762_a_nat @ F2 @ A2 ) @ ( groups5521247699297860762_a_nat @ F2 @ B3 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_462_zero__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% zero_le_power
thf(fact_463_zero__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_le_power
thf(fact_464_power__mono,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% power_mono
thf(fact_465_power__mono,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% power_mono
thf(fact_466_ex__least__nat__le,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K3 )
=> ~ ( P2 @ I4 ) )
& ( P2 @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_467_Diff__infinite__finite,axiom,
! [T2: set_list_list_a,S2: set_list_list_a] :
( ( finite1660835950917165235list_a @ T2 )
=> ( ~ ( finite1660835950917165235list_a @ S2 )
=> ~ ( finite1660835950917165235list_a @ ( minus_5335179877275218001list_a @ S2 @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_468_Diff__infinite__finite,axiom,
! [T2: set_list_a,S2: set_list_a] :
( ( finite_finite_list_a @ T2 )
=> ( ~ ( finite_finite_list_a @ S2 )
=> ~ ( finite_finite_list_a @ ( minus_646659088055828811list_a @ S2 @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_469_Diff__infinite__finite,axiom,
! [T2: set_nat,S2: set_nat] :
( ( finite_finite_nat @ T2 )
=> ( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S2 @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_470_Diff__infinite__finite,axiom,
! [T2: set_int,S2: set_int] :
( ( finite_finite_int @ T2 )
=> ( ~ ( finite_finite_int @ S2 )
=> ~ ( finite_finite_int @ ( minus_minus_set_int @ S2 @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_471_Diff__infinite__finite,axiom,
! [T2: set_a,S2: set_a] :
( ( finite_finite_a @ T2 )
=> ( ~ ( finite_finite_a @ S2 )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S2 @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_472_diff__mono,axiom,
! [A: int,B: int,D2: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ D2 @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D2 ) ) ) ) ).
% diff_mono
thf(fact_473_diff__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_474_diff__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_475_diff__eq__diff__less__eq,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D2 ) )
=> ( ( ord_less_eq_int @ A @ B )
= ( ord_less_eq_int @ C @ D2 ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_476_finite__psubset__induct,axiom,
! [A2: set_list_list_a,P2: set_list_list_a > $o] :
( ( finite1660835950917165235list_a @ A2 )
=> ( ! [A4: set_list_list_a] :
( ( finite1660835950917165235list_a @ A4 )
=> ( ! [B5: set_list_list_a] :
( ( ord_le5338140678153942166list_a @ B5 @ A4 )
=> ( P2 @ B5 ) )
=> ( P2 @ A4 ) ) )
=> ( P2 @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_477_finite__psubset__induct,axiom,
! [A2: set_list_a,P2: set_list_a > $o] :
( ( finite_finite_list_a @ A2 )
=> ( ! [A4: set_list_a] :
( ( finite_finite_list_a @ A4 )
=> ( ! [B5: set_list_a] :
( ( ord_less_set_list_a @ B5 @ A4 )
=> ( P2 @ B5 ) )
=> ( P2 @ A4 ) ) )
=> ( P2 @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_478_finite__psubset__induct,axiom,
! [A2: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ! [B5: set_nat] :
( ( ord_less_set_nat @ B5 @ A4 )
=> ( P2 @ B5 ) )
=> ( P2 @ A4 ) ) )
=> ( P2 @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_479_finite__psubset__induct,axiom,
! [A2: set_int,P2: set_int > $o] :
( ( finite_finite_int @ A2 )
=> ( ! [A4: set_int] :
( ( finite_finite_int @ A4 )
=> ( ! [B5: set_int] :
( ( ord_less_set_int @ B5 @ A4 )
=> ( P2 @ B5 ) )
=> ( P2 @ A4 ) ) )
=> ( P2 @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_480_finite__psubset__induct,axiom,
! [A2: set_a,P2: set_a > $o] :
( ( finite_finite_a @ A2 )
=> ( ! [A4: set_a] :
( ( finite_finite_a @ A4 )
=> ( ! [B5: set_a] :
( ( ord_less_set_a @ B5 @ A4 )
=> ( P2 @ B5 ) )
=> ( P2 @ A4 ) ) )
=> ( P2 @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_481_finite__has__minimal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_eq_nat @ X3 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_482_finite__has__minimal2,axiom,
! [A2: set_int,A: int] :
( ( finite_finite_int @ A2 )
=> ( ( member_int @ A @ A2 )
=> ? [X3: int] :
( ( member_int @ X3 @ A2 )
& ( ord_less_eq_int @ X3 @ A )
& ! [Xa: int] :
( ( member_int @ Xa @ A2 )
=> ( ( ord_less_eq_int @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_483_finite__has__minimal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ( ord_less_eq_set_a @ X3 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_484_finite__has__minimal2,axiom,
! [A2: set_set_list_a,A: set_list_a] :
( ( finite5282473924520328461list_a @ A2 )
=> ( ( member_set_list_a @ A @ A2 )
=> ? [X3: set_list_a] :
( ( member_set_list_a @ X3 @ A2 )
& ( ord_le8861187494160871172list_a @ X3 @ A )
& ! [Xa: set_list_a] :
( ( member_set_list_a @ Xa @ A2 )
=> ( ( ord_le8861187494160871172list_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_485_finite__has__maximal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_eq_nat @ A @ X3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_486_finite__has__maximal2,axiom,
! [A2: set_int,A: int] :
( ( finite_finite_int @ A2 )
=> ( ( member_int @ A @ A2 )
=> ? [X3: int] :
( ( member_int @ X3 @ A2 )
& ( ord_less_eq_int @ A @ X3 )
& ! [Xa: int] :
( ( member_int @ Xa @ A2 )
=> ( ( ord_less_eq_int @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_487_finite__has__maximal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ( ord_less_eq_set_a @ A @ X3 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_488_finite__has__maximal2,axiom,
! [A2: set_set_list_a,A: set_list_a] :
( ( finite5282473924520328461list_a @ A2 )
=> ( ( member_set_list_a @ A @ A2 )
=> ? [X3: set_list_a] :
( ( member_set_list_a @ X3 @ A2 )
& ( ord_le8861187494160871172list_a @ A @ X3 )
& ! [Xa: set_list_a] :
( ( member_set_list_a @ Xa @ A2 )
=> ( ( ord_le8861187494160871172list_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_489_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_490_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_491_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_492_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_493_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: complex,Z2: complex] : ( Y4 = Z2 ) )
= ( ^ [A3: complex,B2: complex] :
( ( minus_minus_complex @ A3 @ B2 )
= zero_zero_complex ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_494_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: int,Z2: int] : ( Y4 = Z2 ) )
= ( ^ [A3: int,B2: int] :
( ( minus_minus_int @ A3 @ B2 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_495_power__decreasing,axiom,
! [N: nat,N3: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N3 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_496_power__decreasing,axiom,
! [N: nat,N3: nat,A: int] :
( ( ord_less_eq_nat @ N @ N3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N3 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_497_sum_Oneutral,axiom,
! [A2: set_nat,G: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( G @ X3 )
= zero_zero_nat ) )
=> ( ( groups3542108847815614940at_nat @ G @ A2 )
= zero_zero_nat ) ) ).
% sum.neutral
thf(fact_498_sum_Oneutral,axiom,
! [A2: set_list_a,G: list_a > nat] :
( ! [X3: list_a] :
( ( member_list_a @ X3 @ A2 )
=> ( ( G @ X3 )
= zero_zero_nat ) )
=> ( ( groups5521247699297860762_a_nat @ G @ A2 )
= zero_zero_nat ) ) ).
% sum.neutral
thf(fact_499_sum_Oneutral,axiom,
! [A2: set_complex,G: complex > complex] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ( G @ X3 )
= zero_zero_complex ) )
=> ( ( groups7754918857620584856omplex @ G @ A2 )
= zero_zero_complex ) ) ).
% sum.neutral
thf(fact_500_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: a > nat,A2: set_a] :
( ( ( groups6334556678337121940_a_nat @ G @ A2 )
!= zero_zero_nat )
=> ~ ! [A5: a] :
( ( member_a @ A5 @ A2 )
=> ( ( G @ A5 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_501_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: a > int,A2: set_a] :
( ( ( groups6332066207828071664_a_int @ G @ A2 )
!= zero_zero_int )
=> ~ ! [A5: a] :
( ( member_a @ A5 @ A2 )
=> ( ( G @ A5 )
= zero_zero_int ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_502_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: a > complex,A2: set_a] :
( ( ( groups8331919209915413362omplex @ G @ A2 )
!= zero_zero_complex )
=> ~ ! [A5: a] :
( ( member_a @ A5 @ A2 )
=> ( ( G @ A5 )
= zero_zero_complex ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_503_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > nat,A2: set_nat] :
( ( ( groups3542108847815614940at_nat @ G @ A2 )
!= zero_zero_nat )
=> ~ ! [A5: nat] :
( ( member_nat @ A5 @ A2 )
=> ( ( G @ A5 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_504_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: complex > complex,A2: set_complex] :
( ( ( groups7754918857620584856omplex @ G @ A2 )
!= zero_zero_complex )
=> ~ ! [A5: complex] :
( ( member_complex @ A5 @ A2 )
=> ( ( G @ A5 )
= zero_zero_complex ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_505_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: set_a > nat,A2: set_set_a] :
( ( ( groups6141743369313575924_a_nat @ G @ A2 )
!= zero_zero_nat )
=> ~ ! [A5: set_a] :
( ( member_set_a @ A5 @ A2 )
=> ( ( G @ A5 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_506_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: list_a > int,A2: set_list_a] :
( ( ( groups5518757228788810486_a_int @ G @ A2 )
!= zero_zero_int )
=> ~ ! [A5: list_a] :
( ( member_list_a @ A5 @ A2 )
=> ( ( G @ A5 )
= zero_zero_int ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_507_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: set_a > int,A2: set_set_a] :
( ( ( groups6139252898804525648_a_int @ G @ A2 )
!= zero_zero_int )
=> ~ ! [A5: set_a] :
( ( member_set_a @ A5 @ A2 )
=> ( ( G @ A5 )
= zero_zero_int ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_508_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: list_a > complex,A2: set_list_a] :
( ( ( groups3853138014179693944omplex @ G @ A2 )
!= zero_zero_complex )
=> ~ ! [A5: list_a] :
( ( member_list_a @ A5 @ A2 )
=> ( ( G @ A5 )
= zero_zero_complex ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_509_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: set_a > complex,A2: set_set_a] :
( ( ( groups6391100438827976914omplex @ G @ A2 )
!= zero_zero_complex )
=> ~ ! [A5: set_a] :
( ( member_set_a @ A5 @ A2 )
=> ( ( G @ A5 )
= zero_zero_complex ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_510_power__eq__imp__eq__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ( power_power_nat @ A @ N )
= ( power_power_nat @ B @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_511_power__eq__imp__eq__base,axiom,
! [A: int,N: nat,B: int] :
( ( ( power_power_int @ A @ N )
= ( power_power_int @ B @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_512_power__eq__iff__eq__base,axiom,
! [N: nat,A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ( power_power_nat @ A @ N )
= ( power_power_nat @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_513_power__eq__iff__eq__base,axiom,
! [N: nat,A: int,B: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ( power_power_int @ A @ N )
= ( power_power_int @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_514_card__le__sym__Diff,axiom,
! [A2: set_set_list_a,B3: set_set_list_a] :
( ( finite5282473924520328461list_a @ A2 )
=> ( ( finite5282473924520328461list_a @ B3 )
=> ( ( ord_less_eq_nat @ ( finite7490188192133390220list_a @ A2 ) @ ( finite7490188192133390220list_a @ B3 ) )
=> ( ord_less_eq_nat @ ( finite7490188192133390220list_a @ ( minus_4782336368215558443list_a @ A2 @ B3 ) ) @ ( finite7490188192133390220list_a @ ( minus_4782336368215558443list_a @ B3 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_515_card__le__sym__Diff,axiom,
! [A2: set_set_a,B3: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( finite_finite_set_a @ B3 )
=> ( ( ord_less_eq_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ B3 ) )
=> ( ord_less_eq_nat @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B3 ) ) @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ B3 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_516_card__le__sym__Diff,axiom,
! [A2: set_list_list_a,B3: set_list_list_a] :
( ( finite1660835950917165235list_a @ A2 )
=> ( ( finite1660835950917165235list_a @ B3 )
=> ( ( ord_less_eq_nat @ ( finite9134805042761151410list_a @ A2 ) @ ( finite9134805042761151410list_a @ B3 ) )
=> ( ord_less_eq_nat @ ( finite9134805042761151410list_a @ ( minus_5335179877275218001list_a @ A2 @ B3 ) ) @ ( finite9134805042761151410list_a @ ( minus_5335179877275218001list_a @ B3 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_517_card__le__sym__Diff,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( finite_finite_list_a @ A2 )
=> ( ( finite_finite_list_a @ B3 )
=> ( ( ord_less_eq_nat @ ( finite_card_list_a @ A2 ) @ ( finite_card_list_a @ B3 ) )
=> ( ord_less_eq_nat @ ( finite_card_list_a @ ( minus_646659088055828811list_a @ A2 @ B3 ) ) @ ( finite_card_list_a @ ( minus_646659088055828811list_a @ B3 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_518_card__le__sym__Diff,axiom,
! [A2: set_nat,B3: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B3 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B3 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B3 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B3 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_519_card__le__sym__Diff,axiom,
! [A2: set_int,B3: set_int] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ B3 )
=> ( ( ord_less_eq_nat @ ( finite_card_int @ A2 ) @ ( finite_card_int @ B3 ) )
=> ( ord_less_eq_nat @ ( finite_card_int @ ( minus_minus_set_int @ A2 @ B3 ) ) @ ( finite_card_int @ ( minus_minus_set_int @ B3 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_520_card__le__sym__Diff,axiom,
! [A2: set_a,B3: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B3 )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B3 ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ B3 ) ) @ ( finite_card_a @ ( minus_minus_set_a @ B3 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_521_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S3: set_int,T3: set_int,S2: set_int,I: int > int,J: int > int,T2: set_int,G: int > nat,H: int > nat] :
( ( finite_finite_int @ S3 )
=> ( ( finite_finite_int @ T3 )
=> ( ! [A5: int] :
( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S3 ) )
=> ( ( I @ ( J @ A5 ) )
= A5 ) )
=> ( ! [A5: int] :
( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S3 ) )
=> ( member_int @ ( J @ A5 ) @ ( minus_minus_set_int @ T2 @ T3 ) ) )
=> ( ! [B4: int] :
( ( member_int @ B4 @ ( minus_minus_set_int @ T2 @ T3 ) )
=> ( ( J @ ( I @ B4 ) )
= B4 ) )
=> ( ! [B4: int] :
( ( member_int @ B4 @ ( minus_minus_set_int @ T2 @ T3 ) )
=> ( member_int @ ( I @ B4 ) @ ( minus_minus_set_int @ S2 @ S3 ) ) )
=> ( ! [A5: int] :
( ( member_int @ A5 @ S3 )
=> ( ( G @ A5 )
= zero_zero_nat ) )
=> ( ! [B4: int] :
( ( member_int @ B4 @ T3 )
=> ( ( H @ B4 )
= zero_zero_nat ) )
=> ( ! [A5: int] :
( ( member_int @ A5 @ S2 )
=> ( ( H @ ( J @ A5 ) )
= ( G @ A5 ) ) )
=> ( ( groups4541462559716669496nt_nat @ G @ S2 )
= ( groups4541462559716669496nt_nat @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_522_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S3: set_int,T3: set_a,S2: set_int,I: a > int,J: int > a,T2: set_a,G: int > nat,H: a > nat] :
( ( finite_finite_int @ S3 )
=> ( ( finite_finite_a @ T3 )
=> ( ! [A5: int] :
( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S3 ) )
=> ( ( I @ ( J @ A5 ) )
= A5 ) )
=> ( ! [A5: int] :
( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S3 ) )
=> ( member_a @ ( J @ A5 ) @ ( minus_minus_set_a @ T2 @ T3 ) ) )
=> ( ! [B4: a] :
( ( member_a @ B4 @ ( minus_minus_set_a @ T2 @ T3 ) )
=> ( ( J @ ( I @ B4 ) )
= B4 ) )
=> ( ! [B4: a] :
( ( member_a @ B4 @ ( minus_minus_set_a @ T2 @ T3 ) )
=> ( member_int @ ( I @ B4 ) @ ( minus_minus_set_int @ S2 @ S3 ) ) )
=> ( ! [A5: int] :
( ( member_int @ A5 @ S3 )
=> ( ( G @ A5 )
= zero_zero_nat ) )
=> ( ! [B4: a] :
( ( member_a @ B4 @ T3 )
=> ( ( H @ B4 )
= zero_zero_nat ) )
=> ( ! [A5: int] :
( ( member_int @ A5 @ S2 )
=> ( ( H @ ( J @ A5 ) )
= ( G @ A5 ) ) )
=> ( ( groups4541462559716669496nt_nat @ G @ S2 )
= ( groups6334556678337121940_a_nat @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_523_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S3: set_a,T3: set_int,S2: set_a,I: int > a,J: a > int,T2: set_int,G: a > nat,H: int > nat] :
( ( finite_finite_a @ S3 )
=> ( ( finite_finite_int @ T3 )
=> ( ! [A5: a] :
( ( member_a @ A5 @ ( minus_minus_set_a @ S2 @ S3 ) )
=> ( ( I @ ( J @ A5 ) )
= A5 ) )
=> ( ! [A5: a] :
( ( member_a @ A5 @ ( minus_minus_set_a @ S2 @ S3 ) )
=> ( member_int @ ( J @ A5 ) @ ( minus_minus_set_int @ T2 @ T3 ) ) )
=> ( ! [B4: int] :
( ( member_int @ B4 @ ( minus_minus_set_int @ T2 @ T3 ) )
=> ( ( J @ ( I @ B4 ) )
= B4 ) )
=> ( ! [B4: int] :
( ( member_int @ B4 @ ( minus_minus_set_int @ T2 @ T3 ) )
=> ( member_a @ ( I @ B4 ) @ ( minus_minus_set_a @ S2 @ S3 ) ) )
=> ( ! [A5: a] :
( ( member_a @ A5 @ S3 )
=> ( ( G @ A5 )
= zero_zero_nat ) )
=> ( ! [B4: int] :
( ( member_int @ B4 @ T3 )
=> ( ( H @ B4 )
= zero_zero_nat ) )
=> ( ! [A5: a] :
( ( member_a @ A5 @ S2 )
=> ( ( H @ ( J @ A5 ) )
= ( G @ A5 ) ) )
=> ( ( groups6334556678337121940_a_nat @ G @ S2 )
= ( groups4541462559716669496nt_nat @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_524_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S3: set_a,T3: set_a,S2: set_a,I: a > a,J: a > a,T2: set_a,G: a > nat,H: a > nat] :
( ( finite_finite_a @ S3 )
=> ( ( finite_finite_a @ T3 )
=> ( ! [A5: a] :
( ( member_a @ A5 @ ( minus_minus_set_a @ S2 @ S3 ) )
=> ( ( I @ ( J @ A5 ) )
= A5 ) )
=> ( ! [A5: a] :
( ( member_a @ A5 @ ( minus_minus_set_a @ S2 @ S3 ) )
=> ( member_a @ ( J @ A5 ) @ ( minus_minus_set_a @ T2 @ T3 ) ) )
=> ( ! [B4: a] :
( ( member_a @ B4 @ ( minus_minus_set_a @ T2 @ T3 ) )
=> ( ( J @ ( I @ B4 ) )
= B4 ) )
=> ( ! [B4: a] :
( ( member_a @ B4 @ ( minus_minus_set_a @ T2 @ T3 ) )
=> ( member_a @ ( I @ B4 ) @ ( minus_minus_set_a @ S2 @ S3 ) ) )
=> ( ! [A5: a] :
( ( member_a @ A5 @ S3 )
=> ( ( G @ A5 )
= zero_zero_nat ) )
=> ( ! [B4: a] :
( ( member_a @ B4 @ T3 )
=> ( ( H @ B4 )
= zero_zero_nat ) )
=> ( ! [A5: a] :
( ( member_a @ A5 @ S2 )
=> ( ( H @ ( J @ A5 ) )
= ( G @ A5 ) ) )
=> ( ( groups6334556678337121940_a_nat @ G @ S2 )
= ( groups6334556678337121940_a_nat @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_525_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S3: set_nat,T3: set_nat,S2: set_nat,I: nat > nat,J: nat > nat,T2: set_nat,G: nat > int,H: nat > int] :
( ( finite_finite_nat @ S3 )
=> ( ( finite_finite_nat @ T3 )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ ( minus_minus_set_nat @ S2 @ S3 ) )
=> ( ( I @ ( J @ A5 ) )
= A5 ) )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ ( minus_minus_set_nat @ S2 @ S3 ) )
=> ( member_nat @ ( J @ A5 ) @ ( minus_minus_set_nat @ T2 @ T3 ) ) )
=> ( ! [B4: nat] :
( ( member_nat @ B4 @ ( minus_minus_set_nat @ T2 @ T3 ) )
=> ( ( J @ ( I @ B4 ) )
= B4 ) )
=> ( ! [B4: nat] :
( ( member_nat @ B4 @ ( minus_minus_set_nat @ T2 @ T3 ) )
=> ( member_nat @ ( I @ B4 ) @ ( minus_minus_set_nat @ S2 @ S3 ) ) )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ S3 )
=> ( ( G @ A5 )
= zero_zero_int ) )
=> ( ! [B4: nat] :
( ( member_nat @ B4 @ T3 )
=> ( ( H @ B4 )
= zero_zero_int ) )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ S2 )
=> ( ( H @ ( J @ A5 ) )
= ( G @ A5 ) ) )
=> ( ( groups3539618377306564664at_int @ G @ S2 )
= ( groups3539618377306564664at_int @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_526_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S3: set_nat,T3: set_int,S2: set_nat,I: int > nat,J: nat > int,T2: set_int,G: nat > int,H: int > int] :
( ( finite_finite_nat @ S3 )
=> ( ( finite_finite_int @ T3 )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ ( minus_minus_set_nat @ S2 @ S3 ) )
=> ( ( I @ ( J @ A5 ) )
= A5 ) )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ ( minus_minus_set_nat @ S2 @ S3 ) )
=> ( member_int @ ( J @ A5 ) @ ( minus_minus_set_int @ T2 @ T3 ) ) )
=> ( ! [B4: int] :
( ( member_int @ B4 @ ( minus_minus_set_int @ T2 @ T3 ) )
=> ( ( J @ ( I @ B4 ) )
= B4 ) )
=> ( ! [B4: int] :
( ( member_int @ B4 @ ( minus_minus_set_int @ T2 @ T3 ) )
=> ( member_nat @ ( I @ B4 ) @ ( minus_minus_set_nat @ S2 @ S3 ) ) )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ S3 )
=> ( ( G @ A5 )
= zero_zero_int ) )
=> ( ! [B4: int] :
( ( member_int @ B4 @ T3 )
=> ( ( H @ B4 )
= zero_zero_int ) )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ S2 )
=> ( ( H @ ( J @ A5 ) )
= ( G @ A5 ) ) )
=> ( ( groups3539618377306564664at_int @ G @ S2 )
= ( groups4538972089207619220nt_int @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_527_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S3: set_nat,T3: set_a,S2: set_nat,I: a > nat,J: nat > a,T2: set_a,G: nat > int,H: a > int] :
( ( finite_finite_nat @ S3 )
=> ( ( finite_finite_a @ T3 )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ ( minus_minus_set_nat @ S2 @ S3 ) )
=> ( ( I @ ( J @ A5 ) )
= A5 ) )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ ( minus_minus_set_nat @ S2 @ S3 ) )
=> ( member_a @ ( J @ A5 ) @ ( minus_minus_set_a @ T2 @ T3 ) ) )
=> ( ! [B4: a] :
( ( member_a @ B4 @ ( minus_minus_set_a @ T2 @ T3 ) )
=> ( ( J @ ( I @ B4 ) )
= B4 ) )
=> ( ! [B4: a] :
( ( member_a @ B4 @ ( minus_minus_set_a @ T2 @ T3 ) )
=> ( member_nat @ ( I @ B4 ) @ ( minus_minus_set_nat @ S2 @ S3 ) ) )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ S3 )
=> ( ( G @ A5 )
= zero_zero_int ) )
=> ( ! [B4: a] :
( ( member_a @ B4 @ T3 )
=> ( ( H @ B4 )
= zero_zero_int ) )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ S2 )
=> ( ( H @ ( J @ A5 ) )
= ( G @ A5 ) ) )
=> ( ( groups3539618377306564664at_int @ G @ S2 )
= ( groups6332066207828071664_a_int @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_528_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S3: set_int,T3: set_nat,S2: set_int,I: nat > int,J: int > nat,T2: set_nat,G: int > int,H: nat > int] :
( ( finite_finite_int @ S3 )
=> ( ( finite_finite_nat @ T3 )
=> ( ! [A5: int] :
( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S3 ) )
=> ( ( I @ ( J @ A5 ) )
= A5 ) )
=> ( ! [A5: int] :
( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S3 ) )
=> ( member_nat @ ( J @ A5 ) @ ( minus_minus_set_nat @ T2 @ T3 ) ) )
=> ( ! [B4: nat] :
( ( member_nat @ B4 @ ( minus_minus_set_nat @ T2 @ T3 ) )
=> ( ( J @ ( I @ B4 ) )
= B4 ) )
=> ( ! [B4: nat] :
( ( member_nat @ B4 @ ( minus_minus_set_nat @ T2 @ T3 ) )
=> ( member_int @ ( I @ B4 ) @ ( minus_minus_set_int @ S2 @ S3 ) ) )
=> ( ! [A5: int] :
( ( member_int @ A5 @ S3 )
=> ( ( G @ A5 )
= zero_zero_int ) )
=> ( ! [B4: nat] :
( ( member_nat @ B4 @ T3 )
=> ( ( H @ B4 )
= zero_zero_int ) )
=> ( ! [A5: int] :
( ( member_int @ A5 @ S2 )
=> ( ( H @ ( J @ A5 ) )
= ( G @ A5 ) ) )
=> ( ( groups4538972089207619220nt_int @ G @ S2 )
= ( groups3539618377306564664at_int @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_529_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S3: set_int,T3: set_int,S2: set_int,I: int > int,J: int > int,T2: set_int,G: int > int,H: int > int] :
( ( finite_finite_int @ S3 )
=> ( ( finite_finite_int @ T3 )
=> ( ! [A5: int] :
( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S3 ) )
=> ( ( I @ ( J @ A5 ) )
= A5 ) )
=> ( ! [A5: int] :
( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S3 ) )
=> ( member_int @ ( J @ A5 ) @ ( minus_minus_set_int @ T2 @ T3 ) ) )
=> ( ! [B4: int] :
( ( member_int @ B4 @ ( minus_minus_set_int @ T2 @ T3 ) )
=> ( ( J @ ( I @ B4 ) )
= B4 ) )
=> ( ! [B4: int] :
( ( member_int @ B4 @ ( minus_minus_set_int @ T2 @ T3 ) )
=> ( member_int @ ( I @ B4 ) @ ( minus_minus_set_int @ S2 @ S3 ) ) )
=> ( ! [A5: int] :
( ( member_int @ A5 @ S3 )
=> ( ( G @ A5 )
= zero_zero_int ) )
=> ( ! [B4: int] :
( ( member_int @ B4 @ T3 )
=> ( ( H @ B4 )
= zero_zero_int ) )
