TPTP Problem File: SLH0737^1.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_00255_009590__15517662_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1646 ( 568 unt; 360 typ; 0 def)
% Number of atoms : 3718 (1225 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 11156 ( 499 ~; 53 |; 280 &;8753 @)
% ( 0 <=>;1571 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 7 avg)
% Number of types : 43 ( 42 usr)
% Number of type conns : 1492 (1492 >; 0 *; 0 +; 0 <<)
% Number of symbols : 321 ( 318 usr; 26 con; 0-7 aty)
% Number of variables : 3919 ( 270 ^;3466 !; 183 ?;3919 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 13:05:41.509
%------------------------------------------------------------------------------
% Could-be-implicit typings (42)
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thf(sy_c_Set__Impl_Oset__aux_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mtf__b_J_J,type,
set_se1939246924176758703um_a_b: set_set_impl > list_l4199846171218662726um_a_b > set_list_Sum_sum_a_b ).
thf(sy_c_Set__Impl_Oset__aux_001t__List__Olist_Itf__a_J,type,
set_set_aux_list_a: set_set_impl > list_list_a > set_list_a ).
thf(sy_c_Set__Impl_Oset__aux_001t__Sum____Type__Osum_Itf__a_Mtf__b_J,type,
set_se7976317804018791967um_a_b: set_set_impl > list_Sum_sum_a_b > set_Sum_sum_a_b ).
thf(sy_c_Set__Impl_Oset__aux_001tf__a,type,
set_set_aux_a: set_set_impl > list_a > set_a ).
thf(sy_c_Set__Linorder_Oord__class_Oset__less__aux_001t__Set__Oset_Itf__a_J,type,
set_or6638329327428242490_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Sum__Type_OInl_001t__Sum____Type__Osum_Itf__a_Mtf__b_J_001t__Sum____Type__Osum_Itf__a_Mtf__b_J,type,
sum_In5992699931424873788um_a_b: sum_sum_a_b > sum_su3067303292148767147um_a_b ).
thf(sy_c_Sum__Type_OInl_001t__Sum____Type__Osum_Itf__a_Mtf__b_J_001tf__a,type,
sum_In6222238883715738344_a_b_a: sum_sum_a_b > sum_su3831877439928360143_a_b_a ).
thf(sy_c_Sum__Type_OInl_001tf__a_001t__Sum____Type__Osum_Itf__a_Mtf__b_J,type,
sum_In2918432097854297526um_a_b: a > sum_su5898878462909468885um_a_b ).
thf(sy_c_Sum__Type_OInl_001tf__a_001tf__a,type,
sum_Inl_a_a: a > sum_sum_a_a ).
thf(sy_c_Sum__Type_OInl_001tf__a_001tf__b,type,
sum_Inl_a_b: a > sum_sum_a_b ).
thf(sy_c_Sum__Type_OInr_001t__Sum____Type__Osum_Itf__a_Mtf__b_J_001t__Sum____Type__Osum_Itf__a_Mtf__b_J,type,
sum_In6272690418125038914um_a_b: sum_sum_a_b > sum_su3067303292148767147um_a_b ).
thf(sy_c_Sum__Type_OInr_001t__Sum____Type__Osum_Itf__a_Mtf__b_J_001tf__a,type,
sum_In2710243301657490530_a_b_a: sum_sum_a_b > sum_su5898878462909468885um_a_b ).
thf(sy_c_Sum__Type_OInr_001tf__a_001t__Sum____Type__Osum_Itf__a_Mtf__b_J,type,
sum_In8629808552650825520um_a_b: a > sum_su3831877439928360143_a_b_a ).
thf(sy_c_Sum__Type_OInr_001tf__a_001tf__a,type,
sum_Inr_a_a: a > sum_sum_a_a ).
thf(sy_c_Sum__Type_OInr_001tf__b_001tf__a,type,
sum_Inr_b_a: b > sum_sum_a_b ).
thf(sy_c_Sum__Type_OPlus_001t__Sum____Type__Osum_Itf__a_Mtf__b_J_001t__Sum____Type__Osum_Itf__a_Mtf__b_J,type,
sum_Pl6362402249328468033um_a_b: set_Sum_sum_a_b > set_Sum_sum_a_b > set_Su2595742659359353697um_a_b ).
thf(sy_c_Sum__Type_OPlus_001t__Sum____Type__Osum_Itf__a_Mtf__b_J_001tf__a,type,
sum_Pl4155217190204358947_a_b_a: set_Sum_sum_a_b > set_a > set_Su6333382036065364015_a_b_a ).
thf(sy_c_Sum__Type_OPlus_001tf__a_001t__Sum____Type__Osum_Itf__a_Mtf__b_J,type,
sum_Pl851410404342918129um_a_b: set_a > set_Sum_sum_a_b > set_Su126771859443611957um_a_b ).
thf(sy_c_Sum__Type_OPlus_001tf__a_001tf__a,type,
sum_Plus_a_a: set_a > set_a > set_Sum_sum_a_a ).
thf(sy_c_Sum__Type_OPlus_001tf__a_001tf__b,type,
sum_Plus_a_b: set_a > set_b > set_Sum_sum_a_b ).
thf(sy_c_Topological__Spaces_Omonoseq_001t__Set__Oset_Itf__a_J,type,
topolo725164666729632753_set_a: ( nat > set_a ) > $o ).
thf(sy_c_Zorn_Ochains_001tf__a,type,
chains_a: set_set_a > set_set_set_a ).
thf(sy_c_member_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mtf__b_J_J,type,
member7701661377270014157um_a_b: list_Sum_sum_a_b > set_list_Sum_sum_a_b > $o ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a2: list_a > set_list_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mtf__b_J_J_J,type,
member1385076861102201347um_a_b: set_list_Sum_sum_a_b > set_se1932129030832874786um_a_b > $o ).
thf(sy_c_member_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
member_set_list_a: set_list_a > set_set_list_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
member_set_set_a2: set_set_a > set_set_set_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mtf__b_J_J,type,
member4060935254435997939um_a_b: set_Sum_sum_a_b > set_set_Sum_sum_a_b > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a2: set_a > set_set_a > $o ).
thf(sy_c_member_001t__Sum____Type__Osum_It__Sum____Type__Osum_Itf__a_Mtf__b_J_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J,type,
member7265535425885436866um_a_b: sum_su3067303292148767147um_a_b > set_Su2595742659359353697um_a_b > $o ).
thf(sy_c_member_001t__Sum____Type__Osum_It__Sum____Type__Osum_Itf__a_Mtf__b_J_Mtf__a_J,type,
member1807623034088458360_a_b_a: sum_su3831877439928360143_a_b_a > set_Su6333382036065364015_a_b_a > $o ).
thf(sy_c_member_001t__Sum____Type__Osum_Itf__a_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J,type,
member3874624057069567102um_a_b: sum_su5898878462909468885um_a_b > set_Su126771859443611957um_a_b > $o ).
thf(sy_c_member_001t__Sum____Type__Osum_Itf__a_Mtf__a_J,type,
member_Sum_sum_a_a: sum_sum_a_a > set_Sum_sum_a_a > $o ).
thf(sy_c_member_001t__Sum____Type__Osum_Itf__a_Mtf__b_J,type,
member_Sum_sum_a_b2: sum_sum_a_b > set_Sum_sum_a_b > $o ).
thf(sy_c_member_001tf__a,type,
member_a2: a > set_a > $o ).
thf(sy_c_member_001tf__b,type,
member_b2: b > set_b > $o ).
thf(sy_v_X,type,
x: set_a ).
thf(sy_v_xs,type,
xs: list_Sum_sum_a_b ).
thf(sy_v_y,type,
y: a ).
thf(sy_v_ys,type,
ys: list_Sum_sum_a_b ).
% Relevant facts (1265)
thf(fact_0_ad__agr__list__comm,axiom,
! [X: set_a,Xs: list_Sum_sum_a_b,Ys: list_Sum_sum_a_b] :
( ( ad_agr_list_a_b @ X @ Xs @ Ys )
=> ( ad_agr_list_a_b @ X @ Ys @ Xs ) ) ).
% ad_agr_list_comm
thf(fact_1_ad__agr__list__refl,axiom,
! [X: set_a,Xs: list_Sum_sum_a_b] : ( ad_agr_list_a_b @ X @ Xs @ Xs ) ).
% ad_agr_list_refl
thf(fact_2_ad__agr__list__trans,axiom,
! [X: set_a,Xs: list_Sum_sum_a_b,Ys: list_Sum_sum_a_b,Zs: list_Sum_sum_a_b] :
( ( ad_agr_list_a_b @ X @ Xs @ Ys )
=> ( ( ad_agr_list_a_b @ X @ Ys @ Zs )
=> ( ad_agr_list_a_b @ X @ Xs @ Zs ) ) ) ).
% ad_agr_list_trans
thf(fact_3_sum_Oinject_I1_J,axiom,
! [X1: sum_sum_a_b,Y1: sum_sum_a_b] :
( ( ( sum_In5992699931424873788um_a_b @ X1 )
= ( sum_In5992699931424873788um_a_b @ Y1 ) )
= ( X1 = Y1 ) ) ).
% sum.inject(1)
thf(fact_4_sum_Oinject_I1_J,axiom,
! [X1: sum_sum_a_b,Y1: sum_sum_a_b] :
( ( ( sum_In6222238883715738344_a_b_a @ X1 )
= ( sum_In6222238883715738344_a_b_a @ Y1 ) )
= ( X1 = Y1 ) ) ).
% sum.inject(1)
thf(fact_5_sum_Oinject_I1_J,axiom,
! [X1: a,Y1: a] :
( ( ( sum_In2918432097854297526um_a_b @ X1 )
= ( sum_In2918432097854297526um_a_b @ Y1 ) )
= ( X1 = Y1 ) ) ).
% sum.inject(1)
thf(fact_6_sum_Oinject_I1_J,axiom,
! [X1: a,Y1: a] :
( ( ( sum_Inl_a_a @ X1 )
= ( sum_Inl_a_a @ Y1 ) )
= ( X1 = Y1 ) ) ).
% sum.inject(1)
thf(fact_7_sum_Oinject_I1_J,axiom,
! [X1: a,Y1: a] :
( ( ( sum_Inl_a_b @ X1 )
= ( sum_Inl_a_b @ Y1 ) )
= ( X1 = Y1 ) ) ).
% sum.inject(1)
thf(fact_8_old_Osum_Oinject_I1_J,axiom,
! [A: sum_sum_a_b,A2: sum_sum_a_b] :
( ( ( sum_In5992699931424873788um_a_b @ A )
= ( sum_In5992699931424873788um_a_b @ A2 ) )
= ( A = A2 ) ) ).
% old.sum.inject(1)
thf(fact_9_old_Osum_Oinject_I1_J,axiom,
! [A: sum_sum_a_b,A2: sum_sum_a_b] :
( ( ( sum_In6222238883715738344_a_b_a @ A )
= ( sum_In6222238883715738344_a_b_a @ A2 ) )
= ( A = A2 ) ) ).
% old.sum.inject(1)
thf(fact_10_old_Osum_Oinject_I1_J,axiom,
! [A: a,A2: a] :
( ( ( sum_In2918432097854297526um_a_b @ A )
= ( sum_In2918432097854297526um_a_b @ A2 ) )
= ( A = A2 ) ) ).
% old.sum.inject(1)
thf(fact_11_old_Osum_Oinject_I1_J,axiom,
! [A: a,A2: a] :
( ( ( sum_Inl_a_a @ A )
= ( sum_Inl_a_a @ A2 ) )
= ( A = A2 ) ) ).
% old.sum.inject(1)
thf(fact_12_old_Osum_Oinject_I1_J,axiom,
! [A: a,A2: a] :
( ( ( sum_Inl_a_b @ A )
= ( sum_Inl_a_b @ A2 ) )
= ( A = A2 ) ) ).
% old.sum.inject(1)
thf(fact_13_Inl__inject,axiom,
! [X2: sum_sum_a_b,Y: sum_sum_a_b] :
( ( ( sum_In5992699931424873788um_a_b @ X2 )
= ( sum_In5992699931424873788um_a_b @ Y ) )
=> ( X2 = Y ) ) ).
% Inl_inject
thf(fact_14_Inl__inject,axiom,
! [X2: sum_sum_a_b,Y: sum_sum_a_b] :
( ( ( sum_In6222238883715738344_a_b_a @ X2 )
= ( sum_In6222238883715738344_a_b_a @ Y ) )
=> ( X2 = Y ) ) ).
% Inl_inject
thf(fact_15_Inl__inject,axiom,
! [X2: a,Y: a] :
( ( ( sum_In2918432097854297526um_a_b @ X2 )
= ( sum_In2918432097854297526um_a_b @ Y ) )
=> ( X2 = Y ) ) ).
% Inl_inject
thf(fact_16_Inl__inject,axiom,
! [X2: a,Y: a] :
( ( ( sum_Inl_a_a @ X2 )
= ( sum_Inl_a_a @ Y ) )
=> ( X2 = Y ) ) ).
% Inl_inject
thf(fact_17_Inl__inject,axiom,
! [X2: a,Y: a] :
( ( ( sum_Inl_a_b @ X2 )
= ( sum_Inl_a_b @ Y ) )
=> ( X2 = Y ) ) ).
% Inl_inject
thf(fact_18_ad__agr__list__rev__mono,axiom,
! [Y2: set_a,X: set_a,Ys: list_S1422458098724880219um_a_b,Xs: list_S1422458098724880219um_a_b] :
( ( ord_less_eq_set_a @ Y2 @ X )
=> ( ( ad_agr9091678679625263750um_a_b @ Y2 @ Ys @ Xs )
=> ( ( ord_less_eq_set_a @ ( vimage6946176754568057234um_a_b @ sum_In2918432097854297526um_a_b @ ( set_Su7489568695747919082um_a_b @ Xs ) ) @ Y2 )
=> ( ( ord_less_eq_set_a @ ( vimage6946176754568057234um_a_b @ sum_In2918432097854297526um_a_b @ ( set_Su7489568695747919082um_a_b @ Ys ) ) @ Y2 )
=> ( ad_agr9091678679625263750um_a_b @ X @ Ys @ Xs ) ) ) ) ) ).
% ad_agr_list_rev_mono
thf(fact_19_ad__agr__list__rev__mono,axiom,
! [Y2: set_a,X: set_a,Ys: list_Sum_sum_a_a,Xs: list_Sum_sum_a_a] :
( ( ord_less_eq_set_a @ Y2 @ X )
=> ( ( ad_agr_list_a_a @ Y2 @ Ys @ Xs )
=> ( ( ord_less_eq_set_a @ ( vimage_a_Sum_sum_a_a @ sum_Inl_a_a @ ( set_Sum_sum_a_a2 @ Xs ) ) @ Y2 )
=> ( ( ord_less_eq_set_a @ ( vimage_a_Sum_sum_a_a @ sum_Inl_a_a @ ( set_Sum_sum_a_a2 @ Ys ) ) @ Y2 )
=> ( ad_agr_list_a_a @ X @ Ys @ Xs ) ) ) ) ) ).
% ad_agr_list_rev_mono
thf(fact_20_ad__agr__list__rev__mono,axiom,
! [Y2: set_Sum_sum_a_b,X: set_Sum_sum_a_b,Ys: list_S6010472233439964091um_a_b,Xs: list_S6010472233439964091um_a_b] :
( ( ord_le9019793522827316924um_a_b @ Y2 @ X )
=> ( ( ad_agr4670348907520218220um_a_b @ Y2 @ Ys @ Xs )
=> ( ( ord_le9019793522827316924um_a_b @ ( vimage3339171716227007374um_a_b @ sum_In5992699931424873788um_a_b @ ( set_Su2529850101214429270um_a_b @ Xs ) ) @ Y2 )
=> ( ( ord_le9019793522827316924um_a_b @ ( vimage3339171716227007374um_a_b @ sum_In5992699931424873788um_a_b @ ( set_Su2529850101214429270um_a_b @ Ys ) ) @ Y2 )
=> ( ad_agr4670348907520218220um_a_b @ X @ Ys @ Xs ) ) ) ) ) ).
% ad_agr_list_rev_mono
thf(fact_21_ad__agr__list__rev__mono,axiom,
! [Y2: set_Sum_sum_a_b,X: set_Sum_sum_a_b,Ys: list_S7629068275346632277_a_b_a,Xs: list_S7629068275346632277_a_b_a] :
( ( ord_le9019793522827316924um_a_b @ Y2 @ X )
=> ( ( ad_agr3172113428631928760_a_b_a @ Y2 @ Ys @ Xs )
=> ( ( ord_le9019793522827316924um_a_b @ ( vimage8170263477503431596_a_b_a @ sum_In6222238883715738344_a_b_a @ ( set_Su5422567672766810340_a_b_a @ Xs ) ) @ Y2 )
=> ( ( ord_le9019793522827316924um_a_b @ ( vimage8170263477503431596_a_b_a @ sum_In6222238883715738344_a_b_a @ ( set_Su5422567672766810340_a_b_a @ Ys ) ) @ Y2 )
=> ( ad_agr3172113428631928760_a_b_a @ X @ Ys @ Xs ) ) ) ) ) ).
% ad_agr_list_rev_mono
thf(fact_22_ad__agr__list__rev__mono,axiom,
! [Y2: set_a,X: set_a,Ys: list_Sum_sum_a_b,Xs: list_Sum_sum_a_b] :
( ( ord_less_eq_set_a @ Y2 @ X )
=> ( ( ad_agr_list_a_b @ Y2 @ Ys @ Xs )
=> ( ( ord_less_eq_set_a @ ( vimage_a_Sum_sum_a_b @ sum_Inl_a_b @ ( set_Sum_sum_a_b2 @ Xs ) ) @ Y2 )
=> ( ( ord_less_eq_set_a @ ( vimage_a_Sum_sum_a_b @ sum_Inl_a_b @ ( set_Sum_sum_a_b2 @ Ys ) ) @ Y2 )
=> ( ad_agr_list_a_b @ X @ Ys @ Xs ) ) ) ) ) ).
% ad_agr_list_rev_mono
thf(fact_23_ad__agr__list__mono,axiom,
! [X: set_a,Y2: set_a,Ys: list_Sum_sum_a_b,Xs: list_Sum_sum_a_b] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ( ad_agr_list_a_b @ Y2 @ Ys @ Xs )
=> ( ad_agr_list_a_b @ X @ Ys @ Xs ) ) ) ).
% ad_agr_list_mono
thf(fact_24_equality__sum__simps_I1_J,axiom,
! [Eq_a: sum_sum_a_b > sum_sum_a_b > $o,Eq_b: sum_sum_a_b > sum_sum_a_b > $o,X2: sum_sum_a_b,Y: sum_sum_a_b] :
( ( equali7701304696083298663um_a_b @ Eq_a @ Eq_b @ ( sum_In5992699931424873788um_a_b @ X2 ) @ ( sum_In5992699931424873788um_a_b @ Y ) )
= ( Eq_a @ X2 @ Y ) ) ).
% equality_sum_simps(1)
thf(fact_25_equality__sum__simps_I1_J,axiom,
! [Eq_a: sum_sum_a_b > sum_sum_a_b > $o,Eq_b: a > a > $o,X2: sum_sum_a_b,Y: sum_sum_a_b] :
( ( equali6768406920669315453_a_b_a @ Eq_a @ Eq_b @ ( sum_In6222238883715738344_a_b_a @ X2 ) @ ( sum_In6222238883715738344_a_b_a @ Y ) )
= ( Eq_a @ X2 @ Y ) ) ).
% equality_sum_simps(1)
thf(fact_26_equality__sum__simps_I1_J,axiom,
! [Eq_a: a > a > $o,Eq_b: sum_sum_a_b > sum_sum_a_b > $o,X2: a,Y: a] :
( ( equali3464600134807874635um_a_b @ Eq_a @ Eq_b @ ( sum_In2918432097854297526um_a_b @ X2 ) @ ( sum_In2918432097854297526um_a_b @ Y ) )
= ( Eq_a @ X2 @ Y ) ) ).
% equality_sum_simps(1)
thf(fact_27_equality__sum__simps_I1_J,axiom,
! [Eq_a: a > a > $o,Eq_b: a > a > $o,X2: a,Y: a] :
( ( equali5972598040643577753um_a_a @ Eq_a @ Eq_b @ ( sum_Inl_a_a @ X2 ) @ ( sum_Inl_a_a @ Y ) )
= ( Eq_a @ X2 @ Y ) ) ).
% equality_sum_simps(1)
thf(fact_28_equality__sum__simps_I1_J,axiom,
! [Eq_a: a > a > $o,Eq_b: b > b > $o,X2: a,Y: a] :
( ( equali5972598040643577754um_a_b @ Eq_a @ Eq_b @ ( sum_Inl_a_b @ X2 ) @ ( sum_Inl_a_b @ Y ) )
= ( Eq_a @ X2 @ Y ) ) ).
% equality_sum_simps(1)
thf(fact_29_in__set__member,axiom,
! [X2: set_Sum_sum_a_b,Xs: list_set_Sum_sum_a_b] :
( ( member4060935254435997939um_a_b @ X2 @ ( set_set_Sum_sum_a_b2 @ Xs ) )
= ( member4336013021441118209um_a_b @ Xs @ X2 ) ) ).
% in_set_member
thf(fact_30_in__set__member,axiom,
! [X2: set_set_a,Xs: list_set_set_a] :
( ( member_set_set_a2 @ X2 @ ( set_set_set_a2 @ Xs ) )
= ( member_set_set_a @ Xs @ X2 ) ) ).
% in_set_member
thf(fact_31_in__set__member,axiom,
! [X2: set_a,Xs: list_set_a] :
( ( member_set_a2 @ X2 @ ( set_set_a2 @ Xs ) )
= ( member_set_a @ Xs @ X2 ) ) ).
% in_set_member
thf(fact_32_in__set__member,axiom,
! [X2: b,Xs: list_b] :
( ( member_b2 @ X2 @ ( set_b2 @ Xs ) )
= ( member_b @ Xs @ X2 ) ) ).
% in_set_member
thf(fact_33_in__set__member,axiom,
! [X2: list_a,Xs: list_list_a] :
( ( member_list_a2 @ X2 @ ( set_list_a2 @ Xs ) )
= ( member_list_a @ Xs @ X2 ) ) ).
% in_set_member
thf(fact_34_in__set__member,axiom,
! [X2: list_Sum_sum_a_b,Xs: list_l4199846171218662726um_a_b] :
( ( member7701661377270014157um_a_b @ X2 @ ( set_list_Sum_sum_a_b2 @ Xs ) )
= ( member9050564485692678363um_a_b @ Xs @ X2 ) ) ).
% in_set_member
thf(fact_35_in__set__member,axiom,
! [X2: a,Xs: list_a] :
( ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
= ( member_a @ Xs @ X2 ) ) ).
% in_set_member
thf(fact_36_in__set__member,axiom,
! [X2: sum_sum_a_b,Xs: list_Sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X2 @ ( set_Sum_sum_a_b2 @ Xs ) )
= ( member_Sum_sum_a_b @ Xs @ X2 ) ) ).
% in_set_member
thf(fact_37_set__aux__def,axiom,
( set_set_aux_a
= ( ^ [Uu: set_set_impl] : set_a2 ) ) ).
% set_aux_def
thf(fact_38_set__aux__def,axiom,
( set_set_aux_list_a
= ( ^ [Uu: set_set_impl] : set_list_a2 ) ) ).
% set_aux_def
thf(fact_39_set__aux__def,axiom,
( set_se1939246924176758703um_a_b
= ( ^ [Uu: set_set_impl] : set_list_Sum_sum_a_b2 ) ) ).
% set_aux_def
thf(fact_40_set__aux__def,axiom,
( set_se7976317804018791967um_a_b
= ( ^ [Uu: set_set_impl] : set_Sum_sum_a_b2 ) ) ).
% set_aux_def
thf(fact_41_setlp_Ocases,axiom,
! [S: sum_su3067303292148767147um_a_b,A: sum_sum_a_b] :
( ( basic_6528986648383318361um_a_b @ S @ A )
=> ( S
= ( sum_In5992699931424873788um_a_b @ A ) ) ) ).
% setlp.cases
thf(fact_42_setlp_Ocases,axiom,
! [S: sum_su3831877439928360143_a_b_a,A: sum_sum_a_b] :
( ( basic_2132064382650842635_a_b_a @ S @ A )
=> ( S
= ( sum_In6222238883715738344_a_b_a @ A ) ) ) ).
% setlp.cases
thf(fact_43_setlp_Ocases,axiom,
! [S: sum_su5898878462909468885um_a_b,A: a] :
( ( basic_8051629633644177625um_a_b @ S @ A )
=> ( S
= ( sum_In2918432097854297526um_a_b @ A ) ) ) ).
% setlp.cases
thf(fact_44_setlp_Ocases,axiom,
! [S: sum_sum_a_a,A: a] :
( ( basic_setlp_a_a @ S @ A )
=> ( S
= ( sum_Inl_a_a @ A ) ) ) ).
% setlp.cases
thf(fact_45_setlp_Ocases,axiom,
! [S: sum_sum_a_b,A: a] :
( ( basic_setlp_a_b @ S @ A )
=> ( S
= ( sum_Inl_a_b @ A ) ) ) ).
% setlp.cases
thf(fact_46_subset__code_I1_J,axiom,
! [Xs: list_set_Sum_sum_a_b,B: set_set_Sum_sum_a_b] :
( ( ord_le1944875106711258738um_a_b @ ( set_set_Sum_sum_a_b2 @ Xs ) @ B )
= ( ! [X3: set_Sum_sum_a_b] :
( ( member4060935254435997939um_a_b @ X3 @ ( set_set_Sum_sum_a_b2 @ Xs ) )
=> ( member4060935254435997939um_a_b @ X3 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_47_subset__code_I1_J,axiom,
! [Xs: list_set_set_a,B: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ ( set_set_set_a2 @ Xs ) @ B )
= ( ! [X3: set_set_a] :
( ( member_set_set_a2 @ X3 @ ( set_set_set_a2 @ Xs ) )
=> ( member_set_set_a2 @ X3 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_48_subset__code_I1_J,axiom,
! [Xs: list_b,B: set_b] :
( ( ord_less_eq_set_b @ ( set_b2 @ Xs ) @ B )
= ( ! [X3: b] :
( ( member_b2 @ X3 @ ( set_b2 @ Xs ) )
=> ( member_b2 @ X3 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_49_subset__code_I1_J,axiom,
! [Xs: list_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ B )
= ( ! [X3: list_a] :
( ( member_list_a2 @ X3 @ ( set_list_a2 @ Xs ) )
=> ( member_list_a2 @ X3 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_50_subset__code_I1_J,axiom,
! [Xs: list_l4199846171218662726um_a_b,B: set_list_Sum_sum_a_b] :
( ( ord_le2472362315733485388um_a_b @ ( set_list_Sum_sum_a_b2 @ Xs ) @ B )
= ( ! [X3: list_Sum_sum_a_b] :
( ( member7701661377270014157um_a_b @ X3 @ ( set_list_Sum_sum_a_b2 @ Xs ) )
=> ( member7701661377270014157um_a_b @ X3 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_51_subset__code_I1_J,axiom,
! [Xs: list_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ Xs ) @ B )
= ( ! [X3: set_a] :
( ( member_set_a2 @ X3 @ ( set_set_a2 @ Xs ) )
=> ( member_set_a2 @ X3 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_52_subset__code_I1_J,axiom,
! [Xs: list_Sum_sum_a_b,B: set_Sum_sum_a_b] :
( ( ord_le9019793522827316924um_a_b @ ( set_Sum_sum_a_b2 @ Xs ) @ B )
= ( ! [X3: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X3 @ ( set_Sum_sum_a_b2 @ Xs ) )
=> ( member_Sum_sum_a_b2 @ X3 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_53_subset__code_I1_J,axiom,
! [Xs: list_a,B: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ B )
= ( ! [X3: a] :
( ( member_a2 @ X3 @ ( set_a2 @ Xs ) )
=> ( member_a2 @ X3 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_54_setlp_Ointros,axiom,
! [S: sum_su3067303292148767147um_a_b,X2: sum_sum_a_b] :
( ( S
= ( sum_In5992699931424873788um_a_b @ X2 ) )
=> ( basic_6528986648383318361um_a_b @ S @ X2 ) ) ).
% setlp.intros
thf(fact_55_setlp_Ointros,axiom,
! [S: sum_su3831877439928360143_a_b_a,X2: sum_sum_a_b] :
( ( S
= ( sum_In6222238883715738344_a_b_a @ X2 ) )
=> ( basic_2132064382650842635_a_b_a @ S @ X2 ) ) ).
% setlp.intros
thf(fact_56_setlp_Ointros,axiom,
! [S: sum_su5898878462909468885um_a_b,X2: a] :
( ( S
= ( sum_In2918432097854297526um_a_b @ X2 ) )
=> ( basic_8051629633644177625um_a_b @ S @ X2 ) ) ).
% setlp.intros
thf(fact_57_setlp_Ointros,axiom,
! [S: sum_sum_a_a,X2: a] :
( ( S
= ( sum_Inl_a_a @ X2 ) )
=> ( basic_setlp_a_a @ S @ X2 ) ) ).
% setlp.intros
thf(fact_58_setlp_Ointros,axiom,
! [S: sum_sum_a_b,X2: a] :
( ( S
= ( sum_Inl_a_b @ X2 ) )
=> ( basic_setlp_a_b @ S @ X2 ) ) ).
% setlp.intros
thf(fact_59_setlp_Osimps,axiom,
( basic_6528986648383318361um_a_b
= ( ^ [S2: sum_su3067303292148767147um_a_b,A3: sum_sum_a_b] :
? [X3: sum_sum_a_b] :
( ( A3 = X3 )
& ( S2
= ( sum_In5992699931424873788um_a_b @ X3 ) ) ) ) ) ).
% setlp.simps
thf(fact_60_setlp_Osimps,axiom,
( basic_2132064382650842635_a_b_a
= ( ^ [S2: sum_su3831877439928360143_a_b_a,A3: sum_sum_a_b] :
? [X3: sum_sum_a_b] :
( ( A3 = X3 )
& ( S2
= ( sum_In6222238883715738344_a_b_a @ X3 ) ) ) ) ) ).
% setlp.simps
thf(fact_61_setlp_Osimps,axiom,
( basic_8051629633644177625um_a_b
= ( ^ [S2: sum_su5898878462909468885um_a_b,A3: a] :
? [X3: a] :
( ( A3 = X3 )
& ( S2
= ( sum_In2918432097854297526um_a_b @ X3 ) ) ) ) ) ).
% setlp.simps
thf(fact_62_setlp_Osimps,axiom,
( basic_setlp_a_a
= ( ^ [S2: sum_sum_a_a,A3: a] :
? [X3: a] :
( ( A3 = X3 )
& ( S2
= ( sum_Inl_a_a @ X3 ) ) ) ) ) ).
% setlp.simps
thf(fact_63_setlp_Osimps,axiom,
( basic_setlp_a_b
= ( ^ [S2: sum_sum_a_b,A3: a] :
? [X3: a] :
( ( A3 = X3 )
& ( S2
= ( sum_Inl_a_b @ X3 ) ) ) ) ) ).
% setlp.simps
thf(fact_64_vimageI,axiom,
! [F: a > a,A: a,B2: a,B: set_a] :
( ( ( F @ A )
= B2 )
=> ( ( member_a2 @ B2 @ B )
=> ( member_a2 @ A @ ( vimage_a_a @ F @ B ) ) ) ) ).
% vimageI
thf(fact_65_vimageI,axiom,
! [F: sum_sum_a_b > a,A: sum_sum_a_b,B2: a,B: set_a] :
( ( ( F @ A )
= B2 )
=> ( ( member_a2 @ B2 @ B )
=> ( member_Sum_sum_a_b2 @ A @ ( vimage_Sum_sum_a_b_a @ F @ B ) ) ) ) ).
% vimageI
thf(fact_66_vimageI,axiom,
! [F: a > sum_sum_a_b,A: a,B2: sum_sum_a_b,B: set_Sum_sum_a_b] :
( ( ( F @ A )
= B2 )
=> ( ( member_Sum_sum_a_b2 @ B2 @ B )
=> ( member_a2 @ A @ ( vimage_a_Sum_sum_a_b @ F @ B ) ) ) ) ).
% vimageI
thf(fact_67_vimageI,axiom,
! [F: sum_sum_a_b > sum_sum_a_b,A: sum_sum_a_b,B2: sum_sum_a_b,B: set_Sum_sum_a_b] :
( ( ( F @ A )
= B2 )
=> ( ( member_Sum_sum_a_b2 @ B2 @ B )
=> ( member_Sum_sum_a_b2 @ A @ ( vimage4749860508033320969um_a_b @ F @ B ) ) ) ) ).
% vimageI
thf(fact_68_vimageI,axiom,
! [F: b > a,A: b,B2: a,B: set_a] :
( ( ( F @ A )
= B2 )
=> ( ( member_a2 @ B2 @ B )
=> ( member_b2 @ A @ ( vimage_b_a @ F @ B ) ) ) ) ).
% vimageI
thf(fact_69_vimageI,axiom,
! [F: a > b,A: a,B2: b,B: set_b] :
( ( ( F @ A )
= B2 )
=> ( ( member_b2 @ B2 @ B )
=> ( member_a2 @ A @ ( vimage_a_b @ F @ B ) ) ) ) ).
% vimageI
thf(fact_70_vimageI,axiom,
! [F: b > b,A: b,B2: b,B: set_b] :
( ( ( F @ A )
= B2 )
=> ( ( member_b2 @ B2 @ B )
=> ( member_b2 @ A @ ( vimage_b_b @ F @ B ) ) ) ) ).
% vimageI
thf(fact_71_vimageI,axiom,
! [F: list_a > a,A: list_a,B2: a,B: set_a] :
( ( ( F @ A )
= B2 )
=> ( ( member_a2 @ B2 @ B )
=> ( member_list_a2 @ A @ ( vimage_list_a_a @ F @ B ) ) ) ) ).
% vimageI
thf(fact_72_vimageI,axiom,
! [F: set_a > a,A: set_a,B2: a,B: set_a] :
( ( ( F @ A )
= B2 )
=> ( ( member_a2 @ B2 @ B )
=> ( member_set_a2 @ A @ ( vimage_set_a_a @ F @ B ) ) ) ) ).
% vimageI
thf(fact_73_vimageI,axiom,
! [F: a > list_a,A: a,B2: list_a,B: set_list_a] :
( ( ( F @ A )
= B2 )
=> ( ( member_list_a2 @ B2 @ B )
=> ( member_a2 @ A @ ( vimage_a_list_a @ F @ B ) ) ) ) ).
% vimageI
thf(fact_74_vimage__eq,axiom,
! [A: a,F: a > a,B: set_a] :
( ( member_a2 @ A @ ( vimage_a_a @ F @ B ) )
= ( member_a2 @ ( F @ A ) @ B ) ) ).
% vimage_eq
thf(fact_75_vimage__eq,axiom,
! [A: sum_sum_a_b,F: sum_sum_a_b > a,B: set_a] :
( ( member_Sum_sum_a_b2 @ A @ ( vimage_Sum_sum_a_b_a @ F @ B ) )
= ( member_a2 @ ( F @ A ) @ B ) ) ).
% vimage_eq
thf(fact_76_vimage__eq,axiom,
! [A: a,F: a > sum_sum_a_b,B: set_Sum_sum_a_b] :
( ( member_a2 @ A @ ( vimage_a_Sum_sum_a_b @ F @ B ) )
= ( member_Sum_sum_a_b2 @ ( F @ A ) @ B ) ) ).
% vimage_eq
thf(fact_77_vimage__eq,axiom,
! [A: sum_sum_a_b,F: sum_sum_a_b > sum_sum_a_b,B: set_Sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ A @ ( vimage4749860508033320969um_a_b @ F @ B ) )
= ( member_Sum_sum_a_b2 @ ( F @ A ) @ B ) ) ).
% vimage_eq
thf(fact_78_vimage__eq,axiom,
! [A: a,F: a > b,B: set_b] :
( ( member_a2 @ A @ ( vimage_a_b @ F @ B ) )
= ( member_b2 @ ( F @ A ) @ B ) ) ).
% vimage_eq
thf(fact_79_vimage__eq,axiom,
! [A: b,F: b > a,B: set_a] :
( ( member_b2 @ A @ ( vimage_b_a @ F @ B ) )
= ( member_a2 @ ( F @ A ) @ B ) ) ).
% vimage_eq
thf(fact_80_vimage__eq,axiom,
! [A: b,F: b > b,B: set_b] :
( ( member_b2 @ A @ ( vimage_b_b @ F @ B ) )
= ( member_b2 @ ( F @ A ) @ B ) ) ).
% vimage_eq
thf(fact_81_vimage__eq,axiom,
! [A: a,F: a > list_a,B: set_list_a] :
( ( member_a2 @ A @ ( vimage_a_list_a @ F @ B ) )
= ( member_list_a2 @ ( F @ A ) @ B ) ) ).
% vimage_eq
thf(fact_82_vimage__eq,axiom,
! [A: a,F: a > set_a,B: set_set_a] :
( ( member_a2 @ A @ ( vimage_a_set_a @ F @ B ) )
= ( member_set_a2 @ ( F @ A ) @ B ) ) ).
% vimage_eq
thf(fact_83_vimage__eq,axiom,
! [A: list_a,F: list_a > a,B: set_a] :
( ( member_list_a2 @ A @ ( vimage_list_a_a @ F @ B ) )
= ( member_a2 @ ( F @ A ) @ B ) ) ).
% vimage_eq
thf(fact_84_subsetI,axiom,
! [A4: set_list_a,B: set_list_a] :
( ! [X4: list_a] :
( ( member_list_a2 @ X4 @ A4 )
=> ( member_list_a2 @ X4 @ B ) )
=> ( ord_le8861187494160871172list_a @ A4 @ B ) ) ).
% subsetI
thf(fact_85_subsetI,axiom,
! [A4: set_set_Sum_sum_a_b,B: set_set_Sum_sum_a_b] :
( ! [X4: set_Sum_sum_a_b] :
( ( member4060935254435997939um_a_b @ X4 @ A4 )
=> ( member4060935254435997939um_a_b @ X4 @ B ) )
=> ( ord_le1944875106711258738um_a_b @ A4 @ B ) ) ).
% subsetI
thf(fact_86_subsetI,axiom,
! [A4: set_set_set_a,B: set_set_set_a] :
( ! [X4: set_set_a] :
( ( member_set_set_a2 @ X4 @ A4 )
=> ( member_set_set_a2 @ X4 @ B ) )
=> ( ord_le5722252365846178494_set_a @ A4 @ B ) ) ).
% subsetI
thf(fact_87_subsetI,axiom,
! [A4: set_b,B: set_b] :
( ! [X4: b] :
( ( member_b2 @ X4 @ A4 )
=> ( member_b2 @ X4 @ B ) )
=> ( ord_less_eq_set_b @ A4 @ B ) ) ).
% subsetI
thf(fact_88_subsetI,axiom,
! [A4: set_set_a,B: set_set_a] :
( ! [X4: set_a] :
( ( member_set_a2 @ X4 @ A4 )
=> ( member_set_a2 @ X4 @ B ) )
=> ( ord_le3724670747650509150_set_a @ A4 @ B ) ) ).
% subsetI
thf(fact_89_subsetI,axiom,
! [A4: set_Sum_sum_a_b,B: set_Sum_sum_a_b] :
( ! [X4: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X4 @ A4 )
=> ( member_Sum_sum_a_b2 @ X4 @ B ) )
=> ( ord_le9019793522827316924um_a_b @ A4 @ B ) ) ).
% subsetI
thf(fact_90_subsetI,axiom,
! [A4: set_a,B: set_a] :
( ! [X4: a] :
( ( member_a2 @ X4 @ A4 )
=> ( member_a2 @ X4 @ B ) )
=> ( ord_less_eq_set_a @ A4 @ B ) ) ).
% subsetI
thf(fact_91_subset__antisym,axiom,
! [A4: set_Sum_sum_a_b,B: set_Sum_sum_a_b] :
( ( ord_le9019793522827316924um_a_b @ A4 @ B )
=> ( ( ord_le9019793522827316924um_a_b @ B @ A4 )
=> ( A4 = B ) ) ) ).
% subset_antisym
thf(fact_92_subset__antisym,axiom,
! [A4: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ A4 )
=> ( A4 = B ) ) ) ).
% subset_antisym
thf(fact_93_subset__antisym,axiom,
! [A4: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A4 @ B )
=> ( ( ord_less_eq_set_a @ B @ A4 )
=> ( A4 = B ) ) ) ).
% subset_antisym
thf(fact_94_order__refl,axiom,
! [X2: set_Sum_sum_a_b] : ( ord_le9019793522827316924um_a_b @ X2 @ X2 ) ).
% order_refl
thf(fact_95_order__refl,axiom,
! [X2: $o > set_a] : ( ord_less_eq_o_set_a @ X2 @ X2 ) ).
% order_refl
thf(fact_96_order__refl,axiom,
! [X2: set_set_a] : ( ord_le3724670747650509150_set_a @ X2 @ X2 ) ).
% order_refl
thf(fact_97_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_98_order__refl,axiom,
! [X2: set_a] : ( ord_less_eq_set_a @ X2 @ X2 ) ).
% order_refl
thf(fact_99_dual__order_Orefl,axiom,
! [A: set_Sum_sum_a_b] : ( ord_le9019793522827316924um_a_b @ A @ A ) ).
% dual_order.refl
thf(fact_100_dual__order_Orefl,axiom,
! [A: $o > set_a] : ( ord_less_eq_o_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_101_dual__order_Orefl,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_102_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_103_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_104_vimage__mono,axiom,
! [A4: set_a,B: set_a,F: sum_sum_a_b > a] :
( ( ord_less_eq_set_a @ A4 @ B )
=> ( ord_le9019793522827316924um_a_b @ ( vimage_Sum_sum_a_b_a @ F @ A4 ) @ ( vimage_Sum_sum_a_b_a @ F @ B ) ) ) ).
% vimage_mono
thf(fact_105_vimage__mono,axiom,
! [A4: set_a,B: set_a,F: set_a > a] :
( ( ord_less_eq_set_a @ A4 @ B )
=> ( ord_le3724670747650509150_set_a @ ( vimage_set_a_a @ F @ A4 ) @ ( vimage_set_a_a @ F @ B ) ) ) ).
% vimage_mono
thf(fact_106_vimage__mono,axiom,
! [A4: set_Sum_sum_a_b,B: set_Sum_sum_a_b,F: sum_sum_a_b > sum_sum_a_b] :
( ( ord_le9019793522827316924um_a_b @ A4 @ B )
=> ( ord_le9019793522827316924um_a_b @ ( vimage4749860508033320969um_a_b @ F @ A4 ) @ ( vimage4749860508033320969um_a_b @ F @ B ) ) ) ).
% vimage_mono
thf(fact_107_vimage__mono,axiom,
! [A4: set_Sum_sum_a_b,B: set_Sum_sum_a_b,F: set_a > sum_sum_a_b] :
( ( ord_le9019793522827316924um_a_b @ A4 @ B )
=> ( ord_le3724670747650509150_set_a @ ( vimage9084226422076968841um_a_b @ F @ A4 ) @ ( vimage9084226422076968841um_a_b @ F @ B ) ) ) ).
% vimage_mono
thf(fact_108_vimage__mono,axiom,
! [A4: set_set_a,B: set_set_a,F: a > set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ B )
=> ( ord_less_eq_set_a @ ( vimage_a_set_a @ F @ A4 ) @ ( vimage_a_set_a @ F @ B ) ) ) ).
% vimage_mono
thf(fact_109_vimage__mono,axiom,
! [A4: set_set_a,B: set_set_a,F: sum_sum_a_b > set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ B )
=> ( ord_le9019793522827316924um_a_b @ ( vimage7202883975058866363_set_a @ F @ A4 ) @ ( vimage7202883975058866363_set_a @ F @ B ) ) ) ).
% vimage_mono
thf(fact_110_vimage__mono,axiom,
! [A4: set_set_a,B: set_set_a,F: set_a > set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ B )
=> ( ord_le3724670747650509150_set_a @ ( vimage_set_a_set_a @ F @ A4 ) @ ( vimage_set_a_set_a @ F @ B ) ) ) ).
% vimage_mono
thf(fact_111_vimage__mono,axiom,
! [A4: set_Sum_sum_a_b,B: set_Sum_sum_a_b,F: a > sum_sum_a_b] :
( ( ord_le9019793522827316924um_a_b @ A4 @ B )
=> ( ord_less_eq_set_a @ ( vimage_a_Sum_sum_a_b @ F @ A4 ) @ ( vimage_a_Sum_sum_a_b @ F @ B ) ) ) ).
% vimage_mono
thf(fact_112_vimage__mono,axiom,
! [A4: set_a,B: set_a,F: a > a] :
( ( ord_less_eq_set_a @ A4 @ B )
=> ( ord_less_eq_set_a @ ( vimage_a_a @ F @ A4 ) @ ( vimage_a_a @ F @ B ) ) ) ).
% vimage_mono
thf(fact_113_subset__vimage__iff,axiom,
! [A4: set_a,F: a > a,B: set_a] :
( ( ord_less_eq_set_a @ A4 @ ( vimage_a_a @ F @ B ) )
= ( ! [X3: a] :
( ( member_a2 @ X3 @ A4 )
=> ( member_a2 @ ( F @ X3 ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_114_subset__vimage__iff,axiom,
! [A4: set_a,F: a > sum_sum_a_b,B: set_Sum_sum_a_b] :
( ( ord_less_eq_set_a @ A4 @ ( vimage_a_Sum_sum_a_b @ F @ B ) )
= ( ! [X3: a] :
( ( member_a2 @ X3 @ A4 )
=> ( member_Sum_sum_a_b2 @ ( F @ X3 ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_115_subset__vimage__iff,axiom,
! [A4: set_a,F: a > b,B: set_b] :
( ( ord_less_eq_set_a @ A4 @ ( vimage_a_b @ F @ B ) )
= ( ! [X3: a] :
( ( member_a2 @ X3 @ A4 )
=> ( member_b2 @ ( F @ X3 ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_116_subset__vimage__iff,axiom,
! [A4: set_a,F: a > list_a,B: set_list_a] :
( ( ord_less_eq_set_a @ A4 @ ( vimage_a_list_a @ F @ B ) )
= ( ! [X3: a] :
( ( member_a2 @ X3 @ A4 )
=> ( member_list_a2 @ ( F @ X3 ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_117_subset__vimage__iff,axiom,
! [A4: set_a,F: a > set_a,B: set_set_a] :
( ( ord_less_eq_set_a @ A4 @ ( vimage_a_set_a @ F @ B ) )
= ( ! [X3: a] :
( ( member_a2 @ X3 @ A4 )
=> ( member_set_a2 @ ( F @ X3 ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_118_subset__vimage__iff,axiom,
! [A4: set_set_a,F: set_a > a,B: set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ ( vimage_set_a_a @ F @ B ) )
= ( ! [X3: set_a] :
( ( member_set_a2 @ X3 @ A4 )
=> ( member_a2 @ ( F @ X3 ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_119_subset__vimage__iff,axiom,
! [A4: set_set_a,F: set_a > b,B: set_b] :
( ( ord_le3724670747650509150_set_a @ A4 @ ( vimage_set_a_b @ F @ B ) )
= ( ! [X3: set_a] :
( ( member_set_a2 @ X3 @ A4 )
=> ( member_b2 @ ( F @ X3 ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_120_subset__vimage__iff,axiom,
! [A4: set_a,F: a > set_set_a,B: set_set_set_a] :
( ( ord_less_eq_set_a @ A4 @ ( vimage_a_set_set_a @ F @ B ) )
= ( ! [X3: a] :
( ( member_a2 @ X3 @ A4 )
=> ( member_set_set_a2 @ ( F @ X3 ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_121_subset__vimage__iff,axiom,
! [A4: set_Sum_sum_a_b,F: sum_sum_a_b > b,B: set_b] :
( ( ord_le9019793522827316924um_a_b @ A4 @ ( vimage_Sum_sum_a_b_b @ F @ B ) )
= ( ! [X3: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X3 @ A4 )
=> ( member_b2 @ ( F @ X3 ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_122_subset__vimage__iff,axiom,
! [A4: set_Sum_sum_a_b,F: sum_sum_a_b > a,B: set_a] :
( ( ord_le9019793522827316924um_a_b @ A4 @ ( vimage_Sum_sum_a_b_a @ F @ B ) )
= ( ! [X3: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X3 @ A4 )
=> ( member_a2 @ ( F @ X3 ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_123_vimageD,axiom,
! [A: a,F: a > a,A4: set_a] :
( ( member_a2 @ A @ ( vimage_a_a @ F @ A4 ) )
=> ( member_a2 @ ( F @ A ) @ A4 ) ) ).
% vimageD
thf(fact_124_vimageD,axiom,
! [A: sum_sum_a_b,F: sum_sum_a_b > a,A4: set_a] :
( ( member_Sum_sum_a_b2 @ A @ ( vimage_Sum_sum_a_b_a @ F @ A4 ) )
=> ( member_a2 @ ( F @ A ) @ A4 ) ) ).
% vimageD
thf(fact_125_vimageD,axiom,
! [A: a,F: a > sum_sum_a_b,A4: set_Sum_sum_a_b] :
( ( member_a2 @ A @ ( vimage_a_Sum_sum_a_b @ F @ A4 ) )
=> ( member_Sum_sum_a_b2 @ ( F @ A ) @ A4 ) ) ).
% vimageD
thf(fact_126_vimageD,axiom,
! [A: sum_sum_a_b,F: sum_sum_a_b > sum_sum_a_b,A4: set_Sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ A @ ( vimage4749860508033320969um_a_b @ F @ A4 ) )
=> ( member_Sum_sum_a_b2 @ ( F @ A ) @ A4 ) ) ).
% vimageD
thf(fact_127_vimageD,axiom,
! [A: a,F: a > b,A4: set_b] :
( ( member_a2 @ A @ ( vimage_a_b @ F @ A4 ) )
=> ( member_b2 @ ( F @ A ) @ A4 ) ) ).
% vimageD
thf(fact_128_vimageD,axiom,
! [A: b,F: b > a,A4: set_a] :
( ( member_b2 @ A @ ( vimage_b_a @ F @ A4 ) )
=> ( member_a2 @ ( F @ A ) @ A4 ) ) ).
% vimageD
thf(fact_129_vimageD,axiom,
! [A: b,F: b > b,A4: set_b] :
( ( member_b2 @ A @ ( vimage_b_b @ F @ A4 ) )
=> ( member_b2 @ ( F @ A ) @ A4 ) ) ).
% vimageD
thf(fact_130_vimageD,axiom,
! [A: a,F: a > list_a,A4: set_list_a] :
( ( member_a2 @ A @ ( vimage_a_list_a @ F @ A4 ) )
=> ( member_list_a2 @ ( F @ A ) @ A4 ) ) ).
% vimageD
thf(fact_131_vimageD,axiom,
! [A: a,F: a > set_a,A4: set_set_a] :
( ( member_a2 @ A @ ( vimage_a_set_a @ F @ A4 ) )
=> ( member_set_a2 @ ( F @ A ) @ A4 ) ) ).
% vimageD
thf(fact_132_vimageD,axiom,
! [A: list_a,F: list_a > a,A4: set_a] :
( ( member_list_a2 @ A @ ( vimage_list_a_a @ F @ A4 ) )
=> ( member_a2 @ ( F @ A ) @ A4 ) ) ).
% vimageD
thf(fact_133_vimageE,axiom,
! [A: a,F: a > a,B: set_a] :
( ( member_a2 @ A @ ( vimage_a_a @ F @ B ) )
=> ( member_a2 @ ( F @ A ) @ B ) ) ).
% vimageE
thf(fact_134_vimageE,axiom,
! [A: sum_sum_a_b,F: sum_sum_a_b > a,B: set_a] :
( ( member_Sum_sum_a_b2 @ A @ ( vimage_Sum_sum_a_b_a @ F @ B ) )
=> ( member_a2 @ ( F @ A ) @ B ) ) ).
% vimageE
thf(fact_135_vimageE,axiom,
! [A: a,F: a > sum_sum_a_b,B: set_Sum_sum_a_b] :
( ( member_a2 @ A @ ( vimage_a_Sum_sum_a_b @ F @ B ) )
=> ( member_Sum_sum_a_b2 @ ( F @ A ) @ B ) ) ).
% vimageE
thf(fact_136_vimageE,axiom,
! [A: sum_sum_a_b,F: sum_sum_a_b > sum_sum_a_b,B: set_Sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ A @ ( vimage4749860508033320969um_a_b @ F @ B ) )
=> ( member_Sum_sum_a_b2 @ ( F @ A ) @ B ) ) ).
% vimageE
thf(fact_137_vimageE,axiom,
! [A: a,F: a > b,B: set_b] :
( ( member_a2 @ A @ ( vimage_a_b @ F @ B ) )
=> ( member_b2 @ ( F @ A ) @ B ) ) ).
% vimageE
thf(fact_138_vimageE,axiom,
! [A: b,F: b > a,B: set_a] :
( ( member_b2 @ A @ ( vimage_b_a @ F @ B ) )
=> ( member_a2 @ ( F @ A ) @ B ) ) ).
% vimageE
thf(fact_139_vimageE,axiom,
! [A: b,F: b > b,B: set_b] :
( ( member_b2 @ A @ ( vimage_b_b @ F @ B ) )
=> ( member_b2 @ ( F @ A ) @ B ) ) ).
