TPTP Problem File: SLH0740^1.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_04493_188082__16181776_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1392 ( 791 unt; 111 typ; 0 def)
% Number of atoms : 3088 (1255 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 9049 ( 287 ~; 47 |; 198 &;7503 @)
% ( 0 <=>;1014 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 5 avg)
% Number of types : 10 ( 9 usr)
% Number of type conns : 384 ( 384 >; 0 *; 0 +; 0 <<)
% Number of symbols : 103 ( 102 usr; 13 con; 0-5 aty)
% Number of variables : 3137 ( 218 ^;2882 !; 37 ?;3137 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 13:12:51.332
%------------------------------------------------------------------------------
% Could-be-implicit typings (9)
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% Explicit typings (102)
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thf(sy_c_Set__Impl_Oset__empty__choose_001tf__a,type,
set_se792222636694734277oose_a: set_a ).
thf(sy_c_Set__Impl_Ounion__monad_001tf__a,type,
set_union_monad_a: set_a > set_a > set_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
set_or2565139667135407898_a_nat: set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_se4330304633200676677_a_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_Itf__a_J,type,
set_or6288561110385358355_set_a: set_a > set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
set_or464376051672557814_a_nat: set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_se4330304633200676677_a_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_Itf__a_J,type,
set_or2348907005316661231_set_a: set_a > set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
set_or2796000563533817904_a_nat: set_li6526943997496501093_a_nat > set_se4330304633200676677_a_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Set__Oset_Itf__a_J,type,
set_or8362275514725411625_set_a: set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
set_or2322826630718027820_a_nat: set_li6526943997496501093_a_nat > set_se4330304633200676677_a_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_Itf__a_J,type,
set_ord_atMost_set_a: set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
set_or4159382470967997621_a_nat: set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_se4330304633200676677_a_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Set__Oset_Itf__a_J,type,
set_or2503527069484367278_set_a: set_a > set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
set_or2112173576855165201_a_nat: set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_se4330304633200676677_a_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Set__Oset_Itf__a_J,type,
set_or6017932776736107018_set_a: set_a > set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_Itf__a_J,type,
set_or5421148953861284865_set_a: set_a > set_set_a ).
thf(sy_c_member_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
member408289922725080238_a_nat: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
member5553968465346197646_a_nat: set_li6526943997496501093_a_nat > set_se4330304633200676677_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_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_AD,type,
ad: set_a ).
thf(sy_v_AD_H,type,
ad2: set_a ).
thf(sy_v_X,type,
x: set_li6526943997496501093_a_nat ).
thf(sy_v_z,type,
z: list_Sum_sum_a_nat ).
% Relevant facts (1277)
thf(fact_0__092_060open_062_092_060And_062ys_Axs_O_A_092_060lbrakk_062ys_A_092_060in_062_Aad__agr__close_A_IAD_A_N_AAD_H_J_Axs_059_Afo__nmlzd_AAD_H_Axs_059_AAD_H_A_092_060inter_062_A_IAD_A_N_AAD_H_J_A_061_A_123_125_092_060rbrakk_062_A_092_060Longrightarrow_062_Afo__nmlzd_A_IAD_H_A_092_060union_062_A_IAD_A_N_AAD_H_J_J_Ays_A_092_060and_062_Aad__agr__list_AAD_H_Axs_Ays_092_060close_062,axiom,
! [Ys: list_Sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ Ys @ ( ad_agr_close_a @ ( minus_minus_set_a @ ad @ ad2 ) @ Xs ) )
=> ( ( fo_nmlzd_a @ ad2 @ Xs )
=> ( ( ( inf_inf_set_a @ ad2 @ ( minus_minus_set_a @ ad @ ad2 ) )
= bot_bot_set_a )
=> ( ( fo_nmlzd_a @ ( sup_sup_set_a @ ad2 @ ( minus_minus_set_a @ ad @ ad2 ) ) @ Ys )
& ( ad_agr_list_a_nat @ ad2 @ Xs @ Ys ) ) ) ) ) ).
% \<open>\<And>ys xs. \<lbrakk>ys \<in> ad_agr_close (AD - AD') xs; fo_nmlzd AD' xs; AD' \<inter> (AD - AD') = {}\<rbrakk> \<Longrightarrow> fo_nmlzd (AD' \<union> (AD - AD')) ys \<and> ad_agr_list AD' xs ys\<close>
thf(fact_1_fo__nmlzd__mono__sub,axiom,
! [X: set_se4330304633200676677_a_nat,X2: set_se4330304633200676677_a_nat,Xs: list_S2547498912977063020at_nat] :
( ( ord_le8138476598237931237_a_nat @ X @ X2 )
=> ( ( fo_nml874434790335044927_a_nat @ X @ Xs )
=> ( fo_nml874434790335044927_a_nat @ X2 @ Xs ) ) ) ).
% fo_nmlzd_mono_sub
thf(fact_2_fo__nmlzd__mono__sub,axiom,
! [X: set_set_a,X2: set_set_a,Xs: list_S1494550720859036901_a_nat] :
( ( ord_le3724670747650509150_set_a @ X @ X2 )
=> ( ( fo_nmlzd_set_a @ X @ Xs )
=> ( fo_nmlzd_set_a @ X2 @ Xs ) ) ) ).
% fo_nmlzd_mono_sub
thf(fact_3_fo__nmlzd__mono__sub,axiom,
! [X: set_li6526943997496501093_a_nat,X2: set_li6526943997496501093_a_nat,Xs: list_S7122517066888371340at_nat] :
( ( ord_le1147066620699065093_a_nat @ X @ X2 )
=> ( ( fo_nml1386993274042258783_a_nat @ X @ Xs )
=> ( fo_nml1386993274042258783_a_nat @ X2 @ Xs ) ) ) ).
% fo_nmlzd_mono_sub
thf(fact_4_fo__nmlzd__mono__sub,axiom,
! [X: set_a,X2: set_a,Xs: list_Sum_sum_a_nat] :
( ( ord_less_eq_set_a @ X @ X2 )
=> ( ( fo_nmlzd_a @ X @ Xs )
=> ( fo_nmlzd_a @ X2 @ Xs ) ) ) ).
% fo_nmlzd_mono_sub
thf(fact_5_DiffI,axiom,
! [C: set_li6526943997496501093_a_nat,A: set_se4330304633200676677_a_nat,B: set_se4330304633200676677_a_nat] :
( ( member5553968465346197646_a_nat @ C @ A )
=> ( ~ ( member5553968465346197646_a_nat @ C @ B )
=> ( member5553968465346197646_a_nat @ C @ ( minus_8004192006823430828_a_nat @ A @ B ) ) ) ) ).
% DiffI
thf(fact_6_DiffI,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ A )
=> ( ~ ( member_set_a @ C @ B )
=> ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A @ B ) ) ) ) ).
% DiffI
thf(fact_7_DiffI,axiom,
! [C: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ A )
=> ( ~ ( member408289922725080238_a_nat @ C @ B )
=> ( member408289922725080238_a_nat @ C @ ( minus_7395159227704179404_a_nat @ A @ B ) ) ) ) ).
% DiffI
thf(fact_8_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_9_Diff__iff,axiom,
! [C: set_li6526943997496501093_a_nat,A: set_se4330304633200676677_a_nat,B: set_se4330304633200676677_a_nat] :
( ( member5553968465346197646_a_nat @ C @ ( minus_8004192006823430828_a_nat @ A @ B ) )
= ( ( member5553968465346197646_a_nat @ C @ A )
& ~ ( member5553968465346197646_a_nat @ C @ B ) ) ) ).
% Diff_iff
thf(fact_10_Diff__iff,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A @ B ) )
= ( ( member_set_a @ C @ A )
& ~ ( member_set_a @ C @ B ) ) ) ).
% Diff_iff
thf(fact_11_Diff__iff,axiom,
! [C: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ ( minus_7395159227704179404_a_nat @ A @ B ) )
= ( ( member408289922725080238_a_nat @ C @ A )
& ~ ( member408289922725080238_a_nat @ C @ B ) ) ) ).
% Diff_iff
thf(fact_12_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_13_Diff__idemp,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( minus_7395159227704179404_a_nat @ ( minus_7395159227704179404_a_nat @ A @ B ) @ B )
= ( minus_7395159227704179404_a_nat @ A @ B ) ) ).
% Diff_idemp
thf(fact_14_Diff__idemp,axiom,
! [A: set_a,B: set_a] :
( ( minus_minus_set_a @ ( minus_minus_set_a @ A @ B ) @ B )
= ( minus_minus_set_a @ A @ B ) ) ).
% Diff_idemp
thf(fact_15_subsetI,axiom,
! [A: set_se4330304633200676677_a_nat,B: set_se4330304633200676677_a_nat] :
( ! [X3: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ X3 @ A )
=> ( member5553968465346197646_a_nat @ X3 @ B ) )
=> ( ord_le8138476598237931237_a_nat @ A @ B ) ) ).
% subsetI
thf(fact_16_subsetI,axiom,
! [A: set_set_a,B: set_set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A )
=> ( member_set_a @ X3 @ B ) )
=> ( ord_le3724670747650509150_set_a @ A @ B ) ) ).
% subsetI
thf(fact_17_subsetI,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ! [X3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ A )
=> ( member408289922725080238_a_nat @ X3 @ B ) )
=> ( ord_le1147066620699065093_a_nat @ A @ B ) ) ).
% subsetI
thf(fact_18_subsetI,axiom,
! [A: set_a,B: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( member_a @ X3 @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% subsetI
thf(fact_19_subset__antisym,axiom,
! [A: set_se4330304633200676677_a_nat,B: set_se4330304633200676677_a_nat] :
( ( ord_le8138476598237931237_a_nat @ A @ B )
=> ( ( ord_le8138476598237931237_a_nat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_20_subset__antisym,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_21_subset__antisym,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A @ B )
=> ( ( ord_le1147066620699065093_a_nat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_22_subset__antisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_23_order__refl,axiom,
! [X4: set_se4330304633200676677_a_nat] : ( ord_le8138476598237931237_a_nat @ X4 @ X4 ) ).
% order_refl
thf(fact_24_order__refl,axiom,
! [X4: set_set_a] : ( ord_le3724670747650509150_set_a @ X4 @ X4 ) ).
% order_refl
thf(fact_25_order__refl,axiom,
! [X4: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ X4 @ X4 ) ).
% order_refl
thf(fact_26_order__refl,axiom,
! [X4: set_a] : ( ord_less_eq_set_a @ X4 @ X4 ) ).
% order_refl
thf(fact_27_dual__order_Orefl,axiom,
! [A2: set_se4330304633200676677_a_nat] : ( ord_le8138476598237931237_a_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_28_dual__order_Orefl,axiom,
! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_29_dual__order_Orefl,axiom,
! [A2: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_30_dual__order_Orefl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_31_Diff__mono,axiom,
! [A: set_se4330304633200676677_a_nat,C2: set_se4330304633200676677_a_nat,D: set_se4330304633200676677_a_nat,B: set_se4330304633200676677_a_nat] :
( ( ord_le8138476598237931237_a_nat @ A @ C2 )
=> ( ( ord_le8138476598237931237_a_nat @ D @ B )
=> ( ord_le8138476598237931237_a_nat @ ( minus_8004192006823430828_a_nat @ A @ B ) @ ( minus_8004192006823430828_a_nat @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_32_Diff__mono,axiom,
! [A: set_set_a,C2: set_set_a,D: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ C2 )
=> ( ( ord_le3724670747650509150_set_a @ D @ B )
=> ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A @ B ) @ ( minus_5736297505244876581_set_a @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_33_Diff__mono,axiom,
! [A: set_a,C2: set_a,D: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ D @ B )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ B ) @ ( minus_minus_set_a @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_34_Diff__mono,axiom,
! [A: set_li6526943997496501093_a_nat,C2: set_li6526943997496501093_a_nat,D: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A @ C2 )
=> ( ( ord_le1147066620699065093_a_nat @ D @ B )
=> ( ord_le1147066620699065093_a_nat @ ( minus_7395159227704179404_a_nat @ A @ B ) @ ( minus_7395159227704179404_a_nat @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_35_Diff__subset,axiom,
! [A: set_se4330304633200676677_a_nat,B: set_se4330304633200676677_a_nat] : ( ord_le8138476598237931237_a_nat @ ( minus_8004192006823430828_a_nat @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_36_Diff__subset,axiom,
! [A: set_set_a,B: set_set_a] : ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_37_Diff__subset,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_38_Diff__subset,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ ( minus_7395159227704179404_a_nat @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_39_double__diff,axiom,
! [A: set_se4330304633200676677_a_nat,B: set_se4330304633200676677_a_nat,C2: set_se4330304633200676677_a_nat] :
( ( ord_le8138476598237931237_a_nat @ A @ B )
=> ( ( ord_le8138476598237931237_a_nat @ B @ C2 )
=> ( ( minus_8004192006823430828_a_nat @ B @ ( minus_8004192006823430828_a_nat @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_40_double__diff,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ C2 )
=> ( ( minus_5736297505244876581_set_a @ B @ ( minus_5736297505244876581_set_a @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_41_double__diff,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ( minus_minus_set_a @ B @ ( minus_minus_set_a @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_42_double__diff,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,C2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A @ B )
=> ( ( ord_le1147066620699065093_a_nat @ B @ C2 )
=> ( ( minus_7395159227704179404_a_nat @ B @ ( minus_7395159227704179404_a_nat @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_43_bot__apply,axiom,
( bot_bot_a_o
= ( ^ [X5: a] : bot_bot_o ) ) ).
% bot_apply
thf(fact_44_bot__apply,axiom,
( bot_bo9042073657639083596_nat_o
= ( ^ [X5: list_Sum_sum_a_nat] : bot_bot_o ) ) ).
% bot_apply
thf(fact_45_empty__iff,axiom,
! [C: set_li6526943997496501093_a_nat] :
~ ( member5553968465346197646_a_nat @ C @ bot_bo3237059034911209905_a_nat ) ).
% empty_iff
thf(fact_46_empty__iff,axiom,
! [C: set_a] :
~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_47_empty__iff,axiom,
! [C: list_Sum_sum_a_nat] :
~ ( member408289922725080238_a_nat @ C @ bot_bo1033123847703346641_a_nat ) ).
% empty_iff
thf(fact_48_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_49_all__not__in__conv,axiom,
! [A: set_se4330304633200676677_a_nat] :
( ( ! [X5: set_li6526943997496501093_a_nat] :
~ ( member5553968465346197646_a_nat @ X5 @ A ) )
= ( A = bot_bo3237059034911209905_a_nat ) ) ).
% all_not_in_conv
thf(fact_50_all__not__in__conv,axiom,
! [A: set_set_a] :
( ( ! [X5: set_a] :
~ ( member_set_a @ X5 @ A ) )
= ( A = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_51_all__not__in__conv,axiom,
! [A: set_li6526943997496501093_a_nat] :
( ( ! [X5: list_Sum_sum_a_nat] :
~ ( member408289922725080238_a_nat @ X5 @ A ) )
= ( A = bot_bo1033123847703346641_a_nat ) ) ).
% all_not_in_conv
thf(fact_52_all__not__in__conv,axiom,
! [A: set_a] :
( ( ! [X5: a] :
~ ( member_a @ X5 @ A ) )
= ( A = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_53_Collect__empty__eq,axiom,
! [P: list_Sum_sum_a_nat > $o] :
( ( ( collec7555443234367654128_a_nat @ P )
= bot_bo1033123847703346641_a_nat )
= ( ! [X5: list_Sum_sum_a_nat] :
~ ( P @ X5 ) ) ) ).
% Collect_empty_eq
thf(fact_54_Collect__empty__eq,axiom,
! [P: set_li6526943997496501093_a_nat > $o] :
( ( ( collec7528627406912015568_a_nat @ P )
= bot_bo3237059034911209905_a_nat )
= ( ! [X5: set_li6526943997496501093_a_nat] :
~ ( P @ X5 ) ) ) ).
% Collect_empty_eq
thf(fact_55_Collect__empty__eq,axiom,
! [P: set_a > $o] :
( ( ( collect_set_a @ P )
= bot_bot_set_set_a )
= ( ! [X5: set_a] :
~ ( P @ X5 ) ) ) ).
% Collect_empty_eq
thf(fact_56_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X5: a] :
~ ( P @ X5 ) ) ) ).
% Collect_empty_eq
thf(fact_57_empty__Collect__eq,axiom,
! [P: list_Sum_sum_a_nat > $o] :
( ( bot_bo1033123847703346641_a_nat
= ( collec7555443234367654128_a_nat @ P ) )
= ( ! [X5: list_Sum_sum_a_nat] :
~ ( P @ X5 ) ) ) ).
% empty_Collect_eq
thf(fact_58_empty__Collect__eq,axiom,
! [P: set_li6526943997496501093_a_nat > $o] :
( ( bot_bo3237059034911209905_a_nat
= ( collec7528627406912015568_a_nat @ P ) )
= ( ! [X5: set_li6526943997496501093_a_nat] :
~ ( P @ X5 ) ) ) ).
% empty_Collect_eq
thf(fact_59_empty__Collect__eq,axiom,
! [P: set_a > $o] :
( ( bot_bot_set_set_a
= ( collect_set_a @ P ) )
= ( ! [X5: set_a] :
~ ( P @ X5 ) ) ) ).
% empty_Collect_eq
thf(fact_60_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X5: a] :
~ ( P @ X5 ) ) ) ).
% empty_Collect_eq
thf(fact_61_sup_Oidem,axiom,
! [A2: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_62_sup_Oidem,axiom,
! [A2: set_set_a] :
( ( sup_sup_set_set_a @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_63_sup_Oidem,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_64_sup__idem,axiom,
! [X4: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ X4 @ X4 )
= X4 ) ).
% sup_idem
thf(fact_65_sup__idem,axiom,
! [X4: set_set_a] :
( ( sup_sup_set_set_a @ X4 @ X4 )
= X4 ) ).
% sup_idem
thf(fact_66_sup__idem,axiom,
! [X4: set_a] :
( ( sup_sup_set_a @ X4 @ X4 )
= X4 ) ).
% sup_idem
thf(fact_67_sup_Oleft__idem,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ A2 @ ( sup_su4083067149120280889_a_nat @ A2 @ B2 ) )
= ( sup_su4083067149120280889_a_nat @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_68_sup_Oleft__idem,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( sup_sup_set_set_a @ A2 @ ( sup_sup_set_set_a @ A2 @ B2 ) )
= ( sup_sup_set_set_a @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_69_sup_Oleft__idem,axiom,
! [A2: set_a,B2: set_a] :
( ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B2 ) )
= ( sup_sup_set_a @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_70_sup__left__idem,axiom,
! [X4: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ X4 @ ( sup_su4083067149120280889_a_nat @ X4 @ Y ) )
= ( sup_su4083067149120280889_a_nat @ X4 @ Y ) ) ).
% sup_left_idem
thf(fact_71_sup__left__idem,axiom,
! [X4: set_set_a,Y: set_set_a] :
( ( sup_sup_set_set_a @ X4 @ ( sup_sup_set_set_a @ X4 @ Y ) )
= ( sup_sup_set_set_a @ X4 @ Y ) ) ).
% sup_left_idem
thf(fact_72_sup__left__idem,axiom,
! [X4: set_a,Y: set_a] :
( ( sup_sup_set_a @ X4 @ ( sup_sup_set_a @ X4 @ Y ) )
= ( sup_sup_set_a @ X4 @ Y ) ) ).
% sup_left_idem
thf(fact_73_sup_Oright__idem,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ ( sup_su4083067149120280889_a_nat @ A2 @ B2 ) @ B2 )
= ( sup_su4083067149120280889_a_nat @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_74_sup_Oright__idem,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ A2 @ B2 ) @ B2 )
= ( sup_sup_set_set_a @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_75_sup_Oright__idem,axiom,
! [A2: set_a,B2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ B2 )
= ( sup_sup_set_a @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_76_inf_Oidem,axiom,
! [A2: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_77_inf_Oidem,axiom,
! [A2: set_se4330304633200676677_a_nat] :
( ( inf_in5367731912061063475_a_nat @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_78_inf_Oidem,axiom,
! [A2: set_set_a] :
( ( inf_inf_set_set_a @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_79_inf_Oidem,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_80_inf__idem,axiom,
! [X4: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ X4 @ X4 )
= X4 ) ).
% inf_idem
thf(fact_81_inf__idem,axiom,
! [X4: set_se4330304633200676677_a_nat] :
( ( inf_in5367731912061063475_a_nat @ X4 @ X4 )
= X4 ) ).
% inf_idem
thf(fact_82_inf__idem,axiom,
! [X4: set_set_a] :
( ( inf_inf_set_set_a @ X4 @ X4 )
= X4 ) ).
% inf_idem
thf(fact_83_inf__idem,axiom,
! [X4: set_a] :
( ( inf_inf_set_a @ X4 @ X4 )
= X4 ) ).
% inf_idem
thf(fact_84_inf_Oleft__idem,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ A2 @ ( inf_in3249246906714053971_a_nat @ A2 @ B2 ) )
= ( inf_in3249246906714053971_a_nat @ A2 @ B2 ) ) ).
% inf.left_idem
thf(fact_85_inf_Oleft__idem,axiom,
! [A2: set_se4330304633200676677_a_nat,B2: set_se4330304633200676677_a_nat] :
( ( inf_in5367731912061063475_a_nat @ A2 @ ( inf_in5367731912061063475_a_nat @ A2 @ B2 ) )
= ( inf_in5367731912061063475_a_nat @ A2 @ B2 ) ) ).
% inf.left_idem
thf(fact_86_inf_Oleft__idem,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( inf_inf_set_set_a @ A2 @ ( inf_inf_set_set_a @ A2 @ B2 ) )
= ( inf_inf_set_set_a @ A2 @ B2 ) ) ).
% inf.left_idem
thf(fact_87_inf_Oleft__idem,axiom,
! [A2: set_a,B2: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ A2 @ B2 ) )
= ( inf_inf_set_a @ A2 @ B2 ) ) ).
% inf.left_idem
thf(fact_88_inf__left__idem,axiom,
! [X4: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ X4 @ ( inf_in3249246906714053971_a_nat @ X4 @ Y ) )
= ( inf_in3249246906714053971_a_nat @ X4 @ Y ) ) ).
% inf_left_idem
thf(fact_89_inf__left__idem,axiom,
! [X4: set_se4330304633200676677_a_nat,Y: set_se4330304633200676677_a_nat] :
( ( inf_in5367731912061063475_a_nat @ X4 @ ( inf_in5367731912061063475_a_nat @ X4 @ Y ) )
= ( inf_in5367731912061063475_a_nat @ X4 @ Y ) ) ).
% inf_left_idem
thf(fact_90_inf__left__idem,axiom,
! [X4: set_set_a,Y: set_set_a] :
( ( inf_inf_set_set_a @ X4 @ ( inf_inf_set_set_a @ X4 @ Y ) )
= ( inf_inf_set_set_a @ X4 @ Y ) ) ).
% inf_left_idem
thf(fact_91_inf__left__idem,axiom,
! [X4: set_a,Y: set_a] :
( ( inf_inf_set_a @ X4 @ ( inf_inf_set_a @ X4 @ Y ) )
= ( inf_inf_set_a @ X4 @ Y ) ) ).
% inf_left_idem
thf(fact_92_inf_Oright__idem,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ ( inf_in3249246906714053971_a_nat @ A2 @ B2 ) @ B2 )
= ( inf_in3249246906714053971_a_nat @ A2 @ B2 ) ) ).
% inf.right_idem
thf(fact_93_inf_Oright__idem,axiom,
! [A2: set_se4330304633200676677_a_nat,B2: set_se4330304633200676677_a_nat] :
( ( inf_in5367731912061063475_a_nat @ ( inf_in5367731912061063475_a_nat @ A2 @ B2 ) @ B2 )
= ( inf_in5367731912061063475_a_nat @ A2 @ B2 ) ) ).
% inf.right_idem
thf(fact_94_inf_Oright__idem,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ A2 @ B2 ) @ B2 )
= ( inf_inf_set_set_a @ A2 @ B2 ) ) ).
% inf.right_idem
thf(fact_95_inf_Oright__idem,axiom,
! [A2: set_a,B2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ B2 )
= ( inf_inf_set_a @ A2 @ B2 ) ) ).
% inf.right_idem
thf(fact_96_inf__right__idem,axiom,
! [X4: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ ( inf_in3249246906714053971_a_nat @ X4 @ Y ) @ Y )
= ( inf_in3249246906714053971_a_nat @ X4 @ Y ) ) ).
% inf_right_idem
thf(fact_97_inf__right__idem,axiom,
! [X4: set_se4330304633200676677_a_nat,Y: set_se4330304633200676677_a_nat] :
( ( inf_in5367731912061063475_a_nat @ ( inf_in5367731912061063475_a_nat @ X4 @ Y ) @ Y )
= ( inf_in5367731912061063475_a_nat @ X4 @ Y ) ) ).
% inf_right_idem
thf(fact_98_inf__right__idem,axiom,
! [X4: set_set_a,Y: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ X4 @ Y ) @ Y )
= ( inf_inf_set_set_a @ X4 @ Y ) ) ).
% inf_right_idem
thf(fact_99_inf__right__idem,axiom,
! [X4: set_a,Y: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X4 @ Y ) @ Y )
= ( inf_inf_set_a @ X4 @ Y ) ) ).
% inf_right_idem
thf(fact_100_UnCI,axiom,
! [C: set_li6526943997496501093_a_nat,B: set_se4330304633200676677_a_nat,A: set_se4330304633200676677_a_nat] :
( ( ~ ( member5553968465346197646_a_nat @ C @ B )
=> ( member5553968465346197646_a_nat @ C @ A ) )
=> ( member5553968465346197646_a_nat @ C @ ( sup_su499249268922660121_a_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_101_UnCI,axiom,
! [C: set_a,B: set_set_a,A: set_set_a] :
( ( ~ ( member_set_a @ C @ B )
=> ( member_set_a @ C @ A ) )
=> ( member_set_a @ C @ ( sup_sup_set_set_a @ A @ B ) ) ) ).
% UnCI
thf(fact_102_UnCI,axiom,
! [C: list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat] :
( ( ~ ( member408289922725080238_a_nat @ C @ B )
=> ( member408289922725080238_a_nat @ C @ A ) )
=> ( member408289922725080238_a_nat @ C @ ( sup_su4083067149120280889_a_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_103_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_104_Un__iff,axiom,
! [C: set_li6526943997496501093_a_nat,A: set_se4330304633200676677_a_nat,B: set_se4330304633200676677_a_nat] :
( ( member5553968465346197646_a_nat @ C @ ( sup_su499249268922660121_a_nat @ A @ B ) )
= ( ( member5553968465346197646_a_nat @ C @ A )
| ( member5553968465346197646_a_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_105_Un__iff,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( sup_sup_set_set_a @ A @ B ) )
= ( ( member_set_a @ C @ A )
| ( member_set_a @ C @ B ) ) ) ).
% Un_iff
thf(fact_106_Un__iff,axiom,
! [C: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ ( sup_su4083067149120280889_a_nat @ A @ B ) )
= ( ( member408289922725080238_a_nat @ C @ A )
| ( member408289922725080238_a_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_107_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_108_IntI,axiom,
! [C: set_li6526943997496501093_a_nat,A: set_se4330304633200676677_a_nat,B: set_se4330304633200676677_a_nat] :
( ( member5553968465346197646_a_nat @ C @ A )
=> ( ( member5553968465346197646_a_nat @ C @ B )
=> ( member5553968465346197646_a_nat @ C @ ( inf_in5367731912061063475_a_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_109_IntI,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ A )
=> ( ( member_set_a @ C @ B )
=> ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B ) ) ) ) ).
% IntI
thf(fact_110_IntI,axiom,
! [C: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ A )
=> ( ( member408289922725080238_a_nat @ C @ B )
=> ( member408289922725080238_a_nat @ C @ ( inf_in3249246906714053971_a_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_111_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_112_Int__iff,axiom,
! [C: set_li6526943997496501093_a_nat,A: set_se4330304633200676677_a_nat,B: set_se4330304633200676677_a_nat] :
( ( member5553968465346197646_a_nat @ C @ ( inf_in5367731912061063475_a_nat @ A @ B ) )
= ( ( member5553968465346197646_a_nat @ C @ A )
& ( member5553968465346197646_a_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_113_Int__iff,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B ) )
= ( ( member_set_a @ C @ A )
& ( member_set_a @ C @ B ) ) ) ).
% Int_iff
thf(fact_114_Int__iff,axiom,
! [C: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ ( inf_in3249246906714053971_a_nat @ A @ B ) )
= ( ( member408289922725080238_a_nat @ C @ A )
& ( member408289922725080238_a_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_115_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_116_sup_Obounded__iff,axiom,
! [B2: set_se4330304633200676677_a_nat,C: set_se4330304633200676677_a_nat,A2: set_se4330304633200676677_a_nat] :
( ( ord_le8138476598237931237_a_nat @ ( sup_su499249268922660121_a_nat @ B2 @ C ) @ A2 )
= ( ( ord_le8138476598237931237_a_nat @ B2 @ A2 )
& ( ord_le8138476598237931237_a_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_117_sup_Obounded__iff,axiom,
! [B2: set_set_a,C: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ B2 @ C ) @ A2 )
= ( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
& ( ord_le3724670747650509150_set_a @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_118_sup_Obounded__iff,axiom,
! [B2: set_a,C: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C ) @ A2 )
= ( ( ord_less_eq_set_a @ B2 @ A2 )
& ( ord_less_eq_set_a @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_119_sup_Obounded__iff,axiom,
! [B2: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ ( sup_su4083067149120280889_a_nat @ B2 @ C ) @ A2 )
= ( ( ord_le1147066620699065093_a_nat @ B2 @ A2 )
& ( ord_le1147066620699065093_a_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_120_le__sup__iff,axiom,
! [X4: set_se4330304633200676677_a_nat,Y: set_se4330304633200676677_a_nat,Z: set_se4330304633200676677_a_nat] :
( ( ord_le8138476598237931237_a_nat @ ( sup_su499249268922660121_a_nat @ X4 @ Y ) @ Z )
= ( ( ord_le8138476598237931237_a_nat @ X4 @ Z )
& ( ord_le8138476598237931237_a_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_121_le__sup__iff,axiom,
! [X4: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ X4 @ Y ) @ Z )
= ( ( ord_le3724670747650509150_set_a @ X4 @ Z )
& ( ord_le3724670747650509150_set_a @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_122_le__sup__iff,axiom,
! [X4: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X4 @ Y ) @ Z )
= ( ( ord_less_eq_set_a @ X4 @ Z )
& ( ord_less_eq_set_a @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_123_le__sup__iff,axiom,
! [X4: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat,Z: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ ( sup_su4083067149120280889_a_nat @ X4 @ Y ) @ Z )
= ( ( ord_le1147066620699065093_a_nat @ X4 @ Z )
& ( ord_le1147066620699065093_a_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_124_inf_Obounded__iff,axiom,
! [A2: set_se4330304633200676677_a_nat,B2: set_se4330304633200676677_a_nat,C: set_se4330304633200676677_a_nat] :
( ( ord_le8138476598237931237_a_nat @ A2 @ ( inf_in5367731912061063475_a_nat @ B2 @ C ) )
= ( ( ord_le8138476598237931237_a_nat @ A2 @ B2 )
& ( ord_le8138476598237931237_a_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_125_inf_Obounded__iff,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( inf_inf_set_set_a @ B2 @ C ) )
= ( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
& ( ord_le3724670747650509150_set_a @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_126_inf_Obounded__iff,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) )
= ( ( ord_less_eq_set_a @ A2 @ B2 )
& ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_127_inf_Obounded__iff,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A2 @ ( inf_in3249246906714053971_a_nat @ B2 @ C ) )
= ( ( ord_le1147066620699065093_a_nat @ A2 @ B2 )
& ( ord_le1147066620699065093_a_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_128_le__inf__iff,axiom,
! [X4: set_se4330304633200676677_a_nat,Y: set_se4330304633200676677_a_nat,Z: set_se4330304633200676677_a_nat] :
( ( ord_le8138476598237931237_a_nat @ X4 @ ( inf_in5367731912061063475_a_nat @ Y @ Z ) )
= ( ( ord_le8138476598237931237_a_nat @ X4 @ Y )
& ( ord_le8138476598237931237_a_nat @ X4 @ Z ) ) ) ).
% le_inf_iff
thf(fact_129_le__inf__iff,axiom,
! [X4: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ ( inf_inf_set_set_a @ Y @ Z ) )
= ( ( ord_le3724670747650509150_set_a @ X4 @ Y )
& ( ord_le3724670747650509150_set_a @ X4 @ Z ) ) ) ).
% le_inf_iff
thf(fact_130_le__inf__iff,axiom,
! [X4: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X4 @ ( inf_inf_set_a @ Y @ Z ) )
= ( ( ord_less_eq_set_a @ X4 @ Y )
& ( ord_less_eq_set_a @ X4 @ Z ) ) ) ).
% le_inf_iff
thf(fact_131_le__inf__iff,axiom,
! [X4: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat,Z: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ X4 @ ( inf_in3249246906714053971_a_nat @ Y @ Z ) )
= ( ( ord_le1147066620699065093_a_nat @ X4 @ Y )
& ( ord_le1147066620699065093_a_nat @ X4 @ Z ) ) ) ).
% le_inf_iff
thf(fact_132_empty__subsetI,axiom,
! [A: set_se4330304633200676677_a_nat] : ( ord_le8138476598237931237_a_nat @ bot_bo3237059034911209905_a_nat @ A ) ).
% empty_subsetI
thf(fact_133_empty__subsetI,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A ) ).
% empty_subsetI
thf(fact_134_empty__subsetI,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% empty_subsetI
thf(fact_135_empty__subsetI,axiom,
! [A: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ bot_bo1033123847703346641_a_nat @ A ) ).
% empty_subsetI
thf(fact_136_subset__empty,axiom,
! [A: set_se4330304633200676677_a_nat] :
( ( ord_le8138476598237931237_a_nat @ A @ bot_bo3237059034911209905_a_nat )
= ( A = bot_bo3237059034911209905_a_nat ) ) ).
% subset_empty
thf(fact_137_subset__empty,axiom,
! [A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ bot_bot_set_set_a )
= ( A = bot_bot_set_set_a ) ) ).
% subset_empty
thf(fact_138_subset__empty,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_139_subset__empty,axiom,
! [A: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A @ bot_bo1033123847703346641_a_nat )
= ( A = bot_bo1033123847703346641_a_nat ) ) ).
% subset_empty
thf(fact_140_sup__bot__left,axiom,
! [X4: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ bot_bo1033123847703346641_a_nat @ X4 )
= X4 ) ).
% sup_bot_left
thf(fact_141_sup__bot__left,axiom,
! [X4: a > $o] :
( ( sup_sup_a_o @ bot_bot_a_o @ X4 )
= X4 ) ).
% sup_bot_left
thf(fact_142_sup__bot__left,axiom,
! [X4: list_Sum_sum_a_nat > $o] :
( ( sup_su1334248866174809316_nat_o @ bot_bo9042073657639083596_nat_o @ X4 )
= X4 ) ).
% sup_bot_left
thf(fact_143_sup__bot__left,axiom,
! [X4: set_se4330304633200676677_a_nat] :
( ( sup_su499249268922660121_a_nat @ bot_bo3237059034911209905_a_nat @ X4 )
= X4 ) ).
% sup_bot_left
thf(fact_144_sup__bot__left,axiom,
! [X4: set_set_a] :
( ( sup_sup_set_set_a @ bot_bot_set_set_a @ X4 )
= X4 ) ).
% sup_bot_left
thf(fact_145_sup__bot__left,axiom,
! [X4: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ X4 )
= X4 ) ).
% sup_bot_left
thf(fact_146_sup__bot__right,axiom,
! [X4: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ X4 @ bot_bo1033123847703346641_a_nat )
= X4 ) ).
% sup_bot_right
thf(fact_147_sup__bot__right,axiom,
! [X4: a > $o] :
( ( sup_sup_a_o @ X4 @ bot_bot_a_o )
= X4 ) ).
% sup_bot_right
thf(fact_148_sup__bot__right,axiom,
! [X4: list_Sum_sum_a_nat > $o] :
( ( sup_su1334248866174809316_nat_o @ X4 @ bot_bo9042073657639083596_nat_o )
= X4 ) ).
% sup_bot_right
thf(fact_149_sup__bot__right,axiom,
! [X4: set_se4330304633200676677_a_nat] :
( ( sup_su499249268922660121_a_nat @ X4 @ bot_bo3237059034911209905_a_nat )
= X4 ) ).
% sup_bot_right
thf(fact_150_sup__bot__right,axiom,
! [X4: set_set_a] :
( ( sup_sup_set_set_a @ X4 @ bot_bot_set_set_a )
= X4 ) ).
% sup_bot_right
thf(fact_151_sup__bot__right,axiom,
! [X4: set_a] :
( ( sup_sup_set_a @ X4 @ bot_bot_set_a )
= X4 ) ).
% sup_bot_right
thf(fact_152_bot__eq__sup__iff,axiom,
! [X4: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat] :
( ( bot_bo1033123847703346641_a_nat
= ( sup_su4083067149120280889_a_nat @ X4 @ Y ) )
= ( ( X4 = bot_bo1033123847703346641_a_nat )
& ( Y = bot_bo1033123847703346641_a_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_153_bot__eq__sup__iff,axiom,
! [X4: a > $o,Y: a > $o] :
( ( bot_bot_a_o
= ( sup_sup_a_o @ X4 @ Y ) )
= ( ( X4 = bot_bot_a_o )
& ( Y = bot_bot_a_o ) ) ) ).
% bot_eq_sup_iff
thf(fact_154_bot__eq__sup__iff,axiom,
! [X4: list_Sum_sum_a_nat > $o,Y: list_Sum_sum_a_nat > $o] :
( ( bot_bo9042073657639083596_nat_o
= ( sup_su1334248866174809316_nat_o @ X4 @ Y ) )
= ( ( X4 = bot_bo9042073657639083596_nat_o )
& ( Y = bot_bo9042073657639083596_nat_o ) ) ) ).