=> ( ! [A5: int] :
( ( member_int @ A5 @ S2 )
=> ( ( H @ ( J @ A5 ) )
= ( G @ A5 ) ) )
=> ( ( groups4538972089207619220nt_int @ G @ S2 )
= ( groups4538972089207619220nt_int @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_530_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S3: set_int,T3: set_a,S2: set_int,I: a > int,J: int > a,T2: set_a,G: int > int,H: a > int] :
( ( finite_finite_int @ S3 )
=> ( ( finite_finite_a @ T3 )
=> ( ! [A5: int] :
( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S3 ) )
=> ( ( I @ ( J @ A5 ) )
= A5 ) )
=> ( ! [A5: int] :
( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S3 ) )
=> ( member_a @ ( J @ A5 ) @ ( minus_minus_set_a @ T2 @ T3 ) ) )
=> ( ! [B4: a] :
( ( member_a @ B4 @ ( minus_minus_set_a @ T2 @ T3 ) )
=> ( ( J @ ( I @ B4 ) )
= B4 ) )
=> ( ! [B4: a] :
( ( member_a @ B4 @ ( minus_minus_set_a @ T2 @ T3 ) )
=> ( member_int @ ( I @ B4 ) @ ( minus_minus_set_int @ S2 @ S3 ) ) )
=> ( ! [A5: int] :
( ( member_int @ A5 @ S3 )
=> ( ( G @ A5 )
= zero_zero_int ) )
=> ( ! [B4: a] :
( ( member_a @ B4 @ T3 )
=> ( ( H @ B4 )
= zero_zero_int ) )
=> ( ! [A5: int] :
( ( member_int @ A5 @ S2 )
=> ( ( H @ ( J @ A5 ) )
= ( G @ A5 ) ) )
=> ( ( groups4538972089207619220nt_int @ G @ S2 )
= ( groups6332066207828071664_a_int @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_531_sum__nonneg__eq__0__iff,axiom,
! [A2: set_int,F2: int > nat] :
( ( finite_finite_int @ A2 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ X3 ) ) )
=> ( ( ( groups4541462559716669496nt_nat @ F2 @ A2 )
= zero_zero_nat )
= ( ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ( ( F2 @ X2 )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_532_sum__nonneg__eq__0__iff,axiom,
! [A2: set_a,F2: a > nat] :
( ( finite_finite_a @ A2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ X3 ) ) )
=> ( ( ( groups6334556678337121940_a_nat @ F2 @ A2 )
= zero_zero_nat )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ( F2 @ X2 )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_533_sum__nonneg__eq__0__iff,axiom,
! [A2: set_nat,F2: nat > int] :
( ( finite_finite_nat @ A2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F2 @ X3 ) ) )
=> ( ( ( groups3539618377306564664at_int @ F2 @ A2 )
= zero_zero_int )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( F2 @ X2 )
= zero_zero_int ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_534_sum__nonneg__eq__0__iff,axiom,
! [A2: set_int,F2: int > int] :
( ( finite_finite_int @ A2 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F2 @ X3 ) ) )
=> ( ( ( groups4538972089207619220nt_int @ F2 @ A2 )
= zero_zero_int )
= ( ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ( ( F2 @ X2 )
= zero_zero_int ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_535_sum__nonneg__eq__0__iff,axiom,
! [A2: set_a,F2: a > int] :
( ( finite_finite_a @ A2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F2 @ X3 ) ) )
=> ( ( ( groups6332066207828071664_a_int @ F2 @ A2 )
= zero_zero_int )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ( F2 @ X2 )
= zero_zero_int ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_536_sum__nonneg__eq__0__iff,axiom,
! [A2: set_nat,F2: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ X3 ) ) )
=> ( ( ( groups3542108847815614940at_nat @ F2 @ A2 )
= zero_zero_nat )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( F2 @ X2 )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_537_sum__nonneg__eq__0__iff,axiom,
! [A2: set_set_a,F2: set_a > nat] :
( ( finite_finite_set_a @ A2 )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ X3 ) ) )
=> ( ( ( groups6141743369313575924_a_nat @ F2 @ A2 )
= zero_zero_nat )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( ( F2 @ X2 )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_538_sum__nonneg__eq__0__iff,axiom,
! [A2: set_set_a,F2: set_a > int] :
( ( finite_finite_set_a @ A2 )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F2 @ X3 ) ) )
=> ( ( ( groups6139252898804525648_a_int @ F2 @ A2 )
= zero_zero_int )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( ( F2 @ X2 )
= zero_zero_int ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_539_sum__nonneg__eq__0__iff,axiom,
! [A2: set_list_a,F2: list_a > int] :
( ( finite_finite_list_a @ A2 )
=> ( ! [X3: list_a] :
( ( member_list_a @ X3 @ A2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F2 @ X3 ) ) )
=> ( ( ( groups5518757228788810486_a_int @ F2 @ A2 )
= zero_zero_int )
= ( ! [X2: list_a] :
( ( member_list_a @ X2 @ A2 )
=> ( ( F2 @ X2 )
= zero_zero_int ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_540_sum__nonneg__eq__0__iff,axiom,
! [A2: set_list_a,F2: list_a > nat] :
( ( finite_finite_list_a @ A2 )
=> ( ! [X3: list_a] :
( ( member_list_a @ X3 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ X3 ) ) )
=> ( ( ( groups5521247699297860762_a_nat @ F2 @ A2 )
= zero_zero_nat )
= ( ! [X2: list_a] :
( ( member_list_a @ X2 @ A2 )
=> ( ( F2 @ X2 )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_541_sum__le__included,axiom,
! [S: set_int,T: set_int,G: int > nat,I: int > int,F2: int > nat] :
( ( finite_finite_int @ S )
=> ( ( finite_finite_int @ T )
=> ( ! [X3: int] :
( ( member_int @ X3 @ T )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( G @ X3 ) ) )
=> ( ! [X3: int] :
( ( member_int @ X3 @ S )
=> ? [Xa: int] :
( ( member_int @ Xa @ T )
& ( ( I @ Xa )
= X3 )
& ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F2 @ S ) @ ( groups4541462559716669496nt_nat @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_542_sum__le__included,axiom,
! [S: set_int,T: set_a,G: a > nat,I: a > int,F2: int > nat] :
( ( finite_finite_int @ S )
=> ( ( finite_finite_a @ T )
=> ( ! [X3: a] :
( ( member_a @ X3 @ T )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( G @ X3 ) ) )
=> ( ! [X3: int] :
( ( member_int @ X3 @ S )
=> ? [Xa: a] :
( ( member_a @ Xa @ T )
& ( ( I @ Xa )
= X3 )
& ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F2 @ S ) @ ( groups6334556678337121940_a_nat @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_543_sum__le__included,axiom,
! [S: set_a,T: set_int,G: int > nat,I: int > a,F2: a > nat] :
( ( finite_finite_a @ S )
=> ( ( finite_finite_int @ T )
=> ( ! [X3: int] :
( ( member_int @ X3 @ T )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( G @ X3 ) ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ S )
=> ? [Xa: int] :
( ( member_int @ Xa @ T )
& ( ( I @ Xa )
= X3 )
& ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_nat @ ( groups6334556678337121940_a_nat @ F2 @ S ) @ ( groups4541462559716669496nt_nat @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_544_sum__le__included,axiom,
! [S: set_a,T: set_a,G: a > nat,I: a > a,F2: a > nat] :
( ( finite_finite_a @ S )
=> ( ( finite_finite_a @ T )
=> ( ! [X3: a] :
( ( member_a @ X3 @ T )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( G @ X3 ) ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ S )
=> ? [Xa: a] :
( ( member_a @ Xa @ T )
& ( ( I @ Xa )
= X3 )
& ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_nat @ ( groups6334556678337121940_a_nat @ F2 @ S ) @ ( groups6334556678337121940_a_nat @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_545_sum__le__included,axiom,
! [S: set_nat,T: set_nat,G: nat > int,I: nat > nat,F2: nat > int] :
( ( finite_finite_nat @ S )
=> ( ( finite_finite_nat @ T )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ T )
=> ( ord_less_eq_int @ zero_zero_int @ ( G @ X3 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ S )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ T )
& ( ( I @ Xa )
= X3 )
& ( ord_less_eq_int @ ( F2 @ X3 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F2 @ S ) @ ( groups3539618377306564664at_int @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_546_sum__le__included,axiom,
! [S: set_nat,T: set_int,G: int > int,I: int > nat,F2: nat > int] :
( ( finite_finite_nat @ S )
=> ( ( finite_finite_int @ T )
=> ( ! [X3: int] :
( ( member_int @ X3 @ T )
=> ( ord_less_eq_int @ zero_zero_int @ ( G @ X3 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ S )
=> ? [Xa: int] :
( ( member_int @ Xa @ T )
& ( ( I @ Xa )
= X3 )
& ( ord_less_eq_int @ ( F2 @ X3 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F2 @ S ) @ ( groups4538972089207619220nt_int @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_547_sum__le__included,axiom,
! [S: set_nat,T: set_a,G: a > int,I: a > nat,F2: nat > int] :
( ( finite_finite_nat @ S )
=> ( ( finite_finite_a @ T )
=> ( ! [X3: a] :
( ( member_a @ X3 @ T )
=> ( ord_less_eq_int @ zero_zero_int @ ( G @ X3 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ S )
=> ? [Xa: a] :
( ( member_a @ Xa @ T )
& ( ( I @ Xa )
= X3 )
& ( ord_less_eq_int @ ( F2 @ X3 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F2 @ S ) @ ( groups6332066207828071664_a_int @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_548_sum__le__included,axiom,
! [S: set_int,T: set_nat,G: nat > int,I: nat > int,F2: int > int] :
( ( finite_finite_int @ S )
=> ( ( finite_finite_nat @ T )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ T )
=> ( ord_less_eq_int @ zero_zero_int @ ( G @ X3 ) ) )
=> ( ! [X3: int] :
( ( member_int @ X3 @ S )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ T )
& ( ( I @ Xa )
= X3 )
& ( ord_less_eq_int @ ( F2 @ X3 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_int @ ( groups4538972089207619220nt_int @ F2 @ S ) @ ( groups3539618377306564664at_int @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_549_sum__le__included,axiom,
! [S: set_int,T: set_int,G: int > int,I: int > int,F2: int > int] :
( ( finite_finite_int @ S )
=> ( ( finite_finite_int @ T )
=> ( ! [X3: int] :
( ( member_int @ X3 @ T )
=> ( ord_less_eq_int @ zero_zero_int @ ( G @ X3 ) ) )
=> ( ! [X3: int] :
( ( member_int @ X3 @ S )
=> ? [Xa: int] :
( ( member_int @ Xa @ T )
& ( ( I @ Xa )
= X3 )
& ( ord_less_eq_int @ ( F2 @ X3 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_int @ ( groups4538972089207619220nt_int @ F2 @ S ) @ ( groups4538972089207619220nt_int @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_550_sum__le__included,axiom,
! [S: set_int,T: set_a,G: a > int,I: a > int,F2: int > int] :
( ( finite_finite_int @ S )
=> ( ( finite_finite_a @ T )
=> ( ! [X3: a] :
( ( member_a @ X3 @ T )
=> ( ord_less_eq_int @ zero_zero_int @ ( G @ X3 ) ) )
=> ( ! [X3: int] :
( ( member_int @ X3 @ S )
=> ? [Xa: a] :
( ( member_a @ Xa @ T )
& ( ( I @ Xa )
= X3 )
& ( ord_less_eq_int @ ( F2 @ X3 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_int @ ( groups4538972089207619220nt_int @ F2 @ S ) @ ( groups6332066207828071664_a_int @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_551_sum_Osetdiff__irrelevant,axiom,
! [A2: set_int,G: int > nat] :
( ( finite_finite_int @ A2 )
=> ( ( groups4541462559716669496nt_nat @ G
@ ( minus_minus_set_int @ A2
@ ( collect_int
@ ^ [X2: int] :
( ( G @ X2 )
= zero_zero_nat ) ) ) )
= ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_552_sum_Osetdiff__irrelevant,axiom,
! [A2: set_a,G: a > nat] :
( ( finite_finite_a @ A2 )
=> ( ( groups6334556678337121940_a_nat @ G
@ ( minus_minus_set_a @ A2
@ ( collect_a
@ ^ [X2: a] :
( ( G @ X2 )
= zero_zero_nat ) ) ) )
= ( groups6334556678337121940_a_nat @ G @ A2 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_553_sum_Osetdiff__irrelevant,axiom,
! [A2: set_nat,G: nat > int] :
( ( finite_finite_nat @ A2 )
=> ( ( groups3539618377306564664at_int @ G
@ ( minus_minus_set_nat @ A2
@ ( collect_nat
@ ^ [X2: nat] :
( ( G @ X2 )
= zero_zero_int ) ) ) )
= ( groups3539618377306564664at_int @ G @ A2 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_554_sum_Osetdiff__irrelevant,axiom,
! [A2: set_int,G: int > int] :
( ( finite_finite_int @ A2 )
=> ( ( groups4538972089207619220nt_int @ G
@ ( minus_minus_set_int @ A2
@ ( collect_int
@ ^ [X2: int] :
( ( G @ X2 )
= zero_zero_int ) ) ) )
= ( groups4538972089207619220nt_int @ G @ A2 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_555_sum_Osetdiff__irrelevant,axiom,
! [A2: set_a,G: a > int] :
( ( finite_finite_a @ A2 )
=> ( ( groups6332066207828071664_a_int @ G
@ ( minus_minus_set_a @ A2
@ ( collect_a
@ ^ [X2: a] :
( ( G @ X2 )
= zero_zero_int ) ) ) )
= ( groups6332066207828071664_a_int @ G @ A2 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_556_sum_Osetdiff__irrelevant,axiom,
! [A2: set_nat,G: nat > complex] :
( ( finite_finite_nat @ A2 )
=> ( ( groups2073611262835488442omplex @ G
@ ( minus_minus_set_nat @ A2
@ ( collect_nat
@ ^ [X2: nat] :
( ( G @ X2 )
= zero_zero_complex ) ) ) )
= ( groups2073611262835488442omplex @ G @ A2 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_557_sum_Osetdiff__irrelevant,axiom,
! [A2: set_int,G: int > complex] :
( ( finite_finite_int @ A2 )
=> ( ( groups3049146728041665814omplex @ G
@ ( minus_minus_set_int @ A2
@ ( collect_int
@ ^ [X2: int] :
( ( G @ X2 )
= zero_zero_complex ) ) ) )
= ( groups3049146728041665814omplex @ G @ A2 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_558_sum_Osetdiff__irrelevant,axiom,
! [A2: set_a,G: a > complex] :
( ( finite_finite_a @ A2 )
=> ( ( groups8331919209915413362omplex @ G
@ ( minus_minus_set_a @ A2
@ ( collect_a
@ ^ [X2: a] :
( ( G @ X2 )
= zero_zero_complex ) ) ) )
= ( groups8331919209915413362omplex @ G @ A2 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_559_sum_Osetdiff__irrelevant,axiom,
! [A2: set_nat,G: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( groups3542108847815614940at_nat @ G
@ ( minus_minus_set_nat @ A2
@ ( collect_nat
@ ^ [X2: nat] :
( ( G @ X2 )
= zero_zero_nat ) ) ) )
= ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_560_sum_Osetdiff__irrelevant,axiom,
! [A2: set_complex,G: complex > complex] :
( ( finite3207457112153483333omplex @ A2 )
=> ( ( groups7754918857620584856omplex @ G
@ ( minus_811609699411566653omplex @ A2
@ ( collect_complex
@ ^ [X2: complex] :
( ( G @ X2 )
= zero_zero_complex ) ) ) )
= ( groups7754918857620584856omplex @ G @ A2 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_561_sum__nonneg__leq__bound,axiom,
! [S: set_int,F2: int > nat,B3: nat,I: int] :
( ( finite_finite_int @ S )
=> ( ! [I5: int] :
( ( member_int @ I5 @ S )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ I5 ) ) )
=> ( ( ( groups4541462559716669496nt_nat @ F2 @ S )
= B3 )
=> ( ( member_int @ I @ S )
=> ( ord_less_eq_nat @ ( F2 @ I ) @ B3 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_562_sum__nonneg__leq__bound,axiom,
! [S: set_a,F2: a > nat,B3: nat,I: a] :
( ( finite_finite_a @ S )
=> ( ! [I5: a] :
( ( member_a @ I5 @ S )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ I5 ) ) )
=> ( ( ( groups6334556678337121940_a_nat @ F2 @ S )
= B3 )
=> ( ( member_a @ I @ S )
=> ( ord_less_eq_nat @ ( F2 @ I ) @ B3 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_563_sum__nonneg__leq__bound,axiom,
! [S: set_nat,F2: nat > int,B3: int,I: nat] :
( ( finite_finite_nat @ S )
=> ( ! [I5: nat] :
( ( member_nat @ I5 @ S )
=> ( ord_less_eq_int @ zero_zero_int @ ( F2 @ I5 ) ) )
=> ( ( ( groups3539618377306564664at_int @ F2 @ S )
= B3 )
=> ( ( member_nat @ I @ S )
=> ( ord_less_eq_int @ ( F2 @ I ) @ B3 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_564_sum__nonneg__leq__bound,axiom,
! [S: set_int,F2: int > int,B3: int,I: int] :
( ( finite_finite_int @ S )
=> ( ! [I5: int] :
( ( member_int @ I5 @ S )
=> ( ord_less_eq_int @ zero_zero_int @ ( F2 @ I5 ) ) )
=> ( ( ( groups4538972089207619220nt_int @ F2 @ S )
= B3 )
=> ( ( member_int @ I @ S )
=> ( ord_less_eq_int @ ( F2 @ I ) @ B3 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_565_sum__nonneg__leq__bound,axiom,
! [S: set_a,F2: a > int,B3: int,I: a] :
( ( finite_finite_a @ S )
=> ( ! [I5: a] :
( ( member_a @ I5 @ S )
=> ( ord_less_eq_int @ zero_zero_int @ ( F2 @ I5 ) ) )
=> ( ( ( groups6332066207828071664_a_int @ F2 @ S )
= B3 )
=> ( ( member_a @ I @ S )
=> ( ord_less_eq_int @ ( F2 @ I ) @ B3 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_566_sum__nonneg__leq__bound,axiom,
! [S: set_nat,F2: nat > nat,B3: nat,I: nat] :
( ( finite_finite_nat @ S )
=> ( ! [I5: nat] :
( ( member_nat @ I5 @ S )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ I5 ) ) )
=> ( ( ( groups3542108847815614940at_nat @ F2 @ S )
= B3 )
=> ( ( member_nat @ I @ S )
=> ( ord_less_eq_nat @ ( F2 @ I ) @ B3 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_567_sum__nonneg__leq__bound,axiom,
! [S: set_set_a,F2: set_a > nat,B3: nat,I: set_a] :
( ( finite_finite_set_a @ S )
=> ( ! [I5: set_a] :
( ( member_set_a @ I5 @ S )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ I5 ) ) )
=> ( ( ( groups6141743369313575924_a_nat @ F2 @ S )
= B3 )
=> ( ( member_set_a @ I @ S )
=> ( ord_less_eq_nat @ ( F2 @ I ) @ B3 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_568_sum__nonneg__leq__bound,axiom,
! [S: set_set_a,F2: set_a > int,B3: int,I: set_a] :
( ( finite_finite_set_a @ S )
=> ( ! [I5: set_a] :
( ( member_set_a @ I5 @ S )
=> ( ord_less_eq_int @ zero_zero_int @ ( F2 @ I5 ) ) )
=> ( ( ( groups6139252898804525648_a_int @ F2 @ S )
= B3 )
=> ( ( member_set_a @ I @ S )
=> ( ord_less_eq_int @ ( F2 @ I ) @ B3 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_569_sum__nonneg__leq__bound,axiom,
! [S: set_list_a,F2: list_a > int,B3: int,I: list_a] :
( ( finite_finite_list_a @ S )
=> ( ! [I5: list_a] :
( ( member_list_a @ I5 @ S )
=> ( ord_less_eq_int @ zero_zero_int @ ( F2 @ I5 ) ) )
=> ( ( ( groups5518757228788810486_a_int @ F2 @ S )
= B3 )
=> ( ( member_list_a @ I @ S )
=> ( ord_less_eq_int @ ( F2 @ I ) @ B3 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_570_sum__nonneg__leq__bound,axiom,
! [S: set_list_a,F2: list_a > nat,B3: nat,I: list_a] :
( ( finite_finite_list_a @ S )
=> ( ! [I5: list_a] :
( ( member_list_a @ I5 @ S )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ I5 ) ) )
=> ( ( ( groups5521247699297860762_a_nat @ F2 @ S )
= B3 )
=> ( ( member_list_a @ I @ S )
=> ( ord_less_eq_nat @ ( F2 @ I ) @ B3 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_571_sum__nonneg__0,axiom,
! [S: set_int,F2: int > nat,I: int] :
( ( finite_finite_int @ S )
=> ( ! [I5: int] :
( ( member_int @ I5 @ S )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ I5 ) ) )
=> ( ( ( groups4541462559716669496nt_nat @ F2 @ S )
= zero_zero_nat )
=> ( ( member_int @ I @ S )
=> ( ( F2 @ I )
= zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_572_sum__nonneg__0,axiom,
! [S: set_a,F2: a > nat,I: a] :
( ( finite_finite_a @ S )
=> ( ! [I5: a] :
( ( member_a @ I5 @ S )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ I5 ) ) )
=> ( ( ( groups6334556678337121940_a_nat @ F2 @ S )
= zero_zero_nat )
=> ( ( member_a @ I @ S )
=> ( ( F2 @ I )
= zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_573_sum__nonneg__0,axiom,
! [S: set_nat,F2: nat > int,I: nat] :
( ( finite_finite_nat @ S )
=> ( ! [I5: nat] :
( ( member_nat @ I5 @ S )
=> ( ord_less_eq_int @ zero_zero_int @ ( F2 @ I5 ) ) )
=> ( ( ( groups3539618377306564664at_int @ F2 @ S )
= zero_zero_int )
=> ( ( member_nat @ I @ S )
=> ( ( F2 @ I )
= zero_zero_int ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_574_sum__nonneg__0,axiom,
! [S: set_int,F2: int > int,I: int] :
( ( finite_finite_int @ S )
=> ( ! [I5: int] :
( ( member_int @ I5 @ S )
=> ( ord_less_eq_int @ zero_zero_int @ ( F2 @ I5 ) ) )
=> ( ( ( groups4538972089207619220nt_int @ F2 @ S )
= zero_zero_int )
=> ( ( member_int @ I @ S )
=> ( ( F2 @ I )
= zero_zero_int ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_575_sum__nonneg__0,axiom,
! [S: set_a,F2: a > int,I: a] :
( ( finite_finite_a @ S )
=> ( ! [I5: a] :
( ( member_a @ I5 @ S )
=> ( ord_less_eq_int @ zero_zero_int @ ( F2 @ I5 ) ) )
=> ( ( ( groups6332066207828071664_a_int @ F2 @ S )
= zero_zero_int )
=> ( ( member_a @ I @ S )
=> ( ( F2 @ I )
= zero_zero_int ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_576_sum__nonneg__0,axiom,
! [S: set_nat,F2: nat > nat,I: nat] :
( ( finite_finite_nat @ S )
=> ( ! [I5: nat] :
( ( member_nat @ I5 @ S )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ I5 ) ) )
=> ( ( ( groups3542108847815614940at_nat @ F2 @ S )
= zero_zero_nat )
=> ( ( member_nat @ I @ S )
=> ( ( F2 @ I )
= zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_577_sum__nonneg__0,axiom,
! [S: set_set_a,F2: set_a > nat,I: set_a] :
( ( finite_finite_set_a @ S )
=> ( ! [I5: set_a] :
( ( member_set_a @ I5 @ S )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ I5 ) ) )
=> ( ( ( groups6141743369313575924_a_nat @ F2 @ S )
= zero_zero_nat )
=> ( ( member_set_a @ I @ S )
=> ( ( F2 @ I )
= zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_578_sum__nonneg__0,axiom,
! [S: set_set_a,F2: set_a > int,I: set_a] :
( ( finite_finite_set_a @ S )
=> ( ! [I5: set_a] :
( ( member_set_a @ I5 @ S )
=> ( ord_less_eq_int @ zero_zero_int @ ( F2 @ I5 ) ) )
=> ( ( ( groups6139252898804525648_a_int @ F2 @ S )
= zero_zero_int )
=> ( ( member_set_a @ I @ S )
=> ( ( F2 @ I )
= zero_zero_int ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_579_sum__nonneg__0,axiom,
! [S: set_list_a,F2: list_a > int,I: list_a] :
( ( finite_finite_list_a @ S )
=> ( ! [I5: list_a] :
( ( member_list_a @ I5 @ S )
=> ( ord_less_eq_int @ zero_zero_int @ ( F2 @ I5 ) ) )
=> ( ( ( groups5518757228788810486_a_int @ F2 @ S )
= zero_zero_int )
=> ( ( member_list_a @ I @ S )
=> ( ( F2 @ I )
= zero_zero_int ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_580_sum__nonneg__0,axiom,
! [S: set_list_a,F2: list_a > nat,I: list_a] :
( ( finite_finite_list_a @ S )
=> ( ! [I5: list_a] :
( ( member_list_a @ I5 @ S )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ I5 ) ) )
=> ( ( ( groups5521247699297860762_a_nat @ F2 @ S )
= zero_zero_nat )
=> ( ( member_list_a @ I @ S )
=> ( ( F2 @ I )
= zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_581_sum__mono,axiom,
! [K4: set_a,F2: a > nat,G: a > nat] :
( ! [I5: a] :
( ( member_a @ I5 @ K4 )
=> ( ord_less_eq_nat @ ( F2 @ I5 ) @ ( G @ I5 ) ) )
=> ( ord_less_eq_nat @ ( groups6334556678337121940_a_nat @ F2 @ K4 ) @ ( groups6334556678337121940_a_nat @ G @ K4 ) ) ) ).
% sum_mono
thf(fact_582_sum__mono,axiom,
! [K4: set_a,F2: a > int,G: a > int] :
( ! [I5: a] :
( ( member_a @ I5 @ K4 )
=> ( ord_less_eq_int @ ( F2 @ I5 ) @ ( G @ I5 ) ) )
=> ( ord_less_eq_int @ ( groups6332066207828071664_a_int @ F2 @ K4 ) @ ( groups6332066207828071664_a_int @ G @ K4 ) ) ) ).
% sum_mono
thf(fact_583_sum__mono,axiom,
! [K4: set_nat,F2: nat > nat,G: nat > nat] :
( ! [I5: nat] :
( ( member_nat @ I5 @ K4 )
=> ( ord_less_eq_nat @ ( F2 @ I5 ) @ ( G @ I5 ) ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F2 @ K4 ) @ ( groups3542108847815614940at_nat @ G @ K4 ) ) ) ).
% sum_mono
thf(fact_584_sum__mono,axiom,
! [K4: set_set_a,F2: set_a > nat,G: set_a > nat] :
( ! [I5: set_a] :
( ( member_set_a @ I5 @ K4 )
=> ( ord_less_eq_nat @ ( F2 @ I5 ) @ ( G @ I5 ) ) )
=> ( ord_less_eq_nat @ ( groups6141743369313575924_a_nat @ F2 @ K4 ) @ ( groups6141743369313575924_a_nat @ G @ K4 ) ) ) ).
% sum_mono
thf(fact_585_sum__mono,axiom,
! [K4: set_list_a,F2: list_a > int,G: list_a > int] :
( ! [I5: list_a] :
( ( member_list_a @ I5 @ K4 )
=> ( ord_less_eq_int @ ( F2 @ I5 ) @ ( G @ I5 ) ) )
=> ( ord_less_eq_int @ ( groups5518757228788810486_a_int @ F2 @ K4 ) @ ( groups5518757228788810486_a_int @ G @ K4 ) ) ) ).
% sum_mono
thf(fact_586_sum__mono,axiom,
! [K4: set_set_a,F2: set_a > int,G: set_a > int] :
( ! [I5: set_a] :
( ( member_set_a @ I5 @ K4 )
=> ( ord_less_eq_int @ ( F2 @ I5 ) @ ( G @ I5 ) ) )
=> ( ord_less_eq_int @ ( groups6139252898804525648_a_int @ F2 @ K4 ) @ ( groups6139252898804525648_a_int @ G @ K4 ) ) ) ).
% sum_mono
thf(fact_587_sum__mono,axiom,
! [K4: set_list_a,F2: list_a > nat,G: list_a > nat] :
( ! [I5: list_a] :
( ( member_list_a @ I5 @ K4 )
=> ( ord_less_eq_nat @ ( F2 @ I5 ) @ ( G @ I5 ) ) )
=> ( ord_less_eq_nat @ ( groups5521247699297860762_a_nat @ F2 @ K4 ) @ ( groups5521247699297860762_a_nat @ G @ K4 ) ) ) ).
% sum_mono
thf(fact_588_sum__mono,axiom,
! [K4: set_set_list_a,F2: set_list_a > nat,G: set_list_a > nat] :
( ! [I5: set_list_a] :
( ( member_set_list_a @ I5 @ K4 )
=> ( ord_less_eq_nat @ ( F2 @ I5 ) @ ( G @ I5 ) ) )
=> ( ord_less_eq_nat @ ( groups5993734322560061562_a_nat @ F2 @ K4 ) @ ( groups5993734322560061562_a_nat @ G @ K4 ) ) ) ).