% vimageE
thf(fact_140_vimageE,axiom,
! [A: a,F: a > list_a,B: set_list_a] :
( ( member_a2 @ A @ ( vimage_a_list_a @ F @ B ) )
=> ( member_list_a2 @ ( F @ A ) @ B ) ) ).
% vimageE
thf(fact_141_vimageE,axiom,
! [A: a,F: a > set_a,B: set_set_a] :
( ( member_a2 @ A @ ( vimage_a_set_a @ F @ B ) )
=> ( member_set_a2 @ ( F @ A ) @ B ) ) ).
% vimageE
thf(fact_142_vimageE,axiom,
! [A: list_a,F: list_a > a,B: set_a] :
( ( member_list_a2 @ A @ ( vimage_list_a_a @ F @ B ) )
=> ( member_a2 @ ( F @ A ) @ B ) ) ).
% vimageE
thf(fact_143_vimageI2,axiom,
! [F: a > a,A: a,A4: set_a] :
( ( member_a2 @ ( F @ A ) @ A4 )
=> ( member_a2 @ A @ ( vimage_a_a @ F @ A4 ) ) ) ).
% vimageI2
thf(fact_144_vimageI2,axiom,
! [F: sum_sum_a_b > a,A: sum_sum_a_b,A4: set_a] :
( ( member_a2 @ ( F @ A ) @ A4 )
=> ( member_Sum_sum_a_b2 @ A @ ( vimage_Sum_sum_a_b_a @ F @ A4 ) ) ) ).
% vimageI2
thf(fact_145_vimageI2,axiom,
! [F: a > sum_sum_a_b,A: a,A4: set_Sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ ( F @ A ) @ A4 )
=> ( member_a2 @ A @ ( vimage_a_Sum_sum_a_b @ F @ A4 ) ) ) ).
% vimageI2
thf(fact_146_vimageI2,axiom,
! [F: sum_sum_a_b > sum_sum_a_b,A: sum_sum_a_b,A4: set_Sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ ( F @ A ) @ A4 )
=> ( member_Sum_sum_a_b2 @ A @ ( vimage4749860508033320969um_a_b @ F @ A4 ) ) ) ).
% vimageI2
thf(fact_147_vimageI2,axiom,
! [F: b > a,A: b,A4: set_a] :
( ( member_a2 @ ( F @ A ) @ A4 )
=> ( member_b2 @ A @ ( vimage_b_a @ F @ A4 ) ) ) ).
% vimageI2
thf(fact_148_vimageI2,axiom,
! [F: a > b,A: a,A4: set_b] :
( ( member_b2 @ ( F @ A ) @ A4 )
=> ( member_a2 @ A @ ( vimage_a_b @ F @ A4 ) ) ) ).
% vimageI2
thf(fact_149_vimageI2,axiom,
! [F: b > b,A: b,A4: set_b] :
( ( member_b2 @ ( F @ A ) @ A4 )
=> ( member_b2 @ A @ ( vimage_b_b @ F @ A4 ) ) ) ).
% vimageI2
thf(fact_150_vimageI2,axiom,
! [F: list_a > a,A: list_a,A4: set_a] :
( ( member_a2 @ ( F @ A ) @ A4 )
=> ( member_list_a2 @ A @ ( vimage_list_a_a @ F @ A4 ) ) ) ).
% vimageI2
thf(fact_151_vimageI2,axiom,
! [F: set_a > a,A: set_a,A4: set_a] :
( ( member_a2 @ ( F @ A ) @ A4 )
=> ( member_set_a2 @ A @ ( vimage_set_a_a @ F @ A4 ) ) ) ).
% vimageI2
thf(fact_152_vimageI2,axiom,
! [F: a > list_a,A: a,A4: set_list_a] :
( ( member_list_a2 @ ( F @ A ) @ A4 )
=> ( member_a2 @ A @ ( vimage_a_list_a @ F @ A4 ) ) ) ).
% vimageI2
thf(fact_153_vimage__Collect,axiom,
! [P: sum_sum_a_b > $o,F: sum_sum_a_b > sum_sum_a_b,Q: sum_sum_a_b > $o] :
( ! [X4: sum_sum_a_b] :
( ( P @ ( F @ X4 ) )
= ( Q @ X4 ) )
=> ( ( vimage4749860508033320969um_a_b @ F @ ( collect_Sum_sum_a_b @ P ) )
= ( collect_Sum_sum_a_b @ Q ) ) ) ).
% vimage_Collect
thf(fact_154_vimage__Collect,axiom,
! [P: a > $o,F: sum_sum_a_b > a,Q: sum_sum_a_b > $o] :
( ! [X4: sum_sum_a_b] :
( ( P @ ( F @ X4 ) )
= ( Q @ X4 ) )
=> ( ( vimage_Sum_sum_a_b_a @ F @ ( collect_a @ P ) )
= ( collect_Sum_sum_a_b @ Q ) ) ) ).
% vimage_Collect
thf(fact_155_vimage__Collect,axiom,
! [P: a > $o,F: a > a,Q: a > $o] :
( ! [X4: a] :
( ( P @ ( F @ X4 ) )
= ( Q @ X4 ) )
=> ( ( vimage_a_a @ F @ ( collect_a @ P ) )
= ( collect_a @ Q ) ) ) ).
% vimage_Collect
thf(fact_156_vimage__Collect,axiom,
! [P: sum_sum_a_b > $o,F: a > sum_sum_a_b,Q: a > $o] :
( ! [X4: a] :
( ( P @ ( F @ X4 ) )
= ( Q @ X4 ) )
=> ( ( vimage_a_Sum_sum_a_b @ F @ ( collect_Sum_sum_a_b @ P ) )
= ( collect_a @ Q ) ) ) ).
% vimage_Collect
thf(fact_157_order__antisym__conv,axiom,
! [Y: set_Sum_sum_a_b,X2: set_Sum_sum_a_b] :
( ( ord_le9019793522827316924um_a_b @ Y @ X2 )
=> ( ( ord_le9019793522827316924um_a_b @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_158_order__antisym__conv,axiom,
! [Y: $o > set_a,X2: $o > set_a] :
( ( ord_less_eq_o_set_a @ Y @ X2 )
=> ( ( ord_less_eq_o_set_a @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_159_order__antisym__conv,axiom,
! [Y: set_set_a,X2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y @ X2 )
=> ( ( ord_le3724670747650509150_set_a @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_160_order__antisym__conv,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_161_order__antisym__conv,axiom,
! [Y: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y @ X2 )
=> ( ( ord_less_eq_set_a @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_162_linorder__le__cases,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_le_cases
thf(fact_163_ord__le__eq__subst,axiom,
! [A: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_164_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_165_ord__le__eq__subst,axiom,
! [A: set_a,B2: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_166_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_167_ord__le__eq__subst,axiom,
! [A: set_set_a,B2: set_set_a,F: set_set_a > nat,C: nat] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_168_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > set_set_a,C: set_set_a] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_169_ord__le__eq__subst,axiom,
! [A: set_a,B2: set_a,F: set_a > set_set_a,C: set_set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_170_ord__le__eq__subst,axiom,
! [A: set_Sum_sum_a_b,B2: set_Sum_sum_a_b,F: set_Sum_sum_a_b > nat,C: nat] :
( ( ord_le9019793522827316924um_a_b @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: set_Sum_sum_a_b,Y3: set_Sum_sum_a_b] :
( ( ord_le9019793522827316924um_a_b @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_171_ord__le__eq__subst,axiom,
! [A: $o > set_a,B2: $o > set_a,F: ( $o > set_a ) > nat,C: nat] :
( ( ord_less_eq_o_set_a @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: $o > set_a,Y3: $o > set_a] :
( ( ord_less_eq_o_set_a @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_172_ord__le__eq__subst,axiom,
! [A: set_set_a,B2: set_set_a,F: set_set_a > set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_173_ord__eq__le__subst,axiom,
! [A: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X4: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_174_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_175_ord__eq__le__subst,axiom,
! [A: nat,F: set_a > nat,B2: set_a,C: set_a] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X4: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_176_ord__eq__le__subst,axiom,
! [A: set_a,F: nat > set_a,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_177_ord__eq__le__subst,axiom,
! [A: nat,F: set_set_a > nat,B2: set_set_a,C: set_set_a] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ! [X4: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_178_ord__eq__le__subst,axiom,
! [A: set_set_a,F: nat > set_set_a,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le3724670747650509150_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_179_ord__eq__le__subst,axiom,
! [A: set_set_a,F: set_a > set_set_a,B2: set_a,C: set_a] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X4: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le3724670747650509150_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_180_ord__eq__le__subst,axiom,
! [A: nat,F: set_Sum_sum_a_b > nat,B2: set_Sum_sum_a_b,C: set_Sum_sum_a_b] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_le9019793522827316924um_a_b @ B2 @ C )
=> ( ! [X4: set_Sum_sum_a_b,Y3: set_Sum_sum_a_b] :
( ( ord_le9019793522827316924um_a_b @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_181_ord__eq__le__subst,axiom,
! [A: nat,F: ( $o > set_a ) > nat,B2: $o > set_a,C: $o > set_a] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_o_set_a @ B2 @ C )
=> ( ! [X4: $o > set_a,Y3: $o > set_a] :
( ( ord_less_eq_o_set_a @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_182_ord__eq__le__subst,axiom,
! [A: set_a,F: set_set_a > set_a,B2: set_set_a,C: set_set_a] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ! [X4: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_183_linorder__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
| ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_linear
thf(fact_184_order__eq__refl,axiom,
! [X2: set_Sum_sum_a_b,Y: set_Sum_sum_a_b] :
( ( X2 = Y )
=> ( ord_le9019793522827316924um_a_b @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_185_order__eq__refl,axiom,
! [X2: $o > set_a,Y: $o > set_a] :
( ( X2 = Y )
=> ( ord_less_eq_o_set_a @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_186_order__eq__refl,axiom,
! [X2: set_set_a,Y: set_set_a] :
( ( X2 = Y )
=> ( ord_le3724670747650509150_set_a @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_187_order__eq__refl,axiom,
! [X2: nat,Y: nat] :
( ( X2 = Y )
=> ( ord_less_eq_nat @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_188_order__eq__refl,axiom,
! [X2: set_a,Y: set_a] :
( ( X2 = Y )
=> ( ord_less_eq_set_a @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_189_order__subst2,axiom,
! [A: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
=> ( ! [X4: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_190_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_191_order__subst2,axiom,
! [A: set_a,B2: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_192_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_193_order__subst2,axiom,
! [A: set_set_a,B2: set_set_a,F: set_set_a > nat,C: nat] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_194_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > set_set_a,C: set_set_a] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_195_order__subst2,axiom,
! [A: set_a,B2: set_a,F: set_a > set_set_a,C: set_set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ ( F @ B2 ) @ C )
=> ( ! [X4: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_196_order__subst2,axiom,
! [A: set_Sum_sum_a_b,B2: set_Sum_sum_a_b,F: set_Sum_sum_a_b > nat,C: nat] :
( ( ord_le9019793522827316924um_a_b @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: set_Sum_sum_a_b,Y3: set_Sum_sum_a_b] :
( ( ord_le9019793522827316924um_a_b @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_197_order__subst2,axiom,
! [A: $o > set_a,B2: $o > set_a,F: ( $o > set_a ) > nat,C: nat] :
( ( ord_less_eq_o_set_a @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: $o > set_a,Y3: $o > set_a] :
( ( ord_less_eq_o_set_a @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_198_order__subst2,axiom,
! [A: set_set_a,B2: set_set_a,F: set_set_a > set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
=> ( ! [X4: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_199_order__subst1,axiom,
! [A: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X4: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_200_order__subst1,axiom,
! [A: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_201_order__subst1,axiom,
! [A: set_a,F: nat > set_a,B2: nat,C: nat] :
( ( ord_less_eq_set_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_202_order__subst1,axiom,
! [A: nat,F: set_a > nat,B2: set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X4: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_203_order__subst1,axiom,
! [A: set_set_a,F: nat > set_set_a,B2: nat,C: nat] :
( ( ord_le3724670747650509150_set_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le3724670747650509150_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_204_order__subst1,axiom,
! [A: nat,F: set_set_a > nat,B2: set_set_a,C: set_set_a] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ! [X4: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_205_order__subst1,axiom,
! [A: set_a,F: set_set_a > set_a,B2: set_set_a,C: set_set_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B2 ) )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ! [X4: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_206_order__subst1,axiom,
! [A: set_Sum_sum_a_b,F: nat > set_Sum_sum_a_b,B2: nat,C: nat] :
( ( ord_le9019793522827316924um_a_b @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_le9019793522827316924um_a_b @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le9019793522827316924um_a_b @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_207_order__subst1,axiom,
! [A: $o > set_a,F: nat > $o > set_a,B2: nat,C: nat] :
( ( ord_less_eq_o_set_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_o_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_o_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_208_order__subst1,axiom,
! [A: set_set_a,F: set_a > set_set_a,B2: set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X4: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le3724670747650509150_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_209_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Sum_sum_a_b,Z: set_Sum_sum_a_b] : ( Y4 = Z ) )
= ( ^ [A3: set_Sum_sum_a_b,B3: set_Sum_sum_a_b] :
( ( ord_le9019793522827316924um_a_b @ A3 @ B3 )
& ( ord_le9019793522827316924um_a_b @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_210_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: $o > set_a,Z: $o > set_a] : ( Y4 = Z ) )
= ( ^ [A3: $o > set_a,B3: $o > set_a] :
( ( ord_less_eq_o_set_a @ A3 @ B3 )
& ( ord_less_eq_o_set_a @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_211_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_set_a,Z: set_set_a] : ( Y4 = Z ) )
= ( ^ [A3: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B3 )
& ( ord_le3724670747650509150_set_a @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_212_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_213_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_a,Z: set_a] : ( Y4 = Z ) )
= ( ^ [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_214_le__fun__def,axiom,
( ord_less_eq_o_set_a
= ( ^ [F2: $o > set_a,G: $o > set_a] :
! [X3: $o] : ( ord_less_eq_set_a @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ).
% le_fun_def
thf(fact_215_le__funI,axiom,
! [F: $o > set_a,G2: $o > set_a] :
( ! [X4: $o] : ( ord_less_eq_set_a @ ( F @ X4 ) @ ( G2 @ X4 ) )
=> ( ord_less_eq_o_set_a @ F @ G2 ) ) ).
% le_funI
thf(fact_216_le__funE,axiom,
! [F: $o > set_a,G2: $o > set_a,X2: $o] :
( ( ord_less_eq_o_set_a @ F @ G2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( G2 @ X2 ) ) ) ).
% le_funE
thf(fact_217_le__funD,axiom,
! [F: $o > set_a,G2: $o > set_a,X2: $o] :
( ( ord_less_eq_o_set_a @ F @ G2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( G2 @ X2 ) ) ) ).
% le_funD
thf(fact_218_mem__Collect__eq,axiom,
! [A: list_a,P: list_a > $o] :
( ( member_list_a2 @ A @ ( collect_list_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_219_mem__Collect__eq,axiom,
! [A: set_Sum_sum_a_b,P: set_Sum_sum_a_b > $o] :
( ( member4060935254435997939um_a_b @ A @ ( collec6703685677538285617um_a_b @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_220_mem__Collect__eq,axiom,
! [A: set_set_a,P: set_set_a > $o] :
( ( member_set_set_a2 @ A @ ( collect_set_set_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_221_mem__Collect__eq,axiom,
! [A: set_a,P: set_a > $o] :
( ( member_set_a2 @ A @ ( collect_set_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_222_mem__Collect__eq,axiom,
! [A: b,P: b > $o] :
( ( member_b2 @ A @ ( collect_b @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_223_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a2 @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_224_mem__Collect__eq,axiom,
! [A: sum_sum_a_b,P: sum_sum_a_b > $o] :
( ( member_Sum_sum_a_b2 @ A @ ( collect_Sum_sum_a_b @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_225_Collect__mem__eq,axiom,
! [A4: set_list_a] :
( ( collect_list_a
@ ^ [X3: list_a] : ( member_list_a2 @ X3 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_226_Collect__mem__eq,axiom,
! [A4: set_set_Sum_sum_a_b] :
( ( collec6703685677538285617um_a_b
@ ^ [X3: set_Sum_sum_a_b] : ( member4060935254435997939um_a_b @ X3 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_227_Collect__mem__eq,axiom,
! [A4: set_set_set_a] :
( ( collect_set_set_a
@ ^ [X3: set_set_a] : ( member_set_set_a2 @ X3 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_228_Collect__mem__eq,axiom,
! [A4: set_set_a] :
( ( collect_set_a
@ ^ [X3: set_a] : ( member_set_a2 @ X3 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_229_Collect__mem__eq,axiom,
! [A4: set_b] :
( ( collect_b
@ ^ [X3: b] : ( member_b2 @ X3 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_230_Collect__mem__eq,axiom,
! [A4: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a2 @ X3 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_231_Collect__mem__eq,axiom,
! [A4: set_Sum_sum_a_b] :
( ( collect_Sum_sum_a_b
@ ^ [X3: sum_sum_a_b] : ( member_Sum_sum_a_b2 @ X3 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_232_Collect__cong,axiom,
! [P: sum_sum_a_b > $o,Q: sum_sum_a_b > $o] :
( ! [X4: sum_sum_a_b] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_Sum_sum_a_b @ P )
= ( collect_Sum_sum_a_b @ Q ) ) ) ).
% Collect_cong
thf(fact_233_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X4: a] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_234_antisym,axiom,
! [A: set_Sum_sum_a_b,B2: set_Sum_sum_a_b] :
( ( ord_le9019793522827316924um_a_b @ A @ B2 )
=> ( ( ord_le9019793522827316924um_a_b @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_235_antisym,axiom,
! [A: $o > set_a,B2: $o > set_a] :
( ( ord_less_eq_o_set_a @ A @ B2 )
=> ( ( ord_less_eq_o_set_a @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_236_antisym,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_237_antisym,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_238_antisym,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_239_dual__order_Otrans,axiom,
! [B2: set_Sum_sum_a_b,A: set_Sum_sum_a_b,C: set_Sum_sum_a_b] :
( ( ord_le9019793522827316924um_a_b @ B2 @ A )
=> ( ( ord_le9019793522827316924um_a_b @ C @ B2 )
=> ( ord_le9019793522827316924um_a_b @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_240_dual__order_Otrans,axiom,
! [B2: $o > set_a,A: $o > set_a,C: $o > set_a] :
( ( ord_less_eq_o_set_a @ B2 @ A )
=> ( ( ord_less_eq_o_set_a @ C @ B2 )
=> ( ord_less_eq_o_set_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_241_dual__order_Otrans,axiom,
! [B2: set_set_a,A: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A )
=> ( ( ord_le3724670747650509150_set_a @ C @ B2 )
=> ( ord_le3724670747650509150_set_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_242_dual__order_Otrans,axiom,
! [B2: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_243_dual__order_Otrans,axiom,
! [B2: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( ord_less_eq_set_a @ C @ B2 )
=> ( ord_less_eq_set_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_244_dual__order_Oantisym,axiom,
! [B2: set_Sum_sum_a_b,A: set_Sum_sum_a_b] :
( ( ord_le9019793522827316924um_a_b @ B2 @ A )
=> ( ( ord_le9019793522827316924um_a_b @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_245_dual__order_Oantisym,axiom,
! [B2: $o > set_a,A: $o > set_a] :
( ( ord_less_eq_o_set_a @ B2 @ A )
=> ( ( ord_less_eq_o_set_a @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_246_dual__order_Oantisym,axiom,
! [B2: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A )
=> ( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_247_dual__order_Oantisym,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_248_dual__order_Oantisym,axiom,
! [B2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( ord_less_eq_set_a @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_249_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_Sum_sum_a_b,Z: set_Sum_sum_a_b] : ( Y4 = Z ) )
= ( ^ [A3: set_Sum_sum_a_b,B3: set_Sum_sum_a_b] :
( ( ord_le9019793522827316924um_a_b @ B3 @ A3 )
& ( ord_le9019793522827316924um_a_b @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_250_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: $o > set_a,Z: $o > set_a] : ( Y4 = Z ) )
= ( ^ [A3: $o > set_a,B3: $o > set_a] :
( ( ord_less_eq_o_set_a @ B3 @ A3 )
& ( ord_less_eq_o_set_a @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_251_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_set_a,Z: set_set_a] : ( Y4 = Z ) )
= ( ^ [A3: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ A3 )
& ( ord_le3724670747650509150_set_a @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_252_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_253_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_a,Z: set_a] : ( Y4 = Z ) )
= ( ^ [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_254_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B2: nat] :
( ! [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
=> ( P @ A5 @ B4 ) )
=> ( ! [A5: nat,B4: nat] :
( ( P @ B4 @ A5 )
=> ( P @ A5 @ B4 ) )
=> ( P @ A @ B2 ) ) ) ).
% linorder_wlog
thf(fact_255_order__trans,axiom,
! [X2: set_Sum_sum_a_b,Y: set_Sum_sum_a_b,Z2: set_Sum_sum_a_b] :
( ( ord_le9019793522827316924um_a_b @ X2 @ Y )
=> ( ( ord_le9019793522827316924um_a_b @ Y @ Z2 )
=> ( ord_le9019793522827316924um_a_b @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_256_order__trans,axiom,
! [X2: $o > set_a,Y: $o > set_a,Z2: $o > set_a] :
( ( ord_less_eq_o_set_a @ X2 @ Y )
=> ( ( ord_less_eq_o_set_a @ Y @ Z2 )
=> ( ord_less_eq_o_set_a @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_257_order__trans,axiom,
! [X2: set_set_a,Y: set_set_a,Z2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y )
=> ( ( ord_le3724670747650509150_set_a @ Y @ Z2 )
=> ( ord_le3724670747650509150_set_a @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_258_order__trans,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_259_order__trans,axiom,
! [X2: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z2 )
=> ( ord_less_eq_set_a @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_260_order_Otrans,axiom,
! [A: set_Sum_sum_a_b,B2: set_Sum_sum_a_b,C: set_Sum_sum_a_b] :
( ( ord_le9019793522827316924um_a_b @ A @ B2 )
=> ( ( ord_le9019793522827316924um_a_b @ B2 @ C )
=> ( ord_le9019793522827316924um_a_b @ A @ C ) ) ) ).
% order.trans
thf(fact_261_order_Otrans,axiom,
! [A: $o > set_a,B2: $o > set_a,C: $o > set_a] :
( ( ord_less_eq_o_set_a @ A @ B2 )
=> ( ( ord_less_eq_o_set_a @ B2 @ C )
=> ( ord_less_eq_o_set_a @ A @ C ) ) ) ).
% order.trans
thf(fact_262_order_Otrans,axiom,
! [A: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ord_le3724670747650509150_set_a @ A @ C ) ) ) ).
% order.trans
thf(fact_263_order_Otrans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_264_order_Otrans,axiom,
! [A: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% order.trans
thf(fact_265_order__antisym,axiom,
! [X2: set_Sum_sum_a_b,Y: set_Sum_sum_a_b] :
( ( ord_le9019793522827316924um_a_b @ X2 @ Y )
=> ( ( ord_le9019793522827316924um_a_b @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_266_order__antisym,axiom,
! [X2: $o > set_a,Y: $o > set_a] :
( ( ord_less_eq_o_set_a @ X2 @ Y )
=> ( ( ord_less_eq_o_set_a @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_267_order__antisym,axiom,
! [X2: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y )
=> ( ( ord_le3724670747650509150_set_a @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_268_order__antisym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_269_order__antisym,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ( ord_less_eq_set_a @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_270_ord__le__eq__trans,axiom,
! [A: set_Sum_sum_a_b,B2: set_Sum_sum_a_b,C: set_Sum_sum_a_b] :
( ( ord_le9019793522827316924um_a_b @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_le9019793522827316924um_a_b @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_271_ord__le__eq__trans,axiom,
! [A: $o > set_a,B2: $o > set_a,C: $o > set_a] :
( ( ord_less_eq_o_set_a @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_o_set_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_272_ord__le__eq__trans,axiom,
! [A: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_le3724670747650509150_set_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_273_ord__le__eq__trans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_274_ord__le__eq__trans,axiom,
! [A: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_275_ord__eq__le__trans,axiom,
! [A: set_Sum_sum_a_b,B2: set_Sum_sum_a_b,C: set_Sum_sum_a_b] :
( ( A = B2 )
=> ( ( ord_le9019793522827316924um_a_b @ B2 @ C )
=> ( ord_le9019793522827316924um_a_b @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_276_ord__eq__le__trans,axiom,
! [A: $o > set_a,B2: $o > set_a,C: $o > set_a] :
( ( A = B2 )
=> ( ( ord_less_eq_o_set_a @ B2 @ C )
=> ( ord_less_eq_o_set_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_277_ord__eq__le__trans,axiom,
! [A: set_set_a,B2: set_set_a,C: set_set_a] :
( ( A = B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ord_le3724670747650509150_set_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_278_ord__eq__le__trans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( A = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_279_ord__eq__le__trans,axiom,
! [A: set_a,B2: set_a,C: set_a] :
( ( A = B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_280_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Sum_sum_a_b,Z: set_Sum_sum_a_b] : ( Y4 = Z ) )
= ( ^ [X3: set_Sum_sum_a_b,Y5: set_Sum_sum_a_b] :
( ( ord_le9019793522827316924um_a_b @ X3 @ Y5 )
& ( ord_le9019793522827316924um_a_b @ Y5 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_281_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: $o > set_a,Z: $o > set_a] : ( Y4 = Z ) )
= ( ^ [X3: $o > set_a,Y5: $o > set_a] :
( ( ord_less_eq_o_set_a @ X3 @ Y5 )
& ( ord_less_eq_o_set_a @ Y5 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_282_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_set_a,Z: set_set_a] : ( Y4 = Z ) )
= ( ^ [X3: set_set_a,Y5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y5 )
& ( ord_le3724670747650509150_set_a @ Y5 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_283_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
& ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_284_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_a,Z: set_a] : ( Y4 = Z ) )
= ( ^ [X3: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y5 )
& ( ord_less_eq_set_a @ Y5 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_285_le__cases3,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_286_nle__le,axiom,
! [A: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A )
& ( B2 != A ) ) ) ).
% nle_le
thf(fact_287_Collect__mono__iff,axiom,
! [P: sum_sum_a_b > $o,Q: sum_sum_a_b > $o] :
( ( ord_le9019793522827316924um_a_b @ ( collect_Sum_sum_a_b @ P ) @ ( collect_Sum_sum_a_b @ Q ) )
= ( ! [X3: sum_sum_a_b] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_288_Collect__mono__iff,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) )
= ( ! [X3: set_a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_289_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_290_set__eq__subset,axiom,
( ( ^ [Y4: set_Sum_sum_a_b,Z: set_Sum_sum_a_b] : ( Y4 = Z ) )
= ( ^ [A6: set_Sum_sum_a_b,B5: set_Sum_sum_a_b] :
( ( ord_le9019793522827316924um_a_b @ A6 @ B5 )
& ( ord_le9019793522827316924um_a_b @ B5 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_291_set__eq__subset,axiom,
( ( ^ [Y4: set_set_a,Z: set_set_a] : ( Y4 = Z ) )
= ( ^ [A6: set_set_a,B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A6 @ B5 )
& ( ord_le3724670747650509150_set_a @ B5 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_292_set__eq__subset,axiom,
( ( ^ [Y4: set_a,Z: set_a] : ( Y4 = Z ) )
= ( ^ [A6: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A6 @ B5 )
& ( ord_less_eq_set_a @ B5 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_293_subset__trans,axiom,
! [A4: set_Sum_sum_a_b,B: set_Sum_sum_a_b,C2: set_Sum_sum_a_b] :
( ( ord_le9019793522827316924um_a_b @ A4 @ B )
=> ( ( ord_le9019793522827316924um_a_b @ B @ C2 )
=> ( ord_le9019793522827316924um_a_b @ A4 @ C2 ) ) ) ).
% subset_trans
thf(fact_294_subset__trans,axiom,
! [A4: set_set_a,B: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ C2 )
=> ( ord_le3724670747650509150_set_a @ A4 @ C2 ) ) ) ).
% subset_trans
thf(fact_295_subset__trans,axiom,
! [A4: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A4 @ B )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_eq_set_a @ A4 @ C2 ) ) ) ).
% subset_trans
thf(fact_296_Collect__mono,axiom,
! [P: sum_sum_a_b > $o,Q: sum_sum_a_b > $o] :
( ! [X4: sum_sum_a_b] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le9019793522827316924um_a_b @ ( collect_Sum_sum_a_b @ P ) @ ( collect_Sum_sum_a_b @ Q ) ) ) ).
% Collect_mono
thf(fact_297_Collect__mono,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ! [X4: set_a] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) ) ) ).
% Collect_mono
thf(fact_298_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X4: a] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_299_subset__refl,axiom,
! [A4: set_Sum_sum_a_b] : ( ord_le9019793522827316924um_a_b @ A4 @ A4 ) ).
% subset_refl
thf(fact_300_subset__refl,axiom,
! [A4: set_set_a] : ( ord_le3724670747650509150_set_a @ A4 @ A4 ) ).
% subset_refl
thf(fact_301_subset__refl,axiom,
! [A4: set_a] : ( ord_less_eq_set_a @ A4 @ A4 ) ).
% subset_refl
thf(fact_302_subset__iff,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A6: set_list_a,B5: set_list_a] :
! [T: list_a] :
( ( member_list_a2 @ T @ A6 )
=> ( member_list_a2 @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_303_subset__iff,axiom,
( ord_le1944875106711258738um_a_b
= ( ^ [A6: set_set_Sum_sum_a_b,B5: set_set_Sum_sum_a_b] :
! [T: set_Sum_sum_a_b] :
( ( member4060935254435997939um_a_b @ T @ A6 )
=> ( member4060935254435997939um_a_b @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_304_subset__iff,axiom,
( ord_le5722252365846178494_set_a
= ( ^ [A6: set_set_set_a,B5: set_set_set_a] :
! [T: set_set_a] :
( ( member_set_set_a2 @ T @ A6 )
=> ( member_set_set_a2 @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_305_subset__iff,axiom,
( ord_less_eq_set_b
= ( ^ [A6: set_b,B5: set_b] :
! [T: b] :
( ( member_b2 @ T @ A6 )
=> ( member_b2 @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_306_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A6: set_set_a,B5: set_set_a] :
! [T: set_a] :
( ( member_set_a2 @ T @ A6 )
=> ( member_set_a2 @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_307_subset__iff,axiom,
( ord_le9019793522827316924um_a_b
= ( ^ [A6: set_Sum_sum_a_b,B5: set_Sum_sum_a_b] :
! [T: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ T @ A6 )
=> ( member_Sum_sum_a_b2 @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_308_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B5: set_a] :
! [T: a] :
( ( member_a2 @ T @ A6 )
=> ( member_a2 @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_309_Set_OequalityD2,axiom,
! [A4: set_Sum_sum_a_b,B: set_Sum_sum_a_b] :
( ( A4 = B )
=> ( ord_le9019793522827316924um_a_b @ B @ A4 ) ) ).
% Set.equalityD2
thf(fact_310_Set_OequalityD2,axiom,
! [A4: set_set_a,B: set_set_a] :
( ( A4 = B )
=> ( ord_le3724670747650509150_set_a @ B @ A4 ) ) ).
% Set.equalityD2
thf(fact_311_Set_OequalityD2,axiom,
! [A4: set_a,B: set_a] :
( ( A4 = B )
=> ( ord_less_eq_set_a @ B @ A4 ) ) ).
% Set.equalityD2
thf(fact_312_equalityD1,axiom,
! [A4: set_Sum_sum_a_b,B: set_Sum_sum_a_b] :
( ( A4 = B )
=> ( ord_le9019793522827316924um_a_b @ A4 @ B ) ) ).
% equalityD1
thf(fact_313_equalityD1,axiom,
! [A4: set_set_a,B: set_set_a] :
( ( A4 = B )
=> ( ord_le3724670747650509150_set_a @ A4 @ B ) ) ).
% equalityD1
thf(fact_314_equalityD1,axiom,
! [A4: set_a,B: set_a] :
( ( A4 = B )
=> ( ord_less_eq_set_a @ A4 @ B ) ) ).
% equalityD1
thf(fact_315_subset__eq,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A6: set_list_a,B5: set_list_a] :
! [X3: list_a] :
( ( member_list_a2 @ X3 @ A6 )
=> ( member_list_a2 @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_316_subset__eq,axiom,
( ord_le1944875106711258738um_a_b
= ( ^ [A6: set_set_Sum_sum_a_b,B5: set_set_Sum_sum_a_b] :
! [X3: set_Sum_sum_a_b] :
( ( member4060935254435997939um_a_b @ X3 @ A6 )
=> ( member4060935254435997939um_a_b @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_317_subset__eq,axiom,
( ord_le5722252365846178494_set_a
= ( ^ [A6: set_set_set_a,B5: set_set_set_a] :
! [X3: set_set_a] :
( ( member_set_set_a2 @ X3 @ A6 )
=> ( member_set_set_a2 @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_318_subset__eq,axiom,
( ord_less_eq_set_b
= ( ^ [A6: set_b,B5: set_b] :
! [X3: b] :
( ( member_b2 @ X3 @ A6 )
=> ( member_b2 @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_319_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A6: set_set_a,B5: set_set_a] :
! [X3: set_a] :
( ( member_set_a2 @ X3 @ A6 )
=> ( member_set_a2 @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_320_subset__eq,axiom,
( ord_le9019793522827316924um_a_b
= ( ^ [A6: set_Sum_sum_a_b,B5: set_Sum_sum_a_b] :
! [X3: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X3 @ A6 )
=> ( member_Sum_sum_a_b2 @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_321_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B5: set_a] :
! [X3: a] :
( ( member_a2 @ X3 @ A6 )
=> ( member_a2 @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_322_equalityE,axiom,
! [A4: set_Sum_sum_a_b,B: set_Sum_sum_a_b] :
( ( A4 = B )
=> ~ ( ( ord_le9019793522827316924um_a_b @ A4 @ B )
=> ~ ( ord_le9019793522827316924um_a_b @ B @ A4 ) ) ) ).
% equalityE
thf(fact_323_equalityE,axiom,
! [A4: set_set_a,B: set_set_a] :
( ( A4 = B )
=> ~ ( ( ord_le3724670747650509150_set_a @ A4 @ B )
=> ~ ( ord_le3724670747650509150_set_a @ B @ A4 ) ) ) ).
% equalityE
thf(fact_324_equalityE,axiom,
! [A4: set_a,B: set_a] :
( ( A4 = B )
=> ~ ( ( ord_less_eq_set_a @ A4 @ B )
=> ~ ( ord_less_eq_set_a @ B @ A4 ) ) ) ).
% equalityE
thf(fact_325_subsetD,axiom,
! [A4: set_list_a,B: set_list_a,C: list_a] :
( ( ord_le8861187494160871172list_a @ A4 @ B )
=> ( ( member_list_a2 @ C @ A4 )
=> ( member_list_a2 @ C @ B ) ) ) ).
% subsetD
thf(fact_326_subsetD,axiom,
! [A4: set_set_Sum_sum_a_b,B: set_set_Sum_sum_a_b,C: set_Sum_sum_a_b] :
( ( ord_le1944875106711258738um_a_b @ A4 @ B )
=> ( ( member4060935254435997939um_a_b @ C @ A4 )
=> ( member4060935254435997939um_a_b @ C @ B ) ) ) ).
% subsetD
thf(fact_327_subsetD,axiom,
! [A4: set_set_set_a,B: set_set_set_a,C: set_set_a] :
( ( ord_le5722252365846178494_set_a @ A4 @ B )
=> ( ( member_set_set_a2 @ C @ A4 )
=> ( member_set_set_a2 @ C @ B ) ) ) ).
% subsetD
thf(fact_328_subsetD,axiom,
! [A4: set_b,B: set_b,C: b] :
( ( ord_less_eq_set_b @ A4 @ B )
=> ( ( member_b2 @ C @ A4 )
=> ( member_b2 @ C @ B ) ) ) ).
% subsetD
thf(fact_329_subsetD,axiom,
! [A4: set_set_a,B: set_set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ B )
=> ( ( member_set_a2 @ C @ A4 )
=> ( member_set_a2 @ C @ B ) ) ) ).
% subsetD
thf(fact_330_subsetD,axiom,
! [A4: set_Sum_sum_a_b,B: set_Sum_sum_a_b,C: sum_sum_a_b] :
( ( ord_le9019793522827316924um_a_b @ A4 @ B )
=> ( ( member_Sum_sum_a_b2 @ C @ A4 )
=> ( member_Sum_sum_a_b2 @ C @ B ) ) ) ).
% subsetD
thf(fact_331_subsetD,axiom,
! [A4: set_a,B: set_a,C: a] :
( ( ord_less_eq_set_a @ A4 @ B )
=> ( ( member_a2 @ C @ A4 )
=> ( member_a2 @ C @ B ) ) ) ).
% subsetD
thf(fact_332_in__mono,axiom,
! [A4: set_list_a,B: set_list_a,X2: list_a] :
( ( ord_le8861187494160871172list_a @ A4 @ B )
=> ( ( member_list_a2 @ X2 @ A4 )
=> ( member_list_a2 @ X2 @ B ) ) ) ).
% in_mono
thf(fact_333_in__mono,axiom,
! [A4: set_set_Sum_sum_a_b,B: set_set_Sum_sum_a_b,X2: set_Sum_sum_a_b] :
( ( ord_le1944875106711258738um_a_b @ A4 @ B )
=> ( ( member4060935254435997939um_a_b @ X2 @ A4 )
=> ( member4060935254435997939um_a_b @ X2 @ B ) ) ) ).
% in_mono
thf(fact_334_in__mono,axiom,
! [A4: set_set_set_a,B: set_set_set_a,X2: set_set_a] :
( ( ord_le5722252365846178494_set_a @ A4 @ B )
=> ( ( member_set_set_a2 @ X2 @ A4 )
=> ( member_set_set_a2 @ X2 @ B ) ) ) ).
% in_mono
thf(fact_335_in__mono,axiom,
! [A4: set_b,B: set_b,X2: b] :
( ( ord_less_eq_set_b @ A4 @ B )
=> ( ( member_b2 @ X2 @ A4 )
=> ( member_b2 @ X2 @ B ) ) ) ).
% in_mono
thf(fact_336_in__mono,axiom,
! [A4: set_set_a,B: set_set_a,X2: set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ B )
=> ( ( member_set_a2 @ X2 @ A4 )
=> ( member_set_a2 @ X2 @ B ) ) ) ).
% in_mono
thf(fact_337_in__mono,axiom,
! [A4: set_Sum_sum_a_b,B: set_Sum_sum_a_b,X2: sum_sum_a_b] :
( ( ord_le9019793522827316924um_a_b @ A4 @ B )
=> ( ( member_Sum_sum_a_b2 @ X2 @ A4 )
=> ( member_Sum_sum_a_b2 @ X2 @ B ) ) ) ).
% in_mono
thf(fact_338_in__mono,axiom,
! [A4: set_a,B: set_a,X2: a] :
( ( ord_less_eq_set_a @ A4 @ B )
=> ( ( member_a2 @ X2 @ A4 )
=> ( member_a2 @ X2 @ B ) ) ) ).
% in_mono
thf(fact_339_Greatest__equality,axiom,
! [P: set_Sum_sum_a_b > $o,X2: set_Sum_sum_a_b] :
( ( P @ X2 )
=> ( ! [Y3: set_Sum_sum_a_b] :
( ( P @ Y3 )
=> ( ord_le9019793522827316924um_a_b @ Y3 @ X2 ) )
=> ( ( order_5783291715448221251um_a_b @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_340_Greatest__equality,axiom,
! [P: ( $o > set_a ) > $o,X2: $o > set_a] :
( ( P @ X2 )
=> ( ! [Y3: $o > set_a] :
( ( P @ Y3 )
=> ( ord_less_eq_o_set_a @ Y3 @ X2 ) )
=> ( ( order_6114237596796908690_set_a @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_341_Greatest__equality,axiom,
! [P: set_set_a > $o,X2: set_set_a] :
( ( P @ X2 )
=> ( ! [Y3: set_set_a] :
( ( P @ Y3 )
=> ( ord_le3724670747650509150_set_a @ Y3 @ X2 ) )
=> ( ( order_3565860530148683671_set_a @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_342_Greatest__equality,axiom,
! [P: nat > $o,X2: nat] :
( ( P @ X2 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X2 ) )
=> ( ( order_Greatest_nat @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_343_Greatest__equality,axiom,
! [P: set_a > $o,X2: set_a] :
( ( P @ X2 )
=> ( ! [Y3: set_a] :
( ( P @ Y3 )
=> ( ord_less_eq_set_a @ Y3 @ X2 ) )
=> ( ( order_Greatest_set_a @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_344_GreatestI2__order,axiom,
! [P: set_Sum_sum_a_b > $o,X2: set_Sum_sum_a_b,Q: set_Sum_sum_a_b > $o] :
( ( P @ X2 )
=> ( ! [Y3: set_Sum_sum_a_b] :
( ( P @ Y3 )
=> ( ord_le9019793522827316924um_a_b @ Y3 @ X2 ) )
=> ( ! [X4: set_Sum_sum_a_b] :
( ( P @ X4 )
=> ( ! [Y6: set_Sum_sum_a_b] :
( ( P @ Y6 )
=> ( ord_le9019793522827316924um_a_b @ Y6 @ X4 ) )
=> ( Q @ X4 ) ) )
=> ( Q @ ( order_5783291715448221251um_a_b @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_345_GreatestI2__order,axiom,
! [P: ( $o > set_a ) > $o,X2: $o > set_a,Q: ( $o > set_a ) > $o] :
( ( P @ X2 )
=> ( ! [Y3: $o > set_a] :
( ( P @ Y3 )
=> ( ord_less_eq_o_set_a @ Y3 @ X2 ) )
=> ( ! [X4: $o > set_a] :
( ( P @ X4 )
=> ( ! [Y6: $o > set_a] :
( ( P @ Y6 )
=> ( ord_less_eq_o_set_a @ Y6 @ X4 ) )
=> ( Q @ X4 ) ) )
=> ( Q @ ( order_6114237596796908690_set_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_346_GreatestI2__order,axiom,
! [P: set_set_a > $o,X2: set_set_a,Q: set_set_a > $o] :
( ( P @ X2 )
=> ( ! [Y3: set_set_a] :
( ( P @ Y3 )
=> ( ord_le3724670747650509150_set_a @ Y3 @ X2 ) )
=> ( ! [X4: set_set_a] :
( ( P @ X4 )
=> ( ! [Y6: set_set_a] :
( ( P @ Y6 )
=> ( ord_le3724670747650509150_set_a @ Y6 @ X4 ) )
=> ( Q @ X4 ) ) )
=> ( Q @ ( order_3565860530148683671_set_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_347_GreatestI2__order,axiom,
! [P: nat > $o,X2: nat,Q: nat > $o] :
( ( P @ X2 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X2 ) )
=> ( ! [X4: nat] :
( ( P @ X4 )
=> ( ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X4 ) )
=> ( Q @ X4 ) ) )
=> ( Q @ ( order_Greatest_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_348_GreatestI2__order,axiom,
! [P: set_a > $o,X2: set_a,Q: set_a > $o] :
( ( P @ X2 )
=> ( ! [Y3: set_a] :
( ( P @ Y3 )
=> ( ord_less_eq_set_a @ Y3 @ X2 ) )
=> ( ! [X4: set_a] :
( ( P @ X4 )
=> ( ! [Y6: set_a] :
( ( P @ Y6 )
=> ( ord_less_eq_set_a @ Y6 @ X4 ) )
=> ( Q @ X4 ) ) )
=> ( Q @ ( order_Greatest_set_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_349_subset__code_I2_J,axiom,
! [A4: set_set_Sum_sum_a_b,Ys: list_set_Sum_sum_a_b] :
( ( ord_le1944875106711258738um_a_b @ A4 @ ( coset_2146284473681853737um_a_b @ Ys ) )
= ( ! [X3: set_Sum_sum_a_b] :
( ( member4060935254435997939um_a_b @ X3 @ ( set_set_Sum_sum_a_b2 @ Ys ) )
=> ~ ( member4060935254435997939um_a_b @ X3 @ A4 ) ) ) ) ).
% subset_code(2)
thf(fact_350_subset__code_I2_J,axiom,
! [A4: set_set_set_a,Ys: list_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A4 @ ( coset_set_set_a @ Ys ) )
= ( ! [X3: set_set_a] :
( ( member_set_set_a2 @ X3 @ ( set_set_set_a2 @ Ys ) )
=> ~ ( member_set_set_a2 @ X3 @ A4 ) ) ) ) ).
% subset_code(2)
thf(fact_351_subset__code_I2_J,axiom,
! [A4: set_b,Ys: list_b] :
( ( ord_less_eq_set_b @ A4 @ ( coset_b @ Ys ) )
= ( ! [X3: b] :
( ( member_b2 @ X3 @ ( set_b2 @ Ys ) )
=> ~ ( member_b2 @ X3 @ A4 ) ) ) ) ).
% subset_code(2)
thf(fact_352_subset__code_I2_J,axiom,
! [A4: set_list_a,Ys: list_list_a] :
( ( ord_le8861187494160871172list_a @ A4 @ ( coset_list_a @ Ys ) )
= ( ! [X3: list_a] :
( ( member_list_a2 @ X3 @ ( set_list_a2 @ Ys ) )
=> ~ ( member_list_a2 @ X3 @ A4 ) ) ) ) ).
% subset_code(2)
thf(fact_353_subset__code_I2_J,axiom,
! [A4: set_list_Sum_sum_a_b,Ys: list_l4199846171218662726um_a_b] :
( ( ord_le2472362315733485388um_a_b @ A4 @ ( coset_6412875586394208771um_a_b @ Ys ) )
= ( ! [X3: list_Sum_sum_a_b] :
( ( member7701661377270014157um_a_b @ X3 @ ( set_list_Sum_sum_a_b2 @ Ys ) )
=> ~ ( member7701661377270014157um_a_b @ X3 @ A4 ) ) ) ) ).
% subset_code(2)
thf(fact_354_subset__code_I2_J,axiom,
! [A4: set_set_a,Ys: list_set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ ( coset_set_a @ Ys ) )
= ( ! [X3: set_a] :
( ( member_set_a2 @ X3 @ ( set_set_a2 @ Ys ) )
=> ~ ( member_set_a2 @ X3 @ A4 ) ) ) ) ).
% subset_code(2)
thf(fact_355_subset__code_I2_J,axiom,
! [A4: set_Sum_sum_a_b,Ys: list_Sum_sum_a_b] :
( ( ord_le9019793522827316924um_a_b @ A4 @ ( coset_Sum_sum_a_b @ Ys ) )
= ( ! [X3: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X3 @ ( set_Sum_sum_a_b2 @ Ys ) )
=> ~ ( member_Sum_sum_a_b2 @ X3 @ A4 ) ) ) ) ).
% subset_code(2)
thf(fact_356_subset__code_I2_J,axiom,
! [A4: set_a,Ys: list_a] :
( ( ord_less_eq_set_a @ A4 @ ( coset_a @ Ys ) )
= ( ! [X3: a] :
( ( member_a2 @ X3 @ ( set_a2 @ Ys ) )
=> ~ ( member_a2 @ X3 @ A4 ) ) ) ) ).
% subset_code(2)
thf(fact_357_le__rel__bool__arg__iff,axiom,
( ord_le4308901180383770373um_a_b
= ( ^ [X5: $o > set_Sum_sum_a_b,Y7: $o > set_Sum_sum_a_b] :
( ( ord_le9019793522827316924um_a_b @ ( X5 @ $false ) @ ( Y7 @ $false ) )
& ( ord_le9019793522827316924um_a_b @ ( X5 @ $true ) @ ( Y7 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_358_le__rel__bool__arg__iff,axiom,
( ord_le5604070792465694036_set_a
= ( ^ [X5: $o > $o > set_a,Y7: $o > $o > set_a] :
( ( ord_less_eq_o_set_a @ ( X5 @ $false ) @ ( Y7 @ $false ) )
& ( ord_less_eq_o_set_a @ ( X5 @ $true ) @ ( Y7 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_359_le__rel__bool__arg__iff,axiom,
( ord_le2411852534652195563_set_a
= ( ^ [X5: $o > set_set_a,Y7: $o > set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( X5 @ $false ) @ ( Y7 @ $false ) )
& ( ord_le3724670747650509150_set_a @ ( X5 @ $true ) @ ( Y7 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_360_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_nat
= ( ^ [X5: $o > nat,Y7: $o > nat] :
( ( ord_less_eq_nat @ ( X5 @ $false ) @ ( Y7 @ $false ) )
& ( ord_less_eq_nat @ ( X5 @ $true ) @ ( Y7 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_361_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_set_a
= ( ^ [X5: $o > set_a,Y7: $o > set_a] :
( ( ord_less_eq_set_a @ ( X5 @ $false ) @ ( Y7 @ $false ) )
& ( ord_less_eq_set_a @ ( X5 @ $true ) @ ( Y7 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_362_verit__la__disequality,axiom,
! [A: nat,B2: nat] :
( ( A = B2 )
| ~ ( ord_less_eq_nat @ A @ B2 )
| ~ ( ord_less_eq_nat @ B2 @ A ) ) ).
% verit_la_disequality
thf(fact_363_wlog__le,axiom,
! [P: nat > nat > $o,B2: nat,A: nat] :
( ! [A5: nat,B4: nat] :
( ( P @ A5 @ B4 )
=> ( P @ B4 @ A5 ) )
=> ( ! [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
=> ( P @ A5 @ B4 ) )
=> ( P @ B2 @ A ) ) ) ).
% wlog_le
thf(fact_364_verit__comp__simplify1_I2_J,axiom,
! [A: set_Sum_sum_a_b] : ( ord_le9019793522827316924um_a_b @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_365_verit__comp__simplify1_I2_J,axiom,
! [A: $o > set_a] : ( ord_less_eq_o_set_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_366_verit__comp__simplify1_I2_J,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_367_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_368_verit__comp__simplify1_I2_J,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_369_equality__sum__simps_I3_J,axiom,
! [Eq_a: sum_sum_a_b > sum_sum_a_b > $o,Eq_b: sum_sum_a_b > sum_sum_a_b > $o,X2: sum_sum_a_b,Y: sum_sum_a_b] :
~ ( equali7701304696083298663um_a_b @ Eq_a @ Eq_b @ ( sum_In6272690418125038914um_a_b @ X2 ) @ ( sum_In5992699931424873788um_a_b @ Y ) ) ).
% equality_sum_simps(3)
thf(fact_370_equality__sum__simps_I3_J,axiom,
! [Eq_a: sum_sum_a_b > sum_sum_a_b > $o,Eq_b: a > a > $o,X2: a,Y: sum_sum_a_b] :
~ ( equali6768406920669315453_a_b_a @ Eq_a @ Eq_b @ ( sum_In8629808552650825520um_a_b @ X2 ) @ ( sum_In6222238883715738344_a_b_a @ Y ) ) ).
% equality_sum_simps(3)
thf(fact_371_equality__sum__simps_I3_J,axiom,
! [Eq_a: a > a > $o,Eq_b: sum_sum_a_b > sum_sum_a_b > $o,X2: sum_sum_a_b,Y: a] :
~ ( equali3464600134807874635um_a_b @ Eq_a @ Eq_b @ ( sum_In2710243301657490530_a_b_a @ X2 ) @ ( sum_In2918432097854297526um_a_b @ Y ) ) ).
% equality_sum_simps(3)
thf(fact_372_equality__sum__simps_I3_J,axiom,
! [Eq_a: a > a > $o,Eq_b: a > a > $o,X2: a,Y: a] :
~ ( equali5972598040643577753um_a_a @ Eq_a @ Eq_b @ ( sum_Inr_a_a @ X2 ) @ ( sum_Inl_a_a @ Y ) ) ).
% equality_sum_simps(3)
thf(fact_373_equality__sum__simps_I3_J,axiom,
! [Eq_a: a > a > $o,Eq_b: b > b > $o,X2: b,Y: a] :
~ ( equali5972598040643577754um_a_b @ Eq_a @ Eq_b @ ( sum_Inr_b_a @ X2 ) @ ( sum_Inl_a_b @ Y ) ) ).
% equality_sum_simps(3)
thf(fact_374_equality__sum__simps_I2_J,axiom,
! [Eq_a: sum_sum_a_b > sum_sum_a_b > $o,Eq_b: sum_sum_a_b > sum_sum_a_b > $o,X2: sum_sum_a_b,Ya: sum_sum_a_b] :
~ ( equali7701304696083298663um_a_b @ Eq_a @ Eq_b @ ( sum_In5992699931424873788um_a_b @ X2 ) @ ( sum_In6272690418125038914um_a_b @ Ya ) ) ).
% equality_sum_simps(2)
thf(fact_375_equality__sum__simps_I2_J,axiom,
! [Eq_a: sum_sum_a_b > sum_sum_a_b > $o,Eq_b: a > a > $o,X2: sum_sum_a_b,Ya: a] :
~ ( equali6768406920669315453_a_b_a @ Eq_a @ Eq_b @ ( sum_In6222238883715738344_a_b_a @ X2 ) @ ( sum_In8629808552650825520um_a_b @ Ya ) ) ).
% equality_sum_simps(2)
thf(fact_376_equality__sum__simps_I2_J,axiom,
! [Eq_a: a > a > $o,Eq_b: sum_sum_a_b > sum_sum_a_b > $o,X2: a,Ya: sum_sum_a_b] :
~ ( equali3464600134807874635um_a_b @ Eq_a @ Eq_b @ ( sum_In2918432097854297526um_a_b @ X2 ) @ ( sum_In2710243301657490530_a_b_a @ Ya ) ) ).
