TPTP Problem File: SLH0469^1.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 : Interpolation_Polynomials_HOL_Algebra/0001_Lagrange_Interpolation/prob_00195_007492__17287384_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1557 ( 368 unt; 279 typ; 0 def)
% Number of atoms : 4887 (1654 equ; 0 cnn)
% Maximal formula atoms : 17 ( 3 avg)
% Number of connectives : 18991 ( 290 ~; 37 |; 444 &;15479 @)
% ( 0 <=>;2741 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 8 avg)
% Number of types : 28 ( 27 usr)
% Number of type conns : 1695 (1695 >; 0 *; 0 +; 0 <<)
% Number of symbols : 255 ( 252 usr; 13 con; 0-4 aty)
% Number of variables : 4344 ( 348 ^;3841 !; 155 ?;4344 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:37:46.120
%------------------------------------------------------------------------------
% Could-be-implicit typings (27)
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finsum3771204255718262726it_int: partia5333488208502193986t_unit > ( int > list_list_list_a ) > set_int > list_list_list_a ).
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finsum3773694726227313002it_nat: partia5333488208502193986t_unit > ( nat > list_list_list_a ) > set_nat > list_list_list_a ).
thf(sy_c_Ring_Ofinsum_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_001t__Product____Type__Ounit_001tf__a,type,
finsum1842526356606396388unit_a: partia5333488208502193986t_unit > ( a > list_list_list_a ) > set_a > list_list_list_a ).
thf(sy_c_Ring_Ofinsum_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit_001t__Int__Oint,type,
finsum3988480971234277964it_int: partia2956882679547061052t_unit > ( int > list_list_a ) > set_int > list_list_a ).
thf(sy_c_Ring_Ofinsum_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit_001t__Nat__Onat,type,
finsum3990971441743328240it_nat: partia2956882679547061052t_unit > ( nat > list_list_a ) > set_nat > list_list_a ).
thf(sy_c_Ring_Ofinsum_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit_001tf__a,type,
finsum463596448938265310unit_a: partia2956882679547061052t_unit > ( a > list_list_a ) > set_a > list_list_a ).
thf(sy_c_Ring_Ofinsum_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J,type,
finsum4426778018909949125list_a: partia2670972154091845814t_unit > ( ( nat > list_a ) > list_a ) > set_nat_list_a > list_a ).
thf(sy_c_Ring_Ofinsum_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001_062_It__Nat__Onat_Mtf__a_J,type,
finsum7881878320310621759_nat_a: partia2670972154091845814t_unit > ( ( nat > a ) > list_a ) > set_nat_a > list_a ).
thf(sy_c_Ring_Ofinsum_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001t__Int__Oint,type,
finsum3495021991707498834it_int: partia2670972154091845814t_unit > ( int > list_a ) > set_int > list_a ).
thf(sy_c_Ring_Ofinsum_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001t__List__Olist_Itf__a_J,type,
finsum8721804980556663006list_a: partia2670972154091845814t_unit > ( list_a > list_a ) > set_list_a > list_a ).
thf(sy_c_Ring_Ofinsum_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001t__Nat__Onat,type,
finsum3497512462216549110it_nat: partia2670972154091845814t_unit > ( nat > list_a ) > set_nat > list_a ).
thf(sy_c_Ring_Ofinsum_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001tf__a,type,
finsum7322697649718157656unit_a: partia2670972154091845814t_unit > ( a > list_a ) > set_a > list_a ).
thf(sy_c_Ring_Ofinsum_001tf__a_001tf__b_001t__Nat__Onat,type,
finsum_a_b_nat: partia2175431115845679010xt_a_b > ( nat > a ) > set_nat > a ).
thf(sy_c_Ring_Oring_Oadd_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_001t__Product____Type__Ounit,type,
add_li5162926044081146114t_unit: partia5333488208502193986t_unit > list_list_list_a > list_list_list_a > list_list_list_a ).
thf(sy_c_Ring_Oring_Oadd_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
add_li174743652000525320t_unit: partia2956882679547061052t_unit > list_list_a > list_list_a > list_list_a ).
thf(sy_c_Ring_Oring_Oadd_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
add_li7652885771158616974t_unit: partia2670972154091845814t_unit > list_a > list_a > list_a ).
thf(sy_c_Ring_Oring_Oadd_001tf__a_001tf__b,type,
add_a_b: partia2175431115845679010xt_a_b > a > a > a ).
thf(sy_c_Ring_Oring_Ozero_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
zero_l347298301471573063t_unit: partia2956882679547061052t_unit > list_list_a ).
thf(sy_c_Ring_Oring_Ozero_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
zero_l4142658623432671053t_unit: partia2670972154091845814t_unit > list_a ).
thf(sy_c_Ring_Oring_Ozero_001tf__a_001tf__b,type,
zero_a_b: partia2175431115845679010xt_a_b > a ).
thf(sy_c_Ring_Osemiring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
semiri2871908745932252451t_unit: partia2670972154091845814t_unit > $o ).
thf(sy_c_Ring_Osemiring_001tf__a_001tf__b,type,
semiring_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring__Divisibility_Oeuclidean__domain_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_e7478897652244013592t_unit: partia2670972154091845814t_unit > ( list_a > nat ) > $o ).
thf(sy_c_Ring__Divisibility_Oprincipal__domain_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_p8098905331641078952t_unit: partia2670972154091845814t_unit > $o ).
thf(sy_c_Ring__Divisibility_Oring__irreducible_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_001t__Product____Type__Ounit,type,
ring_r5224476855413033410t_unit: partia5333488208502193986t_unit > list_list_list_a > $o ).
thf(sy_c_Ring__Divisibility_Oring__irreducible_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_r360171070648044744t_unit: partia2956882679547061052t_unit > list_list_a > $o ).
thf(sy_c_Ring__Divisibility_Oring__irreducible_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_r932985474545269838t_unit: partia2670972154091845814t_unit > list_a > $o ).
thf(sy_c_Ring__Divisibility_Oring__irreducible_001tf__a_001tf__b,type,
ring_r999134135267193926le_a_b: partia2175431115845679010xt_a_b > a > $o ).
thf(sy_c_Ring__Divisibility_Oring__prime_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_r6430282645014804837t_unit: partia2670972154091845814t_unit > list_a > $o ).
thf(sy_c_Ring__Divisibility_Oring__prime_001tf__a_001tf__b,type,
ring_ring_prime_a_b: partia2175431115845679010xt_a_b > a > $o ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J,type,
collect_nat_list_a: ( ( nat > list_a ) > $o ) > set_nat_list_a ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mtf__a_J,type,
collect_nat_a: ( ( nat > a ) > $o ) > set_nat_a ).
thf(sy_c_Set_OCollect_001t__Int__Oint,type,
collect_int: ( int > $o ) > set_int ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
collect_list_list_a: ( list_list_a > $o ) > set_list_list_a ).
thf(sy_c_Set_OCollect_001t__List__Olist_Itf__a_J,type,
collect_list_a: ( list_a > $o ) > set_list_a ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Int__Oint_J,type,
collect_set_int: ( set_int > $o ) > set_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_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J_001tf__a,type,
image_nat_list_a_a: ( ( nat > list_a ) > a ) > set_nat_list_a > set_a ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mtf__a_J_001tf__a,type,
image_nat_a_a: ( ( nat > a ) > a ) > set_nat_a > set_a ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
image_int_int: ( int > int ) > set_int > set_int ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Nat__Onat,type,
image_int_nat: ( int > nat ) > set_int > set_nat ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001tf__a,type,
image_int_a: ( int > a ) > set_int > set_a ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
image_1156962946714028939list_a: ( list_list_a > list_list_list_a ) > set_list_list_a > set_list_list_list_a ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__List__Olist_Itf__a_J_J_001tf__a,type,
image_list_list_a_a: ( list_list_a > a ) > set_list_list_a > set_a ).
thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001t__Int__Oint,type,
image_list_a_int: ( list_a > int ) > set_list_a > set_int ).
thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
image_8260866953997875467list_a: ( list_a > list_list_a ) > set_list_a > set_list_list_a ).
thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
image_list_a_list_a: ( list_a > list_a ) > set_list_a > set_list_a ).
thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001t__Nat__Onat,type,
image_list_a_nat: ( list_a > nat ) > set_list_a > set_nat ).
thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001tf__a,type,
image_list_a_a: ( list_a > a ) > set_list_a > set_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Int__Oint,type,
image_nat_int: ( nat > int ) > set_nat > set_int ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__List__Olist_Itf__a_J,type,
image_nat_list_a: ( nat > list_a ) > set_nat > set_list_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001tf__a,type,
image_nat_a: ( nat > a ) > set_nat > set_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__Int__Oint,type,
image_a_int: ( a > int ) > set_a > set_int ).
thf(sy_c_Set_Oimage_001tf__a_001t__List__Olist_Itf__a_J,type,
image_a_list_a: ( a > list_a ) > set_a > set_list_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__Nat__Onat,type,
image_a_nat: ( a > nat ) > set_a > set_nat ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
set_or1269000886237332187st_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Int__Oint,type,
set_ord_atMost_int: int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
set_ord_atMost_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
set_or6279072120763780779list_a: set_list_a > set_set_list_a ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Nat__Onat_J,type,
set_or4236626031148496127et_nat: set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_Itf__a_J,type,
set_ord_atMost_set_a: set_a > set_set_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_Osubfield_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
subfie4546268998243038636t_unit: set_list_list_a > partia2956882679547061052t_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_UnivPoly_Obound_001t__List__Olist_Itf__a_J,type,
bound_list_a: list_a > nat > ( nat > 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_Itf__a_J_001t__Product____Type__Ounit,type,
up_lis8464167429055313730t_unit: partia2670972154091845814t_unit > set_nat_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_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_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_It__List__Olist_Itf__a_J_J_J,type,
member5342144027231129785list_a: list_list_list_a > set_list_list_list_a > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
member_list_list_a: list_list_a > set_list_list_a > $o ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
member_set_list_a: set_list_a > set_set_list_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_R,type,
r: partia2175431115845679010xt_a_b ).
thf(sy_v_S,type,
s: set_a ).
% Relevant facts (1270)
thf(fact_0_b,axiom,
! [Y: a] :
( ( member_a @ Y @ s )
=> ( member_list_a @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( var_a_b @ r ) @ ( poly_of_const_a_b @ r @ Y ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% b
thf(fact_1_lagrange__basis__polynomial__aux__def,axiom,
! [S: set_a] :
( ( lagran9092808442999052491ux_a_b @ r @ S )
= ( finpro4329226410377213737unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ^ [S2: a] : ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( var_a_b @ r ) @ ( poly_of_const_a_b @ r @ S2 ) )
@ S ) ) ).
% lagrange_basis_polynomial_aux_def
thf(fact_2__092_060open_062_092_060And_062f_O_A_I_092_060And_062x_O_Ax_A_092_060in_062_AS_A_092_060Longrightarrow_062_Af_Ax_A_092_060in_062_Acarrier_A_Ipoly__ring_AR_J_J_A_092_060Longrightarrow_062_Adegree_A_Ifinprod_A_Ipoly__ring_AR_J_Af_AS_J_A_092_060le_062_A_I_092_060Sum_062x_092_060in_062S_O_Adegree_A_If_Ax_J_J_092_060close_062,axiom,
! [F: a > list_a] :
( ! [X: a] :
( ( member_a @ X @ s )
=> ( member_list_a @ ( F @ X ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( finpro4329226410377213737unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ s ) ) @ one_one_nat )
@ ( groups6334556678337121940_a_nat
@ ^ [X2: a] : ( minus_minus_nat @ ( size_size_list_a @ ( F @ X2 ) ) @ one_one_nat )
@ s ) ) ) ).
% \<open>\<And>f. (\<And>x. x \<in> S \<Longrightarrow> f x \<in> carrier (poly_ring R)) \<Longrightarrow> degree (finprod (poly_ring R) f S) \<le> (\<Sum>x\<in>S. degree (f x))\<close>
thf(fact_3__092_060open_062degree_AX_A_092_060le_062_A1_092_060close_062,axiom,
ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( var_a_b @ r ) ) @ one_one_nat ) @ one_one_nat ).
% \<open>degree X \<le> 1\<close>
thf(fact_4_poly__of__const__in__carrier,axiom,
! [S3: a] :
( ( member_a @ S3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_list_a @ ( poly_of_const_a_b @ r @ S3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% poly_of_const_in_carrier
thf(fact_5_assms_I1_J,axiom,
finite_finite_a @ s ).
% assms(1)
thf(fact_6__092_060open_062_092_060And_062y_O_Ay_A_092_060in_062_AS_A_092_060Longrightarrow_062_Adegree_A_Ipoly__of__const_Ay_J_A_092_060le_062_A1_092_060close_062,axiom,
! [Y: a] :
( ( member_a @ Y @ s )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( poly_of_const_a_b @ r @ Y ) ) @ one_one_nat ) @ one_one_nat ) ) ).
% \<open>\<And>y. y \<in> S \<Longrightarrow> degree (poly_of_const y) \<le> 1\<close>
thf(fact_7_domain__axioms,axiom,
domain_a_b @ r ).
% domain_axioms
thf(fact_8_a,axiom,
! [Y: a] :
( ( member_a @ Y @ s )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( var_a_b @ r ) @ ( poly_of_const_a_b @ r @ Y ) ) ) @ one_one_nat ) @ one_one_nat ) ) ).
% a
thf(fact_9_assms_I2_J,axiom,
ord_less_eq_set_a @ s @ ( partia707051561876973205xt_a_b @ r ) ).
% assms(2)
thf(fact_10_poly__sub__degree__le,axiom,
! [X3: list_a,N: nat,Y: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ X3 ) @ one_one_nat ) @ N )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ Y ) @ one_one_nat ) @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Y ) ) @ one_one_nat ) @ N ) ) ) ) ) ).
% poly_sub_degree_le
thf(fact_11_ring_Olagrange__basis__polynomial__aux_Ocong,axiom,
lagran9092808442999052491ux_a_b = lagran9092808442999052491ux_a_b ).
% ring.lagrange_basis_polynomial_aux.cong
thf(fact_12_ring_Olagrange__basis__polynomial__aux_Ocong,axiom,
lagran3534788790333317459t_unit = lagran3534788790333317459t_unit ).
% ring.lagrange_basis_polynomial_aux.cong
thf(fact_13_x_Ominus__closed,axiom,
! [X3: list_a,Y: list_a] :
( ( member_list_a @ X3 @ ( 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 ) ) @ X3 @ Y ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.minus_closed
thf(fact_14_onepideal,axiom,
principalideal_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% onepideal
thf(fact_15_x_Oup__minus__closed,axiom,
! [P: nat > list_a,Q: nat > list_a] :
( ( member_nat_list_a @ P @ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_nat_list_a @ Q @ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_nat_list_a
@ ^ [I: nat] : ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( P @ I ) @ ( Q @ I ) )
@ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.up_minus_closed
thf(fact_16_diff__diff__cancel,axiom,
! [I2: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I2 ) )
= I2 ) ) ).
% diff_diff_cancel
thf(fact_17_x_Osemiring__axioms,axiom,
semiri2871908745932252451t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% x.semiring_axioms
thf(fact_18_sum__subtractf__nat,axiom,
! [A: set_list_a,G: list_a > nat,F: list_a > nat] :
( ! [X: list_a] :
( ( member_list_a @ X @ A )
=> ( ord_less_eq_nat @ ( G @ X ) @ ( F @ X ) ) )
=> ( ( groups5521247699297860762_a_nat
@ ^ [X2: list_a] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
@ A )
= ( minus_minus_nat @ ( groups5521247699297860762_a_nat @ F @ A ) @ ( groups5521247699297860762_a_nat @ G @ A ) ) ) ) ).
% sum_subtractf_nat
thf(fact_19_sum__subtractf__nat,axiom,
! [A: set_nat,G: nat > nat,F: nat > nat] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_nat @ ( G @ X ) @ ( F @ X ) ) )
=> ( ( groups3542108847815614940at_nat
@ ^ [X2: nat] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
@ A )
= ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A ) @ ( groups3542108847815614940at_nat @ G @ A ) ) ) ) ).
% sum_subtractf_nat
thf(fact_20_sum__subtractf__nat,axiom,
! [A: set_nat_list_a,G: ( nat > list_a ) > nat,F: ( nat > list_a ) > nat] :
( ! [X: nat > list_a] :
( ( member_nat_list_a @ X @ A )
=> ( ord_less_eq_nat @ ( G @ X ) @ ( F @ X ) ) )
=> ( ( groups669906071623145473_a_nat
@ ^ [X2: nat > list_a] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
@ A )
= ( minus_minus_nat @ ( groups669906071623145473_a_nat @ F @ A ) @ ( groups669906071623145473_a_nat @ G @ A ) ) ) ) ).
% sum_subtractf_nat
thf(fact_21_sum__subtractf__nat,axiom,
! [A: set_nat_a,G: ( nat > a ) > nat,F: ( nat > a ) > nat] :
( ! [X: nat > a] :
( ( member_nat_a @ X @ A )
=> ( ord_less_eq_nat @ ( G @ X ) @ ( F @ X ) ) )
=> ( ( groups154653438316501755_a_nat
@ ^ [X2: nat > a] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
@ A )
= ( minus_minus_nat @ ( groups154653438316501755_a_nat @ F @ A ) @ ( groups154653438316501755_a_nat @ G @ A ) ) ) ) ).
% sum_subtractf_nat
thf(fact_22_sum__subtractf__nat,axiom,
! [A: set_a,G: a > nat,F: a > nat] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_nat @ ( G @ X ) @ ( F @ X ) ) )
=> ( ( groups6334556678337121940_a_nat
@ ^ [X2: a] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
@ A )
= ( minus_minus_nat @ ( groups6334556678337121940_a_nat @ F @ A ) @ ( groups6334556678337121940_a_nat @ G @ A ) ) ) ) ).
% sum_subtractf_nat
thf(fact_23_poly__add__degree__le,axiom,
! [X3: list_a,N: nat,Y: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ X3 ) @ one_one_nat ) @ N )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ Y ) @ one_one_nat ) @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Y ) ) @ one_one_nat ) @ N ) ) ) ) ) ).
% poly_add_degree_le
thf(fact_24_x_Oonepideal,axiom,
princi8786919440553033881t_unit @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% x.onepideal
thf(fact_25_x_Oabelian__monoid__axioms,axiom,
abelia226231641709521465t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% x.abelian_monoid_axioms
thf(fact_26_ee__length,axiom,
! [As: list_a,Bs: list_a] :
( ( essent8953798148185448568xt_a_b @ r @ As @ Bs )
=> ( ( size_size_list_a @ As )
= ( size_size_list_a @ Bs ) ) ) ).
% ee_length
thf(fact_27_poly__prod__degree__le,axiom,
! [A: set_nat_list_a,F: ( nat > list_a ) > list_a] :
( ( finite7630042315537210004list_a @ A )
=> ( ! [X: nat > list_a] :
( ( member_nat_list_a @ X @ A )
=> ( member_list_a @ ( F @ X ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( finpro5225711790780397718list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A ) ) @ one_one_nat )
@ ( groups669906071623145473_a_nat
@ ^ [X2: nat > list_a] : ( minus_minus_nat @ ( size_size_list_a @ ( F @ X2 ) ) @ one_one_nat )
@ A ) ) ) ) ).
% poly_prod_degree_le
thf(fact_28_poly__prod__degree__le,axiom,
! [A: set_nat_a,F: ( nat > a ) > list_a] :
( ( finite_finite_nat_a @ A )
=> ( ! [X: nat > a] :
( ( member_nat_a @ X @ A )
=> ( member_list_a @ ( F @ X ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( finpro2700224059436637840_nat_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A ) ) @ one_one_nat )
@ ( groups154653438316501755_a_nat
@ ^ [X2: nat > a] : ( minus_minus_nat @ ( size_size_list_a @ ( F @ X2 ) ) @ one_one_nat )
@ A ) ) ) ) ).
% poly_prod_degree_le
thf(fact_29_poly__prod__degree__le,axiom,
! [A: set_list_a,F: list_a > list_a] :
( ( finite_finite_list_a @ A )
=> ( ! [X: list_a] :
( ( member_list_a @ X @ A )
=> ( member_list_a @ ( F @ X ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( finpro738134188688310831list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A ) ) @ one_one_nat )
@ ( groups5521247699297860762_a_nat
@ ^ [X2: list_a] : ( minus_minus_nat @ ( size_size_list_a @ ( F @ X2 ) ) @ one_one_nat )
@ A ) ) ) ) ).
% poly_prod_degree_le
thf(fact_30_poly__prod__degree__le,axiom,
! [A: set_int,F: int > list_a] :
( ( finite_finite_int @ A )
=> ( ! [X: int] :
( ( member_int @ X @ A )
=> ( member_list_a @ ( F @ X ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( finpro1915614264500035905it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A ) ) @ one_one_nat )
@ ( groups4541462559716669496nt_nat
@ ^ [X2: int] : ( minus_minus_nat @ ( size_size_list_a @ ( F @ X2 ) ) @ one_one_nat )
@ A ) ) ) ) ).
% poly_prod_degree_le
thf(fact_31_poly__prod__degree__le,axiom,
! [A: set_nat,F: nat > list_a] :
( ( finite_finite_nat @ A )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_list_a @ ( F @ X ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( finpro1918104735009086181it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A ) ) @ one_one_nat )
@ ( groups3542108847815614940at_nat
@ ^ [X2: nat] : ( minus_minus_nat @ ( size_size_list_a @ ( F @ X2 ) ) @ one_one_nat )
@ A ) ) ) ) ).
% poly_prod_degree_le
thf(fact_32_poly__prod__degree__le,axiom,
! [A: set_a,F: a > list_a] :
( ( finite_finite_a @ A )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_list_a @ ( F @ X ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( finpro4329226410377213737unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A ) ) @ one_one_nat )
@ ( groups6334556678337121940_a_nat
@ ^ [X2: a] : ( minus_minus_nat @ ( size_size_list_a @ ( F @ X2 ) ) @ one_one_nat )
@ A ) ) ) ) ).
% poly_prod_degree_le
thf(fact_33_x_Oadd_Or__cancel,axiom,
! [A2: list_a,C: list_a,B: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ C )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ C ) )
=> ( ( member_list_a @ A2 @ ( 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 ) ) ) )
=> ( A2 = B ) ) ) ) ) ).
% x.add.r_cancel
thf(fact_34_x_Oadd_Om__lcomm,axiom,
! [X3: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Z ) ) ) ) ) ) ).
% x.add.m_lcomm
thf(fact_35_x_Oadd_Om__comm,axiom,
! [X3: list_a,Y: list_a] :
( ( member_list_a @ X3 @ ( 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 ) ) @ X3 @ Y )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X3 ) ) ) ) ).
% x.add.m_comm
thf(fact_36_x_Oadd_Om__assoc,axiom,
! [X3: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Y ) @ Z )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z ) ) ) ) ) ) ).
% x.add.m_assoc
thf(fact_37_x_Oadd_Ol__cancel,axiom,
! [C: list_a,A2: list_a,B: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ A2 )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ B ) )
=> ( ( member_list_a @ A2 @ ( 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 ) ) ) )
=> ( A2 = B ) ) ) ) ) ).
% x.add.l_cancel
thf(fact_38_x_Oup__add__closed,axiom,
! [P: nat > list_a,Q: nat > list_a] :
( ( member_nat_list_a @ P @ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_nat_list_a @ Q @ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_nat_list_a
@ ^ [I: nat] : ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( P @ I ) @ ( Q @ I ) )
@ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.up_add_closed
thf(fact_39_lagrange__aux__poly,axiom,
! [S: set_a] :
( ( finite_finite_a @ S )
=> ( ( ord_less_eq_set_a @ S @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_list_a @ ( lagran9092808442999052491ux_a_b @ r @ S ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% lagrange_aux_poly
thf(fact_40_x_Oadd_Oright__cancel,axiom,
! [X3: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X3 )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Z @ X3 ) )
= ( Y = Z ) ) ) ) ) ).
% x.add.right_cancel
thf(fact_41_x_Oadd_Om__closed,axiom,
! [X3: list_a,Y: list_a] :
( ( member_list_a @ X3 @ ( 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 ) ) @ X3 @ Y ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.add.m_closed
thf(fact_42_sum__diff,axiom,
! [A: set_int,B2: set_int,F: int > int] :
( ( finite_finite_int @ A )
=> ( ( ord_less_eq_set_int @ B2 @ A )
=> ( ( groups4538972089207619220nt_int @ F @ ( minus_minus_set_int @ A @ B2 ) )
= ( minus_minus_int @ ( groups4538972089207619220nt_int @ F @ A ) @ ( groups4538972089207619220nt_int @ F @ B2 ) ) ) ) ) ).
% sum_diff
thf(fact_43_sum__diff,axiom,
! [A: set_a,B2: set_a,F: a > int] :
( ( finite_finite_a @ A )
=> ( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( groups6332066207828071664_a_int @ F @ ( minus_minus_set_a @ A @ B2 ) )
= ( minus_minus_int @ ( groups6332066207828071664_a_int @ F @ A ) @ ( groups6332066207828071664_a_int @ F @ B2 ) ) ) ) ) ).
% sum_diff
thf(fact_44_sum__diff,axiom,
! [A: set_list_a,B2: set_list_a,F: list_a > int] :
( ( finite_finite_list_a @ A )
=> ( ( ord_le8861187494160871172list_a @ B2 @ A )
=> ( ( groups5518757228788810486_a_int @ F @ ( minus_646659088055828811list_a @ A @ B2 ) )
= ( minus_minus_int @ ( groups5518757228788810486_a_int @ F @ A ) @ ( groups5518757228788810486_a_int @ F @ B2 ) ) ) ) ) ).
% sum_diff
thf(fact_45_sum__diff,axiom,
! [A: set_nat,B2: set_nat,F: nat > int] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ B2 @ A )
=> ( ( groups3539618377306564664at_int @ F @ ( minus_minus_set_nat @ A @ B2 ) )
= ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ A ) @ ( groups3539618377306564664at_int @ F @ B2 ) ) ) ) ) ).
% sum_diff
thf(fact_46_sum_Oswap__restrict,axiom,
! [A: set_nat,B2: set_a,G: nat > a > nat,R: nat > a > $o] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_a @ B2 )
=> ( ( groups3542108847815614940at_nat
@ ^ [X2: nat] :
( groups6334556678337121940_a_nat @ ( G @ X2 )
@ ( collect_a
@ ^ [Y2: a] :
( ( member_a @ Y2 @ B2 )
& ( R @ X2 @ Y2 ) ) ) )
@ A )
= ( groups6334556678337121940_a_nat
@ ^ [Y2: a] :
( groups3542108847815614940at_nat
@ ^ [X2: nat] : ( G @ X2 @ Y2 )
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A )
& ( R @ X2 @ Y2 ) ) ) )
@ B2 ) ) ) ) ).
% sum.swap_restrict
thf(fact_47_sum_Oswap__restrict,axiom,
! [A: set_int,B2: set_a,G: int > a > nat,R: int > a > $o] :
( ( finite_finite_int @ A )
=> ( ( finite_finite_a @ B2 )
=> ( ( groups4541462559716669496nt_nat
@ ^ [X2: int] :
( groups6334556678337121940_a_nat @ ( G @ X2 )
@ ( collect_a
@ ^ [Y2: a] :
( ( member_a @ Y2 @ B2 )
& ( R @ X2 @ Y2 ) ) ) )
@ A )
= ( groups6334556678337121940_a_nat
@ ^ [Y2: a] :
( groups4541462559716669496nt_nat
@ ^ [X2: int] : ( G @ X2 @ Y2 )
@ ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ A )
& ( R @ X2 @ Y2 ) ) ) )
@ B2 ) ) ) ) ).
% sum.swap_restrict
thf(fact_48_sum_Oswap__restrict,axiom,
! [A: set_a,B2: set_nat,G: a > nat > nat,R: a > nat > $o] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_nat @ B2 )
=> ( ( groups6334556678337121940_a_nat
@ ^ [X2: a] :
( groups3542108847815614940at_nat @ ( G @ X2 )
@ ( collect_nat
@ ^ [Y2: nat] :
( ( member_nat @ Y2 @ B2 )
& ( R @ X2 @ Y2 ) ) ) )
@ A )
= ( groups3542108847815614940at_nat
@ ^ [Y2: nat] :
( groups6334556678337121940_a_nat
@ ^ [X2: a] : ( G @ X2 @ Y2 )
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A )
& ( R @ X2 @ Y2 ) ) ) )
@ B2 ) ) ) ) ).
% sum.swap_restrict
thf(fact_49_sum_Oswap__restrict,axiom,
! [A: set_a,B2: set_int,G: a > int > nat,R: a > int > $o] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_int @ B2 )
=> ( ( groups6334556678337121940_a_nat
@ ^ [X2: a] :
( groups4541462559716669496nt_nat @ ( G @ X2 )
@ ( collect_int
@ ^ [Y2: int] :
( ( member_int @ Y2 @ B2 )
& ( R @ X2 @ Y2 ) ) ) )
@ A )
= ( groups4541462559716669496nt_nat
@ ^ [Y2: int] :
( groups6334556678337121940_a_nat
@ ^ [X2: a] : ( G @ X2 @ Y2 )
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A )
& ( R @ X2 @ Y2 ) ) ) )
@ B2 ) ) ) ) ).
% sum.swap_restrict
thf(fact_50_sum_Oswap__restrict,axiom,
! [A: set_a,B2: set_a,G: a > a > nat,R: a > a > $o] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B2 )
=> ( ( groups6334556678337121940_a_nat
@ ^ [X2: a] :
( groups6334556678337121940_a_nat @ ( G @ X2 )
@ ( collect_a
@ ^ [Y2: a] :
( ( member_a @ Y2 @ B2 )
& ( R @ X2 @ Y2 ) ) ) )
@ A )
= ( groups6334556678337121940_a_nat
@ ^ [Y2: a] :
( groups6334556678337121940_a_nat
@ ^ [X2: a] : ( G @ X2 @ Y2 )
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A )
& ( R @ X2 @ Y2 ) ) ) )
@ B2 ) ) ) ) ).
% sum.swap_restrict
thf(fact_51_sum_Oswap__restrict,axiom,
! [A: set_list_a,B2: set_a,G: list_a > a > nat,R: list_a > a > $o] :
( ( finite_finite_list_a @ A )
=> ( ( finite_finite_a @ B2 )
=> ( ( groups5521247699297860762_a_nat
@ ^ [X2: list_a] :
( groups6334556678337121940_a_nat @ ( G @ X2 )
@ ( collect_a
@ ^ [Y2: a] :
( ( member_a @ Y2 @ B2 )
& ( R @ X2 @ Y2 ) ) ) )
@ A )
= ( groups6334556678337121940_a_nat
@ ^ [Y2: a] :
( groups5521247699297860762_a_nat
@ ^ [X2: list_a] : ( G @ X2 @ Y2 )
@ ( collect_list_a
@ ^ [X2: list_a] :
( ( member_list_a @ X2 @ A )
& ( R @ X2 @ Y2 ) ) ) )
@ B2 ) ) ) ) ).
% sum.swap_restrict
thf(fact_52_sum_Oswap__restrict,axiom,
! [A: set_a,B2: set_list_a,G: a > list_a > nat,R: a > list_a > $o] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_list_a @ B2 )
=> ( ( groups6334556678337121940_a_nat
@ ^ [X2: a] :
( groups5521247699297860762_a_nat @ ( G @ X2 )
@ ( collect_list_a
@ ^ [Y2: list_a] :
( ( member_list_a @ Y2 @ B2 )
& ( R @ X2 @ Y2 ) ) ) )
@ A )
= ( groups5521247699297860762_a_nat
@ ^ [Y2: list_a] :
( groups6334556678337121940_a_nat
@ ^ [X2: a] : ( G @ X2 @ Y2 )
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A )
& ( R @ X2 @ Y2 ) ) ) )
@ B2 ) ) ) ) ).
% sum.swap_restrict
thf(fact_53_sum_Oswap__restrict,axiom,
! [A: set_nat_a,B2: set_a,G: ( nat > a ) > a > nat,R: ( nat > a ) > a > $o] :
( ( finite_finite_nat_a @ A )
=> ( ( finite_finite_a @ B2 )
=> ( ( groups154653438316501755_a_nat
@ ^ [X2: nat > a] :
( groups6334556678337121940_a_nat @ ( G @ X2 )
@ ( collect_a
@ ^ [Y2: a] :
( ( member_a @ Y2 @ B2 )
& ( R @ X2 @ Y2 ) ) ) )
@ A )
= ( groups6334556678337121940_a_nat
@ ^ [Y2: a] :
( groups154653438316501755_a_nat
@ ^ [X2: nat > a] : ( G @ X2 @ Y2 )
@ ( collect_nat_a
@ ^ [X2: nat > a] :
( ( member_nat_a @ X2 @ A )
& ( R @ X2 @ Y2 ) ) ) )
@ B2 ) ) ) ) ).
% sum.swap_restrict
thf(fact_54_sum_Oswap__restrict,axiom,
! [A: set_list_list_a,B2: set_a,G: list_list_a > a > nat,R: list_list_a > a > $o] :
( ( finite1660835950917165235list_a @ A )
=> ( ( finite_finite_a @ B2 )
=> ( ( groups7548105480907152928_a_nat
@ ^ [X2: list_list_a] :
( groups6334556678337121940_a_nat @ ( G @ X2 )
@ ( collect_a
@ ^ [Y2: a] :
( ( member_a @ Y2 @ B2 )
& ( R @ X2 @ Y2 ) ) ) )
@ A )
= ( groups6334556678337121940_a_nat
@ ^ [Y2: a] :
( groups7548105480907152928_a_nat
@ ^ [X2: list_list_a] : ( G @ X2 @ Y2 )
@ ( collect_list_list_a
@ ^ [X2: list_list_a] :
( ( member_list_list_a @ X2 @ A )
& ( R @ X2 @ Y2 ) ) ) )
@ B2 ) ) ) ) ).
% sum.swap_restrict
thf(fact_55_sum_Oswap__restrict,axiom,
! [A: set_a,B2: set_nat_a,G: a > ( nat > a ) > nat,R: a > ( nat > a ) > $o] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_nat_a @ B2 )
=> ( ( groups6334556678337121940_a_nat
@ ^ [X2: a] :
( groups154653438316501755_a_nat @ ( G @ X2 )
@ ( collect_nat_a
@ ^ [Y2: nat > a] :
( ( member_nat_a @ Y2 @ B2 )
& ( R @ X2 @ Y2 ) ) ) )
@ A )
= ( groups154653438316501755_a_nat
@ ^ [Y2: nat > a] :
( groups6334556678337121940_a_nat
@ ^ [X2: a] : ( G @ X2 @ Y2 )
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A )
& ( R @ X2 @ Y2 ) ) ) )
@ B2 ) ) ) ) ).
% sum.swap_restrict
thf(fact_56_domain_Olagrange__aux__poly,axiom,
! [R: partia2956882679547061052t_unit,S: set_list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( finite1660835950917165235list_a @ S )
=> ( ( ord_le8488217952732425610list_a @ S @ ( partia2464479390973590831t_unit @ R ) )
=> ( member5342144027231129785list_a @ ( lagran8640377047181650765t_unit @ R @ S ) @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ) ) ).
% domain.lagrange_aux_poly
thf(fact_57_domain_Olagrange__aux__poly,axiom,
! [R: partia2175431115845679010xt_a_b,S: set_a] :
( ( domain_a_b @ R )
=> ( ( finite_finite_a @ S )
=> ( ( ord_less_eq_set_a @ S @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_list_a @ ( lagran9092808442999052491ux_a_b @ R @ S ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ) ) ).
% domain.lagrange_aux_poly
thf(fact_58_domain_Olagrange__aux__poly,axiom,
! [R: partia2670972154091845814t_unit,S: set_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( finite_finite_list_a @ S )
=> ( ( ord_le8861187494160871172list_a @ S @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_list_a @ ( lagran3534788790333317459t_unit @ R @ S ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ) ) ).
% domain.lagrange_aux_poly
thf(fact_59_sum__mono__inv,axiom,
! [F: nat > nat,I3: set_nat,G: nat > nat,I2: nat] :
( ( ( groups3542108847815614940at_nat @ F @ I3 )
= ( groups3542108847815614940at_nat @ G @ I3 ) )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ( member_nat @ I2 @ I3 )
=> ( ( finite_finite_nat @ I3 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_60_sum__mono__inv,axiom,
! [F: int > nat,I3: set_int,G: int > nat,I2: int] :
( ( ( groups4541462559716669496nt_nat @ F @ I3 )
= ( groups4541462559716669496nt_nat @ G @ I3 ) )
=> ( ! [I4: int] :
( ( member_int @ I4 @ I3 )
=> ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ( member_int @ I2 @ I3 )
=> ( ( finite_finite_int @ I3 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_61_sum__mono__inv,axiom,
! [F: a > int,I3: set_a,G: a > int,I2: a] :
( ( ( groups6332066207828071664_a_int @ F @ I3 )
= ( groups6332066207828071664_a_int @ G @ I3 ) )
=> ( ! [I4: a] :
( ( member_a @ I4 @ I3 )
=> ( ord_less_eq_int @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ( member_a @ I2 @ I3 )
=> ( ( finite_finite_a @ I3 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_62_sum__mono__inv,axiom,
! [F: nat > int,I3: set_nat,G: nat > int,I2: nat] :
( ( ( groups3539618377306564664at_int @ F @ I3 )
= ( groups3539618377306564664at_int @ G @ I3 ) )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( ord_less_eq_int @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ( member_nat @ I2 @ I3 )
=> ( ( finite_finite_nat @ I3 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_63_sum__mono__inv,axiom,
! [F: int > int,I3: set_int,G: int > int,I2: int] :
( ( ( groups4538972089207619220nt_int @ F @ I3 )
= ( groups4538972089207619220nt_int @ G @ I3 ) )
=> ( ! [I4: int] :
( ( member_int @ I4 @ I3 )
=> ( ord_less_eq_int @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ( member_int @ I2 @ I3 )
=> ( ( finite_finite_int @ I3 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_64_sum__mono__inv,axiom,
! [F: a > nat,I3: set_a,G: a > nat,I2: a] :
( ( ( groups6334556678337121940_a_nat @ F @ I3 )
= ( groups6334556678337121940_a_nat @ G @ I3 ) )
=> ( ! [I4: a] :
( ( member_a @ I4 @ I3 )
=> ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ( member_a @ I2 @ I3 )
=> ( ( finite_finite_a @ I3 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_65_sum__mono__inv,axiom,
! [F: list_a > nat,I3: set_list_a,G: list_a > nat,I2: list_a] :
( ( ( groups5521247699297860762_a_nat @ F @ I3 )
= ( groups5521247699297860762_a_nat @ G @ I3 ) )
=> ( ! [I4: list_a] :
( ( member_list_a @ I4 @ I3 )
=> ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ( member_list_a @ I2 @ I3 )
=> ( ( finite_finite_list_a @ I3 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_66_sum__mono__inv,axiom,
! [F: list_a > int,I3: set_list_a,G: list_a > int,I2: list_a] :
( ( ( groups5518757228788810486_a_int @ F @ I3 )
= ( groups5518757228788810486_a_int @ G @ I3 ) )
=> ( ! [I4: list_a] :
( ( member_list_a @ I4 @ I3 )
=> ( ord_less_eq_int @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ( member_list_a @ I2 @ I3 )
=> ( ( finite_finite_list_a @ I3 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_67_sum__mono__inv,axiom,
! [F: ( nat > a ) > nat,I3: set_nat_a,G: ( nat > a ) > nat,I2: nat > a] :
( ( ( groups154653438316501755_a_nat @ F @ I3 )
= ( groups154653438316501755_a_nat @ G @ I3 ) )
=> ( ! [I4: nat > a] :
( ( member_nat_a @ I4 @ I3 )
=> ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ( member_nat_a @ I2 @ I3 )
=> ( ( finite_finite_nat_a @ I3 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_68_sum__mono__inv,axiom,
! [F: ( nat > a ) > int,I3: set_nat_a,G: ( nat > a ) > int,I2: nat > a] :
( ( ( groups152162967807451479_a_int @ F @ I3 )
= ( groups152162967807451479_a_int @ G @ I3 ) )
=> ( ! [I4: nat > a] :
( ( member_nat_a @ I4 @ I3 )
=> ( ord_less_eq_int @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ( member_nat_a @ I2 @ I3 )
=> ( ( finite_finite_nat_a @ I3 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_69_domain_Opoly__add__degree__le,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_list_a,N: nat,Y: list_list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member5342144027231129785list_a @ X3 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s2403821588304063868list_a @ X3 ) @ one_one_nat ) @ N )
=> ( ( member5342144027231129785list_a @ Y @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s2403821588304063868list_a @ Y ) @ one_one_nat ) @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s2403821588304063868list_a @ ( add_li5162926044081146114t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ X3 @ Y ) ) @ one_one_nat ) @ N ) ) ) ) ) ) ).
% domain.poly_add_degree_le
thf(fact_70_domain_Opoly__add__degree__le,axiom,
! [R: partia2175431115845679010xt_a_b,X3: list_a,N: nat,Y: list_a] :
( ( domain_a_b @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ X3 ) @ one_one_nat ) @ N )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ Y ) @ one_one_nat ) @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ X3 @ Y ) ) @ one_one_nat ) @ N ) ) ) ) ) ) ).
% domain.poly_add_degree_le
thf(fact_71_domain_Opoly__add__degree__le,axiom,
! [R: partia2670972154091845814t_unit,X3: list_list_a,N: nat,Y: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ X3 ) @ one_one_nat ) @ N )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ Y ) @ one_one_nat ) @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ X3 @ Y ) ) @ one_one_nat ) @ N ) ) ) ) ) ) ).
% domain.poly_add_degree_le
thf(fact_72_domain_Opoly__prod__degree__le,axiom,
! [R: partia2175431115845679010xt_a_b,A: set_int,F: int > list_a] :
( ( domain_a_b @ R )
=> ( ( finite_finite_int @ A )
=> ( ! [X: int] :
( ( member_int @ X @ A )
=> ( member_list_a @ ( F @ X ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( finpro1915614264500035905it_int @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ F @ A ) ) @ one_one_nat )
@ ( groups4541462559716669496nt_nat
@ ^ [X2: int] : ( minus_minus_nat @ ( size_size_list_a @ ( F @ X2 ) ) @ one_one_nat )
@ A ) ) ) ) ) ).
% domain.poly_prod_degree_le
thf(fact_73_domain_Opoly__prod__degree__le,axiom,
! [R: partia2175431115845679010xt_a_b,A: set_nat,F: nat > list_a] :
( ( domain_a_b @ R )
=> ( ( finite_finite_nat @ A )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_list_a @ ( F @ X ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( finpro1918104735009086181it_nat @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ F @ A ) ) @ one_one_nat )
@ ( groups3542108847815614940at_nat
@ ^ [X2: nat] : ( minus_minus_nat @ ( size_size_list_a @ ( F @ X2 ) ) @ one_one_nat )
@ A ) ) ) ) ) ).
% domain.poly_prod_degree_le
thf(fact_74_domain_Opoly__prod__degree__le,axiom,
! [R: partia2175431115845679010xt_a_b,A: set_a,F: a > list_a] :
( ( domain_a_b @ R )
=> ( ( finite_finite_a @ A )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_list_a @ ( F @ X ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( finpro4329226410377213737unit_a @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ F @ A ) ) @ one_one_nat )
@ ( groups6334556678337121940_a_nat
@ ^ [X2: a] : ( minus_minus_nat @ ( size_size_list_a @ ( F @ X2 ) ) @ one_one_nat )
@ A ) ) ) ) ) ).
% domain.poly_prod_degree_le
thf(fact_75_domain_Opoly__prod__degree__le,axiom,
! [R: partia2175431115845679010xt_a_b,A: set_list_a,F: list_a > list_a] :
( ( domain_a_b @ R )
=> ( ( finite_finite_list_a @ A )
=> ( ! [X: list_a] :
( ( member_list_a @ X @ A )
=> ( member_list_a @ ( F @ X ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( finpro738134188688310831list_a @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ F @ A ) ) @ one_one_nat )
@ ( groups5521247699297860762_a_nat
@ ^ [X2: list_a] : ( minus_minus_nat @ ( size_size_list_a @ ( F @ X2 ) ) @ one_one_nat )
@ A ) ) ) ) ) ).
% domain.poly_prod_degree_le
thf(fact_76_domain_Opoly__prod__degree__le,axiom,
! [R: partia2670972154091845814t_unit,A: set_nat,F: nat > list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( finite_finite_nat @ A )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_list_list_a @ ( F @ X ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( finpro4561275463894985573it_nat @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ F @ A ) ) @ one_one_nat )
@ ( groups3542108847815614940at_nat
@ ^ [X2: nat] : ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( F @ X2 ) ) @ one_one_nat )
@ A ) ) ) ) ) ).
% domain.poly_prod_degree_le
thf(fact_77_domain_Opoly__prod__degree__le,axiom,
! [R: partia2670972154091845814t_unit,A: set_int,F: int > list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( finite_finite_int @ A )
=> ( ! [X: int] :
( ( member_int @ X @ A )
=> ( member_list_list_a @ ( F @ X ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( finpro4558784993385935297it_int @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ F @ A ) ) @ one_one_nat )
@ ( groups4541462559716669496nt_nat
@ ^ [X2: int] : ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( F @ X2 ) ) @ one_one_nat )
@ A ) ) ) ) ) ).
% domain.poly_prod_degree_le
thf(fact_78_domain_Opoly__prod__degree__le,axiom,
! [R: partia2670972154091845814t_unit,A: set_a,F: a > list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( finite_finite_a @ A )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_list_list_a @ ( F @ X ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( finpro5596966875920909993unit_a @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ F @ A ) ) @ one_one_nat )
@ ( groups6334556678337121940_a_nat
@ ^ [X2: a] : ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( F @ X2 ) ) @ one_one_nat )
@ A ) ) ) ) ) ).
% domain.poly_prod_degree_le
thf(fact_79_domain_Opoly__prod__degree__le,axiom,
! [R: partia2956882679547061052t_unit,A: set_nat,F: nat > list_list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( finite_finite_nat @ A )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member5342144027231129785list_a @ ( F @ X ) @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s2403821588304063868list_a @ ( finpro3906265973382178277it_nat @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ F @ A ) ) @ one_one_nat )
@ ( groups3542108847815614940at_nat
@ ^ [X2: nat] : ( minus_minus_nat @ ( size_s2403821588304063868list_a @ ( F @ X2 ) ) @ one_one_nat )
@ A ) ) ) ) ) ).
% domain.poly_prod_degree_le
thf(fact_80_domain_Opoly__prod__degree__le,axiom,
! [R: partia2956882679547061052t_unit,A: set_int,F: int > list_list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( finite_finite_int @ A )
=> ( ! [X: int] :
( ( member_int @ X @ A )
=> ( member5342144027231129785list_a @ ( F @ X ) @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s2403821588304063868list_a @ ( finpro3903775502873128001it_int @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ F @ A ) ) @ one_one_nat )
@ ( groups4541462559716669496nt_nat
@ ^ [X2: int] : ( minus_minus_nat @ ( size_s2403821588304063868list_a @ ( F @ X2 ) ) @ one_one_nat )
@ A ) ) ) ) ) ).
% domain.poly_prod_degree_le
thf(fact_81_domain_Opoly__prod__degree__le,axiom,
! [R: partia2175431115845679010xt_a_b,A: set_nat_a,F: ( nat > a ) > list_a] :
( ( domain_a_b @ R )
=> ( ( finite_finite_nat_a @ A )
=> ( ! [X: nat > a] :
( ( member_nat_a @ X @ A )
=> ( member_list_a @ ( F @ X ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( finpro2700224059436637840_nat_a @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ F @ A ) ) @ one_one_nat )
@ ( groups154653438316501755_a_nat
@ ^ [X2: nat > a] : ( minus_minus_nat @ ( size_size_list_a @ ( F @ X2 ) ) @ one_one_nat )
@ A ) ) ) ) ) ).
