TPTP Problem File: SLH0468^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Interpolation_Polynomials_HOL_Algebra/0001_Lagrange_Interpolation/prob_00125_004589__17196082_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1630 ( 426 unt; 352 typ; 0 def)
% Number of atoms : 3964 (1105 equ; 0 cnn)
% Maximal formula atoms : 11 ( 3 avg)
% Number of connectives : 15673 ( 238 ~; 40 |; 165 &;13215 @)
% ( 0 <=>;2015 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 7 avg)
% Number of types : 47 ( 46 usr)
% Number of type conns : 1876 (1876 >; 0 *; 0 +; 0 <<)
% Number of symbols : 309 ( 306 usr; 9 con; 0-3 aty)
% Number of variables : 3865 ( 624 ^;3165 !; 76 ?;3865 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:37:04.318
%------------------------------------------------------------------------------
% Could-be-implicit typings (46)
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zero_zero_nat: nat ).
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if_list_a: $o > list_a > list_a > list_a ).
thf(sy_c_If_001tf__a,type,
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inf_inf_nat: nat > nat > nat ).
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sup_sup_nat: nat > nat > nat ).
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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_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
zero_s2910681146719230829t_unit: partia7496981018696276118t_unit > set_list_a ).
thf(sy_c_Ring_Oring_Ozero_001tf__a_001tf__b,type,
zero_a_b: partia2175431115845679010xt_a_b > a ).
thf(sy_c_Ring_Oring_Ozero__update_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
zero_u1196785550890449590t_unit: ( list_a > list_a ) > partia2670972154091845814t_unit > partia2670972154091845814t_unit ).
thf(sy_c_Ring_Oring_Ozero__update_001tf__a_001tf__b,type,
zero_update_a_b: ( a > a ) > partia2175431115845679010xt_a_b > partia2175431115845679010xt_a_b ).
thf(sy_c_Ring_Oring__hom_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001tf__a_001tf__b,type,
ring_h2895973938487309444it_a_b: partia2670972154091845814t_unit > partia2175431115845679010xt_a_b > set_list_a_a ).
thf(sy_c_Ring_Oring__hom_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit_001tf__a_001tf__b,type,
ring_h8906680420194085028it_a_b: partia7496981018696276118t_unit > partia2175431115845679010xt_a_b > set_set_list_a_a ).
thf(sy_c_Ring_Oring__hom__cring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001tf__a_001tf__b,type,
ring_h1547129875642963619it_a_b: partia2670972154091845814t_unit > partia2175431115845679010xt_a_b > ( list_a > a ) > $o ).
thf(sy_c_Ring__Divisibility_Oeuclidean__domain_001tf__a_001tf__b,type,
ring_e8745995371659049232in_a_b: partia2175431115845679010xt_a_b > ( a > nat ) > $o ).
thf(sy_c_Ring__Divisibility_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_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_r5115406448772830318t_unit: partia7496981018696276118t_unit > set_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_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_r5437400583859147359t_unit: partia2956882679547061052t_unit > list_list_a > $o ).
thf(sy_c_Ring__Divisibility_Oring__prime_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_r6430282645014804837t_unit: partia2670972154091845814t_unit > list_a > $o ).
thf(sy_c_Ring__Divisibility_Oring__prime_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_r1091214237498979717t_unit: partia7496981018696276118t_unit > set_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_001_062_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mtf__a_J,type,
collect_set_list_a_a: ( ( set_list_a > a ) > $o ) > set_set_list_a_a ).
thf(sy_c_Set_OCollect_001_062_Itf__a_Mtf__a_J,type,
collect_a_a: ( ( a > a ) > $o ) > set_a_a ).
thf(sy_c_Set_OCollect_001t__List__Olist_Itf__a_J,type,
collect_list_a: ( list_a > $o ) > set_list_a ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
collect_set_list_a: ( set_list_a > $o ) > set_set_list_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_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_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_001tf__a,type,
image_list_a_a: ( list_a > a ) > set_list_a > set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001tf__a,type,
image_set_list_a_a: ( set_list_a > a ) > set_set_list_a > set_a ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set_Oinsert_001t__List__Olist_Itf__a_J,type,
insert_list_a: list_a > set_list_a > set_list_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Set_Othe__elem_001tf__a,type,
the_elem_a: set_a > 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_Osubcring_001tf__a_001tf__b,type,
subcring_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Subrings_Osubdomain_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
subdom7821232466298058046t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_Subrings_Osubdomain_001tf__a_001tf__b,type,
subdomain_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_UnivPoly_Obound_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
bound_list_list_a: list_list_a > nat > ( nat > list_list_a ) > $o ).
thf(sy_c_UnivPoly_Obound_001t__List__Olist_Itf__a_J,type,
bound_list_a: list_a > nat > ( nat > list_a ) > $o ).
thf(sy_c_UnivPoly_Obound_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
bound_set_list_a: set_list_a > nat > ( nat > set_list_a ) > $o ).
thf(sy_c_UnivPoly_Obound_001tf__a,type,
bound_a: a > nat > ( nat > a ) > $o ).
thf(sy_c_UnivPoly_Oup_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
up_lis8963924889346801084t_unit: partia2956882679547061052t_unit > set_nat_list_list_a ).
thf(sy_c_UnivPoly_Oup_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
up_lis8464167429055313730t_unit: partia2670972154091845814t_unit > set_nat_list_a ).
thf(sy_c_UnivPoly_Oup_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
up_set529185716248919906t_unit: partia7496981018696276118t_unit > set_nat_set_list_a ).
thf(sy_c_UnivPoly_Oup_001tf__a_001tf__b,type,
up_a_b: partia2175431115845679010xt_a_b > set_nat_a ).
thf(sy_c_member_001_062_I_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J_Mtf__a_J,type,
member_nat_list_a_a: ( ( nat > list_a ) > a ) > set_nat_list_a_a > $o ).
thf(sy_c_member_001_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J,type,
member_nat_a_a: ( ( nat > a ) > a ) > set_nat_a_a > $o ).
thf(sy_c_member_001_062_I_062_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mtf__a_J_Mtf__a_J,type,
member969817812316227871_a_a_a: ( ( set_list_a > a ) > a ) > set_set_list_a_a_a > $o ).
thf(sy_c_member_001_062_I_062_Itf__a_Mtf__a_J_Mtf__a_J,type,
member_a_a_a: ( ( a > a ) > a ) > set_a_a_a2 > $o ).
thf(sy_c_member_001_062_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
member8231385768148312316list_a: ( list_list_a > list_list_a ) > set_li5608457238520824219list_a > $o ).
thf(sy_c_member_001_062_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__List__Olist_Itf__a_J_J,type,
member7168557129179038582list_a: ( list_list_a > list_a ) > set_li3422455791611400469list_a > $o ).
thf(sy_c_member_001_062_It__List__Olist_It__List__Olist_Itf__a_J_J_Mtf__a_J,type,
member_list_list_a_a: ( list_list_a > a ) > set_list_list_a_a > $o ).
thf(sy_c_member_001_062_It__List__Olist_Itf__a_J_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
member6714375691612171394list_a: ( list_a > list_list_a ) > set_li6773872926390105121list_a > $o ).
thf(sy_c_member_001_062_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
member_list_a_list_a: ( list_a > list_a ) > set_list_a_list_a > $o ).
thf(sy_c_member_001_062_It__List__Olist_Itf__a_J_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
member4263473470251683292list_a: ( list_a > set_list_a ) > set_li1071299071675007611list_a > $o ).
thf(sy_c_member_001_062_It__List__Olist_Itf__a_J_Mtf__a_J,type,
member_list_a_a: ( list_a > a ) > set_list_a_a > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
member8650753269014980122list_a: ( nat > list_list_a ) > set_nat_list_list_a > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J,type,
member_nat_list_a: ( nat > list_a ) > set_nat_list_a > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
member491565700723299188list_a: ( nat > set_list_a ) > set_nat_set_list_a > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mtf__a_J,type,
member_nat_a: ( nat > a ) > set_nat_a > $o ).
thf(sy_c_member_001_062_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mt__List__Olist_Itf__a_J_J,type,
member5910328476188217884list_a: ( set_list_a > list_a ) > set_se5067313844698916539list_a > $o ).
thf(sy_c_member_001_062_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
member5068272912271824380list_a: ( set_list_a > set_list_a ) > set_se1917860372504128155list_a > $o ).
thf(sy_c_member_001_062_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mtf__a_J,type,
member_set_list_a_a: ( set_list_a > a ) > set_set_list_a_a > $o ).
thf(sy_c_member_001_062_Itf__a_M_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J_J,type,
member_a_nat_list_a: ( a > nat > list_a ) > set_a_nat_list_a > $o ).
thf(sy_c_member_001_062_Itf__a_M_062_It__Nat__Onat_Mtf__a_J_J,type,
member_a_nat_a: ( a > nat > a ) > set_a_nat_a > $o ).
thf(sy_c_member_001_062_Itf__a_M_062_Itf__a_Mtf__a_J_J,type,
member_a_a_a2: ( a > a > a ) > set_a_a_a > $o ).
thf(sy_c_member_001_062_Itf__a_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
member_a_list_list_a: ( a > list_list_a ) > set_a_list_list_a > $o ).
thf(sy_c_member_001_062_Itf__a_Mt__List__Olist_Itf__a_J_J,type,
member_a_list_a: ( a > list_a ) > set_a_list_a > $o ).
thf(sy_c_member_001_062_Itf__a_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
member_a_set_list_a: ( a > set_list_a ) > set_a_set_list_a > $o ).
thf(sy_c_member_001_062_Itf__a_Mtf__a_J,type,
member_a_a: ( a > a ) > set_a_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_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 ).
thf(sy_v_x,type,
x: a ).
% Relevant facts (1272)
thf(fact_0_assms_I1_J,axiom,
finite_finite_a @ s ).
% assms(1)
thf(fact_1_assms_I3_J,axiom,
member_a @ x @ ( partia707051561876973205xt_a_b @ r ) ).
% assms(3)
thf(fact_2_assms_I2_J,axiom,
ord_less_eq_set_a @ s @ ( partia707051561876973205xt_a_b @ r ) ).
% assms(2)
thf(fact_3_domain__axioms,axiom,
domain_a_b @ r ).
% domain_axioms
thf(fact_4_minus__closed,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( a_minus_a_b @ r @ X @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% minus_closed
thf(fact_5_onepideal,axiom,
principalideal_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% onepideal
thf(fact_6_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_7_carrier__is__subcring,axiom,
subcring_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% carrier_is_subcring
thf(fact_8_Pi__I,axiom,
! [A: set_nat,F: nat > a,B: nat > set_a] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ ( B @ X2 ) ) )
=> ( member_nat_a @ F @ ( pi_nat_a @ A @ B ) ) ) ).
% Pi_I
thf(fact_9_Pi__I,axiom,
! [A: set_a,F: a > a,B: a > set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ ( B @ X2 ) ) )
=> ( member_a_a @ F @ ( pi_a_a @ A @ B ) ) ) ).
% Pi_I
thf(fact_10_Pi__I,axiom,
! [A: set_nat,F: nat > list_a,B: nat > set_list_a] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_list_a @ ( F @ X2 ) @ ( B @ X2 ) ) )
=> ( member_nat_list_a @ F @ ( pi_nat_list_a @ A @ B ) ) ) ).
% Pi_I
thf(fact_11_Pi__I,axiom,
! [A: set_set_list_a,F: set_list_a > a,B: set_list_a > set_a] :
( ! [X2: set_list_a] :
( ( member_set_list_a @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ ( B @ X2 ) ) )
=> ( member_set_list_a_a @ F @ ( pi_set_list_a_a @ A @ B ) ) ) ).
% Pi_I
thf(fact_12_Pi__I,axiom,
! [A: set_nat_a,F: ( nat > a ) > a,B: ( nat > a ) > set_a] :
( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ ( B @ X2 ) ) )
=> ( member_nat_a_a @ F @ ( pi_nat_a_a @ A @ B ) ) ) ).
% Pi_I
thf(fact_13_Pi__I,axiom,
! [A: set_a_a,F: ( a > a ) > a,B: ( a > a ) > set_a] :
( ! [X2: a > a] :
( ( member_a_a @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ ( B @ X2 ) ) )
=> ( member_a_a_a @ F @ ( pi_a_a_a @ A @ B ) ) ) ).
% Pi_I
thf(fact_14_Pi__I,axiom,
! [A: set_a,F: a > nat > a,B: a > set_nat_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_nat_a @ ( F @ X2 ) @ ( B @ X2 ) ) )
=> ( member_a_nat_a @ F @ ( pi_a_nat_a @ A @ B ) ) ) ).
% Pi_I
thf(fact_15_Pi__I,axiom,
! [A: set_a,F: a > a > a,B: a > set_a_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_a_a @ ( F @ X2 ) @ ( B @ X2 ) ) )
=> ( member_a_a_a2 @ F @ ( pi_a_a_a2 @ A @ B ) ) ) ).
% Pi_I
thf(fact_16_Pi__I,axiom,
! [A: set_nat_list_a,F: ( nat > list_a ) > a,B: ( nat > list_a ) > set_a] :
( ! [X2: nat > list_a] :
( ( member_nat_list_a @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ ( B @ X2 ) ) )
=> ( member_nat_list_a_a @ F @ ( pi_nat_list_a_a @ A @ B ) ) ) ).
% Pi_I
thf(fact_17_Pi__I,axiom,
! [A: set_a,F: a > nat > list_a,B: a > set_nat_list_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_nat_list_a @ ( F @ X2 ) @ ( B @ X2 ) ) )
=> ( member_a_nat_list_a @ F @ ( pi_a_nat_list_a @ A @ B ) ) ) ).
% Pi_I
thf(fact_18_cgenideal__self,axiom,
! [I2: a] :
( ( member_a @ I2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ I2 @ ( cgenid547466209912283029xt_a_b @ r @ I2 ) ) ) ).
% cgenideal_self
thf(fact_19_finsum__closed,axiom,
! [F: set_list_a > a,A: set_set_list_a] :
( ( member_set_list_a_a @ F
@ ( pi_set_list_a_a @ A
@ ^ [Uu: set_list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( member_a @ ( finsum7367453022336983110list_a @ r @ F @ A ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% finsum_closed
thf(fact_20_finsum__closed,axiom,
! [F: nat > a,A: set_nat] :
( ( member_nat_a @ F
@ ( pi_nat_a @ A
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( member_a @ ( finsum_a_b_nat @ r @ F @ A ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% finsum_closed
thf(fact_21_finsum__closed,axiom,
! [F: a > a,A: set_a] :
( ( member_a_a @ F
@ ( pi_a_a @ A
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( member_a @ ( finsum_a_b_a @ r @ F @ A ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% finsum_closed
thf(fact_22_finsum__cong_H,axiom,
! [A: set_set_list_a_a,B: set_set_list_a_a,G: ( set_list_a > a ) > a,F: ( set_list_a > a ) > a] :
( ( A = B )
=> ( ( member969817812316227871_a_a_a @ G
@ ( pi_set_list_a_a_a @ B
@ ^ [Uu: set_list_a > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ! [I3: set_list_a > a] :
( ( member_set_list_a_a @ I3 @ B )
=> ( ( F @ I3 )
= ( G @ I3 ) ) )
=> ( ( finsum7228396637597461149st_a_a @ r @ F @ A )
= ( finsum7228396637597461149st_a_a @ r @ G @ B ) ) ) ) ) ).
% finsum_cong'
thf(fact_23_finsum__cong_H,axiom,
! [A: set_nat_list_a,B: set_nat_list_a,G: ( nat > list_a ) > a,F: ( nat > list_a ) > a] :
( ( A = B )
=> ( ( member_nat_list_a_a @ G
@ ( pi_nat_list_a_a @ B
@ ^ [Uu: nat > list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ! [I3: nat > list_a] :
( ( member_nat_list_a @ I3 @ B )
=> ( ( F @ I3 )
= ( G @ I3 ) ) )
=> ( ( finsum1341700292807219277list_a @ r @ F @ A )
= ( finsum1341700292807219277list_a @ r @ G @ B ) ) ) ) ) ).
% finsum_cong'
thf(fact_24_finsum__cong_H,axiom,
! [A: set_nat_a,B: set_nat_a,G: ( nat > a ) > a,F: ( nat > a ) > a] :
( ( A = B )
=> ( ( member_nat_a_a @ G
@ ( pi_nat_a_a @ B
@ ^ [Uu: nat > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ! [I3: nat > a] :
( ( member_nat_a @ I3 @ B )
=> ( ( F @ I3 )
= ( G @ I3 ) ) )
=> ( ( finsum_a_b_nat_a @ r @ F @ A )
= ( finsum_a_b_nat_a @ r @ G @ B ) ) ) ) ) ).
% finsum_cong'
thf(fact_25_finsum__cong_H,axiom,
! [A: set_a_a,B: set_a_a,G: ( a > a ) > a,F: ( a > a ) > a] :
( ( A = B )
=> ( ( member_a_a_a @ G
@ ( pi_a_a_a @ B
@ ^ [Uu: a > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ! [I3: a > a] :
( ( member_a_a @ I3 @ B )
=> ( ( F @ I3 )
= ( G @ I3 ) ) )
=> ( ( finsum_a_b_a_a @ r @ F @ A )
= ( finsum_a_b_a_a @ r @ G @ B ) ) ) ) ) ).
% finsum_cong'
thf(fact_26_finsum__cong_H,axiom,
! [A: set_set_list_a,B: set_set_list_a,G: set_list_a > a,F: set_list_a > a] :
( ( A = B )
=> ( ( member_set_list_a_a @ G
@ ( pi_set_list_a_a @ B
@ ^ [Uu: set_list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ! [I3: set_list_a] :
( ( member_set_list_a @ I3 @ B )
=> ( ( F @ I3 )
= ( G @ I3 ) ) )
=> ( ( finsum7367453022336983110list_a @ r @ F @ A )
= ( finsum7367453022336983110list_a @ r @ G @ B ) ) ) ) ) ).
% finsum_cong'
thf(fact_27_finsum__cong_H,axiom,
! [A: set_nat,B: set_nat,G: nat > a,F: nat > a] :
( ( A = B )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ B
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ B )
=> ( ( F @ I3 )
= ( G @ I3 ) ) )
=> ( ( finsum_a_b_nat @ r @ F @ A )
= ( finsum_a_b_nat @ r @ G @ B ) ) ) ) ) ).
% finsum_cong'
thf(fact_28_finsum__cong_H,axiom,
! [A: set_a,B: set_a,G: a > a,F: a > a] :
( ( A = B )
=> ( ( member_a_a @ G
@ ( pi_a_a @ B
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ! [I3: a] :
( ( member_a @ I3 @ B )
=> ( ( F @ I3 )
= ( G @ I3 ) ) )
=> ( ( finsum_a_b_a @ r @ F @ A )
= ( finsum_a_b_a @ r @ G @ B ) ) ) ) ) ).
% finsum_cong'
thf(fact_29_finprod__closed,axiom,
! [F: set_list_a > a,A: set_set_list_a] :
( ( member_set_list_a_a @ F
@ ( pi_set_list_a_a @ A
@ ^ [Uu: set_list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( member_a @ ( finpro3826550488720007709list_a @ r @ F @ A ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% finprod_closed
thf(fact_30_finprod__closed,axiom,
! [F: nat > a,A: set_nat] :
( ( member_nat_a @ F
@ ( pi_nat_a @ A
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( member_a @ ( finpro1280035270526425175_b_nat @ r @ F @ A ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% finprod_closed
thf(fact_31_finprod__closed,axiom,
! [F: a > a,A: set_a] :
( ( member_a_a @ F
@ ( pi_a_a @ A
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( member_a @ ( finpro205304725090349623_a_b_a @ r @ F @ A ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% finprod_closed
thf(fact_32_finprod__cong_H,axiom,
! [A: set_set_list_a_a,B: set_set_list_a_a,G: ( set_list_a > a ) > a,F: ( set_list_a > a ) > a] :
( ( A = B )
=> ( ( member969817812316227871_a_a_a @ G
@ ( pi_set_list_a_a_a @ B
@ ^ [Uu: set_list_a > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ! [I3: set_list_a > a] :
( ( member_set_list_a_a @ I3 @ B )
=> ( ( F @ I3 )
= ( G @ I3 ) ) )
=> ( ( finpro4938371440467910406st_a_a @ r @ F @ A )
= ( finpro4938371440467910406st_a_a @ r @ G @ B ) ) ) ) ) ).
% finprod_cong'
thf(fact_33_finprod__cong_H,axiom,
! [A: set_nat_list_a,B: set_nat_list_a,G: ( nat > list_a ) > a,F: ( nat > list_a ) > a] :
( ( A = B )
=> ( ( member_nat_list_a_a @ G
@ ( pi_nat_list_a_a @ B
@ ^ [Uu: nat > list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ! [I3: nat > list_a] :
( ( member_nat_list_a @ I3 @ B )
=> ( ( F @ I3 )
= ( G @ I3 ) ) )
=> ( ( finpro4838020199848830884list_a @ r @ F @ A )
= ( finpro4838020199848830884list_a @ r @ G @ B ) ) ) ) ) ).
% finprod_cong'
thf(fact_34_finprod__cong_H,axiom,
! [A: set_nat_a,B: set_nat_a,G: ( nat > a ) > a,F: ( nat > a ) > a] :
( ( A = B )
=> ( ( member_nat_a_a @ G
@ ( pi_nat_a_a @ B
@ ^ [Uu: nat > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ! [I3: nat > a] :
( ( member_nat_a @ I3 @ B )
=> ( ( F @ I3 )
= ( G @ I3 ) ) )
=> ( ( finpro5839458686994656414_nat_a @ r @ F @ A )
= ( finpro5839458686994656414_nat_a @ r @ G @ B ) ) ) ) ) ).
% finprod_cong'
thf(fact_35_finprod__cong_H,axiom,
! [A: set_a_a,B: set_a_a,G: ( a > a ) > a,F: ( a > a ) > a] :
( ( A = B )
=> ( ( member_a_a_a @ G
@ ( pi_a_a_a @ B
@ ^ [Uu: a > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ! [I3: a > a] :
( ( member_a_a @ I3 @ B )
=> ( ( F @ I3 )
= ( G @ I3 ) ) )
=> ( ( finpro3012607322079259884_b_a_a @ r @ F @ A )
= ( finpro3012607322079259884_b_a_a @ r @ G @ B ) ) ) ) ) ).
% finprod_cong'
thf(fact_36_finprod__cong_H,axiom,
! [A: set_set_list_a,B: set_set_list_a,G: set_list_a > a,F: set_list_a > a] :
( ( A = B )
=> ( ( member_set_list_a_a @ G
@ ( pi_set_list_a_a @ B
@ ^ [Uu: set_list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ! [I3: set_list_a] :
( ( member_set_list_a @ I3 @ B )
=> ( ( F @ I3 )
= ( G @ I3 ) ) )
=> ( ( finpro3826550488720007709list_a @ r @ F @ A )
= ( finpro3826550488720007709list_a @ r @ G @ B ) ) ) ) ) ).
% finprod_cong'
thf(fact_37_finprod__cong_H,axiom,
! [A: set_nat,B: set_nat,G: nat > a,F: nat > a] :
( ( A = B )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ B
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ B )
=> ( ( F @ I3 )
= ( G @ I3 ) ) )
=> ( ( finpro1280035270526425175_b_nat @ r @ F @ A )
= ( finpro1280035270526425175_b_nat @ r @ G @ B ) ) ) ) ) ).
% finprod_cong'
thf(fact_38_finprod__cong_H,axiom,
! [A: set_a,B: set_a,G: a > a,F: a > a] :
( ( A = B )
=> ( ( member_a_a @ G
@ ( pi_a_a @ B
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ! [I3: a] :
( ( member_a @ I3 @ B )
=> ( ( F @ I3 )
= ( G @ I3 ) ) )
=> ( ( finpro205304725090349623_a_b_a @ r @ F @ A )
= ( finpro205304725090349623_a_b_a @ r @ G @ B ) ) ) ) ) ).
% finprod_cong'
thf(fact_39_funcset__carrier,axiom,
! [F: a > a,X3: partia2175431115845679010xt_a_b,Y2: partia2175431115845679010xt_a_b,X: a] :
( ( member_a_a @ F
@ ( pi_a_a @ ( partia707051561876973205xt_a_b @ X3 )
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ Y2 ) ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ X3 ) )
=> ( member_a @ ( F @ X ) @ ( partia707051561876973205xt_a_b @ Y2 ) ) ) ) ).
% funcset_carrier
thf(fact_40_funcset__carrier,axiom,
! [F: a > set_list_a,X3: partia2175431115845679010xt_a_b,Y2: partia1167524930811108609t_unit,X: a] :
( ( member_a_set_list_a @ F
@ ( pi_a_set_list_a @ ( partia707051561876973205xt_a_b @ X3 )
@ ^ [Uu: a] : ( partia5178357399839081912t_unit @ Y2 ) ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ X3 ) )
=> ( member_set_list_a @ ( F @ X ) @ ( partia5178357399839081912t_unit @ Y2 ) ) ) ) ).
% funcset_carrier
thf(fact_41_funcset__carrier,axiom,
! [F: set_list_a > a,X3: partia1167524930811108609t_unit,Y2: partia2175431115845679010xt_a_b,X: set_list_a] :
( ( member_set_list_a_a @ F
@ ( pi_set_list_a_a @ ( partia5178357399839081912t_unit @ X3 )
@ ^ [Uu: set_list_a] : ( partia707051561876973205xt_a_b @ Y2 ) ) )
=> ( ( member_set_list_a @ X @ ( partia5178357399839081912t_unit @ X3 ) )
=> ( member_a @ ( F @ X ) @ ( partia707051561876973205xt_a_b @ Y2 ) ) ) ) ).
% funcset_carrier
thf(fact_42_funcset__carrier,axiom,
! [F: a > list_a,X3: partia2175431115845679010xt_a_b,Y2: partia2670972154091845814t_unit,X: a] :
( ( member_a_list_a @ F
@ ( pi_a_list_a @ ( partia707051561876973205xt_a_b @ X3 )
@ ^ [Uu: a] : ( partia5361259788508890537t_unit @ Y2 ) ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ X3 ) )
=> ( member_list_a @ ( F @ X ) @ ( partia5361259788508890537t_unit @ Y2 ) ) ) ) ).
% funcset_carrier
thf(fact_43_funcset__carrier,axiom,
! [F: list_a > a,X3: partia2670972154091845814t_unit,Y2: partia2175431115845679010xt_a_b,X: list_a] :
( ( member_list_a_a @ F
@ ( pi_list_a_a @ ( partia5361259788508890537t_unit @ X3 )
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ Y2 ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ X3 ) )
=> ( member_a @ ( F @ X ) @ ( partia707051561876973205xt_a_b @ Y2 ) ) ) ) ).
% funcset_carrier
thf(fact_44_funcset__carrier,axiom,
! [F: set_list_a > set_list_a,X3: partia1167524930811108609t_unit,Y2: partia1167524930811108609t_unit,X: set_list_a] :
( ( member5068272912271824380list_a @ F
@ ( pi_set2094285526804900683list_a @ ( partia5178357399839081912t_unit @ X3 )
@ ^ [Uu: set_list_a] : ( partia5178357399839081912t_unit @ Y2 ) ) )
=> ( ( member_set_list_a @ X @ ( partia5178357399839081912t_unit @ X3 ) )
=> ( member_set_list_a @ ( F @ X ) @ ( partia5178357399839081912t_unit @ Y2 ) ) ) ) ).
% funcset_carrier
thf(fact_45_funcset__carrier,axiom,
! [F: list_a > set_list_a,X3: partia2670972154091845814t_unit,Y2: partia1167524930811108609t_unit,X: list_a] :
( ( member4263473470251683292list_a @ F
@ ( pi_list_a_set_list_a @ ( partia5361259788508890537t_unit @ X3 )
@ ^ [Uu: list_a] : ( partia5178357399839081912t_unit @ Y2 ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ X3 ) )
=> ( member_set_list_a @ ( F @ X ) @ ( partia5178357399839081912t_unit @ Y2 ) ) ) ) ).
% funcset_carrier
thf(fact_46_funcset__carrier,axiom,
! [F: set_list_a > list_a,X3: partia1167524930811108609t_unit,Y2: partia2670972154091845814t_unit,X: set_list_a] :
( ( member5910328476188217884list_a @ F
@ ( pi_set_list_a_list_a @ ( partia5178357399839081912t_unit @ X3 )
@ ^ [Uu: set_list_a] : ( partia5361259788508890537t_unit @ Y2 ) ) )
=> ( ( member_set_list_a @ X @ ( partia5178357399839081912t_unit @ X3 ) )
=> ( member_list_a @ ( F @ X ) @ ( partia5361259788508890537t_unit @ Y2 ) ) ) ) ).
% funcset_carrier
thf(fact_47_funcset__carrier,axiom,
! [F: a > list_list_a,X3: partia2175431115845679010xt_a_b,Y2: partia2956882679547061052t_unit,X: a] :
( ( member_a_list_list_a @ F
@ ( pi_a_list_list_a @ ( partia707051561876973205xt_a_b @ X3 )
@ ^ [Uu: a] : ( partia2464479390973590831t_unit @ Y2 ) ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ X3 ) )
=> ( member_list_list_a @ ( F @ X ) @ ( partia2464479390973590831t_unit @ Y2 ) ) ) ) ).
% funcset_carrier
thf(fact_48_funcset__carrier,axiom,
! [F: a > set_list_a,X3: partia2175431115845679010xt_a_b,Y2: partia7496981018696276118t_unit,X: a] :
( ( member_a_set_list_a @ F
@ ( pi_a_set_list_a @ ( partia707051561876973205xt_a_b @ X3 )
@ ^ [Uu: a] : ( partia141011252114345353t_unit @ Y2 ) ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ X3 ) )
=> ( member_set_list_a @ ( F @ X ) @ ( partia141011252114345353t_unit @ Y2 ) ) ) ) ).
% funcset_carrier
thf(fact_49_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_50_Pi__anti__mono,axiom,
! [A2: set_a,A: set_a,B: a > set_a] :
( ( ord_less_eq_set_a @ A2 @ A )
=> ( ord_less_eq_set_a_a @ ( pi_a_a @ A @ B ) @ ( pi_a_a @ A2 @ B ) ) ) ).
% Pi_anti_mono
thf(fact_51_Pi__mono,axiom,
! [A: set_set_list_a_a,B: ( set_list_a > a ) > set_a,C: ( set_list_a > a ) > set_a] :
( ! [X2: set_list_a > a] :
( ( member_set_list_a_a @ X2 @ A )
=> ( ord_less_eq_set_a @ ( B @ X2 ) @ ( C @ X2 ) ) )
=> ( ord_le3681906544784463606_a_a_a @ ( pi_set_list_a_a_a @ A @ B ) @ ( pi_set_list_a_a_a @ A @ C ) ) ) ).
% Pi_mono
thf(fact_52_Pi__mono,axiom,
! [A: set_nat_list_a,B: ( nat > list_a ) > set_a,C: ( nat > list_a ) > set_a] :
( ! [X2: nat > list_a] :
( ( member_nat_list_a @ X2 @ A )
=> ( ord_less_eq_set_a @ ( B @ X2 ) @ ( C @ X2 ) ) )
=> ( ord_le6710563759049263694st_a_a @ ( pi_nat_list_a_a @ A @ B ) @ ( pi_nat_list_a_a @ A @ C ) ) ) ).
% Pi_mono
thf(fact_53_Pi__mono,axiom,
! [A: set_nat_a,B: ( nat > a ) > set_a,C: ( nat > a ) > set_a] :
( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A )
=> ( ord_less_eq_set_a @ ( B @ X2 ) @ ( C @ X2 ) ) )
=> ( ord_le3509452538356653652at_a_a @ ( pi_nat_a_a @ A @ B ) @ ( pi_nat_a_a @ A @ C ) ) ) ).
% Pi_mono
thf(fact_54_Pi__mono,axiom,
! [A: set_a_a,B: ( a > a ) > set_a,C: ( a > a ) > set_a] :
( ! [X2: a > a] :
( ( member_a_a @ X2 @ A )
=> ( ord_less_eq_set_a @ ( B @ X2 ) @ ( C @ X2 ) ) )
=> ( ord_le7181591058469194768_a_a_a @ ( pi_a_a_a @ A @ B ) @ ( pi_a_a_a @ A @ C ) ) ) ).
% Pi_mono
thf(fact_55_Pi__mono,axiom,
! [A: set_a,B: a > set_a,C: a > set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ord_less_eq_set_a @ ( B @ X2 ) @ ( C @ X2 ) ) )
=> ( ord_less_eq_set_a_a @ ( pi_a_a @ A @ B ) @ ( pi_a_a @ A @ C ) ) ) ).
% Pi_mono
thf(fact_56_Pi__mono,axiom,
! [A: set_set_list_a_a,B: ( set_list_a > a ) > set_list_a,C: ( set_list_a > a ) > set_list_a] :
( ! [X2: set_list_a > a] :
( ( member_set_list_a_a @ X2 @ A )
=> ( ord_le8861187494160871172list_a @ ( B @ X2 ) @ ( C @ X2 ) ) )
=> ( ord_le6279516915688846460list_a @ ( pi_set7883948722368885176list_a @ A @ B ) @ ( pi_set7883948722368885176list_a @ A @ C ) ) ) ).
% Pi_mono
thf(fact_57_Pi__mono,axiom,
! [A: set_nat_list_a,B: ( nat > list_a ) > set_list_a,C: ( nat > list_a ) > set_list_a] :
( ! [X2: nat > list_a] :
( ( member_nat_list_a @ X2 @ A )
=> ( ord_le8861187494160871172list_a @ ( B @ X2 ) @ ( C @ X2 ) ) )
=> ( ord_le4586323718659152852list_a @ ( pi_nat_list_a_list_a @ A @ B ) @ ( pi_nat_list_a_list_a @ A @ C ) ) ) ).
% Pi_mono
thf(fact_58_Pi__mono,axiom,
! [A: set_nat_a,B: ( nat > a ) > set_list_a,C: ( nat > a ) > set_list_a] :
( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A )
=> ( ord_le8861187494160871172list_a @ ( B @ X2 ) @ ( C @ X2 ) ) )
=> ( ord_le8592148164225365210list_a @ ( pi_nat_a_list_a @ A @ B ) @ ( pi_nat_a_list_a @ A @ C ) ) ) ).
% Pi_mono
thf(fact_59_Pi__mono,axiom,
! [A: set_a_a,B: ( a > a ) > set_list_a,C: ( a > a ) > set_list_a] :
( ! [X2: a > a] :
( ( member_a_a @ X2 @ A )
=> ( ord_le8861187494160871172list_a @ ( B @ X2 ) @ ( C @ X2 ) ) )
=> ( ord_le4729556964648129686list_a @ ( pi_a_a_list_a @ A @ B ) @ ( pi_a_a_list_a @ A @ C ) ) ) ).
% Pi_mono
thf(fact_60_Pi__mono,axiom,
! [A: set_a,B: a > set_list_a,C: a > set_list_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ord_le8861187494160871172list_a @ ( B @ X2 ) @ ( C @ X2 ) ) )
=> ( ord_le50412136050534657list_a @ ( pi_a_list_a @ A @ B ) @ ( pi_a_list_a @ A @ C ) ) ) ).
% Pi_mono
thf(fact_61_Pi__cong,axiom,
! [A: set_set_list_a,F: set_list_a > a,G: set_list_a > a,B: set_list_a > set_a] :
( ! [W: set_list_a] :
( ( member_set_list_a @ W @ A )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( member_set_list_a_a @ F @ ( pi_set_list_a_a @ A @ B ) )
= ( member_set_list_a_a @ G @ ( pi_set_list_a_a @ A @ B ) ) ) ) ).
% Pi_cong
thf(fact_62_Pi__cong,axiom,
! [A: set_nat,F: nat > list_a,G: nat > list_a,B: nat > set_list_a] :
( ! [W: nat] :
( ( member_nat @ W @ A )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( member_nat_list_a @ F @ ( pi_nat_list_a @ A @ B ) )
= ( member_nat_list_a @ G @ ( pi_nat_list_a @ A @ B ) ) ) ) ).
% Pi_cong
thf(fact_63_Pi__cong,axiom,
! [A: set_nat,F: nat > a,G: nat > a,B: nat > set_a] :
( ! [W: nat] :
( ( member_nat @ W @ A )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( member_nat_a @ F @ ( pi_nat_a @ A @ B ) )
= ( member_nat_a @ G @ ( pi_nat_a @ A @ B ) ) ) ) ).
% Pi_cong
thf(fact_64_Pi__cong,axiom,
! [A: set_a,F: a > a,G: a > a,B: a > set_a] :
( ! [W: a] :
( ( member_a @ W @ A )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( member_a_a @ F @ ( pi_a_a @ A @ B ) )
= ( member_a_a @ G @ ( pi_a_a @ A @ B ) ) ) ) ).
% Pi_cong
thf(fact_65_Pi__mem,axiom,
! [F: nat > a,A: set_nat,B: nat > set_a,X: nat] :
( ( member_nat_a @ F @ ( pi_nat_a @ A @ B ) )
=> ( ( member_nat @ X @ A )
=> ( member_a @ ( F @ X ) @ ( B @ X ) ) ) ) ).
% Pi_mem
thf(fact_66_Pi__mem,axiom,
! [F: a > a,A: set_a,B: a > set_a,X: a] :
( ( member_a_a @ F @ ( pi_a_a @ A @ B ) )
=> ( ( member_a @ X @ A )
=> ( member_a @ ( F @ X ) @ ( B @ X ) ) ) ) ).
% Pi_mem
thf(fact_67_Pi__mem,axiom,
! [F: nat > list_a,A: set_nat,B: nat > set_list_a,X: nat] :
( ( member_nat_list_a @ F @ ( pi_nat_list_a @ A @ B ) )
=> ( ( member_nat @ X @ A )
=> ( member_list_a @ ( F @ X ) @ ( B @ X ) ) ) ) ).
% Pi_mem
thf(fact_68_Pi__mem,axiom,
! [F: ( nat > a ) > a,A: set_nat_a,B: ( nat > a ) > set_a,X: nat > a] :
( ( member_nat_a_a @ F @ ( pi_nat_a_a @ A @ B ) )
=> ( ( member_nat_a @ X @ A )
=> ( member_a @ ( F @ X ) @ ( B @ X ) ) ) ) ).
% Pi_mem
thf(fact_69_Pi__mem,axiom,
! [F: ( a > a ) > a,A: set_a_a,B: ( a > a ) > set_a,X: a > a] :
( ( member_a_a_a @ F @ ( pi_a_a_a @ A @ B ) )
=> ( ( member_a_a @ X @ A )
=> ( member_a @ ( F @ X ) @ ( B @ X ) ) ) ) ).
% Pi_mem
thf(fact_70_Pi__mem,axiom,
! [F: a > nat > a,A: set_a,B: a > set_nat_a,X: a] :
( ( member_a_nat_a @ F @ ( pi_a_nat_a @ A @ B ) )
=> ( ( member_a @ X @ A )
=> ( member_nat_a @ ( F @ X ) @ ( B @ X ) ) ) ) ).
% Pi_mem
thf(fact_71_Pi__mem,axiom,
! [F: a > a > a,A: set_a,B: a > set_a_a,X: a] :
( ( member_a_a_a2 @ F @ ( pi_a_a_a2 @ A @ B ) )
=> ( ( member_a @ X @ A )
=> ( member_a_a @ ( F @ X ) @ ( B @ X ) ) ) ) ).
% Pi_mem
thf(fact_72_Pi__mem,axiom,
! [F: set_list_a > a,A: set_set_list_a,B: set_list_a > set_a,X: set_list_a] :
( ( member_set_list_a_a @ F @ ( pi_set_list_a_a @ A @ B ) )
=> ( ( member_set_list_a @ X @ A )
=> ( member_a @ ( F @ X ) @ ( B @ X ) ) ) ) ).
% Pi_mem
thf(fact_73_Pi__mem,axiom,
! [F: ( nat > list_a ) > a,A: set_nat_list_a,B: ( nat > list_a ) > set_a,X: nat > list_a] :
( ( member_nat_list_a_a @ F @ ( pi_nat_list_a_a @ A @ B ) )
=> ( ( member_nat_list_a @ X @ A )
=> ( member_a @ ( F @ X ) @ ( B @ X ) ) ) ) ).
% Pi_mem
thf(fact_74_Pi__mem,axiom,
! [F: a > nat > list_a,A: set_a,B: a > set_nat_list_a,X: a] :
( ( member_a_nat_list_a @ F @ ( pi_a_nat_list_a @ A @ B ) )
=> ( ( member_a @ X @ A )
=> ( member_nat_list_a @ ( F @ X ) @ ( B @ X ) ) ) ) ).
% Pi_mem
thf(fact_75_Pi__iff,axiom,
! [F: set_list_a > a,I4: set_set_list_a,X3: set_list_a > set_a] :
( ( member_set_list_a_a @ F @ ( pi_set_list_a_a @ I4 @ X3 ) )
= ( ! [X4: set_list_a] :
( ( member_set_list_a @ X4 @ I4 )
=> ( member_a @ ( F @ X4 ) @ ( X3 @ X4 ) ) ) ) ) ).
% Pi_iff
thf(fact_76_Pi__iff,axiom,
! [F: nat > list_a,I4: set_nat,X3: nat > set_list_a] :
( ( member_nat_list_a @ F @ ( pi_nat_list_a @ I4 @ X3 ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ I4 )
=> ( member_list_a @ ( F @ X4 ) @ ( X3 @ X4 ) ) ) ) ) ).
% Pi_iff
thf(fact_77_Pi__iff,axiom,
! [F: nat > a,I4: set_nat,X3: nat > set_a] :
( ( member_nat_a @ F @ ( pi_nat_a @ I4 @ X3 ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ I4 )
=> ( member_a @ ( F @ X4 ) @ ( X3 @ X4 ) ) ) ) ) ).
% Pi_iff
thf(fact_78_Pi__iff,axiom,
! [F: a > a,I4: set_a,X3: a > set_a] :
( ( member_a_a @ F @ ( pi_a_a @ I4 @ X3 ) )
= ( ! [X4: a] :
( ( member_a @ X4 @ I4 )
=> ( member_a @ ( F @ X4 ) @ ( X3 @ X4 ) ) ) ) ) ).
% Pi_iff
thf(fact_79_Pi__I_H,axiom,
! [A: set_nat,F: nat > a,B: nat > set_a] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ ( B @ X2 ) ) )
=> ( member_nat_a @ F @ ( pi_nat_a @ A @ B ) ) ) ).
% Pi_I'
thf(fact_80_Pi__I_H,axiom,
! [A: set_a,F: a > a,B: a > set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ ( B @ X2 ) ) )
=> ( member_a_a @ F @ ( pi_a_a @ A @ B ) ) ) ).
% Pi_I'
thf(fact_81_Pi__I_H,axiom,
! [A: set_nat,F: nat > list_a,B: nat > set_list_a] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_list_a @ ( F @ X2 ) @ ( B @ X2 ) ) )
=> ( member_nat_list_a @ F @ ( pi_nat_list_a @ A @ B ) ) ) ).
% Pi_I'
thf(fact_82_Pi__I_H,axiom,
! [A: set_set_list_a,F: set_list_a > a,B: set_list_a > set_a] :
( ! [X2: set_list_a] :
( ( member_set_list_a @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ ( B @ X2 ) ) )
=> ( member_set_list_a_a @ F @ ( pi_set_list_a_a @ A @ B ) ) ) ).
% Pi_I'
thf(fact_83_Pi__I_H,axiom,
! [A: set_nat_a,F: ( nat > a ) > a,B: ( nat > a ) > set_a] :
( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ ( B @ X2 ) ) )
=> ( member_nat_a_a @ F @ ( pi_nat_a_a @ A @ B ) ) ) ).
% Pi_I'
thf(fact_84_Pi__I_H,axiom,
! [A: set_a_a,F: ( a > a ) > a,B: ( a > a ) > set_a] :
( ! [X2: a > a] :
( ( member_a_a @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ ( B @ X2 ) ) )
=> ( member_a_a_a @ F @ ( pi_a_a_a @ A @ B ) ) ) ).
% Pi_I'
thf(fact_85_Pi__I_H,axiom,
! [A: set_a,F: a > nat > a,B: a > set_nat_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_nat_a @ ( F @ X2 ) @ ( B @ X2 ) ) )
=> ( member_a_nat_a @ F @ ( pi_a_nat_a @ A @ B ) ) ) ).
% Pi_I'
thf(fact_86_Pi__I_H,axiom,
! [A: set_a,F: a > a > a,B: a > set_a_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_a_a @ ( F @ X2 ) @ ( B @ X2 ) ) )
=> ( member_a_a_a2 @ F @ ( pi_a_a_a2 @ A @ B ) ) ) ).
% Pi_I'
thf(fact_87_Pi__I_H,axiom,
! [A: set_nat_list_a,F: ( nat > list_a ) > a,B: ( nat > list_a ) > set_a] :
( ! [X2: nat > list_a] :
( ( member_nat_list_a @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ ( B @ X2 ) ) )
=> ( member_nat_list_a_a @ F @ ( pi_nat_list_a_a @ A @ B ) ) ) ).
% Pi_I'
thf(fact_88_Pi__I_H,axiom,
! [A: set_a,F: a > nat > list_a,B: a > set_nat_list_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_nat_list_a @ ( F @ X2 ) @ ( B @ X2 ) ) )
=> ( member_a_nat_list_a @ F @ ( pi_a_nat_list_a @ A @ B ) ) ) ).
% Pi_I'
thf(fact_89_PiE,axiom,
! [F: nat > a,A: set_nat,B: nat > set_a,X: nat] :
( ( member_nat_a @ F @ ( pi_nat_a @ A @ B ) )
=> ( ~ ( member_a @ ( F @ X ) @ ( B @ X ) )
=> ~ ( member_nat @ X @ A ) ) ) ).
% PiE
thf(fact_90_PiE,axiom,
! [F: a > a,A: set_a,B: a > set_a,X: a] :
( ( member_a_a @ F @ ( pi_a_a @ A @ B ) )
=> ( ~ ( member_a @ ( F @ X ) @ ( B @ X ) )
=> ~ ( member_a @ X @ A ) ) ) ).
% PiE
thf(fact_91_PiE,axiom,
! [F: nat > list_a,A: set_nat,B: nat > set_list_a,X: nat] :
( ( member_nat_list_a @ F @ ( pi_nat_list_a @ A @ B ) )
=> ( ~ ( member_list_a @ ( F @ X ) @ ( B @ X ) )
=> ~ ( member_nat @ X @ A ) ) ) ).
% PiE
thf(fact_92_PiE,axiom,
! [F: a > nat > a,A: set_a,B: a > set_nat_a,X: a] :
( ( member_a_nat_a @ F @ ( pi_a_nat_a @ A @ B ) )
=> ( ~ ( member_nat_a @ ( F @ X ) @ ( B @ X ) )
=> ~ ( member_a @ X @ A ) ) ) ).
% PiE
thf(fact_93_PiE,axiom,
! [F: a > a > a,A: set_a,B: a > set_a_a,X: a] :
( ( member_a_a_a2 @ F @ ( pi_a_a_a2 @ A @ B ) )
=> ( ~ ( member_a_a @ ( F @ X ) @ ( B @ X ) )
=> ~ ( member_a @ X @ A ) ) ) ).
% PiE
thf(fact_94_PiE,axiom,
! [F: ( nat > a ) > a,A: set_nat_a,B: ( nat > a ) > set_a,X: nat > a] :
( ( member_nat_a_a @ F @ ( pi_nat_a_a @ A @ B ) )
=> ( ~ ( member_a @ ( F @ X ) @ ( B @ X ) )
=> ~ ( member_nat_a @ X @ A ) ) ) ).
% PiE
thf(fact_95_PiE,axiom,
! [F: ( a > a ) > a,A: set_a_a,B: ( a > a ) > set_a,X: a > a] :
( ( member_a_a_a @ F @ ( pi_a_a_a @ A @ B ) )
=> ( ~ ( member_a @ ( F @ X ) @ ( B @ X ) )
=> ~ ( member_a_a @ X @ A ) ) ) ).