% equality_sum_simps(2)
thf(fact_377_equality__sum__simps_I2_J,axiom,
! [Eq_a: a > a > $o,Eq_b: a > a > $o,X2: a,Ya: a] :
~ ( equali5972598040643577753um_a_a @ Eq_a @ Eq_b @ ( sum_Inl_a_a @ X2 ) @ ( sum_Inr_a_a @ Ya ) ) ).
% equality_sum_simps(2)
thf(fact_378_equality__sum__simps_I2_J,axiom,
! [Eq_a: a > a > $o,Eq_b: b > b > $o,X2: a,Ya: b] :
~ ( equali5972598040643577754um_a_b @ Eq_a @ Eq_b @ ( sum_Inl_a_b @ X2 ) @ ( sum_Inr_b_a @ Ya ) ) ).
% equality_sum_simps(2)
thf(fact_379_list__ex1__iff,axiom,
( list_e8491526644333077203um_a_b
= ( ^ [P2: set_Sum_sum_a_b > $o,Xs2: list_set_Sum_sum_a_b] :
? [X3: set_Sum_sum_a_b] :
( ( member4060935254435997939um_a_b @ X3 @ ( set_set_Sum_sum_a_b2 @ Xs2 ) )
& ( P2 @ X3 )
& ! [Y5: set_Sum_sum_a_b] :
( ( ( member4060935254435997939um_a_b @ Y5 @ ( set_set_Sum_sum_a_b2 @ Xs2 ) )
& ( P2 @ Y5 ) )
=> ( Y5 = X3 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_380_list__ex1__iff,axiom,
( list_ex1_set_set_a
= ( ^ [P2: set_set_a > $o,Xs2: list_set_set_a] :
? [X3: set_set_a] :
( ( member_set_set_a2 @ X3 @ ( set_set_set_a2 @ Xs2 ) )
& ( P2 @ X3 )
& ! [Y5: set_set_a] :
( ( ( member_set_set_a2 @ Y5 @ ( set_set_set_a2 @ Xs2 ) )
& ( P2 @ Y5 ) )
=> ( Y5 = X3 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_381_list__ex1__iff,axiom,
( list_ex1_set_a
= ( ^ [P2: set_a > $o,Xs2: list_set_a] :
? [X3: set_a] :
( ( member_set_a2 @ X3 @ ( set_set_a2 @ Xs2 ) )
& ( P2 @ X3 )
& ! [Y5: set_a] :
( ( ( member_set_a2 @ Y5 @ ( set_set_a2 @ Xs2 ) )
& ( P2 @ Y5 ) )
=> ( Y5 = X3 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_382_list__ex1__iff,axiom,
( list_ex1_b
= ( ^ [P2: b > $o,Xs2: list_b] :
? [X3: b] :
( ( member_b2 @ X3 @ ( set_b2 @ Xs2 ) )
& ( P2 @ X3 )
& ! [Y5: b] :
( ( ( member_b2 @ Y5 @ ( set_b2 @ Xs2 ) )
& ( P2 @ Y5 ) )
=> ( Y5 = X3 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_383_list__ex1__iff,axiom,
( list_ex1_list_a
= ( ^ [P2: list_a > $o,Xs2: list_list_a] :
? [X3: list_a] :
( ( member_list_a2 @ X3 @ ( set_list_a2 @ Xs2 ) )
& ( P2 @ X3 )
& ! [Y5: list_a] :
( ( ( member_list_a2 @ Y5 @ ( set_list_a2 @ Xs2 ) )
& ( P2 @ Y5 ) )
=> ( Y5 = X3 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_384_list__ex1__iff,axiom,
( list_e4286574564361611949um_a_b
= ( ^ [P2: list_Sum_sum_a_b > $o,Xs2: list_l4199846171218662726um_a_b] :
? [X3: list_Sum_sum_a_b] :
( ( member7701661377270014157um_a_b @ X3 @ ( set_list_Sum_sum_a_b2 @ Xs2 ) )
& ( P2 @ X3 )
& ! [Y5: list_Sum_sum_a_b] :
( ( ( member7701661377270014157um_a_b @ Y5 @ ( set_list_Sum_sum_a_b2 @ Xs2 ) )
& ( P2 @ Y5 ) )
=> ( Y5 = X3 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_385_list__ex1__iff,axiom,
( list_ex1_a
= ( ^ [P2: a > $o,Xs2: list_a] :
? [X3: a] :
( ( member_a2 @ X3 @ ( set_a2 @ Xs2 ) )
& ( P2 @ X3 )
& ! [Y5: a] :
( ( ( member_a2 @ Y5 @ ( set_a2 @ Xs2 ) )
& ( P2 @ Y5 ) )
=> ( Y5 = X3 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_386_list__ex1__iff,axiom,
( list_ex1_Sum_sum_a_b
= ( ^ [P2: sum_sum_a_b > $o,Xs2: list_Sum_sum_a_b] :
? [X3: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X3 @ ( set_Sum_sum_a_b2 @ Xs2 ) )
& ( P2 @ X3 )
& ! [Y5: sum_sum_a_b] :
( ( ( member_Sum_sum_a_b2 @ Y5 @ ( set_Sum_sum_a_b2 @ Xs2 ) )
& ( P2 @ Y5 ) )
=> ( Y5 = X3 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_387_in__set__insert,axiom,
! [X2: set_Sum_sum_a_b,Xs: list_set_Sum_sum_a_b] :
( ( member4060935254435997939um_a_b @ X2 @ ( set_set_Sum_sum_a_b2 @ Xs ) )
=> ( ( insert301144099385869888um_a_b @ X2 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_388_in__set__insert,axiom,
! [X2: set_set_a,Xs: list_set_set_a] :
( ( member_set_set_a2 @ X2 @ ( set_set_set_a2 @ Xs ) )
=> ( ( insert_set_set_a @ X2 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_389_in__set__insert,axiom,
! [X2: set_a,Xs: list_set_a] :
( ( member_set_a2 @ X2 @ ( set_set_a2 @ Xs ) )
=> ( ( insert_set_a @ X2 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_390_in__set__insert,axiom,
! [X2: b,Xs: list_b] :
( ( member_b2 @ X2 @ ( set_b2 @ Xs ) )
=> ( ( insert_b @ X2 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_391_in__set__insert,axiom,
! [X2: list_a,Xs: list_list_a] :
( ( member_list_a2 @ X2 @ ( set_list_a2 @ Xs ) )
=> ( ( insert_list_a @ X2 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_392_in__set__insert,axiom,
! [X2: list_Sum_sum_a_b,Xs: list_l4199846171218662726um_a_b] :
( ( member7701661377270014157um_a_b @ X2 @ ( set_list_Sum_sum_a_b2 @ Xs ) )
=> ( ( insert3356916551913007002um_a_b @ X2 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_393_in__set__insert,axiom,
! [X2: a,Xs: list_a] :
( ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
=> ( ( insert_a @ X2 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_394_in__set__insert,axiom,
! [X2: sum_sum_a_b,Xs: list_Sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X2 @ ( set_Sum_sum_a_b2 @ Xs ) )
=> ( ( insert_Sum_sum_a_b @ X2 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_395_old_Osum_Oinject_I2_J,axiom,
! [B2: b,B6: b] :
( ( ( sum_Inr_b_a @ B2 )
= ( sum_Inr_b_a @ B6 ) )
= ( B2 = B6 ) ) ).
% old.sum.inject(2)
thf(fact_396_old_Osum_Oinject_I2_J,axiom,
! [B2: sum_sum_a_b,B6: sum_sum_a_b] :
( ( ( sum_In6272690418125038914um_a_b @ B2 )
= ( sum_In6272690418125038914um_a_b @ B6 ) )
= ( B2 = B6 ) ) ).
% old.sum.inject(2)
thf(fact_397_old_Osum_Oinject_I2_J,axiom,
! [B2: a,B6: a] :
( ( ( sum_In8629808552650825520um_a_b @ B2 )
= ( sum_In8629808552650825520um_a_b @ B6 ) )
= ( B2 = B6 ) ) ).
% old.sum.inject(2)
thf(fact_398_old_Osum_Oinject_I2_J,axiom,
! [B2: sum_sum_a_b,B6: sum_sum_a_b] :
( ( ( sum_In2710243301657490530_a_b_a @ B2 )
= ( sum_In2710243301657490530_a_b_a @ B6 ) )
= ( B2 = B6 ) ) ).
% old.sum.inject(2)
thf(fact_399_old_Osum_Oinject_I2_J,axiom,
! [B2: a,B6: a] :
( ( ( sum_Inr_a_a @ B2 )
= ( sum_Inr_a_a @ B6 ) )
= ( B2 = B6 ) ) ).
% old.sum.inject(2)
thf(fact_400_sum_Oinject_I2_J,axiom,
! [X22: b,Y22: b] :
( ( ( sum_Inr_b_a @ X22 )
= ( sum_Inr_b_a @ Y22 ) )
= ( X22 = Y22 ) ) ).
% sum.inject(2)
thf(fact_401_sum_Oinject_I2_J,axiom,
! [X22: sum_sum_a_b,Y22: sum_sum_a_b] :
( ( ( sum_In6272690418125038914um_a_b @ X22 )
= ( sum_In6272690418125038914um_a_b @ Y22 ) )
= ( X22 = Y22 ) ) ).
% sum.inject(2)
thf(fact_402_sum_Oinject_I2_J,axiom,
! [X22: a,Y22: a] :
( ( ( sum_In8629808552650825520um_a_b @ X22 )
= ( sum_In8629808552650825520um_a_b @ Y22 ) )
= ( X22 = Y22 ) ) ).
% sum.inject(2)
thf(fact_403_sum_Oinject_I2_J,axiom,
! [X22: sum_sum_a_b,Y22: sum_sum_a_b] :
( ( ( sum_In2710243301657490530_a_b_a @ X22 )
= ( sum_In2710243301657490530_a_b_a @ Y22 ) )
= ( X22 = Y22 ) ) ).
% sum.inject(2)
thf(fact_404_sum_Oinject_I2_J,axiom,
! [X22: a,Y22: a] :
( ( ( sum_Inr_a_a @ X22 )
= ( sum_Inr_a_a @ Y22 ) )
= ( X22 = Y22 ) ) ).
% sum.inject(2)
thf(fact_405_equality__sum__simps_I4_J,axiom,
! [Eq_a: sum_sum_a_b > sum_sum_a_b > $o,Eq_b: sum_sum_a_b > sum_sum_a_b > $o,X2: sum_sum_a_b,Ya: sum_sum_a_b] :
( ( equali7701304696083298663um_a_b @ Eq_a @ Eq_b @ ( sum_In6272690418125038914um_a_b @ X2 ) @ ( sum_In6272690418125038914um_a_b @ Ya ) )
= ( Eq_b @ X2 @ Ya ) ) ).
% equality_sum_simps(4)
thf(fact_406_equality__sum__simps_I4_J,axiom,
! [Eq_a: sum_sum_a_b > sum_sum_a_b > $o,Eq_b: a > a > $o,X2: a,Ya: a] :
( ( equali6768406920669315453_a_b_a @ Eq_a @ Eq_b @ ( sum_In8629808552650825520um_a_b @ X2 ) @ ( sum_In8629808552650825520um_a_b @ Ya ) )
= ( Eq_b @ X2 @ Ya ) ) ).
% equality_sum_simps(4)
thf(fact_407_equality__sum__simps_I4_J,axiom,
! [Eq_a: a > a > $o,Eq_b: sum_sum_a_b > sum_sum_a_b > $o,X2: sum_sum_a_b,Ya: sum_sum_a_b] :
( ( equali3464600134807874635um_a_b @ Eq_a @ Eq_b @ ( sum_In2710243301657490530_a_b_a @ X2 ) @ ( sum_In2710243301657490530_a_b_a @ Ya ) )
= ( Eq_b @ X2 @ Ya ) ) ).
% equality_sum_simps(4)
thf(fact_408_equality__sum__simps_I4_J,axiom,
! [Eq_a: a > a > $o,Eq_b: a > a > $o,X2: a,Ya: a] :
( ( equali5972598040643577753um_a_a @ Eq_a @ Eq_b @ ( sum_Inr_a_a @ X2 ) @ ( sum_Inr_a_a @ Ya ) )
= ( Eq_b @ X2 @ Ya ) ) ).
% equality_sum_simps(4)
thf(fact_409_equality__sum__simps_I4_J,axiom,
! [Eq_a: a > a > $o,Eq_b: b > b > $o,X2: b,Ya: b] :
( ( equali5972598040643577754um_a_b @ Eq_a @ Eq_b @ ( sum_Inr_b_a @ X2 ) @ ( sum_Inr_b_a @ Ya ) )
= ( Eq_b @ X2 @ Ya ) ) ).
% equality_sum_simps(4)
thf(fact_410_Inr__inject,axiom,
! [X2: b,Y: b] :
( ( ( sum_Inr_b_a @ X2 )
= ( sum_Inr_b_a @ Y ) )
=> ( X2 = Y ) ) ).
% Inr_inject
thf(fact_411_Inr__inject,axiom,
! [X2: sum_sum_a_b,Y: sum_sum_a_b] :
( ( ( sum_In6272690418125038914um_a_b @ X2 )
= ( sum_In6272690418125038914um_a_b @ Y ) )
=> ( X2 = Y ) ) ).
% Inr_inject
thf(fact_412_Inr__inject,axiom,
! [X2: a,Y: a] :
( ( ( sum_In8629808552650825520um_a_b @ X2 )
= ( sum_In8629808552650825520um_a_b @ Y ) )
=> ( X2 = Y ) ) ).
% Inr_inject
thf(fact_413_Inr__inject,axiom,
! [X2: sum_sum_a_b,Y: sum_sum_a_b] :
( ( ( sum_In2710243301657490530_a_b_a @ X2 )
= ( sum_In2710243301657490530_a_b_a @ Y ) )
=> ( X2 = Y ) ) ).
% Inr_inject
thf(fact_414_Inr__inject,axiom,
! [X2: a,Y: a] :
( ( ( sum_Inr_a_a @ X2 )
= ( sum_Inr_a_a @ Y ) )
=> ( X2 = Y ) ) ).
% Inr_inject
thf(fact_415_split__sum__all,axiom,
( ( ^ [P3: sum_su3067303292148767147um_a_b > $o] :
! [X6: sum_su3067303292148767147um_a_b] : ( P3 @ X6 ) )
= ( ^ [P2: sum_su3067303292148767147um_a_b > $o] :
( ! [X3: sum_sum_a_b] : ( P2 @ ( sum_In5992699931424873788um_a_b @ X3 ) )
& ! [X3: sum_sum_a_b] : ( P2 @ ( sum_In6272690418125038914um_a_b @ X3 ) ) ) ) ) ).
% split_sum_all
thf(fact_416_split__sum__all,axiom,
( ( ^ [P3: sum_su3831877439928360143_a_b_a > $o] :
! [X6: sum_su3831877439928360143_a_b_a] : ( P3 @ X6 ) )
= ( ^ [P2: sum_su3831877439928360143_a_b_a > $o] :
( ! [X3: sum_sum_a_b] : ( P2 @ ( sum_In6222238883715738344_a_b_a @ X3 ) )
& ! [X3: a] : ( P2 @ ( sum_In8629808552650825520um_a_b @ X3 ) ) ) ) ) ).
% split_sum_all
thf(fact_417_split__sum__all,axiom,
( ( ^ [P3: sum_su5898878462909468885um_a_b > $o] :
! [X6: sum_su5898878462909468885um_a_b] : ( P3 @ X6 ) )
= ( ^ [P2: sum_su5898878462909468885um_a_b > $o] :
( ! [X3: a] : ( P2 @ ( sum_In2918432097854297526um_a_b @ X3 ) )
& ! [X3: sum_sum_a_b] : ( P2 @ ( sum_In2710243301657490530_a_b_a @ X3 ) ) ) ) ) ).
% split_sum_all
thf(fact_418_split__sum__all,axiom,
( ( ^ [P3: sum_sum_a_a > $o] :
! [X6: sum_sum_a_a] : ( P3 @ X6 ) )
= ( ^ [P2: sum_sum_a_a > $o] :
( ! [X3: a] : ( P2 @ ( sum_Inl_a_a @ X3 ) )
& ! [X3: a] : ( P2 @ ( sum_Inr_a_a @ X3 ) ) ) ) ) ).
% split_sum_all
thf(fact_419_split__sum__all,axiom,
( ( ^ [P3: sum_sum_a_b > $o] :
! [X6: sum_sum_a_b] : ( P3 @ X6 ) )
= ( ^ [P2: sum_sum_a_b > $o] :
( ! [X3: a] : ( P2 @ ( sum_Inl_a_b @ X3 ) )
& ! [X3: b] : ( P2 @ ( sum_Inr_b_a @ X3 ) ) ) ) ) ).
% split_sum_all
thf(fact_420_split__sum__ex,axiom,
( ( ^ [P3: sum_su3067303292148767147um_a_b > $o] :
? [X6: sum_su3067303292148767147um_a_b] : ( P3 @ X6 ) )
= ( ^ [P2: sum_su3067303292148767147um_a_b > $o] :
( ? [X3: sum_sum_a_b] : ( P2 @ ( sum_In5992699931424873788um_a_b @ X3 ) )
| ? [X3: sum_sum_a_b] : ( P2 @ ( sum_In6272690418125038914um_a_b @ X3 ) ) ) ) ) ).
% split_sum_ex
thf(fact_421_split__sum__ex,axiom,
( ( ^ [P3: sum_su3831877439928360143_a_b_a > $o] :
? [X6: sum_su3831877439928360143_a_b_a] : ( P3 @ X6 ) )
= ( ^ [P2: sum_su3831877439928360143_a_b_a > $o] :
( ? [X3: sum_sum_a_b] : ( P2 @ ( sum_In6222238883715738344_a_b_a @ X3 ) )
| ? [X3: a] : ( P2 @ ( sum_In8629808552650825520um_a_b @ X3 ) ) ) ) ) ).
% split_sum_ex
thf(fact_422_split__sum__ex,axiom,
( ( ^ [P3: sum_su5898878462909468885um_a_b > $o] :
? [X6: sum_su5898878462909468885um_a_b] : ( P3 @ X6 ) )
= ( ^ [P2: sum_su5898878462909468885um_a_b > $o] :
( ? [X3: a] : ( P2 @ ( sum_In2918432097854297526um_a_b @ X3 ) )
| ? [X3: sum_sum_a_b] : ( P2 @ ( sum_In2710243301657490530_a_b_a @ X3 ) ) ) ) ) ).
% split_sum_ex
thf(fact_423_split__sum__ex,axiom,
( ( ^ [P3: sum_sum_a_a > $o] :
? [X6: sum_sum_a_a] : ( P3 @ X6 ) )
= ( ^ [P2: sum_sum_a_a > $o] :
( ? [X3: a] : ( P2 @ ( sum_Inl_a_a @ X3 ) )
| ? [X3: a] : ( P2 @ ( sum_Inr_a_a @ X3 ) ) ) ) ) ).
% split_sum_ex
thf(fact_424_split__sum__ex,axiom,
( ( ^ [P3: sum_sum_a_b > $o] :
? [X6: sum_sum_a_b] : ( P3 @ X6 ) )
= ( ^ [P2: sum_sum_a_b > $o] :
( ? [X3: a] : ( P2 @ ( sum_Inl_a_b @ X3 ) )
| ? [X3: b] : ( P2 @ ( sum_Inr_b_a @ X3 ) ) ) ) ) ).
% split_sum_ex
thf(fact_425_Inr__not__Inl,axiom,
! [B2: sum_sum_a_b,A: sum_sum_a_b] :
( ( sum_In6272690418125038914um_a_b @ B2 )
!= ( sum_In5992699931424873788um_a_b @ A ) ) ).
% Inr_not_Inl
thf(fact_426_Inr__not__Inl,axiom,
! [B2: a,A: sum_sum_a_b] :
( ( sum_In8629808552650825520um_a_b @ B2 )
!= ( sum_In6222238883715738344_a_b_a @ A ) ) ).
% Inr_not_Inl
thf(fact_427_Inr__not__Inl,axiom,
! [B2: sum_sum_a_b,A: a] :
( ( sum_In2710243301657490530_a_b_a @ B2 )
!= ( sum_In2918432097854297526um_a_b @ A ) ) ).
% Inr_not_Inl
thf(fact_428_Inr__not__Inl,axiom,
! [B2: a,A: a] :
( ( sum_Inr_a_a @ B2 )
!= ( sum_Inl_a_a @ A ) ) ).
% Inr_not_Inl
thf(fact_429_Inr__not__Inl,axiom,
! [B2: b,A: a] :
( ( sum_Inr_b_a @ B2 )
!= ( sum_Inl_a_b @ A ) ) ).
% Inr_not_Inl
thf(fact_430_sumE,axiom,
! [S: sum_su3067303292148767147um_a_b] :
( ! [X4: sum_sum_a_b] :
( S
!= ( sum_In5992699931424873788um_a_b @ X4 ) )
=> ~ ! [Y3: sum_sum_a_b] :
( S
!= ( sum_In6272690418125038914um_a_b @ Y3 ) ) ) ).
% sumE
thf(fact_431_sumE,axiom,
! [S: sum_su3831877439928360143_a_b_a] :
( ! [X4: sum_sum_a_b] :
( S
!= ( sum_In6222238883715738344_a_b_a @ X4 ) )
=> ~ ! [Y3: a] :
( S
!= ( sum_In8629808552650825520um_a_b @ Y3 ) ) ) ).
% sumE
thf(fact_432_sumE,axiom,
! [S: sum_su5898878462909468885um_a_b] :
( ! [X4: a] :
( S
!= ( sum_In2918432097854297526um_a_b @ X4 ) )
=> ~ ! [Y3: sum_sum_a_b] :
( S
!= ( sum_In2710243301657490530_a_b_a @ Y3 ) ) ) ).
% sumE
thf(fact_433_sumE,axiom,
! [S: sum_sum_a_a] :
( ! [X4: a] :
( S
!= ( sum_Inl_a_a @ X4 ) )
=> ~ ! [Y3: a] :
( S
!= ( sum_Inr_a_a @ Y3 ) ) ) ).
% sumE
thf(fact_434_sumE,axiom,
! [S: sum_sum_a_b] :
( ! [X4: a] :
( S
!= ( sum_Inl_a_b @ X4 ) )
=> ~ ! [Y3: b] :
( S
!= ( sum_Inr_b_a @ Y3 ) ) ) ).
% sumE
thf(fact_435_old_Osum_Oexhaust,axiom,
! [Y: sum_su3067303292148767147um_a_b] :
( ! [A5: sum_sum_a_b] :
( Y
!= ( sum_In5992699931424873788um_a_b @ A5 ) )
=> ~ ! [B4: sum_sum_a_b] :
( Y
!= ( sum_In6272690418125038914um_a_b @ B4 ) ) ) ).
% old.sum.exhaust
thf(fact_436_old_Osum_Oexhaust,axiom,
! [Y: sum_su3831877439928360143_a_b_a] :
( ! [A5: sum_sum_a_b] :
( Y
!= ( sum_In6222238883715738344_a_b_a @ A5 ) )
=> ~ ! [B4: a] :
( Y
!= ( sum_In8629808552650825520um_a_b @ B4 ) ) ) ).
% old.sum.exhaust
thf(fact_437_old_Osum_Oexhaust,axiom,
! [Y: sum_su5898878462909468885um_a_b] :
( ! [A5: a] :
( Y
!= ( sum_In2918432097854297526um_a_b @ A5 ) )
=> ~ ! [B4: sum_sum_a_b] :
( Y
!= ( sum_In2710243301657490530_a_b_a @ B4 ) ) ) ).
% old.sum.exhaust
thf(fact_438_old_Osum_Oexhaust,axiom,
! [Y: sum_sum_a_a] :
( ! [A5: a] :
( Y
!= ( sum_Inl_a_a @ A5 ) )
=> ~ ! [B4: a] :
( Y
!= ( sum_Inr_a_a @ B4 ) ) ) ).
% old.sum.exhaust
thf(fact_439_old_Osum_Oexhaust,axiom,
! [Y: sum_sum_a_b] :
( ! [A5: a] :
( Y
!= ( sum_Inl_a_b @ A5 ) )
=> ~ ! [B4: b] :
( Y
!= ( sum_Inr_b_a @ B4 ) ) ) ).
% old.sum.exhaust
thf(fact_440_old_Osum_Odistinct_I1_J,axiom,
! [A: sum_sum_a_b,B6: sum_sum_a_b] :
( ( sum_In5992699931424873788um_a_b @ A )
!= ( sum_In6272690418125038914um_a_b @ B6 ) ) ).
% old.sum.distinct(1)
thf(fact_441_old_Osum_Odistinct_I1_J,axiom,
! [A: sum_sum_a_b,B6: a] :
( ( sum_In6222238883715738344_a_b_a @ A )
!= ( sum_In8629808552650825520um_a_b @ B6 ) ) ).
% old.sum.distinct(1)
thf(fact_442_old_Osum_Odistinct_I1_J,axiom,
! [A: a,B6: sum_sum_a_b] :
( ( sum_In2918432097854297526um_a_b @ A )
!= ( sum_In2710243301657490530_a_b_a @ B6 ) ) ).
% old.sum.distinct(1)
thf(fact_443_old_Osum_Odistinct_I1_J,axiom,
! [A: a,B6: a] :
( ( sum_Inl_a_a @ A )
!= ( sum_Inr_a_a @ B6 ) ) ).
% old.sum.distinct(1)
thf(fact_444_old_Osum_Odistinct_I1_J,axiom,
! [A: a,B6: b] :
( ( sum_Inl_a_b @ A )
!= ( sum_Inr_b_a @ B6 ) ) ).
% old.sum.distinct(1)
thf(fact_445_old_Osum_Odistinct_I2_J,axiom,
! [B6: sum_sum_a_b,A: sum_sum_a_b] :
( ( sum_In6272690418125038914um_a_b @ B6 )
!= ( sum_In5992699931424873788um_a_b @ A ) ) ).
% old.sum.distinct(2)
thf(fact_446_old_Osum_Odistinct_I2_J,axiom,
! [B6: a,A: sum_sum_a_b] :
( ( sum_In8629808552650825520um_a_b @ B6 )
!= ( sum_In6222238883715738344_a_b_a @ A ) ) ).
% old.sum.distinct(2)
thf(fact_447_old_Osum_Odistinct_I2_J,axiom,
! [B6: sum_sum_a_b,A: a] :
( ( sum_In2710243301657490530_a_b_a @ B6 )
!= ( sum_In2918432097854297526um_a_b @ A ) ) ).
% old.sum.distinct(2)
thf(fact_448_old_Osum_Odistinct_I2_J,axiom,
! [B6: a,A: a] :
( ( sum_Inr_a_a @ B6 )
!= ( sum_Inl_a_a @ A ) ) ).
% old.sum.distinct(2)
thf(fact_449_old_Osum_Odistinct_I2_J,axiom,
! [B6: b,A: a] :
( ( sum_Inr_b_a @ B6 )
!= ( sum_Inl_a_b @ A ) ) ).
% old.sum.distinct(2)
thf(fact_450_sum_Odistinct_I1_J,axiom,
! [X1: sum_sum_a_b,X22: sum_sum_a_b] :
( ( sum_In5992699931424873788um_a_b @ X1 )
!= ( sum_In6272690418125038914um_a_b @ X22 ) ) ).
% sum.distinct(1)
thf(fact_451_sum_Odistinct_I1_J,axiom,
! [X1: sum_sum_a_b,X22: a] :
( ( sum_In6222238883715738344_a_b_a @ X1 )
!= ( sum_In8629808552650825520um_a_b @ X22 ) ) ).
% sum.distinct(1)
thf(fact_452_sum_Odistinct_I1_J,axiom,
! [X1: a,X22: sum_sum_a_b] :
( ( sum_In2918432097854297526um_a_b @ X1 )
!= ( sum_In2710243301657490530_a_b_a @ X22 ) ) ).
% sum.distinct(1)
thf(fact_453_sum_Odistinct_I1_J,axiom,
! [X1: a,X22: a] :
( ( sum_Inl_a_a @ X1 )
!= ( sum_Inr_a_a @ X22 ) ) ).
% sum.distinct(1)
thf(fact_454_sum_Odistinct_I1_J,axiom,
! [X1: a,X22: b] :
( ( sum_Inl_a_b @ X1 )
!= ( sum_Inr_b_a @ X22 ) ) ).
% sum.distinct(1)
thf(fact_455_Inr__Inl__False,axiom,
! [X2: sum_sum_a_b,Y: sum_sum_a_b] :
( ( sum_In6272690418125038914um_a_b @ X2 )
!= ( sum_In5992699931424873788um_a_b @ Y ) ) ).
% Inr_Inl_False
thf(fact_456_Inr__Inl__False,axiom,
! [X2: a,Y: sum_sum_a_b] :
( ( sum_In8629808552650825520um_a_b @ X2 )
!= ( sum_In6222238883715738344_a_b_a @ Y ) ) ).
% Inr_Inl_False
thf(fact_457_Inr__Inl__False,axiom,
! [X2: sum_sum_a_b,Y: a] :
( ( sum_In2710243301657490530_a_b_a @ X2 )
!= ( sum_In2918432097854297526um_a_b @ Y ) ) ).
% Inr_Inl_False
thf(fact_458_Inr__Inl__False,axiom,
! [X2: a,Y: a] :
( ( sum_Inr_a_a @ X2 )
!= ( sum_Inl_a_a @ Y ) ) ).
% Inr_Inl_False
thf(fact_459_Inr__Inl__False,axiom,
! [X2: b,Y: a] :
( ( sum_Inr_b_a @ X2 )
!= ( sum_Inl_a_b @ Y ) ) ).
% Inr_Inl_False
thf(fact_460_Inl__Inr__False,axiom,
! [X2: sum_sum_a_b,Y: sum_sum_a_b] :
( ( sum_In5992699931424873788um_a_b @ X2 )
!= ( sum_In6272690418125038914um_a_b @ Y ) ) ).
% Inl_Inr_False
thf(fact_461_Inl__Inr__False,axiom,
! [X2: sum_sum_a_b,Y: a] :
( ( sum_In6222238883715738344_a_b_a @ X2 )
!= ( sum_In8629808552650825520um_a_b @ Y ) ) ).
% Inl_Inr_False
thf(fact_462_Inl__Inr__False,axiom,
! [X2: a,Y: sum_sum_a_b] :
( ( sum_In2918432097854297526um_a_b @ X2 )
!= ( sum_In2710243301657490530_a_b_a @ Y ) ) ).
% Inl_Inr_False
thf(fact_463_Inl__Inr__False,axiom,
! [X2: a,Y: a] :
( ( sum_Inl_a_a @ X2 )
!= ( sum_Inr_a_a @ Y ) ) ).
% Inl_Inr_False
thf(fact_464_Inl__Inr__False,axiom,
! [X2: a,Y: b] :
( ( sum_Inl_a_b @ X2 )
!= ( sum_Inr_b_a @ Y ) ) ).
% Inl_Inr_False
thf(fact_465_obj__sumE,axiom,
! [S: sum_su3067303292148767147um_a_b] :
( ! [X4: sum_sum_a_b] :
( S
!= ( sum_In5992699931424873788um_a_b @ X4 ) )
=> ~ ! [X4: sum_sum_a_b] :
( S
!= ( sum_In6272690418125038914um_a_b @ X4 ) ) ) ).
% obj_sumE
thf(fact_466_obj__sumE,axiom,
! [S: sum_su3831877439928360143_a_b_a] :
( ! [X4: sum_sum_a_b] :
( S
!= ( sum_In6222238883715738344_a_b_a @ X4 ) )
=> ~ ! [X4: a] :
( S
!= ( sum_In8629808552650825520um_a_b @ X4 ) ) ) ).
% obj_sumE
thf(fact_467_obj__sumE,axiom,
! [S: sum_su5898878462909468885um_a_b] :
( ! [X4: a] :
( S
!= ( sum_In2918432097854297526um_a_b @ X4 ) )
=> ~ ! [X4: sum_sum_a_b] :
( S
!= ( sum_In2710243301657490530_a_b_a @ X4 ) ) ) ).
% obj_sumE
thf(fact_468_obj__sumE,axiom,
! [S: sum_sum_a_a] :
( ! [X4: a] :
( S
!= ( sum_Inl_a_a @ X4 ) )
=> ~ ! [X4: a] :
( S
!= ( sum_Inr_a_a @ X4 ) ) ) ).
% obj_sumE
thf(fact_469_obj__sumE,axiom,
! [S: sum_sum_a_b] :
( ! [X4: a] :
( S
!= ( sum_Inl_a_b @ X4 ) )
=> ~ ! [X4: b] :
( S
!= ( sum_Inr_b_a @ X4 ) ) ) ).
% obj_sumE
thf(fact_470_can__select__set__list__ex1,axiom,
! [P: list_a > $o,A4: list_list_a] :
( ( can_select_list_a @ P @ ( set_list_a2 @ A4 ) )
= ( list_ex1_list_a @ P @ A4 ) ) ).
% can_select_set_list_ex1
thf(fact_471_can__select__set__list__ex1,axiom,
! [P: list_Sum_sum_a_b > $o,A4: list_l4199846171218662726um_a_b] :
( ( can_se3726215003699432057um_a_b @ P @ ( set_list_Sum_sum_a_b2 @ A4 ) )
= ( list_e4286574564361611949um_a_b @ P @ A4 ) ) ).
% can_select_set_list_ex1
thf(fact_472_can__select__set__list__ex1,axiom,
! [P: a > $o,A4: list_a] :
( ( can_select_a @ P @ ( set_a2 @ A4 ) )
= ( list_ex1_a @ P @ A4 ) ) ).
% can_select_set_list_ex1
thf(fact_473_can__select__set__list__ex1,axiom,
! [P: sum_sum_a_b > $o,A4: list_Sum_sum_a_b] :
( ( can_se4659341492078941673um_a_b @ P @ ( set_Sum_sum_a_b2 @ A4 ) )
= ( list_ex1_Sum_sum_a_b @ P @ A4 ) ) ).
% can_select_set_list_ex1
thf(fact_474_remove__code_I2_J,axiom,
! [X2: sum_sum_a_b,Xs: list_Sum_sum_a_b] :
( ( remove_Sum_sum_a_b @ X2 @ ( coset_Sum_sum_a_b @ Xs ) )
= ( coset_Sum_sum_a_b @ ( insert_Sum_sum_a_b @ X2 @ Xs ) ) ) ).
% remove_code(2)
thf(fact_475_remove__code_I2_J,axiom,
! [X2: a,Xs: list_a] :
( ( remove_a @ X2 @ ( coset_a @ Xs ) )
= ( coset_a @ ( insert_a @ X2 @ Xs ) ) ) ).
% remove_code(2)
thf(fact_476_subset__code_I3_J,axiom,
~ ( ord_le8861187494160871172list_a @ ( coset_list_a @ nil_list_a ) @ ( set_list_a2 @ nil_list_a ) ) ).
% subset_code(3)
thf(fact_477_subset__code_I3_J,axiom,
~ ( ord_le2472362315733485388um_a_b @ ( coset_6412875586394208771um_a_b @ nil_list_Sum_sum_a_b ) @ ( set_list_Sum_sum_a_b2 @ nil_list_Sum_sum_a_b ) ) ).
% subset_code(3)
thf(fact_478_subset__code_I3_J,axiom,
~ ( ord_le3724670747650509150_set_a @ ( coset_set_a @ nil_set_a ) @ ( set_set_a2 @ nil_set_a ) ) ).
% subset_code(3)
thf(fact_479_subset__code_I3_J,axiom,
~ ( ord_le9019793522827316924um_a_b @ ( coset_Sum_sum_a_b @ nil_Sum_sum_a_b ) @ ( set_Sum_sum_a_b2 @ nil_Sum_sum_a_b ) ) ).
% subset_code(3)
thf(fact_480_subset__code_I3_J,axiom,
~ ( ord_less_eq_set_a @ ( coset_a @ nil_a ) @ ( set_a2 @ nil_a ) ) ).
% subset_code(3)
thf(fact_481_not__arg__cong__Inr,axiom,
! [X2: b,Y: b] :
( ( X2 != Y )
=> ( ( sum_Inr_b_a @ X2 )
!= ( sum_Inr_b_a @ Y ) ) ) ).
% not_arg_cong_Inr
thf(fact_482_not__arg__cong__Inr,axiom,
! [X2: sum_sum_a_b,Y: sum_sum_a_b] :
( ( X2 != Y )
=> ( ( sum_In6272690418125038914um_a_b @ X2 )
!= ( sum_In6272690418125038914um_a_b @ Y ) ) ) ).
% not_arg_cong_Inr
thf(fact_483_not__arg__cong__Inr,axiom,
! [X2: a,Y: a] :
( ( X2 != Y )
=> ( ( sum_In8629808552650825520um_a_b @ X2 )
!= ( sum_In8629808552650825520um_a_b @ Y ) ) ) ).
% not_arg_cong_Inr
thf(fact_484_not__arg__cong__Inr,axiom,
! [X2: sum_sum_a_b,Y: sum_sum_a_b] :
( ( X2 != Y )
=> ( ( sum_In2710243301657490530_a_b_a @ X2 )
!= ( sum_In2710243301657490530_a_b_a @ Y ) ) ) ).
% not_arg_cong_Inr
thf(fact_485_not__arg__cong__Inr,axiom,
! [X2: a,Y: a] :
( ( X2 != Y )
=> ( ( sum_Inr_a_a @ X2 )
!= ( sum_Inr_a_a @ Y ) ) ) ).
% not_arg_cong_Inr
thf(fact_486_not__in__set__insert,axiom,
! [X2: set_Sum_sum_a_b,Xs: list_set_Sum_sum_a_b] :
( ~ ( member4060935254435997939um_a_b @ X2 @ ( set_set_Sum_sum_a_b2 @ Xs ) )
=> ( ( insert301144099385869888um_a_b @ X2 @ Xs )
= ( cons_set_Sum_sum_a_b @ X2 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_487_not__in__set__insert,axiom,
! [X2: set_set_a,Xs: list_set_set_a] :
( ~ ( member_set_set_a2 @ X2 @ ( set_set_set_a2 @ Xs ) )
=> ( ( insert_set_set_a @ X2 @ Xs )
= ( cons_set_set_a @ X2 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_488_not__in__set__insert,axiom,
! [X2: set_a,Xs: list_set_a] :
( ~ ( member_set_a2 @ X2 @ ( set_set_a2 @ Xs ) )
=> ( ( insert_set_a @ X2 @ Xs )
= ( cons_set_a @ X2 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_489_not__in__set__insert,axiom,
! [X2: b,Xs: list_b] :
( ~ ( member_b2 @ X2 @ ( set_b2 @ Xs ) )
=> ( ( insert_b @ X2 @ Xs )
= ( cons_b @ X2 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_490_not__in__set__insert,axiom,
! [X2: list_a,Xs: list_list_a] :
( ~ ( member_list_a2 @ X2 @ ( set_list_a2 @ Xs ) )
=> ( ( insert_list_a @ X2 @ Xs )
= ( cons_list_a @ X2 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_491_not__in__set__insert,axiom,
! [X2: list_Sum_sum_a_b,Xs: list_l4199846171218662726um_a_b] :
( ~ ( member7701661377270014157um_a_b @ X2 @ ( set_list_Sum_sum_a_b2 @ Xs ) )
=> ( ( insert3356916551913007002um_a_b @ X2 @ Xs )
= ( cons_l7912394880171584384um_a_b @ X2 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_492_not__in__set__insert,axiom,
! [X2: a,Xs: list_a] :
( ~ ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
=> ( ( insert_a @ X2 @ Xs )
= ( cons_a @ X2 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_493_not__in__set__insert,axiom,
! [X2: sum_sum_a_b,Xs: list_Sum_sum_a_b] :
( ~ ( member_Sum_sum_a_b2 @ X2 @ ( set_Sum_sum_a_b2 @ Xs ) )
=> ( ( insert_Sum_sum_a_b @ X2 @ Xs )
= ( cons_Sum_sum_a_b @ X2 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_494_list_Oinject,axiom,
! [X21: sum_sum_a_b,X222: list_Sum_sum_a_b,Y21: sum_sum_a_b,Y222: list_Sum_sum_a_b] :
( ( ( cons_Sum_sum_a_b @ X21 @ X222 )
= ( cons_Sum_sum_a_b @ Y21 @ Y222 ) )
= ( ( X21 = Y21 )
& ( X222 = Y222 ) ) ) ).
% list.inject
thf(fact_495_list_Oinject,axiom,
! [X21: a,X222: list_a,Y21: a,Y222: list_a] :
( ( ( cons_a @ X21 @ X222 )
= ( cons_a @ Y21 @ Y222 ) )
= ( ( X21 = Y21 )
& ( X222 = Y222 ) ) ) ).
% list.inject
thf(fact_496_Set_Omember__remove,axiom,
! [X2: list_a,Y: list_a,A4: set_list_a] :
( ( member_list_a2 @ X2 @ ( remove_list_a @ Y @ A4 ) )
= ( ( member_list_a2 @ X2 @ A4 )
& ( X2 != Y ) ) ) ).
% Set.member_remove
thf(fact_497_Set_Omember__remove,axiom,
! [X2: set_Sum_sum_a_b,Y: set_Sum_sum_a_b,A4: set_set_Sum_sum_a_b] :
( ( member4060935254435997939um_a_b @ X2 @ ( remove3542032160739950135um_a_b @ Y @ A4 ) )
= ( ( member4060935254435997939um_a_b @ X2 @ A4 )
& ( X2 != Y ) ) ) ).
% Set.member_remove
thf(fact_498_Set_Omember__remove,axiom,
! [X2: set_set_a,Y: set_set_a,A4: set_set_set_a] :
( ( member_set_set_a2 @ X2 @ ( remove_set_set_a @ Y @ A4 ) )
= ( ( member_set_set_a2 @ X2 @ A4 )
& ( X2 != Y ) ) ) ).
% Set.member_remove
thf(fact_499_Set_Omember__remove,axiom,
! [X2: set_a,Y: set_a,A4: set_set_a] :
( ( member_set_a2 @ X2 @ ( remove_set_a @ Y @ A4 ) )
= ( ( member_set_a2 @ X2 @ A4 )
& ( X2 != Y ) ) ) ).
% Set.member_remove
thf(fact_500_Set_Omember__remove,axiom,
! [X2: b,Y: b,A4: set_b] :
( ( member_b2 @ X2 @ ( remove_b @ Y @ A4 ) )
= ( ( member_b2 @ X2 @ A4 )
& ( X2 != Y ) ) ) ).
% Set.member_remove
thf(fact_501_Set_Omember__remove,axiom,
! [X2: a,Y: a,A4: set_a] :
( ( member_a2 @ X2 @ ( remove_a @ Y @ A4 ) )
= ( ( member_a2 @ X2 @ A4 )
& ( X2 != Y ) ) ) ).
% Set.member_remove
thf(fact_502_Set_Omember__remove,axiom,
! [X2: sum_sum_a_b,Y: sum_sum_a_b,A4: set_Sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X2 @ ( remove_Sum_sum_a_b @ Y @ A4 ) )
= ( ( member_Sum_sum_a_b2 @ X2 @ A4 )
& ( X2 != Y ) ) ) ).
% Set.member_remove
thf(fact_503_list__ex1__simps_I1_J,axiom,
! [P: sum_sum_a_b > $o] :
~ ( list_ex1_Sum_sum_a_b @ P @ nil_Sum_sum_a_b ) ).
% list_ex1_simps(1)
thf(fact_504_list__ex1__simps_I1_J,axiom,
! [P: a > $o] :
~ ( list_ex1_a @ P @ nil_a ) ).
% list_ex1_simps(1)
thf(fact_505_insert__Nil,axiom,
! [X2: sum_sum_a_b] :
( ( insert_Sum_sum_a_b @ X2 @ nil_Sum_sum_a_b )
= ( cons_Sum_sum_a_b @ X2 @ nil_Sum_sum_a_b ) ) ).
% insert_Nil
thf(fact_506_insert__Nil,axiom,
! [X2: a] :
( ( insert_a @ X2 @ nil_a )
= ( cons_a @ X2 @ nil_a ) ) ).
% insert_Nil
thf(fact_507_proper__intrvl_Oexhaustive_Ocases,axiom,
! [X2: list_Sum_sum_a_b] :
( ( X2 != nil_Sum_sum_a_b )
=> ~ ! [X4: sum_sum_a_b,Xs3: list_Sum_sum_a_b] :
( X2
!= ( cons_Sum_sum_a_b @ X4 @ Xs3 ) ) ) ).
% proper_intrvl.exhaustive.cases
thf(fact_508_proper__intrvl_Oexhaustive_Ocases,axiom,
! [X2: list_a] :
( ( X2 != nil_a )
=> ~ ! [X4: a,Xs3: list_a] :
( X2
!= ( cons_a @ X4 @ Xs3 ) ) ) ).
% proper_intrvl.exhaustive.cases
thf(fact_509_can__select__def,axiom,
( can_select_list_a
= ( ^ [P2: list_a > $o,A6: set_list_a] :
? [X3: list_a] :
( ( member_list_a2 @ X3 @ A6 )
& ( P2 @ X3 )
& ! [Y5: list_a] :
( ( ( member_list_a2 @ Y5 @ A6 )
& ( P2 @ Y5 ) )
=> ( Y5 = X3 ) ) ) ) ) ).
% can_select_def
thf(fact_510_can__select__def,axiom,
( can_se4838630876668995743um_a_b
= ( ^ [P2: set_Sum_sum_a_b > $o,A6: set_set_Sum_sum_a_b] :
? [X3: set_Sum_sum_a_b] :
( ( member4060935254435997939um_a_b @ X3 @ A6 )
& ( P2 @ X3 )
& ! [Y5: set_Sum_sum_a_b] :
( ( ( member4060935254435997939um_a_b @ Y5 @ A6 )
& ( P2 @ Y5 ) )
=> ( Y5 = X3 ) ) ) ) ) ).
% can_select_def
thf(fact_511_can__select__def,axiom,
( can_select_set_set_a
= ( ^ [P2: set_set_a > $o,A6: set_set_set_a] :
? [X3: set_set_a] :
( ( member_set_set_a2 @ X3 @ A6 )
& ( P2 @ X3 )
& ! [Y5: set_set_a] :
( ( ( member_set_set_a2 @ Y5 @ A6 )
& ( P2 @ Y5 ) )
=> ( Y5 = X3 ) ) ) ) ) ).
% can_select_def
thf(fact_512_can__select__def,axiom,
( can_select_set_a
= ( ^ [P2: set_a > $o,A6: set_set_a] :
? [X3: set_a] :
( ( member_set_a2 @ X3 @ A6 )
& ( P2 @ X3 )
& ! [Y5: set_a] :
( ( ( member_set_a2 @ Y5 @ A6 )
& ( P2 @ Y5 ) )
=> ( Y5 = X3 ) ) ) ) ) ).
% can_select_def
thf(fact_513_can__select__def,axiom,
( can_select_b
= ( ^ [P2: b > $o,A6: set_b] :
? [X3: b] :
( ( member_b2 @ X3 @ A6 )
& ( P2 @ X3 )
& ! [Y5: b] :
( ( ( member_b2 @ Y5 @ A6 )
& ( P2 @ Y5 ) )
=> ( Y5 = X3 ) ) ) ) ) ).
% can_select_def
thf(fact_514_can__select__def,axiom,
( can_select_a
= ( ^ [P2: a > $o,A6: set_a] :
? [X3: a] :
( ( member_a2 @ X3 @ A6 )
& ( P2 @ X3 )
& ! [Y5: a] :
( ( ( member_a2 @ Y5 @ A6 )
& ( P2 @ Y5 ) )
=> ( Y5 = X3 ) ) ) ) ) ).
% can_select_def
thf(fact_515_can__select__def,axiom,
( can_se4659341492078941673um_a_b
= ( ^ [P2: sum_sum_a_b > $o,A6: set_Sum_sum_a_b] :
? [X3: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X3 @ A6 )
& ( P2 @ X3 )
& ! [Y5: sum_sum_a_b] :
( ( ( member_Sum_sum_a_b2 @ Y5 @ A6 )
& ( P2 @ Y5 ) )
=> ( Y5 = X3 ) ) ) ) ) ).
% can_select_def
thf(fact_516_list_Odistinct_I1_J,axiom,
! [X21: sum_sum_a_b,X222: list_Sum_sum_a_b] :
( nil_Sum_sum_a_b
!= ( cons_Sum_sum_a_b @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_517_list_Odistinct_I1_J,axiom,
! [X21: a,X222: list_a] :
( nil_a
!= ( cons_a @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_518_list_OdiscI,axiom,
! [List: list_Sum_sum_a_b,X21: sum_sum_a_b,X222: list_Sum_sum_a_b] :
( ( List
= ( cons_Sum_sum_a_b @ X21 @ X222 ) )
=> ( List != nil_Sum_sum_a_b ) ) ).
% list.discI
thf(fact_519_list_OdiscI,axiom,
! [List: list_a,X21: a,X222: list_a] :
( ( List
= ( cons_a @ X21 @ X222 ) )
=> ( List != nil_a ) ) ).
% list.discI
thf(fact_520_list_Oexhaust,axiom,
! [Y: list_Sum_sum_a_b] :
( ( Y != nil_Sum_sum_a_b )
=> ~ ! [X212: sum_sum_a_b,X223: list_Sum_sum_a_b] :
( Y
!= ( cons_Sum_sum_a_b @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_521_list_Oexhaust,axiom,
! [Y: list_a] :
( ( Y != nil_a )
=> ~ ! [X212: a,X223: list_a] :
( Y
!= ( cons_a @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_522_transpose_Ocases,axiom,
! [X2: list_l4199846171218662726um_a_b] :
( ( X2 != nil_list_Sum_sum_a_b )
=> ( ! [Xss: list_l4199846171218662726um_a_b] :
( X2
!= ( cons_l7912394880171584384um_a_b @ nil_Sum_sum_a_b @ Xss ) )
=> ~ ! [X4: sum_sum_a_b,Xs3: list_Sum_sum_a_b,Xss: list_l4199846171218662726um_a_b] :
( X2
!= ( cons_l7912394880171584384um_a_b @ ( cons_Sum_sum_a_b @ X4 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_523_transpose_Ocases,axiom,
! [X2: list_list_a] :
( ( X2 != nil_list_a )
=> ( ! [Xss: list_list_a] :
( X2
!= ( cons_list_a @ nil_a @ Xss ) )
=> ~ ! [X4: a,Xs3: list_a,Xss: list_list_a] :
( X2
!= ( cons_list_a @ ( cons_a @ X4 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_524_ord_Oremdups__sorted_Ocases,axiom,
! [X2: list_Sum_sum_a_b] :
( ( X2 != nil_Sum_sum_a_b )
=> ( ! [X4: sum_sum_a_b] :
( X2
!= ( cons_Sum_sum_a_b @ X4 @ nil_Sum_sum_a_b ) )
=> ~ ! [X4: sum_sum_a_b,Y3: sum_sum_a_b,Xs3: list_Sum_sum_a_b] :
( X2
!= ( cons_Sum_sum_a_b @ X4 @ ( cons_Sum_sum_a_b @ Y3 @ Xs3 ) ) ) ) ) ).
% ord.remdups_sorted.cases
thf(fact_525_ord_Oremdups__sorted_Ocases,axiom,
! [X2: list_a] :
( ( X2 != nil_a )
=> ( ! [X4: a] :
( X2
!= ( cons_a @ X4 @ nil_a ) )
=> ~ ! [X4: a,Y3: a,Xs3: list_a] :
( X2
!= ( cons_a @ X4 @ ( cons_a @ Y3 @ Xs3 ) ) ) ) ) ).