% domain.poly_prod_degree_le
thf(fact_82_size__neq__size__imp__neq,axiom,
! [X3: list_a,Y: list_a] :
( ( ( size_size_list_a @ X3 )
!= ( size_size_list_a @ Y ) )
=> ( X3 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_83_size__neq__size__imp__neq,axiom,
! [X3: multiset_list_a,Y: multiset_list_a] :
( ( ( size_s2335926164413107382list_a @ X3 )
!= ( size_s2335926164413107382list_a @ Y ) )
=> ( X3 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_84_size__neq__size__imp__neq,axiom,
! [X3: list_list_a,Y: list_list_a] :
( ( ( size_s349497388124573686list_a @ X3 )
!= ( size_s349497388124573686list_a @ Y ) )
=> ( X3 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_85_size__neq__size__imp__neq,axiom,
! [X3: multiset_a,Y: multiset_a] :
( ( ( size_size_multiset_a @ X3 )
!= ( size_size_multiset_a @ Y ) )
=> ( X3 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_86_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 ) )
=> ? [X: nat] :
( ( P2 @ X )
& ! [Y4: nat] :
( ( P2 @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_87_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_88_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_89_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_90_mem__Collect__eq,axiom,
! [A2: nat > list_a,P2: ( nat > list_a ) > $o] :
( ( member_nat_list_a @ A2 @ ( collect_nat_list_a @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_91_mem__Collect__eq,axiom,
! [A2: nat > a,P2: ( nat > a ) > $o] :
( ( member_nat_a @ A2 @ ( collect_nat_a @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_92_mem__Collect__eq,axiom,
! [A2: a,P2: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_93_mem__Collect__eq,axiom,
! [A2: nat,P2: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_94_mem__Collect__eq,axiom,
! [A2: list_list_a,P2: list_list_a > $o] :
( ( member_list_list_a @ A2 @ ( collect_list_list_a @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_95_mem__Collect__eq,axiom,
! [A2: list_a,P2: list_a > $o] :
( ( member_list_a @ A2 @ ( collect_list_a @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_96_mem__Collect__eq,axiom,
! [A2: int,P2: int > $o] :
( ( member_int @ A2 @ ( collect_int @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_97_Collect__mem__eq,axiom,
! [A: set_nat_list_a] :
( ( collect_nat_list_a
@ ^ [X2: nat > list_a] : ( member_nat_list_a @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_98_Collect__mem__eq,axiom,
! [A: set_nat_a] :
( ( collect_nat_a
@ ^ [X2: nat > a] : ( member_nat_a @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_99_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_100_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_101_Collect__mem__eq,axiom,
! [A: set_list_list_a] :
( ( collect_list_list_a
@ ^ [X2: list_list_a] : ( member_list_list_a @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_102_Collect__mem__eq,axiom,
! [A: set_list_a] :
( ( collect_list_a
@ ^ [X2: list_a] : ( member_list_a @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_103_Collect__mem__eq,axiom,
! [A: set_int] :
( ( collect_int
@ ^ [X2: int] : ( member_int @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_104_Collect__cong,axiom,
! [P2: a > $o,Q2: a > $o] :
( ! [X: a] :
( ( P2 @ X )
= ( Q2 @ X ) )
=> ( ( collect_a @ P2 )
= ( collect_a @ Q2 ) ) ) ).
% Collect_cong
thf(fact_105_Collect__cong,axiom,
! [P2: nat > $o,Q2: nat > $o] :
( ! [X: nat] :
( ( P2 @ X )
= ( Q2 @ X ) )
=> ( ( collect_nat @ P2 )
= ( collect_nat @ Q2 ) ) ) ).
% Collect_cong
thf(fact_106_Collect__cong,axiom,
! [P2: list_list_a > $o,Q2: list_list_a > $o] :
( ! [X: list_list_a] :
( ( P2 @ X )
= ( Q2 @ X ) )
=> ( ( collect_list_list_a @ P2 )
= ( collect_list_list_a @ Q2 ) ) ) ).
% Collect_cong
thf(fact_107_Collect__cong,axiom,
! [P2: list_a > $o,Q2: list_a > $o] :
( ! [X: list_a] :
( ( P2 @ X )
= ( Q2 @ X ) )
=> ( ( collect_list_a @ P2 )
= ( collect_list_a @ Q2 ) ) ) ).
% Collect_cong
thf(fact_108_Collect__cong,axiom,
! [P2: int > $o,Q2: int > $o] :
( ! [X: int] :
( ( P2 @ X )
= ( Q2 @ X ) )
=> ( ( collect_int @ P2 )
= ( collect_int @ Q2 ) ) ) ).
% Collect_cong
thf(fact_109_le__trans,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I2 @ K ) ) ) ).
% le_trans
thf(fact_110_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_111_diff__commute,axiom,
! [I2: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K ) @ J ) ) ).
% diff_commute
thf(fact_112_sum_Oreindex__bij__witness,axiom,
! [S: set_list_a,I2: a > list_a,J: list_a > a,T: set_a,H: a > nat,G: list_a > nat] :
( ! [A3: list_a] :
( ( member_list_a @ A3 @ S )
=> ( ( I2 @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: list_a] :
( ( member_list_a @ A3 @ S )
=> ( member_a @ ( J @ A3 ) @ T ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ T )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ T )
=> ( member_list_a @ ( I2 @ B3 ) @ S ) )
=> ( ! [A3: list_a] :
( ( member_list_a @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups5521247699297860762_a_nat @ G @ S )
= ( groups6334556678337121940_a_nat @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_113_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I2: a > nat,J: nat > a,T: set_a,H: a > nat,G: nat > nat] :
( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( I2 @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( member_a @ ( J @ A3 ) @ T ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ T )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ T )
=> ( member_nat @ ( I2 @ B3 ) @ S ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ S )
= ( groups6334556678337121940_a_nat @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_114_sum_Oreindex__bij__witness,axiom,
! [S: set_nat_list_a,I2: a > nat > list_a,J: ( nat > list_a ) > a,T: set_a,H: a > nat,G: ( nat > list_a ) > nat] :
( ! [A3: nat > list_a] :
( ( member_nat_list_a @ A3 @ S )
=> ( ( I2 @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: nat > list_a] :
( ( member_nat_list_a @ A3 @ S )
=> ( member_a @ ( J @ A3 ) @ T ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ T )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ T )
=> ( member_nat_list_a @ ( I2 @ B3 ) @ S ) )
=> ( ! [A3: nat > list_a] :
( ( member_nat_list_a @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups669906071623145473_a_nat @ G @ S )
= ( groups6334556678337121940_a_nat @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_115_sum_Oreindex__bij__witness,axiom,
! [S: set_nat_a,I2: a > nat > a,J: ( nat > a ) > a,T: set_a,H: a > nat,G: ( nat > a ) > nat] :
( ! [A3: nat > a] :
( ( member_nat_a @ A3 @ S )
=> ( ( I2 @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: nat > a] :
( ( member_nat_a @ A3 @ S )
=> ( member_a @ ( J @ A3 ) @ T ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ T )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ T )
=> ( member_nat_a @ ( I2 @ B3 ) @ S ) )
=> ( ! [A3: nat > a] :
( ( member_nat_a @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups154653438316501755_a_nat @ G @ S )
= ( groups6334556678337121940_a_nat @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_116_sum_Oreindex__bij__witness,axiom,
! [S: set_a,I2: list_a > a,J: a > list_a,T: set_list_a,H: list_a > nat,G: a > nat] :
( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( ( I2 @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( member_list_a @ ( J @ A3 ) @ T ) )
=> ( ! [B3: list_a] :
( ( member_list_a @ B3 @ T )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: list_a] :
( ( member_list_a @ B3 @ T )
=> ( member_a @ ( I2 @ B3 ) @ S ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups6334556678337121940_a_nat @ G @ S )
= ( groups5521247699297860762_a_nat @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_117_sum_Oreindex__bij__witness,axiom,
! [S: set_a,I2: nat > a,J: a > nat,T: set_nat,H: nat > nat,G: a > nat] :
( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( ( I2 @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( member_nat @ ( J @ A3 ) @ T ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( member_a @ ( I2 @ B3 ) @ S ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups6334556678337121940_a_nat @ G @ S )
= ( groups3542108847815614940at_nat @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_118_sum_Oreindex__bij__witness,axiom,
! [S: set_a,I2: ( nat > list_a ) > a,J: a > nat > list_a,T: set_nat_list_a,H: ( nat > list_a ) > nat,G: a > nat] :
( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( ( I2 @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( member_nat_list_a @ ( J @ A3 ) @ T ) )
=> ( ! [B3: nat > list_a] :
( ( member_nat_list_a @ B3 @ T )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: nat > list_a] :
( ( member_nat_list_a @ B3 @ T )
=> ( member_a @ ( I2 @ B3 ) @ S ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups6334556678337121940_a_nat @ G @ S )
= ( groups669906071623145473_a_nat @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_119_sum_Oreindex__bij__witness,axiom,
! [S: set_a,I2: ( nat > a ) > a,J: a > nat > a,T: set_nat_a,H: ( nat > a ) > nat,G: a > nat] :
( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( ( I2 @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( member_nat_a @ ( J @ A3 ) @ T ) )
=> ( ! [B3: nat > a] :
( ( member_nat_a @ B3 @ T )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: nat > a] :
( ( member_nat_a @ B3 @ T )
=> ( member_a @ ( I2 @ B3 ) @ S ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups6334556678337121940_a_nat @ G @ S )
= ( groups154653438316501755_a_nat @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_120_sum_Oreindex__bij__witness,axiom,
! [S: set_a,I2: a > a,J: a > a,T: set_a,H: a > nat,G: a > nat] :
( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( ( I2 @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( member_a @ ( J @ A3 ) @ T ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ T )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ T )
=> ( member_a @ ( I2 @ B3 ) @ S ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups6334556678337121940_a_nat @ G @ S )
= ( groups6334556678337121940_a_nat @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_121_sum_Oeq__general__inverses,axiom,
! [B2: set_a,K: a > list_a,A: set_list_a,H: list_a > a,Gamma: a > nat,Phi: list_a > nat] :
( ! [Y3: a] :
( ( member_a @ Y3 @ B2 )
=> ( ( member_list_a @ ( K @ Y3 ) @ A )
& ( ( H @ ( K @ Y3 ) )
= Y3 ) ) )
=> ( ! [X: list_a] :
( ( member_list_a @ X @ A )
=> ( ( member_a @ ( H @ X ) @ B2 )
& ( ( K @ ( H @ X ) )
= X )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups5521247699297860762_a_nat @ Phi @ A )
= ( groups6334556678337121940_a_nat @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_122_sum_Oeq__general__inverses,axiom,
! [B2: set_a,K: a > nat,A: set_nat,H: nat > a,Gamma: a > nat,Phi: nat > nat] :
( ! [Y3: a] :
( ( member_a @ Y3 @ B2 )
=> ( ( member_nat @ ( K @ Y3 ) @ A )
& ( ( H @ ( K @ Y3 ) )
= Y3 ) ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ( member_a @ ( H @ X ) @ B2 )
& ( ( K @ ( H @ X ) )
= X )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A )
= ( groups6334556678337121940_a_nat @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_123_sum_Oeq__general__inverses,axiom,
! [B2: set_a,K: a > nat > list_a,A: set_nat_list_a,H: ( nat > list_a ) > a,Gamma: a > nat,Phi: ( nat > list_a ) > nat] :
( ! [Y3: a] :
( ( member_a @ Y3 @ B2 )
=> ( ( member_nat_list_a @ ( K @ Y3 ) @ A )
& ( ( H @ ( K @ Y3 ) )
= Y3 ) ) )
=> ( ! [X: nat > list_a] :
( ( member_nat_list_a @ X @ A )
=> ( ( member_a @ ( H @ X ) @ B2 )
& ( ( K @ ( H @ X ) )
= X )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups669906071623145473_a_nat @ Phi @ A )
= ( groups6334556678337121940_a_nat @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_124_sum_Oeq__general__inverses,axiom,
! [B2: set_a,K: a > nat > a,A: set_nat_a,H: ( nat > a ) > a,Gamma: a > nat,Phi: ( nat > a ) > nat] :
( ! [Y3: a] :
( ( member_a @ Y3 @ B2 )
=> ( ( member_nat_a @ ( K @ Y3 ) @ A )
& ( ( H @ ( K @ Y3 ) )
= Y3 ) ) )
=> ( ! [X: nat > a] :
( ( member_nat_a @ X @ A )
=> ( ( member_a @ ( H @ X ) @ B2 )
& ( ( K @ ( H @ X ) )
= X )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups154653438316501755_a_nat @ Phi @ A )
= ( groups6334556678337121940_a_nat @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_125_sum_Oeq__general__inverses,axiom,
! [B2: set_list_a,K: list_a > a,A: set_a,H: a > list_a,Gamma: list_a > nat,Phi: a > nat] :
( ! [Y3: list_a] :
( ( member_list_a @ Y3 @ B2 )
=> ( ( member_a @ ( K @ Y3 ) @ A )
& ( ( H @ ( K @ Y3 ) )
= Y3 ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ( member_list_a @ ( H @ X ) @ B2 )
& ( ( K @ ( H @ X ) )
= X )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups6334556678337121940_a_nat @ Phi @ A )
= ( groups5521247699297860762_a_nat @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_126_sum_Oeq__general__inverses,axiom,
! [B2: set_nat,K: nat > a,A: set_a,H: a > nat,Gamma: nat > nat,Phi: a > nat] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B2 )
=> ( ( member_a @ ( K @ Y3 ) @ A )
& ( ( H @ ( K @ Y3 ) )
= Y3 ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ( member_nat @ ( H @ X ) @ B2 )
& ( ( K @ ( H @ X ) )
= X )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups6334556678337121940_a_nat @ Phi @ A )
= ( groups3542108847815614940at_nat @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_127_sum_Oeq__general__inverses,axiom,
! [B2: set_nat_list_a,K: ( nat > list_a ) > a,A: set_a,H: a > nat > list_a,Gamma: ( nat > list_a ) > nat,Phi: a > nat] :
( ! [Y3: nat > list_a] :
( ( member_nat_list_a @ Y3 @ B2 )
=> ( ( member_a @ ( K @ Y3 ) @ A )
& ( ( H @ ( K @ Y3 ) )
= Y3 ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ( member_nat_list_a @ ( H @ X ) @ B2 )
& ( ( K @ ( H @ X ) )
= X )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups6334556678337121940_a_nat @ Phi @ A )
= ( groups669906071623145473_a_nat @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_128_sum_Oeq__general__inverses,axiom,
! [B2: set_nat_a,K: ( nat > a ) > a,A: set_a,H: a > nat > a,Gamma: ( nat > a ) > nat,Phi: a > nat] :
( ! [Y3: nat > a] :
( ( member_nat_a @ Y3 @ B2 )
=> ( ( member_a @ ( K @ Y3 ) @ A )
& ( ( H @ ( K @ Y3 ) )
= Y3 ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ( member_nat_a @ ( H @ X ) @ B2 )
& ( ( K @ ( H @ X ) )
= X )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups6334556678337121940_a_nat @ Phi @ A )
= ( groups154653438316501755_a_nat @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_129_sum_Oeq__general__inverses,axiom,
! [B2: set_a,K: a > a,A: set_a,H: a > a,Gamma: a > nat,Phi: a > nat] :
( ! [Y3: a] :
( ( member_a @ Y3 @ B2 )
=> ( ( member_a @ ( K @ Y3 ) @ A )
& ( ( H @ ( K @ Y3 ) )
= Y3 ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ( member_a @ ( H @ X ) @ B2 )
& ( ( K @ ( H @ X ) )
= X )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups6334556678337121940_a_nat @ Phi @ A )
= ( groups6334556678337121940_a_nat @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_130_sum_Oeq__general,axiom,
! [B2: set_a,A: set_list_a,H: list_a > a,Gamma: a > nat,Phi: list_a > nat] :
( ! [Y3: a] :
( ( member_a @ Y3 @ B2 )
=> ? [X4: list_a] :
( ( member_list_a @ X4 @ A )
& ( ( H @ X4 )
= Y3 )
& ! [Ya: list_a] :
( ( ( member_list_a @ Ya @ A )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X: list_a] :
( ( member_list_a @ X @ A )
=> ( ( member_a @ ( H @ X ) @ B2 )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups5521247699297860762_a_nat @ Phi @ A )
= ( groups6334556678337121940_a_nat @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_131_sum_Oeq__general,axiom,
! [B2: set_a,A: set_nat,H: nat > a,Gamma: a > nat,Phi: nat > nat] :
( ! [Y3: a] :
( ( member_a @ Y3 @ B2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ( H @ X4 )
= Y3 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ( member_a @ ( H @ X ) @ B2 )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A )
= ( groups6334556678337121940_a_nat @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_132_sum_Oeq__general,axiom,
! [B2: set_a,A: set_nat_list_a,H: ( nat > list_a ) > a,Gamma: a > nat,Phi: ( nat > list_a ) > nat] :
( ! [Y3: a] :
( ( member_a @ Y3 @ B2 )
=> ? [X4: nat > list_a] :
( ( member_nat_list_a @ X4 @ A )
& ( ( H @ X4 )
= Y3 )
& ! [Ya: nat > list_a] :
( ( ( member_nat_list_a @ Ya @ A )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X: nat > list_a] :
( ( member_nat_list_a @ X @ A )
=> ( ( member_a @ ( H @ X ) @ B2 )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups669906071623145473_a_nat @ Phi @ A )
= ( groups6334556678337121940_a_nat @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_133_sum_Oeq__general,axiom,
! [B2: set_a,A: set_nat_a,H: ( nat > a ) > a,Gamma: a > nat,Phi: ( nat > a ) > nat] :
( ! [Y3: a] :
( ( member_a @ Y3 @ B2 )
=> ? [X4: nat > a] :
( ( member_nat_a @ X4 @ A )
& ( ( H @ X4 )
= Y3 )
& ! [Ya: nat > a] :
( ( ( member_nat_a @ Ya @ A )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X: nat > a] :
( ( member_nat_a @ X @ A )
=> ( ( member_a @ ( H @ X ) @ B2 )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups154653438316501755_a_nat @ Phi @ A )
= ( groups6334556678337121940_a_nat @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_134_sum_Oeq__general,axiom,
! [B2: set_list_a,A: set_a,H: a > list_a,Gamma: list_a > nat,Phi: a > nat] :
( ! [Y3: list_a] :
( ( member_list_a @ Y3 @ B2 )
=> ? [X4: a] :
( ( member_a @ X4 @ A )
& ( ( H @ X4 )
= Y3 )
& ! [Ya: a] :
( ( ( member_a @ Ya @ A )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ( member_list_a @ ( H @ X ) @ B2 )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups6334556678337121940_a_nat @ Phi @ A )
= ( groups5521247699297860762_a_nat @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_135_sum_Oeq__general,axiom,
! [B2: set_nat,A: set_a,H: a > nat,Gamma: nat > nat,Phi: a > nat] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B2 )
=> ? [X4: a] :
( ( member_a @ X4 @ A )
& ( ( H @ X4 )
= Y3 )
& ! [Ya: a] :
( ( ( member_a @ Ya @ A )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ( member_nat @ ( H @ X ) @ B2 )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups6334556678337121940_a_nat @ Phi @ A )
= ( groups3542108847815614940at_nat @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_136_sum_Oeq__general,axiom,
! [B2: set_nat_list_a,A: set_a,H: a > nat > list_a,Gamma: ( nat > list_a ) > nat,Phi: a > nat] :
( ! [Y3: nat > list_a] :
( ( member_nat_list_a @ Y3 @ B2 )
=> ? [X4: a] :
( ( member_a @ X4 @ A )
& ( ( H @ X4 )
= Y3 )
& ! [Ya: a] :
( ( ( member_a @ Ya @ A )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ( member_nat_list_a @ ( H @ X ) @ B2 )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups6334556678337121940_a_nat @ Phi @ A )
= ( groups669906071623145473_a_nat @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_137_sum_Oeq__general,axiom,
! [B2: set_nat_a,A: set_a,H: a > nat > a,Gamma: ( nat > a ) > nat,Phi: a > nat] :
( ! [Y3: nat > a] :
( ( member_nat_a @ Y3 @ B2 )
=> ? [X4: a] :
( ( member_a @ X4 @ A )
& ( ( H @ X4 )
= Y3 )
& ! [Ya: a] :
( ( ( member_a @ Ya @ A )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ( member_nat_a @ ( H @ X ) @ B2 )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups6334556678337121940_a_nat @ Phi @ A )
= ( groups154653438316501755_a_nat @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_138_sum_Oeq__general,axiom,
! [B2: set_a,A: set_a,H: a > a,Gamma: a > nat,Phi: a > nat] :
( ! [Y3: a] :
( ( member_a @ Y3 @ B2 )
=> ? [X4: a] :
( ( member_a @ X4 @ A )
& ( ( H @ X4 )
= Y3 )
& ! [Ya: a] :
( ( ( member_a @ Ya @ A )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ( member_a @ ( H @ X ) @ B2 )
& ( ( Gamma @ ( H @ X ) )
= ( Phi @ X ) ) ) )
=> ( ( groups6334556678337121940_a_nat @ Phi @ A )
= ( groups6334556678337121940_a_nat @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_139_sum_Ocong,axiom,
! [A: set_a,B2: set_a,G: a > nat,H: a > nat] :
( ( A = B2 )
=> ( ! [X: a] :
( ( member_a @ X @ B2 )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups6334556678337121940_a_nat @ G @ A )
= ( groups6334556678337121940_a_nat @ H @ B2 ) ) ) ) ).
% sum.cong
thf(fact_140_sum_Oswap,axiom,
! [G: a > a > nat,B2: set_a,A: set_a] :
( ( groups6334556678337121940_a_nat
@ ^ [I: a] : ( groups6334556678337121940_a_nat @ ( G @ I ) @ B2 )
@ A )
= ( groups6334556678337121940_a_nat
@ ^ [J2: a] :
( groups6334556678337121940_a_nat
@ ^ [I: a] : ( G @ I @ J2 )
@ A )
@ B2 ) ) ).
% sum.swap
thf(fact_141_domain_Opoly__sub__degree__le,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_list_a,N: nat,Y: list_list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member5342144027231129785list_a @ X3 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s2403821588304063868list_a @ X3 ) @ one_one_nat ) @ N )
=> ( ( member5342144027231129785list_a @ Y @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s2403821588304063868list_a @ Y ) @ one_one_nat ) @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s2403821588304063868list_a @ ( a_minu4820293213911669576t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ X3 @ Y ) ) @ one_one_nat ) @ N ) ) ) ) ) ) ).
% domain.poly_sub_degree_le
thf(fact_142_domain_Opoly__sub__degree__le,axiom,
! [R: partia2175431115845679010xt_a_b,X3: list_a,N: nat,Y: list_a] :
( ( domain_a_b @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ X3 ) @ one_one_nat ) @ N )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ Y ) @ one_one_nat ) @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ X3 @ Y ) ) @ one_one_nat ) @ N ) ) ) ) ) ) ).
% domain.poly_sub_degree_le
thf(fact_143_domain_Opoly__sub__degree__le,axiom,
! [R: partia2670972154091845814t_unit,X3: list_list_a,N: nat,Y: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ X3 ) @ one_one_nat ) @ N )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ Y ) @ one_one_nat ) @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ X3 @ Y ) ) @ one_one_nat ) @ N ) ) ) ) ) ) ).
% domain.poly_sub_degree_le
thf(fact_144_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_145_le__diff__iff_H,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A2 ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_146_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_147_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_148_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_149_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_150_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_151_sum__mono,axiom,
! [K2: set_list_a,F: list_a > nat,G: list_a > nat] :
( ! [I4: list_a] :
( ( member_list_a @ I4 @ K2 )
=> ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ord_less_eq_nat @ ( groups5521247699297860762_a_nat @ F @ K2 ) @ ( groups5521247699297860762_a_nat @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_152_sum__mono,axiom,
! [K2: set_nat,F: nat > nat,G: nat > nat] :
( ! [I4: nat] :
( ( member_nat @ I4 @ K2 )
=> ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ K2 ) @ ( groups3542108847815614940at_nat @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_153_sum__mono,axiom,
! [K2: set_nat_list_a,F: ( nat > list_a ) > nat,G: ( nat > list_a ) > nat] :
( ! [I4: nat > list_a] :
( ( member_nat_list_a @ I4 @ K2 )
=> ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ord_less_eq_nat @ ( groups669906071623145473_a_nat @ F @ K2 ) @ ( groups669906071623145473_a_nat @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_154_sum__mono,axiom,
! [K2: set_nat_a,F: ( nat > a ) > nat,G: ( nat > a ) > nat] :
( ! [I4: nat > a] :
( ( member_nat_a @ I4 @ K2 )
=> ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ord_less_eq_nat @ ( groups154653438316501755_a_nat @ F @ K2 ) @ ( groups154653438316501755_a_nat @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_155_sum__mono,axiom,
! [K2: set_list_a,F: list_a > int,G: list_a > int] :
( ! [I4: list_a] :
( ( member_list_a @ I4 @ K2 )
=> ( ord_less_eq_int @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ord_less_eq_int @ ( groups5518757228788810486_a_int @ F @ K2 ) @ ( groups5518757228788810486_a_int @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_156_sum__mono,axiom,
! [K2: set_nat,F: nat > int,G: nat > int] :
( ! [I4: nat] :
( ( member_nat @ I4 @ K2 )
=> ( ord_less_eq_int @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ K2 ) @ ( groups3539618377306564664at_int @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_157_sum__mono,axiom,
! [K2: set_nat_list_a,F: ( nat > list_a ) > int,G: ( nat > list_a ) > int] :
( ! [I4: nat > list_a] :
( ( member_nat_list_a @ I4 @ K2 )
=> ( ord_less_eq_int @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ord_less_eq_int @ ( groups667415601114095197_a_int @ F @ K2 ) @ ( groups667415601114095197_a_int @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_158_sum__mono,axiom,
! [K2: set_nat_a,F: ( nat > a ) > int,G: ( nat > a ) > int] :
( ! [I4: nat > a] :
( ( member_nat_a @ I4 @ K2 )
=> ( ord_less_eq_int @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ord_less_eq_int @ ( groups152162967807451479_a_int @ F @ K2 ) @ ( groups152162967807451479_a_int @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_159_sum__mono,axiom,
! [K2: set_a,F: a > int,G: a > int] :
( ! [I4: a] :
( ( member_a @ I4 @ K2 )
=> ( ord_less_eq_int @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ord_less_eq_int @ ( groups6332066207828071664_a_int @ F @ K2 ) @ ( groups6332066207828071664_a_int @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_160_sum__mono,axiom,
! [K2: set_a,F: a > nat,G: a > nat] :
( ! [I4: a] :
( ( member_a @ I4 @ K2 )
=> ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ord_less_eq_nat @ ( groups6334556678337121940_a_nat @ F @ K2 ) @ ( groups6334556678337121940_a_nat @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_161_finite__number__of__roots,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( finite_finite_a @ ( collect_a @ ( polyno4133073214067823460ot_a_b @ r @ P ) ) ) ) ).
% finite_number_of_roots
thf(fact_162_finite__Collect__subsets,axiom,
! [A: set_int] :
( ( finite_finite_int @ A )
=> ( finite6197958912794628473et_int
@ ( collect_set_int
@ ^ [B4: set_int] : ( ord_less_eq_set_int @ B4 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_163_finite__Collect__subsets,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( finite_finite_set_a
@ ( collect_set_a
@ ^ [B4: set_a] : ( ord_less_eq_set_a @ B4 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_164_finite__Collect__subsets,axiom,
! [A: set_list_a] :
( ( finite_finite_list_a @ A )
=> ( finite5282473924520328461list_a
@ ( collect_set_list_a
@ ^ [B4: set_list_a] : ( ord_le8861187494160871172list_a @ B4 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_165_finite__Collect__subsets,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B4: set_nat] : ( ord_less_eq_set_nat @ B4 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_166_poly__sum__degree__le,axiom,
! [A: set_nat_list_a,F: ( nat > list_a ) > list_a,N: nat] :
( ( finite7630042315537210004list_a @ A )
=> ( ! [X: nat > list_a] :
( ( member_nat_list_a @ X @ A )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( F @ X ) ) @ one_one_nat ) @ N ) )
=> ( ! [X: nat > list_a] :
( ( member_nat_list_a @ X @ A )
=> ( member_list_a @ ( F @ X ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( finsum4426778018909949125list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A ) ) @ one_one_nat ) @ N ) ) ) ) ).
% poly_sum_degree_le
thf(fact_167_poly__sum__degree__le,axiom,
! [A: set_nat_a,F: ( nat > a ) > list_a,N: nat] :
( ( finite_finite_nat_a @ A )
=> ( ! [X: nat > a] :
( ( member_nat_a @ X @ A )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( F @ X ) ) @ one_one_nat ) @ N ) )
=> ( ! [X: nat > a] :
( ( member_nat_a @ X @ A )
=> ( member_list_a @ ( F @ X ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( finsum7881878320310621759_nat_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A ) ) @ one_one_nat ) @ N ) ) ) ) ).
% poly_sum_degree_le
thf(fact_168_poly__sum__degree__le,axiom,
! [A: set_a,F: a > list_a,N: nat] :
( ( finite_finite_a @ A )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( F @ X ) ) @ one_one_nat ) @ N ) )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_list_a @ ( F @ X ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A ) ) @ one_one_nat ) @ N ) ) ) ) ).
% poly_sum_degree_le
thf(fact_169_poly__sum__degree__le,axiom,
! [A: set_nat,F: nat > list_a,N: nat] :
( ( finite_finite_nat @ A )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( F @ X ) ) @ one_one_nat ) @ N ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_list_a @ ( F @ X ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( finsum3497512462216549110it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A ) ) @ one_one_nat ) @ N ) ) ) ) ).
% poly_sum_degree_le
thf(fact_170_poly__sum__degree__le,axiom,
! [A: set_list_a,F: list_a > list_a,N: nat] :
( ( finite_finite_list_a @ A )
=> ( ! [X: list_a] :
( ( member_list_a @ X @ A )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( F @ X ) ) @ one_one_nat ) @ N ) )
=> ( ! [X: list_a] :
( ( member_list_a @ X @ A )
=> ( member_list_a @ ( F @ X ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( finsum8721804980556663006list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A ) ) @ one_one_nat ) @ N ) ) ) ) ).
% poly_sum_degree_le
thf(fact_171_poly__sum__degree__le,axiom,
! [A: set_int,F: int > list_a,N: nat] :
( ( finite_finite_int @ A )
=> ( ! [X: int] :
( ( member_int @ X @ A )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( F @ X ) ) @ one_one_nat ) @ N ) )
=> ( ! [X: int] :
( ( member_int @ X @ A )
=> ( member_list_a @ ( F @ X ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( finsum3495021991707498834it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A ) ) @ one_one_nat ) @ N ) ) ) ) ).
% poly_sum_degree_le
thf(fact_172_x_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 ) ) ).
% x.carrier_is_subcring
thf(fact_173_x_Oadd_Oint__pow__mult__distrib,axiom,
! [X3: list_a,Y: list_a,I2: int] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Y )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X3 ) )
=> ( ( member_list_a @ X3 @ ( 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_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I2 @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Y ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I2 @ X3 ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I2 @ Y ) ) ) ) ) ) ).
% x.add.int_pow_mult_distrib
thf(fact_174_x_Oadd_Oint__pow__distrib,axiom,
! [X3: list_a,Y: list_a,I2: int] :
( ( member_list_a @ X3 @ ( 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_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I2 @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Y ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I2 @ X3 ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I2 @ Y ) ) ) ) ) ).
% x.add.int_pow_distrib
thf(fact_175_x_Oup__smult__closed,axiom,
! [A2: list_a,P: nat > list_a] :
( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_nat_list_a @ P @ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_nat_list_a
@ ^ [I: nat] : ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ ( P @ I ) )
@ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.up_smult_closed
thf(fact_176_x_Ocgenideal__is__principalideal,axiom,
! [I2: list_a] :
( ( member_list_a @ I2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( princi8786919440553033881t_unit @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I2 ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.cgenideal_is_principalideal
thf(fact_177_cgenideal__is__principalideal,axiom,
! [I2: a] :
( ( member_a @ I2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( principalideal_a_b @ ( cgenid547466209912283029xt_a_b @ r @ I2 ) @ r ) ) ).
% cgenideal_is_principalideal
thf(fact_178_cgenideal__self,axiom,
! [I2: a] :
( ( member_a @ I2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ I2 @ ( cgenid547466209912283029xt_a_b @ r @ I2 ) ) ) ).
% cgenideal_self
thf(fact_179_x_Om__assoc,axiom,
! [X3: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Y ) @ Z )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z ) ) ) ) ) ) ).
% x.m_assoc
thf(fact_180_x_Om__comm,axiom,
! [X3: list_a,Y: list_a] :
( ( member_list_a @ X3 @ ( 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 ) ) @ X3 @ Y )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X3 ) ) ) ) ).
% x.m_comm
thf(fact_181_x_Om__lcomm,axiom,
! [X3: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z ) )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Z ) ) ) ) ) ) ).
% x.m_lcomm
thf(fact_182_x_Ocgenideal__self,axiom,
! [I2: list_a] :
( ( member_list_a @ I2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ I2 @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I2 ) ) ) ).
% x.cgenideal_self
thf(fact_183_finite__Collect__disjI,axiom,
! [P2: list_list_a > $o,Q2: list_list_a > $o] :
( ( finite1660835950917165235list_a
@ ( collect_list_list_a
@ ^ [X2: list_list_a] :
( ( P2 @ X2 )
| ( Q2 @ X2 ) ) ) )
= ( ( finite1660835950917165235list_a @ ( collect_list_list_a @ P2 ) )
& ( finite1660835950917165235list_a @ ( collect_list_list_a @ Q2 ) ) ) ) ).
% finite_Collect_disjI
thf(fact_184_finite__Collect__disjI,axiom,
! [P2: a > $o,Q2: a > $o] :
( ( finite_finite_a
@ ( collect_a
@ ^ [X2: a] :
( ( P2 @ X2 )
| ( Q2 @ X2 ) ) ) )
= ( ( finite_finite_a @ ( collect_a @ P2 ) )
& ( finite_finite_a @ ( collect_a @ Q2 ) ) ) ) ).
% finite_Collect_disjI
thf(fact_185_finite__Collect__disjI,axiom,
! [P2: nat > $o,Q2: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( P2 @ X2 )
| ( Q2 @ X2 ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
& ( finite_finite_nat @ ( collect_nat @ Q2 ) ) ) ) ).
% finite_Collect_disjI
thf(fact_186_finite__Collect__disjI,axiom,
! [P2: list_a > $o,Q2: list_a > $o] :
( ( finite_finite_list_a
@ ( collect_list_a
@ ^ [X2: list_a] :
( ( P2 @ X2 )
| ( Q2 @ X2 ) ) ) )
= ( ( finite_finite_list_a @ ( collect_list_a @ P2 ) )
& ( finite_finite_list_a @ ( collect_list_a @ Q2 ) ) ) ) ).
% finite_Collect_disjI
thf(fact_187_finite__Collect__disjI,axiom,
! [P2: int > $o,Q2: int > $o] :
( ( finite_finite_int
@ ( collect_int
@ ^ [X2: int] :
( ( P2 @ X2 )
| ( Q2 @ X2 ) ) ) )
= ( ( finite_finite_int @ ( collect_int @ P2 ) )
& ( finite_finite_int @ ( collect_int @ Q2 ) ) ) ) ).
% finite_Collect_disjI
thf(fact_188_finite__Collect__conjI,axiom,
! [P2: list_list_a > $o,Q2: list_list_a > $o] :
( ( ( finite1660835950917165235list_a @ ( collect_list_list_a @ P2 ) )
| ( finite1660835950917165235list_a @ ( collect_list_list_a @ Q2 ) ) )
=> ( finite1660835950917165235list_a
@ ( collect_list_list_a
@ ^ [X2: list_list_a] :
( ( P2 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_189_finite__Collect__conjI,axiom,
! [P2: a > $o,Q2: a > $o] :
( ( ( finite_finite_a @ ( collect_a @ P2 ) )
| ( finite_finite_a @ ( collect_a @ Q2 ) ) )
=> ( finite_finite_a
@ ( collect_a
@ ^ [X2: a] :
( ( P2 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_190_finite__Collect__conjI,axiom,
! [P2: nat > $o,Q2: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
| ( finite_finite_nat @ ( collect_nat @ Q2 ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( P2 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_191_finite__Collect__conjI,axiom,
! [P2: list_a > $o,Q2: list_a > $o] :
( ( ( finite_finite_list_a @ ( collect_list_a @ P2 ) )
| ( finite_finite_list_a @ ( collect_list_a @ Q2 ) ) )
=> ( finite_finite_list_a
@ ( collect_list_a
@ ^ [X2: list_a] :
( ( P2 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_192_finite__Collect__conjI,axiom,
! [P2: int > $o,Q2: int > $o] :
( ( ( finite_finite_int @ ( collect_int @ P2 ) )
| ( finite_finite_int @ ( collect_int @ Q2 ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [X2: int] :
( ( P2 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_193_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_194_x_Ol__distr,axiom,
! [X3: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Y ) @ Z )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Z ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z ) ) ) ) ) ) ).
% x.l_distr
thf(fact_195_x_Or__distr,axiom,
! [X3: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Z @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Y ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Z @ X3 ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Z @ Y ) ) ) ) ) ) ).
% x.r_distr
thf(fact_196_x_Oadd__pow__ldistr__int,axiom,
! [A2: list_a,B: list_a,K: int] :
( ( member_list_a @ A2 @ ( 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 ) ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ A2 ) @ B )
= ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ B ) ) ) ) ) ).
% x.add_pow_ldistr_int
thf(fact_197_x_Oadd__pow__rdistr__int,axiom,
! [A2: list_a,B: list_a,K: int] :
( ( member_list_a @ A2 @ ( 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 ) ) @ A2 @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ B ) )
= ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ B ) ) ) ) ) ).
% x.add_pow_rdistr_int
thf(fact_198_is__root__poly__mult__imp__is__root,axiom,
! [P: list_a,Q: list_a,X3: 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 ) @ X3 )
=> ( ( polyno4133073214067823460ot_a_b @ r @ P @ X3 )
| ( polyno4133073214067823460ot_a_b @ r @ Q @ X3 ) ) ) ) ) ).
% is_root_poly_mult_imp_is_root
thf(fact_199_finite__Diff2,axiom,
! [B2: set_a,A: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( finite_finite_a @ ( minus_minus_set_a @ A @ B2 ) )
= ( finite_finite_a @ A ) ) ) ).
% finite_Diff2
thf(fact_200_finite__Diff2,axiom,
! [B2: set_nat,A: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B2 ) )
= ( finite_finite_nat @ A ) ) ) ).
% finite_Diff2
thf(fact_201_finite__Diff2,axiom,
! [B2: set_list_a,A: set_list_a] :
( ( finite_finite_list_a @ B2 )
=> ( ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A @ B2 ) )
= ( finite_finite_list_a @ A ) ) ) ).
% finite_Diff2
thf(fact_202_finite__Diff2,axiom,
! [B2: set_int,A: set_int] :
( ( finite_finite_int @ B2 )
=> ( ( finite_finite_int @ ( minus_minus_set_int @ A @ B2 ) )
= ( finite_finite_int @ A ) ) ) ).
% finite_Diff2
thf(fact_203_finite__Diff,axiom,
! [A: set_a,B2: set_a] :
( ( finite_finite_a @ A )
=> ( finite_finite_a @ ( minus_minus_set_a @ A @ B2 ) ) ) ).
% finite_Diff
thf(fact_204_finite__Diff,axiom,
! [A: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B2 ) ) ) ).
% finite_Diff
thf(fact_205_finite__Diff,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( finite_finite_list_a @ A )
=> ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A @ B2 ) ) ) ).
% finite_Diff
thf(fact_206_finite__Diff,axiom,
! [A: set_int,B2: set_int] :
( ( finite_finite_int @ A )
=> ( finite_finite_int @ ( minus_minus_set_int @ A @ B2 ) ) ) ).
% finite_Diff
thf(fact_207_x_Om__closed,axiom,
! [X3: list_a,Y: list_a] :
( ( member_list_a @ X3 @ ( 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 ) ) @ X3 @ Y ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.m_closed
thf(fact_208_x_Oadd_Oint__pow__closed,axiom,
! [X3: list_a,I2: int] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I2 @ X3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.add.int_pow_closed
thf(fact_209_Diff__infinite__finite,axiom,
! [T: set_a,S: set_a] :
( ( finite_finite_a @ T )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_210_Diff__infinite__finite,axiom,
! [T: set_nat,S: set_nat] :
( ( finite_finite_nat @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_211_Diff__infinite__finite,axiom,
! [T: set_list_a,S: set_list_a] :
( ( finite_finite_list_a @ T )
=> ( ~ ( finite_finite_list_a @ S )
=> ~ ( finite_finite_list_a @ ( minus_646659088055828811list_a @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_212_Diff__infinite__finite,axiom,
! [T: set_int,S: set_int] :
( ( finite_finite_int @ T )
=> ( ~ ( finite_finite_int @ S )
=> ~ ( finite_finite_int @ ( minus_minus_set_int @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_213_sum__diff__nat,axiom,
! [B2: set_int,A: set_int,F: int > nat] :
( ( finite_finite_int @ B2 )
=> ( ( ord_less_eq_set_int @ B2 @ A )
=> ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A @ B2 ) )
= ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A ) @ ( groups4541462559716669496nt_nat @ F @ B2 ) ) ) ) ) ).
% sum_diff_nat
thf(fact_214_sum__diff__nat,axiom,
! [B2: set_list_a,A: set_list_a,F: list_a > nat] :
( ( finite_finite_list_a @ B2 )
=> ( ( ord_le8861187494160871172list_a @ B2 @ A )
=> ( ( groups5521247699297860762_a_nat @ F @ ( minus_646659088055828811list_a @ A @ B2 ) )
= ( minus_minus_nat @ ( groups5521247699297860762_a_nat @ F @ A ) @ ( groups5521247699297860762_a_nat @ F @ B2 ) ) ) ) ) ).
% sum_diff_nat
thf(fact_215_sum__diff__nat,axiom,
! [B2: set_nat,A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A )
=> ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A @ B2 ) )
= ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A ) @ ( groups3542108847815614940at_nat @ F @ B2 ) ) ) ) ) ).
% sum_diff_nat
thf(fact_216_sum__diff__nat,axiom,
! [B2: set_a,A: set_a,F: a > nat] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( groups6334556678337121940_a_nat @ F @ ( minus_minus_set_a @ A @ B2 ) )
= ( minus_minus_nat @ ( groups6334556678337121940_a_nat @ F @ A ) @ ( groups6334556678337121940_a_nat @ F @ B2 ) ) ) ) ) ).
% sum_diff_nat
thf(fact_217_pigeonhole__infinite__rel,axiom,
! [A: set_a,B2: set_a,R: a > a > $o] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B2 )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ? [Xa: a] :
( ( member_a @ Xa @ B2 )
& ( R @ X @ Xa ) ) )
=> ? [X: a] :
( ( member_a @ X @ B2 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A )
& ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_218_pigeonhole__infinite__rel,axiom,
! [A: set_a,B2: set_nat,R: a > nat > $o] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B2 )
& ( R @ X @ Xa ) ) )
=> ? [X: nat] :
( ( member_nat @ X @ B2 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A )
& ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_219_pigeonhole__infinite__rel,axiom,
! [A: set_a,B2: set_int,R: a > int > $o] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_finite_int @ B2 )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ? [Xa: int] :
( ( member_int @ Xa @ B2 )
& ( R @ X @ Xa ) ) )
=> ? [X: int] :
( ( member_int @ X @ B2 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A )
& ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_220_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B2: set_a,R: nat > a > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_a @ B2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ? [Xa: a] :
( ( member_a @ Xa @ B2 )
& ( R @ X @ Xa ) ) )
=> ? [X: a] :
( ( member_a @ X @ B2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A )
& ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_221_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B2: set_nat,R: nat > nat > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B2 )
& ( R @ X @ Xa ) ) )
=> ? [X: nat] :
( ( member_nat @ X @ B2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A )
& ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_222_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B2: set_int,R: nat > int > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_int @ B2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ? [Xa: int] :
( ( member_int @ Xa @ B2 )
& ( R @ X @ Xa ) ) )
=> ? [X: int] :
( ( member_int @ X @ B2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A )
& ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_223_pigeonhole__infinite__rel,axiom,
! [A: set_int,B2: set_a,R: int > a > $o] :
( ~ ( finite_finite_int @ A )
=> ( ( finite_finite_a @ B2 )
=> ( ! [X: int] :
( ( member_int @ X @ A )
=> ? [Xa: a] :
( ( member_a @ Xa @ B2 )
& ( R @ X @ Xa ) ) )
=> ? [X: a] :
( ( member_a @ X @ B2 )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A4: int] :
( ( member_int @ A4 @ A )
& ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_224_pigeonhole__infinite__rel,axiom,
! [A: set_int,B2: set_nat,R: int > nat > $o] :
( ~ ( finite_finite_int @ A )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X: int] :
( ( member_int @ X @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B2 )
& ( R @ X @ Xa ) ) )
=> ? [X: nat] :
( ( member_nat @ X @ B2 )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A4: int] :
( ( member_int @ A4 @ A )
& ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_225_pigeonhole__infinite__rel,axiom,
! [A: set_int,B2: set_int,R: int > int > $o] :
( ~ ( finite_finite_int @ A )
=> ( ( finite_finite_int @ B2 )
=> ( ! [X: int] :
( ( member_int @ X @ A )
=> ? [Xa: int] :
( ( member_int @ Xa @ B2 )
& ( R @ X @ Xa ) ) )
=> ? [X: int] :
( ( member_int @ X @ B2 )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A4: int] :
( ( member_int @ A4 @ A )
& ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_226_pigeonhole__infinite__rel,axiom,
! [A: set_a,B2: set_list_a,R: a > list_a > $o] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_finite_list_a @ B2 )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ? [Xa: list_a] :
( ( member_list_a @ Xa @ B2 )
& ( R @ X @ Xa ) ) )
=> ? [X: list_a] :
( ( member_list_a @ X @ B2 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A )
& ( R @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_227_not__finite__existsD,axiom,
! [P2: list_list_a > $o] :
( ~ ( finite1660835950917165235list_a @ ( collect_list_list_a @ P2 ) )
=> ? [X_1: list_list_a] : ( P2 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_228_not__finite__existsD,axiom,
! [P2: a > $o] :
( ~ ( finite_finite_a @ ( collect_a @ P2 ) )
=> ? [X_1: a] : ( P2 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_229_not__finite__existsD,axiom,
! [P2: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P2 ) )
=> ? [X_1: nat] : ( P2 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_230_not__finite__existsD,axiom,
! [P2: list_a > $o] :
( ~ ( finite_finite_list_a @ ( collect_list_a @ P2 ) )
=> ? [X_1: list_a] : ( P2 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_231_not__finite__existsD,axiom,
! [P2: int > $o] :
( ~ ( finite_finite_int @ ( collect_int @ P2 ) )
=> ? [X_1: int] : ( P2 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_232_finite__has__maximal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ( ord_less_eq_nat @ A2 @ X )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_233_finite__has__maximal2,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ? [X: set_a] :
( ( member_set_a @ X @ A )
& ( ord_less_eq_set_a @ A2 @ X )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_234_finite__has__maximal2,axiom,
! [A: set_set_list_a,A2: set_list_a] :
( ( finite5282473924520328461list_a @ A )
=> ( ( member_set_list_a @ A2 @ A )
=> ? [X: set_list_a] :
( ( member_set_list_a @ X @ A )
& ( ord_le8861187494160871172list_a @ A2 @ X )
& ! [Xa: set_list_a] :
( ( member_set_list_a @ Xa @ A )
=> ( ( ord_le8861187494160871172list_a @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_235_finite__has__maximal2,axiom,
! [A: set_int,A2: int] :
( ( finite_finite_int @ A )
=> ( ( member_int @ A2 @ A )
=> ? [X: int] :
( ( member_int @ X @ A )
& ( ord_less_eq_int @ A2 @ X )
& ! [Xa: int] :
( ( member_int @ Xa @ A )
=> ( ( ord_less_eq_int @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_236_finite__has__maximal2,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( member_set_nat @ A2 @ A )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ A )
& ( ord_less_eq_set_nat @ A2 @ X )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_237_finite__has__minimal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ( ord_less_eq_nat @ X @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_238_finite__has__minimal2,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ? [X: set_a] :
( ( member_set_a @ X @ A )
& ( ord_less_eq_set_a @ X @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_239_finite__has__minimal2,axiom,
! [A: set_set_list_a,A2: set_list_a] :
( ( finite5282473924520328461list_a @ A )
=> ( ( member_set_list_a @ A2 @ A )
=> ? [X: set_list_a] :
( ( member_set_list_a @ X @ A )
& ( ord_le8861187494160871172list_a @ X @ A2 )
& ! [Xa: set_list_a] :
( ( member_set_list_a @ Xa @ A )
=> ( ( ord_le8861187494160871172list_a @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_240_finite__has__minimal2,axiom,
! [A: set_int,A2: int] :
( ( finite_finite_int @ A )
=> ( ( member_int @ A2 @ A )
=> ? [X: int] :
( ( member_int @ X @ A )
& ( ord_less_eq_int @ X @ A2 )
& ! [Xa: int] :
( ( member_int @ Xa @ A )
=> ( ( ord_less_eq_int @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_241_finite__has__minimal2,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( member_set_nat @ A2 @ A )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ A )
& ( ord_less_eq_set_nat @ X @ A2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_242_rev__finite__subset,axiom,
! [B2: set_int,A: set_int] :
( ( finite_finite_int @ B2 )
=> ( ( ord_less_eq_set_int @ A @ B2 )
=> ( finite_finite_int @ A ) ) ) ).
