TPTP Problem File: SLH0964^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Safe_Range_RC/0021_Relational_Calculus/prob_00309_009502__16556184_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1594 ( 769 unt; 301 typ; 0 def)
% Number of atoms : 3344 (1645 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 10703 ( 372 ~; 62 |; 280 &;8776 @)
% ( 0 <=>;1213 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 5 avg)
% Number of types : 22 ( 21 usr)
% Number of type conns : 1416 (1416 >; 0 *; 0 +; 0 <<)
% Number of symbols : 283 ( 280 usr; 16 con; 0-5 aty)
% Number of variables : 3468 ( 161 ^;3179 !; 128 ?;3468 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:25:05.325
%------------------------------------------------------------------------------
% Could-be-implicit typings (21)
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% Explicit typings (280)
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thf(sy_c_Set_Othe__elem_001t__List__Olist_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
the_el2253512510949844602la_a_b: set_li8334890247616788453la_a_b > list_R8263082107343818799la_a_b ).
thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
the_elem_nat: set_nat > nat ).
thf(sy_c_Set_Othe__elem_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
the_el6350558617753882986la_a_b: set_Re381260168593705685la_a_b > relational_fmla_a_b ).
thf(sy_c_Set_Othe__elem_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
the_el7486773796875720352la_a_b: set_se6865892389300016395la_a_b > set_Re381260168593705685la_a_b ).
thf(sy_c_Set_Ovimage_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
vimage568529607348484603la_a_b: ( relational_fmla_a_b > relational_fmla_a_b ) > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b ).
thf(sy_c_member_001_062_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_M_Eo_J,type,
member3985631836946730265_a_b_o: ( relational_fmla_a_b > $o ) > set_Re4210025064442248784_a_b_o > $o ).
thf(sy_c_member_001t__List__Olist_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
member8712135081189583430la_a_b: list_R8263082107343818799la_a_b > set_li8334890247616788453la_a_b > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat2: nat > set_nat > $o ).
thf(sy_c_member_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
member4680049679412964150la_a_b: relational_fmla_a_b > set_Re381260168593705685la_a_b > $o ).
thf(sy_c_member_001t__Set__Oset_It__List__Olist_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_J,type,
member5650711565134853628la_a_b: set_li8334890247616788453la_a_b > set_se2998258007684708507la_a_b > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
member3481406638322139244la_a_b: set_Re381260168593705685la_a_b > set_se6865892389300016395la_a_b > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_J,type,
member1693404252451860130la_a_b: set_se6865892389300016395la_a_b > set_se1626149436229564481la_a_b > $o ).
thf(sy_v_I,type,
i: product_prod_b_nat > set_list_a ).
thf(sy_v_Q,type,
q: relational_fmla_a_b ).
thf(sy_v__092_060sigma_062,type,
sigma: nat > a ).
thf(sy_v_xs,type,
xs: list_R8263082107343818799la_a_b ).
% Relevant facts (1279)
thf(fact_0_fmla_Oinject_I6_J,axiom,
! [X61: relational_fmla_a_b,X62: relational_fmla_a_b,Y61: relational_fmla_a_b,Y62: relational_fmla_a_b] :
( ( ( relational_Disj_a_b @ X61 @ X62 )
= ( relational_Disj_a_b @ Y61 @ Y62 ) )
= ( ( X61 = Y61 )
& ( X62 = Y62 ) ) ) ).
% fmla.inject(6)
thf(fact_1_sat_Osimps_I6_J,axiom,
! [Phi: relational_fmla_a_b,Psi: relational_fmla_a_b,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
( ( relational_sat_a_b @ ( relational_Disj_a_b @ Phi @ Psi ) @ I @ Sigma )
= ( ( relational_sat_a_b @ Phi @ I @ Sigma )
| ( relational_sat_a_b @ Psi @ I @ Sigma ) ) ) ).
% sat.simps(6)
thf(fact_2_Un__insert__left,axiom,
! [A: list_R8263082107343818799la_a_b,B: set_li8334890247616788453la_a_b,C: set_li8334890247616788453la_a_b] :
( ( sup_su8392741853961112337la_a_b @ ( insert3617074100887873119la_a_b @ A @ B ) @ C )
= ( insert3617074100887873119la_a_b @ A @ ( sup_su8392741853961112337la_a_b @ B @ C ) ) ) ).
% Un_insert_left
thf(fact_3_Un__insert__left,axiom,
! [A: set_Re381260168593705685la_a_b,B: set_se6865892389300016395la_a_b,C: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ ( insert2023870700798818565la_a_b @ A @ B ) @ C )
= ( insert2023870700798818565la_a_b @ A @ ( sup_su4783144482993978935la_a_b @ B @ C ) ) ) ).
% Un_insert_left
thf(fact_4_Un__insert__left,axiom,
! [A: nat,B: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ ( insert_nat2 @ A @ B ) @ C )
= ( insert_nat2 @ A @ ( sup_sup_set_nat @ B @ C ) ) ) ).
% Un_insert_left
thf(fact_5_Un__insert__left,axiom,
! [A: relational_fmla_a_b,B: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ ( insert7010464514620295119la_a_b @ A @ B ) @ C )
= ( insert7010464514620295119la_a_b @ A @ ( sup_su5130108678486352897la_a_b @ B @ C ) ) ) ).
% Un_insert_left
thf(fact_6_Un__insert__right,axiom,
! [A2: set_li8334890247616788453la_a_b,A: list_R8263082107343818799la_a_b,B: set_li8334890247616788453la_a_b] :
( ( sup_su8392741853961112337la_a_b @ A2 @ ( insert3617074100887873119la_a_b @ A @ B ) )
= ( insert3617074100887873119la_a_b @ A @ ( sup_su8392741853961112337la_a_b @ A2 @ B ) ) ) ).
% Un_insert_right
thf(fact_7_Un__insert__right,axiom,
! [A2: set_se6865892389300016395la_a_b,A: set_Re381260168593705685la_a_b,B: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ A2 @ ( insert2023870700798818565la_a_b @ A @ B ) )
= ( insert2023870700798818565la_a_b @ A @ ( sup_su4783144482993978935la_a_b @ A2 @ B ) ) ) ).
% Un_insert_right
thf(fact_8_Un__insert__right,axiom,
! [A2: set_nat,A: nat,B: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( insert_nat2 @ A @ B ) )
= ( insert_nat2 @ A @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% Un_insert_right
thf(fact_9_Un__insert__right,axiom,
! [A2: set_Re381260168593705685la_a_b,A: relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ A @ B ) )
= ( insert7010464514620295119la_a_b @ A @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) ) ) ).
% Un_insert_right
thf(fact_10_Un__empty,axiom,
! [A2: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ( ( sup_su4783144482993978935la_a_b @ A2 @ B )
= bot_bo2891247006866115487la_a_b )
= ( ( A2 = bot_bo2891247006866115487la_a_b )
& ( B = bot_bo2891247006866115487la_a_b ) ) ) ).
% Un_empty
thf(fact_11_Un__empty,axiom,
! [A2: set_nat,B: set_nat] :
( ( ( sup_sup_set_nat @ A2 @ B )
= bot_bot_set_nat )
= ( ( A2 = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% Un_empty
thf(fact_12_Un__empty,axiom,
! [A2: set_li8334890247616788453la_a_b,B: set_li8334890247616788453la_a_b] :
( ( ( sup_su8392741853961112337la_a_b @ A2 @ B )
= bot_bo3230846185698255993la_a_b )
= ( ( A2 = bot_bo3230846185698255993la_a_b )
& ( B = bot_bo3230846185698255993la_a_b ) ) ) ).
% Un_empty
thf(fact_13_Un__empty,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ( sup_su5130108678486352897la_a_b @ A2 @ B )
= bot_bo4495933725496725865la_a_b )
= ( ( A2 = bot_bo4495933725496725865la_a_b )
& ( B = bot_bo4495933725496725865la_a_b ) ) ) ).
% Un_empty
thf(fact_14_sup__bot__left,axiom,
! [X: relational_fmla_a_b > $o] :
( ( sup_su1471977682094119364_a_b_o @ bot_bo8852203127187332700_a_b_o @ X )
= X ) ).
% sup_bot_left
thf(fact_15_sup__bot__left,axiom,
! [X: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ bot_bo2891247006866115487la_a_b @ X )
= X ) ).
% sup_bot_left
thf(fact_16_sup__bot__left,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ X )
= X ) ).
% sup_bot_left
thf(fact_17_sup__bot__left,axiom,
! [X: set_li8334890247616788453la_a_b] :
( ( sup_su8392741853961112337la_a_b @ bot_bo3230846185698255993la_a_b @ X )
= X ) ).
% sup_bot_left
thf(fact_18_sup__bot__left,axiom,
! [X: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ bot_bo4495933725496725865la_a_b @ X )
= X ) ).
% sup_bot_left
thf(fact_19_sup__bot__right,axiom,
! [X: relational_fmla_a_b > $o] :
( ( sup_su1471977682094119364_a_b_o @ X @ bot_bo8852203127187332700_a_b_o )
= X ) ).
% sup_bot_right
thf(fact_20_sup__bot__right,axiom,
! [X: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ X @ bot_bo2891247006866115487la_a_b )
= X ) ).
% sup_bot_right
thf(fact_21_sup__bot__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
= X ) ).
% sup_bot_right
thf(fact_22_sup__bot__right,axiom,
! [X: set_li8334890247616788453la_a_b] :
( ( sup_su8392741853961112337la_a_b @ X @ bot_bo3230846185698255993la_a_b )
= X ) ).
% sup_bot_right
thf(fact_23_sup__bot__right,axiom,
! [X: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ X @ bot_bo4495933725496725865la_a_b )
= X ) ).
% sup_bot_right
thf(fact_24_bot__eq__sup__iff,axiom,
! [X: relational_fmla_a_b > $o,Y: relational_fmla_a_b > $o] :
( ( bot_bo8852203127187332700_a_b_o
= ( sup_su1471977682094119364_a_b_o @ X @ Y ) )
= ( ( X = bot_bo8852203127187332700_a_b_o )
& ( Y = bot_bo8852203127187332700_a_b_o ) ) ) ).
% bot_eq_sup_iff
thf(fact_25_bot__eq__sup__iff,axiom,
! [X: set_se6865892389300016395la_a_b,Y: set_se6865892389300016395la_a_b] :
( ( bot_bo2891247006866115487la_a_b
= ( sup_su4783144482993978935la_a_b @ X @ Y ) )
= ( ( X = bot_bo2891247006866115487la_a_b )
& ( Y = bot_bo2891247006866115487la_a_b ) ) ) ).
% bot_eq_sup_iff
thf(fact_26_bot__eq__sup__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ X @ Y ) )
= ( ( X = bot_bot_set_nat )
& ( Y = bot_bot_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_27_bot__eq__sup__iff,axiom,
! [X: set_li8334890247616788453la_a_b,Y: set_li8334890247616788453la_a_b] :
( ( bot_bo3230846185698255993la_a_b
= ( sup_su8392741853961112337la_a_b @ X @ Y ) )
= ( ( X = bot_bo3230846185698255993la_a_b )
& ( Y = bot_bo3230846185698255993la_a_b ) ) ) ).
% bot_eq_sup_iff
thf(fact_28_bot__eq__sup__iff,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b] :
( ( bot_bo4495933725496725865la_a_b
= ( sup_su5130108678486352897la_a_b @ X @ Y ) )
= ( ( X = bot_bo4495933725496725865la_a_b )
& ( Y = bot_bo4495933725496725865la_a_b ) ) ) ).
% bot_eq_sup_iff
thf(fact_29_sup__eq__bot__iff,axiom,
! [X: relational_fmla_a_b > $o,Y: relational_fmla_a_b > $o] :
( ( ( sup_su1471977682094119364_a_b_o @ X @ Y )
= bot_bo8852203127187332700_a_b_o )
= ( ( X = bot_bo8852203127187332700_a_b_o )
& ( Y = bot_bo8852203127187332700_a_b_o ) ) ) ).
% sup_eq_bot_iff
thf(fact_30_sup__eq__bot__iff,axiom,
! [X: set_se6865892389300016395la_a_b,Y: set_se6865892389300016395la_a_b] :
( ( ( sup_su4783144482993978935la_a_b @ X @ Y )
= bot_bo2891247006866115487la_a_b )
= ( ( X = bot_bo2891247006866115487la_a_b )
& ( Y = bot_bo2891247006866115487la_a_b ) ) ) ).
% sup_eq_bot_iff
thf(fact_31_sup__eq__bot__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( sup_sup_set_nat @ X @ Y )
= bot_bot_set_nat )
= ( ( X = bot_bot_set_nat )
& ( Y = bot_bot_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_32_sup__eq__bot__iff,axiom,
! [X: set_li8334890247616788453la_a_b,Y: set_li8334890247616788453la_a_b] :
( ( ( sup_su8392741853961112337la_a_b @ X @ Y )
= bot_bo3230846185698255993la_a_b )
= ( ( X = bot_bo3230846185698255993la_a_b )
& ( Y = bot_bo3230846185698255993la_a_b ) ) ) ).
% sup_eq_bot_iff
thf(fact_33_sup__eq__bot__iff,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b] :
( ( ( sup_su5130108678486352897la_a_b @ X @ Y )
= bot_bo4495933725496725865la_a_b )
= ( ( X = bot_bo4495933725496725865la_a_b )
& ( Y = bot_bo4495933725496725865la_a_b ) ) ) ).
% sup_eq_bot_iff
thf(fact_34_sup__bot_Oeq__neutr__iff,axiom,
! [A: relational_fmla_a_b > $o,B2: relational_fmla_a_b > $o] :
( ( ( sup_su1471977682094119364_a_b_o @ A @ B2 )
= bot_bo8852203127187332700_a_b_o )
= ( ( A = bot_bo8852203127187332700_a_b_o )
& ( B2 = bot_bo8852203127187332700_a_b_o ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_35_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_se6865892389300016395la_a_b,B2: set_se6865892389300016395la_a_b] :
( ( ( sup_su4783144482993978935la_a_b @ A @ B2 )
= bot_bo2891247006866115487la_a_b )
= ( ( A = bot_bo2891247006866115487la_a_b )
& ( B2 = bot_bo2891247006866115487la_a_b ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_36_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_nat,B2: set_nat] :
( ( ( sup_sup_set_nat @ A @ B2 )
= bot_bot_set_nat )
= ( ( A = bot_bot_set_nat )
& ( B2 = bot_bot_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_37_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_li8334890247616788453la_a_b,B2: set_li8334890247616788453la_a_b] :
( ( ( sup_su8392741853961112337la_a_b @ A @ B2 )
= bot_bo3230846185698255993la_a_b )
= ( ( A = bot_bo3230846185698255993la_a_b )
& ( B2 = bot_bo3230846185698255993la_a_b ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_38_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( ( sup_su5130108678486352897la_a_b @ A @ B2 )
= bot_bo4495933725496725865la_a_b )
= ( ( A = bot_bo4495933725496725865la_a_b )
& ( B2 = bot_bo4495933725496725865la_a_b ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_39_sup__bot_Oleft__neutral,axiom,
! [A: relational_fmla_a_b > $o] :
( ( sup_su1471977682094119364_a_b_o @ bot_bo8852203127187332700_a_b_o @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_40_sup__bot_Oleft__neutral,axiom,
! [A: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ bot_bo2891247006866115487la_a_b @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_41_sup__bot_Oleft__neutral,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_42_sup__bot_Oleft__neutral,axiom,
! [A: set_li8334890247616788453la_a_b] :
( ( sup_su8392741853961112337la_a_b @ bot_bo3230846185698255993la_a_b @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_43_sup__bot_Oleft__neutral,axiom,
! [A: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ bot_bo4495933725496725865la_a_b @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_44_sup__bot_Oneutr__eq__iff,axiom,
! [A: relational_fmla_a_b > $o,B2: relational_fmla_a_b > $o] :
( ( bot_bo8852203127187332700_a_b_o
= ( sup_su1471977682094119364_a_b_o @ A @ B2 ) )
= ( ( A = bot_bo8852203127187332700_a_b_o )
& ( B2 = bot_bo8852203127187332700_a_b_o ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_45_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_se6865892389300016395la_a_b,B2: set_se6865892389300016395la_a_b] :
( ( bot_bo2891247006866115487la_a_b
= ( sup_su4783144482993978935la_a_b @ A @ B2 ) )
= ( ( A = bot_bo2891247006866115487la_a_b )
& ( B2 = bot_bo2891247006866115487la_a_b ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_46_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_nat,B2: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ A @ B2 ) )
= ( ( A = bot_bot_set_nat )
& ( B2 = bot_bot_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_47_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_li8334890247616788453la_a_b,B2: set_li8334890247616788453la_a_b] :
( ( bot_bo3230846185698255993la_a_b
= ( sup_su8392741853961112337la_a_b @ A @ B2 ) )
= ( ( A = bot_bo3230846185698255993la_a_b )
& ( B2 = bot_bo3230846185698255993la_a_b ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_48_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( bot_bo4495933725496725865la_a_b
= ( sup_su5130108678486352897la_a_b @ A @ B2 ) )
= ( ( A = bot_bo4495933725496725865la_a_b )
& ( B2 = bot_bo4495933725496725865la_a_b ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_49_sup__bot_Oright__neutral,axiom,
! [A: relational_fmla_a_b > $o] :
( ( sup_su1471977682094119364_a_b_o @ A @ bot_bo8852203127187332700_a_b_o )
= A ) ).
% sup_bot.right_neutral
thf(fact_50_sup__bot_Oright__neutral,axiom,
! [A: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ A @ bot_bo2891247006866115487la_a_b )
= A ) ).
% sup_bot.right_neutral
thf(fact_51_sup__bot_Oright__neutral,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% sup_bot.right_neutral
thf(fact_52_sup__bot_Oright__neutral,axiom,
! [A: set_li8334890247616788453la_a_b] :
( ( sup_su8392741853961112337la_a_b @ A @ bot_bo3230846185698255993la_a_b )
= A ) ).
% sup_bot.right_neutral
thf(fact_53_sup__bot_Oright__neutral,axiom,
! [A: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ A @ bot_bo4495933725496725865la_a_b )
= A ) ).
% sup_bot.right_neutral
thf(fact_54_singletonI,axiom,
! [A: set_Re381260168593705685la_a_b] : ( member3481406638322139244la_a_b @ A @ ( insert2023870700798818565la_a_b @ A @ bot_bo2891247006866115487la_a_b ) ) ).
% singletonI
thf(fact_55_singletonI,axiom,
! [A: nat] : ( member_nat2 @ A @ ( insert_nat2 @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_56_singletonI,axiom,
! [A: list_R8263082107343818799la_a_b] : ( member8712135081189583430la_a_b @ A @ ( insert3617074100887873119la_a_b @ A @ bot_bo3230846185698255993la_a_b ) ) ).
% singletonI
thf(fact_57_singletonI,axiom,
! [A: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ A @ ( insert7010464514620295119la_a_b @ A @ bot_bo4495933725496725865la_a_b ) ) ).