% ord.remdups_sorted.cases
thf(fact_526_neq__Nil__conv,axiom,
! [Xs: list_Sum_sum_a_b] :
( ( Xs != nil_Sum_sum_a_b )
= ( ? [Y5: sum_sum_a_b,Ys2: list_Sum_sum_a_b] :
( Xs
= ( cons_Sum_sum_a_b @ Y5 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_527_neq__Nil__conv,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
= ( ? [Y5: a,Ys2: list_a] :
( Xs
= ( cons_a @ Y5 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_528_list__induct2_H,axiom,
! [P: list_Sum_sum_a_b > list_Sum_sum_a_b > $o,Xs: list_Sum_sum_a_b,Ys: list_Sum_sum_a_b] :
( ( P @ nil_Sum_sum_a_b @ nil_Sum_sum_a_b )
=> ( ! [X4: sum_sum_a_b,Xs3: list_Sum_sum_a_b] : ( P @ ( cons_Sum_sum_a_b @ X4 @ Xs3 ) @ nil_Sum_sum_a_b )
=> ( ! [Y3: sum_sum_a_b,Ys3: list_Sum_sum_a_b] : ( P @ nil_Sum_sum_a_b @ ( cons_Sum_sum_a_b @ Y3 @ Ys3 ) )
=> ( ! [X4: sum_sum_a_b,Xs3: list_Sum_sum_a_b,Y3: sum_sum_a_b,Ys3: list_Sum_sum_a_b] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_Sum_sum_a_b @ X4 @ Xs3 ) @ ( cons_Sum_sum_a_b @ Y3 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_529_list__induct2_H,axiom,
! [P: list_Sum_sum_a_b > list_a > $o,Xs: list_Sum_sum_a_b,Ys: list_a] :
( ( P @ nil_Sum_sum_a_b @ nil_a )
=> ( ! [X4: sum_sum_a_b,Xs3: list_Sum_sum_a_b] : ( P @ ( cons_Sum_sum_a_b @ X4 @ Xs3 ) @ nil_a )
=> ( ! [Y3: a,Ys3: list_a] : ( P @ nil_Sum_sum_a_b @ ( cons_a @ Y3 @ Ys3 ) )
=> ( ! [X4: sum_sum_a_b,Xs3: list_Sum_sum_a_b,Y3: a,Ys3: list_a] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_Sum_sum_a_b @ X4 @ Xs3 ) @ ( cons_a @ Y3 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_530_list__induct2_H,axiom,
! [P: list_a > list_Sum_sum_a_b > $o,Xs: list_a,Ys: list_Sum_sum_a_b] :
( ( P @ nil_a @ nil_Sum_sum_a_b )
=> ( ! [X4: a,Xs3: list_a] : ( P @ ( cons_a @ X4 @ Xs3 ) @ nil_Sum_sum_a_b )
=> ( ! [Y3: sum_sum_a_b,Ys3: list_Sum_sum_a_b] : ( P @ nil_a @ ( cons_Sum_sum_a_b @ Y3 @ Ys3 ) )
=> ( ! [X4: a,Xs3: list_a,Y3: sum_sum_a_b,Ys3: list_Sum_sum_a_b] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_a @ X4 @ Xs3 ) @ ( cons_Sum_sum_a_b @ Y3 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_531_list__induct2_H,axiom,
! [P: list_a > list_a > $o,Xs: list_a,Ys: list_a] :
( ( P @ nil_a @ nil_a )
=> ( ! [X4: a,Xs3: list_a] : ( P @ ( cons_a @ X4 @ Xs3 ) @ nil_a )
=> ( ! [Y3: a,Ys3: list_a] : ( P @ nil_a @ ( cons_a @ Y3 @ Ys3 ) )
=> ( ! [X4: a,Xs3: list_a,Y3: a,Ys3: list_a] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_a @ X4 @ Xs3 ) @ ( cons_a @ Y3 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_532_not__Cons__self2,axiom,
! [X2: sum_sum_a_b,Xs: list_Sum_sum_a_b] :
( ( cons_Sum_sum_a_b @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_533_not__Cons__self2,axiom,
! [X2: a,Xs: list_a] :
( ( cons_a @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_534_list__nonempty__induct,axiom,
! [Xs: list_Sum_sum_a_b,P: list_Sum_sum_a_b > $o] :
( ( Xs != nil_Sum_sum_a_b )
=> ( ! [X4: sum_sum_a_b] : ( P @ ( cons_Sum_sum_a_b @ X4 @ nil_Sum_sum_a_b ) )
=> ( ! [X4: sum_sum_a_b,Xs3: list_Sum_sum_a_b] :
( ( Xs3 != nil_Sum_sum_a_b )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_Sum_sum_a_b @ X4 @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_535_list__nonempty__induct,axiom,
! [Xs: list_a,P: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X4: a] : ( P @ ( cons_a @ X4 @ nil_a ) )
=> ( ! [X4: a,Xs3: list_a] :
( ( Xs3 != nil_a )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_a @ X4 @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_536_list_Oset__intros_I2_J,axiom,
! [Y: set_Sum_sum_a_b,X222: list_set_Sum_sum_a_b,X21: set_Sum_sum_a_b] :
( ( member4060935254435997939um_a_b @ Y @ ( set_set_Sum_sum_a_b2 @ X222 ) )
=> ( member4060935254435997939um_a_b @ Y @ ( set_set_Sum_sum_a_b2 @ ( cons_set_Sum_sum_a_b @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_537_list_Oset__intros_I2_J,axiom,
! [Y: set_set_a,X222: list_set_set_a,X21: set_set_a] :
( ( member_set_set_a2 @ Y @ ( set_set_set_a2 @ X222 ) )
=> ( member_set_set_a2 @ Y @ ( set_set_set_a2 @ ( cons_set_set_a @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_538_list_Oset__intros_I2_J,axiom,
! [Y: set_a,X222: list_set_a,X21: set_a] :
( ( member_set_a2 @ Y @ ( set_set_a2 @ X222 ) )
=> ( member_set_a2 @ Y @ ( set_set_a2 @ ( cons_set_a @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_539_list_Oset__intros_I2_J,axiom,
! [Y: b,X222: list_b,X21: b] :
( ( member_b2 @ Y @ ( set_b2 @ X222 ) )
=> ( member_b2 @ Y @ ( set_b2 @ ( cons_b @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_540_list_Oset__intros_I2_J,axiom,
! [Y: list_a,X222: list_list_a,X21: list_a] :
( ( member_list_a2 @ Y @ ( set_list_a2 @ X222 ) )
=> ( member_list_a2 @ Y @ ( set_list_a2 @ ( cons_list_a @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_541_list_Oset__intros_I2_J,axiom,
! [Y: list_Sum_sum_a_b,X222: list_l4199846171218662726um_a_b,X21: list_Sum_sum_a_b] :
( ( member7701661377270014157um_a_b @ Y @ ( set_list_Sum_sum_a_b2 @ X222 ) )
=> ( member7701661377270014157um_a_b @ Y @ ( set_list_Sum_sum_a_b2 @ ( cons_l7912394880171584384um_a_b @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_542_list_Oset__intros_I2_J,axiom,
! [Y: a,X222: list_a,X21: a] :
( ( member_a2 @ Y @ ( set_a2 @ X222 ) )
=> ( member_a2 @ Y @ ( set_a2 @ ( cons_a @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_543_list_Oset__intros_I2_J,axiom,
! [Y: sum_sum_a_b,X222: list_Sum_sum_a_b,X21: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ Y @ ( set_Sum_sum_a_b2 @ X222 ) )
=> ( member_Sum_sum_a_b2 @ Y @ ( set_Sum_sum_a_b2 @ ( cons_Sum_sum_a_b @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_544_list_Oset__intros_I1_J,axiom,
! [X21: set_Sum_sum_a_b,X222: list_set_Sum_sum_a_b] : ( member4060935254435997939um_a_b @ X21 @ ( set_set_Sum_sum_a_b2 @ ( cons_set_Sum_sum_a_b @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_545_list_Oset__intros_I1_J,axiom,
! [X21: set_set_a,X222: list_set_set_a] : ( member_set_set_a2 @ X21 @ ( set_set_set_a2 @ ( cons_set_set_a @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_546_list_Oset__intros_I1_J,axiom,
! [X21: set_a,X222: list_set_a] : ( member_set_a2 @ X21 @ ( set_set_a2 @ ( cons_set_a @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_547_list_Oset__intros_I1_J,axiom,
! [X21: b,X222: list_b] : ( member_b2 @ X21 @ ( set_b2 @ ( cons_b @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_548_list_Oset__intros_I1_J,axiom,
! [X21: list_a,X222: list_list_a] : ( member_list_a2 @ X21 @ ( set_list_a2 @ ( cons_list_a @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_549_list_Oset__intros_I1_J,axiom,
! [X21: list_Sum_sum_a_b,X222: list_l4199846171218662726um_a_b] : ( member7701661377270014157um_a_b @ X21 @ ( set_list_Sum_sum_a_b2 @ ( cons_l7912394880171584384um_a_b @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_550_list_Oset__intros_I1_J,axiom,
! [X21: a,X222: list_a] : ( member_a2 @ X21 @ ( set_a2 @ ( cons_a @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_551_list_Oset__intros_I1_J,axiom,
! [X21: sum_sum_a_b,X222: list_Sum_sum_a_b] : ( member_Sum_sum_a_b2 @ X21 @ ( set_Sum_sum_a_b2 @ ( cons_Sum_sum_a_b @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_552_list_Oset__cases,axiom,
! [E: set_Sum_sum_a_b,A: list_set_Sum_sum_a_b] :
( ( member4060935254435997939um_a_b @ E @ ( set_set_Sum_sum_a_b2 @ A ) )
=> ( ! [Z22: list_set_Sum_sum_a_b] :
( A
!= ( cons_set_Sum_sum_a_b @ E @ Z22 ) )
=> ~ ! [Z1: set_Sum_sum_a_b,Z22: list_set_Sum_sum_a_b] :
( ( A
= ( cons_set_Sum_sum_a_b @ Z1 @ Z22 ) )
=> ~ ( member4060935254435997939um_a_b @ E @ ( set_set_Sum_sum_a_b2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_553_list_Oset__cases,axiom,
! [E: set_set_a,A: list_set_set_a] :
( ( member_set_set_a2 @ E @ ( set_set_set_a2 @ A ) )
=> ( ! [Z22: list_set_set_a] :
( A
!= ( cons_set_set_a @ E @ Z22 ) )
=> ~ ! [Z1: set_set_a,Z22: list_set_set_a] :
( ( A
= ( cons_set_set_a @ Z1 @ Z22 ) )
=> ~ ( member_set_set_a2 @ E @ ( set_set_set_a2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_554_list_Oset__cases,axiom,
! [E: set_a,A: list_set_a] :
( ( member_set_a2 @ E @ ( set_set_a2 @ A ) )
=> ( ! [Z22: list_set_a] :
( A
!= ( cons_set_a @ E @ Z22 ) )
=> ~ ! [Z1: set_a,Z22: list_set_a] :
( ( A
= ( cons_set_a @ Z1 @ Z22 ) )
=> ~ ( member_set_a2 @ E @ ( set_set_a2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_555_list_Oset__cases,axiom,
! [E: b,A: list_b] :
( ( member_b2 @ E @ ( set_b2 @ A ) )
=> ( ! [Z22: list_b] :
( A
!= ( cons_b @ E @ Z22 ) )
=> ~ ! [Z1: b,Z22: list_b] :
( ( A
= ( cons_b @ Z1 @ Z22 ) )
=> ~ ( member_b2 @ E @ ( set_b2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_556_list_Oset__cases,axiom,
! [E: list_a,A: list_list_a] :
( ( member_list_a2 @ E @ ( set_list_a2 @ A ) )
=> ( ! [Z22: list_list_a] :
( A
!= ( cons_list_a @ E @ Z22 ) )
=> ~ ! [Z1: list_a,Z22: list_list_a] :
( ( A
= ( cons_list_a @ Z1 @ Z22 ) )
=> ~ ( member_list_a2 @ E @ ( set_list_a2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_557_list_Oset__cases,axiom,
! [E: list_Sum_sum_a_b,A: list_l4199846171218662726um_a_b] :
( ( member7701661377270014157um_a_b @ E @ ( set_list_Sum_sum_a_b2 @ A ) )
=> ( ! [Z22: list_l4199846171218662726um_a_b] :
( A
!= ( cons_l7912394880171584384um_a_b @ E @ Z22 ) )
=> ~ ! [Z1: list_Sum_sum_a_b,Z22: list_l4199846171218662726um_a_b] :
( ( A
= ( cons_l7912394880171584384um_a_b @ Z1 @ Z22 ) )
=> ~ ( member7701661377270014157um_a_b @ E @ ( set_list_Sum_sum_a_b2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_558_list_Oset__cases,axiom,
! [E: a,A: list_a] :
( ( member_a2 @ E @ ( set_a2 @ A ) )
=> ( ! [Z22: list_a] :
( A
!= ( cons_a @ E @ Z22 ) )
=> ~ ! [Z1: a,Z22: list_a] :
( ( A
= ( cons_a @ Z1 @ Z22 ) )
=> ~ ( member_a2 @ E @ ( set_a2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_559_list_Oset__cases,axiom,
! [E: sum_sum_a_b,A: list_Sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ E @ ( set_Sum_sum_a_b2 @ A ) )
=> ( ! [Z22: list_Sum_sum_a_b] :
( A
!= ( cons_Sum_sum_a_b @ E @ Z22 ) )
=> ~ ! [Z1: sum_sum_a_b,Z22: list_Sum_sum_a_b] :
( ( A
= ( cons_Sum_sum_a_b @ Z1 @ Z22 ) )
=> ~ ( member_Sum_sum_a_b2 @ E @ ( set_Sum_sum_a_b2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_560_set__ConsD,axiom,
! [Y: set_Sum_sum_a_b,X2: set_Sum_sum_a_b,Xs: list_set_Sum_sum_a_b] :
( ( member4060935254435997939um_a_b @ Y @ ( set_set_Sum_sum_a_b2 @ ( cons_set_Sum_sum_a_b @ X2 @ Xs ) ) )
=> ( ( Y = X2 )
| ( member4060935254435997939um_a_b @ Y @ ( set_set_Sum_sum_a_b2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_561_set__ConsD,axiom,
! [Y: set_set_a,X2: set_set_a,Xs: list_set_set_a] :
( ( member_set_set_a2 @ Y @ ( set_set_set_a2 @ ( cons_set_set_a @ X2 @ Xs ) ) )
=> ( ( Y = X2 )
| ( member_set_set_a2 @ Y @ ( set_set_set_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_562_set__ConsD,axiom,
! [Y: set_a,X2: set_a,Xs: list_set_a] :
( ( member_set_a2 @ Y @ ( set_set_a2 @ ( cons_set_a @ X2 @ Xs ) ) )
=> ( ( Y = X2 )
| ( member_set_a2 @ Y @ ( set_set_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_563_set__ConsD,axiom,
! [Y: b,X2: b,Xs: list_b] :
( ( member_b2 @ Y @ ( set_b2 @ ( cons_b @ X2 @ Xs ) ) )
=> ( ( Y = X2 )
| ( member_b2 @ Y @ ( set_b2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_564_set__ConsD,axiom,
! [Y: list_a,X2: list_a,Xs: list_list_a] :
( ( member_list_a2 @ Y @ ( set_list_a2 @ ( cons_list_a @ X2 @ Xs ) ) )
=> ( ( Y = X2 )
| ( member_list_a2 @ Y @ ( set_list_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_565_set__ConsD,axiom,
! [Y: list_Sum_sum_a_b,X2: list_Sum_sum_a_b,Xs: list_l4199846171218662726um_a_b] :
( ( member7701661377270014157um_a_b @ Y @ ( set_list_Sum_sum_a_b2 @ ( cons_l7912394880171584384um_a_b @ X2 @ Xs ) ) )
=> ( ( Y = X2 )
| ( member7701661377270014157um_a_b @ Y @ ( set_list_Sum_sum_a_b2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_566_set__ConsD,axiom,
! [Y: a,X2: a,Xs: list_a] :
( ( member_a2 @ Y @ ( set_a2 @ ( cons_a @ X2 @ Xs ) ) )
=> ( ( Y = X2 )
| ( member_a2 @ Y @ ( set_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_567_set__ConsD,axiom,
! [Y: sum_sum_a_b,X2: sum_sum_a_b,Xs: list_Sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ Y @ ( set_Sum_sum_a_b2 @ ( cons_Sum_sum_a_b @ X2 @ Xs ) ) )
=> ( ( Y = X2 )
| ( member_Sum_sum_a_b2 @ Y @ ( set_Sum_sum_a_b2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_568_member__rec_I1_J,axiom,
! [X2: sum_sum_a_b,Xs: list_Sum_sum_a_b,Y: sum_sum_a_b] :
( ( member_Sum_sum_a_b @ ( cons_Sum_sum_a_b @ X2 @ Xs ) @ Y )
= ( ( X2 = Y )
| ( member_Sum_sum_a_b @ Xs @ Y ) ) ) ).
% member_rec(1)
thf(fact_569_member__rec_I1_J,axiom,
! [X2: a,Xs: list_a,Y: a] :
( ( member_a @ ( cons_a @ X2 @ Xs ) @ Y )
= ( ( X2 = Y )
| ( member_a @ Xs @ Y ) ) ) ).
% member_rec(1)
thf(fact_570_member__rec_I2_J,axiom,
! [Y: a] :
~ ( member_a @ nil_a @ Y ) ).
% member_rec(2)
thf(fact_571_member__rec_I2_J,axiom,
! [Y: sum_sum_a_b] :
~ ( member_Sum_sum_a_b @ nil_Sum_sum_a_b @ Y ) ).
% member_rec(2)
thf(fact_572_set__subset__Cons,axiom,
! [Xs: list_list_a,X2: list_a] : ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ ( set_list_a2 @ ( cons_list_a @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_573_set__subset__Cons,axiom,
! [Xs: list_l4199846171218662726um_a_b,X2: list_Sum_sum_a_b] : ( ord_le2472362315733485388um_a_b @ ( set_list_Sum_sum_a_b2 @ Xs ) @ ( set_list_Sum_sum_a_b2 @ ( cons_l7912394880171584384um_a_b @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_574_set__subset__Cons,axiom,
! [Xs: list_set_a,X2: set_a] : ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ Xs ) @ ( set_set_a2 @ ( cons_set_a @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_575_set__subset__Cons,axiom,
! [Xs: list_Sum_sum_a_b,X2: sum_sum_a_b] : ( ord_le9019793522827316924um_a_b @ ( set_Sum_sum_a_b2 @ Xs ) @ ( set_Sum_sum_a_b2 @ ( cons_Sum_sum_a_b @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_576_set__subset__Cons,axiom,
! [Xs: list_a,X2: a] : ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ ( cons_a @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_577_List_Oinsert__def,axiom,
( insert301144099385869888um_a_b
= ( ^ [X3: set_Sum_sum_a_b,Xs2: list_set_Sum_sum_a_b] : ( if_lis1608374263067180582um_a_b @ ( member4060935254435997939um_a_b @ X3 @ ( set_set_Sum_sum_a_b2 @ Xs2 ) ) @ Xs2 @ ( cons_set_Sum_sum_a_b @ X3 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_578_List_Oinsert__def,axiom,
( insert_set_set_a
= ( ^ [X3: set_set_a,Xs2: list_set_set_a] : ( if_list_set_set_a @ ( member_set_set_a2 @ X3 @ ( set_set_set_a2 @ Xs2 ) ) @ Xs2 @ ( cons_set_set_a @ X3 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_579_List_Oinsert__def,axiom,
( insert_set_a
= ( ^ [X3: set_a,Xs2: list_set_a] : ( if_list_set_a @ ( member_set_a2 @ X3 @ ( set_set_a2 @ Xs2 ) ) @ Xs2 @ ( cons_set_a @ X3 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_580_List_Oinsert__def,axiom,
( insert_b
= ( ^ [X3: b,Xs2: list_b] : ( if_list_b @ ( member_b2 @ X3 @ ( set_b2 @ Xs2 ) ) @ Xs2 @ ( cons_b @ X3 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_581_List_Oinsert__def,axiom,
( insert_list_a
= ( ^ [X3: list_a,Xs2: list_list_a] : ( if_list_list_a @ ( member_list_a2 @ X3 @ ( set_list_a2 @ Xs2 ) ) @ Xs2 @ ( cons_list_a @ X3 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_582_List_Oinsert__def,axiom,
( insert3356916551913007002um_a_b
= ( ^ [X3: list_Sum_sum_a_b,Xs2: list_l4199846171218662726um_a_b] : ( if_lis6122350189682124032um_a_b @ ( member7701661377270014157um_a_b @ X3 @ ( set_list_Sum_sum_a_b2 @ Xs2 ) ) @ Xs2 @ ( cons_l7912394880171584384um_a_b @ X3 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_583_List_Oinsert__def,axiom,
( insert_a
= ( ^ [X3: a,Xs2: list_a] : ( if_list_a @ ( member_a2 @ X3 @ ( set_a2 @ Xs2 ) ) @ Xs2 @ ( cons_a @ X3 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_584_List_Oinsert__def,axiom,
( insert_Sum_sum_a_b
= ( ^ [X3: sum_sum_a_b,Xs2: list_Sum_sum_a_b] : ( if_list_Sum_sum_a_b @ ( member_Sum_sum_a_b2 @ X3 @ ( set_Sum_sum_a_b2 @ Xs2 ) ) @ Xs2 @ ( cons_Sum_sum_a_b @ X3 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_585_in__set__simps_I2_J,axiom,
! [X2: set_Sum_sum_a_b,Y: set_Sum_sum_a_b] :
( ( member4060935254435997939um_a_b @ X2 @ ( set_set_Sum_sum_a_b2 @ ( cons_set_Sum_sum_a_b @ Y @ nil_set_Sum_sum_a_b ) ) )
= ( X2 = Y ) ) ).
% in_set_simps(2)
thf(fact_586_in__set__simps_I2_J,axiom,
! [X2: set_set_a,Y: set_set_a] :
( ( member_set_set_a2 @ X2 @ ( set_set_set_a2 @ ( cons_set_set_a @ Y @ nil_set_set_a ) ) )
= ( X2 = Y ) ) ).
% in_set_simps(2)
thf(fact_587_in__set__simps_I2_J,axiom,
! [X2: set_a,Y: set_a] :
( ( member_set_a2 @ X2 @ ( set_set_a2 @ ( cons_set_a @ Y @ nil_set_a ) ) )
= ( X2 = Y ) ) ).
% in_set_simps(2)
thf(fact_588_in__set__simps_I2_J,axiom,
! [X2: b,Y: b] :
( ( member_b2 @ X2 @ ( set_b2 @ ( cons_b @ Y @ nil_b ) ) )
= ( X2 = Y ) ) ).
% in_set_simps(2)
thf(fact_589_in__set__simps_I2_J,axiom,
! [X2: list_a,Y: list_a] :
( ( member_list_a2 @ X2 @ ( set_list_a2 @ ( cons_list_a @ Y @ nil_list_a ) ) )
= ( X2 = Y ) ) ).
% in_set_simps(2)
thf(fact_590_in__set__simps_I2_J,axiom,
! [X2: list_Sum_sum_a_b,Y: list_Sum_sum_a_b] :
( ( member7701661377270014157um_a_b @ X2 @ ( set_list_Sum_sum_a_b2 @ ( cons_l7912394880171584384um_a_b @ Y @ nil_list_Sum_sum_a_b ) ) )
= ( X2 = Y ) ) ).
% in_set_simps(2)
thf(fact_591_in__set__simps_I2_J,axiom,
! [X2: a,Y: a] :
( ( member_a2 @ X2 @ ( set_a2 @ ( cons_a @ Y @ nil_a ) ) )
= ( X2 = Y ) ) ).
% in_set_simps(2)
thf(fact_592_in__set__simps_I2_J,axiom,
! [X2: sum_sum_a_b,Y: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X2 @ ( set_Sum_sum_a_b2 @ ( cons_Sum_sum_a_b @ Y @ nil_Sum_sum_a_b ) ) )
= ( X2 = Y ) ) ).
% in_set_simps(2)
thf(fact_593_in__set__simps_I3_J,axiom,
! [X2: set_Sum_sum_a_b] :
~ ( member4060935254435997939um_a_b @ X2 @ ( set_set_Sum_sum_a_b2 @ nil_set_Sum_sum_a_b ) ) ).
% in_set_simps(3)
thf(fact_594_in__set__simps_I3_J,axiom,
! [X2: set_set_a] :
~ ( member_set_set_a2 @ X2 @ ( set_set_set_a2 @ nil_set_set_a ) ) ).
% in_set_simps(3)
thf(fact_595_in__set__simps_I3_J,axiom,
! [X2: set_a] :
~ ( member_set_a2 @ X2 @ ( set_set_a2 @ nil_set_a ) ) ).
% in_set_simps(3)
thf(fact_596_in__set__simps_I3_J,axiom,
! [X2: b] :
~ ( member_b2 @ X2 @ ( set_b2 @ nil_b ) ) ).
% in_set_simps(3)
thf(fact_597_in__set__simps_I3_J,axiom,
! [X2: list_a] :
~ ( member_list_a2 @ X2 @ ( set_list_a2 @ nil_list_a ) ) ).
% in_set_simps(3)
thf(fact_598_in__set__simps_I3_J,axiom,
! [X2: list_Sum_sum_a_b] :
~ ( member7701661377270014157um_a_b @ X2 @ ( set_list_Sum_sum_a_b2 @ nil_list_Sum_sum_a_b ) ) ).
% in_set_simps(3)
thf(fact_599_in__set__simps_I3_J,axiom,
! [X2: a] :
~ ( member_a2 @ X2 @ ( set_a2 @ nil_a ) ) ).
% in_set_simps(3)
thf(fact_600_in__set__simps_I3_J,axiom,
! [X2: sum_sum_a_b] :
~ ( member_Sum_sum_a_b2 @ X2 @ ( set_Sum_sum_a_b2 @ nil_Sum_sum_a_b ) ) ).
% in_set_simps(3)
thf(fact_601_in__set__simps_I1_J,axiom,
! [X2: set_Sum_sum_a_b,Y: set_Sum_sum_a_b,Z2: set_Sum_sum_a_b,Ys: list_set_Sum_sum_a_b] :
( ( member4060935254435997939um_a_b @ X2 @ ( set_set_Sum_sum_a_b2 @ ( cons_set_Sum_sum_a_b @ Y @ ( cons_set_Sum_sum_a_b @ Z2 @ Ys ) ) ) )
= ( ( X2 = Y )
| ( member4060935254435997939um_a_b @ X2 @ ( set_set_Sum_sum_a_b2 @ ( cons_set_Sum_sum_a_b @ Z2 @ Ys ) ) ) ) ) ).
% in_set_simps(1)
thf(fact_602_in__set__simps_I1_J,axiom,
! [X2: set_set_a,Y: set_set_a,Z2: set_set_a,Ys: list_set_set_a] :
( ( member_set_set_a2 @ X2 @ ( set_set_set_a2 @ ( cons_set_set_a @ Y @ ( cons_set_set_a @ Z2 @ Ys ) ) ) )
= ( ( X2 = Y )
| ( member_set_set_a2 @ X2 @ ( set_set_set_a2 @ ( cons_set_set_a @ Z2 @ Ys ) ) ) ) ) ).
% in_set_simps(1)
thf(fact_603_in__set__simps_I1_J,axiom,
! [X2: set_a,Y: set_a,Z2: set_a,Ys: list_set_a] :
( ( member_set_a2 @ X2 @ ( set_set_a2 @ ( cons_set_a @ Y @ ( cons_set_a @ Z2 @ Ys ) ) ) )
= ( ( X2 = Y )
| ( member_set_a2 @ X2 @ ( set_set_a2 @ ( cons_set_a @ Z2 @ Ys ) ) ) ) ) ).
% in_set_simps(1)
thf(fact_604_in__set__simps_I1_J,axiom,
! [X2: b,Y: b,Z2: b,Ys: list_b] :
( ( member_b2 @ X2 @ ( set_b2 @ ( cons_b @ Y @ ( cons_b @ Z2 @ Ys ) ) ) )
= ( ( X2 = Y )
| ( member_b2 @ X2 @ ( set_b2 @ ( cons_b @ Z2 @ Ys ) ) ) ) ) ).
% in_set_simps(1)
thf(fact_605_in__set__simps_I1_J,axiom,
! [X2: list_a,Y: list_a,Z2: list_a,Ys: list_list_a] :
( ( member_list_a2 @ X2 @ ( set_list_a2 @ ( cons_list_a @ Y @ ( cons_list_a @ Z2 @ Ys ) ) ) )
= ( ( X2 = Y )
| ( member_list_a2 @ X2 @ ( set_list_a2 @ ( cons_list_a @ Z2 @ Ys ) ) ) ) ) ).
% in_set_simps(1)
thf(fact_606_in__set__simps_I1_J,axiom,
! [X2: list_Sum_sum_a_b,Y: list_Sum_sum_a_b,Z2: list_Sum_sum_a_b,Ys: list_l4199846171218662726um_a_b] :
( ( member7701661377270014157um_a_b @ X2 @ ( set_list_Sum_sum_a_b2 @ ( cons_l7912394880171584384um_a_b @ Y @ ( cons_l7912394880171584384um_a_b @ Z2 @ Ys ) ) ) )
= ( ( X2 = Y )
| ( member7701661377270014157um_a_b @ X2 @ ( set_list_Sum_sum_a_b2 @ ( cons_l7912394880171584384um_a_b @ Z2 @ Ys ) ) ) ) ) ).
% in_set_simps(1)
thf(fact_607_in__set__simps_I1_J,axiom,
! [X2: a,Y: a,Z2: a,Ys: list_a] :
( ( member_a2 @ X2 @ ( set_a2 @ ( cons_a @ Y @ ( cons_a @ Z2 @ Ys ) ) ) )
= ( ( X2 = Y )
| ( member_a2 @ X2 @ ( set_a2 @ ( cons_a @ Z2 @ Ys ) ) ) ) ) ).
% in_set_simps(1)
thf(fact_608_in__set__simps_I1_J,axiom,
! [X2: sum_sum_a_b,Y: sum_sum_a_b,Z2: sum_sum_a_b,Ys: list_Sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X2 @ ( set_Sum_sum_a_b2 @ ( cons_Sum_sum_a_b @ Y @ ( cons_Sum_sum_a_b @ Z2 @ Ys ) ) ) )
= ( ( X2 = Y )
| ( member_Sum_sum_a_b2 @ X2 @ ( set_Sum_sum_a_b2 @ ( cons_Sum_sum_a_b @ Z2 @ Ys ) ) ) ) ) ).
% in_set_simps(1)
thf(fact_609_equality__list__simps_I3_J,axiom,
! [Eq_a: sum_sum_a_b > sum_sum_a_b > $o,X2: sum_sum_a_b,Xa: list_Sum_sum_a_b] :
~ ( equali3241580248861235287um_a_b @ Eq_a @ ( cons_Sum_sum_a_b @ X2 @ Xa ) @ nil_Sum_sum_a_b ) ).
% equality_list_simps(3)
thf(fact_610_equality__list__simps_I3_J,axiom,
! [Eq_a: a > a > $o,X2: a,Xa: list_a] :
~ ( equali7354908176174964365list_a @ Eq_a @ ( cons_a @ X2 @ Xa ) @ nil_a ) ).
% equality_list_simps(3)
thf(fact_611_equality__list__simps_I2_J,axiom,
! [Eq_a: sum_sum_a_b > sum_sum_a_b > $o,Y: sum_sum_a_b,Ya: list_Sum_sum_a_b] :
~ ( equali3241580248861235287um_a_b @ Eq_a @ nil_Sum_sum_a_b @ ( cons_Sum_sum_a_b @ Y @ Ya ) ) ).
% equality_list_simps(2)
thf(fact_612_equality__list__simps_I2_J,axiom,
! [Eq_a: a > a > $o,Y: a,Ya: list_a] :
~ ( equali7354908176174964365list_a @ Eq_a @ nil_a @ ( cons_a @ Y @ Ya ) ) ).
% equality_list_simps(2)
thf(fact_613_the__elem__set,axiom,
! [X2: list_a] :
( ( the_elem_list_a @ ( set_list_a2 @ ( cons_list_a @ X2 @ nil_list_a ) ) )
= X2 ) ).
% the_elem_set
thf(fact_614_the__elem__set,axiom,
! [X2: list_Sum_sum_a_b] :
( ( the_el2539774873181534465um_a_b @ ( set_list_Sum_sum_a_b2 @ ( cons_l7912394880171584384um_a_b @ X2 @ nil_list_Sum_sum_a_b ) ) )
= X2 ) ).
% the_elem_set
thf(fact_615_the__elem__set,axiom,
! [X2: a] :
( ( the_elem_a @ ( set_a2 @ ( cons_a @ X2 @ nil_a ) ) )
= X2 ) ).
% the_elem_set
thf(fact_616_the__elem__set,axiom,
! [X2: sum_sum_a_b] :
( ( the_elem_Sum_sum_a_b @ ( set_Sum_sum_a_b2 @ ( cons_Sum_sum_a_b @ X2 @ nil_Sum_sum_a_b ) ) )
= X2 ) ).
% the_elem_set
thf(fact_617_equality__list__simps_I4_J,axiom,
! [Eq_a: sum_sum_a_b > sum_sum_a_b > $o,X2: sum_sum_a_b,Xa: list_Sum_sum_a_b,Y: sum_sum_a_b,Ya: list_Sum_sum_a_b] :
( ( equali3241580248861235287um_a_b @ Eq_a @ ( cons_Sum_sum_a_b @ X2 @ Xa ) @ ( cons_Sum_sum_a_b @ Y @ Ya ) )
= ( ( Eq_a @ X2 @ Y )
& ( equali3241580248861235287um_a_b @ Eq_a @ Xa @ Ya ) ) ) ).
% equality_list_simps(4)
thf(fact_618_equality__list__simps_I4_J,axiom,
! [Eq_a: a > a > $o,X2: a,Xa: list_a,Y: a,Ya: list_a] :
( ( equali7354908176174964365list_a @ Eq_a @ ( cons_a @ X2 @ Xa ) @ ( cons_a @ Y @ Ya ) )
= ( ( Eq_a @ X2 @ Y )
& ( equali7354908176174964365list_a @ Eq_a @ Xa @ Ya ) ) ) ).
% equality_list_simps(4)
thf(fact_619_equality__list__simps_I1_J,axiom,
! [Eq_a: a > a > $o] : ( equali7354908176174964365list_a @ Eq_a @ nil_a @ nil_a ) ).
% equality_list_simps(1)
thf(fact_620_equality__list__simps_I1_J,axiom,
! [Eq_a: sum_sum_a_b > sum_sum_a_b > $o] : ( equali3241580248861235287um_a_b @ Eq_a @ nil_Sum_sum_a_b @ nil_Sum_sum_a_b ) ).
% equality_list_simps(1)
thf(fact_621_product__lists_Osimps_I1_J,axiom,
( ( product_lists_a @ nil_list_a )
= ( cons_list_a @ nil_a @ nil_list_a ) ) ).
% product_lists.simps(1)
thf(fact_622_product__lists_Osimps_I1_J,axiom,
( ( produc7097738601274840055um_a_b @ nil_list_Sum_sum_a_b )
= ( cons_l7912394880171584384um_a_b @ nil_Sum_sum_a_b @ nil_list_Sum_sum_a_b ) ) ).
% product_lists.simps(1)
thf(fact_623_subseqs_Osimps_I1_J,axiom,
( ( subseqs_a @ nil_a )
= ( cons_list_a @ nil_a @ nil_list_a ) ) ).
% subseqs.simps(1)
thf(fact_624_subseqs_Osimps_I1_J,axiom,
( ( subseqs_Sum_sum_a_b @ nil_Sum_sum_a_b )
= ( cons_l7912394880171584384um_a_b @ nil_Sum_sum_a_b @ nil_list_Sum_sum_a_b ) ) ).
% subseqs.simps(1)
thf(fact_625_map__tailrec__rev_Oelims,axiom,
! [X2: sum_sum_a_b > sum_sum_a_b,Xa: list_Sum_sum_a_b,Xb: list_Sum_sum_a_b,Y: list_Sum_sum_a_b] :
( ( ( map_ta5595598354961588547um_a_b @ X2 @ Xa @ Xb )
= Y )
=> ( ( ( Xa = nil_Sum_sum_a_b )
=> ( Y != Xb ) )
=> ~ ! [A5: sum_sum_a_b,As: list_Sum_sum_a_b] :
( ( Xa
= ( cons_Sum_sum_a_b @ A5 @ As ) )
=> ( Y
!= ( map_ta5595598354961588547um_a_b @ X2 @ As @ ( cons_Sum_sum_a_b @ ( X2 @ A5 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_626_map__tailrec__rev_Oelims,axiom,
! [X2: sum_sum_a_b > a,Xa: list_Sum_sum_a_b,Xb: list_a,Y: list_a] :
( ( ( map_ta999682415496933537_a_b_a @ X2 @ Xa @ Xb )
= Y )
=> ( ( ( Xa = nil_Sum_sum_a_b )
=> ( Y != Xb ) )
=> ~ ! [A5: sum_sum_a_b,As: list_Sum_sum_a_b] :
( ( Xa
= ( cons_Sum_sum_a_b @ A5 @ As ) )
=> ( Y
!= ( map_ta999682415496933537_a_b_a @ X2 @ As @ ( cons_a @ ( X2 @ A5 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_627_map__tailrec__rev_Oelims,axiom,
! [X2: a > sum_sum_a_b,Xa: list_a,Xb: list_Sum_sum_a_b,Y: list_Sum_sum_a_b] :
( ( ( map_ta6919247666490268527um_a_b @ X2 @ Xa @ Xb )
= Y )
=> ( ( ( Xa = nil_a )
=> ( Y != Xb ) )
=> ~ ! [A5: a,As: list_a] :
( ( Xa
= ( cons_a @ A5 @ As ) )
=> ( Y
!= ( map_ta6919247666490268527um_a_b @ X2 @ As @ ( cons_Sum_sum_a_b @ ( X2 @ A5 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_628_map__tailrec__rev_Oelims,axiom,
! [X2: a > a,Xa: list_a,Xb: list_a,Y: list_a] :
( ( ( map_tailrec_rev_a_a @ X2 @ Xa @ Xb )
= Y )
=> ( ( ( Xa = nil_a )
=> ( Y != Xb ) )
=> ~ ! [A5: a,As: list_a] :
( ( Xa
= ( cons_a @ A5 @ As ) )
=> ( Y
!= ( map_tailrec_rev_a_a @ X2 @ As @ ( cons_a @ ( X2 @ A5 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_629_listrelp_Osimps,axiom,
( listre4268742318075517069um_a_b
= ( ^ [R: sum_sum_a_b > sum_sum_a_b > $o,A1: list_Sum_sum_a_b,A22: list_Sum_sum_a_b] :
( ( ( A1 = nil_Sum_sum_a_b )
& ( A22 = nil_Sum_sum_a_b ) )
| ? [X3: sum_sum_a_b,Y5: sum_sum_a_b,Xs2: list_Sum_sum_a_b,Ys2: list_Sum_sum_a_b] :
( ( A1
= ( cons_Sum_sum_a_b @ X3 @ Xs2 ) )
& ( A22
= ( cons_Sum_sum_a_b @ Y5 @ Ys2 ) )
& ( R @ X3 @ Y5 )
& ( listre4268742318075517069um_a_b @ R @ Xs2 @ Ys2 ) ) ) ) ) ).
% listrelp.simps
thf(fact_630_listrelp_Osimps,axiom,
( listre65364914291276759_a_b_a
= ( ^ [R: sum_sum_a_b > a > $o,A1: list_Sum_sum_a_b,A22: list_a] :
( ( ( A1 = nil_Sum_sum_a_b )
& ( A22 = nil_a ) )
| ? [X3: sum_sum_a_b,Y5: a,Xs2: list_Sum_sum_a_b,Ys2: list_a] :
( ( A1
= ( cons_Sum_sum_a_b @ X3 @ Xs2 ) )
& ( A22
= ( cons_a @ Y5 @ Ys2 ) )
& ( R @ X3 @ Y5 )
& ( listre65364914291276759_a_b_a @ R @ Xs2 @ Ys2 ) ) ) ) ) ).
% listrelp.simps
thf(fact_631_listrelp_Osimps,axiom,
( listre5984930165284611749um_a_b
= ( ^ [R: a > sum_sum_a_b > $o,A1: list_a,A22: list_Sum_sum_a_b] :
( ( ( A1 = nil_a )
& ( A22 = nil_Sum_sum_a_b ) )
| ? [X3: a,Y5: sum_sum_a_b,Xs2: list_a,Ys2: list_Sum_sum_a_b] :
( ( A1
= ( cons_a @ X3 @ Xs2 ) )
& ( A22
= ( cons_Sum_sum_a_b @ Y5 @ Ys2 ) )
& ( R @ X3 @ Y5 )
& ( listre5984930165284611749um_a_b @ R @ Xs2 @ Ys2 ) ) ) ) ) ).
% listrelp.simps
thf(fact_632_listrelp_Osimps,axiom,
( listrelp_a_a
= ( ^ [R: a > a > $o,A1: list_a,A22: list_a] :
( ( ( A1 = nil_a )
& ( A22 = nil_a ) )
| ? [X3: a,Y5: a,Xs2: list_a,Ys2: list_a] :
( ( A1
= ( cons_a @ X3 @ Xs2 ) )
& ( A22
= ( cons_a @ Y5 @ Ys2 ) )
& ( R @ X3 @ Y5 )
& ( listrelp_a_a @ R @ Xs2 @ Ys2 ) ) ) ) ) ).
% listrelp.simps
thf(fact_633_listrelp_Ocases,axiom,
! [R2: sum_sum_a_b > sum_sum_a_b > $o,A12: list_Sum_sum_a_b,A23: list_Sum_sum_a_b] :
( ( listre4268742318075517069um_a_b @ R2 @ A12 @ A23 )
=> ( ( ( A12 = nil_Sum_sum_a_b )
=> ( A23 != nil_Sum_sum_a_b ) )
=> ~ ! [X4: sum_sum_a_b,Y3: sum_sum_a_b,Xs3: list_Sum_sum_a_b] :
( ( A12
= ( cons_Sum_sum_a_b @ X4 @ Xs3 ) )
=> ! [Ys3: list_Sum_sum_a_b] :
( ( A23
= ( cons_Sum_sum_a_b @ Y3 @ Ys3 ) )
=> ( ( R2 @ X4 @ Y3 )
=> ~ ( listre4268742318075517069um_a_b @ R2 @ Xs3 @ Ys3 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_634_listrelp_Ocases,axiom,
! [R2: sum_sum_a_b > a > $o,A12: list_Sum_sum_a_b,A23: list_a] :
( ( listre65364914291276759_a_b_a @ R2 @ A12 @ A23 )
=> ( ( ( A12 = nil_Sum_sum_a_b )
=> ( A23 != nil_a ) )
=> ~ ! [X4: sum_sum_a_b,Y3: a,Xs3: list_Sum_sum_a_b] :
( ( A12
= ( cons_Sum_sum_a_b @ X4 @ Xs3 ) )
=> ! [Ys3: list_a] :
( ( A23
= ( cons_a @ Y3 @ Ys3 ) )
=> ( ( R2 @ X4 @ Y3 )
=> ~ ( listre65364914291276759_a_b_a @ R2 @ Xs3 @ Ys3 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_635_listrelp_Ocases,axiom,
! [R2: a > sum_sum_a_b > $o,A12: list_a,A23: list_Sum_sum_a_b] :
( ( listre5984930165284611749um_a_b @ R2 @ A12 @ A23 )
=> ( ( ( A12 = nil_a )
=> ( A23 != nil_Sum_sum_a_b ) )
=> ~ ! [X4: a,Y3: sum_sum_a_b,Xs3: list_a] :
( ( A12
= ( cons_a @ X4 @ Xs3 ) )
=> ! [Ys3: list_Sum_sum_a_b] :
( ( A23
= ( cons_Sum_sum_a_b @ Y3 @ Ys3 ) )
=> ( ( R2 @ X4 @ Y3 )
=> ~ ( listre5984930165284611749um_a_b @ R2 @ Xs3 @ Ys3 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_636_listrelp_Ocases,axiom,
! [R2: a > a > $o,A12: list_a,A23: list_a] :
( ( listrelp_a_a @ R2 @ A12 @ A23 )
=> ( ( ( A12 = nil_a )
=> ( A23 != nil_a ) )
=> ~ ! [X4: a,Y3: a,Xs3: list_a] :
( ( A12
= ( cons_a @ X4 @ Xs3 ) )
=> ! [Ys3: list_a] :
( ( A23
= ( cons_a @ Y3 @ Ys3 ) )
=> ( ( R2 @ X4 @ Y3 )
=> ~ ( listrelp_a_a @ R2 @ Xs3 @ Ys3 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_637_ord_Olexordp__eq__simps_I3_J,axiom,
! [Less: sum_sum_a_b > sum_sum_a_b > $o,X2: sum_sum_a_b,Xs: list_Sum_sum_a_b] :
~ ( lexord5773373854852884058um_a_b @ Less @ ( cons_Sum_sum_a_b @ X2 @ Xs ) @ nil_Sum_sum_a_b ) ).
% ord.lexordp_eq_simps(3)
thf(fact_638_ord_Olexordp__eq__simps_I3_J,axiom,
! [Less: a > a > $o,X2: a,Xs: list_a] :
~ ( lexordp_eq_a @ Less @ ( cons_a @ X2 @ Xs ) @ nil_a ) ).
% ord.lexordp_eq_simps(3)
thf(fact_639_ord_Olexordp__eq__simps_I4_J,axiom,
! [Less: sum_sum_a_b > sum_sum_a_b > $o,X2: sum_sum_a_b,Xs: list_Sum_sum_a_b,Y: sum_sum_a_b,Ys: list_Sum_sum_a_b] :
( ( lexord5773373854852884058um_a_b @ Less @ ( cons_Sum_sum_a_b @ X2 @ Xs ) @ ( cons_Sum_sum_a_b @ Y @ Ys ) )
= ( ( Less @ X2 @ Y )
| ( ~ ( Less @ Y @ X2 )
& ( lexord5773373854852884058um_a_b @ Less @ Xs @ Ys ) ) ) ) ).
% ord.lexordp_eq_simps(4)
thf(fact_640_ord_Olexordp__eq__simps_I4_J,axiom,
! [Less: a > a > $o,X2: a,Xs: list_a,Y: a,Ys: list_a] :
( ( lexordp_eq_a @ Less @ ( cons_a @ X2 @ Xs ) @ ( cons_a @ Y @ Ys ) )
= ( ( Less @ X2 @ Y )
| ( ~ ( Less @ Y @ X2 )
& ( lexordp_eq_a @ Less @ Xs @ Ys ) ) ) ) ).
% ord.lexordp_eq_simps(4)
thf(fact_641_ord_Olexordp__eq__simps_I2_J,axiom,
! [Less: a > a > $o,Xs: list_a] :
( ( lexordp_eq_a @ Less @ Xs @ nil_a )
= ( Xs = nil_a ) ) ).
% ord.lexordp_eq_simps(2)
thf(fact_642_ord_Olexordp__eq__simps_I2_J,axiom,
! [Less: sum_sum_a_b > sum_sum_a_b > $o,Xs: list_Sum_sum_a_b] :
( ( lexord5773373854852884058um_a_b @ Less @ Xs @ nil_Sum_sum_a_b )
= ( Xs = nil_Sum_sum_a_b ) ) ).
% ord.lexordp_eq_simps(2)
thf(fact_643_ord_Olexordp__eq__simps_I1_J,axiom,
! [Less: a > a > $o,Ys: list_a] : ( lexordp_eq_a @ Less @ nil_a @ Ys ) ).
% ord.lexordp_eq_simps(1)
thf(fact_644_ord_Olexordp__eq__simps_I1_J,axiom,
! [Less: sum_sum_a_b > sum_sum_a_b > $o,Ys: list_Sum_sum_a_b] : ( lexord5773373854852884058um_a_b @ Less @ nil_Sum_sum_a_b @ Ys ) ).
% ord.lexordp_eq_simps(1)
thf(fact_645_subseqs__refl,axiom,
! [Xs: list_a] : ( member_list_a2 @ Xs @ ( set_list_a2 @ ( subseqs_a @ Xs ) ) ) ).
% subseqs_refl
thf(fact_646_subseqs__refl,axiom,
! [Xs: list_Sum_sum_a_b] : ( member7701661377270014157um_a_b @ Xs @ ( set_list_Sum_sum_a_b2 @ ( subseqs_Sum_sum_a_b @ Xs ) ) ) ).
% subseqs_refl
thf(fact_647_ord_Olexordp__eq_OCons,axiom,
! [Less: sum_sum_a_b > sum_sum_a_b > $o,X2: sum_sum_a_b,Y: sum_sum_a_b,Xs: list_Sum_sum_a_b,Ys: list_Sum_sum_a_b] :
( ( Less @ X2 @ Y )
=> ( lexord5773373854852884058um_a_b @ Less @ ( cons_Sum_sum_a_b @ X2 @ Xs ) @ ( cons_Sum_sum_a_b @ Y @ Ys ) ) ) ).
% ord.lexordp_eq.Cons
thf(fact_648_ord_Olexordp__eq_OCons,axiom,
! [Less: a > a > $o,X2: a,Y: a,Xs: list_a,Ys: list_a] :
( ( Less @ X2 @ Y )
=> ( lexordp_eq_a @ Less @ ( cons_a @ X2 @ Xs ) @ ( cons_a @ Y @ Ys ) ) ) ).
% ord.lexordp_eq.Cons
thf(fact_649_ord_Olexordp__eq_OCons__eq,axiom,
! [Less: sum_sum_a_b > sum_sum_a_b > $o,X2: sum_sum_a_b,Y: sum_sum_a_b,Xs: list_Sum_sum_a_b,Ys: list_Sum_sum_a_b] :
( ~ ( Less @ X2 @ Y )
=> ( ~ ( Less @ Y @ X2 )
=> ( ( lexord5773373854852884058um_a_b @ Less @ Xs @ Ys )
=> ( lexord5773373854852884058um_a_b @ Less @ ( cons_Sum_sum_a_b @ X2 @ Xs ) @ ( cons_Sum_sum_a_b @ Y @ Ys ) ) ) ) ) ).
% ord.lexordp_eq.Cons_eq
thf(fact_650_ord_Olexordp__eq_OCons__eq,axiom,
! [Less: a > a > $o,X2: a,Y: a,Xs: list_a,Ys: list_a] :
( ~ ( Less @ X2 @ Y )
=> ( ~ ( Less @ Y @ X2 )
=> ( ( lexordp_eq_a @ Less @ Xs @ Ys )
=> ( lexordp_eq_a @ Less @ ( cons_a @ X2 @ Xs ) @ ( cons_a @ Y @ Ys ) ) ) ) ) ).
% ord.lexordp_eq.Cons_eq
thf(fact_651_ord_Olexordp__eq_ONil,axiom,
! [Less: a > a > $o,Ys: list_a] : ( lexordp_eq_a @ Less @ nil_a @ Ys ) ).
% ord.lexordp_eq.Nil
thf(fact_652_ord_Olexordp__eq_ONil,axiom,
! [Less: sum_sum_a_b > sum_sum_a_b > $o,Ys: list_Sum_sum_a_b] : ( lexord5773373854852884058um_a_b @ Less @ nil_Sum_sum_a_b @ Ys ) ).
% ord.lexordp_eq.Nil
thf(fact_653_Cons__in__subseqsD,axiom,
! [Y: sum_sum_a_b,Ys: list_Sum_sum_a_b,Xs: list_Sum_sum_a_b] :
( ( member7701661377270014157um_a_b @ ( cons_Sum_sum_a_b @ Y @ Ys ) @ ( set_list_Sum_sum_a_b2 @ ( subseqs_Sum_sum_a_b @ Xs ) ) )
=> ( member7701661377270014157um_a_b @ Ys @ ( set_list_Sum_sum_a_b2 @ ( subseqs_Sum_sum_a_b @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_654_Cons__in__subseqsD,axiom,
! [Y: a,Ys: list_a,Xs: list_a] :
( ( member_list_a2 @ ( cons_a @ Y @ Ys ) @ ( set_list_a2 @ ( subseqs_a @ Xs ) ) )
=> ( member_list_a2 @ Ys @ ( set_list_a2 @ ( subseqs_a @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_655_listrelp_OCons,axiom,
! [R2: sum_sum_a_b > sum_sum_a_b > $o,X2: sum_sum_a_b,Y: sum_sum_a_b,Xs: list_Sum_sum_a_b,Ys: list_Sum_sum_a_b] :
( ( R2 @ X2 @ Y )
=> ( ( listre4268742318075517069um_a_b @ R2 @ Xs @ Ys )
=> ( listre4268742318075517069um_a_b @ R2 @ ( cons_Sum_sum_a_b @ X2 @ Xs ) @ ( cons_Sum_sum_a_b @ Y @ Ys ) ) ) ) ).
% listrelp.Cons
thf(fact_656_listrelp_OCons,axiom,
! [R2: sum_sum_a_b > a > $o,X2: sum_sum_a_b,Y: a,Xs: list_Sum_sum_a_b,Ys: list_a] :
( ( R2 @ X2 @ Y )
=> ( ( listre65364914291276759_a_b_a @ R2 @ Xs @ Ys )
=> ( listre65364914291276759_a_b_a @ R2 @ ( cons_Sum_sum_a_b @ X2 @ Xs ) @ ( cons_a @ Y @ Ys ) ) ) ) ).
% listrelp.Cons
thf(fact_657_listrelp_OCons,axiom,
! [R2: a > sum_sum_a_b > $o,X2: a,Y: sum_sum_a_b,Xs: list_a,Ys: list_Sum_sum_a_b] :
( ( R2 @ X2 @ Y )
=> ( ( listre5984930165284611749um_a_b @ R2 @ Xs @ Ys )
=> ( listre5984930165284611749um_a_b @ R2 @ ( cons_a @ X2 @ Xs ) @ ( cons_Sum_sum_a_b @ Y @ Ys ) ) ) ) ).
% listrelp.Cons
thf(fact_658_listrelp_OCons,axiom,
! [R2: a > a > $o,X2: a,Y: a,Xs: list_a,Ys: list_a] :
( ( R2 @ X2 @ Y )
=> ( ( listrelp_a_a @ R2 @ Xs @ Ys )
=> ( listrelp_a_a @ R2 @ ( cons_a @ X2 @ Xs ) @ ( cons_a @ Y @ Ys ) ) ) ) ).
% listrelp.Cons
thf(fact_659_listrelp_ONil,axiom,
! [R2: a > a > $o] : ( listrelp_a_a @ R2 @ nil_a @ nil_a ) ).
% listrelp.Nil
thf(fact_660_listrelp_ONil,axiom,
! [R2: a > sum_sum_a_b > $o] : ( listre5984930165284611749um_a_b @ R2 @ nil_a @ nil_Sum_sum_a_b ) ).
% listrelp.Nil
thf(fact_661_listrelp_ONil,axiom,
! [R2: sum_sum_a_b > a > $o] : ( listre65364914291276759_a_b_a @ R2 @ nil_Sum_sum_a_b @ nil_a ) ).
