TPTP Problem File: SLH0547^1.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 : Pluennecke_Ruzsa_Inequality/0003_Pluennecke_Ruzsa_Inequality/prob_00357_012679__12203758_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1383 ( 493 unt; 110 typ; 0 def)
% Number of atoms : 3681 (1097 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 12976 ( 376 ~; 53 |; 265 &;10441 @)
% ( 0 <=>;1841 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 7 avg)
% Number of types : 9 ( 8 usr)
% Number of type conns : 863 ( 863 >; 0 *; 0 +; 0 <<)
% Number of symbols : 103 ( 102 usr; 18 con; 0-5 aty)
% Number of variables : 3648 ( 150 ^;3394 !; 104 ?;3648 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:20:50.982
%------------------------------------------------------------------------------
% Could-be-implicit typings (8)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Ounit_J,type,
set_Product_unit: $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_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Product____Type__Ounit,type,
product_unit: $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_tf__a,type,
a: $tType ).
% Explicit typings (102)
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Product____Type__Ounit,type,
finite410649719033368117t_unit: set_Product_unit > nat ).
thf(sy_c_Finite__Set_Ocard_001tf__a,type,
finite_card_a: set_a > nat ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Ounit,type,
finite4290736615968046902t_unit: set_Product_unit > $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__Theory_Oabelian__group_001tf__a,type,
group_201663378560352916roup_a: set_a > ( a > a > a ) > a > $o ).
thf(sy_c_Group__Theory_Ocommutative__monoid_001t__Nat__Onat,type,
group_6791354081887936081id_nat: set_nat > ( nat > nat > nat ) > nat > $o ).
thf(sy_c_Group__Theory_Ocommutative__monoid_001tf__a,type,
group_4866109990395492029noid_a: set_a > ( a > a > a ) > a > $o ).
thf(sy_c_Group__Theory_Ogroup_001t__Nat__Onat,type,
group_group_nat: set_nat > ( nat > nat > nat ) > nat > $o ).
thf(sy_c_Group__Theory_Ogroup_001tf__a,type,
group_group_a: set_a > ( a > a > a ) > a > $o ).
thf(sy_c_Group__Theory_Omonoid_001t__Nat__Onat,type,
group_monoid_nat: set_nat > ( nat > nat > nat ) > nat > $o ).
thf(sy_c_Group__Theory_Omonoid_001tf__a,type,
group_monoid_a: set_a > ( a > a > a ) > a > $o ).
thf(sy_c_Group__Theory_Omonoid_OUnits_001t__Nat__Onat,type,
group_Units_nat: set_nat > ( nat > nat > nat ) > nat > set_nat ).
thf(sy_c_Group__Theory_Omonoid_OUnits_001tf__a,type,
group_Units_a: set_a > ( a > a > a ) > a > set_a ).
thf(sy_c_Group__Theory_Omonoid_Oinverse_001t__Nat__Onat,type,
group_inverse_nat: set_nat > ( nat > nat > nat ) > nat > nat > nat ).
thf(sy_c_Group__Theory_Omonoid_Oinverse_001tf__a,type,
group_inverse_a: set_a > ( a > a > a ) > a > a > a ).
thf(sy_c_Group__Theory_Omonoid_Oinvertible_001t__Nat__Onat,type,
group_invertible_nat: set_nat > ( nat > nat > nat ) > nat > nat > $o ).
thf(sy_c_Group__Theory_Omonoid_Oinvertible_001tf__a,type,
group_invertible_a: set_a > ( a > a > a ) > a > a > $o ).
thf(sy_c_Group__Theory_Osubgroup_001t__Nat__Onat,type,
group_subgroup_nat: set_nat > set_nat > ( nat > nat > nat ) > nat > $o ).
thf(sy_c_Group__Theory_Osubgroup_001tf__a,type,
group_subgroup_a: set_a > set_a > ( a > a > a ) > a > $o ).
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__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Ounit_J,type,
minus_6452836326544984404t_unit: set_Product_unit > set_Product_unit > set_Product_unit ).
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__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_HOL_ONO__MATCH_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
nO_MATCH_set_a_set_a: set_a > set_a > $o ).
thf(sy_c_HOL_Oundefined_001tf__a,type,
undefined_a: a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
inf_inf_nat: nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
inf_inf_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Product____Type__Ounit_J,type,
inf_in4660618365625256667t_unit: set_Product_unit > set_Product_unit > set_Product_unit ).
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_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
sup_sup_set_a: set_a > set_a > set_a ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
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__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Ounit_J,type,
bot_bo3957492148770167129t_unit: set_Product_unit ).
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_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Product____Type__Ounit_J,type,
ord_le8056459307392131481t_unit: set_Product_unit > set_Product_unit > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
ord_less_set_a: set_a > set_a > $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__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Ounit_J,type,
ord_le3507040750410214029t_unit: set_Product_unit > set_Product_unit > $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_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Ounit_J,type,
top_to1996260823553986621t_unit: set_Product_unit ).
thf(sy_c_Pluennecke__Ruzsa__Inequality_Oadditive__abelian__group_001t__Nat__Onat,type,
pluenn2073725187428264546up_nat: set_nat > ( nat > nat > nat ) > nat > $o ).
thf(sy_c_Pluennecke__Ruzsa__Inequality_Oadditive__abelian__group_001t__Product____Type__Ounit,type,
pluenn3635716580025208315t_unit: set_Product_unit > ( product_unit > product_unit > product_unit ) > product_unit > $o ).
thf(sy_c_Pluennecke__Ruzsa__Inequality_Oadditive__abelian__group_001tf__a,type,
pluenn1164192988769422572roup_a: set_a > ( a > a > a ) > a > $o ).
thf(sy_c_Pluennecke__Ruzsa__Inequality_Oadditive__abelian__group_Ominusset_001t__Nat__Onat,type,
pluenn7323955030898006884et_nat: set_nat > ( nat > nat > nat ) > nat > set_nat > set_nat ).
thf(sy_c_Pluennecke__Ruzsa__Inequality_Oadditive__abelian__group_Ominusset_001t__Product____Type__Ounit,type,
pluenn8189473251256886137t_unit: set_Product_unit > ( product_unit > product_unit > product_unit ) > product_unit > set_Product_unit > set_Product_unit ).
thf(sy_c_Pluennecke__Ruzsa__Inequality_Oadditive__abelian__group_Ominusset_001tf__a,type,
pluenn2534204936789923946sset_a: set_a > ( a > a > a ) > a > set_a > set_a ).
thf(sy_c_Pluennecke__Ruzsa__Inequality_Oadditive__abelian__group_Ominussetp_001t__Nat__Onat,type,
pluenn8372939692575285934tp_nat: set_nat > ( nat > nat > nat ) > nat > ( nat > $o ) > nat > $o ).
thf(sy_c_Pluennecke__Ruzsa__Inequality_Oadditive__abelian__group_Ominussetp_001tf__a,type,
pluenn1126946703085653920setp_a: set_a > ( a > a > a ) > a > ( a > $o ) > a > $o ).
thf(sy_c_Pluennecke__Ruzsa__Inequality_Oadditive__abelian__group_Osumset_001t__Nat__Onat,type,
pluenn3669378163024332905et_nat: set_nat > ( nat > nat > nat ) > set_nat > set_nat > set_nat ).
thf(sy_c_Pluennecke__Ruzsa__Inequality_Oadditive__abelian__group_Osumset_001t__Product____Type__Ounit,type,
pluenn1407455289632237236t_unit: set_Product_unit > ( product_unit > product_unit > product_unit ) > set_Product_unit > set_Product_unit > set_Product_unit ).
thf(sy_c_Pluennecke__Ruzsa__Inequality_Oadditive__abelian__group_Osumset_001tf__a,type,
pluenn3038260743871226533mset_a: set_a > ( a > a > a ) > set_a > set_a > set_a ).
thf(sy_c_Pluennecke__Ruzsa__Inequality_Oadditive__abelian__group_Osumset__iterated_001t__Nat__Onat,type,
pluenn7055013279391836755ed_nat: set_nat > ( nat > nat > nat ) > nat > set_nat > nat > set_nat ).
thf(sy_c_Pluennecke__Ruzsa__Inequality_Oadditive__abelian__group_Osumset__iterated_001t__Product____Type__Ounit,type,
pluenn7331107188738702666t_unit: set_Product_unit > ( product_unit > product_unit > product_unit ) > product_unit > set_Product_unit > nat > set_Product_unit ).
thf(sy_c_Pluennecke__Ruzsa__Inequality_Oadditive__abelian__group_Osumset__iterated_001tf__a,type,
pluenn1960970773371692859ated_a: set_a > ( a > a > a ) > a > set_a > nat > set_a ).
thf(sy_c_Pluennecke__Ruzsa__Inequality_Oadditive__abelian__group_Osumsetp_001t__Nat__Onat,type,
pluenn5670965976768739049tp_nat: set_nat > ( nat > nat > nat ) > ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Pluennecke__Ruzsa__Inequality_Oadditive__abelian__group_Osumsetp_001tf__a,type,
pluenn895083305082786853setp_a: set_a > ( a > a > a ) > ( a > $o ) > ( a > $o ) > a > $o ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Product____Type__Ounit,type,
image_8730104196221521654t_unit: ( nat > product_unit ) > set_nat > set_Product_unit ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001tf__a,type,
image_nat_a: ( nat > a ) > set_nat > set_a ).
thf(sy_c_Set_Oimage_001t__Product____Type__Ounit_001t__Product____Type__Ounit,type,
image_405062704495631173t_unit: ( product_unit > product_unit ) > set_Product_unit > set_Product_unit ).
thf(sy_c_Set_Oimage_001t__Product____Type__Ounit_001tf__a,type,
image_Product_unit_a: ( product_unit > a ) > set_Product_unit > set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7916887816326733075et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
image_set_a_set_a: ( set_a > set_a ) > set_set_a > set_set_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__Nat__Onat,type,
image_a_nat: ( a > nat ) > set_a > set_nat ).
thf(sy_c_Set_Oimage_001tf__a_001t__Product____Type__Ounit,type,
image_a_Product_unit: ( a > product_unit ) > set_a > set_Product_unit ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Ounit,type,
insert_Product_unit: product_unit > set_Product_unit > set_Product_unit ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Ounit,type,
member_Product_unit: product_unit > set_Product_unit > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_G,type,
g: set_a ).
thf(sy_v_U,type,
u: set_a ).
thf(sy_v_V,type,
v: set_a ).
thf(sy_v_W,type,
w: set_a ).
thf(sy_v_addition,type,
addition: a > a > a ).
thf(sy_v_u1____,type,
u1: a ).
thf(sy_v_u2____,type,
u2: a ).
thf(sy_v_v____,type,
v2: a > a ).
thf(sy_v_w____,type,
w2: a > a ).
thf(sy_v_x1____,type,
x1: a ).
thf(sy_v_x2____,type,
x2: a ).
thf(sy_v_zero,type,
zero: a ).
% Relevant facts (1272)
thf(fact_0__092_060open_062u1_A_092_060in_062_AG_092_060close_062,axiom,
member_a @ u1 @ g ).
% \<open>u1 \<in> G\<close>
thf(fact_1__092_060open_062u2_A_092_060in_062_AG_092_060close_062,axiom,
member_a @ u2 @ g ).
% \<open>u2 \<in> G\<close>
thf(fact_2__092_060open_062u1_A_092_060in_062_AU_092_060close_062,axiom,
member_a @ u1 @ u ).
% \<open>u1 \<in> U\<close>
thf(fact_3__092_060open_062u2_A_092_060in_062_AU_092_060close_062,axiom,
member_a @ u2 @ u ).
% \<open>u2 \<in> U\<close>
thf(fact_4__092_060open_062_092_060And_062thesis_O_A_I_092_060lbrakk_062u1_A_092_060in_062_AG_059_Au2_A_092_060in_062_AG_059_Ax1_A_092_060in_062_AG_059_Ax2_A_092_060in_062_AG_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ( ( member_a @ u1 @ g )
=> ( ( member_a @ u2 @ g )
=> ( ( member_a @ x1 @ g )
=> ~ ( member_a @ x2 @ g ) ) ) ) ).
% \<open>\<And>thesis. (\<lbrakk>u1 \<in> G; u2 \<in> G; x1 \<in> G; x2 \<in> G\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_5_w,axiom,
( ( addition @ u1 @ ( group_inverse_a @ g @ addition @ zero @ ( w2 @ x1 ) ) )
= ( addition @ u2 @ ( group_inverse_a @ g @ addition @ zero @ ( w2 @ x2 ) ) ) ) ).
% w
thf(fact_6_assms_I5_J,axiom,
finite_finite_a @ w ).
% assms(5)
thf(fact_7_assms_I3_J,axiom,
finite_finite_a @ v ).
% assms(3)
thf(fact_8_v,axiom,
( ( addition @ u1 @ ( group_inverse_a @ g @ addition @ zero @ ( v2 @ x1 ) ) )
= ( addition @ u2 @ ( group_inverse_a @ g @ addition @ zero @ ( v2 @ x2 ) ) ) ) ).
% v
thf(fact_9_commutative,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ g )
=> ( ( member_a @ Y @ g )
=> ( ( addition @ X @ Y )
= ( addition @ Y @ X ) ) ) ) ).
% commutative
thf(fact_10__092_060open_062u1_A_092_060ominus_062_Aw_Ax1_A_092_060ominus_062_A_Iu1_A_092_060ominus_062_Av_Ax1_J_A_061_Au2_A_092_060ominus_062_Aw_Ax2_A_092_060ominus_062_A_Iu2_A_092_060ominus_062_Av_Ax2_J_092_060close_062,axiom,
( ( addition @ ( addition @ u1 @ ( group_inverse_a @ g @ addition @ zero @ ( w2 @ x1 ) ) ) @ ( group_inverse_a @ g @ addition @ zero @ ( addition @ u1 @ ( group_inverse_a @ g @ addition @ zero @ ( v2 @ x1 ) ) ) ) )
= ( addition @ ( addition @ u2 @ ( group_inverse_a @ g @ addition @ zero @ ( w2 @ x2 ) ) ) @ ( group_inverse_a @ g @ addition @ zero @ ( addition @ u2 @ ( group_inverse_a @ g @ addition @ zero @ ( v2 @ x2 ) ) ) ) ) ) ).
% \<open>u1 \<ominus> w x1 \<ominus> (u1 \<ominus> v x1) = u2 \<ominus> w x2 \<ominus> (u2 \<ominus> v x2)\<close>
thf(fact_11_additive__abelian__group_Ominusset_Ocong,axiom,
pluenn2534204936789923946sset_a = pluenn2534204936789923946sset_a ).
% additive_abelian_group.minusset.cong
thf(fact_12_additive__abelian__group_Osumset_Ocong,axiom,
pluenn3038260743871226533mset_a = pluenn3038260743871226533mset_a ).
% additive_abelian_group.sumset.cong
thf(fact_13__092_060open_062v_Ax1_A_092_060ominus_062_Aw_Ax1_A_061_Au1_A_092_060ominus_062_Aw_Ax1_A_092_060ominus_062_A_Iu1_A_092_060ominus_062_Av_Ax1_J_092_060close_062,axiom,
( ( addition @ ( v2 @ x1 ) @ ( group_inverse_a @ g @ addition @ zero @ ( w2 @ x1 ) ) )
= ( addition @ ( addition @ u1 @ ( group_inverse_a @ g @ addition @ zero @ ( w2 @ x1 ) ) ) @ ( group_inverse_a @ g @ addition @ zero @ ( addition @ u1 @ ( group_inverse_a @ g @ addition @ zero @ ( v2 @ x1 ) ) ) ) ) ) ).
% \<open>v x1 \<ominus> w x1 = u1 \<ominus> w x1 \<ominus> (u1 \<ominus> v x1)\<close>
thf(fact_14__092_060open_062u2_A_092_060ominus_062_Aw_Ax2_A_092_060ominus_062_A_Iu2_A_092_060ominus_062_Av_Ax2_J_A_061_Av_Ax2_A_092_060ominus_062_Aw_Ax2_092_060close_062,axiom,
( ( addition @ ( addition @ u2 @ ( group_inverse_a @ g @ addition @ zero @ ( w2 @ x2 ) ) ) @ ( group_inverse_a @ g @ addition @ zero @ ( addition @ u2 @ ( group_inverse_a @ g @ addition @ zero @ ( v2 @ x2 ) ) ) ) )
= ( addition @ ( v2 @ x2 ) @ ( group_inverse_a @ g @ addition @ zero @ ( w2 @ x2 ) ) ) ) ).
% \<open>u2 \<ominus> w x2 \<ominus> (u2 \<ominus> v x2) = v x2 \<ominus> w x2\<close>
thf(fact_15_assms_I6_J,axiom,
ord_less_eq_set_a @ w @ g ).
% assms(6)
thf(fact_16__092_060open_062x1_A_061_Ax2_092_060close_062,axiom,
x1 = x2 ).
% \<open>x1 = x2\<close>
thf(fact_17__092_060open_062x1_A_092_060in_062_AG_092_060close_062,axiom,
member_a @ x1 @ g ).
% \<open>x1 \<in> G\<close>
thf(fact_18__092_060open_062x2_A_092_060in_062_AG_092_060close_062,axiom,
member_a @ x2 @ g ).
% \<open>x2 \<in> G\<close>
thf(fact_19_assms_I1_J,axiom,
finite_finite_a @ u ).
% assms(1)
thf(fact_20_assms_I4_J,axiom,
ord_less_eq_set_a @ v @ g ).
% assms(4)
thf(fact_21_assms_I2_J,axiom,
ord_less_eq_set_a @ u @ g ).
% assms(2)
thf(fact_22_sumset__commute,axiom,
! [A: set_a,B: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B )
= ( pluenn3038260743871226533mset_a @ g @ addition @ B @ A ) ) ).
% sumset_commute
thf(fact_23_sumset__assoc,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) @ C )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( pluenn3038260743871226533mset_a @ g @ addition @ B @ C ) ) ) ).
% sumset_assoc
thf(fact_24_sumset_OsumsetI,axiom,
! [A2: a,A: set_a,B2: a,B: set_a] :
( ( member_a @ A2 @ A )
=> ( ( member_a @ A2 @ g )
=> ( ( member_a @ B2 @ B )
=> ( ( member_a @ B2 @ g )
=> ( member_a @ ( addition @ A2 @ B2 ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) ) ) ) ) ) ).
% sumset.sumsetI
thf(fact_25_sumset_Osimps,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( member_a @ A2 @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) )
= ( ? [A3: a,B3: a] :
( ( A2
= ( addition @ A3 @ B3 ) )
& ( member_a @ A3 @ A )
& ( member_a @ A3 @ g )
& ( member_a @ B3 @ B )
& ( member_a @ B3 @ g ) ) ) ) ).
% sumset.simps
thf(fact_26_sumset_Ocases,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( member_a @ A2 @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) )
=> ~ ! [A4: a,B4: a] :
( ( A2
= ( addition @ A4 @ B4 ) )
=> ( ( member_a @ A4 @ A )
=> ( ( member_a @ A4 @ g )
=> ( ( member_a @ B4 @ B )
=> ~ ( member_a @ B4 @ g ) ) ) ) ) ) ).
% sumset.cases
thf(fact_27_winG,axiom,
! [X: a] :
( ( member_a @ X @ ( pluenn3038260743871226533mset_a @ g @ addition @ v @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ w ) ) )
=> ( member_a @ ( w2 @ X ) @ g ) ) ).
% winG
thf(fact_28_x1,axiom,
member_a @ x1 @ ( pluenn3038260743871226533mset_a @ g @ addition @ v @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ w ) ) ).
% x1
thf(fact_29_finite__sumset,axiom,
! [A: set_a,B: set_a] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) ) ) ) ).
% finite_sumset
thf(fact_30_sumset__mono,axiom,
! [A5: set_a,A: set_a,B5: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A5 @ A )
=> ( ( ord_less_eq_set_a @ B5 @ B )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A5 @ B5 ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) ) ) ) ).
% sumset_mono
thf(fact_31_sumset__subset__carrier,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) @ g ) ).
% sumset_subset_carrier
thf(fact_32_minusset__distrib__sum,axiom,
! [A: set_a,B: set_a] :
( ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) )
= ( pluenn3038260743871226533mset_a @ g @ addition @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A ) @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ B ) ) ) ).
% minusset_distrib_sum
thf(fact_33_local_Oinverse__unique,axiom,
! [U: a,V: a,V2: a] :
( ( ( addition @ U @ V )
= zero )
=> ( ( ( addition @ V2 @ U )
= zero )
=> ( ( member_a @ U @ g )
=> ( ( member_a @ V2 @ g )
=> ( ( member_a @ V @ g )
=> ( V2 = V ) ) ) ) ) ) ).
% local.inverse_unique
thf(fact_34_x2,axiom,
member_a @ x2 @ ( pluenn3038260743871226533mset_a @ g @ addition @ v @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ w ) ) ).
% x2
thf(fact_35_minusset_Ocases,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A ) )
=> ~ ! [A4: a] :
( ( A2
= ( group_inverse_a @ g @ addition @ zero @ A4 ) )
=> ( ( member_a @ A4 @ A )
=> ~ ( member_a @ A4 @ g ) ) ) ) ).
% minusset.cases
thf(fact_36_minusset_OminussetI,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ( ( member_a @ A2 @ g )
=> ( member_a @ ( group_inverse_a @ g @ addition @ zero @ A2 ) @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A ) ) ) ) ).
% minusset.minussetI
thf(fact_37_minusset_Osimps,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A ) )
= ( ? [A3: a] :
( ( A2
= ( group_inverse_a @ g @ addition @ zero @ A3 ) )
& ( member_a @ A3 @ A )
& ( member_a @ A3 @ g ) ) ) ) ).
% minusset.simps
thf(fact_38_inverse__closed,axiom,
! [X: a] :
( ( member_a @ X @ g )
=> ( member_a @ ( group_inverse_a @ g @ addition @ zero @ X ) @ g ) ) ).
% inverse_closed
thf(fact_39_inverse__equality,axiom,
! [U: a,V2: a] :
( ( ( addition @ U @ V2 )
= zero )
=> ( ( ( addition @ V2 @ U )
= zero )
=> ( ( member_a @ U @ g )
=> ( ( member_a @ V2 @ g )
=> ( ( group_inverse_a @ g @ addition @ zero @ U )
= V2 ) ) ) ) ) ).
% inverse_equality
thf(fact_40_finite__differenceset,axiom,
! [A: set_a,B: set_a] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ B ) ) ) ) ) ).
% finite_differenceset
thf(fact_41_finite__minusset,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( finite_finite_a @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A ) ) ) ).
% finite_minusset
thf(fact_42_minusset__subset__carrier,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A ) @ g ) ).
% minusset_subset_carrier
thf(fact_43_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_44_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_47__092_060open_062_092_060And_062x_O_Ax_A_092_060in_062_Adifferenceset_AV_AW_A_092_060Longrightarrow_062_A_092_060exists_062v_Aw_O_Av_A_092_060in_062_AV_A_092_060and_062_Aw_A_092_060in_062_AW_A_092_060and_062_Ax_A_061_Av_A_092_060ominus_062_Aw_092_060close_062,axiom,
! [X: a] :
( ( member_a @ X @ ( pluenn3038260743871226533mset_a @ g @ addition @ v @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ w ) ) )
=> ? [V3: a,W: a] :
( ( member_a @ V3 @ v )
& ( member_a @ W @ w )
& ( X
= ( addition @ V3 @ ( group_inverse_a @ g @ addition @ zero @ W ) ) ) ) ) ).
% \<open>\<And>x. x \<in> differenceset V W \<Longrightarrow> \<exists>v w. v \<in> V \<and> w \<in> W \<and> x = v \<ominus> w\<close>
thf(fact_48__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062v_Aw_O_A_092_060lbrakk_062_092_060And_062x_O_Ax_A_092_060in_062_Adifferenceset_AV_AW_A_092_060Longrightarrow_062_Av_Ax_A_092_060in_062_AV_059_A_092_060And_062x_O_Ax_A_092_060in_062_Adifferenceset_AV_AW_A_092_060Longrightarrow_062_Aw_Ax_A_092_060in_062_AW_059_A_092_060And_062x_O_Ax_A_092_060in_062_Adifferenceset_AV_AW_A_092_060Longrightarrow_062_Av_Ax_A_092_060ominus_062_Aw_Ax_A_061_Ax_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [V3: a > a] :
( ! [X3: a] :
( ( member_a @ X3 @ ( pluenn3038260743871226533mset_a @ g @ addition @ v @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ w ) ) )
=> ( member_a @ ( V3 @ X3 ) @ v ) )
=> ! [W: a > a] :
( ! [X3: a] :
( ( member_a @ X3 @ ( pluenn3038260743871226533mset_a @ g @ addition @ v @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ w ) ) )
=> ( member_a @ ( W @ X3 ) @ w ) )
=> ~ ! [X3: a] :
( ( member_a @ X3 @ ( pluenn3038260743871226533mset_a @ g @ addition @ v @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ w ) ) )
=> ( ( addition @ ( V3 @ X3 ) @ ( group_inverse_a @ g @ addition @ zero @ ( W @ X3 ) ) )
= X3 ) ) ) ) ).
% \<open>\<And>thesis. (\<And>v w. \<lbrakk>\<And>x. x \<in> differenceset V W \<Longrightarrow> v x \<in> V; \<And>x. x \<in> differenceset V W \<Longrightarrow> w x \<in> W; \<And>x. x \<in> differenceset V W \<Longrightarrow> v x \<ominus> w x = x\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_49_fin_I2_J,axiom,
finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ u @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ w ) ) ).
% fin(2)
thf(fact_50_fin_I1_J,axiom,
finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ u @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ v ) ) ).
% fin(1)
thf(fact_51_vinG,axiom,
! [X: a] :
( ( member_a @ X @ ( pluenn3038260743871226533mset_a @ g @ addition @ v @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ w ) ) )
=> ( member_a @ ( v2 @ X ) @ g ) ) ).
% vinG
thf(fact_52_vinV,axiom,
! [X: a] :
( ( member_a @ X @ ( pluenn3038260743871226533mset_a @ g @ addition @ v @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ w ) ) )
=> ( member_a @ ( v2 @ X ) @ v ) ) ).
% vinV
thf(fact_53_winW,axiom,
! [X: a] :
( ( member_a @ X @ ( pluenn3038260743871226533mset_a @ g @ addition @ v @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ w ) ) )
=> ( member_a @ ( w2 @ X ) @ w ) ) ).
% winW
thf(fact_54_composition__closed,axiom,
! [A2: a,B2: a] :
( ( member_a @ A2 @ g )
=> ( ( member_a @ B2 @ g )
=> ( member_a @ ( addition @ A2 @ B2 ) @ g ) ) ) ).
% composition_closed
thf(fact_55_associative,axiom,
! [A2: a,B2: a,C2: a] :
( ( member_a @ A2 @ g )
=> ( ( member_a @ B2 @ g )
=> ( ( member_a @ C2 @ g )
=> ( ( addition @ ( addition @ A2 @ B2 ) @ C2 )
= ( addition @ A2 @ ( addition @ B2 @ C2 ) ) ) ) ) ) ).
% associative
thf(fact_56_unit__closed,axiom,
member_a @ zero @ g ).
% unit_closed
thf(fact_57_vw__eq,axiom,
! [X: a] :
( ( member_a @ X @ ( pluenn3038260743871226533mset_a @ g @ addition @ v @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ w ) ) )
=> ( ( addition @ ( v2 @ X ) @ ( group_inverse_a @ g @ addition @ zero @ ( w2 @ X ) ) )
= X ) ) ).
% vw_eq
thf(fact_58__092_060open_062v_Ax1_A_092_060ominus_062_Aw_Ax1_A_061_Av_Ax2_A_092_060ominus_062_Aw_Ax2_092_060close_062,axiom,
( ( addition @ ( v2 @ x1 ) @ ( group_inverse_a @ g @ addition @ zero @ ( w2 @ x1 ) ) )
= ( addition @ ( v2 @ x2 ) @ ( group_inverse_a @ g @ addition @ zero @ ( w2 @ x2 ) ) ) ) ).
% \<open>v x1 \<ominus> w x1 = v x2 \<ominus> w x2\<close>
thf(fact_59_differenceset__commute,axiom,
! [B: set_a,A: set_a] :
( ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ ( pluenn3038260743871226533mset_a @ g @ addition @ B @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A ) ) )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ B ) ) ) ).
% differenceset_commute
thf(fact_60_left__unit,axiom,
! [A2: a] :
( ( member_a @ A2 @ g )
=> ( ( addition @ zero @ A2 )
= A2 ) ) ).
% left_unit
thf(fact_61_right__unit,axiom,
! [A2: a] :
( ( member_a @ A2 @ g )
=> ( ( addition @ A2 @ zero )
= A2 ) ) ).
% right_unit
thf(fact_62_inverse__unit,axiom,
( ( group_inverse_a @ g @ addition @ zero @ zero )
= zero ) ).
% inverse_unit
thf(fact_63_additive__abelian__group__axioms,axiom,
pluenn1164192988769422572roup_a @ g @ addition @ zero ).
% additive_abelian_group_axioms
thf(fact_64_commutative__monoid__axioms,axiom,
group_4866109990395492029noid_a @ g @ addition @ zero ).
% commutative_monoid_axioms
thf(fact_65_minussetp_Ocases,axiom,
! [A: a > $o,A2: a] :
( ( pluenn1126946703085653920setp_a @ g @ addition @ zero @ A @ A2 )
=> ~ ! [A4: a] :
( ( A2
= ( group_inverse_a @ g @ addition @ zero @ A4 ) )
=> ( ( A @ A4 )
=> ~ ( member_a @ A4 @ g ) ) ) ) ).
% minussetp.cases
thf(fact_66_minussetp_OminussetI,axiom,
! [A: a > $o,A2: a] :
( ( A @ A2 )
=> ( ( member_a @ A2 @ g )
=> ( pluenn1126946703085653920setp_a @ g @ addition @ zero @ A @ ( group_inverse_a @ g @ addition @ zero @ A2 ) ) ) ) ).
% minussetp.minussetI
thf(fact_67_minussetp_Osimps,axiom,
! [A: a > $o,A2: a] :
( ( pluenn1126946703085653920setp_a @ g @ addition @ zero @ A @ A2 )
= ( ? [A3: a] :
( ( A2
= ( group_inverse_a @ g @ addition @ zero @ A3 ) )
& ( A @ A3 )
& ( member_a @ A3 @ g ) ) ) ) ).
% minussetp.simps
thf(fact_68_abelian__group__axioms,axiom,
group_201663378560352916roup_a @ g @ addition @ zero ).
% abelian_group_axioms
thf(fact_69_sumsetp_Ocases,axiom,
! [A: a > $o,B: a > $o,A2: a] :
( ( pluenn895083305082786853setp_a @ g @ addition @ A @ B @ A2 )
=> ~ ! [A4: a,B4: a] :
( ( A2
= ( addition @ A4 @ B4 ) )
=> ( ( A @ A4 )
=> ( ( member_a @ A4 @ g )
=> ( ( B @ B4 )
=> ~ ( member_a @ B4 @ g ) ) ) ) ) ) ).
% sumsetp.cases
thf(fact_70_sumsetp_Osimps,axiom,
! [A: a > $o,B: a > $o,A2: a] :
( ( pluenn895083305082786853setp_a @ g @ addition @ A @ B @ A2 )
= ( ? [A3: a,B3: a] :
( ( A2
= ( addition @ A3 @ B3 ) )
& ( A @ A3 )
& ( member_a @ A3 @ g )
& ( B @ B3 )
& ( member_a @ B3 @ g ) ) ) ) ).
% sumsetp.simps
thf(fact_71_sumsetp_OsumsetI,axiom,
! [A: a > $o,A2: a,B: a > $o,B2: a] :
( ( A @ A2 )
=> ( ( member_a @ A2 @ g )
=> ( ( B @ B2 )
=> ( ( member_a @ B2 @ g )
=> ( pluenn895083305082786853setp_a @ g @ addition @ A @ B @ ( addition @ A2 @ B2 ) ) ) ) ) ) ).
% sumsetp.sumsetI
thf(fact_72_sumset__iterated__subset__carrier,axiom,
! [A: set_a,K: nat] : ( ord_less_eq_set_a @ ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ A @ K ) @ g ) ).
% sumset_iterated_subset_carrier
thf(fact_73_finite__sumset__iterated,axiom,
! [A: set_a,R: nat] :
( ( finite_finite_a @ A )
=> ( finite_finite_a @ ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ A @ R ) ) ) ).
% finite_sumset_iterated
thf(fact_74_minusset__iterated__minusset,axiom,
! [A: set_a,K: nat] :
( ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A ) @ K )
= ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ A @ K ) ) ) ).
% minusset_iterated_minusset
thf(fact_75_additive__abelian__group__def,axiom,
pluenn1164192988769422572roup_a = group_201663378560352916roup_a ).