% singletonI
thf(fact_58_empty__Collect__eq,axiom,
! [P: set_Re381260168593705685la_a_b > $o] :
( ( bot_bo2891247006866115487la_a_b
= ( collec2099942116761351594la_a_b @ P ) )
= ( ! [X2: set_Re381260168593705685la_a_b] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_59_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_60_empty__Collect__eq,axiom,
! [P: list_R8263082107343818799la_a_b > $o] :
( ( bot_bo3230846185698255993la_a_b
= ( collec1121316490678886532la_a_b @ P ) )
= ( ! [X2: list_R8263082107343818799la_a_b] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_61_empty__Collect__eq,axiom,
! [P: relational_fmla_a_b > $o] :
( ( bot_bo4495933725496725865la_a_b
= ( collec3419995626248312948la_a_b @ P ) )
= ( ! [X2: relational_fmla_a_b] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_62_Collect__empty__eq,axiom,
! [P: set_Re381260168593705685la_a_b > $o] :
( ( ( collec2099942116761351594la_a_b @ P )
= bot_bo2891247006866115487la_a_b )
= ( ! [X2: set_Re381260168593705685la_a_b] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_63_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_64_Collect__empty__eq,axiom,
! [P: list_R8263082107343818799la_a_b > $o] :
( ( ( collec1121316490678886532la_a_b @ P )
= bot_bo3230846185698255993la_a_b )
= ( ! [X2: list_R8263082107343818799la_a_b] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_65_Collect__empty__eq,axiom,
! [P: relational_fmla_a_b > $o] :
( ( ( collec3419995626248312948la_a_b @ P )
= bot_bo4495933725496725865la_a_b )
= ( ! [X2: relational_fmla_a_b] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_66_all__not__in__conv,axiom,
! [A2: set_se6865892389300016395la_a_b] :
( ( ! [X2: set_Re381260168593705685la_a_b] :
~ ( member3481406638322139244la_a_b @ X2 @ A2 ) )
= ( A2 = bot_bo2891247006866115487la_a_b ) ) ).
% all_not_in_conv
thf(fact_67_all__not__in__conv,axiom,
! [A2: set_nat] :
( ( ! [X2: nat] :
~ ( member_nat2 @ X2 @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_68_all__not__in__conv,axiom,
! [A2: set_li8334890247616788453la_a_b] :
( ( ! [X2: list_R8263082107343818799la_a_b] :
~ ( member8712135081189583430la_a_b @ X2 @ A2 ) )
= ( A2 = bot_bo3230846185698255993la_a_b ) ) ).
% all_not_in_conv
thf(fact_69_all__not__in__conv,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( ! [X2: relational_fmla_a_b] :
~ ( member4680049679412964150la_a_b @ X2 @ A2 ) )
= ( A2 = bot_bo4495933725496725865la_a_b ) ) ).
% all_not_in_conv
thf(fact_70_empty__iff,axiom,
! [C2: set_Re381260168593705685la_a_b] :
~ ( member3481406638322139244la_a_b @ C2 @ bot_bo2891247006866115487la_a_b ) ).
% empty_iff
thf(fact_71_empty__iff,axiom,
! [C2: nat] :
~ ( member_nat2 @ C2 @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_72_empty__iff,axiom,
! [C2: list_R8263082107343818799la_a_b] :
~ ( member8712135081189583430la_a_b @ C2 @ bot_bo3230846185698255993la_a_b ) ).
% empty_iff
thf(fact_73_empty__iff,axiom,
! [C2: relational_fmla_a_b] :
~ ( member4680049679412964150la_a_b @ C2 @ bot_bo4495933725496725865la_a_b ) ).
% empty_iff
thf(fact_74_insert__absorb2,axiom,
! [X: set_Re381260168593705685la_a_b,A2: set_se6865892389300016395la_a_b] :
( ( insert2023870700798818565la_a_b @ X @ ( insert2023870700798818565la_a_b @ X @ A2 ) )
= ( insert2023870700798818565la_a_b @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_75_insert__absorb2,axiom,
! [X: nat,A2: set_nat] :
( ( insert_nat2 @ X @ ( insert_nat2 @ X @ A2 ) )
= ( insert_nat2 @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_76_insert__absorb2,axiom,
! [X: list_R8263082107343818799la_a_b,A2: set_li8334890247616788453la_a_b] :
( ( insert3617074100887873119la_a_b @ X @ ( insert3617074100887873119la_a_b @ X @ A2 ) )
= ( insert3617074100887873119la_a_b @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_77_insert__absorb2,axiom,
! [X: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( insert7010464514620295119la_a_b @ X @ ( insert7010464514620295119la_a_b @ X @ A2 ) )
= ( insert7010464514620295119la_a_b @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_78_insert__iff,axiom,
! [A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b,A2: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ A @ ( insert2023870700798818565la_a_b @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member3481406638322139244la_a_b @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_79_insert__iff,axiom,
! [A: nat,B2: nat,A2: set_nat] :
( ( member_nat2 @ A @ ( insert_nat2 @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member_nat2 @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_80_insert__iff,axiom,
! [A: list_R8263082107343818799la_a_b,B2: list_R8263082107343818799la_a_b,A2: set_li8334890247616788453la_a_b] :
( ( member8712135081189583430la_a_b @ A @ ( insert3617074100887873119la_a_b @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member8712135081189583430la_a_b @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_81_insert__iff,axiom,
! [A: relational_fmla_a_b,B2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ A @ ( insert7010464514620295119la_a_b @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member4680049679412964150la_a_b @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_82_insertCI,axiom,
! [A: set_Re381260168593705685la_a_b,B: set_se6865892389300016395la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( ~ ( member3481406638322139244la_a_b @ A @ B )
=> ( A = B2 ) )
=> ( member3481406638322139244la_a_b @ A @ ( insert2023870700798818565la_a_b @ B2 @ B ) ) ) ).
% insertCI
thf(fact_83_insertCI,axiom,
! [A: nat,B: set_nat,B2: nat] :
( ( ~ ( member_nat2 @ A @ B )
=> ( A = B2 ) )
=> ( member_nat2 @ A @ ( insert_nat2 @ B2 @ B ) ) ) ).
% insertCI
thf(fact_84_insertCI,axiom,
! [A: list_R8263082107343818799la_a_b,B: set_li8334890247616788453la_a_b,B2: list_R8263082107343818799la_a_b] :
( ( ~ ( member8712135081189583430la_a_b @ A @ B )
=> ( A = B2 ) )
=> ( member8712135081189583430la_a_b @ A @ ( insert3617074100887873119la_a_b @ B2 @ B ) ) ) ).
% insertCI
thf(fact_85_insertCI,axiom,
! [A: relational_fmla_a_b,B: set_Re381260168593705685la_a_b,B2: relational_fmla_a_b] :
( ( ~ ( member4680049679412964150la_a_b @ A @ B )
=> ( A = B2 ) )
=> ( member4680049679412964150la_a_b @ A @ ( insert7010464514620295119la_a_b @ B2 @ B ) ) ) ).
% insertCI
thf(fact_86_sup_Oright__idem,axiom,
! [A: set_se6865892389300016395la_a_b,B2: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ ( sup_su4783144482993978935la_a_b @ A @ B2 ) @ B2 )
= ( sup_su4783144482993978935la_a_b @ A @ B2 ) ) ).
% sup.right_idem
thf(fact_87_sup_Oright__idem,axiom,
! [A: nat,B2: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ A @ B2 ) @ B2 )
= ( sup_sup_nat @ A @ B2 ) ) ).
% sup.right_idem
thf(fact_88_sup_Oright__idem,axiom,
! [A: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B2 ) @ B2 )
= ( sup_sup_set_nat @ A @ B2 ) ) ).
% sup.right_idem
thf(fact_89_sup_Oright__idem,axiom,
! [A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ ( sup_su5130108678486352897la_a_b @ A @ B2 ) @ B2 )
= ( sup_su5130108678486352897la_a_b @ A @ B2 ) ) ).
% sup.right_idem
thf(fact_90_sup__left__idem,axiom,
! [X: set_se6865892389300016395la_a_b,Y: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ X @ ( sup_su4783144482993978935la_a_b @ X @ Y ) )
= ( sup_su4783144482993978935la_a_b @ X @ Y ) ) ).
% sup_left_idem
thf(fact_91_sup__left__idem,axiom,
! [X: nat,Y: nat] :
( ( sup_sup_nat @ X @ ( sup_sup_nat @ X @ Y ) )
= ( sup_sup_nat @ X @ Y ) ) ).
% sup_left_idem
thf(fact_92_sup__left__idem,axiom,
! [X: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) )
= ( sup_sup_set_nat @ X @ Y ) ) ).
% sup_left_idem
thf(fact_93_sup__left__idem,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ X @ ( sup_su5130108678486352897la_a_b @ X @ Y ) )
= ( sup_su5130108678486352897la_a_b @ X @ Y ) ) ).
% sup_left_idem
thf(fact_94_sup_Oleft__idem,axiom,
! [A: set_se6865892389300016395la_a_b,B2: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ A @ ( sup_su4783144482993978935la_a_b @ A @ B2 ) )
= ( sup_su4783144482993978935la_a_b @ A @ B2 ) ) ).
% sup.left_idem
thf(fact_95_sup_Oleft__idem,axiom,
! [A: nat,B2: nat] :
( ( sup_sup_nat @ A @ ( sup_sup_nat @ A @ B2 ) )
= ( sup_sup_nat @ A @ B2 ) ) ).
% sup.left_idem
thf(fact_96_sup_Oleft__idem,axiom,
! [A: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ A @ B2 ) )
= ( sup_sup_set_nat @ A @ B2 ) ) ).
% sup.left_idem
thf(fact_97_sup_Oleft__idem,axiom,
! [A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ A @ ( sup_su5130108678486352897la_a_b @ A @ B2 ) )
= ( sup_su5130108678486352897la_a_b @ A @ B2 ) ) ).
% sup.left_idem
thf(fact_98_sup__idem,axiom,
! [X: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ X @ X )
= X ) ).
% sup_idem
thf(fact_99_sup__idem,axiom,
! [X: nat] :
( ( sup_sup_nat @ X @ X )
= X ) ).
% sup_idem
thf(fact_100_sup__idem,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ X )
= X ) ).
% sup_idem
thf(fact_101_sup__idem,axiom,
! [X: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ X @ X )
= X ) ).
% sup_idem
thf(fact_102_sup_Oidem,axiom,
! [A: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ A @ A )
= A ) ).
% sup.idem
thf(fact_103_sup_Oidem,axiom,
! [A: nat] :
( ( sup_sup_nat @ A @ A )
= A ) ).
% sup.idem
thf(fact_104_sup_Oidem,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ A )
= A ) ).
% sup.idem
thf(fact_105_sup_Oidem,axiom,
! [A: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ A @ A )
= A ) ).
% sup.idem
thf(fact_106_Un__iff,axiom,
! [C2: list_R8263082107343818799la_a_b,A2: set_li8334890247616788453la_a_b,B: set_li8334890247616788453la_a_b] :
( ( member8712135081189583430la_a_b @ C2 @ ( sup_su8392741853961112337la_a_b @ A2 @ B ) )
= ( ( member8712135081189583430la_a_b @ C2 @ A2 )
| ( member8712135081189583430la_a_b @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_107_Un__iff,axiom,
! [C2: set_Re381260168593705685la_a_b,A2: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ C2 @ ( sup_su4783144482993978935la_a_b @ A2 @ B ) )
= ( ( member3481406638322139244la_a_b @ C2 @ A2 )
| ( member3481406638322139244la_a_b @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_108_Un__iff,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat2 @ C2 @ ( sup_sup_set_nat @ A2 @ B ) )
= ( ( member_nat2 @ C2 @ A2 )
| ( member_nat2 @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_109_Un__iff,axiom,
! [C2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C2 @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) )
= ( ( member4680049679412964150la_a_b @ C2 @ A2 )
| ( member4680049679412964150la_a_b @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_110_UnCI,axiom,
! [C2: list_R8263082107343818799la_a_b,B: set_li8334890247616788453la_a_b,A2: set_li8334890247616788453la_a_b] :
( ( ~ ( member8712135081189583430la_a_b @ C2 @ B )
=> ( member8712135081189583430la_a_b @ C2 @ A2 ) )
=> ( member8712135081189583430la_a_b @ C2 @ ( sup_su8392741853961112337la_a_b @ A2 @ B ) ) ) ).
% UnCI
thf(fact_111_UnCI,axiom,
! [C2: set_Re381260168593705685la_a_b,B: set_se6865892389300016395la_a_b,A2: set_se6865892389300016395la_a_b] :
( ( ~ ( member3481406638322139244la_a_b @ C2 @ B )
=> ( member3481406638322139244la_a_b @ C2 @ A2 ) )
=> ( member3481406638322139244la_a_b @ C2 @ ( sup_su4783144482993978935la_a_b @ A2 @ B ) ) ) ).
% UnCI
thf(fact_112_UnCI,axiom,
! [C2: nat,B: set_nat,A2: set_nat] :
( ( ~ ( member_nat2 @ C2 @ B )
=> ( member_nat2 @ C2 @ A2 ) )
=> ( member_nat2 @ C2 @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% UnCI
thf(fact_113_UnCI,axiom,
! [C2: relational_fmla_a_b,B: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b] :
( ( ~ ( member4680049679412964150la_a_b @ C2 @ B )
=> ( member4680049679412964150la_a_b @ C2 @ A2 ) )
=> ( member4680049679412964150la_a_b @ C2 @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) ) ) ).
% UnCI
thf(fact_114_bot__set__def,axiom,
( bot_bo2891247006866115487la_a_b
= ( collec2099942116761351594la_a_b @ bot_bo8397419132826146726_a_b_o ) ) ).
% bot_set_def
thf(fact_115_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_116_bot__set__def,axiom,
( bot_bo3230846185698255993la_a_b
= ( collec1121316490678886532la_a_b @ bot_bo217863987673589068_a_b_o ) ) ).
% bot_set_def
thf(fact_117_bot__set__def,axiom,
( bot_bo4495933725496725865la_a_b
= ( collec3419995626248312948la_a_b @ bot_bo8852203127187332700_a_b_o ) ) ).
% bot_set_def
thf(fact_118_ex__in__conv,axiom,
! [A2: set_se6865892389300016395la_a_b] :
( ( ? [X2: set_Re381260168593705685la_a_b] : ( member3481406638322139244la_a_b @ X2 @ A2 ) )
= ( A2 != bot_bo2891247006866115487la_a_b ) ) ).
% ex_in_conv
thf(fact_119_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X2: nat] : ( member_nat2 @ X2 @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_120_ex__in__conv,axiom,
! [A2: set_li8334890247616788453la_a_b] :
( ( ? [X2: list_R8263082107343818799la_a_b] : ( member8712135081189583430la_a_b @ X2 @ A2 ) )
= ( A2 != bot_bo3230846185698255993la_a_b ) ) ).
% ex_in_conv
thf(fact_121_ex__in__conv,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( ? [X2: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X2 @ A2 ) )
= ( A2 != bot_bo4495933725496725865la_a_b ) ) ).
% ex_in_conv
thf(fact_122_equals0I,axiom,
! [A2: set_se6865892389300016395la_a_b] :
( ! [Y2: set_Re381260168593705685la_a_b] :
~ ( member3481406638322139244la_a_b @ Y2 @ A2 )
=> ( A2 = bot_bo2891247006866115487la_a_b ) ) ).
% equals0I
thf(fact_123_equals0I,axiom,
! [A2: set_nat] :
( ! [Y2: nat] :
~ ( member_nat2 @ Y2 @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_124_equals0I,axiom,
! [A2: set_li8334890247616788453la_a_b] :
( ! [Y2: list_R8263082107343818799la_a_b] :
~ ( member8712135081189583430la_a_b @ Y2 @ A2 )
=> ( A2 = bot_bo3230846185698255993la_a_b ) ) ).
% equals0I
thf(fact_125_equals0I,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ! [Y2: relational_fmla_a_b] :
~ ( member4680049679412964150la_a_b @ Y2 @ A2 )
=> ( A2 = bot_bo4495933725496725865la_a_b ) ) ).
% equals0I
thf(fact_126_equals0D,axiom,
! [A2: set_se6865892389300016395la_a_b,A: set_Re381260168593705685la_a_b] :
( ( A2 = bot_bo2891247006866115487la_a_b )
=> ~ ( member3481406638322139244la_a_b @ A @ A2 ) ) ).
% equals0D
thf(fact_127_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat2 @ A @ A2 ) ) ).
% equals0D
thf(fact_128_equals0D,axiom,
! [A2: set_li8334890247616788453la_a_b,A: list_R8263082107343818799la_a_b] :
( ( A2 = bot_bo3230846185698255993la_a_b )
=> ~ ( member8712135081189583430la_a_b @ A @ A2 ) ) ).
% equals0D
thf(fact_129_equals0D,axiom,
! [A2: set_Re381260168593705685la_a_b,A: relational_fmla_a_b] :
( ( A2 = bot_bo4495933725496725865la_a_b )
=> ~ ( member4680049679412964150la_a_b @ A @ A2 ) ) ).
% equals0D
thf(fact_130_emptyE,axiom,
! [A: set_Re381260168593705685la_a_b] :
~ ( member3481406638322139244la_a_b @ A @ bot_bo2891247006866115487la_a_b ) ).
% emptyE
thf(fact_131_emptyE,axiom,
! [A: nat] :
~ ( member_nat2 @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_132_emptyE,axiom,
! [A: list_R8263082107343818799la_a_b] :
~ ( member8712135081189583430la_a_b @ A @ bot_bo3230846185698255993la_a_b ) ).
% emptyE
thf(fact_133_emptyE,axiom,
! [A: relational_fmla_a_b] :
~ ( member4680049679412964150la_a_b @ A @ bot_bo4495933725496725865la_a_b ) ).