% listrelp.Nil
thf(fact_662_listrelp_ONil,axiom,
! [R2: sum_sum_a_b > sum_sum_a_b > $o] : ( listre4268742318075517069um_a_b @ R2 @ nil_Sum_sum_a_b @ nil_Sum_sum_a_b ) ).
% listrelp.Nil
thf(fact_663_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: sum_sum_a_b > sum_sum_a_b,A: sum_sum_a_b,As2: list_Sum_sum_a_b,Bs: list_Sum_sum_a_b] :
( ( map_ta5595598354961588547um_a_b @ F @ ( cons_Sum_sum_a_b @ A @ As2 ) @ Bs )
= ( map_ta5595598354961588547um_a_b @ F @ As2 @ ( cons_Sum_sum_a_b @ ( F @ A ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_664_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: sum_sum_a_b > a,A: sum_sum_a_b,As2: list_Sum_sum_a_b,Bs: list_a] :
( ( map_ta999682415496933537_a_b_a @ F @ ( cons_Sum_sum_a_b @ A @ As2 ) @ Bs )
= ( map_ta999682415496933537_a_b_a @ F @ As2 @ ( cons_a @ ( F @ A ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_665_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: a > sum_sum_a_b,A: a,As2: list_a,Bs: list_Sum_sum_a_b] :
( ( map_ta6919247666490268527um_a_b @ F @ ( cons_a @ A @ As2 ) @ Bs )
= ( map_ta6919247666490268527um_a_b @ F @ As2 @ ( cons_Sum_sum_a_b @ ( F @ A ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_666_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: a > a,A: a,As2: list_a,Bs: list_a] :
( ( map_tailrec_rev_a_a @ F @ ( cons_a @ A @ As2 ) @ Bs )
= ( map_tailrec_rev_a_a @ F @ As2 @ ( cons_a @ ( F @ A ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_667_ord_Olexordp__eq_Ocases,axiom,
! [Less: sum_sum_a_b > sum_sum_a_b > $o,A12: list_Sum_sum_a_b,A23: list_Sum_sum_a_b] :
( ( lexord5773373854852884058um_a_b @ Less @ A12 @ A23 )
=> ( ( A12 != nil_Sum_sum_a_b )
=> ( ! [X4: sum_sum_a_b] :
( ? [Xs3: list_Sum_sum_a_b] :
( A12
= ( cons_Sum_sum_a_b @ X4 @ Xs3 ) )
=> ! [Y3: sum_sum_a_b] :
( ? [Ys3: list_Sum_sum_a_b] :
( A23
= ( cons_Sum_sum_a_b @ Y3 @ Ys3 ) )
=> ~ ( Less @ X4 @ Y3 ) ) )
=> ~ ! [X4: sum_sum_a_b,Y3: sum_sum_a_b,Xs3: list_Sum_sum_a_b] :
( ( A12
= ( cons_Sum_sum_a_b @ X4 @ Xs3 ) )
=> ! [Ys3: list_Sum_sum_a_b] :
( ( A23
= ( cons_Sum_sum_a_b @ Y3 @ Ys3 ) )
=> ( ~ ( Less @ X4 @ Y3 )
=> ( ~ ( Less @ Y3 @ X4 )
=> ~ ( lexord5773373854852884058um_a_b @ Less @ Xs3 @ Ys3 ) ) ) ) ) ) ) ) ).
% ord.lexordp_eq.cases
thf(fact_668_ord_Olexordp__eq_Ocases,axiom,
! [Less: a > a > $o,A12: list_a,A23: list_a] :
( ( lexordp_eq_a @ Less @ A12 @ A23 )
=> ( ( A12 != nil_a )
=> ( ! [X4: a] :
( ? [Xs3: list_a] :
( A12
= ( cons_a @ X4 @ Xs3 ) )
=> ! [Y3: a] :
( ? [Ys3: list_a] :
( A23
= ( cons_a @ Y3 @ Ys3 ) )
=> ~ ( Less @ X4 @ Y3 ) ) )
=> ~ ! [X4: a,Y3: a,Xs3: list_a] :
( ( A12
= ( cons_a @ X4 @ Xs3 ) )
=> ! [Ys3: list_a] :
( ( A23
= ( cons_a @ Y3 @ Ys3 ) )
=> ( ~ ( Less @ X4 @ Y3 )
=> ( ~ ( Less @ Y3 @ X4 )
=> ~ ( lexordp_eq_a @ Less @ Xs3 @ Ys3 ) ) ) ) ) ) ) ) ).
% ord.lexordp_eq.cases
thf(fact_669_ord_Olexordp__eq_Osimps,axiom,
( lexord5773373854852884058um_a_b
= ( ^ [Less2: sum_sum_a_b > sum_sum_a_b > $o,A1: list_Sum_sum_a_b,A22: list_Sum_sum_a_b] :
( ? [Ys2: list_Sum_sum_a_b] :
( ( A1 = nil_Sum_sum_a_b )
& ( A22 = Ys2 ) )
| ? [X3: sum_sum_a_b,Y5: sum_sum_a_b,Xs2: list_Sum_sum_a_b,Ys2: list_Sum_sum_a_b] :
( ( A1
= ( cons_Sum_sum_a_b @ X3 @ Xs2 ) )
& ( A22
= ( cons_Sum_sum_a_b @ Y5 @ Ys2 ) )
& ( Less2 @ X3 @ Y5 ) )
| ? [X3: sum_sum_a_b,Y5: sum_sum_a_b,Xs2: list_Sum_sum_a_b,Ys2: list_Sum_sum_a_b] :
( ( A1
= ( cons_Sum_sum_a_b @ X3 @ Xs2 ) )
& ( A22
= ( cons_Sum_sum_a_b @ Y5 @ Ys2 ) )
& ~ ( Less2 @ X3 @ Y5 )
& ~ ( Less2 @ Y5 @ X3 )
& ( lexord5773373854852884058um_a_b @ Less2 @ Xs2 @ Ys2 ) ) ) ) ) ).
% ord.lexordp_eq.simps
thf(fact_670_ord_Olexordp__eq_Osimps,axiom,
( lexordp_eq_a
= ( ^ [Less2: a > a > $o,A1: list_a,A22: list_a] :
( ? [Ys2: list_a] :
( ( A1 = nil_a )
& ( A22 = Ys2 ) )
| ? [X3: a,Y5: a,Xs2: list_a,Ys2: list_a] :
( ( A1
= ( cons_a @ X3 @ Xs2 ) )
& ( A22
= ( cons_a @ Y5 @ Ys2 ) )
& ( Less2 @ X3 @ Y5 ) )
| ? [X3: a,Y5: a,Xs2: list_a,Ys2: list_a] :
( ( A1
= ( cons_a @ X3 @ Xs2 ) )
& ( A22
= ( cons_a @ Y5 @ Ys2 ) )
& ~ ( Less2 @ X3 @ Y5 )
& ~ ( Less2 @ Y5 @ X3 )
& ( lexordp_eq_a @ Less2 @ Xs2 @ Ys2 ) ) ) ) ) ).
% ord.lexordp_eq.simps
thf(fact_671_subset__subseqs,axiom,
! [X: set_list_a,Xs: list_list_a] :
( ( ord_le8861187494160871172list_a @ X @ ( set_list_a2 @ Xs ) )
=> ( member_set_list_a @ X @ ( image_432481560377026271list_a @ set_list_a2 @ ( set_list_list_a2 @ ( subseqs_list_a @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_672_subset__subseqs,axiom,
! [X: set_list_Sum_sum_a_b,Xs: list_l4199846171218662726um_a_b] :
( ( ord_le2472362315733485388um_a_b @ X @ ( set_list_Sum_sum_a_b2 @ Xs ) )
=> ( member1385076861102201347um_a_b @ X @ ( image_6061808253777162169um_a_b @ set_list_Sum_sum_a_b2 @ ( set_li2158956922031301489um_a_b @ ( subseq2702345516914364321um_a_b @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_673_subset__subseqs,axiom,
! [X: set_set_a,Xs: list_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ ( set_set_a2 @ Xs ) )
=> ( member_set_set_a2 @ X @ ( image_8804695481318321887_set_a @ set_set_a2 @ ( set_list_set_a2 @ ( subseqs_set_a @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_674_subset__subseqs,axiom,
! [X: set_Sum_sum_a_b,Xs: list_Sum_sum_a_b] :
( ( ord_le9019793522827316924um_a_b @ X @ ( set_Sum_sum_a_b2 @ Xs ) )
=> ( member4060935254435997939um_a_b @ X @ ( image_3885185050486581529um_a_b @ set_Sum_sum_a_b2 @ ( set_list_Sum_sum_a_b2 @ ( subseqs_Sum_sum_a_b @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_675_subset__subseqs,axiom,
! [X: set_a,Xs: list_a] :
( ( ord_less_eq_set_a @ X @ ( set_a2 @ Xs ) )
=> ( member_set_a2 @ X @ ( image_list_a_set_a @ set_a2 @ ( set_list_a2 @ ( subseqs_a @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_676_bind__simps_I1_J,axiom,
! [F: a > list_a] :
( ( bind_a_a @ nil_a @ F )
= nil_a ) ).
% bind_simps(1)
thf(fact_677_bind__simps_I1_J,axiom,
! [F: a > list_Sum_sum_a_b] :
( ( bind_a_Sum_sum_a_b @ nil_a @ F )
= nil_Sum_sum_a_b ) ).
% bind_simps(1)
thf(fact_678_bind__simps_I1_J,axiom,
! [F: sum_sum_a_b > list_a] :
( ( bind_Sum_sum_a_b_a @ nil_Sum_sum_a_b @ F )
= nil_a ) ).
% bind_simps(1)
thf(fact_679_bind__simps_I1_J,axiom,
! [F: sum_sum_a_b > list_Sum_sum_a_b] :
( ( bind_S7382095983525407189um_a_b @ nil_Sum_sum_a_b @ F )
= nil_Sum_sum_a_b ) ).
% bind_simps(1)
thf(fact_680_ShiftD,axiom,
! [Kl: list_Sum_sum_a_b,Kl2: set_list_Sum_sum_a_b,K: sum_sum_a_b] :
( ( member7701661377270014157um_a_b @ Kl @ ( bNF_Gr7310031135006470983um_a_b @ Kl2 @ K ) )
=> ( member7701661377270014157um_a_b @ ( cons_Sum_sum_a_b @ K @ Kl ) @ Kl2 ) ) ).
% ShiftD
thf(fact_681_ShiftD,axiom,
! [Kl: list_a,Kl2: set_list_a,K: a] :
( ( member_list_a2 @ Kl @ ( bNF_Greatest_Shift_a @ Kl2 @ K ) )
=> ( member_list_a2 @ ( cons_a @ K @ Kl ) @ Kl2 ) ) ).
% ShiftD
thf(fact_682_the__elem__code_I1_J,axiom,
! [X2: a] :
( ( the_elem_a @ ( set_Set_Monad_a @ ( cons_a @ X2 @ nil_a ) ) )
= X2 ) ).
% the_elem_code(1)
thf(fact_683_the__elem__code_I1_J,axiom,
! [X2: sum_sum_a_b] :
( ( the_elem_Sum_sum_a_b @ ( set_Se4110677268627701994um_a_b @ ( cons_Sum_sum_a_b @ X2 @ nil_Sum_sum_a_b ) ) )
= X2 ) ).
% the_elem_code(1)
thf(fact_684_image__eqI,axiom,
! [B2: a,F: a > a,X2: a,A4: set_a] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_a2 @ X2 @ A4 )
=> ( member_a2 @ B2 @ ( image_a_a @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_685_image__eqI,axiom,
! [B2: sum_sum_a_b,F: a > sum_sum_a_b,X2: a,A4: set_a] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_a2 @ X2 @ A4 )
=> ( member_Sum_sum_a_b2 @ B2 @ ( image_a_Sum_sum_a_b @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_686_image__eqI,axiom,
! [B2: a,F: sum_sum_a_b > a,X2: sum_sum_a_b,A4: set_Sum_sum_a_b] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_Sum_sum_a_b2 @ X2 @ A4 )
=> ( member_a2 @ B2 @ ( image_Sum_sum_a_b_a @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_687_image__eqI,axiom,
! [B2: sum_sum_a_b,F: sum_sum_a_b > sum_sum_a_b,X2: sum_sum_a_b,A4: set_Sum_sum_a_b] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_Sum_sum_a_b2 @ X2 @ A4 )
=> ( member_Sum_sum_a_b2 @ B2 @ ( image_3358989901043947347um_a_b @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_688_image__eqI,axiom,
! [B2: b,F: a > b,X2: a,A4: set_a] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_a2 @ X2 @ A4 )
=> ( member_b2 @ B2 @ ( image_a_b @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_689_image__eqI,axiom,
! [B2: a,F: b > a,X2: b,A4: set_b] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_b2 @ X2 @ A4 )
=> ( member_a2 @ B2 @ ( image_b_a @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_690_image__eqI,axiom,
! [B2: b,F: b > b,X2: b,A4: set_b] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_b2 @ X2 @ A4 )
=> ( member_b2 @ B2 @ ( image_b_b @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_691_image__eqI,axiom,
! [B2: list_a,F: a > list_a,X2: a,A4: set_a] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_a2 @ X2 @ A4 )
=> ( member_list_a2 @ B2 @ ( image_a_list_a @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_692_image__eqI,axiom,
! [B2: set_a,F: a > set_a,X2: a,A4: set_a] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_a2 @ X2 @ A4 )
=> ( member_set_a2 @ B2 @ ( image_a_set_a @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_693_image__eqI,axiom,
! [B2: a,F: list_a > a,X2: list_a,A4: set_list_a] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_list_a2 @ X2 @ A4 )
=> ( member_a2 @ B2 @ ( image_list_a_a @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_694_rev__image__eqI,axiom,
! [X2: a,A4: set_a,B2: a,F: a > a] :
( ( member_a2 @ X2 @ A4 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_a2 @ B2 @ ( image_a_a @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_695_rev__image__eqI,axiom,
! [X2: a,A4: set_a,B2: sum_sum_a_b,F: a > sum_sum_a_b] :
( ( member_a2 @ X2 @ A4 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_Sum_sum_a_b2 @ B2 @ ( image_a_Sum_sum_a_b @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_696_rev__image__eqI,axiom,
! [X2: sum_sum_a_b,A4: set_Sum_sum_a_b,B2: a,F: sum_sum_a_b > a] :
( ( member_Sum_sum_a_b2 @ X2 @ A4 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_a2 @ B2 @ ( image_Sum_sum_a_b_a @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_697_rev__image__eqI,axiom,
! [X2: sum_sum_a_b,A4: set_Sum_sum_a_b,B2: sum_sum_a_b,F: sum_sum_a_b > sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X2 @ A4 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_Sum_sum_a_b2 @ B2 @ ( image_3358989901043947347um_a_b @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_698_rev__image__eqI,axiom,
! [X2: a,A4: set_a,B2: b,F: a > b] :
( ( member_a2 @ X2 @ A4 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_b2 @ B2 @ ( image_a_b @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_699_rev__image__eqI,axiom,
! [X2: b,A4: set_b,B2: a,F: b > a] :
( ( member_b2 @ X2 @ A4 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_a2 @ B2 @ ( image_b_a @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_700_rev__image__eqI,axiom,
! [X2: b,A4: set_b,B2: b,F: b > b] :
( ( member_b2 @ X2 @ A4 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_b2 @ B2 @ ( image_b_b @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_701_rev__image__eqI,axiom,
! [X2: a,A4: set_a,B2: list_a,F: a > list_a] :
( ( member_a2 @ X2 @ A4 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_list_a2 @ B2 @ ( image_a_list_a @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_702_rev__image__eqI,axiom,
! [X2: a,A4: set_a,B2: set_a,F: a > set_a] :
( ( member_a2 @ X2 @ A4 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_set_a2 @ B2 @ ( image_a_set_a @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_703_rev__image__eqI,axiom,
! [X2: list_a,A4: set_list_a,B2: a,F: list_a > a] :
( ( member_list_a2 @ X2 @ A4 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_a2 @ B2 @ ( image_list_a_a @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_704_ball__imageD,axiom,
! [F: set_Sum_sum_a_b > set_Sum_sum_a_b,A4: set_set_Sum_sum_a_b,P: set_Sum_sum_a_b > $o] :
( ! [X4: set_Sum_sum_a_b] :
( ( member4060935254435997939um_a_b @ X4 @ ( image_7006159782026564799um_a_b @ F @ A4 ) )
=> ( P @ X4 ) )
=> ! [X7: set_Sum_sum_a_b] :
( ( member4060935254435997939um_a_b @ X7 @ A4 )
=> ( P @ ( F @ X7 ) ) ) ) ).
% ball_imageD
thf(fact_705_ball__imageD,axiom,
! [F: set_a > set_a,A4: set_set_a,P: set_a > $o] :
( ! [X4: set_a] :
( ( member_set_a2 @ X4 @ ( image_set_a_set_a @ F @ A4 ) )
=> ( P @ X4 ) )
=> ! [X7: set_a] :
( ( member_set_a2 @ X7 @ A4 )
=> ( P @ ( F @ X7 ) ) ) ) ).
% ball_imageD
thf(fact_706_ball__imageD,axiom,
! [F: b > sum_sum_a_b,A4: set_b,P: sum_sum_a_b > $o] :
( ! [X4: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X4 @ ( image_b_Sum_sum_a_b @ F @ A4 ) )
=> ( P @ X4 ) )
=> ! [X7: b] :
( ( member_b2 @ X7 @ A4 )
=> ( P @ ( F @ X7 ) ) ) ) ).
% ball_imageD
thf(fact_707_ball__imageD,axiom,
! [F: a > sum_sum_a_b,A4: set_a,P: sum_sum_a_b > $o] :
( ! [X4: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X4 @ ( image_a_Sum_sum_a_b @ F @ A4 ) )
=> ( P @ X4 ) )
=> ! [X7: a] :
( ( member_a2 @ X7 @ A4 )
=> ( P @ ( F @ X7 ) ) ) ) ).
% ball_imageD
thf(fact_708_ball__imageD,axiom,
! [F: a > a,A4: set_a,P: a > $o] :
( ! [X4: a] :
( ( member_a2 @ X4 @ ( image_a_a @ F @ A4 ) )
=> ( P @ X4 ) )
=> ! [X7: a] :
( ( member_a2 @ X7 @ A4 )
=> ( P @ ( F @ X7 ) ) ) ) ).
% ball_imageD
thf(fact_709_image__cong,axiom,
! [M: set_a,N: set_a,F: a > sum_sum_a_b,G2: a > sum_sum_a_b] :
( ( M = N )
=> ( ! [X4: a] :
( ( member_a2 @ X4 @ N )
=> ( ( F @ X4 )
= ( G2 @ X4 ) ) )
=> ( ( image_a_Sum_sum_a_b @ F @ M )
= ( image_a_Sum_sum_a_b @ G2 @ N ) ) ) ) ).
% image_cong
thf(fact_710_image__cong,axiom,
! [M: set_a,N: set_a,F: a > a,G2: a > a] :
( ( M = N )
=> ( ! [X4: a] :
( ( member_a2 @ X4 @ N )
=> ( ( F @ X4 )
= ( G2 @ X4 ) ) )
=> ( ( image_a_a @ F @ M )
= ( image_a_a @ G2 @ N ) ) ) ) ).
% image_cong
thf(fact_711_image__cong,axiom,
! [M: set_set_Sum_sum_a_b,N: set_set_Sum_sum_a_b,F: set_Sum_sum_a_b > set_Sum_sum_a_b,G2: set_Sum_sum_a_b > set_Sum_sum_a_b] :
( ( M = N )
=> ( ! [X4: set_Sum_sum_a_b] :
( ( member4060935254435997939um_a_b @ X4 @ N )
=> ( ( F @ X4 )
= ( G2 @ X4 ) ) )
=> ( ( image_7006159782026564799um_a_b @ F @ M )
= ( image_7006159782026564799um_a_b @ G2 @ N ) ) ) ) ).
% image_cong
thf(fact_712_image__cong,axiom,
! [M: set_set_a,N: set_set_a,F: set_a > set_a,G2: set_a > set_a] :
( ( M = N )
=> ( ! [X4: set_a] :
( ( member_set_a2 @ X4 @ N )
=> ( ( F @ X4 )
= ( G2 @ X4 ) ) )
=> ( ( image_set_a_set_a @ F @ M )
= ( image_set_a_set_a @ G2 @ N ) ) ) ) ).
% image_cong
thf(fact_713_image__cong,axiom,
! [M: set_b,N: set_b,F: b > sum_sum_a_b,G2: b > sum_sum_a_b] :
( ( M = N )
=> ( ! [X4: b] :
( ( member_b2 @ X4 @ N )
=> ( ( F @ X4 )
= ( G2 @ X4 ) ) )
=> ( ( image_b_Sum_sum_a_b @ F @ M )
= ( image_b_Sum_sum_a_b @ G2 @ N ) ) ) ) ).
% image_cong
thf(fact_714_bex__imageD,axiom,
! [F: set_Sum_sum_a_b > set_Sum_sum_a_b,A4: set_set_Sum_sum_a_b,P: set_Sum_sum_a_b > $o] :
( ? [X7: set_Sum_sum_a_b] :
( ( member4060935254435997939um_a_b @ X7 @ ( image_7006159782026564799um_a_b @ F @ A4 ) )
& ( P @ X7 ) )
=> ? [X4: set_Sum_sum_a_b] :
( ( member4060935254435997939um_a_b @ X4 @ A4 )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_715_bex__imageD,axiom,
! [F: set_a > set_a,A4: set_set_a,P: set_a > $o] :
( ? [X7: set_a] :
( ( member_set_a2 @ X7 @ ( image_set_a_set_a @ F @ A4 ) )
& ( P @ X7 ) )
=> ? [X4: set_a] :
( ( member_set_a2 @ X4 @ A4 )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_716_bex__imageD,axiom,
! [F: b > sum_sum_a_b,A4: set_b,P: sum_sum_a_b > $o] :
( ? [X7: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X7 @ ( image_b_Sum_sum_a_b @ F @ A4 ) )
& ( P @ X7 ) )
=> ? [X4: b] :
( ( member_b2 @ X4 @ A4 )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_717_bex__imageD,axiom,
! [F: a > sum_sum_a_b,A4: set_a,P: sum_sum_a_b > $o] :
( ? [X7: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X7 @ ( image_a_Sum_sum_a_b @ F @ A4 ) )
& ( P @ X7 ) )
=> ? [X4: a] :
( ( member_a2 @ X4 @ A4 )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_718_bex__imageD,axiom,
! [F: a > a,A4: set_a,P: a > $o] :
( ? [X7: a] :
( ( member_a2 @ X7 @ ( image_a_a @ F @ A4 ) )
& ( P @ X7 ) )
=> ? [X4: a] :
( ( member_a2 @ X4 @ A4 )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_719_image__iff,axiom,
! [Z2: a,F: a > a,A4: set_a] :
( ( member_a2 @ Z2 @ ( image_a_a @ F @ A4 ) )
= ( ? [X3: a] :
( ( member_a2 @ X3 @ A4 )
& ( Z2
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_720_image__iff,axiom,
! [Z2: sum_sum_a_b,F: b > sum_sum_a_b,A4: set_b] :
( ( member_Sum_sum_a_b2 @ Z2 @ ( image_b_Sum_sum_a_b @ F @ A4 ) )
= ( ? [X3: b] :
( ( member_b2 @ X3 @ A4 )
& ( Z2
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_721_image__iff,axiom,
! [Z2: sum_sum_a_b,F: a > sum_sum_a_b,A4: set_a] :
( ( member_Sum_sum_a_b2 @ Z2 @ ( image_a_Sum_sum_a_b @ F @ A4 ) )
= ( ? [X3: a] :
( ( member_a2 @ X3 @ A4 )
& ( Z2
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_722_image__iff,axiom,
! [Z2: set_Sum_sum_a_b,F: set_Sum_sum_a_b > set_Sum_sum_a_b,A4: set_set_Sum_sum_a_b] :
( ( member4060935254435997939um_a_b @ Z2 @ ( image_7006159782026564799um_a_b @ F @ A4 ) )
= ( ? [X3: set_Sum_sum_a_b] :
( ( member4060935254435997939um_a_b @ X3 @ A4 )
& ( Z2
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_723_image__iff,axiom,
! [Z2: set_a,F: set_a > set_a,A4: set_set_a] :
( ( member_set_a2 @ Z2 @ ( image_set_a_set_a @ F @ A4 ) )
= ( ? [X3: set_a] :
( ( member_set_a2 @ X3 @ A4 )
& ( Z2
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_724_imageI,axiom,
! [X2: a,A4: set_a,F: a > a] :
( ( member_a2 @ X2 @ A4 )
=> ( member_a2 @ ( F @ X2 ) @ ( image_a_a @ F @ A4 ) ) ) ).
% imageI
thf(fact_725_imageI,axiom,
! [X2: a,A4: set_a,F: a > sum_sum_a_b] :
( ( member_a2 @ X2 @ A4 )
=> ( member_Sum_sum_a_b2 @ ( F @ X2 ) @ ( image_a_Sum_sum_a_b @ F @ A4 ) ) ) ).
% imageI
thf(fact_726_imageI,axiom,
! [X2: sum_sum_a_b,A4: set_Sum_sum_a_b,F: sum_sum_a_b > a] :
( ( member_Sum_sum_a_b2 @ X2 @ A4 )
=> ( member_a2 @ ( F @ X2 ) @ ( image_Sum_sum_a_b_a @ F @ A4 ) ) ) ).
% imageI
thf(fact_727_imageI,axiom,
! [X2: sum_sum_a_b,A4: set_Sum_sum_a_b,F: sum_sum_a_b > sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X2 @ A4 )
=> ( member_Sum_sum_a_b2 @ ( F @ X2 ) @ ( image_3358989901043947347um_a_b @ F @ A4 ) ) ) ).
% imageI
thf(fact_728_imageI,axiom,
! [X2: a,A4: set_a,F: a > b] :
( ( member_a2 @ X2 @ A4 )
=> ( member_b2 @ ( F @ X2 ) @ ( image_a_b @ F @ A4 ) ) ) ).
% imageI
thf(fact_729_imageI,axiom,
! [X2: b,A4: set_b,F: b > a] :
( ( member_b2 @ X2 @ A4 )
=> ( member_a2 @ ( F @ X2 ) @ ( image_b_a @ F @ A4 ) ) ) ).
% imageI
thf(fact_730_imageI,axiom,
! [X2: b,A4: set_b,F: b > b] :
( ( member_b2 @ X2 @ A4 )
=> ( member_b2 @ ( F @ X2 ) @ ( image_b_b @ F @ A4 ) ) ) ).
% imageI
thf(fact_731_imageI,axiom,
! [X2: a,A4: set_a,F: a > list_a] :
( ( member_a2 @ X2 @ A4 )
=> ( member_list_a2 @ ( F @ X2 ) @ ( image_a_list_a @ F @ A4 ) ) ) ).
% imageI
thf(fact_732_imageI,axiom,
! [X2: a,A4: set_a,F: a > set_a] :
( ( member_a2 @ X2 @ A4 )
=> ( member_set_a2 @ ( F @ X2 ) @ ( image_a_set_a @ F @ A4 ) ) ) ).
% imageI
thf(fact_733_imageI,axiom,
! [X2: list_a,A4: set_list_a,F: list_a > a] :
( ( member_list_a2 @ X2 @ A4 )
=> ( member_a2 @ ( F @ X2 ) @ ( image_list_a_a @ F @ A4 ) ) ) ).
% imageI
thf(fact_734_image__mono,axiom,
! [A4: set_a,B: set_a,F: a > a] :
( ( ord_less_eq_set_a @ A4 @ B )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A4 ) @ ( image_a_a @ F @ B ) ) ) ).
% image_mono
thf(fact_735_image__mono,axiom,
! [A4: set_a,B: set_a,F: a > set_a] :
( ( ord_less_eq_set_a @ A4 @ B )
=> ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ A4 ) @ ( image_a_set_a @ F @ B ) ) ) ).
% image_mono
thf(fact_736_image__mono,axiom,
! [A4: set_set_a,B: set_set_a,F: set_a > a] :
( ( ord_le3724670747650509150_set_a @ A4 @ B )
=> ( ord_less_eq_set_a @ ( image_set_a_a @ F @ A4 ) @ ( image_set_a_a @ F @ B ) ) ) ).
% image_mono
thf(fact_737_image__mono,axiom,
! [A4: set_b,B: set_b,F: b > sum_sum_a_b] :
( ( ord_less_eq_set_b @ A4 @ B )
=> ( ord_le9019793522827316924um_a_b @ ( image_b_Sum_sum_a_b @ F @ A4 ) @ ( image_b_Sum_sum_a_b @ F @ B ) ) ) ).
% image_mono
thf(fact_738_image__mono,axiom,
! [A4: set_a,B: set_a,F: a > sum_sum_a_b] :
( ( ord_less_eq_set_a @ A4 @ B )
=> ( ord_le9019793522827316924um_a_b @ ( image_a_Sum_sum_a_b @ F @ A4 ) @ ( image_a_Sum_sum_a_b @ F @ B ) ) ) ).
% image_mono
thf(fact_739_image__mono,axiom,
! [A4: set_Sum_sum_a_b,B: set_Sum_sum_a_b,F: sum_sum_a_b > a] :
( ( ord_le9019793522827316924um_a_b @ A4 @ B )
=> ( ord_less_eq_set_a @ ( image_Sum_sum_a_b_a @ F @ A4 ) @ ( image_Sum_sum_a_b_a @ F @ B ) ) ) ).
% image_mono
thf(fact_740_image__mono,axiom,
! [A4: set_set_a,B: set_set_a,F: set_a > set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ B )
=> ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a @ F @ A4 ) @ ( image_set_a_set_a @ F @ B ) ) ) ).
% image_mono
thf(fact_741_image__mono,axiom,
! [A4: set_Sum_sum_a_b,B: set_Sum_sum_a_b,F: sum_sum_a_b > set_a] :
( ( ord_le9019793522827316924um_a_b @ A4 @ B )
=> ( ord_le3724670747650509150_set_a @ ( image_7870623470966840305_set_a @ F @ A4 ) @ ( image_7870623470966840305_set_a @ F @ B ) ) ) ).
% image_mono
thf(fact_742_image__mono,axiom,
! [A4: set_set_a,B: set_set_a,F: set_a > sum_sum_a_b] :
( ( ord_le3724670747650509150_set_a @ A4 @ B )
=> ( ord_le9019793522827316924um_a_b @ ( image_528593881130166975um_a_b @ F @ A4 ) @ ( image_528593881130166975um_a_b @ F @ B ) ) ) ).
% image_mono
thf(fact_743_image__mono,axiom,
! [A4: set_Sum_sum_a_b,B: set_Sum_sum_a_b,F: sum_sum_a_b > sum_sum_a_b] :
( ( ord_le9019793522827316924um_a_b @ A4 @ B )
=> ( ord_le9019793522827316924um_a_b @ ( image_3358989901043947347um_a_b @ F @ A4 ) @ ( image_3358989901043947347um_a_b @ F @ B ) ) ) ).
% image_mono
thf(fact_744_image__subsetI,axiom,
! [A4: set_a,F: a > a,B: set_a] :
( ! [X4: a] :
( ( member_a2 @ X4 @ A4 )
=> ( member_a2 @ ( F @ X4 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A4 ) @ B ) ) ).
% image_subsetI
thf(fact_745_image__subsetI,axiom,
! [A4: set_a,F: a > sum_sum_a_b,B: set_Sum_sum_a_b] :
( ! [X4: a] :
( ( member_a2 @ X4 @ A4 )
=> ( member_Sum_sum_a_b2 @ ( F @ X4 ) @ B ) )
=> ( ord_le9019793522827316924um_a_b @ ( image_a_Sum_sum_a_b @ F @ A4 ) @ B ) ) ).
% image_subsetI
thf(fact_746_image__subsetI,axiom,
! [A4: set_Sum_sum_a_b,F: sum_sum_a_b > a,B: set_a] :
( ! [X4: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X4 @ A4 )
=> ( member_a2 @ ( F @ X4 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_Sum_sum_a_b_a @ F @ A4 ) @ B ) ) ).
% image_subsetI
thf(fact_747_image__subsetI,axiom,
! [A4: set_Sum_sum_a_b,F: sum_sum_a_b > sum_sum_a_b,B: set_Sum_sum_a_b] :
( ! [X4: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X4 @ A4 )
=> ( member_Sum_sum_a_b2 @ ( F @ X4 ) @ B ) )
=> ( ord_le9019793522827316924um_a_b @ ( image_3358989901043947347um_a_b @ F @ A4 ) @ B ) ) ).
% image_subsetI
thf(fact_748_image__subsetI,axiom,
! [A4: set_a,F: a > b,B: set_b] :
( ! [X4: a] :
( ( member_a2 @ X4 @ A4 )
=> ( member_b2 @ ( F @ X4 ) @ B ) )
=> ( ord_less_eq_set_b @ ( image_a_b @ F @ A4 ) @ B ) ) ).
% image_subsetI
thf(fact_749_image__subsetI,axiom,
! [A4: set_b,F: b > b,B: set_b] :
( ! [X4: b] :
( ( member_b2 @ X4 @ A4 )
=> ( member_b2 @ ( F @ X4 ) @ B ) )
=> ( ord_less_eq_set_b @ ( image_b_b @ F @ A4 ) @ B ) ) ).
% image_subsetI
thf(fact_750_image__subsetI,axiom,
! [A4: set_b,F: b > a,B: set_a] :
( ! [X4: b] :
( ( member_b2 @ X4 @ A4 )
=> ( member_a2 @ ( F @ X4 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_b_a @ F @ A4 ) @ B ) ) ).
% image_subsetI
thf(fact_751_image__subsetI,axiom,
! [A4: set_a,F: a > list_a,B: set_list_a] :
( ! [X4: a] :
( ( member_a2 @ X4 @ A4 )
=> ( member_list_a2 @ ( F @ X4 ) @ B ) )
=> ( ord_le8861187494160871172list_a @ ( image_a_list_a @ F @ A4 ) @ B ) ) ).
% image_subsetI
thf(fact_752_image__subsetI,axiom,
! [A4: set_list_a,F: list_a > b,B: set_b] :
( ! [X4: list_a] :
( ( member_list_a2 @ X4 @ A4 )
=> ( member_b2 @ ( F @ X4 ) @ B ) )
=> ( ord_less_eq_set_b @ ( image_list_a_b @ F @ A4 ) @ B ) ) ).
% image_subsetI
thf(fact_753_image__subsetI,axiom,
! [A4: set_set_a,F: set_a > b,B: set_b] :
( ! [X4: set_a] :
( ( member_set_a2 @ X4 @ A4 )
=> ( member_b2 @ ( F @ X4 ) @ B ) )
=> ( ord_less_eq_set_b @ ( image_set_a_b @ F @ A4 ) @ B ) ) ).
% image_subsetI
thf(fact_754_subset__imageE,axiom,
! [B: set_a,F: a > a,A4: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A4 ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A4 )
=> ( B
!= ( image_a_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_755_subset__imageE,axiom,
! [B: set_a,F: set_a > a,A4: set_set_a] :
( ( ord_less_eq_set_a @ B @ ( image_set_a_a @ F @ A4 ) )
=> ~ ! [C3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C3 @ A4 )
=> ( B
!= ( image_set_a_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_756_subset__imageE,axiom,
! [B: set_set_a,F: a > set_a,A4: set_a] :
( ( ord_le3724670747650509150_set_a @ B @ ( image_a_set_a @ F @ A4 ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A4 )
=> ( B
!= ( image_a_set_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_757_subset__imageE,axiom,
! [B: set_a,F: sum_sum_a_b > a,A4: set_Sum_sum_a_b] :
( ( ord_less_eq_set_a @ B @ ( image_Sum_sum_a_b_a @ F @ A4 ) )
=> ~ ! [C3: set_Sum_sum_a_b] :
( ( ord_le9019793522827316924um_a_b @ C3 @ A4 )
=> ( B
!= ( image_Sum_sum_a_b_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_758_subset__imageE,axiom,
! [B: set_Sum_sum_a_b,F: b > sum_sum_a_b,A4: set_b] :
( ( ord_le9019793522827316924um_a_b @ B @ ( image_b_Sum_sum_a_b @ F @ A4 ) )
=> ~ ! [C3: set_b] :
( ( ord_less_eq_set_b @ C3 @ A4 )
=> ( B
!= ( image_b_Sum_sum_a_b @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_759_subset__imageE,axiom,
! [B: set_Sum_sum_a_b,F: a > sum_sum_a_b,A4: set_a] :
( ( ord_le9019793522827316924um_a_b @ B @ ( image_a_Sum_sum_a_b @ F @ A4 ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A4 )
=> ( B
!= ( image_a_Sum_sum_a_b @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_760_subset__imageE,axiom,
! [B: set_set_a,F: set_a > set_a,A4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ ( image_set_a_set_a @ F @ A4 ) )
=> ~ ! [C3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C3 @ A4 )
=> ( B
!= ( image_set_a_set_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_761_subset__imageE,axiom,
! [B: set_Sum_sum_a_b,F: set_a > sum_sum_a_b,A4: set_set_a] :
( ( ord_le9019793522827316924um_a_b @ B @ ( image_528593881130166975um_a_b @ F @ A4 ) )
=> ~ ! [C3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C3 @ A4 )
=> ( B
!= ( image_528593881130166975um_a_b @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_762_subset__imageE,axiom,
! [B: set_set_a,F: sum_sum_a_b > set_a,A4: set_Sum_sum_a_b] :
( ( ord_le3724670747650509150_set_a @ B @ ( image_7870623470966840305_set_a @ F @ A4 ) )
=> ~ ! [C3: set_Sum_sum_a_b] :
( ( ord_le9019793522827316924um_a_b @ C3 @ A4 )
=> ( B
!= ( image_7870623470966840305_set_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_763_subset__imageE,axiom,
! [B: set_Sum_sum_a_b,F: sum_sum_a_b > sum_sum_a_b,A4: set_Sum_sum_a_b] :
( ( ord_le9019793522827316924um_a_b @ B @ ( image_3358989901043947347um_a_b @ F @ A4 ) )
=> ~ ! [C3: set_Sum_sum_a_b] :
( ( ord_le9019793522827316924um_a_b @ C3 @ A4 )
=> ( B
!= ( image_3358989901043947347um_a_b @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_764_image__subset__iff,axiom,
! [F: set_Sum_sum_a_b > set_Sum_sum_a_b,A4: set_set_Sum_sum_a_b,B: set_set_Sum_sum_a_b] :
( ( ord_le1944875106711258738um_a_b @ ( image_7006159782026564799um_a_b @ F @ A4 ) @ B )
= ( ! [X3: set_Sum_sum_a_b] :
( ( member4060935254435997939um_a_b @ X3 @ A4 )
=> ( member4060935254435997939um_a_b @ ( F @ X3 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_765_image__subset__iff,axiom,
! [F: a > a,A4: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( image_a_a @ F @ A4 ) @ B )
= ( ! [X3: a] :
( ( member_a2 @ X3 @ A4 )
=> ( member_a2 @ ( F @ X3 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_766_image__subset__iff,axiom,
! [F: b > sum_sum_a_b,A4: set_b,B: set_Sum_sum_a_b] :
( ( ord_le9019793522827316924um_a_b @ ( image_b_Sum_sum_a_b @ F @ A4 ) @ B )
= ( ! [X3: b] :
( ( member_b2 @ X3 @ A4 )
=> ( member_Sum_sum_a_b2 @ ( F @ X3 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_767_image__subset__iff,axiom,
! [F: a > sum_sum_a_b,A4: set_a,B: set_Sum_sum_a_b] :
( ( ord_le9019793522827316924um_a_b @ ( image_a_Sum_sum_a_b @ F @ A4 ) @ B )
= ( ! [X3: a] :
( ( member_a2 @ X3 @ A4 )
=> ( member_Sum_sum_a_b2 @ ( F @ X3 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_768_image__subset__iff,axiom,
! [F: set_a > set_a,A4: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a @ F @ A4 ) @ B )
= ( ! [X3: set_a] :
( ( member_set_a2 @ X3 @ A4 )
=> ( member_set_a2 @ ( F @ X3 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_769_subset__image__iff,axiom,
! [B: set_Sum_sum_a_b,F: a > sum_sum_a_b,A4: set_a] :
( ( ord_le9019793522827316924um_a_b @ B @ ( image_a_Sum_sum_a_b @ F @ A4 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A4 )
& ( B
= ( image_a_Sum_sum_a_b @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_770_subset__image__iff,axiom,
! [B: set_Sum_sum_a_b,F: sum_sum_a_b > sum_sum_a_b,A4: set_Sum_sum_a_b] :
( ( ord_le9019793522827316924um_a_b @ B @ ( image_3358989901043947347um_a_b @ F @ A4 ) )
= ( ? [AA: set_Sum_sum_a_b] :
( ( ord_le9019793522827316924um_a_b @ AA @ A4 )
& ( B
= ( image_3358989901043947347um_a_b @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_771_subset__image__iff,axiom,
! [B: set_Sum_sum_a_b,F: set_a > sum_sum_a_b,A4: set_set_a] :
( ( ord_le9019793522827316924um_a_b @ B @ ( image_528593881130166975um_a_b @ F @ A4 ) )
= ( ? [AA: set_set_a] :
( ( ord_le3724670747650509150_set_a @ AA @ A4 )
& ( B
= ( image_528593881130166975um_a_b @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_772_subset__image__iff,axiom,
! [B: set_set_a,F: a > set_a,A4: set_a] :
( ( ord_le3724670747650509150_set_a @ B @ ( image_a_set_a @ F @ A4 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A4 )
& ( B
= ( image_a_set_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_773_subset__image__iff,axiom,
! [B: set_set_a,F: sum_sum_a_b > set_a,A4: set_Sum_sum_a_b] :
( ( ord_le3724670747650509150_set_a @ B @ ( image_7870623470966840305_set_a @ F @ A4 ) )
= ( ? [AA: set_Sum_sum_a_b] :
( ( ord_le9019793522827316924um_a_b @ AA @ A4 )
& ( B
= ( image_7870623470966840305_set_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_774_subset__image__iff,axiom,
! [B: set_set_a,F: set_a > set_a,A4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ ( image_set_a_set_a @ F @ A4 ) )
= ( ? [AA: set_set_a] :
( ( ord_le3724670747650509150_set_a @ AA @ A4 )
& ( B
= ( image_set_a_set_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_775_subset__image__iff,axiom,
! [B: set_a,F: a > a,A4: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A4 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A4 )
& ( B
= ( image_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_776_Set__Monad__def,axiom,
set_Se4110677268627701994um_a_b = set_Sum_sum_a_b2 ).
% Set_Monad_def
thf(fact_777_image__vimage__subset,axiom,
! [F: a > sum_sum_a_b,A4: set_Sum_sum_a_b] : ( ord_le9019793522827316924um_a_b @ ( image_a_Sum_sum_a_b @ F @ ( vimage_a_Sum_sum_a_b @ F @ A4 ) ) @ A4 ) ).
% image_vimage_subset
thf(fact_778_image__subset__iff__subset__vimage,axiom,
! [F: a > sum_sum_a_b,A4: set_a,B: set_Sum_sum_a_b] :
( ( ord_le9019793522827316924um_a_b @ ( image_a_Sum_sum_a_b @ F @ A4 ) @ B )
= ( ord_less_eq_set_a @ A4 @ ( vimage_a_Sum_sum_a_b @ F @ B ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_779_image__subset__iff__subset__vimage,axiom,
! [F: a > a,A4: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( image_a_a @ F @ A4 ) @ B )
= ( ord_less_eq_set_a @ A4 @ ( vimage_a_a @ F @ B ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_780_all__subset__image,axiom,
! [F: a > a,A4: set_a,P: set_a > $o] :
( ( ! [B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ ( image_a_a @ F @ A4 ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ A4 )
=> ( P @ ( image_a_a @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_781_empty__Shift,axiom,
! [Kl2: set_list_a,K: a] :
( ( member_list_a2 @ nil_a @ Kl2 )
=> ( ( member_a2 @ K @ ( bNF_Greatest_Succ_a @ Kl2 @ nil_a ) )
=> ( member_list_a2 @ nil_a @ ( bNF_Greatest_Shift_a @ Kl2 @ K ) ) ) ) ).
% empty_Shift
thf(fact_782_empty__Shift,axiom,
! [Kl2: set_list_Sum_sum_a_b,K: sum_sum_a_b] :
( ( member7701661377270014157um_a_b @ nil_Sum_sum_a_b @ Kl2 )
=> ( ( member_Sum_sum_a_b2 @ K @ ( bNF_Gr2307127074836149963um_a_b @ Kl2 @ nil_Sum_sum_a_b ) )
=> ( member7701661377270014157um_a_b @ nil_Sum_sum_a_b @ ( bNF_Gr7310031135006470983um_a_b @ Kl2 @ K ) ) ) ) ).
% empty_Shift
thf(fact_783_subseqs__powset,axiom,
! [Xs: list_Sum_sum_a_b] :
( ( image_3885185050486581529um_a_b @ set_Sum_sum_a_b2 @ ( set_list_Sum_sum_a_b2 @ ( subseqs_Sum_sum_a_b @ Xs ) ) )
= ( pow_Sum_sum_a_b @ ( set_Sum_sum_a_b2 @ Xs ) ) ) ).
% subseqs_powset
thf(fact_784_Pow__iff,axiom,
! [A4: set_a,B: set_a] :
( ( member_set_a2 @ A4 @ ( pow_a @ B ) )
= ( ord_less_eq_set_a @ A4 @ B ) ) ).
% Pow_iff
thf(fact_785_PowI,axiom,
! [A4: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A4 @ B )
=> ( member_set_a2 @ A4 @ ( pow_a @ B ) ) ) ).
% PowI
thf(fact_786_Pow__mono,axiom,
! [A4: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A4 @ B )
=> ( ord_le3724670747650509150_set_a @ ( pow_a @ A4 ) @ ( pow_a @ B ) ) ) ).
% Pow_mono
thf(fact_787_PowD,axiom,
! [A4: set_a,B: set_a] :
( ( member_set_a2 @ A4 @ ( pow_a @ B ) )
=> ( ord_less_eq_set_a @ A4 @ B ) ) ).
% PowD
thf(fact_788_SuccI,axiom,
! [Kl: list_a,K: a,Kl2: set_list_a] :
( ( member_list_a2 @ ( append_a @ Kl @ ( cons_a @ K @ nil_a ) ) @ Kl2 )
=> ( member_a2 @ K @ ( bNF_Greatest_Succ_a @ Kl2 @ Kl ) ) ) ).
% SuccI
thf(fact_789_SuccI,axiom,
! [Kl: list_Sum_sum_a_b,K: sum_sum_a_b,Kl2: set_list_Sum_sum_a_b] :
( ( member7701661377270014157um_a_b @ ( append_Sum_sum_a_b @ Kl @ ( cons_Sum_sum_a_b @ K @ nil_Sum_sum_a_b ) ) @ Kl2 )
=> ( member_Sum_sum_a_b2 @ K @ ( bNF_Gr2307127074836149963um_a_b @ Kl2 @ Kl ) ) ) ).
% SuccI
thf(fact_790_SuccD,axiom,
! [K: a,Kl2: set_list_a,Kl: list_a] :
( ( member_a2 @ K @ ( bNF_Greatest_Succ_a @ Kl2 @ Kl ) )
=> ( member_list_a2 @ ( append_a @ Kl @ ( cons_a @ K @ nil_a ) ) @ Kl2 ) ) ).
% SuccD
thf(fact_791_SuccD,axiom,
! [K: sum_sum_a_b,Kl2: set_list_Sum_sum_a_b,Kl: list_Sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ K @ ( bNF_Gr2307127074836149963um_a_b @ Kl2 @ Kl ) )
=> ( member7701661377270014157um_a_b @ ( append_Sum_sum_a_b @ Kl @ ( cons_Sum_sum_a_b @ K @ nil_Sum_sum_a_b ) ) @ Kl2 ) ) ).
% SuccD
thf(fact_792_split__list,axiom,
! [X2: a,Xs: list_a] :
( ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
=> ? [Ys3: list_a,Zs2: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X2 @ Zs2 ) ) ) ) ).
% split_list
thf(fact_793_split__list,axiom,
! [X2: sum_sum_a_b,Xs: list_Sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X2 @ ( set_Sum_sum_a_b2 @ Xs ) )
=> ? [Ys3: list_Sum_sum_a_b,Zs2: list_Sum_sum_a_b] :
( Xs
= ( append_Sum_sum_a_b @ Ys3 @ ( cons_Sum_sum_a_b @ X2 @ Zs2 ) ) ) ) ).
% split_list
thf(fact_794_split__list__last,axiom,
! [X2: a,Xs: list_a] :
( ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
=> ? [Ys3: list_a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X2 @ Zs2 ) ) )
& ~ ( member_a2 @ X2 @ ( set_a2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_795_split__list__last,axiom,
! [X2: sum_sum_a_b,Xs: list_Sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X2 @ ( set_Sum_sum_a_b2 @ Xs ) )
=> ? [Ys3: list_Sum_sum_a_b,Zs2: list_Sum_sum_a_b] :
( ( Xs
= ( append_Sum_sum_a_b @ Ys3 @ ( cons_Sum_sum_a_b @ X2 @ Zs2 ) ) )
& ~ ( member_Sum_sum_a_b2 @ X2 @ ( set_Sum_sum_a_b2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_796_split__list__prop,axiom,
! [Xs: list_Sum_sum_a_b,P: sum_sum_a_b > $o] :
( ? [X7: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X7 @ ( set_Sum_sum_a_b2 @ Xs ) )
& ( P @ X7 ) )
=> ? [Ys3: list_Sum_sum_a_b,X4: sum_sum_a_b] :
( ? [Zs2: list_Sum_sum_a_b] :
( Xs
= ( append_Sum_sum_a_b @ Ys3 @ ( cons_Sum_sum_a_b @ X4 @ Zs2 ) ) )
& ( P @ X4 ) ) ) ).
% split_list_prop
thf(fact_797_split__list__first,axiom,
! [X2: a,Xs: list_a] :
( ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
=> ? [Ys3: list_a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X2 @ Zs2 ) ) )
& ~ ( member_a2 @ X2 @ ( set_a2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_798_split__list__first,axiom,
! [X2: sum_sum_a_b,Xs: list_Sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X2 @ ( set_Sum_sum_a_b2 @ Xs ) )
=> ? [Ys3: list_Sum_sum_a_b,Zs2: list_Sum_sum_a_b] :
( ( Xs
= ( append_Sum_sum_a_b @ Ys3 @ ( cons_Sum_sum_a_b @ X2 @ Zs2 ) ) )
& ~ ( member_Sum_sum_a_b2 @ X2 @ ( set_Sum_sum_a_b2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_799_split__list__propE,axiom,
! [Xs: list_Sum_sum_a_b,P: sum_sum_a_b > $o] :
( ? [X7: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X7 @ ( set_Sum_sum_a_b2 @ Xs ) )
& ( P @ X7 ) )
=> ~ ! [Ys3: list_Sum_sum_a_b,X4: sum_sum_a_b] :
( ? [Zs2: list_Sum_sum_a_b] :
( Xs
= ( append_Sum_sum_a_b @ Ys3 @ ( cons_Sum_sum_a_b @ X4 @ Zs2 ) ) )
=> ~ ( P @ X4 ) ) ) ).