% bot_eq_sup_iff
thf(fact_155_bot__eq__sup__iff,axiom,
! [X4: set_se4330304633200676677_a_nat,Y: set_se4330304633200676677_a_nat] :
( ( bot_bo3237059034911209905_a_nat
= ( sup_su499249268922660121_a_nat @ X4 @ Y ) )
= ( ( X4 = bot_bo3237059034911209905_a_nat )
& ( Y = bot_bo3237059034911209905_a_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_156_bot__eq__sup__iff,axiom,
! [X4: set_set_a,Y: set_set_a] :
( ( bot_bot_set_set_a
= ( sup_sup_set_set_a @ X4 @ Y ) )
= ( ( X4 = bot_bot_set_set_a )
& ( Y = bot_bot_set_set_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_157_bot__eq__sup__iff,axiom,
! [X4: set_a,Y: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ X4 @ Y ) )
= ( ( X4 = bot_bot_set_a )
& ( Y = bot_bot_set_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_158_mem__Collect__eq,axiom,
! [A2: set_li6526943997496501093_a_nat,P: set_li6526943997496501093_a_nat > $o] :
( ( member5553968465346197646_a_nat @ A2 @ ( collec7528627406912015568_a_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_159_mem__Collect__eq,axiom,
! [A2: set_a,P: set_a > $o] :
( ( member_set_a @ A2 @ ( collect_set_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_160_mem__Collect__eq,axiom,
! [A2: list_Sum_sum_a_nat,P: list_Sum_sum_a_nat > $o] :
( ( member408289922725080238_a_nat @ A2 @ ( collec7555443234367654128_a_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_161_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_162_Collect__mem__eq,axiom,
! [A: set_se4330304633200676677_a_nat] :
( ( collec7528627406912015568_a_nat
@ ^ [X5: set_li6526943997496501093_a_nat] : ( member5553968465346197646_a_nat @ X5 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_163_Collect__mem__eq,axiom,
! [A: set_set_a] :
( ( collect_set_a
@ ^ [X5: set_a] : ( member_set_a @ X5 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_164_Collect__mem__eq,axiom,
! [A: set_li6526943997496501093_a_nat] :
( ( collec7555443234367654128_a_nat
@ ^ [X5: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X5 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_165_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X5: a] : ( member_a @ X5 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_166_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_167_Collect__cong,axiom,
! [P: list_Sum_sum_a_nat > $o,Q: list_Sum_sum_a_nat > $o] :
( ! [X3: list_Sum_sum_a_nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collec7555443234367654128_a_nat @ P )
= ( collec7555443234367654128_a_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_168_sup__eq__bot__iff,axiom,
! [X4: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat] :
( ( ( sup_su4083067149120280889_a_nat @ X4 @ Y )
= bot_bo1033123847703346641_a_nat )
= ( ( X4 = bot_bo1033123847703346641_a_nat )
& ( Y = bot_bo1033123847703346641_a_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_169_sup__eq__bot__iff,axiom,
! [X4: a > $o,Y: a > $o] :
( ( ( sup_sup_a_o @ X4 @ Y )
= bot_bot_a_o )
= ( ( X4 = bot_bot_a_o )
& ( Y = bot_bot_a_o ) ) ) ).
% sup_eq_bot_iff
thf(fact_170_sup__eq__bot__iff,axiom,
! [X4: list_Sum_sum_a_nat > $o,Y: list_Sum_sum_a_nat > $o] :
( ( ( sup_su1334248866174809316_nat_o @ X4 @ Y )
= bot_bo9042073657639083596_nat_o )
= ( ( X4 = bot_bo9042073657639083596_nat_o )
& ( Y = bot_bo9042073657639083596_nat_o ) ) ) ).
% sup_eq_bot_iff
thf(fact_171_sup__eq__bot__iff,axiom,
! [X4: set_se4330304633200676677_a_nat,Y: set_se4330304633200676677_a_nat] :
( ( ( sup_su499249268922660121_a_nat @ X4 @ Y )
= bot_bo3237059034911209905_a_nat )
= ( ( X4 = bot_bo3237059034911209905_a_nat )
& ( Y = bot_bo3237059034911209905_a_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_172_sup__eq__bot__iff,axiom,
! [X4: set_set_a,Y: set_set_a] :
( ( ( sup_sup_set_set_a @ X4 @ Y )
= bot_bot_set_set_a )
= ( ( X4 = bot_bot_set_set_a )
& ( Y = bot_bot_set_set_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_173_sup__eq__bot__iff,axiom,
! [X4: set_a,Y: set_a] :
( ( ( sup_sup_set_a @ X4 @ Y )
= bot_bot_set_a )
= ( ( X4 = bot_bot_set_a )
& ( Y = bot_bot_set_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_174_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ( sup_su4083067149120280889_a_nat @ A2 @ B2 )
= bot_bo1033123847703346641_a_nat )
= ( ( A2 = bot_bo1033123847703346641_a_nat )
& ( B2 = bot_bo1033123847703346641_a_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_175_sup__bot_Oeq__neutr__iff,axiom,
! [A2: a > $o,B2: a > $o] :
( ( ( sup_sup_a_o @ A2 @ B2 )
= bot_bot_a_o )
= ( ( A2 = bot_bot_a_o )
& ( B2 = bot_bot_a_o ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_176_sup__bot_Oeq__neutr__iff,axiom,
! [A2: list_Sum_sum_a_nat > $o,B2: list_Sum_sum_a_nat > $o] :
( ( ( sup_su1334248866174809316_nat_o @ A2 @ B2 )
= bot_bo9042073657639083596_nat_o )
= ( ( A2 = bot_bo9042073657639083596_nat_o )
& ( B2 = bot_bo9042073657639083596_nat_o ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_177_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_se4330304633200676677_a_nat,B2: set_se4330304633200676677_a_nat] :
( ( ( sup_su499249268922660121_a_nat @ A2 @ B2 )
= bot_bo3237059034911209905_a_nat )
= ( ( A2 = bot_bo3237059034911209905_a_nat )
& ( B2 = bot_bo3237059034911209905_a_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_178_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ( sup_sup_set_set_a @ A2 @ B2 )
= bot_bot_set_set_a )
= ( ( A2 = bot_bot_set_set_a )
& ( B2 = bot_bot_set_set_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_179_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( ( sup_sup_set_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ( A2 = bot_bot_set_a )
& ( B2 = bot_bot_set_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_180_sup__bot_Oleft__neutral,axiom,
! [A2: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ bot_bo1033123847703346641_a_nat @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_181_sup__bot_Oleft__neutral,axiom,
! [A2: a > $o] :
( ( sup_sup_a_o @ bot_bot_a_o @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_182_sup__bot_Oleft__neutral,axiom,
! [A2: list_Sum_sum_a_nat > $o] :
( ( sup_su1334248866174809316_nat_o @ bot_bo9042073657639083596_nat_o @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_183_sup__bot_Oleft__neutral,axiom,
! [A2: set_se4330304633200676677_a_nat] :
( ( sup_su499249268922660121_a_nat @ bot_bo3237059034911209905_a_nat @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_184_sup__bot_Oleft__neutral,axiom,
! [A2: set_set_a] :
( ( sup_sup_set_set_a @ bot_bot_set_set_a @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_185_sup__bot_Oleft__neutral,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_186_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( bot_bo1033123847703346641_a_nat
= ( sup_su4083067149120280889_a_nat @ A2 @ B2 ) )
= ( ( A2 = bot_bo1033123847703346641_a_nat )
& ( B2 = bot_bo1033123847703346641_a_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_187_sup__bot_Oneutr__eq__iff,axiom,
! [A2: a > $o,B2: a > $o] :
( ( bot_bot_a_o
= ( sup_sup_a_o @ A2 @ B2 ) )
= ( ( A2 = bot_bot_a_o )
& ( B2 = bot_bot_a_o ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_188_sup__bot_Oneutr__eq__iff,axiom,
! [A2: list_Sum_sum_a_nat > $o,B2: list_Sum_sum_a_nat > $o] :
( ( bot_bo9042073657639083596_nat_o
= ( sup_su1334248866174809316_nat_o @ A2 @ B2 ) )
= ( ( A2 = bot_bo9042073657639083596_nat_o )
& ( B2 = bot_bo9042073657639083596_nat_o ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_189_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_se4330304633200676677_a_nat,B2: set_se4330304633200676677_a_nat] :
( ( bot_bo3237059034911209905_a_nat
= ( sup_su499249268922660121_a_nat @ A2 @ B2 ) )
= ( ( A2 = bot_bo3237059034911209905_a_nat )
& ( B2 = bot_bo3237059034911209905_a_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_190_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( bot_bot_set_set_a
= ( sup_sup_set_set_a @ A2 @ B2 ) )
= ( ( A2 = bot_bot_set_set_a )
& ( B2 = bot_bot_set_set_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_191_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ A2 @ B2 ) )
= ( ( A2 = bot_bot_set_a )
& ( B2 = bot_bot_set_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_192_sup__bot_Oright__neutral,axiom,
! [A2: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ A2 @ bot_bo1033123847703346641_a_nat )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_193_sup__bot_Oright__neutral,axiom,
! [A2: a > $o] :
( ( sup_sup_a_o @ A2 @ bot_bot_a_o )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_194_sup__bot_Oright__neutral,axiom,
! [A2: list_Sum_sum_a_nat > $o] :
( ( sup_su1334248866174809316_nat_o @ A2 @ bot_bo9042073657639083596_nat_o )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_195_sup__bot_Oright__neutral,axiom,
! [A2: set_se4330304633200676677_a_nat] :
( ( sup_su499249268922660121_a_nat @ A2 @ bot_bo3237059034911209905_a_nat )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_196_sup__bot_Oright__neutral,axiom,
! [A2: set_set_a] :
( ( sup_sup_set_set_a @ A2 @ bot_bot_set_set_a )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_197_sup__bot_Oright__neutral,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_198_inf__bot__left,axiom,
! [X4: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ bot_bo1033123847703346641_a_nat @ X4 )
= bot_bo1033123847703346641_a_nat ) ).
% inf_bot_left
thf(fact_199_inf__bot__left,axiom,
! [X4: a > $o] :
( ( inf_inf_a_o @ bot_bot_a_o @ X4 )
= bot_bot_a_o ) ).
% inf_bot_left
thf(fact_200_inf__bot__left,axiom,
! [X4: list_Sum_sum_a_nat > $o] :
( ( inf_in954358986474102090_nat_o @ bot_bo9042073657639083596_nat_o @ X4 )
= bot_bo9042073657639083596_nat_o ) ).
% inf_bot_left
thf(fact_201_inf__bot__left,axiom,
! [X4: set_se4330304633200676677_a_nat] :
( ( inf_in5367731912061063475_a_nat @ bot_bo3237059034911209905_a_nat @ X4 )
= bot_bo3237059034911209905_a_nat ) ).
% inf_bot_left
thf(fact_202_inf__bot__left,axiom,
! [X4: set_set_a] :
( ( inf_inf_set_set_a @ bot_bot_set_set_a @ X4 )
= bot_bot_set_set_a ) ).
% inf_bot_left
thf(fact_203_inf__bot__left,axiom,
! [X4: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X4 )
= bot_bot_set_a ) ).
% inf_bot_left
thf(fact_204_inf__bot__right,axiom,
! [X4: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ X4 @ bot_bo1033123847703346641_a_nat )
= bot_bo1033123847703346641_a_nat ) ).
% inf_bot_right
thf(fact_205_inf__bot__right,axiom,
! [X4: a > $o] :
( ( inf_inf_a_o @ X4 @ bot_bot_a_o )
= bot_bot_a_o ) ).
% inf_bot_right
thf(fact_206_inf__bot__right,axiom,
! [X4: list_Sum_sum_a_nat > $o] :
( ( inf_in954358986474102090_nat_o @ X4 @ bot_bo9042073657639083596_nat_o )
= bot_bo9042073657639083596_nat_o ) ).
% inf_bot_right
thf(fact_207_inf__bot__right,axiom,
! [X4: set_se4330304633200676677_a_nat] :
( ( inf_in5367731912061063475_a_nat @ X4 @ bot_bo3237059034911209905_a_nat )
= bot_bo3237059034911209905_a_nat ) ).
% inf_bot_right
thf(fact_208_inf__bot__right,axiom,
! [X4: set_set_a] :
( ( inf_inf_set_set_a @ X4 @ bot_bot_set_set_a )
= bot_bot_set_set_a ) ).
% inf_bot_right
thf(fact_209_inf__bot__right,axiom,
! [X4: set_a] :
( ( inf_inf_set_a @ X4 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% inf_bot_right
thf(fact_210_inf__sup__absorb,axiom,
! [X4: set_se4330304633200676677_a_nat,Y: set_se4330304633200676677_a_nat] :
( ( inf_in5367731912061063475_a_nat @ X4 @ ( sup_su499249268922660121_a_nat @ X4 @ Y ) )
= X4 ) ).
% inf_sup_absorb
thf(fact_211_inf__sup__absorb,axiom,
! [X4: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ X4 @ ( sup_su4083067149120280889_a_nat @ X4 @ Y ) )
= X4 ) ).
% inf_sup_absorb
thf(fact_212_inf__sup__absorb,axiom,
! [X4: set_set_a,Y: set_set_a] :
( ( inf_inf_set_set_a @ X4 @ ( sup_sup_set_set_a @ X4 @ Y ) )
= X4 ) ).
% inf_sup_absorb
thf(fact_213_inf__sup__absorb,axiom,
! [X4: set_a,Y: set_a] :
( ( inf_inf_set_a @ X4 @ ( sup_sup_set_a @ X4 @ Y ) )
= X4 ) ).
% inf_sup_absorb
thf(fact_214_sup__inf__absorb,axiom,
! [X4: set_se4330304633200676677_a_nat,Y: set_se4330304633200676677_a_nat] :
( ( sup_su499249268922660121_a_nat @ X4 @ ( inf_in5367731912061063475_a_nat @ X4 @ Y ) )
= X4 ) ).
% sup_inf_absorb
thf(fact_215_sup__inf__absorb,axiom,
! [X4: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ X4 @ ( inf_in3249246906714053971_a_nat @ X4 @ Y ) )
= X4 ) ).
% sup_inf_absorb
thf(fact_216_sup__inf__absorb,axiom,
! [X4: set_set_a,Y: set_set_a] :
( ( sup_sup_set_set_a @ X4 @ ( inf_inf_set_set_a @ X4 @ Y ) )
= X4 ) ).
% sup_inf_absorb
thf(fact_217_sup__inf__absorb,axiom,
! [X4: set_a,Y: set_a] :
( ( sup_sup_set_a @ X4 @ ( inf_inf_set_a @ X4 @ Y ) )
= X4 ) ).
% sup_inf_absorb
thf(fact_218_Un__empty,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ( sup_su4083067149120280889_a_nat @ A @ B )
= bot_bo1033123847703346641_a_nat )
= ( ( A = bot_bo1033123847703346641_a_nat )
& ( B = bot_bo1033123847703346641_a_nat ) ) ) ).
% Un_empty
thf(fact_219_Un__empty,axiom,
! [A: set_se4330304633200676677_a_nat,B: set_se4330304633200676677_a_nat] :
( ( ( sup_su499249268922660121_a_nat @ A @ B )
= bot_bo3237059034911209905_a_nat )
= ( ( A = bot_bo3237059034911209905_a_nat )
& ( B = bot_bo3237059034911209905_a_nat ) ) ) ).
% Un_empty
thf(fact_220_Un__empty,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ( sup_sup_set_set_a @ A @ B )
= bot_bot_set_set_a )
= ( ( A = bot_bot_set_set_a )
& ( B = bot_bot_set_set_a ) ) ) ).
% Un_empty
thf(fact_221_Un__empty,axiom,
! [A: set_a,B: set_a] :
( ( ( sup_sup_set_a @ A @ B )
= bot_bot_set_a )
= ( ( A = bot_bot_set_a )
& ( B = bot_bot_set_a ) ) ) ).
% Un_empty
thf(fact_222_Un__subset__iff,axiom,
! [A: set_se4330304633200676677_a_nat,B: set_se4330304633200676677_a_nat,C2: set_se4330304633200676677_a_nat] :
( ( ord_le8138476598237931237_a_nat @ ( sup_su499249268922660121_a_nat @ A @ B ) @ C2 )
= ( ( ord_le8138476598237931237_a_nat @ A @ C2 )
& ( ord_le8138476598237931237_a_nat @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_223_Un__subset__iff,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ A @ B ) @ C2 )
= ( ( ord_le3724670747650509150_set_a @ A @ C2 )
& ( ord_le3724670747650509150_set_a @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_224_Un__subset__iff,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ C2 )
= ( ( ord_less_eq_set_a @ A @ C2 )
& ( ord_less_eq_set_a @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_225_Un__subset__iff,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,C2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ ( sup_su4083067149120280889_a_nat @ A @ B ) @ C2 )
= ( ( ord_le1147066620699065093_a_nat @ A @ C2 )
& ( ord_le1147066620699065093_a_nat @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_226_Int__subset__iff,axiom,
! [C2: set_se4330304633200676677_a_nat,A: set_se4330304633200676677_a_nat,B: set_se4330304633200676677_a_nat] :
( ( ord_le8138476598237931237_a_nat @ C2 @ ( inf_in5367731912061063475_a_nat @ A @ B ) )
= ( ( ord_le8138476598237931237_a_nat @ C2 @ A )
& ( ord_le8138476598237931237_a_nat @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_227_Int__subset__iff,axiom,
! [C2: set_set_a,A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C2 @ ( inf_inf_set_set_a @ A @ B ) )
= ( ( ord_le3724670747650509150_set_a @ C2 @ A )
& ( ord_le3724670747650509150_set_a @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_228_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_229_Int__subset__iff,axiom,
! [C2: set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ C2 @ ( inf_in3249246906714053971_a_nat @ A @ B ) )
= ( ( ord_le1147066620699065093_a_nat @ C2 @ A )
& ( ord_le1147066620699065093_a_nat @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_230_Diff__cancel,axiom,
! [A: set_se4330304633200676677_a_nat] :
( ( minus_8004192006823430828_a_nat @ A @ A )
= bot_bo3237059034911209905_a_nat ) ).
% Diff_cancel
thf(fact_231_Diff__cancel,axiom,
! [A: set_set_a] :
( ( minus_5736297505244876581_set_a @ A @ A )
= bot_bot_set_set_a ) ).
% Diff_cancel
thf(fact_232_Diff__cancel,axiom,
! [A: set_a] :
( ( minus_minus_set_a @ A @ A )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_233_Diff__cancel,axiom,
! [A: set_li6526943997496501093_a_nat] :
( ( minus_7395159227704179404_a_nat @ A @ A )
= bot_bo1033123847703346641_a_nat ) ).
% Diff_cancel
thf(fact_234_empty__Diff,axiom,
! [A: set_se4330304633200676677_a_nat] :
( ( minus_8004192006823430828_a_nat @ bot_bo3237059034911209905_a_nat @ A )
= bot_bo3237059034911209905_a_nat ) ).
% empty_Diff
thf(fact_235_empty__Diff,axiom,
! [A: set_set_a] :
( ( minus_5736297505244876581_set_a @ bot_bot_set_set_a @ A )
= bot_bot_set_set_a ) ).
% empty_Diff
thf(fact_236_empty__Diff,axiom,
! [A: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_237_empty__Diff,axiom,
! [A: set_li6526943997496501093_a_nat] :
( ( minus_7395159227704179404_a_nat @ bot_bo1033123847703346641_a_nat @ A )
= bot_bo1033123847703346641_a_nat ) ).
% empty_Diff
thf(fact_238_Diff__empty,axiom,
! [A: set_se4330304633200676677_a_nat] :
( ( minus_8004192006823430828_a_nat @ A @ bot_bo3237059034911209905_a_nat )
= A ) ).
% Diff_empty
thf(fact_239_Diff__empty,axiom,
! [A: set_set_a] :
( ( minus_5736297505244876581_set_a @ A @ bot_bot_set_set_a )
= A ) ).
% Diff_empty
thf(fact_240_Diff__empty,axiom,
! [A: set_a] :
( ( minus_minus_set_a @ A @ bot_bot_set_a )
= A ) ).
% Diff_empty
thf(fact_241_Diff__empty,axiom,
! [A: set_li6526943997496501093_a_nat] :
( ( minus_7395159227704179404_a_nat @ A @ bot_bo1033123847703346641_a_nat )
= A ) ).
% Diff_empty
thf(fact_242_Int__Un__eq_I4_J,axiom,
! [T: set_se4330304633200676677_a_nat,S: set_se4330304633200676677_a_nat] :
( ( sup_su499249268922660121_a_nat @ T @ ( inf_in5367731912061063475_a_nat @ S @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_243_Int__Un__eq_I4_J,axiom,
! [T: set_li6526943997496501093_a_nat,S: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ T @ ( inf_in3249246906714053971_a_nat @ S @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_244_Int__Un__eq_I4_J,axiom,
! [T: set_set_a,S: set_set_a] :
( ( sup_sup_set_set_a @ T @ ( inf_inf_set_set_a @ S @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_245_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_246_Int__Un__eq_I3_J,axiom,
! [S: set_se4330304633200676677_a_nat,T: set_se4330304633200676677_a_nat] :
( ( sup_su499249268922660121_a_nat @ S @ ( inf_in5367731912061063475_a_nat @ S @ T ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_247_Int__Un__eq_I3_J,axiom,
! [S: set_li6526943997496501093_a_nat,T: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ S @ ( inf_in3249246906714053971_a_nat @ S @ T ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_248_Int__Un__eq_I3_J,axiom,
! [S: set_set_a,T: set_set_a] :
( ( sup_sup_set_set_a @ S @ ( inf_inf_set_set_a @ S @ T ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_249_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_250_Int__Un__eq_I2_J,axiom,
! [S: set_se4330304633200676677_a_nat,T: set_se4330304633200676677_a_nat] :
( ( sup_su499249268922660121_a_nat @ ( inf_in5367731912061063475_a_nat @ S @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_251_Int__Un__eq_I2_J,axiom,
! [S: set_li6526943997496501093_a_nat,T: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ ( inf_in3249246906714053971_a_nat @ S @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_252_Int__Un__eq_I2_J,axiom,
! [S: set_set_a,T: set_set_a] :
( ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ S @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_253_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_254_Int__Un__eq_I1_J,axiom,
! [S: set_se4330304633200676677_a_nat,T: set_se4330304633200676677_a_nat] :
( ( sup_su499249268922660121_a_nat @ ( inf_in5367731912061063475_a_nat @ S @ T ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_255_Int__Un__eq_I1_J,axiom,
! [S: set_li6526943997496501093_a_nat,T: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ ( inf_in3249246906714053971_a_nat @ S @ T ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_256_Int__Un__eq_I1_J,axiom,
! [S: set_set_a,T: set_set_a] :
( ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ S @ T ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_257_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_258_Un__Int__eq_I4_J,axiom,
! [T: set_se4330304633200676677_a_nat,S: set_se4330304633200676677_a_nat] :
( ( inf_in5367731912061063475_a_nat @ T @ ( sup_su499249268922660121_a_nat @ S @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_259_Un__Int__eq_I4_J,axiom,
! [T: set_li6526943997496501093_a_nat,S: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ T @ ( sup_su4083067149120280889_a_nat @ S @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_260_Un__Int__eq_I4_J,axiom,
! [T: set_set_a,S: set_set_a] :
( ( inf_inf_set_set_a @ T @ ( sup_sup_set_set_a @ S @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_261_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_262_Un__Int__eq_I3_J,axiom,
! [S: set_se4330304633200676677_a_nat,T: set_se4330304633200676677_a_nat] :
( ( inf_in5367731912061063475_a_nat @ S @ ( sup_su499249268922660121_a_nat @ S @ T ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_263_Un__Int__eq_I3_J,axiom,
! [S: set_li6526943997496501093_a_nat,T: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ S @ ( sup_su4083067149120280889_a_nat @ S @ T ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_264_Un__Int__eq_I3_J,axiom,
! [S: set_set_a,T: set_set_a] :
( ( inf_inf_set_set_a @ S @ ( sup_sup_set_set_a @ S @ T ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_265_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_266_Un__Int__eq_I2_J,axiom,
! [S: set_se4330304633200676677_a_nat,T: set_se4330304633200676677_a_nat] :
( ( inf_in5367731912061063475_a_nat @ ( sup_su499249268922660121_a_nat @ S @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_267_Un__Int__eq_I2_J,axiom,
! [S: set_li6526943997496501093_a_nat,T: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ ( sup_su4083067149120280889_a_nat @ S @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_268_Un__Int__eq_I2_J,axiom,
! [S: set_set_a,T: set_set_a] :
( ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ S @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_269_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_270_Un__Int__eq_I1_J,axiom,
! [S: set_se4330304633200676677_a_nat,T: set_se4330304633200676677_a_nat] :
( ( inf_in5367731912061063475_a_nat @ ( sup_su499249268922660121_a_nat @ S @ T ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_271_Un__Int__eq_I1_J,axiom,
! [S: set_li6526943997496501093_a_nat,T: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ ( sup_su4083067149120280889_a_nat @ S @ T ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_272_Un__Int__eq_I1_J,axiom,
! [S: set_set_a,T: set_set_a] :
( ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ S @ T ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_273_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_274_Un__Diff__cancel2,axiom,
! [B: set_set_a,A: set_set_a] :
( ( sup_sup_set_set_a @ ( minus_5736297505244876581_set_a @ B @ A ) @ A )
= ( sup_sup_set_set_a @ B @ A ) ) ).
% Un_Diff_cancel2
thf(fact_275_Un__Diff__cancel2,axiom,
! [B: set_a,A: set_a] :
( ( sup_sup_set_a @ ( minus_minus_set_a @ B @ A ) @ A )
= ( sup_sup_set_a @ B @ A ) ) ).
% Un_Diff_cancel2
thf(fact_276_Un__Diff__cancel2,axiom,
! [B: set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ ( minus_7395159227704179404_a_nat @ B @ A ) @ A )
= ( sup_su4083067149120280889_a_nat @ B @ A ) ) ).
% Un_Diff_cancel2
thf(fact_277_Un__Diff__cancel,axiom,
! [A: set_set_a,B: set_set_a] :
( ( sup_sup_set_set_a @ A @ ( minus_5736297505244876581_set_a @ B @ A ) )
= ( sup_sup_set_set_a @ A @ B ) ) ).
% Un_Diff_cancel
thf(fact_278_Un__Diff__cancel,axiom,
! [A: set_a,B: set_a] :
( ( sup_sup_set_a @ A @ ( minus_minus_set_a @ B @ A ) )
= ( sup_sup_set_a @ A @ B ) ) ).
% Un_Diff_cancel
thf(fact_279_Un__Diff__cancel,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ A @ ( minus_7395159227704179404_a_nat @ B @ A ) )
= ( sup_su4083067149120280889_a_nat @ A @ B ) ) ).
% Un_Diff_cancel
thf(fact_280_Diff__eq__empty__iff,axiom,
! [A: set_se4330304633200676677_a_nat,B: set_se4330304633200676677_a_nat] :
( ( ( minus_8004192006823430828_a_nat @ A @ B )
= bot_bo3237059034911209905_a_nat )
= ( ord_le8138476598237931237_a_nat @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_281_Diff__eq__empty__iff,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ( minus_5736297505244876581_set_a @ A @ B )
= bot_bot_set_set_a )
= ( ord_le3724670747650509150_set_a @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_282_Diff__eq__empty__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( minus_minus_set_a @ A @ B )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_283_Diff__eq__empty__iff,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ( minus_7395159227704179404_a_nat @ A @ B )
= bot_bo1033123847703346641_a_nat )
= ( ord_le1147066620699065093_a_nat @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_284_Diff__disjoint,axiom,
! [A: set_se4330304633200676677_a_nat,B: set_se4330304633200676677_a_nat] :
( ( inf_in5367731912061063475_a_nat @ A @ ( minus_8004192006823430828_a_nat @ B @ A ) )
= bot_bo3237059034911209905_a_nat ) ).
% Diff_disjoint
thf(fact_285_Diff__disjoint,axiom,
! [A: set_set_a,B: set_set_a] :
( ( inf_inf_set_set_a @ A @ ( minus_5736297505244876581_set_a @ B @ A ) )
= bot_bot_set_set_a ) ).
% Diff_disjoint
thf(fact_286_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_287_Diff__disjoint,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ A @ ( minus_7395159227704179404_a_nat @ B @ A ) )
= bot_bo1033123847703346641_a_nat ) ).
% Diff_disjoint
thf(fact_288_inf__sup__aci_I8_J,axiom,
! [X4: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ X4 @ ( sup_su4083067149120280889_a_nat @ X4 @ Y ) )
= ( sup_su4083067149120280889_a_nat @ X4 @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_289_inf__sup__aci_I8_J,axiom,
! [X4: set_set_a,Y: set_set_a] :
( ( sup_sup_set_set_a @ X4 @ ( sup_sup_set_set_a @ X4 @ Y ) )
= ( sup_sup_set_set_a @ X4 @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_290_inf__sup__aci_I8_J,axiom,
! [X4: set_a,Y: set_a] :
( ( sup_sup_set_a @ X4 @ ( sup_sup_set_a @ X4 @ Y ) )
= ( sup_sup_set_a @ X4 @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_291_inf__sup__aci_I7_J,axiom,
! [X4: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat,Z: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ X4 @ ( sup_su4083067149120280889_a_nat @ Y @ Z ) )
= ( sup_su4083067149120280889_a_nat @ Y @ ( sup_su4083067149120280889_a_nat @ X4 @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_292_inf__sup__aci_I7_J,axiom,
! [X4: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( sup_sup_set_set_a @ X4 @ ( sup_sup_set_set_a @ Y @ Z ) )
= ( sup_sup_set_set_a @ Y @ ( sup_sup_set_set_a @ X4 @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_293_inf__sup__aci_I7_J,axiom,
! [X4: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ X4 @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ Y @ ( sup_sup_set_a @ X4 @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_294_inf__sup__aci_I6_J,axiom,
! [X4: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat,Z: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ ( sup_su4083067149120280889_a_nat @ X4 @ Y ) @ Z )
= ( sup_su4083067149120280889_a_nat @ X4 @ ( sup_su4083067149120280889_a_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_295_inf__sup__aci_I6_J,axiom,
! [X4: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ X4 @ Y ) @ Z )
= ( sup_sup_set_set_a @ X4 @ ( sup_sup_set_set_a @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_296_inf__sup__aci_I6_J,axiom,
! [X4: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X4 @ Y ) @ Z )
= ( sup_sup_set_a @ X4 @ ( sup_sup_set_a @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_297_inf__sup__aci_I5_J,axiom,
( sup_su4083067149120280889_a_nat
= ( ^ [X5: set_li6526943997496501093_a_nat,Y2: set_li6526943997496501093_a_nat] : ( sup_su4083067149120280889_a_nat @ Y2 @ X5 ) ) ) ).
% inf_sup_aci(5)
thf(fact_298_inf__sup__aci_I5_J,axiom,
( sup_sup_set_set_a
= ( ^ [X5: set_set_a,Y2: set_set_a] : ( sup_sup_set_set_a @ Y2 @ X5 ) ) ) ).
% inf_sup_aci(5)
thf(fact_299_inf__sup__aci_I5_J,axiom,
( sup_sup_set_a
= ( ^ [X5: set_a,Y2: set_a] : ( sup_sup_set_a @ Y2 @ X5 ) ) ) ).
% inf_sup_aci(5)
thf(fact_300_inf__sup__aci_I4_J,axiom,
! [X4: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ X4 @ ( inf_in3249246906714053971_a_nat @ X4 @ Y ) )
= ( inf_in3249246906714053971_a_nat @ X4 @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_301_inf__sup__aci_I4_J,axiom,
! [X4: set_se4330304633200676677_a_nat,Y: set_se4330304633200676677_a_nat] :
( ( inf_in5367731912061063475_a_nat @ X4 @ ( inf_in5367731912061063475_a_nat @ X4 @ Y ) )
= ( inf_in5367731912061063475_a_nat @ X4 @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_302_inf__sup__aci_I4_J,axiom,
! [X4: set_set_a,Y: set_set_a] :
( ( inf_inf_set_set_a @ X4 @ ( inf_inf_set_set_a @ X4 @ Y ) )
= ( inf_inf_set_set_a @ X4 @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_303_inf__sup__aci_I4_J,axiom,
! [X4: set_a,Y: set_a] :
( ( inf_inf_set_a @ X4 @ ( inf_inf_set_a @ X4 @ Y ) )
= ( inf_inf_set_a @ X4 @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_304_inf__sup__aci_I3_J,axiom,
! [X4: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat,Z: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ X4 @ ( inf_in3249246906714053971_a_nat @ Y @ Z ) )
= ( inf_in3249246906714053971_a_nat @ Y @ ( inf_in3249246906714053971_a_nat @ X4 @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_305_inf__sup__aci_I3_J,axiom,
! [X4: set_se4330304633200676677_a_nat,Y: set_se4330304633200676677_a_nat,Z: set_se4330304633200676677_a_nat] :
( ( inf_in5367731912061063475_a_nat @ X4 @ ( inf_in5367731912061063475_a_nat @ Y @ Z ) )
= ( inf_in5367731912061063475_a_nat @ Y @ ( inf_in5367731912061063475_a_nat @ X4 @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_306_inf__sup__aci_I3_J,axiom,
! [X4: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( inf_inf_set_set_a @ X4 @ ( inf_inf_set_set_a @ Y @ Z ) )
= ( inf_inf_set_set_a @ Y @ ( inf_inf_set_set_a @ X4 @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_307_inf__sup__aci_I3_J,axiom,
! [X4: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X4 @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X4 @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_308_inf__sup__aci_I2_J,axiom,
! [X4: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat,Z: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ ( inf_in3249246906714053971_a_nat @ X4 @ Y ) @ Z )
= ( inf_in3249246906714053971_a_nat @ X4 @ ( inf_in3249246906714053971_a_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_309_inf__sup__aci_I2_J,axiom,
! [X4: set_se4330304633200676677_a_nat,Y: set_se4330304633200676677_a_nat,Z: set_se4330304633200676677_a_nat] :
( ( inf_in5367731912061063475_a_nat @ ( inf_in5367731912061063475_a_nat @ X4 @ Y ) @ Z )
= ( inf_in5367731912061063475_a_nat @ X4 @ ( inf_in5367731912061063475_a_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_310_inf__sup__aci_I2_J,axiom,
! [X4: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ X4 @ Y ) @ Z )
= ( inf_inf_set_set_a @ X4 @ ( inf_inf_set_set_a @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_311_inf__sup__aci_I2_J,axiom,
! [X4: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X4 @ Y ) @ Z )
= ( inf_inf_set_a @ X4 @ ( inf_inf_set_a @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_312_inf__sup__aci_I1_J,axiom,
( inf_in3249246906714053971_a_nat
= ( ^ [X5: set_li6526943997496501093_a_nat,Y2: set_li6526943997496501093_a_nat] : ( inf_in3249246906714053971_a_nat @ Y2 @ X5 ) ) ) ).
% inf_sup_aci(1)
thf(fact_313_inf__sup__aci_I1_J,axiom,
( inf_in5367731912061063475_a_nat
= ( ^ [X5: set_se4330304633200676677_a_nat,Y2: set_se4330304633200676677_a_nat] : ( inf_in5367731912061063475_a_nat @ Y2 @ X5 ) ) ) ).
% inf_sup_aci(1)
thf(fact_314_inf__sup__aci_I1_J,axiom,
( inf_inf_set_set_a
= ( ^ [X5: set_set_a,Y2: set_set_a] : ( inf_inf_set_set_a @ Y2 @ X5 ) ) ) ).
% inf_sup_aci(1)
thf(fact_315_inf__sup__aci_I1_J,axiom,
( inf_inf_set_a
= ( ^ [X5: set_a,Y2: set_a] : ( inf_inf_set_a @ Y2 @ X5 ) ) ) ).
% inf_sup_aci(1)
thf(fact_316_distrib__imp1,axiom,
! [X4: set_se4330304633200676677_a_nat,Y: set_se4330304633200676677_a_nat,Z: set_se4330304633200676677_a_nat] :
( ! [X3: set_se4330304633200676677_a_nat,Y3: set_se4330304633200676677_a_nat,Z2: set_se4330304633200676677_a_nat] :
( ( inf_in5367731912061063475_a_nat @ X3 @ ( sup_su499249268922660121_a_nat @ Y3 @ Z2 ) )
= ( sup_su499249268922660121_a_nat @ ( inf_in5367731912061063475_a_nat @ X3 @ Y3 ) @ ( inf_in5367731912061063475_a_nat @ X3 @ Z2 ) ) )
=> ( ( sup_su499249268922660121_a_nat @ X4 @ ( inf_in5367731912061063475_a_nat @ Y @ Z ) )
= ( inf_in5367731912061063475_a_nat @ ( sup_su499249268922660121_a_nat @ X4 @ Y ) @ ( sup_su499249268922660121_a_nat @ X4 @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_317_distrib__imp1,axiom,
! [X4: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat,Z: set_li6526943997496501093_a_nat] :
( ! [X3: set_li6526943997496501093_a_nat,Y3: set_li6526943997496501093_a_nat,Z2: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ X3 @ ( sup_su4083067149120280889_a_nat @ Y3 @ Z2 ) )
= ( sup_su4083067149120280889_a_nat @ ( inf_in3249246906714053971_a_nat @ X3 @ Y3 ) @ ( inf_in3249246906714053971_a_nat @ X3 @ Z2 ) ) )
=> ( ( sup_su4083067149120280889_a_nat @ X4 @ ( inf_in3249246906714053971_a_nat @ Y @ Z ) )
= ( inf_in3249246906714053971_a_nat @ ( sup_su4083067149120280889_a_nat @ X4 @ Y ) @ ( sup_su4083067149120280889_a_nat @ X4 @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_318_distrib__imp1,axiom,
! [X4: set_set_a,Y: set_set_a,Z: set_set_a] :
( ! [X3: set_set_a,Y3: set_set_a,Z2: set_set_a] :
( ( inf_inf_set_set_a @ X3 @ ( sup_sup_set_set_a @ Y3 @ Z2 ) )
= ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ X3 @ Y3 ) @ ( inf_inf_set_set_a @ X3 @ Z2 ) ) )
=> ( ( sup_sup_set_set_a @ X4 @ ( inf_inf_set_set_a @ Y @ Z ) )
= ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ X4 @ Y ) @ ( sup_sup_set_set_a @ X4 @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_319_distrib__imp1,axiom,
! [X4: set_a,Y: set_a,Z: set_a] :
( ! [X3: set_a,Y3: set_a,Z2: set_a] :
( ( inf_inf_set_a @ X3 @ ( sup_sup_set_a @ Y3 @ Z2 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X3 @ Y3 ) @ ( inf_inf_set_a @ X3 @ Z2 ) ) )
=> ( ( sup_sup_set_a @ X4 @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X4 @ Y ) @ ( sup_sup_set_a @ X4 @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_320_distrib__imp2,axiom,
! [X4: set_se4330304633200676677_a_nat,Y: set_se4330304633200676677_a_nat,Z: set_se4330304633200676677_a_nat] :
( ! [X3: set_se4330304633200676677_a_nat,Y3: set_se4330304633200676677_a_nat,Z2: set_se4330304633200676677_a_nat] :
( ( sup_su499249268922660121_a_nat @ X3 @ ( inf_in5367731912061063475_a_nat @ Y3 @ Z2 ) )
= ( inf_in5367731912061063475_a_nat @ ( sup_su499249268922660121_a_nat @ X3 @ Y3 ) @ ( sup_su499249268922660121_a_nat @ X3 @ Z2 ) ) )
=> ( ( inf_in5367731912061063475_a_nat @ X4 @ ( sup_su499249268922660121_a_nat @ Y @ Z ) )
= ( sup_su499249268922660121_a_nat @ ( inf_in5367731912061063475_a_nat @ X4 @ Y ) @ ( inf_in5367731912061063475_a_nat @ X4 @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_321_distrib__imp2,axiom,
! [X4: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat,Z: set_li6526943997496501093_a_nat] :
( ! [X3: set_li6526943997496501093_a_nat,Y3: set_li6526943997496501093_a_nat,Z2: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ X3 @ ( inf_in3249246906714053971_a_nat @ Y3 @ Z2 ) )
= ( inf_in3249246906714053971_a_nat @ ( sup_su4083067149120280889_a_nat @ X3 @ Y3 ) @ ( sup_su4083067149120280889_a_nat @ X3 @ Z2 ) ) )
=> ( ( inf_in3249246906714053971_a_nat @ X4 @ ( sup_su4083067149120280889_a_nat @ Y @ Z ) )
= ( sup_su4083067149120280889_a_nat @ ( inf_in3249246906714053971_a_nat @ X4 @ Y ) @ ( inf_in3249246906714053971_a_nat @ X4 @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_322_distrib__imp2,axiom,
! [X4: set_set_a,Y: set_set_a,Z: set_set_a] :
( ! [X3: set_set_a,Y3: set_set_a,Z2: set_set_a] :
( ( sup_sup_set_set_a @ X3 @ ( inf_inf_set_set_a @ Y3 @ Z2 ) )
= ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ X3 @ Y3 ) @ ( sup_sup_set_set_a @ X3 @ Z2 ) ) )
=> ( ( inf_inf_set_set_a @ X4 @ ( sup_sup_set_set_a @ Y @ Z ) )
= ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ X4 @ Y ) @ ( inf_inf_set_set_a @ X4 @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_323_distrib__imp2,axiom,
! [X4: set_a,Y: set_a,Z: set_a] :
( ! [X3: set_a,Y3: set_a,Z2: set_a] :
( ( sup_sup_set_a @ X3 @ ( inf_inf_set_a @ Y3 @ Z2 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X3 @ Y3 ) @ ( sup_sup_set_a @ X3 @ Z2 ) ) )
=> ( ( inf_inf_set_a @ X4 @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X4 @ Y ) @ ( inf_inf_set_a @ X4 @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_324_inf_Oassoc,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ ( inf_in3249246906714053971_a_nat @ A2 @ B2 ) @ C )
= ( inf_in3249246906714053971_a_nat @ A2 @ ( inf_in3249246906714053971_a_nat @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_325_inf_Oassoc,axiom,
! [A2: set_se4330304633200676677_a_nat,B2: set_se4330304633200676677_a_nat,C: set_se4330304633200676677_a_nat] :
( ( inf_in5367731912061063475_a_nat @ ( inf_in5367731912061063475_a_nat @ A2 @ B2 ) @ C )
= ( inf_in5367731912061063475_a_nat @ A2 @ ( inf_in5367731912061063475_a_nat @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_326_inf_Oassoc,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ A2 @ B2 ) @ C )
= ( inf_inf_set_set_a @ A2 @ ( inf_inf_set_set_a @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_327_inf_Oassoc,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C )
= ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_328_inf__assoc,axiom,
! [X4: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat,Z: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ ( inf_in3249246906714053971_a_nat @ X4 @ Y ) @ Z )
= ( inf_in3249246906714053971_a_nat @ X4 @ ( inf_in3249246906714053971_a_nat @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_329_inf__assoc,axiom,
! [X4: set_se4330304633200676677_a_nat,Y: set_se4330304633200676677_a_nat,Z: set_se4330304633200676677_a_nat] :
( ( inf_in5367731912061063475_a_nat @ ( inf_in5367731912061063475_a_nat @ X4 @ Y ) @ Z )
= ( inf_in5367731912061063475_a_nat @ X4 @ ( inf_in5367731912061063475_a_nat @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_330_inf__assoc,axiom,
! [X4: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ X4 @ Y ) @ Z )
= ( inf_inf_set_set_a @ X4 @ ( inf_inf_set_set_a @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_331_inf__assoc,axiom,
! [X4: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X4 @ Y ) @ Z )
= ( inf_inf_set_a @ X4 @ ( inf_inf_set_a @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_332_sup_Oassoc,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ ( sup_su4083067149120280889_a_nat @ A2 @ B2 ) @ C )
= ( sup_su4083067149120280889_a_nat @ A2 @ ( sup_su4083067149120280889_a_nat @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_333_sup_Oassoc,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a] :
( ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ A2 @ B2 ) @ C )
= ( sup_sup_set_set_a @ A2 @ ( sup_sup_set_set_a @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_334_sup_Oassoc,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ C )
= ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_335_sup__assoc,axiom,
! [X4: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat,Z: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ ( sup_su4083067149120280889_a_nat @ X4 @ Y ) @ Z )
= ( sup_su4083067149120280889_a_nat @ X4 @ ( sup_su4083067149120280889_a_nat @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_336_sup__assoc,axiom,
! [X4: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ X4 @ Y ) @ Z )
= ( sup_sup_set_set_a @ X4 @ ( sup_sup_set_set_a @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_337_sup__assoc,axiom,
! [X4: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X4 @ Y ) @ Z )
= ( sup_sup_set_a @ X4 @ ( sup_sup_set_a @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_338_inf_Ocommute,axiom,
( inf_in3249246906714053971_a_nat
= ( ^ [A3: set_li6526943997496501093_a_nat,B3: set_li6526943997496501093_a_nat] : ( inf_in3249246906714053971_a_nat @ B3 @ A3 ) ) ) ).
