TPTP Problem File: SLH0734^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Eval_FO/0005_Ailamazyan/prob_01348_050665__15689324_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1406 ( 663 unt; 138 typ; 0 def)
% Number of atoms : 3087 (1169 equ; 0 cnn)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 11144 ( 320 ~; 44 |; 276 &;9318 @)
% ( 0 <=>;1186 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 20 ( 19 usr)
% Number of type conns : 702 ( 702 >; 0 *; 0 +; 0 <<)
% Number of symbols : 122 ( 119 usr; 11 con; 0-4 aty)
% Number of variables : 3629 ( 303 ^;3232 !; 94 ?;3629 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 13:06:23.616
%------------------------------------------------------------------------------
% Could-be-implicit typings (19)
thf(ty_n_t__Set__Oset_I_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Option__Ooption_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
set_Su6413317981332641185_a_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J_J,type,
set_Su6274467284060765424on_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Option__Ooption_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
set_na5864254054690418060_a_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
set_se4904748513628223167_a_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Option__Ooption_It__Nat__Onat_J_J_J,type,
set_nat_option_nat: $tType ).
thf(ty_n_t__Option__Ooption_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
option_Sum_sum_a_nat: $tType ).
thf(ty_n_t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
list_Sum_sum_a_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
set_Sum_sum_a_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Nat__Onat_J_J,type,
set_option_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
sum_sum_a_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Option__Ooption_It__Nat__Onat_J,type,
option_nat: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (119)
thf(sy_c_Ailamazyan_Ofo__nmlz__rec_001tf__a,type,
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thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
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thf(sy_c_Finite__Set_Ocard_001tf__a,type,
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thf(sy_c_Finite__Set_Ofinite_001_062_It__Nat__Onat_Mt__Option__Ooption_It__Nat__Onat_J_J,type,
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finite5498713997695131394_a_nat: set_Su6413317981332641185_a_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
finite1152437895449049373et_nat: set_set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001t__Nat__Onat,type,
fun_upd_nat_nat: ( nat > nat ) > nat > nat > nat > nat ).
thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001t__Option__Ooption_It__Nat__Onat_J,type,
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thf(sy_c_Fun_Ofun__upd_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Option__Ooption_It__Nat__Onat_J,type,
fun_up4750343006489851234on_nat: ( sum_sum_a_nat > option_nat ) > sum_sum_a_nat > option_nat > sum_sum_a_nat > option_nat ).
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fun_up6086130847573437501_a_nat: ( sum_sum_a_nat > sum_sum_a_nat ) > sum_sum_a_nat > sum_sum_a_nat > sum_sum_a_nat > sum_sum_a_nat ).
thf(sy_c_Fun_Ofun__upd_001tf__a_001t__Nat__Onat,type,
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thf(sy_c_Fun_Ofun__upd_001tf__a_001t__Option__Ooption_It__Nat__Onat_J,type,
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thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Nat__Onat_M_Eo_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001_062_Itf__a_M_Eo_J,type,
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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,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_If_001t__Option__Ooption_It__Nat__Onat_J,type,
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thf(sy_c_If_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Nat__Onat_M_Eo_J,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001_062_Itf__a_M_Eo_J,type,
inf_inf_a_o: ( a > $o ) > ( a > $o ) > 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_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
inf_in7084830621192376909_a_nat: set_Sum_sum_a_nat > set_Sum_sum_a_nat > set_Sum_sum_a_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_001_062_It__Nat__Onat_M_Eo_J,type,
sup_sup_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J,type,
sup_su491480579010597738_nat_o: ( sum_sum_a_nat > $o ) > ( sum_sum_a_nat > $o ) > sum_sum_a_nat > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_Itf__a_M_Eo_J,type,
sup_sup_a_o: ( a > $o ) > ( a > $o ) > a > $o ).
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_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
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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_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
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cons_Sum_sum_a_nat: sum_sum_a_nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat ).
thf(sy_c_List_Olist_OCons_001tf__a,type,
cons_a: a > list_a > list_a ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
set_Sum_sum_a_nat2: list_Sum_sum_a_nat > set_Sum_sum_a_nat ).
thf(sy_c_List_Olist_Oset_001tf__a,type,
set_a2: list_a > set_a ).
thf(sy_c_List_Ounion_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
union_Sum_sum_a_nat: list_Sum_sum_a_nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat ).
thf(sy_c_Map_Odom_001t__Nat__Onat_001t__Nat__Onat,type,
dom_nat_nat: ( nat > option_nat ) > set_nat ).
thf(sy_c_Map_Odom_001t__Nat__Onat_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
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thf(sy_c_Map_Odom_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Nat__Onat,type,
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thf(sy_c_Map_Odom_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
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thf(sy_c_Map_Odom_001tf__a_001t__Nat__Onat,type,
dom_a_nat: ( a > option_nat ) > set_a ).
thf(sy_c_Map_Oran_001t__Nat__Onat_001t__Nat__Onat,type,
ran_nat_nat: ( nat > option_nat ) > set_nat ).
thf(sy_c_Map_Oran_001t__Nat__Onat_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
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thf(sy_c_Nat_OSuc,type,
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thf(sy_c_Option_Ooption_ONone_001t__Nat__Onat,type,
none_nat: option_nat ).
thf(sy_c_Option_Ooption_OSome_001t__Nat__Onat,type,
some_nat: nat > option_nat ).
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,
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thf(sy_c_Set_OCollect_001tf__a,type,
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thf(sy_c_Set_Oimage_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
image_5599399343138760645_a_nat: ( set_Sum_sum_a_nat > set_Sum_sum_a_nat ) > set_se4904748513628223167_a_nat > set_se4904748513628223167_a_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_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Nat__Onat,type,
image_2473878607534554506at_nat: ( sum_sum_a_nat > nat ) > set_Sum_sum_a_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Option__Ooption_It__Nat__Onat_J,type,
image_9069542366050205658on_nat: ( sum_sum_a_nat > option_nat ) > set_Sum_sum_a_nat > set_option_nat ).
thf(sy_c_Set_Oimage_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
image_7142520692256960453_a_nat: ( sum_sum_a_nat > sum_sum_a_nat ) > set_Sum_sum_a_nat > set_Sum_sum_a_nat ).
thf(sy_c_Set_Oimage_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001tf__a,type,
image_6322530041254294468_nat_a: ( sum_sum_a_nat > a ) > set_Sum_sum_a_nat > set_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__Nat__Onat,type,
image_a_nat: ( a > nat ) > set_a > set_nat ).
thf(sy_c_Set_Oimage_001tf__a_001t__Option__Ooption_It__Nat__Onat_J,type,
image_a_option_nat: ( a > option_nat ) > set_a > set_option_nat ).
thf(sy_c_Set_Oimage_001tf__a_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
image_7873763678140191238_a_nat: ( a > sum_sum_a_nat ) > set_a > set_Sum_sum_a_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_001t__Option__Ooption_It__Nat__Onat_J,type,
insert_option_nat: option_nat > set_option_nat > set_option_nat ).
thf(sy_c_Set_Oinsert_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
insert_Sum_sum_a_nat: sum_sum_a_nat > set_Sum_sum_a_nat > set_Sum_sum_a_nat ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(sy_c_Sum__Type_OInl_001tf__a_001t__Nat__Onat,type,
sum_Inl_a_nat: a > sum_sum_a_nat ).
thf(sy_c_Sum__Type_OInr_001t__Nat__Onat_001tf__a,type,
sum_Inr_nat_a: nat > sum_sum_a_nat ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Option__Ooption_It__Nat__Onat_J,type,
member_option_nat: option_nat > set_option_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_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
member8098812455498974984_a_nat: set_Sum_sum_a_nat > set_se4904748513628223167_a_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_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
member_Sum_sum_a_nat: sum_sum_a_nat > set_Sum_sum_a_nat > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_ADa____,type,
aDa: set_a ).
thf(sy_v_ia____,type,
ia: nat ).
thf(sy_v_ma____,type,
ma: sum_sum_a_nat > option_nat ).
thf(sy_v_x____,type,
x: a ).
thf(sy_v_xsa____,type,
xsa: list_Sum_sum_a_nat ).
% Relevant facts (1262)
thf(fact_0_False,axiom,
~ ( member_a @ x @ aDa ) ).
% False
thf(fact_1_None,axiom,
( ( ma @ ( sum_Inl_a_nat @ x ) )
= none_nat ) ).
% None
thf(fact_2__092_060open_062set_A_Ifo__nmlz__rec_A_ISuc_Ai_J_A_Im_IInl_Ax_A_092_060mapsto_062_Ai_J_J_AAD_Axs_J_A_092_060union_062_AInr_A_096_A_123_O_O_060Suc_Ai_125_A_061_Aset_Axs_A_092_060inter_062_AInl_A_096_AAD_A_092_060union_062_AInr_A_096_A_123_O_O_060Suc_Ai_A_L_Acard_A_Iset_Axs_A_N_AInl_A_096_AAD_A_N_Adom_A_Im_IInl_Ax_A_092_060mapsto_062_Ai_J_J_J_125_092_060close_062,axiom,
( ( sup_su6804446743777130803_a_nat @ ( set_Sum_sum_a_nat2 @ ( fo_nmlz_rec_a @ ( suc @ ia ) @ ( fun_up4750343006489851234on_nat @ ma @ ( sum_Inl_a_nat @ x ) @ ( some_nat @ ia ) ) @ aDa @ xsa ) ) @ ( image_7293268710728258664_a_nat @ sum_Inr_nat_a @ ( set_ord_lessThan_nat @ ( suc @ ia ) ) ) )
= ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ ( set_Sum_sum_a_nat2 @ xsa ) @ ( image_7873763678140191238_a_nat @ sum_Inl_a_nat @ aDa ) ) @ ( image_7293268710728258664_a_nat @ sum_Inr_nat_a @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( suc @ ia ) @ ( finite6080979521523705895_a_nat @ ( minus_1134630996077396038_a_nat @ ( minus_1134630996077396038_a_nat @ ( set_Sum_sum_a_nat2 @ xsa ) @ ( image_7873763678140191238_a_nat @ sum_Inl_a_nat @ aDa ) ) @ ( dom_Su2255998037560862461at_nat @ ( fun_up4750343006489851234on_nat @ ma @ ( sum_Inl_a_nat @ x ) @ ( some_nat @ ia ) ) ) ) ) ) ) ) ) ) ).
% \<open>set (fo_nmlz_rec (Suc i) (m(Inl x \<mapsto> i)) AD xs) \<union> Inr ` {..<Suc i} = set xs \<inter> Inl ` AD \<union> Inr ` {..<Suc i + card (set xs - Inl ` AD - dom (m(Inl x \<mapsto> i)))}\<close>
thf(fact_3__092_060open_062x_A_092_060in_062_AAD_A_092_060Longrightarrow_062_Aset_A_Ifo__nmlz__rec_Ai_Am_AAD_Axs_J_A_092_060union_062_AInr_A_096_A_123_O_O_060i_125_A_061_Aset_Axs_A_092_060inter_062_AInl_A_096_AAD_A_092_060union_062_AInr_A_096_A_123_O_O_060i_A_L_Acard_A_Iset_Axs_A_N_AInl_A_096_AAD_A_N_Adom_Am_J_125_092_060close_062,axiom,
( ( member_a @ x @ aDa )
=> ( ( sup_su6804446743777130803_a_nat @ ( set_Sum_sum_a_nat2 @ ( fo_nmlz_rec_a @ ia @ ma @ aDa @ xsa ) ) @ ( image_7293268710728258664_a_nat @ sum_Inr_nat_a @ ( set_ord_lessThan_nat @ ia ) ) )
= ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ ( set_Sum_sum_a_nat2 @ xsa ) @ ( image_7873763678140191238_a_nat @ sum_Inl_a_nat @ aDa ) ) @ ( image_7293268710728258664_a_nat @ sum_Inr_nat_a @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ia @ ( finite6080979521523705895_a_nat @ ( minus_1134630996077396038_a_nat @ ( minus_1134630996077396038_a_nat @ ( set_Sum_sum_a_nat2 @ xsa ) @ ( image_7873763678140191238_a_nat @ sum_Inl_a_nat @ aDa ) ) @ ( dom_Su2255998037560862461at_nat @ ma ) ) ) ) ) ) ) ) ) ).
% \<open>x \<in> AD \<Longrightarrow> set (fo_nmlz_rec i m AD xs) \<union> Inr ` {..<i} = set xs \<inter> Inl ` AD \<union> Inr ` {..<i + card (set xs - Inl ` AD - dom m)}\<close>
thf(fact_4_Suc,axiom,
( ( plus_plus_nat @ ( suc @ ia ) @ ( finite6080979521523705895_a_nat @ ( minus_1134630996077396038_a_nat @ ( minus_1134630996077396038_a_nat @ ( set_Sum_sum_a_nat2 @ xsa ) @ ( image_7873763678140191238_a_nat @ sum_Inl_a_nat @ aDa ) ) @ ( dom_Su2255998037560862461at_nat @ ( fun_up4750343006489851234on_nat @ ma @ ( sum_Inl_a_nat @ x ) @ ( some_nat @ ia ) ) ) ) ) )
= ( plus_plus_nat @ ia @ ( finite6080979521523705895_a_nat @ ( minus_1134630996077396038_a_nat @ ( minus_1134630996077396038_a_nat @ ( set_Sum_sum_a_nat2 @ ( cons_Sum_sum_a_nat @ ( sum_Inl_a_nat @ x ) @ xsa ) ) @ ( image_7873763678140191238_a_nat @ sum_Inl_a_nat @ aDa ) ) @ ( dom_Su2255998037560862461at_nat @ ma ) ) ) ) ) ).
% Suc
thf(fact_5_fin,axiom,
finite502105017643426984_a_nat @ ( minus_1134630996077396038_a_nat @ ( minus_1134630996077396038_a_nat @ ( set_Sum_sum_a_nat2 @ ( cons_Sum_sum_a_nat @ ( sum_Inl_a_nat @ x ) @ xsa ) ) @ ( image_7873763678140191238_a_nat @ sum_Inl_a_nat @ aDa ) ) @ ( dom_Su2255998037560862461at_nat @ ma ) ) ).
% fin
thf(fact_6__C2_OIH_C_I1_J,axiom,
( ( member_a @ x @ aDa )
=> ( ( ord_less_eq_set_nat @ ( ran_Su5179294584019872288at_nat @ ma ) @ ( set_ord_lessThan_nat @ ia ) )
=> ( ( sup_su6804446743777130803_a_nat @ ( set_Sum_sum_a_nat2 @ ( fo_nmlz_rec_a @ ia @ ma @ aDa @ xsa ) ) @ ( image_7293268710728258664_a_nat @ sum_Inr_nat_a @ ( set_ord_lessThan_nat @ ia ) ) )
= ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ ( set_Sum_sum_a_nat2 @ xsa ) @ ( image_7873763678140191238_a_nat @ sum_Inl_a_nat @ aDa ) ) @ ( image_7293268710728258664_a_nat @ sum_Inr_nat_a @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ia @ ( finite6080979521523705895_a_nat @ ( minus_1134630996077396038_a_nat @ ( minus_1134630996077396038_a_nat @ ( set_Sum_sum_a_nat2 @ xsa ) @ ( image_7873763678140191238_a_nat @ sum_Inl_a_nat @ aDa ) ) @ ( dom_Su2255998037560862461at_nat @ ma ) ) ) ) ) ) ) ) ) ) ).
% "2.IH"(1)
thf(fact_7_if__image__distrib,axiom,
! [P: a > $o,F: a > sum_sum_a_nat,G: a > sum_sum_a_nat,S: set_a] :
( ( image_7873763678140191238_a_nat
@ ^ [X: a] : ( if_Sum_sum_a_nat @ ( P @ X ) @ ( F @ X ) @ ( G @ X ) )
@ S )
= ( sup_su6804446743777130803_a_nat @ ( image_7873763678140191238_a_nat @ F @ ( inf_inf_set_a @ S @ ( collect_a @ P ) ) )
@ ( image_7873763678140191238_a_nat @ G
@ ( inf_inf_set_a @ S
@ ( collect_a
@ ^ [X: a] :
~ ( P @ X ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_8_if__image__distrib,axiom,
! [P: nat > $o,F: nat > sum_sum_a_nat,G: nat > sum_sum_a_nat,S: set_nat] :
( ( image_7293268710728258664_a_nat
@ ^ [X: nat] : ( if_Sum_sum_a_nat @ ( P @ X ) @ ( F @ X ) @ ( G @ X ) )
@ S )
= ( sup_su6804446743777130803_a_nat @ ( image_7293268710728258664_a_nat @ F @ ( inf_inf_set_nat @ S @ ( collect_nat @ P ) ) )
@ ( image_7293268710728258664_a_nat @ G
@ ( inf_inf_set_nat @ S
@ ( collect_nat
@ ^ [X: nat] :
~ ( P @ X ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_9_if__image__distrib,axiom,
! [P: sum_sum_a_nat > $o,F: sum_sum_a_nat > sum_sum_a_nat,G: sum_sum_a_nat > sum_sum_a_nat,S: set_Sum_sum_a_nat] :
( ( image_7142520692256960453_a_nat
@ ^ [X: sum_sum_a_nat] : ( if_Sum_sum_a_nat @ ( P @ X ) @ ( F @ X ) @ ( G @ X ) )
@ S )
= ( sup_su6804446743777130803_a_nat @ ( image_7142520692256960453_a_nat @ F @ ( inf_in7084830621192376909_a_nat @ S @ ( collec7073057861543223018_a_nat @ P ) ) )
@ ( image_7142520692256960453_a_nat @ G
@ ( inf_in7084830621192376909_a_nat @ S
@ ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
~ ( P @ X ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_10__C2_OIH_C_I3_J,axiom,
! [X2: nat] :
( ~ ( member_a @ x @ aDa )
=> ( ( ( ma @ ( sum_Inl_a_nat @ x ) )
= ( some_nat @ X2 ) )
=> ( ( ord_less_eq_set_nat @ ( ran_Su5179294584019872288at_nat @ ma ) @ ( set_ord_lessThan_nat @ ia ) )
=> ( ( sup_su6804446743777130803_a_nat @ ( set_Sum_sum_a_nat2 @ ( fo_nmlz_rec_a @ ia @ ma @ aDa @ xsa ) ) @ ( image_7293268710728258664_a_nat @ sum_Inr_nat_a @ ( set_ord_lessThan_nat @ ia ) ) )
= ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ ( set_Sum_sum_a_nat2 @ xsa ) @ ( image_7873763678140191238_a_nat @ sum_Inl_a_nat @ aDa ) ) @ ( image_7293268710728258664_a_nat @ sum_Inr_nat_a @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ia @ ( finite6080979521523705895_a_nat @ ( minus_1134630996077396038_a_nat @ ( minus_1134630996077396038_a_nat @ ( set_Sum_sum_a_nat2 @ xsa ) @ ( image_7873763678140191238_a_nat @ sum_Inl_a_nat @ aDa ) ) @ ( dom_Su2255998037560862461at_nat @ ma ) ) ) ) ) ) ) ) ) ) ) ).
% "2.IH"(3)
thf(fact_11_Inl__Inr__False,axiom,
! [X3: a,Y: nat] :
( ( sum_Inl_a_nat @ X3 )
!= ( sum_Inr_nat_a @ Y ) ) ).
% Inl_Inr_False
thf(fact_12_Inr__Inl__False,axiom,
! [X3: nat,Y: a] :
( ( sum_Inr_nat_a @ X3 )
!= ( sum_Inl_a_nat @ Y ) ) ).
% Inr_Inl_False
thf(fact_13_Un__Diff__cancel,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A @ ( minus_1134630996077396038_a_nat @ B @ A ) )
= ( sup_su6804446743777130803_a_nat @ A @ B ) ) ).
% Un_Diff_cancel
thf(fact_14_Un__Diff__cancel2,axiom,
! [B: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ ( minus_1134630996077396038_a_nat @ B @ A ) @ A )
= ( sup_su6804446743777130803_a_nat @ B @ A ) ) ).
% Un_Diff_cancel2
thf(fact_15_Int__Un__eq_I4_J,axiom,
! [T: set_nat,S: set_nat] :
( ( sup_sup_set_nat @ T @ ( inf_inf_set_nat @ S @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_16_Int__Un__eq_I4_J,axiom,
! [T: set_a,S: set_a] :
( ( sup_sup_set_a @ T @ ( inf_inf_set_a @ S @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_17_Int__Un__eq_I4_J,axiom,
! [T: set_Sum_sum_a_nat,S: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ T @ ( inf_in7084830621192376909_a_nat @ S @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_18_Int__Un__eq_I3_J,axiom,
! [S: set_nat,T: set_nat] :
( ( sup_sup_set_nat @ S @ ( inf_inf_set_nat @ S @ T ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_19_Int__Un__eq_I3_J,axiom,
! [S: set_a,T: set_a] :
( ( sup_sup_set_a @ S @ ( inf_inf_set_a @ S @ T ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_20_Int__Un__eq_I3_J,axiom,
! [S: set_Sum_sum_a_nat,T: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ S @ ( inf_in7084830621192376909_a_nat @ S @ T ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_21_Int__Un__eq_I2_J,axiom,
! [S: set_nat,T: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_22_Int__Un__eq_I2_J,axiom,
! [S: set_a,T: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_23_Int__Un__eq_I2_J,axiom,
! [S: set_Sum_sum_a_nat,T: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ S @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_24__C2_Oprems_C,axiom,
ord_less_eq_set_nat @ ( ran_Su5179294584019872288at_nat @ ma ) @ ( set_ord_lessThan_nat @ ia ) ).
% "2.prems"
thf(fact_25_image__eqI,axiom,
! [B2: sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat,X3: sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_Sum_sum_a_nat @ X3 @ A )
=> ( member_Sum_sum_a_nat @ B2 @ ( image_7142520692256960453_a_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_26_image__eqI,axiom,
! [B2: sum_sum_a_nat,F: a > sum_sum_a_nat,X3: a,A: set_a] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_a @ X3 @ A )
=> ( member_Sum_sum_a_nat @ B2 @ ( image_7873763678140191238_a_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_27_image__eqI,axiom,
! [B2: a,F: a > a,X3: a,A: set_a] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_a @ X3 @ A )
=> ( member_a @ B2 @ ( image_a_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_28_image__eqI,axiom,
! [B2: nat,F: a > nat,X3: a,A: set_a] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_a @ X3 @ A )
=> ( member_nat @ B2 @ ( image_a_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_29_image__eqI,axiom,
! [B2: sum_sum_a_nat,F: nat > sum_sum_a_nat,X3: nat,A: set_nat] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_nat @ X3 @ A )
=> ( member_Sum_sum_a_nat @ B2 @ ( image_7293268710728258664_a_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_30_image__eqI,axiom,
! [B2: a,F: nat > a,X3: nat,A: set_nat] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_nat @ X3 @ A )
=> ( member_a @ B2 @ ( image_nat_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_31_image__eqI,axiom,
! [B2: nat,F: nat > nat,X3: nat,A: set_nat] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_nat @ X3 @ A )
=> ( member_nat @ B2 @ ( image_nat_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_32_subsetI,axiom,
! [A: set_a,B: set_a] :
( ! [X4: a] :
( ( member_a @ X4 @ A )
=> ( member_a @ X4 @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% subsetI
thf(fact_33_subsetI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_nat @ X4 @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% subsetI
thf(fact_34_subset__antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_35_Int__iff,axiom,
! [C: sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C @ ( inf_in7084830621192376909_a_nat @ A @ B ) )
= ( ( member_Sum_sum_a_nat @ C @ A )
& ( member_Sum_sum_a_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_36_Int__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
= ( ( member_nat @ C @ A )
& ( member_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_37_Int__iff,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
= ( ( member_a @ C @ A )
& ( member_a @ C @ B ) ) ) ).
% Int_iff
thf(fact_38_IntI,axiom,
! [C: sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C @ A )
=> ( ( member_Sum_sum_a_nat @ C @ B )
=> ( member_Sum_sum_a_nat @ C @ ( inf_in7084830621192376909_a_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_39_IntI,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ A )
=> ( ( member_nat @ C @ B )
=> ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_40_IntI,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ A )
=> ( ( member_a @ C @ B )
=> ( member_a @ C @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% IntI
thf(fact_41_Un__iff,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A @ B ) )
= ( ( member_a @ C @ A )
| ( member_a @ C @ B ) ) ) ).
% Un_iff
thf(fact_42_Un__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) )
= ( ( member_nat @ C @ A )
| ( member_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_43_Un__iff,axiom,
! [C: sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C @ ( sup_su6804446743777130803_a_nat @ A @ B ) )
= ( ( member_Sum_sum_a_nat @ C @ A )
| ( member_Sum_sum_a_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_44_UnCI,axiom,
! [C: a,B: set_a,A: set_a] :
( ( ~ ( member_a @ C @ B )
=> ( member_a @ C @ A ) )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnCI
thf(fact_45_UnCI,axiom,
! [C: nat,B: set_nat,A: set_nat] :
( ( ~ ( member_nat @ C @ B )
=> ( member_nat @ C @ A ) )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_46_UnCI,axiom,
! [C: sum_sum_a_nat,B: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( ~ ( member_Sum_sum_a_nat @ C @ B )
=> ( member_Sum_sum_a_nat @ C @ A ) )
=> ( member_Sum_sum_a_nat @ C @ ( sup_su6804446743777130803_a_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_47_Diff__idemp,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( minus_1134630996077396038_a_nat @ ( minus_1134630996077396038_a_nat @ A @ B ) @ B )
= ( minus_1134630996077396038_a_nat @ A @ B ) ) ).
% Diff_idemp
thf(fact_48_Diff__iff,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
= ( ( member_a @ C @ A )
& ~ ( member_a @ C @ B ) ) ) ).
% Diff_iff
thf(fact_49_Diff__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
= ( ( member_nat @ C @ A )
& ~ ( member_nat @ C @ B ) ) ) ).
% Diff_iff
thf(fact_50_Diff__iff,axiom,
! [C: sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C @ ( minus_1134630996077396038_a_nat @ A @ B ) )
= ( ( member_Sum_sum_a_nat @ C @ A )
& ~ ( member_Sum_sum_a_nat @ C @ B ) ) ) ).
% Diff_iff
thf(fact_51_DiffI,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ A )
=> ( ~ ( member_a @ C @ B )
=> ( member_a @ C @ ( minus_minus_set_a @ A @ B ) ) ) ) ).
% DiffI
thf(fact_52_DiffI,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ A )
=> ( ~ ( member_nat @ C @ B )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) ) ) ) ).
% DiffI
thf(fact_53_DiffI,axiom,
! [C: sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C @ A )
=> ( ~ ( member_Sum_sum_a_nat @ C @ B )
=> ( member_Sum_sum_a_nat @ C @ ( minus_1134630996077396038_a_nat @ A @ B ) ) ) ) ).
% DiffI
thf(fact_54_image__ident,axiom,
! [Y2: set_Sum_sum_a_nat] :
( ( image_7142520692256960453_a_nat
@ ^ [X: sum_sum_a_nat] : X
@ Y2 )
= Y2 ) ).
% image_ident
thf(fact_55_Int__subset__iff,axiom,
! [C2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ C2 @ ( inf_in7084830621192376909_a_nat @ A @ B ) )
= ( ( ord_le1325389633284124927_a_nat @ C2 @ A )
& ( ord_le1325389633284124927_a_nat @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_56_Int__subset__iff,axiom,
! [C2: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A @ B ) )
= ( ( ord_less_eq_set_a @ C2 @ A )
& ( ord_less_eq_set_a @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_57_Int__subset__iff,axiom,
! [C2: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ ( inf_inf_set_nat @ A @ B ) )
= ( ( ord_less_eq_set_nat @ C2 @ A )
& ( ord_less_eq_set_nat @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_58_Un__subset__iff,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ A @ B ) @ C2 )
= ( ( ord_le1325389633284124927_a_nat @ A @ C2 )
& ( ord_le1325389633284124927_a_nat @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_59_Un__subset__iff,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C2 )
= ( ( ord_less_eq_set_nat @ A @ C2 )
& ( ord_less_eq_set_nat @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_60_Un__Int__eq_I1_J,axiom,
! [S: set_nat,T: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S @ T ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_61_Un__Int__eq_I1_J,axiom,
! [S: set_a,T: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_62_Un__Int__eq_I1_J,axiom,
! [S: set_Sum_sum_a_nat,T: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ ( sup_su6804446743777130803_a_nat @ S @ T ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_63_Un__Int__eq_I2_J,axiom,
! [S: set_nat,T: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_64_Un__Int__eq_I2_J,axiom,
! [S: set_a,T: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_65_Un__Int__eq_I2_J,axiom,
! [S: set_Sum_sum_a_nat,T: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ ( sup_su6804446743777130803_a_nat @ S @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_66_Un__Int__eq_I3_J,axiom,
! [S: set_nat,T: set_nat] :
( ( inf_inf_set_nat @ S @ ( sup_sup_set_nat @ S @ T ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_67_Un__Int__eq_I3_J,axiom,
! [S: set_a,T: set_a] :
( ( inf_inf_set_a @ S @ ( sup_sup_set_a @ S @ T ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_68_Un__Int__eq_I3_J,axiom,
! [S: set_Sum_sum_a_nat,T: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ S @ ( sup_su6804446743777130803_a_nat @ S @ T ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_69_Un__Int__eq_I4_J,axiom,
! [T: set_nat,S: set_nat] :
( ( inf_inf_set_nat @ T @ ( sup_sup_set_nat @ S @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_70_Un__Int__eq_I4_J,axiom,
! [T: set_a,S: set_a] :
( ( inf_inf_set_a @ T @ ( sup_sup_set_a @ S @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_71_Un__Int__eq_I4_J,axiom,
! [T: set_Sum_sum_a_nat,S: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ T @ ( sup_su6804446743777130803_a_nat @ S @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_72_Int__Un__eq_I1_J,axiom,
! [S: set_nat,T: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S @ T ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_73_Int__Un__eq_I1_J,axiom,
! [S: set_a,T: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_74_Int__Un__eq_I1_J,axiom,
! [S: set_Sum_sum_a_nat,T: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ S @ T ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_75_pred,axiom,
ord_less_eq_set_nat @ ( ran_Su5179294584019872288at_nat @ ( fun_up4750343006489851234on_nat @ ma @ ( sum_Inl_a_nat @ x ) @ ( some_nat @ ia ) ) ) @ ( set_ord_lessThan_nat @ ( suc @ ia ) ) ).
% pred
thf(fact_76__C2_OIH_C_I2_J,axiom,
( ~ ( member_a @ x @ aDa )
=> ( ( ( ma @ ( sum_Inl_a_nat @ x ) )
= none_nat )
=> ( ( ord_less_eq_set_nat @ ( ran_Su5179294584019872288at_nat @ ( fun_up4750343006489851234on_nat @ ma @ ( sum_Inl_a_nat @ x ) @ ( some_nat @ ia ) ) ) @ ( set_ord_lessThan_nat @ ( suc @ ia ) ) )
=> ( ( sup_su6804446743777130803_a_nat @ ( set_Sum_sum_a_nat2 @ ( fo_nmlz_rec_a @ ( suc @ ia ) @ ( fun_up4750343006489851234on_nat @ ma @ ( sum_Inl_a_nat @ x ) @ ( some_nat @ ia ) ) @ aDa @ xsa ) ) @ ( image_7293268710728258664_a_nat @ sum_Inr_nat_a @ ( set_ord_lessThan_nat @ ( suc @ ia ) ) ) )
= ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ ( set_Sum_sum_a_nat2 @ xsa ) @ ( image_7873763678140191238_a_nat @ sum_Inl_a_nat @ aDa ) ) @ ( image_7293268710728258664_a_nat @ sum_Inr_nat_a @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( suc @ ia ) @ ( finite6080979521523705895_a_nat @ ( minus_1134630996077396038_a_nat @ ( minus_1134630996077396038_a_nat @ ( set_Sum_sum_a_nat2 @ xsa ) @ ( image_7873763678140191238_a_nat @ sum_Inl_a_nat @ aDa ) ) @ ( dom_Su2255998037560862461at_nat @ ( fun_up4750343006489851234on_nat @ ma @ ( sum_Inl_a_nat @ x ) @ ( some_nat @ ia ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% "2.IH"(2)
thf(fact_77_in__mono,axiom,
! [A: set_a,B: set_a,X3: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ X3 @ A )
=> ( member_a @ X3 @ B ) ) ) ).
% in_mono
thf(fact_78_in__mono,axiom,
! [A: set_nat,B: set_nat,X3: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ X3 @ A )
=> ( member_nat @ X3 @ B ) ) ) ).
% in_mono
thf(fact_79_subsetD,axiom,
! [A: set_a,B: set_a,C: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% subsetD
thf(fact_80_subsetD,axiom,
! [A: set_nat,B: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_81_equalityE,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_nat @ A @ B )
=> ~ ( ord_less_eq_set_nat @ B @ A ) ) ) ).
% equalityE
thf(fact_82_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A2: set_a,B3: set_a] :
! [X: a] :
( ( member_a @ X @ A2 )
=> ( member_a @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_83_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A2: set_nat,B3: set_nat] :
! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_84_equalityD1,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% equalityD1
thf(fact_85_Set_OequalityD2,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% Set.equalityD2
thf(fact_86_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A2: set_a,B3: set_a] :
! [T2: a] :
( ( member_a @ T2 @ A2 )
=> ( member_a @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_87_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A2: set_nat,B3: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A2 )
=> ( member_nat @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_88_subset__refl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% subset_refl
thf(fact_89_mem__Collect__eq,axiom,
! [A3: nat,P: nat > $o] :
( ( member_nat @ A3 @ ( collect_nat @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_90_mem__Collect__eq,axiom,
! [A3: sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ( member_Sum_sum_a_nat @ A3 @ ( collec7073057861543223018_a_nat @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_91_mem__Collect__eq,axiom,
! [A3: a,P: a > $o] :
( ( member_a @ A3 @ ( collect_a @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_92_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_93_Collect__mem__eq,axiom,
! [A: set_Sum_sum_a_nat] :
( ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_94_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X: a] : ( member_a @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_95_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X4: nat] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_96_Collect__cong,axiom,
! [P: sum_sum_a_nat > $o,Q: sum_sum_a_nat > $o] :
( ! [X4: sum_sum_a_nat] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collec7073057861543223018_a_nat @ P )
= ( collec7073057861543223018_a_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_97_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X4: a] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_98_Collect__mono,axiom,
! [P: sum_sum_a_nat > $o,Q: sum_sum_a_nat > $o] :
( ! [X4: sum_sum_a_nat] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le1325389633284124927_a_nat @ ( collec7073057861543223018_a_nat @ P ) @ ( collec7073057861543223018_a_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_99_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X4: a] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_100_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X4: nat] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_101_subset__trans,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_102_set__eq__subset,axiom,
( ( ^ [Y3: set_nat,Z: set_nat] : ( Y3 = Z ) )
= ( ^ [A2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A2 ) ) ) ) ).
% set_eq_subset
thf(fact_103_Collect__subset,axiom,
! [A: set_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ord_le1325389633284124927_a_nat
@ ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_104_Collect__subset,axiom,
! [A: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_105_Collect__subset,axiom,
! [A: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_106_Collect__mono__iff,axiom,
! [P: sum_sum_a_nat > $o,Q: sum_sum_a_nat > $o] :
( ( ord_le1325389633284124927_a_nat @ ( collec7073057861543223018_a_nat @ P ) @ ( collec7073057861543223018_a_nat @ Q ) )
= ( ! [X: sum_sum_a_nat] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_107_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X: a] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_108_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X: nat] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_109_subset__image__iff,axiom,
! [B: set_Sum_sum_a_nat,F: a > sum_sum_a_nat,A: set_a] :
( ( ord_le1325389633284124927_a_nat @ B @ ( image_7873763678140191238_a_nat @ F @ A ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A )
& ( B
= ( image_7873763678140191238_a_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_110_subset__image__iff,axiom,
! [B: set_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B @ ( image_7142520692256960453_a_nat @ F @ A ) )
= ( ? [AA: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ AA @ A )
& ( B
= ( image_7142520692256960453_a_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_111_subset__image__iff,axiom,
! [B: set_Sum_sum_a_nat,F: nat > sum_sum_a_nat,A: set_nat] :
( ( ord_le1325389633284124927_a_nat @ B @ ( image_7293268710728258664_a_nat @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B
= ( image_7293268710728258664_a_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_112_subset__image__iff,axiom,
! [B: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_113_image__subset__iff,axiom,
! [F: nat > sum_sum_a_nat,A: set_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( image_7293268710728258664_a_nat @ F @ A ) @ B )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_Sum_sum_a_nat @ ( F @ X ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_114_image__subset__iff,axiom,
! [F: a > sum_sum_a_nat,A: set_a,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( image_7873763678140191238_a_nat @ F @ A ) @ B )
= ( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_Sum_sum_a_nat @ ( F @ X ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_115_image__subset__iff,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( image_7142520692256960453_a_nat @ F @ A ) @ B )
= ( ! [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ A )
=> ( member_Sum_sum_a_nat @ ( F @ X ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_116_subset__imageE,axiom,
! [B: set_Sum_sum_a_nat,F: a > sum_sum_a_nat,A: set_a] :
( ( ord_le1325389633284124927_a_nat @ B @ ( image_7873763678140191238_a_nat @ F @ A ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A )
=> ( B
!= ( image_7873763678140191238_a_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_117_subset__imageE,axiom,
! [B: set_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B @ ( image_7142520692256960453_a_nat @ F @ A ) )
=> ~ ! [C3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ C3 @ A )
=> ( B
!= ( image_7142520692256960453_a_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_118_subset__imageE,axiom,
! [B: set_Sum_sum_a_nat,F: nat > sum_sum_a_nat,A: set_nat] :
( ( ord_le1325389633284124927_a_nat @ B @ ( image_7293268710728258664_a_nat @ F @ A ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
=> ( B
!= ( image_7293268710728258664_a_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_119_subset__imageE,axiom,
! [B: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
=> ( B
!= ( image_nat_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_120_image__subsetI,axiom,
! [A: set_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ! [X4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X4 @ A )
=> ( member_Sum_sum_a_nat @ ( F @ X4 ) @ B ) )
=> ( ord_le1325389633284124927_a_nat @ ( image_7142520692256960453_a_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_121_image__subsetI,axiom,
! [A: set_a,F: a > sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ! [X4: a] :
( ( member_a @ X4 @ A )
=> ( member_Sum_sum_a_nat @ ( F @ X4 ) @ B ) )
=> ( ord_le1325389633284124927_a_nat @ ( image_7873763678140191238_a_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_122_image__subsetI,axiom,
! [A: set_a,F: a > a,B: set_a] :
( ! [X4: a] :
( ( member_a @ X4 @ A )
=> ( member_a @ ( F @ X4 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_123_image__subsetI,axiom,
! [A: set_nat,F: nat > sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_Sum_sum_a_nat @ ( F @ X4 ) @ B ) )
=> ( ord_le1325389633284124927_a_nat @ ( image_7293268710728258664_a_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_124_image__subsetI,axiom,
! [A: set_nat,F: nat > a,B: set_a] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_a @ ( F @ X4 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_nat_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_125_image__subsetI,axiom,
! [A: set_a,F: a > nat,B: set_nat] :
( ! [X4: a] :
( ( member_a @ X4 @ A )
=> ( member_nat @ ( F @ X4 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_a_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_126_image__subsetI,axiom,
! [A: set_nat,F: nat > nat,B: set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_nat @ ( F @ X4 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_127_image__mono,axiom,
! [A: set_a,B: set_a,F: a > sum_sum_a_nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_le1325389633284124927_a_nat @ ( image_7873763678140191238_a_nat @ F @ A ) @ ( image_7873763678140191238_a_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_128_image__mono,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ B )
=> ( ord_le1325389633284124927_a_nat @ ( image_7142520692256960453_a_nat @ F @ A ) @ ( image_7142520692256960453_a_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_129_image__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > sum_sum_a_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_le1325389633284124927_a_nat @ ( image_7293268710728258664_a_nat @ F @ A ) @ ( image_7293268710728258664_a_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_130_image__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_131_Int__Collect__mono,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,P: sum_sum_a_nat > $o,Q: sum_sum_a_nat > $o] :
( ( ord_le1325389633284124927_a_nat @ A @ B )
=> ( ! [X4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X4 @ A )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ A @ ( collec7073057861543223018_a_nat @ P ) ) @ ( inf_in7084830621192376909_a_nat @ B @ ( collec7073057861543223018_a_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_132_Int__Collect__mono,axiom,
! [A: set_a,B: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ! [X4: a] :
( ( member_a @ X4 @ A )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_133_Int__Collect__mono,axiom,
! [A: set_nat,B: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_134_Int__greatest,axiom,
! [C2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ C2 @ A )
=> ( ( ord_le1325389633284124927_a_nat @ C2 @ B )
=> ( ord_le1325389633284124927_a_nat @ C2 @ ( inf_in7084830621192376909_a_nat @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_135_Int__greatest,axiom,
! [C2: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C2 @ A )
=> ( ( ord_less_eq_set_a @ C2 @ B )
=> ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_136_Int__greatest,axiom,
! [C2: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A )
=> ( ( ord_less_eq_set_nat @ C2 @ B )
=> ( ord_less_eq_set_nat @ C2 @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_137_Int__absorb2,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ B )
=> ( ( inf_in7084830621192376909_a_nat @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_138_Int__absorb2,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( inf_inf_set_a @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_139_Int__absorb2,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( inf_inf_set_nat @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_140_Int__absorb1,axiom,
! [B: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B @ A )
=> ( ( inf_in7084830621192376909_a_nat @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_141_Int__absorb1,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( inf_inf_set_a @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_142_Int__absorb1,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( inf_inf_set_nat @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_143_Int__lower2,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_144_Int__lower2,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_145_Int__lower2,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_146_Int__lower1,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_147_Int__lower1,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_148_Int__lower1,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_149_Int__mono,axiom,
! [A: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,D: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ C2 )
=> ( ( ord_le1325389633284124927_a_nat @ B @ D )
=> ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ A @ B ) @ ( inf_in7084830621192376909_a_nat @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_150_Int__mono,axiom,
! [A: set_a,C2: set_a,B: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_inf_set_a @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_151_Int__mono,axiom,
! [A: set_nat,C2: set_nat,B: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B @ D )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( inf_inf_set_nat @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_152_subset__Un__eq,axiom,
( ord_le1325389633284124927_a_nat
= ( ^ [A2: set_Sum_sum_a_nat,B3: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A2 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_153_subset__Un__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A2: set_nat,B3: set_nat] :
( ( sup_sup_set_nat @ A2 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_154_subset__UnE,axiom,
! [C2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ C2 @ ( sup_su6804446743777130803_a_nat @ A @ B ) )
=> ~ ! [A4: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A4 @ A )
=> ! [B4: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B4 @ B )
=> ( C2
!= ( sup_su6804446743777130803_a_nat @ A4 @ B4 ) ) ) ) ) ).