% additive_abelian_group_def
thf(fact_76_additive__abelian__group_Osumset__iterated_Ocong,axiom,
pluenn1960970773371692859ated_a = pluenn1960970773371692859ated_a ).
% additive_abelian_group.sumset_iterated.cong
thf(fact_77_additive__abelian__group_Osumsetp_OsumsetI,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: nat > $o,A2: nat,B: nat > $o,B2: nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( A @ A2 )
=> ( ( member_nat @ A2 @ G )
=> ( ( B @ B2 )
=> ( ( member_nat @ B2 @ G )
=> ( pluenn5670965976768739049tp_nat @ G @ Addition @ A @ B @ ( Addition @ A2 @ B2 ) ) ) ) ) ) ) ).
% additive_abelian_group.sumsetp.sumsetI
thf(fact_78_additive__abelian__group_Osumsetp_OsumsetI,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: a > $o,A2: a,B: a > $o,B2: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( A @ A2 )
=> ( ( member_a @ A2 @ G )
=> ( ( B @ B2 )
=> ( ( member_a @ B2 @ G )
=> ( pluenn895083305082786853setp_a @ G @ Addition @ A @ B @ ( Addition @ A2 @ B2 ) ) ) ) ) ) ) ).
% additive_abelian_group.sumsetp.sumsetI
thf(fact_79_additive__abelian__group_Osumsetp_Osimps,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: nat > $o,B: nat > $o,A2: nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( pluenn5670965976768739049tp_nat @ G @ Addition @ A @ B @ A2 )
= ( ? [A3: nat,B3: nat] :
( ( A2
= ( Addition @ A3 @ B3 ) )
& ( A @ A3 )
& ( member_nat @ A3 @ G )
& ( B @ B3 )
& ( member_nat @ B3 @ G ) ) ) ) ) ).
% additive_abelian_group.sumsetp.simps
thf(fact_80_additive__abelian__group_Osumsetp_Osimps,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: a > $o,B: a > $o,A2: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn895083305082786853setp_a @ G @ Addition @ A @ B @ A2 )
= ( ? [A3: a,B3: a] :
( ( A2
= ( Addition @ A3 @ B3 ) )
& ( A @ A3 )
& ( member_a @ A3 @ G )
& ( B @ B3 )
& ( member_a @ B3 @ G ) ) ) ) ) ).
% additive_abelian_group.sumsetp.simps
thf(fact_81_additive__abelian__group_Osumsetp_Ocases,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: nat > $o,B: nat > $o,A2: nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( pluenn5670965976768739049tp_nat @ G @ Addition @ A @ B @ A2 )
=> ~ ! [A4: nat,B4: nat] :
( ( A2
= ( Addition @ A4 @ B4 ) )
=> ( ( A @ A4 )
=> ( ( member_nat @ A4 @ G )
=> ( ( B @ B4 )
=> ~ ( member_nat @ B4 @ G ) ) ) ) ) ) ) ).
% additive_abelian_group.sumsetp.cases
thf(fact_82_additive__abelian__group_Osumsetp_Ocases,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: a > $o,B: a > $o,A2: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn895083305082786853setp_a @ G @ Addition @ A @ B @ A2 )
=> ~ ! [A4: a,B4: a] :
( ( A2
= ( Addition @ A4 @ B4 ) )
=> ( ( A @ A4 )
=> ( ( member_a @ A4 @ G )
=> ( ( B @ B4 )
=> ~ ( member_a @ B4 @ G ) ) ) ) ) ) ) ).
% additive_abelian_group.sumsetp.cases
thf(fact_83_additive__abelian__group_Oaxioms,axiom,
! [G: set_a,Addition: a > a > a,Zero: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( group_201663378560352916roup_a @ G @ Addition @ Zero ) ) ).
% additive_abelian_group.axioms
thf(fact_84_additive__abelian__group_Ointro,axiom,
! [G: set_a,Addition: a > a > a,Zero: a] :
( ( group_201663378560352916roup_a @ G @ Addition @ Zero )
=> ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero ) ) ).
% additive_abelian_group.intro
thf(fact_85_additive__abelian__group_Osumsetp_Ocong,axiom,
pluenn895083305082786853setp_a = pluenn895083305082786853setp_a ).
% additive_abelian_group.sumsetp.cong
thf(fact_86_additive__abelian__group_Ominussetp_Ocong,axiom,
pluenn1126946703085653920setp_a = pluenn1126946703085653920setp_a ).
% additive_abelian_group.minussetp.cong
thf(fact_87_additive__abelian__group_Ofinite__sumset__iterated,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: set_nat,R: nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( finite_finite_nat @ A )
=> ( finite_finite_nat @ ( pluenn7055013279391836755ed_nat @ G @ Addition @ Zero @ A @ R ) ) ) ) ).
% additive_abelian_group.finite_sumset_iterated
thf(fact_88_additive__abelian__group_Ofinite__sumset__iterated,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,R: nat] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_finite_a @ A )
=> ( finite_finite_a @ ( pluenn1960970773371692859ated_a @ G @ Addition @ Zero @ A @ R ) ) ) ) ).
% additive_abelian_group.finite_sumset_iterated
thf(fact_89_additive__abelian__group_Osumset__iterated__subset__carrier,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,K: nat] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ord_less_eq_set_a @ ( pluenn1960970773371692859ated_a @ G @ Addition @ Zero @ A @ K ) @ G ) ) ).
% additive_abelian_group.sumset_iterated_subset_carrier
thf(fact_90_additive__abelian__group_Ominusset__iterated__minusset,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,K: nat] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn1960970773371692859ated_a @ G @ Addition @ Zero @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A ) @ K )
= ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ ( pluenn1960970773371692859ated_a @ G @ Addition @ Zero @ A @ K ) ) ) ) ).
% additive_abelian_group.minusset_iterated_minusset
thf(fact_91_additive__abelian__group_Ominussetp_OminussetI,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: nat > $o,A2: nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( A @ A2 )
=> ( ( member_nat @ A2 @ G )
=> ( pluenn8372939692575285934tp_nat @ G @ Addition @ Zero @ A @ ( group_inverse_nat @ G @ Addition @ Zero @ A2 ) ) ) ) ) ).
% additive_abelian_group.minussetp.minussetI
thf(fact_92_additive__abelian__group_Ominussetp_OminussetI,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: a > $o,A2: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( A @ A2 )
=> ( ( member_a @ A2 @ G )
=> ( pluenn1126946703085653920setp_a @ G @ Addition @ Zero @ A @ ( group_inverse_a @ G @ Addition @ Zero @ A2 ) ) ) ) ) ).
% additive_abelian_group.minussetp.minussetI
thf(fact_93_additive__abelian__group_Ominussetp_Osimps,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: nat > $o,A2: nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( pluenn8372939692575285934tp_nat @ G @ Addition @ Zero @ A @ A2 )
= ( ? [A3: nat] :
( ( A2
= ( group_inverse_nat @ G @ Addition @ Zero @ A3 ) )
& ( A @ A3 )
& ( member_nat @ A3 @ G ) ) ) ) ) ).
% additive_abelian_group.minussetp.simps
thf(fact_94_additive__abelian__group_Ominussetp_Osimps,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: a > $o,A2: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn1126946703085653920setp_a @ G @ Addition @ Zero @ A @ A2 )
= ( ? [A3: a] :
( ( A2
= ( group_inverse_a @ G @ Addition @ Zero @ A3 ) )
& ( A @ A3 )
& ( member_a @ A3 @ G ) ) ) ) ) ).
% additive_abelian_group.minussetp.simps
thf(fact_95_additive__abelian__group_Ominussetp_Ocases,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: nat > $o,A2: nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( pluenn8372939692575285934tp_nat @ G @ Addition @ Zero @ A @ A2 )
=> ~ ! [A4: nat] :
( ( A2
= ( group_inverse_nat @ G @ Addition @ Zero @ A4 ) )
=> ( ( A @ A4 )
=> ~ ( member_nat @ A4 @ G ) ) ) ) ) ).
% additive_abelian_group.minussetp.cases
thf(fact_96_additive__abelian__group_Ominussetp_Ocases,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: a > $o,A2: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn1126946703085653920setp_a @ G @ Addition @ Zero @ A @ A2 )
=> ~ ! [A4: a] :
( ( A2
= ( group_inverse_a @ G @ Addition @ Zero @ A4 ) )
=> ( ( A @ A4 )
=> ~ ( member_a @ A4 @ G ) ) ) ) ) ).
% additive_abelian_group.minussetp.cases
thf(fact_97_additive__abelian__group_Osumset__commute,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B )
= ( pluenn3038260743871226533mset_a @ G @ Addition @ B @ A ) ) ) ).
% additive_abelian_group.sumset_commute
thf(fact_98_additive__abelian__group_Osumset_OsumsetI,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: nat,A: set_nat,B2: nat,B: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( member_nat @ A2 @ A )
=> ( ( member_nat @ A2 @ G )
=> ( ( member_nat @ B2 @ B )
=> ( ( member_nat @ B2 @ G )
=> ( member_nat @ ( Addition @ A2 @ B2 ) @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A @ B ) ) ) ) ) ) ) ).
% additive_abelian_group.sumset.sumsetI
thf(fact_99_additive__abelian__group_Osumset_OsumsetI,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: a,A: set_a,B2: a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( member_a @ A2 @ A )
=> ( ( member_a @ A2 @ G )
=> ( ( member_a @ B2 @ B )
=> ( ( member_a @ B2 @ G )
=> ( member_a @ ( Addition @ A2 @ B2 ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) ) ) ) ) ) ) ).
% additive_abelian_group.sumset.sumsetI
thf(fact_100_additive__abelian__group_Osumset__assoc,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a,C: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) @ C )
= ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ ( pluenn3038260743871226533mset_a @ G @ Addition @ B @ C ) ) ) ) ).
% additive_abelian_group.sumset_assoc
thf(fact_101_additive__abelian__group_Osumset_Osimps,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: nat,A: set_nat,B: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( member_nat @ A2 @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A @ B ) )
= ( ? [A3: nat,B3: nat] :
( ( A2
= ( Addition @ A3 @ B3 ) )
& ( member_nat @ A3 @ A )
& ( member_nat @ A3 @ G )
& ( member_nat @ B3 @ B )
& ( member_nat @ B3 @ G ) ) ) ) ) ).
% additive_abelian_group.sumset.simps
thf(fact_102_additive__abelian__group_Osumset_Osimps,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: a,A: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( member_a @ A2 @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) )
= ( ? [A3: a,B3: a] :
( ( A2
= ( Addition @ A3 @ B3 ) )
& ( member_a @ A3 @ A )
& ( member_a @ A3 @ G )
& ( member_a @ B3 @ B )
& ( member_a @ B3 @ G ) ) ) ) ) ).
% additive_abelian_group.sumset.simps
thf(fact_103_additive__abelian__group_Osumset_Ocases,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: nat,A: set_nat,B: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( member_nat @ A2 @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A @ B ) )
=> ~ ! [A4: nat,B4: nat] :
( ( A2
= ( Addition @ A4 @ B4 ) )
=> ( ( member_nat @ A4 @ A )
=> ( ( member_nat @ A4 @ G )
=> ( ( member_nat @ B4 @ B )
=> ~ ( member_nat @ B4 @ G ) ) ) ) ) ) ) ).
% additive_abelian_group.sumset.cases
thf(fact_104_additive__abelian__group_Osumset_Ocases,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: a,A: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( member_a @ A2 @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) )
=> ~ ! [A4: a,B4: a] :
( ( A2
= ( Addition @ A4 @ B4 ) )
=> ( ( member_a @ A4 @ A )
=> ( ( member_a @ A4 @ G )
=> ( ( member_a @ B4 @ B )
=> ~ ( member_a @ B4 @ G ) ) ) ) ) ) ) ).
% additive_abelian_group.sumset.cases
thf(fact_105_additive__abelian__group_Oinverse__closed,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,X: nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( member_nat @ X @ G )
=> ( member_nat @ ( group_inverse_nat @ G @ Addition @ Zero @ X ) @ G ) ) ) ).
% additive_abelian_group.inverse_closed
thf(fact_106_additive__abelian__group_Oinverse__closed,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,X: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( member_a @ X @ G )
=> ( member_a @ ( group_inverse_a @ G @ Addition @ Zero @ X ) @ G ) ) ) ).
% additive_abelian_group.inverse_closed
thf(fact_107_additive__abelian__group_Ofinite__sumset,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: set_nat,B: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A @ B ) ) ) ) ) ).
% additive_abelian_group.finite_sumset
thf(fact_108_additive__abelian__group_Ofinite__sumset,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) ) ) ) ) ).
% additive_abelian_group.finite_sumset
thf(fact_109_additive__abelian__group_Osumset__subset__carrier,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) @ G ) ) ).
% additive_abelian_group.sumset_subset_carrier
thf(fact_110_additive__abelian__group_Osumset__mono,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A5: set_a,A: set_a,B5: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ord_less_eq_set_a @ A5 @ A )
=> ( ( ord_less_eq_set_a @ B5 @ B )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A5 @ B5 ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) ) ) ) ) ).
% additive_abelian_group.sumset_mono
thf(fact_111_additive__abelian__group_Ofinite__minusset,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( finite_finite_nat @ A )
=> ( finite_finite_nat @ ( pluenn7323955030898006884et_nat @ G @ Addition @ Zero @ A ) ) ) ) ).
% additive_abelian_group.finite_minusset
thf(fact_112_additive__abelian__group_Ofinite__minusset,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_finite_a @ A )
=> ( finite_finite_a @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A ) ) ) ) ).
% additive_abelian_group.finite_minusset
thf(fact_113_additive__abelian__group_Ominusset__subset__carrier,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ord_less_eq_set_a @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A ) @ G ) ) ).
% additive_abelian_group.minusset_subset_carrier
thf(fact_114_additive__abelian__group_Odifferenceset__commute,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,B: set_a,A: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ ( pluenn3038260743871226533mset_a @ G @ Addition @ B @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A ) ) )
= ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ B ) ) ) ) ).
% additive_abelian_group.differenceset_commute
thf(fact_115_additive__abelian__group_Ominusset__distrib__sum,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) )
= ( pluenn3038260743871226533mset_a @ G @ Addition @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A ) @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ B ) ) ) ) ).
% additive_abelian_group.minusset_distrib_sum
thf(fact_116_additive__abelian__group_Odiff__minus__set,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A ) @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ B ) )
= ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) ) ) ) ).
% additive_abelian_group.diff_minus_set
thf(fact_117_additive__abelian__group_Ominusset_OminussetI,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: nat,A: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( member_nat @ A2 @ A )
=> ( ( member_nat @ A2 @ G )
=> ( member_nat @ ( group_inverse_nat @ G @ Addition @ Zero @ A2 ) @ ( pluenn7323955030898006884et_nat @ G @ Addition @ Zero @ A ) ) ) ) ) ).
% additive_abelian_group.minusset.minussetI
thf(fact_118_additive__abelian__group_Ominusset_OminussetI,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: a,A: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( member_a @ A2 @ A )
=> ( ( member_a @ A2 @ G )
=> ( member_a @ ( group_inverse_a @ G @ Addition @ Zero @ A2 ) @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A ) ) ) ) ) ).
% additive_abelian_group.minusset.minussetI
thf(fact_119_additive__abelian__group_Ominusset_Osimps,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: nat,A: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( member_nat @ A2 @ ( pluenn7323955030898006884et_nat @ G @ Addition @ Zero @ A ) )
= ( ? [A3: nat] :
( ( A2
= ( group_inverse_nat @ G @ Addition @ Zero @ A3 ) )
& ( member_nat @ A3 @ A )
& ( member_nat @ A3 @ G ) ) ) ) ) ).
% additive_abelian_group.minusset.simps
thf(fact_120_additive__abelian__group_Ominusset_Osimps,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: a,A: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( member_a @ A2 @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A ) )
= ( ? [A3: a] :
( ( A2
= ( group_inverse_a @ G @ Addition @ Zero @ A3 ) )
& ( member_a @ A3 @ A )
& ( member_a @ A3 @ G ) ) ) ) ) ).
% additive_abelian_group.minusset.simps
thf(fact_121_additive__abelian__group_Ominusset_Ocases,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: nat,A: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( member_nat @ A2 @ ( pluenn7323955030898006884et_nat @ G @ Addition @ Zero @ A ) )
=> ~ ! [A4: nat] :
( ( A2
= ( group_inverse_nat @ G @ Addition @ Zero @ A4 ) )
=> ( ( member_nat @ A4 @ A )
=> ~ ( member_nat @ A4 @ G ) ) ) ) ) ).
% additive_abelian_group.minusset.cases
thf(fact_122_additive__abelian__group_Ominusset_Ocases,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: a,A: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( member_a @ A2 @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A ) )
=> ~ ! [A4: a] :
( ( A2
= ( group_inverse_a @ G @ Addition @ Zero @ A4 ) )
=> ( ( member_a @ A4 @ A )
=> ~ ( member_a @ A4 @ G ) ) ) ) ) ).
% additive_abelian_group.minusset.cases
thf(fact_123_additive__abelian__group_Ofinite__differenceset,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: set_nat,B: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A @ ( pluenn7323955030898006884et_nat @ G @ Addition @ Zero @ B ) ) ) ) ) ) ).
% additive_abelian_group.finite_differenceset
thf(fact_124_additive__abelian__group_Ofinite__differenceset,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ B ) ) ) ) ) ) ).
% additive_abelian_group.finite_differenceset
thf(fact_125_card__sumset__iterated__minusset,axiom,
! [A: set_a,K: nat] :
( ( finite_card_a @ ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A ) @ K ) )
= ( finite_card_a @ ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ A @ K ) ) ) ).
% card_sumset_iterated_minusset
thf(fact_126_card__minusset_H,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ g )
=> ( ( finite_card_a @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A ) )
= ( finite_card_a @ A ) ) ) ).
% card_minusset'
thf(fact_127_card__differenceset__commute,axiom,
! [B: set_a,A: set_a] :
( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ B @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A ) ) )
= ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ B ) ) ) ) ).
% card_differenceset_commute
thf(fact_128_group__axioms,axiom,
group_group_a @ g @ addition @ zero ).
% group_axioms
thf(fact_129_sumset__subset__insert_I1_J,axiom,
! [A: set_a,B: set_a,X: a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( insert_a @ X @ B ) ) ) ).
% sumset_subset_insert(1)
thf(fact_130_sumset__subset__insert_I2_J,axiom,
! [A: set_a,B: set_a,X: a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ ( insert_a @ X @ A ) @ B ) ) ).
% sumset_subset_insert(2)
thf(fact_131_finite__sumset_H,axiom,
! [A: set_a,B: set_a] :
( ( finite_finite_a @ ( inf_inf_set_a @ A @ g ) )
=> ( ( finite_finite_a @ ( inf_inf_set_a @ B @ g ) )
=> ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) ) ) ) ).
% finite_sumset'
thf(fact_132_sumset__subset__Un_I1_J,axiom,
! [A: set_a,B: set_a,C: set_a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( sup_sup_set_a @ B @ C ) ) ) ).
% sumset_subset_Un(1)
thf(fact_133_sumset__subset__Un_I2_J,axiom,
! [A: set_a,B: set_a,C: set_a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ ( sup_sup_set_a @ A @ C ) @ B ) ) ).
% sumset_subset_Un(2)
thf(fact_134_invertible__right__inverse2,axiom,
! [U: a,V2: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( ( member_a @ V2 @ g )
=> ( ( addition @ U @ ( addition @ ( group_inverse_a @ g @ addition @ zero @ U ) @ V2 ) )
= V2 ) ) ) ) ).
% invertible_right_inverse2
thf(fact_135_sumset__subset__Un1,axiom,
! [A: set_a,A5: set_a,B: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ ( sup_sup_set_a @ A @ A5 ) @ B )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A5 @ B ) ) ) ).
% sumset_subset_Un1
thf(fact_136_sumset__subset__Un2,axiom,
! [A: set_a,B: set_a,B5: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( sup_sup_set_a @ B @ B5 ) )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B5 ) ) ) ).
% sumset_subset_Un2
thf(fact_137_invertibleE,axiom,
! [U: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ! [V3: a] :
( ( ( ( addition @ U @ V3 )
= zero )
& ( ( addition @ V3 @ U )
= zero ) )
=> ~ ( member_a @ V3 @ g ) )
=> ~ ( member_a @ U @ g ) ) ) ).
% invertibleE
thf(fact_138_invertible__def,axiom,
! [U: a] :
( ( member_a @ U @ g )
=> ( ( group_invertible_a @ g @ addition @ zero @ U )
= ( ? [X2: a] :
( ( member_a @ X2 @ g )
& ( ( addition @ U @ X2 )
= zero )
& ( ( addition @ X2 @ U )
= zero ) ) ) ) ) ).
% invertible_def
thf(fact_139_unit__invertible,axiom,
group_invertible_a @ g @ addition @ zero @ zero ).
% unit_invertible
thf(fact_140_inverse__composition__commute,axiom,
! [X: a,Y: a] :
( ( group_invertible_a @ g @ addition @ zero @ X )
=> ( ( group_invertible_a @ g @ addition @ zero @ Y )
=> ( ( member_a @ X @ g )
=> ( ( member_a @ Y @ g )
=> ( ( group_inverse_a @ g @ addition @ zero @ ( addition @ X @ Y ) )
= ( addition @ ( group_inverse_a @ g @ addition @ zero @ Y ) @ ( group_inverse_a @ g @ addition @ zero @ X ) ) ) ) ) ) ) ).
% inverse_composition_commute
thf(fact_141_invertible__left__inverse2,axiom,
! [U: a,V2: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( ( member_a @ V2 @ g )
=> ( ( addition @ ( group_inverse_a @ g @ addition @ zero @ U ) @ ( addition @ U @ V2 ) )
= V2 ) ) ) ) ).
% invertible_left_inverse2
thf(fact_142_sumset__Int__carrier__eq_I2_J,axiom,
! [A: set_a,B: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ ( inf_inf_set_a @ A @ g ) @ B )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) ) ).
% sumset_Int_carrier_eq(2)
thf(fact_143_sumset__Int__carrier__eq_I1_J,axiom,
! [A: set_a,B: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( inf_inf_set_a @ B @ g ) )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) ) ).
% sumset_Int_carrier_eq(1)
thf(fact_144_sumset__Int__carrier,axiom,
! [A: set_a,B: set_a] :
( ( inf_inf_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) @ g )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) ) ).
% sumset_Int_carrier
thf(fact_145_composition__invertible,axiom,
! [X: a,Y: a] :
( ( group_invertible_a @ g @ addition @ zero @ X )
=> ( ( group_invertible_a @ g @ addition @ zero @ Y )
=> ( ( member_a @ X @ g )
=> ( ( member_a @ Y @ g )
=> ( group_invertible_a @ g @ addition @ zero @ ( addition @ X @ Y ) ) ) ) ) ) ).
% composition_invertible
thf(fact_146_invertible,axiom,
! [U: a] :
( ( member_a @ U @ g )
=> ( group_invertible_a @ g @ addition @ zero @ U ) ) ).
% invertible
thf(fact_147_invertibleI,axiom,
! [U: a,V2: a] :
( ( ( addition @ U @ V2 )
= zero )
=> ( ( ( addition @ V2 @ U )
= zero )
=> ( ( member_a @ U @ g )
=> ( ( member_a @ V2 @ g )
=> ( group_invertible_a @ g @ addition @ zero @ U ) ) ) ) ) ).
% invertibleI
thf(fact_148_invertible__left__cancel,axiom,
! [X: a,Y: a,Z: a] :
( ( group_invertible_a @ g @ addition @ zero @ X )
=> ( ( member_a @ X @ g )
=> ( ( member_a @ Y @ g )
=> ( ( member_a @ Z @ g )
=> ( ( ( addition @ X @ Y )
= ( addition @ X @ Z ) )
= ( Y = Z ) ) ) ) ) ) ).
% invertible_left_cancel
thf(fact_149_invertible__right__cancel,axiom,
! [X: a,Y: a,Z: a] :
( ( group_invertible_a @ g @ addition @ zero @ X )
=> ( ( member_a @ X @ g )
=> ( ( member_a @ Y @ g )
=> ( ( member_a @ Z @ g )
=> ( ( ( addition @ Y @ X )
= ( addition @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ) ).
% invertible_right_cancel
thf(fact_150_invertible__inverse__closed,axiom,
! [U: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( member_a @ ( group_inverse_a @ g @ addition @ zero @ U ) @ g ) ) ) ).
% invertible_inverse_closed
thf(fact_151_invertible__inverse__inverse,axiom,
! [U: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( ( group_inverse_a @ g @ addition @ zero @ ( group_inverse_a @ g @ addition @ zero @ U ) )
= U ) ) ) ).
% invertible_inverse_inverse
thf(fact_152_invertible__inverse__invertible,axiom,
! [U: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( group_invertible_a @ g @ addition @ zero @ ( group_inverse_a @ g @ addition @ zero @ U ) ) ) ) ).
% invertible_inverse_invertible
thf(fact_153_invertible__left__inverse,axiom,
! [U: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( ( addition @ ( group_inverse_a @ g @ addition @ zero @ U ) @ U )
= zero ) ) ) ).
% invertible_left_inverse
thf(fact_154_invertible__right__inverse,axiom,
! [U: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( ( addition @ U @ ( group_inverse_a @ g @ addition @ zero @ U ) )
= zero ) ) ) ).
% invertible_right_inverse
thf(fact_155_minus__minusset,axiom,
! [A: set_a] :
( ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A ) )
= ( inf_inf_set_a @ A @ g ) ) ).
% minus_minusset
thf(fact_156_card__minusset,axiom,
! [A: set_a] :
( ( finite_card_a @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A ) )
= ( finite_card_a @ ( inf_inf_set_a @ A @ g ) ) ) ).
% card_minusset
thf(fact_157_additive__abelian__group_Ocard__minusset,axiom,
! [G: set_Product_unit,Addition: product_unit > product_unit > product_unit,Zero: product_unit,A: set_Product_unit] :
( ( pluenn3635716580025208315t_unit @ G @ Addition @ Zero )
=> ( ( finite410649719033368117t_unit @ ( pluenn8189473251256886137t_unit @ G @ Addition @ Zero @ A ) )
= ( finite410649719033368117t_unit @ ( inf_in4660618365625256667t_unit @ A @ G ) ) ) ) ).
% additive_abelian_group.card_minusset
thf(fact_158_additive__abelian__group_Ocard__minusset,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_card_a @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A ) )
= ( finite_card_a @ ( inf_inf_set_a @ A @ G ) ) ) ) ).
% additive_abelian_group.card_minusset
thf(fact_159_additive__abelian__group_Osumset__subset__Un1,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,A5: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ ( sup_sup_set_a @ A @ A5 ) @ B )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A5 @ B ) ) ) ) ).
% additive_abelian_group.sumset_subset_Un1
thf(fact_160_additive__abelian__group_Osumset__subset__Un2,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a,B5: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ ( sup_sup_set_a @ B @ B5 ) )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B5 ) ) ) ) ).
% additive_abelian_group.sumset_subset_Un2
thf(fact_161_additive__abelian__group_Osumset__Int__carrier__eq_I2_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ ( inf_inf_set_a @ A @ G ) @ B )
= ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) ) ) ).
% additive_abelian_group.sumset_Int_carrier_eq(2)
thf(fact_162_additive__abelian__group_Osumset__Int__carrier__eq_I1_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ ( inf_inf_set_a @ B @ G ) )
= ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) ) ) ).
% additive_abelian_group.sumset_Int_carrier_eq(1)
thf(fact_163_additive__abelian__group_Osumset__Int__carrier,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( inf_inf_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) @ G )
= ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) ) ) ).
% additive_abelian_group.sumset_Int_carrier
thf(fact_164_additive__abelian__group_Ominus__minusset,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A ) )
= ( inf_inf_set_a @ A @ G ) ) ) ).
% additive_abelian_group.minus_minusset
thf(fact_165_additive__abelian__group_Ocard__minusset_H,axiom,
! [G: set_Product_unit,Addition: product_unit > product_unit > product_unit,Zero: product_unit,A: set_Product_unit] :
( ( pluenn3635716580025208315t_unit @ G @ Addition @ Zero )
=> ( ( ord_le3507040750410214029t_unit @ A @ G )
=> ( ( finite410649719033368117t_unit @ ( pluenn8189473251256886137t_unit @ G @ Addition @ Zero @ A ) )
= ( finite410649719033368117t_unit @ A ) ) ) ) ).
% additive_abelian_group.card_minusset'
thf(fact_166_additive__abelian__group_Ocard__minusset_H,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ord_less_eq_set_a @ A @ G )
=> ( ( finite_card_a @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A ) )
= ( finite_card_a @ A ) ) ) ) ).
% additive_abelian_group.card_minusset'
thf(fact_167_additive__abelian__group_Osumset__subset__Un_I2_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a,C: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ ( sup_sup_set_a @ A @ C ) @ B ) ) ) ).
% additive_abelian_group.sumset_subset_Un(2)
thf(fact_168_additive__abelian__group_Osumset__subset__Un_I1_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a,C: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ ( sup_sup_set_a @ B @ C ) ) ) ) ).
% additive_abelian_group.sumset_subset_Un(1)
thf(fact_169_additive__abelian__group_Ofinite__sumset_H,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: set_nat,B: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( finite_finite_nat @ ( inf_inf_set_nat @ A @ G ) )
=> ( ( finite_finite_nat @ ( inf_inf_set_nat @ B @ G ) )
=> ( finite_finite_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A @ B ) ) ) ) ) ).
% additive_abelian_group.finite_sumset'
thf(fact_170_additive__abelian__group_Ofinite__sumset_H,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_finite_a @ ( inf_inf_set_a @ A @ G ) )
=> ( ( finite_finite_a @ ( inf_inf_set_a @ B @ G ) )
=> ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) ) ) ) ) ).
% additive_abelian_group.finite_sumset'
thf(fact_171_additive__abelian__group_Osumset__subset__insert_I2_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a,X: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ ( insert_a @ X @ A ) @ B ) ) ) ).
% additive_abelian_group.sumset_subset_insert(2)
thf(fact_172_additive__abelian__group_Osumset__subset__insert_I1_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a,X: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ ( insert_a @ X @ B ) ) ) ) ).
% additive_abelian_group.sumset_subset_insert(1)
thf(fact_173_additive__abelian__group_Ocard__differenceset__commute,axiom,
! [G: set_Product_unit,Addition: product_unit > product_unit > product_unit,Zero: product_unit,B: set_Product_unit,A: set_Product_unit] :
( ( pluenn3635716580025208315t_unit @ G @ Addition @ Zero )
=> ( ( finite410649719033368117t_unit @ ( pluenn1407455289632237236t_unit @ G @ Addition @ B @ ( pluenn8189473251256886137t_unit @ G @ Addition @ Zero @ A ) ) )
= ( finite410649719033368117t_unit @ ( pluenn1407455289632237236t_unit @ G @ Addition @ A @ ( pluenn8189473251256886137t_unit @ G @ Addition @ Zero @ B ) ) ) ) ) ).
% additive_abelian_group.card_differenceset_commute
thf(fact_174_additive__abelian__group_Ocard__differenceset__commute,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,B: set_a,A: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ B @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A ) ) )
= ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ B ) ) ) ) ) ).
% additive_abelian_group.card_differenceset_commute
thf(fact_175_additive__abelian__group_Ocard__sumset__iterated__minusset,axiom,
! [G: set_Product_unit,Addition: product_unit > product_unit > product_unit,Zero: product_unit,A: set_Product_unit,K: nat] :
( ( pluenn3635716580025208315t_unit @ G @ Addition @ Zero )
=> ( ( finite410649719033368117t_unit @ ( pluenn7331107188738702666t_unit @ G @ Addition @ Zero @ ( pluenn8189473251256886137t_unit @ G @ Addition @ Zero @ A ) @ K ) )
= ( finite410649719033368117t_unit @ ( pluenn7331107188738702666t_unit @ G @ Addition @ Zero @ A @ K ) ) ) ) ).
% additive_abelian_group.card_sumset_iterated_minusset
thf(fact_176_additive__abelian__group_Ocard__sumset__iterated__minusset,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,K: nat] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_card_a @ ( pluenn1960970773371692859ated_a @ G @ Addition @ Zero @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A ) @ K ) )
= ( finite_card_a @ ( pluenn1960970773371692859ated_a @ G @ Addition @ Zero @ A @ K ) ) ) ) ).
% additive_abelian_group.card_sumset_iterated_minusset
thf(fact_177_mem__UnitsD,axiom,
! [U: a] :
( ( member_a @ U @ ( group_Units_a @ g @ addition @ zero ) )
=> ( ( group_invertible_a @ g @ addition @ zero @ U )
& ( member_a @ U @ g ) ) ) ).