% rev_finite_subset
thf(fact_243_rev__finite__subset,axiom,
! [B2: set_a,A: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A @ B2 )
=> ( finite_finite_a @ A ) ) ) ).
% rev_finite_subset
thf(fact_244_rev__finite__subset,axiom,
! [B2: set_list_a,A: set_list_a] :
( ( finite_finite_list_a @ B2 )
=> ( ( ord_le8861187494160871172list_a @ A @ B2 )
=> ( finite_finite_list_a @ A ) ) ) ).
% rev_finite_subset
thf(fact_245_rev__finite__subset,axiom,
! [B2: set_nat,A: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( finite_finite_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_246_infinite__super,axiom,
! [S: set_int,T: set_int] :
( ( ord_less_eq_set_int @ S @ T )
=> ( ~ ( finite_finite_int @ S )
=> ~ ( finite_finite_int @ T ) ) ) ).
% infinite_super
thf(fact_247_infinite__super,axiom,
! [S: set_a,T: set_a] :
( ( ord_less_eq_set_a @ S @ T )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ T ) ) ) ).
% infinite_super
thf(fact_248_infinite__super,axiom,
! [S: set_list_a,T: set_list_a] :
( ( ord_le8861187494160871172list_a @ S @ T )
=> ( ~ ( finite_finite_list_a @ S )
=> ~ ( finite_finite_list_a @ T ) ) ) ).
% infinite_super
thf(fact_249_infinite__super,axiom,
! [S: set_nat,T: set_nat] :
( ( ord_less_eq_set_nat @ S @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T ) ) ) ).
% infinite_super
thf(fact_250_finite__subset,axiom,
! [A: set_int,B2: set_int] :
( ( ord_less_eq_set_int @ A @ B2 )
=> ( ( finite_finite_int @ B2 )
=> ( finite_finite_int @ A ) ) ) ).
% finite_subset
thf(fact_251_finite__subset,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( finite_finite_a @ B2 )
=> ( finite_finite_a @ A ) ) ) ).
% finite_subset
thf(fact_252_finite__subset,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B2 )
=> ( ( finite_finite_list_a @ B2 )
=> ( finite_finite_list_a @ A ) ) ) ).
% finite_subset
thf(fact_253_finite__subset,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( finite_finite_nat @ B2 )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_subset
thf(fact_254_domain_Opoly__sum__degree__le,axiom,
! [R: partia2175431115845679010xt_a_b,A: set_a,F: a > list_a,N: nat] :
( ( domain_a_b @ R )
=> ( ( finite_finite_a @ A )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( F @ X ) ) @ one_one_nat ) @ N ) )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_list_a @ ( F @ X ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ F @ A ) ) @ one_one_nat ) @ N ) ) ) ) ) ).
% domain.poly_sum_degree_le
thf(fact_255_domain_Opoly__sum__degree__le,axiom,
! [R: partia2175431115845679010xt_a_b,A: set_nat,F: nat > list_a,N: nat] :
( ( domain_a_b @ R )
=> ( ( finite_finite_nat @ A )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( F @ X ) ) @ one_one_nat ) @ N ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_list_a @ ( F @ X ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( finsum3497512462216549110it_nat @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ F @ A ) ) @ one_one_nat ) @ N ) ) ) ) ) ).
% domain.poly_sum_degree_le
thf(fact_256_domain_Opoly__sum__degree__le,axiom,
! [R: partia2175431115845679010xt_a_b,A: set_int,F: int > list_a,N: nat] :
( ( domain_a_b @ R )
=> ( ( finite_finite_int @ A )
=> ( ! [X: int] :
( ( member_int @ X @ A )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( F @ X ) ) @ one_one_nat ) @ N ) )
=> ( ! [X: int] :
( ( member_int @ X @ A )
=> ( member_list_a @ ( F @ X ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( finsum3495021991707498834it_int @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ F @ A ) ) @ one_one_nat ) @ N ) ) ) ) ) ).
% domain.poly_sum_degree_le
thf(fact_257_domain_Opoly__sum__degree__le,axiom,
! [R: partia2175431115845679010xt_a_b,A: set_list_a,F: list_a > list_a,N: nat] :
( ( domain_a_b @ R )
=> ( ( finite_finite_list_a @ A )
=> ( ! [X: list_a] :
( ( member_list_a @ X @ A )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( F @ X ) ) @ one_one_nat ) @ N ) )
=> ( ! [X: list_a] :
( ( member_list_a @ X @ A )
=> ( member_list_a @ ( F @ X ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( finsum8721804980556663006list_a @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ F @ A ) ) @ one_one_nat ) @ N ) ) ) ) ) ).
% domain.poly_sum_degree_le
thf(fact_258_domain_Opoly__sum__degree__le,axiom,
! [R: partia2670972154091845814t_unit,A: set_a,F: a > list_list_a,N: nat] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( finite_finite_a @ A )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( F @ X ) ) @ one_one_nat ) @ N ) )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_list_list_a @ ( F @ X ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( finsum463596448938265310unit_a @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ F @ A ) ) @ one_one_nat ) @ N ) ) ) ) ) ).
% domain.poly_sum_degree_le
thf(fact_259_domain_Opoly__sum__degree__le,axiom,
! [R: partia2670972154091845814t_unit,A: set_nat,F: nat > list_list_a,N: nat] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( finite_finite_nat @ A )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( F @ X ) ) @ one_one_nat ) @ N ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_list_list_a @ ( F @ X ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( finsum3990971441743328240it_nat @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ F @ A ) ) @ one_one_nat ) @ N ) ) ) ) ) ).
% domain.poly_sum_degree_le
thf(fact_260_domain_Opoly__sum__degree__le,axiom,
! [R: partia2670972154091845814t_unit,A: set_int,F: int > list_list_a,N: nat] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( finite_finite_int @ A )
=> ( ! [X: int] :
( ( member_int @ X @ A )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( F @ X ) ) @ one_one_nat ) @ N ) )
=> ( ! [X: int] :
( ( member_int @ X @ A )
=> ( member_list_list_a @ ( F @ X ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( finsum3988480971234277964it_int @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ F @ A ) ) @ one_one_nat ) @ N ) ) ) ) ) ).
% domain.poly_sum_degree_le
thf(fact_261_domain_Opoly__sum__degree__le,axiom,
! [R: partia2956882679547061052t_unit,A: set_a,F: a > list_list_list_a,N: nat] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( finite_finite_a @ A )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s2403821588304063868list_a @ ( F @ X ) ) @ one_one_nat ) @ N ) )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( member5342144027231129785list_a @ ( F @ X ) @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s2403821588304063868list_a @ ( finsum1842526356606396388unit_a @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ F @ A ) ) @ one_one_nat ) @ N ) ) ) ) ) ).
% domain.poly_sum_degree_le
thf(fact_262_domain_Opoly__sum__degree__le,axiom,
! [R: partia2956882679547061052t_unit,A: set_nat,F: nat > list_list_list_a,N: nat] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( finite_finite_nat @ A )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s2403821588304063868list_a @ ( F @ X ) ) @ one_one_nat ) @ N ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member5342144027231129785list_a @ ( F @ X ) @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s2403821588304063868list_a @ ( finsum3773694726227313002it_nat @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ F @ A ) ) @ one_one_nat ) @ N ) ) ) ) ) ).
% domain.poly_sum_degree_le
thf(fact_263_domain_Opoly__sum__degree__le,axiom,
! [R: partia2956882679547061052t_unit,A: set_int,F: int > list_list_list_a,N: nat] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( finite_finite_int @ A )
=> ( ! [X: int] :
( ( member_int @ X @ A )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s2403821588304063868list_a @ ( F @ X ) ) @ one_one_nat ) @ N ) )
=> ( ! [X: int] :
( ( member_int @ X @ A )
=> ( member5342144027231129785list_a @ ( F @ X ) @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s2403821588304063868list_a @ ( finsum3771204255718262726it_int @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ F @ A ) ) @ one_one_nat ) @ N ) ) ) ) ) ).
% domain.poly_sum_degree_le
thf(fact_264_x_Omonoid__cancelI,axiom,
( ! [A3: list_a,B3: list_a,C2: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C2 @ A3 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C2 @ B3 ) )
=> ( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( A3 = B3 ) ) ) ) )
=> ( ! [A3: list_a,B3: list_a,C2: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A3 @ C2 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B3 @ C2 ) )
=> ( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( A3 = B3 ) ) ) ) )
=> ( monoid4303264861975686087t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.monoid_cancelI
thf(fact_265_degree__zero__imp__not__is__root,axiom,
! [P: list_a,X3: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= zero_zero_nat )
=> ~ ( polyno4133073214067823460ot_a_b @ r @ P @ X3 ) ) ) ).
% degree_zero_imp_not_is_root
thf(fact_266_poly__mult__degree__le,axiom,
! [X3: list_a,Y: list_a,N: nat,M: nat] :
( ( member_list_a @ X3 @ ( 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 ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ X3 ) @ one_one_nat ) @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ Y ) @ one_one_nat ) @ M )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Y ) ) @ one_one_nat ) @ ( plus_plus_nat @ N @ M ) ) ) ) ) ) ).
% poly_mult_degree_le
thf(fact_267_poly__mult__degree__le__1,axiom,
! [X3: list_a,Y: list_a] :
( ( member_list_a @ X3 @ ( 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 ) ) ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Y ) ) @ one_one_nat ) @ ( plus_plus_nat @ ( minus_minus_nat @ ( size_size_list_a @ X3 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Y ) @ one_one_nat ) ) ) ) ) ).
% poly_mult_degree_le_1
thf(fact_268_domain_Ofinite__number__of__roots,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( finite1660835950917165235list_a @ ( collect_list_list_a @ ( polyno5142720416380192742t_unit @ R @ P ) ) ) ) ) ).
% domain.finite_number_of_roots
thf(fact_269_domain_Ofinite__number__of__roots,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( finite_finite_list_a @ ( collect_list_a @ ( polyno6951661231331188332t_unit @ R @ P ) ) ) ) ) ).
% domain.finite_number_of_roots
thf(fact_270_domain_Ofinite__number__of__roots,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a] :
( ( domain_a_b @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( finite_finite_a @ ( collect_a @ ( polyno4133073214067823460ot_a_b @ R @ P ) ) ) ) ) ).
% domain.finite_number_of_roots
thf(fact_271_x_Osplitted__on__def,axiom,
! [K2: set_list_a,P: list_list_a] :
( ( polyno1986131841096413848t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ P )
= ( ( size_s2335926164413107382list_a @ ( polyno5990348478334826338t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ P ) )
= ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat ) ) ) ).
% x.splitted_on_def
thf(fact_272_domain_Ois__root__poly__mult__imp__is__root,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a,Q: list_list_list_a,X3: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( member5342144027231129785list_a @ Q @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( polyno5142720416380192742t_unit @ R @ ( mult_l3065349954589089105t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ P @ Q ) @ X3 )
=> ( ( polyno5142720416380192742t_unit @ R @ P @ X3 )
| ( polyno5142720416380192742t_unit @ R @ Q @ X3 ) ) ) ) ) ) ).
% domain.is_root_poly_mult_imp_is_root
thf(fact_273_domain_Ois__root__poly__mult__imp__is__root,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,Q: list_list_a,X3: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( polyno6951661231331188332t_unit @ R @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ P @ Q ) @ X3 )
=> ( ( polyno6951661231331188332t_unit @ R @ P @ X3 )
| ( polyno6951661231331188332t_unit @ R @ Q @ X3 ) ) ) ) ) ) ).
% domain.is_root_poly_mult_imp_is_root
thf(fact_274_domain_Ois__root__poly__mult__imp__is__root,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,Q: list_a,X3: a] :
( ( domain_a_b @ R )
=> ( ( 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 ) @ X3 )
=> ( ( polyno4133073214067823460ot_a_b @ R @ P @ X3 )
| ( polyno4133073214067823460ot_a_b @ R @ Q @ X3 ) ) ) ) ) ) ).
% domain.is_root_poly_mult_imp_is_root
thf(fact_275_x_Oa__lcos__m__assoc,axiom,
! [M2: set_list_a,G: list_a,H: list_a] :
( ( ord_le8861187494160871172list_a @ M2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ G @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ H @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ M2 ) )
= ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ H ) @ M2 ) ) ) ) ) ).
% x.a_lcos_m_assoc
thf(fact_276_x_Opoly__of__const__in__carrier,axiom,
! [S3: list_a] :
( ( member_list_a @ S3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_list_a @ ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ S3 ) @ ( 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 ) ) ) ) ) ) ) ).
% x.poly_of_const_in_carrier
thf(fact_277_x_Oline__extension__mem__iff,axiom,
! [U: list_a,K2: set_list_a,A2: list_a,E: set_list_a] :
( ( member_list_a @ U @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ A2 @ E ) )
= ( ? [X2: list_a] :
( ( member_list_a @ X2 @ K2 )
& ? [Y2: list_a] :
( ( member_list_a @ Y2 @ 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 @ A2 ) @ Y2 ) ) ) ) ) ) ).
% x.line_extension_mem_iff
thf(fact_278_x_Oline__extension__in__carrier,axiom,
! [K2: set_list_a,A2: list_a,E: set_list_a] :
( ( ord_le8861187494160871172list_a @ K2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A2 @ ( 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 @ A2 @ E ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.line_extension_in_carrier
thf(fact_279_x_Oa__l__coset__subset__G,axiom,
! [H2: set_list_a,X3: list_a] :
( ( ord_le8861187494160871172list_a @ H2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X3 @ ( 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 ) ) @ X3 @ H2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.a_l_coset_subset_G
thf(fact_280_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_281_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_282_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_283_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_284_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_285_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_286_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_287_diff__diff__left,axiom,
! [I2: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
= ( minus_minus_nat @ I2 @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_288_sum_Oneutral__const,axiom,
! [A: set_a] :
( ( groups6334556678337121940_a_nat
@ ^ [Uu: a] : zero_zero_nat
@ A )
= zero_zero_nat ) ).
% sum.neutral_const
thf(fact_289_sum_Oinfinite,axiom,
! [A: set_nat,G: nat > nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( groups3542108847815614940at_nat @ G @ A )
= zero_zero_nat ) ) ).
% sum.infinite
thf(fact_290_sum_Oinfinite,axiom,
! [A: set_int,G: int > nat] :
( ~ ( finite_finite_int @ A )
=> ( ( groups4541462559716669496nt_nat @ G @ A )
= zero_zero_nat ) ) ).
% sum.infinite
thf(fact_291_sum_Oinfinite,axiom,
! [A: set_a,G: a > int] :
( ~ ( finite_finite_a @ A )
=> ( ( groups6332066207828071664_a_int @ G @ A )
= zero_zero_int ) ) ).
% sum.infinite
thf(fact_292_sum_Oinfinite,axiom,
! [A: set_nat,G: nat > int] :
( ~ ( finite_finite_nat @ A )
=> ( ( groups3539618377306564664at_int @ G @ A )
= zero_zero_int ) ) ).
% sum.infinite
thf(fact_293_sum_Oinfinite,axiom,
! [A: set_int,G: int > int] :
( ~ ( finite_finite_int @ A )
=> ( ( groups4538972089207619220nt_int @ G @ A )
= zero_zero_int ) ) ).
% sum.infinite
thf(fact_294_sum_Oinfinite,axiom,
! [A: set_a,G: a > nat] :
( ~ ( finite_finite_a @ A )
=> ( ( groups6334556678337121940_a_nat @ G @ A )
= zero_zero_nat ) ) ).
% sum.infinite
thf(fact_295_sum_Oinfinite,axiom,
! [A: set_list_a,G: list_a > nat] :
( ~ ( finite_finite_list_a @ A )
=> ( ( groups5521247699297860762_a_nat @ G @ A )
= zero_zero_nat ) ) ).
% sum.infinite
thf(fact_296_sum_Oinfinite,axiom,
! [A: set_a,G: a > multiset_a] :
( ~ ( finite_finite_a @ A )
=> ( ( groups4808324907802680448iset_a @ G @ A )
= zero_zero_multiset_a ) ) ).
% sum.infinite
thf(fact_297_sum_Oinfinite,axiom,
! [A: set_nat,G: nat > multiset_a] :
( ~ ( finite_finite_nat @ A )
=> ( ( groups1580436272196575032iset_a @ G @ A )
= zero_zero_multiset_a ) ) ).
% sum.infinite
thf(fact_298_sum_Oinfinite,axiom,
! [A: set_int,G: int > multiset_a] :
( ~ ( finite_finite_int @ A )
=> ( ( groups3457364905213935068iset_a @ G @ A )
= zero_zero_multiset_a ) ) ).
% sum.infinite
thf(fact_299_sum__eq__0__iff,axiom,
! [F2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ F2 )
=> ( ( ( groups3542108847815614940at_nat @ F @ F2 )
= zero_zero_nat )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ F2 )
=> ( ( F @ X2 )
= zero_zero_nat ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_300_sum__eq__0__iff,axiom,
! [F2: set_list_a,F: list_a > nat] :
( ( finite_finite_list_a @ F2 )
=> ( ( ( groups5521247699297860762_a_nat @ F @ F2 )
= zero_zero_nat )
= ( ! [X2: list_a] :
( ( member_list_a @ X2 @ F2 )
=> ( ( F @ X2 )
= zero_zero_nat ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_301_sum__eq__0__iff,axiom,
! [F2: set_int,F: int > nat] :
( ( finite_finite_int @ F2 )
=> ( ( ( groups4541462559716669496nt_nat @ F @ F2 )
= zero_zero_nat )
= ( ! [X2: int] :
( ( member_int @ X2 @ F2 )
=> ( ( F @ X2 )
= zero_zero_nat ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_302_sum__eq__0__iff,axiom,
! [F2: set_a,F: a > nat] :
( ( finite_finite_a @ F2 )
=> ( ( ( groups6334556678337121940_a_nat @ F @ F2 )
= zero_zero_nat )
= ( ! [X2: a] :
( ( member_a @ X2 @ F2 )
=> ( ( F @ X2 )
= zero_zero_nat ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_303_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_304_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_305_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_306_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_307_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_308_sum_Odelta_H,axiom,
! [S: set_nat,A2: nat,B: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A2 @ S )
=> ( ( groups3542108847815614940at_nat
@ ^ [K3: nat] : ( if_nat @ ( A2 = K3 ) @ ( B @ K3 ) @ zero_zero_nat )
@ S )
= ( B @ A2 ) ) )
& ( ~ ( member_nat @ A2 @ S )
=> ( ( groups3542108847815614940at_nat
@ ^ [K3: nat] : ( if_nat @ ( A2 = K3 ) @ ( B @ K3 ) @ zero_zero_nat )
@ S )
= zero_zero_nat ) ) ) ) ).
% sum.delta'
thf(fact_309_sum_Odelta_H,axiom,
! [S: set_int,A2: int,B: int > nat] :
( ( finite_finite_int @ S )
=> ( ( ( member_int @ A2 @ S )
=> ( ( groups4541462559716669496nt_nat
@ ^ [K3: int] : ( if_nat @ ( A2 = K3 ) @ ( B @ K3 ) @ zero_zero_nat )
@ S )
= ( B @ A2 ) ) )
& ( ~ ( member_int @ A2 @ S )
=> ( ( groups4541462559716669496nt_nat
@ ^ [K3: int] : ( if_nat @ ( A2 = K3 ) @ ( B @ K3 ) @ zero_zero_nat )
@ S )
= zero_zero_nat ) ) ) ) ).
% sum.delta'
thf(fact_310_sum_Odelta_H,axiom,
! [S: set_a,A2: a,B: a > int] :
( ( finite_finite_a @ S )
=> ( ( ( member_a @ A2 @ S )
=> ( ( groups6332066207828071664_a_int
@ ^ [K3: a] : ( if_int @ ( A2 = K3 ) @ ( B @ K3 ) @ zero_zero_int )
@ S )
= ( B @ A2 ) ) )
& ( ~ ( member_a @ A2 @ S )
=> ( ( groups6332066207828071664_a_int
@ ^ [K3: a] : ( if_int @ ( A2 = K3 ) @ ( B @ K3 ) @ zero_zero_int )
@ S )
= zero_zero_int ) ) ) ) ).
% sum.delta'
thf(fact_311_sum_Odelta_H,axiom,
! [S: set_nat,A2: nat,B: nat > int] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A2 @ S )
=> ( ( groups3539618377306564664at_int
@ ^ [K3: nat] : ( if_int @ ( A2 = K3 ) @ ( B @ K3 ) @ zero_zero_int )
@ S )
= ( B @ A2 ) ) )
& ( ~ ( member_nat @ A2 @ S )
=> ( ( groups3539618377306564664at_int
@ ^ [K3: nat] : ( if_int @ ( A2 = K3 ) @ ( B @ K3 ) @ zero_zero_int )
@ S )
= zero_zero_int ) ) ) ) ).
% sum.delta'
thf(fact_312_sum_Odelta_H,axiom,
! [S: set_int,A2: int,B: int > int] :
( ( finite_finite_int @ S )
=> ( ( ( member_int @ A2 @ S )
=> ( ( groups4538972089207619220nt_int
@ ^ [K3: int] : ( if_int @ ( A2 = K3 ) @ ( B @ K3 ) @ zero_zero_int )
@ S )
= ( B @ A2 ) ) )
& ( ~ ( member_int @ A2 @ S )
=> ( ( groups4538972089207619220nt_int
@ ^ [K3: int] : ( if_int @ ( A2 = K3 ) @ ( B @ K3 ) @ zero_zero_int )
@ S )
= zero_zero_int ) ) ) ) ).
% sum.delta'
thf(fact_313_sum_Odelta_H,axiom,
! [S: set_a,A2: a,B: a > nat] :
( ( finite_finite_a @ S )
=> ( ( ( member_a @ A2 @ S )
=> ( ( groups6334556678337121940_a_nat
@ ^ [K3: a] : ( if_nat @ ( A2 = K3 ) @ ( B @ K3 ) @ zero_zero_nat )
@ S )
= ( B @ A2 ) ) )
& ( ~ ( member_a @ A2 @ S )
=> ( ( groups6334556678337121940_a_nat
@ ^ [K3: a] : ( if_nat @ ( A2 = K3 ) @ ( B @ K3 ) @ zero_zero_nat )
@ S )
= zero_zero_nat ) ) ) ) ).
% sum.delta'
thf(fact_314_sum_Odelta_H,axiom,
! [S: set_list_a,A2: list_a,B: list_a > nat] :
( ( finite_finite_list_a @ S )
=> ( ( ( member_list_a @ A2 @ S )
=> ( ( groups5521247699297860762_a_nat
@ ^ [K3: list_a] : ( if_nat @ ( A2 = K3 ) @ ( B @ K3 ) @ zero_zero_nat )
@ S )
= ( B @ A2 ) ) )
& ( ~ ( member_list_a @ A2 @ S )
=> ( ( groups5521247699297860762_a_nat
@ ^ [K3: list_a] : ( if_nat @ ( A2 = K3 ) @ ( B @ K3 ) @ zero_zero_nat )
@ S )
= zero_zero_nat ) ) ) ) ).
% sum.delta'
thf(fact_315_sum_Odelta_H,axiom,
! [S: set_a,A2: a,B: a > multiset_a] :
( ( finite_finite_a @ S )
=> ( ( ( member_a @ A2 @ S )
=> ( ( groups4808324907802680448iset_a
@ ^ [K3: a] : ( if_multiset_a @ ( A2 = K3 ) @ ( B @ K3 ) @ zero_zero_multiset_a )
@ S )
= ( B @ A2 ) ) )
& ( ~ ( member_a @ A2 @ S )
=> ( ( groups4808324907802680448iset_a
@ ^ [K3: a] : ( if_multiset_a @ ( A2 = K3 ) @ ( B @ K3 ) @ zero_zero_multiset_a )
@ S )
= zero_zero_multiset_a ) ) ) ) ).
% sum.delta'
thf(fact_316_sum_Odelta_H,axiom,
! [S: set_nat,A2: nat,B: nat > multiset_a] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A2 @ S )
=> ( ( groups1580436272196575032iset_a
@ ^ [K3: nat] : ( if_multiset_a @ ( A2 = K3 ) @ ( B @ K3 ) @ zero_zero_multiset_a )
@ S )
= ( B @ A2 ) ) )
& ( ~ ( member_nat @ A2 @ S )
=> ( ( groups1580436272196575032iset_a
@ ^ [K3: nat] : ( if_multiset_a @ ( A2 = K3 ) @ ( B @ K3 ) @ zero_zero_multiset_a )
@ S )
= zero_zero_multiset_a ) ) ) ) ).
% sum.delta'
thf(fact_317_sum_Odelta_H,axiom,
! [S: set_int,A2: int,B: int > multiset_a] :
( ( finite_finite_int @ S )
=> ( ( ( member_int @ A2 @ S )
=> ( ( groups3457364905213935068iset_a
@ ^ [K3: int] : ( if_multiset_a @ ( A2 = K3 ) @ ( B @ K3 ) @ zero_zero_multiset_a )
@ S )
= ( B @ A2 ) ) )
& ( ~ ( member_int @ A2 @ S )
=> ( ( groups3457364905213935068iset_a
@ ^ [K3: int] : ( if_multiset_a @ ( A2 = K3 ) @ ( B @ K3 ) @ zero_zero_multiset_a )
@ S )
= zero_zero_multiset_a ) ) ) ) ).
% sum.delta'
thf(fact_318_sum_Odelta,axiom,
! [S: set_nat,A2: nat,B: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A2 @ S )
=> ( ( groups3542108847815614940at_nat
@ ^ [K3: nat] : ( if_nat @ ( K3 = A2 ) @ ( B @ K3 ) @ zero_zero_nat )
@ S )
= ( B @ A2 ) ) )
& ( ~ ( member_nat @ A2 @ S )
=> ( ( groups3542108847815614940at_nat
@ ^ [K3: nat] : ( if_nat @ ( K3 = A2 ) @ ( B @ K3 ) @ zero_zero_nat )
@ S )
= zero_zero_nat ) ) ) ) ).
% sum.delta
thf(fact_319_sum_Odelta,axiom,
! [S: set_int,A2: int,B: int > nat] :
( ( finite_finite_int @ S )
=> ( ( ( member_int @ A2 @ S )
=> ( ( groups4541462559716669496nt_nat
@ ^ [K3: int] : ( if_nat @ ( K3 = A2 ) @ ( B @ K3 ) @ zero_zero_nat )
@ S )
= ( B @ A2 ) ) )
& ( ~ ( member_int @ A2 @ S )
=> ( ( groups4541462559716669496nt_nat
@ ^ [K3: int] : ( if_nat @ ( K3 = A2 ) @ ( B @ K3 ) @ zero_zero_nat )
@ S )
= zero_zero_nat ) ) ) ) ).
% sum.delta
thf(fact_320_sum_Odelta,axiom,
! [S: set_a,A2: a,B: a > int] :
( ( finite_finite_a @ S )
=> ( ( ( member_a @ A2 @ S )
=> ( ( groups6332066207828071664_a_int
@ ^ [K3: a] : ( if_int @ ( K3 = A2 ) @ ( B @ K3 ) @ zero_zero_int )
@ S )
= ( B @ A2 ) ) )
& ( ~ ( member_a @ A2 @ S )
=> ( ( groups6332066207828071664_a_int
@ ^ [K3: a] : ( if_int @ ( K3 = A2 ) @ ( B @ K3 ) @ zero_zero_int )
@ S )
= zero_zero_int ) ) ) ) ).
% sum.delta
thf(fact_321_sum_Odelta,axiom,
! [S: set_nat,A2: nat,B: nat > int] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A2 @ S )
=> ( ( groups3539618377306564664at_int
@ ^ [K3: nat] : ( if_int @ ( K3 = A2 ) @ ( B @ K3 ) @ zero_zero_int )
@ S )
= ( B @ A2 ) ) )
& ( ~ ( member_nat @ A2 @ S )
=> ( ( groups3539618377306564664at_int
@ ^ [K3: nat] : ( if_int @ ( K3 = A2 ) @ ( B @ K3 ) @ zero_zero_int )
@ S )
= zero_zero_int ) ) ) ) ).
% sum.delta
thf(fact_322_sum_Odelta,axiom,
! [S: set_int,A2: int,B: int > int] :
( ( finite_finite_int @ S )
=> ( ( ( member_int @ A2 @ S )
=> ( ( groups4538972089207619220nt_int
@ ^ [K3: int] : ( if_int @ ( K3 = A2 ) @ ( B @ K3 ) @ zero_zero_int )
@ S )
= ( B @ A2 ) ) )
& ( ~ ( member_int @ A2 @ S )
=> ( ( groups4538972089207619220nt_int
@ ^ [K3: int] : ( if_int @ ( K3 = A2 ) @ ( B @ K3 ) @ zero_zero_int )
@ S )
= zero_zero_int ) ) ) ) ).
% sum.delta
thf(fact_323_sum_Odelta,axiom,
! [S: set_a,A2: a,B: a > nat] :
( ( finite_finite_a @ S )
=> ( ( ( member_a @ A2 @ S )
=> ( ( groups6334556678337121940_a_nat
@ ^ [K3: a] : ( if_nat @ ( K3 = A2 ) @ ( B @ K3 ) @ zero_zero_nat )
@ S )
= ( B @ A2 ) ) )
& ( ~ ( member_a @ A2 @ S )
=> ( ( groups6334556678337121940_a_nat
@ ^ [K3: a] : ( if_nat @ ( K3 = A2 ) @ ( B @ K3 ) @ zero_zero_nat )
@ S )
= zero_zero_nat ) ) ) ) ).
% sum.delta
thf(fact_324_sum_Odelta,axiom,
! [S: set_list_a,A2: list_a,B: list_a > nat] :
( ( finite_finite_list_a @ S )
=> ( ( ( member_list_a @ A2 @ S )
=> ( ( groups5521247699297860762_a_nat
@ ^ [K3: list_a] : ( if_nat @ ( K3 = A2 ) @ ( B @ K3 ) @ zero_zero_nat )
@ S )
= ( B @ A2 ) ) )
& ( ~ ( member_list_a @ A2 @ S )
=> ( ( groups5521247699297860762_a_nat
@ ^ [K3: list_a] : ( if_nat @ ( K3 = A2 ) @ ( B @ K3 ) @ zero_zero_nat )
@ S )
= zero_zero_nat ) ) ) ) ).
% sum.delta
thf(fact_325_sum_Odelta,axiom,
! [S: set_a,A2: a,B: a > multiset_a] :
( ( finite_finite_a @ S )
=> ( ( ( member_a @ A2 @ S )
=> ( ( groups4808324907802680448iset_a
@ ^ [K3: a] : ( if_multiset_a @ ( K3 = A2 ) @ ( B @ K3 ) @ zero_zero_multiset_a )
@ S )
= ( B @ A2 ) ) )
& ( ~ ( member_a @ A2 @ S )
=> ( ( groups4808324907802680448iset_a
@ ^ [K3: a] : ( if_multiset_a @ ( K3 = A2 ) @ ( B @ K3 ) @ zero_zero_multiset_a )
@ S )
= zero_zero_multiset_a ) ) ) ) ).
% sum.delta
thf(fact_326_sum_Odelta,axiom,
! [S: set_nat,A2: nat,B: nat > multiset_a] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A2 @ S )
=> ( ( groups1580436272196575032iset_a
@ ^ [K3: nat] : ( if_multiset_a @ ( K3 = A2 ) @ ( B @ K3 ) @ zero_zero_multiset_a )
@ S )
= ( B @ A2 ) ) )
& ( ~ ( member_nat @ A2 @ S )
=> ( ( groups1580436272196575032iset_a
@ ^ [K3: nat] : ( if_multiset_a @ ( K3 = A2 ) @ ( B @ K3 ) @ zero_zero_multiset_a )
@ S )
= zero_zero_multiset_a ) ) ) ) ).
% sum.delta
thf(fact_327_sum_Odelta,axiom,
! [S: set_int,A2: int,B: int > multiset_a] :
( ( finite_finite_int @ S )
=> ( ( ( member_int @ A2 @ S )
=> ( ( groups3457364905213935068iset_a
@ ^ [K3: int] : ( if_multiset_a @ ( K3 = A2 ) @ ( B @ K3 ) @ zero_zero_multiset_a )
@ S )
= ( B @ A2 ) ) )
& ( ~ ( member_int @ A2 @ S )
=> ( ( groups3457364905213935068iset_a
@ ^ [K3: int] : ( if_multiset_a @ ( K3 = A2 ) @ ( B @ K3 ) @ zero_zero_multiset_a )
@ S )
= zero_zero_multiset_a ) ) ) ) ).
% sum.delta
thf(fact_328_ring_Oroots__on_Ocong,axiom,
polyno5990348478334826338t_unit = polyno5990348478334826338t_unit ).
% ring.roots_on.cong
thf(fact_329_ring_Oroots__on_Ocong,axiom,
polyno5714441830345289050on_a_b = polyno5714441830345289050on_a_b ).
% ring.roots_on.cong
thf(fact_330_ring_Osplitted__on_Ocong,axiom,
polyno1986131841096413848t_unit = polyno1986131841096413848t_unit ).
% ring.splitted_on.cong
thf(fact_331_ring_Osplitted__on_Ocong,axiom,
polyno2453258491555121552on_a_b = polyno2453258491555121552on_a_b ).
% ring.splitted_on.cong
thf(fact_332_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_333_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_334_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_335_sum_Ofinite__Collect__op,axiom,
! [I3: set_a,X3: a > nat,Y: a > nat] :
( ( finite_finite_a
@ ( collect_a
@ ^ [I: a] :
( ( member_a @ I @ I3 )
& ( ( X3 @ I )
!= zero_zero_nat ) ) ) )
=> ( ( finite_finite_a
@ ( collect_a
@ ^ [I: a] :
( ( member_a @ I @ I3 )
& ( ( Y @ I )
!= zero_zero_nat ) ) ) )
=> ( finite_finite_a
@ ( collect_a
@ ^ [I: a] :
( ( member_a @ I @ I3 )
& ( ( plus_plus_nat @ ( X3 @ I ) @ ( Y @ I ) )
!= zero_zero_nat ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_336_sum_Ofinite__Collect__op,axiom,
! [I3: set_nat,X3: nat > nat,Y: nat > nat] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I: nat] :
( ( member_nat @ I @ I3 )
& ( ( X3 @ I )
!= zero_zero_nat ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I: nat] :
( ( member_nat @ I @ I3 )
& ( ( Y @ I )
!= zero_zero_nat ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I: nat] :
( ( member_nat @ I @ I3 )
& ( ( plus_plus_nat @ ( X3 @ I ) @ ( Y @ I ) )
!= zero_zero_nat ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_337_sum_Ofinite__Collect__op,axiom,
! [I3: set_int,X3: int > nat,Y: int > nat] :
( ( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( member_int @ I @ I3 )
& ( ( X3 @ I )
!= zero_zero_nat ) ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( member_int @ I @ I3 )
& ( ( Y @ I )
!= zero_zero_nat ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( member_int @ I @ I3 )
& ( ( plus_plus_nat @ ( X3 @ I ) @ ( Y @ I ) )
!= zero_zero_nat ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_338_sum_Ofinite__Collect__op,axiom,
! [I3: set_a,X3: a > int,Y: a > int] :
( ( finite_finite_a
@ ( collect_a
@ ^ [I: a] :
( ( member_a @ I @ I3 )
& ( ( X3 @ I )
!= zero_zero_int ) ) ) )
=> ( ( finite_finite_a
@ ( collect_a
@ ^ [I: a] :
( ( member_a @ I @ I3 )
& ( ( Y @ I )
!= zero_zero_int ) ) ) )
=> ( finite_finite_a
@ ( collect_a
@ ^ [I: a] :
( ( member_a @ I @ I3 )
& ( ( plus_plus_int @ ( X3 @ I ) @ ( Y @ I ) )
!= zero_zero_int ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_339_sum_Ofinite__Collect__op,axiom,
! [I3: set_nat,X3: nat > int,Y: nat > int] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I: nat] :
( ( member_nat @ I @ I3 )
& ( ( X3 @ I )
!= zero_zero_int ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I: nat] :
( ( member_nat @ I @ I3 )
& ( ( Y @ I )
!= zero_zero_int ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I: nat] :
( ( member_nat @ I @ I3 )
& ( ( plus_plus_int @ ( X3 @ I ) @ ( Y @ I ) )
!= zero_zero_int ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_340_sum_Ofinite__Collect__op,axiom,
! [I3: set_int,X3: int > int,Y: int > int] :
( ( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( member_int @ I @ I3 )
& ( ( X3 @ I )
!= zero_zero_int ) ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( member_int @ I @ I3 )
& ( ( Y @ I )
!= zero_zero_int ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( member_int @ I @ I3 )
& ( ( plus_plus_int @ ( X3 @ I ) @ ( Y @ I ) )
!= zero_zero_int ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_341_sum_Ofinite__Collect__op,axiom,
! [I3: set_list_a,X3: list_a > nat,Y: list_a > nat] :
( ( finite_finite_list_a
@ ( collect_list_a
@ ^ [I: list_a] :
( ( member_list_a @ I @ I3 )
& ( ( X3 @ I )
!= zero_zero_nat ) ) ) )
=> ( ( finite_finite_list_a
@ ( collect_list_a
@ ^ [I: list_a] :
( ( member_list_a @ I @ I3 )
& ( ( Y @ I )
!= zero_zero_nat ) ) ) )
=> ( finite_finite_list_a
@ ( collect_list_a
@ ^ [I: list_a] :
( ( member_list_a @ I @ I3 )
& ( ( plus_plus_nat @ ( X3 @ I ) @ ( Y @ I ) )
!= zero_zero_nat ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_342_sum_Ofinite__Collect__op,axiom,
! [I3: set_a,X3: a > multiset_a,Y: a > multiset_a] :
( ( finite_finite_a
@ ( collect_a
@ ^ [I: a] :
( ( member_a @ I @ I3 )
& ( ( X3 @ I )
!= zero_zero_multiset_a ) ) ) )
=> ( ( finite_finite_a
@ ( collect_a
@ ^ [I: a] :
( ( member_a @ I @ I3 )
& ( ( Y @ I )
!= zero_zero_multiset_a ) ) ) )
=> ( finite_finite_a
@ ( collect_a
@ ^ [I: a] :
( ( member_a @ I @ I3 )
& ( ( plus_plus_multiset_a @ ( X3 @ I ) @ ( Y @ I ) )
!= zero_zero_multiset_a ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_343_sum_Ofinite__Collect__op,axiom,
! [I3: set_nat,X3: nat > multiset_a,Y: nat > multiset_a] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I: nat] :
( ( member_nat @ I @ I3 )
& ( ( X3 @ I )
!= zero_zero_multiset_a ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I: nat] :
( ( member_nat @ I @ I3 )
& ( ( Y @ I )
!= zero_zero_multiset_a ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I: nat] :
( ( member_nat @ I @ I3 )
& ( ( plus_plus_multiset_a @ ( X3 @ I ) @ ( Y @ I ) )
!= zero_zero_multiset_a ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_344_sum_Ofinite__Collect__op,axiom,
! [I3: set_int,X3: int > multiset_a,Y: int > multiset_a] :
( ( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( member_int @ I @ I3 )
& ( ( X3 @ I )
!= zero_zero_multiset_a ) ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( member_int @ I @ I3 )
& ( ( Y @ I )
!= zero_zero_multiset_a ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( member_int @ I @ I3 )
& ( ( plus_plus_multiset_a @ ( X3 @ I ) @ ( Y @ I ) )
!= zero_zero_multiset_a ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_345_sum_Orelated,axiom,
! [R: nat > nat > $o,S: set_nat,H: nat > nat,G: nat > nat] :
( ( R @ zero_zero_nat @ zero_zero_nat )
=> ( ! [X1: nat,Y1: nat,X22: nat,Y22: nat] :
( ( ( R @ X1 @ X22 )
& ( R @ Y1 @ Y22 ) )
=> ( R @ ( plus_plus_nat @ X1 @ Y1 ) @ ( plus_plus_nat @ X22 @ Y22 ) ) )
=> ( ( finite_finite_nat @ S )
=> ( ! [X: nat] :
( ( member_nat @ X @ S )
=> ( R @ ( H @ X ) @ ( G @ X ) ) )
=> ( R @ ( groups3542108847815614940at_nat @ H @ S ) @ ( groups3542108847815614940at_nat @ G @ S ) ) ) ) ) ) ).
% sum.related
thf(fact_346_sum_Orelated,axiom,
! [R: nat > nat > $o,S: set_int,H: int > nat,G: int > nat] :
( ( R @ zero_zero_nat @ zero_zero_nat )
=> ( ! [X1: nat,Y1: nat,X22: nat,Y22: nat] :
( ( ( R @ X1 @ X22 )
& ( R @ Y1 @ Y22 ) )
=> ( R @ ( plus_plus_nat @ X1 @ Y1 ) @ ( plus_plus_nat @ X22 @ Y22 ) ) )
=> ( ( finite_finite_int @ S )
=> ( ! [X: int] :
( ( member_int @ X @ S )
=> ( R @ ( H @ X ) @ ( G @ X ) ) )
=> ( R @ ( groups4541462559716669496nt_nat @ H @ S ) @ ( groups4541462559716669496nt_nat @ G @ S ) ) ) ) ) ) ).
% sum.related
thf(fact_347_sum_Orelated,axiom,
! [R: int > int > $o,S: set_a,H: a > int,G: a > int] :
( ( R @ zero_zero_int @ zero_zero_int )
=> ( ! [X1: int,Y1: int,X22: int,Y22: int] :
( ( ( R @ X1 @ X22 )
& ( R @ Y1 @ Y22 ) )
=> ( R @ ( plus_plus_int @ X1 @ Y1 ) @ ( plus_plus_int @ X22 @ Y22 ) ) )
=> ( ( finite_finite_a @ S )
=> ( ! [X: a] :
( ( member_a @ X @ S )
=> ( R @ ( H @ X ) @ ( G @ X ) ) )
=> ( R @ ( groups6332066207828071664_a_int @ H @ S ) @ ( groups6332066207828071664_a_int @ G @ S ) ) ) ) ) ) ).
% sum.related
thf(fact_348_sum_Orelated,axiom,
! [R: int > int > $o,S: set_nat,H: nat > int,G: nat > int] :
( ( R @ zero_zero_int @ zero_zero_int )
=> ( ! [X1: int,Y1: int,X22: int,Y22: int] :
( ( ( R @ X1 @ X22 )
& ( R @ Y1 @ Y22 ) )
=> ( R @ ( plus_plus_int @ X1 @ Y1 ) @ ( plus_plus_int @ X22 @ Y22 ) ) )
=> ( ( finite_finite_nat @ S )
=> ( ! [X: nat] :
( ( member_nat @ X @ S )
=> ( R @ ( H @ X ) @ ( G @ X ) ) )
=> ( R @ ( groups3539618377306564664at_int @ H @ S ) @ ( groups3539618377306564664at_int @ G @ S ) ) ) ) ) ) ).
% sum.related
thf(fact_349_sum_Orelated,axiom,
! [R: int > int > $o,S: set_int,H: int > int,G: int > int] :
( ( R @ zero_zero_int @ zero_zero_int )
=> ( ! [X1: int,Y1: int,X22: int,Y22: int] :
( ( ( R @ X1 @ X22 )
& ( R @ Y1 @ Y22 ) )
=> ( R @ ( plus_plus_int @ X1 @ Y1 ) @ ( plus_plus_int @ X22 @ Y22 ) ) )
=> ( ( finite_finite_int @ S )
=> ( ! [X: int] :
( ( member_int @ X @ S )
=> ( R @ ( H @ X ) @ ( G @ X ) ) )
=> ( R @ ( groups4538972089207619220nt_int @ H @ S ) @ ( groups4538972089207619220nt_int @ G @ S ) ) ) ) ) ) ).
% sum.related
thf(fact_350_sum_Orelated,axiom,
! [R: nat > nat > $o,S: set_a,H: a > nat,G: a > nat] :
( ( R @ zero_zero_nat @ zero_zero_nat )
=> ( ! [X1: nat,Y1: nat,X22: nat,Y22: nat] :
( ( ( R @ X1 @ X22 )
& ( R @ Y1 @ Y22 ) )
=> ( R @ ( plus_plus_nat @ X1 @ Y1 ) @ ( plus_plus_nat @ X22 @ Y22 ) ) )
=> ( ( finite_finite_a @ S )
=> ( ! [X: a] :
( ( member_a @ X @ S )
=> ( R @ ( H @ X ) @ ( G @ X ) ) )
=> ( R @ ( groups6334556678337121940_a_nat @ H @ S ) @ ( groups6334556678337121940_a_nat @ G @ S ) ) ) ) ) ) ).
% sum.related
thf(fact_351_sum_Orelated,axiom,
! [R: nat > nat > $o,S: set_list_a,H: list_a > nat,G: list_a > nat] :
( ( R @ zero_zero_nat @ zero_zero_nat )
=> ( ! [X1: nat,Y1: nat,X22: nat,Y22: nat] :
( ( ( R @ X1 @ X22 )
& ( R @ Y1 @ Y22 ) )
=> ( R @ ( plus_plus_nat @ X1 @ Y1 ) @ ( plus_plus_nat @ X22 @ Y22 ) ) )
=> ( ( finite_finite_list_a @ S )
=> ( ! [X: list_a] :
( ( member_list_a @ X @ S )
=> ( R @ ( H @ X ) @ ( G @ X ) ) )
=> ( R @ ( groups5521247699297860762_a_nat @ H @ S ) @ ( groups5521247699297860762_a_nat @ G @ S ) ) ) ) ) ) ).
% sum.related
thf(fact_352_sum_Orelated,axiom,
! [R: multiset_a > multiset_a > $o,S: set_a,H: a > multiset_a,G: a > multiset_a] :
( ( R @ zero_zero_multiset_a @ zero_zero_multiset_a )
=> ( ! [X1: multiset_a,Y1: multiset_a,X22: multiset_a,Y22: multiset_a] :
( ( ( R @ X1 @ X22 )
& ( R @ Y1 @ Y22 ) )
=> ( R @ ( plus_plus_multiset_a @ X1 @ Y1 ) @ ( plus_plus_multiset_a @ X22 @ Y22 ) ) )
=> ( ( finite_finite_a @ S )
=> ( ! [X: a] :
( ( member_a @ X @ S )
=> ( R @ ( H @ X ) @ ( G @ X ) ) )
=> ( R @ ( groups4808324907802680448iset_a @ H @ S ) @ ( groups4808324907802680448iset_a @ G @ S ) ) ) ) ) ) ).
% sum.related
thf(fact_353_sum_Orelated,axiom,
! [R: multiset_a > multiset_a > $o,S: set_nat,H: nat > multiset_a,G: nat > multiset_a] :
( ( R @ zero_zero_multiset_a @ zero_zero_multiset_a )
=> ( ! [X1: multiset_a,Y1: multiset_a,X22: multiset_a,Y22: multiset_a] :
( ( ( R @ X1 @ X22 )
& ( R @ Y1 @ Y22 ) )
=> ( R @ ( plus_plus_multiset_a @ X1 @ Y1 ) @ ( plus_plus_multiset_a @ X22 @ Y22 ) ) )
=> ( ( finite_finite_nat @ S )
=> ( ! [X: nat] :
( ( member_nat @ X @ S )
=> ( R @ ( H @ X ) @ ( G @ X ) ) )
=> ( R @ ( groups1580436272196575032iset_a @ H @ S ) @ ( groups1580436272196575032iset_a @ G @ S ) ) ) ) ) ) ).