% emptyE
thf(fact_134_mk__disjoint__insert,axiom,
! [A: set_Re381260168593705685la_a_b,A2: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ A @ A2 )
=> ? [B3: set_se6865892389300016395la_a_b] :
( ( A2
= ( insert2023870700798818565la_a_b @ A @ B3 ) )
& ~ ( member3481406638322139244la_a_b @ A @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_135_mk__disjoint__insert,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat2 @ A @ A2 )
=> ? [B3: set_nat] :
( ( A2
= ( insert_nat2 @ A @ B3 ) )
& ~ ( member_nat2 @ A @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_136_mk__disjoint__insert,axiom,
! [A: list_R8263082107343818799la_a_b,A2: set_li8334890247616788453la_a_b] :
( ( member8712135081189583430la_a_b @ A @ A2 )
=> ? [B3: set_li8334890247616788453la_a_b] :
( ( A2
= ( insert3617074100887873119la_a_b @ A @ B3 ) )
& ~ ( member8712135081189583430la_a_b @ A @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_137_mk__disjoint__insert,axiom,
! [A: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ A @ A2 )
=> ? [B3: set_Re381260168593705685la_a_b] :
( ( A2
= ( insert7010464514620295119la_a_b @ A @ B3 ) )
& ~ ( member4680049679412964150la_a_b @ A @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_138_insert__commute,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b,A2: set_se6865892389300016395la_a_b] :
( ( insert2023870700798818565la_a_b @ X @ ( insert2023870700798818565la_a_b @ Y @ A2 ) )
= ( insert2023870700798818565la_a_b @ Y @ ( insert2023870700798818565la_a_b @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_139_insert__commute,axiom,
! [X: nat,Y: nat,A2: set_nat] :
( ( insert_nat2 @ X @ ( insert_nat2 @ Y @ A2 ) )
= ( insert_nat2 @ Y @ ( insert_nat2 @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_140_insert__commute,axiom,
! [X: list_R8263082107343818799la_a_b,Y: list_R8263082107343818799la_a_b,A2: set_li8334890247616788453la_a_b] :
( ( insert3617074100887873119la_a_b @ X @ ( insert3617074100887873119la_a_b @ Y @ A2 ) )
= ( insert3617074100887873119la_a_b @ Y @ ( insert3617074100887873119la_a_b @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_141_insert__commute,axiom,
! [X: relational_fmla_a_b,Y: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( insert7010464514620295119la_a_b @ X @ ( insert7010464514620295119la_a_b @ Y @ A2 ) )
= ( insert7010464514620295119la_a_b @ Y @ ( insert7010464514620295119la_a_b @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_142_insert__eq__iff,axiom,
! [A: set_Re381260168593705685la_a_b,A2: set_se6865892389300016395la_a_b,B2: set_Re381260168593705685la_a_b,B: set_se6865892389300016395la_a_b] :
( ~ ( member3481406638322139244la_a_b @ A @ A2 )
=> ( ~ ( member3481406638322139244la_a_b @ B2 @ B )
=> ( ( ( insert2023870700798818565la_a_b @ A @ A2 )
= ( insert2023870700798818565la_a_b @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C3: set_se6865892389300016395la_a_b] :
( ( A2
= ( insert2023870700798818565la_a_b @ B2 @ C3 ) )
& ~ ( member3481406638322139244la_a_b @ B2 @ C3 )
& ( B
= ( insert2023870700798818565la_a_b @ A @ C3 ) )
& ~ ( member3481406638322139244la_a_b @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_143_insert__eq__iff,axiom,
! [A: nat,A2: set_nat,B2: nat,B: set_nat] :
( ~ ( member_nat2 @ A @ A2 )
=> ( ~ ( member_nat2 @ B2 @ B )
=> ( ( ( insert_nat2 @ A @ A2 )
= ( insert_nat2 @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C3: set_nat] :
( ( A2
= ( insert_nat2 @ B2 @ C3 ) )
& ~ ( member_nat2 @ B2 @ C3 )
& ( B
= ( insert_nat2 @ A @ C3 ) )
& ~ ( member_nat2 @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_144_insert__eq__iff,axiom,
! [A: list_R8263082107343818799la_a_b,A2: set_li8334890247616788453la_a_b,B2: list_R8263082107343818799la_a_b,B: set_li8334890247616788453la_a_b] :
( ~ ( member8712135081189583430la_a_b @ A @ A2 )
=> ( ~ ( member8712135081189583430la_a_b @ B2 @ B )
=> ( ( ( insert3617074100887873119la_a_b @ A @ A2 )
= ( insert3617074100887873119la_a_b @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C3: set_li8334890247616788453la_a_b] :
( ( A2
= ( insert3617074100887873119la_a_b @ B2 @ C3 ) )
& ~ ( member8712135081189583430la_a_b @ B2 @ C3 )
& ( B
= ( insert3617074100887873119la_a_b @ A @ C3 ) )
& ~ ( member8712135081189583430la_a_b @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_145_insert__eq__iff,axiom,
! [A: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B2: relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ~ ( member4680049679412964150la_a_b @ A @ A2 )
=> ( ~ ( member4680049679412964150la_a_b @ B2 @ B )
=> ( ( ( insert7010464514620295119la_a_b @ A @ A2 )
= ( insert7010464514620295119la_a_b @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C3: set_Re381260168593705685la_a_b] :
( ( A2
= ( insert7010464514620295119la_a_b @ B2 @ C3 ) )
& ~ ( member4680049679412964150la_a_b @ B2 @ C3 )
& ( B
= ( insert7010464514620295119la_a_b @ A @ C3 ) )
& ~ ( member4680049679412964150la_a_b @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_146_insert__absorb,axiom,
! [A: set_Re381260168593705685la_a_b,A2: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ A @ A2 )
=> ( ( insert2023870700798818565la_a_b @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_147_insert__absorb,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat2 @ A @ A2 )
=> ( ( insert_nat2 @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_148_insert__absorb,axiom,
! [A: list_R8263082107343818799la_a_b,A2: set_li8334890247616788453la_a_b] :
( ( member8712135081189583430la_a_b @ A @ A2 )
=> ( ( insert3617074100887873119la_a_b @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_149_insert__absorb,axiom,
! [A: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ A @ A2 )
=> ( ( insert7010464514620295119la_a_b @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_150_insert__ident,axiom,
! [X: set_Re381260168593705685la_a_b,A2: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ~ ( member3481406638322139244la_a_b @ X @ A2 )
=> ( ~ ( member3481406638322139244la_a_b @ X @ B )
=> ( ( ( insert2023870700798818565la_a_b @ X @ A2 )
= ( insert2023870700798818565la_a_b @ X @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_151_insert__ident,axiom,
! [X: nat,A2: set_nat,B: set_nat] :
( ~ ( member_nat2 @ X @ A2 )
=> ( ~ ( member_nat2 @ X @ B )
=> ( ( ( insert_nat2 @ X @ A2 )
= ( insert_nat2 @ X @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_152_insert__ident,axiom,
! [X: list_R8263082107343818799la_a_b,A2: set_li8334890247616788453la_a_b,B: set_li8334890247616788453la_a_b] :
( ~ ( member8712135081189583430la_a_b @ X @ A2 )
=> ( ~ ( member8712135081189583430la_a_b @ X @ B )
=> ( ( ( insert3617074100887873119la_a_b @ X @ A2 )
= ( insert3617074100887873119la_a_b @ X @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_153_insert__ident,axiom,
! [X: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ~ ( member4680049679412964150la_a_b @ X @ A2 )
=> ( ~ ( member4680049679412964150la_a_b @ X @ B )
=> ( ( ( insert7010464514620295119la_a_b @ X @ A2 )
= ( insert7010464514620295119la_a_b @ X @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_154_Set_Oset__insert,axiom,
! [X: set_Re381260168593705685la_a_b,A2: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ X @ A2 )
=> ~ ! [B3: set_se6865892389300016395la_a_b] :
( ( A2
= ( insert2023870700798818565la_a_b @ X @ B3 ) )
=> ( member3481406638322139244la_a_b @ X @ B3 ) ) ) ).
% Set.set_insert
thf(fact_155_Set_Oset__insert,axiom,
! [X: nat,A2: set_nat] :
( ( member_nat2 @ X @ A2 )
=> ~ ! [B3: set_nat] :
( ( A2
= ( insert_nat2 @ X @ B3 ) )
=> ( member_nat2 @ X @ B3 ) ) ) ).
% Set.set_insert
thf(fact_156_Set_Oset__insert,axiom,
! [X: list_R8263082107343818799la_a_b,A2: set_li8334890247616788453la_a_b] :
( ( member8712135081189583430la_a_b @ X @ A2 )
=> ~ ! [B3: set_li8334890247616788453la_a_b] :
( ( A2
= ( insert3617074100887873119la_a_b @ X @ B3 ) )
=> ( member8712135081189583430la_a_b @ X @ B3 ) ) ) ).
% Set.set_insert
thf(fact_157_Set_Oset__insert,axiom,
! [X: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ X @ A2 )
=> ~ ! [B3: set_Re381260168593705685la_a_b] :
( ( A2
= ( insert7010464514620295119la_a_b @ X @ B3 ) )
=> ( member4680049679412964150la_a_b @ X @ B3 ) ) ) ).
% Set.set_insert
thf(fact_158_insertI2,axiom,
! [A: set_Re381260168593705685la_a_b,B: set_se6865892389300016395la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ A @ B )
=> ( member3481406638322139244la_a_b @ A @ ( insert2023870700798818565la_a_b @ B2 @ B ) ) ) ).
% insertI2
thf(fact_159_insertI2,axiom,
! [A: nat,B: set_nat,B2: nat] :
( ( member_nat2 @ A @ B )
=> ( member_nat2 @ A @ ( insert_nat2 @ B2 @ B ) ) ) ).
% insertI2
thf(fact_160_insertI2,axiom,
! [A: list_R8263082107343818799la_a_b,B: set_li8334890247616788453la_a_b,B2: list_R8263082107343818799la_a_b] :
( ( member8712135081189583430la_a_b @ A @ B )
=> ( member8712135081189583430la_a_b @ A @ ( insert3617074100887873119la_a_b @ B2 @ B ) ) ) ).
% insertI2
thf(fact_161_insertI2,axiom,
! [A: relational_fmla_a_b,B: set_Re381260168593705685la_a_b,B2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ A @ B )
=> ( member4680049679412964150la_a_b @ A @ ( insert7010464514620295119la_a_b @ B2 @ B ) ) ) ).
% insertI2
thf(fact_162_insertI1,axiom,
! [A: set_Re381260168593705685la_a_b,B: set_se6865892389300016395la_a_b] : ( member3481406638322139244la_a_b @ A @ ( insert2023870700798818565la_a_b @ A @ B ) ) ).
% insertI1
thf(fact_163_insertI1,axiom,
! [A: nat,B: set_nat] : ( member_nat2 @ A @ ( insert_nat2 @ A @ B ) ) ).
% insertI1
thf(fact_164_insertI1,axiom,
! [A: list_R8263082107343818799la_a_b,B: set_li8334890247616788453la_a_b] : ( member8712135081189583430la_a_b @ A @ ( insert3617074100887873119la_a_b @ A @ B ) ) ).
% insertI1
thf(fact_165_insertI1,axiom,
! [A: relational_fmla_a_b,B: set_Re381260168593705685la_a_b] : ( member4680049679412964150la_a_b @ A @ ( insert7010464514620295119la_a_b @ A @ B ) ) ).
% insertI1
thf(fact_166_insertE,axiom,
! [A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b,A2: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ A @ ( insert2023870700798818565la_a_b @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member3481406638322139244la_a_b @ A @ A2 ) ) ) ).
% insertE
thf(fact_167_insertE,axiom,
! [A: nat,B2: nat,A2: set_nat] :
( ( member_nat2 @ A @ ( insert_nat2 @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member_nat2 @ A @ A2 ) ) ) ).
% insertE
thf(fact_168_insertE,axiom,
! [A: list_R8263082107343818799la_a_b,B2: list_R8263082107343818799la_a_b,A2: set_li8334890247616788453la_a_b] :
( ( member8712135081189583430la_a_b @ A @ ( insert3617074100887873119la_a_b @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member8712135081189583430la_a_b @ A @ A2 ) ) ) ).
% insertE
thf(fact_169_insertE,axiom,
! [A: relational_fmla_a_b,B2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ A @ ( insert7010464514620295119la_a_b @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member4680049679412964150la_a_b @ A @ A2 ) ) ) ).
% insertE
thf(fact_170_sup__left__commute,axiom,
! [X: set_se6865892389300016395la_a_b,Y: set_se6865892389300016395la_a_b,Z: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ X @ ( sup_su4783144482993978935la_a_b @ Y @ Z ) )
= ( sup_su4783144482993978935la_a_b @ Y @ ( sup_su4783144482993978935la_a_b @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_171_sup__left__commute,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z ) )
= ( sup_sup_nat @ Y @ ( sup_sup_nat @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_172_sup__left__commute,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) )
= ( sup_sup_set_nat @ Y @ ( sup_sup_set_nat @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_173_sup__left__commute,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b,Z: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ X @ ( sup_su5130108678486352897la_a_b @ Y @ Z ) )
= ( sup_su5130108678486352897la_a_b @ Y @ ( sup_su5130108678486352897la_a_b @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_174_mem__Collect__eq,axiom,
! [A: set_Re381260168593705685la_a_b,P: set_Re381260168593705685la_a_b > $o] :
( ( member3481406638322139244la_a_b @ A @ ( collec2099942116761351594la_a_b @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_175_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat2 @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_176_mem__Collect__eq,axiom,
! [A: list_R8263082107343818799la_a_b,P: list_R8263082107343818799la_a_b > $o] :
( ( member8712135081189583430la_a_b @ A @ ( collec1121316490678886532la_a_b @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_177_mem__Collect__eq,axiom,
! [A: relational_fmla_a_b,P: relational_fmla_a_b > $o] :
( ( member4680049679412964150la_a_b @ A @ ( collec3419995626248312948la_a_b @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_178_Collect__mem__eq,axiom,
! [A2: set_se6865892389300016395la_a_b] :
( ( collec2099942116761351594la_a_b
@ ^ [X2: set_Re381260168593705685la_a_b] : ( member3481406638322139244la_a_b @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_179_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat2 @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_180_Collect__mem__eq,axiom,
! [A2: set_li8334890247616788453la_a_b] :
( ( collec1121316490678886532la_a_b
@ ^ [X2: list_R8263082107343818799la_a_b] : ( member8712135081189583430la_a_b @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_181_Collect__mem__eq,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( collec3419995626248312948la_a_b
@ ^ [X2: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_182_Collect__cong,axiom,
! [P: relational_fmla_a_b > $o,Q: relational_fmla_a_b > $o] :
( ! [X3: relational_fmla_a_b] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collec3419995626248312948la_a_b @ P )
= ( collec3419995626248312948la_a_b @ Q ) ) ) ).
% Collect_cong
thf(fact_183_sup_Oleft__commute,axiom,
! [B2: set_se6865892389300016395la_a_b,A: set_se6865892389300016395la_a_b,C2: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ B2 @ ( sup_su4783144482993978935la_a_b @ A @ C2 ) )
= ( sup_su4783144482993978935la_a_b @ A @ ( sup_su4783144482993978935la_a_b @ B2 @ C2 ) ) ) ).
% sup.left_commute
thf(fact_184_sup_Oleft__commute,axiom,
! [B2: nat,A: nat,C2: nat] :
( ( sup_sup_nat @ B2 @ ( sup_sup_nat @ A @ C2 ) )
= ( sup_sup_nat @ A @ ( sup_sup_nat @ B2 @ C2 ) ) ) ).
% sup.left_commute
thf(fact_185_sup_Oleft__commute,axiom,
! [B2: set_nat,A: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ B2 @ ( sup_sup_set_nat @ A @ C2 ) )
= ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B2 @ C2 ) ) ) ).
% sup.left_commute
thf(fact_186_sup_Oleft__commute,axiom,
! [B2: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b,C2: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ B2 @ ( sup_su5130108678486352897la_a_b @ A @ C2 ) )
= ( sup_su5130108678486352897la_a_b @ A @ ( sup_su5130108678486352897la_a_b @ B2 @ C2 ) ) ) ).
% sup.left_commute
thf(fact_187_sup__commute,axiom,
( sup_su4783144482993978935la_a_b
= ( ^ [X2: set_se6865892389300016395la_a_b,Y3: set_se6865892389300016395la_a_b] : ( sup_su4783144482993978935la_a_b @ Y3 @ X2 ) ) ) ).
% sup_commute
thf(fact_188_sup__commute,axiom,
( sup_sup_nat
= ( ^ [X2: nat,Y3: nat] : ( sup_sup_nat @ Y3 @ X2 ) ) ) ).
% sup_commute
thf(fact_189_sup__commute,axiom,
( sup_sup_set_nat
= ( ^ [X2: set_nat,Y3: set_nat] : ( sup_sup_set_nat @ Y3 @ X2 ) ) ) ).
% sup_commute
thf(fact_190_sup__commute,axiom,
( sup_su5130108678486352897la_a_b
= ( ^ [X2: set_Re381260168593705685la_a_b,Y3: set_Re381260168593705685la_a_b] : ( sup_su5130108678486352897la_a_b @ Y3 @ X2 ) ) ) ).
% sup_commute
thf(fact_191_sup_Ocommute,axiom,
( sup_su4783144482993978935la_a_b
= ( ^ [A3: set_se6865892389300016395la_a_b,B4: set_se6865892389300016395la_a_b] : ( sup_su4783144482993978935la_a_b @ B4 @ A3 ) ) ) ).
% sup.commute
thf(fact_192_sup_Ocommute,axiom,
( sup_sup_nat
= ( ^ [A3: nat,B4: nat] : ( sup_sup_nat @ B4 @ A3 ) ) ) ).
% sup.commute
thf(fact_193_sup_Ocommute,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B4: set_nat] : ( sup_sup_set_nat @ B4 @ A3 ) ) ) ).
% sup.commute
thf(fact_194_sup_Ocommute,axiom,
( sup_su5130108678486352897la_a_b
= ( ^ [A3: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b] : ( sup_su5130108678486352897la_a_b @ B4 @ A3 ) ) ) ).
% sup.commute
thf(fact_195_sup__assoc,axiom,
! [X: set_se6865892389300016395la_a_b,Y: set_se6865892389300016395la_a_b,Z: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ ( sup_su4783144482993978935la_a_b @ X @ Y ) @ Z )
= ( sup_su4783144482993978935la_a_b @ X @ ( sup_su4783144482993978935la_a_b @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_196_sup__assoc,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ X @ Y ) @ Z )
= ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_197_sup__assoc,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ Z )
= ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_198_sup__assoc,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b,Z: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ ( sup_su5130108678486352897la_a_b @ X @ Y ) @ Z )
= ( sup_su5130108678486352897la_a_b @ X @ ( sup_su5130108678486352897la_a_b @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_199_sup_Oassoc,axiom,
! [A: set_se6865892389300016395la_a_b,B2: set_se6865892389300016395la_a_b,C2: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ ( sup_su4783144482993978935la_a_b @ A @ B2 ) @ C2 )
= ( sup_su4783144482993978935la_a_b @ A @ ( sup_su4783144482993978935la_a_b @ B2 @ C2 ) ) ) ).
% sup.assoc
thf(fact_200_sup_Oassoc,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ A @ B2 ) @ C2 )
= ( sup_sup_nat @ A @ ( sup_sup_nat @ B2 @ C2 ) ) ) ).
% sup.assoc
thf(fact_201_sup_Oassoc,axiom,
! [A: set_nat,B2: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B2 ) @ C2 )
= ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B2 @ C2 ) ) ) ).
% sup.assoc
thf(fact_202_sup_Oassoc,axiom,
! [A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b,C2: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ ( sup_su5130108678486352897la_a_b @ A @ B2 ) @ C2 )
= ( sup_su5130108678486352897la_a_b @ A @ ( sup_su5130108678486352897la_a_b @ B2 @ C2 ) ) ) ).