% mem_UnitsD
thf(fact_178_mem__UnitsI,axiom,
! [U: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( member_a @ U @ ( group_Units_a @ g @ addition @ zero ) ) ) ) ).
% mem_UnitsI
thf(fact_179_subgroupI,axiom,
! [G: set_a] :
( ( ord_less_eq_set_a @ G @ g )
=> ( ( member_a @ zero @ G )
=> ( ! [G2: a,H: a] :
( ( member_a @ G2 @ G )
=> ( ( member_a @ H @ G )
=> ( member_a @ ( addition @ G2 @ H ) @ G ) ) )
=> ( ! [G2: a] :
( ( member_a @ G2 @ G )
=> ( group_invertible_a @ g @ addition @ zero @ G2 ) )
=> ( ! [G2: a] :
( ( member_a @ G2 @ G )
=> ( member_a @ ( group_inverse_a @ g @ addition @ zero @ G2 ) @ G ) )
=> ( group_subgroup_a @ G @ g @ addition @ zero ) ) ) ) ) ) ).
% subgroupI
thf(fact_180_card__le__sumset,axiom,
! [A: set_a,A2: a,B: set_a] :
( ( finite_finite_a @ A )
=> ( ( member_a @ A2 @ A )
=> ( ( member_a @ A2 @ g )
=> ( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ B @ g )
=> ( ord_less_eq_nat @ ( finite_card_a @ B ) @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) ) ) ) ) ) ) ) ).
% card_le_sumset
thf(fact_181_card__sumset__0__iff,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ g )
=> ( ( ord_less_eq_set_a @ B @ g )
=> ( ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) )
= zero_zero_nat )
= ( ( ( finite_card_a @ A )
= zero_zero_nat )
| ( ( finite_card_a @ B )
= zero_zero_nat ) ) ) ) ) ).
% card_sumset_0_iff
thf(fact_182_card__sumset__0__iff_H,axiom,
! [A: set_a,B: set_a] :
( ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) )
= zero_zero_nat )
= ( ( ( finite_card_a @ ( inf_inf_set_a @ A @ g ) )
= zero_zero_nat )
| ( ( finite_card_a @ ( inf_inf_set_a @ B @ g ) )
= zero_zero_nat ) ) ) ).
% card_sumset_0_iff'
thf(fact_183_infinite__sumset__aux,axiom,
! [A: set_a,B: set_a] :
( ~ ( finite_finite_a @ ( inf_inf_set_a @ A @ g ) )
=> ( ( ~ ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) ) )
= ( ( inf_inf_set_a @ B @ g )
!= bot_bot_set_a ) ) ) ).
% infinite_sumset_aux
thf(fact_184_infinite__sumset__iff,axiom,
! [A: set_a,B: set_a] :
( ( ~ ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) ) )
= ( ( ~ ( finite_finite_a @ ( inf_inf_set_a @ A @ g ) )
& ( ( inf_inf_set_a @ B @ g )
!= bot_bot_set_a ) )
| ( ( ( inf_inf_set_a @ A @ g )
!= bot_bot_set_a )
& ~ ( finite_finite_a @ ( inf_inf_set_a @ B @ g ) ) ) ) ) ).
% infinite_sumset_iff
thf(fact_185_minusset__eq,axiom,
! [A: set_a] :
( ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A )
= ( image_a_a @ ( group_inverse_a @ g @ addition @ zero ) @ ( inf_inf_set_a @ A @ g ) ) ) ).
% minusset_eq
thf(fact_186_sumset__empty_H_I1_J,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ g )
= bot_bot_set_a )
=> ( ( pluenn3038260743871226533mset_a @ g @ addition @ B @ A )
= bot_bot_set_a ) ) ).
% sumset_empty'(1)
thf(fact_187_sumset__empty_H_I2_J,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ g )
= bot_bot_set_a )
=> ( ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B )
= bot_bot_set_a ) ) ).
% sumset_empty'(2)
thf(fact_188_inverse__subgroupI,axiom,
! [H2: set_a] :
( ( group_subgroup_a @ H2 @ g @ addition @ zero )
=> ( group_subgroup_a @ ( image_a_a @ ( group_inverse_a @ g @ addition @ zero ) @ H2 ) @ g @ addition @ zero ) ) ).
% inverse_subgroupI
thf(fact_189_card__sumset__le,axiom,
! [A: set_a,A2: a] :
( ( finite_finite_a @ A )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A ) ) ) ).
% card_sumset_le
thf(fact_190_card__sumset__singleton__eq,axiom,
! [A: set_a,A2: a] :
( ( finite_finite_a @ A )
=> ( ( ( member_a @ A2 @ g )
=> ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) )
= ( finite_card_a @ ( inf_inf_set_a @ A @ g ) ) ) )
& ( ~ ( member_a @ A2 @ g )
=> ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) )
= zero_zero_nat ) ) ) ) ).
% card_sumset_singleton_eq
thf(fact_191_inverse__subgroupD,axiom,
! [H2: set_a] :
( ( group_subgroup_a @ ( image_a_a @ ( group_inverse_a @ g @ addition @ zero ) @ H2 ) @ g @ addition @ zero )
=> ( ( ord_less_eq_set_a @ H2 @ ( group_Units_a @ g @ addition @ zero ) )
=> ( group_subgroup_a @ H2 @ g @ addition @ zero ) ) ) ).
% inverse_subgroupD
thf(fact_192_sumset__empty_I1_J,axiom,
! [A: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A @ bot_bot_set_a )
= bot_bot_set_a ) ).
% sumset_empty(1)
thf(fact_193_sumset__empty_I2_J,axiom,
! [A: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ bot_bot_set_a @ A )
= bot_bot_set_a ) ).
% sumset_empty(2)
thf(fact_194_sumset__is__empty__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B )
= bot_bot_set_a )
= ( ( ( inf_inf_set_a @ A @ g )
= bot_bot_set_a )
| ( ( inf_inf_set_a @ B @ g )
= bot_bot_set_a ) ) ) ).
% sumset_is_empty_iff
thf(fact_195_minusset__is__empty__iff,axiom,
! [A: set_a] :
( ( ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A )
= bot_bot_set_a )
= ( ( inf_inf_set_a @ A @ g )
= bot_bot_set_a ) ) ).
% minusset_is_empty_iff
thf(fact_196_minusset__triv,axiom,
( ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ ( insert_a @ zero @ bot_bot_set_a ) )
= ( insert_a @ zero @ bot_bot_set_a ) ) ).
% minusset_triv
thf(fact_197_group__of__Units,axiom,
group_group_a @ ( group_Units_a @ g @ addition @ zero ) @ addition @ zero ).
% group_of_Units
thf(fact_198_sumset__iterated__0,axiom,
! [A: set_a] :
( ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ A @ zero_zero_nat )
= ( insert_a @ zero @ bot_bot_set_a ) ) ).
% sumset_iterated_0
thf(fact_199_sumset__D_I1_J,axiom,
! [A: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( insert_a @ zero @ bot_bot_set_a ) )
= ( inf_inf_set_a @ A @ g ) ) ).
% sumset_D(1)
thf(fact_200_sumset__D_I2_J,axiom,
! [A: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ ( insert_a @ zero @ bot_bot_set_a ) @ A )
= ( inf_inf_set_a @ A @ g ) ) ).
% sumset_D(2)
thf(fact_201_additive__abelian__group_Osumset__iterated__0,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn1960970773371692859ated_a @ G @ Addition @ Zero @ A @ zero_zero_nat )
= ( insert_a @ Zero @ bot_bot_set_a ) ) ) ).
% additive_abelian_group.sumset_iterated_0
thf(fact_202_additive__abelian__group_Osumset__empty_I1_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ bot_bot_set_a )
= bot_bot_set_a ) ) ).
% additive_abelian_group.sumset_empty(1)
thf(fact_203_additive__abelian__group_Osumset__empty_I2_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ bot_bot_set_a @ A )
= bot_bot_set_a ) ) ).
% additive_abelian_group.sumset_empty(2)
thf(fact_204_additive__abelian__group_Ocard__sumset__le,axiom,
! [G: set_Product_unit,Addition: product_unit > product_unit > product_unit,Zero: product_unit,A: set_Product_unit,A2: product_unit] :
( ( pluenn3635716580025208315t_unit @ G @ Addition @ Zero )
=> ( ( finite4290736615968046902t_unit @ A )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ ( pluenn1407455289632237236t_unit @ G @ Addition @ A @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) ) @ ( finite410649719033368117t_unit @ A ) ) ) ) ).
% additive_abelian_group.card_sumset_le
thf(fact_205_additive__abelian__group_Ocard__sumset__le,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: set_nat,A2: nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( finite_finite_nat @ A )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A ) ) ) ) ).
% additive_abelian_group.card_sumset_le
thf(fact_206_additive__abelian__group_Ocard__sumset__le,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,A2: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_finite_a @ A )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A ) ) ) ) ).
% additive_abelian_group.card_sumset_le
thf(fact_207_additive__abelian__group_Ocard__sumset__singleton__eq,axiom,
! [G: set_Product_unit,Addition: product_unit > product_unit > product_unit,Zero: product_unit,A: set_Product_unit,A2: product_unit] :
( ( pluenn3635716580025208315t_unit @ G @ Addition @ Zero )
=> ( ( finite4290736615968046902t_unit @ A )
=> ( ( ( member_Product_unit @ A2 @ G )
=> ( ( finite410649719033368117t_unit @ ( pluenn1407455289632237236t_unit @ G @ Addition @ A @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) )
= ( finite410649719033368117t_unit @ ( inf_in4660618365625256667t_unit @ A @ G ) ) ) )
& ( ~ ( member_Product_unit @ A2 @ G )
=> ( ( finite410649719033368117t_unit @ ( pluenn1407455289632237236t_unit @ G @ Addition @ A @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) )
= zero_zero_nat ) ) ) ) ) ).
% additive_abelian_group.card_sumset_singleton_eq
thf(fact_208_additive__abelian__group_Ocard__sumset__singleton__eq,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: set_nat,A2: nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( finite_finite_nat @ A )
=> ( ( ( member_nat @ A2 @ G )
=> ( ( finite_card_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
= ( finite_card_nat @ ( inf_inf_set_nat @ A @ G ) ) ) )
& ( ~ ( member_nat @ A2 @ G )
=> ( ( finite_card_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
= zero_zero_nat ) ) ) ) ) ).
% additive_abelian_group.card_sumset_singleton_eq
thf(fact_209_additive__abelian__group_Ocard__sumset__singleton__eq,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,A2: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_finite_a @ A )
=> ( ( ( member_a @ A2 @ G )
=> ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) )
= ( finite_card_a @ ( inf_inf_set_a @ A @ G ) ) ) )
& ( ~ ( member_a @ A2 @ G )
=> ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) )
= zero_zero_nat ) ) ) ) ) ).
% additive_abelian_group.card_sumset_singleton_eq
thf(fact_210_additive__abelian__group_Osumset__empty_H_I2_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ( inf_inf_set_a @ A @ G )
= bot_bot_set_a )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B )
= bot_bot_set_a ) ) ) ).
% additive_abelian_group.sumset_empty'(2)
thf(fact_211_additive__abelian__group_Osumset__empty_H_I1_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ( inf_inf_set_a @ A @ G )
= bot_bot_set_a )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ B @ A )
= bot_bot_set_a ) ) ) ).
% additive_abelian_group.sumset_empty'(1)
thf(fact_212_additive__abelian__group_Osumset__is__empty__iff,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B )
= bot_bot_set_a )
= ( ( ( inf_inf_set_a @ A @ G )
= bot_bot_set_a )
| ( ( inf_inf_set_a @ B @ G )
= bot_bot_set_a ) ) ) ) ).
% additive_abelian_group.sumset_is_empty_iff
thf(fact_213_additive__abelian__group_Ominusset__triv,axiom,
! [G: set_a,Addition: a > a > a,Zero: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ ( insert_a @ Zero @ bot_bot_set_a ) )
= ( insert_a @ Zero @ bot_bot_set_a ) ) ) ).
% additive_abelian_group.minusset_triv
thf(fact_214_additive__abelian__group_Ominusset__is__empty__iff,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A )
= bot_bot_set_a )
= ( ( inf_inf_set_a @ A @ G )
= bot_bot_set_a ) ) ) ).
% additive_abelian_group.minusset_is_empty_iff
thf(fact_215_additive__abelian__group_Ominusset__eq,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( pluenn7323955030898006884et_nat @ G @ Addition @ Zero @ A )
= ( image_nat_nat @ ( group_inverse_nat @ G @ Addition @ Zero ) @ ( inf_inf_set_nat @ A @ G ) ) ) ) ).
% additive_abelian_group.minusset_eq
thf(fact_216_additive__abelian__group_Ominusset__eq,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A )
= ( image_a_a @ ( group_inverse_a @ G @ Addition @ Zero ) @ ( inf_inf_set_a @ A @ G ) ) ) ) ).
% additive_abelian_group.minusset_eq
thf(fact_217_additive__abelian__group_Oinfinite__sumset__iff,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: set_nat,B: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( ~ ( finite_finite_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A @ B ) ) )
= ( ( ~ ( finite_finite_nat @ ( inf_inf_set_nat @ A @ G ) )
& ( ( inf_inf_set_nat @ B @ G )
!= bot_bot_set_nat ) )
| ( ( ( inf_inf_set_nat @ A @ G )
!= bot_bot_set_nat )
& ~ ( finite_finite_nat @ ( inf_inf_set_nat @ B @ G ) ) ) ) ) ) ).
% additive_abelian_group.infinite_sumset_iff
thf(fact_218_additive__abelian__group_Oinfinite__sumset__iff,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ~ ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) ) )
= ( ( ~ ( finite_finite_a @ ( inf_inf_set_a @ A @ G ) )
& ( ( inf_inf_set_a @ B @ G )
!= bot_bot_set_a ) )
| ( ( ( inf_inf_set_a @ A @ G )
!= bot_bot_set_a )
& ~ ( finite_finite_a @ ( inf_inf_set_a @ B @ G ) ) ) ) ) ) ).
% additive_abelian_group.infinite_sumset_iff
thf(fact_219_additive__abelian__group_Oinfinite__sumset__aux,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: set_nat,B: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ~ ( finite_finite_nat @ ( inf_inf_set_nat @ A @ G ) )
=> ( ( ~ ( finite_finite_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A @ B ) ) )
= ( ( inf_inf_set_nat @ B @ G )
!= bot_bot_set_nat ) ) ) ) ).
% additive_abelian_group.infinite_sumset_aux
thf(fact_220_additive__abelian__group_Oinfinite__sumset__aux,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ~ ( finite_finite_a @ ( inf_inf_set_a @ A @ G ) )
=> ( ( ~ ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) ) )
= ( ( inf_inf_set_a @ B @ G )
!= bot_bot_set_a ) ) ) ) ).
% additive_abelian_group.infinite_sumset_aux
thf(fact_221_additive__abelian__group_Osumset__D_I2_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ ( insert_a @ Zero @ bot_bot_set_a ) @ A )
= ( inf_inf_set_a @ A @ G ) ) ) ).
% additive_abelian_group.sumset_D(2)
thf(fact_222_additive__abelian__group_Osumset__D_I1_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ ( insert_a @ Zero @ bot_bot_set_a ) )
= ( inf_inf_set_a @ A @ G ) ) ) ).
% additive_abelian_group.sumset_D(1)
thf(fact_223_additive__abelian__group_Ocard__sumset__0__iff,axiom,
! [G: set_Product_unit,Addition: product_unit > product_unit > product_unit,Zero: product_unit,A: set_Product_unit,B: set_Product_unit] :
( ( pluenn3635716580025208315t_unit @ G @ Addition @ Zero )
=> ( ( ord_le3507040750410214029t_unit @ A @ G )
=> ( ( ord_le3507040750410214029t_unit @ B @ G )
=> ( ( ( finite410649719033368117t_unit @ ( pluenn1407455289632237236t_unit @ G @ Addition @ A @ B ) )
= zero_zero_nat )
= ( ( ( finite410649719033368117t_unit @ A )
= zero_zero_nat )
| ( ( finite410649719033368117t_unit @ B )
= zero_zero_nat ) ) ) ) ) ) ).
% additive_abelian_group.card_sumset_0_iff
thf(fact_224_additive__abelian__group_Ocard__sumset__0__iff,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ord_less_eq_set_a @ A @ G )
=> ( ( ord_less_eq_set_a @ B @ G )
=> ( ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) )
= zero_zero_nat )
= ( ( ( finite_card_a @ A )
= zero_zero_nat )
| ( ( finite_card_a @ B )
= zero_zero_nat ) ) ) ) ) ) ).
% additive_abelian_group.card_sumset_0_iff
thf(fact_225_additive__abelian__group_Ocard__sumset__0__iff_H,axiom,
! [G: set_Product_unit,Addition: product_unit > product_unit > product_unit,Zero: product_unit,A: set_Product_unit,B: set_Product_unit] :
( ( pluenn3635716580025208315t_unit @ G @ Addition @ Zero )
=> ( ( ( finite410649719033368117t_unit @ ( pluenn1407455289632237236t_unit @ G @ Addition @ A @ B ) )
= zero_zero_nat )
= ( ( ( finite410649719033368117t_unit @ ( inf_in4660618365625256667t_unit @ A @ G ) )
= zero_zero_nat )
| ( ( finite410649719033368117t_unit @ ( inf_in4660618365625256667t_unit @ B @ G ) )
= zero_zero_nat ) ) ) ) ).
% additive_abelian_group.card_sumset_0_iff'
thf(fact_226_additive__abelian__group_Ocard__sumset__0__iff_H,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) )
= zero_zero_nat )
= ( ( ( finite_card_a @ ( inf_inf_set_a @ A @ G ) )
= zero_zero_nat )
| ( ( finite_card_a @ ( inf_inf_set_a @ B @ G ) )
= zero_zero_nat ) ) ) ) ).
% additive_abelian_group.card_sumset_0_iff'
thf(fact_227_additive__abelian__group_Ocard__le__sumset,axiom,
! [G: set_Product_unit,Addition: product_unit > product_unit > product_unit,Zero: product_unit,A: set_Product_unit,A2: product_unit,B: set_Product_unit] :
( ( pluenn3635716580025208315t_unit @ G @ Addition @ Zero )
=> ( ( finite4290736615968046902t_unit @ A )
=> ( ( member_Product_unit @ A2 @ A )
=> ( ( member_Product_unit @ A2 @ G )
=> ( ( finite4290736615968046902t_unit @ B )
=> ( ( ord_le3507040750410214029t_unit @ B @ G )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ B ) @ ( finite410649719033368117t_unit @ ( pluenn1407455289632237236t_unit @ G @ Addition @ A @ B ) ) ) ) ) ) ) ) ) ).
% additive_abelian_group.card_le_sumset
thf(fact_228_additive__abelian__group_Ocard__le__sumset,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: set_nat,A2: nat,B: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ( ( member_nat @ A2 @ G )
=> ( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ G )
=> ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite_card_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A @ B ) ) ) ) ) ) ) ) ) ).
% additive_abelian_group.card_le_sumset
thf(fact_229_additive__abelian__group_Ocard__le__sumset,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,A2: a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_finite_a @ A )
=> ( ( member_a @ A2 @ A )
=> ( ( member_a @ A2 @ G )
=> ( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ B @ G )
=> ( ord_less_eq_nat @ ( finite_card_a @ B ) @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) ) ) ) ) ) ) ) ) ).
% additive_abelian_group.card_le_sumset
thf(fact_230_sumset__insert2,axiom,
! [B: set_a,A: set_a,X: a] :
( ( nO_MATCH_set_a_set_a @ bot_bot_set_a @ B )
=> ( ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( insert_a @ X @ B ) )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( insert_a @ X @ bot_bot_set_a ) ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) ) ) ) ).
% sumset_insert2
thf(fact_231_sumset__insert1,axiom,
! [A: set_a,X: a,B: set_a] :
( ( nO_MATCH_set_a_set_a @ bot_bot_set_a @ A )
=> ( ( pluenn3038260743871226533mset_a @ g @ addition @ ( insert_a @ X @ A ) @ B )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ ( insert_a @ X @ bot_bot_set_a ) @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) ) ) ) ).
% sumset_insert1
thf(fact_232_card__0__eq,axiom,
! [A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( ( finite410649719033368117t_unit @ A )
= zero_zero_nat )
= ( A = bot_bo3957492148770167129t_unit ) ) ) ).
% card_0_eq
thf(fact_233_card__0__eq,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( ( finite_card_nat @ A )
= zero_zero_nat )
= ( A = bot_bot_set_nat ) ) ) ).
% card_0_eq
thf(fact_234_card__0__eq,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( ( ( finite_card_a @ A )
= zero_zero_nat )
= ( A = bot_bot_set_a ) ) ) ).
% card_0_eq
thf(fact_235_card_Oinfinite,axiom,
! [A: set_Product_unit] :
( ~ ( finite4290736615968046902t_unit @ A )
=> ( ( finite410649719033368117t_unit @ A )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_236_card_Oinfinite,axiom,
! [A: set_a] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_card_a @ A )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_237_card_Oinfinite,axiom,
! [A: set_nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_card_nat @ A )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_238_card_Oempty,axiom,
( ( finite410649719033368117t_unit @ bot_bo3957492148770167129t_unit )
= zero_zero_nat ) ).
% card.empty
thf(fact_239_card_Oempty,axiom,
( ( finite_card_a @ bot_bot_set_a )
= zero_zero_nat ) ).
% card.empty
thf(fact_240_insert__disjoint_I1_J,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ B )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A2 @ B )
& ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_241_insert__disjoint_I1_J,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ B )
= bot_bot_set_a )
= ( ~ ( member_a @ A2 @ B )
& ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_242_insert__disjoint_I2_J,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ B ) )
= ( ~ ( member_nat @ A2 @ B )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_243_insert__disjoint_I2_J,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ B ) )
= ( ~ ( member_a @ A2 @ B )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_244_disjoint__insert_I1_J,axiom,
! [B: set_nat,A2: nat,A: set_nat] :
( ( ( inf_inf_set_nat @ B @ ( insert_nat @ A2 @ A ) )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A2 @ B )
& ( ( inf_inf_set_nat @ B @ A )
= bot_bot_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_245_disjoint__insert_I1_J,axiom,
! [B: set_a,A2: a,A: set_a] :
( ( ( inf_inf_set_a @ B @ ( insert_a @ A2 @ A ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A2 @ B )
& ( ( inf_inf_set_a @ B @ A )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_246_image__eqI,axiom,
! [B2: a,F: a > a,X: a,A: set_a] :
( ( B2
= ( F @ X ) )
=> ( ( member_a @ X @ A )
=> ( member_a @ B2 @ ( image_a_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_247_image__eqI,axiom,
! [B2: nat,F: a > nat,X: a,A: set_a] :
( ( B2
= ( F @ X ) )
=> ( ( member_a @ X @ A )
=> ( member_nat @ B2 @ ( image_a_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_248_image__eqI,axiom,
! [B2: a,F: nat > a,X: nat,A: set_nat] :
( ( B2
= ( F @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member_a @ B2 @ ( image_nat_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_249_image__eqI,axiom,
! [B2: nat,F: nat > nat,X: nat,A: set_nat] :
( ( B2
= ( F @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member_nat @ B2 @ ( image_nat_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_250_empty__iff,axiom,
! [C2: nat] :
~ ( member_nat @ C2 @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_251_empty__iff,axiom,
! [C2: a] :
~ ( member_a @ C2 @ bot_bot_set_a ) ).
% empty_iff
thf(fact_252_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X2: nat] :
~ ( member_nat @ X2 @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_253_all__not__in__conv,axiom,
! [A: set_a] :
( ( ! [X2: a] :
~ ( member_a @ X2 @ A ) )
= ( A = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_254_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X2: a] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_255_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X2: a] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_256_subsetI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_nat @ X4 @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% subsetI
thf(fact_257_subsetI,axiom,
! [A: set_a,B: set_a] :
( ! [X4: a] :
( ( member_a @ X4 @ A )
=> ( member_a @ X4 @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% subsetI
thf(fact_258_subset__antisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_259_insertCI,axiom,
! [A2: a,B: set_a,B2: a] :
( ( ~ ( member_a @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_a @ A2 @ ( insert_a @ B2 @ B ) ) ) ).
% insertCI
thf(fact_260_insertCI,axiom,
! [A2: nat,B: set_nat,B2: nat] :
( ( ~ ( member_nat @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_nat @ A2 @ ( insert_nat @ B2 @ B ) ) ) ).
% insertCI
thf(fact_261_insert__iff,axiom,
! [A2: a,B2: a,A: set_a] :
( ( member_a @ A2 @ ( insert_a @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_a @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_262_insert__iff,axiom,
! [A2: nat,B2: nat,A: set_nat] :
( ( member_nat @ A2 @ ( insert_nat @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_nat @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_263_insert__absorb2,axiom,
! [X: a,A: set_a] :
( ( insert_a @ X @ ( insert_a @ X @ A ) )
= ( insert_a @ X @ A ) ) ).
% insert_absorb2
thf(fact_264_IntI,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ A )
=> ( ( member_nat @ C2 @ B )
=> ( member_nat @ C2 @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_265_IntI,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ A )
=> ( ( member_a @ C2 @ B )
=> ( member_a @ C2 @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% IntI
thf(fact_266_Int__iff,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A @ B ) )
= ( ( member_nat @ C2 @ A )
& ( member_nat @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_267_Int__iff,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A @ B ) )
= ( ( member_a @ C2 @ A )
& ( member_a @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_268_UnCI,axiom,
! [C2: nat,B: set_nat,A: set_nat] :
( ( ~ ( member_nat @ C2 @ B )
=> ( member_nat @ C2 @ A ) )
=> ( member_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_269_UnCI,axiom,
! [C2: a,B: set_a,A: set_a] :
( ( ~ ( member_a @ C2 @ B )
=> ( member_a @ C2 @ A ) )
=> ( member_a @ C2 @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnCI
thf(fact_270_Un__iff,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) )
= ( ( member_nat @ C2 @ A )
| ( member_nat @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_271_Un__iff,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ ( sup_sup_set_a @ A @ B ) )
= ( ( member_a @ C2 @ A )
| ( member_a @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_272_image__empty,axiom,
! [F: nat > nat] :
( ( image_nat_nat @ F @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_273_image__empty,axiom,
! [F: a > a] :
( ( image_a_a @ F @ bot_bot_set_a )
= bot_bot_set_a ) ).
% image_empty
thf(fact_274_empty__is__image,axiom,
! [F: nat > nat,A: set_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat @ F @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_275_empty__is__image,axiom,
! [F: a > a,A: set_a] :
( ( bot_bot_set_a
= ( image_a_a @ F @ A ) )
= ( A = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_276_image__is__empty,axiom,
! [F: nat > nat,A: set_nat] :
( ( ( image_nat_nat @ F @ A )
= bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_277_image__is__empty,axiom,
! [F: a > a,A: set_a] :
( ( ( image_a_a @ F @ A )
= bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_278_finite__imageI,axiom,
! [F2: set_a,H3: a > a] :
( ( finite_finite_a @ F2 )
=> ( finite_finite_a @ ( image_a_a @ H3 @ F2 ) ) ) ).
% finite_imageI
thf(fact_279_finite__imageI,axiom,
! [F2: set_a,H3: a > nat] :
( ( finite_finite_a @ F2 )
=> ( finite_finite_nat @ ( image_a_nat @ H3 @ F2 ) ) ) ).
% finite_imageI
thf(fact_280_finite__imageI,axiom,
! [F2: set_nat,H3: nat > a] :
( ( finite_finite_nat @ F2 )
=> ( finite_finite_a @ ( image_nat_a @ H3 @ F2 ) ) ) ).
% finite_imageI
thf(fact_281_finite__imageI,axiom,
! [F2: set_nat,H3: nat > nat] :
( ( finite_finite_nat @ F2 )
=> ( finite_finite_nat @ ( image_nat_nat @ H3 @ F2 ) ) ) ).
% finite_imageI
thf(fact_282_subset__empty,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_283_empty__subsetI,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% empty_subsetI
thf(fact_284_image__insert,axiom,
! [F: nat > nat,A2: nat,B: set_nat] :
( ( image_nat_nat @ F @ ( insert_nat @ A2 @ B ) )
= ( insert_nat @ ( F @ A2 ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_insert
thf(fact_285_image__insert,axiom,
! [F: a > a,A2: a,B: set_a] :
( ( image_a_a @ F @ ( insert_a @ A2 @ B ) )
= ( insert_a @ ( F @ A2 ) @ ( image_a_a @ F @ B ) ) ) ).
% image_insert
thf(fact_286_insert__image,axiom,
! [X: a,A: set_a,F: a > a] :
( ( member_a @ X @ A )
=> ( ( insert_a @ ( F @ X ) @ ( image_a_a @ F @ A ) )
= ( image_a_a @ F @ A ) ) ) ).
% insert_image
thf(fact_287_insert__image,axiom,
! [X: nat,A: set_nat,F: nat > nat] :
( ( member_nat @ X @ A )
=> ( ( insert_nat @ ( F @ X ) @ ( image_nat_nat @ F @ A ) )
= ( image_nat_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_288_insert__image,axiom,
! [X: nat,A: set_nat,F: nat > a] :
( ( member_nat @ X @ A )
=> ( ( insert_a @ ( F @ X ) @ ( image_nat_a @ F @ A ) )
= ( image_nat_a @ F @ A ) ) ) ).
% insert_image
thf(fact_289_singletonI,axiom,
! [A2: nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_290_singletonI,axiom,
! [A2: a] : ( member_a @ A2 @ ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_291_finite__insert,axiom,
! [A2: a,A: set_a] :
( ( finite_finite_a @ ( insert_a @ A2 @ A ) )
= ( finite_finite_a @ A ) ) ).
% finite_insert
thf(fact_292_finite__insert,axiom,
! [A2: nat,A: set_nat] :
( ( finite_finite_nat @ ( insert_nat @ A2 @ A ) )
= ( finite_finite_nat @ A ) ) ).
% finite_insert
thf(fact_293_insert__subset,axiom,
! [X: nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X @ A ) @ B )
= ( ( member_nat @ X @ B )
& ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% insert_subset
thf(fact_294_insert__subset,axiom,
! [X: a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A ) @ B )
= ( ( member_a @ X @ B )
& ( ord_less_eq_set_a @ A @ B ) ) ) ).
% insert_subset
thf(fact_295_finite__Int,axiom,
! [F2: set_nat,G: set_nat] :
( ( ( finite_finite_nat @ F2 )
| ( finite_finite_nat @ G ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_296_finite__Int,axiom,
! [F2: set_a,G: set_a] :
( ( ( finite_finite_a @ F2 )
| ( finite_finite_a @ G ) )
=> ( finite_finite_a @ ( inf_inf_set_a @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_297_Int__subset__iff,axiom,
! [C: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ ( inf_inf_set_a @ A @ B ) )
= ( ( ord_less_eq_set_a @ C @ A )
& ( ord_less_eq_set_a @ C @ B ) ) ) ).
% Int_subset_iff
thf(fact_298_Un__empty,axiom,
! [A: set_a,B: set_a] :
( ( ( sup_sup_set_a @ A @ B )
= bot_bot_set_a )
= ( ( A = bot_bot_set_a )
& ( B = bot_bot_set_a ) ) ) ).
% Un_empty
thf(fact_299_Int__insert__left__if0,axiom,
! [A2: nat,C: set_nat,B: set_nat] :
( ~ ( member_nat @ A2 @ C )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B ) @ C )
= ( inf_inf_set_nat @ B @ C ) ) ) ).
% Int_insert_left_if0
thf(fact_300_Int__insert__left__if0,axiom,
! [A2: a,C: set_a,B: set_a] :
( ~ ( member_a @ A2 @ C )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C )
= ( inf_inf_set_a @ B @ C ) ) ) ).
% Int_insert_left_if0
thf(fact_301_Int__insert__left__if1,axiom,
! [A2: nat,C: set_nat,B: set_nat] :
( ( member_nat @ A2 @ C )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B ) @ C )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ B @ C ) ) ) ) ).
% Int_insert_left_if1
thf(fact_302_Int__insert__left__if1,axiom,
! [A2: a,C: set_a,B: set_a] :
( ( member_a @ A2 @ C )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C )
= ( insert_a @ A2 @ ( inf_inf_set_a @ B @ C ) ) ) ) ).
% Int_insert_left_if1
thf(fact_303_insert__inter__insert,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ ( insert_a @ A2 @ B ) )
= ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B ) ) ) ).
% insert_inter_insert
thf(fact_304_Int__insert__right__if0,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( inf_inf_set_nat @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_305_Int__insert__right__if0,axiom,
! [A2: a,A: set_a,B: set_a] :
( ~ ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_306_Int__insert__right__if1,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_307_Int__insert__right__if1,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_308_finite__Un,axiom,
! [F2: set_nat,G: set_nat] :
( ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G ) )
= ( ( finite_finite_nat @ F2 )
& ( finite_finite_nat @ G ) ) ) ).
