TPTP Problem File: SLH0714^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Finite_Fields/0008_Card_Irreducible_Polynomials_Aux/prob_00775_028152__18414370_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1507 ( 426 unt; 234 typ; 0 def)
% Number of atoms : 4052 (1058 equ; 0 cnn)
% Maximal formula atoms : 11 ( 3 avg)
% Number of connectives : 18339 ( 254 ~; 99 |; 168 &;15870 @)
% ( 0 <=>;1948 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 7 avg)
% Number of types : 27 ( 26 usr)
% Number of type conns : 440 ( 440 >; 0 *; 0 +; 0 <<)
% Number of symbols : 209 ( 208 usr; 14 con; 0-4 aty)
% Number of variables : 2683 ( 31 ^;2604 !; 48 ?;2683 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 13:23:35.443
%------------------------------------------------------------------------------
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thf(sy_c_Set_OCollect_001t__Int__Oint,type,
collect_int: ( int > $o ) > set_int ).
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_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001t__List__Olist_Itf__a_J,type,
image_int_list_a: ( int > list_a ) > set_int > set_list_a ).
thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
image_5464838071766335845list_a: ( list_a > set_list_a ) > set_list_a > set_set_list_a ).
thf(sy_c_Subrings_Osubcring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
subcri7763218559781929323t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
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_It__List__Olist_Itf__a_J_J,type,
bound_list_list_a: list_list_a > nat > ( nat > list_list_a ) > $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_UnivPoly_Obound_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
bound_set_list_a: set_list_a > nat > ( nat > set_list_a ) > $o ).
thf(sy_c_UnivPoly_Obound_001tf__a,type,
bound_a: a > nat > ( nat > a ) > $o ).
thf(sy_c_UnivPoly_Oup_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
up_lis8963924889346801084t_unit: partia2956882679547061052t_unit > set_nat_list_list_a ).
thf(sy_c_UnivPoly_Oup_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
up_lis8464167429055313730t_unit: partia2670972154091845814t_unit > set_nat_list_a ).
thf(sy_c_UnivPoly_Oup_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
up_set529185716248919906t_unit: partia7496981018696276118t_unit > set_nat_set_list_a ).
thf(sy_c_UnivPoly_Oup_001tf__a_001tf__b,type,
up_a_b: partia2175431115845679010xt_a_b > set_nat_a ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
member8650753269014980122list_a: ( nat > list_list_a ) > set_nat_list_list_a > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J,type,
member_nat_list_a: ( nat > list_a ) > set_nat_list_a > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
member491565700723299188list_a: ( nat > set_list_a ) > set_nat_set_list_a > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mtf__a_J,type,
member_nat_a: ( nat > a ) > set_nat_a > $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__Real__Oreal,type,
member_real: real > set_real > $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_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_R,type,
r: partia2175431115845679010xt_a_b ).
thf(sy_v_d____,type,
d: nat ).
thf(sy_v_f,type,
f: list_a ).
thf(sy_v_n,type,
n: nat ).
% Relevant facts (1272)
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_assms_I2_J,axiom,
monic_4919232885364369782ly_a_b @ r @ f ).
% assms(2)
thf(fact_3__092_060open_062f_A_091_094_093_092_060_094bsub_062P_092_060_094esub_062_A2_Apdivides_Agauss__poly_AR_A_Iorder_AR_A_094_An_J_092_060close_062,axiom,
polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ f @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( card_I2373409586816755191ly_a_b @ r @ ( power_power_nat @ ( order_a_ring_ext_a_b @ r ) @ n ) ) ).
% \<open>f [^]\<^bsub>P\<^esub> 2 pdivides gauss_poly R (order R ^ n)\<close>
thf(fact_4_noetherian__domain__axioms,axiom,
ring_n4045954140777738665in_a_b @ r ).
% noetherian_domain_axioms
thf(fact_5_p_Ofactorial__domain__axioms,axiom,
ring_f796907574329358751t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% p.factorial_domain_axioms
thf(fact_6__092_060open_062f_Apdivides_Agauss__poly_AR_A_Iorder_AR_A_094_An_J_092_060close_062,axiom,
polyno5814909790663948098es_a_b @ r @ f @ ( card_I2373409586816755191ly_a_b @ r @ ( power_power_nat @ ( order_a_ring_ext_a_b @ r ) @ n ) ) ).
% \<open>f pdivides gauss_poly R (order R ^ n)\<close>
thf(fact_7_p_Onoetherian__domain__axioms,axiom,
ring_n4705423059119889713t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% p.noetherian_domain_axioms
thf(fact_8_p_Omult__of_Odivisor__chain__condition__monoid__axioms,axiom,
diviso7418750317856897340t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% p.mult_of.divisor_chain_condition_monoid_axioms
thf(fact_9_p_Omult__of_Oprimeness__condition__monoid__axioms,axiom,
primen7481049025794127375t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% p.mult_of.primeness_condition_monoid_axioms
thf(fact_10_principal__domain__axioms,axiom,
ring_p8803135361686045600in_a_b @ r ).
% principal_domain_axioms
thf(fact_11_p_Onoetherian__ring__axioms,axiom,
ring_n5188127996776581661t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% p.noetherian_ring_axioms
thf(fact_12_p_Oprincipal__domain__axioms,axiom,
ring_p8098905331641078952t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% p.principal_domain_axioms
thf(fact_13_onepideal,axiom,
principalideal_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% onepideal
thf(fact_14_noetherian__ring__axioms,axiom,
ring_n3639167112692572309ng_a_b @ r ).
% noetherian_ring_axioms
thf(fact_15_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_16_p_Omult__of_Ogcd__condition__monoid__axioms,axiom,
gcd_co7549656856577097917t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% p.mult_of.gcd_condition_monoid_axioms
thf(fact_17__092_060open_062_092_060not_062_Apmult_Af_A_Igauss__poly_AR_A_Iorder_AR_A_094_An_J_J_A_060_A2_092_060close_062,axiom,
~ ( ord_less_nat @ ( monic_5301438133677370042lt_a_b @ r @ f @ ( card_I2373409586816755191ly_a_b @ r @ ( power_power_nat @ ( order_a_ring_ext_a_b @ r ) @ n ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% \<open>\<not> pmult f (gauss_poly R (order R ^ n)) < 2\<close>
thf(fact_18_p_Omult__of_Omonoid__cancel__axioms,axiom,
monoid5117334421817186628t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% p.mult_of.monoid_cancel_axioms
thf(fact_19_semiring__norm_I85_J,axiom,
! [M: num] :
( ( bit0 @ M )
!= one ) ).
% semiring_norm(85)
thf(fact_20_semiring__norm_I83_J,axiom,
! [N: num] :
( one
!= ( bit0 @ N ) ) ).
% semiring_norm(83)
thf(fact_21_f__carr_I1_J,axiom,
member_list_a @ f @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% f_carr(1)
thf(fact_22_semiring__norm_I87_J,axiom,
! [M: num,N: num] :
( ( ( bit0 @ M )
= ( bit0 @ N ) )
= ( M = N ) ) ).
% semiring_norm(87)
thf(fact_23_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_24_assms_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ n ).
% assms(1)
thf(fact_25_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_26_finite__field_Ogauss__poly__splitted,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_f3414212785825285533t_unit @ R )
=> ( polyno6259083269128200473t_unit @ R @ ( card_I2619780863984422015t_unit @ R @ ( order_3240872107759947550t_unit @ R ) ) ) ) ).
% finite_field.gauss_poly_splitted
thf(fact_27_finite__field_Ogauss__poly__splitted,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_f9002592259703987221ld_a_b @ R )
=> ( polyno8329700637149614481ed_a_b @ R @ ( card_I2373409586816755191ly_a_b @ R @ ( order_a_ring_ext_a_b @ R ) ) ) ) ).
% finite_field.gauss_poly_splitted
thf(fact_28_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_29__092_060open_0622_A_092_060le_062_Apmult_Af_A_Igauss__poly_AR_A_Iorder_AR_A_094_An_J_J_092_060close_062,axiom,
ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( monic_5301438133677370042lt_a_b @ r @ f @ ( card_I2373409586816755191ly_a_b @ r @ ( power_power_nat @ ( order_a_ring_ext_a_b @ r ) @ n ) ) ) ).
% \<open>2 \<le> pmult f (gauss_poly R (order R ^ n))\<close>
thf(fact_30_o21,axiom,
ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( order_a_ring_ext_a_b @ r ) @ n ) ).
% o21
thf(fact_31_o2,axiom,
ord_less_nat @ one_one_nat @ ( power_power_nat @ ( order_a_ring_ext_a_b @ r ) @ n ) ).
% o2
thf(fact_32_pdivides__imp__splitted,axiom,
! [P: list_a,Q: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( Q != nil_a )
=> ( ( polyno8329700637149614481ed_a_b @ r @ Q )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ Q )
=> ( polyno8329700637149614481ed_a_b @ r @ P ) ) ) ) ) ) ).
% pdivides_imp_splitted
thf(fact_33_finite__field_Oorder__pow__eq__self_H,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a,D: nat] :
( ( ring_f1633963914286637719t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( pow_li488931774710091566it_nat @ R @ X @ ( power_power_nat @ ( order_7989348053904188830t_unit @ R ) @ D ) )
= X ) ) ) ).
% finite_field.order_pow_eq_self'
thf(fact_34_finite__field_Oorder__pow__eq__self_H,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,D: nat] :
( ( ring_f3414212785825285533t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( pow_li1142815632869257134it_nat @ R @ X @ ( power_power_nat @ ( order_3240872107759947550t_unit @ R ) @ D ) )
= X ) ) ) ).
% finite_field.order_pow_eq_self'
thf(fact_35_finite__field_Oorder__pow__eq__self_H,axiom,
! [R: partia7496981018696276118t_unit,X: set_list_a,D: nat] :
( ( ring_f902895163656626621t_unit @ R )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
=> ( ( pow_se8252319793075206062it_nat @ R @ X @ ( power_power_nat @ ( order_1351569949434154782t_unit @ R ) @ D ) )
= X ) ) ) ).
% finite_field.order_pow_eq_self'
thf(fact_36_finite__field_Oorder__pow__eq__self_H,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,D: nat] :
( ( ring_f9002592259703987221ld_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( pow_a_1026414303147256608_b_nat @ R @ X @ ( power_power_nat @ ( order_a_ring_ext_a_b @ R ) @ D ) )
= X ) ) ) ).
% finite_field.order_pow_eq_self'
thf(fact_37_polynomial__pow__division,axiom,
! [P: list_a,N: nat,M: nat] :
( ( member_list_a @ P @ ( 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 ) ) @ P @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ M ) ) ) ) ).
% polynomial_pow_division
thf(fact_38_less__exp,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% less_exp
thf(fact_39_p_Odivides__mult__zero,axiom,
! [A: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ ( 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 ) ) ) ) ) ) ).
% p.divides_mult_zero
thf(fact_40_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_41_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_42_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_43_order__dim_I2_J,axiom,
ord_less_nat @ zero_zero_nat @ d ).
% order_dim(2)
thf(fact_44_f__carr_I2_J,axiom,
f != nil_a ).
% f_carr(2)
thf(fact_45_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_46_zero__pdivides,axiom,
! [P: list_a] :
( ( polyno5814909790663948098es_a_b @ r @ nil_a @ P )
= ( P = nil_a ) ) ).
% zero_pdivides
thf(fact_47_zero__pdivides__zero,axiom,
polyno5814909790663948098es_a_b @ r @ nil_a @ nil_a ).
% zero_pdivides_zero
thf(fact_48_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_nat @ M )
= ( numeral_numeral_nat @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_49_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_real @ M )
= ( numeral_numeral_real @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_50_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_int @ M )
= ( numeral_numeral_int @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_51_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_52_mem__Collect__eq,axiom,
! [A: int,P2: int > $o] :
( ( member_int @ A @ ( collect_int @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_53_mem__Collect__eq,axiom,
! [A: nat > list_a,P2: ( nat > list_a ) > $o] :
( ( member_nat_list_a @ A @ ( collect_nat_list_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_54_mem__Collect__eq,axiom,
! [A: nat > a,P2: ( nat > a ) > $o] :
( ( member_nat_a @ A @ ( collect_nat_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_55_mem__Collect__eq,axiom,
! [A: a,P2: a > $o] :
( ( member_a @ A @ ( collect_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_56_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_57_Collect__mem__eq,axiom,
! [A2: set_int] :
( ( collect_int
@ ^ [X2: int] : ( member_int @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_58_Collect__mem__eq,axiom,
! [A2: set_nat_list_a] :
( ( collect_nat_list_a
@ ^ [X2: nat > list_a] : ( member_nat_list_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_59_Collect__mem__eq,axiom,
! [A2: set_nat_a] :
( ( collect_nat_a
@ ^ [X2: nat > a] : ( member_nat_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_60_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_61_finite__field__min__order,axiom,
ord_less_nat @ one_one_nat @ ( order_a_ring_ext_a_b @ r ) ).
% finite_field_min_order
thf(fact_62_order__pow__eq__self_H,axiom,
! [X: a,D: 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 ) @ D ) )
= X ) ) ).
% order_pow_eq_self'
thf(fact_63_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_64_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_65_polynomial__pow__not__zero,axiom,
! [P: list_a,N: nat] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P != nil_a )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ N )
!= nil_a ) ) ) ).
% polynomial_pow_not_zero
thf(fact_66_pdivides__zero,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( polyno5814909790663948098es_a_b @ r @ P @ nil_a ) ) ).
% pdivides_zero
thf(fact_67_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_68_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_69_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_70_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_71_power__one,axiom,
! [N: nat] :
( ( power_power_nat @ one_one_nat @ N )
= one_one_nat ) ).
% power_one
thf(fact_72_power__one,axiom,
! [N: nat] :
( ( power_power_real @ one_one_real @ N )
= one_one_real ) ).
% power_one
thf(fact_73_power__one,axiom,
! [N: nat] :
( ( power_power_int @ one_one_int @ N )
= one_one_int ) ).
% power_one
thf(fact_74_power__one__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_75_power__one__right,axiom,
! [A: real] :
( ( power_power_real @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_76_power__one__right,axiom,
! [A: int] :
( ( power_power_int @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_77_semiring__norm_I78_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(78)
thf(fact_78_semiring__norm_I75_J,axiom,
! [M: num] :
~ ( ord_less_num @ M @ one ) ).
% semiring_norm(75)
thf(fact_79_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_80_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_81_power__inject__exp,axiom,
! [A: real,M: nat,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ( power_power_real @ A @ M )
= ( power_power_real @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_82_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_83_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_nat @ N )
= one_one_nat )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_84_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_real @ N )
= one_one_real )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_85_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_int @ N )
= one_one_int )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_86_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_nat
= ( numeral_numeral_nat @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_87_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_real
= ( numeral_numeral_real @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_88_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_int
= ( numeral_numeral_int @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_89_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
= zero_zero_nat ) ).
% power_zero_numeral
thf(fact_90_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
= zero_zero_real ) ).
% power_zero_numeral
thf(fact_91_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
= zero_zero_int ) ).
% power_zero_numeral
thf(fact_92_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_93_calculation,axiom,
ord_less_eq_nat @ one_one_nat @ ( monic_5301438133677370042lt_a_b @ r @ f @ ( card_I2373409586816755191ly_a_b @ r @ ( power_power_nat @ ( order_a_ring_ext_a_b @ r ) @ n ) ) ) ).
% calculation
thf(fact_94_semiring__norm_I76_J,axiom,
! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).
% semiring_norm(76)
thf(fact_95_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_96_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ one_one_real )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_97_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_98_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ one_one_int )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_99_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_100_power__strict__increasing__iff,axiom,
! [B: real,X: nat,Y: nat] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_101_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_102_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_103_power__eq__0__iff,axiom,
! [A: real,N: nat] :
( ( ( power_power_real @ A @ N )
= zero_zero_real )
= ( ( A = zero_zero_real )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_104_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_105_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_106_power__strict__decreasing__iff,axiom,
! [B: real,M: nat,N: nat] :
( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_real @ B @ one_one_real )
=> ( ( ord_less_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_107_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_108_power__increasing__iff,axiom,
! [B: real,X: nat,Y: nat] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_eq_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_109_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_110_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_111_power__mono__iff,axiom,
! [A: real,B: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
= ( ord_less_eq_real @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_112_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_113_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_114_zero__eq__power2,axiom,
! [A: nat] :
( ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% zero_eq_power2
thf(fact_115_zero__eq__power2,axiom,
! [A: real] :
( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% zero_eq_power2
thf(fact_116_zero__eq__power2,axiom,
! [A: int] :
( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% zero_eq_power2
thf(fact_117_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_118_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_real @ one_one_real @ ( numeral_numeral_real @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_119_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_120_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_121_power__decreasing__iff,axiom,
! [B: real,M: nat,N: nat] :
( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_real @ B @ one_one_real )
=> ( ( ord_less_eq_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_122_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_123_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_124_power2__less__eq__zero__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real )
= ( A = zero_zero_real ) ) ).
% power2_less_eq_zero_iff
thf(fact_125_power2__less__eq__zero__iff,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int )
= ( A = zero_zero_int ) ) ).
% power2_less_eq_zero_iff
thf(fact_126_power2__eq__iff__nonneg,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( X = Y ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_127_power2__eq__iff__nonneg,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( X = Y ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_128_power2__eq__iff__nonneg,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( X = Y ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_129_zero__less__power2,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( A != zero_zero_real ) ) ).
% zero_less_power2
thf(fact_130_zero__less__power2,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( A != zero_zero_int ) ) ).
% zero_less_power2
thf(fact_131_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_132_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_133_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_134_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_135_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% not_numeral_le_zero
thf(fact_136_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_le_zero
thf(fact_137_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% not_numeral_le_zero
thf(fact_138_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% zero_le_numeral
thf(fact_139_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_le_numeral
thf(fact_140_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% zero_le_numeral
thf(fact_141_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N ) ) ).
% one_le_numeral
thf(fact_142_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% one_le_numeral
thf(fact_143_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).
% one_le_numeral
thf(fact_144_power__increasing,axiom,
! [N: nat,N2: nat,A: real] :
( ( ord_less_eq_nat @ N @ N2 )
=> ( ( ord_less_eq_real @ one_one_real @ A )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N2 ) ) ) ) ).
% power_increasing
thf(fact_145_power__increasing,axiom,
! [N: nat,N2: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N2 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N2 ) ) ) ) ).
% power_increasing
thf(fact_146_power__increasing,axiom,
! [N: nat,N2: nat,A: int] :
( ( ord_less_eq_nat @ N @ N2 )
=> ( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N2 ) ) ) ) ).
% power_increasing
thf(fact_147_power__decreasing,axiom,
! [N: nat,N2: nat,A: real] :
( ( ord_less_eq_nat @ N @ N2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_148_power__decreasing,axiom,
! [N: nat,N2: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_149_power__decreasing,axiom,
! [N: nat,N2: nat,A: int] :
( ( ord_less_eq_nat @ N @ N2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_150_zero__le__power,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% zero_le_power
thf(fact_151_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_152_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_153_power__le__one,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ one_one_real ) ) ) ).
% power_le_one
thf(fact_154_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_155_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_156_one__le__power,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ one_one_real @ A )
=> ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).
% one_le_power
thf(fact_157_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_158_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_159_power__mono,axiom,
! [A: real,B: real,N: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% power_mono
thf(fact_160_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_161_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_162_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_163_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N )
= one_one_real ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N )
= zero_zero_real ) ) ) ).
% power_0_left
thf(fact_164_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_165_power__0,axiom,
! [A: nat] :
( ( power_power_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_166_power__0,axiom,
! [A: real] :
( ( power_power_real @ A @ zero_zero_nat )
= one_one_real ) ).
% power_0
thf(fact_167_power__0,axiom,
! [A: int] :
( ( power_power_int @ A @ zero_zero_nat )
= one_one_int ) ).
% power_0
thf(fact_168_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_169_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_170_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_171_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_172_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_173_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_174_power__le__imp__le__exp,axiom,
! [A: real,M: nat,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_eq_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_175_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_176_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_177_self__le__power,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ one_one_real @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_178_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_179_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_180_power__eq__imp__eq__base,axiom,
! [A: real,N: nat,B: real] :
( ( ( power_power_real @ A @ N )
= ( power_power_real @ B @ N ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_181_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_182_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_183_power__eq__iff__eq__base,axiom,
! [N: nat,A: real,B: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ( power_power_real @ A @ N )
= ( power_power_real @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_184_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_185_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_186_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_187_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_188_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_189_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_190_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_191_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_192_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_nat
!= ( numeral_numeral_nat @ N ) ) ).
% zero_neq_numeral
thf(fact_193_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_real
!= ( numeral_numeral_real @ N ) ) ).
% zero_neq_numeral
thf(fact_194_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_int
!= ( numeral_numeral_int @ N ) ) ).
% zero_neq_numeral
thf(fact_195_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_196_power__not__zero,axiom,
! [A: real,N: nat] :
( ( A != zero_zero_real )
=> ( ( power_power_real @ A @ N )
!= zero_zero_real ) ) ).
% power_not_zero
thf(fact_197_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_198_power__less__imp__less__base,axiom,
! [A: real,N: nat,B: real] :
( ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_real @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_199_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_200_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_201_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_202_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_real @ zero_zero_real @ N )
= zero_zero_real ) ) ).
% zero_power
thf(fact_203_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_204_power__strict__mono,axiom,
! [A: real,B: real,N: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_205_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_206_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_207_power__strict__decreasing,axiom,
! [N: nat,N2: nat,A: nat] :
( ( ord_less_nat @ N @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_208_power__strict__decreasing,axiom,
! [N: nat,N2: nat,A: real] :
( ( ord_less_nat @ N @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ A @ one_one_real )
=> ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_209_power__strict__decreasing,axiom,
! [N: nat,N2: nat,A: int] :
( ( ord_less_nat @ N @ N2 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_210_one__less__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_211_one__less__power,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_212_one__less__power,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_213_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).
% not_numeral_less_one
thf(fact_214_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ one_one_real ) ).
% not_numeral_less_one
thf(fact_215_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ).
% not_numeral_less_one
thf(fact_216_numeral__One,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numeral_One
thf(fact_217_numeral__One,axiom,
( ( numeral_numeral_real @ one )
= one_one_real ) ).
% numeral_One
thf(fact_218_numeral__One,axiom,
( ( numeral_numeral_int @ one )
= one_one_int ) ).