% sum.related
thf(fact_354_sum_Orelated,axiom,
! [R: multiset_a > multiset_a > $o,S: set_int,H: int > multiset_a,G: int > multiset_a] :
( ( R @ zero_zero_multiset_a @ zero_zero_multiset_a )
=> ( ! [X1: multiset_a,Y1: multiset_a,X22: multiset_a,Y22: multiset_a] :
( ( ( R @ X1 @ X22 )
& ( R @ Y1 @ Y22 ) )
=> ( R @ ( plus_plus_multiset_a @ X1 @ Y1 ) @ ( plus_plus_multiset_a @ X22 @ Y22 ) ) )
=> ( ( finite_finite_int @ S )
=> ( ! [X: int] :
( ( member_int @ X @ S )
=> ( R @ ( H @ X ) @ ( G @ X ) ) )
=> ( R @ ( groups3457364905213935068iset_a @ H @ S ) @ ( groups3457364905213935068iset_a @ G @ S ) ) ) ) ) ) ).
% sum.related
thf(fact_355_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N2: nat] :
? [K3: nat] :
( N2
= ( plus_plus_nat @ M3 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_356_trans__le__add2,axiom,
! [I2: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_357_trans__le__add1,axiom,
! [I2: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_358_add__le__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_359_add__le__mono,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_360_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_361_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_362_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_363_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_364_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_365_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_366_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_367_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_368_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_369_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_370_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > nat,A: set_nat] :
( ( ( groups3542108847815614940at_nat @ G @ A )
!= zero_zero_nat )
=> ~ ! [A3: nat] :
( ( member_nat @ A3 @ A )
=> ( ( G @ A3 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_371_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > int,A: set_nat] :
( ( ( groups3539618377306564664at_int @ G @ A )
!= zero_zero_int )
=> ~ ! [A3: nat] :
( ( member_nat @ A3 @ A )
=> ( ( G @ A3 )
= zero_zero_int ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_372_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: a > int,A: set_a] :
( ( ( groups6332066207828071664_a_int @ G @ A )
!= zero_zero_int )
=> ~ ! [A3: a] :
( ( member_a @ A3 @ A )
=> ( ( G @ A3 )
= zero_zero_int ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_373_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: a > nat,A: set_a] :
( ( ( groups6334556678337121940_a_nat @ G @ A )
!= zero_zero_nat )
=> ~ ! [A3: a] :
( ( member_a @ A3 @ A )
=> ( ( G @ A3 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_374_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: list_a > nat,A: set_list_a] :
( ( ( groups5521247699297860762_a_nat @ G @ A )
!= zero_zero_nat )
=> ~ ! [A3: list_a] :
( ( member_list_a @ A3 @ A )
=> ( ( G @ A3 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_375_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > multiset_a,A: set_nat] :
( ( ( groups1580436272196575032iset_a @ G @ A )
!= zero_zero_multiset_a )
=> ~ ! [A3: nat] :
( ( member_nat @ A3 @ A )
=> ( ( G @ A3 )
= zero_zero_multiset_a ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_376_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: a > multiset_a,A: set_a] :
( ( ( groups4808324907802680448iset_a @ G @ A )
!= zero_zero_multiset_a )
=> ~ ! [A3: a] :
( ( member_a @ A3 @ A )
=> ( ( G @ A3 )
= zero_zero_multiset_a ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_377_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: list_a > int,A: set_list_a] :
( ( ( groups5518757228788810486_a_int @ G @ A )
!= zero_zero_int )
=> ~ ! [A3: list_a] :
( ( member_list_a @ A3 @ A )
=> ( ( G @ A3 )
= zero_zero_int ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_378_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: ( nat > a ) > nat,A: set_nat_a] :
( ( ( groups154653438316501755_a_nat @ G @ A )
!= zero_zero_nat )
=> ~ ! [A3: nat > a] :
( ( member_nat_a @ A3 @ A )
=> ( ( G @ A3 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_379_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: list_a > multiset_a,A: set_list_a] :
( ( ( groups2539338179767937786iset_a @ G @ A )
!= zero_zero_multiset_a )
=> ~ ! [A3: list_a] :
( ( member_list_a @ A3 @ A )
=> ( ( G @ A3 )
= zero_zero_multiset_a ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_380_sum_Oneutral,axiom,
! [A: set_a,G: a > nat] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( ( G @ X )
= zero_zero_nat ) )
=> ( ( groups6334556678337121940_a_nat @ G @ A )
= zero_zero_nat ) ) ).
% sum.neutral
thf(fact_381_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_382_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_383_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_384_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_385_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_386_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_387_sum_Odistrib,axiom,
! [G: a > nat,H: a > nat,A: set_a] :
( ( groups6334556678337121940_a_nat
@ ^ [X2: a] : ( plus_plus_nat @ ( G @ X2 ) @ ( H @ X2 ) )
@ A )
= ( plus_plus_nat @ ( groups6334556678337121940_a_nat @ G @ A ) @ ( groups6334556678337121940_a_nat @ H @ A ) ) ) ).
% sum.distrib
thf(fact_388_Nat_Ole__imp__diff__is__add,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ( minus_minus_nat @ J @ I2 )
= K )
= ( J
= ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_389_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_390_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
= ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_391_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_392_le__diff__conv,axiom,
! [J: nat,K: nat,I2: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I2 @ K ) ) ) ).
% le_diff_conv
thf(fact_393_sum__nonpos,axiom,
! [A: set_list_a,F: list_a > nat] :
( ! [X: list_a] :
( ( member_list_a @ X @ A )
=> ( ord_less_eq_nat @ ( F @ X ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups5521247699297860762_a_nat @ F @ A ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_394_sum__nonpos,axiom,
! [A: set_nat,F: nat > nat] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_nat @ ( F @ X ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ A ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_395_sum__nonpos,axiom,
! [A: set_nat_list_a,F: ( nat > list_a ) > nat] :
( ! [X: nat > list_a] :
( ( member_nat_list_a @ X @ A )
=> ( ord_less_eq_nat @ ( F @ X ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups669906071623145473_a_nat @ F @ A ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_396_sum__nonpos,axiom,
! [A: set_nat_a,F: ( nat > a ) > nat] :
( ! [X: nat > a] :
( ( member_nat_a @ X @ A )
=> ( ord_less_eq_nat @ ( F @ X ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups154653438316501755_a_nat @ F @ A ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_397_sum__nonpos,axiom,
! [A: set_list_a,F: list_a > int] :
( ! [X: list_a] :
( ( member_list_a @ X @ A )
=> ( ord_less_eq_int @ ( F @ X ) @ zero_zero_int ) )
=> ( ord_less_eq_int @ ( groups5518757228788810486_a_int @ F @ A ) @ zero_zero_int ) ) ).
% sum_nonpos
thf(fact_398_sum__nonpos,axiom,
! [A: set_nat,F: nat > int] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_int @ ( F @ X ) @ zero_zero_int ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ A ) @ zero_zero_int ) ) ).
% sum_nonpos
thf(fact_399_sum__nonpos,axiom,
! [A: set_nat_list_a,F: ( nat > list_a ) > int] :
( ! [X: nat > list_a] :
( ( member_nat_list_a @ X @ A )
=> ( ord_less_eq_int @ ( F @ X ) @ zero_zero_int ) )
=> ( ord_less_eq_int @ ( groups667415601114095197_a_int @ F @ A ) @ zero_zero_int ) ) ).
% sum_nonpos
thf(fact_400_sum__nonpos,axiom,
! [A: set_nat_a,F: ( nat > a ) > int] :
( ! [X: nat > a] :
( ( member_nat_a @ X @ A )
=> ( ord_less_eq_int @ ( F @ X ) @ zero_zero_int ) )
=> ( ord_less_eq_int @ ( groups152162967807451479_a_int @ F @ A ) @ zero_zero_int ) ) ).
% sum_nonpos
thf(fact_401_sum__nonpos,axiom,
! [A: set_a,F: a > int] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_int @ ( F @ X ) @ zero_zero_int ) )
=> ( ord_less_eq_int @ ( groups6332066207828071664_a_int @ F @ A ) @ zero_zero_int ) ) ).
% sum_nonpos
thf(fact_402_sum__nonpos,axiom,
! [A: set_a,F: a > nat] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_nat @ ( F @ X ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups6334556678337121940_a_nat @ F @ A ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_403_sum__nonneg,axiom,
! [A: set_list_a,F: list_a > nat] :
( ! [X: list_a] :
( ( member_list_a @ X @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups5521247699297860762_a_nat @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_404_sum__nonneg,axiom,
! [A: set_nat,F: nat > nat] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups3542108847815614940at_nat @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_405_sum__nonneg,axiom,
! [A: set_nat_list_a,F: ( nat > list_a ) > nat] :
( ! [X: nat > list_a] :
( ( member_nat_list_a @ X @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups669906071623145473_a_nat @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_406_sum__nonneg,axiom,
! [A: set_nat_a,F: ( nat > a ) > nat] :
( ! [X: nat > a] :
( ( member_nat_a @ X @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups154653438316501755_a_nat @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_407_sum__nonneg,axiom,
! [A: set_list_a,F: list_a > int] :
( ! [X: list_a] :
( ( member_list_a @ X @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups5518757228788810486_a_int @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_408_sum__nonneg,axiom,
! [A: set_nat,F: nat > int] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups3539618377306564664at_int @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_409_sum__nonneg,axiom,
! [A: set_nat_list_a,F: ( nat > list_a ) > int] :
( ! [X: nat > list_a] :
( ( member_nat_list_a @ X @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups667415601114095197_a_int @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_410_sum__nonneg,axiom,
! [A: set_nat_a,F: ( nat > a ) > int] :
( ! [X: nat > a] :
( ( member_nat_a @ X @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups152162967807451479_a_int @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_411_sum__nonneg,axiom,
! [A: set_a,F: a > int] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups6332066207828071664_a_int @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_412_sum__nonneg,axiom,
! [A: set_a,F: a > nat] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups6334556678337121940_a_nat @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_413_sum_Ointer__filter,axiom,
! [A: set_nat,G: nat > nat,P2: nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( groups3542108847815614940at_nat @ G
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P2 @ X2 ) ) ) )
= ( groups3542108847815614940at_nat
@ ^ [X2: nat] : ( if_nat @ ( P2 @ X2 ) @ ( G @ X2 ) @ zero_zero_nat )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_414_sum_Ointer__filter,axiom,
! [A: set_int,G: int > nat,P2: int > $o] :
( ( finite_finite_int @ A )
=> ( ( groups4541462559716669496nt_nat @ G
@ ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ A )
& ( P2 @ X2 ) ) ) )
= ( groups4541462559716669496nt_nat
@ ^ [X2: int] : ( if_nat @ ( P2 @ X2 ) @ ( G @ X2 ) @ zero_zero_nat )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_415_sum_Ointer__filter,axiom,
! [A: set_a,G: a > int,P2: a > $o] :
( ( finite_finite_a @ A )
=> ( ( groups6332066207828071664_a_int @ G
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A )
& ( P2 @ X2 ) ) ) )
= ( groups6332066207828071664_a_int
@ ^ [X2: a] : ( if_int @ ( P2 @ X2 ) @ ( G @ X2 ) @ zero_zero_int )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_416_sum_Ointer__filter,axiom,
! [A: set_nat,G: nat > int,P2: nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( groups3539618377306564664at_int @ G
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P2 @ X2 ) ) ) )
= ( groups3539618377306564664at_int
@ ^ [X2: nat] : ( if_int @ ( P2 @ X2 ) @ ( G @ X2 ) @ zero_zero_int )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_417_sum_Ointer__filter,axiom,
! [A: set_int,G: int > int,P2: int > $o] :
( ( finite_finite_int @ A )
=> ( ( groups4538972089207619220nt_int @ G
@ ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ A )
& ( P2 @ X2 ) ) ) )
= ( groups4538972089207619220nt_int
@ ^ [X2: int] : ( if_int @ ( P2 @ X2 ) @ ( G @ X2 ) @ zero_zero_int )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_418_sum_Ointer__filter,axiom,
! [A: set_a,G: a > nat,P2: a > $o] :
( ( finite_finite_a @ A )
=> ( ( groups6334556678337121940_a_nat @ G
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A )
& ( P2 @ X2 ) ) ) )
= ( groups6334556678337121940_a_nat
@ ^ [X2: a] : ( if_nat @ ( P2 @ X2 ) @ ( G @ X2 ) @ zero_zero_nat )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_419_sum_Ointer__filter,axiom,
! [A: set_list_a,G: list_a > nat,P2: list_a > $o] :
( ( finite_finite_list_a @ A )
=> ( ( groups5521247699297860762_a_nat @ G
@ ( collect_list_a
@ ^ [X2: list_a] :
( ( member_list_a @ X2 @ A )
& ( P2 @ X2 ) ) ) )
= ( groups5521247699297860762_a_nat
@ ^ [X2: list_a] : ( if_nat @ ( P2 @ X2 ) @ ( G @ X2 ) @ zero_zero_nat )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_420_sum_Ointer__filter,axiom,
! [A: set_a,G: a > multiset_a,P2: a > $o] :
( ( finite_finite_a @ A )
=> ( ( groups4808324907802680448iset_a @ G
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A )
& ( P2 @ X2 ) ) ) )
= ( groups4808324907802680448iset_a
@ ^ [X2: a] : ( if_multiset_a @ ( P2 @ X2 ) @ ( G @ X2 ) @ zero_zero_multiset_a )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_421_sum_Ointer__filter,axiom,
! [A: set_nat,G: nat > multiset_a,P2: nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( groups1580436272196575032iset_a @ G
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P2 @ X2 ) ) ) )
= ( groups1580436272196575032iset_a
@ ^ [X2: nat] : ( if_multiset_a @ ( P2 @ X2 ) @ ( G @ X2 ) @ zero_zero_multiset_a )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_422_sum_Ointer__filter,axiom,
! [A: set_int,G: int > multiset_a,P2: int > $o] :
( ( finite_finite_int @ A )
=> ( ( groups3457364905213935068iset_a @ G
@ ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ A )
& ( P2 @ X2 ) ) ) )
= ( groups3457364905213935068iset_a
@ ^ [X2: int] : ( if_multiset_a @ ( P2 @ X2 ) @ ( G @ X2 ) @ zero_zero_multiset_a )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_423_sum__nonneg__eq__0__iff,axiom,
! [A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X ) ) )
=> ( ( ( groups3542108847815614940at_nat @ F @ A )
= zero_zero_nat )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( F @ X2 )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_424_sum__nonneg__eq__0__iff,axiom,
! [A: set_int,F: int > nat] :
( ( finite_finite_int @ A )
=> ( ! [X: int] :
( ( member_int @ X @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X ) ) )
=> ( ( ( groups4541462559716669496nt_nat @ F @ A )
= zero_zero_nat )
= ( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( ( F @ X2 )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_425_sum__nonneg__eq__0__iff,axiom,
! [A: set_a,F: a > int] :
( ( finite_finite_a @ A )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X ) ) )
=> ( ( ( groups6332066207828071664_a_int @ F @ A )
= zero_zero_int )
= ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( F @ X2 )
= zero_zero_int ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_426_sum__nonneg__eq__0__iff,axiom,
! [A: set_nat,F: nat > int] :
( ( finite_finite_nat @ A )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X ) ) )
=> ( ( ( groups3539618377306564664at_int @ F @ A )
= zero_zero_int )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( F @ X2 )
= zero_zero_int ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_427_sum__nonneg__eq__0__iff,axiom,
! [A: set_int,F: int > int] :
( ( finite_finite_int @ A )
=> ( ! [X: int] :
( ( member_int @ X @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X ) ) )
=> ( ( ( groups4538972089207619220nt_int @ F @ A )
= zero_zero_int )
= ( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( ( F @ X2 )
= zero_zero_int ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_428_sum__nonneg__eq__0__iff,axiom,
! [A: set_a,F: a > nat] :
( ( finite_finite_a @ A )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X ) ) )
=> ( ( ( groups6334556678337121940_a_nat @ F @ A )
= zero_zero_nat )
= ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( F @ X2 )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_429_sum__nonneg__eq__0__iff,axiom,
! [A: set_list_a,F: list_a > nat] :
( ( finite_finite_list_a @ A )
=> ( ! [X: list_a] :
( ( member_list_a @ X @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X ) ) )
=> ( ( ( groups5521247699297860762_a_nat @ F @ A )
= zero_zero_nat )
= ( ! [X2: list_a] :
( ( member_list_a @ X2 @ A )
=> ( ( F @ X2 )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_430_sum__nonneg__eq__0__iff,axiom,
! [A: set_list_a,F: list_a > int] :
( ( finite_finite_list_a @ A )
=> ( ! [X: list_a] :
( ( member_list_a @ X @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X ) ) )
=> ( ( ( groups5518757228788810486_a_int @ F @ A )
= zero_zero_int )
= ( ! [X2: list_a] :
( ( member_list_a @ X2 @ A )
=> ( ( F @ X2 )
= zero_zero_int ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_431_sum__nonneg__eq__0__iff,axiom,
! [A: set_nat_a,F: ( nat > a ) > nat] :
( ( finite_finite_nat_a @ A )
=> ( ! [X: nat > a] :
( ( member_nat_a @ X @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X ) ) )
=> ( ( ( groups154653438316501755_a_nat @ F @ A )
= zero_zero_nat )
= ( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A )
=> ( ( F @ X2 )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_432_sum__nonneg__eq__0__iff,axiom,
! [A: set_nat_a,F: ( nat > a ) > int] :
( ( finite_finite_nat_a @ A )
=> ( ! [X: nat > a] :
( ( member_nat_a @ X @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X ) ) )
=> ( ( ( groups152162967807451479_a_int @ F @ A )
= zero_zero_int )
= ( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A )
=> ( ( F @ X2 )
= zero_zero_int ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_433_sum__le__included,axiom,
! [S3: set_nat,T2: set_nat,G: nat > nat,I2: nat > nat,F: nat > nat] :
( ( finite_finite_nat @ S3 )
=> ( ( finite_finite_nat @ T2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ T2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( G @ X ) ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ S3 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ T2 )
& ( ( I2 @ Xa )
= X )
& ( ord_less_eq_nat @ ( F @ X ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ S3 ) @ ( groups3542108847815614940at_nat @ G @ T2 ) ) ) ) ) ) ).
% sum_le_included
thf(fact_434_sum__le__included,axiom,
! [S3: set_nat,T2: set_int,G: int > nat,I2: int > nat,F: nat > nat] :
( ( finite_finite_nat @ S3 )
=> ( ( finite_finite_int @ T2 )
=> ( ! [X: int] :
( ( member_int @ X @ T2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( G @ X ) ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ S3 )
=> ? [Xa: int] :
( ( member_int @ Xa @ T2 )
& ( ( I2 @ Xa )
= X )
& ( ord_less_eq_nat @ ( F @ X ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ S3 ) @ ( groups4541462559716669496nt_nat @ G @ T2 ) ) ) ) ) ) ).
% sum_le_included
thf(fact_435_sum__le__included,axiom,
! [S3: set_int,T2: set_nat,G: nat > nat,I2: nat > int,F: int > nat] :
( ( finite_finite_int @ S3 )
=> ( ( finite_finite_nat @ T2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ T2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( G @ X ) ) )
=> ( ! [X: int] :
( ( member_int @ X @ S3 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ T2 )
& ( ( I2 @ Xa )
= X )
& ( ord_less_eq_nat @ ( F @ X ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ S3 ) @ ( groups3542108847815614940at_nat @ G @ T2 ) ) ) ) ) ) ).
% sum_le_included
thf(fact_436_sum__le__included,axiom,
! [S3: set_int,T2: set_int,G: int > nat,I2: int > int,F: int > nat] :
( ( finite_finite_int @ S3 )
=> ( ( finite_finite_int @ T2 )
=> ( ! [X: int] :
( ( member_int @ X @ T2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( G @ X ) ) )
=> ( ! [X: int] :
( ( member_int @ X @ S3 )
=> ? [Xa: int] :
( ( member_int @ Xa @ T2 )
& ( ( I2 @ Xa )
= X )
& ( ord_less_eq_nat @ ( F @ X ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ S3 ) @ ( groups4541462559716669496nt_nat @ G @ T2 ) ) ) ) ) ) ).
% sum_le_included
thf(fact_437_sum__le__included,axiom,
! [S3: set_a,T2: set_a,G: a > int,I2: a > a,F: a > int] :
( ( finite_finite_a @ S3 )
=> ( ( finite_finite_a @ T2 )
=> ( ! [X: a] :
( ( member_a @ X @ T2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( G @ X ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ S3 )
=> ? [Xa: a] :
( ( member_a @ Xa @ T2 )
& ( ( I2 @ Xa )
= X )
& ( ord_less_eq_int @ ( F @ X ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_int @ ( groups6332066207828071664_a_int @ F @ S3 ) @ ( groups6332066207828071664_a_int @ G @ T2 ) ) ) ) ) ) ).
% sum_le_included
thf(fact_438_sum__le__included,axiom,
! [S3: set_a,T2: set_nat,G: nat > int,I2: nat > a,F: a > int] :
( ( finite_finite_a @ S3 )
=> ( ( finite_finite_nat @ T2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ T2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( G @ X ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ S3 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ T2 )
& ( ( I2 @ Xa )
= X )
& ( ord_less_eq_int @ ( F @ X ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_int @ ( groups6332066207828071664_a_int @ F @ S3 ) @ ( groups3539618377306564664at_int @ G @ T2 ) ) ) ) ) ) ).
% sum_le_included
thf(fact_439_sum__le__included,axiom,
! [S3: set_a,T2: set_int,G: int > int,I2: int > a,F: a > int] :
( ( finite_finite_a @ S3 )
=> ( ( finite_finite_int @ T2 )
=> ( ! [X: int] :
( ( member_int @ X @ T2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( G @ X ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ S3 )
=> ? [Xa: int] :
( ( member_int @ Xa @ T2 )
& ( ( I2 @ Xa )
= X )
& ( ord_less_eq_int @ ( F @ X ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_int @ ( groups6332066207828071664_a_int @ F @ S3 ) @ ( groups4538972089207619220nt_int @ G @ T2 ) ) ) ) ) ) ).
% sum_le_included
thf(fact_440_sum__le__included,axiom,
! [S3: set_nat,T2: set_a,G: a > int,I2: a > nat,F: nat > int] :
( ( finite_finite_nat @ S3 )
=> ( ( finite_finite_a @ T2 )
=> ( ! [X: a] :
( ( member_a @ X @ T2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( G @ X ) ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ S3 )
=> ? [Xa: a] :
( ( member_a @ Xa @ T2 )
& ( ( I2 @ Xa )
= X )
& ( ord_less_eq_int @ ( F @ X ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ S3 ) @ ( groups6332066207828071664_a_int @ G @ T2 ) ) ) ) ) ) ).
% sum_le_included
thf(fact_441_sum__le__included,axiom,
! [S3: set_nat,T2: set_nat,G: nat > int,I2: nat > nat,F: nat > int] :
( ( finite_finite_nat @ S3 )
=> ( ( finite_finite_nat @ T2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ T2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( G @ X ) ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ S3 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ T2 )
& ( ( I2 @ Xa )
= X )
& ( ord_less_eq_int @ ( F @ X ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ S3 ) @ ( groups3539618377306564664at_int @ G @ T2 ) ) ) ) ) ) ).
% sum_le_included
thf(fact_442_sum__le__included,axiom,
! [S3: set_nat,T2: set_int,G: int > int,I2: int > nat,F: nat > int] :
( ( finite_finite_nat @ S3 )
=> ( ( finite_finite_int @ T2 )
=> ( ! [X: int] :
( ( member_int @ X @ T2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( G @ X ) ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ S3 )
=> ? [Xa: int] :
( ( member_int @ Xa @ T2 )
& ( ( I2 @ Xa )
= X )
& ( ord_less_eq_int @ ( F @ X ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ S3 ) @ ( groups4538972089207619220nt_int @ G @ T2 ) ) ) ) ) ) ).
% sum_le_included
thf(fact_443_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_nat,T3: set_nat,S: set_nat,I2: nat > nat,J: nat > nat,T: set_nat,G: nat > nat,H: nat > nat] :
( ( finite_finite_nat @ S4 )
=> ( ( finite_finite_nat @ T3 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ ( minus_minus_set_nat @ S @ S4 ) )
=> ( ( I2 @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ ( minus_minus_set_nat @ S @ S4 ) )
=> ( member_nat @ ( J @ A3 ) @ ( minus_minus_set_nat @ T @ T3 ) ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ T @ T3 ) )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ T @ T3 ) )
=> ( member_nat @ ( I2 @ B3 ) @ ( minus_minus_set_nat @ S @ S4 ) ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S4 )
=> ( ( G @ A3 )
= zero_zero_nat ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T3 )
=> ( ( H @ B3 )
= zero_zero_nat ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ S )
= ( groups3542108847815614940at_nat @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_444_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_nat,T3: set_int,S: set_nat,I2: int > nat,J: nat > int,T: set_int,G: nat > nat,H: int > nat] :
( ( finite_finite_nat @ S4 )
=> ( ( finite_finite_int @ T3 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ ( minus_minus_set_nat @ S @ S4 ) )
=> ( ( I2 @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ ( minus_minus_set_nat @ S @ S4 ) )
=> ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T @ T3 ) ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ ( minus_minus_set_int @ T @ T3 ) )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ ( minus_minus_set_int @ T @ T3 ) )
=> ( member_nat @ ( I2 @ B3 ) @ ( minus_minus_set_nat @ S @ S4 ) ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S4 )
=> ( ( G @ A3 )
= zero_zero_nat ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ T3 )
=> ( ( H @ B3 )
= zero_zero_nat ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ S )
= ( groups4541462559716669496nt_nat @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_445_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_int,T3: set_nat,S: set_int,I2: nat > int,J: int > nat,T: set_nat,G: int > nat,H: nat > nat] :
( ( finite_finite_int @ S4 )
=> ( ( finite_finite_nat @ T3 )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ S @ S4 ) )
=> ( ( I2 @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ S @ S4 ) )
=> ( member_nat @ ( J @ A3 ) @ ( minus_minus_set_nat @ T @ T3 ) ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ T @ T3 ) )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ T @ T3 ) )
=> ( member_int @ ( I2 @ B3 ) @ ( minus_minus_set_int @ S @ S4 ) ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S4 )
=> ( ( G @ A3 )
= zero_zero_nat ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T3 )
=> ( ( H @ B3 )
= zero_zero_nat ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups4541462559716669496nt_nat @ G @ S )
= ( groups3542108847815614940at_nat @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_446_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_int,T3: set_int,S: set_int,I2: int > int,J: int > int,T: set_int,G: int > nat,H: int > nat] :
( ( finite_finite_int @ S4 )
=> ( ( finite_finite_int @ T3 )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ S @ S4 ) )
=> ( ( I2 @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ S @ S4 ) )
=> ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T @ T3 ) ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ ( minus_minus_set_int @ T @ T3 ) )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ ( minus_minus_set_int @ T @ T3 ) )
=> ( member_int @ ( I2 @ B3 ) @ ( minus_minus_set_int @ S @ S4 ) ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S4 )
=> ( ( G @ A3 )
= zero_zero_nat ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ T3 )
=> ( ( H @ B3 )
= zero_zero_nat ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups4541462559716669496nt_nat @ G @ S )
= ( groups4541462559716669496nt_nat @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_447_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_a,T3: set_a,S: set_a,I2: a > a,J: a > a,T: set_a,G: a > int,H: a > int] :
( ( finite_finite_a @ S4 )
=> ( ( finite_finite_a @ T3 )
=> ( ! [A3: a] :
( ( member_a @ A3 @ ( minus_minus_set_a @ S @ S4 ) )
=> ( ( I2 @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ ( minus_minus_set_a @ S @ S4 ) )
=> ( member_a @ ( J @ A3 ) @ ( minus_minus_set_a @ T @ T3 ) ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ ( minus_minus_set_a @ T @ T3 ) )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ ( minus_minus_set_a @ T @ T3 ) )
=> ( member_a @ ( I2 @ B3 ) @ ( minus_minus_set_a @ S @ S4 ) ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ S4 )
=> ( ( G @ A3 )
= zero_zero_int ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ T3 )
=> ( ( H @ B3 )
= zero_zero_int ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups6332066207828071664_a_int @ G @ S )
= ( groups6332066207828071664_a_int @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_448_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_a,T3: set_nat,S: set_a,I2: nat > a,J: a > nat,T: set_nat,G: a > int,H: nat > int] :
( ( finite_finite_a @ S4 )
=> ( ( finite_finite_nat @ T3 )
=> ( ! [A3: a] :
( ( member_a @ A3 @ ( minus_minus_set_a @ S @ S4 ) )
=> ( ( I2 @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ ( minus_minus_set_a @ S @ S4 ) )
=> ( member_nat @ ( J @ A3 ) @ ( minus_minus_set_nat @ T @ T3 ) ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ T @ T3 ) )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ T @ T3 ) )
=> ( member_a @ ( I2 @ B3 ) @ ( minus_minus_set_a @ S @ S4 ) ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ S4 )
=> ( ( G @ A3 )
= zero_zero_int ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T3 )
=> ( ( H @ B3 )
= zero_zero_int ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups6332066207828071664_a_int @ G @ S )
= ( groups3539618377306564664at_int @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_449_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_a,T3: set_int,S: set_a,I2: int > a,J: a > int,T: set_int,G: a > int,H: int > int] :
( ( finite_finite_a @ S4 )
=> ( ( finite_finite_int @ T3 )
=> ( ! [A3: a] :
( ( member_a @ A3 @ ( minus_minus_set_a @ S @ S4 ) )
=> ( ( I2 @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ ( minus_minus_set_a @ S @ S4 ) )
=> ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T @ T3 ) ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ ( minus_minus_set_int @ T @ T3 ) )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ ( minus_minus_set_int @ T @ T3 ) )
=> ( member_a @ ( I2 @ B3 ) @ ( minus_minus_set_a @ S @ S4 ) ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ S4 )
=> ( ( G @ A3 )
= zero_zero_int ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ T3 )
=> ( ( H @ B3 )
= zero_zero_int ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups6332066207828071664_a_int @ G @ S )
= ( groups4538972089207619220nt_int @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_450_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_nat,T3: set_a,S: set_nat,I2: a > nat,J: nat > a,T: set_a,G: nat > int,H: a > int] :
( ( finite_finite_nat @ S4 )
=> ( ( finite_finite_a @ T3 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ ( minus_minus_set_nat @ S @ S4 ) )
=> ( ( I2 @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ ( minus_minus_set_nat @ S @ S4 ) )
=> ( member_a @ ( J @ A3 ) @ ( minus_minus_set_a @ T @ T3 ) ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ ( minus_minus_set_a @ T @ T3 ) )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ ( minus_minus_set_a @ T @ T3 ) )
=> ( member_nat @ ( I2 @ B3 ) @ ( minus_minus_set_nat @ S @ S4 ) ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S4 )
=> ( ( G @ A3 )
= zero_zero_int ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ T3 )
=> ( ( H @ B3 )
= zero_zero_int ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups3539618377306564664at_int @ G @ S )
= ( groups6332066207828071664_a_int @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_451_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_nat,T3: set_nat,S: set_nat,I2: nat > nat,J: nat > nat,T: set_nat,G: nat > int,H: nat > int] :
( ( finite_finite_nat @ S4 )
=> ( ( finite_finite_nat @ T3 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ ( minus_minus_set_nat @ S @ S4 ) )
=> ( ( I2 @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ ( minus_minus_set_nat @ S @ S4 ) )
=> ( member_nat @ ( J @ A3 ) @ ( minus_minus_set_nat @ T @ T3 ) ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ T @ T3 ) )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ T @ T3 ) )
=> ( member_nat @ ( I2 @ B3 ) @ ( minus_minus_set_nat @ S @ S4 ) ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S4 )
=> ( ( G @ A3 )
= zero_zero_int ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T3 )
=> ( ( H @ B3 )
= zero_zero_int ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups3539618377306564664at_int @ G @ S )
= ( groups3539618377306564664at_int @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_452_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_nat,T3: set_int,S: set_nat,I2: int > nat,J: nat > int,T: set_int,G: nat > int,H: int > int] :
( ( finite_finite_nat @ S4 )
=> ( ( finite_finite_int @ T3 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ ( minus_minus_set_nat @ S @ S4 ) )
=> ( ( I2 @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ ( minus_minus_set_nat @ S @ S4 ) )
=> ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T @ T3 ) ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ ( minus_minus_set_int @ T @ T3 ) )
=> ( ( J @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ ( minus_minus_set_int @ T @ T3 ) )
=> ( member_nat @ ( I2 @ B3 ) @ ( minus_minus_set_nat @ S @ S4 ) ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S4 )
=> ( ( G @ A3 )
= zero_zero_int ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ T3 )
=> ( ( H @ B3 )
= zero_zero_int ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups3539618377306564664at_int @ G @ S )
= ( groups4538972089207619220nt_int @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_453_sum__eq__1__iff,axiom,
! [A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ( groups3542108847815614940at_nat @ F @ A )
= one_one_nat )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ( F @ X2 )
= one_one_nat )
& ! [Y2: nat] :
( ( member_nat @ Y2 @ A )
=> ( ( X2 != Y2 )
=> ( ( F @ Y2 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_454_sum__eq__1__iff,axiom,
! [A: set_list_a,F: list_a > nat] :
( ( finite_finite_list_a @ A )
=> ( ( ( groups5521247699297860762_a_nat @ F @ A )
= one_one_nat )
= ( ? [X2: list_a] :
( ( member_list_a @ X2 @ A )
& ( ( F @ X2 )
= one_one_nat )
& ! [Y2: list_a] :
( ( member_list_a @ Y2 @ A )
=> ( ( X2 != Y2 )
=> ( ( F @ Y2 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_455_sum__eq__1__iff,axiom,
! [A: set_int,F: int > nat] :
( ( finite_finite_int @ A )
=> ( ( ( groups4541462559716669496nt_nat @ F @ A )
= one_one_nat )
= ( ? [X2: int] :
( ( member_int @ X2 @ A )
& ( ( F @ X2 )
= one_one_nat )
& ! [Y2: int] :
( ( member_int @ Y2 @ A )
=> ( ( X2 != Y2 )
=> ( ( F @ Y2 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_456_sum__eq__1__iff,axiom,
! [A: set_a,F: a > nat] :
( ( finite_finite_a @ A )
=> ( ( ( groups6334556678337121940_a_nat @ F @ A )
= one_one_nat )
= ( ? [X2: a] :
( ( member_a @ X2 @ A )
& ( ( F @ X2 )
= one_one_nat )
& ! [Y2: a] :
( ( member_a @ Y2 @ A )
=> ( ( X2 != Y2 )
=> ( ( F @ Y2 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_457_sum__nonneg__leq__bound,axiom,
! [S3: set_nat,F: nat > nat,B2: nat,I2: nat] :
( ( finite_finite_nat @ S3 )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ S3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
=> ( ( ( groups3542108847815614940at_nat @ F @ S3 )
= B2 )
=> ( ( member_nat @ I2 @ S3 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ B2 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_458_sum__nonneg__leq__bound,axiom,
! [S3: set_int,F: int > nat,B2: nat,I2: int] :
( ( finite_finite_int @ S3 )
=> ( ! [I4: int] :
( ( member_int @ I4 @ S3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
=> ( ( ( groups4541462559716669496nt_nat @ F @ S3 )
= B2 )
=> ( ( member_int @ I2 @ S3 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ B2 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_459_sum__nonneg__leq__bound,axiom,
! [S3: set_a,F: a > int,B2: int,I2: a] :
( ( finite_finite_a @ S3 )
=> ( ! [I4: a] :
( ( member_a @ I4 @ S3 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ I4 ) ) )
=> ( ( ( groups6332066207828071664_a_int @ F @ S3 )
= B2 )
=> ( ( member_a @ I2 @ S3 )
=> ( ord_less_eq_int @ ( F @ I2 ) @ B2 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_460_sum__nonneg__leq__bound,axiom,
! [S3: set_nat,F: nat > int,B2: int,I2: nat] :
( ( finite_finite_nat @ S3 )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ S3 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ I4 ) ) )
=> ( ( ( groups3539618377306564664at_int @ F @ S3 )
= B2 )
=> ( ( member_nat @ I2 @ S3 )
=> ( ord_less_eq_int @ ( F @ I2 ) @ B2 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_461_sum__nonneg__leq__bound,axiom,
! [S3: set_int,F: int > int,B2: int,I2: int] :
( ( finite_finite_int @ S3 )
=> ( ! [I4: int] :
( ( member_int @ I4 @ S3 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ I4 ) ) )
=> ( ( ( groups4538972089207619220nt_int @ F @ S3 )
= B2 )
=> ( ( member_int @ I2 @ S3 )
=> ( ord_less_eq_int @ ( F @ I2 ) @ B2 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_462_sum__nonneg__leq__bound,axiom,
! [S3: set_a,F: a > nat,B2: nat,I2: a] :
( ( finite_finite_a @ S3 )
=> ( ! [I4: a] :
( ( member_a @ I4 @ S3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
=> ( ( ( groups6334556678337121940_a_nat @ F @ S3 )
= B2 )
=> ( ( member_a @ I2 @ S3 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ B2 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_463_sum__nonneg__leq__bound,axiom,
! [S3: set_list_a,F: list_a > nat,B2: nat,I2: list_a] :
( ( finite_finite_list_a @ S3 )
=> ( ! [I4: list_a] :
( ( member_list_a @ I4 @ S3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
=> ( ( ( groups5521247699297860762_a_nat @ F @ S3 )
= B2 )
=> ( ( member_list_a @ I2 @ S3 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ B2 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_464_sum__nonneg__leq__bound,axiom,
! [S3: set_list_a,F: list_a > int,B2: int,I2: list_a] :
( ( finite_finite_list_a @ S3 )
=> ( ! [I4: list_a] :
( ( member_list_a @ I4 @ S3 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ I4 ) ) )
=> ( ( ( groups5518757228788810486_a_int @ F @ S3 )
= B2 )
=> ( ( member_list_a @ I2 @ S3 )
=> ( ord_less_eq_int @ ( F @ I2 ) @ B2 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_465_sum__nonneg__leq__bound,axiom,
! [S3: set_nat_a,F: ( nat > a ) > nat,B2: nat,I2: nat > a] :
( ( finite_finite_nat_a @ S3 )
=> ( ! [I4: nat > a] :
( ( member_nat_a @ I4 @ S3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
=> ( ( ( groups154653438316501755_a_nat @ F @ S3 )
= B2 )
=> ( ( member_nat_a @ I2 @ S3 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ B2 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_466_sum__nonneg__leq__bound,axiom,
! [S3: set_nat_a,F: ( nat > a ) > int,B2: int,I2: nat > a] :
( ( finite_finite_nat_a @ S3 )
=> ( ! [I4: nat > a] :
( ( member_nat_a @ I4 @ S3 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ I4 ) ) )
=> ( ( ( groups152162967807451479_a_int @ F @ S3 )
= B2 )
=> ( ( member_nat_a @ I2 @ S3 )
=> ( ord_less_eq_int @ ( F @ I2 ) @ B2 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_467_sum__nonneg__0,axiom,
! [S3: set_nat,F: nat > nat,I2: nat] :
( ( finite_finite_nat @ S3 )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ S3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
=> ( ( ( groups3542108847815614940at_nat @ F @ S3 )
= zero_zero_nat )
=> ( ( member_nat @ I2 @ S3 )
=> ( ( F @ I2 )
= zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_468_sum__nonneg__0,axiom,
! [S3: set_int,F: int > nat,I2: int] :
( ( finite_finite_int @ S3 )
=> ( ! [I4: int] :
( ( member_int @ I4 @ S3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
=> ( ( ( groups4541462559716669496nt_nat @ F @ S3 )
= zero_zero_nat )
=> ( ( member_int @ I2 @ S3 )
=> ( ( F @ I2 )
= zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_469_sum__nonneg__0,axiom,
! [S3: set_a,F: a > int,I2: a] :
( ( finite_finite_a @ S3 )
=> ( ! [I4: a] :
( ( member_a @ I4 @ S3 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ I4 ) ) )
=> ( ( ( groups6332066207828071664_a_int @ F @ S3 )
= zero_zero_int )
=> ( ( member_a @ I2 @ S3 )
=> ( ( F @ I2 )
= zero_zero_int ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_470_sum__nonneg__0,axiom,
! [S3: set_nat,F: nat > int,I2: nat] :
( ( finite_finite_nat @ S3 )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ S3 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ I4 ) ) )
=> ( ( ( groups3539618377306564664at_int @ F @ S3 )
= zero_zero_int )
=> ( ( member_nat @ I2 @ S3 )
=> ( ( F @ I2 )
= zero_zero_int ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_471_sum__nonneg__0,axiom,
! [S3: set_int,F: int > int,I2: int] :
( ( finite_finite_int @ S3 )
=> ( ! [I4: int] :
( ( member_int @ I4 @ S3 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ I4 ) ) )
=> ( ( ( groups4538972089207619220nt_int @ F @ S3 )
= zero_zero_int )
=> ( ( member_int @ I2 @ S3 )
=> ( ( F @ I2 )
= zero_zero_int ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_472_sum__nonneg__0,axiom,
! [S3: set_a,F: a > nat,I2: a] :
( ( finite_finite_a @ S3 )
=> ( ! [I4: a] :
( ( member_a @ I4 @ S3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
=> ( ( ( groups6334556678337121940_a_nat @ F @ S3 )
= zero_zero_nat )
=> ( ( member_a @ I2 @ S3 )
=> ( ( F @ I2 )
= zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_473_sum__nonneg__0,axiom,
! [S3: set_list_a,F: list_a > nat,I2: list_a] :
( ( finite_finite_list_a @ S3 )
=> ( ! [I4: list_a] :
( ( member_list_a @ I4 @ S3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
=> ( ( ( groups5521247699297860762_a_nat @ F @ S3 )
= zero_zero_nat )
=> ( ( member_list_a @ I2 @ S3 )
=> ( ( F @ I2 )
= zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_474_sum__nonneg__0,axiom,
! [S3: set_list_a,F: list_a > int,I2: list_a] :
( ( finite_finite_list_a @ S3 )
=> ( ! [I4: list_a] :
( ( member_list_a @ I4 @ S3 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ I4 ) ) )
=> ( ( ( groups5518757228788810486_a_int @ F @ S3 )
= zero_zero_int )
=> ( ( member_list_a @ I2 @ S3 )
=> ( ( F @ I2 )
= zero_zero_int ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_475_sum__nonneg__0,axiom,
! [S3: set_nat_a,F: ( nat > a ) > nat,I2: nat > a] :
( ( finite_finite_nat_a @ S3 )
=> ( ! [I4: nat > a] :
( ( member_nat_a @ I4 @ S3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
=> ( ( ( groups154653438316501755_a_nat @ F @ S3 )
= zero_zero_nat )
=> ( ( member_nat_a @ I2 @ S3 )
=> ( ( F @ I2 )
= zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_476_sum__nonneg__0,axiom,
! [S3: set_nat_a,F: ( nat > a ) > int,I2: nat > a] :
( ( finite_finite_nat_a @ S3 )
=> ( ! [I4: nat > a] :
( ( member_nat_a @ I4 @ S3 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ I4 ) ) )
=> ( ( ( groups152162967807451479_a_int @ F @ S3 )
= zero_zero_int )
=> ( ( member_nat_a @ I2 @ S3 )
=> ( ( F @ I2 )
= zero_zero_int ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_477_sum_Osetdiff__irrelevant,axiom,
! [A: set_nat,G: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( groups3542108847815614940at_nat @ G
@ ( minus_minus_set_nat @ A
@ ( collect_nat
@ ^ [X2: nat] :
( ( G @ X2 )
= zero_zero_nat ) ) ) )
= ( groups3542108847815614940at_nat @ G @ A ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_478_sum_Osetdiff__irrelevant,axiom,
! [A: set_int,G: int > nat] :
( ( finite_finite_int @ A )
=> ( ( groups4541462559716669496nt_nat @ G
@ ( minus_minus_set_int @ A
@ ( collect_int
@ ^ [X2: int] :
( ( G @ X2 )
= zero_zero_nat ) ) ) )
= ( groups4541462559716669496nt_nat @ G @ A ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_479_sum_Osetdiff__irrelevant,axiom,
! [A: set_a,G: a > int] :
( ( finite_finite_a @ A )
=> ( ( groups6332066207828071664_a_int @ G
@ ( minus_minus_set_a @ A
@ ( collect_a
@ ^ [X2: a] :
( ( G @ X2 )
= zero_zero_int ) ) ) )
= ( groups6332066207828071664_a_int @ G @ A ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_480_sum_Osetdiff__irrelevant,axiom,
! [A: set_nat,G: nat > int] :
( ( finite_finite_nat @ A )
=> ( ( groups3539618377306564664at_int @ G
@ ( minus_minus_set_nat @ A
@ ( collect_nat
@ ^ [X2: nat] :
( ( G @ X2 )
= zero_zero_int ) ) ) )
= ( groups3539618377306564664at_int @ G @ A ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_481_sum_Osetdiff__irrelevant,axiom,
! [A: set_int,G: int > int] :
( ( finite_finite_int @ A )
=> ( ( groups4538972089207619220nt_int @ G
@ ( minus_minus_set_int @ A
@ ( collect_int
@ ^ [X2: int] :
( ( G @ X2 )
= zero_zero_int ) ) ) )
= ( groups4538972089207619220nt_int @ G @ A ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_482_sum_Osetdiff__irrelevant,axiom,
! [A: set_a,G: a > nat] :
( ( finite_finite_a @ A )
=> ( ( groups6334556678337121940_a_nat @ G
@ ( minus_minus_set_a @ A
@ ( collect_a
@ ^ [X2: a] :
( ( G @ X2 )
= zero_zero_nat ) ) ) )
= ( groups6334556678337121940_a_nat @ G @ A ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_483_sum_Osetdiff__irrelevant,axiom,
! [A: set_list_a,G: list_a > nat] :
( ( finite_finite_list_a @ A )
=> ( ( groups5521247699297860762_a_nat @ G
@ ( minus_646659088055828811list_a @ A
@ ( collect_list_a
@ ^ [X2: list_a] :
( ( G @ X2 )
= zero_zero_nat ) ) ) )
= ( groups5521247699297860762_a_nat @ G @ A ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_484_sum_Osetdiff__irrelevant,axiom,
! [A: set_a,G: a > multiset_a] :
( ( finite_finite_a @ A )
=> ( ( groups4808324907802680448iset_a @ G
@ ( minus_minus_set_a @ A
@ ( collect_a
@ ^ [X2: a] :
( ( G @ X2 )
= zero_zero_multiset_a ) ) ) )
= ( groups4808324907802680448iset_a @ G @ A ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_485_sum_Osetdiff__irrelevant,axiom,
! [A: set_nat,G: nat > multiset_a] :
( ( finite_finite_nat @ A )
=> ( ( groups1580436272196575032iset_a @ G
@ ( minus_minus_set_nat @ A
@ ( collect_nat
@ ^ [X2: nat] :
( ( G @ X2 )
= zero_zero_multiset_a ) ) ) )
= ( groups1580436272196575032iset_a @ G @ A ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_486_sum_Osetdiff__irrelevant,axiom,
! [A: set_int,G: int > multiset_a] :
( ( finite_finite_int @ A )
=> ( ( groups3457364905213935068iset_a @ G
@ ( minus_minus_set_int @ A
@ ( collect_int
@ ^ [X2: int] :
( ( G @ X2 )
= zero_zero_multiset_a ) ) ) )
= ( groups3457364905213935068iset_a @ G @ A ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_487_ring_Ois__root_Ocong,axiom,
polyno4133073214067823460ot_a_b = polyno4133073214067823460ot_a_b ).
