TPTP Problem File: SLH0785^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Pluennecke_Ruzsa_Inequality/0003_Pluennecke_Ruzsa_Inequality/prob_00323_010950__12174690_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1352 ( 521 unt; 83 typ; 0 def)
% Number of atoms : 3565 (1100 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 11985 ( 391 ~; 64 |; 254 &;9606 @)
% ( 0 <=>;1670 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 7 avg)
% Number of types : 7 ( 6 usr)
% Number of type conns : 684 ( 684 >; 0 *; 0 +; 0 <<)
% Number of symbols : 78 ( 77 usr; 11 con; 0-5 aty)
% Number of variables : 3548 ( 155 ^;3301 !; 92 ?;3548 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:20:28.556
%------------------------------------------------------------------------------
% Could-be-implicit typings (6)
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__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__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 (77)
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > 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__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_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_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_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_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_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__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_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_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
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_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_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_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_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_001tf__a,type,
image_nat_a: ( nat > a ) > set_nat > 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_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_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__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_zero,type,
zero: a ).
% Relevant facts (1268)
thf(fact_0_commutative,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ g )
=> ( ( member_a @ Y @ g )
=> ( ( addition @ X @ Y )
= ( addition @ Y @ X ) ) ) ) ).
% commutative
thf(fact_1_sumset_Ocases,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( member_a @ A @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) )
=> ~ ! [A3: a,B2: a] :
( ( A
= ( addition @ A3 @ B2 ) )
=> ( ( member_a @ A3 @ A2 )
=> ( ( member_a @ A3 @ g )
=> ( ( member_a @ B2 @ B )
=> ~ ( member_a @ B2 @ g ) ) ) ) ) ) ).
% sumset.cases
thf(fact_2_sumset_Osimps,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( member_a @ A @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) )
= ( ? [A4: a,B3: a] :
( ( A
= ( addition @ A4 @ B3 ) )
& ( member_a @ A4 @ A2 )
& ( member_a @ A4 @ g )
& ( member_a @ B3 @ B )
& ( member_a @ B3 @ g ) ) ) ) ).
% sumset.simps
thf(fact_3_sumset_OsumsetI,axiom,
! [A: a,A2: set_a,B4: a,B: set_a] :
( ( member_a @ A @ A2 )
=> ( ( member_a @ A @ g )
=> ( ( member_a @ B4 @ B )
=> ( ( member_a @ B4 @ g )
=> ( member_a @ ( addition @ A @ B4 ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ) ) ) ) ).
% sumset.sumsetI
thf(fact_4_sumset__assoc,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) @ C )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( pluenn3038260743871226533mset_a @ g @ addition @ B @ C ) ) ) ).
% sumset_assoc
thf(fact_5_sumset__commute,axiom,
! [A2: set_a,B: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B )
= ( pluenn3038260743871226533mset_a @ g @ addition @ B @ A2 ) ) ).
% sumset_commute
thf(fact_6_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_7_minusset__distrib__sum,axiom,
! [A2: set_a,B: set_a] :
( ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) )
= ( pluenn3038260743871226533mset_a @ g @ addition @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ B ) ) ) ).
% minusset_distrib_sum
thf(fact_8_card__differenceset__commute,axiom,
! [B: set_a,A2: set_a] :
( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ B @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) ) )
= ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ B ) ) ) ) ).
% card_differenceset_commute
thf(fact_9_associative,axiom,
! [A: a,B4: a,C2: a] :
( ( member_a @ A @ g )
=> ( ( member_a @ B4 @ g )
=> ( ( member_a @ C2 @ g )
=> ( ( addition @ ( addition @ A @ B4 ) @ C2 )
= ( addition @ A @ ( addition @ B4 @ C2 ) ) ) ) ) ) ).
% associative
thf(fact_10_composition__closed,axiom,
! [A: a,B4: a] :
( ( member_a @ A @ g )
=> ( ( member_a @ B4 @ g )
=> ( member_a @ ( addition @ A @ B4 ) @ g ) ) ) ).
% composition_closed
thf(fact_11_unit__closed,axiom,
member_a @ zero @ g ).
% unit_closed
thf(fact_12_assms_I5_J,axiom,
finite_finite_a @ w ).
% assms(5)
thf(fact_13_assms_I3_J,axiom,
finite_finite_a @ v ).
% assms(3)
thf(fact_14_assms_I1_J,axiom,
finite_finite_a @ u ).
% assms(1)
thf(fact_15_assms_I6_J,axiom,
ord_less_eq_set_a @ w @ g ).
% assms(6)
thf(fact_16_assms_I4_J,axiom,
ord_less_eq_set_a @ v @ g ).
% assms(4)
thf(fact_17_assms_I2_J,axiom,
ord_less_eq_set_a @ u @ g ).
% assms(2)
thf(fact_18_left__unit,axiom,
! [A: a] :
( ( member_a @ A @ g )
=> ( ( addition @ zero @ A )
= A ) ) ).
% left_unit
thf(fact_19_right__unit,axiom,
! [A: a] :
( ( member_a @ A @ g )
=> ( ( addition @ A @ zero )
= A ) ) ).
% right_unit
thf(fact_20_differenceset__commute,axiom,
! [B: set_a,A2: set_a] :
( ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ ( pluenn3038260743871226533mset_a @ g @ addition @ B @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) ) )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ B ) ) ) ).
% differenceset_commute
thf(fact_21_additive__abelian__group__axioms,axiom,
pluenn1164192988769422572roup_a @ g @ addition @ zero ).
% additive_abelian_group_axioms
thf(fact_22_commutative__monoid__axioms,axiom,
group_4866109990395492029noid_a @ g @ addition @ zero ).
% commutative_monoid_axioms
thf(fact_23_additive__abelian__group_Osumset_Ocong,axiom,
pluenn3038260743871226533mset_a = pluenn3038260743871226533mset_a ).
% additive_abelian_group.sumset.cong
thf(fact_24_additive__abelian__group_Ominusset_Ocong,axiom,
pluenn2534204936789923946sset_a = pluenn2534204936789923946sset_a ).
% additive_abelian_group.minusset.cong
thf(fact_25_abelian__group__axioms,axiom,
group_201663378560352916roup_a @ g @ addition @ zero ).
% abelian_group_axioms
thf(fact_26_sumsetp_Ocases,axiom,
! [A2: a > $o,B: a > $o,A: a] :
( ( pluenn895083305082786853setp_a @ g @ addition @ A2 @ B @ A )
=> ~ ! [A3: a,B2: a] :
( ( A
= ( addition @ A3 @ B2 ) )
=> ( ( A2 @ A3 )
=> ( ( member_a @ A3 @ g )
=> ( ( B @ B2 )
=> ~ ( member_a @ B2 @ g ) ) ) ) ) ) ).
% sumsetp.cases
thf(fact_27_sumsetp_Osimps,axiom,
! [A2: a > $o,B: a > $o,A: a] :
( ( pluenn895083305082786853setp_a @ g @ addition @ A2 @ B @ A )
= ( ? [A4: a,B3: a] :
( ( A
= ( addition @ A4 @ B3 ) )
& ( A2 @ A4 )
& ( member_a @ A4 @ g )
& ( B @ B3 )
& ( member_a @ B3 @ g ) ) ) ) ).
% sumsetp.simps
thf(fact_28_sumsetp_OsumsetI,axiom,
! [A2: a > $o,A: a,B: a > $o,B4: a] :
( ( A2 @ A )
=> ( ( member_a @ A @ g )
=> ( ( B @ B4 )
=> ( ( member_a @ B4 @ g )
=> ( pluenn895083305082786853setp_a @ g @ addition @ A2 @ B @ ( addition @ A @ B4 ) ) ) ) ) ) ).
% sumsetp.sumsetI
thf(fact_29_card__sumset__iterated__minusset,axiom,
! [A2: set_a,K: nat] :
( ( finite_card_a @ ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) @ K ) )
= ( finite_card_a @ ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ A2 @ K ) ) ) ).
% card_sumset_iterated_minusset
thf(fact_30_minusset__iterated__minusset,axiom,
! [A2: set_a,K: nat] :
( ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) @ K )
= ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ A2 @ K ) ) ) ).
% minusset_iterated_minusset
thf(fact_31_card__minusset_H,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ g )
=> ( ( finite_card_a @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) )
= ( finite_card_a @ A2 ) ) ) ).
% card_minusset'
thf(fact_32_finite__differenceset,axiom,
! [A2: set_a,B: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ B ) ) ) ) ) ).
% finite_differenceset
thf(fact_33_minusset__subset__carrier,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) @ g ) ).
% minusset_subset_carrier
thf(fact_34_group__axioms,axiom,
group_group_a @ g @ addition @ zero ).
% group_axioms
thf(fact_35_finite__minusset,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) ) ) ).
% finite_minusset
thf(fact_36_finite__sumset,axiom,
! [A2: set_a,B: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ) ) ).
% finite_sumset
thf(fact_37_sumset__subset__carrier,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) @ g ) ).
% sumset_subset_carrier
thf(fact_38_sumset__mono,axiom,
! [A5: set_a,A2: set_a,B5: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A5 @ A2 )
=> ( ( ord_less_eq_set_a @ B5 @ B )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A5 @ B5 ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ) ) ).
% sumset_mono
thf(fact_39_finite__sumset__iterated,axiom,
! [A2: set_a,R: nat] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ A2 @ R ) ) ) ).
% finite_sumset_iterated
thf(fact_40_sumset__iterated__subset__carrier,axiom,
! [A2: set_a,K: nat] : ( ord_less_eq_set_a @ ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ A2 @ K ) @ g ) ).
% sumset_iterated_subset_carrier
thf(fact_41_card__le__sumset,axiom,
! [A2: set_a,A: a,B: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( member_a @ A @ A2 )
=> ( ( member_a @ A @ 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 @ A2 @ B ) ) ) ) ) ) ) ) ).
% card_le_sumset
thf(fact_42_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_43_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_44_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_47_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_48_additive__abelian__group_Osumsetp_Ocases,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: nat > $o,B: nat > $o,A: nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( pluenn5670965976768739049tp_nat @ G @ Addition @ A2 @ B @ A )
=> ~ ! [A3: nat,B2: nat] :
( ( A
= ( Addition @ A3 @ B2 ) )
=> ( ( A2 @ A3 )
=> ( ( member_nat @ A3 @ G )
=> ( ( B @ B2 )
=> ~ ( member_nat @ B2 @ G ) ) ) ) ) ) ) ).
% additive_abelian_group.sumsetp.cases
thf(fact_49_additive__abelian__group_Osumsetp_Ocases,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: a > $o,B: a > $o,A: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn895083305082786853setp_a @ G @ Addition @ A2 @ B @ A )
=> ~ ! [A3: a,B2: a] :
( ( A
= ( Addition @ A3 @ B2 ) )
=> ( ( A2 @ A3 )
=> ( ( member_a @ A3 @ G )
=> ( ( B @ B2 )
=> ~ ( member_a @ B2 @ G ) ) ) ) ) ) ) ).
% additive_abelian_group.sumsetp.cases
thf(fact_50_additive__abelian__group_Osumsetp_Osimps,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: nat > $o,B: nat > $o,A: nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( pluenn5670965976768739049tp_nat @ G @ Addition @ A2 @ B @ A )
= ( ? [A4: nat,B3: nat] :
( ( A
= ( Addition @ A4 @ B3 ) )
& ( A2 @ A4 )
& ( member_nat @ A4 @ G )
& ( B @ B3 )
& ( member_nat @ B3 @ G ) ) ) ) ) ).
% additive_abelian_group.sumsetp.simps
thf(fact_51_additive__abelian__group_Osumsetp_Osimps,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: a > $o,B: a > $o,A: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn895083305082786853setp_a @ G @ Addition @ A2 @ B @ A )
= ( ? [A4: a,B3: a] :
( ( A
= ( Addition @ A4 @ B3 ) )
& ( A2 @ A4 )
& ( member_a @ A4 @ G )
& ( B @ B3 )
& ( member_a @ B3 @ G ) ) ) ) ) ).
% additive_abelian_group.sumsetp.simps
thf(fact_52_additive__abelian__group_Osumsetp_OsumsetI,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: nat > $o,A: nat,B: nat > $o,B4: nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( A2 @ A )
=> ( ( member_nat @ A @ G )
=> ( ( B @ B4 )
=> ( ( member_nat @ B4 @ G )
=> ( pluenn5670965976768739049tp_nat @ G @ Addition @ A2 @ B @ ( Addition @ A @ B4 ) ) ) ) ) ) ) ).
% additive_abelian_group.sumsetp.sumsetI
thf(fact_53_additive__abelian__group_Osumsetp_OsumsetI,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: a > $o,A: a,B: a > $o,B4: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( A2 @ A )
=> ( ( member_a @ A @ G )
=> ( ( B @ B4 )
=> ( ( member_a @ B4 @ G )
=> ( pluenn895083305082786853setp_a @ G @ Addition @ A2 @ B @ ( Addition @ A @ B4 ) ) ) ) ) ) ) ).
% additive_abelian_group.sumsetp.sumsetI
thf(fact_54_additive__abelian__group_Osumset__iterated_Ocong,axiom,
pluenn1960970773371692859ated_a = pluenn1960970773371692859ated_a ).
% additive_abelian_group.sumset_iterated.cong
thf(fact_55_additive__abelian__group_Ofinite__sumset__iterated,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat,R: nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( pluenn7055013279391836755ed_nat @ G @ Addition @ Zero @ A2 @ R ) ) ) ) ).
% additive_abelian_group.finite_sumset_iterated
thf(fact_56_additive__abelian__group_Ofinite__sumset__iterated,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,R: nat] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( pluenn1960970773371692859ated_a @ G @ Addition @ Zero @ A2 @ R ) ) ) ) ).
% additive_abelian_group.finite_sumset_iterated
thf(fact_57_additive__abelian__group_Osumset__iterated__subset__carrier,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,K: nat] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ord_less_eq_set_a @ ( pluenn1960970773371692859ated_a @ G @ Addition @ Zero @ A2 @ K ) @ G ) ) ).
% additive_abelian_group.sumset_iterated_subset_carrier
thf(fact_58_additive__abelian__group__def,axiom,
pluenn1164192988769422572roup_a = group_201663378560352916roup_a ).
% additive_abelian_group_def
thf(fact_59_additive__abelian__group_Ofinite__sumset,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat,B: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A2 @ B ) ) ) ) ) ).
% additive_abelian_group.finite_sumset
thf(fact_60_additive__abelian__group_Ofinite__sumset,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) ) ) ) ).
% additive_abelian_group.finite_sumset
thf(fact_61_additive__abelian__group_Osumsetp_Ocong,axiom,
pluenn895083305082786853setp_a = pluenn895083305082786853setp_a ).
% additive_abelian_group.sumsetp.cong
thf(fact_62_additive__abelian__group_Osumset__subset__carrier,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) @ G ) ) ).
% additive_abelian_group.sumset_subset_carrier
thf(fact_63_additive__abelian__group_Osumset__mono,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A5: set_a,A2: set_a,B5: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ord_less_eq_set_a @ A5 @ A2 )
=> ( ( ord_less_eq_set_a @ B5 @ B )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A5 @ B5 ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) ) ) ) ).
% additive_abelian_group.sumset_mono
thf(fact_64_additive__abelian__group_Ofinite__minusset,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( pluenn7323955030898006884et_nat @ G @ Addition @ Zero @ A2 ) ) ) ) ).
% additive_abelian_group.finite_minusset
thf(fact_65_additive__abelian__group_Ofinite__minusset,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A2 ) ) ) ) ).
% additive_abelian_group.finite_minusset
thf(fact_66_additive__abelian__group_Ominusset__subset__carrier,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ord_less_eq_set_a @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A2 ) @ G ) ) ).
% additive_abelian_group.minusset_subset_carrier
thf(fact_67_additive__abelian__group_Ominusset__iterated__minusset,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,K: nat] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn1960970773371692859ated_a @ G @ Addition @ Zero @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A2 ) @ K )
= ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ ( pluenn1960970773371692859ated_a @ G @ Addition @ Zero @ A2 @ K ) ) ) ) ).
% additive_abelian_group.minusset_iterated_minusset
thf(fact_68_additive__abelian__group_Ocard__le__sumset,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat,A: nat,B: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ( ( member_nat @ A @ 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 @ A2 @ B ) ) ) ) ) ) ) ) ) ).
% additive_abelian_group.card_le_sumset
thf(fact_69_additive__abelian__group_Ocard__le__sumset,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,A: a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_finite_a @ A2 )
=> ( ( member_a @ A @ A2 )
=> ( ( member_a @ A @ 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 @ A2 @ B ) ) ) ) ) ) ) ) ) ).
% additive_abelian_group.card_le_sumset
thf(fact_70_additive__abelian__group_Ocard__minusset_H,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ord_less_eq_set_a @ A2 @ G )
=> ( ( finite_card_a @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A2 ) )
= ( finite_card_a @ A2 ) ) ) ) ).
% additive_abelian_group.card_minusset'
thf(fact_71_additive__abelian__group_Ofinite__differenceset,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat,B: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A2 @ ( pluenn7323955030898006884et_nat @ G @ Addition @ Zero @ B ) ) ) ) ) ) ).
% additive_abelian_group.finite_differenceset
thf(fact_72_additive__abelian__group_Ofinite__differenceset,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ B ) ) ) ) ) ) ).
% additive_abelian_group.finite_differenceset
thf(fact_73_additive__abelian__group_Osumset__commute,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B )
= ( pluenn3038260743871226533mset_a @ G @ Addition @ B @ A2 ) ) ) ).
% additive_abelian_group.sumset_commute
thf(fact_74_additive__abelian__group_Osumset_OsumsetI,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: nat,A2: set_nat,B4: nat,B: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( member_nat @ A @ A2 )
=> ( ( member_nat @ A @ G )
=> ( ( member_nat @ B4 @ B )
=> ( ( member_nat @ B4 @ G )
=> ( member_nat @ ( Addition @ A @ B4 ) @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A2 @ B ) ) ) ) ) ) ) ).
% additive_abelian_group.sumset.sumsetI
thf(fact_75_additive__abelian__group_Osumset_OsumsetI,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: a,A2: set_a,B4: a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( member_a @ A @ A2 )
=> ( ( member_a @ A @ G )
=> ( ( member_a @ B4 @ B )
=> ( ( member_a @ B4 @ G )
=> ( member_a @ ( Addition @ A @ B4 ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) ) ) ) ) ) ).
% additive_abelian_group.sumset.sumsetI
thf(fact_76_additive__abelian__group_Osumset__assoc,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a,C: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) @ C )
= ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( pluenn3038260743871226533mset_a @ G @ Addition @ B @ C ) ) ) ) ).
% additive_abelian_group.sumset_assoc
thf(fact_77_additive__abelian__group_Osumset_Osimps,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: nat,A2: set_nat,B: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( member_nat @ A @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A2 @ B ) )
= ( ? [A4: nat,B3: nat] :
( ( A
= ( Addition @ A4 @ B3 ) )
& ( member_nat @ A4 @ A2 )
& ( member_nat @ A4 @ G )
& ( member_nat @ B3 @ B )
& ( member_nat @ B3 @ G ) ) ) ) ) ).
% additive_abelian_group.sumset.simps
thf(fact_78_additive__abelian__group_Osumset_Osimps,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( member_a @ A @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) )
= ( ? [A4: a,B3: a] :
( ( A
= ( Addition @ A4 @ B3 ) )
& ( member_a @ A4 @ A2 )
& ( member_a @ A4 @ G )
& ( member_a @ B3 @ B )
& ( member_a @ B3 @ G ) ) ) ) ) ).
% additive_abelian_group.sumset.simps
thf(fact_79_additive__abelian__group_Osumset_Ocases,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: nat,A2: set_nat,B: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( member_nat @ A @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A2 @ B ) )
=> ~ ! [A3: nat,B2: nat] :
( ( A
= ( Addition @ A3 @ B2 ) )
=> ( ( member_nat @ A3 @ A2 )
=> ( ( member_nat @ A3 @ G )
=> ( ( member_nat @ B2 @ B )
=> ~ ( member_nat @ B2 @ G ) ) ) ) ) ) ) ).
% additive_abelian_group.sumset.cases
thf(fact_80_additive__abelian__group_Osumset_Ocases,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( member_a @ A @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) )
=> ~ ! [A3: a,B2: a] :
( ( A
= ( Addition @ A3 @ B2 ) )
=> ( ( member_a @ A3 @ A2 )
=> ( ( member_a @ A3 @ G )
=> ( ( member_a @ B2 @ B )
=> ~ ( member_a @ B2 @ G ) ) ) ) ) ) ) ).
% additive_abelian_group.sumset.cases
thf(fact_81_additive__abelian__group_Ocard__sumset__iterated__minusset,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,K: nat] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_card_a @ ( pluenn1960970773371692859ated_a @ G @ Addition @ Zero @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A2 ) @ K ) )
= ( finite_card_a @ ( pluenn1960970773371692859ated_a @ G @ Addition @ Zero @ A2 @ K ) ) ) ) ).
% additive_abelian_group.card_sumset_iterated_minusset
thf(fact_82_additive__abelian__group_Odifferenceset__commute,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,B: set_a,A2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ ( pluenn3038260743871226533mset_a @ G @ Addition @ B @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A2 ) ) )
= ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ B ) ) ) ) ).
% additive_abelian_group.differenceset_commute
thf(fact_83_additive__abelian__group_Ominusset__distrib__sum,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) )
= ( pluenn3038260743871226533mset_a @ G @ Addition @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A2 ) @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ B ) ) ) ) ).
% additive_abelian_group.minusset_distrib_sum
thf(fact_84_additive__abelian__group_Odiff__minus__set,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A2 ) @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ B ) )
= ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) ) ) ).
% additive_abelian_group.diff_minus_set
thf(fact_85_additive__abelian__group_Ocard__differenceset__commute,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,B: set_a,A2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ B @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A2 ) ) )
= ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ B ) ) ) ) ) ).
% additive_abelian_group.card_differenceset_commute
thf(fact_86_card__sumset__0__iff,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ g )
=> ( ( ord_less_eq_set_a @ B @ g )
=> ( ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) )
= zero_zero_nat )
= ( ( ( finite_card_a @ A2 )
= zero_zero_nat )
| ( ( finite_card_a @ B )
= zero_zero_nat ) ) ) ) ) ).
% card_sumset_0_iff
thf(fact_87_group__of__Units,axiom,
group_group_a @ ( group_Units_a @ g @ addition @ zero ) @ addition @ zero ).
% group_of_Units
thf(fact_88_sumset__subset__insert_I1_J,axiom,
! [A2: set_a,B: set_a,X: a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( insert_a @ X @ B ) ) ) ).
% sumset_subset_insert(1)
thf(fact_89_sumset__subset__insert_I2_J,axiom,
! [A2: set_a,B: set_a,X: a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ ( insert_a @ X @ A2 ) @ B ) ) ).
% sumset_subset_insert(2)
thf(fact_90_finite__sumset_H,axiom,
! [A2: set_a,B: set_a] :
( ( finite_finite_a @ ( inf_inf_set_a @ A2 @ g ) )
=> ( ( finite_finite_a @ ( inf_inf_set_a @ B @ g ) )
=> ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ) ) ).
% finite_sumset'
thf(fact_91_sumset__subset__Un_I1_J,axiom,
! [A2: set_a,B: set_a,C: set_a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( sup_sup_set_a @ B @ C ) ) ) ).
% sumset_subset_Un(1)
thf(fact_92_sumset__subset__Un_I2_J,axiom,
! [A2: set_a,B: set_a,C: set_a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ ( sup_sup_set_a @ A2 @ C ) @ B ) ) ).
% sumset_subset_Un(2)
thf(fact_93_finite__if__finite__subsets__card__bdd,axiom,
! [F: set_nat,C: nat] :
( ! [G2: set_nat] :
( ( ord_less_eq_set_nat @ G2 @ F )
=> ( ( finite_finite_nat @ G2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G2 ) @ C ) ) )
=> ( ( finite_finite_nat @ F )
& ( ord_less_eq_nat @ ( finite_card_nat @ F ) @ C ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_94_finite__if__finite__subsets__card__bdd,axiom,
! [F: set_a,C: nat] :
( ! [G2: set_a] :
( ( ord_less_eq_set_a @ G2 @ F )
=> ( ( finite_finite_a @ G2 )
=> ( ord_less_eq_nat @ ( finite_card_a @ G2 ) @ C ) ) )
=> ( ( finite_finite_a @ F )
& ( ord_less_eq_nat @ ( finite_card_a @ F ) @ C ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_95_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_nat @ S ) )
=> ~ ! [T: set_nat] :
( ( ord_less_eq_set_nat @ T @ S )
=> ( ( ( finite_card_nat @ T )
= N )
=> ~ ( finite_finite_nat @ T ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_96_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_a] :
( ( ord_less_eq_nat @ N @ ( finite_card_a @ S ) )
=> ~ ! [T: set_a] :
( ( ord_less_eq_set_a @ T @ S )
=> ( ( ( finite_card_a @ T )
= N )
=> ~ ( finite_finite_a @ T ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_97_exists__subset__between,axiom,
! [A2: set_nat,N: nat,C: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ C ) )
=> ( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ( finite_finite_nat @ C )
=> ? [B6: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B6 )
& ( ord_less_eq_set_nat @ B6 @ C )
& ( ( finite_card_nat @ B6 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_98_exists__subset__between,axiom,
! [A2: set_a,N: nat,C: set_a] :
( ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_a @ C ) )
=> ( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( finite_finite_a @ C )
=> ? [B6: set_a] :
( ( ord_less_eq_set_a @ A2 @ B6 )
& ( ord_less_eq_set_a @ B6 @ C )
& ( ( finite_card_a @ B6 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_99_card__seteq,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite_card_nat @ A2 ) )
=> ( A2 = B ) ) ) ) ).
% card_seteq
thf(fact_100_card__seteq,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ B ) @ ( finite_card_a @ A2 ) )
=> ( A2 = B ) ) ) ) ).
% card_seteq
thf(fact_101_sumset__subset__Un1,axiom,
! [A2: set_a,A5: set_a,B: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ ( sup_sup_set_a @ A2 @ A5 ) @ B )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A5 @ B ) ) ) ).
% sumset_subset_Un1
thf(fact_102_sumset__subset__Un2,axiom,
! [A2: set_a,B: set_a,B5: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( sup_sup_set_a @ B @ B5 ) )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B5 ) ) ) ).
% sumset_subset_Un2
thf(fact_103_card__sumset__0__iff_H,axiom,
! [A2: set_a,B: set_a] :
( ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) )
= zero_zero_nat )
= ( ( ( finite_card_a @ ( inf_inf_set_a @ A2 @ g ) )
= zero_zero_nat )
| ( ( finite_card_a @ ( inf_inf_set_a @ B @ g ) )
= zero_zero_nat ) ) ) ).
% card_sumset_0_iff'
thf(fact_104_finite__insert,axiom,
! [A: a,A2: set_a] :
( ( finite_finite_a @ ( insert_a @ A @ A2 ) )
= ( finite_finite_a @ A2 ) ) ).
% finite_insert
thf(fact_105_finite__insert,axiom,
! [A: nat,A2: set_nat] :
( ( finite_finite_nat @ ( insert_nat @ A @ A2 ) )
= ( finite_finite_nat @ A2 ) ) ).
