TPTP Problem File: SLH0899^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Interpolation_Polynomials_HOL_Algebra/0000_Bounded_Degree_Polynomials/prob_00157_006401__17087584_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1390 ( 496 unt; 107 typ; 0 def)
% Number of atoms : 3772 (1341 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 13136 ( 271 ~; 54 |; 183 &;10701 @)
% ( 0 <=>;1927 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 7 avg)
% Number of types : 12 ( 11 usr)
% Number of type conns : 273 ( 273 >; 0 *; 0 +; 0 <<)
% Number of symbols : 99 ( 96 usr; 14 con; 0-4 aty)
% Number of variables : 3338 ( 78 ^;3167 !; 93 ?;3338 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:36:16.332
%------------------------------------------------------------------------------
% Could-be-implicit typings (11)
thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_Itf__a_Mt__Group__Omonoid__Omonoid____ext_Itf__a_Mt__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J_J_J,type,
partia2175431115845679010xt_a_b: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
set_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
set_a_a: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (96)
thf(sy_c_AbelCoset_OA__RCOSETS_001tf__a_001tf__b,type,
a_RCOSETS_a_b: partia2175431115845679010xt_a_b > set_a > set_set_a ).
thf(sy_c_AbelCoset_Oa__l__coset_001tf__a_001tf__b,type,
a_l_coset_a_b: partia2175431115845679010xt_a_b > a > set_a > set_a ).
thf(sy_c_AbelCoset_Oa__r__coset_001tf__a_001tf__b,type,
a_r_coset_a_b: partia2175431115845679010xt_a_b > set_a > a > set_a ).
thf(sy_c_AbelCoset_Oadditive__subgroup_001tf__a_001tf__b,type,
additi2834746164131130830up_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_AbelCoset_Oset__add_001tf__a_001tf__b,type,
set_add_a_b: partia2175431115845679010xt_a_b > set_a > set_a > set_a ).
thf(sy_c_Bounded__Degree__Polynomials_Obounded__degree__polynomials_001tf__a_001tf__b,type,
bounde2262800523058855161ls_a_b: partia2175431115845679010xt_a_b > nat > set_list_a ).
thf(sy_c_Congruence_Opartial__object_Ocarrier_001tf__a_001t__Group__Omonoid__Omonoid____ext_Itf__a_Mt__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J_J,type,
partia707051561876973205xt_a_b: partia2175431115845679010xt_a_b > set_a ).
thf(sy_c_Coset_Oorder_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
order_a_ring_ext_a_b: partia2175431115845679010xt_a_b > nat ).
thf(sy_c_Divisibility_Omonoid__cancel_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
monoid5798828371819920185xt_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Divisibility_Oproperfactor_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
proper19828929941537682xt_a_b: partia2175431115845679010xt_a_b > a > a > $o ).
thf(sy_c_Embedded__Algebras_Oring_Odimension_001tf__a_001tf__b,type,
embedd2795209813406577254on_a_b: partia2175431115845679010xt_a_b > nat > set_a > set_a > $o ).
thf(sy_c_Embedded__Algebras_Oring_Ofinite__dimension_001tf__a_001tf__b,type,
embedd8708762675212832759on_a_b: partia2175431115845679010xt_a_b > set_a > set_a > $o ).
thf(sy_c_Embedded__Algebras_Oring_Oline__extension_001tf__a_001tf__b,type,
embedd971793762689825387on_a_b: partia2175431115845679010xt_a_b > set_a > a > set_a > set_a ).
thf(sy_c_Embedded__Algebras_Osubalgebra_001tf__a_001tf__b,type,
embedd9027525575939734154ra_a_b: set_a > set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Finite__Set_Ocard_001t__List__Olist_Itf__a_J,type,
finite_card_list_a: set_list_a > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_Itf__a_J,type,
finite_card_set_a: set_set_a > nat ).
thf(sy_c_Finite__Set_Ocard_001tf__a,type,
finite_card_a: set_a > nat ).
thf(sy_c_Finite__Set_Ofinite_001t__Int__Oint,type,
finite_finite_int: set_int > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_Itf__a_J,type,
finite_finite_list_a: set_list_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__a_J,type,
finite_finite_set_a: set_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_Group_Ogroup_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
group_a_ring_ext_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Group_Om__inv_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
m_inv_a_ring_ext_a_b: partia2175431115845679010xt_a_b > a > a ).
thf(sy_c_Group_Omonoid_Omult_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
mult_a_ring_ext_a_b: partia2175431115845679010xt_a_b > a > a > a ).
thf(sy_c_Group_Omonoid_Oone_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
one_a_ring_ext_a_b: partia2175431115845679010xt_a_b > a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
minus_646659088055828811list_a: set_list_a > set_list_a > set_list_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
minus_5736297505244876581_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
uminus7925729386456332763list_a: set_list_a > set_list_a ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
uminus6103902357914783669_set_a: set_set_a > set_set_a ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_Itf__a_J,type,
uminus_uminus_set_a: set_a > set_a ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Ideal_Ocgenideal_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
cgenid547466209912283029xt_a_b: partia2175431115845679010xt_a_b > a > set_a ).
thf(sy_c_Ideal_Ogenideal_001tf__a_001tf__b,type,
genideal_a_b: partia2175431115845679010xt_a_b > set_a > set_a ).
thf(sy_c_Ideal_Oprincipalideal_001tf__a_001tf__b,type,
principalideal_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_If_001t__Int__Oint,type,
if_int: $o > int > int > int ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
inf_inf_set_a: set_a > set_a > set_a ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__List__Olist_Itf__a_J_M_Eo_J,type,
bot_bot_list_a_o: list_a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
bot_bot_a_o: a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
bot_bot_set_int: set_int ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
bot_bot_set_list_a: set_list_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
bot_bot_set_set_a: set_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
ord_le8861187494160871172list_a: set_list_a > set_list_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_le3724670747650509150_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
power_power_int: int > nat > int ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_QuotRing_Oring__iso_001tf__a_001tf__b_001tf__a_001tf__b,type,
ring_iso_a_b_a_b: partia2175431115845679010xt_a_b > partia2175431115845679010xt_a_b > set_a_a ).
thf(sy_c_Ring_Oa__inv_001tf__a_001tf__b,type,
a_inv_a_b: partia2175431115845679010xt_a_b > a > a ).
thf(sy_c_Ring_Oa__minus_001tf__a_001tf__b,type,
a_minus_a_b: partia2175431115845679010xt_a_b > a > a > a ).
thf(sy_c_Ring_Oadd__pow_001tf__a_001tf__b_001t__Int__Oint,type,
add_pow_a_b_int: partia2175431115845679010xt_a_b > int > a > a ).
thf(sy_c_Ring_Oadd__pow_001tf__a_001tf__b_001t__Nat__Onat,type,
add_pow_a_b_nat: partia2175431115845679010xt_a_b > nat > a > a ).
thf(sy_c_Ring_Oring_001tf__a_001tf__b,type,
ring_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring_Oring_Oadd_001tf__a_001tf__b,type,
add_a_b: partia2175431115845679010xt_a_b > a > a > a ).
thf(sy_c_Ring_Oring_Ozero_001tf__a_001tf__b,type,
zero_a_b: partia2175431115845679010xt_a_b > a ).
thf(sy_c_Ring_Oring__hom_001tf__a_001tf__b_001tf__a_001tf__b,type,
ring_hom_a_b_a_b: partia2175431115845679010xt_a_b > partia2175431115845679010xt_a_b > set_a_a ).
thf(sy_c_Ring_Osemiring_001tf__a_001tf__b,type,
semiring_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring_Osemiring__axioms_001tf__a_001tf__b,type,
semiring_axioms_a_b: partia2175431115845679010xt_a_b > $o ).
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__Set__Oset_Itf__a_J,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oinsert_001t__List__Olist_Itf__a_J,type,
insert_list_a: list_a > set_list_a > set_list_a ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_Itf__a_J,type,
insert_set_a: set_a > set_set_a > set_set_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Subrings_Osubcring_001tf__a_001tf__b,type,
subcring_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Subrings_Osubdomain_001tf__a_001tf__b,type,
subdomain_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Subrings_Osubfield_001tf__a_001tf__b,type,
subfield_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Subrings_Osubring_001tf__a_001tf__b,type,
subring_a_b: set_a > partia2175431115845679010xt_a_b > $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__Int__Oint,type,
member_int: int > set_int > $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_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_R,type,
r: partia2175431115845679010xt_a_b ).
thf(sy_v_n,type,
n: nat ).
% Relevant facts (1277)
thf(fact_0_local_Oring__axioms,axiom,
ring_a_b @ r ).
% local.ring_axioms
thf(fact_1_onepideal,axiom,
principalideal_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% onepideal
thf(fact_2_cgenideal__self,axiom,
! [I: a] :
( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ I @ ( cgenid547466209912283029xt_a_b @ r @ I ) ) ) ).
% cgenideal_self
thf(fact_3_non__empty__bounded__degree__polynomials,axiom,
! [K: nat] :
( ( bounde2262800523058855161ls_a_b @ r @ K )
!= bot_bot_set_list_a ) ).
% non_empty_bounded_degree_polynomials
thf(fact_4_carrier__not__empty,axiom,
( ( partia707051561876973205xt_a_b @ r )
!= bot_bot_set_a ) ).
% carrier_not_empty
thf(fact_5_local_Osemiring__axioms,axiom,
semiring_a_b @ r ).
% local.semiring_axioms
thf(fact_6_add_Ol__cancel,axiom,
! [C: a,A: a,B: a] :
( ( ( add_a_b @ r @ C @ A )
= ( add_a_b @ r @ C @ B ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A = B ) ) ) ) ) ).
% add.l_cancel
thf(fact_7_add_Or__cancel,axiom,
! [A: a,C: a,B: a] :
( ( ( add_a_b @ r @ A @ C )
= ( add_a_b @ r @ B @ C ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A = B ) ) ) ) ) ).
% add.r_cancel
thf(fact_8_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_9_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_10_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_11_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_12_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_13_ring_Onon__empty__bounded__degree__polynomials,axiom,
! [R: partia2175431115845679010xt_a_b,K: nat] :
( ( ring_a_b @ R )
=> ( ( bounde2262800523058855161ls_a_b @ R @ K )
!= bot_bot_set_list_a ) ) ).
% ring.non_empty_bounded_degree_polynomials
thf(fact_14_ring_Oonepideal,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( principalideal_a_b @ ( partia707051561876973205xt_a_b @ R ) @ R ) ) ).
% ring.onepideal
thf(fact_15_add_Oint__pow__mult__distrib,axiom,
! [X: a,Y: a,I: int] :
( ( ( add_a_b @ r @ X @ Y )
= ( add_a_b @ r @ Y @ X ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_int @ r @ I @ ( add_a_b @ r @ X @ Y ) )
= ( add_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X ) @ ( add_pow_a_b_int @ r @ I @ Y ) ) ) ) ) ) ).
% add.int_pow_mult_distrib
thf(fact_16_add_Oint__pow__distrib,axiom,
! [X: a,Y: a,I: int] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_int @ r @ I @ ( add_a_b @ r @ X @ Y ) )
= ( add_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X ) @ ( add_pow_a_b_int @ r @ I @ Y ) ) ) ) ) ).
% add.int_pow_distrib
thf(fact_17_ring__iso__memE_I3_J,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( member_a_a @ H @ ( ring_iso_a_b_a_b @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H @ ( add_a_b @ R @ X @ Y ) )
= ( add_a_b @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_18_semiring_Osemiring__simprules_I12_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( semiring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(12)
thf(fact_19_semiring_Osemiring__simprules_I7_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( semiring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ).
% semiring.semiring_simprules(7)
thf(fact_20_semiring_Osemiring__simprules_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( semiring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(5)
thf(fact_21_semiring_Osemiring__simprules_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( semiring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ).
% semiring.semiring_simprules(1)
thf(fact_22_ring_Ocgenideal__self,axiom,
! [R: partia2175431115845679010xt_a_b,I: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ I @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ I @ ( cgenid547466209912283029xt_a_b @ R @ I ) ) ) ) ).
% ring.cgenideal_self
thf(fact_23_ring__hom__add,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( member_a_a @ H @ ( ring_hom_a_b_a_b @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H @ ( add_a_b @ R @ X @ Y ) )
= ( add_a_b @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_24_add_Oint__pow__closed,axiom,
! [X: a,I: int] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( add_pow_a_b_int @ r @ I @ X ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% add.int_pow_closed
thf(fact_25_principalideal_Ois__principalideal,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b] :
( ( principalideal_a_b @ I2 @ R )
=> ( principalideal_a_b @ I2 @ R ) ) ).
% principalideal.is_principalideal
thf(fact_26_ring__hom__closed,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X: a] :
( ( member_a_a @ H @ ( ring_hom_a_b_a_b @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( H @ X ) @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_27_ring__iso__memE_I1_J,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X: a] :
( ( member_a_a @ H @ ( ring_iso_a_b_a_b @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( H @ X ) @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_28_ring_Oring__simprules_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ).
% ring.ring_simprules(1)
thf(fact_29_ring_Oring__simprules_I7_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( ring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ) ) ).
% ring.ring_simprules(7)
thf(fact_30_ring_Oring__simprules_I10_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ).
% ring.ring_simprules(10)
thf(fact_31_ring_Oring__simprules_I22_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( ring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ) ) ).
% ring.ring_simprules(22)
thf(fact_32_add_Oint__pow__mult,axiom,
! [X: a,I: int,J: int] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_int @ r @ ( plus_plus_int @ I @ J ) @ X )
= ( add_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X ) @ ( add_pow_a_b_int @ r @ J @ X ) ) ) ) ).
% add.int_pow_mult
thf(fact_33_add_Oint__pow__pow,axiom,
! [X: a,M: int,N: int] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_int @ r @ M @ ( add_pow_a_b_int @ r @ N @ X ) )
= ( add_pow_a_b_int @ r @ ( times_times_int @ N @ M ) @ X ) ) ) ).
% add.int_pow_pow
thf(fact_34_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_35_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_36_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_37_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_38_local_Ominus__unique,axiom,
! [Y: a,X: a,Y2: a] :
( ( ( add_a_b @ r @ Y @ X )
= ( zero_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X @ Y2 )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% local.minus_unique
thf(fact_39_add__pow__ldistr__int,axiom,
! [A: a,B: a,K: int] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( add_pow_a_b_int @ r @ K @ A ) @ B )
= ( add_pow_a_b_int @ r @ K @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ).
% add_pow_ldistr_int
thf(fact_40_add__pow__rdistr__int,axiom,
! [A: a,B: a,K: int] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ A @ ( add_pow_a_b_int @ r @ K @ B ) )
= ( add_pow_a_b_int @ r @ K @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ).
% add_pow_rdistr_int
thf(fact_41_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_42_mem__Collect__eq,axiom,
! [A: set_a,P: set_a > $o] :
( ( member_set_a @ A @ ( collect_set_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_43_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_44_Collect__mem__eq,axiom,
! [A2: set_set_a] :
( ( collect_set_a
@ ^ [X3: set_a] : ( member_set_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_45_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_46_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_47_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_48_zero__closed,axiom,
member_a @ ( zero_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% zero_closed
thf(fact_49_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_50_add_Oint__pow__one,axiom,
! [Z: int] :
( ( add_pow_a_b_int @ r @ Z @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ).
% add.int_pow_one
thf(fact_51_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_52_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_53_add_Or__cancel__one_H,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X
= ( add_a_b @ r @ A @ X ) )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.r_cancel_one'
thf(fact_54_add_Or__cancel__one,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ A @ X )
= X )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.r_cancel_one
thf(fact_55_add_Ol__cancel__one_H,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X
= ( add_a_b @ r @ X @ A ) )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.l_cancel_one'
thf(fact_56_add_Ol__cancel__one,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X @ A )
= X )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.l_cancel_one
thf(fact_57_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_58_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_59_ring_Oring__simprules_I25_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ X @ ( zero_a_b @ R ) )
= ( zero_a_b @ R ) ) ) ) ).
% ring.ring_simprules(25)
thf(fact_60_ring_Oring__simprules_I24_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( zero_a_b @ R ) @ X )
= ( zero_a_b @ R ) ) ) ) ).
% ring.ring_simprules(24)
thf(fact_61_semiring_Ol__null,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( zero_a_b @ R ) @ X )
= ( zero_a_b @ R ) ) ) ) ).
% semiring.l_null
thf(fact_62_semiring_Or__null,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ X @ ( zero_a_b @ R ) )
= ( zero_a_b @ R ) ) ) ) ).
% semiring.r_null
thf(fact_63_ring_Oring__simprules_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% ring.ring_simprules(2)
thf(fact_64_ring_Oring__simprules_I11_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( ring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ) ) ).
% ring.ring_simprules(11)
thf(fact_65_ring_Oring__simprules_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ).
% ring.ring_simprules(5)
thf(fact_66_ring__hom__mult,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( member_a_a @ H @ ( ring_hom_a_b_a_b @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) )
= ( mult_a_ring_ext_a_b @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_hom_mult
thf(fact_67_semiring_Osemiring__simprules_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_68_ring__hom__zero,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b] :
( ( member_a_a @ H @ ( ring_hom_a_b_a_b @ R @ S ) )
=> ( ( ring_a_b @ R )
=> ( ( ring_a_b @ S )
=> ( ( H @ ( zero_a_b @ R ) )
= ( zero_a_b @ S ) ) ) ) ) ).
% ring_hom_zero
thf(fact_69_semiring_Osemiring__simprules_I8_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( semiring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(8)
thf(fact_70_semiring_Osemiring__simprules_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( semiring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ).
% semiring.semiring_simprules(3)
thf(fact_71_ring__iso__memE_I2_J,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( member_a_a @ H @ ( ring_iso_a_b_a_b @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) )
= ( mult_a_ring_ext_a_b @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_72_ring_Oring__simprules_I15_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ ( zero_a_b @ R ) )
= X ) ) ) ).
% ring.ring_simprules(15)
thf(fact_73_ring_Oring__simprules_I8_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X )
= X ) ) ) ).
% ring.ring_simprules(8)
thf(fact_74_ring_Oring__simprules_I23_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( ring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ) ) ).
% ring.ring_simprules(23)
thf(fact_75_ring_Oring__simprules_I13_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( ring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ) ) ).
% ring.ring_simprules(13)
thf(fact_76_ring_Oadd__pow__ldistr__int,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a,K: int] :
( ( ring_a_b @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( add_pow_a_b_int @ R @ K @ A ) @ B )
= ( add_pow_a_b_int @ R @ K @ ( mult_a_ring_ext_a_b @ R @ A @ B ) ) ) ) ) ) ).
% ring.add_pow_ldistr_int
thf(fact_77_ring_Oadd__pow__rdistr__int,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a,K: int] :
( ( ring_a_b @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ A @ ( add_pow_a_b_int @ R @ K @ B ) )
= ( add_pow_a_b_int @ R @ K @ ( mult_a_ring_ext_a_b @ R @ A @ B ) ) ) ) ) ) ).
% ring.add_pow_rdistr_int
thf(fact_78_semiring_Osemiring__simprules_I11_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ ( zero_a_b @ R ) )
= X ) ) ) ).
% semiring.semiring_simprules(11)
thf(fact_79_semiring_Osemiring__simprules_I6_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(6)
thf(fact_80_semiring_Ol__distr,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( semiring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ) ) ).
% semiring.l_distr
thf(fact_81_semiring_Or__distr,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( semiring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ) ) ).
% semiring.r_distr
thf(fact_82_monoid__cancelI,axiom,
( ! [A3: a,B2: a,C2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ C2 @ A3 )
= ( mult_a_ring_ext_a_b @ r @ C2 @ B2 ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A3 = B2 ) ) ) ) )
=> ( ! [A3: a,B2: a,C2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A3 @ C2 )
= ( mult_a_ring_ext_a_b @ r @ B2 @ C2 ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A3 = B2 ) ) ) ) )
=> ( monoid5798828371819920185xt_a_b @ r ) ) ) ).
% monoid_cancelI
thf(fact_83_zeropideal,axiom,
principalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeropideal
thf(fact_84_line__extension__mem__iff,axiom,
! [U: a,K2: set_a,A: a,E: set_a] :
( ( member_a @ U @ ( embedd971793762689825387on_a_b @ r @ K2 @ A @ E ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ K2 )
& ? [Y3: a] :
( ( member_a @ Y3 @ E )
& ( U
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X3 @ A ) @ Y3 ) ) ) ) ) ) ).
% line_extension_mem_iff
thf(fact_85_properfactor__prod__r,axiom,
! [A: a,B: a,C: a] :
( ( proper19828929941537682xt_a_b @ r @ A @ B )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( proper19828929941537682xt_a_b @ r @ A @ ( mult_a_ring_ext_a_b @ r @ B @ C ) ) ) ) ) ) ).
% properfactor_prod_r
thf(fact_86_r__neg,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ ( a_inv_a_b @ r @ X ) )
= ( zero_a_b @ r ) ) ) ).
% r_neg
thf(fact_87_minus__equality,axiom,
! [Y: a,X: a] :
( ( ( add_a_b @ r @ Y @ X )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ X )
= Y ) ) ) ) ).
% minus_equality
thf(fact_88_l__neg,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ X )
= ( zero_a_b @ r ) ) ) ).
% l_neg
thf(fact_89_semiring__axioms_Ointro,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ! [X2: a,Y4: a,Z2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( add_a_b @ R @ X2 @ Y4 ) @ Z2 )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X2 @ Z2 ) @ ( mult_a_ring_ext_a_b @ R @ Y4 @ Z2 ) ) ) ) ) )
=> ( ! [X2: a,Y4: a,Z2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ Z2 @ ( add_a_b @ R @ X2 @ Y4 ) )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ Z2 @ X2 ) @ ( mult_a_ring_ext_a_b @ R @ Z2 @ Y4 ) ) ) ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( zero_a_b @ R ) @ X2 )
= ( zero_a_b @ R ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ X2 @ ( zero_a_b @ R ) )
= ( zero_a_b @ R ) ) )
=> ( semiring_axioms_a_b @ R ) ) ) ) ) ).
% semiring_axioms.intro
thf(fact_90_semiring__axioms__def,axiom,
( semiring_axioms_a_b
= ( ^ [R2: partia2175431115845679010xt_a_b] :
( ! [X3: a,Y3: a,Z3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Z3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( mult_a_ring_ext_a_b @ R2 @ ( add_a_b @ R2 @ X3 @ Y3 ) @ Z3 )
= ( add_a_b @ R2 @ ( mult_a_ring_ext_a_b @ R2 @ X3 @ Z3 ) @ ( mult_a_ring_ext_a_b @ R2 @ Y3 @ Z3 ) ) ) ) ) )
& ! [X3: a,Y3: a,Z3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Z3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( mult_a_ring_ext_a_b @ R2 @ Z3 @ ( add_a_b @ R2 @ X3 @ Y3 ) )
= ( add_a_b @ R2 @ ( mult_a_ring_ext_a_b @ R2 @ Z3 @ X3 ) @ ( mult_a_ring_ext_a_b @ R2 @ Z3 @ Y3 ) ) ) ) ) )
& ! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( mult_a_ring_ext_a_b @ R2 @ ( zero_a_b @ R2 ) @ X3 )
= ( zero_a_b @ R2 ) ) )
& ! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( mult_a_ring_ext_a_b @ R2 @ X3 @ ( zero_a_b @ R2 ) )
= ( zero_a_b @ R2 ) ) ) ) ) ) ).
% semiring_axioms_def
thf(fact_91_add__pow__rdistr,axiom,
! [A: a,B: a,K: nat] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ A @ ( add_pow_a_b_nat @ r @ K @ B ) )
= ( add_pow_a_b_nat @ r @ K @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ).
% add_pow_rdistr
thf(fact_92_add__pow__ldistr,axiom,
! [A: a,B: a,K: nat] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( add_pow_a_b_nat @ r @ K @ A ) @ B )
= ( add_pow_a_b_nat @ r @ K @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ).
% add_pow_ldistr
thf(fact_93_add_Oinv__mult__group,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ ( add_a_b @ r @ X @ Y ) )
= ( add_a_b @ r @ ( a_inv_a_b @ r @ Y ) @ ( a_inv_a_b @ r @ X ) ) ) ) ) ).
% add.inv_mult_group
thf(fact_94_add_Oinv__solve__left,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A
= ( add_a_b @ r @ ( a_inv_a_b @ r @ B ) @ C ) )
= ( C
= ( add_a_b @ r @ B @ A ) ) ) ) ) ) ).
% add.inv_solve_left
thf(fact_95_add_Oinv__solve__left_H,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ ( a_inv_a_b @ r @ B ) @ C )
= A )
= ( C
= ( add_a_b @ r @ B @ A ) ) ) ) ) ) ).
% add.inv_solve_left'
thf(fact_96_add_Oinv__solve__right,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A
= ( add_a_b @ r @ B @ ( a_inv_a_b @ r @ C ) ) )
= ( B
= ( add_a_b @ r @ A @ C ) ) ) ) ) ) ).
% add.inv_solve_right
thf(fact_97_add_Oinv__solve__right_H,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ B @ ( a_inv_a_b @ r @ C ) )
= A )
= ( B
= ( add_a_b @ r @ A @ C ) ) ) ) ) ) ).
% add.inv_solve_right'
thf(fact_98_a__transpose__inv,axiom,
! [X: a,Y: a,Z: a] :
( ( ( add_a_b @ r @ X @ Y )
= Z )
=> ( ( 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 @ ( a_inv_a_b @ r @ X ) @ Z )
= Y ) ) ) ) ) ).
% a_transpose_inv
thf(fact_99_local_Ominus__add,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ ( add_a_b @ r @ X @ Y ) )
= ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ ( a_inv_a_b @ r @ Y ) ) ) ) ) ).
% local.minus_add
thf(fact_100_r__neg1,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ ( add_a_b @ r @ X @ Y ) )
= Y ) ) ) ).
% r_neg1
thf(fact_101_r__neg2,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Y ) )
= Y ) ) ) ).
% r_neg2
thf(fact_102_add_Ogroup__commutes__pow,axiom,
! [X: a,Y: a,N: nat] :
( ( ( add_a_b @ r @ X @ Y )
= ( add_a_b @ r @ Y @ X ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ N @ X ) @ Y )
= ( add_a_b @ r @ Y @ ( add_pow_a_b_nat @ r @ N @ X ) ) ) ) ) ) ).
% add.group_commutes_pow
thf(fact_103_add_Onat__pow__comm,axiom,
! [X: a,N: nat,M: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ N @ X ) @ ( add_pow_a_b_nat @ r @ M @ X ) )
= ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ M @ X ) @ ( add_pow_a_b_nat @ r @ N @ X ) ) ) ) ).
% add.nat_pow_comm
thf(fact_104_add_Onat__pow__distrib,axiom,
! [X: a,Y: a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_nat @ r @ N @ ( add_a_b @ r @ X @ Y ) )
= ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ N @ X ) @ ( add_pow_a_b_nat @ r @ N @ Y ) ) ) ) ) ).
% add.nat_pow_distrib
thf(fact_105_add_Opow__mult__distrib,axiom,
! [X: a,Y: a,N: nat] :
( ( ( add_a_b @ r @ X @ Y )
= ( add_a_b @ r @ Y @ X ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_nat @ r @ N @ ( add_a_b @ r @ X @ Y ) )
= ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ N @ X ) @ ( add_pow_a_b_nat @ r @ N @ Y ) ) ) ) ) ) ).