% sup.assoc
thf(fact_203_inf__sup__aci_I5_J,axiom,
( sup_su4783144482993978935la_a_b
= ( ^ [X2: set_se6865892389300016395la_a_b,Y3: set_se6865892389300016395la_a_b] : ( sup_su4783144482993978935la_a_b @ Y3 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_204_inf__sup__aci_I5_J,axiom,
( sup_sup_nat
= ( ^ [X2: nat,Y3: nat] : ( sup_sup_nat @ Y3 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_205_inf__sup__aci_I5_J,axiom,
( sup_sup_set_nat
= ( ^ [X2: set_nat,Y3: set_nat] : ( sup_sup_set_nat @ Y3 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_206_inf__sup__aci_I5_J,axiom,
( sup_su5130108678486352897la_a_b
= ( ^ [X2: set_Re381260168593705685la_a_b,Y3: set_Re381260168593705685la_a_b] : ( sup_su5130108678486352897la_a_b @ Y3 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_207_inf__sup__aci_I6_J,axiom,
! [X: set_se6865892389300016395la_a_b,Y: set_se6865892389300016395la_a_b,Z: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ ( sup_su4783144482993978935la_a_b @ X @ Y ) @ Z )
= ( sup_su4783144482993978935la_a_b @ X @ ( sup_su4783144482993978935la_a_b @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_208_inf__sup__aci_I6_J,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ X @ Y ) @ Z )
= ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_209_inf__sup__aci_I6_J,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ Z )
= ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_210_inf__sup__aci_I6_J,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b,Z: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ ( sup_su5130108678486352897la_a_b @ X @ Y ) @ Z )
= ( sup_su5130108678486352897la_a_b @ X @ ( sup_su5130108678486352897la_a_b @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_211_inf__sup__aci_I7_J,axiom,
! [X: set_se6865892389300016395la_a_b,Y: set_se6865892389300016395la_a_b,Z: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ X @ ( sup_su4783144482993978935la_a_b @ Y @ Z ) )
= ( sup_su4783144482993978935la_a_b @ Y @ ( sup_su4783144482993978935la_a_b @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_212_inf__sup__aci_I7_J,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z ) )
= ( sup_sup_nat @ Y @ ( sup_sup_nat @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_213_inf__sup__aci_I7_J,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) )
= ( sup_sup_set_nat @ Y @ ( sup_sup_set_nat @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_214_inf__sup__aci_I7_J,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b,Z: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ X @ ( sup_su5130108678486352897la_a_b @ Y @ Z ) )
= ( sup_su5130108678486352897la_a_b @ Y @ ( sup_su5130108678486352897la_a_b @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_215_inf__sup__aci_I8_J,axiom,
! [X: set_se6865892389300016395la_a_b,Y: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ X @ ( sup_su4783144482993978935la_a_b @ X @ Y ) )
= ( sup_su4783144482993978935la_a_b @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_216_inf__sup__aci_I8_J,axiom,
! [X: nat,Y: nat] :
( ( sup_sup_nat @ X @ ( sup_sup_nat @ X @ Y ) )
= ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_217_inf__sup__aci_I8_J,axiom,
! [X: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) )
= ( sup_sup_set_nat @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_218_inf__sup__aci_I8_J,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ X @ ( sup_su5130108678486352897la_a_b @ X @ Y ) )
= ( sup_su5130108678486352897la_a_b @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_219_Un__left__commute,axiom,
! [A2: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b,C: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ A2 @ ( sup_su4783144482993978935la_a_b @ B @ C ) )
= ( sup_su4783144482993978935la_a_b @ B @ ( sup_su4783144482993978935la_a_b @ A2 @ C ) ) ) ).
% Un_left_commute
thf(fact_220_Un__left__commute,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B @ C ) )
= ( sup_sup_set_nat @ B @ ( sup_sup_set_nat @ A2 @ C ) ) ) ).
% Un_left_commute
thf(fact_221_Un__left__commute,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ A2 @ ( sup_su5130108678486352897la_a_b @ B @ C ) )
= ( sup_su5130108678486352897la_a_b @ B @ ( sup_su5130108678486352897la_a_b @ A2 @ C ) ) ) ).
% Un_left_commute
thf(fact_222_Un__left__absorb,axiom,
! [A2: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ A2 @ ( sup_su4783144482993978935la_a_b @ A2 @ B ) )
= ( sup_su4783144482993978935la_a_b @ A2 @ B ) ) ).
% Un_left_absorb
thf(fact_223_Un__left__absorb,axiom,
! [A2: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B ) )
= ( sup_sup_set_nat @ A2 @ B ) ) ).
% Un_left_absorb
thf(fact_224_Un__left__absorb,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ A2 @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) )
= ( sup_su5130108678486352897la_a_b @ A2 @ B ) ) ).
% Un_left_absorb
thf(fact_225_Un__commute,axiom,
( sup_su4783144482993978935la_a_b
= ( ^ [A4: set_se6865892389300016395la_a_b,B5: set_se6865892389300016395la_a_b] : ( sup_su4783144482993978935la_a_b @ B5 @ A4 ) ) ) ).
% Un_commute
thf(fact_226_Un__commute,axiom,
( sup_sup_set_nat
= ( ^ [A4: set_nat,B5: set_nat] : ( sup_sup_set_nat @ B5 @ A4 ) ) ) ).
% Un_commute
thf(fact_227_Un__commute,axiom,
( sup_su5130108678486352897la_a_b
= ( ^ [A4: set_Re381260168593705685la_a_b,B5: set_Re381260168593705685la_a_b] : ( sup_su5130108678486352897la_a_b @ B5 @ A4 ) ) ) ).
% Un_commute
thf(fact_228_Un__absorb,axiom,
! [A2: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_229_Un__absorb,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_230_Un__absorb,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_231_Un__assoc,axiom,
! [A2: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b,C: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ ( sup_su4783144482993978935la_a_b @ A2 @ B ) @ C )
= ( sup_su4783144482993978935la_a_b @ A2 @ ( sup_su4783144482993978935la_a_b @ B @ C ) ) ) ).
% Un_assoc
thf(fact_232_Un__assoc,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ C )
= ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B @ C ) ) ) ).
% Un_assoc
thf(fact_233_Un__assoc,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) @ C )
= ( sup_su5130108678486352897la_a_b @ A2 @ ( sup_su5130108678486352897la_a_b @ B @ C ) ) ) ).
% Un_assoc
thf(fact_234_ball__Un,axiom,
! [A2: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b,P: set_Re381260168593705685la_a_b > $o] :
( ( ! [X2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X2 @ ( sup_su4783144482993978935la_a_b @ A2 @ B ) )
=> ( P @ X2 ) ) )
= ( ! [X2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X2 @ A2 )
=> ( P @ X2 ) )
& ! [X2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X2 @ B )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_235_ball__Un,axiom,
! [A2: set_nat,B: set_nat,P: nat > $o] :
( ( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( sup_sup_set_nat @ A2 @ B ) )
=> ( P @ X2 ) ) )
= ( ! [X2: nat] :
( ( member_nat2 @ X2 @ A2 )
=> ( P @ X2 ) )
& ! [X2: nat] :
( ( member_nat2 @ X2 @ B )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_236_ball__Un,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,P: relational_fmla_a_b > $o] :
( ( ! [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) )
=> ( P @ X2 ) ) )
= ( ! [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ A2 )
=> ( P @ X2 ) )
& ! [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ B )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_237_bex__Un,axiom,
! [A2: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b,P: set_Re381260168593705685la_a_b > $o] :
( ( ? [X2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X2 @ ( sup_su4783144482993978935la_a_b @ A2 @ B ) )
& ( P @ X2 ) ) )
= ( ? [X2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X2 @ A2 )
& ( P @ X2 ) )
| ? [X2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X2 @ B )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_238_bex__Un,axiom,
! [A2: set_nat,B: set_nat,P: nat > $o] :
( ( ? [X2: nat] :
( ( member_nat2 @ X2 @ ( sup_sup_set_nat @ A2 @ B ) )
& ( P @ X2 ) ) )
= ( ? [X2: nat] :
( ( member_nat2 @ X2 @ A2 )
& ( P @ X2 ) )
| ? [X2: nat] :
( ( member_nat2 @ X2 @ B )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_239_bex__Un,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,P: relational_fmla_a_b > $o] :
( ( ? [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) )
& ( P @ X2 ) ) )
= ( ? [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ A2 )
& ( P @ X2 ) )
| ? [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ B )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_240_UnI2,axiom,
! [C2: list_R8263082107343818799la_a_b,B: set_li8334890247616788453la_a_b,A2: set_li8334890247616788453la_a_b] :
( ( member8712135081189583430la_a_b @ C2 @ B )
=> ( member8712135081189583430la_a_b @ C2 @ ( sup_su8392741853961112337la_a_b @ A2 @ B ) ) ) ).
% UnI2
thf(fact_241_UnI2,axiom,
! [C2: set_Re381260168593705685la_a_b,B: set_se6865892389300016395la_a_b,A2: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ C2 @ B )
=> ( member3481406638322139244la_a_b @ C2 @ ( sup_su4783144482993978935la_a_b @ A2 @ B ) ) ) ).
% UnI2
thf(fact_242_UnI2,axiom,
! [C2: nat,B: set_nat,A2: set_nat] :
( ( member_nat2 @ C2 @ B )
=> ( member_nat2 @ C2 @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% UnI2
thf(fact_243_UnI2,axiom,
! [C2: relational_fmla_a_b,B: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C2 @ B )
=> ( member4680049679412964150la_a_b @ C2 @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) ) ) ).
% UnI2
thf(fact_244_UnI1,axiom,
! [C2: list_R8263082107343818799la_a_b,A2: set_li8334890247616788453la_a_b,B: set_li8334890247616788453la_a_b] :
( ( member8712135081189583430la_a_b @ C2 @ A2 )
=> ( member8712135081189583430la_a_b @ C2 @ ( sup_su8392741853961112337la_a_b @ A2 @ B ) ) ) ).
% UnI1
thf(fact_245_UnI1,axiom,
! [C2: set_Re381260168593705685la_a_b,A2: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ C2 @ A2 )
=> ( member3481406638322139244la_a_b @ C2 @ ( sup_su4783144482993978935la_a_b @ A2 @ B ) ) ) ).
% UnI1
thf(fact_246_UnI1,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat2 @ C2 @ A2 )
=> ( member_nat2 @ C2 @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% UnI1
thf(fact_247_UnI1,axiom,
! [C2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C2 @ A2 )
=> ( member4680049679412964150la_a_b @ C2 @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) ) ) ).
% UnI1
thf(fact_248_UnE,axiom,
! [C2: list_R8263082107343818799la_a_b,A2: set_li8334890247616788453la_a_b,B: set_li8334890247616788453la_a_b] :
( ( member8712135081189583430la_a_b @ C2 @ ( sup_su8392741853961112337la_a_b @ A2 @ B ) )
=> ( ~ ( member8712135081189583430la_a_b @ C2 @ A2 )
=> ( member8712135081189583430la_a_b @ C2 @ B ) ) ) ).
% UnE
thf(fact_249_UnE,axiom,
! [C2: set_Re381260168593705685la_a_b,A2: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ C2 @ ( sup_su4783144482993978935la_a_b @ A2 @ B ) )
=> ( ~ ( member3481406638322139244la_a_b @ C2 @ A2 )
=> ( member3481406638322139244la_a_b @ C2 @ B ) ) ) ).
% UnE
thf(fact_250_UnE,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat2 @ C2 @ ( sup_sup_set_nat @ A2 @ B ) )
=> ( ~ ( member_nat2 @ C2 @ A2 )
=> ( member_nat2 @ C2 @ B ) ) ) ).
% UnE
thf(fact_251_UnE,axiom,
! [C2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C2 @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) )
=> ( ~ ( member4680049679412964150la_a_b @ C2 @ A2 )
=> ( member4680049679412964150la_a_b @ C2 @ B ) ) ) ).
% UnE
thf(fact_252_singleton__inject,axiom,
! [A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( ( insert2023870700798818565la_a_b @ A @ bot_bo2891247006866115487la_a_b )
= ( insert2023870700798818565la_a_b @ B2 @ bot_bo2891247006866115487la_a_b ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_253_singleton__inject,axiom,
! [A: nat,B2: nat] :
( ( ( insert_nat2 @ A @ bot_bot_set_nat )
= ( insert_nat2 @ B2 @ bot_bot_set_nat ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_254_singleton__inject,axiom,
! [A: list_R8263082107343818799la_a_b,B2: list_R8263082107343818799la_a_b] :
( ( ( insert3617074100887873119la_a_b @ A @ bot_bo3230846185698255993la_a_b )
= ( insert3617074100887873119la_a_b @ B2 @ bot_bo3230846185698255993la_a_b ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_255_singleton__inject,axiom,
! [A: relational_fmla_a_b,B2: relational_fmla_a_b] :
( ( ( insert7010464514620295119la_a_b @ A @ bot_bo4495933725496725865la_a_b )
= ( insert7010464514620295119la_a_b @ B2 @ bot_bo4495933725496725865la_a_b ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_256_insert__not__empty,axiom,
! [A: set_Re381260168593705685la_a_b,A2: set_se6865892389300016395la_a_b] :
( ( insert2023870700798818565la_a_b @ A @ A2 )
!= bot_bo2891247006866115487la_a_b ) ).
% insert_not_empty
thf(fact_257_insert__not__empty,axiom,
! [A: nat,A2: set_nat] :
( ( insert_nat2 @ A @ A2 )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_258_insert__not__empty,axiom,
! [A: list_R8263082107343818799la_a_b,A2: set_li8334890247616788453la_a_b] :
( ( insert3617074100887873119la_a_b @ A @ A2 )
!= bot_bo3230846185698255993la_a_b ) ).
% insert_not_empty
thf(fact_259_insert__not__empty,axiom,
! [A: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( insert7010464514620295119la_a_b @ A @ A2 )
!= bot_bo4495933725496725865la_a_b ) ).
% insert_not_empty
thf(fact_260_doubleton__eq__iff,axiom,
! [A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b,C2: set_Re381260168593705685la_a_b,D: set_Re381260168593705685la_a_b] :
( ( ( insert2023870700798818565la_a_b @ A @ ( insert2023870700798818565la_a_b @ B2 @ bot_bo2891247006866115487la_a_b ) )
= ( insert2023870700798818565la_a_b @ C2 @ ( insert2023870700798818565la_a_b @ D @ bot_bo2891247006866115487la_a_b ) ) )
= ( ( ( A = C2 )
& ( B2 = D ) )
| ( ( A = D )
& ( B2 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_261_doubleton__eq__iff,axiom,
! [A: nat,B2: nat,C2: nat,D: nat] :
( ( ( insert_nat2 @ A @ ( insert_nat2 @ B2 @ bot_bot_set_nat ) )
= ( insert_nat2 @ C2 @ ( insert_nat2 @ D @ bot_bot_set_nat ) ) )
= ( ( ( A = C2 )
& ( B2 = D ) )
| ( ( A = D )
& ( B2 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_262_doubleton__eq__iff,axiom,
! [A: list_R8263082107343818799la_a_b,B2: list_R8263082107343818799la_a_b,C2: list_R8263082107343818799la_a_b,D: list_R8263082107343818799la_a_b] :
( ( ( insert3617074100887873119la_a_b @ A @ ( insert3617074100887873119la_a_b @ B2 @ bot_bo3230846185698255993la_a_b ) )
= ( insert3617074100887873119la_a_b @ C2 @ ( insert3617074100887873119la_a_b @ D @ bot_bo3230846185698255993la_a_b ) ) )
= ( ( ( A = C2 )
& ( B2 = D ) )
| ( ( A = D )
& ( B2 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_263_doubleton__eq__iff,axiom,
! [A: relational_fmla_a_b,B2: relational_fmla_a_b,C2: relational_fmla_a_b,D: relational_fmla_a_b] :
( ( ( insert7010464514620295119la_a_b @ A @ ( insert7010464514620295119la_a_b @ B2 @ bot_bo4495933725496725865la_a_b ) )
= ( insert7010464514620295119la_a_b @ C2 @ ( insert7010464514620295119la_a_b @ D @ bot_bo4495933725496725865la_a_b ) ) )
= ( ( ( A = C2 )
& ( B2 = D ) )
| ( ( A = D )
& ( B2 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_264_singleton__iff,axiom,
! [B2: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ B2 @ ( insert2023870700798818565la_a_b @ A @ bot_bo2891247006866115487la_a_b ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_265_singleton__iff,axiom,
! [B2: nat,A: nat] :
( ( member_nat2 @ B2 @ ( insert_nat2 @ A @ bot_bot_set_nat ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_266_singleton__iff,axiom,
! [B2: list_R8263082107343818799la_a_b,A: list_R8263082107343818799la_a_b] :
( ( member8712135081189583430la_a_b @ B2 @ ( insert3617074100887873119la_a_b @ A @ bot_bo3230846185698255993la_a_b ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_267_singleton__iff,axiom,
! [B2: relational_fmla_a_b,A: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ B2 @ ( insert7010464514620295119la_a_b @ A @ bot_bo4495933725496725865la_a_b ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_268_singletonD,axiom,
! [B2: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ B2 @ ( insert2023870700798818565la_a_b @ A @ bot_bo2891247006866115487la_a_b ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_269_singletonD,axiom,
! [B2: nat,A: nat] :
( ( member_nat2 @ B2 @ ( insert_nat2 @ A @ bot_bot_set_nat ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_270_singletonD,axiom,
! [B2: list_R8263082107343818799la_a_b,A: list_R8263082107343818799la_a_b] :
( ( member8712135081189583430la_a_b @ B2 @ ( insert3617074100887873119la_a_b @ A @ bot_bo3230846185698255993la_a_b ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_271_singletonD,axiom,
! [B2: relational_fmla_a_b,A: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ B2 @ ( insert7010464514620295119la_a_b @ A @ bot_bo4495933725496725865la_a_b ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_272_Un__empty__right,axiom,
! [A2: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ A2 @ bot_bo2891247006866115487la_a_b )
= A2 ) ).
% Un_empty_right
thf(fact_273_Un__empty__right,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% Un_empty_right
thf(fact_274_Un__empty__right,axiom,
! [A2: set_li8334890247616788453la_a_b] :
( ( sup_su8392741853961112337la_a_b @ A2 @ bot_bo3230846185698255993la_a_b )
= A2 ) ).
% Un_empty_right
thf(fact_275_Un__empty__right,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ A2 @ bot_bo4495933725496725865la_a_b )
= A2 ) ).
% Un_empty_right
thf(fact_276_Un__empty__left,axiom,
! [B: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ bot_bo2891247006866115487la_a_b @ B )
= B ) ).
% Un_empty_left
thf(fact_277_Un__empty__left,axiom,
! [B: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ B )
= B ) ).
% Un_empty_left
thf(fact_278_Un__empty__left,axiom,
! [B: set_li8334890247616788453la_a_b] :
( ( sup_su8392741853961112337la_a_b @ bot_bo3230846185698255993la_a_b @ B )
= B ) ).