% numeral_One
thf(fact_219_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_less_zero
thf(fact_220_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% not_numeral_less_zero
thf(fact_221_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% not_numeral_less_zero
thf(fact_222_zero__less__numeral,axiom,
! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_less_numeral
thf(fact_223_zero__less__numeral,axiom,
! [N: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% zero_less_numeral
thf(fact_224_zero__less__numeral,axiom,
! [N: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% zero_less_numeral
thf(fact_225_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_226_zero__less__power,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% zero_less_power
thf(fact_227_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_228_numerals_I1_J,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numerals(1)
thf(fact_229_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_230_zero__le__power2,axiom,
! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% zero_le_power2
thf(fact_231_zero__le__power2,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% zero_le_power2
thf(fact_232_power2__eq__imp__eq,axiom,
! [X: real,Y: real] :
( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( X = Y ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_233_power2__eq__imp__eq,axiom,
! [X: nat,Y: nat] :
( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( X = Y ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_234_power2__eq__imp__eq,axiom,
! [X: int,Y: int] :
( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( X = Y ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_235_power2__le__imp__le,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ X @ Y ) ) ) ).
% power2_le_imp_le
thf(fact_236_power2__le__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power2_le_imp_le
thf(fact_237_power2__le__imp__le,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ord_less_eq_int @ X @ Y ) ) ) ).
% power2_le_imp_le
thf(fact_238_power2__less__imp__less,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_real @ X @ Y ) ) ) ).
% power2_less_imp_less
thf(fact_239_power2__less__imp__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ord_less_nat @ X @ Y ) ) ) ).
% power2_less_imp_less
thf(fact_240_power2__less__imp__less,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ord_less_int @ X @ Y ) ) ) ).
% power2_less_imp_less
thf(fact_241_power__strict__increasing,axiom,
! [N: nat,N2: nat,A: nat] :
( ( ord_less_nat @ N @ N2 )
=> ( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N2 ) ) ) ) ).
% power_strict_increasing
thf(fact_242_power__strict__increasing,axiom,
! [N: nat,N2: nat,A: real] :
( ( ord_less_nat @ N @ N2 )
=> ( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N2 ) ) ) ) ).
% power_strict_increasing
thf(fact_243_power__strict__increasing,axiom,
! [N: nat,N2: nat,A: int] :
( ( ord_less_nat @ N @ N2 )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N2 ) ) ) ) ).
% power_strict_increasing
thf(fact_244_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_245_power__less__imp__less__exp,axiom,
! [A: real,M: nat,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_246_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_247_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_248_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_249_one__power2,axiom,
( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ).
% one_power2
thf(fact_250_one__power2,axiom,
( ( power_power_real @ one_one_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_real ) ).
% one_power2
thf(fact_251_one__power2,axiom,
( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_int ) ).
% one_power2
thf(fact_252_self__le__ge2__pow,axiom,
! [K: nat,M: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ( ord_less_eq_nat @ M @ ( power_power_nat @ K @ M ) ) ) ).
% self_le_ge2_pow
thf(fact_253_power2__nat__le__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% power2_nat_le_eq_le
thf(fact_254_power2__nat__le__imp__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% power2_nat_le_imp_le
thf(fact_255_zero__power2,axiom,
( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% zero_power2
thf(fact_256_zero__power2,axiom,
( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_real ) ).
% zero_power2
thf(fact_257_zero__power2,axiom,
( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% zero_power2
thf(fact_258_power2__less__0,axiom,
! [A: real] :
~ ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real ) ).
% power2_less_0
thf(fact_259_power2__less__0,axiom,
! [A: int] :
~ ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int ) ).
% power2_less_0
thf(fact_260_finite__field_Oorder__pow__eq__self,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a] :
( ( ring_f1633963914286637719t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( pow_li488931774710091566it_nat @ R @ X @ ( order_7989348053904188830t_unit @ R ) )
= X ) ) ) ).
% finite_field.order_pow_eq_self
thf(fact_261_finite__field_Oorder__pow__eq__self,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( ring_f3414212785825285533t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( pow_li1142815632869257134it_nat @ R @ X @ ( order_3240872107759947550t_unit @ R ) )
= X ) ) ) ).
% finite_field.order_pow_eq_self
thf(fact_262_finite__field_Oorder__pow__eq__self,axiom,
! [R: partia7496981018696276118t_unit,X: set_list_a] :
( ( ring_f902895163656626621t_unit @ R )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
=> ( ( pow_se8252319793075206062it_nat @ R @ X @ ( order_1351569949434154782t_unit @ R ) )
= X ) ) ) ).
% finite_field.order_pow_eq_self
thf(fact_263_finite__field_Oorder__pow__eq__self,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( ring_f9002592259703987221ld_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( pow_a_1026414303147256608_b_nat @ R @ X @ ( order_a_ring_ext_a_b @ R ) )
= X ) ) ) ).
% finite_field.order_pow_eq_self
thf(fact_264_p_OboundD__carrier,axiom,
! [N: nat,F: nat > list_a,M: nat] :
( ( bound_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ N @ F )
=> ( ( ord_less_nat @ N @ M )
=> ( member_list_a @ ( F @ M ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.boundD_carrier
thf(fact_265_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_266_p_Oring__primeE_I1_J,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
=> ( P
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.ring_primeE(1)
thf(fact_267_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_268_univ__poly__zero__closed,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a] : ( member_list_list_a @ nil_list_a @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K2 ) ) ) ).
% univ_poly_zero_closed
thf(fact_269_univ__poly__zero__closed,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a] : ( member_list_a @ nil_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K2 ) ) ) ).
% univ_poly_zero_closed
thf(fact_270_p_Ozero__is__prime_I2_J,axiom,
prime_769485436051073893t_unit @ ( ring_m2863707994090333347t_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(2)
thf(fact_271_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_272_divides__monic__poly,axiom,
! [F: list_a,G: list_a] :
( ( monic_3145109188698636716ly_a_b @ r @ F )
=> ( ( monic_3145109188698636716ly_a_b @ r @ G )
=> ( ! [D2: list_a] :
( ( monic_4919232885364369782ly_a_b @ r @ D2 )
=> ( ord_less_eq_nat @ ( monic_5301438133677370042lt_a_b @ r @ D2 @ F ) @ ( monic_5301438133677370042lt_a_b @ r @ D2 @ G ) ) )
=> ( polyno5814909790663948098es_a_b @ r @ F @ G ) ) ) ) ).
% divides_monic_poly
thf(fact_273_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_274_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_275_pos2,axiom,
ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).
% pos2
thf(fact_276_p_Opolynomial__pow__not__zero,axiom,
! [P: list_list_a,N: nat] :
( ( member_list_list_a @ P @ ( 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 != 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 ) ) ) ) @ P @ N )
!= nil_list_a ) ) ) ).
% p.polynomial_pow_not_zero
thf(fact_277_monic__poly__carr,axiom,
! [F: list_a] :
( ( monic_3145109188698636716ly_a_b @ r @ F )
=> ( member_list_a @ F @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% monic_poly_carr
thf(fact_278_monic__poly__pow,axiom,
! [F: list_a,N: nat] :
( ( monic_3145109188698636716ly_a_b @ r @ F )
=> ( monic_3145109188698636716ly_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ N ) ) ) ).
% monic_poly_pow
thf(fact_279_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_280_p_Oring__primeE_I3_J,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
=> ( prime_2011924034616061926t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) ) ).
% p.ring_primeE(3)
thf(fact_281_p_Oring__primeI,axiom,
! [P: list_a] :
( ( P
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( prime_2011924034616061926t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
=> ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) ) ).
% p.ring_primeI
thf(fact_282_p_Oring__primeE_I2_J,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
=> ( prime_769485436051073893t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ P ) ) ) ).
% p.ring_primeE(2)
thf(fact_283_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_284_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_285_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_286_p_Oprime__eq__prime__mult,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( prime_2011924034616061926t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
= ( prime_769485436051073893t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ P ) ) ) ).
% p.prime_eq_prime_mult
thf(fact_287_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_288_semiring__norm_I71_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(71)
thf(fact_289_semiring__norm_I68_J,axiom,
! [N: num] : ( ord_less_eq_num @ one @ N ) ).
% semiring_norm(68)
thf(fact_290_semiring__norm_I69_J,axiom,
! [M: num] :
~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).
% semiring_norm(69)
thf(fact_291_order__dim_I1_J,axiom,
( ( order_a_ring_ext_a_b @ r )
= ( power_power_nat @ ( ring_char_a_b @ r ) @ d ) ) ).
% order_dim(1)
thf(fact_292_dvd__antisym,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ N )
=> ( ( dvd_dvd_nat @ N @ M )
=> ( M = N ) ) ) ).
% dvd_antisym
thf(fact_293_realpow__pos__nth__unique,axiom,
! [N: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
& ( ( power_power_real @ X3 @ N )
= A )
& ! [Y2: real] :
( ( ( ord_less_real @ zero_zero_real @ Y2 )
& ( ( power_power_real @ Y2 @ N )
= A ) )
=> ( Y2 = X3 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_294_realpow__pos__nth,axiom,
! [N: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [R2: real] :
( ( ord_less_real @ zero_zero_real @ R2 )
& ( ( power_power_real @ R2 @ N )
= A ) ) ) ) ).
% realpow_pos_nth
thf(fact_295_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_296_dvd__power__same,axiom,
! [X: real,Y: real,N: nat] :
( ( dvd_dvd_real @ X @ Y )
=> ( dvd_dvd_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y @ N ) ) ) ).
% dvd_power_same
thf(fact_297_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_298_le__num__One__iff,axiom,
! [X: num] :
( ( ord_less_eq_num @ X @ one )
= ( X = one ) ) ).
% le_num_One_iff
thf(fact_299_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_300_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_301_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_302_dvd__power__le,axiom,
! [X: real,Y: real,N: nat,M: nat] :
( ( dvd_dvd_real @ X @ Y )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y @ M ) ) ) ) ).
% dvd_power_le
thf(fact_303_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_304_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_305_power__le__dvd,axiom,
! [A: real,N: nat,B: real,M: nat] :
( ( dvd_dvd_real @ ( power_power_real @ A @ N ) @ B )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ B ) ) ) ).
% power_le_dvd
thf(fact_306_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_307_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_308_le__imp__power__dvd,axiom,
! [M: nat,N: nat,A: real] :
( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ).
% le_imp_power_dvd
thf(fact_309_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_310_is__unit__power__iff,axiom,
! [A: nat,N: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ one_one_nat )
= ( ( dvd_dvd_nat @ A @ one_one_nat )
| ( N = zero_zero_nat ) ) ) ).
% is_unit_power_iff
thf(fact_311_is__unit__power__iff,axiom,
! [A: int,N: nat] :
( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ one_one_int )
= ( ( dvd_dvd_int @ A @ one_one_int )
| ( N = zero_zero_nat ) ) ) ).
% is_unit_power_iff
thf(fact_312_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_313_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_314_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_315_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_316_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_317_nat__less__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ( P2 @ M2 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ N ) ) ).
% nat_less_induct
thf(fact_318_infinite__descent,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P2 @ N3 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
& ~ ( P2 @ M2 ) ) )
=> ( P2 @ N ) ) ).
% infinite_descent
thf(fact_319_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_320_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_321_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_322_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_323_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_324_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_325_Nat_Oex__has__greatest__nat,axiom,
! [P2: nat > $o,K: nat,B: nat] :
( ( P2 @ K )
=> ( ! [Y3: nat] :
( ( P2 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X3: nat] :
( ( P2 @ X3 )
& ! [Y2: nat] :
( ( P2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_326_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_327_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_328_dvd__power,axiom,
! [N: nat,X: nat] :
( ( ( ord_less_nat @ zero_zero_nat @ N )
| ( X = one_one_nat ) )
=> ( dvd_dvd_nat @ X @ ( power_power_nat @ X @ N ) ) ) ).
% dvd_power
thf(fact_329_dvd__power,axiom,
! [N: nat,X: real] :
( ( ( ord_less_nat @ zero_zero_nat @ N )
| ( X = one_one_real ) )
=> ( dvd_dvd_real @ X @ ( power_power_real @ X @ N ) ) ) ).
% dvd_power
thf(fact_330_dvd__power,axiom,
! [N: nat,X: int] :
( ( ( ord_less_nat @ zero_zero_nat @ N )
| ( X = one_one_int ) )
=> ( dvd_dvd_int @ X @ ( power_power_int @ X @ N ) ) ) ).
% dvd_power
thf(fact_331_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_332_dvd__power__iff__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% dvd_power_iff_le
thf(fact_333_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_334_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_335_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_336_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_337_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_338_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_339_infinite__descent0,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P2 @ N3 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
& ~ ( P2 @ M2 ) ) ) )
=> ( P2 @ N ) ) ) ).
% infinite_descent0
thf(fact_340_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_341_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_342_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_343_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_344_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N4: nat] :
( ( ord_less_eq_nat @ M3 @ N4 )
& ( M3 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_345_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_346_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_347_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_348_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_349_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_350_univ__poly__zero,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a] :
( ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ R @ K2 ) )
= nil_a ) ).
% univ_poly_zero
thf(fact_351_univ__poly__zero,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a] :
( ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ R @ K2 ) )
= nil_list_a ) ).
% univ_poly_zero
thf(fact_352_ex__least__nat__le,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K3 )
=> ~ ( P2 @ I3 ) )
& ( P2 @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_353_power__le__zero__eq__numeral,axiom,
! [A: real,W: num] :
( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_real )
= ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
& ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( ord_less_eq_real @ A @ zero_zero_real ) )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( A = zero_zero_real ) ) ) ) ) ).
% power_le_zero_eq_numeral
thf(fact_354_power__le__zero__eq__numeral,axiom,
! [A: int,W: num] :
( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
= ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
& ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( ord_less_eq_int @ A @ zero_zero_int ) )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( A = zero_zero_int ) ) ) ) ) ).
% power_le_zero_eq_numeral
thf(fact_355_zero__less__power__eq__numeral,axiom,
! [A: real,W: num] :
( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) )
= ( ( ( numeral_numeral_nat @ W )
= zero_zero_nat )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( A != zero_zero_real ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).
% zero_less_power_eq_numeral
thf(fact_356_zero__less__power__eq__numeral,axiom,
! [A: int,W: num] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
= ( ( ( numeral_numeral_nat @ W )
= zero_zero_nat )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( A != zero_zero_int ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% zero_less_power_eq_numeral
thf(fact_357_power__less__zero__eq__numeral,axiom,
! [A: real,W: num] :
( ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_real )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% power_less_zero_eq_numeral
thf(fact_358_power__less__zero__eq__numeral,axiom,
! [A: int,W: num] :
( ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% power_less_zero_eq_numeral
thf(fact_359_power__less__zero__eq,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ ( power_power_real @ A @ N ) @ zero_zero_real )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% power_less_zero_eq
thf(fact_360_power__less__zero__eq,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% power_less_zero_eq
thf(fact_361_zero__le__power__eq__numeral,axiom,
! [A: real,W: num] :
( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).
% zero_le_power_eq_numeral
thf(fact_362_zero__le__power__eq__numeral,axiom,
! [A: int,W: num] :
( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% zero_le_power_eq_numeral
thf(fact_363_even__power,axiom,
! [A: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ A @ N ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% even_power
thf(fact_364_even__power,axiom,
! [A: int,N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ A @ N ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% even_power
thf(fact_365_pow__divides__pow__iff,axiom,
! [N: nat,A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
= ( dvd_dvd_nat @ A @ B ) ) ) ).
% pow_divides_pow_iff
thf(fact_366_pow__divides__pow__iff,axiom,
! [N: nat,A: int,B: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
= ( dvd_dvd_int @ A @ B ) ) ) ).
% pow_divides_pow_iff
thf(fact_367_power__le__zero__eq,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ zero_zero_real )
= ( ( ord_less_nat @ zero_zero_nat @ N )
& ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_eq_real @ A @ zero_zero_real ) )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( A = zero_zero_real ) ) ) ) ) ).
% power_le_zero_eq
thf(fact_368_power__le__zero__eq,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
= ( ( ord_less_nat @ zero_zero_nat @ N )
& ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_eq_int @ A @ zero_zero_int ) )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( A = zero_zero_int ) ) ) ) ) ).
% power_le_zero_eq
thf(fact_369_zero__less__power__eq,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
= ( ( N = zero_zero_nat )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( A != zero_zero_real ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).
% zero_less_power_eq
thf(fact_370_zero__less__power__eq,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
= ( ( N = zero_zero_nat )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( A != zero_zero_int ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% zero_less_power_eq
thf(fact_371__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062d_O_A_092_060lbrakk_062order_AR_A_061_Achar_AR_A_094_Ad_059_A0_A_060_Ad_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [D2: nat] :
( ( ( order_a_ring_ext_a_b @ r )
= ( power_power_nat @ ( ring_char_a_b @ r ) @ D2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ D2 ) ) ).
% \<open>\<And>thesis. (\<And>d. \<lbrakk>order R = char R ^ d; 0 < d\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_372_finite__field__order,axiom,
? [N3: nat] :
( ( ( order_a_ring_ext_a_b @ r )
= ( power_power_nat @ ( ring_char_a_b @ r ) @ N3 ) )
& ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).
% finite_field_order
thf(fact_373_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_374_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_375_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_376_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_377_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_378_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_379_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_380_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_381_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_382_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_383_gcd__nat_Oirrefl,axiom,
! [A: nat] :
~ ( ( dvd_dvd_nat @ A @ A )
& ( A != A ) ) ).
% gcd_nat.irrefl
thf(fact_384_gcd__nat_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [A3: nat,B2: nat] :
( ( dvd_dvd_nat @ A3 @ B2 )
& ( dvd_dvd_nat @ B2 @ A3 ) ) ) ) ).
% gcd_nat.eq_iff
thf(fact_385_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_386_gcd__nat_Orefl,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).
% gcd_nat.refl
thf(fact_387_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_388_gcd__nat_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
=> ( A = zero_zero_nat ) ) ).
% gcd_nat.extremum_uniqueI
thf(fact_389_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_390_gcd__nat_Oextremum__unique,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
= ( A = zero_zero_nat ) ) ).
% gcd_nat.extremum_unique
thf(fact_391_gcd__nat_Oextremum__strict,axiom,
! [A: nat] :
~ ( ( dvd_dvd_nat @ zero_zero_nat @ A )
& ( zero_zero_nat != A ) ) ).
% gcd_nat.extremum_strict
thf(fact_392_gcd__nat_Oextremum,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% gcd_nat.extremum
thf(fact_393_finite__field_Ofinite__field__order,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_f3414212785825285533t_unit @ R )
=> ? [N3: nat] :
( ( ( order_3240872107759947550t_unit @ R )
= ( power_power_nat @ ( ring_c500279861223467766t_unit @ R ) @ N3 ) )
& ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% finite_field.finite_field_order
thf(fact_394_finite__field_Ofinite__field__order,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_f9002592259703987221ld_a_b @ R )
=> ? [N3: nat] :
( ( ( order_a_ring_ext_a_b @ R )
= ( power_power_nat @ ( ring_char_a_b @ R ) @ N3 ) )
& ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% finite_field.finite_field_order
thf(fact_395_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_396_even__numeral,axiom,
! [N: num] : ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ).
% even_numeral
thf(fact_397_even__numeral,axiom,
! [N: num] : ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ).
% even_numeral
thf(fact_398_even__zero,axiom,
dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).
% even_zero
thf(fact_399_even__zero,axiom,
dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).
% even_zero
thf(fact_400_odd__one,axiom,
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ one_one_nat ) ).
% odd_one
thf(fact_401_odd__one,axiom,
~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ one_one_int ) ).
% odd_one
thf(fact_402_power__mono__odd,axiom,
! [N: nat,A: real,B: real] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% power_mono_odd
thf(fact_403_power__mono__odd,axiom,
! [N: nat,A: int,B: int] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% power_mono_odd
thf(fact_404_odd__pos,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% odd_pos
thf(fact_405_zero__le__power__eq,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).
% zero_le_power_eq
thf(fact_406_zero__le__power__eq,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% zero_le_power_eq
thf(fact_407_zero__le__odd__power,axiom,
! [N: nat,A: real] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ).
% zero_le_odd_power
thf(fact_408_zero__le__odd__power,axiom,
! [N: nat,A: int] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% zero_le_odd_power
thf(fact_409_zero__le__even__power,axiom,
! [N: nat,A: real] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% zero_le_even_power
thf(fact_410_zero__le__even__power,axiom,
! [N: nat,A: int] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_le_even_power
thf(fact_411_p_Obound__upD,axiom,
! [F: nat > list_a] :
( ( member_nat_list_a @ F @ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ? [N3: nat] : ( bound_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ N3 @ F ) ) ).
% p.bound_upD
thf(fact_412_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_413_freshmans__dream__ext,axiom,
! [X: a,Y: a,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( ring_char_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( add_a_b @ r @ X @ Y ) @ ( power_power_nat @ ( ring_char_a_b @ r ) @ M ) )
= ( add_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ ( power_power_nat @ ( ring_char_a_b @ r ) @ M ) ) @ ( pow_a_1026414303147256608_b_nat @ r @ Y @ ( power_power_nat @ ( ring_char_a_b @ r ) @ M ) ) ) ) ) ) ) ).
% freshmans_dream_ext
thf(fact_414_p_Opolynomial__pow__division,axiom,
! [P: list_list_a,N: nat,M: nat] :
( ( member_list_list_a @ P @ ( 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 ) ) ) ) @ P @ 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 ) ) ) ) @ P @ M ) ) ) ) ).
% p.polynomial_pow_division
thf(fact_415_bound_Ointro,axiom,
! [N: nat,F: nat > list_a,Z2: list_a] :
( ! [M5: nat] :
( ( ord_less_nat @ N @ M5 )
=> ( ( F @ M5 )
= Z2 ) )
=> ( bound_list_a @ Z2 @ N @ F ) ) ).
% bound.intro
thf(fact_416_bound_Ointro,axiom,
! [N: nat,F: nat > a,Z2: a] :
( ! [M5: nat] :
( ( ord_less_nat @ N @ M5 )
=> ( ( F @ M5 )
= Z2 ) )
=> ( bound_a @ Z2 @ N @ F ) ) ).
% bound.intro
thf(fact_417__092_060open_0620_A_060_Ad_A_K_An_092_060close_062,axiom,
ord_less_nat @ zero_zero_nat @ ( times_times_nat @ d @ n ) ).