% add.pow_mult_distrib
thf(fact_106_l__minus,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 @ ( a_inv_a_b @ r @ X ) @ Y )
= ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) ) ) ) ) ).
% l_minus
thf(fact_107_r__minus,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 @ ( a_inv_a_b @ r @ Y ) )
= ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) ) ) ) ) ).
% r_minus
thf(fact_108_add_Onat__pow__inv,axiom,
! [X: a,I: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_nat @ r @ I @ ( a_inv_a_b @ r @ X ) )
= ( a_inv_a_b @ r @ ( add_pow_a_b_nat @ r @ I @ X ) ) ) ) ).
% add.nat_pow_inv
thf(fact_109_add_Oint__pow__inv,axiom,
! [X: a,I: int] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_int @ r @ I @ ( a_inv_a_b @ r @ X ) )
= ( a_inv_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X ) ) ) ) ).
% add.int_pow_inv
thf(fact_110_a__inv__closed,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( a_inv_a_b @ r @ X ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% a_inv_closed
thf(fact_111_local_Ominus__minus,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ ( a_inv_a_b @ r @ X ) )
= X ) ) ).
% local.minus_minus
thf(fact_112_add_Onat__pow__closed,axiom,
! [X: a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( add_pow_a_b_nat @ r @ N @ X ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% add.nat_pow_closed
thf(fact_113_local_Ominus__zero,axiom,
( ( a_inv_a_b @ r @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ).
% local.minus_zero
thf(fact_114_add_Onat__pow__one,axiom,
! [N: nat] :
( ( add_pow_a_b_nat @ r @ N @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ).
% add.nat_pow_one
thf(fact_115_add_Oinv__eq__1__iff,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( a_inv_a_b @ r @ X )
= ( zero_a_b @ r ) )
= ( X
= ( zero_a_b @ r ) ) ) ) ).
% add.inv_eq_1_iff
thf(fact_116_ring_Oring__simprules_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( a_inv_a_b @ R @ X ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ).
% ring.ring_simprules(3)
thf(fact_117_ring_Oring__simprules_I20_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( a_inv_a_b @ R @ ( a_inv_a_b @ R @ X ) )
= X ) ) ) ).
% ring.ring_simprules(20)
thf(fact_118_ring_Ominus__zero,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( ( a_inv_a_b @ R @ ( zero_a_b @ R ) )
= ( zero_a_b @ R ) ) ) ).
% ring.minus_zero
thf(fact_119_ring_Oring__simprules_I17_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ ( add_a_b @ R @ ( a_inv_a_b @ R @ X ) @ Y ) )
= Y ) ) ) ) ).
% ring.ring_simprules(17)
thf(fact_120_ring_Oring__simprules_I18_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( a_inv_a_b @ R @ X ) @ ( add_a_b @ R @ X @ Y ) )
= Y ) ) ) ) ).
% ring.ring_simprules(18)
thf(fact_121_ring_Oring__simprules_I19_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( a_inv_a_b @ R @ ( add_a_b @ R @ X @ Y ) )
= ( add_a_b @ R @ ( a_inv_a_b @ R @ X ) @ ( a_inv_a_b @ R @ Y ) ) ) ) ) ) ).
% ring.ring_simprules(19)
thf(fact_122_ring_Or__minus,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ X @ ( a_inv_a_b @ R @ Y ) )
= ( a_inv_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) ) ) ) ) ) ).
% ring.r_minus
thf(fact_123_ring_Ol__minus,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( a_inv_a_b @ R @ X ) @ Y )
= ( a_inv_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) ) ) ) ) ) ).
% ring.l_minus
thf(fact_124_semiring_Oaxioms_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( semiring_axioms_a_b @ R ) ) ).
% semiring.axioms(3)
thf(fact_125_ring_Oring__simprules_I9_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( a_inv_a_b @ R @ X ) @ X )
= ( zero_a_b @ R ) ) ) ) ).
% ring.ring_simprules(9)
thf(fact_126_ring_Oring__simprules_I16_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ ( a_inv_a_b @ R @ X ) )
= ( zero_a_b @ R ) ) ) ) ).
% ring.ring_simprules(16)
thf(fact_127_ring_Ozeropideal,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( principalideal_a_b @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) @ R ) ) ).
% ring.zeropideal
thf(fact_128_semiring_Oadd__pow__rdistr,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a,K: nat] :
( ( semiring_a_b @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ A @ ( add_pow_a_b_nat @ R @ K @ B ) )
= ( add_pow_a_b_nat @ R @ K @ ( mult_a_ring_ext_a_b @ R @ A @ B ) ) ) ) ) ) ).
% semiring.add_pow_rdistr
thf(fact_129_semiring_Oadd__pow__ldistr,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a,K: nat] :
( ( semiring_a_b @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( add_pow_a_b_nat @ R @ K @ A ) @ B )
= ( add_pow_a_b_nat @ R @ K @ ( mult_a_ring_ext_a_b @ R @ A @ B ) ) ) ) ) ) ).
% semiring.add_pow_ldistr
thf(fact_130_minus__eq,axiom,
! [X: a,Y: a] :
( ( a_minus_a_b @ r @ X @ Y )
= ( add_a_b @ r @ X @ ( a_inv_a_b @ r @ Y ) ) ) ).
% minus_eq
thf(fact_131_add_Oone__in__subset,axiom,
! [H2: set_a] :
( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( H2 != bot_bot_set_a )
=> ( ! [X2: a] :
( ( member_a @ X2 @ H2 )
=> ( member_a @ ( a_inv_a_b @ r @ X2 ) @ H2 ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ H2 )
=> ! [Xa: a] :
( ( member_a @ Xa @ H2 )
=> ( member_a @ ( add_a_b @ r @ X2 @ Xa ) @ H2 ) ) )
=> ( member_a @ ( zero_a_b @ r ) @ H2 ) ) ) ) ) ).
% add.one_in_subset
thf(fact_132_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_133_genideal__self_H,axiom,
! [I: a] :
( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ I @ ( genideal_a_b @ r @ ( insert_a @ I @ bot_bot_set_a ) ) ) ) ).
% genideal_self'
thf(fact_134_singletonI,axiom,
! [A: set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) ).
% singletonI
thf(fact_135_singletonI,axiom,
! [A: list_a] : ( member_list_a @ A @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) ).
% singletonI
thf(fact_136_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_137_add_Opower__order__eq__one,axiom,
! [A: a] :
( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_nat @ r @ ( finite_card_a @ ( partia707051561876973205xt_a_b @ r ) ) @ A )
= ( zero_a_b @ r ) ) ) ) ).
% add.power_order_eq_one
thf(fact_138_add_Oint__pow__diff,axiom,
! [X: a,N: int,M: int] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_int @ r @ ( minus_minus_int @ N @ M ) @ X )
= ( add_a_b @ r @ ( add_pow_a_b_int @ r @ N @ X ) @ ( a_inv_a_b @ r @ ( add_pow_a_b_int @ r @ M @ X ) ) ) ) ) ).
% add.int_pow_diff
thf(fact_139_monoid__cancel_Oproperfactor__mult__l,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a,C: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( proper19828929941537682xt_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ C @ A ) @ ( mult_a_ring_ext_a_b @ G @ C @ B ) )
= ( proper19828929941537682xt_a_b @ G @ A @ B ) ) ) ) ) ) ).
% monoid_cancel.properfactor_mult_l
thf(fact_140_monoid__cancel_Oproperfactor__mult__lI,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a,C: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( proper19828929941537682xt_a_b @ G @ A @ B )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( proper19828929941537682xt_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ C @ A ) @ ( mult_a_ring_ext_a_b @ G @ C @ B ) ) ) ) ) ) ).
% monoid_cancel.properfactor_mult_lI
thf(fact_141_ring_Oline__extension__mem__iff,axiom,
! [R: partia2175431115845679010xt_a_b,U: a,K2: set_a,A: a,E: set_a] :
( ( ring_a_b @ R )
=> ( ( member_a @ U @ ( embedd971793762689825387on_a_b @ R @ K2 @ A @ E ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ K2 )
& ? [Y3: a] :
( ( member_a @ Y3 @ E )
& ( U
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X3 @ A ) @ Y3 ) ) ) ) ) ) ) ).
% ring.line_extension_mem_iff
thf(fact_142_subsetI,axiom,
! [A2: set_set_a,B3: set_set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( member_set_a @ X2 @ B3 ) )
=> ( ord_le3724670747650509150_set_a @ A2 @ B3 ) ) ).
% subsetI
thf(fact_143_subsetI,axiom,
! [A2: set_a,B3: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_a @ X2 @ B3 ) )
=> ( ord_less_eq_set_a @ A2 @ B3 ) ) ).
% subsetI
thf(fact_144_subset__antisym,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% subset_antisym
thf(fact_145_empty__iff,axiom,
! [C: set_a] :
~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_146_empty__iff,axiom,
! [C: list_a] :
~ ( member_list_a @ C @ bot_bot_set_list_a ) ).
% empty_iff
thf(fact_147_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_148_all__not__in__conv,axiom,
! [A2: set_set_a] :
( ( ! [X3: set_a] :
~ ( member_set_a @ X3 @ A2 ) )
= ( A2 = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_149_all__not__in__conv,axiom,
! [A2: set_list_a] :
( ( ! [X3: list_a] :
~ ( member_list_a @ X3 @ A2 ) )
= ( A2 = bot_bot_set_list_a ) ) ).
% all_not_in_conv
thf(fact_150_all__not__in__conv,axiom,
! [A2: set_a] :
( ( ! [X3: a] :
~ ( member_a @ X3 @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_151_Collect__empty__eq,axiom,
! [P: list_a > $o] :
( ( ( collect_list_a @ P )
= bot_bot_set_list_a )
= ( ! [X3: list_a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_152_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_153_empty__Collect__eq,axiom,
! [P: list_a > $o] :
( ( bot_bot_set_list_a
= ( collect_list_a @ P ) )
= ( ! [X3: list_a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_154_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_155_insertCI,axiom,
! [A: a,B3: set_a,B: a] :
( ( ~ ( member_a @ A @ B3 )
=> ( A = B ) )
=> ( member_a @ A @ ( insert_a @ B @ B3 ) ) ) ).
% insertCI
thf(fact_156_insertCI,axiom,
! [A: set_a,B3: set_set_a,B: set_a] :
( ( ~ ( member_set_a @ A @ B3 )
=> ( A = B ) )
=> ( member_set_a @ A @ ( insert_set_a @ B @ B3 ) ) ) ).
% insertCI
thf(fact_157_insert__iff,axiom,
! [A: a,B: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A2 ) )
= ( ( A = B )
| ( member_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_158_insert__iff,axiom,
! [A: set_a,B: set_a,A2: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B @ A2 ) )
= ( ( A = B )
| ( member_set_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_159_insert__absorb2,axiom,
! [X: a,A2: set_a] :
( ( insert_a @ X @ ( insert_a @ X @ A2 ) )
= ( insert_a @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_160_line__extension__in__carrier,axiom,
! [K2: set_a,A: a,E: set_a] :
( ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedd971793762689825387on_a_b @ r @ K2 @ A @ E ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% line_extension_in_carrier
thf(fact_161_subset__Idl__subset,axiom,
! [I2: set_a,H2: set_a] :
( ( ord_less_eq_set_a @ I2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ H2 @ I2 )
=> ( ord_less_eq_set_a @ ( genideal_a_b @ r @ H2 ) @ ( genideal_a_b @ r @ I2 ) ) ) ) ).
% subset_Idl_subset
thf(fact_162_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_163_Idl__subset__ideal_H,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( genideal_a_b @ r @ ( insert_a @ A @ bot_bot_set_a ) ) @ ( genideal_a_b @ r @ ( insert_a @ B @ bot_bot_set_a ) ) )
= ( member_a @ A @ ( genideal_a_b @ r @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ) ) ).
% Idl_subset_ideal'
thf(fact_164_subset__empty,axiom,
! [A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ bot_bot_set_list_a )
= ( A2 = bot_bot_set_list_a ) ) ).
% subset_empty
thf(fact_165_subset__empty,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_166_empty__subsetI,axiom,
! [A2: set_list_a] : ( ord_le8861187494160871172list_a @ bot_bot_set_list_a @ A2 ) ).
% empty_subsetI
thf(fact_167_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_168_insert__subset,axiom,
! [X: set_a,A2: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X @ A2 ) @ B3 )
= ( ( member_set_a @ X @ B3 )
& ( ord_le3724670747650509150_set_a @ A2 @ B3 ) ) ) ).
% insert_subset
thf(fact_169_insert__subset,axiom,
! [X: a,A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A2 ) @ B3 )
= ( ( member_a @ X @ B3 )
& ( ord_less_eq_set_a @ A2 @ B3 ) ) ) ).
% insert_subset
thf(fact_170_singleton__insert__inj__eq,axiom,
! [B: list_a,A: list_a,A2: set_list_a] :
( ( ( insert_list_a @ B @ bot_bot_set_list_a )
= ( insert_list_a @ A @ A2 ) )
= ( ( A = B )
& ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ B @ bot_bot_set_list_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_171_singleton__insert__inj__eq,axiom,
! [B: a,A: a,A2: set_a] :
( ( ( insert_a @ B @ bot_bot_set_a )
= ( insert_a @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_172_singleton__insert__inj__eq_H,axiom,
! [A: list_a,A2: set_list_a,B: list_a] :
( ( ( insert_list_a @ A @ A2 )
= ( insert_list_a @ B @ bot_bot_set_list_a ) )
= ( ( A = B )
& ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ B @ bot_bot_set_list_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_173_singleton__insert__inj__eq_H,axiom,
! [A: a,A2: set_a,B: a] :
( ( ( insert_a @ A @ A2 )
= ( insert_a @ B @ bot_bot_set_a ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_174_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_175_r__right__minus__eq,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( a_minus_a_b @ r @ A @ B )
= ( zero_a_b @ r ) )
= ( A = B ) ) ) ) ).
% r_right_minus_eq
thf(fact_176_in__mono,axiom,
! [A2: set_set_a,B3: set_set_a,X: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B3 )
=> ( ( member_set_a @ X @ A2 )
=> ( member_set_a @ X @ B3 ) ) ) ).
% in_mono
thf(fact_177_in__mono,axiom,
! [A2: set_a,B3: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( member_a @ X @ A2 )
=> ( member_a @ X @ B3 ) ) ) ).
% in_mono
thf(fact_178_subsetD,axiom,
! [A2: set_set_a,B3: set_set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B3 )
=> ( ( member_set_a @ C @ A2 )
=> ( member_set_a @ C @ B3 ) ) ) ).
% subsetD
thf(fact_179_subsetD,axiom,
! [A2: set_a,B3: set_a,C: a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B3 ) ) ) ).
% subsetD
thf(fact_180_equalityE,axiom,
! [A2: set_a,B3: set_a] :
( ( A2 = B3 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B3 )
=> ~ ( ord_less_eq_set_a @ B3 @ A2 ) ) ) ).
% equalityE
thf(fact_181_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B4: set_set_a] :
! [X3: set_a] :
( ( member_set_a @ X3 @ A4 )
=> ( member_set_a @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_182_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A4 )
=> ( member_a @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_183_equalityD1,axiom,
! [A2: set_a,B3: set_a] :
( ( A2 = B3 )
=> ( ord_less_eq_set_a @ A2 @ B3 ) ) ).
% equalityD1
thf(fact_184_equalityD2,axiom,
! [A2: set_a,B3: set_a] :
( ( A2 = B3 )
=> ( ord_less_eq_set_a @ B3 @ A2 ) ) ).
% equalityD2
thf(fact_185_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B4: set_set_a] :
! [T: set_a] :
( ( member_set_a @ T @ A4 )
=> ( member_set_a @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_186_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
! [T: a] :
( ( member_a @ T @ A4 )
=> ( member_a @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_187_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_188_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X2: a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_189_subset__trans,axiom,
! [A2: set_a,B3: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C3 )
=> ( ord_less_eq_set_a @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_190_set__eq__subset,axiom,
( ( ^ [Y5: set_a,Z4: set_a] : ( Y5 = Z4 ) )
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
& ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_191_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_192_insert__mono,axiom,
! [C3: set_a,D: set_a,A: a] :
( ( ord_less_eq_set_a @ C3 @ D )
=> ( ord_less_eq_set_a @ ( insert_a @ A @ C3 ) @ ( insert_a @ A @ D ) ) ) ).
% insert_mono
thf(fact_193_subset__insert,axiom,
! [X: set_a,A2: set_set_a,B3: set_set_a] :
( ~ ( member_set_a @ X @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ X @ B3 ) )
= ( ord_le3724670747650509150_set_a @ A2 @ B3 ) ) ) ).
% subset_insert
thf(fact_194_subset__insert,axiom,
! [X: a,A2: set_a,B3: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B3 ) )
= ( ord_less_eq_set_a @ A2 @ B3 ) ) ) ).
% subset_insert
thf(fact_195_subset__insertI,axiom,
! [B3: set_a,A: a] : ( ord_less_eq_set_a @ B3 @ ( insert_a @ A @ B3 ) ) ).
% subset_insertI
thf(fact_196_subset__insertI2,axiom,
! [A2: set_a,B3: set_a,B: a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ B3 ) ) ) ).
% subset_insertI2
thf(fact_197_ring_Ogenideal__self,axiom,
! [R: partia2175431115845679010xt_a_b,S: set_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ S @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ord_less_eq_set_a @ S @ ( genideal_a_b @ R @ S ) ) ) ) ).
% ring.genideal_self
thf(fact_198_ring_Osubset__Idl__subset,axiom,
! [R: partia2175431115845679010xt_a_b,I2: set_a,H2: set_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ I2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ H2 @ I2 )
=> ( ord_less_eq_set_a @ ( genideal_a_b @ R @ H2 ) @ ( genideal_a_b @ R @ I2 ) ) ) ) ) ).
% ring.subset_Idl_subset
thf(fact_199_subset__singletonD,axiom,
! [A2: set_list_a,X: list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ X @ bot_bot_set_list_a ) )
=> ( ( A2 = bot_bot_set_list_a )
| ( A2
= ( insert_list_a @ X @ bot_bot_set_list_a ) ) ) ) ).
% subset_singletonD
thf(fact_200_subset__singletonD,axiom,
! [A2: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) )
=> ( ( A2 = bot_bot_set_a )
| ( A2
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_201_subset__singleton__iff,axiom,
! [X4: set_list_a,A: list_a] :
( ( ord_le8861187494160871172list_a @ X4 @ ( insert_list_a @ A @ bot_bot_set_list_a ) )
= ( ( X4 = bot_bot_set_list_a )
| ( X4
= ( insert_list_a @ A @ bot_bot_set_list_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_202_subset__singleton__iff,axiom,
! [X4: set_a,A: a] :
( ( ord_less_eq_set_a @ X4 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( ( X4 = bot_bot_set_a )
| ( X4
= ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_203_ring_Ofin__degree__bounded,axiom,
! [R: partia2175431115845679010xt_a_b,N: nat] :
( ( ring_a_b @ R )
=> ( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ R ) )
=> ( finite_finite_list_a @ ( bounde2262800523058855161ls_a_b @ R @ N ) ) ) ) ).
% ring.fin_degree_bounded
thf(fact_204_ring_OIdl__subset__ideal_H,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( genideal_a_b @ R @ ( insert_a @ A @ bot_bot_set_a ) ) @ ( genideal_a_b @ R @ ( insert_a @ B @ bot_bot_set_a ) ) )
= ( member_a @ A @ ( genideal_a_b @ R @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ) ) ) ).
% ring.Idl_subset_ideal'
thf(fact_205_ring_Oline__extension__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,A: a,E: set_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ord_less_eq_set_a @ ( embedd971793762689825387on_a_b @ R @ K2 @ A @ E ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ) ).
% ring.line_extension_in_carrier
thf(fact_206_emptyE,axiom,
! [A: set_a] :
~ ( member_set_a @ A @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_207_emptyE,axiom,
! [A: list_a] :
~ ( member_list_a @ A @ bot_bot_set_list_a ) ).
% emptyE
thf(fact_208_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_209_equals0D,axiom,
! [A2: set_set_a,A: set_a] :
( ( A2 = bot_bot_set_set_a )
=> ~ ( member_set_a @ A @ A2 ) ) ).
% equals0D
thf(fact_210_equals0D,axiom,
! [A2: set_list_a,A: list_a] :
( ( A2 = bot_bot_set_list_a )
=> ~ ( member_list_a @ A @ A2 ) ) ).
% equals0D
thf(fact_211_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_212_equals0I,axiom,
! [A2: set_set_a] :
( ! [Y4: set_a] :
~ ( member_set_a @ Y4 @ A2 )
=> ( A2 = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_213_equals0I,axiom,
! [A2: set_list_a] :
( ! [Y4: list_a] :
~ ( member_list_a @ Y4 @ A2 )
=> ( A2 = bot_bot_set_list_a ) ) ).
% equals0I
thf(fact_214_equals0I,axiom,
! [A2: set_a] :
( ! [Y4: a] :
~ ( member_a @ Y4 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_215_ex__in__conv,axiom,
! [A2: set_set_a] :
( ( ? [X3: set_a] : ( member_set_a @ X3 @ A2 ) )
= ( A2 != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_216_ex__in__conv,axiom,
! [A2: set_list_a] :
( ( ? [X3: list_a] : ( member_list_a @ X3 @ A2 ) )
= ( A2 != bot_bot_set_list_a ) ) ).
% ex_in_conv
thf(fact_217_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X3: a] : ( member_a @ X3 @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_218_ring_Oring__simprules_I4_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ).
% ring.ring_simprules(4)
thf(fact_219_insertE,axiom,
! [A: a,B: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A2 ) )
=> ( ( A != B )
=> ( member_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_220_insertE,axiom,
! [A: set_a,B: set_a,A2: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B @ A2 ) )
=> ( ( A != B )
=> ( member_set_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_221_insertI1,axiom,
! [A: a,B3: set_a] : ( member_a @ A @ ( insert_a @ A @ B3 ) ) ).
% insertI1
thf(fact_222_insertI1,axiom,
! [A: set_a,B3: set_set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ B3 ) ) ).
% insertI1
thf(fact_223_insertI2,axiom,
! [A: a,B3: set_a,B: a] :
( ( member_a @ A @ B3 )
=> ( member_a @ A @ ( insert_a @ B @ B3 ) ) ) ).
% insertI2
thf(fact_224_insertI2,axiom,
! [A: set_a,B3: set_set_a,B: set_a] :
( ( member_set_a @ A @ B3 )
=> ( member_set_a @ A @ ( insert_set_a @ B @ B3 ) ) ) ).
% insertI2
thf(fact_225_Set_Oset__insert,axiom,
! [X: a,A2: set_a] :
( ( member_a @ X @ A2 )
=> ~ ! [B5: set_a] :
( ( A2
= ( insert_a @ X @ B5 ) )
=> ( member_a @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_226_Set_Oset__insert,axiom,
! [X: set_a,A2: set_set_a] :
( ( member_set_a @ X @ A2 )
=> ~ ! [B5: set_set_a] :
( ( A2
= ( insert_set_a @ X @ B5 ) )
=> ( member_set_a @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_227_insert__ident,axiom,
! [X: a,A2: set_a,B3: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ~ ( member_a @ X @ B3 )
=> ( ( ( insert_a @ X @ A2 )
= ( insert_a @ X @ B3 ) )
= ( A2 = B3 ) ) ) ) ).
% insert_ident
thf(fact_228_insert__ident,axiom,
! [X: set_a,A2: set_set_a,B3: set_set_a] :
( ~ ( member_set_a @ X @ A2 )
=> ( ~ ( member_set_a @ X @ B3 )
=> ( ( ( insert_set_a @ X @ A2 )
= ( insert_set_a @ X @ B3 ) )
= ( A2 = B3 ) ) ) ) ).
% insert_ident
thf(fact_229_insert__absorb,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_230_insert__absorb,axiom,
! [A: set_a,A2: set_set_a] :
( ( member_set_a @ A @ A2 )
=> ( ( insert_set_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_231_insert__eq__iff,axiom,
! [A: a,A2: set_a,B: a,B3: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ~ ( member_a @ B @ B3 )
=> ( ( ( insert_a @ A @ A2 )
= ( insert_a @ B @ B3 ) )
= ( ( ( A = B )
=> ( A2 = B3 ) )
& ( ( A != B )
=> ? [C4: set_a] :
( ( A2
= ( insert_a @ B @ C4 ) )
& ~ ( member_a @ B @ C4 )
& ( B3
= ( insert_a @ A @ C4 ) )
& ~ ( member_a @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_232_insert__eq__iff,axiom,
! [A: set_a,A2: set_set_a,B: set_a,B3: set_set_a] :
( ~ ( member_set_a @ A @ A2 )
=> ( ~ ( member_set_a @ B @ B3 )
=> ( ( ( insert_set_a @ A @ A2 )
= ( insert_set_a @ B @ B3 ) )
= ( ( ( A = B )
=> ( A2 = B3 ) )
& ( ( A != B )
=> ? [C4: set_set_a] :
( ( A2
= ( insert_set_a @ B @ C4 ) )
& ~ ( member_set_a @ B @ C4 )
& ( B3
= ( insert_set_a @ A @ C4 ) )
& ~ ( member_set_a @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_233_insert__commute,axiom,
! [X: a,Y: a,A2: set_a] :
( ( insert_a @ X @ ( insert_a @ Y @ A2 ) )
= ( insert_a @ Y @ ( insert_a @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_234_mk__disjoint__insert,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ? [B5: set_a] :
( ( A2
= ( insert_a @ A @ B5 ) )
& ~ ( member_a @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_235_mk__disjoint__insert,axiom,
! [A: set_a,A2: set_set_a] :
( ( member_set_a @ A @ A2 )
=> ? [B5: set_set_a] :
( ( A2
= ( insert_set_a @ A @ B5 ) )
& ~ ( member_set_a @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_236_a__minus__def,axiom,
( a_minus_a_b
= ( ^ [R2: partia2175431115845679010xt_a_b,X3: a,Y3: a] : ( add_a_b @ R2 @ X3 @ ( a_inv_a_b @ R2 @ Y3 ) ) ) ) ).
% a_minus_def
thf(fact_237_ring_Oline__extension_Ocong,axiom,
embedd971793762689825387on_a_b = embedd971793762689825387on_a_b ).
% ring.line_extension.cong
thf(fact_238_monoid__cancel_Ois__monoid__cancel,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( monoid5798828371819920185xt_a_b @ G ) ) ).
% monoid_cancel.is_monoid_cancel
thf(fact_239_ring_Oring__simprules_I14_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( a_minus_a_b @ R @ X @ Y )
= ( add_a_b @ R @ X @ ( a_inv_a_b @ R @ Y ) ) ) ) ).
% ring.ring_simprules(14)
thf(fact_240_ring_Ogenideal__self_H,axiom,
! [R: partia2175431115845679010xt_a_b,I: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ I @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ I @ ( genideal_a_b @ R @ ( insert_a @ I @ bot_bot_set_a ) ) ) ) ) ).
% ring.genideal_self'
thf(fact_241_ring_Ogenideal__zero,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( ( genideal_a_b @ R @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) )
= ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) ) ).