% sum_mono
thf(fact_589_sum__mono,axiom,
! [K4: set_set_list_a,F2: set_list_a > int,G: set_list_a > int] :
( ! [I5: set_list_a] :
( ( member_set_list_a @ I5 @ K4 )
=> ( ord_less_eq_int @ ( F2 @ I5 ) @ ( G @ I5 ) ) )
=> ( ord_less_eq_int @ ( groups5991243852051011286_a_int @ F2 @ K4 ) @ ( groups5991243852051011286_a_int @ G @ K4 ) ) ) ).
% sum_mono
thf(fact_590_sum__mono,axiom,
! [K4: set_li3422455791611400469list_a,F2: ( list_list_a > list_a ) > nat,G: ( list_list_a > list_a ) > nat] :
( ! [I5: list_list_a > list_a] :
( ( member7168557129179038582list_a @ I5 @ K4 )
=> ( ord_less_eq_nat @ ( F2 @ I5 ) @ ( G @ I5 ) ) )
=> ( ord_less_eq_nat @ ( groups5819086642190997227_a_nat @ F2 @ K4 ) @ ( groups5819086642190997227_a_nat @ G @ K4 ) ) ) ).
% sum_mono
thf(fact_591_sum_OG__def,axiom,
( groups2985770679641041633nt_nat
= ( ^ [P: int > nat,I6: set_int] :
( if_nat
@ ( finite_finite_int
@ ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ I6 )
& ( ( P @ X2 )
!= zero_zero_nat ) ) ) )
@ ( groups4541462559716669496nt_nat @ P
@ ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ I6 )
& ( ( P @ X2 )
!= zero_zero_nat ) ) ) )
@ zero_zero_nat ) ) ) ).
% sum.G_def
thf(fact_592_sum_OG__def,axiom,
( groups7493686093766423595_a_nat
= ( ^ [P: a > nat,I6: set_a] :
( if_nat
@ ( finite_finite_a
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ I6 )
& ( ( P @ X2 )
!= zero_zero_nat ) ) ) )
@ ( groups6334556678337121940_a_nat @ P
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ I6 )
& ( ( P @ X2 )
!= zero_zero_nat ) ) ) )
@ zero_zero_nat ) ) ) ).
% sum.G_def
thf(fact_593_sum_OG__def,axiom,
( groups1983926497230936801at_int
= ( ^ [P: nat > int,I6: set_nat] :
( if_int
@ ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ I6 )
& ( ( P @ X2 )
!= zero_zero_int ) ) ) )
@ ( groups3539618377306564664at_int @ P
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ I6 )
& ( ( P @ X2 )
!= zero_zero_int ) ) ) )
@ zero_zero_int ) ) ) ).
% sum.G_def
thf(fact_594_sum_OG__def,axiom,
( groups2983280209131991357nt_int
= ( ^ [P: int > int,I6: set_int] :
( if_int
@ ( finite_finite_int
@ ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ I6 )
& ( ( P @ X2 )
!= zero_zero_int ) ) ) )
@ ( groups4538972089207619220nt_int @ P
@ ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ I6 )
& ( ( P @ X2 )
!= zero_zero_int ) ) ) )
@ zero_zero_int ) ) ) ).
% sum.G_def
thf(fact_595_sum_OG__def,axiom,
( groups7491195623257373319_a_int
= ( ^ [P: a > int,I6: set_a] :
( if_int
@ ( finite_finite_a
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ I6 )
& ( ( P @ X2 )
!= zero_zero_int ) ) ) )
@ ( groups6332066207828071664_a_int @ P
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ I6 )
& ( ( P @ X2 )
!= zero_zero_int ) ) ) )
@ zero_zero_int ) ) ) ).
% sum.G_def
thf(fact_596_sum_OG__def,axiom,
( groups8515261248781899619omplex
= ( ^ [P: nat > complex,I6: set_nat] :
( if_complex
@ ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ I6 )
& ( ( P @ X2 )
!= zero_zero_complex ) ) ) )
@ ( groups2073611262835488442omplex @ P
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ I6 )
& ( ( P @ X2 )
!= zero_zero_complex ) ) ) )
@ zero_zero_complex ) ) ) ).
% sum.G_def
thf(fact_597_sum_OG__def,axiom,
( groups267424677133301183omplex
= ( ^ [P: int > complex,I6: set_int] :
( if_complex
@ ( finite_finite_int
@ ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ I6 )
& ( ( P @ X2 )
!= zero_zero_complex ) ) ) )
@ ( groups3049146728041665814omplex @ P
@ ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ I6 )
& ( ( P @ X2 )
!= zero_zero_complex ) ) ) )
@ zero_zero_complex ) ) ) ).
% sum.G_def
thf(fact_598_sum_OG__def,axiom,
( groups3888493890042768137omplex
= ( ^ [P: a > complex,I6: set_a] :
( if_complex
@ ( finite_finite_a
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ I6 )
& ( ( P @ X2 )
!= zero_zero_complex ) ) ) )
@ ( groups8331919209915413362omplex @ P
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ I6 )
& ( ( P @ X2 )
!= zero_zero_complex ) ) ) )
@ zero_zero_complex ) ) ) ).
% sum.G_def
thf(fact_599_sum_OG__def,axiom,
( groups1986416967739987077at_nat
= ( ^ [P: nat > nat,I6: set_nat] :
( if_nat
@ ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ I6 )
& ( ( P @ X2 )
!= zero_zero_nat ) ) ) )
@ ( groups3542108847815614940at_nat @ P
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ I6 )
& ( ( P @ X2 )
!= zero_zero_nat ) ) ) )
@ zero_zero_nat ) ) ) ).
% sum.G_def
thf(fact_600_sum_OG__def,axiom,
( groups808145749697022017omplex
= ( ^ [P: complex > complex,I6: set_complex] :
( if_complex
@ ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X2: complex] :
( ( member_complex @ X2 @ I6 )
& ( ( P @ X2 )
!= zero_zero_complex ) ) ) )
@ ( groups7754918857620584856omplex @ P
@ ( collect_complex
@ ^ [X2: complex] :
( ( member_complex @ X2 @ I6 )
& ( ( P @ X2 )
!= zero_zero_complex ) ) ) )
@ zero_zero_complex ) ) ) ).
% sum.G_def
thf(fact_601_power__strict__mono,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_602_power__strict__mono,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_603_power__increasing,axiom,
! [N: nat,N3: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N3 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N3 ) ) ) ) ).
% power_increasing
thf(fact_604_power__increasing,axiom,
! [N: nat,N3: nat,A: int] :
( ( ord_less_eq_nat @ N @ N3 )
=> ( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N3 ) ) ) ) ).
% power_increasing
thf(fact_605_mult__less__le__imp__less,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_606_mult__less__le__imp__less,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_607_mult__le__less__imp__less,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_608_mult__le__less__imp__less,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D2 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_609_mult__right__le__imp__le,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_610_mult__right__le__imp__le,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_611_mult__left__le__imp__le,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_612_mult__left__le__imp__le,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_613_mult__le__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_614_mult__le__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_615_mult__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_616_mult__strict__mono_H,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_617_mult__strict__mono_H,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_618_mult__right__less__imp__less,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_619_mult__right__less__imp__less,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_620_mult__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_621_mult__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_622_mult__strict__mono,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D2 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_623_mult__left__less__imp__less,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_624_mult__left__less__imp__less,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_625_mult__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_626_mult__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_627_mult__left__le__one__le,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ Y @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_628_mult__right__le__one__le,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ X @ Y ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_629_mult__le__one,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_630_mult__le__one,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ B @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ) ).
% mult_le_one
thf(fact_631_mult__left__le,axiom,
! [C: nat,A: nat] :
( ( ord_less_eq_nat @ C @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_632_mult__left__le,axiom,
! [C: int,A: int] :
( ( ord_less_eq_int @ C @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_633_power__less__imp__less__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_634_power__less__imp__less__base,axiom,
! [A: int,N: nat,B: int] :
( ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_635_power__le__one,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ one_one_nat ) ) ) ).
% power_le_one
thf(fact_636_power__le__one,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ one_one_int ) ) ) ).
% power_le_one
thf(fact_637_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= one_one_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ) ).
% power_0_left
thf(fact_638_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= one_one_int ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ) ).
% power_0_left
thf(fact_639_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_complex @ zero_zero_complex @ N )
= one_one_complex ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_complex @ zero_zero_complex @ N )
= zero_zero_complex ) ) ) ).
% power_0_left
thf(fact_640_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ).
% zero_power
thf(fact_641_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ).
% zero_power
thf(fact_642_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_complex @ zero_zero_complex @ N )
= zero_zero_complex ) ) ).
% zero_power
thf(fact_643_dvd__imp__le,axiom,
! [K: nat,N: nat] :
( ( dvd_dvd_nat @ K @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ) ).
% dvd_imp_le
thf(fact_644_diff__card__le__card__Diff,axiom,
! [B3: set_set_list_a,A2: set_set_list_a] :
( ( finite5282473924520328461list_a @ B3 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite7490188192133390220list_a @ A2 ) @ ( finite7490188192133390220list_a @ B3 ) ) @ ( finite7490188192133390220list_a @ ( minus_4782336368215558443list_a @ A2 @ B3 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_645_diff__card__le__card__Diff,axiom,
! [B3: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B3 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ B3 ) ) @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B3 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_646_diff__card__le__card__Diff,axiom,
! [B3: set_list_list_a,A2: set_list_list_a] :
( ( finite1660835950917165235list_a @ B3 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite9134805042761151410list_a @ A2 ) @ ( finite9134805042761151410list_a @ B3 ) ) @ ( finite9134805042761151410list_a @ ( minus_5335179877275218001list_a @ A2 @ B3 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_647_diff__card__le__card__Diff,axiom,
! [B3: set_list_a,A2: set_list_a] :
( ( finite_finite_list_a @ B3 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_list_a @ A2 ) @ ( finite_card_list_a @ B3 ) ) @ ( finite_card_list_a @ ( minus_646659088055828811list_a @ A2 @ B3 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_648_diff__card__le__card__Diff,axiom,
! [B3: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B3 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B3 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B3 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_649_diff__card__le__card__Diff,axiom,
! [B3: set_int,A2: set_int] :
( ( finite_finite_int @ B3 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_int @ A2 ) @ ( finite_card_int @ B3 ) ) @ ( finite_card_int @ ( minus_minus_set_int @ A2 @ B3 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_650_diff__card__le__card__Diff,axiom,
! [B3: set_a,A2: set_a] :
( ( finite_finite_a @ B3 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B3 ) ) @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ B3 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_651_sum__pos2,axiom,
! [I3: set_int,I: int,F2: int > nat] :
( ( finite_finite_int @ I3 )
=> ( ( member_int @ I @ I3 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F2 @ I ) )
=> ( ! [I5: int] :
( ( member_int @ I5 @ I3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ I5 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F2 @ I3 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_652_sum__pos2,axiom,
! [I3: set_a,I: a,F2: a > nat] :
( ( finite_finite_a @ I3 )
=> ( ( member_a @ I @ I3 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F2 @ I ) )
=> ( ! [I5: a] :
( ( member_a @ I5 @ I3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ I5 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups6334556678337121940_a_nat @ F2 @ I3 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_653_sum__pos2,axiom,
! [I3: set_nat,I: nat,F2: nat > int] :
( ( finite_finite_nat @ I3 )
=> ( ( member_nat @ I @ I3 )
=> ( ( ord_less_int @ zero_zero_int @ ( F2 @ I ) )
=> ( ! [I5: nat] :
( ( member_nat @ I5 @ I3 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F2 @ I5 ) ) )
=> ( ord_less_int @ zero_zero_int @ ( groups3539618377306564664at_int @ F2 @ I3 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_654_sum__pos2,axiom,
! [I3: set_int,I: int,F2: int > int] :
( ( finite_finite_int @ I3 )
=> ( ( member_int @ I @ I3 )
=> ( ( ord_less_int @ zero_zero_int @ ( F2 @ I ) )
=> ( ! [I5: int] :
( ( member_int @ I5 @ I3 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F2 @ I5 ) ) )
=> ( ord_less_int @ zero_zero_int @ ( groups4538972089207619220nt_int @ F2 @ I3 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_655_sum__pos2,axiom,
! [I3: set_a,I: a,F2: a > int] :
( ( finite_finite_a @ I3 )
=> ( ( member_a @ I @ I3 )
=> ( ( ord_less_int @ zero_zero_int @ ( F2 @ I ) )
=> ( ! [I5: a] :
( ( member_a @ I5 @ I3 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F2 @ I5 ) ) )
=> ( ord_less_int @ zero_zero_int @ ( groups6332066207828071664_a_int @ F2 @ I3 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_656_sum__pos2,axiom,
! [I3: set_nat,I: nat,F2: nat > nat] :
( ( finite_finite_nat @ I3 )
=> ( ( member_nat @ I @ I3 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F2 @ I ) )
=> ( ! [I5: nat] :
( ( member_nat @ I5 @ I3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ I5 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups3542108847815614940at_nat @ F2 @ I3 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_657_sum__pos2,axiom,
! [I3: set_set_a,I: set_a,F2: set_a > nat] :
( ( finite_finite_set_a @ I3 )
=> ( ( member_set_a @ I @ I3 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F2 @ I ) )
=> ( ! [I5: set_a] :
( ( member_set_a @ I5 @ I3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ I5 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups6141743369313575924_a_nat @ F2 @ I3 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_658_sum__pos2,axiom,
! [I3: set_set_a,I: set_a,F2: set_a > int] :
( ( finite_finite_set_a @ I3 )
=> ( ( member_set_a @ I @ I3 )
=> ( ( ord_less_int @ zero_zero_int @ ( F2 @ I ) )
=> ( ! [I5: set_a] :
( ( member_set_a @ I5 @ I3 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F2 @ I5 ) ) )
=> ( ord_less_int @ zero_zero_int @ ( groups6139252898804525648_a_int @ F2 @ I3 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_659_sum__pos2,axiom,
! [I3: set_list_a,I: list_a,F2: list_a > int] :
( ( finite_finite_list_a @ I3 )
=> ( ( member_list_a @ I @ I3 )
=> ( ( ord_less_int @ zero_zero_int @ ( F2 @ I ) )
=> ( ! [I5: list_a] :
( ( member_list_a @ I5 @ I3 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F2 @ I5 ) ) )
=> ( ord_less_int @ zero_zero_int @ ( groups5518757228788810486_a_int @ F2 @ I3 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_660_sum__pos2,axiom,
! [I3: set_list_a,I: list_a,F2: list_a > nat] :
( ( finite_finite_list_a @ I3 )
=> ( ( member_list_a @ I @ I3 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F2 @ I ) )
=> ( ! [I5: list_a] :
( ( member_list_a @ I5 @ I3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ I5 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups5521247699297860762_a_nat @ F2 @ I3 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_661_less__mono__imp__le__mono,axiom,
! [F2: nat > nat,I: nat,J: nat] :
( ! [I5: nat,J3: nat] :
( ( ord_less_nat @ I5 @ J3 )
=> ( ord_less_nat @ ( F2 @ I5 ) @ ( F2 @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F2 @ I ) @ ( F2 @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_662_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_663_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_664_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N4: nat] :
( ( ord_less_nat @ M3 @ N4 )
| ( M3 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_665_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_666_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N4: nat] :
( ( ord_less_eq_nat @ M3 @ N4 )
& ( M3 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_667_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_668_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_669_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_670_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_671_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_672_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_673_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_674_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_675_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_676_zero__neq__one,axiom,
zero_zero_complex != one_one_complex ).
% zero_neq_one
thf(fact_677_mult__not__zero,axiom,
! [A: complex,B: complex] :
( ( ( times_times_complex @ A @ B )
!= zero_zero_complex )
=> ( ( A != zero_zero_complex )
& ( B != zero_zero_complex ) ) ) ).
% mult_not_zero
thf(fact_678_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_679_mult__not__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
!= zero_zero_int )
=> ( ( A != zero_zero_int )
& ( B != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_680_divisors__zero,axiom,
! [A: complex,B: complex] :
( ( ( times_times_complex @ A @ B )
= zero_zero_complex )
=> ( ( A = zero_zero_complex )
| ( B = zero_zero_complex ) ) ) ).
% divisors_zero
thf(fact_681_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_682_divisors__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
=> ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_683_no__zero__divisors,axiom,
! [A: complex,B: complex] :
( ( A != zero_zero_complex )
=> ( ( B != zero_zero_complex )
=> ( ( times_times_complex @ A @ B )
!= zero_zero_complex ) ) ) ).
% no_zero_divisors
thf(fact_684_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_685_no__zero__divisors,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( ( times_times_int @ A @ B )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_686_mult__left__cancel,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( ( times_times_complex @ C @ A )
= ( times_times_complex @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_687_mult__left__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_688_mult__left__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_689_mult__right__cancel,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( ( times_times_complex @ A @ C )
= ( times_times_complex @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_690_mult__right__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_691_mult__right__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_692_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_693_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_694_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_695_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_696_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_697_power__not__zero,axiom,
! [A: nat,N: nat] :
( ( A != zero_zero_nat )
=> ( ( power_power_nat @ A @ N )
!= zero_zero_nat ) ) ).
% power_not_zero
thf(fact_698_power__not__zero,axiom,
! [A: int,N: nat] :
( ( A != zero_zero_int )
=> ( ( power_power_int @ A @ N )
!= zero_zero_int ) ) ).
% power_not_zero
thf(fact_699_power__not__zero,axiom,
! [A: complex,N: nat] :
( ( A != zero_zero_complex )
=> ( ( power_power_complex @ A @ N )
!= zero_zero_complex ) ) ).
% power_not_zero
thf(fact_700_dvd__0__left,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
=> ( A = zero_zero_nat ) ) ).
% dvd_0_left
thf(fact_701_dvd__0__left,axiom,
! [A: int] :
( ( dvd_dvd_int @ zero_zero_int @ A )
=> ( A = zero_zero_int ) ) ).
% dvd_0_left
thf(fact_702_dvd__0__left,axiom,
! [A: complex] :
( ( dvd_dvd_complex @ zero_zero_complex @ A )
=> ( A = zero_zero_complex ) ) ).
% dvd_0_left
thf(fact_703_infinite__descent0,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P2 @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P2 @ M2 ) ) ) )
=> ( P2 @ N ) ) ) ).
% infinite_descent0
thf(fact_704_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_705_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_706_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_707_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_708_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_709_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_710_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_711_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_712_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_713_sum__mono__inv,axiom,
! [F2: int > nat,I3: set_int,G: int > nat,I: int] :
( ( ( groups4541462559716669496nt_nat @ F2 @ I3 )
= ( groups4541462559716669496nt_nat @ G @ I3 ) )
=> ( ! [I5: int] :
( ( member_int @ I5 @ I3 )
=> ( ord_less_eq_nat @ ( F2 @ I5 ) @ ( G @ I5 ) ) )
=> ( ( member_int @ I @ I3 )
=> ( ( finite_finite_int @ I3 )
=> ( ( F2 @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_714_sum__mono__inv,axiom,
! [F2: a > nat,I3: set_a,G: a > nat,I: a] :
( ( ( groups6334556678337121940_a_nat @ F2 @ I3 )
= ( groups6334556678337121940_a_nat @ G @ I3 ) )
=> ( ! [I5: a] :
( ( member_a @ I5 @ I3 )
=> ( ord_less_eq_nat @ ( F2 @ I5 ) @ ( G @ I5 ) ) )
=> ( ( member_a @ I @ I3 )
=> ( ( finite_finite_a @ I3 )
=> ( ( F2 @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_715_sum__mono__inv,axiom,
! [F2: nat > int,I3: set_nat,G: nat > int,I: nat] :
( ( ( groups3539618377306564664at_int @ F2 @ I3 )
= ( groups3539618377306564664at_int @ G @ I3 ) )
=> ( ! [I5: nat] :
( ( member_nat @ I5 @ I3 )
=> ( ord_less_eq_int @ ( F2 @ I5 ) @ ( G @ I5 ) ) )
=> ( ( member_nat @ I @ I3 )
=> ( ( finite_finite_nat @ I3 )
=> ( ( F2 @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_716_sum__mono__inv,axiom,
! [F2: int > int,I3: set_int,G: int > int,I: int] :
( ( ( groups4538972089207619220nt_int @ F2 @ I3 )
= ( groups4538972089207619220nt_int @ G @ I3 ) )
=> ( ! [I5: int] :
( ( member_int @ I5 @ I3 )
=> ( ord_less_eq_int @ ( F2 @ I5 ) @ ( G @ I5 ) ) )
=> ( ( member_int @ I @ I3 )
=> ( ( finite_finite_int @ I3 )
=> ( ( F2 @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_717_sum__mono__inv,axiom,
! [F2: a > int,I3: set_a,G: a > int,I: a] :
( ( ( groups6332066207828071664_a_int @ F2 @ I3 )
= ( groups6332066207828071664_a_int @ G @ I3 ) )
=> ( ! [I5: a] :
( ( member_a @ I5 @ I3 )
=> ( ord_less_eq_int @ ( F2 @ I5 ) @ ( G @ I5 ) ) )
=> ( ( member_a @ I @ I3 )
=> ( ( finite_finite_a @ I3 )
=> ( ( F2 @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_718_sum__mono__inv,axiom,
! [F2: nat > nat,I3: set_nat,G: nat > nat,I: nat] :
( ( ( groups3542108847815614940at_nat @ F2 @ I3 )
= ( groups3542108847815614940at_nat @ G @ I3 ) )
=> ( ! [I5: nat] :
( ( member_nat @ I5 @ I3 )
=> ( ord_less_eq_nat @ ( F2 @ I5 ) @ ( G @ I5 ) ) )
=> ( ( member_nat @ I @ I3 )
=> ( ( finite_finite_nat @ I3 )
=> ( ( F2 @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_719_sum__mono__inv,axiom,
! [F2: set_a > nat,I3: set_set_a,G: set_a > nat,I: set_a] :
( ( ( groups6141743369313575924_a_nat @ F2 @ I3 )
= ( groups6141743369313575924_a_nat @ G @ I3 ) )
=> ( ! [I5: set_a] :
( ( member_set_a @ I5 @ I3 )
=> ( ord_less_eq_nat @ ( F2 @ I5 ) @ ( G @ I5 ) ) )
=> ( ( member_set_a @ I @ I3 )
=> ( ( finite_finite_set_a @ I3 )
=> ( ( F2 @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_720_sum__mono__inv,axiom,
! [F2: set_a > int,I3: set_set_a,G: set_a > int,I: set_a] :
( ( ( groups6139252898804525648_a_int @ F2 @ I3 )
= ( groups6139252898804525648_a_int @ G @ I3 ) )
=> ( ! [I5: set_a] :
( ( member_set_a @ I5 @ I3 )
=> ( ord_less_eq_int @ ( F2 @ I5 ) @ ( G @ I5 ) ) )
=> ( ( member_set_a @ I @ I3 )
=> ( ( finite_finite_set_a @ I3 )
=> ( ( F2 @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_721_sum__mono__inv,axiom,
! [F2: list_a > int,I3: set_list_a,G: list_a > int,I: list_a] :
( ( ( groups5518757228788810486_a_int @ F2 @ I3 )
= ( groups5518757228788810486_a_int @ G @ I3 ) )
=> ( ! [I5: list_a] :
( ( member_list_a @ I5 @ I3 )
=> ( ord_less_eq_int @ ( F2 @ I5 ) @ ( G @ I5 ) ) )
=> ( ( member_list_a @ I @ I3 )
=> ( ( finite_finite_list_a @ I3 )
=> ( ( F2 @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_722_sum__mono__inv,axiom,
! [F2: list_a > nat,I3: set_list_a,G: list_a > nat,I: list_a] :
( ( ( groups5521247699297860762_a_nat @ F2 @ I3 )
= ( groups5521247699297860762_a_nat @ G @ I3 ) )
=> ( ! [I5: list_a] :
( ( member_list_a @ I5 @ I3 )
=> ( ord_less_eq_nat @ ( F2 @ I5 ) @ ( G @ I5 ) ) )
=> ( ( member_list_a @ I @ I3 )
=> ( ( finite_finite_list_a @ I3 )
=> ( ( F2 @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_723_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A3: int,B2: int] : ( ord_less_int @ ( minus_minus_int @ A3 @ B2 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_724_power__le__imp__le__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_725_power__le__imp__le__exp,axiom,
! [A: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_726_mult__le__cancel__left1,axiom,
! [C: int,B: int] :
( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ one_one_int @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_727_mult__le__cancel__left2,axiom,
! [C: int,A: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ C )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ one_one_int ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_728_mult__le__cancel__right1,axiom,
! [C: int,B: int] :
( ( ord_less_eq_int @ C @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ one_one_int @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_729_mult__le__cancel__right2,axiom,
! [A: int,C: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ C )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ one_one_int ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_730_mult__less__cancel__left1,axiom,
! [C: int,B: int] :
( ( ord_less_int @ C @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_731_mult__less__cancel__left2,axiom,
! [C: int,A: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_732_mult__less__cancel__right1,axiom,
! [C: int,B: int] :
( ( ord_less_int @ C @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_733_mult__less__cancel__right2,axiom,
! [A: int,C: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_734_dvd__power__iff,axiom,
! [X: nat,M: nat,N: nat] :
( ( X != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ X @ M ) @ ( power_power_nat @ X @ N ) )
= ( ( dvd_dvd_nat @ X @ one_one_nat )
| ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% dvd_power_iff
thf(fact_735_dvd__power__iff,axiom,
! [X: int,M: nat,N: nat] :
( ( X != zero_zero_int )
=> ( ( dvd_dvd_int @ ( power_power_int @ X @ M ) @ ( power_power_int @ X @ N ) )
= ( ( dvd_dvd_int @ X @ one_one_int )
| ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% dvd_power_iff
thf(fact_736_self__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_737_self__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ one_one_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_738_strict__subset__divisors__dvd,axiom,
! [A: nat,B: nat] :
( ( ord_less_set_nat
@ ( collect_nat
@ ^ [C2: nat] : ( dvd_dvd_nat @ C2 @ A ) )
@ ( collect_nat
@ ^ [C2: nat] : ( dvd_dvd_nat @ C2 @ B ) ) )
= ( ( dvd_dvd_nat @ A @ B )
& ~ ( dvd_dvd_nat @ B @ A ) ) ) ).
% strict_subset_divisors_dvd
thf(fact_739_strict__subset__divisors__dvd,axiom,
! [A: int,B: int] :
( ( ord_less_set_int
@ ( collect_int
@ ^ [C2: int] : ( dvd_dvd_int @ C2 @ A ) )
@ ( collect_int
@ ^ [C2: int] : ( dvd_dvd_int @ C2 @ B ) ) )
= ( ( dvd_dvd_int @ A @ B )
& ~ ( dvd_dvd_int @ B @ A ) ) ) ).
% strict_subset_divisors_dvd
thf(fact_740_lambda__zero,axiom,
( ( ^ [H2: complex] : zero_zero_complex )
= ( times_times_complex @ zero_zero_complex ) ) ).
% lambda_zero
thf(fact_741_lambda__zero,axiom,
( ( ^ [H2: nat] : zero_zero_nat )
= ( times_times_nat @ zero_zero_nat ) ) ).
% lambda_zero
thf(fact_742_lambda__zero,axiom,
( ( ^ [H2: int] : zero_zero_int )
= ( times_times_int @ zero_zero_int ) ) ).
% lambda_zero
thf(fact_743_sum_Ointer__filter,axiom,
! [A2: set_int,G: int > nat,P2: int > $o] :
( ( finite_finite_int @ A2 )
=> ( ( groups4541462559716669496nt_nat @ G
@ ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ A2 )
& ( P2 @ X2 ) ) ) )
= ( groups4541462559716669496nt_nat
@ ^ [X2: int] : ( if_nat @ ( P2 @ X2 ) @ ( G @ X2 ) @ zero_zero_nat )
@ A2 ) ) ) ).
% sum.inter_filter
thf(fact_744_sum_Ointer__filter,axiom,
! [A2: set_a,G: a > nat,P2: a > $o] :
( ( finite_finite_a @ A2 )
=> ( ( groups6334556678337121940_a_nat @ G
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A2 )
& ( P2 @ X2 ) ) ) )
= ( groups6334556678337121940_a_nat
@ ^ [X2: a] : ( if_nat @ ( P2 @ X2 ) @ ( G @ X2 ) @ zero_zero_nat )
@ A2 ) ) ) ).