% finite_insert
thf(fact_106_finite__Int,axiom,
! [F: set_nat,G: set_nat] :
( ( ( finite_finite_nat @ F )
| ( finite_finite_nat @ G ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ F @ G ) ) ) ).
% finite_Int
thf(fact_107_finite__Int,axiom,
! [F: set_a,G: set_a] :
( ( ( finite_finite_a @ F )
| ( finite_finite_a @ G ) )
=> ( finite_finite_a @ ( inf_inf_set_a @ F @ G ) ) ) ).
% finite_Int
thf(fact_108_finite__Un,axiom,
! [F: set_nat,G: set_nat] :
( ( finite_finite_nat @ ( sup_sup_set_nat @ F @ G ) )
= ( ( finite_finite_nat @ F )
& ( finite_finite_nat @ G ) ) ) ).
% finite_Un
thf(fact_109_finite__Un,axiom,
! [F: set_a,G: set_a] :
( ( finite_finite_a @ ( sup_sup_set_a @ F @ G ) )
= ( ( finite_finite_a @ F )
& ( finite_finite_a @ G ) ) ) ).
% finite_Un
thf(fact_110_card_Oinfinite,axiom,
! [A2: set_a] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_card_a @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_111_card_Oinfinite,axiom,
! [A2: set_nat] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_card_nat @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_112_sumset__Int__carrier__eq_I2_J,axiom,
! [A2: set_a,B: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ ( inf_inf_set_a @ A2 @ g ) @ B )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ).
% sumset_Int_carrier_eq(2)
thf(fact_113_sumset__Int__carrier__eq_I1_J,axiom,
! [A2: set_a,B: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( inf_inf_set_a @ B @ g ) )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ).
% sumset_Int_carrier_eq(1)
thf(fact_114_sumset__Int__carrier,axiom,
! [A2: set_a,B: set_a] :
( ( inf_inf_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) @ g )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ).
% sumset_Int_carrier
thf(fact_115_minus__minusset,axiom,
! [A2: set_a] :
( ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) )
= ( inf_inf_set_a @ A2 @ g ) ) ).
% minus_minusset
thf(fact_116_card__minusset,axiom,
! [A2: set_a] :
( ( finite_card_a @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) )
= ( finite_card_a @ ( inf_inf_set_a @ A2 @ g ) ) ) ).
% card_minusset
thf(fact_117_infinite__Un,axiom,
! [S: set_nat,T2: set_nat] :
( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T2 ) ) )
= ( ~ ( finite_finite_nat @ S )
| ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_Un
thf(fact_118_infinite__Un,axiom,
! [S: set_a,T2: set_a] :
( ( ~ ( finite_finite_a @ ( sup_sup_set_a @ S @ T2 ) ) )
= ( ~ ( finite_finite_a @ S )
| ~ ( finite_finite_a @ T2 ) ) ) ).
% infinite_Un
thf(fact_119_Un__infinite,axiom,
! [S: set_nat,T2: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T2 ) ) ) ).
% Un_infinite
thf(fact_120_Un__infinite,axiom,
! [S: set_a,T2: set_a] :
( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( sup_sup_set_a @ S @ T2 ) ) ) ).
% Un_infinite
thf(fact_121_finite__UnI,axiom,
! [F: set_nat,G: set_nat] :
( ( finite_finite_nat @ F )
=> ( ( finite_finite_nat @ G )
=> ( finite_finite_nat @ ( sup_sup_set_nat @ F @ G ) ) ) ) ).
% finite_UnI
thf(fact_122_finite__UnI,axiom,
! [F: set_a,G: set_a] :
( ( finite_finite_a @ F )
=> ( ( finite_finite_a @ G )
=> ( finite_finite_a @ ( sup_sup_set_a @ F @ G ) ) ) ) ).
% finite_UnI
thf(fact_123_finite_OinsertI,axiom,
! [A2: set_a,A: a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( insert_a @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_124_finite_OinsertI,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( insert_nat @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_125_card__insert__le,axiom,
! [A2: set_a,X: a] : ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ ( insert_a @ X @ A2 ) ) ) ).
% card_insert_le
thf(fact_126_additive__abelian__group_Ocard__sumset__0__iff_H,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) )
= zero_zero_nat )
= ( ( ( finite_card_a @ ( inf_inf_set_a @ A2 @ 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_127_additive__abelian__group_Osumset__subset__Un1,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,A5: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ ( sup_sup_set_a @ A2 @ A5 ) @ B )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A5 @ B ) ) ) ) ).
% additive_abelian_group.sumset_subset_Un1
thf(fact_128_additive__abelian__group_Osumset__subset__Un2,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a,B5: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( sup_sup_set_a @ B @ B5 ) )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B5 ) ) ) ) ).
% additive_abelian_group.sumset_subset_Un2
thf(fact_129_additive__abelian__group_Osumset__Int__carrier__eq_I2_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ ( inf_inf_set_a @ A2 @ G ) @ B )
= ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) ) ).
% additive_abelian_group.sumset_Int_carrier_eq(2)
thf(fact_130_additive__abelian__group_Osumset__Int__carrier__eq_I1_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( inf_inf_set_a @ B @ G ) )
= ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) ) ).
% additive_abelian_group.sumset_Int_carrier_eq(1)
thf(fact_131_additive__abelian__group_Osumset__Int__carrier,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( inf_inf_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) @ G )
= ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) ) ).
% additive_abelian_group.sumset_Int_carrier
thf(fact_132_additive__abelian__group_Ominus__minusset,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A2 ) )
= ( inf_inf_set_a @ A2 @ G ) ) ) ).
% additive_abelian_group.minus_minusset
thf(fact_133_additive__abelian__group_Osumset__subset__Un_I1_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a,C: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( sup_sup_set_a @ B @ C ) ) ) ) ).
% additive_abelian_group.sumset_subset_Un(1)
thf(fact_134_additive__abelian__group_Osumset__subset__Un_I2_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a,C: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ ( sup_sup_set_a @ A2 @ C ) @ B ) ) ) ).
% additive_abelian_group.sumset_subset_Un(2)
thf(fact_135_additive__abelian__group_Ofinite__sumset_H,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat,B: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( finite_finite_nat @ ( inf_inf_set_nat @ A2 @ G ) )
=> ( ( finite_finite_nat @ ( inf_inf_set_nat @ B @ G ) )
=> ( finite_finite_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A2 @ B ) ) ) ) ) ).
% additive_abelian_group.finite_sumset'
thf(fact_136_additive__abelian__group_Ofinite__sumset_H,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_finite_a @ ( inf_inf_set_a @ A2 @ G ) )
=> ( ( finite_finite_a @ ( inf_inf_set_a @ B @ G ) )
=> ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) ) ) ) ).
% additive_abelian_group.finite_sumset'
thf(fact_137_additive__abelian__group_Osumset__subset__insert_I1_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a,X: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( insert_a @ X @ B ) ) ) ) ).
% additive_abelian_group.sumset_subset_insert(1)
thf(fact_138_additive__abelian__group_Osumset__subset__insert_I2_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a,X: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ ( insert_a @ X @ A2 ) @ B ) ) ) ).
% additive_abelian_group.sumset_subset_insert(2)
thf(fact_139_additive__abelian__group_Ocard__minusset,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_card_a @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A2 ) )
= ( finite_card_a @ ( inf_inf_set_a @ A2 @ G ) ) ) ) ).
% additive_abelian_group.card_minusset
thf(fact_140_additive__abelian__group_Ocard__sumset__0__iff,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ord_less_eq_set_a @ A2 @ G )
=> ( ( ord_less_eq_set_a @ B @ G )
=> ( ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) )
= zero_zero_nat )
= ( ( ( finite_card_a @ A2 )
= zero_zero_nat )
| ( ( finite_card_a @ B )
= zero_zero_nat ) ) ) ) ) ) ).
% additive_abelian_group.card_sumset_0_iff
thf(fact_141_finite__has__maximal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_eq_nat @ A @ X3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_142_finite__has__maximal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ( ord_less_eq_set_a @ A @ X3 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_143_finite__has__minimal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_eq_nat @ X3 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_144_finite__has__minimal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ( ord_less_eq_set_a @ X3 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_145_rev__finite__subset,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( finite_finite_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_146_rev__finite__subset,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A2 @ B )
=> ( finite_finite_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_147_infinite__super,axiom,
! [S: set_nat,T2: set_nat] :
( ( ord_less_eq_set_nat @ S @ T2 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_148_infinite__super,axiom,
! [S: set_a,T2: set_a] :
( ( ord_less_eq_set_a @ S @ T2 )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ T2 ) ) ) ).
% infinite_super
thf(fact_149_finite__subset,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_150_finite__subset,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ A2 ) ) ) ).
% finite_subset
thf(fact_151_infinite__arbitrarily__large,axiom,
! [A2: set_nat,N: nat] :
( ~ ( finite_finite_nat @ A2 )
=> ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ( finite_card_nat @ B6 )
= N )
& ( ord_less_eq_set_nat @ B6 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_152_infinite__arbitrarily__large,axiom,
! [A2: set_a,N: nat] :
( ~ ( finite_finite_a @ A2 )
=> ? [B6: set_a] :
( ( finite_finite_a @ B6 )
& ( ( finite_card_a @ B6 )
= N )
& ( ord_less_eq_set_a @ B6 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_153_card__subset__eq,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ( finite_card_nat @ A2 )
= ( finite_card_nat @ B ) )
=> ( A2 = B ) ) ) ) ).
% card_subset_eq
thf(fact_154_card__subset__eq,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ( finite_card_a @ A2 )
= ( finite_card_a @ B ) )
=> ( A2 = B ) ) ) ) ).
% card_subset_eq
thf(fact_155_card__mono,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B ) ) ) ) ).
% card_mono
thf(fact_156_card__mono,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A2 @ B )
=> ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B ) ) ) ) ).
% card_mono
thf(fact_157_card__sumset__singleton__eq,axiom,
! [A2: set_a,A: a] :
( ( finite_finite_a @ A2 )
=> ( ( ( member_a @ A @ g )
=> ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( finite_card_a @ ( inf_inf_set_a @ A2 @ g ) ) ) )
& ( ~ ( member_a @ A @ g )
=> ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= zero_zero_nat ) ) ) ) ).
% card_sumset_singleton_eq
thf(fact_158_card__sumset__le,axiom,
! [A2: set_a,A: a] :
( ( finite_finite_a @ A2 )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A2 ) ) ) ).
% card_sumset_le
thf(fact_159_infinite__sumset__aux,axiom,
! [A2: set_a,B: set_a] :
( ~ ( finite_finite_a @ ( inf_inf_set_a @ A2 @ g ) )
=> ( ( ~ ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) )
= ( ( inf_inf_set_a @ B @ g )
!= bot_bot_set_a ) ) ) ).
% infinite_sumset_aux
thf(fact_160_infinite__sumset__iff,axiom,
! [A2: set_a,B: set_a] :
( ( ~ ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) )
= ( ( ~ ( finite_finite_a @ ( inf_inf_set_a @ A2 @ g ) )
& ( ( inf_inf_set_a @ B @ g )
!= bot_bot_set_a ) )
| ( ( ( inf_inf_set_a @ A2 @ g )
!= bot_bot_set_a )
& ~ ( finite_finite_a @ ( inf_inf_set_a @ B @ g ) ) ) ) ) ).
% infinite_sumset_iff
thf(fact_161_Un__Int__eq_I1_J,axiom,
! [S: set_a,T2: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_162_Un__Int__eq_I2_J,axiom,
! [S: set_a,T2: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_163_Un__Int__eq_I3_J,axiom,
! [S: set_a,T2: set_a] :
( ( inf_inf_set_a @ S @ ( sup_sup_set_a @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_164_Un__Int__eq_I4_J,axiom,
! [T2: set_a,S: set_a] :
( ( inf_inf_set_a @ T2 @ ( sup_sup_set_a @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_165_Int__Un__eq_I1_J,axiom,
! [S: set_a,T2: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_166_Int__Un__eq_I2_J,axiom,
! [S: set_a,T2: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_167_empty__iff,axiom,
! [C2: nat] :
~ ( member_nat @ C2 @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_168_empty__iff,axiom,
! [C2: a] :
~ ( member_a @ C2 @ bot_bot_set_a ) ).
% empty_iff
thf(fact_169_all__not__in__conv,axiom,
! [A2: set_nat] :
( ( ! [X2: nat] :
~ ( member_nat @ X2 @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_170_all__not__in__conv,axiom,
! [A2: set_a] :
( ( ! [X2: a] :
~ ( member_a @ X2 @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_171_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X2: a] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_172_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X2: a] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_173_subsetI,axiom,
! [A2: set_nat,B: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat @ X3 @ B ) )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% subsetI
thf(fact_174_subsetI,axiom,
! [A2: set_a,B: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member_a @ X3 @ B ) )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ).
% subsetI
thf(fact_175_subset__antisym,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_176_insertCI,axiom,
! [A: a,B: set_a,B4: a] :
( ( ~ ( member_a @ A @ B )
=> ( A = B4 ) )
=> ( member_a @ A @ ( insert_a @ B4 @ B ) ) ) ).
% insertCI
thf(fact_177_insertCI,axiom,
! [A: nat,B: set_nat,B4: nat] :
( ( ~ ( member_nat @ A @ B )
=> ( A = B4 ) )
=> ( member_nat @ A @ ( insert_nat @ B4 @ B ) ) ) ).
% insertCI
thf(fact_178_insert__iff,axiom,
! [A: a,B4: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B4 @ A2 ) )
= ( ( A = B4 )
| ( member_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_179_insert__iff,axiom,
! [A: nat,B4: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B4 @ A2 ) )
= ( ( A = B4 )
| ( member_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_180_insert__absorb2,axiom,
! [X: a,A2: set_a] :
( ( insert_a @ X @ ( insert_a @ X @ A2 ) )
= ( insert_a @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_181_IntI,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ A2 )
=> ( ( member_nat @ C2 @ B )
=> ( member_nat @ C2 @ ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_182_IntI,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ A2 )
=> ( ( member_a @ C2 @ B )
=> ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_183_Int__iff,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A2 @ B ) )
= ( ( member_nat @ C2 @ A2 )
& ( member_nat @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_184_Int__iff,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B ) )
= ( ( member_a @ C2 @ A2 )
& ( member_a @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_185_UnCI,axiom,
! [C2: nat,B: set_nat,A2: set_nat] :
( ( ~ ( member_nat @ C2 @ B )
=> ( member_nat @ C2 @ A2 ) )
=> ( member_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% UnCI
thf(fact_186_UnCI,axiom,
! [C2: a,B: set_a,A2: set_a] :
( ( ~ ( member_a @ C2 @ B )
=> ( member_a @ C2 @ A2 ) )
=> ( member_a @ C2 @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% UnCI
thf(fact_187_Un__iff,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B ) )
= ( ( member_nat @ C2 @ A2 )
| ( member_nat @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_188_Un__iff,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( sup_sup_set_a @ A2 @ B ) )
= ( ( member_a @ C2 @ A2 )
| ( member_a @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_189_sumset__empty_H_I1_J,axiom,
! [A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A2 @ g )
= bot_bot_set_a )
=> ( ( pluenn3038260743871226533mset_a @ g @ addition @ B @ A2 )
= bot_bot_set_a ) ) ).
% sumset_empty'(1)
thf(fact_190_sumset__empty_H_I2_J,axiom,
! [A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A2 @ g )
= bot_bot_set_a )
=> ( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B )
= bot_bot_set_a ) ) ).
% sumset_empty'(2)
thf(fact_191_subset__empty,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_192_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_193_singletonI,axiom,
! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_194_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_195_insert__subset,axiom,
! [X: nat,A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X @ A2 ) @ B )
= ( ( member_nat @ X @ B )
& ( ord_less_eq_set_nat @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_196_insert__subset,axiom,
! [X: a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A2 ) @ B )
= ( ( member_a @ X @ B )
& ( ord_less_eq_set_a @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_197_Int__subset__iff,axiom,
! [C: set_a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ ( inf_inf_set_a @ A2 @ B ) )
= ( ( ord_less_eq_set_a @ C @ A2 )
& ( ord_less_eq_set_a @ C @ B ) ) ) ).
% Int_subset_iff
thf(fact_198_Un__empty,axiom,
! [A2: set_a,B: set_a] :
( ( ( sup_sup_set_a @ A2 @ B )
= bot_bot_set_a )
= ( ( A2 = bot_bot_set_a )
& ( B = bot_bot_set_a ) ) ) ).
% Un_empty
thf(fact_199_Int__insert__left__if0,axiom,
! [A: nat,C: set_nat,B: set_nat] :
( ~ ( member_nat @ A @ C )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B ) @ C )
= ( inf_inf_set_nat @ B @ C ) ) ) ).
% Int_insert_left_if0
thf(fact_200_Int__insert__left__if0,axiom,
! [A: a,C: set_a,B: set_a] :
( ~ ( member_a @ A @ C )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B ) @ C )
= ( inf_inf_set_a @ B @ C ) ) ) ).
% Int_insert_left_if0
thf(fact_201_Int__insert__left__if1,axiom,
! [A: nat,C: set_nat,B: set_nat] :
( ( member_nat @ A @ C )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B ) @ C )
= ( insert_nat @ A @ ( inf_inf_set_nat @ B @ C ) ) ) ) ).
% Int_insert_left_if1
thf(fact_202_Int__insert__left__if1,axiom,
! [A: a,C: set_a,B: set_a] :
( ( member_a @ A @ C )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B ) @ C )
= ( insert_a @ A @ ( inf_inf_set_a @ B @ C ) ) ) ) ).
% Int_insert_left_if1
thf(fact_203_insert__inter__insert,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ ( insert_a @ A @ B ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B ) ) ) ).
% insert_inter_insert
thf(fact_204_Int__insert__right__if0,axiom,
! [A: nat,A2: set_nat,B: set_nat] :
( ~ ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B ) )
= ( inf_inf_set_nat @ A2 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_205_Int__insert__right__if0,axiom,
! [A: a,A2: set_a,B: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( inf_inf_set_a @ A2 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_206_Int__insert__right__if1,axiom,
! [A: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B ) )
= ( insert_nat @ A @ ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_207_Int__insert__right__if1,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_208_Un__subset__iff,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B ) @ C )
= ( ( ord_less_eq_set_a @ A2 @ C )
& ( ord_less_eq_set_a @ B @ C ) ) ) ).
% Un_subset_iff
thf(fact_209_Un__insert__left,axiom,
! [A: a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ ( insert_a @ A @ B ) @ C )
= ( insert_a @ A @ ( sup_sup_set_a @ B @ C ) ) ) ).
% Un_insert_left
thf(fact_210_Un__insert__right,axiom,
! [A2: set_a,A: a,B: set_a] :
( ( sup_sup_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( insert_a @ A @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% Un_insert_right
thf(fact_211_Int__Un__eq_I4_J,axiom,
! [T2: set_a,S: set_a] :
( ( sup_sup_set_a @ T2 @ ( inf_inf_set_a @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_212_Int__Un__eq_I3_J,axiom,
! [S: set_a,T2: set_a] :
( ( sup_sup_set_a @ S @ ( inf_inf_set_a @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_213_singleton__insert__inj__eq_H,axiom,
! [A: a,A2: set_a,B4: a] :
( ( ( insert_a @ A @ A2 )
= ( insert_a @ B4 @ bot_bot_set_a ) )
= ( ( A = B4 )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B4 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_214_singleton__insert__inj__eq,axiom,
! [B4: a,A: a,A2: set_a] :
( ( ( insert_a @ B4 @ bot_bot_set_a )
= ( insert_a @ A @ A2 ) )
= ( ( A = B4 )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B4 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_215_disjoint__insert_I2_J,axiom,
! [A2: set_nat,B4: nat,B: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ A2 @ ( insert_nat @ B4 @ B ) ) )
= ( ~ ( member_nat @ B4 @ A2 )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_216_disjoint__insert_I2_J,axiom,
! [A2: set_a,B4: a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ ( insert_a @ B4 @ B ) ) )
= ( ~ ( member_a @ B4 @ A2 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_217_disjoint__insert_I1_J,axiom,
! [B: set_nat,A: nat,A2: set_nat] :
( ( ( inf_inf_set_nat @ B @ ( insert_nat @ A @ A2 ) )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A @ B )
& ( ( inf_inf_set_nat @ B @ A2 )
= bot_bot_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_218_disjoint__insert_I1_J,axiom,
! [B: set_a,A: a,A2: set_a] :
( ( ( inf_inf_set_a @ B @ ( insert_a @ A @ A2 ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B )
& ( ( inf_inf_set_a @ B @ A2 )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_219_insert__disjoint_I2_J,axiom,
! [A: nat,A2: set_nat,B: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ ( insert_nat @ A @ A2 ) @ B ) )
= ( ~ ( member_nat @ A @ B )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_220_insert__disjoint_I2_J,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ B ) )
= ( ~ ( member_a @ A @ B )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_221_insert__disjoint_I1_J,axiom,
! [A: nat,A2: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ ( insert_nat @ A @ A2 ) @ B )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A @ B )
& ( ( inf_inf_set_nat @ A2 @ B )
= bot_bot_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_222_insert__disjoint_I1_J,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ B )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B )
& ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_223_card_Oempty,axiom,
( ( finite_card_a @ bot_bot_set_a )
= zero_zero_nat ) ).
% card.empty
thf(fact_224_card__0__eq,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( finite_card_nat @ A2 )
= zero_zero_nat )
= ( A2 = bot_bot_set_nat ) ) ) ).
% card_0_eq
thf(fact_225_card__0__eq,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( ( finite_card_a @ A2 )
= zero_zero_nat )
= ( A2 = bot_bot_set_a ) ) ) ).
% card_0_eq
thf(fact_226_sumset__empty_I1_J,axiom,
! [A2: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% sumset_empty(1)
thf(fact_227_sumset__empty_I2_J,axiom,
! [A2: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ bot_bot_set_a @ A2 )
= bot_bot_set_a ) ).
% sumset_empty(2)
thf(fact_228_sumset__is__empty__iff,axiom,
! [A2: set_a,B: set_a] :
( ( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B )
= bot_bot_set_a )
= ( ( ( inf_inf_set_a @ A2 @ g )
= bot_bot_set_a )
| ( ( inf_inf_set_a @ B @ g )
= bot_bot_set_a ) ) ) ).
% sumset_is_empty_iff
thf(fact_229_minusset__is__empty__iff,axiom,
! [A2: set_a] :
( ( ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 )
= bot_bot_set_a )
= ( ( inf_inf_set_a @ A2 @ g )
= bot_bot_set_a ) ) ).
% minusset_is_empty_iff
thf(fact_230_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_231_sumset__iterated__0,axiom,
! [A2: set_a] :
( ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ A2 @ zero_zero_nat )
= ( insert_a @ zero @ bot_bot_set_a ) ) ).
% sumset_iterated_0
thf(fact_232_sumset__D_I1_J,axiom,
! [A2: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( insert_a @ zero @ bot_bot_set_a ) )
= ( inf_inf_set_a @ A2 @ g ) ) ).
% sumset_D(1)
thf(fact_233_sumset__D_I2_J,axiom,
! [A2: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ ( insert_a @ zero @ bot_bot_set_a ) @ A2 )
= ( inf_inf_set_a @ A2 @ g ) ) ).
% sumset_D(2)
thf(fact_234_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_235_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_236_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_237_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_238_equals0I,axiom,
! [A2: set_nat] :
( ! [Y2: nat] :
~ ( member_nat @ Y2 @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_239_equals0I,axiom,
! [A2: set_a] :
( ! [Y2: a] :
~ ( member_a @ Y2 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_240_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X2: nat] : ( member_nat @ X2 @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_241_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X2: a] : ( member_a @ X2 @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_242_singletonD,axiom,
! [B4: nat,A: nat] :
( ( member_nat @ B4 @ ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( B4 = A ) ) ).
% singletonD
thf(fact_243_singletonD,axiom,
! [B4: a,A: a] :
( ( member_a @ B4 @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B4 = A ) ) ).
% singletonD
thf(fact_244_singleton__iff,axiom,
! [B4: nat,A: nat] :
( ( member_nat @ B4 @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( B4 = A ) ) ).
% singleton_iff
thf(fact_245_singleton__iff,axiom,
! [B4: a,A: a] :
( ( member_a @ B4 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B4 = A ) ) ).
% singleton_iff
thf(fact_246_doubleton__eq__iff,axiom,
! [A: a,B4: a,C2: a,D: a] :
( ( ( insert_a @ A @ ( insert_a @ B4 @ bot_bot_set_a ) )
= ( insert_a @ C2 @ ( insert_a @ D @ bot_bot_set_a ) ) )
= ( ( ( A = C2 )
& ( B4 = D ) )
| ( ( A = D )
& ( B4 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_247_insert__not__empty,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ A2 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_248_singleton__inject,axiom,
! [A: a,B4: a] :
( ( ( insert_a @ A @ bot_bot_set_a )
= ( insert_a @ B4 @ bot_bot_set_a ) )
=> ( A = B4 ) ) ).
% singleton_inject
thf(fact_249_Int__emptyI,axiom,
! [A2: set_nat,B: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ~ ( member_nat @ X3 @ B ) )
=> ( ( inf_inf_set_nat @ A2 @ B )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_250_Int__emptyI,axiom,
! [A2: set_a,B: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ~ ( member_a @ X3 @ B ) )
=> ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_251_disjoint__iff,axiom,
! [A2: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B )
= bot_bot_set_nat )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ~ ( member_nat @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_252_disjoint__iff,axiom,
! [A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ~ ( member_a @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_253_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_254_Int__empty__right,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% Int_empty_right
thf(fact_255_disjoint__iff__not__equal,axiom,
! [A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ! [Y3: a] :
( ( member_a @ Y3 @ B )
=> ( X2 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_256_Un__empty__left,axiom,
! [B: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ B )
= B ) ).
% Un_empty_left
thf(fact_257_Un__empty__right,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% Un_empty_right
thf(fact_258_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_259_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_260_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_261_infinite__imp__nonempty,axiom,
! [S: set_a] :
( ~ ( finite_finite_a @ S )
=> ( S != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_262_subset__singleton__iff,axiom,
! [X4: set_a,A: a] :
( ( ord_less_eq_set_a @ X4 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( ( X4 = bot_bot_set_a )
| ( X4
= ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_263_subset__singletonD,axiom,
! [A2: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) )
=> ( ( A2 = bot_bot_set_a )
| ( A2
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_264_insert__is__Un,axiom,
( insert_a
= ( ^ [A4: a] : ( sup_sup_set_a @ ( insert_a @ A4 @ bot_bot_set_a ) ) ) ) ).