% ring.genideal_zero
thf(fact_242_principalideal_Ogenerate,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b] :
( ( principalideal_a_b @ I2 @ R )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
& ( I2
= ( genideal_a_b @ R @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ) ).
% principalideal.generate
thf(fact_243_singletonD,axiom,
! [B: set_a,A: set_a] :
( ( member_set_a @ B @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_244_singletonD,axiom,
! [B: list_a,A: list_a] :
( ( member_list_a @ B @ ( insert_list_a @ A @ bot_bot_set_list_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_245_singletonD,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_246_singleton__iff,axiom,
! [B: set_a,A: set_a] :
( ( member_set_a @ B @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_247_singleton__iff,axiom,
! [B: list_a,A: list_a] :
( ( member_list_a @ B @ ( insert_list_a @ A @ bot_bot_set_list_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_248_singleton__iff,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_249_doubleton__eq__iff,axiom,
! [A: list_a,B: list_a,C: list_a,D2: list_a] :
( ( ( insert_list_a @ A @ ( insert_list_a @ B @ bot_bot_set_list_a ) )
= ( insert_list_a @ C @ ( insert_list_a @ D2 @ bot_bot_set_list_a ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_250_doubleton__eq__iff,axiom,
! [A: a,B: a,C: a,D2: a] :
( ( ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) )
= ( insert_a @ C @ ( insert_a @ D2 @ bot_bot_set_a ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_251_insert__not__empty,axiom,
! [A: list_a,A2: set_list_a] :
( ( insert_list_a @ A @ A2 )
!= bot_bot_set_list_a ) ).
% insert_not_empty
thf(fact_252_insert__not__empty,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ A2 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_253_singleton__inject,axiom,
! [A: list_a,B: list_a] :
( ( ( insert_list_a @ A @ bot_bot_set_list_a )
= ( insert_list_a @ B @ bot_bot_set_list_a ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_254_singleton__inject,axiom,
! [A: a,B: a] :
( ( ( insert_a @ A @ bot_bot_set_a )
= ( insert_a @ B @ bot_bot_set_a ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_255_monoid__cancel_Or__cancel,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,C: a,B: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( ( mult_a_ring_ext_a_b @ G @ A @ C )
= ( mult_a_ring_ext_a_b @ G @ B @ C ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( A = B ) ) ) ) ) ) ).
% monoid_cancel.r_cancel
thf(fact_256_monoid__cancel_Ol__cancel,axiom,
! [G: partia2175431115845679010xt_a_b,C: a,A: a,B: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( ( mult_a_ring_ext_a_b @ G @ C @ A )
= ( mult_a_ring_ext_a_b @ G @ C @ B ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( A = B ) ) ) ) ) ) ).
% monoid_cancel.l_cancel
thf(fact_257_a__card__cosets__equal,axiom,
! [C: set_a,H2: set_a] :
( ( member_set_a @ C @ ( a_RCOSETS_a_b @ r @ H2 ) )
=> ( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( finite_card_a @ C )
= ( finite_card_a @ H2 ) ) ) ) ) ).
% a_card_cosets_equal
thf(fact_258_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_259_le__add__diff__inverse2,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_260_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_261_le__add__diff__inverse,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_262_a__lcos__m__assoc,axiom,
! [M2: set_a,G2: a,H: a] :
( ( ord_less_eq_set_a @ M2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ G2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_l_coset_a_b @ r @ G2 @ ( a_l_coset_a_b @ r @ H @ M2 ) )
= ( a_l_coset_a_b @ r @ ( add_a_b @ r @ G2 @ H ) @ M2 ) ) ) ) ) ).
% a_lcos_m_assoc
thf(fact_263_a__lcos__mult__one,axiom,
! [M2: set_a] :
( ( ord_less_eq_set_a @ M2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_l_coset_a_b @ r @ ( zero_a_b @ r ) @ M2 )
= M2 ) ) ).
% a_lcos_mult_one
thf(fact_264_finite__insert,axiom,
! [A: a,A2: set_a] :
( ( finite_finite_a @ ( insert_a @ A @ A2 ) )
= ( finite_finite_a @ A2 ) ) ).
% finite_insert
thf(fact_265_finite__insert,axiom,
! [A: list_a,A2: set_list_a] :
( ( finite_finite_list_a @ ( insert_list_a @ A @ A2 ) )
= ( finite_finite_list_a @ A2 ) ) ).
% finite_insert
thf(fact_266_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_267_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_268_add__diff__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_269_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_270_add_Opow__eq__div2,axiom,
! [X: a,M: nat,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_pow_a_b_nat @ r @ M @ X )
= ( add_pow_a_b_nat @ r @ N @ X ) )
=> ( ( add_pow_a_b_nat @ r @ ( minus_minus_nat @ M @ N ) @ X )
= ( zero_a_b @ r ) ) ) ) ).
% add.pow_eq_div2
thf(fact_271_add_Onat__pow__mult,axiom,
! [X: a,N: nat,M: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ N @ X ) @ ( add_pow_a_b_nat @ r @ M @ X ) )
= ( add_pow_a_b_nat @ r @ ( plus_plus_nat @ N @ M ) @ X ) ) ) ).
% add.nat_pow_mult
thf(fact_272_add__right__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_273_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_274_add__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_275_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_276_Diff__cancel,axiom,
! [A2: set_list_a] :
( ( minus_646659088055828811list_a @ A2 @ A2 )
= bot_bot_set_list_a ) ).
% Diff_cancel
thf(fact_277_Diff__cancel,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ A2 )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_278_empty__Diff,axiom,
! [A2: set_list_a] :
( ( minus_646659088055828811list_a @ bot_bot_set_list_a @ A2 )
= bot_bot_set_list_a ) ).
% empty_Diff
thf(fact_279_empty__Diff,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A2 )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_280_Diff__empty,axiom,
! [A2: set_list_a] :
( ( minus_646659088055828811list_a @ A2 @ bot_bot_set_list_a )
= A2 ) ).
% Diff_empty
thf(fact_281_Diff__empty,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% Diff_empty
thf(fact_282_finite__Diff2,axiom,
! [B3: set_list_a,A2: set_list_a] :
( ( finite_finite_list_a @ B3 )
=> ( ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A2 @ B3 ) )
= ( finite_finite_list_a @ A2 ) ) ) ).
% finite_Diff2
thf(fact_283_finite__Diff2,axiom,
! [B3: set_a,A2: set_a] :
( ( finite_finite_a @ B3 )
=> ( ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B3 ) )
= ( finite_finite_a @ A2 ) ) ) ).
% finite_Diff2
thf(fact_284_finite__Diff,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( finite_finite_list_a @ A2 )
=> ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A2 @ B3 ) ) ) ).
% finite_Diff
thf(fact_285_finite__Diff,axiom,
! [A2: set_a,B3: set_a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B3 ) ) ) ).
% finite_Diff
thf(fact_286_insert__Diff1,axiom,
! [X: set_a,B3: set_set_a,A2: set_set_a] :
( ( member_set_a @ X @ B3 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A2 ) @ B3 )
= ( minus_5736297505244876581_set_a @ A2 @ B3 ) ) ) ).
% insert_Diff1
thf(fact_287_insert__Diff1,axiom,
! [X: a,B3: set_a,A2: set_a] :
( ( member_a @ X @ B3 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B3 )
= ( minus_minus_set_a @ A2 @ B3 ) ) ) ).
% insert_Diff1
thf(fact_288_Diff__insert0,axiom,
! [X: set_a,A2: set_set_a,B3: set_set_a] :
( ~ ( member_set_a @ X @ A2 )
=> ( ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ B3 ) )
= ( minus_5736297505244876581_set_a @ A2 @ B3 ) ) ) ).
% Diff_insert0
thf(fact_289_Diff__insert0,axiom,
! [X: a,A2: set_a,B3: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( minus_minus_set_a @ A2 @ ( insert_a @ X @ B3 ) )
= ( minus_minus_set_a @ A2 @ B3 ) ) ) ).
% Diff_insert0
thf(fact_290_a__l__coset__subset__G,axiom,
! [H2: set_a,X: a] :
( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( a_l_coset_a_b @ r @ X @ H2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% a_l_coset_subset_G
thf(fact_291_fin__degree__bounded,axiom,
! [N: nat] :
( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) )
=> ( finite_finite_list_a @ ( bounde2262800523058855161ls_a_b @ r @ N ) ) ) ).
% fin_degree_bounded
thf(fact_292_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_293_add__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_294_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_295_add__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_296_add__diff__cancel__right_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_297_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_298_add__diff__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_299_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_300_add__diff__cancel__left_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_301_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_302_Diff__eq__empty__iff,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( ( minus_646659088055828811list_a @ A2 @ B3 )
= bot_bot_set_list_a )
= ( ord_le8861187494160871172list_a @ A2 @ B3 ) ) ).
% Diff_eq_empty_iff
thf(fact_303_Diff__eq__empty__iff,axiom,
! [A2: set_a,B3: set_a] :
( ( ( minus_minus_set_a @ A2 @ B3 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A2 @ B3 ) ) ).
% Diff_eq_empty_iff
thf(fact_304_insert__Diff__single,axiom,
! [A: list_a,A2: set_list_a] :
( ( insert_list_a @ A @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) )
= ( insert_list_a @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_305_insert__Diff__single,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( insert_a @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_306_finite__Diff__insert,axiom,
! [A2: set_list_a,A: list_a,B3: set_list_a] :
( ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A @ B3 ) ) )
= ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A2 @ B3 ) ) ) ).
% finite_Diff_insert
thf(fact_307_finite__Diff__insert,axiom,
! [A2: set_a,A: a,B3: set_a] :
( ( finite_finite_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B3 ) ) )
= ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B3 ) ) ) ).
% finite_Diff_insert
thf(fact_308_Diff__infinite__finite,axiom,
! [T2: set_list_a,S: set_list_a] :
( ( finite_finite_list_a @ T2 )
=> ( ~ ( finite_finite_list_a @ S )
=> ~ ( finite_finite_list_a @ ( minus_646659088055828811list_a @ S @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_309_Diff__infinite__finite,axiom,
! [T2: set_a,S: set_a] :
( ( finite_finite_a @ T2 )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_310_diff__card__le__card__Diff,axiom,
! [B3: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B3 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ B3 ) ) @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B3 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_311_diff__card__le__card__Diff,axiom,
! [B3: set_list_a,A2: set_list_a] :
( ( finite_finite_list_a @ B3 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_list_a @ A2 ) @ ( finite_card_list_a @ B3 ) ) @ ( finite_card_list_a @ ( minus_646659088055828811list_a @ A2 @ B3 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_312_diff__card__le__card__Diff,axiom,
! [B3: set_a,A2: set_a] :
( ( finite_finite_a @ B3 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B3 ) ) @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ B3 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_313_card__le__sym__Diff,axiom,
! [A2: set_set_a,B3: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( finite_finite_set_a @ B3 )
=> ( ( ord_less_eq_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ B3 ) )
=> ( ord_less_eq_nat @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B3 ) ) @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ B3 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_314_card__le__sym__Diff,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( finite_finite_list_a @ A2 )
=> ( ( finite_finite_list_a @ B3 )
=> ( ( ord_less_eq_nat @ ( finite_card_list_a @ A2 ) @ ( finite_card_list_a @ B3 ) )
=> ( ord_less_eq_nat @ ( finite_card_list_a @ ( minus_646659088055828811list_a @ A2 @ B3 ) ) @ ( finite_card_list_a @ ( minus_646659088055828811list_a @ B3 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_315_card__le__sym__Diff,axiom,
! [A2: set_a,B3: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B3 )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B3 ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ B3 ) ) @ ( finite_card_a @ ( minus_minus_set_a @ B3 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_316_card__Diff__subset,axiom,
! [B3: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B3 )
=> ( ( ord_le3724670747650509150_set_a @ B3 @ A2 )
=> ( ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B3 ) )
= ( minus_minus_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ B3 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_317_card__Diff__subset,axiom,
! [B3: set_list_a,A2: set_list_a] :
( ( finite_finite_list_a @ B3 )
=> ( ( ord_le8861187494160871172list_a @ B3 @ A2 )
=> ( ( finite_card_list_a @ ( minus_646659088055828811list_a @ A2 @ B3 ) )
= ( minus_minus_nat @ ( finite_card_list_a @ A2 ) @ ( finite_card_list_a @ B3 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_318_card__Diff__subset,axiom,
! [B3: set_a,A2: set_a] :
( ( finite_finite_a @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ B3 ) )
= ( minus_minus_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B3 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_319_card__Diff1__le,axiom,
! [A2: set_set_a,X: set_a] : ( ord_less_eq_nat @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) @ ( finite_card_set_a @ A2 ) ) ).
% card_Diff1_le
thf(fact_320_card__Diff1__le,axiom,
! [A2: set_list_a,X: list_a] : ( ord_less_eq_nat @ ( finite_card_list_a @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ X @ bot_bot_set_list_a ) ) ) @ ( finite_card_list_a @ A2 ) ) ).
% card_Diff1_le
thf(fact_321_card__Diff1__le,axiom,
! [A2: set_a,X: a] : ( ord_less_eq_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A2 ) ) ).
% card_Diff1_le
thf(fact_322_double__diff,axiom,
! [A2: set_a,B3: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C3 )
=> ( ( minus_minus_set_a @ B3 @ ( minus_minus_set_a @ C3 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_323_Diff__subset,axiom,
! [A2: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B3 ) @ A2 ) ).
% Diff_subset
thf(fact_324_Diff__mono,axiom,
! [A2: set_a,C3: set_a,D: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ C3 )
=> ( ( ord_less_eq_set_a @ D @ B3 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B3 ) @ ( minus_minus_set_a @ C3 @ D ) ) ) ) ).
% Diff_mono
thf(fact_325_insert__Diff__if,axiom,
! [X: set_a,B3: set_set_a,A2: set_set_a] :
( ( ( member_set_a @ X @ B3 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A2 ) @ B3 )
= ( minus_5736297505244876581_set_a @ A2 @ B3 ) ) )
& ( ~ ( member_set_a @ X @ B3 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A2 ) @ B3 )
= ( insert_set_a @ X @ ( minus_5736297505244876581_set_a @ A2 @ B3 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_326_insert__Diff__if,axiom,
! [X: a,B3: set_a,A2: set_a] :
( ( ( member_a @ X @ B3 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B3 )
= ( minus_minus_set_a @ A2 @ B3 ) ) )
& ( ~ ( member_a @ X @ B3 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B3 )
= ( insert_a @ X @ ( minus_minus_set_a @ A2 @ B3 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_327_card__insert__le,axiom,
! [A2: set_list_a,X: list_a] : ( ord_less_eq_nat @ ( finite_card_list_a @ A2 ) @ ( finite_card_list_a @ ( insert_list_a @ X @ A2 ) ) ) ).
% card_insert_le
thf(fact_328_card__insert__le,axiom,
! [A2: set_a,X: a] : ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ ( insert_a @ X @ A2 ) ) ) ).
% card_insert_le
thf(fact_329_card__insert__le,axiom,
! [A2: set_set_a,X: set_a] : ( ord_less_eq_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ ( insert_set_a @ X @ A2 ) ) ) ).
% card_insert_le
thf(fact_330_finite__empty__induct,axiom,
! [A2: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ A2 )
=> ( ( P @ A2 )
=> ( ! [A3: set_a,A5: set_set_a] :
( ( finite_finite_set_a @ A5 )
=> ( ( member_set_a @ A3 @ A5 )
=> ( ( P @ A5 )
=> ( P @ ( minus_5736297505244876581_set_a @ A5 @ ( insert_set_a @ A3 @ bot_bot_set_set_a ) ) ) ) ) )
=> ( P @ bot_bot_set_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_331_finite__empty__induct,axiom,
! [A2: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ A2 )
=> ( ( P @ A2 )
=> ( ! [A3: list_a,A5: set_list_a] :
( ( finite_finite_list_a @ A5 )
=> ( ( member_list_a @ A3 @ A5 )
=> ( ( P @ A5 )
=> ( P @ ( minus_646659088055828811list_a @ A5 @ ( insert_list_a @ A3 @ bot_bot_set_list_a ) ) ) ) ) )
=> ( P @ bot_bot_set_list_a ) ) ) ) ).
% finite_empty_induct
thf(fact_332_finite__empty__induct,axiom,
! [A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ A2 )
=> ( ( P @ A2 )
=> ( ! [A3: a,A5: set_a] :
( ( finite_finite_a @ A5 )
=> ( ( member_a @ A3 @ A5 )
=> ( ( P @ A5 )
=> ( P @ ( minus_minus_set_a @ A5 @ ( insert_a @ A3 @ bot_bot_set_a ) ) ) ) ) )
=> ( P @ bot_bot_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_333_infinite__coinduct,axiom,
! [X4: set_list_a > $o,A2: set_list_a] :
( ( X4 @ A2 )
=> ( ! [A5: set_list_a] :
( ( X4 @ A5 )
=> ? [X5: list_a] :
( ( member_list_a @ X5 @ A5 )
& ( ( X4 @ ( minus_646659088055828811list_a @ A5 @ ( insert_list_a @ X5 @ bot_bot_set_list_a ) ) )
| ~ ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A5 @ ( insert_list_a @ X5 @ bot_bot_set_list_a ) ) ) ) ) )
=> ~ ( finite_finite_list_a @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_334_infinite__coinduct,axiom,
! [X4: set_a > $o,A2: set_a] :
( ( X4 @ A2 )
=> ( ! [A5: set_a] :
( ( X4 @ A5 )
=> ? [X5: a] :
( ( member_a @ X5 @ A5 )
& ( ( X4 @ ( minus_minus_set_a @ A5 @ ( insert_a @ X5 @ bot_bot_set_a ) ) )
| ~ ( finite_finite_a @ ( minus_minus_set_a @ A5 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) ) ) )
=> ~ ( finite_finite_a @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_335_infinite__remove,axiom,
! [S: set_list_a,A: list_a] :
( ~ ( finite_finite_list_a @ S )
=> ~ ( finite_finite_list_a @ ( minus_646659088055828811list_a @ S @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) ) ) ).
% infinite_remove
thf(fact_336_infinite__remove,axiom,
! [S: set_a,A: a] :
( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% infinite_remove
thf(fact_337_subset__Diff__insert,axiom,
! [A2: set_set_a,B3: set_set_a,X: set_a,C3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( minus_5736297505244876581_set_a @ B3 @ ( insert_set_a @ X @ C3 ) ) )
= ( ( ord_le3724670747650509150_set_a @ A2 @ ( minus_5736297505244876581_set_a @ B3 @ C3 ) )
& ~ ( member_set_a @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_338_subset__Diff__insert,axiom,
! [A2: set_a,B3: set_a,X: a,C3: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B3 @ ( insert_a @ X @ C3 ) ) )
= ( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B3 @ C3 ) )
& ~ ( member_a @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_339_Diff__insert__absorb,axiom,
! [X: set_a,A2: set_set_a] :
( ~ ( member_set_a @ X @ A2 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A2 ) @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_340_Diff__insert__absorb,axiom,
! [X: list_a,A2: set_list_a] :
( ~ ( member_list_a @ X @ A2 )
=> ( ( minus_646659088055828811list_a @ ( insert_list_a @ X @ A2 ) @ ( insert_list_a @ X @ bot_bot_set_list_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_341_Diff__insert__absorb,axiom,
! [X: a,A2: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ ( insert_a @ X @ bot_bot_set_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_342_Diff__insert2,axiom,
! [A2: set_list_a,A: list_a,B3: set_list_a] :
( ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A @ B3 ) )
= ( minus_646659088055828811list_a @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) @ B3 ) ) ).
% Diff_insert2
thf(fact_343_Diff__insert2,axiom,
! [A2: set_a,A: a,B3: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B3 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) @ B3 ) ) ).
% Diff_insert2
thf(fact_344_insert__Diff,axiom,
! [A: set_a,A2: set_set_a] :
( ( member_set_a @ A @ A2 )
=> ( ( insert_set_a @ A @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_345_insert__Diff,axiom,
! [A: list_a,A2: set_list_a] :
( ( member_list_a @ A @ A2 )
=> ( ( insert_list_a @ A @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_346_insert__Diff,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_347_Diff__insert,axiom,
! [A2: set_list_a,A: list_a,B3: set_list_a] :
( ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A @ B3 ) )
= ( minus_646659088055828811list_a @ ( minus_646659088055828811list_a @ A2 @ B3 ) @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) ) ).
% Diff_insert
thf(fact_348_Diff__insert,axiom,
! [A2: set_a,A: a,B3: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B3 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ B3 ) @ ( insert_a @ A @ bot_bot_set_a ) ) ) ).
% Diff_insert
thf(fact_349_finite__if__finite__subsets__card__bdd,axiom,
! [F: set_set_a,C3: nat] :
( ! [G3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ G3 @ F )
=> ( ( finite_finite_set_a @ G3 )
=> ( ord_less_eq_nat @ ( finite_card_set_a @ G3 ) @ C3 ) ) )
=> ( ( finite_finite_set_a @ F )
& ( ord_less_eq_nat @ ( finite_card_set_a @ F ) @ C3 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_350_finite__if__finite__subsets__card__bdd,axiom,
! [F: set_list_a,C3: nat] :
( ! [G3: set_list_a] :
( ( ord_le8861187494160871172list_a @ G3 @ F )
=> ( ( finite_finite_list_a @ G3 )
=> ( ord_less_eq_nat @ ( finite_card_list_a @ G3 ) @ C3 ) ) )
=> ( ( finite_finite_list_a @ F )
& ( ord_less_eq_nat @ ( finite_card_list_a @ F ) @ C3 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_351_finite__if__finite__subsets__card__bdd,axiom,
! [F: set_a,C3: nat] :
( ! [G3: set_a] :
( ( ord_less_eq_set_a @ G3 @ F )
=> ( ( finite_finite_a @ G3 )
=> ( ord_less_eq_nat @ ( finite_card_a @ G3 ) @ C3 ) ) )
=> ( ( finite_finite_a @ F )
& ( ord_less_eq_nat @ ( finite_card_a @ F ) @ C3 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_352_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_set_a] :
( ( ord_less_eq_nat @ N @ ( finite_card_set_a @ S ) )
=> ~ ! [T3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ T3 @ S )
=> ( ( ( finite_card_set_a @ T3 )
= N )
=> ~ ( finite_finite_set_a @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_353_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_list_a] :
( ( ord_less_eq_nat @ N @ ( finite_card_list_a @ S ) )
=> ~ ! [T3: set_list_a] :
( ( ord_le8861187494160871172list_a @ T3 @ S )
=> ( ( ( finite_card_list_a @ T3 )
= N )
=> ~ ( finite_finite_list_a @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_354_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_a] :
( ( ord_less_eq_nat @ N @ ( finite_card_a @ S ) )
=> ~ ! [T3: set_a] :
( ( ord_less_eq_set_a @ T3 @ S )
=> ( ( ( finite_card_a @ T3 )
= N )
=> ~ ( finite_finite_a @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_355_exists__subset__between,axiom,
! [A2: set_set_a,N: nat,C3: set_set_a] :
( ( ord_less_eq_nat @ ( finite_card_set_a @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_set_a @ C3 ) )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ C3 )
=> ( ( finite_finite_set_a @ C3 )
=> ? [B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B5 )
& ( ord_le3724670747650509150_set_a @ B5 @ C3 )
& ( ( finite_card_set_a @ B5 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_356_exists__subset__between,axiom,
! [A2: set_list_a,N: nat,C3: set_list_a] :
( ( ord_less_eq_nat @ ( finite_card_list_a @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_list_a @ C3 ) )
=> ( ( ord_le8861187494160871172list_a @ A2 @ C3 )
=> ( ( finite_finite_list_a @ C3 )
=> ? [B5: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B5 )
& ( ord_le8861187494160871172list_a @ B5 @ C3 )
& ( ( finite_card_list_a @ B5 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_357_exists__subset__between,axiom,
! [A2: set_a,N: nat,C3: set_a] :
( ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_a @ C3 ) )
=> ( ( ord_less_eq_set_a @ A2 @ C3 )
=> ( ( finite_finite_a @ C3 )
=> ? [B5: set_a] :
( ( ord_less_eq_set_a @ A2 @ B5 )
& ( ord_less_eq_set_a @ B5 @ C3 )
& ( ( finite_card_a @ B5 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_358_card__seteq,axiom,
! [B3: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B3 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( finite_card_set_a @ B3 ) @ ( finite_card_set_a @ A2 ) )
=> ( A2 = B3 ) ) ) ) ).
% card_seteq
thf(fact_359_card__seteq,axiom,
! [B3: set_list_a,A2: set_list_a] :
( ( finite_finite_list_a @ B3 )
=> ( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( finite_card_list_a @ B3 ) @ ( finite_card_list_a @ A2 ) )
=> ( A2 = B3 ) ) ) ) ).
% card_seteq
thf(fact_360_card__seteq,axiom,
! [B3: set_a,A2: set_a] :
( ( finite_finite_a @ B3 )
=> ( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ B3 ) @ ( finite_card_a @ A2 ) )
=> ( A2 = B3 ) ) ) ) ).
% card_seteq
thf(fact_361_card__mono,axiom,
! [B3: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B3 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B3 )
=> ( ord_less_eq_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ B3 ) ) ) ) ).
% card_mono
thf(fact_362_card__mono,axiom,
! [B3: set_list_a,A2: set_list_a] :
( ( finite_finite_list_a @ B3 )
=> ( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( ord_less_eq_nat @ ( finite_card_list_a @ A2 ) @ ( finite_card_list_a @ B3 ) ) ) ) ).
% card_mono
thf(fact_363_card__mono,axiom,
! [B3: set_a,A2: set_a] :
( ( finite_finite_a @ B3 )
=> ( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B3 ) ) ) ) ).
% card_mono
thf(fact_364_remove__induct,axiom,
! [P: set_set_a > $o,B3: set_set_a] :
( ( P @ bot_bot_set_set_a )
=> ( ( ~ ( finite_finite_set_a @ B3 )
=> ( P @ B3 ) )
=> ( ! [A5: set_set_a] :
( ( finite_finite_set_a @ A5 )
=> ( ( A5 != bot_bot_set_set_a )
=> ( ( ord_le3724670747650509150_set_a @ A5 @ B3 )
=> ( ! [X5: set_a] :
( ( member_set_a @ X5 @ A5 )
=> ( P @ ( minus_5736297505244876581_set_a @ A5 @ ( insert_set_a @ X5 @ bot_bot_set_set_a ) ) ) )
=> ( P @ A5 ) ) ) ) )
=> ( P @ B3 ) ) ) ) ).
% remove_induct
thf(fact_365_remove__induct,axiom,
! [P: set_list_a > $o,B3: set_list_a] :
( ( P @ bot_bot_set_list_a )
=> ( ( ~ ( finite_finite_list_a @ B3 )
=> ( P @ B3 ) )
=> ( ! [A5: set_list_a] :
( ( finite_finite_list_a @ A5 )
=> ( ( A5 != bot_bot_set_list_a )
=> ( ( ord_le8861187494160871172list_a @ A5 @ B3 )
=> ( ! [X5: list_a] :
( ( member_list_a @ X5 @ A5 )
=> ( P @ ( minus_646659088055828811list_a @ A5 @ ( insert_list_a @ X5 @ bot_bot_set_list_a ) ) ) )
=> ( P @ A5 ) ) ) ) )
=> ( P @ B3 ) ) ) ) ).