% ring.is_root.cong
thf(fact_488_sum_Osubset__diff,axiom,
! [B2: set_int,A: set_int,G: int > nat] :
( ( ord_less_eq_set_int @ B2 @ A )
=> ( ( finite_finite_int @ A )
=> ( ( groups4541462559716669496nt_nat @ G @ A )
= ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G @ ( minus_minus_set_int @ A @ B2 ) ) @ ( groups4541462559716669496nt_nat @ G @ B2 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_489_sum_Osubset__diff,axiom,
! [B2: set_int,A: set_int,G: int > int] :
( ( ord_less_eq_set_int @ B2 @ A )
=> ( ( finite_finite_int @ A )
=> ( ( groups4538972089207619220nt_int @ G @ A )
= ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ ( minus_minus_set_int @ A @ B2 ) ) @ ( groups4538972089207619220nt_int @ G @ B2 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_490_sum_Osubset__diff,axiom,
! [B2: set_a,A: set_a,G: a > int] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( finite_finite_a @ A )
=> ( ( groups6332066207828071664_a_int @ G @ A )
= ( plus_plus_int @ ( groups6332066207828071664_a_int @ G @ ( minus_minus_set_a @ A @ B2 ) ) @ ( groups6332066207828071664_a_int @ G @ B2 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_491_sum_Osubset__diff,axiom,
! [B2: set_list_a,A: set_list_a,G: list_a > nat] :
( ( ord_le8861187494160871172list_a @ B2 @ A )
=> ( ( finite_finite_list_a @ A )
=> ( ( groups5521247699297860762_a_nat @ G @ A )
= ( plus_plus_nat @ ( groups5521247699297860762_a_nat @ G @ ( minus_646659088055828811list_a @ A @ B2 ) ) @ ( groups5521247699297860762_a_nat @ G @ B2 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_492_sum_Osubset__diff,axiom,
! [B2: set_list_a,A: set_list_a,G: list_a > int] :
( ( ord_le8861187494160871172list_a @ B2 @ A )
=> ( ( finite_finite_list_a @ A )
=> ( ( groups5518757228788810486_a_int @ G @ A )
= ( plus_plus_int @ ( groups5518757228788810486_a_int @ G @ ( minus_646659088055828811list_a @ A @ B2 ) ) @ ( groups5518757228788810486_a_int @ G @ B2 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_493_sum_Osubset__diff,axiom,
! [B2: set_nat,A: set_nat,G: nat > nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( ( finite_finite_nat @ A )
=> ( ( groups3542108847815614940at_nat @ G @ A )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( minus_minus_set_nat @ A @ B2 ) ) @ ( groups3542108847815614940at_nat @ G @ B2 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_494_sum_Osubset__diff,axiom,
! [B2: set_nat,A: set_nat,G: nat > int] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( ( finite_finite_nat @ A )
=> ( ( groups3539618377306564664at_int @ G @ A )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( minus_minus_set_nat @ A @ B2 ) ) @ ( groups3539618377306564664at_int @ G @ B2 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_495_sum_Osubset__diff,axiom,
! [B2: set_a,A: set_a,G: a > nat] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( finite_finite_a @ A )
=> ( ( groups6334556678337121940_a_nat @ G @ A )
= ( plus_plus_nat @ ( groups6334556678337121940_a_nat @ G @ ( minus_minus_set_a @ A @ B2 ) ) @ ( groups6334556678337121940_a_nat @ G @ B2 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_496_sum_Osame__carrier,axiom,
! [C3: set_int,A: set_int,B2: set_int,G: int > nat,H: int > nat] :
( ( finite_finite_int @ C3 )
=> ( ( ord_less_eq_set_int @ A @ C3 )
=> ( ( ord_less_eq_set_int @ B2 @ C3 )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ C3 @ A ) )
=> ( ( G @ A3 )
= zero_zero_nat ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ ( minus_minus_set_int @ C3 @ B2 ) )
=> ( ( H @ B3 )
= zero_zero_nat ) )
=> ( ( ( groups4541462559716669496nt_nat @ G @ A )
= ( groups4541462559716669496nt_nat @ H @ B2 ) )
= ( ( groups4541462559716669496nt_nat @ G @ C3 )
= ( groups4541462559716669496nt_nat @ H @ C3 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_497_sum_Osame__carrier,axiom,
! [C3: set_int,A: set_int,B2: set_int,G: int > int,H: int > int] :
( ( finite_finite_int @ C3 )
=> ( ( ord_less_eq_set_int @ A @ C3 )
=> ( ( ord_less_eq_set_int @ B2 @ C3 )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ C3 @ A ) )
=> ( ( G @ A3 )
= zero_zero_int ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ ( minus_minus_set_int @ C3 @ B2 ) )
=> ( ( H @ B3 )
= zero_zero_int ) )
=> ( ( ( groups4538972089207619220nt_int @ G @ A )
= ( groups4538972089207619220nt_int @ H @ B2 ) )
= ( ( groups4538972089207619220nt_int @ G @ C3 )
= ( groups4538972089207619220nt_int @ H @ C3 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_498_sum_Osame__carrier,axiom,
! [C3: set_a,A: set_a,B2: set_a,G: a > int,H: a > int] :
( ( finite_finite_a @ C3 )
=> ( ( ord_less_eq_set_a @ A @ C3 )
=> ( ( ord_less_eq_set_a @ B2 @ C3 )
=> ( ! [A3: a] :
( ( member_a @ A3 @ ( minus_minus_set_a @ C3 @ A ) )
=> ( ( G @ A3 )
= zero_zero_int ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ ( minus_minus_set_a @ C3 @ B2 ) )
=> ( ( H @ B3 )
= zero_zero_int ) )
=> ( ( ( groups6332066207828071664_a_int @ G @ A )
= ( groups6332066207828071664_a_int @ H @ B2 ) )
= ( ( groups6332066207828071664_a_int @ G @ C3 )
= ( groups6332066207828071664_a_int @ H @ C3 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_499_sum_Osame__carrier,axiom,
! [C3: set_nat,A: set_nat,B2: set_nat,G: nat > nat,H: nat > nat] :
( ( finite_finite_nat @ C3 )
=> ( ( ord_less_eq_set_nat @ A @ C3 )
=> ( ( ord_less_eq_set_nat @ B2 @ C3 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ ( minus_minus_set_nat @ C3 @ A ) )
=> ( ( G @ A3 )
= zero_zero_nat ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ C3 @ B2 ) )
=> ( ( H @ B3 )
= zero_zero_nat ) )
=> ( ( ( groups3542108847815614940at_nat @ G @ A )
= ( groups3542108847815614940at_nat @ H @ B2 ) )
= ( ( groups3542108847815614940at_nat @ G @ C3 )
= ( groups3542108847815614940at_nat @ H @ C3 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_500_sum_Osame__carrier,axiom,
! [C3: set_nat,A: set_nat,B2: set_nat,G: nat > int,H: nat > int] :
( ( finite_finite_nat @ C3 )
=> ( ( ord_less_eq_set_nat @ A @ C3 )
=> ( ( ord_less_eq_set_nat @ B2 @ C3 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ ( minus_minus_set_nat @ C3 @ A ) )
=> ( ( G @ A3 )
= zero_zero_int ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ C3 @ B2 ) )
=> ( ( H @ B3 )
= zero_zero_int ) )
=> ( ( ( groups3539618377306564664at_int @ G @ A )
= ( groups3539618377306564664at_int @ H @ B2 ) )
= ( ( groups3539618377306564664at_int @ G @ C3 )
= ( groups3539618377306564664at_int @ H @ C3 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_501_sum_Osame__carrier,axiom,
! [C3: set_a,A: set_a,B2: set_a,G: a > nat,H: a > nat] :
( ( finite_finite_a @ C3 )
=> ( ( ord_less_eq_set_a @ A @ C3 )
=> ( ( ord_less_eq_set_a @ B2 @ C3 )
=> ( ! [A3: a] :
( ( member_a @ A3 @ ( minus_minus_set_a @ C3 @ A ) )
=> ( ( G @ A3 )
= zero_zero_nat ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ ( minus_minus_set_a @ C3 @ B2 ) )
=> ( ( H @ B3 )
= zero_zero_nat ) )
=> ( ( ( groups6334556678337121940_a_nat @ G @ A )
= ( groups6334556678337121940_a_nat @ H @ B2 ) )
= ( ( groups6334556678337121940_a_nat @ G @ C3 )
= ( groups6334556678337121940_a_nat @ H @ C3 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_502_sum_Osame__carrier,axiom,
! [C3: set_int,A: set_int,B2: set_int,G: int > multiset_a,H: int > multiset_a] :
( ( finite_finite_int @ C3 )
=> ( ( ord_less_eq_set_int @ A @ C3 )
=> ( ( ord_less_eq_set_int @ B2 @ C3 )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ C3 @ A ) )
=> ( ( G @ A3 )
= zero_zero_multiset_a ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ ( minus_minus_set_int @ C3 @ B2 ) )
=> ( ( H @ B3 )
= zero_zero_multiset_a ) )
=> ( ( ( groups3457364905213935068iset_a @ G @ A )
= ( groups3457364905213935068iset_a @ H @ B2 ) )
= ( ( groups3457364905213935068iset_a @ G @ C3 )
= ( groups3457364905213935068iset_a @ H @ C3 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_503_sum_Osame__carrier,axiom,
! [C3: set_a,A: set_a,B2: set_a,G: a > multiset_a,H: a > multiset_a] :
( ( finite_finite_a @ C3 )
=> ( ( ord_less_eq_set_a @ A @ C3 )
=> ( ( ord_less_eq_set_a @ B2 @ C3 )
=> ( ! [A3: a] :
( ( member_a @ A3 @ ( minus_minus_set_a @ C3 @ A ) )
=> ( ( G @ A3 )
= zero_zero_multiset_a ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ ( minus_minus_set_a @ C3 @ B2 ) )
=> ( ( H @ B3 )
= zero_zero_multiset_a ) )
=> ( ( ( groups4808324907802680448iset_a @ G @ A )
= ( groups4808324907802680448iset_a @ H @ B2 ) )
= ( ( groups4808324907802680448iset_a @ G @ C3 )
= ( groups4808324907802680448iset_a @ H @ C3 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_504_sum_Osame__carrier,axiom,
! [C3: set_list_a,A: set_list_a,B2: set_list_a,G: list_a > nat,H: list_a > nat] :
( ( finite_finite_list_a @ C3 )
=> ( ( ord_le8861187494160871172list_a @ A @ C3 )
=> ( ( ord_le8861187494160871172list_a @ B2 @ C3 )
=> ( ! [A3: list_a] :
( ( member_list_a @ A3 @ ( minus_646659088055828811list_a @ C3 @ A ) )
=> ( ( G @ A3 )
= zero_zero_nat ) )
=> ( ! [B3: list_a] :
( ( member_list_a @ B3 @ ( minus_646659088055828811list_a @ C3 @ B2 ) )
=> ( ( H @ B3 )
= zero_zero_nat ) )
=> ( ( ( groups5521247699297860762_a_nat @ G @ A )
= ( groups5521247699297860762_a_nat @ H @ B2 ) )
= ( ( groups5521247699297860762_a_nat @ G @ C3 )
= ( groups5521247699297860762_a_nat @ H @ C3 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_505_sum_Osame__carrier,axiom,
! [C3: set_list_a,A: set_list_a,B2: set_list_a,G: list_a > int,H: list_a > int] :
( ( finite_finite_list_a @ C3 )
=> ( ( ord_le8861187494160871172list_a @ A @ C3 )
=> ( ( ord_le8861187494160871172list_a @ B2 @ C3 )
=> ( ! [A3: list_a] :
( ( member_list_a @ A3 @ ( minus_646659088055828811list_a @ C3 @ A ) )
=> ( ( G @ A3 )
= zero_zero_int ) )
=> ( ! [B3: list_a] :
( ( member_list_a @ B3 @ ( minus_646659088055828811list_a @ C3 @ B2 ) )
=> ( ( H @ B3 )
= zero_zero_int ) )
=> ( ( ( groups5518757228788810486_a_int @ G @ A )
= ( groups5518757228788810486_a_int @ H @ B2 ) )
= ( ( groups5518757228788810486_a_int @ G @ C3 )
= ( groups5518757228788810486_a_int @ H @ C3 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_506_sum_Osame__carrierI,axiom,
! [C3: set_int,A: set_int,B2: set_int,G: int > nat,H: int > nat] :
( ( finite_finite_int @ C3 )
=> ( ( ord_less_eq_set_int @ A @ C3 )
=> ( ( ord_less_eq_set_int @ B2 @ C3 )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ C3 @ A ) )
=> ( ( G @ A3 )
= zero_zero_nat ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ ( minus_minus_set_int @ C3 @ B2 ) )
=> ( ( H @ B3 )
= zero_zero_nat ) )
=> ( ( ( groups4541462559716669496nt_nat @ G @ C3 )
= ( groups4541462559716669496nt_nat @ H @ C3 ) )
=> ( ( groups4541462559716669496nt_nat @ G @ A )
= ( groups4541462559716669496nt_nat @ H @ B2 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_507_sum_Osame__carrierI,axiom,
! [C3: set_int,A: set_int,B2: set_int,G: int > int,H: int > int] :
( ( finite_finite_int @ C3 )
=> ( ( ord_less_eq_set_int @ A @ C3 )
=> ( ( ord_less_eq_set_int @ B2 @ C3 )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ C3 @ A ) )
=> ( ( G @ A3 )
= zero_zero_int ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ ( minus_minus_set_int @ C3 @ B2 ) )
=> ( ( H @ B3 )
= zero_zero_int ) )
=> ( ( ( groups4538972089207619220nt_int @ G @ C3 )
= ( groups4538972089207619220nt_int @ H @ C3 ) )
=> ( ( groups4538972089207619220nt_int @ G @ A )
= ( groups4538972089207619220nt_int @ H @ B2 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_508_sum_Osame__carrierI,axiom,
! [C3: set_a,A: set_a,B2: set_a,G: a > int,H: a > int] :
( ( finite_finite_a @ C3 )
=> ( ( ord_less_eq_set_a @ A @ C3 )
=> ( ( ord_less_eq_set_a @ B2 @ C3 )
=> ( ! [A3: a] :
( ( member_a @ A3 @ ( minus_minus_set_a @ C3 @ A ) )
=> ( ( G @ A3 )
= zero_zero_int ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ ( minus_minus_set_a @ C3 @ B2 ) )
=> ( ( H @ B3 )
= zero_zero_int ) )
=> ( ( ( groups6332066207828071664_a_int @ G @ C3 )
= ( groups6332066207828071664_a_int @ H @ C3 ) )
=> ( ( groups6332066207828071664_a_int @ G @ A )
= ( groups6332066207828071664_a_int @ H @ B2 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_509_sum_Osame__carrierI,axiom,
! [C3: set_nat,A: set_nat,B2: set_nat,G: nat > nat,H: nat > nat] :
( ( finite_finite_nat @ C3 )
=> ( ( ord_less_eq_set_nat @ A @ C3 )
=> ( ( ord_less_eq_set_nat @ B2 @ C3 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ ( minus_minus_set_nat @ C3 @ A ) )
=> ( ( G @ A3 )
= zero_zero_nat ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ C3 @ B2 ) )
=> ( ( H @ B3 )
= zero_zero_nat ) )
=> ( ( ( groups3542108847815614940at_nat @ G @ C3 )
= ( groups3542108847815614940at_nat @ H @ C3 ) )
=> ( ( groups3542108847815614940at_nat @ G @ A )
= ( groups3542108847815614940at_nat @ H @ B2 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_510_sum_Osame__carrierI,axiom,
! [C3: set_nat,A: set_nat,B2: set_nat,G: nat > int,H: nat > int] :
( ( finite_finite_nat @ C3 )
=> ( ( ord_less_eq_set_nat @ A @ C3 )
=> ( ( ord_less_eq_set_nat @ B2 @ C3 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ ( minus_minus_set_nat @ C3 @ A ) )
=> ( ( G @ A3 )
= zero_zero_int ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ C3 @ B2 ) )
=> ( ( H @ B3 )
= zero_zero_int ) )
=> ( ( ( groups3539618377306564664at_int @ G @ C3 )
= ( groups3539618377306564664at_int @ H @ C3 ) )
=> ( ( groups3539618377306564664at_int @ G @ A )
= ( groups3539618377306564664at_int @ H @ B2 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_511_sum_Osame__carrierI,axiom,
! [C3: set_a,A: set_a,B2: set_a,G: a > nat,H: a > nat] :
( ( finite_finite_a @ C3 )
=> ( ( ord_less_eq_set_a @ A @ C3 )
=> ( ( ord_less_eq_set_a @ B2 @ C3 )
=> ( ! [A3: a] :
( ( member_a @ A3 @ ( minus_minus_set_a @ C3 @ A ) )
=> ( ( G @ A3 )
= zero_zero_nat ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ ( minus_minus_set_a @ C3 @ B2 ) )
=> ( ( H @ B3 )
= zero_zero_nat ) )
=> ( ( ( groups6334556678337121940_a_nat @ G @ C3 )
= ( groups6334556678337121940_a_nat @ H @ C3 ) )
=> ( ( groups6334556678337121940_a_nat @ G @ A )
= ( groups6334556678337121940_a_nat @ H @ B2 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_512_sum_Osame__carrierI,axiom,
! [C3: set_int,A: set_int,B2: set_int,G: int > multiset_a,H: int > multiset_a] :
( ( finite_finite_int @ C3 )
=> ( ( ord_less_eq_set_int @ A @ C3 )
=> ( ( ord_less_eq_set_int @ B2 @ C3 )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ C3 @ A ) )
=> ( ( G @ A3 )
= zero_zero_multiset_a ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ ( minus_minus_set_int @ C3 @ B2 ) )
=> ( ( H @ B3 )
= zero_zero_multiset_a ) )
=> ( ( ( groups3457364905213935068iset_a @ G @ C3 )
= ( groups3457364905213935068iset_a @ H @ C3 ) )
=> ( ( groups3457364905213935068iset_a @ G @ A )
= ( groups3457364905213935068iset_a @ H @ B2 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_513_sum_Osame__carrierI,axiom,
! [C3: set_a,A: set_a,B2: set_a,G: a > multiset_a,H: a > multiset_a] :
( ( finite_finite_a @ C3 )
=> ( ( ord_less_eq_set_a @ A @ C3 )
=> ( ( ord_less_eq_set_a @ B2 @ C3 )
=> ( ! [A3: a] :
( ( member_a @ A3 @ ( minus_minus_set_a @ C3 @ A ) )
=> ( ( G @ A3 )
= zero_zero_multiset_a ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ ( minus_minus_set_a @ C3 @ B2 ) )
=> ( ( H @ B3 )
= zero_zero_multiset_a ) )
=> ( ( ( groups4808324907802680448iset_a @ G @ C3 )
= ( groups4808324907802680448iset_a @ H @ C3 ) )
=> ( ( groups4808324907802680448iset_a @ G @ A )
= ( groups4808324907802680448iset_a @ H @ B2 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_514_sum_Osame__carrierI,axiom,
! [C3: set_list_a,A: set_list_a,B2: set_list_a,G: list_a > nat,H: list_a > nat] :
( ( finite_finite_list_a @ C3 )
=> ( ( ord_le8861187494160871172list_a @ A @ C3 )
=> ( ( ord_le8861187494160871172list_a @ B2 @ C3 )
=> ( ! [A3: list_a] :
( ( member_list_a @ A3 @ ( minus_646659088055828811list_a @ C3 @ A ) )
=> ( ( G @ A3 )
= zero_zero_nat ) )
=> ( ! [B3: list_a] :
( ( member_list_a @ B3 @ ( minus_646659088055828811list_a @ C3 @ B2 ) )
=> ( ( H @ B3 )
= zero_zero_nat ) )
=> ( ( ( groups5521247699297860762_a_nat @ G @ C3 )
= ( groups5521247699297860762_a_nat @ H @ C3 ) )
=> ( ( groups5521247699297860762_a_nat @ G @ A )
= ( groups5521247699297860762_a_nat @ H @ B2 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_515_sum_Osame__carrierI,axiom,
! [C3: set_list_a,A: set_list_a,B2: set_list_a,G: list_a > int,H: list_a > int] :
( ( finite_finite_list_a @ C3 )
=> ( ( ord_le8861187494160871172list_a @ A @ C3 )
=> ( ( ord_le8861187494160871172list_a @ B2 @ C3 )
=> ( ! [A3: list_a] :
( ( member_list_a @ A3 @ ( minus_646659088055828811list_a @ C3 @ A ) )
=> ( ( G @ A3 )
= zero_zero_int ) )
=> ( ! [B3: list_a] :
( ( member_list_a @ B3 @ ( minus_646659088055828811list_a @ C3 @ B2 ) )
=> ( ( H @ B3 )
= zero_zero_int ) )
=> ( ( ( groups5518757228788810486_a_int @ G @ C3 )
= ( groups5518757228788810486_a_int @ H @ C3 ) )
=> ( ( groups5518757228788810486_a_int @ G @ A )
= ( groups5518757228788810486_a_int @ H @ B2 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_516_sum_Omono__neutral__left,axiom,
! [T: set_int,S: set_int,G: int > nat] :
( ( finite_finite_int @ T )
=> ( ( ord_less_eq_set_int @ S @ T )
=> ( ! [X: int] :
( ( member_int @ X @ ( minus_minus_set_int @ T @ S ) )
=> ( ( G @ X )
= zero_zero_nat ) )
=> ( ( groups4541462559716669496nt_nat @ G @ S )
= ( groups4541462559716669496nt_nat @ G @ T ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_517_sum_Omono__neutral__left,axiom,
! [T: set_int,S: set_int,G: int > int] :
( ( finite_finite_int @ T )
=> ( ( ord_less_eq_set_int @ S @ T )
=> ( ! [X: int] :
( ( member_int @ X @ ( minus_minus_set_int @ T @ S ) )
=> ( ( G @ X )
= zero_zero_int ) )
=> ( ( groups4538972089207619220nt_int @ G @ S )
= ( groups4538972089207619220nt_int @ G @ T ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_518_sum_Omono__neutral__left,axiom,
! [T: set_a,S: set_a,G: a > int] :
( ( finite_finite_a @ T )
=> ( ( ord_less_eq_set_a @ S @ T )
=> ( ! [X: a] :
( ( member_a @ X @ ( minus_minus_set_a @ T @ S ) )
=> ( ( G @ X )
= zero_zero_int ) )
=> ( ( groups6332066207828071664_a_int @ G @ S )
= ( groups6332066207828071664_a_int @ G @ T ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_519_sum_Omono__neutral__left,axiom,
! [T: set_nat,S: set_nat,G: nat > nat] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X: nat] :
( ( member_nat @ X @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( G @ X )
= zero_zero_nat ) )
=> ( ( groups3542108847815614940at_nat @ G @ S )
= ( groups3542108847815614940at_nat @ G @ T ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_520_sum_Omono__neutral__left,axiom,
! [T: set_nat,S: set_nat,G: nat > int] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X: nat] :
( ( member_nat @ X @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( G @ X )
= zero_zero_int ) )
=> ( ( groups3539618377306564664at_int @ G @ S )
= ( groups3539618377306564664at_int @ G @ T ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_521_sum_Omono__neutral__left,axiom,
! [T: set_a,S: set_a,G: a > nat] :
( ( finite_finite_a @ T )
=> ( ( ord_less_eq_set_a @ S @ T )
=> ( ! [X: a] :
( ( member_a @ X @ ( minus_minus_set_a @ T @ S ) )
=> ( ( G @ X )
= zero_zero_nat ) )
=> ( ( groups6334556678337121940_a_nat @ G @ S )
= ( groups6334556678337121940_a_nat @ G @ T ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_522_sum_Omono__neutral__left,axiom,
! [T: set_int,S: set_int,G: int > multiset_a] :
( ( finite_finite_int @ T )
=> ( ( ord_less_eq_set_int @ S @ T )
=> ( ! [X: int] :
( ( member_int @ X @ ( minus_minus_set_int @ T @ S ) )
=> ( ( G @ X )
= zero_zero_multiset_a ) )
=> ( ( groups3457364905213935068iset_a @ G @ S )
= ( groups3457364905213935068iset_a @ G @ T ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_523_sum_Omono__neutral__left,axiom,
! [T: set_a,S: set_a,G: a > multiset_a] :
( ( finite_finite_a @ T )
=> ( ( ord_less_eq_set_a @ S @ T )
=> ( ! [X: a] :
( ( member_a @ X @ ( minus_minus_set_a @ T @ S ) )
=> ( ( G @ X )
= zero_zero_multiset_a ) )
=> ( ( groups4808324907802680448iset_a @ G @ S )
= ( groups4808324907802680448iset_a @ G @ T ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_524_sum_Omono__neutral__left,axiom,
! [T: set_list_a,S: set_list_a,G: list_a > nat] :
( ( finite_finite_list_a @ T )
=> ( ( ord_le8861187494160871172list_a @ S @ T )
=> ( ! [X: list_a] :
( ( member_list_a @ X @ ( minus_646659088055828811list_a @ T @ S ) )
=> ( ( G @ X )
= zero_zero_nat ) )
=> ( ( groups5521247699297860762_a_nat @ G @ S )
= ( groups5521247699297860762_a_nat @ G @ T ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_525_sum_Omono__neutral__left,axiom,
! [T: set_list_a,S: set_list_a,G: list_a > int] :
( ( finite_finite_list_a @ T )
=> ( ( ord_le8861187494160871172list_a @ S @ T )
=> ( ! [X: list_a] :
( ( member_list_a @ X @ ( minus_646659088055828811list_a @ T @ S ) )
=> ( ( G @ X )
= zero_zero_int ) )
=> ( ( groups5518757228788810486_a_int @ G @ S )
= ( groups5518757228788810486_a_int @ G @ T ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_526_sum_Omono__neutral__right,axiom,
! [T: set_int,S: set_int,G: int > nat] :
( ( finite_finite_int @ T )
=> ( ( ord_less_eq_set_int @ S @ T )
=> ( ! [X: int] :
( ( member_int @ X @ ( minus_minus_set_int @ T @ S ) )
=> ( ( G @ X )
= zero_zero_nat ) )
=> ( ( groups4541462559716669496nt_nat @ G @ T )
= ( groups4541462559716669496nt_nat @ G @ S ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_527_sum_Omono__neutral__right,axiom,
! [T: set_int,S: set_int,G: int > int] :
( ( finite_finite_int @ T )
=> ( ( ord_less_eq_set_int @ S @ T )
=> ( ! [X: int] :
( ( member_int @ X @ ( minus_minus_set_int @ T @ S ) )
=> ( ( G @ X )
= zero_zero_int ) )
=> ( ( groups4538972089207619220nt_int @ G @ T )
= ( groups4538972089207619220nt_int @ G @ S ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_528_sum_Omono__neutral__right,axiom,
! [T: set_a,S: set_a,G: a > int] :
( ( finite_finite_a @ T )
=> ( ( ord_less_eq_set_a @ S @ T )
=> ( ! [X: a] :
( ( member_a @ X @ ( minus_minus_set_a @ T @ S ) )
=> ( ( G @ X )
= zero_zero_int ) )
=> ( ( groups6332066207828071664_a_int @ G @ T )
= ( groups6332066207828071664_a_int @ G @ S ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_529_sum_Omono__neutral__right,axiom,
! [T: set_nat,S: set_nat,G: nat > nat] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X: nat] :
( ( member_nat @ X @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( G @ X )
= zero_zero_nat ) )
=> ( ( groups3542108847815614940at_nat @ G @ T )
= ( groups3542108847815614940at_nat @ G @ S ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_530_sum_Omono__neutral__right,axiom,
! [T: set_nat,S: set_nat,G: nat > int] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X: nat] :
( ( member_nat @ X @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( G @ X )
= zero_zero_int ) )
=> ( ( groups3539618377306564664at_int @ G @ T )
= ( groups3539618377306564664at_int @ G @ S ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_531_sum_Omono__neutral__right,axiom,
! [T: set_a,S: set_a,G: a > nat] :
( ( finite_finite_a @ T )
=> ( ( ord_less_eq_set_a @ S @ T )
=> ( ! [X: a] :
( ( member_a @ X @ ( minus_minus_set_a @ T @ S ) )
=> ( ( G @ X )
= zero_zero_nat ) )
=> ( ( groups6334556678337121940_a_nat @ G @ T )
= ( groups6334556678337121940_a_nat @ G @ S ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_532_sum_Omono__neutral__right,axiom,
! [T: set_int,S: set_int,G: int > multiset_a] :
( ( finite_finite_int @ T )
=> ( ( ord_less_eq_set_int @ S @ T )
=> ( ! [X: int] :
( ( member_int @ X @ ( minus_minus_set_int @ T @ S ) )
=> ( ( G @ X )
= zero_zero_multiset_a ) )
=> ( ( groups3457364905213935068iset_a @ G @ T )
= ( groups3457364905213935068iset_a @ G @ S ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_533_sum_Omono__neutral__right,axiom,
! [T: set_a,S: set_a,G: a > multiset_a] :
( ( finite_finite_a @ T )
=> ( ( ord_less_eq_set_a @ S @ T )
=> ( ! [X: a] :
( ( member_a @ X @ ( minus_minus_set_a @ T @ S ) )
=> ( ( G @ X )
= zero_zero_multiset_a ) )
=> ( ( groups4808324907802680448iset_a @ G @ T )
= ( groups4808324907802680448iset_a @ G @ S ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_534_sum_Omono__neutral__right,axiom,
! [T: set_list_a,S: set_list_a,G: list_a > nat] :
( ( finite_finite_list_a @ T )
=> ( ( ord_le8861187494160871172list_a @ S @ T )
=> ( ! [X: list_a] :
( ( member_list_a @ X @ ( minus_646659088055828811list_a @ T @ S ) )
=> ( ( G @ X )
= zero_zero_nat ) )
=> ( ( groups5521247699297860762_a_nat @ G @ T )
= ( groups5521247699297860762_a_nat @ G @ S ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_535_sum_Omono__neutral__right,axiom,
! [T: set_list_a,S: set_list_a,G: list_a > int] :
( ( finite_finite_list_a @ T )
=> ( ( ord_le8861187494160871172list_a @ S @ T )
=> ( ! [X: list_a] :
( ( member_list_a @ X @ ( minus_646659088055828811list_a @ T @ S ) )
=> ( ( G @ X )
= zero_zero_int ) )
=> ( ( groups5518757228788810486_a_int @ G @ T )
= ( groups5518757228788810486_a_int @ G @ S ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_536_sum_Omono__neutral__cong__left,axiom,
! [T: set_int,S: set_int,H: int > nat,G: int > nat] :
( ( finite_finite_int @ T )
=> ( ( ord_less_eq_set_int @ S @ T )
=> ( ! [X: int] :
( ( member_int @ X @ ( minus_minus_set_int @ T @ S ) )
=> ( ( H @ X )
= zero_zero_nat ) )
=> ( ! [X: int] :
( ( member_int @ X @ S )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups4541462559716669496nt_nat @ G @ S )
= ( groups4541462559716669496nt_nat @ H @ T ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_537_sum_Omono__neutral__cong__left,axiom,
! [T: set_int,S: set_int,H: int > int,G: int > int] :
( ( finite_finite_int @ T )
=> ( ( ord_less_eq_set_int @ S @ T )
=> ( ! [X: int] :
( ( member_int @ X @ ( minus_minus_set_int @ T @ S ) )
=> ( ( H @ X )
= zero_zero_int ) )
=> ( ! [X: int] :
( ( member_int @ X @ S )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups4538972089207619220nt_int @ G @ S )
= ( groups4538972089207619220nt_int @ H @ T ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_538_sum_Omono__neutral__cong__left,axiom,
! [T: set_a,S: set_a,H: a > int,G: a > int] :
( ( finite_finite_a @ T )
=> ( ( ord_less_eq_set_a @ S @ T )
=> ( ! [X: a] :
( ( member_a @ X @ ( minus_minus_set_a @ T @ S ) )
=> ( ( H @ X )
= zero_zero_int ) )
=> ( ! [X: a] :
( ( member_a @ X @ S )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups6332066207828071664_a_int @ G @ S )
= ( groups6332066207828071664_a_int @ H @ T ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_539_sum_Omono__neutral__cong__left,axiom,
! [T: set_nat,S: set_nat,H: nat > nat,G: nat > nat] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X: nat] :
( ( member_nat @ X @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( H @ X )
= zero_zero_nat ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ S )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ S )
= ( groups3542108847815614940at_nat @ H @ T ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_540_sum_Omono__neutral__cong__left,axiom,
! [T: set_nat,S: set_nat,H: nat > int,G: nat > int] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X: nat] :
( ( member_nat @ X @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( H @ X )
= zero_zero_int ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ S )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups3539618377306564664at_int @ G @ S )
= ( groups3539618377306564664at_int @ H @ T ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_541_sum_Omono__neutral__cong__left,axiom,
! [T: set_a,S: set_a,H: a > nat,G: a > nat] :
( ( finite_finite_a @ T )
=> ( ( ord_less_eq_set_a @ S @ T )
=> ( ! [X: a] :
( ( member_a @ X @ ( minus_minus_set_a @ T @ S ) )
=> ( ( H @ X )
= zero_zero_nat ) )
=> ( ! [X: a] :
( ( member_a @ X @ S )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups6334556678337121940_a_nat @ G @ S )
= ( groups6334556678337121940_a_nat @ H @ T ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_542_sum_Omono__neutral__cong__left,axiom,
! [T: set_int,S: set_int,H: int > multiset_a,G: int > multiset_a] :
( ( finite_finite_int @ T )
=> ( ( ord_less_eq_set_int @ S @ T )
=> ( ! [X: int] :
( ( member_int @ X @ ( minus_minus_set_int @ T @ S ) )
=> ( ( H @ X )
= zero_zero_multiset_a ) )
=> ( ! [X: int] :
( ( member_int @ X @ S )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups3457364905213935068iset_a @ G @ S )
= ( groups3457364905213935068iset_a @ H @ T ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_543_sum_Omono__neutral__cong__left,axiom,
! [T: set_a,S: set_a,H: a > multiset_a,G: a > multiset_a] :
( ( finite_finite_a @ T )
=> ( ( ord_less_eq_set_a @ S @ T )
=> ( ! [X: a] :
( ( member_a @ X @ ( minus_minus_set_a @ T @ S ) )
=> ( ( H @ X )
= zero_zero_multiset_a ) )
=> ( ! [X: a] :
( ( member_a @ X @ S )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups4808324907802680448iset_a @ G @ S )
= ( groups4808324907802680448iset_a @ H @ T ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_544_sum_Omono__neutral__cong__left,axiom,
! [T: set_list_a,S: set_list_a,H: list_a > nat,G: list_a > nat] :
( ( finite_finite_list_a @ T )
=> ( ( ord_le8861187494160871172list_a @ S @ T )
=> ( ! [X: list_a] :
( ( member_list_a @ X @ ( minus_646659088055828811list_a @ T @ S ) )
=> ( ( H @ X )
= zero_zero_nat ) )
=> ( ! [X: list_a] :
( ( member_list_a @ X @ S )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups5521247699297860762_a_nat @ G @ S )
= ( groups5521247699297860762_a_nat @ H @ T ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_545_sum_Omono__neutral__cong__left,axiom,
! [T: set_list_a,S: set_list_a,H: list_a > int,G: list_a > int] :
( ( finite_finite_list_a @ T )
=> ( ( ord_le8861187494160871172list_a @ S @ T )
=> ( ! [X: list_a] :
( ( member_list_a @ X @ ( minus_646659088055828811list_a @ T @ S ) )
=> ( ( H @ X )
= zero_zero_int ) )
=> ( ! [X: list_a] :
( ( member_list_a @ X @ S )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups5518757228788810486_a_int @ G @ S )
= ( groups5518757228788810486_a_int @ H @ T ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_546_sum_Omono__neutral__cong__right,axiom,
! [T: set_int,S: set_int,G: int > nat,H: int > nat] :
( ( finite_finite_int @ T )
=> ( ( ord_less_eq_set_int @ S @ T )
=> ( ! [X: int] :
( ( member_int @ X @ ( minus_minus_set_int @ T @ S ) )
=> ( ( G @ X )
= zero_zero_nat ) )
=> ( ! [X: int] :
( ( member_int @ X @ S )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups4541462559716669496nt_nat @ G @ T )
= ( groups4541462559716669496nt_nat @ H @ S ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_547_sum_Omono__neutral__cong__right,axiom,
! [T: set_int,S: set_int,G: int > int,H: int > int] :
( ( finite_finite_int @ T )
=> ( ( ord_less_eq_set_int @ S @ T )
=> ( ! [X: int] :
( ( member_int @ X @ ( minus_minus_set_int @ T @ S ) )
=> ( ( G @ X )
= zero_zero_int ) )
=> ( ! [X: int] :
( ( member_int @ X @ S )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups4538972089207619220nt_int @ G @ T )
= ( groups4538972089207619220nt_int @ H @ S ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_548_sum_Omono__neutral__cong__right,axiom,
! [T: set_a,S: set_a,G: a > int,H: a > int] :
( ( finite_finite_a @ T )
=> ( ( ord_less_eq_set_a @ S @ T )
=> ( ! [X: a] :
( ( member_a @ X @ ( minus_minus_set_a @ T @ S ) )
=> ( ( G @ X )
= zero_zero_int ) )
=> ( ! [X: a] :
( ( member_a @ X @ S )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups6332066207828071664_a_int @ G @ T )
= ( groups6332066207828071664_a_int @ H @ S ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_549_sum_Omono__neutral__cong__right,axiom,
! [T: set_nat,S: set_nat,G: nat > nat,H: nat > nat] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X: nat] :
( ( member_nat @ X @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( G @ X )
= zero_zero_nat ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ S )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ T )
= ( groups3542108847815614940at_nat @ H @ S ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_550_sum_Omono__neutral__cong__right,axiom,
! [T: set_nat,S: set_nat,G: nat > int,H: nat > int] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X: nat] :
( ( member_nat @ X @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( G @ X )
= zero_zero_int ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ S )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups3539618377306564664at_int @ G @ T )
= ( groups3539618377306564664at_int @ H @ S ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_551_sum_Omono__neutral__cong__right,axiom,
! [T: set_a,S: set_a,G: a > nat,H: a > nat] :
( ( finite_finite_a @ T )
=> ( ( ord_less_eq_set_a @ S @ T )
=> ( ! [X: a] :
( ( member_a @ X @ ( minus_minus_set_a @ T @ S ) )
=> ( ( G @ X )
= zero_zero_nat ) )
=> ( ! [X: a] :
( ( member_a @ X @ S )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups6334556678337121940_a_nat @ G @ T )
= ( groups6334556678337121940_a_nat @ H @ S ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_552_sum_Omono__neutral__cong__right,axiom,
! [T: set_int,S: set_int,G: int > multiset_a,H: int > multiset_a] :
( ( finite_finite_int @ T )
=> ( ( ord_less_eq_set_int @ S @ T )
=> ( ! [X: int] :
( ( member_int @ X @ ( minus_minus_set_int @ T @ S ) )
=> ( ( G @ X )
= zero_zero_multiset_a ) )
=> ( ! [X: int] :
( ( member_int @ X @ S )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups3457364905213935068iset_a @ G @ T )
= ( groups3457364905213935068iset_a @ H @ S ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_553_sum_Omono__neutral__cong__right,axiom,
! [T: set_a,S: set_a,G: a > multiset_a,H: a > multiset_a] :
( ( finite_finite_a @ T )
=> ( ( ord_less_eq_set_a @ S @ T )
=> ( ! [X: a] :
( ( member_a @ X @ ( minus_minus_set_a @ T @ S ) )
=> ( ( G @ X )
= zero_zero_multiset_a ) )
=> ( ! [X: a] :
( ( member_a @ X @ S )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups4808324907802680448iset_a @ G @ T )
= ( groups4808324907802680448iset_a @ H @ S ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_554_sum_Omono__neutral__cong__right,axiom,
! [T: set_list_a,S: set_list_a,G: list_a > nat,H: list_a > nat] :
( ( finite_finite_list_a @ T )
=> ( ( ord_le8861187494160871172list_a @ S @ T )
=> ( ! [X: list_a] :
( ( member_list_a @ X @ ( minus_646659088055828811list_a @ T @ S ) )
=> ( ( G @ X )
= zero_zero_nat ) )
=> ( ! [X: list_a] :
( ( member_list_a @ X @ S )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups5521247699297860762_a_nat @ G @ T )
= ( groups5521247699297860762_a_nat @ H @ S ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_555_sum_Omono__neutral__cong__right,axiom,
! [T: set_list_a,S: set_list_a,G: list_a > int,H: list_a > int] :
( ( finite_finite_list_a @ T )
=> ( ( ord_le8861187494160871172list_a @ S @ T )
=> ( ! [X: list_a] :
( ( member_list_a @ X @ ( minus_646659088055828811list_a @ T @ S ) )
=> ( ( G @ X )
= zero_zero_int ) )
=> ( ! [X: list_a] :
( ( member_list_a @ X @ S )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( groups5518757228788810486_a_int @ G @ T )
= ( groups5518757228788810486_a_int @ H @ S ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_556_sum__mono2,axiom,
! [B2: set_int,A: set_int,F: int > nat] :
( ( finite_finite_int @ B2 )
=> ( ( ord_less_eq_set_int @ A @ B2 )
=> ( ! [B3: int] :
( ( member_int @ B3 @ ( minus_minus_set_int @ B2 @ A ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B3 ) ) )
=> ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ A ) @ ( groups4541462559716669496nt_nat @ F @ B2 ) ) ) ) ) ).
% sum_mono2
thf(fact_557_sum__mono2,axiom,
! [B2: set_int,A: set_int,F: int > int] :
( ( finite_finite_int @ B2 )
=> ( ( ord_less_eq_set_int @ A @ B2 )
=> ( ! [B3: int] :
( ( member_int @ B3 @ ( minus_minus_set_int @ B2 @ A ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ B3 ) ) )
=> ( ord_less_eq_int @ ( groups4538972089207619220nt_int @ F @ A ) @ ( groups4538972089207619220nt_int @ F @ B2 ) ) ) ) ) ).
% sum_mono2
thf(fact_558_sum__mono2,axiom,
! [B2: set_a,A: set_a,F: a > int] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A @ B2 )
=> ( ! [B3: a] :
( ( member_a @ B3 @ ( minus_minus_set_a @ B2 @ A ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ B3 ) ) )
=> ( ord_less_eq_int @ ( groups6332066207828071664_a_int @ F @ A ) @ ( groups6332066207828071664_a_int @ F @ B2 ) ) ) ) ) ).
% sum_mono2
thf(fact_559_sum__mono2,axiom,
! [B2: set_nat,A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ B2 @ A ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B3 ) ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ A ) @ ( groups3542108847815614940at_nat @ F @ B2 ) ) ) ) ) ).
% sum_mono2
thf(fact_560_sum__mono2,axiom,
! [B2: set_nat,A: set_nat,F: nat > int] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ B2 @ A ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ B3 ) ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ A ) @ ( groups3539618377306564664at_int @ F @ B2 ) ) ) ) ) ).
% sum_mono2
thf(fact_561_sum__mono2,axiom,
! [B2: set_a,A: set_a,F: a > nat] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A @ B2 )
=> ( ! [B3: a] :
( ( member_a @ B3 @ ( minus_minus_set_a @ B2 @ A ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B3 ) ) )
=> ( ord_less_eq_nat @ ( groups6334556678337121940_a_nat @ F @ A ) @ ( groups6334556678337121940_a_nat @ F @ B2 ) ) ) ) ) ).
% sum_mono2
thf(fact_562_sum__mono2,axiom,
! [B2: set_list_a,A: set_list_a,F: list_a > nat] :
( ( finite_finite_list_a @ B2 )
=> ( ( ord_le8861187494160871172list_a @ A @ B2 )
=> ( ! [B3: list_a] :
( ( member_list_a @ B3 @ ( minus_646659088055828811list_a @ B2 @ A ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B3 ) ) )
=> ( ord_less_eq_nat @ ( groups5521247699297860762_a_nat @ F @ A ) @ ( groups5521247699297860762_a_nat @ F @ B2 ) ) ) ) ) ).
% sum_mono2
thf(fact_563_sum__mono2,axiom,
! [B2: set_list_a,A: set_list_a,F: list_a > int] :
( ( finite_finite_list_a @ B2 )
=> ( ( ord_le8861187494160871172list_a @ A @ B2 )
=> ( ! [B3: list_a] :
( ( member_list_a @ B3 @ ( minus_646659088055828811list_a @ B2 @ A ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ B3 ) ) )
=> ( ord_less_eq_int @ ( groups5518757228788810486_a_int @ F @ A ) @ ( groups5518757228788810486_a_int @ F @ B2 ) ) ) ) ) ).
% sum_mono2
thf(fact_564_sum__mono2,axiom,
! [B2: set_nat_a,A: set_nat_a,F: ( nat > a ) > nat] :
( ( finite_finite_nat_a @ B2 )
=> ( ( ord_le871467723717165285_nat_a @ A @ B2 )
=> ( ! [B3: nat > a] :
( ( member_nat_a @ B3 @ ( minus_490503922182417452_nat_a @ B2 @ A ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B3 ) ) )
=> ( ord_less_eq_nat @ ( groups154653438316501755_a_nat @ F @ A ) @ ( groups154653438316501755_a_nat @ F @ B2 ) ) ) ) ) ).
% sum_mono2
thf(fact_565_sum__mono2,axiom,
! [B2: set_nat_a,A: set_nat_a,F: ( nat > a ) > int] :
( ( finite_finite_nat_a @ B2 )
=> ( ( ord_le871467723717165285_nat_a @ A @ B2 )
=> ( ! [B3: nat > a] :
( ( member_nat_a @ B3 @ ( minus_490503922182417452_nat_a @ B2 @ A ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ B3 ) ) )
=> ( ord_less_eq_int @ ( groups152162967807451479_a_int @ F @ A ) @ ( groups152162967807451479_a_int @ F @ B2 ) ) ) ) ) ).
% sum_mono2
thf(fact_566_domain_Odegree__zero__imp__not__is__root,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a,X3: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_s2403821588304063868list_a @ P ) @ one_one_nat )
= zero_zero_nat )
=> ~ ( polyno5142720416380192742t_unit @ R @ P @ X3 ) ) ) ) ).
% domain.degree_zero_imp_not_is_root
thf(fact_567_domain_Odegree__zero__imp__not__is__root,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,X3: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat )
= zero_zero_nat )
=> ~ ( polyno6951661231331188332t_unit @ R @ P @ X3 ) ) ) ) ).
% domain.degree_zero_imp_not_is_root
thf(fact_568_domain_Odegree__zero__imp__not__is__root,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,X3: a] :
( ( domain_a_b @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= zero_zero_nat )
=> ~ ( polyno4133073214067823460ot_a_b @ R @ P @ X3 ) ) ) ) ).
% domain.degree_zero_imp_not_is_root
thf(fact_569_domain_Opoly__mult__degree__le__1,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_list_a,Y: list_list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member5342144027231129785list_a @ X3 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( member5342144027231129785list_a @ Y @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s2403821588304063868list_a @ ( mult_l3065349954589089105t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ X3 @ Y ) ) @ one_one_nat ) @ ( plus_plus_nat @ ( minus_minus_nat @ ( size_s2403821588304063868list_a @ X3 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s2403821588304063868list_a @ Y ) @ one_one_nat ) ) ) ) ) ) ).
% domain.poly_mult_degree_le_1
thf(fact_570_domain_Opoly__mult__degree__le__1,axiom,
! [R: partia2175431115845679010xt_a_b,X3: list_a,Y: list_a] :
( ( domain_a_b @ R )
=> ( ( member_list_a @ X3 @ ( 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 ) ) ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ X3 @ Y ) ) @ one_one_nat ) @ ( plus_plus_nat @ ( minus_minus_nat @ ( size_size_list_a @ X3 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Y ) @ one_one_nat ) ) ) ) ) ) ).
% domain.poly_mult_degree_le_1
thf(fact_571_domain_Opoly__mult__degree__le__1,axiom,
! [R: partia2670972154091845814t_unit,X3: list_list_a,Y: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ X3 @ Y ) ) @ one_one_nat ) @ ( plus_plus_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ X3 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ Y ) @ one_one_nat ) ) ) ) ) ) ).
% domain.poly_mult_degree_le_1
thf(fact_572_domain_Opoly__mult__degree__le,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_list_a,Y: list_list_list_a,N: nat,M: nat] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member5342144027231129785list_a @ X3 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( member5342144027231129785list_a @ Y @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s2403821588304063868list_a @ X3 ) @ one_one_nat ) @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s2403821588304063868list_a @ Y ) @ one_one_nat ) @ M )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s2403821588304063868list_a @ ( mult_l3065349954589089105t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ X3 @ Y ) ) @ one_one_nat ) @ ( plus_plus_nat @ N @ M ) ) ) ) ) ) ) ).