% finite_Un
thf(fact_309_finite__Un,axiom,
! [F2: set_a,G: set_a] :
( ( finite_finite_a @ ( sup_sup_set_a @ F2 @ G ) )
= ( ( finite_finite_a @ F2 )
& ( finite_finite_a @ G ) ) ) ).
% finite_Un
thf(fact_310_Un__subset__iff,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ C )
= ( ( ord_less_eq_set_a @ A @ C )
& ( ord_less_eq_set_a @ B @ C ) ) ) ).
% Un_subset_iff
thf(fact_311_Un__insert__left,axiom,
! [A2: a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ ( insert_a @ A2 @ B ) @ C )
= ( insert_a @ A2 @ ( sup_sup_set_a @ B @ C ) ) ) ).
% Un_insert_left
thf(fact_312_Un__insert__right,axiom,
! [A: set_a,A2: a,B: set_a] :
( ( sup_sup_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( insert_a @ A2 @ ( sup_sup_set_a @ A @ B ) ) ) ).
% Un_insert_right
thf(fact_313_Int__Un__eq_I4_J,axiom,
! [T: set_a,S: set_a] :
( ( sup_sup_set_a @ T @ ( inf_inf_set_a @ S @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_314_Int__Un__eq_I3_J,axiom,
! [S: set_a,T: set_a] :
( ( sup_sup_set_a @ S @ ( inf_inf_set_a @ S @ T ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_315_Int__Un__eq_I2_J,axiom,
! [S: set_a,T: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_316_Int__Un__eq_I1_J,axiom,
! [S: set_a,T: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_317_Un__Int__eq_I4_J,axiom,
! [T: set_a,S: set_a] :
( ( inf_inf_set_a @ T @ ( sup_sup_set_a @ S @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_318_Un__Int__eq_I3_J,axiom,
! [S: set_a,T: set_a] :
( ( inf_inf_set_a @ S @ ( sup_sup_set_a @ S @ T ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_319_Un__Int__eq_I2_J,axiom,
! [S: set_a,T: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_320_Un__Int__eq_I1_J,axiom,
! [S: set_a,T: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_321_singleton__insert__inj__eq_H,axiom,
! [A2: a,A: set_a,B2: a] :
( ( ( insert_a @ A2 @ A )
= ( insert_a @ B2 @ bot_bot_set_a ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_a @ A @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_322_singleton__insert__inj__eq,axiom,
! [B2: a,A2: a,A: set_a] :
( ( ( insert_a @ B2 @ bot_bot_set_a )
= ( insert_a @ A2 @ A ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_a @ A @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_323_disjoint__insert_I2_J,axiom,
! [A: set_nat,B2: nat,B: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ ( insert_nat @ B2 @ B ) ) )
= ( ~ ( member_nat @ B2 @ A )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_324_disjoint__insert_I2_J,axiom,
! [A: set_a,B2: a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A @ ( insert_a @ B2 @ B ) ) )
= ( ~ ( member_a @ B2 @ A )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_325_imageI,axiom,
! [X: a,A: set_a,F: a > a] :
( ( member_a @ X @ A )
=> ( member_a @ ( F @ X ) @ ( image_a_a @ F @ A ) ) ) ).
% imageI
thf(fact_326_imageI,axiom,
! [X: a,A: set_a,F: a > nat] :
( ( member_a @ X @ A )
=> ( member_nat @ ( F @ X ) @ ( image_a_nat @ F @ A ) ) ) ).
% imageI
thf(fact_327_imageI,axiom,
! [X: nat,A: set_nat,F: nat > a] :
( ( member_nat @ X @ A )
=> ( member_a @ ( F @ X ) @ ( image_nat_a @ F @ A ) ) ) ).
% imageI
thf(fact_328_imageI,axiom,
! [X: nat,A: set_nat,F: nat > nat] :
( ( member_nat @ X @ A )
=> ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ A ) ) ) ).
% imageI
thf(fact_329_image__iff,axiom,
! [Z: a,F: a > a,A: set_a] :
( ( member_a @ Z @ ( image_a_a @ F @ A ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_330_image__iff,axiom,
! [Z: nat,F: nat > nat,A: set_nat] :
( ( member_nat @ Z @ ( image_nat_nat @ F @ A ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_331_bex__imageD,axiom,
! [F: a > a,A: set_a,P: a > $o] :
( ? [X3: a] :
( ( member_a @ X3 @ ( image_a_a @ F @ A ) )
& ( P @ X3 ) )
=> ? [X4: a] :
( ( member_a @ X4 @ A )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_332_bex__imageD,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ? [X3: nat] :
( ( member_nat @ X3 @ ( image_nat_nat @ F @ A ) )
& ( P @ X3 ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_333_image__cong,axiom,
! [M: set_a,N: set_a,F: a > a,G3: a > a] :
( ( M = N )
=> ( ! [X4: a] :
( ( member_a @ X4 @ N )
=> ( ( F @ X4 )
= ( G3 @ X4 ) ) )
=> ( ( image_a_a @ F @ M )
= ( image_a_a @ G3 @ N ) ) ) ) ).
% image_cong
thf(fact_334_image__cong,axiom,
! [M: set_nat,N: set_nat,F: nat > nat,G3: nat > nat] :
( ( M = N )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ N )
=> ( ( F @ X4 )
= ( G3 @ X4 ) ) )
=> ( ( image_nat_nat @ F @ M )
= ( image_nat_nat @ G3 @ N ) ) ) ) ).
% image_cong
thf(fact_335_ball__imageD,axiom,
! [F: a > a,A: set_a,P: a > $o] :
( ! [X4: a] :
( ( member_a @ X4 @ ( image_a_a @ F @ A ) )
=> ( P @ X4 ) )
=> ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( P @ ( F @ X3 ) ) ) ) ).
% ball_imageD
thf(fact_336_ball__imageD,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ! [X4: nat] :
( ( member_nat @ X4 @ ( image_nat_nat @ F @ A ) )
=> ( P @ X4 ) )
=> ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( P @ ( F @ X3 ) ) ) ) ).
% ball_imageD
thf(fact_337_rev__image__eqI,axiom,
! [X: a,A: set_a,B2: a,F: a > a] :
( ( member_a @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member_a @ B2 @ ( image_a_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_338_rev__image__eqI,axiom,
! [X: a,A: set_a,B2: nat,F: a > nat] :
( ( member_a @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member_nat @ B2 @ ( image_a_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_339_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B2: a,F: nat > a] :
( ( member_nat @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member_a @ B2 @ ( image_nat_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_340_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B2: nat,F: nat > nat] :
( ( member_nat @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member_nat @ B2 @ ( image_nat_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_341_emptyE,axiom,
! [A2: nat] :
~ ( member_nat @ A2 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_342_emptyE,axiom,
! [A2: a] :
~ ( member_a @ A2 @ bot_bot_set_a ) ).
% emptyE
thf(fact_343_equals0D,axiom,
! [A: set_nat,A2: nat] :
( ( A = bot_bot_set_nat )
=> ~ ( member_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_344_equals0D,axiom,
! [A: set_a,A2: a] :
( ( A = bot_bot_set_a )
=> ~ ( member_a @ A2 @ A ) ) ).
% equals0D
thf(fact_345_equals0I,axiom,
! [A: set_nat] :
( ! [Y2: nat] :
~ ( member_nat @ Y2 @ A )
=> ( A = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_346_equals0I,axiom,
! [A: set_a] :
( ! [Y2: a] :
~ ( member_a @ Y2 @ A )
=> ( A = bot_bot_set_a ) ) ).
% equals0I
thf(fact_347_ex__in__conv,axiom,
! [A: set_nat] :
( ( ? [X2: nat] : ( member_nat @ X2 @ A ) )
= ( A != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_348_ex__in__conv,axiom,
! [A: set_a] :
( ( ? [X2: a] : ( member_a @ X2 @ A ) )
= ( A != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_349_in__mono,axiom,
! [A: set_nat,B: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ X @ A )
=> ( member_nat @ X @ B ) ) ) ).
% in_mono
thf(fact_350_in__mono,axiom,
! [A: set_a,B: set_a,X: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ X @ A )
=> ( member_a @ X @ B ) ) ) ).
% in_mono
thf(fact_351_subsetD,axiom,
! [A: set_nat,B: set_nat,C2: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ C2 @ A )
=> ( member_nat @ C2 @ B ) ) ) ).
% subsetD
thf(fact_352_subsetD,axiom,
! [A: set_a,B: set_a,C2: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ C2 @ A )
=> ( member_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_353_equalityE,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_a @ A @ B )
=> ~ ( ord_less_eq_set_a @ B @ A ) ) ) ).
% equalityE
thf(fact_354_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
! [X2: nat] :
( ( member_nat @ X2 @ A6 )
=> ( member_nat @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_355_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
! [X2: a] :
( ( member_a @ X2 @ A6 )
=> ( member_a @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_356_equalityD1,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% equalityD1
thf(fact_357_equalityD2,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% equalityD2
thf(fact_358_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A6 )
=> ( member_nat @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_359_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
! [T2: a] :
( ( member_a @ T2 @ A6 )
=> ( member_a @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_360_subset__refl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% subset_refl
thf(fact_361_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X4: a] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_362_subset__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% subset_trans
thf(fact_363_set__eq__subset,axiom,
( ( ^ [Y3: set_a,Z2: set_a] : ( Y3 = Z2 ) )
= ( ^ [A6: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A6 @ B6 )
& ( ord_less_eq_set_a @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_364_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X2: a] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_365_insertE,axiom,
! [A2: a,B2: a,A: set_a] :
( ( member_a @ A2 @ ( insert_a @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_a @ A2 @ A ) ) ) ).
% insertE
thf(fact_366_insertE,axiom,
! [A2: nat,B2: nat,A: set_nat] :
( ( member_nat @ A2 @ ( insert_nat @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_nat @ A2 @ A ) ) ) ).
% insertE
thf(fact_367_insertI1,axiom,
! [A2: a,B: set_a] : ( member_a @ A2 @ ( insert_a @ A2 @ B ) ) ).
% insertI1
thf(fact_368_insertI1,axiom,
! [A2: nat,B: set_nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ B ) ) ).
% insertI1
thf(fact_369_insertI2,axiom,
! [A2: a,B: set_a,B2: a] :
( ( member_a @ A2 @ B )
=> ( member_a @ A2 @ ( insert_a @ B2 @ B ) ) ) ).
% insertI2
thf(fact_370_insertI2,axiom,
! [A2: nat,B: set_nat,B2: nat] :
( ( member_nat @ A2 @ B )
=> ( member_nat @ A2 @ ( insert_nat @ B2 @ B ) ) ) ).
% insertI2
thf(fact_371_Set_Oset__insert,axiom,
! [X: a,A: set_a] :
( ( member_a @ X @ A )
=> ~ ! [B7: set_a] :
( ( A
= ( insert_a @ X @ B7 ) )
=> ( member_a @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_372_Set_Oset__insert,axiom,
! [X: nat,A: set_nat] :
( ( member_nat @ X @ A )
=> ~ ! [B7: set_nat] :
( ( A
= ( insert_nat @ X @ B7 ) )
=> ( member_nat @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_373_insert__ident,axiom,
! [X: a,A: set_a,B: set_a] :
( ~ ( member_a @ X @ A )
=> ( ~ ( member_a @ X @ B )
=> ( ( ( insert_a @ X @ A )
= ( insert_a @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_374_insert__ident,axiom,
! [X: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ X @ A )
=> ( ~ ( member_nat @ X @ B )
=> ( ( ( insert_nat @ X @ A )
= ( insert_nat @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_375_insert__absorb,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ( ( insert_a @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_376_insert__absorb,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( insert_nat @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_377_insert__eq__iff,axiom,
! [A2: a,A: set_a,B2: a,B: set_a] :
( ~ ( member_a @ A2 @ A )
=> ( ~ ( member_a @ B2 @ B )
=> ( ( ( insert_a @ A2 @ A )
= ( insert_a @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C3: set_a] :
( ( A
= ( insert_a @ B2 @ C3 ) )
& ~ ( member_a @ B2 @ C3 )
& ( B
= ( insert_a @ A2 @ C3 ) )
& ~ ( member_a @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_378_insert__eq__iff,axiom,
! [A2: nat,A: set_nat,B2: nat,B: set_nat] :
( ~ ( member_nat @ A2 @ A )
=> ( ~ ( member_nat @ B2 @ B )
=> ( ( ( insert_nat @ A2 @ A )
= ( insert_nat @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C3: set_nat] :
( ( A
= ( insert_nat @ B2 @ C3 ) )
& ~ ( member_nat @ B2 @ C3 )
& ( B
= ( insert_nat @ A2 @ C3 ) )
& ~ ( member_nat @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_379_insert__commute,axiom,
! [X: a,Y: a,A: set_a] :
( ( insert_a @ X @ ( insert_a @ Y @ A ) )
= ( insert_a @ Y @ ( insert_a @ X @ A ) ) ) ).
% insert_commute
thf(fact_380_mk__disjoint__insert,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ? [B7: set_a] :
( ( A
= ( insert_a @ A2 @ B7 ) )
& ~ ( member_a @ A2 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_381_mk__disjoint__insert,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ? [B7: set_nat] :
( ( A
= ( insert_nat @ A2 @ B7 ) )
& ~ ( member_nat @ A2 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_382_IntE,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A @ B ) )
=> ~ ( ( member_nat @ C2 @ A )
=> ~ ( member_nat @ C2 @ B ) ) ) ).
% IntE
thf(fact_383_IntE,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A @ B ) )
=> ~ ( ( member_a @ C2 @ A )
=> ~ ( member_a @ C2 @ B ) ) ) ).
% IntE
thf(fact_384_IntD1,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A @ B ) )
=> ( member_nat @ C2 @ A ) ) ).
% IntD1
thf(fact_385_IntD1,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A @ B ) )
=> ( member_a @ C2 @ A ) ) ).
% IntD1
thf(fact_386_IntD2,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A @ B ) )
=> ( member_nat @ C2 @ B ) ) ).
% IntD2
thf(fact_387_IntD2,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A @ B ) )
=> ( member_a @ C2 @ B ) ) ).
% IntD2
thf(fact_388_Int__assoc,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B ) @ C )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B @ C ) ) ) ).
% Int_assoc
thf(fact_389_Int__absorb,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ A )
= A ) ).
% Int_absorb
thf(fact_390_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A6: set_a,B6: set_a] : ( inf_inf_set_a @ B6 @ A6 ) ) ) ).
% Int_commute
thf(fact_391_Int__left__absorb,axiom,
! [A: set_a,B: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ A @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ).
% Int_left_absorb
thf(fact_392_Int__left__commute,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B @ C ) )
= ( inf_inf_set_a @ B @ ( inf_inf_set_a @ A @ C ) ) ) ).
% Int_left_commute
thf(fact_393_UnE,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) )
=> ( ~ ( member_nat @ C2 @ A )
=> ( member_nat @ C2 @ B ) ) ) ).
% UnE
thf(fact_394_UnE,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ ( sup_sup_set_a @ A @ B ) )
=> ( ~ ( member_a @ C2 @ A )
=> ( member_a @ C2 @ B ) ) ) ).
% UnE
thf(fact_395_UnI1,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ A )
=> ( member_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_396_UnI1,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ A )
=> ( member_a @ C2 @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnI1
thf(fact_397_UnI2,axiom,
! [C2: nat,B: set_nat,A: set_nat] :
( ( member_nat @ C2 @ B )
=> ( member_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_398_UnI2,axiom,
! [C2: a,B: set_a,A: set_a] :
( ( member_a @ C2 @ B )
=> ( member_a @ C2 @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnI2
thf(fact_399_bex__Un,axiom,
! [A: set_a,B: set_a,P: a > $o] :
( ( ? [X2: a] :
( ( member_a @ X2 @ ( sup_sup_set_a @ A @ B ) )
& ( P @ X2 ) ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A )
& ( P @ X2 ) )
| ? [X2: a] :
( ( member_a @ X2 @ B )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_400_ball__Un,axiom,
! [A: set_a,B: set_a,P: a > $o] :
( ( ! [X2: a] :
( ( member_a @ X2 @ ( sup_sup_set_a @ A @ B ) )
=> ( P @ X2 ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( P @ X2 ) )
& ! [X2: a] :
( ( member_a @ X2 @ B )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_401_Un__assoc,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A @ B ) @ C )
= ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B @ C ) ) ) ).
% Un_assoc
thf(fact_402_Un__absorb,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ A )
= A ) ).
% Un_absorb
thf(fact_403_Un__commute,axiom,
( sup_sup_set_a
= ( ^ [A6: set_a,B6: set_a] : ( sup_sup_set_a @ B6 @ A6 ) ) ) ).
% Un_commute
thf(fact_404_Un__left__absorb,axiom,
! [A: set_a,B: set_a] :
( ( sup_sup_set_a @ A @ ( sup_sup_set_a @ A @ B ) )
= ( sup_sup_set_a @ A @ B ) ) ).
% Un_left_absorb
thf(fact_405_Un__left__commute,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B @ C ) )
= ( sup_sup_set_a @ B @ ( sup_sup_set_a @ A @ C ) ) ) ).
% Un_left_commute
thf(fact_406_finite__has__maximal2,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ A )
& ( ord_less_eq_set_a @ A2 @ X4 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_407_finite__has__maximal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ord_less_eq_nat @ A2 @ X4 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_408_finite__has__minimal2,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ A )
& ( ord_less_eq_set_a @ X4 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_409_finite__has__minimal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ord_less_eq_nat @ X4 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_410_all__subset__image,axiom,
! [F: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ! [B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat @ F @ A ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ A )
=> ( P @ ( image_nat_nat @ F @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_411_all__subset__image,axiom,
! [F: a > a,A: set_a,P: set_a > $o] :
( ( ! [B6: set_a] :
( ( ord_less_eq_set_a @ B6 @ ( image_a_a @ F @ A ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_a] :
( ( ord_less_eq_set_a @ B6 @ A )
=> ( P @ ( image_a_a @ F @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_412_subset__image__iff,axiom,
! [B: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_413_subset__image__iff,axiom,
! [B: set_a,F: a > a,A: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A )
& ( B
= ( image_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_414_image__subset__iff,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_415_image__subset__iff,axiom,
! [F: a > a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ B )
= ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_416_subset__imageE,axiom,
! [B: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
=> ~ ! [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A )
=> ( B
!= ( image_nat_nat @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_417_subset__imageE,axiom,
! [B: set_a,F: a > a,A: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A ) )
=> ~ ! [C4: set_a] :
( ( ord_less_eq_set_a @ C4 @ A )
=> ( B
!= ( image_a_a @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_418_image__subsetI,axiom,
! [A: set_a,F: a > nat,B: set_nat] :
( ! [X4: a] :
( ( member_a @ X4 @ A )
=> ( member_nat @ ( F @ X4 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_a_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_419_image__subsetI,axiom,
! [A: set_nat,F: nat > nat,B: set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_nat @ ( F @ X4 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_420_image__subsetI,axiom,
! [A: set_a,F: a > a,B: set_a] :
( ! [X4: a] :
( ( member_a @ X4 @ A )
=> ( member_a @ ( F @ X4 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_421_image__subsetI,axiom,
! [A: set_nat,F: nat > a,B: set_a] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_a @ ( F @ X4 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_nat_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_422_image__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_423_image__mono,axiom,
! [A: set_a,B: set_a,F: a > a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ ( image_a_a @ F @ B ) ) ) ).
% image_mono
thf(fact_424_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_425_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_426_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_427_infinite__imp__nonempty,axiom,
! [S: set_a] :
( ~ ( finite_finite_a @ S )
=> ( S != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_428_finite__subset,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_subset
thf(fact_429_finite__subset,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ A ) ) ) ).
% finite_subset
thf(fact_430_infinite__super,axiom,
! [S: set_nat,T: set_nat] :
( ( ord_less_eq_set_nat @ S @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T ) ) ) ).
% infinite_super
thf(fact_431_infinite__super,axiom,
! [S: set_a,T: set_a] :
( ( ord_less_eq_set_a @ S @ T )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ T ) ) ) ).
% infinite_super
thf(fact_432_rev__finite__subset,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_433_rev__finite__subset,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( finite_finite_a @ A ) ) ) ).
% rev_finite_subset
thf(fact_434_singletonD,axiom,
! [B2: nat,A2: nat] :
( ( member_nat @ B2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_435_singletonD,axiom,
! [B2: a,A2: a] :
( ( member_a @ B2 @ ( insert_a @ A2 @ bot_bot_set_a ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_436_singleton__iff,axiom,
! [B2: nat,A2: nat] :
( ( member_nat @ B2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_437_singleton__iff,axiom,
! [B2: a,A2: a] :
( ( member_a @ B2 @ ( insert_a @ A2 @ bot_bot_set_a ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_438_doubleton__eq__iff,axiom,
! [A2: a,B2: a,C2: a,D: a] :
( ( ( insert_a @ A2 @ ( insert_a @ B2 @ bot_bot_set_a ) )
= ( insert_a @ C2 @ ( insert_a @ D @ bot_bot_set_a ) ) )
= ( ( ( A2 = C2 )
& ( B2 = D ) )
| ( ( A2 = D )
& ( B2 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_439_insert__not__empty,axiom,
! [A2: a,A: set_a] :
( ( insert_a @ A2 @ A )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_440_singleton__inject,axiom,
! [A2: a,B2: a] :
( ( ( insert_a @ A2 @ bot_bot_set_a )
= ( insert_a @ B2 @ bot_bot_set_a ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_441_finite_OinsertI,axiom,
! [A: set_a,A2: a] :
( ( finite_finite_a @ A )
=> ( finite_finite_a @ ( insert_a @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_442_finite_OinsertI,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( finite_finite_nat @ ( insert_nat @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_443_insert__mono,axiom,
! [C: set_a,D2: set_a,A2: a] :
( ( ord_less_eq_set_a @ C @ D2 )
=> ( ord_less_eq_set_a @ ( insert_a @ A2 @ C ) @ ( insert_a @ A2 @ D2 ) ) ) ).
% insert_mono
thf(fact_444_subset__insert,axiom,
! [X: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ X @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X @ B ) )
= ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% subset_insert
thf(fact_445_subset__insert,axiom,
! [X: a,A: set_a,B: set_a] :
( ~ ( member_a @ X @ A )
=> ( ( ord_less_eq_set_a @ A @ ( insert_a @ X @ B ) )
= ( ord_less_eq_set_a @ A @ B ) ) ) ).
% subset_insert
thf(fact_446_subset__insertI,axiom,
! [B: set_a,A2: a] : ( ord_less_eq_set_a @ B @ ( insert_a @ A2 @ B ) ) ).
% subset_insertI
thf(fact_447_subset__insertI2,axiom,
! [A: set_a,B: set_a,B2: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ A @ ( insert_a @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_448_Int__emptyI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ~ ( member_nat @ X4 @ B ) )
=> ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_449_Int__emptyI,axiom,
! [A: set_a,B: set_a] :
( ! [X4: a] :
( ( member_a @ X4 @ A )
=> ~ ( member_a @ X4 @ B ) )
=> ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_450_disjoint__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ~ ( member_nat @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_451_disjoint__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
= ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ~ ( member_a @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_452_Int__empty__left,axiom,
! [B: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ B )
= bot_bot_set_a ) ).
% Int_empty_left
thf(fact_453_Int__empty__right,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ bot_bot_set_a )
= bot_bot_set_a ) ).
% Int_empty_right
thf(fact_454_disjoint__iff__not__equal,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
= ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ! [Y4: a] :
( ( member_a @ Y4 @ B )
=> ( X2 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_455_Int__mono,axiom,
! [A: set_a,C: set_a,B: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ B @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_inf_set_a @ C @ D2 ) ) ) ) ).
% Int_mono
thf(fact_456_Int__lower1,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_457_Int__lower2,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_458_Int__absorb1,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( inf_inf_set_a @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_459_Int__absorb2,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( inf_inf_set_a @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_460_Int__greatest,axiom,
! [C: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ A )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_461_Int__Collect__mono,axiom,
! [A: set_nat,B: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_462_Int__Collect__mono,axiom,
! [A: set_a,B: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ! [X4: a] :
( ( member_a @ X4 @ A )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_463_image__Un,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( image_nat_nat @ F @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_sup_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_Un
thf(fact_464_image__Un,axiom,
! [F: a > a,A: set_a,B: set_a] :
( ( image_a_a @ F @ ( sup_sup_set_a @ A @ B ) )
= ( sup_sup_set_a @ ( image_a_a @ F @ A ) @ ( image_a_a @ F @ B ) ) ) ).
% image_Un
thf(fact_465_Un__empty__left,axiom,
! [B: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ B )
= B ) ).
% Un_empty_left
thf(fact_466_Un__empty__right,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ bot_bot_set_a )
= A ) ).
% Un_empty_right
thf(fact_467_Int__insert__left,axiom,
! [A2: nat,C: set_nat,B: set_nat] :
( ( ( member_nat @ A2 @ C )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B ) @ C )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ B @ C ) ) ) )
& ( ~ ( member_nat @ A2 @ C )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B ) @ C )
= ( inf_inf_set_nat @ B @ C ) ) ) ) ).
% Int_insert_left
thf(fact_468_Int__insert__left,axiom,
! [A2: a,C: set_a,B: set_a] :
( ( ( member_a @ A2 @ C )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C )
= ( insert_a @ A2 @ ( inf_inf_set_a @ B @ C ) ) ) )
& ( ~ ( member_a @ A2 @ C )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C )
= ( inf_inf_set_a @ B @ C ) ) ) ) ).
% Int_insert_left
thf(fact_469_Int__insert__right,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B ) ) ) )
& ( ~ ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_470_Int__insert__right,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B ) ) ) )
& ( ~ ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_471_finite__UnI,axiom,
! [F2: set_nat,G: set_nat] :
( ( finite_finite_nat @ F2 )
=> ( ( finite_finite_nat @ G )
=> ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G ) ) ) ) ).
% finite_UnI
thf(fact_472_finite__UnI,axiom,
! [F2: set_a,G: set_a] :
( ( finite_finite_a @ F2 )
=> ( ( finite_finite_a @ G )
=> ( finite_finite_a @ ( sup_sup_set_a @ F2 @ G ) ) ) ) ).
% finite_UnI
thf(fact_473_Un__infinite,axiom,
! [S: set_nat,T: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T ) ) ) ).
% Un_infinite
thf(fact_474_Un__infinite,axiom,
! [S: set_a,T: set_a] :
( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( sup_sup_set_a @ S @ T ) ) ) ).
% Un_infinite
thf(fact_475_infinite__Un,axiom,
! [S: set_nat,T: set_nat] :
( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T ) ) )
= ( ~ ( finite_finite_nat @ S )
| ~ ( finite_finite_nat @ T ) ) ) ).
% infinite_Un
thf(fact_476_infinite__Un,axiom,
! [S: set_a,T: set_a] :
( ( ~ ( finite_finite_a @ ( sup_sup_set_a @ S @ T ) ) )
= ( ~ ( finite_finite_a @ S )
| ~ ( finite_finite_a @ T ) ) ) ).
% infinite_Un
thf(fact_477_Un__mono,axiom,
! [A: set_a,C: set_a,B: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ B @ D2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ ( sup_sup_set_a @ C @ D2 ) ) ) ) ).
% Un_mono
thf(fact_478_Un__least,axiom,
! [A: set_a,C: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ C ) ) ) ).
% Un_least
thf(fact_479_Un__upper1,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ A @ B ) ) ).
% Un_upper1
thf(fact_480_Un__upper2,axiom,
! [B: set_a,A: set_a] : ( ord_less_eq_set_a @ B @ ( sup_sup_set_a @ A @ B ) ) ).
% Un_upper2
thf(fact_481_Un__absorb1,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( sup_sup_set_a @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_482_Un__absorb2,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( sup_sup_set_a @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_483_subset__UnE,axiom,
! [C: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A @ B ) )
=> ~ ! [A7: set_a] :
( ( ord_less_eq_set_a @ A7 @ A )
=> ! [B8: set_a] :
( ( ord_less_eq_set_a @ B8 @ B )
=> ( C
!= ( sup_sup_set_a @ A7 @ B8 ) ) ) ) ) ).
% subset_UnE
thf(fact_484_subset__Un__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
( ( sup_sup_set_a @ A6 @ B6 )
= B6 ) ) ) ).
% subset_Un_eq
thf(fact_485_additive__abelian__group_Osumset__insert1,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,X: a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( nO_MATCH_set_a_set_a @ bot_bot_set_a @ A )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ ( insert_a @ X @ A ) @ B )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ ( insert_a @ X @ bot_bot_set_a ) @ B ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) ) ) ) ) ).
% additive_abelian_group.sumset_insert1
thf(fact_486_additive__abelian__group_Osumset__insert2,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,B: set_a,A: set_a,X: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( nO_MATCH_set_a_set_a @ bot_bot_set_a @ B )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ ( insert_a @ X @ B ) )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ ( insert_a @ X @ bot_bot_set_a ) ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) ) ) ) ) ).
% additive_abelian_group.sumset_insert2
thf(fact_487_Un__Int__crazy,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_inf_set_a @ B @ C ) ) @ ( inf_inf_set_a @ C @ A ) )
= ( inf_inf_set_a @ ( inf_inf_set_a @ ( sup_sup_set_a @ A @ B ) @ ( sup_sup_set_a @ B @ C ) ) @ ( sup_sup_set_a @ C @ A ) ) ) ).
% Un_Int_crazy
thf(fact_488_Int__Un__distrib,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( inf_inf_set_a @ A @ ( sup_sup_set_a @ B @ C ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_inf_set_a @ A @ C ) ) ) ).
% Int_Un_distrib
thf(fact_489_Un__Int__distrib,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ A @ ( inf_inf_set_a @ B @ C ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ A @ B ) @ ( sup_sup_set_a @ A @ C ) ) ) ).
% Un_Int_distrib
thf(fact_490_Int__Un__distrib2,axiom,
! [B: set_a,C: set_a,A: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ B @ C ) @ A )
= ( sup_sup_set_a @ ( inf_inf_set_a @ B @ A ) @ ( inf_inf_set_a @ C @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_491_Un__Int__distrib2,axiom,
! [B: set_a,C: set_a,A: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ B @ C ) @ A )
= ( inf_inf_set_a @ ( sup_sup_set_a @ B @ A ) @ ( sup_sup_set_a @ C @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_492_finite__has__maximal,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_493_finite__has__maximal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_494_finite__has__minimal,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_495_finite__has__minimal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_496_finite__surj,axiom,
! [A: set_a,B: set_nat,F: a > nat] :
( ( finite_finite_a @ A )
=> ( ( ord_less_eq_set_nat @ B @ ( image_a_nat @ F @ A ) )
=> ( finite_finite_nat @ B ) ) ) ).
% finite_surj
thf(fact_497_finite__surj,axiom,
! [A: set_nat,B: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
=> ( finite_finite_nat @ B ) ) ) ).
% finite_surj
thf(fact_498_finite__surj,axiom,
! [A: set_a,B: set_a,F: a > a] :
( ( finite_finite_a @ A )
=> ( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A ) )
=> ( finite_finite_a @ B ) ) ) ).
% finite_surj
thf(fact_499_finite__surj,axiom,
! [A: set_nat,B: set_a,F: nat > a] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_a @ B @ ( image_nat_a @ F @ A ) )
=> ( finite_finite_a @ B ) ) ) ).