% PiE
thf(fact_96_PiE,axiom,
! [F: set_list_a > a,A: set_set_list_a,B: set_list_a > set_a,X: set_list_a] :
( ( member_set_list_a_a @ F @ ( pi_set_list_a_a @ A @ B ) )
=> ( ~ ( member_a @ ( F @ X ) @ ( B @ X ) )
=> ~ ( member_set_list_a @ X @ A ) ) ) ).
% PiE
thf(fact_97_PiE,axiom,
! [F: a > nat > list_a,A: set_a,B: a > set_nat_list_a,X: a] :
( ( member_a_nat_list_a @ F @ ( pi_a_nat_list_a @ A @ B ) )
=> ( ~ ( member_nat_list_a @ ( F @ X ) @ ( B @ X ) )
=> ~ ( member_a @ X @ A ) ) ) ).
% PiE
thf(fact_98_PiE,axiom,
! [F: ( nat > list_a ) > a,A: set_nat_list_a,B: ( nat > list_a ) > set_a,X: nat > list_a] :
( ( member_nat_list_a_a @ F @ ( pi_nat_list_a_a @ A @ B ) )
=> ( ~ ( member_a @ ( F @ X ) @ ( B @ X ) )
=> ~ ( member_nat_list_a @ X @ A ) ) ) ).
% PiE
thf(fact_99_funcset__mem,axiom,
! [F: nat > a,A: set_nat,B: set_a,X: nat] :
( ( member_nat_a @ F
@ ( pi_nat_a @ A
@ ^ [Uu: nat] : B ) )
=> ( ( member_nat @ X @ A )
=> ( member_a @ ( F @ X ) @ B ) ) ) ).
% funcset_mem
thf(fact_100_funcset__mem,axiom,
! [F: a > a,A: set_a,B: set_a,X: a] :
( ( member_a_a @ F
@ ( pi_a_a @ A
@ ^ [Uu: a] : B ) )
=> ( ( member_a @ X @ A )
=> ( member_a @ ( F @ X ) @ B ) ) ) ).
% funcset_mem
thf(fact_101_funcset__mem,axiom,
! [F: nat > list_a,A: set_nat,B: set_list_a,X: nat] :
( ( member_nat_list_a @ F
@ ( pi_nat_list_a @ A
@ ^ [Uu: nat] : B ) )
=> ( ( member_nat @ X @ A )
=> ( member_list_a @ ( F @ X ) @ B ) ) ) ).
% funcset_mem
thf(fact_102_funcset__mem,axiom,
! [F: ( nat > a ) > a,A: set_nat_a,B: set_a,X: nat > a] :
( ( member_nat_a_a @ F
@ ( pi_nat_a_a @ A
@ ^ [Uu: nat > a] : B ) )
=> ( ( member_nat_a @ X @ A )
=> ( member_a @ ( F @ X ) @ B ) ) ) ).
% funcset_mem
thf(fact_103_funcset__mem,axiom,
! [F: ( a > a ) > a,A: set_a_a,B: set_a,X: a > a] :
( ( member_a_a_a @ F
@ ( pi_a_a_a @ A
@ ^ [Uu: a > a] : B ) )
=> ( ( member_a_a @ X @ A )
=> ( member_a @ ( F @ X ) @ B ) ) ) ).
% funcset_mem
thf(fact_104_funcset__mem,axiom,
! [F: a > nat > a,A: set_a,B: set_nat_a,X: a] :
( ( member_a_nat_a @ F
@ ( pi_a_nat_a @ A
@ ^ [Uu: a] : B ) )
=> ( ( member_a @ X @ A )
=> ( member_nat_a @ ( F @ X ) @ B ) ) ) ).
% funcset_mem
thf(fact_105_funcset__mem,axiom,
! [F: a > a > a,A: set_a,B: set_a_a,X: a] :
( ( member_a_a_a2 @ F
@ ( pi_a_a_a2 @ A
@ ^ [Uu: a] : B ) )
=> ( ( member_a @ X @ A )
=> ( member_a_a @ ( F @ X ) @ B ) ) ) ).
% funcset_mem
thf(fact_106_funcset__mem,axiom,
! [F: set_list_a > a,A: set_set_list_a,B: set_a,X: set_list_a] :
( ( member_set_list_a_a @ F
@ ( pi_set_list_a_a @ A
@ ^ [Uu: set_list_a] : B ) )
=> ( ( member_set_list_a @ X @ A )
=> ( member_a @ ( F @ X ) @ B ) ) ) ).
% funcset_mem
thf(fact_107_funcset__mem,axiom,
! [F: ( nat > list_a ) > a,A: set_nat_list_a,B: set_a,X: nat > list_a] :
( ( member_nat_list_a_a @ F
@ ( pi_nat_list_a_a @ A
@ ^ [Uu: nat > list_a] : B ) )
=> ( ( member_nat_list_a @ X @ A )
=> ( member_a @ ( F @ X ) @ B ) ) ) ).
% funcset_mem
thf(fact_108_funcset__mem,axiom,
! [F: a > nat > list_a,A: set_a,B: set_nat_list_a,X: a] :
( ( member_a_nat_list_a @ F
@ ( pi_a_nat_list_a @ A
@ ^ [Uu: a] : B ) )
=> ( ( member_a @ X @ A )
=> ( member_nat_list_a @ ( F @ X ) @ B ) ) ) ).
% funcset_mem
thf(fact_109_funcset__id,axiom,
! [A: set_a] :
( member_a_a
@ ^ [X4: a] : X4
@ ( pi_a_a @ A
@ ^ [Uu: a] : A ) ) ).
% funcset_id
thf(fact_110_funcsetI,axiom,
! [A: set_nat,F: nat > a,B: set_a] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ B ) )
=> ( member_nat_a @ F
@ ( pi_nat_a @ A
@ ^ [Uu: nat] : B ) ) ) ).
% funcsetI
thf(fact_111_funcsetI,axiom,
! [A: set_a,F: a > a,B: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ B ) )
=> ( member_a_a @ F
@ ( pi_a_a @ A
@ ^ [Uu: a] : B ) ) ) ).
% funcsetI
thf(fact_112_funcsetI,axiom,
! [A: set_nat,F: nat > list_a,B: set_list_a] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_list_a @ ( F @ X2 ) @ B ) )
=> ( member_nat_list_a @ F
@ ( pi_nat_list_a @ A
@ ^ [Uu: nat] : B ) ) ) ).
% funcsetI
thf(fact_113_funcsetI,axiom,
! [A: set_set_list_a,F: set_list_a > a,B: set_a] :
( ! [X2: set_list_a] :
( ( member_set_list_a @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ B ) )
=> ( member_set_list_a_a @ F
@ ( pi_set_list_a_a @ A
@ ^ [Uu: set_list_a] : B ) ) ) ).
% funcsetI
thf(fact_114_funcsetI,axiom,
! [A: set_nat_a,F: ( nat > a ) > a,B: set_a] :
( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ B ) )
=> ( member_nat_a_a @ F
@ ( pi_nat_a_a @ A
@ ^ [Uu: nat > a] : B ) ) ) ).
% funcsetI
thf(fact_115_funcsetI,axiom,
! [A: set_a_a,F: ( a > a ) > a,B: set_a] :
( ! [X2: a > a] :
( ( member_a_a @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ B ) )
=> ( member_a_a_a @ F
@ ( pi_a_a_a @ A
@ ^ [Uu: a > a] : B ) ) ) ).
% funcsetI
thf(fact_116_funcsetI,axiom,
! [A: set_a,F: a > nat > a,B: set_nat_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_nat_a @ ( F @ X2 ) @ B ) )
=> ( member_a_nat_a @ F
@ ( pi_a_nat_a @ A
@ ^ [Uu: a] : B ) ) ) ).
% funcsetI
thf(fact_117_funcsetI,axiom,
! [A: set_a,F: a > a > a,B: set_a_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_a_a @ ( F @ X2 ) @ B ) )
=> ( member_a_a_a2 @ F
@ ( pi_a_a_a2 @ A
@ ^ [Uu: a] : B ) ) ) ).
% funcsetI
thf(fact_118_funcsetI,axiom,
! [A: set_nat_list_a,F: ( nat > list_a ) > a,B: set_a] :
( ! [X2: nat > list_a] :
( ( member_nat_list_a @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ B ) )
=> ( member_nat_list_a_a @ F
@ ( pi_nat_list_a_a @ A
@ ^ [Uu: nat > list_a] : B ) ) ) ).
% funcsetI
thf(fact_119_funcsetI,axiom,
! [A: set_a,F: a > nat > list_a,B: set_nat_list_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_nat_list_a @ ( F @ X2 ) @ B ) )
=> ( member_a_nat_list_a @ F
@ ( pi_a_nat_list_a @ A
@ ^ [Uu: a] : B ) ) ) ).
% funcsetI
thf(fact_120_funcset__carrier_H,axiom,
! [F: a > a,A: partia2175431115845679010xt_a_b,X: a] :
( ( member_a_a @ F
@ ( pi_a_a @ ( partia707051561876973205xt_a_b @ A )
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ A ) ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ A ) )
=> ( member_a @ ( F @ X ) @ ( partia707051561876973205xt_a_b @ A ) ) ) ) ).
% funcset_carrier'
thf(fact_121_funcset__carrier_H,axiom,
! [F: list_a > list_a,A: partia2670972154091845814t_unit,X: list_a] :
( ( member_list_a_list_a @ F
@ ( pi_list_a_list_a @ ( partia5361259788508890537t_unit @ A )
@ ^ [Uu: list_a] : ( partia5361259788508890537t_unit @ A ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ A ) )
=> ( member_list_a @ ( F @ X ) @ ( partia5361259788508890537t_unit @ A ) ) ) ) ).
% funcset_carrier'
thf(fact_122_funcset__carrier_H,axiom,
! [F: list_list_a > list_list_a,A: partia2956882679547061052t_unit,X: list_list_a] :
( ( member8231385768148312316list_a @ F
@ ( pi_lis7180132755996294475list_a @ ( partia2464479390973590831t_unit @ A )
@ ^ [Uu: list_list_a] : ( partia2464479390973590831t_unit @ A ) ) )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ A ) )
=> ( member_list_list_a @ ( F @ X ) @ ( partia2464479390973590831t_unit @ A ) ) ) ) ).
% funcset_carrier'
thf(fact_123_funcset__carrier_H,axiom,
! [F: set_list_a > set_list_a,A: partia1167524930811108609t_unit,X: set_list_a] :
( ( member5068272912271824380list_a @ F
@ ( pi_set2094285526804900683list_a @ ( partia5178357399839081912t_unit @ A )
@ ^ [Uu: set_list_a] : ( partia5178357399839081912t_unit @ A ) ) )
=> ( ( member_set_list_a @ X @ ( partia5178357399839081912t_unit @ A ) )
=> ( member_set_list_a @ ( F @ X ) @ ( partia5178357399839081912t_unit @ A ) ) ) ) ).
% funcset_carrier'
thf(fact_124_funcset__carrier_H,axiom,
! [F: set_list_a > set_list_a,A: partia7496981018696276118t_unit,X: set_list_a] :
( ( member5068272912271824380list_a @ F
@ ( pi_set2094285526804900683list_a @ ( partia141011252114345353t_unit @ A )
@ ^ [Uu: set_list_a] : ( partia141011252114345353t_unit @ A ) ) )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ A ) )
=> ( member_set_list_a @ ( F @ X ) @ ( partia141011252114345353t_unit @ A ) ) ) ) ).
% funcset_carrier'
thf(fact_125_finite__Collect__subsets,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B2: set_nat] : ( ord_less_eq_set_nat @ B2 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_126_finite__Collect__subsets,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( finite_finite_set_a
@ ( collect_set_a
@ ^ [B2: set_a] : ( ord_less_eq_set_a @ B2 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_127_finite__Collect__subsets,axiom,
! [A: set_list_a] :
( ( finite_finite_list_a @ A )
=> ( finite5282473924520328461list_a
@ ( collect_set_list_a
@ ^ [B2: set_list_a] : ( ord_le8861187494160871172list_a @ B2 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_128_finsum__singleton,axiom,
! [I2: set_list_a,A: set_set_list_a,F: set_list_a > a] :
( ( member_set_list_a @ I2 @ A )
=> ( ( finite5282473924520328461list_a @ A )
=> ( ( member_set_list_a_a @ F
@ ( pi_set_list_a_a @ A
@ ^ [Uu: set_list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum7367453022336983110list_a @ r
@ ^ [J: set_list_a] : ( if_a @ ( I2 = J ) @ ( F @ J ) @ ( zero_a_b @ r ) )
@ A )
= ( F @ I2 ) ) ) ) ) ).
% finsum_singleton
thf(fact_129_finsum__singleton,axiom,
! [I2: set_list_a > a,A: set_set_list_a_a,F: ( set_list_a > a ) > a] :
( ( member_set_list_a_a @ I2 @ A )
=> ( ( finite6385009043124570134st_a_a @ A )
=> ( ( member969817812316227871_a_a_a @ F
@ ( pi_set_list_a_a_a @ A
@ ^ [Uu: set_list_a > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum7228396637597461149st_a_a @ r
@ ^ [J: set_list_a > a] : ( if_a @ ( I2 = J ) @ ( F @ J ) @ ( zero_a_b @ r ) )
@ A )
= ( F @ I2 ) ) ) ) ) ).
% finsum_singleton
thf(fact_130_finsum__singleton,axiom,
! [I2: nat > list_a,A: set_nat_list_a,F: ( nat > list_a ) > a] :
( ( member_nat_list_a @ I2 @ A )
=> ( ( finite7630042315537210004list_a @ A )
=> ( ( member_nat_list_a_a @ F
@ ( pi_nat_list_a_a @ A
@ ^ [Uu: nat > list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum1341700292807219277list_a @ r
@ ^ [J: nat > list_a] : ( if_a @ ( I2 = J ) @ ( F @ J ) @ ( zero_a_b @ r ) )
@ A )
= ( F @ I2 ) ) ) ) ) ).
% finsum_singleton
thf(fact_131_finsum__singleton,axiom,
! [I2: nat > a,A: set_nat_a,F: ( nat > a ) > a] :
( ( member_nat_a @ I2 @ A )
=> ( ( finite_finite_nat_a @ A )
=> ( ( member_nat_a_a @ F
@ ( pi_nat_a_a @ A
@ ^ [Uu: nat > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum_a_b_nat_a @ r
@ ^ [J: nat > a] : ( if_a @ ( I2 = J ) @ ( F @ J ) @ ( zero_a_b @ r ) )
@ A )
= ( F @ I2 ) ) ) ) ) ).
% finsum_singleton
thf(fact_132_finsum__singleton,axiom,
! [I2: a > a,A: set_a_a,F: ( a > a ) > a] :
( ( member_a_a @ I2 @ A )
=> ( ( finite_finite_a_a @ A )
=> ( ( member_a_a_a @ F
@ ( pi_a_a_a @ A
@ ^ [Uu: a > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum_a_b_a_a @ r
@ ^ [J: a > a] : ( if_a @ ( I2 = J ) @ ( F @ J ) @ ( zero_a_b @ r ) )
@ A )
= ( F @ I2 ) ) ) ) ) ).
% finsum_singleton
thf(fact_133_finsum__singleton,axiom,
! [I2: a,A: set_a,F: a > a] :
( ( member_a @ I2 @ A )
=> ( ( finite_finite_a @ A )
=> ( ( member_a_a @ F
@ ( pi_a_a @ A
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum_a_b_a @ r
@ ^ [J: a] : ( if_a @ ( I2 = J ) @ ( F @ J ) @ ( zero_a_b @ r ) )
@ A )
= ( F @ I2 ) ) ) ) ) ).
% finsum_singleton
thf(fact_134_finsum__singleton,axiom,
! [I2: nat,A: set_nat,F: nat > a] :
( ( member_nat @ I2 @ A )
=> ( ( finite_finite_nat @ A )
=> ( ( member_nat_a @ F
@ ( pi_nat_a @ A
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum_a_b_nat @ r
@ ^ [J: nat] : ( if_a @ ( I2 = J ) @ ( F @ J ) @ ( zero_a_b @ r ) )
@ A )
= ( F @ I2 ) ) ) ) ) ).
% finsum_singleton
thf(fact_135_finsum__singleton,axiom,
! [I2: list_a,A: set_list_a,F: list_a > a] :
( ( member_list_a @ I2 @ A )
=> ( ( finite_finite_list_a @ A )
=> ( ( member_list_a_a @ F
@ ( pi_list_a_a @ A
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum_a_b_list_a @ r
@ ^ [J: list_a] : ( if_a @ ( I2 = J ) @ ( F @ J ) @ ( zero_a_b @ r ) )
@ A )
= ( F @ I2 ) ) ) ) ) ).
% finsum_singleton
thf(fact_136_add_Ofinprod__singleton__swap,axiom,
! [I2: set_list_a,A: set_set_list_a,F: set_list_a > a] :
( ( member_set_list_a @ I2 @ A )
=> ( ( finite5282473924520328461list_a @ A )
=> ( ( member_set_list_a_a @ F
@ ( pi_set_list_a_a @ A
@ ^ [Uu: set_list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum7367453022336983110list_a @ r
@ ^ [J: set_list_a] : ( if_a @ ( J = I2 ) @ ( F @ J ) @ ( zero_a_b @ r ) )
@ A )
= ( F @ I2 ) ) ) ) ) ).
% add.finprod_singleton_swap
thf(fact_137_add_Ofinprod__singleton__swap,axiom,
! [I2: set_list_a > a,A: set_set_list_a_a,F: ( set_list_a > a ) > a] :
( ( member_set_list_a_a @ I2 @ A )
=> ( ( finite6385009043124570134st_a_a @ A )
=> ( ( member969817812316227871_a_a_a @ F
@ ( pi_set_list_a_a_a @ A
@ ^ [Uu: set_list_a > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum7228396637597461149st_a_a @ r
@ ^ [J: set_list_a > a] : ( if_a @ ( J = I2 ) @ ( F @ J ) @ ( zero_a_b @ r ) )
@ A )
= ( F @ I2 ) ) ) ) ) ).
% add.finprod_singleton_swap
thf(fact_138_add_Ofinprod__singleton__swap,axiom,
! [I2: nat > list_a,A: set_nat_list_a,F: ( nat > list_a ) > a] :
( ( member_nat_list_a @ I2 @ A )
=> ( ( finite7630042315537210004list_a @ A )
=> ( ( member_nat_list_a_a @ F
@ ( pi_nat_list_a_a @ A
@ ^ [Uu: nat > list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum1341700292807219277list_a @ r
@ ^ [J: nat > list_a] : ( if_a @ ( J = I2 ) @ ( F @ J ) @ ( zero_a_b @ r ) )
@ A )
= ( F @ I2 ) ) ) ) ) ).
% add.finprod_singleton_swap
thf(fact_139_add_Ofinprod__singleton__swap,axiom,
! [I2: nat > a,A: set_nat_a,F: ( nat > a ) > a] :
( ( member_nat_a @ I2 @ A )
=> ( ( finite_finite_nat_a @ A )
=> ( ( member_nat_a_a @ F
@ ( pi_nat_a_a @ A
@ ^ [Uu: nat > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum_a_b_nat_a @ r
@ ^ [J: nat > a] : ( if_a @ ( J = I2 ) @ ( F @ J ) @ ( zero_a_b @ r ) )
@ A )
= ( F @ I2 ) ) ) ) ) ).
% add.finprod_singleton_swap
thf(fact_140_add_Ofinprod__singleton__swap,axiom,
! [I2: a > a,A: set_a_a,F: ( a > a ) > a] :
( ( member_a_a @ I2 @ A )
=> ( ( finite_finite_a_a @ A )
=> ( ( member_a_a_a @ F
@ ( pi_a_a_a @ A
@ ^ [Uu: a > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum_a_b_a_a @ r
@ ^ [J: a > a] : ( if_a @ ( J = I2 ) @ ( F @ J ) @ ( zero_a_b @ r ) )
@ A )
= ( F @ I2 ) ) ) ) ) ).
% add.finprod_singleton_swap
thf(fact_141_add_Ofinprod__singleton__swap,axiom,
! [I2: a,A: set_a,F: a > a] :
( ( member_a @ I2 @ A )
=> ( ( finite_finite_a @ A )
=> ( ( member_a_a @ F
@ ( pi_a_a @ A
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum_a_b_a @ r
@ ^ [J: a] : ( if_a @ ( J = I2 ) @ ( F @ J ) @ ( zero_a_b @ r ) )
@ A )
= ( F @ I2 ) ) ) ) ) ).
% add.finprod_singleton_swap
thf(fact_142_add_Ofinprod__singleton__swap,axiom,
! [I2: nat,A: set_nat,F: nat > a] :
( ( member_nat @ I2 @ A )
=> ( ( finite_finite_nat @ A )
=> ( ( member_nat_a @ F
@ ( pi_nat_a @ A
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum_a_b_nat @ r
@ ^ [J: nat] : ( if_a @ ( J = I2 ) @ ( F @ J ) @ ( zero_a_b @ r ) )
@ A )
= ( F @ I2 ) ) ) ) ) ).
% add.finprod_singleton_swap
thf(fact_143_add_Ofinprod__singleton__swap,axiom,
! [I2: list_a,A: set_list_a,F: list_a > a] :
( ( member_list_a @ I2 @ A )
=> ( ( finite_finite_list_a @ A )
=> ( ( member_list_a_a @ F
@ ( pi_list_a_a @ A
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum_a_b_list_a @ r
@ ^ [J: list_a] : ( if_a @ ( J = I2 ) @ ( F @ J ) @ ( zero_a_b @ r ) )
@ A )
= ( F @ I2 ) ) ) ) ) ).
% add.finprod_singleton_swap
thf(fact_144_finsum__rdistr,axiom,
! [A: set_set_list_a,A3: a,F: set_list_a > a] :
( ( finite5282473924520328461list_a @ A )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_set_list_a_a @ F
@ ( pi_set_list_a_a @ A
@ ^ [Uu: set_list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ A3 @ ( finsum7367453022336983110list_a @ r @ F @ A ) )
= ( finsum7367453022336983110list_a @ r
@ ^ [I: set_list_a] : ( mult_a_ring_ext_a_b @ r @ A3 @ ( F @ I ) )
@ A ) ) ) ) ) ).
% finsum_rdistr
thf(fact_145_finsum__rdistr,axiom,
! [A: set_a,A3: a,F: a > a] :
( ( finite_finite_a @ A )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a_a @ F
@ ( pi_a_a @ A
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ A3 @ ( finsum_a_b_a @ r @ F @ A ) )
= ( finsum_a_b_a @ r
@ ^ [I: a] : ( mult_a_ring_ext_a_b @ r @ A3 @ ( F @ I ) )
@ A ) ) ) ) ) ).
% finsum_rdistr
thf(fact_146_finsum__rdistr,axiom,
! [A: set_nat,A3: a,F: nat > a] :
( ( finite_finite_nat @ A )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_nat_a @ F
@ ( pi_nat_a @ A
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ A3 @ ( finsum_a_b_nat @ r @ F @ A ) )
= ( finsum_a_b_nat @ r
@ ^ [I: nat] : ( mult_a_ring_ext_a_b @ r @ A3 @ ( F @ I ) )
@ A ) ) ) ) ) ).
% finsum_rdistr
thf(fact_147_finsum__rdistr,axiom,
! [A: set_list_a,A3: a,F: list_a > a] :
( ( finite_finite_list_a @ A )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_list_a_a @ F
@ ( pi_list_a_a @ A
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ A3 @ ( finsum_a_b_list_a @ r @ F @ A ) )
= ( finsum_a_b_list_a @ r
@ ^ [I: list_a] : ( mult_a_ring_ext_a_b @ r @ A3 @ ( F @ I ) )
@ A ) ) ) ) ) ).
% finsum_rdistr
thf(fact_148_finsum__ldistr,axiom,
! [A: set_set_list_a,A3: a,F: set_list_a > a] :
( ( finite5282473924520328461list_a @ A )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_set_list_a_a @ F
@ ( pi_set_list_a_a @ A
@ ^ [Uu: set_list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( finsum7367453022336983110list_a @ r @ F @ A ) @ A3 )
= ( finsum7367453022336983110list_a @ r
@ ^ [I: set_list_a] : ( mult_a_ring_ext_a_b @ r @ ( F @ I ) @ A3 )
@ A ) ) ) ) ) ).
% finsum_ldistr
thf(fact_149_finsum__ldistr,axiom,
! [A: set_a,A3: a,F: a > a] :
( ( finite_finite_a @ A )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a_a @ F
@ ( pi_a_a @ A
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( finsum_a_b_a @ r @ F @ A ) @ A3 )
= ( finsum_a_b_a @ r
@ ^ [I: a] : ( mult_a_ring_ext_a_b @ r @ ( F @ I ) @ A3 )
@ A ) ) ) ) ) ).
% finsum_ldistr
thf(fact_150_finsum__ldistr,axiom,
! [A: set_nat,A3: a,F: nat > a] :
( ( finite_finite_nat @ A )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_nat_a @ F
@ ( pi_nat_a @ A
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( finsum_a_b_nat @ r @ F @ A ) @ A3 )
= ( finsum_a_b_nat @ r
@ ^ [I: nat] : ( mult_a_ring_ext_a_b @ r @ ( F @ I ) @ A3 )
@ A ) ) ) ) ) ).
% finsum_ldistr
thf(fact_151_finsum__ldistr,axiom,
! [A: set_list_a,A3: a,F: list_a > a] :
( ( finite_finite_list_a @ A )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_list_a_a @ F
@ ( pi_list_a_a @ A
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( finsum_a_b_list_a @ r @ F @ A ) @ A3 )
= ( finsum_a_b_list_a @ r
@ ^ [I: list_a] : ( mult_a_ring_ext_a_b @ r @ ( F @ I ) @ A3 )
@ A ) ) ) ) ) ).
% finsum_ldistr
thf(fact_152_finprod__singleton__swap,axiom,
! [I2: set_list_a,A: set_set_list_a,F: set_list_a > a] :
( ( member_set_list_a @ I2 @ A )
=> ( ( finite5282473924520328461list_a @ A )
=> ( ( member_set_list_a_a @ F
@ ( pi_set_list_a_a @ A
@ ^ [Uu: set_list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro3826550488720007709list_a @ r
@ ^ [J: set_list_a] : ( if_a @ ( J = I2 ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
@ A )
= ( F @ I2 ) ) ) ) ) ).
% finprod_singleton_swap
thf(fact_153_finprod__singleton__swap,axiom,
! [I2: set_list_a > a,A: set_set_list_a_a,F: ( set_list_a > a ) > a] :
( ( member_set_list_a_a @ I2 @ A )
=> ( ( finite6385009043124570134st_a_a @ A )
=> ( ( member969817812316227871_a_a_a @ F
@ ( pi_set_list_a_a_a @ A
@ ^ [Uu: set_list_a > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro4938371440467910406st_a_a @ r
@ ^ [J: set_list_a > a] : ( if_a @ ( J = I2 ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
@ A )
= ( F @ I2 ) ) ) ) ) ).
% finprod_singleton_swap
thf(fact_154_finprod__singleton__swap,axiom,
! [I2: nat > list_a,A: set_nat_list_a,F: ( nat > list_a ) > a] :
( ( member_nat_list_a @ I2 @ A )
=> ( ( finite7630042315537210004list_a @ A )
=> ( ( member_nat_list_a_a @ F
@ ( pi_nat_list_a_a @ A
@ ^ [Uu: nat > list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro4838020199848830884list_a @ r
@ ^ [J: nat > list_a] : ( if_a @ ( J = I2 ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
@ A )
= ( F @ I2 ) ) ) ) ) ).
% finprod_singleton_swap
thf(fact_155_finprod__singleton__swap,axiom,
! [I2: nat > a,A: set_nat_a,F: ( nat > a ) > a] :
( ( member_nat_a @ I2 @ A )
=> ( ( finite_finite_nat_a @ A )
=> ( ( member_nat_a_a @ F
@ ( pi_nat_a_a @ A
@ ^ [Uu: nat > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro5839458686994656414_nat_a @ r
@ ^ [J: nat > a] : ( if_a @ ( J = I2 ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
@ A )
= ( F @ I2 ) ) ) ) ) ).
% finprod_singleton_swap
thf(fact_156_finprod__singleton__swap,axiom,
! [I2: a > a,A: set_a_a,F: ( a > a ) > a] :
( ( member_a_a @ I2 @ A )
=> ( ( finite_finite_a_a @ A )
=> ( ( member_a_a_a @ F
@ ( pi_a_a_a @ A
@ ^ [Uu: a > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro3012607322079259884_b_a_a @ r
@ ^ [J: a > a] : ( if_a @ ( J = I2 ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
@ A )
= ( F @ I2 ) ) ) ) ) ).
% finprod_singleton_swap
thf(fact_157_finprod__singleton__swap,axiom,
! [I2: a,A: set_a,F: a > a] :
( ( member_a @ I2 @ A )
=> ( ( finite_finite_a @ A )
=> ( ( member_a_a @ F
@ ( pi_a_a @ A
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro205304725090349623_a_b_a @ r
@ ^ [J: a] : ( if_a @ ( J = I2 ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
@ A )
= ( F @ I2 ) ) ) ) ) ).
% finprod_singleton_swap
thf(fact_158_finprod__singleton__swap,axiom,
! [I2: nat,A: set_nat,F: nat > a] :
( ( member_nat @ I2 @ A )
=> ( ( finite_finite_nat @ A )
=> ( ( member_nat_a @ F
@ ( pi_nat_a @ A
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro1280035270526425175_b_nat @ r
@ ^ [J: nat] : ( if_a @ ( J = I2 ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
@ A )
= ( F @ I2 ) ) ) ) ) ).
% finprod_singleton_swap
thf(fact_159_finprod__singleton__swap,axiom,
! [I2: list_a,A: set_list_a,F: list_a > a] :
( ( member_list_a @ I2 @ A )
=> ( ( finite_finite_list_a @ A )
=> ( ( member_list_a_a @ F
@ ( pi_list_a_a @ A
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro6052973074229812797list_a @ r
@ ^ [J: list_a] : ( if_a @ ( J = I2 ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
@ A )
= ( F @ I2 ) ) ) ) ) ).
% finprod_singleton_swap
thf(fact_160_finprod__singleton,axiom,
! [I2: set_list_a,A: set_set_list_a,F: set_list_a > a] :
( ( member_set_list_a @ I2 @ A )
=> ( ( finite5282473924520328461list_a @ A )
=> ( ( member_set_list_a_a @ F
@ ( pi_set_list_a_a @ A
@ ^ [Uu: set_list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro3826550488720007709list_a @ r
@ ^ [J: set_list_a] : ( if_a @ ( I2 = J ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
@ A )
= ( F @ I2 ) ) ) ) ) ).
% finprod_singleton
thf(fact_161_finprod__singleton,axiom,
! [I2: set_list_a > a,A: set_set_list_a_a,F: ( set_list_a > a ) > a] :
( ( member_set_list_a_a @ I2 @ A )
=> ( ( finite6385009043124570134st_a_a @ A )
=> ( ( member969817812316227871_a_a_a @ F
@ ( pi_set_list_a_a_a @ A
@ ^ [Uu: set_list_a > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro4938371440467910406st_a_a @ r
@ ^ [J: set_list_a > a] : ( if_a @ ( I2 = J ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
@ A )
= ( F @ I2 ) ) ) ) ) ).
% finprod_singleton
thf(fact_162_finprod__singleton,axiom,
! [I2: nat > list_a,A: set_nat_list_a,F: ( nat > list_a ) > a] :
( ( member_nat_list_a @ I2 @ A )
=> ( ( finite7630042315537210004list_a @ A )
=> ( ( member_nat_list_a_a @ F
@ ( pi_nat_list_a_a @ A
@ ^ [Uu: nat > list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro4838020199848830884list_a @ r
@ ^ [J: nat > list_a] : ( if_a @ ( I2 = J ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
@ A )
= ( F @ I2 ) ) ) ) ) ).
% finprod_singleton
thf(fact_163_finprod__singleton,axiom,
! [I2: nat > a,A: set_nat_a,F: ( nat > a ) > a] :
( ( member_nat_a @ I2 @ A )
=> ( ( finite_finite_nat_a @ A )
=> ( ( member_nat_a_a @ F
@ ( pi_nat_a_a @ A
@ ^ [Uu: nat > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro5839458686994656414_nat_a @ r
@ ^ [J: nat > a] : ( if_a @ ( I2 = J ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
@ A )
= ( F @ I2 ) ) ) ) ) ).
% finprod_singleton
thf(fact_164_finprod__singleton,axiom,
! [I2: a > a,A: set_a_a,F: ( a > a ) > a] :
( ( member_a_a @ I2 @ A )
=> ( ( finite_finite_a_a @ A )
=> ( ( member_a_a_a @ F
@ ( pi_a_a_a @ A
@ ^ [Uu: a > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro3012607322079259884_b_a_a @ r
@ ^ [J: a > a] : ( if_a @ ( I2 = J ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
@ A )
= ( F @ I2 ) ) ) ) ) ).
% finprod_singleton
thf(fact_165_finprod__singleton,axiom,
! [I2: a,A: set_a,F: a > a] :
( ( member_a @ I2 @ A )
=> ( ( finite_finite_a @ A )
=> ( ( member_a_a @ F
@ ( pi_a_a @ A
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro205304725090349623_a_b_a @ r
@ ^ [J: a] : ( if_a @ ( I2 = J ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
@ A )
= ( F @ I2 ) ) ) ) ) ).
% finprod_singleton
thf(fact_166_finprod__singleton,axiom,
! [I2: nat,A: set_nat,F: nat > a] :
( ( member_nat @ I2 @ A )
=> ( ( finite_finite_nat @ A )
=> ( ( member_nat_a @ F
@ ( pi_nat_a @ A
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro1280035270526425175_b_nat @ r
@ ^ [J: nat] : ( if_a @ ( I2 = J ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
@ A )
= ( F @ I2 ) ) ) ) ) ).
% finprod_singleton
thf(fact_167_finprod__singleton,axiom,
! [I2: list_a,A: set_list_a,F: list_a > a] :
( ( member_list_a @ I2 @ A )
=> ( ( finite_finite_list_a @ A )
=> ( ( member_list_a_a @ F
@ ( pi_list_a_a @ A
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro6052973074229812797list_a @ r
@ ^ [J: list_a] : ( if_a @ ( I2 = J ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
@ A )
= ( F @ I2 ) ) ) ) ) ).
% finprod_singleton
thf(fact_168_subalgebra__in__carrier,axiom,
! [K: set_a,V: set_a] :
( ( embedd9027525575939734154ra_a_b @ K @ V @ r )
=> ( ord_less_eq_set_a @ V @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% subalgebra_in_carrier
thf(fact_169_carrier__is__subalgebra,axiom,
! [K: set_a] :
( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
=> ( embedd9027525575939734154ra_a_b @ K @ ( partia707051561876973205xt_a_b @ r ) @ r ) ) ).
% carrier_is_subalgebra
thf(fact_170_a__l__coset__subset__G,axiom,
! [H: set_a,X: a] :
( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( a_l_coset_a_b @ r @ X @ H ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% a_l_coset_subset_G
thf(fact_171_subset__Idl__subset,axiom,
! [I4: set_a,H: set_a] :
( ( ord_less_eq_set_a @ I4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ H @ I4 )
=> ( ord_less_eq_set_a @ ( genideal_a_b @ r @ H ) @ ( genideal_a_b @ r @ I4 ) ) ) ) ).
% subset_Idl_subset
thf(fact_172_genideal__self,axiom,
! [S: set_a] :
( ( ord_less_eq_set_a @ S @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ S @ ( genideal_a_b @ r @ S ) ) ) ).
% genideal_self
thf(fact_173_m__assoc,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( 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 @ X @ Y ) @ Z )
= ( mult_a_ring_ext_a_b @ r @ X @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% m_assoc
thf(fact_174_m__comm,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ X ) ) ) ) ).
% m_comm
thf(fact_175_m__lcomm,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( 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 @ X @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) )
= ( mult_a_ring_ext_a_b @ r @ Y @ ( mult_a_ring_ext_a_b @ r @ X @ Z ) ) ) ) ) ) ).
% m_lcomm
thf(fact_176_zero__not__one,axiom,
( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) ) ).
% zero_not_one
thf(fact_177_mem__Collect__eq,axiom,
! [A3: set_list_a > a,P2: ( set_list_a > a ) > $o] :
( ( member_set_list_a_a @ A3 @ ( collect_set_list_a_a @ P2 ) )
= ( P2 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_178_mem__Collect__eq,axiom,
! [A3: nat > list_a,P2: ( nat > list_a ) > $o] :
( ( member_nat_list_a @ A3 @ ( collect_nat_list_a @ P2 ) )
= ( P2 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_179_mem__Collect__eq,axiom,
! [A3: nat > a,P2: ( nat > a ) > $o] :
( ( member_nat_a @ A3 @ ( collect_nat_a @ P2 ) )
= ( P2 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_180_mem__Collect__eq,axiom,
! [A3: a > a,P2: ( a > a ) > $o] :
( ( member_a_a @ A3 @ ( collect_a_a @ P2 ) )
= ( P2 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_181_mem__Collect__eq,axiom,
! [A3: nat,P2: nat > $o] :
( ( member_nat @ A3 @ ( collect_nat @ P2 ) )
= ( P2 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_182_mem__Collect__eq,axiom,
! [A3: a,P2: a > $o] :
( ( member_a @ A3 @ ( collect_a @ P2 ) )
= ( P2 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_183_Collect__mem__eq,axiom,
! [A: set_set_list_a_a] :
( ( collect_set_list_a_a
@ ^ [X4: set_list_a > a] : ( member_set_list_a_a @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_184_Collect__mem__eq,axiom,
! [A: set_nat_list_a] :
( ( collect_nat_list_a
@ ^ [X4: nat > list_a] : ( member_nat_list_a @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_185_Collect__mem__eq,axiom,
! [A: set_nat_a] :
( ( collect_nat_a
@ ^ [X4: nat > a] : ( member_nat_a @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_186_Collect__mem__eq,axiom,
! [A: set_a_a] :
( ( collect_a_a
@ ^ [X4: a > a] : ( member_a_a @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_187_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X4: nat] : ( member_nat @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_188_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X4: a] : ( member_a @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_189_Collect__cong,axiom,
! [P2: nat > $o,Q2: nat > $o] :
( ! [X2: nat] :
( ( P2 @ X2 )
= ( Q2 @ X2 ) )
=> ( ( collect_nat @ P2 )
= ( collect_nat @ Q2 ) ) ) ).
% Collect_cong
thf(fact_190_Collect__cong,axiom,
! [P2: a > $o,Q2: a > $o] :
( ! [X2: a] :
( ( P2 @ X2 )
= ( Q2 @ X2 ) )
=> ( ( collect_a @ P2 )
= ( collect_a @ Q2 ) ) ) ).
% Collect_cong
thf(fact_191_add_Ofinprod__one__eqI,axiom,
! [A: set_set_list_a_a,F: ( set_list_a > a ) > a] :
( ! [X2: set_list_a > a] :
( ( member_set_list_a_a @ X2 @ A )
=> ( ( F @ X2 )
= ( zero_a_b @ r ) ) )
=> ( ( finsum7228396637597461149st_a_a @ r @ F @ A )
= ( zero_a_b @ r ) ) ) ).
% add.finprod_one_eqI
thf(fact_192_add_Ofinprod__one__eqI,axiom,
! [A: set_nat_list_a,F: ( nat > list_a ) > a] :
( ! [X2: nat > list_a] :
( ( member_nat_list_a @ X2 @ A )
=> ( ( F @ X2 )
= ( zero_a_b @ r ) ) )
=> ( ( finsum1341700292807219277list_a @ r @ F @ A )
= ( zero_a_b @ r ) ) ) ).
% add.finprod_one_eqI
thf(fact_193_add_Ofinprod__one__eqI,axiom,
! [A: set_nat_a,F: ( nat > a ) > a] :
( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A )
=> ( ( F @ X2 )
= ( zero_a_b @ r ) ) )
=> ( ( finsum_a_b_nat_a @ r @ F @ A )
= ( zero_a_b @ r ) ) ) ).
% add.finprod_one_eqI
thf(fact_194_add_Ofinprod__one__eqI,axiom,
! [A: set_a_a,F: ( a > a ) > a] :
( ! [X2: a > a] :
( ( member_a_a @ X2 @ A )
=> ( ( F @ X2 )
= ( zero_a_b @ r ) ) )
=> ( ( finsum_a_b_a_a @ r @ F @ A )
= ( zero_a_b @ r ) ) ) ).
% add.finprod_one_eqI
thf(fact_195_add_Ofinprod__one__eqI,axiom,
! [A: set_a,F: a > a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( F @ X2 )
= ( zero_a_b @ r ) ) )
=> ( ( finsum_a_b_a @ r @ F @ A )
= ( zero_a_b @ r ) ) ) ).
% add.finprod_one_eqI
thf(fact_196_finprod__one__eqI,axiom,
! [A: set_set_list_a_a,F: ( set_list_a > a ) > a] :
( ! [X2: set_list_a > a] :
( ( member_set_list_a_a @ X2 @ A )
=> ( ( F @ X2 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ( finpro4938371440467910406st_a_a @ r @ F @ A )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% finprod_one_eqI
thf(fact_197_finprod__one__eqI,axiom,
! [A: set_nat_list_a,F: ( nat > list_a ) > a] :
( ! [X2: nat > list_a] :
( ( member_nat_list_a @ X2 @ A )
=> ( ( F @ X2 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ( finpro4838020199848830884list_a @ r @ F @ A )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% finprod_one_eqI
thf(fact_198_finprod__one__eqI,axiom,
! [A: set_nat_a,F: ( nat > a ) > a] :
( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A )
=> ( ( F @ X2 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ( finpro5839458686994656414_nat_a @ r @ F @ A )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% finprod_one_eqI
thf(fact_199_finprod__one__eqI,axiom,
! [A: set_a_a,F: ( a > a ) > a] :
( ! [X2: a > a] :
( ( member_a_a @ X2 @ A )
=> ( ( F @ X2 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ( finpro3012607322079259884_b_a_a @ r @ F @ A )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% finprod_one_eqI
thf(fact_200_finprod__one__eqI,axiom,
! [A: set_a,F: a > a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( F @ X2 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ( finpro205304725090349623_a_b_a @ r @ F @ A )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% finprod_one_eqI
thf(fact_201_local_Ointegral,axiom,
! [A3: a,B3: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A3 @ B3 )
= ( zero_a_b @ r ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A3
= ( zero_a_b @ r ) )
| ( B3
= ( zero_a_b @ r ) ) ) ) ) ) ).
% local.integral
thf(fact_202_integral__iff,axiom,
! [A3: a,B3: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ A3 @ B3 )
= ( zero_a_b @ r ) )
= ( ( A3
= ( zero_a_b @ r ) )
| ( B3
= ( zero_a_b @ r ) ) ) ) ) ) ).
% integral_iff
thf(fact_203_m__lcancel,axiom,
! [A3: a,B3: a,C2: a] :
( ( A3
!= ( zero_a_b @ r ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ A3 @ B3 )
= ( mult_a_ring_ext_a_b @ r @ A3 @ C2 ) )
= ( B3 = C2 ) ) ) ) ) ) ).
% m_lcancel
thf(fact_204_m__rcancel,axiom,
! [A3: a,B3: a,C2: a] :
( ( A3
!= ( zero_a_b @ r ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ B3 @ A3 )
= ( mult_a_ring_ext_a_b @ r @ C2 @ A3 ) )
= ( B3 = C2 ) ) ) ) ) ) ).
% m_rcancel
thf(fact_205_inv__unique,axiom,
! [Y: a,X: a,Y3: a] :
( ( ( mult_a_ring_ext_a_b @ r @ Y @ X )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y3 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( Y = Y3 ) ) ) ) ) ) ).
% inv_unique
thf(fact_206_one__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ U @ X2 )
= X2 ) )
=> ( U
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% one_unique
thf(fact_207_up__smult__closed,axiom,
! [A3: a,P: nat > a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_nat_a @ P @ ( up_a_b @ r ) )
=> ( member_nat_a
@ ^ [I: nat] : ( mult_a_ring_ext_a_b @ r @ A3 @ ( P @ I ) )
@ ( up_a_b @ r ) ) ) ) ).
% up_smult_closed
thf(fact_208_finprod__zero__iff,axiom,
! [A: set_set_list_a_a,F: ( set_list_a > a ) > a] :
( ( finite6385009043124570134st_a_a @ A )
=> ( ! [A4: set_list_a > a] :
( ( member_set_list_a_a @ A4 @ A )
=> ( member_a @ ( F @ A4 ) @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( finpro4938371440467910406st_a_a @ r @ F @ A )
= ( zero_a_b @ r ) )
= ( ? [X4: set_list_a > a] :
( ( member_set_list_a_a @ X4 @ A )
& ( ( F @ X4 )
= ( zero_a_b @ r ) ) ) ) ) ) ) ).
% finprod_zero_iff
thf(fact_209_finprod__zero__iff,axiom,
! [A: set_nat_list_a,F: ( nat > list_a ) > a] :
( ( finite7630042315537210004list_a @ A )
=> ( ! [A4: nat > list_a] :
( ( member_nat_list_a @ A4 @ A )
=> ( member_a @ ( F @ A4 ) @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( finpro4838020199848830884list_a @ r @ F @ A )
= ( zero_a_b @ r ) )
= ( ? [X4: nat > list_a] :
( ( member_nat_list_a @ X4 @ A )
& ( ( F @ X4 )
= ( zero_a_b @ r ) ) ) ) ) ) ) ).
% finprod_zero_iff
thf(fact_210_finprod__zero__iff,axiom,
! [A: set_nat_a,F: ( nat > a ) > a] :
( ( finite_finite_nat_a @ A )
=> ( ! [A4: nat > a] :
( ( member_nat_a @ A4 @ A )
=> ( member_a @ ( F @ A4 ) @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( finpro5839458686994656414_nat_a @ r @ F @ A )
= ( zero_a_b @ r ) )
= ( ? [X4: nat > a] :
( ( member_nat_a @ X4 @ A )
& ( ( F @ X4 )
= ( zero_a_b @ r ) ) ) ) ) ) ) ).
% finprod_zero_iff
thf(fact_211_finprod__zero__iff,axiom,
! [A: set_a_a,F: ( a > a ) > a] :
( ( finite_finite_a_a @ A )
=> ( ! [A4: a > a] :
( ( member_a_a @ A4 @ A )
=> ( member_a @ ( F @ A4 ) @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( finpro3012607322079259884_b_a_a @ r @ F @ A )
= ( zero_a_b @ r ) )
= ( ? [X4: a > a] :
( ( member_a_a @ X4 @ A )
& ( ( F @ X4 )
= ( zero_a_b @ r ) ) ) ) ) ) ) ).
% finprod_zero_iff
thf(fact_212_finprod__zero__iff,axiom,
! [A: set_a,F: a > a] :
( ( finite_finite_a @ A )
=> ( ! [A4: a] :
( ( member_a @ A4 @ A )
=> ( member_a @ ( F @ A4 ) @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( finpro205304725090349623_a_b_a @ r @ F @ A )
= ( zero_a_b @ r ) )
= ( ? [X4: a] :
( ( member_a @ X4 @ A )
& ( ( F @ X4 )
= ( zero_a_b @ r ) ) ) ) ) ) ) ).
% finprod_zero_iff
thf(fact_213_finprod__zero__iff,axiom,
! [A: set_nat,F: nat > a] :
( ( finite_finite_nat @ A )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A )
=> ( member_a @ ( F @ A4 ) @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( finpro1280035270526425175_b_nat @ r @ F @ A )
= ( zero_a_b @ r ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ( F @ X4 )
= ( zero_a_b @ r ) ) ) ) ) ) ) ).
% finprod_zero_iff
thf(fact_214_finprod__zero__iff,axiom,
! [A: set_list_a,F: list_a > a] :
( ( finite_finite_list_a @ A )
=> ( ! [A4: list_a] :
( ( member_list_a @ A4 @ A )
=> ( member_a @ ( F @ A4 ) @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( finpro6052973074229812797list_a @ r @ F @ A )
= ( zero_a_b @ r ) )
= ( ? [X4: list_a] :
( ( member_list_a @ X4 @ A )
& ( ( F @ X4 )
= ( zero_a_b @ r ) ) ) ) ) ) ) ).
% finprod_zero_iff
thf(fact_215_a__lcos__mult__one,axiom,
! [M: set_a] :
( ( ord_less_eq_set_a @ M @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_l_coset_a_b @ r @ ( zero_a_b @ r ) @ M )
= M ) ) ).