% insert_is_Un
thf(fact_265_Un__singleton__iff,axiom,
! [A2: set_a,B: set_a,X: a] :
( ( ( sup_sup_set_a @ A2 @ B )
= ( insert_a @ X @ bot_bot_set_a ) )
= ( ( ( A2 = bot_bot_set_a )
& ( B
= ( insert_a @ X @ bot_bot_set_a ) ) )
| ( ( A2
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B = bot_bot_set_a ) )
| ( ( A2
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_266_singleton__Un__iff,axiom,
! [X: a,A2: set_a,B: set_a] :
( ( ( insert_a @ X @ bot_bot_set_a )
= ( sup_sup_set_a @ A2 @ B ) )
= ( ( ( A2 = bot_bot_set_a )
& ( B
= ( insert_a @ X @ bot_bot_set_a ) ) )
| ( ( A2
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B = bot_bot_set_a ) )
| ( ( A2
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_267_finite__has__maximal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_268_finite__has__maximal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_269_finite__has__minimal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_270_finite__has__minimal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_271_infinite__finite__induct,axiom,
! [P: set_nat > $o,A2: set_nat] :
( ! [A6: set_nat] :
( ~ ( finite_finite_nat @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X3: nat,F2: set_nat] :
( ( finite_finite_nat @ F2 )
=> ( ~ ( member_nat @ X3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_nat @ X3 @ F2 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_272_infinite__finite__induct,axiom,
! [P: set_a > $o,A2: set_a] :
( ! [A6: set_a] :
( ~ ( finite_finite_a @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X3: a,F2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ~ ( member_a @ X3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_a @ X3 @ F2 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_273_finite__ne__induct,axiom,
! [F: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F )
=> ( ( F != bot_bot_set_nat )
=> ( ! [X3: nat] : ( P @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
=> ( ! [X3: nat,F2: set_nat] :
( ( finite_finite_nat @ F2 )
=> ( ( F2 != bot_bot_set_nat )
=> ( ~ ( member_nat @ X3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_nat @ X3 @ F2 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_ne_induct
thf(fact_274_finite__ne__induct,axiom,
! [F: set_a,P: set_a > $o] :
( ( finite_finite_a @ F )
=> ( ( F != bot_bot_set_a )
=> ( ! [X3: a] : ( P @ ( insert_a @ X3 @ bot_bot_set_a ) )
=> ( ! [X3: a,F2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ( F2 != bot_bot_set_a )
=> ( ~ ( member_a @ X3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_a @ X3 @ F2 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_ne_induct
thf(fact_275_finite__induct,axiom,
! [F: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X3: nat,F2: set_nat] :
( ( finite_finite_nat @ F2 )
=> ( ~ ( member_nat @ X3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_nat @ X3 @ F2 ) ) ) ) )
=> ( P @ F ) ) ) ) ).
% finite_induct
thf(fact_276_finite__induct,axiom,
! [F: set_a,P: set_a > $o] :
( ( finite_finite_a @ F )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X3: a,F2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ~ ( member_a @ X3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_a @ X3 @ F2 ) ) ) ) )
=> ( P @ F ) ) ) ) ).
% finite_induct
thf(fact_277_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A4: set_nat] :
( ( A4 = bot_bot_set_nat )
| ? [A7: set_nat,B3: nat] :
( ( A4
= ( insert_nat @ B3 @ A7 ) )
& ( finite_finite_nat @ A7 ) ) ) ) ) ).
% finite.simps
thf(fact_278_finite_Osimps,axiom,
( finite_finite_a
= ( ^ [A4: set_a] :
( ( A4 = bot_bot_set_a )
| ? [A7: set_a,B3: a] :
( ( A4
= ( insert_a @ B3 @ A7 ) )
& ( finite_finite_a @ A7 ) ) ) ) ) ).
% finite.simps
thf(fact_279_finite_Ocases,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ~ ! [A6: set_nat] :
( ? [A3: nat] :
( A
= ( insert_nat @ A3 @ A6 ) )
=> ~ ( finite_finite_nat @ A6 ) ) ) ) ).
% finite.cases
thf(fact_280_finite_Ocases,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( ( A != bot_bot_set_a )
=> ~ ! [A6: set_a] :
( ? [A3: a] :
( A
= ( insert_a @ A3 @ A6 ) )
=> ~ ( finite_finite_a @ A6 ) ) ) ) ).
% finite.cases
thf(fact_281_additive__abelian__group_Osumset__empty_I1_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ bot_bot_set_a )
= bot_bot_set_a ) ) ).
% additive_abelian_group.sumset_empty(1)
thf(fact_282_additive__abelian__group_Osumset__empty_I2_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ bot_bot_set_a @ A2 )
= bot_bot_set_a ) ) ).
% additive_abelian_group.sumset_empty(2)
thf(fact_283_finite__subset__induct,axiom,
! [F: set_nat,A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F )
=> ( ( ord_less_eq_set_nat @ F @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A3: nat,F2: set_nat] :
( ( finite_finite_nat @ F2 )
=> ( ( member_nat @ A3 @ A2 )
=> ( ~ ( member_nat @ A3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_nat @ A3 @ F2 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct
thf(fact_284_finite__subset__induct,axiom,
! [F: set_a,A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F )
=> ( ( ord_less_eq_set_a @ F @ A2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A3: a,F2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ( member_a @ A3 @ A2 )
=> ( ~ ( member_a @ A3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_a @ A3 @ F2 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct
thf(fact_285_finite__subset__induct_H,axiom,
! [F: set_nat,A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F )
=> ( ( ord_less_eq_set_nat @ F @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A3: nat,F2: set_nat] :
( ( finite_finite_nat @ F2 )
=> ( ( member_nat @ A3 @ A2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A2 )
=> ( ~ ( member_nat @ A3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_nat @ A3 @ F2 ) ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_286_finite__subset__induct_H,axiom,
! [F: set_a,A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F )
=> ( ( ord_less_eq_set_a @ F @ A2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A3: a,F2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ( member_a @ A3 @ A2 )
=> ( ( ord_less_eq_set_a @ F2 @ A2 )
=> ( ~ ( member_a @ A3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_a @ A3 @ F2 ) ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_287_in__mono,axiom,
! [A2: set_nat,B: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ X @ B ) ) ) ).
% in_mono
thf(fact_288_in__mono,axiom,
! [A2: set_a,B: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( member_a @ X @ A2 )
=> ( member_a @ X @ B ) ) ) ).
% in_mono
thf(fact_289_subsetD,axiom,
! [A2: set_nat,B: set_nat,C2: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( member_nat @ C2 @ A2 )
=> ( member_nat @ C2 @ B ) ) ) ).
% subsetD
thf(fact_290_subsetD,axiom,
! [A2: set_a,B: set_a,C2: a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( member_a @ C2 @ A2 )
=> ( member_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_291_equalityE,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B )
=> ~ ( ord_less_eq_set_a @ B @ A2 ) ) ) ).
% equalityE
thf(fact_292_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A7: set_nat,B7: set_nat] :
! [X2: nat] :
( ( member_nat @ X2 @ A7 )
=> ( member_nat @ X2 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_293_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A7: set_a,B7: set_a] :
! [X2: a] :
( ( member_a @ X2 @ A7 )
=> ( member_a @ X2 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_294_equalityD1,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ).
% equalityD1
thf(fact_295_equalityD2,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ( ord_less_eq_set_a @ B @ A2 ) ) ).
% equalityD2
thf(fact_296_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A7: set_nat,B7: set_nat] :
! [T3: nat] :
( ( member_nat @ T3 @ A7 )
=> ( member_nat @ T3 @ B7 ) ) ) ) ).
% subset_iff
thf(fact_297_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A7: set_a,B7: set_a] :
! [T3: a] :
( ( member_a @ T3 @ A7 )
=> ( member_a @ T3 @ B7 ) ) ) ) ).
% subset_iff
thf(fact_298_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_299_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_300_subset__trans,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% subset_trans
thf(fact_301_set__eq__subset,axiom,
( ( ^ [Y4: set_a,Z: set_a] : ( Y4 = Z ) )
= ( ^ [A7: set_a,B7: set_a] :
( ( ord_less_eq_set_a @ A7 @ B7 )
& ( ord_less_eq_set_a @ B7 @ A7 ) ) ) ) ).
% set_eq_subset
thf(fact_302_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_303_card__eq__0__iff,axiom,
! [A2: set_nat] :
( ( ( finite_card_nat @ A2 )
= zero_zero_nat )
= ( ( A2 = bot_bot_set_nat )
| ~ ( finite_finite_nat @ A2 ) ) ) ).
% card_eq_0_iff
thf(fact_304_card__eq__0__iff,axiom,
! [A2: set_a] :
( ( ( finite_card_a @ A2 )
= zero_zero_nat )
= ( ( A2 = bot_bot_set_a )
| ~ ( finite_finite_a @ A2 ) ) ) ).
% card_eq_0_iff
thf(fact_305_insertE,axiom,
! [A: a,B4: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B4 @ A2 ) )
=> ( ( A != B4 )
=> ( member_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_306_insertE,axiom,
! [A: nat,B4: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B4 @ A2 ) )
=> ( ( A != B4 )
=> ( member_nat @ A @ A2 ) ) ) ).
% insertE
thf(fact_307_insertI1,axiom,
! [A: a,B: set_a] : ( member_a @ A @ ( insert_a @ A @ B ) ) ).
% insertI1
thf(fact_308_insertI1,axiom,
! [A: nat,B: set_nat] : ( member_nat @ A @ ( insert_nat @ A @ B ) ) ).
% insertI1
thf(fact_309_insertI2,axiom,
! [A: a,B: set_a,B4: a] :
( ( member_a @ A @ B )
=> ( member_a @ A @ ( insert_a @ B4 @ B ) ) ) ).
% insertI2
thf(fact_310_insertI2,axiom,
! [A: nat,B: set_nat,B4: nat] :
( ( member_nat @ A @ B )
=> ( member_nat @ A @ ( insert_nat @ B4 @ B ) ) ) ).
% insertI2
thf(fact_311_Set_Oset__insert,axiom,
! [X: a,A2: set_a] :
( ( member_a @ X @ A2 )
=> ~ ! [B6: set_a] :
( ( A2
= ( insert_a @ X @ B6 ) )
=> ( member_a @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_312_Set_Oset__insert,axiom,
! [X: nat,A2: set_nat] :
( ( member_nat @ X @ A2 )
=> ~ ! [B6: set_nat] :
( ( A2
= ( insert_nat @ X @ B6 ) )
=> ( member_nat @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_313_insert__ident,axiom,
! [X: a,A2: set_a,B: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ~ ( member_a @ X @ B )
=> ( ( ( insert_a @ X @ A2 )
= ( insert_a @ X @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_314_insert__ident,axiom,
! [X: nat,A2: set_nat,B: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ~ ( member_nat @ X @ B )
=> ( ( ( insert_nat @ X @ A2 )
= ( insert_nat @ X @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_315_insert__absorb,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_316_insert__absorb,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( insert_nat @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_317_insert__eq__iff,axiom,
! [A: a,A2: set_a,B4: a,B: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ~ ( member_a @ B4 @ B )
=> ( ( ( insert_a @ A @ A2 )
= ( insert_a @ B4 @ B ) )
= ( ( ( A = B4 )
=> ( A2 = B ) )
& ( ( A != B4 )
=> ? [C3: set_a] :
( ( A2
= ( insert_a @ B4 @ C3 ) )
& ~ ( member_a @ B4 @ C3 )
& ( B
= ( insert_a @ A @ C3 ) )
& ~ ( member_a @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_318_insert__eq__iff,axiom,
! [A: nat,A2: set_nat,B4: nat,B: set_nat] :
( ~ ( member_nat @ A @ A2 )
=> ( ~ ( member_nat @ B4 @ B )
=> ( ( ( insert_nat @ A @ A2 )
= ( insert_nat @ B4 @ B ) )
= ( ( ( A = B4 )
=> ( A2 = B ) )
& ( ( A != B4 )
=> ? [C3: set_nat] :
( ( A2
= ( insert_nat @ B4 @ C3 ) )
& ~ ( member_nat @ B4 @ C3 )
& ( B
= ( insert_nat @ A @ C3 ) )
& ~ ( member_nat @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_319_insert__commute,axiom,
! [X: a,Y: a,A2: set_a] :
( ( insert_a @ X @ ( insert_a @ Y @ A2 ) )
= ( insert_a @ Y @ ( insert_a @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_320_mk__disjoint__insert,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ? [B6: set_a] :
( ( A2
= ( insert_a @ A @ B6 ) )
& ~ ( member_a @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_321_mk__disjoint__insert,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ? [B6: set_nat] :
( ( A2
= ( insert_nat @ A @ B6 ) )
& ~ ( member_nat @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_322_IntE,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A2 @ B ) )
=> ~ ( ( member_nat @ C2 @ A2 )
=> ~ ( member_nat @ C2 @ B ) ) ) ).
% IntE
thf(fact_323_IntE,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B ) )
=> ~ ( ( member_a @ C2 @ A2 )
=> ~ ( member_a @ C2 @ B ) ) ) ).
% IntE
thf(fact_324_IntD1,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A2 @ B ) )
=> ( member_nat @ C2 @ A2 ) ) ).
% IntD1
thf(fact_325_IntD1,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B ) )
=> ( member_a @ C2 @ A2 ) ) ).
% IntD1
thf(fact_326_IntD2,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A2 @ B ) )
=> ( member_nat @ C2 @ B ) ) ).
% IntD2
thf(fact_327_IntD2,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B ) )
=> ( member_a @ C2 @ B ) ) ).
% IntD2
thf(fact_328_Int__assoc,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B ) @ C )
= ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B @ C ) ) ) ).
% Int_assoc
thf(fact_329_Int__absorb,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_330_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A7: set_a,B7: set_a] : ( inf_inf_set_a @ B7 @ A7 ) ) ) ).
% Int_commute
thf(fact_331_Int__left__absorb,axiom,
! [A2: set_a,B: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ A2 @ B ) )
= ( inf_inf_set_a @ A2 @ B ) ) ).
% Int_left_absorb
thf(fact_332_Int__left__commute,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B @ C ) )
= ( inf_inf_set_a @ B @ ( inf_inf_set_a @ A2 @ C ) ) ) ).
% Int_left_commute
thf(fact_333_UnE,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B ) )
=> ( ~ ( member_nat @ C2 @ A2 )
=> ( member_nat @ C2 @ B ) ) ) ).
% UnE
thf(fact_334_UnE,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( sup_sup_set_a @ A2 @ B ) )
=> ( ~ ( member_a @ C2 @ A2 )
=> ( member_a @ C2 @ B ) ) ) ).
% UnE
thf(fact_335_UnI1,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ A2 )
=> ( member_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% UnI1
thf(fact_336_UnI1,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ A2 )
=> ( member_a @ C2 @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% UnI1
thf(fact_337_UnI2,axiom,
! [C2: nat,B: set_nat,A2: set_nat] :
( ( member_nat @ C2 @ B )
=> ( member_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% UnI2
thf(fact_338_UnI2,axiom,
! [C2: a,B: set_a,A2: set_a] :
( ( member_a @ C2 @ B )
=> ( member_a @ C2 @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% UnI2
thf(fact_339_bex__Un,axiom,
! [A2: set_a,B: set_a,P: a > $o] :
( ( ? [X2: a] :
( ( member_a @ X2 @ ( sup_sup_set_a @ A2 @ B ) )
& ( P @ X2 ) ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ( P @ X2 ) )
| ? [X2: a] :
( ( member_a @ X2 @ B )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_340_ball__Un,axiom,
! [A2: set_a,B: set_a,P: a > $o] :
( ( ! [X2: a] :
( ( member_a @ X2 @ ( sup_sup_set_a @ A2 @ B ) )
=> ( P @ X2 ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( P @ X2 ) )
& ! [X2: a] :
( ( member_a @ X2 @ B )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_341_Un__assoc,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A2 @ B ) @ C )
= ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B @ C ) ) ) ).
% Un_assoc
thf(fact_342_Un__absorb,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_343_Un__commute,axiom,
( sup_sup_set_a
= ( ^ [A7: set_a,B7: set_a] : ( sup_sup_set_a @ B7 @ A7 ) ) ) ).
% Un_commute
thf(fact_344_Un__left__absorb,axiom,
! [A2: set_a,B: set_a] :
( ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B ) )
= ( sup_sup_set_a @ A2 @ B ) ) ).
% Un_left_absorb
thf(fact_345_Un__left__commute,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B @ C ) )
= ( sup_sup_set_a @ B @ ( sup_sup_set_a @ A2 @ C ) ) ) ).
% Un_left_commute
thf(fact_346_additive__abelian__group_Osumset__empty_H_I2_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ( inf_inf_set_a @ A2 @ G )
= bot_bot_set_a )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B )
= bot_bot_set_a ) ) ) ).
% additive_abelian_group.sumset_empty'(2)
thf(fact_347_additive__abelian__group_Osumset__empty_H_I1_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ( inf_inf_set_a @ A2 @ G )
= bot_bot_set_a )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ B @ A2 )
= bot_bot_set_a ) ) ) ).
% additive_abelian_group.sumset_empty'(1)
thf(fact_348_additive__abelian__group_Osumset__is__empty__iff,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B )
= bot_bot_set_a )
= ( ( ( inf_inf_set_a @ A2 @ G )
= bot_bot_set_a )
| ( ( inf_inf_set_a @ B @ G )
= bot_bot_set_a ) ) ) ) ).
% additive_abelian_group.sumset_is_empty_iff
thf(fact_349_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_350_additive__abelian__group_Ominusset__is__empty__iff,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A2 )
= bot_bot_set_a )
= ( ( inf_inf_set_a @ A2 @ G )
= bot_bot_set_a ) ) ) ).
% additive_abelian_group.minusset_is_empty_iff
thf(fact_351_additive__abelian__group_Oinfinite__sumset__aux,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat,B: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ~ ( finite_finite_nat @ ( inf_inf_set_nat @ A2 @ G ) )
=> ( ( ~ ( finite_finite_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A2 @ B ) ) )
= ( ( inf_inf_set_nat @ B @ G )
!= bot_bot_set_nat ) ) ) ) ).
% additive_abelian_group.infinite_sumset_aux
thf(fact_352_additive__abelian__group_Oinfinite__sumset__aux,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ~ ( finite_finite_a @ ( inf_inf_set_a @ A2 @ G ) )
=> ( ( ~ ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) )
= ( ( inf_inf_set_a @ B @ G )
!= bot_bot_set_a ) ) ) ) ).
% additive_abelian_group.infinite_sumset_aux
thf(fact_353_additive__abelian__group_Oinfinite__sumset__iff,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat,B: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( ~ ( finite_finite_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A2 @ B ) ) )
= ( ( ~ ( finite_finite_nat @ ( inf_inf_set_nat @ A2 @ G ) )
& ( ( inf_inf_set_nat @ B @ G )
!= bot_bot_set_nat ) )
| ( ( ( inf_inf_set_nat @ A2 @ G )
!= bot_bot_set_nat )
& ~ ( finite_finite_nat @ ( inf_inf_set_nat @ B @ G ) ) ) ) ) ) ).
% additive_abelian_group.infinite_sumset_iff
thf(fact_354_additive__abelian__group_Oinfinite__sumset__iff,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ~ ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) )
= ( ( ~ ( finite_finite_a @ ( inf_inf_set_a @ A2 @ G ) )
& ( ( inf_inf_set_a @ B @ G )
!= bot_bot_set_a ) )
| ( ( ( inf_inf_set_a @ A2 @ G )
!= bot_bot_set_a )
& ~ ( finite_finite_a @ ( inf_inf_set_a @ B @ G ) ) ) ) ) ) ).
% additive_abelian_group.infinite_sumset_iff
thf(fact_355_additive__abelian__group_Osumset__D_I2_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ ( insert_a @ Zero @ bot_bot_set_a ) @ A2 )
= ( inf_inf_set_a @ A2 @ G ) ) ) ).
% additive_abelian_group.sumset_D(2)
thf(fact_356_additive__abelian__group_Osumset__D_I1_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( insert_a @ Zero @ bot_bot_set_a ) )
= ( inf_inf_set_a @ A2 @ G ) ) ) ).
% additive_abelian_group.sumset_D(1)
thf(fact_357_additive__abelian__group_Osumset__iterated__0,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn1960970773371692859ated_a @ G @ Addition @ Zero @ A2 @ zero_zero_nat )
= ( insert_a @ Zero @ bot_bot_set_a ) ) ) ).
% additive_abelian_group.sumset_iterated_0
thf(fact_358_additive__abelian__group_Ocard__sumset__le,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat,A: nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( finite_finite_nat @ A2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) ) ) ) ).
% additive_abelian_group.card_sumset_le
thf(fact_359_additive__abelian__group_Ocard__sumset__le,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,A: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_finite_a @ A2 )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A2 ) ) ) ) ).
% additive_abelian_group.card_sumset_le
thf(fact_360_insert__mono,axiom,
! [C: set_a,D2: set_a,A: a] :
( ( ord_less_eq_set_a @ C @ D2 )
=> ( ord_less_eq_set_a @ ( insert_a @ A @ C ) @ ( insert_a @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_361_subset__insert,axiom,
! [X: nat,A2: set_nat,B: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B ) )
= ( ord_less_eq_set_nat @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_362_subset__insert,axiom,
! [X: a,A2: set_a,B: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B ) )
= ( ord_less_eq_set_a @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_363_subset__insertI,axiom,
! [B: set_a,A: a] : ( ord_less_eq_set_a @ B @ ( insert_a @ A @ B ) ) ).
% subset_insertI
thf(fact_364_subset__insertI2,axiom,
! [A2: set_a,B: set_a,B4: a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ B4 @ B ) ) ) ).
% subset_insertI2
thf(fact_365_Int__mono,axiom,
! [A2: set_a,C: set_a,B: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ ( inf_inf_set_a @ C @ D2 ) ) ) ) ).
% Int_mono
thf(fact_366_Int__lower1,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ A2 ) ).
% Int_lower1
thf(fact_367_Int__lower2,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ B ) ).
% Int_lower2
thf(fact_368_Int__absorb1,axiom,
! [B: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B @ A2 )
=> ( ( inf_inf_set_a @ A2 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_369_Int__absorb2,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( inf_inf_set_a @ A2 @ B )
= A2 ) ) ).
% Int_absorb2
thf(fact_370_Int__greatest,axiom,
! [C: set_a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ A2 )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% Int_greatest
thf(fact_371_Int__Collect__mono,axiom,
! [A2: set_nat,B: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_372_Int__Collect__mono,axiom,
! [A2: set_a,B: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_373_Int__insert__left,axiom,
! [A: nat,C: set_nat,B: set_nat] :
( ( ( member_nat @ A @ C )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B ) @ C )
= ( insert_nat @ A @ ( inf_inf_set_nat @ B @ C ) ) ) )
& ( ~ ( member_nat @ A @ C )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B ) @ C )
= ( inf_inf_set_nat @ B @ C ) ) ) ) ).
% Int_insert_left
thf(fact_374_Int__insert__left,axiom,
! [A: a,C: set_a,B: set_a] :
( ( ( member_a @ A @ C )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B ) @ C )
= ( insert_a @ A @ ( inf_inf_set_a @ B @ C ) ) ) )
& ( ~ ( member_a @ A @ C )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B ) @ C )
= ( inf_inf_set_a @ B @ C ) ) ) ) ).
% Int_insert_left
thf(fact_375_Int__insert__right,axiom,
! [A: nat,A2: set_nat,B: set_nat] :
( ( ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B ) )
= ( insert_nat @ A @ ( inf_inf_set_nat @ A2 @ B ) ) ) )
& ( ~ ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B ) )
= ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_376_Int__insert__right,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B ) ) ) )
& ( ~ ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_377_Un__mono,axiom,
! [A2: set_a,C: set_a,B: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B @ D2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B ) @ ( sup_sup_set_a @ C @ D2 ) ) ) ) ).
% Un_mono
thf(fact_378_Un__least,axiom,
! [A2: set_a,C: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B ) @ C ) ) ) ).
% Un_least
thf(fact_379_Un__upper1,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B ) ) ).
% Un_upper1
thf(fact_380_Un__upper2,axiom,
! [B: set_a,A2: set_a] : ( ord_less_eq_set_a @ B @ ( sup_sup_set_a @ A2 @ B ) ) ).
% Un_upper2
thf(fact_381_Un__absorb1,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( sup_sup_set_a @ A2 @ B )
= B ) ) ).
% Un_absorb1
thf(fact_382_Un__absorb2,axiom,
! [B: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B @ A2 )
=> ( ( sup_sup_set_a @ A2 @ B )
= A2 ) ) ).
% Un_absorb2
thf(fact_383_subset__UnE,axiom,
! [C: set_a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A2 @ B ) )
=> ~ ! [A8: set_a] :
( ( ord_less_eq_set_a @ A8 @ A2 )
=> ! [B8: set_a] :
( ( ord_less_eq_set_a @ B8 @ B )
=> ( C
!= ( sup_sup_set_a @ A8 @ B8 ) ) ) ) ) ).
% subset_UnE
thf(fact_384_subset__Un__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A7: set_a,B7: set_a] :
( ( sup_sup_set_a @ A7 @ B7 )
= B7 ) ) ) ).
% subset_Un_eq
thf(fact_385_Un__Int__crazy,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ A2 @ B ) @ ( inf_inf_set_a @ B @ C ) ) @ ( inf_inf_set_a @ C @ A2 ) )
= ( inf_inf_set_a @ ( inf_inf_set_a @ ( sup_sup_set_a @ A2 @ B ) @ ( sup_sup_set_a @ B @ C ) ) @ ( sup_sup_set_a @ C @ A2 ) ) ) ).
% Un_Int_crazy
thf(fact_386_Int__Un__distrib,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( inf_inf_set_a @ A2 @ ( sup_sup_set_a @ B @ C ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ A2 @ B ) @ ( inf_inf_set_a @ A2 @ C ) ) ) ).
% Int_Un_distrib
thf(fact_387_Un__Int__distrib,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ A2 @ ( inf_inf_set_a @ B @ C ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ A2 @ B ) @ ( sup_sup_set_a @ A2 @ C ) ) ) ).
% Un_Int_distrib
thf(fact_388_Int__Un__distrib2,axiom,
! [B: set_a,C: set_a,A2: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ B @ C ) @ A2 )
= ( sup_sup_set_a @ ( inf_inf_set_a @ B @ A2 ) @ ( inf_inf_set_a @ C @ A2 ) ) ) ).
% Int_Un_distrib2
thf(fact_389_Un__Int__distrib2,axiom,
! [B: set_a,C: set_a,A2: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ B @ C ) @ A2 )
= ( inf_inf_set_a @ ( sup_sup_set_a @ B @ A2 ) @ ( sup_sup_set_a @ C @ A2 ) ) ) ).
% Un_Int_distrib2
thf(fact_390_additive__abelian__group_Ocard__sumset__singleton__eq,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat,A: nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( finite_finite_nat @ A2 )
=> ( ( ( member_nat @ A @ G )
=> ( ( finite_card_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( finite_card_nat @ ( inf_inf_set_nat @ A2 @ G ) ) ) )
& ( ~ ( member_nat @ A @ G )
=> ( ( finite_card_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= zero_zero_nat ) ) ) ) ) ).
% additive_abelian_group.card_sumset_singleton_eq
thf(fact_391_additive__abelian__group_Ocard__sumset__singleton__eq,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,A: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_finite_a @ A2 )
=> ( ( ( member_a @ A @ G )
=> ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( finite_card_a @ ( inf_inf_set_a @ A2 @ G ) ) ) )
& ( ~ ( member_a @ A @ G )
=> ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= zero_zero_nat ) ) ) ) ) ).