% remove_induct
thf(fact_366_remove__induct,axiom,
! [P: set_a > $o,B3: set_a] :
( ( P @ bot_bot_set_a )
=> ( ( ~ ( finite_finite_a @ B3 )
=> ( P @ B3 ) )
=> ( ! [A5: set_a] :
( ( finite_finite_a @ A5 )
=> ( ( A5 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A5 @ B3 )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A5 )
=> ( P @ ( minus_minus_set_a @ A5 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) )
=> ( P @ A5 ) ) ) ) )
=> ( P @ B3 ) ) ) ) ).
% remove_induct
thf(fact_367_finite__remove__induct,axiom,
! [B3: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ B3 )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [A5: set_set_a] :
( ( finite_finite_set_a @ A5 )
=> ( ( A5 != bot_bot_set_set_a )
=> ( ( ord_le3724670747650509150_set_a @ A5 @ B3 )
=> ( ! [X5: set_a] :
( ( member_set_a @ X5 @ A5 )
=> ( P @ ( minus_5736297505244876581_set_a @ A5 @ ( insert_set_a @ X5 @ bot_bot_set_set_a ) ) ) )
=> ( P @ A5 ) ) ) ) )
=> ( P @ B3 ) ) ) ) ).
% finite_remove_induct
thf(fact_368_finite__remove__induct,axiom,
! [B3: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ B3 )
=> ( ( P @ bot_bot_set_list_a )
=> ( ! [A5: set_list_a] :
( ( finite_finite_list_a @ A5 )
=> ( ( A5 != bot_bot_set_list_a )
=> ( ( ord_le8861187494160871172list_a @ A5 @ B3 )
=> ( ! [X5: list_a] :
( ( member_list_a @ X5 @ A5 )
=> ( P @ ( minus_646659088055828811list_a @ A5 @ ( insert_list_a @ X5 @ bot_bot_set_list_a ) ) ) )
=> ( P @ A5 ) ) ) ) )
=> ( P @ B3 ) ) ) ) ).
% finite_remove_induct
thf(fact_369_finite__remove__induct,axiom,
! [B3: set_a,P: set_a > $o] :
( ( finite_finite_a @ B3 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A5: set_a] :
( ( finite_finite_a @ A5 )
=> ( ( A5 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A5 @ B3 )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A5 )
=> ( P @ ( minus_minus_set_a @ A5 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) )
=> ( P @ A5 ) ) ) ) )
=> ( P @ B3 ) ) ) ) ).
% finite_remove_induct
thf(fact_370_Diff__single__insert,axiom,
! [A2: set_list_a,X: list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ X @ bot_bot_set_list_a ) ) @ B3 )
=> ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ X @ B3 ) ) ) ).
% Diff_single_insert
thf(fact_371_Diff__single__insert,axiom,
! [A2: set_a,X: a,B3: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B3 )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B3 ) ) ) ).
% Diff_single_insert
thf(fact_372_subset__insert__iff,axiom,
! [A2: set_set_a,X: set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ X @ B3 ) )
= ( ( ( member_set_a @ X @ A2 )
=> ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) @ B3 ) )
& ( ~ ( member_set_a @ X @ A2 )
=> ( ord_le3724670747650509150_set_a @ A2 @ B3 ) ) ) ) ).
% subset_insert_iff
thf(fact_373_subset__insert__iff,axiom,
! [A2: set_list_a,X: list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ X @ B3 ) )
= ( ( ( member_list_a @ X @ A2 )
=> ( ord_le8861187494160871172list_a @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ X @ bot_bot_set_list_a ) ) @ B3 ) )
& ( ~ ( member_list_a @ X @ A2 )
=> ( ord_le8861187494160871172list_a @ A2 @ B3 ) ) ) ) ).
% subset_insert_iff
thf(fact_374_subset__insert__iff,axiom,
! [A2: set_a,X: a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B3 ) )
= ( ( ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B3 ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ A2 @ B3 ) ) ) ) ).
% subset_insert_iff
thf(fact_375_mult_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_376_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_377_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A6: int,B6: int] : ( times_times_int @ B6 @ A6 ) ) ) ).
% mult.commute
thf(fact_378_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A6: nat,B6: nat] : ( times_times_nat @ B6 @ A6 ) ) ) ).
% mult.commute
thf(fact_379_mult_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.assoc
thf(fact_380_mult_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_381_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_382_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_383_add__right__imp__eq,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_384_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_385_add__left__imp__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_386_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_387_add_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.left_commute
thf(fact_388_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_389_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A6: int,B6: int] : ( plus_plus_int @ B6 @ A6 ) ) ) ).
% add.commute
thf(fact_390_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A6: nat,B6: nat] : ( plus_plus_nat @ B6 @ A6 ) ) ) ).
% add.commute
thf(fact_391_group__add__class_Oadd_Oright__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% group_add_class.add.right_cancel
thf(fact_392_add_Oleft__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_393_add_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.assoc
thf(fact_394_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_395_group__cancel_Oadd2,axiom,
! [B3: int,K: int,B: int,A: int] :
( ( B3
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B3 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_396_group__cancel_Oadd2,axiom,
! [B3: nat,K: nat,B: nat,A: nat] :
( ( B3
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B3 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_397_group__cancel_Oadd1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_398_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_399_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_int @ I @ K )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_400_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_401_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_402_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_403_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_404_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_405_diff__eq__diff__eq,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D2 ) )
=> ( ( A = B )
= ( C = D2 ) ) ) ).
% diff_eq_diff_eq
thf(fact_406_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_407_add__le__imp__le__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_408_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_409_add__le__imp__le__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_410_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A6: nat,B6: nat] :
? [C5: nat] :
( B6
= ( plus_plus_nat @ A6 @ C5 ) ) ) ) ).
% le_iff_add
thf(fact_411_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_412_add__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_right_mono
thf(fact_413_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C2: nat] :
( B
!= ( plus_plus_nat @ A @ C2 ) ) ) ).
% less_eqE
thf(fact_414_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_415_add__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_left_mono
thf(fact_416_add__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_417_add__mono,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_418_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_419_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_420_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_421_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_422_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_423_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_424_diff__eq__diff__less__eq,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D2 ) )
=> ( ( ord_less_eq_int @ A @ B )
= ( ord_less_eq_int @ C @ D2 ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_425_diff__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_426_diff__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_427_diff__mono,axiom,
! [A: int,B: int,D2: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ D2 @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D2 ) ) ) ) ).
% diff_mono
thf(fact_428_finite__has__minimal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ( ord_less_eq_set_a @ X2 @ A )
& ! [Xa2: set_a] :
( ( member_set_a @ Xa2 @ A2 )
=> ( ( ord_less_eq_set_a @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_429_finite__has__minimal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( ord_less_eq_nat @ X2 @ A )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A2 )
=> ( ( ord_less_eq_nat @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_430_finite__has__minimal2,axiom,
! [A2: set_int,A: int] :
( ( finite_finite_int @ A2 )
=> ( ( member_int @ A @ A2 )
=> ? [X2: int] :
( ( member_int @ X2 @ A2 )
& ( ord_less_eq_int @ X2 @ A )
& ! [Xa2: int] :
( ( member_int @ Xa2 @ A2 )
=> ( ( ord_less_eq_int @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_431_finite__has__maximal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ( ord_less_eq_set_a @ A @ X2 )
& ! [Xa2: set_a] :
( ( member_set_a @ Xa2 @ A2 )
=> ( ( ord_less_eq_set_a @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_432_finite__has__maximal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( ord_less_eq_nat @ A @ X2 )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A2 )
=> ( ( ord_less_eq_nat @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_433_finite__has__maximal2,axiom,
! [A2: set_int,A: int] :
( ( finite_finite_int @ A2 )
=> ( ( member_int @ A @ A2 )
=> ? [X2: int] :
( ( member_int @ X2 @ A2 )
& ( ord_less_eq_int @ A @ X2 )
& ! [Xa2: int] :
( ( member_int @ Xa2 @ A2 )
=> ( ( ord_less_eq_int @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_434_combine__common__factor,axiom,
! [A: int,E2: int,B: int,C: int] :
( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ C ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E2 ) @ C ) ) ).
% combine_common_factor
thf(fact_435_combine__common__factor,axiom,
! [A: nat,E2: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A @ E2 ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E2 ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E2 ) @ C ) ) ).
% combine_common_factor
thf(fact_436_distrib__right,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% distrib_right
thf(fact_437_distrib__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% distrib_right
thf(fact_438_distrib__left,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% distrib_left
thf(fact_439_distrib__left,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% distrib_left
thf(fact_440_comm__semiring__class_Odistrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_441_comm__semiring__class_Odistrib,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_442_ring__class_Oring__distribs_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_443_ring__class_Oring__distribs_I2_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_444_right__diff__distrib_H,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_445_right__diff__distrib_H,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_446_left__diff__distrib_H,axiom,
! [B: int,C: int,A: int] :
( ( times_times_int @ ( minus_minus_int @ B @ C ) @ A )
= ( minus_minus_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_447_left__diff__distrib_H,axiom,
! [B: nat,C: nat,A: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
= ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_448_right__diff__distrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% right_diff_distrib
thf(fact_449_left__diff__distrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_450_diff__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_451_diff__diff__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_452_add__implies__diff,axiom,
! [C: int,B: int,A: int] :
( ( ( plus_plus_int @ C @ B )
= A )
=> ( C
= ( minus_minus_int @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_453_add__implies__diff,axiom,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B )
= A )
=> ( C
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_454_diff__add__eq__diff__diff__swap,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_455_diff__add__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_456_diff__diff__eq2,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_457_add__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_458_eq__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( A
= ( minus_minus_int @ C @ B ) )
= ( ( plus_plus_int @ A @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_459_diff__eq__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( minus_minus_int @ A @ B )
= C )
= ( A
= ( plus_plus_int @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_460_group__cancel_Osub1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( minus_minus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_461_rev__finite__subset,axiom,
! [B3: set_list_a,A2: set_list_a] :
( ( finite_finite_list_a @ B3 )
=> ( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( finite_finite_list_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_462_rev__finite__subset,axiom,
! [B3: set_a,A2: set_a] :
( ( finite_finite_a @ B3 )
=> ( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( finite_finite_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_463_infinite__super,axiom,
! [S: set_list_a,T2: set_list_a] :
( ( ord_le8861187494160871172list_a @ S @ T2 )
=> ( ~ ( finite_finite_list_a @ S )
=> ~ ( finite_finite_list_a @ T2 ) ) ) ).
% infinite_super
thf(fact_464_infinite__super,axiom,
! [S: set_a,T2: set_a] :
( ( ord_less_eq_set_a @ S @ T2 )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ T2 ) ) ) ).
% infinite_super
thf(fact_465_finite__subset,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( ( finite_finite_list_a @ B3 )
=> ( finite_finite_list_a @ A2 ) ) ) ).
% finite_subset
thf(fact_466_finite__subset,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( finite_finite_a @ B3 )
=> ( finite_finite_a @ A2 ) ) ) ).
% finite_subset
thf(fact_467_infinite__imp__nonempty,axiom,
! [S: set_list_a] :
( ~ ( finite_finite_list_a @ S )
=> ( S != bot_bot_set_list_a ) ) ).
% infinite_imp_nonempty
thf(fact_468_infinite__imp__nonempty,axiom,
! [S: set_a] :
( ~ ( finite_finite_a @ S )
=> ( S != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_469_finite_OemptyI,axiom,
finite_finite_list_a @ bot_bot_set_list_a ).
% finite.emptyI
thf(fact_470_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_471_finite_OinsertI,axiom,
! [A2: set_a,A: a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( insert_a @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_472_finite_OinsertI,axiom,
! [A2: set_list_a,A: list_a] :
( ( finite_finite_list_a @ A2 )
=> ( finite_finite_list_a @ ( insert_list_a @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_473_diff__le__eq,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C )
= ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% diff_le_eq
thf(fact_474_le__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B ) )
= ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% le_diff_eq
thf(fact_475_add__le__imp__le__diff,axiom,
! [I: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_476_add__le__imp__le__diff,axiom,
! [I: int,K: int,N: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_477_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_478_add__le__add__imp__diff__le,axiom,
! [I: nat,K: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_479_add__le__add__imp__diff__le,axiom,
! [I: int,K: int,N: int,J: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_480_le__add__diff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% le_add_diff
thf(fact_481_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_482_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_483_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_484_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_485_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_486_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_487_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_488_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_489_finite__has__minimal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ! [Xa2: set_a] :
( ( member_set_a @ Xa2 @ A2 )
=> ( ( ord_less_eq_set_a @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_490_finite__has__minimal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A2 )
=> ( ( ord_less_eq_nat @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_491_finite__has__minimal,axiom,
! [A2: set_int] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ? [X2: int] :
( ( member_int @ X2 @ A2 )
& ! [Xa2: int] :
( ( member_int @ Xa2 @ A2 )
=> ( ( ord_less_eq_int @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_492_finite__has__maximal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ! [Xa2: set_a] :
( ( member_set_a @ Xa2 @ A2 )
=> ( ( ord_less_eq_set_a @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_493_finite__has__maximal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A2 )
=> ( ( ord_less_eq_nat @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_494_finite__has__maximal,axiom,
! [A2: set_int] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ? [X2: int] :
( ( member_int @ X2 @ A2 )
& ! [Xa2: int] :
( ( member_int @ Xa2 @ A2 )
=> ( ( ord_less_eq_int @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_495_square__diff__square__factored,axiom,
! [X: int,Y: int] :
( ( minus_minus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
= ( times_times_int @ ( plus_plus_int @ X @ Y ) @ ( minus_minus_int @ X @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_496_eq__add__iff2,axiom,
! [A: int,E2: int,C: int,B: int,D2: int] :
( ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C )
= ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D2 ) )
= ( C
= ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D2 ) ) ) ).
% eq_add_iff2
thf(fact_497_eq__add__iff1,axiom,
! [A: int,E2: int,C: int,B: int,D2: int] :
( ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C )
= ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D2 ) )
= ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C )
= D2 ) ) ).
% eq_add_iff1
thf(fact_498_infinite__finite__induct,axiom,
! [P: set_set_a > $o,A2: set_set_a] :
( ! [A5: set_set_a] :
( ~ ( finite_finite_set_a @ A5 )
=> ( P @ A5 ) )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [X2: set_a,F2: set_set_a] :
( ( finite_finite_set_a @ F2 )
=> ( ~ ( member_set_a @ X2 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_set_a @ X2 @ F2 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_499_infinite__finite__induct,axiom,
! [P: set_list_a > $o,A2: set_list_a] :
( ! [A5: set_list_a] :
( ~ ( finite_finite_list_a @ A5 )
=> ( P @ A5 ) )
=> ( ( P @ bot_bot_set_list_a )
=> ( ! [X2: list_a,F2: set_list_a] :
( ( finite_finite_list_a @ F2 )
=> ( ~ ( member_list_a @ X2 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_list_a @ X2 @ F2 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_500_infinite__finite__induct,axiom,
! [P: set_a > $o,A2: set_a] :
( ! [A5: set_a] :
( ~ ( finite_finite_a @ A5 )
=> ( P @ A5 ) )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X2: a,F2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ~ ( member_a @ X2 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_a @ X2 @ F2 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_501_finite__ne__induct,axiom,
! [F: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F )
=> ( ( F != bot_bot_set_set_a )
=> ( ! [X2: set_a] : ( P @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) )
=> ( ! [X2: set_a,F2: set_set_a] :
( ( finite_finite_set_a @ F2 )
=> ( ( F2 != bot_bot_set_set_a )
=> ( ~ ( member_set_a @ X2 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_set_a @ X2 @ F2 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_ne_induct
thf(fact_502_finite__ne__induct,axiom,
! [F: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ F )
=> ( ( F != bot_bot_set_list_a )
=> ( ! [X2: list_a] : ( P @ ( insert_list_a @ X2 @ bot_bot_set_list_a ) )
=> ( ! [X2: list_a,F2: set_list_a] :
( ( finite_finite_list_a @ F2 )
=> ( ( F2 != bot_bot_set_list_a )
=> ( ~ ( member_list_a @ X2 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_list_a @ X2 @ F2 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_ne_induct
thf(fact_503_finite__ne__induct,axiom,
! [F: set_a,P: set_a > $o] :
( ( finite_finite_a @ F )
=> ( ( F != bot_bot_set_a )
=> ( ! [X2: a] : ( P @ ( insert_a @ X2 @ bot_bot_set_a ) )
=> ( ! [X2: a,F2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ( F2 != bot_bot_set_a )
=> ( ~ ( member_a @ X2 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_a @ X2 @ F2 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_ne_induct
thf(fact_504_finite__induct,axiom,
! [F: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [X2: set_a,F2: set_set_a] :
( ( finite_finite_set_a @ F2 )
=> ( ~ ( member_set_a @ X2 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_set_a @ X2 @ F2 ) ) ) ) )
=> ( P @ F ) ) ) ) ).
% finite_induct
thf(fact_505_finite__induct,axiom,
! [F: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ F )
=> ( ( P @ bot_bot_set_list_a )
=> ( ! [X2: list_a,F2: set_list_a] :
( ( finite_finite_list_a @ F2 )
=> ( ~ ( member_list_a @ X2 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_list_a @ X2 @ F2 ) ) ) ) )
=> ( P @ F ) ) ) ) ).
% finite_induct
thf(fact_506_finite__induct,axiom,
! [F: set_a,P: set_a > $o] :
( ( finite_finite_a @ F )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X2: a,F2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ~ ( member_a @ X2 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_a @ X2 @ F2 ) ) ) ) )
=> ( P @ F ) ) ) ) ).
% finite_induct
thf(fact_507_finite_Osimps,axiom,
( finite_finite_list_a
= ( ^ [A6: set_list_a] :
( ( A6 = bot_bot_set_list_a )
| ? [A4: set_list_a,B6: list_a] :
( ( A6
= ( insert_list_a @ B6 @ A4 ) )
& ( finite_finite_list_a @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_508_finite_Osimps,axiom,
( finite_finite_a
= ( ^ [A6: set_a] :
( ( A6 = bot_bot_set_a )
| ? [A4: set_a,B6: a] :
( ( A6
= ( insert_a @ B6 @ A4 ) )
& ( finite_finite_a @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_509_finite_Ocases,axiom,
! [A: set_list_a] :
( ( finite_finite_list_a @ A )
=> ( ( A != bot_bot_set_list_a )
=> ~ ! [A5: set_list_a] :
( ? [A3: list_a] :
( A
= ( insert_list_a @ A3 @ A5 ) )
=> ~ ( finite_finite_list_a @ A5 ) ) ) ) ).
% finite.cases
thf(fact_510_finite_Ocases,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( ( A != bot_bot_set_a )
=> ~ ! [A5: set_a] :
( ? [A3: a] :
( A
= ( insert_a @ A3 @ A5 ) )
=> ~ ( finite_finite_a @ A5 ) ) ) ) ).
% finite.cases
thf(fact_511_infinite__arbitrarily__large,axiom,
! [A2: set_set_a,N: nat] :
( ~ ( finite_finite_set_a @ A2 )
=> ? [B5: set_set_a] :
( ( finite_finite_set_a @ B5 )
& ( ( finite_card_set_a @ B5 )
= N )
& ( ord_le3724670747650509150_set_a @ B5 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_512_infinite__arbitrarily__large,axiom,
! [A2: set_list_a,N: nat] :
( ~ ( finite_finite_list_a @ A2 )
=> ? [B5: set_list_a] :
( ( finite_finite_list_a @ B5 )
& ( ( finite_card_list_a @ B5 )
= N )
& ( ord_le8861187494160871172list_a @ B5 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_513_infinite__arbitrarily__large,axiom,
! [A2: set_a,N: nat] :
( ~ ( finite_finite_a @ A2 )
=> ? [B5: set_a] :
( ( finite_finite_a @ B5 )
& ( ( finite_card_a @ B5 )
= N )
& ( ord_less_eq_set_a @ B5 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_514_card__subset__eq,axiom,
! [B3: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B3 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B3 )
=> ( ( ( finite_card_set_a @ A2 )
= ( finite_card_set_a @ B3 ) )
=> ( A2 = B3 ) ) ) ) ).
% card_subset_eq
thf(fact_515_card__subset__eq,axiom,
! [B3: set_list_a,A2: set_list_a] :
( ( finite_finite_list_a @ B3 )
=> ( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( ( ( finite_card_list_a @ A2 )
= ( finite_card_list_a @ B3 ) )
=> ( A2 = B3 ) ) ) ) ).
% card_subset_eq
thf(fact_516_card__subset__eq,axiom,
! [B3: set_a,A2: set_a] :
( ( finite_finite_a @ B3 )
=> ( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ( finite_card_a @ A2 )
= ( finite_card_a @ B3 ) )
=> ( A2 = B3 ) ) ) ) ).
% card_subset_eq
thf(fact_517_ordered__ring__class_Ole__add__iff2,axiom,
! [A: int,E2: int,C: int,B: int,D2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D2 ) )
= ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D2 ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_518_ordered__ring__class_Ole__add__iff1,axiom,
! [A: int,E2: int,C: int,B: int,D2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D2 ) )
= ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C ) @ D2 ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_519_finite__subset__induct_H,axiom,
! [F: set_set_a,A2: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F )
=> ( ( ord_le3724670747650509150_set_a @ F @ A2 )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [A3: set_a,F2: set_set_a] :
( ( finite_finite_set_a @ F2 )
=> ( ( member_set_a @ A3 @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ F2 @ A2 )
=> ( ~ ( member_set_a @ A3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_set_a @ A3 @ F2 ) ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_520_finite__subset__induct_H,axiom,
! [F: set_list_a,A2: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ F )
=> ( ( ord_le8861187494160871172list_a @ F @ A2 )
=> ( ( P @ bot_bot_set_list_a )
=> ( ! [A3: list_a,F2: set_list_a] :
( ( finite_finite_list_a @ F2 )
=> ( ( member_list_a @ A3 @ A2 )
=> ( ( ord_le8861187494160871172list_a @ F2 @ A2 )
=> ( ~ ( member_list_a @ A3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_list_a @ A3 @ F2 ) ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_521_finite__subset__induct_H,axiom,
! [F: set_a,A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F )
=> ( ( ord_less_eq_set_a @ F @ A2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A3: a,F2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ( member_a @ A3 @ A2 )
=> ( ( ord_less_eq_set_a @ F2 @ A2 )
=> ( ~ ( member_a @ A3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_a @ A3 @ F2 ) ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_522_finite__subset__induct,axiom,
! [F: set_set_a,A2: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F )
=> ( ( ord_le3724670747650509150_set_a @ F @ A2 )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [A3: set_a,F2: set_set_a] :
( ( finite_finite_set_a @ F2 )
=> ( ( member_set_a @ A3 @ A2 )
=> ( ~ ( member_set_a @ A3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_set_a @ A3 @ F2 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct
thf(fact_523_finite__subset__induct,axiom,
! [F: set_list_a,A2: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ F )
=> ( ( ord_le8861187494160871172list_a @ F @ A2 )
=> ( ( P @ bot_bot_set_list_a )
=> ( ! [A3: list_a,F2: set_list_a] :
( ( finite_finite_list_a @ F2 )
=> ( ( member_list_a @ A3 @ A2 )
=> ( ~ ( member_list_a @ A3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_list_a @ A3 @ F2 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct
thf(fact_524_finite__subset__induct,axiom,
! [F: set_a,A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F )
=> ( ( ord_less_eq_set_a @ F @ A2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A3: a,F2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ( member_a @ A3 @ A2 )
=> ( ~ ( member_a @ A3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_a @ A3 @ F2 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct
thf(fact_525_subfield__m__inv__simprule,axiom,
! [K2: set_a,K: a,A: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_a @ K @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ r @ K @ A ) @ K2 )
=> ( member_a @ A @ K2 ) ) ) ) ) ).
% subfield_m_inv_simprule
thf(fact_526_carrier__is__subalgebra,axiom,
! [K2: set_a] :
( ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( embedd9027525575939734154ra_a_b @ K2 @ ( partia707051561876973205xt_a_b @ r ) @ r ) ) ).
% carrier_is_subalgebra
thf(fact_527_subalgebra__in__carrier,axiom,
! [K2: set_a,V: set_a] :
( ( embedd9027525575939734154ra_a_b @ K2 @ V @ r )
=> ( ord_less_eq_set_a @ V @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% subalgebra_in_carrier
thf(fact_528_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_529_ring_Or__right__minus__eq,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ( a_minus_a_b @ R @ A @ B )
= ( zero_a_b @ R ) )
= ( A = B ) ) ) ) ) ).
% ring.r_right_minus_eq
thf(fact_530_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_531_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_532_subring__props_I2_J,axiom,
! [K2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( member_a @ ( zero_a_b @ r ) @ K2 ) ) ).
% subring_props(2)
thf(fact_533_subring__props_I7_J,axiom,
! [K2: set_a,H1: a,H22: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_a @ H1 @ K2 )
=> ( ( member_a @ H22 @ K2 )
=> ( member_a @ ( add_a_b @ r @ H1 @ H22 ) @ K2 ) ) ) ) ).
% subring_props(7)
thf(fact_534_add_Onat__pow__pow,axiom,
! [X: a,M: nat,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_nat @ r @ M @ ( add_pow_a_b_nat @ r @ N @ X ) )
= ( add_pow_a_b_nat @ r @ ( times_times_nat @ N @ M ) @ X ) ) ) ).
% add.nat_pow_pow
thf(fact_535_subring__props_I6_J,axiom,
! [K2: set_a,H1: a,H22: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_a @ H1 @ K2 )
=> ( ( member_a @ H22 @ K2 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ H1 @ H22 ) @ K2 ) ) ) ) ).
% subring_props(6)
thf(fact_536_subring__props_I4_J,axiom,
! [K2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( K2 != bot_bot_set_a ) ) ).
% subring_props(4)
thf(fact_537_subring__props_I5_J,axiom,
! [K2: set_a,H: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_a @ H @ K2 )
=> ( member_a @ ( a_inv_a_b @ r @ H ) @ K2 ) ) ) ).
% subring_props(5)
thf(fact_538_subring__props_I3_J,axiom,
! [K2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( member_a @ ( one_a_ring_ext_a_b @ r ) @ K2 ) ) ).
% subring_props(3)
thf(fact_539_subring__props_I1_J,axiom,
! [K2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% subring_props(1)
thf(fact_540_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_541_inv__unique,axiom,
! [Y: a,X: a,Y2: 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 @ Y2 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% inv_unique
thf(fact_542_DiffI,axiom,
! [C: set_a,A2: set_set_a,B3: set_set_a] :
( ( member_set_a @ C @ A2 )
=> ( ~ ( member_set_a @ C @ B3 )
=> ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A2 @ B3 ) ) ) ) ).
% DiffI
thf(fact_543_DiffI,axiom,
! [C: a,A2: set_a,B3: set_a] :
( ( member_a @ C @ A2 )
=> ( ~ ( member_a @ C @ B3 )
=> ( member_a @ C @ ( minus_minus_set_a @ A2 @ B3 ) ) ) ) ).
% DiffI
thf(fact_544_Diff__iff,axiom,
! [C: set_a,A2: set_set_a,B3: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A2 @ B3 ) )
= ( ( member_set_a @ C @ A2 )
& ~ ( member_set_a @ C @ B3 ) ) ) ).