% split_list_propE
thf(fact_800_append__Cons__eq__iff,axiom,
! [X2: a,Xs: list_a,Ys: list_a,Xs4: list_a,Ys4: list_a] :
( ~ ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
=> ( ~ ( member_a2 @ X2 @ ( set_a2 @ Ys ) )
=> ( ( ( append_a @ Xs @ ( cons_a @ X2 @ Ys ) )
= ( append_a @ Xs4 @ ( cons_a @ X2 @ Ys4 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys4 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_801_append__Cons__eq__iff,axiom,
! [X2: sum_sum_a_b,Xs: list_Sum_sum_a_b,Ys: list_Sum_sum_a_b,Xs4: list_Sum_sum_a_b,Ys4: list_Sum_sum_a_b] :
( ~ ( member_Sum_sum_a_b2 @ X2 @ ( set_Sum_sum_a_b2 @ Xs ) )
=> ( ~ ( member_Sum_sum_a_b2 @ X2 @ ( set_Sum_sum_a_b2 @ Ys ) )
=> ( ( ( append_Sum_sum_a_b @ Xs @ ( cons_Sum_sum_a_b @ X2 @ Ys ) )
= ( append_Sum_sum_a_b @ Xs4 @ ( cons_Sum_sum_a_b @ X2 @ Ys4 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys4 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_802_in__set__conv__decomp,axiom,
! [X2: a,Xs: list_a] :
( ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
= ( ? [Ys2: list_a,Zs3: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_803_in__set__conv__decomp,axiom,
! [X2: sum_sum_a_b,Xs: list_Sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X2 @ ( set_Sum_sum_a_b2 @ Xs ) )
= ( ? [Ys2: list_Sum_sum_a_b,Zs3: list_Sum_sum_a_b] :
( Xs
= ( append_Sum_sum_a_b @ Ys2 @ ( cons_Sum_sum_a_b @ X2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_804_split__list__last__prop,axiom,
! [Xs: list_Sum_sum_a_b,P: sum_sum_a_b > $o] :
( ? [X7: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X7 @ ( set_Sum_sum_a_b2 @ Xs ) )
& ( P @ X7 ) )
=> ? [Ys3: list_Sum_sum_a_b,X4: sum_sum_a_b,Zs2: list_Sum_sum_a_b] :
( ( Xs
= ( append_Sum_sum_a_b @ Ys3 @ ( cons_Sum_sum_a_b @ X4 @ Zs2 ) ) )
& ( P @ X4 )
& ! [Xa2: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ Xa2 @ ( set_Sum_sum_a_b2 @ Zs2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_805_split__list__first__prop,axiom,
! [Xs: list_Sum_sum_a_b,P: sum_sum_a_b > $o] :
( ? [X7: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X7 @ ( set_Sum_sum_a_b2 @ Xs ) )
& ( P @ X7 ) )
=> ? [Ys3: list_Sum_sum_a_b,X4: sum_sum_a_b] :
( ? [Zs2: list_Sum_sum_a_b] :
( Xs
= ( append_Sum_sum_a_b @ Ys3 @ ( cons_Sum_sum_a_b @ X4 @ Zs2 ) ) )
& ( P @ X4 )
& ! [Xa2: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ Xa2 @ ( set_Sum_sum_a_b2 @ Ys3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_806_split__list__last__propE,axiom,
! [Xs: list_Sum_sum_a_b,P: sum_sum_a_b > $o] :
( ? [X7: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X7 @ ( set_Sum_sum_a_b2 @ Xs ) )
& ( P @ X7 ) )
=> ~ ! [Ys3: list_Sum_sum_a_b,X4: sum_sum_a_b,Zs2: list_Sum_sum_a_b] :
( ( Xs
= ( append_Sum_sum_a_b @ Ys3 @ ( cons_Sum_sum_a_b @ X4 @ Zs2 ) ) )
=> ( ( P @ X4 )
=> ~ ! [Xa2: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ Xa2 @ ( set_Sum_sum_a_b2 @ Zs2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_807_split__list__first__propE,axiom,
! [Xs: list_Sum_sum_a_b,P: sum_sum_a_b > $o] :
( ? [X7: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X7 @ ( set_Sum_sum_a_b2 @ Xs ) )
& ( P @ X7 ) )
=> ~ ! [Ys3: list_Sum_sum_a_b,X4: sum_sum_a_b] :
( ? [Zs2: list_Sum_sum_a_b] :
( Xs
= ( append_Sum_sum_a_b @ Ys3 @ ( cons_Sum_sum_a_b @ X4 @ Zs2 ) ) )
=> ( ( P @ X4 )
=> ~ ! [Xa2: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ Xa2 @ ( set_Sum_sum_a_b2 @ Ys3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_808_in__set__conv__decomp__last,axiom,
! [X2: a,Xs: list_a] :
( ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
= ( ? [Ys2: list_a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs3 ) ) )
& ~ ( member_a2 @ X2 @ ( set_a2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_809_in__set__conv__decomp__last,axiom,
! [X2: sum_sum_a_b,Xs: list_Sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X2 @ ( set_Sum_sum_a_b2 @ Xs ) )
= ( ? [Ys2: list_Sum_sum_a_b,Zs3: list_Sum_sum_a_b] :
( ( Xs
= ( append_Sum_sum_a_b @ Ys2 @ ( cons_Sum_sum_a_b @ X2 @ Zs3 ) ) )
& ~ ( member_Sum_sum_a_b2 @ X2 @ ( set_Sum_sum_a_b2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_810_in__set__conv__decomp__first,axiom,
! [X2: a,Xs: list_a] :
( ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
= ( ? [Ys2: list_a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs3 ) ) )
& ~ ( member_a2 @ X2 @ ( set_a2 @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_811_in__set__conv__decomp__first,axiom,
! [X2: sum_sum_a_b,Xs: list_Sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X2 @ ( set_Sum_sum_a_b2 @ Xs ) )
= ( ? [Ys2: list_Sum_sum_a_b,Zs3: list_Sum_sum_a_b] :
( ( Xs
= ( append_Sum_sum_a_b @ Ys2 @ ( cons_Sum_sum_a_b @ X2 @ Zs3 ) ) )
& ~ ( member_Sum_sum_a_b2 @ X2 @ ( set_Sum_sum_a_b2 @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_812_split__list__last__prop__iff,axiom,
! [Xs: list_Sum_sum_a_b,P: sum_sum_a_b > $o] :
( ( ? [X3: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X3 @ ( set_Sum_sum_a_b2 @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys2: list_Sum_sum_a_b,X3: sum_sum_a_b,Zs3: list_Sum_sum_a_b] :
( ( Xs
= ( append_Sum_sum_a_b @ Ys2 @ ( cons_Sum_sum_a_b @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Y5: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ Y5 @ ( set_Sum_sum_a_b2 @ Zs3 ) )
=> ~ ( P @ Y5 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_813_split__list__first__prop__iff,axiom,
! [Xs: list_Sum_sum_a_b,P: sum_sum_a_b > $o] :
( ( ? [X3: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X3 @ ( set_Sum_sum_a_b2 @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys2: list_Sum_sum_a_b,X3: sum_sum_a_b] :
( ? [Zs3: list_Sum_sum_a_b] :
( Xs
= ( append_Sum_sum_a_b @ Ys2 @ ( cons_Sum_sum_a_b @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Y5: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ Y5 @ ( set_Sum_sum_a_b2 @ Ys2 ) )
=> ~ ( P @ Y5 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_814_psubsetI,axiom,
! [A4: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A4 @ B )
=> ( ( A4 != B )
=> ( ord_less_set_a @ A4 @ B ) ) ) ).
% psubsetI
thf(fact_815_leD,axiom,
! [Y: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y @ X2 )
=> ~ ( ord_less_set_a @ X2 @ Y ) ) ).
% leD
thf(fact_816_nless__le,axiom,
! [A: set_a,B2: set_a] :
( ( ~ ( ord_less_set_a @ A @ B2 ) )
= ( ~ ( ord_less_eq_set_a @ A @ B2 )
| ( A = B2 ) ) ) ).
% nless_le
thf(fact_817_antisym__conv1,axiom,
! [X2: set_a,Y: set_a] :
( ~ ( ord_less_set_a @ X2 @ Y )
=> ( ( ord_less_eq_set_a @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_818_antisym__conv2,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ( ~ ( ord_less_set_a @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_819_less__le__not__le,axiom,
( ord_less_set_a
= ( ^ [X3: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y5 )
& ~ ( ord_less_eq_set_a @ Y5 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_820_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_821_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_822_order_Ostrict__trans1,axiom,
! [A: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_823_order_Ostrict__trans2,axiom,
! [A: set_a,B2: set_a,C: set_a] :
( ( ord_less_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_824_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_825_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_826_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_827_dual__order_Ostrict__trans1,axiom,
! [B2: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( ord_less_set_a @ C @ B2 )
=> ( ord_less_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_828_dual__order_Ostrict__trans2,axiom,
! [B2: set_a,A: set_a,C: set_a] :
( ( ord_less_set_a @ B2 @ A )
=> ( ( ord_less_eq_set_a @ C @ B2 )
=> ( ord_less_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_829_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_830_order_Ostrict__implies__order,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_set_a @ A @ B2 )
=> ( ord_less_eq_set_a @ A @ B2 ) ) ).
% order.strict_implies_order
thf(fact_831_dual__order_Ostrict__implies__order,axiom,
! [B2: set_a,A: set_a] :
( ( ord_less_set_a @ B2 @ A )
=> ( ord_less_eq_set_a @ B2 @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_832_order__le__less,axiom,
( ord_less_eq_set_a
= ( ^ [X3: set_a,Y5: set_a] :
( ( ord_less_set_a @ X3 @ Y5 )
| ( X3 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_833_order__less__le,axiom,
( ord_less_set_a
= ( ^ [X3: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y5 )
& ( X3 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_834_order__less__imp__le,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_set_a @ X2 @ Y )
=> ( ord_less_eq_set_a @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_835_order__le__neq__trans,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( A != B2 )
=> ( ord_less_set_a @ A @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_836_order__neq__le__trans,axiom,
! [A: set_a,B2: set_a] :
( ( A != B2 )
=> ( ( ord_less_eq_set_a @ A @ B2 )
=> ( ord_less_set_a @ A @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_837_order__le__less__trans,axiom,
! [X2: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ( ord_less_set_a @ Y @ Z2 )
=> ( ord_less_set_a @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_838_order__less__le__trans,axiom,
! [X2: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_set_a @ X2 @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z2 )
=> ( ord_less_set_a @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_839_order__le__less__subst2,axiom,
! [A: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_set_a @ ( F @ B2 ) @ C )
=> ( ! [X4: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_840_order__less__le__subst1,axiom,
! [A: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( ord_less_set_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X4: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_841_order__le__imp__less__or__eq,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ( ord_less_set_a @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_842_psubsetE,axiom,
! [A4: set_a,B: set_a] :
( ( ord_less_set_a @ A4 @ B )
=> ~ ( ( ord_less_eq_set_a @ A4 @ B )
=> ( ord_less_eq_set_a @ B @ A4 ) ) ) ).
% psubsetE
thf(fact_843_psubset__eq,axiom,
( ord_less_set_a
= ( ^ [A6: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A6 @ B5 )
& ( A6 != B5 ) ) ) ) ).
% psubset_eq
thf(fact_844_psubset__imp__subset,axiom,
! [A4: set_a,B: set_a] :
( ( ord_less_set_a @ A4 @ B )
=> ( ord_less_eq_set_a @ A4 @ B ) ) ).
% psubset_imp_subset
thf(fact_845_psubset__subset__trans,axiom,
! [A4: set_a,B: set_a,C2: set_a] :
( ( ord_less_set_a @ A4 @ B )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_set_a @ A4 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_846_subset__not__subset__eq,axiom,
( ord_less_set_a
= ( ^ [A6: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A6 @ B5 )
& ~ ( ord_less_eq_set_a @ B5 @ A6 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_847_subset__psubset__trans,axiom,
! [A4: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A4 @ B )
=> ( ( ord_less_set_a @ B @ C2 )
=> ( ord_less_set_a @ A4 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_848_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B5: set_a] :
( ( ord_less_set_a @ A6 @ B5 )
| ( A6 = B5 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_849_in__set__butlastD,axiom,
! [X2: a,Xs: list_a] :
( ( member_a2 @ X2 @ ( set_a2 @ ( butlast_a @ Xs ) ) )
=> ( member_a2 @ X2 @ ( set_a2 @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_850_in__set__butlastD,axiom,
! [X2: sum_sum_a_b,Xs: list_Sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X2 @ ( set_Sum_sum_a_b2 @ ( butlast_Sum_sum_a_b @ Xs ) ) )
=> ( member_Sum_sum_a_b2 @ X2 @ ( set_Sum_sum_a_b2 @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_851_in__set__butlast__appendI,axiom,
! [X2: a,Xs: list_a,Ys: list_a] :
( ( ( member_a2 @ X2 @ ( set_a2 @ ( butlast_a @ Xs ) ) )
| ( member_a2 @ X2 @ ( set_a2 @ ( butlast_a @ Ys ) ) ) )
=> ( member_a2 @ X2 @ ( set_a2 @ ( butlast_a @ ( append_a @ Xs @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_852_in__set__butlast__appendI,axiom,
! [X2: sum_sum_a_b,Xs: list_Sum_sum_a_b,Ys: list_Sum_sum_a_b] :
( ( ( member_Sum_sum_a_b2 @ X2 @ ( set_Sum_sum_a_b2 @ ( butlast_Sum_sum_a_b @ Xs ) ) )
| ( member_Sum_sum_a_b2 @ X2 @ ( set_Sum_sum_a_b2 @ ( butlast_Sum_sum_a_b @ Ys ) ) ) )
=> ( member_Sum_sum_a_b2 @ X2 @ ( set_Sum_sum_a_b2 @ ( butlast_Sum_sum_a_b @ ( append_Sum_sum_a_b @ Xs @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_853_psubsetD,axiom,
! [A4: set_a,B: set_a,C: a] :
( ( ord_less_set_a @ A4 @ B )
=> ( ( member_a2 @ C @ A4 )
=> ( member_a2 @ C @ B ) ) ) ).
% psubsetD
thf(fact_854_psubsetD,axiom,
! [A4: set_Sum_sum_a_b,B: set_Sum_sum_a_b,C: sum_sum_a_b] :
( ( ord_le415258996318255280um_a_b @ A4 @ B )
=> ( ( member_Sum_sum_a_b2 @ C @ A4 )
=> ( member_Sum_sum_a_b2 @ C @ B ) ) ) ).
% psubsetD
thf(fact_855_Fpow__mono,axiom,
! [A4: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A4 @ B )
=> ( ord_le3724670747650509150_set_a @ ( finite_Fpow_a @ A4 ) @ ( finite_Fpow_a @ B ) ) ) ).
% Fpow_mono
thf(fact_856_set__rotate1,axiom,
! [Xs: list_Sum_sum_a_b] :
( ( set_Sum_sum_a_b2 @ ( rotate1_Sum_sum_a_b @ Xs ) )
= ( set_Sum_sum_a_b2 @ Xs ) ) ).
% set_rotate1
thf(fact_857_last__in__set,axiom,
! [As2: list_a] :
( ( As2 != nil_a )
=> ( member_a2 @ ( last_a @ As2 ) @ ( set_a2 @ As2 ) ) ) ).
% last_in_set
thf(fact_858_last__in__set,axiom,
! [As2: list_Sum_sum_a_b] :
( ( As2 != nil_Sum_sum_a_b )
=> ( member_Sum_sum_a_b2 @ ( last_Sum_sum_a_b @ As2 ) @ ( set_Sum_sum_a_b2 @ As2 ) ) ) ).
% last_in_set
thf(fact_859_sorted__wrt__mono__rel,axiom,
! [Xs: list_a,P: a > a > $o,Q: a > a > $o] :
( ! [X4: a,Y3: a] :
( ( member_a2 @ X4 @ ( set_a2 @ Xs ) )
=> ( ( member_a2 @ Y3 @ ( set_a2 @ Xs ) )
=> ( ( P @ X4 @ Y3 )
=> ( Q @ X4 @ Y3 ) ) ) )
=> ( ( sorted_wrt_a @ P @ Xs )
=> ( sorted_wrt_a @ Q @ Xs ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_860_sorted__wrt__mono__rel,axiom,
! [Xs: list_Sum_sum_a_b,P: sum_sum_a_b > sum_sum_a_b > $o,Q: sum_sum_a_b > sum_sum_a_b > $o] :
( ! [X4: sum_sum_a_b,Y3: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X4 @ ( set_Sum_sum_a_b2 @ Xs ) )
=> ( ( member_Sum_sum_a_b2 @ Y3 @ ( set_Sum_sum_a_b2 @ Xs ) )
=> ( ( P @ X4 @ Y3 )
=> ( Q @ X4 @ Y3 ) ) ) )
=> ( ( sorted4121554950369191289um_a_b @ P @ Xs )
=> ( sorted4121554950369191289um_a_b @ Q @ Xs ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_861_sorted__wrt__append,axiom,
! [P: sum_sum_a_b > sum_sum_a_b > $o,Xs: list_Sum_sum_a_b,Ys: list_Sum_sum_a_b] :
( ( sorted4121554950369191289um_a_b @ P @ ( append_Sum_sum_a_b @ Xs @ Ys ) )
= ( ( sorted4121554950369191289um_a_b @ P @ Xs )
& ( sorted4121554950369191289um_a_b @ P @ Ys )
& ! [X3: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X3 @ ( set_Sum_sum_a_b2 @ Xs ) )
=> ! [Y5: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ Y5 @ ( set_Sum_sum_a_b2 @ Ys ) )
=> ( P @ X3 @ Y5 ) ) ) ) ) ).
% sorted_wrt_append
thf(fact_862_distinct_Osimps_I2_J,axiom,
! [X2: a,Xs: list_a] :
( ( distinct_a @ ( cons_a @ X2 @ Xs ) )
= ( ~ ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
& ( distinct_a @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_863_distinct_Osimps_I2_J,axiom,
! [X2: sum_sum_a_b,Xs: list_Sum_sum_a_b] :
( ( distinct_Sum_sum_a_b @ ( cons_Sum_sum_a_b @ X2 @ Xs ) )
= ( ~ ( member_Sum_sum_a_b2 @ X2 @ ( set_Sum_sum_a_b2 @ Xs ) )
& ( distinct_Sum_sum_a_b @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_864_not__distinct__conv__prefix,axiom,
! [As2: list_a] :
( ( ~ ( distinct_a @ As2 ) )
= ( ? [Xs2: list_a,Y5: a,Ys2: list_a] :
( ( member_a2 @ Y5 @ ( set_a2 @ Xs2 ) )
& ( distinct_a @ Xs2 )
& ( As2
= ( append_a @ Xs2 @ ( cons_a @ Y5 @ Ys2 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_865_not__distinct__conv__prefix,axiom,
! [As2: list_Sum_sum_a_b] :
( ( ~ ( distinct_Sum_sum_a_b @ As2 ) )
= ( ? [Xs2: list_Sum_sum_a_b,Y5: sum_sum_a_b,Ys2: list_Sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ Y5 @ ( set_Sum_sum_a_b2 @ Xs2 ) )
& ( distinct_Sum_sum_a_b @ Xs2 )
& ( As2
= ( append_Sum_sum_a_b @ Xs2 @ ( cons_Sum_sum_a_b @ Y5 @ Ys2 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_866_is__empty__set,axiom,
! [Xs: list_Sum_sum_a_b] :
( ( is_empty_Sum_sum_a_b @ ( set_Sum_sum_a_b2 @ Xs ) )
= ( null_Sum_sum_a_b @ Xs ) ) ).
% is_empty_set
thf(fact_867_insertCI,axiom,
! [A: a,B: set_a,B2: a] :
( ( ~ ( member_a2 @ A @ B )
=> ( A = B2 ) )
=> ( member_a2 @ A @ ( insert_a2 @ B2 @ B ) ) ) ).
% insertCI
thf(fact_868_insertCI,axiom,
! [A: sum_sum_a_b,B: set_Sum_sum_a_b,B2: sum_sum_a_b] :
( ( ~ ( member_Sum_sum_a_b2 @ A @ B )
=> ( A = B2 ) )
=> ( member_Sum_sum_a_b2 @ A @ ( insert_Sum_sum_a_b2 @ B2 @ B ) ) ) ).
% insertCI
thf(fact_869_insert__iff,axiom,
! [A: a,B2: a,A4: set_a] :
( ( member_a2 @ A @ ( insert_a2 @ B2 @ A4 ) )
= ( ( A = B2 )
| ( member_a2 @ A @ A4 ) ) ) ).
% insert_iff
thf(fact_870_insert__iff,axiom,
! [A: sum_sum_a_b,B2: sum_sum_a_b,A4: set_Sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ A @ ( insert_Sum_sum_a_b2 @ B2 @ A4 ) )
= ( ( A = B2 )
| ( member_Sum_sum_a_b2 @ A @ A4 ) ) ) ).
% insert_iff
thf(fact_871_insert__subset,axiom,
! [X2: sum_sum_a_b,A4: set_Sum_sum_a_b,B: set_Sum_sum_a_b] :
( ( ord_le9019793522827316924um_a_b @ ( insert_Sum_sum_a_b2 @ X2 @ A4 ) @ B )
= ( ( member_Sum_sum_a_b2 @ X2 @ B )
& ( ord_le9019793522827316924um_a_b @ A4 @ B ) ) ) ).
% insert_subset
thf(fact_872_insert__subset,axiom,
! [X2: a,A4: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( insert_a2 @ X2 @ A4 ) @ B )
= ( ( member_a2 @ X2 @ B )
& ( ord_less_eq_set_a @ A4 @ B ) ) ) ).
% insert_subset
thf(fact_873_list_Osimps_I15_J,axiom,
! [X21: sum_sum_a_b,X222: list_Sum_sum_a_b] :
( ( set_Sum_sum_a_b2 @ ( cons_Sum_sum_a_b @ X21 @ X222 ) )
= ( insert_Sum_sum_a_b2 @ X21 @ ( set_Sum_sum_a_b2 @ X222 ) ) ) ).
% list.simps(15)
thf(fact_874_List_Oset__insert,axiom,
! [X2: sum_sum_a_b,Xs: list_Sum_sum_a_b] :
( ( set_Sum_sum_a_b2 @ ( insert_Sum_sum_a_b @ X2 @ Xs ) )
= ( insert_Sum_sum_a_b2 @ X2 @ ( set_Sum_sum_a_b2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_875_insertE,axiom,
! [A: a,B2: a,A4: set_a] :
( ( member_a2 @ A @ ( insert_a2 @ B2 @ A4 ) )
=> ( ( A != B2 )
=> ( member_a2 @ A @ A4 ) ) ) ).
% insertE
thf(fact_876_insertE,axiom,
! [A: sum_sum_a_b,B2: sum_sum_a_b,A4: set_Sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ A @ ( insert_Sum_sum_a_b2 @ B2 @ A4 ) )
=> ( ( A != B2 )
=> ( member_Sum_sum_a_b2 @ A @ A4 ) ) ) ).
% insertE
thf(fact_877_insertI1,axiom,
! [A: a,B: set_a] : ( member_a2 @ A @ ( insert_a2 @ A @ B ) ) ).
% insertI1
thf(fact_878_insertI1,axiom,
! [A: sum_sum_a_b,B: set_Sum_sum_a_b] : ( member_Sum_sum_a_b2 @ A @ ( insert_Sum_sum_a_b2 @ A @ B ) ) ).
% insertI1
thf(fact_879_insertI2,axiom,
! [A: a,B: set_a,B2: a] :
( ( member_a2 @ A @ B )
=> ( member_a2 @ A @ ( insert_a2 @ B2 @ B ) ) ) ).
% insertI2
thf(fact_880_insertI2,axiom,
! [A: sum_sum_a_b,B: set_Sum_sum_a_b,B2: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ A @ B )
=> ( member_Sum_sum_a_b2 @ A @ ( insert_Sum_sum_a_b2 @ B2 @ B ) ) ) ).
% insertI2
thf(fact_881_Set_Oset__insert,axiom,
! [X2: a,A4: set_a] :
( ( member_a2 @ X2 @ A4 )
=> ~ ! [B7: set_a] :
( ( A4
= ( insert_a2 @ X2 @ B7 ) )
=> ( member_a2 @ X2 @ B7 ) ) ) ).
% Set.set_insert
thf(fact_882_Set_Oset__insert,axiom,
! [X2: sum_sum_a_b,A4: set_Sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X2 @ A4 )
=> ~ ! [B7: set_Sum_sum_a_b] :
( ( A4
= ( insert_Sum_sum_a_b2 @ X2 @ B7 ) )
=> ( member_Sum_sum_a_b2 @ X2 @ B7 ) ) ) ).
% Set.set_insert
thf(fact_883_insert__ident,axiom,
! [X2: a,A4: set_a,B: set_a] :
( ~ ( member_a2 @ X2 @ A4 )
=> ( ~ ( member_a2 @ X2 @ B )
=> ( ( ( insert_a2 @ X2 @ A4 )
= ( insert_a2 @ X2 @ B ) )
= ( A4 = B ) ) ) ) ).
% insert_ident
thf(fact_884_insert__ident,axiom,
! [X2: sum_sum_a_b,A4: set_Sum_sum_a_b,B: set_Sum_sum_a_b] :
( ~ ( member_Sum_sum_a_b2 @ X2 @ A4 )
=> ( ~ ( member_Sum_sum_a_b2 @ X2 @ B )
=> ( ( ( insert_Sum_sum_a_b2 @ X2 @ A4 )
= ( insert_Sum_sum_a_b2 @ X2 @ B ) )
= ( A4 = B ) ) ) ) ).
% insert_ident
thf(fact_885_insert__absorb,axiom,
! [A: a,A4: set_a] :
( ( member_a2 @ A @ A4 )
=> ( ( insert_a2 @ A @ A4 )
= A4 ) ) ).
% insert_absorb
thf(fact_886_insert__absorb,axiom,
! [A: sum_sum_a_b,A4: set_Sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ A @ A4 )
=> ( ( insert_Sum_sum_a_b2 @ A @ A4 )
= A4 ) ) ).
% insert_absorb
thf(fact_887_insert__eq__iff,axiom,
! [A: a,A4: set_a,B2: a,B: set_a] :
( ~ ( member_a2 @ A @ A4 )
=> ( ~ ( member_a2 @ B2 @ B )
=> ( ( ( insert_a2 @ A @ A4 )
= ( insert_a2 @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A4 = B ) )
& ( ( A != B2 )
=> ? [C4: set_a] :
( ( A4
= ( insert_a2 @ B2 @ C4 ) )
& ~ ( member_a2 @ B2 @ C4 )
& ( B
= ( insert_a2 @ A @ C4 ) )
& ~ ( member_a2 @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_888_insert__eq__iff,axiom,
! [A: sum_sum_a_b,A4: set_Sum_sum_a_b,B2: sum_sum_a_b,B: set_Sum_sum_a_b] :
( ~ ( member_Sum_sum_a_b2 @ A @ A4 )
=> ( ~ ( member_Sum_sum_a_b2 @ B2 @ B )
=> ( ( ( insert_Sum_sum_a_b2 @ A @ A4 )
= ( insert_Sum_sum_a_b2 @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A4 = B ) )
& ( ( A != B2 )
=> ? [C4: set_Sum_sum_a_b] :
( ( A4
= ( insert_Sum_sum_a_b2 @ B2 @ C4 ) )
& ~ ( member_Sum_sum_a_b2 @ B2 @ C4 )
& ( B
= ( insert_Sum_sum_a_b2 @ A @ C4 ) )
& ~ ( member_Sum_sum_a_b2 @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_889_mk__disjoint__insert,axiom,
! [A: a,A4: set_a] :
( ( member_a2 @ A @ A4 )
=> ? [B7: set_a] :
( ( A4
= ( insert_a2 @ A @ B7 ) )
& ~ ( member_a2 @ A @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_890_mk__disjoint__insert,axiom,
! [A: sum_sum_a_b,A4: set_Sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ A @ A4 )
=> ? [B7: set_Sum_sum_a_b] :
( ( A4
= ( insert_Sum_sum_a_b2 @ A @ B7 ) )
& ~ ( member_Sum_sum_a_b2 @ A @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_891_insert__mono,axiom,
! [C2: set_a,D: set_a,A: a] :
( ( ord_less_eq_set_a @ C2 @ D )
=> ( ord_less_eq_set_a @ ( insert_a2 @ A @ C2 ) @ ( insert_a2 @ A @ D ) ) ) ).
% insert_mono
thf(fact_892_subset__insert,axiom,
! [X2: sum_sum_a_b,A4: set_Sum_sum_a_b,B: set_Sum_sum_a_b] :
( ~ ( member_Sum_sum_a_b2 @ X2 @ A4 )
=> ( ( ord_le9019793522827316924um_a_b @ A4 @ ( insert_Sum_sum_a_b2 @ X2 @ B ) )
= ( ord_le9019793522827316924um_a_b @ A4 @ B ) ) ) ).
% subset_insert
thf(fact_893_subset__insert,axiom,
! [X2: a,A4: set_a,B: set_a] :
( ~ ( member_a2 @ X2 @ A4 )
=> ( ( ord_less_eq_set_a @ A4 @ ( insert_a2 @ X2 @ B ) )
= ( ord_less_eq_set_a @ A4 @ B ) ) ) ).
% subset_insert
thf(fact_894_subset__insertI,axiom,
! [B: set_a,A: a] : ( ord_less_eq_set_a @ B @ ( insert_a2 @ A @ B ) ) ).
% subset_insertI
thf(fact_895_subset__insertI2,axiom,
! [A4: set_a,B: set_a,B2: a] :
( ( ord_less_eq_set_a @ A4 @ B )
=> ( ord_less_eq_set_a @ A4 @ ( insert_a2 @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_896_ord_Oset__quicksort,axiom,
! [Less: sum_sum_a_b > sum_sum_a_b > $o,Xs: list_Sum_sum_a_b] :
( ( set_Sum_sum_a_b2 @ ( set_qu7630589951769768151um_a_b @ Less @ Xs ) )
= ( set_Sum_sum_a_b2 @ Xs ) ) ).
% ord.set_quicksort
thf(fact_897_insert__subsetI,axiom,
! [X2: sum_sum_a_b,A4: set_Sum_sum_a_b,X: set_Sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X2 @ A4 )
=> ( ( ord_le9019793522827316924um_a_b @ X @ A4 )
=> ( ord_le9019793522827316924um_a_b @ ( insert_Sum_sum_a_b2 @ X2 @ X ) @ A4 ) ) ) ).
% insert_subsetI
thf(fact_898_insert__subsetI,axiom,
! [X2: a,A4: set_a,X: set_a] :
( ( member_a2 @ X2 @ A4 )
=> ( ( ord_less_eq_set_a @ X @ A4 )
=> ( ord_less_eq_set_a @ ( insert_a2 @ X2 @ X ) @ A4 ) ) ) ).
% insert_subsetI
thf(fact_899_sorted__wrt_Oelims_I2_J,axiom,
! [X2: sum_sum_a_b > sum_sum_a_b > $o,Xa: list_Sum_sum_a_b] :
( ( sorted4121554950369191289um_a_b @ X2 @ Xa )
=> ( ( Xa != nil_Sum_sum_a_b )
=> ~ ! [X4: sum_sum_a_b,Ys3: list_Sum_sum_a_b] :
( ( Xa
= ( cons_Sum_sum_a_b @ X4 @ Ys3 ) )
=> ~ ( ! [Xa2: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ Xa2 @ ( set_Sum_sum_a_b2 @ Ys3 ) )
=> ( X2 @ X4 @ Xa2 ) )
& ( sorted4121554950369191289um_a_b @ X2 @ Ys3 ) ) ) ) ) ).
% sorted_wrt.elims(2)
thf(fact_900_List_Ofinite__set,axiom,
! [Xs: list_Sum_sum_a_b] : ( finite51705151567313725um_a_b @ ( set_Sum_sum_a_b2 @ Xs ) ) ).
% List.finite_set
thf(fact_901_finite__has__maximal2,axiom,
! [A4: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A4 )
=> ( ( member_set_a2 @ A @ A4 )
=> ? [X4: set_a] :
( ( member_set_a2 @ X4 @ A4 )
& ( ord_less_eq_set_a @ A @ X4 )
& ! [Xa2: set_a] :
( ( member_set_a2 @ Xa2 @ A4 )
=> ( ( ord_less_eq_set_a @ X4 @ Xa2 )
=> ( X4 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_902_finite__has__minimal2,axiom,
! [A4: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A4 )
=> ( ( member_set_a2 @ A @ A4 )
=> ? [X4: set_a] :
( ( member_set_a2 @ X4 @ A4 )
& ( ord_less_eq_set_a @ X4 @ A )
& ! [Xa2: set_a] :
( ( member_set_a2 @ Xa2 @ A4 )
=> ( ( ord_less_eq_set_a @ Xa2 @ X4 )
=> ( X4 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_903_finite__subset,axiom,
! [A4: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A4 @ B )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ A4 ) ) ) ).
% finite_subset
thf(fact_904_infinite__super,axiom,
! [S3: set_a,T2: set_a] :
( ( ord_less_eq_set_a @ S3 @ T2 )
=> ( ~ ( finite_finite_a @ S3 )
=> ~ ( finite_finite_a @ T2 ) ) ) ).
% infinite_super
thf(fact_905_rev__finite__subset,axiom,
! [B: set_a,A4: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A4 @ B )
=> ( finite_finite_a @ A4 ) ) ) ).
% rev_finite_subset
thf(fact_906_finite__list,axiom,
! [A4: set_Sum_sum_a_b] :
( ( finite51705151567313725um_a_b @ A4 )
=> ? [Xs3: list_Sum_sum_a_b] :
( ( set_Sum_sum_a_b2 @ Xs3 )
= A4 ) ) ).
% finite_list
thf(fact_907_pretty__sets,axiom,
set_code_post_set ).
% pretty_sets
thf(fact_908_Ball__def,axiom,
( ball_a
= ( ^ [A6: set_a,P2: a > $o] :
! [X3: a] :
( ( member_a2 @ X3 @ A6 )
=> ( P2 @ X3 ) ) ) ) ).
% Ball_def
thf(fact_909_Ball__def,axiom,
( ball_Sum_sum_a_b
= ( ^ [A6: set_Sum_sum_a_b,P2: sum_sum_a_b > $o] :
! [X3: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X3 @ A6 )
=> ( P2 @ X3 ) ) ) ) ).
% Ball_def
thf(fact_910_in__set__simps_I6_J,axiom,
! [X2: sum_sum_a_b,Y: sum_sum_a_b,Zs: list_Sum_sum_a_b,P: sum_sum_a_b > $o] :
( ( ! [X3: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X3 @ ( set_Sum_sum_a_b2 @ ( cons_Sum_sum_a_b @ X2 @ ( cons_Sum_sum_a_b @ Y @ Zs ) ) ) )
=> ( P @ X3 ) ) )
= ( ( P @ X2 )
& ! [X3: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X3 @ ( set_Sum_sum_a_b2 @ ( cons_Sum_sum_a_b @ Y @ Zs ) ) )
=> ( P @ X3 ) ) ) ) ).
% in_set_simps(6)
thf(fact_911_in__set__simps_I4_J,axiom,
! [P: sum_sum_a_b > $o,X7: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X7 @ ( set_Sum_sum_a_b2 @ nil_Sum_sum_a_b ) )
=> ( P @ X7 ) ) ).
% in_set_simps(4)
thf(fact_912_all__finite__subset__image,axiom,
! [F: a > a,A4: set_a,P: set_a > $o] :
( ( ! [B5: set_a] :
( ( ( finite_finite_a @ B5 )
& ( ord_less_eq_set_a @ B5 @ ( image_a_a @ F @ A4 ) ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_a] :
( ( ( finite_finite_a @ B5 )
& ( ord_less_eq_set_a @ B5 @ A4 ) )
=> ( P @ ( image_a_a @ F @ B5 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_913_ex__finite__subset__image,axiom,
! [F: a > a,A4: set_a,P: set_a > $o] :
( ( ? [B5: set_a] :
( ( finite_finite_a @ B5 )
& ( ord_less_eq_set_a @ B5 @ ( image_a_a @ F @ A4 ) )
& ( P @ B5 ) ) )
= ( ? [B5: set_a] :
( ( finite_finite_a @ B5 )
& ( ord_less_eq_set_a @ B5 @ A4 )
& ( P @ ( image_a_a @ F @ B5 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_914_finite__subset__image,axiom,
! [B: set_a,F: a > a,A4: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A4 ) )
=> ? [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A4 )
& ( finite_finite_a @ C3 )
& ( B
= ( image_a_a @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_915_finite__distinct__list,axiom,
! [A4: set_Sum_sum_a_b] :
( ( finite51705151567313725um_a_b @ A4 )
=> ? [Xs3: list_Sum_sum_a_b] :
( ( ( set_Sum_sum_a_b2 @ Xs3 )
= A4 )
& ( distinct_Sum_sum_a_b @ Xs3 ) ) ) ).
% finite_distinct_list
thf(fact_916_in__set__simps_I5_J,axiom,
! [X2: sum_sum_a_b,P: sum_sum_a_b > $o] :
( ( ! [X3: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X3 @ ( set_Sum_sum_a_b2 @ ( cons_Sum_sum_a_b @ X2 @ nil_Sum_sum_a_b ) ) )
=> ( P @ X3 ) ) )
= ( P @ X2 ) ) ).
% in_set_simps(5)
thf(fact_917_sorted__wrt_Oelims_I3_J,axiom,
! [X2: sum_sum_a_b > sum_sum_a_b > $o,Xa: list_Sum_sum_a_b] :
( ~ ( sorted4121554950369191289um_a_b @ X2 @ Xa )
=> ~ ! [X4: sum_sum_a_b,Ys3: list_Sum_sum_a_b] :
( ( Xa
= ( cons_Sum_sum_a_b @ X4 @ Ys3 ) )
=> ( ! [Xa3: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ Xa3 @ ( set_Sum_sum_a_b2 @ Ys3 ) )
=> ( X2 @ X4 @ Xa3 ) )
& ( sorted4121554950369191289um_a_b @ X2 @ Ys3 ) ) ) ) ).
% sorted_wrt.elims(3)
thf(fact_918_sorted__wrt_Osimps_I2_J,axiom,
! [P: sum_sum_a_b > sum_sum_a_b > $o,X2: sum_sum_a_b,Ys: list_Sum_sum_a_b] :
( ( sorted4121554950369191289um_a_b @ P @ ( cons_Sum_sum_a_b @ X2 @ Ys ) )
= ( ! [X3: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X3 @ ( set_Sum_sum_a_b2 @ Ys ) )
=> ( P @ X2 @ X3 ) )
& ( sorted4121554950369191289um_a_b @ P @ Ys ) ) ) ).
% sorted_wrt.simps(2)
thf(fact_919_sorted__wrt_Oelims_I1_J,axiom,
! [X2: sum_sum_a_b > sum_sum_a_b > $o,Xa: list_Sum_sum_a_b,Y: $o] :
( ( ( sorted4121554950369191289um_a_b @ X2 @ Xa )
= Y )
=> ( ( ( Xa = nil_Sum_sum_a_b )
=> ~ Y )
=> ~ ! [X4: sum_sum_a_b,Ys3: list_Sum_sum_a_b] :
( ( Xa
= ( cons_Sum_sum_a_b @ X4 @ Ys3 ) )
=> ( Y
= ( ~ ( ! [Y5: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ Y5 @ ( set_Sum_sum_a_b2 @ Ys3 ) )
=> ( X2 @ X4 @ Y5 ) )
& ( sorted4121554950369191289um_a_b @ X2 @ Ys3 ) ) ) ) ) ) ) ).
% sorted_wrt.elims(1)
thf(fact_920_Ball__Collect,axiom,
( ball_a
= ( ^ [A6: set_a,P2: a > $o] : ( ord_less_eq_set_a @ A6 @ ( collect_a @ P2 ) ) ) ) ).
% Ball_Collect
thf(fact_921_finite__vimageD_H,axiom,
! [F: a > sum_sum_a_b,A4: set_Sum_sum_a_b] :
( ( finite_finite_a @ ( vimage_a_Sum_sum_a_b @ F @ A4 ) )
=> ( ( ord_le9019793522827316924um_a_b @ A4 @ ( image_a_Sum_sum_a_b @ F @ top_top_set_a ) )
=> ( finite51705151567313725um_a_b @ A4 ) ) ) ).
% finite_vimageD'
thf(fact_922_inf__img__fin__dom,axiom,
! [F: a > sum_sum_a_b,A4: set_a] :
( ( finite51705151567313725um_a_b @ ( image_a_Sum_sum_a_b @ F @ A4 ) )
=> ( ~ ( finite_finite_a @ A4 )
=> ? [X4: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X4 @ ( image_a_Sum_sum_a_b @ F @ A4 ) )
& ~ ( finite_finite_a @ ( vimage_a_Sum_sum_a_b @ F @ ( insert_Sum_sum_a_b2 @ X4 @ bot_bo8744036662862057712um_a_b ) ) ) ) ) ) ).
% inf_img_fin_dom
thf(fact_923_inf__img__fin__domE,axiom,
! [F: a > sum_sum_a_b,A4: set_a] :
( ( finite51705151567313725um_a_b @ ( image_a_Sum_sum_a_b @ F @ A4 ) )
=> ( ~ ( finite_finite_a @ A4 )
=> ~ ! [Y3: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ Y3 @ ( image_a_Sum_sum_a_b @ F @ A4 ) )
=> ( finite_finite_a @ ( vimage_a_Sum_sum_a_b @ F @ ( insert_Sum_sum_a_b2 @ Y3 @ bot_bo8744036662862057712um_a_b ) ) ) ) ) ) ).
% inf_img_fin_domE
thf(fact_924_UNIV__I,axiom,
! [X2: a] : ( member_a2 @ X2 @ top_top_set_a ) ).
% UNIV_I
thf(fact_925_UNIV__I,axiom,
! [X2: sum_sum_a_b] : ( member_Sum_sum_a_b2 @ X2 @ top_to8919940040651885836um_a_b ) ).
% UNIV_I
thf(fact_926_all__not__in__conv,axiom,
! [A4: set_a] :
( ( ! [X3: a] :
~ ( member_a2 @ X3 @ A4 ) )
= ( A4 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_927_all__not__in__conv,axiom,
! [A4: set_Sum_sum_a_b] :
( ( ! [X3: sum_sum_a_b] :
~ ( member_Sum_sum_a_b2 @ X3 @ A4 ) )
= ( A4 = bot_bo8744036662862057712um_a_b ) ) ).
% all_not_in_conv
thf(fact_928_empty__iff,axiom,
! [C: a] :
~ ( member_a2 @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_929_empty__iff,axiom,
! [C: sum_sum_a_b] :
~ ( member_Sum_sum_a_b2 @ C @ bot_bo8744036662862057712um_a_b ) ).
% empty_iff
thf(fact_930_empty__subsetI,axiom,
! [A4: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A4 ) ).
% empty_subsetI
thf(fact_931_subset__empty,axiom,
! [A4: set_a] :
( ( ord_less_eq_set_a @ A4 @ bot_bot_set_a )
= ( A4 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_932_singletonI,axiom,
! [A: a] : ( member_a2 @ A @ ( insert_a2 @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_933_singletonI,axiom,
! [A: sum_sum_a_b] : ( member_Sum_sum_a_b2 @ A @ ( insert_Sum_sum_a_b2 @ A @ bot_bo8744036662862057712um_a_b ) ) ).
% singletonI
thf(fact_934_vimage__UNIV,axiom,
! [F: a > sum_sum_a_b] :
( ( vimage_a_Sum_sum_a_b @ F @ top_to8919940040651885836um_a_b )
= top_top_set_a ) ).
% vimage_UNIV
thf(fact_935_vimage__empty,axiom,
! [F: a > sum_sum_a_b] :
( ( vimage_a_Sum_sum_a_b @ F @ bot_bo8744036662862057712um_a_b )
= bot_bot_set_a ) ).
% vimage_empty
thf(fact_936_List_Oset__empty,axiom,
! [Xs: list_Sum_sum_a_b] :
( ( ( set_Sum_sum_a_b2 @ Xs )
= bot_bo8744036662862057712um_a_b )
= ( Xs = nil_Sum_sum_a_b ) ) ).
% List.set_empty
thf(fact_937_set__empty2,axiom,
! [Xs: list_Sum_sum_a_b] :
( ( bot_bo8744036662862057712um_a_b
= ( set_Sum_sum_a_b2 @ Xs ) )
= ( Xs = nil_Sum_sum_a_b ) ) ).
% set_empty2
thf(fact_938_singleton__insert__inj__eq_H,axiom,
! [A: a,A4: set_a,B2: a] :
( ( ( insert_a2 @ A @ A4 )
= ( insert_a2 @ B2 @ bot_bot_set_a ) )
= ( ( A = B2 )
& ( ord_less_eq_set_a @ A4 @ ( insert_a2 @ B2 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_939_singleton__insert__inj__eq,axiom,
! [B2: a,A: a,A4: set_a] :
( ( ( insert_a2 @ B2 @ bot_bot_set_a )
= ( insert_a2 @ A @ A4 ) )
= ( ( A = B2 )
& ( ord_less_eq_set_a @ A4 @ ( insert_a2 @ B2 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_940_singletonD,axiom,
! [B2: a,A: a] :
( ( member_a2 @ B2 @ ( insert_a2 @ A @ bot_bot_set_a ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_941_singletonD,axiom,
! [B2: sum_sum_a_b,A: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ B2 @ ( insert_Sum_sum_a_b2 @ A @ bot_bo8744036662862057712um_a_b ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_942_singleton__iff,axiom,
! [B2: a,A: a] :
( ( member_a2 @ B2 @ ( insert_a2 @ A @ bot_bot_set_a ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_943_singleton__iff,axiom,
! [B2: sum_sum_a_b,A: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ B2 @ ( insert_Sum_sum_a_b2 @ A @ bot_bo8744036662862057712um_a_b ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_944_bot_Oextremum__uniqueI,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
=> ( A = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_945_bot_Oextremum__unique,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_946_bot_Oextremum,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% bot.extremum
thf(fact_947_top_Oextremum__uniqueI,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A )
=> ( A = top_top_set_a ) ) ).
% top.extremum_uniqueI
thf(fact_948_top_Oextremum__unique,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A )
= ( A = top_top_set_a ) ) ).
% top.extremum_unique
thf(fact_949_top__greatest,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ top_top_set_a ) ).
% top_greatest
thf(fact_950_UNIV__witness,axiom,
? [X4: a] : ( member_a2 @ X4 @ top_top_set_a ) ).
% UNIV_witness
thf(fact_951_UNIV__witness,axiom,
? [X4: sum_sum_a_b] : ( member_Sum_sum_a_b2 @ X4 @ top_to8919940040651885836um_a_b ) ).
% UNIV_witness
thf(fact_952_ex__in__conv,axiom,
! [A4: set_a] :
( ( ? [X3: a] : ( member_a2 @ X3 @ A4 ) )
= ( A4 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_953_ex__in__conv,axiom,
! [A4: set_Sum_sum_a_b] :
( ( ? [X3: sum_sum_a_b] : ( member_Sum_sum_a_b2 @ X3 @ A4 ) )
= ( A4 != bot_bo8744036662862057712um_a_b ) ) ).
% ex_in_conv
thf(fact_954_UNIV__eq__I,axiom,
! [A4: set_a] :
( ! [X4: a] : ( member_a2 @ X4 @ A4 )
=> ( top_top_set_a = A4 ) ) ).
% UNIV_eq_I
thf(fact_955_UNIV__eq__I,axiom,
! [A4: set_Sum_sum_a_b] :
( ! [X4: sum_sum_a_b] : ( member_Sum_sum_a_b2 @ X4 @ A4 )
=> ( top_to8919940040651885836um_a_b = A4 ) ) ).
% UNIV_eq_I
thf(fact_956_equals0I,axiom,
! [A4: set_a] :
( ! [Y3: a] :
~ ( member_a2 @ Y3 @ A4 )
=> ( A4 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_957_equals0I,axiom,
! [A4: set_Sum_sum_a_b] :
( ! [Y3: sum_sum_a_b] :
~ ( member_Sum_sum_a_b2 @ Y3 @ A4 )
=> ( A4 = bot_bo8744036662862057712um_a_b ) ) ).
% equals0I
thf(fact_958_equals0D,axiom,
! [A4: set_a,A: a] :
( ( A4 = bot_bot_set_a )
=> ~ ( member_a2 @ A @ A4 ) ) ).
% equals0D
thf(fact_959_equals0D,axiom,
! [A4: set_Sum_sum_a_b,A: sum_sum_a_b] :
( ( A4 = bot_bo8744036662862057712um_a_b )
=> ~ ( member_Sum_sum_a_b2 @ A @ A4 ) ) ).
% equals0D
thf(fact_960_emptyE,axiom,
! [A: a] :
~ ( member_a2 @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_961_emptyE,axiom,
! [A: sum_sum_a_b] :
~ ( member_Sum_sum_a_b2 @ A @ bot_bo8744036662862057712um_a_b ) ).
% emptyE
thf(fact_962_subset__UNIV,axiom,
! [A4: set_a] : ( ord_less_eq_set_a @ A4 @ top_top_set_a ) ).
% subset_UNIV
thf(fact_963_subset__emptyI,axiom,
! [A4: set_Sum_sum_a_b] :
( ! [X4: sum_sum_a_b] :
~ ( member_Sum_sum_a_b2 @ X4 @ A4 )
=> ( ord_le9019793522827316924um_a_b @ A4 @ bot_bo8744036662862057712um_a_b ) ) ).
% subset_emptyI
thf(fact_964_subset__emptyI,axiom,
! [A4: set_a] :
( ! [X4: a] :
~ ( member_a2 @ X4 @ A4 )
=> ( ord_less_eq_set_a @ A4 @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_965_Pow__set_I1_J,axiom,
( ( pow_Sum_sum_a_b @ ( set_Sum_sum_a_b2 @ nil_Sum_sum_a_b ) )
= ( insert1219653501429402764um_a_b @ bot_bo8744036662862057712um_a_b @ bot_bo1722945631228016806um_a_b ) ) ).
% Pow_set(1)
thf(fact_966_finite__has__minimal,axiom,
! [A4: set_set_a] :
( ( finite_finite_set_a @ A4 )
=> ( ( A4 != bot_bot_set_set_a )
=> ? [X4: set_a] :
( ( member_set_a2 @ X4 @ A4 )
& ! [Xa2: set_a] :
( ( member_set_a2 @ Xa2 @ A4 )
=> ( ( ord_less_eq_set_a @ Xa2 @ X4 )
=> ( X4 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_967_finite__has__maximal,axiom,
! [A4: set_set_a] :
( ( finite_finite_set_a @ A4 )
=> ( ( A4 != bot_bot_set_set_a )
=> ? [X4: set_a] :
( ( member_set_a2 @ X4 @ A4 )
& ! [Xa2: set_a] :
( ( member_set_a2 @ Xa2 @ A4 )
=> ( ( ord_less_eq_set_a @ X4 @ Xa2 )
=> ( X4 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_968_empty__set,axiom,
( bot_bo8744036662862057712um_a_b
= ( set_Sum_sum_a_b2 @ nil_Sum_sum_a_b ) ) ).
% empty_set
thf(fact_969_subset__singleton__iff,axiom,
! [X: set_a,A: a] :
( ( ord_less_eq_set_a @ X @ ( insert_a2 @ A @ bot_bot_set_a ) )
= ( ( X = bot_bot_set_a )
| ( X
= ( insert_a2 @ A @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_970_subset__singletonD,axiom,
! [A4: set_a,X2: a] :
( ( ord_less_eq_set_a @ A4 @ ( insert_a2 @ X2 @ bot_bot_set_a ) )
=> ( ( A4 = bot_bot_set_a )
| ( A4
= ( insert_a2 @ X2 @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_971_vimage__singleton__eq,axiom,
! [A: a,F: a > sum_sum_a_b,B2: sum_sum_a_b] :
( ( member_a2 @ A @ ( vimage_a_Sum_sum_a_b @ F @ ( insert_Sum_sum_a_b2 @ B2 @ bot_bo8744036662862057712um_a_b ) ) )
= ( ( F @ A )
= B2 ) ) ).
% vimage_singleton_eq
thf(fact_972_finite__subset__induct,axiom,
! [F3: set_Sum_sum_a_b,A4: set_Sum_sum_a_b,P: set_Sum_sum_a_b > $o] :
( ( finite51705151567313725um_a_b @ F3 )
=> ( ( ord_le9019793522827316924um_a_b @ F3 @ A4 )
=> ( ( P @ bot_bo8744036662862057712um_a_b )
=> ( ! [A5: sum_sum_a_b,F4: set_Sum_sum_a_b] :
( ( finite51705151567313725um_a_b @ F4 )
=> ( ( member_Sum_sum_a_b2 @ A5 @ A4 )
=> ( ~ ( member_Sum_sum_a_b2 @ A5 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_Sum_sum_a_b2 @ A5 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_973_finite__subset__induct,axiom,
! [F3: set_a,A4: set_a,P: set_a > $o] :
( ( finite_finite_a @ F3 )
=> ( ( ord_less_eq_set_a @ F3 @ A4 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A5: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ( member_a2 @ A5 @ A4 )
=> ( ~ ( member_a2 @ A5 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_a2 @ A5 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_974_finite__subset__induct_H,axiom,
! [F3: set_Sum_sum_a_b,A4: set_Sum_sum_a_b,P: set_Sum_sum_a_b > $o] :
( ( finite51705151567313725um_a_b @ F3 )
=> ( ( ord_le9019793522827316924um_a_b @ F3 @ A4 )
=> ( ( P @ bot_bo8744036662862057712um_a_b )
=> ( ! [A5: sum_sum_a_b,F4: set_Sum_sum_a_b] :
( ( finite51705151567313725um_a_b @ F4 )
=> ( ( member_Sum_sum_a_b2 @ A5 @ A4 )
=> ( ( ord_le9019793522827316924um_a_b @ F4 @ A4 )
=> ( ~ ( member_Sum_sum_a_b2 @ A5 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_Sum_sum_a_b2 @ A5 @ F4 ) ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_975_finite__subset__induct_H,axiom,
! [F3: set_a,A4: set_a,P: set_a > $o] :
( ( finite_finite_a @ F3 )
=> ( ( ord_less_eq_set_a @ F3 @ A4 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A5: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ( member_a2 @ A5 @ A4 )
=> ( ( ord_less_eq_set_a @ F4 @ A4 )
=> ( ~ ( member_a2 @ A5 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_a2 @ A5 @ F4 ) ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_976_finite__vimageD,axiom,
! [H: a > sum_sum_a_b,F3: set_Sum_sum_a_b] :
( ( finite_finite_a @ ( vimage_a_Sum_sum_a_b @ H @ F3 ) )
=> ( ( ( image_a_Sum_sum_a_b @ H @ top_top_set_a )
= top_to8919940040651885836um_a_b )
=> ( finite51705151567313725um_a_b @ F3 ) ) ) ).
% finite_vimageD
thf(fact_977_vimage__subsetD,axiom,
! [F: a > sum_sum_a_b,B: set_Sum_sum_a_b,A4: set_a] :
( ( ( image_a_Sum_sum_a_b @ F @ top_top_set_a )
= top_to8919940040651885836um_a_b )
=> ( ( ord_less_eq_set_a @ ( vimage_a_Sum_sum_a_b @ F @ B ) @ A4 )
=> ( ord_le9019793522827316924um_a_b @ B @ ( image_a_Sum_sum_a_b @ F @ A4 ) ) ) ) ).