% subset_UnE
thf(fact_155_subset__UnE,axiom,
! [C2: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) )
=> ~ ! [A4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ A )
=> ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ B )
=> ( C2
!= ( sup_sup_set_nat @ A4 @ B4 ) ) ) ) ) ).
% subset_UnE
thf(fact_156_Un__absorb2,axiom,
! [B: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B @ A )
=> ( ( sup_su6804446743777130803_a_nat @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_157_Un__absorb2,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( sup_sup_set_nat @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_158_Un__absorb1,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ B )
=> ( ( sup_su6804446743777130803_a_nat @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_159_Un__absorb1,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( sup_sup_set_nat @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_160_Un__upper2,axiom,
! [B: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ B @ ( sup_su6804446743777130803_a_nat @ A @ B ) ) ).
% Un_upper2
thf(fact_161_Un__upper2,axiom,
! [B: set_nat,A: set_nat] : ( ord_less_eq_set_nat @ B @ ( sup_sup_set_nat @ A @ B ) ) ).
% Un_upper2
thf(fact_162_Un__upper1,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ A @ ( sup_su6804446743777130803_a_nat @ A @ B ) ) ).
% Un_upper1
thf(fact_163_Un__upper1,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) ) ).
% Un_upper1
thf(fact_164_Un__least,axiom,
! [A: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ C2 )
=> ( ( ord_le1325389633284124927_a_nat @ B @ C2 )
=> ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_165_Un__least,axiom,
! [A: set_nat,C2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_166_Un__mono,axiom,
! [A: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,D: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ C2 )
=> ( ( ord_le1325389633284124927_a_nat @ B @ D )
=> ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ A @ B ) @ ( sup_su6804446743777130803_a_nat @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_167_Un__mono,axiom,
! [A: set_nat,C2: set_nat,B: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B @ D )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sup_sup_set_nat @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_168_double__diff,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ B )
=> ( ( ord_le1325389633284124927_a_nat @ B @ C2 )
=> ( ( minus_1134630996077396038_a_nat @ B @ ( minus_1134630996077396038_a_nat @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_169_double__diff,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ( minus_minus_set_nat @ B @ ( minus_minus_set_nat @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_170_Diff__subset,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( minus_1134630996077396038_a_nat @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_171_Diff__subset,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_172_Diff__mono,axiom,
! [A: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat,D: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ C2 )
=> ( ( ord_le1325389633284124927_a_nat @ D @ B )
=> ( ord_le1325389633284124927_a_nat @ ( minus_1134630996077396038_a_nat @ A @ B ) @ ( minus_1134630996077396038_a_nat @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_173_Diff__mono,axiom,
! [A: set_nat,C2: set_nat,D: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ D @ B )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B ) @ ( minus_minus_set_nat @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_174_image__Int__subset,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( image_7142520692256960453_a_nat @ F @ ( inf_in7084830621192376909_a_nat @ A @ B ) ) @ ( inf_in7084830621192376909_a_nat @ ( image_7142520692256960453_a_nat @ F @ A ) @ ( image_7142520692256960453_a_nat @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_175_image__Int__subset,axiom,
! [F: sum_sum_a_nat > a,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] : ( ord_less_eq_set_a @ ( image_6322530041254294468_nat_a @ F @ ( inf_in7084830621192376909_a_nat @ A @ B ) ) @ ( inf_inf_set_a @ ( image_6322530041254294468_nat_a @ F @ A ) @ ( image_6322530041254294468_nat_a @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_176_image__Int__subset,axiom,
! [F: nat > sum_sum_a_nat,A: set_nat,B: set_nat] : ( ord_le1325389633284124927_a_nat @ ( image_7293268710728258664_a_nat @ F @ ( inf_inf_set_nat @ A @ B ) ) @ ( inf_in7084830621192376909_a_nat @ ( image_7293268710728258664_a_nat @ F @ A ) @ ( image_7293268710728258664_a_nat @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_177_image__Int__subset,axiom,
! [F: nat > a,A: set_nat,B: set_nat] : ( ord_less_eq_set_a @ ( image_nat_a @ F @ ( inf_inf_set_nat @ A @ B ) ) @ ( inf_inf_set_a @ ( image_nat_a @ F @ A ) @ ( image_nat_a @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_178_image__Int__subset,axiom,
! [F: a > sum_sum_a_nat,A: set_a,B: set_a] : ( ord_le1325389633284124927_a_nat @ ( image_7873763678140191238_a_nat @ F @ ( inf_inf_set_a @ A @ B ) ) @ ( inf_in7084830621192376909_a_nat @ ( image_7873763678140191238_a_nat @ F @ A ) @ ( image_7873763678140191238_a_nat @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_179_image__Int__subset,axiom,
! [F: a > a,A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( image_a_a @ F @ ( inf_inf_set_a @ A @ B ) ) @ ( inf_inf_set_a @ ( image_a_a @ F @ A ) @ ( image_a_a @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_180_image__Int__subset,axiom,
! [F: sum_sum_a_nat > nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] : ( ord_less_eq_set_nat @ ( image_2473878607534554506at_nat @ F @ ( inf_in7084830621192376909_a_nat @ A @ B ) ) @ ( inf_inf_set_nat @ ( image_2473878607534554506at_nat @ F @ A ) @ ( image_2473878607534554506at_nat @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_181_image__Int__subset,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ ( inf_inf_set_nat @ A @ B ) ) @ ( inf_inf_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_182_image__Int__subset,axiom,
! [F: a > nat,A: set_a,B: set_a] : ( ord_less_eq_set_nat @ ( image_a_nat @ F @ ( inf_inf_set_a @ A @ B ) ) @ ( inf_inf_set_nat @ ( image_a_nat @ F @ A ) @ ( image_a_nat @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_183_image__diff__subset,axiom,
! [F: nat > sum_sum_a_nat,A: set_nat,B: set_nat] : ( ord_le1325389633284124927_a_nat @ ( minus_1134630996077396038_a_nat @ ( image_7293268710728258664_a_nat @ F @ A ) @ ( image_7293268710728258664_a_nat @ F @ B ) ) @ ( image_7293268710728258664_a_nat @ F @ ( minus_minus_set_nat @ A @ B ) ) ) ).
% image_diff_subset
thf(fact_184_image__diff__subset,axiom,
! [F: a > sum_sum_a_nat,A: set_a,B: set_a] : ( ord_le1325389633284124927_a_nat @ ( minus_1134630996077396038_a_nat @ ( image_7873763678140191238_a_nat @ F @ A ) @ ( image_7873763678140191238_a_nat @ F @ B ) ) @ ( image_7873763678140191238_a_nat @ F @ ( minus_minus_set_a @ A @ B ) ) ) ).
% image_diff_subset
thf(fact_185_image__diff__subset,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( minus_1134630996077396038_a_nat @ ( image_7142520692256960453_a_nat @ F @ A ) @ ( image_7142520692256960453_a_nat @ F @ B ) ) @ ( image_7142520692256960453_a_nat @ F @ ( minus_1134630996077396038_a_nat @ A @ B ) ) ) ).
% image_diff_subset
thf(fact_186_image__diff__subset,axiom,
! [F: sum_sum_a_nat > nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_2473878607534554506at_nat @ F @ A ) @ ( image_2473878607534554506at_nat @ F @ B ) ) @ ( image_2473878607534554506at_nat @ F @ ( minus_1134630996077396038_a_nat @ A @ B ) ) ) ).
% image_diff_subset
thf(fact_187_Un__Int__assoc__eq,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ( sup_sup_set_a @ ( inf_inf_set_a @ A @ B ) @ C2 )
= ( inf_inf_set_a @ A @ ( sup_sup_set_a @ B @ C2 ) ) )
= ( ord_less_eq_set_a @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_188_Un__Int__assoc__eq,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
( ( ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ A @ B ) @ C2 )
= ( inf_in7084830621192376909_a_nat @ A @ ( sup_su6804446743777130803_a_nat @ B @ C2 ) ) )
= ( ord_le1325389633284124927_a_nat @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_189_Un__Int__assoc__eq,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C2 )
= ( inf_inf_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) ) )
= ( ord_less_eq_set_nat @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_190_Diff__subset__conv,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( minus_1134630996077396038_a_nat @ A @ B ) @ C2 )
= ( ord_le1325389633284124927_a_nat @ A @ ( sup_su6804446743777130803_a_nat @ B @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_191_Diff__subset__conv,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B ) @ C2 )
= ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_192_Diff__partition,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ B )
=> ( ( sup_su6804446743777130803_a_nat @ A @ ( minus_1134630996077396038_a_nat @ B @ A ) )
= B ) ) ).
% Diff_partition
thf(fact_193_Diff__partition,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( sup_sup_set_nat @ A @ ( minus_minus_set_nat @ B @ A ) )
= B ) ) ).
% Diff_partition
thf(fact_194_rev__image__eqI,axiom,
! [X3: sum_sum_a_nat,A: set_Sum_sum_a_nat,B2: sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_Sum_sum_a_nat @ B2 @ ( image_7142520692256960453_a_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_195_rev__image__eqI,axiom,
! [X3: a,A: set_a,B2: sum_sum_a_nat,F: a > sum_sum_a_nat] :
( ( member_a @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_Sum_sum_a_nat @ B2 @ ( image_7873763678140191238_a_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_196_rev__image__eqI,axiom,
! [X3: a,A: set_a,B2: a,F: a > a] :
( ( member_a @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_a @ B2 @ ( image_a_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_197_rev__image__eqI,axiom,
! [X3: a,A: set_a,B2: nat,F: a > nat] :
( ( member_a @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_nat @ B2 @ ( image_a_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_198_rev__image__eqI,axiom,
! [X3: nat,A: set_nat,B2: sum_sum_a_nat,F: nat > sum_sum_a_nat] :
( ( member_nat @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_Sum_sum_a_nat @ B2 @ ( image_7293268710728258664_a_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_199_rev__image__eqI,axiom,
! [X3: nat,A: set_nat,B2: a,F: nat > a] :
( ( member_nat @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_a @ B2 @ ( image_nat_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_200_rev__image__eqI,axiom,
! [X3: nat,A: set_nat,B2: nat,F: nat > nat] :
( ( member_nat @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_nat @ B2 @ ( image_nat_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_201_ball__imageD,axiom,
! [F: nat > sum_sum_a_nat,A: set_nat,P: sum_sum_a_nat > $o] :
( ! [X4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X4 @ ( image_7293268710728258664_a_nat @ F @ A ) )
=> ( P @ X4 ) )
=> ! [X5: nat] :
( ( member_nat @ X5 @ A )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_202_ball__imageD,axiom,
! [F: a > sum_sum_a_nat,A: set_a,P: sum_sum_a_nat > $o] :
( ! [X4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X4 @ ( image_7873763678140191238_a_nat @ F @ A ) )
=> ( P @ X4 ) )
=> ! [X5: a] :
( ( member_a @ X5 @ A )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_203_ball__imageD,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,A: set_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ! [X4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X4 @ ( image_7142520692256960453_a_nat @ F @ A ) )
=> ( P @ X4 ) )
=> ! [X5: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X5 @ A )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_204_image__cong,axiom,
! [M: set_Sum_sum_a_nat,N: set_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat,G: sum_sum_a_nat > sum_sum_a_nat] :
( ( M = N )
=> ( ! [X4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X4 @ N )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image_7142520692256960453_a_nat @ F @ M )
= ( image_7142520692256960453_a_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_205_image__cong,axiom,
! [M: set_a,N: set_a,F: a > sum_sum_a_nat,G: a > sum_sum_a_nat] :
( ( M = N )
=> ( ! [X4: a] :
( ( member_a @ X4 @ N )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image_7873763678140191238_a_nat @ F @ M )
= ( image_7873763678140191238_a_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_206_image__cong,axiom,
! [M: set_nat,N: set_nat,F: nat > sum_sum_a_nat,G: nat > sum_sum_a_nat] :
( ( M = N )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ N )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image_7293268710728258664_a_nat @ F @ M )
= ( image_7293268710728258664_a_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_207_bex__imageD,axiom,
! [F: nat > sum_sum_a_nat,A: set_nat,P: sum_sum_a_nat > $o] :
( ? [X5: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X5 @ ( image_7293268710728258664_a_nat @ F @ A ) )
& ( P @ X5 ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_208_bex__imageD,axiom,
! [F: a > sum_sum_a_nat,A: set_a,P: sum_sum_a_nat > $o] :
( ? [X5: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X5 @ ( image_7873763678140191238_a_nat @ F @ A ) )
& ( P @ X5 ) )
=> ? [X4: a] :
( ( member_a @ X4 @ A )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_209_bex__imageD,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,A: set_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ? [X5: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X5 @ ( image_7142520692256960453_a_nat @ F @ A ) )
& ( P @ X5 ) )
=> ? [X4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X4 @ A )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_210_image__iff,axiom,
! [Z2: sum_sum_a_nat,F: nat > sum_sum_a_nat,A: set_nat] :
( ( member_Sum_sum_a_nat @ Z2 @ ( image_7293268710728258664_a_nat @ F @ A ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A )
& ( Z2
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_211_image__iff,axiom,
! [Z2: sum_sum_a_nat,F: a > sum_sum_a_nat,A: set_a] :
( ( member_Sum_sum_a_nat @ Z2 @ ( image_7873763678140191238_a_nat @ F @ A ) )
= ( ? [X: a] :
( ( member_a @ X @ A )
& ( Z2
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_212_image__iff,axiom,
! [Z2: sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ Z2 @ ( image_7142520692256960453_a_nat @ F @ A ) )
= ( ? [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ A )
& ( Z2
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_213_imageI,axiom,
! [X3: sum_sum_a_nat,A: set_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X3 @ A )
=> ( member_Sum_sum_a_nat @ ( F @ X3 ) @ ( image_7142520692256960453_a_nat @ F @ A ) ) ) ).
% imageI
thf(fact_214_imageI,axiom,
! [X3: a,A: set_a,F: a > sum_sum_a_nat] :
( ( member_a @ X3 @ A )
=> ( member_Sum_sum_a_nat @ ( F @ X3 ) @ ( image_7873763678140191238_a_nat @ F @ A ) ) ) ).
% imageI
thf(fact_215_imageI,axiom,
! [X3: a,A: set_a,F: a > a] :
( ( member_a @ X3 @ A )
=> ( member_a @ ( F @ X3 ) @ ( image_a_a @ F @ A ) ) ) ).
% imageI
thf(fact_216_imageI,axiom,
! [X3: a,A: set_a,F: a > nat] :
( ( member_a @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ ( image_a_nat @ F @ A ) ) ) ).
% imageI
thf(fact_217_imageI,axiom,
! [X3: nat,A: set_nat,F: nat > sum_sum_a_nat] :
( ( member_nat @ X3 @ A )
=> ( member_Sum_sum_a_nat @ ( F @ X3 ) @ ( image_7293268710728258664_a_nat @ F @ A ) ) ) ).
% imageI
thf(fact_218_imageI,axiom,
! [X3: nat,A: set_nat,F: nat > a] :
( ( member_nat @ X3 @ A )
=> ( member_a @ ( F @ X3 ) @ ( image_nat_a @ F @ A ) ) ) ).
% imageI
thf(fact_219_imageI,axiom,
! [X3: nat,A: set_nat,F: nat > nat] :
( ( member_nat @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ ( image_nat_nat @ F @ A ) ) ) ).
% imageI
thf(fact_220_Int__left__commute,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ A @ ( inf_in7084830621192376909_a_nat @ B @ C2 ) )
= ( inf_in7084830621192376909_a_nat @ B @ ( inf_in7084830621192376909_a_nat @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_221_Int__left__commute,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ B @ C2 ) )
= ( inf_inf_set_nat @ B @ ( inf_inf_set_nat @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_222_Int__left__commute,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B @ C2 ) )
= ( inf_inf_set_a @ B @ ( inf_inf_set_a @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_223_Int__left__absorb,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ A @ ( inf_in7084830621192376909_a_nat @ A @ B ) )
= ( inf_in7084830621192376909_a_nat @ A @ B ) ) ).
% Int_left_absorb
thf(fact_224_Int__left__absorb,axiom,
! [A: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ A @ B ) )
= ( inf_inf_set_nat @ A @ B ) ) ).
% Int_left_absorb
thf(fact_225_Int__left__absorb,axiom,
! [A: set_a,B: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ A @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ).
% Int_left_absorb
thf(fact_226_Int__commute,axiom,
( inf_in7084830621192376909_a_nat
= ( ^ [A2: set_Sum_sum_a_nat,B3: set_Sum_sum_a_nat] : ( inf_in7084830621192376909_a_nat @ B3 @ A2 ) ) ) ).
% Int_commute
thf(fact_227_Int__commute,axiom,
( inf_inf_set_nat
= ( ^ [A2: set_nat,B3: set_nat] : ( inf_inf_set_nat @ B3 @ A2 ) ) ) ).
% Int_commute
thf(fact_228_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A2: set_a,B3: set_a] : ( inf_inf_set_a @ B3 @ A2 ) ) ) ).
% Int_commute
thf(fact_229_Int__absorb,axiom,
! [A: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ A @ A )
= A ) ).
% Int_absorb
thf(fact_230_Int__absorb,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ A )
= A ) ).
% Int_absorb
thf(fact_231_Int__absorb,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ A )
= A ) ).
% Int_absorb
thf(fact_232_Int__assoc,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ ( inf_in7084830621192376909_a_nat @ A @ B ) @ C2 )
= ( inf_in7084830621192376909_a_nat @ A @ ( inf_in7084830621192376909_a_nat @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_233_Int__assoc,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C2 )
= ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_234_Int__assoc,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B ) @ C2 )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_235_IntD2,axiom,
! [C: sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C @ ( inf_in7084830621192376909_a_nat @ A @ B ) )
=> ( member_Sum_sum_a_nat @ C @ B ) ) ).
% IntD2
thf(fact_236_IntD2,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
=> ( member_nat @ C @ B ) ) ).
% IntD2
thf(fact_237_IntD2,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
=> ( member_a @ C @ B ) ) ).
% IntD2
thf(fact_238_IntD1,axiom,
! [C: sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C @ ( inf_in7084830621192376909_a_nat @ A @ B ) )
=> ( member_Sum_sum_a_nat @ C @ A ) ) ).
% IntD1
thf(fact_239_IntD1,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
=> ( member_nat @ C @ A ) ) ).
% IntD1
thf(fact_240_IntD1,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
=> ( member_a @ C @ A ) ) ).
% IntD1
thf(fact_241_IntE,axiom,
! [C: sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C @ ( inf_in7084830621192376909_a_nat @ A @ B ) )
=> ~ ( ( member_Sum_sum_a_nat @ C @ A )
=> ~ ( member_Sum_sum_a_nat @ C @ B ) ) ) ).
% IntE
thf(fact_242_IntE,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
=> ~ ( ( member_nat @ C @ A )
=> ~ ( member_nat @ C @ B ) ) ) ).
% IntE
thf(fact_243_IntE,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
=> ~ ( ( member_a @ C @ A )
=> ~ ( member_a @ C @ B ) ) ) ).
% IntE
thf(fact_244_Un__left__commute,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A @ ( sup_su6804446743777130803_a_nat @ B @ C2 ) )
= ( sup_su6804446743777130803_a_nat @ B @ ( sup_su6804446743777130803_a_nat @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_245_Un__left__absorb,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A @ ( sup_su6804446743777130803_a_nat @ A @ B ) )
= ( sup_su6804446743777130803_a_nat @ A @ B ) ) ).
% Un_left_absorb
thf(fact_246_Un__commute,axiom,
( sup_su6804446743777130803_a_nat
= ( ^ [A2: set_Sum_sum_a_nat,B3: set_Sum_sum_a_nat] : ( sup_su6804446743777130803_a_nat @ B3 @ A2 ) ) ) ).
% Un_commute
thf(fact_247_Un__absorb,axiom,
! [A: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A @ A )
= A ) ).
% Un_absorb
thf(fact_248_Un__assoc,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ ( sup_su6804446743777130803_a_nat @ A @ B ) @ C2 )
= ( sup_su6804446743777130803_a_nat @ A @ ( sup_su6804446743777130803_a_nat @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_249_ball__Un,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ( ! [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ ( sup_su6804446743777130803_a_nat @ A @ B ) )
=> ( P @ X ) ) )
= ( ! [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ A )
=> ( P @ X ) )
& ! [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ B )
=> ( P @ X ) ) ) ) ).
% ball_Un
thf(fact_250_bex__Un,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ( ? [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ ( sup_su6804446743777130803_a_nat @ A @ B ) )
& ( P @ X ) ) )
= ( ? [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ A )
& ( P @ X ) )
| ? [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ B )
& ( P @ X ) ) ) ) ).
% bex_Un
thf(fact_251_UnI2,axiom,
! [C: a,B: set_a,A: set_a] :
( ( member_a @ C @ B )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnI2
thf(fact_252_UnI2,axiom,
! [C: nat,B: set_nat,A: set_nat] :
( ( member_nat @ C @ B )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_253_UnI2,axiom,
! [C: sum_sum_a_nat,B: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C @ B )
=> ( member_Sum_sum_a_nat @ C @ ( sup_su6804446743777130803_a_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_254_UnI1,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ A )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnI1
thf(fact_255_UnI1,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ A )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_256_UnI1,axiom,
! [C: sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C @ A )
=> ( member_Sum_sum_a_nat @ C @ ( sup_su6804446743777130803_a_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_257_UnE,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A @ B ) )
=> ( ~ ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% UnE
thf(fact_258_UnE,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) )
=> ( ~ ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% UnE
thf(fact_259_UnE,axiom,
! [C: sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C @ ( sup_su6804446743777130803_a_nat @ A @ B ) )
=> ( ~ ( member_Sum_sum_a_nat @ C @ A )
=> ( member_Sum_sum_a_nat @ C @ B ) ) ) ).
% UnE
thf(fact_260_DiffD2,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
=> ~ ( member_a @ C @ B ) ) ).
% DiffD2
thf(fact_261_DiffD2,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
=> ~ ( member_nat @ C @ B ) ) ).
% DiffD2
thf(fact_262_DiffD2,axiom,
! [C: sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C @ ( minus_1134630996077396038_a_nat @ A @ B ) )
=> ~ ( member_Sum_sum_a_nat @ C @ B ) ) ).
% DiffD2
thf(fact_263_DiffD1,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
=> ( member_a @ C @ A ) ) ).
% DiffD1
thf(fact_264_DiffD1,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
=> ( member_nat @ C @ A ) ) ).
% DiffD1
thf(fact_265_DiffD1,axiom,
! [C: sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C @ ( minus_1134630996077396038_a_nat @ A @ B ) )
=> ( member_Sum_sum_a_nat @ C @ A ) ) ).
% DiffD1
thf(fact_266_DiffE,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
=> ~ ( ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% DiffE
thf(fact_267_DiffE,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
=> ~ ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% DiffE
thf(fact_268_DiffE,axiom,
! [C: sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C @ ( minus_1134630996077396038_a_nat @ A @ B ) )
=> ~ ( ( member_Sum_sum_a_nat @ C @ A )
=> ( member_Sum_sum_a_nat @ C @ B ) ) ) ).
% DiffE
thf(fact_269_Compr__image__eq,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( image_nat_nat @ F @ A ) )
& ( P @ X ) ) )
= ( image_nat_nat @ F
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_270_Compr__image__eq,axiom,
! [F: sum_sum_a_nat > nat,A: set_Sum_sum_a_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( image_2473878607534554506at_nat @ F @ A ) )
& ( P @ X ) ) )
= ( image_2473878607534554506at_nat @ F
@ ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_271_Compr__image__eq,axiom,
! [F: a > nat,A: set_a,P: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( image_a_nat @ F @ A ) )
& ( P @ X ) ) )
= ( image_a_nat @ F
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_272_Compr__image__eq,axiom,
! [F: nat > sum_sum_a_nat,A: set_nat,P: sum_sum_a_nat > $o] :
( ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ ( image_7293268710728258664_a_nat @ F @ A ) )
& ( P @ X ) ) )
= ( image_7293268710728258664_a_nat @ F
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_273_Compr__image__eq,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,A: set_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ ( image_7142520692256960453_a_nat @ F @ A ) )
& ( P @ X ) ) )
= ( image_7142520692256960453_a_nat @ F
@ ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_274_Compr__image__eq,axiom,
! [F: a > sum_sum_a_nat,A: set_a,P: sum_sum_a_nat > $o] :
( ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ ( image_7873763678140191238_a_nat @ F @ A ) )
& ( P @ X ) ) )
= ( image_7873763678140191238_a_nat @ F
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_275_Compr__image__eq,axiom,
! [F: nat > a,A: set_nat,P: a > $o] :
( ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( image_nat_a @ F @ A ) )
& ( P @ X ) ) )
= ( image_nat_a @ F
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_276_Compr__image__eq,axiom,
! [F: sum_sum_a_nat > a,A: set_Sum_sum_a_nat,P: a > $o] :
( ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( image_6322530041254294468_nat_a @ F @ A ) )
& ( P @ X ) ) )
= ( image_6322530041254294468_nat_a @ F
@ ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_277_Compr__image__eq,axiom,
! [F: a > a,A: set_a,P: a > $o] :
( ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( image_a_a @ F @ A ) )
& ( P @ X ) ) )
= ( image_a_a @ F
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_278_image__image,axiom,
! [F: nat > sum_sum_a_nat,G: nat > nat,A: set_nat] :
( ( image_7293268710728258664_a_nat @ F @ ( image_nat_nat @ G @ A ) )
= ( image_7293268710728258664_a_nat
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ A ) ) ).
% image_image
thf(fact_279_image__image,axiom,
! [F: nat > sum_sum_a_nat,G: a > nat,A: set_a] :
( ( image_7293268710728258664_a_nat @ F @ ( image_a_nat @ G @ A ) )
= ( image_7873763678140191238_a_nat
@ ^ [X: a] : ( F @ ( G @ X ) )
@ A ) ) ).
% image_image
thf(fact_280_image__image,axiom,
! [F: nat > sum_sum_a_nat,G: sum_sum_a_nat > nat,A: set_Sum_sum_a_nat] :
( ( image_7293268710728258664_a_nat @ F @ ( image_2473878607534554506at_nat @ G @ A ) )
= ( image_7142520692256960453_a_nat
@ ^ [X: sum_sum_a_nat] : ( F @ ( G @ X ) )
@ A ) ) ).
% image_image
thf(fact_281_image__image,axiom,
! [F: a > sum_sum_a_nat,G: nat > a,A: set_nat] :
( ( image_7873763678140191238_a_nat @ F @ ( image_nat_a @ G @ A ) )
= ( image_7293268710728258664_a_nat
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ A ) ) ).
% image_image
thf(fact_282_image__image,axiom,
! [F: a > sum_sum_a_nat,G: a > a,A: set_a] :
( ( image_7873763678140191238_a_nat @ F @ ( image_a_a @ G @ A ) )
= ( image_7873763678140191238_a_nat
@ ^ [X: a] : ( F @ ( G @ X ) )
@ A ) ) ).
% image_image
thf(fact_283_image__image,axiom,
! [F: a > sum_sum_a_nat,G: sum_sum_a_nat > a,A: set_Sum_sum_a_nat] :
( ( image_7873763678140191238_a_nat @ F @ ( image_6322530041254294468_nat_a @ G @ A ) )
= ( image_7142520692256960453_a_nat
@ ^ [X: sum_sum_a_nat] : ( F @ ( G @ X ) )
@ A ) ) ).
% image_image
thf(fact_284_image__image,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,G: nat > sum_sum_a_nat,A: set_nat] :
( ( image_7142520692256960453_a_nat @ F @ ( image_7293268710728258664_a_nat @ G @ A ) )
= ( image_7293268710728258664_a_nat
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ A ) ) ).
% image_image
thf(fact_285_image__image,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,G: a > sum_sum_a_nat,A: set_a] :
( ( image_7142520692256960453_a_nat @ F @ ( image_7873763678140191238_a_nat @ G @ A ) )
= ( image_7873763678140191238_a_nat
@ ^ [X: a] : ( F @ ( G @ X ) )
@ A ) ) ).
% image_image
thf(fact_286_image__image,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,G: sum_sum_a_nat > sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( image_7142520692256960453_a_nat @ F @ ( image_7142520692256960453_a_nat @ G @ A ) )
= ( image_7142520692256960453_a_nat
@ ^ [X: sum_sum_a_nat] : ( F @ ( G @ X ) )
@ A ) ) ).
% image_image
thf(fact_287_imageE,axiom,
! [B2: sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ B2 @ ( image_7142520692256960453_a_nat @ F @ A ) )
=> ~ ! [X4: sum_sum_a_nat] :
( ( B2
= ( F @ X4 ) )
=> ~ ( member_Sum_sum_a_nat @ X4 @ A ) ) ) ).
% imageE
thf(fact_288_imageE,axiom,
! [B2: sum_sum_a_nat,F: a > sum_sum_a_nat,A: set_a] :
( ( member_Sum_sum_a_nat @ B2 @ ( image_7873763678140191238_a_nat @ F @ A ) )
=> ~ ! [X4: a] :
( ( B2
= ( F @ X4 ) )
=> ~ ( member_a @ X4 @ A ) ) ) ).
% imageE
thf(fact_289_imageE,axiom,
! [B2: sum_sum_a_nat,F: nat > sum_sum_a_nat,A: set_nat] :
( ( member_Sum_sum_a_nat @ B2 @ ( image_7293268710728258664_a_nat @ F @ A ) )
=> ~ ! [X4: nat] :
( ( B2
= ( F @ X4 ) )
=> ~ ( member_nat @ X4 @ A ) ) ) ).
% imageE
thf(fact_290_imageE,axiom,
! [B2: a,F: a > a,A: set_a] :
( ( member_a @ B2 @ ( image_a_a @ F @ A ) )
=> ~ ! [X4: a] :
( ( B2
= ( F @ X4 ) )
=> ~ ( member_a @ X4 @ A ) ) ) ).
% imageE
thf(fact_291_imageE,axiom,
! [B2: a,F: nat > a,A: set_nat] :
( ( member_a @ B2 @ ( image_nat_a @ F @ A ) )
=> ~ ! [X4: nat] :
( ( B2
= ( F @ X4 ) )
=> ~ ( member_nat @ X4 @ A ) ) ) ).
% imageE
thf(fact_292_imageE,axiom,
! [B2: nat,F: a > nat,A: set_a] :
( ( member_nat @ B2 @ ( image_a_nat @ F @ A ) )
=> ~ ! [X4: a] :
( ( B2
= ( F @ X4 ) )
=> ~ ( member_a @ X4 @ A ) ) ) ).
% imageE
thf(fact_293_imageE,axiom,
! [B2: nat,F: nat > nat,A: set_nat] :
( ( member_nat @ B2 @ ( image_nat_nat @ F @ A ) )
=> ~ ! [X4: nat] :
( ( B2
= ( F @ X4 ) )
=> ~ ( member_nat @ X4 @ A ) ) ) ).
% imageE
thf(fact_294_Collect__conj__eq,axiom,
! [P: sum_sum_a_nat > $o,Q: sum_sum_a_nat > $o] :
( ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( P @ X )
& ( Q @ X ) ) )
= ( inf_in7084830621192376909_a_nat @ ( collec7073057861543223018_a_nat @ P ) @ ( collec7073057861543223018_a_nat @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_295_Collect__conj__eq,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( P @ X )
& ( Q @ X ) ) )
= ( inf_inf_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_296_Collect__conj__eq,axiom,
! [P: a > $o,Q: a > $o] :
( ( collect_a
@ ^ [X: a] :
( ( P @ X )
& ( Q @ X ) ) )
= ( inf_inf_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_297_Int__Collect,axiom,
! [X3: sum_sum_a_nat,A: set_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ( member_Sum_sum_a_nat @ X3 @ ( inf_in7084830621192376909_a_nat @ A @ ( collec7073057861543223018_a_nat @ P ) ) )
= ( ( member_Sum_sum_a_nat @ X3 @ A )
& ( P @ X3 ) ) ) ).
% Int_Collect
thf(fact_298_Int__Collect,axiom,
! [X3: nat,A: set_nat,P: nat > $o] :
( ( member_nat @ X3 @ ( inf_inf_set_nat @ A @ ( collect_nat @ P ) ) )
= ( ( member_nat @ X3 @ A )
& ( P @ X3 ) ) ) ).
% Int_Collect
thf(fact_299_Int__Collect,axiom,
! [X3: a,A: set_a,P: a > $o] :
( ( member_a @ X3 @ ( inf_inf_set_a @ A @ ( collect_a @ P ) ) )
= ( ( member_a @ X3 @ A )
& ( P @ X3 ) ) ) ).
% Int_Collect
thf(fact_300_Int__def,axiom,
( inf_in7084830621192376909_a_nat
= ( ^ [A2: set_Sum_sum_a_nat,B3: set_Sum_sum_a_nat] :
( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ A2 )
& ( member_Sum_sum_a_nat @ X @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_301_Int__def,axiom,
( inf_inf_set_nat
= ( ^ [A2: set_nat,B3: set_nat] :
( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A2 )
& ( member_nat @ X @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_302_Int__def,axiom,
( inf_inf_set_a
= ( ^ [A2: set_a,B3: set_a] :
( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A2 )
& ( member_a @ X @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_303_Collect__disj__eq,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( P @ X )
| ( Q @ X ) ) )
= ( sup_sup_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_304_Collect__disj__eq,axiom,
! [P: a > $o,Q: a > $o] :
( ( collect_a
@ ^ [X: a] :
( ( P @ X )
| ( Q @ X ) ) )
= ( sup_sup_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_305_Collect__disj__eq,axiom,
! [P: sum_sum_a_nat > $o,Q: sum_sum_a_nat > $o] :
( ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( P @ X )
| ( Q @ X ) ) )
= ( sup_su6804446743777130803_a_nat @ ( collec7073057861543223018_a_nat @ P ) @ ( collec7073057861543223018_a_nat @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_306_Un__def,axiom,
( sup_sup_set_nat
= ( ^ [A2: set_nat,B3: set_nat] :
( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A2 )
| ( member_nat @ X @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_307_Un__def,axiom,
( sup_sup_set_a
= ( ^ [A2: set_a,B3: set_a] :
( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A2 )
| ( member_a @ X @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_308_Un__def,axiom,
( sup_su6804446743777130803_a_nat
= ( ^ [A2: set_Sum_sum_a_nat,B3: set_Sum_sum_a_nat] :
( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ A2 )
| ( member_Sum_sum_a_nat @ X @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_309_set__diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [A2: set_nat,B3: set_nat] :
( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A2 )
& ~ ( member_nat @ X @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_310_set__diff__eq,axiom,
( minus_minus_set_a
= ( ^ [A2: set_a,B3: set_a] :
( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A2 )
& ~ ( member_a @ X @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_311_set__diff__eq,axiom,
( minus_1134630996077396038_a_nat
= ( ^ [A2: set_Sum_sum_a_nat,B3: set_Sum_sum_a_nat] :
( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ A2 )
& ~ ( member_Sum_sum_a_nat @ X @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_312_image__Un,axiom,
! [F: nat > sum_sum_a_nat,A: set_nat,B: set_nat] :
( ( image_7293268710728258664_a_nat @ F @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_su6804446743777130803_a_nat @ ( image_7293268710728258664_a_nat @ F @ A ) @ ( image_7293268710728258664_a_nat @ F @ B ) ) ) ).
% image_Un
thf(fact_313_image__Un,axiom,
! [F: a > sum_sum_a_nat,A: set_a,B: set_a] :
( ( image_7873763678140191238_a_nat @ F @ ( sup_sup_set_a @ A @ B ) )
= ( sup_su6804446743777130803_a_nat @ ( image_7873763678140191238_a_nat @ F @ A ) @ ( image_7873763678140191238_a_nat @ F @ B ) ) ) ).
% image_Un
thf(fact_314_image__Un,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( image_7142520692256960453_a_nat @ F @ ( sup_su6804446743777130803_a_nat @ A @ B ) )
= ( sup_su6804446743777130803_a_nat @ ( image_7142520692256960453_a_nat @ F @ A ) @ ( image_7142520692256960453_a_nat @ F @ B ) ) ) ).
% image_Un
thf(fact_315_Un__Int__distrib2,axiom,
! [B: set_nat,C2: set_nat,A: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ B @ C2 ) @ A )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ B @ A ) @ ( sup_sup_set_nat @ C2 @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_316_Un__Int__distrib2,axiom,
! [B: set_a,C2: set_a,A: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ B @ C2 ) @ A )
= ( inf_inf_set_a @ ( sup_sup_set_a @ B @ A ) @ ( sup_sup_set_a @ C2 @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_317_Un__Int__distrib2,axiom,
! [B: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ B @ C2 ) @ A )
= ( inf_in7084830621192376909_a_nat @ ( sup_su6804446743777130803_a_nat @ B @ A ) @ ( sup_su6804446743777130803_a_nat @ C2 @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_318_Int__Un__distrib2,axiom,
! [B: set_nat,C2: set_nat,A: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ B @ C2 ) @ A )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ B @ A ) @ ( inf_inf_set_nat @ C2 @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_319_Int__Un__distrib2,axiom,
! [B: set_a,C2: set_a,A: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ B @ C2 ) @ A )
= ( sup_sup_set_a @ ( inf_inf_set_a @ B @ A ) @ ( inf_inf_set_a @ C2 @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_320_Int__Un__distrib2,axiom,
! [B: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ ( sup_su6804446743777130803_a_nat @ B @ C2 ) @ A )
= ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ B @ A ) @ ( inf_in7084830621192376909_a_nat @ C2 @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_321_Un__Int__distrib,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ A @ ( inf_inf_set_nat @ B @ C2 ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sup_sup_set_nat @ A @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_322_Un__Int__distrib,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( sup_sup_set_a @ A @ ( inf_inf_set_a @ B @ C2 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ A @ B ) @ ( sup_sup_set_a @ A @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_323_Un__Int__distrib,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A @ ( inf_in7084830621192376909_a_nat @ B @ C2 ) )
= ( inf_in7084830621192376909_a_nat @ ( sup_su6804446743777130803_a_nat @ A @ B ) @ ( sup_su6804446743777130803_a_nat @ A @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_324_Int__Un__distrib,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( inf_inf_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( inf_inf_set_nat @ A @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_325_Int__Un__distrib,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( inf_inf_set_a @ A @ ( sup_sup_set_a @ B @ C2 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_inf_set_a @ A @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_326_Int__Un__distrib,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ A @ ( sup_su6804446743777130803_a_nat @ B @ C2 ) )
= ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ A @ B ) @ ( inf_in7084830621192376909_a_nat @ A @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_327_Un__Int__crazy,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( inf_inf_set_nat @ B @ C2 ) ) @ ( inf_inf_set_nat @ C2 @ A ) )
= ( inf_inf_set_nat @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sup_sup_set_nat @ B @ C2 ) ) @ ( sup_sup_set_nat @ C2 @ A ) ) ) ).
% Un_Int_crazy
thf(fact_328_Un__Int__crazy,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_inf_set_a @ B @ C2 ) ) @ ( inf_inf_set_a @ C2 @ A ) )
= ( inf_inf_set_a @ ( inf_inf_set_a @ ( sup_sup_set_a @ A @ B ) @ ( sup_sup_set_a @ B @ C2 ) ) @ ( sup_sup_set_a @ C2 @ A ) ) ) ).
% Un_Int_crazy
thf(fact_329_Un__Int__crazy,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ A @ B ) @ ( inf_in7084830621192376909_a_nat @ B @ C2 ) ) @ ( inf_in7084830621192376909_a_nat @ C2 @ A ) )
= ( inf_in7084830621192376909_a_nat @ ( inf_in7084830621192376909_a_nat @ ( sup_su6804446743777130803_a_nat @ A @ B ) @ ( sup_su6804446743777130803_a_nat @ B @ C2 ) ) @ ( sup_su6804446743777130803_a_nat @ C2 @ A ) ) ) ).
% Un_Int_crazy
thf(fact_330_Diff__Int__distrib2,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( inf_inf_set_nat @ ( minus_minus_set_nat @ A @ B ) @ C2 )
= ( minus_minus_set_nat @ ( inf_inf_set_nat @ A @ C2 ) @ ( inf_inf_set_nat @ B @ C2 ) ) ) ).
% Diff_Int_distrib2
thf(fact_331_Diff__Int__distrib2,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( minus_minus_set_a @ A @ B ) @ C2 )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A @ C2 ) @ ( inf_inf_set_a @ B @ C2 ) ) ) ).
% Diff_Int_distrib2
thf(fact_332_Diff__Int__distrib2,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ ( minus_1134630996077396038_a_nat @ A @ B ) @ C2 )
= ( minus_1134630996077396038_a_nat @ ( inf_in7084830621192376909_a_nat @ A @ C2 ) @ ( inf_in7084830621192376909_a_nat @ B @ C2 ) ) ) ).
% Diff_Int_distrib2
thf(fact_333_Diff__Int__distrib,axiom,
! [C2: set_nat,A: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ C2 @ ( minus_minus_set_nat @ A @ B ) )
= ( minus_minus_set_nat @ ( inf_inf_set_nat @ C2 @ A ) @ ( inf_inf_set_nat @ C2 @ B ) ) ) ).
% Diff_Int_distrib
thf(fact_334_Diff__Int__distrib,axiom,
! [C2: set_a,A: set_a,B: set_a] :
( ( inf_inf_set_a @ C2 @ ( minus_minus_set_a @ A @ B ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ C2 @ A ) @ ( inf_inf_set_a @ C2 @ B ) ) ) ).
% Diff_Int_distrib
thf(fact_335_Diff__Int__distrib,axiom,
! [C2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ C2 @ ( minus_1134630996077396038_a_nat @ A @ B ) )
= ( minus_1134630996077396038_a_nat @ ( inf_in7084830621192376909_a_nat @ C2 @ A ) @ ( inf_in7084830621192376909_a_nat @ C2 @ B ) ) ) ).