% \<open>0 < d * n\<close>
thf(fact_418_freshmans__dream,axiom,
! [X: a,Y: a] :
( ( ord_less_nat @ zero_zero_nat @ ( ring_char_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( add_a_b @ r @ X @ Y ) @ ( ring_char_a_b @ r ) )
= ( add_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ ( ring_char_a_b @ r ) ) @ ( pow_a_1026414303147256608_b_nat @ r @ Y @ ( ring_char_a_b @ r ) ) ) ) ) ) ) ).
% freshmans_dream
thf(fact_419_a__lcomm,axiom,
! [X: a,Y: a,Z2: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ ( add_a_b @ r @ Y @ Z2 ) )
= ( add_a_b @ r @ Y @ ( add_a_b @ r @ X @ Z2 ) ) ) ) ) ) ).
% a_lcomm
thf(fact_420_a__comm,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ Y )
= ( add_a_b @ r @ Y @ X ) ) ) ) ).
% a_comm
thf(fact_421_a__assoc,axiom,
! [X: a,Y: a,Z2: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( add_a_b @ r @ X @ Y ) @ Z2 )
= ( add_a_b @ r @ X @ ( add_a_b @ r @ Y @ Z2 ) ) ) ) ) ) ).
% a_assoc
thf(fact_422_add_Or__cancel,axiom,
! [A: a,C: a,B: a] :
( ( ( add_a_b @ r @ A @ C )
= ( add_a_b @ r @ B @ C ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A = B ) ) ) ) ) ).
% add.r_cancel
thf(fact_423_add_Ol__cancel,axiom,
! [C: a,A: a,B: a] :
( ( ( add_a_b @ r @ C @ A )
= ( add_a_b @ r @ C @ B ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A = B ) ) ) ) ) ).
% add.l_cancel
thf(fact_424_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_425_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_426_p_Ozero__pdivides,axiom,
! [P: list_list_a] :
( ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ nil_list_a @ P )
= ( P = nil_list_a ) ) ).
% p.zero_pdivides
thf(fact_427_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_428_p_Oa__l__coset__subset__G,axiom,
! [H: set_list_a,X: list_a] :
( ( ord_le8861187494160871172list_a @ H @ ( 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 @ H ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.a_l_coset_subset_G
thf(fact_429_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z2: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W ) @ Z2 ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W ) ) @ Z2 ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_430_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z2: real] :
( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Z2 ) )
= ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Z2 ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_431_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z2: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Z2 ) )
= ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Z2 ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_432_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_times_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_433_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
= ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_434_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_435_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_436_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_437_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_438_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_439_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_440_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_441_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_442_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_443_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_444_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_445_a__closed,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( add_a_b @ r @ X @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% a_closed
thf(fact_446_local_Oadd_Oright__cancel,axiom,
! [X: a,Y: a,Z2: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ Y @ X )
= ( add_a_b @ r @ Z2 @ X ) )
= ( Y = Z2 ) ) ) ) ) ).
% local.add.right_cancel
thf(fact_447_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_448_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_449_mem__upI,axiom,
! [F: nat > list_list_a,R: partia2956882679547061052t_unit] :
( ! [N3: nat] : ( member_list_list_a @ ( F @ N3 ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ? [N5: nat] : ( bound_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ N5 @ F )
=> ( member8650753269014980122list_a @ F @ ( up_lis8963924889346801084t_unit @ R ) ) ) ) ).
% mem_upI
thf(fact_450_mem__upI,axiom,
! [F: nat > list_a,R: partia2670972154091845814t_unit] :
( ! [N3: nat] : ( member_list_a @ ( F @ N3 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ? [N5: nat] : ( bound_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ N5 @ F )
=> ( member_nat_list_a @ F @ ( up_lis8464167429055313730t_unit @ R ) ) ) ) ).
% mem_upI
thf(fact_451_mem__upI,axiom,
! [F: nat > set_list_a,R: partia7496981018696276118t_unit] :
( ! [N3: nat] : ( member_set_list_a @ ( F @ N3 ) @ ( partia141011252114345353t_unit @ R ) )
=> ( ? [N5: nat] : ( bound_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ N5 @ F )
=> ( member491565700723299188list_a @ F @ ( up_set529185716248919906t_unit @ R ) ) ) ) ).
% mem_upI
thf(fact_452_mem__upI,axiom,
! [F: nat > a,R: partia2175431115845679010xt_a_b] :
( ! [N3: nat] : ( member_a @ ( F @ N3 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ? [N5: nat] : ( bound_a @ ( zero_a_b @ R ) @ N5 @ F )
=> ( member_nat_a @ F @ ( up_a_b @ R ) ) ) ) ).
% mem_upI
thf(fact_453_even__mult__iff,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ A @ B ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
| ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_mult_iff
thf(fact_454_even__mult__iff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ A @ B ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
| ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_mult_iff
thf(fact_455_mem__upD,axiom,
! [F: nat > list_a,R: partia2670972154091845814t_unit,N: nat] :
( ( member_nat_list_a @ F @ ( up_lis8464167429055313730t_unit @ R ) )
=> ( member_list_a @ ( F @ N ) @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% mem_upD
thf(fact_456_mem__upD,axiom,
! [F: nat > set_list_a,R: partia7496981018696276118t_unit,N: nat] :
( ( member491565700723299188list_a @ F @ ( up_set529185716248919906t_unit @ R ) )
=> ( member_set_list_a @ ( F @ N ) @ ( partia141011252114345353t_unit @ R ) ) ) ).
% mem_upD
thf(fact_457_mem__upD,axiom,
! [F: nat > a,R: partia2175431115845679010xt_a_b,N: nat] :
( ( member_nat_a @ F @ ( up_a_b @ R ) )
=> ( member_a @ ( F @ N ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% mem_upD
thf(fact_458_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_459_power__commuting__commutes,axiom,
! [X: real,Y: real,N: nat] :
( ( ( times_times_real @ X @ Y )
= ( times_times_real @ Y @ X ) )
=> ( ( times_times_real @ ( power_power_real @ X @ N ) @ Y )
= ( times_times_real @ Y @ ( power_power_real @ X @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_460_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_461_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_462_power__mult__distrib,axiom,
! [A: real,B: real,N: nat] :
( ( power_power_real @ ( times_times_real @ A @ B ) @ N )
= ( times_times_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_463_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_464_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_465_power__commutes,axiom,
! [A: real,N: nat] :
( ( times_times_real @ ( power_power_real @ A @ N ) @ A )
= ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).
% power_commutes
thf(fact_466_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_467_division__decomp,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
=> ? [B3: nat,C2: nat] :
( ( A
= ( times_times_nat @ B3 @ C2 ) )
& ( dvd_dvd_nat @ B3 @ B )
& ( dvd_dvd_nat @ C2 @ C ) ) ) ).
% division_decomp
thf(fact_468_division__decomp,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
=> ? [B3: int,C2: int] :
( ( A
= ( times_times_int @ B3 @ C2 ) )
& ( dvd_dvd_int @ B3 @ B )
& ( dvd_dvd_int @ C2 @ C ) ) ) ).
% division_decomp
thf(fact_469_dvd__productE,axiom,
! [P: nat,A: nat,B: nat] :
( ( dvd_dvd_nat @ P @ ( times_times_nat @ A @ B ) )
=> ~ ! [X3: nat,Y3: nat] :
( ( P
= ( times_times_nat @ X3 @ Y3 ) )
=> ( ( dvd_dvd_nat @ X3 @ A )
=> ~ ( dvd_dvd_nat @ Y3 @ B ) ) ) ) ).
% dvd_productE
thf(fact_470_dvd__productE,axiom,
! [P: int,A: int,B: int] :
( ( dvd_dvd_int @ P @ ( times_times_int @ A @ B ) )
=> ~ ! [X3: int,Y3: int] :
( ( P
= ( times_times_int @ X3 @ Y3 ) )
=> ( ( dvd_dvd_int @ X3 @ A )
=> ~ ( dvd_dvd_int @ Y3 @ B ) ) ) ) ).
% dvd_productE
thf(fact_471_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_472_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_473_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_474_power__mult,axiom,
! [A: real,M: nat,N: nat] :
( ( power_power_real @ A @ ( times_times_nat @ M @ N ) )
= ( power_power_real @ ( power_power_real @ A @ M ) @ N ) ) ).
% power_mult
thf(fact_475_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_476_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_477_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_478_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_479_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_480_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_481_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_482_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_483_mult__numeral__1__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_484_mult__numeral__1__right,axiom,
! [A: real] :
( ( times_times_real @ A @ ( numeral_numeral_real @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_485_mult__numeral__1__right,axiom,
! [A: int] :
( ( times_times_int @ A @ ( numeral_numeral_int @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_486_mult__numeral__1,axiom,
! [A: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_487_mult__numeral__1,axiom,
! [A: real] :
( ( times_times_real @ ( numeral_numeral_real @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_488_mult__numeral__1,axiom,
! [A: int] :
( ( times_times_int @ ( numeral_numeral_int @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_489_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_490_left__right__inverse__power,axiom,
! [X: real,Y: real,N: nat] :
( ( ( times_times_real @ X @ Y )
= one_one_real )
=> ( ( times_times_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y @ N ) )
= one_one_real ) ) ).
% left_right_inverse_power
thf(fact_491_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_492_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_493_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_494_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_495_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_496_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_497_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_498_power__gt1__lemma,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_499_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_500_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_501_power__less__power__Suc,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ ( power_power_real @ A @ N ) @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_502_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_503_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_504_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_505_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_506_power__Suc__less,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) @ ( power_power_nat @ A @ N ) ) ) ) ).
% power_Suc_less
thf(fact_507_power__Suc__less,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ A @ one_one_real )
=> ( ord_less_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) @ ( power_power_real @ A @ N ) ) ) ) ).
% power_Suc_less
thf(fact_508_power__Suc__less,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) @ ( power_power_int @ A @ N ) ) ) ) ).
% power_Suc_less
thf(fact_509_evenE,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ~ ! [B4: nat] :
( A
!= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B4 ) ) ) ).
% evenE
thf(fact_510_evenE,axiom,
! [A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ~ ! [B4: int] :
( A
!= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B4 ) ) ) ).
% evenE
thf(fact_511_power2__eq__square,axiom,
! [A: nat] :
( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_nat @ A @ A ) ) ).
% power2_eq_square
thf(fact_512_power2__eq__square,axiom,
! [A: real] :
( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_real @ A @ A ) ) ).
% power2_eq_square
thf(fact_513_power2__eq__square,axiom,
! [A: int] :
( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_int @ A @ A ) ) ).
% power2_eq_square
thf(fact_514_power4__eq__xxxx,axiom,
! [X: nat] :
( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( times_times_nat @ ( times_times_nat @ ( times_times_nat @ X @ X ) @ X ) @ X ) ) ).
% power4_eq_xxxx
thf(fact_515_power4__eq__xxxx,axiom,
! [X: real] :
( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( times_times_real @ ( times_times_real @ ( times_times_real @ X @ X ) @ X ) @ X ) ) ).
% power4_eq_xxxx
thf(fact_516_power4__eq__xxxx,axiom,
! [X: int] :
( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( times_times_int @ ( times_times_int @ ( times_times_int @ X @ X ) @ X ) @ X ) ) ).
% power4_eq_xxxx
thf(fact_517_power__even__eq,axiom,
! [A: nat,N: nat] :
( ( power_power_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_power_nat @ ( power_power_nat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power_even_eq
thf(fact_518_power__even__eq,axiom,
! [A: real,N: nat] :
( ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_power_real @ ( power_power_real @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power_even_eq
thf(fact_519_power__even__eq,axiom,
! [A: int,N: nat] :
( ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_power_int @ ( power_power_int @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power_even_eq
thf(fact_520_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_521_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_522_zero__le__even__power_H,axiom,
! [A: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% zero_le_even_power'
thf(fact_523_zero__le__even__power_H,axiom,
! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% zero_le_even_power'
thf(fact_524_bound_Obound,axiom,
! [Z2: list_a,N: nat,F: nat > list_a,M: nat] :
( ( bound_list_a @ Z2 @ N @ F )
=> ( ( ord_less_nat @ N @ M )
=> ( ( F @ M )
= Z2 ) ) ) ).
% bound.bound
thf(fact_525_bound_Obound,axiom,
! [Z2: a,N: nat,F: nat > a,M: nat] :
( ( bound_a @ Z2 @ N @ F )
=> ( ( ord_less_nat @ N @ M )
=> ( ( F @ M )
= Z2 ) ) ) ).
% bound.bound
thf(fact_526_bound__def,axiom,
( bound_list_a
= ( ^ [Z3: list_a,N4: nat,F2: nat > list_a] :
! [M3: nat] :
( ( ord_less_nat @ N4 @ M3 )
=> ( ( F2 @ M3 )
= Z3 ) ) ) ) ).
% bound_def
thf(fact_527_bound__def,axiom,
( bound_a
= ( ^ [Z3: a,N4: nat,F2: nat > a] :
! [M3: nat] :
( ( ord_less_nat @ N4 @ M3 )
=> ( ( F2 @ M3 )
= Z3 ) ) ) ) ).
% bound_def
thf(fact_528_bound__below,axiom,
! [Z2: list_a,M: nat,F: nat > list_a,N: nat] :
( ( bound_list_a @ Z2 @ M @ F )
=> ( ( ( F @ N )
!= Z2 )
=> ( ord_less_eq_nat @ N @ M ) ) ) ).
% bound_below
thf(fact_529_bound__below,axiom,
! [Z2: a,M: nat,F: nat > a,N: nat] :
( ( bound_a @ Z2 @ M @ F )
=> ( ( ( F @ N )
!= Z2 )
=> ( ord_less_eq_nat @ N @ M ) ) ) ).
% bound_below
thf(fact_530_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_531_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_532_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_533_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_534_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_535_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_536_dvd__mult__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( dvd_dvd_real @ A @ B ) ) ) ).
% dvd_mult_cancel_right
thf(fact_537_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_538_dvd__mult__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( dvd_dvd_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( dvd_dvd_real @ A @ B ) ) ) ).
% dvd_mult_cancel_left
thf(fact_539_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_540_mult__cancel__left1,axiom,
! [C: real,B: real] :
( ( C
= ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_left1
thf(fact_541_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_542_mult__cancel__left2,axiom,
! [C: real,A: real] :
( ( ( times_times_real @ C @ A )
= C )
= ( ( C = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_left2
thf(fact_543_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_544_bound__upD,axiom,
! [F: nat > a] :
( ( member_nat_a @ F @ ( up_a_b @ r ) )
=> ? [N3: nat] : ( bound_a @ ( zero_a_b @ r ) @ N3 @ F ) ) ).
% bound_upD
thf(fact_545_boundD__carrier,axiom,
! [N: nat,F: nat > a,M: nat] :
( ( bound_a @ ( zero_a_b @ r ) @ N @ F )
=> ( ( ord_less_nat @ N @ M )
=> ( member_a @ ( F @ M ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% boundD_carrier
thf(fact_546_add_Oinv__comm,axiom,
! [X: a,Y: a] :
( ( ( add_a_b @ r @ X @ Y )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ Y @ X )
= ( zero_a_b @ r ) ) ) ) ) ).
% add.inv_comm
thf(fact_547_add_Ol__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( add_a_b @ r @ X3 @ X )
= ( zero_a_b @ r ) ) ) ) ).
% add.l_inv_ex
thf(fact_548_add_Oone__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ U @ X3 )
= X3 ) )
=> ( U
= ( zero_a_b @ r ) ) ) ) ).
% add.one_unique
thf(fact_549_add_Or__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( add_a_b @ r @ X @ X3 )
= ( zero_a_b @ r ) ) ) ) ).
% add.r_inv_ex
thf(fact_550_local_Ominus__unique,axiom,
! [Y: a,X: a,Y5: a] :
( ( ( add_a_b @ r @ Y @ X )
= ( zero_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X @ Y5 )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y5 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( Y = Y5 ) ) ) ) ) ) ).
% local.minus_unique
thf(fact_551_pow__non__zero,axiom,
! [X: a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X
!= ( zero_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ X @ N )
!= ( zero_a_b @ r ) ) ) ) ).
% pow_non_zero
thf(fact_552_nat__pow__zero,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( zero_a_b @ r ) @ N )
= ( zero_a_b @ r ) ) ) ).
% nat_pow_zero
thf(fact_553_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_554_mult__zero__left,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_555_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_556_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_557_mult__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_558_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_559_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_560_mult__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_561_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_562_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_563_mult__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_564_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_565_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_566_mult__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ( times_times_real @ A @ C )
= ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_567_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_568_dvd__0__left__iff,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
= ( A = zero_zero_nat ) ) ).
% dvd_0_left_iff
thf(fact_569_dvd__0__left__iff,axiom,
! [A: real] :
( ( dvd_dvd_real @ zero_zero_real @ A )
= ( A = zero_zero_real ) ) ).
% dvd_0_left_iff
thf(fact_570_dvd__0__left__iff,axiom,
! [A: int] :
( ( dvd_dvd_int @ zero_zero_int @ A )
= ( A = zero_zero_int ) ) ).
% dvd_0_left_iff
thf(fact_571_dvd__0__right,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% dvd_0_right
thf(fact_572_dvd__0__right,axiom,
! [A: real] : ( dvd_dvd_real @ A @ zero_zero_real ) ).
% dvd_0_right
thf(fact_573_dvd__0__right,axiom,
! [A: int] : ( dvd_dvd_int @ A @ zero_zero_int ) ).
% dvd_0_right
thf(fact_574_not__real__square__gt__zero,axiom,
! [X: real] :
( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) ) )
= ( X = zero_zero_real ) ) ).
% not_real_square_gt_zero
thf(fact_575_semiring__norm_I13_J,axiom,
! [M: num,N: num] :
( ( times_times_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ).
% semiring_norm(13)
thf(fact_576_semiring__norm_I12_J,axiom,
! [N: num] :
( ( times_times_num @ one @ N )
= N ) ).
% semiring_norm(12)
thf(fact_577_semiring__norm_I11_J,axiom,
! [M: num] :
( ( times_times_num @ M @ one )
= M ) ).
% semiring_norm(11)
thf(fact_578_zero__closed,axiom,
member_a @ ( zero_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% zero_closed
thf(fact_579_mult__cancel__right2,axiom,
! [A: real,C: real] :
( ( ( times_times_real @ A @ C )
= C )
= ( ( C = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_right2
thf(fact_580_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_581_mult__cancel__right1,axiom,
! [C: real,B: real] :
( ( C
= ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_right1
thf(fact_582_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_583_num__double,axiom,
! [N: num] :
( ( times_times_num @ ( bit0 @ one ) @ N )
= ( bit0 @ N ) ) ).
% num_double
thf(fact_584_power__mult__numeral,axiom,
! [A: nat,M: num,N: num] :
( ( power_power_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
= ( power_power_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% power_mult_numeral
thf(fact_585_power__mult__numeral,axiom,
! [A: real,M: num,N: num] :
( ( power_power_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
= ( power_power_real @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% power_mult_numeral
thf(fact_586_power__mult__numeral,axiom,
! [A: int,M: num,N: num] :
( ( power_power_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
= ( power_power_int @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% power_mult_numeral
thf(fact_587_add_Ol__cancel__one,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X @ A )
= X )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.l_cancel_one
thf(fact_588_add_Ol__cancel__one_H,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X
= ( add_a_b @ r @ X @ A ) )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.l_cancel_one'
thf(fact_589_add_Or__cancel__one,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ A @ X )
= X )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.r_cancel_one
thf(fact_590_add_Or__cancel__one_H,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X
= ( add_a_b @ r @ A @ X ) )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.r_cancel_one'
thf(fact_591_l__zero,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( zero_a_b @ r ) @ X )
= X ) ) ).
% l_zero
thf(fact_592_r__zero,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ ( zero_a_b @ r ) )
= X ) ) ).
% r_zero
thf(fact_593_linorder__neqE__linordered__idom,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_594_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_595_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_596_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_597_dvd__refl,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).
% dvd_refl
thf(fact_598_dvd__refl,axiom,
! [A: int] : ( dvd_dvd_int @ A @ A ) ).
% dvd_refl
thf(fact_599_four__x__squared,axiom,
! [X: real] :
( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( power_power_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% four_x_squared
thf(fact_600_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_601_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_602_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_603_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_604_mult__right__cancel,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A @ C )
= ( times_times_real @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_605_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_606_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_607_mult__left__cancel,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_608_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_609_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_610_no__zero__divisors,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( times_times_real @ A @ B )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_611_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_612_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_613_divisors__zero,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
=> ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_614_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_615_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_616_mult__not__zero,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
!= zero_zero_real )
=> ( ( A != zero_zero_real )
& ( B != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_617_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_618_dvd__0__left,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
=> ( A = zero_zero_nat ) ) ).
% dvd_0_left
thf(fact_619_dvd__0__left,axiom,
! [A: real] :
( ( dvd_dvd_real @ zero_zero_real @ A )
=> ( A = zero_zero_real ) ) ).
% dvd_0_left
thf(fact_620_dvd__0__left,axiom,
! [A: int] :
( ( dvd_dvd_int @ zero_zero_int @ A )
=> ( A = zero_zero_int ) ) ).
% dvd_0_left
thf(fact_621_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_622_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_623_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_624_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_625_one__dvd,axiom,
! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).
% one_dvd
thf(fact_626_one__dvd,axiom,
! [A: int] : ( dvd_dvd_int @ one_one_int @ A ) ).
% one_dvd
thf(fact_627_one__dvd,axiom,
! [A: real] : ( dvd_dvd_real @ one_one_real @ A ) ).
% one_dvd
thf(fact_628_dvd__triv__right,axiom,
! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ A ) ) ).
% dvd_triv_right
thf(fact_629_dvd__triv__right,axiom,
! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ B @ A ) ) ).
% dvd_triv_right
thf(fact_630_dvd__triv__right,axiom,
! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ B @ A ) ) ).
% dvd_triv_right
thf(fact_631_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_632_dvd__mult__right,axiom,
! [A: real,B: real,C: real] :
( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C )
=> ( dvd_dvd_real @ B @ C ) ) ).
% dvd_mult_right
thf(fact_633_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_634_mult__dvd__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ C @ D )
=> ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_635_mult__dvd__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( dvd_dvd_real @ A @ B )
=> ( ( dvd_dvd_real @ C @ D )
=> ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_636_mult__dvd__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ C @ D )
=> ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_637_dvd__triv__left,axiom,
! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ A @ B ) ) ).