% additive_abelian_group.card_sumset_singleton_eq
thf(fact_392_Un__Int__assoc__eq,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( ( sup_sup_set_a @ ( inf_inf_set_a @ A2 @ B ) @ C )
= ( inf_inf_set_a @ A2 @ ( sup_sup_set_a @ B @ C ) ) )
= ( ord_less_eq_set_a @ C @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_393_sumset__insert2,axiom,
! [B: set_a,A2: set_a,X: a] :
( ( nO_MATCH_set_a_set_a @ bot_bot_set_a @ B )
=> ( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( insert_a @ X @ B ) )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ) ) ).
% sumset_insert2
thf(fact_394_sumset__insert1,axiom,
! [A2: set_a,X: a,B: set_a] :
( ( nO_MATCH_set_a_set_a @ bot_bot_set_a @ A2 )
=> ( ( pluenn3038260743871226533mset_a @ g @ addition @ ( insert_a @ X @ A2 ) @ B )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ ( insert_a @ X @ bot_bot_set_a ) @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ) ) ).
% sumset_insert1
thf(fact_395_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_396_sumsetdiff__sing,axiom,
! [A2: set_a,B: set_a,X: a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ ( minus_minus_set_a @ A2 @ B ) @ ( insert_a @ X @ bot_bot_set_a ) )
= ( minus_minus_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ B @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% sumsetdiff_sing
thf(fact_397_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_398_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_399_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_400_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_401_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_402_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_403_inf_Oright__idem,axiom,
! [A: set_a,B4: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B4 ) @ B4 )
= ( inf_inf_set_a @ A @ B4 ) ) ).
% inf.right_idem
thf(fact_404_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_405_inf_Oleft__idem,axiom,
! [A: set_a,B4: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ A @ B4 ) )
= ( inf_inf_set_a @ A @ B4 ) ) ).
% inf.left_idem
thf(fact_406_inf__idem,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ X )
= X ) ).
% inf_idem
thf(fact_407_inf_Oidem,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ A )
= A ) ).
% inf.idem
thf(fact_408_sup_Oright__idem,axiom,
! [A: set_a,B4: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A @ B4 ) @ B4 )
= ( sup_sup_set_a @ A @ B4 ) ) ).
% sup.right_idem
thf(fact_409_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_410_sup_Oleft__idem,axiom,
! [A: set_a,B4: set_a] :
( ( sup_sup_set_a @ A @ ( sup_sup_set_a @ A @ B4 ) )
= ( sup_sup_set_a @ A @ B4 ) ) ).
% sup.left_idem
thf(fact_411_sup__idem,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ X )
= X ) ).
% sup_idem
thf(fact_412_sup_Oidem,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ A )
= A ) ).
% sup.idem
thf(fact_413_DiffI,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ A2 )
=> ( ~ ( member_nat @ C2 @ B )
=> ( member_nat @ C2 @ ( minus_minus_set_nat @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_414_DiffI,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ A2 )
=> ( ~ ( member_a @ C2 @ B )
=> ( member_a @ C2 @ ( minus_minus_set_a @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_415_Diff__iff,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A2 @ B ) )
= ( ( member_nat @ C2 @ A2 )
& ~ ( member_nat @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_416_Diff__iff,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A2 @ B ) )
= ( ( member_a @ C2 @ A2 )
& ~ ( member_a @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_417_Diff__idemp,axiom,
! [A2: set_a,B: set_a] :
( ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ B ) @ B )
= ( minus_minus_set_a @ A2 @ B ) ) ).
% Diff_idemp
thf(fact_418_le__inf__iff,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z2 ) )
= ( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_419_le__inf__iff,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) )
= ( ( ord_less_eq_set_a @ X @ Y )
& ( ord_less_eq_set_a @ X @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_420_inf_Obounded__iff,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B4 @ C2 ) )
= ( ( ord_less_eq_nat @ A @ B4 )
& ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_421_inf_Obounded__iff,axiom,
! [A: set_a,B4: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B4 @ C2 ) )
= ( ( ord_less_eq_set_a @ A @ B4 )
& ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_422_le__sup__iff,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z2 )
= ( ( ord_less_eq_nat @ X @ Z2 )
& ( ord_less_eq_nat @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_423_le__sup__iff,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z2 )
= ( ( ord_less_eq_set_a @ X @ Z2 )
& ( ord_less_eq_set_a @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_424_sup_Obounded__iff,axiom,
! [B4: nat,C2: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B4 @ C2 ) @ A )
= ( ( ord_less_eq_nat @ B4 @ A )
& ( ord_less_eq_nat @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_425_sup_Obounded__iff,axiom,
! [B4: set_a,C2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B4 @ C2 ) @ A )
= ( ( ord_less_eq_set_a @ B4 @ A )
& ( ord_less_eq_set_a @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_426_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_427_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_428_sup__bot_Oright__neutral,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ bot_bot_set_a )
= A ) ).
% sup_bot.right_neutral
thf(fact_429_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_a,B4: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ A @ B4 ) )
= ( ( A = bot_bot_set_a )
& ( B4 = bot_bot_set_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_430_sup__bot_Oleft__neutral,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_431_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_a,B4: set_a] :
( ( ( sup_sup_set_a @ A @ B4 )
= bot_bot_set_a )
= ( ( A = bot_bot_set_a )
& ( B4 = bot_bot_set_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_432_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_433_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_434_sup__bot__right,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ bot_bot_set_a )
= X ) ).
% sup_bot_right
thf(fact_435_sup__bot__left,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ X )
= X ) ).
% sup_bot_left
thf(fact_436_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_437_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_438_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_439_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_440_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_441_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_442_Diff__cancel,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ A2 )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_443_empty__Diff,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A2 )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_444_Diff__empty,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% Diff_empty
thf(fact_445_finite__Diff2,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B ) )
= ( finite_finite_nat @ A2 ) ) ) ).
% finite_Diff2
thf(fact_446_finite__Diff2,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B ) )
= ( finite_finite_a @ A2 ) ) ) ).
% finite_Diff2
thf(fact_447_finite__Diff,axiom,
! [A2: set_nat,B: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B ) ) ) ).
% finite_Diff
thf(fact_448_finite__Diff,axiom,
! [A2: set_a,B: set_a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B ) ) ) ).
% finite_Diff
thf(fact_449_insert__Diff1,axiom,
! [X: nat,B: set_nat,A2: set_nat] :
( ( member_nat @ X @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B )
= ( minus_minus_set_nat @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_450_insert__Diff1,axiom,
! [X: a,B: set_a,A2: set_a] :
( ( member_a @ X @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B )
= ( minus_minus_set_a @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_451_Diff__insert0,axiom,
! [X: nat,A2: set_nat,B: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ B ) )
= ( minus_minus_set_nat @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_452_Diff__insert0,axiom,
! [X: a,A2: set_a,B: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( minus_minus_set_a @ A2 @ ( insert_a @ X @ B ) )
= ( minus_minus_set_a @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_453_Un__Diff__cancel2,axiom,
! [B: set_a,A2: set_a] :
( ( sup_sup_set_a @ ( minus_minus_set_a @ B @ A2 ) @ A2 )
= ( sup_sup_set_a @ B @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_454_Un__Diff__cancel,axiom,
! [A2: set_a,B: set_a] :
( ( sup_sup_set_a @ A2 @ ( minus_minus_set_a @ B @ A2 ) )
= ( sup_sup_set_a @ A2 @ B ) ) ).
% Un_Diff_cancel
thf(fact_455_Diff__eq__empty__iff,axiom,
! [A2: set_a,B: set_a] :
( ( ( minus_minus_set_a @ A2 @ B )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A2 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_456_insert__Diff__single,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( insert_a @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_457_finite__Diff__insert,axiom,
! [A2: set_nat,A: nat,B: set_nat] :
( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B ) ) )
= ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B ) ) ) ).
% finite_Diff_insert
thf(fact_458_finite__Diff__insert,axiom,
! [A2: set_a,A: a,B: set_a] :
( ( finite_finite_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B ) ) )
= ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B ) ) ) ).
% finite_Diff_insert
thf(fact_459_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_460_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_461_Diff__disjoint,axiom,
! [A2: set_a,B: set_a] :
( ( inf_inf_set_a @ A2 @ ( minus_minus_set_a @ B @ A2 ) )
= bot_bot_set_a ) ).
% Diff_disjoint
thf(fact_462_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_463_DiffE,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A2 @ B ) )
=> ~ ( ( member_nat @ C2 @ A2 )
=> ( member_nat @ C2 @ B ) ) ) ).
% DiffE
thf(fact_464_DiffE,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A2 @ B ) )
=> ~ ( ( member_a @ C2 @ A2 )
=> ( member_a @ C2 @ B ) ) ) ).
% DiffE
thf(fact_465_DiffD1,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A2 @ B ) )
=> ( member_nat @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_466_DiffD1,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A2 @ B ) )
=> ( member_a @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_467_DiffD2,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A2 @ B ) )
=> ~ ( member_nat @ C2 @ B ) ) ).
% DiffD2
thf(fact_468_DiffD2,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A2 @ B ) )
=> ~ ( member_a @ C2 @ B ) ) ).
% DiffD2
thf(fact_469_less__infI1,axiom,
! [A: set_a,X: set_a,B4: set_a] :
( ( ord_less_set_a @ A @ X )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A @ B4 ) @ X ) ) ).
% less_infI1
thf(fact_470_less__infI1,axiom,
! [A: nat,X: nat,B4: nat] :
( ( ord_less_nat @ A @ X )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B4 ) @ X ) ) ).
% less_infI1
thf(fact_471_less__infI2,axiom,
! [B4: set_a,X: set_a,A: set_a] :
( ( ord_less_set_a @ B4 @ X )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A @ B4 ) @ X ) ) ).
% less_infI2
thf(fact_472_less__infI2,axiom,
! [B4: nat,X: nat,A: nat] :
( ( ord_less_nat @ B4 @ X )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B4 ) @ X ) ) ).
% less_infI2
thf(fact_473_inf_Oabsorb3,axiom,
! [A: set_a,B4: set_a] :
( ( ord_less_set_a @ A @ B4 )
=> ( ( inf_inf_set_a @ A @ B4 )
= A ) ) ).
% inf.absorb3
thf(fact_474_inf_Oabsorb3,axiom,
! [A: nat,B4: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ( inf_inf_nat @ A @ B4 )
= A ) ) ).
% inf.absorb3
thf(fact_475_inf_Oabsorb4,axiom,
! [B4: set_a,A: set_a] :
( ( ord_less_set_a @ B4 @ A )
=> ( ( inf_inf_set_a @ A @ B4 )
= B4 ) ) ).
% inf.absorb4
thf(fact_476_inf_Oabsorb4,axiom,
! [B4: nat,A: nat] :
( ( ord_less_nat @ B4 @ A )
=> ( ( inf_inf_nat @ A @ B4 )
= B4 ) ) ).
% inf.absorb4
thf(fact_477_inf_Ostrict__boundedE,axiom,
! [A: set_a,B4: set_a,C2: set_a] :
( ( ord_less_set_a @ A @ ( inf_inf_set_a @ B4 @ C2 ) )
=> ~ ( ( ord_less_set_a @ A @ B4 )
=> ~ ( ord_less_set_a @ A @ C2 ) ) ) ).
% inf.strict_boundedE
thf(fact_478_inf_Ostrict__boundedE,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_nat @ A @ ( inf_inf_nat @ B4 @ C2 ) )
=> ~ ( ( ord_less_nat @ A @ B4 )
=> ~ ( ord_less_nat @ A @ C2 ) ) ) ).
% inf.strict_boundedE
thf(fact_479_inf_Ostrict__order__iff,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( A4
= ( inf_inf_set_a @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_480_inf_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] :
( ( A4
= ( inf_inf_nat @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_481_inf_Ostrict__coboundedI1,axiom,
! [A: set_a,C2: set_a,B4: set_a] :
( ( ord_less_set_a @ A @ C2 )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A @ B4 ) @ C2 ) ) ).
% inf.strict_coboundedI1
thf(fact_482_inf_Ostrict__coboundedI1,axiom,
! [A: nat,C2: nat,B4: nat] :
( ( ord_less_nat @ A @ C2 )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B4 ) @ C2 ) ) ).
% inf.strict_coboundedI1
thf(fact_483_inf_Ostrict__coboundedI2,axiom,
! [B4: set_a,C2: set_a,A: set_a] :
( ( ord_less_set_a @ B4 @ C2 )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A @ B4 ) @ C2 ) ) ).
% inf.strict_coboundedI2
thf(fact_484_inf_Ostrict__coboundedI2,axiom,
! [B4: nat,C2: nat,A: nat] :
( ( ord_less_nat @ B4 @ C2 )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B4 ) @ C2 ) ) ).
% inf.strict_coboundedI2
thf(fact_485_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_486_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_487_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_488_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_489_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_490_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_491_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_492_less__supI1,axiom,
! [X: set_a,A: set_a,B4: set_a] :
( ( ord_less_set_a @ X @ A )
=> ( ord_less_set_a @ X @ ( sup_sup_set_a @ A @ B4 ) ) ) ).
% less_supI1
thf(fact_493_less__supI1,axiom,
! [X: nat,A: nat,B4: nat] :
( ( ord_less_nat @ X @ A )
=> ( ord_less_nat @ X @ ( sup_sup_nat @ A @ B4 ) ) ) ).
% less_supI1
thf(fact_494_less__supI2,axiom,
! [X: set_a,B4: set_a,A: set_a] :
( ( ord_less_set_a @ X @ B4 )
=> ( ord_less_set_a @ X @ ( sup_sup_set_a @ A @ B4 ) ) ) ).
% less_supI2
thf(fact_495_less__supI2,axiom,
! [X: nat,B4: nat,A: nat] :
( ( ord_less_nat @ X @ B4 )
=> ( ord_less_nat @ X @ ( sup_sup_nat @ A @ B4 ) ) ) ).
% less_supI2
thf(fact_496_sup_Oabsorb3,axiom,
! [B4: set_a,A: set_a] :
( ( ord_less_set_a @ B4 @ A )
=> ( ( sup_sup_set_a @ A @ B4 )
= A ) ) ).
% sup.absorb3
thf(fact_497_sup_Oabsorb3,axiom,
! [B4: nat,A: nat] :
( ( ord_less_nat @ B4 @ A )
=> ( ( sup_sup_nat @ A @ B4 )
= A ) ) ).
% sup.absorb3
thf(fact_498_sup_Oabsorb4,axiom,
! [A: set_a,B4: set_a] :
( ( ord_less_set_a @ A @ B4 )
=> ( ( sup_sup_set_a @ A @ B4 )
= B4 ) ) ).
% sup.absorb4
thf(fact_499_sup_Oabsorb4,axiom,
! [A: nat,B4: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ( sup_sup_nat @ A @ B4 )
= B4 ) ) ).
% sup.absorb4
thf(fact_500_sup_Ostrict__boundedE,axiom,
! [B4: set_a,C2: set_a,A: set_a] :
( ( ord_less_set_a @ ( sup_sup_set_a @ B4 @ C2 ) @ A )
=> ~ ( ( ord_less_set_a @ B4 @ A )
=> ~ ( ord_less_set_a @ C2 @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_501_sup_Ostrict__boundedE,axiom,
! [B4: nat,C2: nat,A: nat] :
( ( ord_less_nat @ ( sup_sup_nat @ B4 @ C2 ) @ A )
=> ~ ( ( ord_less_nat @ B4 @ A )
=> ~ ( ord_less_nat @ C2 @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_502_sup_Ostrict__order__iff,axiom,
( ord_less_set_a
= ( ^ [B3: set_a,A4: set_a] :
( ( A4
= ( sup_sup_set_a @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_503_sup_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B3: nat,A4: nat] :
( ( A4
= ( sup_sup_nat @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_504_sup_Ostrict__coboundedI1,axiom,
! [C2: set_a,A: set_a,B4: set_a] :
( ( ord_less_set_a @ C2 @ A )
=> ( ord_less_set_a @ C2 @ ( sup_sup_set_a @ A @ B4 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_505_sup_Ostrict__coboundedI1,axiom,
! [C2: nat,A: nat,B4: nat] :
( ( ord_less_nat @ C2 @ A )
=> ( ord_less_nat @ C2 @ ( sup_sup_nat @ A @ B4 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_506_sup_Ostrict__coboundedI2,axiom,
! [C2: set_a,B4: set_a,A: set_a] :
( ( ord_less_set_a @ C2 @ B4 )
=> ( ord_less_set_a @ C2 @ ( sup_sup_set_a @ A @ B4 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_507_sup_Ostrict__coboundedI2,axiom,
! [C2: nat,B4: nat,A: nat] :
( ( ord_less_nat @ C2 @ B4 )
=> ( ord_less_nat @ C2 @ ( sup_sup_nat @ A @ B4 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_508_less__mono__imp__le__mono,axiom,
! [F3: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F3 @ I2 ) @ ( F3 @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F3 @ I ) @ ( F3 @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_509_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_510_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_511_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
| ( M3 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_512_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_513_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
& ( M3 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_514_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_515_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_516_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_517_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_518_less__not__refl3,axiom,
! [S2: nat,T4: nat] :
( ( ord_less_nat @ S2 @ T4 )
=> ( S2 != T4 ) ) ).
% less_not_refl3
thf(fact_519_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_520_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_521_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_522_card__less__sym__Diff,axiom,
! [A2: set_nat,B: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B ) )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_523_card__less__sym__Diff,axiom,
! [A2: set_a,B: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B )
=> ( ( ord_less_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B ) )
=> ( ord_less_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ B ) ) @ ( finite_card_a @ ( minus_minus_set_a @ B @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_524_Diff__infinite__finite,axiom,
! [T2: set_nat,S: set_nat] :
( ( finite_finite_nat @ T2 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_525_Diff__infinite__finite,axiom,
! [T2: set_a,S: set_a] :
( ( finite_finite_a @ T2 )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_526_double__diff,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ( minus_minus_set_a @ B @ ( minus_minus_set_a @ C @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_527_Diff__subset,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B ) @ A2 ) ).
% Diff_subset
thf(fact_528_Diff__mono,axiom,
! [A2: set_a,C: set_a,D2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ D2 @ B )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B ) @ ( minus_minus_set_a @ C @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_529_insert__Diff__if,axiom,
! [X: nat,B: set_nat,A2: set_nat] :
( ( ( member_nat @ X @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B )
= ( minus_minus_set_nat @ A2 @ B ) ) )
& ( ~ ( member_nat @ X @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B )
= ( insert_nat @ X @ ( minus_minus_set_nat @ A2 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_530_insert__Diff__if,axiom,
! [X: a,B: set_a,A2: set_a] :
( ( ( member_a @ X @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B )
= ( minus_minus_set_a @ A2 @ B ) ) )
& ( ~ ( member_a @ X @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B )
= ( insert_a @ X @ ( minus_minus_set_a @ A2 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_531_Diff__Int__distrib2,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( inf_inf_set_a @ ( minus_minus_set_a @ A2 @ B ) @ C )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ C ) @ ( inf_inf_set_a @ B @ C ) ) ) ).
% Diff_Int_distrib2
thf(fact_532_Diff__Int__distrib,axiom,
! [C: set_a,A2: set_a,B: set_a] :
( ( inf_inf_set_a @ C @ ( minus_minus_set_a @ A2 @ B ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ C @ A2 ) @ ( inf_inf_set_a @ C @ B ) ) ) ).
% Diff_Int_distrib
thf(fact_533_Diff__Diff__Int,axiom,
! [A2: set_a,B: set_a] :
( ( minus_minus_set_a @ A2 @ ( minus_minus_set_a @ A2 @ B ) )
= ( inf_inf_set_a @ A2 @ B ) ) ).
% Diff_Diff_Int
thf(fact_534_Diff__Int2,axiom,
! [A2: set_a,C: set_a,B: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ C ) @ ( inf_inf_set_a @ B @ C ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ C ) @ B ) ) ).
% Diff_Int2
thf(fact_535_Int__Diff,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ B ) @ C )
= ( inf_inf_set_a @ A2 @ ( minus_minus_set_a @ B @ C ) ) ) ).
% Int_Diff
thf(fact_536_Un__Diff,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( minus_minus_set_a @ ( sup_sup_set_a @ A2 @ B ) @ C )
= ( sup_sup_set_a @ ( minus_minus_set_a @ A2 @ C ) @ ( minus_minus_set_a @ B @ C ) ) ) ).
% Un_Diff
thf(fact_537_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_538_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_539_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_540_card__Diff1__less,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) ) ) ) ).
% card_Diff1_less
thf(fact_541_card__Diff1__less,axiom,
! [A2: set_a,X: a] :
( ( finite_finite_a @ A2 )
=> ( ( member_a @ X @ A2 )
=> ( ord_less_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A2 ) ) ) ) ).
% card_Diff1_less
thf(fact_542_card__Diff2__less,axiom,
! [A2: set_nat,X: nat,Y: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ ( insert_nat @ Y @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_543_card__Diff2__less,axiom,
! [A2: set_a,X: a,Y: a] :
( ( finite_finite_a @ A2 )
=> ( ( member_a @ X @ A2 )
=> ( ( member_a @ Y @ A2 )
=> ( ord_less_nat @ ( finite_card_a @ ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ ( insert_a @ Y @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A2 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_544_card__Diff1__less__iff,axiom,
! [A2: set_nat,X: nat] :
( ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) )
= ( ( finite_finite_nat @ A2 )
& ( member_nat @ X @ A2 ) ) ) ).
% card_Diff1_less_iff
thf(fact_545_card__Diff1__less__iff,axiom,
! [A2: set_a,X: a] :
( ( ord_less_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A2 ) )
= ( ( finite_finite_a @ A2 )
& ( member_a @ X @ A2 ) ) ) ).
% card_Diff1_less_iff
thf(fact_546_Diff__insert,axiom,
! [A2: set_a,A: a,B: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ B ) @ ( insert_a @ A @ bot_bot_set_a ) ) ) ).
% Diff_insert
thf(fact_547_insert__Diff,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_548_insert__Diff,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_549_Diff__insert2,axiom,
! [A2: set_a,A: a,B: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) @ B ) ) ).
% Diff_insert2
thf(fact_550_Diff__insert__absorb,axiom,
! [X: nat,A2: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_551_Diff__insert__absorb,axiom,
! [X: a,A2: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ ( insert_a @ X @ bot_bot_set_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_552_subset__Diff__insert,axiom,
! [A2: set_nat,B: set_nat,X: nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B @ ( insert_nat @ X @ C ) ) )
= ( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B @ C ) )
& ~ ( member_nat @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_553_subset__Diff__insert,axiom,
! [A2: set_a,B: set_a,X: a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B @ ( insert_a @ X @ C ) ) )
= ( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B @ C ) )
& ~ ( member_a @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_554_Diff__triv,axiom,
! [A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a )
=> ( ( minus_minus_set_a @ A2 @ B )
= A2 ) ) ).
% Diff_triv
thf(fact_555_Int__Diff__disjoint,axiom,
! [A2: set_a,B: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B ) @ ( minus_minus_set_a @ A2 @ B ) )
= bot_bot_set_a ) ).
% Int_Diff_disjoint
thf(fact_556_Diff__partition,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( sup_sup_set_a @ A2 @ ( minus_minus_set_a @ B @ A2 ) )
= B ) ) ).
% Diff_partition
thf(fact_557_Diff__subset__conv,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B ) @ C )
= ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ B @ C ) ) ) ).
% Diff_subset_conv
thf(fact_558_Un__Diff__Int,axiom,
! [A2: set_a,B: set_a] :
( ( sup_sup_set_a @ ( minus_minus_set_a @ A2 @ B ) @ ( inf_inf_set_a @ A2 @ B ) )
= A2 ) ).
% Un_Diff_Int
thf(fact_559_Int__Diff__Un,axiom,
! [A2: set_a,B: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ A2 @ B ) @ ( minus_minus_set_a @ A2 @ B ) )
= A2 ) ).
% Int_Diff_Un
thf(fact_560_Diff__Int,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( minus_minus_set_a @ A2 @ ( inf_inf_set_a @ B @ C ) )
= ( sup_sup_set_a @ ( minus_minus_set_a @ A2 @ B ) @ ( minus_minus_set_a @ A2 @ C ) ) ) ).
% Diff_Int
thf(fact_561_Diff__Un,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( minus_minus_set_a @ A2 @ ( sup_sup_set_a @ B @ C ) )
= ( inf_inf_set_a @ ( minus_minus_set_a @ A2 @ B ) @ ( minus_minus_set_a @ A2 @ C ) ) ) ).
% Diff_Un
thf(fact_562_infinite__remove,axiom,
! [S: set_nat,A: nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% infinite_remove
thf(fact_563_infinite__remove,axiom,
! [S: set_a,A: a] :
( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% infinite_remove
thf(fact_564_infinite__coinduct,axiom,
! [X4: set_nat > $o,A2: set_nat] :
( ( X4 @ A2 )
=> ( ! [A6: set_nat] :
( ( X4 @ A6 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A6 )
& ( ( X4 @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) )
| ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) ) ) )
=> ~ ( finite_finite_nat @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_565_infinite__coinduct,axiom,
! [X4: set_a > $o,A2: set_a] :
( ( X4 @ A2 )
=> ( ! [A6: set_a] :
( ( X4 @ A6 )
=> ? [X5: a] :
( ( member_a @ X5 @ A6 )
& ( ( X4 @ ( minus_minus_set_a @ A6 @ ( insert_a @ X5 @ bot_bot_set_a ) ) )
| ~ ( finite_finite_a @ ( minus_minus_set_a @ A6 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) ) ) )
=> ~ ( finite_finite_a @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_566_finite__empty__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( P @ A2 )
=> ( ! [A3: nat,A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ( member_nat @ A3 @ A6 )
=> ( ( P @ A6 )
=> ( P @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) ) ) ) )
=> ( P @ bot_bot_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_567_finite__empty__induct,axiom,
! [A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ A2 )
=> ( ( P @ A2 )
=> ( ! [A3: a,A6: set_a] :
( ( finite_finite_a @ A6 )
=> ( ( member_a @ A3 @ A6 )
=> ( ( P @ A6 )
=> ( P @ ( minus_minus_set_a @ A6 @ ( insert_a @ A3 @ bot_bot_set_a ) ) ) ) ) )
=> ( P @ bot_bot_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_568_subset__insert__iff,axiom,
! [A2: set_nat,X: nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B ) )
= ( ( ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B ) )
& ( ~ ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_569_subset__insert__iff,axiom,
! [A2: set_a,X: a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B ) )
= ( ( ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_570_Diff__single__insert,axiom,
! [A2: set_a,X: a,B: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B ) ) ) ).
% Diff_single_insert
thf(fact_571_card__le__sym__Diff,axiom,
! [A2: set_nat,B: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_572_card__le__sym__Diff,axiom,
! [A2: set_a,B: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ B ) ) @ ( finite_card_a @ ( minus_minus_set_a @ B @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_573_card__ge__0__finite,axiom,
! [A2: set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ A2 ) )
=> ( finite_finite_a @ A2 ) ) ).
% card_ge_0_finite
thf(fact_574_card__ge__0__finite,axiom,
! [A2: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A2 ) )
=> ( finite_finite_nat @ A2 ) ) ).