% inf.commute
thf(fact_339_inf_Ocommute,axiom,
( inf_in5367731912061063475_a_nat
= ( ^ [A3: set_se4330304633200676677_a_nat,B3: set_se4330304633200676677_a_nat] : ( inf_in5367731912061063475_a_nat @ B3 @ A3 ) ) ) ).
% inf.commute
thf(fact_340_inf_Ocommute,axiom,
( inf_inf_set_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] : ( inf_inf_set_set_a @ B3 @ A3 ) ) ) ).
% inf.commute
thf(fact_341_inf_Ocommute,axiom,
( inf_inf_set_a
= ( ^ [A3: set_a,B3: set_a] : ( inf_inf_set_a @ B3 @ A3 ) ) ) ).
% inf.commute
thf(fact_342_inf__commute,axiom,
( inf_in3249246906714053971_a_nat
= ( ^ [X5: set_li6526943997496501093_a_nat,Y2: set_li6526943997496501093_a_nat] : ( inf_in3249246906714053971_a_nat @ Y2 @ X5 ) ) ) ).
% inf_commute
thf(fact_343_inf__commute,axiom,
( inf_in5367731912061063475_a_nat
= ( ^ [X5: set_se4330304633200676677_a_nat,Y2: set_se4330304633200676677_a_nat] : ( inf_in5367731912061063475_a_nat @ Y2 @ X5 ) ) ) ).
% inf_commute
thf(fact_344_inf__commute,axiom,
( inf_inf_set_set_a
= ( ^ [X5: set_set_a,Y2: set_set_a] : ( inf_inf_set_set_a @ Y2 @ X5 ) ) ) ).
% inf_commute
thf(fact_345_inf__commute,axiom,
( inf_inf_set_a
= ( ^ [X5: set_a,Y2: set_a] : ( inf_inf_set_a @ Y2 @ X5 ) ) ) ).
% inf_commute
thf(fact_346_sup_Ocommute,axiom,
( sup_su4083067149120280889_a_nat
= ( ^ [A3: set_li6526943997496501093_a_nat,B3: set_li6526943997496501093_a_nat] : ( sup_su4083067149120280889_a_nat @ B3 @ A3 ) ) ) ).
% sup.commute
thf(fact_347_sup_Ocommute,axiom,
( sup_sup_set_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] : ( sup_sup_set_set_a @ B3 @ A3 ) ) ) ).
% sup.commute
thf(fact_348_sup_Ocommute,axiom,
( sup_sup_set_a
= ( ^ [A3: set_a,B3: set_a] : ( sup_sup_set_a @ B3 @ A3 ) ) ) ).
% sup.commute
thf(fact_349_sup__commute,axiom,
( sup_su4083067149120280889_a_nat
= ( ^ [X5: set_li6526943997496501093_a_nat,Y2: set_li6526943997496501093_a_nat] : ( sup_su4083067149120280889_a_nat @ Y2 @ X5 ) ) ) ).
% sup_commute
thf(fact_350_sup__commute,axiom,
( sup_sup_set_set_a
= ( ^ [X5: set_set_a,Y2: set_set_a] : ( sup_sup_set_set_a @ Y2 @ X5 ) ) ) ).
% sup_commute
thf(fact_351_sup__commute,axiom,
( sup_sup_set_a
= ( ^ [X5: set_a,Y2: set_a] : ( sup_sup_set_a @ Y2 @ X5 ) ) ) ).
% sup_commute
thf(fact_352_inf__sup__distrib1,axiom,
! [X4: set_se4330304633200676677_a_nat,Y: set_se4330304633200676677_a_nat,Z: set_se4330304633200676677_a_nat] :
( ( inf_in5367731912061063475_a_nat @ X4 @ ( sup_su499249268922660121_a_nat @ Y @ Z ) )
= ( sup_su499249268922660121_a_nat @ ( inf_in5367731912061063475_a_nat @ X4 @ Y ) @ ( inf_in5367731912061063475_a_nat @ X4 @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_353_inf__sup__distrib1,axiom,
! [X4: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat,Z: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ X4 @ ( sup_su4083067149120280889_a_nat @ Y @ Z ) )
= ( sup_su4083067149120280889_a_nat @ ( inf_in3249246906714053971_a_nat @ X4 @ Y ) @ ( inf_in3249246906714053971_a_nat @ X4 @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_354_inf__sup__distrib1,axiom,
! [X4: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( inf_inf_set_set_a @ X4 @ ( sup_sup_set_set_a @ Y @ Z ) )
= ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ X4 @ Y ) @ ( inf_inf_set_set_a @ X4 @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_355_inf__sup__distrib1,axiom,
! [X4: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X4 @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X4 @ Y ) @ ( inf_inf_set_a @ X4 @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_356_inf__sup__distrib2,axiom,
! [Y: set_se4330304633200676677_a_nat,Z: set_se4330304633200676677_a_nat,X4: set_se4330304633200676677_a_nat] :
( ( inf_in5367731912061063475_a_nat @ ( sup_su499249268922660121_a_nat @ Y @ Z ) @ X4 )
= ( sup_su499249268922660121_a_nat @ ( inf_in5367731912061063475_a_nat @ Y @ X4 ) @ ( inf_in5367731912061063475_a_nat @ Z @ X4 ) ) ) ).
% inf_sup_distrib2
thf(fact_357_inf__sup__distrib2,axiom,
! [Y: set_li6526943997496501093_a_nat,Z: set_li6526943997496501093_a_nat,X4: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ ( sup_su4083067149120280889_a_nat @ Y @ Z ) @ X4 )
= ( sup_su4083067149120280889_a_nat @ ( inf_in3249246906714053971_a_nat @ Y @ X4 ) @ ( inf_in3249246906714053971_a_nat @ Z @ X4 ) ) ) ).
% inf_sup_distrib2
thf(fact_358_inf__sup__distrib2,axiom,
! [Y: set_set_a,Z: set_set_a,X4: set_set_a] :
( ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ Y @ Z ) @ X4 )
= ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ Y @ X4 ) @ ( inf_inf_set_set_a @ Z @ X4 ) ) ) ).
% inf_sup_distrib2
thf(fact_359_inf__sup__distrib2,axiom,
! [Y: set_a,Z: set_a,X4: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ Z ) @ X4 )
= ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ X4 ) @ ( inf_inf_set_a @ Z @ X4 ) ) ) ).
% inf_sup_distrib2
thf(fact_360_sup__inf__distrib1,axiom,
! [X4: set_se4330304633200676677_a_nat,Y: set_se4330304633200676677_a_nat,Z: set_se4330304633200676677_a_nat] :
( ( sup_su499249268922660121_a_nat @ X4 @ ( inf_in5367731912061063475_a_nat @ Y @ Z ) )
= ( inf_in5367731912061063475_a_nat @ ( sup_su499249268922660121_a_nat @ X4 @ Y ) @ ( sup_su499249268922660121_a_nat @ X4 @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_361_sup__inf__distrib1,axiom,
! [X4: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat,Z: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ X4 @ ( inf_in3249246906714053971_a_nat @ Y @ Z ) )
= ( inf_in3249246906714053971_a_nat @ ( sup_su4083067149120280889_a_nat @ X4 @ Y ) @ ( sup_su4083067149120280889_a_nat @ X4 @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_362_sup__inf__distrib1,axiom,
! [X4: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( sup_sup_set_set_a @ X4 @ ( inf_inf_set_set_a @ Y @ Z ) )
= ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ X4 @ Y ) @ ( sup_sup_set_set_a @ X4 @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_363_sup__inf__distrib1,axiom,
! [X4: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ X4 @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X4 @ Y ) @ ( sup_sup_set_a @ X4 @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_364_sup__inf__distrib2,axiom,
! [Y: set_se4330304633200676677_a_nat,Z: set_se4330304633200676677_a_nat,X4: set_se4330304633200676677_a_nat] :
( ( sup_su499249268922660121_a_nat @ ( inf_in5367731912061063475_a_nat @ Y @ Z ) @ X4 )
= ( inf_in5367731912061063475_a_nat @ ( sup_su499249268922660121_a_nat @ Y @ X4 ) @ ( sup_su499249268922660121_a_nat @ Z @ X4 ) ) ) ).
% sup_inf_distrib2
thf(fact_365_sup__inf__distrib2,axiom,
! [Y: set_li6526943997496501093_a_nat,Z: set_li6526943997496501093_a_nat,X4: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ ( inf_in3249246906714053971_a_nat @ Y @ Z ) @ X4 )
= ( inf_in3249246906714053971_a_nat @ ( sup_su4083067149120280889_a_nat @ Y @ X4 ) @ ( sup_su4083067149120280889_a_nat @ Z @ X4 ) ) ) ).
% sup_inf_distrib2
thf(fact_366_sup__inf__distrib2,axiom,
! [Y: set_set_a,Z: set_set_a,X4: set_set_a] :
( ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ Y @ Z ) @ X4 )
= ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ Y @ X4 ) @ ( sup_sup_set_set_a @ Z @ X4 ) ) ) ).
% sup_inf_distrib2
thf(fact_367_sup__inf__distrib2,axiom,
! [Y: set_a,Z: set_a,X4: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ Z ) @ X4 )
= ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ X4 ) @ ( sup_sup_set_a @ Z @ X4 ) ) ) ).
% sup_inf_distrib2
thf(fact_368_inf_Oleft__commute,axiom,
! [B2: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ B2 @ ( inf_in3249246906714053971_a_nat @ A2 @ C ) )
= ( inf_in3249246906714053971_a_nat @ A2 @ ( inf_in3249246906714053971_a_nat @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_369_inf_Oleft__commute,axiom,
! [B2: set_se4330304633200676677_a_nat,A2: set_se4330304633200676677_a_nat,C: set_se4330304633200676677_a_nat] :
( ( inf_in5367731912061063475_a_nat @ B2 @ ( inf_in5367731912061063475_a_nat @ A2 @ C ) )
= ( inf_in5367731912061063475_a_nat @ A2 @ ( inf_in5367731912061063475_a_nat @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_370_inf_Oleft__commute,axiom,
! [B2: set_set_a,A2: set_set_a,C: set_set_a] :
( ( inf_inf_set_set_a @ B2 @ ( inf_inf_set_set_a @ A2 @ C ) )
= ( inf_inf_set_set_a @ A2 @ ( inf_inf_set_set_a @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_371_inf_Oleft__commute,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( inf_inf_set_a @ B2 @ ( inf_inf_set_a @ A2 @ C ) )
= ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_372_inf__left__commute,axiom,
! [X4: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat,Z: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ X4 @ ( inf_in3249246906714053971_a_nat @ Y @ Z ) )
= ( inf_in3249246906714053971_a_nat @ Y @ ( inf_in3249246906714053971_a_nat @ X4 @ Z ) ) ) ).
% inf_left_commute
thf(fact_373_inf__left__commute,axiom,
! [X4: set_se4330304633200676677_a_nat,Y: set_se4330304633200676677_a_nat,Z: set_se4330304633200676677_a_nat] :
( ( inf_in5367731912061063475_a_nat @ X4 @ ( inf_in5367731912061063475_a_nat @ Y @ Z ) )
= ( inf_in5367731912061063475_a_nat @ Y @ ( inf_in5367731912061063475_a_nat @ X4 @ Z ) ) ) ).
% inf_left_commute
thf(fact_374_inf__left__commute,axiom,
! [X4: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( inf_inf_set_set_a @ X4 @ ( inf_inf_set_set_a @ Y @ Z ) )
= ( inf_inf_set_set_a @ Y @ ( inf_inf_set_set_a @ X4 @ Z ) ) ) ).
% inf_left_commute
thf(fact_375_inf__left__commute,axiom,
! [X4: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X4 @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X4 @ Z ) ) ) ).
% inf_left_commute
thf(fact_376_sup_Oleft__commute,axiom,
! [B2: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ B2 @ ( sup_su4083067149120280889_a_nat @ A2 @ C ) )
= ( sup_su4083067149120280889_a_nat @ A2 @ ( sup_su4083067149120280889_a_nat @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_377_sup_Oleft__commute,axiom,
! [B2: set_set_a,A2: set_set_a,C: set_set_a] :
( ( sup_sup_set_set_a @ B2 @ ( sup_sup_set_set_a @ A2 @ C ) )
= ( sup_sup_set_set_a @ A2 @ ( sup_sup_set_set_a @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_378_sup_Oleft__commute,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( sup_sup_set_a @ B2 @ ( sup_sup_set_a @ A2 @ C ) )
= ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_379_sup__left__commute,axiom,
! [X4: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat,Z: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ X4 @ ( sup_su4083067149120280889_a_nat @ Y @ Z ) )
= ( sup_su4083067149120280889_a_nat @ Y @ ( sup_su4083067149120280889_a_nat @ X4 @ Z ) ) ) ).
% sup_left_commute
thf(fact_380_sup__left__commute,axiom,
! [X4: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( sup_sup_set_set_a @ X4 @ ( sup_sup_set_set_a @ Y @ Z ) )
= ( sup_sup_set_set_a @ Y @ ( sup_sup_set_set_a @ X4 @ Z ) ) ) ).
% sup_left_commute
thf(fact_381_sup__left__commute,axiom,
! [X4: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ X4 @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ Y @ ( sup_sup_set_a @ X4 @ Z ) ) ) ).
% sup_left_commute
thf(fact_382_UnE,axiom,
! [C: set_li6526943997496501093_a_nat,A: set_se4330304633200676677_a_nat,B: set_se4330304633200676677_a_nat] :
( ( member5553968465346197646_a_nat @ C @ ( sup_su499249268922660121_a_nat @ A @ B ) )
=> ( ~ ( member5553968465346197646_a_nat @ C @ A )
=> ( member5553968465346197646_a_nat @ C @ B ) ) ) ).
% UnE
thf(fact_383_UnE,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( sup_sup_set_set_a @ A @ B ) )
=> ( ~ ( member_set_a @ C @ A )
=> ( member_set_a @ C @ B ) ) ) ).
% UnE
thf(fact_384_UnE,axiom,
! [C: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ ( sup_su4083067149120280889_a_nat @ A @ B ) )
=> ( ~ ( member408289922725080238_a_nat @ C @ A )
=> ( member408289922725080238_a_nat @ C @ B ) ) ) ).
% UnE
thf(fact_385_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_386_IntE,axiom,
! [C: set_li6526943997496501093_a_nat,A: set_se4330304633200676677_a_nat,B: set_se4330304633200676677_a_nat] :
( ( member5553968465346197646_a_nat @ C @ ( inf_in5367731912061063475_a_nat @ A @ B ) )
=> ~ ( ( member5553968465346197646_a_nat @ C @ A )
=> ~ ( member5553968465346197646_a_nat @ C @ B ) ) ) ).
% IntE
thf(fact_387_IntE,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B ) )
=> ~ ( ( member_set_a @ C @ A )
=> ~ ( member_set_a @ C @ B ) ) ) ).
% IntE
thf(fact_388_IntE,axiom,
! [C: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ ( inf_in3249246906714053971_a_nat @ A @ B ) )
=> ~ ( ( member408289922725080238_a_nat @ C @ A )
=> ~ ( member408289922725080238_a_nat @ C @ B ) ) ) ).
% IntE
thf(fact_389_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_390_UnI1,axiom,
! [C: set_li6526943997496501093_a_nat,A: set_se4330304633200676677_a_nat,B: set_se4330304633200676677_a_nat] :
( ( member5553968465346197646_a_nat @ C @ A )
=> ( member5553968465346197646_a_nat @ C @ ( sup_su499249268922660121_a_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_391_UnI1,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ A )
=> ( member_set_a @ C @ ( sup_sup_set_set_a @ A @ B ) ) ) ).
% UnI1
thf(fact_392_UnI1,axiom,
! [C: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ A )
=> ( member408289922725080238_a_nat @ C @ ( sup_su4083067149120280889_a_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_393_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_394_UnI2,axiom,
! [C: set_li6526943997496501093_a_nat,B: set_se4330304633200676677_a_nat,A: set_se4330304633200676677_a_nat] :
( ( member5553968465346197646_a_nat @ C @ B )
=> ( member5553968465346197646_a_nat @ C @ ( sup_su499249268922660121_a_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_395_UnI2,axiom,
! [C: set_a,B: set_set_a,A: set_set_a] :
( ( member_set_a @ C @ B )
=> ( member_set_a @ C @ ( sup_sup_set_set_a @ A @ B ) ) ) ).
% UnI2
thf(fact_396_UnI2,axiom,
! [C: list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ B )
=> ( member408289922725080238_a_nat @ C @ ( sup_su4083067149120280889_a_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_397_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_398_IntD1,axiom,
! [C: set_li6526943997496501093_a_nat,A: set_se4330304633200676677_a_nat,B: set_se4330304633200676677_a_nat] :
( ( member5553968465346197646_a_nat @ C @ ( inf_in5367731912061063475_a_nat @ A @ B ) )
=> ( member5553968465346197646_a_nat @ C @ A ) ) ).
% IntD1
thf(fact_399_IntD1,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B ) )
=> ( member_set_a @ C @ A ) ) ).
% IntD1
thf(fact_400_IntD1,axiom,
! [C: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ ( inf_in3249246906714053971_a_nat @ A @ B ) )
=> ( member408289922725080238_a_nat @ C @ A ) ) ).
% IntD1
thf(fact_401_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_402_IntD2,axiom,
! [C: set_li6526943997496501093_a_nat,A: set_se4330304633200676677_a_nat,B: set_se4330304633200676677_a_nat] :
( ( member5553968465346197646_a_nat @ C @ ( inf_in5367731912061063475_a_nat @ A @ B ) )
=> ( member5553968465346197646_a_nat @ C @ B ) ) ).
% IntD2
thf(fact_403_IntD2,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B ) )
=> ( member_set_a @ C @ B ) ) ).
% IntD2
thf(fact_404_IntD2,axiom,
! [C: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ ( inf_in3249246906714053971_a_nat @ A @ B ) )
=> ( member408289922725080238_a_nat @ C @ B ) ) ).
% IntD2
thf(fact_405_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_406_bex__Un,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,P: list_Sum_sum_a_nat > $o] :
( ( ? [X5: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X5 @ ( sup_su4083067149120280889_a_nat @ A @ B ) )
& ( P @ X5 ) ) )
= ( ? [X5: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X5 @ A )
& ( P @ X5 ) )
| ? [X5: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X5 @ B )
& ( P @ X5 ) ) ) ) ).
% bex_Un
thf(fact_407_bex__Un,axiom,
! [A: set_set_a,B: set_set_a,P: set_a > $o] :
( ( ? [X5: set_a] :
( ( member_set_a @ X5 @ ( sup_sup_set_set_a @ A @ B ) )
& ( P @ X5 ) ) )
= ( ? [X5: set_a] :
( ( member_set_a @ X5 @ A )
& ( P @ X5 ) )
| ? [X5: set_a] :
( ( member_set_a @ X5 @ B )
& ( P @ X5 ) ) ) ) ).
% bex_Un
thf(fact_408_bex__Un,axiom,
! [A: set_a,B: set_a,P: a > $o] :
( ( ? [X5: a] :
( ( member_a @ X5 @ ( sup_sup_set_a @ A @ B ) )
& ( P @ X5 ) ) )
= ( ? [X5: a] :
( ( member_a @ X5 @ A )
& ( P @ X5 ) )
| ? [X5: a] :
( ( member_a @ X5 @ B )
& ( P @ X5 ) ) ) ) ).
% bex_Un
thf(fact_409_emptyE,axiom,
! [A2: set_li6526943997496501093_a_nat] :
~ ( member5553968465346197646_a_nat @ A2 @ bot_bo3237059034911209905_a_nat ) ).
% emptyE
thf(fact_410_emptyE,axiom,
! [A2: set_a] :
~ ( member_set_a @ A2 @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_411_emptyE,axiom,
! [A2: list_Sum_sum_a_nat] :
~ ( member408289922725080238_a_nat @ A2 @ bot_bo1033123847703346641_a_nat ) ).
% emptyE
thf(fact_412_emptyE,axiom,
! [A2: a] :
~ ( member_a @ A2 @ bot_bot_set_a ) ).
% emptyE
thf(fact_413_ball__Un,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,P: list_Sum_sum_a_nat > $o] :
( ( ! [X5: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X5 @ ( sup_su4083067149120280889_a_nat @ A @ B ) )
=> ( P @ X5 ) ) )
= ( ! [X5: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X5 @ A )
=> ( P @ X5 ) )
& ! [X5: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X5 @ B )
=> ( P @ X5 ) ) ) ) ).
% ball_Un
thf(fact_414_ball__Un,axiom,
! [A: set_set_a,B: set_set_a,P: set_a > $o] :
( ( ! [X5: set_a] :
( ( member_set_a @ X5 @ ( sup_sup_set_set_a @ A @ B ) )
=> ( P @ X5 ) ) )
= ( ! [X5: set_a] :
( ( member_set_a @ X5 @ A )
=> ( P @ X5 ) )
& ! [X5: set_a] :
( ( member_set_a @ X5 @ B )
=> ( P @ X5 ) ) ) ) ).
% ball_Un
thf(fact_415_ball__Un,axiom,
! [A: set_a,B: set_a,P: a > $o] :
( ( ! [X5: a] :
( ( member_a @ X5 @ ( sup_sup_set_a @ A @ B ) )
=> ( P @ X5 ) ) )
= ( ! [X5: a] :
( ( member_a @ X5 @ A )
=> ( P @ X5 ) )
& ! [X5: a] :
( ( member_a @ X5 @ B )
=> ( P @ X5 ) ) ) ) ).
% ball_Un
thf(fact_416_Un__assoc,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,C2: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ ( sup_su4083067149120280889_a_nat @ A @ B ) @ C2 )
= ( sup_su4083067149120280889_a_nat @ A @ ( sup_su4083067149120280889_a_nat @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_417_Un__assoc,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ A @ B ) @ C2 )
= ( sup_sup_set_set_a @ A @ ( sup_sup_set_set_a @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_418_Un__assoc,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A @ B ) @ C2 )
= ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_419_equals0D,axiom,
! [A: set_se4330304633200676677_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( A = bot_bo3237059034911209905_a_nat )
=> ~ ( member5553968465346197646_a_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_420_equals0D,axiom,
! [A: set_set_a,A2: set_a] :
( ( A = bot_bot_set_set_a )
=> ~ ( member_set_a @ A2 @ A ) ) ).
% equals0D
thf(fact_421_equals0D,axiom,
! [A: set_li6526943997496501093_a_nat,A2: list_Sum_sum_a_nat] :
( ( A = bot_bo1033123847703346641_a_nat )
=> ~ ( member408289922725080238_a_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_422_equals0D,axiom,
! [A: set_a,A2: a] :
( ( A = bot_bot_set_a )
=> ~ ( member_a @ A2 @ A ) ) ).
% equals0D
thf(fact_423_equals0I,axiom,
! [A: set_se4330304633200676677_a_nat] :
( ! [Y3: set_li6526943997496501093_a_nat] :
~ ( member5553968465346197646_a_nat @ Y3 @ A )
=> ( A = bot_bo3237059034911209905_a_nat ) ) ).
% equals0I
thf(fact_424_equals0I,axiom,
! [A: set_set_a] :
( ! [Y3: set_a] :
~ ( member_set_a @ Y3 @ A )
=> ( A = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_425_equals0I,axiom,
! [A: set_li6526943997496501093_a_nat] :
( ! [Y3: list_Sum_sum_a_nat] :
~ ( member408289922725080238_a_nat @ Y3 @ A )
=> ( A = bot_bo1033123847703346641_a_nat ) ) ).
% equals0I
thf(fact_426_equals0I,axiom,
! [A: set_a] :
( ! [Y3: a] :
~ ( member_a @ Y3 @ A )
=> ( A = bot_bot_set_a ) ) ).
% equals0I
thf(fact_427_Int__assoc,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,C2: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ ( inf_in3249246906714053971_a_nat @ A @ B ) @ C2 )
= ( inf_in3249246906714053971_a_nat @ A @ ( inf_in3249246906714053971_a_nat @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_428_Int__assoc,axiom,
! [A: set_se4330304633200676677_a_nat,B: set_se4330304633200676677_a_nat,C2: set_se4330304633200676677_a_nat] :
( ( inf_in5367731912061063475_a_nat @ ( inf_in5367731912061063475_a_nat @ A @ B ) @ C2 )
= ( inf_in5367731912061063475_a_nat @ A @ ( inf_in5367731912061063475_a_nat @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_429_Int__assoc,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ A @ B ) @ C2 )
= ( inf_inf_set_set_a @ A @ ( inf_inf_set_set_a @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_430_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_431_Un__absorb,axiom,
! [A: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ A @ A )
= A ) ).
% Un_absorb
thf(fact_432_Un__absorb,axiom,
! [A: set_set_a] :
( ( sup_sup_set_set_a @ A @ A )
= A ) ).
% Un_absorb
thf(fact_433_Un__absorb,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ A )
= A ) ).
% Un_absorb
thf(fact_434_Int__absorb,axiom,
! [A: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ A @ A )
= A ) ).
% Int_absorb
thf(fact_435_Int__absorb,axiom,
! [A: set_se4330304633200676677_a_nat] :
( ( inf_in5367731912061063475_a_nat @ A @ A )
= A ) ).
% Int_absorb
thf(fact_436_Int__absorb,axiom,
! [A: set_set_a] :
( ( inf_inf_set_set_a @ A @ A )
= A ) ).
% Int_absorb
thf(fact_437_Int__absorb,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ A )
= A ) ).
% Int_absorb
thf(fact_438_Int__emptyI,axiom,
! [A: set_se4330304633200676677_a_nat,B: set_se4330304633200676677_a_nat] :
( ! [X3: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ X3 @ A )
=> ~ ( member5553968465346197646_a_nat @ X3 @ B ) )
=> ( ( inf_in5367731912061063475_a_nat @ A @ B )
= bot_bo3237059034911209905_a_nat ) ) ).
% Int_emptyI
thf(fact_439_Int__emptyI,axiom,
! [A: set_set_a,B: set_set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A )
=> ~ ( member_set_a @ X3 @ B ) )
=> ( ( inf_inf_set_set_a @ A @ B )
= bot_bot_set_set_a ) ) ).
% Int_emptyI
thf(fact_440_Int__emptyI,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ! [X3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ A )
=> ~ ( member408289922725080238_a_nat @ X3 @ B ) )
=> ( ( inf_in3249246906714053971_a_nat @ A @ B )
= bot_bo1033123847703346641_a_nat ) ) ).
% Int_emptyI
thf(fact_441_Int__emptyI,axiom,
! [A: set_a,B: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ~ ( member_a @ X3 @ B ) )
=> ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_442_Un__commute,axiom,
( sup_su4083067149120280889_a_nat
= ( ^ [A4: set_li6526943997496501093_a_nat,B4: set_li6526943997496501093_a_nat] : ( sup_su4083067149120280889_a_nat @ B4 @ A4 ) ) ) ).
% Un_commute
thf(fact_443_Un__commute,axiom,
( sup_sup_set_set_a
= ( ^ [A4: set_set_a,B4: set_set_a] : ( sup_sup_set_set_a @ B4 @ A4 ) ) ) ).
% Un_commute
thf(fact_444_Un__commute,axiom,
( sup_sup_set_a
= ( ^ [A4: set_a,B4: set_a] : ( sup_sup_set_a @ B4 @ A4 ) ) ) ).
% Un_commute
thf(fact_445_ex__in__conv,axiom,
! [A: set_se4330304633200676677_a_nat] :
( ( ? [X5: set_li6526943997496501093_a_nat] : ( member5553968465346197646_a_nat @ X5 @ A ) )
= ( A != bot_bo3237059034911209905_a_nat ) ) ).
% ex_in_conv
thf(fact_446_ex__in__conv,axiom,
! [A: set_set_a] :
( ( ? [X5: set_a] : ( member_set_a @ X5 @ A ) )
= ( A != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_447_ex__in__conv,axiom,
! [A: set_li6526943997496501093_a_nat] :
( ( ? [X5: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X5 @ A ) )
= ( A != bot_bo1033123847703346641_a_nat ) ) ).
% ex_in_conv
thf(fact_448_ex__in__conv,axiom,
! [A: set_a] :
( ( ? [X5: a] : ( member_a @ X5 @ A ) )
= ( A != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_449_Int__commute,axiom,
( inf_in3249246906714053971_a_nat
= ( ^ [A4: set_li6526943997496501093_a_nat,B4: set_li6526943997496501093_a_nat] : ( inf_in3249246906714053971_a_nat @ B4 @ A4 ) ) ) ).
% Int_commute
thf(fact_450_Int__commute,axiom,
( inf_in5367731912061063475_a_nat
= ( ^ [A4: set_se4330304633200676677_a_nat,B4: set_se4330304633200676677_a_nat] : ( inf_in5367731912061063475_a_nat @ B4 @ A4 ) ) ) ).
% Int_commute
thf(fact_451_Int__commute,axiom,
( inf_inf_set_set_a
= ( ^ [A4: set_set_a,B4: set_set_a] : ( inf_inf_set_set_a @ B4 @ A4 ) ) ) ).
% Int_commute
thf(fact_452_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A4: set_a,B4: set_a] : ( inf_inf_set_a @ B4 @ A4 ) ) ) ).
% Int_commute
thf(fact_453_Un__Int__crazy,axiom,
! [A: set_se4330304633200676677_a_nat,B: set_se4330304633200676677_a_nat,C2: set_se4330304633200676677_a_nat] :
( ( sup_su499249268922660121_a_nat @ ( sup_su499249268922660121_a_nat @ ( inf_in5367731912061063475_a_nat @ A @ B ) @ ( inf_in5367731912061063475_a_nat @ B @ C2 ) ) @ ( inf_in5367731912061063475_a_nat @ C2 @ A ) )
= ( inf_in5367731912061063475_a_nat @ ( inf_in5367731912061063475_a_nat @ ( sup_su499249268922660121_a_nat @ A @ B ) @ ( sup_su499249268922660121_a_nat @ B @ C2 ) ) @ ( sup_su499249268922660121_a_nat @ C2 @ A ) ) ) ).
% Un_Int_crazy
thf(fact_454_Un__Int__crazy,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,C2: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ ( sup_su4083067149120280889_a_nat @ ( inf_in3249246906714053971_a_nat @ A @ B ) @ ( inf_in3249246906714053971_a_nat @ B @ C2 ) ) @ ( inf_in3249246906714053971_a_nat @ C2 @ A ) )
= ( inf_in3249246906714053971_a_nat @ ( inf_in3249246906714053971_a_nat @ ( sup_su4083067149120280889_a_nat @ A @ B ) @ ( sup_su4083067149120280889_a_nat @ B @ C2 ) ) @ ( sup_su4083067149120280889_a_nat @ C2 @ A ) ) ) ).
% Un_Int_crazy
thf(fact_455_Un__Int__crazy,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ A @ B ) @ ( inf_inf_set_set_a @ B @ C2 ) ) @ ( inf_inf_set_set_a @ C2 @ A ) )
= ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ A @ B ) @ ( sup_sup_set_set_a @ B @ C2 ) ) @ ( sup_sup_set_set_a @ C2 @ A ) ) ) ).
% Un_Int_crazy
thf(fact_456_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_457_disjoint__iff,axiom,
! [A: set_se4330304633200676677_a_nat,B: set_se4330304633200676677_a_nat] :
( ( ( inf_in5367731912061063475_a_nat @ A @ B )
= bot_bo3237059034911209905_a_nat )
= ( ! [X5: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ X5 @ A )
=> ~ ( member5553968465346197646_a_nat @ X5 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_458_disjoint__iff,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ( inf_inf_set_set_a @ A @ B )
= bot_bot_set_set_a )
= ( ! [X5: set_a] :
( ( member_set_a @ X5 @ A )
=> ~ ( member_set_a @ X5 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_459_disjoint__iff,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ( inf_in3249246906714053971_a_nat @ A @ B )
= bot_bo1033123847703346641_a_nat )
= ( ! [X5: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X5 @ A )
=> ~ ( member408289922725080238_a_nat @ X5 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_460_disjoint__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
= ( ! [X5: a] :
( ( member_a @ X5 @ A )
=> ~ ( member_a @ X5 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_461_Un__empty__left,axiom,
! [B: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ bot_bo1033123847703346641_a_nat @ B )
= B ) ).
% Un_empty_left
thf(fact_462_Un__empty__left,axiom,
! [B: set_se4330304633200676677_a_nat] :
( ( sup_su499249268922660121_a_nat @ bot_bo3237059034911209905_a_nat @ B )
= B ) ).
% Un_empty_left
thf(fact_463_Un__empty__left,axiom,
! [B: set_set_a] :
( ( sup_sup_set_set_a @ bot_bot_set_set_a @ B )
= B ) ).
% Un_empty_left
thf(fact_464_Un__empty__left,axiom,
! [B: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ B )
= B ) ).
% Un_empty_left
thf(fact_465_Int__Un__distrib,axiom,
! [A: set_se4330304633200676677_a_nat,B: set_se4330304633200676677_a_nat,C2: set_se4330304633200676677_a_nat] :
( ( inf_in5367731912061063475_a_nat @ A @ ( sup_su499249268922660121_a_nat @ B @ C2 ) )
= ( sup_su499249268922660121_a_nat @ ( inf_in5367731912061063475_a_nat @ A @ B ) @ ( inf_in5367731912061063475_a_nat @ A @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_466_Int__Un__distrib,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,C2: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ A @ ( sup_su4083067149120280889_a_nat @ B @ C2 ) )
= ( sup_su4083067149120280889_a_nat @ ( inf_in3249246906714053971_a_nat @ A @ B ) @ ( inf_in3249246906714053971_a_nat @ A @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_467_Int__Un__distrib,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( inf_inf_set_set_a @ A @ ( sup_sup_set_set_a @ B @ C2 ) )
= ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ A @ B ) @ ( inf_inf_set_set_a @ A @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_468_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_469_Int__empty__left,axiom,
! [B: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ bot_bo1033123847703346641_a_nat @ B )
= bot_bo1033123847703346641_a_nat ) ).
% Int_empty_left
thf(fact_470_Int__empty__left,axiom,
! [B: set_se4330304633200676677_a_nat] :
( ( inf_in5367731912061063475_a_nat @ bot_bo3237059034911209905_a_nat @ B )
= bot_bo3237059034911209905_a_nat ) ).
% Int_empty_left
thf(fact_471_Int__empty__left,axiom,
! [B: set_set_a] :
( ( inf_inf_set_set_a @ bot_bot_set_set_a @ B )
= bot_bot_set_set_a ) ).
% Int_empty_left
thf(fact_472_Int__empty__left,axiom,
! [B: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ B )
= bot_bot_set_a ) ).
% Int_empty_left
thf(fact_473_Un__Int__distrib,axiom,
! [A: set_se4330304633200676677_a_nat,B: set_se4330304633200676677_a_nat,C2: set_se4330304633200676677_a_nat] :
( ( sup_su499249268922660121_a_nat @ A @ ( inf_in5367731912061063475_a_nat @ B @ C2 ) )
= ( inf_in5367731912061063475_a_nat @ ( sup_su499249268922660121_a_nat @ A @ B ) @ ( sup_su499249268922660121_a_nat @ A @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_474_Un__Int__distrib,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,C2: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ A @ ( inf_in3249246906714053971_a_nat @ B @ C2 ) )
= ( inf_in3249246906714053971_a_nat @ ( sup_su4083067149120280889_a_nat @ A @ B ) @ ( sup_su4083067149120280889_a_nat @ A @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_475_Un__Int__distrib,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( sup_sup_set_set_a @ A @ ( inf_inf_set_set_a @ B @ C2 ) )
= ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ A @ B ) @ ( sup_sup_set_set_a @ A @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_476_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_477_Un__empty__right,axiom,
! [A: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ A @ bot_bo1033123847703346641_a_nat )
= A ) ).