% dvd_triv_left
thf(fact_638_dvd__triv__left,axiom,
! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ A @ B ) ) ).
% dvd_triv_left
thf(fact_639_dvd__triv__left,axiom,
! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ A @ B ) ) ).
% dvd_triv_left
thf(fact_640_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_641_dvd__mult__left,axiom,
! [A: real,B: real,C: real] :
( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C )
=> ( dvd_dvd_real @ A @ C ) ) ).
% dvd_mult_left
thf(fact_642_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_643_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_644_dvd__mult2,axiom,
! [A: real,B: real,C: real] :
( ( dvd_dvd_real @ A @ B )
=> ( dvd_dvd_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% dvd_mult2
thf(fact_645_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_646_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_647_dvd__mult,axiom,
! [A: real,C: real,B: real] :
( ( dvd_dvd_real @ A @ C )
=> ( dvd_dvd_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% dvd_mult
thf(fact_648_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_649_dvd__def,axiom,
( dvd_dvd_nat
= ( ^ [B2: nat,A3: nat] :
? [K4: nat] :
( A3
= ( times_times_nat @ B2 @ K4 ) ) ) ) ).
% dvd_def
thf(fact_650_dvd__def,axiom,
( dvd_dvd_real
= ( ^ [B2: real,A3: real] :
? [K4: real] :
( A3
= ( times_times_real @ B2 @ K4 ) ) ) ) ).
% dvd_def
thf(fact_651_dvd__def,axiom,
( dvd_dvd_int
= ( ^ [B2: int,A3: int] :
? [K4: int] :
( A3
= ( times_times_int @ B2 @ K4 ) ) ) ) ).
% dvd_def
thf(fact_652_dvdI,axiom,
! [A: nat,B: nat,K: nat] :
( ( A
= ( times_times_nat @ B @ K ) )
=> ( dvd_dvd_nat @ B @ A ) ) ).
% dvdI
thf(fact_653_dvdI,axiom,
! [A: real,B: real,K: real] :
( ( A
= ( times_times_real @ B @ K ) )
=> ( dvd_dvd_real @ B @ A ) ) ).
% dvdI
thf(fact_654_dvdI,axiom,
! [A: int,B: int,K: int] :
( ( A
= ( times_times_int @ B @ K ) )
=> ( dvd_dvd_int @ B @ A ) ) ).
% dvdI
thf(fact_655_dvdE,axiom,
! [B: nat,A: nat] :
( ( dvd_dvd_nat @ B @ A )
=> ~ ! [K3: nat] :
( A
!= ( times_times_nat @ B @ K3 ) ) ) ).
% dvdE
thf(fact_656_dvdE,axiom,
! [B: real,A: real] :
( ( dvd_dvd_real @ B @ A )
=> ~ ! [K3: real] :
( A
!= ( times_times_real @ B @ K3 ) ) ) ).
% dvdE
thf(fact_657_dvdE,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ A )
=> ~ ! [K3: int] :
( A
!= ( times_times_int @ B @ K3 ) ) ) ).
% dvdE
thf(fact_658_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_659_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_660_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_661_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_662_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_663_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_664_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% zero_less_one_class.zero_le_one
thf(fact_665_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_666_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_667_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_668_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_669_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_670_zero__le__mult__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% zero_le_mult_iff
thf(fact_671_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_672_mult__nonneg__nonpos2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_673_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_674_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_675_mult__nonpos__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_nonpos_nonneg
thf(fact_676_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_677_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_678_mult__nonneg__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos
thf(fact_679_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_680_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_681_mult__nonneg__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_682_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_683_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_684_split__mult__neg__le,axiom,
! [A: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ).
% split_mult_neg_le
thf(fact_685_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_686_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_687_mult__le__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_688_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_689_mult__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_690_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_691_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_692_mult__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_693_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_694_mult__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_695_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_696_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_697_mult__nonpos__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_698_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_699_mult__left__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_700_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_701_split__mult__pos__le,axiom,
! [A: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_702_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_703_zero__le__square,axiom,
! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% zero_le_square
thf(fact_704_zero__le__square,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% zero_le_square
thf(fact_705_mult__mono_H,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_706_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ 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 @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_707_mult__mono_H,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( 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 @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_708_mult__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_709_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ 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 @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_710_mult__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( 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 @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_711_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_712_zero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one
thf(fact_713_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_714_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_715_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_716_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_717_mult__neg__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_718_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_719_not__square__less__zero,axiom,
! [A: real] :
~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).
% not_square_less_zero
thf(fact_720_not__square__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_721_mult__less__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B @ zero_zero_real ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_722_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_723_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_724_mult__neg__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_neg_pos
thf(fact_725_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_726_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_727_mult__pos__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_pos_neg
thf(fact_728_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_729_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_730_mult__pos__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_731_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_732_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_733_mult__pos__neg2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% mult_pos_neg2
thf(fact_734_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_735_zero__less__mult__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ zero_zero_real @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% zero_less_mult_iff
thf(fact_736_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_737_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_738_zero__less__mult__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_739_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_740_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_741_zero__less__mult__pos2,axiom,
! [B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B @ A ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_742_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_743_mult__less__cancel__left__neg,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_real @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_744_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_745_mult__less__cancel__left__pos,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_real @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_746_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_747_mult__strict__left__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_748_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_749_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_750_mult__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_751_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_752_mult__less__cancel__left__disj,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_753_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_754_mult__strict__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_755_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_756_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_757_mult__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_758_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_759_mult__less__cancel__right__disj,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_760_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_761_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_762_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_763_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_764_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_765_less__1__mult,axiom,
! [M: real,N: real] :
( ( ord_less_real @ one_one_real @ M )
=> ( ( ord_less_real @ one_one_real @ N )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_766_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_767_not__is__unit__0,axiom,
~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).
% not_is_unit_0
thf(fact_768_not__is__unit__0,axiom,
~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).
% not_is_unit_0
thf(fact_769_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_770_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_771_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_772_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_773_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_774_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_775_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_776_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_777_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_778_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_779_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_780_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_781_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_782_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_783_mult__le__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_784_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_785_mult__le__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_786_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_787_mult__left__less__imp__less,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_788_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_789_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_790_mult__strict__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_791_mult__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( 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 @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_792_mult__strict__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( 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 @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_793_mult__less__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_794_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_795_mult__right__less__imp__less,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_796_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_797_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_798_mult__strict__mono_H,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_799_mult__strict__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( 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 @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_800_mult__strict__mono_H,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( 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 @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_801_mult__less__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_802_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_803_mult__le__cancel__left__neg,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_804_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_805_mult__le__cancel__left__pos,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_806_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_807_mult__left__le__imp__le,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_808_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_809_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_810_mult__right__le__imp__le,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_811_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_812_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_813_mult__le__less__imp__less,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_814_mult__le__less__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( 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 @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_815_mult__le__less__imp__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( 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 @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_816_mult__less__le__imp__less,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_817_mult__less__le__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( 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 @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_818_mult__less__le__imp__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( 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 @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_819_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_820_ring__primeE_I1_J,axiom,
! [P: a] :
( ( member_a @ P @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_ring_prime_a_b @ r @ P )
=> ( P
!= ( zero_a_b @ r ) ) ) ) ).
% ring_primeE(1)
thf(fact_821_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_822_p_Ocarrier__is__subalgebra,axiom,
! [K2: set_list_a] :
( ( ord_le8861187494160871172list_a @ K2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( embedd1768981623711841426t_unit @ K2 @ ( 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_823_p_Osubalgebra__in__carrier,axiom,
! [K2: set_list_a,V2: set_list_a] :
( ( embedd1768981623711841426t_unit @ K2 @ 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_824_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_825_p_Ogenideal__self,axiom,
! [S2: set_list_a] :
( ( ord_le8861187494160871172list_a @ S2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ S2 @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ S2 ) ) ) ).
% p.genideal_self
thf(fact_826_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_827_p_Oadd_Om__lcomm,axiom,
! [X: list_a,Y: list_a,Z2: 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 @ Z2 @ ( 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 @ Z2 ) )
= ( 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 @ Z2 ) ) ) ) ) ) ).
% p.add.m_lcomm
thf(fact_828_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_829_p_Oadd_Om__assoc,axiom,
! [X: list_a,Y: list_a,Z2: 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 @ Z2 @ ( 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 ) @ Z2 )
= ( 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 @ Z2 ) ) ) ) ) ) ).
% p.add.m_assoc
thf(fact_830_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_831_p_Ominus__unique,axiom,
! [Y: list_a,X: list_a,Y5: 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 @ Y5 )
= ( 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 @ Y5 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( Y = Y5 ) ) ) ) ) ) ).
% p.minus_unique
thf(fact_832_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_833_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_834_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_835_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_836_p_Osubset__Idl__subset,axiom,
! [I4: set_list_a,H: set_list_a] :
( ( ord_le8861187494160871172list_a @ I4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ H @ I4 )
=> ( ord_le8861187494160871172list_a @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H ) @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I4 ) ) ) ) ).
% p.subset_Idl_subset
thf(fact_837_p_Oa__lcos__m__assoc,axiom,
! [M4: set_list_a,G: list_a,H2: 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 @ H2 @ ( 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 ) ) @ H2 @ 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 @ H2 ) @ M4 ) ) ) ) ) ).
% p.a_lcos_m_assoc
thf(fact_838_p_Oadd_Oright__cancel,axiom,
! [X: list_a,Y: list_a,Z2: 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 @ Z2 @ ( 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 ) ) @ Z2 @ X ) )
= ( Y = Z2 ) ) ) ) ) ).
% p.add.right_cancel
thf(fact_839_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_840_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_841_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_842_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_843_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_844_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_845_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_846_primeness__condition,axiom,
! [P: a] :
( ( member_a @ P @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ P )
= ( ring_ring_prime_a_b @ r @ P ) ) ) ).
% primeness_condition
thf(fact_847_ring__primeI,axiom,
! [P: a] :
( ( P
!= ( zero_a_b @ r ) )
=> ( ( prime_a_ring_ext_a_b @ r @ P )
=> ( ring_ring_prime_a_b @ r @ P ) ) ) ).
% ring_primeI
thf(fact_848_ring__primeE_I3_J,axiom,
! [P: a] :
( ( member_a @ P @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_ring_prime_a_b @ r @ P )
=> ( prime_a_ring_ext_a_b @ r @ P ) ) ) ).
% ring_primeE(3)
thf(fact_849_ring__irreducibleE_I1_J,axiom,
! [R3: a] :
( ( member_a @ R3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R3 )
=> ( R3
!= ( zero_a_b @ r ) ) ) ) ).
% ring_irreducibleE(1)
thf(fact_850_zero__is__prime_I1_J,axiom,
prime_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) ).
% zero_is_prime(1)
thf(fact_851_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_852_p_Oline__extension__in__carrier,axiom,
! [K2: set_list_a,A: list_a,E: set_list_a] :
( ( ord_le8861187494160871172list_a @ K2 @ ( 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 ) ) @ K2 @ A @ E ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% p.line_extension_in_carrier
thf(fact_853_char__dvd__order,axiom,
dvd_dvd_int @ ( semiri1314217659103216013at_int @ ( ring_char_a_b @ r ) ) @ ( semiri1314217659103216013at_int @ ( power_power_nat @ ( order_a_ring_ext_a_b @ r ) @ n ) ) ).
% char_dvd_order
thf(fact_854_p_Osubring__polynomial__pow__not__zero,axiom,
! [K2: set_list_a,P: list_list_a,N: nat] :
( ( subrin6918843898125473962t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( P != nil_list_a )
=> ( ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ P @ N )
!= nil_list_a ) ) ) ) ).
% p.subring_polynomial_pow_not_zero
thf(fact_855_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_856_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_857_p_Opdivides__zero,axiom,
! [K2: set_list_a,P: list_list_a] :
( ( subrin6918843898125473962t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ nil_list_a ) ) ) ).
% p.pdivides_zero
thf(fact_858_p_Ocarrier__polynomial__shell,axiom,
! [K2: set_list_a,P: list_list_a] :
( ( subrin6918843898125473962t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( member_list_list_a @ P @ ( 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_859_p_Ouniv__poly__a__minus__consistent,axiom,
! [K2: set_list_a,Q: list_list_a,P: list_list_a] :
( ( subrin6918843898125473962t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ P @ Q )
= ( 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 ) ) ) ) @ P @ Q ) ) ) ) ).
% p.univ_poly_a_minus_consistent
thf(fact_860_int__dvd__int__iff,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( dvd_dvd_nat @ M @ N ) ) ).
% int_dvd_int_iff
thf(fact_861_p_OsubdomainI_H,axiom,
! [H: set_list_a] :
( ( subrin6918843898125473962t_unit @ H @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( subdom7821232466298058046t_unit @ H @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% p.subdomainI'
thf(fact_862_p_OpirreducibleE_I1_J,axiom,
! [K2: set_list_a,P: list_list_a] :
( ( subrin6918843898125473962t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ P )
=> ( P != nil_list_a ) ) ) ) ).
% p.pirreducibleE(1)
thf(fact_863_finite__carrier,axiom,
finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) ).
% finite_carrier
thf(fact_864_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_865_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_866_int__eq__iff__numeral,axiom,
! [M: nat,V: num] :
( ( ( semiri1314217659103216013at_int @ M )
= ( numeral_numeral_int @ V ) )
= ( M
= ( numeral_numeral_nat @ V ) ) ) ).
% int_eq_iff_numeral
thf(fact_867_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_868_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( K
!= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% nonneg_int_cases
thf(fact_869_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_870_zdvd__imp__le,axiom,
! [Z2: int,N: int] :
( ( dvd_dvd_int @ Z2 @ N )
=> ( ( ord_less_int @ zero_zero_int @ N )
=> ( ord_less_eq_int @ Z2 @ N ) ) ) ).
% zdvd_imp_le
thf(fact_871_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_872_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% zle_int
thf(fact_873_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_874_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% pos_int_cases
thf(fact_875_zmult__zless__mono2__lemma,axiom,
! [I: int,J: int,K: nat] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_876_real__of__nat__less__numeral__iff,axiom,
! [N: nat,W: num] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( numeral_numeral_real @ W ) )
= ( ord_less_nat @ N @ ( numeral_numeral_nat @ W ) ) ) ).
% real_of_nat_less_numeral_iff
thf(fact_877_numeral__less__real__of__nat__iff,axiom,
! [W: num,N: nat] :
( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ ( numeral_numeral_nat @ W ) @ N ) ) ).
% numeral_less_real_of_nat_iff
thf(fact_878_numeral__le__real__of__nat__iff,axiom,
! [N: num,M: nat] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( semiri5074537144036343181t_real @ M ) )
= ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ M ) ) ).
% numeral_le_real_of_nat_iff
thf(fact_879_p_Ochar__bound_I1_J,axiom,
! [X: nat] :
( ( ord_less_nat @ zero_zero_nat @ X )
=> ( ( ( ring_i7845631192508594762t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( semiri1314217659103216013at_int @ X ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_less_eq_nat @ ( ring_c500279861223467766t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X ) ) ) ).
% p.char_bound(1)
thf(fact_880_p_Oint__embed__closed,axiom,
! [K: int] : ( member_list_a @ ( ring_i7845631192508594762t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% p.int_embed_closed
thf(fact_881_p_Oint__embed__range,axiom,
! [K2: set_list_a,K: int] :
( ( subrin6918843898125473962t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( member_list_a @ ( ring_i7845631192508594762t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ K2 ) ) ).
% p.int_embed_range
thf(fact_882_p_Oint__embed__zero,axiom,
( ( ring_i7845631192508594762t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ zero_zero_int )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% p.int_embed_zero
thf(fact_883_p_Oembed__char__eq__0,axiom,
( ( ring_i7845631192508594762t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( semiri1314217659103216013at_int @ ( ring_c500279861223467766t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% p.embed_char_eq_0
thf(fact_884_p_Oembed__char__eq__0__iff,axiom,
! [N: int] :
( ( ( ring_i7845631192508594762t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ ( ring_c500279861223467766t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ N ) ) ).
% p.embed_char_eq_0_iff
thf(fact_885_p_Ochar__bound_I2_J,axiom,
! [X: nat] :
( ( ord_less_nat @ zero_zero_nat @ X )
=> ( ( ( ring_i7845631192508594762t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( semiri1314217659103216013at_int @ X ) )
= ( zero_l4142658623432671053t_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.char_bound(2)
thf(fact_886_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_887_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_888_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_889_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_890_int__one__le__iff__zero__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ one_one_int @ Z2 )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% int_one_le_iff_zero_less
thf(fact_891_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_892_complete__real,axiom,
! [S2: set_real] :
( ? [X4: real] : ( member_real @ X4 @ S2 )
=> ( ? [Z4: real] :
! [X3: real] :
( ( member_real @ X3 @ S2 )
=> ( ord_less_eq_real @ X3 @ Z4 ) )
=> ? [Y3: real] :
( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ord_less_eq_real @ X4 @ Y3 ) )
& ! [Z4: real] :
( ! [X3: real] :
( ( member_real @ X3 @ S2 )
=> ( ord_less_eq_real @ X3 @ Z4 ) )
=> ( ord_less_eq_real @ Y3 @ Z4 ) ) ) ) ) ).
% complete_real
thf(fact_893_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y6: real] :
( ( ord_less_real @ X2 @ Y6 )
| ( X2 = Y6 ) ) ) ) ).
% less_eq_real_def
thf(fact_894_real__arch__pow,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ one_one_real @ X )
=> ? [N3: nat] : ( ord_less_real @ Y @ ( power_power_real @ X @ N3 ) ) ) ).
% real_arch_pow
thf(fact_895_reals__Archimedean3,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ! [Y2: real] :
? [N3: nat] : ( ord_less_real @ Y2 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X ) ) ) ).
% reals_Archimedean3
thf(fact_896_real__arch__pow__inv,axiom,
! [Y: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ X @ one_one_real )
=> ? [N3: nat] : ( ord_less_real @ ( power_power_real @ X @ N3 ) @ Y ) ) ) ).
% real_arch_pow_inv
thf(fact_897_real__archimedian__rdiv__eq__0,axiom,
! [X: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ! [M5: nat] :
( ( ord_less_nat @ zero_zero_nat @ M5 )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M5 ) @ X ) @ C ) )
=> ( X = zero_zero_real ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_898_two__realpow__ge__one,axiom,
! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).
% two_realpow_ge_one
thf(fact_899_p_Opderiv__carr,axiom,
! [K2: set_list_a,F: list_list_a] :
( ( subrin6918843898125473962t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ F @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( member_list_list_a @ ( formal6075833236969493044t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) ) ) ) ).
% p.pderiv_carr
thf(fact_900_p_Onoetherian__ringI,axiom,
( ! [I5: set_list_a] :
( ( ideal_8896367198367571637t_unit @ I5 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ? [A4: set_list_a] :
( ( ord_le8861187494160871172list_a @ A4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( finite_finite_list_a @ A4 )
& ( I5
= ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A4 ) ) ) )
=> ( ring_n5188127996776581661t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% p.noetherian_ringI
thf(fact_901_p_Oconst__term__simprules__shell_I1_J,axiom,
! [K2: set_list_a,P: list_list_a] :
( ( subrin6918843898125473962t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( member_list_a @ ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) @ K2 ) ) ) ).
% p.const_term_simprules_shell(1)
thf(fact_902_p_Oring__irreducibleE_I1_J,axiom,
! [R3: list_a] :
( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ R3 )
=> ( R3
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.ring_irreducibleE(1)
thf(fact_903_p_Ooneideal,axiom,
ideal_8896367198367571637t_unit @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% p.oneideal
thf(fact_904_p_Oprimeness__condition,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
= ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) ) ).
% p.primeness_condition
thf(fact_905_p_Opderiv__zero,axiom,
! [K2: 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 ) ) @ K2 ) ) )
= ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) ) ).
% p.pderiv_zero
thf(fact_906_p_Oconst__term__not__zero,axiom,
! [P: list_list_a] :
( ( ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( P != nil_list_a ) ) ).
% p.const_term_not_zero
thf(fact_907_p_Ogenideal__minimal,axiom,
! [I4: set_list_a,S2: set_list_a] :
( ( ideal_8896367198367571637t_unit @ I4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ S2 @ I4 )
=> ( ord_le8861187494160871172list_a @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ S2 ) @ I4 ) ) ) ).
% p.genideal_minimal
thf(fact_908_p_OIdl__subset__ideal,axiom,
! [I4: set_list_a,H: set_list_a] :
( ( ideal_8896367198367571637t_unit @ I4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H ) @ I4 )
= ( ord_le8861187494160871172list_a @ H @ I4 ) ) ) ) ).
% p.Idl_subset_ideal
thf(fact_909_p_Ogenideal__ideal,axiom,
! [S2: set_list_a] :
( ( ord_le8861187494160871172list_a @ S2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ideal_8896367198367571637t_unit @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ S2 ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% p.genideal_ideal
thf(fact_910_p_Oideal__is__subalgebra,axiom,
! [K2: set_list_a,I4: set_list_a] :
( ( ord_le8861187494160871172list_a @ K2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ideal_8896367198367571637t_unit @ I4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( embedd1768981623711841426t_unit @ K2 @ I4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.ideal_is_subalgebra
thf(fact_911_p_Ofinetely__gen,axiom,
! [I4: set_list_a] :
( ( ideal_8896367198367571637t_unit @ I4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ? [A5: set_list_a] :
( ( ord_le8861187494160871172list_a @ A5 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( finite_finite_list_a @ A5 )
& ( I4
= ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A5 ) ) ) ) ).
% p.finetely_gen
thf(fact_912_p_Oconst__term__simprules__shell_I3_J,axiom,
! [K2: set_list_a,P: list_list_a,Q: list_list_a] :
( ( subrin6918843898125473962t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( 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 ) ) @ K2 ) @ P @ Q ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) @ ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q ) ) ) ) ) ) ).
% p.const_term_simprules_shell(3)
thf(fact_913_p_Oquot__carr,axiom,
! [I4: set_list_a,Y: set_list_a] :
( ( ideal_8896367198367571637t_unit @ I4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I4 ) ) )
=> ( ord_le8861187494160871172list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.quot_carr
thf(fact_914_genideal__minimal,axiom,
! [I4: set_a,S2: set_a] :
( ( ideal_a_b @ I4 @ r )
=> ( ( ord_less_eq_set_a @ S2 @ I4 )
=> ( ord_less_eq_set_a @ ( genideal_a_b @ r @ S2 ) @ I4 ) ) ) ).