% Diff_iff
thf(fact_545_Diff__iff,axiom,
! [C: a,A2: set_a,B3: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B3 ) )
= ( ( member_a @ C @ A2 )
& ~ ( member_a @ C @ B3 ) ) ) ).
% Diff_iff
thf(fact_546_Diff__idemp,axiom,
! [A2: set_a,B3: set_a] :
( ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ B3 ) @ B3 )
= ( minus_minus_set_a @ A2 @ B3 ) ) ).
% Diff_idemp
thf(fact_547_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_548_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_549_line__extension__smult__closed,axiom,
! [K2: set_a,E: set_a,A: a,K: a,U: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ! [K3: a,V2: a] :
( ( member_a @ K3 @ K2 )
=> ( ( member_a @ V2 @ E )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K3 @ V2 ) @ E ) ) )
=> ( ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ K @ K2 )
=> ( ( member_a @ U @ ( embedd971793762689825387on_a_b @ r @ K2 @ A @ E ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K @ U ) @ ( embedd971793762689825387on_a_b @ r @ K2 @ A @ E ) ) ) ) ) ) ) ) ).
% line_extension_smult_closed
thf(fact_550_one__closed,axiom,
member_a @ ( one_a_ring_ext_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% one_closed
thf(fact_551_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_552_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_553_DiffE,axiom,
! [C: set_a,A2: set_set_a,B3: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A2 @ B3 ) )
=> ~ ( ( member_set_a @ C @ A2 )
=> ( member_set_a @ C @ B3 ) ) ) ).
% DiffE
thf(fact_554_DiffE,axiom,
! [C: a,A2: set_a,B3: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B3 ) )
=> ~ ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B3 ) ) ) ).
% DiffE
thf(fact_555_DiffD1,axiom,
! [C: set_a,A2: set_set_a,B3: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A2 @ B3 ) )
=> ( member_set_a @ C @ A2 ) ) ).
% DiffD1
thf(fact_556_DiffD1,axiom,
! [C: a,A2: set_a,B3: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B3 ) )
=> ( member_a @ C @ A2 ) ) ).
% DiffD1
thf(fact_557_DiffD2,axiom,
! [C: set_a,A2: set_set_a,B3: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A2 @ B3 ) )
=> ~ ( member_set_a @ C @ B3 ) ) ).
% DiffD2
thf(fact_558_DiffD2,axiom,
! [C: a,A2: set_a,B3: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B3 ) )
=> ~ ( member_a @ C @ B3 ) ) ).
% DiffD2
thf(fact_559_ring_Osubring__props_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( member_a @ ( one_a_ring_ext_a_b @ R ) @ K2 ) ) ) ).
% ring.subring_props(3)
thf(fact_560_ring__hom__one,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b] :
( ( member_a_a @ H @ ( ring_hom_a_b_a_b @ R @ S ) )
=> ( ( H @ ( one_a_ring_ext_a_b @ R ) )
= ( one_a_ring_ext_a_b @ S ) ) ) ).
% ring_hom_one
thf(fact_561_ring__iso__memE_I4_J,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b] :
( ( member_a_a @ H @ ( ring_iso_a_b_a_b @ R @ S ) )
=> ( ( H @ ( one_a_ring_ext_a_b @ R ) )
= ( one_a_ring_ext_a_b @ S ) ) ) ).
% ring_iso_memE(4)
thf(fact_562_subalgebra_Osmult__closed,axiom,
! [K2: set_a,V: set_a,R: partia2175431115845679010xt_a_b,K: a,V3: a] :
( ( embedd9027525575939734154ra_a_b @ K2 @ V @ R )
=> ( ( member_a @ K @ K2 )
=> ( ( member_a @ V3 @ V )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ K @ V3 ) @ V ) ) ) ) ).
% subalgebra.smult_closed
thf(fact_563_ring_Osubring__props_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( member_a @ ( zero_a_b @ R ) @ K2 ) ) ) ).
% ring.subring_props(2)
thf(fact_564_ring_Osubring__props_I7_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,H1: a,H22: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( member_a @ H1 @ K2 )
=> ( ( member_a @ H22 @ K2 )
=> ( member_a @ ( add_a_b @ R @ H1 @ H22 ) @ K2 ) ) ) ) ) ).
% ring.subring_props(7)
thf(fact_565_ring_Osubring__props_I4_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( K2 != bot_bot_set_a ) ) ) ).
% ring.subring_props(4)
thf(fact_566_ring_Osubring__props_I6_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,H1: a,H22: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( member_a @ H1 @ K2 )
=> ( ( member_a @ H22 @ K2 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ H1 @ H22 ) @ K2 ) ) ) ) ) ).
% ring.subring_props(6)
thf(fact_567_ring_Oring__simprules_I6_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( member_a @ ( one_a_ring_ext_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% ring.ring_simprules(6)
thf(fact_568_ring_Osubring__props_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,H: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( member_a @ H @ K2 )
=> ( member_a @ ( a_inv_a_b @ R @ H ) @ K2 ) ) ) ) ).
% ring.subring_props(5)
thf(fact_569_semiring_Osemiring__simprules_I4_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( member_a @ ( one_a_ring_ext_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% semiring.semiring_simprules(4)
thf(fact_570_ring_Osubring__props_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ).
% ring.subring_props(1)
thf(fact_571_ring_Oring__simprules_I12_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( one_a_ring_ext_a_b @ R ) @ X )
= X ) ) ) ).
% ring.ring_simprules(12)
thf(fact_572_semiring_Osemiring__simprules_I9_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( one_a_ring_ext_a_b @ R ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(9)
thf(fact_573_ring__hom__memI,axiom,
! [R: partia2175431115845679010xt_a_b,H: a > a,S: partia2175431115845679010xt_a_b] :
( ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( H @ X2 ) @ ( partia707051561876973205xt_a_b @ S ) ) )
=> ( ! [X2: a,Y4: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H @ ( mult_a_ring_ext_a_b @ R @ X2 @ Y4 ) )
= ( mult_a_ring_ext_a_b @ S @ ( H @ X2 ) @ ( H @ Y4 ) ) ) ) )
=> ( ! [X2: a,Y4: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H @ ( add_a_b @ R @ X2 @ Y4 ) )
= ( add_a_b @ S @ ( H @ X2 ) @ ( H @ Y4 ) ) ) ) )
=> ( ( ( H @ ( one_a_ring_ext_a_b @ R ) )
= ( one_a_ring_ext_a_b @ S ) )
=> ( member_a_a @ H @ ( ring_hom_a_b_a_b @ R @ S ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_574_ring_Ocarrier__is__subalgebra,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( embedd9027525575939734154ra_a_b @ K2 @ ( partia707051561876973205xt_a_b @ R ) @ R ) ) ) ).
% ring.carrier_is_subalgebra
thf(fact_575_ring_Osubalgebra__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,V: set_a] :
( ( ring_a_b @ R )
=> ( ( embedd9027525575939734154ra_a_b @ K2 @ V @ R )
=> ( ord_less_eq_set_a @ V @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ).
% ring.subalgebra_in_carrier
thf(fact_576_ring_Oline__extension__smult__closed,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,E: set_a,A: a,K: a,U: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ! [K3: a,V2: a] :
( ( member_a @ K3 @ K2 )
=> ( ( member_a @ V2 @ E )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ K3 @ V2 ) @ E ) ) )
=> ( ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ K @ K2 )
=> ( ( member_a @ U @ ( embedd971793762689825387on_a_b @ R @ K2 @ A @ E ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ K @ U ) @ ( embedd971793762689825387on_a_b @ R @ K2 @ A @ E ) ) ) ) ) ) ) ) ) ).
% ring.line_extension_smult_closed
thf(fact_577_ring_Ogenideal__one,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( ( genideal_a_b @ R @ ( insert_a @ ( one_a_ring_ext_a_b @ R ) @ bot_bot_set_a ) )
= ( partia707051561876973205xt_a_b @ R ) ) ) ).
% ring.genideal_one
thf(fact_578_semiring_Oone__zeroD,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( ( ( 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 ) ) ) ) ).
% semiring.one_zeroD
thf(fact_579_semiring_Oone__zeroI,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( ( ( 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 ) ) ) ) ).
% semiring.one_zeroI
thf(fact_580_semiring_Ocarrier__one__zero,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( ( ( 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 ) ) ) ) ).
% semiring.carrier_one_zero
thf(fact_581_semiring_Ocarrier__one__not__zero,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( ( ( 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 ) ) ) ) ).
% semiring.carrier_one_not_zero
thf(fact_582_ring_Osubfield__m__inv__simprule,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,K: a,A: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( member_a @ K @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ R @ K @ A ) @ K2 )
=> ( member_a @ A @ K2 ) ) ) ) ) ) ).
% ring.subfield_m_inv_simprule
thf(fact_583_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_584_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_585_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_586_subfield__m__inv_I3_J,axiom,
! [K2: set_a,K: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_a @ K @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( m_inv_a_ring_ext_a_b @ r @ K ) @ K )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% subfield_m_inv(3)
thf(fact_587_inv__char,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 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ Y @ X )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( m_inv_a_ring_ext_a_b @ r @ X )
= Y ) ) ) ) ) ).
% inv_char
thf(fact_588_inv__unique_H,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 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ Y @ X )
= ( one_a_ring_ext_a_b @ r ) )
=> ( Y
= ( m_inv_a_ring_ext_a_b @ r @ X ) ) ) ) ) ) ).
% inv_unique'
thf(fact_589_subfield__m__inv_I1_J,axiom,
! [K2: set_a,K: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_a @ K @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( member_a @ ( m_inv_a_ring_ext_a_b @ r @ K ) @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ) ) ).
% subfield_m_inv(1)
thf(fact_590_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_591_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_592_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_593_subfield__m__inv_I2_J,axiom,
! [K2: set_a,K: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_a @ K @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ K @ ( m_inv_a_ring_ext_a_b @ r @ K ) )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% subfield_m_inv(2)
thf(fact_594_inv__one,axiom,
( ( m_inv_a_ring_ext_a_b @ r @ ( one_a_ring_ext_a_b @ r ) )
= ( one_a_ring_ext_a_b @ r ) ) ).
% inv_one
thf(fact_595_inv__neg__one,axiom,
( ( m_inv_a_ring_ext_a_b @ r @ ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) )
= ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) ) ).
% inv_neg_one
thf(fact_596_ring_Oinv__neg__one,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( ( m_inv_a_ring_ext_a_b @ R @ ( a_inv_a_b @ R @ ( one_a_ring_ext_a_b @ R ) ) )
= ( a_inv_a_b @ R @ ( one_a_ring_ext_a_b @ R ) ) ) ) ).
% ring.inv_neg_one
thf(fact_597_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_598_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_599_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_600_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_601_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_602_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_603_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_604_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_605_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_606_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_607_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_608_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_609_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_610_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_611_ring_Osubfield__m__inv_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,K: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( member_a @ K @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ( member_a @ ( m_inv_a_ring_ext_a_b @ R @ K ) @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) ) ) ) ) ).
% ring.subfield_m_inv(1)
thf(fact_612_ring_Osubfield__m__inv_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,K: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( member_a @ K @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ( ( mult_a_ring_ext_a_b @ R @ K @ ( m_inv_a_ring_ext_a_b @ R @ K ) )
= ( one_a_ring_ext_a_b @ R ) ) ) ) ) ).
% ring.subfield_m_inv(2)
thf(fact_613_ring_Osubfield__m__inv_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,K: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( member_a @ K @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( m_inv_a_ring_ext_a_b @ R @ K ) @ K )
= ( one_a_ring_ext_a_b @ R ) ) ) ) ) ).
% ring.subfield_m_inv(3)
thf(fact_614_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_615_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_616_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_617_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_618_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_619_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_620_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_621_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_622_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_623_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_624_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_625_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_626_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_627_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_628_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_629_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_630_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_631_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
? [K4: nat] :
( N3
= ( plus_plus_nat @ M3 @ K4 ) ) ) ) ).
% nat_le_iff_add
thf(fact_632_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_633_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_634_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_635_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_636_subfieldE_I4_J,axiom,
! [K2: set_a,R: partia2175431115845679010xt_a_b,K1: a,K22: a] :
( ( subfield_a_b @ K2 @ R )
=> ( ( member_a @ K1 @ K2 )
=> ( ( member_a @ K22 @ K2 )
=> ( ( mult_a_ring_ext_a_b @ R @ K1 @ K22 )
= ( mult_a_ring_ext_a_b @ R @ K22 @ K1 ) ) ) ) ) ).
% subfieldE(4)
thf(fact_637_subfieldE_I3_J,axiom,
! [K2: set_a,R: partia2175431115845679010xt_a_b] :
( ( subfield_a_b @ K2 @ R )
=> ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% subfieldE(3)
thf(fact_638_subfieldE_I5_J,axiom,
! [K2: set_a,R: partia2175431115845679010xt_a_b,K1: a,K22: a] :
( ( subfield_a_b @ K2 @ R )
=> ( ( member_a @ K1 @ K2 )
=> ( ( member_a @ K22 @ K2 )
=> ( ( ( mult_a_ring_ext_a_b @ R @ K1 @ K22 )
= ( zero_a_b @ R ) )
=> ( ( K1
= ( zero_a_b @ R ) )
| ( K22
= ( zero_a_b @ R ) ) ) ) ) ) ) ).
% subfieldE(5)
thf(fact_639_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_640_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_641_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_642_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_643_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_644_subfieldE_I6_J,axiom,
! [K2: set_a,R: partia2175431115845679010xt_a_b] :
( ( subfield_a_b @ K2 @ R )
=> ( ( one_a_ring_ext_a_b @ R )
!= ( zero_a_b @ R ) ) ) ).
% subfieldE(6)
thf(fact_645_space__subgroup__props_I6_J,axiom,
! [K2: set_a,N: nat,E: set_a,K: a,A: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E )
=> ( ( member_a @ K @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ r @ K @ A ) @ E )
=> ( member_a @ A @ E ) ) ) ) ) ) ).
% space_subgroup_props(6)
thf(fact_646_subalbegra__incl__imp__finite__dimension,axiom,
! [K2: set_a,E: set_a,V: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K2 @ E )
=> ( ( embedd9027525575939734154ra_a_b @ K2 @ V @ r )
=> ( ( ord_less_eq_set_a @ V @ E )
=> ( embedd8708762675212832759on_a_b @ r @ K2 @ V ) ) ) ) ) ).
% subalbegra_incl_imp_finite_dimension
thf(fact_647_subringI,axiom,
! [H2: set_a] :
( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ ( one_a_ring_ext_a_b @ r ) @ H2 )
=> ( ! [H3: a] :
( ( member_a @ H3 @ H2 )
=> ( member_a @ ( a_inv_a_b @ r @ H3 ) @ H2 ) )
=> ( ! [H12: a,H23: a] :
( ( member_a @ H12 @ H2 )
=> ( ( member_a @ H23 @ H2 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ H12 @ H23 ) @ H2 ) ) )
=> ( ! [H12: a,H23: a] :
( ( member_a @ H12 @ H2 )
=> ( ( member_a @ H23 @ H2 )
=> ( member_a @ ( add_a_b @ r @ H12 @ H23 ) @ H2 ) ) )
=> ( subring_a_b @ H2 @ r ) ) ) ) ) ) ).
% subringI
thf(fact_648_nat__eq__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_649_nat__eq__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( M
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_650_carrier__is__subring,axiom,
subring_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% carrier_is_subring
thf(fact_651_dimension__is__inj,axiom,
! [K2: set_a,N: nat,E: set_a,M: nat] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E )
=> ( ( embedd2795209813406577254on_a_b @ r @ M @ K2 @ E )
=> ( N = M ) ) ) ) ).
% dimension_is_inj
thf(fact_652_telescopic__base__dim_I1_J,axiom,
! [K2: set_a,F: set_a,E: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( subfield_a_b @ F @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K2 @ F )
=> ( ( embedd8708762675212832759on_a_b @ r @ F @ E )
=> ( embedd8708762675212832759on_a_b @ r @ K2 @ E ) ) ) ) ) ).
% telescopic_base_dim(1)
thf(fact_653_finite__dimensionE_H,axiom,
! [K2: set_a,E: set_a] :
( ( embedd8708762675212832759on_a_b @ r @ K2 @ E )
=> ~ ! [N2: nat] :
~ ( embedd2795209813406577254on_a_b @ r @ N2 @ K2 @ E ) ) ).
% finite_dimensionE'
thf(fact_654_finite__dimensionI,axiom,
! [N: nat,K2: set_a,E: set_a] :
( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E )
=> ( embedd8708762675212832759on_a_b @ r @ K2 @ E ) ) ).
% finite_dimensionI
thf(fact_655_finite__dimension__def,axiom,
! [K2: set_a,E: set_a] :
( ( embedd8708762675212832759on_a_b @ r @ K2 @ E )
= ( ? [N3: nat] : ( embedd2795209813406577254on_a_b @ r @ N3 @ K2 @ E ) ) ) ).
% finite_dimension_def
thf(fact_656_space__subgroup__props_I2_J,axiom,
! [K2: set_a,N: nat,E: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E )
=> ( member_a @ ( zero_a_b @ r ) @ E ) ) ) ).
% space_subgroup_props(2)
thf(fact_657_space__subgroup__props_I3_J,axiom,
! [K2: set_a,N: nat,E: set_a,V1: a,V22: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E )
=> ( ( member_a @ V1 @ E )
=> ( ( member_a @ V22 @ E )
=> ( member_a @ ( add_a_b @ r @ V1 @ V22 ) @ E ) ) ) ) ) ).
% space_subgroup_props(3)
thf(fact_658_telescopic__base,axiom,
! [K2: set_a,F: set_a,N: nat,M: nat,E: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( subfield_a_b @ F @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ F )
=> ( ( embedd2795209813406577254on_a_b @ r @ M @ F @ E )
=> ( embedd2795209813406577254on_a_b @ r @ ( times_times_nat @ N @ M ) @ K2 @ E ) ) ) ) ) ).
% telescopic_base
thf(fact_659_space__subgroup__props_I5_J,axiom,
! [K2: set_a,N: nat,E: set_a,K: a,V3: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E )
=> ( ( member_a @ K @ K2 )
=> ( ( member_a @ V3 @ E )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K @ V3 ) @ E ) ) ) ) ) ).
% space_subgroup_props(5)
thf(fact_660_space__subgroup__props_I4_J,axiom,
! [K2: set_a,N: nat,E: set_a,V3: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E )
=> ( ( member_a @ V3 @ E )
=> ( member_a @ ( a_inv_a_b @ r @ V3 ) @ E ) ) ) ) ).
% space_subgroup_props(4)
thf(fact_661_unique__dimension,axiom,
! [K2: set_a,E: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K2 @ E )
=> ? [X2: nat] :
( ( embedd2795209813406577254on_a_b @ r @ X2 @ K2 @ E )
& ! [Y6: nat] :
( ( embedd2795209813406577254on_a_b @ r @ Y6 @ K2 @ E )
=> ( Y6 = X2 ) ) ) ) ) ).
% unique_dimension
thf(fact_662_finite__dimension__imp__subalgebra,axiom,
! [K2: set_a,E: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K2 @ E )
=> ( embedd9027525575939734154ra_a_b @ K2 @ E @ r ) ) ) ).
% finite_dimension_imp_subalgebra
thf(fact_663_space__subgroup__props_I1_J,axiom,
! [K2: set_a,N: nat,E: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E )
=> ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% space_subgroup_props(1)
thf(fact_664_subringE_I4_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b] :
( ( subring_a_b @ H2 @ R )
=> ( H2 != bot_bot_set_a ) ) ).
% subringE(4)
thf(fact_665_subringE_I6_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b,H1: a,H22: a] :
( ( subring_a_b @ H2 @ R )
=> ( ( member_a @ H1 @ H2 )
=> ( ( member_a @ H22 @ H2 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ H1 @ H22 ) @ H2 ) ) ) ) ).
% subringE(6)
thf(fact_666_subringE_I5_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b,H: a] :
( ( subring_a_b @ H2 @ R )
=> ( ( member_a @ H @ H2 )
=> ( member_a @ ( a_inv_a_b @ R @ H ) @ H2 ) ) ) ).
% subringE(5)
thf(fact_667_subringE_I7_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b,H1: a,H22: a] :
( ( subring_a_b @ H2 @ R )
=> ( ( member_a @ H1 @ H2 )
=> ( ( member_a @ H22 @ H2 )
=> ( member_a @ ( add_a_b @ R @ H1 @ H22 ) @ H2 ) ) ) ) ).
% subringE(7)
thf(fact_668_subringE_I2_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b] :
( ( subring_a_b @ H2 @ R )
=> ( member_a @ ( zero_a_b @ R ) @ H2 ) ) ).
% subringE(2)
thf(fact_669_ring_Ounique__dimension,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,E: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd8708762675212832759on_a_b @ R @ K2 @ E )
=> ? [X2: nat] :
( ( embedd2795209813406577254on_a_b @ R @ X2 @ K2 @ E )
& ! [Y6: nat] :
( ( embedd2795209813406577254on_a_b @ R @ Y6 @ K2 @ E )
=> ( Y6 = X2 ) ) ) ) ) ) ).
% ring.unique_dimension
thf(fact_670_ring_Ofinite__dimension__def,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,E: set_a] :
( ( ring_a_b @ R )
=> ( ( embedd8708762675212832759on_a_b @ R @ K2 @ E )
= ( ? [N3: nat] : ( embedd2795209813406577254on_a_b @ R @ N3 @ K2 @ E ) ) ) ) ).
% ring.finite_dimension_def
thf(fact_671_ring_Ofinite__dimensionE_H,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,E: set_a] :
( ( ring_a_b @ R )
=> ( ( embedd8708762675212832759on_a_b @ R @ K2 @ E )
=> ~ ! [N2: nat] :
~ ( embedd2795209813406577254on_a_b @ R @ N2 @ K2 @ E ) ) ) ).
% ring.finite_dimensionE'
thf(fact_672_ring_Ofinite__dimensionI,axiom,
! [R: partia2175431115845679010xt_a_b,N: nat,K2: set_a,E: set_a] :
( ( ring_a_b @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ E )
=> ( embedd8708762675212832759on_a_b @ R @ K2 @ E ) ) ) ).
% ring.finite_dimensionI
thf(fact_673_ring_Ofinite__dimension_Ocong,axiom,
embedd8708762675212832759on_a_b = embedd8708762675212832759on_a_b ).
% ring.finite_dimension.cong
thf(fact_674_ring_Odimension_Ocong,axiom,
embedd2795209813406577254on_a_b = embedd2795209813406577254on_a_b ).
% ring.dimension.cong
thf(fact_675_subringE_I1_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b] :
( ( subring_a_b @ H2 @ R )
=> ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% subringE(1)
thf(fact_676_ring_Ocarrier__is__subring,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( subring_a_b @ ( partia707051561876973205xt_a_b @ R ) @ R ) ) ).
% ring.carrier_is_subring
thf(fact_677_ring_Odimension__is__inj,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,N: nat,E: set_a,M: nat] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ E )
=> ( ( embedd2795209813406577254on_a_b @ R @ M @ K2 @ E )
=> ( N = M ) ) ) ) ) ).
% ring.dimension_is_inj
thf(fact_678_ring_Otelescopic__base__dim_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,F: set_a,E: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( subfield_a_b @ F @ R )
=> ( ( embedd8708762675212832759on_a_b @ R @ K2 @ F )
=> ( ( embedd8708762675212832759on_a_b @ R @ F @ E )
=> ( embedd8708762675212832759on_a_b @ R @ K2 @ E ) ) ) ) ) ) ).
% ring.telescopic_base_dim(1)
thf(fact_679_ring_Ospace__subgroup__props_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,N: nat,E: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ E )
=> ( member_a @ ( zero_a_b @ R ) @ E ) ) ) ) ).
% ring.space_subgroup_props(2)
thf(fact_680_ring_Ospace__subgroup__props_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,N: nat,E: set_a,V1: a,V22: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ E )
=> ( ( member_a @ V1 @ E )
=> ( ( member_a @ V22 @ E )
=> ( member_a @ ( add_a_b @ R @ V1 @ V22 ) @ E ) ) ) ) ) ) ).
% ring.space_subgroup_props(3)
thf(fact_681_ring_Ospace__subgroup__props_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,N: nat,E: set_a,K: a,V3: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ E )
=> ( ( member_a @ K @ K2 )
=> ( ( member_a @ V3 @ E )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ K @ V3 ) @ E ) ) ) ) ) ) ).
% ring.space_subgroup_props(5)
thf(fact_682_ring_Otelescopic__base,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,F: set_a,N: nat,M: nat,E: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( subfield_a_b @ F @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ F )
=> ( ( embedd2795209813406577254on_a_b @ R @ M @ F @ E )
=> ( embedd2795209813406577254on_a_b @ R @ ( times_times_nat @ N @ M ) @ K2 @ E ) ) ) ) ) ) ).
% ring.telescopic_base
thf(fact_683_ring_Ospace__subgroup__props_I4_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,N: nat,E: set_a,V3: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ E )
=> ( ( member_a @ V3 @ E )
=> ( member_a @ ( a_inv_a_b @ R @ V3 ) @ E ) ) ) ) ) ).
% ring.space_subgroup_props(4)
thf(fact_684_ring_Ospace__subgroup__props_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,N: nat,E: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ E )
=> ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% ring.space_subgroup_props(1)
thf(fact_685_ring_Ofinite__dimension__imp__subalgebra,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,E: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd8708762675212832759on_a_b @ R @ K2 @ E )
=> ( embedd9027525575939734154ra_a_b @ K2 @ E @ R ) ) ) ) ).
% ring.finite_dimension_imp_subalgebra
thf(fact_686_ring_Osubalbegra__incl__imp__finite__dimension,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,E: set_a,V: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd8708762675212832759on_a_b @ R @ K2 @ E )
=> ( ( embedd9027525575939734154ra_a_b @ K2 @ V @ R )
=> ( ( ord_less_eq_set_a @ V @ E )
=> ( embedd8708762675212832759on_a_b @ R @ K2 @ V ) ) ) ) ) ) ).
% ring.subalbegra_incl_imp_finite_dimension
thf(fact_687_ring_OsubringI,axiom,
! [R: partia2175431115845679010xt_a_b,H2: set_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ ( one_a_ring_ext_a_b @ R ) @ H2 )
=> ( ! [H3: a] :
( ( member_a @ H3 @ H2 )
=> ( member_a @ ( a_inv_a_b @ R @ H3 ) @ H2 ) )
=> ( ! [H12: a,H23: a] :
( ( member_a @ H12 @ H2 )
=> ( ( member_a @ H23 @ H2 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ H12 @ H23 ) @ H2 ) ) )
=> ( ! [H12: a,H23: a] :
( ( member_a @ H12 @ H2 )
=> ( ( member_a @ H23 @ H2 )
=> ( member_a @ ( add_a_b @ R @ H12 @ H23 ) @ H2 ) ) )
=> ( subring_a_b @ H2 @ R ) ) ) ) ) ) ) ).
% ring.subringI
thf(fact_688_ring_Ospace__subgroup__props_I6_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,N: nat,E: set_a,K: a,A: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ E )
=> ( ( member_a @ K @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ R @ K @ A ) @ E )
=> ( member_a @ A @ E ) ) ) ) ) ) ) ).