% a_lcos_mult_one
thf(fact_216_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
@ ^ [X4: a] :
( ( P2 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_217_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
@ ^ [X4: nat] :
( ( P2 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_218_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
@ ^ [X4: list_a] :
( ( P2 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_219_finite__Collect__disjI,axiom,
! [P2: a > $o,Q2: a > $o] :
( ( finite_finite_a
@ ( collect_a
@ ^ [X4: a] :
( ( P2 @ X4 )
| ( Q2 @ X4 ) ) ) )
= ( ( finite_finite_a @ ( collect_a @ P2 ) )
& ( finite_finite_a @ ( collect_a @ Q2 ) ) ) ) ).
% finite_Collect_disjI
thf(fact_220_finite__Collect__disjI,axiom,
! [P2: nat > $o,Q2: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( P2 @ X4 )
| ( Q2 @ X4 ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
& ( finite_finite_nat @ ( collect_nat @ Q2 ) ) ) ) ).
% finite_Collect_disjI
thf(fact_221_finite__Collect__disjI,axiom,
! [P2: list_a > $o,Q2: list_a > $o] :
( ( finite_finite_list_a
@ ( collect_list_a
@ ^ [X4: list_a] :
( ( P2 @ X4 )
| ( Q2 @ X4 ) ) ) )
= ( ( finite_finite_list_a @ ( collect_list_a @ P2 ) )
& ( finite_finite_list_a @ ( collect_list_a @ Q2 ) ) ) ) ).
% finite_Collect_disjI
thf(fact_222_zero__closed,axiom,
member_a @ ( zero_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% zero_closed
thf(fact_223_m__closed,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% m_closed
thf(fact_224_one__closed,axiom,
member_a @ ( one_a_ring_ext_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% one_closed
thf(fact_225_l__null,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) @ X )
= ( zero_a_b @ r ) ) ) ).
% l_null
thf(fact_226_r__null,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ) ).
% r_null
thf(fact_227_l__one,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( one_a_ring_ext_a_b @ r ) @ X )
= X ) ) ).
% l_one
thf(fact_228_r__one,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ ( one_a_ring_ext_a_b @ r ) )
= X ) ) ).
% r_one
thf(fact_229_finsum__infinite,axiom,
! [A: set_a,F: a > a] :
( ~ ( finite_finite_a @ A )
=> ( ( finsum_a_b_a @ r @ F @ A )
= ( zero_a_b @ r ) ) ) ).
% finsum_infinite
thf(fact_230_finsum__infinite,axiom,
! [A: set_nat,F: nat > a] :
( ~ ( finite_finite_nat @ A )
=> ( ( finsum_a_b_nat @ r @ F @ A )
= ( zero_a_b @ r ) ) ) ).
% finsum_infinite
thf(fact_231_finsum__infinite,axiom,
! [A: set_list_a,F: list_a > a] :
( ~ ( finite_finite_list_a @ A )
=> ( ( finsum_a_b_list_a @ r @ F @ A )
= ( zero_a_b @ r ) ) ) ).
% finsum_infinite
thf(fact_232_finprod__infinite,axiom,
! [A: set_a,F: a > a] :
( ~ ( finite_finite_a @ A )
=> ( ( finpro205304725090349623_a_b_a @ r @ F @ A )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% finprod_infinite
thf(fact_233_finprod__infinite,axiom,
! [A: set_nat,F: nat > a] :
( ~ ( finite_finite_nat @ A )
=> ( ( finpro1280035270526425175_b_nat @ r @ F @ A )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% finprod_infinite
thf(fact_234_finprod__infinite,axiom,
! [A: set_list_a,F: list_a > a] :
( ~ ( finite_finite_list_a @ A )
=> ( ( finpro6052973074229812797list_a @ r @ F @ A )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% finprod_infinite
thf(fact_235_r__right__minus__eq,axiom,
! [A3: a,B3: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( a_minus_a_b @ r @ A3 @ B3 )
= ( zero_a_b @ r ) )
= ( A3 = B3 ) ) ) ) ).
% r_right_minus_eq
thf(fact_236_finprod__multf,axiom,
! [F: set_list_a > a,A: set_set_list_a,G: set_list_a > a] :
( ( member_set_list_a_a @ F
@ ( pi_set_list_a_a @ A
@ ^ [Uu: set_list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_set_list_a_a @ G
@ ( pi_set_list_a_a @ A
@ ^ [Uu: set_list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro3826550488720007709list_a @ r
@ ^ [X4: set_list_a] : ( mult_a_ring_ext_a_b @ r @ ( F @ X4 ) @ ( G @ X4 ) )
@ A )
= ( mult_a_ring_ext_a_b @ r @ ( finpro3826550488720007709list_a @ r @ F @ A ) @ ( finpro3826550488720007709list_a @ r @ G @ A ) ) ) ) ) ).
% finprod_multf
thf(fact_237_finprod__multf,axiom,
! [F: nat > a,A: set_nat,G: nat > a] :
( ( member_nat_a @ F
@ ( pi_nat_a @ A
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ A
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro1280035270526425175_b_nat @ r
@ ^ [X4: nat] : ( mult_a_ring_ext_a_b @ r @ ( F @ X4 ) @ ( G @ X4 ) )
@ A )
= ( mult_a_ring_ext_a_b @ r @ ( finpro1280035270526425175_b_nat @ r @ F @ A ) @ ( finpro1280035270526425175_b_nat @ r @ G @ A ) ) ) ) ) ).
% finprod_multf
thf(fact_238_finprod__multf,axiom,
! [F: a > a,A: set_a,G: a > a] :
( ( member_a_a @ F
@ ( pi_a_a @ A
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_a_a @ G
@ ( pi_a_a @ A
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro205304725090349623_a_b_a @ r
@ ^ [X4: a] : ( mult_a_ring_ext_a_b @ r @ ( F @ X4 ) @ ( G @ X4 ) )
@ A )
= ( mult_a_ring_ext_a_b @ r @ ( finpro205304725090349623_a_b_a @ r @ F @ A ) @ ( finpro205304725090349623_a_b_a @ r @ G @ A ) ) ) ) ) ).
% finprod_multf
thf(fact_239_domain_Ofinprod__zero__iff,axiom,
! [R: partia2175431115845679010xt_a_b,A: set_a,F: a > a] :
( ( domain_a_b @ R )
=> ( ( finite_finite_a @ A )
=> ( ! [A4: a] :
( ( member_a @ A4 @ A )
=> ( member_a @ ( F @ A4 ) @ ( partia707051561876973205xt_a_b @ R ) ) )
=> ( ( ( finpro205304725090349623_a_b_a @ R @ F @ A )
= ( zero_a_b @ R ) )
= ( ? [X4: a] :
( ( member_a @ X4 @ A )
& ( ( F @ X4 )
= ( zero_a_b @ R ) ) ) ) ) ) ) ) ).
% domain.finprod_zero_iff
thf(fact_240_domain_Ofinprod__zero__iff,axiom,
! [R: partia2175431115845679010xt_a_b,A: set_nat,F: nat > a] :
( ( domain_a_b @ R )
=> ( ( finite_finite_nat @ A )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A )
=> ( member_a @ ( F @ A4 ) @ ( partia707051561876973205xt_a_b @ R ) ) )
=> ( ( ( finpro1280035270526425175_b_nat @ R @ F @ A )
= ( zero_a_b @ R ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ( F @ X4 )
= ( zero_a_b @ R ) ) ) ) ) ) ) ) ).
% domain.finprod_zero_iff
thf(fact_241_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 )
=> ( ! [A4: list_a] :
( ( member_list_a @ A4 @ A )
=> ( member_a @ ( F @ A4 ) @ ( partia707051561876973205xt_a_b @ R ) ) )
=> ( ( ( finpro6052973074229812797list_a @ R @ F @ A )
= ( zero_a_b @ R ) )
= ( ? [X4: list_a] :
( ( member_list_a @ X4 @ A )
& ( ( F @ X4 )
= ( zero_a_b @ R ) ) ) ) ) ) ) ) ).
% domain.finprod_zero_iff
thf(fact_242_domain_Ofinprod__zero__iff,axiom,
! [R: partia2670972154091845814t_unit,A: set_nat,F: nat > list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( finite_finite_nat @ A )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A )
=> ( member_list_a @ ( F @ A4 ) @ ( partia5361259788508890537t_unit @ R ) ) )
=> ( ( ( finpro1918104735009086181it_nat @ R @ F @ A )
= ( zero_l4142658623432671053t_unit @ R ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ( F @ X4 )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ) ) ) ).
% domain.finprod_zero_iff
thf(fact_243_domain_Ofinprod__zero__iff,axiom,
! [R: partia2670972154091845814t_unit,A: set_a,F: a > list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( finite_finite_a @ A )
=> ( ! [A4: a] :
( ( member_a @ A4 @ A )
=> ( member_list_a @ ( F @ A4 ) @ ( partia5361259788508890537t_unit @ R ) ) )
=> ( ( ( finpro4329226410377213737unit_a @ R @ F @ A )
= ( zero_l4142658623432671053t_unit @ R ) )
= ( ? [X4: a] :
( ( member_a @ X4 @ A )
& ( ( F @ X4 )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ) ) ) ).
% domain.finprod_zero_iff
thf(fact_244_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 )
=> ( ! [A4: nat > a] :
( ( member_nat_a @ A4 @ A )
=> ( member_a @ ( F @ A4 ) @ ( partia707051561876973205xt_a_b @ R ) ) )
=> ( ( ( finpro5839458686994656414_nat_a @ R @ F @ A )
= ( zero_a_b @ R ) )
= ( ? [X4: nat > a] :
( ( member_nat_a @ X4 @ A )
& ( ( F @ X4 )
= ( zero_a_b @ R ) ) ) ) ) ) ) ) ).
% domain.finprod_zero_iff
thf(fact_245_domain_Ofinprod__zero__iff,axiom,
! [R: partia2175431115845679010xt_a_b,A: set_a_a,F: ( a > a ) > a] :
( ( domain_a_b @ R )
=> ( ( finite_finite_a_a @ A )
=> ( ! [A4: a > a] :
( ( member_a_a @ A4 @ A )
=> ( member_a @ ( F @ A4 ) @ ( partia707051561876973205xt_a_b @ R ) ) )
=> ( ( ( finpro3012607322079259884_b_a_a @ R @ F @ A )
= ( zero_a_b @ R ) )
= ( ? [X4: a > a] :
( ( member_a_a @ X4 @ A )
& ( ( F @ X4 )
= ( zero_a_b @ R ) ) ) ) ) ) ) ) ).
% domain.finprod_zero_iff
thf(fact_246_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 )
=> ( ! [A4: list_a] :
( ( member_list_a @ A4 @ A )
=> ( member_list_a @ ( F @ A4 ) @ ( partia5361259788508890537t_unit @ R ) ) )
=> ( ( ( finpro738134188688310831list_a @ R @ F @ A )
= ( zero_l4142658623432671053t_unit @ R ) )
= ( ? [X4: list_a] :
( ( member_list_a @ X4 @ A )
& ( ( F @ X4 )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ) ) ) ).
% domain.finprod_zero_iff
thf(fact_247_domain_Ofinprod__zero__iff,axiom,
! [R: partia2956882679547061052t_unit,A: set_a,F: a > list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( finite_finite_a @ A )
=> ( ! [A4: a] :
( ( member_a @ A4 @ A )
=> ( member_list_list_a @ ( F @ A4 ) @ ( partia2464479390973590831t_unit @ R ) ) )
=> ( ( ( finpro5596966875920909993unit_a @ R @ F @ A )
= ( zero_l347298301471573063t_unit @ R ) )
= ( ? [X4: a] :
( ( member_a @ X4 @ A )
& ( ( F @ X4 )
= ( zero_l347298301471573063t_unit @ R ) ) ) ) ) ) ) ) ).
% domain.finprod_zero_iff
thf(fact_248_domain_Ofinprod__zero__iff,axiom,
! [R: partia2956882679547061052t_unit,A: set_nat,F: nat > list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( finite_finite_nat @ A )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A )
=> ( member_list_list_a @ ( F @ A4 ) @ ( partia2464479390973590831t_unit @ R ) ) )
=> ( ( ( finpro4561275463894985573it_nat @ R @ F @ A )
= ( zero_l347298301471573063t_unit @ R ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ( F @ X4 )
= ( zero_l347298301471573063t_unit @ R ) ) ) ) ) ) ) ) ).
% domain.finprod_zero_iff
thf(fact_249_not__finite__existsD,axiom,
! [P2: a > $o] :
( ~ ( finite_finite_a @ ( collect_a @ P2 ) )
=> ? [X_1: a] : ( P2 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_250_not__finite__existsD,axiom,
! [P2: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P2 ) )
=> ? [X_1: nat] : ( P2 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_251_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_252_pigeonhole__infinite__rel,axiom,
! [A: set_a,B: set_a,R: a > a > $o] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ? [Xa: a] :
( ( member_a @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: a] :
( ( member_a @ X2 @ B )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A5: a] :
( ( member_a @ A5 @ A )
& ( R @ A5 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_253_pigeonhole__infinite__rel,axiom,
! [A: set_a,B: set_nat,R: a > nat > $o] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A5: a] :
( ( member_a @ A5 @ A )
& ( R @ A5 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_254_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B: set_a,R: nat > a > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_a @ B )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ? [Xa: a] :
( ( member_a @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: a] :
( ( member_a @ X2 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A5: nat] :
( ( member_nat @ A5 @ A )
& ( R @ A5 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_255_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B: set_nat,R: nat > nat > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A5: nat] :
( ( member_nat @ A5 @ A )
& ( R @ A5 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_256_pigeonhole__infinite__rel,axiom,
! [A: set_a,B: set_list_a,R: a > list_a > $o] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_finite_list_a @ B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ? [Xa: list_a] :
( ( member_list_a @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ B )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A5: a] :
( ( member_a @ A5 @ A )
& ( R @ A5 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_257_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B: set_list_a,R: nat > list_a > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_list_a @ B )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ? [Xa: list_a] :
( ( member_list_a @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A5: nat] :
( ( member_nat @ A5 @ A )
& ( R @ A5 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_258_pigeonhole__infinite__rel,axiom,
! [A: set_list_a,B: set_a,R: list_a > a > $o] :
( ~ ( finite_finite_list_a @ A )
=> ( ( finite_finite_a @ B )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ A )
=> ? [Xa: a] :
( ( member_a @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: a] :
( ( member_a @ X2 @ B )
& ~ ( finite_finite_list_a
@ ( collect_list_a
@ ^ [A5: list_a] :
( ( member_list_a @ A5 @ A )
& ( R @ A5 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_259_pigeonhole__infinite__rel,axiom,
! [A: set_list_a,B: set_nat,R: list_a > nat > $o] :
( ~ ( finite_finite_list_a @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ~ ( finite_finite_list_a
@ ( collect_list_a
@ ^ [A5: list_a] :
( ( member_list_a @ A5 @ A )
& ( R @ A5 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_260_pigeonhole__infinite__rel,axiom,
! [A: set_nat_a,B: set_a,R: ( nat > a ) > a > $o] :
( ~ ( finite_finite_nat_a @ A )
=> ( ( finite_finite_a @ B )
=> ( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A )
=> ? [Xa: a] :
( ( member_a @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: a] :
( ( member_a @ X2 @ B )
& ~ ( finite_finite_nat_a
@ ( collect_nat_a
@ ^ [A5: nat > a] :
( ( member_nat_a @ A5 @ A )
& ( R @ A5 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_261_pigeonhole__infinite__rel,axiom,
! [A: set_a_a,B: set_a,R: ( a > a ) > a > $o] :
( ~ ( finite_finite_a_a @ A )
=> ( ( finite_finite_a @ B )
=> ( ! [X2: a > a] :
( ( member_a_a @ X2 @ A )
=> ? [Xa: a] :
( ( member_a @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: a] :
( ( member_a @ X2 @ B )
& ~ ( finite_finite_a_a
@ ( collect_a_a
@ ^ [A5: a > a] :
( ( member_a_a @ A5 @ A )
& ( R @ A5 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_262_finite__has__minimal2,axiom,
! [A: set_set_a,A3: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A3 @ A )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ( ord_less_eq_set_a @ X2 @ A3 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_263_finite__has__minimal2,axiom,
! [A: set_nat,A3: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A3 @ A )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ord_less_eq_nat @ X2 @ A3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_264_finite__has__minimal2,axiom,
! [A: set_set_list_a,A3: set_list_a] :
( ( finite5282473924520328461list_a @ A )
=> ( ( member_set_list_a @ A3 @ A )
=> ? [X2: set_list_a] :
( ( member_set_list_a @ X2 @ A )
& ( ord_le8861187494160871172list_a @ X2 @ A3 )
& ! [Xa: set_list_a] :
( ( member_set_list_a @ Xa @ A )
=> ( ( ord_le8861187494160871172list_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_265_finite__has__maximal2,axiom,
! [A: set_set_a,A3: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A3 @ A )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ( ord_less_eq_set_a @ A3 @ X2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_266_finite__has__maximal2,axiom,
! [A: set_nat,A3: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A3 @ A )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ord_less_eq_nat @ A3 @ X2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_267_finite__has__maximal2,axiom,
! [A: set_set_list_a,A3: set_list_a] :
( ( finite5282473924520328461list_a @ A )
=> ( ( member_set_list_a @ A3 @ A )
=> ? [X2: set_list_a] :
( ( member_set_list_a @ X2 @ A )
& ( ord_le8861187494160871172list_a @ A3 @ X2 )
& ! [Xa: set_list_a] :
( ( member_set_list_a @ Xa @ A )
=> ( ( ord_le8861187494160871172list_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_268_rev__finite__subset,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_269_rev__finite__subset,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( finite_finite_a @ A ) ) ) ).
% rev_finite_subset
thf(fact_270_rev__finite__subset,axiom,
! [B: set_list_a,A: set_list_a] :
( ( finite_finite_list_a @ B )
=> ( ( ord_le8861187494160871172list_a @ A @ B )
=> ( finite_finite_list_a @ A ) ) ) ).
% rev_finite_subset
thf(fact_271_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_272_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_273_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_274_finite__subset,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_subset
thf(fact_275_finite__subset,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ A ) ) ) ).
% finite_subset
thf(fact_276_finite__subset,axiom,
! [A: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( finite_finite_list_a @ B )
=> ( finite_finite_list_a @ A ) ) ) ).
% finite_subset
thf(fact_277_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_278_bound__upD,axiom,
! [F: nat > a] :
( ( member_nat_a @ F @ ( up_a_b @ r ) )
=> ? [N: nat] : ( bound_a @ ( zero_a_b @ r ) @ N @ F ) ) ).
% bound_upD
thf(fact_279_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_280_subdomainI,axiom,
! [H: set_a] :
( ( subcring_a_b @ H @ r )
=> ( ( ( one_a_ring_ext_a_b @ r )
!= ( zero_a_b @ r ) )
=> ( ! [H1: a,H2: a] :
( ( member_a @ H1 @ H )
=> ( ( member_a @ H2 @ H )
=> ( ( ( mult_a_ring_ext_a_b @ r @ H1 @ H2 )
= ( zero_a_b @ r ) )
=> ( ( H1
= ( zero_a_b @ r ) )
| ( H2
= ( zero_a_b @ r ) ) ) ) ) )
=> ( subdomain_a_b @ H @ r ) ) ) ) ).
% subdomainI
thf(fact_281_ring__iso__imp__img__domain,axiom,
! [H3: a > a,S: partia2175431115845679010xt_a_b] :
( ( member_a_a @ H3 @ ( ring_iso_a_b_a_b @ r @ S ) )
=> ( domain_a_b
@ ( zero_update_a_b
@ ^ [Uu: a] : ( H3 @ ( zero_a_b @ r ) )
@ S ) ) ) ).
% ring_iso_imp_img_domain
thf(fact_282_ring__iso__imp__img__domain,axiom,
! [H3: a > list_a,S: partia2670972154091845814t_unit] :
( ( member_a_list_a @ H3 @ ( ring_i4557880751517319194t_unit @ r @ S ) )
=> ( domain6553523120543210313t_unit
@ ( zero_u1196785550890449590t_unit
@ ^ [Uu: list_a] : ( H3 @ ( zero_a_b @ r ) )
@ S ) ) ) ).
% ring_iso_imp_img_domain
thf(fact_283_finprod__mono__neutral__cong__left,axiom,
! [B: set_set_list_a,A: set_set_list_a,H3: set_list_a > a,G: set_list_a > a] :
( ( finite5282473924520328461list_a @ B )
=> ( ( ord_le8877086941679407844list_a @ A @ B )
=> ( ! [I3: set_list_a] :
( ( member_set_list_a @ I3 @ ( minus_4782336368215558443list_a @ B @ A ) )
=> ( ( H3 @ I3 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: set_list_a] :
( ( member_set_list_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( member_set_list_a_a @ H3
@ ( pi_set_list_a_a @ B
@ ^ [Uu: set_list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro3826550488720007709list_a @ r @ G @ A )
= ( finpro3826550488720007709list_a @ r @ H3 @ B ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong_left
thf(fact_284_finprod__mono__neutral__cong__left,axiom,
! [B: set_set_list_a_a,A: set_set_list_a_a,H3: ( set_list_a > a ) > a,G: ( set_list_a > a ) > a] :
( ( finite6385009043124570134st_a_a @ B )
=> ( ( ord_le4799719167512954133st_a_a @ A @ B )
=> ( ! [I3: set_list_a > a] :
( ( member_set_list_a_a @ I3 @ ( minus_5613498140476352782st_a_a @ B @ A ) )
=> ( ( H3 @ I3 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: set_list_a > a] :
( ( member_set_list_a_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( member969817812316227871_a_a_a @ H3
@ ( pi_set_list_a_a_a @ B
@ ^ [Uu: set_list_a > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro4938371440467910406st_a_a @ r @ G @ A )
= ( finpro4938371440467910406st_a_a @ r @ H3 @ B ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong_left
thf(fact_285_finprod__mono__neutral__cong__left,axiom,
! [B: set_nat_list_a,A: set_nat_list_a,H3: ( nat > list_a ) > a,G: ( nat > list_a ) > a] :
( ( finite7630042315537210004list_a @ B )
=> ( ( ord_le2145805922479659755list_a @ A @ B )
=> ( ! [I3: nat > list_a] :
( ( member_nat_list_a @ I3 @ ( minus_4169782841487898290list_a @ B @ A ) )
=> ( ( H3 @ I3 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: nat > list_a] :
( ( member_nat_list_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( member_nat_list_a_a @ H3
@ ( pi_nat_list_a_a @ B
@ ^ [Uu: nat > list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro4838020199848830884list_a @ r @ G @ A )
= ( finpro4838020199848830884list_a @ r @ H3 @ B ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong_left
thf(fact_286_finprod__mono__neutral__cong__left,axiom,
! [B: set_nat_a,A: set_nat_a,H3: ( nat > a ) > a,G: ( nat > a ) > a] :
( ( finite_finite_nat_a @ B )
=> ( ( ord_le871467723717165285_nat_a @ A @ B )
=> ( ! [I3: nat > a] :
( ( member_nat_a @ I3 @ ( minus_490503922182417452_nat_a @ B @ A ) )
=> ( ( H3 @ I3 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( member_nat_a_a @ H3
@ ( pi_nat_a_a @ B
@ ^ [Uu: nat > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro5839458686994656414_nat_a @ r @ G @ A )
= ( finpro5839458686994656414_nat_a @ r @ H3 @ B ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong_left
thf(fact_287_finprod__mono__neutral__cong__left,axiom,
! [B: set_a_a,A: set_a_a,H3: ( a > a ) > a,G: ( a > a ) > a] :
( ( finite_finite_a_a @ B )
=> ( ( ord_less_eq_set_a_a @ A @ B )
=> ( ! [I3: a > a] :
( ( member_a_a @ I3 @ ( minus_minus_set_a_a @ B @ A ) )
=> ( ( H3 @ I3 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: a > a] :
( ( member_a_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( member_a_a_a @ H3
@ ( pi_a_a_a @ B
@ ^ [Uu: a > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro3012607322079259884_b_a_a @ r @ G @ A )
= ( finpro3012607322079259884_b_a_a @ r @ H3 @ B ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong_left
thf(fact_288_finprod__mono__neutral__cong__left,axiom,
! [B: set_nat,A: set_nat,H3: nat > a,G: nat > a] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ ( minus_minus_set_nat @ B @ A ) )
=> ( ( H3 @ I3 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( G @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( member_nat_a @ H3
@ ( pi_nat_a @ B
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro1280035270526425175_b_nat @ r @ G @ A )
= ( finpro1280035270526425175_b_nat @ r @ H3 @ B ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong_left
thf(fact_289_finprod__mono__neutral__cong__left,axiom,
! [B: set_a,A: set_a,H3: a > a,G: a > a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ! [I3: a] :
( ( member_a @ I3 @ ( minus_minus_set_a @ B @ A ) )
=> ( ( H3 @ I3 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( member_a_a @ H3
@ ( pi_a_a @ B
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro205304725090349623_a_b_a @ r @ G @ A )
= ( finpro205304725090349623_a_b_a @ r @ H3 @ B ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong_left
thf(fact_290_finprod__mono__neutral__cong__left,axiom,
! [B: set_list_a,A: set_list_a,H3: list_a > a,G: list_a > a] :
( ( finite_finite_list_a @ B )
=> ( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ! [I3: list_a] :
( ( member_list_a @ I3 @ ( minus_646659088055828811list_a @ B @ A ) )
=> ( ( H3 @ I3 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( member_list_a_a @ H3
@ ( pi_list_a_a @ B
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro6052973074229812797list_a @ r @ G @ A )
= ( finpro6052973074229812797list_a @ r @ H3 @ B ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong_left
thf(fact_291_finprod__mono__neutral__cong__right,axiom,
! [B: set_set_list_a,A: set_set_list_a,G: set_list_a > a,H3: set_list_a > a] :
( ( finite5282473924520328461list_a @ B )
=> ( ( ord_le8877086941679407844list_a @ A @ B )
=> ( ! [I3: set_list_a] :
( ( member_set_list_a @ I3 @ ( minus_4782336368215558443list_a @ B @ A ) )
=> ( ( G @ I3 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: set_list_a] :
( ( member_set_list_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( member_set_list_a_a @ G
@ ( pi_set_list_a_a @ B
@ ^ [Uu: set_list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro3826550488720007709list_a @ r @ G @ B )
= ( finpro3826550488720007709list_a @ r @ H3 @ A ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong_right
thf(fact_292_finprod__mono__neutral__cong__right,axiom,
! [B: set_set_list_a_a,A: set_set_list_a_a,G: ( set_list_a > a ) > a,H3: ( set_list_a > a ) > a] :
( ( finite6385009043124570134st_a_a @ B )
=> ( ( ord_le4799719167512954133st_a_a @ A @ B )
=> ( ! [I3: set_list_a > a] :
( ( member_set_list_a_a @ I3 @ ( minus_5613498140476352782st_a_a @ B @ A ) )
=> ( ( G @ I3 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: set_list_a > a] :
( ( member_set_list_a_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( member969817812316227871_a_a_a @ G
@ ( pi_set_list_a_a_a @ B
@ ^ [Uu: set_list_a > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro4938371440467910406st_a_a @ r @ G @ B )
= ( finpro4938371440467910406st_a_a @ r @ H3 @ A ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong_right
thf(fact_293_finprod__mono__neutral__cong__right,axiom,
! [B: set_nat_list_a,A: set_nat_list_a,G: ( nat > list_a ) > a,H3: ( nat > list_a ) > a] :
( ( finite7630042315537210004list_a @ B )
=> ( ( ord_le2145805922479659755list_a @ A @ B )
=> ( ! [I3: nat > list_a] :
( ( member_nat_list_a @ I3 @ ( minus_4169782841487898290list_a @ B @ A ) )
=> ( ( G @ I3 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: nat > list_a] :
( ( member_nat_list_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( member_nat_list_a_a @ G
@ ( pi_nat_list_a_a @ B
@ ^ [Uu: nat > list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro4838020199848830884list_a @ r @ G @ B )
= ( finpro4838020199848830884list_a @ r @ H3 @ A ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong_right
thf(fact_294_finprod__mono__neutral__cong__right,axiom,
! [B: set_nat_a,A: set_nat_a,G: ( nat > a ) > a,H3: ( nat > a ) > a] :
( ( finite_finite_nat_a @ B )
=> ( ( ord_le871467723717165285_nat_a @ A @ B )
=> ( ! [I3: nat > a] :
( ( member_nat_a @ I3 @ ( minus_490503922182417452_nat_a @ B @ A ) )
=> ( ( G @ I3 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( member_nat_a_a @ G
@ ( pi_nat_a_a @ B
@ ^ [Uu: nat > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro5839458686994656414_nat_a @ r @ G @ B )
= ( finpro5839458686994656414_nat_a @ r @ H3 @ A ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong_right
thf(fact_295_finprod__mono__neutral__cong__right,axiom,
! [B: set_a_a,A: set_a_a,G: ( a > a ) > a,H3: ( a > a ) > a] :
( ( finite_finite_a_a @ B )
=> ( ( ord_less_eq_set_a_a @ A @ B )
=> ( ! [I3: a > a] :
( ( member_a_a @ I3 @ ( minus_minus_set_a_a @ B @ A ) )
=> ( ( G @ I3 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: a > a] :
( ( member_a_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( member_a_a_a @ G
@ ( pi_a_a_a @ B
@ ^ [Uu: a > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro3012607322079259884_b_a_a @ r @ G @ B )
= ( finpro3012607322079259884_b_a_a @ r @ H3 @ A ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong_right
thf(fact_296_finprod__mono__neutral__cong__right,axiom,
! [B: set_nat,A: set_nat,G: nat > a,H3: nat > a] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ ( minus_minus_set_nat @ B @ A ) )
=> ( ( G @ I3 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( G @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ B
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro1280035270526425175_b_nat @ r @ G @ B )
= ( finpro1280035270526425175_b_nat @ r @ H3 @ A ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong_right
thf(fact_297_finprod__mono__neutral__cong__right,axiom,
! [B: set_a,A: set_a,G: a > a,H3: a > a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ! [I3: a] :
( ( member_a @ I3 @ ( minus_minus_set_a @ B @ A ) )
=> ( ( G @ I3 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( member_a_a @ G
@ ( pi_a_a @ B
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro205304725090349623_a_b_a @ r @ G @ B )
= ( finpro205304725090349623_a_b_a @ r @ H3 @ A ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong_right
thf(fact_298_finprod__mono__neutral__cong__right,axiom,
! [B: set_list_a,A: set_list_a,G: list_a > a,H3: list_a > a] :
( ( finite_finite_list_a @ B )
=> ( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ! [I3: list_a] :
( ( member_list_a @ I3 @ ( minus_646659088055828811list_a @ B @ A ) )
=> ( ( G @ I3 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( member_list_a_a @ G
@ ( pi_list_a_a @ B
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro6052973074229812797list_a @ r @ G @ B )
= ( finpro6052973074229812797list_a @ r @ H3 @ A ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong_right
thf(fact_299_add_Ofinprod__mono__neutral__cong__left,axiom,
! [B: set_set_list_a,A: set_set_list_a,H3: set_list_a > a,G: set_list_a > a] :
( ( finite5282473924520328461list_a @ B )
=> ( ( ord_le8877086941679407844list_a @ A @ B )
=> ( ! [I3: set_list_a] :
( ( member_set_list_a @ I3 @ ( minus_4782336368215558443list_a @ B @ A ) )
=> ( ( H3 @ I3 )
= ( zero_a_b @ r ) ) )
=> ( ! [X2: set_list_a] :
( ( member_set_list_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( member_set_list_a_a @ H3
@ ( pi_set_list_a_a @ B
@ ^ [Uu: set_list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum7367453022336983110list_a @ r @ G @ A )
= ( finsum7367453022336983110list_a @ r @ H3 @ B ) ) ) ) ) ) ) ).
% add.finprod_mono_neutral_cong_left
thf(fact_300_add_Ofinprod__mono__neutral__cong__left,axiom,
! [B: set_set_list_a_a,A: set_set_list_a_a,H3: ( set_list_a > a ) > a,G: ( set_list_a > a ) > a] :
( ( finite6385009043124570134st_a_a @ B )
=> ( ( ord_le4799719167512954133st_a_a @ A @ B )
=> ( ! [I3: set_list_a > a] :
( ( member_set_list_a_a @ I3 @ ( minus_5613498140476352782st_a_a @ B @ A ) )
=> ( ( H3 @ I3 )
= ( zero_a_b @ r ) ) )
=> ( ! [X2: set_list_a > a] :
( ( member_set_list_a_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( member969817812316227871_a_a_a @ H3
@ ( pi_set_list_a_a_a @ B
@ ^ [Uu: set_list_a > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum7228396637597461149st_a_a @ r @ G @ A )
= ( finsum7228396637597461149st_a_a @ r @ H3 @ B ) ) ) ) ) ) ) ).
% add.finprod_mono_neutral_cong_left
thf(fact_301_add_Ofinprod__mono__neutral__cong__left,axiom,
! [B: set_nat_list_a,A: set_nat_list_a,H3: ( nat > list_a ) > a,G: ( nat > list_a ) > a] :
( ( finite7630042315537210004list_a @ B )
=> ( ( ord_le2145805922479659755list_a @ A @ B )
=> ( ! [I3: nat > list_a] :
( ( member_nat_list_a @ I3 @ ( minus_4169782841487898290list_a @ B @ A ) )
=> ( ( H3 @ I3 )
= ( zero_a_b @ r ) ) )
=> ( ! [X2: nat > list_a] :
( ( member_nat_list_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( member_nat_list_a_a @ H3
@ ( pi_nat_list_a_a @ B
@ ^ [Uu: nat > list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum1341700292807219277list_a @ r @ G @ A )
= ( finsum1341700292807219277list_a @ r @ H3 @ B ) ) ) ) ) ) ) ).
% add.finprod_mono_neutral_cong_left
thf(fact_302_add_Ofinprod__mono__neutral__cong__left,axiom,
! [B: set_nat_a,A: set_nat_a,H3: ( nat > a ) > a,G: ( nat > a ) > a] :
( ( finite_finite_nat_a @ B )
=> ( ( ord_le871467723717165285_nat_a @ A @ B )
=> ( ! [I3: nat > a] :
( ( member_nat_a @ I3 @ ( minus_490503922182417452_nat_a @ B @ A ) )
=> ( ( H3 @ I3 )
= ( zero_a_b @ r ) ) )
=> ( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( member_nat_a_a @ H3
@ ( pi_nat_a_a @ B
@ ^ [Uu: nat > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum_a_b_nat_a @ r @ G @ A )
= ( finsum_a_b_nat_a @ r @ H3 @ B ) ) ) ) ) ) ) ).
% add.finprod_mono_neutral_cong_left
thf(fact_303_add_Ofinprod__mono__neutral__cong__left,axiom,
! [B: set_a_a,A: set_a_a,H3: ( a > a ) > a,G: ( a > a ) > a] :
( ( finite_finite_a_a @ B )
=> ( ( ord_less_eq_set_a_a @ A @ B )
=> ( ! [I3: a > a] :
( ( member_a_a @ I3 @ ( minus_minus_set_a_a @ B @ A ) )
=> ( ( H3 @ I3 )
= ( zero_a_b @ r ) ) )
=> ( ! [X2: a > a] :
( ( member_a_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( member_a_a_a @ H3
@ ( pi_a_a_a @ B
@ ^ [Uu: a > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum_a_b_a_a @ r @ G @ A )
= ( finsum_a_b_a_a @ r @ H3 @ B ) ) ) ) ) ) ) ).
% add.finprod_mono_neutral_cong_left
thf(fact_304_add_Ofinprod__mono__neutral__cong__left,axiom,
! [B: set_nat,A: set_nat,H3: nat > a,G: nat > a] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ ( minus_minus_set_nat @ B @ A ) )
=> ( ( H3 @ I3 )
= ( zero_a_b @ r ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( G @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( member_nat_a @ H3
@ ( pi_nat_a @ B
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum_a_b_nat @ r @ G @ A )
= ( finsum_a_b_nat @ r @ H3 @ B ) ) ) ) ) ) ) ).
% add.finprod_mono_neutral_cong_left
thf(fact_305_add_Ofinprod__mono__neutral__cong__left,axiom,
! [B: set_a,A: set_a,H3: a > a,G: a > a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ! [I3: a] :
( ( member_a @ I3 @ ( minus_minus_set_a @ B @ A ) )
=> ( ( H3 @ I3 )
= ( zero_a_b @ r ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( member_a_a @ H3
@ ( pi_a_a @ B
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum_a_b_a @ r @ G @ A )
= ( finsum_a_b_a @ r @ H3 @ B ) ) ) ) ) ) ) ).
% add.finprod_mono_neutral_cong_left
thf(fact_306_add_Ofinprod__mono__neutral__cong__left,axiom,
! [B: set_list_a,A: set_list_a,H3: list_a > a,G: list_a > a] :
( ( finite_finite_list_a @ B )
=> ( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ! [I3: list_a] :
( ( member_list_a @ I3 @ ( minus_646659088055828811list_a @ B @ A ) )
=> ( ( H3 @ I3 )
= ( zero_a_b @ r ) ) )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( member_list_a_a @ H3
@ ( pi_list_a_a @ B
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum_a_b_list_a @ r @ G @ A )
= ( finsum_a_b_list_a @ r @ H3 @ B ) ) ) ) ) ) ) ).
% add.finprod_mono_neutral_cong_left
thf(fact_307_add_Ofinprod__mono__neutral__cong__right,axiom,
! [B: set_set_list_a,A: set_set_list_a,G: set_list_a > a,H3: set_list_a > a] :
( ( finite5282473924520328461list_a @ B )
=> ( ( ord_le8877086941679407844list_a @ A @ B )
=> ( ! [I3: set_list_a] :
( ( member_set_list_a @ I3 @ ( minus_4782336368215558443list_a @ B @ A ) )
=> ( ( G @ I3 )
= ( zero_a_b @ r ) ) )
=> ( ! [X2: set_list_a] :
( ( member_set_list_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( member_set_list_a_a @ G
@ ( pi_set_list_a_a @ B
@ ^ [Uu: set_list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum7367453022336983110list_a @ r @ G @ B )
= ( finsum7367453022336983110list_a @ r @ H3 @ A ) ) ) ) ) ) ) ).
% add.finprod_mono_neutral_cong_right
thf(fact_308_add_Ofinprod__mono__neutral__cong__right,axiom,
! [B: set_set_list_a_a,A: set_set_list_a_a,G: ( set_list_a > a ) > a,H3: ( set_list_a > a ) > a] :
( ( finite6385009043124570134st_a_a @ B )
=> ( ( ord_le4799719167512954133st_a_a @ A @ B )
=> ( ! [I3: set_list_a > a] :
( ( member_set_list_a_a @ I3 @ ( minus_5613498140476352782st_a_a @ B @ A ) )
=> ( ( G @ I3 )
= ( zero_a_b @ r ) ) )
=> ( ! [X2: set_list_a > a] :
( ( member_set_list_a_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( member969817812316227871_a_a_a @ G
@ ( pi_set_list_a_a_a @ B
@ ^ [Uu: set_list_a > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum7228396637597461149st_a_a @ r @ G @ B )
= ( finsum7228396637597461149st_a_a @ r @ H3 @ A ) ) ) ) ) ) ) ).
% add.finprod_mono_neutral_cong_right
thf(fact_309_add_Ofinprod__mono__neutral__cong__right,axiom,
! [B: set_nat_list_a,A: set_nat_list_a,G: ( nat > list_a ) > a,H3: ( nat > list_a ) > a] :
( ( finite7630042315537210004list_a @ B )
=> ( ( ord_le2145805922479659755list_a @ A @ B )
=> ( ! [I3: nat > list_a] :
( ( member_nat_list_a @ I3 @ ( minus_4169782841487898290list_a @ B @ A ) )
=> ( ( G @ I3 )
= ( zero_a_b @ r ) ) )
=> ( ! [X2: nat > list_a] :
( ( member_nat_list_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( member_nat_list_a_a @ G
@ ( pi_nat_list_a_a @ B
@ ^ [Uu: nat > list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum1341700292807219277list_a @ r @ G @ B )
= ( finsum1341700292807219277list_a @ r @ H3 @ A ) ) ) ) ) ) ) ).
% add.finprod_mono_neutral_cong_right
thf(fact_310_add_Ofinprod__mono__neutral__cong__right,axiom,
! [B: set_nat_a,A: set_nat_a,G: ( nat > a ) > a,H3: ( nat > a ) > a] :
( ( finite_finite_nat_a @ B )
=> ( ( ord_le871467723717165285_nat_a @ A @ B )
=> ( ! [I3: nat > a] :
( ( member_nat_a @ I3 @ ( minus_490503922182417452_nat_a @ B @ A ) )
=> ( ( G @ I3 )
= ( zero_a_b @ r ) ) )
=> ( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( member_nat_a_a @ G
@ ( pi_nat_a_a @ B
@ ^ [Uu: nat > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum_a_b_nat_a @ r @ G @ B )
= ( finsum_a_b_nat_a @ r @ H3 @ A ) ) ) ) ) ) ) ).
% add.finprod_mono_neutral_cong_right
thf(fact_311_add_Ofinprod__mono__neutral__cong__right,axiom,
! [B: set_a_a,A: set_a_a,G: ( a > a ) > a,H3: ( a > a ) > a] :
( ( finite_finite_a_a @ B )
=> ( ( ord_less_eq_set_a_a @ A @ B )
=> ( ! [I3: a > a] :
( ( member_a_a @ I3 @ ( minus_minus_set_a_a @ B @ A ) )
=> ( ( G @ I3 )
= ( zero_a_b @ r ) ) )
=> ( ! [X2: a > a] :
( ( member_a_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( member_a_a_a @ G
@ ( pi_a_a_a @ B
@ ^ [Uu: a > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum_a_b_a_a @ r @ G @ B )
= ( finsum_a_b_a_a @ r @ H3 @ A ) ) ) ) ) ) ) ).
% add.finprod_mono_neutral_cong_right
thf(fact_312_add_Ofinprod__mono__neutral__cong__right,axiom,
! [B: set_nat,A: set_nat,G: nat > a,H3: nat > a] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ ( minus_minus_set_nat @ B @ A ) )
=> ( ( G @ I3 )
= ( zero_a_b @ r ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( G @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ B
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum_a_b_nat @ r @ G @ B )
= ( finsum_a_b_nat @ r @ H3 @ A ) ) ) ) ) ) ) ).
% add.finprod_mono_neutral_cong_right
thf(fact_313_add_Ofinprod__mono__neutral__cong__right,axiom,
! [B: set_a,A: set_a,G: a > a,H3: a > a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ! [I3: a] :
( ( member_a @ I3 @ ( minus_minus_set_a @ B @ A ) )
=> ( ( G @ I3 )
= ( zero_a_b @ r ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( member_a_a @ G
@ ( pi_a_a @ B
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum_a_b_a @ r @ G @ B )
= ( finsum_a_b_a @ r @ H3 @ A ) ) ) ) ) ) ) ).
% add.finprod_mono_neutral_cong_right
thf(fact_314_add_Ofinprod__mono__neutral__cong__right,axiom,
! [B: set_list_a,A: set_list_a,G: list_a > a,H3: list_a > a] :
( ( finite_finite_list_a @ B )
=> ( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ! [I3: list_a] :
( ( member_list_a @ I3 @ ( minus_646659088055828811list_a @ B @ A ) )
=> ( ( G @ I3 )
= ( zero_a_b @ r ) ) )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( member_list_a_a @ G
@ ( pi_list_a_a @ B
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum_a_b_list_a @ r @ G @ B )
= ( finsum_a_b_list_a @ r @ H3 @ A ) ) ) ) ) ) ) ).
% add.finprod_mono_neutral_cong_right
thf(fact_315_monoid__cancelI,axiom,
( ! [A4: a,B4: a,C3: a] :
( ( ( mult_a_ring_ext_a_b @ r @ C3 @ A4 )
= ( mult_a_ring_ext_a_b @ r @ C3 @ B4 ) )
=> ( ( member_a @ A4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A4 = B4 ) ) ) ) )
=> ( ! [A4: a,B4: a,C3: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A4 @ C3 )
= ( mult_a_ring_ext_a_b @ r @ B4 @ C3 ) )
=> ( ( member_a @ A4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A4 = B4 ) ) ) ) )
=> ( monoid5798828371819920185xt_a_b @ r ) ) ) ).
% monoid_cancelI
thf(fact_316_zero__is__prime_I1_J,axiom,
prime_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) ).
% zero_is_prime(1)
thf(fact_317_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_318_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_319_finite__Diff,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) ) ) ).
% finite_Diff
thf(fact_320_finite__Diff,axiom,
! [A: set_a,B: set_a] :
( ( finite_finite_a @ A )
=> ( finite_finite_a @ ( minus_minus_set_a @ A @ B ) ) ) ).
% finite_Diff
thf(fact_321_finite__Diff,axiom,
! [A: set_list_a,B: set_list_a] :
( ( finite_finite_list_a @ A )
=> ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A @ B ) ) ) ).
% finite_Diff
thf(fact_322_finite__Diff2,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) )
= ( finite_finite_nat @ A ) ) ) ).
% finite_Diff2
thf(fact_323_finite__Diff2,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( finite_finite_a @ ( minus_minus_set_a @ A @ B ) )
= ( finite_finite_a @ A ) ) ) ).
% finite_Diff2
thf(fact_324_finite__Diff2,axiom,
! [B: set_list_a,A: set_list_a] :
( ( finite_finite_list_a @ B )
=> ( ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A @ B ) )
= ( finite_finite_list_a @ A ) ) ) ).
% finite_Diff2
thf(fact_325_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_326_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_327_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_328_mem__upI,axiom,
! [F: nat > a,R: partia2175431115845679010xt_a_b] :
( ! [N: nat] : ( member_a @ ( F @ N ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ? [N2: nat] : ( bound_a @ ( zero_a_b @ R ) @ N2 @ F )
=> ( member_nat_a @ F @ ( up_a_b @ R ) ) ) ) ).
% mem_upI
thf(fact_329_mem__upI,axiom,
! [F: nat > list_a,R: partia2670972154091845814t_unit] :
( ! [N: nat] : ( member_list_a @ ( F @ N ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ? [N2: nat] : ( bound_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ N2 @ F )
=> ( member_nat_list_a @ F @ ( up_lis8464167429055313730t_unit @ R ) ) ) ) ).
% mem_upI
thf(fact_330_mem__upI,axiom,
! [F: nat > list_list_a,R: partia2956882679547061052t_unit] :
( ! [N: nat] : ( member_list_list_a @ ( F @ N ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ? [N2: nat] : ( bound_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ N2 @ F )
=> ( member8650753269014980122list_a @ F @ ( up_lis8963924889346801084t_unit @ R ) ) ) ) ).
% mem_upI
thf(fact_331_mem__upI,axiom,
! [F: nat > set_list_a,R: partia7496981018696276118t_unit] :
( ! [N: nat] : ( member_set_list_a @ ( F @ N ) @ ( partia141011252114345353t_unit @ R ) )
=> ( ? [N2: nat] : ( bound_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ N2 @ F )
=> ( member491565700723299188list_a @ F @ ( up_set529185716248919906t_unit @ R ) ) ) ) ).