% sum.inter_filter
thf(fact_745_sum_Ointer__filter,axiom,
! [A2: set_nat,G: nat > int,P2: nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( groups3539618377306564664at_int @ G
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P2 @ X2 ) ) ) )
= ( groups3539618377306564664at_int
@ ^ [X2: nat] : ( if_int @ ( P2 @ X2 ) @ ( G @ X2 ) @ zero_zero_int )
@ A2 ) ) ) ).
% sum.inter_filter
thf(fact_746_sum_Ointer__filter,axiom,
! [A2: set_int,G: int > int,P2: int > $o] :
( ( finite_finite_int @ A2 )
=> ( ( groups4538972089207619220nt_int @ G
@ ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ A2 )
& ( P2 @ X2 ) ) ) )
= ( groups4538972089207619220nt_int
@ ^ [X2: int] : ( if_int @ ( P2 @ X2 ) @ ( G @ X2 ) @ zero_zero_int )
@ A2 ) ) ) ).
% sum.inter_filter
thf(fact_747_sum_Ointer__filter,axiom,
! [A2: set_a,G: a > int,P2: a > $o] :
( ( finite_finite_a @ A2 )
=> ( ( groups6332066207828071664_a_int @ G
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A2 )
& ( P2 @ X2 ) ) ) )
= ( groups6332066207828071664_a_int
@ ^ [X2: a] : ( if_int @ ( P2 @ X2 ) @ ( G @ X2 ) @ zero_zero_int )
@ A2 ) ) ) ).
% sum.inter_filter
thf(fact_748_sum_Ointer__filter,axiom,
! [A2: set_nat,G: nat > complex,P2: nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( groups2073611262835488442omplex @ G
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P2 @ X2 ) ) ) )
= ( groups2073611262835488442omplex
@ ^ [X2: nat] : ( if_complex @ ( P2 @ X2 ) @ ( G @ X2 ) @ zero_zero_complex )
@ A2 ) ) ) ).
% sum.inter_filter
thf(fact_749_sum_Ointer__filter,axiom,
! [A2: set_int,G: int > complex,P2: int > $o] :
( ( finite_finite_int @ A2 )
=> ( ( groups3049146728041665814omplex @ G
@ ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ A2 )
& ( P2 @ X2 ) ) ) )
= ( groups3049146728041665814omplex
@ ^ [X2: int] : ( if_complex @ ( P2 @ X2 ) @ ( G @ X2 ) @ zero_zero_complex )
@ A2 ) ) ) ).
% sum.inter_filter
thf(fact_750_sum_Ointer__filter,axiom,
! [A2: set_a,G: a > complex,P2: a > $o] :
( ( finite_finite_a @ A2 )
=> ( ( groups8331919209915413362omplex @ G
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A2 )
& ( P2 @ X2 ) ) ) )
= ( groups8331919209915413362omplex
@ ^ [X2: a] : ( if_complex @ ( P2 @ X2 ) @ ( G @ X2 ) @ zero_zero_complex )
@ A2 ) ) ) ).
% sum.inter_filter
thf(fact_751_sum_Ointer__filter,axiom,
! [A2: set_nat,G: nat > nat,P2: nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( groups3542108847815614940at_nat @ G
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P2 @ X2 ) ) ) )
= ( groups3542108847815614940at_nat
@ ^ [X2: nat] : ( if_nat @ ( P2 @ X2 ) @ ( G @ X2 ) @ zero_zero_nat )
@ A2 ) ) ) ).
% sum.inter_filter
thf(fact_752_sum_Ointer__filter,axiom,
! [A2: set_complex,G: complex > complex,P2: complex > $o] :
( ( finite3207457112153483333omplex @ A2 )
=> ( ( groups7754918857620584856omplex @ G
@ ( collect_complex
@ ^ [X2: complex] :
( ( member_complex @ X2 @ A2 )
& ( P2 @ X2 ) ) ) )
= ( groups7754918857620584856omplex
@ ^ [X2: complex] : ( if_complex @ ( P2 @ X2 ) @ ( G @ X2 ) @ zero_zero_complex )
@ A2 ) ) ) ).
% sum.inter_filter
thf(fact_753_sum__subtractf__nat,axiom,
! [A2: set_set_list_a,G: set_list_a > nat,F2: set_list_a > nat] :
( ! [X3: set_list_a] :
( ( member_set_list_a @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( G @ X3 ) @ ( F2 @ X3 ) ) )
=> ( ( groups5993734322560061562_a_nat
@ ^ [X2: set_list_a] : ( minus_minus_nat @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ A2 )
= ( minus_minus_nat @ ( groups5993734322560061562_a_nat @ F2 @ A2 ) @ ( groups5993734322560061562_a_nat @ G @ A2 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_754_sum__subtractf__nat,axiom,
! [A2: set_set_a,G: set_a > nat,F2: set_a > nat] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( G @ X3 ) @ ( F2 @ X3 ) ) )
=> ( ( groups6141743369313575924_a_nat
@ ^ [X2: set_a] : ( minus_minus_nat @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ A2 )
= ( minus_minus_nat @ ( groups6141743369313575924_a_nat @ F2 @ A2 ) @ ( groups6141743369313575924_a_nat @ G @ A2 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_755_sum__subtractf__nat,axiom,
! [A2: set_li3422455791611400469list_a,G: ( list_list_a > list_a ) > nat,F2: ( list_list_a > list_a ) > nat] :
( ! [X3: list_list_a > list_a] :
( ( member7168557129179038582list_a @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( G @ X3 ) @ ( F2 @ X3 ) ) )
=> ( ( groups5819086642190997227_a_nat
@ ^ [X2: list_list_a > list_a] : ( minus_minus_nat @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ A2 )
= ( minus_minus_nat @ ( groups5819086642190997227_a_nat @ F2 @ A2 ) @ ( groups5819086642190997227_a_nat @ G @ A2 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_756_sum__subtractf__nat,axiom,
! [A2: set_a,G: a > nat,F2: a > nat] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( G @ X3 ) @ ( F2 @ X3 ) ) )
=> ( ( groups6334556678337121940_a_nat
@ ^ [X2: a] : ( minus_minus_nat @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ A2 )
= ( minus_minus_nat @ ( groups6334556678337121940_a_nat @ F2 @ A2 ) @ ( groups6334556678337121940_a_nat @ G @ A2 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_757_sum__subtractf__nat,axiom,
! [A2: set_nat,G: nat > nat,F2: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( G @ X3 ) @ ( F2 @ X3 ) ) )
=> ( ( groups3542108847815614940at_nat
@ ^ [X2: nat] : ( minus_minus_nat @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ A2 )
= ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F2 @ A2 ) @ ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_758_sum__subtractf__nat,axiom,
! [A2: set_list_a,G: list_a > nat,F2: list_a > nat] :
( ! [X3: list_a] :
( ( member_list_a @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( G @ X3 ) @ ( F2 @ X3 ) ) )
=> ( ( groups5521247699297860762_a_nat
@ ^ [X2: list_a] : ( minus_minus_nat @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ A2 )
= ( minus_minus_nat @ ( groups5521247699297860762_a_nat @ F2 @ A2 ) @ ( groups5521247699297860762_a_nat @ G @ A2 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_759_sum__strict__mono__ex1,axiom,
! [A2: set_list_list_a,F2: list_list_a > nat,G: list_list_a > nat] :
( ( finite1660835950917165235list_a @ A2 )
=> ( ! [X3: list_list_a] :
( ( member_list_list_a @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
=> ( ? [X4: list_list_a] :
( ( member_list_list_a @ X4 @ A2 )
& ( ord_less_nat @ ( F2 @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_nat @ ( groups7548105480907152928_a_nat @ F2 @ A2 ) @ ( groups7548105480907152928_a_nat @ G @ A2 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_760_sum__strict__mono__ex1,axiom,
! [A2: set_int,F2: int > nat,G: int > nat] :
( ( finite_finite_int @ A2 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
=> ( ? [X4: int] :
( ( member_int @ X4 @ A2 )
& ( ord_less_nat @ ( F2 @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F2 @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_761_sum__strict__mono__ex1,axiom,
! [A2: set_a,F2: a > nat,G: a > nat] :
( ( finite_finite_a @ A2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
=> ( ? [X4: a] :
( ( member_a @ X4 @ A2 )
& ( ord_less_nat @ ( F2 @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_nat @ ( groups6334556678337121940_a_nat @ F2 @ A2 ) @ ( groups6334556678337121940_a_nat @ G @ A2 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_762_sum__strict__mono__ex1,axiom,
! [A2: set_list_list_a,F2: list_list_a > int,G: list_list_a > int] :
( ( finite1660835950917165235list_a @ A2 )
=> ( ! [X3: list_list_a] :
( ( member_list_list_a @ X3 @ A2 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
=> ( ? [X4: list_list_a] :
( ( member_list_list_a @ X4 @ A2 )
& ( ord_less_int @ ( F2 @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_int @ ( groups7545615010398102652_a_int @ F2 @ A2 ) @ ( groups7545615010398102652_a_int @ G @ A2 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_763_sum__strict__mono__ex1,axiom,
! [A2: set_list_a,F2: list_a > int,G: list_a > int] :
( ( finite_finite_list_a @ A2 )
=> ( ! [X3: list_a] :
( ( member_list_a @ X3 @ A2 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
=> ( ? [X4: list_a] :
( ( member_list_a @ X4 @ A2 )
& ( ord_less_int @ ( F2 @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_int @ ( groups5518757228788810486_a_int @ F2 @ A2 ) @ ( groups5518757228788810486_a_int @ G @ A2 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_764_sum__strict__mono__ex1,axiom,
! [A2: set_nat,F2: nat > int,G: nat > int] :
( ( finite_finite_nat @ A2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
=> ( ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_less_int @ ( F2 @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_int @ ( groups3539618377306564664at_int @ F2 @ A2 ) @ ( groups3539618377306564664at_int @ G @ A2 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_765_sum__strict__mono__ex1,axiom,
! [A2: set_int,F2: int > int,G: int > int] :
( ( finite_finite_int @ A2 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
=> ( ? [X4: int] :
( ( member_int @ X4 @ A2 )
& ( ord_less_int @ ( F2 @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_int @ ( groups4538972089207619220nt_int @ F2 @ A2 ) @ ( groups4538972089207619220nt_int @ G @ A2 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_766_sum__strict__mono__ex1,axiom,
! [A2: set_a,F2: a > int,G: a > int] :
( ( finite_finite_a @ A2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
=> ( ? [X4: a] :
( ( member_a @ X4 @ A2 )
& ( ord_less_int @ ( F2 @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_int @ ( groups6332066207828071664_a_int @ F2 @ A2 ) @ ( groups6332066207828071664_a_int @ G @ A2 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_767_sum__strict__mono__ex1,axiom,
! [A2: set_nat,F2: nat > nat,G: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
=> ( ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_less_nat @ ( F2 @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_nat @ ( groups3542108847815614940at_nat @ F2 @ A2 ) @ ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_768_sum__strict__mono__ex1,axiom,
! [A2: set_list_a,F2: list_a > nat,G: list_a > nat] :
( ( finite_finite_list_a @ A2 )
=> ( ! [X3: list_a] :
( ( member_list_a @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
=> ( ? [X4: list_a] :
( ( member_list_a @ X4 @ A2 )
& ( ord_less_nat @ ( F2 @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_nat @ ( groups5521247699297860762_a_nat @ F2 @ A2 ) @ ( groups5521247699297860762_a_nat @ G @ A2 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_769_card__less__sym__Diff,axiom,
! [A2: set_set_list_a,B3: set_set_list_a] :
( ( finite5282473924520328461list_a @ A2 )
=> ( ( finite5282473924520328461list_a @ B3 )
=> ( ( ord_less_nat @ ( finite7490188192133390220list_a @ A2 ) @ ( finite7490188192133390220list_a @ B3 ) )
=> ( ord_less_nat @ ( finite7490188192133390220list_a @ ( minus_4782336368215558443list_a @ A2 @ B3 ) ) @ ( finite7490188192133390220list_a @ ( minus_4782336368215558443list_a @ B3 @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_770_card__less__sym__Diff,axiom,
! [A2: set_set_a,B3: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( finite_finite_set_a @ B3 )
=> ( ( ord_less_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ B3 ) )
=> ( ord_less_nat @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B3 ) ) @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ B3 @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_771_card__less__sym__Diff,axiom,
! [A2: set_list_list_a,B3: set_list_list_a] :
( ( finite1660835950917165235list_a @ A2 )
=> ( ( finite1660835950917165235list_a @ B3 )
=> ( ( ord_less_nat @ ( finite9134805042761151410list_a @ A2 ) @ ( finite9134805042761151410list_a @ B3 ) )
=> ( ord_less_nat @ ( finite9134805042761151410list_a @ ( minus_5335179877275218001list_a @ A2 @ B3 ) ) @ ( finite9134805042761151410list_a @ ( minus_5335179877275218001list_a @ B3 @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_772_card__less__sym__Diff,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( finite_finite_list_a @ A2 )
=> ( ( finite_finite_list_a @ B3 )
=> ( ( ord_less_nat @ ( finite_card_list_a @ A2 ) @ ( finite_card_list_a @ B3 ) )
=> ( ord_less_nat @ ( finite_card_list_a @ ( minus_646659088055828811list_a @ A2 @ B3 ) ) @ ( finite_card_list_a @ ( minus_646659088055828811list_a @ B3 @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_773_card__less__sym__Diff,axiom,
! [A2: set_nat,B3: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B3 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B3 ) )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B3 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B3 @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_774_card__less__sym__Diff,axiom,
! [A2: set_int,B3: set_int] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ B3 )
=> ( ( ord_less_nat @ ( finite_card_int @ A2 ) @ ( finite_card_int @ B3 ) )
=> ( ord_less_nat @ ( finite_card_int @ ( minus_minus_set_int @ A2 @ B3 ) ) @ ( finite_card_int @ ( minus_minus_set_int @ B3 @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_775_card__less__sym__Diff,axiom,
! [A2: set_a,B3: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B3 )
=> ( ( ord_less_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B3 ) )
=> ( ord_less_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ B3 ) ) @ ( finite_card_a @ ( minus_minus_set_a @ B3 @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_776_psubset__card__mono,axiom,
! [B3: set_set_list_a,A2: set_set_list_a] :
( ( finite5282473924520328461list_a @ B3 )
=> ( ( ord_le1620366983259561968list_a @ A2 @ B3 )
=> ( ord_less_nat @ ( finite7490188192133390220list_a @ A2 ) @ ( finite7490188192133390220list_a @ B3 ) ) ) ) ).
% psubset_card_mono
thf(fact_777_psubset__card__mono,axiom,
! [B3: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B3 )
=> ( ( ord_less_set_set_a @ A2 @ B3 )
=> ( ord_less_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ B3 ) ) ) ) ).
% psubset_card_mono
thf(fact_778_psubset__card__mono,axiom,
! [B3: set_list_list_a,A2: set_list_list_a] :
( ( finite1660835950917165235list_a @ B3 )
=> ( ( ord_le5338140678153942166list_a @ A2 @ B3 )
=> ( ord_less_nat @ ( finite9134805042761151410list_a @ A2 ) @ ( finite9134805042761151410list_a @ B3 ) ) ) ) ).
% psubset_card_mono
thf(fact_779_psubset__card__mono,axiom,
! [B3: set_list_a,A2: set_list_a] :
( ( finite_finite_list_a @ B3 )
=> ( ( ord_less_set_list_a @ A2 @ B3 )
=> ( ord_less_nat @ ( finite_card_list_a @ A2 ) @ ( finite_card_list_a @ B3 ) ) ) ) ).
% psubset_card_mono
thf(fact_780_psubset__card__mono,axiom,
! [B3: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B3 )
=> ( ( ord_less_set_nat @ A2 @ B3 )
=> ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B3 ) ) ) ) ).
% psubset_card_mono
thf(fact_781_psubset__card__mono,axiom,
! [B3: set_int,A2: set_int] :
( ( finite_finite_int @ B3 )
=> ( ( ord_less_set_int @ A2 @ B3 )
=> ( ord_less_nat @ ( finite_card_int @ A2 ) @ ( finite_card_int @ B3 ) ) ) ) ).
% psubset_card_mono
thf(fact_782_psubset__card__mono,axiom,
! [B3: set_a,A2: set_a] :
( ( finite_finite_a @ B3 )
=> ( ( ord_less_set_a @ A2 @ B3 )
=> ( ord_less_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B3 ) ) ) ) ).
% psubset_card_mono
thf(fact_783_one__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ).
% one_le_power
thf(fact_784_one__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ).
% one_le_power
thf(fact_785_dvd__power__le,axiom,
! [X: nat,Y: nat,N: nat,M: nat] :
( ( dvd_dvd_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ M ) ) ) ) ).
% dvd_power_le
thf(fact_786_dvd__power__le,axiom,
! [X: int,Y: int,N: nat,M: nat] :
( ( dvd_dvd_int @ X @ Y )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ M ) ) ) ) ).
% dvd_power_le
thf(fact_787_dvd__power__le,axiom,
! [X: complex,Y: complex,N: nat,M: nat] :
( ( dvd_dvd_complex @ X @ Y )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y @ M ) ) ) ) ).
% dvd_power_le
thf(fact_788_power__le__dvd,axiom,
! [A: nat,N: nat,B: nat,M: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ B )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ B ) ) ) ).
% power_le_dvd
thf(fact_789_power__le__dvd,axiom,
! [A: int,N: nat,B: int,M: nat] :
( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ B )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ B ) ) ) ).
% power_le_dvd
thf(fact_790_power__le__dvd,axiom,
! [A: complex,N: nat,B: complex,M: nat] :
( ( dvd_dvd_complex @ ( power_power_complex @ A @ N ) @ B )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ B ) ) ) ).
% power_le_dvd
thf(fact_791_le__imp__power__dvd,axiom,
! [M: nat,N: nat,A: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).
% le_imp_power_dvd
thf(fact_792_le__imp__power__dvd,axiom,
! [M: nat,N: nat,A: int] :
( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).
% le_imp_power_dvd
thf(fact_793_le__imp__power__dvd,axiom,
! [M: nat,N: nat,A: complex] :
( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ).
% le_imp_power_dvd
thf(fact_794_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_795_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_796_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_797_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_798_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_799_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_800_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_801_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_802_mult__less__cancel__right__disj,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_803_mult__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_804_mult__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_805_mult__strict__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_806_mult__less__cancel__left__disj,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_807_mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_808_mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_809_mult__strict__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_810_mult__less__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_811_mult__less__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_812_zero__less__mult__pos2,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_813_zero__less__mult__pos2,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_814_zero__less__mult__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_815_zero__less__mult__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_816_zero__less__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ zero_zero_int @ B ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_817_mult__pos__neg2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_818_mult__pos__neg2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_pos_neg2
thf(fact_819_mult__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 @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_820_mult__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_821_mult__pos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_822_mult__pos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_pos_neg
thf(fact_823_mult__neg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_824_mult__neg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_neg_pos
thf(fact_825_mult__less__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ B @ zero_zero_int ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_826_not__square__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_827_mult__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_828_dvd__diffD,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
=> ( ( dvd_dvd_nat @ K @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_nat @ K @ M ) ) ) ) ).
% dvd_diffD
thf(fact_829_dvd__diffD1,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
=> ( ( dvd_dvd_nat @ K @ M )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_nat @ K @ N ) ) ) ) ).
% dvd_diffD1
thf(fact_830_less__eq__dvd__minus,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( dvd_dvd_nat @ M @ N )
= ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% less_eq_dvd_minus
thf(fact_831_card__ge__0__finite,axiom,
! [A2: set_set_list_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite7490188192133390220list_a @ A2 ) )
=> ( finite5282473924520328461list_a @ A2 ) ) ).
% card_ge_0_finite
thf(fact_832_card__ge__0__finite,axiom,
! [A2: set_set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_set_a @ A2 ) )
=> ( finite_finite_set_a @ A2 ) ) ).
% card_ge_0_finite
thf(fact_833_card__ge__0__finite,axiom,
! [A2: set_list_list_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite9134805042761151410list_a @ A2 ) )
=> ( finite1660835950917165235list_a @ A2 ) ) ).
% card_ge_0_finite
thf(fact_834_card__ge__0__finite,axiom,
! [A2: set_list_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_list_a @ A2 ) )
=> ( finite_finite_list_a @ A2 ) ) ).
% card_ge_0_finite
thf(fact_835_card__ge__0__finite,axiom,
! [A2: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A2 ) )
=> ( finite_finite_nat @ A2 ) ) ).
% card_ge_0_finite
thf(fact_836_card__ge__0__finite,axiom,
! [A2: set_int] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_int @ A2 ) )
=> ( finite_finite_int @ A2 ) ) ).
% card_ge_0_finite
thf(fact_837_card__ge__0__finite,axiom,
! [A2: set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ A2 ) )
=> ( finite_finite_a @ A2 ) ) ).
% card_ge_0_finite
thf(fact_838_zero__less__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% zero_less_power
thf(fact_839_zero__less__power,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_less_power
thf(fact_840_not__is__unit__0,axiom,
~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).
% not_is_unit_0
thf(fact_841_not__is__unit__0,axiom,
~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).
% not_is_unit_0
thf(fact_842_power__0,axiom,
! [A: nat] :
( ( power_power_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_843_power__0,axiom,
! [A: int] :
( ( power_power_int @ A @ zero_zero_nat )
= one_one_int ) ).
% power_0
thf(fact_844_power__0,axiom,
! [A: complex] :
( ( power_power_complex @ A @ zero_zero_nat )
= one_one_complex ) ).
% power_0
thf(fact_845_sum__eq__1__iff,axiom,
! [A2: set_list_list_a,F2: list_list_a > nat] :
( ( finite1660835950917165235list_a @ A2 )
=> ( ( ( groups7548105480907152928_a_nat @ F2 @ A2 )
= one_one_nat )
= ( ? [X2: list_list_a] :
( ( member_list_list_a @ X2 @ A2 )
& ( ( F2 @ X2 )
= one_one_nat )
& ! [Y5: list_list_a] :
( ( member_list_list_a @ Y5 @ A2 )
=> ( ( X2 != Y5 )
=> ( ( F2 @ Y5 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_846_sum__eq__1__iff,axiom,
! [A2: set_int,F2: int > nat] :
( ( finite_finite_int @ A2 )
=> ( ( ( groups4541462559716669496nt_nat @ F2 @ A2 )
= one_one_nat )
= ( ? [X2: int] :
( ( member_int @ X2 @ A2 )
& ( ( F2 @ X2 )
= one_one_nat )
& ! [Y5: int] :
( ( member_int @ Y5 @ A2 )
=> ( ( X2 != Y5 )
=> ( ( F2 @ Y5 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_847_sum__eq__1__iff,axiom,
! [A2: set_a,F2: a > nat] :
( ( finite_finite_a @ A2 )
=> ( ( ( groups6334556678337121940_a_nat @ F2 @ A2 )
= one_one_nat )
= ( ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ( ( F2 @ X2 )
= one_one_nat )
& ! [Y5: a] :
( ( member_a @ Y5 @ A2 )
=> ( ( X2 != Y5 )
=> ( ( F2 @ Y5 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_848_sum__eq__1__iff,axiom,
! [A2: set_nat,F2: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( groups3542108847815614940at_nat @ F2 @ A2 )
= one_one_nat )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( ( F2 @ X2 )
= one_one_nat )
& ! [Y5: nat] :
( ( member_nat @ Y5 @ A2 )
=> ( ( X2 != Y5 )
=> ( ( F2 @ Y5 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_849_sum__eq__1__iff,axiom,
! [A2: set_list_a,F2: list_a > nat] :
( ( finite_finite_list_a @ A2 )
=> ( ( ( groups5521247699297860762_a_nat @ F2 @ A2 )
= one_one_nat )
= ( ? [X2: list_a] :
( ( member_list_a @ X2 @ A2 )
& ( ( F2 @ X2 )
= one_one_nat )
& ! [Y5: list_a] :
( ( member_list_a @ Y5 @ A2 )
=> ( ( X2 != Y5 )
=> ( ( F2 @ Y5 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_850_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_851_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_852_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_853_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_854_nat__dvd__not__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ M @ N )
=> ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% nat_dvd_not_less
thf(fact_855_nat__power__less__imp__less,axiom,
! [I: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ I )
=> ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_856_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_857_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_858_one__reorient,axiom,
! [X: complex] :
( ( one_one_complex = X )
= ( X = one_one_complex ) ) ).
% one_reorient
thf(fact_859_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_860_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_861_mult_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_862_mult_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.assoc
thf(fact_863_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A3: nat,B2: nat] : ( times_times_nat @ B2 @ A3 ) ) ) ).
% mult.commute
thf(fact_864_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A3: int,B2: int] : ( times_times_int @ B2 @ A3 ) ) ) ).
% mult.commute
thf(fact_865_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_866_mult_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_867_diff__eq__diff__eq,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D2 ) )
=> ( ( A = B )
= ( C = D2 ) ) ) ).
% diff_eq_diff_eq
thf(fact_868_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_869_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_870_dvd__mult__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( dvd_dvd_nat @ M @ N ) ) ) ).
% dvd_mult_cancel
thf(fact_871_power__dvd__imp__le,axiom,
! [I: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
=> ( ( ord_less_nat @ one_one_nat @ I )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_dvd_imp_le
thf(fact_872_dvd__mult__cancel1,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ M @ N ) @ M )
= ( N = one_one_nat ) ) ) ).
% dvd_mult_cancel1
thf(fact_873_dvd__mult__cancel2,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ N @ M ) @ M )
= ( N = one_one_nat ) ) ) ).
% dvd_mult_cancel2
thf(fact_874__092_060open_062degree_A_Igauss__poly_AR_A_Iorder_AR_A_094_An_J_J_A_061_Asum_H_A_I_092_060lambda_062d_O_Apmult_Ad_A_Igauss__poly_AR_A_Iorder_AR_A_094_An_J_J_A_K_Adegree_Ad_J_A_123f_O_Am__i__p_AR_Af_125_092_060close_062,axiom,
( ( minus_minus_nat @ ( size_size_list_a @ ( card_I2373409586816755191ly_a_b @ r @ ( power_power_nat @ ( order_a_ring_ext_a_b @ r ) @ n ) ) ) @ one_one_nat )
= ( groups1083796956622547633_a_nat
@ ^ [D: list_a] : ( times_times_nat @ ( monic_5301438133677370042lt_a_b @ r @ D @ ( card_I2373409586816755191ly_a_b @ r @ ( power_power_nat @ ( order_a_ring_ext_a_b @ r ) @ n ) ) ) @ ( minus_minus_nat @ ( size_size_list_a @ D ) @ one_one_nat ) )
@ ( collect_list_a @ ( monic_4919232885364369782ly_a_b @ r ) ) ) ) ).
% \<open>degree (gauss_poly R (order R ^ n)) = sum' (\<lambda>d. pmult d (gauss_poly R (order R ^ n)) * degree d) {f. m_i_p R f}\<close>
thf(fact_875_degree__monic__poly_H,axiom,
! [F2: list_a] :
( ( monic_3145109188698636716ly_a_b @ r @ F2 )
=> ( ( groups1083796956622547633_a_nat
@ ^ [D: list_a] : ( times_times_nat @ ( monic_5301438133677370042lt_a_b @ r @ D @ F2 ) @ ( minus_minus_nat @ ( size_size_list_a @ D ) @ one_one_nat ) )
@ ( collect_list_a @ ( monic_4919232885364369782ly_a_b @ r ) ) )
= ( minus_minus_nat @ ( size_size_list_a @ F2 ) @ one_one_nat ) ) ) ).
% degree_monic_poly'
thf(fact_876_factor__monic__poly__fin,axiom,
! [A: list_a] :
( ( monic_3145109188698636716ly_a_b @ r @ A )
=> ( finite_finite_list_a
@ ( collect_list_a
@ ^ [D: list_a] :
( ( monic_4919232885364369782ly_a_b @ r @ D )
& ( ord_less_nat @ zero_zero_nat @ ( monic_5301438133677370042lt_a_b @ r @ D @ A ) ) ) ) ) ) ).
% factor_monic_poly_fin
thf(fact_877_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_878_div__gauss__poly__iff,axiom,
! [N: nat,F2: list_a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( monic_4919232885364369782ly_a_b @ r @ F2 )
=> ( ( polyno5814909790663948098es_a_b @ r @ F2 @ ( card_I2373409586816755191ly_a_b @ r @ ( power_power_nat @ ( order_a_ring_ext_a_b @ r ) @ N ) ) )
= ( dvd_dvd_nat @ ( minus_minus_nat @ ( size_size_list_a @ F2 ) @ one_one_nat ) @ N ) ) ) ) ).