% ring.space_subgroup_props(6)
thf(fact_689_nat__diff__add__eq2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_690_nat__diff__add__eq1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_691_nat__le__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_692_nat__le__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_693_dimension__zero,axiom,
! [K2: set_a,E: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ zero_zero_nat @ K2 @ E )
=> ( E
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ) ).
% dimension_zero
thf(fact_694_zero__dim,axiom,
! [K2: set_a] : ( embedd2795209813406577254on_a_b @ r @ zero_zero_nat @ K2 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ).
% zero_dim
thf(fact_695_dimension__backwards,axiom,
! [K2: set_a,N: nat,E: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ ( suc @ N ) @ K2 @ E )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
& ? [E3: set_a] :
( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E3 )
& ~ ( member_a @ X2 @ E3 )
& ( E
= ( embedd971793762689825387on_a_b @ r @ K2 @ X2 @ E3 ) ) ) ) ) ) ).
% dimension_backwards
thf(fact_696_subcringI,axiom,
! [H2: set_a] :
( ( subring_a_b @ H2 @ r )
=> ( ! [H12: a,H23: a] :
( ( member_a @ H12 @ H2 )
=> ( ( member_a @ H23 @ H2 )
=> ( ( mult_a_ring_ext_a_b @ r @ H12 @ H23 )
= ( mult_a_ring_ext_a_b @ r @ H23 @ H12 ) ) ) )
=> ( subcring_a_b @ H2 @ r ) ) ) ).
% subcringI
thf(fact_697_add_Onat__pow__Suc2,axiom,
! [X: a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_nat @ r @ ( suc @ N ) @ X )
= ( add_a_b @ r @ X @ ( add_pow_a_b_nat @ r @ N @ X ) ) ) ) ).
% add.nat_pow_Suc2
thf(fact_698_Suc__dim,axiom,
! [V3: a,E: set_a,N: nat,K2: set_a] :
( ( member_a @ V3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ~ ( member_a @ V3 @ E )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E )
=> ( embedd2795209813406577254on_a_b @ r @ ( suc @ N ) @ K2 @ ( embedd971793762689825387on_a_b @ r @ K2 @ V3 @ E ) ) ) ) ) ).
% Suc_dim
thf(fact_699_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_700_mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_701_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_702_mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_703_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_704_mult__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
= ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_705_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_706_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_707_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_708_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_709_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_710_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_711_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_712_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_713_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_714_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_715_add__cancel__right__right,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ A @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_716_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_717_add__cancel__right__left,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ B @ A ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_718_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_719_add__cancel__left__right,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_720_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_721_add__cancel__left__left,axiom,
! [B: int,A: int] :
( ( ( plus_plus_int @ B @ A )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_722_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_723_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_724_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_725_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_726_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_727_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_728_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_729_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_730_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_731_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_732_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_733_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_734_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_735_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_736_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_737_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_738_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_739_dimension_Osimps,axiom,
! [A1: nat,A22: set_a,A32: set_a] :
( ( embedd2795209813406577254on_a_b @ r @ A1 @ A22 @ A32 )
= ( ? [K5: set_a] :
( ( A1 = zero_zero_nat )
& ( A22 = K5 )
& ( A32
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
| ? [V4: a,E4: set_a,N3: nat,K5: set_a] :
( ( A1
= ( suc @ N3 ) )
& ( A22 = K5 )
& ( A32
= ( embedd971793762689825387on_a_b @ r @ K5 @ V4 @ E4 ) )
& ( member_a @ V4 @ ( partia707051561876973205xt_a_b @ r ) )
& ~ ( member_a @ V4 @ E4 )
& ( embedd2795209813406577254on_a_b @ r @ N3 @ K5 @ E4 ) ) ) ) ).
% dimension.simps
thf(fact_740_dimension_Ocases,axiom,
! [A1: nat,A22: set_a,A32: set_a] :
( ( embedd2795209813406577254on_a_b @ r @ A1 @ A22 @ A32 )
=> ( ( ( A1 = zero_zero_nat )
=> ( A32
!= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ~ ! [V2: a,E5: set_a,N2: nat] :
( ( A1
= ( suc @ N2 ) )
=> ( ( A32
= ( embedd971793762689825387on_a_b @ r @ A22 @ V2 @ E5 ) )
=> ( ( member_a @ V2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ~ ( member_a @ V2 @ E5 )
=> ~ ( embedd2795209813406577254on_a_b @ r @ N2 @ A22 @ E5 ) ) ) ) ) ) ) ).
% dimension.cases
thf(fact_741_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_742_add__le__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_743_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_744_add__le__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_745_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_746_le__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_747_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_748_le__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_749_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_750_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_751_diff__ge__0__iff__ge,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_752_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_753_card_Oempty,axiom,
( ( finite_card_set_a @ bot_bot_set_set_a )
= zero_zero_nat ) ).
% card.empty
thf(fact_754_card_Oempty,axiom,
( ( finite_card_list_a @ bot_bot_set_list_a )
= zero_zero_nat ) ).
% card.empty
thf(fact_755_card_Oempty,axiom,
( ( finite_card_a @ bot_bot_set_a )
= zero_zero_nat ) ).
% card.empty
thf(fact_756_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_757_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_758_card_Oinfinite,axiom,
! [A2: set_set_a] :
( ~ ( finite_finite_set_a @ A2 )
=> ( ( finite_card_set_a @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_759_card_Oinfinite,axiom,
! [A2: set_a] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_card_a @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_760_card_Oinfinite,axiom,
! [A2: set_list_a] :
( ~ ( finite_finite_list_a @ A2 )
=> ( ( finite_card_list_a @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_761_card__0__eq,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( ( finite_card_set_a @ A2 )
= zero_zero_nat )
= ( A2 = bot_bot_set_set_a ) ) ) ).
% card_0_eq
thf(fact_762_card__0__eq,axiom,
! [A2: set_list_a] :
( ( finite_finite_list_a @ A2 )
=> ( ( ( finite_card_list_a @ A2 )
= zero_zero_nat )
= ( A2 = bot_bot_set_list_a ) ) ) ).
% card_0_eq
thf(fact_763_card__0__eq,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( ( finite_card_a @ A2 )
= zero_zero_nat )
= ( A2 = bot_bot_set_a ) ) ) ).
% card_0_eq
thf(fact_764_card__insert__disjoint,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ~ ( member_set_a @ X @ A2 )
=> ( ( finite_card_set_a @ ( insert_set_a @ X @ A2 ) )
= ( suc @ ( finite_card_set_a @ A2 ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_765_card__insert__disjoint,axiom,
! [A2: set_a,X: a] :
( ( finite_finite_a @ A2 )
=> ( ~ ( member_a @ X @ A2 )
=> ( ( finite_card_a @ ( insert_a @ X @ A2 ) )
= ( suc @ ( finite_card_a @ A2 ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_766_card__insert__disjoint,axiom,
! [A2: set_list_a,X: list_a] :
( ( finite_finite_list_a @ A2 )
=> ( ~ ( member_list_a @ X @ A2 )
=> ( ( finite_card_list_a @ ( insert_list_a @ X @ A2 ) )
= ( suc @ ( finite_card_list_a @ A2 ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_767_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_768_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_769_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_770_add_Onat__pow__0,axiom,
! [X: a] :
( ( add_pow_a_b_nat @ r @ zero_zero_nat @ X )
= ( zero_a_b @ r ) ) ).
% add.nat_pow_0
thf(fact_771_add_Onat__pow__Suc,axiom,
! [N: nat,X: a] :
( ( add_pow_a_b_nat @ r @ ( suc @ N ) @ X )
= ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ N @ X ) @ X ) ) ).
% add.nat_pow_Suc
thf(fact_772_mult__right__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_773_mult__right__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_774_mult__left__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_775_mult__left__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_776_no__zero__divisors,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( ( times_times_int @ A @ B )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_777_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_778_divisors__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
=> ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_779_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_780_mult__not__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
!= zero_zero_int )
=> ( ( A != zero_zero_int )
& ( B != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_781_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_782_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_783_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_784_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_785_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_786_comm__monoid__add__class_Oadd__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_787_eq__iff__diff__eq__0,axiom,
( ( ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
= ( ^ [A6: int,B6: int] :
( ( minus_minus_int @ A6 @ B6 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_788_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_789_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_790_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_791_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_792_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_793_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_794_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_795_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X2: nat] : ( R @ X2 @ X2 )
=> ( ! [X2: nat,Y4: nat,Z2: nat] :
( ( R @ X2 @ Y4 )
=> ( ( R @ Y4 @ Z2 )
=> ( R @ X2 @ Z2 ) ) )
=> ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_796_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_797_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_798_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M4: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N2 )
=> ( P @ M4 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_799_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_800_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_801_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_802_Suc__le__D,axiom,
! [N: nat,M5: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M5 )
=> ? [M6: nat] :
( M5
= ( suc @ M6 ) ) ) ).
% Suc_le_D
thf(fact_803_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_804_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_805_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_806_subcringE_I2_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b] :
( ( subcring_a_b @ H2 @ R )
=> ( member_a @ ( zero_a_b @ R ) @ H2 ) ) ).
% subcringE(2)
thf(fact_807_subcringE_I7_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b,H1: a,H22: a] :
( ( subcring_a_b @ H2 @ R )
=> ( ( member_a @ H1 @ H2 )
=> ( ( member_a @ H22 @ H2 )
=> ( member_a @ ( add_a_b @ R @ H1 @ H22 ) @ H2 ) ) ) ) ).
% subcringE(7)
thf(fact_808_subcringE_I4_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b] :
( ( subcring_a_b @ H2 @ R )
=> ( H2 != bot_bot_set_a ) ) ).
% subcringE(4)
thf(fact_809_subcringE_I6_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b,H1: a,H22: a] :
( ( subcring_a_b @ H2 @ R )
=> ( ( member_a @ H1 @ H2 )
=> ( ( member_a @ H22 @ H2 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ H1 @ H22 ) @ H2 ) ) ) ) ).
% subcringE(6)
thf(fact_810_subcring_Osub__m__comm,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b,H1: a,H22: a] :
( ( subcring_a_b @ H2 @ R )
=> ( ( member_a @ H1 @ H2 )
=> ( ( member_a @ H22 @ H2 )
=> ( ( mult_a_ring_ext_a_b @ R @ H1 @ H22 )
= ( mult_a_ring_ext_a_b @ R @ H22 @ H1 ) ) ) ) ) ).
% subcring.sub_m_comm
thf(fact_811_card__Suc__eq,axiom,
! [A2: set_set_a,K: nat] :
( ( ( finite_card_set_a @ A2 )
= ( suc @ K ) )
= ( ? [B6: set_a,B4: set_set_a] :
( ( A2
= ( insert_set_a @ B6 @ B4 ) )
& ~ ( member_set_a @ B6 @ B4 )
& ( ( finite_card_set_a @ B4 )
= K )
& ( ( K = zero_zero_nat )
=> ( B4 = bot_bot_set_set_a ) ) ) ) ) ).
% card_Suc_eq
thf(fact_812_card__Suc__eq,axiom,
! [A2: set_list_a,K: nat] :
( ( ( finite_card_list_a @ A2 )
= ( suc @ K ) )
= ( ? [B6: list_a,B4: set_list_a] :
( ( A2
= ( insert_list_a @ B6 @ B4 ) )
& ~ ( member_list_a @ B6 @ B4 )
& ( ( finite_card_list_a @ B4 )
= K )
& ( ( K = zero_zero_nat )
=> ( B4 = bot_bot_set_list_a ) ) ) ) ) ).
% card_Suc_eq
thf(fact_813_card__Suc__eq,axiom,
! [A2: set_a,K: nat] :
( ( ( finite_card_a @ A2 )
= ( suc @ K ) )
= ( ? [B6: a,B4: set_a] :
( ( A2
= ( insert_a @ B6 @ B4 ) )
& ~ ( member_a @ B6 @ B4 )
& ( ( finite_card_a @ B4 )
= K )
& ( ( K = zero_zero_nat )
=> ( B4 = bot_bot_set_a ) ) ) ) ) ).
% card_Suc_eq
thf(fact_814_card__eq__SucD,axiom,
! [A2: set_set_a,K: nat] :
( ( ( finite_card_set_a @ A2 )
= ( suc @ K ) )
=> ? [B2: set_a,B5: set_set_a] :
( ( A2
= ( insert_set_a @ B2 @ B5 ) )
& ~ ( member_set_a @ B2 @ B5 )
& ( ( finite_card_set_a @ B5 )
= K )
& ( ( K = zero_zero_nat )
=> ( B5 = bot_bot_set_set_a ) ) ) ) ).
% card_eq_SucD
thf(fact_815_card__eq__SucD,axiom,
! [A2: set_list_a,K: nat] :
( ( ( finite_card_list_a @ A2 )
= ( suc @ K ) )
=> ? [B2: list_a,B5: set_list_a] :
( ( A2
= ( insert_list_a @ B2 @ B5 ) )
& ~ ( member_list_a @ B2 @ B5 )
& ( ( finite_card_list_a @ B5 )
= K )
& ( ( K = zero_zero_nat )
=> ( B5 = bot_bot_set_list_a ) ) ) ) ).
% card_eq_SucD
thf(fact_816_card__eq__SucD,axiom,
! [A2: set_a,K: nat] :
( ( ( finite_card_a @ A2 )
= ( suc @ K ) )
=> ? [B2: a,B5: set_a] :
( ( A2
= ( insert_a @ B2 @ B5 ) )
& ~ ( member_a @ B2 @ B5 )
& ( ( finite_card_a @ B5 )
= K )
& ( ( K = zero_zero_nat )
=> ( B5 = bot_bot_set_a ) ) ) ) ).
% card_eq_SucD
thf(fact_817_card__1__singleton__iff,axiom,
! [A2: set_set_a] :
( ( ( finite_card_set_a @ A2 )
= ( suc @ zero_zero_nat ) )
= ( ? [X3: set_a] :
( A2
= ( insert_set_a @ X3 @ bot_bot_set_set_a ) ) ) ) ).
% card_1_singleton_iff
thf(fact_818_card__1__singleton__iff,axiom,
! [A2: set_list_a] :
( ( ( finite_card_list_a @ A2 )
= ( suc @ zero_zero_nat ) )
= ( ? [X3: list_a] :
( A2
= ( insert_list_a @ X3 @ bot_bot_set_list_a ) ) ) ) ).
% card_1_singleton_iff
thf(fact_819_card__1__singleton__iff,axiom,
! [A2: set_a] :
( ( ( finite_card_a @ A2 )
= ( suc @ zero_zero_nat ) )
= ( ? [X3: a] :
( A2
= ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ).
% card_1_singleton_iff
thf(fact_820_subcringE_I5_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b,H: a] :
( ( subcring_a_b @ H2 @ R )
=> ( ( member_a @ H @ H2 )
=> ( member_a @ ( a_inv_a_b @ R @ H ) @ H2 ) ) ) ).
% subcringE(5)
thf(fact_821_card__le__Suc0__iff__eq,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( ord_less_eq_nat @ ( finite_card_set_a @ A2 ) @ ( suc @ zero_zero_nat ) )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ! [Y3: set_a] :
( ( member_set_a @ Y3 @ A2 )
=> ( X3 = Y3 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_822_card__le__Suc0__iff__eq,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( suc @ zero_zero_nat ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ! [Y3: a] :
( ( member_a @ Y3 @ A2 )
=> ( X3 = Y3 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_823_card__le__Suc0__iff__eq,axiom,
! [A2: set_list_a] :
( ( finite_finite_list_a @ A2 )
=> ( ( ord_less_eq_nat @ ( finite_card_list_a @ A2 ) @ ( suc @ zero_zero_nat ) )
= ( ! [X3: list_a] :
( ( member_list_a @ X3 @ A2 )
=> ! [Y3: list_a] :
( ( member_list_a @ Y3 @ A2 )
=> ( X3 = Y3 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_824_lift__Suc__antimono__le,axiom,
! [F3: nat > set_a,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_set_a @ ( F3 @ ( suc @ N2 ) ) @ ( F3 @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_set_a @ ( F3 @ N4 ) @ ( F3 @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_825_lift__Suc__antimono__le,axiom,
! [F3: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F3 @ ( suc @ N2 ) ) @ ( F3 @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F3 @ N4 ) @ ( F3 @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_826_lift__Suc__antimono__le,axiom,
! [F3: nat > int,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_int @ ( F3 @ ( suc @ N2 ) ) @ ( F3 @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_int @ ( F3 @ N4 ) @ ( F3 @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_827_lift__Suc__mono__le,axiom,
! [F3: nat > set_a,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_set_a @ ( F3 @ N2 ) @ ( F3 @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_set_a @ ( F3 @ N ) @ ( F3 @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_828_lift__Suc__mono__le,axiom,
! [F3: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F3 @ N2 ) @ ( F3 @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F3 @ N ) @ ( F3 @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_829_lift__Suc__mono__le,axiom,
! [F3: nat > int,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_int @ ( F3 @ N2 ) @ ( F3 @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_int @ ( F3 @ N ) @ ( F3 @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_830_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_831_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_832_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_833_mult__mono,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_834_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_835_mult__mono_H,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_836_zero__le__square,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% zero_le_square
thf(fact_837_split__mult__pos__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_838_mult__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_839_mult__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_840_mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_841_mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_842_mult__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_843_mult__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_844_mult__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_845_mult__le__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_846_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_847_split__mult__neg__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_848_mult__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_849_mult__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_850_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_851_mult__nonneg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_852_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_853_mult__nonpos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_854_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_855_mult__nonneg__nonpos2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_856_zero__le__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_857_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_858_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_859_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_860_add__decreasing,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_861_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_862_add__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_863_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_864_add__decreasing2,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_865_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_866_add__increasing2,axiom,
! [C: int,B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_867_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_868_add__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_869_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_870_add__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_871_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_872_add__nonneg__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_873_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_874_add__nonpos__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_875_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A6: int,B6: int] : ( ord_less_eq_int @ ( minus_minus_int @ A6 @ B6 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_876_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_877_subcringE_I1_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b] :
( ( subcring_a_b @ H2 @ R )
=> ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% subcringE(1)
thf(fact_878_card__Suc__eq__finite,axiom,
! [A2: set_set_a,K: nat] :
( ( ( finite_card_set_a @ A2 )
= ( suc @ K ) )
= ( ? [B6: set_a,B4: set_set_a] :
( ( A2
= ( insert_set_a @ B6 @ B4 ) )
& ~ ( member_set_a @ B6 @ B4 )
& ( ( finite_card_set_a @ B4 )
= K )
& ( finite_finite_set_a @ B4 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_879_card__Suc__eq__finite,axiom,
! [A2: set_a,K: nat] :
( ( ( finite_card_a @ A2 )
= ( suc @ K ) )
= ( ? [B6: a,B4: set_a] :
( ( A2
= ( insert_a @ B6 @ B4 ) )
& ~ ( member_a @ B6 @ B4 )
& ( ( finite_card_a @ B4 )
= K )
& ( finite_finite_a @ B4 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_880_card__Suc__eq__finite,axiom,
! [A2: set_list_a,K: nat] :
( ( ( finite_card_list_a @ A2 )
= ( suc @ K ) )
= ( ? [B6: list_a,B4: set_list_a] :
( ( A2
= ( insert_list_a @ B6 @ B4 ) )
& ~ ( member_list_a @ B6 @ B4 )
& ( ( finite_card_list_a @ B4 )
= K )
& ( finite_finite_list_a @ B4 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_881_card__insert__if,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( ( member_set_a @ X @ A2 )
=> ( ( finite_card_set_a @ ( insert_set_a @ X @ A2 ) )
= ( finite_card_set_a @ A2 ) ) )
& ( ~ ( member_set_a @ X @ A2 )
=> ( ( finite_card_set_a @ ( insert_set_a @ X @ A2 ) )
= ( suc @ ( finite_card_set_a @ A2 ) ) ) ) ) ) ).
% card_insert_if
thf(fact_882_card__insert__if,axiom,
! [A2: set_a,X: a] :
( ( finite_finite_a @ A2 )
=> ( ( ( member_a @ X @ A2 )
=> ( ( finite_card_a @ ( insert_a @ X @ A2 ) )
= ( finite_card_a @ A2 ) ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ( finite_card_a @ ( insert_a @ X @ A2 ) )
= ( suc @ ( finite_card_a @ A2 ) ) ) ) ) ) ).
% card_insert_if
thf(fact_883_card__insert__if,axiom,
! [A2: set_list_a,X: list_a] :
( ( finite_finite_list_a @ A2 )
=> ( ( ( member_list_a @ X @ A2 )
=> ( ( finite_card_list_a @ ( insert_list_a @ X @ A2 ) )
= ( finite_card_list_a @ A2 ) ) )
& ( ~ ( member_list_a @ X @ A2 )
=> ( ( finite_card_list_a @ ( insert_list_a @ X @ A2 ) )
= ( suc @ ( finite_card_list_a @ A2 ) ) ) ) ) ) ).
% card_insert_if
thf(fact_884_sum__squares__ge__zero,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) ) ).
% sum_squares_ge_zero
thf(fact_885_card__eq__0__iff,axiom,
! [A2: set_set_a] :
( ( ( finite_card_set_a @ A2 )
= zero_zero_nat )
= ( ( A2 = bot_bot_set_set_a )
| ~ ( finite_finite_set_a @ A2 ) ) ) ).
% card_eq_0_iff
thf(fact_886_card__eq__0__iff,axiom,
! [A2: set_list_a] :
( ( ( finite_card_list_a @ A2 )
= zero_zero_nat )
= ( ( A2 = bot_bot_set_list_a )
| ~ ( finite_finite_list_a @ A2 ) ) ) ).
% card_eq_0_iff
thf(fact_887_card__eq__0__iff,axiom,
! [A2: set_a] :
( ( ( finite_card_a @ A2 )
= zero_zero_nat )
= ( ( A2 = bot_bot_set_a )
| ~ ( finite_finite_a @ A2 ) ) ) ).
% card_eq_0_iff
thf(fact_888_ring_Odimension_Osimps,axiom,
! [R: partia2175431115845679010xt_a_b,A1: nat,A22: set_a,A32: set_a] :
( ( ring_a_b @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ A1 @ A22 @ A32 )
= ( ? [K5: set_a] :
( ( A1 = zero_zero_nat )
& ( A22 = K5 )
& ( A32
= ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
| ? [V4: a,E4: set_a,N3: nat,K5: set_a] :
( ( A1
= ( suc @ N3 ) )
& ( A22 = K5 )
& ( A32
= ( embedd971793762689825387on_a_b @ R @ K5 @ V4 @ E4 ) )
& ( member_a @ V4 @ ( partia707051561876973205xt_a_b @ R ) )
& ~ ( member_a @ V4 @ E4 )
& ( embedd2795209813406577254on_a_b @ R @ N3 @ K5 @ E4 ) ) ) ) ) ).
% ring.dimension.simps
thf(fact_889_ring_Odimension_Ocases,axiom,
! [R: partia2175431115845679010xt_a_b,A1: nat,A22: set_a,A32: set_a] :
( ( ring_a_b @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ A1 @ A22 @ A32 )
=> ( ( ( A1 = zero_zero_nat )
=> ( A32
!= ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ~ ! [V2: a,E5: set_a,N2: nat] :
( ( A1
= ( suc @ N2 ) )
=> ( ( A32
= ( embedd971793762689825387on_a_b @ R @ A22 @ V2 @ E5 ) )
=> ( ( member_a @ V2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ~ ( member_a @ V2 @ E5 )
=> ~ ( embedd2795209813406577254on_a_b @ R @ N2 @ A22 @ E5 ) ) ) ) ) ) ) ) ).
% ring.dimension.cases
thf(fact_890_ring_OsubcringI,axiom,
! [R: partia2175431115845679010xt_a_b,H2: set_a] :
( ( ring_a_b @ R )
=> ( ( subring_a_b @ H2 @ R )
=> ( ! [H12: a,H23: a] :
( ( member_a @ H12 @ H2 )
=> ( ( member_a @ H23 @ H2 )
=> ( ( mult_a_ring_ext_a_b @ R @ H12 @ H23 )
= ( mult_a_ring_ext_a_b @ R @ H23 @ H12 ) ) ) )
=> ( subcring_a_b @ H2 @ R ) ) ) ) ).
% ring.subcringI
thf(fact_891_card__le__Suc__iff,axiom,
! [N: nat,A2: set_set_a] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_set_a @ A2 ) )
= ( ? [A6: set_a,B4: set_set_a] :
( ( A2
= ( insert_set_a @ A6 @ B4 ) )
& ~ ( member_set_a @ A6 @ B4 )
& ( ord_less_eq_nat @ N @ ( finite_card_set_a @ B4 ) )
& ( finite_finite_set_a @ B4 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_892_card__le__Suc__iff,axiom,
! [N: nat,A2: set_a] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_a @ A2 ) )
= ( ? [A6: a,B4: set_a] :
( ( A2
= ( insert_a @ A6 @ B4 ) )
& ~ ( member_a @ A6 @ B4 )
& ( ord_less_eq_nat @ N @ ( finite_card_a @ B4 ) )
& ( finite_finite_a @ B4 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_893_card__le__Suc__iff,axiom,
! [N: nat,A2: set_list_a] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_list_a @ A2 ) )
= ( ? [A6: list_a,B4: set_list_a] :
( ( A2
= ( insert_list_a @ A6 @ B4 ) )
& ~ ( member_list_a @ A6 @ B4 )
& ( ord_less_eq_nat @ N @ ( finite_card_list_a @ B4 ) )
& ( finite_finite_list_a @ B4 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_894_ring_OSuc__dim,axiom,
! [R: partia2175431115845679010xt_a_b,V3: a,E: set_a,N: nat,K2: set_a] :
( ( ring_a_b @ R )
=> ( ( member_a @ V3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ~ ( member_a @ V3 @ E )
=> ( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ E )
=> ( embedd2795209813406577254on_a_b @ R @ ( suc @ N ) @ K2 @ ( embedd971793762689825387on_a_b @ R @ K2 @ V3 @ E ) ) ) ) ) ) ).
% ring.Suc_dim
thf(fact_895_card__Suc__Diff1,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ X @ A2 )
=> ( ( suc @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) )
= ( finite_card_set_a @ A2 ) ) ) ) ).
% card_Suc_Diff1
thf(fact_896_card__Suc__Diff1,axiom,
! [A2: set_list_a,X: list_a] :
( ( finite_finite_list_a @ A2 )
=> ( ( member_list_a @ X @ A2 )
=> ( ( suc @ ( finite_card_list_a @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ X @ bot_bot_set_list_a ) ) ) )
= ( finite_card_list_a @ A2 ) ) ) ) ).
% card_Suc_Diff1
thf(fact_897_card__Suc__Diff1,axiom,
! [A2: set_a,X: a] :
( ( finite_finite_a @ A2 )
=> ( ( member_a @ X @ A2 )
=> ( ( suc @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) ) )
= ( finite_card_a @ A2 ) ) ) ) ).
% card_Suc_Diff1
thf(fact_898_card_Oinsert__remove,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( finite_card_set_a @ ( insert_set_a @ X @ A2 ) )
= ( suc @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) ) ) ) ).
% card.insert_remove
thf(fact_899_card_Oinsert__remove,axiom,
! [A2: set_list_a,X: list_a] :
( ( finite_finite_list_a @ A2 )
=> ( ( finite_card_list_a @ ( insert_list_a @ X @ A2 ) )
= ( suc @ ( finite_card_list_a @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ X @ bot_bot_set_list_a ) ) ) ) ) ) ).