% Un_empty_left
thf(fact_279_Un__empty__left,axiom,
! [B: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ bot_bo4495933725496725865la_a_b @ B )
= B ) ).
% Un_empty_left
thf(fact_280_singleton__Un__iff,axiom,
! [X: set_Re381260168593705685la_a_b,A2: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ( ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b )
= ( sup_su4783144482993978935la_a_b @ A2 @ B ) )
= ( ( ( A2 = bot_bo2891247006866115487la_a_b )
& ( B
= ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) ) )
| ( ( A2
= ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) )
& ( B = bot_bo2891247006866115487la_a_b ) )
| ( ( A2
= ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) )
& ( B
= ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_281_singleton__Un__iff,axiom,
! [X: nat,A2: set_nat,B: set_nat] :
( ( ( insert_nat2 @ X @ bot_bot_set_nat )
= ( sup_sup_set_nat @ A2 @ B ) )
= ( ( ( A2 = bot_bot_set_nat )
& ( B
= ( insert_nat2 @ X @ bot_bot_set_nat ) ) )
| ( ( A2
= ( insert_nat2 @ X @ bot_bot_set_nat ) )
& ( B = bot_bot_set_nat ) )
| ( ( A2
= ( insert_nat2 @ X @ bot_bot_set_nat ) )
& ( B
= ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_282_singleton__Un__iff,axiom,
! [X: list_R8263082107343818799la_a_b,A2: set_li8334890247616788453la_a_b,B: set_li8334890247616788453la_a_b] :
( ( ( insert3617074100887873119la_a_b @ X @ bot_bo3230846185698255993la_a_b )
= ( sup_su8392741853961112337la_a_b @ A2 @ B ) )
= ( ( ( A2 = bot_bo3230846185698255993la_a_b )
& ( B
= ( insert3617074100887873119la_a_b @ X @ bot_bo3230846185698255993la_a_b ) ) )
| ( ( A2
= ( insert3617074100887873119la_a_b @ X @ bot_bo3230846185698255993la_a_b ) )
& ( B = bot_bo3230846185698255993la_a_b ) )
| ( ( A2
= ( insert3617074100887873119la_a_b @ X @ bot_bo3230846185698255993la_a_b ) )
& ( B
= ( insert3617074100887873119la_a_b @ X @ bot_bo3230846185698255993la_a_b ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_283_singleton__Un__iff,axiom,
! [X: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b )
= ( sup_su5130108678486352897la_a_b @ A2 @ B ) )
= ( ( ( A2 = bot_bo4495933725496725865la_a_b )
& ( B
= ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) )
| ( ( A2
= ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) )
& ( B = bot_bo4495933725496725865la_a_b ) )
| ( ( A2
= ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) )
& ( B
= ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_284_Un__singleton__iff,axiom,
! [A2: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b,X: set_Re381260168593705685la_a_b] :
( ( ( sup_su4783144482993978935la_a_b @ A2 @ B )
= ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) )
= ( ( ( A2 = bot_bo2891247006866115487la_a_b )
& ( B
= ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) ) )
| ( ( A2
= ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) )
& ( B = bot_bo2891247006866115487la_a_b ) )
| ( ( A2
= ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) )
& ( B
= ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_285_Un__singleton__iff,axiom,
! [A2: set_nat,B: set_nat,X: nat] :
( ( ( sup_sup_set_nat @ A2 @ B )
= ( insert_nat2 @ X @ bot_bot_set_nat ) )
= ( ( ( A2 = bot_bot_set_nat )
& ( B
= ( insert_nat2 @ X @ bot_bot_set_nat ) ) )
| ( ( A2
= ( insert_nat2 @ X @ bot_bot_set_nat ) )
& ( B = bot_bot_set_nat ) )
| ( ( A2
= ( insert_nat2 @ X @ bot_bot_set_nat ) )
& ( B
= ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_286_Un__singleton__iff,axiom,
! [A2: set_li8334890247616788453la_a_b,B: set_li8334890247616788453la_a_b,X: list_R8263082107343818799la_a_b] :
( ( ( sup_su8392741853961112337la_a_b @ A2 @ B )
= ( insert3617074100887873119la_a_b @ X @ bot_bo3230846185698255993la_a_b ) )
= ( ( ( A2 = bot_bo3230846185698255993la_a_b )
& ( B
= ( insert3617074100887873119la_a_b @ X @ bot_bo3230846185698255993la_a_b ) ) )
| ( ( A2
= ( insert3617074100887873119la_a_b @ X @ bot_bo3230846185698255993la_a_b ) )
& ( B = bot_bo3230846185698255993la_a_b ) )
| ( ( A2
= ( insert3617074100887873119la_a_b @ X @ bot_bo3230846185698255993la_a_b ) )
& ( B
= ( insert3617074100887873119la_a_b @ X @ bot_bo3230846185698255993la_a_b ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_287_Un__singleton__iff,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,X: relational_fmla_a_b] :
( ( ( sup_su5130108678486352897la_a_b @ A2 @ B )
= ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) )
= ( ( ( A2 = bot_bo4495933725496725865la_a_b )
& ( B
= ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) )
| ( ( A2
= ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) )
& ( B = bot_bo4495933725496725865la_a_b ) )
| ( ( A2
= ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) )
& ( B
= ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_288_insert__is__Un,axiom,
( insert2023870700798818565la_a_b
= ( ^ [A3: set_Re381260168593705685la_a_b] : ( sup_su4783144482993978935la_a_b @ ( insert2023870700798818565la_a_b @ A3 @ bot_bo2891247006866115487la_a_b ) ) ) ) ).
% insert_is_Un
thf(fact_289_insert__is__Un,axiom,
( insert_nat2
= ( ^ [A3: nat] : ( sup_sup_set_nat @ ( insert_nat2 @ A3 @ bot_bot_set_nat ) ) ) ) ).
% insert_is_Un
thf(fact_290_insert__is__Un,axiom,
( insert3617074100887873119la_a_b
= ( ^ [A3: list_R8263082107343818799la_a_b] : ( sup_su8392741853961112337la_a_b @ ( insert3617074100887873119la_a_b @ A3 @ bot_bo3230846185698255993la_a_b ) ) ) ) ).
% insert_is_Un
thf(fact_291_insert__is__Un,axiom,
( insert7010464514620295119la_a_b
= ( ^ [A3: relational_fmla_a_b] : ( sup_su5130108678486352897la_a_b @ ( insert7010464514620295119la_a_b @ A3 @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% insert_is_Un
thf(fact_292_the__elem__eq,axiom,
! [X: set_Re381260168593705685la_a_b] :
( ( the_el7486773796875720352la_a_b @ ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) )
= X ) ).
% the_elem_eq
thf(fact_293_the__elem__eq,axiom,
! [X: nat] :
( ( the_elem_nat @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
= X ) ).
% the_elem_eq
thf(fact_294_the__elem__eq,axiom,
! [X: list_R8263082107343818799la_a_b] :
( ( the_el2253512510949844602la_a_b @ ( insert3617074100887873119la_a_b @ X @ bot_bo3230846185698255993la_a_b ) )
= X ) ).
% the_elem_eq
thf(fact_295_the__elem__eq,axiom,
! [X: relational_fmla_a_b] :
( ( the_el6350558617753882986la_a_b @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) )
= X ) ).
% the_elem_eq
thf(fact_296_bot__apply,axiom,
( bot_bo8852203127187332700_a_b_o
= ( ^ [X2: relational_fmla_a_b] : bot_bot_o ) ) ).
% bot_apply
thf(fact_297_foldr__cong,axiom,
! [A: relational_fmla_a_b,B2: relational_fmla_a_b,L: list_R8263082107343818799la_a_b,K: list_R8263082107343818799la_a_b,F: relational_fmla_a_b > relational_fmla_a_b > relational_fmla_a_b,G: relational_fmla_a_b > relational_fmla_a_b > relational_fmla_a_b] :
( ( A = B2 )
=> ( ( L = K )
=> ( ! [A5: relational_fmla_a_b,X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ ( set_Re9104216502384355786la_a_b @ L ) )
=> ( ( F @ X3 @ A5 )
= ( G @ X3 @ A5 ) ) )
=> ( ( foldr_789212930732525799la_a_b @ F @ L @ A )
= ( foldr_789212930732525799la_a_b @ G @ K @ B2 ) ) ) ) ) ).
% foldr_cong
thf(fact_298_is__singletonI,axiom,
! [X: set_Re381260168593705685la_a_b] : ( is_sin1114528679824004833la_a_b @ ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) ) ).
% is_singletonI
thf(fact_299_is__singletonI,axiom,
! [X: nat] : ( is_singleton_nat @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ).
% is_singletonI
thf(fact_300_is__singletonI,axiom,
! [X: list_R8263082107343818799la_a_b] : ( is_sin7113676573447591483la_a_b @ ( insert3617074100887873119la_a_b @ X @ bot_bo3230846185698255993la_a_b ) ) ).
% is_singletonI
thf(fact_301_is__singletonI,axiom,
! [X: relational_fmla_a_b] : ( is_sin6594375743535830443la_a_b @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) ).
% is_singletonI
thf(fact_302_boolean__algebra_Odisj__zero__right,axiom,
! [X: relational_fmla_a_b > $o] :
( ( sup_su1471977682094119364_a_b_o @ X @ bot_bo8852203127187332700_a_b_o )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_303_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ X @ bot_bo2891247006866115487la_a_b )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_304_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_305_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_li8334890247616788453la_a_b] :
( ( sup_su8392741853961112337la_a_b @ X @ bot_bo3230846185698255993la_a_b )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_306_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ X @ bot_bo4495933725496725865la_a_b )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_307_Set_Ois__empty__def,axiom,
( is_emp2970527627064493955la_a_b
= ( ^ [A4: set_se6865892389300016395la_a_b] : ( A4 = bot_bo2891247006866115487la_a_b ) ) ) ).
% Set.is_empty_def
thf(fact_308_Set_Ois__empty__def,axiom,
( is_empty_nat
= ( ^ [A4: set_nat] : ( A4 = bot_bot_set_nat ) ) ) ).
% Set.is_empty_def
thf(fact_309_Set_Ois__empty__def,axiom,
( is_emp3097910472067716573la_a_b
= ( ^ [A4: set_li8334890247616788453la_a_b] : ( A4 = bot_bo3230846185698255993la_a_b ) ) ) ).
% Set.is_empty_def
thf(fact_310_Set_Ois__empty__def,axiom,
( is_emp6953259385542938189la_a_b
= ( ^ [A4: set_Re381260168593705685la_a_b] : ( A4 = bot_bo4495933725496725865la_a_b ) ) ) ).
% Set.is_empty_def
thf(fact_311_is__singleton__def,axiom,
( is_sin1114528679824004833la_a_b
= ( ^ [A4: set_se6865892389300016395la_a_b] :
? [X2: set_Re381260168593705685la_a_b] :
( A4
= ( insert2023870700798818565la_a_b @ X2 @ bot_bo2891247006866115487la_a_b ) ) ) ) ).
% is_singleton_def
thf(fact_312_is__singleton__def,axiom,
( is_singleton_nat
= ( ^ [A4: set_nat] :
? [X2: nat] :
( A4
= ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_def
thf(fact_313_is__singleton__def,axiom,
( is_sin7113676573447591483la_a_b
= ( ^ [A4: set_li8334890247616788453la_a_b] :
? [X2: list_R8263082107343818799la_a_b] :
( A4
= ( insert3617074100887873119la_a_b @ X2 @ bot_bo3230846185698255993la_a_b ) ) ) ) ).
% is_singleton_def
thf(fact_314_is__singleton__def,axiom,
( is_sin6594375743535830443la_a_b
= ( ^ [A4: set_Re381260168593705685la_a_b] :
? [X2: relational_fmla_a_b] :
( A4
= ( insert7010464514620295119la_a_b @ X2 @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% is_singleton_def
thf(fact_315_is__singletonE,axiom,
! [A2: set_se6865892389300016395la_a_b] :
( ( is_sin1114528679824004833la_a_b @ A2 )
=> ~ ! [X3: set_Re381260168593705685la_a_b] :
( A2
!= ( insert2023870700798818565la_a_b @ X3 @ bot_bo2891247006866115487la_a_b ) ) ) ).
% is_singletonE
thf(fact_316_is__singletonE,axiom,
! [A2: set_nat] :
( ( is_singleton_nat @ A2 )
=> ~ ! [X3: nat] :
( A2
!= ( insert_nat2 @ X3 @ bot_bot_set_nat ) ) ) ).
% is_singletonE
thf(fact_317_is__singletonE,axiom,
! [A2: set_li8334890247616788453la_a_b] :
( ( is_sin7113676573447591483la_a_b @ A2 )
=> ~ ! [X3: list_R8263082107343818799la_a_b] :
( A2
!= ( insert3617074100887873119la_a_b @ X3 @ bot_bo3230846185698255993la_a_b ) ) ) ).
% is_singletonE
thf(fact_318_is__singletonE,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( is_sin6594375743535830443la_a_b @ A2 )
=> ~ ! [X3: relational_fmla_a_b] :
( A2
!= ( insert7010464514620295119la_a_b @ X3 @ bot_bo4495933725496725865la_a_b ) ) ) ).
% is_singletonE
thf(fact_319_set__union,axiom,
! [Xs: list_l3216422893793986623la_a_b,Ys: list_l3216422893793986623la_a_b] :
( ( set_li8988895567068466906la_a_b @ ( union_2282237785388101813la_a_b @ Xs @ Ys ) )
= ( sup_su8392741853961112337la_a_b @ ( set_li8988895567068466906la_a_b @ Xs ) @ ( set_li8988895567068466906la_a_b @ Ys ) ) ) ).
% set_union
thf(fact_320_set__union,axiom,
! [Xs: list_s3671366213839843941la_a_b,Ys: list_s3671366213839843941la_a_b] :
( ( set_se5111286776497155840la_a_b @ ( union_7072582408929332827la_a_b @ Xs @ Ys ) )
= ( sup_su4783144482993978935la_a_b @ ( set_se5111286776497155840la_a_b @ Xs ) @ ( set_se5111286776497155840la_a_b @ Ys ) ) ) ).
% set_union
thf(fact_321_set__union,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( set_nat2 @ ( union_nat @ Xs @ Ys ) )
= ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) ) ) ).
% set_union
thf(fact_322_set__union,axiom,
! [Xs: list_R8263082107343818799la_a_b,Ys: list_R8263082107343818799la_a_b] :
( ( set_Re9104216502384355786la_a_b @ ( union_5669641164172981541la_a_b @ Xs @ Ys ) )
= ( sup_su5130108678486352897la_a_b @ ( set_Re9104216502384355786la_a_b @ Xs ) @ ( set_Re9104216502384355786la_a_b @ Ys ) ) ) ).
% set_union
thf(fact_323_boolean__algebra__cancel_Osup2,axiom,
! [B: set_se6865892389300016395la_a_b,K: set_se6865892389300016395la_a_b,B2: set_se6865892389300016395la_a_b,A: set_se6865892389300016395la_a_b] :
( ( B
= ( sup_su4783144482993978935la_a_b @ K @ B2 ) )
=> ( ( sup_su4783144482993978935la_a_b @ A @ B )
= ( sup_su4783144482993978935la_a_b @ K @ ( sup_su4783144482993978935la_a_b @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_324_boolean__algebra__cancel_Osup2,axiom,
! [B: nat,K: nat,B2: nat,A: nat] :
( ( B
= ( sup_sup_nat @ K @ B2 ) )
=> ( ( sup_sup_nat @ A @ B )
= ( sup_sup_nat @ K @ ( sup_sup_nat @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_325_boolean__algebra__cancel_Osup2,axiom,
! [B: set_nat,K: set_nat,B2: set_nat,A: set_nat] :
( ( B
= ( sup_sup_set_nat @ K @ B2 ) )
=> ( ( sup_sup_set_nat @ A @ B )
= ( sup_sup_set_nat @ K @ ( sup_sup_set_nat @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_326_boolean__algebra__cancel_Osup2,axiom,
! [B: set_Re381260168593705685la_a_b,K: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b] :
( ( B
= ( sup_su5130108678486352897la_a_b @ K @ B2 ) )
=> ( ( sup_su5130108678486352897la_a_b @ A @ B )
= ( sup_su5130108678486352897la_a_b @ K @ ( sup_su5130108678486352897la_a_b @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_327_boolean__algebra__cancel_Osup1,axiom,
! [A2: set_se6865892389300016395la_a_b,K: set_se6865892389300016395la_a_b,A: set_se6865892389300016395la_a_b,B2: set_se6865892389300016395la_a_b] :
( ( A2
= ( sup_su4783144482993978935la_a_b @ K @ A ) )
=> ( ( sup_su4783144482993978935la_a_b @ A2 @ B2 )
= ( sup_su4783144482993978935la_a_b @ K @ ( sup_su4783144482993978935la_a_b @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_328_boolean__algebra__cancel_Osup1,axiom,
! [A2: nat,K: nat,A: nat,B2: nat] :
( ( A2
= ( sup_sup_nat @ K @ A ) )
=> ( ( sup_sup_nat @ A2 @ B2 )
= ( sup_sup_nat @ K @ ( sup_sup_nat @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_329_boolean__algebra__cancel_Osup1,axiom,
! [A2: set_nat,K: set_nat,A: set_nat,B2: set_nat] :
( ( A2
= ( sup_sup_set_nat @ K @ A ) )
=> ( ( sup_sup_set_nat @ A2 @ B2 )
= ( sup_sup_set_nat @ K @ ( sup_sup_set_nat @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_330_boolean__algebra__cancel_Osup1,axiom,
! [A2: set_Re381260168593705685la_a_b,K: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( A2
= ( sup_su5130108678486352897la_a_b @ K @ A ) )
=> ( ( sup_su5130108678486352897la_a_b @ A2 @ B2 )
= ( sup_su5130108678486352897la_a_b @ K @ ( sup_su5130108678486352897la_a_b @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_331_Collect__empty__eq__bot,axiom,
! [P: set_Re381260168593705685la_a_b > $o] :
( ( ( collec2099942116761351594la_a_b @ P )
= bot_bo2891247006866115487la_a_b )
= ( P = bot_bo8397419132826146726_a_b_o ) ) ).
% Collect_empty_eq_bot
thf(fact_332_Collect__empty__eq__bot,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( P = bot_bot_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_333_Collect__empty__eq__bot,axiom,
! [P: list_R8263082107343818799la_a_b > $o] :
( ( ( collec1121316490678886532la_a_b @ P )
= bot_bo3230846185698255993la_a_b )
= ( P = bot_bo217863987673589068_a_b_o ) ) ).
% Collect_empty_eq_bot
thf(fact_334_Collect__empty__eq__bot,axiom,
! [P: relational_fmla_a_b > $o] :
( ( ( collec3419995626248312948la_a_b @ P )
= bot_bo4495933725496725865la_a_b )
= ( P = bot_bo8852203127187332700_a_b_o ) ) ).