% genideal_minimal
thf(fact_915_genideal__ideal,axiom,
! [S2: set_a] :
( ( ord_less_eq_set_a @ S2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ideal_a_b @ ( genideal_a_b @ r @ S2 ) @ r ) ) ).
% genideal_ideal
thf(fact_916_Idl__subset__ideal,axiom,
! [I4: set_a,H: set_a] :
( ( ideal_a_b @ I4 @ r )
=> ( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( genideal_a_b @ r @ H ) @ I4 )
= ( ord_less_eq_set_a @ H @ I4 ) ) ) ) ).
% Idl_subset_ideal
thf(fact_917_oneideal,axiom,
ideal_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% oneideal
thf(fact_918_subset__Idl__subset,axiom,
! [I4: set_a,H: set_a] :
( ( ord_less_eq_set_a @ I4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ H @ I4 )
=> ( ord_less_eq_set_a @ ( genideal_a_b @ r @ H ) @ ( genideal_a_b @ r @ I4 ) ) ) ) ).
% subset_Idl_subset
thf(fact_919_genideal__self,axiom,
! [S2: set_a] :
( ( ord_less_eq_set_a @ S2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ S2 @ ( genideal_a_b @ r @ S2 ) ) ) ).
% genideal_self
thf(fact_920_const__term__not__zero,axiom,
! [P: list_a] :
( ( ( const_term_a_b @ r @ P )
!= ( zero_a_b @ r ) )
=> ( P != nil_a ) ) ).
% const_term_not_zero
thf(fact_921_finetely__gen,axiom,
! [I4: set_a] :
( ( ideal_a_b @ I4 @ r )
=> ? [A5: set_a] :
( ( ord_less_eq_set_a @ A5 @ ( partia707051561876973205xt_a_b @ r ) )
& ( finite_finite_a @ A5 )
& ( I4
= ( genideal_a_b @ r @ A5 ) ) ) ) ).
% finetely_gen
thf(fact_922_noetherian__ringI,axiom,
( ! [I5: set_a] :
( ( ideal_a_b @ I5 @ r )
=> ? [A4: set_a] :
( ( ord_less_eq_set_a @ A4 @ ( partia707051561876973205xt_a_b @ r ) )
& ( finite_finite_a @ A4 )
& ( I5
= ( genideal_a_b @ r @ A4 ) ) ) )
=> ( ring_n3639167112692572309ng_a_b @ r ) ) ).
% noetherian_ringI
thf(fact_923_p_Opderiv__add,axiom,
! [K2: set_list_a,F: list_list_a,G: list_list_a] :
( ( subrin6918843898125473962t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ F @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ G @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( 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 ) ) @ K2 ) @ F @ G ) )
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ ( formal6075833236969493044t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F ) @ ( formal6075833236969493044t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G ) ) ) ) ) ) ).
% p.pderiv_add
thf(fact_924_p_Ocanonical__proj__vimage__in__carrier,axiom,
! [I4: set_list_a,J3: set_set_list_a] :
( ( ideal_8896367198367571637t_unit @ I4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8877086941679407844list_a @ J3 @ ( partia141011252114345353t_unit @ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I4 ) ) )
=> ( ord_le8861187494160871172list_a @ ( comple6928918032620976721list_a @ J3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.canonical_proj_vimage_in_carrier
thf(fact_925_p_Opderiv__mult,axiom,
! [K2: set_list_a,F: list_list_a,G: list_list_a] :
( ( subrin6918843898125473962t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ F @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ G @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( 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 ) ) @ K2 ) @ F @ G ) )
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ ( formal6075833236969493044t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F ) @ G ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ F @ ( formal6075833236969493044t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G ) ) ) ) ) ) ) ).
% p.pderiv_mult
thf(fact_926_p_Osubring__polynomial__pow__division,axiom,
! [K2: set_list_a,P: list_list_a,N: nat,M: nat] :
( ( subrin6918843898125473962t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( factor6954119973539764400t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ P @ N ) @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ P @ M ) ) ) ) ) ).
% p.subring_polynomial_pow_division
thf(fact_927_p_Oquot__ideal__imp__ring__ideal,axiom,
! [I4: set_list_a,J3: set_set_list_a] :
( ( ideal_8896367198367571637t_unit @ I4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ideal_5294671857925479125t_unit @ J3 @ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I4 ) )
=> ( ideal_8896367198367571637t_unit @ ( comple6928918032620976721list_a @ J3 ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.quot_ideal_imp_ring_ideal
thf(fact_928_p_Ois__root__poly__mult__imp__is__root,axiom,
! [P: list_list_a,Q: list_list_a,X: list_a] :
( ( member_list_list_a @ P @ ( 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 @ Q @ ( 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 ) ) ) ) @ P @ Q ) @ X )
=> ( ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ X )
| ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q @ X ) ) ) ) ) ).
% p.is_root_poly_mult_imp_is_root
thf(fact_929_p_Opirreducible__pow__pdivides__iff,axiom,
! [K2: set_list_a,P: list_list_a,Q: list_list_a,R3: list_list_a,N: nat] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ R3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ P )
=> ( ~ ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q )
=> ( ( 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 ) ) @ K2 ) @ P @ N ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ Q @ R3 ) )
= ( 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 ) ) @ K2 ) @ P @ N ) @ R3 ) ) ) ) ) ) ) ) ).
% p.pirreducible_pow_pdivides_iff
thf(fact_930_mult__of_Odivides__prod__l,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ ( mult_a_ring_ext_a_b @ r @ C @ B ) ) ) ) ) ) ).
% mult_of.divides_prod_l
thf(fact_931_mult__of_Odivides__prod__r,axiom,
! [A: a,B: a,C: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ ( mult_a_ring_ext_a_b @ r @ B @ C ) ) ) ) ) ).
% mult_of.divides_prod_r
thf(fact_932_mult__of_Odivides__trans,axiom,
! [A: a,B: a,C: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ B @ C )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ C ) ) ) ) ).
% mult_of.divides_trans
thf(fact_933_mult__of_Or__cancel,axiom,
! [A: a,C: a,B: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A @ C )
= ( mult_a_ring_ext_a_b @ r @ B @ C ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( A = B ) ) ) ) ) ).
% mult_of.r_cancel
thf(fact_934_mult__of_Om__lcomm,axiom,
! [X: a,Y: a,Z2: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Z2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ ( mult_a_ring_ext_a_b @ r @ Y @ Z2 ) )
= ( mult_a_ring_ext_a_b @ r @ Y @ ( mult_a_ring_ext_a_b @ r @ X @ Z2 ) ) ) ) ) ) ).
% mult_of.m_lcomm
thf(fact_935_mult__of_Om__comm,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ X ) ) ) ) ).
% mult_of.m_comm
thf(fact_936_mult__of_Om__assoc,axiom,
! [X: a,Y: a,Z2: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Z2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ Z2 )
= ( mult_a_ring_ext_a_b @ r @ X @ ( mult_a_ring_ext_a_b @ r @ Y @ Z2 ) ) ) ) ) ) ).
% mult_of.m_assoc
thf(fact_937_mult__of_Ol__cancel,axiom,
! [C: a,A: a,B: a] :
( ( ( mult_a_ring_ext_a_b @ r @ C @ A )
= ( mult_a_ring_ext_a_b @ r @ C @ B ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( A = B ) ) ) ) ) ).
% mult_of.l_cancel
thf(fact_938_local_Odivides__mult,axiom,
! [A: a,C: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( factor8216151070175719842xt_a_b @ r @ A @ B )
=> ( factor8216151070175719842xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ C @ A ) @ ( mult_a_ring_ext_a_b @ r @ C @ B ) ) ) ) ) ).
% local.divides_mult
thf(fact_939_divides__prod__l,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( factor8216151070175719842xt_a_b @ r @ A @ B )
=> ( factor8216151070175719842xt_a_b @ r @ A @ ( mult_a_ring_ext_a_b @ r @ C @ B ) ) ) ) ) ) ).
% divides_prod_l
thf(fact_940_divides__prod__r,axiom,
! [A: a,B: a,C: a] :
( ( factor8216151070175719842xt_a_b @ r @ A @ B )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ A @ ( mult_a_ring_ext_a_b @ r @ B @ C ) ) ) ) ) ).
% divides_prod_r
thf(fact_941_divides__trans,axiom,
! [A: a,B: a,C: a] :
( ( factor8216151070175719842xt_a_b @ r @ A @ B )
=> ( ( factor8216151070175719842xt_a_b @ r @ B @ C )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ A @ C ) ) ) ) ).
% divides_trans
thf(fact_942_m__lcomm,axiom,
! [X: a,Y: a,Z2: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ ( mult_a_ring_ext_a_b @ r @ Y @ Z2 ) )
= ( mult_a_ring_ext_a_b @ r @ Y @ ( mult_a_ring_ext_a_b @ r @ X @ Z2 ) ) ) ) ) ) ).
% m_lcomm
thf(fact_943_m__comm,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ X ) ) ) ) ).
% m_comm
thf(fact_944_m__assoc,axiom,
! [X: a,Y: a,Z2: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ Z2 )
= ( mult_a_ring_ext_a_b @ r @ X @ ( mult_a_ring_ext_a_b @ r @ Y @ Z2 ) ) ) ) ) ) ).
% m_assoc
thf(fact_945_zero__divides,axiom,
! [A: a] :
( ( factor8216151070175719842xt_a_b @ r @ ( zero_a_b @ r ) @ A )
= ( A
= ( zero_a_b @ r ) ) ) ).
% zero_divides
thf(fact_946_divides__mult__zero,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ ( zero_a_b @ r ) )
=> ( A
= ( zero_a_b @ r ) ) ) ) ).
% divides_mult_zero
thf(fact_947_divides__zero,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ A @ ( zero_a_b @ r ) ) ) ).
% divides_zero
thf(fact_948_m__rcancel,axiom,
! [A: a,B: a,C: a] :
( ( A
!= ( zero_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ B @ A )
= ( mult_a_ring_ext_a_b @ r @ C @ A ) )
= ( B = C ) ) ) ) ) ) ).
% m_rcancel
thf(fact_949_m__lcancel,axiom,
! [A: a,B: a,C: a] :
( ( A
!= ( zero_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ A @ B )
= ( mult_a_ring_ext_a_b @ r @ A @ C ) )
= ( B = C ) ) ) ) ) ) ).
% m_lcancel
thf(fact_950_integral__iff,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ A @ B )
= ( zero_a_b @ r ) )
= ( ( A
= ( zero_a_b @ r ) )
| ( B
= ( zero_a_b @ r ) ) ) ) ) ) ).
% integral_iff
thf(fact_951_local_Ointegral,axiom,
! [A: a,B: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A @ B )
= ( zero_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A
= ( zero_a_b @ r ) )
| ( B
= ( zero_a_b @ r ) ) ) ) ) ) ).
% local.integral
thf(fact_952_f__comm__group__1,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X
!= ( zero_a_b @ r ) )
=> ( ( Y
!= ( zero_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
!= ( zero_a_b @ r ) ) ) ) ) ) ).
% f_comm_group_1
thf(fact_953_div__sum__iff,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( factor8216151070175719842xt_a_b @ r @ A @ B )
=> ( ( factor8216151070175719842xt_a_b @ r @ A @ ( add_a_b @ r @ B @ C ) )
= ( factor8216151070175719842xt_a_b @ r @ A @ C ) ) ) ) ) ) ).
% div_sum_iff
thf(fact_954_div__sum,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( factor8216151070175719842xt_a_b @ r @ A @ B )
=> ( ( factor8216151070175719842xt_a_b @ r @ A @ C )
=> ( factor8216151070175719842xt_a_b @ r @ A @ ( add_a_b @ r @ B @ C ) ) ) ) ) ) ) ).
% div_sum
thf(fact_955_r__distr,axiom,
! [X: a,Y: a,Z2: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ Z2 @ ( add_a_b @ r @ X @ Y ) )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ Z2 @ X ) @ ( mult_a_ring_ext_a_b @ r @ Z2 @ Y ) ) ) ) ) ) ).
% r_distr
thf(fact_956_l__distr,axiom,
! [X: a,Y: a,Z2: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ X @ Y ) @ Z2 )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Z2 ) @ ( mult_a_ring_ext_a_b @ r @ Y @ Z2 ) ) ) ) ) ) ).
% l_distr
thf(fact_957_pow__mult__distrib,axiom,
! [X: a,Y: a,N: nat] :
( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ X ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ N )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ Y @ N ) ) ) ) ) ) ).
% pow_mult_distrib
thf(fact_958_nat__pow__distrib,axiom,
! [X: a,Y: a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ N )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ Y @ N ) ) ) ) ) ).
% nat_pow_distrib
thf(fact_959_nat__pow__comm,axiom,
! [X: a,N: nat,M: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ X @ M ) )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ M ) @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) ) ) ) ).
% nat_pow_comm
thf(fact_960_group__commutes__pow,axiom,
! [X: a,Y: a,N: nat] :
( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ X ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) ) ) ) ) ) ).
% group_commutes_pow
thf(fact_961_p_Odivides__prod__r,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B )
=> ( ( member_list_a @ A @ ( 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 ) ) ) )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ C ) ) ) ) ) ).
% p.divides_prod_r
thf(fact_962_p_Odivides__prod__l,axiom,
! [A: list_a,B: list_a,C: 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 ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ B ) ) ) ) ) ) ).
% p.divides_prod_l
thf(fact_963_p_Odivides__mult,axiom,
! [A: list_a,C: list_a,B: list_a] :
( ( member_list_a @ A @ ( 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 ) ) ) )
=> ( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ A ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ B ) ) ) ) ) ).
% p.divides_mult
thf(fact_964_p_Odivides__trans,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B )
=> ( ( factor1757716651909850160t_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 ) ) ) )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ C ) ) ) ) ).
% p.divides_trans
thf(fact_965_p_Om__lcomm,axiom,
! [X: list_a,Y: list_a,Z2: 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 @ Z2 @ ( 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 @ Z2 ) )
= ( 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 @ Z2 ) ) ) ) ) ) ).
% p.m_lcomm
thf(fact_966_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_967_p_Om__assoc,axiom,
! [X: list_a,Y: list_a,Z2: 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 @ Z2 @ ( 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 ) @ Z2 )
= ( 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 @ Z2 ) ) ) ) ) ) ).
% p.m_assoc
thf(fact_968_p_Omult__of_Or__cancel,axiom,
! [A: list_a,C: list_a,B: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ C )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ C ) )
=> ( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ C @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( A = B ) ) ) ) ) ).
% p.mult_of.r_cancel
thf(fact_969_p_Omult__of_Om__lcomm,axiom,
! [X: list_a,Y: list_a,Z2: list_a] :
( ( member_list_a @ X @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ Y @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ Z2 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_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 @ Z2 ) )
= ( 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 @ Z2 ) ) ) ) ) ) ).
% p.mult_of.m_lcomm
thf(fact_970_p_Omult__of_Om__comm,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ Y @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_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.mult_of.m_comm
thf(fact_971_p_Omult__of_Om__assoc,axiom,
! [X: list_a,Y: list_a,Z2: list_a] :
( ( member_list_a @ X @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ Y @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ Z2 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_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 ) @ Z2 )
= ( 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 @ Z2 ) ) ) ) ) ) ).
% p.mult_of.m_assoc
thf(fact_972_p_Omult__of_Ol__cancel,axiom,
! [C: list_a,A: list_a,B: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ A )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ B ) )
=> ( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ C @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( A = B ) ) ) ) ) ).
% p.mult_of.l_cancel
thf(fact_973_p_Ozero__divides,axiom,
! [A: list_a] :
( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A )
= ( A
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.zero_divides
thf(fact_974_p_Osubring__props_I6_J,axiom,
! [K2: set_list_a,H1: list_a,H22: list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a @ H1 @ K2 )
=> ( ( member_list_a @ H22 @ K2 )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H1 @ H22 ) @ K2 ) ) ) ) ).
% p.subring_props(6)
thf(fact_975_monic__poly__mult,axiom,
! [F: list_a,G: list_a] :
( ( monic_3145109188698636716ly_a_b @ r @ F )
=> ( ( monic_3145109188698636716ly_a_b @ r @ G )
=> ( monic_3145109188698636716ly_a_b @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ G ) ) ) ) ).
% monic_poly_mult
thf(fact_976_p_Odivides__zero,axiom,
! [A: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.divides_zero
thf(fact_977_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_978_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_979_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_980_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_981_p_Odiv__sum__iff,axiom,
! [A: list_a,B: list_a,C: 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 ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B )
=> ( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ C ) )
= ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ C ) ) ) ) ) ) ).
% p.div_sum_iff
thf(fact_982_p_Odiv__sum,axiom,
! [A: list_a,B: list_a,C: 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 ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B )
=> ( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ C )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ C ) ) ) ) ) ) ) ).
% p.div_sum
thf(fact_983_p_Or__distr,axiom,
! [X: list_a,Y: list_a,Z2: 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 @ Z2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Z2 @ ( 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 ) ) @ Z2 @ X ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Z2 @ Y ) ) ) ) ) ) ).
% p.r_distr
thf(fact_984_p_Ol__distr,axiom,
! [X: list_a,Y: list_a,Z2: 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 @ Z2 @ ( 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 ) @ Z2 )
= ( 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 @ Z2 ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z2 ) ) ) ) ) ) ).
% p.l_distr
thf(fact_985_p_Osubring__props_I2_J,axiom,
! [K2: set_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ K2 ) ) ).
% p.subring_props(2)
thf(fact_986_p_Osubring__props_I7_J,axiom,
! [K2: set_list_a,H1: list_a,H22: list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a @ H1 @ K2 )
=> ( ( member_list_a @ H22 @ K2 )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H1 @ H22 ) @ K2 ) ) ) ) ).
% p.subring_props(7)
thf(fact_987_p_Omult__of_Odivides__prod__r,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ B )
=> ( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ C @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ C ) ) ) ) ) ).
% p.mult_of.divides_prod_r
thf(fact_988_p_Omult__of_Odivides__prod__l,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ C @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ B )
=> ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ B ) ) ) ) ) ) ).
% p.mult_of.divides_prod_l
thf(fact_989_p_Omult__of_Odivides__trans,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ B )
=> ( ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ B @ C )
=> ( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ C ) ) ) ) ).
% p.mult_of.divides_trans
thf(fact_990_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_991_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_992_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_993_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_994_p_Ouniv__poly__is__principal,axiom,
! [K2: set_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ring_p715737262848045090t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) ) ).
% p.univ_poly_is_principal
thf(fact_995_pdivides__mult__r,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ C @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ C ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ C ) )
= ( polyno5814909790663948098es_a_b @ r @ A @ B ) ) ) ) ) ).
% pdivides_mult_r
thf(fact_996_p_Oint__embed__mult,axiom,
! [X: int,Y: int] :
( ( ring_i7845631192508594762t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( times_times_int @ X @ Y ) )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( ring_i7845631192508594762t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) @ ( ring_i7845631192508594762t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y ) ) ) ).
% p.int_embed_mult
thf(fact_997_monic__poly__carr__2,axiom,
! [F: list_a] :
( ( monic_3145109188698636716ly_a_b @ r @ F )
=> ( member_list_a @ F @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% monic_poly_carr_2
thf(fact_998_p_Oline__extension__mem__iff,axiom,
! [U: list_a,K2: 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 ) ) @ K2 @ A @ E ) )
= ( ? [X2: list_a] :
( ( member_list_a @ X2 @ K2 )
& ? [Y6: list_a] :
( ( member_list_a @ Y6 @ 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 ) @ Y6 ) ) ) ) ) ) ).
% p.line_extension_mem_iff
thf(fact_999_mult__of_Olcmof__exists,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ? [C3: a] :
( ( member_a @ C3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( islcm_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ C3 @ A @ B ) ) ) ) ).
% mult_of.lcmof_exists
thf(fact_1000_p_Omult__of_Omonoid__cancelI,axiom,
( ! [A6: list_a,B4: list_a,C3: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C3 @ A6 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C3 @ B4 ) )
=> ( ( member_list_a @ A6 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B4 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ C3 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( A6 = B4 ) ) ) ) )
=> ( ! [A6: list_a,B4: list_a,C3: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A6 @ C3 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B4 @ C3 ) )
=> ( ( member_list_a @ A6 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B4 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ C3 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( A6 = B4 ) ) ) ) )
=> ( monoid5117334421817186628t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.mult_of.monoid_cancelI
thf(fact_1001_mult__of_Odivides__mult__l,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ C @ A ) @ ( mult_a_ring_ext_a_b @ r @ C @ B ) )
= ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B ) ) ) ) ) ).
% mult_of.divides_mult_l
thf(fact_1002_mult__of_Odivides__mult__lI,axiom,
! [A: a,B: a,C: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ C @ A ) @ ( mult_a_ring_ext_a_b @ r @ C @ B ) ) ) ) ) ).
% mult_of.divides_mult_lI
thf(fact_1003_mult__of_Odivides__mult__r,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ A @ C ) @ ( mult_a_ring_ext_a_b @ r @ B @ C ) )
= ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B ) ) ) ) ) ).
% mult_of.divides_mult_r
thf(fact_1004_mult__of_Odivides__mult__rI,axiom,
! [A: a,B: a,C: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ A @ C ) @ ( mult_a_ring_ext_a_b @ r @ B @ C ) ) ) ) ) ) ).
% mult_of.divides_mult_rI
thf(fact_1005_mult__of_Odivides__refl,axiom,
! [A: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ A ) ) ).
% mult_of.divides_refl
thf(fact_1006_mult__of_Oright__cancel,axiom,
! [X: a,Y: a,Z2: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Z2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ Y @ X )
= ( mult_a_ring_ext_a_b @ r @ Z2 @ X ) )
= ( Y = Z2 ) ) ) ) ) ).
% mult_of.right_cancel
thf(fact_1007_mult__of_Om__closed,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ).
% mult_of.m_closed
thf(fact_1008_p_Osubring__props_I1_J,axiom,
! [K2: set_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ord_le8861187494160871172list_a @ K2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.subring_props(1)
thf(fact_1009_p_Oint__embed__mult__aux,axiom,
! [X: int,Y: nat] :
( ( ring_i7845631192508594762t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( times_times_int @ X @ ( semiri1314217659103216013at_int @ Y ) ) )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( ring_i7845631192508594762t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) @ ( ring_i7845631192508594762t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( semiri1314217659103216013at_int @ Y ) ) ) ) ).