% Diff_Int_distrib
thf(fact_336_Diff__Diff__Int,axiom,
! [A: set_nat,B: set_nat] :
( ( minus_minus_set_nat @ A @ ( minus_minus_set_nat @ A @ B ) )
= ( inf_inf_set_nat @ A @ B ) ) ).
% Diff_Diff_Int
thf(fact_337_Diff__Diff__Int,axiom,
! [A: set_a,B: set_a] :
( ( minus_minus_set_a @ A @ ( minus_minus_set_a @ A @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ).
% Diff_Diff_Int
thf(fact_338_Diff__Diff__Int,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( minus_1134630996077396038_a_nat @ A @ ( minus_1134630996077396038_a_nat @ A @ B ) )
= ( inf_in7084830621192376909_a_nat @ A @ B ) ) ).
% Diff_Diff_Int
thf(fact_339_Diff__Int2,axiom,
! [A: set_nat,C2: set_nat,B: set_nat] :
( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A @ C2 ) @ ( inf_inf_set_nat @ B @ C2 ) )
= ( minus_minus_set_nat @ ( inf_inf_set_nat @ A @ C2 ) @ B ) ) ).
% Diff_Int2
thf(fact_340_Diff__Int2,axiom,
! [A: set_a,C2: set_a,B: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A @ C2 ) @ ( inf_inf_set_a @ B @ C2 ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A @ C2 ) @ B ) ) ).
% Diff_Int2
thf(fact_341_Diff__Int2,axiom,
! [A: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( minus_1134630996077396038_a_nat @ ( inf_in7084830621192376909_a_nat @ A @ C2 ) @ ( inf_in7084830621192376909_a_nat @ B @ C2 ) )
= ( minus_1134630996077396038_a_nat @ ( inf_in7084830621192376909_a_nat @ A @ C2 ) @ B ) ) ).
% Diff_Int2
thf(fact_342_Int__Diff,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C2 )
= ( inf_inf_set_nat @ A @ ( minus_minus_set_nat @ B @ C2 ) ) ) ).
% Int_Diff
thf(fact_343_Int__Diff,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A @ B ) @ C2 )
= ( inf_inf_set_a @ A @ ( minus_minus_set_a @ B @ C2 ) ) ) ).
% Int_Diff
thf(fact_344_Int__Diff,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
( ( minus_1134630996077396038_a_nat @ ( inf_in7084830621192376909_a_nat @ A @ B ) @ C2 )
= ( inf_in7084830621192376909_a_nat @ A @ ( minus_1134630996077396038_a_nat @ B @ C2 ) ) ) ).
% Int_Diff
thf(fact_345_Un__Diff,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
( ( minus_1134630996077396038_a_nat @ ( sup_su6804446743777130803_a_nat @ A @ B ) @ C2 )
= ( sup_su6804446743777130803_a_nat @ ( minus_1134630996077396038_a_nat @ A @ C2 ) @ ( minus_1134630996077396038_a_nat @ B @ C2 ) ) ) ).
% Un_Diff
thf(fact_346_Un__Diff__Int,axiom,
! [A: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ ( minus_minus_set_nat @ A @ B ) @ ( inf_inf_set_nat @ A @ B ) )
= A ) ).
% Un_Diff_Int
thf(fact_347_Un__Diff__Int,axiom,
! [A: set_a,B: set_a] :
( ( sup_sup_set_a @ ( minus_minus_set_a @ A @ B ) @ ( inf_inf_set_a @ A @ B ) )
= A ) ).
% Un_Diff_Int
thf(fact_348_Un__Diff__Int,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ ( minus_1134630996077396038_a_nat @ A @ B ) @ ( inf_in7084830621192376909_a_nat @ A @ B ) )
= A ) ).
% Un_Diff_Int
thf(fact_349_Int__Diff__Un,axiom,
! [A: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( minus_minus_set_nat @ A @ B ) )
= A ) ).
% Int_Diff_Un
thf(fact_350_Int__Diff__Un,axiom,
! [A: set_a,B: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ A @ B ) @ ( minus_minus_set_a @ A @ B ) )
= A ) ).
% Int_Diff_Un
thf(fact_351_Int__Diff__Un,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ A @ B ) @ ( minus_1134630996077396038_a_nat @ A @ B ) )
= A ) ).
% Int_Diff_Un
thf(fact_352_Diff__Int,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( minus_minus_set_nat @ A @ ( inf_inf_set_nat @ B @ C2 ) )
= ( sup_sup_set_nat @ ( minus_minus_set_nat @ A @ B ) @ ( minus_minus_set_nat @ A @ C2 ) ) ) ).
% Diff_Int
thf(fact_353_Diff__Int,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( minus_minus_set_a @ A @ ( inf_inf_set_a @ B @ C2 ) )
= ( sup_sup_set_a @ ( minus_minus_set_a @ A @ B ) @ ( minus_minus_set_a @ A @ C2 ) ) ) ).
% Diff_Int
thf(fact_354_Diff__Int,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
( ( minus_1134630996077396038_a_nat @ A @ ( inf_in7084830621192376909_a_nat @ B @ C2 ) )
= ( sup_su6804446743777130803_a_nat @ ( minus_1134630996077396038_a_nat @ A @ B ) @ ( minus_1134630996077396038_a_nat @ A @ C2 ) ) ) ).
% Diff_Int
thf(fact_355_Diff__Un,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( minus_minus_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) )
= ( inf_inf_set_nat @ ( minus_minus_set_nat @ A @ B ) @ ( minus_minus_set_nat @ A @ C2 ) ) ) ).
% Diff_Un
thf(fact_356_Diff__Un,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( minus_minus_set_a @ A @ ( sup_sup_set_a @ B @ C2 ) )
= ( inf_inf_set_a @ ( minus_minus_set_a @ A @ B ) @ ( minus_minus_set_a @ A @ C2 ) ) ) ).
% Diff_Un
thf(fact_357_Diff__Un,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
( ( minus_1134630996077396038_a_nat @ A @ ( sup_su6804446743777130803_a_nat @ B @ C2 ) )
= ( inf_in7084830621192376909_a_nat @ ( minus_1134630996077396038_a_nat @ A @ B ) @ ( minus_1134630996077396038_a_nat @ A @ C2 ) ) ) ).
% Diff_Un
thf(fact_358_fun__upd__None__if__notin__dom,axiom,
! [K: a,M2: a > option_nat] :
( ~ ( member_a @ K @ ( dom_a_nat @ M2 ) )
=> ( ( fun_upd_a_option_nat @ M2 @ K @ none_nat )
= M2 ) ) ).
% fun_upd_None_if_notin_dom
thf(fact_359_fun__upd__None__if__notin__dom,axiom,
! [K: nat,M2: nat > option_nat] :
( ~ ( member_nat @ K @ ( dom_nat_nat @ M2 ) )
=> ( ( fun_up1493157387958331631on_nat @ M2 @ K @ none_nat )
= M2 ) ) ).
% fun_upd_None_if_notin_dom
thf(fact_360_fun__upd__None__if__notin__dom,axiom,
! [K: sum_sum_a_nat,M2: sum_sum_a_nat > option_nat] :
( ~ ( member_Sum_sum_a_nat @ K @ ( dom_Su2255998037560862461at_nat @ M2 ) )
=> ( ( fun_up4750343006489851234on_nat @ M2 @ K @ none_nat )
= M2 ) ) ).
% fun_upd_None_if_notin_dom
thf(fact_361_lessThan__subset__iff,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X3 ) @ ( set_ord_lessThan_nat @ Y ) )
= ( ord_less_eq_nat @ X3 @ Y ) ) ).
% lessThan_subset_iff
thf(fact_362_le__add__diff__inverse2,axiom,
! [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A3 @ B2 ) @ B2 )
= A3 ) ) ).
% le_add_diff_inverse2
thf(fact_363_le__add__diff__inverse,axiom,
! [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
=> ( ( plus_plus_nat @ B2 @ ( minus_minus_nat @ A3 @ B2 ) )
= A3 ) ) ).
% le_add_diff_inverse
thf(fact_364_linorder__finite__Map__induct,axiom,
! [M2: nat > option_nat,P: ( nat > option_nat ) > $o] :
( ( finite_finite_nat @ ( dom_nat_nat @ M2 ) )
=> ( ( P
@ ^ [X: nat] : none_nat )
=> ( ! [K2: nat,V: nat,M3: nat > option_nat] :
( ( finite_finite_nat @ ( dom_nat_nat @ M3 ) )
=> ( ~ ( member_nat @ K2 @ ( dom_nat_nat @ M3 ) )
=> ( ! [K3: nat] :
( ( member_nat @ K3 @ ( dom_nat_nat @ M3 ) )
=> ( ord_less_eq_nat @ K3 @ K2 ) )
=> ( ( P @ M3 )
=> ( P @ ( fun_up1493157387958331631on_nat @ M3 @ K2 @ ( some_nat @ V ) ) ) ) ) ) )
=> ( P @ M2 ) ) ) ) ).
% linorder_finite_Map_induct
thf(fact_365_empty__upd__none,axiom,
! [X3: sum_sum_a_nat] :
( ( fun_up4750343006489851234on_nat
@ ^ [X: sum_sum_a_nat] : none_nat
@ X3
@ none_nat )
= ( ^ [X: sum_sum_a_nat] : none_nat ) ) ).
% empty_upd_none
thf(fact_366_finite__Collect__subsets,axiom,
! [A: set_Sum_sum_a_nat] :
( ( finite502105017643426984_a_nat @ A )
=> ( finite4842507993062306312_a_nat
@ ( collec4049389696321283146_a_nat
@ ^ [B3: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ B3 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_367_finite__Collect__subsets,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B3: set_nat] : ( ord_less_eq_set_nat @ B3 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_368_finite__Map__induct,axiom,
! [M2: a > option_nat,P: ( a > option_nat ) > $o] :
( ( finite_finite_a @ ( dom_a_nat @ M2 ) )
=> ( ( P
@ ^ [X: a] : none_nat )
=> ( ! [K2: a,V: nat,M3: a > option_nat] :
( ( finite_finite_a @ ( dom_a_nat @ M3 ) )
=> ( ~ ( member_a @ K2 @ ( dom_a_nat @ M3 ) )
=> ( ( P @ M3 )
=> ( P @ ( fun_upd_a_option_nat @ M3 @ K2 @ ( some_nat @ V ) ) ) ) ) )
=> ( P @ M2 ) ) ) ) ).
% finite_Map_induct
thf(fact_369_finite__Map__induct,axiom,
! [M2: sum_sum_a_nat > option_nat,P: ( sum_sum_a_nat > option_nat ) > $o] :
( ( finite502105017643426984_a_nat @ ( dom_Su2255998037560862461at_nat @ M2 ) )
=> ( ( P
@ ^ [X: sum_sum_a_nat] : none_nat )
=> ( ! [K2: sum_sum_a_nat,V: nat,M3: sum_sum_a_nat > option_nat] :
( ( finite502105017643426984_a_nat @ ( dom_Su2255998037560862461at_nat @ M3 ) )
=> ( ~ ( member_Sum_sum_a_nat @ K2 @ ( dom_Su2255998037560862461at_nat @ M3 ) )
=> ( ( P @ M3 )
=> ( P @ ( fun_up4750343006489851234on_nat @ M3 @ K2 @ ( some_nat @ V ) ) ) ) ) )
=> ( P @ M2 ) ) ) ) ).
% finite_Map_induct
thf(fact_370_finite__Map__induct,axiom,
! [M2: nat > option_nat,P: ( nat > option_nat ) > $o] :
( ( finite_finite_nat @ ( dom_nat_nat @ M2 ) )
=> ( ( P
@ ^ [X: nat] : none_nat )
=> ( ! [K2: nat,V: nat,M3: nat > option_nat] :
( ( finite_finite_nat @ ( dom_nat_nat @ M3 ) )
=> ( ~ ( member_nat @ K2 @ ( dom_nat_nat @ M3 ) )
=> ( ( P @ M3 )
=> ( P @ ( fun_up1493157387958331631on_nat @ M3 @ K2 @ ( some_nat @ V ) ) ) ) ) )
=> ( P @ M2 ) ) ) ) ).
% finite_Map_induct
thf(fact_371_image__map__upd,axiom,
! [X3: a,A: set_a,M2: a > option_nat,Y: nat] :
( ~ ( member_a @ X3 @ A )
=> ( ( image_a_option_nat @ ( fun_upd_a_option_nat @ M2 @ X3 @ ( some_nat @ Y ) ) @ A )
= ( image_a_option_nat @ M2 @ A ) ) ) ).
% image_map_upd
thf(fact_372_image__map__upd,axiom,
! [X3: nat,A: set_nat,M2: nat > option_nat,Y: nat] :
( ~ ( member_nat @ X3 @ A )
=> ( ( image_nat_option_nat @ ( fun_up1493157387958331631on_nat @ M2 @ X3 @ ( some_nat @ Y ) ) @ A )
= ( image_nat_option_nat @ M2 @ A ) ) ) ).
% image_map_upd
thf(fact_373_image__map__upd,axiom,
! [X3: sum_sum_a_nat,A: set_Sum_sum_a_nat,M2: sum_sum_a_nat > option_nat,Y: nat] :
( ~ ( member_Sum_sum_a_nat @ X3 @ A )
=> ( ( image_9069542366050205658on_nat @ ( fun_up4750343006489851234on_nat @ M2 @ X3 @ ( some_nat @ Y ) ) @ A )
= ( image_9069542366050205658on_nat @ M2 @ A ) ) ) ).
% image_map_upd
thf(fact_374_card__Un__Int,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ( plus_plus_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) )
= ( plus_plus_nat @ ( finite_card_nat @ ( sup_sup_set_nat @ A @ B ) ) @ ( finite_card_nat @ ( inf_inf_set_nat @ A @ B ) ) ) ) ) ) ).
% card_Un_Int
thf(fact_375_card__Un__Int,axiom,
! [A: set_a,B: set_a] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B )
=> ( ( plus_plus_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B ) )
= ( plus_plus_nat @ ( finite_card_a @ ( sup_sup_set_a @ A @ B ) ) @ ( finite_card_a @ ( inf_inf_set_a @ A @ B ) ) ) ) ) ) ).
% card_Un_Int
thf(fact_376_card__Un__Int,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( finite502105017643426984_a_nat @ A )
=> ( ( finite502105017643426984_a_nat @ B )
=> ( ( plus_plus_nat @ ( finite6080979521523705895_a_nat @ A ) @ ( finite6080979521523705895_a_nat @ B ) )
= ( plus_plus_nat @ ( finite6080979521523705895_a_nat @ ( sup_su6804446743777130803_a_nat @ A @ B ) ) @ ( finite6080979521523705895_a_nat @ ( inf_in7084830621192376909_a_nat @ A @ B ) ) ) ) ) ) ).
% card_Un_Int
thf(fact_377_finite__Diff2,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) )
= ( finite_finite_nat @ A ) ) ) ).
% finite_Diff2
thf(fact_378_finite__Diff2,axiom,
! [B: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( finite502105017643426984_a_nat @ B )
=> ( ( finite502105017643426984_a_nat @ ( minus_1134630996077396038_a_nat @ A @ B ) )
= ( finite502105017643426984_a_nat @ A ) ) ) ).
% finite_Diff2
thf(fact_379_finite__Diff,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) ) ) ).
% finite_Diff
thf(fact_380_finite__Diff,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( finite502105017643426984_a_nat @ A )
=> ( finite502105017643426984_a_nat @ ( minus_1134630996077396038_a_nat @ A @ B ) ) ) ).
% finite_Diff
thf(fact_381_lessThan__eq__iff,axiom,
! [X3: nat,Y: nat] :
( ( ( set_ord_lessThan_nat @ X3 )
= ( set_ord_lessThan_nat @ Y ) )
= ( X3 = Y ) ) ).
% lessThan_eq_iff
thf(fact_382_card__lessThan,axiom,
! [U: nat] :
( ( finite_card_nat @ ( set_ord_lessThan_nat @ U ) )
= U ) ).
% card_lessThan
thf(fact_383_finite__lessThan,axiom,
! [K: nat] : ( finite_finite_nat @ ( set_ord_lessThan_nat @ K ) ) ).
% finite_lessThan
thf(fact_384_finite__Collect__disjI,axiom,
! [P: a > $o,Q: a > $o] :
( ( finite_finite_a
@ ( collect_a
@ ^ [X: a] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite_finite_a @ ( collect_a @ P ) )
& ( finite_finite_a @ ( collect_a @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_385_finite__Collect__disjI,axiom,
! [P: sum_sum_a_nat > $o,Q: sum_sum_a_nat > $o] :
( ( finite502105017643426984_a_nat
@ ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite502105017643426984_a_nat @ ( collec7073057861543223018_a_nat @ P ) )
& ( finite502105017643426984_a_nat @ ( collec7073057861543223018_a_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_386_finite__Collect__disjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P ) )
& ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_387_finite__Collect__conjI,axiom,
! [P: a > $o,Q: a > $o] :
( ( ( finite_finite_a @ ( collect_a @ P ) )
| ( finite_finite_a @ ( collect_a @ Q ) ) )
=> ( finite_finite_a
@ ( collect_a
@ ^ [X: a] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_388_finite__Collect__conjI,axiom,
! [P: sum_sum_a_nat > $o,Q: sum_sum_a_nat > $o] :
( ( ( finite502105017643426984_a_nat @ ( collec7073057861543223018_a_nat @ P ) )
| ( finite502105017643426984_a_nat @ ( collec7073057861543223018_a_nat @ Q ) ) )
=> ( finite502105017643426984_a_nat
@ ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_389_finite__Collect__conjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P ) )
| ( finite_finite_nat @ ( collect_nat @ Q ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_390_card__Collect__le__nat,axiom,
! [N2: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_eq_nat @ I @ N2 ) ) )
= ( suc @ N2 ) ) ).
% card_Collect_le_nat
thf(fact_391_finite__imageI,axiom,
! [F2: set_a,H: a > sum_sum_a_nat] :
( ( finite_finite_a @ F2 )
=> ( finite502105017643426984_a_nat @ ( image_7873763678140191238_a_nat @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_392_finite__imageI,axiom,
! [F2: set_Sum_sum_a_nat,H: sum_sum_a_nat > sum_sum_a_nat] :
( ( finite502105017643426984_a_nat @ F2 )
=> ( finite502105017643426984_a_nat @ ( image_7142520692256960453_a_nat @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_393_finite__imageI,axiom,
! [F2: set_Sum_sum_a_nat,H: sum_sum_a_nat > nat] :
( ( finite502105017643426984_a_nat @ F2 )
=> ( finite_finite_nat @ ( image_2473878607534554506at_nat @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_394_finite__imageI,axiom,
! [F2: set_nat,H: nat > sum_sum_a_nat] :
( ( finite_finite_nat @ F2 )
=> ( finite502105017643426984_a_nat @ ( image_7293268710728258664_a_nat @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_395_finite__imageI,axiom,
! [F2: set_nat,H: nat > nat] :
( ( finite_finite_nat @ F2 )
=> ( finite_finite_nat @ ( image_nat_nat @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_396_finite__Int,axiom,
! [F2: set_Sum_sum_a_nat,G2: set_Sum_sum_a_nat] :
( ( ( finite502105017643426984_a_nat @ F2 )
| ( finite502105017643426984_a_nat @ G2 ) )
=> ( finite502105017643426984_a_nat @ ( inf_in7084830621192376909_a_nat @ F2 @ G2 ) ) ) ).
% finite_Int
thf(fact_397_finite__Int,axiom,
! [F2: set_nat,G2: set_nat] :
( ( ( finite_finite_nat @ F2 )
| ( finite_finite_nat @ G2 ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ F2 @ G2 ) ) ) ).
% finite_Int
thf(fact_398_finite__Int,axiom,
! [F2: set_a,G2: set_a] :
( ( ( finite_finite_a @ F2 )
| ( finite_finite_a @ G2 ) )
=> ( finite_finite_a @ ( inf_inf_set_a @ F2 @ G2 ) ) ) ).
% finite_Int
thf(fact_399_finite__Un,axiom,
! [F2: set_nat,G2: set_nat] :
( ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G2 ) )
= ( ( finite_finite_nat @ F2 )
& ( finite_finite_nat @ G2 ) ) ) ).
% finite_Un
thf(fact_400_finite__Un,axiom,
! [F2: set_Sum_sum_a_nat,G2: set_Sum_sum_a_nat] :
( ( finite502105017643426984_a_nat @ ( sup_su6804446743777130803_a_nat @ F2 @ G2 ) )
= ( ( finite502105017643426984_a_nat @ F2 )
& ( finite502105017643426984_a_nat @ G2 ) ) ) ).
% finite_Un
thf(fact_401_diff__card__le__card__Diff,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ B ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_402_diff__card__le__card__Diff,axiom,
! [B: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( finite502105017643426984_a_nat @ B )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite6080979521523705895_a_nat @ A ) @ ( finite6080979521523705895_a_nat @ B ) ) @ ( finite6080979521523705895_a_nat @ ( minus_1134630996077396038_a_nat @ A @ B ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_403_card__le__if__inj__on__rel,axiom,
! [B: set_a,A: set_a,R: a > a > $o] :
( ( finite_finite_a @ B )
=> ( ! [A5: a] :
( ( member_a @ A5 @ A )
=> ? [B5: a] :
( ( member_a @ B5 @ B )
& ( R @ A5 @ B5 ) ) )
=> ( ! [A1: a,A22: a,B6: a] :
( ( member_a @ A1 @ A )
=> ( ( member_a @ A22 @ A )
=> ( ( member_a @ B6 @ B )
=> ( ( R @ A1 @ B6 )
=> ( ( R @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_404_card__le__if__inj__on__rel,axiom,
! [B: set_a,A: set_Sum_sum_a_nat,R: sum_sum_a_nat > a > $o] :
( ( finite_finite_a @ B )
=> ( ! [A5: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ A5 @ A )
=> ? [B5: a] :
( ( member_a @ B5 @ B )
& ( R @ A5 @ B5 ) ) )
=> ( ! [A1: sum_sum_a_nat,A22: sum_sum_a_nat,B6: a] :
( ( member_Sum_sum_a_nat @ A1 @ A )
=> ( ( member_Sum_sum_a_nat @ A22 @ A )
=> ( ( member_a @ B6 @ B )
=> ( ( R @ A1 @ B6 )
=> ( ( R @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite6080979521523705895_a_nat @ A ) @ ( finite_card_a @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_405_card__le__if__inj__on__rel,axiom,
! [B: set_a,A: set_nat,R: nat > a > $o] :
( ( finite_finite_a @ B )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ A )
=> ? [B5: a] :
( ( member_a @ B5 @ B )
& ( R @ A5 @ B5 ) ) )
=> ( ! [A1: nat,A22: nat,B6: a] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member_a @ B6 @ B )
=> ( ( R @ A1 @ B6 )
=> ( ( R @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_a @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_406_card__le__if__inj__on__rel,axiom,
! [B: set_Sum_sum_a_nat,A: set_a,R: a > sum_sum_a_nat > $o] :
( ( finite502105017643426984_a_nat @ B )
=> ( ! [A5: a] :
( ( member_a @ A5 @ A )
=> ? [B5: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ B5 @ B )
& ( R @ A5 @ B5 ) ) )
=> ( ! [A1: a,A22: a,B6: sum_sum_a_nat] :
( ( member_a @ A1 @ A )
=> ( ( member_a @ A22 @ A )
=> ( ( member_Sum_sum_a_nat @ B6 @ B )
=> ( ( R @ A1 @ B6 )
=> ( ( R @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite6080979521523705895_a_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_407_card__le__if__inj__on__rel,axiom,
! [B: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat,R: sum_sum_a_nat > sum_sum_a_nat > $o] :
( ( finite502105017643426984_a_nat @ B )
=> ( ! [A5: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ A5 @ A )
=> ? [B5: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ B5 @ B )
& ( R @ A5 @ B5 ) ) )
=> ( ! [A1: sum_sum_a_nat,A22: sum_sum_a_nat,B6: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ A1 @ A )
=> ( ( member_Sum_sum_a_nat @ A22 @ A )
=> ( ( member_Sum_sum_a_nat @ B6 @ B )
=> ( ( R @ A1 @ B6 )
=> ( ( R @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite6080979521523705895_a_nat @ A ) @ ( finite6080979521523705895_a_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_408_card__le__if__inj__on__rel,axiom,
! [B: set_Sum_sum_a_nat,A: set_nat,R: nat > sum_sum_a_nat > $o] :
( ( finite502105017643426984_a_nat @ B )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ A )
=> ? [B5: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ B5 @ B )
& ( R @ A5 @ B5 ) ) )
=> ( ! [A1: nat,A22: nat,B6: sum_sum_a_nat] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member_Sum_sum_a_nat @ B6 @ B )
=> ( ( R @ A1 @ B6 )
=> ( ( R @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite6080979521523705895_a_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_409_card__le__if__inj__on__rel,axiom,
! [B: set_nat,A: set_a,R: a > nat > $o] :
( ( finite_finite_nat @ B )
=> ( ! [A5: a] :
( ( member_a @ A5 @ A )
=> ? [B5: nat] :
( ( member_nat @ B5 @ B )
& ( R @ A5 @ B5 ) ) )
=> ( ! [A1: a,A22: a,B6: nat] :
( ( member_a @ A1 @ A )
=> ( ( member_a @ A22 @ A )
=> ( ( member_nat @ B6 @ B )
=> ( ( R @ A1 @ B6 )
=> ( ( R @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_410_card__le__if__inj__on__rel,axiom,
! [B: set_nat,A: set_Sum_sum_a_nat,R: sum_sum_a_nat > nat > $o] :
( ( finite_finite_nat @ B )
=> ( ! [A5: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ A5 @ A )
=> ? [B5: nat] :
( ( member_nat @ B5 @ B )
& ( R @ A5 @ B5 ) ) )
=> ( ! [A1: sum_sum_a_nat,A22: sum_sum_a_nat,B6: nat] :
( ( member_Sum_sum_a_nat @ A1 @ A )
=> ( ( member_Sum_sum_a_nat @ A22 @ A )
=> ( ( member_nat @ B6 @ B )
=> ( ( R @ A1 @ B6 )
=> ( ( R @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite6080979521523705895_a_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_411_card__le__if__inj__on__rel,axiom,
! [B: set_nat,A: set_nat,R: nat > nat > $o] :
( ( finite_finite_nat @ B )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ A )
=> ? [B5: nat] :
( ( member_nat @ B5 @ B )
& ( R @ A5 @ B5 ) ) )
=> ( ! [A1: nat,A22: nat,B6: nat] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member_nat @ B6 @ B )
=> ( ( R @ A1 @ B6 )
=> ( ( R @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_412_minus__set__def,axiom,
( minus_minus_set_nat
= ( ^ [A2: set_nat,B3: set_nat] :
( collect_nat
@ ( minus_minus_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A2 )
@ ^ [X: nat] : ( member_nat @ X @ B3 ) ) ) ) ) ).
% minus_set_def
thf(fact_413_minus__set__def,axiom,
( minus_minus_set_a
= ( ^ [A2: set_a,B3: set_a] :
( collect_a
@ ( minus_minus_a_o
@ ^ [X: a] : ( member_a @ X @ A2 )
@ ^ [X: a] : ( member_a @ X @ B3 ) ) ) ) ) ).
% minus_set_def
thf(fact_414_minus__set__def,axiom,
( minus_1134630996077396038_a_nat
= ( ^ [A2: set_Sum_sum_a_nat,B3: set_Sum_sum_a_nat] :
( collec7073057861543223018_a_nat
@ ( minus_2198590880655318551_nat_o
@ ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ A2 )
@ ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ B3 ) ) ) ) ) ).
% minus_set_def
thf(fact_415_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A2: set_a,B3: set_a] :
( ord_less_eq_a_o
@ ^ [X: a] : ( member_a @ X @ A2 )
@ ^ [X: a] : ( member_a @ X @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_416_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A2: set_nat,B3: set_nat] :
( ord_less_eq_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A2 )
@ ^ [X: nat] : ( member_nat @ X @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_417_card__image__le,axiom,
! [A: set_a,F: a > sum_sum_a_nat] :
( ( finite_finite_a @ A )
=> ( ord_less_eq_nat @ ( finite6080979521523705895_a_nat @ ( image_7873763678140191238_a_nat @ F @ A ) ) @ ( finite_card_a @ A ) ) ) ).
% card_image_le
thf(fact_418_card__image__le,axiom,
! [A: set_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat] :
( ( finite502105017643426984_a_nat @ A )
=> ( ord_less_eq_nat @ ( finite6080979521523705895_a_nat @ ( image_7142520692256960453_a_nat @ F @ A ) ) @ ( finite6080979521523705895_a_nat @ A ) ) ) ).
% card_image_le
thf(fact_419_card__image__le,axiom,
! [A: set_Sum_sum_a_nat,F: sum_sum_a_nat > nat] :
( ( finite502105017643426984_a_nat @ A )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_2473878607534554506at_nat @ F @ A ) ) @ ( finite6080979521523705895_a_nat @ A ) ) ) ).
% card_image_le
thf(fact_420_card__image__le,axiom,
! [A: set_nat,F: nat > sum_sum_a_nat] :
( ( finite_finite_nat @ A )
=> ( ord_less_eq_nat @ ( finite6080979521523705895_a_nat @ ( image_7293268710728258664_a_nat @ F @ A ) ) @ ( finite_card_nat @ A ) ) ) ).
% card_image_le
thf(fact_421_card__image__le,axiom,
! [A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_nat_nat @ F @ A ) ) @ ( finite_card_nat @ A ) ) ) ).
% card_image_le
thf(fact_422_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_Sum_sum_a_nat,C2: nat] :
( ! [G3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ G3 @ F2 )
=> ( ( finite502105017643426984_a_nat @ G3 )
=> ( ord_less_eq_nat @ ( finite6080979521523705895_a_nat @ G3 ) @ C2 ) ) )
=> ( ( finite502105017643426984_a_nat @ F2 )
& ( ord_less_eq_nat @ ( finite6080979521523705895_a_nat @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_423_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_nat,C2: nat] :
( ! [G3: set_nat] :
( ( ord_less_eq_set_nat @ G3 @ F2 )
=> ( ( finite_finite_nat @ G3 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G3 ) @ C2 ) ) )
=> ( ( finite_finite_nat @ F2 )
& ( ord_less_eq_nat @ ( finite_card_nat @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_424_obtain__subset__with__card__n,axiom,
! [N2: nat,S: set_Sum_sum_a_nat] :
( ( ord_less_eq_nat @ N2 @ ( finite6080979521523705895_a_nat @ S ) )
=> ~ ! [T3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ T3 @ S )
=> ( ( ( finite6080979521523705895_a_nat @ T3 )
= N2 )
=> ~ ( finite502105017643426984_a_nat @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_425_obtain__subset__with__card__n,axiom,
! [N2: nat,S: set_nat] :
( ( ord_less_eq_nat @ N2 @ ( finite_card_nat @ S ) )
=> ~ ! [T3: set_nat] :
( ( ord_less_eq_set_nat @ T3 @ S )
=> ( ( ( finite_card_nat @ T3 )
= N2 )
=> ~ ( finite_finite_nat @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_426_exists__subset__between,axiom,
! [A: set_Sum_sum_a_nat,N2: nat,C2: set_Sum_sum_a_nat] :
( ( ord_less_eq_nat @ ( finite6080979521523705895_a_nat @ A ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( finite6080979521523705895_a_nat @ C2 ) )
=> ( ( ord_le1325389633284124927_a_nat @ A @ C2 )
=> ( ( finite502105017643426984_a_nat @ C2 )
=> ? [B7: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ B7 )
& ( ord_le1325389633284124927_a_nat @ B7 @ C2 )
& ( ( finite6080979521523705895_a_nat @ B7 )
= N2 ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_427_exists__subset__between,axiom,
! [A: set_nat,N2: nat,C2: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( finite_card_nat @ C2 ) )
=> ( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( finite_finite_nat @ C2 )
=> ? [B7: set_nat] :
( ( ord_less_eq_set_nat @ A @ B7 )
& ( ord_less_eq_set_nat @ B7 @ C2 )
& ( ( finite_card_nat @ B7 )
= N2 ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_428_card__seteq,axiom,
! [B: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( finite502105017643426984_a_nat @ B )
=> ( ( ord_le1325389633284124927_a_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite6080979521523705895_a_nat @ B ) @ ( finite6080979521523705895_a_nat @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_429_card__seteq,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite_card_nat @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_430_card__mono,axiom,
! [B: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( finite502105017643426984_a_nat @ B )
=> ( ( ord_le1325389633284124927_a_nat @ A @ B )
=> ( ord_less_eq_nat @ ( finite6080979521523705895_a_nat @ A ) @ ( finite6080979521523705895_a_nat @ B ) ) ) ) ).
% card_mono
thf(fact_431_card__mono,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ).
% card_mono
thf(fact_432_card__le__sym__Diff,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ B ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B @ A ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_433_card__le__sym__Diff,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( finite502105017643426984_a_nat @ A )
=> ( ( finite502105017643426984_a_nat @ B )
=> ( ( ord_less_eq_nat @ ( finite6080979521523705895_a_nat @ A ) @ ( finite6080979521523705895_a_nat @ B ) )
=> ( ord_less_eq_nat @ ( finite6080979521523705895_a_nat @ ( minus_1134630996077396038_a_nat @ A @ B ) ) @ ( finite6080979521523705895_a_nat @ ( minus_1134630996077396038_a_nat @ B @ A ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_434_card__Un__le,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( sup_sup_set_nat @ A @ B ) ) @ ( plus_plus_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ).
% card_Un_le
thf(fact_435_card__Un__le,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] : ( ord_less_eq_nat @ ( finite6080979521523705895_a_nat @ ( sup_su6804446743777130803_a_nat @ A @ B ) ) @ ( plus_plus_nat @ ( finite6080979521523705895_a_nat @ A ) @ ( finite6080979521523705895_a_nat @ B ) ) ) ).
% card_Un_le
thf(fact_436_card__Diff__subset,axiom,
! [B: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( finite502105017643426984_a_nat @ B )
=> ( ( ord_le1325389633284124927_a_nat @ B @ A )
=> ( ( finite6080979521523705895_a_nat @ ( minus_1134630996077396038_a_nat @ A @ B ) )
= ( minus_minus_nat @ ( finite6080979521523705895_a_nat @ A ) @ ( finite6080979521523705895_a_nat @ B ) ) ) ) ) ).
% card_Diff_subset
thf(fact_437_card__Diff__subset,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A @ B ) )
= ( minus_minus_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_Diff_subset
thf(fact_438_card__Diff__subset__Int,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ ( inf_inf_set_nat @ A @ B ) )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A @ B ) )
= ( minus_minus_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ ( inf_inf_set_nat @ A @ B ) ) ) ) ) ).
% card_Diff_subset_Int
thf(fact_439_card__Diff__subset__Int,axiom,
! [A: set_a,B: set_a] :
( ( finite_finite_a @ ( inf_inf_set_a @ A @ B ) )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A @ B ) )
= ( minus_minus_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ ( inf_inf_set_a @ A @ B ) ) ) ) ) ).
% card_Diff_subset_Int
thf(fact_440_card__Diff__subset__Int,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( finite502105017643426984_a_nat @ ( inf_in7084830621192376909_a_nat @ A @ B ) )
=> ( ( finite6080979521523705895_a_nat @ ( minus_1134630996077396038_a_nat @ A @ B ) )
= ( minus_minus_nat @ ( finite6080979521523705895_a_nat @ A ) @ ( finite6080979521523705895_a_nat @ ( inf_in7084830621192376909_a_nat @ A @ B ) ) ) ) ) ).
% card_Diff_subset_Int
thf(fact_441_surj__card__le,axiom,
! [A: set_a,B: set_Sum_sum_a_nat,F: a > sum_sum_a_nat] :
( ( finite_finite_a @ A )
=> ( ( ord_le1325389633284124927_a_nat @ B @ ( image_7873763678140191238_a_nat @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite6080979521523705895_a_nat @ B ) @ ( finite_card_a @ A ) ) ) ) ).
% surj_card_le
thf(fact_442_surj__card__le,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat] :
( ( finite502105017643426984_a_nat @ A )
=> ( ( ord_le1325389633284124927_a_nat @ B @ ( image_7142520692256960453_a_nat @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite6080979521523705895_a_nat @ B ) @ ( finite6080979521523705895_a_nat @ A ) ) ) ) ).
% surj_card_le
thf(fact_443_surj__card__le,axiom,
! [A: set_nat,B: set_Sum_sum_a_nat,F: nat > sum_sum_a_nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_le1325389633284124927_a_nat @ B @ ( image_7293268710728258664_a_nat @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite6080979521523705895_a_nat @ B ) @ ( finite_card_nat @ A ) ) ) ) ).
% surj_card_le
thf(fact_444_surj__card__le,axiom,
! [A: set_Sum_sum_a_nat,B: set_nat,F: sum_sum_a_nat > nat] :
( ( finite502105017643426984_a_nat @ A )
=> ( ( ord_less_eq_set_nat @ B @ ( image_2473878607534554506at_nat @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite6080979521523705895_a_nat @ A ) ) ) ) ).
% surj_card_le
thf(fact_445_surj__card__le,axiom,
! [A: set_nat,B: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite_card_nat @ A ) ) ) ) ).
% surj_card_le
thf(fact_446_pigeonhole__infinite__rel,axiom,
! [A: set_a,B: set_Sum_sum_a_nat,R2: a > sum_sum_a_nat > $o] :
( ~ ( finite_finite_a @ A )
=> ( ( finite502105017643426984_a_nat @ B )
=> ( ! [X4: a] :
( ( member_a @ X4 @ A )
=> ? [Xa: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ Xa @ B )
& ( R2 @ X4 @ Xa ) ) )
=> ? [X4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X4 @ B )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A6: a] :
( ( member_a @ A6 @ A )
& ( R2 @ A6 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_447_pigeonhole__infinite__rel,axiom,
! [A: set_a,B: set_nat,R2: a > nat > $o] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X4: a] :
( ( member_a @ X4 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R2 @ X4 @ Xa ) ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A6: a] :
( ( member_a @ A6 @ A )
& ( R2 @ A6 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_448_pigeonhole__infinite__rel,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,R2: sum_sum_a_nat > sum_sum_a_nat > $o] :
( ~ ( finite502105017643426984_a_nat @ A )
=> ( ( finite502105017643426984_a_nat @ B )
=> ( ! [X4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X4 @ A )
=> ? [Xa: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ Xa @ B )
& ( R2 @ X4 @ Xa ) ) )
=> ? [X4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X4 @ B )
& ~ ( finite502105017643426984_a_nat
@ ( collec7073057861543223018_a_nat
@ ^ [A6: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ A6 @ A )
& ( R2 @ A6 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_449_pigeonhole__infinite__rel,axiom,
! [A: set_Sum_sum_a_nat,B: set_nat,R2: sum_sum_a_nat > nat > $o] :
( ~ ( finite502105017643426984_a_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X4 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R2 @ X4 @ Xa ) ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B )
& ~ ( finite502105017643426984_a_nat
@ ( collec7073057861543223018_a_nat
@ ^ [A6: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ A6 @ A )
& ( R2 @ A6 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_450_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B: set_Sum_sum_a_nat,R2: nat > sum_sum_a_nat > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite502105017643426984_a_nat @ B )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ? [Xa: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ Xa @ B )
& ( R2 @ X4 @ Xa ) ) )
=> ? [X4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X4 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A6: nat] :
( ( member_nat @ A6 @ A )
& ( R2 @ A6 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_451_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B: set_nat,R2: nat > nat > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R2 @ X4 @ Xa ) ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A6: nat] :
( ( member_nat @ A6 @ A )
& ( R2 @ A6 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_452_not__finite__existsD,axiom,
! [P: a > $o] :
( ~ ( finite_finite_a @ ( collect_a @ P ) )
=> ? [X_1: a] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_453_not__finite__existsD,axiom,
! [P: sum_sum_a_nat > $o] :
( ~ ( finite502105017643426984_a_nat @ ( collec7073057861543223018_a_nat @ P ) )
=> ? [X_1: sum_sum_a_nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_454_not__finite__existsD,axiom,
! [P: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
=> ? [X_1: nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_455_finite__has__minimal2,axiom,
! [A: set_set_nat,A3: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( member_set_nat @ A3 @ A )
=> ? [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
& ( ord_less_eq_set_nat @ X4 @ A3 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_456_finite__has__minimal2,axiom,
! [A: set_nat,A3: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A3 @ A )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ord_less_eq_nat @ X4 @ A3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_457_finite__has__maximal2,axiom,
! [A: set_set_nat,A3: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( member_set_nat @ A3 @ A )
=> ? [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
& ( ord_less_eq_set_nat @ A3 @ X4 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_458_finite__has__maximal2,axiom,
! [A: set_nat,A3: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A3 @ A )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ord_less_eq_nat @ A3 @ X4 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_459_all__subset__image,axiom,
! [F: a > sum_sum_a_nat,A: set_a,P: set_Sum_sum_a_nat > $o] :
( ( ! [B3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B3 @ ( image_7873763678140191238_a_nat @ F @ A ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A )
=> ( P @ ( image_7873763678140191238_a_nat @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_460_all__subset__image,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,A: set_Sum_sum_a_nat,P: set_Sum_sum_a_nat > $o] :
( ( ! [B3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B3 @ ( image_7142520692256960453_a_nat @ F @ A ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B3 @ A )
=> ( P @ ( image_7142520692256960453_a_nat @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_461_all__subset__image,axiom,
! [F: nat > sum_sum_a_nat,A: set_nat,P: set_Sum_sum_a_nat > $o] :
( ( ! [B3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B3 @ ( image_7293268710728258664_a_nat @ F @ A ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A )
=> ( P @ ( image_7293268710728258664_a_nat @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_462_all__subset__image,axiom,
! [F: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A )
=> ( P @ ( image_nat_nat @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_463_rev__finite__subset,axiom,
! [B: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( finite502105017643426984_a_nat @ B )
=> ( ( ord_le1325389633284124927_a_nat @ A @ B )
=> ( finite502105017643426984_a_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_464_rev__finite__subset,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_465_infinite__super,axiom,
! [S: set_Sum_sum_a_nat,T: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ S @ T )
=> ( ~ ( finite502105017643426984_a_nat @ S )
=> ~ ( finite502105017643426984_a_nat @ T ) ) ) ).