% Collect_empty_eq_bot
thf(fact_335_is__singleton__the__elem,axiom,
( is_sin1114528679824004833la_a_b
= ( ^ [A4: set_se6865892389300016395la_a_b] :
( A4
= ( insert2023870700798818565la_a_b @ ( the_el7486773796875720352la_a_b @ A4 ) @ bot_bo2891247006866115487la_a_b ) ) ) ) ).
% is_singleton_the_elem
thf(fact_336_is__singleton__the__elem,axiom,
( is_singleton_nat
= ( ^ [A4: set_nat] :
( A4
= ( insert_nat2 @ ( the_elem_nat @ A4 ) @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_337_is__singleton__the__elem,axiom,
( is_sin7113676573447591483la_a_b
= ( ^ [A4: set_li8334890247616788453la_a_b] :
( A4
= ( insert3617074100887873119la_a_b @ ( the_el2253512510949844602la_a_b @ A4 ) @ bot_bo3230846185698255993la_a_b ) ) ) ) ).
% is_singleton_the_elem
thf(fact_338_is__singleton__the__elem,axiom,
( is_sin6594375743535830443la_a_b
= ( ^ [A4: set_Re381260168593705685la_a_b] :
( A4
= ( insert7010464514620295119la_a_b @ ( the_el6350558617753882986la_a_b @ A4 ) @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% is_singleton_the_elem
thf(fact_339_is__singletonI_H,axiom,
! [A2: set_se6865892389300016395la_a_b] :
( ( A2 != bot_bo2891247006866115487la_a_b )
=> ( ! [X3: set_Re381260168593705685la_a_b,Y2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X3 @ A2 )
=> ( ( member3481406638322139244la_a_b @ Y2 @ A2 )
=> ( X3 = Y2 ) ) )
=> ( is_sin1114528679824004833la_a_b @ A2 ) ) ) ).
% is_singletonI'
thf(fact_340_is__singletonI_H,axiom,
! [A2: set_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ! [X3: nat,Y2: nat] :
( ( member_nat2 @ X3 @ A2 )
=> ( ( member_nat2 @ Y2 @ A2 )
=> ( X3 = Y2 ) ) )
=> ( is_singleton_nat @ A2 ) ) ) ).
% is_singletonI'
thf(fact_341_is__singletonI_H,axiom,
! [A2: set_li8334890247616788453la_a_b] :
( ( A2 != bot_bo3230846185698255993la_a_b )
=> ( ! [X3: list_R8263082107343818799la_a_b,Y2: list_R8263082107343818799la_a_b] :
( ( member8712135081189583430la_a_b @ X3 @ A2 )
=> ( ( member8712135081189583430la_a_b @ Y2 @ A2 )
=> ( X3 = Y2 ) ) )
=> ( is_sin7113676573447591483la_a_b @ A2 ) ) ) ).
% is_singletonI'
thf(fact_342_is__singletonI_H,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( A2 != bot_bo4495933725496725865la_a_b )
=> ( ! [X3: relational_fmla_a_b,Y2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A2 )
=> ( ( member4680049679412964150la_a_b @ Y2 @ A2 )
=> ( X3 = Y2 ) ) )
=> ( is_sin6594375743535830443la_a_b @ A2 ) ) ) ).
% is_singletonI'
thf(fact_343_bot__fun__def,axiom,
( bot_bo8852203127187332700_a_b_o
= ( ^ [X2: relational_fmla_a_b] : bot_bot_o ) ) ).
% bot_fun_def
thf(fact_344_bot__empty__eq,axiom,
( bot_bo8397419132826146726_a_b_o
= ( ^ [X2: set_Re381260168593705685la_a_b] : ( member3481406638322139244la_a_b @ X2 @ bot_bo2891247006866115487la_a_b ) ) ) ).
% bot_empty_eq
thf(fact_345_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X2: nat] : ( member_nat2 @ X2 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_346_bot__empty__eq,axiom,
( bot_bo217863987673589068_a_b_o
= ( ^ [X2: list_R8263082107343818799la_a_b] : ( member8712135081189583430la_a_b @ X2 @ bot_bo3230846185698255993la_a_b ) ) ) ).
% bot_empty_eq
thf(fact_347_bot__empty__eq,axiom,
( bot_bo8852203127187332700_a_b_o
= ( ^ [X2: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X2 @ bot_bo4495933725496725865la_a_b ) ) ) ).
% bot_empty_eq
thf(fact_348_is__empty__set,axiom,
! [Xs: list_s3671366213839843941la_a_b] :
( ( is_emp2970527627064493955la_a_b @ ( set_se5111286776497155840la_a_b @ Xs ) )
= ( null_s6470977797459811143la_a_b @ Xs ) ) ).
% is_empty_set
thf(fact_349_is__empty__set,axiom,
! [Xs: list_l3216422893793986623la_a_b] :
( ( is_emp3097910472067716573la_a_b @ ( set_li8988895567068466906la_a_b @ Xs ) )
= ( null_l5966648429877118881la_a_b @ Xs ) ) ).
% is_empty_set
thf(fact_350_is__empty__set,axiom,
! [Xs: list_nat] :
( ( is_empty_nat @ ( set_nat2 @ Xs ) )
= ( null_nat @ Xs ) ) ).
% is_empty_set
thf(fact_351_is__empty__set,axiom,
! [Xs: list_R8263082107343818799la_a_b] :
( ( is_emp6953259385542938189la_a_b @ ( set_Re9104216502384355786la_a_b @ Xs ) )
= ( null_R4120973431801868817la_a_b @ Xs ) ) ).
% is_empty_set
thf(fact_352_List_Oset__insert,axiom,
! [X: set_Re381260168593705685la_a_b,Xs: list_s3671366213839843941la_a_b] :
( ( set_se5111286776497155840la_a_b @ ( insert443281300761040569la_a_b @ X @ Xs ) )
= ( insert2023870700798818565la_a_b @ X @ ( set_se5111286776497155840la_a_b @ Xs ) ) ) ).
% List.set_insert
thf(fact_353_List_Oset__insert,axiom,
! [X: list_R8263082107343818799la_a_b,Xs: list_l3216422893793986623la_a_b] :
( ( set_li8988895567068466906la_a_b @ ( insert3209172818022656531la_a_b @ X @ Xs ) )
= ( insert3617074100887873119la_a_b @ X @ ( set_li8988895567068466906la_a_b @ Xs ) ) ) ).
% List.set_insert
thf(fact_354_List_Oset__insert,axiom,
! [X: nat,Xs: list_nat] :
( ( set_nat2 @ ( insert_nat @ X @ Xs ) )
= ( insert_nat2 @ X @ ( set_nat2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_355_List_Oset__insert,axiom,
! [X: relational_fmla_a_b,Xs: list_R8263082107343818799la_a_b] :
( ( set_Re9104216502384355786la_a_b @ ( insert5052544347734373763la_a_b @ X @ Xs ) )
= ( insert7010464514620295119la_a_b @ X @ ( set_Re9104216502384355786la_a_b @ Xs ) ) ) ).
% List.set_insert
thf(fact_356_in__set__member,axiom,
! [X: set_Re381260168593705685la_a_b,Xs: list_s3671366213839843941la_a_b] :
( ( member3481406638322139244la_a_b @ X @ ( set_se5111286776497155840la_a_b @ Xs ) )
= ( member1547870223199108474la_a_b @ Xs @ X ) ) ).
% in_set_member
thf(fact_357_in__set__member,axiom,
! [X: list_R8263082107343818799la_a_b,Xs: list_l3216422893793986623la_a_b] :
( ( member8712135081189583430la_a_b @ X @ ( set_li8988895567068466906la_a_b @ Xs ) )
= ( member7957436874562977876la_a_b @ Xs @ X ) ) ).
% in_set_member
thf(fact_358_in__set__member,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
= ( member_nat @ Xs @ X ) ) ).
% in_set_member
thf(fact_359_in__set__member,axiom,
! [X: relational_fmla_a_b,Xs: list_R8263082107343818799la_a_b] :
( ( member4680049679412964150la_a_b @ X @ ( set_Re9104216502384355786la_a_b @ Xs ) )
= ( member3779204850511880260la_a_b @ Xs @ X ) ) ).
% in_set_member
thf(fact_360_sup__bot_Osemilattice__neutr__axioms,axiom,
semila6368083915264326098_a_b_o @ sup_su1471977682094119364_a_b_o @ bot_bo8852203127187332700_a_b_o ).
% sup_bot.semilattice_neutr_axioms
thf(fact_361_sup__bot_Osemilattice__neutr__axioms,axiom,
semila4821588818088870057la_a_b @ sup_su4783144482993978935la_a_b @ bot_bo2891247006866115487la_a_b ).
% sup_bot.semilattice_neutr_axioms
thf(fact_362_sup__bot_Osemilattice__neutr__axioms,axiom,
semila1241773964035338532et_nat @ sup_sup_set_nat @ bot_bot_set_nat ).
% sup_bot.semilattice_neutr_axioms
thf(fact_363_sup__bot_Osemilattice__neutr__axioms,axiom,
semila5676999670054696067la_a_b @ sup_su8392741853961112337la_a_b @ bot_bo3230846185698255993la_a_b ).
% sup_bot.semilattice_neutr_axioms
thf(fact_364_sup__bot_Osemilattice__neutr__axioms,axiom,
semila8128863122405174899la_a_b @ sup_su5130108678486352897la_a_b @ bot_bo4495933725496725865la_a_b ).
% sup_bot.semilattice_neutr_axioms
thf(fact_365_union__set__fold,axiom,
! [Xs: list_l3216422893793986623la_a_b,A2: set_li8334890247616788453la_a_b] :
( ( sup_su8392741853961112337la_a_b @ ( set_li8988895567068466906la_a_b @ Xs ) @ A2 )
= ( fold_l8426049599603076129la_a_b @ insert3617074100887873119la_a_b @ Xs @ A2 ) ) ).
% union_set_fold
thf(fact_366_union__set__fold,axiom,
! [Xs: list_s3671366213839843941la_a_b,A2: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ ( set_se5111286776497155840la_a_b @ Xs ) @ A2 )
= ( fold_s8280270096837327341la_a_b @ insert2023870700798818565la_a_b @ Xs @ A2 ) ) ).
% union_set_fold
thf(fact_367_union__set__fold,axiom,
! [Xs: list_nat,A2: set_nat] :
( ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ A2 )
= ( fold_nat_set_nat @ insert_nat2 @ Xs @ A2 ) ) ).
% union_set_fold
thf(fact_368_union__set__fold,axiom,
! [Xs: list_R8263082107343818799la_a_b,A2: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ ( set_Re9104216502384355786la_a_b @ Xs ) @ A2 )
= ( fold_R3800324640475171969la_a_b @ insert7010464514620295119la_a_b @ Xs @ A2 ) ) ).
% union_set_fold
thf(fact_369_sup__bot_Omonoid__axioms,axiom,
monoid1149870907623488113_a_b_o @ sup_su1471977682094119364_a_b_o @ bot_bo8852203127187332700_a_b_o ).
% sup_bot.monoid_axioms
thf(fact_370_sup__bot_Omonoid__axioms,axiom,
monoid1791658389517411146la_a_b @ sup_su4783144482993978935la_a_b @ bot_bo2891247006866115487la_a_b ).
% sup_bot.monoid_axioms
thf(fact_371_sup__bot_Omonoid__axioms,axiom,
monoid_set_nat @ sup_sup_set_nat @ bot_bot_set_nat ).
% sup_bot.monoid_axioms
thf(fact_372_sup__bot_Omonoid__axioms,axiom,
monoid1701094794961311396la_a_b @ sup_su8392741853961112337la_a_b @ bot_bo3230846185698255993la_a_b ).
% sup_bot.monoid_axioms
thf(fact_373_sup__bot_Omonoid__axioms,axiom,
monoid7681897595190940948la_a_b @ sup_su5130108678486352897la_a_b @ bot_bo4495933725496725865la_a_b ).
% sup_bot.monoid_axioms
thf(fact_374_pairwise__singleton,axiom,
! [P: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b > $o,A2: set_Re381260168593705685la_a_b] : ( pairwi8740917440443160746la_a_b @ P @ ( insert2023870700798818565la_a_b @ A2 @ bot_bo2891247006866115487la_a_b ) ) ).
% pairwise_singleton
thf(fact_375_pairwise__singleton,axiom,
! [P: nat > nat > $o,A2: nat] : ( pairwise_nat @ P @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) ).
% pairwise_singleton
thf(fact_376_pairwise__singleton,axiom,
! [P: list_R8263082107343818799la_a_b > list_R8263082107343818799la_a_b > $o,A2: list_R8263082107343818799la_a_b] : ( pairwi51927932823847812la_a_b @ P @ ( insert3617074100887873119la_a_b @ A2 @ bot_bo3230846185698255993la_a_b ) ) ).
% pairwise_singleton
thf(fact_377_pairwise__singleton,axiom,
! [P: relational_fmla_a_b > relational_fmla_a_b > $o,A2: relational_fmla_a_b] : ( pairwi7405196478771624820la_a_b @ P @ ( insert7010464514620295119la_a_b @ A2 @ bot_bo4495933725496725865la_a_b ) ) ).
% pairwise_singleton
thf(fact_378_in__set__insert,axiom,
! [X: set_Re381260168593705685la_a_b,Xs: list_s3671366213839843941la_a_b] :
( ( member3481406638322139244la_a_b @ X @ ( set_se5111286776497155840la_a_b @ Xs ) )
=> ( ( insert443281300761040569la_a_b @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_379_in__set__insert,axiom,
! [X: list_R8263082107343818799la_a_b,Xs: list_l3216422893793986623la_a_b] :
( ( member8712135081189583430la_a_b @ X @ ( set_li8988895567068466906la_a_b @ Xs ) )
=> ( ( insert3209172818022656531la_a_b @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_380_in__set__insert,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ( insert_nat @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_381_in__set__insert,axiom,
! [X: relational_fmla_a_b,Xs: list_R8263082107343818799la_a_b] :
( ( member4680049679412964150la_a_b @ X @ ( set_Re9104216502384355786la_a_b @ Xs ) )
=> ( ( insert5052544347734373763la_a_b @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_382_list__ex1__iff,axiom,
( list_e6968285794140641868la_a_b
= ( ^ [P2: set_Re381260168593705685la_a_b > $o,Xs2: list_s3671366213839843941la_a_b] :
? [X2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X2 @ ( set_se5111286776497155840la_a_b @ Xs2 ) )
& ( P2 @ X2 )
& ! [Y3: set_Re381260168593705685la_a_b] :
( ( ( member3481406638322139244la_a_b @ Y3 @ ( set_se5111286776497155840la_a_b @ Xs2 ) )
& ( P2 @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_383_list__ex1__iff,axiom,
( list_e3575657358457426982la_a_b
= ( ^ [P2: list_R8263082107343818799la_a_b > $o,Xs2: list_l3216422893793986623la_a_b] :
? [X2: list_R8263082107343818799la_a_b] :
( ( member8712135081189583430la_a_b @ X2 @ ( set_li8988895567068466906la_a_b @ Xs2 ) )
& ( P2 @ X2 )
& ! [Y3: list_R8263082107343818799la_a_b] :
( ( ( member8712135081189583430la_a_b @ Y3 @ ( set_li8988895567068466906la_a_b @ Xs2 ) )
& ( P2 @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_384_list__ex1__iff,axiom,
( list_ex1_nat
= ( ^ [P2: nat > $o,Xs2: list_nat] :
? [X2: nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs2 ) )
& ( P2 @ X2 )
& ! [Y3: nat] :
( ( ( member_nat2 @ Y3 @ ( set_nat2 @ Xs2 ) )
& ( P2 @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_385_list__ex1__iff,axiom,
( list_e1892966175763416342la_a_b
= ( ^ [P2: relational_fmla_a_b > $o,Xs2: list_R8263082107343818799la_a_b] :
? [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ ( set_Re9104216502384355786la_a_b @ Xs2 ) )
& ( P2 @ X2 )
& ! [Y3: relational_fmla_a_b] :
( ( ( member4680049679412964150la_a_b @ Y3 @ ( set_Re9104216502384355786la_a_b @ Xs2 ) )
& ( P2 @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_386_pairwise__def,axiom,
( pairwi7405196478771624820la_a_b
= ( ^ [R: relational_fmla_a_b > relational_fmla_a_b > $o,S: set_Re381260168593705685la_a_b] :
! [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ S )
=> ! [Y3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Y3 @ S )
=> ( ( X2 != Y3 )
=> ( R @ X2 @ Y3 ) ) ) ) ) ) ).
% pairwise_def
thf(fact_387_pairwiseI,axiom,
! [S2: set_se6865892389300016395la_a_b,R2: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b > $o] :
( ! [X3: set_Re381260168593705685la_a_b,Y2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X3 @ S2 )
=> ( ( member3481406638322139244la_a_b @ Y2 @ S2 )
=> ( ( X3 != Y2 )
=> ( R2 @ X3 @ Y2 ) ) ) )
=> ( pairwi8740917440443160746la_a_b @ R2 @ S2 ) ) ).
% pairwiseI
thf(fact_388_pairwiseI,axiom,
! [S2: set_nat,R2: nat > nat > $o] :
( ! [X3: nat,Y2: nat] :
( ( member_nat2 @ X3 @ S2 )
=> ( ( member_nat2 @ Y2 @ S2 )
=> ( ( X3 != Y2 )
=> ( R2 @ X3 @ Y2 ) ) ) )
=> ( pairwise_nat @ R2 @ S2 ) ) ).
% pairwiseI
thf(fact_389_pairwiseI,axiom,
! [S2: set_li8334890247616788453la_a_b,R2: list_R8263082107343818799la_a_b > list_R8263082107343818799la_a_b > $o] :
( ! [X3: list_R8263082107343818799la_a_b,Y2: list_R8263082107343818799la_a_b] :
( ( member8712135081189583430la_a_b @ X3 @ S2 )
=> ( ( member8712135081189583430la_a_b @ Y2 @ S2 )
=> ( ( X3 != Y2 )
=> ( R2 @ X3 @ Y2 ) ) ) )
=> ( pairwi51927932823847812la_a_b @ R2 @ S2 ) ) ).
% pairwiseI
thf(fact_390_pairwiseI,axiom,
! [S2: set_Re381260168593705685la_a_b,R2: relational_fmla_a_b > relational_fmla_a_b > $o] :
( ! [X3: relational_fmla_a_b,Y2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ S2 )
=> ( ( member4680049679412964150la_a_b @ Y2 @ S2 )
=> ( ( X3 != Y2 )
=> ( R2 @ X3 @ Y2 ) ) ) )
=> ( pairwi7405196478771624820la_a_b @ R2 @ S2 ) ) ).
% pairwiseI
thf(fact_391_pairwiseD,axiom,
! [R2: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b > $o,S2: set_se6865892389300016395la_a_b,X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b] :
( ( pairwi8740917440443160746la_a_b @ R2 @ S2 )
=> ( ( member3481406638322139244la_a_b @ X @ S2 )
=> ( ( member3481406638322139244la_a_b @ Y @ S2 )
=> ( ( X != Y )
=> ( R2 @ X @ Y ) ) ) ) ) ).