% p.int_embed_mult_aux
thf(fact_1010_p_Omult__of_Oprime__divides,axiom,
! [A: list_a,B: list_a,P: list_a] :
( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( prime_769485436051073893t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ P )
=> ( ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ P @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B ) )
=> ( ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ P @ A )
| ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ P @ B ) ) ) ) ) ) ).
% p.mult_of.prime_divides
thf(fact_1011_p_OpprimeE_I1_J,axiom,
! [K2: set_list_a,P: list_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ P )
=> ( P != nil_list_a ) ) ) ) ).
% p.pprimeE(1)
thf(fact_1012_p_Oline__extension__smult__closed,axiom,
! [K2: set_list_a,E: set_list_a,A: list_a,K: list_a,U: list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ! [K3: list_a,V3: list_a] :
( ( member_list_a @ K3 @ K2 )
=> ( ( member_list_a @ V3 @ E )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K3 @ V3 ) @ 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 @ K2 )
=> ( ( member_list_a @ U @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ 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 ) ) @ K2 @ A @ E ) ) ) ) ) ) ) ) ).
% p.line_extension_smult_closed
thf(fact_1013_p_Opprime__iff__pirreducible,axiom,
! [K2: set_list_a,P: list_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ P )
= ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ P ) ) ) ) ).
% p.pprime_iff_pirreducible
thf(fact_1014_p_OpprimeE_I3_J,axiom,
! [K2: set_list_a,P: list_list_a,Q: list_list_a,R3: list_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ P )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ R3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ Q @ R3 ) )
=> ( ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q )
| ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ R3 ) ) ) ) ) ) ) ) ).
% p.pprimeE(3)
thf(fact_1015_p_Oconst__term__simprules__shell_I2_J,axiom,
! [K2: set_list_a,P: list_list_a,Q: list_list_a] :
( ( subrin6918843898125473962t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( 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 ) ) @ K2 ) @ P @ Q ) )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) @ ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q ) ) ) ) ) ) ).
% p.const_term_simprules_shell(2)
thf(fact_1016_p_Opdivides__iff__shell,axiom,
! [K2: set_list_a,P: list_list_a,Q: list_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q )
= ( factor6954119973539764400t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ P @ Q ) ) ) ) ) ).
% p.pdivides_iff_shell
thf(fact_1017_divides__mult__lI,axiom,
! [A: a,B: a,C: a] :
( ( factor8216151070175719842xt_a_b @ r @ A @ B )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ C @ A ) @ ( mult_a_ring_ext_a_b @ r @ C @ B ) ) ) ) ) ).
% divides_mult_lI
thf(fact_1018_divides__mult__rI,axiom,
! [A: a,B: a,C: a] :
( ( factor8216151070175719842xt_a_b @ r @ A @ B )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A @ C ) @ ( mult_a_ring_ext_a_b @ r @ B @ C ) ) ) ) ) ) ).
% divides_mult_rI
thf(fact_1019_divides__refl,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ A @ A ) ) ).
% divides_refl
thf(fact_1020_m__closed,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% m_closed
thf(fact_1021_r__null,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ) ).
% r_null
thf(fact_1022_l__null,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) @ X )
= ( zero_a_b @ r ) ) ) ).
% l_null
thf(fact_1023_p_Omult__of_Olcmof__exists,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ? [C3: list_a] :
( ( member_list_a @ C3 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
& ( islcm_4429780156966476082t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ C3 @ A @ B ) ) ) ) ).
% p.mult_of.lcmof_exists
thf(fact_1024_p_Odivides__mult__rI,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ 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 ) ) ) )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ C ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ C ) ) ) ) ) ) ).
% p.divides_mult_rI
thf(fact_1025_p_Odivides__mult__lI,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B )
=> ( ( member_list_a @ A @ ( 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 ) ) ) )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ A ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ B ) ) ) ) ) ).
% p.divides_mult_lI
thf(fact_1026_p_Odivides__refl,axiom,
! [A: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ A ) ) ).
% p.divides_refl
thf(fact_1027_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_1028_p_Omult__of_Om__closed,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ Y @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_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 ) @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% p.mult_of.m_closed
thf(fact_1029_mult__of_Odivisor__chain__condition__monoid__axioms,axiom,
diviso6259607970152342594t_unit @ ( ring_mult_of_a_b @ r ) ).
% mult_of.divisor_chain_condition_monoid_axioms
thf(fact_1030_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_1031_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_1032_p_Omult__of_Odivides__mult__rI,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ B )
=> ( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ C @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ C ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ C ) ) ) ) ) ) ).
% p.mult_of.divides_mult_rI
thf(fact_1033_p_Omult__of_Odivides__mult__r,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ C @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ C ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ C ) )
= ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ B ) ) ) ) ) ).
% p.mult_of.divides_mult_r
thf(fact_1034_p_Omult__of_Odivides__mult__lI,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ B )
=> ( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ C @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ A ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ B ) ) ) ) ) ).
% p.mult_of.divides_mult_lI
thf(fact_1035_p_Omult__of_Odivides__mult__l,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ C @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ A ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ B ) )
= ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ B ) ) ) ) ) ).
% p.mult_of.divides_mult_l
thf(fact_1036_p_Omult__of_Odivides__refl,axiom,
! [A: list_a] :
( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ A ) ) ).
% p.mult_of.divides_refl
thf(fact_1037_mult__of_Oprimeness__condition__monoid__axioms,axiom,
primen965786292471834261t_unit @ ( ring_mult_of_a_b @ r ) ).
% mult_of.primeness_condition_monoid_axioms
thf(fact_1038_p_Omult__of_Odivides__fcount,axiom,
! [A: list_a,B: list_a] :
( ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ B )
=> ( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ord_less_eq_nat @ ( factor8927536889056116518t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A ) @ ( factor8927536889056116518t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ B ) ) ) ) ) ).
% p.mult_of.divides_fcount
thf(fact_1039_p_Oexists__unique__long__division,axiom,
! [K2: set_list_a,P: list_list_a,Q: list_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( Q != nil_list_a )
=> ? [X3: produc7709606177366032167list_a] :
( ( polyno6947042923167803568t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q @ X3 )
& ! [Y2: produc7709606177366032167list_a] :
( ( polyno6947042923167803568t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q @ Y2 )
=> ( Y2 = X3 ) ) ) ) ) ) ) ).
% p.exists_unique_long_division
thf(fact_1040_p_Oalg__mult__gt__zero__iff__is__root,axiom,
! [P: list_list_a,X: list_a] :
( ( member_list_list_a @ P @ ( 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 ) ) @ P @ X ) )
= ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ X ) ) ) ).
% p.alg_mult_gt_zero_iff_is_root
thf(fact_1041_mult__of_Omonoid__cancel__axioms,axiom,
monoid1999574367301118026t_unit @ ( ring_mult_of_a_b @ r ) ).
% mult_of.monoid_cancel_axioms
thf(fact_1042_mult__of_Ogcd__condition__monoid__axioms,axiom,
gcd_co701944698663231555t_unit @ ( ring_mult_of_a_b @ r ) ).
% mult_of.gcd_condition_monoid_axioms
thf(fact_1043_monoid__cancelI,axiom,
( ! [A6: a,B4: a,C3: a] :
( ( ( mult_a_ring_ext_a_b @ r @ C3 @ A6 )
= ( mult_a_ring_ext_a_b @ r @ C3 @ B4 ) )
=> ( ( member_a @ A6 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A6 = B4 ) ) ) ) )
=> ( ! [A6: a,B4: a,C3: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A6 @ C3 )
= ( mult_a_ring_ext_a_b @ r @ B4 @ C3 ) )
=> ( ( member_a @ A6 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A6 = B4 ) ) ) ) )
=> ( monoid5798828371819920185xt_a_b @ r ) ) ) ).
% monoid_cancelI
thf(fact_1044_mult__of_Omonoid__cancelI,axiom,
( ! [A6: a,B4: a,C3: a] :
( ( ( mult_a_ring_ext_a_b @ r @ C3 @ A6 )
= ( mult_a_ring_ext_a_b @ r @ C3 @ B4 ) )
=> ( ( member_a @ A6 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B4 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( A6 = B4 ) ) ) ) )
=> ( ! [A6: a,B4: a,C3: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A6 @ C3 )
= ( mult_a_ring_ext_a_b @ r @ B4 @ C3 ) )
=> ( ( member_a @ A6 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B4 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( A6 = B4 ) ) ) ) )
=> ( monoid1999574367301118026t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ).
% mult_of.monoid_cancelI
thf(fact_1045_p_Omonoid__cancelI,axiom,
( ! [A6: list_a,B4: list_a,C3: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C3 @ A6 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C3 @ B4 ) )
=> ( ( member_list_a @ A6 @ ( 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 ) ) ) )
=> ( A6 = B4 ) ) ) ) )
=> ( ! [A6: list_a,B4: list_a,C3: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A6 @ C3 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B4 @ C3 ) )
=> ( ( member_list_a @ A6 @ ( 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 ) ) ) )
=> ( A6 = B4 ) ) ) ) )
=> ( monoid4303264861975686087t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.monoid_cancelI
thf(fact_1046_mult__of_Odivides__fcount,axiom,
! [A: a,B: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ord_less_eq_nat @ ( factor4067924603488134956t_unit @ ( ring_mult_of_a_b @ r ) @ A ) @ ( factor4067924603488134956t_unit @ ( ring_mult_of_a_b @ r ) @ B ) ) ) ) ) ).
% mult_of.divides_fcount
thf(fact_1047_p_Omult__of_Oorder__gt__0__iff__finite,axiom,
( ( ord_less_nat @ zero_zero_nat @ ( order_3723632364699938349t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
= ( finite_finite_list_a @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.mult_of.order_gt_0_iff_finite
thf(fact_1048_p_Oisgcd__divides__l,axiom,
! [A: list_a,B: list_a] :
( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ 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 ) ) ) )
=> ( isgcd_1118609098697428327t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ A @ B ) ) ) ) ).
% p.isgcd_divides_l
thf(fact_1049_p_Oisgcd__divides__r,axiom,
! [B: list_a,A: list_a] :
( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ 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 ) ) ) )
=> ( isgcd_1118609098697428327t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ A @ B ) ) ) ) ).
% p.isgcd_divides_r
thf(fact_1050_mult__of_Oorder__gt__0__iff__finite,axiom,
( ( ord_less_nat @ zero_zero_nat @ ( order_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
= ( finite_finite_a @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ).
% mult_of.order_gt_0_iff_finite
thf(fact_1051_p_Omult__of_Oisgcd__divides__r,axiom,
! [B: list_a,A: list_a] :
( ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ B @ A )
=> ( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( isgcd_454116541068447652t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ B @ A @ B ) ) ) ) ).
% p.mult_of.isgcd_divides_r
thf(fact_1052_mult__of_Ogcdof__exists,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ? [C3: a] :
( ( member_a @ C3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( isgcd_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ C3 @ A @ B ) ) ) ) ).
% mult_of.gcdof_exists
thf(fact_1053_isgcd__divides__l,axiom,
! [A: a,B: a] :
( ( factor8216151070175719842xt_a_b @ r @ A @ B )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( isgcd_a_ring_ext_a_b @ r @ A @ A @ B ) ) ) ) ).
% isgcd_divides_l
thf(fact_1054_isgcd__divides__r,axiom,
! [B: a,A: a] :
( ( factor8216151070175719842xt_a_b @ r @ B @ A )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( isgcd_a_ring_ext_a_b @ r @ B @ A @ B ) ) ) ) ).
% isgcd_divides_r
thf(fact_1055_mult__of_Oisgcd__divides__r,axiom,
! [B: a,A: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ B @ A )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( isgcd_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ B @ A @ B ) ) ) ) ).
% mult_of.isgcd_divides_r
thf(fact_1056_mult__of_Oisgcd__divides__l,axiom,
! [A: a,B: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( isgcd_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ A @ A @ B ) ) ) ) ).
% mult_of.isgcd_divides_l
thf(fact_1057_p_Omult__of_Ogcdof__exists,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ? [C3: list_a] :
( ( member_list_a @ C3 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
& ( isgcd_454116541068447652t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ C3 @ A @ B ) ) ) ) ).
% p.mult_of.gcdof_exists
thf(fact_1058_p_Omult__of_Oisgcd__divides__l,axiom,
! [A: list_a,B: list_a] :
( ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ B )
=> ( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( isgcd_454116541068447652t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ A @ B ) ) ) ) ).
% p.mult_of.isgcd_divides_l
thf(fact_1059_p_Olong__division__add_I1_J,axiom,
! [K2: set_list_a,A: list_list_a,B: list_list_a,Q: list_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( 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 ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( 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 ) ) @ K2 ) @ A @ B ) @ Q )
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ Q ) @ ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ Q ) ) ) ) ) ) ) ).
% p.long_division_add(1)
thf(fact_1060_p_Olong__division__zero_I1_J,axiom,
! [K2: set_list_a,Q: list_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ nil_list_a @ Q )
= nil_list_a ) ) ) ).
% p.long_division_zero(1)
thf(fact_1061_p_Olong__division__closed_I1_J,axiom,
! [K2: set_list_a,P: list_list_a,Q: list_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( member_list_list_a @ ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) ) ) ) ) ).
% p.long_division_closed(1)
thf(fact_1062_p_Opdiv__pmod,axiom,
! [K2: set_list_a,P: list_list_a,Q: list_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( P
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ Q @ ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q ) ) @ ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q ) ) ) ) ) ) ).
% p.pdiv_pmod
thf(fact_1063_mult__of_Oprime__divides,axiom,
! [A: a,B: a,P: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( prime_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ P )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ P @ ( mult_a_ring_ext_a_b @ r @ A @ B ) )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ P @ A )
| ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ P @ B ) ) ) ) ) ) ).
% mult_of.prime_divides
thf(fact_1064_zero__is__prime_I2_J,axiom,
prime_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ ( zero_a_b @ r ) ).
% zero_is_prime(2)
thf(fact_1065_ring__primeE_I2_J,axiom,
! [P: a] :
( ( member_a @ P @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_ring_prime_a_b @ r @ P )
=> ( prime_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ P ) ) ) ).
% ring_primeE(2)
thf(fact_1066_prime__eq__prime__mult,axiom,
! [P: a] :
( ( member_a @ P @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( prime_a_ring_ext_a_b @ r @ P )
= ( prime_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ P ) ) ) ).
% prime_eq_prime_mult
thf(fact_1067_p_Olong__division__closed_I2_J,axiom,
! [K2: set_list_a,P: list_list_a,Q: list_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( member_list_list_a @ ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) ) ) ) ) ).
% p.long_division_closed(2)
thf(fact_1068_p_Olong__division__zero_I2_J,axiom,
! [K2: set_list_a,Q: list_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ nil_list_a @ Q )
= nil_list_a ) ) ) ).
% p.long_division_zero(2)
thf(fact_1069_p_Olong__division__add_I2_J,axiom,
! [K2: set_list_a,A: list_list_a,B: list_list_a,Q: list_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( 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 ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( 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 ) ) @ K2 ) @ A @ B ) @ Q )
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ Q ) @ ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ Q ) ) ) ) ) ) ) ).
% p.long_division_add(2)
thf(fact_1070_p_Olong__division__add__iff,axiom,
! [K2: set_list_a,A: list_list_a,B: list_list_a,C: list_list_a,Q: list_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( 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 ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ C @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ Q )
= ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ Q ) )
= ( ( 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 ) ) @ K2 ) @ A @ C ) @ Q )
= ( 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 ) ) @ K2 ) @ B @ C ) @ Q ) ) ) ) ) ) ) ) ).
% p.long_division_add_iff
thf(fact_1071_p_Opmod__zero__iff__pdivides,axiom,
! [K2: set_list_a,P: list_list_a,Q: list_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q )
= nil_list_a )
= ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q @ P ) ) ) ) ) ).
% p.pmod_zero_iff_pdivides
thf(fact_1072_p_Osame__pmod__iff__pdivides,axiom,
! [K2: set_list_a,A: list_list_a,B: list_list_a,Q: list_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( 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 ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ Q )
= ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ Q ) )
= ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q @ ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ A @ B ) ) ) ) ) ) ) ).
% p.same_pmod_iff_pdivides
thf(fact_1073_p_Olong__divisionI,axiom,
! [K2: set_list_a,P: list_list_a,Q: list_list_a,B: list_list_a,R3: list_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( Q != nil_list_a )
=> ( ( polyno6947042923167803568t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q @ ( produc8696003437204565271list_a @ B @ R3 ) )
=> ( ( produc8696003437204565271list_a @ B @ R3 )
= ( produc8696003437204565271list_a @ ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q ) @ ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q ) ) ) ) ) ) ) ) ).
% p.long_divisionI
thf(fact_1074_p_Olong__divisionE,axiom,
! [K2: set_list_a,P: list_list_a,Q: list_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( Q != nil_list_a )
=> ( polyno6947042923167803568t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q @ ( produc8696003437204565271list_a @ ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q ) @ ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q ) ) ) ) ) ) ) ).
% p.long_divisionE
thf(fact_1075_p_Opoly__add_Ocases,axiom,
! [X: produc7709606177366032167list_a] :
~ ! [P1: list_list_a,P22: list_list_a] :
( X
!= ( produc8696003437204565271list_a @ P1 @ P22 ) ) ).
% p.poly_add.cases
thf(fact_1076_p_Oexists__long__division,axiom,
! [K2: set_list_a,P: list_list_a,Q: list_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( Q != 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 ) ) @ K2 ) ) )
=> ! [R2: list_list_a] :
( ( member_list_list_a @ R2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ~ ( polyno6947042923167803568t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q @ ( produc8696003437204565271list_a @ B4 @ R2 ) ) ) ) ) ) ) ) ).
% p.exists_long_division
thf(fact_1077_is__root__poly__mult__imp__is__root,axiom,
! [P: list_a,Q: list_a,X: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q @ ( 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 ) ) @ P @ Q ) @ X )
=> ( ( polyno4133073214067823460ot_a_b @ r @ P @ X )
| ( polyno4133073214067823460ot_a_b @ r @ Q @ X ) ) ) ) ) ).
% is_root_poly_mult_imp_is_root
thf(fact_1078_p_OpprimeI,axiom,
! [K2: set_list_a,P: list_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( P != nil_list_a )
=> ( ~ ( member_list_list_a @ P @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ! [Q2: list_list_a,R2: list_list_a] :
( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ R2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ Q2 @ R2 ) )
=> ( ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q2 )
| ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ R2 ) ) ) ) )
=> ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ P ) ) ) ) ) ) ).
% p.pprimeI
thf(fact_1079_p_OpirreducibleE_I2_J,axiom,
! [K2: set_list_a,P: list_list_a] :
( ( subrin6918843898125473962t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ P )
=> ~ ( member_list_list_a @ P @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) ) ) ) ) ).
% p.pirreducibleE(2)
thf(fact_1080_p_OpprimeE_I2_J,axiom,
! [K2: set_list_a,P: list_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ P )
=> ~ ( member_list_list_a @ P @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) ) ) ) ) ).
% p.pprimeE(2)
thf(fact_1081_p_OpirreducibleE_I3_J,axiom,
! [K2: set_list_a,P: list_list_a,Q: list_list_a,R3: list_list_a] :
( ( subrin6918843898125473962t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ P )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ R3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( P
= ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ Q @ R3 ) )
=> ( ( member_list_list_a @ Q @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
| ( member_list_list_a @ R3 @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) ) ) ) ) ) ) ) ) ).
% p.pirreducibleE(3)
thf(fact_1082_p_OpirreducibleI,axiom,
! [K2: set_list_a,P: list_list_a] :
( ( subrin6918843898125473962t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( P != nil_list_a )
=> ( ~ ( member_list_list_a @ P @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ! [Q2: list_list_a,R2: list_list_a] :
( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ R2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( P
= ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ Q2 @ R2 ) )
=> ( ( member_list_list_a @ Q2 @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
| ( member_list_list_a @ R2 @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) ) ) ) ) )
=> ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ P ) ) ) ) ) ) ).
% p.pirreducibleI
thf(fact_1083_alg__mult__gt__zero__iff__is__root,axiom,
! [P: list_a,X: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( polyno4422430861927485590lt_a_b @ r @ P @ X ) )
= ( polyno4133073214067823460ot_a_b @ r @ P @ X ) ) ) ).
% alg_mult_gt_zero_iff_is_root
thf(fact_1084_p_Ocanonical__proj__vimage__mem__iff,axiom,
! [I4: set_list_a,J3: set_set_list_a,A: list_a] :
( ( ideal_8896367198367571637t_unit @ I4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8877086941679407844list_a @ J3 @ ( partia141011252114345353t_unit @ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I4 ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( comple6928918032620976721list_a @ J3 ) )
= ( member_set_list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I4 @ A ) @ J3 ) ) ) ) ) ).
% p.canonical_proj_vimage_mem_iff
thf(fact_1085_Units__closed,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% Units_closed
thf(fact_1086_Units__pow__closed,axiom,
! [X: a,D: nat] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ ( pow_a_1026414303147256608_b_nat @ r @ X @ D ) @ ( units_a_ring_ext_a_b @ r ) ) ) ).
% Units_pow_closed
thf(fact_1087_prod__unit__l,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% prod_unit_l
thf(fact_1088_prod__unit__r,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% prod_unit_r
thf(fact_1089_unit__factor,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ).
% unit_factor
thf(fact_1090_mult__of_OUnits__closed,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ).
% mult_of.Units_closed
thf(fact_1091_divides__unit,axiom,
! [A: a,U: a] :
( ( factor8216151070175719842xt_a_b @ r @ A @ U )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ U @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ).
% divides_unit
thf(fact_1092_unit__divides,axiom,
! [U: a,A: a] :
( ( member_a @ U @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ U @ A ) ) ) ).
% unit_divides
thf(fact_1093_ring__irreducibleE_I4_J,axiom,
! [R3: a] :
( ( member_a @ R3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R3 )
=> ~ ( member_a @ R3 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% ring_irreducibleE(4)
thf(fact_1094_p_OUnits__closed,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_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 ) ) ) ) ) ).
% p.Units_closed
thf(fact_1095_mult__of_OUnits,axiom,
ord_less_eq_set_a @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ).