% mem_upI
thf(fact_332_finprod__mono__neutral__cong,axiom,
! [B: set_set_list_a,A: set_set_list_a,H3: set_list_a > a,G: set_list_a > a] :
( ( finite5282473924520328461list_a @ B )
=> ( ( finite5282473924520328461list_a @ A )
=> ( ! [I3: set_list_a] :
( ( member_set_list_a @ I3 @ ( minus_4782336368215558443list_a @ B @ A ) )
=> ( ( H3 @ I3 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [I3: set_list_a] :
( ( member_set_list_a @ I3 @ ( minus_4782336368215558443list_a @ A @ B ) )
=> ( ( G @ I3 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: set_list_a] :
( ( member_set_list_a @ X2 @ ( inf_in4657809108759609906list_a @ A @ B ) )
=> ( ( G @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( member_set_list_a_a @ G
@ ( pi_set_list_a_a @ A
@ ^ [Uu: set_list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_set_list_a_a @ H3
@ ( pi_set_list_a_a @ B
@ ^ [Uu: set_list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro3826550488720007709list_a @ r @ G @ A )
= ( finpro3826550488720007709list_a @ r @ H3 @ B ) ) ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong
thf(fact_333_finprod__mono__neutral__cong,axiom,
! [B: set_set_list_a_a,A: set_set_list_a_a,H3: ( set_list_a > a ) > a,G: ( set_list_a > a ) > a] :
( ( finite6385009043124570134st_a_a @ B )
=> ( ( finite6385009043124570134st_a_a @ A )
=> ( ! [I3: set_list_a > a] :
( ( member_set_list_a_a @ I3 @ ( minus_5613498140476352782st_a_a @ B @ A ) )
=> ( ( H3 @ I3 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [I3: set_list_a > a] :
( ( member_set_list_a_a @ I3 @ ( minus_5613498140476352782st_a_a @ A @ B ) )
=> ( ( G @ I3 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: set_list_a > a] :
( ( member_set_list_a_a @ X2 @ ( inf_in6568206481208318535st_a_a @ A @ B ) )
=> ( ( G @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( member969817812316227871_a_a_a @ G
@ ( pi_set_list_a_a_a @ A
@ ^ [Uu: set_list_a > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member969817812316227871_a_a_a @ H3
@ ( pi_set_list_a_a_a @ B
@ ^ [Uu: set_list_a > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro4938371440467910406st_a_a @ r @ G @ A )
= ( finpro4938371440467910406st_a_a @ r @ H3 @ B ) ) ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong
thf(fact_334_finprod__mono__neutral__cong,axiom,
! [B: set_nat_list_a,A: set_nat_list_a,H3: ( nat > list_a ) > a,G: ( nat > list_a ) > a] :
( ( finite7630042315537210004list_a @ B )
=> ( ( finite7630042315537210004list_a @ A )
=> ( ! [I3: nat > list_a] :
( ( member_nat_list_a @ I3 @ ( minus_4169782841487898290list_a @ B @ A ) )
=> ( ( H3 @ I3 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [I3: nat > list_a] :
( ( member_nat_list_a @ I3 @ ( minus_4169782841487898290list_a @ A @ B ) )
=> ( ( G @ I3 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: nat > list_a] :
( ( member_nat_list_a @ X2 @ ( inf_in6652419485960844601list_a @ A @ B ) )
=> ( ( G @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( member_nat_list_a_a @ G
@ ( pi_nat_list_a_a @ A
@ ^ [Uu: nat > list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_nat_list_a_a @ H3
@ ( pi_nat_list_a_a @ B
@ ^ [Uu: nat > list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro4838020199848830884list_a @ r @ G @ A )
= ( finpro4838020199848830884list_a @ r @ H3 @ B ) ) ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong
thf(fact_335_finprod__mono__neutral__cong,axiom,
! [B: set_nat_a,A: set_nat_a,H3: ( nat > a ) > a,G: ( nat > a ) > a] :
( ( finite_finite_nat_a @ B )
=> ( ( finite_finite_nat_a @ A )
=> ( ! [I3: nat > a] :
( ( member_nat_a @ I3 @ ( minus_490503922182417452_nat_a @ B @ A ) )
=> ( ( H3 @ I3 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [I3: nat > a] :
( ( member_nat_a @ I3 @ ( minus_490503922182417452_nat_a @ A @ B ) )
=> ( ( G @ I3 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ ( inf_inf_set_nat_a @ A @ B ) )
=> ( ( G @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( member_nat_a_a @ G
@ ( pi_nat_a_a @ A
@ ^ [Uu: nat > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_nat_a_a @ H3
@ ( pi_nat_a_a @ B
@ ^ [Uu: nat > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro5839458686994656414_nat_a @ r @ G @ A )
= ( finpro5839458686994656414_nat_a @ r @ H3 @ B ) ) ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong
thf(fact_336_finprod__mono__neutral__cong,axiom,
! [B: set_a_a,A: set_a_a,H3: ( a > a ) > a,G: ( a > a ) > a] :
( ( finite_finite_a_a @ B )
=> ( ( finite_finite_a_a @ A )
=> ( ! [I3: a > a] :
( ( member_a_a @ I3 @ ( minus_minus_set_a_a @ B @ A ) )
=> ( ( H3 @ I3 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [I3: a > a] :
( ( member_a_a @ I3 @ ( minus_minus_set_a_a @ A @ B ) )
=> ( ( G @ I3 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: a > a] :
( ( member_a_a @ X2 @ ( inf_inf_set_a_a @ A @ B ) )
=> ( ( G @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( member_a_a_a @ G
@ ( pi_a_a_a @ A
@ ^ [Uu: a > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_a_a_a @ H3
@ ( pi_a_a_a @ B
@ ^ [Uu: a > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro3012607322079259884_b_a_a @ r @ G @ A )
= ( finpro3012607322079259884_b_a_a @ r @ H3 @ B ) ) ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong
thf(fact_337_finprod__mono__neutral__cong,axiom,
! [B: set_nat,A: set_nat,H3: nat > a,G: nat > a] :
( ( finite_finite_nat @ B )
=> ( ( finite_finite_nat @ A )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ ( minus_minus_set_nat @ B @ A ) )
=> ( ( H3 @ I3 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ ( minus_minus_set_nat @ A @ B ) )
=> ( ( G @ I3 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( inf_inf_set_nat @ A @ B ) )
=> ( ( G @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ A
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_nat_a @ H3
@ ( pi_nat_a @ B
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro1280035270526425175_b_nat @ r @ G @ A )
= ( finpro1280035270526425175_b_nat @ r @ H3 @ B ) ) ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong
thf(fact_338_finprod__mono__neutral__cong,axiom,
! [B: set_a,A: set_a,H3: a > a,G: a > a] :
( ( finite_finite_a @ B )
=> ( ( finite_finite_a @ A )
=> ( ! [I3: a] :
( ( member_a @ I3 @ ( minus_minus_set_a @ B @ A ) )
=> ( ( H3 @ I3 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [I3: a] :
( ( member_a @ I3 @ ( minus_minus_set_a @ A @ B ) )
=> ( ( G @ I3 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( inf_inf_set_a @ A @ B ) )
=> ( ( G @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( member_a_a @ G
@ ( pi_a_a @ A
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_a_a @ H3
@ ( pi_a_a @ B
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro205304725090349623_a_b_a @ r @ G @ A )
= ( finpro205304725090349623_a_b_a @ r @ H3 @ B ) ) ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong
thf(fact_339_finprod__mono__neutral__cong,axiom,
! [B: set_list_a,A: set_list_a,H3: list_a > a,G: list_a > a] :
( ( finite_finite_list_a @ B )
=> ( ( finite_finite_list_a @ A )
=> ( ! [I3: list_a] :
( ( member_list_a @ I3 @ ( minus_646659088055828811list_a @ B @ A ) )
=> ( ( H3 @ I3 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [I3: list_a] :
( ( member_list_a @ I3 @ ( minus_646659088055828811list_a @ A @ B ) )
=> ( ( G @ I3 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( inf_inf_set_list_a @ A @ B ) )
=> ( ( G @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( member_list_a_a @ G
@ ( pi_list_a_a @ A
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a_a @ H3
@ ( pi_list_a_a @ B
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro6052973074229812797list_a @ r @ G @ A )
= ( finpro6052973074229812797list_a @ r @ H3 @ B ) ) ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong
thf(fact_340_add_Ofinprod__mono__neutral__cong,axiom,
! [B: set_set_list_a,A: set_set_list_a,H3: set_list_a > a,G: set_list_a > a] :
( ( finite5282473924520328461list_a @ B )
=> ( ( finite5282473924520328461list_a @ A )
=> ( ! [I3: set_list_a] :
( ( member_set_list_a @ I3 @ ( minus_4782336368215558443list_a @ B @ A ) )
=> ( ( H3 @ I3 )
= ( zero_a_b @ r ) ) )
=> ( ! [I3: set_list_a] :
( ( member_set_list_a @ I3 @ ( minus_4782336368215558443list_a @ A @ B ) )
=> ( ( G @ I3 )
= ( zero_a_b @ r ) ) )
=> ( ! [X2: set_list_a] :
( ( member_set_list_a @ X2 @ ( inf_in4657809108759609906list_a @ A @ B ) )
=> ( ( G @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( member_set_list_a_a @ G
@ ( pi_set_list_a_a @ A
@ ^ [Uu: set_list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_set_list_a_a @ H3
@ ( pi_set_list_a_a @ B
@ ^ [Uu: set_list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum7367453022336983110list_a @ r @ G @ A )
= ( finsum7367453022336983110list_a @ r @ H3 @ B ) ) ) ) ) ) ) ) ) ).
% add.finprod_mono_neutral_cong
thf(fact_341_add_Ofinprod__mono__neutral__cong,axiom,
! [B: set_set_list_a_a,A: set_set_list_a_a,H3: ( set_list_a > a ) > a,G: ( set_list_a > a ) > a] :
( ( finite6385009043124570134st_a_a @ B )
=> ( ( finite6385009043124570134st_a_a @ A )
=> ( ! [I3: set_list_a > a] :
( ( member_set_list_a_a @ I3 @ ( minus_5613498140476352782st_a_a @ B @ A ) )
=> ( ( H3 @ I3 )
= ( zero_a_b @ r ) ) )
=> ( ! [I3: set_list_a > a] :
( ( member_set_list_a_a @ I3 @ ( minus_5613498140476352782st_a_a @ A @ B ) )
=> ( ( G @ I3 )
= ( zero_a_b @ r ) ) )
=> ( ! [X2: set_list_a > a] :
( ( member_set_list_a_a @ X2 @ ( inf_in6568206481208318535st_a_a @ A @ B ) )
=> ( ( G @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( member969817812316227871_a_a_a @ G
@ ( pi_set_list_a_a_a @ A
@ ^ [Uu: set_list_a > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member969817812316227871_a_a_a @ H3
@ ( pi_set_list_a_a_a @ B
@ ^ [Uu: set_list_a > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum7228396637597461149st_a_a @ r @ G @ A )
= ( finsum7228396637597461149st_a_a @ r @ H3 @ B ) ) ) ) ) ) ) ) ) ).
% add.finprod_mono_neutral_cong
thf(fact_342_add_Ofinprod__mono__neutral__cong,axiom,
! [B: set_nat_list_a,A: set_nat_list_a,H3: ( nat > list_a ) > a,G: ( nat > list_a ) > a] :
( ( finite7630042315537210004list_a @ B )
=> ( ( finite7630042315537210004list_a @ A )
=> ( ! [I3: nat > list_a] :
( ( member_nat_list_a @ I3 @ ( minus_4169782841487898290list_a @ B @ A ) )
=> ( ( H3 @ I3 )
= ( zero_a_b @ r ) ) )
=> ( ! [I3: nat > list_a] :
( ( member_nat_list_a @ I3 @ ( minus_4169782841487898290list_a @ A @ B ) )
=> ( ( G @ I3 )
= ( zero_a_b @ r ) ) )
=> ( ! [X2: nat > list_a] :
( ( member_nat_list_a @ X2 @ ( inf_in6652419485960844601list_a @ A @ B ) )
=> ( ( G @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( member_nat_list_a_a @ G
@ ( pi_nat_list_a_a @ A
@ ^ [Uu: nat > list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_nat_list_a_a @ H3
@ ( pi_nat_list_a_a @ B
@ ^ [Uu: nat > list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum1341700292807219277list_a @ r @ G @ A )
= ( finsum1341700292807219277list_a @ r @ H3 @ B ) ) ) ) ) ) ) ) ) ).
% add.finprod_mono_neutral_cong
thf(fact_343_add_Ofinprod__mono__neutral__cong,axiom,
! [B: set_nat_a,A: set_nat_a,H3: ( nat > a ) > a,G: ( nat > a ) > a] :
( ( finite_finite_nat_a @ B )
=> ( ( finite_finite_nat_a @ A )
=> ( ! [I3: nat > a] :
( ( member_nat_a @ I3 @ ( minus_490503922182417452_nat_a @ B @ A ) )
=> ( ( H3 @ I3 )
= ( zero_a_b @ r ) ) )
=> ( ! [I3: nat > a] :
( ( member_nat_a @ I3 @ ( minus_490503922182417452_nat_a @ A @ B ) )
=> ( ( G @ I3 )
= ( zero_a_b @ r ) ) )
=> ( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ ( inf_inf_set_nat_a @ A @ B ) )
=> ( ( G @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( member_nat_a_a @ G
@ ( pi_nat_a_a @ A
@ ^ [Uu: nat > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_nat_a_a @ H3
@ ( pi_nat_a_a @ B
@ ^ [Uu: nat > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum_a_b_nat_a @ r @ G @ A )
= ( finsum_a_b_nat_a @ r @ H3 @ B ) ) ) ) ) ) ) ) ) ).
% add.finprod_mono_neutral_cong
thf(fact_344_add_Ofinprod__mono__neutral__cong,axiom,
! [B: set_a_a,A: set_a_a,H3: ( a > a ) > a,G: ( a > a ) > a] :
( ( finite_finite_a_a @ B )
=> ( ( finite_finite_a_a @ A )
=> ( ! [I3: a > a] :
( ( member_a_a @ I3 @ ( minus_minus_set_a_a @ B @ A ) )
=> ( ( H3 @ I3 )
= ( zero_a_b @ r ) ) )
=> ( ! [I3: a > a] :
( ( member_a_a @ I3 @ ( minus_minus_set_a_a @ A @ B ) )
=> ( ( G @ I3 )
= ( zero_a_b @ r ) ) )
=> ( ! [X2: a > a] :
( ( member_a_a @ X2 @ ( inf_inf_set_a_a @ A @ B ) )
=> ( ( G @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( member_a_a_a @ G
@ ( pi_a_a_a @ A
@ ^ [Uu: a > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_a_a_a @ H3
@ ( pi_a_a_a @ B
@ ^ [Uu: a > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum_a_b_a_a @ r @ G @ A )
= ( finsum_a_b_a_a @ r @ H3 @ B ) ) ) ) ) ) ) ) ) ).
% add.finprod_mono_neutral_cong
thf(fact_345_add_Ofinprod__mono__neutral__cong,axiom,
! [B: set_nat,A: set_nat,H3: nat > a,G: nat > a] :
( ( finite_finite_nat @ B )
=> ( ( finite_finite_nat @ A )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ ( minus_minus_set_nat @ B @ A ) )
=> ( ( H3 @ I3 )
= ( zero_a_b @ r ) ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ ( minus_minus_set_nat @ A @ B ) )
=> ( ( G @ I3 )
= ( zero_a_b @ r ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( inf_inf_set_nat @ A @ B ) )
=> ( ( G @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ A
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_nat_a @ H3
@ ( pi_nat_a @ B
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum_a_b_nat @ r @ G @ A )
= ( finsum_a_b_nat @ r @ H3 @ B ) ) ) ) ) ) ) ) ) ).
% add.finprod_mono_neutral_cong
thf(fact_346_add_Ofinprod__mono__neutral__cong,axiom,
! [B: set_a,A: set_a,H3: a > a,G: a > a] :
( ( finite_finite_a @ B )
=> ( ( finite_finite_a @ A )
=> ( ! [I3: a] :
( ( member_a @ I3 @ ( minus_minus_set_a @ B @ A ) )
=> ( ( H3 @ I3 )
= ( zero_a_b @ r ) ) )
=> ( ! [I3: a] :
( ( member_a @ I3 @ ( minus_minus_set_a @ A @ B ) )
=> ( ( G @ I3 )
= ( zero_a_b @ r ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( inf_inf_set_a @ A @ B ) )
=> ( ( G @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( member_a_a @ G
@ ( pi_a_a @ A
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_a_a @ H3
@ ( pi_a_a @ B
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum_a_b_a @ r @ G @ A )
= ( finsum_a_b_a @ r @ H3 @ B ) ) ) ) ) ) ) ) ) ).
% add.finprod_mono_neutral_cong
thf(fact_347_add_Ofinprod__mono__neutral__cong,axiom,
! [B: set_list_a,A: set_list_a,H3: list_a > a,G: list_a > a] :
( ( finite_finite_list_a @ B )
=> ( ( finite_finite_list_a @ A )
=> ( ! [I3: list_a] :
( ( member_list_a @ I3 @ ( minus_646659088055828811list_a @ B @ A ) )
=> ( ( H3 @ I3 )
= ( zero_a_b @ r ) ) )
=> ( ! [I3: list_a] :
( ( member_list_a @ I3 @ ( minus_646659088055828811list_a @ A @ B ) )
=> ( ( G @ I3 )
= ( zero_a_b @ r ) ) )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( inf_inf_set_list_a @ A @ B ) )
=> ( ( G @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( member_list_a_a @ G
@ ( pi_list_a_a @ A
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a_a @ H3
@ ( pi_list_a_a @ B
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum_a_b_list_a @ r @ G @ A )
= ( finsum_a_b_list_a @ r @ H3 @ B ) ) ) ) ) ) ) ) ) ).
% add.finprod_mono_neutral_cong
thf(fact_348_up__one__closed,axiom,
( member_nat_a
@ ^ [N3: nat] : ( if_a @ ( N3 = zero_zero_nat ) @ ( one_a_ring_ext_a_b @ r ) @ ( zero_a_b @ r ) )
@ ( up_a_b @ r ) ) ).
% up_one_closed
thf(fact_349_domain_Oring__iso__imp__img__domain,axiom,
! [R: partia2175431115845679010xt_a_b,H3: a > a,S: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R )
=> ( ( member_a_a @ H3 @ ( ring_iso_a_b_a_b @ R @ S ) )
=> ( domain_a_b
@ ( zero_update_a_b
@ ^ [Uu: a] : ( H3 @ ( zero_a_b @ R ) )
@ S ) ) ) ) ).
% domain.ring_iso_imp_img_domain
thf(fact_350_domain_Oring__iso__imp__img__domain,axiom,
! [R: partia2175431115845679010xt_a_b,H3: a > list_a,S: partia2670972154091845814t_unit] :
( ( domain_a_b @ R )
=> ( ( member_a_list_a @ H3 @ ( ring_i4557880751517319194t_unit @ R @ S ) )
=> ( domain6553523120543210313t_unit
@ ( zero_u1196785550890449590t_unit
@ ^ [Uu: list_a] : ( H3 @ ( zero_a_b @ R ) )
@ S ) ) ) ) ).
% domain.ring_iso_imp_img_domain
thf(fact_351_domain_Oring__iso__imp__img__domain,axiom,
! [R: partia2670972154091845814t_unit,H3: list_a > a,S: partia2175431115845679010xt_a_b] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a_a @ H3 @ ( ring_i7048835797181109658it_a_b @ R @ S ) )
=> ( domain_a_b
@ ( zero_update_a_b
@ ^ [Uu: a] : ( H3 @ ( zero_l4142658623432671053t_unit @ R ) )
@ S ) ) ) ) ).
% domain.ring_iso_imp_img_domain
thf(fact_352_domain_Oring__iso__imp__img__domain,axiom,
! [R: partia2670972154091845814t_unit,H3: list_a > list_a,S: partia2670972154091845814t_unit] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a_list_a @ H3 @ ( ring_i7414513579304222626t_unit @ R @ S ) )
=> ( domain6553523120543210313t_unit
@ ( zero_u1196785550890449590t_unit
@ ^ [Uu: list_a] : ( H3 @ ( zero_l4142658623432671053t_unit @ R ) )
@ S ) ) ) ) ).
% domain.ring_iso_imp_img_domain
thf(fact_353_domain_Oring__iso__imp__img__domain,axiom,
! [R: partia7496981018696276118t_unit,H3: set_list_a > a,S: partia2175431115845679010xt_a_b] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member_set_list_a_a @ H3 @ ( ring_i8122894263081988538it_a_b @ R @ S ) )
=> ( domain_a_b
@ ( zero_update_a_b
@ ^ [Uu: a] : ( H3 @ ( zero_s2910681146719230829t_unit @ R ) )
@ S ) ) ) ) ).
% domain.ring_iso_imp_img_domain
thf(fact_354_domain_Oring__primeE_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,P: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ P @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_ring_prime_a_b @ R @ P )
=> ( prime_a_ring_ext_a_b @ R @ P ) ) ) ) ).
% domain.ring_primeE(3)
thf(fact_355_domain_Oring__primeE_I3_J,axiom,
! [R: partia2670972154091845814t_unit,P: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ring_r6430282645014804837t_unit @ R @ P )
=> ( prime_2011924034616061926t_unit @ R @ P ) ) ) ) ).
% domain.ring_primeE(3)
thf(fact_356_domain_Oring__primeE_I3_J,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ring_r5437400583859147359t_unit @ R @ P )
=> ( prime_1232919612140715622t_unit @ R @ P ) ) ) ) ).
% domain.ring_primeE(3)
thf(fact_357_domain_Oring__primeE_I3_J,axiom,
! [R: partia7496981018696276118t_unit,P: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member_set_list_a @ P @ ( partia141011252114345353t_unit @ R ) )
=> ( ( ring_r1091214237498979717t_unit @ R @ P )
=> ( prime_5738381090551951334t_unit @ R @ P ) ) ) ) ).
% domain.ring_primeE(3)
thf(fact_358_cring__fieldI2,axiom,
( ( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) )
=> ( ! [A4: a] :
( ( member_a @ A4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A4
!= ( zero_a_b @ r ) )
=> ? [X5: a] :
( ( member_a @ X5 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ A4 @ X5 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) )
=> ( field_a_b @ r ) ) ) ).
% cring_fieldI2
thf(fact_359_boundD__carrier,axiom,
! [N4: nat,F: nat > a,M2: nat] :
( ( bound_a @ ( zero_a_b @ r ) @ N4 @ F )
=> ( ( ord_less_nat @ N4 @ M2 )
=> ( member_a @ ( F @ M2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% boundD_carrier
thf(fact_360_domain_Oring__primeE_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,P: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ P @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_ring_prime_a_b @ R @ P )
=> ( P
!= ( zero_a_b @ R ) ) ) ) ) ).
% domain.ring_primeE(1)
thf(fact_361_domain_Oring__primeE_I1_J,axiom,
! [R: partia2670972154091845814t_unit,P: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ring_r6430282645014804837t_unit @ R @ P )
=> ( P
!= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ).
% domain.ring_primeE(1)
thf(fact_362_domain_Oring__primeE_I1_J,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ring_r5437400583859147359t_unit @ R @ P )
=> ( P
!= ( zero_l347298301471573063t_unit @ R ) ) ) ) ) ).
% domain.ring_primeE(1)
thf(fact_363_domain_Oring__primeE_I1_J,axiom,
! [R: partia7496981018696276118t_unit,P: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member_set_list_a @ P @ ( partia141011252114345353t_unit @ R ) )
=> ( ( ring_r1091214237498979717t_unit @ R @ P )
=> ( P
!= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ) ).
% domain.ring_primeE(1)
thf(fact_364_domain_Oring__irreducibleE_I1_J,axiom,
! [R: partia2670972154091845814t_unit,R2: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ring_r932985474545269838t_unit @ R @ R2 )
=> ( R2
!= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ).
% domain.ring_irreducibleE(1)
thf(fact_365_domain_Oring__irreducibleE_I1_J,axiom,
! [R: partia2956882679547061052t_unit,R2: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member_list_list_a @ R2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ring_r360171070648044744t_unit @ R @ R2 )
=> ( R2
!= ( zero_l347298301471573063t_unit @ R ) ) ) ) ) ).
% domain.ring_irreducibleE(1)
thf(fact_366_domain_Oring__irreducibleE_I1_J,axiom,
! [R: partia7496981018696276118t_unit,R2: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member_set_list_a @ R2 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( ring_r5115406448772830318t_unit @ R @ R2 )
=> ( R2
!= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ) ).
% domain.ring_irreducibleE(1)
thf(fact_367_domain_Oring__irreducibleE_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,R2: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_r999134135267193926le_a_b @ R @ R2 )
=> ( R2
!= ( zero_a_b @ R ) ) ) ) ) ).
% domain.ring_irreducibleE(1)
thf(fact_368_subalgebra__inter,axiom,
! [K: set_a,V: set_a,V2: set_a] :
( ( embedd9027525575939734154ra_a_b @ K @ V @ r )
=> ( ( embedd9027525575939734154ra_a_b @ K @ V2 @ r )
=> ( embedd9027525575939734154ra_a_b @ K @ ( inf_inf_set_a @ V @ V2 ) @ r ) ) ) ).
% subalgebra_inter
thf(fact_369_subcring__inter,axiom,
! [I4: set_a,J2: set_a] :
( ( subcring_a_b @ I4 @ r )
=> ( ( subcring_a_b @ J2 @ r )
=> ( subcring_a_b @ ( inf_inf_set_a @ I4 @ J2 ) @ r ) ) ) ).
% subcring_inter
thf(fact_370_finite__Collect__less__nat,axiom,
! [K2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N3: nat] : ( ord_less_nat @ N3 @ K2 ) ) ) ).
% finite_Collect_less_nat
thf(fact_371_finite__Int,axiom,
! [F2: set_nat,G2: set_nat] :
( ( ( finite_finite_nat @ F2 )
| ( finite_finite_nat @ G2 ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ F2 @ G2 ) ) ) ).
% finite_Int
thf(fact_372_finite__Int,axiom,
! [F2: set_a,G2: set_a] :
( ( ( finite_finite_a @ F2 )
| ( finite_finite_a @ G2 ) )
=> ( finite_finite_a @ ( inf_inf_set_a @ F2 @ G2 ) ) ) ).
% finite_Int
thf(fact_373_finite__Int,axiom,
! [F2: set_list_a,G2: set_list_a] :
( ( ( finite_finite_list_a @ F2 )
| ( finite_finite_list_a @ G2 ) )
=> ( finite_finite_list_a @ ( inf_inf_set_list_a @ F2 @ G2 ) ) ) ).
% finite_Int
thf(fact_374_bound_Ointro,axiom,
! [N4: nat,F: nat > a,Z: a] :
( ! [M3: nat] :
( ( ord_less_nat @ N4 @ M3 )
=> ( ( F @ M3 )
= Z ) )
=> ( bound_a @ Z @ N4 @ F ) ) ).
% bound.intro
thf(fact_375_bound_Ointro,axiom,
! [N4: nat,F: nat > list_a,Z: list_a] :
( ! [M3: nat] :
( ( ord_less_nat @ N4 @ M3 )
=> ( ( F @ M3 )
= Z ) )
=> ( bound_list_a @ Z @ N4 @ F ) ) ).
% bound.intro
thf(fact_376_funcset__Int__left,axiom,
! [F: set_list_a > a,A: set_set_list_a,C: set_a,B: set_set_list_a] :
( ( member_set_list_a_a @ F
@ ( pi_set_list_a_a @ A
@ ^ [Uu: set_list_a] : C ) )
=> ( ( member_set_list_a_a @ F
@ ( pi_set_list_a_a @ B
@ ^ [Uu: set_list_a] : C ) )
=> ( member_set_list_a_a @ F
@ ( pi_set_list_a_a @ ( inf_in4657809108759609906list_a @ A @ B )
@ ^ [Uu: set_list_a] : C ) ) ) ) ).
% funcset_Int_left
thf(fact_377_funcset__Int__left,axiom,
! [F: nat > list_a,A: set_nat,C: set_list_a,B: set_nat] :
( ( member_nat_list_a @ F
@ ( pi_nat_list_a @ A
@ ^ [Uu: nat] : C ) )
=> ( ( member_nat_list_a @ F
@ ( pi_nat_list_a @ B
@ ^ [Uu: nat] : C ) )
=> ( member_nat_list_a @ F
@ ( pi_nat_list_a @ ( inf_inf_set_nat @ A @ B )
@ ^ [Uu: nat] : C ) ) ) ) ).
% funcset_Int_left
thf(fact_378_funcset__Int__left,axiom,
! [F: nat > a,A: set_nat,C: set_a,B: set_nat] :
( ( member_nat_a @ F
@ ( pi_nat_a @ A
@ ^ [Uu: nat] : C ) )
=> ( ( member_nat_a @ F
@ ( pi_nat_a @ B
@ ^ [Uu: nat] : C ) )
=> ( member_nat_a @ F
@ ( pi_nat_a @ ( inf_inf_set_nat @ A @ B )
@ ^ [Uu: nat] : C ) ) ) ) ).
% funcset_Int_left
thf(fact_379_funcset__Int__left,axiom,
! [F: a > a,A: set_a,C: set_a,B: set_a] :
( ( member_a_a @ F
@ ( pi_a_a @ A
@ ^ [Uu: a] : C ) )
=> ( ( member_a_a @ F
@ ( pi_a_a @ B
@ ^ [Uu: a] : C ) )
=> ( member_a_a @ F
@ ( pi_a_a @ ( inf_inf_set_a @ A @ B )
@ ^ [Uu: a] : C ) ) ) ) ).
% funcset_Int_left
thf(fact_380_Pi__Int,axiom,
! [I4: set_a,E: a > set_a,F2: a > set_a] :
( ( inf_inf_set_a_a @ ( pi_a_a @ I4 @ E ) @ ( pi_a_a @ I4 @ F2 ) )
= ( pi_a_a @ I4
@ ^ [I: a] : ( inf_inf_set_a @ ( E @ I ) @ ( F2 @ I ) ) ) ) ).
% Pi_Int
thf(fact_381_bound__def,axiom,
( bound_a
= ( ^ [Z2: a,N3: nat,F3: nat > a] :
! [M4: nat] :
( ( ord_less_nat @ N3 @ M4 )
=> ( ( F3 @ M4 )
= Z2 ) ) ) ) ).
% bound_def
thf(fact_382_bound__def,axiom,
( bound_list_a
= ( ^ [Z2: list_a,N3: nat,F3: nat > list_a] :
! [M4: nat] :
( ( ord_less_nat @ N3 @ M4 )
=> ( ( F3 @ M4 )
= Z2 ) ) ) ) ).
% bound_def
thf(fact_383_bound_Obound,axiom,
! [Z: a,N4: nat,F: nat > a,M2: nat] :
( ( bound_a @ Z @ N4 @ F )
=> ( ( ord_less_nat @ N4 @ M2 )
=> ( ( F @ M2 )
= Z ) ) ) ).
% bound.bound
thf(fact_384_bound_Obound,axiom,
! [Z: list_a,N4: nat,F: nat > list_a,M2: nat] :
( ( bound_list_a @ Z @ N4 @ F )
=> ( ( ord_less_nat @ N4 @ M2 )
=> ( ( F @ M2 )
= Z ) ) ) ).
% bound.bound
thf(fact_385_field_Oring__iso__imp__img__field,axiom,
! [R: partia2175431115845679010xt_a_b,H3: a > list_a,S: partia2670972154091845814t_unit] :
( ( field_a_b @ R )
=> ( ( member_a_list_a @ H3 @ ( ring_i4557880751517319194t_unit @ R @ S ) )
=> ( field_6388047844668329575t_unit
@ ( zero_u1196785550890449590t_unit
@ ^ [Uu: list_a] : ( H3 @ ( zero_a_b @ R ) )
@ S ) ) ) ) ).
% field.ring_iso_imp_img_field
thf(fact_386_field_Oring__iso__imp__img__field,axiom,
! [R: partia2175431115845679010xt_a_b,H3: a > a,S: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R )
=> ( ( member_a_a @ H3 @ ( ring_iso_a_b_a_b @ R @ S ) )
=> ( field_a_b
@ ( zero_update_a_b
@ ^ [Uu: a] : ( H3 @ ( zero_a_b @ R ) )
@ S ) ) ) ) ).
% field.ring_iso_imp_img_field
thf(fact_387_field_Oring__iso__imp__img__field,axiom,
! [R: partia2670972154091845814t_unit,H3: list_a > a,S: partia2175431115845679010xt_a_b] :
( ( field_6388047844668329575t_unit @ R )
=> ( ( member_list_a_a @ H3 @ ( ring_i7048835797181109658it_a_b @ R @ S ) )
=> ( field_a_b
@ ( zero_update_a_b
@ ^ [Uu: a] : ( H3 @ ( zero_l4142658623432671053t_unit @ R ) )
@ S ) ) ) ) ).
% field.ring_iso_imp_img_field
thf(fact_388_field_Oring__iso__imp__img__field,axiom,
! [R: partia2670972154091845814t_unit,H3: list_a > list_a,S: partia2670972154091845814t_unit] :
( ( field_6388047844668329575t_unit @ R )
=> ( ( member_list_a_list_a @ H3 @ ( ring_i7414513579304222626t_unit @ R @ S ) )
=> ( field_6388047844668329575t_unit
@ ( zero_u1196785550890449590t_unit
@ ^ [Uu: list_a] : ( H3 @ ( zero_l4142658623432671053t_unit @ R ) )
@ S ) ) ) ) ).
% field.ring_iso_imp_img_field
thf(fact_389_field_Oring__iso__imp__img__field,axiom,
! [R: partia7496981018696276118t_unit,H3: set_list_a > a,S: partia2175431115845679010xt_a_b] :
( ( field_26233345952514695t_unit @ R )
=> ( ( member_set_list_a_a @ H3 @ ( ring_i8122894263081988538it_a_b @ R @ S ) )
=> ( field_a_b
@ ( zero_update_a_b
@ ^ [Uu: a] : ( H3 @ ( zero_s2910681146719230829t_unit @ R ) )
@ S ) ) ) ) ).
% field.ring_iso_imp_img_field
thf(fact_390_bound__below,axiom,
! [Z: a,M2: nat,F: nat > a,N4: nat] :
( ( bound_a @ Z @ M2 @ F )
=> ( ( ( F @ N4 )
!= Z )
=> ( ord_less_eq_nat @ N4 @ M2 ) ) ) ).
% bound_below
thf(fact_391_bound__below,axiom,
! [Z: list_a,M2: nat,F: nat > list_a,N4: nat] :
( ( bound_list_a @ Z @ M2 @ F )
=> ( ( ( F @ N4 )
!= Z )
=> ( ord_less_eq_nat @ N4 @ M2 ) ) ) ).
% bound_below
thf(fact_392_mem__upD,axiom,
! [F: nat > a,R: partia2175431115845679010xt_a_b,N4: nat] :
( ( member_nat_a @ F @ ( up_a_b @ R ) )
=> ( member_a @ ( F @ N4 ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% mem_upD
thf(fact_393_mem__upD,axiom,
! [F: nat > list_a,R: partia2670972154091845814t_unit,N4: nat] :
( ( member_nat_list_a @ F @ ( up_lis8464167429055313730t_unit @ R ) )
=> ( member_list_a @ ( F @ N4 ) @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% mem_upD
thf(fact_394_mem__upD,axiom,
! [F: nat > list_list_a,R: partia2956882679547061052t_unit,N4: nat] :
( ( member8650753269014980122list_a @ F @ ( up_lis8963924889346801084t_unit @ R ) )
=> ( member_list_list_a @ ( F @ N4 ) @ ( partia2464479390973590831t_unit @ R ) ) ) ).
% mem_upD
thf(fact_395_mem__upD,axiom,
! [F: nat > set_list_a,R: partia7496981018696276118t_unit,N4: nat] :
( ( member491565700723299188list_a @ F @ ( up_set529185716248919906t_unit @ R ) )
=> ( member_set_list_a @ ( F @ N4 ) @ ( partia141011252114345353t_unit @ R ) ) ) ).
% mem_upD
thf(fact_396_ring__iso__memE_I1_J,axiom,
! [H3: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X: a] :
( ( member_a_a @ H3 @ ( ring_iso_a_b_a_b @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( H3 @ X ) @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_397_ring__iso__memE_I1_J,axiom,
! [H3: a > list_a,R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,X: a] :
( ( member_a_list_a @ H3 @ ( ring_i4557880751517319194t_unit @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_list_a @ ( H3 @ X ) @ ( partia5361259788508890537t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_398_ring__iso__memE_I1_J,axiom,
! [H3: list_a > a,R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,X: list_a] :
( ( member_list_a_a @ H3 @ ( ring_i7048835797181109658it_a_b @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_a @ ( H3 @ X ) @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_399_ring__iso__memE_I1_J,axiom,
! [H3: a > list_list_a,R: partia2175431115845679010xt_a_b,S: partia2956882679547061052t_unit,X: a] :
( ( member_a_list_list_a @ H3 @ ( ring_i4464730343205239444t_unit @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_list_list_a @ ( H3 @ X ) @ ( partia2464479390973590831t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_400_ring__iso__memE_I1_J,axiom,
! [H3: a > set_list_a,R: partia2175431115845679010xt_a_b,S: partia7496981018696276118t_unit,X: a] :
( ( member_a_set_list_a @ H3 @ ( ring_i5325512697602418746t_unit @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_set_list_a @ ( H3 @ X ) @ ( partia141011252114345353t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_401_ring__iso__memE_I1_J,axiom,
! [H3: list_a > list_a,R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,X: list_a] :
( ( member_list_a_list_a @ H3 @ ( ring_i7414513579304222626t_unit @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( H3 @ X ) @ ( partia5361259788508890537t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_402_ring__iso__memE_I1_J,axiom,
! [H3: list_list_a > a,R: partia2956882679547061052t_unit,S: partia2175431115845679010xt_a_b,X: list_list_a] :
( ( member_list_list_a_a @ H3 @ ( ring_i5684343068699926420it_a_b @ R @ S ) )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_a @ ( H3 @ X ) @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_403_ring__iso__memE_I1_J,axiom,
! [H3: set_list_a > a,R: partia7496981018696276118t_unit,S: partia2175431115845679010xt_a_b,X: set_list_a] :
( ( member_set_list_a_a @ H3 @ ( ring_i8122894263081988538it_a_b @ R @ S ) )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
=> ( member_a @ ( H3 @ X ) @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_404_ring__iso__memE_I1_J,axiom,
! [H3: list_a > list_list_a,R: partia2670972154091845814t_unit,S: partia2956882679547061052t_unit,X: list_a] :
( ( member6714375691612171394list_a @ H3 @ ( ring_i7582117978422105628t_unit @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_list_a @ ( H3 @ X ) @ ( partia2464479390973590831t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_405_ring__iso__memE_I1_J,axiom,
! [H3: list_a > set_list_a,R: partia2670972154091845814t_unit,S: partia7496981018696276118t_unit,X: list_a] :
( ( member4263473470251683292list_a @ H3 @ ( ring_i6716952747341498306t_unit @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_set_list_a @ ( H3 @ X ) @ ( partia141011252114345353t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_406_ring__iso__memE_I4_J,axiom,
! [H3: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b] :
( ( member_a_a @ H3 @ ( ring_iso_a_b_a_b @ R @ S ) )
=> ( ( H3 @ ( one_a_ring_ext_a_b @ R ) )
= ( one_a_ring_ext_a_b @ S ) ) ) ).
% ring_iso_memE(4)
thf(fact_407_ring__iso__memE_I4_J,axiom,
! [H3: a > list_a,R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit] :
( ( member_a_list_a @ H3 @ ( ring_i4557880751517319194t_unit @ R @ S ) )
=> ( ( H3 @ ( one_a_ring_ext_a_b @ R ) )
= ( one_li8328186300101108157t_unit @ S ) ) ) ).
% ring_iso_memE(4)
thf(fact_408_ring__iso__memE_I4_J,axiom,
! [H3: list_a > a,R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b] :
( ( member_list_a_a @ H3 @ ( ring_i7048835797181109658it_a_b @ R @ S ) )
=> ( ( H3 @ ( one_li8328186300101108157t_unit @ R ) )
= ( one_a_ring_ext_a_b @ S ) ) ) ).
% ring_iso_memE(4)
thf(fact_409_ring__iso__memE_I4_J,axiom,
! [H3: list_a > list_a,R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit] :
( ( member_list_a_list_a @ H3 @ ( ring_i7414513579304222626t_unit @ R @ S ) )
=> ( ( H3 @ ( one_li8328186300101108157t_unit @ R ) )
= ( one_li8328186300101108157t_unit @ S ) ) ) ).
% ring_iso_memE(4)
thf(fact_410_ring__iso__memE_I4_J,axiom,
! [H3: set_list_a > a,R: partia7496981018696276118t_unit,S: partia2175431115845679010xt_a_b] :
( ( member_set_list_a_a @ H3 @ ( ring_i8122894263081988538it_a_b @ R @ S ) )
=> ( ( H3 @ ( one_se1127990129394575805t_unit @ R ) )
= ( one_a_ring_ext_a_b @ S ) ) ) ).
% ring_iso_memE(4)
thf(fact_411_ring__iso__memE_I2_J,axiom,
! [H3: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( member_a_a @ H3 @ ( ring_iso_a_b_a_b @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H3 @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) )
= ( mult_a_ring_ext_a_b @ S @ ( H3 @ X ) @ ( H3 @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_412_ring__iso__memE_I2_J,axiom,
! [H3: a > list_a,R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,X: a,Y: a] :
( ( member_a_list_a @ H3 @ ( ring_i4557880751517319194t_unit @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H3 @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) )
= ( mult_l7073676228092353617t_unit @ S @ ( H3 @ X ) @ ( H3 @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_413_ring__iso__memE_I2_J,axiom,
! [H3: list_a > a,R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,X: list_a,Y: list_a] :
( ( member_list_a_a @ H3 @ ( ring_i7048835797181109658it_a_b @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H3 @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) )
= ( mult_a_ring_ext_a_b @ S @ ( H3 @ X ) @ ( H3 @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_414_ring__iso__memE_I2_J,axiom,
! [H3: list_a > list_a,R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( member_list_a_list_a @ H3 @ ( ring_i7414513579304222626t_unit @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H3 @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) )
= ( mult_l7073676228092353617t_unit @ S @ ( H3 @ X ) @ ( H3 @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_415_ring__iso__memE_I2_J,axiom,
! [H3: list_list_a > a,R: partia2956882679547061052t_unit,S: partia2175431115845679010xt_a_b,X: list_list_a,Y: list_list_a] :
( ( member_list_list_a_a @ H3 @ ( ring_i5684343068699926420it_a_b @ R @ S ) )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( H3 @ ( mult_l4853965630390486993t_unit @ R @ X @ Y ) )
= ( mult_a_ring_ext_a_b @ S @ ( H3 @ X ) @ ( H3 @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_416_ring__iso__memE_I2_J,axiom,
! [H3: list_list_a > list_a,R: partia2956882679547061052t_unit,S: partia2670972154091845814t_unit,X: list_list_a,Y: list_list_a] :
( ( member7168557129179038582list_a @ H3 @ ( ring_i4611353245267337884t_unit @ R @ S ) )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( H3 @ ( mult_l4853965630390486993t_unit @ R @ X @ Y ) )
= ( mult_l7073676228092353617t_unit @ S @ ( H3 @ X ) @ ( H3 @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_417_ring__iso__memE_I2_J,axiom,
! [H3: set_list_a > list_a,R: partia7496981018696276118t_unit,S: partia2670972154091845814t_unit,X: set_list_a,Y: set_list_a] :
( ( member5910328476188217884list_a @ H3 @ ( ring_i8566987394125245378t_unit @ R @ S ) )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ R ) )
=> ( ( H3 @ ( mult_s7802724872828879953t_unit @ R @ X @ Y ) )
= ( mult_l7073676228092353617t_unit @ S @ ( H3 @ X ) @ ( H3 @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_418_ring__iso__memE_I2_J,axiom,
! [H3: set_list_a > a,R: partia7496981018696276118t_unit,S: partia2175431115845679010xt_a_b,X: set_list_a,Y: set_list_a] :
( ( member_set_list_a_a @ H3 @ ( ring_i8122894263081988538it_a_b @ R @ S ) )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ R ) )
=> ( ( H3 @ ( mult_s7802724872828879953t_unit @ R @ X @ Y ) )
= ( mult_a_ring_ext_a_b @ S @ ( H3 @ X ) @ ( H3 @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_419_domain_Ozero__is__prime_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R )
=> ( prime_a_ring_ext_a_b @ R @ ( zero_a_b @ R ) ) ) ).
% domain.zero_is_prime(1)
thf(fact_420_domain_Ozero__is__prime_I1_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( domain6553523120543210313t_unit @ R )
=> ( prime_2011924034616061926t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) ) ) ).
% domain.zero_is_prime(1)
thf(fact_421_ring__prime__def,axiom,
( ring_ring_prime_a_b
= ( ^ [R3: partia2175431115845679010xt_a_b,A5: a] :
( ( A5
!= ( zero_a_b @ R3 ) )
& ( prime_a_ring_ext_a_b @ R3 @ A5 ) ) ) ) ).
% ring_prime_def
thf(fact_422_ring__prime__def,axiom,
( ring_r6430282645014804837t_unit
= ( ^ [R3: partia2670972154091845814t_unit,A5: list_a] :
( ( A5
!= ( zero_l4142658623432671053t_unit @ R3 ) )
& ( prime_2011924034616061926t_unit @ R3 @ A5 ) ) ) ) ).
% ring_prime_def
thf(fact_423_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_424_finprod__Un__Int,axiom,
! [A: set_set_list_a,B: set_set_list_a,G: set_list_a > a] :
( ( finite5282473924520328461list_a @ A )
=> ( ( finite5282473924520328461list_a @ B )
=> ( ( member_set_list_a_a @ G
@ ( pi_set_list_a_a @ A
@ ^ [Uu: set_list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_set_list_a_a @ G
@ ( pi_set_list_a_a @ B
@ ^ [Uu: set_list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( finpro3826550488720007709list_a @ r @ G @ ( sup_su4537662296134749976list_a @ A @ B ) ) @ ( finpro3826550488720007709list_a @ r @ G @ ( inf_in4657809108759609906list_a @ A @ B ) ) )
= ( mult_a_ring_ext_a_b @ r @ ( finpro3826550488720007709list_a @ r @ G @ A ) @ ( finpro3826550488720007709list_a @ r @ G @ B ) ) ) ) ) ) ) ).
% finprod_Un_Int
thf(fact_425_finprod__Un__Int,axiom,
! [A: set_nat,B: set_nat,G: nat > a] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ A
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ B
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( finpro1280035270526425175_b_nat @ r @ G @ ( sup_sup_set_nat @ A @ B ) ) @ ( finpro1280035270526425175_b_nat @ r @ G @ ( inf_inf_set_nat @ A @ B ) ) )
= ( mult_a_ring_ext_a_b @ r @ ( finpro1280035270526425175_b_nat @ r @ G @ A ) @ ( finpro1280035270526425175_b_nat @ r @ G @ B ) ) ) ) ) ) ) ).
% finprod_Un_Int
thf(fact_426_finprod__Un__Int,axiom,
! [A: set_a,B: set_a,G: a > a] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B )
=> ( ( member_a_a @ G
@ ( pi_a_a @ A
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_a_a @ G
@ ( pi_a_a @ B
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( finpro205304725090349623_a_b_a @ r @ G @ ( sup_sup_set_a @ A @ B ) ) @ ( finpro205304725090349623_a_b_a @ r @ G @ ( inf_inf_set_a @ A @ B ) ) )
= ( mult_a_ring_ext_a_b @ r @ ( finpro205304725090349623_a_b_a @ r @ G @ A ) @ ( finpro205304725090349623_a_b_a @ r @ G @ B ) ) ) ) ) ) ) ).
% finprod_Un_Int
thf(fact_427_finprod__Un__Int,axiom,
! [A: set_list_a,B: set_list_a,G: list_a > a] :
( ( finite_finite_list_a @ A )
=> ( ( finite_finite_list_a @ B )
=> ( ( member_list_a_a @ G
@ ( pi_list_a_a @ A
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a_a @ G
@ ( pi_list_a_a @ B
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( finpro6052973074229812797list_a @ r @ G @ ( sup_sup_set_list_a @ A @ B ) ) @ ( finpro6052973074229812797list_a @ r @ G @ ( inf_inf_set_list_a @ A @ B ) ) )
= ( mult_a_ring_ext_a_b @ r @ ( finpro6052973074229812797list_a @ r @ G @ A ) @ ( finpro6052973074229812797list_a @ r @ G @ B ) ) ) ) ) ) ) ).
% finprod_Un_Int
thf(fact_428_less__nat__zero__code,axiom,
! [N4: nat] :
~ ( ord_less_nat @ N4 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_429_neq0__conv,axiom,
! [N4: nat] :
( ( N4 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N4 ) ) ).