% Un_empty_right
thf(fact_478_Un__empty__right,axiom,
! [A: set_se4330304633200676677_a_nat] :
( ( sup_su499249268922660121_a_nat @ A @ bot_bo3237059034911209905_a_nat )
= A ) ).
% Un_empty_right
thf(fact_479_Un__empty__right,axiom,
! [A: set_set_a] :
( ( sup_sup_set_set_a @ A @ bot_bot_set_set_a )
= A ) ).
% Un_empty_right
thf(fact_480_Un__empty__right,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ bot_bot_set_a )
= A ) ).
% Un_empty_right
thf(fact_481_Un__left__absorb,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ A @ ( sup_su4083067149120280889_a_nat @ A @ B ) )
= ( sup_su4083067149120280889_a_nat @ A @ B ) ) ).
% Un_left_absorb
thf(fact_482_Un__left__absorb,axiom,
! [A: set_set_a,B: set_set_a] :
( ( sup_sup_set_set_a @ A @ ( sup_sup_set_set_a @ A @ B ) )
= ( sup_sup_set_set_a @ A @ B ) ) ).
% Un_left_absorb
thf(fact_483_Un__left__absorb,axiom,
! [A: set_a,B: set_a] :
( ( sup_sup_set_a @ A @ ( sup_sup_set_a @ A @ B ) )
= ( sup_sup_set_a @ A @ B ) ) ).
% Un_left_absorb
thf(fact_484_Int__Un__distrib2,axiom,
! [B: set_se4330304633200676677_a_nat,C2: set_se4330304633200676677_a_nat,A: set_se4330304633200676677_a_nat] :
( ( inf_in5367731912061063475_a_nat @ ( sup_su499249268922660121_a_nat @ B @ C2 ) @ A )
= ( sup_su499249268922660121_a_nat @ ( inf_in5367731912061063475_a_nat @ B @ A ) @ ( inf_in5367731912061063475_a_nat @ C2 @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_485_Int__Un__distrib2,axiom,
! [B: set_li6526943997496501093_a_nat,C2: set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ ( sup_su4083067149120280889_a_nat @ B @ C2 ) @ A )
= ( sup_su4083067149120280889_a_nat @ ( inf_in3249246906714053971_a_nat @ B @ A ) @ ( inf_in3249246906714053971_a_nat @ C2 @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_486_Int__Un__distrib2,axiom,
! [B: set_set_a,C2: set_set_a,A: set_set_a] :
( ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ B @ C2 ) @ A )
= ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ B @ A ) @ ( inf_inf_set_set_a @ C2 @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_487_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_488_Int__empty__right,axiom,
! [A: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ A @ bot_bo1033123847703346641_a_nat )
= bot_bo1033123847703346641_a_nat ) ).
% Int_empty_right
thf(fact_489_Int__empty__right,axiom,
! [A: set_se4330304633200676677_a_nat] :
( ( inf_in5367731912061063475_a_nat @ A @ bot_bo3237059034911209905_a_nat )
= bot_bo3237059034911209905_a_nat ) ).
% Int_empty_right
thf(fact_490_Int__empty__right,axiom,
! [A: set_set_a] :
( ( inf_inf_set_set_a @ A @ bot_bot_set_set_a )
= bot_bot_set_set_a ) ).
% Int_empty_right
thf(fact_491_Int__empty__right,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ bot_bot_set_a )
= bot_bot_set_a ) ).
% Int_empty_right
thf(fact_492_Int__left__absorb,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ A @ ( inf_in3249246906714053971_a_nat @ A @ B ) )
= ( inf_in3249246906714053971_a_nat @ A @ B ) ) ).
% Int_left_absorb
thf(fact_493_Int__left__absorb,axiom,
! [A: set_se4330304633200676677_a_nat,B: set_se4330304633200676677_a_nat] :
( ( inf_in5367731912061063475_a_nat @ A @ ( inf_in5367731912061063475_a_nat @ A @ B ) )
= ( inf_in5367731912061063475_a_nat @ A @ B ) ) ).
% Int_left_absorb
thf(fact_494_Int__left__absorb,axiom,
! [A: set_set_a,B: set_set_a] :
( ( inf_inf_set_set_a @ A @ ( inf_inf_set_set_a @ A @ B ) )
= ( inf_inf_set_set_a @ A @ B ) ) ).
% Int_left_absorb
thf(fact_495_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_496_Un__Int__distrib2,axiom,
! [B: set_se4330304633200676677_a_nat,C2: set_se4330304633200676677_a_nat,A: set_se4330304633200676677_a_nat] :
( ( sup_su499249268922660121_a_nat @ ( inf_in5367731912061063475_a_nat @ B @ C2 ) @ A )
= ( inf_in5367731912061063475_a_nat @ ( sup_su499249268922660121_a_nat @ B @ A ) @ ( sup_su499249268922660121_a_nat @ C2 @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_497_Un__Int__distrib2,axiom,
! [B: set_li6526943997496501093_a_nat,C2: set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ ( inf_in3249246906714053971_a_nat @ B @ C2 ) @ A )
= ( inf_in3249246906714053971_a_nat @ ( sup_su4083067149120280889_a_nat @ B @ A ) @ ( sup_su4083067149120280889_a_nat @ C2 @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_498_Un__Int__distrib2,axiom,
! [B: set_set_a,C2: set_set_a,A: set_set_a] :
( ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ B @ C2 ) @ A )
= ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ B @ A ) @ ( sup_sup_set_set_a @ C2 @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_499_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_500_Un__left__commute,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,C2: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ A @ ( sup_su4083067149120280889_a_nat @ B @ C2 ) )
= ( sup_su4083067149120280889_a_nat @ B @ ( sup_su4083067149120280889_a_nat @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_501_Un__left__commute,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( sup_sup_set_set_a @ A @ ( sup_sup_set_set_a @ B @ C2 ) )
= ( sup_sup_set_set_a @ B @ ( sup_sup_set_set_a @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_502_Un__left__commute,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B @ C2 ) )
= ( sup_sup_set_a @ B @ ( sup_sup_set_a @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_503_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_504_disjoint__iff__not__equal,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
= ( ! [X5: a] :
( ( member_a @ X5 @ A )
=> ! [Y2: a] :
( ( member_a @ Y2 @ B )
=> ( X5 != Y2 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_505_distrib__sup__le,axiom,
! [X4: set_a,Y: set_a,Z: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ X4 @ ( inf_inf_set_a @ Y @ Z ) ) @ ( inf_inf_set_a @ ( sup_sup_set_a @ X4 @ Y ) @ ( sup_sup_set_a @ X4 @ Z ) ) ) ).
% distrib_sup_le
thf(fact_506_distrib__sup__le,axiom,
! [X4: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat,Z: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ ( sup_su4083067149120280889_a_nat @ X4 @ ( inf_in3249246906714053971_a_nat @ Y @ Z ) ) @ ( inf_in3249246906714053971_a_nat @ ( sup_su4083067149120280889_a_nat @ X4 @ Y ) @ ( sup_su4083067149120280889_a_nat @ X4 @ Z ) ) ) ).
% distrib_sup_le
thf(fact_507_distrib__inf__le,axiom,
! [X4: set_a,Y: set_a,Z: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ X4 @ Y ) @ ( inf_inf_set_a @ X4 @ Z ) ) @ ( inf_inf_set_a @ X4 @ ( sup_sup_set_a @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_508_distrib__inf__le,axiom,
! [X4: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat,Z: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ ( sup_su4083067149120280889_a_nat @ ( inf_in3249246906714053971_a_nat @ X4 @ Y ) @ ( inf_in3249246906714053971_a_nat @ X4 @ Z ) ) @ ( inf_in3249246906714053971_a_nat @ X4 @ ( sup_su4083067149120280889_a_nat @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_509_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_510_Un__Int__assoc__eq,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,C2: set_li6526943997496501093_a_nat] :
( ( ( sup_su4083067149120280889_a_nat @ ( inf_in3249246906714053971_a_nat @ A @ B ) @ C2 )
= ( inf_in3249246906714053971_a_nat @ A @ ( sup_su4083067149120280889_a_nat @ B @ C2 ) ) )
= ( ord_le1147066620699065093_a_nat @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_511_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_512_Un__Diff__Int,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ ( minus_7395159227704179404_a_nat @ A @ B ) @ ( inf_in3249246906714053971_a_nat @ A @ B ) )
= A ) ).
% Un_Diff_Int
thf(fact_513_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_514_Int__Diff__Un,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ ( inf_in3249246906714053971_a_nat @ A @ B ) @ ( minus_7395159227704179404_a_nat @ A @ B ) )
= A ) ).
% Int_Diff_Un
thf(fact_515_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_516_Diff__Int,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,C2: set_li6526943997496501093_a_nat] :
( ( minus_7395159227704179404_a_nat @ A @ ( inf_in3249246906714053971_a_nat @ B @ C2 ) )
= ( sup_su4083067149120280889_a_nat @ ( minus_7395159227704179404_a_nat @ A @ B ) @ ( minus_7395159227704179404_a_nat @ A @ C2 ) ) ) ).
% Diff_Int
thf(fact_517_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_518_Diff__Un,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,C2: set_li6526943997496501093_a_nat] :
( ( minus_7395159227704179404_a_nat @ A @ ( sup_su4083067149120280889_a_nat @ B @ C2 ) )
= ( inf_in3249246906714053971_a_nat @ ( minus_7395159227704179404_a_nat @ A @ B ) @ ( minus_7395159227704179404_a_nat @ A @ C2 ) ) ) ).
% Diff_Un
thf(fact_519_Int__Diff__disjoint,axiom,
! [A: set_a,B: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B ) @ ( minus_minus_set_a @ A @ B ) )
= bot_bot_set_a ) ).
% Int_Diff_disjoint
thf(fact_520_Int__Diff__disjoint,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ ( inf_in3249246906714053971_a_nat @ A @ B ) @ ( minus_7395159227704179404_a_nat @ A @ B ) )
= bot_bo1033123847703346641_a_nat ) ).
% Int_Diff_disjoint
thf(fact_521_Diff__triv,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
=> ( ( minus_minus_set_a @ A @ B )
= A ) ) ).
% Diff_triv
thf(fact_522_Diff__triv,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ( inf_in3249246906714053971_a_nat @ A @ B )
= bot_bo1033123847703346641_a_nat )
=> ( ( minus_7395159227704179404_a_nat @ A @ B )
= A ) ) ).
% Diff_triv
thf(fact_523_ad__agr__close__complete,axiom,
! [X: set_a,Y4: set_a,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ( inf_inf_set_a @ X @ Y4 )
= bot_bot_set_a )
=> ( ( fo_nmlzd_a @ X @ Xs )
=> ( ( fo_nmlzd_a @ ( sup_sup_set_a @ X @ Y4 ) @ Ys )
=> ( ( ad_agr_list_a_nat @ X @ Xs @ Ys )
=> ( member408289922725080238_a_nat @ Ys @ ( ad_agr_close_a @ Y4 @ Xs ) ) ) ) ) ) ).
% ad_agr_close_complete
thf(fact_524_ad__agr__close__sound,axiom,
! [Ys: list_Sum_sum_a_nat,Y4: set_a,Xs: list_Sum_sum_a_nat,X: set_a] :
( ( member408289922725080238_a_nat @ Ys @ ( ad_agr_close_a @ Y4 @ Xs ) )
=> ( ( fo_nmlzd_a @ X @ Xs )
=> ( ( ( inf_inf_set_a @ X @ Y4 )
= bot_bot_set_a )
=> ( ( fo_nmlzd_a @ ( sup_sup_set_a @ X @ Y4 ) @ Ys )
& ( ad_agr_list_a_nat @ X @ Xs @ Ys ) ) ) ) ) ).
% ad_agr_close_sound
thf(fact_525_bot_Oextremum__uniqueI,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
=> ( A2 = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_526_bot_Oextremum__uniqueI,axiom,
! [A2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A2 @ bot_bo1033123847703346641_a_nat )
=> ( A2 = bot_bo1033123847703346641_a_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_527_bot_Oextremum__unique,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_528_bot_Oextremum__unique,axiom,
! [A2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A2 @ bot_bo1033123847703346641_a_nat )
= ( A2 = bot_bo1033123847703346641_a_nat ) ) ).
% bot.extremum_unique
thf(fact_529_bot_Oextremum,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% bot.extremum
thf(fact_530_bot_Oextremum,axiom,
! [A2: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ bot_bo1033123847703346641_a_nat @ A2 ) ).
% bot.extremum
thf(fact_531_sup_OcoboundedI2,axiom,
! [C: set_a,B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ C @ B2 )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_532_sup_OcoboundedI2,axiom,
! [C: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ C @ B2 )
=> ( ord_le1147066620699065093_a_nat @ C @ ( sup_su4083067149120280889_a_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_533_sup_OcoboundedI1,axiom,
! [C: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C @ A2 )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_534_sup_OcoboundedI1,axiom,
! [C: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ C @ A2 )
=> ( ord_le1147066620699065093_a_nat @ C @ ( sup_su4083067149120280889_a_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_535_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( sup_sup_set_a @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_536_sup_Oabsorb__iff2,axiom,
( ord_le1147066620699065093_a_nat
= ( ^ [A3: set_li6526943997496501093_a_nat,B3: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_537_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A3: set_a] :
( ( sup_sup_set_a @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_538_sup_Oabsorb__iff1,axiom,
( ord_le1147066620699065093_a_nat
= ( ^ [B3: set_li6526943997496501093_a_nat,A3: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_539_sup_Ocobounded2,axiom,
! [B2: set_a,A2: set_a] : ( ord_less_eq_set_a @ B2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_540_sup_Ocobounded2,axiom,
! [B2: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ B2 @ ( sup_su4083067149120280889_a_nat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_541_sup_Ocobounded1,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_542_sup_Ocobounded1,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ A2 @ ( sup_su4083067149120280889_a_nat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_543_sup_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A3: set_a] :
( A3
= ( sup_sup_set_a @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_544_sup_Oorder__iff,axiom,
( ord_le1147066620699065093_a_nat
= ( ^ [B3: set_li6526943997496501093_a_nat,A3: set_li6526943997496501093_a_nat] :
( A3
= ( sup_su4083067149120280889_a_nat @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_545_sup_OboundedI,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( ord_less_eq_set_a @ C @ A2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_546_sup_OboundedI,axiom,
! [B2: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ B2 @ A2 )
=> ( ( ord_le1147066620699065093_a_nat @ C @ A2 )
=> ( ord_le1147066620699065093_a_nat @ ( sup_su4083067149120280889_a_nat @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_547_sup_OboundedE,axiom,
! [B2: set_a,C: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ~ ( ord_less_eq_set_a @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_548_sup_OboundedE,axiom,
! [B2: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ ( sup_su4083067149120280889_a_nat @ B2 @ C ) @ A2 )
=> ~ ( ( ord_le1147066620699065093_a_nat @ B2 @ A2 )
=> ~ ( ord_le1147066620699065093_a_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_549_sup__absorb2,axiom,
! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ( sup_sup_set_a @ X4 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_550_sup__absorb2,axiom,
! [X4: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ X4 @ Y )
=> ( ( sup_su4083067149120280889_a_nat @ X4 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_551_sup__absorb1,axiom,
! [Y: set_a,X4: set_a] :
( ( ord_less_eq_set_a @ Y @ X4 )
=> ( ( sup_sup_set_a @ X4 @ Y )
= X4 ) ) ).
% sup_absorb1
thf(fact_552_sup__absorb1,axiom,
! [Y: set_li6526943997496501093_a_nat,X4: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ Y @ X4 )
=> ( ( sup_su4083067149120280889_a_nat @ X4 @ Y )
= X4 ) ) ).
% sup_absorb1
thf(fact_553_sup_Oabsorb2,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( sup_sup_set_a @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_554_sup_Oabsorb2,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A2 @ B2 )
=> ( ( sup_su4083067149120280889_a_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_555_sup_Oabsorb1,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( sup_sup_set_a @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_556_sup_Oabsorb1,axiom,
! [B2: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ B2 @ A2 )
=> ( ( sup_su4083067149120280889_a_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_557_sup__unique,axiom,
! [F: set_a > set_a > set_a,X4: set_a,Y: set_a] :
( ! [X3: set_a,Y3: set_a] : ( ord_less_eq_set_a @ X3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: set_a,Y3: set_a] : ( ord_less_eq_set_a @ Y3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: set_a,Y3: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X3 )
=> ( ( ord_less_eq_set_a @ Z2 @ X3 )
=> ( ord_less_eq_set_a @ ( F @ Y3 @ Z2 ) @ X3 ) ) )
=> ( ( sup_sup_set_a @ X4 @ Y )
= ( F @ X4 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_558_sup__unique,axiom,
! [F: set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,X4: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat] :
( ! [X3: set_li6526943997496501093_a_nat,Y3: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ X3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: set_li6526943997496501093_a_nat,Y3: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ Y3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: set_li6526943997496501093_a_nat,Y3: set_li6526943997496501093_a_nat,Z2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ Y3 @ X3 )
=> ( ( ord_le1147066620699065093_a_nat @ Z2 @ X3 )
=> ( ord_le1147066620699065093_a_nat @ ( F @ Y3 @ Z2 ) @ X3 ) ) )
=> ( ( sup_su4083067149120280889_a_nat @ X4 @ Y )
= ( F @ X4 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_559_sup_OorderI,axiom,
! [A2: set_a,B2: set_a] :
( ( A2
= ( sup_sup_set_a @ A2 @ B2 ) )
=> ( ord_less_eq_set_a @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_560_sup_OorderI,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( A2
= ( sup_su4083067149120280889_a_nat @ A2 @ B2 ) )
=> ( ord_le1147066620699065093_a_nat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_561_sup_OorderE,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( A2
= ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_562_sup_OorderE,axiom,
! [B2: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ B2 @ A2 )
=> ( A2
= ( sup_su4083067149120280889_a_nat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_563_le__iff__sup,axiom,
( ord_less_eq_set_a
= ( ^ [X5: set_a,Y2: set_a] :
( ( sup_sup_set_a @ X5 @ Y2 )
= Y2 ) ) ) ).
% le_iff_sup
thf(fact_564_le__iff__sup,axiom,
( ord_le1147066620699065093_a_nat
= ( ^ [X5: set_li6526943997496501093_a_nat,Y2: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ X5 @ Y2 )
= Y2 ) ) ) ).
% le_iff_sup
thf(fact_565_sup__least,axiom,
! [Y: set_a,X4: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ Y @ X4 )
=> ( ( ord_less_eq_set_a @ Z @ X4 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ Y @ Z ) @ X4 ) ) ) ).
% sup_least
thf(fact_566_sup__least,axiom,
! [Y: set_li6526943997496501093_a_nat,X4: set_li6526943997496501093_a_nat,Z: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ Y @ X4 )
=> ( ( ord_le1147066620699065093_a_nat @ Z @ X4 )
=> ( ord_le1147066620699065093_a_nat @ ( sup_su4083067149120280889_a_nat @ Y @ Z ) @ X4 ) ) ) ).
% sup_least
thf(fact_567_sup__mono,axiom,
! [A2: set_a,C: set_a,B2: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B2 @ D2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ ( sup_sup_set_a @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_568_sup__mono,axiom,
! [A2: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,D2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A2 @ C )
=> ( ( ord_le1147066620699065093_a_nat @ B2 @ D2 )
=> ( ord_le1147066620699065093_a_nat @ ( sup_su4083067149120280889_a_nat @ A2 @ B2 ) @ ( sup_su4083067149120280889_a_nat @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_569_sup_Omono,axiom,
! [C: set_a,A2: set_a,D2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C @ A2 )
=> ( ( ord_less_eq_set_a @ D2 @ B2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ C @ D2 ) @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_570_sup_Omono,axiom,
! [C: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat,D2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ C @ A2 )
=> ( ( ord_le1147066620699065093_a_nat @ D2 @ B2 )
=> ( ord_le1147066620699065093_a_nat @ ( sup_su4083067149120280889_a_nat @ C @ D2 ) @ ( sup_su4083067149120280889_a_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_571_le__supI2,axiom,
! [X4: set_a,B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ X4 @ B2 )
=> ( ord_less_eq_set_a @ X4 @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_572_le__supI2,axiom,
! [X4: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ X4 @ B2 )
=> ( ord_le1147066620699065093_a_nat @ X4 @ ( sup_su4083067149120280889_a_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_573_le__supI1,axiom,
! [X4: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X4 @ A2 )
=> ( ord_less_eq_set_a @ X4 @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_574_le__supI1,axiom,
! [X4: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ X4 @ A2 )
=> ( ord_le1147066620699065093_a_nat @ X4 @ ( sup_su4083067149120280889_a_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_575_sup__ge2,axiom,
! [Y: set_a,X4: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X4 @ Y ) ) ).
% sup_ge2
thf(fact_576_sup__ge2,axiom,
! [Y: set_li6526943997496501093_a_nat,X4: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ Y @ ( sup_su4083067149120280889_a_nat @ X4 @ Y ) ) ).
% sup_ge2
thf(fact_577_sup__ge1,axiom,
! [X4: set_a,Y: set_a] : ( ord_less_eq_set_a @ X4 @ ( sup_sup_set_a @ X4 @ Y ) ) ).
% sup_ge1
thf(fact_578_sup__ge1,axiom,
! [X4: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ X4 @ ( sup_su4083067149120280889_a_nat @ X4 @ Y ) ) ).
% sup_ge1
thf(fact_579_le__supI,axiom,
! [A2: set_a,X4: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ X4 )
=> ( ( ord_less_eq_set_a @ B2 @ X4 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ X4 ) ) ) ).
% le_supI
thf(fact_580_le__supI,axiom,
! [A2: set_li6526943997496501093_a_nat,X4: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A2 @ X4 )
=> ( ( ord_le1147066620699065093_a_nat @ B2 @ X4 )
=> ( ord_le1147066620699065093_a_nat @ ( sup_su4083067149120280889_a_nat @ A2 @ B2 ) @ X4 ) ) ) ).
% le_supI
thf(fact_581_le__supE,axiom,
! [A2: set_a,B2: set_a,X4: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ X4 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ X4 )
=> ~ ( ord_less_eq_set_a @ B2 @ X4 ) ) ) ).
% le_supE
thf(fact_582_le__supE,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,X4: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ ( sup_su4083067149120280889_a_nat @ A2 @ B2 ) @ X4 )
=> ~ ( ( ord_le1147066620699065093_a_nat @ A2 @ X4 )
=> ~ ( ord_le1147066620699065093_a_nat @ B2 @ X4 ) ) ) ).
% le_supE
thf(fact_583_inf__sup__ord_I3_J,axiom,
! [X4: set_a,Y: set_a] : ( ord_less_eq_set_a @ X4 @ ( sup_sup_set_a @ X4 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_584_inf__sup__ord_I3_J,axiom,
! [X4: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ X4 @ ( sup_su4083067149120280889_a_nat @ X4 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_585_inf__sup__ord_I4_J,axiom,
! [Y: set_a,X4: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X4 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_586_inf__sup__ord_I4_J,axiom,
! [Y: set_li6526943997496501093_a_nat,X4: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ Y @ ( sup_su4083067149120280889_a_nat @ X4 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_587_inf_OcoboundedI2,axiom,
! [B2: set_a,C: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_588_inf_OcoboundedI2,axiom,
! [B2: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ B2 @ C )
=> ( ord_le1147066620699065093_a_nat @ ( inf_in3249246906714053971_a_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_589_inf_OcoboundedI1,axiom,
! [A2: set_a,C: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_590_inf_OcoboundedI1,axiom,
! [A2: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A2 @ C )
=> ( ord_le1147066620699065093_a_nat @ ( inf_in3249246906714053971_a_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_591_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A3: set_a] :
( ( inf_inf_set_a @ A3 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_592_inf_Oabsorb__iff2,axiom,
( ord_le1147066620699065093_a_nat
= ( ^ [B3: set_li6526943997496501093_a_nat,A3: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ A3 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_593_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( inf_inf_set_a @ A3 @ B3 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_594_inf_Oabsorb__iff1,axiom,
( ord_le1147066620699065093_a_nat
= ( ^ [A3: set_li6526943997496501093_a_nat,B3: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ A3 @ B3 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_595_inf_Ocobounded2,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_596_inf_Ocobounded2,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ ( inf_in3249246906714053971_a_nat @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_597_inf_Ocobounded1,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_598_inf_Ocobounded1,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ ( inf_in3249246906714053971_a_nat @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_599_inf_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
( A3
= ( inf_inf_set_a @ A3 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_600_inf_Oorder__iff,axiom,
( ord_le1147066620699065093_a_nat
= ( ^ [A3: set_li6526943997496501093_a_nat,B3: set_li6526943997496501093_a_nat] :
( A3
= ( inf_in3249246906714053971_a_nat @ A3 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_601_inf__greatest,axiom,
! [X4: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ( ord_less_eq_set_a @ X4 @ Z )
=> ( ord_less_eq_set_a @ X4 @ ( inf_inf_set_a @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_602_inf__greatest,axiom,
! [X4: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat,Z: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ X4 @ Y )
=> ( ( ord_le1147066620699065093_a_nat @ X4 @ Z )
=> ( ord_le1147066620699065093_a_nat @ X4 @ ( inf_in3249246906714053971_a_nat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_603_inf_OboundedI,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ C )
=> ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_604_inf_OboundedI,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A2 @ B2 )
=> ( ( ord_le1147066620699065093_a_nat @ A2 @ C )
=> ( ord_le1147066620699065093_a_nat @ A2 @ ( inf_in3249246906714053971_a_nat @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_605_inf_OboundedE,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ~ ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_606_inf_OboundedE,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A2 @ ( inf_in3249246906714053971_a_nat @ B2 @ C ) )
=> ~ ( ( ord_le1147066620699065093_a_nat @ A2 @ B2 )
=> ~ ( ord_le1147066620699065093_a_nat @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_607_inf__absorb2,axiom,
! [Y: set_a,X4: set_a] :
( ( ord_less_eq_set_a @ Y @ X4 )
=> ( ( inf_inf_set_a @ X4 @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_608_inf__absorb2,axiom,
! [Y: set_li6526943997496501093_a_nat,X4: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ Y @ X4 )
=> ( ( inf_in3249246906714053971_a_nat @ X4 @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_609_inf__absorb1,axiom,
! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ( inf_inf_set_a @ X4 @ Y )
= X4 ) ) ).
% inf_absorb1
thf(fact_610_inf__absorb1,axiom,
! [X4: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ X4 @ Y )
=> ( ( inf_in3249246906714053971_a_nat @ X4 @ Y )
= X4 ) ) ).
% inf_absorb1
thf(fact_611_inf_Oabsorb2,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_612_inf_Oabsorb2,axiom,
! [B2: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ B2 @ A2 )
=> ( ( inf_in3249246906714053971_a_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_613_inf_Oabsorb1,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_614_inf_Oabsorb1,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A2 @ B2 )
=> ( ( inf_in3249246906714053971_a_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_615_le__iff__inf,axiom,
( ord_less_eq_set_a
= ( ^ [X5: set_a,Y2: set_a] :
( ( inf_inf_set_a @ X5 @ Y2 )
= X5 ) ) ) ).
% le_iff_inf
thf(fact_616_le__iff__inf,axiom,
( ord_le1147066620699065093_a_nat
= ( ^ [X5: set_li6526943997496501093_a_nat,Y2: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ X5 @ Y2 )
= X5 ) ) ) ).
% le_iff_inf
thf(fact_617_inf__unique,axiom,
! [F: set_a > set_a > set_a,X4: set_a,Y: set_a] :
( ! [X3: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( F @ X3 @ Y3 ) @ X3 )
=> ( ! [X3: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( F @ X3 @ Y3 ) @ Y3 )
=> ( ! [X3: set_a,Y3: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ( ord_less_eq_set_a @ X3 @ Z2 )
=> ( ord_less_eq_set_a @ X3 @ ( F @ Y3 @ Z2 ) ) ) )
=> ( ( inf_inf_set_a @ X4 @ Y )
= ( F @ X4 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_618_inf__unique,axiom,
! [F: set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,X4: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat] :
( ! [X3: set_li6526943997496501093_a_nat,Y3: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ ( F @ X3 @ Y3 ) @ X3 )
=> ( ! [X3: set_li6526943997496501093_a_nat,Y3: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ ( F @ X3 @ Y3 ) @ Y3 )
=> ( ! [X3: set_li6526943997496501093_a_nat,Y3: set_li6526943997496501093_a_nat,Z2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ X3 @ Y3 )
=> ( ( ord_le1147066620699065093_a_nat @ X3 @ Z2 )
=> ( ord_le1147066620699065093_a_nat @ X3 @ ( F @ Y3 @ Z2 ) ) ) )
=> ( ( inf_in3249246906714053971_a_nat @ X4 @ Y )
= ( F @ X4 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_619_inf_OorderI,axiom,
! [A2: set_a,B2: set_a] :
( ( A2
= ( inf_inf_set_a @ A2 @ B2 ) )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_620_inf_OorderI,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( A2
= ( inf_in3249246906714053971_a_nat @ A2 @ B2 ) )
=> ( ord_le1147066620699065093_a_nat @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_621_inf_OorderE,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( A2
= ( inf_inf_set_a @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_622_inf_OorderE,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A2 @ B2 )
=> ( A2
= ( inf_in3249246906714053971_a_nat @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_623_le__infI2,axiom,
! [B2: set_a,X4: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ X4 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ X4 ) ) ).
% le_infI2
thf(fact_624_le__infI2,axiom,
! [B2: set_li6526943997496501093_a_nat,X4: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ B2 @ X4 )
=> ( ord_le1147066620699065093_a_nat @ ( inf_in3249246906714053971_a_nat @ A2 @ B2 ) @ X4 ) ) ).
% le_infI2
thf(fact_625_le__infI1,axiom,
! [A2: set_a,X4: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ X4 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ X4 ) ) ).
% le_infI1
thf(fact_626_le__infI1,axiom,
! [A2: set_li6526943997496501093_a_nat,X4: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A2 @ X4 )
=> ( ord_le1147066620699065093_a_nat @ ( inf_in3249246906714053971_a_nat @ A2 @ B2 ) @ X4 ) ) ).
% le_infI1
thf(fact_627_inf__mono,axiom,
! [A2: set_a,C: set_a,B2: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B2 @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ ( inf_inf_set_a @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_628_inf__mono,axiom,
! [A2: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,D2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A2 @ C )
=> ( ( ord_le1147066620699065093_a_nat @ B2 @ D2 )
=> ( ord_le1147066620699065093_a_nat @ ( inf_in3249246906714053971_a_nat @ A2 @ B2 ) @ ( inf_in3249246906714053971_a_nat @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_629_le__infI,axiom,
! [X4: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X4 @ A2 )
=> ( ( ord_less_eq_set_a @ X4 @ B2 )
=> ( ord_less_eq_set_a @ X4 @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_630_le__infI,axiom,
! [X4: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ X4 @ A2 )
=> ( ( ord_le1147066620699065093_a_nat @ X4 @ B2 )
=> ( ord_le1147066620699065093_a_nat @ X4 @ ( inf_in3249246906714053971_a_nat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_631_le__infE,axiom,
! [X4: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X4 @ ( inf_inf_set_a @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_set_a @ X4 @ A2 )
=> ~ ( ord_less_eq_set_a @ X4 @ B2 ) ) ) ).
% le_infE
thf(fact_632_le__infE,axiom,
! [X4: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ X4 @ ( inf_in3249246906714053971_a_nat @ A2 @ B2 ) )
=> ~ ( ( ord_le1147066620699065093_a_nat @ X4 @ A2 )
=> ~ ( ord_le1147066620699065093_a_nat @ X4 @ B2 ) ) ) ).
% le_infE
thf(fact_633_inf__le2,axiom,
! [X4: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X4 @ Y ) @ Y ) ).
% inf_le2
thf(fact_634_inf__le2,axiom,
! [X4: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ ( inf_in3249246906714053971_a_nat @ X4 @ Y ) @ Y ) ).
% inf_le2
thf(fact_635_inf__le1,axiom,
! [X4: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X4 @ Y ) @ X4 ) ).
% inf_le1
thf(fact_636_inf__le1,axiom,
! [X4: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ ( inf_in3249246906714053971_a_nat @ X4 @ Y ) @ X4 ) ).
% inf_le1
thf(fact_637_inf__sup__ord_I1_J,axiom,
! [X4: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X4 @ Y ) @ X4 ) ).
% inf_sup_ord(1)
thf(fact_638_inf__sup__ord_I1_J,axiom,
! [X4: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ ( inf_in3249246906714053971_a_nat @ X4 @ Y ) @ X4 ) ).
% inf_sup_ord(1)
thf(fact_639_inf__sup__ord_I2_J,axiom,
! [X4: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X4 @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_640_inf__sup__ord_I2_J,axiom,
! [X4: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ ( inf_in3249246906714053971_a_nat @ X4 @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_641_subset__Un__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( sup_sup_set_a @ A4 @ B4 )
= B4 ) ) ) ).
% subset_Un_eq
thf(fact_642_subset__Un__eq,axiom,
( ord_le1147066620699065093_a_nat
= ( ^ [A4: set_li6526943997496501093_a_nat,B4: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ A4 @ B4 )
= B4 ) ) ) ).
% subset_Un_eq
thf(fact_643_subset__UnE,axiom,
! [C2: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A @ B ) )
=> ~ ! [A5: set_a] :
( ( ord_less_eq_set_a @ A5 @ A )
=> ! [B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ B )
=> ( C2
!= ( sup_sup_set_a @ A5 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_644_subset__UnE,axiom,
! [C2: set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ C2 @ ( sup_su4083067149120280889_a_nat @ A @ B ) )
=> ~ ! [A5: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A5 @ A )
=> ! [B5: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ B5 @ B )
=> ( C2
!= ( sup_su4083067149120280889_a_nat @ A5 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_645_Un__absorb2,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( sup_sup_set_a @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_646_Un__absorb2,axiom,
! [B: set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ B @ A )
=> ( ( sup_su4083067149120280889_a_nat @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_647_Un__absorb1,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( sup_sup_set_a @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_648_Un__absorb1,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A @ B )
=> ( ( sup_su4083067149120280889_a_nat @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_649_Un__upper2,axiom,
! [B: set_a,A: set_a] : ( ord_less_eq_set_a @ B @ ( sup_sup_set_a @ A @ B ) ) ).
% Un_upper2
thf(fact_650_Un__upper2,axiom,
! [B: set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ B @ ( sup_su4083067149120280889_a_nat @ A @ B ) ) ).
% Un_upper2
thf(fact_651_Un__upper1,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ A @ B ) ) ).
% Un_upper1
thf(fact_652_Un__upper1,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ A @ ( sup_su4083067149120280889_a_nat @ A @ B ) ) ).
% Un_upper1
thf(fact_653_Un__least,axiom,
! [A: set_a,C2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_654_Un__least,axiom,
! [A: set_li6526943997496501093_a_nat,C2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A @ C2 )
=> ( ( ord_le1147066620699065093_a_nat @ B @ C2 )
=> ( ord_le1147066620699065093_a_nat @ ( sup_su4083067149120280889_a_nat @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_655_Un__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 @ ( sup_sup_set_a @ A @ B ) @ ( sup_sup_set_a @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_656_Un__mono,axiom,
! [A: set_li6526943997496501093_a_nat,C2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,D: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A @ C2 )
=> ( ( ord_le1147066620699065093_a_nat @ B @ D )
=> ( ord_le1147066620699065093_a_nat @ ( sup_su4083067149120280889_a_nat @ A @ B ) @ ( sup_su4083067149120280889_a_nat @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_657_Int__Collect__mono,axiom,
! [A: set_a,B: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( 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_658_Int__Collect__mono,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,P: list_Sum_sum_a_nat > $o,Q: list_Sum_sum_a_nat > $o] :
( ( ord_le1147066620699065093_a_nat @ A @ B )
=> ( ! [X3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le1147066620699065093_a_nat @ ( inf_in3249246906714053971_a_nat @ A @ ( collec7555443234367654128_a_nat @ P ) ) @ ( inf_in3249246906714053971_a_nat @ B @ ( collec7555443234367654128_a_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_659_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_660_Int__greatest,axiom,
! [C2: set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ C2 @ A )
=> ( ( ord_le1147066620699065093_a_nat @ C2 @ B )
=> ( ord_le1147066620699065093_a_nat @ C2 @ ( inf_in3249246906714053971_a_nat @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_661_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_662_Int__absorb2,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A @ B )
=> ( ( inf_in3249246906714053971_a_nat @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_663_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_664_Int__absorb1,axiom,
! [B: set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ B @ A )
=> ( ( inf_in3249246906714053971_a_nat @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_665_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_666_Int__lower2,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ ( inf_in3249246906714053971_a_nat @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_667_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_668_Int__lower1,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ ( inf_in3249246906714053971_a_nat @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_669_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_670_Int__mono,axiom,
! [A: set_li6526943997496501093_a_nat,C2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,D: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A @ C2 )
=> ( ( ord_le1147066620699065093_a_nat @ B @ D )
=> ( ord_le1147066620699065093_a_nat @ ( inf_in3249246906714053971_a_nat @ A @ B ) @ ( inf_in3249246906714053971_a_nat @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_671_ad__agr__list__trans,axiom,
! [X: set_a,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Zs: list_Sum_sum_a_nat] :
( ( ad_agr_list_a_nat @ X @ Xs @ Ys )
=> ( ( ad_agr_list_a_nat @ X @ Ys @ Zs )
=> ( ad_agr_list_a_nat @ X @ Xs @ Zs ) ) ) ).
% ad_agr_list_trans
thf(fact_672_ad__agr__list__refl,axiom,
! [X: set_a,Xs: list_Sum_sum_a_nat] : ( ad_agr_list_a_nat @ X @ Xs @ Xs ) ).
% ad_agr_list_refl
thf(fact_673_ad__agr__list__comm,axiom,
! [X: set_a,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ad_agr_list_a_nat @ X @ Xs @ Ys )
=> ( ad_agr_list_a_nat @ X @ Ys @ Xs ) ) ).
% ad_agr_list_comm
thf(fact_674_Un__Diff,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( minus_minus_set_a @ ( sup_sup_set_a @ A @ B ) @ C2 )
= ( sup_sup_set_a @ ( minus_minus_set_a @ A @ C2 ) @ ( minus_minus_set_a @ B @ C2 ) ) ) ).
% Un_Diff
thf(fact_675_Un__Diff,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,C2: set_li6526943997496501093_a_nat] :
( ( minus_7395159227704179404_a_nat @ ( sup_su4083067149120280889_a_nat @ A @ B ) @ C2 )
= ( sup_su4083067149120280889_a_nat @ ( minus_7395159227704179404_a_nat @ A @ C2 ) @ ( minus_7395159227704179404_a_nat @ B @ C2 ) ) ) ).
% Un_Diff
thf(fact_676_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_677_Diff__Int__distrib2,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,C2: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ ( minus_7395159227704179404_a_nat @ A @ B ) @ C2 )
= ( minus_7395159227704179404_a_nat @ ( inf_in3249246906714053971_a_nat @ A @ C2 ) @ ( inf_in3249246906714053971_a_nat @ B @ C2 ) ) ) ).