% infinite_super
thf(fact_466_infinite__super,axiom,
! [S: set_nat,T: set_nat] :
( ( ord_less_eq_set_nat @ S @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T ) ) ) ).
% infinite_super
thf(fact_467_finite__subset,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ B )
=> ( ( finite502105017643426984_a_nat @ B )
=> ( finite502105017643426984_a_nat @ A ) ) ) ).
% finite_subset
thf(fact_468_finite__subset,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_subset
thf(fact_469_finite__UnI,axiom,
! [F2: set_nat,G2: set_nat] :
( ( finite_finite_nat @ F2 )
=> ( ( finite_finite_nat @ G2 )
=> ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G2 ) ) ) ) ).
% finite_UnI
thf(fact_470_finite__UnI,axiom,
! [F2: set_Sum_sum_a_nat,G2: set_Sum_sum_a_nat] :
( ( finite502105017643426984_a_nat @ F2 )
=> ( ( finite502105017643426984_a_nat @ G2 )
=> ( finite502105017643426984_a_nat @ ( sup_su6804446743777130803_a_nat @ F2 @ G2 ) ) ) ) ).
% finite_UnI
thf(fact_471_Un__infinite,axiom,
! [S: set_nat,T: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T ) ) ) ).
% Un_infinite
thf(fact_472_Un__infinite,axiom,
! [S: set_Sum_sum_a_nat,T: set_Sum_sum_a_nat] :
( ~ ( finite502105017643426984_a_nat @ S )
=> ~ ( finite502105017643426984_a_nat @ ( sup_su6804446743777130803_a_nat @ S @ T ) ) ) ).
% Un_infinite
thf(fact_473_infinite__Un,axiom,
! [S: set_nat,T: set_nat] :
( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T ) ) )
= ( ~ ( finite_finite_nat @ S )
| ~ ( finite_finite_nat @ T ) ) ) ).
% infinite_Un
thf(fact_474_infinite__Un,axiom,
! [S: set_Sum_sum_a_nat,T: set_Sum_sum_a_nat] :
( ( ~ ( finite502105017643426984_a_nat @ ( sup_su6804446743777130803_a_nat @ S @ T ) ) )
= ( ~ ( finite502105017643426984_a_nat @ S )
| ~ ( finite502105017643426984_a_nat @ T ) ) ) ).
% infinite_Un
thf(fact_475_Diff__infinite__finite,axiom,
! [T: set_nat,S: set_nat] :
( ( finite_finite_nat @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_476_Diff__infinite__finite,axiom,
! [T: set_Sum_sum_a_nat,S: set_Sum_sum_a_nat] :
( ( finite502105017643426984_a_nat @ T )
=> ( ~ ( finite502105017643426984_a_nat @ S )
=> ~ ( finite502105017643426984_a_nat @ ( minus_1134630996077396038_a_nat @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_477_map__upd__Some__unfold,axiom,
! [M2: sum_sum_a_nat > option_nat,A3: sum_sum_a_nat,B2: nat,X3: sum_sum_a_nat,Y: nat] :
( ( ( fun_up4750343006489851234on_nat @ M2 @ A3 @ ( some_nat @ B2 ) @ X3 )
= ( some_nat @ Y ) )
= ( ( ( X3 = A3 )
& ( B2 = Y ) )
| ( ( X3 != A3 )
& ( ( M2 @ X3 )
= ( some_nat @ Y ) ) ) ) ) ).
% map_upd_Some_unfold
thf(fact_478_map__upd__triv,axiom,
! [T4: sum_sum_a_nat > option_nat,K: sum_sum_a_nat,X3: nat] :
( ( ( T4 @ K )
= ( some_nat @ X3 ) )
=> ( ( fun_up4750343006489851234on_nat @ T4 @ K @ ( some_nat @ X3 ) )
= T4 ) ) ).
% map_upd_triv
thf(fact_479_map__upd__eqD1,axiom,
! [M2: sum_sum_a_nat > option_nat,A3: sum_sum_a_nat,X3: nat,N2: sum_sum_a_nat > option_nat,Y: nat] :
( ( ( fun_up4750343006489851234on_nat @ M2 @ A3 @ ( some_nat @ X3 ) )
= ( fun_up4750343006489851234on_nat @ N2 @ A3 @ ( some_nat @ Y ) ) )
=> ( X3 = Y ) ) ).
% map_upd_eqD1
thf(fact_480_domD,axiom,
! [A3: sum_sum_a_nat,M2: sum_sum_a_nat > option_nat] :
( ( member_Sum_sum_a_nat @ A3 @ ( dom_Su2255998037560862461at_nat @ M2 ) )
=> ? [B6: nat] :
( ( M2 @ A3 )
= ( some_nat @ B6 ) ) ) ).
% domD
thf(fact_481_domD,axiom,
! [A3: a,M2: a > option_nat] :
( ( member_a @ A3 @ ( dom_a_nat @ M2 ) )
=> ? [B6: nat] :
( ( M2 @ A3 )
= ( some_nat @ B6 ) ) ) ).
% domD
thf(fact_482_domD,axiom,
! [A3: nat,M2: nat > option_nat] :
( ( member_nat @ A3 @ ( dom_nat_nat @ M2 ) )
=> ? [B6: nat] :
( ( M2 @ A3 )
= ( some_nat @ B6 ) ) ) ).
% domD
thf(fact_483_domI,axiom,
! [M2: sum_sum_a_nat > option_nat,A3: sum_sum_a_nat,B2: nat] :
( ( ( M2 @ A3 )
= ( some_nat @ B2 ) )
=> ( member_Sum_sum_a_nat @ A3 @ ( dom_Su2255998037560862461at_nat @ M2 ) ) ) ).
% domI
thf(fact_484_domI,axiom,
! [M2: a > option_nat,A3: a,B2: nat] :
( ( ( M2 @ A3 )
= ( some_nat @ B2 ) )
=> ( member_a @ A3 @ ( dom_a_nat @ M2 ) ) ) ).
% domI
thf(fact_485_domI,axiom,
! [M2: nat > option_nat,A3: nat,B2: nat] :
( ( ( M2 @ A3 )
= ( some_nat @ B2 ) )
=> ( member_nat @ A3 @ ( dom_nat_nat @ M2 ) ) ) ).
% domI
thf(fact_486_domIff,axiom,
! [A3: sum_sum_a_nat,M2: sum_sum_a_nat > option_nat] :
( ( member_Sum_sum_a_nat @ A3 @ ( dom_Su2255998037560862461at_nat @ M2 ) )
= ( ( M2 @ A3 )
!= none_nat ) ) ).
% domIff
thf(fact_487_domIff,axiom,
! [A3: a,M2: a > option_nat] :
( ( member_a @ A3 @ ( dom_a_nat @ M2 ) )
= ( ( M2 @ A3 )
!= none_nat ) ) ).
% domIff
thf(fact_488_domIff,axiom,
! [A3: nat,M2: nat > option_nat] :
( ( member_nat @ A3 @ ( dom_nat_nat @ M2 ) )
= ( ( M2 @ A3 )
!= none_nat ) ) ).
% domIff
thf(fact_489_ranI,axiom,
! [M2: sum_sum_a_nat > option_nat,A3: sum_sum_a_nat,B2: nat] :
( ( ( M2 @ A3 )
= ( some_nat @ B2 ) )
=> ( member_nat @ B2 @ ( ran_Su5179294584019872288at_nat @ M2 ) ) ) ).
% ranI
thf(fact_490_pigeonhole__infinite,axiom,
! [A: set_a,F: a > sum_sum_a_nat] :
( ~ ( finite_finite_a @ A )
=> ( ( finite502105017643426984_a_nat @ ( image_7873763678140191238_a_nat @ F @ A ) )
=> ? [X4: a] :
( ( member_a @ X4 @ A )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A6: a] :
( ( member_a @ A6 @ A )
& ( ( F @ A6 )
= ( F @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_491_pigeonhole__infinite,axiom,
! [A: set_a,F: a > nat] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_finite_nat @ ( image_a_nat @ F @ A ) )
=> ? [X4: a] :
( ( member_a @ X4 @ A )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A6: a] :
( ( member_a @ A6 @ A )
& ( ( F @ A6 )
= ( F @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_492_pigeonhole__infinite,axiom,
! [A: set_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat] :
( ~ ( finite502105017643426984_a_nat @ A )
=> ( ( finite502105017643426984_a_nat @ ( image_7142520692256960453_a_nat @ F @ A ) )
=> ? [X4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X4 @ A )
& ~ ( finite502105017643426984_a_nat
@ ( collec7073057861543223018_a_nat
@ ^ [A6: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ A6 @ A )
& ( ( F @ A6 )
= ( F @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_493_pigeonhole__infinite,axiom,
! [A: set_Sum_sum_a_nat,F: sum_sum_a_nat > nat] :
( ~ ( finite502105017643426984_a_nat @ A )
=> ( ( finite_finite_nat @ ( image_2473878607534554506at_nat @ F @ A ) )
=> ? [X4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X4 @ A )
& ~ ( finite502105017643426984_a_nat
@ ( collec7073057861543223018_a_nat
@ ^ [A6: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ A6 @ A )
& ( ( F @ A6 )
= ( F @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_494_pigeonhole__infinite,axiom,
! [A: set_nat,F: nat > sum_sum_a_nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite502105017643426984_a_nat @ ( image_7293268710728258664_a_nat @ F @ A ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A6: nat] :
( ( member_nat @ A6 @ A )
& ( ( F @ A6 )
= ( F @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_495_pigeonhole__infinite,axiom,
! [A: set_nat,F: nat > nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ ( image_nat_nat @ F @ A ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A6: nat] :
( ( member_nat @ A6 @ A )
& ( ( F @ A6 )
= ( F @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_496_finite__update__induct,axiom,
! [F: sum_sum_a_nat > option_nat,C: option_nat,P: ( sum_sum_a_nat > option_nat ) > $o] :
( ( finite502105017643426984_a_nat
@ ( collec7073057861543223018_a_nat
@ ^ [A6: sum_sum_a_nat] :
( ( F @ A6 )
!= C ) ) )
=> ( ( P
@ ^ [A6: sum_sum_a_nat] : C )
=> ( ! [A5: sum_sum_a_nat,B6: option_nat,F3: sum_sum_a_nat > option_nat] :
( ( finite502105017643426984_a_nat
@ ( collec7073057861543223018_a_nat
@ ^ [C4: sum_sum_a_nat] :
( ( F3 @ C4 )
!= C ) ) )
=> ( ( ( F3 @ A5 )
= C )
=> ( ( B6 != C )
=> ( ( P @ F3 )
=> ( P @ ( fun_up4750343006489851234on_nat @ F3 @ A5 @ B6 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ).
% finite_update_induct
thf(fact_497_dom__def,axiom,
( dom_nat_nat
= ( ^ [M4: nat > option_nat] :
( collect_nat
@ ^ [A6: nat] :
( ( M4 @ A6 )
!= none_nat ) ) ) ) ).
% dom_def
thf(fact_498_dom__def,axiom,
( dom_Su2255998037560862461at_nat
= ( ^ [M4: sum_sum_a_nat > option_nat] :
( collec7073057861543223018_a_nat
@ ^ [A6: sum_sum_a_nat] :
( ( M4 @ A6 )
!= none_nat ) ) ) ) ).
% dom_def
thf(fact_499_dom__def,axiom,
( dom_a_nat
= ( ^ [M4: a > option_nat] :
( collect_a
@ ^ [A6: a] :
( ( M4 @ A6 )
!= none_nat ) ) ) ) ).
% dom_def
thf(fact_500_add__le__imp__le__diff,axiom,
! [I2: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N2 )
=> ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ N2 @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_501_add__le__add__imp__diff__le,axiom,
! [I2: nat,K: nat,N2: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N2 @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_502_all__finite__subset__image,axiom,
! [F: a > sum_sum_a_nat,A: set_a,P: set_Sum_sum_a_nat > $o] :
( ( ! [B3: set_Sum_sum_a_nat] :
( ( ( finite502105017643426984_a_nat @ B3 )
& ( ord_le1325389633284124927_a_nat @ B3 @ ( image_7873763678140191238_a_nat @ F @ A ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_a] :
( ( ( finite_finite_a @ B3 )
& ( ord_less_eq_set_a @ B3 @ A ) )
=> ( P @ ( image_7873763678140191238_a_nat @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_503_all__finite__subset__image,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,A: set_Sum_sum_a_nat,P: set_Sum_sum_a_nat > $o] :
( ( ! [B3: set_Sum_sum_a_nat] :
( ( ( finite502105017643426984_a_nat @ B3 )
& ( ord_le1325389633284124927_a_nat @ B3 @ ( image_7142520692256960453_a_nat @ F @ A ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_Sum_sum_a_nat] :
( ( ( finite502105017643426984_a_nat @ B3 )
& ( ord_le1325389633284124927_a_nat @ B3 @ A ) )
=> ( P @ ( image_7142520692256960453_a_nat @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_504_all__finite__subset__image,axiom,
! [F: nat > sum_sum_a_nat,A: set_nat,P: set_Sum_sum_a_nat > $o] :
( ( ! [B3: set_Sum_sum_a_nat] :
( ( ( finite502105017643426984_a_nat @ B3 )
& ( ord_le1325389633284124927_a_nat @ B3 @ ( image_7293268710728258664_a_nat @ F @ A ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A ) )
=> ( P @ ( image_7293268710728258664_a_nat @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_505_all__finite__subset__image,axiom,
! [F: sum_sum_a_nat > nat,A: set_Sum_sum_a_nat,P: set_nat > $o] :
( ( ! [B3: set_nat] :
( ( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ ( image_2473878607534554506at_nat @ F @ A ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_Sum_sum_a_nat] :
( ( ( finite502105017643426984_a_nat @ B3 )
& ( ord_le1325389633284124927_a_nat @ B3 @ A ) )
=> ( P @ ( image_2473878607534554506at_nat @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_506_all__finite__subset__image,axiom,
! [F: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ! [B3: set_nat] :
( ( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A ) )
=> ( P @ ( image_nat_nat @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_507_ex__finite__subset__image,axiom,
! [F: a > sum_sum_a_nat,A: set_a,P: set_Sum_sum_a_nat > $o] :
( ( ? [B3: set_Sum_sum_a_nat] :
( ( finite502105017643426984_a_nat @ B3 )
& ( ord_le1325389633284124927_a_nat @ B3 @ ( image_7873763678140191238_a_nat @ F @ A ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_a] :
( ( finite_finite_a @ B3 )
& ( ord_less_eq_set_a @ B3 @ A )
& ( P @ ( image_7873763678140191238_a_nat @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_508_ex__finite__subset__image,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,A: set_Sum_sum_a_nat,P: set_Sum_sum_a_nat > $o] :
( ( ? [B3: set_Sum_sum_a_nat] :
( ( finite502105017643426984_a_nat @ B3 )
& ( ord_le1325389633284124927_a_nat @ B3 @ ( image_7142520692256960453_a_nat @ F @ A ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_Sum_sum_a_nat] :
( ( finite502105017643426984_a_nat @ B3 )
& ( ord_le1325389633284124927_a_nat @ B3 @ A )
& ( P @ ( image_7142520692256960453_a_nat @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_509_ex__finite__subset__image,axiom,
! [F: nat > sum_sum_a_nat,A: set_nat,P: set_Sum_sum_a_nat > $o] :
( ( ? [B3: set_Sum_sum_a_nat] :
( ( finite502105017643426984_a_nat @ B3 )
& ( ord_le1325389633284124927_a_nat @ B3 @ ( image_7293268710728258664_a_nat @ F @ A ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_nat] :
( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A )
& ( P @ ( image_7293268710728258664_a_nat @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_510_ex__finite__subset__image,axiom,
! [F: sum_sum_a_nat > nat,A: set_Sum_sum_a_nat,P: set_nat > $o] :
( ( ? [B3: set_nat] :
( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ ( image_2473878607534554506at_nat @ F @ A ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_Sum_sum_a_nat] :
( ( finite502105017643426984_a_nat @ B3 )
& ( ord_le1325389633284124927_a_nat @ B3 @ A )
& ( P @ ( image_2473878607534554506at_nat @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_511_ex__finite__subset__image,axiom,
! [F: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ? [B3: set_nat] :
( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_nat] :
( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A )
& ( P @ ( image_nat_nat @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_512_finite__subset__image,axiom,
! [B: set_Sum_sum_a_nat,F: a > sum_sum_a_nat,A: set_a] :
( ( finite502105017643426984_a_nat @ B )
=> ( ( ord_le1325389633284124927_a_nat @ B @ ( image_7873763678140191238_a_nat @ F @ A ) )
=> ? [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A )
& ( finite_finite_a @ C3 )
& ( B
= ( image_7873763678140191238_a_nat @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_513_finite__subset__image,axiom,
! [B: set_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( finite502105017643426984_a_nat @ B )
=> ( ( ord_le1325389633284124927_a_nat @ B @ ( image_7142520692256960453_a_nat @ F @ A ) )
=> ? [C3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ C3 @ A )
& ( finite502105017643426984_a_nat @ C3 )
& ( B
= ( image_7142520692256960453_a_nat @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_514_finite__subset__image,axiom,
! [B: set_Sum_sum_a_nat,F: nat > sum_sum_a_nat,A: set_nat] :
( ( finite502105017643426984_a_nat @ B )
=> ( ( ord_le1325389633284124927_a_nat @ B @ ( image_7293268710728258664_a_nat @ F @ A ) )
=> ? [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
& ( finite_finite_nat @ C3 )
& ( B
= ( image_7293268710728258664_a_nat @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_515_finite__subset__image,axiom,
! [B: set_nat,F: sum_sum_a_nat > nat,A: set_Sum_sum_a_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ ( image_2473878607534554506at_nat @ F @ A ) )
=> ? [C3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ C3 @ A )
& ( finite502105017643426984_a_nat @ C3 )
& ( B
= ( image_2473878607534554506at_nat @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_516_finite__subset__image,axiom,
! [B: set_nat,F: nat > nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
=> ? [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
& ( finite_finite_nat @ C3 )
& ( B
= ( image_nat_nat @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_517_finite__surj,axiom,
! [A: set_a,B: set_Sum_sum_a_nat,F: a > sum_sum_a_nat] :
( ( finite_finite_a @ A )
=> ( ( ord_le1325389633284124927_a_nat @ B @ ( image_7873763678140191238_a_nat @ F @ A ) )
=> ( finite502105017643426984_a_nat @ B ) ) ) ).
% finite_surj
thf(fact_518_finite__surj,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat] :
( ( finite502105017643426984_a_nat @ A )
=> ( ( ord_le1325389633284124927_a_nat @ B @ ( image_7142520692256960453_a_nat @ F @ A ) )
=> ( finite502105017643426984_a_nat @ B ) ) ) ).
% finite_surj
thf(fact_519_finite__surj,axiom,
! [A: set_nat,B: set_Sum_sum_a_nat,F: nat > sum_sum_a_nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_le1325389633284124927_a_nat @ B @ ( image_7293268710728258664_a_nat @ F @ A ) )
=> ( finite502105017643426984_a_nat @ B ) ) ) ).
% finite_surj
thf(fact_520_finite__surj,axiom,
! [A: set_Sum_sum_a_nat,B: set_nat,F: sum_sum_a_nat > nat] :
( ( finite502105017643426984_a_nat @ A )
=> ( ( ord_less_eq_set_nat @ B @ ( image_2473878607534554506at_nat @ F @ A ) )
=> ( finite_finite_nat @ B ) ) ) ).
% finite_surj
thf(fact_521_finite__surj,axiom,
! [A: set_nat,B: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
=> ( finite_finite_nat @ B ) ) ) ).
% finite_surj
thf(fact_522_infinite__arbitrarily__large,axiom,
! [A: set_Sum_sum_a_nat,N2: nat] :
( ~ ( finite502105017643426984_a_nat @ A )
=> ? [B7: set_Sum_sum_a_nat] :
( ( finite502105017643426984_a_nat @ B7 )
& ( ( finite6080979521523705895_a_nat @ B7 )
= N2 )
& ( ord_le1325389633284124927_a_nat @ B7 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_523_infinite__arbitrarily__large,axiom,
! [A: set_nat,N2: nat] :
( ~ ( finite_finite_nat @ A )
=> ? [B7: set_nat] :
( ( finite_finite_nat @ B7 )
& ( ( finite_card_nat @ B7 )
= N2 )
& ( ord_less_eq_set_nat @ B7 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_524_card__subset__eq,axiom,
! [B: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( finite502105017643426984_a_nat @ B )
=> ( ( ord_le1325389633284124927_a_nat @ A @ B )
=> ( ( ( finite6080979521523705895_a_nat @ A )
= ( finite6080979521523705895_a_nat @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_525_card__subset__eq,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( finite_card_nat @ A )
= ( finite_card_nat @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_526_map__upd__nonempty,axiom,
! [T4: sum_sum_a_nat > option_nat,K: sum_sum_a_nat,X3: nat] :
( ( fun_up4750343006489851234on_nat @ T4 @ K @ ( some_nat @ X3 ) )
!= ( ^ [X: sum_sum_a_nat] : none_nat ) ) ).
% map_upd_nonempty
thf(fact_527_finite__ran,axiom,
! [P2: sum_sum_a_nat > option_Sum_sum_a_nat] :
( ( finite502105017643426984_a_nat @ ( dom_Su2558219634521952658_a_nat @ P2 ) )
=> ( finite502105017643426984_a_nat @ ( ran_Su6606007654835531183_a_nat @ P2 ) ) ) ).
% finite_ran
thf(fact_528_finite__ran,axiom,
! [P2: sum_sum_a_nat > option_nat] :
( ( finite502105017643426984_a_nat @ ( dom_Su2255998037560862461at_nat @ P2 ) )
=> ( finite_finite_nat @ ( ran_Su5179294584019872288at_nat @ P2 ) ) ) ).
% finite_ran
thf(fact_529_finite__ran,axiom,
! [P2: nat > option_Sum_sum_a_nat] :
( ( finite_finite_nat @ ( dom_na7075388140754566619_a_nat @ P2 ) )
=> ( finite502105017643426984_a_nat @ ( ran_na775312650358800638_a_nat @ P2 ) ) ) ).
% finite_ran
thf(fact_530_finite__ran,axiom,
! [P2: nat > option_nat] :
( ( finite_finite_nat @ ( dom_nat_nat @ P2 ) )
=> ( finite_finite_nat @ ( ran_nat_nat @ P2 ) ) ) ).
% finite_ran
thf(fact_531_dom__if,axiom,
! [P: sum_sum_a_nat > $o,F: sum_sum_a_nat > option_nat,G: sum_sum_a_nat > option_nat] :
( ( dom_Su2255998037560862461at_nat
@ ^ [X: sum_sum_a_nat] : ( if_option_nat @ ( P @ X ) @ ( F @ X ) @ ( G @ X ) ) )
= ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ ( dom_Su2255998037560862461at_nat @ F ) @ ( collec7073057861543223018_a_nat @ P ) )
@ ( inf_in7084830621192376909_a_nat @ ( dom_Su2255998037560862461at_nat @ G )
@ ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
~ ( P @ X ) ) ) ) ) ).
% dom_if
thf(fact_532_finite__set__of__finite__maps,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( finite502105017643426984_a_nat @ A )
=> ( ( finite502105017643426984_a_nat @ B )
=> ( finite5498713997695131394_a_nat
@ ( collec6598233257319602752_a_nat
@ ^ [M4: sum_sum_a_nat > option_Sum_sum_a_nat] :
( ( ( dom_Su2558219634521952658_a_nat @ M4 )
= A )
& ( ord_le1325389633284124927_a_nat @ ( ran_Su6606007654835531183_a_nat @ M4 ) @ B ) ) ) ) ) ) ).
% finite_set_of_finite_maps
thf(fact_533_finite__set__of__finite__maps,axiom,
! [A: set_nat,B: set_Sum_sum_a_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite502105017643426984_a_nat @ B )
=> ( finite1751948946651059029_a_nat
@ ( collec7363285504596670359_a_nat
@ ^ [M4: nat > option_Sum_sum_a_nat] :
( ( ( dom_na7075388140754566619_a_nat @ M4 )
= A )
& ( ord_le1325389633284124927_a_nat @ ( ran_na775312650358800638_a_nat @ M4 ) @ B ) ) ) ) ) ) ).
% finite_set_of_finite_maps
thf(fact_534_finite__set__of__finite__maps,axiom,
! [A: set_Sum_sum_a_nat,B: set_nat] :
( ( finite502105017643426984_a_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( finite952674058959131833on_nat
@ ( collec6564010616904743163on_nat
@ ^ [M4: sum_sum_a_nat > option_nat] :
( ( ( dom_Su2255998037560862461at_nat @ M4 )
= A )
& ( ord_less_eq_set_nat @ ( ran_Su5179294584019872288at_nat @ M4 ) @ B ) ) ) ) ) ) ).
% finite_set_of_finite_maps
thf(fact_535_finite__set__of__finite__maps,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( finite2741051152680492326on_nat
@ ( collec6270343374860597604on_nat
@ ^ [M4: nat > option_nat] :
( ( ( dom_nat_nat @ M4 )
= A )
& ( ord_less_eq_set_nat @ ( ran_nat_nat @ M4 ) @ B ) ) ) ) ) ) ).
% finite_set_of_finite_maps
thf(fact_536_inf__sup__absorb,axiom,
! [X3: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ X3 @ ( sup_sup_set_nat @ X3 @ Y ) )
= X3 ) ).
% inf_sup_absorb
thf(fact_537_inf__sup__absorb,axiom,
! [X3: set_a,Y: set_a] :
( ( inf_inf_set_a @ X3 @ ( sup_sup_set_a @ X3 @ Y ) )
= X3 ) ).
% inf_sup_absorb
thf(fact_538_inf__sup__absorb,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ X3 @ ( sup_su6804446743777130803_a_nat @ X3 @ Y ) )
= X3 ) ).
% inf_sup_absorb
thf(fact_539_sup__inf__absorb,axiom,
! [X3: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( inf_inf_set_nat @ X3 @ Y ) )
= X3 ) ).
% sup_inf_absorb
thf(fact_540_sup__inf__absorb,axiom,
! [X3: set_a,Y: set_a] :
( ( sup_sup_set_a @ X3 @ ( inf_inf_set_a @ X3 @ Y ) )
= X3 ) ).
% sup_inf_absorb
thf(fact_541_sup__inf__absorb,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ X3 @ ( inf_in7084830621192376909_a_nat @ X3 @ Y ) )
= X3 ) ).
% sup_inf_absorb
thf(fact_542_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I2 )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_543_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I2 @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_544_add__Suc__right,axiom,
! [M2: nat,N2: nat] :
( ( plus_plus_nat @ M2 @ ( suc @ N2 ) )
= ( suc @ ( plus_plus_nat @ M2 @ N2 ) ) ) ).
% add_Suc_right
thf(fact_545_List_Ofinite__set,axiom,
! [Xs: list_Sum_sum_a_nat] : ( finite502105017643426984_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) ) ).
% List.finite_set
thf(fact_546_List_Ofinite__set,axiom,
! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).
% List.finite_set
thf(fact_547_not__Some__eq,axiom,
! [X3: option_nat] :
( ( ! [Y4: nat] :
( X3
!= ( some_nat @ Y4 ) ) )
= ( X3 = none_nat ) ) ).
% not_Some_eq
thf(fact_548_not__None__eq,axiom,
! [X3: option_nat] :
( ( X3 != none_nat )
= ( ? [Y4: nat] :
( X3
= ( some_nat @ Y4 ) ) ) ) ).
% not_None_eq
thf(fact_549_le__inf__iff,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X3 @ ( inf_in7084830621192376909_a_nat @ Y @ Z2 ) )
= ( ( ord_le1325389633284124927_a_nat @ X3 @ Y )
& ( ord_le1325389633284124927_a_nat @ X3 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_550_le__inf__iff,axiom,
! [X3: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X3 @ ( inf_inf_set_a @ Y @ Z2 ) )
= ( ( ord_less_eq_set_a @ X3 @ Y )
& ( ord_less_eq_set_a @ X3 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_551_le__inf__iff,axiom,
! [X3: set_nat,Y: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ ( inf_inf_set_nat @ Y @ Z2 ) )
= ( ( ord_less_eq_set_nat @ X3 @ Y )
& ( ord_less_eq_set_nat @ X3 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_552_le__inf__iff,axiom,
! [X3: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ Y @ Z2 ) )
= ( ( ord_less_eq_nat @ X3 @ Y )
& ( ord_less_eq_nat @ X3 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_553_list_Oinject,axiom,
! [X21: sum_sum_a_nat,X22: list_Sum_sum_a_nat,Y21: sum_sum_a_nat,Y22: list_Sum_sum_a_nat] :
( ( ( cons_Sum_sum_a_nat @ X21 @ X22 )
= ( cons_Sum_sum_a_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_554_option_Oinject,axiom,
! [X2: nat,Y23: nat] :
( ( ( some_nat @ X2 )
= ( some_nat @ Y23 ) )
= ( X2 = Y23 ) ) ).
% option.inject
thf(fact_555_nat_Oinject,axiom,
! [X2: nat,Y23: nat] :
( ( ( suc @ X2 )
= ( suc @ Y23 ) )
= ( X2 = Y23 ) ) ).
% nat.inject
thf(fact_556_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_557_sup_Oright__idem,axiom,
! [A3: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ ( sup_su6804446743777130803_a_nat @ A3 @ B2 ) @ B2 )
= ( sup_su6804446743777130803_a_nat @ A3 @ B2 ) ) ).
% sup.right_idem
thf(fact_558_sup__left__idem,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ X3 @ ( sup_su6804446743777130803_a_nat @ X3 @ Y ) )
= ( sup_su6804446743777130803_a_nat @ X3 @ Y ) ) ).
% sup_left_idem
thf(fact_559_sup_Oleft__idem,axiom,
! [A3: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A3 @ ( sup_su6804446743777130803_a_nat @ A3 @ B2 ) )
= ( sup_su6804446743777130803_a_nat @ A3 @ B2 ) ) ).
% sup.left_idem
thf(fact_560_sup__idem,axiom,
! [X3: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ X3 @ X3 )
= X3 ) ).
% sup_idem
thf(fact_561_sup_Oidem,axiom,
! [A3: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A3 @ A3 )
= A3 ) ).
% sup.idem
thf(fact_562_inf__right__idem,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ ( inf_in7084830621192376909_a_nat @ X3 @ Y ) @ Y )
= ( inf_in7084830621192376909_a_nat @ X3 @ Y ) ) ).
% inf_right_idem
thf(fact_563_inf__right__idem,axiom,
! [X3: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X3 @ Y ) @ Y )
= ( inf_inf_set_nat @ X3 @ Y ) ) ).
% inf_right_idem
thf(fact_564_inf__right__idem,axiom,
! [X3: set_a,Y: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X3 @ Y ) @ Y )
= ( inf_inf_set_a @ X3 @ Y ) ) ).
% inf_right_idem
thf(fact_565_inf_Oright__idem,axiom,
! [A3: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ ( inf_in7084830621192376909_a_nat @ A3 @ B2 ) @ B2 )
= ( inf_in7084830621192376909_a_nat @ A3 @ B2 ) ) ).
% inf.right_idem
thf(fact_566_inf_Oright__idem,axiom,
! [A3: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A3 @ B2 ) @ B2 )
= ( inf_inf_set_nat @ A3 @ B2 ) ) ).
% inf.right_idem
thf(fact_567_inf_Oright__idem,axiom,
! [A3: set_a,B2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A3 @ B2 ) @ B2 )
= ( inf_inf_set_a @ A3 @ B2 ) ) ).
% inf.right_idem
thf(fact_568_inf__left__idem,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ X3 @ ( inf_in7084830621192376909_a_nat @ X3 @ Y ) )
= ( inf_in7084830621192376909_a_nat @ X3 @ Y ) ) ).
% inf_left_idem
thf(fact_569_inf__left__idem,axiom,
! [X3: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ X3 @ ( inf_inf_set_nat @ X3 @ Y ) )
= ( inf_inf_set_nat @ X3 @ Y ) ) ).
% inf_left_idem
thf(fact_570_inf__left__idem,axiom,
! [X3: set_a,Y: set_a] :
( ( inf_inf_set_a @ X3 @ ( inf_inf_set_a @ X3 @ Y ) )
= ( inf_inf_set_a @ X3 @ Y ) ) ).
% inf_left_idem
thf(fact_571_inf_Oleft__idem,axiom,
! [A3: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ A3 @ ( inf_in7084830621192376909_a_nat @ A3 @ B2 ) )
= ( inf_in7084830621192376909_a_nat @ A3 @ B2 ) ) ).
% inf.left_idem
thf(fact_572_inf_Oleft__idem,axiom,
! [A3: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ A3 @ ( inf_inf_set_nat @ A3 @ B2 ) )
= ( inf_inf_set_nat @ A3 @ B2 ) ) ).
% inf.left_idem
thf(fact_573_inf_Oleft__idem,axiom,
! [A3: set_a,B2: set_a] :
( ( inf_inf_set_a @ A3 @ ( inf_inf_set_a @ A3 @ B2 ) )
= ( inf_inf_set_a @ A3 @ B2 ) ) ).
% inf.left_idem
thf(fact_574_inf__idem,axiom,
! [X3: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ X3 @ X3 )
= X3 ) ).
% inf_idem
thf(fact_575_inf__idem,axiom,
! [X3: set_nat] :
( ( inf_inf_set_nat @ X3 @ X3 )
= X3 ) ).
% inf_idem
thf(fact_576_inf__idem,axiom,
! [X3: set_a] :
( ( inf_inf_set_a @ X3 @ X3 )
= X3 ) ).
% inf_idem
thf(fact_577_inf_Oidem,axiom,
! [A3: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ A3 @ A3 )
= A3 ) ).
% inf.idem
thf(fact_578_inf_Oidem,axiom,
! [A3: set_nat] :
( ( inf_inf_set_nat @ A3 @ A3 )
= A3 ) ).
% inf.idem
thf(fact_579_inf_Oidem,axiom,
! [A3: set_a] :
( ( inf_inf_set_a @ A3 @ A3 )
= A3 ) ).
% inf.idem
thf(fact_580_sup_Obounded__iff,axiom,
! [B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat,A3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ B2 @ C ) @ A3 )
= ( ( ord_le1325389633284124927_a_nat @ B2 @ A3 )
& ( ord_le1325389633284124927_a_nat @ C @ A3 ) ) ) ).
% sup.bounded_iff
thf(fact_581_sup_Obounded__iff,axiom,
! [B2: set_nat,C: set_nat,A3: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C ) @ A3 )
= ( ( ord_less_eq_set_nat @ B2 @ A3 )
& ( ord_less_eq_set_nat @ C @ A3 ) ) ) ).
% sup.bounded_iff
thf(fact_582_sup_Obounded__iff,axiom,
! [B2: nat,C: nat,A3: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A3 )
= ( ( ord_less_eq_nat @ B2 @ A3 )
& ( ord_less_eq_nat @ C @ A3 ) ) ) ).
% sup.bounded_iff
thf(fact_583_le__sup__iff,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ X3 @ Y ) @ Z2 )
= ( ( ord_le1325389633284124927_a_nat @ X3 @ Z2 )
& ( ord_le1325389633284124927_a_nat @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_584_le__sup__iff,axiom,
! [X3: set_nat,Y: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X3 @ Y ) @ Z2 )
= ( ( ord_less_eq_set_nat @ X3 @ Z2 )
& ( ord_less_eq_set_nat @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_585_le__sup__iff,axiom,
! [X3: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X3 @ Y ) @ Z2 )
= ( ( ord_less_eq_nat @ X3 @ Z2 )
& ( ord_less_eq_nat @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_586_inf_Obounded__iff,axiom,
! [A3: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A3 @ ( inf_in7084830621192376909_a_nat @ B2 @ C ) )
= ( ( ord_le1325389633284124927_a_nat @ A3 @ B2 )
& ( ord_le1325389633284124927_a_nat @ A3 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_587_inf_Obounded__iff,axiom,
! [A3: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A3 @ ( inf_inf_set_a @ B2 @ C ) )
= ( ( ord_less_eq_set_a @ A3 @ B2 )
& ( ord_less_eq_set_a @ A3 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_588_inf_Obounded__iff,axiom,
! [A3: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ ( inf_inf_set_nat @ B2 @ C ) )
= ( ( ord_less_eq_set_nat @ A3 @ B2 )
& ( ord_less_eq_set_nat @ A3 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_589_inf_Obounded__iff,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ ( inf_inf_nat @ B2 @ C ) )
= ( ( ord_less_eq_nat @ A3 @ B2 )
& ( ord_less_eq_nat @ A3 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_590_Suc__le__mono,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( suc @ M2 ) )
= ( ord_less_eq_nat @ N2 @ M2 ) ) ).
% Suc_le_mono
thf(fact_591_diff__Suc__Suc,axiom,
! [M2: nat,N2: nat] :
( ( minus_minus_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ).
% diff_Suc_Suc
thf(fact_592_Suc__diff__diff,axiom,
! [M2: nat,N2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N2 ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ N2 ) @ K ) ) ).
% Suc_diff_diff
thf(fact_593_nat__add__left__cancel__le,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% nat_add_left_cancel_le
thf(fact_594_diff__diff__cancel,axiom,
! [I2: nat,N2: nat] :
( ( ord_less_eq_nat @ I2 @ N2 )
=> ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I2 ) )
= I2 ) ) ).
% diff_diff_cancel
thf(fact_595_diff__diff__left,axiom,
! [I2: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
= ( minus_minus_nat @ I2 @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_596_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N3: nat] : ( ord_less_eq_nat @ N3 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_597_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_598_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_599_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_600_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_601_le__trans,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I2 @ K ) ) ) ).
% le_trans
thf(fact_602_eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( M2 = N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% eq_imp_le
thf(fact_603_le__antisym,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M2 )
=> ( M2 = N2 ) ) ) ).
% le_antisym
thf(fact_604_eq__diff__iff,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ( minus_minus_nat @ M2 @ K )
= ( minus_minus_nat @ N2 @ K ) )
= ( M2 = N2 ) ) ) ) ).
% eq_diff_iff
thf(fact_605_le__diff__iff,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ) ) ).
% le_diff_iff
thf(fact_606_diff__commute,axiom,
! [I2: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K ) @ J ) ) ).
% diff_commute
thf(fact_607_Nat_Odiff__diff__eq,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_608_diff__le__mono,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).
% diff_le_mono
thf(fact_609_diff__le__self,axiom,
! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N2 ) @ M2 ) ).
% diff_le_self
thf(fact_610_le__diff__iff_H,axiom,
! [A3: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ C )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A3 ) @ ( minus_minus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).
% le_diff_iff'
thf(fact_611_diff__le__mono2,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_612_nat__le__linear,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
| ( ord_less_eq_nat @ N2 @ M2 ) ) ).
% nat_le_linear
thf(fact_613_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ B2 ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_614_finite__less__ub,axiom,
! [F: nat > nat,U: nat] :
( ! [N4: nat] : ( ord_less_eq_nat @ N4 @ ( F @ N4 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_615_bounded__Max__nat,axiom,
! [P: nat > $o,X3: nat,M: nat] :
( ( P @ X3 )
=> ( ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M ) )
=> ~ ! [M3: nat] :
( ( P @ M3 )
=> ~ ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_616_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M4: nat] :
! [X: nat] :
( ( member_nat @ X @ N5 )
=> ( ord_less_eq_nat @ X @ M4 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_617_sup__set__def,axiom,
( sup_sup_set_nat
= ( ^ [A2: set_nat,B3: set_nat] :
( collect_nat
@ ( sup_sup_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A2 )
@ ^ [X: nat] : ( member_nat @ X @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_618_sup__set__def,axiom,
( sup_sup_set_a
= ( ^ [A2: set_a,B3: set_a] :
( collect_a
@ ( sup_sup_a_o
@ ^ [X: a] : ( member_a @ X @ A2 )
@ ^ [X: a] : ( member_a @ X @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_619_sup__set__def,axiom,
( sup_su6804446743777130803_a_nat
= ( ^ [A2: set_Sum_sum_a_nat,B3: set_Sum_sum_a_nat] :
( collec7073057861543223018_a_nat
@ ( sup_su491480579010597738_nat_o
@ ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ A2 )
@ ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_620_inf__set__def,axiom,
( inf_in7084830621192376909_a_nat
= ( ^ [A2: set_Sum_sum_a_nat,B3: set_Sum_sum_a_nat] :
( collec7073057861543223018_a_nat
@ ( inf_in5242522483218605776_nat_o
@ ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ A2 )
@ ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_621_inf__set__def,axiom,
( inf_inf_set_nat
= ( ^ [A2: set_nat,B3: set_nat] :
( collect_nat
@ ( inf_inf_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A2 )
@ ^ [X: nat] : ( member_nat @ X @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_622_inf__set__def,axiom,
( inf_inf_set_a
= ( ^ [A2: set_a,B3: set_a] :
( collect_a
@ ( inf_inf_a_o
@ ^ [X: a] : ( member_a @ X @ A2 )
@ ^ [X: a] : ( member_a @ X @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_623_not__Cons__self2,axiom,
! [X3: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( cons_Sum_sum_a_nat @ X3 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_624_Suc__inject,axiom,
! [X3: nat,Y: nat] :
( ( ( suc @ X3 )
= ( suc @ Y ) )
=> ( X3 = Y ) ) ).
% Suc_inject
thf(fact_625_n__not__Suc__n,axiom,
! [N2: nat] :
( N2
!= ( suc @ N2 ) ) ).
% n_not_Suc_n
thf(fact_626_Suc__leD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% Suc_leD
thf(fact_627_le__SucE,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
=> ( ~ ( ord_less_eq_nat @ M2 @ N2 )
=> ( M2
= ( suc @ N2 ) ) ) ) ).
% le_SucE
thf(fact_628_le__SucI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) ) ) ).
% le_SucI
thf(fact_629_Suc__le__D,axiom,
! [N2: nat,M5: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ M5 )
=> ? [M3: nat] :
( M5
= ( suc @ M3 ) ) ) ).
% Suc_le_D
thf(fact_630_le__Suc__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
= ( ( ord_less_eq_nat @ M2 @ N2 )
| ( M2
= ( suc @ N2 ) ) ) ) ).