% finite_surj
thf(fact_500_finite__subset__image,axiom,
! [B: set_nat,F: nat > nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
=> ? [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A )
& ( finite_finite_nat @ C4 )
& ( B
= ( image_nat_nat @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_501_finite__subset__image,axiom,
! [B: set_nat,F: a > nat,A: set_a] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ ( image_a_nat @ F @ A ) )
=> ? [C4: set_a] :
( ( ord_less_eq_set_a @ C4 @ A )
& ( finite_finite_a @ C4 )
& ( B
= ( image_a_nat @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_502_finite__subset__image,axiom,
! [B: set_a,F: nat > a,A: set_nat] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ B @ ( image_nat_a @ F @ A ) )
=> ? [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A )
& ( finite_finite_nat @ C4 )
& ( B
= ( image_nat_a @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_503_finite__subset__image,axiom,
! [B: set_a,F: a > a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A ) )
=> ? [C4: set_a] :
( ( ord_less_eq_set_a @ C4 @ A )
& ( finite_finite_a @ C4 )
& ( B
= ( image_a_a @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_504_ex__finite__subset__image,axiom,
! [F: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat @ F @ A ) )
& ( P @ B6 ) ) )
= ( ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A )
& ( P @ ( image_nat_nat @ F @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_505_ex__finite__subset__image,axiom,
! [F: a > nat,A: set_a,P: set_nat > $o] :
( ( ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ ( image_a_nat @ F @ A ) )
& ( P @ B6 ) ) )
= ( ? [B6: set_a] :
( ( finite_finite_a @ B6 )
& ( ord_less_eq_set_a @ B6 @ A )
& ( P @ ( image_a_nat @ F @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_506_ex__finite__subset__image,axiom,
! [F: nat > a,A: set_nat,P: set_a > $o] :
( ( ? [B6: set_a] :
( ( finite_finite_a @ B6 )
& ( ord_less_eq_set_a @ B6 @ ( image_nat_a @ F @ A ) )
& ( P @ B6 ) ) )
= ( ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A )
& ( P @ ( image_nat_a @ F @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_507_ex__finite__subset__image,axiom,
! [F: a > a,A: set_a,P: set_a > $o] :
( ( ? [B6: set_a] :
( ( finite_finite_a @ B6 )
& ( ord_less_eq_set_a @ B6 @ ( image_a_a @ F @ A ) )
& ( P @ B6 ) ) )
= ( ? [B6: set_a] :
( ( finite_finite_a @ B6 )
& ( ord_less_eq_set_a @ B6 @ A )
& ( P @ ( image_a_a @ F @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_508_all__finite__subset__image,axiom,
! [F: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ! [B6: set_nat] :
( ( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat @ F @ A ) ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_nat] :
( ( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A ) )
=> ( P @ ( image_nat_nat @ F @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_509_all__finite__subset__image,axiom,
! [F: a > nat,A: set_a,P: set_nat > $o] :
( ( ! [B6: set_nat] :
( ( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ ( image_a_nat @ F @ A ) ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_a] :
( ( ( finite_finite_a @ B6 )
& ( ord_less_eq_set_a @ B6 @ A ) )
=> ( P @ ( image_a_nat @ F @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_510_all__finite__subset__image,axiom,
! [F: nat > a,A: set_nat,P: set_a > $o] :
( ( ! [B6: set_a] :
( ( ( finite_finite_a @ B6 )
& ( ord_less_eq_set_a @ B6 @ ( image_nat_a @ F @ A ) ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_nat] :
( ( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A ) )
=> ( P @ ( image_nat_a @ F @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_511_all__finite__subset__image,axiom,
! [F: a > a,A: set_a,P: set_a > $o] :
( ( ! [B6: set_a] :
( ( ( finite_finite_a @ B6 )
& ( ord_less_eq_set_a @ B6 @ ( image_a_a @ F @ A ) ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_a] :
( ( ( finite_finite_a @ B6 )
& ( ord_less_eq_set_a @ B6 @ A ) )
=> ( P @ ( image_a_a @ F @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_512_infinite__finite__induct,axiom,
! [P: set_nat > $o,A: set_nat] :
( ! [A8: set_nat] :
( ~ ( finite_finite_nat @ A8 )
=> ( P @ A8 ) )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X4: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X4 @ F3 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_513_infinite__finite__induct,axiom,
! [P: set_a > $o,A: set_a] :
( ! [A8: set_a] :
( ~ ( finite_finite_a @ A8 )
=> ( P @ A8 ) )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X4: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X4 @ F3 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_514_finite__ne__induct,axiom,
! [F2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( F2 != bot_bot_set_nat )
=> ( ! [X4: nat] : ( P @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
=> ( ! [X4: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( F3 != bot_bot_set_nat )
=> ( ~ ( member_nat @ X4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X4 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_515_finite__ne__induct,axiom,
! [F2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( F2 != bot_bot_set_a )
=> ( ! [X4: a] : ( P @ ( insert_a @ X4 @ bot_bot_set_a ) )
=> ( ! [X4: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( F3 != bot_bot_set_a )
=> ( ~ ( member_a @ X4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X4 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_516_finite__induct,axiom,
! [F2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X4: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X4 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_517_finite__induct,axiom,
! [F2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X4: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X4 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_518_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A3: set_nat] :
( ( A3 = bot_bot_set_nat )
| ? [A6: set_nat,B3: nat] :
( ( A3
= ( insert_nat @ B3 @ A6 ) )
& ( finite_finite_nat @ A6 ) ) ) ) ) ).
% finite.simps
thf(fact_519_finite_Osimps,axiom,
( finite_finite_a
= ( ^ [A3: set_a] :
( ( A3 = bot_bot_set_a )
| ? [A6: set_a,B3: a] :
( ( A3
= ( insert_a @ B3 @ A6 ) )
& ( finite_finite_a @ A6 ) ) ) ) ) ).
% finite.simps
thf(fact_520_finite_Ocases,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ~ ! [A8: set_nat] :
( ? [A4: nat] :
( A2
= ( insert_nat @ A4 @ A8 ) )
=> ~ ( finite_finite_nat @ A8 ) ) ) ) ).
% finite.cases
thf(fact_521_finite_Ocases,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( A2 != bot_bot_set_a )
=> ~ ! [A8: set_a] :
( ? [A4: a] :
( A2
= ( insert_a @ A4 @ A8 ) )
=> ~ ( finite_finite_a @ A8 ) ) ) ) ).
% finite.cases
thf(fact_522_subset__singleton__iff,axiom,
! [X5: set_a,A2: a] :
( ( ord_less_eq_set_a @ X5 @ ( insert_a @ A2 @ bot_bot_set_a ) )
= ( ( X5 = bot_bot_set_a )
| ( X5
= ( insert_a @ A2 @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_523_subset__singletonD,axiom,
! [A: set_a,X: a] :
( ( ord_less_eq_set_a @ A @ ( insert_a @ X @ bot_bot_set_a ) )
=> ( ( A = bot_bot_set_a )
| ( A
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_524_image__Int__subset,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ ( inf_inf_set_nat @ A @ B ) ) @ ( inf_inf_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_525_image__Int__subset,axiom,
! [F: a > a,A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( image_a_a @ F @ ( inf_inf_set_a @ A @ B ) ) @ ( inf_inf_set_a @ ( image_a_a @ F @ A ) @ ( image_a_a @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_526_card__subset__eq,axiom,
! [B: set_Product_unit,A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ B )
=> ( ( ord_le3507040750410214029t_unit @ A @ B )
=> ( ( ( finite410649719033368117t_unit @ A )
= ( finite410649719033368117t_unit @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_527_card__subset__eq,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( finite_card_nat @ A )
= ( finite_card_nat @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_528_card__subset__eq,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( finite_card_a @ A )
= ( finite_card_a @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_529_infinite__arbitrarily__large,axiom,
! [A: set_Product_unit,N2: nat] :
( ~ ( finite4290736615968046902t_unit @ A )
=> ? [B7: set_Product_unit] :
( ( finite4290736615968046902t_unit @ B7 )
& ( ( finite410649719033368117t_unit @ B7 )
= N2 )
& ( ord_le3507040750410214029t_unit @ B7 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_530_infinite__arbitrarily__large,axiom,
! [A: set_nat,N2: nat] :
( ~ ( finite_finite_nat @ A )
=> ? [B7: set_nat] :
( ( finite_finite_nat @ B7 )
& ( ( finite_card_nat @ B7 )
= N2 )
& ( ord_less_eq_set_nat @ B7 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_531_infinite__arbitrarily__large,axiom,
! [A: set_a,N2: nat] :
( ~ ( finite_finite_a @ A )
=> ? [B7: set_a] :
( ( finite_finite_a @ B7 )
& ( ( finite_card_a @ B7 )
= N2 )
& ( ord_less_eq_set_a @ B7 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_532_insert__is__Un,axiom,
( insert_a
= ( ^ [A3: a] : ( sup_sup_set_a @ ( insert_a @ A3 @ bot_bot_set_a ) ) ) ) ).
% insert_is_Un
thf(fact_533_Un__singleton__iff,axiom,
! [A: set_a,B: set_a,X: a] :
( ( ( sup_sup_set_a @ A @ B )
= ( insert_a @ X @ bot_bot_set_a ) )
= ( ( ( A = bot_bot_set_a )
& ( B
= ( insert_a @ X @ bot_bot_set_a ) ) )
| ( ( A
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B = bot_bot_set_a ) )
| ( ( A
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_534_singleton__Un__iff,axiom,
! [X: a,A: set_a,B: set_a] :
( ( ( insert_a @ X @ bot_bot_set_a )
= ( sup_sup_set_a @ A @ B ) )
= ( ( ( A = bot_bot_set_a )
& ( B
= ( insert_a @ X @ bot_bot_set_a ) ) )
| ( ( A
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B = bot_bot_set_a ) )
| ( ( A
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_535_card__insert__le,axiom,
! [A: set_a,X: a] : ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ ( insert_a @ X @ A ) ) ) ).
% card_insert_le
thf(fact_536_card__insert__le,axiom,
! [A: set_Product_unit,X: product_unit] : ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite410649719033368117t_unit @ ( insert_Product_unit @ X @ A ) ) ) ).
% card_insert_le
thf(fact_537_Un__Int__assoc__eq,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ( sup_sup_set_a @ ( inf_inf_set_a @ A @ B ) @ C )
= ( inf_inf_set_a @ A @ ( sup_sup_set_a @ B @ C ) ) )
= ( ord_less_eq_set_a @ C @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_538_finite__subset__induct,axiom,
! [F2: set_nat,A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A4: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A4 @ A )
=> ( ~ ( member_nat @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ A4 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_539_finite__subset__induct,axiom,
! [F2: set_a,A: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( ord_less_eq_set_a @ F2 @ A )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A4: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A4 @ A )
=> ( ~ ( member_a @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ A4 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_540_finite__subset__induct_H,axiom,
! [F2: set_nat,A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A4: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A4 @ A )
=> ( ( ord_less_eq_set_nat @ F3 @ A )
=> ( ~ ( member_nat @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ A4 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_541_finite__subset__induct_H,axiom,
! [F2: set_a,A: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( ord_less_eq_set_a @ F2 @ A )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A4: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A4 @ A )
=> ( ( ord_less_eq_set_a @ F3 @ A )
=> ( ~ ( member_a @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ A4 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_542_card__eq__0__iff,axiom,
! [A: set_Product_unit] :
( ( ( finite410649719033368117t_unit @ A )
= zero_zero_nat )
= ( ( A = bot_bo3957492148770167129t_unit )
| ~ ( finite4290736615968046902t_unit @ A ) ) ) ).
% card_eq_0_iff
thf(fact_543_card__eq__0__iff,axiom,
! [A: set_nat] :
( ( ( finite_card_nat @ A )
= zero_zero_nat )
= ( ( A = bot_bot_set_nat )
| ~ ( finite_finite_nat @ A ) ) ) ).
% card_eq_0_iff
thf(fact_544_card__eq__0__iff,axiom,
! [A: set_a] :
( ( ( finite_card_a @ A )
= zero_zero_nat )
= ( ( A = bot_bot_set_a )
| ~ ( finite_finite_a @ A ) ) ) ).
% card_eq_0_iff
thf(fact_545_card__image__le,axiom,
! [A: set_Product_unit,F: product_unit > a] :
( ( finite4290736615968046902t_unit @ A )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( image_Product_unit_a @ F @ A ) ) @ ( finite410649719033368117t_unit @ A ) ) ) ).
% card_image_le
thf(fact_546_card__image__le,axiom,
! [A: set_Product_unit,F: product_unit > product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ ( image_405062704495631173t_unit @ F @ A ) ) @ ( finite410649719033368117t_unit @ A ) ) ) ).
% card_image_le
thf(fact_547_card__image__le,axiom,
! [A: set_a,F: a > a] :
( ( finite_finite_a @ A )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( image_a_a @ F @ A ) ) @ ( finite_card_a @ A ) ) ) ).
% card_image_le
thf(fact_548_card__image__le,axiom,
! [A: set_a,F: a > product_unit] :
( ( finite_finite_a @ A )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ ( image_a_Product_unit @ F @ A ) ) @ ( finite_card_a @ A ) ) ) ).
% card_image_le
thf(fact_549_card__image__le,axiom,
! [A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_nat_nat @ F @ A ) ) @ ( finite_card_nat @ A ) ) ) ).
% card_image_le
thf(fact_550_card__image__le,axiom,
! [A: set_nat,F: nat > a] :
( ( finite_finite_nat @ A )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( image_nat_a @ F @ A ) ) @ ( finite_card_nat @ A ) ) ) ).
% card_image_le
thf(fact_551_card__image__le,axiom,
! [A: set_nat,F: nat > product_unit] :
( ( finite_finite_nat @ A )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ ( image_8730104196221521654t_unit @ F @ A ) ) @ ( finite_card_nat @ A ) ) ) ).
% card_image_le
thf(fact_552_card__mono,axiom,
! [B: set_Product_unit,A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ B )
=> ( ( ord_le3507040750410214029t_unit @ A @ B )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite410649719033368117t_unit @ B ) ) ) ) ).
% card_mono
thf(fact_553_card__mono,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ).
% card_mono
thf(fact_554_card__mono,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B ) ) ) ) ).
% card_mono
thf(fact_555_card__seteq,axiom,
! [B: set_Product_unit,A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ B )
=> ( ( ord_le3507040750410214029t_unit @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ B ) @ ( finite410649719033368117t_unit @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_556_card__seteq,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite_card_nat @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_557_card__seteq,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ B ) @ ( finite_card_a @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_558_exists__subset__between,axiom,
! [A: set_Product_unit,N2: nat,C: set_Product_unit] :
( ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( finite410649719033368117t_unit @ C ) )
=> ( ( ord_le3507040750410214029t_unit @ A @ C )
=> ( ( finite4290736615968046902t_unit @ C )
=> ? [B7: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A @ B7 )
& ( ord_le3507040750410214029t_unit @ B7 @ C )
& ( ( finite410649719033368117t_unit @ B7 )
= N2 ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_559_exists__subset__between,axiom,
! [A: set_nat,N2: nat,C: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( finite_card_nat @ C ) )
=> ( ( ord_less_eq_set_nat @ A @ C )
=> ( ( finite_finite_nat @ C )
=> ? [B7: set_nat] :
( ( ord_less_eq_set_nat @ A @ B7 )
& ( ord_less_eq_set_nat @ B7 @ C )
& ( ( finite_card_nat @ B7 )
= N2 ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_560_exists__subset__between,axiom,
! [A: set_a,N2: nat,C: set_a] :
( ( ord_less_eq_nat @ ( finite_card_a @ A ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( finite_card_a @ C ) )
=> ( ( ord_less_eq_set_a @ A @ C )
=> ( ( finite_finite_a @ C )
=> ? [B7: set_a] :
( ( ord_less_eq_set_a @ A @ B7 )
& ( ord_less_eq_set_a @ B7 @ C )
& ( ( finite_card_a @ B7 )
= N2 ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_561_obtain__subset__with__card__n,axiom,
! [N2: nat,S: set_Product_unit] :
( ( ord_less_eq_nat @ N2 @ ( finite410649719033368117t_unit @ S ) )
=> ~ ! [T3: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ T3 @ S )
=> ( ( ( finite410649719033368117t_unit @ T3 )
= N2 )
=> ~ ( finite4290736615968046902t_unit @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_562_obtain__subset__with__card__n,axiom,
! [N2: nat,S: set_nat] :
( ( ord_less_eq_nat @ N2 @ ( finite_card_nat @ S ) )
=> ~ ! [T3: set_nat] :
( ( ord_less_eq_set_nat @ T3 @ S )
=> ( ( ( finite_card_nat @ T3 )
= N2 )
=> ~ ( finite_finite_nat @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_563_obtain__subset__with__card__n,axiom,
! [N2: nat,S: set_a] :
( ( ord_less_eq_nat @ N2 @ ( finite_card_a @ S ) )
=> ~ ! [T3: set_a] :
( ( ord_less_eq_set_a @ T3 @ S )
=> ( ( ( finite_card_a @ T3 )
= N2 )
=> ~ ( finite_finite_a @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_564_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_Product_unit,C: nat] :
( ! [G4: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ G4 @ F2 )
=> ( ( finite4290736615968046902t_unit @ G4 )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ G4 ) @ C ) ) )
=> ( ( finite4290736615968046902t_unit @ F2 )
& ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ F2 ) @ C ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_565_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_nat,C: nat] :
( ! [G4: set_nat] :
( ( ord_less_eq_set_nat @ G4 @ F2 )
=> ( ( finite_finite_nat @ G4 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G4 ) @ C ) ) )
=> ( ( finite_finite_nat @ F2 )
& ( ord_less_eq_nat @ ( finite_card_nat @ F2 ) @ C ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_566_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_a,C: nat] :
( ! [G4: set_a] :
( ( ord_less_eq_set_a @ G4 @ F2 )
=> ( ( finite_finite_a @ G4 )
=> ( ord_less_eq_nat @ ( finite_card_a @ G4 ) @ C ) ) )
=> ( ( finite_finite_a @ F2 )
& ( ord_less_eq_nat @ ( finite_card_a @ F2 ) @ C ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_567_surj__card__le,axiom,
! [A: set_Product_unit,B: set_Product_unit,F: product_unit > product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( ord_le3507040750410214029t_unit @ B @ ( image_405062704495631173t_unit @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ B ) @ ( finite410649719033368117t_unit @ A ) ) ) ) ).
% surj_card_le
thf(fact_568_surj__card__le,axiom,
! [A: set_a,B: set_Product_unit,F: a > product_unit] :
( ( finite_finite_a @ A )
=> ( ( ord_le3507040750410214029t_unit @ B @ ( image_a_Product_unit @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ B ) @ ( finite_card_a @ A ) ) ) ) ).
% surj_card_le
thf(fact_569_surj__card__le,axiom,
! [A: set_nat,B: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite_card_nat @ A ) ) ) ) ).
% surj_card_le
thf(fact_570_surj__card__le,axiom,
! [A: set_nat,B: set_Product_unit,F: nat > product_unit] :
( ( finite_finite_nat @ A )
=> ( ( ord_le3507040750410214029t_unit @ B @ ( image_8730104196221521654t_unit @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ B ) @ ( finite_card_nat @ A ) ) ) ) ).
% surj_card_le
thf(fact_571_surj__card__le,axiom,
! [A: set_Product_unit,B: set_a,F: product_unit > a] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( ord_less_eq_set_a @ B @ ( image_Product_unit_a @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ B ) @ ( finite410649719033368117t_unit @ A ) ) ) ) ).
% surj_card_le
thf(fact_572_surj__card__le,axiom,
! [A: set_a,B: set_a,F: a > a] :
( ( finite_finite_a @ A )
=> ( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ B ) @ ( finite_card_a @ A ) ) ) ) ).
% surj_card_le
thf(fact_573_surj__card__le,axiom,
! [A: set_nat,B: set_a,F: nat > a] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_a @ B @ ( image_nat_a @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ B ) @ ( finite_card_nat @ A ) ) ) ) ).
% surj_card_le
thf(fact_574_group_Oinverse__subgroupD,axiom,
! [G: set_nat,Composition: nat > nat > nat,Unit: nat,H2: set_nat] :
( ( group_group_nat @ G @ Composition @ Unit )
=> ( ( group_subgroup_nat @ ( image_nat_nat @ ( group_inverse_nat @ G @ Composition @ Unit ) @ H2 ) @ G @ Composition @ Unit )
=> ( ( ord_less_eq_set_nat @ H2 @ ( group_Units_nat @ G @ Composition @ Unit ) )
=> ( group_subgroup_nat @ H2 @ G @ Composition @ Unit ) ) ) ) ).
% group.inverse_subgroupD
thf(fact_575_group_Oinverse__subgroupD,axiom,
! [G: set_a,Composition: a > a > a,Unit: a,H2: set_a] :
( ( group_group_a @ G @ Composition @ Unit )
=> ( ( group_subgroup_a @ ( image_a_a @ ( group_inverse_a @ G @ Composition @ Unit ) @ H2 ) @ G @ Composition @ Unit )
=> ( ( ord_less_eq_set_a @ H2 @ ( group_Units_a @ G @ Composition @ Unit ) )
=> ( group_subgroup_a @ H2 @ G @ Composition @ Unit ) ) ) ) ).
% group.inverse_subgroupD
thf(fact_576_sumset__iterated__empty,axiom,
! [R: nat] :
( ( ord_less_nat @ zero_zero_nat @ R )
=> ( ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ bot_bot_set_a @ R )
= bot_bot_set_a ) ) ).
% sumset_iterated_empty
thf(fact_577_sumsetdiff__sing,axiom,
! [A: set_a,B: set_a,X: a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ ( minus_minus_set_a @ A @ B ) @ ( insert_a @ X @ bot_bot_set_a ) )
= ( minus_minus_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( insert_a @ X @ bot_bot_set_a ) ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ B @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% sumsetdiff_sing
thf(fact_578_inf__sup__absorb,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_579_sup__inf__absorb,axiom,
! [X: set_a,Y: set_a] :
( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_580_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_581_le0,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% le0
thf(fact_582_sup__bot__left,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ X )
= X ) ).
% sup_bot_left
thf(fact_583_inf__right__idem,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y )
= ( inf_inf_set_a @ X @ Y ) ) ).
% inf_right_idem
thf(fact_584_inf_Oright__idem,axiom,
! [A2: set_a,B2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ B2 )
= ( inf_inf_set_a @ A2 @ B2 ) ) ).
% inf.right_idem
thf(fact_585_inf__left__idem,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ X @ Y ) )
= ( inf_inf_set_a @ X @ Y ) ) ).
% inf_left_idem
thf(fact_586_inf_Oleft__idem,axiom,
! [A2: set_a,B2: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ A2 @ B2 ) )
= ( inf_inf_set_a @ A2 @ B2 ) ) ).
% inf.left_idem
thf(fact_587_inf__idem,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ X )
= X ) ).
% inf_idem
thf(fact_588_inf_Oidem,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_589_sup_Oright__idem,axiom,
! [A2: set_a,B2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ B2 )
= ( sup_sup_set_a @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_590_sup__left__idem,axiom,
! [X: set_a,Y: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ X @ Y ) )
= ( sup_sup_set_a @ X @ Y ) ) ).
% sup_left_idem
thf(fact_591_sup_Oleft__idem,axiom,
! [A2: set_a,B2: set_a] :
( ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B2 ) )
= ( sup_sup_set_a @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_592_sup__idem,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ X )
= X ) ).
% sup_idem
thf(fact_593_sup_Oidem,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_594_DiffI,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ A )
=> ( ~ ( member_nat @ C2 @ B )
=> ( member_nat @ C2 @ ( minus_minus_set_nat @ A @ B ) ) ) ) ).
% DiffI
thf(fact_595_DiffI,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ A )
=> ( ~ ( member_a @ C2 @ B )
=> ( member_a @ C2 @ ( minus_minus_set_a @ A @ B ) ) ) ) ).
% DiffI
thf(fact_596_Diff__iff,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A @ B ) )
= ( ( member_nat @ C2 @ A )
& ~ ( member_nat @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_597_Diff__iff,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A @ B ) )
= ( ( member_a @ C2 @ A )
& ~ ( member_a @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_598_Diff__idemp,axiom,
! [A: set_a,B: set_a] :
( ( minus_minus_set_a @ ( minus_minus_set_a @ A @ B ) @ B )
= ( minus_minus_set_a @ A @ B ) ) ).
% Diff_idemp
thf(fact_599_inf_Obounded__iff,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C2 ) )
= ( ( ord_less_eq_set_a @ A2 @ B2 )
& ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_600_inf_Obounded__iff,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C2 ) )
= ( ( ord_less_eq_nat @ A2 @ B2 )
& ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_601_le__inf__iff,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( ( ord_less_eq_set_a @ X @ Y )
& ( ord_less_eq_set_a @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_602_le__inf__iff,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
= ( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_603_sup_Obounded__iff,axiom,
! [B2: set_a,C2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C2 ) @ A2 )
= ( ( ord_less_eq_set_a @ B2 @ A2 )
& ( ord_less_eq_set_a @ C2 @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_604_sup_Obounded__iff,axiom,
! [B2: nat,C2: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C2 ) @ A2 )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ C2 @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_605_le__sup__iff,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z )
= ( ( ord_less_eq_set_a @ X @ Z )
& ( ord_less_eq_set_a @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_606_le__sup__iff,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z )
= ( ( ord_less_eq_nat @ X @ Z )
& ( ord_less_eq_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_607_inf__bot__right,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ bot_bot_set_a )
= bot_bot_set_a ) ).
% inf_bot_right
thf(fact_608_inf__bot__left,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X )
= bot_bot_set_a ) ).
% inf_bot_left
thf(fact_609_sup__bot_Oright__neutral,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_610_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ A2 @ B2 ) )
= ( ( A2 = bot_bot_set_a )
& ( B2 = bot_bot_set_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_611_sup__bot_Oleft__neutral,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_612_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( ( sup_sup_set_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ( A2 = bot_bot_set_a )
& ( B2 = bot_bot_set_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_613_sup__eq__bot__iff,axiom,
! [X: set_a,Y: set_a] :
( ( ( sup_sup_set_a @ X @ Y )
= bot_bot_set_a )
= ( ( X = bot_bot_set_a )
& ( Y = bot_bot_set_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_614_bot__eq__sup__iff,axiom,
! [X: set_a,Y: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ X @ Y ) )
= ( ( X = bot_bot_set_a )
& ( Y = bot_bot_set_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_615_sup__bot__right,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ bot_bot_set_a )
= X ) ).
% sup_bot_right
thf(fact_616_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_617_neq0__conv,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% neq0_conv
thf(fact_618_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_619_Diff__cancel,axiom,
! [A: set_a] :
( ( minus_minus_set_a @ A @ A )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_620_empty__Diff,axiom,
! [A: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_621_Diff__empty,axiom,
! [A: set_a] :
( ( minus_minus_set_a @ A @ bot_bot_set_a )
= A ) ).
% Diff_empty
thf(fact_622_finite__Diff2,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) )
= ( finite_finite_nat @ A ) ) ) ).
% finite_Diff2
thf(fact_623_finite__Diff2,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( finite_finite_a @ ( minus_minus_set_a @ A @ B ) )
= ( finite_finite_a @ A ) ) ) ).
% finite_Diff2
thf(fact_624_finite__Diff,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) ) ) ).
% finite_Diff
thf(fact_625_finite__Diff,axiom,
! [A: set_a,B: set_a] :
( ( finite_finite_a @ A )
=> ( finite_finite_a @ ( minus_minus_set_a @ A @ B ) ) ) ).
% finite_Diff
thf(fact_626_insert__Diff1,axiom,
! [X: nat,B: set_nat,A: set_nat] :
( ( member_nat @ X @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ B )
= ( minus_minus_set_nat @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_627_insert__Diff1,axiom,
! [X: a,B: set_a,A: set_a] :
( ( member_a @ X @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A ) @ B )
= ( minus_minus_set_a @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_628_Diff__insert0,axiom,
! [X: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ X @ A )
=> ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ B ) )
= ( minus_minus_set_nat @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_629_Diff__insert0,axiom,
! [X: a,A: set_a,B: set_a] :
( ~ ( member_a @ X @ A )
=> ( ( minus_minus_set_a @ A @ ( insert_a @ X @ B ) )
= ( minus_minus_set_a @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_630_Un__Diff__cancel2,axiom,
! [B: set_a,A: set_a] :
( ( sup_sup_set_a @ ( minus_minus_set_a @ B @ A ) @ A )
= ( sup_sup_set_a @ B @ A ) ) ).
% Un_Diff_cancel2
thf(fact_631_Un__Diff__cancel,axiom,
! [A: set_a,B: set_a] :
( ( sup_sup_set_a @ A @ ( minus_minus_set_a @ B @ A ) )
= ( sup_sup_set_a @ A @ B ) ) ).
% Un_Diff_cancel
thf(fact_632_Diff__eq__empty__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( minus_minus_set_a @ A @ B )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_633_insert__Diff__single,axiom,
! [A2: a,A: set_a] :
( ( insert_a @ A2 @ ( minus_minus_set_a @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) )
= ( insert_a @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_634_finite__Diff__insert,axiom,
! [A: set_nat,A2: nat,B: set_nat] :
( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ B ) ) )
= ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) ) ) ).
% finite_Diff_insert
thf(fact_635_finite__Diff__insert,axiom,
! [A: set_a,A2: a,B: set_a] :
( ( finite_finite_a @ ( minus_minus_set_a @ A @ ( insert_a @ A2 @ B ) ) )
= ( finite_finite_a @ ( minus_minus_set_a @ A @ B ) ) ) ).
% finite_Diff_insert
thf(fact_636_Diff__disjoint,axiom,
! [A: set_a,B: set_a] :
( ( inf_inf_set_a @ A @ ( minus_minus_set_a @ B @ A ) )
= bot_bot_set_a ) ).
% Diff_disjoint
thf(fact_637_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_638_DiffE,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A @ B ) )
=> ~ ( ( member_nat @ C2 @ A )
=> ( member_nat @ C2 @ B ) ) ) ).
% DiffE
thf(fact_639_DiffE,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A @ B ) )
=> ~ ( ( member_a @ C2 @ A )
=> ( member_a @ C2 @ B ) ) ) ).
% DiffE
thf(fact_640_DiffD1,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A @ B ) )
=> ( member_nat @ C2 @ A ) ) ).
% DiffD1
thf(fact_641_DiffD1,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A @ B ) )
=> ( member_a @ C2 @ A ) ) ).
% DiffD1
thf(fact_642_DiffD2,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A @ B ) )
=> ~ ( member_nat @ C2 @ B ) ) ).
% DiffD2
thf(fact_643_DiffD2,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A @ B ) )
=> ~ ( member_a @ C2 @ B ) ) ).
% DiffD2
thf(fact_644_less__supI1,axiom,
! [X: set_a,A2: set_a,B2: set_a] :
( ( ord_less_set_a @ X @ A2 )
=> ( ord_less_set_a @ X @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% less_supI1
thf(fact_645_less__supI1,axiom,
! [X: nat,A2: nat,B2: nat] :
( ( ord_less_nat @ X @ A2 )
=> ( ord_less_nat @ X @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% less_supI1
thf(fact_646_less__supI2,axiom,
! [X: set_a,B2: set_a,A2: set_a] :
( ( ord_less_set_a @ X @ B2 )
=> ( ord_less_set_a @ X @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% less_supI2
thf(fact_647_less__supI2,axiom,
! [X: nat,B2: nat,A2: nat] :
( ( ord_less_nat @ X @ B2 )
=> ( ord_less_nat @ X @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% less_supI2
thf(fact_648_sup_Oabsorb3,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_set_a @ B2 @ A2 )
=> ( ( sup_sup_set_a @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb3
thf(fact_649_sup_Oabsorb3,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb3
thf(fact_650_sup_Oabsorb4,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( sup_sup_set_a @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb4
thf(fact_651_sup_Oabsorb4,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb4
thf(fact_652_sup_Ostrict__boundedE,axiom,
! [B2: set_a,C2: set_a,A2: set_a] :
( ( ord_less_set_a @ ( sup_sup_set_a @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_less_set_a @ B2 @ A2 )
=> ~ ( ord_less_set_a @ C2 @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_653_sup_Ostrict__boundedE,axiom,
! [B2: nat,C2: nat,A2: nat] :
( ( ord_less_nat @ ( sup_sup_nat @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_less_nat @ B2 @ A2 )
=> ~ ( ord_less_nat @ C2 @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_654_sup_Ostrict__order__iff,axiom,
( ord_less_set_a
= ( ^ [B3: set_a,A3: set_a] :
( ( A3
= ( sup_sup_set_a @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_655_sup_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( A3
= ( sup_sup_nat @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_656_sup_Ostrict__coboundedI1,axiom,
! [C2: set_a,A2: set_a,B2: set_a] :
( ( ord_less_set_a @ C2 @ A2 )
=> ( ord_less_set_a @ C2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_657_sup_Ostrict__coboundedI1,axiom,
! [C2: nat,A2: nat,B2: nat] :
( ( ord_less_nat @ C2 @ A2 )
=> ( ord_less_nat @ C2 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_658_sup_Ostrict__coboundedI2,axiom,
! [C2: set_a,B2: set_a,A2: set_a] :
( ( ord_less_set_a @ C2 @ B2 )
=> ( ord_less_set_a @ C2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_659_sup_Ostrict__coboundedI2,axiom,
! [C2: nat,B2: nat,A2: nat] :
( ( ord_less_nat @ C2 @ B2 )
=> ( ord_less_nat @ C2 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_660_less__infI1,axiom,
! [A2: set_a,X: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ X )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ X ) ) ).
% less_infI1
thf(fact_661_less__infI1,axiom,
! [A2: nat,X: nat,B2: nat] :
( ( ord_less_nat @ A2 @ X )
=> ( ord_less_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X ) ) ).
% less_infI1
thf(fact_662_less__infI2,axiom,
! [B2: set_a,X: set_a,A2: set_a] :
( ( ord_less_set_a @ B2 @ X )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ X ) ) ).
% less_infI2
thf(fact_663_less__infI2,axiom,
! [B2: nat,X: nat,A2: nat] :
( ( ord_less_nat @ B2 @ X )
=> ( ord_less_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X ) ) ).