% Diff_Int_distrib2
thf(fact_678_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_679_Diff__Int__distrib,axiom,
! [C2: set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ C2 @ ( minus_7395159227704179404_a_nat @ A @ B ) )
= ( minus_7395159227704179404_a_nat @ ( inf_in3249246906714053971_a_nat @ C2 @ A ) @ ( inf_in3249246906714053971_a_nat @ C2 @ B ) ) ) ).
% Diff_Int_distrib
thf(fact_680_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_681_Diff__Diff__Int,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( minus_7395159227704179404_a_nat @ A @ ( minus_7395159227704179404_a_nat @ A @ B ) )
= ( inf_in3249246906714053971_a_nat @ A @ B ) ) ).
% Diff_Diff_Int
thf(fact_682_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_683_Diff__Int2,axiom,
! [A: set_li6526943997496501093_a_nat,C2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( minus_7395159227704179404_a_nat @ ( inf_in3249246906714053971_a_nat @ A @ C2 ) @ ( inf_in3249246906714053971_a_nat @ B @ C2 ) )
= ( minus_7395159227704179404_a_nat @ ( inf_in3249246906714053971_a_nat @ A @ C2 ) @ B ) ) ).
% Diff_Int2
thf(fact_684_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_685_Int__Diff,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,C2: set_li6526943997496501093_a_nat] :
( ( minus_7395159227704179404_a_nat @ ( inf_in3249246906714053971_a_nat @ A @ B ) @ C2 )
= ( inf_in3249246906714053971_a_nat @ A @ ( minus_7395159227704179404_a_nat @ B @ C2 ) ) ) ).
% Int_Diff
thf(fact_686_order__antisym__conv,axiom,
! [Y: set_a,X4: set_a] :
( ( ord_less_eq_set_a @ Y @ X4 )
=> ( ( ord_less_eq_set_a @ X4 @ Y )
= ( X4 = Y ) ) ) ).
% order_antisym_conv
thf(fact_687_order__antisym__conv,axiom,
! [Y: set_li6526943997496501093_a_nat,X4: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ Y @ X4 )
=> ( ( ord_le1147066620699065093_a_nat @ X4 @ Y )
= ( X4 = Y ) ) ) ).
% order_antisym_conv
thf(fact_688_ord__le__eq__subst,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_689_ord__le__eq__subst,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_le1147066620699065093_a_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le1147066620699065093_a_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_690_ord__le__eq__subst,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,F: set_li6526943997496501093_a_nat > set_a,C: set_a] :
( ( ord_le1147066620699065093_a_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_li6526943997496501093_a_nat,Y3: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_691_ord__le__eq__subst,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,F: set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_li6526943997496501093_a_nat,Y3: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ X3 @ Y3 )
=> ( ord_le1147066620699065093_a_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le1147066620699065093_a_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_692_ord__eq__le__subst,axiom,
! [A2: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_693_ord__eq__le__subst,axiom,
! [A2: set_li6526943997496501093_a_nat,F: set_a > set_li6526943997496501093_a_nat,B2: set_a,C: set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_le1147066620699065093_a_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le1147066620699065093_a_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_694_ord__eq__le__subst,axiom,
! [A2: set_a,F: set_li6526943997496501093_a_nat > set_a,B2: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le1147066620699065093_a_nat @ B2 @ C )
=> ( ! [X3: set_li6526943997496501093_a_nat,Y3: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_695_ord__eq__le__subst,axiom,
! [A2: set_li6526943997496501093_a_nat,F: set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le1147066620699065093_a_nat @ B2 @ C )
=> ( ! [X3: set_li6526943997496501093_a_nat,Y3: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ X3 @ Y3 )
=> ( ord_le1147066620699065093_a_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le1147066620699065093_a_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_696_order__eq__refl,axiom,
! [X4: set_a,Y: set_a] :
( ( X4 = Y )
=> ( ord_less_eq_set_a @ X4 @ Y ) ) ).
% order_eq_refl
thf(fact_697_order__eq__refl,axiom,
! [X4: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat] :
( ( X4 = Y )
=> ( ord_le1147066620699065093_a_nat @ X4 @ Y ) ) ).
% order_eq_refl
thf(fact_698_order__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_699_order__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_le1147066620699065093_a_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_le1147066620699065093_a_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le1147066620699065093_a_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_700_order__subst2,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,F: set_li6526943997496501093_a_nat > set_a,C: set_a] :
( ( ord_le1147066620699065093_a_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
=> ( ! [X3: set_li6526943997496501093_a_nat,Y3: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_701_order__subst2,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,F: set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A2 @ B2 )
=> ( ( ord_le1147066620699065093_a_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: set_li6526943997496501093_a_nat,Y3: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ X3 @ Y3 )
=> ( ord_le1147066620699065093_a_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le1147066620699065093_a_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_702_order__subst1,axiom,
! [A2: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_703_order__subst1,axiom,
! [A2: set_a,F: set_li6526943997496501093_a_nat > set_a,B2: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_le1147066620699065093_a_nat @ B2 @ C )
=> ( ! [X3: set_li6526943997496501093_a_nat,Y3: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_704_order__subst1,axiom,
! [A2: set_li6526943997496501093_a_nat,F: set_a > set_li6526943997496501093_a_nat,B2: set_a,C: set_a] :
( ( ord_le1147066620699065093_a_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_le1147066620699065093_a_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le1147066620699065093_a_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_705_order__subst1,axiom,
! [A2: set_li6526943997496501093_a_nat,F: set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_le1147066620699065093_a_nat @ B2 @ C )
=> ( ! [X3: set_li6526943997496501093_a_nat,Y3: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ X3 @ Y3 )
=> ( ord_le1147066620699065093_a_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le1147066620699065093_a_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_706_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_a,Z3: set_a] : ( Y5 = Z3 ) )
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
& ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_707_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_li6526943997496501093_a_nat,Z3: set_li6526943997496501093_a_nat] : ( Y5 = Z3 ) )
= ( ^ [A3: set_li6526943997496501093_a_nat,B3: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A3 @ B3 )
& ( ord_le1147066620699065093_a_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_708_antisym,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_709_antisym,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A2 @ B2 )
=> ( ( ord_le1147066620699065093_a_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_710_dual__order_Otrans,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( ord_less_eq_set_a @ C @ B2 )
=> ( ord_less_eq_set_a @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_711_dual__order_Otrans,axiom,
! [B2: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ B2 @ A2 )
=> ( ( ord_le1147066620699065093_a_nat @ C @ B2 )
=> ( ord_le1147066620699065093_a_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_712_dual__order_Oantisym,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_713_dual__order_Oantisym,axiom,
! [B2: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ B2 @ A2 )
=> ( ( ord_le1147066620699065093_a_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_714_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_a,Z3: set_a] : ( Y5 = Z3 ) )
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A3 )
& ( ord_less_eq_set_a @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_715_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_li6526943997496501093_a_nat,Z3: set_li6526943997496501093_a_nat] : ( Y5 = Z3 ) )
= ( ^ [A3: set_li6526943997496501093_a_nat,B3: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ B3 @ A3 )
& ( ord_le1147066620699065093_a_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_716_order__trans,axiom,
! [X4: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z )
=> ( ord_less_eq_set_a @ X4 @ Z ) ) ) ).
% order_trans
thf(fact_717_order__trans,axiom,
! [X4: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat,Z: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ X4 @ Y )
=> ( ( ord_le1147066620699065093_a_nat @ Y @ Z )
=> ( ord_le1147066620699065093_a_nat @ X4 @ Z ) ) ) ).
% order_trans
thf(fact_718_order_Otrans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% order.trans
thf(fact_719_order_Otrans,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A2 @ B2 )
=> ( ( ord_le1147066620699065093_a_nat @ B2 @ C )
=> ( ord_le1147066620699065093_a_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_720_order__antisym,axiom,
! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ( ord_less_eq_set_a @ Y @ X4 )
=> ( X4 = Y ) ) ) ).
% order_antisym
thf(fact_721_order__antisym,axiom,
! [X4: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ X4 @ Y )
=> ( ( ord_le1147066620699065093_a_nat @ Y @ X4 )
=> ( X4 = Y ) ) ) ).
% order_antisym
thf(fact_722_ord__le__eq__trans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_723_ord__le__eq__trans,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_le1147066620699065093_a_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_724_ord__eq__le__trans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( A2 = B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_725_ord__eq__le__trans,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat] :
( ( A2 = B2 )
=> ( ( ord_le1147066620699065093_a_nat @ B2 @ C )
=> ( ord_le1147066620699065093_a_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_726_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_a,Z3: set_a] : ( Y5 = Z3 ) )
= ( ^ [X5: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y2 )
& ( ord_less_eq_set_a @ Y2 @ X5 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_727_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_li6526943997496501093_a_nat,Z3: set_li6526943997496501093_a_nat] : ( Y5 = Z3 ) )
= ( ^ [X5: set_li6526943997496501093_a_nat,Y2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ X5 @ Y2 )
& ( ord_le1147066620699065093_a_nat @ Y2 @ X5 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_728_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X5: a] :
( ( P @ X5 )
=> ( Q @ X5 ) ) ) ) ).
% Collect_mono_iff
thf(fact_729_Collect__mono__iff,axiom,
! [P: list_Sum_sum_a_nat > $o,Q: list_Sum_sum_a_nat > $o] :
( ( ord_le1147066620699065093_a_nat @ ( collec7555443234367654128_a_nat @ P ) @ ( collec7555443234367654128_a_nat @ Q ) )
= ( ! [X5: list_Sum_sum_a_nat] :
( ( P @ X5 )
=> ( Q @ X5 ) ) ) ) ).
% Collect_mono_iff
thf(fact_730_set__eq__subset,axiom,
( ( ^ [Y5: set_a,Z3: set_a] : ( Y5 = Z3 ) )
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
& ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_731_set__eq__subset,axiom,
( ( ^ [Y5: set_li6526943997496501093_a_nat,Z3: set_li6526943997496501093_a_nat] : ( Y5 = Z3 ) )
= ( ^ [A4: set_li6526943997496501093_a_nat,B4: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A4 @ B4 )
& ( ord_le1147066620699065093_a_nat @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_732_subset__trans,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_733_subset__trans,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,C2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A @ B )
=> ( ( ord_le1147066620699065093_a_nat @ B @ C2 )
=> ( ord_le1147066620699065093_a_nat @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_734_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_735_Collect__mono,axiom,
! [P: list_Sum_sum_a_nat > $o,Q: list_Sum_sum_a_nat > $o] :
( ! [X3: list_Sum_sum_a_nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le1147066620699065093_a_nat @ ( collec7555443234367654128_a_nat @ P ) @ ( collec7555443234367654128_a_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_736_subset__refl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% subset_refl
thf(fact_737_subset__refl,axiom,
! [A: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ A @ A ) ).
% subset_refl
thf(fact_738_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
! [T2: a] :
( ( member_a @ T2 @ A4 )
=> ( member_a @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_739_subset__iff,axiom,
( ord_le1147066620699065093_a_nat
= ( ^ [A4: set_li6526943997496501093_a_nat,B4: set_li6526943997496501093_a_nat] :
! [T2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ T2 @ A4 )
=> ( member408289922725080238_a_nat @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_740_Set_OequalityD2,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% Set.equalityD2
thf(fact_741_Set_OequalityD2,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( A = B )
=> ( ord_le1147066620699065093_a_nat @ B @ A ) ) ).
% Set.equalityD2
thf(fact_742_equalityD1,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% equalityD1
thf(fact_743_equalityD1,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( A = B )
=> ( ord_le1147066620699065093_a_nat @ A @ B ) ) ).
% equalityD1
thf(fact_744_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
! [X5: a] :
( ( member_a @ X5 @ A4 )
=> ( member_a @ X5 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_745_subset__eq,axiom,
( ord_le1147066620699065093_a_nat
= ( ^ [A4: set_li6526943997496501093_a_nat,B4: set_li6526943997496501093_a_nat] :
! [X5: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X5 @ A4 )
=> ( member408289922725080238_a_nat @ X5 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_746_equalityE,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_a @ A @ B )
=> ~ ( ord_less_eq_set_a @ B @ A ) ) ) ).
% equalityE
thf(fact_747_equalityE,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( A = B )
=> ~ ( ( ord_le1147066620699065093_a_nat @ A @ B )
=> ~ ( ord_le1147066620699065093_a_nat @ B @ A ) ) ) ).
% equalityE
thf(fact_748_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_749_subsetD,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,C: list_Sum_sum_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A @ B )
=> ( ( member408289922725080238_a_nat @ C @ A )
=> ( member408289922725080238_a_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_750_in__mono,axiom,
! [A: set_a,B: set_a,X4: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ X4 @ A )
=> ( member_a @ X4 @ B ) ) ) ).
% in_mono
thf(fact_751_in__mono,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,X4: list_Sum_sum_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A @ B )
=> ( ( member408289922725080238_a_nat @ X4 @ A )
=> ( member408289922725080238_a_nat @ X4 @ B ) ) ) ).
% in_mono
thf(fact_752_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_753_DiffD2,axiom,
! [C: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ ( minus_7395159227704179404_a_nat @ A @ B ) )
=> ~ ( member408289922725080238_a_nat @ C @ B ) ) ).
% DiffD2
thf(fact_754_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_755_DiffD1,axiom,
! [C: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ ( minus_7395159227704179404_a_nat @ A @ B ) )
=> ( member408289922725080238_a_nat @ C @ A ) ) ).
% DiffD1
thf(fact_756_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_757_DiffE,axiom,
! [C: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ ( minus_7395159227704179404_a_nat @ A @ B ) )
=> ~ ( ( member408289922725080238_a_nat @ C @ A )
=> ( member408289922725080238_a_nat @ C @ B ) ) ) ).
% DiffE
thf(fact_758_Diff__subset__conv,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ B ) @ C2 )
= ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ B @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_759_Diff__subset__conv,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,C2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ ( minus_7395159227704179404_a_nat @ A @ B ) @ C2 )
= ( ord_le1147066620699065093_a_nat @ A @ ( sup_su4083067149120280889_a_nat @ B @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_760_Diff__partition,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( sup_sup_set_a @ A @ ( minus_minus_set_a @ B @ A ) )
= B ) ) ).
% Diff_partition
thf(fact_761_Diff__partition,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A @ B )
=> ( ( sup_su4083067149120280889_a_nat @ A @ ( minus_7395159227704179404_a_nat @ B @ A ) )
= B ) ) ).
% Diff_partition
thf(fact_762_ad__agr__list__fo__nmlzd,axiom,
! [AD: set_a,Vs: list_Sum_sum_a_nat,Vs2: list_Sum_sum_a_nat] :
( ( ad_agr_list_a_nat @ AD @ Vs @ Vs2 )
=> ( ( fo_nmlzd_a @ AD @ Vs )
=> ( ( fo_nmlzd_a @ AD @ Vs2 )
=> ( Vs = Vs2 ) ) ) ) ).
% ad_agr_list_fo_nmlzd
thf(fact_763_ad__agr__list__mono,axiom,
! [X: set_a,Y4: set_a,Ys: list_Sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( ord_less_eq_set_a @ X @ Y4 )
=> ( ( ad_agr_list_a_nat @ Y4 @ Ys @ Xs )
=> ( ad_agr_list_a_nat @ X @ Ys @ Xs ) ) ) ).
% ad_agr_list_mono
thf(fact_764_boolean__algebra_Oconj__zero__left,axiom,
! [X4: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X4 )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_765_boolean__algebra_Oconj__zero__right,axiom,
! [X4: set_a] :
( ( inf_inf_set_a @ X4 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_766_diff__shunt__var,axiom,
! [X4: set_a,Y: set_a] :
( ( ( minus_minus_set_a @ X4 @ Y )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ X4 @ Y ) ) ).
% diff_shunt_var
thf(fact_767_diff__shunt__var,axiom,
! [X4: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat] :
( ( ( minus_7395159227704179404_a_nat @ X4 @ Y )
= bot_bo1033123847703346641_a_nat )
= ( ord_le1147066620699065093_a_nat @ X4 @ Y ) ) ).
% diff_shunt_var
thf(fact_768_boolean__algebra_Oconj__disj__distrib,axiom,
! [X4: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X4 @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X4 @ Y ) @ ( inf_inf_set_a @ X4 @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_769_boolean__algebra_Odisj__conj__distrib,axiom,
! [X4: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ X4 @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X4 @ Y ) @ ( sup_sup_set_a @ X4 @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_770_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_a,Z: set_a,X4: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ Z ) @ X4 )
= ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ X4 ) @ ( inf_inf_set_a @ Z @ X4 ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_771_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_a,Z: set_a,X4: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ Z ) @ X4 )
= ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ X4 ) @ ( sup_sup_set_a @ Z @ X4 ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_772_boolean__algebra_Odisj__zero__right,axiom,
! [X4: set_a] :
( ( sup_sup_set_a @ X4 @ bot_bot_set_a )
= X4 ) ).
% boolean_algebra.disj_zero_right
thf(fact_773_subset__emptyI,axiom,
! [A: set_a] :
( ! [X3: a] :
~ ( member_a @ X3 @ A )
=> ( ord_less_eq_set_a @ A @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_774_subset__emptyI,axiom,
! [A: set_li6526943997496501093_a_nat] :
( ! [X3: list_Sum_sum_a_nat] :
~ ( member408289922725080238_a_nat @ X3 @ A )
=> ( ord_le1147066620699065093_a_nat @ A @ bot_bo1033123847703346641_a_nat ) ) ).
% subset_emptyI
thf(fact_775_ad__agr__close__empty,axiom,
! [X: set_a,Xs: list_Sum_sum_a_nat] :
( ( fo_nmlzd_a @ X @ Xs )
=> ( ( ad_agr_close_a @ bot_bot_set_a @ Xs )
= ( insert2950094090816004437_a_nat @ Xs @ bot_bo1033123847703346641_a_nat ) ) ) ).
% ad_agr_close_empty
thf(fact_776_Greatest__equality,axiom,
! [P: set_a > $o,X4: set_a] :
( ( P @ X4 )
=> ( ! [Y3: set_a] :
( ( P @ Y3 )
=> ( ord_less_eq_set_a @ Y3 @ X4 ) )
=> ( ( order_Greatest_set_a @ P )
= X4 ) ) ) ).
% Greatest_equality
thf(fact_777_Greatest__equality,axiom,
! [P: set_li6526943997496501093_a_nat > $o,X4: set_li6526943997496501093_a_nat] :
( ( P @ X4 )
=> ( ! [Y3: set_li6526943997496501093_a_nat] :
( ( P @ Y3 )
=> ( ord_le1147066620699065093_a_nat @ Y3 @ X4 ) )
=> ( ( order_5197389636244612926_a_nat @ P )
= X4 ) ) ) ).
% Greatest_equality
thf(fact_778_GreatestI2__order,axiom,
! [P: set_a > $o,X4: set_a,Q: set_a > $o] :
( ( P @ X4 )
=> ( ! [Y3: set_a] :
( ( P @ Y3 )
=> ( ord_less_eq_set_a @ Y3 @ X4 ) )
=> ( ! [X3: set_a] :
( ( P @ X3 )
=> ( ! [Y6: set_a] :
( ( P @ Y6 )
=> ( ord_less_eq_set_a @ Y6 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_Greatest_set_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_779_GreatestI2__order,axiom,
! [P: set_li6526943997496501093_a_nat > $o,X4: set_li6526943997496501093_a_nat,Q: set_li6526943997496501093_a_nat > $o] :
( ( P @ X4 )
=> ( ! [Y3: set_li6526943997496501093_a_nat] :
( ( P @ Y3 )
=> ( ord_le1147066620699065093_a_nat @ Y3 @ X4 ) )
=> ( ! [X3: set_li6526943997496501093_a_nat] :
( ( P @ X3 )
=> ( ! [Y6: set_li6526943997496501093_a_nat] :
( ( P @ Y6 )
=> ( ord_le1147066620699065093_a_nat @ Y6 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_5197389636244612926_a_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_780_insertCI,axiom,
! [A2: list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat,B2: list_Sum_sum_a_nat] :
( ( ~ ( member408289922725080238_a_nat @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member408289922725080238_a_nat @ A2 @ ( insert2950094090816004437_a_nat @ B2 @ B ) ) ) ).
% insertCI
thf(fact_781_insertCI,axiom,
! [A2: a,B: set_a,B2: a] :
( ( ~ ( member_a @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_a @ A2 @ ( insert_a @ B2 @ B ) ) ) ).
% insertCI
thf(fact_782_insert__iff,axiom,
! [A2: list_Sum_sum_a_nat,B2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ A2 @ ( insert2950094090816004437_a_nat @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member408289922725080238_a_nat @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_783_insert__iff,axiom,
! [A2: a,B2: a,A: set_a] :
( ( member_a @ A2 @ ( insert_a @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_a @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_784_singletonI,axiom,
! [A2: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ A2 @ ( insert2950094090816004437_a_nat @ A2 @ bot_bo1033123847703346641_a_nat ) ) ).
% singletonI
thf(fact_785_singletonI,axiom,
! [A2: a] : ( member_a @ A2 @ ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_786_insert__subset,axiom,
! [X4: a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X4 @ A ) @ B )
= ( ( member_a @ X4 @ B )
& ( ord_less_eq_set_a @ A @ B ) ) ) ).
% insert_subset
thf(fact_787_insert__subset,axiom,
! [X4: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ ( insert2950094090816004437_a_nat @ X4 @ A ) @ B )
= ( ( member408289922725080238_a_nat @ X4 @ B )
& ( ord_le1147066620699065093_a_nat @ A @ B ) ) ) ).
% insert_subset
thf(fact_788_Un__insert__right,axiom,
! [A: set_a,A2: a,B: set_a] :
( ( sup_sup_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( insert_a @ A2 @ ( sup_sup_set_a @ A @ B ) ) ) ).
% Un_insert_right
thf(fact_789_Un__insert__left,axiom,
! [A2: a,B: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( insert_a @ A2 @ ( sup_sup_set_a @ B @ C2 ) ) ) ).
% Un_insert_left
thf(fact_790_Int__insert__right__if1,axiom,
! [A2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ A2 @ A )
=> ( ( inf_in3249246906714053971_a_nat @ A @ ( insert2950094090816004437_a_nat @ A2 @ B ) )
= ( insert2950094090816004437_a_nat @ A2 @ ( inf_in3249246906714053971_a_nat @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_791_Int__insert__right__if1,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_792_Int__insert__right__if0,axiom,
! [A2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ~ ( member408289922725080238_a_nat @ A2 @ A )
=> ( ( inf_in3249246906714053971_a_nat @ A @ ( insert2950094090816004437_a_nat @ A2 @ B ) )
= ( inf_in3249246906714053971_a_nat @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_793_Int__insert__right__if0,axiom,
! [A2: a,A: set_a,B: set_a] :
( ~ ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_794_insert__inter__insert,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ ( insert_a @ A2 @ B ) )
= ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B ) ) ) ).
% insert_inter_insert
thf(fact_795_Int__insert__left__if1,axiom,
! [A2: list_Sum_sum_a_nat,C2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ A2 @ C2 )
=> ( ( inf_in3249246906714053971_a_nat @ ( insert2950094090816004437_a_nat @ A2 @ B ) @ C2 )
= ( insert2950094090816004437_a_nat @ A2 @ ( inf_in3249246906714053971_a_nat @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_796_Int__insert__left__if1,axiom,
! [A2: a,C2: set_a,B: set_a] :
( ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( insert_a @ A2 @ ( inf_inf_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_797_Int__insert__left__if0,axiom,
! [A2: list_Sum_sum_a_nat,C2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ~ ( member408289922725080238_a_nat @ A2 @ C2 )
=> ( ( inf_in3249246906714053971_a_nat @ ( insert2950094090816004437_a_nat @ A2 @ B ) @ C2 )
= ( inf_in3249246906714053971_a_nat @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_798_Int__insert__left__if0,axiom,
! [A2: a,C2: set_a,B: set_a] :
( ~ ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( inf_inf_set_a @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_799_Diff__insert0,axiom,
! [X4: a,A: set_a,B: set_a] :
( ~ ( member_a @ X4 @ A )
=> ( ( minus_minus_set_a @ A @ ( insert_a @ X4 @ B ) )
= ( minus_minus_set_a @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_800_Diff__insert0,axiom,
! [X4: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ~ ( member408289922725080238_a_nat @ X4 @ A )
=> ( ( minus_7395159227704179404_a_nat @ A @ ( insert2950094090816004437_a_nat @ X4 @ B ) )
= ( minus_7395159227704179404_a_nat @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_801_insert__Diff1,axiom,
! [X4: a,B: set_a,A: set_a] :
( ( member_a @ X4 @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X4 @ A ) @ B )
= ( minus_minus_set_a @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_802_insert__Diff1,axiom,
! [X4: list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ B )
=> ( ( minus_7395159227704179404_a_nat @ ( insert2950094090816004437_a_nat @ X4 @ A ) @ B )
= ( minus_7395159227704179404_a_nat @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_803_singleton__insert__inj__eq,axiom,
! [B2: a,A2: a,A: set_a] :
( ( ( insert_a @ B2 @ bot_bot_set_a )
= ( insert_a @ A2 @ A ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_a @ A @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_804_singleton__insert__inj__eq,axiom,
! [B2: list_Sum_sum_a_nat,A2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( ( insert2950094090816004437_a_nat @ B2 @ bot_bo1033123847703346641_a_nat )
= ( insert2950094090816004437_a_nat @ A2 @ A ) )
= ( ( A2 = B2 )
& ( ord_le1147066620699065093_a_nat @ A @ ( insert2950094090816004437_a_nat @ B2 @ bot_bo1033123847703346641_a_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_805_singleton__insert__inj__eq_H,axiom,
! [A2: a,A: set_a,B2: a] :
( ( ( insert_a @ A2 @ A )
= ( insert_a @ B2 @ bot_bot_set_a ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_a @ A @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_806_singleton__insert__inj__eq_H,axiom,
! [A2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B2: list_Sum_sum_a_nat] :
( ( ( insert2950094090816004437_a_nat @ A2 @ A )
= ( insert2950094090816004437_a_nat @ B2 @ bot_bo1033123847703346641_a_nat ) )
= ( ( A2 = B2 )
& ( ord_le1147066620699065093_a_nat @ A @ ( insert2950094090816004437_a_nat @ B2 @ bot_bo1033123847703346641_a_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_807_insert__disjoint_I1_J,axiom,
! [A2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ( inf_in3249246906714053971_a_nat @ ( insert2950094090816004437_a_nat @ A2 @ A ) @ B )
= bot_bo1033123847703346641_a_nat )
= ( ~ ( member408289922725080238_a_nat @ A2 @ B )
& ( ( inf_in3249246906714053971_a_nat @ A @ B )
= bot_bo1033123847703346641_a_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_808_insert__disjoint_I1_J,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ B )
= bot_bot_set_a )
= ( ~ ( member_a @ A2 @ B )
& ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_809_insert__disjoint_I2_J,axiom,
! [A2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( bot_bo1033123847703346641_a_nat
= ( inf_in3249246906714053971_a_nat @ ( insert2950094090816004437_a_nat @ A2 @ A ) @ B ) )
= ( ~ ( member408289922725080238_a_nat @ A2 @ B )
& ( bot_bo1033123847703346641_a_nat
= ( inf_in3249246906714053971_a_nat @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_810_insert__disjoint_I2_J,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ B ) )
= ( ~ ( member_a @ A2 @ B )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_811_disjoint__insert_I1_J,axiom,
! [B: set_li6526943997496501093_a_nat,A2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( ( inf_in3249246906714053971_a_nat @ B @ ( insert2950094090816004437_a_nat @ A2 @ A ) )
= bot_bo1033123847703346641_a_nat )
= ( ~ ( member408289922725080238_a_nat @ A2 @ B )
& ( ( inf_in3249246906714053971_a_nat @ B @ A )
= bot_bo1033123847703346641_a_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_812_disjoint__insert_I1_J,axiom,
! [B: set_a,A2: a,A: set_a] :
( ( ( inf_inf_set_a @ B @ ( insert_a @ A2 @ A ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A2 @ B )
& ( ( inf_inf_set_a @ B @ A )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_813_disjoint__insert_I2_J,axiom,
! [A: set_li6526943997496501093_a_nat,B2: list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat] :
( ( bot_bo1033123847703346641_a_nat
= ( inf_in3249246906714053971_a_nat @ A @ ( insert2950094090816004437_a_nat @ B2 @ B ) ) )
= ( ~ ( member408289922725080238_a_nat @ B2 @ A )
& ( bot_bo1033123847703346641_a_nat
= ( inf_in3249246906714053971_a_nat @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_814_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_815_insert__Diff__single,axiom,
! [A2: a,A: set_a] :
( ( insert_a @ A2 @ ( minus_minus_set_a @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) )
= ( insert_a @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_816_insert__Diff__single,axiom,
! [A2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( insert2950094090816004437_a_nat @ A2 @ ( minus_7395159227704179404_a_nat @ A @ ( insert2950094090816004437_a_nat @ A2 @ bot_bo1033123847703346641_a_nat ) ) )
= ( insert2950094090816004437_a_nat @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_817_insertE,axiom,
! [A2: list_Sum_sum_a_nat,B2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ A2 @ ( insert2950094090816004437_a_nat @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member408289922725080238_a_nat @ A2 @ A ) ) ) ).
% insertE
thf(fact_818_insertE,axiom,
! [A2: a,B2: a,A: set_a] :
( ( member_a @ A2 @ ( insert_a @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_a @ A2 @ A ) ) ) ).
% insertE
thf(fact_819_insertI1,axiom,
! [A2: list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat] : ( member408289922725080238_a_nat @ A2 @ ( insert2950094090816004437_a_nat @ A2 @ B ) ) ).
% insertI1
thf(fact_820_insertI1,axiom,
! [A2: a,B: set_a] : ( member_a @ A2 @ ( insert_a @ A2 @ B ) ) ).
% insertI1
thf(fact_821_insertI2,axiom,
! [A2: list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat,B2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ A2 @ B )
=> ( member408289922725080238_a_nat @ A2 @ ( insert2950094090816004437_a_nat @ B2 @ B ) ) ) ).
% insertI2
thf(fact_822_insertI2,axiom,
! [A2: a,B: set_a,B2: a] :
( ( member_a @ A2 @ B )
=> ( member_a @ A2 @ ( insert_a @ B2 @ B ) ) ) ).
% insertI2
thf(fact_823_Set_Oset__insert,axiom,
! [X4: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ A )
=> ~ ! [B6: set_li6526943997496501093_a_nat] :
( ( A
= ( insert2950094090816004437_a_nat @ X4 @ B6 ) )
=> ( member408289922725080238_a_nat @ X4 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_824_Set_Oset__insert,axiom,
! [X4: a,A: set_a] :
( ( member_a @ X4 @ A )
=> ~ ! [B6: set_a] :
( ( A
= ( insert_a @ X4 @ B6 ) )
=> ( member_a @ X4 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_825_insert__ident,axiom,
! [X4: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ~ ( member408289922725080238_a_nat @ X4 @ A )
=> ( ~ ( member408289922725080238_a_nat @ X4 @ B )
=> ( ( ( insert2950094090816004437_a_nat @ X4 @ A )
= ( insert2950094090816004437_a_nat @ X4 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_826_insert__ident,axiom,
! [X4: a,A: set_a,B: set_a] :
( ~ ( member_a @ X4 @ A )
=> ( ~ ( member_a @ X4 @ B )
=> ( ( ( insert_a @ X4 @ A )
= ( insert_a @ X4 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_827_insert__absorb,axiom,
! [A2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ A2 @ A )
=> ( ( insert2950094090816004437_a_nat @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_828_insert__absorb,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ( ( insert_a @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_829_insert__eq__iff,axiom,
! [A2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B2: list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat] :
( ~ ( member408289922725080238_a_nat @ A2 @ A )
=> ( ~ ( member408289922725080238_a_nat @ B2 @ B )
=> ( ( ( insert2950094090816004437_a_nat @ A2 @ A )
= ( insert2950094090816004437_a_nat @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C3: set_li6526943997496501093_a_nat] :
( ( A
= ( insert2950094090816004437_a_nat @ B2 @ C3 ) )
& ~ ( member408289922725080238_a_nat @ B2 @ C3 )
& ( B
= ( insert2950094090816004437_a_nat @ A2 @ C3 ) )
& ~ ( member408289922725080238_a_nat @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_830_insert__eq__iff,axiom,
! [A2: a,A: set_a,B2: a,B: set_a] :
( ~ ( member_a @ A2 @ A )
=> ( ~ ( member_a @ B2 @ B )
=> ( ( ( insert_a @ A2 @ A )
= ( insert_a @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C3: set_a] :
( ( A
= ( insert_a @ B2 @ C3 ) )
& ~ ( member_a @ B2 @ C3 )
& ( B
= ( insert_a @ A2 @ C3 ) )
& ~ ( member_a @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_831_mk__disjoint__insert,axiom,
! [A2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ A2 @ A )
=> ? [B6: set_li6526943997496501093_a_nat] :
( ( A
= ( insert2950094090816004437_a_nat @ A2 @ B6 ) )
& ~ ( member408289922725080238_a_nat @ A2 @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_832_mk__disjoint__insert,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ? [B6: set_a] :
( ( A
= ( insert_a @ A2 @ B6 ) )
& ~ ( member_a @ A2 @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_833_insert__subsetI,axiom,
! [X4: a,A: set_a,X: set_a] :
( ( member_a @ X4 @ A )
=> ( ( ord_less_eq_set_a @ X @ A )
=> ( ord_less_eq_set_a @ ( insert_a @ X4 @ X ) @ A ) ) ) ).
% insert_subsetI
thf(fact_834_insert__subsetI,axiom,
! [X4: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,X: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ A )
=> ( ( ord_le1147066620699065093_a_nat @ X @ A )
=> ( ord_le1147066620699065093_a_nat @ ( insert2950094090816004437_a_nat @ X4 @ X ) @ A ) ) ) ).
% insert_subsetI
thf(fact_835_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_836_singleton__inject,axiom,
! [A2: a,B2: a] :
( ( ( insert_a @ A2 @ bot_bot_set_a )
= ( insert_a @ B2 @ bot_bot_set_a ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_837_insert__not__empty,axiom,
! [A2: a,A: set_a] :
( ( insert_a @ A2 @ A )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_838_doubleton__eq__iff,axiom,
! [A2: a,B2: a,C: a,D2: a] :
( ( ( insert_a @ A2 @ ( insert_a @ B2 @ bot_bot_set_a ) )
= ( insert_a @ C @ ( insert_a @ D2 @ bot_bot_set_a ) ) )
= ( ( ( A2 = C )
& ( B2 = D2 ) )
| ( ( A2 = D2 )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_839_singleton__iff,axiom,
! [B2: list_Sum_sum_a_nat,A2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ B2 @ ( insert2950094090816004437_a_nat @ A2 @ bot_bo1033123847703346641_a_nat ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_840_singleton__iff,axiom,
! [B2: a,A2: a] :
( ( member_a @ B2 @ ( insert_a @ A2 @ bot_bot_set_a ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_841_singletonD,axiom,
! [B2: list_Sum_sum_a_nat,A2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ B2 @ ( insert2950094090816004437_a_nat @ A2 @ bot_bo1033123847703346641_a_nat ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_842_singletonD,axiom,
! [B2: a,A2: a] :
( ( member_a @ B2 @ ( insert_a @ A2 @ bot_bot_set_a ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_843_insert__mono,axiom,
! [C2: set_a,D: set_a,A2: a] :
( ( ord_less_eq_set_a @ C2 @ D )
=> ( ord_less_eq_set_a @ ( insert_a @ A2 @ C2 ) @ ( insert_a @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_844_insert__mono,axiom,
! [C2: set_li6526943997496501093_a_nat,D: set_li6526943997496501093_a_nat,A2: list_Sum_sum_a_nat] :
( ( ord_le1147066620699065093_a_nat @ C2 @ D )
=> ( ord_le1147066620699065093_a_nat @ ( insert2950094090816004437_a_nat @ A2 @ C2 ) @ ( insert2950094090816004437_a_nat @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_845_subset__insert,axiom,
! [X4: a,A: set_a,B: set_a] :
( ~ ( member_a @ X4 @ A )
=> ( ( ord_less_eq_set_a @ A @ ( insert_a @ X4 @ B ) )
= ( ord_less_eq_set_a @ A @ B ) ) ) ).
% subset_insert
thf(fact_846_subset__insert,axiom,
! [X4: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ~ ( member408289922725080238_a_nat @ X4 @ A )
=> ( ( ord_le1147066620699065093_a_nat @ A @ ( insert2950094090816004437_a_nat @ X4 @ B ) )
= ( ord_le1147066620699065093_a_nat @ A @ B ) ) ) ).
% subset_insert
thf(fact_847_subset__insertI,axiom,
! [B: set_a,A2: a] : ( ord_less_eq_set_a @ B @ ( insert_a @ A2 @ B ) ) ).
% subset_insertI
thf(fact_848_subset__insertI,axiom,
! [B: set_li6526943997496501093_a_nat,A2: list_Sum_sum_a_nat] : ( ord_le1147066620699065093_a_nat @ B @ ( insert2950094090816004437_a_nat @ A2 @ B ) ) ).
% subset_insertI
thf(fact_849_subset__insertI2,axiom,
! [A: set_a,B: set_a,B2: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ A @ ( insert_a @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_850_subset__insertI2,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,B2: list_Sum_sum_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A @ B )
=> ( ord_le1147066620699065093_a_nat @ A @ ( insert2950094090816004437_a_nat @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_851_Int__insert__right,axiom,
! [A2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ( member408289922725080238_a_nat @ A2 @ A )
=> ( ( inf_in3249246906714053971_a_nat @ A @ ( insert2950094090816004437_a_nat @ A2 @ B ) )
= ( insert2950094090816004437_a_nat @ A2 @ ( inf_in3249246906714053971_a_nat @ A @ B ) ) ) )
& ( ~ ( member408289922725080238_a_nat @ A2 @ A )
=> ( ( inf_in3249246906714053971_a_nat @ A @ ( insert2950094090816004437_a_nat @ A2 @ B ) )
= ( inf_in3249246906714053971_a_nat @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_852_Int__insert__right,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B ) ) ) )
& ( ~ ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_853_Int__insert__left,axiom,
! [A2: list_Sum_sum_a_nat,C2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ( member408289922725080238_a_nat @ A2 @ C2 )
=> ( ( inf_in3249246906714053971_a_nat @ ( insert2950094090816004437_a_nat @ A2 @ B ) @ C2 )
= ( insert2950094090816004437_a_nat @ A2 @ ( inf_in3249246906714053971_a_nat @ B @ C2 ) ) ) )
& ( ~ ( member408289922725080238_a_nat @ A2 @ C2 )
=> ( ( inf_in3249246906714053971_a_nat @ ( insert2950094090816004437_a_nat @ A2 @ B ) @ C2 )
= ( inf_in3249246906714053971_a_nat @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_854_Int__insert__left,axiom,
! [A2: a,C2: set_a,B: set_a] :
( ( ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( insert_a @ A2 @ ( inf_inf_set_a @ B @ C2 ) ) ) )
& ( ~ ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( inf_inf_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_855_insert__Diff__if,axiom,
! [X4: a,B: set_a,A: set_a] :
( ( ( member_a @ X4 @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X4 @ A ) @ B )
= ( minus_minus_set_a @ A @ B ) ) )
& ( ~ ( member_a @ X4 @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X4 @ A ) @ B )
= ( insert_a @ X4 @ ( minus_minus_set_a @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_856_insert__Diff__if,axiom,
! [X4: list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat] :
( ( ( member408289922725080238_a_nat @ X4 @ B )
=> ( ( minus_7395159227704179404_a_nat @ ( insert2950094090816004437_a_nat @ X4 @ A ) @ B )
= ( minus_7395159227704179404_a_nat @ A @ B ) ) )
& ( ~ ( member408289922725080238_a_nat @ X4 @ B )
=> ( ( minus_7395159227704179404_a_nat @ ( insert2950094090816004437_a_nat @ X4 @ A ) @ B )
= ( insert2950094090816004437_a_nat @ X4 @ ( minus_7395159227704179404_a_nat @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_857_subset__singletonD,axiom,
! [A: set_a,X4: a] :
( ( ord_less_eq_set_a @ A @ ( insert_a @ X4 @ bot_bot_set_a ) )
=> ( ( A = bot_bot_set_a )
| ( A
= ( insert_a @ X4 @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_858_subset__singletonD,axiom,
! [A: set_li6526943997496501093_a_nat,X4: list_Sum_sum_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A @ ( insert2950094090816004437_a_nat @ X4 @ bot_bo1033123847703346641_a_nat ) )
=> ( ( A = bot_bo1033123847703346641_a_nat )
| ( A
= ( insert2950094090816004437_a_nat @ X4 @ bot_bo1033123847703346641_a_nat ) ) ) ) ).