% neq0_conv
thf(fact_430_bot__nat__0_Onot__eq__extremum,axiom,
! [A3: nat] :
( ( A3 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A3 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_431_Int__subset__iff,axiom,
! [C: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ ( inf_inf_set_a @ A @ B ) )
= ( ( ord_less_eq_set_a @ C @ A )
& ( ord_less_eq_set_a @ C @ B ) ) ) ).
% Int_subset_iff
thf(fact_432_Int__subset__iff,axiom,
! [C: set_list_a,A: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ C @ ( inf_inf_set_list_a @ A @ B ) )
= ( ( ord_le8861187494160871172list_a @ C @ A )
& ( ord_le8861187494160871172list_a @ C @ B ) ) ) ).
% Int_subset_iff
thf(fact_433_inf_Obounded__iff,axiom,
! [A3: set_a,B3: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A3 @ ( inf_inf_set_a @ B3 @ C2 ) )
= ( ( ord_less_eq_set_a @ A3 @ B3 )
& ( ord_less_eq_set_a @ A3 @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_434_inf_Obounded__iff,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ ( inf_inf_nat @ B3 @ C2 ) )
= ( ( ord_less_eq_nat @ A3 @ B3 )
& ( ord_less_eq_nat @ A3 @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_435_inf_Obounded__iff,axiom,
! [A3: set_list_a,B3: set_list_a,C2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A3 @ ( inf_inf_set_list_a @ B3 @ C2 ) )
= ( ( ord_le8861187494160871172list_a @ A3 @ B3 )
& ( ord_le8861187494160871172list_a @ A3 @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_436_psubsetI,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_a @ A @ B ) ) ) ).
% psubsetI
thf(fact_437_psubsetI,axiom,
! [A: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_list_a @ A @ B ) ) ) ).
% psubsetI
thf(fact_438_subsetI,axiom,
! [A: set_set_list_a_a,B: set_set_list_a_a] :
( ! [X2: set_list_a > a] :
( ( member_set_list_a_a @ X2 @ A )
=> ( member_set_list_a_a @ X2 @ B ) )
=> ( ord_le4799719167512954133st_a_a @ A @ B ) ) ).
% subsetI
thf(fact_439_subsetI,axiom,
! [A: set_nat_list_a,B: set_nat_list_a] :
( ! [X2: nat > list_a] :
( ( member_nat_list_a @ X2 @ A )
=> ( member_nat_list_a @ X2 @ B ) )
=> ( ord_le2145805922479659755list_a @ A @ B ) ) ).
% subsetI
thf(fact_440_subsetI,axiom,
! [A: set_nat_a,B: set_nat_a] :
( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A )
=> ( member_nat_a @ X2 @ B ) )
=> ( ord_le871467723717165285_nat_a @ A @ B ) ) ).
% subsetI
thf(fact_441_subsetI,axiom,
! [A: set_a_a,B: set_a_a] :
( ! [X2: a > a] :
( ( member_a_a @ X2 @ A )
=> ( member_a_a @ X2 @ B ) )
=> ( ord_less_eq_set_a_a @ A @ B ) ) ).
% subsetI
thf(fact_442_subsetI,axiom,
! [A: set_a,B: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_a @ X2 @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% subsetI
thf(fact_443_subsetI,axiom,
! [A: set_list_a,B: set_list_a] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ A )
=> ( member_list_a @ X2 @ B ) )
=> ( ord_le8861187494160871172list_a @ A @ B ) ) ).
% subsetI
thf(fact_444_subset__antisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_445_subset__antisym,axiom,
! [A: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( ord_le8861187494160871172list_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_446_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_447_diff__0__eq__0,axiom,
! [N4: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N4 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_448_inf_Oidem,axiom,
! [A3: set_a] :
( ( inf_inf_set_a @ A3 @ A3 )
= A3 ) ).
% inf.idem
thf(fact_449_inf_Oidem,axiom,
! [A3: set_list_a] :
( ( inf_inf_set_list_a @ A3 @ A3 )
= A3 ) ).
% inf.idem
thf(fact_450_inf__idem,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ X )
= X ) ).
% inf_idem
thf(fact_451_inf__idem,axiom,
! [X: set_list_a] :
( ( inf_inf_set_list_a @ X @ X )
= X ) ).
% inf_idem
thf(fact_452_inf_Oleft__idem,axiom,
! [A3: set_a,B3: set_a] :
( ( inf_inf_set_a @ A3 @ ( inf_inf_set_a @ A3 @ B3 ) )
= ( inf_inf_set_a @ A3 @ B3 ) ) ).
% inf.left_idem
thf(fact_453_inf_Oleft__idem,axiom,
! [A3: set_list_a,B3: set_list_a] :
( ( inf_inf_set_list_a @ A3 @ ( inf_inf_set_list_a @ A3 @ B3 ) )
= ( inf_inf_set_list_a @ A3 @ B3 ) ) ).
% inf.left_idem
thf(fact_454_inf__left__idem,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ X @ Y ) )
= ( inf_inf_set_a @ X @ Y ) ) ).
% inf_left_idem
thf(fact_455_inf__left__idem,axiom,
! [X: set_list_a,Y: set_list_a] :
( ( inf_inf_set_list_a @ X @ ( inf_inf_set_list_a @ X @ Y ) )
= ( inf_inf_set_list_a @ X @ Y ) ) ).
% inf_left_idem
thf(fact_456_inf_Oright__idem,axiom,
! [A3: set_a,B3: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ B3 )
= ( inf_inf_set_a @ A3 @ B3 ) ) ).
% inf.right_idem
thf(fact_457_inf_Oright__idem,axiom,
! [A3: set_list_a,B3: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ A3 @ B3 ) @ B3 )
= ( inf_inf_set_list_a @ A3 @ B3 ) ) ).
% inf.right_idem
thf(fact_458_inf__right__idem,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y )
= ( inf_inf_set_a @ X @ Y ) ) ).
% inf_right_idem
thf(fact_459_inf__right__idem,axiom,
! [X: set_list_a,Y: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ X @ Y ) @ Y )
= ( inf_inf_set_list_a @ X @ Y ) ) ).
% inf_right_idem
thf(fact_460_diff__diff__cancel,axiom,
! [I2: nat,N4: nat] :
( ( ord_less_eq_nat @ I2 @ N4 )
=> ( ( minus_minus_nat @ N4 @ ( minus_minus_nat @ N4 @ I2 ) )
= I2 ) ) ).
% diff_diff_cancel
thf(fact_461_IntI,axiom,
! [C2: set_list_a > a,A: set_set_list_a_a,B: set_set_list_a_a] :
( ( member_set_list_a_a @ C2 @ A )
=> ( ( member_set_list_a_a @ C2 @ B )
=> ( member_set_list_a_a @ C2 @ ( inf_in6568206481208318535st_a_a @ A @ B ) ) ) ) ).
% IntI
thf(fact_462_IntI,axiom,
! [C2: nat > list_a,A: set_nat_list_a,B: set_nat_list_a] :
( ( member_nat_list_a @ C2 @ A )
=> ( ( member_nat_list_a @ C2 @ B )
=> ( member_nat_list_a @ C2 @ ( inf_in6652419485960844601list_a @ A @ B ) ) ) ) ).
% IntI
thf(fact_463_IntI,axiom,
! [C2: nat > a,A: set_nat_a,B: set_nat_a] :
( ( member_nat_a @ C2 @ A )
=> ( ( member_nat_a @ C2 @ B )
=> ( member_nat_a @ C2 @ ( inf_inf_set_nat_a @ A @ B ) ) ) ) ).
% IntI
thf(fact_464_IntI,axiom,
! [C2: a > a,A: set_a_a,B: set_a_a] :
( ( member_a_a @ C2 @ A )
=> ( ( member_a_a @ C2 @ B )
=> ( member_a_a @ C2 @ ( inf_inf_set_a_a @ A @ B ) ) ) ) ).
% IntI
thf(fact_465_IntI,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ A )
=> ( ( member_a @ C2 @ B )
=> ( member_a @ C2 @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% IntI
thf(fact_466_IntI,axiom,
! [C2: list_a,A: set_list_a,B: set_list_a] :
( ( member_list_a @ C2 @ A )
=> ( ( member_list_a @ C2 @ B )
=> ( member_list_a @ C2 @ ( inf_inf_set_list_a @ A @ B ) ) ) ) ).
% IntI
thf(fact_467_Int__iff,axiom,
! [C2: set_list_a > a,A: set_set_list_a_a,B: set_set_list_a_a] :
( ( member_set_list_a_a @ C2 @ ( inf_in6568206481208318535st_a_a @ A @ B ) )
= ( ( member_set_list_a_a @ C2 @ A )
& ( member_set_list_a_a @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_468_Int__iff,axiom,
! [C2: nat > list_a,A: set_nat_list_a,B: set_nat_list_a] :
( ( member_nat_list_a @ C2 @ ( inf_in6652419485960844601list_a @ A @ B ) )
= ( ( member_nat_list_a @ C2 @ A )
& ( member_nat_list_a @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_469_Int__iff,axiom,
! [C2: nat > a,A: set_nat_a,B: set_nat_a] :
( ( member_nat_a @ C2 @ ( inf_inf_set_nat_a @ A @ B ) )
= ( ( member_nat_a @ C2 @ A )
& ( member_nat_a @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_470_Int__iff,axiom,
! [C2: a > a,A: set_a_a,B: set_a_a] :
( ( member_a_a @ C2 @ ( inf_inf_set_a_a @ A @ B ) )
= ( ( member_a_a @ C2 @ A )
& ( member_a_a @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_471_Int__iff,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A @ B ) )
= ( ( member_a @ C2 @ A )
& ( member_a @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_472_Int__iff,axiom,
! [C2: list_a,A: set_list_a,B: set_list_a] :
( ( member_list_a @ C2 @ ( inf_inf_set_list_a @ A @ B ) )
= ( ( member_list_a @ C2 @ A )
& ( member_list_a @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_473_DiffI,axiom,
! [C2: set_list_a > a,A: set_set_list_a_a,B: set_set_list_a_a] :
( ( member_set_list_a_a @ C2 @ A )
=> ( ~ ( member_set_list_a_a @ C2 @ B )
=> ( member_set_list_a_a @ C2 @ ( minus_5613498140476352782st_a_a @ A @ B ) ) ) ) ).
% DiffI
thf(fact_474_DiffI,axiom,
! [C2: nat > list_a,A: set_nat_list_a,B: set_nat_list_a] :
( ( member_nat_list_a @ C2 @ A )
=> ( ~ ( member_nat_list_a @ C2 @ B )
=> ( member_nat_list_a @ C2 @ ( minus_4169782841487898290list_a @ A @ B ) ) ) ) ).
% DiffI
thf(fact_475_DiffI,axiom,
! [C2: nat > a,A: set_nat_a,B: set_nat_a] :
( ( member_nat_a @ C2 @ A )
=> ( ~ ( member_nat_a @ C2 @ B )
=> ( member_nat_a @ C2 @ ( minus_490503922182417452_nat_a @ A @ B ) ) ) ) ).
% DiffI
thf(fact_476_DiffI,axiom,
! [C2: a > a,A: set_a_a,B: set_a_a] :
( ( member_a_a @ C2 @ A )
=> ( ~ ( member_a_a @ C2 @ B )
=> ( member_a_a @ C2 @ ( minus_minus_set_a_a @ A @ B ) ) ) ) ).
% DiffI
thf(fact_477_DiffI,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ A )
=> ( ~ ( member_a @ C2 @ B )
=> ( member_a @ C2 @ ( minus_minus_set_a @ A @ B ) ) ) ) ).
% DiffI
thf(fact_478_DiffI,axiom,
! [C2: list_a,A: set_list_a,B: set_list_a] :
( ( member_list_a @ C2 @ A )
=> ( ~ ( member_list_a @ C2 @ B )
=> ( member_list_a @ C2 @ ( minus_646659088055828811list_a @ A @ B ) ) ) ) ).
% DiffI
thf(fact_479_Diff__iff,axiom,
! [C2: set_list_a > a,A: set_set_list_a_a,B: set_set_list_a_a] :
( ( member_set_list_a_a @ C2 @ ( minus_5613498140476352782st_a_a @ A @ B ) )
= ( ( member_set_list_a_a @ C2 @ A )
& ~ ( member_set_list_a_a @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_480_Diff__iff,axiom,
! [C2: nat > list_a,A: set_nat_list_a,B: set_nat_list_a] :
( ( member_nat_list_a @ C2 @ ( minus_4169782841487898290list_a @ A @ B ) )
= ( ( member_nat_list_a @ C2 @ A )
& ~ ( member_nat_list_a @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_481_Diff__iff,axiom,
! [C2: nat > a,A: set_nat_a,B: set_nat_a] :
( ( member_nat_a @ C2 @ ( minus_490503922182417452_nat_a @ A @ B ) )
= ( ( member_nat_a @ C2 @ A )
& ~ ( member_nat_a @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_482_Diff__iff,axiom,
! [C2: a > a,A: set_a_a,B: set_a_a] :
( ( member_a_a @ C2 @ ( minus_minus_set_a_a @ A @ B ) )
= ( ( member_a_a @ C2 @ A )
& ~ ( member_a_a @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_483_Diff__iff,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A @ B ) )
= ( ( member_a @ C2 @ A )
& ~ ( member_a @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_484_Diff__iff,axiom,
! [C2: list_a,A: set_list_a,B: set_list_a] :
( ( member_list_a @ C2 @ ( minus_646659088055828811list_a @ A @ B ) )
= ( ( member_list_a @ C2 @ A )
& ~ ( member_list_a @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_485_Diff__idemp,axiom,
! [A: set_a,B: set_a] :
( ( minus_minus_set_a @ ( minus_minus_set_a @ A @ B ) @ B )
= ( minus_minus_set_a @ A @ B ) ) ).
% Diff_idemp
thf(fact_486_Diff__idemp,axiom,
! [A: set_list_a,B: set_list_a] :
( ( minus_646659088055828811list_a @ ( minus_646659088055828811list_a @ A @ B ) @ B )
= ( minus_646659088055828811list_a @ A @ B ) ) ).
% Diff_idemp
thf(fact_487_UnCI,axiom,
! [C2: set_list_a > a,B: set_set_list_a_a,A: set_set_list_a_a] :
( ( ~ ( member_set_list_a_a @ C2 @ B )
=> ( member_set_list_a_a @ C2 @ A ) )
=> ( member_set_list_a_a @ C2 @ ( sup_su8833226608330050529st_a_a @ A @ B ) ) ) ).
% UnCI
thf(fact_488_UnCI,axiom,
! [C2: nat > list_a,B: set_nat_list_a,A: set_nat_list_a] :
( ( ~ ( member_nat_list_a @ C2 @ B )
=> ( member_nat_list_a @ C2 @ A ) )
=> ( member_nat_list_a @ C2 @ ( sup_su5649930751583389983list_a @ A @ B ) ) ) ).
% UnCI
thf(fact_489_UnCI,axiom,
! [C2: nat > a,B: set_nat_a,A: set_nat_a] :
( ( ~ ( member_nat_a @ C2 @ B )
=> ( member_nat_a @ C2 @ A ) )
=> ( member_nat_a @ C2 @ ( sup_sup_set_nat_a @ A @ B ) ) ) ).
% UnCI
thf(fact_490_UnCI,axiom,
! [C2: a > a,B: set_a_a,A: set_a_a] :
( ( ~ ( member_a_a @ C2 @ B )
=> ( member_a_a @ C2 @ A ) )
=> ( member_a_a @ C2 @ ( sup_sup_set_a_a @ A @ B ) ) ) ).
% UnCI
thf(fact_491_UnCI,axiom,
! [C2: a,B: set_a,A: set_a] :
( ( ~ ( member_a @ C2 @ B )
=> ( member_a @ C2 @ A ) )
=> ( member_a @ C2 @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnCI
thf(fact_492_Un__iff,axiom,
! [C2: set_list_a > a,A: set_set_list_a_a,B: set_set_list_a_a] :
( ( member_set_list_a_a @ C2 @ ( sup_su8833226608330050529st_a_a @ A @ B ) )
= ( ( member_set_list_a_a @ C2 @ A )
| ( member_set_list_a_a @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_493_Un__iff,axiom,
! [C2: nat > list_a,A: set_nat_list_a,B: set_nat_list_a] :
( ( member_nat_list_a @ C2 @ ( sup_su5649930751583389983list_a @ A @ B ) )
= ( ( member_nat_list_a @ C2 @ A )
| ( member_nat_list_a @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_494_Un__iff,axiom,
! [C2: nat > a,A: set_nat_a,B: set_nat_a] :
( ( member_nat_a @ C2 @ ( sup_sup_set_nat_a @ A @ B ) )
= ( ( member_nat_a @ C2 @ A )
| ( member_nat_a @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_495_Un__iff,axiom,
! [C2: a > a,A: set_a_a,B: set_a_a] :
( ( member_a_a @ C2 @ ( sup_sup_set_a_a @ A @ B ) )
= ( ( member_a_a @ C2 @ A )
| ( member_a_a @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_496_Un__iff,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ ( sup_sup_set_a @ A @ B ) )
= ( ( member_a @ C2 @ A )
| ( member_a @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_497_le__zero__eq,axiom,
! [N4: nat] :
( ( ord_less_eq_nat @ N4 @ zero_zero_nat )
= ( N4 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_498_not__gr__zero,axiom,
! [N4: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N4 ) )
= ( N4 = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_499_zero__diff,axiom,
! [A3: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A3 )
= zero_zero_nat ) ).
% zero_diff
thf(fact_500_diff__zero,axiom,
! [A3: nat] :
( ( minus_minus_nat @ A3 @ zero_zero_nat )
= A3 ) ).
% diff_zero
thf(fact_501_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A3: nat] :
( ( minus_minus_nat @ A3 @ A3 )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_502_le__inf__iff,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( ( ord_less_eq_set_a @ X @ Y )
& ( ord_less_eq_set_a @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_503_le__inf__iff,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
= ( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_504_le__inf__iff,axiom,
! [X: set_list_a,Y: set_list_a,Z: set_list_a] :
( ( ord_le8861187494160871172list_a @ X @ ( inf_inf_set_list_a @ Y @ Z ) )
= ( ( ord_le8861187494160871172list_a @ X @ Y )
& ( ord_le8861187494160871172list_a @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_505_le__sup__iff,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z )
= ( ( ord_less_eq_set_a @ X @ Z )
& ( ord_less_eq_set_a @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_506_le__sup__iff,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z )
= ( ( ord_less_eq_nat @ X @ Z )
& ( ord_less_eq_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_507_le__sup__iff,axiom,
! [X: set_list_a,Y: set_list_a,Z: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ X @ Y ) @ Z )
= ( ( ord_le8861187494160871172list_a @ X @ Z )
& ( ord_le8861187494160871172list_a @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_508_sup_Obounded__iff,axiom,
! [B3: set_a,C2: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B3 @ C2 ) @ A3 )
= ( ( ord_less_eq_set_a @ B3 @ A3 )
& ( ord_less_eq_set_a @ C2 @ A3 ) ) ) ).
% sup.bounded_iff
thf(fact_509_sup_Obounded__iff,axiom,
! [B3: nat,C2: nat,A3: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B3 @ C2 ) @ A3 )
= ( ( ord_less_eq_nat @ B3 @ A3 )
& ( ord_less_eq_nat @ C2 @ A3 ) ) ) ).
% sup.bounded_iff
thf(fact_510_sup_Obounded__iff,axiom,
! [B3: set_list_a,C2: set_list_a,A3: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ B3 @ C2 ) @ A3 )
= ( ( ord_le8861187494160871172list_a @ B3 @ A3 )
& ( ord_le8861187494160871172list_a @ C2 @ A3 ) ) ) ).
% sup.bounded_iff
thf(fact_511_zero__less__diff,axiom,
! [N4: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N4 @ M2 ) )
= ( ord_less_nat @ M2 @ N4 ) ) ).
% zero_less_diff
thf(fact_512_diff__is__0__eq_H,axiom,
! [M2: nat,N4: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
=> ( ( minus_minus_nat @ M2 @ N4 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_513_diff__is__0__eq,axiom,
! [M2: nat,N4: nat] :
( ( ( minus_minus_nat @ M2 @ N4 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N4 ) ) ).
% diff_is_0_eq
thf(fact_514_le0,axiom,
! [N4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N4 ) ).
% le0
thf(fact_515_bot__nat__0_Oextremum,axiom,
! [A3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A3 ) ).
% bot_nat_0.extremum
thf(fact_516_Un__subset__iff,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ C )
= ( ( ord_less_eq_set_a @ A @ C )
& ( ord_less_eq_set_a @ B @ C ) ) ) ).
% Un_subset_iff
thf(fact_517_Un__subset__iff,axiom,
! [A: set_list_a,B: set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ A @ B ) @ C )
= ( ( ord_le8861187494160871172list_a @ A @ C )
& ( ord_le8861187494160871172list_a @ B @ C ) ) ) ).
% Un_subset_iff
thf(fact_518_inf__sup__absorb,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_519_inf__sup__absorb,axiom,
! [X: set_list_a,Y: set_list_a] :
( ( inf_inf_set_list_a @ X @ ( sup_sup_set_list_a @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_520_sup__inf__absorb,axiom,
! [X: set_a,Y: set_a] :
( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_521_sup__inf__absorb,axiom,
! [X: set_list_a,Y: set_list_a] :
( ( sup_sup_set_list_a @ X @ ( inf_inf_set_list_a @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_522_finite__Un,axiom,
! [F2: set_a,G2: set_a] :
( ( finite_finite_a @ ( sup_sup_set_a @ F2 @ G2 ) )
= ( ( finite_finite_a @ F2 )
& ( finite_finite_a @ G2 ) ) ) ).
% finite_Un
thf(fact_523_finite__Un,axiom,
! [F2: set_nat,G2: set_nat] :
( ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G2 ) )
= ( ( finite_finite_nat @ F2 )
& ( finite_finite_nat @ G2 ) ) ) ).
% finite_Un
thf(fact_524_finite__Un,axiom,
! [F2: set_list_a,G2: set_list_a] :
( ( finite_finite_list_a @ ( sup_sup_set_list_a @ F2 @ G2 ) )
= ( ( finite_finite_list_a @ F2 )
& ( finite_finite_list_a @ G2 ) ) ) ).
% finite_Un
thf(fact_525_Int__Un__eq_I4_J,axiom,
! [T: set_a,S: set_a] :
( ( sup_sup_set_a @ T @ ( inf_inf_set_a @ S @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_526_Int__Un__eq_I4_J,axiom,
! [T: set_list_a,S: set_list_a] :
( ( sup_sup_set_list_a @ T @ ( inf_inf_set_list_a @ S @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_527_Int__Un__eq_I3_J,axiom,
! [S: set_a,T: set_a] :
( ( sup_sup_set_a @ S @ ( inf_inf_set_a @ S @ T ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_528_Int__Un__eq_I3_J,axiom,
! [S: set_list_a,T: set_list_a] :
( ( sup_sup_set_list_a @ S @ ( inf_inf_set_list_a @ S @ T ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_529_Int__Un__eq_I2_J,axiom,
! [S: set_a,T: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_530_Int__Un__eq_I2_J,axiom,
! [S: set_list_a,T: set_list_a] :
( ( sup_sup_set_list_a @ ( inf_inf_set_list_a @ S @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_531_Int__Un__eq_I1_J,axiom,
! [S: set_a,T: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_532_Int__Un__eq_I1_J,axiom,
! [S: set_list_a,T: set_list_a] :
( ( sup_sup_set_list_a @ ( inf_inf_set_list_a @ S @ T ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_533_Un__Int__eq_I4_J,axiom,
! [T: set_a,S: set_a] :
( ( inf_inf_set_a @ T @ ( sup_sup_set_a @ S @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_534_Un__Int__eq_I4_J,axiom,
! [T: set_list_a,S: set_list_a] :
( ( inf_inf_set_list_a @ T @ ( sup_sup_set_list_a @ S @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_535_Un__Int__eq_I3_J,axiom,
! [S: set_a,T: set_a] :
( ( inf_inf_set_a @ S @ ( sup_sup_set_a @ S @ T ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_536_Un__Int__eq_I3_J,axiom,
! [S: set_list_a,T: set_list_a] :
( ( inf_inf_set_list_a @ S @ ( sup_sup_set_list_a @ S @ T ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_537_Un__Int__eq_I2_J,axiom,
! [S: set_a,T: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_538_Un__Int__eq_I2_J,axiom,
! [S: set_list_a,T: set_list_a] :
( ( inf_inf_set_list_a @ ( sup_sup_set_list_a @ S @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_539_Un__Int__eq_I1_J,axiom,
! [S: set_a,T: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_540_Un__Int__eq_I1_J,axiom,
! [S: set_list_a,T: set_list_a] :
( ( inf_inf_set_list_a @ ( sup_sup_set_list_a @ S @ T ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_541_Un__Diff__cancel,axiom,
! [A: set_a,B: set_a] :
( ( sup_sup_set_a @ A @ ( minus_minus_set_a @ B @ A ) )
= ( sup_sup_set_a @ A @ B ) ) ).
% Un_Diff_cancel
thf(fact_542_Un__Diff__cancel,axiom,
! [A: set_list_a,B: set_list_a] :
( ( sup_sup_set_list_a @ A @ ( minus_646659088055828811list_a @ B @ A ) )
= ( sup_sup_set_list_a @ A @ B ) ) ).
% Un_Diff_cancel
thf(fact_543_Un__Diff__cancel2,axiom,
! [B: set_a,A: set_a] :
( ( sup_sup_set_a @ ( minus_minus_set_a @ B @ A ) @ A )
= ( sup_sup_set_a @ B @ A ) ) ).
% Un_Diff_cancel2
thf(fact_544_Un__Diff__cancel2,axiom,
! [B: set_list_a,A: set_list_a] :
( ( sup_sup_set_list_a @ ( minus_646659088055828811list_a @ B @ A ) @ A )
= ( sup_sup_set_list_a @ B @ A ) ) ).
% Un_Diff_cancel2
thf(fact_545_Pi__split__domain,axiom,
! [X: set_list_a > a,I4: set_set_list_a,J2: set_set_list_a,X3: set_list_a > set_a] :
( ( member_set_list_a_a @ X @ ( pi_set_list_a_a @ ( sup_su4537662296134749976list_a @ I4 @ J2 ) @ X3 ) )
= ( ( member_set_list_a_a @ X @ ( pi_set_list_a_a @ I4 @ X3 ) )
& ( member_set_list_a_a @ X @ ( pi_set_list_a_a @ J2 @ X3 ) ) ) ) ).
% Pi_split_domain
thf(fact_546_Pi__split__domain,axiom,
! [X: nat > list_a,I4: set_nat,J2: set_nat,X3: nat > set_list_a] :
( ( member_nat_list_a @ X @ ( pi_nat_list_a @ ( sup_sup_set_nat @ I4 @ J2 ) @ X3 ) )
= ( ( member_nat_list_a @ X @ ( pi_nat_list_a @ I4 @ X3 ) )
& ( member_nat_list_a @ X @ ( pi_nat_list_a @ J2 @ X3 ) ) ) ) ).
% Pi_split_domain
thf(fact_547_Pi__split__domain,axiom,
! [X: nat > a,I4: set_nat,J2: set_nat,X3: nat > set_a] :
( ( member_nat_a @ X @ ( pi_nat_a @ ( sup_sup_set_nat @ I4 @ J2 ) @ X3 ) )
= ( ( member_nat_a @ X @ ( pi_nat_a @ I4 @ X3 ) )
& ( member_nat_a @ X @ ( pi_nat_a @ J2 @ X3 ) ) ) ) ).
% Pi_split_domain
thf(fact_548_Pi__split__domain,axiom,
! [X: a > a,I4: set_a,J2: set_a,X3: a > set_a] :
( ( member_a_a @ X @ ( pi_a_a @ ( sup_sup_set_a @ I4 @ J2 ) @ X3 ) )
= ( ( member_a_a @ X @ ( pi_a_a @ I4 @ X3 ) )
& ( member_a_a @ X @ ( pi_a_a @ J2 @ X3 ) ) ) ) ).
% Pi_split_domain
thf(fact_549_funcset__Un__left,axiom,
! [F: set_list_a > a,A: set_set_list_a,B: set_set_list_a,C: set_a] :
( ( member_set_list_a_a @ F
@ ( pi_set_list_a_a @ ( sup_su4537662296134749976list_a @ A @ B )
@ ^ [Uu: set_list_a] : C ) )
= ( ( member_set_list_a_a @ F
@ ( pi_set_list_a_a @ A
@ ^ [Uu: set_list_a] : C ) )
& ( member_set_list_a_a @ F
@ ( pi_set_list_a_a @ B
@ ^ [Uu: set_list_a] : C ) ) ) ) ).
% funcset_Un_left
thf(fact_550_funcset__Un__left,axiom,
! [F: nat > list_a,A: set_nat,B: set_nat,C: set_list_a] :
( ( member_nat_list_a @ F
@ ( pi_nat_list_a @ ( sup_sup_set_nat @ A @ B )
@ ^ [Uu: nat] : C ) )
= ( ( member_nat_list_a @ F
@ ( pi_nat_list_a @ A
@ ^ [Uu: nat] : C ) )
& ( member_nat_list_a @ F
@ ( pi_nat_list_a @ B
@ ^ [Uu: nat] : C ) ) ) ) ).
% funcset_Un_left
thf(fact_551_funcset__Un__left,axiom,
! [F: nat > a,A: set_nat,B: set_nat,C: set_a] :
( ( member_nat_a @ F
@ ( pi_nat_a @ ( sup_sup_set_nat @ A @ B )
@ ^ [Uu: nat] : C ) )
= ( ( member_nat_a @ F
@ ( pi_nat_a @ A
@ ^ [Uu: nat] : C ) )
& ( member_nat_a @ F
@ ( pi_nat_a @ B
@ ^ [Uu: nat] : C ) ) ) ) ).
% funcset_Un_left
thf(fact_552_funcset__Un__left,axiom,
! [F: a > a,A: set_a,B: set_a,C: set_a] :
( ( member_a_a @ F
@ ( pi_a_a @ ( sup_sup_set_a @ A @ B )
@ ^ [Uu: a] : C ) )
= ( ( member_a_a @ F
@ ( pi_a_a @ A
@ ^ [Uu: a] : C ) )
& ( member_a_a @ F
@ ( pi_a_a @ B
@ ^ [Uu: a] : C ) ) ) ) ).
% funcset_Un_left
thf(fact_553_finite__Collect__le__nat,axiom,
! [K2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N3: nat] : ( ord_less_eq_nat @ N3 @ K2 ) ) ) ).
% finite_Collect_le_nat
thf(fact_554_sup_Ostrict__coboundedI2,axiom,
! [C2: nat,B3: nat,A3: nat] :
( ( ord_less_nat @ C2 @ B3 )
=> ( ord_less_nat @ C2 @ ( sup_sup_nat @ A3 @ B3 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_555_sup_Ostrict__coboundedI1,axiom,
! [C2: nat,A3: nat,B3: nat] :
( ( ord_less_nat @ C2 @ A3 )
=> ( ord_less_nat @ C2 @ ( sup_sup_nat @ A3 @ B3 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_556_sup_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B5: nat,A5: nat] :
( ( A5
= ( sup_sup_nat @ A5 @ B5 ) )
& ( A5 != B5 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_557_sup_Ostrict__boundedE,axiom,
! [B3: nat,C2: nat,A3: nat] :
( ( ord_less_nat @ ( sup_sup_nat @ B3 @ C2 ) @ A3 )
=> ~ ( ( ord_less_nat @ B3 @ A3 )
=> ~ ( ord_less_nat @ C2 @ A3 ) ) ) ).
% sup.strict_boundedE
thf(fact_558_sup__inf__distrib2,axiom,
! [Y: set_a,Z: set_a,X: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ Z ) @ X )
= ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ X ) @ ( sup_sup_set_a @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_559_sup__inf__distrib2,axiom,
! [Y: set_list_a,Z: set_list_a,X: set_list_a] :
( ( sup_sup_set_list_a @ ( inf_inf_set_list_a @ Y @ Z ) @ X )
= ( inf_inf_set_list_a @ ( sup_sup_set_list_a @ Y @ X ) @ ( sup_sup_set_list_a @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_560_sup__inf__distrib1,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_561_sup__inf__distrib1,axiom,
! [X: set_list_a,Y: set_list_a,Z: set_list_a] :
( ( sup_sup_set_list_a @ X @ ( inf_inf_set_list_a @ Y @ Z ) )
= ( inf_inf_set_list_a @ ( sup_sup_set_list_a @ X @ Y ) @ ( sup_sup_set_list_a @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_562_inf__sup__distrib2,axiom,
! [Y: set_a,Z: set_a,X: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ Z ) @ X )
= ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ X ) @ ( inf_inf_set_a @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_563_inf__sup__distrib2,axiom,
! [Y: set_list_a,Z: set_list_a,X: set_list_a] :
( ( inf_inf_set_list_a @ ( sup_sup_set_list_a @ Y @ Z ) @ X )
= ( sup_sup_set_list_a @ ( inf_inf_set_list_a @ Y @ X ) @ ( inf_inf_set_list_a @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_564_inf__sup__distrib1,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_565_inf__sup__distrib1,axiom,
! [X: set_list_a,Y: set_list_a,Z: set_list_a] :
( ( inf_inf_set_list_a @ X @ ( sup_sup_set_list_a @ Y @ Z ) )
= ( sup_sup_set_list_a @ ( inf_inf_set_list_a @ X @ Y ) @ ( inf_inf_set_list_a @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_566_sup_Oabsorb4,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( sup_sup_nat @ A3 @ B3 )
= B3 ) ) ).
% sup.absorb4
thf(fact_567_sup_Oabsorb3,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_nat @ B3 @ A3 )
=> ( ( sup_sup_nat @ A3 @ B3 )
= A3 ) ) ).
% sup.absorb3
thf(fact_568_less__supI2,axiom,
! [X: nat,B3: nat,A3: nat] :
( ( ord_less_nat @ X @ B3 )
=> ( ord_less_nat @ X @ ( sup_sup_nat @ A3 @ B3 ) ) ) ).
% less_supI2
thf(fact_569_less__supI1,axiom,
! [X: nat,A3: nat,B3: nat] :
( ( ord_less_nat @ X @ A3 )
=> ( ord_less_nat @ X @ ( sup_sup_nat @ A3 @ B3 ) ) ) ).
% less_supI1
thf(fact_570_distrib__imp2,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ! [X2: set_a,Y4: set_a,Z3: set_a] :
( ( sup_sup_set_a @ X2 @ ( inf_inf_set_a @ Y4 @ Z3 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X2 @ Y4 ) @ ( sup_sup_set_a @ X2 @ Z3 ) ) )
=> ( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_571_distrib__imp2,axiom,
! [X: set_list_a,Y: set_list_a,Z: set_list_a] :
( ! [X2: set_list_a,Y4: set_list_a,Z3: set_list_a] :
( ( sup_sup_set_list_a @ X2 @ ( inf_inf_set_list_a @ Y4 @ Z3 ) )
= ( inf_inf_set_list_a @ ( sup_sup_set_list_a @ X2 @ Y4 ) @ ( sup_sup_set_list_a @ X2 @ Z3 ) ) )
=> ( ( inf_inf_set_list_a @ X @ ( sup_sup_set_list_a @ Y @ Z ) )
= ( sup_sup_set_list_a @ ( inf_inf_set_list_a @ X @ Y ) @ ( inf_inf_set_list_a @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_572_distrib__imp1,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ! [X2: set_a,Y4: set_a,Z3: set_a] :
( ( inf_inf_set_a @ X2 @ ( sup_sup_set_a @ Y4 @ Z3 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X2 @ Y4 ) @ ( inf_inf_set_a @ X2 @ Z3 ) ) )
=> ( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_573_distrib__imp1,axiom,
! [X: set_list_a,Y: set_list_a,Z: set_list_a] :
( ! [X2: set_list_a,Y4: set_list_a,Z3: set_list_a] :
( ( inf_inf_set_list_a @ X2 @ ( sup_sup_set_list_a @ Y4 @ Z3 ) )
= ( sup_sup_set_list_a @ ( inf_inf_set_list_a @ X2 @ Y4 ) @ ( inf_inf_set_list_a @ X2 @ Z3 ) ) )
=> ( ( sup_sup_set_list_a @ X @ ( inf_inf_set_list_a @ Y @ Z ) )
= ( inf_inf_set_list_a @ ( sup_sup_set_list_a @ X @ Y ) @ ( sup_sup_set_list_a @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_574_subset__Un__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B2: set_a] :
( ( sup_sup_set_a @ A6 @ B2 )
= B2 ) ) ) ).
% subset_Un_eq
thf(fact_575_subset__Un__eq,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A6: set_list_a,B2: set_list_a] :
( ( sup_sup_set_list_a @ A6 @ B2 )
= B2 ) ) ) ).
% subset_Un_eq
thf(fact_576_subset__UnE,axiom,
! [C: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A @ B ) )
=> ~ ! [A7: set_a] :
( ( ord_less_eq_set_a @ A7 @ A )
=> ! [B6: set_a] :
( ( ord_less_eq_set_a @ B6 @ B )
=> ( C
!= ( sup_sup_set_a @ A7 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_577_subset__UnE,axiom,
! [C: set_list_a,A: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ C @ ( sup_sup_set_list_a @ A @ B ) )
=> ~ ! [A7: set_list_a] :
( ( ord_le8861187494160871172list_a @ A7 @ A )
=> ! [B6: set_list_a] :
( ( ord_le8861187494160871172list_a @ B6 @ B )
=> ( C
!= ( sup_sup_set_list_a @ A7 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_578_Un__absorb2,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( sup_sup_set_a @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_579_Un__absorb2,axiom,
! [B: set_list_a,A: set_list_a] :
( ( ord_le8861187494160871172list_a @ B @ A )
=> ( ( sup_sup_set_list_a @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_580_Un__absorb1,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( sup_sup_set_a @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_581_Un__absorb1,axiom,
! [A: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( sup_sup_set_list_a @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_582_Un__upper2,axiom,
! [B: set_a,A: set_a] : ( ord_less_eq_set_a @ B @ ( sup_sup_set_a @ A @ B ) ) ).
% Un_upper2
thf(fact_583_Un__upper2,axiom,
! [B: set_list_a,A: set_list_a] : ( ord_le8861187494160871172list_a @ B @ ( sup_sup_set_list_a @ A @ B ) ) ).
% Un_upper2
thf(fact_584_Un__upper1,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ A @ B ) ) ).
% Un_upper1
thf(fact_585_Un__upper1,axiom,
! [A: set_list_a,B: set_list_a] : ( ord_le8861187494160871172list_a @ A @ ( sup_sup_set_list_a @ A @ B ) ) ).
% Un_upper1
thf(fact_586_Un__least,axiom,
! [A: set_a,C: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ C ) ) ) ).
% Un_least
thf(fact_587_Un__least,axiom,
! [A: set_list_a,C: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ C )
=> ( ( ord_le8861187494160871172list_a @ B @ C )
=> ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ A @ B ) @ C ) ) ) ).
% Un_least
thf(fact_588_Un__mono,axiom,
! [A: set_a,C: set_a,B: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ B @ D )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ ( sup_sup_set_a @ C @ D ) ) ) ) ).
% Un_mono
thf(fact_589_Un__mono,axiom,
! [A: set_list_a,C: set_list_a,B: set_list_a,D: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ C )
=> ( ( ord_le8861187494160871172list_a @ B @ D )
=> ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ A @ B ) @ ( sup_sup_set_list_a @ C @ D ) ) ) ) ).
% Un_mono
thf(fact_590_UnE,axiom,
! [C2: set_list_a > a,A: set_set_list_a_a,B: set_set_list_a_a] :
( ( member_set_list_a_a @ C2 @ ( sup_su8833226608330050529st_a_a @ A @ B ) )
=> ( ~ ( member_set_list_a_a @ C2 @ A )
=> ( member_set_list_a_a @ C2 @ B ) ) ) ).
% UnE
thf(fact_591_UnE,axiom,
! [C2: nat > list_a,A: set_nat_list_a,B: set_nat_list_a] :
( ( member_nat_list_a @ C2 @ ( sup_su5649930751583389983list_a @ A @ B ) )
=> ( ~ ( member_nat_list_a @ C2 @ A )
=> ( member_nat_list_a @ C2 @ B ) ) ) ).
% UnE
thf(fact_592_UnE,axiom,
! [C2: nat > a,A: set_nat_a,B: set_nat_a] :
( ( member_nat_a @ C2 @ ( sup_sup_set_nat_a @ A @ B ) )
=> ( ~ ( member_nat_a @ C2 @ A )
=> ( member_nat_a @ C2 @ B ) ) ) ).
% UnE
thf(fact_593_UnE,axiom,
! [C2: a > a,A: set_a_a,B: set_a_a] :
( ( member_a_a @ C2 @ ( sup_sup_set_a_a @ A @ B ) )
=> ( ~ ( member_a_a @ C2 @ A )
=> ( member_a_a @ C2 @ B ) ) ) ).
% UnE
thf(fact_594_UnE,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ ( sup_sup_set_a @ A @ B ) )
=> ( ~ ( member_a @ C2 @ A )
=> ( member_a @ C2 @ B ) ) ) ).
% UnE
thf(fact_595_UnI1,axiom,
! [C2: set_list_a > a,A: set_set_list_a_a,B: set_set_list_a_a] :
( ( member_set_list_a_a @ C2 @ A )
=> ( member_set_list_a_a @ C2 @ ( sup_su8833226608330050529st_a_a @ A @ B ) ) ) ).
% UnI1
thf(fact_596_UnI1,axiom,
! [C2: nat > list_a,A: set_nat_list_a,B: set_nat_list_a] :
( ( member_nat_list_a @ C2 @ A )
=> ( member_nat_list_a @ C2 @ ( sup_su5649930751583389983list_a @ A @ B ) ) ) ).
% UnI1
thf(fact_597_UnI1,axiom,
! [C2: nat > a,A: set_nat_a,B: set_nat_a] :
( ( member_nat_a @ C2 @ A )
=> ( member_nat_a @ C2 @ ( sup_sup_set_nat_a @ A @ B ) ) ) ).
% UnI1
thf(fact_598_UnI1,axiom,
! [C2: a > a,A: set_a_a,B: set_a_a] :
( ( member_a_a @ C2 @ A )
=> ( member_a_a @ C2 @ ( sup_sup_set_a_a @ A @ B ) ) ) ).
% UnI1
thf(fact_599_UnI1,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ A )
=> ( member_a @ C2 @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnI1
thf(fact_600_UnI2,axiom,
! [C2: set_list_a > a,B: set_set_list_a_a,A: set_set_list_a_a] :
( ( member_set_list_a_a @ C2 @ B )
=> ( member_set_list_a_a @ C2 @ ( sup_su8833226608330050529st_a_a @ A @ B ) ) ) ).
% UnI2
thf(fact_601_UnI2,axiom,
! [C2: nat > list_a,B: set_nat_list_a,A: set_nat_list_a] :
( ( member_nat_list_a @ C2 @ B )
=> ( member_nat_list_a @ C2 @ ( sup_su5649930751583389983list_a @ A @ B ) ) ) ).
% UnI2
thf(fact_602_UnI2,axiom,
! [C2: nat > a,B: set_nat_a,A: set_nat_a] :
( ( member_nat_a @ C2 @ B )
=> ( member_nat_a @ C2 @ ( sup_sup_set_nat_a @ A @ B ) ) ) ).
% UnI2
thf(fact_603_UnI2,axiom,
! [C2: a > a,B: set_a_a,A: set_a_a] :
( ( member_a_a @ C2 @ B )
=> ( member_a_a @ C2 @ ( sup_sup_set_a_a @ A @ B ) ) ) ).
% UnI2
thf(fact_604_UnI2,axiom,
! [C2: a,B: set_a,A: set_a] :
( ( member_a @ C2 @ B )
=> ( member_a @ C2 @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnI2
thf(fact_605_Un__def,axiom,
( sup_su8833226608330050529st_a_a
= ( ^ [A6: set_set_list_a_a,B2: set_set_list_a_a] :
( collect_set_list_a_a
@ ^ [X4: set_list_a > a] :
( ( member_set_list_a_a @ X4 @ A6 )
| ( member_set_list_a_a @ X4 @ B2 ) ) ) ) ) ).
% Un_def
thf(fact_606_Un__def,axiom,
( sup_su5649930751583389983list_a
= ( ^ [A6: set_nat_list_a,B2: set_nat_list_a] :
( collect_nat_list_a
@ ^ [X4: nat > list_a] :
( ( member_nat_list_a @ X4 @ A6 )
| ( member_nat_list_a @ X4 @ B2 ) ) ) ) ) ).
% Un_def
thf(fact_607_Un__def,axiom,
( sup_sup_set_nat_a
= ( ^ [A6: set_nat_a,B2: set_nat_a] :
( collect_nat_a
@ ^ [X4: nat > a] :
( ( member_nat_a @ X4 @ A6 )
| ( member_nat_a @ X4 @ B2 ) ) ) ) ) ).
% Un_def
thf(fact_608_Un__def,axiom,
( sup_sup_set_a_a
= ( ^ [A6: set_a_a,B2: set_a_a] :
( collect_a_a
@ ^ [X4: a > a] :
( ( member_a_a @ X4 @ A6 )
| ( member_a_a @ X4 @ B2 ) ) ) ) ) ).
% Un_def
thf(fact_609_Un__def,axiom,
( sup_sup_set_nat
= ( ^ [A6: set_nat,B2: set_nat] :
( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A6 )
| ( member_nat @ X4 @ B2 ) ) ) ) ) ).
% Un_def
thf(fact_610_Un__def,axiom,
( sup_sup_set_a
= ( ^ [A6: set_a,B2: set_a] :
( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ A6 )
| ( member_a @ X4 @ B2 ) ) ) ) ) ).
% Un_def
thf(fact_611_Collect__disj__eq,axiom,
! [P2: nat > $o,Q2: nat > $o] :
( ( collect_nat
@ ^ [X4: nat] :
( ( P2 @ X4 )
| ( Q2 @ X4 ) ) )
= ( sup_sup_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q2 ) ) ) ).
% Collect_disj_eq
thf(fact_612_Collect__disj__eq,axiom,
! [P2: a > $o,Q2: a > $o] :
( ( collect_a
@ ^ [X4: a] :
( ( P2 @ X4 )
| ( Q2 @ X4 ) ) )
= ( sup_sup_set_a @ ( collect_a @ P2 ) @ ( collect_a @ Q2 ) ) ) ).
% Collect_disj_eq
thf(fact_613_sup_OcoboundedI2,axiom,
! [C2: set_a,B3: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ C2 @ B3 )
=> ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A3 @ B3 ) ) ) ).
% sup.coboundedI2
thf(fact_614_sup_OcoboundedI2,axiom,
! [C2: nat,B3: nat,A3: nat] :
( ( ord_less_eq_nat @ C2 @ B3 )
=> ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A3 @ B3 ) ) ) ).
% sup.coboundedI2
thf(fact_615_sup_OcoboundedI2,axiom,
! [C2: set_list_a,B3: set_list_a,A3: set_list_a] :
( ( ord_le8861187494160871172list_a @ C2 @ B3 )
=> ( ord_le8861187494160871172list_a @ C2 @ ( sup_sup_set_list_a @ A3 @ B3 ) ) ) ).
% sup.coboundedI2
thf(fact_616_sup_OcoboundedI1,axiom,
! [C2: set_a,A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ C2 @ A3 )
=> ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A3 @ B3 ) ) ) ).
% sup.coboundedI1
thf(fact_617_sup_OcoboundedI1,axiom,
! [C2: nat,A3: nat,B3: nat] :
( ( ord_less_eq_nat @ C2 @ A3 )
=> ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A3 @ B3 ) ) ) ).
% sup.coboundedI1
thf(fact_618_sup_OcoboundedI1,axiom,
! [C2: set_list_a,A3: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ C2 @ A3 )
=> ( ord_le8861187494160871172list_a @ C2 @ ( sup_sup_set_list_a @ A3 @ B3 ) ) ) ).