% pairwiseD
thf(fact_392_pairwiseD,axiom,
! [R2: nat > nat > $o,S2: set_nat,X: nat,Y: nat] :
( ( pairwise_nat @ R2 @ S2 )
=> ( ( member_nat2 @ X @ S2 )
=> ( ( member_nat2 @ Y @ S2 )
=> ( ( X != Y )
=> ( R2 @ X @ Y ) ) ) ) ) ).
% pairwiseD
thf(fact_393_pairwiseD,axiom,
! [R2: list_R8263082107343818799la_a_b > list_R8263082107343818799la_a_b > $o,S2: set_li8334890247616788453la_a_b,X: list_R8263082107343818799la_a_b,Y: list_R8263082107343818799la_a_b] :
( ( pairwi51927932823847812la_a_b @ R2 @ S2 )
=> ( ( member8712135081189583430la_a_b @ X @ S2 )
=> ( ( member8712135081189583430la_a_b @ Y @ S2 )
=> ( ( X != Y )
=> ( R2 @ X @ Y ) ) ) ) ) ).
% pairwiseD
thf(fact_394_pairwiseD,axiom,
! [R2: relational_fmla_a_b > relational_fmla_a_b > $o,S2: set_Re381260168593705685la_a_b,X: relational_fmla_a_b,Y: relational_fmla_a_b] :
( ( pairwi7405196478771624820la_a_b @ R2 @ S2 )
=> ( ( member4680049679412964150la_a_b @ X @ S2 )
=> ( ( member4680049679412964150la_a_b @ Y @ S2 )
=> ( ( X != Y )
=> ( R2 @ X @ Y ) ) ) ) ) ).
% pairwiseD
thf(fact_395_fold__invariant,axiom,
! [Xs: list_R8263082107343818799la_a_b,Q: relational_fmla_a_b > $o,P: set_Re381260168593705685la_a_b > $o,S3: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b] :
( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ ( set_Re9104216502384355786la_a_b @ Xs ) )
=> ( Q @ X3 ) )
=> ( ( P @ S3 )
=> ( ! [X3: relational_fmla_a_b,S4: set_Re381260168593705685la_a_b] :
( ( Q @ X3 )
=> ( ( P @ S4 )
=> ( P @ ( F @ X3 @ S4 ) ) ) )
=> ( P @ ( fold_R3800324640475171969la_a_b @ F @ Xs @ S3 ) ) ) ) ) ).
% fold_invariant
thf(fact_396_fold__invariant,axiom,
! [Xs: list_R8263082107343818799la_a_b,Q: relational_fmla_a_b > $o,P: relational_fmla_a_b > $o,S3: relational_fmla_a_b,F: relational_fmla_a_b > relational_fmla_a_b > relational_fmla_a_b] :
( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ ( set_Re9104216502384355786la_a_b @ Xs ) )
=> ( Q @ X3 ) )
=> ( ( P @ S3 )
=> ( ! [X3: relational_fmla_a_b,S4: relational_fmla_a_b] :
( ( Q @ X3 )
=> ( ( P @ S4 )
=> ( P @ ( F @ X3 @ S4 ) ) ) )
=> ( P @ ( fold_R509364916538903371la_a_b @ F @ Xs @ S3 ) ) ) ) ) ).
% fold_invariant
thf(fact_397_fold__invariant,axiom,
! [Xs: list_s3671366213839843941la_a_b,Q: set_Re381260168593705685la_a_b > $o,P: set_Re381260168593705685la_a_b > $o,S3: set_Re381260168593705685la_a_b,F: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b] :
( ! [X3: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X3 @ ( set_se5111286776497155840la_a_b @ Xs ) )
=> ( Q @ X3 ) )
=> ( ( P @ S3 )
=> ( ! [X3: set_Re381260168593705685la_a_b,S4: set_Re381260168593705685la_a_b] :
( ( Q @ X3 )
=> ( ( P @ S4 )
=> ( P @ ( F @ X3 @ S4 ) ) ) )
=> ( P @ ( fold_s1495082921969268663la_a_b @ F @ Xs @ S3 ) ) ) ) ) ).
% fold_invariant
thf(fact_398_fold__invariant,axiom,
! [Xs: list_nat,Q: nat > $o,P: nat > $o,S3: nat,F: nat > nat > nat] :
( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ( Q @ X3 ) )
=> ( ( P @ S3 )
=> ( ! [X3: nat,S4: nat] :
( ( Q @ X3 )
=> ( ( P @ S4 )
=> ( P @ ( F @ X3 @ S4 ) ) ) )
=> ( P @ ( fold_nat_nat @ F @ Xs @ S3 ) ) ) ) ) ).
% fold_invariant
thf(fact_399_List_Ofold__cong,axiom,
! [A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b,Xs: list_R8263082107343818799la_a_b,Ys: list_R8263082107343818799la_a_b,F: relational_fmla_a_b > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b,G: relational_fmla_a_b > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b] :
( ( A = B2 )
=> ( ( Xs = Ys )
=> ( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ ( set_Re9104216502384355786la_a_b @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( fold_R3800324640475171969la_a_b @ F @ Xs @ A )
= ( fold_R3800324640475171969la_a_b @ G @ Ys @ B2 ) ) ) ) ) ).
% List.fold_cong
thf(fact_400_List_Ofold__cong,axiom,
! [A: relational_fmla_a_b,B2: relational_fmla_a_b,Xs: list_R8263082107343818799la_a_b,Ys: list_R8263082107343818799la_a_b,F: relational_fmla_a_b > relational_fmla_a_b > relational_fmla_a_b,G: relational_fmla_a_b > relational_fmla_a_b > relational_fmla_a_b] :
( ( A = B2 )
=> ( ( Xs = Ys )
=> ( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ ( set_Re9104216502384355786la_a_b @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( fold_R509364916538903371la_a_b @ F @ Xs @ A )
= ( fold_R509364916538903371la_a_b @ G @ Ys @ B2 ) ) ) ) ) ).
% List.fold_cong
thf(fact_401_List_Ofold__cong,axiom,
! [A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b,Xs: list_s3671366213839843941la_a_b,Ys: list_s3671366213839843941la_a_b,F: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b,G: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b] :
( ( A = B2 )
=> ( ( Xs = Ys )
=> ( ! [X3: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X3 @ ( set_se5111286776497155840la_a_b @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( fold_s1495082921969268663la_a_b @ F @ Xs @ A )
= ( fold_s1495082921969268663la_a_b @ G @ Ys @ B2 ) ) ) ) ) ).
% List.fold_cong
thf(fact_402_List_Ofold__cong,axiom,
! [A: nat,B2: nat,Xs: list_nat,Ys: list_nat,F: nat > nat > nat,G: nat > nat > nat] :
( ( A = B2 )
=> ( ( Xs = Ys )
=> ( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( fold_nat_nat @ F @ Xs @ A )
= ( fold_nat_nat @ G @ Ys @ B2 ) ) ) ) ) ).
% List.fold_cong
thf(fact_403_pairwise__empty,axiom,
! [P: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b > $o] : ( pairwi8740917440443160746la_a_b @ P @ bot_bo2891247006866115487la_a_b ) ).
% pairwise_empty
thf(fact_404_pairwise__empty,axiom,
! [P: nat > nat > $o] : ( pairwise_nat @ P @ bot_bot_set_nat ) ).
% pairwise_empty
thf(fact_405_pairwise__empty,axiom,
! [P: list_R8263082107343818799la_a_b > list_R8263082107343818799la_a_b > $o] : ( pairwi51927932823847812la_a_b @ P @ bot_bo3230846185698255993la_a_b ) ).
% pairwise_empty
thf(fact_406_pairwise__empty,axiom,
! [P: relational_fmla_a_b > relational_fmla_a_b > $o] : ( pairwi7405196478771624820la_a_b @ P @ bot_bo4495933725496725865la_a_b ) ).
% pairwise_empty
thf(fact_407_pairwise__insert,axiom,
! [R3: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b > $o,X: set_Re381260168593705685la_a_b,S3: set_se6865892389300016395la_a_b] :
( ( pairwi8740917440443160746la_a_b @ R3 @ ( insert2023870700798818565la_a_b @ X @ S3 ) )
= ( ! [Y3: set_Re381260168593705685la_a_b] :
( ( ( member3481406638322139244la_a_b @ Y3 @ S3 )
& ( Y3 != X ) )
=> ( ( R3 @ X @ Y3 )
& ( R3 @ Y3 @ X ) ) )
& ( pairwi8740917440443160746la_a_b @ R3 @ S3 ) ) ) ).
% pairwise_insert
thf(fact_408_pairwise__insert,axiom,
! [R3: nat > nat > $o,X: nat,S3: set_nat] :
( ( pairwise_nat @ R3 @ ( insert_nat2 @ X @ S3 ) )
= ( ! [Y3: nat] :
( ( ( member_nat2 @ Y3 @ S3 )
& ( Y3 != X ) )
=> ( ( R3 @ X @ Y3 )
& ( R3 @ Y3 @ X ) ) )
& ( pairwise_nat @ R3 @ S3 ) ) ) ).
% pairwise_insert
thf(fact_409_pairwise__insert,axiom,
! [R3: list_R8263082107343818799la_a_b > list_R8263082107343818799la_a_b > $o,X: list_R8263082107343818799la_a_b,S3: set_li8334890247616788453la_a_b] :
( ( pairwi51927932823847812la_a_b @ R3 @ ( insert3617074100887873119la_a_b @ X @ S3 ) )
= ( ! [Y3: list_R8263082107343818799la_a_b] :
( ( ( member8712135081189583430la_a_b @ Y3 @ S3 )
& ( Y3 != X ) )
=> ( ( R3 @ X @ Y3 )
& ( R3 @ Y3 @ X ) ) )
& ( pairwi51927932823847812la_a_b @ R3 @ S3 ) ) ) ).
% pairwise_insert
thf(fact_410_pairwise__insert,axiom,
! [R3: relational_fmla_a_b > relational_fmla_a_b > $o,X: relational_fmla_a_b,S3: set_Re381260168593705685la_a_b] :
( ( pairwi7405196478771624820la_a_b @ R3 @ ( insert7010464514620295119la_a_b @ X @ S3 ) )
= ( ! [Y3: relational_fmla_a_b] :
( ( ( member4680049679412964150la_a_b @ Y3 @ S3 )
& ( Y3 != X ) )
=> ( ( R3 @ X @ Y3 )
& ( R3 @ Y3 @ X ) ) )
& ( pairwi7405196478771624820la_a_b @ R3 @ S3 ) ) ) ).
% pairwise_insert
thf(fact_411_can__select__set__list__ex1,axiom,
! [P: set_Re381260168593705685la_a_b > $o,A2: list_s3671366213839843941la_a_b] :
( ( can_se982673022169791512la_a_b @ P @ ( set_se5111286776497155840la_a_b @ A2 ) )
= ( list_e6968285794140641868la_a_b @ P @ A2 ) ) ).
% can_select_set_list_ex1
thf(fact_412_can__select__set__list__ex1,axiom,
! [P: list_R8263082107343818799la_a_b > $o,A2: list_l3216422893793986623la_a_b] :
( ( can_se4691468443538722802la_a_b @ P @ ( set_li8988895567068466906la_a_b @ A2 ) )
= ( list_e3575657358457426982la_a_b @ P @ A2 ) ) ).
% can_select_set_list_ex1
thf(fact_413_can__select__set__list__ex1,axiom,
! [P: nat > $o,A2: list_nat] :
( ( can_select_nat @ P @ ( set_nat2 @ A2 ) )
= ( list_ex1_nat @ P @ A2 ) ) ).
% can_select_set_list_ex1
thf(fact_414_can__select__set__list__ex1,axiom,
! [P: relational_fmla_a_b > $o,A2: list_R8263082107343818799la_a_b] :
( ( can_se1623252625881650914la_a_b @ P @ ( set_Re9104216502384355786la_a_b @ A2 ) )
= ( list_e1892966175763416342la_a_b @ P @ A2 ) ) ).
% can_select_set_list_ex1
thf(fact_415_monoid_Oleft__neutral,axiom,
! [F: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b,Z: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b] :
( ( monoid7681897595190940948la_a_b @ F @ Z )
=> ( ( F @ Z @ A )
= A ) ) ).
% monoid.left_neutral
thf(fact_416_monoid_Oright__neutral,axiom,
! [F: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b,Z: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b] :
( ( monoid7681897595190940948la_a_b @ F @ Z )
=> ( ( F @ A @ Z )
= A ) ) ).
% monoid.right_neutral
thf(fact_417_Sup__set__fold,axiom,
! [Xs: list_R2386395423656516982_a_b_o] :
( ( comple3423763510802972509_a_b_o @ ( set_Re5970316886255677317_a_b_o @ Xs ) )
= ( fold_R6147292597075160331_a_b_o @ sup_su1471977682094119364_a_b_o @ Xs @ bot_bo8852203127187332700_a_b_o ) ) ).
% Sup_set_fold
thf(fact_418_Sup__set__fold,axiom,
! [Xs: list_s5517785546295487899la_a_b] :
( ( comple7774258788699406174la_a_b @ ( set_se4873105269812759862la_a_b @ Xs ) )
= ( fold_s5321508111014094371la_a_b @ sup_su4783144482993978935la_a_b @ Xs @ bot_bo2891247006866115487la_a_b ) ) ).
% Sup_set_fold
thf(fact_419_Sup__set__fold,axiom,
! [Xs: list_set_nat] :
( ( comple7399068483239264473et_nat @ ( set_set_nat2 @ Xs ) )
= ( fold_set_nat_set_nat @ sup_sup_set_nat @ Xs @ bot_bot_set_nat ) ) ).
% Sup_set_fold
thf(fact_420_Sup__set__fold,axiom,
! [Xs: list_s6159904150117900533la_a_b] :
( ( comple4129898094138812600la_a_b @ ( set_se7917458537561623184la_a_b @ Xs ) )
= ( fold_s2770086089136257751la_a_b @ sup_su8392741853961112337la_a_b @ Xs @ bot_bo3230846185698255993la_a_b ) ) ).
% Sup_set_fold
thf(fact_421_Sup__set__fold,axiom,
! [Xs: list_s3671366213839843941la_a_b] :
( ( comple8442120529048846632la_a_b @ ( set_se5111286776497155840la_a_b @ Xs ) )
= ( fold_s1495082921969268663la_a_b @ sup_su5130108678486352897la_a_b @ Xs @ bot_bo4495933725496725865la_a_b ) ) ).
% Sup_set_fold
thf(fact_422_pairwise__alt,axiom,
( pairwi51927932823847812la_a_b
= ( ^ [R: list_R8263082107343818799la_a_b > list_R8263082107343818799la_a_b > $o,S: set_li8334890247616788453la_a_b] :
! [X2: list_R8263082107343818799la_a_b] :
( ( member8712135081189583430la_a_b @ X2 @ S )
=> ! [Y3: list_R8263082107343818799la_a_b] :
( ( member8712135081189583430la_a_b @ Y3 @ ( minus_4977183416030855486la_a_b @ S @ ( insert3617074100887873119la_a_b @ X2 @ bot_bo3230846185698255993la_a_b ) ) )
=> ( R @ X2 @ Y3 ) ) ) ) ) ).
% pairwise_alt
thf(fact_423_pairwise__alt,axiom,
( pairwi8740917440443160746la_a_b
= ( ^ [R: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b > $o,S: set_se6865892389300016395la_a_b] :
! [X2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X2 @ S )
=> ! [Y3: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ Y3 @ ( minus_4705846553145473764la_a_b @ S @ ( insert2023870700798818565la_a_b @ X2 @ bot_bo2891247006866115487la_a_b ) ) )
=> ( R @ X2 @ Y3 ) ) ) ) ) ).
% pairwise_alt
thf(fact_424_pairwise__alt,axiom,
( pairwise_nat
= ( ^ [R: nat > nat > $o,S: set_nat] :
! [X2: nat] :
( ( member_nat2 @ X2 @ S )
=> ! [Y3: nat] :
( ( member_nat2 @ Y3 @ ( minus_minus_set_nat @ S @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) )
=> ( R @ X2 @ Y3 ) ) ) ) ) ).
% pairwise_alt
thf(fact_425_pairwise__alt,axiom,
( pairwi7405196478771624820la_a_b
= ( ^ [R: relational_fmla_a_b > relational_fmla_a_b > $o,S: set_Re381260168593705685la_a_b] :
! [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ S )
=> ! [Y3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Y3 @ ( minus_4077726661957047470la_a_b @ S @ ( insert7010464514620295119la_a_b @ X2 @ bot_bo4495933725496725865la_a_b ) ) )
=> ( R @ X2 @ Y3 ) ) ) ) ) ).
% pairwise_alt
thf(fact_426_List_Ounion__def,axiom,
( union_5669641164172981541la_a_b
= ( fold_R6874853119116082139la_a_b @ insert5052544347734373763la_a_b ) ) ).
% List.union_def
thf(fact_427_foldr__fold,axiom,
! [Xs: list_R8263082107343818799la_a_b,F: relational_fmla_a_b > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b] :
( ! [X3: relational_fmla_a_b,Y2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ ( set_Re9104216502384355786la_a_b @ Xs ) )
=> ( ( member4680049679412964150la_a_b @ Y2 @ ( set_Re9104216502384355786la_a_b @ Xs ) )
=> ( ( comp_s8660012457442781028la_a_b @ ( F @ Y2 ) @ ( F @ X3 ) )
= ( comp_s8660012457442781028la_a_b @ ( F @ X3 ) @ ( F @ Y2 ) ) ) ) )
=> ( ( foldr_7424894414501665309la_a_b @ F @ Xs )
= ( fold_R3800324640475171969la_a_b @ F @ Xs ) ) ) ).
% foldr_fold
thf(fact_428_foldr__fold,axiom,
! [Xs: list_s3671366213839843941la_a_b,F: set_Re381260168593705685la_a_b > relational_fmla_a_b > relational_fmla_a_b] :
( ! [X3: set_Re381260168593705685la_a_b,Y2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X3 @ ( set_se5111286776497155840la_a_b @ Xs ) )
=> ( ( member3481406638322139244la_a_b @ Y2 @ ( set_se5111286776497155840la_a_b @ Xs ) )
=> ( ( comp_R8862297837303855554la_a_b @ ( F @ Y2 ) @ ( F @ X3 ) )
= ( comp_R8862297837303855554la_a_b @ ( F @ X3 ) @ ( F @ Y2 ) ) ) ) )
=> ( ( foldr_1179560620697312285la_a_b @ F @ Xs )
= ( fold_s6778362883525594753la_a_b @ F @ Xs ) ) ) ).