% div_gauss_poly_iff
thf(fact_879_nat__mult__dvd__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( dvd_dvd_nat @ M @ N ) ) ) ).
% nat_mult_dvd_cancel_disj
thf(fact_880_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_881__092_060open_062sum_H_A_I_092_060lambda_062d_O_Aof__bool_A_Idegree_Ad_Advd_An_J_A_K_Adegree_Ad_J_A_123f_O_Am__i__p_AR_Af_125_A_061_Asum_H_Adegree_A_123f_O_Am__i__p_AR_Af_A_092_060and_062_Adegree_Af_Advd_An_125_092_060close_062,axiom,
( ( groups1083796956622547633_a_nat
@ ^ [D: list_a] : ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( minus_minus_nat @ ( size_size_list_a @ D ) @ one_one_nat ) @ n ) ) @ ( minus_minus_nat @ ( size_size_list_a @ D ) @ one_one_nat ) )
@ ( collect_list_a @ ( monic_4919232885364369782ly_a_b @ r ) ) )
= ( groups1083796956622547633_a_nat
@ ^ [D: list_a] : ( minus_minus_nat @ ( size_size_list_a @ D ) @ one_one_nat )
@ ( collect_list_a
@ ^ [F: list_a] :
( ( monic_4919232885364369782ly_a_b @ r @ F )
& ( dvd_dvd_nat @ ( minus_minus_nat @ ( size_size_list_a @ F ) @ one_one_nat ) @ n ) ) ) ) ) ).
% \<open>sum' (\<lambda>d. of_bool (degree d dvd n) * degree d) {f. m_i_p R f} = sum' degree {f. m_i_p R f \<and> degree f dvd n}\<close>
thf(fact_882_divides__monic__poly,axiom,
! [F2: list_a,G: list_a] :
( ( monic_3145109188698636716ly_a_b @ r @ F2 )
=> ( ( monic_3145109188698636716ly_a_b @ r @ G )
=> ( ! [D3: list_a] :
( ( monic_4919232885364369782ly_a_b @ r @ D3 )
=> ( ord_less_eq_nat @ ( monic_5301438133677370042lt_a_b @ r @ D3 @ F2 ) @ ( monic_5301438133677370042lt_a_b @ r @ D3 @ G ) ) )
=> ( polyno5814909790663948098es_a_b @ r @ F2 @ G ) ) ) ) ).
% divides_monic_poly
thf(fact_883_gauss__poly__div__gauss__poly__iff,axiom,
! [M: nat,N: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ one_one_nat @ A )
=> ( ( polyno5814909790663948098es_a_b @ r @ ( card_I2373409586816755191ly_a_b @ r @ ( power_power_nat @ A @ N ) ) @ ( card_I2373409586816755191ly_a_b @ r @ ( power_power_nat @ A @ M ) ) )
= ( dvd_dvd_nat @ N @ M ) ) ) ) ) ).
% gauss_poly_div_gauss_poly_iff
thf(fact_884_multiplicity__of__factor__of__gauss__poly,axiom,
! [N: nat,F2: list_a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( monic_4919232885364369782ly_a_b @ r @ F2 )
=> ( ( monic_5301438133677370042lt_a_b @ r @ F2 @ ( card_I2373409586816755191ly_a_b @ r @ ( power_power_nat @ ( order_a_ring_ext_a_b @ r ) @ N ) ) )
= ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( minus_minus_nat @ ( size_size_list_a @ F2 ) @ one_one_nat ) @ N ) ) ) ) ) ).
% multiplicity_of_factor_of_gauss_poly
thf(fact_885__092_060open_062sum_H_A_I_092_060lambda_062d_O_Apmult_Ad_A_Igauss__poly_AR_A_Iorder_AR_A_094_An_J_J_A_K_Adegree_Ad_J_A_123f_O_Am__i__p_AR_Af_125_A_061_Asum_H_A_I_092_060lambda_062d_O_Aof__bool_A_Idegree_Ad_Advd_An_J_A_K_Adegree_Ad_J_A_123f_O_Am__i__p_AR_Af_125_092_060close_062,axiom,
( ( groups1083796956622547633_a_nat
@ ^ [D: list_a] : ( times_times_nat @ ( monic_5301438133677370042lt_a_b @ r @ D @ ( card_I2373409586816755191ly_a_b @ r @ ( power_power_nat @ ( order_a_ring_ext_a_b @ r ) @ n ) ) ) @ ( minus_minus_nat @ ( size_size_list_a @ D ) @ one_one_nat ) )
@ ( collect_list_a @ ( monic_4919232885364369782ly_a_b @ r ) ) )
= ( groups1083796956622547633_a_nat
@ ^ [D: list_a] : ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( minus_minus_nat @ ( size_size_list_a @ D ) @ one_one_nat ) @ n ) ) @ ( minus_minus_nat @ ( size_size_list_a @ D ) @ one_one_nat ) )
@ ( collect_list_a @ ( monic_4919232885364369782ly_a_b @ r ) ) ) ) ).
% \<open>sum' (\<lambda>d. pmult d (gauss_poly R (order R ^ n)) * degree d) {f. m_i_p R f} = sum' (\<lambda>d. of_bool (degree d dvd n) * degree d) {f. m_i_p R f}\<close>
thf(fact_886_finite__divisors__int,axiom,
! [I: int] :
( ( I != zero_zero_int )
=> ( finite_finite_int
@ ( collect_int
@ ^ [D: int] : ( dvd_dvd_int @ D @ I ) ) ) ) ).
% finite_divisors_int
thf(fact_887_gcd__nat_Onot__eq__order__implies__strict,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) ) ) ) ).
% gcd_nat.not_eq_order_implies_strict
thf(fact_888_gcd__nat_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
=> ( A != B ) ) ).
% gcd_nat.strict_implies_not_eq
thf(fact_889_gcd__nat_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
=> ( dvd_dvd_nat @ A @ B ) ) ).
% gcd_nat.strict_implies_order
thf(fact_890_gcd__nat_Ostrict__iff__order,axiom,
! [A: nat,B: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
= ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) ) ) ).
% gcd_nat.strict_iff_order
thf(fact_891_gcd__nat_Oorder__iff__strict,axiom,
( dvd_dvd_nat
= ( ^ [A3: nat,B2: nat] :
( ( ( dvd_dvd_nat @ A3 @ B2 )
& ( A3 != B2 ) )
| ( A3 = B2 ) ) ) ) ).
% gcd_nat.order_iff_strict
thf(fact_892_gcd__nat_Ostrict__iff__not,axiom,
! [A: nat,B: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
= ( ( dvd_dvd_nat @ A @ B )
& ~ ( dvd_dvd_nat @ B @ A ) ) ) ).
% gcd_nat.strict_iff_not
thf(fact_893_gcd__nat_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
=> ( ( dvd_dvd_nat @ B @ C )
=> ( ( dvd_dvd_nat @ A @ C )
& ( A != C ) ) ) ) ).
% gcd_nat.strict_trans2
thf(fact_894_gcd__nat_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( ( dvd_dvd_nat @ B @ C )
& ( B != C ) )
=> ( ( dvd_dvd_nat @ A @ C )
& ( A != C ) ) ) ) ).
% gcd_nat.strict_trans1
thf(fact_895_gcd__nat_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
=> ( ( ( dvd_dvd_nat @ B @ C )
& ( B != C ) )
=> ( ( dvd_dvd_nat @ A @ C )
& ( A != C ) ) ) ) ).
% gcd_nat.strict_trans
thf(fact_896_gcd__nat_Oantisym,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ B @ A )
=> ( A = B ) ) ) ).
% gcd_nat.antisym
thf(fact_897_gcd__nat_Oirrefl,axiom,
! [A: nat] :
~ ( ( dvd_dvd_nat @ A @ A )
& ( A != A ) ) ).
% gcd_nat.irrefl
thf(fact_898_gcd__nat_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [A3: nat,B2: nat] :
( ( dvd_dvd_nat @ A3 @ B2 )
& ( dvd_dvd_nat @ B2 @ A3 ) ) ) ) ).
% gcd_nat.eq_iff
thf(fact_899_gcd__nat_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ B @ C )
=> ( dvd_dvd_nat @ A @ C ) ) ) ).
% gcd_nat.trans
thf(fact_900_gcd__nat_Orefl,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).
% gcd_nat.refl
thf(fact_901_gcd__nat_Oasym,axiom,
! [A: nat,B: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
=> ~ ( ( dvd_dvd_nat @ B @ A )
& ( B != A ) ) ) ).
% gcd_nat.asym
thf(fact_902_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_903_gcd__nat_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
=> ( A = zero_zero_nat ) ) ).
% gcd_nat.extremum_uniqueI
thf(fact_904_gcd__nat_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ( dvd_dvd_nat @ A @ zero_zero_nat )
& ( A != zero_zero_nat ) ) ) ).
% gcd_nat.not_eq_extremum
thf(fact_905_gcd__nat_Oextremum__unique,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
= ( A = zero_zero_nat ) ) ).
% gcd_nat.extremum_unique
thf(fact_906_gcd__nat_Oextremum__strict,axiom,
! [A: nat] :
~ ( ( dvd_dvd_nat @ zero_zero_nat @ A )
& ( zero_zero_nat != A ) ) ).
% gcd_nat.extremum_strict
thf(fact_907_gcd__nat_Oextremum,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% gcd_nat.extremum
thf(fact_908_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_909_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_910_dvd__pos__nat,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_nat @ M @ N )
=> ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).
% dvd_pos_nat
thf(fact_911_bezout1__nat,axiom,
! [A: nat,B: nat] :
? [D3: nat,X3: nat,Y2: nat] :
( ( dvd_dvd_nat @ D3 @ A )
& ( dvd_dvd_nat @ D3 @ B )
& ( ( ( minus_minus_nat @ ( times_times_nat @ A @ X3 ) @ ( times_times_nat @ B @ Y2 ) )
= D3 )
| ( ( minus_minus_nat @ ( times_times_nat @ B @ X3 ) @ ( times_times_nat @ A @ Y2 ) )
= D3 ) ) ) ).
% bezout1_nat
thf(fact_912_finite__divisors__nat,axiom,
! [M: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [D: nat] : ( dvd_dvd_nat @ D @ M ) ) ) ) ).
% finite_divisors_nat
thf(fact_913_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_914_nat__mult__dvd__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( dvd_dvd_nat @ M @ N ) ) ) ).
% nat_mult_dvd_cancel1
thf(fact_915_zle__diff1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z @ one_one_int ) )
= ( ord_less_int @ W @ Z ) ) ).
% zle_diff1_eq
thf(fact_916_int_Onat__pow__0,axiom,
! [X: int] :
( ( power_power_int @ X @ zero_zero_nat )
= one_one_int ) ).
% int.nat_pow_0
thf(fact_917_int_Onat__pow__one,axiom,
! [N: nat] :
( ( power_power_int @ one_one_int @ N )
= one_one_int ) ).
% int.nat_pow_one
thf(fact_918_finite__lessThan,axiom,
! [K: nat] : ( finite_finite_nat @ ( set_ord_lessThan_nat @ K ) ) ).
% finite_lessThan
thf(fact_919_card__lessThan,axiom,
! [U: nat] :
( ( finite_card_nat @ ( set_ord_lessThan_nat @ U ) )
= U ) ).
% card_lessThan
thf(fact_920_finite__interval__int1,axiom,
! [A: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I2: int] :
( ( ord_less_eq_int @ A @ I2 )
& ( ord_less_eq_int @ I2 @ B ) ) ) ) ).
% finite_interval_int1
thf(fact_921_finite__interval__int4,axiom,
! [A: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I2: int] :
( ( ord_less_int @ A @ I2 )
& ( ord_less_int @ I2 @ B ) ) ) ) ).
% finite_interval_int4
thf(fact_922_finite__interval__int3,axiom,
! [A: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I2: int] :
( ( ord_less_int @ A @ I2 )
& ( ord_less_eq_int @ I2 @ B ) ) ) ) ).
% finite_interval_int3
thf(fact_923_finite__interval__int2,axiom,
! [A: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I2: int] :
( ( ord_less_eq_int @ A @ I2 )
& ( ord_less_int @ I2 @ B ) ) ) ) ).
% finite_interval_int2
thf(fact_924_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_925_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_926_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_927_bounded__Max__nat,axiom,
! [P2: nat > $o,X: nat,M4: nat] :
( ( P2 @ X )
=> ( ! [X3: nat] :
( ( P2 @ X3 )
=> ( ord_less_eq_nat @ X3 @ M4 ) )
=> ~ ! [M5: nat] :
( ( P2 @ M5 )
=> ~ ! [X4: nat] :
( ( P2 @ X4 )
=> ( ord_less_eq_nat @ X4 @ M5 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_928_int_Ozero__not__one,axiom,
zero_zero_int != one_one_int ).
% int.zero_not_one
thf(fact_929_zmult__zless__mono2,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_930_int_Olless__eq,axiom,
( ord_less_int
= ( ^ [X2: int,Y5: int] :
( ( ord_less_eq_int @ X2 @ Y5 )
& ( X2 != Y5 ) ) ) ) ).
% int.lless_eq
thf(fact_931_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_932_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_933_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(3)
thf(fact_934_int__distrib_I4_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
= ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_935_zdvd__antisym__nonneg,axiom,
! [M: int,N: int] :
( ( ord_less_eq_int @ zero_zero_int @ M )
=> ( ( ord_less_eq_int @ zero_zero_int @ N )
=> ( ( dvd_dvd_int @ M @ N )
=> ( ( dvd_dvd_int @ N @ M )
=> ( M = N ) ) ) ) ) ).
% zdvd_antisym_nonneg
thf(fact_936_zdvd__mult__cancel,axiom,
! [K: int,M: int,N: int] :
( ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) )
=> ( ( K != zero_zero_int )
=> ( dvd_dvd_int @ M @ N ) ) ) ).
% zdvd_mult_cancel
thf(fact_937_int__one__le__iff__zero__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ one_one_int @ Z )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% int_one_le_iff_zero_less
thf(fact_938_pos__zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( M = one_one_int )
& ( N = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_939_int__le__induct,axiom,
! [I: int,K: int,P2: int > $o] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P2 @ K )
=> ( ! [I5: int] :
( ( ord_less_eq_int @ I5 @ K )
=> ( ( P2 @ I5 )
=> ( P2 @ ( minus_minus_int @ I5 @ one_one_int ) ) ) )
=> ( P2 @ I ) ) ) ) ).
% int_le_induct
thf(fact_940_int_Onat__pow__zero,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ).
% int.nat_pow_zero
thf(fact_941_zdvd__not__zless,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ord_less_int @ M @ N )
=> ~ ( dvd_dvd_int @ N @ M ) ) ) ).
% zdvd_not_zless
thf(fact_942_zdvd__imp__le,axiom,
! [Z: int,N: int] :
( ( dvd_dvd_int @ Z @ N )
=> ( ( ord_less_int @ zero_zero_int @ N )
=> ( ord_less_eq_int @ Z @ N ) ) ) ).
% zdvd_imp_le
thf(fact_943_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M3: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ N5 )
=> ( ord_less_nat @ X2 @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_944_bounded__nat__set__is__finite,axiom,
! [N3: set_nat,N: nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ N3 )
=> ( ord_less_nat @ X3 @ N ) )
=> ( finite_finite_nat @ N3 ) ) ).
% bounded_nat_set_is_finite
thf(fact_945_zdvd__zdiffD,axiom,
! [K: int,M: int,N: int] :
( ( dvd_dvd_int @ K @ ( minus_minus_int @ M @ N ) )
=> ( ( dvd_dvd_int @ K @ N )
=> ( dvd_dvd_int @ K @ M ) ) ) ).
% zdvd_zdiffD
thf(fact_946_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M3: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ N5 )
=> ( ord_less_eq_nat @ X2 @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_947_finite__M__bounded__by__nat,axiom,
! [P2: nat > $o,I: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K2: nat] :
( ( P2 @ K2 )
& ( ord_less_nat @ K2 @ I ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_948_finite__less__ub,axiom,
! [F2: nat > nat,U: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( F2 @ N2 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_eq_nat @ ( F2 @ N4 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_949_int__less__induct,axiom,
! [I: int,K: int,P2: int > $o] :
( ( ord_less_int @ I @ K )
=> ( ( P2 @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I5: int] :
( ( ord_less_int @ I5 @ K )
=> ( ( P2 @ I5 )
=> ( P2 @ ( minus_minus_int @ I5 @ one_one_int ) ) ) )
=> ( P2 @ I ) ) ) ) ).
% int_less_induct
thf(fact_950_finite__nth__roots,axiom,
! [N: nat,C: complex] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [Z3: complex] :
( ( power_power_complex @ Z3 @ N )
= C ) ) ) ) ).
% finite_nth_roots
thf(fact_951_splitted__on__def,axiom,
! [K4: set_a,P3: list_a] :
( ( polyno2453258491555121552on_a_b @ r @ K4 @ P3 )
= ( ( size_size_multiset_a @ ( polyno5714441830345289050on_a_b @ r @ K4 @ P3 ) )
= ( minus_minus_nat @ ( size_size_list_a @ P3 ) @ one_one_nat ) ) ) ).
% splitted_on_def
thf(fact_952_card__nth__roots,axiom,
! [C: complex,N: nat] :
( ( C != zero_zero_complex )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( finite_card_complex
@ ( collect_complex
@ ^ [Z3: complex] :
( ( power_power_complex @ Z3 @ N )
= C ) ) )
= N ) ) ) ).
% card_nth_roots
thf(fact_953_card__roots__unity__eq,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( finite_card_complex
@ ( collect_complex
@ ^ [Z3: complex] :
( ( power_power_complex @ Z3 @ N )
= one_one_complex ) ) )
= N ) ) ).
% card_roots_unity_eq
thf(fact_954_sum__nth__roots,axiom,
! [N: nat,C: complex] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( groups7754918857620584856omplex
@ ^ [X2: complex] : X2
@ ( collect_complex
@ ^ [Z3: complex] :
( ( power_power_complex @ Z3 @ N )
= C ) ) )
= zero_zero_complex ) ) ).
% sum_nth_roots
thf(fact_955_sum__roots__unity,axiom,
! [N: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( groups7754918857620584856omplex
@ ^ [X2: complex] : X2
@ ( collect_complex
@ ^ [Z3: complex] :
( ( power_power_complex @ Z3 @ N )
= one_one_complex ) ) )
= zero_zero_complex ) ) ).
% sum_roots_unity
thf(fact_956_splitted__def,axiom,
! [P3: list_a] :
( ( polyno8329700637149614481ed_a_b @ r @ P3 )
= ( ( size_size_multiset_a @ ( polynomial_roots_a_b @ r @ P3 ) )
= ( minus_minus_nat @ ( size_size_list_a @ P3 ) @ one_one_nat ) ) ) ).
% splitted_def
thf(fact_957_finite__field__order,axiom,
? [N2: nat] :
( ( ( order_a_ring_ext_a_b @ r )
= ( power_power_nat @ ( ring_char_a_b @ r ) @ N2 ) )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% finite_field_order
thf(fact_958_decr__mult__lemma,axiom,
! [D2: int,P2: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ! [X3: int] :
( ( P2 @ X3 )
=> ( P2 @ ( minus_minus_int @ X3 @ D2 ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X4: int] :
( ( P2 @ X4 )
=> ( P2 @ ( minus_minus_int @ X4 @ ( times_times_int @ K @ D2 ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_959_imp__le__cong,axiom,
! [X: int,X5: int,P2: $o,P4: $o] :
( ( X = X5 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X5 )
=> ( P2 = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P2 )
= ( ( ord_less_eq_int @ zero_zero_int @ X5 )
=> P4 ) ) ) ) ).
% imp_le_cong
thf(fact_960_conj__le__cong,axiom,
! [X: int,X5: int,P2: $o,P4: $o] :
( ( X = X5 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X5 )
=> ( P2 = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P2 )
= ( ( ord_less_eq_int @ zero_zero_int @ X5 )
& P4 ) ) ) ) ).
% conj_le_cong
thf(fact_961_zdvd__mono,axiom,
! [K: int,M: int,T: int] :
( ( K != zero_zero_int )
=> ( ( dvd_dvd_int @ M @ T )
= ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ T ) ) ) ) ).
% zdvd_mono
thf(fact_962_plusinfinity,axiom,
! [D2: int,P4: int > $o,P2: int > $o] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ! [X3: int,K3: int] :
( ( P4 @ X3 )
= ( P4 @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D2 ) ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( P2 @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [X_1: int] : ( P4 @ X_1 )
=> ? [X_12: int] : ( P2 @ X_12 ) ) ) ) ) ).
% plusinfinity
thf(fact_963_minusinfinity,axiom,
! [D2: int,P1: int > $o,P2: int > $o] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ! [X3: int,K3: int] :
( ( P1 @ X3 )
= ( P1 @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D2 ) ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( P2 @ X3 )
= ( P1 @ X3 ) ) )
=> ( ? [X_1: int] : ( P1 @ X_1 )
=> ? [X_12: int] : ( P2 @ X_12 ) ) ) ) ) ).
% minusinfinity
thf(fact_964_order__gt__0__iff__finite,axiom,
( ( ord_less_nat @ zero_zero_nat @ ( order_a_ring_ext_a_b @ r ) )
= ( finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% order_gt_0_iff_finite
thf(fact_965_finite__carr__imp__char__ge__0,axiom,
( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_nat @ zero_zero_nat @ ( ring_char_a_b @ r ) ) ) ).
% finite_carr_imp_char_ge_0
thf(fact_966_finite__carrier,axiom,
finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) ).
% finite_carrier
thf(fact_967_onepideal,axiom,
principalideal_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% onepideal
thf(fact_968_a__card__cosets__equal,axiom,
! [C: set_a,H3: set_a] :
( ( member_set_a @ C @ ( a_RCOSETS_a_b @ r @ H3 ) )
=> ( ( ord_less_eq_set_a @ H3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( finite_card_a @ C )
= ( finite_card_a @ H3 ) ) ) ) ) ).
% a_card_cosets_equal
thf(fact_969_a__lagrange,axiom,
! [H3: set_a] :
( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( additi2834746164131130830up_a_b @ H3 @ r )
=> ( ( times_times_nat @ ( finite_card_set_a @ ( a_RCOSETS_a_b @ r @ H3 ) ) @ ( finite_card_a @ H3 ) )
= ( order_a_ring_ext_a_b @ r ) ) ) ) ).
% a_lagrange
thf(fact_970_size__roots__le__degree,axiom,
! [P3: list_a] :
( ( member_list_a @ P3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_less_eq_nat @ ( size_size_multiset_a @ ( polynomial_roots_a_b @ r @ P3 ) ) @ ( minus_minus_nat @ ( size_size_list_a @ P3 ) @ one_one_nat ) ) ) ).
% size_roots_le_degree
thf(fact_971_finite__poly_I2_J,axiom,
! [N: nat] :
( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) )
=> ( finite_finite_list_a
@ ( collect_list_a
@ ^ [F: list_a] :
( ( member_list_a @ F @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ F ) @ one_one_nat ) @ N ) ) ) ) ) ).
% finite_poly(2)
thf(fact_972_monic__poly__carr,axiom,
! [F2: list_a] :
( ( monic_3145109188698636716ly_a_b @ r @ F2 )
=> ( member_list_a @ F2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% monic_poly_carr
thf(fact_973_gauss__poly__carr,axiom,
! [N: nat] : ( member_list_a @ ( card_I2373409586816755191ly_a_b @ r @ N ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% gauss_poly_carr
thf(fact_974_p_Ofinite__carr__imp__char__ge__0,axiom,
( ( finite_finite_list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( ring_c500279861223467766t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.finite_carr_imp_char_ge_0
thf(fact_975_p_Oorder__gt__0__iff__finite,axiom,
( ( ord_less_nat @ zero_zero_nat @ ( order_3240872107759947550t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( finite_finite_list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.order_gt_0_iff_finite
thf(fact_976_degree__one__imp__splitted,axiom,
! [P3: list_a] :
( ( member_list_a @ P3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P3 ) @ one_one_nat )
= one_one_nat )
=> ( polyno8329700637149614481ed_a_b @ r @ P3 ) ) ) ).
% degree_one_imp_splitted
thf(fact_977_degree__zero__imp__empty__roots,axiom,
! [P3: list_a] :
( ( member_list_a @ P3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P3 ) @ one_one_nat )
= zero_zero_nat )
=> ( ( polynomial_roots_a_b @ r @ P3 )
= zero_zero_multiset_a ) ) ) ).
% degree_zero_imp_empty_roots
thf(fact_978_degree__zero__imp__splitted,axiom,
! [P3: list_a] :
( ( member_list_a @ P3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P3 ) @ one_one_nat )
= zero_zero_nat )
=> ( polyno8329700637149614481ed_a_b @ r @ P3 ) ) ) ).
% degree_zero_imp_splitted
thf(fact_979_finite__poly_I1_J,axiom,
! [N: nat] :
( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) )
=> ( finite_finite_list_a
@ ( collect_list_a
@ ^ [F: list_a] :
( ( member_list_a @ F @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( ( minus_minus_nat @ ( size_size_list_a @ F ) @ one_one_nat )
= N ) ) ) ) ) ).
% finite_poly(1)
thf(fact_980_p_Ofactorial__domain__axioms,axiom,
ring_f796907574329358751t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% p.factorial_domain_axioms
thf(fact_981_p_Onoetherian__domain__axioms,axiom,
ring_n4705423059119889713t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% p.noetherian_domain_axioms
thf(fact_982_p_Oprincipal__domain__axioms,axiom,
ring_p8098905331641078952t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% p.principal_domain_axioms
thf(fact_983_p_Onoetherian__ring__axioms,axiom,
ring_n5188127996776581661t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% p.noetherian_ring_axioms
thf(fact_984_p_Oonepideal,axiom,
princi8786919440553033881t_unit @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% p.onepideal
thf(fact_985_degree__zero__imp__not__is__root,axiom,
! [P3: list_a,X: a] :
( ( member_list_a @ P3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P3 ) @ one_one_nat )
= zero_zero_nat )
=> ~ ( polyno4133073214067823460ot_a_b @ r @ P3 @ X ) ) ) ).
% degree_zero_imp_not_is_root
thf(fact_986_p_Oa__card__cosets__equal,axiom,
! [C: set_list_a,H3: set_list_a] :
( ( member_set_list_a @ C @ ( a_RCOS6220190738183020281t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H3 ) )
=> ( ( ord_le8861187494160871172list_a @ H3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( finite_finite_list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( finite_card_list_a @ C )
= ( finite_card_list_a @ H3 ) ) ) ) ) ).
% p.a_card_cosets_equal
thf(fact_987_pdivides__imp__degree__le,axiom,
! [P3: list_a,Q2: list_a] :
( ( member_list_a @ P3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( Q2 != nil_a )
=> ( ( polyno5814909790663948098es_a_b @ r @ P3 @ Q2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ P3 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q2 ) @ one_one_nat ) ) ) ) ) ) ).
% pdivides_imp_degree_le
thf(fact_988_zero__pdivides,axiom,
! [P3: list_a] :
( ( polyno5814909790663948098es_a_b @ r @ nil_a @ P3 )
= ( P3 = nil_a ) ) ).
% zero_pdivides
thf(fact_989_zero__pdivides__zero,axiom,
polyno5814909790663948098es_a_b @ r @ nil_a @ nil_a ).
% zero_pdivides_zero
thf(fact_990_pdivides__zero,axiom,
! [P3: list_a] :
( ( member_list_a @ P3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( polyno5814909790663948098es_a_b @ r @ P3 @ nil_a ) ) ).
% pdivides_zero
thf(fact_991_finite__number__of__roots,axiom,
! [P3: list_a] :
( ( member_list_a @ P3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( finite_finite_a @ ( collect_a @ ( polyno4133073214067823460ot_a_b @ r @ P3 ) ) ) ) ).
% finite_number_of_roots
thf(fact_992_pdivides__imp__splitted,axiom,
! [P3: list_a,Q2: list_a] :
( ( member_list_a @ P3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( Q2 != nil_a )
=> ( ( polyno8329700637149614481ed_a_b @ r @ Q2 )
=> ( ( polyno5814909790663948098es_a_b @ r @ P3 @ Q2 )
=> ( polyno8329700637149614481ed_a_b @ r @ P3 ) ) ) ) ) ) ).
% pdivides_imp_splitted
thf(fact_993_alg__mult__gt__zero__iff__is__root,axiom,
! [P3: list_a,X: a] :
( ( member_list_a @ P3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( polyno4422430861927485590lt_a_b @ r @ P3 @ X ) )
= ( polyno4133073214067823460ot_a_b @ r @ P3 @ X ) ) ) ).