% less_infI2
thf(fact_664_inf_Oabsorb3,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb3
thf(fact_665_inf_Oabsorb3,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( inf_inf_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb3
thf(fact_666_inf_Oabsorb4,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_set_a @ B2 @ A2 )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb4
thf(fact_667_inf_Oabsorb4,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( inf_inf_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb4
thf(fact_668_inf_Ostrict__boundedE,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C2 ) )
=> ~ ( ( ord_less_set_a @ A2 @ B2 )
=> ~ ( ord_less_set_a @ A2 @ C2 ) ) ) ).
% inf.strict_boundedE
thf(fact_669_inf_Ostrict__boundedE,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A2 @ ( inf_inf_nat @ B2 @ C2 ) )
=> ~ ( ( ord_less_nat @ A2 @ B2 )
=> ~ ( ord_less_nat @ A2 @ C2 ) ) ) ).
% inf.strict_boundedE
thf(fact_670_inf_Ostrict__order__iff,axiom,
( ord_less_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( A3
= ( inf_inf_set_a @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_671_inf_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( A3
= ( inf_inf_nat @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_672_inf_Ostrict__coboundedI1,axiom,
! [A2: set_a,C2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ C2 )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C2 ) ) ).
% inf.strict_coboundedI1
thf(fact_673_inf_Ostrict__coboundedI1,axiom,
! [A2: nat,C2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ C2 )
=> ( ord_less_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C2 ) ) ).
% inf.strict_coboundedI1
thf(fact_674_inf_Ostrict__coboundedI2,axiom,
! [B2: set_a,C2: set_a,A2: set_a] :
( ( ord_less_set_a @ B2 @ C2 )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C2 ) ) ).
% inf.strict_coboundedI2
thf(fact_675_inf_Ostrict__coboundedI2,axiom,
! [B2: nat,C2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ C2 )
=> ( ord_less_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C2 ) ) ).
% inf.strict_coboundedI2
thf(fact_676_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_677_gr0I,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr0I
thf(fact_678_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_679_not__less0,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less0
thf(fact_680_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_681_gr__implies__not0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_682_infinite__descent0,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N2 ) ) ) ).
% infinite_descent0
thf(fact_683_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M4: nat,N4: nat] :
( ( ord_less_eq_nat @ M4 @ N4 )
& ( M4 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_684_less__imp__le__nat,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_imp_le_nat
thf(fact_685_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N4: nat] :
( ( ord_less_nat @ M4 @ N4 )
| ( M4 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_686_less__or__eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_687_le__neq__implies__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( M2 != N2 )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_688_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_689_card__less__sym__Diff,axiom,
! [A: set_Product_unit,B: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( finite4290736615968046902t_unit @ B )
=> ( ( ord_less_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite410649719033368117t_unit @ B ) )
=> ( ord_less_nat @ ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A @ B ) ) @ ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ B @ A ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_690_card__less__sym__Diff,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ( ord_less_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ B ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B @ A ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_691_card__less__sym__Diff,axiom,
! [A: set_a,B: set_a] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B )
=> ( ( ord_less_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B ) )
=> ( ord_less_nat @ ( finite_card_a @ ( minus_minus_set_a @ A @ B ) ) @ ( finite_card_a @ ( minus_minus_set_a @ B @ A ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_692_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_693_Diff__infinite__finite,axiom,
! [T: set_nat,S: set_nat] :
( ( finite_finite_nat @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_694_Diff__infinite__finite,axiom,
! [T: set_a,S: set_a] :
( ( finite_finite_a @ T )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_695_Diff__mono,axiom,
! [A: set_a,C: set_a,D2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ D2 @ B )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ B ) @ ( minus_minus_set_a @ C @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_696_Diff__subset,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_697_double__diff,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ( minus_minus_set_a @ B @ ( minus_minus_set_a @ C @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_698_insert__Diff__if,axiom,
! [X: nat,B: set_nat,A: set_nat] :
( ( ( member_nat @ X @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ B )
= ( minus_minus_set_nat @ A @ B ) ) )
& ( ~ ( member_nat @ X @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ B )
= ( insert_nat @ X @ ( minus_minus_set_nat @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_699_insert__Diff__if,axiom,
! [X: a,B: set_a,A: set_a] :
( ( ( member_a @ X @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A ) @ B )
= ( minus_minus_set_a @ A @ B ) ) )
& ( ~ ( member_a @ X @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A ) @ B )
= ( insert_a @ X @ ( minus_minus_set_a @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_700_Int__Diff,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A @ B ) @ C )
= ( inf_inf_set_a @ A @ ( minus_minus_set_a @ B @ C ) ) ) ).
% Int_Diff
thf(fact_701_Diff__Int2,axiom,
! [A: set_a,C: set_a,B: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A @ C ) @ ( inf_inf_set_a @ B @ C ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A @ C ) @ B ) ) ).
% Diff_Int2
thf(fact_702_Diff__Diff__Int,axiom,
! [A: set_a,B: set_a] :
( ( minus_minus_set_a @ A @ ( minus_minus_set_a @ A @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ).
% Diff_Diff_Int
thf(fact_703_Diff__Int__distrib,axiom,
! [C: set_a,A: set_a,B: set_a] :
( ( inf_inf_set_a @ C @ ( minus_minus_set_a @ A @ B ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ C @ A ) @ ( inf_inf_set_a @ C @ B ) ) ) ).
% Diff_Int_distrib
thf(fact_704_Diff__Int__distrib2,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( inf_inf_set_a @ ( minus_minus_set_a @ A @ B ) @ C )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A @ C ) @ ( inf_inf_set_a @ B @ C ) ) ) ).
% Diff_Int_distrib2
thf(fact_705_Un__Diff,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( minus_minus_set_a @ ( sup_sup_set_a @ A @ B ) @ C )
= ( sup_sup_set_a @ ( minus_minus_set_a @ A @ C ) @ ( minus_minus_set_a @ B @ C ) ) ) ).
% Un_Diff
thf(fact_706_ex__least__nat__le,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N2 )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_707_card__Diff1__less,axiom,
! [A: set_Product_unit,X: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( member_Product_unit @ X @ A )
=> ( ord_less_nat @ ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) @ ( finite410649719033368117t_unit @ A ) ) ) ) ).
% card_Diff1_less
thf(fact_708_card__Diff1__less,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X @ A )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A ) ) ) ) ).
% card_Diff1_less
thf(fact_709_card__Diff1__less,axiom,
! [A: set_a,X: a] :
( ( finite_finite_a @ A )
=> ( ( member_a @ X @ A )
=> ( ord_less_nat @ ( finite_card_a @ ( minus_minus_set_a @ A @ ( insert_a @ X @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A ) ) ) ) ).
% card_Diff1_less
thf(fact_710_card__Diff2__less,axiom,
! [A: set_Product_unit,X: product_unit,Y: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( member_Product_unit @ X @ A )
=> ( ( member_Product_unit @ Y @ A )
=> ( ord_less_nat @ ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) @ ( insert_Product_unit @ Y @ bot_bo3957492148770167129t_unit ) ) ) @ ( finite410649719033368117t_unit @ A ) ) ) ) ) ).
% card_Diff2_less
thf(fact_711_card__Diff2__less,axiom,
! [A: set_nat,X: nat,Y: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ ( insert_nat @ Y @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A ) ) ) ) ) ).
% card_Diff2_less
thf(fact_712_card__Diff2__less,axiom,
! [A: set_a,X: a,Y: a] :
( ( finite_finite_a @ A )
=> ( ( member_a @ X @ A )
=> ( ( member_a @ Y @ A )
=> ( ord_less_nat @ ( finite_card_a @ ( minus_minus_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ X @ bot_bot_set_a ) ) @ ( insert_a @ Y @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A ) ) ) ) ) ).
% card_Diff2_less
thf(fact_713_card__Diff1__less__iff,axiom,
! [A: set_Product_unit,X: product_unit] :
( ( ord_less_nat @ ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) @ ( finite410649719033368117t_unit @ A ) )
= ( ( finite4290736615968046902t_unit @ A )
& ( member_Product_unit @ X @ A ) ) ) ).
% card_Diff1_less_iff
thf(fact_714_card__Diff1__less__iff,axiom,
! [A: set_nat,X: nat] :
( ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A ) )
= ( ( finite_finite_nat @ A )
& ( member_nat @ X @ A ) ) ) ).
% card_Diff1_less_iff
thf(fact_715_card__Diff1__less__iff,axiom,
! [A: set_a,X: a] :
( ( ord_less_nat @ ( finite_card_a @ ( minus_minus_set_a @ A @ ( insert_a @ X @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A ) )
= ( ( finite_finite_a @ A )
& ( member_a @ X @ A ) ) ) ).
% card_Diff1_less_iff
thf(fact_716_image__diff__subset,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B ) ) @ ( image_nat_nat @ F @ ( minus_minus_set_nat @ A @ B ) ) ) ).
% image_diff_subset
thf(fact_717_image__diff__subset,axiom,
! [F: a > a,A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ ( image_a_a @ F @ A ) @ ( image_a_a @ F @ B ) ) @ ( image_a_a @ F @ ( minus_minus_set_a @ A @ B ) ) ) ).
% image_diff_subset
thf(fact_718_Diff__insert,axiom,
! [A: set_a,A2: a,B: set_a] :
( ( minus_minus_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A @ B ) @ ( insert_a @ A2 @ bot_bot_set_a ) ) ) ).
% Diff_insert
thf(fact_719_insert__Diff,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( insert_nat @ A2 @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
= A ) ) ).
% insert_Diff
thf(fact_720_insert__Diff,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ( ( insert_a @ A2 @ ( minus_minus_set_a @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) )
= A ) ) ).
% insert_Diff
thf(fact_721_Diff__insert2,axiom,
! [A: set_a,A2: a,B: set_a] :
( ( minus_minus_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) @ B ) ) ).
% Diff_insert2
thf(fact_722_Diff__insert__absorb,axiom,
! [X: nat,A: set_nat] :
( ~ ( member_nat @ X @ A )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_723_Diff__insert__absorb,axiom,
! [X: a,A: set_a] :
( ~ ( member_a @ X @ A )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A ) @ ( insert_a @ X @ bot_bot_set_a ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_724_in__image__insert__iff,axiom,
! [B: set_set_nat,X: nat,A: set_nat] :
( ! [C4: set_nat] :
( ( member_set_nat @ C4 @ B )
=> ~ ( member_nat @ X @ C4 ) )
=> ( ( member_set_nat @ A @ ( image_7916887816326733075et_nat @ ( insert_nat @ X ) @ B ) )
= ( ( member_nat @ X @ A )
& ( member_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_725_in__image__insert__iff,axiom,
! [B: set_set_a,X: a,A: set_a] :
( ! [C4: set_a] :
( ( member_set_a @ C4 @ B )
=> ~ ( member_a @ X @ C4 ) )
=> ( ( member_set_a @ A @ ( image_set_a_set_a @ ( insert_a @ X ) @ B ) )
= ( ( member_a @ X @ A )
& ( member_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ X @ bot_bot_set_a ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_726_subset__Diff__insert,axiom,
! [A: set_nat,B: set_nat,X: nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B @ ( insert_nat @ X @ C ) ) )
= ( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B @ C ) )
& ~ ( member_nat @ X @ A ) ) ) ).
% subset_Diff_insert
thf(fact_727_subset__Diff__insert,axiom,
! [A: set_a,B: set_a,X: a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( minus_minus_set_a @ B @ ( insert_a @ X @ C ) ) )
= ( ( ord_less_eq_set_a @ A @ ( minus_minus_set_a @ B @ C ) )
& ~ ( member_a @ X @ A ) ) ) ).
% subset_Diff_insert
thf(fact_728_Diff__triv,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
=> ( ( minus_minus_set_a @ A @ B )
= A ) ) ).
% Diff_triv
thf(fact_729_Int__Diff__disjoint,axiom,
! [A: set_a,B: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B ) @ ( minus_minus_set_a @ A @ B ) )
= bot_bot_set_a ) ).
% Int_Diff_disjoint
thf(fact_730_Diff__partition,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( sup_sup_set_a @ A @ ( minus_minus_set_a @ B @ A ) )
= B ) ) ).
% Diff_partition
thf(fact_731_Diff__subset__conv,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ B ) @ C )
= ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ B @ C ) ) ) ).
% Diff_subset_conv
thf(fact_732_Un__Diff__Int,axiom,
! [A: set_a,B: set_a] :
( ( sup_sup_set_a @ ( minus_minus_set_a @ A @ B ) @ ( inf_inf_set_a @ A @ B ) )
= A ) ).
% Un_Diff_Int
thf(fact_733_Int__Diff__Un,axiom,
! [A: set_a,B: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ A @ B ) @ ( minus_minus_set_a @ A @ B ) )
= A ) ).
% Int_Diff_Un
thf(fact_734_Diff__Int,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( minus_minus_set_a @ A @ ( inf_inf_set_a @ B @ C ) )
= ( sup_sup_set_a @ ( minus_minus_set_a @ A @ B ) @ ( minus_minus_set_a @ A @ C ) ) ) ).
% Diff_Int
thf(fact_735_Diff__Un,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( minus_minus_set_a @ A @ ( sup_sup_set_a @ B @ C ) )
= ( inf_inf_set_a @ ( minus_minus_set_a @ A @ B ) @ ( minus_minus_set_a @ A @ C ) ) ) ).
% Diff_Un
thf(fact_736_infinite__remove,axiom,
! [S: set_nat,A2: nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).
% infinite_remove
thf(fact_737_infinite__remove,axiom,
! [S: set_a,A2: a] :
( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ ( insert_a @ A2 @ bot_bot_set_a ) ) ) ) ).
% infinite_remove
thf(fact_738_infinite__coinduct,axiom,
! [X5: set_nat > $o,A: set_nat] :
( ( X5 @ A )
=> ( ! [A8: set_nat] :
( ( X5 @ A8 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A8 )
& ( ( X5 @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) )
| ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) )
=> ~ ( finite_finite_nat @ A ) ) ) ).
% infinite_coinduct
thf(fact_739_infinite__coinduct,axiom,
! [X5: set_a > $o,A: set_a] :
( ( X5 @ A )
=> ( ! [A8: set_a] :
( ( X5 @ A8 )
=> ? [X3: a] :
( ( member_a @ X3 @ A8 )
& ( ( X5 @ ( minus_minus_set_a @ A8 @ ( insert_a @ X3 @ bot_bot_set_a ) ) )
| ~ ( finite_finite_a @ ( minus_minus_set_a @ A8 @ ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ) )
=> ~ ( finite_finite_a @ A ) ) ) ).
% infinite_coinduct
thf(fact_740_finite__empty__induct,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( P @ A )
=> ( ! [A4: nat,A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ( member_nat @ A4 @ A8 )
=> ( ( P @ A8 )
=> ( P @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ A4 @ bot_bot_set_nat ) ) ) ) ) )
=> ( P @ bot_bot_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_741_finite__empty__induct,axiom,
! [A: set_a,P: set_a > $o] :
( ( finite_finite_a @ A )
=> ( ( P @ A )
=> ( ! [A4: a,A8: set_a] :
( ( finite_finite_a @ A8 )
=> ( ( member_a @ A4 @ A8 )
=> ( ( P @ A8 )
=> ( P @ ( minus_minus_set_a @ A8 @ ( insert_a @ A4 @ bot_bot_set_a ) ) ) ) ) )
=> ( P @ bot_bot_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_742_subset__insert__iff,axiom,
! [A: set_nat,X: nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X @ B ) )
= ( ( ( member_nat @ X @ A )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B ) )
& ( ~ ( member_nat @ X @ A )
=> ( ord_less_eq_set_nat @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_743_subset__insert__iff,axiom,
! [A: set_a,X: a,B: set_a] :
( ( ord_less_eq_set_a @ A @ ( insert_a @ X @ B ) )
= ( ( ( member_a @ X @ A )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ X @ bot_bot_set_a ) ) @ B ) )
& ( ~ ( member_a @ X @ A )
=> ( ord_less_eq_set_a @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_744_Diff__single__insert,axiom,
! [A: set_a,X: a,B: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ X @ bot_bot_set_a ) ) @ B )
=> ( ord_less_eq_set_a @ A @ ( insert_a @ X @ B ) ) ) ).
% Diff_single_insert
thf(fact_745_card__le__sym__Diff,axiom,
! [A: set_Product_unit,B: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( finite4290736615968046902t_unit @ B )
=> ( ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite410649719033368117t_unit @ B ) )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A @ B ) ) @ ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ B @ A ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_746_card__le__sym__Diff,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ B ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B @ A ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_747_card__le__sym__Diff,axiom,
! [A: set_a,B: set_a] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( minus_minus_set_a @ A @ B ) ) @ ( finite_card_a @ ( minus_minus_set_a @ B @ A ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_748_card__ge__0__finite,axiom,
! [A: set_Product_unit] :
( ( ord_less_nat @ zero_zero_nat @ ( finite410649719033368117t_unit @ A ) )
=> ( finite4290736615968046902t_unit @ A ) ) ).
% card_ge_0_finite
thf(fact_749_card__ge__0__finite,axiom,
! [A: set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ A ) )
=> ( finite_finite_a @ A ) ) ).
% card_ge_0_finite
thf(fact_750_card__ge__0__finite,axiom,
! [A: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A ) )
=> ( finite_finite_nat @ A ) ) ).
% card_ge_0_finite
thf(fact_751_inf__left__commute,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_752_inf_Oleft__commute,axiom,
! [B2: set_a,A2: set_a,C2: set_a] :
( ( inf_inf_set_a @ B2 @ ( inf_inf_set_a @ A2 @ C2 ) )
= ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C2 ) ) ) ).
% inf.left_commute
thf(fact_753_inf__commute,axiom,
( inf_inf_set_a
= ( ^ [X2: set_a,Y4: set_a] : ( inf_inf_set_a @ Y4 @ X2 ) ) ) ).
% inf_commute
thf(fact_754_inf_Ocommute,axiom,
( inf_inf_set_a
= ( ^ [A3: set_a,B3: set_a] : ( inf_inf_set_a @ B3 @ A3 ) ) ) ).
% inf.commute
thf(fact_755_inf__assoc,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z )
= ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_756_inf_Oassoc,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C2 )
= ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C2 ) ) ) ).
% inf.assoc
thf(fact_757_inf__sup__aci_I1_J,axiom,
( inf_inf_set_a
= ( ^ [X2: set_a,Y4: set_a] : ( inf_inf_set_a @ Y4 @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_758_inf__sup__aci_I2_J,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z )
= ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_759_inf__sup__aci_I3_J,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_760_inf__sup__aci_I4_J,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ X @ Y ) )
= ( inf_inf_set_a @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_761_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B2 ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_762_nat__le__linear,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
| ( ord_less_eq_nat @ N2 @ M2 ) ) ).
% nat_le_linear
thf(fact_763_le__antisym,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M2 )
=> ( M2 = N2 ) ) ) ).
% le_antisym
thf(fact_764_eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( M2 = N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% eq_imp_le
thf(fact_765_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_766_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_767_sup__left__commute,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ Y @ ( sup_sup_set_a @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_768_sup_Oleft__commute,axiom,
! [B2: set_a,A2: set_a,C2: set_a] :
( ( sup_sup_set_a @ B2 @ ( sup_sup_set_a @ A2 @ C2 ) )
= ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B2 @ C2 ) ) ) ).
% sup.left_commute
thf(fact_769_sup__commute,axiom,
( sup_sup_set_a
= ( ^ [X2: set_a,Y4: set_a] : ( sup_sup_set_a @ Y4 @ X2 ) ) ) ).
% sup_commute
thf(fact_770_sup_Ocommute,axiom,
( sup_sup_set_a
= ( ^ [A3: set_a,B3: set_a] : ( sup_sup_set_a @ B3 @ A3 ) ) ) ).
% sup.commute
thf(fact_771_sup__assoc,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z )
= ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_772_sup_Oassoc,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ C2 )
= ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B2 @ C2 ) ) ) ).
% sup.assoc
thf(fact_773_inf__sup__aci_I5_J,axiom,
( sup_sup_set_a
= ( ^ [X2: set_a,Y4: set_a] : ( sup_sup_set_a @ Y4 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_774_inf__sup__aci_I6_J,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z )
= ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_775_inf__sup__aci_I7_J,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ Y @ ( sup_sup_set_a @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_776_inf__sup__aci_I8_J,axiom,
! [X: set_a,Y: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ X @ Y ) )
= ( sup_sup_set_a @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_777_monoid_Oinverse_Ocong,axiom,
group_inverse_a = group_inverse_a ).
% monoid.inverse.cong
thf(fact_778_monoid_Oinvertible_Ocong,axiom,
group_invertible_a = group_invertible_a ).
% monoid.invertible.cong
thf(fact_779_subgroup__transitive,axiom,
! [K3: set_a,H2: set_a,Composition: a > a > a,Unit: a,G: set_a] :
( ( group_subgroup_a @ K3 @ H2 @ Composition @ Unit )
=> ( ( group_subgroup_a @ H2 @ G @ Composition @ Unit )
=> ( group_subgroup_a @ K3 @ G @ Composition @ Unit ) ) ) ).
% subgroup_transitive
thf(fact_780_remove__induct,axiom,
! [P: set_nat > $o,B: set_nat] :
( ( P @ bot_bot_set_nat )
=> ( ( ~ ( finite_finite_nat @ B )
=> ( P @ B ) )
=> ( ! [A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ( A8 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A8 @ B )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A8 )
=> ( P @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_781_remove__induct,axiom,
! [P: set_a > $o,B: set_a] :
( ( P @ bot_bot_set_a )
=> ( ( ~ ( finite_finite_a @ B )
=> ( P @ B ) )
=> ( ! [A8: set_a] :
( ( finite_finite_a @ A8 )
=> ( ( A8 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A8 @ B )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A8 )
=> ( P @ ( minus_minus_set_a @ A8 @ ( insert_a @ X3 @ bot_bot_set_a ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_782_finite__remove__induct,axiom,
! [B: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ B )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ( A8 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A8 @ B )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A8 )
=> ( P @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_783_finite__remove__induct,axiom,
! [B: set_a,P: set_a > $o] :
( ( finite_finite_a @ B )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A8: set_a] :
( ( finite_finite_a @ A8 )
=> ( ( A8 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A8 @ B )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A8 )
=> ( P @ ( minus_minus_set_a @ A8 @ ( insert_a @ X3 @ bot_bot_set_a ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_784_card__Diff1__le,axiom,
! [A: set_Product_unit,X: product_unit] : ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) @ ( finite410649719033368117t_unit @ A ) ) ).
% card_Diff1_le
thf(fact_785_card__Diff1__le,axiom,
! [A: set_a,X: a] : ( ord_less_eq_nat @ ( finite_card_a @ ( minus_minus_set_a @ A @ ( insert_a @ X @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A ) ) ).
% card_Diff1_le
thf(fact_786_card__gt__0__iff,axiom,
! [A: set_Product_unit] :
( ( ord_less_nat @ zero_zero_nat @ ( finite410649719033368117t_unit @ A ) )
= ( ( A != bot_bo3957492148770167129t_unit )
& ( finite4290736615968046902t_unit @ A ) ) ) ).
% card_gt_0_iff
thf(fact_787_card__gt__0__iff,axiom,
! [A: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A ) )
= ( ( A != bot_bot_set_nat )
& ( finite_finite_nat @ A ) ) ) ).
% card_gt_0_iff
thf(fact_788_card__gt__0__iff,axiom,
! [A: set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ A ) )
= ( ( A != bot_bot_set_a )
& ( finite_finite_a @ A ) ) ) ).
% card_gt_0_iff
thf(fact_789_additive__abelian__group_Osumsetdiff__sing,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a,X: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ ( minus_minus_set_a @ A @ B ) @ ( insert_a @ X @ bot_bot_set_a ) )
= ( minus_minus_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ ( insert_a @ X @ bot_bot_set_a ) ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ B @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% additive_abelian_group.sumsetdiff_sing
thf(fact_790_monoid_OUnits_Ocong,axiom,
group_Units_a = group_Units_a ).
% monoid.Units.cong
thf(fact_791_additive__abelian__group_Osumset__iterated__empty,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,R: nat] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ord_less_nat @ zero_zero_nat @ R )
=> ( ( pluenn1960970773371692859ated_a @ G @ Addition @ Zero @ bot_bot_set_a @ R )
= bot_bot_set_a ) ) ) ).
% additive_abelian_group.sumset_iterated_empty
thf(fact_792_commutative__monoid_Ocommutative,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,X: nat,Y: nat] :
( ( group_6791354081887936081id_nat @ M @ Composition @ Unit )
=> ( ( member_nat @ X @ M )
=> ( ( member_nat @ Y @ M )
=> ( ( Composition @ X @ Y )
= ( Composition @ Y @ X ) ) ) ) ) ).
% commutative_monoid.commutative
thf(fact_793_commutative__monoid_Ocommutative,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,X: a,Y: a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( member_a @ X @ M )
=> ( ( member_a @ Y @ M )
=> ( ( Composition @ X @ Y )
= ( Composition @ Y @ X ) ) ) ) ) ).
% commutative_monoid.commutative
thf(fact_794_inf_OcoboundedI2,axiom,
! [B2: set_a,C2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_795_inf_OcoboundedI2,axiom,
! [B2: nat,C2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_796_inf_OcoboundedI1,axiom,
! [A2: set_a,C2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_797_inf_OcoboundedI1,axiom,
! [A2: nat,C2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ C2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_798_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A3: set_a] :
( ( inf_inf_set_a @ A3 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_799_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( ( inf_inf_nat @ A3 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_800_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( inf_inf_set_a @ A3 @ B3 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_801_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( ( inf_inf_nat @ A3 @ B3 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_802_inf_Ocobounded2,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_803_inf_Ocobounded2,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_804_inf_Ocobounded1,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_805_inf_Ocobounded1,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_806_inf_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
( A3
= ( inf_inf_set_a @ A3 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_807_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( A3
= ( inf_inf_nat @ A3 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_808_inf__greatest,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ X @ Z )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_809_inf__greatest,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Z )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_810_inf_OboundedI,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_811_inf_OboundedI,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ A2 @ C2 )
=> ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_812_inf_OboundedE,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C2 ) )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ~ ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% inf.boundedE
thf(fact_813_inf_OboundedE,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C2 ) )
=> ~ ( ( ord_less_eq_nat @ A2 @ B2 )
=> ~ ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% inf.boundedE
thf(fact_814_inf__absorb2,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( inf_inf_set_a @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_815_inf__absorb2,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( inf_inf_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_816_inf__absorb1,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( inf_inf_set_a @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_817_inf__absorb1,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( inf_inf_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_818_inf_Oabsorb2,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_819_inf_Oabsorb2,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( inf_inf_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_820_inf_Oabsorb1,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_821_inf_Oabsorb1,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( inf_inf_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_822_le__iff__inf,axiom,
( ord_less_eq_set_a
= ( ^ [X2: set_a,Y4: set_a] :
( ( inf_inf_set_a @ X2 @ Y4 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_823_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y4: nat] :
( ( inf_inf_nat @ X2 @ Y4 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_824_inf__unique,axiom,
! [F: set_a > set_a > set_a,X: set_a,Y: set_a] :
( ! [X4: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( F @ X4 @ Y2 ) @ X4 )
=> ( ! [X4: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( F @ X4 @ Y2 ) @ Y2 )
=> ( ! [X4: set_a,Y2: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y2 )
=> ( ( ord_less_eq_set_a @ X4 @ Z3 )
=> ( ord_less_eq_set_a @ X4 @ ( F @ Y2 @ Z3 ) ) ) )
=> ( ( inf_inf_set_a @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_825_inf__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X4: nat,Y2: nat] : ( ord_less_eq_nat @ ( F @ X4 @ Y2 ) @ X4 )
=> ( ! [X4: nat,Y2: nat] : ( ord_less_eq_nat @ ( F @ X4 @ Y2 ) @ Y2 )
=> ( ! [X4: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ( ord_less_eq_nat @ X4 @ Z3 )
=> ( ord_less_eq_nat @ X4 @ ( F @ Y2 @ Z3 ) ) ) )
=> ( ( inf_inf_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_826_inf_OorderI,axiom,
! [A2: set_a,B2: set_a] :
( ( A2
= ( inf_inf_set_a @ A2 @ B2 ) )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_827_inf_OorderI,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( inf_inf_nat @ A2 @ B2 ) )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_828_inf_OorderE,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( A2
= ( inf_inf_set_a @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_829_inf_OorderE,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2
= ( inf_inf_nat @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_830_le__infI2,axiom,
! [B2: set_a,X: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_831_le__infI2,axiom,
! [B2: nat,X: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_832_le__infI1,axiom,
! [A2: set_a,X: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_833_le__infI1,axiom,
! [A2: nat,X: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_834_inf__mono,axiom,
! [A2: set_a,C2: set_a,B2: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( ord_less_eq_set_a @ B2 @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ ( inf_inf_set_a @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_835_inf__mono,axiom,
! [A2: nat,C2: nat,B2: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ C2 )
=> ( ( ord_less_eq_nat @ B2 @ D )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ ( inf_inf_nat @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_836_le__infI,axiom,
! [X: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X @ A2 )
=> ( ( ord_less_eq_set_a @ X @ B2 )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_837_le__infI,axiom,
! [X: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X @ A2 )
=> ( ( ord_less_eq_nat @ X @ B2 )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_838_le__infE,axiom,
! [X: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_set_a @ X @ A2 )
=> ~ ( ord_less_eq_set_a @ X @ B2 ) ) ) ).
% le_infE
thf(fact_839_le__infE,axiom,
! [X: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_nat @ X @ A2 )
=> ~ ( ord_less_eq_nat @ X @ B2 ) ) ) ).
% le_infE
thf(fact_840_inf__le2,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_841_inf__le2,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_842_inf__le1,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_843_inf__le1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_844_inf__sup__ord_I1_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_845_inf__sup__ord_I1_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_846_inf__sup__ord_I2_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_847_inf__sup__ord_I2_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_848_sup_OcoboundedI2,axiom,
! [C2: set_a,B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ C2 @ B2 )
=> ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_849_sup_OcoboundedI2,axiom,
! [C2: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ C2 @ B2 )
=> ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_850_sup_OcoboundedI1,axiom,
! [C2: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C2 @ A2 )
=> ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_851_sup_OcoboundedI1,axiom,
! [C2: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ C2 @ A2 )
=> ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_852_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( sup_sup_set_a @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_853_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( ( sup_sup_nat @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_854_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A3: set_a] :
( ( sup_sup_set_a @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_855_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( ( sup_sup_nat @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_856_sup_Ocobounded2,axiom,
! [B2: set_a,A2: set_a] : ( ord_less_eq_set_a @ B2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_857_sup_Ocobounded2,axiom,
! [B2: nat,A2: nat] : ( ord_less_eq_nat @ B2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_858_sup_Ocobounded1,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_859_sup_Ocobounded1,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ A2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_860_sup_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A3: set_a] :
( A3
= ( sup_sup_set_a @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_861_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( A3
= ( sup_sup_nat @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_862_sup_OboundedI,axiom,
! [B2: set_a,A2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( ord_less_eq_set_a @ C2 @ A2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C2 ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_863_sup_OboundedI,axiom,
! [B2: nat,A2: nat,C2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C2 @ A2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C2 ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_864_sup_OboundedE,axiom,
! [B2: set_a,C2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ~ ( ord_less_eq_set_a @ C2 @ A2 ) ) ) ).
% sup.boundedE
thf(fact_865_sup_OboundedE,axiom,
! [B2: nat,C2: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_less_eq_nat @ B2 @ A2 )
=> ~ ( ord_less_eq_nat @ C2 @ A2 ) ) ) ).