% domain.poly_mult_degree_le
thf(fact_573_domain_Opoly__mult__degree__le,axiom,
! [R: partia2175431115845679010xt_a_b,X3: list_a,Y: list_a,N: nat,M: nat] :
( ( domain_a_b @ R )
=> ( ( member_list_a @ X3 @ ( 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 ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ X3 ) @ one_one_nat ) @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ Y ) @ one_one_nat ) @ M )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ X3 @ Y ) ) @ one_one_nat ) @ ( plus_plus_nat @ N @ M ) ) ) ) ) ) ) ).
% domain.poly_mult_degree_le
thf(fact_574_domain_Opoly__mult__degree__le,axiom,
! [R: partia2670972154091845814t_unit,X3: list_list_a,Y: list_list_a,N: nat,M: nat] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ X3 ) @ one_one_nat ) @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ Y ) @ one_one_nat ) @ M )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ X3 @ Y ) ) @ one_one_nat ) @ ( plus_plus_nat @ N @ M ) ) ) ) ) ) ) ).
% domain.poly_mult_degree_le
thf(fact_575_degree__zero__imp__splitted,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= zero_zero_nat )
=> ( polyno8329700637149614481ed_a_b @ r @ P ) ) ) ).
% degree_zero_imp_splitted
thf(fact_576_x_Olagrange__basis__polynomial__aux__def,axiom,
! [S: set_list_a] :
( ( lagran3534788790333317459t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ S )
= ( finpro3417560807142560175list_a @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
@ ^ [S2: list_a] : ( 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 ) ) ) ) @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ S2 ) )
@ S ) ) ).
% x.lagrange_basis_polynomial_aux_def
thf(fact_577_diff__add__zero,axiom,
! [A2: multiset_a,B: multiset_a] :
( ( minus_3765977307040488491iset_a @ A2 @ ( plus_plus_multiset_a @ A2 @ B ) )
= zero_zero_multiset_a ) ).
% diff_add_zero
thf(fact_578_diff__add__zero,axiom,
! [A2: nat,B: nat] :
( ( minus_minus_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_579_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_580_le__add__diff__inverse2,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A2 @ B ) @ B )
= A2 ) ) ).
% le_add_diff_inverse2
thf(fact_581_le__add__diff__inverse2,axiom,
! [B: int,A2: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ( plus_plus_int @ ( minus_minus_int @ A2 @ B ) @ B )
= A2 ) ) ).
% le_add_diff_inverse2
thf(fact_582_le__add__diff__inverse,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A2 @ B ) )
= A2 ) ) ).
% le_add_diff_inverse
thf(fact_583_le__add__diff__inverse,axiom,
! [B: int,A2: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A2 @ B ) )
= A2 ) ) ).
% le_add_diff_inverse
thf(fact_584_add__le__same__cancel1,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A2 ) @ B )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_585_add__le__same__cancel1,axiom,
! [B: int,A2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A2 ) @ B )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_586_add__le__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_587_add__le__same__cancel2,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B ) @ B )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_588_le__add__same__cancel1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_589_le__add__same__cancel1,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ ( plus_plus_int @ A2 @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_590_splitted__on__def,axiom,
! [K2: set_a,P: list_a] :
( ( polyno2453258491555121552on_a_b @ r @ K2 @ P )
= ( ( size_size_multiset_a @ ( polyno5714441830345289050on_a_b @ r @ K2 @ P ) )
= ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) ) ) ).
% splitted_on_def
thf(fact_591_add__right__cancel,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= ( plus_plus_nat @ C @ A2 ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_592_add__right__cancel,axiom,
! [B: int,A2: int,C: int] :
( ( ( plus_plus_int @ B @ A2 )
= ( plus_plus_int @ C @ A2 ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_593_add__left__cancel,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ A2 @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_594_add__left__cancel,axiom,
! [A2: int,B: int,C: int] :
( ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ A2 @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_595_x_Oadd_Oint__pow__mult,axiom,
! [X3: list_a,I2: int,J: int] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( plus_plus_int @ I2 @ J ) @ X3 )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I2 @ X3 ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ J @ X3 ) ) ) ) ).
% x.add.int_pow_mult
thf(fact_596_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_597_add__le__cancel__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_cancel_left
thf(fact_598_add__le__cancel__left,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_eq_int @ A2 @ B ) ) ).
% add_le_cancel_left
thf(fact_599_add__le__cancel__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_cancel_right
thf(fact_600_add__le__cancel__right,axiom,
! [A2: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_eq_int @ A2 @ B ) ) ).
% add_le_cancel_right
thf(fact_601_diff__self,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ A2 )
= zero_zero_int ) ).
% diff_self
thf(fact_602_diff__0__right,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ zero_zero_int )
= A2 ) ).
% diff_0_right
thf(fact_603_zero__diff,axiom,
! [A2: multiset_a] :
( ( minus_3765977307040488491iset_a @ zero_zero_multiset_a @ A2 )
= zero_zero_multiset_a ) ).
% zero_diff
thf(fact_604_zero__diff,axiom,
! [A2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% zero_diff
thf(fact_605_diff__zero,axiom,
! [A2: multiset_a] :
( ( minus_3765977307040488491iset_a @ A2 @ zero_zero_multiset_a )
= A2 ) ).
% diff_zero
thf(fact_606_diff__zero,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% diff_zero
thf(fact_607_diff__zero,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ zero_zero_int )
= A2 ) ).
% diff_zero
thf(fact_608_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: multiset_a] :
( ( minus_3765977307040488491iset_a @ A2 @ A2 )
= zero_zero_multiset_a ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_609_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ A2 )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_610_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ A2 )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_611_add__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% add_0
thf(fact_612_add__0,axiom,
! [A2: multiset_a] :
( ( plus_plus_multiset_a @ zero_zero_multiset_a @ A2 )
= A2 ) ).
% add_0
thf(fact_613_add__0,axiom,
! [A2: int] :
( ( plus_plus_int @ zero_zero_int @ A2 )
= A2 ) ).
% add_0
thf(fact_614_zero__eq__add__iff__both__eq__0,axiom,
! [X3: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X3 @ Y ) )
= ( ( X3 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_615_add__eq__0__iff__both__eq__0,axiom,
! [X3: nat,Y: nat] :
( ( ( plus_plus_nat @ X3 @ Y )
= zero_zero_nat )
= ( ( X3 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_616_add__cancel__right__right,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ A2 @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_617_add__cancel__right__right,axiom,
! [A2: multiset_a,B: multiset_a] :
( ( A2
= ( plus_plus_multiset_a @ A2 @ B ) )
= ( B = zero_zero_multiset_a ) ) ).
% add_cancel_right_right
thf(fact_618_add__cancel__right__right,axiom,
! [A2: int,B: int] :
( ( A2
= ( plus_plus_int @ A2 @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_619_add__cancel__right__left,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ B @ A2 ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_620_add__cancel__right__left,axiom,
! [A2: multiset_a,B: multiset_a] :
( ( A2
= ( plus_plus_multiset_a @ B @ A2 ) )
= ( B = zero_zero_multiset_a ) ) ).
% add_cancel_right_left
thf(fact_621_add__cancel__right__left,axiom,
! [A2: int,B: int] :
( ( A2
= ( plus_plus_int @ B @ A2 ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_622_add__cancel__left__right,axiom,
! [A2: nat,B: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= A2 )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_623_add__cancel__left__right,axiom,
! [A2: multiset_a,B: multiset_a] :
( ( ( plus_plus_multiset_a @ A2 @ B )
= A2 )
= ( B = zero_zero_multiset_a ) ) ).
% add_cancel_left_right
thf(fact_624_add__cancel__left__right,axiom,
! [A2: int,B: int] :
( ( ( plus_plus_int @ A2 @ B )
= A2 )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_625_add__cancel__left__left,axiom,
! [B: nat,A2: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= A2 )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_626_add__cancel__left__left,axiom,
! [B: multiset_a,A2: multiset_a] :
( ( ( plus_plus_multiset_a @ B @ A2 )
= A2 )
= ( B = zero_zero_multiset_a ) ) ).
% add_cancel_left_left
thf(fact_627_add__cancel__left__left,axiom,
! [B: int,A2: int] :
( ( ( plus_plus_int @ B @ A2 )
= A2 )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_628_double__zero__sym,axiom,
! [A2: int] :
( ( zero_zero_int
= ( plus_plus_int @ A2 @ A2 ) )
= ( A2 = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_629_add_Oright__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.right_neutral
thf(fact_630_add_Oright__neutral,axiom,
! [A2: multiset_a] :
( ( plus_plus_multiset_a @ A2 @ zero_zero_multiset_a )
= A2 ) ).
% add.right_neutral
thf(fact_631_add_Oright__neutral,axiom,
! [A2: int] :
( ( plus_plus_int @ A2 @ zero_zero_int )
= A2 ) ).
% add.right_neutral
thf(fact_632_add__diff__cancel,axiom,
! [A2: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A2 @ B ) @ B )
= A2 ) ).
% add_diff_cancel
thf(fact_633_diff__add__cancel,axiom,
! [A2: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A2 @ B ) @ B )
= A2 ) ).
% diff_add_cancel
thf(fact_634_add__diff__cancel__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A2 @ B ) ) ).
% add_diff_cancel_left
thf(fact_635_add__diff__cancel__left,axiom,
! [C: int,A2: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) )
= ( minus_minus_int @ A2 @ B ) ) ).
% add_diff_cancel_left
thf(fact_636_add__diff__cancel__left_H,axiom,
! [A2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B ) @ A2 )
= B ) ).
% add_diff_cancel_left'
thf(fact_637_add__diff__cancel__left_H,axiom,
! [A2: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A2 @ B ) @ A2 )
= B ) ).
% add_diff_cancel_left'
thf(fact_638_add__diff__cancel__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A2 @ B ) ) ).
% add_diff_cancel_right
thf(fact_639_add__diff__cancel__right,axiom,
! [A2: int,C: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ A2 @ B ) ) ).
% add_diff_cancel_right
thf(fact_640_add__diff__cancel__right_H,axiom,
! [A2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
= A2 ) ).
% add_diff_cancel_right'
thf(fact_641_add__diff__cancel__right_H,axiom,
! [A2: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A2 @ B ) @ B )
= A2 ) ).
% add_diff_cancel_right'
thf(fact_642_diff__ge__0__iff__ge,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B ) )
= ( ord_less_eq_int @ B @ A2 ) ) ).
% diff_ge_0_iff_ge
thf(fact_643_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A2 @ A2 ) )
= ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_644_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ A2 ) @ zero_zero_int )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_645_le__add__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ B @ A2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_646_le__add__same__cancel2,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ ( plus_plus_int @ B @ A2 ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_647_x_Oadd_Oint__pow__1,axiom,
! [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ one_one_int @ X3 )
= X3 ) ) ).
% x.add.int_pow_1
thf(fact_648_ring_Osplitted_Ocong,axiom,
polyno8329700637149614481ed_a_b = polyno8329700637149614481ed_a_b ).
% ring.splitted.cong
thf(fact_649_ring_Osplitted_Ocong,axiom,
polyno6259083269128200473t_unit = polyno6259083269128200473t_unit ).
% ring.splitted.cong
thf(fact_650_zero__reorient,axiom,
! [X3: nat] :
( ( zero_zero_nat = X3 )
= ( X3 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_651_zero__reorient,axiom,
! [X3: multiset_a] :
( ( zero_zero_multiset_a = X3 )
= ( X3 = zero_zero_multiset_a ) ) ).
% zero_reorient
thf(fact_652_zero__reorient,axiom,
! [X3: int] :
( ( zero_zero_int = X3 )
= ( X3 = zero_zero_int ) ) ).
% zero_reorient
thf(fact_653_one__reorient,axiom,
! [X3: nat] :
( ( one_one_nat = X3 )
= ( X3 = one_one_nat ) ) ).
% one_reorient
thf(fact_654_one__reorient,axiom,
! [X3: int] :
( ( one_one_int = X3 )
= ( X3 = one_one_int ) ) ).
% one_reorient
thf(fact_655_diff__eq__diff__eq,axiom,
! [A2: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A2 @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( A2 = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_656_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A2: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_657_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A2: int,C: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A2 @ C ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A2 @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_658_add__right__imp__eq,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= ( plus_plus_nat @ C @ A2 ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_659_add__right__imp__eq,axiom,
! [B: int,A2: int,C: int] :
( ( ( plus_plus_int @ B @ A2 )
= ( plus_plus_int @ C @ A2 ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_660_add__left__imp__eq,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ A2 @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_661_add__left__imp__eq,axiom,
! [A2: int,B: int,C: int] :
( ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ A2 @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_662_add_Oleft__commute,axiom,
! [B: nat,A2: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A2 @ C ) )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_663_add_Oleft__commute,axiom,
! [B: int,A2: int,C: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A2 @ C ) )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B @ C ) ) ) ).
% add.left_commute
thf(fact_664_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B5: nat] : ( plus_plus_nat @ B5 @ A4 ) ) ) ).
% add.commute
thf(fact_665_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A4: int,B5: int] : ( plus_plus_int @ B5 @ A4 ) ) ) ).
% add.commute
thf(fact_666_group__add__class_Oadd_Oright__cancel,axiom,
! [B: int,A2: int,C: int] :
( ( ( plus_plus_int @ B @ A2 )
= ( plus_plus_int @ C @ A2 ) )
= ( B = C ) ) ).
% group_add_class.add.right_cancel
thf(fact_667_add_Oleft__cancel,axiom,
! [A2: int,B: int,C: int] :
( ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ A2 @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_668_add_Oassoc,axiom,
! [A2: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_669_add_Oassoc,axiom,
! [A2: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A2 @ B ) @ C )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B @ C ) ) ) ).
% add.assoc
thf(fact_670_group__cancel_Oadd2,axiom,
! [B2: nat,K: nat,B: nat,A2: nat] :
( ( B2
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A2 @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_671_group__cancel_Oadd2,axiom,
! [B2: int,K: int,B: int,A2: int] :
( ( B2
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A2 @ B2 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A2 @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_672_group__cancel_Oadd1,axiom,
! [A: nat,K: nat,A2: nat,B: nat] :
( ( A
= ( plus_plus_nat @ K @ A2 ) )
=> ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_673_group__cancel_Oadd1,axiom,
! [A: int,K: int,A2: int,B: int] :
( ( A
= ( plus_plus_int @ K @ A2 ) )
=> ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A2 @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_674_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( I2 = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I2 @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_675_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: int,J: int,K: int,L: int] :
( ( ( I2 = J )
& ( K = L ) )
=> ( ( plus_plus_int @ I2 @ K )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_676_is__num__normalize_I1_J,axiom,
! [A2: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A2 @ B ) @ C )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_677_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_678_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A2 @ B ) @ C )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_679_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_680_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_681_zero__le,axiom,
! [X3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X3 ) ).
% zero_le
thf(fact_682_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_683_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_684_diff__mono,axiom,
! [A2: int,B: int,D: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ D @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_685_diff__left__mono,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ord_less_eq_int @ ( minus_minus_int @ C @ A2 ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_686_diff__right__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ord_less_eq_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_687_diff__eq__diff__less__eq,axiom,
! [A2: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A2 @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_eq_int @ A2 @ B )
= ( ord_less_eq_int @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_688_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_689_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I2 @ J )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_690_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( I2 = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_691_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: int,J: int,K: int,L: int] :
( ( ( I2 = J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_692_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_693_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I2 @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_694_add__mono,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_695_add__mono,axiom,
! [A2: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_mono
thf(fact_696_add__left__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_697_add__left__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_left_mono
thf(fact_698_less__eqE,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ~ ! [C2: nat] :
( B
!= ( plus_plus_nat @ A2 @ C2 ) ) ) ).
% less_eqE
thf(fact_699_add__right__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_700_add__right__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_right_mono
thf(fact_701_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B5: nat] :
? [C4: nat] :
( B5
= ( plus_plus_nat @ A4 @ C4 ) ) ) ) ).
% le_iff_add
thf(fact_702_add__le__imp__le__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_imp_le_left
thf(fact_703_add__le__imp__le__left,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_eq_int @ A2 @ B ) ) ).
% add_le_imp_le_left
thf(fact_704_add__le__imp__le__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_imp_le_right
thf(fact_705_add__le__imp__le__right,axiom,
! [A2: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_eq_int @ A2 @ B ) ) ).
% add_le_imp_le_right
thf(fact_706_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_707_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_708_eq__iff__diff__eq__0,axiom,
( ( ^ [Y5: int,Z2: int] : ( Y5 = Z2 ) )
= ( ^ [A4: int,B5: int] :
( ( minus_minus_int @ A4 @ B5 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_709_add_Ogroup__left__neutral,axiom,
! [A2: int] :
( ( plus_plus_int @ zero_zero_int @ A2 )
= A2 ) ).
% add.group_left_neutral
thf(fact_710_add_Ocomm__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.comm_neutral
thf(fact_711_add_Ocomm__neutral,axiom,
! [A2: multiset_a] :
( ( plus_plus_multiset_a @ A2 @ zero_zero_multiset_a )
= A2 ) ).
% add.comm_neutral
thf(fact_712_add_Ocomm__neutral,axiom,
! [A2: int] :
( ( plus_plus_int @ A2 @ zero_zero_int )
= A2 ) ).
% add.comm_neutral
thf(fact_713_comm__monoid__add__class_Oadd__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_714_comm__monoid__add__class_Oadd__0,axiom,
! [A2: multiset_a] :
( ( plus_plus_multiset_a @ zero_zero_multiset_a @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_715_comm__monoid__add__class_Oadd__0,axiom,
! [A2: int] :
( ( plus_plus_int @ zero_zero_int @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_716_group__cancel_Osub1,axiom,
! [A: int,K: int,A2: int,B: int] :
( ( A
= ( plus_plus_int @ K @ A2 ) )
=> ( ( minus_minus_int @ A @ B )
= ( plus_plus_int @ K @ ( minus_minus_int @ A2 @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_717_diff__eq__eq,axiom,
! [A2: int,B: int,C: int] :
( ( ( minus_minus_int @ A2 @ B )
= C )
= ( A2
= ( plus_plus_int @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_718_eq__diff__eq,axiom,
! [A2: int,C: int,B: int] :
( ( A2
= ( minus_minus_int @ C @ B ) )
= ( ( plus_plus_int @ A2 @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_719_add__diff__eq,axiom,
! [A2: int,B: int,C: int] :
( ( plus_plus_int @ A2 @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A2 @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_720_diff__diff__eq2,axiom,
! [A2: int,B: int,C: int] :
( ( minus_minus_int @ A2 @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_721_diff__add__eq,axiom,
! [A2: int,B: int,C: int] :
( ( plus_plus_int @ ( minus_minus_int @ A2 @ B ) @ C )
= ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_722_diff__add__eq__diff__diff__swap,axiom,
! [A2: int,B: int,C: int] :
( ( minus_minus_int @ A2 @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ ( minus_minus_int @ A2 @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_723_add__implies__diff,axiom,
! [C: nat,B: nat,A2: nat] :
( ( ( plus_plus_nat @ C @ B )
= A2 )
=> ( C
= ( minus_minus_nat @ A2 @ B ) ) ) ).
% add_implies_diff
thf(fact_724_add__implies__diff,axiom,
! [C: int,B: int,A2: int] :
( ( ( plus_plus_int @ C @ B )
= A2 )
=> ( C
= ( minus_minus_int @ A2 @ B ) ) ) ).
% add_implies_diff
thf(fact_725_diff__diff__eq,axiom,
! [A2: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B ) @ C )
= ( minus_minus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_726_diff__diff__eq,axiom,
! [A2: int,B: int,C: int] :
( ( minus_minus_int @ ( minus_minus_int @ A2 @ B ) @ C )
= ( minus_minus_int @ A2 @ ( plus_plus_int @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_727_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_728_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_729_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_730_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_731_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_732_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_733_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B5: int] : ( ord_less_eq_int @ ( minus_minus_int @ A4 @ B5 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_734_add__nonpos__eq__0__iff,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X3 @ Y )
= zero_zero_nat )
= ( ( X3 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_735_add__nonpos__eq__0__iff,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( ( ( plus_plus_int @ X3 @ Y )
= zero_zero_int )
= ( ( X3 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_736_add__nonneg__eq__0__iff,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X3 @ Y )
= zero_zero_nat )
= ( ( X3 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_737_add__nonneg__eq__0__iff,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( plus_plus_int @ X3 @ Y )
= zero_zero_int )
= ( ( X3 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_738_add__nonpos__nonpos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_739_add__nonpos__nonpos,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_740_add__nonneg__nonneg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_741_add__nonneg__nonneg,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_742_add__increasing2,axiom,
! [C: nat,B: nat,A2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A2 )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_increasing2
thf(fact_743_add__increasing2,axiom,
! [C: int,B: int,A2: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ B @ A2 )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% add_increasing2
thf(fact_744_add__decreasing2,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_745_add__decreasing2,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ A2 @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_746_add__increasing,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_increasing
thf(fact_747_add__increasing,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% add_increasing
thf(fact_748_add__decreasing,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_749_add__decreasing,axiom,
! [A2: int,C: int,B: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_750_diff__le__eq,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ ( minus_minus_int @ A2 @ B ) @ C )
= ( ord_less_eq_int @ A2 @ ( plus_plus_int @ C @ B ) ) ) ).
% diff_le_eq
thf(fact_751_le__diff__eq,axiom,
! [A2: int,C: int,B: int] :
( ( ord_less_eq_int @ A2 @ ( minus_minus_int @ C @ B ) )
= ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B ) @ C ) ) ).
% le_diff_eq
thf(fact_752_add__le__imp__le__diff,axiom,
! [I2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N )
=> ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_753_add__le__imp__le__diff,axiom,
! [I2: int,K: int,N: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ N )
=> ( ord_less_eq_int @ I2 @ ( minus_minus_int @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_754_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A2 ) @ A2 )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_755_add__le__add__imp__diff__le,axiom,
! [I2: nat,K: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_756_add__le__add__imp__diff__le,axiom,
! [I2: int,K: int,N: int,J: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_757_le__add__diff,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A2 ) ) ) ).
% le_add_diff
thf(fact_758_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_759_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_760_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A2 )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_761_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A2 ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_762_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A2 )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A2 ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_763_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_764_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( plus_plus_nat @ A2 @ ( minus_minus_nat @ B @ A2 ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_765_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( minus_minus_nat @ B @ A2 )
= C )
= ( B
= ( plus_plus_nat @ C @ A2 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_766_domain_Odegree__zero__imp__splitted,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_s2403821588304063868list_a @ P ) @ one_one_nat )
= zero_zero_nat )
=> ( polyno5970451904377802771t_unit @ R @ P ) ) ) ) ).
% domain.degree_zero_imp_splitted
thf(fact_767_domain_Odegree__zero__imp__splitted,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a] :
( ( domain_a_b @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= zero_zero_nat )
=> ( polyno8329700637149614481ed_a_b @ R @ P ) ) ) ) ).
% domain.degree_zero_imp_splitted
thf(fact_768_domain_Odegree__zero__imp__splitted,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat )
= zero_zero_nat )
=> ( polyno6259083269128200473t_unit @ R @ P ) ) ) ) ).
% domain.degree_zero_imp_splitted
thf(fact_769_degree__zero__imp__empty__roots,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= zero_zero_nat )
=> ( ( polynomial_roots_a_b @ r @ P )
= zero_zero_multiset_a ) ) ) ).
% degree_zero_imp_empty_roots
thf(fact_770_x_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 ) ) ) ) ).
% x.carrier_is_subalgebra
thf(fact_771_x_Osubalgebra__in__carrier,axiom,
! [K2: set_list_a,V: set_list_a] :
( ( embedd1768981623711841426t_unit @ K2 @ V @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ord_le8861187494160871172list_a @ V @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.subalgebra_in_carrier
thf(fact_772_splitted__def,axiom,
! [P: list_a] :
( ( polyno8329700637149614481ed_a_b @ r @ P )
= ( ( size_size_multiset_a @ ( polynomial_roots_a_b @ r @ P ) )
= ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) ) ) ).
% splitted_def
thf(fact_773_x_Ogenideal__self,axiom,
! [S: set_list_a] :
( ( ord_le8861187494160871172list_a @ S @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ S @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ S ) ) ) ).
% x.genideal_self
thf(fact_774_x_Osubset__Idl__subset,axiom,
! [I3: set_list_a,H2: set_list_a] :
( ( ord_le8861187494160871172list_a @ I3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ H2 @ I3 )
=> ( ord_le8861187494160871172list_a @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H2 ) @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I3 ) ) ) ) ).
% x.subset_Idl_subset
thf(fact_775_size__union,axiom,
! [M2: multiset_list_a,N4: multiset_list_a] :
( ( size_s2335926164413107382list_a @ ( plus_p690419498615200257list_a @ M2 @ N4 ) )
= ( plus_plus_nat @ ( size_s2335926164413107382list_a @ M2 ) @ ( size_s2335926164413107382list_a @ N4 ) ) ) ).
% size_union
thf(fact_776_size__union,axiom,
! [M2: multiset_a,N4: multiset_a] :
( ( size_size_multiset_a @ ( plus_plus_multiset_a @ M2 @ N4 ) )
= ( plus_plus_nat @ ( size_size_multiset_a @ M2 ) @ ( size_size_multiset_a @ N4 ) ) ) ).
% size_union
thf(fact_777_size__empty,axiom,
( ( size_s2335926164413107382list_a @ zero_z4454100511807792257list_a )
= zero_zero_nat ) ).
% size_empty
thf(fact_778_size__empty,axiom,
( ( size_size_multiset_a @ zero_zero_multiset_a )
= zero_zero_nat ) ).
% size_empty
thf(fact_779_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
! [X3: multiset_a,Y: multiset_a] :
( ( ( plus_plus_multiset_a @ X3 @ Y )
= zero_zero_multiset_a )
= ( ( X3 = zero_zero_multiset_a )
& ( Y = zero_zero_multiset_a ) ) ) ).
% subset_mset.add_eq_0_iff_both_eq_0
thf(fact_780_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
! [X3: multiset_a,Y: multiset_a] :
( ( zero_zero_multiset_a
= ( plus_plus_multiset_a @ X3 @ Y ) )
= ( ( X3 = zero_zero_multiset_a )
& ( Y = zero_zero_multiset_a ) ) ) ).
% subset_mset.zero_eq_add_iff_both_eq_0
thf(fact_781_empty__eq__union,axiom,
! [M2: multiset_a,N4: multiset_a] :
( ( zero_zero_multiset_a
= ( plus_plus_multiset_a @ M2 @ N4 ) )
= ( ( M2 = zero_zero_multiset_a )
& ( N4 = zero_zero_multiset_a ) ) ) ).
% empty_eq_union
thf(fact_782_union__eq__empty,axiom,
! [M2: multiset_a,N4: multiset_a] :
( ( ( plus_plus_multiset_a @ M2 @ N4 )
= zero_zero_multiset_a )
= ( ( M2 = zero_zero_multiset_a )
& ( N4 = zero_zero_multiset_a ) ) ) ).
% union_eq_empty
thf(fact_783_size__eq__0__iff__empty,axiom,
! [M2: multiset_list_a] :
( ( ( size_s2335926164413107382list_a @ M2 )
= zero_zero_nat )
= ( M2 = zero_z4454100511807792257list_a ) ) ).
% size_eq_0_iff_empty
thf(fact_784_size__eq__0__iff__empty,axiom,
! [M2: multiset_a] :
( ( ( size_size_multiset_a @ M2 )
= zero_zero_nat )
= ( M2 = zero_zero_multiset_a ) ) ).
% size_eq_0_iff_empty
thf(fact_785_Multiset_Odiff__cancel,axiom,
! [A: multiset_a] :
( ( minus_3765977307040488491iset_a @ A @ A )
= zero_zero_multiset_a ) ).
% Multiset.diff_cancel
thf(fact_786_diff__empty,axiom,
! [M2: multiset_a] :
( ( ( minus_3765977307040488491iset_a @ M2 @ zero_zero_multiset_a )
= M2 )
& ( ( minus_3765977307040488491iset_a @ zero_zero_multiset_a @ M2 )
= zero_zero_multiset_a ) ) ).
% diff_empty
thf(fact_787_empty__neutral_I2_J,axiom,
! [X3: multiset_a] :
( ( plus_plus_multiset_a @ X3 @ zero_zero_multiset_a )
= X3 ) ).
% empty_neutral(2)
thf(fact_788_empty__neutral_I1_J,axiom,
! [X3: multiset_a] :
( ( plus_plus_multiset_a @ zero_zero_multiset_a @ X3 )
= X3 ) ).
% empty_neutral(1)
thf(fact_789_ring_Oroots_Ocong,axiom,
polynomial_roots_a_b = polynomial_roots_a_b ).
% ring.roots.cong
thf(fact_790_ring_Oroots_Ocong,axiom,
polyno7858422826990252003t_unit = polyno7858422826990252003t_unit ).
% ring.roots.cong
thf(fact_791_sum__eq__empty__iff,axiom,
! [A: set_a,F: a > multiset_a] :
( ( finite_finite_a @ A )
=> ( ( ( groups4808324907802680448iset_a @ F @ A )
= zero_zero_multiset_a )
= ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( F @ X2 )
= zero_zero_multiset_a ) ) ) ) ) ).
% sum_eq_empty_iff
thf(fact_792_sum__eq__empty__iff,axiom,
! [A: set_nat,F: nat > multiset_a] :
( ( finite_finite_nat @ A )
=> ( ( ( groups1580436272196575032iset_a @ F @ A )
= zero_zero_multiset_a )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( F @ X2 )
= zero_zero_multiset_a ) ) ) ) ) ).
% sum_eq_empty_iff
thf(fact_793_sum__eq__empty__iff,axiom,
! [A: set_list_a,F: list_a > multiset_a] :
( ( finite_finite_list_a @ A )
=> ( ( ( groups2539338179767937786iset_a @ F @ A )
= zero_zero_multiset_a )
= ( ! [X2: list_a] :
( ( member_list_a @ X2 @ A )
=> ( ( F @ X2 )
= zero_zero_multiset_a ) ) ) ) ) ).
% sum_eq_empty_iff
thf(fact_794_sum__eq__empty__iff,axiom,
! [A: set_int,F: int > multiset_a] :
( ( finite_finite_int @ A )
=> ( ( ( groups3457364905213935068iset_a @ F @ A )
= zero_zero_multiset_a )
= ( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( ( F @ X2 )
= zero_zero_multiset_a ) ) ) ) ) ).
% sum_eq_empty_iff
thf(fact_795_domain_Odegree__zero__imp__empty__roots,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_s2403821588304063868list_a @ P ) @ one_one_nat )
= zero_zero_nat )
=> ( ( polyno3707469075594375645t_unit @ R @ P )
= zero_z1542645121299710087list_a ) ) ) ) ).
% domain.degree_zero_imp_empty_roots
thf(fact_796_domain_Odegree__zero__imp__empty__roots,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a] :
( ( domain_a_b @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= zero_zero_nat )
=> ( ( polynomial_roots_a_b @ R @ P )
= zero_zero_multiset_a ) ) ) ) ).
% domain.degree_zero_imp_empty_roots
thf(fact_797_domain_Odegree__zero__imp__empty__roots,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat )
= zero_zero_nat )
=> ( ( polyno7858422826990252003t_unit @ R @ P )
= zero_z4454100511807792257list_a ) ) ) ) ).
% domain.degree_zero_imp_empty_roots
thf(fact_798_diff__size__le__size__Diff,axiom,
! [M2: multiset_list_a,M4: multiset_list_a] : ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s2335926164413107382list_a @ M2 ) @ ( size_s2335926164413107382list_a @ M4 ) ) @ ( size_s2335926164413107382list_a @ ( minus_7431248565939055793list_a @ M2 @ M4 ) ) ) ).
% diff_size_le_size_Diff
thf(fact_799_diff__size__le__size__Diff,axiom,
! [M2: multiset_a,M4: multiset_a] : ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_multiset_a @ M2 ) @ ( size_size_multiset_a @ M4 ) ) @ ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ M2 @ M4 ) ) ) ).
% diff_size_le_size_Diff
thf(fact_800_x_Osplitted__def,axiom,
! [P: list_list_a] :
( ( polyno6259083269128200473t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
= ( ( size_s2335926164413107382list_a @ ( polyno7858422826990252003t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) )
= ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat ) ) ) ).
% x.splitted_def
thf(fact_801_pirreducible__roots,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 )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
!= one_one_nat )
=> ( ( polynomial_roots_a_b @ r @ P )
= zero_zero_multiset_a ) ) ) ) ).
% pirreducible_roots
thf(fact_802_x_Oa__lcos__mult__one,axiom,
! [M2: set_list_a] :
( ( ord_le8861187494160871172list_a @ M2 @ ( 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 ) ) ) @ M2 )
= M2 ) ) ).
% x.a_lcos_mult_one
thf(fact_803_x_Oline__extension__smult__closed,axiom,
! [K2: set_list_a,E: set_list_a,A2: list_a,K: list_a,U: list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ! [K4: list_a,V2: list_a] :
( ( member_list_a @ K4 @ K2 )
=> ( ( member_list_a @ V2 @ E )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 @ V2 ) @ E ) ) )
=> ( ( ord_le8861187494160871172list_a @ E @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A2 @ ( 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 @ A2 @ 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 @ A2 @ E ) ) ) ) ) ) ) ) ).
% x.line_extension_smult_closed
thf(fact_804_x_Oee__length,axiom,
! [As: list_list_a,Bs: list_list_a] :
( ( essent703981920984620806t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ As @ Bs )
=> ( ( size_s349497388124573686list_a @ As )
= ( size_s349497388124573686list_a @ Bs ) ) ) ).
% x.ee_length
thf(fact_805_x_Oup__mult__closed,axiom,
! [P: nat > list_a,Q: nat > list_a] :
( ( member_nat_list_a @ P @ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_nat_list_a @ Q @ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_nat_list_a
@ ^ [N2: nat] :
( finsum3497512462216549110it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ^ [I: nat] : ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( P @ I ) @ ( Q @ ( minus_minus_nat @ N2 @ I ) ) )
@ ( set_ord_atMost_nat @ N2 ) )
@ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.up_mult_closed
thf(fact_806_x_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 ) ) ) ) ).
% x.subring_props(7)
thf(fact_807_x_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 ) ) ).
% x.subring_props(2)
thf(fact_808_x_Oadd_Ofinprod__one__eqI,axiom,
! [A: set_list_a,F: list_a > list_a] :
( ! [X: list_a] :
( ( member_list_a @ X @ A )
=> ( ( F @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( finsum8721804980556663006list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.add.finprod_one_eqI
thf(fact_809_x_Oadd_Ofinprod__one__eqI,axiom,
! [A: set_nat_list_a,F: ( nat > list_a ) > list_a] :
( ! [X: nat > list_a] :
( ( member_nat_list_a @ X @ A )
=> ( ( F @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( finsum4426778018909949125list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.add.finprod_one_eqI
thf(fact_810_x_Oadd_Ofinprod__one__eqI,axiom,
! [A: set_nat_a,F: ( nat > a ) > list_a] :
( ! [X: nat > a] :
( ( member_nat_a @ X @ A )
=> ( ( F @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( finsum7881878320310621759_nat_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.add.finprod_one_eqI
thf(fact_811_x_Oadd_Ofinprod__one__eqI,axiom,
! [A: set_a,F: a > list_a] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( ( F @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.add.finprod_one_eqI
thf(fact_812_x_Oadd_Ofinprod__one__eqI,axiom,
! [A: set_nat,F: nat > list_a] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ( F @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( finsum3497512462216549110it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.add.finprod_one_eqI
thf(fact_813_x_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 ) ) ) ) ).
% x.subring_props(6)
thf(fact_814_x_Oadd_Oinv__comm,axiom,
! [X3: list_a,Y: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Y )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X3 @ ( 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 @ X3 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.add.inv_comm
thf(fact_815_x_Oadd_Ol__inv__ex,axiom,
! [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ? [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 @ X3 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.add.l_inv_ex
thf(fact_816_x_Oadd_Oone__unique,axiom,
! [U: list_a] :
( ( member_list_a @ U @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [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 ) ) @ U @ X )
= X ) )
=> ( U
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.add.one_unique
thf(fact_817_x_Oadd_Or__inv__ex,axiom,
! [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ? [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 ) ) @ X3 @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.add.r_inv_ex
thf(fact_818_x_Ominus__unique,axiom,
! [Y: list_a,X3: list_a,Y6: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X3 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Y6 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y6 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( Y = Y6 ) ) ) ) ) ) ).
% x.minus_unique
thf(fact_819_x_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 ) ) ) ) ) ).
% x.subring_props(1)
thf(fact_820_x_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 ) ) ) ).
% x.zero_closed
thf(fact_821_x_Oadd_Oint__pow__one,axiom,
! [Z: int] :
( ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Z @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.add.int_pow_one
thf(fact_822_x_Ofinsum__zero,axiom,
! [A: set_nat] :
( ( finsum3497512462216549110it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ^ [I: nat] : ( 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 ) ) ) ) ).
% x.finsum_zero
thf(fact_823_x_Oadd_Ol__cancel__one,axiom,
! [X3: list_a,A2: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ A2 )
= X3 )
= ( A2
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.add.l_cancel_one
thf(fact_824_x_Oadd_Ol__cancel__one_H,axiom,
! [X3: list_a,A2: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( X3
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ A2 ) )
= ( A2
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.add.l_cancel_one'
thf(fact_825_x_Oadd_Or__cancel__one,axiom,
! [X3: list_a,A2: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ X3 )
= X3 )
= ( A2
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.add.r_cancel_one
thf(fact_826_x_Oadd_Or__cancel__one_H,axiom,
! [X3: list_a,A2: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( X3
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ X3 ) )
= ( A2
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.add.r_cancel_one'
thf(fact_827_x_Ol__zero,axiom,
! [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 ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X3 )
= X3 ) ) ).
% x.l_zero
thf(fact_828_x_Or__zero,axiom,
! [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 @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= X3 ) ) ).
% x.r_zero
thf(fact_829_x_Ol__null,axiom,
! [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X3 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.l_null
thf(fact_830_x_Or__null,axiom,
! [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.r_null
thf(fact_831_x_Ofinsum__infinite,axiom,
! [A: set_a,F: a > list_a] :
( ~ ( finite_finite_a @ A )
=> ( ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.finsum_infinite
thf(fact_832_x_Ofinsum__infinite,axiom,
! [A: set_nat,F: nat > list_a] :
( ~ ( finite_finite_nat @ A )
=> ( ( finsum3497512462216549110it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.finsum_infinite
thf(fact_833_x_Ofinsum__infinite,axiom,
! [A: set_list_a,F: list_a > list_a] :
( ~ ( finite_finite_list_a @ A )
=> ( ( finsum8721804980556663006list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.finsum_infinite
thf(fact_834_x_Ofinsum__infinite,axiom,
! [A: set_int,F: int > list_a] :
( ~ ( finite_finite_int @ A )
=> ( ( finsum3495021991707498834it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.finsum_infinite
thf(fact_835_x_Or__right__minus__eq,axiom,
! [A2: list_a,B: list_a] :
( ( member_list_a @ A2 @ ( 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 ) ) @ A2 @ B )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( A2 = B ) ) ) ) ).
% x.r_right_minus_eq
thf(fact_836_domain_Odegree__one__imp__pirreducible,axiom,
! [R: partia2956882679547061052t_unit,K2: set_list_list_a,P: list_list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( subfie4546268998243038636t_unit @ K2 @ R )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K2 ) ) )
=> ( ( ( minus_minus_nat @ ( size_s2403821588304063868list_a @ P ) @ one_one_nat )
= one_one_nat )
=> ( ring_r5224476855413033410t_unit @ ( univ_p2250591967980070728t_unit @ R @ K2 ) @ P ) ) ) ) ) ).
% domain.degree_one_imp_pirreducible
thf(fact_837_domain_Odegree__one__imp__pirreducible,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,P: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K2 ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= one_one_nat )
=> ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R @ K2 ) @ P ) ) ) ) ) ).
% domain.degree_one_imp_pirreducible
thf(fact_838_domain_Odegree__one__imp__pirreducible,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K2 ) ) )
=> ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat )
= one_one_nat )
=> ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R @ K2 ) @ P ) ) ) ) ) ).
% domain.degree_one_imp_pirreducible
thf(fact_839_domain_Ofinprod__zero__iff,axiom,
! [R: partia2175431115845679010xt_a_b,A: set_a,F: a > a] :
( ( domain_a_b @ R )
=> ( ( finite_finite_a @ A )
=> ( ! [A3: a] :
( ( member_a @ A3 @ A )
=> ( member_a @ ( F @ A3 ) @ ( partia707051561876973205xt_a_b @ R ) ) )
=> ( ( ( finpro205304725090349623_a_b_a @ R @ F @ A )
= ( zero_a_b @ R ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A )
& ( ( F @ X2 )
= ( zero_a_b @ R ) ) ) ) ) ) ) ) ).
% domain.finprod_zero_iff
thf(fact_840_domain_Ofinprod__zero__iff,axiom,
! [R: partia2175431115845679010xt_a_b,A: set_int,F: int > a] :
( ( domain_a_b @ R )
=> ( ( finite_finite_int @ A )
=> ( ! [A3: int] :
( ( member_int @ A3 @ A )
=> ( member_a @ ( F @ A3 ) @ ( partia707051561876973205xt_a_b @ R ) ) )
=> ( ( ( finpro1277544800017374899_b_int @ R @ F @ A )
= ( zero_a_b @ R ) )
= ( ? [X2: int] :
( ( member_int @ X2 @ A )
& ( ( F @ X2 )
= ( zero_a_b @ R ) ) ) ) ) ) ) ) ).
% domain.finprod_zero_iff
thf(fact_841_domain_Ofinprod__zero__iff,axiom,
! [R: partia2175431115845679010xt_a_b,A: set_nat,F: nat > a] :
( ( domain_a_b @ R )
=> ( ( finite_finite_nat @ A )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A )
=> ( member_a @ ( F @ A3 ) @ ( partia707051561876973205xt_a_b @ R ) ) )
=> ( ( ( finpro1280035270526425175_b_nat @ R @ F @ A )
= ( zero_a_b @ R ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ( F @ X2 )
= ( zero_a_b @ R ) ) ) ) ) ) ) ) ).
% domain.finprod_zero_iff
thf(fact_842_domain_Ofinprod__zero__iff,axiom,
! [R: partia2175431115845679010xt_a_b,A: set_list_a,F: list_a > a] :
( ( domain_a_b @ R )
=> ( ( finite_finite_list_a @ A )
=> ( ! [A3: list_a] :
( ( member_list_a @ A3 @ A )
=> ( member_a @ ( F @ A3 ) @ ( partia707051561876973205xt_a_b @ R ) ) )
=> ( ( ( finpro6052973074229812797list_a @ R @ F @ A )
= ( zero_a_b @ R ) )
= ( ? [X2: list_a] :
( ( member_list_a @ X2 @ A )
& ( ( F @ X2 )
= ( zero_a_b @ R ) ) ) ) ) ) ) ) ).
% domain.finprod_zero_iff
thf(fact_843_domain_Ofinprod__zero__iff,axiom,
! [R: partia2670972154091845814t_unit,A: set_int,F: int > list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( finite_finite_int @ A )
=> ( ! [A3: int] :
( ( member_int @ A3 @ A )
=> ( member_list_a @ ( F @ A3 ) @ ( partia5361259788508890537t_unit @ R ) ) )
=> ( ( ( finpro1915614264500035905it_int @ R @ F @ A )
= ( zero_l4142658623432671053t_unit @ R ) )
= ( ? [X2: int] :
( ( member_int @ X2 @ A )
& ( ( F @ X2 )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ) ) ) ).
% domain.finprod_zero_iff
thf(fact_844_domain_Ofinprod__zero__iff,axiom,
! [R: partia2670972154091845814t_unit,A: set_a,F: a > list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( finite_finite_a @ A )
=> ( ! [A3: a] :
( ( member_a @ A3 @ A )
=> ( member_list_a @ ( F @ A3 ) @ ( partia5361259788508890537t_unit @ R ) ) )
=> ( ( ( finpro4329226410377213737unit_a @ R @ F @ A )
= ( zero_l4142658623432671053t_unit @ R ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A )
& ( ( F @ X2 )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ) ) ) ).
% domain.finprod_zero_iff
thf(fact_845_domain_Ofinprod__zero__iff,axiom,
! [R: partia2670972154091845814t_unit,A: set_nat,F: nat > list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( finite_finite_nat @ A )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A )
=> ( member_list_a @ ( F @ A3 ) @ ( partia5361259788508890537t_unit @ R ) ) )
=> ( ( ( finpro1918104735009086181it_nat @ R @ F @ A )
= ( zero_l4142658623432671053t_unit @ R ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ( F @ X2 )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ) ) ) ).
% domain.finprod_zero_iff
thf(fact_846_domain_Ofinprod__zero__iff,axiom,
! [R: partia2175431115845679010xt_a_b,A: set_nat_a,F: ( nat > a ) > a] :
( ( domain_a_b @ R )
=> ( ( finite_finite_nat_a @ A )
=> ( ! [A3: nat > a] :
( ( member_nat_a @ A3 @ A )
=> ( member_a @ ( F @ A3 ) @ ( partia707051561876973205xt_a_b @ R ) ) )
=> ( ( ( finpro5839458686994656414_nat_a @ R @ F @ A )
= ( zero_a_b @ R ) )
= ( ? [X2: nat > a] :
( ( member_nat_a @ X2 @ A )
& ( ( F @ X2 )
= ( zero_a_b @ R ) ) ) ) ) ) ) ) ).
% domain.finprod_zero_iff
thf(fact_847_domain_Ofinprod__zero__iff,axiom,
! [R: partia2670972154091845814t_unit,A: set_list_a,F: list_a > list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( finite_finite_list_a @ A )
=> ( ! [A3: list_a] :
( ( member_list_a @ A3 @ A )
=> ( member_list_a @ ( F @ A3 ) @ ( partia5361259788508890537t_unit @ R ) ) )
=> ( ( ( finpro738134188688310831list_a @ R @ F @ A )
= ( zero_l4142658623432671053t_unit @ R ) )
= ( ? [X2: list_a] :
( ( member_list_a @ X2 @ A )
& ( ( F @ X2 )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ) ) ) ).
% domain.finprod_zero_iff
thf(fact_848_domain_Ofinprod__zero__iff,axiom,
! [R: partia2956882679547061052t_unit,A: set_a,F: a > list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( finite_finite_a @ A )
=> ( ! [A3: a] :
( ( member_a @ A3 @ A )
=> ( member_list_list_a @ ( F @ A3 ) @ ( partia2464479390973590831t_unit @ R ) ) )
=> ( ( ( finpro5596966875920909993unit_a @ R @ F @ A )
= ( zero_l347298301471573063t_unit @ R ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A )
& ( ( F @ X2 )
= ( zero_l347298301471573063t_unit @ R ) ) ) ) ) ) ) ) ).
% domain.finprod_zero_iff
thf(fact_849_domain_Opirreducible__roots,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( ring_r5224476855413033410t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ P )
=> ( ( ( minus_minus_nat @ ( size_s2403821588304063868list_a @ P ) @ one_one_nat )
!= one_one_nat )
=> ( ( polyno3707469075594375645t_unit @ R @ P )
= zero_z1542645121299710087list_a ) ) ) ) ) ).
% domain.pirreducible_roots
thf(fact_850_domain_Opirreducible__roots,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a] :
( ( domain_a_b @ R )
=> ( ( 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 )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
!= one_one_nat )
=> ( ( polynomial_roots_a_b @ R @ P )
= zero_zero_multiset_a ) ) ) ) ) ).
% domain.pirreducible_roots
thf(fact_851_domain_Opirreducible__roots,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ P )
=> ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat )
!= one_one_nat )
=> ( ( polyno7858422826990252003t_unit @ R @ P )
= zero_z4454100511807792257list_a ) ) ) ) ) ).
% domain.pirreducible_roots
thf(fact_852_x_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 ) ) ).
% x.bound_upD
thf(fact_853_x_Opirreducible__degree,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_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ P )
=> ( ord_less_eq_nat @ one_one_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat ) ) ) ) ) ).
% x.pirreducible_degree
thf(fact_854_x_Opoly__of__const__over__subfield,axiom,
! [K2: set_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( image_8260866953997875467list_a @ ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ K2 )
= ( collect_list_list_a
@ ^ [P3: list_list_a] :
( ( member_list_list_a @ P3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
& ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P3 ) @ one_one_nat )
= zero_zero_nat ) ) ) ) ) ).