% mult_of.Units
thf(fact_1096_p_OUnits__pow__closed,axiom,
! [X: list_a,D: nat] :
( ( member_list_a @ X @ ( units_2932844235741507942t_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 @ D ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.Units_pow_closed
thf(fact_1097_mult__of_Oprod__unit__l,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ B @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ) ) ).
% mult_of.prod_unit_l
thf(fact_1098_mult__of_Oprod__unit__r,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ A @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ) ) ).
% mult_of.prod_unit_r
thf(fact_1099_mult__of_Ounit__factor,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ A @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ) ).
% mult_of.unit_factor
thf(fact_1100_mult__of_Odivides__unit,axiom,
! [A: a,U: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ U )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ U @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ A @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ) ).
% mult_of.divides_unit
thf(fact_1101_mult__of_Ounit__divides,axiom,
! [U: a,A: a] :
( ( member_a @ U @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ U @ A ) ) ) ).
% mult_of.unit_divides
thf(fact_1102_ring__irreducibleE_I5_J,axiom,
! [R3: a,A: a,B: a] :
( ( member_a @ R3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R3 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( R3
= ( mult_a_ring_ext_a_b @ r @ A @ B ) )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
| ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ) ) ).
% ring_irreducibleE(5)
thf(fact_1103_p_Ounit__factor,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B ) @ ( units_2932844235741507942t_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 @ A @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% p.unit_factor
thf(fact_1104_p_Oprod__unit__r,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( units_2932844235741507942t_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 @ A @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ).
% p.prod_unit_r
thf(fact_1105_p_Oprod__unit__l,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( units_2932844235741507942t_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 @ B @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ).
% p.prod_unit_l
thf(fact_1106_p_Omult__of_OUnits__closed,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( member_list_a @ X @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.mult_of.Units_closed
thf(fact_1107_p_Ounit__divides,axiom,
! [U: list_a,A: list_a] :
( ( member_list_a @ U @ ( units_2932844235741507942t_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 ) ) ) )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ U @ A ) ) ) ).
% p.unit_divides
thf(fact_1108_p_Odivides__unit,axiom,
! [A: list_a,U: list_a] :
( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ U )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ U @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ A @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% p.divides_unit
thf(fact_1109_p_Oring__irreducibleE_I4_J,axiom,
! [R3: list_a] :
( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ R3 )
=> ~ ( member_list_a @ R3 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.ring_irreducibleE(4)
thf(fact_1110_p_Oa__rcos__zero,axiom,
! [I4: set_list_a,I: list_a] :
( ( ideal_8896367198367571637t_unit @ I4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a @ I @ I4 )
=> ( ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I4 @ I )
= I4 ) ) ) ).
% p.a_rcos_zero
thf(fact_1111_p_Omult__of_Oprod__unit__l,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B ) @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ A @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( member_list_a @ B @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ) ).
% p.mult_of.prod_unit_l
thf(fact_1112_p_Omult__of_Oprod__unit__r,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B ) @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( member_list_a @ A @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ) ).
% p.mult_of.prod_unit_r
thf(fact_1113_p_Omult__of_Ounit__factor,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B ) @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( member_list_a @ A @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ).
% p.mult_of.unit_factor
thf(fact_1114_p_Omult__of_Ounit__divides,axiom,
! [U: list_a,A: list_a] :
( ( member_list_a @ U @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ U @ A ) ) ) ).
% p.mult_of.unit_divides
thf(fact_1115_p_Omult__of_Odivides__unit,axiom,
! [A: list_a,U: list_a] :
( ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ U )
=> ( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ U @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( member_list_a @ A @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ).
% p.mult_of.divides_unit
thf(fact_1116_p_Oa__r__coset__subset__G,axiom,
! [H: set_list_a,X: list_a] :
( ( ord_le8861187494160871172list_a @ H @ ( 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_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ X ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.a_r_coset_subset_G
thf(fact_1117_p_Oring__irreducibleE_I5_J,axiom,
! [R3: list_a,A: list_a,B: list_a] :
( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ R3 )
=> ( ( 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 ) ) ) )
=> ( ( R3
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B ) )
=> ( ( member_list_a @ A @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
| ( member_list_a @ B @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ) ) ).
% p.ring_irreducibleE(5)
thf(fact_1118_p_Oa__rcosI,axiom,
! [H2: list_a,H: set_list_a,X: list_a] :
( ( member_list_a @ H2 @ H )
=> ( ( ord_le8861187494160871172list_a @ H @ ( 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 @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H2 @ X ) @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ X ) ) ) ) ) ).
% p.a_rcosI
thf(fact_1119_p_Oa__coset__add__assoc,axiom,
! [M4: set_list_a,G: list_a,H2: 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 @ H2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ M4 @ G ) @ H2 )
= ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ M4 @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ H2 ) ) ) ) ) ) ).
% p.a_coset_add_assoc
thf(fact_1120_Units__mult__eq__Units,axiom,
( ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) )
= ( units_a_ring_ext_a_b @ r ) ) ).
% Units_mult_eq_Units
thf(fact_1121_Units__m__closed,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ Y @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_m_closed
thf(fact_1122_Units__l__cancel,axiom,
! [X: a,Y: a,Z2: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( mult_a_ring_ext_a_b @ r @ X @ Z2 ) )
= ( Y = Z2 ) ) ) ) ) ).
% Units_l_cancel
thf(fact_1123_mult__of_OUnits__eq,axiom,
( ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) )
= ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ).
% mult_of.Units_eq
thf(fact_1124_mult__of_OUnits__m__closed,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ).
% mult_of.Units_m_closed
thf(fact_1125_finite__ring__finite__units,axiom,
( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) )
=> ( finite_finite_a @ ( units_a_ring_ext_a_b @ r ) ) ) ).
% finite_ring_finite_units
thf(fact_1126_p_OUnits__mult__eq__Units,axiom,
( ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% p.Units_mult_eq_Units
thf(fact_1127_p_OUnits__m__closed,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( units_2932844235741507942t_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 ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.Units_m_closed
thf(fact_1128_mult__of_OUnits__l__cancel,axiom,
! [X: a,Y: a,Z2: a] :
( ( member_a @ X @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Z2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( mult_a_ring_ext_a_b @ r @ X @ Z2 ) )
= ( Y = Z2 ) ) ) ) ) ).
% mult_of.Units_l_cancel
thf(fact_1129_p_OUnits__l__cancel,axiom,
! [X: list_a,Y: list_a,Z2: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_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 @ Z2 @ ( 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 ) ) @ X @ Z2 ) )
= ( Y = Z2 ) ) ) ) ) ).
% p.Units_l_cancel
thf(fact_1130_p_Omult__of_OUnits__m__closed,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ Y @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_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 ) @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% p.mult_of.Units_m_closed
thf(fact_1131_p_Ofinite__ring__finite__units,axiom,
( ( finite_finite_list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( finite_finite_list_a @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.finite_ring_finite_units
thf(fact_1132_p_Omult__of_OUnits__l__cancel,axiom,
! [X: list_a,Y: list_a,Z2: list_a] :
( ( member_list_a @ X @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ Y @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ Z2 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_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 ) ) @ X @ Z2 ) )
= ( Y = Z2 ) ) ) ) ) ).
% p.mult_of.Units_l_cancel
thf(fact_1133_p_Oa__coset__add__zero,axiom,
! [M4: set_list_a] :
( ( ord_le8861187494160871172list_a @ M4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ M4 @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= M4 ) ) ).
% p.a_coset_add_zero
thf(fact_1134_p_Oa__rcosetsI,axiom,
! [H: set_list_a,X: list_a] :
( ( ord_le8861187494160871172list_a @ H @ ( 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_set_list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ X ) @ ( a_RCOS6220190738183020281t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H ) ) ) ) ).
% p.a_rcosetsI
thf(fact_1135_p_Oquotient__eq__iff__same__a__r__cos,axiom,
! [I4: set_list_a,A: list_a,B: list_a] :
( ( ideal_8896367198367571637t_unit @ I4 @ ( 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 @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B ) @ I4 )
= ( ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I4 @ A )
= ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I4 @ B ) ) ) ) ) ) ).
% p.quotient_eq_iff_same_a_r_cos
thf(fact_1136_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_1137_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_1138_p_Omult__of_Ogcd__isgcd,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( isgcd_454116541068447652t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( somegc4251160090145361146t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ B ) @ A @ B ) ) ) ).
% p.mult_of.gcd_isgcd
thf(fact_1139_p_Osubalbegra__incl__imp__finite__dimension,axiom,
! [K2: set_list_a,E: set_list_a,V2: set_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ E )
=> ( ( embedd1768981623711841426t_unit @ K2 @ 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 ) ) @ K2 @ V2 ) ) ) ) ) ).
% p.subalbegra_incl_imp_finite_dimension
thf(fact_1140_p_Otelescopic__base__dim_I1_J,axiom,
! [K2: set_list_a,F3: set_list_a,E: set_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( 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 ) ) @ K2 @ 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 ) ) @ K2 @ E ) ) ) ) ) ).
% p.telescopic_base_dim(1)
thf(fact_1141_p_Omult__of_Ogcd__exists,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( member_list_a @ ( somegc4251160090145361146t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ B ) @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% p.mult_of.gcd_exists
thf(fact_1142_p_Omult__of_Ogcd__closed,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( member_list_a @ ( somegc4251160090145361146t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ B ) @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% p.mult_of.gcd_closed
thf(fact_1143_p_Ofinite__dimension__imp__subalgebra,axiom,
! [K2: set_list_a,E: set_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ E )
=> ( embedd1768981623711841426t_unit @ K2 @ E @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.finite_dimension_imp_subalgebra
thf(fact_1144_p_Omult__of_Ogcd__divides__r,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( somegc4251160090145361146t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ B ) @ B ) ) ) ).
% p.mult_of.gcd_divides_r
thf(fact_1145_p_Omult__of_Ogcd__divides__l,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( somegc4251160090145361146t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ B ) @ A ) ) ) ).
% p.mult_of.gcd_divides_l
thf(fact_1146_p_Omult__of_Ogcd__divides,axiom,
! [Z2: list_a,X: list_a,Y: list_a] :
( ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ Z2 @ X )
=> ( ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ Z2 @ Y )
=> ( ( member_list_a @ X @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ Y @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ Z2 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ Z2 @ ( somegc4251160090145361146t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X @ Y ) ) ) ) ) ) ) ).
% p.mult_of.gcd_divides
thf(fact_1147_pprimeI,axiom,
! [K2: set_a,P: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( P != nil_a )
=> ( ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ! [Q2: list_a,R2: list_a] :
( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K2 ) @ Q2 @ R2 ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ Q2 )
| ( polyno5814909790663948098es_a_b @ r @ P @ R2 ) ) ) ) )
=> ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K2 ) @ P ) ) ) ) ) ) ).
% pprimeI
thf(fact_1148_p_Oring__ideal__imp__quot__ideal,axiom,
! [I4: set_list_a,J3: set_list_a] :
( ( ideal_8896367198367571637t_unit @ I4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ideal_8896367198367571637t_unit @ J3 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ideal_5294671857925479125t_unit @ ( image_5464838071766335845list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I4 ) @ J3 ) @ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I4 ) ) ) ) ).
% p.ring_ideal_imp_quot_ideal
thf(fact_1149_mult__of_Ogcd__exists,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ).
% mult_of.gcd_exists
thf(fact_1150_mult__of_Ogcd__closed,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ).
% mult_of.gcd_closed
thf(fact_1151_polynomial__ring__assms,axiom,
subfield_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% polynomial_ring_assms
thf(fact_1152_subring__props_I2_J,axiom,
! [K2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( member_a @ ( zero_a_b @ r ) @ K2 ) ) ).
% subring_props(2)
thf(fact_1153_subring__props_I7_J,axiom,
! [K2: set_a,H1: a,H22: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_a @ H1 @ K2 )
=> ( ( member_a @ H22 @ K2 )
=> ( member_a @ ( add_a_b @ r @ H1 @ H22 ) @ K2 ) ) ) ) ).
% subring_props(7)
thf(fact_1154_subring__props_I6_J,axiom,
! [K2: set_a,H1: a,H22: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_a @ H1 @ K2 )
=> ( ( member_a @ H22 @ K2 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ H1 @ H22 ) @ K2 ) ) ) ) ).
% subring_props(6)
thf(fact_1155_subring__props_I1_J,axiom,
! [K2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% subring_props(1)
thf(fact_1156_mult__of_Ogcd__divides__r,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B ) @ B ) ) ) ).
% mult_of.gcd_divides_r
thf(fact_1157_mult__of_Ogcd__divides__l,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B ) @ A ) ) ) ).
% mult_of.gcd_divides_l
thf(fact_1158_mult__of_Ogcd__divides,axiom,
! [Z2: a,X: a,Y: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ Z2 @ X )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ Z2 @ Y )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Z2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ Z2 @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ X @ Y ) ) ) ) ) ) ) ).
% mult_of.gcd_divides
thf(fact_1159_mult__of_Ogcd__isgcd,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( isgcd_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B ) @ A @ B ) ) ) ).
% mult_of.gcd_isgcd
thf(fact_1160_univ__poly__is__principal,axiom,
! [K2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ring_p8098905331641078952t_unit @ ( univ_poly_a_b @ r @ K2 ) ) ) ).
% univ_poly_is_principal
thf(fact_1161_pdivides__iff__shell,axiom,
! [K2: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ Q )
= ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ K2 ) @ P @ Q ) ) ) ) ) ).
% pdivides_iff_shell
thf(fact_1162_pprimeE_I1_J,axiom,
! [K2: set_a,P: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K2 ) @ P )
=> ( P != nil_a ) ) ) ) ).
% pprimeE(1)
thf(fact_1163_pprimeE_I2_J,axiom,
! [K2: set_a,P: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K2 ) @ P )
=> ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K2 ) ) ) ) ) ) ).
% pprimeE(2)
thf(fact_1164_pprime__iff__pirreducible,axiom,
! [K2: set_a,P: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K2 ) @ P )
= ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K2 ) @ P ) ) ) ) ).
% pprime_iff_pirreducible
thf(fact_1165_pprimeE_I3_J,axiom,
! [K2: set_a,P: list_a,Q: list_a,R3: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K2 ) @ P )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K2 ) @ Q @ R3 ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ Q )
| ( polyno5814909790663948098es_a_b @ r @ P @ R3 ) ) ) ) ) ) ) ) ).
% pprimeE(3)
thf(fact_1166_pirreducible__pow__pdivides__iff,axiom,
! [K2: set_a,P: list_a,Q: list_a,R3: list_a,N: nat] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K2 ) @ P )
=> ( ~ ( polyno5814909790663948098es_a_b @ r @ P @ Q )
=> ( ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K2 ) @ P @ N ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K2 ) @ Q @ R3 ) )
= ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K2 ) @ P @ N ) @ R3 ) ) ) ) ) ) ) ) ).
% pirreducible_pow_pdivides_iff
thf(fact_1167_exists__unique__long__division,axiom,
! [K2: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( Q != nil_a )
=> ? [X3: produc9164743771328383783list_a] :
( ( polyno2806191415236617128es_a_b @ r @ P @ Q @ X3 )
& ! [Y2: produc9164743771328383783list_a] :
( ( polyno2806191415236617128es_a_b @ r @ P @ Q @ Y2 )
=> ( Y2 = X3 ) ) ) ) ) ) ) ).
% exists_unique_long_division
thf(fact_1168_long__division__zero_I1_J,axiom,
! [K2: set_a,Q: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( polynomial_pdiv_a_b @ r @ nil_a @ Q )
= nil_a ) ) ) ).
% long_division_zero(1)
thf(fact_1169_long__division__closed_I1_J,axiom,
! [K2: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( member_list_a @ ( polynomial_pdiv_a_b @ r @ P @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) ) ) ) ) ).
% long_division_closed(1)
thf(fact_1170_long__division__add_I1_J,axiom,
! [K2: set_a,A: list_a,B: list_a,Q: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( polynomial_pdiv_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K2 ) @ A @ B ) @ Q )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K2 ) @ ( polynomial_pdiv_a_b @ r @ A @ Q ) @ ( polynomial_pdiv_a_b @ r @ B @ Q ) ) ) ) ) ) ) ).
% long_division_add(1)
thf(fact_1171_p_Ochar__ring__is__subfield,axiom,
( ( ord_less_nat @ zero_zero_nat @ ( ring_c500279861223467766t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( subfie1779122896746047282t_unit @ ( image_int_list_a @ ( ring_i7845631192508594762t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ top_top_set_int ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% p.char_ring_is_subfield
thf(fact_1172_pdiv__pmod,axiom,
! [K2: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( P
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K2 ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K2 ) @ Q @ ( polynomial_pdiv_a_b @ r @ P @ Q ) ) @ ( polynomial_pmod_a_b @ r @ P @ Q ) ) ) ) ) ) ).
% pdiv_pmod
thf(fact_1173_long__division__closed_I2_J,axiom,
! [K2: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( member_list_a @ ( polynomial_pmod_a_b @ r @ P @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) ) ) ) ) ).
% long_division_closed(2)
thf(fact_1174_long__division__add_I2_J,axiom,
! [K2: set_a,A: list_a,B: list_a,Q: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( polynomial_pmod_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K2 ) @ A @ B ) @ Q )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K2 ) @ ( polynomial_pmod_a_b @ r @ A @ Q ) @ ( polynomial_pmod_a_b @ r @ B @ Q ) ) ) ) ) ) ) ).
% long_division_add(2)
thf(fact_1175_long__division__add__iff,axiom,
! [K2: set_a,A: list_a,B: list_a,C: list_a,Q: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( ( polynomial_pmod_a_b @ r @ A @ Q )
= ( polynomial_pmod_a_b @ r @ B @ Q ) )
= ( ( polynomial_pmod_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K2 ) @ A @ C ) @ Q )
= ( polynomial_pmod_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K2 ) @ B @ C ) @ Q ) ) ) ) ) ) ) ) ).
% long_division_add_iff
thf(fact_1176_long__division__zero_I2_J,axiom,
! [K2: set_a,Q: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( polynomial_pmod_a_b @ r @ nil_a @ Q )
= nil_a ) ) ) ).
% long_division_zero(2)
thf(fact_1177_pmod__zero__iff__pdivides,axiom,
! [K2: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( ( polynomial_pmod_a_b @ r @ P @ Q )
= nil_a )
= ( polyno5814909790663948098es_a_b @ r @ Q @ P ) ) ) ) ) ).
% pmod_zero_iff_pdivides
thf(fact_1178_p_Ochar__ring__is__subring,axiom,
subrin6918843898125473962t_unit @ ( image_int_list_a @ ( ring_i7845631192508594762t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ top_top_set_int ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% p.char_ring_is_subring
thf(fact_1179_same__pmod__iff__pdivides,axiom,
! [K2: set_a,A: list_a,B: list_a,Q: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( ( polynomial_pmod_a_b @ r @ A @ Q )
= ( polynomial_pmod_a_b @ r @ B @ Q ) )
= ( polyno5814909790663948098es_a_b @ r @ Q @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ K2 ) @ A @ B ) ) ) ) ) ) ) ).
% same_pmod_iff_pdivides
thf(fact_1180_p_Ochar__ring__is__subdomain,axiom,
subdom7821232466298058046t_unit @ ( image_int_list_a @ ( ring_i7845631192508594762t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ top_top_set_int ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% p.char_ring_is_subdomain
thf(fact_1181_int_Ol__one,axiom,
! [X: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( times_times_int @ one_one_int @ X )
= X ) ) ).
% int.l_one
thf(fact_1182_int_Or__one,axiom,
! [X: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( times_times_int @ X @ one_one_int )
= X ) ) ).
% int.r_one
thf(fact_1183_int_Onat__pow__0,axiom,
! [X: int] :
( ( power_power_int @ X @ zero_zero_nat )
= one_one_int ) ).
% int.nat_pow_0
thf(fact_1184_int_Om__closed,axiom,
! [X: int,Y: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( member_int @ ( times_times_int @ X @ Y ) @ top_top_set_int ) ) ) ).
% int.m_closed
thf(fact_1185_int_Ole__refl,axiom,
! [X: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ord_less_eq_int @ X @ X ) ) ).
% int.le_refl
thf(fact_1186_int_Ole__antisym,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ X )
=> ( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( X = Y ) ) ) ) ) ).
% int.le_antisym
thf(fact_1187_int_Or__null,axiom,
! [X: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( times_times_int @ X @ zero_zero_int )
= zero_zero_int ) ) ).
% int.r_null
thf(fact_1188_int_Ol__null,axiom,
! [X: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( times_times_int @ zero_zero_int @ X )
= zero_zero_int ) ) ).
% int.l_null
thf(fact_1189_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_1190_int_Onat__pow__comm,axiom,
! [X: int,N: nat,M: nat] :
( ( member_int @ X @ top_top_set_int )
=> ( ( times_times_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ X @ M ) )
= ( times_times_int @ ( power_power_int @ X @ M ) @ ( power_power_int @ X @ N ) ) ) ) ).
% int.nat_pow_comm
thf(fact_1191_int_Onat__pow__distrib,axiom,
! [X: int,Y: int,N: nat] :
( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( power_power_int @ ( times_times_int @ X @ Y ) @ N )
= ( times_times_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ N ) ) ) ) ) ).
% int.nat_pow_distrib
thf(fact_1192_int_Opow__mult__distrib,axiom,
! [X: int,Y: int,N: nat] :
( ( ( times_times_int @ X @ Y )
= ( times_times_int @ Y @ X ) )
=> ( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( power_power_int @ ( times_times_int @ X @ Y ) @ N )
= ( times_times_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ N ) ) ) ) ) ) ).
% int.pow_mult_distrib
thf(fact_1193_int_Ogroup__commutes__pow,axiom,
! [X: int,Y: int,N: nat] :
( ( ( times_times_int @ X @ Y )
= ( times_times_int @ Y @ X ) )
=> ( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( times_times_int @ ( power_power_int @ X @ N ) @ Y )
= ( times_times_int @ Y @ ( power_power_int @ X @ N ) ) ) ) ) ) ).
% int.group_commutes_pow
thf(fact_1194_int_Om__comm,axiom,
! [X: int,Y: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( times_times_int @ X @ Y )
= ( times_times_int @ Y @ X ) ) ) ) ).