% sup.coboundedI1
thf(fact_619_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ( sup_sup_set_a @ A5 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_620_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
( ( sup_sup_nat @ A5 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_621_sup_Oabsorb__iff2,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A5: set_list_a,B5: set_list_a] :
( ( sup_sup_set_list_a @ A5 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_622_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [B5: set_a,A5: set_a] :
( ( sup_sup_set_a @ A5 @ B5 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_623_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A5: nat] :
( ( sup_sup_nat @ A5 @ B5 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_624_sup_Oabsorb__iff1,axiom,
( ord_le8861187494160871172list_a
= ( ^ [B5: set_list_a,A5: set_list_a] :
( ( sup_sup_set_list_a @ A5 @ B5 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_625_sup_Ocobounded2,axiom,
! [B3: set_a,A3: set_a] : ( ord_less_eq_set_a @ B3 @ ( sup_sup_set_a @ A3 @ B3 ) ) ).
% sup.cobounded2
thf(fact_626_sup_Ocobounded2,axiom,
! [B3: nat,A3: nat] : ( ord_less_eq_nat @ B3 @ ( sup_sup_nat @ A3 @ B3 ) ) ).
% sup.cobounded2
thf(fact_627_sup_Ocobounded2,axiom,
! [B3: set_list_a,A3: set_list_a] : ( ord_le8861187494160871172list_a @ B3 @ ( sup_sup_set_list_a @ A3 @ B3 ) ) ).
% sup.cobounded2
thf(fact_628_sup_Ocobounded1,axiom,
! [A3: set_a,B3: set_a] : ( ord_less_eq_set_a @ A3 @ ( sup_sup_set_a @ A3 @ B3 ) ) ).
% sup.cobounded1
thf(fact_629_sup_Ocobounded1,axiom,
! [A3: nat,B3: nat] : ( ord_less_eq_nat @ A3 @ ( sup_sup_nat @ A3 @ B3 ) ) ).
% sup.cobounded1
thf(fact_630_sup_Ocobounded1,axiom,
! [A3: set_list_a,B3: set_list_a] : ( ord_le8861187494160871172list_a @ A3 @ ( sup_sup_set_list_a @ A3 @ B3 ) ) ).
% sup.cobounded1
thf(fact_631_sup_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [B5: set_a,A5: set_a] :
( A5
= ( sup_sup_set_a @ A5 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_632_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A5: nat] :
( A5
= ( sup_sup_nat @ A5 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_633_sup_Oorder__iff,axiom,
( ord_le8861187494160871172list_a
= ( ^ [B5: set_list_a,A5: set_list_a] :
( A5
= ( sup_sup_set_list_a @ A5 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_634_sup_OboundedI,axiom,
! [B3: set_a,A3: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ B3 @ A3 )
=> ( ( ord_less_eq_set_a @ C2 @ A3 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ B3 @ C2 ) @ A3 ) ) ) ).
% sup.boundedI
thf(fact_635_sup_OboundedI,axiom,
! [B3: nat,A3: nat,C2: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
=> ( ( ord_less_eq_nat @ C2 @ A3 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B3 @ C2 ) @ A3 ) ) ) ).
% sup.boundedI
thf(fact_636_sup_OboundedI,axiom,
! [B3: set_list_a,A3: set_list_a,C2: set_list_a] :
( ( ord_le8861187494160871172list_a @ B3 @ A3 )
=> ( ( ord_le8861187494160871172list_a @ C2 @ A3 )
=> ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ B3 @ C2 ) @ A3 ) ) ) ).
% sup.boundedI
thf(fact_637_sup_OboundedE,axiom,
! [B3: set_a,C2: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B3 @ C2 ) @ A3 )
=> ~ ( ( ord_less_eq_set_a @ B3 @ A3 )
=> ~ ( ord_less_eq_set_a @ C2 @ A3 ) ) ) ).
% sup.boundedE
thf(fact_638_sup_OboundedE,axiom,
! [B3: nat,C2: nat,A3: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B3 @ C2 ) @ A3 )
=> ~ ( ( ord_less_eq_nat @ B3 @ A3 )
=> ~ ( ord_less_eq_nat @ C2 @ A3 ) ) ) ).
% sup.boundedE
thf(fact_639_sup_OboundedE,axiom,
! [B3: set_list_a,C2: set_list_a,A3: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ B3 @ C2 ) @ A3 )
=> ~ ( ( ord_le8861187494160871172list_a @ B3 @ A3 )
=> ~ ( ord_le8861187494160871172list_a @ C2 @ A3 ) ) ) ).
% sup.boundedE
thf(fact_640_sup__absorb2,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( sup_sup_set_a @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_641_sup__absorb2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( sup_sup_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_642_sup__absorb2,axiom,
! [X: set_list_a,Y: set_list_a] :
( ( ord_le8861187494160871172list_a @ X @ Y )
=> ( ( sup_sup_set_list_a @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_643_sup__absorb1,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( sup_sup_set_a @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_644_sup__absorb1,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( sup_sup_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_645_sup__absorb1,axiom,
! [Y: set_list_a,X: set_list_a] :
( ( ord_le8861187494160871172list_a @ Y @ X )
=> ( ( sup_sup_set_list_a @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_646_sup_Oabsorb2,axiom,
! [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( sup_sup_set_a @ A3 @ B3 )
= B3 ) ) ).
% sup.absorb2
thf(fact_647_sup_Oabsorb2,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( sup_sup_nat @ A3 @ B3 )
= B3 ) ) ).
% sup.absorb2
thf(fact_648_sup_Oabsorb2,axiom,
! [A3: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ A3 @ B3 )
=> ( ( sup_sup_set_list_a @ A3 @ B3 )
= B3 ) ) ).
% sup.absorb2
thf(fact_649_sup_Oabsorb1,axiom,
! [B3: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A3 )
=> ( ( sup_sup_set_a @ A3 @ B3 )
= A3 ) ) ).
% sup.absorb1
thf(fact_650_sup_Oabsorb1,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
=> ( ( sup_sup_nat @ A3 @ B3 )
= A3 ) ) ).
% sup.absorb1
thf(fact_651_sup_Oabsorb1,axiom,
! [B3: set_list_a,A3: set_list_a] :
( ( ord_le8861187494160871172list_a @ B3 @ A3 )
=> ( ( sup_sup_set_list_a @ A3 @ B3 )
= A3 ) ) ).
% sup.absorb1
thf(fact_652_sup__unique,axiom,
! [F: set_a > set_a > set_a,X: set_a,Y: set_a] :
( ! [X2: set_a,Y4: set_a] : ( ord_less_eq_set_a @ X2 @ ( F @ X2 @ Y4 ) )
=> ( ! [X2: set_a,Y4: set_a] : ( ord_less_eq_set_a @ Y4 @ ( F @ X2 @ Y4 ) )
=> ( ! [X2: set_a,Y4: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ Y4 @ X2 )
=> ( ( ord_less_eq_set_a @ Z3 @ X2 )
=> ( ord_less_eq_set_a @ ( F @ Y4 @ Z3 ) @ X2 ) ) )
=> ( ( sup_sup_set_a @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_653_sup__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X2: nat,Y4: nat] : ( ord_less_eq_nat @ X2 @ ( F @ X2 @ Y4 ) )
=> ( ! [X2: nat,Y4: nat] : ( ord_less_eq_nat @ Y4 @ ( F @ X2 @ Y4 ) )
=> ( ! [X2: nat,Y4: nat,Z3: nat] :
( ( ord_less_eq_nat @ Y4 @ X2 )
=> ( ( ord_less_eq_nat @ Z3 @ X2 )
=> ( ord_less_eq_nat @ ( F @ Y4 @ Z3 ) @ X2 ) ) )
=> ( ( sup_sup_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_654_sup__unique,axiom,
! [F: set_list_a > set_list_a > set_list_a,X: set_list_a,Y: set_list_a] :
( ! [X2: set_list_a,Y4: set_list_a] : ( ord_le8861187494160871172list_a @ X2 @ ( F @ X2 @ Y4 ) )
=> ( ! [X2: set_list_a,Y4: set_list_a] : ( ord_le8861187494160871172list_a @ Y4 @ ( F @ X2 @ Y4 ) )
=> ( ! [X2: set_list_a,Y4: set_list_a,Z3: set_list_a] :
( ( ord_le8861187494160871172list_a @ Y4 @ X2 )
=> ( ( ord_le8861187494160871172list_a @ Z3 @ X2 )
=> ( ord_le8861187494160871172list_a @ ( F @ Y4 @ Z3 ) @ X2 ) ) )
=> ( ( sup_sup_set_list_a @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_655_sup_OorderI,axiom,
! [A3: set_a,B3: set_a] :
( ( A3
= ( sup_sup_set_a @ A3 @ B3 ) )
=> ( ord_less_eq_set_a @ B3 @ A3 ) ) ).
% sup.orderI
thf(fact_656_sup_OorderI,axiom,
! [A3: nat,B3: nat] :
( ( A3
= ( sup_sup_nat @ A3 @ B3 ) )
=> ( ord_less_eq_nat @ B3 @ A3 ) ) ).
% sup.orderI
thf(fact_657_sup_OorderI,axiom,
! [A3: set_list_a,B3: set_list_a] :
( ( A3
= ( sup_sup_set_list_a @ A3 @ B3 ) )
=> ( ord_le8861187494160871172list_a @ B3 @ A3 ) ) ).
% sup.orderI
thf(fact_658_sup_OorderE,axiom,
! [B3: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A3 )
=> ( A3
= ( sup_sup_set_a @ A3 @ B3 ) ) ) ).
% sup.orderE
thf(fact_659_sup_OorderE,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
=> ( A3
= ( sup_sup_nat @ A3 @ B3 ) ) ) ).
% sup.orderE
thf(fact_660_sup_OorderE,axiom,
! [B3: set_list_a,A3: set_list_a] :
( ( ord_le8861187494160871172list_a @ B3 @ A3 )
=> ( A3
= ( sup_sup_set_list_a @ A3 @ B3 ) ) ) ).
% sup.orderE
thf(fact_661_le__iff__sup,axiom,
( ord_less_eq_set_a
= ( ^ [X4: set_a,Y5: set_a] :
( ( sup_sup_set_a @ X4 @ Y5 )
= Y5 ) ) ) ).
% le_iff_sup
thf(fact_662_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y5: nat] :
( ( sup_sup_nat @ X4 @ Y5 )
= Y5 ) ) ) ).
% le_iff_sup
thf(fact_663_le__iff__sup,axiom,
( ord_le8861187494160871172list_a
= ( ^ [X4: set_list_a,Y5: set_list_a] :
( ( sup_sup_set_list_a @ X4 @ Y5 )
= Y5 ) ) ) ).
% le_iff_sup
thf(fact_664_sup__least,axiom,
! [Y: set_a,X: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( ord_less_eq_set_a @ Z @ X )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_665_sup__least,axiom,
! [Y: nat,X: nat,Z: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ Z @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_666_sup__least,axiom,
! [Y: set_list_a,X: set_list_a,Z: set_list_a] :
( ( ord_le8861187494160871172list_a @ Y @ X )
=> ( ( ord_le8861187494160871172list_a @ Z @ X )
=> ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_667_sup__mono,axiom,
! [A3: set_a,C2: set_a,B3: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A3 @ C2 )
=> ( ( ord_less_eq_set_a @ B3 @ D2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A3 @ B3 ) @ ( sup_sup_set_a @ C2 @ D2 ) ) ) ) ).
% sup_mono
thf(fact_668_sup__mono,axiom,
! [A3: nat,C2: nat,B3: nat,D2: nat] :
( ( ord_less_eq_nat @ A3 @ C2 )
=> ( ( ord_less_eq_nat @ B3 @ D2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A3 @ B3 ) @ ( sup_sup_nat @ C2 @ D2 ) ) ) ) ).
% sup_mono
thf(fact_669_sup__mono,axiom,
! [A3: set_list_a,C2: set_list_a,B3: set_list_a,D2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A3 @ C2 )
=> ( ( ord_le8861187494160871172list_a @ B3 @ D2 )
=> ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ A3 @ B3 ) @ ( sup_sup_set_list_a @ C2 @ D2 ) ) ) ) ).
% sup_mono
thf(fact_670_sup_Omono,axiom,
! [C2: set_a,A3: set_a,D2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ C2 @ A3 )
=> ( ( ord_less_eq_set_a @ D2 @ B3 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ C2 @ D2 ) @ ( sup_sup_set_a @ A3 @ B3 ) ) ) ) ).
% sup.mono
thf(fact_671_sup_Omono,axiom,
! [C2: nat,A3: nat,D2: nat,B3: nat] :
( ( ord_less_eq_nat @ C2 @ A3 )
=> ( ( ord_less_eq_nat @ D2 @ B3 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C2 @ D2 ) @ ( sup_sup_nat @ A3 @ B3 ) ) ) ) ).
% sup.mono
thf(fact_672_sup_Omono,axiom,
! [C2: set_list_a,A3: set_list_a,D2: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ C2 @ A3 )
=> ( ( ord_le8861187494160871172list_a @ D2 @ B3 )
=> ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ C2 @ D2 ) @ ( sup_sup_set_list_a @ A3 @ B3 ) ) ) ) ).
% sup.mono
thf(fact_673_le__supI2,axiom,
! [X: set_a,B3: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ X @ B3 )
=> ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A3 @ B3 ) ) ) ).
% le_supI2
thf(fact_674_le__supI2,axiom,
! [X: nat,B3: nat,A3: nat] :
( ( ord_less_eq_nat @ X @ B3 )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A3 @ B3 ) ) ) ).
% le_supI2
thf(fact_675_le__supI2,axiom,
! [X: set_list_a,B3: set_list_a,A3: set_list_a] :
( ( ord_le8861187494160871172list_a @ X @ B3 )
=> ( ord_le8861187494160871172list_a @ X @ ( sup_sup_set_list_a @ A3 @ B3 ) ) ) ).
% le_supI2
thf(fact_676_le__supI1,axiom,
! [X: set_a,A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ X @ A3 )
=> ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A3 @ B3 ) ) ) ).
% le_supI1
thf(fact_677_le__supI1,axiom,
! [X: nat,A3: nat,B3: nat] :
( ( ord_less_eq_nat @ X @ A3 )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A3 @ B3 ) ) ) ).
% le_supI1
thf(fact_678_le__supI1,axiom,
! [X: set_list_a,A3: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ X @ A3 )
=> ( ord_le8861187494160871172list_a @ X @ ( sup_sup_set_list_a @ A3 @ B3 ) ) ) ).
% le_supI1
thf(fact_679_sup__ge2,axiom,
! [Y: set_a,X: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X @ Y ) ) ).
% sup_ge2
thf(fact_680_sup__ge2,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_681_sup__ge2,axiom,
! [Y: set_list_a,X: set_list_a] : ( ord_le8861187494160871172list_a @ Y @ ( sup_sup_set_list_a @ X @ Y ) ) ).
% sup_ge2
thf(fact_682_sup__ge1,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ X @ Y ) ) ).
% sup_ge1
thf(fact_683_sup__ge1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_684_sup__ge1,axiom,
! [X: set_list_a,Y: set_list_a] : ( ord_le8861187494160871172list_a @ X @ ( sup_sup_set_list_a @ X @ Y ) ) ).
% sup_ge1
thf(fact_685_le__supI,axiom,
! [A3: set_a,X: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ X )
=> ( ( ord_less_eq_set_a @ B3 @ X )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A3 @ B3 ) @ X ) ) ) ).
% le_supI
thf(fact_686_le__supI,axiom,
! [A3: nat,X: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ X )
=> ( ( ord_less_eq_nat @ B3 @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A3 @ B3 ) @ X ) ) ) ).
% le_supI
thf(fact_687_le__supI,axiom,
! [A3: set_list_a,X: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ A3 @ X )
=> ( ( ord_le8861187494160871172list_a @ B3 @ X )
=> ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ A3 @ B3 ) @ X ) ) ) ).
% le_supI
thf(fact_688_le__supE,axiom,
! [A3: set_a,B3: set_a,X: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A3 @ B3 ) @ X )
=> ~ ( ( ord_less_eq_set_a @ A3 @ X )
=> ~ ( ord_less_eq_set_a @ B3 @ X ) ) ) ).
% le_supE
thf(fact_689_le__supE,axiom,
! [A3: nat,B3: nat,X: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A3 @ B3 ) @ X )
=> ~ ( ( ord_less_eq_nat @ A3 @ X )
=> ~ ( ord_less_eq_nat @ B3 @ X ) ) ) ).
% le_supE
thf(fact_690_le__supE,axiom,
! [A3: set_list_a,B3: set_list_a,X: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ A3 @ B3 ) @ X )
=> ~ ( ( ord_le8861187494160871172list_a @ A3 @ X )
=> ~ ( ord_le8861187494160871172list_a @ B3 @ X ) ) ) ).
% le_supE
thf(fact_691_inf__sup__ord_I3_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_692_inf__sup__ord_I3_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_693_inf__sup__ord_I3_J,axiom,
! [X: set_list_a,Y: set_list_a] : ( ord_le8861187494160871172list_a @ X @ ( sup_sup_set_list_a @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_694_inf__sup__ord_I4_J,axiom,
! [Y: set_a,X: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_695_inf__sup__ord_I4_J,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_696_inf__sup__ord_I4_J,axiom,
! [Y: set_list_a,X: set_list_a] : ( ord_le8861187494160871172list_a @ Y @ ( sup_sup_set_list_a @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_697_Nat_Oex__has__greatest__nat,axiom,
! [P2: nat > $o,K2: nat,B3: nat] :
( ( P2 @ K2 )
=> ( ! [Y4: nat] :
( ( P2 @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B3 ) )
=> ? [X2: nat] :
( ( P2 @ X2 )
& ! [Y6: nat] :
( ( P2 @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_698_nat__le__linear,axiom,
! [M2: nat,N4: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
| ( ord_less_eq_nat @ N4 @ M2 ) ) ).
% nat_le_linear
thf(fact_699_diff__le__mono2,axiom,
! [M2: nat,N4: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N4 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_700_le__diff__iff_H,axiom,
! [A3: nat,C2: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ C2 )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A3 ) @ ( minus_minus_nat @ C2 @ B3 ) )
= ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% le_diff_iff'
thf(fact_701_diff__le__self,axiom,
! [M2: nat,N4: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N4 ) @ M2 ) ).
% diff_le_self
thf(fact_702_diff__le__mono,axiom,
! [M2: nat,N4: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N4 @ L ) ) ) ).
% diff_le_mono
thf(fact_703_Nat_Odiff__diff__eq,axiom,
! [K2: nat,M2: nat,N4: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N4 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N4 @ K2 ) )
= ( minus_minus_nat @ M2 @ N4 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_704_le__diff__iff,axiom,
! [K2: nat,M2: nat,N4: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N4 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N4 @ K2 ) )
= ( ord_less_eq_nat @ M2 @ N4 ) ) ) ) ).
% le_diff_iff
thf(fact_705_eq__diff__iff,axiom,
! [K2: nat,M2: nat,N4: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N4 )
=> ( ( ( minus_minus_nat @ M2 @ K2 )
= ( minus_minus_nat @ N4 @ K2 ) )
= ( M2 = N4 ) ) ) ) ).
% eq_diff_iff
thf(fact_706_le__antisym,axiom,
! [M2: nat,N4: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
=> ( ( ord_less_eq_nat @ N4 @ M2 )
=> ( M2 = N4 ) ) ) ).
% le_antisym
thf(fact_707_eq__imp__le,axiom,
! [M2: nat,N4: nat] :
( ( M2 = N4 )
=> ( ord_less_eq_nat @ M2 @ N4 ) ) ).
% eq_imp_le
thf(fact_708_le__trans,axiom,
! [I2: nat,J3: nat,K2: nat] :
( ( ord_less_eq_nat @ I2 @ J3 )
=> ( ( ord_less_eq_nat @ J3 @ K2 )
=> ( ord_less_eq_nat @ I2 @ K2 ) ) ) ).
% le_trans
thf(fact_709_le__refl,axiom,
! [N4: nat] : ( ord_less_eq_nat @ N4 @ N4 ) ).
% le_refl
thf(fact_710_psubsetE,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ~ ( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ B @ A ) ) ) ).
% psubsetE
thf(fact_711_psubsetE,axiom,
! [A: set_list_a,B: set_list_a] :
( ( ord_less_set_list_a @ A @ B )
=> ~ ( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ord_le8861187494160871172list_a @ B @ A ) ) ) ).
% psubsetE
thf(fact_712_psubset__eq,axiom,
( ord_less_set_a
= ( ^ [A6: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A6 @ B2 )
& ( A6 != B2 ) ) ) ) ).
% psubset_eq
thf(fact_713_psubset__eq,axiom,
( ord_less_set_list_a
= ( ^ [A6: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A6 @ B2 )
& ( A6 != B2 ) ) ) ) ).
% psubset_eq
thf(fact_714_psubset__imp__subset,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% psubset_imp_subset
thf(fact_715_psubset__imp__subset,axiom,
! [A: set_list_a,B: set_list_a] :
( ( ord_less_set_list_a @ A @ B )
=> ( ord_le8861187494160871172list_a @ A @ B ) ) ).
% psubset_imp_subset
thf(fact_716_psubset__subset__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% psubset_subset_trans
thf(fact_717_psubset__subset__trans,axiom,
! [A: set_list_a,B: set_list_a,C: set_list_a] :
( ( ord_less_set_list_a @ A @ B )
=> ( ( ord_le8861187494160871172list_a @ B @ C )
=> ( ord_less_set_list_a @ A @ C ) ) ) ).
% psubset_subset_trans
thf(fact_718_subset__not__subset__eq,axiom,
( ord_less_set_a
= ( ^ [A6: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A6 @ B2 )
& ~ ( ord_less_eq_set_a @ B2 @ A6 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_719_subset__not__subset__eq,axiom,
( ord_less_set_list_a
= ( ^ [A6: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A6 @ B2 )
& ~ ( ord_le8861187494160871172list_a @ B2 @ A6 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_720_subset__psubset__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_set_a @ B @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% subset_psubset_trans
thf(fact_721_subset__psubset__trans,axiom,
! [A: set_list_a,B: set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( ord_less_set_list_a @ B @ C )
=> ( ord_less_set_list_a @ A @ C ) ) ) ).
% subset_psubset_trans
thf(fact_722_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B2: set_a] :
( ( ord_less_set_a @ A6 @ B2 )
| ( A6 = B2 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_723_subset__iff__psubset__eq,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A6: set_list_a,B2: set_list_a] :
( ( ord_less_set_list_a @ A6 @ B2 )
| ( A6 = B2 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_724_diffs0__imp__equal,axiom,
! [M2: nat,N4: nat] :
( ( ( minus_minus_nat @ M2 @ N4 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N4 @ M2 )
= zero_zero_nat )
=> ( M2 = N4 ) ) ) ).
% diffs0_imp_equal
thf(fact_725_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_726_Un__Int__crazy,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_inf_set_a @ B @ C ) ) @ ( inf_inf_set_a @ C @ A ) )
= ( inf_inf_set_a @ ( inf_inf_set_a @ ( sup_sup_set_a @ A @ B ) @ ( sup_sup_set_a @ B @ C ) ) @ ( sup_sup_set_a @ C @ A ) ) ) ).
% Un_Int_crazy
thf(fact_727_Un__Int__crazy,axiom,
! [A: set_list_a,B: set_list_a,C: set_list_a] :
( ( sup_sup_set_list_a @ ( sup_sup_set_list_a @ ( inf_inf_set_list_a @ A @ B ) @ ( inf_inf_set_list_a @ B @ C ) ) @ ( inf_inf_set_list_a @ C @ A ) )
= ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ ( sup_sup_set_list_a @ A @ B ) @ ( sup_sup_set_list_a @ B @ C ) ) @ ( sup_sup_set_list_a @ C @ A ) ) ) ).
% Un_Int_crazy
thf(fact_728_Int__Un__distrib,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( inf_inf_set_a @ A @ ( sup_sup_set_a @ B @ C ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_inf_set_a @ A @ C ) ) ) ).
% Int_Un_distrib
thf(fact_729_Int__Un__distrib,axiom,
! [A: set_list_a,B: set_list_a,C: set_list_a] :
( ( inf_inf_set_list_a @ A @ ( sup_sup_set_list_a @ B @ C ) )
= ( sup_sup_set_list_a @ ( inf_inf_set_list_a @ A @ B ) @ ( inf_inf_set_list_a @ A @ C ) ) ) ).
% Int_Un_distrib
thf(fact_730_Un__Int__distrib,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ A @ ( inf_inf_set_a @ B @ C ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ A @ B ) @ ( sup_sup_set_a @ A @ C ) ) ) ).
% Un_Int_distrib
thf(fact_731_Un__Int__distrib,axiom,
! [A: set_list_a,B: set_list_a,C: set_list_a] :
( ( sup_sup_set_list_a @ A @ ( inf_inf_set_list_a @ B @ C ) )
= ( inf_inf_set_list_a @ ( sup_sup_set_list_a @ A @ B ) @ ( sup_sup_set_list_a @ A @ C ) ) ) ).
% Un_Int_distrib
thf(fact_732_Int__Un__distrib2,axiom,
! [B: set_a,C: set_a,A: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ B @ C ) @ A )
= ( sup_sup_set_a @ ( inf_inf_set_a @ B @ A ) @ ( inf_inf_set_a @ C @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_733_Int__Un__distrib2,axiom,
! [B: set_list_a,C: set_list_a,A: set_list_a] :
( ( inf_inf_set_list_a @ ( sup_sup_set_list_a @ B @ C ) @ A )
= ( sup_sup_set_list_a @ ( inf_inf_set_list_a @ B @ A ) @ ( inf_inf_set_list_a @ C @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_734_Un__Int__distrib2,axiom,
! [B: set_a,C: set_a,A: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ B @ C ) @ A )
= ( inf_inf_set_a @ ( sup_sup_set_a @ B @ A ) @ ( sup_sup_set_a @ C @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_735_Un__Int__distrib2,axiom,
! [B: set_list_a,C: set_list_a,A: set_list_a] :
( ( sup_sup_set_list_a @ ( inf_inf_set_list_a @ B @ C ) @ A )
= ( inf_inf_set_list_a @ ( sup_sup_set_list_a @ B @ A ) @ ( sup_sup_set_list_a @ C @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_736_Un__Diff,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( minus_minus_set_a @ ( sup_sup_set_a @ A @ B ) @ C )
= ( sup_sup_set_a @ ( minus_minus_set_a @ A @ C ) @ ( minus_minus_set_a @ B @ C ) ) ) ).
% Un_Diff
thf(fact_737_Un__Diff,axiom,
! [A: set_list_a,B: set_list_a,C: set_list_a] :
( ( minus_646659088055828811list_a @ ( sup_sup_set_list_a @ A @ B ) @ C )
= ( sup_sup_set_list_a @ ( minus_646659088055828811list_a @ A @ C ) @ ( minus_646659088055828811list_a @ B @ C ) ) ) ).
% Un_Diff
thf(fact_738_less__imp__diff__less,axiom,
! [J3: nat,K2: nat,N4: nat] :
( ( ord_less_nat @ J3 @ K2 )
=> ( ord_less_nat @ ( minus_minus_nat @ J3 @ N4 ) @ K2 ) ) ).
% less_imp_diff_less
thf(fact_739_diff__less__mono2,axiom,
! [M2: nat,N4: nat,L: nat] :
( ( ord_less_nat @ M2 @ N4 )
=> ( ( ord_less_nat @ M2 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N4 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_740_diff__less__mono,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ C2 @ A3 )
=> ( ord_less_nat @ ( minus_minus_nat @ A3 @ C2 ) @ ( minus_minus_nat @ B3 @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_741_less__diff__iff,axiom,
! [K2: nat,M2: nat,N4: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N4 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N4 @ K2 ) )
= ( ord_less_nat @ M2 @ N4 ) ) ) ) ).
% less_diff_iff
thf(fact_742_psubset__imp__ex__mem,axiom,
! [A: set_set_list_a_a,B: set_set_list_a_a] :
( ( ord_le3077272015153833225st_a_a @ A @ B )
=> ? [B4: set_list_a > a] : ( member_set_list_a_a @ B4 @ ( minus_5613498140476352782st_a_a @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_743_psubset__imp__ex__mem,axiom,
! [A: set_nat_list_a,B: set_nat_list_a] :
( ( ord_le975800131162592759list_a @ A @ B )
=> ? [B4: nat > list_a] : ( member_nat_list_a @ B4 @ ( minus_4169782841487898290list_a @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_744_psubset__imp__ex__mem,axiom,
! [A: set_nat_a,B: set_nat_a] :
( ( ord_less_set_nat_a @ A @ B )
=> ? [B4: nat > a] : ( member_nat_a @ B4 @ ( minus_490503922182417452_nat_a @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_745_psubset__imp__ex__mem,axiom,
! [A: set_a_a,B: set_a_a] :
( ( ord_less_set_a_a @ A @ B )
=> ? [B4: a > a] : ( member_a_a @ B4 @ ( minus_minus_set_a_a @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_746_psubset__imp__ex__mem,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ? [B4: a] : ( member_a @ B4 @ ( minus_minus_set_a @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_747_psubset__imp__ex__mem,axiom,
! [A: set_list_a,B: set_list_a] :
( ( ord_less_set_list_a @ A @ B )
=> ? [B4: list_a] : ( member_list_a @ B4 @ ( minus_646659088055828811list_a @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_748_inf__set__def,axiom,
( inf_in6568206481208318535st_a_a
= ( ^ [A6: set_set_list_a_a,B2: set_set_list_a_a] :
( collect_set_list_a_a
@ ( inf_in1325899312859132158_a_a_o
@ ^ [X4: set_list_a > a] : ( member_set_list_a_a @ X4 @ A6 )
@ ^ [X4: set_list_a > a] : ( member_set_list_a_a @ X4 @ B2 ) ) ) ) ) ).
% inf_set_def
thf(fact_749_inf__set__def,axiom,
( inf_in6652419485960844601list_a
= ( ^ [A6: set_nat_list_a,B2: set_nat_list_a] :
( collect_nat_list_a
@ ( inf_inf_nat_list_a_o
@ ^ [X4: nat > list_a] : ( member_nat_list_a @ X4 @ A6 )
@ ^ [X4: nat > list_a] : ( member_nat_list_a @ X4 @ B2 ) ) ) ) ) ).
% inf_set_def
thf(fact_750_inf__set__def,axiom,
( inf_inf_set_nat_a
= ( ^ [A6: set_nat_a,B2: set_nat_a] :
( collect_nat_a
@ ( inf_inf_nat_a_o
@ ^ [X4: nat > a] : ( member_nat_a @ X4 @ A6 )
@ ^ [X4: nat > a] : ( member_nat_a @ X4 @ B2 ) ) ) ) ) ).
% inf_set_def
thf(fact_751_inf__set__def,axiom,
( inf_inf_set_a_a
= ( ^ [A6: set_a_a,B2: set_a_a] :
( collect_a_a
@ ( inf_inf_a_a_o
@ ^ [X4: a > a] : ( member_a_a @ X4 @ A6 )
@ ^ [X4: a > a] : ( member_a_a @ X4 @ B2 ) ) ) ) ) ).
% inf_set_def
thf(fact_752_inf__set__def,axiom,
( inf_inf_set_nat
= ( ^ [A6: set_nat,B2: set_nat] :
( collect_nat
@ ( inf_inf_nat_o
@ ^ [X4: nat] : ( member_nat @ X4 @ A6 )
@ ^ [X4: nat] : ( member_nat @ X4 @ B2 ) ) ) ) ) ).
% inf_set_def
thf(fact_753_inf__set__def,axiom,
( inf_inf_set_a
= ( ^ [A6: set_a,B2: set_a] :
( collect_a
@ ( inf_inf_a_o
@ ^ [X4: a] : ( member_a @ X4 @ A6 )
@ ^ [X4: a] : ( member_a @ X4 @ B2 ) ) ) ) ) ).
% inf_set_def
thf(fact_754_inf__set__def,axiom,
( inf_inf_set_list_a
= ( ^ [A6: set_list_a,B2: set_list_a] :
( collect_list_a
@ ( inf_inf_list_a_o
@ ^ [X4: list_a] : ( member_list_a @ X4 @ A6 )
@ ^ [X4: list_a] : ( member_list_a @ X4 @ B2 ) ) ) ) ) ).
% inf_set_def
thf(fact_755_minus__set__def,axiom,
( minus_5613498140476352782st_a_a
= ( ^ [A6: set_set_list_a_a,B2: set_set_list_a_a] :
( collect_set_list_a_a
@ ( minus_2237239494651866359_a_a_o
@ ^ [X4: set_list_a > a] : ( member_set_list_a_a @ X4 @ A6 )
@ ^ [X4: set_list_a > a] : ( member_set_list_a_a @ X4 @ B2 ) ) ) ) ) ).
% minus_set_def
thf(fact_756_minus__set__def,axiom,
( minus_4169782841487898290list_a
= ( ^ [A6: set_nat_list_a,B2: set_nat_list_a] :
( collect_nat_list_a
@ ( minus_2157385651697719467st_a_o
@ ^ [X4: nat > list_a] : ( member_nat_list_a @ X4 @ A6 )
@ ^ [X4: nat > list_a] : ( member_nat_list_a @ X4 @ B2 ) ) ) ) ) ).
% minus_set_def
thf(fact_757_minus__set__def,axiom,
( minus_490503922182417452_nat_a
= ( ^ [A6: set_nat_a,B2: set_nat_a] :
( collect_nat_a
@ ( minus_minus_nat_a_o
@ ^ [X4: nat > a] : ( member_nat_a @ X4 @ A6 )
@ ^ [X4: nat > a] : ( member_nat_a @ X4 @ B2 ) ) ) ) ) ).
% minus_set_def
thf(fact_758_minus__set__def,axiom,
( minus_minus_set_a_a
= ( ^ [A6: set_a_a,B2: set_a_a] :
( collect_a_a
@ ( minus_minus_a_a_o
@ ^ [X4: a > a] : ( member_a_a @ X4 @ A6 )
@ ^ [X4: a > a] : ( member_a_a @ X4 @ B2 ) ) ) ) ) ).
% minus_set_def
thf(fact_759_minus__set__def,axiom,
( minus_minus_set_nat
= ( ^ [A6: set_nat,B2: set_nat] :
( collect_nat
@ ( minus_minus_nat_o
@ ^ [X4: nat] : ( member_nat @ X4 @ A6 )
@ ^ [X4: nat] : ( member_nat @ X4 @ B2 ) ) ) ) ) ).
% minus_set_def
thf(fact_760_minus__set__def,axiom,
( minus_minus_set_a
= ( ^ [A6: set_a,B2: set_a] :
( collect_a
@ ( minus_minus_a_o
@ ^ [X4: a] : ( member_a @ X4 @ A6 )
@ ^ [X4: a] : ( member_a @ X4 @ B2 ) ) ) ) ) ).
% minus_set_def
thf(fact_761_minus__set__def,axiom,
( minus_646659088055828811list_a
= ( ^ [A6: set_list_a,B2: set_list_a] :
( collect_list_a
@ ( minus_minus_list_a_o
@ ^ [X4: list_a] : ( member_list_a @ X4 @ A6 )
@ ^ [X4: list_a] : ( member_list_a @ X4 @ B2 ) ) ) ) ) ).
% minus_set_def
thf(fact_762_finite__UnI,axiom,
! [F2: set_a,G2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ( finite_finite_a @ G2 )
=> ( finite_finite_a @ ( sup_sup_set_a @ F2 @ G2 ) ) ) ) ).
% finite_UnI
thf(fact_763_finite__UnI,axiom,
! [F2: set_nat,G2: set_nat] :
( ( finite_finite_nat @ F2 )
=> ( ( finite_finite_nat @ G2 )
=> ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G2 ) ) ) ) ).
% finite_UnI
thf(fact_764_finite__UnI,axiom,
! [F2: set_list_a,G2: set_list_a] :
( ( finite_finite_list_a @ F2 )
=> ( ( finite_finite_list_a @ G2 )
=> ( finite_finite_list_a @ ( sup_sup_set_list_a @ F2 @ G2 ) ) ) ) ).
% finite_UnI
thf(fact_765_Un__infinite,axiom,
! [S: set_a,T: set_a] :
( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( sup_sup_set_a @ S @ T ) ) ) ).
% Un_infinite
thf(fact_766_Un__infinite,axiom,
! [S: set_nat,T: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T ) ) ) ).
% Un_infinite
thf(fact_767_Un__infinite,axiom,
! [S: set_list_a,T: set_list_a] :
( ~ ( finite_finite_list_a @ S )
=> ~ ( finite_finite_list_a @ ( sup_sup_set_list_a @ S @ T ) ) ) ).
% Un_infinite
thf(fact_768_infinite__Un,axiom,
! [S: set_a,T: set_a] :
( ( ~ ( finite_finite_a @ ( sup_sup_set_a @ S @ T ) ) )
= ( ~ ( finite_finite_a @ S )
| ~ ( finite_finite_a @ T ) ) ) ).
% infinite_Un
thf(fact_769_infinite__Un,axiom,
! [S: set_nat,T: set_nat] :
( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T ) ) )
= ( ~ ( finite_finite_nat @ S )
| ~ ( finite_finite_nat @ T ) ) ) ).
% infinite_Un
thf(fact_770_infinite__Un,axiom,
! [S: set_list_a,T: set_list_a] :
( ( ~ ( finite_finite_list_a @ ( sup_sup_set_list_a @ S @ T ) ) )
= ( ~ ( finite_finite_list_a @ S )
| ~ ( finite_finite_list_a @ T ) ) ) ).
% infinite_Un
thf(fact_771_finite__psubset__induct,axiom,
! [A: set_a,P2: set_a > $o] :
( ( finite_finite_a @ A )
=> ( ! [A8: set_a] :
( ( finite_finite_a @ A8 )
=> ( ! [B7: set_a] :
( ( ord_less_set_a @ B7 @ A8 )
=> ( P2 @ B7 ) )
=> ( P2 @ A8 ) ) )
=> ( P2 @ A ) ) ) ).
% finite_psubset_induct
thf(fact_772_finite__psubset__induct,axiom,
! [A: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ! [A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ! [B7: set_nat] :
( ( ord_less_set_nat @ B7 @ A8 )
=> ( P2 @ B7 ) )
=> ( P2 @ A8 ) ) )
=> ( P2 @ A ) ) ) ).
% finite_psubset_induct
thf(fact_773_finite__psubset__induct,axiom,
! [A: set_list_a,P2: set_list_a > $o] :
( ( finite_finite_list_a @ A )
=> ( ! [A8: set_list_a] :
( ( finite_finite_list_a @ A8 )
=> ( ! [B7: set_list_a] :
( ( ord_less_set_list_a @ B7 @ A8 )
=> ( P2 @ B7 ) )
=> ( P2 @ A8 ) ) )
=> ( P2 @ A ) ) ) ).
% finite_psubset_induct
thf(fact_774_distrib__sup__le,axiom,
! [X: set_a,Y: set_a,Z: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) @ ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_775_distrib__sup__le,axiom,
! [X: nat,Y: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_776_distrib__sup__le,axiom,
! [X: set_list_a,Y: set_list_a,Z: set_list_a] : ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ X @ ( inf_inf_set_list_a @ Y @ Z ) ) @ ( inf_inf_set_list_a @ ( sup_sup_set_list_a @ X @ Y ) @ ( sup_sup_set_list_a @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_777_distrib__inf__le,axiom,
! [X: set_a,Y: set_a,Z: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z ) ) @ ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_778_distrib__inf__le,axiom,
! [X: nat,Y: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z ) ) @ ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_779_distrib__inf__le,axiom,
! [X: set_list_a,Y: set_list_a,Z: set_list_a] : ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ ( inf_inf_set_list_a @ X @ Y ) @ ( inf_inf_set_list_a @ X @ Z ) ) @ ( inf_inf_set_list_a @ X @ ( sup_sup_set_list_a @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_780_Un__Int__assoc__eq,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ( sup_sup_set_a @ ( inf_inf_set_a @ A @ B ) @ C )
= ( inf_inf_set_a @ A @ ( sup_sup_set_a @ B @ C ) ) )
= ( ord_less_eq_set_a @ C @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_781_Un__Int__assoc__eq,axiom,
! [A: set_list_a,B: set_list_a,C: set_list_a] :
( ( ( sup_sup_set_list_a @ ( inf_inf_set_list_a @ A @ B ) @ C )
= ( inf_inf_set_list_a @ A @ ( sup_sup_set_list_a @ B @ C ) ) )
= ( ord_le8861187494160871172list_a @ C @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_782_Diff__partition,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( sup_sup_set_a @ A @ ( minus_minus_set_a @ B @ A ) )
= B ) ) ).
% Diff_partition
thf(fact_783_Diff__partition,axiom,
! [A: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( sup_sup_set_list_a @ A @ ( minus_646659088055828811list_a @ B @ A ) )
= B ) ) ).
% Diff_partition
thf(fact_784_Diff__subset__conv,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ B ) @ C )
= ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ B @ C ) ) ) ).
% Diff_subset_conv
thf(fact_785_Diff__subset__conv,axiom,
! [A: set_list_a,B: set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( minus_646659088055828811list_a @ A @ B ) @ C )
= ( ord_le8861187494160871172list_a @ A @ ( sup_sup_set_list_a @ B @ C ) ) ) ).
% Diff_subset_conv
thf(fact_786_Un__Diff__Int,axiom,
! [A: set_a,B: set_a] :
( ( sup_sup_set_a @ ( minus_minus_set_a @ A @ B ) @ ( inf_inf_set_a @ A @ B ) )
= A ) ).
% Un_Diff_Int
thf(fact_787_Un__Diff__Int,axiom,
! [A: set_list_a,B: set_list_a] :
( ( sup_sup_set_list_a @ ( minus_646659088055828811list_a @ A @ B ) @ ( inf_inf_set_list_a @ A @ B ) )
= A ) ).
% Un_Diff_Int
thf(fact_788_Int__Diff__Un,axiom,
! [A: set_a,B: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ A @ B ) @ ( minus_minus_set_a @ A @ B ) )
= A ) ).
% Int_Diff_Un
thf(fact_789_Int__Diff__Un,axiom,
! [A: set_list_a,B: set_list_a] :
( ( sup_sup_set_list_a @ ( inf_inf_set_list_a @ A @ B ) @ ( minus_646659088055828811list_a @ A @ B ) )
= A ) ).
% Int_Diff_Un
thf(fact_790_Diff__Int,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( minus_minus_set_a @ A @ ( inf_inf_set_a @ B @ C ) )
= ( sup_sup_set_a @ ( minus_minus_set_a @ A @ B ) @ ( minus_minus_set_a @ A @ C ) ) ) ).
% Diff_Int
thf(fact_791_Diff__Int,axiom,
! [A: set_list_a,B: set_list_a,C: set_list_a] :
( ( minus_646659088055828811list_a @ A @ ( inf_inf_set_list_a @ B @ C ) )
= ( sup_sup_set_list_a @ ( minus_646659088055828811list_a @ A @ B ) @ ( minus_646659088055828811list_a @ A @ C ) ) ) ).
% Diff_Int
thf(fact_792_Diff__Un,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( minus_minus_set_a @ A @ ( sup_sup_set_a @ B @ C ) )
= ( inf_inf_set_a @ ( minus_minus_set_a @ A @ B ) @ ( minus_minus_set_a @ A @ C ) ) ) ).
% Diff_Un
thf(fact_793_Diff__Un,axiom,
! [A: set_list_a,B: set_list_a,C: set_list_a] :
( ( minus_646659088055828811list_a @ A @ ( sup_sup_set_list_a @ B @ C ) )
= ( inf_inf_set_list_a @ ( minus_646659088055828811list_a @ A @ B ) @ ( minus_646659088055828811list_a @ A @ C ) ) ) ).
% Diff_Un
thf(fact_794_diff__less,axiom,
! [N4: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( minus_minus_nat @ M2 @ N4 ) @ M2 ) ) ) ).
% diff_less
thf(fact_795_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_796_in__mono,axiom,
! [A: set_set_list_a_a,B: set_set_list_a_a,X: set_list_a > a] :
( ( ord_le4799719167512954133st_a_a @ A @ B )
=> ( ( member_set_list_a_a @ X @ A )
=> ( member_set_list_a_a @ X @ B ) ) ) ).
% in_mono
thf(fact_797_in__mono,axiom,
! [A: set_nat_list_a,B: set_nat_list_a,X: nat > list_a] :
( ( ord_le2145805922479659755list_a @ A @ B )
=> ( ( member_nat_list_a @ X @ A )
=> ( member_nat_list_a @ X @ B ) ) ) ).
% in_mono
thf(fact_798_in__mono,axiom,
! [A: set_nat_a,B: set_nat_a,X: nat > a] :
( ( ord_le871467723717165285_nat_a @ A @ B )
=> ( ( member_nat_a @ X @ A )
=> ( member_nat_a @ X @ B ) ) ) ).
% in_mono
thf(fact_799_in__mono,axiom,
! [A: set_a_a,B: set_a_a,X: a > a] :
( ( ord_less_eq_set_a_a @ A @ B )
=> ( ( member_a_a @ X @ A )
=> ( member_a_a @ X @ B ) ) ) ).
% in_mono
thf(fact_800_in__mono,axiom,
! [A: set_a,B: set_a,X: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ X @ A )
=> ( member_a @ X @ B ) ) ) ).
% in_mono
thf(fact_801_in__mono,axiom,
! [A: set_list_a,B: set_list_a,X: list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( member_list_a @ X @ A )
=> ( member_list_a @ X @ B ) ) ) ).
% in_mono
thf(fact_802_subsetD,axiom,
! [A: set_set_list_a_a,B: set_set_list_a_a,C2: set_list_a > a] :
( ( ord_le4799719167512954133st_a_a @ A @ B )
=> ( ( member_set_list_a_a @ C2 @ A )
=> ( member_set_list_a_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_803_subsetD,axiom,
! [A: set_nat_list_a,B: set_nat_list_a,C2: nat > list_a] :
( ( ord_le2145805922479659755list_a @ A @ B )
=> ( ( member_nat_list_a @ C2 @ A )
=> ( member_nat_list_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_804_subsetD,axiom,
! [A: set_nat_a,B: set_nat_a,C2: nat > a] :
( ( ord_le871467723717165285_nat_a @ A @ B )
=> ( ( member_nat_a @ C2 @ A )
=> ( member_nat_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_805_subsetD,axiom,
! [A: set_a_a,B: set_a_a,C2: a > a] :
( ( ord_less_eq_set_a_a @ A @ B )
=> ( ( member_a_a @ C2 @ A )
=> ( member_a_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_806_subsetD,axiom,
! [A: set_a,B: set_a,C2: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ C2 @ A )
=> ( member_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_807_subsetD,axiom,
! [A: set_list_a,B: set_list_a,C2: list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( member_list_a @ C2 @ A )
=> ( member_list_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_808_equalityE,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_a @ A @ B )
=> ~ ( ord_less_eq_set_a @ B @ A ) ) ) ).
% equalityE
thf(fact_809_equalityE,axiom,
! [A: set_list_a,B: set_list_a] :
( ( A = B )
=> ~ ( ( ord_le8861187494160871172list_a @ A @ B )
=> ~ ( ord_le8861187494160871172list_a @ B @ A ) ) ) ).
% equalityE
thf(fact_810_subset__eq,axiom,
( ord_le4799719167512954133st_a_a
= ( ^ [A6: set_set_list_a_a,B2: set_set_list_a_a] :
! [X4: set_list_a > a] :
( ( member_set_list_a_a @ X4 @ A6 )
=> ( member_set_list_a_a @ X4 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_811_subset__eq,axiom,
( ord_le2145805922479659755list_a
= ( ^ [A6: set_nat_list_a,B2: set_nat_list_a] :
! [X4: nat > list_a] :
( ( member_nat_list_a @ X4 @ A6 )
=> ( member_nat_list_a @ X4 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_812_subset__eq,axiom,
( ord_le871467723717165285_nat_a
= ( ^ [A6: set_nat_a,B2: set_nat_a] :
! [X4: nat > a] :
( ( member_nat_a @ X4 @ A6 )
=> ( member_nat_a @ X4 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_813_subset__eq,axiom,
( ord_less_eq_set_a_a
= ( ^ [A6: set_a_a,B2: set_a_a] :
! [X4: a > a] :
( ( member_a_a @ X4 @ A6 )
=> ( member_a_a @ X4 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_814_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B2: set_a] :
! [X4: a] :
( ( member_a @ X4 @ A6 )
=> ( member_a @ X4 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_815_subset__eq,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A6: set_list_a,B2: set_list_a] :
! [X4: list_a] :
( ( member_list_a @ X4 @ A6 )
=> ( member_list_a @ X4 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_816_equalityD1,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% equalityD1
thf(fact_817_equalityD1,axiom,
! [A: set_list_a,B: set_list_a] :
( ( A = B )
=> ( ord_le8861187494160871172list_a @ A @ B ) ) ).