% x.poly_of_const_over_subfield
thf(fact_855_atMost__subset__iff,axiom,
! [X3: set_a,Y: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_ord_atMost_set_a @ X3 ) @ ( set_ord_atMost_set_a @ Y ) )
= ( ord_less_eq_set_a @ X3 @ Y ) ) ).
% atMost_subset_iff
thf(fact_856_atMost__subset__iff,axiom,
! [X3: set_list_a,Y: set_list_a] :
( ( ord_le8877086941679407844list_a @ ( set_or6279072120763780779list_a @ X3 ) @ ( set_or6279072120763780779list_a @ Y ) )
= ( ord_le8861187494160871172list_a @ X3 @ Y ) ) ).
% atMost_subset_iff
thf(fact_857_atMost__subset__iff,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ X3 ) @ ( set_ord_atMost_int @ Y ) )
= ( ord_less_eq_int @ X3 @ Y ) ) ).
% atMost_subset_iff
thf(fact_858_atMost__subset__iff,axiom,
! [X3: set_nat,Y: set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_or4236626031148496127et_nat @ X3 ) @ ( set_or4236626031148496127et_nat @ Y ) )
= ( ord_less_eq_set_nat @ X3 @ Y ) ) ).
% atMost_subset_iff
thf(fact_859_atMost__subset__iff,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X3 ) @ ( set_ord_atMost_nat @ Y ) )
= ( ord_less_eq_nat @ X3 @ Y ) ) ).
% atMost_subset_iff
thf(fact_860_x_Osubalbegra__incl__imp__finite__dimension,axiom,
! [K2: set_list_a,E: set_list_a,V: 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 @ V @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ V @ E )
=> ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ V ) ) ) ) ) ).
% x.subalbegra_incl_imp_finite_dimension
thf(fact_861_pirreducible__degree,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_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K2 ) @ P )
=> ( ord_less_eq_nat @ one_one_nat @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) ) ) ) ) ).
% pirreducible_degree
thf(fact_862_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_863_finprod__zero__iff,axiom,
! [A: set_nat_list_a,F: ( nat > list_a ) > a] :
( ( finite7630042315537210004list_a @ A )
=> ( ! [A3: nat > list_a] :
( ( member_nat_list_a @ A3 @ A )
=> ( member_a @ ( F @ A3 ) @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( finpro4838020199848830884list_a @ r @ F @ A )
= ( zero_a_b @ r ) )
= ( ? [X2: nat > list_a] :
( ( member_nat_list_a @ X2 @ A )
& ( ( F @ X2 )
= ( zero_a_b @ r ) ) ) ) ) ) ) ).
% finprod_zero_iff
thf(fact_864_finprod__zero__iff,axiom,
! [A: set_nat_a,F: ( nat > a ) > a] :
( ( finite_finite_nat_a @ A )
=> ( ! [A3: nat > a] :
( ( member_nat_a @ A3 @ A )
=> ( member_a @ ( F @ A3 ) @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( finpro5839458686994656414_nat_a @ r @ F @ A )
= ( zero_a_b @ r ) )
= ( ? [X2: nat > a] :
( ( member_nat_a @ X2 @ A )
& ( ( F @ X2 )
= ( zero_a_b @ r ) ) ) ) ) ) ) ).
% finprod_zero_iff
thf(fact_865_finprod__zero__iff,axiom,
! [A: set_a,F: a > a] :
( ( finite_finite_a @ A )
=> ( ! [A3: a] :
( ( member_a @ A3 @ A )
=> ( member_a @ ( F @ A3 ) @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( finpro205304725090349623_a_b_a @ r @ F @ A )
= ( zero_a_b @ r ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A )
& ( ( F @ X2 )
= ( zero_a_b @ r ) ) ) ) ) ) ) ).
% finprod_zero_iff
thf(fact_866_finprod__zero__iff,axiom,
! [A: set_list_a,F: list_a > a] :
( ( finite_finite_list_a @ A )
=> ( ! [A3: list_a] :
( ( member_list_a @ A3 @ A )
=> ( member_a @ ( F @ A3 ) @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( finpro6052973074229812797list_a @ r @ F @ A )
= ( zero_a_b @ r ) )
= ( ? [X2: list_a] :
( ( member_list_a @ X2 @ A )
& ( ( F @ X2 )
= ( zero_a_b @ r ) ) ) ) ) ) ) ).
% finprod_zero_iff
thf(fact_867_finprod__zero__iff,axiom,
! [A: set_int,F: int > a] :
( ( finite_finite_int @ A )
=> ( ! [A3: int] :
( ( member_int @ A3 @ A )
=> ( member_a @ ( F @ A3 ) @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( finpro1277544800017374899_b_int @ r @ F @ A )
= ( zero_a_b @ r ) )
= ( ? [X2: int] :
( ( member_int @ X2 @ A )
& ( ( F @ X2 )
= ( zero_a_b @ r ) ) ) ) ) ) ) ).
% finprod_zero_iff
thf(fact_868_finprod__zero__iff,axiom,
! [A: set_nat,F: nat > a] :
( ( finite_finite_nat @ A )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A )
=> ( member_a @ ( F @ A3 ) @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( finpro1280035270526425175_b_nat @ r @ F @ A )
= ( zero_a_b @ r ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ( F @ X2 )
= ( zero_a_b @ r ) ) ) ) ) ) ) ).
% finprod_zero_iff
thf(fact_869_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_870_x_Otelescopic__base__dim_I1_J,axiom,
! [K2: set_list_a,F2: set_list_a,E: set_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( subfie1779122896746047282t_unit @ F2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ F2 )
=> ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F2 @ E )
=> ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ E ) ) ) ) ) ).
% x.telescopic_base_dim(1)
thf(fact_871_x_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 ) ) ) ) ) ).
% x.finite_dimension_imp_subalgebra
thf(fact_872_finite__imageI,axiom,
! [F2: set_a,H: a > a] :
( ( finite_finite_a @ F2 )
=> ( finite_finite_a @ ( image_a_a @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_873_finite__imageI,axiom,
! [F2: set_a,H: a > nat] :
( ( finite_finite_a @ F2 )
=> ( finite_finite_nat @ ( image_a_nat @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_874_finite__imageI,axiom,
! [F2: set_a,H: a > int] :
( ( finite_finite_a @ F2 )
=> ( finite_finite_int @ ( image_a_int @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_875_finite__imageI,axiom,
! [F2: set_nat,H: nat > a] :
( ( finite_finite_nat @ F2 )
=> ( finite_finite_a @ ( image_nat_a @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_876_finite__imageI,axiom,
! [F2: set_nat,H: nat > nat] :
( ( finite_finite_nat @ F2 )
=> ( finite_finite_nat @ ( image_nat_nat @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_877_finite__imageI,axiom,
! [F2: set_nat,H: nat > int] :
( ( finite_finite_nat @ F2 )
=> ( finite_finite_int @ ( image_nat_int @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_878_finite__imageI,axiom,
! [F2: set_int,H: int > a] :
( ( finite_finite_int @ F2 )
=> ( finite_finite_a @ ( image_int_a @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_879_finite__imageI,axiom,
! [F2: set_int,H: int > nat] :
( ( finite_finite_int @ F2 )
=> ( finite_finite_nat @ ( image_int_nat @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_880_finite__imageI,axiom,
! [F2: set_int,H: int > int] :
( ( finite_finite_int @ F2 )
=> ( finite_finite_int @ ( image_int_int @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_881_finite__imageI,axiom,
! [F2: set_a,H: a > list_a] :
( ( finite_finite_a @ F2 )
=> ( finite_finite_list_a @ ( image_a_list_a @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_882_atMost__iff,axiom,
! [I2: set_a,K: set_a] :
( ( member_set_a @ I2 @ ( set_ord_atMost_set_a @ K ) )
= ( ord_less_eq_set_a @ I2 @ K ) ) ).
% atMost_iff
thf(fact_883_atMost__iff,axiom,
! [I2: set_list_a,K: set_list_a] :
( ( member_set_list_a @ I2 @ ( set_or6279072120763780779list_a @ K ) )
= ( ord_le8861187494160871172list_a @ I2 @ K ) ) ).
% atMost_iff
thf(fact_884_atMost__iff,axiom,
! [I2: int,K: int] :
( ( member_int @ I2 @ ( set_ord_atMost_int @ K ) )
= ( ord_less_eq_int @ I2 @ K ) ) ).
% atMost_iff
thf(fact_885_atMost__iff,axiom,
! [I2: set_nat,K: set_nat] :
( ( member_set_nat @ I2 @ ( set_or4236626031148496127et_nat @ K ) )
= ( ord_less_eq_set_nat @ I2 @ K ) ) ).
% atMost_iff
thf(fact_886_atMost__iff,axiom,
! [I2: nat,K: nat] :
( ( member_nat @ I2 @ ( set_ord_atMost_nat @ K ) )
= ( ord_less_eq_nat @ I2 @ K ) ) ).
% atMost_iff
thf(fact_887_degree__one__imp__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 ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= one_one_nat )
=> ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K2 ) @ P ) ) ) ) ).
% degree_one_imp_pirreducible
thf(fact_888_finite__atMost,axiom,
! [K: nat] : ( finite_finite_nat @ ( set_ord_atMost_nat @ K ) ) ).
% finite_atMost
thf(fact_889_x_Ouniv__poly__subfield__of__consts,axiom,
! [K2: set_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( subfie4546268998243038636t_unit @ ( image_8260866953997875467list_a @ ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ K2 ) @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) ) ).
% x.univ_poly_subfield_of_consts
thf(fact_890_zero__closed,axiom,
member_a @ ( zero_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% zero_closed
thf(fact_891_image__add__atMost,axiom,
! [C: int,A2: int] :
( ( image_int_int @ ( plus_plus_int @ C ) @ ( set_ord_atMost_int @ A2 ) )
= ( set_ord_atMost_int @ ( plus_plus_int @ C @ A2 ) ) ) ).
% image_add_atMost
thf(fact_892_univ__poly__is__euclidean,axiom,
! [K2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ring_e7478897652244013592t_unit @ ( univ_poly_a_b @ r @ K2 )
@ ^ [P3: list_a] : ( minus_minus_nat @ ( size_size_list_a @ P3 ) @ one_one_nat ) ) ) ).
% univ_poly_is_euclidean
thf(fact_893_all__subset__image,axiom,
! [F: list_a > list_list_a,A: set_list_a,P2: set_list_list_a > $o] :
( ( ! [B4: set_list_list_a] :
( ( ord_le8488217952732425610list_a @ B4 @ ( image_8260866953997875467list_a @ F @ A ) )
=> ( P2 @ B4 ) ) )
= ( ! [B4: set_list_a] :
( ( ord_le8861187494160871172list_a @ B4 @ A )
=> ( P2 @ ( image_8260866953997875467list_a @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_894_all__subset__image,axiom,
! [F: a > a,A: set_a,P2: set_a > $o] :
( ( ! [B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ ( image_a_a @ F @ A ) )
=> ( P2 @ B4 ) ) )
= ( ! [B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( P2 @ ( image_a_a @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_895_all__subset__image,axiom,
! [F: list_a > a,A: set_list_a,P2: set_a > $o] :
( ( ! [B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ ( image_list_a_a @ F @ A ) )
=> ( P2 @ B4 ) ) )
= ( ! [B4: set_list_a] :
( ( ord_le8861187494160871172list_a @ B4 @ A )
=> ( P2 @ ( image_list_a_a @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_896_all__subset__image,axiom,
! [F: nat > a,A: set_nat,P2: set_a > $o] :
( ( ! [B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ ( image_nat_a @ F @ A ) )
=> ( P2 @ B4 ) ) )
= ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A )
=> ( P2 @ ( image_nat_a @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_897_all__subset__image,axiom,
! [F: a > list_a,A: set_a,P2: set_list_a > $o] :
( ( ! [B4: set_list_a] :
( ( ord_le8861187494160871172list_a @ B4 @ ( image_a_list_a @ F @ A ) )
=> ( P2 @ B4 ) ) )
= ( ! [B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( P2 @ ( image_a_list_a @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_898_all__subset__image,axiom,
! [F: list_a > list_a,A: set_list_a,P2: set_list_a > $o] :
( ( ! [B4: set_list_a] :
( ( ord_le8861187494160871172list_a @ B4 @ ( image_list_a_list_a @ F @ A ) )
=> ( P2 @ B4 ) ) )
= ( ! [B4: set_list_a] :
( ( ord_le8861187494160871172list_a @ B4 @ A )
=> ( P2 @ ( image_list_a_list_a @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_899_all__subset__image,axiom,
! [F: nat > list_a,A: set_nat,P2: set_list_a > $o] :
( ( ! [B4: set_list_a] :
( ( ord_le8861187494160871172list_a @ B4 @ ( image_nat_list_a @ F @ A ) )
=> ( P2 @ B4 ) ) )
= ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A )
=> ( P2 @ ( image_nat_list_a @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_900_all__subset__image,axiom,
! [F: a > nat,A: set_a,P2: set_nat > $o] :
( ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ ( image_a_nat @ F @ A ) )
=> ( P2 @ B4 ) ) )
= ( ! [B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( P2 @ ( image_a_nat @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_901_all__subset__image,axiom,
! [F: list_a > nat,A: set_list_a,P2: set_nat > $o] :
( ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ ( image_list_a_nat @ F @ A ) )
=> ( P2 @ B4 ) ) )
= ( ! [B4: set_list_a] :
( ( ord_le8861187494160871172list_a @ B4 @ A )
=> ( P2 @ ( image_list_a_nat @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_902_all__subset__image,axiom,
! [F: nat > nat,A: set_nat,P2: set_nat > $o] :
( ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F @ A ) )
=> ( P2 @ B4 ) ) )
= ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A )
=> ( P2 @ ( image_nat_nat @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_903_pigeonhole__infinite,axiom,
! [A: set_a,F: a > a] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_finite_a @ ( image_a_a @ F @ A ) )
=> ? [X: a] :
( ( member_a @ X @ A )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A )
& ( ( F @ A4 )
= ( F @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_904_pigeonhole__infinite,axiom,
! [A: set_a,F: a > nat] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_finite_nat @ ( image_a_nat @ F @ A ) )
=> ? [X: a] :
( ( member_a @ X @ A )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A )
& ( ( F @ A4 )
= ( F @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_905_pigeonhole__infinite,axiom,
! [A: set_a,F: a > int] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_finite_int @ ( image_a_int @ F @ A ) )
=> ? [X: a] :
( ( member_a @ X @ A )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A )
& ( ( F @ A4 )
= ( F @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_906_pigeonhole__infinite,axiom,
! [A: set_nat,F: nat > a] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_a @ ( image_nat_a @ F @ A ) )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A )
& ( ( F @ A4 )
= ( F @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_907_pigeonhole__infinite,axiom,
! [A: set_nat,F: nat > nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ ( image_nat_nat @ F @ A ) )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A )
& ( ( F @ A4 )
= ( F @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_908_pigeonhole__infinite,axiom,
! [A: set_nat,F: nat > int] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_int @ ( image_nat_int @ F @ A ) )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A )
& ( ( F @ A4 )
= ( F @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_909_pigeonhole__infinite,axiom,
! [A: set_int,F: int > a] :
( ~ ( finite_finite_int @ A )
=> ( ( finite_finite_a @ ( image_int_a @ F @ A ) )
=> ? [X: int] :
( ( member_int @ X @ A )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A4: int] :
( ( member_int @ A4 @ A )
& ( ( F @ A4 )
= ( F @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_910_pigeonhole__infinite,axiom,
! [A: set_int,F: int > nat] :
( ~ ( finite_finite_int @ A )
=> ( ( finite_finite_nat @ ( image_int_nat @ F @ A ) )
=> ? [X: int] :
( ( member_int @ X @ A )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A4: int] :
( ( member_int @ A4 @ A )
& ( ( F @ A4 )
= ( F @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_911_pigeonhole__infinite,axiom,
! [A: set_int,F: int > int] :
( ~ ( finite_finite_int @ A )
=> ( ( finite_finite_int @ ( image_int_int @ F @ A ) )
=> ? [X: int] :
( ( member_int @ X @ A )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A4: int] :
( ( member_int @ A4 @ A )
& ( ( F @ A4 )
= ( F @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_912_pigeonhole__infinite,axiom,
! [A: set_a,F: a > list_a] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_finite_list_a @ ( image_a_list_a @ F @ A ) )
=> ? [X: a] :
( ( member_a @ X @ A )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A )
& ( ( F @ A4 )
= ( F @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_913_finite__surj,axiom,
! [A: set_a,B2: set_int,F: a > int] :
( ( finite_finite_a @ A )
=> ( ( ord_less_eq_set_int @ B2 @ ( image_a_int @ F @ A ) )
=> ( finite_finite_int @ B2 ) ) ) ).
% finite_surj
thf(fact_914_finite__surj,axiom,
! [A: set_nat,B2: set_int,F: nat > int] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_int @ B2 @ ( image_nat_int @ F @ A ) )
=> ( finite_finite_int @ B2 ) ) ) ).
% finite_surj
thf(fact_915_finite__surj,axiom,
! [A: set_int,B2: set_int,F: int > int] :
( ( finite_finite_int @ A )
=> ( ( ord_less_eq_set_int @ B2 @ ( image_int_int @ F @ A ) )
=> ( finite_finite_int @ B2 ) ) ) ).
% finite_surj
thf(fact_916_finite__surj,axiom,
! [A: set_a,B2: set_a,F: a > a] :
( ( finite_finite_a @ A )
=> ( ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F @ A ) )
=> ( finite_finite_a @ B2 ) ) ) ).
% finite_surj
thf(fact_917_finite__surj,axiom,
! [A: set_nat,B2: set_a,F: nat > a] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_a @ B2 @ ( image_nat_a @ F @ A ) )
=> ( finite_finite_a @ B2 ) ) ) ).
% finite_surj
thf(fact_918_finite__surj,axiom,
! [A: set_int,B2: set_a,F: int > a] :
( ( finite_finite_int @ A )
=> ( ( ord_less_eq_set_a @ B2 @ ( image_int_a @ F @ A ) )
=> ( finite_finite_a @ B2 ) ) ) ).
% finite_surj
thf(fact_919_finite__surj,axiom,
! [A: set_a,B2: set_nat,F: a > nat] :
( ( finite_finite_a @ A )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_a_nat @ F @ A ) )
=> ( finite_finite_nat @ B2 ) ) ) ).
% finite_surj
thf(fact_920_finite__surj,axiom,
! [A: set_nat,B2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A ) )
=> ( finite_finite_nat @ B2 ) ) ) ).
% finite_surj
thf(fact_921_finite__surj,axiom,
! [A: set_int,B2: set_nat,F: int > nat] :
( ( finite_finite_int @ A )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_int_nat @ F @ A ) )
=> ( finite_finite_nat @ B2 ) ) ) ).
% finite_surj
thf(fact_922_finite__surj,axiom,
! [A: set_list_a,B2: set_int,F: list_a > int] :
( ( finite_finite_list_a @ A )
=> ( ( ord_less_eq_set_int @ B2 @ ( image_list_a_int @ F @ A ) )
=> ( finite_finite_int @ B2 ) ) ) ).
% finite_surj
thf(fact_923_finite__subset__image,axiom,
! [B2: set_int,F: int > int,A: set_int] :
( ( finite_finite_int @ B2 )
=> ( ( ord_less_eq_set_int @ B2 @ ( image_int_int @ F @ A ) )
=> ? [C5: set_int] :
( ( ord_less_eq_set_int @ C5 @ A )
& ( finite_finite_int @ C5 )
& ( B2
= ( image_int_int @ F @ C5 ) ) ) ) ) ).
% finite_subset_image
thf(fact_924_finite__subset__image,axiom,
! [B2: set_int,F: a > int,A: set_a] :
( ( finite_finite_int @ B2 )
=> ( ( ord_less_eq_set_int @ B2 @ ( image_a_int @ F @ A ) )
=> ? [C5: set_a] :
( ( ord_less_eq_set_a @ C5 @ A )
& ( finite_finite_a @ C5 )
& ( B2
= ( image_a_int @ F @ C5 ) ) ) ) ) ).
% finite_subset_image
thf(fact_925_finite__subset__image,axiom,
! [B2: set_int,F: nat > int,A: set_nat] :
( ( finite_finite_int @ B2 )
=> ( ( ord_less_eq_set_int @ B2 @ ( image_nat_int @ F @ A ) )
=> ? [C5: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ A )
& ( finite_finite_nat @ C5 )
& ( B2
= ( image_nat_int @ F @ C5 ) ) ) ) ) ).
% finite_subset_image
thf(fact_926_finite__subset__image,axiom,
! [B2: set_a,F: int > a,A: set_int] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ ( image_int_a @ F @ A ) )
=> ? [C5: set_int] :
( ( ord_less_eq_set_int @ C5 @ A )
& ( finite_finite_int @ C5 )
& ( B2
= ( image_int_a @ F @ C5 ) ) ) ) ) ).
% finite_subset_image
thf(fact_927_finite__subset__image,axiom,
! [B2: set_a,F: a > a,A: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F @ A ) )
=> ? [C5: set_a] :
( ( ord_less_eq_set_a @ C5 @ A )
& ( finite_finite_a @ C5 )
& ( B2
= ( image_a_a @ F @ C5 ) ) ) ) ) ).
% finite_subset_image
thf(fact_928_finite__subset__image,axiom,
! [B2: set_a,F: nat > a,A: set_nat] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ ( image_nat_a @ F @ A ) )
=> ? [C5: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ A )
& ( finite_finite_nat @ C5 )
& ( B2
= ( image_nat_a @ F @ C5 ) ) ) ) ) ).
% finite_subset_image
thf(fact_929_finite__subset__image,axiom,
! [B2: set_nat,F: int > nat,A: set_int] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_int_nat @ F @ A ) )
=> ? [C5: set_int] :
( ( ord_less_eq_set_int @ C5 @ A )
& ( finite_finite_int @ C5 )
& ( B2
= ( image_int_nat @ F @ C5 ) ) ) ) ) ).
% finite_subset_image
thf(fact_930_finite__subset__image,axiom,
! [B2: set_nat,F: a > nat,A: set_a] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_a_nat @ F @ A ) )
=> ? [C5: set_a] :
( ( ord_less_eq_set_a @ C5 @ A )
& ( finite_finite_a @ C5 )
& ( B2
= ( image_a_nat @ F @ C5 ) ) ) ) ) ).
% finite_subset_image
thf(fact_931_finite__subset__image,axiom,
! [B2: set_nat,F: nat > nat,A: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A ) )
=> ? [C5: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ A )
& ( finite_finite_nat @ C5 )
& ( B2
= ( image_nat_nat @ F @ C5 ) ) ) ) ) ).
% finite_subset_image
thf(fact_932_finite__subset__image,axiom,
! [B2: set_int,F: list_a > int,A: set_list_a] :
( ( finite_finite_int @ B2 )
=> ( ( ord_less_eq_set_int @ B2 @ ( image_list_a_int @ F @ A ) )
=> ? [C5: set_list_a] :
( ( ord_le8861187494160871172list_a @ C5 @ A )
& ( finite_finite_list_a @ C5 )
& ( B2
= ( image_list_a_int @ F @ C5 ) ) ) ) ) ).
% finite_subset_image
thf(fact_933_ex__finite__subset__image,axiom,
! [F: int > int,A: set_int,P2: set_int > $o] :
( ( ? [B4: set_int] :
( ( finite_finite_int @ B4 )
& ( ord_less_eq_set_int @ B4 @ ( image_int_int @ F @ A ) )
& ( P2 @ B4 ) ) )
= ( ? [B4: set_int] :
( ( finite_finite_int @ B4 )
& ( ord_less_eq_set_int @ B4 @ A )
& ( P2 @ ( image_int_int @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_934_ex__finite__subset__image,axiom,
! [F: a > int,A: set_a,P2: set_int > $o] :
( ( ? [B4: set_int] :
( ( finite_finite_int @ B4 )
& ( ord_less_eq_set_int @ B4 @ ( image_a_int @ F @ A ) )
& ( P2 @ B4 ) ) )
= ( ? [B4: set_a] :
( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ A )
& ( P2 @ ( image_a_int @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_935_ex__finite__subset__image,axiom,
! [F: nat > int,A: set_nat,P2: set_int > $o] :
( ( ? [B4: set_int] :
( ( finite_finite_int @ B4 )
& ( ord_less_eq_set_int @ B4 @ ( image_nat_int @ F @ A ) )
& ( P2 @ B4 ) ) )
= ( ? [B4: set_nat] :
( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A )
& ( P2 @ ( image_nat_int @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_936_ex__finite__subset__image,axiom,
! [F: int > a,A: set_int,P2: set_a > $o] :
( ( ? [B4: set_a] :
( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ ( image_int_a @ F @ A ) )
& ( P2 @ B4 ) ) )
= ( ? [B4: set_int] :
( ( finite_finite_int @ B4 )
& ( ord_less_eq_set_int @ B4 @ A )
& ( P2 @ ( image_int_a @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_937_ex__finite__subset__image,axiom,
! [F: a > a,A: set_a,P2: set_a > $o] :
( ( ? [B4: set_a] :
( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ ( image_a_a @ F @ A ) )
& ( P2 @ B4 ) ) )
= ( ? [B4: set_a] :
( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ A )
& ( P2 @ ( image_a_a @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_938_ex__finite__subset__image,axiom,
! [F: nat > a,A: set_nat,P2: set_a > $o] :
( ( ? [B4: set_a] :
( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ ( image_nat_a @ F @ A ) )
& ( P2 @ B4 ) ) )
= ( ? [B4: set_nat] :
( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A )
& ( P2 @ ( image_nat_a @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_939_ex__finite__subset__image,axiom,
! [F: int > nat,A: set_int,P2: set_nat > $o] :
( ( ? [B4: set_nat] :
( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ ( image_int_nat @ F @ A ) )
& ( P2 @ B4 ) ) )
= ( ? [B4: set_int] :
( ( finite_finite_int @ B4 )
& ( ord_less_eq_set_int @ B4 @ A )
& ( P2 @ ( image_int_nat @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_940_ex__finite__subset__image,axiom,
! [F: a > nat,A: set_a,P2: set_nat > $o] :
( ( ? [B4: set_nat] :
( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ ( image_a_nat @ F @ A ) )
& ( P2 @ B4 ) ) )
= ( ? [B4: set_a] :
( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ A )
& ( P2 @ ( image_a_nat @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_941_ex__finite__subset__image,axiom,
! [F: nat > nat,A: set_nat,P2: set_nat > $o] :
( ( ? [B4: set_nat] :
( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F @ A ) )
& ( P2 @ B4 ) ) )
= ( ? [B4: set_nat] :
( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A )
& ( P2 @ ( image_nat_nat @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_942_ex__finite__subset__image,axiom,
! [F: list_a > int,A: set_list_a,P2: set_int > $o] :
( ( ? [B4: set_int] :
( ( finite_finite_int @ B4 )
& ( ord_less_eq_set_int @ B4 @ ( image_list_a_int @ F @ A ) )
& ( P2 @ B4 ) ) )
= ( ? [B4: set_list_a] :
( ( finite_finite_list_a @ B4 )
& ( ord_le8861187494160871172list_a @ B4 @ A )
& ( P2 @ ( image_list_a_int @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_943_all__finite__subset__image,axiom,
! [F: int > int,A: set_int,P2: set_int > $o] :
( ( ! [B4: set_int] :
( ( ( finite_finite_int @ B4 )
& ( ord_less_eq_set_int @ B4 @ ( image_int_int @ F @ A ) ) )
=> ( P2 @ B4 ) ) )
= ( ! [B4: set_int] :
( ( ( finite_finite_int @ B4 )
& ( ord_less_eq_set_int @ B4 @ A ) )
=> ( P2 @ ( image_int_int @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_944_all__finite__subset__image,axiom,
! [F: a > int,A: set_a,P2: set_int > $o] :
( ( ! [B4: set_int] :
( ( ( finite_finite_int @ B4 )
& ( ord_less_eq_set_int @ B4 @ ( image_a_int @ F @ A ) ) )
=> ( P2 @ B4 ) ) )
= ( ! [B4: set_a] :
( ( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ A ) )
=> ( P2 @ ( image_a_int @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_945_all__finite__subset__image,axiom,
! [F: nat > int,A: set_nat,P2: set_int > $o] :
( ( ! [B4: set_int] :
( ( ( finite_finite_int @ B4 )
& ( ord_less_eq_set_int @ B4 @ ( image_nat_int @ F @ A ) ) )
=> ( P2 @ B4 ) ) )
= ( ! [B4: set_nat] :
( ( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A ) )
=> ( P2 @ ( image_nat_int @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_946_all__finite__subset__image,axiom,
! [F: int > a,A: set_int,P2: set_a > $o] :
( ( ! [B4: set_a] :
( ( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ ( image_int_a @ F @ A ) ) )
=> ( P2 @ B4 ) ) )
= ( ! [B4: set_int] :
( ( ( finite_finite_int @ B4 )
& ( ord_less_eq_set_int @ B4 @ A ) )
=> ( P2 @ ( image_int_a @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_947_all__finite__subset__image,axiom,
! [F: a > a,A: set_a,P2: set_a > $o] :
( ( ! [B4: set_a] :
( ( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ ( image_a_a @ F @ A ) ) )
=> ( P2 @ B4 ) ) )
= ( ! [B4: set_a] :
( ( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ A ) )
=> ( P2 @ ( image_a_a @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_948_all__finite__subset__image,axiom,
! [F: nat > a,A: set_nat,P2: set_a > $o] :
( ( ! [B4: set_a] :
( ( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ ( image_nat_a @ F @ A ) ) )
=> ( P2 @ B4 ) ) )
= ( ! [B4: set_nat] :
( ( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A ) )
=> ( P2 @ ( image_nat_a @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_949_all__finite__subset__image,axiom,
! [F: int > nat,A: set_int,P2: set_nat > $o] :
( ( ! [B4: set_nat] :
( ( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ ( image_int_nat @ F @ A ) ) )
=> ( P2 @ B4 ) ) )
= ( ! [B4: set_int] :
( ( ( finite_finite_int @ B4 )
& ( ord_less_eq_set_int @ B4 @ A ) )
=> ( P2 @ ( image_int_nat @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_950_all__finite__subset__image,axiom,
! [F: a > nat,A: set_a,P2: set_nat > $o] :
( ( ! [B4: set_nat] :
( ( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ ( image_a_nat @ F @ A ) ) )
=> ( P2 @ B4 ) ) )
= ( ! [B4: set_a] :
( ( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ A ) )
=> ( P2 @ ( image_a_nat @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_951_all__finite__subset__image,axiom,
! [F: nat > nat,A: set_nat,P2: set_nat > $o] :
( ( ! [B4: set_nat] :
( ( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F @ A ) ) )
=> ( P2 @ B4 ) ) )
= ( ! [B4: set_nat] :
( ( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A ) )
=> ( P2 @ ( image_nat_nat @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_952_all__finite__subset__image,axiom,
! [F: list_a > int,A: set_list_a,P2: set_int > $o] :
( ( ! [B4: set_int] :
( ( ( finite_finite_int @ B4 )
& ( ord_less_eq_set_int @ B4 @ ( image_list_a_int @ F @ A ) ) )
=> ( P2 @ B4 ) ) )
= ( ! [B4: set_list_a] :
( ( ( finite_finite_list_a @ B4 )
& ( ord_le8861187494160871172list_a @ B4 @ A ) )
=> ( P2 @ ( image_list_a_int @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_953_sum_Oimage__gen,axiom,
! [S: set_nat_list_a,H: ( nat > list_a ) > nat,G: ( nat > list_a ) > a] :
( ( finite7630042315537210004list_a @ S )
=> ( ( groups669906071623145473_a_nat @ H @ S )
= ( groups6334556678337121940_a_nat
@ ^ [Y2: a] :
( groups669906071623145473_a_nat @ H
@ ( collect_nat_list_a
@ ^ [X2: nat > list_a] :
( ( member_nat_list_a @ X2 @ S )
& ( ( G @ X2 )
= Y2 ) ) ) )
@ ( image_nat_list_a_a @ G @ S ) ) ) ) ).
% sum.image_gen
thf(fact_954_sum_Oimage__gen,axiom,
! [S: set_nat_a,H: ( nat > a ) > nat,G: ( nat > a ) > a] :
( ( finite_finite_nat_a @ S )
=> ( ( groups154653438316501755_a_nat @ H @ S )
= ( groups6334556678337121940_a_nat
@ ^ [Y2: a] :
( groups154653438316501755_a_nat @ H
@ ( collect_nat_a
@ ^ [X2: nat > a] :
( ( member_nat_a @ X2 @ S )
& ( ( G @ X2 )
= Y2 ) ) ) )
@ ( image_nat_a_a @ G @ S ) ) ) ) ).
% sum.image_gen
thf(fact_955_sum_Oimage__gen,axiom,
! [S: set_list_list_a,H: list_list_a > nat,G: list_list_a > a] :
( ( finite1660835950917165235list_a @ S )
=> ( ( groups7548105480907152928_a_nat @ H @ S )
= ( groups6334556678337121940_a_nat
@ ^ [Y2: a] :
( groups7548105480907152928_a_nat @ H
@ ( collect_list_list_a
@ ^ [X2: list_list_a] :
( ( member_list_list_a @ X2 @ S )
& ( ( G @ X2 )
= Y2 ) ) ) )
@ ( image_list_list_a_a @ G @ S ) ) ) ) ).
% sum.image_gen
thf(fact_956_sum_Oimage__gen,axiom,
! [S: set_nat,H: nat > nat,G: nat > a] :
( ( finite_finite_nat @ S )
=> ( ( groups3542108847815614940at_nat @ H @ S )
= ( groups6334556678337121940_a_nat
@ ^ [Y2: a] :
( groups3542108847815614940at_nat @ H
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ S )
& ( ( G @ X2 )
= Y2 ) ) ) )
@ ( image_nat_a @ G @ S ) ) ) ) ).
% sum.image_gen
thf(fact_957_sum_Oimage__gen,axiom,
! [S: set_list_a,H: list_a > nat,G: list_a > a] :
( ( finite_finite_list_a @ S )
=> ( ( groups5521247699297860762_a_nat @ H @ S )
= ( groups6334556678337121940_a_nat
@ ^ [Y2: a] :
( groups5521247699297860762_a_nat @ H
@ ( collect_list_a
@ ^ [X2: list_a] :
( ( member_list_a @ X2 @ S )
& ( ( G @ X2 )
= Y2 ) ) ) )
@ ( image_list_a_a @ G @ S ) ) ) ) ).
% sum.image_gen
thf(fact_958_sum_Oimage__gen,axiom,
! [S: set_int,H: int > nat,G: int > a] :
( ( finite_finite_int @ S )
=> ( ( groups4541462559716669496nt_nat @ H @ S )
= ( groups6334556678337121940_a_nat
@ ^ [Y2: a] :
( groups4541462559716669496nt_nat @ H
@ ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ S )
& ( ( G @ X2 )
= Y2 ) ) ) )
@ ( image_int_a @ G @ S ) ) ) ) ).
% sum.image_gen
thf(fact_959_sum_Oimage__gen,axiom,
! [S: set_a,H: a > nat,G: a > list_a] :
( ( finite_finite_a @ S )
=> ( ( groups6334556678337121940_a_nat @ H @ S )
= ( groups5521247699297860762_a_nat
@ ^ [Y2: list_a] :
( groups6334556678337121940_a_nat @ H
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ S )
& ( ( G @ X2 )
= Y2 ) ) ) )
@ ( image_a_list_a @ G @ S ) ) ) ) ).
% sum.image_gen
thf(fact_960_sum_Oimage__gen,axiom,
! [S: set_a,H: a > nat,G: a > a] :
( ( finite_finite_a @ S )
=> ( ( groups6334556678337121940_a_nat @ H @ S )
= ( groups6334556678337121940_a_nat
@ ^ [Y2: a] :
( groups6334556678337121940_a_nat @ H
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ S )
& ( ( G @ X2 )
= Y2 ) ) ) )
@ ( image_a_a @ G @ S ) ) ) ) ).
% sum.image_gen
thf(fact_961_domain_Ouniv__poly__infinite__dimension,axiom,
! [R: partia2956882679547061052t_unit,K2: set_list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( subfie4546268998243038636t_unit @ K2 @ R )
=> ~ ( embedd5776004836630637299t_unit @ ( univ_p2250591967980070728t_unit @ R @ K2 ) @ ( image_1156962946714028939list_a @ ( poly_o1617770581224298896t_unit @ R ) @ K2 ) @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K2 ) ) ) ) ) ).
% domain.univ_poly_infinite_dimension
thf(fact_962_domain_Ouniv__poly__infinite__dimension,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ~ ( embedd2411333406617385593t_unit @ ( univ_p7953238456130426574t_unit @ R @ K2 ) @ ( image_8260866953997875467list_a @ ( poly_o8716471131768098070t_unit @ R ) @ K2 ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K2 ) ) ) ) ) ).
% domain.univ_poly_infinite_dimension
thf(fact_963_domain_Ouniv__poly__infinite__dimension,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ~ ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ R @ K2 ) @ ( image_a_list_a @ ( poly_of_const_a_b @ R ) @ K2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K2 ) ) ) ) ) ).
% domain.univ_poly_infinite_dimension
thf(fact_964_sum_Ogroup,axiom,
! [S: set_a,T: set_int,G: a > int,H: a > nat] :
( ( finite_finite_a @ S )
=> ( ( finite_finite_int @ T )
=> ( ( ord_less_eq_set_int @ ( image_a_int @ G @ S ) @ T )
=> ( ( groups4541462559716669496nt_nat
@ ^ [Y2: int] :
( groups6334556678337121940_a_nat @ H
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ S )
& ( ( G @ X2 )
= Y2 ) ) ) )
@ T )
= ( groups6334556678337121940_a_nat @ H @ S ) ) ) ) ) ).
% sum.group
thf(fact_965_sum_Ogroup,axiom,
! [S: set_a,T: set_list_a,G: a > list_a,H: a > nat] :
( ( finite_finite_a @ S )
=> ( ( finite_finite_list_a @ T )
=> ( ( ord_le8861187494160871172list_a @ ( image_a_list_a @ G @ S ) @ T )
=> ( ( groups5521247699297860762_a_nat
@ ^ [Y2: list_a] :
( groups6334556678337121940_a_nat @ H
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ S )
& ( ( G @ X2 )
= Y2 ) ) ) )
@ T )
= ( groups6334556678337121940_a_nat @ H @ S ) ) ) ) ) ).
% sum.group
thf(fact_966_sum_Ogroup,axiom,
! [S: set_a,T: set_nat,G: a > nat,H: a > nat] :
( ( finite_finite_a @ S )
=> ( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ ( image_a_nat @ G @ S ) @ T )
=> ( ( groups3542108847815614940at_nat
@ ^ [Y2: nat] :
( groups6334556678337121940_a_nat @ H
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ S )
& ( ( G @ X2 )
= Y2 ) ) ) )
@ T )
= ( groups6334556678337121940_a_nat @ H @ S ) ) ) ) ) ).
% sum.group
thf(fact_967_sum_Ogroup,axiom,
! [S: set_nat_list_a,T: set_a,G: ( nat > list_a ) > a,H: ( nat > list_a ) > nat] :
( ( finite7630042315537210004list_a @ S )
=> ( ( finite_finite_a @ T )
=> ( ( ord_less_eq_set_a @ ( image_nat_list_a_a @ G @ S ) @ T )
=> ( ( groups6334556678337121940_a_nat
@ ^ [Y2: a] :
( groups669906071623145473_a_nat @ H
@ ( collect_nat_list_a
@ ^ [X2: nat > list_a] :
( ( member_nat_list_a @ X2 @ S )
& ( ( G @ X2 )
= Y2 ) ) ) )
@ T )
= ( groups669906071623145473_a_nat @ H @ S ) ) ) ) ) ).
% sum.group
thf(fact_968_sum_Ogroup,axiom,
! [S: set_nat_a,T: set_a,G: ( nat > a ) > a,H: ( nat > a ) > nat] :
( ( finite_finite_nat_a @ S )
=> ( ( finite_finite_a @ T )
=> ( ( ord_less_eq_set_a @ ( image_nat_a_a @ G @ S ) @ T )
=> ( ( groups6334556678337121940_a_nat
@ ^ [Y2: a] :
( groups154653438316501755_a_nat @ H
@ ( collect_nat_a
@ ^ [X2: nat > a] :
( ( member_nat_a @ X2 @ S )
& ( ( G @ X2 )
= Y2 ) ) ) )
@ T )
= ( groups154653438316501755_a_nat @ H @ S ) ) ) ) ) ).
% sum.group
thf(fact_969_sum_Ogroup,axiom,
! [S: set_list_list_a,T: set_a,G: list_list_a > a,H: list_list_a > nat] :
( ( finite1660835950917165235list_a @ S )
=> ( ( finite_finite_a @ T )
=> ( ( ord_less_eq_set_a @ ( image_list_list_a_a @ G @ S ) @ T )
=> ( ( groups6334556678337121940_a_nat
@ ^ [Y2: a] :
( groups7548105480907152928_a_nat @ H
@ ( collect_list_list_a
@ ^ [X2: list_list_a] :
( ( member_list_list_a @ X2 @ S )
& ( ( G @ X2 )
= Y2 ) ) ) )
@ T )
= ( groups7548105480907152928_a_nat @ H @ S ) ) ) ) ) ).
% sum.group
thf(fact_970_sum_Ogroup,axiom,
! [S: set_nat,T: set_a,G: nat > a,H: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( finite_finite_a @ T )
=> ( ( ord_less_eq_set_a @ ( image_nat_a @ G @ S ) @ T )
=> ( ( groups6334556678337121940_a_nat
@ ^ [Y2: a] :
( groups3542108847815614940at_nat @ H
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ S )
& ( ( G @ X2 )
= Y2 ) ) ) )
@ T )
= ( groups3542108847815614940at_nat @ H @ S ) ) ) ) ) ).
% sum.group
thf(fact_971_sum_Ogroup,axiom,
! [S: set_list_a,T: set_a,G: list_a > a,H: list_a > nat] :
( ( finite_finite_list_a @ S )
=> ( ( finite_finite_a @ T )
=> ( ( ord_less_eq_set_a @ ( image_list_a_a @ G @ S ) @ T )
=> ( ( groups6334556678337121940_a_nat
@ ^ [Y2: a] :
( groups5521247699297860762_a_nat @ H
@ ( collect_list_a
@ ^ [X2: list_a] :
( ( member_list_a @ X2 @ S )
& ( ( G @ X2 )
= Y2 ) ) ) )
@ T )
= ( groups5521247699297860762_a_nat @ H @ S ) ) ) ) ) ).
% sum.group
thf(fact_972_sum_Ogroup,axiom,
! [S: set_int,T: set_a,G: int > a,H: int > nat] :
( ( finite_finite_int @ S )
=> ( ( finite_finite_a @ T )
=> ( ( ord_less_eq_set_a @ ( image_int_a @ G @ S ) @ T )
=> ( ( groups6334556678337121940_a_nat
@ ^ [Y2: a] :
( groups4541462559716669496nt_nat @ H
@ ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ S )
& ( ( G @ X2 )
= Y2 ) ) ) )
@ T )
= ( groups4541462559716669496nt_nat @ H @ S ) ) ) ) ) ).
% sum.group
thf(fact_973_sum_Ogroup,axiom,
! [S: set_a,T: set_a,G: a > a,H: a > nat] :
( ( finite_finite_a @ S )
=> ( ( finite_finite_a @ T )
=> ( ( ord_less_eq_set_a @ ( image_a_a @ G @ S ) @ T )
=> ( ( groups6334556678337121940_a_nat
@ ^ [Y2: a] :
( groups6334556678337121940_a_nat @ H
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ S )
& ( ( G @ X2 )
= Y2 ) ) ) )
@ T )
= ( groups6334556678337121940_a_nat @ H @ S ) ) ) ) ) ).
% sum.group
thf(fact_974_bounded__Max__nat,axiom,
! [P2: nat > $o,X3: nat,M2: nat] :
( ( P2 @ X3 )
=> ( ! [X: nat] :
( ( P2 @ X )
=> ( ord_less_eq_nat @ X @ M2 ) )
=> ~ ! [M5: nat] :
( ( P2 @ M5 )
=> ~ ! [X4: nat] :
( ( P2 @ X4 )
=> ( ord_less_eq_nat @ X4 @ M5 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_975_infinite__Iic,axiom,
! [A2: int] :
~ ( finite_finite_int @ ( set_ord_atMost_int @ A2 ) ) ).
% infinite_Iic
thf(fact_976_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M3: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ N5 )
=> ( ord_less_eq_nat @ X2 @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_977_atMost__def,axiom,
( set_ord_atMost_int
= ( ^ [U2: int] :
( collect_int
@ ^ [X2: int] : ( ord_less_eq_int @ X2 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_978_atMost__def,axiom,
( set_or4236626031148496127et_nat
= ( ^ [U2: set_nat] :
( collect_set_nat
@ ^ [X2: set_nat] : ( ord_less_eq_set_nat @ X2 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_979_atMost__def,axiom,
( set_ord_atMost_nat
= ( ^ [U2: nat] :
( collect_nat
@ ^ [X2: nat] : ( ord_less_eq_nat @ X2 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_980_finite__less__ub,axiom,
! [F: nat > nat,U: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ N3 @ ( F @ N3 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_981_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_982_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_983_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_984_long__division__add_I1_J,axiom,
! [K2: set_a,A2: list_a,B: list_a,Q: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ A2 @ ( 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 ) @ A2 @ B ) @ Q )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K2 ) @ ( polynomial_pdiv_a_b @ r @ A2 @ Q ) @ ( polynomial_pdiv_a_b @ r @ B @ Q ) ) ) ) ) ) ) ).
% long_division_add(1)
thf(fact_985_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_986_x_Oadd_Osurj__const__mult,axiom,
! [A2: list_a] :
( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( image_list_a_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.add.surj_const_mult
thf(fact_987_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_988_univ__poly__subfield__of__consts,axiom,
! [K2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( subfie1779122896746047282t_unit @ ( image_a_list_a @ ( poly_of_const_a_b @ r ) @ K2 ) @ ( univ_poly_a_b @ r @ K2 ) ) ) ).
% univ_poly_subfield_of_consts
thf(fact_989_univ__poly__infinite__dimension,axiom,
! [K2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ~ ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ K2 ) @ ( image_a_list_a @ ( poly_of_const_a_b @ r ) @ K2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) ) ) ).
% univ_poly_infinite_dimension
thf(fact_990_poly__of__const__over__subfield,axiom,
! [K2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( image_a_list_a @ ( poly_of_const_a_b @ r ) @ K2 )
= ( collect_list_a
@ ^ [P3: list_a] :
( ( member_list_a @ P3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
& ( ( minus_minus_nat @ ( size_size_list_a @ P3 ) @ one_one_nat )
= zero_zero_nat ) ) ) ) ) ).
% poly_of_const_over_subfield
thf(fact_991_x_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 ) ) ) ).
% x.ring_primeI
thf(fact_992_ring__irreducibleE_I1_J,axiom,
! [R2: a] :
( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R2 )
=> ( R2
!= ( zero_a_b @ r ) ) ) ) ).
% ring_irreducibleE(1)
thf(fact_993_subfield__long__division__theorem__shell,axiom,
! [K2: set_a,P: list_a,B: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( B
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ? [Q3: list_a,R3: list_a] :
( ( member_list_a @ Q3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
& ( member_list_a @ R3 @ ( 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 ) @ B @ Q3 ) @ R3 ) )
& ( ( R3
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ R3 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ B ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% subfield_long_division_theorem_shell
thf(fact_994_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_995_x_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 ) ) ) ) ) ) ).
% x.boundD_carrier
thf(fact_996_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_997_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_998_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_999_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1000_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_nat @ N2 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_1001_x_Osubfield__long__division__theorem__shell,axiom,
! [K2: set_list_a,P: list_list_a,B: 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 @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( B
!= ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ? [Q3: list_list_a,R3: list_list_a] :
( ( member_list_list_a @ Q3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
& ( member_list_list_a @ R3 @ ( 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 ) @ B @ Q3 ) @ R3 ) )
& ( ( R3
= ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ R3 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ B ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% x.subfield_long_division_theorem_shell
thf(fact_1002_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_1003_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1004_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1005_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1006_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_1007_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1008_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_1009_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1010_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1011_infinite__descent0,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P2 @ N3 )
=> ? [M6: nat] :
( ( ord_less_nat @ M6 @ N3 )
& ~ ( P2 @ M6 ) ) ) )
=> ( P2 @ N ) ) ) ).