% vimage_subsetD
thf(fact_978_vimage__subsetD,axiom,
! [F: a > a,B: set_a,A4: set_a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
=> ( ( ord_less_eq_set_a @ ( vimage_a_a @ F @ B ) @ A4 )
=> ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A4 ) ) ) ) ).
% vimage_subsetD
thf(fact_979_surj__vimage__empty,axiom,
! [F: a > sum_sum_a_b,A4: set_Sum_sum_a_b] :
( ( ( image_a_Sum_sum_a_b @ F @ top_top_set_a )
= top_to8919940040651885836um_a_b )
=> ( ( ( vimage_a_Sum_sum_a_b @ F @ A4 )
= bot_bot_set_a )
= ( A4 = bot_bo8744036662862057712um_a_b ) ) ) ).
% surj_vimage_empty
thf(fact_980_surj__image__vimage__eq,axiom,
! [F: a > sum_sum_a_b,A4: set_Sum_sum_a_b] :
( ( ( image_a_Sum_sum_a_b @ F @ top_top_set_a )
= top_to8919940040651885836um_a_b )
=> ( ( image_a_Sum_sum_a_b @ F @ ( vimage_a_Sum_sum_a_b @ F @ A4 ) )
= A4 ) ) ).
% surj_image_vimage_eq
thf(fact_981_Un__iff,axiom,
! [C: a,A4: set_a,B: set_a] :
( ( member_a2 @ C @ ( sup_sup_set_a @ A4 @ B ) )
= ( ( member_a2 @ C @ A4 )
| ( member_a2 @ C @ B ) ) ) ).
% Un_iff
thf(fact_982_Un__iff,axiom,
! [C: sum_sum_a_b,A4: set_Sum_sum_a_b,B: set_Sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ C @ ( sup_su5708271937873308552um_a_b @ A4 @ B ) )
= ( ( member_Sum_sum_a_b2 @ C @ A4 )
| ( member_Sum_sum_a_b2 @ C @ B ) ) ) ).
% Un_iff
thf(fact_983_UnCI,axiom,
! [C: a,B: set_a,A4: set_a] :
( ( ~ ( member_a2 @ C @ B )
=> ( member_a2 @ C @ A4 ) )
=> ( member_a2 @ C @ ( sup_sup_set_a @ A4 @ B ) ) ) ).
% UnCI
thf(fact_984_UnCI,axiom,
! [C: sum_sum_a_b,B: set_Sum_sum_a_b,A4: set_Sum_sum_a_b] :
( ( ~ ( member_Sum_sum_a_b2 @ C @ B )
=> ( member_Sum_sum_a_b2 @ C @ A4 ) )
=> ( member_Sum_sum_a_b2 @ C @ ( sup_su5708271937873308552um_a_b @ A4 @ B ) ) ) ).
% UnCI
thf(fact_985_Un__subset__iff,axiom,
! [A4: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A4 @ B ) @ C2 )
= ( ( ord_less_eq_set_a @ A4 @ C2 )
& ( ord_less_eq_set_a @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_986_vimage__Un,axiom,
! [F: a > sum_sum_a_b,A4: set_Sum_sum_a_b,B: set_Sum_sum_a_b] :
( ( vimage_a_Sum_sum_a_b @ F @ ( sup_su5708271937873308552um_a_b @ A4 @ B ) )
= ( sup_sup_set_a @ ( vimage_a_Sum_sum_a_b @ F @ A4 ) @ ( vimage_a_Sum_sum_a_b @ F @ B ) ) ) ).
% vimage_Un
thf(fact_987_set__append,axiom,
! [Xs: list_Sum_sum_a_b,Ys: list_Sum_sum_a_b] :
( ( set_Sum_sum_a_b2 @ ( append_Sum_sum_a_b @ Xs @ Ys ) )
= ( sup_su5708271937873308552um_a_b @ ( set_Sum_sum_a_b2 @ Xs ) @ ( set_Sum_sum_a_b2 @ Ys ) ) ) ).
% set_append
thf(fact_988_set__union,axiom,
! [Xs: list_Sum_sum_a_b,Ys: list_Sum_sum_a_b] :
( ( set_Sum_sum_a_b2 @ ( union_Sum_sum_a_b @ Xs @ Ys ) )
= ( sup_su5708271937873308552um_a_b @ ( set_Sum_sum_a_b2 @ Xs ) @ ( set_Sum_sum_a_b2 @ Ys ) ) ) ).
% set_union
thf(fact_989_subset__Un__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B5: set_a] :
( ( sup_sup_set_a @ A6 @ B5 )
= B5 ) ) ) ).
% subset_Un_eq
thf(fact_990_subset__UnE,axiom,
! [C2: set_a,A4: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A4 @ B ) )
=> ~ ! [A7: set_a] :
( ( ord_less_eq_set_a @ A7 @ A4 )
=> ! [B8: set_a] :
( ( ord_less_eq_set_a @ B8 @ B )
=> ( C2
!= ( sup_sup_set_a @ A7 @ B8 ) ) ) ) ) ).
% subset_UnE
thf(fact_991_Un__absorb2,axiom,
! [B: set_a,A4: set_a] :
( ( ord_less_eq_set_a @ B @ A4 )
=> ( ( sup_sup_set_a @ A4 @ B )
= A4 ) ) ).
% Un_absorb2
thf(fact_992_Un__absorb1,axiom,
! [A4: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A4 @ B )
=> ( ( sup_sup_set_a @ A4 @ B )
= B ) ) ).
% Un_absorb1
thf(fact_993_Un__upper2,axiom,
! [B: set_a,A4: set_a] : ( ord_less_eq_set_a @ B @ ( sup_sup_set_a @ A4 @ B ) ) ).
% Un_upper2
thf(fact_994_Un__upper1,axiom,
! [A4: set_a,B: set_a] : ( ord_less_eq_set_a @ A4 @ ( sup_sup_set_a @ A4 @ B ) ) ).
% Un_upper1
thf(fact_995_Un__least,axiom,
! [A4: set_a,C2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A4 @ C2 )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A4 @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_996_Un__mono,axiom,
! [A4: set_a,C2: set_a,B: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A4 @ C2 )
=> ( ( ord_less_eq_set_a @ B @ D )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A4 @ B ) @ ( sup_sup_set_a @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_997_UnI2,axiom,
! [C: a,B: set_a,A4: set_a] :
( ( member_a2 @ C @ B )
=> ( member_a2 @ C @ ( sup_sup_set_a @ A4 @ B ) ) ) ).
% UnI2
thf(fact_998_UnI2,axiom,
! [C: sum_sum_a_b,B: set_Sum_sum_a_b,A4: set_Sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ C @ B )
=> ( member_Sum_sum_a_b2 @ C @ ( sup_su5708271937873308552um_a_b @ A4 @ B ) ) ) ).
% UnI2
thf(fact_999_UnI1,axiom,
! [C: a,A4: set_a,B: set_a] :
( ( member_a2 @ C @ A4 )
=> ( member_a2 @ C @ ( sup_sup_set_a @ A4 @ B ) ) ) ).
% UnI1
thf(fact_1000_UnI1,axiom,
! [C: sum_sum_a_b,A4: set_Sum_sum_a_b,B: set_Sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ C @ A4 )
=> ( member_Sum_sum_a_b2 @ C @ ( sup_su5708271937873308552um_a_b @ A4 @ B ) ) ) ).
% UnI1
thf(fact_1001_UnE,axiom,
! [C: a,A4: set_a,B: set_a] :
( ( member_a2 @ C @ ( sup_sup_set_a @ A4 @ B ) )
=> ( ~ ( member_a2 @ C @ A4 )
=> ( member_a2 @ C @ B ) ) ) ).
% UnE
thf(fact_1002_UnE,axiom,
! [C: sum_sum_a_b,A4: set_Sum_sum_a_b,B: set_Sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ C @ ( sup_su5708271937873308552um_a_b @ A4 @ B ) )
=> ( ~ ( member_Sum_sum_a_b2 @ C @ A4 )
=> ( member_Sum_sum_a_b2 @ C @ B ) ) ) ).
% UnE
thf(fact_1003_ord_Oset__quicksort__acc,axiom,
! [Less: sum_sum_a_b > sum_sum_a_b > $o,Ac: list_Sum_sum_a_b,Xs: list_Sum_sum_a_b] :
( ( set_Sum_sum_a_b2 @ ( set_qu4809963270459086488um_a_b @ Less @ Ac @ Xs ) )
= ( sup_su5708271937873308552um_a_b @ ( set_Sum_sum_a_b2 @ Ac ) @ ( set_Sum_sum_a_b2 @ Xs ) ) ) ).
% ord.set_quicksort_acc
thf(fact_1004_is__singletonI_H,axiom,
! [A4: set_a] :
( ( A4 != bot_bot_set_a )
=> ( ! [X4: a,Y3: a] :
( ( member_a2 @ X4 @ A4 )
=> ( ( member_a2 @ Y3 @ A4 )
=> ( X4 = Y3 ) ) )
=> ( is_singleton_a @ A4 ) ) ) ).
% is_singletonI'
thf(fact_1005_is__singletonI_H,axiom,
! [A4: set_Sum_sum_a_b] :
( ( A4 != bot_bo8744036662862057712um_a_b )
=> ( ! [X4: sum_sum_a_b,Y3: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X4 @ A4 )
=> ( ( member_Sum_sum_a_b2 @ Y3 @ A4 )
=> ( X4 = Y3 ) ) )
=> ( is_sin5238331724580605874um_a_b @ A4 ) ) ) ).
% is_singletonI'
thf(fact_1006_UNIV__sum,axiom,
( top_to8919940040651885836um_a_b
= ( sup_su5708271937873308552um_a_b @ ( image_a_Sum_sum_a_b @ sum_Inl_a_b @ top_top_set_a ) @ ( image_b_Sum_sum_a_b @ sum_Inr_b_a @ top_top_set_b ) ) ) ).
% UNIV_sum
thf(fact_1007_vimage__insert,axiom,
! [F: a > sum_sum_a_b,A: sum_sum_a_b,B: set_Sum_sum_a_b] :
( ( vimage_a_Sum_sum_a_b @ F @ ( insert_Sum_sum_a_b2 @ A @ B ) )
= ( sup_sup_set_a @ ( vimage_a_Sum_sum_a_b @ F @ ( insert_Sum_sum_a_b2 @ A @ bot_bo8744036662862057712um_a_b ) ) @ ( vimage_a_Sum_sum_a_b @ F @ B ) ) ) ).
% vimage_insert
thf(fact_1008_ord_Oset__quicksort__part,axiom,
! [Less: sum_sum_a_b > sum_sum_a_b > $o,Ac: list_Sum_sum_a_b,X2: sum_sum_a_b,Lts: list_Sum_sum_a_b,Eqs: list_Sum_sum_a_b,Gts: list_Sum_sum_a_b,Zs: list_Sum_sum_a_b] :
( ( set_Sum_sum_a_b2 @ ( set_qu1023382790995444362um_a_b @ Less @ Ac @ X2 @ Lts @ Eqs @ Gts @ Zs ) )
= ( sup_su5708271937873308552um_a_b @ ( sup_su5708271937873308552um_a_b @ ( sup_su5708271937873308552um_a_b @ ( sup_su5708271937873308552um_a_b @ ( sup_su5708271937873308552um_a_b @ ( set_Sum_sum_a_b2 @ Ac ) @ ( insert_Sum_sum_a_b2 @ X2 @ bot_bo8744036662862057712um_a_b ) ) @ ( set_Sum_sum_a_b2 @ Lts ) ) @ ( set_Sum_sum_a_b2 @ Eqs ) ) @ ( set_Sum_sum_a_b2 @ Gts ) ) @ ( set_Sum_sum_a_b2 @ Zs ) ) ) ).
% ord.set_quicksort_part
thf(fact_1009_le__sup__iff,axiom,
! [X2: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X2 @ Y ) @ Z2 )
= ( ( ord_less_eq_set_a @ X2 @ Z2 )
& ( ord_less_eq_set_a @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_1010_sup_Obounded__iff,axiom,
! [B2: set_a,C: set_a,A: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C ) @ A )
= ( ( ord_less_eq_set_a @ B2 @ A )
& ( ord_less_eq_set_a @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_1011_distinct__concat,axiom,
! [Xs: list_l4199846171218662726um_a_b] :
( ( distin853461317451161469um_a_b @ Xs )
=> ( ! [Ys3: list_Sum_sum_a_b] :
( ( member7701661377270014157um_a_b @ Ys3 @ ( set_list_Sum_sum_a_b2 @ Xs ) )
=> ( distinct_Sum_sum_a_b @ Ys3 ) )
=> ( ! [Ys3: list_Sum_sum_a_b,Zs2: list_Sum_sum_a_b] :
( ( member7701661377270014157um_a_b @ Ys3 @ ( set_list_Sum_sum_a_b2 @ Xs ) )
=> ( ( member7701661377270014157um_a_b @ Zs2 @ ( set_list_Sum_sum_a_b2 @ Xs ) )
=> ( ( Ys3 != Zs2 )
=> ( ( inf_in4290284198306014446um_a_b @ ( set_Sum_sum_a_b2 @ Ys3 ) @ ( set_Sum_sum_a_b2 @ Zs2 ) )
= bot_bo8744036662862057712um_a_b ) ) ) )
=> ( distinct_Sum_sum_a_b @ ( concat_Sum_sum_a_b @ Xs ) ) ) ) ) ).
% distinct_concat
thf(fact_1012_IntI,axiom,
! [C: a,A4: set_a,B: set_a] :
( ( member_a2 @ C @ A4 )
=> ( ( member_a2 @ C @ B )
=> ( member_a2 @ C @ ( inf_inf_set_a @ A4 @ B ) ) ) ) ).
% IntI
thf(fact_1013_IntI,axiom,
! [C: sum_sum_a_b,A4: set_Sum_sum_a_b,B: set_Sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ C @ A4 )
=> ( ( member_Sum_sum_a_b2 @ C @ B )
=> ( member_Sum_sum_a_b2 @ C @ ( inf_in4290284198306014446um_a_b @ A4 @ B ) ) ) ) ).
% IntI
thf(fact_1014_Int__iff,axiom,
! [C: a,A4: set_a,B: set_a] :
( ( member_a2 @ C @ ( inf_inf_set_a @ A4 @ B ) )
= ( ( member_a2 @ C @ A4 )
& ( member_a2 @ C @ B ) ) ) ).
% Int_iff
thf(fact_1015_Int__iff,axiom,
! [C: sum_sum_a_b,A4: set_Sum_sum_a_b,B: set_Sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ C @ ( inf_in4290284198306014446um_a_b @ A4 @ B ) )
= ( ( member_Sum_sum_a_b2 @ C @ A4 )
& ( member_Sum_sum_a_b2 @ C @ B ) ) ) ).
% Int_iff
thf(fact_1016_inf_Obounded__iff,axiom,
! [A: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B2 @ C ) )
= ( ( ord_less_eq_set_a @ A @ B2 )
& ( ord_less_eq_set_a @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_1017_le__inf__iff,axiom,
! [X2: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X2 @ ( inf_inf_set_a @ Y @ Z2 ) )
= ( ( ord_less_eq_set_a @ X2 @ Y )
& ( ord_less_eq_set_a @ X2 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_1018_Int__subset__iff,axiom,
! [C2: set_a,A4: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A4 @ B ) )
= ( ( ord_less_eq_set_a @ C2 @ A4 )
& ( ord_less_eq_set_a @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_1019_Int__insert__left__if0,axiom,
! [A: a,C2: set_a,B: set_a] :
( ~ ( member_a2 @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a2 @ A @ B ) @ C2 )
= ( inf_inf_set_a @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_1020_Int__insert__left__if0,axiom,
! [A: sum_sum_a_b,C2: set_Sum_sum_a_b,B: set_Sum_sum_a_b] :
( ~ ( member_Sum_sum_a_b2 @ A @ C2 )
=> ( ( inf_in4290284198306014446um_a_b @ ( insert_Sum_sum_a_b2 @ A @ B ) @ C2 )
= ( inf_in4290284198306014446um_a_b @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_1021_Int__insert__left__if1,axiom,
! [A: a,C2: set_a,B: set_a] :
( ( member_a2 @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a2 @ A @ B ) @ C2 )
= ( insert_a2 @ A @ ( inf_inf_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1022_Int__insert__left__if1,axiom,
! [A: sum_sum_a_b,C2: set_Sum_sum_a_b,B: set_Sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ A @ C2 )
=> ( ( inf_in4290284198306014446um_a_b @ ( insert_Sum_sum_a_b2 @ A @ B ) @ C2 )
= ( insert_Sum_sum_a_b2 @ A @ ( inf_in4290284198306014446um_a_b @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1023_Int__insert__right__if0,axiom,
! [A: a,A4: set_a,B: set_a] :
( ~ ( member_a2 @ A @ A4 )
=> ( ( inf_inf_set_a @ A4 @ ( insert_a2 @ A @ B ) )
= ( inf_inf_set_a @ A4 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_1024_Int__insert__right__if0,axiom,
! [A: sum_sum_a_b,A4: set_Sum_sum_a_b,B: set_Sum_sum_a_b] :
( ~ ( member_Sum_sum_a_b2 @ A @ A4 )
=> ( ( inf_in4290284198306014446um_a_b @ A4 @ ( insert_Sum_sum_a_b2 @ A @ B ) )
= ( inf_in4290284198306014446um_a_b @ A4 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_1025_Int__insert__right__if1,axiom,
! [A: a,A4: set_a,B: set_a] :
( ( member_a2 @ A @ A4 )
=> ( ( inf_inf_set_a @ A4 @ ( insert_a2 @ A @ B ) )
= ( insert_a2 @ A @ ( inf_inf_set_a @ A4 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1026_Int__insert__right__if1,axiom,
! [A: sum_sum_a_b,A4: set_Sum_sum_a_b,B: set_Sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ A @ A4 )
=> ( ( inf_in4290284198306014446um_a_b @ A4 @ ( insert_Sum_sum_a_b2 @ A @ B ) )
= ( insert_Sum_sum_a_b2 @ A @ ( inf_in4290284198306014446um_a_b @ A4 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1027_vimage__Int,axiom,
! [F: a > sum_sum_a_b,A4: set_Sum_sum_a_b,B: set_Sum_sum_a_b] :
( ( vimage_a_Sum_sum_a_b @ F @ ( inf_in4290284198306014446um_a_b @ A4 @ B ) )
= ( inf_inf_set_a @ ( vimage_a_Sum_sum_a_b @ F @ A4 ) @ ( vimage_a_Sum_sum_a_b @ F @ B ) ) ) ).
% vimage_Int
thf(fact_1028_insert__disjoint_I1_J,axiom,
! [A: a,A4: set_a,B: set_a] :
( ( ( inf_inf_set_a @ ( insert_a2 @ A @ A4 ) @ B )
= bot_bot_set_a )
= ( ~ ( member_a2 @ A @ B )
& ( ( inf_inf_set_a @ A4 @ B )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_1029_insert__disjoint_I1_J,axiom,
! [A: sum_sum_a_b,A4: set_Sum_sum_a_b,B: set_Sum_sum_a_b] :
( ( ( inf_in4290284198306014446um_a_b @ ( insert_Sum_sum_a_b2 @ A @ A4 ) @ B )
= bot_bo8744036662862057712um_a_b )
= ( ~ ( member_Sum_sum_a_b2 @ A @ B )
& ( ( inf_in4290284198306014446um_a_b @ A4 @ B )
= bot_bo8744036662862057712um_a_b ) ) ) ).
% insert_disjoint(1)
thf(fact_1030_insert__disjoint_I2_J,axiom,
! [A: a,A4: set_a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a2 @ A @ A4 ) @ B ) )
= ( ~ ( member_a2 @ A @ B )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A4 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1031_insert__disjoint_I2_J,axiom,
! [A: sum_sum_a_b,A4: set_Sum_sum_a_b,B: set_Sum_sum_a_b] :
( ( bot_bo8744036662862057712um_a_b
= ( inf_in4290284198306014446um_a_b @ ( insert_Sum_sum_a_b2 @ A @ A4 ) @ B ) )
= ( ~ ( member_Sum_sum_a_b2 @ A @ B )
& ( bot_bo8744036662862057712um_a_b
= ( inf_in4290284198306014446um_a_b @ A4 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1032_disjoint__insert_I1_J,axiom,
! [B: set_a,A: a,A4: set_a] :
( ( ( inf_inf_set_a @ B @ ( insert_a2 @ A @ A4 ) )
= bot_bot_set_a )
= ( ~ ( member_a2 @ A @ B )
& ( ( inf_inf_set_a @ B @ A4 )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_1033_disjoint__insert_I1_J,axiom,
! [B: set_Sum_sum_a_b,A: sum_sum_a_b,A4: set_Sum_sum_a_b] :
( ( ( inf_in4290284198306014446um_a_b @ B @ ( insert_Sum_sum_a_b2 @ A @ A4 ) )
= bot_bo8744036662862057712um_a_b )
= ( ~ ( member_Sum_sum_a_b2 @ A @ B )
& ( ( inf_in4290284198306014446um_a_b @ B @ A4 )
= bot_bo8744036662862057712um_a_b ) ) ) ).
% disjoint_insert(1)
thf(fact_1034_disjoint__insert_I2_J,axiom,
! [A4: set_a,B2: a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A4 @ ( insert_a2 @ B2 @ B ) ) )
= ( ~ ( member_a2 @ B2 @ A4 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A4 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1035_disjoint__insert_I2_J,axiom,
! [A4: set_Sum_sum_a_b,B2: sum_sum_a_b,B: set_Sum_sum_a_b] :
( ( bot_bo8744036662862057712um_a_b
= ( inf_in4290284198306014446um_a_b @ A4 @ ( insert_Sum_sum_a_b2 @ B2 @ B ) ) )
= ( ~ ( member_Sum_sum_a_b2 @ B2 @ A4 )
& ( bot_bo8744036662862057712um_a_b
= ( inf_in4290284198306014446um_a_b @ A4 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1036_image__vimage__eq,axiom,
! [F: a > sum_sum_a_b,A4: set_Sum_sum_a_b] :
( ( image_a_Sum_sum_a_b @ F @ ( vimage_a_Sum_sum_a_b @ F @ A4 ) )
= ( inf_in4290284198306014446um_a_b @ A4 @ ( image_a_Sum_sum_a_b @ F @ top_top_set_a ) ) ) ).
% image_vimage_eq
thf(fact_1037_distinct__append,axiom,
! [Xs: list_Sum_sum_a_b,Ys: list_Sum_sum_a_b] :
( ( distinct_Sum_sum_a_b @ ( append_Sum_sum_a_b @ Xs @ Ys ) )
= ( ( distinct_Sum_sum_a_b @ Xs )
& ( distinct_Sum_sum_a_b @ Ys )
& ( ( inf_in4290284198306014446um_a_b @ ( set_Sum_sum_a_b2 @ Xs ) @ ( set_Sum_sum_a_b2 @ Ys ) )
= bot_bo8744036662862057712um_a_b ) ) ) ).
% distinct_append
thf(fact_1038_disjoint__iff,axiom,
! [A4: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A4 @ B )
= bot_bot_set_a )
= ( ! [X3: a] :
( ( member_a2 @ X3 @ A4 )
=> ~ ( member_a2 @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_1039_disjoint__iff,axiom,
! [A4: set_Sum_sum_a_b,B: set_Sum_sum_a_b] :
( ( ( inf_in4290284198306014446um_a_b @ A4 @ B )
= bot_bo8744036662862057712um_a_b )
= ( ! [X3: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X3 @ A4 )
=> ~ ( member_Sum_sum_a_b2 @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_1040_Int__emptyI,axiom,
! [A4: set_a,B: set_a] :
( ! [X4: a] :
( ( member_a2 @ X4 @ A4 )
=> ~ ( member_a2 @ X4 @ B ) )
=> ( ( inf_inf_set_a @ A4 @ B )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_1041_Int__emptyI,axiom,
! [A4: set_Sum_sum_a_b,B: set_Sum_sum_a_b] :
( ! [X4: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X4 @ A4 )
=> ~ ( member_Sum_sum_a_b2 @ X4 @ B ) )
=> ( ( inf_in4290284198306014446um_a_b @ A4 @ B )
= bot_bo8744036662862057712um_a_b ) ) ).
% Int_emptyI
thf(fact_1042_distrib__inf__le,axiom,
! [X2: set_a,Y: set_a,Z2: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ X2 @ Y ) @ ( inf_inf_set_a @ X2 @ Z2 ) ) @ ( inf_inf_set_a @ X2 @ ( sup_sup_set_a @ Y @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_1043_distrib__sup__le,axiom,
! [X2: set_a,Y: set_a,Z2: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ X2 @ ( inf_inf_set_a @ Y @ Z2 ) ) @ ( inf_inf_set_a @ ( sup_sup_set_a @ X2 @ Y ) @ ( sup_sup_set_a @ X2 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_1044_vimage__inter__cong,axiom,
! [S3: set_a,F: a > sum_sum_a_b,G2: a > sum_sum_a_b,Y: set_Sum_sum_a_b] :
( ! [W: a] :
( ( member_a2 @ W @ S3 )
=> ( ( F @ W )
= ( G2 @ W ) ) )
=> ( ( inf_inf_set_a @ ( vimage_a_Sum_sum_a_b @ F @ Y ) @ S3 )
= ( inf_inf_set_a @ ( vimage_a_Sum_sum_a_b @ G2 @ Y ) @ S3 ) ) ) ).
% vimage_inter_cong
thf(fact_1045_inf_OcoboundedI2,axiom,
! [B2: set_a,C: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_1046_inf_OcoboundedI1,axiom,
! [A: set_a,C: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_1047_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_1048_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_1049_inf_Ocobounded2,axiom,
! [A: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_1050_inf_Ocobounded1,axiom,
! [A: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ A ) ).
% inf.cobounded1
thf(fact_1051_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_1052_inf__greatest,axiom,
! [X2: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ( ord_less_eq_set_a @ X2 @ Z2 )
=> ( ord_less_eq_set_a @ X2 @ ( inf_inf_set_a @ Y @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_1053_inf_OboundedI,axiom,
! [A: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ A @ C )
=> ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_1054_inf_OboundedE,axiom,
! [A: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B2 @ C ) )
=> ~ ( ( ord_less_eq_set_a @ A @ B2 )
=> ~ ( ord_less_eq_set_a @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_1055_inf__absorb2,axiom,
! [Y: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y @ X2 )
=> ( ( inf_inf_set_a @ X2 @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_1056_inf__absorb1,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ( inf_inf_set_a @ X2 @ Y )
= X2 ) ) ).
% inf_absorb1
thf(fact_1057_inf_Oabsorb2,axiom,
! [B2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( inf_inf_set_a @ A @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_1058_inf_Oabsorb1,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( inf_inf_set_a @ A @ B2 )
= A ) ) ).
% inf.absorb1
thf(fact_1059_le__iff__inf,axiom,
( ord_less_eq_set_a
= ( ^ [X3: set_a,Y5: set_a] :
( ( inf_inf_set_a @ X3 @ Y5 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_1060_inf__unique,axiom,
! [F: set_a > set_a > set_a,X2: set_a,Y: set_a] :
( ! [X4: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( F @ X4 @ Y3 ) @ X4 )
=> ( ! [X4: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( F @ X4 @ Y3 ) @ Y3 )
=> ( ! [X4: set_a,Y3: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ( ord_less_eq_set_a @ X4 @ Z3 )
=> ( ord_less_eq_set_a @ X4 @ ( F @ Y3 @ Z3 ) ) ) )
=> ( ( inf_inf_set_a @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_1061_inf_OorderI,axiom,
! [A: set_a,B2: set_a] :
( ( A
= ( inf_inf_set_a @ A @ B2 ) )
=> ( ord_less_eq_set_a @ A @ B2 ) ) ).
% inf.orderI
thf(fact_1062_inf_OorderE,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( A
= ( inf_inf_set_a @ A @ B2 ) ) ) ).
% inf.orderE
thf(fact_1063_le__infI2,axiom,
! [B2: set_a,X2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B2 @ X2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ X2 ) ) ).
% le_infI2
thf(fact_1064_le__infI1,axiom,
! [A: set_a,X2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ X2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ X2 ) ) ).
% le_infI1
thf(fact_1065_inf__mono,axiom,
! [A: set_a,C: set_a,B2: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ B2 @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ ( inf_inf_set_a @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_1066_le__infI,axiom,
! [X2: set_a,A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X2 @ A )
=> ( ( ord_less_eq_set_a @ X2 @ B2 )
=> ( ord_less_eq_set_a @ X2 @ ( inf_inf_set_a @ A @ B2 ) ) ) ) ).
% le_infI
thf(fact_1067_le__infE,axiom,
! [X2: set_a,A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X2 @ ( inf_inf_set_a @ A @ B2 ) )
=> ~ ( ( ord_less_eq_set_a @ X2 @ A )
=> ~ ( ord_less_eq_set_a @ X2 @ B2 ) ) ) ).
% le_infE
thf(fact_1068_inf__le2,axiom,
! [X2: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X2 @ Y ) @ Y ) ).
% inf_le2
thf(fact_1069_inf__le1,axiom,
! [X2: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X2 @ Y ) @ X2 ) ).
% inf_le1
thf(fact_1070_inf__sup__ord_I1_J,axiom,
! [X2: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X2 @ Y ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_1071_inf__sup__ord_I2_J,axiom,
! [X2: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X2 @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_1072_IntE,axiom,
! [C: a,A4: set_a,B: set_a] :
( ( member_a2 @ C @ ( inf_inf_set_a @ A4 @ B ) )
=> ~ ( ( member_a2 @ C @ A4 )
=> ~ ( member_a2 @ C @ B ) ) ) ).
% IntE
thf(fact_1073_IntE,axiom,
! [C: sum_sum_a_b,A4: set_Sum_sum_a_b,B: set_Sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ C @ ( inf_in4290284198306014446um_a_b @ A4 @ B ) )
=> ~ ( ( member_Sum_sum_a_b2 @ C @ A4 )
=> ~ ( member_Sum_sum_a_b2 @ C @ B ) ) ) ).
% IntE
thf(fact_1074_IntD1,axiom,
! [C: a,A4: set_a,B: set_a] :
( ( member_a2 @ C @ ( inf_inf_set_a @ A4 @ B ) )
=> ( member_a2 @ C @ A4 ) ) ).
% IntD1
thf(fact_1075_IntD1,axiom,
! [C: sum_sum_a_b,A4: set_Sum_sum_a_b,B: set_Sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ C @ ( inf_in4290284198306014446um_a_b @ A4 @ B ) )
=> ( member_Sum_sum_a_b2 @ C @ A4 ) ) ).
% IntD1
thf(fact_1076_IntD2,axiom,
! [C: a,A4: set_a,B: set_a] :
( ( member_a2 @ C @ ( inf_inf_set_a @ A4 @ B ) )
=> ( member_a2 @ C @ B ) ) ).
% IntD2
thf(fact_1077_IntD2,axiom,
! [C: sum_sum_a_b,A4: set_Sum_sum_a_b,B: set_Sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ C @ ( inf_in4290284198306014446um_a_b @ A4 @ B ) )
=> ( member_Sum_sum_a_b2 @ C @ B ) ) ).
% IntD2
thf(fact_1078_Int__mono,axiom,
! [A4: set_a,C2: set_a,B: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A4 @ C2 )
=> ( ( ord_less_eq_set_a @ B @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A4 @ B ) @ ( inf_inf_set_a @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_1079_Int__lower1,axiom,
! [A4: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A4 @ B ) @ A4 ) ).
% Int_lower1
thf(fact_1080_Int__lower2,axiom,
! [A4: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A4 @ B ) @ B ) ).
% Int_lower2
thf(fact_1081_Int__absorb1,axiom,
! [B: set_a,A4: set_a] :
( ( ord_less_eq_set_a @ B @ A4 )
=> ( ( inf_inf_set_a @ A4 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_1082_Int__absorb2,axiom,
! [A4: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A4 @ B )
=> ( ( inf_inf_set_a @ A4 @ B )
= A4 ) ) ).
% Int_absorb2
thf(fact_1083_Int__greatest,axiom,
! [C2: set_a,A4: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C2 @ A4 )
=> ( ( ord_less_eq_set_a @ C2 @ B )
=> ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A4 @ B ) ) ) ) ).
% Int_greatest
thf(fact_1084_Int__Collect__mono,axiom,
! [A4: set_Sum_sum_a_b,B: set_Sum_sum_a_b,P: sum_sum_a_b > $o,Q: sum_sum_a_b > $o] :
( ( ord_le9019793522827316924um_a_b @ A4 @ B )
=> ( ! [X4: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X4 @ A4 )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_le9019793522827316924um_a_b @ ( inf_in4290284198306014446um_a_b @ A4 @ ( collect_Sum_sum_a_b @ P ) ) @ ( inf_in4290284198306014446um_a_b @ B @ ( collect_Sum_sum_a_b @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1085_Int__Collect__mono,axiom,
! [A4: set_a,B: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A4 @ B )
=> ( ! [X4: a] :
( ( member_a2 @ X4 @ A4 )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A4 @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1086_Int__insert__left,axiom,
! [A: a,C2: set_a,B: set_a] :
( ( ( member_a2 @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a2 @ A @ B ) @ C2 )
= ( insert_a2 @ A @ ( inf_inf_set_a @ B @ C2 ) ) ) )
& ( ~ ( member_a2 @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a2 @ A @ B ) @ C2 )
= ( inf_inf_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_1087_Int__insert__left,axiom,
! [A: sum_sum_a_b,C2: set_Sum_sum_a_b,B: set_Sum_sum_a_b] :
( ( ( member_Sum_sum_a_b2 @ A @ C2 )
=> ( ( inf_in4290284198306014446um_a_b @ ( insert_Sum_sum_a_b2 @ A @ B ) @ C2 )
= ( insert_Sum_sum_a_b2 @ A @ ( inf_in4290284198306014446um_a_b @ B @ C2 ) ) ) )
& ( ~ ( member_Sum_sum_a_b2 @ A @ C2 )
=> ( ( inf_in4290284198306014446um_a_b @ ( insert_Sum_sum_a_b2 @ A @ B ) @ C2 )
= ( inf_in4290284198306014446um_a_b @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_1088_Int__insert__right,axiom,
! [A: a,A4: set_a,B: set_a] :
( ( ( member_a2 @ A @ A4 )
=> ( ( inf_inf_set_a @ A4 @ ( insert_a2 @ A @ B ) )
= ( insert_a2 @ A @ ( inf_inf_set_a @ A4 @ B ) ) ) )
& ( ~ ( member_a2 @ A @ A4 )
=> ( ( inf_inf_set_a @ A4 @ ( insert_a2 @ A @ B ) )
= ( inf_inf_set_a @ A4 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_1089_Int__insert__right,axiom,
! [A: sum_sum_a_b,A4: set_Sum_sum_a_b,B: set_Sum_sum_a_b] :
( ( ( member_Sum_sum_a_b2 @ A @ A4 )
=> ( ( inf_in4290284198306014446um_a_b @ A4 @ ( insert_Sum_sum_a_b2 @ A @ B ) )
= ( insert_Sum_sum_a_b2 @ A @ ( inf_in4290284198306014446um_a_b @ A4 @ B ) ) ) )
& ( ~ ( member_Sum_sum_a_b2 @ A @ A4 )
=> ( ( inf_in4290284198306014446um_a_b @ A4 @ ( insert_Sum_sum_a_b2 @ A @ B ) )
= ( inf_in4290284198306014446um_a_b @ A4 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_1090_Un__Int__assoc__eq,axiom,
! [A4: set_a,B: set_a,C2: set_a] :
( ( ( sup_sup_set_a @ ( inf_inf_set_a @ A4 @ B ) @ C2 )
= ( inf_inf_set_a @ A4 @ ( sup_sup_set_a @ B @ C2 ) ) )
= ( ord_less_eq_set_a @ C2 @ A4 ) ) ).
% Un_Int_assoc_eq
thf(fact_1091_finite__finite__vimage__IntI,axiom,
! [F3: set_Sum_sum_a_b,H: a > sum_sum_a_b,A4: set_a] :
( ( finite51705151567313725um_a_b @ F3 )
=> ( ! [Y3: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ Y3 @ F3 )
=> ( finite_finite_a @ ( inf_inf_set_a @ ( vimage_a_Sum_sum_a_b @ H @ ( insert_Sum_sum_a_b2 @ Y3 @ bot_bo8744036662862057712um_a_b ) ) @ A4 ) ) )
=> ( finite_finite_a @ ( inf_inf_set_a @ ( vimage_a_Sum_sum_a_b @ H @ F3 ) @ A4 ) ) ) ) ).
% finite_finite_vimage_IntI
thf(fact_1092_inf__img__fin__domE_H,axiom,
! [F: a > sum_sum_a_b,A4: set_a] :
( ( finite51705151567313725um_a_b @ ( image_a_Sum_sum_a_b @ F @ A4 ) )
=> ( ~ ( finite_finite_a @ A4 )
=> ~ ! [Y3: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ Y3 @ ( image_a_Sum_sum_a_b @ F @ A4 ) )
=> ( finite_finite_a @ ( inf_inf_set_a @ ( vimage_a_Sum_sum_a_b @ F @ ( insert_Sum_sum_a_b2 @ Y3 @ bot_bo8744036662862057712um_a_b ) ) @ A4 ) ) ) ) ) ).
% inf_img_fin_domE'
thf(fact_1093_inf__img__fin__dom_H,axiom,
! [F: a > sum_sum_a_b,A4: set_a] :
( ( finite51705151567313725um_a_b @ ( image_a_Sum_sum_a_b @ F @ A4 ) )
=> ( ~ ( finite_finite_a @ A4 )
=> ? [X4: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X4 @ ( image_a_Sum_sum_a_b @ F @ A4 ) )
& ~ ( finite_finite_a @ ( inf_inf_set_a @ ( vimage_a_Sum_sum_a_b @ F @ ( insert_Sum_sum_a_b2 @ X4 @ bot_bo8744036662862057712um_a_b ) ) @ A4 ) ) ) ) ) ).
% inf_img_fin_dom'
thf(fact_1094_sup_OcoboundedI2,axiom,
! [C: set_a,B2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ C @ B2 )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_1095_sup_OcoboundedI1,axiom,
! [C: set_a,A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C @ A )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_1096_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_1097_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_1098_sup_Ocobounded2,axiom,
! [B2: set_a,A: set_a] : ( ord_less_eq_set_a @ B2 @ ( sup_sup_set_a @ A @ B2 ) ) ).
% sup.cobounded2
thf(fact_1099_sup_Ocobounded1,axiom,
! [A: set_a,B2: set_a] : ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ A @ B2 ) ) ).
% sup.cobounded1
thf(fact_1100_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_1101_sup_OboundedI,axiom,
! [B2: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( ord_less_eq_set_a @ C @ A )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_1102_sup_OboundedE,axiom,
! [B2: set_a,C: set_a,A: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C ) @ A )
=> ~ ( ( ord_less_eq_set_a @ B2 @ A )
=> ~ ( ord_less_eq_set_a @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_1103_sup__absorb2,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ( sup_sup_set_a @ X2 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_1104_sup__absorb1,axiom,
! [Y: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y @ X2 )
=> ( ( sup_sup_set_a @ X2 @ Y )
= X2 ) ) ).
% sup_absorb1
thf(fact_1105_sup_Oabsorb2,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( sup_sup_set_a @ A @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_1106_sup_Oabsorb1,axiom,
! [B2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( sup_sup_set_a @ A @ B2 )
= A ) ) ).
% sup.absorb1
thf(fact_1107_sup__unique,axiom,
! [F: set_a > set_a > set_a,X2: set_a,Y: set_a] :
( ! [X4: set_a,Y3: set_a] : ( ord_less_eq_set_a @ X4 @ ( F @ X4 @ Y3 ) )
=> ( ! [X4: set_a,Y3: set_a] : ( ord_less_eq_set_a @ Y3 @ ( F @ X4 @ Y3 ) )
=> ( ! [X4: set_a,Y3: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X4 )
=> ( ( ord_less_eq_set_a @ Z3 @ X4 )
=> ( ord_less_eq_set_a @ ( F @ Y3 @ Z3 ) @ X4 ) ) )
=> ( ( sup_sup_set_a @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_1108_sup_OorderI,axiom,
! [A: set_a,B2: set_a] :
( ( A
= ( sup_sup_set_a @ A @ B2 ) )
=> ( ord_less_eq_set_a @ B2 @ A ) ) ).
% sup.orderI
thf(fact_1109_sup_OorderE,axiom,
! [B2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( A
= ( sup_sup_set_a @ A @ B2 ) ) ) ).
% sup.orderE
thf(fact_1110_le__iff__sup,axiom,
( ord_less_eq_set_a
= ( ^ [X3: set_a,Y5: set_a] :
( ( sup_sup_set_a @ X3 @ Y5 )
= Y5 ) ) ) ).
% le_iff_sup
thf(fact_1111_sup__least,axiom,
! [Y: set_a,X2: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ Y @ X2 )
=> ( ( ord_less_eq_set_a @ Z2 @ X2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ Y @ Z2 ) @ X2 ) ) ) ).
% sup_least
thf(fact_1112_sup__mono,axiom,
! [A: set_a,C: set_a,B2: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ B2 @ D2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B2 ) @ ( sup_sup_set_a @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_1113_sup_Omono,axiom,
! [C: set_a,A: set_a,D2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C @ A )
=> ( ( ord_less_eq_set_a @ D2 @ B2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ C @ D2 ) @ ( sup_sup_set_a @ A @ B2 ) ) ) ) ).
% sup.mono
thf(fact_1114_le__supI2,axiom,
! [X2: set_a,B2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ X2 @ B2 )
=> ( ord_less_eq_set_a @ X2 @ ( sup_sup_set_a @ A @ B2 ) ) ) ).
% le_supI2
thf(fact_1115_le__supI1,axiom,
! [X2: set_a,A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X2 @ A )
=> ( ord_less_eq_set_a @ X2 @ ( sup_sup_set_a @ A @ B2 ) ) ) ).
% le_supI1
thf(fact_1116_sup__ge2,axiom,
! [Y: set_a,X2: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X2 @ Y ) ) ).
% sup_ge2
thf(fact_1117_sup__ge1,axiom,
! [X2: set_a,Y: set_a] : ( ord_less_eq_set_a @ X2 @ ( sup_sup_set_a @ X2 @ Y ) ) ).
% sup_ge1
thf(fact_1118_le__supI,axiom,
! [A: set_a,X2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ X2 )
=> ( ( ord_less_eq_set_a @ B2 @ X2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B2 ) @ X2 ) ) ) ).
% le_supI
thf(fact_1119_le__supE,axiom,
! [A: set_a,B2: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B2 ) @ X2 )
=> ~ ( ( ord_less_eq_set_a @ A @ X2 )
=> ~ ( ord_less_eq_set_a @ B2 @ X2 ) ) ) ).
% le_supE
thf(fact_1120_inf__sup__ord_I3_J,axiom,
! [X2: set_a,Y: set_a] : ( ord_less_eq_set_a @ X2 @ ( sup_sup_set_a @ X2 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_1121_inf__sup__ord_I4_J,axiom,
! [Y: set_a,X2: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X2 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_1122_distinct__concat__iff,axiom,
! [Xs: list_l4199846171218662726um_a_b] :
( ( distinct_Sum_sum_a_b @ ( concat_Sum_sum_a_b @ Xs ) )
= ( ( distin853461317451161469um_a_b @ ( remove4044100378980913834um_a_b @ nil_Sum_sum_a_b @ Xs ) )
& ! [Ys2: list_Sum_sum_a_b] :
( ( member7701661377270014157um_a_b @ Ys2 @ ( set_list_Sum_sum_a_b2 @ Xs ) )
=> ( distinct_Sum_sum_a_b @ Ys2 ) )
& ! [Ys2: list_Sum_sum_a_b,Zs3: list_Sum_sum_a_b] :
( ( ( member7701661377270014157um_a_b @ Ys2 @ ( set_list_Sum_sum_a_b2 @ Xs ) )
& ( member7701661377270014157um_a_b @ Zs3 @ ( set_list_Sum_sum_a_b2 @ Xs ) )
& ( Ys2 != Zs3 ) )
=> ( ( inf_in4290284198306014446um_a_b @ ( set_Sum_sum_a_b2 @ Ys2 ) @ ( set_Sum_sum_a_b2 @ Zs3 ) )
= bot_bo8744036662862057712um_a_b ) ) ) ) ).
% distinct_concat_iff
thf(fact_1123_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_1124_removeAll__id,axiom,
! [X2: a,Xs: list_a] :
( ~ ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
=> ( ( removeAll_a @ X2 @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_1125_removeAll__id,axiom,
! [X2: sum_sum_a_b,Xs: list_Sum_sum_a_b] :
( ~ ( member_Sum_sum_a_b2 @ X2 @ ( set_Sum_sum_a_b2 @ Xs ) )
=> ( ( remove4403382267521460890um_a_b @ X2 @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_1126_remove__code_I1_J,axiom,
! [X2: sum_sum_a_b,Xs: list_Sum_sum_a_b] :
( ( remove_Sum_sum_a_b @ X2 @ ( set_Sum_sum_a_b2 @ Xs ) )
= ( set_Sum_sum_a_b2 @ ( remove4403382267521460890um_a_b @ X2 @ Xs ) ) ) ).
% remove_code(1)
thf(fact_1127_chains__extend,axiom,
! [C: set_set_a,S3: set_set_a,Z2: set_a] :
( ( member_set_set_a2 @ C @ ( chains_a @ S3 ) )
=> ( ( member_set_a2 @ Z2 @ S3 )
=> ( ! [X4: set_a] :
( ( member_set_a2 @ X4 @ C )
=> ( ord_less_eq_set_a @ X4 @ Z2 ) )
=> ( member_set_set_a2 @ ( sup_sup_set_set_a @ ( insert_set_a2 @ Z2 @ bot_bot_set_set_a ) @ C ) @ ( chains_a @ S3 ) ) ) ) ) ).
% chains_extend
thf(fact_1128_DiffI,axiom,
! [C: a,A4: set_a,B: set_a] :
( ( member_a2 @ C @ A4 )
=> ( ~ ( member_a2 @ C @ B )
=> ( member_a2 @ C @ ( minus_minus_set_a @ A4 @ B ) ) ) ) ).
% DiffI
thf(fact_1129_DiffI,axiom,
! [C: sum_sum_a_b,A4: set_Sum_sum_a_b,B: set_Sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ C @ A4 )
=> ( ~ ( member_Sum_sum_a_b2 @ C @ B )
=> ( member_Sum_sum_a_b2 @ C @ ( minus_380166324194722613um_a_b @ A4 @ B ) ) ) ) ).
% DiffI
thf(fact_1130_Diff__iff,axiom,
! [C: a,A4: set_a,B: set_a] :
( ( member_a2 @ C @ ( minus_minus_set_a @ A4 @ B ) )
= ( ( member_a2 @ C @ A4 )
& ~ ( member_a2 @ C @ B ) ) ) ).
% Diff_iff
thf(fact_1131_Diff__iff,axiom,
! [C: sum_sum_a_b,A4: set_Sum_sum_a_b,B: set_Sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ C @ ( minus_380166324194722613um_a_b @ A4 @ B ) )
= ( ( member_Sum_sum_a_b2 @ C @ A4 )
& ~ ( member_Sum_sum_a_b2 @ C @ B ) ) ) ).
% Diff_iff
thf(fact_1132_Diff__insert0,axiom,
! [X2: a,A4: set_a,B: set_a] :
( ~ ( member_a2 @ X2 @ A4 )
=> ( ( minus_minus_set_a @ A4 @ ( insert_a2 @ X2 @ B ) )
= ( minus_minus_set_a @ A4 @ B ) ) ) ).
% Diff_insert0
thf(fact_1133_Diff__insert0,axiom,
! [X2: sum_sum_a_b,A4: set_Sum_sum_a_b,B: set_Sum_sum_a_b] :
( ~ ( member_Sum_sum_a_b2 @ X2 @ A4 )
=> ( ( minus_380166324194722613um_a_b @ A4 @ ( insert_Sum_sum_a_b2 @ X2 @ B ) )
= ( minus_380166324194722613um_a_b @ A4 @ B ) ) ) ).
% Diff_insert0
thf(fact_1134_insert__Diff1,axiom,
! [X2: a,B: set_a,A4: set_a] :
( ( member_a2 @ X2 @ B )
=> ( ( minus_minus_set_a @ ( insert_a2 @ X2 @ A4 ) @ B )
= ( minus_minus_set_a @ A4 @ B ) ) ) ).
% insert_Diff1
thf(fact_1135_insert__Diff1,axiom,
! [X2: sum_sum_a_b,B: set_Sum_sum_a_b,A4: set_Sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X2 @ B )
=> ( ( minus_380166324194722613um_a_b @ ( insert_Sum_sum_a_b2 @ X2 @ A4 ) @ B )
= ( minus_380166324194722613um_a_b @ A4 @ B ) ) ) ).
% insert_Diff1
thf(fact_1136_Diff__eq__empty__iff,axiom,
! [A4: set_a,B: set_a] :
( ( ( minus_minus_set_a @ A4 @ B )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A4 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_1137_set__removeAll,axiom,
! [X2: sum_sum_a_b,Xs: list_Sum_sum_a_b] :
( ( set_Sum_sum_a_b2 @ ( remove4403382267521460890um_a_b @ X2 @ Xs ) )
= ( minus_380166324194722613um_a_b @ ( set_Sum_sum_a_b2 @ Xs ) @ ( insert_Sum_sum_a_b2 @ X2 @ bot_bo8744036662862057712um_a_b ) ) ) ).
% set_removeAll
thf(fact_1138_insert__Diff,axiom,
! [A: a,A4: set_a] :
( ( member_a2 @ A @ A4 )
=> ( ( insert_a2 @ A @ ( minus_minus_set_a @ A4 @ ( insert_a2 @ A @ bot_bot_set_a ) ) )
= A4 ) ) ).
% insert_Diff
thf(fact_1139_insert__Diff,axiom,
! [A: sum_sum_a_b,A4: set_Sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ A @ A4 )
=> ( ( insert_Sum_sum_a_b2 @ A @ ( minus_380166324194722613um_a_b @ A4 @ ( insert_Sum_sum_a_b2 @ A @ bot_bo8744036662862057712um_a_b ) ) )
= A4 ) ) ).
% insert_Diff
thf(fact_1140_Diff__insert__absorb,axiom,
! [X2: a,A4: set_a] :
( ~ ( member_a2 @ X2 @ A4 )
=> ( ( minus_minus_set_a @ ( insert_a2 @ X2 @ A4 ) @ ( insert_a2 @ X2 @ bot_bot_set_a ) )
= A4 ) ) ).
% Diff_insert_absorb
thf(fact_1141_Diff__insert__absorb,axiom,
! [X2: sum_sum_a_b,A4: set_Sum_sum_a_b] :
( ~ ( member_Sum_sum_a_b2 @ X2 @ A4 )
=> ( ( minus_380166324194722613um_a_b @ ( insert_Sum_sum_a_b2 @ X2 @ A4 ) @ ( insert_Sum_sum_a_b2 @ X2 @ bot_bo8744036662862057712um_a_b ) )
= A4 ) ) ).
% Diff_insert_absorb
thf(fact_1142_subset__Diff__insert,axiom,
! [A4: set_Sum_sum_a_b,B: set_Sum_sum_a_b,X2: sum_sum_a_b,C2: set_Sum_sum_a_b] :
( ( ord_le9019793522827316924um_a_b @ A4 @ ( minus_380166324194722613um_a_b @ B @ ( insert_Sum_sum_a_b2 @ X2 @ C2 ) ) )
= ( ( ord_le9019793522827316924um_a_b @ A4 @ ( minus_380166324194722613um_a_b @ B @ C2 ) )
& ~ ( member_Sum_sum_a_b2 @ X2 @ A4 ) ) ) ).
% subset_Diff_insert
thf(fact_1143_subset__Diff__insert,axiom,
! [A4: set_a,B: set_a,X2: a,C2: set_a] :
( ( ord_less_eq_set_a @ A4 @ ( minus_minus_set_a @ B @ ( insert_a2 @ X2 @ C2 ) ) )
= ( ( ord_less_eq_set_a @ A4 @ ( minus_minus_set_a @ B @ C2 ) )
& ~ ( member_a2 @ X2 @ A4 ) ) ) ).
% subset_Diff_insert
thf(fact_1144_Diff__subset__conv,axiom,
! [A4: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A4 @ B ) @ C2 )
= ( ord_less_eq_set_a @ A4 @ ( sup_sup_set_a @ B @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_1145_Diff__partition,axiom,
! [A4: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A4 @ B )
=> ( ( sup_sup_set_a @ A4 @ ( minus_minus_set_a @ B @ A4 ) )
= B ) ) ).
% Diff_partition
thf(fact_1146_vimage__Diff,axiom,
! [F: a > sum_sum_a_b,A4: set_Sum_sum_a_b,B: set_Sum_sum_a_b] :
( ( vimage_a_Sum_sum_a_b @ F @ ( minus_380166324194722613um_a_b @ A4 @ B ) )
= ( minus_minus_set_a @ ( vimage_a_Sum_sum_a_b @ F @ A4 ) @ ( vimage_a_Sum_sum_a_b @ F @ B ) ) ) ).
% vimage_Diff
thf(fact_1147_DiffE,axiom,
! [C: a,A4: set_a,B: set_a] :
( ( member_a2 @ C @ ( minus_minus_set_a @ A4 @ B ) )
=> ~ ( ( member_a2 @ C @ A4 )
=> ( member_a2 @ C @ B ) ) ) ).