% alg_mult_gt_zero_iff_is_root
thf(fact_994_p_Oa__lagrange,axiom,
! [H3: set_list_a] :
( ( finite_finite_list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( additi4714453376129182166t_unit @ H3 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( times_times_nat @ ( finite7490188192133390220list_a @ ( a_RCOS6220190738183020281t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H3 ) ) @ ( finite_card_list_a @ H3 ) )
= ( order_3240872107759947550t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.a_lagrange
thf(fact_995_p_Ozero__pdivides__zero,axiom,
polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ nil_list_a @ nil_list_a ).
% p.zero_pdivides_zero
thf(fact_996_p_Ozero__pdivides,axiom,
! [P3: list_list_a] :
( ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ nil_list_a @ P3 )
= ( P3 = nil_list_a ) ) ).
% p.zero_pdivides
thf(fact_997_splitted__imp__trivial__factors,axiom,
! [P3: list_a,Q2: list_a] :
( ( member_list_a @ P3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P3 != nil_a )
=> ( ( polyno8329700637149614481ed_a_b @ r @ P3 )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q2 )
=> ( ( polyno5814909790663948098es_a_b @ r @ Q2 @ P3 )
=> ( ( minus_minus_nat @ ( size_size_list_a @ Q2 ) @ one_one_nat )
= one_one_nat ) ) ) ) ) ) ) ).
% splitted_imp_trivial_factors
thf(fact_998_trivial__factors__imp__splitted,axiom,
! [P3: list_a] :
( ( member_list_a @ P3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [Q3: list_a] :
( ( member_list_a @ Q3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q3 )
=> ( ( polyno5814909790663948098es_a_b @ r @ Q3 @ P3 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ Q3 ) @ one_one_nat ) @ one_one_nat ) ) ) )
=> ( polyno8329700637149614481ed_a_b @ r @ P3 ) ) ) ).
% trivial_factors_imp_splitted
thf(fact_999_p_Osplitted__on__def,axiom,
! [K4: set_list_a,P3: list_list_a] :
( ( polyno1986131841096413848t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 @ P3 )
= ( ( size_s2335926164413107382list_a @ ( polyno5990348478334826338t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 @ P3 ) )
= ( minus_minus_nat @ ( size_s349497388124573686list_a @ P3 ) @ one_one_nat ) ) ) ).
% p.splitted_on_def
thf(fact_1000_degree__one__imp__pirreducible,axiom,
! [P3: list_a] :
( ( member_list_a @ P3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P3 ) @ one_one_nat )
= one_one_nat )
=> ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 ) ) ) ).
% degree_one_imp_pirreducible
thf(fact_1001_pirreducible__degree,axiom,
! [P3: list_a] :
( ( member_list_a @ P3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 )
=> ( ord_less_eq_nat @ one_one_nat @ ( minus_minus_nat @ ( size_size_list_a @ P3 ) @ one_one_nat ) ) ) ) ).
% pirreducible_degree
thf(fact_1002_pirreducible__imp__not__splitted,axiom,
! [P3: list_a] :
( ( member_list_a @ P3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P3 ) @ one_one_nat )
!= one_one_nat )
=> ~ ( polyno8329700637149614481ed_a_b @ r @ P3 ) ) ) ) ).
% pirreducible_imp_not_splitted
thf(fact_1003_pirreducible__roots,axiom,
! [P3: list_a] :
( ( member_list_a @ P3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P3 ) @ one_one_nat )
!= one_one_nat )
=> ( ( polynomial_roots_a_b @ r @ P3 )
= zero_zero_multiset_a ) ) ) ) ).
% pirreducible_roots
thf(fact_1004_p_Oprimeness__condition,axiom,
! [P3: list_a] :
( ( member_list_a @ P3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 )
= ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 ) ) ) ).
% p.primeness_condition
thf(fact_1005_p_Osplitted__def,axiom,
! [P3: list_list_a] :
( ( polyno6259083269128200473t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 )
= ( ( size_s2335926164413107382list_a @ ( polyno7858422826990252003t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 ) )
= ( minus_minus_nat @ ( size_s349497388124573686list_a @ P3 ) @ one_one_nat ) ) ) ).
% p.splitted_def
thf(fact_1006_monic__poly__add__distinct,axiom,
! [F2: list_a,G: list_a] :
( ( monic_3145109188698636716ly_a_b @ r @ F2 )
=> ( ( member_list_a @ G @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ G ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ F2 ) @ one_one_nat ) )
=> ( monic_3145109188698636716ly_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F2 @ G ) ) ) ) ) ).
% monic_poly_add_distinct
thf(fact_1007_p_Oadd_Ol__cancel,axiom,
! [C: list_a,A: list_a,B: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ A )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ B ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( A = B ) ) ) ) ) ).
% p.add.l_cancel
thf(fact_1008_p_Oadd_Om__assoc,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ Z )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z ) ) ) ) ) ) ).
% p.add.m_assoc
thf(fact_1009_p_Oadd_Om__comm,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X ) ) ) ) ).
% p.add.m_comm
thf(fact_1010_p_Oadd_Om__lcomm,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Z ) ) ) ) ) ) ).
% p.add.m_lcomm
thf(fact_1011_p_Oadd_Or__cancel,axiom,
! [A: list_a,C: list_a,B: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ C )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ C ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( A = B ) ) ) ) ) ).
% p.add.r_cancel
thf(fact_1012_p_Oadd_Om__closed,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.add.m_closed
thf(fact_1013_p_Oadd_Oright__cancel,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ).
% p.add.right_cancel
thf(fact_1014_p_Oring__primeE_I3_J,axiom,
! [P3: list_a] :
( ( member_list_a @ P3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 )
=> ( prime_2011924034616061926t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 ) ) ) ).
% p.ring_primeE(3)
thf(fact_1015_p_Odegree__zero__imp__splitted,axiom,
! [P3: list_list_a] :
( ( member_list_list_a @ P3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P3 ) @ one_one_nat )
= zero_zero_nat )
=> ( polyno6259083269128200473t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 ) ) ) ).
% p.degree_zero_imp_splitted
thf(fact_1016_p_Odegree__zero__imp__empty__roots,axiom,
! [P3: list_list_a] :
( ( member_list_list_a @ P3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P3 ) @ one_one_nat )
= zero_zero_nat )
=> ( ( polyno7858422826990252003t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 )
= zero_z4454100511807792257list_a ) ) ) ).
% p.degree_zero_imp_empty_roots
thf(fact_1017_p_Odegree__zero__imp__not__is__root,axiom,
! [P3: list_list_a,X: list_a] :
( ( member_list_list_a @ P3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P3 ) @ one_one_nat )
= zero_zero_nat )
=> ~ ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 @ X ) ) ) ).
% p.degree_zero_imp_not_is_root
thf(fact_1018_p_Ofinite__number__of__roots,axiom,
! [P3: list_list_a] :
( ( member_list_list_a @ P3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( finite_finite_list_a @ ( collect_list_a @ ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 ) ) ) ) ).
% p.finite_number_of_roots
thf(fact_1019_p_Opirreducible__roots,axiom,
! [P3: list_list_a] :
( ( member_list_list_a @ P3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ P3 )
=> ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P3 ) @ one_one_nat )
!= one_one_nat )
=> ( ( polyno7858422826990252003t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 )
= zero_z4454100511807792257list_a ) ) ) ) ).
% p.pirreducible_roots
thf(fact_1020_p_Oalg__mult__gt__zero__iff__is__root,axiom,
! [P3: list_list_a,X: list_a] :
( ( member_list_list_a @ P3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( polyno4259638811958763678t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 @ X ) )
= ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 @ X ) ) ) ).
% p.alg_mult_gt_zero_iff_is_root
thf(fact_1021_p_Opdivides__imp__degree__le,axiom,
! [K4: set_list_a,P3: list_list_a,Q2: list_list_a] :
( ( subrin6918843898125473962t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( Q2 != nil_list_a )
=> ( ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 @ Q2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P3 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ Q2 ) @ one_one_nat ) ) ) ) ) ) ) ).
% p.pdivides_imp_degree_le
thf(fact_1022_p_Ofinite__poly_I2_J,axiom,
! [K4: set_list_a,N: nat] :
( ( subrin6918843898125473962t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finite_finite_list_a @ K4 )
=> ( finite1660835950917165235list_a
@ ( collect_list_list_a
@ ^ [F: list_list_a] :
( ( member_list_list_a @ F @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
& ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ F ) @ one_one_nat ) @ N ) ) ) ) ) ) ).
% p.finite_poly(2)
thf(fact_1023_p_Ocarrier__is__subring,axiom,
subrin6918843898125473962t_unit @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% p.carrier_is_subring
thf(fact_1024_p_Opdivides__zero,axiom,
! [K4: set_list_a,P3: list_list_a] :
( ( subrin6918843898125473962t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 @ nil_list_a ) ) ) ).
% p.pdivides_zero
thf(fact_1025_p_OpirreducibleE_I1_J,axiom,
! [K4: set_list_a,P3: list_list_a] :
( ( subrin6918843898125473962t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) @ P3 )
=> ( P3 != nil_list_a ) ) ) ) ).
% p.pirreducibleE(1)
thf(fact_1026_p_Ofinite__poly_I1_J,axiom,
! [K4: set_list_a,N: nat] :
( ( subrin6918843898125473962t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finite_finite_list_a @ K4 )
=> ( finite1660835950917165235list_a
@ ( collect_list_list_a
@ ^ [F: list_list_a] :
( ( member_list_list_a @ F @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
& ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ F ) @ one_one_nat )
= N ) ) ) ) ) ) ).
% p.finite_poly(1)
thf(fact_1027_p_Ocarrier__polynomial__shell,axiom,
! [K4: set_list_a,P3: list_list_a] :
( ( subrin6918843898125473962t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( member_list_list_a @ P3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ).
% p.carrier_polynomial_shell
thf(fact_1028_p_Ouniv__poly__a__minus__consistent,axiom,
! [K4: set_list_a,Q2: list_list_a,P3: list_list_a] :
( ( subrin6918843898125473962t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) @ P3 @ Q2 )
= ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ P3 @ Q2 ) ) ) ) ).
% p.univ_poly_a_minus_consistent
thf(fact_1029_p_Opirreducible__degree,axiom,
! [K4: set_list_a,P3: list_list_a] :
( ( subfie1779122896746047282t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) @ P3 )
=> ( ord_less_eq_nat @ one_one_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P3 ) @ one_one_nat ) ) ) ) ) ).
% p.pirreducible_degree
thf(fact_1030_p_Opdivides__imp__roots__incl,axiom,
! [P3: list_list_a,Q2: list_list_a] :
( ( member_list_list_a @ P3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( Q2 != nil_list_a )
=> ( ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 @ Q2 )
=> ( subseteq_mset_list_a @ ( polyno7858422826990252003t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 ) @ ( polyno7858422826990252003t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q2 ) ) ) ) ) ) ).
% p.pdivides_imp_roots_incl
thf(fact_1031_p_Osubring__props_I7_J,axiom,
! [K4: set_list_a,H1: list_a,H22: list_a] :
( ( subfie1779122896746047282t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a @ H1 @ K4 )
=> ( ( member_list_a @ H22 @ K4 )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H1 @ H22 ) @ K4 ) ) ) ) ).
% p.subring_props(7)
thf(fact_1032_p_Osubring__props_I1_J,axiom,
! [K4: set_list_a] :
( ( subfie1779122896746047282t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ord_le8861187494160871172list_a @ K4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.subring_props(1)
thf(fact_1033_p_Odegree__one__imp__pirreducible,axiom,
! [K4: set_list_a,P3: list_list_a] :
( ( subfie1779122896746047282t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P3 ) @ one_one_nat )
= one_one_nat )
=> ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) @ P3 ) ) ) ) ).
% p.degree_one_imp_pirreducible
thf(fact_1034_p_Ouniv__poly__is__euclidean,axiom,
! [K4: set_list_a] :
( ( subfie1779122896746047282t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ring_e6434146001954145682t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 )
@ ^ [P: list_list_a] : ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat ) ) ) ).
% p.univ_poly_is_euclidean
thf(fact_1035_p_Ouniv__poly__is__principal,axiom,
! [K4: set_list_a] :
( ( subfie1779122896746047282t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ring_p715737262848045090t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) ) ).
% p.univ_poly_is_principal
thf(fact_1036_p_Oexists__unique__long__division,axiom,
! [K4: set_list_a,P3: list_list_a,Q2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( Q2 != nil_list_a )
=> ? [X3: produc7709606177366032167list_a] :
( ( polyno6947042923167803568t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 @ Q2 @ X3 )
& ! [Y3: produc7709606177366032167list_a] :
( ( polyno6947042923167803568t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 @ Q2 @ Y3 )
=> ( Y3 = X3 ) ) ) ) ) ) ) ).
% p.exists_unique_long_division
thf(fact_1037_p_Opprime__iff__pirreducible,axiom,
! [K4: set_list_a,P3: list_list_a] :
( ( subfie1779122896746047282t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) @ P3 )
= ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) @ P3 ) ) ) ) ).
% p.pprime_iff_pirreducible
thf(fact_1038_pdivides__imp__roots__incl,axiom,
! [P3: list_a,Q2: list_a] :
( ( member_list_a @ P3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( Q2 != nil_a )
=> ( ( polyno5814909790663948098es_a_b @ r @ P3 @ Q2 )
=> ( subseteq_mset_a @ ( polynomial_roots_a_b @ r @ P3 ) @ ( polynomial_roots_a_b @ r @ Q2 ) ) ) ) ) ) ).
% pdivides_imp_roots_incl
thf(fact_1039_p_OpprimeE_I1_J,axiom,
! [K4: set_list_a,P3: list_list_a] :
( ( subfie1779122896746047282t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) @ P3 )
=> ( P3 != nil_list_a ) ) ) ) ).
% p.pprimeE(1)
thf(fact_1040_p_Opmod__const_I1_J,axiom,
! [K4: set_list_a,P3: list_list_a,Q2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( ord_less_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P3 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ Q2 ) @ one_one_nat ) )
=> ( ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 @ Q2 )
= nil_list_a ) ) ) ) ) ).
% p.pmod_const(1)
thf(fact_1041_p_Orupture__order,axiom,
! [K4: set_list_a,F2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ F2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ F2 ) @ one_one_nat ) )
=> ( ( order_78496787366231454t_unit @ ( polyno859807163042199155t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 @ F2 ) )
= ( power_power_nat @ ( finite_card_list_a @ K4 ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ F2 ) @ one_one_nat ) ) ) ) ) ) ).
% p.rupture_order
thf(fact_1042_p_Orupture__char,axiom,
! [K4: set_list_a,F2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ F2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ F2 ) @ one_one_nat ) )
=> ( ( ring_c8395554250859618576t_unit @ ( polyno859807163042199155t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 @ F2 ) )
= ( ring_c500279861223467766t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% p.rupture_char
thf(fact_1043_p_Olong__division__closed_I1_J,axiom,
! [K4: set_list_a,P3: list_list_a,Q2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( member_list_list_a @ ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 @ Q2 ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) ) ) ) ) ).
% p.long_division_closed(1)
thf(fact_1044_p_Olong__division__zero_I1_J,axiom,
! [K4: set_list_a,Q2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ nil_list_a @ Q2 )
= nil_list_a ) ) ) ).
% p.long_division_zero(1)
thf(fact_1045_p_Opmod__degree,axiom,
! [K4: set_list_a,P3: list_list_a,Q2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( Q2 != nil_list_a )
=> ( ( ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 @ Q2 )
= nil_list_a )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 @ Q2 ) ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ Q2 ) @ one_one_nat ) ) ) ) ) ) ) ).
% p.pmod_degree
thf(fact_1046_p_Opmod__const_I2_J,axiom,
! [K4: set_list_a,P3: list_list_a,Q2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( ord_less_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P3 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ Q2 ) @ one_one_nat ) )
=> ( ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 @ Q2 )
= P3 ) ) ) ) ) ).
% p.pmod_const(2)
thf(fact_1047_p_Osame__pmod__iff__pdivides,axiom,
! [K4: set_list_a,A: list_list_a,B: list_list_a,Q2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ Q2 )
= ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ Q2 ) )
= ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q2 @ ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) @ A @ B ) ) ) ) ) ) ) ).
% p.same_pmod_iff_pdivides
thf(fact_1048_p_Olong__division__closed_I2_J,axiom,
! [K4: set_list_a,P3: list_list_a,Q2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( member_list_list_a @ ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 @ Q2 ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) ) ) ) ) ).
% p.long_division_closed(2)
thf(fact_1049_p_Olong__division__zero_I2_J,axiom,
! [K4: set_list_a,Q2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ nil_list_a @ Q2 )
= nil_list_a ) ) ) ).
% p.long_division_zero(2)
thf(fact_1050_p_Opmod__zero__iff__pdivides,axiom,
! [K4: set_list_a,P3: list_list_a,Q2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 @ Q2 )
= nil_list_a )
= ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q2 @ P3 ) ) ) ) ) ).
% p.pmod_zero_iff_pdivides
thf(fact_1051_p_Olong__divisionE,axiom,
! [K4: set_list_a,P3: list_list_a,Q2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( Q2 != nil_list_a )
=> ( polyno6947042923167803568t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 @ Q2 @ ( produc8696003437204565271list_a @ ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 @ Q2 ) @ ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 @ Q2 ) ) ) ) ) ) ) ).
% p.long_divisionE
thf(fact_1052_p_Olong__divisionI,axiom,
! [K4: set_list_a,P3: list_list_a,Q2: list_list_a,B: list_list_a,R2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( Q2 != nil_list_a )
=> ( ( polyno6947042923167803568t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 @ Q2 @ ( produc8696003437204565271list_a @ B @ R2 ) )
=> ( ( produc8696003437204565271list_a @ B @ R2 )
= ( produc8696003437204565271list_a @ ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 @ Q2 ) @ ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 @ Q2 ) ) ) ) ) ) ) ) ).
% p.long_divisionI
thf(fact_1053_p_Opmod__image__characterization,axiom,
! [K4: set_list_a,P3: list_list_a] :
( ( subfie1779122896746047282t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( P3 != nil_list_a )
=> ( ( image_5446147394702115205list_a
@ ^ [Q4: list_list_a] : ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q4 @ P3 )
@ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
= ( collect_list_list_a
@ ^ [Q4: list_list_a] :
( ( member_list_list_a @ Q4 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
& ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ Q4 ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P3 ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% p.pmod_image_characterization
thf(fact_1054_p_Opoly__add_Ocases,axiom,
! [X: produc7709606177366032167list_a] :
~ ! [P12: list_list_a,P22: list_list_a] :
( X
!= ( produc8696003437204565271list_a @ P12 @ P22 ) ) ).
% p.poly_add.cases
thf(fact_1055_p_Oexists__long__division,axiom,
! [K4: set_list_a,P3: list_list_a,Q2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( Q2 != nil_list_a )
=> ~ ! [B4: list_list_a] :
( ( member_list_list_a @ B4 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ! [R3: list_list_a] :
( ( member_list_list_a @ R3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ~ ( polyno6947042923167803568t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 @ Q2 @ ( produc8696003437204565271list_a @ B4 @ R3 ) ) ) ) ) ) ) ) ).
% p.exists_long_division
thf(fact_1056_p_Oroots__inclI_H,axiom,
! [P3: list_list_a,M: multiset_list_a] :
( ( member_list_list_a @ P3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ! [A5: list_a] :
( ( member_list_a @ A5 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P3 != nil_list_a )
=> ( ord_less_eq_nat @ ( polyno4259638811958763678t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 @ A5 ) @ ( count_list_a @ M @ A5 ) ) ) )
=> ( subseteq_mset_list_a @ ( polyno7858422826990252003t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 ) @ M ) ) ) ).
% p.roots_inclI'
thf(fact_1057_p_Oa__lcos__m__assoc,axiom,
! [M4: set_list_a,G: list_a,H: list_a] :
( ( ord_le8861187494160871172list_a @ M4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ G @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ H @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ M4 ) )
= ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ H ) @ M4 ) ) ) ) ) ).
% p.a_lcos_m_assoc
thf(fact_1058_p_Ouniv__poly__units_H,axiom,
! [K4: set_list_a,P3: list_list_a] :
( ( subfie1779122896746047282t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P3 @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
= ( ( member_list_list_a @ P3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
& ( P3 != nil_list_a )
& ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P3 ) @ one_one_nat )
= zero_zero_nat ) ) ) ) ).
% p.univ_poly_units'
thf(fact_1059_p_Oadd_Osurj__const__mult,axiom,
! [A: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( image_list_a_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.add.surj_const_mult
thf(fact_1060_p_Oa__l__coset__subset__G,axiom,
! [H3: set_list_a,X: list_a] :
( ( ord_le8861187494160871172list_a @ H3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ H3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.a_l_coset_subset_G
thf(fact_1061_p_OpirreducibleE_I2_J,axiom,
! [K4: set_list_a,P3: list_list_a] :
( ( subrin6918843898125473962t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) @ P3 )
=> ~ ( member_list_list_a @ P3 @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) ) ) ) ) ).
% p.pirreducibleE(2)
thf(fact_1062_p_OpprimeE_I2_J,axiom,
! [K4: set_list_a,P3: list_list_a] :
( ( subfie1779122896746047282t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) @ P3 )
=> ~ ( member_list_list_a @ P3 @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) ) ) ) ) ).
% p.pprimeE(2)
thf(fact_1063_p_Oalg__mult__eq__count__roots,axiom,
! [P3: list_list_a] :
( ( member_list_list_a @ P3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( polyno4259638811958763678t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 )
= ( count_list_a @ ( polyno7858422826990252003t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 ) ) ) ) ).
% p.alg_mult_eq_count_roots
thf(fact_1064_p_Opoly__of__const__over__subfield,axiom,
! [K4: set_list_a] :
( ( subfie1779122896746047282t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( image_8260866953997875467list_a @ ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ K4 )
= ( collect_list_list_a
@ ^ [P: list_list_a] :
( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
& ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat )
= zero_zero_nat ) ) ) ) ) ).
% p.poly_of_const_over_subfield
thf(fact_1065_p_OpprimeI,axiom,
! [K4: set_list_a,P3: list_list_a] :
( ( subfie1779122896746047282t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( P3 != nil_list_a )
=> ( ~ ( member_list_list_a @ P3 @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ! [Q3: list_list_a,R3: list_list_a] :
( ( member_list_list_a @ Q3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( member_list_list_a @ R3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) @ Q3 @ R3 ) )
=> ( ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 @ Q3 )
| ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 @ R3 ) ) ) ) )
=> ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) @ P3 ) ) ) ) ) ) ).
% p.pprimeI
thf(fact_1066_alg__mult__eq__count__roots,axiom,
! [P3: list_a] :
( ( member_list_a @ P3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( polyno4422430861927485590lt_a_b @ r @ P3 )
= ( count_a @ ( polynomial_roots_a_b @ r @ P3 ) ) ) ) ).
% alg_mult_eq_count_roots
thf(fact_1067_p_Ois__root__poly__mult__imp__is__root,axiom,
! [P3: list_list_a,Q2: list_list_a,X: list_a] :
( ( member_list_list_a @ P3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ P3 @ Q2 ) @ X )
=> ( ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 @ X )
| ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q2 @ X ) ) ) ) ) ).
% p.is_root_poly_mult_imp_is_root
thf(fact_1068_p_OpirreducibleE_I3_J,axiom,
! [K4: set_list_a,P3: list_list_a,Q2: list_list_a,R2: list_list_a] :
( ( subrin6918843898125473962t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) @ P3 )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( member_list_list_a @ R2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( P3
= ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) @ Q2 @ R2 ) )
=> ( ( member_list_list_a @ Q2 @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
| ( member_list_list_a @ R2 @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) ) ) ) ) ) ) ) ) ).
% p.pirreducibleE(3)
thf(fact_1069_p_OpprimeE_I3_J,axiom,
! [K4: set_list_a,P3: list_list_a,Q2: list_list_a,R2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) @ P3 )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( member_list_list_a @ R2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) @ Q2 @ R2 ) )
=> ( ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 @ Q2 )
| ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 @ R2 ) ) ) ) ) ) ) ) ).
% p.pprimeE(3)
thf(fact_1070_roots__inclI_H,axiom,
! [P3: list_a,M: multiset_a] :
( ( member_list_a @ P3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [A5: a] :
( ( member_a @ A5 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( P3 != nil_a )
=> ( ord_less_eq_nat @ ( polyno4422430861927485590lt_a_b @ r @ P3 @ A5 ) @ ( count_a @ M @ A5 ) ) ) )
=> ( subseteq_mset_a @ ( polynomial_roots_a_b @ r @ P3 ) @ M ) ) ) ).
% roots_inclI'
thf(fact_1071_p_OpirreducibleI,axiom,
! [K4: set_list_a,P3: list_list_a] :
( ( subrin6918843898125473962t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( P3 != nil_list_a )
=> ( ~ ( member_list_list_a @ P3 @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ! [Q3: list_list_a,R3: list_list_a] :
( ( member_list_list_a @ Q3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( member_list_list_a @ R3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( P3
= ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) @ Q3 @ R3 ) )
=> ( ( member_list_list_a @ Q3 @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
| ( member_list_list_a @ R3 @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) ) ) ) ) )
=> ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) @ P3 ) ) ) ) ) ) ).
% p.pirreducibleI
thf(fact_1072_p_Ouniv__poly__infinite__dimension,axiom,
! [K4: set_list_a] :
( ( subfie1779122896746047282t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ~ ( embedd2411333406617385593t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) @ ( image_8260866953997875467list_a @ ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ K4 ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) ) ) ).
% p.univ_poly_infinite_dimension
thf(fact_1073_p_Olong__dividesI,axiom,
! [B: list_list_a,R2: list_list_a,P3: list_list_a,Q2: list_list_a] :
( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( member_list_list_a @ R2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( P3
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ Q2 @ B ) @ R2 ) )
=> ( ( ( R2 = nil_list_a )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ R2 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ Q2 ) @ one_one_nat ) ) )
=> ( polyno6947042923167803568t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 @ Q2 @ ( produc8696003437204565271list_a @ B @ R2 ) ) ) ) ) ) ).
% p.long_dividesI
thf(fact_1074_p_Olong__division__add__iff,axiom,
! [K4: set_list_a,A: list_list_a,B: list_list_a,C: list_list_a,Q2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( member_list_list_a @ C @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ Q2 )
= ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ Q2 ) )
= ( ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) @ A @ C ) @ Q2 )
= ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) @ B @ C ) @ Q2 ) ) ) ) ) ) ) ) ).
% p.long_division_add_iff
thf(fact_1075_p_Olong__division__add_I2_J,axiom,
! [K4: set_list_a,A: list_list_a,B: list_list_a,Q2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) @ A @ B ) @ Q2 )
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) @ ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ Q2 ) @ ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ Q2 ) ) ) ) ) ) ) ).
% p.long_division_add(2)
thf(fact_1076_p_Olong__division__add_I1_J,axiom,
! [K4: set_list_a,A: list_list_a,B: list_list_a,Q2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) @ A @ B ) @ Q2 )
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) @ ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ Q2 ) @ ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ Q2 ) ) ) ) ) ) ) ).
% p.long_division_add(1)
thf(fact_1077_p_Opdiv__pmod,axiom,
! [K4: set_list_a,P3: list_list_a,Q2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( P3
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) @ Q2 @ ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 @ Q2 ) ) @ ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 @ Q2 ) ) ) ) ) ) ).
% p.pdiv_pmod
thf(fact_1078_p_Obounded__degree__dimension,axiom,
! [K4: set_list_a,N: nat] :
( ( subfie1779122896746047282t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( embedd2305571234642070248t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) @ N @ ( image_8260866953997875467list_a @ ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ K4 )
@ ( collect_list_list_a
@ ^ [Q4: list_list_a] :
( ( member_list_list_a @ Q4 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
& ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ Q4 ) @ N ) ) ) ) ) ).
% p.bounded_degree_dimension
thf(fact_1079_p_Osubfield__long__division__theorem__shell,axiom,
! [K4: set_list_a,P3: list_list_a,B: list_list_a] :
( ( subfie1779122896746047282t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( B
!= ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ? [Q3: list_list_a,R3: list_list_a] :
( ( member_list_list_a @ Q3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
& ( member_list_list_a @ R3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
& ( P3
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) @ B @ Q3 ) @ R3 ) )
& ( ( R3
= ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ R3 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ B ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% p.subfield_long_division_theorem_shell
thf(fact_1080_p_Opderiv__const,axiom,
! [X: list_list_a,K4: set_list_a] :
( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ X ) @ one_one_nat )
= zero_zero_nat )
=> ( ( formal6075833236969493044t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X )
= ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) ) ) ).