% sup.boundedE
thf(fact_866_sup__absorb2,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( sup_sup_set_a @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_867_sup__absorb2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( sup_sup_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_868_sup__absorb1,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( sup_sup_set_a @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_869_sup__absorb1,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( sup_sup_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_870_sup_Oabsorb2,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( sup_sup_set_a @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_871_sup_Oabsorb2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_872_sup_Oabsorb1,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( sup_sup_set_a @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_873_sup_Oabsorb1,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_874_sup__unique,axiom,
! [F: set_a > set_a > set_a,X: set_a,Y: set_a] :
( ! [X4: set_a,Y2: set_a] : ( ord_less_eq_set_a @ X4 @ ( F @ X4 @ Y2 ) )
=> ( ! [X4: set_a,Y2: set_a] : ( ord_less_eq_set_a @ Y2 @ ( F @ X4 @ Y2 ) )
=> ( ! [X4: set_a,Y2: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X4 )
=> ( ( ord_less_eq_set_a @ Z3 @ X4 )
=> ( ord_less_eq_set_a @ ( F @ Y2 @ Z3 ) @ X4 ) ) )
=> ( ( sup_sup_set_a @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_875_sup__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X4: nat,Y2: nat] : ( ord_less_eq_nat @ X4 @ ( F @ X4 @ Y2 ) )
=> ( ! [X4: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ ( F @ X4 @ Y2 ) )
=> ( ! [X4: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ Y2 @ X4 )
=> ( ( ord_less_eq_nat @ Z3 @ X4 )
=> ( ord_less_eq_nat @ ( F @ Y2 @ Z3 ) @ X4 ) ) )
=> ( ( sup_sup_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_876_sup_OorderI,axiom,
! [A2: set_a,B2: set_a] :
( ( A2
= ( sup_sup_set_a @ A2 @ B2 ) )
=> ( ord_less_eq_set_a @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_877_sup_OorderI,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( sup_sup_nat @ A2 @ B2 ) )
=> ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_878_sup_OorderE,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( A2
= ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_879_sup_OorderE,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2
= ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_880_le__iff__sup,axiom,
( ord_less_eq_set_a
= ( ^ [X2: set_a,Y4: set_a] :
( ( sup_sup_set_a @ X2 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_881_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y4: nat] :
( ( sup_sup_nat @ X2 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_882_sup__least,axiom,
! [Y: set_a,X: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( ord_less_eq_set_a @ Z @ X )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_883_sup__least,axiom,
! [Y: nat,X: nat,Z: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ Z @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_884_sup__mono,axiom,
! [A2: set_a,C2: set_a,B2: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( ord_less_eq_set_a @ B2 @ D )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ ( sup_sup_set_a @ C2 @ D ) ) ) ) ).
% sup_mono
thf(fact_885_sup__mono,axiom,
! [A2: nat,C2: nat,B2: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ C2 )
=> ( ( ord_less_eq_nat @ B2 @ D )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ ( sup_sup_nat @ C2 @ D ) ) ) ) ).
% sup_mono
thf(fact_886_sup_Omono,axiom,
! [C2: set_a,A2: set_a,D: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C2 @ A2 )
=> ( ( ord_less_eq_set_a @ D @ B2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ C2 @ D ) @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_887_sup_Omono,axiom,
! [C2: nat,A2: nat,D: nat,B2: nat] :
( ( ord_less_eq_nat @ C2 @ A2 )
=> ( ( ord_less_eq_nat @ D @ B2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C2 @ D ) @ ( sup_sup_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_888_le__supI2,axiom,
! [X: set_a,B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ X @ B2 )
=> ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_889_le__supI2,axiom,
! [X: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ X @ B2 )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_890_le__supI1,axiom,
! [X: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X @ A2 )
=> ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_891_le__supI1,axiom,
! [X: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X @ A2 )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_892_sup__ge2,axiom,
! [Y: set_a,X: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X @ Y ) ) ).
% sup_ge2
thf(fact_893_sup__ge2,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_894_sup__ge1,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ X @ Y ) ) ).
% sup_ge1
thf(fact_895_sup__ge1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_896_le__supI,axiom,
! [A2: set_a,X: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ X )
=> ( ( ord_less_eq_set_a @ B2 @ X )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_897_le__supI,axiom,
! [A2: nat,X: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ X )
=> ( ( ord_less_eq_nat @ B2 @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_898_le__supE,axiom,
! [A2: set_a,B2: set_a,X: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ X )
=> ~ ( ( ord_less_eq_set_a @ A2 @ X )
=> ~ ( ord_less_eq_set_a @ B2 @ X ) ) ) ).
% le_supE
thf(fact_899_le__supE,axiom,
! [A2: nat,B2: nat,X: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ X )
=> ~ ( ( ord_less_eq_nat @ A2 @ X )
=> ~ ( ord_less_eq_nat @ B2 @ X ) ) ) ).
% le_supE
thf(fact_900_inf__sup__ord_I3_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_901_inf__sup__ord_I3_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_902_inf__sup__ord_I4_J,axiom,
! [Y: set_a,X: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_903_inf__sup__ord_I4_J,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_904_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_905_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_906_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_907_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_908_sup__inf__distrib2,axiom,
! [Y: set_a,Z: set_a,X: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ Z ) @ X )
= ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ X ) @ ( sup_sup_set_a @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_909_sup__inf__distrib1,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_910_inf__sup__distrib2,axiom,
! [Y: set_a,Z: set_a,X: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ Z ) @ X )
= ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ X ) @ ( inf_inf_set_a @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_911_inf__sup__distrib1,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_912_distrib__imp2,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ! [X4: set_a,Y2: set_a,Z3: set_a] :
( ( sup_sup_set_a @ X4 @ ( inf_inf_set_a @ Y2 @ Z3 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X4 @ Y2 ) @ ( sup_sup_set_a @ X4 @ Z3 ) ) )
=> ( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_913_distrib__imp1,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ! [X4: set_a,Y2: set_a,Z3: set_a] :
( ( inf_inf_set_a @ X4 @ ( sup_sup_set_a @ Y2 @ Z3 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X4 @ Y2 ) @ ( inf_inf_set_a @ X4 @ Z3 ) ) )
=> ( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_914_subgroup_Osubgroup__inverse__equality,axiom,
! [G: set_nat,M: set_nat,Composition: nat > nat > nat,Unit: nat,U: nat] :
( ( group_subgroup_nat @ G @ M @ Composition @ Unit )
=> ( ( member_nat @ U @ G )
=> ( ( group_inverse_nat @ M @ Composition @ Unit @ U )
= ( group_inverse_nat @ G @ Composition @ Unit @ U ) ) ) ) ).
% subgroup.subgroup_inverse_equality
thf(fact_915_subgroup_Osubgroup__inverse__equality,axiom,
! [G: set_a,M: set_a,Composition: a > a > a,Unit: a,U: a] :
( ( group_subgroup_a @ G @ M @ Composition @ Unit )
=> ( ( member_a @ U @ G )
=> ( ( group_inverse_a @ M @ Composition @ Unit @ U )
= ( group_inverse_a @ G @ Composition @ Unit @ U ) ) ) ) ).
% subgroup.subgroup_inverse_equality
thf(fact_916_group_Oinvertible,axiom,
! [G: set_nat,Composition: nat > nat > nat,Unit: nat,U: nat] :
( ( group_group_nat @ G @ Composition @ Unit )
=> ( ( member_nat @ U @ G )
=> ( group_invertible_nat @ G @ Composition @ Unit @ U ) ) ) ).
% group.invertible
thf(fact_917_group_Oinvertible,axiom,
! [G: set_a,Composition: a > a > a,Unit: a,U: a] :
( ( group_group_a @ G @ Composition @ Unit )
=> ( ( member_a @ U @ G )
=> ( group_invertible_a @ G @ Composition @ Unit @ U ) ) ) ).
% group.invertible
thf(fact_918_subgroup_Oaxioms_I2_J,axiom,
! [G: set_a,M: set_a,Composition: a > a > a,Unit: a] :
( ( group_subgroup_a @ G @ M @ Composition @ Unit )
=> ( group_group_a @ G @ Composition @ Unit ) ) ).
% subgroup.axioms(2)
thf(fact_919_abelian__group_Oaxioms_I1_J,axiom,
! [G: set_a,Composition: a > a > a,Unit: a] :
( ( group_201663378560352916roup_a @ G @ Composition @ Unit )
=> ( group_group_a @ G @ Composition @ Unit ) ) ).
% abelian_group.axioms(1)
thf(fact_920_abelian__group_Oaxioms_I2_J,axiom,
! [G: set_a,Composition: a > a > a,Unit: a] :
( ( group_201663378560352916roup_a @ G @ Composition @ Unit )
=> ( group_4866109990395492029noid_a @ G @ Composition @ Unit ) ) ).
% abelian_group.axioms(2)
thf(fact_921_distrib__inf__le,axiom,
! [X: set_a,Y: set_a,Z: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z ) ) @ ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_922_distrib__inf__le,axiom,
! [X: nat,Y: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z ) ) @ ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_923_distrib__sup__le,axiom,
! [X: set_a,Y: set_a,Z: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) @ ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_924_distrib__sup__le,axiom,
! [X: nat,Y: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_925_subgroup_Oimage__of__inverse,axiom,
! [G: set_nat,M: set_nat,Composition: nat > nat > nat,Unit: nat,X: nat] :
( ( group_subgroup_nat @ G @ M @ Composition @ Unit )
=> ( ( member_nat @ X @ G )
=> ( member_nat @ X @ ( image_nat_nat @ ( group_inverse_nat @ M @ Composition @ Unit ) @ G ) ) ) ) ).
% subgroup.image_of_inverse
thf(fact_926_subgroup_Oimage__of__inverse,axiom,
! [G: set_a,M: set_a,Composition: a > a > a,Unit: a,X: a] :
( ( group_subgroup_a @ G @ M @ Composition @ Unit )
=> ( ( member_a @ X @ G )
=> ( member_a @ X @ ( image_a_a @ ( group_inverse_a @ M @ Composition @ Unit ) @ G ) ) ) ) ).
% subgroup.image_of_inverse
thf(fact_927_subgroup_Osubgroup__inverse__iff,axiom,
! [G: set_nat,M: set_nat,Composition: nat > nat > nat,Unit: nat,X: nat] :
( ( group_subgroup_nat @ G @ M @ Composition @ Unit )
=> ( ( group_invertible_nat @ M @ Composition @ Unit @ X )
=> ( ( member_nat @ X @ M )
=> ( ( member_nat @ ( group_inverse_nat @ M @ Composition @ Unit @ X ) @ G )
= ( member_nat @ X @ G ) ) ) ) ) ).
% subgroup.subgroup_inverse_iff
thf(fact_928_subgroup_Osubgroup__inverse__iff,axiom,
! [G: set_a,M: set_a,Composition: a > a > a,Unit: a,X: a] :
( ( group_subgroup_a @ G @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ X )
=> ( ( member_a @ X @ M )
=> ( ( member_a @ ( group_inverse_a @ M @ Composition @ Unit @ X ) @ G )
= ( member_a @ X @ G ) ) ) ) ) ).
% subgroup.subgroup_inverse_iff
thf(fact_929_abelian__group__def,axiom,
( group_201663378560352916roup_a
= ( ^ [G5: set_a,Composition2: a > a > a,Unit2: a] :
( ( group_group_a @ G5 @ Composition2 @ Unit2 )
& ( group_4866109990395492029noid_a @ G5 @ Composition2 @ Unit2 ) ) ) ) ).
% abelian_group_def
thf(fact_930_abelian__group_Ointro,axiom,
! [G: set_a,Composition: a > a > a,Unit: a] :
( ( group_group_a @ G @ Composition @ Unit )
=> ( ( group_4866109990395492029noid_a @ G @ Composition @ Unit )
=> ( group_201663378560352916roup_a @ G @ Composition @ Unit ) ) ) ).
% abelian_group.intro
thf(fact_931_group_Oinverse__subgroupI,axiom,
! [G: set_nat,Composition: nat > nat > nat,Unit: nat,H2: set_nat] :
( ( group_group_nat @ G @ Composition @ Unit )
=> ( ( group_subgroup_nat @ H2 @ G @ Composition @ Unit )
=> ( group_subgroup_nat @ ( image_nat_nat @ ( group_inverse_nat @ G @ Composition @ Unit ) @ H2 ) @ G @ Composition @ Unit ) ) ) ).
% group.inverse_subgroupI
thf(fact_932_group_Oinverse__subgroupI,axiom,
! [G: set_a,Composition: a > a > a,Unit: a,H2: set_a] :
( ( group_group_a @ G @ Composition @ Unit )
=> ( ( group_subgroup_a @ H2 @ G @ Composition @ Unit )
=> ( group_subgroup_a @ ( image_a_a @ ( group_inverse_a @ G @ Composition @ Unit ) @ H2 ) @ G @ Composition @ Unit ) ) ) ).
% group.inverse_subgroupI
thf(fact_933_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ bot_bot_set_a )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_934_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_935_psubsetI,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_a @ A @ B ) ) ) ).
% psubsetI
thf(fact_936_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_937_diff__0__eq__0,axiom,
! [N2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_938_diff__diff__cancel,axiom,
! [I: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_939_le__zero__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_940_not__gr__zero,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_941_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ A2 )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_942_diff__zero,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% diff_zero
thf(fact_943_zero__diff,axiom,
! [A2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% zero_diff
thf(fact_944_zero__less__diff,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% zero_less_diff
thf(fact_945_diff__is__0__eq,axiom,
! [M2: nat,N2: nat] :
( ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% diff_is_0_eq
thf(fact_946_diff__is__0__eq_H,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_947_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_948_diffs0__imp__equal,axiom,
! [M2: nat,N2: nat] :
( ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N2 @ M2 )
= zero_zero_nat )
=> ( M2 = N2 ) ) ) ).
% diffs0_imp_equal
thf(fact_949_eq__diff__iff,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ( minus_minus_nat @ M2 @ K )
= ( minus_minus_nat @ N2 @ K ) )
= ( M2 = N2 ) ) ) ) ).
% eq_diff_iff
thf(fact_950_le__diff__iff,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ) ) ).
% le_diff_iff
thf(fact_951_Nat_Odiff__diff__eq,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_952_diff__le__mono,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).
% diff_le_mono
thf(fact_953_diff__le__self,axiom,
! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N2 ) @ M2 ) ).
% diff_le_self
thf(fact_954_le__diff__iff_H,axiom,
! [A2: nat,C2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ C2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A2 ) @ ( minus_minus_nat @ C2 @ B2 ) )
= ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_955_diff__le__mono2,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_956_not__psubset__empty,axiom,
! [A: set_a] :
~ ( ord_less_set_a @ A @ bot_bot_set_a ) ).
% not_psubset_empty
thf(fact_957_finite__psubset__induct,axiom,
! [A: set_a,P: set_a > $o] :
( ( finite_finite_a @ A )
=> ( ! [A8: set_a] :
( ( finite_finite_a @ A8 )
=> ( ! [B9: set_a] :
( ( ord_less_set_a @ B9 @ A8 )
=> ( P @ B9 ) )
=> ( P @ A8 ) ) )
=> ( P @ A ) ) ) ).
% finite_psubset_induct
thf(fact_958_finite__psubset__induct,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ! [A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ! [B9: set_nat] :
( ( ord_less_set_nat @ B9 @ A8 )
=> ( P @ B9 ) )
=> ( P @ A8 ) ) )
=> ( P @ A ) ) ) ).
% finite_psubset_induct
thf(fact_959_psubsetE,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ~ ( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ B @ A ) ) ) ).
% psubsetE
thf(fact_960_psubset__eq,axiom,
( ord_less_set_a
= ( ^ [A6: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A6 @ B6 )
& ( A6 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_961_psubset__imp__subset,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% psubset_imp_subset
thf(fact_962_psubset__subset__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% psubset_subset_trans
thf(fact_963_subset__not__subset__eq,axiom,
( ord_less_set_a
= ( ^ [A6: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A6 @ B6 )
& ~ ( ord_less_eq_set_a @ B6 @ A6 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_964_subset__psubset__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_set_a @ B @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% subset_psubset_trans
thf(fact_965_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
( ( ord_less_set_a @ A6 @ B6 )
| ( A6 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_966_psubset__imp__ex__mem,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ? [B4: nat] : ( member_nat @ B4 @ ( minus_minus_set_nat @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_967_psubset__imp__ex__mem,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ? [B4: a] : ( member_a @ B4 @ ( minus_minus_set_a @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_968_diff__less,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( minus_minus_nat @ M2 @ N2 ) @ M2 ) ) ) ).
% diff_less
thf(fact_969_diff__less__mono,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C2 @ A2 )
=> ( ord_less_nat @ ( minus_minus_nat @ A2 @ C2 ) @ ( minus_minus_nat @ B2 @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_970_less__diff__iff,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_nat @ M2 @ N2 ) ) ) ) ).
% less_diff_iff
thf(fact_971_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_972_psubset__card__mono,axiom,
! [B: set_Product_unit,A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ B )
=> ( ( ord_le8056459307392131481t_unit @ A @ B )
=> ( ord_less_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite410649719033368117t_unit @ B ) ) ) ) ).
% psubset_card_mono
thf(fact_973_psubset__card__mono,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_set_a @ A @ B )
=> ( ord_less_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B ) ) ) ) ).
% psubset_card_mono
thf(fact_974_psubset__card__mono,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_set_nat @ A @ B )
=> ( ord_less_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ).
% psubset_card_mono
thf(fact_975_boolean__algebra__cancel_Oinf1,axiom,
! [A: set_a,K: set_a,A2: set_a,B2: set_a] :
( ( A
= ( inf_inf_set_a @ K @ A2 ) )
=> ( ( inf_inf_set_a @ A @ B2 )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_976_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_a,K: set_a,B2: set_a,A2: set_a] :
( ( B
= ( inf_inf_set_a @ K @ B2 ) )
=> ( ( inf_inf_set_a @ A2 @ B )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_977_boolean__algebra__cancel_Osup1,axiom,
! [A: set_a,K: set_a,A2: set_a,B2: set_a] :
( ( A
= ( sup_sup_set_a @ K @ A2 ) )
=> ( ( sup_sup_set_a @ A @ B2 )
= ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_978_boolean__algebra__cancel_Osup2,axiom,
! [B: set_a,K: set_a,B2: set_a,A2: set_a] :
( ( B
= ( sup_sup_set_a @ K @ B2 ) )
=> ( ( sup_sup_set_a @ A2 @ B )
= ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_979_card__Diff__subset,axiom,
! [B: set_Product_unit,A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ B )
=> ( ( ord_le3507040750410214029t_unit @ B @ A )
=> ( ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A @ B ) )
= ( minus_minus_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite410649719033368117t_unit @ B ) ) ) ) ) ).
% card_Diff_subset
thf(fact_980_card__Diff__subset,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A @ B ) )
= ( minus_minus_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_Diff_subset
thf(fact_981_card__Diff__subset,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A @ B ) )
= ( minus_minus_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B ) ) ) ) ) ).
% card_Diff_subset
thf(fact_982_diff__card__le__card__Diff,axiom,
! [B: set_Product_unit,A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ B )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite410649719033368117t_unit @ B ) ) @ ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A @ B ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_983_diff__card__le__card__Diff,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ B ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_984_diff__card__le__card__Diff,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B ) ) @ ( finite_card_a @ ( minus_minus_set_a @ A @ B ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_985_finite__induct__select,axiom,
! [S: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [T3: set_nat] :
( ( ord_less_set_nat @ T3 @ S )
=> ( ( P @ T3 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ S @ T3 ) )
& ( P @ ( insert_nat @ X3 @ T3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_986_finite__induct__select,axiom,
! [S: set_a,P: set_a > $o] :
( ( finite_finite_a @ S )
=> ( ( P @ bot_bot_set_a )
=> ( ! [T3: set_a] :
( ( ord_less_set_a @ T3 @ S )
=> ( ( P @ T3 )
=> ? [X3: a] :
( ( member_a @ X3 @ ( minus_minus_set_a @ S @ T3 ) )
& ( P @ ( insert_a @ X3 @ T3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_987_psubset__insert__iff,axiom,
! [A: set_nat,X: nat,B: set_nat] :
( ( ord_less_set_nat @ A @ ( insert_nat @ X @ B ) )
= ( ( ( member_nat @ X @ B )
=> ( ord_less_set_nat @ A @ B ) )
& ( ~ ( member_nat @ X @ B )
=> ( ( ( member_nat @ X @ A )
=> ( ord_less_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B ) )
& ( ~ ( member_nat @ X @ A )
=> ( ord_less_eq_set_nat @ A @ B ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_988_psubset__insert__iff,axiom,
! [A: set_a,X: a,B: set_a] :
( ( ord_less_set_a @ A @ ( insert_a @ X @ B ) )
= ( ( ( member_a @ X @ B )
=> ( ord_less_set_a @ A @ B ) )
& ( ~ ( member_a @ X @ B )
=> ( ( ( member_a @ X @ A )
=> ( ord_less_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ X @ bot_bot_set_a ) ) @ B ) )
& ( ~ ( member_a @ X @ A )
=> ( ord_less_eq_set_a @ A @ B ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_989_card__psubset,axiom,
! [B: set_Product_unit,A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ B )
=> ( ( ord_le3507040750410214029t_unit @ A @ B )
=> ( ( ord_less_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite410649719033368117t_unit @ B ) )
=> ( ord_le8056459307392131481t_unit @ A @ B ) ) ) ) ).
% card_psubset
thf(fact_990_card__psubset,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) )
=> ( ord_less_set_nat @ A @ B ) ) ) ) ).
% card_psubset
thf(fact_991_card__psubset,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B ) )
=> ( ord_less_set_a @ A @ B ) ) ) ) ).
% card_psubset
thf(fact_992_card__Diff__subset__Int,axiom,
! [A: set_Product_unit,B: set_Product_unit] :
( ( finite4290736615968046902t_unit @ ( inf_in4660618365625256667t_unit @ A @ B ) )
=> ( ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A @ B ) )
= ( minus_minus_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite410649719033368117t_unit @ ( inf_in4660618365625256667t_unit @ A @ B ) ) ) ) ) ).
% card_Diff_subset_Int
thf(fact_993_card__Diff__subset__Int,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ ( inf_inf_set_nat @ A @ B ) )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A @ B ) )
= ( minus_minus_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ ( inf_inf_set_nat @ A @ B ) ) ) ) ) ).
% card_Diff_subset_Int
thf(fact_994_card__Diff__subset__Int,axiom,
! [A: set_a,B: set_a] :
( ( finite_finite_a @ ( inf_inf_set_a @ A @ B ) )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A @ B ) )
= ( minus_minus_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ ( inf_inf_set_a @ A @ B ) ) ) ) ) ).
% card_Diff_subset_Int
thf(fact_995_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_996_zero__less__iff__neq__zero,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( N2 != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_997_gr__implies__not__zero,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_998_not__less__zero,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less_zero
thf(fact_999_gr__zeroI,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr_zeroI
thf(fact_1000_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ bot_bot_set_a )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_1001_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_1002_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_1003_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_a,Z: set_a,X: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ Z ) @ X )
= ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ X ) @ ( inf_inf_set_a @ Z @ X ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_1004_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_a,Z: set_a,X: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ Z ) @ X )
= ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ X ) @ ( sup_sup_set_a @ Z @ X ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_1005_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_1006_sumset__iterated__r,axiom,
! [R: nat,A: set_a] :
( ( ord_less_nat @ zero_zero_nat @ R )
=> ( ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ A @ R )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ A @ ( minus_minus_nat @ R @ one_one_nat ) ) ) ) ) ).
% sumset_iterated_r
thf(fact_1007_finite__linorder__min__induct,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B4: nat,A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A8 )
=> ( ord_less_nat @ B4 @ X3 ) )
=> ( ( P @ A8 )
=> ( P @ ( insert_nat @ B4 @ A8 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_1008_dual__order_Orefl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_1009_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_1010_order__refl,axiom,
! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% order_refl
thf(fact_1011_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_1012_less__one,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ one_one_nat )
= ( N2 = zero_zero_nat ) ) ).
% less_one
thf(fact_1013_card__Diff__insert,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ~ ( member_nat @ A2 @ B )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ B ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ B ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_1014_card__Diff__insert,axiom,
! [A2: product_unit,A: set_Product_unit,B: set_Product_unit] :
( ( member_Product_unit @ A2 @ A )
=> ( ~ ( member_Product_unit @ A2 @ B )
=> ( ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ A2 @ B ) ) )
= ( minus_minus_nat @ ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A @ B ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_1015_card__Diff__insert,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( member_a @ A2 @ A )
=> ( ~ ( member_a @ A2 @ B )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A @ ( insert_a @ A2 @ B ) ) )
= ( minus_minus_nat @ ( finite_card_a @ ( minus_minus_set_a @ A @ B ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_1016_psubsetD,axiom,
! [A: set_a,B: set_a,C2: a] :
( ( ord_less_set_a @ A @ B )
=> ( ( member_a @ C2 @ A )
=> ( member_a @ C2 @ B ) ) ) ).
% psubsetD
thf(fact_1017_psubsetD,axiom,
! [A: set_nat,B: set_nat,C2: nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( member_nat @ C2 @ A )
=> ( member_nat @ C2 @ B ) ) ) ).
% psubsetD
thf(fact_1018_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_1019_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_1020_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_1021_ord__le__eq__subst,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X4: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1022_ord__le__eq__subst,axiom,
! [A2: set_a,B2: set_a,F: set_a > nat,C2: nat] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X4: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1023_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > set_a,C2: set_a] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1024_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1025_ord__eq__le__subst,axiom,
! [A2: set_a,F: set_a > set_a,B2: set_a,C2: set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ! [X4: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1026_ord__eq__le__subst,axiom,
! [A2: nat,F: set_a > nat,B2: set_a,C2: set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ! [X4: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1027_ord__eq__le__subst,axiom,
! [A2: set_a,F: nat > set_a,B2: nat,C2: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1028_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B2: nat,C2: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1029_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_1030_order__eq__refl,axiom,
! [X: set_a,Y: set_a] :
( ( X = Y )
=> ( ord_less_eq_set_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_1031_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_1032_order__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C2 )
=> ( ! [X4: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_1033_order__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > nat,C2: nat] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X4: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_1034_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > set_a,C2: set_a] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C2 )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_1035_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_1036_order__subst1,axiom,
! [A2: set_a,F: set_a > set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ! [X4: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_1037_order__subst1,axiom,
! [A2: set_a,F: nat > set_a,B2: nat,C2: nat] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_1038_order__subst1,axiom,
! [A2: nat,F: set_a > nat,B2: set_a,C2: set_a] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ! [X4: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_1039_order__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_1040_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_a,Z2: set_a] : ( Y3 = Z2 ) )
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
& ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_1041_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_1042_antisym,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_1043_antisym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_1044_dual__order_Otrans,axiom,
! [B2: set_a,A2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( ord_less_eq_set_a @ C2 @ B2 )
=> ( ord_less_eq_set_a @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_1045_dual__order_Otrans,axiom,
! [B2: nat,A2: nat,C2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C2 @ B2 )
=> ( ord_less_eq_nat @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_1046_dual__order_Oantisym,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_1047_dual__order_Oantisym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_1048_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_a,Z2: set_a] : ( Y3 = Z2 ) )
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A3 )
& ( ord_less_eq_set_a @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_1049_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_1050_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_1051_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_1052_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_1053_order_Otrans,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_1054_order_Otrans,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_1055_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_1056_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_1057_ord__le__eq__trans,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_1058_ord__le__eq__trans,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_1059_ord__eq__le__trans,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( A2 = B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_1060_ord__eq__le__trans,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_1061_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_a,Z2: set_a] : ( Y3 = Z2 ) )
= ( ^ [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
& ( ord_less_eq_set_a @ Y4 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_1062_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_1063_le__cases3,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z ) )
=> ( ( ( ord_less_eq_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_1064_nle__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_1065_card__1__singletonE,axiom,
! [A: set_Product_unit] :
( ( ( finite410649719033368117t_unit @ A )
= one_one_nat )
=> ~ ! [X4: product_unit] :
( A
!= ( insert_Product_unit @ X4 @ bot_bo3957492148770167129t_unit ) ) ) ).
% card_1_singletonE
thf(fact_1066_card__1__singletonE,axiom,
! [A: set_a] :
( ( ( finite_card_a @ A )
= one_one_nat )
=> ~ ! [X4: a] :
( A
!= ( insert_a @ X4 @ bot_bot_set_a ) ) ) ).
% card_1_singletonE
thf(fact_1067_card__Diff__singleton,axiom,
! [X: nat,A: set_nat] :
( ( member_nat @ X @ A )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ A ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_1068_card__Diff__singleton,axiom,
! [X: product_unit,A: set_Product_unit] :
( ( member_Product_unit @ X @ A )
=> ( ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) )
= ( minus_minus_nat @ ( finite410649719033368117t_unit @ A ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_1069_card__Diff__singleton,axiom,
! [X: a,A: set_a] :
( ( member_a @ X @ A )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A @ ( insert_a @ X @ bot_bot_set_a ) ) )
= ( minus_minus_nat @ ( finite_card_a @ A ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_1070_card__Diff__singleton__if,axiom,
! [X: nat,A: set_nat] :
( ( ( member_nat @ X @ A )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ A ) @ one_one_nat ) ) )
& ( ~ ( member_nat @ X @ A )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
= ( finite_card_nat @ A ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_1071_card__Diff__singleton__if,axiom,
! [X: product_unit,A: set_Product_unit] :
( ( ( member_Product_unit @ X @ A )
=> ( ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) )
= ( minus_minus_nat @ ( finite410649719033368117t_unit @ A ) @ one_one_nat ) ) )
& ( ~ ( member_Product_unit @ X @ A )
=> ( ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) )
= ( finite410649719033368117t_unit @ A ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_1072_card__Diff__singleton__if,axiom,
! [X: a,A: set_a] :
( ( ( member_a @ X @ A )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A @ ( insert_a @ X @ bot_bot_set_a ) ) )
= ( minus_minus_nat @ ( finite_card_a @ A ) @ one_one_nat ) ) )
& ( ~ ( member_a @ X @ A )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A @ ( insert_a @ X @ bot_bot_set_a ) ) )
= ( finite_card_a @ A ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_1073_card__insert__le__m1,axiom,
! [N2: nat,Y: set_a,X: a] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ Y ) @ ( minus_minus_nat @ N2 @ one_one_nat ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( insert_a @ X @ Y ) ) @ N2 ) ) ) ).
% card_insert_le_m1
thf(fact_1074_card__insert__le__m1,axiom,
! [N2: nat,Y: set_Product_unit,X: product_unit] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ Y ) @ ( minus_minus_nat @ N2 @ one_one_nat ) )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ ( insert_Product_unit @ X @ Y ) ) @ N2 ) ) ) ).
% card_insert_le_m1
thf(fact_1075_additive__abelian__group_Osumset__iterated__r,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,R: nat,A: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ord_less_nat @ zero_zero_nat @ R )
=> ( ( pluenn1960970773371692859ated_a @ G @ Addition @ Zero @ A @ R )
= ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ ( pluenn1960970773371692859ated_a @ G @ Addition @ Zero @ A @ ( minus_minus_nat @ R @ one_one_nat ) ) ) ) ) ) ).
% additive_abelian_group.sumset_iterated_r
thf(fact_1076_order__le__imp__less__or__eq,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_set_a @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1077_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1078_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_1079_order__less__le__subst2,axiom,
! [A2: nat,B2: nat,F: nat > set_a,C2: set_a] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C2 )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ( ord_less_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_1080_order__less__le__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_1081_order__less__le__subst1,axiom,
! [A2: set_a,F: set_a > set_a,B2: set_a,C2: set_a] :
( ( ord_less_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ! [X4: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1082_order__less__le__subst1,axiom,
! [A2: nat,F: set_a > nat,B2: set_a,C2: set_a] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ! [X4: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1083_order__less__le__subst1,axiom,
! [A2: set_a,F: nat > set_a,B2: nat,C2: nat] :
( ( ord_less_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1084_order__less__le__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1085_order__le__less__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ ( F @ B2 ) @ C2 )
=> ( ! [X4: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1086_order__le__less__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > nat,C2: nat] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X4: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1087_order__le__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > set_a,C2: set_a] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_set_a @ ( F @ B2 ) @ C2 )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1088_order__le__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1089_order__le__less__subst1,axiom,
! [A2: set_a,F: nat > set_a,B2: nat,C2: nat] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ( ord_less_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1090_order__le__less__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1091_order__less__le__trans,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z )
=> ( ord_less_set_a @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_1092_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_1093_order__le__less__trans,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_set_a @ Y @ Z )
=> ( ord_less_set_a @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_1094_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_1095_order__neq__le__trans,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 != B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_set_a @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_1096_order__neq__le__trans,axiom,
! [A2: nat,B2: nat] :
( ( A2 != B2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_nat @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_1097_order__le__neq__trans,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_a @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_1098_order__le__neq__trans,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_nat @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_1099_order__less__imp__le,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( ord_less_eq_set_a @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_1100_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_1101_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_1102_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_1103_order__less__le,axiom,
( ord_less_set_a
= ( ^ [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
& ( X2 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_1104_order__less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
& ( X2 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_1105_order__le__less,axiom,
( ord_less_eq_set_a
= ( ^ [X2: set_a,Y4: set_a] :
( ( ord_less_set_a @ X2 @ Y4 )
| ( X2 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_1106_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y4: nat] :
( ( ord_less_nat @ X2 @ Y4 )
| ( X2 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_1107_dual__order_Ostrict__implies__order,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_set_a @ B2 @ A2 )
=> ( ord_less_eq_set_a @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_1108_dual__order_Ostrict__implies__order,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_1109_order_Ostrict__implies__order,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_1110_order_Ostrict__implies__order,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_1111_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [B3: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A3 )
& ~ ( ord_less_eq_set_a @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1112_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ~ ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1113_dual__order_Ostrict__trans2,axiom,
! [B2: set_a,A2: set_a,C2: set_a] :
( ( ord_less_set_a @ B2 @ A2 )
=> ( ( ord_less_eq_set_a @ C2 @ B2 )
=> ( ord_less_set_a @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_1114_dual__order_Ostrict__trans2,axiom,
! [B2: nat,A2: nat,C2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C2 @ B2 )
=> ( ord_less_nat @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_1115_dual__order_Ostrict__trans1,axiom,
! [B2: set_a,A2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( ord_less_set_a @ C2 @ B2 )
=> ( ord_less_set_a @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_1116_dual__order_Ostrict__trans1,axiom,
! [B2: nat,A2: nat,C2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_nat @ C2 @ B2 )
=> ( ord_less_nat @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_1117_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [B3: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1118_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1119_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A3: set_a] :
( ( ord_less_set_a @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1120_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_nat @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1121_order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
& ~ ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1122_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ~ ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1123_order_Ostrict__trans2,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_set_a @ A2 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_1124_order_Ostrict__trans2,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_1125_order_Ostrict__trans1,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ B2 @ C2 )
=> ( ord_less_set_a @ A2 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_1126_order_Ostrict__trans1,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_1127_order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1128_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1129_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_set_a @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1130_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1131_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_1132_less__le__not__le,axiom,
( ord_less_set_a
= ( ^ [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
& ~ ( ord_less_eq_set_a @ Y4 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_1133_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
& ~ ( ord_less_eq_nat @ Y4 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_1134_antisym__conv2,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ~ ( ord_less_set_a @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_1135_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_1136_antisym__conv1,axiom,
! [X: set_a,Y: set_a] :
( ~ ( ord_less_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_1137_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_1138_nless__le,axiom,
! [A2: set_a,B2: set_a] :
( ( ~ ( ord_less_set_a @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_set_a @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_1139_nless__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_nat @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_1140_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_1141_leD,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ~ ( ord_less_set_a @ X @ Y ) ) ).