% card_ge_0_finite
thf(fact_575_inf__left__commute,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) )
= ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_576_inf_Oleft__commute,axiom,
! [B4: set_a,A: set_a,C2: set_a] :
( ( inf_inf_set_a @ B4 @ ( inf_inf_set_a @ A @ C2 ) )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B4 @ C2 ) ) ) ).
% inf.left_commute
thf(fact_577_inf__commute,axiom,
( inf_inf_set_a
= ( ^ [X2: set_a,Y3: set_a] : ( inf_inf_set_a @ Y3 @ X2 ) ) ) ).
% inf_commute
thf(fact_578_inf_Ocommute,axiom,
( inf_inf_set_a
= ( ^ [A4: set_a,B3: set_a] : ( inf_inf_set_a @ B3 @ A4 ) ) ) ).
% inf.commute
thf(fact_579_inf__assoc,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z2 )
= ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) ) ) ).
% inf_assoc
thf(fact_580_inf_Oassoc,axiom,
! [A: set_a,B4: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B4 ) @ C2 )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B4 @ C2 ) ) ) ).
% inf.assoc
thf(fact_581_inf__sup__aci_I1_J,axiom,
( inf_inf_set_a
= ( ^ [X2: set_a,Y3: set_a] : ( inf_inf_set_a @ Y3 @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_582_inf__sup__aci_I2_J,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z2 )
= ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_583_inf__sup__aci_I3_J,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) )
= ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_584_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_585_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_586_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_587_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_588_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_589_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_590_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B4: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B4 ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_591_sup__left__commute,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z2 ) )
= ( sup_sup_set_a @ Y @ ( sup_sup_set_a @ X @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_592_sup_Oleft__commute,axiom,
! [B4: set_a,A: set_a,C2: set_a] :
( ( sup_sup_set_a @ B4 @ ( sup_sup_set_a @ A @ C2 ) )
= ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B4 @ C2 ) ) ) ).
% sup.left_commute
thf(fact_593_sup__commute,axiom,
( sup_sup_set_a
= ( ^ [X2: set_a,Y3: set_a] : ( sup_sup_set_a @ Y3 @ X2 ) ) ) ).
% sup_commute
thf(fact_594_sup_Ocommute,axiom,
( sup_sup_set_a
= ( ^ [A4: set_a,B3: set_a] : ( sup_sup_set_a @ B3 @ A4 ) ) ) ).
% sup.commute
thf(fact_595_sup__assoc,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z2 )
= ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z2 ) ) ) ).
% sup_assoc
thf(fact_596_sup_Oassoc,axiom,
! [A: set_a,B4: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A @ B4 ) @ C2 )
= ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B4 @ C2 ) ) ) ).
% sup.assoc
thf(fact_597_inf__sup__aci_I5_J,axiom,
( sup_sup_set_a
= ( ^ [X2: set_a,Y3: set_a] : ( sup_sup_set_a @ Y3 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_598_inf__sup__aci_I6_J,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z2 )
= ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_599_inf__sup__aci_I7_J,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z2 ) )
= ( sup_sup_set_a @ Y @ ( sup_sup_set_a @ X @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_600_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_601_remove__induct,axiom,
! [P: set_nat > $o,B: set_nat] :
( ( P @ bot_bot_set_nat )
=> ( ( ~ ( finite_finite_nat @ B )
=> ( P @ B ) )
=> ( ! [A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ( A6 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A6 @ B )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A6 )
=> ( P @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_602_remove__induct,axiom,
! [P: set_a > $o,B: set_a] :
( ( P @ bot_bot_set_a )
=> ( ( ~ ( finite_finite_a @ B )
=> ( P @ B ) )
=> ( ! [A6: set_a] :
( ( finite_finite_a @ A6 )
=> ( ( A6 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A6 @ B )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A6 )
=> ( P @ ( minus_minus_set_a @ A6 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_603_finite__remove__induct,axiom,
! [B: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ B )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ( A6 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A6 @ B )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A6 )
=> ( P @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_604_finite__remove__induct,axiom,
! [B: set_a,P: set_a > $o] :
( ( finite_finite_a @ B )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A6: set_a] :
( ( finite_finite_a @ A6 )
=> ( ( A6 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A6 @ B )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A6 )
=> ( P @ ( minus_minus_set_a @ A6 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_605_card__Diff1__le,axiom,
! [A2: set_a,X: a] : ( ord_less_eq_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A2 ) ) ).
% card_Diff1_le
thf(fact_606_card__gt__0__iff,axiom,
! [A2: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A2 ) )
= ( ( A2 != bot_bot_set_nat )
& ( finite_finite_nat @ A2 ) ) ) ).
% card_gt_0_iff
thf(fact_607_card__gt__0__iff,axiom,
! [A2: set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ A2 ) )
= ( ( A2 != bot_bot_set_a )
& ( finite_finite_a @ A2 ) ) ) ).
% card_gt_0_iff
thf(fact_608_additive__abelian__group_Osumsetdiff__sing,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a,X: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ ( minus_minus_set_a @ A2 @ B ) @ ( insert_a @ X @ bot_bot_set_a ) )
= ( minus_minus_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( 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_609_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_610_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_611_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_612_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_613_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_614_inf__le1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_615_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_616_inf__le2,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_617_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_618_le__infE,axiom,
! [X: nat,A: nat,B4: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B4 ) )
=> ~ ( ( ord_less_eq_nat @ X @ A )
=> ~ ( ord_less_eq_nat @ X @ B4 ) ) ) ).
% le_infE
thf(fact_619_le__infE,axiom,
! [X: set_a,A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A @ B4 ) )
=> ~ ( ( ord_less_eq_set_a @ X @ A )
=> ~ ( ord_less_eq_set_a @ X @ B4 ) ) ) ).
% le_infE
thf(fact_620_le__infI,axiom,
! [X: nat,A: nat,B4: nat] :
( ( ord_less_eq_nat @ X @ A )
=> ( ( ord_less_eq_nat @ X @ B4 )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B4 ) ) ) ) ).
% le_infI
thf(fact_621_le__infI,axiom,
! [X: set_a,A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ X @ A )
=> ( ( ord_less_eq_set_a @ X @ B4 )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A @ B4 ) ) ) ) ).
% le_infI
thf(fact_622_inf__mono,axiom,
! [A: nat,C2: nat,B4: nat,D: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B4 @ D )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B4 ) @ ( inf_inf_nat @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_623_inf__mono,axiom,
! [A: set_a,C2: set_a,B4: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B4 @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B4 ) @ ( inf_inf_set_a @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_624_le__infI1,axiom,
! [A: nat,X: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B4 ) @ X ) ) ).
% le_infI1
thf(fact_625_le__infI1,axiom,
! [A: set_a,X: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B4 ) @ X ) ) ).
% le_infI1
thf(fact_626_le__infI2,axiom,
! [B4: nat,X: nat,A: nat] :
( ( ord_less_eq_nat @ B4 @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B4 ) @ X ) ) ).
% le_infI2
thf(fact_627_le__infI2,axiom,
! [B4: set_a,X: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B4 @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B4 ) @ X ) ) ).
% le_infI2
thf(fact_628_inf_OorderE,axiom,
! [A: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( A
= ( inf_inf_nat @ A @ B4 ) ) ) ).
% inf.orderE
thf(fact_629_inf_OorderE,axiom,
! [A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( A
= ( inf_inf_set_a @ A @ B4 ) ) ) ).
% inf.orderE
thf(fact_630_inf_OorderI,axiom,
! [A: nat,B4: nat] :
( ( A
= ( inf_inf_nat @ A @ B4 ) )
=> ( ord_less_eq_nat @ A @ B4 ) ) ).
% inf.orderI
thf(fact_631_inf_OorderI,axiom,
! [A: set_a,B4: set_a] :
( ( A
= ( inf_inf_set_a @ A @ B4 ) )
=> ( ord_less_eq_set_a @ A @ B4 ) ) ).
% inf.orderI
thf(fact_632_inf__unique,axiom,
! [F3: nat > nat > nat,X: nat,Y: nat] :
( ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ ( F3 @ X3 @ Y2 ) @ X3 )
=> ( ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ ( F3 @ X3 @ Y2 ) @ Y2 )
=> ( ! [X3: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ( ord_less_eq_nat @ X3 @ Z3 )
=> ( ord_less_eq_nat @ X3 @ ( F3 @ Y2 @ Z3 ) ) ) )
=> ( ( inf_inf_nat @ X @ Y )
= ( F3 @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_633_inf__unique,axiom,
! [F3: set_a > set_a > set_a,X: set_a,Y: set_a] :
( ! [X3: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( F3 @ X3 @ Y2 ) @ X3 )
=> ( ! [X3: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( F3 @ X3 @ Y2 ) @ Y2 )
=> ( ! [X3: set_a,Y2: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ( ord_less_eq_set_a @ X3 @ Z3 )
=> ( ord_less_eq_set_a @ X3 @ ( F3 @ Y2 @ Z3 ) ) ) )
=> ( ( inf_inf_set_a @ X @ Y )
= ( F3 @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_634_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y3: nat] :
( ( inf_inf_nat @ X2 @ Y3 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_635_le__iff__inf,axiom,
( ord_less_eq_set_a
= ( ^ [X2: set_a,Y3: set_a] :
( ( inf_inf_set_a @ X2 @ Y3 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_636_inf_Oabsorb1,axiom,
! [A: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( inf_inf_nat @ A @ B4 )
= A ) ) ).
% inf.absorb1
thf(fact_637_inf_Oabsorb1,axiom,
! [A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( inf_inf_set_a @ A @ B4 )
= A ) ) ).
% inf.absorb1
thf(fact_638_inf_Oabsorb2,axiom,
! [B4: nat,A: nat] :
( ( ord_less_eq_nat @ B4 @ A )
=> ( ( inf_inf_nat @ A @ B4 )
= B4 ) ) ).
% inf.absorb2
thf(fact_639_inf_Oabsorb2,axiom,
! [B4: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( ( inf_inf_set_a @ A @ B4 )
= B4 ) ) ).
% inf.absorb2
thf(fact_640_inf__absorb1,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( inf_inf_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_641_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_642_inf__absorb2,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( inf_inf_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_643_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_644_inf_OboundedE,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B4 @ C2 ) )
=> ~ ( ( ord_less_eq_nat @ A @ B4 )
=> ~ ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% inf.boundedE
thf(fact_645_inf_OboundedE,axiom,
! [A: set_a,B4: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B4 @ C2 ) )
=> ~ ( ( ord_less_eq_set_a @ A @ B4 )
=> ~ ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% inf.boundedE
thf(fact_646_inf_OboundedI,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ A @ C2 )
=> ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B4 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_647_inf_OboundedI,axiom,
! [A: set_a,B4: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( ord_less_eq_set_a @ A @ C2 )
=> ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B4 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_648_inf__greatest,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Z2 )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_649_inf__greatest,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ X @ Z2 )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_650_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( A4
= ( inf_inf_nat @ A4 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_651_inf_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B3: set_a] :
( A4
= ( inf_inf_set_a @ A4 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_652_inf_Ocobounded1,axiom,
! [A: nat,B4: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B4 ) @ A ) ).
% inf.cobounded1
thf(fact_653_inf_Ocobounded1,axiom,
! [A: set_a,B4: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B4 ) @ A ) ).
% inf.cobounded1
thf(fact_654_inf_Ocobounded2,axiom,
! [A: nat,B4: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B4 ) @ B4 ) ).
% inf.cobounded2
thf(fact_655_inf_Ocobounded2,axiom,
! [A: set_a,B4: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B4 ) @ B4 ) ).
% inf.cobounded2
thf(fact_656_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( ( inf_inf_nat @ A4 @ B3 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_657_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( inf_inf_set_a @ A4 @ B3 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_658_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( ( inf_inf_nat @ A4 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_659_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A4: set_a] :
( ( inf_inf_set_a @ A4 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_660_inf_OcoboundedI1,axiom,
! [A: nat,C2: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B4 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_661_inf_OcoboundedI1,axiom,
! [A: set_a,C2: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B4 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_662_inf_OcoboundedI2,axiom,
! [B4: nat,C2: nat,A: nat] :
( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B4 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_663_inf_OcoboundedI2,axiom,
! [B4: set_a,C2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B4 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_664_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_665_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_666_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_667_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_668_le__supE,axiom,
! [A: nat,B4: nat,X: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B4 ) @ X )
=> ~ ( ( ord_less_eq_nat @ A @ X )
=> ~ ( ord_less_eq_nat @ B4 @ X ) ) ) ).
% le_supE
thf(fact_669_le__supE,axiom,
! [A: set_a,B4: set_a,X: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B4 ) @ X )
=> ~ ( ( ord_less_eq_set_a @ A @ X )
=> ~ ( ord_less_eq_set_a @ B4 @ X ) ) ) ).
% le_supE
thf(fact_670_le__supI,axiom,
! [A: nat,X: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ X )
=> ( ( ord_less_eq_nat @ B4 @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B4 ) @ X ) ) ) ).
% le_supI
thf(fact_671_le__supI,axiom,
! [A: set_a,X: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A @ X )
=> ( ( ord_less_eq_set_a @ B4 @ X )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B4 ) @ X ) ) ) ).
% le_supI
thf(fact_672_sup__ge1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_673_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_674_sup__ge2,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_675_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_676_le__supI1,axiom,
! [X: nat,A: nat,B4: nat] :
( ( ord_less_eq_nat @ X @ A )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B4 ) ) ) ).
% le_supI1
thf(fact_677_le__supI1,axiom,
! [X: set_a,A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ X @ A )
=> ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A @ B4 ) ) ) ).
% le_supI1
thf(fact_678_le__supI2,axiom,
! [X: nat,B4: nat,A: nat] :
( ( ord_less_eq_nat @ X @ B4 )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B4 ) ) ) ).
% le_supI2
thf(fact_679_le__supI2,axiom,
! [X: set_a,B4: set_a,A: set_a] :
( ( ord_less_eq_set_a @ X @ B4 )
=> ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A @ B4 ) ) ) ).
% le_supI2
thf(fact_680_sup_Omono,axiom,
! [C2: nat,A: nat,D: nat,B4: nat] :
( ( ord_less_eq_nat @ C2 @ A )
=> ( ( ord_less_eq_nat @ D @ B4 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C2 @ D ) @ ( sup_sup_nat @ A @ B4 ) ) ) ) ).
% sup.mono
thf(fact_681_sup_Omono,axiom,
! [C2: set_a,A: set_a,D: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ C2 @ A )
=> ( ( ord_less_eq_set_a @ D @ B4 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ C2 @ D ) @ ( sup_sup_set_a @ A @ B4 ) ) ) ) ).
% sup.mono
thf(fact_682_sup__mono,axiom,
! [A: nat,C2: nat,B4: nat,D: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B4 @ D )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B4 ) @ ( sup_sup_nat @ C2 @ D ) ) ) ) ).
% sup_mono
thf(fact_683_sup__mono,axiom,
! [A: set_a,C2: set_a,B4: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B4 @ D )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B4 ) @ ( sup_sup_set_a @ C2 @ D ) ) ) ) ).
% sup_mono
thf(fact_684_sup__least,axiom,
! [Y: nat,X: nat,Z2: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ Z2 @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z2 ) @ X ) ) ) ).
% sup_least
thf(fact_685_sup__least,axiom,
! [Y: set_a,X: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( ord_less_eq_set_a @ Z2 @ X )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ Y @ Z2 ) @ X ) ) ) ).
% sup_least
thf(fact_686_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y3: nat] :
( ( sup_sup_nat @ X2 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_687_le__iff__sup,axiom,
( ord_less_eq_set_a
= ( ^ [X2: set_a,Y3: set_a] :
( ( sup_sup_set_a @ X2 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_688_sup_OorderE,axiom,
! [B4: nat,A: nat] :
( ( ord_less_eq_nat @ B4 @ A )
=> ( A
= ( sup_sup_nat @ A @ B4 ) ) ) ).
% sup.orderE
thf(fact_689_sup_OorderE,axiom,
! [B4: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( A
= ( sup_sup_set_a @ A @ B4 ) ) ) ).
% sup.orderE
thf(fact_690_sup_OorderI,axiom,
! [A: nat,B4: nat] :
( ( A
= ( sup_sup_nat @ A @ B4 ) )
=> ( ord_less_eq_nat @ B4 @ A ) ) ).
% sup.orderI
thf(fact_691_sup_OorderI,axiom,
! [A: set_a,B4: set_a] :
( ( A
= ( sup_sup_set_a @ A @ B4 ) )
=> ( ord_less_eq_set_a @ B4 @ A ) ) ).
% sup.orderI
thf(fact_692_sup__unique,axiom,
! [F3: nat > nat > nat,X: nat,Y: nat] :
( ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ X3 @ ( F3 @ X3 @ Y2 ) )
=> ( ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ ( F3 @ X3 @ Y2 ) )
=> ( ! [X3: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ Y2 @ X3 )
=> ( ( ord_less_eq_nat @ Z3 @ X3 )
=> ( ord_less_eq_nat @ ( F3 @ Y2 @ Z3 ) @ X3 ) ) )
=> ( ( sup_sup_nat @ X @ Y )
= ( F3 @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_693_sup__unique,axiom,
! [F3: set_a > set_a > set_a,X: set_a,Y: set_a] :
( ! [X3: set_a,Y2: set_a] : ( ord_less_eq_set_a @ X3 @ ( F3 @ X3 @ Y2 ) )
=> ( ! [X3: set_a,Y2: set_a] : ( ord_less_eq_set_a @ Y2 @ ( F3 @ X3 @ Y2 ) )
=> ( ! [X3: set_a,Y2: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X3 )
=> ( ( ord_less_eq_set_a @ Z3 @ X3 )
=> ( ord_less_eq_set_a @ ( F3 @ Y2 @ Z3 ) @ X3 ) ) )
=> ( ( sup_sup_set_a @ X @ Y )
= ( F3 @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_694_sup_Oabsorb1,axiom,
! [B4: nat,A: nat] :
( ( ord_less_eq_nat @ B4 @ A )
=> ( ( sup_sup_nat @ A @ B4 )
= A ) ) ).
% sup.absorb1
thf(fact_695_sup_Oabsorb1,axiom,
! [B4: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( ( sup_sup_set_a @ A @ B4 )
= A ) ) ).
% sup.absorb1
thf(fact_696_sup_Oabsorb2,axiom,
! [A: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( sup_sup_nat @ A @ B4 )
= B4 ) ) ).
% sup.absorb2
thf(fact_697_sup_Oabsorb2,axiom,
! [A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( sup_sup_set_a @ A @ B4 )
= B4 ) ) ).
% sup.absorb2
thf(fact_698_sup__absorb1,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( sup_sup_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_699_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_700_sup__absorb2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( sup_sup_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_701_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_702_sup_OboundedE,axiom,
! [B4: nat,C2: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B4 @ C2 ) @ A )
=> ~ ( ( ord_less_eq_nat @ B4 @ A )
=> ~ ( ord_less_eq_nat @ C2 @ A ) ) ) ).
% sup.boundedE
thf(fact_703_sup_OboundedE,axiom,
! [B4: set_a,C2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B4 @ C2 ) @ A )
=> ~ ( ( ord_less_eq_set_a @ B4 @ A )
=> ~ ( ord_less_eq_set_a @ C2 @ A ) ) ) ).
% sup.boundedE
thf(fact_704_sup_OboundedI,axiom,
! [B4: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B4 @ A )
=> ( ( ord_less_eq_nat @ C2 @ A )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B4 @ C2 ) @ A ) ) ) ).
% sup.boundedI
thf(fact_705_sup_OboundedI,axiom,
! [B4: set_a,A: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( ( ord_less_eq_set_a @ C2 @ A )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ B4 @ C2 ) @ A ) ) ) ).
% sup.boundedI
thf(fact_706_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( A4
= ( sup_sup_nat @ A4 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_707_sup_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A4: set_a] :
( A4
= ( sup_sup_set_a @ A4 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_708_sup_Ocobounded1,axiom,
! [A: nat,B4: nat] : ( ord_less_eq_nat @ A @ ( sup_sup_nat @ A @ B4 ) ) ).
% sup.cobounded1
thf(fact_709_sup_Ocobounded1,axiom,
! [A: set_a,B4: set_a] : ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ A @ B4 ) ) ).
% sup.cobounded1
thf(fact_710_sup_Ocobounded2,axiom,
! [B4: nat,A: nat] : ( ord_less_eq_nat @ B4 @ ( sup_sup_nat @ A @ B4 ) ) ).
% sup.cobounded2
thf(fact_711_sup_Ocobounded2,axiom,
! [B4: set_a,A: set_a] : ( ord_less_eq_set_a @ B4 @ ( sup_sup_set_a @ A @ B4 ) ) ).
% sup.cobounded2
thf(fact_712_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( ( sup_sup_nat @ A4 @ B3 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_713_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A4: set_a] :
( ( sup_sup_set_a @ A4 @ B3 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_714_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( ( sup_sup_nat @ A4 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_715_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( sup_sup_set_a @ A4 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_716_sup_OcoboundedI1,axiom,
! [C2: nat,A: nat,B4: nat] :
( ( ord_less_eq_nat @ C2 @ A )
=> ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A @ B4 ) ) ) ).
% sup.coboundedI1
thf(fact_717_sup_OcoboundedI1,axiom,
! [C2: set_a,A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ C2 @ A )
=> ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A @ B4 ) ) ) ).
% sup.coboundedI1
thf(fact_718_sup_OcoboundedI2,axiom,
! [C2: nat,B4: nat,A: nat] :
( ( ord_less_eq_nat @ C2 @ B4 )
=> ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A @ B4 ) ) ) ).
% sup.coboundedI2
thf(fact_719_sup_OcoboundedI2,axiom,
! [C2: set_a,B4: set_a,A: set_a] :
( ( ord_less_eq_set_a @ C2 @ B4 )
=> ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A @ B4 ) ) ) ).
% sup.coboundedI2
thf(fact_720_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_721_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_722_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_723_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_724_sup__inf__distrib2,axiom,
! [Y: set_a,Z2: set_a,X: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ Z2 ) @ X )
= ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ X ) @ ( sup_sup_set_a @ Z2 @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_725_sup__inf__distrib1,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z2 ) ) ) ).
% sup_inf_distrib1
thf(fact_726_inf__sup__distrib2,axiom,
! [Y: set_a,Z2: set_a,X: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ Z2 ) @ X )
= ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ X ) @ ( inf_inf_set_a @ Z2 @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_727_inf__sup__distrib1,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z2 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z2 ) ) ) ).
% inf_sup_distrib1
thf(fact_728_distrib__imp2,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ! [X3: set_a,Y2: set_a,Z3: set_a] :
( ( sup_sup_set_a @ X3 @ ( inf_inf_set_a @ Y2 @ Z3 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X3 @ Y2 ) @ ( sup_sup_set_a @ X3 @ Z3 ) ) )
=> ( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z2 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z2 ) ) ) ) ).
% distrib_imp2
thf(fact_729_distrib__imp1,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ! [X3: set_a,Y2: set_a,Z3: set_a] :
( ( inf_inf_set_a @ X3 @ ( sup_sup_set_a @ Y2 @ Z3 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X3 @ Y2 ) @ ( inf_inf_set_a @ X3 @ Z3 ) ) )
=> ( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z2 ) ) ) ) ).
% distrib_imp1
thf(fact_730_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_731_additive__abelian__group_Osumset__insert2,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,B: set_a,A2: set_a,X: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( nO_MATCH_set_a_set_a @ bot_bot_set_a @ B )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( insert_a @ X @ B ) )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) ) ) ) ).
% additive_abelian_group.sumset_insert2
thf(fact_732_additive__abelian__group_Osumset__insert1,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,X: a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( nO_MATCH_set_a_set_a @ bot_bot_set_a @ A2 )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ ( insert_a @ X @ A2 ) @ B )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ ( insert_a @ X @ bot_bot_set_a ) @ B ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) ) ) ) ).
% additive_abelian_group.sumset_insert1
thf(fact_733_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_734_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_735_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_736_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_737_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_738_distrib__inf__le,axiom,
! [X: nat,Y: nat,Z2: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z2 ) ) @ ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_739_distrib__inf__le,axiom,
! [X: set_a,Y: set_a,Z2: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z2 ) ) @ ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_740_distrib__sup__le,axiom,
! [X: nat,Y: nat,Z2: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z2 ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_741_distrib__sup__le,axiom,
! [X: set_a,Y: set_a,Z2: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) ) @ ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_742_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_743_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_744_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_745_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_746_psubsetI,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_set_a @ A2 @ B ) ) ) ).
% psubsetI
thf(fact_747_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_748_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_749_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_750_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_751_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_752_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_753_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_754_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_755_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_756_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_757_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_758_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_759_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_760_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_761_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_762_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_763_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_764_not__psubset__empty,axiom,
! [A2: set_a] :
~ ( ord_less_set_a @ A2 @ bot_bot_set_a ) ).
% not_psubset_empty
thf(fact_765_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_766_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_767_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_768_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_769_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_770_le__diff__iff_H,axiom,
! [A: nat,C2: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A ) @ ( minus_minus_nat @ C2 @ B4 ) )
= ( ord_less_eq_nat @ B4 @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_771_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_772_finite__psubset__induct,axiom,
! [A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ A2 )
=> ( ! [A6: set_a] :
( ( finite_finite_a @ A6 )
=> ( ! [B9: set_a] :
( ( ord_less_set_a @ B9 @ A6 )
=> ( P @ B9 ) )
=> ( P @ A6 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_773_finite__psubset__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ! [A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ! [B9: set_nat] :
( ( ord_less_set_nat @ B9 @ A6 )
=> ( P @ B9 ) )
=> ( P @ A6 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_774_psubsetE,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_set_a @ A2 @ B )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B )
=> ( ord_less_eq_set_a @ B @ A2 ) ) ) ).
% psubsetE
thf(fact_775_psubset__eq,axiom,
( ord_less_set_a
= ( ^ [A7: set_a,B7: set_a] :
( ( ord_less_eq_set_a @ A7 @ B7 )
& ( A7 != B7 ) ) ) ) ).
% psubset_eq
thf(fact_776_psubset__imp__subset,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_set_a @ A2 @ B )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ).
% psubset_imp_subset
thf(fact_777_psubset__subset__trans,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_set_a @ A2 @ C ) ) ) ).
% psubset_subset_trans
thf(fact_778_subset__not__subset__eq,axiom,
( ord_less_set_a
= ( ^ [A7: set_a,B7: set_a] :
( ( ord_less_eq_set_a @ A7 @ B7 )
& ~ ( ord_less_eq_set_a @ B7 @ A7 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_779_subset__psubset__trans,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_set_a @ B @ C )
=> ( ord_less_set_a @ A2 @ C ) ) ) ).
% subset_psubset_trans
thf(fact_780_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A7: set_a,B7: set_a] :
( ( ord_less_set_a @ A7 @ B7 )
| ( A7 = B7 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_781_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_782_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_783_psubset__imp__ex__mem,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ? [B2: nat] : ( member_nat @ B2 @ ( minus_minus_set_nat @ B @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_784_psubset__imp__ex__mem,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_set_a @ A2 @ B )
=> ? [B2: a] : ( member_a @ B2 @ ( minus_minus_set_a @ B @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_785_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_786_diff__less__mono,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ C2 @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C2 ) @ ( minus_minus_nat @ B4 @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_787_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_788_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_789_psubset__card__mono,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_set_a @ A2 @ B )
=> ( ord_less_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B ) ) ) ) ).