% subset_singletonD
thf(fact_859_subset__singleton__iff,axiom,
! [X: set_a,A2: a] :
( ( ord_less_eq_set_a @ X @ ( insert_a @ A2 @ bot_bot_set_a ) )
= ( ( X = bot_bot_set_a )
| ( X
= ( insert_a @ A2 @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_860_subset__singleton__iff,axiom,
! [X: set_li6526943997496501093_a_nat,A2: list_Sum_sum_a_nat] :
( ( ord_le1147066620699065093_a_nat @ X @ ( insert2950094090816004437_a_nat @ A2 @ bot_bo1033123847703346641_a_nat ) )
= ( ( X = bot_bo1033123847703346641_a_nat )
| ( X
= ( insert2950094090816004437_a_nat @ A2 @ bot_bo1033123847703346641_a_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_861_singleton__Un__iff,axiom,
! [X4: a,A: set_a,B: set_a] :
( ( ( insert_a @ X4 @ bot_bot_set_a )
= ( sup_sup_set_a @ A @ B ) )
= ( ( ( A = bot_bot_set_a )
& ( B
= ( insert_a @ X4 @ bot_bot_set_a ) ) )
| ( ( A
= ( insert_a @ X4 @ bot_bot_set_a ) )
& ( B = bot_bot_set_a ) )
| ( ( A
= ( insert_a @ X4 @ bot_bot_set_a ) )
& ( B
= ( insert_a @ X4 @ bot_bot_set_a ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_862_Un__singleton__iff,axiom,
! [A: set_a,B: set_a,X4: a] :
( ( ( sup_sup_set_a @ A @ B )
= ( insert_a @ X4 @ bot_bot_set_a ) )
= ( ( ( A = bot_bot_set_a )
& ( B
= ( insert_a @ X4 @ bot_bot_set_a ) ) )
| ( ( A
= ( insert_a @ X4 @ bot_bot_set_a ) )
& ( B = bot_bot_set_a ) )
| ( ( A
= ( insert_a @ X4 @ bot_bot_set_a ) )
& ( B
= ( insert_a @ X4 @ bot_bot_set_a ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_863_insert__is__Un,axiom,
( insert_a
= ( ^ [A3: a] : ( sup_sup_set_a @ ( insert_a @ A3 @ bot_bot_set_a ) ) ) ) ).
% insert_is_Un
thf(fact_864_Diff__insert,axiom,
! [A: set_a,A2: a,B: set_a] :
( ( minus_minus_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A @ B ) @ ( insert_a @ A2 @ bot_bot_set_a ) ) ) ).
% Diff_insert
thf(fact_865_Diff__insert,axiom,
! [A: set_li6526943997496501093_a_nat,A2: list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat] :
( ( minus_7395159227704179404_a_nat @ A @ ( insert2950094090816004437_a_nat @ A2 @ B ) )
= ( minus_7395159227704179404_a_nat @ ( minus_7395159227704179404_a_nat @ A @ B ) @ ( insert2950094090816004437_a_nat @ A2 @ bot_bo1033123847703346641_a_nat ) ) ) ).
% Diff_insert
thf(fact_866_insert__Diff,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ( ( insert_a @ A2 @ ( minus_minus_set_a @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) )
= A ) ) ).
% insert_Diff
thf(fact_867_insert__Diff,axiom,
! [A2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ A2 @ A )
=> ( ( insert2950094090816004437_a_nat @ A2 @ ( minus_7395159227704179404_a_nat @ A @ ( insert2950094090816004437_a_nat @ A2 @ bot_bo1033123847703346641_a_nat ) ) )
= A ) ) ).
% insert_Diff
thf(fact_868_Diff__insert2,axiom,
! [A: set_a,A2: a,B: set_a] :
( ( minus_minus_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) @ B ) ) ).
% Diff_insert2
thf(fact_869_Diff__insert2,axiom,
! [A: set_li6526943997496501093_a_nat,A2: list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat] :
( ( minus_7395159227704179404_a_nat @ A @ ( insert2950094090816004437_a_nat @ A2 @ B ) )
= ( minus_7395159227704179404_a_nat @ ( minus_7395159227704179404_a_nat @ A @ ( insert2950094090816004437_a_nat @ A2 @ bot_bo1033123847703346641_a_nat ) ) @ B ) ) ).
% Diff_insert2
thf(fact_870_Diff__insert__absorb,axiom,
! [X4: a,A: set_a] :
( ~ ( member_a @ X4 @ A )
=> ( ( minus_minus_set_a @ ( insert_a @ X4 @ A ) @ ( insert_a @ X4 @ bot_bot_set_a ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_871_Diff__insert__absorb,axiom,
! [X4: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ~ ( member408289922725080238_a_nat @ X4 @ A )
=> ( ( minus_7395159227704179404_a_nat @ ( insert2950094090816004437_a_nat @ X4 @ A ) @ ( insert2950094090816004437_a_nat @ X4 @ bot_bo1033123847703346641_a_nat ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_872_subset__Diff__insert,axiom,
! [A: set_a,B: set_a,X4: a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ ( minus_minus_set_a @ B @ ( insert_a @ X4 @ C2 ) ) )
= ( ( ord_less_eq_set_a @ A @ ( minus_minus_set_a @ B @ C2 ) )
& ~ ( member_a @ X4 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_873_subset__Diff__insert,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,X4: list_Sum_sum_a_nat,C2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A @ ( minus_7395159227704179404_a_nat @ B @ ( insert2950094090816004437_a_nat @ X4 @ C2 ) ) )
= ( ( ord_le1147066620699065093_a_nat @ A @ ( minus_7395159227704179404_a_nat @ B @ C2 ) )
& ~ ( member408289922725080238_a_nat @ X4 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_874_subset__insert__iff,axiom,
! [A: set_a,X4: a,B: set_a] :
( ( ord_less_eq_set_a @ A @ ( insert_a @ X4 @ B ) )
= ( ( ( member_a @ X4 @ A )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ X4 @ bot_bot_set_a ) ) @ B ) )
& ( ~ ( member_a @ X4 @ A )
=> ( ord_less_eq_set_a @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_875_subset__insert__iff,axiom,
! [A: set_li6526943997496501093_a_nat,X4: list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A @ ( insert2950094090816004437_a_nat @ X4 @ B ) )
= ( ( ( member408289922725080238_a_nat @ X4 @ A )
=> ( ord_le1147066620699065093_a_nat @ ( minus_7395159227704179404_a_nat @ A @ ( insert2950094090816004437_a_nat @ X4 @ bot_bo1033123847703346641_a_nat ) ) @ B ) )
& ( ~ ( member408289922725080238_a_nat @ X4 @ A )
=> ( ord_le1147066620699065093_a_nat @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_876_Diff__single__insert,axiom,
! [A: set_a,X4: a,B: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ X4 @ bot_bot_set_a ) ) @ B )
=> ( ord_less_eq_set_a @ A @ ( insert_a @ X4 @ B ) ) ) ).
% Diff_single_insert
thf(fact_877_Diff__single__insert,axiom,
! [A: set_li6526943997496501093_a_nat,X4: list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ ( minus_7395159227704179404_a_nat @ A @ ( insert2950094090816004437_a_nat @ X4 @ bot_bo1033123847703346641_a_nat ) ) @ B )
=> ( ord_le1147066620699065093_a_nat @ A @ ( insert2950094090816004437_a_nat @ X4 @ B ) ) ) ).
% Diff_single_insert
thf(fact_878_boolean__algebra__cancel_Osup2,axiom,
! [B: set_a,K: set_a,B2: set_a,A2: set_a] :
( ( B
= ( sup_sup_set_a @ K @ B2 ) )
=> ( ( sup_sup_set_a @ A2 @ B )
= ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_879_boolean__algebra__cancel_Osup1,axiom,
! [A: set_a,K: set_a,A2: set_a,B2: set_a] :
( ( A
= ( sup_sup_set_a @ K @ A2 ) )
=> ( ( sup_sup_set_a @ A @ B2 )
= ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_880_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_a,K: set_a,B2: set_a,A2: set_a] :
( ( B
= ( inf_inf_set_a @ K @ B2 ) )
=> ( ( inf_inf_set_a @ A2 @ B )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_881_boolean__algebra__cancel_Oinf1,axiom,
! [A: set_a,K: set_a,A2: set_a,B2: set_a] :
( ( A
= ( inf_inf_set_a @ K @ A2 ) )
=> ( ( inf_inf_set_a @ A @ B2 )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_882_is__singletonI,axiom,
! [X4: a] : ( is_singleton_a @ ( insert_a @ X4 @ bot_bot_set_a ) ) ).
% is_singletonI
thf(fact_883_is__singleton__def,axiom,
( is_singleton_a
= ( ^ [A4: set_a] :
? [X5: a] :
( A4
= ( insert_a @ X5 @ bot_bot_set_a ) ) ) ) ).
% is_singleton_def
thf(fact_884_is__singletonE,axiom,
! [A: set_a] :
( ( is_singleton_a @ A )
=> ~ ! [X3: a] :
( A
!= ( insert_a @ X3 @ bot_bot_set_a ) ) ) ).
% is_singletonE
thf(fact_885_Set_Oremove__def,axiom,
( remove_a
= ( ^ [X5: a,A4: set_a] : ( minus_minus_set_a @ A4 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) ) ).
% Set.remove_def
thf(fact_886_Set_Oremove__def,axiom,
( remove5086202153292001386_a_nat
= ( ^ [X5: list_Sum_sum_a_nat,A4: set_li6526943997496501093_a_nat] : ( minus_7395159227704179404_a_nat @ A4 @ ( insert2950094090816004437_a_nat @ X5 @ bot_bo1033123847703346641_a_nat ) ) ) ) ).
% Set.remove_def
thf(fact_887_the__elem__eq,axiom,
! [X4: a] :
( ( the_elem_a @ ( insert_a @ X4 @ bot_bot_set_a ) )
= X4 ) ).
% the_elem_eq
thf(fact_888_psubset__insert__iff,axiom,
! [A: set_a,X4: a,B: set_a] :
( ( ord_less_set_a @ A @ ( insert_a @ X4 @ B ) )
= ( ( ( member_a @ X4 @ B )
=> ( ord_less_set_a @ A @ B ) )
& ( ~ ( member_a @ X4 @ B )
=> ( ( ( member_a @ X4 @ A )
=> ( ord_less_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ X4 @ bot_bot_set_a ) ) @ B ) )
& ( ~ ( member_a @ X4 @ A )
=> ( ord_less_eq_set_a @ A @ B ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_889_psubset__insert__iff,axiom,
! [A: set_li6526943997496501093_a_nat,X4: list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ord_le5291801191193052689_a_nat @ A @ ( insert2950094090816004437_a_nat @ X4 @ B ) )
= ( ( ( member408289922725080238_a_nat @ X4 @ B )
=> ( ord_le5291801191193052689_a_nat @ A @ B ) )
& ( ~ ( member408289922725080238_a_nat @ X4 @ B )
=> ( ( ( member408289922725080238_a_nat @ X4 @ A )
=> ( ord_le5291801191193052689_a_nat @ ( minus_7395159227704179404_a_nat @ A @ ( insert2950094090816004437_a_nat @ X4 @ bot_bo1033123847703346641_a_nat ) ) @ B ) )
& ( ~ ( member408289922725080238_a_nat @ X4 @ A )
=> ( ord_le1147066620699065093_a_nat @ A @ B ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_890_Set_Omember__remove,axiom,
! [X4: list_Sum_sum_a_nat,Y: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ ( remove5086202153292001386_a_nat @ Y @ A ) )
= ( ( member408289922725080238_a_nat @ X4 @ A )
& ( X4 != Y ) ) ) ).
% Set.member_remove
thf(fact_891_Set_Omember__remove,axiom,
! [X4: a,Y: a,A: set_a] :
( ( member_a @ X4 @ ( remove_a @ Y @ A ) )
= ( ( member_a @ X4 @ A )
& ( X4 != Y ) ) ) ).
% Set.member_remove
thf(fact_892_psubsetI,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_a @ A @ B ) ) ) ).
% psubsetI
thf(fact_893_psubsetI,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A @ B )
=> ( ( A != B )
=> ( ord_le5291801191193052689_a_nat @ A @ B ) ) ) ).
% psubsetI
thf(fact_894_psubsetD,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,C: list_Sum_sum_a_nat] :
( ( ord_le5291801191193052689_a_nat @ A @ B )
=> ( ( member408289922725080238_a_nat @ C @ A )
=> ( member408289922725080238_a_nat @ C @ B ) ) ) ).
% psubsetD
thf(fact_895_psubsetD,axiom,
! [A: set_a,B: set_a,C: a] :
( ( ord_less_set_a @ A @ B )
=> ( ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% psubsetD
thf(fact_896_order__le__imp__less__or__eq,axiom,
! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ( ord_less_set_a @ X4 @ Y )
| ( X4 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_897_order__le__imp__less__or__eq,axiom,
! [X4: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ X4 @ Y )
=> ( ( ord_le5291801191193052689_a_nat @ X4 @ Y )
| ( X4 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_898_order__less__le__subst1,axiom,
! [A2: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_899_order__less__le__subst1,axiom,
! [A2: set_li6526943997496501093_a_nat,F: set_a > set_li6526943997496501093_a_nat,B2: set_a,C: set_a] :
( ( ord_le5291801191193052689_a_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_le1147066620699065093_a_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le5291801191193052689_a_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_900_order__less__le__subst1,axiom,
! [A2: set_a,F: set_li6526943997496501093_a_nat > set_a,B2: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat] :
( ( ord_less_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_le1147066620699065093_a_nat @ B2 @ C )
=> ( ! [X3: set_li6526943997496501093_a_nat,Y3: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_901_order__less__le__subst1,axiom,
! [A2: set_li6526943997496501093_a_nat,F: set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat] :
( ( ord_le5291801191193052689_a_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_le1147066620699065093_a_nat @ B2 @ C )
=> ( ! [X3: set_li6526943997496501093_a_nat,Y3: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ X3 @ Y3 )
=> ( ord_le1147066620699065093_a_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le5291801191193052689_a_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_902_order__le__less__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ ( F @ B2 ) @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_903_order__le__less__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_le5291801191193052689_a_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_le1147066620699065093_a_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le5291801191193052689_a_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_904_order__le__less__subst2,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,F: set_li6526943997496501093_a_nat > set_a,C: set_a] :
( ( ord_le1147066620699065093_a_nat @ A2 @ B2 )
=> ( ( ord_less_set_a @ ( F @ B2 ) @ C )
=> ( ! [X3: set_li6526943997496501093_a_nat,Y3: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_905_order__le__less__subst2,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,F: set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A2 @ B2 )
=> ( ( ord_le5291801191193052689_a_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: set_li6526943997496501093_a_nat,Y3: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ X3 @ Y3 )
=> ( ord_le1147066620699065093_a_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le5291801191193052689_a_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_906_order__less__le__trans,axiom,
! [X4: set_a,Y: set_a,Z: set_a] :
( ( ord_less_set_a @ X4 @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z )
=> ( ord_less_set_a @ X4 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_907_order__less__le__trans,axiom,
! [X4: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat,Z: set_li6526943997496501093_a_nat] :
( ( ord_le5291801191193052689_a_nat @ X4 @ Y )
=> ( ( ord_le1147066620699065093_a_nat @ Y @ Z )
=> ( ord_le5291801191193052689_a_nat @ X4 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_908_order__le__less__trans,axiom,
! [X4: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ( ord_less_set_a @ Y @ Z )
=> ( ord_less_set_a @ X4 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_909_order__le__less__trans,axiom,
! [X4: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat,Z: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ X4 @ Y )
=> ( ( ord_le5291801191193052689_a_nat @ Y @ Z )
=> ( ord_le5291801191193052689_a_nat @ X4 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_910_order__neq__le__trans,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 != B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_set_a @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_911_order__neq__le__trans,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( A2 != B2 )
=> ( ( ord_le1147066620699065093_a_nat @ A2 @ B2 )
=> ( ord_le5291801191193052689_a_nat @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_912_order__le__neq__trans,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_a @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_913_order__le__neq__trans,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_le5291801191193052689_a_nat @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_914_order__less__imp__le,axiom,
! [X4: set_a,Y: set_a] :
( ( ord_less_set_a @ X4 @ Y )
=> ( ord_less_eq_set_a @ X4 @ Y ) ) ).
% order_less_imp_le
thf(fact_915_order__less__imp__le,axiom,
! [X4: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat] :
( ( ord_le5291801191193052689_a_nat @ X4 @ Y )
=> ( ord_le1147066620699065093_a_nat @ X4 @ Y ) ) ).
% order_less_imp_le
thf(fact_916_order__less__le,axiom,
( ord_less_set_a
= ( ^ [X5: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y2 )
& ( X5 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_917_order__less__le,axiom,
( ord_le5291801191193052689_a_nat
= ( ^ [X5: set_li6526943997496501093_a_nat,Y2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ X5 @ Y2 )
& ( X5 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_918_order__le__less,axiom,
( ord_less_eq_set_a
= ( ^ [X5: set_a,Y2: set_a] :
( ( ord_less_set_a @ X5 @ Y2 )
| ( X5 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_919_order__le__less,axiom,
( ord_le1147066620699065093_a_nat
= ( ^ [X5: set_li6526943997496501093_a_nat,Y2: set_li6526943997496501093_a_nat] :
( ( ord_le5291801191193052689_a_nat @ X5 @ Y2 )
| ( X5 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_920_dual__order_Ostrict__implies__order,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_set_a @ B2 @ A2 )
=> ( ord_less_eq_set_a @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_921_dual__order_Ostrict__implies__order,axiom,
! [B2: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( ord_le5291801191193052689_a_nat @ B2 @ A2 )
=> ( ord_le1147066620699065093_a_nat @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_922_order_Ostrict__implies__order,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_923_order_Ostrict__implies__order,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ord_le5291801191193052689_a_nat @ A2 @ B2 )
=> ( ord_le1147066620699065093_a_nat @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_924_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [B3: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A3 )
& ~ ( ord_less_eq_set_a @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_925_dual__order_Ostrict__iff__not,axiom,
( ord_le5291801191193052689_a_nat
= ( ^ [B3: set_li6526943997496501093_a_nat,A3: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ B3 @ A3 )
& ~ ( ord_le1147066620699065093_a_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_926_dual__order_Ostrict__trans2,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( ord_less_set_a @ B2 @ A2 )
=> ( ( ord_less_eq_set_a @ C @ B2 )
=> ( ord_less_set_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_927_dual__order_Ostrict__trans2,axiom,
! [B2: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat] :
( ( ord_le5291801191193052689_a_nat @ B2 @ A2 )
=> ( ( ord_le1147066620699065093_a_nat @ C @ B2 )
=> ( ord_le5291801191193052689_a_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_928_dual__order_Ostrict__trans1,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( ord_less_set_a @ C @ B2 )
=> ( ord_less_set_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_929_dual__order_Ostrict__trans1,axiom,
! [B2: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ B2 @ A2 )
=> ( ( ord_le5291801191193052689_a_nat @ C @ B2 )
=> ( ord_le5291801191193052689_a_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_930_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [B3: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_931_dual__order_Ostrict__iff__order,axiom,
( ord_le5291801191193052689_a_nat
= ( ^ [B3: set_li6526943997496501093_a_nat,A3: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_932_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A3: set_a] :
( ( ord_less_set_a @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_933_dual__order_Oorder__iff__strict,axiom,
( ord_le1147066620699065093_a_nat
= ( ^ [B3: set_li6526943997496501093_a_nat,A3: set_li6526943997496501093_a_nat] :
( ( ord_le5291801191193052689_a_nat @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_934_order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
& ~ ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_935_order_Ostrict__iff__not,axiom,
( ord_le5291801191193052689_a_nat
= ( ^ [A3: set_li6526943997496501093_a_nat,B3: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A3 @ B3 )
& ~ ( ord_le1147066620699065093_a_nat @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_936_order_Ostrict__trans2,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_set_a @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_937_order_Ostrict__trans2,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat] :
( ( ord_le5291801191193052689_a_nat @ A2 @ B2 )
=> ( ( ord_le1147066620699065093_a_nat @ B2 @ C )
=> ( ord_le5291801191193052689_a_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_938_order_Ostrict__trans1,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ord_less_set_a @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_939_order_Ostrict__trans1,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A2 @ B2 )
=> ( ( ord_le5291801191193052689_a_nat @ B2 @ C )
=> ( ord_le5291801191193052689_a_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_940_order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_941_order_Ostrict__iff__order,axiom,
( ord_le5291801191193052689_a_nat
= ( ^ [A3: set_li6526943997496501093_a_nat,B3: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_942_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_set_a @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_943_order_Oorder__iff__strict,axiom,
( ord_le1147066620699065093_a_nat
= ( ^ [A3: set_li6526943997496501093_a_nat,B3: set_li6526943997496501093_a_nat] :
( ( ord_le5291801191193052689_a_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_944_less__le__not__le,axiom,
( ord_less_set_a
= ( ^ [X5: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y2 )
& ~ ( ord_less_eq_set_a @ Y2 @ X5 ) ) ) ) ).
% less_le_not_le
thf(fact_945_less__le__not__le,axiom,
( ord_le5291801191193052689_a_nat
= ( ^ [X5: set_li6526943997496501093_a_nat,Y2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ X5 @ Y2 )
& ~ ( ord_le1147066620699065093_a_nat @ Y2 @ X5 ) ) ) ) ).
% less_le_not_le
thf(fact_946_antisym__conv2,axiom,
! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ( ~ ( ord_less_set_a @ X4 @ Y ) )
= ( X4 = Y ) ) ) ).
% antisym_conv2
thf(fact_947_antisym__conv2,axiom,
! [X4: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ X4 @ Y )
=> ( ( ~ ( ord_le5291801191193052689_a_nat @ X4 @ Y ) )
= ( X4 = Y ) ) ) ).
% antisym_conv2
thf(fact_948_antisym__conv1,axiom,
! [X4: set_a,Y: set_a] :
( ~ ( ord_less_set_a @ X4 @ Y )
=> ( ( ord_less_eq_set_a @ X4 @ Y )
= ( X4 = Y ) ) ) ).
% antisym_conv1
thf(fact_949_antisym__conv1,axiom,
! [X4: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat] :
( ~ ( ord_le5291801191193052689_a_nat @ X4 @ Y )
=> ( ( ord_le1147066620699065093_a_nat @ X4 @ Y )
= ( X4 = Y ) ) ) ).
% antisym_conv1
thf(fact_950_nless__le,axiom,
! [A2: set_a,B2: set_a] :
( ( ~ ( ord_less_set_a @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_set_a @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_951_nless__le,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ~ ( ord_le5291801191193052689_a_nat @ A2 @ B2 ) )
= ( ~ ( ord_le1147066620699065093_a_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_952_leD,axiom,
! [Y: set_a,X4: set_a] :
( ( ord_less_eq_set_a @ Y @ X4 )
=> ~ ( ord_less_set_a @ X4 @ Y ) ) ).
% leD
thf(fact_953_leD,axiom,
! [Y: set_li6526943997496501093_a_nat,X4: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ Y @ X4 )
=> ~ ( ord_le5291801191193052689_a_nat @ X4 @ Y ) ) ).
% leD
thf(fact_954_bot_Oextremum__strict,axiom,
! [A2: set_a] :
~ ( ord_less_set_a @ A2 @ bot_bot_set_a ) ).
% bot.extremum_strict
thf(fact_955_bot_Onot__eq__extremum,axiom,
! [A2: set_a] :
( ( A2 != bot_bot_set_a )
= ( ord_less_set_a @ bot_bot_set_a @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_956_not__psubset__empty,axiom,
! [A: set_a] :
~ ( ord_less_set_a @ A @ bot_bot_set_a ) ).
% not_psubset_empty
thf(fact_957_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_set_a @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_958_subset__iff__psubset__eq,axiom,
( ord_le1147066620699065093_a_nat
= ( ^ [A4: set_li6526943997496501093_a_nat,B4: set_li6526943997496501093_a_nat] :
( ( ord_le5291801191193052689_a_nat @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_959_subset__psubset__trans,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_set_a @ B @ C2 )
=> ( ord_less_set_a @ A @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_960_subset__psubset__trans,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,C2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A @ B )
=> ( ( ord_le5291801191193052689_a_nat @ B @ C2 )
=> ( ord_le5291801191193052689_a_nat @ A @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_961_subset__not__subset__eq,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
& ~ ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_962_subset__not__subset__eq,axiom,
( ord_le5291801191193052689_a_nat
= ( ^ [A4: set_li6526943997496501093_a_nat,B4: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A4 @ B4 )
& ~ ( ord_le1147066620699065093_a_nat @ B4 @ A4 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_963_psubset__subset__trans,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_set_a @ A @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_964_psubset__subset__trans,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,C2: set_li6526943997496501093_a_nat] :
( ( ord_le5291801191193052689_a_nat @ A @ B )
=> ( ( ord_le1147066620699065093_a_nat @ B @ C2 )
=> ( ord_le5291801191193052689_a_nat @ A @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_965_psubset__imp__subset,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% psubset_imp_subset
thf(fact_966_psubset__imp__subset,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ord_le5291801191193052689_a_nat @ A @ B )
=> ( ord_le1147066620699065093_a_nat @ A @ B ) ) ).
% psubset_imp_subset
thf(fact_967_psubset__eq,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% psubset_eq
thf(fact_968_psubset__eq,axiom,
( ord_le5291801191193052689_a_nat
= ( ^ [A4: set_li6526943997496501093_a_nat,B4: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% psubset_eq
thf(fact_969_psubsetE,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ~ ( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ B @ A ) ) ) ).
% psubsetE
thf(fact_970_psubsetE,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ord_le5291801191193052689_a_nat @ A @ B )
=> ~ ( ( ord_le1147066620699065093_a_nat @ A @ B )
=> ( ord_le1147066620699065093_a_nat @ B @ A ) ) ) ).
% psubsetE
thf(fact_971_less__supI1,axiom,
! [X4: set_a,A2: set_a,B2: set_a] :
( ( ord_less_set_a @ X4 @ A2 )
=> ( ord_less_set_a @ X4 @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% less_supI1
thf(fact_972_less__supI2,axiom,
! [X4: set_a,B2: set_a,A2: set_a] :
( ( ord_less_set_a @ X4 @ B2 )
=> ( ord_less_set_a @ X4 @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% less_supI2
thf(fact_973_sup_Oabsorb3,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_set_a @ B2 @ A2 )
=> ( ( sup_sup_set_a @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb3
thf(fact_974_sup_Oabsorb4,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( sup_sup_set_a @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb4
thf(fact_975_sup_Ostrict__boundedE,axiom,
! [B2: set_a,C: set_a,A2: set_a] :
( ( ord_less_set_a @ ( sup_sup_set_a @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_set_a @ B2 @ A2 )
=> ~ ( ord_less_set_a @ C @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_976_sup_Ostrict__order__iff,axiom,
( ord_less_set_a
= ( ^ [B3: set_a,A3: set_a] :
( ( A3
= ( sup_sup_set_a @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_977_sup_Ostrict__coboundedI1,axiom,
! [C: set_a,A2: set_a,B2: set_a] :
( ( ord_less_set_a @ C @ A2 )
=> ( ord_less_set_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_978_sup_Ostrict__coboundedI2,axiom,
! [C: set_a,B2: set_a,A2: set_a] :
( ( ord_less_set_a @ C @ B2 )
=> ( ord_less_set_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_979_less__infI1,axiom,
! [A2: set_a,X4: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ X4 )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ X4 ) ) ).
% less_infI1
thf(fact_980_less__infI2,axiom,
! [B2: set_a,X4: set_a,A2: set_a] :
( ( ord_less_set_a @ B2 @ X4 )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ X4 ) ) ).
% less_infI2
thf(fact_981_inf_Oabsorb3,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb3
thf(fact_982_inf_Oabsorb4,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_set_a @ B2 @ A2 )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb4
thf(fact_983_inf_Ostrict__boundedE,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) )
=> ~ ( ( ord_less_set_a @ A2 @ B2 )
=> ~ ( ord_less_set_a @ A2 @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_984_inf_Ostrict__order__iff,axiom,
( ord_less_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( A3
= ( inf_inf_set_a @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_985_inf_Ostrict__coboundedI1,axiom,
! [A2: set_a,C: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ C )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_986_inf_Ostrict__coboundedI2,axiom,
! [B2: set_a,C: set_a,A2: set_a] :
( ( ord_less_set_a @ B2 @ C )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_987_psubset__imp__ex__mem,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ? [B7: a] : ( member_a @ B7 @ ( minus_minus_set_a @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_988_psubset__imp__ex__mem,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ord_le5291801191193052689_a_nat @ A @ B )
=> ? [B7: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ B7 @ ( minus_7395159227704179404_a_nat @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_989_is__singleton__the__elem,axiom,
( is_singleton_a
= ( ^ [A4: set_a] :
( A4
= ( insert_a @ ( the_elem_a @ A4 ) @ bot_bot_set_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_990_is__singletonI_H,axiom,
! [A: set_li6526943997496501093_a_nat] :
( ( A != bot_bo1033123847703346641_a_nat )
=> ( ! [X3: list_Sum_sum_a_nat,Y3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ A )
=> ( ( member408289922725080238_a_nat @ Y3 @ A )
=> ( X3 = Y3 ) ) )
=> ( is_sin2231188923920309881_a_nat @ A ) ) ) ).
% is_singletonI'
thf(fact_991_is__singletonI_H,axiom,
! [A: set_a] :
( ( A != bot_bot_set_a )
=> ( ! [X3: a,Y3: a] :
( ( member_a @ X3 @ A )
=> ( ( member_a @ Y3 @ A )
=> ( X3 = Y3 ) ) )
=> ( is_singleton_a @ A ) ) ) ).
% is_singletonI'
thf(fact_992_bot__empty__eq,axiom,
( bot_bo9042073657639083596_nat_o
= ( ^ [X5: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X5 @ bot_bo1033123847703346641_a_nat ) ) ) ).
% bot_empty_eq
thf(fact_993_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X5: a] : ( member_a @ X5 @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_994_Collect__empty__eq__bot,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( P = bot_bot_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_995_pairwise__alt,axiom,
( pairwise_a
= ( ^ [R: a > a > $o,S2: set_a] :
! [X5: a] :
( ( member_a @ X5 @ S2 )
=> ! [Y2: a] :
( ( member_a @ Y2 @ ( minus_minus_set_a @ S2 @ ( insert_a @ X5 @ bot_bot_set_a ) ) )
=> ( R @ X5 @ Y2 ) ) ) ) ) ).
% pairwise_alt
thf(fact_996_pairwise__alt,axiom,
( pairwi4897900009783174640_a_nat
= ( ^ [R: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o,S2: set_li6526943997496501093_a_nat] :
! [X5: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X5 @ S2 )
=> ! [Y2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ Y2 @ ( minus_7395159227704179404_a_nat @ S2 @ ( insert2950094090816004437_a_nat @ X5 @ bot_bo1033123847703346641_a_nat ) ) )
=> ( R @ X5 @ Y2 ) ) ) ) ) ).
% pairwise_alt
thf(fact_997_subset__Compl__singleton,axiom,
! [A: set_a,B2: a] :
( ( ord_less_eq_set_a @ A @ ( uminus_uminus_set_a @ ( insert_a @ B2 @ bot_bot_set_a ) ) )
= ( ~ ( member_a @ B2 @ A ) ) ) ).
% subset_Compl_singleton
thf(fact_998_subset__Compl__singleton,axiom,
! [A: set_li6526943997496501093_a_nat,B2: list_Sum_sum_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A @ ( uminus2192744996606729052_a_nat @ ( insert2950094090816004437_a_nat @ B2 @ bot_bo1033123847703346641_a_nat ) ) )
= ( ~ ( member408289922725080238_a_nat @ B2 @ A ) ) ) ).
% subset_Compl_singleton
thf(fact_999_Compl__iff,axiom,
! [C: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ ( uminus2192744996606729052_a_nat @ A ) )
= ( ~ ( member408289922725080238_a_nat @ C @ A ) ) ) ).
% Compl_iff
thf(fact_1000_Compl__iff,axiom,
! [C: a,A: set_a] :
( ( member_a @ C @ ( uminus_uminus_set_a @ A ) )
= ( ~ ( member_a @ C @ A ) ) ) ).
% Compl_iff
thf(fact_1001_ComplI,axiom,
! [C: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ~ ( member408289922725080238_a_nat @ C @ A )
=> ( member408289922725080238_a_nat @ C @ ( uminus2192744996606729052_a_nat @ A ) ) ) ).
% ComplI
thf(fact_1002_ComplI,axiom,
! [C: a,A: set_a] :
( ~ ( member_a @ C @ A )
=> ( member_a @ C @ ( uminus_uminus_set_a @ A ) ) ) ).
% ComplI
thf(fact_1003_compl__le__compl__iff,axiom,
! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ X4 ) @ ( uminus_uminus_set_a @ Y ) )
= ( ord_less_eq_set_a @ Y @ X4 ) ) ).
% compl_le_compl_iff
thf(fact_1004_compl__le__compl__iff,axiom,
! [X4: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ ( uminus2192744996606729052_a_nat @ X4 ) @ ( uminus2192744996606729052_a_nat @ Y ) )
= ( ord_le1147066620699065093_a_nat @ Y @ X4 ) ) ).
% compl_le_compl_iff
thf(fact_1005_Compl__subset__Compl__iff,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ A ) @ ( uminus_uminus_set_a @ B ) )
= ( ord_less_eq_set_a @ B @ A ) ) ).
% Compl_subset_Compl_iff
thf(fact_1006_Compl__subset__Compl__iff,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ ( uminus2192744996606729052_a_nat @ A ) @ ( uminus2192744996606729052_a_nat @ B ) )
= ( ord_le1147066620699065093_a_nat @ B @ A ) ) ).
% Compl_subset_Compl_iff
thf(fact_1007_Compl__anti__mono,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ B ) @ ( uminus_uminus_set_a @ A ) ) ) ).
% Compl_anti_mono
thf(fact_1008_Compl__anti__mono,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A @ B )
=> ( ord_le1147066620699065093_a_nat @ ( uminus2192744996606729052_a_nat @ B ) @ ( uminus2192744996606729052_a_nat @ A ) ) ) ).
% Compl_anti_mono
thf(fact_1009_inf__compl__bot__left1,axiom,
! [X4: set_a,Y: set_a] :
( ( inf_inf_set_a @ ( uminus_uminus_set_a @ X4 ) @ ( inf_inf_set_a @ X4 @ Y ) )
= bot_bot_set_a ) ).
% inf_compl_bot_left1
thf(fact_1010_inf__compl__bot__left2,axiom,
! [X4: set_a,Y: set_a] :
( ( inf_inf_set_a @ X4 @ ( inf_inf_set_a @ ( uminus_uminus_set_a @ X4 ) @ Y ) )
= bot_bot_set_a ) ).
% inf_compl_bot_left2
thf(fact_1011_inf__compl__bot__right,axiom,
! [X4: set_a,Y: set_a] :
( ( inf_inf_set_a @ X4 @ ( inf_inf_set_a @ Y @ ( uminus_uminus_set_a @ X4 ) ) )
= bot_bot_set_a ) ).
% inf_compl_bot_right
thf(fact_1012_boolean__algebra_Oconj__cancel__left,axiom,
! [X4: set_a] :
( ( inf_inf_set_a @ ( uminus_uminus_set_a @ X4 ) @ X4 )
= bot_bot_set_a ) ).
% boolean_algebra.conj_cancel_left
thf(fact_1013_boolean__algebra_Oconj__cancel__right,axiom,
! [X4: set_a] :
( ( inf_inf_set_a @ X4 @ ( uminus_uminus_set_a @ X4 ) )
= bot_bot_set_a ) ).
% boolean_algebra.conj_cancel_right
thf(fact_1014_boolean__algebra_Ode__Morgan__conj,axiom,
! [X4: set_a,Y: set_a] :
( ( uminus_uminus_set_a @ ( inf_inf_set_a @ X4 @ Y ) )
= ( sup_sup_set_a @ ( uminus_uminus_set_a @ X4 ) @ ( uminus_uminus_set_a @ Y ) ) ) ).
% boolean_algebra.de_Morgan_conj
thf(fact_1015_boolean__algebra_Ode__Morgan__disj,axiom,
! [X4: set_a,Y: set_a] :
( ( uminus_uminus_set_a @ ( sup_sup_set_a @ X4 @ Y ) )
= ( inf_inf_set_a @ ( uminus_uminus_set_a @ X4 ) @ ( uminus_uminus_set_a @ Y ) ) ) ).
% boolean_algebra.de_Morgan_disj
thf(fact_1016_Compl__disjoint,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ ( uminus_uminus_set_a @ A ) )
= bot_bot_set_a ) ).
% Compl_disjoint
thf(fact_1017_Compl__disjoint2,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ ( uminus_uminus_set_a @ A ) @ A )
= bot_bot_set_a ) ).
% Compl_disjoint2
thf(fact_1018_Compl__Diff__eq,axiom,
! [A: set_a,B: set_a] :
( ( uminus_uminus_set_a @ ( minus_minus_set_a @ A @ B ) )
= ( sup_sup_set_a @ ( uminus_uminus_set_a @ A ) @ B ) ) ).
% Compl_Diff_eq
thf(fact_1019_Compl__Diff__eq,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( uminus2192744996606729052_a_nat @ ( minus_7395159227704179404_a_nat @ A @ B ) )
= ( sup_su4083067149120280889_a_nat @ ( uminus2192744996606729052_a_nat @ A ) @ B ) ) ).
% Compl_Diff_eq
thf(fact_1020_Diff__Compl,axiom,
! [A: set_a,B: set_a] :
( ( minus_minus_set_a @ A @ ( uminus_uminus_set_a @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ).
% Diff_Compl
thf(fact_1021_Diff__Compl,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( minus_7395159227704179404_a_nat @ A @ ( uminus2192744996606729052_a_nat @ B ) )
= ( inf_in3249246906714053971_a_nat @ A @ B ) ) ).
% Diff_Compl
thf(fact_1022_compl__mono,axiom,
! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ Y ) @ ( uminus_uminus_set_a @ X4 ) ) ) ).