% int.m_comm
thf(fact_1195_int_Om__assoc,axiom,
! [X: int,Y: int,Z2: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( member_int @ Z2 @ top_top_set_int )
=> ( ( times_times_int @ ( times_times_int @ X @ Y ) @ Z2 )
= ( times_times_int @ X @ ( times_times_int @ Y @ Z2 ) ) ) ) ) ) ).
% int.m_assoc
thf(fact_1196_int_Om__lcomm,axiom,
! [X: int,Y: int,Z2: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( member_int @ Z2 @ top_top_set_int )
=> ( ( times_times_int @ X @ ( times_times_int @ Y @ Z2 ) )
= ( times_times_int @ Y @ ( times_times_int @ X @ Z2 ) ) ) ) ) ) ).
% int.m_lcomm
thf(fact_1197_int_Ole__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z2 )
=> ( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( member_int @ Z2 @ top_top_set_int )
=> ( ord_less_eq_int @ X @ Z2 ) ) ) ) ) ) ).
% int.le_trans
thf(fact_1198_int_Ototal__order__total,axiom,
! [X: int,Y: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ) ) ).
% int.total_order_total
thf(fact_1199_int_Onat__pow__pow,axiom,
! [X: int,N: nat,M: nat] :
( ( member_int @ X @ top_top_set_int )
=> ( ( power_power_int @ ( power_power_int @ X @ N ) @ M )
= ( power_power_int @ X @ ( times_times_nat @ N @ M ) ) ) ) ).
% int.nat_pow_pow
thf(fact_1200_int_Olless__eq,axiom,
( ord_less_int
= ( ^ [X2: int,Y6: int] :
( ( ord_less_eq_int @ X2 @ Y6 )
& ( X2 != Y6 ) ) ) ) ).
% int.lless_eq
thf(fact_1201_int_Ointegral__iff,axiom,
! [A: int,B: int] :
( ( member_int @ A @ top_top_set_int )
=> ( ( member_int @ B @ top_top_set_int )
=> ( ( ( times_times_int @ A @ B )
= zero_zero_int )
= ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ) ) ).
% int.integral_iff
thf(fact_1202_int_Om__rcancel,axiom,
! [A: int,B: int,C: int] :
( ( A != zero_zero_int )
=> ( ( member_int @ A @ top_top_set_int )
=> ( ( member_int @ B @ top_top_set_int )
=> ( ( member_int @ C @ top_top_set_int )
=> ( ( ( times_times_int @ B @ A )
= ( times_times_int @ C @ A ) )
= ( B = C ) ) ) ) ) ) ).
% int.m_rcancel
thf(fact_1203_int_Om__lcancel,axiom,
! [A: int,B: int,C: int] :
( ( A != zero_zero_int )
=> ( ( member_int @ A @ top_top_set_int )
=> ( ( member_int @ B @ top_top_set_int )
=> ( ( member_int @ C @ top_top_set_int )
=> ( ( ( times_times_int @ A @ B )
= ( times_times_int @ A @ C ) )
= ( B = C ) ) ) ) ) ) ).
% int.m_lcancel
thf(fact_1204_int_Ointegral,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
=> ( ( member_int @ A @ top_top_set_int )
=> ( ( member_int @ B @ top_top_set_int )
=> ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ) ) ).
% int.integral
thf(fact_1205_int_Oone__unique,axiom,
! [U: int] :
( ( member_int @ U @ top_top_set_int )
=> ( ! [X3: int] :
( ( member_int @ X3 @ top_top_set_int )
=> ( ( times_times_int @ U @ X3 )
= X3 ) )
=> ( U = one_one_int ) ) ) ).
% int.one_unique
thf(fact_1206_int_Oinv__unique,axiom,
! [Y: int,X: int,Y5: int] :
( ( ( times_times_int @ Y @ X )
= one_one_int )
=> ( ( ( times_times_int @ X @ Y5 )
= one_one_int )
=> ( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( member_int @ Y5 @ top_top_set_int )
=> ( Y = Y5 ) ) ) ) ) ) ).
% int.inv_unique
thf(fact_1207_long__divisionE,axiom,
! [K2: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( Q != nil_a )
=> ( polyno2806191415236617128es_a_b @ r @ P @ Q @ ( produc6837034575241423639list_a @ ( polynomial_pdiv_a_b @ r @ P @ Q ) @ ( polynomial_pmod_a_b @ r @ P @ Q ) ) ) ) ) ) ) ).
% long_divisionE
thf(fact_1208_poly__add_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
~ ! [P1: list_a,P22: list_a] :
( X
!= ( produc6837034575241423639list_a @ P1 @ P22 ) ) ).
% poly_add.cases
thf(fact_1209_exists__long__division,axiom,
! [K2: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( Q != nil_a )
=> ~ ! [B4: list_a] :
( ( member_list_a @ B4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ! [R2: list_a] :
( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ~ ( polyno2806191415236617128es_a_b @ r @ P @ Q @ ( produc6837034575241423639list_a @ B4 @ R2 ) ) ) ) ) ) ) ) ).
% exists_long_division
thf(fact_1210_long__divisionI,axiom,
! [K2: set_a,P: list_a,Q: list_a,B: list_a,R3: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( Q != nil_a )
=> ( ( polyno2806191415236617128es_a_b @ r @ P @ Q @ ( produc6837034575241423639list_a @ B @ R3 ) )
=> ( ( produc6837034575241423639list_a @ B @ R3 )
= ( produc6837034575241423639list_a @ ( polynomial_pdiv_a_b @ r @ P @ Q ) @ ( polynomial_pmod_a_b @ r @ P @ Q ) ) ) ) ) ) ) ) ).
% long_divisionI
thf(fact_1211_p_Ochar__ring__is__subcring,axiom,
subcri7763218559781929323t_unit @ ( image_int_list_a @ ( ring_i7845631192508594762t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ top_top_set_int ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% p.char_ring_is_subcring
thf(fact_1212_not__exp__less__eq__0__int,axiom,
! [N: nat] :
~ ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ zero_zero_int ) ).
% not_exp_less_eq_0_int
thf(fact_1213_p_Ocarrier__is__subcring,axiom,
subcri7763218559781929323t_unit @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% p.carrier_is_subcring
thf(fact_1214_p_OsubcringI_H,axiom,
! [H: set_list_a] :
( ( subrin6918843898125473962t_unit @ H @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( subcri7763218559781929323t_unit @ H @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% p.subcringI'
thf(fact_1215_p_OsubcringI,axiom,
! [H: set_list_a] :
( ( subrin6918843898125473962t_unit @ H @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ! [H12: list_a,H23: list_a] :
( ( member_list_a @ H12 @ H )
=> ( ( member_list_a @ H23 @ H )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H12 @ H23 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H23 @ H12 ) ) ) )
=> ( subcri7763218559781929323t_unit @ H @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.subcringI
thf(fact_1216_p_Odivides__pirreducible__condition,axiom,
! [K2: set_list_a,Q: list_list_a,P: list_list_a] :
( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ Q )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( factor6954119973539764400t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ P @ Q )
=> ( ( member_list_list_a @ P @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
| ( associ5603075271488036121t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ P @ Q ) ) ) ) ) ).
% p.divides_pirreducible_condition
thf(fact_1217_pderiv__mult,axiom,
! [F: list_a,G: list_a] :
( ( member_list_a @ F @ ( 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 ) ) @ F @ 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 @ F ) @ G ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ ( formal4452980811800949548iv_a_b @ r @ G ) ) ) ) ) ) ).
% pderiv_mult
thf(fact_1218_pderiv__zero,axiom,
! [K2: set_a] :
( ( formal4452980811800949548iv_a_b @ r @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ K2 ) ) ) ).
% pderiv_zero
thf(fact_1219_pderiv__carr,axiom,
! [F: list_a] :
( ( member_list_a @ F @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( formal4452980811800949548iv_a_b @ r @ F ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% pderiv_carr
thf(fact_1220_pderiv__add,axiom,
! [F: list_a,G: list_a] :
( ( member_list_a @ F @ ( 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 ) ) @ F @ G ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( formal4452980811800949548iv_a_b @ r @ F ) @ ( formal4452980811800949548iv_a_b @ r @ G ) ) ) ) ) ).
% pderiv_add
thf(fact_1221_p_Oassociated__polynomials__imp__same__is__root,axiom,
! [P: list_list_a,Q: list_list_a,X: list_a] :
( ( member_list_list_a @ P @ ( 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 @ Q @ ( 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 ) ) ) ) ) )
=> ( ( associ5603075271488036121t_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 @ Q )
=> ( ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ X )
= ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q @ X ) ) ) ) ) ).
% p.associated_polynomials_imp_same_is_root
thf(fact_1222_pderiv__pow,axiom,
! [N: nat,F: list_a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( member_list_a @ F @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( formal4452980811800949548iv_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ N ) )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( ring_i7845631192508594762t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( semiri1314217659103216013at_int @ N ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ ( minus_minus_nat @ N @ one_one_nat ) ) ) @ ( formal4452980811800949548iv_a_b @ r @ F ) ) ) ) ) ).
% pderiv_pow
thf(fact_1223_p_Oassociated__polynomials__imp__same__roots,axiom,
! [P: list_list_a,Q: list_list_a] :
( ( member_list_list_a @ P @ ( 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 @ Q @ ( 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 ) ) ) ) ) )
=> ( ( associ5603075271488036121t_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 @ Q )
=> ( ( polyno7858422826990252003t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
= ( polyno7858422826990252003t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q ) ) ) ) ) ).
% p.associated_polynomials_imp_same_roots
thf(fact_1224_associated__sym,axiom,
! [A: a,B: a] :
( ( associ5860276527279195403xt_a_b @ r @ A @ B )
=> ( associ5860276527279195403xt_a_b @ r @ B @ A ) ) ).
% associated_sym
thf(fact_1225_associated__trans,axiom,
! [A: a,B: a,C: a] :
( ( associ5860276527279195403xt_a_b @ r @ A @ B )
=> ( ( associ5860276527279195403xt_a_b @ r @ B @ C )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A @ C ) ) ) ) ) ).
% associated_trans
thf(fact_1226_assoc__subst,axiom,
! [A: a,B: a,F: a > a] :
( ( associ5860276527279195403xt_a_b @ r @ A @ B )
=> ( ! [A6: a,B4: a] :
( ( ( member_a @ A6 @ ( partia707051561876973205xt_a_b @ r ) )
& ( member_a @ B4 @ ( partia707051561876973205xt_a_b @ r ) )
& ( associ5860276527279195403xt_a_b @ r @ A6 @ B4 ) )
=> ( ( member_a @ ( F @ A6 ) @ ( partia707051561876973205xt_a_b @ r ) )
& ( member_a @ ( F @ B4 ) @ ( partia707051561876973205xt_a_b @ r ) )
& ( associ5860276527279195403xt_a_b @ r @ ( F @ A6 ) @ ( F @ B4 ) ) ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ ( F @ A ) @ ( F @ B ) ) ) ) ) ) ).
% assoc_subst
thf(fact_1227_mult__of_Oassociated__sym,axiom,
! [A: a,B: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ B @ A ) ) ).
% mult_of.associated_sym
thf(fact_1228_Units__assoc,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A @ B ) ) ) ).
% Units_assoc
thf(fact_1229_p_Oassociated__sym,axiom,
! [A: list_a,B: list_a] :
( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ A ) ) ).
% p.associated_sym
thf(fact_1230_assoc__iff__assoc__mult,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( associ5860276527279195403xt_a_b @ r @ A @ B )
= ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B ) ) ) ) ).
% assoc_iff_assoc_mult
thf(fact_1231_mult__cong__l,axiom,
! [A: a,A7: a,B: a] :
( ( associ5860276527279195403xt_a_b @ r @ A @ A7 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A7 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( mult_a_ring_ext_a_b @ r @ A7 @ B ) ) ) ) ) ) ).
% mult_cong_l
thf(fact_1232_mult__cong__r,axiom,
! [B: a,B5: a,A: a] :
( ( associ5860276527279195403xt_a_b @ r @ B @ B5 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B5 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( mult_a_ring_ext_a_b @ r @ A @ B5 ) ) ) ) ) ) ).
% mult_cong_r
thf(fact_1233_mult__of_Oassoc__subst,axiom,
! [A: a,B: a,F: a > a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( ! [A6: a,B4: a] :
( ( ( member_a @ A6 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( member_a @ B4 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A6 @ B4 ) )
=> ( ( member_a @ ( F @ A6 ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( member_a @ ( F @ B4 ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( F @ A6 ) @ ( F @ B4 ) ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( F @ A ) @ ( F @ B ) ) ) ) ) ) ).
% mult_of.assoc_subst
thf(fact_1234_mult__of_Oassociated__trans,axiom,
! [A: a,B: a,C: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ B @ C )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ C ) ) ) ) ) ).
% mult_of.associated_trans
thf(fact_1235_Units__cong,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( associ5860276527279195403xt_a_b @ r @ A @ B )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ).
% Units_cong
thf(fact_1236_divides__cong__l,axiom,
! [X: a,X5: a,Y: a] :
( ( associ5860276527279195403xt_a_b @ r @ X @ X5 )
=> ( ( factor8216151070175719842xt_a_b @ r @ X5 @ Y )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ X @ Y ) ) ) ) ).
% divides_cong_l
thf(fact_1237_divides__cong__r,axiom,
! [X: a,Y: a,Y5: a] :
( ( factor8216151070175719842xt_a_b @ r @ X @ Y )
=> ( ( associ5860276527279195403xt_a_b @ r @ Y @ Y5 )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ X @ Y5 ) ) ) ) ).
% divides_cong_r
thf(fact_1238_mult__of_OUnits__assoc,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B ) ) ) ).
% mult_of.Units_assoc
thf(fact_1239_p_Oassociated__trans,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B )
=> ( ( associ8407585678920448409t_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 @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ C ) ) ) ) ) ).
% p.associated_trans
thf(fact_1240_p_Oassoc__subst,axiom,
! [A: list_a,B: list_a,F: list_a > list_a] :
( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B )
=> ( ! [A6: list_a,B4: list_a] :
( ( ( member_list_a @ A6 @ ( 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 ) ) ) )
& ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A6 @ B4 ) )
=> ( ( member_list_a @ ( F @ A6 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( member_list_a @ ( F @ B4 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( F @ A6 ) @ ( F @ B4 ) ) ) )
=> ( ( 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 ) ) ) )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( F @ A ) @ ( F @ B ) ) ) ) ) ) ).
% p.assoc_subst
thf(fact_1241_p_Omult__of_Oassociated__sym,axiom,
! [A: list_a,B: list_a] :
( ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ B )
=> ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ B @ A ) ) ).
% p.mult_of.associated_sym
thf(fact_1242_p_OUnits__assoc,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ A @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B ) ) ) ).
% p.Units_assoc
thf(fact_1243_mult__of_Oassoc__l__cancel,axiom,
! [A: a,B: a,B5: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B5 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( mult_a_ring_ext_a_b @ r @ A @ B5 ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ B @ B5 ) ) ) ) ) ).
% mult_of.assoc_l_cancel
thf(fact_1244_mult__of_Oassoc__r__cancel,axiom,
! [A: a,B: a,A7: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( mult_a_ring_ext_a_b @ r @ A7 @ B ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A7 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ A7 ) ) ) ) ) ).
% mult_of.assoc_r_cancel
thf(fact_1245_mult__of_Omult__cong__l,axiom,
! [A: a,A7: a,B: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ A7 )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A7 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( mult_a_ring_ext_a_b @ r @ A7 @ B ) ) ) ) ) ) ).
% mult_of.mult_cong_l
thf(fact_1246_mult__of_Omult__cong__r,axiom,
! [B: a,B5: a,A: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ B @ B5 )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B5 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( mult_a_ring_ext_a_b @ r @ A @ B5 ) ) ) ) ) ) ).
% mult_of.mult_cong_r
thf(fact_1247_ring__associated__iff,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( associ5860276527279195403xt_a_b @ r @ A @ B )
= ( ? [X2: a] :
( ( member_a @ X2 @ ( units_a_ring_ext_a_b @ r ) )
& ( A
= ( mult_a_ring_ext_a_b @ r @ X2 @ B ) ) ) ) ) ) ) ).
% ring_associated_iff
thf(fact_1248_associatedI2_H,axiom,
! [A: a,B: a,U: a] :
( ( A
= ( mult_a_ring_ext_a_b @ r @ B @ U ) )
=> ( ( member_a @ U @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A @ B ) ) ) ) ).
% associatedI2'
thf(fact_1249_associatedI2,axiom,
! [U: a,A: a,B: a] :
( ( member_a @ U @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( A
= ( mult_a_ring_ext_a_b @ r @ B @ U ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A @ B ) ) ) ) ).
% associatedI2
thf(fact_1250_mult__of_Odivides__cong__l,axiom,
! [X: a,X5: a,Y: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ X @ X5 )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ X5 @ Y )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ X @ Y ) ) ) ) ).
% mult_of.divides_cong_l
thf(fact_1251_mult__of_Odivides__cong__r,axiom,
! [X: a,Y: a,Y5: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ X @ Y )
=> ( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ Y @ Y5 )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ X @ Y5 ) ) ) ) ).
% mult_of.divides_cong_r
thf(fact_1252_mult__of_Oassoc__unit__l,axiom,
! [A: a,B: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( ( member_a @ B @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ A @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ) ).
% mult_of.assoc_unit_l
thf(fact_1253_mult__of_Oassoc__unit__r,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ B @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ) ).
% mult_of.assoc_unit_r
thf(fact_1254_mult__of_Ogcd__assoc,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B ) @ C ) @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ A @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ B @ C ) ) ) ) ) ) ).
% mult_of.gcd_assoc
thf(fact_1255_mult__of_Ogcd__cong__l,axiom,
! [X: a,X5: a,Y: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ X @ X5 )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ X5 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ X @ Y ) @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ X5 @ Y ) ) ) ) ) ) ).
% mult_of.gcd_cong_l
thf(fact_1256_mult__of_Ogcd__cong__r,axiom,
! [Y: a,Y5: a,X: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ Y @ Y5 )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y5 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ X @ Y ) @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ X @ Y5 ) ) ) ) ) ) ).
% mult_of.gcd_cong_r
thf(fact_1257_mult__of_Oprime__cong,axiom,
! [P: a,P3: a] :
( ( prime_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ P )
=> ( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ P @ P3 )
=> ( ( member_a @ P @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ P3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( prime_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ P3 ) ) ) ) ) ).
% mult_of.prime_cong
thf(fact_1258_mult__of_Ogcdof__cong__l,axiom,
! [A7: a,A: a,B: a,C: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A7 @ A )
=> ( ( isgcd_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ A @ B @ C )
=> ( ( member_a @ A7 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( isgcd_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ A7 @ B @ C ) ) ) ) ) ) ) ).
% mult_of.gcdof_cong_l
thf(fact_1259_mult__of_Oassociated__fcount,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( ( factor4067924603488134956t_unit @ ( ring_mult_of_a_b @ r ) @ A )
= ( factor4067924603488134956t_unit @ ( ring_mult_of_a_b @ r ) @ B ) ) ) ) ) ).
% mult_of.associated_fcount
thf(fact_1260_p_Oassoc__iff__assoc__mult,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 ) ) ) )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B )
= ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ B ) ) ) ) ).
% p.assoc_iff_assoc_mult
thf(fact_1261_p_Omult__cong__r,axiom,
! [B: list_a,B5: list_a,A: list_a] :
( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ B5 )
=> ( ( 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 @ B5 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( associ8407585678920448409t_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 @ B5 ) ) ) ) ) ) ).
% p.mult_cong_r
thf(fact_1262_p_Omult__cong__l,axiom,
! [A: list_a,A7: list_a,B: list_a] :
( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ A7 )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A7 @ ( 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 ) ) ) )
=> ( associ8407585678920448409t_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 ) ) @ A7 @ B ) ) ) ) ) ) ).
% p.mult_cong_l
thf(fact_1263_p_Omult__of_Oassociated__trans,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ B )
=> ( ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ B @ C )
=> ( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ C @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ C ) ) ) ) ) ).
% p.mult_of.associated_trans
thf(fact_1264_p_Omult__of_Oassoc__subst,axiom,
! [A: list_a,B: list_a,F: list_a > list_a] :
( ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ B )
=> ( ! [A6: list_a,B4: list_a] :
( ( ( member_list_a @ A6 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
& ( member_list_a @ B4 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
& ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A6 @ B4 ) )
=> ( ( member_list_a @ ( F @ A6 ) @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
& ( member_list_a @ ( F @ B4 ) @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
& ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( F @ A6 ) @ ( F @ B4 ) ) ) )
=> ( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( F @ A ) @ ( F @ B ) ) ) ) ) ) ).
% p.mult_of.assoc_subst
thf(fact_1265_p_Omult__of_OUnits__assoc,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ A @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ B ) ) ) ).
% p.mult_of.Units_assoc
thf(fact_1266_p_OUnits__cong,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ A @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ B @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% p.Units_cong
thf(fact_1267_p_Odivides__cong__r,axiom,
! [X: list_a,Y: list_a,Y5: list_a] :
( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Y5 )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y5 ) ) ) ) ).
% p.divides_cong_r
thf(fact_1268_p_Odivides__cong__l,axiom,
! [X: list_a,X5: list_a,Y: list_a] :
( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ X5 )
=> ( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X5 @ Y )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) ) ) ) ).
% p.divides_cong_l
thf(fact_1269_monic__poly__not__assoc,axiom,
! [F: list_a,G: list_a] :
( ( monic_3145109188698636716ly_a_b @ r @ F )
=> ( ( monic_3145109188698636716ly_a_b @ r @ G )
=> ( ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ F @ G )
=> ( F = G ) ) ) ) ).
% monic_poly_not_assoc
thf(fact_1270_mult__of_Oassociated__iff,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
= ( ? [X2: a] :
( ( member_a @ X2 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( A
= ( mult_a_ring_ext_a_b @ r @ B @ X2 ) ) ) ) ) ) ) ).
% mult_of.associated_iff
thf(fact_1271_mult__of_OassociatedI2_H,axiom,
! [A: a,B: a,U: a] :
( ( A
= ( mult_a_ring_ext_a_b @ r @ B @ U ) )
=> ( ( member_a @ U @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B ) ) ) ) ).
% mult_of.associatedI2'
% Conjectures (1)
thf(conj_0,conjecture,
factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ f @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( card_I2373409586816755191ly_a_b @ r @ ( power_power_nat @ ( order_a_ring_ext_a_b @ r ) @ n ) ) ).
%------------------------------------------------------------------------------