% psubset_card_mono
thf(fact_790_psubset__card__mono,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_set_nat @ A2 @ B )
=> ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B ) ) ) ) ).
% psubset_card_mono
thf(fact_791_mult_Oleft__commute,axiom,
! [B4: nat,A: nat,C2: nat] :
( ( times_times_nat @ B4 @ ( times_times_nat @ A @ C2 ) )
= ( times_times_nat @ A @ ( times_times_nat @ B4 @ C2 ) ) ) ).
% mult.left_commute
thf(fact_792_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A4: nat,B3: nat] : ( times_times_nat @ B3 @ A4 ) ) ) ).
% mult.commute
thf(fact_793_mult_Oassoc,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B4 ) @ C2 )
= ( times_times_nat @ A @ ( times_times_nat @ B4 @ C2 ) ) ) ).
% mult.assoc
thf(fact_794_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B4 ) @ C2 )
= ( times_times_nat @ A @ ( times_times_nat @ B4 @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_795_diff__right__commute,axiom,
! [A: nat,C2: nat,B4: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C2 ) @ B4 )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B4 ) @ C2 ) ) ).
% diff_right_commute
thf(fact_796_boolean__algebra__cancel_Oinf1,axiom,
! [A2: set_a,K: set_a,A: set_a,B4: set_a] :
( ( A2
= ( inf_inf_set_a @ K @ A ) )
=> ( ( inf_inf_set_a @ A2 @ B4 )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A @ B4 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_797_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_a,K: set_a,B4: set_a,A: set_a] :
( ( B
= ( inf_inf_set_a @ K @ B4 ) )
=> ( ( inf_inf_set_a @ A @ B )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A @ B4 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_798_boolean__algebra__cancel_Osup1,axiom,
! [A2: set_a,K: set_a,A: set_a,B4: set_a] :
( ( A2
= ( sup_sup_set_a @ K @ A ) )
=> ( ( sup_sup_set_a @ A2 @ B4 )
= ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A @ B4 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_799_boolean__algebra__cancel_Osup2,axiom,
! [B: set_a,K: set_a,B4: set_a,A: set_a] :
( ( B
= ( sup_sup_set_a @ K @ B4 ) )
=> ( ( sup_sup_set_a @ A @ B )
= ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A @ B4 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_800_card__Diff__subset,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B ) )
= ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_Diff_subset
thf(fact_801_card__Diff__subset,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ B @ A2 )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ B ) )
= ( minus_minus_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B ) ) ) ) ) ).
% card_Diff_subset
thf(fact_802_diff__card__le__card__Diff,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_803_diff__card__le__card__Diff,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B ) ) @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ B ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_804_finite__induct__select,axiom,
! [S: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [T: set_nat] :
( ( ord_less_set_nat @ T @ S )
=> ( ( P @ T )
=> ? [X5: nat] :
( ( member_nat @ X5 @ ( minus_minus_set_nat @ S @ T ) )
& ( P @ ( insert_nat @ X5 @ T ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_805_finite__induct__select,axiom,
! [S: set_a,P: set_a > $o] :
( ( finite_finite_a @ S )
=> ( ( P @ bot_bot_set_a )
=> ( ! [T: set_a] :
( ( ord_less_set_a @ T @ S )
=> ( ( P @ T )
=> ? [X5: a] :
( ( member_a @ X5 @ ( minus_minus_set_a @ S @ T ) )
& ( P @ ( insert_a @ X5 @ T ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_806_psubset__insert__iff,axiom,
! [A2: set_nat,X: nat,B: set_nat] :
( ( ord_less_set_nat @ A2 @ ( insert_nat @ X @ B ) )
= ( ( ( member_nat @ X @ B )
=> ( ord_less_set_nat @ A2 @ B ) )
& ( ~ ( member_nat @ X @ B )
=> ( ( ( member_nat @ X @ A2 )
=> ( ord_less_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B ) )
& ( ~ ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_807_psubset__insert__iff,axiom,
! [A2: set_a,X: a,B: set_a] :
( ( ord_less_set_a @ A2 @ ( insert_a @ X @ B ) )
= ( ( ( member_a @ X @ B )
=> ( ord_less_set_a @ A2 @ B ) )
& ( ~ ( member_a @ X @ B )
=> ( ( ( member_a @ X @ A2 )
=> ( ord_less_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_808_card__psubset,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B ) )
=> ( ord_less_set_nat @ A2 @ B ) ) ) ) ).
% card_psubset
thf(fact_809_card__psubset,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B ) )
=> ( ord_less_set_a @ A2 @ B ) ) ) ) ).
% card_psubset
thf(fact_810_card__Diff__subset__Int,axiom,
! [A2: set_nat,B: set_nat] :
( ( finite_finite_nat @ ( inf_inf_set_nat @ A2 @ B ) )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B ) )
= ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ ( inf_inf_set_nat @ A2 @ B ) ) ) ) ) ).
% card_Diff_subset_Int
thf(fact_811_card__Diff__subset__Int,axiom,
! [A2: set_a,B: set_a] :
( ( finite_finite_a @ ( inf_inf_set_a @ A2 @ B ) )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ B ) )
= ( minus_minus_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ) ).
% card_Diff_subset_Int
thf(fact_812_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_813_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_814_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_815_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_816_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_817_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_818_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z2 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z2 ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_819_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z2 ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_820_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_a,Z2: set_a,X: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ Z2 ) @ X )
= ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ X ) @ ( inf_inf_set_a @ Z2 @ X ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_821_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_a,Z2: set_a,X: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ Z2 ) @ X )
= ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ X ) @ ( sup_sup_set_a @ Z2 @ X ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_822_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_823_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_824_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_825_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_826_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_827_sumset__iterated__r,axiom,
! [R: nat,A2: set_a] :
( ( ord_less_nat @ zero_zero_nat @ R )
=> ( ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ A2 @ R )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ A2 @ ( minus_minus_nat @ R @ one_one_nat ) ) ) ) ) ).
% sumset_iterated_r
thf(fact_828_mult__cancel__right,axiom,
! [A: nat,C2: nat,B4: nat] :
( ( ( times_times_nat @ A @ C2 )
= ( times_times_nat @ B4 @ C2 ) )
= ( ( C2 = zero_zero_nat )
| ( A = B4 ) ) ) ).
% mult_cancel_right
thf(fact_829_mult__cancel__left,axiom,
! [C2: nat,A: nat,B4: nat] :
( ( ( times_times_nat @ C2 @ A )
= ( times_times_nat @ C2 @ B4 ) )
= ( ( C2 = zero_zero_nat )
| ( A = B4 ) ) ) ).
% mult_cancel_left
thf(fact_830_mult__eq__0__iff,axiom,
! [A: nat,B4: nat] :
( ( ( times_times_nat @ A @ B4 )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B4 = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_831_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_832_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_833_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_834_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_835_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_836_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_837_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_838_card__Diff__insert,axiom,
! [A: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ~ ( member_nat @ A @ B )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_839_card__Diff__insert,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( member_a @ A @ A2 )
=> ( ~ ( member_a @ A @ B )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B ) ) )
= ( minus_minus_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ B ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_840_psubsetD,axiom,
! [A2: set_a,B: set_a,C2: a] :
( ( ord_less_set_a @ A2 @ B )
=> ( ( member_a @ C2 @ A2 )
=> ( member_a @ C2 @ B ) ) ) ).
% psubsetD
thf(fact_841_psubsetD,axiom,
! [A2: set_nat,B: set_nat,C2: nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ( member_nat @ C2 @ A2 )
=> ( member_nat @ C2 @ B ) ) ) ).
% psubsetD
thf(fact_842_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_843_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_844_comm__monoid__mult__class_Omult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_845_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_846_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_847_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_848_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_849_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_850_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_851_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_852_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_853_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_854_less__1__mult,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ M )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_855_mult__le__one,axiom,
! [A: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B4 )
=> ( ( ord_less_eq_nat @ B4 @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B4 ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_856_mult__left__le,axiom,
! [C2: nat,A: nat] :
( ( ord_less_eq_nat @ C2 @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ A ) ) ) ).
% mult_left_le
thf(fact_857_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_858_card__1__singletonE,axiom,
! [A2: set_a] :
( ( ( finite_card_a @ A2 )
= one_one_nat )
=> ~ ! [X3: a] :
( A2
!= ( insert_a @ X3 @ bot_bot_set_a ) ) ) ).
% card_1_singletonE
thf(fact_859_card__Diff__singleton,axiom,
! [X: nat,A2: set_nat] :
( ( member_nat @ X @ A2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_860_card__Diff__singleton,axiom,
! [X: a,A2: set_a] :
( ( member_a @ X @ A2 )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) )
= ( minus_minus_nat @ ( finite_card_a @ A2 ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_861_card__Diff__singleton__if,axiom,
! [X: nat,A2: set_nat] :
( ( ( member_nat @ X @ A2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ one_one_nat ) ) )
& ( ~ ( member_nat @ X @ A2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
= ( finite_card_nat @ A2 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_862_card__Diff__singleton__if,axiom,
! [X: a,A2: set_a] :
( ( ( member_a @ X @ A2 )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) )
= ( minus_minus_nat @ ( finite_card_a @ A2 ) @ one_one_nat ) ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) )
= ( finite_card_a @ A2 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_863_card__insert__le__m1,axiom,
! [N: nat,Y: set_a,X: a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ Y ) @ ( minus_minus_nat @ N @ one_one_nat ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( insert_a @ X @ Y ) ) @ N ) ) ) ).
% card_insert_le_m1
thf(fact_864_additive__abelian__group_Osumset__iterated__r,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,R: nat,A2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ord_less_nat @ zero_zero_nat @ R )
=> ( ( pluenn1960970773371692859ated_a @ G @ Addition @ Zero @ A2 @ R )
= ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( pluenn1960970773371692859ated_a @ G @ Addition @ Zero @ A2 @ ( minus_minus_nat @ R @ one_one_nat ) ) ) ) ) ) ).
% additive_abelian_group.sumset_iterated_r
thf(fact_865_mult__not__zero,axiom,
! [A: nat,B4: nat] :
( ( ( times_times_nat @ A @ B4 )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B4 != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_866_divisors__zero,axiom,
! [A: nat,B4: nat] :
( ( ( times_times_nat @ A @ B4 )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B4 = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_867_no__zero__divisors,axiom,
! [A: nat,B4: nat] :
( ( A != zero_zero_nat )
=> ( ( B4 != zero_zero_nat )
=> ( ( times_times_nat @ A @ B4 )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_868_mult__left__cancel,axiom,
! [C2: nat,A: nat,B4: nat] :
( ( C2 != zero_zero_nat )
=> ( ( ( times_times_nat @ C2 @ A )
= ( times_times_nat @ C2 @ B4 ) )
= ( A = B4 ) ) ) ).
% mult_left_cancel
thf(fact_869_mult__right__cancel,axiom,
! [C2: nat,A: nat,B4: nat] :
( ( C2 != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C2 )
= ( times_times_nat @ B4 @ C2 ) )
= ( A = B4 ) ) ) ).
% mult_right_cancel
thf(fact_870_right__diff__distrib_H,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( times_times_nat @ A @ ( minus_minus_nat @ B4 @ C2 ) )
= ( minus_minus_nat @ ( times_times_nat @ A @ B4 ) @ ( times_times_nat @ A @ C2 ) ) ) ).
% right_diff_distrib'
thf(fact_871_left__diff__distrib_H,axiom,
! [B4: nat,C2: nat,A: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B4 @ C2 ) @ A )
= ( minus_minus_nat @ ( times_times_nat @ B4 @ A ) @ ( times_times_nat @ C2 @ A ) ) ) ).
% left_diff_distrib'
thf(fact_872_mult__mono,axiom,
! [A: nat,B4: nat,C2: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ C2 @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B4 @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_873_mult__mono_H,axiom,
! [A: nat,B4: nat,C2: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ C2 @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B4 @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_874_mult__left__mono,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B4 ) ) ) ) ).
% mult_left_mono
thf(fact_875_mult__right__mono,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B4 @ C2 ) ) ) ) ).
% mult_right_mono
thf(fact_876_split__mult__neg__le,axiom,
! [A: nat,B4: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B4 @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B4 ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B4 ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_877_mult__nonneg__nonneg,axiom,
! [A: nat,B4: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B4 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B4 ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_878_mult__nonneg__nonpos,axiom,
! [A: nat,B4: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B4 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B4 ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_879_mult__nonpos__nonneg,axiom,
! [A: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B4 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B4 ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_880_mult__nonneg__nonpos2,axiom,
! [A: nat,B4: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B4 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B4 @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_881_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B4 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_882_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B4 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_883_mult__strict__right__mono,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B4 @ C2 ) ) ) ) ).
% mult_strict_right_mono
thf(fact_884_mult__strict__left__mono,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B4 ) ) ) ) ).
% mult_strict_left_mono
thf(fact_885_zero__less__mult__pos2,axiom,
! [B4: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B4 @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B4 ) ) ) ).
% zero_less_mult_pos2
thf(fact_886_zero__less__mult__pos,axiom,
! [A: nat,B4: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B4 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B4 ) ) ) ).
% zero_less_mult_pos
thf(fact_887_mult__pos__neg2,axiom,
! [A: nat,B4: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B4 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B4 @ A ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_888_mult__pos__pos,axiom,
! [A: nat,B4: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B4 )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B4 ) ) ) ) ).
% mult_pos_pos
thf(fact_889_mult__pos__neg,axiom,
! [A: nat,B4: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B4 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B4 ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_890_mult__neg__pos,axiom,
! [A: nat,B4: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B4 )
=> ( ord_less_nat @ ( times_times_nat @ A @ B4 ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_891_mult__left__less__imp__less,axiom,
! [C2: nat,A: nat,B4: nat] :
( ( ord_less_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B4 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ A @ B4 ) ) ) ).
% mult_left_less_imp_less
thf(fact_892_mult__strict__mono,axiom,
! [A: nat,B4: nat,C2: nat,D: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ( ord_less_nat @ C2 @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ B4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B4 @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_893_mult__right__less__imp__less,axiom,
! [A: nat,C2: nat,B4: nat] :
( ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B4 @ C2 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ A @ B4 ) ) ) ).
% mult_right_less_imp_less
thf(fact_894_mult__strict__mono_H,axiom,
! [A: nat,B4: nat,C2: nat,D: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ( ord_less_nat @ C2 @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B4 @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_895_mult__left__le__imp__le,axiom,
! [C2: nat,A: nat,B4: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B4 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ A @ B4 ) ) ) ).
% mult_left_le_imp_le
thf(fact_896_mult__right__le__imp__le,axiom,
! [A: nat,C2: nat,B4: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B4 @ C2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ A @ B4 ) ) ) ).
% mult_right_le_imp_le
thf(fact_897_mult__le__less__imp__less,axiom,
! [A: nat,B4: nat,C2: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_nat @ C2 @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B4 @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_898_mult__less__le__imp__less,axiom,
! [A: nat,B4: nat,C2: nat,D: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ C2 @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B4 @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_899_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_900_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_901_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_902_order__refl,axiom,
! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% order_refl
thf(fact_903_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_904_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_905_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_906_ord__le__eq__subst,axiom,
! [A: nat,B4: nat,F3: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ( F3 @ B4 )
= C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F3 @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_907_ord__le__eq__subst,axiom,
! [A: nat,B4: nat,F3: nat > set_a,C2: set_a] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ( F3 @ B4 )
= C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F3 @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_908_ord__le__eq__subst,axiom,
! [A: set_a,B4: set_a,F3: set_a > nat,C2: nat] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( ( F3 @ B4 )
= C2 )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F3 @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_909_ord__le__eq__subst,axiom,
! [A: set_a,B4: set_a,F3: set_a > set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( ( F3 @ B4 )
= C2 )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F3 @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_910_ord__eq__le__subst,axiom,
! [A: nat,F3: nat > nat,B4: nat,C2: nat] :
( ( A
= ( F3 @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F3 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_911_ord__eq__le__subst,axiom,
! [A: set_a,F3: nat > set_a,B4: nat,C2: nat] :
( ( A
= ( F3 @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F3 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_912_ord__eq__le__subst,axiom,
! [A: nat,F3: set_a > nat,B4: set_a,C2: set_a] :
( ( A
= ( F3 @ B4 ) )
=> ( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F3 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_913_ord__eq__le__subst,axiom,
! [A: set_a,F3: set_a > set_a,B4: set_a,C2: set_a] :
( ( A
= ( F3 @ B4 ) )
=> ( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F3 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_914_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_915_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_916_order__eq__refl,axiom,
! [X: set_a,Y: set_a] :
( ( X = Y )
=> ( ord_less_eq_set_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_917_order__subst2,axiom,
! [A: nat,B4: nat,F3: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ ( F3 @ B4 ) @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F3 @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_918_order__subst2,axiom,
! [A: nat,B4: nat,F3: nat > set_a,C2: set_a] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_eq_set_a @ ( F3 @ B4 ) @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F3 @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_919_order__subst2,axiom,
! [A: set_a,B4: set_a,F3: set_a > nat,C2: nat] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( ord_less_eq_nat @ ( F3 @ B4 ) @ C2 )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F3 @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_920_order__subst2,axiom,
! [A: set_a,B4: set_a,F3: set_a > set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( ord_less_eq_set_a @ ( F3 @ B4 ) @ C2 )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F3 @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_921_order__subst1,axiom,
! [A: nat,F3: nat > nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( F3 @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F3 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_922_order__subst1,axiom,
! [A: nat,F3: set_a > nat,B4: set_a,C2: set_a] :
( ( ord_less_eq_nat @ A @ ( F3 @ B4 ) )
=> ( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F3 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_923_order__subst1,axiom,
! [A: set_a,F3: nat > set_a,B4: nat,C2: nat] :
( ( ord_less_eq_set_a @ A @ ( F3 @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F3 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_924_order__subst1,axiom,
! [A: set_a,F3: set_a > set_a,B4: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ ( F3 @ B4 ) )
=> ( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F3 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_925_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_926_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_a,Z: set_a] : ( Y4 = Z ) )
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A4 @ B3 )
& ( ord_less_eq_set_a @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_927_antisym,axiom,
! [A: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ B4 @ A )
=> ( A = B4 ) ) ) ).
% antisym
thf(fact_928_antisym,axiom,
! [A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( ord_less_eq_set_a @ B4 @ A )
=> ( A = B4 ) ) ) ).
% antisym
thf(fact_929_dual__order_Otrans,axiom,
! [B4: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B4 @ A )
=> ( ( ord_less_eq_nat @ C2 @ B4 )
=> ( ord_less_eq_nat @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_930_dual__order_Otrans,axiom,
! [B4: set_a,A: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( ( ord_less_eq_set_a @ C2 @ B4 )
=> ( ord_less_eq_set_a @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_931_dual__order_Oantisym,axiom,
! [B4: nat,A: nat] :
( ( ord_less_eq_nat @ B4 @ A )
=> ( ( ord_less_eq_nat @ A @ B4 )
=> ( A = B4 ) ) ) ).
% dual_order.antisym
thf(fact_932_dual__order_Oantisym,axiom,
! [B4: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( ( ord_less_eq_set_a @ A @ B4 )
=> ( A = B4 ) ) ) ).
% dual_order.antisym
thf(fact_933_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_934_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_a,Z: set_a] : ( Y4 = Z ) )
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A4 )
& ( ord_less_eq_set_a @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_935_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B4: nat] :
( ! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: nat,B2: nat] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B4 ) ) ) ).
% linorder_wlog
thf(fact_936_order__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_937_order__trans,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z2 )
=> ( ord_less_eq_set_a @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_938_order_Otrans,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% order.trans
thf(fact_939_order_Otrans,axiom,
! [A: set_a,B4: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% order.trans
thf(fact_940_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_941_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_942_ord__le__eq__trans,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( B4 = C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_943_ord__le__eq__trans,axiom,
! [A: set_a,B4: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( B4 = C2 )
=> ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_944_ord__eq__le__trans,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( A = B4 )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_945_ord__eq__le__trans,axiom,
! [A: set_a,B4: set_a,C2: set_a] :
( ( A = B4 )
=> ( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_946_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
& ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_947_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_a,Z: set_a] : ( Y4 = Z ) )
= ( ^ [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
& ( ord_less_eq_set_a @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_948_le__cases3,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_949_nle__le,axiom,
! [A: nat,B4: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B4 ) )
= ( ( ord_less_eq_nat @ B4 @ A )
& ( B4 != A ) ) ) ).
% nle_le
thf(fact_950_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_951_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_952_order_Oasym,axiom,
! [A: nat,B4: nat] :
( ( ord_less_nat @ A @ B4 )
=> ~ ( ord_less_nat @ B4 @ A ) ) ).
% order.asym
thf(fact_953_ord__eq__less__trans,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( A = B4 )
=> ( ( ord_less_nat @ B4 @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_954_ord__less__eq__trans,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ( B4 = C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_955_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_956_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_957_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_958_dual__order_Oasym,axiom,
! [B4: nat,A: nat] :
( ( ord_less_nat @ B4 @ A )
=> ~ ( ord_less_nat @ A @ B4 ) ) ).
% dual_order.asym
thf(fact_959_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_960_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X6: nat] : ( P2 @ X6 ) )
= ( ^ [P3: nat > $o] :
? [N3: nat] :
( ( P3 @ N3 )
& ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ~ ( P3 @ M3 ) ) ) ) ) ).
% exists_least_iff
thf(fact_961_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B4: nat] :
( ! [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: nat] : ( P @ A3 @ A3 )
=> ( ! [A3: nat,B2: nat] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B4 ) ) ) ) ).
% linorder_less_wlog
thf(fact_962_order_Ostrict__trans,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ( ord_less_nat @ B4 @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans
thf(fact_963_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_964_dual__order_Ostrict__trans,axiom,
! [B4: nat,A: nat,C2: nat] :
( ( ord_less_nat @ B4 @ A )
=> ( ( ord_less_nat @ C2 @ B4 )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_965_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B4: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( A != B4 ) ) ).
% order.strict_implies_not_eq
thf(fact_966_dual__order_Ostrict__implies__not__eq,axiom,
! [B4: nat,A: nat] :
( ( ord_less_nat @ B4 @ A )
=> ( A != B4 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_967_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_968_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_969_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_970_order__less__asym_H,axiom,
! [A: nat,B4: nat] :
( ( ord_less_nat @ A @ B4 )
=> ~ ( ord_less_nat @ B4 @ A ) ) ).
% order_less_asym'
thf(fact_971_order__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_972_ord__eq__less__subst,axiom,
! [A: nat,F3: nat > nat,B4: nat,C2: nat] :
( ( A
= ( F3 @ B4 ) )
=> ( ( ord_less_nat @ B4 @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F3 @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_973_ord__less__eq__subst,axiom,
! [A: nat,B4: nat,F3: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ( ( F3 @ B4 )
= C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less_nat @ ( F3 @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_974_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_975_order__less__subst1,axiom,
! [A: nat,F3: nat > nat,B4: nat,C2: nat] :
( ( ord_less_nat @ A @ ( F3 @ B4 ) )
=> ( ( ord_less_nat @ B4 @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F3 @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_976_order__less__subst2,axiom,
! [A: nat,B4: nat,F3: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ( ord_less_nat @ ( F3 @ B4 ) @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less_nat @ ( F3 @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_977_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_978_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_979_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_980_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_981_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_982_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_983_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_984_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_985_leD,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ~ ( ord_less_set_a @ X @ Y ) ) ).
% leD
thf(fact_986_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_987_nless__le,axiom,
! [A: nat,B4: nat] :
( ( ~ ( ord_less_nat @ A @ B4 ) )
= ( ~ ( ord_less_eq_nat @ A @ B4 )
| ( A = B4 ) ) ) ).
% nless_le
thf(fact_988_nless__le,axiom,
! [A: set_a,B4: set_a] :
( ( ~ ( ord_less_set_a @ A @ B4 ) )
= ( ~ ( ord_less_eq_set_a @ A @ B4 )
| ( A = B4 ) ) ) ).
% nless_le
thf(fact_989_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_990_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_991_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_992_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_993_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
& ~ ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_994_less__le__not__le,axiom,
( ord_less_set_a
= ( ^ [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
& ~ ( ord_less_eq_set_a @ Y3 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_995_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_996_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_nat @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_997_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_set_a @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_998_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_999_order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1000_order_Ostrict__trans1,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_nat @ B4 @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_1001_order_Ostrict__trans1,axiom,
! [A: set_a,B4: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( ord_less_set_a @ B4 @ C2 )
=> ( ord_less_set_a @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_1002_order_Ostrict__trans2,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_1003_order_Ostrict__trans2,axiom,
! [A: set_a,B4: set_a,C2: set_a] :
( ( ord_less_set_a @ A @ B4 )
=> ( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ord_less_set_a @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_1004_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ~ ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1005_order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A4 @ B3 )
& ~ ( ord_less_eq_set_a @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1006_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_nat @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1007_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A4: set_a] :
( ( ord_less_set_a @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1008_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1009_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [B3: set_a,A4: set_a] :
( ( ord_less_eq_set_a @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1010_dual__order_Ostrict__trans1,axiom,
! [B4: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B4 @ A )
=> ( ( ord_less_nat @ C2 @ B4 )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_1011_dual__order_Ostrict__trans1,axiom,
! [B4: set_a,A: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( ( ord_less_set_a @ C2 @ B4 )
=> ( ord_less_set_a @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_1012_dual__order_Ostrict__trans2,axiom,
! [B4: nat,A: nat,C2: nat] :
( ( ord_less_nat @ B4 @ A )
=> ( ( ord_less_eq_nat @ C2 @ B4 )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_1013_dual__order_Ostrict__trans2,axiom,
! [B4: set_a,A: set_a,C2: set_a] :
( ( ord_less_set_a @ B4 @ A )
=> ( ( ord_less_eq_set_a @ C2 @ B4 )
=> ( ord_less_set_a @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_1014_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ~ ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1015_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [B3: set_a,A4: set_a] :
( ( ord_less_eq_set_a @ B3 @ A4 )
& ~ ( ord_less_eq_set_a @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1016_order_Ostrict__implies__order,axiom,
! [A: nat,B4: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ord_less_eq_nat @ A @ B4 ) ) ).
% order.strict_implies_order
thf(fact_1017_order_Ostrict__implies__order,axiom,
! [A: set_a,B4: set_a] :
( ( ord_less_set_a @ A @ B4 )
=> ( ord_less_eq_set_a @ A @ B4 ) ) ).
% order.strict_implies_order
thf(fact_1018_dual__order_Ostrict__implies__order,axiom,
! [B4: nat,A: nat] :
( ( ord_less_nat @ B4 @ A )
=> ( ord_less_eq_nat @ B4 @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_1019_dual__order_Ostrict__implies__order,axiom,
! [B4: set_a,A: set_a] :
( ( ord_less_set_a @ B4 @ A )
=> ( ord_less_eq_set_a @ B4 @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_1020_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_1021_order__le__less,axiom,
( ord_less_eq_set_a
= ( ^ [X2: set_a,Y3: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_1022_order__less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_1023_order__less__le,axiom,
( ord_less_set_a
= ( ^ [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_1024_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_1025_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_1026_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_1027_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_1028_order__le__neq__trans,axiom,
! [A: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( A != B4 )
=> ( ord_less_nat @ A @ B4 ) ) ) ).