% equalityD1
thf(fact_818_equalityD2,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% equalityD2
thf(fact_819_equalityD2,axiom,
! [A: set_list_a,B: set_list_a] :
( ( A = B )
=> ( ord_le8861187494160871172list_a @ B @ A ) ) ).
% equalityD2
thf(fact_820_subset__iff,axiom,
( ord_le4799719167512954133st_a_a
= ( ^ [A6: set_set_list_a_a,B2: set_set_list_a_a] :
! [T2: set_list_a > a] :
( ( member_set_list_a_a @ T2 @ A6 )
=> ( member_set_list_a_a @ T2 @ B2 ) ) ) ) ).
% subset_iff
thf(fact_821_subset__iff,axiom,
( ord_le2145805922479659755list_a
= ( ^ [A6: set_nat_list_a,B2: set_nat_list_a] :
! [T2: nat > list_a] :
( ( member_nat_list_a @ T2 @ A6 )
=> ( member_nat_list_a @ T2 @ B2 ) ) ) ) ).
% subset_iff
thf(fact_822_subset__iff,axiom,
( ord_le871467723717165285_nat_a
= ( ^ [A6: set_nat_a,B2: set_nat_a] :
! [T2: nat > a] :
( ( member_nat_a @ T2 @ A6 )
=> ( member_nat_a @ T2 @ B2 ) ) ) ) ).
% subset_iff
thf(fact_823_subset__iff,axiom,
( ord_less_eq_set_a_a
= ( ^ [A6: set_a_a,B2: set_a_a] :
! [T2: a > a] :
( ( member_a_a @ T2 @ A6 )
=> ( member_a_a @ T2 @ B2 ) ) ) ) ).
% subset_iff
thf(fact_824_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B2: set_a] :
! [T2: a] :
( ( member_a @ T2 @ A6 )
=> ( member_a @ T2 @ B2 ) ) ) ) ).
% subset_iff
thf(fact_825_subset__iff,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A6: set_list_a,B2: set_list_a] :
! [T2: list_a] :
( ( member_list_a @ T2 @ A6 )
=> ( member_list_a @ T2 @ B2 ) ) ) ) ).
% subset_iff
thf(fact_826_subset__refl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% subset_refl
thf(fact_827_subset__refl,axiom,
! [A: set_list_a] : ( ord_le8861187494160871172list_a @ A @ A ) ).
% subset_refl
thf(fact_828_Collect__mono,axiom,
! [P2: nat > $o,Q2: nat > $o] :
( ! [X2: nat] :
( ( P2 @ X2 )
=> ( Q2 @ X2 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q2 ) ) ) ).
% Collect_mono
thf(fact_829_Collect__mono,axiom,
! [P2: a > $o,Q2: a > $o] :
( ! [X2: a] :
( ( P2 @ X2 )
=> ( Q2 @ X2 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P2 ) @ ( collect_a @ Q2 ) ) ) ).
% Collect_mono
thf(fact_830_Collect__mono,axiom,
! [P2: list_a > $o,Q2: list_a > $o] :
( ! [X2: list_a] :
( ( P2 @ X2 )
=> ( Q2 @ X2 ) )
=> ( ord_le8861187494160871172list_a @ ( collect_list_a @ P2 ) @ ( collect_list_a @ Q2 ) ) ) ).
% Collect_mono
thf(fact_831_subset__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% subset_trans
thf(fact_832_subset__trans,axiom,
! [A: set_list_a,B: set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( ord_le8861187494160871172list_a @ B @ C )
=> ( ord_le8861187494160871172list_a @ A @ C ) ) ) ).
% subset_trans
thf(fact_833_set__eq__subset,axiom,
( ( ^ [Y7: set_a,Z4: set_a] : ( Y7 = Z4 ) )
= ( ^ [A6: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A6 @ B2 )
& ( ord_less_eq_set_a @ B2 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_834_set__eq__subset,axiom,
( ( ^ [Y7: set_list_a,Z4: set_list_a] : ( Y7 = Z4 ) )
= ( ^ [A6: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A6 @ B2 )
& ( ord_le8861187494160871172list_a @ B2 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_835_Collect__mono__iff,axiom,
! [P2: nat > $o,Q2: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q2 ) )
= ( ! [X4: nat] :
( ( P2 @ X4 )
=> ( Q2 @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_836_Collect__mono__iff,axiom,
! [P2: a > $o,Q2: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P2 ) @ ( collect_a @ Q2 ) )
= ( ! [X4: a] :
( ( P2 @ X4 )
=> ( Q2 @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_837_Collect__mono__iff,axiom,
! [P2: list_a > $o,Q2: list_a > $o] :
( ( ord_le8861187494160871172list_a @ ( collect_list_a @ P2 ) @ ( collect_list_a @ Q2 ) )
= ( ! [X4: list_a] :
( ( P2 @ X4 )
=> ( Q2 @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_838_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A3: nat,C2: nat,B3: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A3 @ C2 ) @ B3 )
= ( minus_minus_nat @ ( minus_minus_nat @ A3 @ B3 ) @ C2 ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_839_le__0__eq,axiom,
! [N4: nat] :
( ( ord_less_eq_nat @ N4 @ zero_zero_nat )
= ( N4 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_840_bot__nat__0_Oextremum__uniqueI,axiom,
! [A3: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( A3 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_841_bot__nat__0_Oextremum__unique,axiom,
! [A3: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
= ( A3 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_842_less__eq__nat_Osimps_I1_J,axiom,
! [N4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N4 ) ).
% less_eq_nat.simps(1)
thf(fact_843_nat__neq__iff,axiom,
! [M2: nat,N4: nat] :
( ( M2 != N4 )
= ( ( ord_less_nat @ M2 @ N4 )
| ( ord_less_nat @ N4 @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_844_less__not__refl,axiom,
! [N4: nat] :
~ ( ord_less_nat @ N4 @ N4 ) ).
% less_not_refl
thf(fact_845_less__not__refl2,axiom,
! [N4: nat,M2: nat] :
( ( ord_less_nat @ N4 @ M2 )
=> ( M2 != N4 ) ) ).
% less_not_refl2
thf(fact_846_less__not__refl3,axiom,
! [S2: nat,T3: nat] :
( ( ord_less_nat @ S2 @ T3 )
=> ( S2 != T3 ) ) ).
% less_not_refl3
thf(fact_847_less__irrefl__nat,axiom,
! [N4: nat] :
~ ( ord_less_nat @ N4 @ N4 ) ).
% less_irrefl_nat
thf(fact_848_nat__less__induct,axiom,
! [P2: nat > $o,N4: nat] :
( ! [N: nat] :
( ! [M5: nat] :
( ( ord_less_nat @ M5 @ N )
=> ( P2 @ M5 ) )
=> ( P2 @ N ) )
=> ( P2 @ N4 ) ) ).
% nat_less_induct
thf(fact_849_infinite__descent,axiom,
! [P2: nat > $o,N4: nat] :
( ! [N: nat] :
( ~ ( P2 @ N )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N )
& ~ ( P2 @ M5 ) ) )
=> ( P2 @ N4 ) ) ).
% infinite_descent
thf(fact_850_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_851_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I2: nat,J3: nat] :
( ! [I3: nat,J4: nat] :
( ( ord_less_nat @ I3 @ J4 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J4 ) ) )
=> ( ( ord_less_eq_nat @ I2 @ J3 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J3 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_852_le__neq__implies__less,axiom,
! [M2: nat,N4: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
=> ( ( M2 != N4 )
=> ( ord_less_nat @ M2 @ N4 ) ) ) ).
% le_neq_implies_less
thf(fact_853_less__or__eq__imp__le,axiom,
! [M2: nat,N4: nat] :
( ( ( ord_less_nat @ M2 @ N4 )
| ( M2 = N4 ) )
=> ( ord_less_eq_nat @ M2 @ N4 ) ) ).
% less_or_eq_imp_le
thf(fact_854_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ N3 )
| ( M4 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_855_less__imp__le__nat,axiom,
! [M2: nat,N4: nat] :
( ( ord_less_nat @ M2 @ N4 )
=> ( ord_less_eq_nat @ M2 @ N4 ) ) ).
% less_imp_le_nat
thf(fact_856_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M4: nat,N3: nat] :
( ( ord_less_eq_nat @ M4 @ N3 )
& ( M4 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_857_inf__sup__aci_I4_J,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ X @ Y ) )
= ( inf_inf_set_a @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_858_inf__sup__aci_I4_J,axiom,
! [X: set_list_a,Y: set_list_a] :
( ( inf_inf_set_list_a @ X @ ( inf_inf_set_list_a @ X @ Y ) )
= ( inf_inf_set_list_a @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_859_inf__sup__aci_I3_J,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_860_inf__sup__aci_I3_J,axiom,
! [X: set_list_a,Y: set_list_a,Z: set_list_a] :
( ( inf_inf_set_list_a @ X @ ( inf_inf_set_list_a @ Y @ Z ) )
= ( inf_inf_set_list_a @ Y @ ( inf_inf_set_list_a @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_861_inf__sup__aci_I2_J,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z )
= ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_862_inf__sup__aci_I2_J,axiom,
! [X: set_list_a,Y: set_list_a,Z: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ X @ Y ) @ Z )
= ( inf_inf_set_list_a @ X @ ( inf_inf_set_list_a @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_863_inf__sup__aci_I1_J,axiom,
( inf_inf_set_a
= ( ^ [X4: set_a,Y5: set_a] : ( inf_inf_set_a @ Y5 @ X4 ) ) ) ).
% inf_sup_aci(1)
thf(fact_864_inf__sup__aci_I1_J,axiom,
( inf_inf_set_list_a
= ( ^ [X4: set_list_a,Y5: set_list_a] : ( inf_inf_set_list_a @ Y5 @ X4 ) ) ) ).
% inf_sup_aci(1)
thf(fact_865_inf_Oassoc,axiom,
! [A3: set_a,B3: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ C2 )
= ( inf_inf_set_a @ A3 @ ( inf_inf_set_a @ B3 @ C2 ) ) ) ).
% inf.assoc
thf(fact_866_inf_Oassoc,axiom,
! [A3: set_list_a,B3: set_list_a,C2: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ A3 @ B3 ) @ C2 )
= ( inf_inf_set_list_a @ A3 @ ( inf_inf_set_list_a @ B3 @ C2 ) ) ) ).
% inf.assoc
thf(fact_867_inf__assoc,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z )
= ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_868_inf__assoc,axiom,
! [X: set_list_a,Y: set_list_a,Z: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ X @ Y ) @ Z )
= ( inf_inf_set_list_a @ X @ ( inf_inf_set_list_a @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_869_inf_Ocommute,axiom,
( inf_inf_set_a
= ( ^ [A5: set_a,B5: set_a] : ( inf_inf_set_a @ B5 @ A5 ) ) ) ).
% inf.commute
thf(fact_870_inf_Ocommute,axiom,
( inf_inf_set_list_a
= ( ^ [A5: set_list_a,B5: set_list_a] : ( inf_inf_set_list_a @ B5 @ A5 ) ) ) ).
% inf.commute
thf(fact_871_inf__commute,axiom,
( inf_inf_set_a
= ( ^ [X4: set_a,Y5: set_a] : ( inf_inf_set_a @ Y5 @ X4 ) ) ) ).
% inf_commute
thf(fact_872_inf__commute,axiom,
( inf_inf_set_list_a
= ( ^ [X4: set_list_a,Y5: set_list_a] : ( inf_inf_set_list_a @ Y5 @ X4 ) ) ) ).
% inf_commute
thf(fact_873_inf_Oleft__commute,axiom,
! [B3: set_a,A3: set_a,C2: set_a] :
( ( inf_inf_set_a @ B3 @ ( inf_inf_set_a @ A3 @ C2 ) )
= ( inf_inf_set_a @ A3 @ ( inf_inf_set_a @ B3 @ C2 ) ) ) ).
% inf.left_commute
thf(fact_874_inf_Oleft__commute,axiom,
! [B3: set_list_a,A3: set_list_a,C2: set_list_a] :
( ( inf_inf_set_list_a @ B3 @ ( inf_inf_set_list_a @ A3 @ C2 ) )
= ( inf_inf_set_list_a @ A3 @ ( inf_inf_set_list_a @ B3 @ C2 ) ) ) ).
% inf.left_commute
thf(fact_875_inf__left__commute,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_876_inf__left__commute,axiom,
! [X: set_list_a,Y: set_list_a,Z: set_list_a] :
( ( inf_inf_set_list_a @ X @ ( inf_inf_set_list_a @ Y @ Z ) )
= ( inf_inf_set_list_a @ Y @ ( inf_inf_set_list_a @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_877_IntE,axiom,
! [C2: set_list_a > a,A: set_set_list_a_a,B: set_set_list_a_a] :
( ( member_set_list_a_a @ C2 @ ( inf_in6568206481208318535st_a_a @ A @ B ) )
=> ~ ( ( member_set_list_a_a @ C2 @ A )
=> ~ ( member_set_list_a_a @ C2 @ B ) ) ) ).
% IntE
thf(fact_878_IntE,axiom,
! [C2: nat > list_a,A: set_nat_list_a,B: set_nat_list_a] :
( ( member_nat_list_a @ C2 @ ( inf_in6652419485960844601list_a @ A @ B ) )
=> ~ ( ( member_nat_list_a @ C2 @ A )
=> ~ ( member_nat_list_a @ C2 @ B ) ) ) ).
% IntE
thf(fact_879_IntE,axiom,
! [C2: nat > a,A: set_nat_a,B: set_nat_a] :
( ( member_nat_a @ C2 @ ( inf_inf_set_nat_a @ A @ B ) )
=> ~ ( ( member_nat_a @ C2 @ A )
=> ~ ( member_nat_a @ C2 @ B ) ) ) ).
% IntE
thf(fact_880_IntE,axiom,
! [C2: a > a,A: set_a_a,B: set_a_a] :
( ( member_a_a @ C2 @ ( inf_inf_set_a_a @ A @ B ) )
=> ~ ( ( member_a_a @ C2 @ A )
=> ~ ( member_a_a @ C2 @ B ) ) ) ).
% IntE
thf(fact_881_IntE,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A @ B ) )
=> ~ ( ( member_a @ C2 @ A )
=> ~ ( member_a @ C2 @ B ) ) ) ).
% IntE
thf(fact_882_IntE,axiom,
! [C2: list_a,A: set_list_a,B: set_list_a] :
( ( member_list_a @ C2 @ ( inf_inf_set_list_a @ A @ B ) )
=> ~ ( ( member_list_a @ C2 @ A )
=> ~ ( member_list_a @ C2 @ B ) ) ) ).
% IntE
thf(fact_883_IntD1,axiom,
! [C2: set_list_a > a,A: set_set_list_a_a,B: set_set_list_a_a] :
( ( member_set_list_a_a @ C2 @ ( inf_in6568206481208318535st_a_a @ A @ B ) )
=> ( member_set_list_a_a @ C2 @ A ) ) ).
% IntD1
thf(fact_884_IntD1,axiom,
! [C2: nat > list_a,A: set_nat_list_a,B: set_nat_list_a] :
( ( member_nat_list_a @ C2 @ ( inf_in6652419485960844601list_a @ A @ B ) )
=> ( member_nat_list_a @ C2 @ A ) ) ).
% IntD1
thf(fact_885_IntD1,axiom,
! [C2: nat > a,A: set_nat_a,B: set_nat_a] :
( ( member_nat_a @ C2 @ ( inf_inf_set_nat_a @ A @ B ) )
=> ( member_nat_a @ C2 @ A ) ) ).
% IntD1
thf(fact_886_IntD1,axiom,
! [C2: a > a,A: set_a_a,B: set_a_a] :
( ( member_a_a @ C2 @ ( inf_inf_set_a_a @ A @ B ) )
=> ( member_a_a @ C2 @ A ) ) ).
% IntD1
thf(fact_887_IntD1,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A @ B ) )
=> ( member_a @ C2 @ A ) ) ).
% IntD1
thf(fact_888_IntD1,axiom,
! [C2: list_a,A: set_list_a,B: set_list_a] :
( ( member_list_a @ C2 @ ( inf_inf_set_list_a @ A @ B ) )
=> ( member_list_a @ C2 @ A ) ) ).
% IntD1
thf(fact_889_IntD2,axiom,
! [C2: set_list_a > a,A: set_set_list_a_a,B: set_set_list_a_a] :
( ( member_set_list_a_a @ C2 @ ( inf_in6568206481208318535st_a_a @ A @ B ) )
=> ( member_set_list_a_a @ C2 @ B ) ) ).
% IntD2
thf(fact_890_IntD2,axiom,
! [C2: nat > list_a,A: set_nat_list_a,B: set_nat_list_a] :
( ( member_nat_list_a @ C2 @ ( inf_in6652419485960844601list_a @ A @ B ) )
=> ( member_nat_list_a @ C2 @ B ) ) ).
% IntD2
thf(fact_891_IntD2,axiom,
! [C2: nat > a,A: set_nat_a,B: set_nat_a] :
( ( member_nat_a @ C2 @ ( inf_inf_set_nat_a @ A @ B ) )
=> ( member_nat_a @ C2 @ B ) ) ).
% IntD2
thf(fact_892_IntD2,axiom,
! [C2: a > a,A: set_a_a,B: set_a_a] :
( ( member_a_a @ C2 @ ( inf_inf_set_a_a @ A @ B ) )
=> ( member_a_a @ C2 @ B ) ) ).
% IntD2
thf(fact_893_IntD2,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A @ B ) )
=> ( member_a @ C2 @ B ) ) ).
% IntD2
thf(fact_894_IntD2,axiom,
! [C2: list_a,A: set_list_a,B: set_list_a] :
( ( member_list_a @ C2 @ ( inf_inf_set_list_a @ A @ B ) )
=> ( member_list_a @ C2 @ B ) ) ).
% IntD2
thf(fact_895_Int__assoc,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B ) @ C )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B @ C ) ) ) ).
% Int_assoc
thf(fact_896_Int__assoc,axiom,
! [A: set_list_a,B: set_list_a,C: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ A @ B ) @ C )
= ( inf_inf_set_list_a @ A @ ( inf_inf_set_list_a @ B @ C ) ) ) ).
% Int_assoc
thf(fact_897_Int__absorb,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ A )
= A ) ).
% Int_absorb
thf(fact_898_Int__absorb,axiom,
! [A: set_list_a] :
( ( inf_inf_set_list_a @ A @ A )
= A ) ).
% Int_absorb
thf(fact_899_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A6: set_a,B2: set_a] : ( inf_inf_set_a @ B2 @ A6 ) ) ) ).
% Int_commute
thf(fact_900_Int__commute,axiom,
( inf_inf_set_list_a
= ( ^ [A6: set_list_a,B2: set_list_a] : ( inf_inf_set_list_a @ B2 @ A6 ) ) ) ).
% Int_commute
thf(fact_901_Int__left__absorb,axiom,
! [A: set_a,B: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ A @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ).
% Int_left_absorb
thf(fact_902_Int__left__absorb,axiom,
! [A: set_list_a,B: set_list_a] :
( ( inf_inf_set_list_a @ A @ ( inf_inf_set_list_a @ A @ B ) )
= ( inf_inf_set_list_a @ A @ B ) ) ).
% Int_left_absorb
thf(fact_903_Int__left__commute,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B @ C ) )
= ( inf_inf_set_a @ B @ ( inf_inf_set_a @ A @ C ) ) ) ).
% Int_left_commute
thf(fact_904_Int__left__commute,axiom,
! [A: set_list_a,B: set_list_a,C: set_list_a] :
( ( inf_inf_set_list_a @ A @ ( inf_inf_set_list_a @ B @ C ) )
= ( inf_inf_set_list_a @ B @ ( inf_inf_set_list_a @ A @ C ) ) ) ).
% Int_left_commute
thf(fact_905_DiffE,axiom,
! [C2: set_list_a > a,A: set_set_list_a_a,B: set_set_list_a_a] :
( ( member_set_list_a_a @ C2 @ ( minus_5613498140476352782st_a_a @ A @ B ) )
=> ~ ( ( member_set_list_a_a @ C2 @ A )
=> ( member_set_list_a_a @ C2 @ B ) ) ) ).
% DiffE
thf(fact_906_DiffE,axiom,
! [C2: nat > list_a,A: set_nat_list_a,B: set_nat_list_a] :
( ( member_nat_list_a @ C2 @ ( minus_4169782841487898290list_a @ A @ B ) )
=> ~ ( ( member_nat_list_a @ C2 @ A )
=> ( member_nat_list_a @ C2 @ B ) ) ) ).
% DiffE
thf(fact_907_DiffE,axiom,
! [C2: nat > a,A: set_nat_a,B: set_nat_a] :
( ( member_nat_a @ C2 @ ( minus_490503922182417452_nat_a @ A @ B ) )
=> ~ ( ( member_nat_a @ C2 @ A )
=> ( member_nat_a @ C2 @ B ) ) ) ).
% DiffE
thf(fact_908_DiffE,axiom,
! [C2: a > a,A: set_a_a,B: set_a_a] :
( ( member_a_a @ C2 @ ( minus_minus_set_a_a @ A @ B ) )
=> ~ ( ( member_a_a @ C2 @ A )
=> ( member_a_a @ C2 @ B ) ) ) ).
% DiffE
thf(fact_909_DiffE,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A @ B ) )
=> ~ ( ( member_a @ C2 @ A )
=> ( member_a @ C2 @ B ) ) ) ).
% DiffE
thf(fact_910_DiffE,axiom,
! [C2: list_a,A: set_list_a,B: set_list_a] :
( ( member_list_a @ C2 @ ( minus_646659088055828811list_a @ A @ B ) )
=> ~ ( ( member_list_a @ C2 @ A )
=> ( member_list_a @ C2 @ B ) ) ) ).
% DiffE
thf(fact_911_DiffD1,axiom,
! [C2: set_list_a > a,A: set_set_list_a_a,B: set_set_list_a_a] :
( ( member_set_list_a_a @ C2 @ ( minus_5613498140476352782st_a_a @ A @ B ) )
=> ( member_set_list_a_a @ C2 @ A ) ) ).
% DiffD1
thf(fact_912_DiffD1,axiom,
! [C2: nat > list_a,A: set_nat_list_a,B: set_nat_list_a] :
( ( member_nat_list_a @ C2 @ ( minus_4169782841487898290list_a @ A @ B ) )
=> ( member_nat_list_a @ C2 @ A ) ) ).
% DiffD1
thf(fact_913_DiffD1,axiom,
! [C2: nat > a,A: set_nat_a,B: set_nat_a] :
( ( member_nat_a @ C2 @ ( minus_490503922182417452_nat_a @ A @ B ) )
=> ( member_nat_a @ C2 @ A ) ) ).
% DiffD1
thf(fact_914_DiffD1,axiom,
! [C2: a > a,A: set_a_a,B: set_a_a] :
( ( member_a_a @ C2 @ ( minus_minus_set_a_a @ A @ B ) )
=> ( member_a_a @ C2 @ A ) ) ).
% DiffD1
thf(fact_915_DiffD1,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A @ B ) )
=> ( member_a @ C2 @ A ) ) ).
% DiffD1
thf(fact_916_DiffD1,axiom,
! [C2: list_a,A: set_list_a,B: set_list_a] :
( ( member_list_a @ C2 @ ( minus_646659088055828811list_a @ A @ B ) )
=> ( member_list_a @ C2 @ A ) ) ).
% DiffD1
thf(fact_917_DiffD2,axiom,
! [C2: set_list_a > a,A: set_set_list_a_a,B: set_set_list_a_a] :
( ( member_set_list_a_a @ C2 @ ( minus_5613498140476352782st_a_a @ A @ B ) )
=> ~ ( member_set_list_a_a @ C2 @ B ) ) ).
% DiffD2
thf(fact_918_DiffD2,axiom,
! [C2: nat > list_a,A: set_nat_list_a,B: set_nat_list_a] :
( ( member_nat_list_a @ C2 @ ( minus_4169782841487898290list_a @ A @ B ) )
=> ~ ( member_nat_list_a @ C2 @ B ) ) ).
% DiffD2
thf(fact_919_DiffD2,axiom,
! [C2: nat > a,A: set_nat_a,B: set_nat_a] :
( ( member_nat_a @ C2 @ ( minus_490503922182417452_nat_a @ A @ B ) )
=> ~ ( member_nat_a @ C2 @ B ) ) ).
% DiffD2
thf(fact_920_DiffD2,axiom,
! [C2: a > a,A: set_a_a,B: set_a_a] :
( ( member_a_a @ C2 @ ( minus_minus_set_a_a @ A @ B ) )
=> ~ ( member_a_a @ C2 @ B ) ) ).
% DiffD2
thf(fact_921_DiffD2,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A @ B ) )
=> ~ ( member_a @ C2 @ B ) ) ).
% DiffD2
thf(fact_922_DiffD2,axiom,
! [C2: list_a,A: set_list_a,B: set_list_a] :
( ( member_list_a @ C2 @ ( minus_646659088055828811list_a @ A @ B ) )
=> ~ ( member_list_a @ C2 @ B ) ) ).
% DiffD2
thf(fact_923_less__eq__set__def,axiom,
( ord_le4799719167512954133st_a_a
= ( ^ [A6: set_set_list_a_a,B2: set_set_list_a_a] :
( ord_le6553425858663066544_a_a_o
@ ^ [X4: set_list_a > a] : ( member_set_list_a_a @ X4 @ A6 )
@ ^ [X4: set_list_a > a] : ( member_set_list_a_a @ X4 @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_924_less__eq__set__def,axiom,
( ord_le2145805922479659755list_a
= ( ^ [A6: set_nat_list_a,B2: set_nat_list_a] :
( ord_le4184171100712167858st_a_o
@ ^ [X4: nat > list_a] : ( member_nat_list_a @ X4 @ A6 )
@ ^ [X4: nat > list_a] : ( member_nat_list_a @ X4 @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_925_less__eq__set__def,axiom,
( ord_le871467723717165285_nat_a
= ( ^ [A6: set_nat_a,B2: set_nat_a] :
( ord_less_eq_nat_a_o
@ ^ [X4: nat > a] : ( member_nat_a @ X4 @ A6 )
@ ^ [X4: nat > a] : ( member_nat_a @ X4 @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_926_less__eq__set__def,axiom,
( ord_less_eq_set_a_a
= ( ^ [A6: set_a_a,B2: set_a_a] :
( ord_less_eq_a_a_o
@ ^ [X4: a > a] : ( member_a_a @ X4 @ A6 )
@ ^ [X4: a > a] : ( member_a_a @ X4 @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_927_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B2: set_a] :
( ord_less_eq_a_o
@ ^ [X4: a] : ( member_a @ X4 @ A6 )
@ ^ [X4: a] : ( member_a @ X4 @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_928_less__eq__set__def,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A6: set_list_a,B2: set_list_a] :
( ord_less_eq_list_a_o
@ ^ [X4: list_a] : ( member_list_a @ X4 @ A6 )
@ ^ [X4: list_a] : ( member_list_a @ X4 @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_929_Collect__subset,axiom,
! [A: set_set_list_a_a,P2: ( set_list_a > a ) > $o] :
( ord_le4799719167512954133st_a_a
@ ( collect_set_list_a_a
@ ^ [X4: set_list_a > a] :
( ( member_set_list_a_a @ X4 @ A )
& ( P2 @ X4 ) ) )
@ A ) ).
% Collect_subset
thf(fact_930_Collect__subset,axiom,
! [A: set_nat_list_a,P2: ( nat > list_a ) > $o] :
( ord_le2145805922479659755list_a
@ ( collect_nat_list_a
@ ^ [X4: nat > list_a] :
( ( member_nat_list_a @ X4 @ A )
& ( P2 @ X4 ) ) )
@ A ) ).
% Collect_subset
thf(fact_931_Collect__subset,axiom,
! [A: set_nat_a,P2: ( nat > a ) > $o] :
( ord_le871467723717165285_nat_a
@ ( collect_nat_a
@ ^ [X4: nat > a] :
( ( member_nat_a @ X4 @ A )
& ( P2 @ X4 ) ) )
@ A ) ).
% Collect_subset
thf(fact_932_Collect__subset,axiom,
! [A: set_a_a,P2: ( a > a ) > $o] :
( ord_less_eq_set_a_a
@ ( collect_a_a
@ ^ [X4: a > a] :
( ( member_a_a @ X4 @ A )
& ( P2 @ X4 ) ) )
@ A ) ).
% Collect_subset
thf(fact_933_Collect__subset,axiom,
! [A: set_nat,P2: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A )
& ( P2 @ X4 ) ) )
@ A ) ).
% Collect_subset
thf(fact_934_Collect__subset,axiom,
! [A: set_a,P2: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ A )
& ( P2 @ X4 ) ) )
@ A ) ).
% Collect_subset
thf(fact_935_Collect__subset,axiom,
! [A: set_list_a,P2: list_a > $o] :
( ord_le8861187494160871172list_a
@ ( collect_list_a
@ ^ [X4: list_a] :
( ( member_list_a @ X4 @ A )
& ( P2 @ X4 ) ) )
@ A ) ).
% Collect_subset
thf(fact_936_Int__def,axiom,
( inf_in6568206481208318535st_a_a
= ( ^ [A6: set_set_list_a_a,B2: set_set_list_a_a] :
( collect_set_list_a_a
@ ^ [X4: set_list_a > a] :
( ( member_set_list_a_a @ X4 @ A6 )
& ( member_set_list_a_a @ X4 @ B2 ) ) ) ) ) ).
% Int_def
thf(fact_937_Int__def,axiom,
( inf_in6652419485960844601list_a
= ( ^ [A6: set_nat_list_a,B2: set_nat_list_a] :
( collect_nat_list_a
@ ^ [X4: nat > list_a] :
( ( member_nat_list_a @ X4 @ A6 )
& ( member_nat_list_a @ X4 @ B2 ) ) ) ) ) ).
% Int_def
thf(fact_938_Int__def,axiom,
( inf_inf_set_nat_a
= ( ^ [A6: set_nat_a,B2: set_nat_a] :
( collect_nat_a
@ ^ [X4: nat > a] :
( ( member_nat_a @ X4 @ A6 )
& ( member_nat_a @ X4 @ B2 ) ) ) ) ) ).
% Int_def
thf(fact_939_Int__def,axiom,
( inf_inf_set_a_a
= ( ^ [A6: set_a_a,B2: set_a_a] :
( collect_a_a
@ ^ [X4: a > a] :
( ( member_a_a @ X4 @ A6 )
& ( member_a_a @ X4 @ B2 ) ) ) ) ) ).
% Int_def
thf(fact_940_Int__def,axiom,
( inf_inf_set_nat
= ( ^ [A6: set_nat,B2: set_nat] :
( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A6 )
& ( member_nat @ X4 @ B2 ) ) ) ) ) ).
% Int_def
thf(fact_941_Int__def,axiom,
( inf_inf_set_a
= ( ^ [A6: set_a,B2: set_a] :
( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ A6 )
& ( member_a @ X4 @ B2 ) ) ) ) ) ).
% Int_def
thf(fact_942_Int__def,axiom,
( inf_inf_set_list_a
= ( ^ [A6: set_list_a,B2: set_list_a] :
( collect_list_a
@ ^ [X4: list_a] :
( ( member_list_a @ X4 @ A6 )
& ( member_list_a @ X4 @ B2 ) ) ) ) ) ).
% Int_def
thf(fact_943_Int__Collect,axiom,
! [X: set_list_a > a,A: set_set_list_a_a,P2: ( set_list_a > a ) > $o] :
( ( member_set_list_a_a @ X @ ( inf_in6568206481208318535st_a_a @ A @ ( collect_set_list_a_a @ P2 ) ) )
= ( ( member_set_list_a_a @ X @ A )
& ( P2 @ X ) ) ) ).
% Int_Collect
thf(fact_944_Int__Collect,axiom,
! [X: nat > list_a,A: set_nat_list_a,P2: ( nat > list_a ) > $o] :
( ( member_nat_list_a @ X @ ( inf_in6652419485960844601list_a @ A @ ( collect_nat_list_a @ P2 ) ) )
= ( ( member_nat_list_a @ X @ A )
& ( P2 @ X ) ) ) ).
% Int_Collect
thf(fact_945_Int__Collect,axiom,
! [X: nat > a,A: set_nat_a,P2: ( nat > a ) > $o] :
( ( member_nat_a @ X @ ( inf_inf_set_nat_a @ A @ ( collect_nat_a @ P2 ) ) )
= ( ( member_nat_a @ X @ A )
& ( P2 @ X ) ) ) ).
% Int_Collect
thf(fact_946_Int__Collect,axiom,
! [X: a > a,A: set_a_a,P2: ( a > a ) > $o] :
( ( member_a_a @ X @ ( inf_inf_set_a_a @ A @ ( collect_a_a @ P2 ) ) )
= ( ( member_a_a @ X @ A )
& ( P2 @ X ) ) ) ).
% Int_Collect
thf(fact_947_Int__Collect,axiom,
! [X: nat,A: set_nat,P2: nat > $o] :
( ( member_nat @ X @ ( inf_inf_set_nat @ A @ ( collect_nat @ P2 ) ) )
= ( ( member_nat @ X @ A )
& ( P2 @ X ) ) ) ).
% Int_Collect
thf(fact_948_Int__Collect,axiom,
! [X: a,A: set_a,P2: a > $o] :
( ( member_a @ X @ ( inf_inf_set_a @ A @ ( collect_a @ P2 ) ) )
= ( ( member_a @ X @ A )
& ( P2 @ X ) ) ) ).
% Int_Collect
thf(fact_949_Int__Collect,axiom,
! [X: list_a,A: set_list_a,P2: list_a > $o] :
( ( member_list_a @ X @ ( inf_inf_set_list_a @ A @ ( collect_list_a @ P2 ) ) )
= ( ( member_list_a @ X @ A )
& ( P2 @ X ) ) ) ).
% Int_Collect
thf(fact_950_Collect__conj__eq,axiom,
! [P2: nat > $o,Q2: nat > $o] :
( ( collect_nat
@ ^ [X4: nat] :
( ( P2 @ X4 )
& ( Q2 @ X4 ) ) )
= ( inf_inf_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q2 ) ) ) ).
% Collect_conj_eq
thf(fact_951_Collect__conj__eq,axiom,
! [P2: a > $o,Q2: a > $o] :
( ( collect_a
@ ^ [X4: a] :
( ( P2 @ X4 )
& ( Q2 @ X4 ) ) )
= ( inf_inf_set_a @ ( collect_a @ P2 ) @ ( collect_a @ Q2 ) ) ) ).
% Collect_conj_eq
thf(fact_952_Collect__conj__eq,axiom,
! [P2: list_a > $o,Q2: list_a > $o] :
( ( collect_list_a
@ ^ [X4: list_a] :
( ( P2 @ X4 )
& ( Q2 @ X4 ) ) )
= ( inf_inf_set_list_a @ ( collect_list_a @ P2 ) @ ( collect_list_a @ Q2 ) ) ) ).
% Collect_conj_eq
thf(fact_953_set__diff__eq,axiom,
( minus_5613498140476352782st_a_a
= ( ^ [A6: set_set_list_a_a,B2: set_set_list_a_a] :
( collect_set_list_a_a
@ ^ [X4: set_list_a > a] :
( ( member_set_list_a_a @ X4 @ A6 )
& ~ ( member_set_list_a_a @ X4 @ B2 ) ) ) ) ) ).
% set_diff_eq
thf(fact_954_set__diff__eq,axiom,
( minus_4169782841487898290list_a
= ( ^ [A6: set_nat_list_a,B2: set_nat_list_a] :
( collect_nat_list_a
@ ^ [X4: nat > list_a] :
( ( member_nat_list_a @ X4 @ A6 )
& ~ ( member_nat_list_a @ X4 @ B2 ) ) ) ) ) ).
% set_diff_eq
thf(fact_955_set__diff__eq,axiom,
( minus_490503922182417452_nat_a
= ( ^ [A6: set_nat_a,B2: set_nat_a] :
( collect_nat_a
@ ^ [X4: nat > a] :
( ( member_nat_a @ X4 @ A6 )
& ~ ( member_nat_a @ X4 @ B2 ) ) ) ) ) ).
% set_diff_eq
thf(fact_956_set__diff__eq,axiom,
( minus_minus_set_a_a
= ( ^ [A6: set_a_a,B2: set_a_a] :
( collect_a_a
@ ^ [X4: a > a] :
( ( member_a_a @ X4 @ A6 )
& ~ ( member_a_a @ X4 @ B2 ) ) ) ) ) ).
% set_diff_eq
thf(fact_957_set__diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [A6: set_nat,B2: set_nat] :
( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A6 )
& ~ ( member_nat @ X4 @ B2 ) ) ) ) ) ).
% set_diff_eq
thf(fact_958_set__diff__eq,axiom,
( minus_minus_set_a
= ( ^ [A6: set_a,B2: set_a] :
( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ A6 )
& ~ ( member_a @ X4 @ B2 ) ) ) ) ) ).
% set_diff_eq
thf(fact_959_set__diff__eq,axiom,
( minus_646659088055828811list_a
= ( ^ [A6: set_list_a,B2: set_list_a] :
( collect_list_a
@ ^ [X4: list_a] :
( ( member_list_a @ X4 @ A6 )
& ~ ( member_list_a @ X4 @ B2 ) ) ) ) ) ).
% set_diff_eq
thf(fact_960_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_961_gr__zeroI,axiom,
! [N4: nat] :
( ( N4 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N4 ) ) ).
% gr_zeroI
thf(fact_962_not__less__zero,axiom,
! [N4: nat] :
~ ( ord_less_nat @ N4 @ zero_zero_nat ) ).
% not_less_zero
thf(fact_963_gr__implies__not__zero,axiom,
! [M2: nat,N4: nat] :
( ( ord_less_nat @ M2 @ N4 )
=> ( N4 != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_964_zero__less__iff__neq__zero,axiom,
! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
= ( N4 != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_965_inf_OcoboundedI2,axiom,
! [B3: set_a,C2: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B3 @ C2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_966_inf_OcoboundedI2,axiom,
! [B3: nat,C2: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A3 @ B3 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_967_inf_OcoboundedI2,axiom,
! [B3: set_list_a,C2: set_list_a,A3: set_list_a] :
( ( ord_le8861187494160871172list_a @ B3 @ C2 )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A3 @ B3 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_968_inf_OcoboundedI1,axiom,
! [A3: set_a,C2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ C2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_969_inf_OcoboundedI1,axiom,
! [A3: nat,C2: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ C2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A3 @ B3 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_970_inf_OcoboundedI1,axiom,
! [A3: set_list_a,C2: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ A3 @ C2 )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A3 @ B3 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_971_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [B5: set_a,A5: set_a] :
( ( inf_inf_set_a @ A5 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_972_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A5: nat] :
( ( inf_inf_nat @ A5 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_973_inf_Oabsorb__iff2,axiom,
( ord_le8861187494160871172list_a
= ( ^ [B5: set_list_a,A5: set_list_a] :
( ( inf_inf_set_list_a @ A5 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_974_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ( inf_inf_set_a @ A5 @ B5 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_975_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
( ( inf_inf_nat @ A5 @ B5 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_976_inf_Oabsorb__iff1,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A5: set_list_a,B5: set_list_a] :
( ( inf_inf_set_list_a @ A5 @ B5 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_977_inf_Ocobounded2,axiom,
! [A3: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ B3 ) ).
% inf.cobounded2
thf(fact_978_inf_Ocobounded2,axiom,
! [A3: nat,B3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A3 @ B3 ) @ B3 ) ).
% inf.cobounded2
thf(fact_979_inf_Ocobounded2,axiom,
! [A3: set_list_a,B3: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A3 @ B3 ) @ B3 ) ).
% inf.cobounded2
thf(fact_980_inf_Ocobounded1,axiom,
! [A3: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ A3 ) ).
% inf.cobounded1
thf(fact_981_inf_Ocobounded1,axiom,
! [A3: nat,B3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A3 @ B3 ) @ A3 ) ).
% inf.cobounded1
thf(fact_982_inf_Ocobounded1,axiom,
! [A3: set_list_a,B3: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A3 @ B3 ) @ A3 ) ).
% inf.cobounded1
thf(fact_983_inf_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
( A5
= ( inf_inf_set_a @ A5 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_984_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
( A5
= ( inf_inf_nat @ A5 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_985_inf_Oorder__iff,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A5: set_list_a,B5: set_list_a] :
( A5
= ( inf_inf_set_list_a @ A5 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_986_inf__greatest,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ X @ Z )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_987_inf__greatest,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Z )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_988_inf__greatest,axiom,
! [X: set_list_a,Y: set_list_a,Z: set_list_a] :
( ( ord_le8861187494160871172list_a @ X @ Y )
=> ( ( ord_le8861187494160871172list_a @ X @ Z )
=> ( ord_le8861187494160871172list_a @ X @ ( inf_inf_set_list_a @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_989_inf_OboundedI,axiom,
! [A3: set_a,B3: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( ord_less_eq_set_a @ A3 @ C2 )
=> ( ord_less_eq_set_a @ A3 @ ( inf_inf_set_a @ B3 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_990_inf_OboundedI,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ A3 @ C2 )
=> ( ord_less_eq_nat @ A3 @ ( inf_inf_nat @ B3 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_991_inf_OboundedI,axiom,
! [A3: set_list_a,B3: set_list_a,C2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A3 @ B3 )
=> ( ( ord_le8861187494160871172list_a @ A3 @ C2 )
=> ( ord_le8861187494160871172list_a @ A3 @ ( inf_inf_set_list_a @ B3 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_992_inf_OboundedE,axiom,
! [A3: set_a,B3: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A3 @ ( inf_inf_set_a @ B3 @ C2 ) )
=> ~ ( ( ord_less_eq_set_a @ A3 @ B3 )
=> ~ ( ord_less_eq_set_a @ A3 @ C2 ) ) ) ).
% inf.boundedE
thf(fact_993_inf_OboundedE,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ ( inf_inf_nat @ B3 @ C2 ) )
=> ~ ( ( ord_less_eq_nat @ A3 @ B3 )
=> ~ ( ord_less_eq_nat @ A3 @ C2 ) ) ) ).
% inf.boundedE
thf(fact_994_inf_OboundedE,axiom,
! [A3: set_list_a,B3: set_list_a,C2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A3 @ ( inf_inf_set_list_a @ B3 @ C2 ) )
=> ~ ( ( ord_le8861187494160871172list_a @ A3 @ B3 )
=> ~ ( ord_le8861187494160871172list_a @ A3 @ C2 ) ) ) ).
% inf.boundedE
thf(fact_995_inf__absorb2,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( inf_inf_set_a @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_996_inf__absorb2,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( inf_inf_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_997_inf__absorb2,axiom,
! [Y: set_list_a,X: set_list_a] :
( ( ord_le8861187494160871172list_a @ Y @ X )
=> ( ( inf_inf_set_list_a @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_998_inf__absorb1,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( inf_inf_set_a @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_999_inf__absorb1,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( inf_inf_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_1000_inf__absorb1,axiom,
! [X: set_list_a,Y: set_list_a] :
( ( ord_le8861187494160871172list_a @ X @ Y )
=> ( ( inf_inf_set_list_a @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_1001_inf_Oabsorb2,axiom,
! [B3: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A3 )
=> ( ( inf_inf_set_a @ A3 @ B3 )
= B3 ) ) ).
% inf.absorb2
thf(fact_1002_inf_Oabsorb2,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
=> ( ( inf_inf_nat @ A3 @ B3 )
= B3 ) ) ).
% inf.absorb2
thf(fact_1003_inf_Oabsorb2,axiom,
! [B3: set_list_a,A3: set_list_a] :
( ( ord_le8861187494160871172list_a @ B3 @ A3 )
=> ( ( inf_inf_set_list_a @ A3 @ B3 )
= B3 ) ) ).
% inf.absorb2
thf(fact_1004_inf_Oabsorb1,axiom,
! [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( inf_inf_set_a @ A3 @ B3 )
= A3 ) ) ).
% inf.absorb1
thf(fact_1005_inf_Oabsorb1,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( inf_inf_nat @ A3 @ B3 )
= A3 ) ) ).
% inf.absorb1
thf(fact_1006_inf_Oabsorb1,axiom,
! [A3: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ A3 @ B3 )
=> ( ( inf_inf_set_list_a @ A3 @ B3 )
= A3 ) ) ).
% inf.absorb1
thf(fact_1007_le__iff__inf,axiom,
( ord_less_eq_set_a
= ( ^ [X4: set_a,Y5: set_a] :
( ( inf_inf_set_a @ X4 @ Y5 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_1008_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y5: nat] :
( ( inf_inf_nat @ X4 @ Y5 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_1009_le__iff__inf,axiom,
( ord_le8861187494160871172list_a
= ( ^ [X4: set_list_a,Y5: set_list_a] :
( ( inf_inf_set_list_a @ X4 @ Y5 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_1010_inf__unique,axiom,
! [F: set_a > set_a > set_a,X: set_a,Y: set_a] :
( ! [X2: set_a,Y4: set_a] : ( ord_less_eq_set_a @ ( F @ X2 @ Y4 ) @ X2 )
=> ( ! [X2: set_a,Y4: set_a] : ( ord_less_eq_set_a @ ( F @ X2 @ Y4 ) @ Y4 )
=> ( ! [X2: set_a,Y4: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ( ord_less_eq_set_a @ X2 @ Z3 )
=> ( ord_less_eq_set_a @ X2 @ ( F @ Y4 @ Z3 ) ) ) )
=> ( ( inf_inf_set_a @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_1011_inf__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X2: nat,Y4: nat] : ( ord_less_eq_nat @ ( F @ X2 @ Y4 ) @ X2 )
=> ( ! [X2: nat,Y4: nat] : ( ord_less_eq_nat @ ( F @ X2 @ Y4 ) @ Y4 )
=> ( ! [X2: nat,Y4: nat,Z3: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ( ord_less_eq_nat @ X2 @ Z3 )
=> ( ord_less_eq_nat @ X2 @ ( F @ Y4 @ Z3 ) ) ) )
=> ( ( inf_inf_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_1012_inf__unique,axiom,
! [F: set_list_a > set_list_a > set_list_a,X: set_list_a,Y: set_list_a] :
( ! [X2: set_list_a,Y4: set_list_a] : ( ord_le8861187494160871172list_a @ ( F @ X2 @ Y4 ) @ X2 )
=> ( ! [X2: set_list_a,Y4: set_list_a] : ( ord_le8861187494160871172list_a @ ( F @ X2 @ Y4 ) @ Y4 )
=> ( ! [X2: set_list_a,Y4: set_list_a,Z3: set_list_a] :
( ( ord_le8861187494160871172list_a @ X2 @ Y4 )
=> ( ( ord_le8861187494160871172list_a @ X2 @ Z3 )
=> ( ord_le8861187494160871172list_a @ X2 @ ( F @ Y4 @ Z3 ) ) ) )
=> ( ( inf_inf_set_list_a @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_1013_inf_OorderI,axiom,
! [A3: set_a,B3: set_a] :
( ( A3
= ( inf_inf_set_a @ A3 @ B3 ) )
=> ( ord_less_eq_set_a @ A3 @ B3 ) ) ).
% inf.orderI
thf(fact_1014_inf_OorderI,axiom,
! [A3: nat,B3: nat] :
( ( A3
= ( inf_inf_nat @ A3 @ B3 ) )
=> ( ord_less_eq_nat @ A3 @ B3 ) ) ).
% inf.orderI
thf(fact_1015_inf_OorderI,axiom,
! [A3: set_list_a,B3: set_list_a] :
( ( A3
= ( inf_inf_set_list_a @ A3 @ B3 ) )
=> ( ord_le8861187494160871172list_a @ A3 @ B3 ) ) ).
% inf.orderI
thf(fact_1016_inf_OorderE,axiom,
! [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( A3
= ( inf_inf_set_a @ A3 @ B3 ) ) ) ).
% inf.orderE
thf(fact_1017_inf_OorderE,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( A3
= ( inf_inf_nat @ A3 @ B3 ) ) ) ).
% inf.orderE
thf(fact_1018_inf_OorderE,axiom,
! [A3: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ A3 @ B3 )
=> ( A3
= ( inf_inf_set_list_a @ A3 @ B3 ) ) ) ).