% le_Suc_eq
thf(fact_631_Suc__n__not__le__n,axiom,
! [N2: nat] :
~ ( ord_less_eq_nat @ ( suc @ N2 ) @ N2 ) ).
% Suc_n_not_le_n
thf(fact_632_not__less__eq__eq,axiom,
! [M2: nat,N2: nat] :
( ( ~ ( ord_less_eq_nat @ M2 @ N2 ) )
= ( ord_less_eq_nat @ ( suc @ N2 ) @ M2 ) ) ).
% not_less_eq_eq
thf(fact_633_full__nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N4: nat] :
( ! [M6: nat] :
( ( ord_less_eq_nat @ ( suc @ M6 ) @ N4 )
=> ( P @ M6 ) )
=> ( P @ N4 ) )
=> ( P @ N2 ) ) ).
% full_nat_induct
thf(fact_634_nat__induct__at__least,axiom,
! [M2: nat,N2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( P @ M2 )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_at_least
thf(fact_635_transitive__stepwise__le,axiom,
! [M2: nat,N2: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ! [X4: nat] : ( R2 @ X4 @ X4 )
=> ( ! [X4: nat,Y5: nat,Z3: nat] :
( ( R2 @ X4 @ Y5 )
=> ( ( R2 @ Y5 @ Z3 )
=> ( R2 @ X4 @ Z3 ) ) )
=> ( ! [N4: nat] : ( R2 @ N4 @ ( suc @ N4 ) )
=> ( R2 @ M2 @ N2 ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_636_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I2: nat] :
( ( P @ K )
=> ( ! [N4: nat] :
( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) )
=> ( P @ ( minus_minus_nat @ K @ I2 ) ) ) ) ).
% zero_induct_lemma
thf(fact_637_Suc__diff__le,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_eq_nat @ N2 @ M2 )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N2 )
= ( suc @ ( minus_minus_nat @ M2 @ N2 ) ) ) ) ).
% Suc_diff_le
thf(fact_638_sup__left__commute,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ X3 @ ( sup_su6804446743777130803_a_nat @ Y @ Z2 ) )
= ( sup_su6804446743777130803_a_nat @ Y @ ( sup_su6804446743777130803_a_nat @ X3 @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_639_sup_Oleft__commute,axiom,
! [B2: set_Sum_sum_a_nat,A3: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ B2 @ ( sup_su6804446743777130803_a_nat @ A3 @ C ) )
= ( sup_su6804446743777130803_a_nat @ A3 @ ( sup_su6804446743777130803_a_nat @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_640_sup__commute,axiom,
( sup_su6804446743777130803_a_nat
= ( ^ [X: set_Sum_sum_a_nat,Y4: set_Sum_sum_a_nat] : ( sup_su6804446743777130803_a_nat @ Y4 @ X ) ) ) ).
% sup_commute
thf(fact_641_sup_Ocommute,axiom,
( sup_su6804446743777130803_a_nat
= ( ^ [A6: set_Sum_sum_a_nat,B8: set_Sum_sum_a_nat] : ( sup_su6804446743777130803_a_nat @ B8 @ A6 ) ) ) ).
% sup.commute
thf(fact_642_sup__assoc,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ ( sup_su6804446743777130803_a_nat @ X3 @ Y ) @ Z2 )
= ( sup_su6804446743777130803_a_nat @ X3 @ ( sup_su6804446743777130803_a_nat @ Y @ Z2 ) ) ) ).
% sup_assoc
thf(fact_643_sup_Oassoc,axiom,
! [A3: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ ( sup_su6804446743777130803_a_nat @ A3 @ B2 ) @ C )
= ( sup_su6804446743777130803_a_nat @ A3 @ ( sup_su6804446743777130803_a_nat @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_644_inf__sup__aci_I5_J,axiom,
( sup_su6804446743777130803_a_nat
= ( ^ [X: set_Sum_sum_a_nat,Y4: set_Sum_sum_a_nat] : ( sup_su6804446743777130803_a_nat @ Y4 @ X ) ) ) ).
% inf_sup_aci(5)
thf(fact_645_inf__sup__aci_I6_J,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ ( sup_su6804446743777130803_a_nat @ X3 @ Y ) @ Z2 )
= ( sup_su6804446743777130803_a_nat @ X3 @ ( sup_su6804446743777130803_a_nat @ Y @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_646_inf__sup__aci_I7_J,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ X3 @ ( sup_su6804446743777130803_a_nat @ Y @ Z2 ) )
= ( sup_su6804446743777130803_a_nat @ Y @ ( sup_su6804446743777130803_a_nat @ X3 @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_647_inf__sup__aci_I8_J,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ X3 @ ( sup_su6804446743777130803_a_nat @ X3 @ Y ) )
= ( sup_su6804446743777130803_a_nat @ X3 @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_648_inf__left__commute,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z2: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ X3 @ ( inf_in7084830621192376909_a_nat @ Y @ Z2 ) )
= ( inf_in7084830621192376909_a_nat @ Y @ ( inf_in7084830621192376909_a_nat @ X3 @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_649_inf__left__commute,axiom,
! [X3: set_nat,Y: set_nat,Z2: set_nat] :
( ( inf_inf_set_nat @ X3 @ ( inf_inf_set_nat @ Y @ Z2 ) )
= ( inf_inf_set_nat @ Y @ ( inf_inf_set_nat @ X3 @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_650_inf__left__commute,axiom,
! [X3: set_a,Y: set_a,Z2: set_a] :
( ( inf_inf_set_a @ X3 @ ( inf_inf_set_a @ Y @ Z2 ) )
= ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X3 @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_651_inf_Oleft__commute,axiom,
! [B2: set_Sum_sum_a_nat,A3: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ B2 @ ( inf_in7084830621192376909_a_nat @ A3 @ C ) )
= ( inf_in7084830621192376909_a_nat @ A3 @ ( inf_in7084830621192376909_a_nat @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_652_inf_Oleft__commute,axiom,
! [B2: set_nat,A3: set_nat,C: set_nat] :
( ( inf_inf_set_nat @ B2 @ ( inf_inf_set_nat @ A3 @ C ) )
= ( inf_inf_set_nat @ A3 @ ( inf_inf_set_nat @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_653_inf_Oleft__commute,axiom,
! [B2: set_a,A3: set_a,C: set_a] :
( ( inf_inf_set_a @ B2 @ ( inf_inf_set_a @ A3 @ C ) )
= ( inf_inf_set_a @ A3 @ ( inf_inf_set_a @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_654_inf__commute,axiom,
( inf_in7084830621192376909_a_nat
= ( ^ [X: set_Sum_sum_a_nat,Y4: set_Sum_sum_a_nat] : ( inf_in7084830621192376909_a_nat @ Y4 @ X ) ) ) ).
% inf_commute
thf(fact_655_inf__commute,axiom,
( inf_inf_set_nat
= ( ^ [X: set_nat,Y4: set_nat] : ( inf_inf_set_nat @ Y4 @ X ) ) ) ).
% inf_commute
thf(fact_656_inf__commute,axiom,
( inf_inf_set_a
= ( ^ [X: set_a,Y4: set_a] : ( inf_inf_set_a @ Y4 @ X ) ) ) ).
% inf_commute
thf(fact_657_inf_Ocommute,axiom,
( inf_in7084830621192376909_a_nat
= ( ^ [A6: set_Sum_sum_a_nat,B8: set_Sum_sum_a_nat] : ( inf_in7084830621192376909_a_nat @ B8 @ A6 ) ) ) ).
% inf.commute
thf(fact_658_inf_Ocommute,axiom,
( inf_inf_set_nat
= ( ^ [A6: set_nat,B8: set_nat] : ( inf_inf_set_nat @ B8 @ A6 ) ) ) ).
% inf.commute
thf(fact_659_inf_Ocommute,axiom,
( inf_inf_set_a
= ( ^ [A6: set_a,B8: set_a] : ( inf_inf_set_a @ B8 @ A6 ) ) ) ).
% inf.commute
thf(fact_660_inf__assoc,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z2: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ ( inf_in7084830621192376909_a_nat @ X3 @ Y ) @ Z2 )
= ( inf_in7084830621192376909_a_nat @ X3 @ ( inf_in7084830621192376909_a_nat @ Y @ Z2 ) ) ) ).
% inf_assoc
thf(fact_661_inf__assoc,axiom,
! [X3: set_nat,Y: set_nat,Z2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X3 @ Y ) @ Z2 )
= ( inf_inf_set_nat @ X3 @ ( inf_inf_set_nat @ Y @ Z2 ) ) ) ).
% inf_assoc
thf(fact_662_inf__assoc,axiom,
! [X3: set_a,Y: set_a,Z2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X3 @ Y ) @ Z2 )
= ( inf_inf_set_a @ X3 @ ( inf_inf_set_a @ Y @ Z2 ) ) ) ).
% inf_assoc
thf(fact_663_inf_Oassoc,axiom,
! [A3: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ ( inf_in7084830621192376909_a_nat @ A3 @ B2 ) @ C )
= ( inf_in7084830621192376909_a_nat @ A3 @ ( inf_in7084830621192376909_a_nat @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_664_inf_Oassoc,axiom,
! [A3: set_nat,B2: set_nat,C: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A3 @ B2 ) @ C )
= ( inf_inf_set_nat @ A3 @ ( inf_inf_set_nat @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_665_inf_Oassoc,axiom,
! [A3: set_a,B2: set_a,C: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A3 @ B2 ) @ C )
= ( inf_inf_set_a @ A3 @ ( inf_inf_set_a @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_666_inf__sup__aci_I1_J,axiom,
( inf_in7084830621192376909_a_nat
= ( ^ [X: set_Sum_sum_a_nat,Y4: set_Sum_sum_a_nat] : ( inf_in7084830621192376909_a_nat @ Y4 @ X ) ) ) ).
% inf_sup_aci(1)
thf(fact_667_inf__sup__aci_I1_J,axiom,
( inf_inf_set_nat
= ( ^ [X: set_nat,Y4: set_nat] : ( inf_inf_set_nat @ Y4 @ X ) ) ) ).
% inf_sup_aci(1)
thf(fact_668_inf__sup__aci_I1_J,axiom,
( inf_inf_set_a
= ( ^ [X: set_a,Y4: set_a] : ( inf_inf_set_a @ Y4 @ X ) ) ) ).
% inf_sup_aci(1)
thf(fact_669_inf__sup__aci_I2_J,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z2: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ ( inf_in7084830621192376909_a_nat @ X3 @ Y ) @ Z2 )
= ( inf_in7084830621192376909_a_nat @ X3 @ ( inf_in7084830621192376909_a_nat @ Y @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_670_inf__sup__aci_I2_J,axiom,
! [X3: set_nat,Y: set_nat,Z2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X3 @ Y ) @ Z2 )
= ( inf_inf_set_nat @ X3 @ ( inf_inf_set_nat @ Y @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_671_inf__sup__aci_I2_J,axiom,
! [X3: set_a,Y: set_a,Z2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X3 @ Y ) @ Z2 )
= ( inf_inf_set_a @ X3 @ ( inf_inf_set_a @ Y @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_672_inf__sup__aci_I3_J,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z2: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ X3 @ ( inf_in7084830621192376909_a_nat @ Y @ Z2 ) )
= ( inf_in7084830621192376909_a_nat @ Y @ ( inf_in7084830621192376909_a_nat @ X3 @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_673_inf__sup__aci_I3_J,axiom,
! [X3: set_nat,Y: set_nat,Z2: set_nat] :
( ( inf_inf_set_nat @ X3 @ ( inf_inf_set_nat @ Y @ Z2 ) )
= ( inf_inf_set_nat @ Y @ ( inf_inf_set_nat @ X3 @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_674_inf__sup__aci_I3_J,axiom,
! [X3: set_a,Y: set_a,Z2: set_a] :
( ( inf_inf_set_a @ X3 @ ( inf_inf_set_a @ Y @ Z2 ) )
= ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X3 @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_675_inf__sup__aci_I4_J,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ X3 @ ( inf_in7084830621192376909_a_nat @ X3 @ Y ) )
= ( inf_in7084830621192376909_a_nat @ X3 @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_676_inf__sup__aci_I4_J,axiom,
! [X3: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ X3 @ ( inf_inf_set_nat @ X3 @ Y ) )
= ( inf_inf_set_nat @ X3 @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_677_inf__sup__aci_I4_J,axiom,
! [X3: set_a,Y: set_a] :
( ( inf_inf_set_a @ X3 @ ( inf_inf_set_a @ X3 @ Y ) )
= ( inf_inf_set_a @ X3 @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_678_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N3: nat] :
? [K4: nat] :
( N3
= ( plus_plus_nat @ M4 @ K4 ) ) ) ) ).
% nat_le_iff_add
thf(fact_679_trans__le__add2,axiom,
! [I2: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_le_add2
thf(fact_680_trans__le__add1,axiom,
! [I2: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_le_add1
thf(fact_681_add__le__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_682_add__le__mono,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_683_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N4: nat] :
( L
= ( plus_plus_nat @ K @ N4 ) ) ) ).
% le_Suc_ex
thf(fact_684_add__leD2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
=> ( ord_less_eq_nat @ K @ N2 ) ) ).
% add_leD2
thf(fact_685_add__leD1,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% add_leD1
thf(fact_686_le__add2,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M2 @ N2 ) ) ).
% le_add2
thf(fact_687_le__add1,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M2 ) ) ).
% le_add1
thf(fact_688_add__leE,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
=> ~ ( ( ord_less_eq_nat @ M2 @ N2 )
=> ~ ( ord_less_eq_nat @ K @ N2 ) ) ) ).
% add_leE
thf(fact_689_diff__add__inverse2,axiom,
! [M2: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N2 ) @ N2 )
= M2 ) ).
% diff_add_inverse2
thf(fact_690_diff__add__inverse,axiom,
! [N2: nat,M2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N2 @ M2 ) @ N2 )
= M2 ) ).
% diff_add_inverse
thf(fact_691_diff__cancel2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N2 @ K ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ).
% diff_cancel2
thf(fact_692_Nat_Odiff__cancel,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ).
% Nat.diff_cancel
thf(fact_693_Nat_Ole__imp__diff__is__add,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ( minus_minus_nat @ J @ I2 )
= K )
= ( J
= ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_694_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_695_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
= ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_696_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_697_le__diff__conv,axiom,
! [J: nat,K: nat,I2: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I2 @ K ) ) ) ).
% le_diff_conv
thf(fact_698_lift__Suc__mono__le,axiom,
! [F: nat > set_nat,N2: nat,N6: nat] :
( ! [N4: nat] : ( ord_less_eq_set_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N6 )
=> ( ord_less_eq_set_nat @ ( F @ N2 ) @ ( F @ N6 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_699_lift__Suc__mono__le,axiom,
! [F: nat > nat,N2: nat,N6: nat] :
( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N6 )
=> ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ N6 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_700_lift__Suc__antimono__le,axiom,
! [F: nat > set_nat,N2: nat,N6: nat] :
( ! [N4: nat] : ( ord_less_eq_set_nat @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
=> ( ( ord_less_eq_nat @ N2 @ N6 )
=> ( ord_less_eq_set_nat @ ( F @ N6 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_701_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N2: nat,N6: nat] :
( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
=> ( ( ord_less_eq_nat @ N2 @ N6 )
=> ( ord_less_eq_nat @ ( F @ N6 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_702_sup_OcoboundedI2,axiom,
! [C: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,A3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ C @ B2 )
=> ( ord_le1325389633284124927_a_nat @ C @ ( sup_su6804446743777130803_a_nat @ A3 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_703_sup_OcoboundedI2,axiom,
! [C: set_nat,B2: set_nat,A3: set_nat] :
( ( ord_less_eq_set_nat @ C @ B2 )
=> ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A3 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_704_sup_OcoboundedI2,axiom,
! [C: nat,B2: nat,A3: nat] :
( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A3 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_705_sup_OcoboundedI1,axiom,
! [C: set_Sum_sum_a_nat,A3: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ C @ A3 )
=> ( ord_le1325389633284124927_a_nat @ C @ ( sup_su6804446743777130803_a_nat @ A3 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_706_sup_OcoboundedI1,axiom,
! [C: set_nat,A3: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C @ A3 )
=> ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A3 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_707_sup_OcoboundedI1,axiom,
! [C: nat,A3: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ A3 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A3 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_708_sup_Oabsorb__iff2,axiom,
( ord_le1325389633284124927_a_nat
= ( ^ [A6: set_Sum_sum_a_nat,B8: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A6 @ B8 )
= B8 ) ) ) ).
% sup.absorb_iff2
thf(fact_709_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B8: set_nat] :
( ( sup_sup_set_nat @ A6 @ B8 )
= B8 ) ) ) ).
% sup.absorb_iff2
thf(fact_710_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A6: nat,B8: nat] :
( ( sup_sup_nat @ A6 @ B8 )
= B8 ) ) ) ).
% sup.absorb_iff2
thf(fact_711_sup_Oabsorb__iff1,axiom,
( ord_le1325389633284124927_a_nat
= ( ^ [B8: set_Sum_sum_a_nat,A6: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A6 @ B8 )
= A6 ) ) ) ).
% sup.absorb_iff1
thf(fact_712_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [B8: set_nat,A6: set_nat] :
( ( sup_sup_set_nat @ A6 @ B8 )
= A6 ) ) ) ).
% sup.absorb_iff1
thf(fact_713_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B8: nat,A6: nat] :
( ( sup_sup_nat @ A6 @ B8 )
= A6 ) ) ) ).
% sup.absorb_iff1
thf(fact_714_sup_Ocobounded2,axiom,
! [B2: set_Sum_sum_a_nat,A3: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ B2 @ ( sup_su6804446743777130803_a_nat @ A3 @ B2 ) ) ).
% sup.cobounded2
thf(fact_715_sup_Ocobounded2,axiom,
! [B2: set_nat,A3: set_nat] : ( ord_less_eq_set_nat @ B2 @ ( sup_sup_set_nat @ A3 @ B2 ) ) ).
% sup.cobounded2
thf(fact_716_sup_Ocobounded2,axiom,
! [B2: nat,A3: nat] : ( ord_less_eq_nat @ B2 @ ( sup_sup_nat @ A3 @ B2 ) ) ).
% sup.cobounded2
thf(fact_717_sup_Ocobounded1,axiom,
! [A3: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ A3 @ ( sup_su6804446743777130803_a_nat @ A3 @ B2 ) ) ).
% sup.cobounded1
thf(fact_718_sup_Ocobounded1,axiom,
! [A3: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ A3 @ ( sup_sup_set_nat @ A3 @ B2 ) ) ).
% sup.cobounded1
thf(fact_719_sup_Ocobounded1,axiom,
! [A3: nat,B2: nat] : ( ord_less_eq_nat @ A3 @ ( sup_sup_nat @ A3 @ B2 ) ) ).
% sup.cobounded1
thf(fact_720_sup_Oorder__iff,axiom,
( ord_le1325389633284124927_a_nat
= ( ^ [B8: set_Sum_sum_a_nat,A6: set_Sum_sum_a_nat] :
( A6
= ( sup_su6804446743777130803_a_nat @ A6 @ B8 ) ) ) ) ).
% sup.order_iff
thf(fact_721_sup_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [B8: set_nat,A6: set_nat] :
( A6
= ( sup_sup_set_nat @ A6 @ B8 ) ) ) ) ).
% sup.order_iff
thf(fact_722_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B8: nat,A6: nat] :
( A6
= ( sup_sup_nat @ A6 @ B8 ) ) ) ) ).
% sup.order_iff
thf(fact_723_sup_OboundedI,axiom,
! [B2: set_Sum_sum_a_nat,A3: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B2 @ A3 )
=> ( ( ord_le1325389633284124927_a_nat @ C @ A3 )
=> ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ B2 @ C ) @ A3 ) ) ) ).
% sup.boundedI
thf(fact_724_sup_OboundedI,axiom,
! [B2: set_nat,A3: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A3 )
=> ( ( ord_less_eq_set_nat @ C @ A3 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C ) @ A3 ) ) ) ).
% sup.boundedI
thf(fact_725_sup_OboundedI,axiom,
! [B2: nat,A3: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
=> ( ( ord_less_eq_nat @ C @ A3 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A3 ) ) ) ).
% sup.boundedI
thf(fact_726_sup_OboundedE,axiom,
! [B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat,A3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ B2 @ C ) @ A3 )
=> ~ ( ( ord_le1325389633284124927_a_nat @ B2 @ A3 )
=> ~ ( ord_le1325389633284124927_a_nat @ C @ A3 ) ) ) ).
% sup.boundedE
thf(fact_727_sup_OboundedE,axiom,
! [B2: set_nat,C: set_nat,A3: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C ) @ A3 )
=> ~ ( ( ord_less_eq_set_nat @ B2 @ A3 )
=> ~ ( ord_less_eq_set_nat @ C @ A3 ) ) ) ).
% sup.boundedE
thf(fact_728_sup_OboundedE,axiom,
! [B2: nat,C: nat,A3: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A3 )
=> ~ ( ( ord_less_eq_nat @ B2 @ A3 )
=> ~ ( ord_less_eq_nat @ C @ A3 ) ) ) ).
% sup.boundedE
thf(fact_729_sup__absorb2,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X3 @ Y )
=> ( ( sup_su6804446743777130803_a_nat @ X3 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_730_sup__absorb2,axiom,
! [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ( sup_sup_set_nat @ X3 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_731_sup__absorb2,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( sup_sup_nat @ X3 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_732_sup__absorb1,axiom,
! [Y: set_Sum_sum_a_nat,X3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ Y @ X3 )
=> ( ( sup_su6804446743777130803_a_nat @ X3 @ Y )
= X3 ) ) ).
% sup_absorb1
thf(fact_733_sup__absorb1,axiom,
! [Y: set_nat,X3: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X3 )
=> ( ( sup_sup_set_nat @ X3 @ Y )
= X3 ) ) ).
% sup_absorb1
thf(fact_734_sup__absorb1,axiom,
! [Y: nat,X3: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ( ( sup_sup_nat @ X3 @ Y )
= X3 ) ) ).
% sup_absorb1
thf(fact_735_sup_Oabsorb2,axiom,
! [A3: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A3 @ B2 )
=> ( ( sup_su6804446743777130803_a_nat @ A3 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_736_sup_Oabsorb2,axiom,
! [A3: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B2 )
=> ( ( sup_sup_set_nat @ A3 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_737_sup_Oabsorb2,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( sup_sup_nat @ A3 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_738_sup_Oabsorb1,axiom,
! [B2: set_Sum_sum_a_nat,A3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B2 @ A3 )
=> ( ( sup_su6804446743777130803_a_nat @ A3 @ B2 )
= A3 ) ) ).
% sup.absorb1
thf(fact_739_sup_Oabsorb1,axiom,
! [B2: set_nat,A3: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A3 )
=> ( ( sup_sup_set_nat @ A3 @ B2 )
= A3 ) ) ).
% sup.absorb1
thf(fact_740_sup_Oabsorb1,axiom,
! [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
=> ( ( sup_sup_nat @ A3 @ B2 )
= A3 ) ) ).
% sup.absorb1
thf(fact_741_sup__unique,axiom,
! [F: set_Sum_sum_a_nat > set_Sum_sum_a_nat > set_Sum_sum_a_nat,X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
( ! [X4: set_Sum_sum_a_nat,Y5: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ X4 @ ( F @ X4 @ Y5 ) )
=> ( ! [X4: set_Sum_sum_a_nat,Y5: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ Y5 @ ( F @ X4 @ Y5 ) )
=> ( ! [X4: set_Sum_sum_a_nat,Y5: set_Sum_sum_a_nat,Z3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ Y5 @ X4 )
=> ( ( ord_le1325389633284124927_a_nat @ Z3 @ X4 )
=> ( ord_le1325389633284124927_a_nat @ ( F @ Y5 @ Z3 ) @ X4 ) ) )
=> ( ( sup_su6804446743777130803_a_nat @ X3 @ Y )
= ( F @ X3 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_742_sup__unique,axiom,
! [F: set_nat > set_nat > set_nat,X3: set_nat,Y: set_nat] :
( ! [X4: set_nat,Y5: set_nat] : ( ord_less_eq_set_nat @ X4 @ ( F @ X4 @ Y5 ) )
=> ( ! [X4: set_nat,Y5: set_nat] : ( ord_less_eq_set_nat @ Y5 @ ( F @ X4 @ Y5 ) )
=> ( ! [X4: set_nat,Y5: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ Y5 @ X4 )
=> ( ( ord_less_eq_set_nat @ Z3 @ X4 )
=> ( ord_less_eq_set_nat @ ( F @ Y5 @ Z3 ) @ X4 ) ) )
=> ( ( sup_sup_set_nat @ X3 @ Y )
= ( F @ X3 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_743_sup__unique,axiom,
! [F: nat > nat > nat,X3: nat,Y: nat] :
( ! [X4: nat,Y5: nat] : ( ord_less_eq_nat @ X4 @ ( F @ X4 @ Y5 ) )
=> ( ! [X4: nat,Y5: nat] : ( ord_less_eq_nat @ Y5 @ ( F @ X4 @ Y5 ) )
=> ( ! [X4: nat,Y5: nat,Z3: nat] :
( ( ord_less_eq_nat @ Y5 @ X4 )
=> ( ( ord_less_eq_nat @ Z3 @ X4 )
=> ( ord_less_eq_nat @ ( F @ Y5 @ Z3 ) @ X4 ) ) )
=> ( ( sup_sup_nat @ X3 @ Y )
= ( F @ X3 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_744_sup_OorderI,axiom,
! [A3: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( A3
= ( sup_su6804446743777130803_a_nat @ A3 @ B2 ) )
=> ( ord_le1325389633284124927_a_nat @ B2 @ A3 ) ) ).
% sup.orderI
thf(fact_745_sup_OorderI,axiom,
! [A3: set_nat,B2: set_nat] :
( ( A3
= ( sup_sup_set_nat @ A3 @ B2 ) )
=> ( ord_less_eq_set_nat @ B2 @ A3 ) ) ).
% sup.orderI
thf(fact_746_sup_OorderI,axiom,
! [A3: nat,B2: nat] :
( ( A3
= ( sup_sup_nat @ A3 @ B2 ) )
=> ( ord_less_eq_nat @ B2 @ A3 ) ) ).
% sup.orderI
thf(fact_747_sup_OorderE,axiom,
! [B2: set_Sum_sum_a_nat,A3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B2 @ A3 )
=> ( A3
= ( sup_su6804446743777130803_a_nat @ A3 @ B2 ) ) ) ).
% sup.orderE
thf(fact_748_sup_OorderE,axiom,
! [B2: set_nat,A3: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A3 )
=> ( A3
= ( sup_sup_set_nat @ A3 @ B2 ) ) ) ).
% sup.orderE
thf(fact_749_sup_OorderE,axiom,
! [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
=> ( A3
= ( sup_sup_nat @ A3 @ B2 ) ) ) ).
% sup.orderE
thf(fact_750_le__iff__sup,axiom,
( ord_le1325389633284124927_a_nat
= ( ^ [X: set_Sum_sum_a_nat,Y4: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ X @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_751_le__iff__sup,axiom,
( ord_less_eq_set_nat
= ( ^ [X: set_nat,Y4: set_nat] :
( ( sup_sup_set_nat @ X @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_752_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y4: nat] :
( ( sup_sup_nat @ X @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_753_sup__least,axiom,
! [Y: set_Sum_sum_a_nat,X3: set_Sum_sum_a_nat,Z2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ Y @ X3 )
=> ( ( ord_le1325389633284124927_a_nat @ Z2 @ X3 )
=> ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ Y @ Z2 ) @ X3 ) ) ) ).
% sup_least
thf(fact_754_sup__least,axiom,
! [Y: set_nat,X3: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X3 )
=> ( ( ord_less_eq_set_nat @ Z2 @ X3 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ Y @ Z2 ) @ X3 ) ) ) ).
% sup_least
thf(fact_755_sup__least,axiom,
! [Y: nat,X3: nat,Z2: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ( ( ord_less_eq_nat @ Z2 @ X3 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z2 ) @ X3 ) ) ) ).
% sup_least
thf(fact_756_sup__mono,axiom,
! [A3: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,D2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A3 @ C )
=> ( ( ord_le1325389633284124927_a_nat @ B2 @ D2 )
=> ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ A3 @ B2 ) @ ( sup_su6804446743777130803_a_nat @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_757_sup__mono,axiom,
! [A3: set_nat,C: set_nat,B2: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ C )
=> ( ( ord_less_eq_set_nat @ B2 @ D2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A3 @ B2 ) @ ( sup_sup_set_nat @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_758_sup__mono,axiom,
! [A3: nat,C: nat,B2: nat,D2: nat] :
( ( ord_less_eq_nat @ A3 @ C )
=> ( ( ord_less_eq_nat @ B2 @ D2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A3 @ B2 ) @ ( sup_sup_nat @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_759_sup_Omono,axiom,
! [C: set_Sum_sum_a_nat,A3: set_Sum_sum_a_nat,D2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ C @ A3 )
=> ( ( ord_le1325389633284124927_a_nat @ D2 @ B2 )
=> ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ C @ D2 ) @ ( sup_su6804446743777130803_a_nat @ A3 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_760_sup_Omono,axiom,
! [C: set_nat,A3: set_nat,D2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C @ A3 )
=> ( ( ord_less_eq_set_nat @ D2 @ B2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ C @ D2 ) @ ( sup_sup_set_nat @ A3 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_761_sup_Omono,axiom,
! [C: nat,A3: nat,D2: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ A3 )
=> ( ( ord_less_eq_nat @ D2 @ B2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D2 ) @ ( sup_sup_nat @ A3 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_762_le__supI2,axiom,
! [X3: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,A3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X3 @ B2 )
=> ( ord_le1325389633284124927_a_nat @ X3 @ ( sup_su6804446743777130803_a_nat @ A3 @ B2 ) ) ) ).
% le_supI2
thf(fact_763_le__supI2,axiom,
! [X3: set_nat,B2: set_nat,A3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ B2 )
=> ( ord_less_eq_set_nat @ X3 @ ( sup_sup_set_nat @ A3 @ B2 ) ) ) ).
% le_supI2
thf(fact_764_le__supI2,axiom,
! [X3: nat,B2: nat,A3: nat] :
( ( ord_less_eq_nat @ X3 @ B2 )
=> ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ A3 @ B2 ) ) ) ).
% le_supI2
thf(fact_765_le__supI1,axiom,
! [X3: set_Sum_sum_a_nat,A3: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X3 @ A3 )
=> ( ord_le1325389633284124927_a_nat @ X3 @ ( sup_su6804446743777130803_a_nat @ A3 @ B2 ) ) ) ).
% le_supI1
thf(fact_766_le__supI1,axiom,
! [X3: set_nat,A3: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ A3 )
=> ( ord_less_eq_set_nat @ X3 @ ( sup_sup_set_nat @ A3 @ B2 ) ) ) ).
% le_supI1
thf(fact_767_le__supI1,axiom,
! [X3: nat,A3: nat,B2: nat] :
( ( ord_less_eq_nat @ X3 @ A3 )
=> ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ A3 @ B2 ) ) ) ).
% le_supI1
thf(fact_768_sup__ge2,axiom,
! [Y: set_Sum_sum_a_nat,X3: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ Y @ ( sup_su6804446743777130803_a_nat @ X3 @ Y ) ) ).
% sup_ge2
thf(fact_769_sup__ge2,axiom,
! [Y: set_nat,X3: set_nat] : ( ord_less_eq_set_nat @ Y @ ( sup_sup_set_nat @ X3 @ Y ) ) ).
% sup_ge2
thf(fact_770_sup__ge2,axiom,
! [Y: nat,X3: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X3 @ Y ) ) ).
% sup_ge2
thf(fact_771_sup__ge1,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ X3 @ ( sup_su6804446743777130803_a_nat @ X3 @ Y ) ) ).
% sup_ge1
thf(fact_772_sup__ge1,axiom,
! [X3: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ X3 @ ( sup_sup_set_nat @ X3 @ Y ) ) ).
% sup_ge1
thf(fact_773_sup__ge1,axiom,
! [X3: nat,Y: nat] : ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ X3 @ Y ) ) ).
% sup_ge1
thf(fact_774_le__supI,axiom,
! [A3: set_Sum_sum_a_nat,X3: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A3 @ X3 )
=> ( ( ord_le1325389633284124927_a_nat @ B2 @ X3 )
=> ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ A3 @ B2 ) @ X3 ) ) ) ).
% le_supI
thf(fact_775_le__supI,axiom,
! [A3: set_nat,X3: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ X3 )
=> ( ( ord_less_eq_set_nat @ B2 @ X3 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A3 @ B2 ) @ X3 ) ) ) ).
% le_supI
thf(fact_776_le__supI,axiom,
! [A3: nat,X3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ X3 )
=> ( ( ord_less_eq_nat @ B2 @ X3 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A3 @ B2 ) @ X3 ) ) ) ).
% le_supI
thf(fact_777_le__supE,axiom,
! [A3: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,X3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ A3 @ B2 ) @ X3 )
=> ~ ( ( ord_le1325389633284124927_a_nat @ A3 @ X3 )
=> ~ ( ord_le1325389633284124927_a_nat @ B2 @ X3 ) ) ) ).
% le_supE
thf(fact_778_le__supE,axiom,
! [A3: set_nat,B2: set_nat,X3: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A3 @ B2 ) @ X3 )
=> ~ ( ( ord_less_eq_set_nat @ A3 @ X3 )
=> ~ ( ord_less_eq_set_nat @ B2 @ X3 ) ) ) ).
% le_supE
thf(fact_779_le__supE,axiom,
! [A3: nat,B2: nat,X3: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A3 @ B2 ) @ X3 )
=> ~ ( ( ord_less_eq_nat @ A3 @ X3 )
=> ~ ( ord_less_eq_nat @ B2 @ X3 ) ) ) ).
% le_supE
thf(fact_780_inf__sup__ord_I3_J,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ X3 @ ( sup_su6804446743777130803_a_nat @ X3 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_781_inf__sup__ord_I3_J,axiom,
! [X3: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ X3 @ ( sup_sup_set_nat @ X3 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_782_inf__sup__ord_I3_J,axiom,
! [X3: nat,Y: nat] : ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ X3 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_783_inf__sup__ord_I4_J,axiom,
! [Y: set_Sum_sum_a_nat,X3: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ Y @ ( sup_su6804446743777130803_a_nat @ X3 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_784_inf__sup__ord_I4_J,axiom,
! [Y: set_nat,X3: set_nat] : ( ord_less_eq_set_nat @ Y @ ( sup_sup_set_nat @ X3 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_785_inf__sup__ord_I4_J,axiom,
! [Y: nat,X3: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X3 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_786_inf_OcoboundedI2,axiom,
! [B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat,A3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B2 @ C )
=> ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ A3 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_787_inf_OcoboundedI2,axiom,
! [B2: set_a,C: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_788_inf_OcoboundedI2,axiom,
! [B2: set_nat,C: set_nat,A3: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A3 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_789_inf_OcoboundedI2,axiom,
! [B2: nat,C: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A3 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_790_inf_OcoboundedI1,axiom,
! [A3: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A3 @ C )
=> ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ A3 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_791_inf_OcoboundedI1,axiom,
! [A3: set_a,C: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A3 @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_792_inf_OcoboundedI1,axiom,
! [A3: set_nat,C: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ C )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A3 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_793_inf_OcoboundedI1,axiom,
! [A3: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A3 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_794_inf_Oabsorb__iff2,axiom,
( ord_le1325389633284124927_a_nat
= ( ^ [B8: set_Sum_sum_a_nat,A6: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ A6 @ B8 )
= B8 ) ) ) ).
% inf.absorb_iff2
thf(fact_795_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [B8: set_a,A6: set_a] :
( ( inf_inf_set_a @ A6 @ B8 )
= B8 ) ) ) ).
% inf.absorb_iff2
thf(fact_796_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [B8: set_nat,A6: set_nat] :
( ( inf_inf_set_nat @ A6 @ B8 )
= B8 ) ) ) ).
% inf.absorb_iff2
thf(fact_797_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B8: nat,A6: nat] :
( ( inf_inf_nat @ A6 @ B8 )
= B8 ) ) ) ).
% inf.absorb_iff2
thf(fact_798_inf_Oabsorb__iff1,axiom,
( ord_le1325389633284124927_a_nat
= ( ^ [A6: set_Sum_sum_a_nat,B8: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ A6 @ B8 )
= A6 ) ) ) ).
% inf.absorb_iff1
thf(fact_799_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B8: set_a] :
( ( inf_inf_set_a @ A6 @ B8 )
= A6 ) ) ) ).
% inf.absorb_iff1
thf(fact_800_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B8: set_nat] :
( ( inf_inf_set_nat @ A6 @ B8 )
= A6 ) ) ) ).
% inf.absorb_iff1
thf(fact_801_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A6: nat,B8: nat] :
( ( inf_inf_nat @ A6 @ B8 )
= A6 ) ) ) ).
% inf.absorb_iff1
thf(fact_802_inf_Ocobounded2,axiom,
! [A3: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ A3 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_803_inf_Ocobounded2,axiom,
! [A3: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_804_inf_Ocobounded2,axiom,
! [A3: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A3 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_805_inf_Ocobounded2,axiom,
! [A3: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A3 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_806_inf_Ocobounded1,axiom,
! [A3: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ A3 @ B2 ) @ A3 ) ).
% inf.cobounded1
thf(fact_807_inf_Ocobounded1,axiom,
! [A3: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B2 ) @ A3 ) ).
% inf.cobounded1
thf(fact_808_inf_Ocobounded1,axiom,
! [A3: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A3 @ B2 ) @ A3 ) ).
% inf.cobounded1
thf(fact_809_inf_Ocobounded1,axiom,
! [A3: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A3 @ B2 ) @ A3 ) ).
% inf.cobounded1
thf(fact_810_inf_Oorder__iff,axiom,
( ord_le1325389633284124927_a_nat
= ( ^ [A6: set_Sum_sum_a_nat,B8: set_Sum_sum_a_nat] :
( A6
= ( inf_in7084830621192376909_a_nat @ A6 @ B8 ) ) ) ) ).
% inf.order_iff
thf(fact_811_inf_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B8: set_a] :
( A6
= ( inf_inf_set_a @ A6 @ B8 ) ) ) ) ).
% inf.order_iff
thf(fact_812_inf_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B8: set_nat] :
( A6
= ( inf_inf_set_nat @ A6 @ B8 ) ) ) ) ).
% inf.order_iff
thf(fact_813_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A6: nat,B8: nat] :
( A6
= ( inf_inf_nat @ A6 @ B8 ) ) ) ) ).
% inf.order_iff
thf(fact_814_inf__greatest,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X3 @ Y )
=> ( ( ord_le1325389633284124927_a_nat @ X3 @ Z2 )
=> ( ord_le1325389633284124927_a_nat @ X3 @ ( inf_in7084830621192376909_a_nat @ Y @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_815_inf__greatest,axiom,
! [X3: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y )
=> ( ( ord_less_eq_set_a @ X3 @ Z2 )
=> ( ord_less_eq_set_a @ X3 @ ( inf_inf_set_a @ Y @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_816_inf__greatest,axiom,
! [X3: set_nat,Y: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ( ord_less_eq_set_nat @ X3 @ Z2 )
=> ( ord_less_eq_set_nat @ X3 @ ( inf_inf_set_nat @ Y @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_817_inf__greatest,axiom,
! [X3: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ X3 @ Z2 )
=> ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ Y @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_818_inf_OboundedI,axiom,
! [A3: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A3 @ B2 )
=> ( ( ord_le1325389633284124927_a_nat @ A3 @ C )
=> ( ord_le1325389633284124927_a_nat @ A3 @ ( inf_in7084830621192376909_a_nat @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_819_inf_OboundedI,axiom,
! [A3: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A3 @ B2 )
=> ( ( ord_less_eq_set_a @ A3 @ C )
=> ( ord_less_eq_set_a @ A3 @ ( inf_inf_set_a @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_820_inf_OboundedI,axiom,
! [A3: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B2 )
=> ( ( ord_less_eq_set_nat @ A3 @ C )
=> ( ord_less_eq_set_nat @ A3 @ ( inf_inf_set_nat @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_821_inf_OboundedI,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ord_less_eq_nat @ A3 @ C )
=> ( ord_less_eq_nat @ A3 @ ( inf_inf_nat @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_822_inf_OboundedE,axiom,
! [A3: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A3 @ ( inf_in7084830621192376909_a_nat @ B2 @ C ) )
=> ~ ( ( ord_le1325389633284124927_a_nat @ A3 @ B2 )
=> ~ ( ord_le1325389633284124927_a_nat @ A3 @ C ) ) ) ).
% inf.boundedE
thf(fact_823_inf_OboundedE,axiom,
! [A3: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A3 @ ( inf_inf_set_a @ B2 @ C ) )
=> ~ ( ( ord_less_eq_set_a @ A3 @ B2 )
=> ~ ( ord_less_eq_set_a @ A3 @ C ) ) ) ).
% inf.boundedE
thf(fact_824_inf_OboundedE,axiom,
! [A3: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ ( inf_inf_set_nat @ B2 @ C ) )
=> ~ ( ( ord_less_eq_set_nat @ A3 @ B2 )
=> ~ ( ord_less_eq_set_nat @ A3 @ C ) ) ) ).
% inf.boundedE
thf(fact_825_inf_OboundedE,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ ( inf_inf_nat @ B2 @ C ) )
=> ~ ( ( ord_less_eq_nat @ A3 @ B2 )
=> ~ ( ord_less_eq_nat @ A3 @ C ) ) ) ).
% inf.boundedE
thf(fact_826_inf__absorb2,axiom,
! [Y: set_Sum_sum_a_nat,X3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ Y @ X3 )
=> ( ( inf_in7084830621192376909_a_nat @ X3 @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_827_inf__absorb2,axiom,
! [Y: set_a,X3: set_a] :
( ( ord_less_eq_set_a @ Y @ X3 )
=> ( ( inf_inf_set_a @ X3 @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_828_inf__absorb2,axiom,
! [Y: set_nat,X3: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X3 )
=> ( ( inf_inf_set_nat @ X3 @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_829_inf__absorb2,axiom,
! [Y: nat,X3: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ( ( inf_inf_nat @ X3 @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_830_inf__absorb1,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X3 @ Y )
=> ( ( inf_in7084830621192376909_a_nat @ X3 @ Y )
= X3 ) ) ).