% card.insert_remove
thf(fact_900_card_Oinsert__remove,axiom,
! [A2: set_a,X: a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_card_a @ ( insert_a @ X @ A2 ) )
= ( suc @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ) ).
% card.insert_remove
thf(fact_901_card_Oremove,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ X @ A2 )
=> ( ( finite_card_set_a @ A2 )
= ( suc @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) ) ) ) ) ).
% card.remove
thf(fact_902_card_Oremove,axiom,
! [A2: set_list_a,X: list_a] :
( ( finite_finite_list_a @ A2 )
=> ( ( member_list_a @ X @ A2 )
=> ( ( finite_card_list_a @ A2 )
= ( suc @ ( finite_card_list_a @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ X @ bot_bot_set_list_a ) ) ) ) ) ) ) ).
% card.remove
thf(fact_903_card_Oremove,axiom,
! [A2: set_a,X: a] :
( ( finite_finite_a @ A2 )
=> ( ( member_a @ X @ A2 )
=> ( ( finite_card_a @ A2 )
= ( suc @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ) ) ).
% card.remove
thf(fact_904_ring_Odimension__backwards,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,N: nat,E: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ ( suc @ N ) @ K2 @ E )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
& ? [E3: set_a] :
( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ E3 )
& ~ ( member_a @ X2 @ E3 )
& ( E
= ( embedd971793762689825387on_a_b @ R @ K2 @ X2 @ E3 ) ) ) ) ) ) ) ).
% ring.dimension_backwards
thf(fact_905_ring_Ozero__dim,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a] :
( ( ring_a_b @ R )
=> ( embedd2795209813406577254on_a_b @ R @ zero_zero_nat @ K2 @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) ) ).
% ring.zero_dim
thf(fact_906_ring_Odimension__zero,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,E: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ zero_zero_nat @ K2 @ E )
=> ( E
= ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) ) ) ) ).
% ring.dimension_zero
thf(fact_907_subdomainI,axiom,
! [H2: set_a] :
( ( subcring_a_b @ H2 @ r )
=> ( ( ( one_a_ring_ext_a_b @ r )
!= ( zero_a_b @ r ) )
=> ( ! [H12: a,H23: a] :
( ( member_a @ H12 @ H2 )
=> ( ( member_a @ H23 @ H2 )
=> ( ( ( mult_a_ring_ext_a_b @ r @ H12 @ H23 )
= ( zero_a_b @ r ) )
=> ( ( H12
= ( zero_a_b @ r ) )
| ( H23
= ( zero_a_b @ r ) ) ) ) ) )
=> ( subdomain_a_b @ H2 @ r ) ) ) ) ).
% subdomainI
thf(fact_908_nat__power__eq__Suc__0__iff,axiom,
! [X: nat,M: nat] :
( ( ( power_power_nat @ X @ M )
= ( suc @ zero_zero_nat ) )
= ( ( M = zero_zero_nat )
| ( X
= ( suc @ zero_zero_nat ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_909_power__Suc__0,axiom,
! [N: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
= ( suc @ zero_zero_nat ) ) ).
% power_Suc_0
thf(fact_910_power__Suc0__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_911_sum__squares__eq__zero__iff,axiom,
! [X: int,Y: int] :
( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_912_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
= zero_zero_int ) ).
% power_0_Suc
thf(fact_913_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
= zero_zero_nat ) ).
% power_0_Suc
thf(fact_914_subdomainE_I2_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b] :
( ( subdomain_a_b @ H2 @ R )
=> ( member_a @ ( zero_a_b @ R ) @ H2 ) ) ).
% subdomainE(2)
thf(fact_915_subdomainE_I7_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b,H1: a,H22: a] :
( ( subdomain_a_b @ H2 @ R )
=> ( ( member_a @ H1 @ H2 )
=> ( ( member_a @ H22 @ H2 )
=> ( member_a @ ( add_a_b @ R @ H1 @ H22 ) @ H2 ) ) ) ) ).
% subdomainE(7)
thf(fact_916_subdomainE_I4_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b] :
( ( subdomain_a_b @ H2 @ R )
=> ( H2 != bot_bot_set_a ) ) ).
% subdomainE(4)
thf(fact_917_subdomainE_I6_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b,H1: a,H22: a] :
( ( subdomain_a_b @ H2 @ R )
=> ( ( member_a @ H1 @ H2 )
=> ( ( member_a @ H22 @ H2 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ H1 @ H22 ) @ H2 ) ) ) ) ).
% subdomainE(6)
thf(fact_918_subdomainE_I8_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b,H1: a,H22: a] :
( ( subdomain_a_b @ H2 @ R )
=> ( ( member_a @ H1 @ H2 )
=> ( ( member_a @ H22 @ H2 )
=> ( ( mult_a_ring_ext_a_b @ R @ H1 @ H22 )
= ( mult_a_ring_ext_a_b @ R @ H22 @ H1 ) ) ) ) ) ).
% subdomainE(8)
thf(fact_919_subdomainE_I5_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b,H: a] :
( ( subdomain_a_b @ H2 @ R )
=> ( ( member_a @ H @ H2 )
=> ( member_a @ ( a_inv_a_b @ R @ H ) @ H2 ) ) ) ).
% subdomainE(5)
thf(fact_920_subdomainE_I1_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b] :
( ( subdomain_a_b @ H2 @ R )
=> ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% subdomainE(1)
thf(fact_921_subdomain_Osubintegral,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b,H1: a,H22: a] :
( ( subdomain_a_b @ H2 @ R )
=> ( ( member_a @ H1 @ H2 )
=> ( ( member_a @ H22 @ H2 )
=> ( ( ( mult_a_ring_ext_a_b @ R @ H1 @ H22 )
= ( zero_a_b @ R ) )
=> ( ( H1
= ( zero_a_b @ R ) )
| ( H22
= ( zero_a_b @ R ) ) ) ) ) ) ) ).
% subdomain.subintegral
thf(fact_922_subdomain_Osub__one__not__zero,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b] :
( ( subdomain_a_b @ H2 @ R )
=> ( ( one_a_ring_ext_a_b @ R )
!= ( zero_a_b @ R ) ) ) ).
% subdomain.sub_one_not_zero
thf(fact_923_power__not__zero,axiom,
! [A: int,N: nat] :
( ( A != zero_zero_int )
=> ( ( power_power_int @ A @ N )
!= zero_zero_int ) ) ).
% power_not_zero
thf(fact_924_power__not__zero,axiom,
! [A: nat,N: nat] :
( ( A != zero_zero_nat )
=> ( ( power_power_nat @ A @ N )
!= zero_zero_nat ) ) ).
% power_not_zero
thf(fact_925_power__commuting__commutes,axiom,
! [X: int,Y: int,N: nat] :
( ( ( times_times_int @ X @ Y )
= ( times_times_int @ Y @ X ) )
=> ( ( times_times_int @ ( power_power_int @ X @ N ) @ Y )
= ( times_times_int @ Y @ ( power_power_int @ X @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_926_power__commuting__commutes,axiom,
! [X: nat,Y: nat,N: nat] :
( ( ( times_times_nat @ X @ Y )
= ( times_times_nat @ Y @ X ) )
=> ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ Y )
= ( times_times_nat @ Y @ ( power_power_nat @ X @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_927_power__mult__distrib,axiom,
! [A: int,B: int,N: nat] :
( ( power_power_int @ ( times_times_int @ A @ B ) @ N )
= ( times_times_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_928_power__mult__distrib,axiom,
! [A: nat,B: nat,N: nat] :
( ( power_power_nat @ ( times_times_nat @ A @ B ) @ N )
= ( times_times_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_929_power__commutes,axiom,
! [A: int,N: nat] :
( ( times_times_int @ ( power_power_int @ A @ N ) @ A )
= ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% power_commutes
thf(fact_930_power__commutes,axiom,
! [A: nat,N: nat] :
( ( times_times_nat @ ( power_power_nat @ A @ N ) @ A )
= ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% power_commutes
thf(fact_931_power__mult,axiom,
! [A: nat,M: nat,N: nat] :
( ( power_power_nat @ A @ ( times_times_nat @ M @ N ) )
= ( power_power_nat @ ( power_power_nat @ A @ M ) @ N ) ) ).
% power_mult
thf(fact_932_ring_OsubdomainI,axiom,
! [R: partia2175431115845679010xt_a_b,H2: set_a] :
( ( ring_a_b @ R )
=> ( ( subcring_a_b @ H2 @ R )
=> ( ( ( one_a_ring_ext_a_b @ R )
!= ( zero_a_b @ R ) )
=> ( ! [H12: a,H23: a] :
( ( member_a @ H12 @ H2 )
=> ( ( member_a @ H23 @ H2 )
=> ( ( ( mult_a_ring_ext_a_b @ R @ H12 @ H23 )
= ( zero_a_b @ R ) )
=> ( ( H12
= ( zero_a_b @ R ) )
| ( H23
= ( zero_a_b @ R ) ) ) ) ) )
=> ( subdomain_a_b @ H2 @ R ) ) ) ) ) ).
% ring.subdomainI
thf(fact_933_zero__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% zero_le_power
thf(fact_934_zero__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_le_power
thf(fact_935_power__mono,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% power_mono
thf(fact_936_power__mono,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% power_mono
thf(fact_937_power__Suc2,axiom,
! [A: int,N: nat] :
( ( power_power_int @ A @ ( suc @ N ) )
= ( times_times_int @ ( power_power_int @ A @ N ) @ A ) ) ).
% power_Suc2
thf(fact_938_power__Suc2,axiom,
! [A: nat,N: nat] :
( ( power_power_nat @ A @ ( suc @ N ) )
= ( times_times_nat @ ( power_power_nat @ A @ N ) @ A ) ) ).
% power_Suc2
thf(fact_939_power__Suc,axiom,
! [A: int,N: nat] :
( ( power_power_int @ A @ ( suc @ N ) )
= ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% power_Suc
thf(fact_940_power__Suc,axiom,
! [A: nat,N: nat] :
( ( power_power_nat @ A @ ( suc @ N ) )
= ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% power_Suc
thf(fact_941_power__add,axiom,
! [A: int,M: nat,N: nat] :
( ( power_power_int @ A @ ( plus_plus_nat @ M @ N ) )
= ( times_times_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).
% power_add
thf(fact_942_power__add,axiom,
! [A: nat,M: nat,N: nat] :
( ( power_power_nat @ A @ ( plus_plus_nat @ M @ N ) )
= ( times_times_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).
% power_add
thf(fact_943_sum__squares__le__zero__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_944_power__inject__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ( power_power_nat @ A @ ( suc @ N ) )
= ( power_power_nat @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( A = B ) ) ) ) ).
% power_inject_base
thf(fact_945_power__inject__base,axiom,
! [A: int,N: nat,B: int] :
( ( ( power_power_int @ A @ ( suc @ N ) )
= ( power_power_int @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( A = B ) ) ) ) ).
% power_inject_base
thf(fact_946_power__le__imp__le__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ ( power_power_nat @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% power_le_imp_le_base
thf(fact_947_power__le__imp__le__base,axiom,
! [A: int,N: nat,B: int] :
( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ ( power_power_int @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% power_le_imp_le_base
thf(fact_948_nat__one__le__power,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N ) ) ) ).
% nat_one_le_power
thf(fact_949_double__eq__0__iff,axiom,
! [A: int] :
( ( ( plus_plus_int @ A @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_950_finite__ranking__induct,axiom,
! [S: set_set_a,P: set_set_a > $o,F3: set_a > nat] :
( ( finite_finite_set_a @ S )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [X2: set_a,S2: set_set_a] :
( ( finite_finite_set_a @ S2 )
=> ( ! [Y6: set_a] :
( ( member_set_a @ Y6 @ S2 )
=> ( ord_less_eq_nat @ ( F3 @ Y6 ) @ ( F3 @ X2 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_set_a @ X2 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_951_finite__ranking__induct,axiom,
! [S: set_list_a,P: set_list_a > $o,F3: list_a > nat] :
( ( finite_finite_list_a @ S )
=> ( ( P @ bot_bot_set_list_a )
=> ( ! [X2: list_a,S2: set_list_a] :
( ( finite_finite_list_a @ S2 )
=> ( ! [Y6: list_a] :
( ( member_list_a @ Y6 @ S2 )
=> ( ord_less_eq_nat @ ( F3 @ Y6 ) @ ( F3 @ X2 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_list_a @ X2 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_952_finite__ranking__induct,axiom,
! [S: set_a,P: set_a > $o,F3: a > nat] :
( ( finite_finite_a @ S )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X2: a,S2: set_a] :
( ( finite_finite_a @ S2 )
=> ( ! [Y6: a] :
( ( member_a @ Y6 @ S2 )
=> ( ord_less_eq_nat @ ( F3 @ Y6 ) @ ( F3 @ X2 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_a @ X2 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_953_finite__ranking__induct,axiom,
! [S: set_set_a,P: set_set_a > $o,F3: set_a > int] :
( ( finite_finite_set_a @ S )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [X2: set_a,S2: set_set_a] :
( ( finite_finite_set_a @ S2 )
=> ( ! [Y6: set_a] :
( ( member_set_a @ Y6 @ S2 )
=> ( ord_less_eq_int @ ( F3 @ Y6 ) @ ( F3 @ X2 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_set_a @ X2 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_954_finite__ranking__induct,axiom,
! [S: set_list_a,P: set_list_a > $o,F3: list_a > int] :
( ( finite_finite_list_a @ S )
=> ( ( P @ bot_bot_set_list_a )
=> ( ! [X2: list_a,S2: set_list_a] :
( ( finite_finite_list_a @ S2 )
=> ( ! [Y6: list_a] :
( ( member_list_a @ Y6 @ S2 )
=> ( ord_less_eq_int @ ( F3 @ Y6 ) @ ( F3 @ X2 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_list_a @ X2 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_955_finite__ranking__induct,axiom,
! [S: set_a,P: set_a > $o,F3: a > int] :
( ( finite_finite_a @ S )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X2: a,S2: set_a] :
( ( finite_finite_a @ S2 )
=> ( ! [Y6: a] :
( ( member_a @ Y6 @ S2 )
=> ( ord_less_eq_int @ ( F3 @ Y6 ) @ ( F3 @ X2 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_a @ X2 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_956_add_Oint__pow__neg,axiom,
! [X: a,I: int] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_int @ r @ ( uminus_uminus_int @ I ) @ X )
= ( a_inv_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X ) ) ) ) ).
% add.int_pow_neg
thf(fact_957_neg__le__iff__le,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_958_mult__minus__left,axiom,
! [A: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
= ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% mult_minus_left
thf(fact_959_minus__mult__minus,axiom,
! [A: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
= ( times_times_int @ A @ B ) ) ).
% minus_mult_minus
thf(fact_960_mult__minus__right,axiom,
! [A: int,B: int] :
( ( times_times_int @ A @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% mult_minus_right
thf(fact_961_add__minus__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_962_minus__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_963_minus__add__distrib,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).
% minus_add_distrib
thf(fact_964_minus__diff__eq,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
= ( minus_minus_int @ B @ A ) ) ).
% minus_diff_eq
thf(fact_965_neg__0__le__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% neg_0_le_iff_le
thf(fact_966_neg__le__0__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_le_0_iff_le
thf(fact_967_less__eq__neg__nonpos,axiom,
! [A: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% less_eq_neg_nonpos
thf(fact_968_neg__less__eq__nonneg,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_969_add_Oright__inverse,axiom,
! [A: int] :
( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
= zero_zero_int ) ).
% add.right_inverse
thf(fact_970_ab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_left_minus
thf(fact_971_diff__0,axiom,
! [A: int] :
( ( minus_minus_int @ zero_zero_int @ A )
= ( uminus_uminus_int @ A ) ) ).
% diff_0
thf(fact_972_diff__minus__eq__add,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
= ( plus_plus_int @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_973_uminus__add__conv__diff,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
= ( minus_minus_int @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_974_minus__int__code_I2_J,axiom,
! [L: int] :
( ( minus_minus_int @ zero_zero_int @ L )
= ( uminus_uminus_int @ L ) ) ).
% minus_int_code(2)
thf(fact_975_minus__diff__commute,axiom,
! [B: int,A: int] :
( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
= ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_976_add_Oinverse__distrib__swap,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_977_group__cancel_Oneg1,axiom,
! [A2: int,K: int,A: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( uminus_uminus_int @ A2 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_978_minus__mult__commute,axiom,
! [A: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
= ( times_times_int @ A @ ( uminus_uminus_int @ B ) ) ) ).
% minus_mult_commute
thf(fact_979_square__eq__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ A )
= ( times_times_int @ B @ B ) )
= ( ( A = B )
| ( A
= ( uminus_uminus_int @ B ) ) ) ) ).
% square_eq_iff
thf(fact_980_le__imp__neg__le,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% le_imp_neg_le
thf(fact_981_minus__le__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_982_le__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% le_minus_iff
thf(fact_983_add__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= zero_zero_int )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% add_eq_0_iff
thf(fact_984_ab__group__add__class_Oab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_group_add_class.ab_left_minus
thf(fact_985_add_Oinverse__unique,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= zero_zero_int )
=> ( ( uminus_uminus_int @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_986_eq__neg__iff__add__eq__0,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( ( plus_plus_int @ A @ B )
= zero_zero_int ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_987_neg__eq__iff__add__eq__0,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( plus_plus_int @ A @ B )
= zero_zero_int ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_988_group__cancel_Osub2,axiom,
! [B3: int,K: int,B: int,A: int] :
( ( B3
= ( plus_plus_int @ K @ B ) )
=> ( ( minus_minus_int @ A @ B3 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_989_diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A6: int,B6: int] : ( plus_plus_int @ A6 @ ( uminus_uminus_int @ B6 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_990_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A6: int,B6: int] : ( plus_plus_int @ A6 @ ( uminus_uminus_int @ B6 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_991_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_992_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_993_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_994_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_995_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_996_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(1)
thf(fact_997_int__distrib_I2_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z22 ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(2)
thf(fact_998_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(3)
thf(fact_999_int__distrib_I4_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
= ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_1000_verit__minus__simplify_I3_J,axiom,
! [B: int] :
( ( minus_minus_int @ zero_zero_int @ B )
= ( uminus_uminus_int @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_1001_add_Oint__pow__neg__int,axiom,
! [X: a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_int @ r @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ X )
= ( a_inv_a_b @ r @ ( add_pow_a_b_nat @ r @ N @ X ) ) ) ) ).
% add.int_pow_neg_int
thf(fact_1002_compl__le__compl__iff,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ X ) @ ( uminus_uminus_set_a @ Y ) )
= ( ord_less_eq_set_a @ Y @ X ) ) ).
% compl_le_compl_iff
thf(fact_1003_Compl__anti__mono,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ B3 ) @ ( uminus_uminus_set_a @ A2 ) ) ) ).
% Compl_anti_mono
thf(fact_1004_Compl__subset__Compl__iff,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ A2 ) @ ( uminus_uminus_set_a @ B3 ) )
= ( ord_less_eq_set_a @ B3 @ A2 ) ) ).
% Compl_subset_Compl_iff
thf(fact_1005_negative__zle,axiom,
! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zle
thf(fact_1006_subset__Compl__singleton,axiom,
! [A2: set_set_a,B: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( uminus6103902357914783669_set_a @ ( insert_set_a @ B @ bot_bot_set_set_a ) ) )
= ( ~ ( member_set_a @ B @ A2 ) ) ) ).
% subset_Compl_singleton
thf(fact_1007_subset__Compl__singleton,axiom,
! [A2: set_list_a,B: list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ ( uminus7925729386456332763list_a @ ( insert_list_a @ B @ bot_bot_set_list_a ) ) )
= ( ~ ( member_list_a @ B @ A2 ) ) ) ).
% subset_Compl_singleton
thf(fact_1008_subset__Compl__singleton,axiom,
! [A2: set_a,B: a] :
( ( ord_less_eq_set_a @ A2 @ ( uminus_uminus_set_a @ ( insert_a @ B @ bot_bot_set_a ) ) )
= ( ~ ( member_a @ B @ A2 ) ) ) ).
% subset_Compl_singleton
thf(fact_1009_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_1010_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_1011_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_add
thf(fact_1012_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_add
thf(fact_1013_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_mult
thf(fact_1014_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_mult
thf(fact_1015_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ( semiri1316708129612266289at_nat @ X )
= ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
= ( X
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_1016_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ( semiri1314217659103216013at_int @ X )
= ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
= ( X
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_1017_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W )
= ( semiri1316708129612266289at_nat @ X ) )
= ( ( power_power_nat @ B @ W )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_1018_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W )
= ( semiri1314217659103216013at_int @ X ) )
= ( ( power_power_nat @ B @ W )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_1019_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( power_power_nat @ M @ N ) )
= ( power_power_nat @ ( semiri1316708129612266289at_nat @ M ) @ N ) ) ).
% of_nat_power
thf(fact_1020_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( power_power_nat @ M @ N ) )
= ( power_power_int @ ( semiri1314217659103216013at_int @ M ) @ N ) ) ).
% of_nat_power
thf(fact_1021_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_1022_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_1023_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_1024_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_1025_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_1026_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_1027_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_1028_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_1029_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( K
!= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% nonneg_int_cases
thf(fact_1030_nonpos__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ~ ! [N2: nat] :
( K
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% nonpos_int_cases
thf(fact_1031_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( K
= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_1032_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% zle_int
thf(fact_1033_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W2: int,Z3: int] :
? [N3: nat] :
( Z3
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_1034_int__diff__cases,axiom,
! [Z: int] :
~ ! [M6: nat,N2: nat] :
( Z
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M6 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% int_diff_cases
thf(fact_1035_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_1036_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_1037_mult__of__nat__commute,axiom,
! [X: nat,Y: nat] :
( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X ) @ Y )
= ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_1038_mult__of__nat__commute,axiom,
! [X: nat,Y: int] :
( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y )
= ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_1039_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A6: nat,B6: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A6 ) @ ( semiri1314217659103216013at_int @ B6 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_1040_int__ops_I7_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(7)
thf(fact_1041_int__plus,axiom,
! [N: nat,M: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% int_plus
thf(fact_1042_int__ops_I5_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(5)
thf(fact_1043_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% of_nat_0_le_iff
thf(fact_1044_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% of_nat_0_le_iff
thf(fact_1045_subset__Compl__self__eq,axiom,
! [A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ ( uminus7925729386456332763list_a @ A2 ) )
= ( A2 = bot_bot_set_list_a ) ) ).
% subset_Compl_self_eq
thf(fact_1046_subset__Compl__self__eq,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( uminus_uminus_set_a @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% subset_Compl_self_eq
thf(fact_1047_int__zle__neg,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_1048_not__zle__0__negative,axiom,
! [N: nat] :
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ).
% not_zle_0_negative
thf(fact_1049_zadd__int__left,axiom,
! [M: nat,N: nat,Z: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z ) ) ).
% zadd_int_left
thf(fact_1050_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% of_nat_diff
thf(fact_1051_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% of_nat_diff
thf(fact_1052_Compl__insert,axiom,
! [X: list_a,A2: set_list_a] :
( ( uminus7925729386456332763list_a @ ( insert_list_a @ X @ A2 ) )
= ( minus_646659088055828811list_a @ ( uminus7925729386456332763list_a @ A2 ) @ ( insert_list_a @ X @ bot_bot_set_list_a ) ) ) ).
% Compl_insert
thf(fact_1053_Compl__insert,axiom,
! [X: a,A2: set_a] :
( ( uminus_uminus_set_a @ ( insert_a @ X @ A2 ) )
= ( minus_minus_set_a @ ( uminus_uminus_set_a @ A2 ) @ ( insert_a @ X @ bot_bot_set_a ) ) ) ).
% Compl_insert
thf(fact_1054_verit__comp__simplify1_I2_J,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1055_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1056_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1057_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_1058_verit__la__disequality,axiom,
! [A: int,B: int] :
( ( A = B )
| ~ ( ord_less_eq_int @ A @ B )
| ~ ( ord_less_eq_int @ B @ A ) ) ).
% verit_la_disequality
thf(fact_1059_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_1060_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_1061_compl__le__swap2,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ Y ) @ X )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ X ) @ Y ) ) ).
% compl_le_swap2
thf(fact_1062_compl__le__swap1,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ ( uminus_uminus_set_a @ X ) )
=> ( ord_less_eq_set_a @ X @ ( uminus_uminus_set_a @ Y ) ) ) ).
% compl_le_swap1
thf(fact_1063_compl__mono,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ Y ) @ ( uminus_uminus_set_a @ X ) ) ) ).
% compl_mono
thf(fact_1064_diff__shunt__var,axiom,
! [X: set_list_a,Y: set_list_a] :
( ( ( minus_646659088055828811list_a @ X @ Y )
= bot_bot_set_list_a )
= ( ord_le8861187494160871172list_a @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_1065_diff__shunt__var,axiom,
! [X: set_a,Y: set_a] :
( ( ( minus_minus_set_a @ X @ Y )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_1066_telescopic__base__aux,axiom,
! [K2: set_a,F: set_a,N: nat,E: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( subfield_a_b @ F @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ F )
=> ( ( embedd2795209813406577254on_a_b @ r @ one_one_nat @ F @ E )
=> ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E ) ) ) ) ) ).
% telescopic_base_aux
thf(fact_1067_a__coset__add__inv2,axiom,
! [M2: set_a,X: a,Y: a] :
( ( ( a_r_coset_a_b @ r @ M2 @ X )
= ( a_r_coset_a_b @ r @ M2 @ Y ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ M2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_r_coset_a_b @ r @ M2 @ ( add_a_b @ r @ X @ ( a_inv_a_b @ r @ Y ) ) )
= M2 ) ) ) ) ) ).
% a_coset_add_inv2
thf(fact_1068_a__coset__add__inv1,axiom,
! [M2: set_a,X: a,Y: a] :
( ( ( a_r_coset_a_b @ r @ M2 @ ( add_a_b @ r @ X @ ( a_inv_a_b @ r @ Y ) ) )
= M2 )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ M2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_r_coset_a_b @ r @ M2 @ X )
= ( a_r_coset_a_b @ r @ M2 @ Y ) ) ) ) ) ) ).
% a_coset_add_inv1
thf(fact_1069_a__r__coset__subset__G,axiom,
! [H2: set_a,X: a] :
( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( a_r_coset_a_b @ r @ H2 @ X ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% a_r_coset_subset_G
thf(fact_1070_ComplI,axiom,
! [C: a,A2: set_a] :
( ~ ( member_a @ C @ A2 )
=> ( member_a @ C @ ( uminus_uminus_set_a @ A2 ) ) ) ).
% ComplI
thf(fact_1071_ComplI,axiom,
! [C: set_a,A2: set_set_a] :
( ~ ( member_set_a @ C @ A2 )
=> ( member_set_a @ C @ ( uminus6103902357914783669_set_a @ A2 ) ) ) ).
% ComplI
thf(fact_1072_Compl__iff,axiom,
! [C: a,A2: set_a] :
( ( member_a @ C @ ( uminus_uminus_set_a @ A2 ) )
= ( ~ ( member_a @ C @ A2 ) ) ) ).