% inf.orderE
thf(fact_1019_le__infI2,axiom,
! [B3: set_a,X: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B3 @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ X ) ) ).
% le_infI2
thf(fact_1020_le__infI2,axiom,
! [B3: nat,X: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A3 @ B3 ) @ X ) ) ).
% le_infI2
thf(fact_1021_le__infI2,axiom,
! [B3: set_list_a,X: set_list_a,A3: set_list_a] :
( ( ord_le8861187494160871172list_a @ B3 @ X )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A3 @ B3 ) @ X ) ) ).
% le_infI2
thf(fact_1022_le__infI1,axiom,
! [A3: set_a,X: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ X ) ) ).
% le_infI1
thf(fact_1023_le__infI1,axiom,
! [A3: nat,X: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A3 @ B3 ) @ X ) ) ).
% le_infI1
thf(fact_1024_le__infI1,axiom,
! [A3: set_list_a,X: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ A3 @ X )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A3 @ B3 ) @ X ) ) ).
% le_infI1
thf(fact_1025_inf__mono,axiom,
! [A3: set_a,C2: set_a,B3: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A3 @ C2 )
=> ( ( ord_less_eq_set_a @ B3 @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ ( inf_inf_set_a @ C2 @ D2 ) ) ) ) ).
% inf_mono
thf(fact_1026_inf__mono,axiom,
! [A3: nat,C2: nat,B3: nat,D2: nat] :
( ( ord_less_eq_nat @ A3 @ C2 )
=> ( ( ord_less_eq_nat @ B3 @ D2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A3 @ B3 ) @ ( inf_inf_nat @ C2 @ D2 ) ) ) ) ).
% inf_mono
thf(fact_1027_inf__mono,axiom,
! [A3: set_list_a,C2: set_list_a,B3: set_list_a,D2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A3 @ C2 )
=> ( ( ord_le8861187494160871172list_a @ B3 @ D2 )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A3 @ B3 ) @ ( inf_inf_set_list_a @ C2 @ D2 ) ) ) ) ).
% inf_mono
thf(fact_1028_le__infI,axiom,
! [X: set_a,A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ X @ A3 )
=> ( ( ord_less_eq_set_a @ X @ B3 )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A3 @ B3 ) ) ) ) ).
% le_infI
thf(fact_1029_le__infI,axiom,
! [X: nat,A3: nat,B3: nat] :
( ( ord_less_eq_nat @ X @ A3 )
=> ( ( ord_less_eq_nat @ X @ B3 )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A3 @ B3 ) ) ) ) ).
% le_infI
thf(fact_1030_le__infI,axiom,
! [X: set_list_a,A3: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ X @ A3 )
=> ( ( ord_le8861187494160871172list_a @ X @ B3 )
=> ( ord_le8861187494160871172list_a @ X @ ( inf_inf_set_list_a @ A3 @ B3 ) ) ) ) ).
% le_infI
thf(fact_1031_le__infE,axiom,
! [X: set_a,A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A3 @ B3 ) )
=> ~ ( ( ord_less_eq_set_a @ X @ A3 )
=> ~ ( ord_less_eq_set_a @ X @ B3 ) ) ) ).
% le_infE
thf(fact_1032_le__infE,axiom,
! [X: nat,A3: nat,B3: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A3 @ B3 ) )
=> ~ ( ( ord_less_eq_nat @ X @ A3 )
=> ~ ( ord_less_eq_nat @ X @ B3 ) ) ) ).
% le_infE
thf(fact_1033_le__infE,axiom,
! [X: set_list_a,A3: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ X @ ( inf_inf_set_list_a @ A3 @ B3 ) )
=> ~ ( ( ord_le8861187494160871172list_a @ X @ A3 )
=> ~ ( ord_le8861187494160871172list_a @ X @ B3 ) ) ) ).
% le_infE
thf(fact_1034_inf__le2,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_1035_inf__le2,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_1036_inf__le2,axiom,
! [X: set_list_a,Y: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_1037_inf__le1,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_1038_inf__le1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_1039_inf__le1,axiom,
! [X: set_list_a,Y: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_1040_inf__sup__ord_I1_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_1041_inf__sup__ord_I1_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_1042_inf__sup__ord_I1_J,axiom,
! [X: set_list_a,Y: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_1043_inf__sup__ord_I2_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_1044_inf__sup__ord_I2_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_1045_inf__sup__ord_I2_J,axiom,
! [X: set_list_a,Y: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_1046_less__infI1,axiom,
! [A3: set_a,X: set_a,B3: set_a] :
( ( ord_less_set_a @ A3 @ X )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ X ) ) ).
% less_infI1
thf(fact_1047_less__infI1,axiom,
! [A3: set_list_a,X: set_list_a,B3: set_list_a] :
( ( ord_less_set_list_a @ A3 @ X )
=> ( ord_less_set_list_a @ ( inf_inf_set_list_a @ A3 @ B3 ) @ X ) ) ).
% less_infI1
thf(fact_1048_less__infI1,axiom,
! [A3: nat,X: nat,B3: nat] :
( ( ord_less_nat @ A3 @ X )
=> ( ord_less_nat @ ( inf_inf_nat @ A3 @ B3 ) @ X ) ) ).
% less_infI1
thf(fact_1049_less__infI2,axiom,
! [B3: set_a,X: set_a,A3: set_a] :
( ( ord_less_set_a @ B3 @ X )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ X ) ) ).
% less_infI2
thf(fact_1050_less__infI2,axiom,
! [B3: set_list_a,X: set_list_a,A3: set_list_a] :
( ( ord_less_set_list_a @ B3 @ X )
=> ( ord_less_set_list_a @ ( inf_inf_set_list_a @ A3 @ B3 ) @ X ) ) ).
% less_infI2
thf(fact_1051_less__infI2,axiom,
! [B3: nat,X: nat,A3: nat] :
( ( ord_less_nat @ B3 @ X )
=> ( ord_less_nat @ ( inf_inf_nat @ A3 @ B3 ) @ X ) ) ).
% less_infI2
thf(fact_1052_inf_Oabsorb3,axiom,
! [A3: set_a,B3: set_a] :
( ( ord_less_set_a @ A3 @ B3 )
=> ( ( inf_inf_set_a @ A3 @ B3 )
= A3 ) ) ).
% inf.absorb3
thf(fact_1053_inf_Oabsorb3,axiom,
! [A3: set_list_a,B3: set_list_a] :
( ( ord_less_set_list_a @ A3 @ B3 )
=> ( ( inf_inf_set_list_a @ A3 @ B3 )
= A3 ) ) ).
% inf.absorb3
thf(fact_1054_inf_Oabsorb3,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( inf_inf_nat @ A3 @ B3 )
= A3 ) ) ).
% inf.absorb3
thf(fact_1055_inf_Oabsorb4,axiom,
! [B3: set_a,A3: set_a] :
( ( ord_less_set_a @ B3 @ A3 )
=> ( ( inf_inf_set_a @ A3 @ B3 )
= B3 ) ) ).
% inf.absorb4
thf(fact_1056_inf_Oabsorb4,axiom,
! [B3: set_list_a,A3: set_list_a] :
( ( ord_less_set_list_a @ B3 @ A3 )
=> ( ( inf_inf_set_list_a @ A3 @ B3 )
= B3 ) ) ).
% inf.absorb4
thf(fact_1057_inf_Oabsorb4,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_nat @ B3 @ A3 )
=> ( ( inf_inf_nat @ A3 @ B3 )
= B3 ) ) ).
% inf.absorb4
thf(fact_1058_bot__nat__0_Oextremum__strict,axiom,
! [A3: nat] :
~ ( ord_less_nat @ A3 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1059_gr0I,axiom,
! [N4: nat] :
( ( N4 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N4 ) ) ).
% gr0I
thf(fact_1060_not__gr0,axiom,
! [N4: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N4 ) )
= ( N4 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1061_not__less0,axiom,
! [N4: nat] :
~ ( ord_less_nat @ N4 @ zero_zero_nat ) ).
% not_less0
thf(fact_1062_less__zeroE,axiom,
! [N4: nat] :
~ ( ord_less_nat @ N4 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1063_gr__implies__not0,axiom,
! [M2: nat,N4: nat] :
( ( ord_less_nat @ M2 @ N4 )
=> ( N4 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1064_infinite__descent0,axiom,
! [P2: nat > $o,N4: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ~ ( P2 @ N )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N )
& ~ ( P2 @ M5 ) ) ) )
=> ( P2 @ N4 ) ) ) ).
% infinite_descent0
thf(fact_1065_ex__least__nat__le,axiom,
! [P2: nat > $o,N4: nat] :
( ( P2 @ N4 )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N4 )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ K3 )
=> ~ ( P2 @ I5 ) )
& ( P2 @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1066_local_Oring__axioms,axiom,
ring_a_b @ r ).
% local.ring_axioms
thf(fact_1067_carrier__not__empty,axiom,
( ( partia707051561876973205xt_a_b @ r )
!= bot_bot_set_a ) ).
% carrier_not_empty
thf(fact_1068_x_Oring__axioms,axiom,
ring_l6212528067271185461t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% x.ring_axioms
thf(fact_1069_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 ) ) ) )
=> ? [N: nat] : ( bound_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ N @ F ) ) ).
% x.bound_upD
thf(fact_1070_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_1071_add_Ol__cancel,axiom,
! [C2: a,A3: a,B3: a] :
( ( ( add_a_b @ r @ C2 @ A3 )
= ( add_a_b @ r @ C2 @ B3 ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A3 = B3 ) ) ) ) ) ).
% add.l_cancel
thf(fact_1072_add_Or__cancel,axiom,
! [A3: a,C2: a,B3: a] :
( ( ( add_a_b @ r @ A3 @ C2 )
= ( add_a_b @ r @ B3 @ C2 ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A3 = B3 ) ) ) ) ) ).
% add.r_cancel
thf(fact_1073_a__assoc,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( 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 @ X @ Y ) @ Z )
= ( add_a_b @ r @ X @ ( add_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% a_assoc
thf(fact_1074_a__comm,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ Y )
= ( add_a_b @ r @ Y @ X ) ) ) ) ).
% a_comm
thf(fact_1075_a__lcomm,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ ( add_a_b @ r @ Y @ Z ) )
= ( add_a_b @ r @ Y @ ( add_a_b @ r @ X @ Z ) ) ) ) ) ) ).
% a_lcomm
thf(fact_1076_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_1077_x_Ocarrier__not__empty,axiom,
( ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
!= bot_bot_set_list_a ) ).
% x.carrier_not_empty
thf(fact_1078_add_Oinv__comm,axiom,
! [X: a,Y: a] :
( ( ( add_a_b @ r @ X @ Y )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ Y @ X )
= ( zero_a_b @ r ) ) ) ) ) ).
% add.inv_comm
thf(fact_1079_add_Ol__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( add_a_b @ r @ X2 @ X )
= ( zero_a_b @ r ) ) ) ) ).
% add.l_inv_ex
thf(fact_1080_add_Oone__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ U @ X2 )
= X2 ) )
=> ( U
= ( zero_a_b @ r ) ) ) ) ).
% add.one_unique
thf(fact_1081_add_Or__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( add_a_b @ r @ X @ X2 )
= ( zero_a_b @ r ) ) ) ) ).
% add.r_inv_ex
thf(fact_1082_local_Ominus__unique,axiom,
! [Y: a,X: a,Y3: a] :
( ( ( add_a_b @ r @ Y @ X )
= ( zero_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X @ Y3 )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( Y = Y3 ) ) ) ) ) ) ).
% local.minus_unique
thf(fact_1083_l__distr,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( 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 @ X @ Y ) @ Z )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Z ) @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% l_distr
thf(fact_1084_r__distr,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( 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 @ X @ Y ) )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ Z @ X ) @ ( mult_a_ring_ext_a_b @ r @ Z @ Y ) ) ) ) ) ) ).
% r_distr
thf(fact_1085_x_OboundD__carrier,axiom,
! [N4: nat,F: nat > list_a,M2: nat] :
( ( bound_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ N4 @ F )
=> ( ( ord_less_nat @ N4 @ M2 )
=> ( member_list_a @ ( F @ M2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.boundD_carrier
thf(fact_1086_x_Om__assoc,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ Z )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z ) ) ) ) ) ) ).
% x.m_assoc
thf(fact_1087_x_Om__comm,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X ) ) ) ) ).
% x.m_comm
thf(fact_1088_x_Om__lcomm,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z ) )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Z ) ) ) ) ) ) ).
% x.m_lcomm
thf(fact_1089_a__lcos__m__assoc,axiom,
! [M: set_a,G: a,H3: a] :
( ( ord_less_eq_set_a @ M @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ G @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ H3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_l_coset_a_b @ r @ G @ ( a_l_coset_a_b @ r @ H3 @ M ) )
= ( a_l_coset_a_b @ r @ ( add_a_b @ r @ G @ H3 ) @ M ) ) ) ) ) ).
% a_lcos_m_assoc
thf(fact_1090_x_Oup__smult__closed,axiom,
! [A3: list_a,P: nat > list_a] :
( ( member_list_a @ A3 @ ( 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 ) ) @ A3 @ ( P @ I ) )
@ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.up_smult_closed
thf(fact_1091_x_Oup__one__closed,axiom,
( member_nat_list_a
@ ^ [N3: nat] : ( if_list_a @ ( N3 = zero_zero_nat ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
@ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.up_one_closed
thf(fact_1092_x_Oinv__unique,axiom,
! [Y: list_a,X: list_a,Y3: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y3 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( Y = Y3 ) ) ) ) ) ) ).
% x.inv_unique
thf(fact_1093_x_Oone__unique,axiom,
! [U: list_a] :
( ( member_list_a @ U @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ U @ X2 )
= X2 ) )
=> ( U
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.one_unique
thf(fact_1094_x_Ocring__fieldI2,axiom,
( ( ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
!= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [A4: list_a] :
( ( member_list_a @ A4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( A4
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ? [X5: list_a] :
( ( member_list_a @ X5 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A4 @ X5 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) )
=> ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.cring_fieldI2
thf(fact_1095_local_Oadd_Oright__cancel,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ Y @ X )
= ( add_a_b @ r @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ).
% local.add.right_cancel
thf(fact_1096_a__closed,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( add_a_b @ r @ X @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% a_closed
thf(fact_1097_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_1098_add_Ol__cancel__one,axiom,
! [X: a,A3: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X @ A3 )
= X )
= ( A3
= ( zero_a_b @ r ) ) ) ) ) ).
% add.l_cancel_one
thf(fact_1099_add_Ol__cancel__one_H,axiom,
! [X: a,A3: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X
= ( add_a_b @ r @ X @ A3 ) )
= ( A3
= ( zero_a_b @ r ) ) ) ) ) ).
% add.l_cancel_one'
thf(fact_1100_add_Or__cancel__one,axiom,
! [X: a,A3: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ A3 @ X )
= X )
= ( A3
= ( zero_a_b @ r ) ) ) ) ) ).
% add.r_cancel_one
thf(fact_1101_add_Or__cancel__one_H,axiom,
! [X: a,A3: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X
= ( add_a_b @ r @ A3 @ X ) )
= ( A3
= ( zero_a_b @ r ) ) ) ) ) ).
% add.r_cancel_one'
thf(fact_1102_l__zero,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( zero_a_b @ r ) @ X )
= X ) ) ).
% l_zero
thf(fact_1103_r__zero,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ ( zero_a_b @ r ) )
= X ) ) ).
% r_zero
thf(fact_1104_x_Om__closed,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.m_closed
thf(fact_1105_x_Ol__null,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.l_null
thf(fact_1106_x_Or__null,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.r_null
thf(fact_1107_x_Oone__closed,axiom,
member_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% x.one_closed
thf(fact_1108_x_Ol__one,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X )
= X ) ) ).
% x.l_one
thf(fact_1109_x_Or__one,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= X ) ) ).
% x.r_one
thf(fact_1110_diff__commute,axiom,
! [I2: nat,J3: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J3 ) @ K2 )
= ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K2 ) @ J3 ) ) ).
% diff_commute
thf(fact_1111_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_1112_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_1113_x_Omonoid__cancelI,axiom,
( ! [A4: list_a,B4: list_a,C3: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C3 @ A4 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C3 @ B4 ) )
=> ( ( member_list_a @ A4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( A4 = B4 ) ) ) ) )
=> ( ! [A4: list_a,B4: list_a,C3: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A4 @ C3 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B4 @ C3 ) )
=> ( ( member_list_a @ A4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( A4 = B4 ) ) ) ) )
=> ( monoid4303264861975686087t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.monoid_cancelI
thf(fact_1114_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_1115_x_Oadd_Or__cancel,axiom,
! [A3: list_a,C2: list_a,B3: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A3 @ C2 )
= ( add_li7652885771158616974t_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 ) ) ) ) ) ).
% x.add.r_cancel
thf(fact_1116_x_Oadd_Om__lcomm,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Z ) ) ) ) ) ) ).
% x.add.m_lcomm
thf(fact_1117_x_Oadd_Om__comm,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X ) ) ) ) ).
% x.add.m_comm
thf(fact_1118_x_Oadd_Om__assoc,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ Z )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z ) ) ) ) ) ) ).
% x.add.m_assoc
thf(fact_1119_x_Oadd_Ol__cancel,axiom,
! [C2: list_a,A3: list_a,B3: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C2 @ A3 )
= ( add_li7652885771158616974t_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 ) ) ) ) ) ).
% x.add.l_cancel
thf(fact_1120_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_1121_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_1122_x_Ominus__unique,axiom,
! [Y: list_a,X: list_a,Y3: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y3 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( Y = Y3 ) ) ) ) ) ) ).
% x.minus_unique
thf(fact_1123_x_Oadd_Or__inv__ex,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ X2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.add.r_inv_ex
thf(fact_1124_x_Oadd_Oone__unique,axiom,
! [U: list_a] :
( ( member_list_a @ U @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ U @ X2 )
= X2 ) )
=> ( U
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.add.one_unique
thf(fact_1125_x_Oadd_Ol__inv__ex,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.add.l_inv_ex
thf(fact_1126_x_Oadd_Oinv__comm,axiom,
! [X: list_a,Y: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.add.inv_comm
thf(fact_1127_x_Or__distr,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Z @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Z @ X ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Z @ Y ) ) ) ) ) ) ).
% x.r_distr
thf(fact_1128_x_Ol__distr,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ Z )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Z ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z ) ) ) ) ) ) ).
% x.l_distr
thf(fact_1129_is__root__poly__mult__imp__is__root,axiom,
! [P: list_a,Q: list_a,X: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( polyno4133073214067823460ot_a_b @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q ) @ X )
=> ( ( polyno4133073214067823460ot_a_b @ r @ P @ X )
| ( polyno4133073214067823460ot_a_b @ r @ Q @ X ) ) ) ) ) ).
% is_root_poly_mult_imp_is_root
thf(fact_1130_x_Oadd_Oright__cancel,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ).
% x.add.right_cancel
thf(fact_1131_x_Oadd_Om__closed,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.add.m_closed
thf(fact_1132_x_Ominus__closed,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.minus_closed
thf(fact_1133_x_Or__zero,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= X ) ) ).
% x.r_zero
thf(fact_1134_x_Ol__zero,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X )
= X ) ) ).
% x.l_zero
thf(fact_1135_x_Oadd_Or__cancel__one_H,axiom,
! [X: list_a,A3: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( X
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A3 @ X ) )
= ( A3
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.add.r_cancel_one'
thf(fact_1136_x_Oadd_Or__cancel__one,axiom,
! [X: list_a,A3: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A3 @ X )
= X )
= ( A3
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.add.r_cancel_one
thf(fact_1137_x_Oadd_Ol__cancel__one_H,axiom,
! [X: list_a,A3: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( X
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ A3 ) )
= ( A3
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.add.l_cancel_one'
thf(fact_1138_x_Oadd_Ol__cancel__one,axiom,
! [X: list_a,A3: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ A3 )
= X )
= ( A3
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.add.l_cancel_one
thf(fact_1139_x_Or__right__minus__eq,axiom,
! [A3: list_a,B3: list_a] :
( ( 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 ) ) ) )
=> ( ( ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A3 @ B3 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( A3 = B3 ) ) ) ) ).
% x.r_right_minus_eq
thf(fact_1140_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1141_alg__mult__gt__zero__iff__is__root,axiom,
! [P: list_a,X: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( polyno4422430861927485590lt_a_b @ r @ P @ X ) )
= ( polyno4133073214067823460ot_a_b @ r @ P @ X ) ) ) ).
% alg_mult_gt_zero_iff_is_root
thf(fact_1142_x_Osubcring__inter,axiom,
! [I4: set_list_a,J2: set_list_a] :
( ( subcri7763218559781929323t_unit @ I4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( subcri7763218559781929323t_unit @ J2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( subcri7763218559781929323t_unit @ ( inf_inf_set_list_a @ I4 @ J2 ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.subcring_inter
thf(fact_1143_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_1144_x_OsubdomainI,axiom,
! [H: set_list_a] :
( ( subcri7763218559781929323t_unit @ H @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [H1: list_a,H2: list_a] :
( ( member_list_a @ H1 @ H )
=> ( ( member_list_a @ H2 @ H )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H1 @ H2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( H1
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
| ( H2
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) )
=> ( subdom7821232466298058046t_unit @ H @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.subdomainI
thf(fact_1145_x_Oa__lcos__m__assoc,axiom,
! [M: set_list_a,G: list_a,H3: list_a] :
( ( ord_le8861187494160871172list_a @ M @ ( 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 @ H3 @ ( 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 ) ) @ H3 @ M ) )
= ( 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 @ H3 ) @ M ) ) ) ) ) ).
% x.a_lcos_m_assoc
thf(fact_1146_x_Oa__lcos__mult__one,axiom,
! [M: set_list_a] :
( ( ord_le8861187494160871172list_a @ M @ ( 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 ) ) ) @ M )
= M ) ) ).
% x.a_lcos_mult_one
thf(fact_1147_x_Oa__l__coset__subset__G,axiom,
! [H: set_list_a,X: list_a] :
( ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ H ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.a_l_coset_subset_G
thf(fact_1148_x_Osubalgebra__in__carrier,axiom,
! [K: set_list_a,V: set_list_a] :
( ( embedd1768981623711841426t_unit @ K @ 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_1149_x_Ocarrier__is__subalgebra,axiom,
! [K: set_list_a] :
( ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( embedd1768981623711841426t_unit @ K @ ( 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_1150_x_Osubset__Idl__subset,axiom,
! [I4: set_list_a,H: set_list_a] :
( ( ord_le8861187494160871172list_a @ I4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ H @ I4 )
=> ( ord_le8861187494160871172list_a @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H ) @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I4 ) ) ) ) ).
% x.subset_Idl_subset
thf(fact_1151_x_Osubalgebra__inter,axiom,
! [K: set_list_a,V: set_list_a,V2: set_list_a] :
( ( embedd1768981623711841426t_unit @ K @ V @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd1768981623711841426t_unit @ K @ V2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( embedd1768981623711841426t_unit @ K @ ( inf_inf_set_list_a @ V @ V2 ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.subalgebra_inter
thf(fact_1152_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_1153_x_Ogenideal__one,axiom,
( ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) )
= ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.genideal_one
thf(fact_1154_x_OIdl__subset__ideal_H,axiom,
! [A3: list_a,B3: list_a] :
( ( 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 ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ A3 @ bot_bot_set_list_a ) ) @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ B3 @ bot_bot_set_list_a ) ) )
= ( member_list_a @ A3 @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ B3 @ bot_bot_set_list_a ) ) ) ) ) ) ).
% x.Idl_subset_ideal'
thf(fact_1155_x_Opoly__of__const__in__carrier,axiom,
! [S2: list_a] :
( ( member_list_a @ S2 @ ( 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 ) ) @ S2 ) @ ( 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_1156_x_Ogenideal__self_H,axiom,
! [I2: list_a] :
( ( member_list_a @ I2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ I2 @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ I2 @ bot_bot_set_list_a ) ) ) ) ).
% x.genideal_self'
thf(fact_1157_x_Ogenideal__zero,axiom,
( ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) ).
% x.genideal_zero
thf(fact_1158_x_Ozeropideal,axiom,
princi8786919440553033881t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% x.zeropideal
thf(fact_1159_x_Ocarrier__one__not__zero,axiom,
( ( ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
!= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) )
= ( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.carrier_one_not_zero
thf(fact_1160_x_Ocarrier__one__zero,axiom,
( ( ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) )
= ( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.carrier_one_zero
thf(fact_1161_x_Oone__zeroD,axiom,
( ( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) ) ).
% x.one_zeroD
thf(fact_1162_x_Oone__zeroI,axiom,
( ( ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) )
=> ( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.one_zeroI
thf(fact_1163_x_Ozeromaximalideal__eq__field,axiom,
( ( maxima6585700282301356660t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.zeromaximalideal_eq_field
thf(fact_1164_x_Ozeromaximalideal__fieldI,axiom,
( ( maxima6585700282301356660t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.zeromaximalideal_fieldI
thf(fact_1165_x_Odomain__eq__zeroprimeideal,axiom,
( ( domain6553523120543210313t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( primei6309817859076077608t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.domain_eq_zeroprimeideal
thf(fact_1166_poly__of__const__in__carrier,axiom,
! [S2: a] :
( ( member_a @ S2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_list_a @ ( poly_of_const_a_b @ r @ S2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% poly_of_const_in_carrier
thf(fact_1167_genideal__self_H,axiom,
! [I2: a] :
( ( member_a @ I2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ I2 @ ( genideal_a_b @ r @ ( insert_a @ I2 @ bot_bot_set_a ) ) ) ) ).
% genideal_self'
thf(fact_1168_genideal__zero,axiom,
( ( genideal_a_b @ r @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ).
% genideal_zero
thf(fact_1169_zeropideal,axiom,
principalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeropideal
thf(fact_1170_carrier__one__not__zero,axiom,
( ( ( partia707051561876973205xt_a_b @ r )
!= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
= ( ( one_a_ring_ext_a_b @ r )
!= ( zero_a_b @ r ) ) ) ).
% carrier_one_not_zero
thf(fact_1171_carrier__one__zero,axiom,
( ( ( partia707051561876973205xt_a_b @ r )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
= ( ( one_a_ring_ext_a_b @ r )
= ( zero_a_b @ r ) ) ) ).
% carrier_one_zero
thf(fact_1172_one__zeroD,axiom,
( ( ( one_a_ring_ext_a_b @ r )
= ( zero_a_b @ r ) )
=> ( ( partia707051561876973205xt_a_b @ r )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ).
% one_zeroD
thf(fact_1173_one__zeroI,axiom,
( ( ( partia707051561876973205xt_a_b @ r )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
=> ( ( one_a_ring_ext_a_b @ r )
= ( zero_a_b @ r ) ) ) ).
% one_zeroI
thf(fact_1174_Idl__subset__ideal_H,axiom,
! [A3: a,B3: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( genideal_a_b @ r @ ( insert_a @ A3 @ bot_bot_set_a ) ) @ ( genideal_a_b @ r @ ( insert_a @ B3 @ bot_bot_set_a ) ) )
= ( member_a @ A3 @ ( genideal_a_b @ r @ ( insert_a @ B3 @ bot_bot_set_a ) ) ) ) ) ) ).
% Idl_subset_ideal'
thf(fact_1175_genideal__one,axiom,
( ( genideal_a_b @ r @ ( insert_a @ ( one_a_ring_ext_a_b @ r ) @ bot_bot_set_a ) )
= ( partia707051561876973205xt_a_b @ r ) ) ).
% genideal_one
thf(fact_1176_cgenideal__eq__genideal,axiom,
! [I2: a] :
( ( member_a @ I2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( cgenid547466209912283029xt_a_b @ r @ I2 )
= ( genideal_a_b @ r @ ( insert_a @ I2 @ bot_bot_set_a ) ) ) ) ).
% cgenideal_eq_genideal
thf(fact_1177_x_Omaximalideal__prime,axiom,
! [I4: set_list_a] :
( ( maxima6585700282301356660t_unit @ I4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( primei6309817859076077608t_unit @ I4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.maximalideal_prime
thf(fact_1178_x_Ozeroprimeideal__domainI,axiom,
( ( primei6309817859076077608t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( domain6553523120543210313t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.zeroprimeideal_domainI
thf(fact_1179_euclidean__domainI,axiom,
! [Phi: a > nat] :
( ! [A4: a,B4: a] :
( ( member_a @ A4 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( member_a @ B4 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ? [Q3: a,R4: a] :
( ( member_a @ Q3 @ ( partia707051561876973205xt_a_b @ r ) )
& ( member_a @ R4 @ ( partia707051561876973205xt_a_b @ r ) )
& ( A4
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ B4 @ Q3 ) @ R4 ) )
& ( ( R4
= ( zero_a_b @ r ) )
| ( ord_less_nat @ ( Phi @ R4 ) @ ( Phi @ B4 ) ) ) ) ) )
=> ( ring_e8745995371659049232in_a_b @ r @ Phi ) ) ).
% euclidean_domainI
thf(fact_1180_zeromaximalideal__fieldI,axiom,
( ( maximalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r )
=> ( field_a_b @ r ) ) ).
% zeromaximalideal_fieldI
thf(fact_1181_zeromaximalideal__eq__field,axiom,
( ( maximalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r )
= ( field_a_b @ r ) ) ).
% zeromaximalideal_eq_field
thf(fact_1182_maximalideal__prime,axiom,
! [I4: set_a] :
( ( maximalideal_a_b @ I4 @ r )
=> ( primeideal_a_b @ I4 @ r ) ) ).
% maximalideal_prime
thf(fact_1183_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_1184_zeroprimeideal,axiom,
primeideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeroprimeideal
thf(fact_1185_domain__eq__zeroprimeideal,axiom,
( ( domain_a_b @ r )
= ( primeideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ) ) ).
% domain_eq_zeroprimeideal
thf(fact_1186_zeroprimeideal__domainI,axiom,
( ( primeideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r )
=> ( domain_a_b @ r ) ) ).
% zeroprimeideal_domainI
thf(fact_1187_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_1188_x_Ocgenideal__eq__genideal,axiom,
! [I2: list_a] :
( ( member_list_a @ I2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I2 )
= ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ I2 @ bot_bot_set_list_a ) ) ) ) ).
% x.cgenideal_eq_genideal
thf(fact_1189_primeideal__iff__prime,axiom,
! [P: a] :
( ( member_a @ P @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( primeideal_a_b @ ( cgenid547466209912283029xt_a_b @ r @ P ) @ r )
= ( ring_ring_prime_a_b @ r @ P ) ) ) ).
% primeideal_iff_prime
thf(fact_1190_calculation_I1_J,axiom,
! [A3: a] :
( ( member_a @ A3 @ 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 @ A3 ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% calculation(1)
thf(fact_1191_x_Ofield__intro2,axiom,
( ( ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
!= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ( member_list_a @ X2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.field_intro2
thf(fact_1192_x_OUnits__closed,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.Units_closed
thf(fact_1193_x_Oprod__unit__l,axiom,
! [A3: list_a,B3: list_a] :
( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A3 @ B3 ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A3 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( 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 @ B3 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ).
% x.prod_unit_l
thf(fact_1194_x_Oprod__unit__r,axiom,
! [A3: list_a,B3: list_a] :
( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A3 @ B3 ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B3 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( 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 @ A3 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ).
% x.prod_unit_r
thf(fact_1195_x_Ounit__factor,axiom,
! [A3: list_a,B3: list_a] :
( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A3 @ B3 ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( 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 @ A3 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.unit_factor
thf(fact_1196_x_OUnits__inv__comm,axiom,
! [X: list_a,Y: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.Units_inv_comm
thf(fact_1197_x_Oideal__eq__carrier__iff,axiom,
! [A3: list_a] :
( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A3 ) )
= ( member_list_a @ A3 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.ideal_eq_carrier_iff
thf(fact_1198_x_OUnits__l__inv__ex,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ X )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.Units_l_inv_ex
thf(fact_1199_x_OUnits__r__inv__ex,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ X2 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.Units_r_inv_ex
thf(fact_1200_x_Ocring__fieldI,axiom,
( ( ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.cring_fieldI
thf(fact_1201_x_OUnits__m__closed,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.Units_m_closed
thf(fact_1202_x_OUnits__one__closed,axiom,
member_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% x.Units_one_closed
thf(fact_1203_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 ) )
@ ^ [S3: a] : ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( var_a_b @ r ) @ ( poly_of_const_a_b @ r @ S3 ) )
@ S ) ) ).
% lagrange_basis_polynomial_aux_def
thf(fact_1204_x_OUnits__l__cancel,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Z ) )
= ( Y = Z ) ) ) ) ) ).
% x.Units_l_cancel
thf(fact_1205_x_Ofinite__ring__finite__units,axiom,
( ( finite_finite_list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( finite_finite_list_a @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.finite_ring_finite_units
thf(fact_1206_calculation_I2_J,axiom,
! [S2: a] :
( ( member_a @ S2 @ s )
=> ( ( eval_a_b @ r @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( var_a_b @ r ) @ ( poly_of_const_a_b @ r @ S2 ) ) @ x )
= ( a_minus_a_b @ r @ x @ S2 ) ) ) ).
% calculation(2)
thf(fact_1207_Units__closed,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% Units_closed
thf(fact_1208_unit__factor,axiom,
! [A3: a,B3: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A3 @ B3 ) @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ A3 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ).
% unit_factor
thf(fact_1209_prod__unit__r,axiom,
! [A3: a,B3: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A3 @ B3 ) @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ B3 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ A3 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% prod_unit_r
thf(fact_1210_prod__unit__l,axiom,
! [A3: a,B3: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A3 @ B3 ) @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A3 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ B3 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% prod_unit_l
thf(fact_1211_Units__inv__comm,axiom,
! [X: a,Y: a] :
( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ Y @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ Y @ X )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% Units_inv_comm
thf(fact_1212_ideal__eq__carrier__iff,axiom,
! [A3: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( partia707051561876973205xt_a_b @ r )
= ( cgenid547466209912283029xt_a_b @ r @ A3 ) )
= ( member_a @ A3 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% ideal_eq_carrier_iff
thf(fact_1213_ring__irreducibleE_I4_J,axiom,
! [R2: a] :
( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R2 )
=> ~ ( member_a @ R2 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% ring_irreducibleE(4)
thf(fact_1214_Units__r__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ X @ X2 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_r_inv_ex
thf(fact_1215_Units__l__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ X2 @ X )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_l_inv_ex
thf(fact_1216_eval__poly__of__const,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( poly_of_const_a_b @ r @ X ) @ Y )
= X ) ) ).
% eval_poly_of_const
thf(fact_1217_eval__var,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( var_a_b @ r ) @ X )
= X ) ) ).
% eval_var
thf(fact_1218_ring__irreducibleE_I5_J,axiom,
! [R2: a,A3: a,B3: a] :
( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R2 )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( R2
= ( mult_a_ring_ext_a_b @ r @ A3 @ B3 ) )
=> ( ( member_a @ A3 @ ( units_a_ring_ext_a_b @ r ) )
| ( member_a @ B3 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ) ) ).
% ring_irreducibleE(5)
thf(fact_1219_eval__in__carrier__2,axiom,
! [X: list_a,Y: a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( eval_a_b @ r @ X @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% eval_in_carrier_2
thf(fact_1220_x_Oring_Ozero__closed,axiom,
member_a @ ( eval_a_b @ r @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ x ) @ ( partia707051561876973205xt_a_b @ r ) ).
% x.ring.zero_closed
thf(fact_1221_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 ) ) ) )
@ ^ [S3: 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 ) ) @ S3 ) )
@ S ) ) ).
% x.lagrange_basis_polynomial_aux_def
thf(fact_1222_cring__fieldI,axiom,
( ( ( units_a_ring_ext_a_b @ r )
= ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( field_a_b @ r ) ) ).
% cring_fieldI
thf(fact_1223_field__intro2,axiom,
( ( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( member_a @ X2 @ ( units_a_ring_ext_a_b @ r ) ) )
=> ( field_a_b @ r ) ) ) ).
% field_intro2
thf(fact_1224_ring__irreducibleI,axiom,
! [R2: a] :
( ( member_a @ R2 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ~ ( member_a @ R2 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ! [A4: a,B4: a] :
( ( member_a @ A4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( R2
= ( mult_a_ring_ext_a_b @ r @ A4 @ B4 ) )
=> ( ( member_a @ A4 @ ( units_a_ring_ext_a_b @ r ) )
| ( member_a @ B4 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) )
=> ( ring_r999134135267193926le_a_b @ r @ R2 ) ) ) ) ).
% ring_irreducibleI
thf(fact_1225_Units__m__closed,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ Y @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_m_closed
thf(fact_1226_Units__one__closed,axiom,
member_a @ ( one_a_ring_ext_a_b @ r ) @ ( units_a_ring_ext_a_b @ r ) ).
% Units_one_closed
thf(fact_1227_Units__l__cancel,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( mult_a_ring_ext_a_b @ r @ X @ Z ) )
= ( Y = Z ) ) ) ) ) ).
% Units_l_cancel
thf(fact_1228_finite__ring__finite__units,axiom,
( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) )
=> ( finite_finite_a @ ( units_a_ring_ext_a_b @ r ) ) ) ).
% finite_ring_finite_units
thf(fact_1229_x_Oring_Ohomh,axiom,
( member_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ ( ring_h2895973938487309444it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r ) ) ).
% x.ring.homh
thf(fact_1230_x_Oring_Ohom__closed,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_a @ ( eval_a_b @ r @ X @ x ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% x.ring.hom_closed
thf(fact_1231_x_Oring_Ohom__zero,axiom,
( ( eval_a_b @ r @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ x )
= ( zero_a_b @ r ) ) ).
% x.ring.hom_zero
thf(fact_1232_x_Oring_Ohom__one,axiom,
( ( eval_a_b @ r @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ x )
= ( one_a_ring_ext_a_b @ r ) ) ).
% x.ring.hom_one
thf(fact_1233_x_Oring_Ohom__add,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( eval_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ x )
= ( add_a_b @ r @ ( eval_a_b @ r @ X @ x ) @ ( eval_a_b @ r @ Y @ x ) ) ) ) ) ).
% x.ring.hom_add
thf(fact_1234_x_Oring_Ohom__mult,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( eval_a_b @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ x )
= ( mult_a_ring_ext_a_b @ r @ ( eval_a_b @ r @ X @ x ) @ ( eval_a_b @ r @ Y @ x ) ) ) ) ) ).
% x.ring.hom_mult
thf(fact_1235_x_Ohom__sub,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( eval_a_b @ r @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ x )
= ( a_minus_a_b @ r @ ( eval_a_b @ r @ X @ x ) @ ( eval_a_b @ r @ Y @ x ) ) ) ) ) ).
% x.hom_sub
thf(fact_1236_x_Oring__hom__cring__axioms,axiom,
( ring_h1547129875642963619it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) ) ).
% x.ring_hom_cring_axioms
thf(fact_1237_x_Oring_Ois__abelian__group__hom,axiom,
( abelia8217020544048703197it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) ) ).
% x.ring.is_abelian_group_hom
thf(fact_1238_x_Oeval__poly__of__const,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) @ Y )
= X ) ) ).
% x.eval_poly_of_const
thf(fact_1239_x_Oeval__var,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X )
= X ) ) ).
% x.eval_var
thf(fact_1240_x_Oeval__in__carrier__2,axiom,
! [X: list_list_a,Y: list_a] :
( ( member_list_list_a @ X @ ( 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_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.eval_in_carrier_2
thf(fact_1241_x_Oring_Oimg__is__subalgebra,axiom,
! [K: set_list_a,V: set_list_a] :
( ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( embedd1768981623711841426t_unit @ K @ V @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( embedd9027525575939734154ra_a_b
@ ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ K )
@ ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ V )
@ r ) ) ) ).
% x.ring.img_is_subalgebra
thf(fact_1242_x_Oring_Onon__trivial__field__hom__imp__inj,axiom,
( ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
!= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
=> ( inj_on_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.ring.non_trivial_field_hom_imp_inj
thf(fact_1243_x_Oring_Otrivial__hom__iff,axiom,
( ( ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
= ( ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) )
= ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.ring.trivial_hom_iff
thf(fact_1244_add_Osurj__const__mult,axiom,
! [A3: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( image_a_a @ ( add_a_b @ r @ A3 ) @ ( partia707051561876973205xt_a_b @ r ) )
= ( partia707051561876973205xt_a_b @ r ) ) ) ).
% add.surj_const_mult
thf(fact_1245_x_Oadd_Osurj__const__mult,axiom,
! [A3: list_a] :
( ( member_list_a @ A3 @ ( 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 ) ) @ A3 ) @ ( 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_1246_x_Oring_Oinj__on__domain,axiom,
( ( inj_on_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( domain_a_b @ r )
=> ( domain6553523120543210313t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.ring.inj_on_domain
thf(fact_1247_x_Oring_Oinj__iff__trivial__ker,axiom,
( ( inj_on_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) ) ).
% x.ring.inj_iff_trivial_ker
thf(fact_1248_x_Oring_Otrivial__ker__imp__inj,axiom,
( ( ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) )
=> ( inj_on_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.ring.trivial_ker_imp_inj
thf(fact_1249_x_Oring_OA__FactGroup__nonempty,axiom,
! [X3: set_list_a] :
( ( member_set_list_a @ X3
@ ( partia5178357399839081912t_unit
@ ( a_Fact452226231247776317t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) ) ) ) )
=> ( X3 != bot_bot_set_list_a ) ) ).
% x.ring.A_FactGroup_nonempty
thf(fact_1250_add_Oinj__on__multc,axiom,
! [C2: a] :
( ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( inj_on_a_a
@ ^ [X4: a] : ( add_a_b @ r @ X4 @ C2 )
@ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% add.inj_on_multc
thf(fact_1251_add_Oinj__on__cmult,axiom,
! [C2: a] :
( ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( inj_on_a_a @ ( add_a_b @ r @ C2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% add.inj_on_cmult
thf(fact_1252_add_Oinj__on__g,axiom,
! [H: set_a,A3: a] :
( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( inj_on_a_a
@ ^ [Y5: a] : ( add_a_b @ r @ Y5 @ A3 )
@ H ) ) ) ).
% add.inj_on_g
thf(fact_1253_x_Oadd_Oinj__on__multc,axiom,
! [C2: list_a] :
( ( member_list_a @ C2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( inj_on_list_a_list_a
@ ^ [X4: list_a] : ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X4 @ C2 )
@ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.add.inj_on_multc
thf(fact_1254_x_Oadd_Oinj__on__cmult,axiom,
! [C2: list_a] :
( ( member_list_a @ C2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( inj_on_list_a_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.add.inj_on_cmult
thf(fact_1255_x_Oadd_Oinj__on__g,axiom,
! [H: set_list_a,A3: list_a] :
( ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( inj_on_list_a_list_a
@ ^ [Y5: list_a] : ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y5 @ A3 )
@ H ) ) ) ).
% x.add.inj_on_g
thf(fact_1256_inj__on__diff__nat,axiom,
! [N5: set_nat,K2: nat] :
( ! [N: nat] :
( ( member_nat @ N @ N5 )
=> ( ord_less_eq_nat @ K2 @ N ) )
=> ( inj_on_nat_nat
@ ^ [N3: nat] : ( minus_minus_nat @ N3 @ K2 )
@ N5 ) ) ).
% inj_on_diff_nat
thf(fact_1257_x_Oring_OA__FactGroup__onto,axiom,
( ( ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( partia707051561876973205xt_a_b @ r ) )
=> ( ( image_set_list_a_a
@ ^ [X6: set_list_a] :
( the_elem_a
@ ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ X6 ) )
@ ( partia5178357399839081912t_unit
@ ( a_Fact452226231247776317t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) ) ) ) )
= ( partia707051561876973205xt_a_b @ r ) ) ) ).
% x.ring.A_FactGroup_onto
thf(fact_1258_x_Oring_OA__FactGroup__inj__on,axiom,
( inj_on_set_list_a_a
@ ^ [X6: set_list_a] :
( the_elem_a
@ ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ X6 ) )
@ ( partia5178357399839081912t_unit
@ ( a_Fact452226231247776317t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) ) ) ) ) ).
% x.ring.A_FactGroup_inj_on
thf(fact_1259_x_Oring_OFactGroup__the__elem__mem,axiom,
! [X3: set_list_a] :
( ( member_set_list_a @ X3
@ ( partia5178357399839081912t_unit
@ ( a_Fact452226231247776317t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) ) ) ) )
=> ( member_a
@ ( the_elem_a
@ ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ X3 ) )
@ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% x.ring.FactGroup_the_elem_mem
thf(fact_1260_x_Oring_Othe__elem__surj,axiom,
( ( image_set_list_a_a
@ ^ [X6: set_list_a] :
( the_elem_a
@ ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ X6 ) )
@ ( partia141011252114345353t_unit
@ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) ) ) ) )
= ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.ring.the_elem_surj
thf(fact_1261_x_Oring_Oadditive__subgroup__a__kernel,axiom,
( additi4714453376129182166t_unit
@ ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) )
@ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% x.ring.additive_subgroup_a_kernel
thf(fact_1262_x_Oring_Othe__elem__hom,axiom,
( member_set_list_a_a
@ ^ [X6: set_list_a] :
( the_elem_a
@ ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ X6 ) )
@ ( ring_h8906680420194085028it_a_b
@ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) ) )
@ r ) ) ).
% x.ring.the_elem_hom
thf(fact_1263_x_Oring_OFactRing__iso__set,axiom,
( ( ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( partia707051561876973205xt_a_b @ r ) )
=> ( member_set_list_a_a
@ ^ [X6: set_list_a] :
( the_elem_a
@ ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ X6 ) )
@ ( ring_i8122894263081988538it_a_b
@ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) ) )
@ r ) ) ) ).
% x.ring.FactRing_iso_set
thf(fact_1264_x_Oring_Othe__elem__inj,axiom,
! [X3: set_list_a,Y2: set_list_a] :
( ( member_set_list_a @ X3
@ ( partia141011252114345353t_unit
@ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) ) ) ) )
=> ( ( member_set_list_a @ Y2
@ ( partia141011252114345353t_unit
@ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) ) ) ) )
=> ( ( ( the_elem_a
@ ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ X3 ) )
= ( the_elem_a
@ ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ Y2 ) ) )
=> ( X3 = Y2 ) ) ) ) ).
% x.ring.the_elem_inj
thf(fact_1265_x_Oring_Othe__elem__wf,axiom,
! [X3: set_list_a] :
( ( member_set_list_a @ X3
@ ( partia141011252114345353t_unit
@ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) ) ) ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ X3 )
= ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ).
% x.ring.the_elem_wf
thf(fact_1266_x_Oring_Othe__elem__wf_H,axiom,
! [X3: set_list_a] :
( ( member_set_list_a @ X3
@ ( partia141011252114345353t_unit
@ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) ) ) ) )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ X3 )
= ( insert_a @ ( eval_a_b @ r @ X2 @ x ) @ bot_bot_set_a ) ) ) ) ).
% x.ring.the_elem_wf'
thf(fact_1267_x_OFactRing__zeroideal_I2_J,axiom,
is_rin2993610189962786360t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) ).
% x.FactRing_zeroideal(2)
thf(fact_1268_x_OFactRing__zeroideal_I1_J,axiom,
is_rin4843644836746533432t_unit @ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% x.FactRing_zeroideal(1)
thf(fact_1269_x_Oring_OFactRing__iso,axiom,
( ( ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( partia707051561876973205xt_a_b @ r ) )
=> ( is_rin5597148638330396976it_a_b
@ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) ) )
@ r ) ) ).
% x.ring.FactRing_iso
thf(fact_1270_FactRing__zeroideal_I2_J,axiom,
is_rin9099215527551818550t_unit @ r @ ( factRing_a_b @ r @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ).
% FactRing_zeroideal(2)
thf(fact_1271_FactRing__zeroideal_I1_J,axiom,
is_rin6001486760346555702it_a_b @ ( factRing_a_b @ r @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) @ r ).
% FactRing_zeroideal(1)
% Helper facts (5)
thf(help_If_2_1_If_001tf__a_T,axiom,
! [X: a,Y: a] :
( ( if_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001tf__a_T,axiom,
! [X: a,Y: a] :
( ( if_a @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X: list_a,Y: list_a] :
( ( if_list_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X: list_a,Y: list_a] :
( ( if_list_a @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( member_a_a @ ( a_minus_a_b @ r @ x )
@ ( pi_a_a @ s
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) ) ).
%------------------------------------------------------------------------------