% p.pderiv_const
thf(fact_1081__092_060open_062sum_Adegree_A_I_092_060Union_062d_092_060in_062_123d_O_Ad_Advd_An_125_O_A_123f_O_Am__i__p_AR_Af_A_092_060and_062_Adegree_Af_A_061_Ad_125_J_A_061_A_I_092_060Sum_062d_A_124_Ad_Advd_An_O_Asum_Adegree_A_123f_O_Am__i__p_AR_Af_A_092_060and_062_Adegree_Af_A_061_Ad_125_J_092_060close_062,axiom,
( ( groups5521247699297860762_a_nat
@ ^ [F: list_a] : ( minus_minus_nat @ ( size_size_list_a @ F ) @ one_one_nat )
@ ( comple6928918032620976721list_a
@ ( image_nat_set_list_a
@ ^ [D: nat] :
( collect_list_a
@ ^ [F: list_a] :
( ( monic_4919232885364369782ly_a_b @ r @ F )
& ( ( minus_minus_nat @ ( size_size_list_a @ F ) @ one_one_nat )
= D ) ) )
@ ( collect_nat
@ ^ [D: nat] : ( dvd_dvd_nat @ D @ n ) ) ) ) )
= ( groups3542108847815614940at_nat
@ ^ [D: nat] :
( groups5521247699297860762_a_nat
@ ^ [P: list_a] : ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
@ ( collect_list_a
@ ^ [F: list_a] :
( ( monic_4919232885364369782ly_a_b @ r @ F )
& ( ( minus_minus_nat @ ( size_size_list_a @ F ) @ one_one_nat )
= D ) ) ) )
@ ( collect_nat
@ ^ [D: nat] : ( dvd_dvd_nat @ D @ n ) ) ) ) ).
% \<open>sum degree (\<Union>d\<in>{d. d dvd n}. {f. m_i_p R f \<and> degree f = d}) = (\<Sum>d | d dvd n. sum degree {f. m_i_p R f \<and> degree f = d})\<close>
thf(fact_1082_p_Opderiv__zero,axiom,
! [K4: set_list_a] :
( ( formal6075833236969493044t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
= ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) ) ).
% p.pderiv_zero
thf(fact_1083_p_Opderiv__carr,axiom,
! [K4: set_list_a,F2: list_list_a] :
( ( subrin6918843898125473962t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ F2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( member_list_list_a @ ( formal6075833236969493044t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F2 ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) ) ) ) ).
% p.pderiv_carr
thf(fact_1084_p_Opderiv__add,axiom,
! [K4: set_list_a,F2: list_list_a,G: list_list_a] :
( ( subrin6918843898125473962t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ F2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( member_list_list_a @ G @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( formal6075833236969493044t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) @ F2 @ G ) )
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) @ ( formal6075833236969493044t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F2 ) @ ( formal6075833236969493044t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G ) ) ) ) ) ) ).
% p.pderiv_add
thf(fact_1085_p_Opderiv__mult,axiom,
! [K4: set_list_a,F2: list_list_a,G: list_list_a] :
( ( subrin6918843898125473962t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ F2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( member_list_list_a @ G @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( formal6075833236969493044t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) @ F2 @ G ) )
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) @ ( formal6075833236969493044t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F2 ) @ G ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) @ F2 @ ( formal6075833236969493044t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G ) ) ) ) ) ) ) ).
% p.pderiv_mult
thf(fact_1086__092_060open_062sum_Adegree_A_123f_O_Am__i__p_AR_Af_A_092_060and_062_Adegree_Af_Advd_An_125_A_061_Asum_Adegree_A_I_092_060Union_062d_092_060in_062_123d_O_Ad_Advd_An_125_O_A_123f_O_Am__i__p_AR_Af_A_092_060and_062_Adegree_Af_A_061_Ad_125_J_092_060close_062,axiom,
( ( groups5521247699297860762_a_nat
@ ^ [D: list_a] : ( minus_minus_nat @ ( size_size_list_a @ D ) @ one_one_nat )
@ ( collect_list_a
@ ^ [F: list_a] :
( ( monic_4919232885364369782ly_a_b @ r @ F )
& ( dvd_dvd_nat @ ( minus_minus_nat @ ( size_size_list_a @ F ) @ one_one_nat ) @ n ) ) ) )
= ( groups5521247699297860762_a_nat
@ ^ [F: list_a] : ( minus_minus_nat @ ( size_size_list_a @ F ) @ one_one_nat )
@ ( comple6928918032620976721list_a
@ ( image_nat_set_list_a
@ ^ [D: nat] :
( collect_list_a
@ ^ [F: list_a] :
( ( monic_4919232885364369782ly_a_b @ r @ F )
& ( ( minus_minus_nat @ ( size_size_list_a @ F ) @ one_one_nat )
= D ) ) )
@ ( collect_nat
@ ^ [D: nat] : ( dvd_dvd_nat @ D @ n ) ) ) ) ) ) ).
% \<open>sum degree {f. m_i_p R f \<and> degree f dvd n} = sum degree (\<Union>d\<in>{d. d dvd n}. {f. m_i_p R f \<and> degree f = d})\<close>
thf(fact_1087_p_Ofield__long__division__theorem,axiom,
! [K4: set_list_a,P3: list_list_a,B: list_list_a] :
( ( subfie1779122896746047282t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 @ P3 )
=> ( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 @ B )
=> ( ( B != nil_list_a )
=> ? [Q3: list_list_a,R3: list_list_a] :
( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 @ Q3 )
& ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 @ R3 )
& ( P3
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) @ B @ Q3 ) @ R3 ) )
& ( ( R3 = nil_list_a )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ R3 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ B ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% p.field_long_division_theorem
thf(fact_1088_p_Oconst__term__simprules__shell_I3_J,axiom,
! [K4: set_list_a,P3: list_list_a,Q2: list_list_a] :
( ( subrin6918843898125473962t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) @ P3 @ Q2 ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 ) @ ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q2 ) ) ) ) ) ) ).
% p.const_term_simprules_shell(3)
thf(fact_1089_p_Oconst__term__simprules__shell_I1_J,axiom,
! [K4: set_list_a,P3: list_list_a] :
( ( subrin6918843898125473962t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( member_list_a @ ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 ) @ K4 ) ) ) ).
% p.const_term_simprules_shell(1)
thf(fact_1090_p_Ozero__is__polynomial,axiom,
! [K4: set_list_a] : ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 @ nil_list_a ) ).
% p.zero_is_polynomial
thf(fact_1091_p_Ocarrier__polynomial,axiom,
! [K4: set_list_a,P3: list_list_a] :
( ( subrin6918843898125473962t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 @ P3 )
=> ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ P3 ) ) ) ).
% p.carrier_polynomial
thf(fact_1092_poly__of__const__over__carrier,axiom,
( ( image_a_list_a @ ( poly_of_const_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) )
= ( collect_list_a
@ ^ [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= zero_zero_nat ) ) ) ) ).
% poly_of_const_over_carrier
thf(fact_1093_p_OsubdomainI_H,axiom,
! [H3: set_list_a] :
( ( subrin6918843898125473962t_unit @ H3 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( subdom7821232466298058046t_unit @ H3 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% p.subdomainI'
thf(fact_1094_univ__poly__carrier__subfield__of__consts,axiom,
subfie1779122896746047282t_unit @ ( image_a_list_a @ ( poly_of_const_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% univ_poly_carrier_subfield_of_consts
thf(fact_1095_poly__of__const__over__subfield,axiom,
! [K4: set_a] :
( ( subfield_a_b @ K4 @ r )
=> ( ( image_a_list_a @ ( poly_of_const_a_b @ r ) @ K4 )
= ( collect_list_a
@ ^ [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
& ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= zero_zero_nat ) ) ) ) ) ).
% poly_of_const_over_subfield
thf(fact_1096_p_Oconst__term__simprules__shell_I2_J,axiom,
! [K4: set_list_a,P3: list_list_a,Q2: list_list_a] :
( ( subrin6918843898125473962t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) @ P3 @ Q2 ) )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 ) @ ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q2 ) ) ) ) ) ) ).
% p.const_term_simprules_shell(2)
thf(fact_1097_polynomial__ring__assms,axiom,
subfield_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% polynomial_ring_assms
thf(fact_1098_subring__props_I1_J,axiom,
! [K4: set_a] :
( ( subfield_a_b @ K4 @ r )
=> ( ord_less_eq_set_a @ K4 @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% subring_props(1)
thf(fact_1099_p_Om__assoc,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ Z )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z ) ) ) ) ) ) ).
% p.m_assoc
thf(fact_1100_p_Om__comm,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X ) ) ) ) ).
% p.m_comm
thf(fact_1101_p_Om__lcomm,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z ) )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Z ) ) ) ) ) ) ).
% p.m_lcomm
thf(fact_1102_p_Osubring__props_I6_J,axiom,
! [K4: set_list_a,H1: list_a,H22: list_a] :
( ( subfie1779122896746047282t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a @ H1 @ K4 )
=> ( ( member_list_a @ H22 @ K4 )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H1 @ H22 ) @ K4 ) ) ) ) ).
% p.subring_props(6)
thf(fact_1103_monic__poly__mult,axiom,
! [F2: list_a,G: list_a] :
( ( monic_3145109188698636716ly_a_b @ r @ F2 )
=> ( ( monic_3145109188698636716ly_a_b @ r @ G )
=> ( monic_3145109188698636716ly_a_b @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F2 @ G ) ) ) ) ).
% monic_poly_mult
thf(fact_1104_univ__poly__is__principal,axiom,
! [K4: set_a] :
( ( subfield_a_b @ K4 @ r )
=> ( ring_p8098905331641078952t_unit @ ( univ_poly_a_b @ r @ K4 ) ) ) ).
% univ_poly_is_principal
thf(fact_1105_p_Ol__distr,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ Z )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Z ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z ) ) ) ) ) ) ).
% p.l_distr
thf(fact_1106_p_Or__distr,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Z @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Z @ X ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Z @ Y ) ) ) ) ) ) ).
% p.r_distr
thf(fact_1107_is__root__poly__mult__imp__is__root,axiom,
! [P3: list_a,Q2: list_a,X: a] :
( ( member_list_a @ P3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( polyno4133073214067823460ot_a_b @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 @ Q2 ) @ X )
=> ( ( polyno4133073214067823460ot_a_b @ r @ P3 @ X )
| ( polyno4133073214067823460ot_a_b @ r @ Q2 @ X ) ) ) ) ) ).
% is_root_poly_mult_imp_is_root
thf(fact_1108_pprimeE_I1_J,axiom,
! [K4: set_a,P3: list_a] :
( ( subfield_a_b @ K4 @ r )
=> ( ( member_list_a @ P3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K4 ) @ P3 )
=> ( P3 != nil_a ) ) ) ) ).
% pprimeE(1)
thf(fact_1109_pprime__iff__pirreducible,axiom,
! [K4: set_a,P3: list_a] :
( ( subfield_a_b @ K4 @ r )
=> ( ( member_list_a @ P3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K4 ) @ P3 )
= ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K4 ) @ P3 ) ) ) ) ).
% pprime_iff_pirreducible
thf(fact_1110_univ__poly__subfield__of__consts,axiom,
! [K4: set_a] :
( ( subfield_a_b @ K4 @ r )
=> ( subfie1779122896746047282t_unit @ ( image_a_list_a @ ( poly_of_const_a_b @ r ) @ K4 ) @ ( univ_poly_a_b @ r @ K4 ) ) ) ).
% univ_poly_subfield_of_consts
thf(fact_1111_pprimeE_I3_J,axiom,
! [K4: set_a,P3: list_a,Q2: list_a,R2: list_a] :
( ( subfield_a_b @ K4 @ r )
=> ( ( member_list_a @ P3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K4 ) @ P3 )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
=> ( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P3 @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K4 ) @ Q2 @ R2 ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P3 @ Q2 )
| ( polyno5814909790663948098es_a_b @ r @ P3 @ R2 ) ) ) ) ) ) ) ) ).
% pprimeE(3)
thf(fact_1112_no__roots__imp__same__roots,axiom,
! [P3: list_a,Q2: list_a] :
( ( member_list_a @ P3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P3 != nil_a )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( polynomial_roots_a_b @ r @ P3 )
= zero_zero_multiset_a )
=> ( ( polynomial_roots_a_b @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 @ Q2 ) )
= ( polynomial_roots_a_b @ r @ Q2 ) ) ) ) ) ) ).
% no_roots_imp_same_roots
thf(fact_1113_p_Om__closed,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.m_closed
thf(fact_1114_univ__poly__is__euclidean,axiom,
! [K4: set_a] :
( ( subfield_a_b @ K4 @ r )
=> ( ring_e7478897652244013592t_unit @ ( univ_poly_a_b @ r @ K4 )
@ ^ [P: list_a] : ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) ) ) ).
% univ_poly_is_euclidean
thf(fact_1115_exists__unique__long__division,axiom,
! [K4: set_a,P3: list_a,Q2: list_a] :
( ( subfield_a_b @ K4 @ r )
=> ( ( member_list_a @ P3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
=> ( ( Q2 != nil_a )
=> ? [X3: produc9164743771328383783list_a] :
( ( polyno2806191415236617128es_a_b @ r @ P3 @ Q2 @ X3 )
& ! [Y3: produc9164743771328383783list_a] :
( ( polyno2806191415236617128es_a_b @ r @ P3 @ Q2 @ Y3 )
=> ( Y3 = X3 ) ) ) ) ) ) ) ).
% exists_unique_long_division
thf(fact_1116_pmod__const_I1_J,axiom,
! [K4: set_a,P3: list_a,Q2: list_a] :
( ( subfield_a_b @ K4 @ r )
=> ( ( member_list_a @ P3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
=> ( ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ P3 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q2 ) @ one_one_nat ) )
=> ( ( polynomial_pdiv_a_b @ r @ P3 @ Q2 )
= nil_a ) ) ) ) ) ).
% pmod_const(1)
thf(fact_1117_long__division__closed_I1_J,axiom,
! [K4: set_a,P3: list_a,Q2: list_a] :
( ( subfield_a_b @ K4 @ r )
=> ( ( member_list_a @ P3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
=> ( member_list_a @ ( polynomial_pdiv_a_b @ r @ P3 @ Q2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) ) ) ) ) ).
% long_division_closed(1)
thf(fact_1118_long__division__add_I1_J,axiom,
! [K4: set_a,A: list_a,B: list_a,Q2: list_a] :
( ( subfield_a_b @ K4 @ r )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
=> ( ( polynomial_pdiv_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K4 ) @ A @ B ) @ Q2 )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K4 ) @ ( polynomial_pdiv_a_b @ r @ A @ Q2 ) @ ( polynomial_pdiv_a_b @ r @ B @ Q2 ) ) ) ) ) ) ) ).
% long_division_add(1)
thf(fact_1119_long__division__zero_I1_J,axiom,
! [K4: set_a,Q2: list_a] :
( ( subfield_a_b @ K4 @ r )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
=> ( ( polynomial_pdiv_a_b @ r @ nil_a @ Q2 )
= nil_a ) ) ) ).
% long_division_zero(1)
thf(fact_1120_rupture__order,axiom,
! [K4: set_a,F2: list_a] :
( ( subfield_a_b @ K4 @ r )
=> ( ( member_list_a @ F2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( size_size_list_a @ F2 ) @ one_one_nat ) )
=> ( ( order_1351569949434154782t_unit @ ( polyno5459750281392823787re_a_b @ r @ K4 @ F2 ) )
= ( power_power_nat @ ( finite_card_a @ K4 ) @ ( minus_minus_nat @ ( size_size_list_a @ F2 ) @ one_one_nat ) ) ) ) ) ) ).
% rupture_order
thf(fact_1121_rupture__char,axiom,
! [K4: set_a,F2: list_a] :
( ( subfield_a_b @ K4 @ r )
=> ( ( member_list_a @ F2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( size_size_list_a @ F2 ) @ one_one_nat ) )
=> ( ( ring_c6053888738502451990t_unit @ ( polyno5459750281392823787re_a_b @ r @ K4 @ F2 ) )
= ( ring_char_a_b @ r ) ) ) ) ) ).
% rupture_char
thf(fact_1122_long__dividesI,axiom,
! [B: list_a,R2: list_a,P3: list_a,Q2: list_a] :
( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P3
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q2 @ B ) @ R2 ) )
=> ( ( ( R2 = nil_a )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ R2 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q2 ) @ one_one_nat ) ) )
=> ( polyno2806191415236617128es_a_b @ r @ P3 @ Q2 @ ( produc6837034575241423639list_a @ B @ R2 ) ) ) ) ) ) ).
% long_dividesI
thf(fact_1123_pmod__image__characterization,axiom,
! [K4: set_a,P3: list_a] :
( ( subfield_a_b @ K4 @ r )
=> ( ( member_list_a @ P3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
=> ( ( P3 != nil_a )
=> ( ( image_list_a_list_a
@ ^ [Q4: list_a] : ( polynomial_pmod_a_b @ r @ Q4 @ P3 )
@ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
= ( collect_list_a
@ ^ [Q4: list_a] :
( ( member_list_a @ Q4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
& ( ord_less_eq_nat @ ( size_size_list_a @ Q4 ) @ ( minus_minus_nat @ ( size_size_list_a @ P3 ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% pmod_image_characterization
thf(fact_1124_poly__add_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
~ ! [P12: list_a,P22: list_a] :
( X
!= ( produc6837034575241423639list_a @ P12 @ P22 ) ) ).
% poly_add.cases
thf(fact_1125_long__division__closed_I2_J,axiom,
! [K4: set_a,P3: list_a,Q2: list_a] :
( ( subfield_a_b @ K4 @ r )
=> ( ( member_list_a @ P3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
=> ( member_list_a @ ( polynomial_pmod_a_b @ r @ P3 @ Q2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) ) ) ) ) ).
% long_division_closed(2)
thf(fact_1126_long__division__add_I2_J,axiom,
! [K4: set_a,A: list_a,B: list_a,Q2: list_a] :
( ( subfield_a_b @ K4 @ r )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
=> ( ( polynomial_pmod_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K4 ) @ A @ B ) @ Q2 )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K4 ) @ ( polynomial_pmod_a_b @ r @ A @ Q2 ) @ ( polynomial_pmod_a_b @ r @ B @ Q2 ) ) ) ) ) ) ) ).
% long_division_add(2)
thf(fact_1127_long__division__add__iff,axiom,
! [K4: set_a,A: list_a,B: list_a,C: list_a,Q2: list_a] :
( ( subfield_a_b @ K4 @ r )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
=> ( ( ( polynomial_pmod_a_b @ r @ A @ Q2 )
= ( polynomial_pmod_a_b @ r @ B @ Q2 ) )
= ( ( polynomial_pmod_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K4 ) @ A @ C ) @ Q2 )
= ( polynomial_pmod_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K4 ) @ B @ C ) @ Q2 ) ) ) ) ) ) ) ) ).
% long_division_add_iff
thf(fact_1128_long__division__zero_I2_J,axiom,
! [K4: set_a,Q2: list_a] :
( ( subfield_a_b @ K4 @ r )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
=> ( ( polynomial_pmod_a_b @ r @ nil_a @ Q2 )
= nil_a ) ) ) ).
% long_division_zero(2)
thf(fact_1129_pmod__zero__iff__pdivides,axiom,
! [K4: set_a,P3: list_a,Q2: list_a] :
( ( subfield_a_b @ K4 @ r )
=> ( ( member_list_a @ P3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
=> ( ( ( polynomial_pmod_a_b @ r @ P3 @ Q2 )
= nil_a )
= ( polyno5814909790663948098es_a_b @ r @ Q2 @ P3 ) ) ) ) ) ).
% pmod_zero_iff_pdivides
thf(fact_1130_exists__long__division,axiom,
! [K4: set_a,P3: list_a,Q2: list_a] :
( ( subfield_a_b @ K4 @ r )
=> ( ( member_list_a @ P3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
=> ( ( Q2 != nil_a )
=> ~ ! [B4: list_a] :
( ( member_list_a @ B4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
=> ! [R3: list_a] :
( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
=> ~ ( polyno2806191415236617128es_a_b @ r @ P3 @ Q2 @ ( produc6837034575241423639list_a @ B4 @ R3 ) ) ) ) ) ) ) ) ).
% exists_long_division
thf(fact_1131_pdiv__pmod,axiom,
! [K4: set_a,P3: list_a,Q2: list_a] :
( ( subfield_a_b @ K4 @ r )
=> ( ( member_list_a @ P3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
=> ( P3
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K4 ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K4 ) @ Q2 @ ( polynomial_pdiv_a_b @ r @ P3 @ Q2 ) ) @ ( polynomial_pmod_a_b @ r @ P3 @ Q2 ) ) ) ) ) ) ).
% pdiv_pmod
thf(fact_1132_pmod__const_I2_J,axiom,
! [K4: set_a,P3: list_a,Q2: list_a] :
( ( subfield_a_b @ K4 @ r )
=> ( ( member_list_a @ P3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
=> ( ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ P3 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q2 ) @ one_one_nat ) )
=> ( ( polynomial_pmod_a_b @ r @ P3 @ Q2 )
= P3 ) ) ) ) ) ).
% pmod_const(2)
thf(fact_1133_pmod__degree,axiom,
! [K4: set_a,P3: list_a,Q2: list_a] :
( ( subfield_a_b @ K4 @ r )
=> ( ( member_list_a @ P3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
=> ( ( Q2 != nil_a )
=> ( ( ( polynomial_pmod_a_b @ r @ P3 @ Q2 )
= nil_a )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( polynomial_pmod_a_b @ r @ P3 @ Q2 ) ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q2 ) @ one_one_nat ) ) ) ) ) ) ) ).
% pmod_degree
thf(fact_1134_long__divisionE,axiom,
! [K4: set_a,P3: list_a,Q2: list_a] :
( ( subfield_a_b @ K4 @ r )
=> ( ( member_list_a @ P3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
=> ( ( Q2 != nil_a )
=> ( polyno2806191415236617128es_a_b @ r @ P3 @ Q2 @ ( produc6837034575241423639list_a @ ( polynomial_pdiv_a_b @ r @ P3 @ Q2 ) @ ( polynomial_pmod_a_b @ r @ P3 @ Q2 ) ) ) ) ) ) ) ).
% long_divisionE
thf(fact_1135_long__divisionI,axiom,
! [K4: set_a,P3: list_a,Q2: list_a,B: list_a,R2: list_a] :
( ( subfield_a_b @ K4 @ r )
=> ( ( member_list_a @ P3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
=> ( ( Q2 != nil_a )
=> ( ( polyno2806191415236617128es_a_b @ r @ P3 @ Q2 @ ( produc6837034575241423639list_a @ B @ R2 ) )
=> ( ( produc6837034575241423639list_a @ B @ R2 )
= ( produc6837034575241423639list_a @ ( polynomial_pdiv_a_b @ r @ P3 @ Q2 ) @ ( polynomial_pmod_a_b @ r @ P3 @ Q2 ) ) ) ) ) ) ) ) ).
% long_divisionI
thf(fact_1136_p_Omonoid__cancelI,axiom,
( ! [A5: list_a,B4: list_a,C3: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C3 @ A5 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C3 @ B4 ) )
=> ( ( member_list_a @ A5 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( A5 = B4 ) ) ) ) )
=> ( ! [A5: list_a,B4: list_a,C3: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A5 @ C3 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B4 @ C3 ) )
=> ( ( member_list_a @ A5 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( A5 = B4 ) ) ) ) )
=> ( monoid4303264861975686087t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.monoid_cancelI
thf(fact_1137_subfield__long__division__theorem__shell,axiom,
! [K4: set_a,P3: list_a,B: list_a] :
( ( subfield_a_b @ K4 @ r )
=> ( ( member_list_a @ P3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
=> ( ( B
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
=> ? [Q3: list_a,R3: list_a] :
( ( member_list_a @ Q3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
& ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
& ( P3
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K4 ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K4 ) @ B @ Q3 ) @ R3 ) )
& ( ( R3
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ R3 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ B ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% subfield_long_division_theorem_shell
thf(fact_1138_p_Osubring__props_I2_J,axiom,
! [K4: set_list_a] :
( ( subfie1779122896746047282t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ K4 ) ) ).
% p.subring_props(2)
thf(fact_1139_p_Ozero__is__prime_I1_J,axiom,
prime_2011924034616061926t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% p.zero_is_prime(1)
thf(fact_1140_p_Oadd_Oinv__comm,axiom,
! [X: list_a,Y: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% p.add.inv_comm
thf(fact_1141_p_Oadd_Ol__inv__ex,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ? [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.add.l_inv_ex
thf(fact_1142_p_Oadd_Oone__unique,axiom,
! [U: list_a] :
( ( member_list_a @ U @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ U @ X3 )
= X3 ) )
=> ( U
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.add.one_unique
thf(fact_1143_p_Oadd_Or__inv__ex,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ? [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ X3 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.add.r_inv_ex
thf(fact_1144_p_Ominus__unique,axiom,
! [Y: list_a,X: list_a,Y6: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y6 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y6 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( Y = Y6 ) ) ) ) ) ) ).
% p.minus_unique
thf(fact_1145_p_Ointegral,axiom,
! [A: list_a,B: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( A
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
| ( B
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ).
% p.integral
thf(fact_1146_p_Ointegral__iff,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( ( A
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
| ( B
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ).
% p.integral_iff
thf(fact_1147_p_Om__lcancel,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( A
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ C ) )
= ( B = C ) ) ) ) ) ) ).
% p.m_lcancel
thf(fact_1148_p_Om__rcancel,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( A
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ A )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ A ) )
= ( B = C ) ) ) ) ) ) ).
% p.m_rcancel
thf(fact_1149_p_Oring__irreducibleE_I1_J,axiom,
! [R2: list_a] :
( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ R2 )
=> ( R2
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.ring_irreducibleE(1)
thf(fact_1150_p_Oring__primeE_I1_J,axiom,
! [P3: list_a] :
( ( member_list_a @ P3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 )
=> ( P3
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.ring_primeE(1)
thf(fact_1151_p_Oconst__term__not__zero,axiom,
! [P3: list_list_a] :
( ( ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 )
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( P3 != nil_list_a ) ) ).
% p.const_term_not_zero
thf(fact_1152_p_Oring__primeI,axiom,
! [P3: list_a] :
( ( P3
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( prime_2011924034616061926t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 )
=> ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 ) ) ) ).
% p.ring_primeI
thf(fact_1153_p_Oa__lcos__mult__one,axiom,
! [M4: set_list_a] :
( ( ord_le8861187494160871172list_a @ M4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ M4 )
= M4 ) ) ).
% p.a_lcos_mult_one
thf(fact_1154_gauss__poly__not__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( card_I2373409586816755191ly_a_b @ r @ N )
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% gauss_poly_not_zero
thf(fact_1155_p_Ozero__closed,axiom,
member_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% p.zero_closed
thf(fact_1156_p_Oadd_Ol__cancel__one,axiom,
! [X: list_a,A: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ A )
= X )
= ( A
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% p.add.l_cancel_one
thf(fact_1157_p_Oadd_Ol__cancel__one_H,axiom,
! [X: list_a,A: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( X
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ A ) )
= ( A
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% p.add.l_cancel_one'
thf(fact_1158_p_Oadd_Or__cancel__one,axiom,
! [X: list_a,A: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ X )
= X )
= ( A
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% p.add.r_cancel_one
thf(fact_1159_p_Oadd_Or__cancel__one_H,axiom,
! [X: list_a,A: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( X
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ X ) )
= ( A
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% p.add.r_cancel_one'
thf(fact_1160_p_Ol__zero,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X )
= X ) ) ).
% p.l_zero
thf(fact_1161_p_Or__zero,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= X ) ) ).
% p.r_zero
thf(fact_1162_p_Or__null,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.r_null
thf(fact_1163_p_Ol__null,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.l_null
thf(fact_1164_p_OboundD__carrier,axiom,
! [N: nat,F2: nat > list_a,M: nat] :
( ( bound_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ N @ F2 )
=> ( ( ord_less_nat @ N @ M )
=> ( member_list_a @ ( F2 @ M ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.boundD_carrier
thf(fact_1165_p_Odegree__oneE,axiom,
! [P3: list_list_a,K4: set_list_a] :
( ( member_list_list_a @ P3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P3 ) @ one_one_nat )
= one_one_nat )
=> ~ ! [A5: list_a] :
( ( member_list_a @ A5 @ K4 )
=> ( ( A5
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ! [B4: list_a] :
( ( member_list_a @ B4 @ K4 )
=> ( P3
!= ( cons_list_a @ A5 @ ( cons_list_a @ B4 @ nil_list_a ) ) ) ) ) ) ) ) ).