% Compl_iff
thf(fact_1073_Compl__iff,axiom,
! [C: set_a,A2: set_set_a] :
( ( member_set_a @ C @ ( uminus6103902357914783669_set_a @ A2 ) )
= ( ~ ( member_set_a @ C @ A2 ) ) ) ).
% Compl_iff
thf(fact_1074_a__coset__add__assoc,axiom,
! [M2: set_a,G2: a,H: a] :
( ( ord_less_eq_set_a @ M2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ G2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_r_coset_a_b @ r @ ( a_r_coset_a_b @ r @ M2 @ G2 ) @ H )
= ( a_r_coset_a_b @ r @ M2 @ ( add_a_b @ r @ G2 @ H ) ) ) ) ) ) ).
% a_coset_add_assoc
thf(fact_1075_a__rcosI,axiom,
! [H: a,H2: set_a,X: a] :
( ( member_a @ H @ H2 )
=> ( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( add_a_b @ r @ H @ X ) @ ( a_r_coset_a_b @ r @ H2 @ X ) ) ) ) ) ).
% a_rcosI
thf(fact_1076_a__rcosetsI,axiom,
! [H2: set_a,X: a] :
( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_set_a @ ( a_r_coset_a_b @ r @ H2 @ X ) @ ( a_RCOSETS_a_b @ r @ H2 ) ) ) ) ).
% a_rcosetsI
thf(fact_1077_mult_Oright__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.right_neutral
thf(fact_1078_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_1079_mult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% mult_1
thf(fact_1080_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_1081_power__one,axiom,
! [N: nat] :
( ( power_power_int @ one_one_int @ N )
= one_one_int ) ).
% power_one
thf(fact_1082_power__one,axiom,
! [N: nat] :
( ( power_power_nat @ one_one_nat @ N )
= one_one_nat ) ).
% power_one
thf(fact_1083_power__one__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_1084_mult__cancel__right2,axiom,
! [A: int,C: int] :
( ( ( times_times_int @ A @ C )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_1085_mult__cancel__right1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_1086_mult__cancel__left2,axiom,
! [C: int,A: int] :
( ( ( times_times_int @ C @ A )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_1087_mult__cancel__left1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_1088_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_1089_left__minus__one__mult__self,axiom,
! [N: nat,A: int] :
( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ A ) )
= A ) ).
% left_minus_one_mult_self
thf(fact_1090_minus__one__mult__self,axiom,
! [N: nat] :
( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) )
= one_one_int ) ).
% minus_one_mult_self
thf(fact_1091_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
= ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).
% of_nat_Suc
thf(fact_1092_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ M ) )
= ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% of_nat_Suc
thf(fact_1093_add_Onat__pow__eone,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_nat @ r @ one_one_nat @ X )
= X ) ) ).
% add.nat_pow_eone
thf(fact_1094_dimension__one,axiom,
! [K2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( embedd2795209813406577254on_a_b @ r @ one_one_nat @ K2 @ K2 ) ) ).
% dimension_one
thf(fact_1095_card__Diff__insert,axiom,
! [A: list_a,A2: set_list_a,B3: set_list_a] :
( ( member_list_a @ A @ A2 )
=> ( ~ ( member_list_a @ A @ B3 )
=> ( ( finite_card_list_a @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A @ B3 ) ) )
= ( minus_minus_nat @ ( finite_card_list_a @ ( minus_646659088055828811list_a @ A2 @ B3 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_1096_card__Diff__insert,axiom,
! [A: set_a,A2: set_set_a,B3: set_set_a] :
( ( member_set_a @ A @ A2 )
=> ( ~ ( member_set_a @ A @ B3 )
=> ( ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ A @ B3 ) ) )
= ( minus_minus_nat @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B3 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_1097_card__Diff__insert,axiom,
! [A: a,A2: set_a,B3: set_a] :
( ( member_a @ A @ A2 )
=> ( ~ ( member_a @ A @ B3 )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B3 ) ) )
= ( minus_minus_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ B3 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_1098_a__coset__add__zero,axiom,
! [M2: set_a] :
( ( ord_less_eq_set_a @ M2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_r_coset_a_b @ r @ M2 @ ( zero_a_b @ r ) )
= M2 ) ) ).
% a_coset_add_zero
thf(fact_1099_ComplD,axiom,
! [C: a,A2: set_a] :
( ( member_a @ C @ ( uminus_uminus_set_a @ A2 ) )
=> ~ ( member_a @ C @ A2 ) ) ).
% ComplD
thf(fact_1100_ComplD,axiom,
! [C: set_a,A2: set_set_a] :
( ( member_set_a @ C @ ( uminus6103902357914783669_set_a @ A2 ) )
=> ~ ( member_set_a @ C @ A2 ) ) ).
% ComplD
thf(fact_1101_power__eq__if,axiom,
( power_power_int
= ( ^ [P2: int,M3: nat] : ( if_int @ ( M3 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P2 @ ( power_power_int @ P2 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_1102_power__eq__if,axiom,
( power_power_nat
= ( ^ [P2: nat,M3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P2 @ ( power_power_nat @ P2 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_1103_verit__la__generic,axiom,
! [A: int,X: int] :
( ( ord_less_eq_int @ A @ X )
| ( A = X )
| ( ord_less_eq_int @ X @ A ) ) ).
% verit_la_generic
thf(fact_1104_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1105_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_1106_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_1107_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1108_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1109_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_1110_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_le_one
thf(fact_1111_one__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ).
% one_le_power
thf(fact_1112_one__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ).
% one_le_power
thf(fact_1113_square__eq__1__iff,axiom,
! [X: int] :
( ( ( times_times_int @ X @ X )
= one_one_int )
= ( ( X = one_one_int )
| ( X
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% square_eq_1_iff
thf(fact_1114_left__right__inverse__power,axiom,
! [X: int,Y: int,N: nat] :
( ( ( times_times_int @ X @ Y )
= one_one_int )
=> ( ( times_times_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ N ) )
= one_one_int ) ) ).
% left_right_inverse_power
thf(fact_1115_left__right__inverse__power,axiom,
! [X: nat,Y: nat,N: nat] :
( ( ( times_times_nat @ X @ Y )
= one_one_nat )
=> ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ N ) )
= one_one_nat ) ) ).
% left_right_inverse_power
thf(fact_1116_power__0,axiom,
! [A: int] :
( ( power_power_int @ A @ zero_zero_nat )
= one_one_int ) ).
% power_0
thf(fact_1117_power__0,axiom,
! [A: nat] :
( ( power_power_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_1118_bot__set__def,axiom,
( bot_bot_set_list_a
= ( collect_list_a @ bot_bot_list_a_o ) ) ).
% bot_set_def
thf(fact_1119_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_1120_mult_Ocomm__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.comm_neutral
thf(fact_1121_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_1122_comm__monoid__mult__class_Omult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_1123_comm__monoid__mult__class_Omult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_1124_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1125_mult__left__le__one__le,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ Y @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_1126_mult__right__le__one__le,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ X @ Y ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_1127_mult__le__one,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_1128_mult__le__one,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ B @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ) ).
% mult_le_one
thf(fact_1129_mult__left__le,axiom,
! [C: nat,A: nat] :
( ( ord_less_eq_nat @ C @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_1130_mult__left__le,axiom,
! [C: int,A: int] :
( ( ord_less_eq_int @ C @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_1131_power__le__one,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ one_one_nat ) ) ) ).
% power_le_one
thf(fact_1132_power__le__one,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ one_one_int ) ) ) ).
% power_le_one
thf(fact_1133_square__diff__one__factored,axiom,
! [X: int] :
( ( minus_minus_int @ ( times_times_int @ X @ X ) @ one_one_int )
= ( times_times_int @ ( plus_plus_int @ X @ one_one_int ) @ ( minus_minus_int @ X @ one_one_int ) ) ) ).
% square_diff_one_factored
thf(fact_1134_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= one_one_int ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ) ).
% power_0_left
thf(fact_1135_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= one_one_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ) ).
% power_0_left
thf(fact_1136_power__increasing,axiom,
! [N: nat,N5: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N5 ) ) ) ) ).
% power_increasing
thf(fact_1137_power__increasing,axiom,
! [N: nat,N5: nat,A: int] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N5 ) ) ) ) ).
% power_increasing
thf(fact_1138_power__minus,axiom,
! [A: int,N: nat] :
( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
= ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ A @ N ) ) ) ).
% power_minus
thf(fact_1139_card__1__singletonE,axiom,
! [A2: set_set_a] :
( ( ( finite_card_set_a @ A2 )
= one_one_nat )
=> ~ ! [X2: set_a] :
( A2
!= ( insert_set_a @ X2 @ bot_bot_set_set_a ) ) ) ).
% card_1_singletonE
thf(fact_1140_card__1__singletonE,axiom,
! [A2: set_list_a] :
( ( ( finite_card_list_a @ A2 )
= one_one_nat )
=> ~ ! [X2: list_a] :
( A2
!= ( insert_list_a @ X2 @ bot_bot_set_list_a ) ) ) ).
% card_1_singletonE
thf(fact_1141_card__1__singletonE,axiom,
! [A2: set_a] :
( ( ( finite_card_a @ A2 )
= one_one_nat )
=> ~ ! [X2: a] :
( A2
!= ( insert_a @ X2 @ bot_bot_set_a ) ) ) ).
% card_1_singletonE
thf(fact_1142_ring_Odimension__one,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( embedd2795209813406577254on_a_b @ R @ one_one_nat @ K2 @ K2 ) ) ) ).
% ring.dimension_one
thf(fact_1143_ring_Otelescopic__base__aux,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,F: set_a,N: nat,E: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( subfield_a_b @ F @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ F )
=> ( ( embedd2795209813406577254on_a_b @ R @ one_one_nat @ F @ E )
=> ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ E ) ) ) ) ) ) ).
% ring.telescopic_base_aux
thf(fact_1144_convex__bound__le,axiom,
! [X: int,A: int,Y: int,U: int,V3: int] :
( ( ord_less_eq_int @ X @ A )
=> ( ( ord_less_eq_int @ Y @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( ord_less_eq_int @ zero_zero_int @ V3 )
=> ( ( ( plus_plus_int @ U @ V3 )
= one_one_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V3 @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_1145_power__Suc__le__self,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_1146_power__Suc__le__self,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_1147_power__decreasing,axiom,
! [N: nat,N5: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N5 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_1148_power__decreasing,axiom,
! [N: nat,N5: nat,A: int] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N5 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_1149_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M3: nat,N3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N3 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% add_eq_if
thf(fact_1150_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M3: nat,N3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N3 @ ( times_times_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1151_card__Diff__singleton__if,axiom,
! [X: set_a,A2: set_set_a] :
( ( ( member_set_a @ X @ A2 )
=> ( ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) )
= ( minus_minus_nat @ ( finite_card_set_a @ A2 ) @ one_one_nat ) ) )
& ( ~ ( member_set_a @ X @ A2 )
=> ( ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) )
= ( finite_card_set_a @ A2 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_1152_card__Diff__singleton__if,axiom,
! [X: list_a,A2: set_list_a] :
( ( ( member_list_a @ X @ A2 )
=> ( ( finite_card_list_a @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ X @ bot_bot_set_list_a ) ) )
= ( minus_minus_nat @ ( finite_card_list_a @ A2 ) @ one_one_nat ) ) )
& ( ~ ( member_list_a @ X @ A2 )
=> ( ( finite_card_list_a @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ X @ bot_bot_set_list_a ) ) )
= ( finite_card_list_a @ A2 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_1153_card__Diff__singleton__if,axiom,
! [X: a,A2: set_a] :
( ( ( member_a @ X @ A2 )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) )
= ( minus_minus_nat @ ( finite_card_a @ A2 ) @ one_one_nat ) ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) )
= ( finite_card_a @ A2 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_1154_card__Diff__singleton,axiom,
! [X: set_a,A2: set_set_a] :
( ( member_set_a @ X @ A2 )
=> ( ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) )
= ( minus_minus_nat @ ( finite_card_set_a @ A2 ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_1155_card__Diff__singleton,axiom,
! [X: list_a,A2: set_list_a] :
( ( member_list_a @ X @ A2 )
=> ( ( finite_card_list_a @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ X @ bot_bot_set_list_a ) ) )
= ( minus_minus_nat @ ( finite_card_list_a @ A2 ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_1156_card__Diff__singleton,axiom,
! [X: a,A2: set_a] :
( ( member_a @ X @ A2 )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) )
= ( minus_minus_nat @ ( finite_card_a @ A2 ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_1157_diff__numeral__special_I12_J,axiom,
( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% diff_numeral_special(12)
thf(fact_1158_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
= zero_zero_int ) ).
% add_neg_numeral_special(8)
thf(fact_1159_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_1160_mult__minus1__right,axiom,
! [Z: int] :
( ( times_times_int @ Z @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ Z ) ) ).
% mult_minus1_right
thf(fact_1161_mult__minus1,axiom,
! [Z: int] :
( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z )
= ( uminus_uminus_int @ Z ) ) ).
% mult_minus1
thf(fact_1162_add_Oint__pow__1,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_int @ r @ one_one_int @ X )
= X ) ) ).
% add.int_pow_1
thf(fact_1163_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% add_neg_numeral_special(7)
thf(fact_1164_int__ge__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_1165_int__le__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ I3 @ K )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_le_induct
thf(fact_1166_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_1167_zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( ( M = one_one_int )
& ( N = one_one_int ) )
| ( ( M
= ( uminus_uminus_int @ one_one_int ) )
& ( N
= ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% zmult_eq_1_iff
thf(fact_1168_pos__zmult__eq__1__iff__lemma,axiom,
! [M: int,N: int] :
( ( ( times_times_int @ M @ N )
= one_one_int )
=> ( ( M = one_one_int )
| ( M
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff_lemma
thf(fact_1169_is__num__normalize_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_1170_int__ops_I4_J,axiom,
! [A: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ A ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).
% int_ops(4)
thf(fact_1171_int__Suc,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).
% int_Suc
thf(fact_1172_int__induct,axiom,
! [P: int > $o,K: int,I: int] :
( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ I3 @ K )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_induct
thf(fact_1173_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_1174_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_1175_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_1176_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_1177_is__num__normalize_I8_J,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_1178_le__minus__one__simps_I2_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% le_minus_one_simps(2)
thf(fact_1179_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(4)
thf(fact_1180_le__minus__one__simps_I3_J,axiom,
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(3)
thf(fact_1181_le__minus__one__simps_I1_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% le_minus_one_simps(1)
thf(fact_1182_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N @ K ) )
= ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_1183_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_1184_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_1185_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_1186_order__refl,axiom,
! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% order_refl
thf(fact_1187_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_1188_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_1189_order__antisym__conv,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( ord_less_eq_set_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_1190_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_1191_order__antisym__conv,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_1192_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_1193_linorder__le__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_1194_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F3: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F3 @ B )
= C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F3 @ X2 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F3 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1195_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F3: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F3 @ B )
= C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F3 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1196_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F3: set_a > int,C: int] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F3 @ B )
= C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_int @ ( F3 @ X2 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F3 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1197_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F3: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F3 @ B )
= C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F3 @ X2 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F3 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1198_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F3: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F3 @ B )
= C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F3 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1199_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F3: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F3 @ B )
= C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_int @ ( F3 @ X2 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F3 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1200_ord__le__eq__subst,axiom,
! [A: int,B: int,F3: int > set_a,C: set_a] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F3 @ B )
= C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F3 @ X2 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F3 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1201_ord__le__eq__subst,axiom,
! [A: int,B: int,F3: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F3 @ B )
= C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F3 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1202_ord__le__eq__subst,axiom,
! [A: int,B: int,F3: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F3 @ B )
= C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
=> ( ord_less_eq_int @ ( F3 @ X2 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F3 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1203_ord__eq__le__subst,axiom,
! [A: set_a,F3: set_a > set_a,B: set_a,C: set_a] :
( ( A
= ( F3 @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F3 @ X2 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F3 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1204_ord__eq__le__subst,axiom,
! [A: nat,F3: set_a > nat,B: set_a,C: set_a] :
( ( A
= ( F3 @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F3 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1205_ord__eq__le__subst,axiom,
! [A: int,F3: set_a > int,B: set_a,C: set_a] :
( ( A
= ( F3 @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_int @ ( F3 @ X2 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F3 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1206_ord__eq__le__subst,axiom,
! [A: set_a,F3: nat > set_a,B: nat,C: nat] :
( ( A
= ( F3 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F3 @ X2 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F3 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1207_ord__eq__le__subst,axiom,
! [A: nat,F3: nat > nat,B: nat,C: nat] :
( ( A
= ( F3 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F3 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1208_ord__eq__le__subst,axiom,
! [A: int,F3: nat > int,B: nat,C: nat] :
( ( A
= ( F3 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_int @ ( F3 @ X2 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F3 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1209_ord__eq__le__subst,axiom,
! [A: set_a,F3: int > set_a,B: int,C: int] :
( ( A
= ( F3 @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F3 @ X2 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F3 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1210_ord__eq__le__subst,axiom,
! [A: nat,F3: int > nat,B: int,C: int] :
( ( A
= ( F3 @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F3 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1211_ord__eq__le__subst,axiom,
! [A: int,F3: int > int,B: int,C: int] :
( ( A
= ( F3 @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
=> ( ord_less_eq_int @ ( F3 @ X2 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F3 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1212_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_1213_linorder__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_linear
thf(fact_1214_order__eq__refl,axiom,
! [X: set_a,Y: set_a] :
( ( X = Y )
=> ( ord_less_eq_set_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_1215_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_1216_order__eq__refl,axiom,
! [X: int,Y: int] :
( ( X = Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_1217_order__subst2,axiom,
! [A: set_a,B: set_a,F3: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F3 @ B ) @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F3 @ X2 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F3 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_1218_order__subst2,axiom,
! [A: set_a,B: set_a,F3: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( F3 @ B ) @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F3 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_1219_order__subst2,axiom,
! [A: set_a,B: set_a,F3: set_a > int,C: int] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_int @ ( F3 @ B ) @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_int @ ( F3 @ X2 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F3 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_1220_order__subst2,axiom,
! [A: nat,B: nat,F3: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F3 @ B ) @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F3 @ X2 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F3 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_1221_order__subst2,axiom,
! [A: nat,B: nat,F3: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F3 @ B ) @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F3 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_1222_order__subst2,axiom,
! [A: nat,B: nat,F3: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F3 @ B ) @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_int @ ( F3 @ X2 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F3 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_1223_order__subst2,axiom,
! [A: int,B: int,F3: int > set_a,C: set_a] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F3 @ B ) @ C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F3 @ X2 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F3 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_1224_order__subst2,axiom,
! [A: int,B: int,F3: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F3 @ B ) @ C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F3 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_1225_order__subst2,axiom,
! [A: int,B: int,F3: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F3 @ B ) @ C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
=> ( ord_less_eq_int @ ( F3 @ X2 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F3 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_1226_order__subst1,axiom,
! [A: set_a,F3: set_a > set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( F3 @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F3 @ X2 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F3 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_1227_order__subst1,axiom,
! [A: set_a,F3: nat > set_a,B: nat,C: nat] :
( ( ord_less_eq_set_a @ A @ ( F3 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F3 @ X2 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F3 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_1228_order__subst1,axiom,
! [A: set_a,F3: int > set_a,B: int,C: int] :
( ( ord_less_eq_set_a @ A @ ( F3 @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F3 @ X2 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F3 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_1229_order__subst1,axiom,
! [A: nat,F3: set_a > nat,B: set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ ( F3 @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F3 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_1230_order__subst1,axiom,
! [A: nat,F3: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F3 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F3 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_1231_order__subst1,axiom,
! [A: nat,F3: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F3 @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F3 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_1232_order__subst1,axiom,
! [A: int,F3: set_a > int,B: set_a,C: set_a] :
( ( ord_less_eq_int @ A @ ( F3 @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_int @ ( F3 @ X2 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F3 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_1233_order__subst1,axiom,
! [A: int,F3: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F3 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_int @ ( F3 @ X2 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F3 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_1234_order__subst1,axiom,
! [A: int,F3: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F3 @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
=> ( ord_less_eq_int @ ( F3 @ X2 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F3 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_1235_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_a,Z4: set_a] : ( Y5 = Z4 ) )
= ( ^ [A6: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A6 @ B6 )
& ( ord_less_eq_set_a @ B6 @ A6 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_1236_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
= ( ^ [A6: nat,B6: nat] :
( ( ord_less_eq_nat @ A6 @ B6 )
& ( ord_less_eq_nat @ B6 @ A6 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_1237_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
= ( ^ [A6: int,B6: int] :
( ( ord_less_eq_int @ A6 @ B6 )
& ( ord_less_eq_int @ B6 @ A6 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_1238_antisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_1239_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_1240_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_1241_dual__order_Otrans,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_1242_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_1243_dual__order_Otrans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_1244_dual__order_Oantisym,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_1245_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_1246_dual__order_Oantisym,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_1247_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_a,Z4: set_a] : ( Y5 = Z4 ) )
= ( ^ [A6: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ B6 @ A6 )
& ( ord_less_eq_set_a @ A6 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_1248_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
= ( ^ [A6: nat,B6: nat] :
( ( ord_less_eq_nat @ B6 @ A6 )
& ( ord_less_eq_nat @ A6 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_1249_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
= ( ^ [A6: int,B6: int] :
( ( ord_less_eq_int @ B6 @ A6 )
& ( ord_less_eq_int @ A6 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_1250_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: nat,B2: nat] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_1251_linorder__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: int,B2: int] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_1252_order__trans,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z )
=> ( ord_less_eq_set_a @ X @ Z ) ) ) ).
% order_trans
thf(fact_1253_order__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_1254_order__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z )
=> ( ord_less_eq_int @ X @ Z ) ) ) ).
% order_trans
thf(fact_1255_order_Otrans,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 ) ) ) ).
% order.trans
thf(fact_1256_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_1257_order_Otrans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% order.trans
thf(fact_1258_order__antisym,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_1259_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_1260_order__antisym,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_1261_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_1262_ord__le__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_1263_a__rcos__assoc__lcos,axiom,
! [H2: set_a,K2: set_a,X: a] :
( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( set_add_a_b @ r @ ( a_r_coset_a_b @ r @ H2 @ X ) @ K2 )
= ( set_add_a_b @ r @ H2 @ ( a_l_coset_a_b @ r @ X @ K2 ) ) ) ) ) ) ).
% a_rcos_assoc_lcos
thf(fact_1264_a__setmult__rcos__assoc,axiom,
! [H2: set_a,K2: set_a,X: a] :
( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( set_add_a_b @ r @ H2 @ ( a_r_coset_a_b @ r @ K2 @ X ) )
= ( a_r_coset_a_b @ r @ ( set_add_a_b @ r @ H2 @ K2 ) @ X ) ) ) ) ) ).
% a_setmult_rcos_assoc
thf(fact_1265_setadd__subset__G,axiom,
! [H2: set_a,K2: set_a] :
( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_add_a_b @ r @ H2 @ K2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% setadd_subset_G
thf(fact_1266_set__add__comm,axiom,
! [I2: set_a,J2: set_a] :
( ( ord_less_eq_set_a @ I2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ J2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( set_add_a_b @ r @ I2 @ J2 )
= ( set_add_a_b @ r @ J2 @ I2 ) ) ) ) ).
% set_add_comm
thf(fact_1267_set__add__closed,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_add_a_b @ r @ A2 @ B3 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% set_add_closed
thf(fact_1268_sum__space__dim_I1_J,axiom,
! [K2: set_a,E: set_a,F: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K2 @ E )
=> ( ( embedd8708762675212832759on_a_b @ r @ K2 @ F )
=> ( embedd8708762675212832759on_a_b @ r @ K2 @ ( set_add_a_b @ r @ E @ F ) ) ) ) ) ).
% sum_space_dim(1)
thf(fact_1269_dimension__direct__sum__space,axiom,
! [K2: set_a,N: nat,E: set_a,M: nat,F: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E )
=> ( ( embedd2795209813406577254on_a_b @ r @ M @ K2 @ F )
=> ( ( ( inf_inf_set_a @ E @ F )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
=> ( embedd2795209813406577254on_a_b @ r @ ( plus_plus_nat @ N @ M ) @ K2 @ ( set_add_a_b @ r @ E @ F ) ) ) ) ) ) ).
% dimension_direct_sum_space
thf(fact_1270_dimension__sum__space,axiom,
! [K2: set_a,N: nat,E: set_a,M: nat,F: set_a,K: nat] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E )
=> ( ( embedd2795209813406577254on_a_b @ r @ M @ K2 @ F )
=> ( ( embedd2795209813406577254on_a_b @ r @ K @ K2 @ ( inf_inf_set_a @ E @ F ) )
=> ( embedd2795209813406577254on_a_b @ r @ ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ K ) @ K2 @ ( set_add_a_b @ r @ E @ F ) ) ) ) ) ) ).
% dimension_sum_space
thf(fact_1271_subring__inter,axiom,
! [I2: set_a,J2: set_a] :
( ( subring_a_b @ I2 @ r )
=> ( ( subring_a_b @ J2 @ r )
=> ( subring_a_b @ ( inf_inf_set_a @ I2 @ J2 ) @ r ) ) ) ).
% subring_inter
thf(fact_1272_subalgebra__inter,axiom,
! [K2: set_a,V: set_a,V5: set_a] :
( ( embedd9027525575939734154ra_a_b @ K2 @ V @ r )
=> ( ( embedd9027525575939734154ra_a_b @ K2 @ V5 @ r )
=> ( embedd9027525575939734154ra_a_b @ K2 @ ( inf_inf_set_a @ V @ V5 ) @ r ) ) ) ).
% subalgebra_inter
thf(fact_1273_subcring__inter,axiom,
! [I2: set_a,J2: set_a] :
( ( subcring_a_b @ I2 @ r )
=> ( ( subcring_a_b @ J2 @ r )
=> ( subcring_a_b @ ( inf_inf_set_a @ I2 @ J2 ) @ r ) ) ) ).
% subcring_inter
thf(fact_1274_add__additive__subgroups,axiom,
! [H2: set_a,K2: set_a] :
( ( additi2834746164131130830up_a_b @ H2 @ r )
=> ( ( additi2834746164131130830up_a_b @ K2 @ r )
=> ( additi2834746164131130830up_a_b @ ( set_add_a_b @ r @ H2 @ K2 ) @ r ) ) ) ).
% add_additive_subgroups
thf(fact_1275_group__l__invI,axiom,
( ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [Xa2: a] :
( ( member_a @ Xa2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ Xa2 @ X2 )
= ( one_a_ring_ext_a_b @ r ) ) ) )
=> ( group_a_ring_ext_a_b @ r ) ) ).
% group_l_invI
thf(fact_1276_a__lagrange,axiom,
! [H2: set_a] :
( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( additi2834746164131130830up_a_b @ H2 @ r )
=> ( ( times_times_nat @ ( finite_card_set_a @ ( a_RCOSETS_a_b @ r @ H2 ) ) @ ( finite_card_a @ H2 ) )
= ( order_a_ring_ext_a_b @ r ) ) ) ) ).
% a_lagrange
% Helper facts (5)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( finite_card_list_a @ ( bounde2262800523058855161ls_a_b @ r @ n ) )
= ( power_power_nat @ ( finite_card_a @ ( partia707051561876973205xt_a_b @ r ) ) @ n ) ) ).
%------------------------------------------------------------------------------