% inf_absorb1
thf(fact_831_inf__absorb1,axiom,
! [X3: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y )
=> ( ( inf_inf_set_a @ X3 @ Y )
= X3 ) ) ).
% inf_absorb1
thf(fact_832_inf__absorb1,axiom,
! [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ( inf_inf_set_nat @ X3 @ Y )
= X3 ) ) ).
% inf_absorb1
thf(fact_833_inf__absorb1,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( inf_inf_nat @ X3 @ Y )
= X3 ) ) ).
% inf_absorb1
thf(fact_834_inf_Oabsorb2,axiom,
! [B2: set_Sum_sum_a_nat,A3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B2 @ A3 )
=> ( ( inf_in7084830621192376909_a_nat @ A3 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_835_inf_Oabsorb2,axiom,
! [B2: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B2 @ A3 )
=> ( ( inf_inf_set_a @ A3 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_836_inf_Oabsorb2,axiom,
! [B2: set_nat,A3: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A3 )
=> ( ( inf_inf_set_nat @ A3 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_837_inf_Oabsorb2,axiom,
! [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
=> ( ( inf_inf_nat @ A3 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_838_inf_Oabsorb1,axiom,
! [A3: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A3 @ B2 )
=> ( ( inf_in7084830621192376909_a_nat @ A3 @ B2 )
= A3 ) ) ).
% inf.absorb1
thf(fact_839_inf_Oabsorb1,axiom,
! [A3: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B2 )
=> ( ( inf_inf_set_a @ A3 @ B2 )
= A3 ) ) ).
% inf.absorb1
thf(fact_840_inf_Oabsorb1,axiom,
! [A3: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B2 )
=> ( ( inf_inf_set_nat @ A3 @ B2 )
= A3 ) ) ).
% inf.absorb1
thf(fact_841_inf_Oabsorb1,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( inf_inf_nat @ A3 @ B2 )
= A3 ) ) ).
% inf.absorb1
thf(fact_842_le__iff__inf,axiom,
( ord_le1325389633284124927_a_nat
= ( ^ [X: set_Sum_sum_a_nat,Y4: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ X @ Y4 )
= X ) ) ) ).
% le_iff_inf
thf(fact_843_le__iff__inf,axiom,
( ord_less_eq_set_a
= ( ^ [X: set_a,Y4: set_a] :
( ( inf_inf_set_a @ X @ Y4 )
= X ) ) ) ).
% le_iff_inf
thf(fact_844_le__iff__inf,axiom,
( ord_less_eq_set_nat
= ( ^ [X: set_nat,Y4: set_nat] :
( ( inf_inf_set_nat @ X @ Y4 )
= X ) ) ) ).
% le_iff_inf
thf(fact_845_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y4: nat] :
( ( inf_inf_nat @ X @ Y4 )
= X ) ) ) ).
% le_iff_inf
thf(fact_846_inf__unique,axiom,
! [F: set_Sum_sum_a_nat > set_Sum_sum_a_nat > set_Sum_sum_a_nat,X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
( ! [X4: set_Sum_sum_a_nat,Y5: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( F @ X4 @ Y5 ) @ X4 )
=> ( ! [X4: set_Sum_sum_a_nat,Y5: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( F @ X4 @ Y5 ) @ Y5 )
=> ( ! [X4: set_Sum_sum_a_nat,Y5: set_Sum_sum_a_nat,Z3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X4 @ Y5 )
=> ( ( ord_le1325389633284124927_a_nat @ X4 @ Z3 )
=> ( ord_le1325389633284124927_a_nat @ X4 @ ( F @ Y5 @ Z3 ) ) ) )
=> ( ( inf_in7084830621192376909_a_nat @ X3 @ Y )
= ( F @ X3 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_847_inf__unique,axiom,
! [F: set_a > set_a > set_a,X3: set_a,Y: set_a] :
( ! [X4: set_a,Y5: set_a] : ( ord_less_eq_set_a @ ( F @ X4 @ Y5 ) @ X4 )
=> ( ! [X4: set_a,Y5: set_a] : ( ord_less_eq_set_a @ ( F @ X4 @ Y5 ) @ Y5 )
=> ( ! [X4: set_a,Y5: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y5 )
=> ( ( ord_less_eq_set_a @ X4 @ Z3 )
=> ( ord_less_eq_set_a @ X4 @ ( F @ Y5 @ Z3 ) ) ) )
=> ( ( inf_inf_set_a @ X3 @ Y )
= ( F @ X3 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_848_inf__unique,axiom,
! [F: set_nat > set_nat > set_nat,X3: set_nat,Y: set_nat] :
( ! [X4: set_nat,Y5: set_nat] : ( ord_less_eq_set_nat @ ( F @ X4 @ Y5 ) @ X4 )
=> ( ! [X4: set_nat,Y5: set_nat] : ( ord_less_eq_set_nat @ ( F @ X4 @ Y5 ) @ Y5 )
=> ( ! [X4: set_nat,Y5: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y5 )
=> ( ( ord_less_eq_set_nat @ X4 @ Z3 )
=> ( ord_less_eq_set_nat @ X4 @ ( F @ Y5 @ Z3 ) ) ) )
=> ( ( inf_inf_set_nat @ X3 @ Y )
= ( F @ X3 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_849_inf__unique,axiom,
! [F: nat > nat > nat,X3: nat,Y: nat] :
( ! [X4: nat,Y5: nat] : ( ord_less_eq_nat @ ( F @ X4 @ Y5 ) @ X4 )
=> ( ! [X4: nat,Y5: nat] : ( ord_less_eq_nat @ ( F @ X4 @ Y5 ) @ Y5 )
=> ( ! [X4: nat,Y5: nat,Z3: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
=> ( ( ord_less_eq_nat @ X4 @ Z3 )
=> ( ord_less_eq_nat @ X4 @ ( F @ Y5 @ Z3 ) ) ) )
=> ( ( inf_inf_nat @ X3 @ Y )
= ( F @ X3 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_850_inf_OorderI,axiom,
! [A3: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( A3
= ( inf_in7084830621192376909_a_nat @ A3 @ B2 ) )
=> ( ord_le1325389633284124927_a_nat @ A3 @ B2 ) ) ).
% inf.orderI
thf(fact_851_inf_OorderI,axiom,
! [A3: set_a,B2: set_a] :
( ( A3
= ( inf_inf_set_a @ A3 @ B2 ) )
=> ( ord_less_eq_set_a @ A3 @ B2 ) ) ).
% inf.orderI
thf(fact_852_inf_OorderI,axiom,
! [A3: set_nat,B2: set_nat] :
( ( A3
= ( inf_inf_set_nat @ A3 @ B2 ) )
=> ( ord_less_eq_set_nat @ A3 @ B2 ) ) ).
% inf.orderI
thf(fact_853_inf_OorderI,axiom,
! [A3: nat,B2: nat] :
( ( A3
= ( inf_inf_nat @ A3 @ B2 ) )
=> ( ord_less_eq_nat @ A3 @ B2 ) ) ).
% inf.orderI
thf(fact_854_inf_OorderE,axiom,
! [A3: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A3 @ B2 )
=> ( A3
= ( inf_in7084830621192376909_a_nat @ A3 @ B2 ) ) ) ).
% inf.orderE
thf(fact_855_inf_OorderE,axiom,
! [A3: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B2 )
=> ( A3
= ( inf_inf_set_a @ A3 @ B2 ) ) ) ).
% inf.orderE
thf(fact_856_inf_OorderE,axiom,
! [A3: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B2 )
=> ( A3
= ( inf_inf_set_nat @ A3 @ B2 ) ) ) ).
% inf.orderE
thf(fact_857_inf_OorderE,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( A3
= ( inf_inf_nat @ A3 @ B2 ) ) ) ).
% inf.orderE
thf(fact_858_le__infI2,axiom,
! [B2: set_Sum_sum_a_nat,X3: set_Sum_sum_a_nat,A3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B2 @ X3 )
=> ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ A3 @ B2 ) @ X3 ) ) ).
% le_infI2
thf(fact_859_le__infI2,axiom,
! [B2: set_a,X3: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B2 @ X3 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B2 ) @ X3 ) ) ).
% le_infI2
thf(fact_860_le__infI2,axiom,
! [B2: set_nat,X3: set_nat,A3: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ X3 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A3 @ B2 ) @ X3 ) ) ).
% le_infI2
thf(fact_861_le__infI2,axiom,
! [B2: nat,X3: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ X3 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A3 @ B2 ) @ X3 ) ) ).
% le_infI2
thf(fact_862_le__infI1,axiom,
! [A3: set_Sum_sum_a_nat,X3: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A3 @ X3 )
=> ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ A3 @ B2 ) @ X3 ) ) ).
% le_infI1
thf(fact_863_le__infI1,axiom,
! [A3: set_a,X3: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A3 @ X3 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B2 ) @ X3 ) ) ).
% le_infI1
thf(fact_864_le__infI1,axiom,
! [A3: set_nat,X3: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ X3 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A3 @ B2 ) @ X3 ) ) ).
% le_infI1
thf(fact_865_le__infI1,axiom,
! [A3: nat,X3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ X3 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A3 @ B2 ) @ X3 ) ) ).
% le_infI1
thf(fact_866_inf__mono,axiom,
! [A3: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,D2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A3 @ C )
=> ( ( ord_le1325389633284124927_a_nat @ B2 @ D2 )
=> ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ A3 @ B2 ) @ ( inf_in7084830621192376909_a_nat @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_867_inf__mono,axiom,
! [A3: set_a,C: set_a,B2: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A3 @ C )
=> ( ( ord_less_eq_set_a @ B2 @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B2 ) @ ( inf_inf_set_a @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_868_inf__mono,axiom,
! [A3: set_nat,C: set_nat,B2: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ C )
=> ( ( ord_less_eq_set_nat @ B2 @ D2 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A3 @ B2 ) @ ( inf_inf_set_nat @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_869_inf__mono,axiom,
! [A3: nat,C: nat,B2: nat,D2: nat] :
( ( ord_less_eq_nat @ A3 @ C )
=> ( ( ord_less_eq_nat @ B2 @ D2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A3 @ B2 ) @ ( inf_inf_nat @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_870_le__infI,axiom,
! [X3: set_Sum_sum_a_nat,A3: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X3 @ A3 )
=> ( ( ord_le1325389633284124927_a_nat @ X3 @ B2 )
=> ( ord_le1325389633284124927_a_nat @ X3 @ ( inf_in7084830621192376909_a_nat @ A3 @ B2 ) ) ) ) ).
% le_infI
thf(fact_871_le__infI,axiom,
! [X3: set_a,A3: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X3 @ A3 )
=> ( ( ord_less_eq_set_a @ X3 @ B2 )
=> ( ord_less_eq_set_a @ X3 @ ( inf_inf_set_a @ A3 @ B2 ) ) ) ) ).
% le_infI
thf(fact_872_le__infI,axiom,
! [X3: set_nat,A3: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ A3 )
=> ( ( ord_less_eq_set_nat @ X3 @ B2 )
=> ( ord_less_eq_set_nat @ X3 @ ( inf_inf_set_nat @ A3 @ B2 ) ) ) ) ).
% le_infI
thf(fact_873_le__infI,axiom,
! [X3: nat,A3: nat,B2: nat] :
( ( ord_less_eq_nat @ X3 @ A3 )
=> ( ( ord_less_eq_nat @ X3 @ B2 )
=> ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ A3 @ B2 ) ) ) ) ).
% le_infI
thf(fact_874_le__infE,axiom,
! [X3: set_Sum_sum_a_nat,A3: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X3 @ ( inf_in7084830621192376909_a_nat @ A3 @ B2 ) )
=> ~ ( ( ord_le1325389633284124927_a_nat @ X3 @ A3 )
=> ~ ( ord_le1325389633284124927_a_nat @ X3 @ B2 ) ) ) ).
% le_infE
thf(fact_875_le__infE,axiom,
! [X3: set_a,A3: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X3 @ ( inf_inf_set_a @ A3 @ B2 ) )
=> ~ ( ( ord_less_eq_set_a @ X3 @ A3 )
=> ~ ( ord_less_eq_set_a @ X3 @ B2 ) ) ) ).
% le_infE
thf(fact_876_le__infE,axiom,
! [X3: set_nat,A3: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ ( inf_inf_set_nat @ A3 @ B2 ) )
=> ~ ( ( ord_less_eq_set_nat @ X3 @ A3 )
=> ~ ( ord_less_eq_set_nat @ X3 @ B2 ) ) ) ).
% le_infE
thf(fact_877_le__infE,axiom,
! [X3: nat,A3: nat,B2: nat] :
( ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ A3 @ B2 ) )
=> ~ ( ( ord_less_eq_nat @ X3 @ A3 )
=> ~ ( ord_less_eq_nat @ X3 @ B2 ) ) ) ).
% le_infE
thf(fact_878_inf__le2,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ X3 @ Y ) @ Y ) ).
% inf_le2
thf(fact_879_inf__le2,axiom,
! [X3: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X3 @ Y ) @ Y ) ).
% inf_le2
thf(fact_880_inf__le2,axiom,
! [X3: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X3 @ Y ) @ Y ) ).
% inf_le2
thf(fact_881_inf__le2,axiom,
! [X3: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y ) @ Y ) ).
% inf_le2
thf(fact_882_inf__le1,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ X3 @ Y ) @ X3 ) ).
% inf_le1
thf(fact_883_inf__le1,axiom,
! [X3: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X3 @ Y ) @ X3 ) ).
% inf_le1
thf(fact_884_inf__le1,axiom,
! [X3: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X3 @ Y ) @ X3 ) ).
% inf_le1
thf(fact_885_inf__le1,axiom,
! [X3: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y ) @ X3 ) ).
% inf_le1
thf(fact_886_inf__sup__ord_I1_J,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ X3 @ Y ) @ X3 ) ).
% inf_sup_ord(1)
thf(fact_887_inf__sup__ord_I1_J,axiom,
! [X3: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X3 @ Y ) @ X3 ) ).
% inf_sup_ord(1)
thf(fact_888_inf__sup__ord_I1_J,axiom,
! [X3: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X3 @ Y ) @ X3 ) ).
% inf_sup_ord(1)
thf(fact_889_inf__sup__ord_I1_J,axiom,
! [X3: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y ) @ X3 ) ).
% inf_sup_ord(1)
thf(fact_890_inf__sup__ord_I2_J,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ X3 @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_891_inf__sup__ord_I2_J,axiom,
! [X3: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X3 @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_892_inf__sup__ord_I2_J,axiom,
! [X3: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X3 @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_893_inf__sup__ord_I2_J,axiom,
! [X3: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_894_set__ConsD,axiom,
! [Y: a,X3: a,Xs: list_a] :
( ( member_a @ Y @ ( set_a2 @ ( cons_a @ X3 @ Xs ) ) )
=> ( ( Y = X3 )
| ( member_a @ Y @ ( set_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_895_set__ConsD,axiom,
! [Y: nat,X3: nat,Xs: list_nat] :
( ( member_nat @ Y @ ( set_nat2 @ ( cons_nat @ X3 @ Xs ) ) )
=> ( ( Y = X3 )
| ( member_nat @ Y @ ( set_nat2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_896_set__ConsD,axiom,
! [Y: sum_sum_a_nat,X3: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ Y @ ( set_Sum_sum_a_nat2 @ ( cons_Sum_sum_a_nat @ X3 @ Xs ) ) )
=> ( ( Y = X3 )
| ( member_Sum_sum_a_nat @ Y @ ( set_Sum_sum_a_nat2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_897_list_Oset__cases,axiom,
! [E: a,A3: list_a] :
( ( member_a @ E @ ( set_a2 @ A3 ) )
=> ( ! [Z22: list_a] :
( A3
!= ( cons_a @ E @ Z22 ) )
=> ~ ! [Z1: a,Z22: list_a] :
( ( A3
= ( cons_a @ Z1 @ Z22 ) )
=> ~ ( member_a @ E @ ( set_a2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_898_list_Oset__cases,axiom,
! [E: nat,A3: list_nat] :
( ( member_nat @ E @ ( set_nat2 @ A3 ) )
=> ( ! [Z22: list_nat] :
( A3
!= ( cons_nat @ E @ Z22 ) )
=> ~ ! [Z1: nat,Z22: list_nat] :
( ( A3
= ( cons_nat @ Z1 @ Z22 ) )
=> ~ ( member_nat @ E @ ( set_nat2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_899_list_Oset__cases,axiom,
! [E: sum_sum_a_nat,A3: list_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ E @ ( set_Sum_sum_a_nat2 @ A3 ) )
=> ( ! [Z22: list_Sum_sum_a_nat] :
( A3
!= ( cons_Sum_sum_a_nat @ E @ Z22 ) )
=> ~ ! [Z1: sum_sum_a_nat,Z22: list_Sum_sum_a_nat] :
( ( A3
= ( cons_Sum_sum_a_nat @ Z1 @ Z22 ) )
=> ~ ( member_Sum_sum_a_nat @ E @ ( set_Sum_sum_a_nat2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_900_list_Oset__intros_I1_J,axiom,
! [X21: a,X22: list_a] : ( member_a @ X21 @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_901_list_Oset__intros_I1_J,axiom,
! [X21: nat,X22: list_nat] : ( member_nat @ X21 @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_902_list_Oset__intros_I1_J,axiom,
! [X21: sum_sum_a_nat,X22: list_Sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X21 @ ( set_Sum_sum_a_nat2 @ ( cons_Sum_sum_a_nat @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_903_list_Oset__intros_I2_J,axiom,
! [Y: a,X22: list_a,X21: a] :
( ( member_a @ Y @ ( set_a2 @ X22 ) )
=> ( member_a @ Y @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_904_list_Oset__intros_I2_J,axiom,
! [Y: nat,X22: list_nat,X21: nat] :
( ( member_nat @ Y @ ( set_nat2 @ X22 ) )
=> ( member_nat @ Y @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_905_list_Oset__intros_I2_J,axiom,
! [Y: sum_sum_a_nat,X22: list_Sum_sum_a_nat,X21: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ Y @ ( set_Sum_sum_a_nat2 @ X22 ) )
=> ( member_Sum_sum_a_nat @ Y @ ( set_Sum_sum_a_nat2 @ ( cons_Sum_sum_a_nat @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_906_option_Odistinct_I1_J,axiom,
! [X2: nat] :
( none_nat
!= ( some_nat @ X2 ) ) ).
% option.distinct(1)
thf(fact_907_option_OdiscI,axiom,
! [Option: option_nat,X2: nat] :
( ( Option
= ( some_nat @ X2 ) )
=> ( Option != none_nat ) ) ).
% option.discI
thf(fact_908_option_Oexhaust,axiom,
! [Y: option_nat] :
( ( Y != none_nat )
=> ~ ! [X23: nat] :
( Y
!= ( some_nat @ X23 ) ) ) ).
% option.exhaust
thf(fact_909_split__option__ex,axiom,
( ( ^ [P3: option_nat > $o] :
? [X6: option_nat] : ( P3 @ X6 ) )
= ( ^ [P4: option_nat > $o] :
( ( P4 @ none_nat )
| ? [X: nat] : ( P4 @ ( some_nat @ X ) ) ) ) ) ).
% split_option_ex
thf(fact_910_split__option__all,axiom,
( ( ^ [P3: option_nat > $o] :
! [X6: option_nat] : ( P3 @ X6 ) )
= ( ^ [P4: option_nat > $o] :
( ( P4 @ none_nat )
& ! [X: nat] : ( P4 @ ( some_nat @ X ) ) ) ) ) ).
% split_option_all
thf(fact_911_combine__options__cases,axiom,
! [X3: option_nat,P: option_nat > option_nat > $o,Y: option_nat] :
( ( ( X3 = none_nat )
=> ( P @ X3 @ Y ) )
=> ( ( ( Y = none_nat )
=> ( P @ X3 @ Y ) )
=> ( ! [A5: nat,B6: nat] :
( ( X3
= ( some_nat @ A5 ) )
=> ( ( Y
= ( some_nat @ B6 ) )
=> ( P @ X3 @ Y ) ) )
=> ( P @ X3 @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_912_finite__list,axiom,
! [A: set_Sum_sum_a_nat] :
( ( finite502105017643426984_a_nat @ A )
=> ? [Xs2: list_Sum_sum_a_nat] :
( ( set_Sum_sum_a_nat2 @ Xs2 )
= A ) ) ).
% finite_list
thf(fact_913_finite__list,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ? [Xs2: list_nat] :
( ( set_nat2 @ Xs2 )
= A ) ) ).
% finite_list
thf(fact_914_subset__code_I1_J,axiom,
! [Xs: list_a,B: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ B )
= ( ! [X: a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( member_a @ X @ B ) ) ) ) ).
% subset_code(1)
thf(fact_915_subset__code_I1_J,axiom,
! [Xs: list_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) @ B )
= ( ! [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ( member_Sum_sum_a_nat @ X @ B ) ) ) ) ).
% subset_code(1)
thf(fact_916_subset__code_I1_J,axiom,
! [Xs: list_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( member_nat @ X @ B ) ) ) ) ).
% subset_code(1)
thf(fact_917_add__Suc__shift,axiom,
! [M2: nat,N2: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N2 )
= ( plus_plus_nat @ M2 @ ( suc @ N2 ) ) ) ).
% add_Suc_shift
thf(fact_918_add__Suc,axiom,
! [M2: nat,N2: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N2 )
= ( suc @ ( plus_plus_nat @ M2 @ N2 ) ) ) ).
% add_Suc
thf(fact_919_nat__arith_Osuc1,axiom,
! [A: nat,K: nat,A3: nat] :
( ( A
= ( plus_plus_nat @ K @ A3 ) )
=> ( ( suc @ A )
= ( plus_plus_nat @ K @ ( suc @ A3 ) ) ) ) ).
% nat_arith.suc1
thf(fact_920_sup__inf__distrib2,axiom,
! [Y: set_nat,Z2: set_nat,X3: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y @ Z2 ) @ X3 )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y @ X3 ) @ ( sup_sup_set_nat @ Z2 @ X3 ) ) ) ).
% sup_inf_distrib2
thf(fact_921_sup__inf__distrib2,axiom,
! [Y: set_a,Z2: set_a,X3: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ Z2 ) @ X3 )
= ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ X3 ) @ ( sup_sup_set_a @ Z2 @ X3 ) ) ) ).
% sup_inf_distrib2
thf(fact_922_sup__inf__distrib2,axiom,
! [Y: set_Sum_sum_a_nat,Z2: set_Sum_sum_a_nat,X3: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ Y @ Z2 ) @ X3 )
= ( inf_in7084830621192376909_a_nat @ ( sup_su6804446743777130803_a_nat @ Y @ X3 ) @ ( sup_su6804446743777130803_a_nat @ Z2 @ X3 ) ) ) ).
% sup_inf_distrib2
thf(fact_923_sup__inf__distrib1,axiom,
! [X3: set_nat,Y: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( inf_inf_set_nat @ Y @ Z2 ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X3 @ Y ) @ ( sup_sup_set_nat @ X3 @ Z2 ) ) ) ).
% sup_inf_distrib1
thf(fact_924_sup__inf__distrib1,axiom,
! [X3: set_a,Y: set_a,Z2: set_a] :
( ( sup_sup_set_a @ X3 @ ( inf_inf_set_a @ Y @ Z2 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X3 @ Y ) @ ( sup_sup_set_a @ X3 @ Z2 ) ) ) ).
% sup_inf_distrib1
thf(fact_925_sup__inf__distrib1,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ X3 @ ( inf_in7084830621192376909_a_nat @ Y @ Z2 ) )
= ( inf_in7084830621192376909_a_nat @ ( sup_su6804446743777130803_a_nat @ X3 @ Y ) @ ( sup_su6804446743777130803_a_nat @ X3 @ Z2 ) ) ) ).
% sup_inf_distrib1
thf(fact_926_inf__sup__distrib2,axiom,
! [Y: set_nat,Z2: set_nat,X3: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y @ Z2 ) @ X3 )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y @ X3 ) @ ( inf_inf_set_nat @ Z2 @ X3 ) ) ) ).
% inf_sup_distrib2
thf(fact_927_inf__sup__distrib2,axiom,
! [Y: set_a,Z2: set_a,X3: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ Z2 ) @ X3 )
= ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ X3 ) @ ( inf_inf_set_a @ Z2 @ X3 ) ) ) ).
% inf_sup_distrib2
thf(fact_928_inf__sup__distrib2,axiom,
! [Y: set_Sum_sum_a_nat,Z2: set_Sum_sum_a_nat,X3: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ ( sup_su6804446743777130803_a_nat @ Y @ Z2 ) @ X3 )
= ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ Y @ X3 ) @ ( inf_in7084830621192376909_a_nat @ Z2 @ X3 ) ) ) ).
% inf_sup_distrib2
thf(fact_929_inf__sup__distrib1,axiom,
! [X3: set_nat,Y: set_nat,Z2: set_nat] :
( ( inf_inf_set_nat @ X3 @ ( sup_sup_set_nat @ Y @ Z2 ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X3 @ Y ) @ ( inf_inf_set_nat @ X3 @ Z2 ) ) ) ).
% inf_sup_distrib1
thf(fact_930_inf__sup__distrib1,axiom,
! [X3: set_a,Y: set_a,Z2: set_a] :
( ( inf_inf_set_a @ X3 @ ( sup_sup_set_a @ Y @ Z2 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X3 @ Y ) @ ( inf_inf_set_a @ X3 @ Z2 ) ) ) ).
% inf_sup_distrib1
thf(fact_931_inf__sup__distrib1,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z2: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ X3 @ ( sup_su6804446743777130803_a_nat @ Y @ Z2 ) )
= ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ X3 @ Y ) @ ( inf_in7084830621192376909_a_nat @ X3 @ Z2 ) ) ) ).
% inf_sup_distrib1
thf(fact_932_distrib__imp2,axiom,
! [X3: set_nat,Y: set_nat,Z2: set_nat] :
( ! [X4: set_nat,Y5: set_nat,Z3: set_nat] :
( ( sup_sup_set_nat @ X4 @ ( inf_inf_set_nat @ Y5 @ Z3 ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X4 @ Y5 ) @ ( sup_sup_set_nat @ X4 @ Z3 ) ) )
=> ( ( inf_inf_set_nat @ X3 @ ( sup_sup_set_nat @ Y @ Z2 ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X3 @ Y ) @ ( inf_inf_set_nat @ X3 @ Z2 ) ) ) ) ).
% distrib_imp2
thf(fact_933_distrib__imp2,axiom,
! [X3: set_a,Y: set_a,Z2: set_a] :
( ! [X4: set_a,Y5: set_a,Z3: set_a] :
( ( sup_sup_set_a @ X4 @ ( inf_inf_set_a @ Y5 @ Z3 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X4 @ Y5 ) @ ( sup_sup_set_a @ X4 @ Z3 ) ) )
=> ( ( inf_inf_set_a @ X3 @ ( sup_sup_set_a @ Y @ Z2 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X3 @ Y ) @ ( inf_inf_set_a @ X3 @ Z2 ) ) ) ) ).
% distrib_imp2
thf(fact_934_distrib__imp2,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z2: set_Sum_sum_a_nat] :
( ! [X4: set_Sum_sum_a_nat,Y5: set_Sum_sum_a_nat,Z3: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ X4 @ ( inf_in7084830621192376909_a_nat @ Y5 @ Z3 ) )
= ( inf_in7084830621192376909_a_nat @ ( sup_su6804446743777130803_a_nat @ X4 @ Y5 ) @ ( sup_su6804446743777130803_a_nat @ X4 @ Z3 ) ) )
=> ( ( inf_in7084830621192376909_a_nat @ X3 @ ( sup_su6804446743777130803_a_nat @ Y @ Z2 ) )
= ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ X3 @ Y ) @ ( inf_in7084830621192376909_a_nat @ X3 @ Z2 ) ) ) ) ).
% distrib_imp2
thf(fact_935_distrib__imp1,axiom,
! [X3: set_nat,Y: set_nat,Z2: set_nat] :
( ! [X4: set_nat,Y5: set_nat,Z3: set_nat] :
( ( inf_inf_set_nat @ X4 @ ( sup_sup_set_nat @ Y5 @ Z3 ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X4 @ Y5 ) @ ( inf_inf_set_nat @ X4 @ Z3 ) ) )
=> ( ( sup_sup_set_nat @ X3 @ ( inf_inf_set_nat @ Y @ Z2 ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X3 @ Y ) @ ( sup_sup_set_nat @ X3 @ Z2 ) ) ) ) ).
% distrib_imp1
thf(fact_936_distrib__imp1,axiom,
! [X3: set_a,Y: set_a,Z2: set_a] :
( ! [X4: set_a,Y5: set_a,Z3: set_a] :
( ( inf_inf_set_a @ X4 @ ( sup_sup_set_a @ Y5 @ Z3 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X4 @ Y5 ) @ ( inf_inf_set_a @ X4 @ Z3 ) ) )
=> ( ( sup_sup_set_a @ X3 @ ( inf_inf_set_a @ Y @ Z2 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X3 @ Y ) @ ( sup_sup_set_a @ X3 @ Z2 ) ) ) ) ).
% distrib_imp1
thf(fact_937_distrib__imp1,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z2: set_Sum_sum_a_nat] :
( ! [X4: set_Sum_sum_a_nat,Y5: set_Sum_sum_a_nat,Z3: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ X4 @ ( sup_su6804446743777130803_a_nat @ Y5 @ Z3 ) )
= ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ X4 @ Y5 ) @ ( inf_in7084830621192376909_a_nat @ X4 @ Z3 ) ) )
=> ( ( sup_su6804446743777130803_a_nat @ X3 @ ( inf_in7084830621192376909_a_nat @ Y @ Z2 ) )
= ( inf_in7084830621192376909_a_nat @ ( sup_su6804446743777130803_a_nat @ X3 @ Y ) @ ( sup_su6804446743777130803_a_nat @ X3 @ Z2 ) ) ) ) ).
% distrib_imp1
thf(fact_938_distrib__sup__le,axiom,
! [X3: set_a,Y: set_a,Z2: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ X3 @ ( inf_inf_set_a @ Y @ Z2 ) ) @ ( inf_inf_set_a @ ( sup_sup_set_a @ X3 @ Y ) @ ( sup_sup_set_a @ X3 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_939_distrib__sup__le,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z2: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ X3 @ ( inf_in7084830621192376909_a_nat @ Y @ Z2 ) ) @ ( inf_in7084830621192376909_a_nat @ ( sup_su6804446743777130803_a_nat @ X3 @ Y ) @ ( sup_su6804446743777130803_a_nat @ X3 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_940_distrib__sup__le,axiom,
! [X3: set_nat,Y: set_nat,Z2: set_nat] : ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X3 @ ( inf_inf_set_nat @ Y @ Z2 ) ) @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ X3 @ Y ) @ ( sup_sup_set_nat @ X3 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_941_distrib__sup__le,axiom,
! [X3: nat,Y: nat,Z2: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X3 @ ( inf_inf_nat @ Y @ Z2 ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X3 @ Y ) @ ( sup_sup_nat @ X3 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_942_distrib__inf__le,axiom,
! [X3: set_a,Y: set_a,Z2: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ X3 @ Y ) @ ( inf_inf_set_a @ X3 @ Z2 ) ) @ ( inf_inf_set_a @ X3 @ ( sup_sup_set_a @ Y @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_943_distrib__inf__le,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z2: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ X3 @ Y ) @ ( inf_in7084830621192376909_a_nat @ X3 @ Z2 ) ) @ ( inf_in7084830621192376909_a_nat @ X3 @ ( sup_su6804446743777130803_a_nat @ Y @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_944_distrib__inf__le,axiom,
! [X3: set_nat,Y: set_nat,Z2: set_nat] : ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ X3 @ Y ) @ ( inf_inf_set_nat @ X3 @ Z2 ) ) @ ( inf_inf_set_nat @ X3 @ ( sup_sup_set_nat @ Y @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_945_distrib__inf__le,axiom,
! [X3: nat,Y: nat,Z2: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X3 @ Y ) @ ( inf_inf_nat @ X3 @ Z2 ) ) @ ( inf_inf_nat @ X3 @ ( sup_sup_nat @ Y @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_946_set__subset__Cons,axiom,
! [Xs: list_Sum_sum_a_nat,X3: sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) @ ( set_Sum_sum_a_nat2 @ ( cons_Sum_sum_a_nat @ X3 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_947_set__subset__Cons,axiom,
! [Xs: list_nat,X3: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ ( cons_nat @ X3 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_948_None__notin__image__Some,axiom,
! [A: set_nat] :
~ ( member_option_nat @ none_nat @ ( image_nat_option_nat @ some_nat @ A ) ) ).
% None_notin_image_Some
thf(fact_949_add__diff__cancel__right_H,axiom,
! [A3: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A3 @ B2 ) @ B2 )
= A3 ) ).
% add_diff_cancel_right'
thf(fact_950_add__diff__cancel__right,axiom,
! [A3: nat,C: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( minus_minus_nat @ A3 @ B2 ) ) ).
% add_diff_cancel_right
thf(fact_951_add__diff__cancel__left_H,axiom,
! [A3: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A3 @ B2 ) @ A3 )
= B2 ) ).
% add_diff_cancel_left'
thf(fact_952_add__diff__cancel__left,axiom,
! [C: nat,A3: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B2 ) )
= ( minus_minus_nat @ A3 @ B2 ) ) ).
% add_diff_cancel_left
thf(fact_953_add__le__cancel__right,axiom,
! [A3: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( ord_less_eq_nat @ A3 @ B2 ) ) ).
% add_le_cancel_right
thf(fact_954_add__left__cancel,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ( plus_plus_nat @ A3 @ B2 )
= ( plus_plus_nat @ A3 @ C ) )
= ( B2 = C ) ) ).
% add_left_cancel
thf(fact_955_add__right__cancel,axiom,
! [B2: nat,A3: nat,C: nat] :
( ( ( plus_plus_nat @ B2 @ A3 )
= ( plus_plus_nat @ C @ A3 ) )
= ( B2 = C ) ) ).
% add_right_cancel
thf(fact_956_add__le__cancel__left,axiom,
! [C: nat,A3: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ A3 @ B2 ) ) ).
% add_le_cancel_left
thf(fact_957_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A3 @ B2 ) @ C )
= ( plus_plus_nat @ A3 @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_958_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( I2 = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I2 @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_959_group__cancel_Oadd1,axiom,
! [A: nat,K: nat,A3: nat,B2: nat] :
( ( A
= ( plus_plus_nat @ K @ A3 ) )
=> ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A3 @ B2 ) ) ) ) ).
% group_cancel.add1
thf(fact_960_group__cancel_Oadd2,axiom,
! [B: nat,K: nat,B2: nat,A3: nat] :
( ( B
= ( plus_plus_nat @ K @ B2 ) )
=> ( ( plus_plus_nat @ A3 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A3 @ B2 ) ) ) ) ).
% group_cancel.add2
thf(fact_961_add_Oassoc,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A3 @ B2 ) @ C )
= ( plus_plus_nat @ A3 @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add.assoc
thf(fact_962_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A6: nat,B8: nat] : ( plus_plus_nat @ B8 @ A6 ) ) ) ).
% add.commute
thf(fact_963_add_Oleft__commute,axiom,
! [B2: nat,A3: nat,C: nat] :
( ( plus_plus_nat @ B2 @ ( plus_plus_nat @ A3 @ C ) )
= ( plus_plus_nat @ A3 @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add.left_commute
thf(fact_964_add__left__imp__eq,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ( plus_plus_nat @ A3 @ B2 )
= ( plus_plus_nat @ A3 @ C ) )
=> ( B2 = C ) ) ).
% add_left_imp_eq
thf(fact_965_add__right__imp__eq,axiom,
! [B2: nat,A3: nat,C: nat] :
( ( ( plus_plus_nat @ B2 @ A3 )
= ( plus_plus_nat @ C @ A3 ) )
=> ( B2 = C ) ) ).
% add_right_imp_eq
thf(fact_966_diff__right__commute,axiom,
! [A3: nat,C: nat,B2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A3 @ C ) @ B2 )
= ( minus_minus_nat @ ( minus_minus_nat @ A3 @ B2 ) @ C ) ) ).
% diff_right_commute
thf(fact_967_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_968_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( I2 = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_969_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_970_add__mono,axiom,
! [A3: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B2 @ D2 ) ) ) ) ).
% add_mono
thf(fact_971_add__left__mono,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).
% add_left_mono
thf(fact_972_less__eqE,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ~ ! [C5: nat] :
( B2
!= ( plus_plus_nat @ A3 @ C5 ) ) ) ).
% less_eqE
thf(fact_973_add__right__mono,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add_right_mono
thf(fact_974_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A6: nat,B8: nat] :
? [C4: nat] :
( B8
= ( plus_plus_nat @ A6 @ C4 ) ) ) ) ).
% le_iff_add
thf(fact_975_add__le__imp__le__left,axiom,
! [C: nat,A3: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B2 ) )
=> ( ord_less_eq_nat @ A3 @ B2 ) ) ).
% add_le_imp_le_left
thf(fact_976_add__le__imp__le__right,axiom,
! [A3: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
=> ( ord_less_eq_nat @ A3 @ B2 ) ) ).
% add_le_imp_le_right
thf(fact_977_add__implies__diff,axiom,
! [C: nat,B2: nat,A3: nat] :
( ( ( plus_plus_nat @ C @ B2 )
= A3 )
=> ( C
= ( minus_minus_nat @ A3 @ B2 ) ) ) ).
% add_implies_diff
thf(fact_978_diff__diff__eq,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A3 @ B2 ) @ C )
= ( minus_minus_nat @ A3 @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% diff_diff_eq
thf(fact_979_diff__add,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A3 ) @ A3 )
= B2 ) ) ).
% diff_add
thf(fact_980_le__add__diff,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A3 ) ) ) ).
% le_add_diff
thf(fact_981_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B2 @ A3 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A3 ) @ B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_982_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B2 @ A3 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B2 ) @ A3 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_983_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B2 ) @ A3 )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B2 @ A3 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_984_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A3 ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A3 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_985_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A3 )
= ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A3 ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_986_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B2 @ A3 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A3 ) @ B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_987_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( plus_plus_nat @ A3 @ ( minus_minus_nat @ B2 @ A3 ) )
= B2 ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_988_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ( minus_minus_nat @ B2 @ A3 )
= C )
= ( B2
= ( plus_plus_nat @ C @ A3 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_989__092_060open_062set_A_IInl_Ax_A_D_Axs_J_A_N_AInl_A_096_AAD_A_N_Adom_Am_A_061_A_123Inl_Ax_125_A_092_060union_062_A_Iset_Axs_A_N_AInl_A_096_AAD_A_N_Adom_A_Im_IInl_Ax_A_092_060mapsto_062_Ai_J_J_J_092_060close_062,axiom,
( ( minus_1134630996077396038_a_nat @ ( minus_1134630996077396038_a_nat @ ( set_Sum_sum_a_nat2 @ ( cons_Sum_sum_a_nat @ ( sum_Inl_a_nat @ x ) @ xsa ) ) @ ( image_7873763678140191238_a_nat @ sum_Inl_a_nat @ aDa ) ) @ ( dom_Su2255998037560862461at_nat @ ma ) )
= ( sup_su6804446743777130803_a_nat @ ( insert_Sum_sum_a_nat @ ( sum_Inl_a_nat @ x ) @ bot_bo3438331934148233675_a_nat ) @ ( minus_1134630996077396038_a_nat @ ( minus_1134630996077396038_a_nat @ ( set_Sum_sum_a_nat2 @ xsa ) @ ( image_7873763678140191238_a_nat @ sum_Inl_a_nat @ aDa ) ) @ ( dom_Su2255998037560862461at_nat @ ( fun_up4750343006489851234on_nat @ ma @ ( sum_Inl_a_nat @ x ) @ ( some_nat @ ia ) ) ) ) ) ) ).
% \<open>set (Inl x # xs) - Inl ` AD - dom m = {Inl x} \<union> (set xs - Inl ` AD - dom (m(Inl x \<mapsto> i)))\<close>
thf(fact_990_set__union,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( set_Sum_sum_a_nat2 @ ( union_Sum_sum_a_nat @ Xs @ Ys ) )
= ( sup_su6804446743777130803_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) @ ( set_Sum_sum_a_nat2 @ Ys ) ) ) ).
% set_union
thf(fact_991_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X: nat] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_992_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X: a] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_993_empty__Collect__eq,axiom,
! [P: sum_sum_a_nat > $o] :
( ( bot_bo3438331934148233675_a_nat
= ( collec7073057861543223018_a_nat @ P ) )
= ( ! [X: sum_sum_a_nat] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_994_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X: nat] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_995_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X: a] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_996_Collect__empty__eq,axiom,
! [P: sum_sum_a_nat > $o] :
( ( ( collec7073057861543223018_a_nat @ P )
= bot_bo3438331934148233675_a_nat )
= ( ! [X: sum_sum_a_nat] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_997_all__not__in__conv,axiom,
! [A: set_a] :
( ( ! [X: a] :
~ ( member_a @ X @ A ) )
= ( A = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_998_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X: nat] :
~ ( member_nat @ X @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_999_all__not__in__conv,axiom,
! [A: set_Sum_sum_a_nat] :
( ( ! [X: sum_sum_a_nat] :
~ ( member_Sum_sum_a_nat @ X @ A ) )
= ( A = bot_bo3438331934148233675_a_nat ) ) ).
% all_not_in_conv
thf(fact_1000_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_1001_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_1002_empty__iff,axiom,
! [C: sum_sum_a_nat] :
~ ( member_Sum_sum_a_nat @ C @ bot_bo3438331934148233675_a_nat ) ).
% empty_iff
thf(fact_1003_insert__absorb2,axiom,
! [X3: sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( insert_Sum_sum_a_nat @ X3 @ ( insert_Sum_sum_a_nat @ X3 @ A ) )
= ( insert_Sum_sum_a_nat @ X3 @ A ) ) ).
% insert_absorb2
thf(fact_1004_insert__absorb2,axiom,
! [X3: nat,A: set_nat] :
( ( insert_nat @ X3 @ ( insert_nat @ X3 @ A ) )
= ( insert_nat @ X3 @ A ) ) ).