% order_le_neq_trans
thf(fact_1029_order__le__neq__trans,axiom,
! [A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( A != B4 )
=> ( ord_less_set_a @ A @ B4 ) ) ) ).
% order_le_neq_trans
thf(fact_1030_order__neq__le__trans,axiom,
! [A: nat,B4: nat] :
( ( A != B4 )
=> ( ( ord_less_eq_nat @ A @ B4 )
=> ( ord_less_nat @ A @ B4 ) ) ) ).
% order_neq_le_trans
thf(fact_1031_order__neq__le__trans,axiom,
! [A: set_a,B4: set_a] :
( ( A != B4 )
=> ( ( ord_less_eq_set_a @ A @ B4 )
=> ( ord_less_set_a @ A @ B4 ) ) ) ).
% order_neq_le_trans
thf(fact_1032_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_1033_order__le__less__trans,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_set_a @ Y @ Z2 )
=> ( ord_less_set_a @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_1034_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_1035_order__less__le__trans,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z2 )
=> ( ord_less_set_a @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_1036_order__le__less__subst1,axiom,
! [A: nat,F3: nat > nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( F3 @ B4 ) )
=> ( ( ord_less_nat @ B4 @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F3 @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1037_order__le__less__subst1,axiom,
! [A: set_a,F3: nat > set_a,B4: nat,C2: nat] :
( ( ord_less_eq_set_a @ A @ ( F3 @ B4 ) )
=> ( ( ord_less_nat @ B4 @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_set_a @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less_set_a @ A @ ( F3 @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1038_order__le__less__subst2,axiom,
! [A: nat,B4: nat,F3: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_nat @ ( F3 @ B4 ) @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less_nat @ ( F3 @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1039_order__le__less__subst2,axiom,
! [A: nat,B4: nat,F3: nat > set_a,C2: set_a] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_set_a @ ( F3 @ B4 ) @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less_set_a @ ( F3 @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1040_order__le__less__subst2,axiom,
! [A: set_a,B4: set_a,F3: set_a > nat,C2: nat] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( ord_less_nat @ ( F3 @ B4 ) @ C2 )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less_nat @ ( F3 @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1041_order__le__less__subst2,axiom,
! [A: set_a,B4: set_a,F3: set_a > set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( ord_less_set_a @ ( F3 @ B4 ) @ C2 )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less_set_a @ ( F3 @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1042_order__less__le__subst1,axiom,
! [A: nat,F3: nat > nat,B4: nat,C2: nat] :
( ( ord_less_nat @ A @ ( F3 @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F3 @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1043_order__less__le__subst1,axiom,
! [A: set_a,F3: nat > set_a,B4: nat,C2: nat] :
( ( ord_less_set_a @ A @ ( F3 @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less_set_a @ A @ ( F3 @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1044_order__less__le__subst1,axiom,
! [A: nat,F3: set_a > nat,B4: set_a,C2: set_a] :
( ( ord_less_nat @ A @ ( F3 @ B4 ) )
=> ( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F3 @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1045_order__less__le__subst1,axiom,
! [A: set_a,F3: set_a > set_a,B4: set_a,C2: set_a] :
( ( ord_less_set_a @ A @ ( F3 @ B4 ) )
=> ( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less_set_a @ A @ ( F3 @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1046_order__less__le__subst2,axiom,
! [A: nat,B4: nat,F3: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ ( F3 @ B4 ) @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less_nat @ ( F3 @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_1047_order__less__le__subst2,axiom,
! [A: nat,B4: nat,F3: nat > set_a,C2: set_a] :
( ( ord_less_nat @ A @ B4 )
=> ( ( ord_less_eq_set_a @ ( F3 @ B4 ) @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_set_a @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less_set_a @ ( F3 @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_1048_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_1049_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_1050_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_1051_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_1052_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_1053_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_1054_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_1055_bot_Oextremum__uniqueI,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
=> ( A = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_1056_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_1057_bot_Oextremum__unique,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_1058_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_1059_bot_Oextremum,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% bot.extremum
thf(fact_1060_bot_Onot__eq__extremum,axiom,
! [A: set_a] :
( ( A != bot_bot_set_a )
= ( ord_less_set_a @ bot_bot_set_a @ A ) ) ).
% bot.not_eq_extremum
thf(fact_1061_bot_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_1062_bot_Oextremum__strict,axiom,
! [A: set_a] :
~ ( ord_less_set_a @ A @ bot_bot_set_a ) ).
% bot.extremum_strict
thf(fact_1063_bot_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_1064_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_1065_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_1066_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_1067_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_1068_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_1069_unit__invertible,axiom,
group_invertible_a @ g @ addition @ zero @ zero ).
% unit_invertible
thf(fact_1070_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_1071_invertible,axiom,
! [U: a] :
( ( member_a @ U @ g )
=> ( group_invertible_a @ g @ addition @ zero @ U ) ) ).
% invertible
thf(fact_1072_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_1073_invertible__left__cancel,axiom,
! [X: a,Y: a,Z2: a] :
( ( group_invertible_a @ g @ addition @ zero @ X )
=> ( ( member_a @ X @ g )
=> ( ( member_a @ Y @ g )
=> ( ( member_a @ Z2 @ g )
=> ( ( ( addition @ X @ Y )
= ( addition @ X @ Z2 ) )
= ( Y = Z2 ) ) ) ) ) ) ).
% invertible_left_cancel
thf(fact_1074_invertible__right__cancel,axiom,
! [X: a,Y: a,Z2: a] :
( ( group_invertible_a @ g @ addition @ zero @ X )
=> ( ( member_a @ X @ g )
=> ( ( member_a @ Y @ g )
=> ( ( member_a @ Z2 @ g )
=> ( ( ( addition @ Y @ X )
= ( addition @ Z2 @ X ) )
= ( Y = Z2 ) ) ) ) ) ) ).
% invertible_right_cancel
thf(fact_1075_abelian__group__def,axiom,
( group_201663378560352916roup_a
= ( ^ [G3: set_a,Composition: a > a > a,Unit: a] :
( ( group_group_a @ G3 @ Composition @ Unit )
& ( group_4866109990395492029noid_a @ G3 @ Composition @ Unit ) ) ) ) ).
% abelian_group_def
thf(fact_1076_abelian__group_Ointro,axiom,
! [G: set_a,Composition2: a > a > a,Unit2: a] :
( ( group_group_a @ G @ Composition2 @ Unit2 )
=> ( ( group_4866109990395492029noid_a @ G @ Composition2 @ Unit2 )
=> ( group_201663378560352916roup_a @ G @ Composition2 @ Unit2 ) ) ) ).
% abelian_group.intro
thf(fact_1077_monoid_OUnits_Ocong,axiom,
group_Units_a = group_Units_a ).
% monoid.Units.cong
thf(fact_1078_commutative__monoid_Ocommutative,axiom,
! [M4: set_nat,Composition2: nat > nat > nat,Unit2: nat,X: nat,Y: nat] :
( ( group_6791354081887936081id_nat @ M4 @ Composition2 @ Unit2 )
=> ( ( member_nat @ X @ M4 )
=> ( ( member_nat @ Y @ M4 )
=> ( ( Composition2 @ X @ Y )
= ( Composition2 @ Y @ X ) ) ) ) ) ).
% commutative_monoid.commutative
thf(fact_1079_commutative__monoid_Ocommutative,axiom,
! [M4: set_a,Composition2: a > a > a,Unit2: a,X: a,Y: a] :
( ( group_4866109990395492029noid_a @ M4 @ Composition2 @ Unit2 )
=> ( ( member_a @ X @ M4 )
=> ( ( member_a @ Y @ M4 )
=> ( ( Composition2 @ X @ Y )
= ( Composition2 @ Y @ X ) ) ) ) ) ).
% commutative_monoid.commutative
thf(fact_1080_group_Oinvertible,axiom,
! [G: set_nat,Composition2: nat > nat > nat,Unit2: nat,U: nat] :
( ( group_group_nat @ G @ Composition2 @ Unit2 )
=> ( ( member_nat @ U @ G )
=> ( group_invertible_nat @ G @ Composition2 @ Unit2 @ U ) ) ) ).
% group.invertible
thf(fact_1081_group_Oinvertible,axiom,
! [G: set_a,Composition2: a > a > a,Unit2: a,U: a] :
( ( group_group_a @ G @ Composition2 @ Unit2 )
=> ( ( member_a @ U @ G )
=> ( group_invertible_a @ G @ Composition2 @ Unit2 @ U ) ) ) ).
% group.invertible
thf(fact_1082_abelian__group_Oaxioms_I1_J,axiom,
! [G: set_a,Composition2: a > a > a,Unit2: a] :
( ( group_201663378560352916roup_a @ G @ Composition2 @ Unit2 )
=> ( group_group_a @ G @ Composition2 @ Unit2 ) ) ).
% abelian_group.axioms(1)
thf(fact_1083_abelian__group_Oaxioms_I2_J,axiom,
! [G: set_a,Composition2: a > a > a,Unit2: a] :
( ( group_201663378560352916roup_a @ G @ Composition2 @ Unit2 )
=> ( group_4866109990395492029noid_a @ G @ Composition2 @ Unit2 ) ) ).
% abelian_group.axioms(2)
thf(fact_1084_finite__linorder__min__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B2: nat,A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A6 )
=> ( ord_less_nat @ B2 @ X5 ) )
=> ( ( P @ A6 )
=> ( P @ ( insert_nat @ B2 @ A6 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_1085_finite__linorder__max__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B2: nat,A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A6 )
=> ( ord_less_nat @ X5 @ B2 ) )
=> ( ( P @ A6 )
=> ( P @ ( insert_nat @ B2 @ A6 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_1086_ex__min__if__finite,axiom,
! [S: set_nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ S )
& ~ ? [Xa: nat] :
( ( member_nat @ Xa @ S )
& ( ord_less_nat @ Xa @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_1087_infinite__growing,axiom,
! [X4: set_nat] :
( ( X4 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X4 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ X4 )
& ( ord_less_nat @ X3 @ Xa ) ) )
=> ~ ( finite_finite_nat @ X4 ) ) ) ).
% infinite_growing
thf(fact_1088_finite__ranking__induct,axiom,
! [S: set_nat,P: set_nat > $o,F3: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X3: nat,S3: set_nat] :
( ( finite_finite_nat @ S3 )
=> ( ! [Y5: nat] :
( ( member_nat @ Y5 @ S3 )
=> ( ord_less_eq_nat @ ( F3 @ Y5 ) @ ( F3 @ X3 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_nat @ X3 @ S3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1089_finite__ranking__induct,axiom,
! [S: set_a,P: set_a > $o,F3: a > nat] :
( ( finite_finite_a @ S )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X3: a,S3: set_a] :
( ( finite_finite_a @ S3 )
=> ( ! [Y5: a] :
( ( member_a @ Y5 @ S3 )
=> ( ord_less_eq_nat @ ( F3 @ Y5 ) @ ( F3 @ X3 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_a @ X3 @ S3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1090_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_1091_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_1092_inverse__closed,axiom,
! [X: a] :
( ( member_a @ X @ g )
=> ( member_a @ ( group_inverse_a @ g @ addition @ zero @ X ) @ g ) ) ).
% inverse_closed
thf(fact_1093_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_1094_minusset_Ocases,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) )
=> ~ ! [A3: a] :
( ( A
= ( group_inverse_a @ g @ addition @ zero @ A3 ) )
=> ( ( member_a @ A3 @ A2 )
=> ~ ( member_a @ A3 @ g ) ) ) ) ).
% minusset.cases
thf(fact_1095_minusset_OminussetI,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( member_a @ A @ g )
=> ( member_a @ ( group_inverse_a @ g @ addition @ zero @ A ) @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) ) ) ) ).
% minusset.minussetI
thf(fact_1096_minusset_Osimps,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) )
= ( ? [A4: a] :
( ( A
= ( group_inverse_a @ g @ addition @ zero @ A4 ) )
& ( member_a @ A4 @ A2 )
& ( member_a @ A4 @ g ) ) ) ) ).
% minusset.simps
thf(fact_1097_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_1098_inverse__unit,axiom,
( ( group_inverse_a @ g @ addition @ zero @ zero )
= zero ) ).
% inverse_unit
thf(fact_1099_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_1100_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_1101_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_1102_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_1103_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_1104_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_1105_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_1106_additive__abelian__group_Ominusset_OminussetI,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: nat,A2: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( member_nat @ A @ A2 )
=> ( ( member_nat @ A @ G )
=> ( member_nat @ ( group_inverse_nat @ G @ Addition @ Zero @ A ) @ ( pluenn7323955030898006884et_nat @ G @ Addition @ Zero @ A2 ) ) ) ) ) ).
% additive_abelian_group.minusset.minussetI
thf(fact_1107_additive__abelian__group_Ominusset_OminussetI,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: a,A2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( member_a @ A @ A2 )
=> ( ( member_a @ A @ G )
=> ( member_a @ ( group_inverse_a @ G @ Addition @ Zero @ A ) @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A2 ) ) ) ) ) ).
% additive_abelian_group.minusset.minussetI
thf(fact_1108_additive__abelian__group_Ominusset_Osimps,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: nat,A2: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( member_nat @ A @ ( pluenn7323955030898006884et_nat @ G @ Addition @ Zero @ A2 ) )
= ( ? [A4: nat] :
( ( A
= ( group_inverse_nat @ G @ Addition @ Zero @ A4 ) )
& ( member_nat @ A4 @ A2 )
& ( member_nat @ A4 @ G ) ) ) ) ) ).
% additive_abelian_group.minusset.simps
thf(fact_1109_additive__abelian__group_Ominusset_Osimps,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: a,A2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( member_a @ A @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A2 ) )
= ( ? [A4: a] :
( ( A
= ( group_inverse_a @ G @ Addition @ Zero @ A4 ) )
& ( member_a @ A4 @ A2 )
& ( member_a @ A4 @ G ) ) ) ) ) ).
% additive_abelian_group.minusset.simps
thf(fact_1110_additive__abelian__group_Ominusset_Ocases,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: nat,A2: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( member_nat @ A @ ( pluenn7323955030898006884et_nat @ G @ Addition @ Zero @ A2 ) )
=> ~ ! [A3: nat] :
( ( A
= ( group_inverse_nat @ G @ Addition @ Zero @ A3 ) )
=> ( ( member_nat @ A3 @ A2 )
=> ~ ( member_nat @ A3 @ G ) ) ) ) ) ).
% additive_abelian_group.minusset.cases
thf(fact_1111_additive__abelian__group_Ominusset_Ocases,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: a,A2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( member_a @ A @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A2 ) )
=> ~ ! [A3: a] :
( ( A
= ( group_inverse_a @ G @ Addition @ Zero @ A3 ) )
=> ( ( member_a @ A3 @ A2 )
=> ~ ( member_a @ A3 @ G ) ) ) ) ) ).
% additive_abelian_group.minusset.cases
thf(fact_1112_minussetp_Ocases,axiom,
! [A2: a > $o,A: a] :
( ( pluenn1126946703085653920setp_a @ g @ addition @ zero @ A2 @ A )
=> ~ ! [A3: a] :
( ( A
= ( group_inverse_a @ g @ addition @ zero @ A3 ) )
=> ( ( A2 @ A3 )
=> ~ ( member_a @ A3 @ g ) ) ) ) ).
% minussetp.cases
thf(fact_1113_minussetp_Osimps,axiom,
! [A2: a > $o,A: a] :
( ( pluenn1126946703085653920setp_a @ g @ addition @ zero @ A2 @ A )
= ( ? [A4: a] :
( ( A
= ( group_inverse_a @ g @ addition @ zero @ A4 ) )
& ( A2 @ A4 )
& ( member_a @ A4 @ g ) ) ) ) ).
% minussetp.simps
thf(fact_1114_minussetp_OminussetI,axiom,
! [A2: a > $o,A: a] :
( ( A2 @ A )
=> ( ( member_a @ A @ g )
=> ( pluenn1126946703085653920setp_a @ g @ addition @ zero @ A2 @ ( group_inverse_a @ g @ addition @ zero @ A ) ) ) ) ).
% minussetp.minussetI
thf(fact_1115_additive__abelian__group_Ominussetp_Ocong,axiom,
pluenn1126946703085653920setp_a = pluenn1126946703085653920setp_a ).
% additive_abelian_group.minussetp.cong
thf(fact_1116_additive__abelian__group_Ominussetp_OminussetI,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: nat > $o,A: nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( A2 @ A )
=> ( ( member_nat @ A @ G )
=> ( pluenn8372939692575285934tp_nat @ G @ Addition @ Zero @ A2 @ ( group_inverse_nat @ G @ Addition @ Zero @ A ) ) ) ) ) ).
% additive_abelian_group.minussetp.minussetI
thf(fact_1117_additive__abelian__group_Ominussetp_OminussetI,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: a > $o,A: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( A2 @ A )
=> ( ( member_a @ A @ G )
=> ( pluenn1126946703085653920setp_a @ G @ Addition @ Zero @ A2 @ ( group_inverse_a @ G @ Addition @ Zero @ A ) ) ) ) ) ).
% additive_abelian_group.minussetp.minussetI
thf(fact_1118_additive__abelian__group_Ominussetp_Osimps,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: nat > $o,A: nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( pluenn8372939692575285934tp_nat @ G @ Addition @ Zero @ A2 @ A )
= ( ? [A4: nat] :
( ( A
= ( group_inverse_nat @ G @ Addition @ Zero @ A4 ) )
& ( A2 @ A4 )
& ( member_nat @ A4 @ G ) ) ) ) ) ).
% additive_abelian_group.minussetp.simps
thf(fact_1119_additive__abelian__group_Ominussetp_Osimps,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: a > $o,A: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn1126946703085653920setp_a @ G @ Addition @ Zero @ A2 @ A )
= ( ? [A4: a] :
( ( A
= ( group_inverse_a @ G @ Addition @ Zero @ A4 ) )
& ( A2 @ A4 )
& ( member_a @ A4 @ G ) ) ) ) ) ).
% additive_abelian_group.minussetp.simps
thf(fact_1120_additive__abelian__group_Ominussetp_Ocases,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: nat > $o,A: nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( pluenn8372939692575285934tp_nat @ G @ Addition @ Zero @ A2 @ A )
=> ~ ! [A3: nat] :
( ( A
= ( group_inverse_nat @ G @ Addition @ Zero @ A3 ) )
=> ( ( A2 @ A3 )
=> ~ ( member_nat @ A3 @ G ) ) ) ) ) ).
% additive_abelian_group.minussetp.cases
thf(fact_1121_additive__abelian__group_Ominussetp_Ocases,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: a > $o,A: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn1126946703085653920setp_a @ G @ Addition @ Zero @ A2 @ A )
=> ~ ! [A3: a] :
( ( A
= ( group_inverse_a @ G @ Addition @ Zero @ A3 ) )
=> ( ( A2 @ A3 )
=> ~ ( member_a @ A3 @ G ) ) ) ) ) ).
% additive_abelian_group.minussetp.cases
thf(fact_1122_subgroupI,axiom,
! [G: set_a] :
( ( ord_less_eq_set_a @ G @ g )
=> ( ( member_a @ zero @ G )
=> ( ! [G4: a,H: a] :
( ( member_a @ G4 @ G )
=> ( ( member_a @ H @ G )
=> ( member_a @ ( addition @ G4 @ H ) @ G ) ) )
=> ( ! [G4: a] :
( ( member_a @ G4 @ G )
=> ( group_invertible_a @ g @ addition @ zero @ G4 ) )
=> ( ! [G4: a] :
( ( member_a @ G4 @ G )
=> ( member_a @ ( group_inverse_a @ g @ addition @ zero @ G4 ) @ G ) )
=> ( group_subgroup_a @ G @ g @ addition @ zero ) ) ) ) ) ) ).
% subgroupI
thf(fact_1123_minusset__eq,axiom,
! [A2: set_a] :
( ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 )
= ( image_a_a @ ( group_inverse_a @ g @ addition @ zero ) @ ( inf_inf_set_a @ A2 @ g ) ) ) ).
% minusset_eq
thf(fact_1124_image__eqI,axiom,
! [B4: a,F3: a > a,X: a,A2: set_a] :
( ( B4
= ( F3 @ X ) )
=> ( ( member_a @ X @ A2 )
=> ( member_a @ B4 @ ( image_a_a @ F3 @ A2 ) ) ) ) ).
% image_eqI
thf(fact_1125_image__eqI,axiom,
! [B4: nat,F3: a > nat,X: a,A2: set_a] :
( ( B4
= ( F3 @ X ) )
=> ( ( member_a @ X @ A2 )
=> ( member_nat @ B4 @ ( image_a_nat @ F3 @ A2 ) ) ) ) ).
% image_eqI
thf(fact_1126_image__eqI,axiom,
! [B4: a,F3: nat > a,X: nat,A2: set_nat] :
( ( B4
= ( F3 @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_a @ B4 @ ( image_nat_a @ F3 @ A2 ) ) ) ) ).
% image_eqI
thf(fact_1127_image__eqI,axiom,
! [B4: nat,F3: nat > nat,X: nat,A2: set_nat] :
( ( B4
= ( F3 @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ B4 @ ( image_nat_nat @ F3 @ A2 ) ) ) ) ).
% image_eqI
thf(fact_1128_image__empty,axiom,
! [F3: nat > nat] :
( ( image_nat_nat @ F3 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_1129_image__empty,axiom,
! [F3: a > a] :
( ( image_a_a @ F3 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% image_empty
thf(fact_1130_empty__is__image,axiom,
! [F3: nat > nat,A2: set_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat @ F3 @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_1131_empty__is__image,axiom,
! [F3: a > a,A2: set_a] :
( ( bot_bot_set_a
= ( image_a_a @ F3 @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_1132_image__is__empty,axiom,
! [F3: nat > nat,A2: set_nat] :
( ( ( image_nat_nat @ F3 @ A2 )
= bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_1133_image__is__empty,axiom,
! [F3: a > a,A2: set_a] :
( ( ( image_a_a @ F3 @ A2 )
= bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_1134_finite__imageI,axiom,
! [F: set_a,H2: a > a] :
( ( finite_finite_a @ F )
=> ( finite_finite_a @ ( image_a_a @ H2 @ F ) ) ) ).
% finite_imageI
thf(fact_1135_finite__imageI,axiom,
! [F: set_a,H2: a > nat] :
( ( finite_finite_a @ F )
=> ( finite_finite_nat @ ( image_a_nat @ H2 @ F ) ) ) ).
% finite_imageI
thf(fact_1136_finite__imageI,axiom,
! [F: set_nat,H2: nat > a] :
( ( finite_finite_nat @ F )
=> ( finite_finite_a @ ( image_nat_a @ H2 @ F ) ) ) ).
% finite_imageI
thf(fact_1137_finite__imageI,axiom,
! [F: set_nat,H2: nat > nat] :
( ( finite_finite_nat @ F )
=> ( finite_finite_nat @ ( image_nat_nat @ H2 @ F ) ) ) ).
% finite_imageI
thf(fact_1138_image__insert,axiom,
! [F3: nat > nat,A: nat,B: set_nat] :
( ( image_nat_nat @ F3 @ ( insert_nat @ A @ B ) )
= ( insert_nat @ ( F3 @ A ) @ ( image_nat_nat @ F3 @ B ) ) ) ).
% image_insert
thf(fact_1139_image__insert,axiom,
! [F3: a > a,A: a,B: set_a] :
( ( image_a_a @ F3 @ ( insert_a @ A @ B ) )
= ( insert_a @ ( F3 @ A ) @ ( image_a_a @ F3 @ B ) ) ) ).
% image_insert
thf(fact_1140_insert__image,axiom,
! [X: a,A2: set_a,F3: a > a] :
( ( member_a @ X @ A2 )
=> ( ( insert_a @ ( F3 @ X ) @ ( image_a_a @ F3 @ A2 ) )
= ( image_a_a @ F3 @ A2 ) ) ) ).
% insert_image
thf(fact_1141_insert__image,axiom,
! [X: nat,A2: set_nat,F3: nat > nat] :
( ( member_nat @ X @ A2 )
=> ( ( insert_nat @ ( F3 @ X ) @ ( image_nat_nat @ F3 @ A2 ) )
= ( image_nat_nat @ F3 @ A2 ) ) ) ).
% insert_image
thf(fact_1142_insert__image,axiom,
! [X: nat,A2: set_nat,F3: nat > a] :
( ( member_nat @ X @ A2 )
=> ( ( insert_a @ ( F3 @ X ) @ ( image_nat_a @ F3 @ A2 ) )
= ( image_nat_a @ F3 @ A2 ) ) ) ).
% insert_image
thf(fact_1143_inverse__subgroupI,axiom,
! [H3: set_a] :
( ( group_subgroup_a @ H3 @ g @ addition @ zero )
=> ( group_subgroup_a @ ( image_a_a @ ( group_inverse_a @ g @ addition @ zero ) @ H3 ) @ g @ addition @ zero ) ) ).
% inverse_subgroupI
thf(fact_1144_inverse__subgroupD,axiom,
! [H3: set_a] :
( ( group_subgroup_a @ ( image_a_a @ ( group_inverse_a @ g @ addition @ zero ) @ H3 ) @ g @ addition @ zero )
=> ( ( ord_less_eq_set_a @ H3 @ ( group_Units_a @ g @ addition @ zero ) )
=> ( group_subgroup_a @ H3 @ g @ addition @ zero ) ) ) ).
% inverse_subgroupD
thf(fact_1145_group_Oinverse__subgroupI,axiom,
! [G: set_nat,Composition2: nat > nat > nat,Unit2: nat,H3: set_nat] :
( ( group_group_nat @ G @ Composition2 @ Unit2 )
=> ( ( group_subgroup_nat @ H3 @ G @ Composition2 @ Unit2 )
=> ( group_subgroup_nat @ ( image_nat_nat @ ( group_inverse_nat @ G @ Composition2 @ Unit2 ) @ H3 ) @ G @ Composition2 @ Unit2 ) ) ) ).
% group.inverse_subgroupI
thf(fact_1146_group_Oinverse__subgroupI,axiom,
! [G: set_a,Composition2: a > a > a,Unit2: a,H3: set_a] :
( ( group_group_a @ G @ Composition2 @ Unit2 )
=> ( ( group_subgroup_a @ H3 @ G @ Composition2 @ Unit2 )
=> ( group_subgroup_a @ ( image_a_a @ ( group_inverse_a @ G @ Composition2 @ Unit2 ) @ H3 ) @ G @ Composition2 @ Unit2 ) ) ) ).
% group.inverse_subgroupI
thf(fact_1147_group_Oinverse__subgroupD,axiom,
! [G: set_nat,Composition2: nat > nat > nat,Unit2: nat,H3: set_nat] :
( ( group_group_nat @ G @ Composition2 @ Unit2 )
=> ( ( group_subgroup_nat @ ( image_nat_nat @ ( group_inverse_nat @ G @ Composition2 @ Unit2 ) @ H3 ) @ G @ Composition2 @ Unit2 )
=> ( ( ord_less_eq_set_nat @ H3 @ ( group_Units_nat @ G @ Composition2 @ Unit2 ) )
=> ( group_subgroup_nat @ H3 @ G @ Composition2 @ Unit2 ) ) ) ) ).