% p.degree_oneE
thf(fact_1166_p_Onormalize_Ocases,axiom,
! [X: list_list_a] :
( ( X != nil_list_a )
=> ~ ! [V: list_a,Va: list_list_a] :
( X
!= ( cons_list_a @ V @ Va ) ) ) ).
% p.normalize.cases
thf(fact_1167_p_Ocombine_Ocases,axiom,
! [X: produc7709606177366032167list_a] :
( ! [K3: list_a,Ks: list_list_a,U2: list_a,Us: list_list_a] :
( X
!= ( produc8696003437204565271list_a @ ( cons_list_a @ K3 @ Ks ) @ ( cons_list_a @ U2 @ Us ) ) )
=> ( ! [Us: list_list_a] :
( X
!= ( produc8696003437204565271list_a @ nil_list_a @ Us ) )
=> ~ ! [Ks: list_list_a] :
( X
!= ( produc8696003437204565271list_a @ Ks @ nil_list_a ) ) ) ) ).
% p.combine.cases
thf(fact_1168_p_Opoly__mult_Ocases,axiom,
! [X: produc7709606177366032167list_a] :
( ! [P22: list_list_a] :
( X
!= ( produc8696003437204565271list_a @ nil_list_a @ P22 ) )
=> ~ ! [V: list_a,Va: list_list_a,P22: list_list_a] :
( X
!= ( produc8696003437204565271list_a @ ( cons_list_a @ V @ Va ) @ P22 ) ) ) ).
% p.poly_mult.cases
thf(fact_1169_p_Oline__extension__smult__closed,axiom,
! [K4: set_list_a,E: set_list_a,A: list_a,K: list_a,U: list_a] :
( ( subfie1779122896746047282t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ! [K3: list_a,V: list_a] :
( ( member_list_a @ K3 @ K4 )
=> ( ( member_list_a @ V @ E )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K3 @ V ) @ E ) ) )
=> ( ( ord_le8861187494160871172list_a @ E @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ K @ K4 )
=> ( ( member_list_a @ U @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 @ A @ E ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ U ) @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 @ A @ E ) ) ) ) ) ) ) ) ).
% p.line_extension_smult_closed
thf(fact_1170_same__pmod__iff__pdivides,axiom,
! [K4: set_a,A: list_a,B: list_a,Q2: list_a] :
( ( subfield_a_b @ K4 @ r )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
=> ( ( ( polynomial_pmod_a_b @ r @ A @ Q2 )
= ( polynomial_pmod_a_b @ r @ B @ Q2 ) )
= ( polyno5814909790663948098es_a_b @ r @ Q2 @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ K4 ) @ A @ B ) ) ) ) ) ) ) ).
% same_pmod_iff_pdivides
thf(fact_1171_normalize_Ocases,axiom,
! [X: list_a] :
( ( X != nil_a )
=> ~ ! [V: a,Va: list_a] :
( X
!= ( cons_a @ V @ Va ) ) ) ).
% normalize.cases
thf(fact_1172_combine_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
( ! [K3: a,Ks: list_a,U2: a,Us: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ K3 @ Ks ) @ ( cons_a @ U2 @ Us ) ) )
=> ( ! [Us: list_a] :
( X
!= ( produc6837034575241423639list_a @ nil_a @ Us ) )
=> ~ ! [Ks: list_a] :
( X
!= ( produc6837034575241423639list_a @ Ks @ nil_a ) ) ) ) ).
% combine.cases
thf(fact_1173_poly__mult_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
( ! [P22: list_a] :
( X
!= ( produc6837034575241423639list_a @ nil_a @ P22 ) )
=> ~ ! [V: a,Va: list_a,P22: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ V @ Va ) @ P22 ) ) ) ).
% poly_mult.cases
thf(fact_1174_p_Oline__extension__in__carrier,axiom,
! [K4: set_list_a,A: list_a,E: set_list_a] :
( ( ord_le8861187494160871172list_a @ K4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ E @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 @ A @ E ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% p.line_extension_in_carrier
thf(fact_1175_p_Oline__extension__mem__iff,axiom,
! [U: list_a,K4: set_list_a,A: list_a,E: set_list_a] :
( ( member_list_a @ U @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 @ A @ E ) )
= ( ? [X2: list_a] :
( ( member_list_a @ X2 @ K4 )
& ? [Y5: list_a] :
( ( member_list_a @ Y5 @ E )
& ( U
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ A ) @ Y5 ) ) ) ) ) ) ).
% p.line_extension_mem_iff
thf(fact_1176_p_Ominus__closed,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.minus_closed
thf(fact_1177_p_Or__right__minus__eq,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( A = B ) ) ) ) ).
% p.r_right_minus_eq
thf(fact_1178_pderiv__const,axiom,
! [X: list_a,K4: set_a] :
( ( ( minus_minus_nat @ ( size_size_list_a @ X ) @ one_one_nat )
= zero_zero_nat )
=> ( ( formal4452980811800949548iv_a_b @ r @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ K4 ) ) ) ) ).
% pderiv_const
thf(fact_1179_p_Oconst__term__zero,axiom,
! [K4: set_list_a,P3: list_list_a] :
( ( subrin6918843898125473962t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 @ P3 )
=> ( ( P3 != nil_list_a )
=> ( ( ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ~ ! [P5: list_list_a] :
( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 @ P5 )
=> ( ( P5 != nil_list_a )
=> ( P3
!= ( append_list_a @ P5 @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ nil_list_a ) ) ) ) ) ) ) ) ) ).
% p.const_term_zero
thf(fact_1180_pderiv__zero,axiom,
! [K4: set_a] :
( ( formal4452980811800949548iv_a_b @ r @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ K4 ) ) ) ).
% pderiv_zero
thf(fact_1181_pderiv__carr,axiom,
! [F2: list_a] :
( ( member_list_a @ F2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( formal4452980811800949548iv_a_b @ r @ F2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% pderiv_carr
thf(fact_1182_pderiv__add,axiom,
! [F2: list_a,G: list_a] :
( ( member_list_a @ F2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ G @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( formal4452980811800949548iv_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F2 @ G ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( formal4452980811800949548iv_a_b @ r @ F2 ) @ ( formal4452980811800949548iv_a_b @ r @ G ) ) ) ) ) ).
% pderiv_add
thf(fact_1183_pderiv__mult,axiom,
! [F2: list_a,G: list_a] :
( ( member_list_a @ F2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ G @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( formal4452980811800949548iv_a_b @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F2 @ G ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( formal4452980811800949548iv_a_b @ r @ F2 ) @ G ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F2 @ ( formal4452980811800949548iv_a_b @ r @ G ) ) ) ) ) ) ).
% pderiv_mult
thf(fact_1184_p_Opoly__mult__var,axiom,
! [K4: set_list_a,P3: list_list_a] :
( ( subrin6918843898125473962t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( ( P3 = nil_list_a )
=> ( ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) @ P3 @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= nil_list_a ) )
& ( ( P3 != nil_list_a )
=> ( ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) @ P3 @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( append_list_a @ P3 @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ nil_list_a ) ) ) ) ) ) ) ).
% p.poly_mult_var
thf(fact_1185_p_Oconst__term__explicit,axiom,
! [P3: list_list_a,A: list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P3 != nil_list_a )
=> ( ( ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 )
= A )
=> ~ ! [P5: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P5 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( P3
!= ( append_list_a @ P5 @ ( cons_list_a @ A @ nil_list_a ) ) ) ) ) ) ) ).
% p.const_term_explicit
thf(fact_1186_p_Opolynomial__incl,axiom,
! [K4: set_list_a,P3: list_list_a] :
( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 @ P3 )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P3 ) @ K4 ) ) ).
% p.polynomial_incl
thf(fact_1187_p_Ovar__closed_I2_J,axiom,
! [K4: set_list_a] :
( ( subrin6918843898125473962t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.var_closed(2)
thf(fact_1188_p_Oconst__term__simprules_I1_J,axiom,
! [P3: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.const_term_simprules(1)
thf(fact_1189_p_Odegree__var,axiom,
( ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ one_one_nat )
= one_one_nat ) ).
% p.degree_var
thf(fact_1190_p_Ovar__closed_I1_J,axiom,
! [K4: set_list_a] :
( ( subrin6918843898125473962t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( member_list_list_a @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) ) ) ).
% p.var_closed(1)
thf(fact_1191_p_Oconst__term__eq__last,axiom,
! [P3: list_list_a,A: list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ P3 @ ( cons_list_a @ A @ nil_list_a ) ) )
= A ) ) ) ).
% p.const_term_eq_last
thf(fact_1192_p_Opolynomial__in__carrier,axiom,
! [K4: set_list_a,P3: list_list_a] :
( ( subrin6918843898125473962t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 @ P3 )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.polynomial_in_carrier
thf(fact_1193_p_Oexp__base__closed,axiom,
! [X: list_a,N: nat] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ ( polyno3522816881121920896t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ N ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.exp_base_closed
thf(fact_1194_p_Ofactors__mult,axiom,
! [Fa: list_list_a,A: list_a,Fb: list_list_a,B: list_a] :
( ( factor7181967632740204193t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Fa @ A )
=> ( ( factor7181967632740204193t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Fb @ B )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Fa ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Fb ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( factor7181967632740204193t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ Fa @ Fb ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B ) ) ) ) ) ) ).
% p.factors_mult
thf(fact_1195_p_Ofactors__closed,axiom,
! [Fs: list_list_a,A: list_a] :
( ( factor7181967632740204193t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Fs @ A )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Fs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.factors_closed
thf(fact_1196_exp__base__closed,axiom,
! [X: a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( polyno2922411391617481336se_a_b @ r @ X @ N ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% exp_base_closed
thf(fact_1197_p_Oee__sym,axiom,
! [As: list_list_a,Bs: list_list_a] :
( ( essent703981920984620806t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ As @ Bs )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ As ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Bs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( essent703981920984620806t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Bs @ As ) ) ) ) ).
% p.ee_sym
thf(fact_1198_p_Oee__length,axiom,
! [As: list_list_a,Bs: list_list_a] :
( ( essent703981920984620806t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ As @ Bs )
=> ( ( size_s349497388124573686list_a @ As )
= ( size_s349497388124573686list_a @ Bs ) ) ) ).
% p.ee_length
thf(fact_1199_p_Oee__trans,axiom,
! [As: list_list_a,Bs: list_list_a,Cs: list_list_a] :
( ( essent703981920984620806t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ As @ Bs )
=> ( ( essent703981920984620806t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Bs @ Cs )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ As ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Bs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Cs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( essent703981920984620806t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ As @ Cs ) ) ) ) ) ) ).
% p.ee_trans
thf(fact_1200_p_Oee__refl,axiom,
! [As: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ As ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( essent703981920984620806t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ As @ As ) ) ).
% p.ee_refl
thf(fact_1201_p_Oeval__append__aux,axiom,
! [P3: list_list_a,B: list_a,A: list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ P3 @ ( cons_list_a @ B @ nil_list_a ) ) @ A )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 @ A ) @ A ) @ B ) ) ) ) ) ).
% p.eval_append_aux
thf(fact_1202_univ__poly__infinite__dimension,axiom,
! [K4: set_a] :
( ( subfield_a_b @ K4 @ r )
=> ~ ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ K4 ) @ ( image_a_list_a @ ( poly_of_const_a_b @ r ) @ K4 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) ) ) ).
% univ_poly_infinite_dimension
thf(fact_1203_p_Otelescopic__base__dim_I1_J,axiom,
! [K4: set_list_a,F3: set_list_a,E: set_list_a] :
( ( subfie1779122896746047282t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( subfie1779122896746047282t_unit @ F3 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 @ F3 )
=> ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F3 @ E )
=> ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 @ E ) ) ) ) ) ).
% p.telescopic_base_dim(1)
thf(fact_1204_p_Oeval_Osimps_I1_J,axiom,
( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ nil_list_a )
= ( ^ [Uu: list_a] : ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.eval.simps(1)
thf(fact_1205_p_Oeval__var,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X )
= X ) ) ).
% p.eval_var
thf(fact_1206_p_Oconst__term__def,axiom,
! [P3: list_list_a] :
( ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 )
= ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.const_term_def
thf(fact_1207_p_Oeval__in__carrier,axiom,
! [P3: list_list_a,X: list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 @ X ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.eval_in_carrier
thf(fact_1208_p_Oeval__poly__in__carrier,axiom,
! [K4: set_list_a,P3: list_list_a,X: list_a] :
( ( subrin6918843898125473962t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 @ P3 )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 @ X ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% p.eval_poly_in_carrier
thf(fact_1209_p_Ois__root__def,axiom,
! [P3: list_list_a,X: list_a] :
( ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 @ X )
= ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( P3 != nil_list_a ) ) ) ).
% p.is_root_def
thf(fact_1210_p_Opdivides__imp__root__sharing,axiom,
! [P3: list_list_a,Q2: list_list_a,A: list_a] :
( ( member_list_list_a @ P3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 @ Q2 )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 @ A )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q2 @ A )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ).
% p.pdivides_imp_root_sharing
thf(fact_1211_p_Oeval__cring__hom,axiom,
! [K4: set_list_a,A: list_a] :
( ( subrin6918843898125473962t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ring_h453377649743177125t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ^ [P: list_list_a] : ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ A ) ) ) ) ).
% p.eval_cring_hom
thf(fact_1212_p_Osubalbegra__incl__imp__finite__dimension,axiom,
! [K4: set_list_a,E: set_list_a,V2: set_list_a] :
( ( subfie1779122896746047282t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 @ E )
=> ( ( embedd1768981623711841426t_unit @ K4 @ V2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ V2 @ E )
=> ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 @ V2 ) ) ) ) ) ).
% p.subalbegra_incl_imp_finite_dimension
thf(fact_1213_p_Ocarrier__is__subalgebra,axiom,
! [K4: set_list_a] :
( ( ord_le8861187494160871172list_a @ K4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( embedd1768981623711841426t_unit @ K4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% p.carrier_is_subalgebra
thf(fact_1214_p_Osubalgebra__in__carrier,axiom,
! [K4: set_list_a,V2: set_list_a] :
( ( embedd1768981623711841426t_unit @ K4 @ V2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ord_le8861187494160871172list_a @ V2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.subalgebra_in_carrier
thf(fact_1215_p_Ofinite__dimension__imp__subalgebra,axiom,
! [K4: set_list_a,E: set_list_a] :
( ( subfie1779122896746047282t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 @ E )
=> ( embedd1768981623711841426t_unit @ K4 @ E @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.finite_dimension_imp_subalgebra
thf(fact_1216_p_Oeval__is__hom,axiom,
! [K4: set_list_a,A: list_a] :
( ( subrin6918843898125473962t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member7168557129179038582list_a
@ ^ [P: list_list_a] : ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ A )
@ ( ring_h5031276006722532742t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.eval_is_hom
thf(fact_1217_p_Oeval__append,axiom,
! [P3: list_list_a,Q2: list_list_a,A: list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Q2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ P3 @ Q2 ) @ A )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 @ A ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ ( size_s349497388124573686list_a @ Q2 ) ) ) @ ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q2 @ A ) ) ) ) ) ) ).
% p.eval_append
thf(fact_1218_monic__poly__pow,axiom,
! [F2: list_a,N: nat] :
( ( monic_3145109188698636716ly_a_b @ r @ F2 )
=> ( monic_3145109188698636716ly_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F2 @ N ) ) ) ).
% monic_poly_pow
thf(fact_1219_p_Opow__non__zero,axiom,
! [X: list_a,N: nat] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( X
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ N )
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.pow_non_zero
thf(fact_1220_p_Onat__pow__zero,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ N )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.nat_pow_zero
thf(fact_1221_p_Onat__pow__pow,axiom,
! [X: list_a,N: nat,M: nat] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ N ) @ M )
= ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( times_times_nat @ N @ M ) ) ) ) ).
% p.nat_pow_pow
thf(fact_1222_p_Opow__mult__distrib,axiom,
! [X: list_a,Y: list_a,N: nat] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ N )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ N ) ) ) ) ) ) ).
% p.pow_mult_distrib
thf(fact_1223_p_Onat__pow__distrib,axiom,
! [X: list_a,Y: list_a,N: nat] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ N )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ N ) ) ) ) ) ).
% p.nat_pow_distrib
thf(fact_1224_p_Onat__pow__comm,axiom,
! [X: list_a,N: nat,M: nat] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ M ) )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ M ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ N ) ) ) ) ).
% p.nat_pow_comm
thf(fact_1225_p_Ogroup__commutes__pow,axiom,
! [X: list_a,Y: list_a,N: nat] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ N ) @ Y )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ N ) ) ) ) ) ) ).
% p.group_commutes_pow
thf(fact_1226_polynomial__pow__not__zero,axiom,
! [P3: list_a,N: nat] :
( ( member_list_a @ P3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P3 != nil_a )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 @ N )
!= nil_a ) ) ) ).
% polynomial_pow_not_zero
thf(fact_1227_polynomial__pow__division,axiom,
! [P3: list_a,N: nat,M: nat] :
( ( member_list_a @ P3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 @ M ) ) ) ) ).
% polynomial_pow_division
thf(fact_1228_pirreducible__pow__pdivides__iff,axiom,
! [K4: set_a,P3: list_a,Q2: list_a,R2: list_a,N: nat] :
( ( subfield_a_b @ K4 @ r )
=> ( ( member_list_a @ P3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
=> ( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K4 ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K4 ) @ P3 )
=> ( ~ ( polyno5814909790663948098es_a_b @ r @ P3 @ Q2 )
=> ( ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K4 ) @ P3 @ N ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K4 ) @ Q2 @ R2 ) )
= ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K4 ) @ P3 @ N ) @ R2 ) ) ) ) ) ) ) ) ).
% pirreducible_pow_pdivides_iff
thf(fact_1229_polynomial__pow__degree,axiom,
! [P3: list_a,N: nat] :
( ( member_list_a @ P3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( minus_minus_nat @ ( size_size_list_a @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 @ N ) ) @ one_one_nat )
= ( times_times_nat @ N @ ( minus_minus_nat @ ( size_size_list_a @ P3 ) @ one_one_nat ) ) ) ) ).
% polynomial_pow_degree
thf(fact_1230_p_Ofreshmans__dream,axiom,
! [X: list_a,Y: list_a] :
( ( ord_less_nat @ zero_zero_nat @ ( ring_c500279861223467766t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ ( ring_c500279861223467766t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( ring_c500279861223467766t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ ( ring_c500279861223467766t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ) ).
% p.freshmans_dream
thf(fact_1231_p_Ofreshmans__dream__ext,axiom,
! [X: list_a,Y: list_a,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( ring_c500279861223467766t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ ( power_power_nat @ ( ring_c500279861223467766t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ M ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( power_power_nat @ ( ring_c500279861223467766t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ M ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ ( power_power_nat @ ( ring_c500279861223467766t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ M ) ) ) ) ) ) ) ).
% p.freshmans_dream_ext
thf(fact_1232_p_Onat__pow__closed,axiom,
! [X: list_a,N: nat] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ N ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.nat_pow_closed
thf(fact_1233_p_Onat__pow__eone,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ one_one_nat )
= X ) ) ).
% p.nat_pow_eone
thf(fact_1234_p_Ofrobenius__hom,axiom,
! [M: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( ring_c500279861223467766t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( M
= ( power_power_nat @ ( ring_c500279861223467766t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ K ) )
=> ( ring_h8282015026914974507t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ^ [X2: list_a] : ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ M ) ) ) ) ).
% p.frobenius_hom
thf(fact_1235_p_Oeval__monom,axiom,
! [B: list_a,A: list_a,N: nat] :
( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( monom_7446464087056152608t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ N ) @ A )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ N ) ) ) ) ) ).
% p.eval_monom
thf(fact_1236_monic__poly__var,axiom,
monic_3145109188698636716ly_a_b @ r @ ( var_a_b @ r ) ).
% monic_poly_var
thf(fact_1237_nat__pow__pow,axiom,
! [X: a,N: nat,M: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ M )
= ( pow_a_1026414303147256608_b_nat @ r @ X @ ( times_times_nat @ N @ M ) ) ) ) ).
% nat_pow_pow
thf(fact_1238_order__pow__eq__self,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ X @ ( order_a_ring_ext_a_b @ r ) )
= X ) ) ).
% order_pow_eq_self
thf(fact_1239_var__closed_I1_J,axiom,
member_list_a @ ( var_a_b @ r ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% var_closed(1)
thf(fact_1240_var__neq__zero,axiom,
( ( var_a_b @ r )
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% var_neq_zero
thf(fact_1241_degree__var,axiom,
( ( minus_minus_nat @ ( size_size_list_a @ ( var_a_b @ r ) ) @ one_one_nat )
= one_one_nat ) ).
% degree_var
thf(fact_1242_order__pow__eq__self_H,axiom,
! [X: a,D2: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ X @ ( power_power_nat @ ( order_a_ring_ext_a_b @ r ) @ D2 ) )
= X ) ) ).
% order_pow_eq_self'
thf(fact_1243_var__pow__closed,axiom,
! [N: nat] : ( member_list_a @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( var_a_b @ r ) @ N ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% var_pow_closed
thf(fact_1244_p_Opolynomial__pow__not__zero,axiom,
! [P3: list_list_a,N: nat] :
( ( member_list_list_a @ P3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( P3 != nil_list_a )
=> ( ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ P3 @ N )
!= nil_list_a ) ) ) ).
% p.polynomial_pow_not_zero
thf(fact_1245_var__pow__degree,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( size_size_list_a @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( var_a_b @ r ) @ N ) ) @ one_one_nat )
= N ) ).
% var_pow_degree
thf(fact_1246_p_Osubring__polynomial__pow__not__zero,axiom,
! [K4: set_list_a,P3: list_list_a,N: nat] :
( ( subrin6918843898125473962t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( P3 != nil_list_a )
=> ( ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) @ P3 @ N )
!= nil_list_a ) ) ) ) ).
% p.subring_polynomial_pow_not_zero
thf(fact_1247_p_Ovar__pow__closed,axiom,
! [K4: set_list_a,N: nat] :
( ( subrin6918843898125473962t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( member_list_list_a @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ N ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) ) ) ).
% p.var_pow_closed
thf(fact_1248_p_Opolynomial__pow__division,axiom,
! [P3: list_list_a,N: nat,M: nat] :
( ( member_list_list_a @ P3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ P3 @ N ) @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ P3 @ M ) ) ) ) ).
% p.polynomial_pow_division
thf(fact_1249_nat__pow__closed,axiom,
! [X: a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% nat_pow_closed
thf(fact_1250_p_Ovar__pow__degree,axiom,
! [K4: set_list_a,N: nat] :
( ( subrin6918843898125473962t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ N ) ) @ one_one_nat )
= N ) ) ).
% p.var_pow_degree
thf(fact_1251_p_Opirreducible__pow__pdivides__iff,axiom,
! [K4: set_list_a,P3: list_list_a,Q2: list_list_a,R2: list_list_a,N: nat] :
( ( subfie1779122896746047282t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( member_list_list_a @ R2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) @ P3 )
=> ( ~ ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P3 @ Q2 )
=> ( ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) @ P3 @ N ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) @ Q2 @ R2 ) )
= ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) @ P3 @ N ) @ R2 ) ) ) ) ) ) ) ) ).
% p.pirreducible_pow_pdivides_iff
thf(fact_1252_p_Opolynomial__pow__degree,axiom,
! [P3: list_list_a,N: nat] :
( ( member_list_list_a @ P3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ P3 @ N ) ) @ one_one_nat )
= ( times_times_nat @ N @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P3 ) @ one_one_nat ) ) ) ) ).
% p.polynomial_pow_degree
thf(fact_1253_p_Osubring__polynomial__pow__degree,axiom,
! [K4: set_list_a,P3: list_list_a,N: nat] :
( ( subrin6918843898125473962t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) ) )
=> ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) @ P3 @ N ) ) @ one_one_nat )
= ( times_times_nat @ N @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P3 ) @ one_one_nat ) ) ) ) ) ).
% p.subring_polynomial_pow_degree
thf(fact_1254_nat__pow__eone,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ X @ one_one_nat )
= X ) ) ).
% nat_pow_eone
thf(fact_1255_p_Omonom__in__carrier,axiom,
! [A: list_a,N: nat] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ ( monom_7446464087056152608t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ N ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.monom_in_carrier
thf(fact_1256_frobenius__hom,axiom,
! [M: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( ring_char_a_b @ r ) )
=> ( ( M
= ( power_power_nat @ ( ring_char_a_b @ r ) @ K ) )
=> ( ring_h661254511236296859_b_a_b @ r @ r
@ ^ [X2: a] : ( pow_a_1026414303147256608_b_nat @ r @ X2 @ M ) ) ) ) ).
% frobenius_hom
thf(fact_1257_gauss__poly__factor,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( card_I2373409586816755191ly_a_b @ r @ N )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( var_a_b @ r ) @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ ( var_a_b @ r ) ) ) ) ).
% gauss_poly_factor
thf(fact_1258_p_Ozero__not__one,axiom,
( ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
!= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% p.zero_not_one
thf(fact_1259_p_Osubring__props_I3_J,axiom,
! [K4: set_list_a] :
( ( subfie1779122896746047282t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( member_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ K4 ) ) ).
% p.subring_props(3)
thf(fact_1260_monic__poly__one,axiom,
monic_3145109188698636716ly_a_b @ r @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% monic_poly_one
thf(fact_1261_pderiv__var,axiom,
! [K4: set_a] :
( ( formal4452980811800949548iv_a_b @ r @ ( var_a_b @ r ) )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ K4 ) ) ) ).
% pderiv_var
thf(fact_1262_p_Oone__unique,axiom,
! [U: list_a] :
( ( member_list_a @ U @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ U @ X3 )
= X3 ) )
=> ( U
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.one_unique
thf(fact_1263_p_Oinv__unique,axiom,
! [Y: list_a,X: list_a,Y6: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y6 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y6 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( Y = Y6 ) ) ) ) ) ) ).
% p.inv_unique
thf(fact_1264_degree__one,axiom,
! [K4: set_a] :
( ( minus_minus_nat @ ( size_size_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ K4 ) ) ) @ one_one_nat )
= zero_zero_nat ) ).
% degree_one
thf(fact_1265_var__pow__eq__one__iff,axiom,
! [K: nat] :
( ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( var_a_b @ r ) @ K )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( K = zero_zero_nat ) ) ).
% var_pow_eq_one_iff
thf(fact_1266_p_Ounitary__monom__eq__var__pow,axiom,
! [K4: set_list_a,N: nat] :
( ( subrin6918843898125473962t_unit @ K4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( monom_7446464087056152608t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ N )
= ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 ) @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ N ) ) ) ).
% p.unitary_monom_eq_var_pow
thf(fact_1267_p_Onum__roots__le__deg,axiom,
! [D2: nat] :
( ( finite_finite_list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( D2 != zero_zero_nat )
=> ( ord_less_eq_nat
@ ( finite_card_list_a
@ ( collect_list_a
@ ^ [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ D2 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) )
@ D2 ) ) ) ).
% p.num_roots_le_deg
thf(fact_1268_gauss__poly__div__gauss__poly__iff__1,axiom,
! [L: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ L )
=> ( ( polyno5814909790663948098es_a_b @ r @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( var_a_b @ r ) @ L ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( var_a_b @ r ) @ M ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
= ( dvd_dvd_nat @ L @ M ) ) ) ).
% gauss_poly_div_gauss_poly_iff_1
% Helper facts (7)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__Complex__Ocomplex_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
! [X: complex,Y: complex] :
( ( if_complex @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
! [X: complex,Y: complex] :
( ( if_complex @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( power_power_nat @ ( order_a_ring_ext_a_b @ r ) @ n )
= ( groups3542108847815614940at_nat
@ ^ [D: nat] :
( times_times_nat @ D
@ ( finite_card_list_a
@ ( collect_list_a
@ ^ [F: list_a] :
( ( monic_4919232885364369782ly_a_b @ r @ F )
& ( ( minus_minus_nat @ ( size_size_list_a @ F ) @ one_one_nat )
= D ) ) ) ) )
@ ( collect_nat
@ ^ [D: nat] : ( dvd_dvd_nat @ D @ n ) ) ) ) ).
%------------------------------------------------------------------------------