% insert_absorb2
thf(fact_1005_insert__iff,axiom,
! [A3: sum_sum_a_nat,B2: sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ A3 @ ( insert_Sum_sum_a_nat @ B2 @ A ) )
= ( ( A3 = B2 )
| ( member_Sum_sum_a_nat @ A3 @ A ) ) ) ).
% insert_iff
thf(fact_1006_insert__iff,axiom,
! [A3: a,B2: a,A: set_a] :
( ( member_a @ A3 @ ( insert_a @ B2 @ A ) )
= ( ( A3 = B2 )
| ( member_a @ A3 @ A ) ) ) ).
% insert_iff
thf(fact_1007_insert__iff,axiom,
! [A3: nat,B2: nat,A: set_nat] :
( ( member_nat @ A3 @ ( insert_nat @ B2 @ A ) )
= ( ( A3 = B2 )
| ( member_nat @ A3 @ A ) ) ) ).
% insert_iff
thf(fact_1008_insertCI,axiom,
! [A3: sum_sum_a_nat,B: set_Sum_sum_a_nat,B2: sum_sum_a_nat] :
( ( ~ ( member_Sum_sum_a_nat @ A3 @ B )
=> ( A3 = B2 ) )
=> ( member_Sum_sum_a_nat @ A3 @ ( insert_Sum_sum_a_nat @ B2 @ B ) ) ) ).
% insertCI
thf(fact_1009_insertCI,axiom,
! [A3: a,B: set_a,B2: a] :
( ( ~ ( member_a @ A3 @ B )
=> ( A3 = B2 ) )
=> ( member_a @ A3 @ ( insert_a @ B2 @ B ) ) ) ).
% insertCI
thf(fact_1010_insertCI,axiom,
! [A3: nat,B: set_nat,B2: nat] :
( ( ~ ( member_nat @ A3 @ B )
=> ( A3 = B2 ) )
=> ( member_nat @ A3 @ ( insert_nat @ B2 @ B ) ) ) ).
% insertCI
thf(fact_1011_fun__upd__upd,axiom,
! [F: sum_sum_a_nat > option_nat,X3: sum_sum_a_nat,Y: option_nat,Z2: option_nat] :
( ( fun_up4750343006489851234on_nat @ ( fun_up4750343006489851234on_nat @ F @ X3 @ Y ) @ X3 @ Z2 )
= ( fun_up4750343006489851234on_nat @ F @ X3 @ Z2 ) ) ).
% fun_upd_upd
thf(fact_1012_fun__upd__triv,axiom,
! [F: sum_sum_a_nat > option_nat,X3: sum_sum_a_nat] :
( ( fun_up4750343006489851234on_nat @ F @ X3 @ ( F @ X3 ) )
= F ) ).
% fun_upd_triv
thf(fact_1013_fun__upd__apply,axiom,
( fun_up4750343006489851234on_nat
= ( ^ [F4: sum_sum_a_nat > option_nat,X: sum_sum_a_nat,Y4: option_nat,Z4: sum_sum_a_nat] : ( if_option_nat @ ( Z4 = X ) @ Y4 @ ( F4 @ Z4 ) ) ) ) ).
% fun_upd_apply
thf(fact_1014_image__is__empty,axiom,
! [F: nat > sum_sum_a_nat,A: set_nat] :
( ( ( image_7293268710728258664_a_nat @ F @ A )
= bot_bo3438331934148233675_a_nat )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_1015_image__is__empty,axiom,
! [F: a > sum_sum_a_nat,A: set_a] :
( ( ( image_7873763678140191238_a_nat @ F @ A )
= bot_bo3438331934148233675_a_nat )
= ( A = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_1016_image__is__empty,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( ( image_7142520692256960453_a_nat @ F @ A )
= bot_bo3438331934148233675_a_nat )
= ( A = bot_bo3438331934148233675_a_nat ) ) ).
% image_is_empty
thf(fact_1017_empty__is__image,axiom,
! [F: nat > sum_sum_a_nat,A: set_nat] :
( ( bot_bo3438331934148233675_a_nat
= ( image_7293268710728258664_a_nat @ F @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_1018_empty__is__image,axiom,
! [F: a > sum_sum_a_nat,A: set_a] :
( ( bot_bo3438331934148233675_a_nat
= ( image_7873763678140191238_a_nat @ F @ A ) )
= ( A = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_1019_empty__is__image,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( bot_bo3438331934148233675_a_nat
= ( image_7142520692256960453_a_nat @ F @ A ) )
= ( A = bot_bo3438331934148233675_a_nat ) ) ).
% empty_is_image
thf(fact_1020_image__empty,axiom,
! [F: nat > sum_sum_a_nat] :
( ( image_7293268710728258664_a_nat @ F @ bot_bot_set_nat )
= bot_bo3438331934148233675_a_nat ) ).
% image_empty
thf(fact_1021_image__empty,axiom,
! [F: a > sum_sum_a_nat] :
( ( image_7873763678140191238_a_nat @ F @ bot_bot_set_a )
= bot_bo3438331934148233675_a_nat ) ).
% image_empty
thf(fact_1022_image__empty,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat] :
( ( image_7142520692256960453_a_nat @ F @ bot_bo3438331934148233675_a_nat )
= bot_bo3438331934148233675_a_nat ) ).
% image_empty
thf(fact_1023_subset__empty,axiom,
! [A: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ bot_bo3438331934148233675_a_nat )
= ( A = bot_bo3438331934148233675_a_nat ) ) ).
% subset_empty
thf(fact_1024_subset__empty,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_1025_empty__subsetI,axiom,
! [A: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ bot_bo3438331934148233675_a_nat @ A ) ).
% empty_subsetI
thf(fact_1026_empty__subsetI,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% empty_subsetI
thf(fact_1027_insert__image,axiom,
! [X3: sum_sum_a_nat,A: set_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X3 @ A )
=> ( ( insert_Sum_sum_a_nat @ ( F @ X3 ) @ ( image_7142520692256960453_a_nat @ F @ A ) )
= ( image_7142520692256960453_a_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_1028_insert__image,axiom,
! [X3: a,A: set_a,F: a > sum_sum_a_nat] :
( ( member_a @ X3 @ A )
=> ( ( insert_Sum_sum_a_nat @ ( F @ X3 ) @ ( image_7873763678140191238_a_nat @ F @ A ) )
= ( image_7873763678140191238_a_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_1029_insert__image,axiom,
! [X3: a,A: set_a,F: a > nat] :
( ( member_a @ X3 @ A )
=> ( ( insert_nat @ ( F @ X3 ) @ ( image_a_nat @ F @ A ) )
= ( image_a_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_1030_insert__image,axiom,
! [X3: nat,A: set_nat,F: nat > sum_sum_a_nat] :
( ( member_nat @ X3 @ A )
=> ( ( insert_Sum_sum_a_nat @ ( F @ X3 ) @ ( image_7293268710728258664_a_nat @ F @ A ) )
= ( image_7293268710728258664_a_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_1031_insert__image,axiom,
! [X3: nat,A: set_nat,F: nat > nat] :
( ( member_nat @ X3 @ A )
=> ( ( insert_nat @ ( F @ X3 ) @ ( image_nat_nat @ F @ A ) )
= ( image_nat_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_1032_image__insert,axiom,
! [F: a > sum_sum_a_nat,A3: a,B: set_a] :
( ( image_7873763678140191238_a_nat @ F @ ( insert_a @ A3 @ B ) )
= ( insert_Sum_sum_a_nat @ ( F @ A3 ) @ ( image_7873763678140191238_a_nat @ F @ B ) ) ) ).
% image_insert
thf(fact_1033_image__insert,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,A3: sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( image_7142520692256960453_a_nat @ F @ ( insert_Sum_sum_a_nat @ A3 @ B ) )
= ( insert_Sum_sum_a_nat @ ( F @ A3 ) @ ( image_7142520692256960453_a_nat @ F @ B ) ) ) ).
% image_insert
thf(fact_1034_image__insert,axiom,
! [F: sum_sum_a_nat > nat,A3: sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( image_2473878607534554506at_nat @ F @ ( insert_Sum_sum_a_nat @ A3 @ B ) )
= ( insert_nat @ ( F @ A3 ) @ ( image_2473878607534554506at_nat @ F @ B ) ) ) ).
% image_insert
thf(fact_1035_image__insert,axiom,
! [F: nat > sum_sum_a_nat,A3: nat,B: set_nat] :
( ( image_7293268710728258664_a_nat @ F @ ( insert_nat @ A3 @ B ) )
= ( insert_Sum_sum_a_nat @ ( F @ A3 ) @ ( image_7293268710728258664_a_nat @ F @ B ) ) ) ).
% image_insert
thf(fact_1036_image__insert,axiom,
! [F: nat > nat,A3: nat,B: set_nat] :
( ( image_nat_nat @ F @ ( insert_nat @ A3 @ B ) )
= ( insert_nat @ ( F @ A3 ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_insert
thf(fact_1037_sup__bot_Oright__neutral,axiom,
! [A3: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A3 @ bot_bo3438331934148233675_a_nat )
= A3 ) ).
% sup_bot.right_neutral
thf(fact_1038_sup__bot_Oneutr__eq__iff,axiom,
! [A3: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( bot_bo3438331934148233675_a_nat
= ( sup_su6804446743777130803_a_nat @ A3 @ B2 ) )
= ( ( A3 = bot_bo3438331934148233675_a_nat )
& ( B2 = bot_bo3438331934148233675_a_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_1039_sup__bot_Oleft__neutral,axiom,
! [A3: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ bot_bo3438331934148233675_a_nat @ A3 )
= A3 ) ).
% sup_bot.left_neutral
thf(fact_1040_sup__bot_Oeq__neutr__iff,axiom,
! [A3: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ( sup_su6804446743777130803_a_nat @ A3 @ B2 )
= bot_bo3438331934148233675_a_nat )
= ( ( A3 = bot_bo3438331934148233675_a_nat )
& ( B2 = bot_bo3438331934148233675_a_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_1041_sup__eq__bot__iff,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
( ( ( sup_su6804446743777130803_a_nat @ X3 @ Y )
= bot_bo3438331934148233675_a_nat )
= ( ( X3 = bot_bo3438331934148233675_a_nat )
& ( Y = bot_bo3438331934148233675_a_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_1042_bot__eq__sup__iff,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
( ( bot_bo3438331934148233675_a_nat
= ( sup_su6804446743777130803_a_nat @ X3 @ Y ) )
= ( ( X3 = bot_bo3438331934148233675_a_nat )
& ( Y = bot_bo3438331934148233675_a_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_1043_sup__bot__right,axiom,
! [X3: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ X3 @ bot_bo3438331934148233675_a_nat )
= X3 ) ).
% sup_bot_right
thf(fact_1044_sup__bot__left,axiom,
! [X3: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ bot_bo3438331934148233675_a_nat @ X3 )
= X3 ) ).
% sup_bot_left
thf(fact_1045_boolean__algebra_Oconj__zero__right,axiom,
! [X3: set_nat] :
( ( inf_inf_set_nat @ X3 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_1046_boolean__algebra_Oconj__zero__right,axiom,
! [X3: set_a] :
( ( inf_inf_set_a @ X3 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_1047_boolean__algebra_Oconj__zero__right,axiom,
! [X3: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ X3 @ bot_bo3438331934148233675_a_nat )
= bot_bo3438331934148233675_a_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_1048_boolean__algebra_Oconj__zero__left,axiom,
! [X3: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X3 )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_1049_boolean__algebra_Oconj__zero__left,axiom,
! [X3: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X3 )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_1050_boolean__algebra_Oconj__zero__left,axiom,
! [X3: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ bot_bo3438331934148233675_a_nat @ X3 )
= bot_bo3438331934148233675_a_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_1051_inf__bot__right,axiom,
! [X3: set_nat] :
( ( inf_inf_set_nat @ X3 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% inf_bot_right
thf(fact_1052_inf__bot__right,axiom,
! [X3: set_a] :
( ( inf_inf_set_a @ X3 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% inf_bot_right
thf(fact_1053_inf__bot__right,axiom,
! [X3: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ X3 @ bot_bo3438331934148233675_a_nat )
= bot_bo3438331934148233675_a_nat ) ).
% inf_bot_right
thf(fact_1054_inf__bot__left,axiom,
! [X3: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X3 )
= bot_bot_set_nat ) ).
% inf_bot_left
thf(fact_1055_inf__bot__left,axiom,
! [X3: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X3 )
= bot_bot_set_a ) ).
% inf_bot_left
thf(fact_1056_inf__bot__left,axiom,
! [X3: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ bot_bo3438331934148233675_a_nat @ X3 )
= bot_bo3438331934148233675_a_nat ) ).
% inf_bot_left
thf(fact_1057_singletonI,axiom,
! [A3: a] : ( member_a @ A3 @ ( insert_a @ A3 @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_1058_singletonI,axiom,
! [A3: nat] : ( member_nat @ A3 @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_1059_singletonI,axiom,
! [A3: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ A3 @ ( insert_Sum_sum_a_nat @ A3 @ bot_bo3438331934148233675_a_nat ) ) ).
% singletonI
thf(fact_1060_finite__insert,axiom,
! [A3: sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( finite502105017643426984_a_nat @ ( insert_Sum_sum_a_nat @ A3 @ A ) )
= ( finite502105017643426984_a_nat @ A ) ) ).
% finite_insert
thf(fact_1061_finite__insert,axiom,
! [A3: nat,A: set_nat] :
( ( finite_finite_nat @ ( insert_nat @ A3 @ A ) )
= ( finite_finite_nat @ A ) ) ).
% finite_insert
thf(fact_1062_insert__subset,axiom,
! [X3: sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( insert_Sum_sum_a_nat @ X3 @ A ) @ B )
= ( ( member_Sum_sum_a_nat @ X3 @ B )
& ( ord_le1325389633284124927_a_nat @ A @ B ) ) ) ).
% insert_subset
thf(fact_1063_insert__subset,axiom,
! [X3: a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X3 @ A ) @ B )
= ( ( member_a @ X3 @ B )
& ( ord_less_eq_set_a @ A @ B ) ) ) ).
% insert_subset
thf(fact_1064_insert__subset,axiom,
! [X3: nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X3 @ A ) @ B )
= ( ( member_nat @ X3 @ B )
& ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% insert_subset
thf(fact_1065_Un__empty,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ( sup_su6804446743777130803_a_nat @ A @ B )
= bot_bo3438331934148233675_a_nat )
= ( ( A = bot_bo3438331934148233675_a_nat )
& ( B = bot_bo3438331934148233675_a_nat ) ) ) ).
% Un_empty
thf(fact_1066_Diff__cancel,axiom,
! [A: set_Sum_sum_a_nat] :
( ( minus_1134630996077396038_a_nat @ A @ A )
= bot_bo3438331934148233675_a_nat ) ).
% Diff_cancel
thf(fact_1067_empty__Diff,axiom,
! [A: set_Sum_sum_a_nat] :
( ( minus_1134630996077396038_a_nat @ bot_bo3438331934148233675_a_nat @ A )
= bot_bo3438331934148233675_a_nat ) ).
% empty_Diff
thf(fact_1068_Diff__empty,axiom,
! [A: set_Sum_sum_a_nat] :
( ( minus_1134630996077396038_a_nat @ A @ bot_bo3438331934148233675_a_nat )
= A ) ).
% Diff_empty
thf(fact_1069_Int__insert__right__if1,axiom,
! [A3: sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ A3 @ A )
=> ( ( inf_in7084830621192376909_a_nat @ A @ ( insert_Sum_sum_a_nat @ A3 @ B ) )
= ( insert_Sum_sum_a_nat @ A3 @ ( inf_in7084830621192376909_a_nat @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1070_Int__insert__right__if1,axiom,
! [A3: nat,A: set_nat,B: set_nat] :
( ( member_nat @ A3 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A3 @ B ) )
= ( insert_nat @ A3 @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1071_Int__insert__right__if1,axiom,
! [A3: a,A: set_a,B: set_a] :
( ( member_a @ A3 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A3 @ B ) )
= ( insert_a @ A3 @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1072_Int__insert__right__if0,axiom,
! [A3: sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ~ ( member_Sum_sum_a_nat @ A3 @ A )
=> ( ( inf_in7084830621192376909_a_nat @ A @ ( insert_Sum_sum_a_nat @ A3 @ B ) )
= ( inf_in7084830621192376909_a_nat @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_1073_Int__insert__right__if0,axiom,
! [A3: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ A3 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A3 @ B ) )
= ( inf_inf_set_nat @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_1074_Int__insert__right__if0,axiom,
! [A3: a,A: set_a,B: set_a] :
( ~ ( member_a @ A3 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A3 @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_1075_insert__inter__insert,axiom,
! [A3: sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ ( insert_Sum_sum_a_nat @ A3 @ A ) @ ( insert_Sum_sum_a_nat @ A3 @ B ) )
= ( insert_Sum_sum_a_nat @ A3 @ ( inf_in7084830621192376909_a_nat @ A @ B ) ) ) ).
% insert_inter_insert
thf(fact_1076_insert__inter__insert,axiom,
! [A3: nat,A: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ ( insert_nat @ A3 @ A ) @ ( insert_nat @ A3 @ B ) )
= ( insert_nat @ A3 @ ( inf_inf_set_nat @ A @ B ) ) ) ).
% insert_inter_insert
thf(fact_1077_insert__inter__insert,axiom,
! [A3: a,A: set_a,B: set_a] :
( ( inf_inf_set_a @ ( insert_a @ A3 @ A ) @ ( insert_a @ A3 @ B ) )
= ( insert_a @ A3 @ ( inf_inf_set_a @ A @ B ) ) ) ).
% insert_inter_insert
thf(fact_1078_Int__insert__left__if1,axiom,
! [A3: sum_sum_a_nat,C2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ A3 @ C2 )
=> ( ( inf_in7084830621192376909_a_nat @ ( insert_Sum_sum_a_nat @ A3 @ B ) @ C2 )
= ( insert_Sum_sum_a_nat @ A3 @ ( inf_in7084830621192376909_a_nat @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1079_Int__insert__left__if1,axiom,
! [A3: nat,C2: set_nat,B: set_nat] :
( ( member_nat @ A3 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A3 @ B ) @ C2 )
= ( insert_nat @ A3 @ ( inf_inf_set_nat @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1080_Int__insert__left__if1,axiom,
! [A3: a,C2: set_a,B: set_a] :
( ( member_a @ A3 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A3 @ B ) @ C2 )
= ( insert_a @ A3 @ ( inf_inf_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1081_Int__insert__left__if0,axiom,
! [A3: sum_sum_a_nat,C2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ~ ( member_Sum_sum_a_nat @ A3 @ C2 )
=> ( ( inf_in7084830621192376909_a_nat @ ( insert_Sum_sum_a_nat @ A3 @ B ) @ C2 )
= ( inf_in7084830621192376909_a_nat @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_1082_Int__insert__left__if0,axiom,
! [A3: nat,C2: set_nat,B: set_nat] :
( ~ ( member_nat @ A3 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A3 @ B ) @ C2 )
= ( inf_inf_set_nat @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_1083_Int__insert__left__if0,axiom,
! [A3: a,C2: set_a,B: set_a] :
( ~ ( member_a @ A3 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A3 @ B ) @ C2 )
= ( inf_inf_set_a @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_1084_Un__insert__right,axiom,
! [A: set_nat,A3: nat,B: set_nat] :
( ( sup_sup_set_nat @ A @ ( insert_nat @ A3 @ B ) )
= ( insert_nat @ A3 @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% Un_insert_right
thf(fact_1085_Un__insert__right,axiom,
! [A: set_Sum_sum_a_nat,A3: sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A @ ( insert_Sum_sum_a_nat @ A3 @ B ) )
= ( insert_Sum_sum_a_nat @ A3 @ ( sup_su6804446743777130803_a_nat @ A @ B ) ) ) ).
% Un_insert_right
thf(fact_1086_Un__insert__left,axiom,
! [A3: nat,B: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( insert_nat @ A3 @ B ) @ C2 )
= ( insert_nat @ A3 @ ( sup_sup_set_nat @ B @ C2 ) ) ) ).
% Un_insert_left
thf(fact_1087_Un__insert__left,axiom,
! [A3: sum_sum_a_nat,B: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ ( insert_Sum_sum_a_nat @ A3 @ B ) @ C2 )
= ( insert_Sum_sum_a_nat @ A3 @ ( sup_su6804446743777130803_a_nat @ B @ C2 ) ) ) ).
% Un_insert_left
thf(fact_1088_insert__Diff1,axiom,
! [X3: a,B: set_a,A: set_a] :
( ( member_a @ X3 @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X3 @ A ) @ B )
= ( minus_minus_set_a @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_1089_insert__Diff1,axiom,
! [X3: nat,B: set_nat,A: set_nat] :
( ( member_nat @ X3 @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X3 @ A ) @ B )
= ( minus_minus_set_nat @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_1090_insert__Diff1,axiom,
! [X3: sum_sum_a_nat,B: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X3 @ B )
=> ( ( minus_1134630996077396038_a_nat @ ( insert_Sum_sum_a_nat @ X3 @ A ) @ B )
= ( minus_1134630996077396038_a_nat @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_1091_Diff__insert0,axiom,
! [X3: a,A: set_a,B: set_a] :
( ~ ( member_a @ X3 @ A )
=> ( ( minus_minus_set_a @ A @ ( insert_a @ X3 @ B ) )
= ( minus_minus_set_a @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_1092_Diff__insert0,axiom,
! [X3: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ X3 @ A )
=> ( ( minus_minus_set_nat @ A @ ( insert_nat @ X3 @ B ) )
= ( minus_minus_set_nat @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_1093_Diff__insert0,axiom,
! [X3: sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ~ ( member_Sum_sum_a_nat @ X3 @ A )
=> ( ( minus_1134630996077396038_a_nat @ A @ ( insert_Sum_sum_a_nat @ X3 @ B ) )
= ( minus_1134630996077396038_a_nat @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_1094_singleton__conv,axiom,
! [A3: nat] :
( ( collect_nat
@ ^ [X: nat] : ( X = A3 ) )
= ( insert_nat @ A3 @ bot_bot_set_nat ) ) ).
% singleton_conv
thf(fact_1095_singleton__conv,axiom,
! [A3: a] :
( ( collect_a
@ ^ [X: a] : ( X = A3 ) )
= ( insert_a @ A3 @ bot_bot_set_a ) ) ).
% singleton_conv
thf(fact_1096_singleton__conv,axiom,
! [A3: sum_sum_a_nat] :
( ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] : ( X = A3 ) )
= ( insert_Sum_sum_a_nat @ A3 @ bot_bo3438331934148233675_a_nat ) ) ).
% singleton_conv
thf(fact_1097_singleton__conv2,axiom,
! [A3: nat] :
( ( collect_nat
@ ( ^ [Y3: nat,Z: nat] : ( Y3 = Z )
@ A3 ) )
= ( insert_nat @ A3 @ bot_bot_set_nat ) ) ).
% singleton_conv2
thf(fact_1098_singleton__conv2,axiom,
! [A3: a] :
( ( collect_a
@ ( ^ [Y3: a,Z: a] : ( Y3 = Z )
@ A3 ) )
= ( insert_a @ A3 @ bot_bot_set_a ) ) ).
% singleton_conv2
thf(fact_1099_singleton__conv2,axiom,
! [A3: sum_sum_a_nat] :
( ( collec7073057861543223018_a_nat
@ ( ^ [Y3: sum_sum_a_nat,Z: sum_sum_a_nat] : ( Y3 = Z )
@ A3 ) )
= ( insert_Sum_sum_a_nat @ A3 @ bot_bo3438331934148233675_a_nat ) ) ).
% singleton_conv2
thf(fact_1100_list_Osimps_I15_J,axiom,
! [X21: nat,X22: list_nat] :
( ( set_nat2 @ ( cons_nat @ X21 @ X22 ) )
= ( insert_nat @ X21 @ ( set_nat2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_1101_list_Osimps_I15_J,axiom,
! [X21: sum_sum_a_nat,X22: list_Sum_sum_a_nat] :
( ( set_Sum_sum_a_nat2 @ ( cons_Sum_sum_a_nat @ X21 @ X22 ) )
= ( insert_Sum_sum_a_nat @ X21 @ ( set_Sum_sum_a_nat2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_1102_singleton__insert__inj__eq,axiom,
! [B2: sum_sum_a_nat,A3: sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( ( insert_Sum_sum_a_nat @ B2 @ bot_bo3438331934148233675_a_nat )
= ( insert_Sum_sum_a_nat @ A3 @ A ) )
= ( ( A3 = B2 )
& ( ord_le1325389633284124927_a_nat @ A @ ( insert_Sum_sum_a_nat @ B2 @ bot_bo3438331934148233675_a_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_1103_singleton__insert__inj__eq,axiom,
! [B2: nat,A3: nat,A: set_nat] :
( ( ( insert_nat @ B2 @ bot_bot_set_nat )
= ( insert_nat @ A3 @ A ) )
= ( ( A3 = B2 )
& ( ord_less_eq_set_nat @ A @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_1104_singleton__insert__inj__eq_H,axiom,
! [A3: sum_sum_a_nat,A: set_Sum_sum_a_nat,B2: sum_sum_a_nat] :
( ( ( insert_Sum_sum_a_nat @ A3 @ A )
= ( insert_Sum_sum_a_nat @ B2 @ bot_bo3438331934148233675_a_nat ) )
= ( ( A3 = B2 )
& ( ord_le1325389633284124927_a_nat @ A @ ( insert_Sum_sum_a_nat @ B2 @ bot_bo3438331934148233675_a_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_1105_singleton__insert__inj__eq_H,axiom,
! [A3: nat,A: set_nat,B2: nat] :
( ( ( insert_nat @ A3 @ A )
= ( insert_nat @ B2 @ bot_bot_set_nat ) )
= ( ( A3 = B2 )
& ( ord_less_eq_set_nat @ A @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_1106_insert__disjoint_I1_J,axiom,
! [A3: nat,A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ ( insert_nat @ A3 @ A ) @ B )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A3 @ B )
& ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_1107_insert__disjoint_I1_J,axiom,
! [A3: a,A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A3 @ A ) @ B )
= bot_bot_set_a )
= ( ~ ( member_a @ A3 @ B )
& ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_1108_insert__disjoint_I1_J,axiom,
! [A3: sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ( inf_in7084830621192376909_a_nat @ ( insert_Sum_sum_a_nat @ A3 @ A ) @ B )
= bot_bo3438331934148233675_a_nat )
= ( ~ ( member_Sum_sum_a_nat @ A3 @ B )
& ( ( inf_in7084830621192376909_a_nat @ A @ B )
= bot_bo3438331934148233675_a_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_1109_insert__disjoint_I2_J,axiom,
! [A3: nat,A: set_nat,B: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ ( insert_nat @ A3 @ A ) @ B ) )
= ( ~ ( member_nat @ A3 @ B )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1110_insert__disjoint_I2_J,axiom,
! [A3: a,A: set_a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A3 @ A ) @ B ) )
= ( ~ ( member_a @ A3 @ B )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1111_insert__disjoint_I2_J,axiom,
! [A3: sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( bot_bo3438331934148233675_a_nat
= ( inf_in7084830621192376909_a_nat @ ( insert_Sum_sum_a_nat @ A3 @ A ) @ B ) )
= ( ~ ( member_Sum_sum_a_nat @ A3 @ B )
& ( bot_bo3438331934148233675_a_nat
= ( inf_in7084830621192376909_a_nat @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1112_disjoint__insert_I1_J,axiom,
! [B: set_nat,A3: nat,A: set_nat] :
( ( ( inf_inf_set_nat @ B @ ( insert_nat @ A3 @ A ) )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A3 @ B )
& ( ( inf_inf_set_nat @ B @ A )
= bot_bot_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_1113_disjoint__insert_I1_J,axiom,
! [B: set_a,A3: a,A: set_a] :
( ( ( inf_inf_set_a @ B @ ( insert_a @ A3 @ A ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A3 @ B )
& ( ( inf_inf_set_a @ B @ A )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_1114_disjoint__insert_I1_J,axiom,
! [B: set_Sum_sum_a_nat,A3: sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( ( inf_in7084830621192376909_a_nat @ B @ ( insert_Sum_sum_a_nat @ A3 @ A ) )
= bot_bo3438331934148233675_a_nat )
= ( ~ ( member_Sum_sum_a_nat @ A3 @ B )
& ( ( inf_in7084830621192376909_a_nat @ B @ A )
= bot_bo3438331934148233675_a_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_1115_disjoint__insert_I2_J,axiom,
! [A: set_nat,B2: nat,B: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ ( insert_nat @ B2 @ B ) ) )
= ( ~ ( member_nat @ B2 @ A )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1116_disjoint__insert_I2_J,axiom,
! [A: set_a,B2: a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A @ ( insert_a @ B2 @ B ) ) )
= ( ~ ( member_a @ B2 @ A )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1117_disjoint__insert_I2_J,axiom,
! [A: set_Sum_sum_a_nat,B2: sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( bot_bo3438331934148233675_a_nat
= ( inf_in7084830621192376909_a_nat @ A @ ( insert_Sum_sum_a_nat @ B2 @ B ) ) )
= ( ~ ( member_Sum_sum_a_nat @ B2 @ A )
& ( bot_bo3438331934148233675_a_nat
= ( inf_in7084830621192376909_a_nat @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1118_Diff__eq__empty__iff,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ( minus_1134630996077396038_a_nat @ A @ B )
= bot_bo3438331934148233675_a_nat )
= ( ord_le1325389633284124927_a_nat @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_1119_Diff__eq__empty__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( minus_minus_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_1120_insert__Diff__single,axiom,
! [A3: nat,A: set_nat] :
( ( insert_nat @ A3 @ ( minus_minus_set_nat @ A @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) )
= ( insert_nat @ A3 @ A ) ) ).
% insert_Diff_single
thf(fact_1121_insert__Diff__single,axiom,
! [A3: sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( insert_Sum_sum_a_nat @ A3 @ ( minus_1134630996077396038_a_nat @ A @ ( insert_Sum_sum_a_nat @ A3 @ bot_bo3438331934148233675_a_nat ) ) )
= ( insert_Sum_sum_a_nat @ A3 @ A ) ) ).
% insert_Diff_single
thf(fact_1122_finite__Diff__insert,axiom,
! [A: set_nat,A3: nat,B: set_nat] :
( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ A3 @ B ) ) )
= ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) ) ) ).
% finite_Diff_insert
thf(fact_1123_finite__Diff__insert,axiom,
! [A: set_Sum_sum_a_nat,A3: sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( finite502105017643426984_a_nat @ ( minus_1134630996077396038_a_nat @ A @ ( insert_Sum_sum_a_nat @ A3 @ B ) ) )
= ( finite502105017643426984_a_nat @ ( minus_1134630996077396038_a_nat @ A @ B ) ) ) ).
% finite_Diff_insert
thf(fact_1124_Diff__disjoint,axiom,
! [A: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ A @ ( minus_minus_set_nat @ B @ A ) )
= bot_bot_set_nat ) ).
% Diff_disjoint
thf(fact_1125_Diff__disjoint,axiom,
! [A: set_a,B: set_a] :
( ( inf_inf_set_a @ A @ ( minus_minus_set_a @ B @ A ) )
= bot_bot_set_a ) ).
% Diff_disjoint
thf(fact_1126_Diff__disjoint,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ A @ ( minus_1134630996077396038_a_nat @ B @ A ) )
= bot_bo3438331934148233675_a_nat ) ).
% Diff_disjoint
thf(fact_1127_dom__eq__empty__conv,axiom,
! [F: sum_sum_a_nat > option_nat] :
( ( ( dom_Su2255998037560862461at_nat @ F )
= bot_bo3438331934148233675_a_nat )
= ( F
= ( ^ [X: sum_sum_a_nat] : none_nat ) ) ) ).
% dom_eq_empty_conv
thf(fact_1128_dom__empty,axiom,
( ( dom_Su2255998037560862461at_nat
@ ^ [X: sum_sum_a_nat] : none_nat )
= bot_bo3438331934148233675_a_nat ) ).
% dom_empty
thf(fact_1129_ran__empty,axiom,
( ( ran_Su5179294584019872288at_nat
@ ^ [X: sum_sum_a_nat] : none_nat )
= bot_bot_set_nat ) ).
% ran_empty
thf(fact_1130_card__insert__disjoint,axiom,
! [A: set_a,X3: a] :
( ( finite_finite_a @ A )
=> ( ~ ( member_a @ X3 @ A )
=> ( ( finite_card_a @ ( insert_a @ X3 @ A ) )
= ( suc @ ( finite_card_a @ A ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_1131_card__insert__disjoint,axiom,
! [A: set_Sum_sum_a_nat,X3: sum_sum_a_nat] :
( ( finite502105017643426984_a_nat @ A )
=> ( ~ ( member_Sum_sum_a_nat @ X3 @ A )
=> ( ( finite6080979521523705895_a_nat @ ( insert_Sum_sum_a_nat @ X3 @ A ) )
= ( suc @ ( finite6080979521523705895_a_nat @ A ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_1132_card__insert__disjoint,axiom,
! [A: set_nat,X3: nat] :
( ( finite_finite_nat @ A )
=> ( ~ ( member_nat @ X3 @ A )
=> ( ( finite_card_nat @ ( insert_nat @ X3 @ A ) )
= ( suc @ ( finite_card_nat @ A ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_1133_single__Diff__lessThan,axiom,
! [K: nat] :
( ( minus_minus_set_nat @ ( insert_nat @ K @ bot_bot_set_nat ) @ ( set_ord_lessThan_nat @ K ) )
= ( insert_nat @ K @ bot_bot_set_nat ) ) ).
% single_Diff_lessThan
thf(fact_1134_ran__map__upd,axiom,
! [M2: sum_sum_a_nat > option_nat,A3: sum_sum_a_nat,B2: nat] :
( ( ( M2 @ A3 )
= none_nat )
=> ( ( ran_Su5179294584019872288at_nat @ ( fun_up4750343006489851234on_nat @ M2 @ A3 @ ( some_nat @ B2 ) ) )
= ( insert_nat @ B2 @ ( ran_Su5179294584019872288at_nat @ M2 ) ) ) ) ).
% ran_map_upd
thf(fact_1135_dom__fun__upd,axiom,
! [Y: option_nat,F: nat > option_nat,X3: nat] :
( ( ( Y = none_nat )
=> ( ( dom_nat_nat @ ( fun_up1493157387958331631on_nat @ F @ X3 @ Y ) )
= ( minus_minus_set_nat @ ( dom_nat_nat @ F ) @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) )
& ( ( Y != none_nat )
=> ( ( dom_nat_nat @ ( fun_up1493157387958331631on_nat @ F @ X3 @ Y ) )
= ( insert_nat @ X3 @ ( dom_nat_nat @ F ) ) ) ) ) ).
% dom_fun_upd
thf(fact_1136_dom__fun__upd,axiom,
! [Y: option_nat,F: sum_sum_a_nat > option_nat,X3: sum_sum_a_nat] :
( ( ( Y = none_nat )
=> ( ( dom_Su2255998037560862461at_nat @ ( fun_up4750343006489851234on_nat @ F @ X3 @ Y ) )
= ( minus_1134630996077396038_a_nat @ ( dom_Su2255998037560862461at_nat @ F ) @ ( insert_Sum_sum_a_nat @ X3 @ bot_bo3438331934148233675_a_nat ) ) ) )
& ( ( Y != none_nat )
=> ( ( dom_Su2255998037560862461at_nat @ ( fun_up4750343006489851234on_nat @ F @ X3 @ Y ) )
= ( insert_Sum_sum_a_nat @ X3 @ ( dom_Su2255998037560862461at_nat @ F ) ) ) ) ) ).
% dom_fun_upd
thf(fact_1137_singleton__Un__iff,axiom,
! [X3: nat,A: set_nat,B: set_nat] :
( ( ( insert_nat @ X3 @ bot_bot_set_nat )
= ( sup_sup_set_nat @ A @ B ) )
= ( ( ( A = bot_bot_set_nat )
& ( B
= ( insert_nat @ X3 @ bot_bot_set_nat ) ) )
| ( ( A
= ( insert_nat @ X3 @ bot_bot_set_nat ) )
& ( B = bot_bot_set_nat ) )
| ( ( A
= ( insert_nat @ X3 @ bot_bot_set_nat ) )
& ( B
= ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_1138_singleton__Un__iff,axiom,
! [X3: sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ( insert_Sum_sum_a_nat @ X3 @ bot_bo3438331934148233675_a_nat )
= ( sup_su6804446743777130803_a_nat @ A @ B ) )
= ( ( ( A = bot_bo3438331934148233675_a_nat )
& ( B
= ( insert_Sum_sum_a_nat @ X3 @ bot_bo3438331934148233675_a_nat ) ) )
| ( ( A
= ( insert_Sum_sum_a_nat @ X3 @ bot_bo3438331934148233675_a_nat ) )
& ( B = bot_bo3438331934148233675_a_nat ) )
| ( ( A
= ( insert_Sum_sum_a_nat @ X3 @ bot_bo3438331934148233675_a_nat ) )
& ( B
= ( insert_Sum_sum_a_nat @ X3 @ bot_bo3438331934148233675_a_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_1139_Un__singleton__iff,axiom,
! [A: set_nat,B: set_nat,X3: nat] :
( ( ( sup_sup_set_nat @ A @ B )
= ( insert_nat @ X3 @ bot_bot_set_nat ) )
= ( ( ( A = bot_bot_set_nat )
& ( B
= ( insert_nat @ X3 @ bot_bot_set_nat ) ) )
| ( ( A
= ( insert_nat @ X3 @ bot_bot_set_nat ) )
& ( B = bot_bot_set_nat ) )
| ( ( A
= ( insert_nat @ X3 @ bot_bot_set_nat ) )
& ( B
= ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_1140_Un__singleton__iff,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,X3: sum_sum_a_nat] :
( ( ( sup_su6804446743777130803_a_nat @ A @ B )
= ( insert_Sum_sum_a_nat @ X3 @ bot_bo3438331934148233675_a_nat ) )
= ( ( ( A = bot_bo3438331934148233675_a_nat )
& ( B
= ( insert_Sum_sum_a_nat @ X3 @ bot_bo3438331934148233675_a_nat ) ) )
| ( ( A
= ( insert_Sum_sum_a_nat @ X3 @ bot_bo3438331934148233675_a_nat ) )
& ( B = bot_bo3438331934148233675_a_nat ) )
| ( ( A
= ( insert_Sum_sum_a_nat @ X3 @ bot_bo3438331934148233675_a_nat ) )
& ( B
= ( insert_Sum_sum_a_nat @ X3 @ bot_bo3438331934148233675_a_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_1141_insert__is__Un,axiom,
( insert_nat
= ( ^ [A6: nat] : ( sup_sup_set_nat @ ( insert_nat @ A6 @ bot_bot_set_nat ) ) ) ) ).
% insert_is_Un
thf(fact_1142_insert__is__Un,axiom,
( insert_Sum_sum_a_nat
= ( ^ [A6: sum_sum_a_nat] : ( sup_su6804446743777130803_a_nat @ ( insert_Sum_sum_a_nat @ A6 @ bot_bo3438331934148233675_a_nat ) ) ) ) ).
% insert_is_Un
thf(fact_1143_in__image__insert__iff,axiom,
! [B: set_set_a,X3: a,A: set_a] :
( ! [C3: set_a] :
( ( member_set_a @ C3 @ B )
=> ~ ( member_a @ X3 @ C3 ) )
=> ( ( member_set_a @ A @ ( image_set_a_set_a @ ( insert_a @ X3 ) @ B ) )
= ( ( member_a @ X3 @ A )
& ( member_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ X3 @ bot_bot_set_a ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_1144_in__image__insert__iff,axiom,
! [B: set_set_nat,X3: nat,A: set_nat] :
( ! [C3: set_nat] :
( ( member_set_nat @ C3 @ B )
=> ~ ( member_nat @ X3 @ C3 ) )
=> ( ( member_set_nat @ A @ ( image_7916887816326733075et_nat @ ( insert_nat @ X3 ) @ B ) )
= ( ( member_nat @ X3 @ A )
& ( member_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_1145_in__image__insert__iff,axiom,
! [B: set_se4904748513628223167_a_nat,X3: sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ! [C3: set_Sum_sum_a_nat] :
( ( member8098812455498974984_a_nat @ C3 @ B )
=> ~ ( member_Sum_sum_a_nat @ X3 @ C3 ) )
=> ( ( member8098812455498974984_a_nat @ A @ ( image_5599399343138760645_a_nat @ ( insert_Sum_sum_a_nat @ X3 ) @ B ) )
= ( ( member_Sum_sum_a_nat @ X3 @ A )
& ( member8098812455498974984_a_nat @ ( minus_1134630996077396038_a_nat @ A @ ( insert_Sum_sum_a_nat @ X3 @ bot_bo3438331934148233675_a_nat ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_1146_Diff__insert__absorb,axiom,
! [X3: a,A: set_a] :
( ~ ( member_a @ X3 @ A )
=> ( ( minus_minus_set_a @ ( insert_a @ X3 @ A ) @ ( insert_a @ X3 @ bot_bot_set_a ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_1147_Diff__insert__absorb,axiom,
! [X3: nat,A: set_nat] :
( ~ ( member_nat @ X3 @ A )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X3 @ A ) @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_1148_Diff__insert__absorb,axiom,
! [X3: sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ~ ( member_Sum_sum_a_nat @ X3 @ A )
=> ( ( minus_1134630996077396038_a_nat @ ( insert_Sum_sum_a_nat @ X3 @ A ) @ ( insert_Sum_sum_a_nat @ X3 @ bot_bo3438331934148233675_a_nat ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_1149_Diff__insert2,axiom,
! [A: set_nat,A3: nat,B: set_nat] :
( ( minus_minus_set_nat @ A @ ( insert_nat @ A3 @ B ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) @ B ) ) ).
% Diff_insert2
thf(fact_1150_Diff__insert2,axiom,
! [A: set_Sum_sum_a_nat,A3: sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( minus_1134630996077396038_a_nat @ A @ ( insert_Sum_sum_a_nat @ A3 @ B ) )
= ( minus_1134630996077396038_a_nat @ ( minus_1134630996077396038_a_nat @ A @ ( insert_Sum_sum_a_nat @ A3 @ bot_bo3438331934148233675_a_nat ) ) @ B ) ) ).