% DiffE
thf(fact_1148_DiffE,axiom,
! [C: sum_sum_a_b,A4: set_Sum_sum_a_b,B: set_Sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ C @ ( minus_380166324194722613um_a_b @ A4 @ B ) )
=> ~ ( ( member_Sum_sum_a_b2 @ C @ A4 )
=> ( member_Sum_sum_a_b2 @ C @ B ) ) ) ).
% DiffE
thf(fact_1149_DiffD1,axiom,
! [C: a,A4: set_a,B: set_a] :
( ( member_a2 @ C @ ( minus_minus_set_a @ A4 @ B ) )
=> ( member_a2 @ C @ A4 ) ) ).
% DiffD1
thf(fact_1150_DiffD1,axiom,
! [C: sum_sum_a_b,A4: set_Sum_sum_a_b,B: set_Sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ C @ ( minus_380166324194722613um_a_b @ A4 @ B ) )
=> ( member_Sum_sum_a_b2 @ C @ A4 ) ) ).
% DiffD1
thf(fact_1151_DiffD2,axiom,
! [C: a,A4: set_a,B: set_a] :
( ( member_a2 @ C @ ( minus_minus_set_a @ A4 @ B ) )
=> ~ ( member_a2 @ C @ B ) ) ).
% DiffD2
thf(fact_1152_DiffD2,axiom,
! [C: sum_sum_a_b,A4: set_Sum_sum_a_b,B: set_Sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ C @ ( minus_380166324194722613um_a_b @ A4 @ B ) )
=> ~ ( member_Sum_sum_a_b2 @ C @ B ) ) ).
% DiffD2
thf(fact_1153_Diff__mono,axiom,
! [A4: set_a,C2: set_a,D: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A4 @ C2 )
=> ( ( ord_less_eq_set_a @ D @ B )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A4 @ B ) @ ( minus_minus_set_a @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_1154_Diff__subset,axiom,
! [A4: set_a,B: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A4 @ B ) @ A4 ) ).
% Diff_subset
thf(fact_1155_double__diff,axiom,
! [A4: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A4 @ B )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ( minus_minus_set_a @ B @ ( minus_minus_set_a @ C2 @ A4 ) )
= A4 ) ) ) ).
% double_diff
thf(fact_1156_chainsD,axiom,
! [C: set_set_a,S3: set_set_a,X2: set_a,Y: set_a] :
( ( member_set_set_a2 @ C @ ( chains_a @ S3 ) )
=> ( ( member_set_a2 @ X2 @ C )
=> ( ( member_set_a2 @ Y @ C )
=> ( ( ord_less_eq_set_a @ X2 @ Y )
| ( ord_less_eq_set_a @ Y @ X2 ) ) ) ) ) ).
% chainsD
thf(fact_1157_Zorn__Lemma2,axiom,
! [A4: set_set_a] :
( ! [X4: set_set_a] :
( ( member_set_set_a2 @ X4 @ ( chains_a @ A4 ) )
=> ? [Xa2: set_a] :
( ( member_set_a2 @ Xa2 @ A4 )
& ! [Xb2: set_a] :
( ( member_set_a2 @ Xb2 @ X4 )
=> ( ord_less_eq_set_a @ Xb2 @ Xa2 ) ) ) )
=> ? [X4: set_a] :
( ( member_set_a2 @ X4 @ A4 )
& ! [Xa2: set_a] :
( ( member_set_a2 @ Xa2 @ A4 )
=> ( ( ord_less_eq_set_a @ X4 @ Xa2 )
=> ( Xa2 = X4 ) ) ) ) ) ).
% Zorn_Lemma2
thf(fact_1158_psubset__imp__ex__mem,axiom,
! [A4: set_a,B: set_a] :
( ( ord_less_set_a @ A4 @ B )
=> ? [B4: a] : ( member_a2 @ B4 @ ( minus_minus_set_a @ B @ A4 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1159_psubset__imp__ex__mem,axiom,
! [A4: set_Sum_sum_a_b,B: set_Sum_sum_a_b] :
( ( ord_le415258996318255280um_a_b @ A4 @ B )
=> ? [B4: sum_sum_a_b] : ( member_Sum_sum_a_b2 @ B4 @ ( minus_380166324194722613um_a_b @ B @ A4 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1160_insert__Diff__if,axiom,
! [X2: a,B: set_a,A4: set_a] :
( ( ( member_a2 @ X2 @ B )
=> ( ( minus_minus_set_a @ ( insert_a2 @ X2 @ A4 ) @ B )
= ( minus_minus_set_a @ A4 @ B ) ) )
& ( ~ ( member_a2 @ X2 @ B )
=> ( ( minus_minus_set_a @ ( insert_a2 @ X2 @ A4 ) @ B )
= ( insert_a2 @ X2 @ ( minus_minus_set_a @ A4 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1161_insert__Diff__if,axiom,
! [X2: sum_sum_a_b,B: set_Sum_sum_a_b,A4: set_Sum_sum_a_b] :
( ( ( member_Sum_sum_a_b2 @ X2 @ B )
=> ( ( minus_380166324194722613um_a_b @ ( insert_Sum_sum_a_b2 @ X2 @ A4 ) @ B )
= ( minus_380166324194722613um_a_b @ A4 @ B ) ) )
& ( ~ ( member_Sum_sum_a_b2 @ X2 @ B )
=> ( ( minus_380166324194722613um_a_b @ ( insert_Sum_sum_a_b2 @ X2 @ A4 ) @ B )
= ( insert_Sum_sum_a_b2 @ X2 @ ( minus_380166324194722613um_a_b @ A4 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1162_subset__insert__iff,axiom,
! [A4: set_Sum_sum_a_b,X2: sum_sum_a_b,B: set_Sum_sum_a_b] :
( ( ord_le9019793522827316924um_a_b @ A4 @ ( insert_Sum_sum_a_b2 @ X2 @ B ) )
= ( ( ( member_Sum_sum_a_b2 @ X2 @ A4 )
=> ( ord_le9019793522827316924um_a_b @ ( minus_380166324194722613um_a_b @ A4 @ ( insert_Sum_sum_a_b2 @ X2 @ bot_bo8744036662862057712um_a_b ) ) @ B ) )
& ( ~ ( member_Sum_sum_a_b2 @ X2 @ A4 )
=> ( ord_le9019793522827316924um_a_b @ A4 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1163_subset__insert__iff,axiom,
! [A4: set_a,X2: a,B: set_a] :
( ( ord_less_eq_set_a @ A4 @ ( insert_a2 @ X2 @ B ) )
= ( ( ( member_a2 @ X2 @ A4 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A4 @ ( insert_a2 @ X2 @ bot_bot_set_a ) ) @ B ) )
& ( ~ ( member_a2 @ X2 @ A4 )
=> ( ord_less_eq_set_a @ A4 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1164_Diff__single__insert,axiom,
! [A4: set_a,X2: a,B: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A4 @ ( insert_a2 @ X2 @ bot_bot_set_a ) ) @ B )
=> ( ord_less_eq_set_a @ A4 @ ( insert_a2 @ X2 @ B ) ) ) ).
% Diff_single_insert
thf(fact_1165_in__image__insert__iff,axiom,
! [B: set_set_a,X2: a,A4: set_a] :
( ! [C3: set_a] :
( ( member_set_a2 @ C3 @ B )
=> ~ ( member_a2 @ X2 @ C3 ) )
=> ( ( member_set_a2 @ A4 @ ( image_set_a_set_a @ ( insert_a2 @ X2 ) @ B ) )
= ( ( member_a2 @ X2 @ A4 )
& ( member_set_a2 @ ( minus_minus_set_a @ A4 @ ( insert_a2 @ X2 @ bot_bot_set_a ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_1166_in__image__insert__iff,axiom,
! [B: set_set_Sum_sum_a_b,X2: sum_sum_a_b,A4: set_Sum_sum_a_b] :
( ! [C3: set_Sum_sum_a_b] :
( ( member4060935254435997939um_a_b @ C3 @ B )
=> ~ ( member_Sum_sum_a_b2 @ X2 @ C3 ) )
=> ( ( member4060935254435997939um_a_b @ A4 @ ( image_7006159782026564799um_a_b @ ( insert_Sum_sum_a_b2 @ X2 ) @ B ) )
= ( ( member_Sum_sum_a_b2 @ X2 @ A4 )
& ( member4060935254435997939um_a_b @ ( minus_380166324194722613um_a_b @ A4 @ ( insert_Sum_sum_a_b2 @ X2 @ bot_bo8744036662862057712um_a_b ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_1167_remove__induct,axiom,
! [P: set_Sum_sum_a_b > $o,B: set_Sum_sum_a_b] :
( ( P @ bot_bo8744036662862057712um_a_b )
=> ( ( ~ ( finite51705151567313725um_a_b @ B )
=> ( P @ B ) )
=> ( ! [A8: set_Sum_sum_a_b] :
( ( finite51705151567313725um_a_b @ A8 )
=> ( ( A8 != bot_bo8744036662862057712um_a_b )
=> ( ( ord_le9019793522827316924um_a_b @ A8 @ B )
=> ( ! [X7: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X7 @ A8 )
=> ( P @ ( minus_380166324194722613um_a_b @ A8 @ ( insert_Sum_sum_a_b2 @ X7 @ bot_bo8744036662862057712um_a_b ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_1168_remove__induct,axiom,
! [P: set_a > $o,B: set_a] :
( ( P @ bot_bot_set_a )
=> ( ( ~ ( finite_finite_a @ B )
=> ( P @ B ) )
=> ( ! [A8: set_a] :
( ( finite_finite_a @ A8 )
=> ( ( A8 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A8 @ B )
=> ( ! [X7: a] :
( ( member_a2 @ X7 @ A8 )
=> ( P @ ( minus_minus_set_a @ A8 @ ( insert_a2 @ X7 @ bot_bot_set_a ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_1169_finite__remove__induct,axiom,
! [B: set_Sum_sum_a_b,P: set_Sum_sum_a_b > $o] :
( ( finite51705151567313725um_a_b @ B )
=> ( ( P @ bot_bo8744036662862057712um_a_b )
=> ( ! [A8: set_Sum_sum_a_b] :
( ( finite51705151567313725um_a_b @ A8 )
=> ( ( A8 != bot_bo8744036662862057712um_a_b )
=> ( ( ord_le9019793522827316924um_a_b @ A8 @ B )
=> ( ! [X7: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X7 @ A8 )
=> ( P @ ( minus_380166324194722613um_a_b @ A8 @ ( insert_Sum_sum_a_b2 @ X7 @ bot_bo8744036662862057712um_a_b ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_1170_finite__remove__induct,axiom,
! [B: set_a,P: set_a > $o] :
( ( finite_finite_a @ B )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A8: set_a] :
( ( finite_finite_a @ A8 )
=> ( ( A8 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A8 @ B )
=> ( ! [X7: a] :
( ( member_a2 @ X7 @ A8 )
=> ( P @ ( minus_minus_set_a @ A8 @ ( insert_a2 @ X7 @ bot_bot_set_a ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_1171_psubset__insert__iff,axiom,
! [A4: set_Sum_sum_a_b,X2: sum_sum_a_b,B: set_Sum_sum_a_b] :
( ( ord_le415258996318255280um_a_b @ A4 @ ( insert_Sum_sum_a_b2 @ X2 @ B ) )
= ( ( ( member_Sum_sum_a_b2 @ X2 @ B )
=> ( ord_le415258996318255280um_a_b @ A4 @ B ) )
& ( ~ ( member_Sum_sum_a_b2 @ X2 @ B )
=> ( ( ( member_Sum_sum_a_b2 @ X2 @ A4 )
=> ( ord_le415258996318255280um_a_b @ ( minus_380166324194722613um_a_b @ A4 @ ( insert_Sum_sum_a_b2 @ X2 @ bot_bo8744036662862057712um_a_b ) ) @ B ) )
& ( ~ ( member_Sum_sum_a_b2 @ X2 @ A4 )
=> ( ord_le9019793522827316924um_a_b @ A4 @ B ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1172_psubset__insert__iff,axiom,
! [A4: set_a,X2: a,B: set_a] :
( ( ord_less_set_a @ A4 @ ( insert_a2 @ X2 @ B ) )
= ( ( ( member_a2 @ X2 @ B )
=> ( ord_less_set_a @ A4 @ B ) )
& ( ~ ( member_a2 @ X2 @ B )
=> ( ( ( member_a2 @ X2 @ A4 )
=> ( ord_less_set_a @ ( minus_minus_set_a @ A4 @ ( insert_a2 @ X2 @ bot_bot_set_a ) ) @ B ) )
& ( ~ ( member_a2 @ X2 @ A4 )
=> ( ord_less_eq_set_a @ A4 @ B ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1173_diff__shunt__var,axiom,
! [X2: set_a,Y: set_a] :
( ( ( minus_minus_set_a @ X2 @ Y )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ X2 @ Y ) ) ).
% diff_shunt_var
thf(fact_1174_set__remove1__eq,axiom,
! [Xs: list_Sum_sum_a_b,X2: sum_sum_a_b] :
( ( distinct_Sum_sum_a_b @ Xs )
=> ( ( set_Sum_sum_a_b2 @ ( remove1_Sum_sum_a_b @ X2 @ Xs ) )
= ( minus_380166324194722613um_a_b @ ( set_Sum_sum_a_b2 @ Xs ) @ ( insert_Sum_sum_a_b2 @ X2 @ bot_bo8744036662862057712um_a_b ) ) ) ) ).
% set_remove1_eq
thf(fact_1175_in__set__remove1,axiom,
! [A: a,B2: a,Xs: list_a] :
( ( A != B2 )
=> ( ( member_a2 @ A @ ( set_a2 @ ( remove1_a @ B2 @ Xs ) ) )
= ( member_a2 @ A @ ( set_a2 @ Xs ) ) ) ) ).
% in_set_remove1
thf(fact_1176_in__set__remove1,axiom,
! [A: sum_sum_a_b,B2: sum_sum_a_b,Xs: list_Sum_sum_a_b] :
( ( A != B2 )
=> ( ( member_Sum_sum_a_b2 @ A @ ( set_Sum_sum_a_b2 @ ( remove1_Sum_sum_a_b @ B2 @ Xs ) ) )
= ( member_Sum_sum_a_b2 @ A @ ( set_Sum_sum_a_b2 @ Xs ) ) ) ) ).
% in_set_remove1
thf(fact_1177_set__remove1__subset,axiom,
! [X2: sum_sum_a_b,Xs: list_Sum_sum_a_b] : ( ord_le9019793522827316924um_a_b @ ( set_Sum_sum_a_b2 @ ( remove1_Sum_sum_a_b @ X2 @ Xs ) ) @ ( set_Sum_sum_a_b2 @ Xs ) ) ).
% set_remove1_subset
thf(fact_1178_set__remove1__subset,axiom,
! [X2: a,Xs: list_a] : ( ord_less_eq_set_a @ ( set_a2 @ ( remove1_a @ X2 @ Xs ) ) @ ( set_a2 @ Xs ) ) ).
% set_remove1_subset
thf(fact_1179_remove1__append,axiom,
! [X2: a,Xs: list_a,Ys: list_a] :
( ( ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
=> ( ( remove1_a @ X2 @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( remove1_a @ X2 @ Xs ) @ Ys ) ) )
& ( ~ ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
=> ( ( remove1_a @ X2 @ ( append_a @ Xs @ Ys ) )
= ( append_a @ Xs @ ( remove1_a @ X2 @ Ys ) ) ) ) ) ).
% remove1_append
thf(fact_1180_remove1__append,axiom,
! [X2: sum_sum_a_b,Xs: list_Sum_sum_a_b,Ys: list_Sum_sum_a_b] :
( ( ( member_Sum_sum_a_b2 @ X2 @ ( set_Sum_sum_a_b2 @ Xs ) )
=> ( ( remove1_Sum_sum_a_b @ X2 @ ( append_Sum_sum_a_b @ Xs @ Ys ) )
= ( append_Sum_sum_a_b @ ( remove1_Sum_sum_a_b @ X2 @ Xs ) @ Ys ) ) )
& ( ~ ( member_Sum_sum_a_b2 @ X2 @ ( set_Sum_sum_a_b2 @ Xs ) )
=> ( ( remove1_Sum_sum_a_b @ X2 @ ( append_Sum_sum_a_b @ Xs @ Ys ) )
= ( append_Sum_sum_a_b @ Xs @ ( remove1_Sum_sum_a_b @ X2 @ Ys ) ) ) ) ) ).
% remove1_append
thf(fact_1181_notin__set__remove1,axiom,
! [X2: a,Xs: list_a,Y: a] :
( ~ ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
=> ~ ( member_a2 @ X2 @ ( set_a2 @ ( remove1_a @ Y @ Xs ) ) ) ) ).
% notin_set_remove1
thf(fact_1182_notin__set__remove1,axiom,
! [X2: sum_sum_a_b,Xs: list_Sum_sum_a_b,Y: sum_sum_a_b] :
( ~ ( member_Sum_sum_a_b2 @ X2 @ ( set_Sum_sum_a_b2 @ Xs ) )
=> ~ ( member_Sum_sum_a_b2 @ X2 @ ( set_Sum_sum_a_b2 @ ( remove1_Sum_sum_a_b @ Y @ Xs ) ) ) ) ).
% notin_set_remove1
thf(fact_1183_remove1__idem,axiom,
! [X2: a,Xs: list_a] :
( ~ ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
=> ( ( remove1_a @ X2 @ Xs )
= Xs ) ) ).
% remove1_idem
thf(fact_1184_remove1__idem,axiom,
! [X2: sum_sum_a_b,Xs: list_Sum_sum_a_b] :
( ~ ( member_Sum_sum_a_b2 @ X2 @ ( set_Sum_sum_a_b2 @ Xs ) )
=> ( ( remove1_Sum_sum_a_b @ X2 @ Xs )
= Xs ) ) ).
% remove1_idem
thf(fact_1185_remove1__split,axiom,
! [A: a,Xs: list_a,Ys: list_a] :
( ( member_a2 @ A @ ( set_a2 @ Xs ) )
=> ( ( ( remove1_a @ A @ Xs )
= Ys )
= ( ? [Ls: list_a,Rs: list_a] :
( ( Xs
= ( append_a @ Ls @ ( cons_a @ A @ Rs ) ) )
& ~ ( member_a2 @ A @ ( set_a2 @ Ls ) )
& ( Ys
= ( append_a @ Ls @ Rs ) ) ) ) ) ) ).
% remove1_split
thf(fact_1186_remove1__split,axiom,
! [A: sum_sum_a_b,Xs: list_Sum_sum_a_b,Ys: list_Sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ A @ ( set_Sum_sum_a_b2 @ Xs ) )
=> ( ( ( remove1_Sum_sum_a_b @ A @ Xs )
= Ys )
= ( ? [Ls: list_Sum_sum_a_b,Rs: list_Sum_sum_a_b] :
( ( Xs
= ( append_Sum_sum_a_b @ Ls @ ( cons_Sum_sum_a_b @ A @ Rs ) ) )
& ~ ( member_Sum_sum_a_b2 @ A @ ( set_Sum_sum_a_b2 @ Ls ) )
& ( Ys
= ( append_Sum_sum_a_b @ Ls @ Rs ) ) ) ) ) ) ).
% remove1_split
thf(fact_1187_set__less__aux__def,axiom,
( set_or6638329327428242490_set_a
= ( ^ [A6: set_set_a,B5: set_set_a] :
( ( finite_finite_set_a @ A6 )
& ( finite_finite_set_a @ B5 )
& ? [X3: set_a] :
( ( member_set_a2 @ X3 @ ( minus_5736297505244876581_set_a @ B5 @ A6 ) )
& ! [Y5: set_a] :
( ( member_set_a2 @ Y5 @ ( sup_sup_set_set_a @ ( minus_5736297505244876581_set_a @ A6 @ B5 ) @ ( minus_5736297505244876581_set_a @ B5 @ A6 ) ) )
=> ( ( ord_less_eq_set_a @ X3 @ Y5 )
& ( ( ord_less_eq_set_a @ Y5 @ X3 )
=> ( X3 = Y5 ) ) ) ) ) ) ) ) ).
% set_less_aux_def
thf(fact_1188_Plus__def,axiom,
( sum_Plus_a_b
= ( ^ [A6: set_a,B5: set_b] : ( sup_su5708271937873308552um_a_b @ ( image_a_Sum_sum_a_b @ sum_Inl_a_b @ A6 ) @ ( image_b_Sum_sum_a_b @ sum_Inr_b_a @ B5 ) ) ) ) ).
% Plus_def
thf(fact_1189_InlI,axiom,
! [A: a,A4: set_a,B: set_b] :
( ( member_a2 @ A @ A4 )
=> ( member_Sum_sum_a_b2 @ ( sum_Inl_a_b @ A ) @ ( sum_Plus_a_b @ A4 @ B ) ) ) ).
% InlI
thf(fact_1190_InrI,axiom,
! [B2: b,B: set_b,A4: set_a] :
( ( member_b2 @ B2 @ B )
=> ( member_Sum_sum_a_b2 @ ( sum_Inr_b_a @ B2 ) @ ( sum_Plus_a_b @ A4 @ B ) ) ) ).
% InrI
thf(fact_1191_mono__Int,axiom,
! [F: set_a > set_a,A4: set_a,B: set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ord_less_eq_set_a @ ( F @ ( inf_inf_set_a @ A4 @ B ) ) @ ( inf_inf_set_a @ ( F @ A4 ) @ ( F @ B ) ) ) ) ).
% mono_Int
thf(fact_1192_mono__Un,axiom,
! [F: set_a > set_a,A4: set_a,B: set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( F @ A4 ) @ ( F @ B ) ) @ ( F @ ( sup_sup_set_a @ A4 @ B ) ) ) ) ).
% mono_Un
thf(fact_1193_mono__inf,axiom,
! [F: set_a > set_a,A4: set_a,B: set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ord_less_eq_set_a @ ( F @ ( inf_inf_set_a @ A4 @ B ) ) @ ( inf_inf_set_a @ ( F @ A4 ) @ ( F @ B ) ) ) ) ).
% mono_inf
thf(fact_1194_mono__sup,axiom,
! [F: set_a > set_a,A4: set_a,B: set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( F @ A4 ) @ ( F @ B ) ) @ ( F @ ( sup_sup_set_a @ A4 @ B ) ) ) ) ).
% mono_sup
thf(fact_1195_PlusE,axiom,
! [U: sum_sum_a_a,A4: set_a,B: set_a] :
( ( member_Sum_sum_a_a @ U @ ( sum_Plus_a_a @ A4 @ B ) )
=> ( ! [X4: a] :
( ( member_a2 @ X4 @ A4 )
=> ( U
!= ( sum_Inl_a_a @ X4 ) ) )
=> ~ ! [Y3: a] :
( ( member_a2 @ Y3 @ B )
=> ( U
!= ( sum_Inr_a_a @ Y3 ) ) ) ) ) ).
% PlusE
thf(fact_1196_PlusE,axiom,
! [U: sum_su5898878462909468885um_a_b,A4: set_a,B: set_Sum_sum_a_b] :
( ( member3874624057069567102um_a_b @ U @ ( sum_Pl851410404342918129um_a_b @ A4 @ B ) )
=> ( ! [X4: a] :
( ( member_a2 @ X4 @ A4 )
=> ( U
!= ( sum_In2918432097854297526um_a_b @ X4 ) ) )
=> ~ ! [Y3: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ Y3 @ B )
=> ( U
!= ( sum_In2710243301657490530_a_b_a @ Y3 ) ) ) ) ) ).
% PlusE
thf(fact_1197_PlusE,axiom,
! [U: sum_su3831877439928360143_a_b_a,A4: set_Sum_sum_a_b,B: set_a] :
( ( member1807623034088458360_a_b_a @ U @ ( sum_Pl4155217190204358947_a_b_a @ A4 @ B ) )
=> ( ! [X4: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X4 @ A4 )
=> ( U
!= ( sum_In6222238883715738344_a_b_a @ X4 ) ) )
=> ~ ! [Y3: a] :
( ( member_a2 @ Y3 @ B )
=> ( U
!= ( sum_In8629808552650825520um_a_b @ Y3 ) ) ) ) ) ).
% PlusE
thf(fact_1198_PlusE,axiom,
! [U: sum_su3067303292148767147um_a_b,A4: set_Sum_sum_a_b,B: set_Sum_sum_a_b] :
( ( member7265535425885436866um_a_b @ U @ ( sum_Pl6362402249328468033um_a_b @ A4 @ B ) )
=> ( ! [X4: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ X4 @ A4 )
=> ( U
!= ( sum_In5992699931424873788um_a_b @ X4 ) ) )
=> ~ ! [Y3: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ Y3 @ B )
=> ( U
!= ( sum_In6272690418125038914um_a_b @ Y3 ) ) ) ) ) ).
% PlusE
thf(fact_1199_PlusE,axiom,
! [U: sum_sum_a_b,A4: set_a,B: set_b] :
( ( member_Sum_sum_a_b2 @ U @ ( sum_Plus_a_b @ A4 @ B ) )
=> ( ! [X4: a] :
( ( member_a2 @ X4 @ A4 )
=> ( U
!= ( sum_Inl_a_b @ X4 ) ) )
=> ~ ! [Y3: b] :
( ( member_b2 @ Y3 @ B )
=> ( U
!= ( sum_Inr_b_a @ Y3 ) ) ) ) ) ).
% PlusE
thf(fact_1200_mono__on__subset,axiom,
! [A4: set_set_a,F: set_a > set_a,B: set_set_a] :
( ( monoto7172710143293369831_set_a @ A4 @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ( ord_le3724670747650509150_set_a @ B @ A4 )
=> ( monoto7172710143293369831_set_a @ B @ ord_less_eq_set_a @ ord_less_eq_set_a @ F ) ) ) ).
% mono_on_subset
thf(fact_1201_ord_Omono__on__subset,axiom,
! [A4: set_a,Less_eq: a > a > $o,F: a > set_a,B: set_a] :
( ( monotone_on_a_set_a @ A4 @ Less_eq @ ord_less_eq_set_a @ F )
=> ( ( ord_less_eq_set_a @ B @ A4 )
=> ( monotone_on_a_set_a @ B @ Less_eq @ ord_less_eq_set_a @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_1202_ord_Omono__onD,axiom,
! [A4: set_a,Less_eq: a > a > $o,F: a > set_a,R2: a,S: a] :
( ( monotone_on_a_set_a @ A4 @ Less_eq @ ord_less_eq_set_a @ F )
=> ( ( member_a2 @ R2 @ A4 )
=> ( ( member_a2 @ S @ A4 )
=> ( ( Less_eq @ R2 @ S )
=> ( ord_less_eq_set_a @ ( F @ R2 ) @ ( F @ S ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_1203_ord_Omono__onD,axiom,
! [A4: set_Sum_sum_a_b,Less_eq: sum_sum_a_b > sum_sum_a_b > $o,F: sum_sum_a_b > set_a,R2: sum_sum_a_b,S: sum_sum_a_b] :
( ( monoto4311748246464393487_set_a @ A4 @ Less_eq @ ord_less_eq_set_a @ F )
=> ( ( member_Sum_sum_a_b2 @ R2 @ A4 )
=> ( ( member_Sum_sum_a_b2 @ S @ A4 )
=> ( ( Less_eq @ R2 @ S )
=> ( ord_less_eq_set_a @ ( F @ R2 ) @ ( F @ S ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_1204_ord_Omono__onI,axiom,
! [A4: set_a,Less_eq: a > a > $o,F: a > set_a] :
( ! [R3: a,S4: a] :
( ( member_a2 @ R3 @ A4 )
=> ( ( member_a2 @ S4 @ A4 )
=> ( ( Less_eq @ R3 @ S4 )
=> ( ord_less_eq_set_a @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) )
=> ( monotone_on_a_set_a @ A4 @ Less_eq @ ord_less_eq_set_a @ F ) ) ).
% ord.mono_onI
thf(fact_1205_ord_Omono__onI,axiom,
! [A4: set_Sum_sum_a_b,Less_eq: sum_sum_a_b > sum_sum_a_b > $o,F: sum_sum_a_b > set_a] :
( ! [R3: sum_sum_a_b,S4: sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ R3 @ A4 )
=> ( ( member_Sum_sum_a_b2 @ S4 @ A4 )
=> ( ( Less_eq @ R3 @ S4 )
=> ( ord_less_eq_set_a @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) )
=> ( monoto4311748246464393487_set_a @ A4 @ Less_eq @ ord_less_eq_set_a @ F ) ) ).
% ord.mono_onI
thf(fact_1206_ord_Omono__on__def,axiom,
! [A4: set_a,Less_eq: a > a > $o,F: a > set_a] :
( ( monotone_on_a_set_a @ A4 @ Less_eq @ ord_less_eq_set_a @ F )
= ( ! [R: a,S2: a] :
( ( ( member_a2 @ R @ A4 )
& ( member_a2 @ S2 @ A4 )
& ( Less_eq @ R @ S2 ) )
=> ( ord_less_eq_set_a @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_1207_ord_Omono__on__def,axiom,
! [A4: set_Sum_sum_a_b,Less_eq: sum_sum_a_b > sum_sum_a_b > $o,F: sum_sum_a_b > set_a] :
( ( monoto4311748246464393487_set_a @ A4 @ Less_eq @ ord_less_eq_set_a @ F )
= ( ! [R: sum_sum_a_b,S2: sum_sum_a_b] :
( ( ( member_Sum_sum_a_b2 @ R @ A4 )
& ( member_Sum_sum_a_b2 @ S2 @ A4 )
& ( Less_eq @ R @ S2 ) )
=> ( ord_less_eq_set_a @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_1208_mono__onD,axiom,
! [A4: set_set_a,F: set_a > set_a,R2: set_a,S: set_a] :
( ( monoto7172710143293369831_set_a @ A4 @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ( member_set_a2 @ R2 @ A4 )
=> ( ( member_set_a2 @ S @ A4 )
=> ( ( ord_less_eq_set_a @ R2 @ S )
=> ( ord_less_eq_set_a @ ( F @ R2 ) @ ( F @ S ) ) ) ) ) ) ).
% mono_onD
thf(fact_1209_mono__onI,axiom,
! [A4: set_set_a,F: set_a > set_a] :
( ! [R3: set_a,S4: set_a] :
( ( member_set_a2 @ R3 @ A4 )
=> ( ( member_set_a2 @ S4 @ A4 )
=> ( ( ord_less_eq_set_a @ R3 @ S4 )
=> ( ord_less_eq_set_a @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) )
=> ( monoto7172710143293369831_set_a @ A4 @ ord_less_eq_set_a @ ord_less_eq_set_a @ F ) ) ).
% mono_onI
thf(fact_1210_mono__imp__mono__on,axiom,
! [F: set_a > set_a,A4: set_set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( monoto7172710143293369831_set_a @ A4 @ ord_less_eq_set_a @ ord_less_eq_set_a @ F ) ) ).
% mono_imp_mono_on
thf(fact_1211_monoI,axiom,
! [F: set_a > set_a] :
( ! [X4: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F ) ) ).
% monoI
thf(fact_1212_monoE,axiom,
! [F: set_a > set_a,X2: set_a,Y: set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) ) ) ).
% monoE
thf(fact_1213_monoD,axiom,
! [F: set_a > set_a,X2: set_a,Y: set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) ) ) ).
% monoD
thf(fact_1214_strict__mono__mono,axiom,
! [F: set_a > set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_set_a @ ord_less_set_a @ F )
=> ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F ) ) ).
% strict_mono_mono
thf(fact_1215_strict__mono__leD,axiom,
! [R2: set_a > set_a,M2: set_a,N2: set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_set_a @ ord_less_set_a @ R2 )
=> ( ( ord_less_eq_set_a @ M2 @ N2 )
=> ( ord_less_eq_set_a @ ( R2 @ M2 ) @ ( R2 @ N2 ) ) ) ) ).
% strict_mono_leD
thf(fact_1216_incseqD,axiom,
! [F: nat > set_a,I: nat,J: nat] :
( ( monoto723715495973462885_set_a @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_set_a @ F )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_set_a @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% incseqD
thf(fact_1217_incseq__def,axiom,
! [X: nat > set_a] :
( ( monoto723715495973462885_set_a @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_set_a @ X )
= ( ! [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
=> ( ord_less_eq_set_a @ ( X @ M3 ) @ ( X @ N3 ) ) ) ) ) ).
% incseq_def
thf(fact_1218_coinduct__set,axiom,
! [F: set_Sum_sum_a_b > set_Sum_sum_a_b,A: sum_sum_a_b,X: set_Sum_sum_a_b] :
( ( monoto3557223508407861281um_a_b @ top_to4377558644481197250um_a_b @ ord_le9019793522827316924um_a_b @ ord_le9019793522827316924um_a_b @ F )
=> ( ( member_Sum_sum_a_b2 @ A @ X )
=> ( ( ord_le9019793522827316924um_a_b @ X @ ( F @ ( sup_su5708271937873308552um_a_b @ X @ ( comple7622384381621123038um_a_b @ F ) ) ) )
=> ( member_Sum_sum_a_b2 @ A @ ( comple7622384381621123038um_a_b @ F ) ) ) ) ) ).
% coinduct_set
thf(fact_1219_coinduct__set,axiom,
! [F: set_a > set_a,A: a,X: set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ( member_a2 @ A @ X )
=> ( ( ord_less_eq_set_a @ X @ ( F @ ( sup_sup_set_a @ X @ ( comple3341859861669737308_set_a @ F ) ) ) )
=> ( member_a2 @ A @ ( comple3341859861669737308_set_a @ F ) ) ) ) ) ).
% coinduct_set
thf(fact_1220_gfp__fun__UnI2,axiom,
! [F: set_Sum_sum_a_b > set_Sum_sum_a_b,A: sum_sum_a_b,X: set_Sum_sum_a_b] :
( ( monoto3557223508407861281um_a_b @ top_to4377558644481197250um_a_b @ ord_le9019793522827316924um_a_b @ ord_le9019793522827316924um_a_b @ F )
=> ( ( member_Sum_sum_a_b2 @ A @ ( comple7622384381621123038um_a_b @ F ) )
=> ( member_Sum_sum_a_b2 @ A @ ( F @ ( sup_su5708271937873308552um_a_b @ X @ ( comple7622384381621123038um_a_b @ F ) ) ) ) ) ) ).
% gfp_fun_UnI2
thf(fact_1221_gfp__fun__UnI2,axiom,
! [F: set_a > set_a,A: a,X: set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ( member_a2 @ A @ ( comple3341859861669737308_set_a @ F ) )
=> ( member_a2 @ A @ ( F @ ( sup_sup_set_a @ X @ ( comple3341859861669737308_set_a @ F ) ) ) ) ) ) ).
% gfp_fun_UnI2
thf(fact_1222_weak__coinduct__image,axiom,
! [A: a,X: set_a,G2: a > sum_sum_a_b,F: set_Sum_sum_a_b > set_Sum_sum_a_b] :
( ( member_a2 @ A @ X )
=> ( ( ord_le9019793522827316924um_a_b @ ( image_a_Sum_sum_a_b @ G2 @ X ) @ ( F @ ( image_a_Sum_sum_a_b @ G2 @ X ) ) )
=> ( member_Sum_sum_a_b2 @ ( G2 @ A ) @ ( comple7622384381621123038um_a_b @ F ) ) ) ) ).
% weak_coinduct_image
thf(fact_1223_weak__coinduct__image,axiom,
! [A: sum_sum_a_b,X: set_Sum_sum_a_b,G2: sum_sum_a_b > sum_sum_a_b,F: set_Sum_sum_a_b > set_Sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ A @ X )
=> ( ( ord_le9019793522827316924um_a_b @ ( image_3358989901043947347um_a_b @ G2 @ X ) @ ( F @ ( image_3358989901043947347um_a_b @ G2 @ X ) ) )
=> ( member_Sum_sum_a_b2 @ ( G2 @ A ) @ ( comple7622384381621123038um_a_b @ F ) ) ) ) ).
% weak_coinduct_image
thf(fact_1224_weak__coinduct__image,axiom,
! [A: a,X: set_a,G2: a > a,F: set_a > set_a] :
( ( member_a2 @ A @ X )
=> ( ( ord_less_eq_set_a @ ( image_a_a @ G2 @ X ) @ ( F @ ( image_a_a @ G2 @ X ) ) )
=> ( member_a2 @ ( G2 @ A ) @ ( comple3341859861669737308_set_a @ F ) ) ) ) ).
% weak_coinduct_image
thf(fact_1225_weak__coinduct__image,axiom,
! [A: sum_sum_a_b,X: set_Sum_sum_a_b,G2: sum_sum_a_b > a,F: set_a > set_a] :
( ( member_Sum_sum_a_b2 @ A @ X )
=> ( ( ord_less_eq_set_a @ ( image_Sum_sum_a_b_a @ G2 @ X ) @ ( F @ ( image_Sum_sum_a_b_a @ G2 @ X ) ) )
=> ( member_a2 @ ( G2 @ A ) @ ( comple3341859861669737308_set_a @ F ) ) ) ) ).
% weak_coinduct_image
thf(fact_1226_gfp__least,axiom,
! [F: set_a > set_a,X: set_a] :
( ! [U2: set_a] :
( ( ord_less_eq_set_a @ U2 @ ( F @ U2 ) )
=> ( ord_less_eq_set_a @ U2 @ X ) )
=> ( ord_less_eq_set_a @ ( comple3341859861669737308_set_a @ F ) @ X ) ) ).
% gfp_least
thf(fact_1227_gfp__upperbound,axiom,
! [X: set_a,F: set_a > set_a] :
( ( ord_less_eq_set_a @ X @ ( F @ X ) )
=> ( ord_less_eq_set_a @ X @ ( comple3341859861669737308_set_a @ F ) ) ) ).
% gfp_upperbound
thf(fact_1228_gfp__mono,axiom,
! [F: set_a > set_a,G2: set_a > set_a] :
( ! [Z4: set_a] : ( ord_less_eq_set_a @ ( F @ Z4 ) @ ( G2 @ Z4 ) )
=> ( ord_less_eq_set_a @ ( comple3341859861669737308_set_a @ F ) @ ( comple3341859861669737308_set_a @ G2 ) ) ) ).
% gfp_mono
thf(fact_1229_weak__coinduct,axiom,
! [A: sum_sum_a_b,X: set_Sum_sum_a_b,F: set_Sum_sum_a_b > set_Sum_sum_a_b] :
( ( member_Sum_sum_a_b2 @ A @ X )
=> ( ( ord_le9019793522827316924um_a_b @ X @ ( F @ X ) )
=> ( member_Sum_sum_a_b2 @ A @ ( comple7622384381621123038um_a_b @ F ) ) ) ) ).
% weak_coinduct
thf(fact_1230_weak__coinduct,axiom,
! [A: a,X: set_a,F: set_a > set_a] :
( ( member_a2 @ A @ X )
=> ( ( ord_less_eq_set_a @ X @ ( F @ X ) )
=> ( member_a2 @ A @ ( comple3341859861669737308_set_a @ F ) ) ) ) ).
% weak_coinduct
thf(fact_1231_def__gfp__unfold,axiom,
! [A4: set_a,F: set_a > set_a] :
( ( A4
= ( comple3341859861669737308_set_a @ F ) )
=> ( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( A4
= ( F @ A4 ) ) ) ) ).
% def_gfp_unfold
thf(fact_1232_gfp__fixpoint,axiom,
! [F: set_a > set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ( F @ ( comple3341859861669737308_set_a @ F ) )
= ( comple3341859861669737308_set_a @ F ) ) ) ).
% gfp_fixpoint
thf(fact_1233_gfp__unfold,axiom,
! [F: set_a > set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ( comple3341859861669737308_set_a @ F )
= ( F @ ( comple3341859861669737308_set_a @ F ) ) ) ) ).
% gfp_unfold
thf(fact_1234_gfp__eqI,axiom,
! [F3: set_a > set_a,X2: set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F3 )
=> ( ( ( F3 @ X2 )
= X2 )
=> ( ! [Z3: set_a] :
( ( ( F3 @ Z3 )
= Z3 )
=> ( ord_less_eq_set_a @ Z3 @ X2 ) )
=> ( ( comple3341859861669737308_set_a @ F3 )
= X2 ) ) ) ) ).
% gfp_eqI
thf(fact_1235_coinduct__lemma,axiom,
! [X: set_a,F: set_a > set_a] :
( ( ord_less_eq_set_a @ X @ ( F @ ( sup_sup_set_a @ X @ ( comple3341859861669737308_set_a @ F ) ) ) )
=> ( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ ( comple3341859861669737308_set_a @ F ) ) @ ( F @ ( sup_sup_set_a @ X @ ( comple3341859861669737308_set_a @ F ) ) ) ) ) ) ).
% coinduct_lemma
thf(fact_1236_def__coinduct,axiom,
! [A4: set_a,F: set_a > set_a,X: set_a] :
( ( A4
= ( comple3341859861669737308_set_a @ F ) )
=> ( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ( ord_less_eq_set_a @ X @ ( F @ ( sup_sup_set_a @ X @ A4 ) ) )
=> ( ord_less_eq_set_a @ X @ A4 ) ) ) ) ).
% def_coinduct
thf(fact_1237_coinduct,axiom,
! [F: set_a > set_a,X: set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ( ord_less_eq_set_a @ X @ ( F @ ( sup_sup_set_a @ X @ ( comple3341859861669737308_set_a @ F ) ) ) )
=> ( ord_less_eq_set_a @ X @ ( comple3341859861669737308_set_a @ F ) ) ) ) ).
% coinduct
thf(fact_1238_def__coinduct__set,axiom,
! [A4: set_Sum_sum_a_b,F: set_Sum_sum_a_b > set_Sum_sum_a_b,A: sum_sum_a_b,X: set_Sum_sum_a_b] :
( ( A4
= ( comple7622384381621123038um_a_b @ F ) )
=> ( ( monoto3557223508407861281um_a_b @ top_to4377558644481197250um_a_b @ ord_le9019793522827316924um_a_b @ ord_le9019793522827316924um_a_b @ F )
=> ( ( member_Sum_sum_a_b2 @ A @ X )
=> ( ( ord_le9019793522827316924um_a_b @ X @ ( F @ ( sup_su5708271937873308552um_a_b @ X @ A4 ) ) )
=> ( member_Sum_sum_a_b2 @ A @ A4 ) ) ) ) ) ).
% def_coinduct_set
thf(fact_1239_def__coinduct__set,axiom,
! [A4: set_a,F: set_a > set_a,A: a,X: set_a] :
( ( A4
= ( comple3341859861669737308_set_a @ F ) )
=> ( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ( member_a2 @ A @ X )
=> ( ( ord_less_eq_set_a @ X @ ( F @ ( sup_sup_set_a @ X @ A4 ) ) )
=> ( member_a2 @ A @ A4 ) ) ) ) ) ).
% def_coinduct_set
thf(fact_1240_incseq__imp__monoseq,axiom,
! [X: nat > set_a] :
( ( monoto723715495973462885_set_a @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_set_a @ X )
=> ( topolo725164666729632753_set_a @ X ) ) ).
% incseq_imp_monoseq
thf(fact_1241_lfp__le__gfp,axiom,
! [F: set_a > set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ord_less_eq_set_a @ ( comple1558298011288954135_set_a @ F ) @ ( comple3341859861669737308_set_a @ F ) ) ) ).
% lfp_le_gfp
thf(fact_1242_monoseq__def,axiom,
( topolo725164666729632753_set_a
= ( ^ [X5: nat > set_a] :
( ! [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
=> ( ord_less_eq_set_a @ ( X5 @ M3 ) @ ( X5 @ N3 ) ) )
| ! [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
=> ( ord_less_eq_set_a @ ( X5 @ N3 ) @ ( X5 @ M3 ) ) ) ) ) ) ).
% monoseq_def
thf(fact_1243_monoI2,axiom,
! [X: nat > set_a] :
( ! [M4: nat,N4: nat] :
( ( ord_less_eq_nat @ M4 @ N4 )
=> ( ord_less_eq_set_a @ ( X @ N4 ) @ ( X @ M4 ) ) )
=> ( topolo725164666729632753_set_a @ X ) ) ).
% monoI2
thf(fact_1244_monoI1,axiom,
! [X: nat > set_a] :
( ! [M4: nat,N4: nat] :
( ( ord_less_eq_nat @ M4 @ N4 )
=> ( ord_less_eq_set_a @ ( X @ M4 ) @ ( X @ N4 ) ) )
=> ( topolo725164666729632753_set_a @ X ) ) ).
% monoI1
thf(fact_1245_lfp__greatest,axiom,
! [F: set_a > set_a,A4: set_a] :
( ! [U2: set_a] :
( ( ord_less_eq_set_a @ ( F @ U2 ) @ U2 )
=> ( ord_less_eq_set_a @ A4 @ U2 ) )
=> ( ord_less_eq_set_a @ A4 @ ( comple1558298011288954135_set_a @ F ) ) ) ).
% lfp_greatest
thf(fact_1246_lfp__lowerbound,axiom,
! [F: set_a > set_a,A4: set_a] :
( ( ord_less_eq_set_a @ ( F @ A4 ) @ A4 )
=> ( ord_less_eq_set_a @ ( comple1558298011288954135_set_a @ F ) @ A4 ) ) ).
% lfp_lowerbound
thf(fact_1247_lfp__mono,axiom,
! [F: set_a > set_a,G2: set_a > set_a] :
( ! [Z4: set_a] : ( ord_less_eq_set_a @ ( F @ Z4 ) @ ( G2 @ Z4 ) )
=> ( ord_less_eq_set_a @ ( comple1558298011288954135_set_a @ F ) @ ( comple1558298011288954135_set_a @ G2 ) ) ) ).
% lfp_mono
thf(fact_1248_lfp__eqI,axiom,
! [F3: set_a > set_a,X2: set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F3 )
=> ( ( ( F3 @ X2 )
= X2 )
=> ( ! [Z3: set_a] :
( ( ( F3 @ Z3 )
= Z3 )
=> ( ord_less_eq_set_a @ X2 @ Z3 ) )
=> ( ( comple1558298011288954135_set_a @ F3 )
= X2 ) ) ) ) ).
% lfp_eqI
thf(fact_1249_lfp__unfold,axiom,
! [F: set_a > set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ( comple1558298011288954135_set_a @ F )
= ( F @ ( comple1558298011288954135_set_a @ F ) ) ) ) ).
% lfp_unfold
thf(fact_1250_lfp__fixpoint,axiom,
! [F: set_a > set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ( F @ ( comple1558298011288954135_set_a @ F ) )
= ( comple1558298011288954135_set_a @ F ) ) ) ).
% lfp_fixpoint
thf(fact_1251_def__lfp__unfold,axiom,
! [H: set_a,F: set_a > set_a] :
( ( H
= ( comple1558298011288954135_set_a @ F ) )
=> ( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( H
= ( F @ H ) ) ) ) ).
% def_lfp_unfold
thf(fact_1252_def__lfp__induct,axiom,
! [A4: set_a,F: set_a > set_a,P: set_a] :
( ( A4
= ( comple1558298011288954135_set_a @ F ) )
=> ( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ( ord_less_eq_set_a @ ( F @ ( inf_inf_set_a @ A4 @ P ) ) @ P )
=> ( ord_less_eq_set_a @ A4 @ P ) ) ) ) ).
% def_lfp_induct
thf(fact_1253_lfp__induct,axiom,
! [F: set_a > set_a,P: set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ( ord_less_eq_set_a @ ( F @ ( inf_inf_set_a @ ( comple1558298011288954135_set_a @ F ) @ P ) ) @ P )
=> ( ord_less_eq_set_a @ ( comple1558298011288954135_set_a @ F ) @ P ) ) ) ).
% lfp_induct
thf(fact_1254_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_1255_lfp__eq__fixp,axiom,
! [F: set_a > set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ( comple1558298011288954135_set_a @ F )
= ( comple6813827801316615403_set_a @ F ) ) ) ).
% lfp_eq_fixp
thf(fact_1256_fixp__lowerbound,axiom,
! [F: set_a > set_a,Z2: set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ( ord_less_eq_set_a @ ( F @ Z2 ) @ Z2 )
=> ( ord_less_eq_set_a @ ( comple6813827801316615403_set_a @ F ) @ Z2 ) ) ) ).
% fixp_lowerbound
thf(fact_1257_fixp__unfold,axiom,
! [F: set_a > set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ( comple6813827801316615403_set_a @ F )
= ( F @ ( comple6813827801316615403_set_a @ F ) ) ) ) ).
% fixp_unfold
thf(fact_1258_iterates__fixp,axiom,
! [F: set_a > set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( member_set_a2 @ ( comple6813827801316615403_set_a @ F ) @ ( comple4964449497533277997_set_a @ F ) ) ) ).
% iterates_fixp
thf(fact_1259_sum__set__simps_I3_J,axiom,
! [X2: a] :
( ( basic_setr_a_b @ ( sum_Inl_a_b @ X2 ) )
= bot_bot_set_b ) ).
% sum_set_simps(3)
thf(fact_1260_iterates__le__f,axiom,
! [X2: set_a,F: set_a > set_a] :
( ( member_set_a2 @ X2 @ ( comple4964449497533277997_set_a @ F ) )
=> ( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ord_less_eq_set_a @ X2 @ ( F @ X2 ) ) ) ) ).
% iterates_le_f
thf(fact_1261_sum__set__simps_I1_J,axiom,
! [X2: a] :
( ( basic_setl_a_b @ ( sum_Inl_a_b @ X2 ) )
= ( insert_a2 @ X2 @ bot_bot_set_a ) ) ).
% sum_set_simps(1)
thf(fact_1262_setl_Ointros,axiom,
! [S: sum_sum_a_b,X2: a] :
( ( S
= ( sum_Inl_a_b @ X2 ) )
=> ( member_a2 @ X2 @ ( basic_setl_a_b @ S ) ) ) ).
% setl.intros
thf(fact_1263_setl_Osimps,axiom,
! [A: a,S: sum_sum_a_b] :
( ( member_a2 @ A @ ( basic_setl_a_b @ S ) )
= ( ? [X3: a] :
( ( A = X3 )
& ( S
= ( sum_Inl_a_b @ X3 ) ) ) ) ) ).
% setl.simps
thf(fact_1264_setl_Ocases,axiom,
! [A: a,S: sum_sum_a_b] :
( ( member_a2 @ A @ ( basic_setl_a_b @ S ) )
=> ( S
= ( sum_Inl_a_b @ A ) ) ) ).
% setl.cases
% Helper facts (17)
thf(help_If_2_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X2: list_a,Y: list_a] :
( ( if_list_a @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X2: list_a,Y: list_a] :
( ( if_list_a @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_2_1_If_001t__List__Olist_Itf__b_J_T,axiom,
! [X2: list_b,Y: list_b] :
( ( if_list_b @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_Itf__b_J_T,axiom,
! [X2: list_b,Y: list_b] :
( ( if_list_b @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Set__Oset_Itf__a_J_J_T,axiom,
! [X2: list_set_a,Y: list_set_a] :
( ( if_list_set_a @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Set__Oset_Itf__a_J_J_T,axiom,
! [X2: list_set_a,Y: list_set_a] :
( ( if_list_set_a @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_2_1_If_001t__List__Olist_It__List__Olist_Itf__a_J_J_T,axiom,
! [X2: list_list_a,Y: list_list_a] :
( ( if_list_list_a @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__List__Olist_Itf__a_J_J_T,axiom,
! [X2: list_list_a,Y: list_list_a] :
( ( if_list_list_a @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mtf__b_J_J_T,axiom,
! [X2: list_Sum_sum_a_b,Y: list_Sum_sum_a_b] :
( ( if_list_Sum_sum_a_b @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mtf__b_J_J_T,axiom,
! [X2: list_Sum_sum_a_b,Y: list_Sum_sum_a_b] :
( ( if_list_Sum_sum_a_b @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J_T,axiom,
! [X2: list_set_set_a,Y: list_set_set_a] :
( ( if_list_set_set_a @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J_T,axiom,
! [X2: list_set_set_a,Y: list_set_set_a] :
( ( if_list_set_set_a @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Set__Oset_It__Sum____Type__Osum_Itf__a_Mtf__b_J_J_J_T,axiom,
! [X2: list_set_Sum_sum_a_b,Y: list_set_Sum_sum_a_b] :
( ( if_lis1608374263067180582um_a_b @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Set__Oset_It__Sum____Type__Osum_Itf__a_Mtf__b_J_J_J_T,axiom,
! [X2: list_set_Sum_sum_a_b,Y: list_set_Sum_sum_a_b] :
( ( if_lis1608374263067180582um_a_b @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_3_1_If_001t__List__Olist_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mtf__b_J_J_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mtf__b_J_J_J_T,axiom,
! [X2: list_l4199846171218662726um_a_b,Y: list_l4199846171218662726um_a_b] :
( ( if_lis6122350189682124032um_a_b @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mtf__b_J_J_J_T,axiom,
! [X2: list_l4199846171218662726um_a_b,Y: list_l4199846171218662726um_a_b] :
( ( if_lis6122350189682124032um_a_b @ $true @ X2 @ Y )
= X2 ) ).
% Conjectures (4)
thf(conj_0,hypothesis,
ad_agr_list_a_b @ x @ xs @ ys ).
thf(conj_1,hypothesis,
member_a2 @ y @ x ).
thf(conj_2,hypothesis,
member_Sum_sum_a_b2 @ ( sum_Inl_a_b @ y ) @ ( set_Sum_sum_a_b2 @ ys ) ).
thf(conj_3,conjecture,
member_Sum_sum_a_b2 @ ( sum_Inl_a_b @ y ) @ ( set_Sum_sum_a_b2 @ xs ) ).
%------------------------------------------------------------------------------