% compl_mono
thf(fact_1023_compl__mono,axiom,
! [X4: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ X4 @ Y )
=> ( ord_le1147066620699065093_a_nat @ ( uminus2192744996606729052_a_nat @ Y ) @ ( uminus2192744996606729052_a_nat @ X4 ) ) ) ).
% compl_mono
thf(fact_1024_compl__le__swap1,axiom,
! [Y: set_a,X4: set_a] :
( ( ord_less_eq_set_a @ Y @ ( uminus_uminus_set_a @ X4 ) )
=> ( ord_less_eq_set_a @ X4 @ ( uminus_uminus_set_a @ Y ) ) ) ).
% compl_le_swap1
thf(fact_1025_compl__le__swap1,axiom,
! [Y: set_li6526943997496501093_a_nat,X4: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ Y @ ( uminus2192744996606729052_a_nat @ X4 ) )
=> ( ord_le1147066620699065093_a_nat @ X4 @ ( uminus2192744996606729052_a_nat @ Y ) ) ) ).
% compl_le_swap1
thf(fact_1026_compl__le__swap2,axiom,
! [Y: set_a,X4: set_a] :
( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ Y ) @ X4 )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ X4 ) @ Y ) ) ).
% compl_le_swap2
thf(fact_1027_compl__le__swap2,axiom,
! [Y: set_li6526943997496501093_a_nat,X4: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ ( uminus2192744996606729052_a_nat @ Y ) @ X4 )
=> ( ord_le1147066620699065093_a_nat @ ( uminus2192744996606729052_a_nat @ X4 ) @ Y ) ) ).
% compl_le_swap2
thf(fact_1028_pairwiseI,axiom,
! [S: set_li6526943997496501093_a_nat,R2: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o] :
( ! [X3: list_Sum_sum_a_nat,Y3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ S )
=> ( ( member408289922725080238_a_nat @ Y3 @ S )
=> ( ( X3 != Y3 )
=> ( R2 @ X3 @ Y3 ) ) ) )
=> ( pairwi4897900009783174640_a_nat @ R2 @ S ) ) ).
% pairwiseI
thf(fact_1029_pairwiseI,axiom,
! [S: set_a,R2: a > a > $o] :
( ! [X3: a,Y3: a] :
( ( member_a @ X3 @ S )
=> ( ( member_a @ Y3 @ S )
=> ( ( X3 != Y3 )
=> ( R2 @ X3 @ Y3 ) ) ) )
=> ( pairwise_a @ R2 @ S ) ) ).
% pairwiseI
thf(fact_1030_pairwiseD,axiom,
! [R2: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o,S: set_li6526943997496501093_a_nat,X4: list_Sum_sum_a_nat,Y: list_Sum_sum_a_nat] :
( ( pairwi4897900009783174640_a_nat @ R2 @ S )
=> ( ( member408289922725080238_a_nat @ X4 @ S )
=> ( ( member408289922725080238_a_nat @ Y @ S )
=> ( ( X4 != Y )
=> ( R2 @ X4 @ Y ) ) ) ) ) ).
% pairwiseD
thf(fact_1031_pairwiseD,axiom,
! [R2: a > a > $o,S: set_a,X4: a,Y: a] :
( ( pairwise_a @ R2 @ S )
=> ( ( member_a @ X4 @ S )
=> ( ( member_a @ Y @ S )
=> ( ( X4 != Y )
=> ( R2 @ X4 @ Y ) ) ) ) ) ).
% pairwiseD
thf(fact_1032_ComplD,axiom,
! [C: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ ( uminus2192744996606729052_a_nat @ A ) )
=> ~ ( member408289922725080238_a_nat @ C @ A ) ) ).
% ComplD
thf(fact_1033_ComplD,axiom,
! [C: a,A: set_a] :
( ( member_a @ C @ ( uminus_uminus_set_a @ A ) )
=> ~ ( member_a @ C @ A ) ) ).
% ComplD
thf(fact_1034_pairwise__empty,axiom,
! [P: a > a > $o] : ( pairwise_a @ P @ bot_bot_set_a ) ).
% pairwise_empty
thf(fact_1035_pairwise__subset,axiom,
! [P: a > a > $o,S: set_a,T: set_a] :
( ( pairwise_a @ P @ S )
=> ( ( ord_less_eq_set_a @ T @ S )
=> ( pairwise_a @ P @ T ) ) ) ).
% pairwise_subset
thf(fact_1036_pairwise__subset,axiom,
! [P: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o,S: set_li6526943997496501093_a_nat,T: set_li6526943997496501093_a_nat] :
( ( pairwi4897900009783174640_a_nat @ P @ S )
=> ( ( ord_le1147066620699065093_a_nat @ T @ S )
=> ( pairwi4897900009783174640_a_nat @ P @ T ) ) ) ).
% pairwise_subset
thf(fact_1037_pairwise__mono,axiom,
! [P: a > a > $o,A: set_a,Q: a > a > $o,B: set_a] :
( ( pairwise_a @ P @ A )
=> ( ! [X3: a,Y3: a] :
( ( P @ X3 @ Y3 )
=> ( Q @ X3 @ Y3 ) )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( pairwise_a @ Q @ B ) ) ) ) ).
% pairwise_mono
thf(fact_1038_pairwise__mono,axiom,
! [P: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o,A: set_li6526943997496501093_a_nat,Q: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o,B: set_li6526943997496501093_a_nat] :
( ( pairwi4897900009783174640_a_nat @ P @ A )
=> ( ! [X3: list_Sum_sum_a_nat,Y3: list_Sum_sum_a_nat] :
( ( P @ X3 @ Y3 )
=> ( Q @ X3 @ Y3 ) )
=> ( ( ord_le1147066620699065093_a_nat @ B @ A )
=> ( pairwi4897900009783174640_a_nat @ Q @ B ) ) ) ) ).
% pairwise_mono
thf(fact_1039_pairwise__insert,axiom,
! [R3: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o,X4: list_Sum_sum_a_nat,S3: set_li6526943997496501093_a_nat] :
( ( pairwi4897900009783174640_a_nat @ R3 @ ( insert2950094090816004437_a_nat @ X4 @ S3 ) )
= ( ! [Y2: list_Sum_sum_a_nat] :
( ( ( member408289922725080238_a_nat @ Y2 @ S3 )
& ( Y2 != X4 ) )
=> ( ( R3 @ X4 @ Y2 )
& ( R3 @ Y2 @ X4 ) ) )
& ( pairwi4897900009783174640_a_nat @ R3 @ S3 ) ) ) ).
% pairwise_insert
thf(fact_1040_pairwise__insert,axiom,
! [R3: a > a > $o,X4: a,S3: set_a] :
( ( pairwise_a @ R3 @ ( insert_a @ X4 @ S3 ) )
= ( ! [Y2: a] :
( ( ( member_a @ Y2 @ S3 )
& ( Y2 != X4 ) )
=> ( ( R3 @ X4 @ Y2 )
& ( R3 @ Y2 @ X4 ) ) )
& ( pairwise_a @ R3 @ S3 ) ) ) ).
% pairwise_insert
thf(fact_1041_diff__eq,axiom,
( minus_minus_set_a
= ( ^ [X5: set_a,Y2: set_a] : ( inf_inf_set_a @ X5 @ ( uminus_uminus_set_a @ Y2 ) ) ) ) ).
% diff_eq
thf(fact_1042_diff__eq,axiom,
( minus_7395159227704179404_a_nat
= ( ^ [X5: set_li6526943997496501093_a_nat,Y2: set_li6526943997496501093_a_nat] : ( inf_in3249246906714053971_a_nat @ X5 @ ( uminus2192744996606729052_a_nat @ Y2 ) ) ) ) ).
% diff_eq
thf(fact_1043_inf__cancel__left1,axiom,
! [X4: set_a,A2: set_a,B2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X4 @ A2 ) @ ( inf_inf_set_a @ ( uminus_uminus_set_a @ X4 ) @ B2 ) )
= bot_bot_set_a ) ).
% inf_cancel_left1
thf(fact_1044_inf__cancel__left2,axiom,
! [X4: set_a,A2: set_a,B2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ ( uminus_uminus_set_a @ X4 ) @ A2 ) @ ( inf_inf_set_a @ X4 @ B2 ) )
= bot_bot_set_a ) ).
% inf_cancel_left2
thf(fact_1045_subset__Compl__self__eq,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ ( uminus_uminus_set_a @ A ) )
= ( A = bot_bot_set_a ) ) ).
% subset_Compl_self_eq
thf(fact_1046_subset__Compl__self__eq,axiom,
! [A: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A @ ( uminus2192744996606729052_a_nat @ A ) )
= ( A = bot_bo1033123847703346641_a_nat ) ) ).
% subset_Compl_self_eq
thf(fact_1047_Compl__Un,axiom,
! [A: set_a,B: set_a] :
( ( uminus_uminus_set_a @ ( sup_sup_set_a @ A @ B ) )
= ( inf_inf_set_a @ ( uminus_uminus_set_a @ A ) @ ( uminus_uminus_set_a @ B ) ) ) ).
% Compl_Un
thf(fact_1048_Compl__Int,axiom,
! [A: set_a,B: set_a] :
( ( uminus_uminus_set_a @ ( inf_inf_set_a @ A @ B ) )
= ( sup_sup_set_a @ ( uminus_uminus_set_a @ A ) @ ( uminus_uminus_set_a @ B ) ) ) ).
% Compl_Int
thf(fact_1049_Diff__eq,axiom,
( minus_minus_set_a
= ( ^ [A4: set_a,B4: set_a] : ( inf_inf_set_a @ A4 @ ( uminus_uminus_set_a @ B4 ) ) ) ) ).
% Diff_eq
thf(fact_1050_Diff__eq,axiom,
( minus_7395159227704179404_a_nat
= ( ^ [A4: set_li6526943997496501093_a_nat,B4: set_li6526943997496501093_a_nat] : ( inf_in3249246906714053971_a_nat @ A4 @ ( uminus2192744996606729052_a_nat @ B4 ) ) ) ) ).
% Diff_eq
thf(fact_1051_pairwise__singleton,axiom,
! [P: a > a > $o,A: a] : ( pairwise_a @ P @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% pairwise_singleton
thf(fact_1052_inf__shunt,axiom,
! [X4: set_a,Y: set_a] :
( ( ( inf_inf_set_a @ X4 @ Y )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ X4 @ ( uminus_uminus_set_a @ Y ) ) ) ).
% inf_shunt
thf(fact_1053_inf__shunt,axiom,
! [X4: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat] :
( ( ( inf_in3249246906714053971_a_nat @ X4 @ Y )
= bot_bo1033123847703346641_a_nat )
= ( ord_le1147066620699065093_a_nat @ X4 @ ( uminus2192744996606729052_a_nat @ Y ) ) ) ).
% inf_shunt
thf(fact_1054_shunt1,axiom,
! [X4: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ ( inf_inf_set_a @ X4 @ Y ) @ Z )
= ( ord_less_eq_set_a @ X4 @ ( sup_sup_set_a @ ( uminus_uminus_set_a @ Y ) @ Z ) ) ) ).
% shunt1
thf(fact_1055_shunt1,axiom,
! [X4: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat,Z: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ ( inf_in3249246906714053971_a_nat @ X4 @ Y ) @ Z )
= ( ord_le1147066620699065093_a_nat @ X4 @ ( sup_su4083067149120280889_a_nat @ ( uminus2192744996606729052_a_nat @ Y ) @ Z ) ) ) ).
% shunt1
thf(fact_1056_shunt2,axiom,
! [X4: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ ( inf_inf_set_a @ X4 @ ( uminus_uminus_set_a @ Y ) ) @ Z )
= ( ord_less_eq_set_a @ X4 @ ( sup_sup_set_a @ Y @ Z ) ) ) ).
% shunt2
thf(fact_1057_shunt2,axiom,
! [X4: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat,Z: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ ( inf_in3249246906714053971_a_nat @ X4 @ ( uminus2192744996606729052_a_nat @ Y ) ) @ Z )
= ( ord_le1147066620699065093_a_nat @ X4 @ ( sup_su4083067149120280889_a_nat @ Y @ Z ) ) ) ).
% shunt2
thf(fact_1058_sup__neg__inf,axiom,
! [P2: set_a,Q2: set_a,R3: set_a] :
( ( ord_less_eq_set_a @ P2 @ ( sup_sup_set_a @ Q2 @ R3 ) )
= ( ord_less_eq_set_a @ ( inf_inf_set_a @ P2 @ ( uminus_uminus_set_a @ Q2 ) ) @ R3 ) ) ).
% sup_neg_inf
thf(fact_1059_sup__neg__inf,axiom,
! [P2: set_li6526943997496501093_a_nat,Q2: set_li6526943997496501093_a_nat,R3: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ P2 @ ( sup_su4083067149120280889_a_nat @ Q2 @ R3 ) )
= ( ord_le1147066620699065093_a_nat @ ( inf_in3249246906714053971_a_nat @ P2 @ ( uminus2192744996606729052_a_nat @ Q2 ) ) @ R3 ) ) ).
% sup_neg_inf
thf(fact_1060_disjoint__eq__subset__Compl,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A @ ( uminus_uminus_set_a @ B ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_1061_disjoint__eq__subset__Compl,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ( inf_in3249246906714053971_a_nat @ A @ B )
= bot_bo1033123847703346641_a_nat )
= ( ord_le1147066620699065093_a_nat @ A @ ( uminus2192744996606729052_a_nat @ B ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_1062_Compl__insert,axiom,
! [X4: a,A: set_a] :
( ( uminus_uminus_set_a @ ( insert_a @ X4 @ A ) )
= ( minus_minus_set_a @ ( uminus_uminus_set_a @ A ) @ ( insert_a @ X4 @ bot_bot_set_a ) ) ) ).
% Compl_insert
thf(fact_1063_Compl__insert,axiom,
! [X4: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( uminus2192744996606729052_a_nat @ ( insert2950094090816004437_a_nat @ X4 @ A ) )
= ( minus_7395159227704179404_a_nat @ ( uminus2192744996606729052_a_nat @ A ) @ ( insert2950094090816004437_a_nat @ X4 @ bot_bo1033123847703346641_a_nat ) ) ) ).
% Compl_insert
thf(fact_1064_verit__comp__simplify1_I2_J,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_1065_verit__comp__simplify1_I2_J,axiom,
! [A2: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_1066_Set__minus__code_I1_J,axiom,
( minus_minus_set_a
= ( ^ [A4: set_a,B4: set_a] : ( inf_inf_set_a @ A4 @ ( uminus_uminus_set_a @ B4 ) ) ) ) ).
% Set_minus_code(1)
thf(fact_1067_Set__minus__code_I1_J,axiom,
( minus_7395159227704179404_a_nat
= ( ^ [A4: set_li6526943997496501093_a_nat,B4: set_li6526943997496501093_a_nat] : ( inf_in3249246906714053971_a_nat @ A4 @ ( uminus2192744996606729052_a_nat @ B4 ) ) ) ) ).
% Set_minus_code(1)
thf(fact_1068_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
! [X4: set_a,Y: set_a] :
( ( ( inf_inf_set_a @ X4 @ Y )
= bot_bot_set_a )
=> ( ( ( sup_sup_set_a @ X4 @ Y )
= top_top_set_a )
=> ( ( uminus_uminus_set_a @ X4 )
= Y ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_1069_inf_Osemilattice__order__axioms,axiom,
semila4706084620769370446_set_a @ inf_inf_set_a @ ord_less_eq_set_a @ ord_less_set_a ).
% inf.semilattice_order_axioms
thf(fact_1070_inf_Osemilattice__order__axioms,axiom,
semila5844162609975154773_a_nat @ inf_in3249246906714053971_a_nat @ ord_le1147066620699065093_a_nat @ ord_le5291801191193052689_a_nat ).
% inf.semilattice_order_axioms
thf(fact_1071_Int__UNIV,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= top_top_set_a )
= ( ( A = top_top_set_a )
& ( B = top_top_set_a ) ) ) ).
% Int_UNIV
thf(fact_1072_sup__top__left,axiom,
! [X4: set_a] :
( ( sup_sup_set_a @ top_top_set_a @ X4 )
= top_top_set_a ) ).
% sup_top_left
thf(fact_1073_sup__top__right,axiom,
! [X4: set_a] :
( ( sup_sup_set_a @ X4 @ top_top_set_a )
= top_top_set_a ) ).
% sup_top_right
thf(fact_1074_boolean__algebra_Odisj__one__left,axiom,
! [X4: set_a] :
( ( sup_sup_set_a @ top_top_set_a @ X4 )
= top_top_set_a ) ).
% boolean_algebra.disj_one_left
thf(fact_1075_boolean__algebra_Odisj__one__right,axiom,
! [X4: set_a] :
( ( sup_sup_set_a @ X4 @ top_top_set_a )
= top_top_set_a ) ).
% boolean_algebra.disj_one_right
thf(fact_1076_inf__top__left,axiom,
! [X4: set_a] :
( ( inf_inf_set_a @ top_top_set_a @ X4 )
= X4 ) ).
% inf_top_left
thf(fact_1077_inf__top__right,axiom,
! [X4: set_a] :
( ( inf_inf_set_a @ X4 @ top_top_set_a )
= X4 ) ).
% inf_top_right
thf(fact_1078_inf__eq__top__iff,axiom,
! [X4: set_a,Y: set_a] :
( ( ( inf_inf_set_a @ X4 @ Y )
= top_top_set_a )
= ( ( X4 = top_top_set_a )
& ( Y = top_top_set_a ) ) ) ).
% inf_eq_top_iff
thf(fact_1079_top__eq__inf__iff,axiom,
! [X4: set_a,Y: set_a] :
( ( top_top_set_a
= ( inf_inf_set_a @ X4 @ Y ) )
= ( ( X4 = top_top_set_a )
& ( Y = top_top_set_a ) ) ) ).
% top_eq_inf_iff
thf(fact_1080_inf__top_Oeq__neutr__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ A2 @ B2 )
= top_top_set_a )
= ( ( A2 = top_top_set_a )
& ( B2 = top_top_set_a ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_1081_inf__top_Oleft__neutral,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ top_top_set_a @ A2 )
= A2 ) ).
% inf_top.left_neutral
thf(fact_1082_inf__top_Oneutr__eq__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( top_top_set_a
= ( inf_inf_set_a @ A2 @ B2 ) )
= ( ( A2 = top_top_set_a )
& ( B2 = top_top_set_a ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_1083_inf__top_Oright__neutral,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ top_top_set_a )
= A2 ) ).
% inf_top.right_neutral
thf(fact_1084_Diff__UNIV,axiom,
! [A: set_a] :
( ( minus_minus_set_a @ A @ top_top_set_a )
= bot_bot_set_a ) ).
% Diff_UNIV
thf(fact_1085_Diff__UNIV,axiom,
! [A: set_li6526943997496501093_a_nat] :
( ( minus_7395159227704179404_a_nat @ A @ top_to6433055325616222389_a_nat )
= bot_bo1033123847703346641_a_nat ) ).
% Diff_UNIV
thf(fact_1086_boolean__algebra_Ocompl__one,axiom,
( ( uminus_uminus_set_a @ top_top_set_a )
= bot_bot_set_a ) ).
% boolean_algebra.compl_one
thf(fact_1087_boolean__algebra_Ocompl__zero,axiom,
( ( uminus_uminus_set_a @ bot_bot_set_a )
= top_top_set_a ) ).
% boolean_algebra.compl_zero
thf(fact_1088_boolean__algebra_Odisj__cancel__right,axiom,
! [X4: set_a] :
( ( sup_sup_set_a @ X4 @ ( uminus_uminus_set_a @ X4 ) )
= top_top_set_a ) ).
% boolean_algebra.disj_cancel_right
thf(fact_1089_boolean__algebra_Odisj__cancel__left,axiom,
! [X4: set_a] :
( ( sup_sup_set_a @ ( uminus_uminus_set_a @ X4 ) @ X4 )
= top_top_set_a ) ).
% boolean_algebra.disj_cancel_left
thf(fact_1090_sup__compl__top__left2,axiom,
! [X4: set_a,Y: set_a] :
( ( sup_sup_set_a @ X4 @ ( sup_sup_set_a @ ( uminus_uminus_set_a @ X4 ) @ Y ) )
= top_top_set_a ) ).
% sup_compl_top_left2
thf(fact_1091_sup__compl__top__left1,axiom,
! [X4: set_a,Y: set_a] :
( ( sup_sup_set_a @ ( uminus_uminus_set_a @ X4 ) @ ( sup_sup_set_a @ X4 @ Y ) )
= top_top_set_a ) ).
% sup_compl_top_left1
thf(fact_1092_top_Oextremum__uniqueI,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A2 )
=> ( A2 = top_top_set_a ) ) ).
% top.extremum_uniqueI
thf(fact_1093_top_Oextremum__uniqueI,axiom,
! [A2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ top_to6433055325616222389_a_nat @ A2 )
=> ( A2 = top_to6433055325616222389_a_nat ) ) ).
% top.extremum_uniqueI
thf(fact_1094_top_Oextremum__unique,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A2 )
= ( A2 = top_top_set_a ) ) ).
% top.extremum_unique
thf(fact_1095_top_Oextremum__unique,axiom,
! [A2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ top_to6433055325616222389_a_nat @ A2 )
= ( A2 = top_to6433055325616222389_a_nat ) ) ).
% top.extremum_unique
thf(fact_1096_top__greatest,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ top_top_set_a ) ).
% top_greatest
thf(fact_1097_top__greatest,axiom,
! [A2: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ A2 @ top_to6433055325616222389_a_nat ) ).
% top_greatest
thf(fact_1098_Int__UNIV__left,axiom,
! [B: set_a] :
( ( inf_inf_set_a @ top_top_set_a @ B )
= B ) ).
% Int_UNIV_left
thf(fact_1099_Int__UNIV__right,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ top_top_set_a )
= A ) ).
% Int_UNIV_right
thf(fact_1100_Un__UNIV__left,axiom,
! [B: set_a] :
( ( sup_sup_set_a @ top_top_set_a @ B )
= top_top_set_a ) ).
% Un_UNIV_left
thf(fact_1101_Un__UNIV__right,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ top_top_set_a )
= top_top_set_a ) ).
% Un_UNIV_right
thf(fact_1102_subset__UNIV,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ top_top_set_a ) ).
% subset_UNIV
thf(fact_1103_subset__UNIV,axiom,
! [A: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ A @ top_to6433055325616222389_a_nat ) ).
% subset_UNIV
thf(fact_1104_empty__not__UNIV,axiom,
bot_bot_set_a != top_top_set_a ).
% empty_not_UNIV
thf(fact_1105_boolean__algebra_Oconj__one__right,axiom,
! [X4: set_a] :
( ( inf_inf_set_a @ X4 @ top_top_set_a )
= X4 ) ).
% boolean_algebra.conj_one_right
thf(fact_1106_UNIV__code,axiom,
( top_top_set_a
= ( uminus_uminus_set_a @ bot_bot_set_a ) ) ).
% UNIV_code
thf(fact_1107_sup__cancel__left2,axiom,
! [X4: set_a,A2: set_a,B2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ ( uminus_uminus_set_a @ X4 ) @ A2 ) @ ( sup_sup_set_a @ X4 @ B2 ) )
= top_top_set_a ) ).
% sup_cancel_left2
thf(fact_1108_sup__cancel__left1,axiom,
! [X4: set_a,A2: set_a,B2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X4 @ A2 ) @ ( sup_sup_set_a @ ( uminus_uminus_set_a @ X4 ) @ B2 ) )
= top_top_set_a ) ).
% sup_cancel_left1
thf(fact_1109_Compl__UNIV__eq,axiom,
( ( uminus_uminus_set_a @ top_top_set_a )
= bot_bot_set_a ) ).
% Compl_UNIV_eq
thf(fact_1110_Compl__partition,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ ( uminus_uminus_set_a @ A ) )
= top_top_set_a ) ).
% Compl_partition
thf(fact_1111_Compl__partition2,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ ( uminus_uminus_set_a @ A ) @ A )
= top_top_set_a ) ).
% Compl_partition2
thf(fact_1112_Compl__eq__Diff__UNIV,axiom,
( uminus_uminus_set_a
= ( minus_minus_set_a @ top_top_set_a ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_1113_Compl__eq__Diff__UNIV,axiom,
( uminus2192744996606729052_a_nat
= ( minus_7395159227704179404_a_nat @ top_to6433055325616222389_a_nat ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_1114_sup__shunt,axiom,
! [X4: set_a,Y: set_a] :
( ( ( sup_sup_set_a @ X4 @ Y )
= top_top_set_a )
= ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ X4 ) @ Y ) ) ).
% sup_shunt
thf(fact_1115_sup__shunt,axiom,
! [X4: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat] :
( ( ( sup_su4083067149120280889_a_nat @ X4 @ Y )
= top_to6433055325616222389_a_nat )
= ( ord_le1147066620699065093_a_nat @ ( uminus2192744996606729052_a_nat @ X4 ) @ Y ) ) ).
% sup_shunt
thf(fact_1116_boolean__algebra_Ocomplement__unique,axiom,
! [A2: set_a,X4: set_a,Y: set_a] :
( ( ( inf_inf_set_a @ A2 @ X4 )
= bot_bot_set_a )
=> ( ( ( sup_sup_set_a @ A2 @ X4 )
= top_top_set_a )
=> ( ( ( inf_inf_set_a @ A2 @ Y )
= bot_bot_set_a )
=> ( ( ( sup_sup_set_a @ A2 @ Y )
= top_top_set_a )
=> ( X4 = Y ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_1117_Compl__eq__empty__iff,axiom,
! [A: set_a] :
( ( ( uminus_uminus_set_a @ A )
= bot_bot_set_a )
= ( A = top_top_set_a ) ) ).
% Compl_eq_empty_iff
thf(fact_1118_inf__top_Osemilattice__neutr__order__axioms,axiom,
semila2496817875450240012_set_a @ inf_inf_set_a @ top_top_set_a @ ord_less_eq_set_a @ ord_less_set_a ).
% inf_top.semilattice_neutr_order_axioms
thf(fact_1119_inf__top_Osemilattice__neutr__order__axioms,axiom,
semila5895130476532002579_a_nat @ inf_in3249246906714053971_a_nat @ top_to6433055325616222389_a_nat @ ord_le1147066620699065093_a_nat @ ord_le5291801191193052689_a_nat ).
% inf_top.semilattice_neutr_order_axioms
thf(fact_1120_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
boolea6678413348699952596_set_a @ inf_inf_set_a @ sup_sup_set_a @ uminus_uminus_set_a @ bot_bot_set_a @ top_top_set_a ).
% boolean_algebra.abstract_boolean_algebra_axioms
thf(fact_1121_UNIV__I,axiom,
! [X4: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X4 @ top_to6433055325616222389_a_nat ) ).
% UNIV_I
thf(fact_1122_UNIV__I,axiom,
! [X4: a] : ( member_a @ X4 @ top_top_set_a ) ).
% UNIV_I
thf(fact_1123_UNIV__eq__I,axiom,
! [A: set_li6526943997496501093_a_nat] :
( ! [X3: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X3 @ A )
=> ( top_to6433055325616222389_a_nat = A ) ) ).
% UNIV_eq_I
thf(fact_1124_UNIV__eq__I,axiom,
! [A: set_a] :
( ! [X3: a] : ( member_a @ X3 @ A )
=> ( top_top_set_a = A ) ) ).
% UNIV_eq_I
thf(fact_1125_UNIV__witness,axiom,
? [X3: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X3 @ top_to6433055325616222389_a_nat ) ).
% UNIV_witness
thf(fact_1126_UNIV__witness,axiom,
? [X3: a] : ( member_a @ X3 @ top_top_set_a ) ).
% UNIV_witness
thf(fact_1127_top__empty__eq,axiom,
( top_to7646693901462700136_nat_o
= ( ^ [X5: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X5 @ top_to6433055325616222389_a_nat ) ) ) ).
% top_empty_eq
thf(fact_1128_top__empty__eq,axiom,
( top_top_a_o
= ( ^ [X5: a] : ( member_a @ X5 @ top_top_set_a ) ) ) ).
% top_empty_eq
thf(fact_1129_top_Oordering__top__axioms,axiom,
ordering_top_set_a @ ord_less_eq_set_a @ ord_less_set_a @ top_top_set_a ).
% top.ordering_top_axioms
thf(fact_1130_top_Oordering__top__axioms,axiom,
orderi2442089198978906902_a_nat @ ord_le1147066620699065093_a_nat @ ord_le5291801191193052689_a_nat @ top_to6433055325616222389_a_nat ).
% top.ordering_top_axioms
thf(fact_1131_set__empty__choose__def,axiom,
set_se792222636694734277oose_a = bot_bot_set_a ).
% set_empty_choose_def
thf(fact_1132_inf__top_Osemilattice__neutr__axioms,axiom,
semila5061130151665174080_set_a @ inf_inf_set_a @ top_top_set_a ).
% inf_top.semilattice_neutr_axioms
thf(fact_1133_sup__bot_Osemilattice__neutr__axioms,axiom,
semila5061130151665174080_set_a @ sup_sup_set_a @ bot_bot_set_a ).
% sup_bot.semilattice_neutr_axioms
thf(fact_1134_order_Oordering__axioms,axiom,
ordering_set_a @ ord_less_eq_set_a @ ord_less_set_a ).
% order.ordering_axioms
thf(fact_1135_order_Oordering__axioms,axiom,
orderi3914186766824710397_a_nat @ ord_le1147066620699065093_a_nat @ ord_le5291801191193052689_a_nat ).
% order.ordering_axioms
thf(fact_1136_union__monad__def,axiom,
set_union_monad_a = sup_sup_set_a ).
% union_monad_def
thf(fact_1137_order_Opartial__preordering__axioms,axiom,
partia6602192050731689876_set_a @ ord_less_eq_set_a ).
% order.partial_preordering_axioms
thf(fact_1138_order_Opartial__preordering__axioms,axiom,
partia6328371389090499227_a_nat @ ord_le1147066620699065093_a_nat ).
% order.partial_preordering_axioms
thf(fact_1139_greaterThanAtMost__iff,axiom,
! [I: set_a,L: set_a,U: set_a] :
( ( member_set_a @ I @ ( set_or2503527069484367278_set_a @ L @ U ) )
= ( ( ord_less_set_a @ L @ I )
& ( ord_less_eq_set_a @ I @ U ) ) ) ).
% greaterThanAtMost_iff
thf(fact_1140_greaterThanAtMost__iff,axiom,
! [I: set_li6526943997496501093_a_nat,L: set_li6526943997496501093_a_nat,U: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ I @ ( set_or4159382470967997621_a_nat @ L @ U ) )
= ( ( ord_le5291801191193052689_a_nat @ L @ I )
& ( ord_le1147066620699065093_a_nat @ I @ U ) ) ) ).
% greaterThanAtMost_iff
thf(fact_1141_greaterThanAtMost__empty,axiom,
! [L: set_a,K: set_a] :
( ( ord_less_eq_set_a @ L @ K )
=> ( ( set_or2503527069484367278_set_a @ K @ L )
= bot_bot_set_set_a ) ) ).
% greaterThanAtMost_empty
thf(fact_1142_greaterThanAtMost__empty,axiom,
! [L: set_li6526943997496501093_a_nat,K: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ L @ K )
=> ( ( set_or4159382470967997621_a_nat @ K @ L )
= bot_bo3237059034911209905_a_nat ) ) ).
% greaterThanAtMost_empty
thf(fact_1143_atLeastAtMost__iff,axiom,
! [I: set_a,L: set_a,U: set_a] :
( ( member_set_a @ I @ ( set_or6288561110385358355_set_a @ L @ U ) )
= ( ( ord_less_eq_set_a @ L @ I )
& ( ord_less_eq_set_a @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_1144_atLeastAtMost__iff,axiom,
! [I: set_li6526943997496501093_a_nat,L: set_li6526943997496501093_a_nat,U: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ I @ ( set_or2565139667135407898_a_nat @ L @ U ) )
= ( ( ord_le1147066620699065093_a_nat @ L @ I )
& ( ord_le1147066620699065093_a_nat @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_1145_Icc__eq__Icc,axiom,
! [L: set_a,H: set_a,L2: set_a,H2: set_a] :
( ( ( set_or6288561110385358355_set_a @ L @ H )
= ( set_or6288561110385358355_set_a @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_set_a @ L @ H )
& ~ ( ord_less_eq_set_a @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_1146_Icc__eq__Icc,axiom,
! [L: set_li6526943997496501093_a_nat,H: set_li6526943997496501093_a_nat,L2: set_li6526943997496501093_a_nat,H2: set_li6526943997496501093_a_nat] :
( ( ( set_or2565139667135407898_a_nat @ L @ H )
= ( set_or2565139667135407898_a_nat @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_le1147066620699065093_a_nat @ L @ H )
& ~ ( ord_le1147066620699065093_a_nat @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_1147_atLeastatMost__empty__iff2,axiom,
! [A2: set_a,B2: set_a] :
( ( bot_bot_set_set_a
= ( set_or6288561110385358355_set_a @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_1148_atLeastatMost__empty__iff2,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( bot_bo3237059034911209905_a_nat
= ( set_or2565139667135407898_a_nat @ A2 @ B2 ) )
= ( ~ ( ord_le1147066620699065093_a_nat @ A2 @ B2 ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_1149_atLeastatMost__empty__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( ( set_or6288561110385358355_set_a @ A2 @ B2 )
= bot_bot_set_set_a )
= ( ~ ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_1150_atLeastatMost__empty__iff,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ( set_or2565139667135407898_a_nat @ A2 @ B2 )
= bot_bo3237059034911209905_a_nat )
= ( ~ ( ord_le1147066620699065093_a_nat @ A2 @ B2 ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_1151_atLeastatMost__subset__iff,axiom,
! [A2: set_a,B2: set_a,C: set_a,D2: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_or6288561110385358355_set_a @ A2 @ B2 ) @ ( set_or6288561110385358355_set_a @ C @ D2 ) )
= ( ~ ( ord_less_eq_set_a @ A2 @ B2 )
| ( ( ord_less_eq_set_a @ C @ A2 )
& ( ord_less_eq_set_a @ B2 @ D2 ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_1152_atLeastatMost__subset__iff,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat,D2: set_li6526943997496501093_a_nat] :
( ( ord_le8138476598237931237_a_nat @ ( set_or2565139667135407898_a_nat @ A2 @ B2 ) @ ( set_or2565139667135407898_a_nat @ C @ D2 ) )
= ( ~ ( ord_le1147066620699065093_a_nat @ A2 @ B2 )
| ( ( ord_le1147066620699065093_a_nat @ C @ A2 )
& ( ord_le1147066620699065093_a_nat @ B2 @ D2 ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_1153_greaterThanLessThan__empty,axiom,
! [L: set_a,K: set_a] :
( ( ord_less_eq_set_a @ L @ K )
=> ( ( set_or6017932776736107018_set_a @ K @ L )
= bot_bot_set_set_a ) ) ).
% greaterThanLessThan_empty
thf(fact_1154_greaterThanLessThan__empty,axiom,
! [L: set_li6526943997496501093_a_nat,K: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ L @ K )
=> ( ( set_or2112173576855165201_a_nat @ K @ L )
= bot_bo3237059034911209905_a_nat ) ) ).
% greaterThanLessThan_empty
thf(fact_1155_atLeastAtMost__eq__UNIV__iff,axiom,
! [X4: set_a,Y: set_a] :
( ( ( set_or6288561110385358355_set_a @ X4 @ Y )
= top_top_set_set_a )
= ( ( X4 = bot_bot_set_a )
& ( Y = top_top_set_a ) ) ) ).
% atLeastAtMost_eq_UNIV_iff
thf(fact_1156_atLeastatMost__psubset__iff,axiom,
! [A2: set_a,B2: set_a,C: set_a,D2: set_a] :
( ( ord_less_set_set_a @ ( set_or6288561110385358355_set_a @ A2 @ B2 ) @ ( set_or6288561110385358355_set_a @ C @ D2 ) )
= ( ( ~ ( ord_less_eq_set_a @ A2 @ B2 )
| ( ( ord_less_eq_set_a @ C @ A2 )
& ( ord_less_eq_set_a @ B2 @ D2 )
& ( ( ord_less_set_a @ C @ A2 )
| ( ord_less_set_a @ B2 @ D2 ) ) ) )
& ( ord_less_eq_set_a @ C @ D2 ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_1157_atLeastatMost__psubset__iff,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat,D2: set_li6526943997496501093_a_nat] :
( ( ord_le81696049219912177_a_nat @ ( set_or2565139667135407898_a_nat @ A2 @ B2 ) @ ( set_or2565139667135407898_a_nat @ C @ D2 ) )
= ( ( ~ ( ord_le1147066620699065093_a_nat @ A2 @ B2 )
| ( ( ord_le1147066620699065093_a_nat @ C @ A2 )
& ( ord_le1147066620699065093_a_nat @ B2 @ D2 )
& ( ( ord_le5291801191193052689_a_nat @ C @ A2 )
| ( ord_le5291801191193052689_a_nat @ B2 @ D2 ) ) ) )
& ( ord_le1147066620699065093_a_nat @ C @ D2 ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_1158_atLeastLessThan__iff,axiom,
! [I: set_a,L: set_a,U: set_a] :
( ( member_set_a @ I @ ( set_or2348907005316661231_set_a @ L @ U ) )
= ( ( ord_less_eq_set_a @ L @ I )
& ( ord_less_set_a @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_1159_atLeastLessThan__iff,axiom,
! [I: set_li6526943997496501093_a_nat,L: set_li6526943997496501093_a_nat,U: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ I @ ( set_or464376051672557814_a_nat @ L @ U ) )
= ( ( ord_le1147066620699065093_a_nat @ L @ I )
& ( ord_le5291801191193052689_a_nat @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_1160_atLeastLessThan__empty,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( set_or2348907005316661231_set_a @ A2 @ B2 )
= bot_bot_set_set_a ) ) ).
% atLeastLessThan_empty
thf(fact_1161_atLeastLessThan__empty,axiom,
! [B2: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ B2 @ A2 )
=> ( ( set_or464376051672557814_a_nat @ A2 @ B2 )
= bot_bo3237059034911209905_a_nat ) ) ).
% atLeastLessThan_empty
thf(fact_1162_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A2: set_a,B2: set_a,C: set_a,D2: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_or6288561110385358355_set_a @ A2 @ B2 ) @ ( set_or2348907005316661231_set_a @ C @ D2 ) )
= ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ C @ A2 )
& ( ord_less_set_a @ B2 @ D2 ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_1163_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat,D2: set_li6526943997496501093_a_nat] :
( ( ord_le8138476598237931237_a_nat @ ( set_or2565139667135407898_a_nat @ A2 @ B2 ) @ ( set_or464376051672557814_a_nat @ C @ D2 ) )
= ( ( ord_le1147066620699065093_a_nat @ A2 @ B2 )
=> ( ( ord_le1147066620699065093_a_nat @ C @ A2 )
& ( ord_le5291801191193052689_a_nat @ B2 @ D2 ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_1164_Icc__subset__Ici__iff,axiom,
! [L: set_a,H: set_a,L2: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_or6288561110385358355_set_a @ L @ H ) @ ( set_or8362275514725411625_set_a @ L2 ) )
= ( ~ ( ord_less_eq_set_a @ L @ H )
| ( ord_less_eq_set_a @ L2 @ L ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_1165_Icc__subset__Ici__iff,axiom,
! [L: set_li6526943997496501093_a_nat,H: set_li6526943997496501093_a_nat,L2: set_li6526943997496501093_a_nat] :
( ( ord_le8138476598237931237_a_nat @ ( set_or2565139667135407898_a_nat @ L @ H ) @ ( set_or2796000563533817904_a_nat @ L2 ) )
= ( ~ ( ord_le1147066620699065093_a_nat @ L @ H )
| ( ord_le1147066620699065093_a_nat @ L2 @ L ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_1166_Icc__subset__Iic__iff,axiom,
! [L: set_a,H: set_a,H2: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_or6288561110385358355_set_a @ L @ H ) @ ( set_ord_atMost_set_a @ H2 ) )
= ( ~ ( ord_less_eq_set_a @ L @ H )
| ( ord_less_eq_set_a @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_1167_Icc__subset__Iic__iff,axiom,
! [L: set_li6526943997496501093_a_nat,H: set_li6526943997496501093_a_nat,H2: set_li6526943997496501093_a_nat] :
( ( ord_le8138476598237931237_a_nat @ ( set_or2565139667135407898_a_nat @ L @ H ) @ ( set_or2322826630718027820_a_nat @ H2 ) )
= ( ~ ( ord_le1147066620699065093_a_nat @ L @ H )
| ( ord_le1147066620699065093_a_nat @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_1168_atLeast__empty__triv,axiom,
( ( set_or8362275514725411625_set_a @ bot_bot_set_a )
= top_top_set_set_a ) ).