% leD
thf(fact_1142_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_1143_bot_Oextremum,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% bot.extremum
thf(fact_1144_bot_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).
% bot.extremum
thf(fact_1145_bot_Oextremum__unique,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_1146_bot_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
= ( A2 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_1147_bot_Oextremum__uniqueI,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
=> ( A2 = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_1148_bot_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
=> ( A2 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_1149_bot_Onot__eq__extremum,axiom,
! [A2: set_a] :
( ( A2 != bot_bot_set_a )
= ( ord_less_set_a @ bot_bot_set_a @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_1150_bot_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_1151_bot_Oextremum__strict,axiom,
! [A2: set_a] :
~ ( ord_less_set_a @ A2 @ bot_bot_set_a ) ).
% bot.extremum_strict
thf(fact_1152_bot_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_1153_ex__min__if__finite,axiom,
! [S: set_nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ? [X4: nat] :
( ( member_nat @ X4 @ S )
& ~ ? [Xa: nat] :
( ( member_nat @ Xa @ S )
& ( ord_less_nat @ Xa @ X4 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_1154_infinite__growing,axiom,
! [X5: set_nat] :
( ( X5 != bot_bot_set_nat )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ X5 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ X5 )
& ( ord_less_nat @ X4 @ Xa ) ) )
=> ~ ( finite_finite_nat @ X5 ) ) ) ).
% infinite_growing
thf(fact_1155_finite__ranking__induct,axiom,
! [S: set_nat,P: set_nat > $o,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X4: nat,S2: set_nat] :
( ( finite_finite_nat @ S2 )
=> ( ! [Y5: nat] :
( ( member_nat @ Y5 @ S2 )
=> ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X4 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_nat @ X4 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1156_finite__ranking__induct,axiom,
! [S: set_a,P: set_a > $o,F: a > nat] :
( ( finite_finite_a @ S )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X4: a,S2: set_a] :
( ( finite_finite_a @ S2 )
=> ( ! [Y5: a] :
( ( member_a @ Y5 @ S2 )
=> ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X4 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_a @ X4 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1157_finite__linorder__max__induct,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B4: nat,A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A8 )
=> ( ord_less_nat @ X3 @ B4 ) )
=> ( ( P @ A8 )
=> ( P @ ( insert_nat @ B4 @ A8 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_1158_card__le__if__inj__on__rel,axiom,
! [B: set_Product_unit,A: set_nat,R: nat > product_unit > $o] :
( ( finite4290736615968046902t_unit @ B )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A )
=> ? [B10: product_unit] :
( ( member_Product_unit @ B10 @ B )
& ( R @ A4 @ B10 ) ) )
=> ( ! [A1: nat,A22: nat,B4: product_unit] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member_Product_unit @ B4 @ B )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite410649719033368117t_unit @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1159_card__le__if__inj__on__rel,axiom,
! [B: set_Product_unit,A: set_a,R: a > product_unit > $o] :
( ( finite4290736615968046902t_unit @ B )
=> ( ! [A4: a] :
( ( member_a @ A4 @ A )
=> ? [B10: product_unit] :
( ( member_Product_unit @ B10 @ B )
& ( R @ A4 @ B10 ) ) )
=> ( ! [A1: a,A22: a,B4: product_unit] :
( ( member_a @ A1 @ A )
=> ( ( member_a @ A22 @ A )
=> ( ( member_Product_unit @ B4 @ B )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite410649719033368117t_unit @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1160_card__le__if__inj__on__rel,axiom,
! [B: set_Product_unit,A: set_Product_unit,R: product_unit > product_unit > $o] :
( ( finite4290736615968046902t_unit @ B )
=> ( ! [A4: product_unit] :
( ( member_Product_unit @ A4 @ A )
=> ? [B10: product_unit] :
( ( member_Product_unit @ B10 @ B )
& ( R @ A4 @ B10 ) ) )
=> ( ! [A1: product_unit,A22: product_unit,B4: product_unit] :
( ( member_Product_unit @ A1 @ A )
=> ( ( member_Product_unit @ A22 @ A )
=> ( ( member_Product_unit @ B4 @ B )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite410649719033368117t_unit @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1161_card__le__if__inj__on__rel,axiom,
! [B: set_a,A: set_nat,R: nat > a > $o] :
( ( finite_finite_a @ B )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A )
=> ? [B10: a] :
( ( member_a @ B10 @ B )
& ( R @ A4 @ B10 ) ) )
=> ( ! [A1: nat,A22: nat,B4: a] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member_a @ B4 @ B )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_a @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1162_card__le__if__inj__on__rel,axiom,
! [B: set_a,A: set_a,R: a > a > $o] :
( ( finite_finite_a @ B )
=> ( ! [A4: a] :
( ( member_a @ A4 @ A )
=> ? [B10: a] :
( ( member_a @ B10 @ B )
& ( R @ A4 @ B10 ) ) )
=> ( ! [A1: a,A22: a,B4: a] :
( ( member_a @ A1 @ A )
=> ( ( member_a @ A22 @ A )
=> ( ( member_a @ B4 @ B )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1163_card__le__if__inj__on__rel,axiom,
! [B: set_a,A: set_Product_unit,R: product_unit > a > $o] :
( ( finite_finite_a @ B )
=> ( ! [A4: product_unit] :
( ( member_Product_unit @ A4 @ A )
=> ? [B10: a] :
( ( member_a @ B10 @ B )
& ( R @ A4 @ B10 ) ) )
=> ( ! [A1: product_unit,A22: product_unit,B4: a] :
( ( member_Product_unit @ A1 @ A )
=> ( ( member_Product_unit @ A22 @ A )
=> ( ( member_a @ B4 @ B )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite_card_a @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1164_card__le__if__inj__on__rel,axiom,
! [B: set_nat,A: set_nat,R: nat > nat > $o] :
( ( finite_finite_nat @ B )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A )
=> ? [B10: nat] :
( ( member_nat @ B10 @ B )
& ( R @ A4 @ B10 ) ) )
=> ( ! [A1: nat,A22: nat,B4: nat] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member_nat @ B4 @ B )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1165_card__le__if__inj__on__rel,axiom,
! [B: set_nat,A: set_a,R: a > nat > $o] :
( ( finite_finite_nat @ B )
=> ( ! [A4: a] :
( ( member_a @ A4 @ A )
=> ? [B10: nat] :
( ( member_nat @ B10 @ B )
& ( R @ A4 @ B10 ) ) )
=> ( ! [A1: a,A22: a,B4: nat] :
( ( member_a @ A1 @ A )
=> ( ( member_a @ A22 @ A )
=> ( ( member_nat @ B4 @ B )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1166_card__le__if__inj__on__rel,axiom,
! [B: set_nat,A: set_Product_unit,R: product_unit > nat > $o] :
( ( finite_finite_nat @ B )
=> ( ! [A4: product_unit] :
( ( member_Product_unit @ A4 @ A )
=> ? [B10: nat] :
( ( member_nat @ B10 @ B )
& ( R @ A4 @ B10 ) ) )
=> ( ! [A1: product_unit,A22: product_unit,B4: nat] :
( ( member_Product_unit @ A1 @ A )
=> ( ( member_Product_unit @ A22 @ A )
=> ( ( member_nat @ B4 @ B )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1167_monoid__axioms,axiom,
group_monoid_a @ g @ addition @ zero ).
% monoid_axioms
thf(fact_1168_monoid_Oinverse__unit,axiom,
! [M: set_a,Composition: a > a > a,Unit: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_inverse_a @ M @ Composition @ Unit @ Unit )
= Unit ) ) ).
% monoid.inverse_unit
thf(fact_1169_monoid_Oinverse__equality,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,U: nat,V2: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( ( Composition @ U @ V2 )
= Unit )
=> ( ( ( Composition @ V2 @ U )
= Unit )
=> ( ( member_nat @ U @ M )
=> ( ( member_nat @ V2 @ M )
=> ( ( group_inverse_nat @ M @ Composition @ Unit @ U )
= V2 ) ) ) ) ) ) ).
% monoid.inverse_equality
thf(fact_1170_monoid_Oinverse__equality,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a,V2: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( ( Composition @ U @ V2 )
= Unit )
=> ( ( ( Composition @ V2 @ U )
= Unit )
=> ( ( member_a @ U @ M )
=> ( ( member_a @ V2 @ M )
=> ( ( group_inverse_a @ M @ Composition @ Unit @ U )
= V2 ) ) ) ) ) ) ).
% monoid.inverse_equality
thf(fact_1171_commutative__monoid_Oaxioms_I1_J,axiom,
! [M: set_a,Composition: a > a > a,Unit: a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( group_monoid_a @ M @ Composition @ Unit ) ) ).
% commutative_monoid.axioms(1)
thf(fact_1172_Group__Theory_Ogroup_Oaxioms_I1_J,axiom,
! [G: set_a,Composition: a > a > a,Unit: a] :
( ( group_group_a @ G @ Composition @ Unit )
=> ( group_monoid_a @ G @ Composition @ Unit ) ) ).
% Group_Theory.group.axioms(1)
thf(fact_1173_monoid_OinvertibleE,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,U: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( group_invertible_nat @ M @ Composition @ Unit @ U )
=> ( ! [V3: nat] :
( ( ( ( Composition @ U @ V3 )
= Unit )
& ( ( Composition @ V3 @ U )
= Unit ) )
=> ~ ( member_nat @ V3 @ M ) )
=> ~ ( member_nat @ U @ M ) ) ) ) ).
% monoid.invertibleE
thf(fact_1174_monoid_OinvertibleE,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ U )
=> ( ! [V3: a] :
( ( ( ( Composition @ U @ V3 )
= Unit )
& ( ( Composition @ V3 @ U )
= Unit ) )
=> ~ ( member_a @ V3 @ M ) )
=> ~ ( member_a @ U @ M ) ) ) ) ).
% monoid.invertibleE
thf(fact_1175_monoid_OinvertibleI,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,U: nat,V2: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( ( Composition @ U @ V2 )
= Unit )
=> ( ( ( Composition @ V2 @ U )
= Unit )
=> ( ( member_nat @ U @ M )
=> ( ( member_nat @ V2 @ M )
=> ( group_invertible_nat @ M @ Composition @ Unit @ U ) ) ) ) ) ) ).
% monoid.invertibleI
thf(fact_1176_monoid_OinvertibleI,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a,V2: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( ( Composition @ U @ V2 )
= Unit )
=> ( ( ( Composition @ V2 @ U )
= Unit )
=> ( ( member_a @ U @ M )
=> ( ( member_a @ V2 @ M )
=> ( group_invertible_a @ M @ Composition @ Unit @ U ) ) ) ) ) ) ).
% monoid.invertibleI
thf(fact_1177_monoid_Oinvertible__def,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,U: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( member_nat @ U @ M )
=> ( ( group_invertible_nat @ M @ Composition @ Unit @ U )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ M )
& ( ( Composition @ U @ X2 )
= Unit )
& ( ( Composition @ X2 @ U )
= Unit ) ) ) ) ) ) ).
% monoid.invertible_def
thf(fact_1178_monoid_Oinvertible__def,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( member_a @ U @ M )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ U )
= ( ? [X2: a] :
( ( member_a @ X2 @ M )
& ( ( Composition @ U @ X2 )
= Unit )
& ( ( Composition @ X2 @ U )
= Unit ) ) ) ) ) ) ).
% monoid.invertible_def
thf(fact_1179_monoid_Ounit__invertible,axiom,
! [M: set_a,Composition: a > a > a,Unit: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( group_invertible_a @ M @ Composition @ Unit @ Unit ) ) ).
% monoid.unit_invertible
thf(fact_1180_monoid_Ocomposition__invertible,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,X: nat,Y: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( group_invertible_nat @ M @ Composition @ Unit @ X )
=> ( ( group_invertible_nat @ M @ Composition @ Unit @ Y )
=> ( ( member_nat @ X @ M )
=> ( ( member_nat @ Y @ M )
=> ( group_invertible_nat @ M @ Composition @ Unit @ ( Composition @ X @ Y ) ) ) ) ) ) ) ).
% monoid.composition_invertible
thf(fact_1181_monoid_Ocomposition__invertible,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,X: a,Y: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ X )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ Y )
=> ( ( member_a @ X @ M )
=> ( ( member_a @ Y @ M )
=> ( group_invertible_a @ M @ Composition @ Unit @ ( Composition @ X @ Y ) ) ) ) ) ) ) ).
% monoid.composition_invertible
thf(fact_1182_monoid_Oinvertible__left__cancel,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,X: nat,Y: nat,Z: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( group_invertible_nat @ M @ Composition @ Unit @ X )
=> ( ( member_nat @ X @ M )
=> ( ( member_nat @ Y @ M )
=> ( ( member_nat @ Z @ M )
=> ( ( ( Composition @ X @ Y )
= ( Composition @ X @ Z ) )
= ( Y = Z ) ) ) ) ) ) ) ).
% monoid.invertible_left_cancel
thf(fact_1183_monoid_Oinvertible__left__cancel,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,X: a,Y: a,Z: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ X )
=> ( ( member_a @ X @ M )
=> ( ( member_a @ Y @ M )
=> ( ( member_a @ Z @ M )
=> ( ( ( Composition @ X @ Y )
= ( Composition @ X @ Z ) )
= ( Y = Z ) ) ) ) ) ) ) ).
% monoid.invertible_left_cancel
thf(fact_1184_monoid_Oinvertible__right__cancel,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,X: nat,Y: nat,Z: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( group_invertible_nat @ M @ Composition @ Unit @ X )
=> ( ( member_nat @ X @ M )
=> ( ( member_nat @ Y @ M )
=> ( ( member_nat @ Z @ M )
=> ( ( ( Composition @ Y @ X )
= ( Composition @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ) ) ).
% monoid.invertible_right_cancel
thf(fact_1185_monoid_Oinvertible__right__cancel,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,X: a,Y: a,Z: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ X )
=> ( ( member_a @ X @ M )
=> ( ( member_a @ Y @ M )
=> ( ( member_a @ Z @ M )
=> ( ( ( Composition @ Y @ X )
= ( Composition @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ) ) ).
% monoid.invertible_right_cancel
thf(fact_1186_monoid_Oinvertible__inverse__invertible,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,U: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( group_invertible_nat @ M @ Composition @ Unit @ U )
=> ( ( member_nat @ U @ M )
=> ( group_invertible_nat @ M @ Composition @ Unit @ ( group_inverse_nat @ M @ Composition @ Unit @ U ) ) ) ) ) ).
% monoid.invertible_inverse_invertible
thf(fact_1187_monoid_Oinvertible__inverse__invertible,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ U )
=> ( ( member_a @ U @ M )
=> ( group_invertible_a @ M @ Composition @ Unit @ ( group_inverse_a @ M @ Composition @ Unit @ U ) ) ) ) ) ).
% monoid.invertible_inverse_invertible
thf(fact_1188_monoid_Oinverse__composition__commute,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,X: nat,Y: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( group_invertible_nat @ M @ Composition @ Unit @ X )
=> ( ( group_invertible_nat @ M @ Composition @ Unit @ Y )
=> ( ( member_nat @ X @ M )
=> ( ( member_nat @ Y @ M )
=> ( ( group_inverse_nat @ M @ Composition @ Unit @ ( Composition @ X @ Y ) )
= ( Composition @ ( group_inverse_nat @ M @ Composition @ Unit @ Y ) @ ( group_inverse_nat @ M @ Composition @ Unit @ X ) ) ) ) ) ) ) ) ).
% monoid.inverse_composition_commute
thf(fact_1189_monoid_Oinverse__composition__commute,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,X: a,Y: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ X )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ Y )
=> ( ( member_a @ X @ M )
=> ( ( member_a @ Y @ M )
=> ( ( group_inverse_a @ M @ Composition @ Unit @ ( Composition @ X @ Y ) )
= ( Composition @ ( group_inverse_a @ M @ Composition @ Unit @ Y ) @ ( group_inverse_a @ M @ Composition @ Unit @ X ) ) ) ) ) ) ) ) ).
% monoid.inverse_composition_commute
thf(fact_1190_monoid_Oinvertible__inverse__inverse,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,U: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( group_invertible_nat @ M @ Composition @ Unit @ U )
=> ( ( member_nat @ U @ M )
=> ( ( group_inverse_nat @ M @ Composition @ Unit @ ( group_inverse_nat @ M @ Composition @ Unit @ U ) )
= U ) ) ) ) ).
% monoid.invertible_inverse_inverse
thf(fact_1191_monoid_Oinvertible__inverse__inverse,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ U )
=> ( ( member_a @ U @ M )
=> ( ( group_inverse_a @ M @ Composition @ Unit @ ( group_inverse_a @ M @ Composition @ Unit @ U ) )
= U ) ) ) ) ).
% monoid.invertible_inverse_inverse
thf(fact_1192_monoid_Oinvertible__right__inverse2,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,U: nat,V2: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( group_invertible_nat @ M @ Composition @ Unit @ U )
=> ( ( member_nat @ U @ M )
=> ( ( member_nat @ V2 @ M )
=> ( ( Composition @ U @ ( Composition @ ( group_inverse_nat @ M @ Composition @ Unit @ U ) @ V2 ) )
= V2 ) ) ) ) ) ).
% monoid.invertible_right_inverse2
thf(fact_1193_monoid_Oinvertible__right__inverse2,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a,V2: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ U )
=> ( ( member_a @ U @ M )
=> ( ( member_a @ V2 @ M )
=> ( ( Composition @ U @ ( Composition @ ( group_inverse_a @ M @ Composition @ Unit @ U ) @ V2 ) )
= V2 ) ) ) ) ) ).
% monoid.invertible_right_inverse2
thf(fact_1194_monoid_Oinvertible__inverse__closed,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,U: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( group_invertible_nat @ M @ Composition @ Unit @ U )
=> ( ( member_nat @ U @ M )
=> ( member_nat @ ( group_inverse_nat @ M @ Composition @ Unit @ U ) @ M ) ) ) ) ).
% monoid.invertible_inverse_closed
thf(fact_1195_monoid_Oinvertible__inverse__closed,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ U )
=> ( ( member_a @ U @ M )
=> ( member_a @ ( group_inverse_a @ M @ Composition @ Unit @ U ) @ M ) ) ) ) ).
% monoid.invertible_inverse_closed
thf(fact_1196_monoid_Oinvertible__right__inverse,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,U: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( group_invertible_nat @ M @ Composition @ Unit @ U )
=> ( ( member_nat @ U @ M )
=> ( ( Composition @ U @ ( group_inverse_nat @ M @ Composition @ Unit @ U ) )
= Unit ) ) ) ) ).
% monoid.invertible_right_inverse
thf(fact_1197_monoid_Oinvertible__right__inverse,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ U )
=> ( ( member_a @ U @ M )
=> ( ( Composition @ U @ ( group_inverse_a @ M @ Composition @ Unit @ U ) )
= Unit ) ) ) ) ).
% monoid.invertible_right_inverse
thf(fact_1198_monoid_Oinvertible__left__inverse2,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,U: nat,V2: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( group_invertible_nat @ M @ Composition @ Unit @ U )
=> ( ( member_nat @ U @ M )
=> ( ( member_nat @ V2 @ M )
=> ( ( Composition @ ( group_inverse_nat @ M @ Composition @ Unit @ U ) @ ( Composition @ U @ V2 ) )
= V2 ) ) ) ) ) ).
% monoid.invertible_left_inverse2
thf(fact_1199_monoid_Oinvertible__left__inverse2,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a,V2: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ U )
=> ( ( member_a @ U @ M )
=> ( ( member_a @ V2 @ M )
=> ( ( Composition @ ( group_inverse_a @ M @ Composition @ Unit @ U ) @ ( Composition @ U @ V2 ) )
= V2 ) ) ) ) ) ).
% monoid.invertible_left_inverse2
thf(fact_1200_monoid_Oinvertible__left__inverse,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,U: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( group_invertible_nat @ M @ Composition @ Unit @ U )
=> ( ( member_nat @ U @ M )
=> ( ( Composition @ ( group_inverse_nat @ M @ Composition @ Unit @ U ) @ U )
= Unit ) ) ) ) ).
% monoid.invertible_left_inverse
thf(fact_1201_monoid_Oinvertible__left__inverse,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ U )
=> ( ( member_a @ U @ M )
=> ( ( Composition @ ( group_inverse_a @ M @ Composition @ Unit @ U ) @ U )
= Unit ) ) ) ) ).
% monoid.invertible_left_inverse
thf(fact_1202_monoid_Omem__UnitsI,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,U: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( group_invertible_nat @ M @ Composition @ Unit @ U )
=> ( ( member_nat @ U @ M )
=> ( member_nat @ U @ ( group_Units_nat @ M @ Composition @ Unit ) ) ) ) ) ).
% monoid.mem_UnitsI
thf(fact_1203_monoid_Omem__UnitsI,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ U )
=> ( ( member_a @ U @ M )
=> ( member_a @ U @ ( group_Units_a @ M @ Composition @ Unit ) ) ) ) ) ).
% monoid.mem_UnitsI
thf(fact_1204_monoid_Omem__UnitsD,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,U: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( member_nat @ U @ ( group_Units_nat @ M @ Composition @ Unit ) )
=> ( ( group_invertible_nat @ M @ Composition @ Unit @ U )
& ( member_nat @ U @ M ) ) ) ) ).
% monoid.mem_UnitsD
thf(fact_1205_monoid_Omem__UnitsD,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( member_a @ U @ ( group_Units_a @ M @ Composition @ Unit ) )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ U )
& ( member_a @ U @ M ) ) ) ) ).
% monoid.mem_UnitsD
thf(fact_1206_monoid_Ogroup__of__Units,axiom,
! [M: set_a,Composition: a > a > a,Unit: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( group_group_a @ ( group_Units_a @ M @ Composition @ Unit ) @ Composition @ Unit ) ) ).
% monoid.group_of_Units
thf(fact_1207_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M: nat] :
( ( P @ X )
=> ( ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M ) )
=> ~ ! [M5: nat] :
( ( P @ M5 )
=> ~ ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M5 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1208_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M4: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ N5 )
=> ( ord_less_eq_nat @ X2 @ M4 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1209_monoid_OsubgroupI,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,G: set_nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( ord_less_eq_set_nat @ G @ M )
=> ( ( member_nat @ Unit @ G )
=> ( ! [G2: nat,H: nat] :
( ( member_nat @ G2 @ G )
=> ( ( member_nat @ H @ G )
=> ( member_nat @ ( Composition @ G2 @ H ) @ G ) ) )
=> ( ! [G2: nat] :
( ( member_nat @ G2 @ G )
=> ( group_invertible_nat @ M @ Composition @ Unit @ G2 ) )
=> ( ! [G2: nat] :
( ( member_nat @ G2 @ G )
=> ( member_nat @ ( group_inverse_nat @ M @ Composition @ Unit @ G2 ) @ G ) )
=> ( group_subgroup_nat @ G @ M @ Composition @ Unit ) ) ) ) ) ) ) ).
% monoid.subgroupI
thf(fact_1210_monoid_OsubgroupI,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,G: set_a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( ord_less_eq_set_a @ G @ M )
=> ( ( member_a @ Unit @ G )
=> ( ! [G2: a,H: a] :
( ( member_a @ G2 @ G )
=> ( ( member_a @ H @ G )
=> ( member_a @ ( Composition @ G2 @ H ) @ G ) ) )
=> ( ! [G2: a] :
( ( member_a @ G2 @ G )
=> ( group_invertible_a @ M @ Composition @ Unit @ G2 ) )
=> ( ! [G2: a] :
( ( member_a @ G2 @ G )
=> ( member_a @ ( group_inverse_a @ M @ Composition @ Unit @ G2 ) @ G ) )
=> ( group_subgroup_a @ G @ M @ Composition @ Unit ) ) ) ) ) ) ) ).
% monoid.subgroupI
thf(fact_1211_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_1212_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_1213_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_1214_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_1215_sumset__iterated__Suc,axiom,
! [A: set_a,K: nat] :
( ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ A @ ( suc @ K ) )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ A @ K ) ) ) ).
% sumset_iterated_Suc
thf(fact_1216_Suc__le__mono,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( suc @ M2 ) )
= ( ord_less_eq_nat @ N2 @ M2 ) ) ).
% Suc_le_mono
thf(fact_1217_less__Suc0,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ ( suc @ zero_zero_nat ) )
= ( N2 = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1218_zero__less__Suc,axiom,
! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N2 ) ) ).
% zero_less_Suc
thf(fact_1219_Suc__pred,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( suc @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) )
= N2 ) ) ).
% Suc_pred
thf(fact_1220_Suc__diff__1,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) )
= N2 ) ) ).
% Suc_diff_1
thf(fact_1221_not0__implies__Suc,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ? [M5: nat] :
( N2
= ( suc @ M5 ) ) ) ).
% not0_implies_Suc
thf(fact_1222_Zero__not__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_1223_Zero__neq__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_1224_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1225_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1226_diff__induct,axiom,
! [P: nat > nat > $o,M2: nat,N2: nat] :
( ! [X4: nat] : ( P @ X4 @ zero_zero_nat )
=> ( ! [Y2: nat] : ( P @ zero_zero_nat @ ( suc @ Y2 ) )
=> ( ! [X4: nat,Y2: nat] :
( ( P @ X4 @ Y2 )
=> ( P @ ( suc @ X4 ) @ ( suc @ Y2 ) ) )
=> ( P @ M2 @ N2 ) ) ) ) ).
% diff_induct
thf(fact_1227_nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N2 ) ) ) ).
% nat_induct
thf(fact_1228_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat: nat] :
( Y
!= ( suc @ Nat ) ) ) ).
% old.nat.exhaust
thf(fact_1229_nat_OdiscI,axiom,
! [Nat2: nat,X22: nat] :
( ( Nat2
= ( suc @ X22 ) )
=> ( Nat2 != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1230_old_Onat_Odistinct_I1_J,axiom,
! [Nat3: nat] :
( zero_zero_nat
!= ( suc @ Nat3 ) ) ).
% old.nat.distinct(1)
thf(fact_1231_old_Onat_Odistinct_I2_J,axiom,
! [Nat3: nat] :
( ( suc @ Nat3 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1232_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_1233_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1234_Suc__diff__le,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_eq_nat @ N2 @ M2 )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N2 )
= ( suc @ ( minus_minus_nat @ M2 @ N2 ) ) ) ) ).
% Suc_diff_le
thf(fact_1235_Suc__leI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 ) ) ).
% Suc_leI
thf(fact_1236_Suc__le__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_le_eq
thf(fact_1237_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_1238_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_1239_Suc__le__lessD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_le_lessD
thf(fact_1240_le__less__Suc__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M2 ) )
= ( N2 = M2 ) ) ) ).
% le_less_Suc_eq
thf(fact_1241_less__Suc__eq__le,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_Suc_eq_le
thf(fact_1242_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1243_le__imp__less__Suc,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_nat @ M2 @ ( suc @ N2 ) ) ) ).
% le_imp_less_Suc
thf(fact_1244_Ex__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N2 ) )
& ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
& ( P @ ( suc @ I4 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1245_gr0__conv__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( ? [M4: nat] :
( N2
= ( suc @ M4 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1246_All__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N2 ) )
=> ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ( P @ ( suc @ I4 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1247_gr0__implies__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ? [M5: nat] :
( N2
= ( suc @ M5 ) ) ) ).
% gr0_implies_Suc
thf(fact_1248_less__Suc__eq__0__disj,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
= ( ( M2 = zero_zero_nat )
| ? [J3: nat] :
( ( M2
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N2 ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1249_zero__notin__Suc__image,axiom,
! [A: set_nat] :
~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A ) ) ).
% zero_notin_Suc_image
thf(fact_1250_transitive__stepwise__le,axiom,
! [M2: nat,N2: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ! [X4: nat] : ( R2 @ X4 @ X4 )
=> ( ! [X4: nat,Y2: nat,Z3: nat] :
( ( R2 @ X4 @ Y2 )
=> ( ( R2 @ Y2 @ Z3 )
=> ( R2 @ X4 @ Z3 ) ) )
=> ( ! [N3: nat] : ( R2 @ N3 @ ( suc @ N3 ) )
=> ( R2 @ M2 @ N2 ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1251_nat__induct__at__least,axiom,
! [M2: nat,N2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( P @ M2 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_at_least
thf(fact_1252_full__nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N2 ) ) ).
% full_nat_induct
thf(fact_1253_not__less__eq__eq,axiom,
! [M2: nat,N2: nat] :
( ( ~ ( ord_less_eq_nat @ M2 @ N2 ) )
= ( ord_less_eq_nat @ ( suc @ N2 ) @ M2 ) ) ).
% not_less_eq_eq
thf(fact_1254_Suc__n__not__le__n,axiom,
! [N2: nat] :
~ ( ord_less_eq_nat @ ( suc @ N2 ) @ N2 ) ).
% Suc_n_not_le_n
thf(fact_1255_le__Suc__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
= ( ( ord_less_eq_nat @ M2 @ N2 )
| ( M2
= ( suc @ N2 ) ) ) ) ).
% le_Suc_eq
thf(fact_1256_Suc__le__D,axiom,
! [N2: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ M6 )
=> ? [M5: nat] :
( M6
= ( suc @ M5 ) ) ) ).
% Suc_le_D
thf(fact_1257_le__SucI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) ) ) ).
% le_SucI
thf(fact_1258_le__SucE,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
=> ( ~ ( ord_less_eq_nat @ M2 @ N2 )
=> ( M2
= ( suc @ N2 ) ) ) ) ).
% le_SucE
thf(fact_1259_Suc__leD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% Suc_leD
thf(fact_1260_ex__least__nat__less,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N2 )
& ! [I3: nat] :
( ( ord_less_eq_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1261_diff__Suc__less,axiom,
! [N2: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ ( minus_minus_nat @ N2 @ ( suc @ I ) ) @ N2 ) ) ).
% diff_Suc_less
thf(fact_1262_nat__induct__non__zero,axiom,
! [N2: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1263_Suc__diff__eq__diff__pred,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N2 )
= ( minus_minus_nat @ M2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1264_Suc__pred_H,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( N2
= ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_1265_inverse__undefined,axiom,
! [U: a] :
( ~ ( member_a @ U @ g )
=> ( ( group_inverse_a @ g @ addition @ zero @ U )
= undefined_a ) ) ).
% inverse_undefined
thf(fact_1266_nat__descend__induct,axiom,
! [N2: nat,P: nat > $o,M2: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N2 @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ! [I3: nat] :
( ( ord_less_nat @ K2 @ I3 )
=> ( P @ I3 ) )
=> ( P @ K2 ) ) )
=> ( P @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_1267_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_1: nat] : ( P @ X_1 )
=> ? [N3: nat] :
( ~ ( P @ N3 )
& ( P @ ( suc @ N3 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_1268_card__UNIV__unit,axiom,
( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
= one_one_nat ) ).
% card_UNIV_unit
thf(fact_1269_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_1270_add__is__0,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N2 = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1271_nat__add__left__cancel__le,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% nat_add_left_cancel_le
% Conjectures (1)
thf(conj_0,conjecture,
u1 = u2 ).
%------------------------------------------------------------------------------