% infinite_descent0
thf(fact_1012_linorder__neqE__nat,axiom,
! [X3: nat,Y: nat] :
( ( X3 != Y )
=> ( ~ ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ Y @ X3 ) ) ) ).
% linorder_neqE_nat
thf(fact_1013_infinite__descent,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P2 @ N3 )
=> ? [M6: nat] :
( ( ord_less_nat @ M6 @ N3 )
& ~ ( P2 @ M6 ) ) )
=> ( P2 @ N ) ) ).
% infinite_descent
thf(fact_1014_nat__less__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M6: nat] :
( ( ord_less_nat @ M6 @ N3 )
=> ( P2 @ M6 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ N ) ) ).
% nat_less_induct
thf(fact_1015_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_1016_less__not__refl3,axiom,
! [S3: nat,T2: nat] :
( ( ord_less_nat @ S3 @ T2 )
=> ( S3 != T2 ) ) ).
% less_not_refl3
thf(fact_1017_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_1018_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_1019_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_1020_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I2: nat,J: nat] :
( ! [I4: nat,J3: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ord_less_nat @ ( F @ I4 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1021_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_1022_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_1023_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
| ( M3 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1024_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_1025_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
& ( M3 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_1026_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1027_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_1028_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_1029_trans__less__add2,axiom,
! [I2: nat,J: nat,M: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_1030_trans__less__add1,axiom,
! [I2: nat,J: nat,M: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_1031_add__less__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_1032_not__add__less2,axiom,
! [J: nat,I2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I2 ) @ I2 ) ).
% not_add_less2
thf(fact_1033_not__add__less1,axiom,
! [I2: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ I2 ) ).
% not_add_less1
thf(fact_1034_add__less__mono,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1035_add__lessD1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
=> ( ord_less_nat @ I2 @ K ) ) ).
% add_lessD1
thf(fact_1036_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M3: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ N5 )
=> ( ord_less_nat @ X2 @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_1037_bounded__nat__set__is__finite,axiom,
! [N4: set_nat,N: nat] :
( ! [X: nat] :
( ( member_nat @ X @ N4 )
=> ( ord_less_nat @ X @ N ) )
=> ( finite_finite_nat @ N4 ) ) ).
% bounded_nat_set_is_finite
thf(fact_1038_finite__M__bounded__by__nat,axiom,
! [P2: nat > $o,I2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( P2 @ K3 )
& ( ord_less_nat @ K3 @ I2 ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_1039_ex__least__nat__le,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K4: nat] :
( ( ord_less_eq_nat @ K4 @ N )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ K4 )
=> ~ ( P2 @ I5 ) )
& ( P2 @ K4 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1040_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_1041_less__imp__add__positive,axiom,
! [I2: nat,J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ? [K4: nat] :
( ( ord_less_nat @ zero_zero_nat @ K4 )
& ( ( plus_plus_nat @ I2 @ K4 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1042_diff__less__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_nat @ ( minus_minus_nat @ A2 @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1043_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1044_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M5: nat,N3: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ( ord_less_nat @ ( F @ M5 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1045_less__diff__conv,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1046_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_1047_nat__diff__split,axiom,
! [P2: nat > $o,A2: nat,B: nat] :
( ( P2 @ ( minus_minus_nat @ A2 @ B ) )
= ( ( ( ord_less_nat @ A2 @ B )
=> ( P2 @ zero_zero_nat ) )
& ! [D2: nat] :
( ( A2
= ( plus_plus_nat @ B @ D2 ) )
=> ( P2 @ D2 ) ) ) ) ).
% nat_diff_split
thf(fact_1048_nat__diff__split__asm,axiom,
! [P2: nat > $o,A2: nat,B: nat] :
( ( P2 @ ( minus_minus_nat @ A2 @ B ) )
= ( ~ ( ( ( ord_less_nat @ A2 @ B )
& ~ ( P2 @ zero_zero_nat ) )
| ? [D2: nat] :
( ( A2
= ( plus_plus_nat @ B @ D2 ) )
& ~ ( P2 @ D2 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1049_less__diff__conv2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I2 @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1050_x_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 ) ) ) ) ) ).
% x.order_gt_0_iff_finite
thf(fact_1051_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_1052_alg__mult__gt__zero__iff__is__root,axiom,
! [P: list_a,X3: 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 @ X3 ) )
= ( polyno4133073214067823460ot_a_b @ r @ P @ X3 ) ) ) ).
% alg_mult_gt_zero_iff_is_root
thf(fact_1053_pmod__const_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 ) ) )
=> ( ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q ) @ one_one_nat ) )
=> ( ( polynomial_pdiv_a_b @ r @ P @ Q )
= nil_a ) ) ) ) ) ).
% pmod_const(1)
thf(fact_1054_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_1055_abelian__monoid__axioms,axiom,
abelian_monoid_a_b @ r ).
% abelian_monoid_axioms
thf(fact_1056_zero__is__prime_I1_J,axiom,
prime_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) ).
% zero_is_prime(1)
thf(fact_1057_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_1058_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_1059_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_1060_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 )
=> ? [X: produc9164743771328383783list_a] :
( ( polyno2806191415236617128es_a_b @ r @ P @ Q @ X )
& ! [Y4: produc9164743771328383783list_a] :
( ( polyno2806191415236617128es_a_b @ r @ P @ Q @ Y4 )
=> ( Y4 = X ) ) ) ) ) ) ) ).
% exists_unique_long_division
thf(fact_1061_pmod__image__characterization,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 )
=> ( ( image_list_a_list_a
@ ^ [Q4: list_a] : ( polynomial_pmod_a_b @ r @ Q4 @ P )
@ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
= ( collect_list_a
@ ^ [Q4: list_a] :
( ( member_list_a @ Q4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
& ( ord_less_eq_nat @ ( size_size_list_a @ Q4 ) @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% pmod_image_characterization
thf(fact_1062_pmod__degree,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 )
=> ( ( ( polynomial_pmod_a_b @ r @ P @ Q )
= nil_a )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( polynomial_pmod_a_b @ r @ P @ Q ) ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q ) @ one_one_nat ) ) ) ) ) ) ) ).
% pmod_degree
thf(fact_1063_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_1064_long__division__add__iff,axiom,
! [K2: set_a,A2: list_a,B: list_a,C: list_a,Q: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ A2 @ ( 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 @ A2 @ Q )
= ( polynomial_pmod_a_b @ r @ B @ Q ) )
= ( ( polynomial_pmod_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K2 ) @ A2 @ 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_1065_long__division__add_I2_J,axiom,
! [K2: set_a,A2: list_a,B: list_a,Q: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ A2 @ ( 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 ) @ A2 @ B ) @ Q )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K2 ) @ ( polynomial_pmod_a_b @ r @ A2 @ Q ) @ ( polynomial_pmod_a_b @ r @ B @ Q ) ) ) ) ) ) ) ).
% long_division_add(2)
thf(fact_1066_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_1067_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_1068_pmod__const_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 ) ) )
=> ( ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q ) @ one_one_nat ) )
=> ( ( polynomial_pmod_a_b @ r @ P @ Q )
= P ) ) ) ) ) ).
% pmod_const(2)
thf(fact_1069_long__dividesI,axiom,
! [B: list_a,R2: list_a,P: list_a,Q: list_a] :
( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q @ B ) @ R2 ) )
=> ( ( ( R2 = nil_a )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ R2 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q ) @ one_one_nat ) ) )
=> ( polyno2806191415236617128es_a_b @ r @ P @ Q @ ( produc6837034575241423639list_a @ B @ R2 ) ) ) ) ) ) ).
% long_dividesI
thf(fact_1070_x_Odegree__oneE,axiom,
! [P: list_list_a,K2: set_list_a] :
( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat )
= one_one_nat )
=> ~ ! [A3: list_a] :
( ( member_list_a @ A3 @ K2 )
=> ( ( A3
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ! [B3: list_a] :
( ( member_list_a @ B3 @ K2 )
=> ( P
!= ( cons_list_a @ A3 @ ( cons_list_a @ B3 @ nil_list_a ) ) ) ) ) ) ) ) ).
% x.degree_oneE
thf(fact_1071_long__divisionI,axiom,
! [K2: set_a,P: list_a,Q: list_a,B: list_a,R2: 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 @ R2 ) )
=> ( ( produc6837034575241423639list_a @ B @ R2 )
= ( produc6837034575241423639list_a @ ( polynomial_pdiv_a_b @ r @ P @ Q ) @ ( polynomial_pmod_a_b @ r @ P @ Q ) ) ) ) ) ) ) ) ).
% long_divisionI
thf(fact_1072_poly__add_Ocases,axiom,
! [X3: produc9164743771328383783list_a] :
~ ! [P1: list_a,P22: list_a] :
( X3
!= ( produc6837034575241423639list_a @ P1 @ P22 ) ) ).
% poly_add.cases
thf(fact_1073_x_Onormalize_Ocases,axiom,
! [X3: list_list_a] :
( ( X3 != nil_list_a )
=> ~ ! [V2: list_a,Va: list_list_a] :
( X3
!= ( cons_list_a @ V2 @ Va ) ) ) ).
% x.normalize.cases
thf(fact_1074_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 )
=> ~ ! [B3: list_a] :
( ( member_list_a @ B3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ! [R3: list_a] :
( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ~ ( polyno2806191415236617128es_a_b @ r @ P @ Q @ ( produc6837034575241423639list_a @ B3 @ R3 ) ) ) ) ) ) ) ) ).
% exists_long_division
thf(fact_1075_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_1076_x_Olong__dividesI,axiom,
! [B: list_list_a,R2: list_list_a,P: list_list_a,Q: list_list_a] :
( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( member_list_list_a @ R2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( P
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ Q @ B ) @ R2 ) )
=> ( ( ( R2 = nil_list_a )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ R2 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ Q ) @ one_one_nat ) ) )
=> ( polyno6947042923167803568t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q @ ( produc8696003437204565271list_a @ B @ R2 ) ) ) ) ) ) ).
% x.long_dividesI
thf(fact_1077_zle__add1__eq__le,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ord_less_eq_int @ W @ Z ) ) ).
% zle_add1_eq_le
thf(fact_1078_normalize_Ocases,axiom,
! [X3: list_a] :
( ( X3 != nil_a )
=> ~ ! [V2: a,Va: list_a] :
( X3
!= ( cons_a @ V2 @ Va ) ) ) ).
% normalize.cases
thf(fact_1079_x_Opoly__add_Ocases,axiom,
! [X3: produc7709606177366032167list_a] :
~ ! [P1: list_list_a,P22: list_list_a] :
( X3
!= ( produc8696003437204565271list_a @ P1 @ P22 ) ) ).
% x.poly_add.cases
thf(fact_1080_combine_Ocases,axiom,
! [X3: produc9164743771328383783list_a] :
( ! [K4: a,Ks: list_a,U3: a,Us: list_a] :
( X3
!= ( produc6837034575241423639list_a @ ( cons_a @ K4 @ Ks ) @ ( cons_a @ U3 @ Us ) ) )
=> ( ! [Us: list_a] :
( X3
!= ( produc6837034575241423639list_a @ nil_a @ Us ) )
=> ~ ! [Ks: list_a] :
( X3
!= ( produc6837034575241423639list_a @ Ks @ nil_a ) ) ) ) ).
% combine.cases
thf(fact_1081_poly__mult_Ocases,axiom,
! [X3: produc9164743771328383783list_a] :
( ! [P22: list_a] :
( X3
!= ( produc6837034575241423639list_a @ nil_a @ P22 ) )
=> ~ ! [V2: a,Va: list_a,P22: list_a] :
( X3
!= ( produc6837034575241423639list_a @ ( cons_a @ V2 @ Va ) @ P22 ) ) ) ).
% poly_mult.cases
thf(fact_1082_x_Ocombine_Ocases,axiom,
! [X3: produc7709606177366032167list_a] :
( ! [K4: list_a,Ks: list_list_a,U3: list_a,Us: list_list_a] :
( X3
!= ( produc8696003437204565271list_a @ ( cons_list_a @ K4 @ Ks ) @ ( cons_list_a @ U3 @ Us ) ) )
=> ( ! [Us: list_list_a] :
( X3
!= ( produc8696003437204565271list_a @ nil_list_a @ Us ) )
=> ~ ! [Ks: list_list_a] :
( X3
!= ( produc8696003437204565271list_a @ Ks @ nil_list_a ) ) ) ) ).
% x.combine.cases
thf(fact_1083_x_Opoly__mult_Ocases,axiom,
! [X3: produc7709606177366032167list_a] :
( ! [P22: list_list_a] :
( X3
!= ( produc8696003437204565271list_a @ nil_list_a @ P22 ) )
=> ~ ! [V2: list_a,Va: list_list_a,P22: list_list_a] :
( X3
!= ( produc8696003437204565271list_a @ ( cons_list_a @ V2 @ Va ) @ P22 ) ) ) ).
% x.poly_mult.cases
thf(fact_1084_finite__interval__int4,axiom,
! [A2: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( ord_less_int @ A2 @ I )
& ( ord_less_int @ I @ B ) ) ) ) ).
% finite_interval_int4
thf(fact_1085_degree__oneE,axiom,
! [P: list_a,K2: set_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= one_one_nat )
=> ~ ! [A3: a] :
( ( member_a @ A3 @ K2 )
=> ( ( A3
!= ( zero_a_b @ r ) )
=> ! [B3: a] :
( ( member_a @ B3 @ K2 )
=> ( P
!= ( cons_a @ A3 @ ( cons_a @ B3 @ nil_a ) ) ) ) ) ) ) ) ).
% degree_oneE
thf(fact_1086_finite__interval__int2,axiom,
! [A2: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( ord_less_eq_int @ A2 @ I )
& ( ord_less_int @ I @ B ) ) ) ) ).
% finite_interval_int2
thf(fact_1087_finite__interval__int3,axiom,
! [A2: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( ord_less_int @ A2 @ I )
& ( ord_less_eq_int @ I @ B ) ) ) ) ).
% finite_interval_int3
thf(fact_1088_zle__diff1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z @ one_one_int ) )
= ( ord_less_int @ W @ Z ) ) ).
% zle_diff1_eq
thf(fact_1089_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_1090_int__less__induct,axiom,
! [I2: int,K: int,P2: int > $o] :
( ( ord_less_int @ I2 @ K )
=> ( ( P2 @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I4: int] :
( ( ord_less_int @ I4 @ K )
=> ( ( P2 @ I4 )
=> ( P2 @ ( minus_minus_int @ I4 @ one_one_int ) ) ) )
=> ( P2 @ I2 ) ) ) ) ).
% int_less_induct
thf(fact_1091_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_1092_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_1093_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_1094_int__le__induct,axiom,
! [I2: int,K: int,P2: int > $o] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P2 @ K )
=> ( ! [I4: int] :
( ( ord_less_eq_int @ I4 @ K )
=> ( ( P2 @ I4 )
=> ( P2 @ ( minus_minus_int @ I4 @ one_one_int ) ) ) )
=> ( P2 @ I2 ) ) ) ) ).
% int_le_induct
thf(fact_1095_int__one__le__iff__zero__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ one_one_int @ Z )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% int_one_le_iff_zero_less
thf(fact_1096_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_1097_int__gr__induct,axiom,
! [K: int,I2: int,P2: int > $o] :
( ( ord_less_int @ K @ I2 )
=> ( ( P2 @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I4: int] :
( ( ord_less_int @ K @ I4 )
=> ( ( P2 @ I4 )
=> ( P2 @ ( plus_plus_int @ I4 @ one_one_int ) ) ) )
=> ( P2 @ I2 ) ) ) ) ).
% int_gr_induct
thf(fact_1098_zless__add1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ( ord_less_int @ W @ Z )
| ( W = Z ) ) ) ).
% zless_add1_eq
thf(fact_1099_odd__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_1100_int__induct,axiom,
! [P2: int > $o,K: int,I2: int] :
( ( P2 @ K )
=> ( ! [I4: int] :
( ( ord_less_eq_int @ K @ I4 )
=> ( ( P2 @ I4 )
=> ( P2 @ ( plus_plus_int @ I4 @ one_one_int ) ) ) )
=> ( ! [I4: int] :
( ( ord_less_eq_int @ I4 @ K )
=> ( ( P2 @ I4 )
=> ( P2 @ ( minus_minus_int @ I4 @ one_one_int ) ) ) )
=> ( P2 @ I2 ) ) ) ) ).
% int_induct
thf(fact_1101_int__ge__induct,axiom,
! [K: int,I2: int,P2: int > $o] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P2 @ K )
=> ( ! [I4: int] :
( ( ord_less_eq_int @ K @ I4 )
=> ( ( P2 @ I4 )
=> ( P2 @ ( plus_plus_int @ I4 @ one_one_int ) ) ) )
=> ( P2 @ I2 ) ) ) ) ).
% int_ge_induct
thf(fact_1102_add1__zle__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z )
= ( ord_less_int @ W @ Z ) ) ).
% add1_zle_eq
thf(fact_1103_le__imp__0__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).
% le_imp_0_less
thf(fact_1104_zless__imp__add1__zle,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ Z )
=> ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z ) ) ).
% zless_imp_add1_zle
thf(fact_1105_x_Ocauchy__product,axiom,
! [N: nat,F: nat > list_a,M: nat,G: nat > list_a] :
( ( bound_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ N @ F )
=> ( ( bound_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ M @ G )
=> ( ( member_nat_list_a @ F
@ ( pi_nat_list_a @ ( set_ord_atMost_nat @ N )
@ ^ [Uu: nat] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_nat_list_a @ G
@ ( pi_nat_list_a @ ( set_ord_atMost_nat @ M )
@ ^ [Uu: nat] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( finsum3497512462216549110it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ^ [K3: nat] :
( finsum3497512462216549110it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ^ [I: nat] : ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( F @ I ) @ ( G @ ( minus_minus_nat @ K3 @ I ) ) )
@ ( set_ord_atMost_nat @ K3 ) )
@ ( set_ord_atMost_nat @ ( plus_plus_nat @ N @ M ) ) )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( finsum3497512462216549110it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ ( set_ord_atMost_nat @ N ) ) @ ( finsum3497512462216549110it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ ( set_ord_atMost_nat @ M ) ) ) ) ) ) ) ) ).
% x.cauchy_product
thf(fact_1106_up__minus__closed,axiom,
! [P: nat > a,Q: nat > a] :
( ( member_nat_a @ P @ ( up_a_b @ r ) )
=> ( ( member_nat_a @ Q @ ( up_a_b @ r ) )
=> ( member_nat_a
@ ^ [I: nat] : ( a_minus_a_b @ r @ ( P @ I ) @ ( Q @ I ) )
@ ( up_a_b @ r ) ) ) ) ).
% up_minus_closed
thf(fact_1107_field__long__division__theorem,axiom,
! [K2: set_a,P: list_a,B: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P )
=> ( ( polynomial_a_b @ r @ K2 @ B )
=> ( ( B != nil_a )
=> ? [Q3: list_a,R3: list_a] :
( ( polynomial_a_b @ r @ K2 @ Q3 )
& ( polynomial_a_b @ r @ K2 @ R3 )
& ( P
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K2 ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K2 ) @ B @ Q3 ) @ R3 ) )
& ( ( R3 = nil_a )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ R3 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ B ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% field_long_division_theorem
thf(fact_1108_x_Odiagonal__sum,axiom,
! [F: nat > list_a,N: nat,M: nat,G: nat > list_a] :
( ( member_nat_list_a @ F
@ ( pi_nat_list_a @ ( set_ord_atMost_nat @ ( plus_plus_nat @ N @ M ) )
@ ^ [Uu: nat] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_nat_list_a @ G
@ ( pi_nat_list_a @ ( set_ord_atMost_nat @ ( plus_plus_nat @ N @ M ) )
@ ^ [Uu: nat] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( finsum3497512462216549110it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ^ [K3: nat] :
( finsum3497512462216549110it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ^ [I: nat] : ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( F @ I ) @ ( G @ ( minus_minus_nat @ K3 @ I ) ) )
@ ( set_ord_atMost_nat @ K3 ) )
@ ( set_ord_atMost_nat @ ( plus_plus_nat @ N @ M ) ) )
= ( finsum3497512462216549110it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ^ [K3: nat] :
( finsum3497512462216549110it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ^ [I: nat] : ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( F @ K3 ) @ ( G @ I ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ K3 ) ) )
@ ( set_ord_atMost_nat @ ( plus_plus_nat @ N @ M ) ) ) ) ) ) ).
% x.diagonal_sum
thf(fact_1109_zero__is__polynomial,axiom,
! [K2: set_a] : ( polynomial_a_b @ r @ K2 @ nil_a ) ).
% zero_is_polynomial
thf(fact_1110_minus__closed,axiom,
! [X3: a,Y: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( a_minus_a_b @ r @ X3 @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% minus_closed
thf(fact_1111_r__right__minus__eq,axiom,
! [A2: a,B: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( a_minus_a_b @ r @ A2 @ B )
= ( zero_a_b @ r ) )
= ( A2 = B ) ) ) ) ).
% r_right_minus_eq
thf(fact_1112_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_1113_x_Ofinsum__Suc2,axiom,
! [F: nat > list_a,N: nat] :
( ( member_nat_list_a @ F
@ ( pi_nat_list_a @ ( set_ord_atMost_nat @ ( suc @ N ) )
@ ^ [Uu: nat] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( finsum3497512462216549110it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ( finsum3497512462216549110it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ^ [I: nat] : ( F @ ( suc @ I ) )
@ ( set_ord_atMost_nat @ N ) )
@ ( F @ zero_zero_nat ) ) ) ) ).
% x.finsum_Suc2
thf(fact_1114_x_Ofinprod__Suc2,axiom,
! [F: nat > list_a,N: nat] :
( ( member_nat_list_a @ F
@ ( pi_nat_list_a @ ( set_ord_atMost_nat @ ( suc @ N ) )
@ ^ [Uu: nat] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( finpro1918104735009086181it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ( finpro1918104735009086181it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ^ [I: nat] : ( F @ ( suc @ I ) )
@ ( set_ord_atMost_nat @ N ) )
@ ( F @ zero_zero_nat ) ) ) ) ).
% x.finprod_Suc2
thf(fact_1115_x_Osubalgebra__inter,axiom,
! [K2: set_list_a,V: set_list_a,V3: set_list_a] :
( ( embedd1768981623711841426t_unit @ K2 @ V @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd1768981623711841426t_unit @ K2 @ V3 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( embedd1768981623711841426t_unit @ K2 @ ( inf_inf_set_list_a @ V @ V3 ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.subalgebra_inter
thf(fact_1116_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_1117_nat_Oinject,axiom,
! [X23: nat,Y23: nat] :
( ( ( suc @ X23 )
= ( suc @ Y23 ) )
= ( X23 = Y23 ) ) ).
% nat.inject
thf(fact_1118_x_Osubcring__inter,axiom,
! [I3: set_list_a,J4: set_list_a] :
( ( subcri7763218559781929323t_unit @ I3 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( subcri7763218559781929323t_unit @ J4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( subcri7763218559781929323t_unit @ ( inf_inf_set_list_a @ I3 @ J4 ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.subcring_inter
thf(fact_1119_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_1120_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_1121_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_1122_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_1123_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_1124_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_1125_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_1126_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1127_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_1128_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_1129_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_1130_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I2 )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1131_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I2 @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1132_x_Ozero__is__polynomial,axiom,
! [K2: set_list_a] : ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ nil_list_a ) ).
% x.zero_is_polynomial
thf(fact_1133_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_1134_x_Ofinsum__Suc,axiom,
! [F: nat > list_a,N: nat] :
( ( member_nat_list_a @ F
@ ( pi_nat_list_a @ ( set_ord_atMost_nat @ ( suc @ N ) )
@ ^ [Uu: nat] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( finsum3497512462216549110it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( F @ ( suc @ N ) ) @ ( finsum3497512462216549110it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ ( set_ord_atMost_nat @ N ) ) ) ) ) ).
% x.finsum_Suc
thf(fact_1135_x_Ofinprod__Suc,axiom,
! [F: nat > list_a,N: nat] :
( ( member_nat_list_a @ F
@ ( pi_nat_list_a @ ( set_ord_atMost_nat @ ( suc @ N ) )
@ ^ [Uu: nat] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( finpro1918104735009086181it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( F @ ( suc @ N ) ) @ ( finpro1918104735009086181it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ ( set_ord_atMost_nat @ N ) ) ) ) ) ).
% x.finprod_Suc
thf(fact_1136_Nat_OlessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( ( K
!= ( suc @ I2 ) )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1137_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_1138_Suc__lessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ K )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ).
% Suc_lessE
thf(fact_1139_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_1140_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_1141_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_1142_Ex__less__Suc,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
& ( P2 @ I ) ) )
= ( ( P2 @ N )
| ? [I: nat] :
( ( ord_less_nat @ I @ N )
& ( P2 @ I ) ) ) ) ).
% Ex_less_Suc
thf(fact_1143_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_1144_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_1145_All__less__Suc,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
=> ( P2 @ I ) ) )
= ( ( P2 @ N )
& ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( P2 @ I ) ) ) ) ).
% All_less_Suc
thf(fact_1146_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M7: nat] :
( ( M
= ( suc @ M7 ) )
& ( ord_less_nat @ N @ M7 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1147_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_1148_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_1149_less__trans__Suc,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I2 ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_1150_less__Suc__induct,axiom,
! [I2: nat,J: nat,P2: nat > nat > $o] :
( ( ord_less_nat @ I2 @ J )
=> ( ! [I4: nat] : ( P2 @ I4 @ ( suc @ I4 ) )
=> ( ! [I4: nat,J3: nat,K4: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ( ord_less_nat @ J3 @ K4 )
=> ( ( P2 @ I4 @ J3 )
=> ( ( P2 @ J3 @ K4 )
=> ( P2 @ I4 @ K4 ) ) ) ) )
=> ( P2 @ I2 @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_1151_strict__inc__induct,axiom,
! [I2: nat,J: nat,P2: nat > $o] :
( ( ord_less_nat @ I2 @ J )
=> ( ! [I4: nat] :
( ( J
= ( suc @ I4 ) )
=> ( P2 @ I4 ) )
=> ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ J )
=> ( ( P2 @ ( suc @ I4 ) )
=> ( P2 @ I4 ) ) )
=> ( P2 @ I2 ) ) ) ) ).
% strict_inc_induct
thf(fact_1152_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_1153_zero__induct__lemma,axiom,
! [P2: nat > $o,K: nat,I2: nat] :
( ( P2 @ K )
=> ( ! [N3: nat] :
( ( P2 @ ( suc @ N3 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ ( minus_minus_nat @ K @ I2 ) ) ) ) ).
% zero_induct_lemma
thf(fact_1154_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ).
% not0_implies_Suc
thf(fact_1155_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_1156_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_1157_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1158_zero__induct,axiom,
! [P2: nat > $o,K: nat] :
( ( P2 @ K )
=> ( ! [N3: nat] :
( ( P2 @ ( suc @ N3 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1159_diff__induct,axiom,
! [P2: nat > nat > $o,M: nat,N: nat] :
( ! [X: nat] : ( P2 @ X @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P2 @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X: nat,Y3: nat] :
( ( P2 @ X @ Y3 )
=> ( P2 @ ( suc @ X ) @ ( suc @ Y3 ) ) )
=> ( P2 @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_1160_nat__induct,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) )
=> ( P2 @ N ) ) ) ).
% nat_induct
thf(fact_1161_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1162_nat_OdiscI,axiom,
! [Nat: nat,X23: nat] :
( ( Nat
= ( suc @ X23 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1163_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1164_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1165_nat_Odistinct_I1_J,axiom,
! [X23: nat] :
( zero_zero_nat
!= ( suc @ X23 ) ) ).
% nat.distinct(1)
thf(fact_1166_zero__notin__Suc__image,axiom,
! [A: set_nat] :
~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A ) ) ).
% zero_notin_Suc_image
thf(fact_1167_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_1168_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_1169_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1170_Suc__le__D,axiom,
! [N: nat,M8: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M8 )
=> ? [M5: nat] :
( M8
= ( suc @ M5 ) ) ) ).
% Suc_le_D
thf(fact_1171_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_1172_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1173_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_1174_full__nat__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M6: nat] :
( ( ord_less_eq_nat @ ( suc @ M6 ) @ N3 )
=> ( P2 @ M6 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ N ) ) ).
% full_nat_induct
thf(fact_1175_nat__induct__at__least,axiom,
! [M: nat,N: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P2 @ M )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) ) )
=> ( P2 @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_1176_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X: nat] : ( R @ X @ X )
=> ( ! [X: nat,Y3: nat,Z3: nat] :
( ( R @ X @ Y3 )
=> ( ( R @ Y3 @ Z3 )
=> ( R @ X @ Z3 ) ) )
=> ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1177_nat__arith_Osuc1,axiom,
! [A: nat,K: nat,A2: nat] :
( ( A
= ( plus_plus_nat @ K @ A2 ) )
=> ( ( suc @ A )
= ( plus_plus_nat @ K @ ( suc @ A2 ) ) ) ) ).
% nat_arith.suc1
thf(fact_1178_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_1179_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_1180_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_1181_Suc__inject,axiom,
! [X3: nat,Y: nat] :
( ( ( suc @ X3 )
= ( suc @ Y ) )
=> ( X3 = Y ) ) ).
% Suc_inject
thf(fact_1182_Ex__less__Suc2,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
& ( P2 @ I ) ) )
= ( ( P2 @ zero_zero_nat )
| ? [I: nat] :
( ( ord_less_nat @ I @ N )
& ( P2 @ ( suc @ I ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1183_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1184_All__less__Suc2,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
=> ( P2 @ I ) ) )
= ( ( P2 @ zero_zero_nat )
& ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( P2 @ ( suc @ I ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1185_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ).
% gr0_implies_Suc
thf(fact_1186_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J2: nat] :
( ( M
= ( suc @ J2 ) )
& ( ord_less_nat @ J2 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1187_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1188_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_1189_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1190_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_1191_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_1192_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_1193_inc__induct,axiom,
! [I2: nat,J: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( P2 @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I2 @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P2 @ ( suc @ N3 ) )
=> ( P2 @ N3 ) ) ) )
=> ( P2 @ I2 ) ) ) ) ).
% inc_induct
thf(fact_1194_dec__induct,axiom,
! [I2: nat,J: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( P2 @ I2 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I2 @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) ) ) )
=> ( P2 @ J ) ) ) ) ).
% dec_induct
thf(fact_1195_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_1196_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_1197_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1198_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1199_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_1200_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_1201_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_1202_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q3: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q3 ) ) ) ) ).
% less_natE
thf(fact_1203_less__add__Suc1,axiom,
! [I2: nat,M: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ I2 @ M ) ) ) ).
% less_add_Suc1
thf(fact_1204_less__add__Suc2,axiom,
! [I2: nat,M: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ M @ I2 ) ) ) ).
% less_add_Suc2
thf(fact_1205_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M3: nat,N2: nat] :
? [K3: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M3 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1206_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K4: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K4 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1207_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1208_Suc__eq__plus1,axiom,
( suc
= ( ^ [N2: nat] : ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1209_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1210_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1211_ex__least__nat__less,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K4: nat] :
( ( ord_less_nat @ K4 @ N )
& ! [I5: nat] :
( ( ord_less_eq_nat @ I5 @ K4 )
=> ~ ( P2 @ I5 ) )
& ( P2 @ ( suc @ K4 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1212_diff__Suc__less,axiom,
! [N: nat,I2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I2 ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_1213_nat__induct__non__zero,axiom,
! [N: nat,P2: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P2 @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) ) )
=> ( P2 @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1214_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1215_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_1216_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M3: nat,N2: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N2 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% add_eq_if
thf(fact_1217_x_Ofinprod__0_H,axiom,
! [F: nat > list_a,N: nat] :
( ( member_nat_list_a @ F
@ ( pi_nat_list_a @ ( set_ord_atMost_nat @ N )
@ ^ [Uu: nat] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( F @ zero_zero_nat ) @ ( finpro1918104735009086181it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
= ( finpro1918104735009086181it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% x.finprod_0'
thf(fact_1218_add_Ol__cancel,axiom,
! [C: a,A2: a,B: a] :
( ( ( add_a_b @ r @ C @ A2 )
= ( add_a_b @ r @ C @ B ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A2 = B ) ) ) ) ) ).
% add.l_cancel
thf(fact_1219_add_Or__cancel,axiom,
! [A2: a,C: a,B: a] :
( ( ( add_a_b @ r @ A2 @ C )
= ( add_a_b @ r @ B @ C ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A2 = B ) ) ) ) ) ).
% add.r_cancel
thf(fact_1220_a__assoc,axiom,
! [X3: a,Y: a,Z: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( add_a_b @ r @ X3 @ Y ) @ Z )
= ( add_a_b @ r @ X3 @ ( add_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% a_assoc
thf(fact_1221_a__comm,axiom,
! [X3: a,Y: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X3 @ Y )
= ( add_a_b @ r @ Y @ X3 ) ) ) ) ).
% a_comm
thf(fact_1222_a__lcomm,axiom,
! [X3: a,Y: a,Z: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X3 @ ( add_a_b @ r @ Y @ Z ) )
= ( add_a_b @ r @ Y @ ( add_a_b @ r @ X3 @ Z ) ) ) ) ) ) ).
% a_lcomm
thf(fact_1223_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_1224_add_Osurj__const__mult,axiom,
! [A2: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( image_a_a @ ( add_a_b @ r @ A2 ) @ ( partia707051561876973205xt_a_b @ r ) )
= ( partia707051561876973205xt_a_b @ r ) ) ) ).
% add.surj_const_mult
thf(fact_1225_up__add__closed,axiom,
! [P: nat > a,Q: nat > a] :
( ( member_nat_a @ P @ ( up_a_b @ r ) )
=> ( ( member_nat_a @ Q @ ( up_a_b @ r ) )
=> ( member_nat_a
@ ^ [I: nat] : ( add_a_b @ r @ ( P @ I ) @ ( Q @ I ) )
@ ( up_a_b @ r ) ) ) ) ).
% up_add_closed
thf(fact_1226_add_Oinv__comm,axiom,
! [X3: a,Y: a] :
( ( ( add_a_b @ r @ X3 @ Y )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ Y @ X3 )
= ( zero_a_b @ r ) ) ) ) ) ).
% add.inv_comm
thf(fact_1227_add_Ol__inv__ex,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( add_a_b @ r @ X @ X3 )
= ( zero_a_b @ r ) ) ) ) ).
% add.l_inv_ex
thf(fact_1228_add_Oone__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ U @ X )
= X ) )
=> ( U
= ( zero_a_b @ r ) ) ) ) ).
% add.one_unique
thf(fact_1229_add_Or__inv__ex,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( add_a_b @ r @ X3 @ X )
= ( zero_a_b @ r ) ) ) ) ).
% add.r_inv_ex
thf(fact_1230_local_Ominus__unique,axiom,
! [Y: a,X3: a,Y6: a] :
( ( ( add_a_b @ r @ Y @ X3 )
= ( zero_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X3 @ Y6 )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y6 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( Y = Y6 ) ) ) ) ) ) ).
% local.minus_unique
thf(fact_1231_finsum__Suc2,axiom,
! [F: nat > a,N: nat] :
( ( member_nat_a @ F
@ ( pi_nat_a @ ( set_ord_atMost_nat @ ( suc @ N ) )
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum_a_b_nat @ r @ F @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( add_a_b @ r
@ ( finsum_a_b_nat @ r
@ ^ [I: nat] : ( F @ ( suc @ I ) )
@ ( set_ord_atMost_nat @ N ) )
@ ( F @ zero_zero_nat ) ) ) ) ).
% finsum_Suc2
thf(fact_1232_add_Ofinprod__0_H,axiom,
! [F: nat > a,N: nat] :
( ( member_nat_a @ F
@ ( pi_nat_a @ ( set_ord_atMost_nat @ N )
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( add_a_b @ r @ ( F @ zero_zero_nat ) @ ( finsum_a_b_nat @ r @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
= ( finsum_a_b_nat @ r @ F @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% add.finprod_0'
thf(fact_1233_finite__atLeastAtMost,axiom,
! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or1269000886237332187st_nat @ L @ U ) ) ).
% finite_atLeastAtMost
thf(fact_1234_local_Oadd_Oright__cancel,axiom,
! [X3: a,Y: a,Z: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ Y @ X3 )
= ( add_a_b @ r @ Z @ X3 ) )
= ( Y = Z ) ) ) ) ) ).
% local.add.right_cancel
thf(fact_1235_a__closed,axiom,
! [X3: a,Y: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( add_a_b @ r @ X3 @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% a_closed
thf(fact_1236_x_Oadd_Ofinprod__0_H,axiom,
! [F: nat > list_a,N: nat] :
( ( member_nat_list_a @ F
@ ( pi_nat_list_a @ ( set_ord_atMost_nat @ N )
@ ^ [Uu: nat] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( F @ zero_zero_nat ) @ ( finsum3497512462216549110it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
= ( finsum3497512462216549110it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% x.add.finprod_0'
thf(fact_1237_add_Ol__cancel__one,axiom,
! [X3: a,A2: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X3 @ A2 )
= X3 )
= ( A2
= ( zero_a_b @ r ) ) ) ) ) ).
% add.l_cancel_one
thf(fact_1238_add_Ol__cancel__one_H,axiom,
! [X3: a,A2: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X3
= ( add_a_b @ r @ X3 @ A2 ) )
= ( A2
= ( zero_a_b @ r ) ) ) ) ) ).
% add.l_cancel_one'
thf(fact_1239_add_Or__cancel__one,axiom,
! [X3: a,A2: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ A2 @ X3 )
= X3 )
= ( A2
= ( zero_a_b @ r ) ) ) ) ) ).
% add.r_cancel_one
thf(fact_1240_add_Or__cancel__one_H,axiom,
! [X3: a,A2: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X3
= ( add_a_b @ r @ A2 @ X3 ) )
= ( A2
= ( zero_a_b @ r ) ) ) ) ) ).
% add.r_cancel_one'
thf(fact_1241_l__zero,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( zero_a_b @ r ) @ X3 )
= X3 ) ) ).
% l_zero
thf(fact_1242_r__zero,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X3 @ ( zero_a_b @ r ) )
= X3 ) ) ).
% r_zero
thf(fact_1243_finsum__Suc,axiom,
! [F: nat > a,N: nat] :
( ( member_nat_a @ F
@ ( pi_nat_a @ ( set_ord_atMost_nat @ ( suc @ N ) )
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum_a_b_nat @ r @ F @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( add_a_b @ r @ ( F @ ( suc @ N ) ) @ ( finsum_a_b_nat @ r @ F @ ( set_ord_atMost_nat @ N ) ) ) ) ) ).
% finsum_Suc
thf(fact_1244_subset__eq__atLeast0__atMost__finite,axiom,
! [N4: set_nat,N: nat] :
( ( ord_less_eq_set_nat @ N4 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
=> ( finite_finite_nat @ N4 ) ) ).
% subset_eq_atLeast0_atMost_finite
thf(fact_1245_atMost__atLeast0,axiom,
( set_ord_atMost_nat
= ( set_or1269000886237332187st_nat @ zero_zero_nat ) ) ).
% atMost_atLeast0
thf(fact_1246_all__nat__less,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [M3: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( P2 @ M3 ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
=> ( P2 @ X2 ) ) ) ) ).
% all_nat_less
thf(fact_1247_ex__nat__less,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [M3: nat] :
( ( ord_less_eq_nat @ M3 @ N )
& ( P2 @ M3 ) ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
& ( P2 @ X2 ) ) ) ) ).
% ex_nat_less
thf(fact_1248_image__Suc__atMost,axiom,
! [N: nat] :
( ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) )
= ( set_or1269000886237332187st_nat @ one_one_nat @ ( suc @ N ) ) ) ).
% image_Suc_atMost
thf(fact_1249_local_Osemiring__axioms,axiom,
semiring_a_b @ r ).
% local.semiring_axioms
thf(fact_1250_m__lcomm,axiom,
! [X3: a,Y: a,Z: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X3 @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) )
= ( mult_a_ring_ext_a_b @ r @ Y @ ( mult_a_ring_ext_a_b @ r @ X3 @ Z ) ) ) ) ) ) ).
% m_lcomm
thf(fact_1251_m__comm,axiom,
! [X3: a,Y: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X3 @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ X3 ) ) ) ) ).
% m_comm
thf(fact_1252_m__assoc,axiom,
! [X3: a,Y: a,Z: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X3 @ Y ) @ Z )
= ( mult_a_ring_ext_a_b @ r @ X3 @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% m_assoc
thf(fact_1253_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_1254_m__rcancel,axiom,
! [A2: a,B: a,C: a] :
( ( A2
!= ( zero_a_b @ r ) )
=> ( ( member_a @ A2 @ ( 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 @ A2 )
= ( mult_a_ring_ext_a_b @ r @ C @ A2 ) )
= ( B = C ) ) ) ) ) ) ).
% m_rcancel
thf(fact_1255_m__lcancel,axiom,
! [A2: a,B: a,C: a] :
( ( A2
!= ( zero_a_b @ r ) )
=> ( ( member_a @ A2 @ ( 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 @ A2 @ B )
= ( mult_a_ring_ext_a_b @ r @ A2 @ C ) )
= ( B = C ) ) ) ) ) ) ).
% m_lcancel
thf(fact_1256_integral__iff,axiom,
! [A2: a,B: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ A2 @ B )
= ( zero_a_b @ r ) )
= ( ( A2
= ( zero_a_b @ r ) )
| ( B
= ( zero_a_b @ r ) ) ) ) ) ) ).
% integral_iff
thf(fact_1257_local_Ointegral,axiom,
! [A2: a,B: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A2 @ B )
= ( zero_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A2
= ( zero_a_b @ r ) )
| ( B
= ( zero_a_b @ r ) ) ) ) ) ) ).
% local.integral
thf(fact_1258_l__distr,axiom,
! [X3: a,Y: a,Z: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ X3 @ Y ) @ Z )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X3 @ Z ) @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% l_distr
thf(fact_1259_r__distr,axiom,
! [X3: a,Y: a,Z: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ Z @ ( add_a_b @ r @ X3 @ Y ) )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ Z @ X3 ) @ ( mult_a_ring_ext_a_b @ r @ Z @ Y ) ) ) ) ) ) ).
% r_distr
thf(fact_1260_up__smult__closed,axiom,
! [A2: a,P: nat > a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_nat_a @ P @ ( up_a_b @ r ) )
=> ( member_nat_a
@ ^ [I: nat] : ( mult_a_ring_ext_a_b @ r @ A2 @ ( P @ I ) )
@ ( up_a_b @ r ) ) ) ) ).
% up_smult_closed
thf(fact_1261_up__mult__closed,axiom,
! [P: nat > a,Q: nat > a] :
( ( member_nat_a @ P @ ( up_a_b @ r ) )
=> ( ( member_nat_a @ Q @ ( up_a_b @ r ) )
=> ( member_nat_a
@ ^ [N2: nat] :
( finsum_a_b_nat @ r
@ ^ [I: nat] : ( mult_a_ring_ext_a_b @ r @ ( P @ I ) @ ( Q @ ( minus_minus_nat @ N2 @ I ) ) )
@ ( set_ord_atMost_nat @ N2 ) )
@ ( up_a_b @ r ) ) ) ) ).
% up_mult_closed
thf(fact_1262_finprod__Suc2,axiom,
! [F: nat > a,N: nat] :
( ( member_nat_a @ F
@ ( pi_nat_a @ ( set_ord_atMost_nat @ ( suc @ N ) )
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro1280035270526425175_b_nat @ r @ F @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( mult_a_ring_ext_a_b @ r
@ ( finpro1280035270526425175_b_nat @ r
@ ^ [I: nat] : ( F @ ( suc @ I ) )
@ ( set_ord_atMost_nat @ N ) )
@ ( F @ zero_zero_nat ) ) ) ) ).
% finprod_Suc2
thf(fact_1263_finprod__0_H,axiom,
! [F: nat > a,N: nat] :
( ( member_nat_a @ F
@ ( pi_nat_a @ ( set_ord_atMost_nat @ N )
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( F @ zero_zero_nat ) @ ( finpro1280035270526425175_b_nat @ r @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
= ( finpro1280035270526425175_b_nat @ r @ F @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% finprod_0'
thf(fact_1264_diagonal__sum,axiom,
! [F: nat > a,N: nat,M: nat,G: nat > a] :
( ( member_nat_a @ F
@ ( pi_nat_a @ ( set_ord_atMost_nat @ ( plus_plus_nat @ N @ M ) )
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ ( set_ord_atMost_nat @ ( plus_plus_nat @ N @ M ) )
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum_a_b_nat @ r
@ ^ [K3: nat] :
( finsum_a_b_nat @ r
@ ^ [I: nat] : ( mult_a_ring_ext_a_b @ r @ ( F @ I ) @ ( G @ ( minus_minus_nat @ K3 @ I ) ) )
@ ( set_ord_atMost_nat @ K3 ) )
@ ( set_ord_atMost_nat @ ( plus_plus_nat @ N @ M ) ) )
= ( finsum_a_b_nat @ r
@ ^ [K3: nat] :
( finsum_a_b_nat @ r
@ ^ [I: nat] : ( mult_a_ring_ext_a_b @ r @ ( F @ K3 ) @ ( G @ I ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ K3 ) ) )
@ ( set_ord_atMost_nat @ ( plus_plus_nat @ N @ M ) ) ) ) ) ) ).
% diagonal_sum
thf(fact_1265_m__closed,axiom,
! [X3: a,Y: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X3 @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% m_closed
thf(fact_1266_cauchy__product,axiom,
! [N: nat,F: nat > a,M: nat,G: nat > a] :
( ( bound_a @ ( zero_a_b @ r ) @ N @ F )
=> ( ( bound_a @ ( zero_a_b @ r ) @ M @ G )
=> ( ( member_nat_a @ F
@ ( pi_nat_a @ ( set_ord_atMost_nat @ N )
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ ( set_ord_atMost_nat @ M )
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum_a_b_nat @ r
@ ^ [K3: nat] :
( finsum_a_b_nat @ r
@ ^ [I: nat] : ( mult_a_ring_ext_a_b @ r @ ( F @ I ) @ ( G @ ( minus_minus_nat @ K3 @ I ) ) )
@ ( set_ord_atMost_nat @ K3 ) )
@ ( set_ord_atMost_nat @ ( plus_plus_nat @ N @ M ) ) )
= ( mult_a_ring_ext_a_b @ r @ ( finsum_a_b_nat @ r @ F @ ( set_ord_atMost_nat @ N ) ) @ ( finsum_a_b_nat @ r @ G @ ( set_ord_atMost_nat @ M ) ) ) ) ) ) ) ) ).
% cauchy_product
thf(fact_1267_r__null,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X3 @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ) ).
% r_null
thf(fact_1268_l__null,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) @ X3 )
= ( zero_a_b @ r ) ) ) ).
% l_null
thf(fact_1269_finprod__Suc,axiom,
! [F: nat > a,N: nat] :
( ( member_nat_a @ F
@ ( pi_nat_a @ ( set_ord_atMost_nat @ ( suc @ N ) )
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro1280035270526425175_b_nat @ r @ F @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( mult_a_ring_ext_a_b @ r @ ( F @ ( suc @ N ) ) @ ( finpro1280035270526425175_b_nat @ r @ F @ ( set_ord_atMost_nat @ N ) ) ) ) ) ).
% finprod_Suc
% Helper facts (7)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X3: int,Y: int] :
( ( if_int @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X3: int,Y: int] :
( ( if_int @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X3: nat,Y: nat] :
( ( if_nat @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X3: nat,Y: nat] :
( ( if_nat @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_3_1_If_001t__Multiset__Omultiset_Itf__a_J_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001t__Multiset__Omultiset_Itf__a_J_T,axiom,
! [X3: multiset_a,Y: multiset_a] :
( ( if_multiset_a @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Multiset__Omultiset_Itf__a_J_T,axiom,
! [X3: multiset_a,Y: multiset_a] :
( ( if_multiset_a @ $true @ X3 @ Y )
= X3 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( lagran9092808442999052491ux_a_b @ r @ s ) ) @ one_one_nat )
@ ( groups6334556678337121940_a_nat
@ ^ [Y2: a] : ( minus_minus_nat @ ( size_size_list_a @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( var_a_b @ r ) @ ( poly_of_const_a_b @ r @ Y2 ) ) ) @ one_one_nat )
@ s ) ) ).
%------------------------------------------------------------------------------