% atLeast_empty_triv
thf(fact_1169_atMost__iff,axiom,
! [I: set_a,K: set_a] :
( ( member_set_a @ I @ ( set_ord_atMost_set_a @ K ) )
= ( ord_less_eq_set_a @ I @ K ) ) ).
% atMost_iff
thf(fact_1170_atMost__iff,axiom,
! [I: set_li6526943997496501093_a_nat,K: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ I @ ( set_or2322826630718027820_a_nat @ K ) )
= ( ord_le1147066620699065093_a_nat @ I @ K ) ) ).
% atMost_iff
thf(fact_1171_atLeast__iff,axiom,
! [I: set_a,K: set_a] :
( ( member_set_a @ I @ ( set_or8362275514725411625_set_a @ K ) )
= ( ord_less_eq_set_a @ K @ I ) ) ).
% atLeast_iff
thf(fact_1172_atLeast__iff,axiom,
! [I: set_li6526943997496501093_a_nat,K: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ I @ ( set_or2796000563533817904_a_nat @ K ) )
= ( ord_le1147066620699065093_a_nat @ K @ I ) ) ).
% atLeast_iff
thf(fact_1173_atMost__subset__iff,axiom,
! [X4: set_a,Y: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_ord_atMost_set_a @ X4 ) @ ( set_ord_atMost_set_a @ Y ) )
= ( ord_less_eq_set_a @ X4 @ Y ) ) ).
% atMost_subset_iff
thf(fact_1174_atMost__subset__iff,axiom,
! [X4: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat] :
( ( ord_le8138476598237931237_a_nat @ ( set_or2322826630718027820_a_nat @ X4 ) @ ( set_or2322826630718027820_a_nat @ Y ) )
= ( ord_le1147066620699065093_a_nat @ X4 @ Y ) ) ).
% atMost_subset_iff
thf(fact_1175_atLeast__subset__iff,axiom,
! [X4: set_a,Y: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_or8362275514725411625_set_a @ X4 ) @ ( set_or8362275514725411625_set_a @ Y ) )
= ( ord_less_eq_set_a @ Y @ X4 ) ) ).
% atLeast_subset_iff
thf(fact_1176_atLeast__subset__iff,axiom,
! [X4: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat] :
( ( ord_le8138476598237931237_a_nat @ ( set_or2796000563533817904_a_nat @ X4 ) @ ( set_or2796000563533817904_a_nat @ Y ) )
= ( ord_le1147066620699065093_a_nat @ Y @ X4 ) ) ).
% atLeast_subset_iff
thf(fact_1177_atLeast__eq__UNIV__iff,axiom,
! [X4: set_a] :
( ( ( set_or8362275514725411625_set_a @ X4 )
= top_top_set_set_a )
= ( X4 = bot_bot_set_a ) ) ).
% atLeast_eq_UNIV_iff
thf(fact_1178_Inf__atMostLessThan,axiom,
! [X4: set_a] :
( ( ord_less_set_a @ top_top_set_a @ X4 )
=> ( ( comple6135023378680113637_set_a @ ( set_or5421148953861284865_set_a @ X4 ) )
= bot_bot_set_a ) ) ).
% Inf_atMostLessThan
thf(fact_1179_cInf__atLeastAtMost,axiom,
! [Y: set_a,X4: set_a] :
( ( ord_less_eq_set_a @ Y @ X4 )
=> ( ( comple6135023378680113637_set_a @ ( set_or6288561110385358355_set_a @ Y @ X4 ) )
= Y ) ) ).
% cInf_atLeastAtMost
thf(fact_1180_cInf__atLeastAtMost,axiom,
! [Y: set_li6526943997496501093_a_nat,X4: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ Y @ X4 )
=> ( ( comple852897431680229100_a_nat @ ( set_or2565139667135407898_a_nat @ Y @ X4 ) )
= Y ) ) ).
% cInf_atLeastAtMost
thf(fact_1181_Inf__atLeastAtMost,axiom,
! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ( comple6135023378680113637_set_a @ ( set_or6288561110385358355_set_a @ X4 @ Y ) )
= X4 ) ) ).
% Inf_atLeastAtMost
thf(fact_1182_Inf__atLeastAtMost,axiom,
! [X4: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ X4 @ Y )
=> ( ( comple852897431680229100_a_nat @ ( set_or2565139667135407898_a_nat @ X4 @ Y ) )
= X4 ) ) ).
% Inf_atLeastAtMost
thf(fact_1183_Inf__atMost,axiom,
! [X4: set_a] :
( ( comple6135023378680113637_set_a @ ( set_ord_atMost_set_a @ X4 ) )
= bot_bot_set_a ) ).
% Inf_atMost
thf(fact_1184_cInf__eq__non__empty,axiom,
! [X: set_set_a,A2: set_a] :
( ( X != bot_bot_set_set_a )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ X )
=> ( ord_less_eq_set_a @ A2 @ X3 ) )
=> ( ! [Y3: set_a] :
( ! [X6: set_a] :
( ( member_set_a @ X6 @ X )
=> ( ord_less_eq_set_a @ Y3 @ X6 ) )
=> ( ord_less_eq_set_a @ Y3 @ A2 ) )
=> ( ( comple6135023378680113637_set_a @ X )
= A2 ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_1185_cInf__eq__non__empty,axiom,
! [X: set_se4330304633200676677_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( X != bot_bo3237059034911209905_a_nat )
=> ( ! [X3: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ X3 @ X )
=> ( ord_le1147066620699065093_a_nat @ A2 @ X3 ) )
=> ( ! [Y3: set_li6526943997496501093_a_nat] :
( ! [X6: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ X6 @ X )
=> ( ord_le1147066620699065093_a_nat @ Y3 @ X6 ) )
=> ( ord_le1147066620699065093_a_nat @ Y3 @ A2 ) )
=> ( ( comple852897431680229100_a_nat @ X )
= A2 ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_1186_cInf__greatest,axiom,
! [X: set_set_a,Z: set_a] :
( ( X != bot_bot_set_set_a )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ X )
=> ( ord_less_eq_set_a @ Z @ X3 ) )
=> ( ord_less_eq_set_a @ Z @ ( comple6135023378680113637_set_a @ X ) ) ) ) ).
% cInf_greatest
thf(fact_1187_cInf__greatest,axiom,
! [X: set_se4330304633200676677_a_nat,Z: set_li6526943997496501093_a_nat] :
( ( X != bot_bo3237059034911209905_a_nat )
=> ( ! [X3: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ X3 @ X )
=> ( ord_le1147066620699065093_a_nat @ Z @ X3 ) )
=> ( ord_le1147066620699065093_a_nat @ Z @ ( comple852897431680229100_a_nat @ X ) ) ) ) ).
% cInf_greatest
thf(fact_1188_cInf__eq__minimum,axiom,
! [Z: set_a,X: set_set_a] :
( ( member_set_a @ Z @ X )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ X )
=> ( ord_less_eq_set_a @ Z @ X3 ) )
=> ( ( comple6135023378680113637_set_a @ X )
= Z ) ) ) ).
% cInf_eq_minimum
thf(fact_1189_cInf__eq__minimum,axiom,
! [Z: set_li6526943997496501093_a_nat,X: set_se4330304633200676677_a_nat] :
( ( member5553968465346197646_a_nat @ Z @ X )
=> ( ! [X3: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ X3 @ X )
=> ( ord_le1147066620699065093_a_nat @ Z @ X3 ) )
=> ( ( comple852897431680229100_a_nat @ X )
= Z ) ) ) ).
% cInf_eq_minimum
thf(fact_1190_Inf__insert,axiom,
! [A2: set_a,A: set_set_a] :
( ( comple6135023378680113637_set_a @ ( insert_set_a @ A2 @ A ) )
= ( inf_inf_set_a @ A2 @ ( comple6135023378680113637_set_a @ A ) ) ) ).
% Inf_insert
thf(fact_1191_Inf__UNIV,axiom,
( ( comple6135023378680113637_set_a @ top_top_set_set_a )
= bot_bot_set_a ) ).
% Inf_UNIV
thf(fact_1192_Inter__subset,axiom,
! [A: set_set_a,B: set_a] :
( ! [X7: set_a] :
( ( member_set_a @ X7 @ A )
=> ( ord_less_eq_set_a @ X7 @ B ) )
=> ( ( A != bot_bot_set_set_a )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A ) @ B ) ) ) ).
% Inter_subset
thf(fact_1193_Inter__subset,axiom,
! [A: set_se4330304633200676677_a_nat,B: set_li6526943997496501093_a_nat] :
( ! [X7: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ X7 @ A )
=> ( ord_le1147066620699065093_a_nat @ X7 @ B ) )
=> ( ( A != bot_bo3237059034911209905_a_nat )
=> ( ord_le1147066620699065093_a_nat @ ( comple852897431680229100_a_nat @ A ) @ B ) ) ) ).
% Inter_subset
thf(fact_1194_Inter__anti__mono,axiom,
! [B: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A ) @ ( comple6135023378680113637_set_a @ B ) ) ) ).
% Inter_anti_mono
thf(fact_1195_Inter__anti__mono,axiom,
! [B: set_se4330304633200676677_a_nat,A: set_se4330304633200676677_a_nat] :
( ( ord_le8138476598237931237_a_nat @ B @ A )
=> ( ord_le1147066620699065093_a_nat @ ( comple852897431680229100_a_nat @ A ) @ ( comple852897431680229100_a_nat @ B ) ) ) ).
% Inter_anti_mono
thf(fact_1196_Inter__greatest,axiom,
! [A: set_set_a,C2: set_a] :
( ! [X7: set_a] :
( ( member_set_a @ X7 @ A )
=> ( ord_less_eq_set_a @ C2 @ X7 ) )
=> ( ord_less_eq_set_a @ C2 @ ( comple6135023378680113637_set_a @ A ) ) ) ).
% Inter_greatest
thf(fact_1197_Inter__greatest,axiom,
! [A: set_se4330304633200676677_a_nat,C2: set_li6526943997496501093_a_nat] :
( ! [X7: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ X7 @ A )
=> ( ord_le1147066620699065093_a_nat @ C2 @ X7 ) )
=> ( ord_le1147066620699065093_a_nat @ C2 @ ( comple852897431680229100_a_nat @ A ) ) ) ).
% Inter_greatest
thf(fact_1198_Inter__lower,axiom,
! [B: set_a,A: set_set_a] :
( ( member_set_a @ B @ A )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A ) @ B ) ) ).
% Inter_lower
thf(fact_1199_Inter__lower,axiom,
! [B: set_li6526943997496501093_a_nat,A: set_se4330304633200676677_a_nat] :
( ( member5553968465346197646_a_nat @ B @ A )
=> ( ord_le1147066620699065093_a_nat @ ( comple852897431680229100_a_nat @ A ) @ B ) ) ).
% Inter_lower
thf(fact_1200_Inter__Un__distrib,axiom,
! [A: set_set_a,B: set_set_a] :
( ( comple6135023378680113637_set_a @ ( sup_sup_set_set_a @ A @ B ) )
= ( inf_inf_set_a @ ( comple6135023378680113637_set_a @ A ) @ ( comple6135023378680113637_set_a @ B ) ) ) ).
% Inter_Un_distrib
thf(fact_1201_Inter__insert,axiom,
! [A2: set_a,B: set_set_a] :
( ( comple6135023378680113637_set_a @ ( insert_set_a @ A2 @ B ) )
= ( inf_inf_set_a @ A2 @ ( comple6135023378680113637_set_a @ B ) ) ) ).
% Inter_insert
thf(fact_1202_Inf__greatest,axiom,
! [A: set_set_a,Z: set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A )
=> ( ord_less_eq_set_a @ Z @ X3 ) )
=> ( ord_less_eq_set_a @ Z @ ( comple6135023378680113637_set_a @ A ) ) ) ).
% Inf_greatest
thf(fact_1203_Inf__greatest,axiom,
! [A: set_se4330304633200676677_a_nat,Z: set_li6526943997496501093_a_nat] :
( ! [X3: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ X3 @ A )
=> ( ord_le1147066620699065093_a_nat @ Z @ X3 ) )
=> ( ord_le1147066620699065093_a_nat @ Z @ ( comple852897431680229100_a_nat @ A ) ) ) ).
% Inf_greatest
thf(fact_1204_le__Inf__iff,axiom,
! [B2: set_a,A: set_set_a] :
( ( ord_less_eq_set_a @ B2 @ ( comple6135023378680113637_set_a @ A ) )
= ( ! [X5: set_a] :
( ( member_set_a @ X5 @ A )
=> ( ord_less_eq_set_a @ B2 @ X5 ) ) ) ) ).
% le_Inf_iff
thf(fact_1205_le__Inf__iff,axiom,
! [B2: set_li6526943997496501093_a_nat,A: set_se4330304633200676677_a_nat] :
( ( ord_le1147066620699065093_a_nat @ B2 @ ( comple852897431680229100_a_nat @ A ) )
= ( ! [X5: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ X5 @ A )
=> ( ord_le1147066620699065093_a_nat @ B2 @ X5 ) ) ) ) ).
% le_Inf_iff
thf(fact_1206_Inf__lower2,axiom,
! [U: set_a,A: set_set_a,V: set_a] :
( ( member_set_a @ U @ A )
=> ( ( ord_less_eq_set_a @ U @ V )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A ) @ V ) ) ) ).
% Inf_lower2
thf(fact_1207_Inf__lower2,axiom,
! [U: set_li6526943997496501093_a_nat,A: set_se4330304633200676677_a_nat,V: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ U @ A )
=> ( ( ord_le1147066620699065093_a_nat @ U @ V )
=> ( ord_le1147066620699065093_a_nat @ ( comple852897431680229100_a_nat @ A ) @ V ) ) ) ).
% Inf_lower2
thf(fact_1208_Inf__lower,axiom,
! [X4: set_a,A: set_set_a] :
( ( member_set_a @ X4 @ A )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A ) @ X4 ) ) ).
% Inf_lower
thf(fact_1209_Inf__lower,axiom,
! [X4: set_li6526943997496501093_a_nat,A: set_se4330304633200676677_a_nat] :
( ( member5553968465346197646_a_nat @ X4 @ A )
=> ( ord_le1147066620699065093_a_nat @ ( comple852897431680229100_a_nat @ A ) @ X4 ) ) ).
% Inf_lower
thf(fact_1210_Inf__mono,axiom,
! [B: set_set_a,A: set_set_a] :
( ! [B7: set_a] :
( ( member_set_a @ B7 @ B )
=> ? [X6: set_a] :
( ( member_set_a @ X6 @ A )
& ( ord_less_eq_set_a @ X6 @ B7 ) ) )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A ) @ ( comple6135023378680113637_set_a @ B ) ) ) ).
% Inf_mono
thf(fact_1211_Inf__mono,axiom,
! [B: set_se4330304633200676677_a_nat,A: set_se4330304633200676677_a_nat] :
( ! [B7: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ B7 @ B )
=> ? [X6: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ X6 @ A )
& ( ord_le1147066620699065093_a_nat @ X6 @ B7 ) ) )
=> ( ord_le1147066620699065093_a_nat @ ( comple852897431680229100_a_nat @ A ) @ ( comple852897431680229100_a_nat @ B ) ) ) ).
% Inf_mono
thf(fact_1212_Inf__eqI,axiom,
! [A: set_set_a,X4: set_a] :
( ! [I2: set_a] :
( ( member_set_a @ I2 @ A )
=> ( ord_less_eq_set_a @ X4 @ I2 ) )
=> ( ! [Y3: set_a] :
( ! [I3: set_a] :
( ( member_set_a @ I3 @ A )
=> ( ord_less_eq_set_a @ Y3 @ I3 ) )
=> ( ord_less_eq_set_a @ Y3 @ X4 ) )
=> ( ( comple6135023378680113637_set_a @ A )
= X4 ) ) ) ).
% Inf_eqI
thf(fact_1213_Inf__eqI,axiom,
! [A: set_se4330304633200676677_a_nat,X4: set_li6526943997496501093_a_nat] :
( ! [I2: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ I2 @ A )
=> ( ord_le1147066620699065093_a_nat @ X4 @ I2 ) )
=> ( ! [Y3: set_li6526943997496501093_a_nat] :
( ! [I3: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ I3 @ A )
=> ( ord_le1147066620699065093_a_nat @ Y3 @ I3 ) )
=> ( ord_le1147066620699065093_a_nat @ Y3 @ X4 ) )
=> ( ( comple852897431680229100_a_nat @ A )
= X4 ) ) ) ).
% Inf_eqI
thf(fact_1214_Inter__Un__subset,axiom,
! [A: set_set_a,B: set_set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( comple6135023378680113637_set_a @ A ) @ ( comple6135023378680113637_set_a @ B ) ) @ ( comple6135023378680113637_set_a @ ( inf_inf_set_set_a @ A @ B ) ) ) ).
% Inter_Un_subset
thf(fact_1215_Inter__Un__subset,axiom,
! [A: set_se4330304633200676677_a_nat,B: set_se4330304633200676677_a_nat] : ( ord_le1147066620699065093_a_nat @ ( sup_su4083067149120280889_a_nat @ ( comple852897431680229100_a_nat @ A ) @ ( comple852897431680229100_a_nat @ B ) ) @ ( comple852897431680229100_a_nat @ ( inf_in5367731912061063475_a_nat @ A @ B ) ) ) ).
% Inter_Un_subset
thf(fact_1216_Inter__UNIV,axiom,
( ( comple6135023378680113637_set_a @ top_top_set_set_a )
= bot_bot_set_a ) ).
% Inter_UNIV
thf(fact_1217_Inf__less__eq,axiom,
! [A: set_set_a,U: set_a] :
( ! [V2: set_a] :
( ( member_set_a @ V2 @ A )
=> ( ord_less_eq_set_a @ V2 @ U ) )
=> ( ( A != bot_bot_set_set_a )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A ) @ U ) ) ) ).
% Inf_less_eq
thf(fact_1218_Inf__less__eq,axiom,
! [A: set_se4330304633200676677_a_nat,U: set_li6526943997496501093_a_nat] :
( ! [V2: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ V2 @ A )
=> ( ord_le1147066620699065093_a_nat @ V2 @ U ) )
=> ( ( A != bot_bo3237059034911209905_a_nat )
=> ( ord_le1147066620699065093_a_nat @ ( comple852897431680229100_a_nat @ A ) @ U ) ) ) ).
% Inf_less_eq
thf(fact_1219_Inf__superset__mono,axiom,
! [B: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A ) @ ( comple6135023378680113637_set_a @ B ) ) ) ).
% Inf_superset_mono
thf(fact_1220_Inf__superset__mono,axiom,
! [B: set_se4330304633200676677_a_nat,A: set_se4330304633200676677_a_nat] :
( ( ord_le8138476598237931237_a_nat @ B @ A )
=> ( ord_le1147066620699065093_a_nat @ ( comple852897431680229100_a_nat @ A ) @ ( comple852897431680229100_a_nat @ B ) ) ) ).
% Inf_superset_mono
thf(fact_1221_Inf__union__distrib,axiom,
! [A: set_set_a,B: set_set_a] :
( ( comple6135023378680113637_set_a @ ( sup_sup_set_set_a @ A @ B ) )
= ( inf_inf_set_a @ ( comple6135023378680113637_set_a @ A ) @ ( comple6135023378680113637_set_a @ B ) ) ) ).
% Inf_union_distrib
thf(fact_1222_less__eq__Inf__inter,axiom,
! [A: set_set_a,B: set_set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( comple6135023378680113637_set_a @ A ) @ ( comple6135023378680113637_set_a @ B ) ) @ ( comple6135023378680113637_set_a @ ( inf_inf_set_set_a @ A @ B ) ) ) ).
% less_eq_Inf_inter
thf(fact_1223_less__eq__Inf__inter,axiom,
! [A: set_se4330304633200676677_a_nat,B: set_se4330304633200676677_a_nat] : ( ord_le1147066620699065093_a_nat @ ( sup_su4083067149120280889_a_nat @ ( comple852897431680229100_a_nat @ A ) @ ( comple852897431680229100_a_nat @ B ) ) @ ( comple852897431680229100_a_nat @ ( inf_in5367731912061063475_a_nat @ A @ B ) ) ) ).
% less_eq_Inf_inter
thf(fact_1224_Inf__sup__eq__top__iff,axiom,
! [B: set_set_a,A2: set_a] :
( ( ( sup_sup_set_a @ ( comple6135023378680113637_set_a @ B ) @ A2 )
= top_top_set_a )
= ( ! [X5: set_a] :
( ( member_set_a @ X5 @ B )
=> ( ( sup_sup_set_a @ X5 @ A2 )
= top_top_set_a ) ) ) ) ).
% Inf_sup_eq_top_iff
thf(fact_1225_cInf__le__finite,axiom,
! [X: set_set_a,X4: set_a] :
( ( finite_finite_set_a @ X )
=> ( ( member_set_a @ X4 @ X )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ X ) @ X4 ) ) ) ).
% cInf_le_finite
thf(fact_1226_cInf__le__finite,axiom,
! [X: set_se4330304633200676677_a_nat,X4: set_li6526943997496501093_a_nat] :
( ( finite8331804442905961358_a_nat @ X )
=> ( ( member5553968465346197646_a_nat @ X4 @ X )
=> ( ord_le1147066620699065093_a_nat @ ( comple852897431680229100_a_nat @ X ) @ X4 ) ) ) ).
% cInf_le_finite
thf(fact_1227_finite__Diff__insert,axiom,
! [A: set_a,A2: a,B: set_a] :
( ( finite_finite_a @ ( minus_minus_set_a @ A @ ( insert_a @ A2 @ B ) ) )
= ( finite_finite_a @ ( minus_minus_set_a @ A @ B ) ) ) ).
% finite_Diff_insert
thf(fact_1228_finite__Diff__insert,axiom,
! [A: set_li6526943997496501093_a_nat,A2: list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat] :
( ( finite1487985464145237934_a_nat @ ( minus_7395159227704179404_a_nat @ A @ ( insert2950094090816004437_a_nat @ A2 @ B ) ) )
= ( finite1487985464145237934_a_nat @ ( minus_7395159227704179404_a_nat @ A @ B ) ) ) ).
% finite_Diff_insert
thf(fact_1229_finite__Un,axiom,
! [F2: set_a,G: set_a] :
( ( finite_finite_a @ ( sup_sup_set_a @ F2 @ G ) )
= ( ( finite_finite_a @ F2 )
& ( finite_finite_a @ G ) ) ) ).
% finite_Un
thf(fact_1230_finite__Int,axiom,
! [F2: set_a,G: set_a] :
( ( ( finite_finite_a @ F2 )
| ( finite_finite_a @ G ) )
=> ( finite_finite_a @ ( inf_inf_set_a @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_1231_finite__Diff2,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( finite_finite_a @ ( minus_minus_set_a @ A @ B ) )
= ( finite_finite_a @ A ) ) ) ).
% finite_Diff2
thf(fact_1232_finite__Diff2,axiom,
! [B: set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat] :
( ( finite1487985464145237934_a_nat @ B )
=> ( ( finite1487985464145237934_a_nat @ ( minus_7395159227704179404_a_nat @ A @ B ) )
= ( finite1487985464145237934_a_nat @ A ) ) ) ).
% finite_Diff2
thf(fact_1233_finite__Diff,axiom,
! [A: set_a,B: set_a] :
( ( finite_finite_a @ A )
=> ( finite_finite_a @ ( minus_minus_set_a @ A @ B ) ) ) ).
% finite_Diff
thf(fact_1234_finite__Diff,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( finite1487985464145237934_a_nat @ A )
=> ( finite1487985464145237934_a_nat @ ( minus_7395159227704179404_a_nat @ A @ B ) ) ) ).
% finite_Diff
thf(fact_1235_finite__has__minimal2,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A )
& ( ord_less_eq_set_a @ X3 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1236_finite__has__minimal2,axiom,
! [A: set_se4330304633200676677_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( finite8331804442905961358_a_nat @ A )
=> ( ( member5553968465346197646_a_nat @ A2 @ A )
=> ? [X3: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ X3 @ A )
& ( ord_le1147066620699065093_a_nat @ X3 @ A2 )
& ! [Xa: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ Xa @ A )
=> ( ( ord_le1147066620699065093_a_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1237_finite__has__maximal2,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A )
& ( ord_less_eq_set_a @ A2 @ X3 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1238_finite__has__maximal2,axiom,
! [A: set_se4330304633200676677_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( finite8331804442905961358_a_nat @ A )
=> ( ( member5553968465346197646_a_nat @ A2 @ A )
=> ? [X3: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ X3 @ A )
& ( ord_le1147066620699065093_a_nat @ A2 @ X3 )
& ! [Xa: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ Xa @ A )
=> ( ( ord_le1147066620699065093_a_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1239_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_1240_infinite__imp__nonempty,axiom,
! [S: set_a] :
( ~ ( finite_finite_a @ S )
=> ( S != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_1241_rev__finite__subset,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( finite_finite_a @ A ) ) ) ).
% rev_finite_subset
thf(fact_1242_rev__finite__subset,axiom,
! [B: set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat] :
( ( finite1487985464145237934_a_nat @ B )
=> ( ( ord_le1147066620699065093_a_nat @ A @ B )
=> ( finite1487985464145237934_a_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_1243_infinite__super,axiom,
! [S: set_a,T: set_a] :
( ( ord_less_eq_set_a @ S @ T )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ T ) ) ) ).
% infinite_super
thf(fact_1244_infinite__super,axiom,
! [S: set_li6526943997496501093_a_nat,T: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ S @ T )
=> ( ~ ( finite1487985464145237934_a_nat @ S )
=> ~ ( finite1487985464145237934_a_nat @ T ) ) ) ).
% infinite_super
thf(fact_1245_finite__subset,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ A ) ) ) ).
% finite_subset
thf(fact_1246_finite__subset,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A @ B )
=> ( ( finite1487985464145237934_a_nat @ B )
=> ( finite1487985464145237934_a_nat @ A ) ) ) ).
% finite_subset
thf(fact_1247_finite__UnI,axiom,
! [F2: set_a,G: set_a] :
( ( finite_finite_a @ F2 )
=> ( ( finite_finite_a @ G )
=> ( finite_finite_a @ ( sup_sup_set_a @ F2 @ G ) ) ) ) ).
% finite_UnI
thf(fact_1248_Un__infinite,axiom,
! [S: set_a,T: set_a] :
( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( sup_sup_set_a @ S @ T ) ) ) ).
% Un_infinite
thf(fact_1249_infinite__Un,axiom,
! [S: set_a,T: set_a] :
( ( ~ ( finite_finite_a @ ( sup_sup_set_a @ S @ T ) ) )
= ( ~ ( finite_finite_a @ S )
| ~ ( finite_finite_a @ T ) ) ) ).
% infinite_Un
thf(fact_1250_Diff__infinite__finite,axiom,
! [T: set_a,S: set_a] :
( ( finite_finite_a @ T )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_1251_Diff__infinite__finite,axiom,
! [T: set_li6526943997496501093_a_nat,S: set_li6526943997496501093_a_nat] :
( ( finite1487985464145237934_a_nat @ T )
=> ( ~ ( finite1487985464145237934_a_nat @ S )
=> ~ ( finite1487985464145237934_a_nat @ ( minus_7395159227704179404_a_nat @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_1252_finite__has__maximal,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1253_finite__has__maximal,axiom,
! [A: set_se4330304633200676677_a_nat] :
( ( finite8331804442905961358_a_nat @ A )
=> ( ( A != bot_bo3237059034911209905_a_nat )
=> ? [X3: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ X3 @ A )
& ! [Xa: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ Xa @ A )
=> ( ( ord_le1147066620699065093_a_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1254_finite__has__minimal,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1255_finite__has__minimal,axiom,
! [A: set_se4330304633200676677_a_nat] :
( ( finite8331804442905961358_a_nat @ A )
=> ( ( A != bot_bo3237059034911209905_a_nat )
=> ? [X3: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ X3 @ A )
& ! [Xa: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ Xa @ A )
=> ( ( ord_le1147066620699065093_a_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1256_finite_Ocases,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( A2 != bot_bot_set_a )
=> ~ ! [A6: set_a] :
( ? [A7: a] :
( A2
= ( insert_a @ A7 @ A6 ) )
=> ~ ( finite_finite_a @ A6 ) ) ) ) ).
% finite.cases
thf(fact_1257_finite_Osimps,axiom,
( finite_finite_a
= ( ^ [A3: set_a] :
( ( A3 = bot_bot_set_a )
| ? [A4: set_a,B3: a] :
( ( A3
= ( insert_a @ B3 @ A4 ) )
& ( finite_finite_a @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_1258_finite__induct,axiom,
! [F2: set_li6526943997496501093_a_nat,P: set_li6526943997496501093_a_nat > $o] :
( ( finite1487985464145237934_a_nat @ F2 )
=> ( ( P @ bot_bo1033123847703346641_a_nat )
=> ( ! [X3: list_Sum_sum_a_nat,F3: set_li6526943997496501093_a_nat] :
( ( finite1487985464145237934_a_nat @ F3 )
=> ( ~ ( member408289922725080238_a_nat @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert2950094090816004437_a_nat @ X3 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_1259_finite__induct,axiom,
! [F2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X3: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X3 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_1260_finite__ne__induct,axiom,
! [F2: set_li6526943997496501093_a_nat,P: set_li6526943997496501093_a_nat > $o] :
( ( finite1487985464145237934_a_nat @ F2 )
=> ( ( F2 != bot_bo1033123847703346641_a_nat )
=> ( ! [X3: list_Sum_sum_a_nat] : ( P @ ( insert2950094090816004437_a_nat @ X3 @ bot_bo1033123847703346641_a_nat ) )
=> ( ! [X3: list_Sum_sum_a_nat,F3: set_li6526943997496501093_a_nat] :
( ( finite1487985464145237934_a_nat @ F3 )
=> ( ( F3 != bot_bo1033123847703346641_a_nat )
=> ( ~ ( member408289922725080238_a_nat @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert2950094090816004437_a_nat @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_1261_finite__ne__induct,axiom,
! [F2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( F2 != bot_bot_set_a )
=> ( ! [X3: a] : ( P @ ( insert_a @ X3 @ bot_bot_set_a ) )
=> ( ! [X3: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( F3 != bot_bot_set_a )
=> ( ~ ( member_a @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_1262_infinite__finite__induct,axiom,
! [P: set_li6526943997496501093_a_nat > $o,A: set_li6526943997496501093_a_nat] :
( ! [A6: set_li6526943997496501093_a_nat] :
( ~ ( finite1487985464145237934_a_nat @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bo1033123847703346641_a_nat )
=> ( ! [X3: list_Sum_sum_a_nat,F3: set_li6526943997496501093_a_nat] :
( ( finite1487985464145237934_a_nat @ F3 )
=> ( ~ ( member408289922725080238_a_nat @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert2950094090816004437_a_nat @ X3 @ F3 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_1263_infinite__finite__induct,axiom,
! [P: set_a > $o,A: set_a] :
( ! [A6: set_a] :
( ~ ( finite_finite_a @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X3: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X3 @ F3 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_1264_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 )
=> ( ! [A7: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A7 @ A )
=> ( ( ord_less_eq_set_a @ F3 @ A )
=> ( ~ ( member_a @ A7 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ A7 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1265_finite__subset__induct_H,axiom,
! [F2: set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat,P: set_li6526943997496501093_a_nat > $o] :
( ( finite1487985464145237934_a_nat @ F2 )
=> ( ( ord_le1147066620699065093_a_nat @ F2 @ A )
=> ( ( P @ bot_bo1033123847703346641_a_nat )
=> ( ! [A7: list_Sum_sum_a_nat,F3: set_li6526943997496501093_a_nat] :
( ( finite1487985464145237934_a_nat @ F3 )
=> ( ( member408289922725080238_a_nat @ A7 @ A )
=> ( ( ord_le1147066620699065093_a_nat @ F3 @ A )
=> ( ~ ( member408289922725080238_a_nat @ A7 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert2950094090816004437_a_nat @ A7 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1266_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 )
=> ( ! [A7: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A7 @ A )
=> ( ~ ( member_a @ A7 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ A7 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1267_finite__subset__induct,axiom,
! [F2: set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat,P: set_li6526943997496501093_a_nat > $o] :
( ( finite1487985464145237934_a_nat @ F2 )
=> ( ( ord_le1147066620699065093_a_nat @ F2 @ A )
=> ( ( P @ bot_bo1033123847703346641_a_nat )
=> ( ! [A7: list_Sum_sum_a_nat,F3: set_li6526943997496501093_a_nat] :
( ( finite1487985464145237934_a_nat @ F3 )
=> ( ( member408289922725080238_a_nat @ A7 @ A )
=> ( ~ ( member408289922725080238_a_nat @ A7 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert2950094090816004437_a_nat @ A7 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1268_finite__empty__induct,axiom,
! [A: set_a,P: set_a > $o] :
( ( finite_finite_a @ A )
=> ( ( P @ A )
=> ( ! [A7: a,A6: set_a] :
( ( finite_finite_a @ A6 )
=> ( ( member_a @ A7 @ A6 )
=> ( ( P @ A6 )
=> ( P @ ( minus_minus_set_a @ A6 @ ( insert_a @ A7 @ bot_bot_set_a ) ) ) ) ) )
=> ( P @ bot_bot_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_1269_finite__empty__induct,axiom,
! [A: set_li6526943997496501093_a_nat,P: set_li6526943997496501093_a_nat > $o] :
( ( finite1487985464145237934_a_nat @ A )
=> ( ( P @ A )
=> ( ! [A7: list_Sum_sum_a_nat,A6: set_li6526943997496501093_a_nat] :
( ( finite1487985464145237934_a_nat @ A6 )
=> ( ( member408289922725080238_a_nat @ A7 @ A6 )
=> ( ( P @ A6 )
=> ( P @ ( minus_7395159227704179404_a_nat @ A6 @ ( insert2950094090816004437_a_nat @ A7 @ bot_bo1033123847703346641_a_nat ) ) ) ) ) )
=> ( P @ bot_bo1033123847703346641_a_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_1270_infinite__coinduct,axiom,
! [X: set_a > $o,A: set_a] :
( ( X @ A )
=> ( ! [A6: set_a] :
( ( X @ A6 )
=> ? [X6: a] :
( ( member_a @ X6 @ A6 )
& ( ( X @ ( minus_minus_set_a @ A6 @ ( insert_a @ X6 @ bot_bot_set_a ) ) )
| ~ ( finite_finite_a @ ( minus_minus_set_a @ A6 @ ( insert_a @ X6 @ bot_bot_set_a ) ) ) ) ) )
=> ~ ( finite_finite_a @ A ) ) ) ).
% infinite_coinduct
thf(fact_1271_infinite__coinduct,axiom,
! [X: set_li6526943997496501093_a_nat > $o,A: set_li6526943997496501093_a_nat] :
( ( X @ A )
=> ( ! [A6: set_li6526943997496501093_a_nat] :
( ( X @ A6 )
=> ? [X6: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X6 @ A6 )
& ( ( X @ ( minus_7395159227704179404_a_nat @ A6 @ ( insert2950094090816004437_a_nat @ X6 @ bot_bo1033123847703346641_a_nat ) ) )
| ~ ( finite1487985464145237934_a_nat @ ( minus_7395159227704179404_a_nat @ A6 @ ( insert2950094090816004437_a_nat @ X6 @ bot_bo1033123847703346641_a_nat ) ) ) ) ) )
=> ~ ( finite1487985464145237934_a_nat @ A ) ) ) ).
% infinite_coinduct
thf(fact_1272_infinite__remove,axiom,
! [S: set_a,A2: a] :
( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ ( insert_a @ A2 @ bot_bot_set_a ) ) ) ) ).
% infinite_remove
thf(fact_1273_infinite__remove,axiom,
! [S: set_li6526943997496501093_a_nat,A2: list_Sum_sum_a_nat] :
( ~ ( finite1487985464145237934_a_nat @ S )
=> ~ ( finite1487985464145237934_a_nat @ ( minus_7395159227704179404_a_nat @ S @ ( insert2950094090816004437_a_nat @ A2 @ bot_bo1033123847703346641_a_nat ) ) ) ) ).
% infinite_remove
thf(fact_1274_finite__Inf__in,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ! [X3: set_a,Y3: set_a] :
( ( member_set_a @ X3 @ A )
=> ( ( member_set_a @ Y3 @ A )
=> ( member_set_a @ ( inf_inf_set_a @ X3 @ Y3 ) @ A ) ) )
=> ( member_set_a @ ( comple6135023378680113637_set_a @ A ) @ A ) ) ) ) ).
% finite_Inf_in
thf(fact_1275_finite__remove__induct,axiom,
! [B: set_a,P: set_a > $o] :
( ( finite_finite_a @ B )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A6: set_a] :
( ( finite_finite_a @ A6 )
=> ( ( A6 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A6 @ B )
=> ( ! [X6: a] :
( ( member_a @ X6 @ A6 )
=> ( P @ ( minus_minus_set_a @ A6 @ ( insert_a @ X6 @ bot_bot_set_a ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_1276_finite__remove__induct,axiom,
! [B: set_li6526943997496501093_a_nat,P: set_li6526943997496501093_a_nat > $o] :
( ( finite1487985464145237934_a_nat @ B )
=> ( ( P @ bot_bo1033123847703346641_a_nat )
=> ( ! [A6: set_li6526943997496501093_a_nat] :
( ( finite1487985464145237934_a_nat @ A6 )
=> ( ( A6 != bot_bo1033123847703346641_a_nat )
=> ( ( ord_le1147066620699065093_a_nat @ A6 @ B )
=> ( ! [X6: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X6 @ A6 )
=> ( P @ ( minus_7395159227704179404_a_nat @ A6 @ ( insert2950094090816004437_a_nat @ X6 @ bot_bo1033123847703346641_a_nat ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
% Conjectures (4)
thf(conj_0,hypothesis,
member408289922725080238_a_nat @ z @ ( ad_agr_close_set_a @ ( minus_minus_set_a @ ad @ ad2 ) @ x ) ).
thf(conj_1,hypothesis,
! [X6: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X6 @ x )
=> ( fo_nmlzd_a @ ad2 @ X6 ) ) ).
thf(conj_2,hypothesis,
ord_less_eq_set_a @ ad2 @ ad ).
thf(conj_3,conjecture,
fo_nmlzd_a @ ad @ z ).
%------------------------------------------------------------------------------