% Diff_insert2
thf(fact_1151_insert__Diff,axiom,
! [A3: a,A: set_a] :
( ( member_a @ A3 @ A )
=> ( ( insert_a @ A3 @ ( minus_minus_set_a @ A @ ( insert_a @ A3 @ bot_bot_set_a ) ) )
= A ) ) ).
% insert_Diff
thf(fact_1152_insert__Diff,axiom,
! [A3: nat,A: set_nat] :
( ( member_nat @ A3 @ A )
=> ( ( insert_nat @ A3 @ ( minus_minus_set_nat @ A @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) )
= A ) ) ).
% insert_Diff
thf(fact_1153_insert__Diff,axiom,
! [A3: sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ A3 @ A )
=> ( ( insert_Sum_sum_a_nat @ A3 @ ( minus_1134630996077396038_a_nat @ A @ ( insert_Sum_sum_a_nat @ A3 @ bot_bo3438331934148233675_a_nat ) ) )
= A ) ) ).
% insert_Diff
thf(fact_1154_Diff__insert,axiom,
! [A: set_nat,A3: nat,B: set_nat] :
( ( minus_minus_set_nat @ A @ ( insert_nat @ A3 @ B ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ B ) @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) ) ).
% Diff_insert
thf(fact_1155_Diff__insert,axiom,
! [A: set_Sum_sum_a_nat,A3: sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( minus_1134630996077396038_a_nat @ A @ ( insert_Sum_sum_a_nat @ A3 @ B ) )
= ( minus_1134630996077396038_a_nat @ ( minus_1134630996077396038_a_nat @ A @ B ) @ ( insert_Sum_sum_a_nat @ A3 @ bot_bo3438331934148233675_a_nat ) ) ) ).
% Diff_insert
thf(fact_1156_boolean__algebra_Odisj__zero__right,axiom,
! [X3: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ X3 @ bot_bo3438331934148233675_a_nat )
= X3 ) ).
% boolean_algebra.disj_zero_right
thf(fact_1157_finite__subset__induct_H,axiom,
! [F2: set_a,A: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( ord_less_eq_set_a @ F2 @ A )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A5: a,F5: set_a] :
( ( finite_finite_a @ F5 )
=> ( ( member_a @ A5 @ A )
=> ( ( ord_less_eq_set_a @ F5 @ A )
=> ( ~ ( member_a @ A5 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert_a @ A5 @ F5 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1158_finite__subset__induct_H,axiom,
! [F2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat,P: set_Sum_sum_a_nat > $o] :
( ( finite502105017643426984_a_nat @ F2 )
=> ( ( ord_le1325389633284124927_a_nat @ F2 @ A )
=> ( ( P @ bot_bo3438331934148233675_a_nat )
=> ( ! [A5: sum_sum_a_nat,F5: set_Sum_sum_a_nat] :
( ( finite502105017643426984_a_nat @ F5 )
=> ( ( member_Sum_sum_a_nat @ A5 @ A )
=> ( ( ord_le1325389633284124927_a_nat @ F5 @ A )
=> ( ~ ( member_Sum_sum_a_nat @ A5 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert_Sum_sum_a_nat @ A5 @ F5 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1159_finite__subset__induct_H,axiom,
! [F2: set_nat,A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A5: nat,F5: set_nat] :
( ( finite_finite_nat @ F5 )
=> ( ( member_nat @ A5 @ A )
=> ( ( ord_less_eq_set_nat @ F5 @ A )
=> ( ~ ( member_nat @ A5 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert_nat @ A5 @ F5 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1160_finite__subset__induct,axiom,
! [F2: set_a,A: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( ord_less_eq_set_a @ F2 @ A )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A5: a,F5: set_a] :
( ( finite_finite_a @ F5 )
=> ( ( member_a @ A5 @ A )
=> ( ~ ( member_a @ A5 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert_a @ A5 @ F5 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1161_finite__subset__induct,axiom,
! [F2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat,P: set_Sum_sum_a_nat > $o] :
( ( finite502105017643426984_a_nat @ F2 )
=> ( ( ord_le1325389633284124927_a_nat @ F2 @ A )
=> ( ( P @ bot_bo3438331934148233675_a_nat )
=> ( ! [A5: sum_sum_a_nat,F5: set_Sum_sum_a_nat] :
( ( finite502105017643426984_a_nat @ F5 )
=> ( ( member_Sum_sum_a_nat @ A5 @ A )
=> ( ~ ( member_Sum_sum_a_nat @ A5 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert_Sum_sum_a_nat @ A5 @ F5 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1162_finite__subset__induct,axiom,
! [F2: set_nat,A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A5: nat,F5: set_nat] :
( ( finite_finite_nat @ F5 )
=> ( ( member_nat @ A5 @ A )
=> ( ~ ( member_nat @ A5 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert_nat @ A5 @ F5 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1163_finite__empty__induct,axiom,
! [A: set_a,P: set_a > $o] :
( ( finite_finite_a @ A )
=> ( ( P @ A )
=> ( ! [A5: a,A7: set_a] :
( ( finite_finite_a @ A7 )
=> ( ( member_a @ A5 @ A7 )
=> ( ( P @ A7 )
=> ( P @ ( minus_minus_set_a @ A7 @ ( insert_a @ A5 @ bot_bot_set_a ) ) ) ) ) )
=> ( P @ bot_bot_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_1164_finite__empty__induct,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( P @ A )
=> ( ! [A5: nat,A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ( member_nat @ A5 @ A7 )
=> ( ( P @ A7 )
=> ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ A5 @ bot_bot_set_nat ) ) ) ) ) )
=> ( P @ bot_bot_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_1165_finite__empty__induct,axiom,
! [A: set_Sum_sum_a_nat,P: set_Sum_sum_a_nat > $o] :
( ( finite502105017643426984_a_nat @ A )
=> ( ( P @ A )
=> ( ! [A5: sum_sum_a_nat,A7: set_Sum_sum_a_nat] :
( ( finite502105017643426984_a_nat @ A7 )
=> ( ( member_Sum_sum_a_nat @ A5 @ A7 )
=> ( ( P @ A7 )
=> ( P @ ( minus_1134630996077396038_a_nat @ A7 @ ( insert_Sum_sum_a_nat @ A5 @ bot_bo3438331934148233675_a_nat ) ) ) ) ) )
=> ( P @ bot_bo3438331934148233675_a_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_1166_infinite__coinduct,axiom,
! [X7: set_nat > $o,A: set_nat] :
( ( X7 @ A )
=> ( ! [A7: set_nat] :
( ( X7 @ A7 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A7 )
& ( ( X7 @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) )
| ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) ) ) )
=> ~ ( finite_finite_nat @ A ) ) ) ).
% infinite_coinduct
thf(fact_1167_infinite__coinduct,axiom,
! [X7: set_Sum_sum_a_nat > $o,A: set_Sum_sum_a_nat] :
( ( X7 @ A )
=> ( ! [A7: set_Sum_sum_a_nat] :
( ( X7 @ A7 )
=> ? [X5: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X5 @ A7 )
& ( ( X7 @ ( minus_1134630996077396038_a_nat @ A7 @ ( insert_Sum_sum_a_nat @ X5 @ bot_bo3438331934148233675_a_nat ) ) )
| ~ ( finite502105017643426984_a_nat @ ( minus_1134630996077396038_a_nat @ A7 @ ( insert_Sum_sum_a_nat @ X5 @ bot_bo3438331934148233675_a_nat ) ) ) ) ) )
=> ~ ( finite502105017643426984_a_nat @ A ) ) ) ).
% infinite_coinduct
thf(fact_1168_infinite__remove,axiom,
! [S: set_nat,A3: nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) ) ) ).
% infinite_remove
thf(fact_1169_infinite__remove,axiom,
! [S: set_Sum_sum_a_nat,A3: sum_sum_a_nat] :
( ~ ( finite502105017643426984_a_nat @ S )
=> ~ ( finite502105017643426984_a_nat @ ( minus_1134630996077396038_a_nat @ S @ ( insert_Sum_sum_a_nat @ A3 @ bot_bo3438331934148233675_a_nat ) ) ) ) ).
% infinite_remove
thf(fact_1170_Diff__single__insert,axiom,
! [A: set_Sum_sum_a_nat,X3: sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( minus_1134630996077396038_a_nat @ A @ ( insert_Sum_sum_a_nat @ X3 @ bot_bo3438331934148233675_a_nat ) ) @ B )
=> ( ord_le1325389633284124927_a_nat @ A @ ( insert_Sum_sum_a_nat @ X3 @ B ) ) ) ).
% Diff_single_insert
thf(fact_1171_Diff__single__insert,axiom,
! [A: set_nat,X3: nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) @ B )
=> ( ord_less_eq_set_nat @ A @ ( insert_nat @ X3 @ B ) ) ) ).
% Diff_single_insert
thf(fact_1172_subset__insert__iff,axiom,
! [A: set_a,X3: a,B: set_a] :
( ( ord_less_eq_set_a @ A @ ( insert_a @ X3 @ B ) )
= ( ( ( member_a @ X3 @ A )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ X3 @ bot_bot_set_a ) ) @ B ) )
& ( ~ ( member_a @ X3 @ A )
=> ( ord_less_eq_set_a @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1173_subset__insert__iff,axiom,
! [A: set_Sum_sum_a_nat,X3: sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ ( insert_Sum_sum_a_nat @ X3 @ B ) )
= ( ( ( member_Sum_sum_a_nat @ X3 @ A )
=> ( ord_le1325389633284124927_a_nat @ ( minus_1134630996077396038_a_nat @ A @ ( insert_Sum_sum_a_nat @ X3 @ bot_bo3438331934148233675_a_nat ) ) @ B ) )
& ( ~ ( member_Sum_sum_a_nat @ X3 @ A )
=> ( ord_le1325389633284124927_a_nat @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1174_subset__insert__iff,axiom,
! [A: set_nat,X3: nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X3 @ B ) )
= ( ( ( member_nat @ X3 @ A )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) @ B ) )
& ( ~ ( member_nat @ X3 @ A )
=> ( ord_less_eq_set_nat @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1175_finite_OinsertI,axiom,
! [A: set_Sum_sum_a_nat,A3: sum_sum_a_nat] :
( ( finite502105017643426984_a_nat @ A )
=> ( finite502105017643426984_a_nat @ ( insert_Sum_sum_a_nat @ A3 @ A ) ) ) ).
% finite.insertI
thf(fact_1176_finite_OinsertI,axiom,
! [A: set_nat,A3: nat] :
( ( finite_finite_nat @ A )
=> ( finite_finite_nat @ ( insert_nat @ A3 @ A ) ) ) ).
% finite.insertI
thf(fact_1177_subset__insertI2,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,B2: sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ B )
=> ( ord_le1325389633284124927_a_nat @ A @ ( insert_Sum_sum_a_nat @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_1178_subset__insertI2,axiom,
! [A: set_nat,B: set_nat,B2: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ A @ ( insert_nat @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_1179_subset__insertI,axiom,
! [B: set_Sum_sum_a_nat,A3: sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ B @ ( insert_Sum_sum_a_nat @ A3 @ B ) ) ).
% subset_insertI
thf(fact_1180_subset__insertI,axiom,
! [B: set_nat,A3: nat] : ( ord_less_eq_set_nat @ B @ ( insert_nat @ A3 @ B ) ) ).
% subset_insertI
thf(fact_1181_subset__insert,axiom,
! [X3: sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ~ ( member_Sum_sum_a_nat @ X3 @ A )
=> ( ( ord_le1325389633284124927_a_nat @ A @ ( insert_Sum_sum_a_nat @ X3 @ B ) )
= ( ord_le1325389633284124927_a_nat @ A @ B ) ) ) ).
% subset_insert
thf(fact_1182_subset__insert,axiom,
! [X3: a,A: set_a,B: set_a] :
( ~ ( member_a @ X3 @ A )
=> ( ( ord_less_eq_set_a @ A @ ( insert_a @ X3 @ B ) )
= ( ord_less_eq_set_a @ A @ B ) ) ) ).
% subset_insert
thf(fact_1183_subset__insert,axiom,
! [X3: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ X3 @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X3 @ B ) )
= ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% subset_insert
thf(fact_1184_insert__mono,axiom,
! [C2: set_Sum_sum_a_nat,D: set_Sum_sum_a_nat,A3: sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ C2 @ D )
=> ( ord_le1325389633284124927_a_nat @ ( insert_Sum_sum_a_nat @ A3 @ C2 ) @ ( insert_Sum_sum_a_nat @ A3 @ D ) ) ) ).
% insert_mono
thf(fact_1185_insert__mono,axiom,
! [C2: set_nat,D: set_nat,A3: nat] :
( ( ord_less_eq_set_nat @ C2 @ D )
=> ( ord_less_eq_set_nat @ ( insert_nat @ A3 @ C2 ) @ ( insert_nat @ A3 @ D ) ) ) ).
% insert_mono
thf(fact_1186_fun__upd__image,axiom,
! [X3: a,A: set_a,F: a > sum_sum_a_nat,Y: sum_sum_a_nat] :
( ( ( member_a @ X3 @ A )
=> ( ( image_7873763678140191238_a_nat @ ( fun_up8976196915827720830_a_nat @ F @ X3 @ Y ) @ A )
= ( insert_Sum_sum_a_nat @ Y @ ( image_7873763678140191238_a_nat @ F @ ( minus_minus_set_a @ A @ ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ) )
& ( ~ ( member_a @ X3 @ A )
=> ( ( image_7873763678140191238_a_nat @ ( fun_up8976196915827720830_a_nat @ F @ X3 @ Y ) @ A )
= ( image_7873763678140191238_a_nat @ F @ A ) ) ) ) ).
% fun_upd_image
thf(fact_1187_fun__upd__image,axiom,
! [X3: a,A: set_a,F: a > nat,Y: nat] :
( ( ( member_a @ X3 @ A )
=> ( ( image_a_nat @ ( fun_upd_a_nat @ F @ X3 @ Y ) @ A )
= ( insert_nat @ Y @ ( image_a_nat @ F @ ( minus_minus_set_a @ A @ ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ) )
& ( ~ ( member_a @ X3 @ A )
=> ( ( image_a_nat @ ( fun_upd_a_nat @ F @ X3 @ Y ) @ A )
= ( image_a_nat @ F @ A ) ) ) ) ).
% fun_upd_image
thf(fact_1188_fun__upd__image,axiom,
! [X3: nat,A: set_nat,F: nat > sum_sum_a_nat,Y: sum_sum_a_nat] :
( ( ( member_nat @ X3 @ A )
=> ( ( image_7293268710728258664_a_nat @ ( fun_up180537416982607344_a_nat @ F @ X3 @ Y ) @ A )
= ( insert_Sum_sum_a_nat @ Y @ ( image_7293268710728258664_a_nat @ F @ ( minus_minus_set_nat @ A @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) )
& ( ~ ( member_nat @ X3 @ A )
=> ( ( image_7293268710728258664_a_nat @ ( fun_up180537416982607344_a_nat @ F @ X3 @ Y ) @ A )
= ( image_7293268710728258664_a_nat @ F @ A ) ) ) ) ).
% fun_upd_image
thf(fact_1189_fun__upd__image,axiom,
! [X3: nat,A: set_nat,F: nat > nat,Y: nat] :
( ( ( member_nat @ X3 @ A )
=> ( ( image_nat_nat @ ( fun_upd_nat_nat @ F @ X3 @ Y ) @ A )
= ( insert_nat @ Y @ ( image_nat_nat @ F @ ( minus_minus_set_nat @ A @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) )
& ( ~ ( member_nat @ X3 @ A )
=> ( ( image_nat_nat @ ( fun_upd_nat_nat @ F @ X3 @ Y ) @ A )
= ( image_nat_nat @ F @ A ) ) ) ) ).
% fun_upd_image
thf(fact_1190_fun__upd__image,axiom,
! [X3: sum_sum_a_nat,A: set_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat,Y: sum_sum_a_nat] :
( ( ( member_Sum_sum_a_nat @ X3 @ A )
=> ( ( image_7142520692256960453_a_nat @ ( fun_up6086130847573437501_a_nat @ F @ X3 @ Y ) @ A )
= ( insert_Sum_sum_a_nat @ Y @ ( image_7142520692256960453_a_nat @ F @ ( minus_1134630996077396038_a_nat @ A @ ( insert_Sum_sum_a_nat @ X3 @ bot_bo3438331934148233675_a_nat ) ) ) ) ) )
& ( ~ ( member_Sum_sum_a_nat @ X3 @ A )
=> ( ( image_7142520692256960453_a_nat @ ( fun_up6086130847573437501_a_nat @ F @ X3 @ Y ) @ A )
= ( image_7142520692256960453_a_nat @ F @ A ) ) ) ) ).
% fun_upd_image
thf(fact_1191_fun__upd__image,axiom,
! [X3: sum_sum_a_nat,A: set_Sum_sum_a_nat,F: sum_sum_a_nat > nat,Y: nat] :
( ( ( member_Sum_sum_a_nat @ X3 @ A )
=> ( ( image_2473878607534554506at_nat @ ( fun_up4584519350643678994at_nat @ F @ X3 @ Y ) @ A )
= ( insert_nat @ Y @ ( image_2473878607534554506at_nat @ F @ ( minus_1134630996077396038_a_nat @ A @ ( insert_Sum_sum_a_nat @ X3 @ bot_bo3438331934148233675_a_nat ) ) ) ) ) )
& ( ~ ( member_Sum_sum_a_nat @ X3 @ A )
=> ( ( image_2473878607534554506at_nat @ ( fun_up4584519350643678994at_nat @ F @ X3 @ Y ) @ A )
= ( image_2473878607534554506at_nat @ F @ A ) ) ) ) ).
% fun_upd_image
thf(fact_1192_fun__upd__image,axiom,
! [X3: sum_sum_a_nat,A: set_Sum_sum_a_nat,F: sum_sum_a_nat > option_nat,Y: option_nat] :
( ( ( member_Sum_sum_a_nat @ X3 @ A )
=> ( ( image_9069542366050205658on_nat @ ( fun_up4750343006489851234on_nat @ F @ X3 @ Y ) @ A )
= ( insert_option_nat @ Y @ ( image_9069542366050205658on_nat @ F @ ( minus_1134630996077396038_a_nat @ A @ ( insert_Sum_sum_a_nat @ X3 @ bot_bo3438331934148233675_a_nat ) ) ) ) ) )
& ( ~ ( member_Sum_sum_a_nat @ X3 @ A )
=> ( ( image_9069542366050205658on_nat @ ( fun_up4750343006489851234on_nat @ F @ X3 @ Y ) @ A )
= ( image_9069542366050205658on_nat @ F @ A ) ) ) ) ).
% fun_upd_image
thf(fact_1193_Int__insert__left,axiom,
! [A3: sum_sum_a_nat,C2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ( member_Sum_sum_a_nat @ A3 @ C2 )
=> ( ( inf_in7084830621192376909_a_nat @ ( insert_Sum_sum_a_nat @ A3 @ B ) @ C2 )
= ( insert_Sum_sum_a_nat @ A3 @ ( inf_in7084830621192376909_a_nat @ B @ C2 ) ) ) )
& ( ~ ( member_Sum_sum_a_nat @ A3 @ C2 )
=> ( ( inf_in7084830621192376909_a_nat @ ( insert_Sum_sum_a_nat @ A3 @ B ) @ C2 )
= ( inf_in7084830621192376909_a_nat @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_1194_Int__insert__left,axiom,
! [A3: nat,C2: set_nat,B: set_nat] :
( ( ( member_nat @ A3 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A3 @ B ) @ C2 )
= ( insert_nat @ A3 @ ( inf_inf_set_nat @ B @ C2 ) ) ) )
& ( ~ ( member_nat @ A3 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A3 @ B ) @ C2 )
= ( inf_inf_set_nat @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_1195_Int__insert__left,axiom,
! [A3: a,C2: set_a,B: set_a] :
( ( ( member_a @ A3 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A3 @ B ) @ C2 )
= ( insert_a @ A3 @ ( inf_inf_set_a @ B @ C2 ) ) ) )
& ( ~ ( member_a @ A3 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A3 @ B ) @ C2 )
= ( inf_inf_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_1196_Int__insert__right,axiom,
! [A3: sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ( member_Sum_sum_a_nat @ A3 @ A )
=> ( ( inf_in7084830621192376909_a_nat @ A @ ( insert_Sum_sum_a_nat @ A3 @ B ) )
= ( insert_Sum_sum_a_nat @ A3 @ ( inf_in7084830621192376909_a_nat @ A @ B ) ) ) )
& ( ~ ( member_Sum_sum_a_nat @ A3 @ A )
=> ( ( inf_in7084830621192376909_a_nat @ A @ ( insert_Sum_sum_a_nat @ A3 @ B ) )
= ( inf_in7084830621192376909_a_nat @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_1197_Int__insert__right,axiom,
! [A3: nat,A: set_nat,B: set_nat] :
( ( ( member_nat @ A3 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A3 @ B ) )
= ( insert_nat @ A3 @ ( inf_inf_set_nat @ A @ B ) ) ) )
& ( ~ ( member_nat @ A3 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A3 @ B ) )
= ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_1198_Int__insert__right,axiom,
! [A3: a,A: set_a,B: set_a] :
( ( ( member_a @ A3 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A3 @ B ) )
= ( insert_a @ A3 @ ( inf_inf_set_a @ A @ B ) ) ) )
& ( ~ ( member_a @ A3 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A3 @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_1199_insert__Diff__if,axiom,
! [X3: a,B: set_a,A: set_a] :
( ( ( member_a @ X3 @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X3 @ A ) @ B )
= ( minus_minus_set_a @ A @ B ) ) )
& ( ~ ( member_a @ X3 @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X3 @ A ) @ B )
= ( insert_a @ X3 @ ( minus_minus_set_a @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1200_insert__Diff__if,axiom,
! [X3: nat,B: set_nat,A: set_nat] :
( ( ( member_nat @ X3 @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X3 @ A ) @ B )
= ( minus_minus_set_nat @ A @ B ) ) )
& ( ~ ( member_nat @ X3 @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X3 @ A ) @ B )
= ( insert_nat @ X3 @ ( minus_minus_set_nat @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1201_insert__Diff__if,axiom,
! [X3: sum_sum_a_nat,B: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( ( member_Sum_sum_a_nat @ X3 @ B )
=> ( ( minus_1134630996077396038_a_nat @ ( insert_Sum_sum_a_nat @ X3 @ A ) @ B )
= ( minus_1134630996077396038_a_nat @ A @ B ) ) )
& ( ~ ( member_Sum_sum_a_nat @ X3 @ B )
=> ( ( minus_1134630996077396038_a_nat @ ( insert_Sum_sum_a_nat @ X3 @ A ) @ B )
= ( insert_Sum_sum_a_nat @ X3 @ ( minus_1134630996077396038_a_nat @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1202_Iio__eq__empty__iff,axiom,
! [N2: nat] :
( ( ( set_ord_lessThan_nat @ N2 )
= bot_bot_set_nat )
= ( N2 = bot_bot_nat ) ) ).
% Iio_eq_empty_iff
thf(fact_1203_mk__disjoint__insert,axiom,
! [A3: sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ A3 @ A )
=> ? [B7: set_Sum_sum_a_nat] :
( ( A
= ( insert_Sum_sum_a_nat @ A3 @ B7 ) )
& ~ ( member_Sum_sum_a_nat @ A3 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_1204_mk__disjoint__insert,axiom,
! [A3: a,A: set_a] :
( ( member_a @ A3 @ A )
=> ? [B7: set_a] :
( ( A
= ( insert_a @ A3 @ B7 ) )
& ~ ( member_a @ A3 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_1205_mk__disjoint__insert,axiom,
! [A3: nat,A: set_nat] :
( ( member_nat @ A3 @ A )
=> ? [B7: set_nat] :
( ( A
= ( insert_nat @ A3 @ B7 ) )
& ~ ( member_nat @ A3 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_1206_singleton__inject,axiom,
! [A3: nat,B2: nat] :
( ( ( insert_nat @ A3 @ bot_bot_set_nat )
= ( insert_nat @ B2 @ bot_bot_set_nat ) )
=> ( A3 = B2 ) ) ).
% singleton_inject
thf(fact_1207_singleton__inject,axiom,
! [A3: sum_sum_a_nat,B2: sum_sum_a_nat] :
( ( ( insert_Sum_sum_a_nat @ A3 @ bot_bo3438331934148233675_a_nat )
= ( insert_Sum_sum_a_nat @ B2 @ bot_bo3438331934148233675_a_nat ) )
=> ( A3 = B2 ) ) ).
% singleton_inject
thf(fact_1208_insert__not__empty,axiom,
! [A3: nat,A: set_nat] :
( ( insert_nat @ A3 @ A )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_1209_insert__not__empty,axiom,
! [A3: sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( insert_Sum_sum_a_nat @ A3 @ A )
!= bot_bo3438331934148233675_a_nat ) ).
% insert_not_empty
thf(fact_1210_doubleton__eq__iff,axiom,
! [A3: nat,B2: nat,C: nat,D2: nat] :
( ( ( insert_nat @ A3 @ ( insert_nat @ B2 @ bot_bot_set_nat ) )
= ( insert_nat @ C @ ( insert_nat @ D2 @ bot_bot_set_nat ) ) )
= ( ( ( A3 = C )
& ( B2 = D2 ) )
| ( ( A3 = D2 )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_1211_doubleton__eq__iff,axiom,
! [A3: sum_sum_a_nat,B2: sum_sum_a_nat,C: sum_sum_a_nat,D2: sum_sum_a_nat] :
( ( ( insert_Sum_sum_a_nat @ A3 @ ( insert_Sum_sum_a_nat @ B2 @ bot_bo3438331934148233675_a_nat ) )
= ( insert_Sum_sum_a_nat @ C @ ( insert_Sum_sum_a_nat @ D2 @ bot_bo3438331934148233675_a_nat ) ) )
= ( ( ( A3 = C )
& ( B2 = D2 ) )
| ( ( A3 = D2 )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_1212_insert__commute,axiom,
! [X3: sum_sum_a_nat,Y: sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( insert_Sum_sum_a_nat @ X3 @ ( insert_Sum_sum_a_nat @ Y @ A ) )
= ( insert_Sum_sum_a_nat @ Y @ ( insert_Sum_sum_a_nat @ X3 @ A ) ) ) ).
% insert_commute
thf(fact_1213_insert__commute,axiom,
! [X3: nat,Y: nat,A: set_nat] :
( ( insert_nat @ X3 @ ( insert_nat @ Y @ A ) )
= ( insert_nat @ Y @ ( insert_nat @ X3 @ A ) ) ) ).
% insert_commute
thf(fact_1214_singleton__iff,axiom,
! [B2: a,A3: a] :
( ( member_a @ B2 @ ( insert_a @ A3 @ bot_bot_set_a ) )
= ( B2 = A3 ) ) ).
% singleton_iff
thf(fact_1215_singleton__iff,axiom,
! [B2: nat,A3: nat] :
( ( member_nat @ B2 @ ( insert_nat @ A3 @ bot_bot_set_nat ) )
= ( B2 = A3 ) ) ).
% singleton_iff
thf(fact_1216_singleton__iff,axiom,
! [B2: sum_sum_a_nat,A3: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ B2 @ ( insert_Sum_sum_a_nat @ A3 @ bot_bo3438331934148233675_a_nat ) )
= ( B2 = A3 ) ) ).
% singleton_iff
thf(fact_1217_insert__eq__iff,axiom,
! [A3: sum_sum_a_nat,A: set_Sum_sum_a_nat,B2: sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ~ ( member_Sum_sum_a_nat @ A3 @ A )
=> ( ~ ( member_Sum_sum_a_nat @ B2 @ B )
=> ( ( ( insert_Sum_sum_a_nat @ A3 @ A )
= ( insert_Sum_sum_a_nat @ B2 @ B ) )
= ( ( ( A3 = B2 )
=> ( A = B ) )
& ( ( A3 != B2 )
=> ? [C6: set_Sum_sum_a_nat] :
( ( A
= ( insert_Sum_sum_a_nat @ B2 @ C6 ) )
& ~ ( member_Sum_sum_a_nat @ B2 @ C6 )
& ( B
= ( insert_Sum_sum_a_nat @ A3 @ C6 ) )
& ~ ( member_Sum_sum_a_nat @ A3 @ C6 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_1218_insert__eq__iff,axiom,
! [A3: a,A: set_a,B2: a,B: set_a] :
( ~ ( member_a @ A3 @ A )
=> ( ~ ( member_a @ B2 @ B )
=> ( ( ( insert_a @ A3 @ A )
= ( insert_a @ B2 @ B ) )
= ( ( ( A3 = B2 )
=> ( A = B ) )
& ( ( A3 != B2 )
=> ? [C6: set_a] :
( ( A
= ( insert_a @ B2 @ C6 ) )
& ~ ( member_a @ B2 @ C6 )
& ( B
= ( insert_a @ A3 @ C6 ) )
& ~ ( member_a @ A3 @ C6 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_1219_insert__eq__iff,axiom,
! [A3: nat,A: set_nat,B2: nat,B: set_nat] :
( ~ ( member_nat @ A3 @ A )
=> ( ~ ( member_nat @ B2 @ B )
=> ( ( ( insert_nat @ A3 @ A )
= ( insert_nat @ B2 @ B ) )
= ( ( ( A3 = B2 )
=> ( A = B ) )
& ( ( A3 != B2 )
=> ? [C6: set_nat] :
( ( A
= ( insert_nat @ B2 @ C6 ) )
& ~ ( member_nat @ B2 @ C6 )
& ( B
= ( insert_nat @ A3 @ C6 ) )
& ~ ( member_nat @ A3 @ C6 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_1220_insert__absorb,axiom,
! [A3: sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ A3 @ A )
=> ( ( insert_Sum_sum_a_nat @ A3 @ A )
= A ) ) ).
% insert_absorb
thf(fact_1221_insert__absorb,axiom,
! [A3: a,A: set_a] :
( ( member_a @ A3 @ A )
=> ( ( insert_a @ A3 @ A )
= A ) ) ).
% insert_absorb
thf(fact_1222_insert__absorb,axiom,
! [A3: nat,A: set_nat] :
( ( member_nat @ A3 @ A )
=> ( ( insert_nat @ A3 @ A )
= A ) ) ).
% insert_absorb
thf(fact_1223_insert__ident,axiom,
! [X3: sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ~ ( member_Sum_sum_a_nat @ X3 @ A )
=> ( ~ ( member_Sum_sum_a_nat @ X3 @ B )
=> ( ( ( insert_Sum_sum_a_nat @ X3 @ A )
= ( insert_Sum_sum_a_nat @ X3 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_1224_insert__ident,axiom,
! [X3: a,A: set_a,B: set_a] :
( ~ ( member_a @ X3 @ A )
=> ( ~ ( member_a @ X3 @ B )
=> ( ( ( insert_a @ X3 @ A )
= ( insert_a @ X3 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_1225_insert__ident,axiom,
! [X3: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ X3 @ A )
=> ( ~ ( member_nat @ X3 @ B )
=> ( ( ( insert_nat @ X3 @ A )
= ( insert_nat @ X3 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_1226_singletonD,axiom,
! [B2: a,A3: a] :
( ( member_a @ B2 @ ( insert_a @ A3 @ bot_bot_set_a ) )
=> ( B2 = A3 ) ) ).
% singletonD
thf(fact_1227_singletonD,axiom,
! [B2: nat,A3: nat] :
( ( member_nat @ B2 @ ( insert_nat @ A3 @ bot_bot_set_nat ) )
=> ( B2 = A3 ) ) ).
% singletonD
thf(fact_1228_singletonD,axiom,
! [B2: sum_sum_a_nat,A3: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ B2 @ ( insert_Sum_sum_a_nat @ A3 @ bot_bo3438331934148233675_a_nat ) )
=> ( B2 = A3 ) ) ).
% singletonD
thf(fact_1229_Set_Oset__insert,axiom,
! [X3: sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X3 @ A )
=> ~ ! [B7: set_Sum_sum_a_nat] :
( ( A
= ( insert_Sum_sum_a_nat @ X3 @ B7 ) )
=> ( member_Sum_sum_a_nat @ X3 @ B7 ) ) ) ).
% Set.set_insert
thf(fact_1230_Set_Oset__insert,axiom,
! [X3: a,A: set_a] :
( ( member_a @ X3 @ A )
=> ~ ! [B7: set_a] :
( ( A
= ( insert_a @ X3 @ B7 ) )
=> ( member_a @ X3 @ B7 ) ) ) ).
% Set.set_insert
thf(fact_1231_Set_Oset__insert,axiom,
! [X3: nat,A: set_nat] :
( ( member_nat @ X3 @ A )
=> ~ ! [B7: set_nat] :
( ( A
= ( insert_nat @ X3 @ B7 ) )
=> ( member_nat @ X3 @ B7 ) ) ) ).
% Set.set_insert
thf(fact_1232_ex__in__conv,axiom,
! [A: set_a] :
( ( ? [X: a] : ( member_a @ X @ A ) )
= ( A != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_1233_ex__in__conv,axiom,
! [A: set_nat] :
( ( ? [X: nat] : ( member_nat @ X @ A ) )
= ( A != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_1234_ex__in__conv,axiom,
! [A: set_Sum_sum_a_nat] :
( ( ? [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ A ) )
= ( A != bot_bo3438331934148233675_a_nat ) ) ).
% ex_in_conv
thf(fact_1235_insertI2,axiom,
! [A3: sum_sum_a_nat,B: set_Sum_sum_a_nat,B2: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ A3 @ B )
=> ( member_Sum_sum_a_nat @ A3 @ ( insert_Sum_sum_a_nat @ B2 @ B ) ) ) ).
% insertI2
thf(fact_1236_insertI2,axiom,
! [A3: a,B: set_a,B2: a] :
( ( member_a @ A3 @ B )
=> ( member_a @ A3 @ ( insert_a @ B2 @ B ) ) ) ).
% insertI2
thf(fact_1237_insertI2,axiom,
! [A3: nat,B: set_nat,B2: nat] :
( ( member_nat @ A3 @ B )
=> ( member_nat @ A3 @ ( insert_nat @ B2 @ B ) ) ) ).
% insertI2
thf(fact_1238_insertI1,axiom,
! [A3: sum_sum_a_nat,B: set_Sum_sum_a_nat] : ( member_Sum_sum_a_nat @ A3 @ ( insert_Sum_sum_a_nat @ A3 @ B ) ) ).
% insertI1
thf(fact_1239_insertI1,axiom,
! [A3: a,B: set_a] : ( member_a @ A3 @ ( insert_a @ A3 @ B ) ) ).
% insertI1
thf(fact_1240_insertI1,axiom,
! [A3: nat,B: set_nat] : ( member_nat @ A3 @ ( insert_nat @ A3 @ B ) ) ).
% insertI1
thf(fact_1241_equals0I,axiom,
! [A: set_a] :
( ! [Y5: a] :
~ ( member_a @ Y5 @ A )
=> ( A = bot_bot_set_a ) ) ).
% equals0I
thf(fact_1242_equals0I,axiom,
! [A: set_nat] :
( ! [Y5: nat] :
~ ( member_nat @ Y5 @ A )
=> ( A = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_1243_equals0I,axiom,
! [A: set_Sum_sum_a_nat] :
( ! [Y5: sum_sum_a_nat] :
~ ( member_Sum_sum_a_nat @ Y5 @ A )
=> ( A = bot_bo3438331934148233675_a_nat ) ) ).
% equals0I
thf(fact_1244_equals0D,axiom,
! [A: set_a,A3: a] :
( ( A = bot_bot_set_a )
=> ~ ( member_a @ A3 @ A ) ) ).
% equals0D
thf(fact_1245_equals0D,axiom,
! [A: set_nat,A3: nat] :
( ( A = bot_bot_set_nat )
=> ~ ( member_nat @ A3 @ A ) ) ).
% equals0D
thf(fact_1246_equals0D,axiom,
! [A: set_Sum_sum_a_nat,A3: sum_sum_a_nat] :
( ( A = bot_bo3438331934148233675_a_nat )
=> ~ ( member_Sum_sum_a_nat @ A3 @ A ) ) ).
% equals0D
thf(fact_1247_insertE,axiom,
! [A3: sum_sum_a_nat,B2: sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ A3 @ ( insert_Sum_sum_a_nat @ B2 @ A ) )
=> ( ( A3 != B2 )
=> ( member_Sum_sum_a_nat @ A3 @ A ) ) ) ).
% insertE
thf(fact_1248_insertE,axiom,
! [A3: a,B2: a,A: set_a] :
( ( member_a @ A3 @ ( insert_a @ B2 @ A ) )
=> ( ( A3 != B2 )
=> ( member_a @ A3 @ A ) ) ) ).
% insertE
thf(fact_1249_insertE,axiom,
! [A3: nat,B2: nat,A: set_nat] :
( ( member_nat @ A3 @ ( insert_nat @ B2 @ A ) )
=> ( ( A3 != B2 )
=> ( member_nat @ A3 @ A ) ) ) ).
% insertE
thf(fact_1250_emptyE,axiom,
! [A3: a] :
~ ( member_a @ A3 @ bot_bot_set_a ) ).
% emptyE
thf(fact_1251_emptyE,axiom,
! [A3: nat] :
~ ( member_nat @ A3 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_1252_emptyE,axiom,
! [A3: sum_sum_a_nat] :
~ ( member_Sum_sum_a_nat @ A3 @ bot_bo3438331934148233675_a_nat ) ).
% emptyE
thf(fact_1253_image__constant,axiom,
! [X3: a,A: set_a,C: nat] :
( ( member_a @ X3 @ A )
=> ( ( image_a_nat
@ ^ [X: a] : C
@ A )
= ( insert_nat @ C @ bot_bot_set_nat ) ) ) ).
% image_constant
thf(fact_1254_image__constant,axiom,
! [X3: nat,A: set_nat,C: nat] :
( ( member_nat @ X3 @ A )
=> ( ( image_nat_nat
@ ^ [X: nat] : C
@ A )
= ( insert_nat @ C @ bot_bot_set_nat ) ) ) ).
% image_constant
thf(fact_1255_image__constant,axiom,
! [X3: sum_sum_a_nat,A: set_Sum_sum_a_nat,C: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X3 @ A )
=> ( ( image_7142520692256960453_a_nat
@ ^ [X: sum_sum_a_nat] : C
@ A )
= ( insert_Sum_sum_a_nat @ C @ bot_bo3438331934148233675_a_nat ) ) ) ).
% image_constant
thf(fact_1256_image__constant,axiom,
! [X3: a,A: set_a,C: sum_sum_a_nat] :
( ( member_a @ X3 @ A )
=> ( ( image_7873763678140191238_a_nat
@ ^ [X: a] : C
@ A )
= ( insert_Sum_sum_a_nat @ C @ bot_bo3438331934148233675_a_nat ) ) ) ).
% image_constant
thf(fact_1257_image__constant,axiom,
! [X3: nat,A: set_nat,C: sum_sum_a_nat] :
( ( member_nat @ X3 @ A )
=> ( ( image_7293268710728258664_a_nat
@ ^ [X: nat] : C
@ A )
= ( insert_Sum_sum_a_nat @ C @ bot_bo3438331934148233675_a_nat ) ) ) ).
% image_constant
thf(fact_1258_image__constant__conv,axiom,
! [A: set_Sum_sum_a_nat,C: sum_sum_a_nat] :
( ( ( A = bot_bo3438331934148233675_a_nat )
=> ( ( image_7142520692256960453_a_nat
@ ^ [X: sum_sum_a_nat] : C
@ A )
= bot_bo3438331934148233675_a_nat ) )
& ( ( A != bot_bo3438331934148233675_a_nat )
=> ( ( image_7142520692256960453_a_nat
@ ^ [X: sum_sum_a_nat] : C
@ A )
= ( insert_Sum_sum_a_nat @ C @ bot_bo3438331934148233675_a_nat ) ) ) ) ).
% image_constant_conv
thf(fact_1259_lessThan__Suc,axiom,
! [K: nat] :
( ( set_ord_lessThan_nat @ ( suc @ K ) )
= ( insert_nat @ K @ ( set_ord_lessThan_nat @ K ) ) ) ).
% lessThan_Suc
thf(fact_1260_inj__Suc,axiom,
! [N: set_nat] : ( inj_on_nat_nat @ suc @ N ) ).
% inj_Suc
thf(fact_1261_inj__on__diff__nat,axiom,
! [N: set_nat,K: nat] :
( ! [N4: nat] :
( ( member_nat @ N4 @ N )
=> ( ord_less_eq_nat @ K @ N4 ) )
=> ( inj_on_nat_nat
@ ^ [N3: nat] : ( minus_minus_nat @ N3 @ K )
@ N ) ) ).
% inj_on_diff_nat
% Helper facts (5)
thf(help_If_2_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
! [X3: option_nat,Y: option_nat] :
( ( if_option_nat @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
! [X3: option_nat,Y: option_nat] :
( ( if_option_nat @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_3_1_If_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_T,axiom,
! [X3: sum_sum_a_nat,Y: sum_sum_a_nat] :
( ( if_Sum_sum_a_nat @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_T,axiom,
! [X3: sum_sum_a_nat,Y: sum_sum_a_nat] :
( ( if_Sum_sum_a_nat @ $true @ X3 @ Y )
= X3 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( sup_su6804446743777130803_a_nat @ ( set_Sum_sum_a_nat2 @ ( fo_nmlz_rec_a @ ia @ ma @ aDa @ ( cons_Sum_sum_a_nat @ ( sum_Inl_a_nat @ x ) @ xsa ) ) ) @ ( image_7293268710728258664_a_nat @ sum_Inr_nat_a @ ( set_ord_lessThan_nat @ ia ) ) )
= ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ ( set_Sum_sum_a_nat2 @ ( cons_Sum_sum_a_nat @ ( sum_Inl_a_nat @ x ) @ xsa ) ) @ ( image_7873763678140191238_a_nat @ sum_Inl_a_nat @ aDa ) ) @ ( image_7293268710728258664_a_nat @ sum_Inr_nat_a @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ia @ ( finite6080979521523705895_a_nat @ ( minus_1134630996077396038_a_nat @ ( minus_1134630996077396038_a_nat @ ( set_Sum_sum_a_nat2 @ ( cons_Sum_sum_a_nat @ ( sum_Inl_a_nat @ x ) @ xsa ) ) @ ( image_7873763678140191238_a_nat @ sum_Inl_a_nat @ aDa ) ) @ ( dom_Su2255998037560862461at_nat @ ma ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------