% group.inverse_subgroupD
thf(fact_1148_group_Oinverse__subgroupD,axiom,
! [G: set_a,Composition2: a > a > a,Unit2: a,H3: set_a] :
( ( group_group_a @ G @ Composition2 @ Unit2 )
=> ( ( group_subgroup_a @ ( image_a_a @ ( group_inverse_a @ G @ Composition2 @ Unit2 ) @ H3 ) @ G @ Composition2 @ Unit2 )
=> ( ( ord_less_eq_set_a @ H3 @ ( group_Units_a @ G @ Composition2 @ Unit2 ) )
=> ( group_subgroup_a @ H3 @ G @ Composition2 @ Unit2 ) ) ) ) ).
% group.inverse_subgroupD
thf(fact_1149_card__image__le,axiom,
! [A2: set_a,F3: a > a] :
( ( finite_finite_a @ A2 )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( image_a_a @ F3 @ A2 ) ) @ ( finite_card_a @ A2 ) ) ) ).
% card_image_le
thf(fact_1150_card__image__le,axiom,
! [A2: set_nat,F3: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_nat_nat @ F3 @ A2 ) ) @ ( finite_card_nat @ A2 ) ) ) ).
% card_image_le
thf(fact_1151_card__image__le,axiom,
! [A2: set_nat,F3: nat > a] :
( ( finite_finite_nat @ A2 )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( image_nat_a @ F3 @ A2 ) ) @ ( finite_card_nat @ A2 ) ) ) ).
% card_image_le
thf(fact_1152_subgroup_Oaxioms_I2_J,axiom,
! [G: set_a,M4: set_a,Composition2: a > a > a,Unit2: a] :
( ( group_subgroup_a @ G @ M4 @ Composition2 @ Unit2 )
=> ( group_group_a @ G @ Composition2 @ Unit2 ) ) ).
% subgroup.axioms(2)
thf(fact_1153_image__Un,axiom,
! [F3: nat > nat,A2: set_nat,B: set_nat] :
( ( image_nat_nat @ F3 @ ( sup_sup_set_nat @ A2 @ B ) )
= ( sup_sup_set_nat @ ( image_nat_nat @ F3 @ A2 ) @ ( image_nat_nat @ F3 @ B ) ) ) ).
% image_Un
thf(fact_1154_image__Un,axiom,
! [F3: a > a,A2: set_a,B: set_a] :
( ( image_a_a @ F3 @ ( sup_sup_set_a @ A2 @ B ) )
= ( sup_sup_set_a @ ( image_a_a @ F3 @ A2 ) @ ( image_a_a @ F3 @ B ) ) ) ).
% image_Un
thf(fact_1155_imageI,axiom,
! [X: a,A2: set_a,F3: a > a] :
( ( member_a @ X @ A2 )
=> ( member_a @ ( F3 @ X ) @ ( image_a_a @ F3 @ A2 ) ) ) ).
% imageI
thf(fact_1156_imageI,axiom,
! [X: a,A2: set_a,F3: a > nat] :
( ( member_a @ X @ A2 )
=> ( member_nat @ ( F3 @ X ) @ ( image_a_nat @ F3 @ A2 ) ) ) ).
% imageI
thf(fact_1157_imageI,axiom,
! [X: nat,A2: set_nat,F3: nat > a] :
( ( member_nat @ X @ A2 )
=> ( member_a @ ( F3 @ X ) @ ( image_nat_a @ F3 @ A2 ) ) ) ).
% imageI
thf(fact_1158_imageI,axiom,
! [X: nat,A2: set_nat,F3: nat > nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat @ ( F3 @ X ) @ ( image_nat_nat @ F3 @ A2 ) ) ) ).
% imageI
thf(fact_1159_image__iff,axiom,
! [Z2: a,F3: a > a,A2: set_a] :
( ( member_a @ Z2 @ ( image_a_a @ F3 @ A2 ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ( Z2
= ( F3 @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_1160_image__iff,axiom,
! [Z2: nat,F3: nat > nat,A2: set_nat] :
( ( member_nat @ Z2 @ ( image_nat_nat @ F3 @ A2 ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( Z2
= ( F3 @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_1161_bex__imageD,axiom,
! [F3: a > a,A2: set_a,P: a > $o] :
( ? [X5: a] :
( ( member_a @ X5 @ ( image_a_a @ F3 @ A2 ) )
& ( P @ X5 ) )
=> ? [X3: a] :
( ( member_a @ X3 @ A2 )
& ( P @ ( F3 @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_1162_bex__imageD,axiom,
! [F3: nat > nat,A2: set_nat,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat @ X5 @ ( image_nat_nat @ F3 @ A2 ) )
& ( P @ X5 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ ( F3 @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_1163_image__cong,axiom,
! [M4: set_a,N4: set_a,F3: a > a,G5: a > a] :
( ( M4 = N4 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ N4 )
=> ( ( F3 @ X3 )
= ( G5 @ X3 ) ) )
=> ( ( image_a_a @ F3 @ M4 )
= ( image_a_a @ G5 @ N4 ) ) ) ) ).
% image_cong
thf(fact_1164_image__cong,axiom,
! [M4: set_nat,N4: set_nat,F3: nat > nat,G5: nat > nat] :
( ( M4 = N4 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ N4 )
=> ( ( F3 @ X3 )
= ( G5 @ X3 ) ) )
=> ( ( image_nat_nat @ F3 @ M4 )
= ( image_nat_nat @ G5 @ N4 ) ) ) ) ).
% image_cong
thf(fact_1165_ball__imageD,axiom,
! [F3: a > a,A2: set_a,P: a > $o] :
( ! [X3: a] :
( ( member_a @ X3 @ ( image_a_a @ F3 @ A2 ) )
=> ( P @ X3 ) )
=> ! [X5: a] :
( ( member_a @ X5 @ A2 )
=> ( P @ ( F3 @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_1166_ball__imageD,axiom,
! [F3: nat > nat,A2: set_nat,P: nat > $o] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( image_nat_nat @ F3 @ A2 ) )
=> ( P @ X3 ) )
=> ! [X5: nat] :
( ( member_nat @ X5 @ A2 )
=> ( P @ ( F3 @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_1167_rev__image__eqI,axiom,
! [X: a,A2: set_a,B4: a,F3: a > a] :
( ( member_a @ X @ A2 )
=> ( ( B4
= ( F3 @ X ) )
=> ( member_a @ B4 @ ( image_a_a @ F3 @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_1168_rev__image__eqI,axiom,
! [X: a,A2: set_a,B4: nat,F3: a > nat] :
( ( member_a @ X @ A2 )
=> ( ( B4
= ( F3 @ X ) )
=> ( member_nat @ B4 @ ( image_a_nat @ F3 @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_1169_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B4: a,F3: nat > a] :
( ( member_nat @ X @ A2 )
=> ( ( B4
= ( F3 @ X ) )
=> ( member_a @ B4 @ ( image_nat_a @ F3 @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_1170_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B4: nat,F3: nat > nat] :
( ( member_nat @ X @ A2 )
=> ( ( B4
= ( F3 @ X ) )
=> ( member_nat @ B4 @ ( image_nat_nat @ F3 @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_1171_all__subset__image,axiom,
! [F3: nat > nat,A2: set_nat,P: set_nat > $o] :
( ( ! [B7: set_nat] :
( ( ord_less_eq_set_nat @ B7 @ ( image_nat_nat @ F3 @ A2 ) )
=> ( P @ B7 ) ) )
= ( ! [B7: set_nat] :
( ( ord_less_eq_set_nat @ B7 @ A2 )
=> ( P @ ( image_nat_nat @ F3 @ B7 ) ) ) ) ) ).
% all_subset_image
thf(fact_1172_all__subset__image,axiom,
! [F3: a > a,A2: set_a,P: set_a > $o] :
( ( ! [B7: set_a] :
( ( ord_less_eq_set_a @ B7 @ ( image_a_a @ F3 @ A2 ) )
=> ( P @ B7 ) ) )
= ( ! [B7: set_a] :
( ( ord_less_eq_set_a @ B7 @ A2 )
=> ( P @ ( image_a_a @ F3 @ B7 ) ) ) ) ) ).
% all_subset_image
thf(fact_1173_subset__image__iff,axiom,
! [B: set_nat,F3: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F3 @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B
= ( image_nat_nat @ F3 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1174_subset__image__iff,axiom,
! [B: set_a,F3: a > a,A2: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a @ F3 @ A2 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A2 )
& ( B
= ( image_a_a @ F3 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1175_image__subset__iff,axiom,
! [F3: nat > nat,A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F3 @ A2 ) @ B )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat @ ( F3 @ X2 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_1176_image__subset__iff,axiom,
! [F3: a > a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( image_a_a @ F3 @ A2 ) @ B )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_a @ ( F3 @ X2 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_1177_subset__imageE,axiom,
! [B: set_nat,F3: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F3 @ A2 ) )
=> ~ ! [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A2 )
=> ( B
!= ( image_nat_nat @ F3 @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_1178_subset__imageE,axiom,
! [B: set_a,F3: a > a,A2: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a @ F3 @ A2 ) )
=> ~ ! [C4: set_a] :
( ( ord_less_eq_set_a @ C4 @ A2 )
=> ( B
!= ( image_a_a @ F3 @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_1179_image__subsetI,axiom,
! [A2: set_a,F3: a > nat,B: set_nat] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member_nat @ ( F3 @ X3 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_a_nat @ F3 @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_1180_image__subsetI,axiom,
! [A2: set_nat,F3: nat > nat,B: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat @ ( F3 @ X3 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F3 @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_1181_image__subsetI,axiom,
! [A2: set_a,F3: a > a,B: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member_a @ ( F3 @ X3 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F3 @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_1182_image__subsetI,axiom,
! [A2: set_nat,F3: nat > a,B: set_a] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_a @ ( F3 @ X3 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_nat_a @ F3 @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_1183_image__mono,axiom,
! [A2: set_nat,B: set_nat,F3: nat > nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F3 @ A2 ) @ ( image_nat_nat @ F3 @ B ) ) ) ).
% image_mono
thf(fact_1184_image__mono,axiom,
! [A2: set_a,B: set_a,F3: a > a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ord_less_eq_set_a @ ( image_a_a @ F3 @ A2 ) @ ( image_a_a @ F3 @ B ) ) ) ).
% image_mono
thf(fact_1185_image__diff__subset,axiom,
! [F3: nat > nat,A2: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_nat_nat @ F3 @ A2 ) @ ( image_nat_nat @ F3 @ B ) ) @ ( image_nat_nat @ F3 @ ( minus_minus_set_nat @ A2 @ B ) ) ) ).
% image_diff_subset
thf(fact_1186_image__diff__subset,axiom,
! [F3: a > a,A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ ( image_a_a @ F3 @ A2 ) @ ( image_a_a @ F3 @ B ) ) @ ( image_a_a @ F3 @ ( minus_minus_set_a @ A2 @ B ) ) ) ).
% image_diff_subset
thf(fact_1187_image__Int__subset,axiom,
! [F3: nat > nat,A2: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( image_nat_nat @ F3 @ ( inf_inf_set_nat @ A2 @ B ) ) @ ( inf_inf_set_nat @ ( image_nat_nat @ F3 @ A2 ) @ ( image_nat_nat @ F3 @ B ) ) ) ).
% image_Int_subset
thf(fact_1188_image__Int__subset,axiom,
! [F3: a > a,A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( image_a_a @ F3 @ ( inf_inf_set_a @ A2 @ B ) ) @ ( inf_inf_set_a @ ( image_a_a @ F3 @ A2 ) @ ( image_a_a @ F3 @ B ) ) ) ).
% image_Int_subset
thf(fact_1189_all__finite__subset__image,axiom,
! [F3: nat > nat,A2: set_nat,P: set_nat > $o] :
( ( ! [B7: set_nat] :
( ( ( finite_finite_nat @ B7 )
& ( ord_less_eq_set_nat @ B7 @ ( image_nat_nat @ F3 @ A2 ) ) )
=> ( P @ B7 ) ) )
= ( ! [B7: set_nat] :
( ( ( finite_finite_nat @ B7 )
& ( ord_less_eq_set_nat @ B7 @ A2 ) )
=> ( P @ ( image_nat_nat @ F3 @ B7 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1190_all__finite__subset__image,axiom,
! [F3: a > nat,A2: set_a,P: set_nat > $o] :
( ( ! [B7: set_nat] :
( ( ( finite_finite_nat @ B7 )
& ( ord_less_eq_set_nat @ B7 @ ( image_a_nat @ F3 @ A2 ) ) )
=> ( P @ B7 ) ) )
= ( ! [B7: set_a] :
( ( ( finite_finite_a @ B7 )
& ( ord_less_eq_set_a @ B7 @ A2 ) )
=> ( P @ ( image_a_nat @ F3 @ B7 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1191_all__finite__subset__image,axiom,
! [F3: nat > a,A2: set_nat,P: set_a > $o] :
( ( ! [B7: set_a] :
( ( ( finite_finite_a @ B7 )
& ( ord_less_eq_set_a @ B7 @ ( image_nat_a @ F3 @ A2 ) ) )
=> ( P @ B7 ) ) )
= ( ! [B7: set_nat] :
( ( ( finite_finite_nat @ B7 )
& ( ord_less_eq_set_nat @ B7 @ A2 ) )
=> ( P @ ( image_nat_a @ F3 @ B7 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1192_all__finite__subset__image,axiom,
! [F3: a > a,A2: set_a,P: set_a > $o] :
( ( ! [B7: set_a] :
( ( ( finite_finite_a @ B7 )
& ( ord_less_eq_set_a @ B7 @ ( image_a_a @ F3 @ A2 ) ) )
=> ( P @ B7 ) ) )
= ( ! [B7: set_a] :
( ( ( finite_finite_a @ B7 )
& ( ord_less_eq_set_a @ B7 @ A2 ) )
=> ( P @ ( image_a_a @ F3 @ B7 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1193_ex__finite__subset__image,axiom,
! [F3: nat > nat,A2: set_nat,P: set_nat > $o] :
( ( ? [B7: set_nat] :
( ( finite_finite_nat @ B7 )
& ( ord_less_eq_set_nat @ B7 @ ( image_nat_nat @ F3 @ A2 ) )
& ( P @ B7 ) ) )
= ( ? [B7: set_nat] :
( ( finite_finite_nat @ B7 )
& ( ord_less_eq_set_nat @ B7 @ A2 )
& ( P @ ( image_nat_nat @ F3 @ B7 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1194_ex__finite__subset__image,axiom,
! [F3: a > nat,A2: set_a,P: set_nat > $o] :
( ( ? [B7: set_nat] :
( ( finite_finite_nat @ B7 )
& ( ord_less_eq_set_nat @ B7 @ ( image_a_nat @ F3 @ A2 ) )
& ( P @ B7 ) ) )
= ( ? [B7: set_a] :
( ( finite_finite_a @ B7 )
& ( ord_less_eq_set_a @ B7 @ A2 )
& ( P @ ( image_a_nat @ F3 @ B7 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1195_ex__finite__subset__image,axiom,
! [F3: nat > a,A2: set_nat,P: set_a > $o] :
( ( ? [B7: set_a] :
( ( finite_finite_a @ B7 )
& ( ord_less_eq_set_a @ B7 @ ( image_nat_a @ F3 @ A2 ) )
& ( P @ B7 ) ) )
= ( ? [B7: set_nat] :
( ( finite_finite_nat @ B7 )
& ( ord_less_eq_set_nat @ B7 @ A2 )
& ( P @ ( image_nat_a @ F3 @ B7 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1196_ex__finite__subset__image,axiom,
! [F3: a > a,A2: set_a,P: set_a > $o] :
( ( ? [B7: set_a] :
( ( finite_finite_a @ B7 )
& ( ord_less_eq_set_a @ B7 @ ( image_a_a @ F3 @ A2 ) )
& ( P @ B7 ) ) )
= ( ? [B7: set_a] :
( ( finite_finite_a @ B7 )
& ( ord_less_eq_set_a @ B7 @ A2 )
& ( P @ ( image_a_a @ F3 @ B7 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1197_finite__subset__image,axiom,
! [B: set_nat,F3: nat > nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F3 @ A2 ) )
=> ? [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A2 )
& ( finite_finite_nat @ C4 )
& ( B
= ( image_nat_nat @ F3 @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1198_finite__subset__image,axiom,
! [B: set_nat,F3: a > nat,A2: set_a] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ ( image_a_nat @ F3 @ A2 ) )
=> ? [C4: set_a] :
( ( ord_less_eq_set_a @ C4 @ A2 )
& ( finite_finite_a @ C4 )
& ( B
= ( image_a_nat @ F3 @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1199_finite__subset__image,axiom,
! [B: set_a,F3: nat > a,A2: set_nat] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ B @ ( image_nat_a @ F3 @ A2 ) )
=> ? [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A2 )
& ( finite_finite_nat @ C4 )
& ( B
= ( image_nat_a @ F3 @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1200_finite__subset__image,axiom,
! [B: set_a,F3: a > a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ B @ ( image_a_a @ F3 @ A2 ) )
=> ? [C4: set_a] :
( ( ord_less_eq_set_a @ C4 @ A2 )
& ( finite_finite_a @ C4 )
& ( B
= ( image_a_a @ F3 @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1201_finite__surj,axiom,
! [A2: set_a,B: set_nat,F3: a > nat] :
( ( finite_finite_a @ A2 )
=> ( ( ord_less_eq_set_nat @ B @ ( image_a_nat @ F3 @ A2 ) )
=> ( finite_finite_nat @ B ) ) ) ).
% finite_surj
thf(fact_1202_finite__surj,axiom,
! [A2: set_nat,B: set_nat,F3: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F3 @ A2 ) )
=> ( finite_finite_nat @ B ) ) ) ).
% finite_surj
thf(fact_1203_finite__surj,axiom,
! [A2: set_a,B: set_a,F3: a > a] :
( ( finite_finite_a @ A2 )
=> ( ( ord_less_eq_set_a @ B @ ( image_a_a @ F3 @ A2 ) )
=> ( finite_finite_a @ B ) ) ) ).
% finite_surj
thf(fact_1204_finite__surj,axiom,
! [A2: set_nat,B: set_a,F3: nat > a] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_set_a @ B @ ( image_nat_a @ F3 @ A2 ) )
=> ( finite_finite_a @ B ) ) ) ).
% finite_surj
thf(fact_1205_surj__card__le,axiom,
! [A2: set_nat,B: set_nat,F3: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F3 @ A2 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite_card_nat @ A2 ) ) ) ) ).
% surj_card_le
thf(fact_1206_surj__card__le,axiom,
! [A2: set_a,B: set_a,F3: a > a] :
( ( finite_finite_a @ A2 )
=> ( ( ord_less_eq_set_a @ B @ ( image_a_a @ F3 @ A2 ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ B ) @ ( finite_card_a @ A2 ) ) ) ) ).
% surj_card_le
thf(fact_1207_surj__card__le,axiom,
! [A2: set_nat,B: set_a,F3: nat > a] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_set_a @ B @ ( image_nat_a @ F3 @ A2 ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ B ) @ ( finite_card_nat @ A2 ) ) ) ) ).
% surj_card_le
thf(fact_1208_additive__abelian__group_Ominusset__eq,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( pluenn7323955030898006884et_nat @ G @ Addition @ Zero @ A2 )
= ( image_nat_nat @ ( group_inverse_nat @ G @ Addition @ Zero ) @ ( inf_inf_set_nat @ A2 @ G ) ) ) ) ).
% additive_abelian_group.minusset_eq
thf(fact_1209_additive__abelian__group_Ominusset__eq,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A2 )
= ( image_a_a @ ( group_inverse_a @ G @ Addition @ Zero ) @ ( inf_inf_set_a @ A2 @ G ) ) ) ) ).
% additive_abelian_group.minusset_eq
thf(fact_1210_monoid__axioms,axiom,
group_monoid_a @ g @ addition @ zero ).
% monoid_axioms
thf(fact_1211_card__le__if__inj__on__rel,axiom,
! [B: set_a,A2: set_nat,R: nat > a > $o] :
( ( finite_finite_a @ B )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A2 )
=> ? [B10: a] :
( ( member_a @ B10 @ B )
& ( R @ A3 @ B10 ) ) )
=> ( ! [A1: nat,A22: nat,B2: a] :
( ( member_nat @ A1 @ A2 )
=> ( ( member_nat @ A22 @ A2 )
=> ( ( member_a @ B2 @ B )
=> ( ( R @ A1 @ B2 )
=> ( ( R @ A22 @ B2 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_a @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1212_card__le__if__inj__on__rel,axiom,
! [B: set_a,A2: set_a,R: a > a > $o] :
( ( finite_finite_a @ B )
=> ( ! [A3: a] :
( ( member_a @ A3 @ A2 )
=> ? [B10: a] :
( ( member_a @ B10 @ B )
& ( R @ A3 @ B10 ) ) )
=> ( ! [A1: a,A22: a,B2: a] :
( ( member_a @ A1 @ A2 )
=> ( ( member_a @ A22 @ A2 )
=> ( ( member_a @ B2 @ B )
=> ( ( R @ A1 @ B2 )
=> ( ( R @ A22 @ B2 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1213_card__le__if__inj__on__rel,axiom,
! [B: set_nat,A2: set_nat,R: nat > nat > $o] :
( ( finite_finite_nat @ B )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A2 )
=> ? [B10: nat] :
( ( member_nat @ B10 @ B )
& ( R @ A3 @ B10 ) ) )
=> ( ! [A1: nat,A22: nat,B2: nat] :
( ( member_nat @ A1 @ A2 )
=> ( ( member_nat @ A22 @ A2 )
=> ( ( member_nat @ B2 @ B )
=> ( ( R @ A1 @ B2 )
=> ( ( R @ A22 @ B2 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1214_card__le__if__inj__on__rel,axiom,
! [B: set_nat,A2: set_a,R: a > nat > $o] :
( ( finite_finite_nat @ B )
=> ( ! [A3: a] :
( ( member_a @ A3 @ A2 )
=> ? [B10: nat] :
( ( member_nat @ B10 @ B )
& ( R @ A3 @ B10 ) ) )
=> ( ! [A1: a,A22: a,B2: nat] :
( ( member_a @ A1 @ A2 )
=> ( ( member_a @ A22 @ A2 )
=> ( ( member_nat @ B2 @ B )
=> ( ( R @ A1 @ B2 )
=> ( ( R @ A22 @ B2 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1215_in__image__insert__iff,axiom,
! [B: set_set_nat,X: nat,A2: set_nat] :
( ! [C4: set_nat] :
( ( member_set_nat @ C4 @ B )
=> ~ ( member_nat @ X @ C4 ) )
=> ( ( member_set_nat @ A2 @ ( image_7916887816326733075et_nat @ ( insert_nat @ X ) @ B ) )
= ( ( member_nat @ X @ A2 )
& ( member_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_1216_in__image__insert__iff,axiom,
! [B: set_set_a,X: a,A2: set_a] :
( ! [C4: set_a] :
( ( member_set_a @ C4 @ B )
=> ~ ( member_a @ X @ C4 ) )
=> ( ( member_set_a @ A2 @ ( image_set_a_set_a @ ( insert_a @ X ) @ B ) )
= ( ( member_a @ X @ A2 )
& ( member_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_1217_monoid_Omem__UnitsD,axiom,
! [M4: set_nat,Composition2: nat > nat > nat,Unit2: nat,U: nat] :
( ( group_monoid_nat @ M4 @ Composition2 @ Unit2 )
=> ( ( member_nat @ U @ ( group_Units_nat @ M4 @ Composition2 @ Unit2 ) )
=> ( ( group_invertible_nat @ M4 @ Composition2 @ Unit2 @ U )
& ( member_nat @ U @ M4 ) ) ) ) ).
% monoid.mem_UnitsD
thf(fact_1218_monoid_Omem__UnitsD,axiom,
! [M4: set_a,Composition2: a > a > a,Unit2: a,U: a] :
( ( group_monoid_a @ M4 @ Composition2 @ Unit2 )
=> ( ( member_a @ U @ ( group_Units_a @ M4 @ Composition2 @ Unit2 ) )
=> ( ( group_invertible_a @ M4 @ Composition2 @ Unit2 @ U )
& ( member_a @ U @ M4 ) ) ) ) ).
% monoid.mem_UnitsD
thf(fact_1219_monoid_Omem__UnitsI,axiom,
! [M4: set_a,Composition2: a > a > a,Unit2: a,U: a] :
( ( group_monoid_a @ M4 @ Composition2 @ Unit2 )
=> ( ( group_invertible_a @ M4 @ Composition2 @ Unit2 @ U )
=> ( ( member_a @ U @ M4 )
=> ( member_a @ U @ ( group_Units_a @ M4 @ Composition2 @ Unit2 ) ) ) ) ) ).
% monoid.mem_UnitsI
thf(fact_1220_bounded__nat__set__is__finite,axiom,
! [N4: set_nat,N: nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ N4 )
=> ( ord_less_nat @ X3 @ N ) )
=> ( finite_finite_nat @ N4 ) ) ).
% bounded_nat_set_is_finite
thf(fact_1221_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M3: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ N5 )
=> ( ord_less_nat @ X2 @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_1222_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M4: nat] :
( ( P @ X )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M4 ) )
=> ~ ! [M5: nat] :
( ( P @ M5 )
=> ~ ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M5 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1223_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M3: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ N5 )
=> ( ord_less_eq_nat @ X2 @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1224_sumset__iterated__Suc,axiom,
! [A2: set_a,K: nat] :
( ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ A2 @ ( suc @ K ) )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ A2 @ K ) ) ) ).
% sumset_iterated_Suc
thf(fact_1225_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_1226_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_1227_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_1228_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_1229_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_1230_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_1231_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_1232_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_1233_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1234_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_1235_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_1236_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M @ N ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_1237_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_1238_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_1239_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_1240_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_1241_zero__notin__Suc__image,axiom,
! [A2: set_nat] :
~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A2 ) ) ).
% zero_notin_Suc_image
thf(fact_1242_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less_eq_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1243_diff__Suc__less,axiom,
! [N: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_1244_one__less__mult,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% one_less_mult
thf(fact_1245_n__less__m__mult__n,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1246_n__less__n__mult__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1247_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1248_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_1249_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1250_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_1251_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_1252_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_1253_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_1254_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_1255_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_1256_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_1257_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1258_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_1259_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_1260_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_1261_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_1262_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_1263_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ N )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P @ I4 ) ) ) ) ).
% Ex_less_Suc
thf(fact_1264_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_1265_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_1266_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ I4 ) ) ) ) ).
% All_less_Suc
thf(fact_1267_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M6: nat] :
( ( M
= ( suc @ M6 ) )
& ( ord_less_nat @ N @ M6 ) ) ) ) ).
% Suc_less_eq2
% Conjectures (1)
thf(conj_0,conjecture,
ord_less_eq_nat @ ( times_times_nat @ ( finite_card_a @ u ) @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ v @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ w ) ) ) ) @ ( times_times_nat @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ u @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ v ) ) ) @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ u @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ w ) ) ) ) ).
%------------------------------------------------------------------------------