% foldr_fold
thf(fact_429_foldr__fold,axiom,
! [Xs: list_s3671366213839843941la_a_b,F: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b] :
( ! [X3: set_Re381260168593705685la_a_b,Y2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X3 @ ( set_se5111286776497155840la_a_b @ Xs ) )
=> ( ( member3481406638322139244la_a_b @ Y2 @ ( set_se5111286776497155840la_a_b @ Xs ) )
=> ( ( comp_s8660012457442781028la_a_b @ ( F @ Y2 ) @ ( F @ X3 ) )
= ( comp_s8660012457442781028la_a_b @ ( F @ X3 ) @ ( F @ Y2 ) ) ) ) )
=> ( ( foldr_5041855395594655059la_a_b @ F @ Xs )
= ( fold_s1495082921969268663la_a_b @ F @ Xs ) ) ) ).
% foldr_fold
thf(fact_430_foldr__fold,axiom,
! [Xs: list_l3216422893793986623la_a_b,F: list_R8263082107343818799la_a_b > relational_fmla_a_b > relational_fmla_a_b] :
( ! [X3: list_R8263082107343818799la_a_b,Y2: list_R8263082107343818799la_a_b] :
( ( member8712135081189583430la_a_b @ X3 @ ( set_li8988895567068466906la_a_b @ Xs ) )
=> ( ( member8712135081189583430la_a_b @ Y2 @ ( set_li8988895567068466906la_a_b @ Xs ) )
=> ( ( comp_R8862297837303855554la_a_b @ ( F @ Y2 ) @ ( F @ X3 ) )
= ( comp_R8862297837303855554la_a_b @ ( F @ X3 ) @ ( F @ Y2 ) ) ) ) )
=> ( ( foldr_7800551455570250615la_a_b @ F @ Xs )
= ( fold_l4061290520114711515la_a_b @ F @ Xs ) ) ) ).
% foldr_fold
thf(fact_431_foldr__fold,axiom,
! [Xs: list_nat,F: nat > relational_fmla_a_b > relational_fmla_a_b] :
( ! [X3: nat,Y2: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( member_nat2 @ Y2 @ ( set_nat2 @ Xs ) )
=> ( ( comp_R8862297837303855554la_a_b @ ( F @ Y2 ) @ ( F @ X3 ) )
= ( comp_R8862297837303855554la_a_b @ ( F @ X3 ) @ ( F @ Y2 ) ) ) ) )
=> ( ( foldr_3281238248259268120la_a_b @ F @ Xs )
= ( fold_n2260362839379620988la_a_b @ F @ Xs ) ) ) ).
% foldr_fold
thf(fact_432_foldr__fold,axiom,
! [Xs: list_nat,F: nat > nat > nat] :
( ! [X3: nat,Y2: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( member_nat2 @ Y2 @ ( set_nat2 @ Xs ) )
=> ( ( comp_nat_nat_nat @ ( F @ Y2 ) @ ( F @ X3 ) )
= ( comp_nat_nat_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) ) ) )
=> ( ( foldr_nat_nat @ F @ Xs )
= ( fold_nat_nat @ F @ Xs ) ) ) ).
% foldr_fold
thf(fact_433_foldr__fold,axiom,
! [Xs: list_R8263082107343818799la_a_b,F: relational_fmla_a_b > relational_fmla_a_b > relational_fmla_a_b] :
( ! [X3: relational_fmla_a_b,Y2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ ( set_Re9104216502384355786la_a_b @ Xs ) )
=> ( ( member4680049679412964150la_a_b @ Y2 @ ( set_Re9104216502384355786la_a_b @ Xs ) )
=> ( ( comp_R8862297837303855554la_a_b @ ( F @ Y2 ) @ ( F @ X3 ) )
= ( comp_R8862297837303855554la_a_b @ ( F @ X3 ) @ ( F @ Y2 ) ) ) ) )
=> ( ( foldr_789212930732525799la_a_b @ F @ Xs )
= ( fold_R509364916538903371la_a_b @ F @ Xs ) ) ) ).
% foldr_fold
thf(fact_434_not__in__set__insert,axiom,
! [X: list_R8263082107343818799la_a_b,Xs: list_l3216422893793986623la_a_b] :
( ~ ( member8712135081189583430la_a_b @ X @ ( set_li8988895567068466906la_a_b @ Xs ) )
=> ( ( insert3209172818022656531la_a_b @ X @ Xs )
= ( cons_l6716632503304403961la_a_b @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_435_not__in__set__insert,axiom,
! [X: set_Re381260168593705685la_a_b,Xs: list_s3671366213839843941la_a_b] :
( ~ ( member3481406638322139244la_a_b @ X @ ( set_se5111286776497155840la_a_b @ Xs ) )
=> ( ( insert443281300761040569la_a_b @ X @ Xs )
= ( cons_s2104095074707585951la_a_b @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_436_not__in__set__insert,axiom,
! [X: nat,Xs: list_nat] :
( ~ ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ( insert_nat @ X @ Xs )
= ( cons_nat @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_437_not__in__set__insert,axiom,
! [X: relational_fmla_a_b,Xs: list_R8263082107343818799la_a_b] :
( ~ ( member4680049679412964150la_a_b @ X @ ( set_Re9104216502384355786la_a_b @ Xs ) )
=> ( ( insert5052544347734373763la_a_b @ X @ Xs )
= ( cons_R4161027894713053801la_a_b @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_438_totalp__on__singleton,axiom,
! [X: set_Re381260168593705685la_a_b,R2: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b > $o] : ( totalp3012131419089278729la_a_b @ ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) @ R2 ) ).
% totalp_on_singleton
thf(fact_439_totalp__on__singleton,axiom,
! [X: nat,R2: nat > nat > $o] : ( totalp_on_nat @ ( insert_nat2 @ X @ bot_bot_set_nat ) @ R2 ) ).
% totalp_on_singleton
thf(fact_440_totalp__on__singleton,axiom,
! [X: list_R8263082107343818799la_a_b,R2: list_R8263082107343818799la_a_b > list_R8263082107343818799la_a_b > $o] : ( totalp7313527317358576227la_a_b @ ( insert3617074100887873119la_a_b @ X @ bot_bo3230846185698255993la_a_b ) @ R2 ) ).
% totalp_on_singleton
thf(fact_441_totalp__on__singleton,axiom,
! [X: relational_fmla_a_b,R2: relational_fmla_a_b > relational_fmla_a_b > $o] : ( totalp1437913959296628179la_a_b @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) @ R2 ) ).
% totalp_on_singleton
thf(fact_442_list_Oinject,axiom,
! [X21: relational_fmla_a_b,X22: list_R8263082107343818799la_a_b,Y21: relational_fmla_a_b,Y22: list_R8263082107343818799la_a_b] :
( ( ( cons_R4161027894713053801la_a_b @ X21 @ X22 )
= ( cons_R4161027894713053801la_a_b @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_443_list_Oinject,axiom,
! [X21: set_Re381260168593705685la_a_b,X22: list_s3671366213839843941la_a_b,Y21: set_Re381260168593705685la_a_b,Y22: list_s3671366213839843941la_a_b] :
( ( ( cons_s2104095074707585951la_a_b @ X21 @ X22 )
= ( cons_s2104095074707585951la_a_b @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_444_list_Oinject,axiom,
! [X21: nat,X22: list_nat,Y21: nat,Y22: list_nat] :
( ( ( cons_nat @ X21 @ X22 )
= ( cons_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_445_DiffI,axiom,
! [C2: list_R8263082107343818799la_a_b,A2: set_li8334890247616788453la_a_b,B: set_li8334890247616788453la_a_b] :
( ( member8712135081189583430la_a_b @ C2 @ A2 )
=> ( ~ ( member8712135081189583430la_a_b @ C2 @ B )
=> ( member8712135081189583430la_a_b @ C2 @ ( minus_4977183416030855486la_a_b @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_446_DiffI,axiom,
! [C2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C2 @ A2 )
=> ( ~ ( member4680049679412964150la_a_b @ C2 @ B )
=> ( member4680049679412964150la_a_b @ C2 @ ( minus_4077726661957047470la_a_b @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_447_DiffI,axiom,
! [C2: set_Re381260168593705685la_a_b,A2: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ C2 @ A2 )
=> ( ~ ( member3481406638322139244la_a_b @ C2 @ B )
=> ( member3481406638322139244la_a_b @ C2 @ ( minus_4705846553145473764la_a_b @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_448_DiffI,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat2 @ C2 @ A2 )
=> ( ~ ( member_nat2 @ C2 @ B )
=> ( member_nat2 @ C2 @ ( minus_minus_set_nat @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_449_Diff__iff,axiom,
! [C2: list_R8263082107343818799la_a_b,A2: set_li8334890247616788453la_a_b,B: set_li8334890247616788453la_a_b] :
( ( member8712135081189583430la_a_b @ C2 @ ( minus_4977183416030855486la_a_b @ A2 @ B ) )
= ( ( member8712135081189583430la_a_b @ C2 @ A2 )
& ~ ( member8712135081189583430la_a_b @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_450_Diff__iff,axiom,
! [C2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C2 @ ( minus_4077726661957047470la_a_b @ A2 @ B ) )
= ( ( member4680049679412964150la_a_b @ C2 @ A2 )
& ~ ( member4680049679412964150la_a_b @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_451_Diff__iff,axiom,
! [C2: set_Re381260168593705685la_a_b,A2: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ C2 @ ( minus_4705846553145473764la_a_b @ A2 @ B ) )
= ( ( member3481406638322139244la_a_b @ C2 @ A2 )
& ~ ( member3481406638322139244la_a_b @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_452_Diff__iff,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat2 @ C2 @ ( minus_minus_set_nat @ A2 @ B ) )
= ( ( member_nat2 @ C2 @ A2 )
& ~ ( member_nat2 @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_453_Diff__idemp,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( minus_4077726661957047470la_a_b @ ( minus_4077726661957047470la_a_b @ A2 @ B ) @ B )
= ( minus_4077726661957047470la_a_b @ A2 @ B ) ) ).
% Diff_idemp
thf(fact_454_Diff__idemp,axiom,
! [A2: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ( minus_4705846553145473764la_a_b @ ( minus_4705846553145473764la_a_b @ A2 @ B ) @ B )
= ( minus_4705846553145473764la_a_b @ A2 @ B ) ) ).
% Diff_idemp
thf(fact_455_Diff__idemp,axiom,
! [A2: set_nat,B: set_nat] :
( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B ) @ B )
= ( minus_minus_set_nat @ A2 @ B ) ) ).
% Diff_idemp
thf(fact_456_Diff__empty,axiom,
! [A2: set_li8334890247616788453la_a_b] :
( ( minus_4977183416030855486la_a_b @ A2 @ bot_bo3230846185698255993la_a_b )
= A2 ) ).
% Diff_empty
thf(fact_457_Diff__empty,axiom,
! [A2: set_se6865892389300016395la_a_b] :
( ( minus_4705846553145473764la_a_b @ A2 @ bot_bo2891247006866115487la_a_b )
= A2 ) ).
% Diff_empty
thf(fact_458_Diff__empty,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% Diff_empty
thf(fact_459_Diff__empty,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( minus_4077726661957047470la_a_b @ A2 @ bot_bo4495933725496725865la_a_b )
= A2 ) ).
% Diff_empty
thf(fact_460_empty__Diff,axiom,
! [A2: set_li8334890247616788453la_a_b] :
( ( minus_4977183416030855486la_a_b @ bot_bo3230846185698255993la_a_b @ A2 )
= bot_bo3230846185698255993la_a_b ) ).
% empty_Diff
thf(fact_461_empty__Diff,axiom,
! [A2: set_se6865892389300016395la_a_b] :
( ( minus_4705846553145473764la_a_b @ bot_bo2891247006866115487la_a_b @ A2 )
= bot_bo2891247006866115487la_a_b ) ).
% empty_Diff
thf(fact_462_empty__Diff,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A2 )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_463_empty__Diff,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( minus_4077726661957047470la_a_b @ bot_bo4495933725496725865la_a_b @ A2 )
= bot_bo4495933725496725865la_a_b ) ).
% empty_Diff
thf(fact_464_Diff__cancel,axiom,
! [A2: set_li8334890247616788453la_a_b] :
( ( minus_4977183416030855486la_a_b @ A2 @ A2 )
= bot_bo3230846185698255993la_a_b ) ).
% Diff_cancel
thf(fact_465_Diff__cancel,axiom,
! [A2: set_se6865892389300016395la_a_b] :
( ( minus_4705846553145473764la_a_b @ A2 @ A2 )
= bot_bo2891247006866115487la_a_b ) ).
% Diff_cancel
thf(fact_466_Diff__cancel,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ A2 )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_467_Diff__cancel,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( minus_4077726661957047470la_a_b @ A2 @ A2 )
= bot_bo4495933725496725865la_a_b ) ).
% Diff_cancel
thf(fact_468_Diff__insert0,axiom,
! [X: list_R8263082107343818799la_a_b,A2: set_li8334890247616788453la_a_b,B: set_li8334890247616788453la_a_b] :
( ~ ( member8712135081189583430la_a_b @ X @ A2 )
=> ( ( minus_4977183416030855486la_a_b @ A2 @ ( insert3617074100887873119la_a_b @ X @ B ) )
= ( minus_4977183416030855486la_a_b @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_469_Diff__insert0,axiom,
! [X: set_Re381260168593705685la_a_b,A2: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ~ ( member3481406638322139244la_a_b @ X @ A2 )
=> ( ( minus_4705846553145473764la_a_b @ A2 @ ( insert2023870700798818565la_a_b @ X @ B ) )
= ( minus_4705846553145473764la_a_b @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_470_Diff__insert0,axiom,
! [X: nat,A2: set_nat,B: set_nat] :
( ~ ( member_nat2 @ X @ A2 )
=> ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ B ) )
= ( minus_minus_set_nat @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_471_Diff__insert0,axiom,
! [X: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ~ ( member4680049679412964150la_a_b @ X @ A2 )
=> ( ( minus_4077726661957047470la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ X @ B ) )
= ( minus_4077726661957047470la_a_b @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_472_insert__Diff1,axiom,
! [X: list_R8263082107343818799la_a_b,B: set_li8334890247616788453la_a_b,A2: set_li8334890247616788453la_a_b] :
( ( member8712135081189583430la_a_b @ X @ B )
=> ( ( minus_4977183416030855486la_a_b @ ( insert3617074100887873119la_a_b @ X @ A2 ) @ B )
= ( minus_4977183416030855486la_a_b @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_473_insert__Diff1,axiom,
! [X: set_Re381260168593705685la_a_b,B: set_se6865892389300016395la_a_b,A2: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ X @ B )
=> ( ( minus_4705846553145473764la_a_b @ ( insert2023870700798818565la_a_b @ X @ A2 ) @ B )
= ( minus_4705846553145473764la_a_b @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_474_insert__Diff1,axiom,
! [X: nat,B: set_nat,A2: set_nat] :
( ( member_nat2 @ X @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat2 @ X @ A2 ) @ B )
= ( minus_minus_set_nat @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_475_insert__Diff1,axiom,
! [X: relational_fmla_a_b,B: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ X @ B )
=> ( ( minus_4077726661957047470la_a_b @ ( insert7010464514620295119la_a_b @ X @ A2 ) @ B )
= ( minus_4077726661957047470la_a_b @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_476_Un__Diff__cancel,axiom,
! [A2: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ A2 @ ( minus_4705846553145473764la_a_b @ B @ A2 ) )
= ( sup_su4783144482993978935la_a_b @ A2 @ B ) ) ).
% Un_Diff_cancel
thf(fact_477_Un__Diff__cancel,axiom,
! [A2: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( minus_minus_set_nat @ B @ A2 ) )
= ( sup_sup_set_nat @ A2 @ B ) ) ).
% Un_Diff_cancel
thf(fact_478_Un__Diff__cancel,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ A2 @ ( minus_4077726661957047470la_a_b @ B @ A2 ) )
= ( sup_su5130108678486352897la_a_b @ A2 @ B ) ) ).
% Un_Diff_cancel
thf(fact_479_Un__Diff__cancel2,axiom,
! [B: set_se6865892389300016395la_a_b,A2: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ ( minus_4705846553145473764la_a_b @ B @ A2 ) @ A2 )
= ( sup_su4783144482993978935la_a_b @ B @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_480_Un__Diff__cancel2,axiom,
! [B: set_nat,A2: set_nat] :
( ( sup_sup_set_nat @ ( minus_minus_set_nat @ B @ A2 ) @ A2 )
= ( sup_sup_set_nat @ B @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_481_Un__Diff__cancel2,axiom,
! [B: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ ( minus_4077726661957047470la_a_b @ B @ A2 ) @ A2 )
= ( sup_su5130108678486352897la_a_b @ B @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_482_list_Osimps_I15_J,axiom,
! [X21: list_R8263082107343818799la_a_b,X22: list_l3216422893793986623la_a_b] :
( ( set_li8988895567068466906la_a_b @ ( cons_l6716632503304403961la_a_b @ X21 @ X22 ) )
= ( insert3617074100887873119la_a_b @ X21 @ ( set_li8988895567068466906la_a_b @ X22 ) ) ) ).
% list.simps(15)
thf(fact_483_list_Osimps_I15_J,axiom,
! [X21: set_Re381260168593705685la_a_b,X22: list_s3671366213839843941la_a_b] :
( ( set_se5111286776497155840la_a_b @ ( cons_s2104095074707585951la_a_b @ X21 @ X22 ) )
= ( insert2023870700798818565la_a_b @ X21 @ ( set_se5111286776497155840la_a_b @ X22 ) ) ) ).
% list.simps(15)
thf(fact_484_list_Osimps_I15_J,axiom,
! [X21: nat,X22: list_nat] :
( ( set_nat2 @ ( cons_nat @ X21 @ X22 ) )
= ( insert_nat2 @ X21 @ ( set_nat2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_485_list_Osimps_I15_J,axiom,
! [X21: relational_fmla_a_b,X22: list_R8263082107343818799la_a_b] :
( ( set_Re9104216502384355786la_a_b @ ( cons_R4161027894713053801la_a_b @ X21 @ X22 ) )
= ( insert7010464514620295119la_a_b @ X21 @ ( set_Re9104216502384355786la_a_b @ X22 ) ) ) ).
% list.simps(15)
thf(fact_486_insert__Diff__single,axiom,
! [A: list_R8263082107343818799la_a_b,A2: set_li8334890247616788453la_a_b] :
( ( insert3617074100887873119la_a_b @ A @ ( minus_4977183416030855486la_a_b @ A2 @ ( insert3617074100887873119la_a_b @ A @ bot_bo3230846185698255993la_a_b ) ) )
= ( insert3617074100887873119la_a_b @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_487_insert__Diff__single,axiom,
! [A: set_Re381260168593705685la_a_b,A2: set_se6865892389300016395la_a_b] :
( ( insert2023870700798818565la_a_b @ A @ ( minus_4705846553145473764la_a_b @ A2 @ ( insert2023870700798818565la_a_b @ A @ bot_bo2891247006866115487la_a_b ) ) )
= ( insert2023870700798818565la_a_b @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_488_insert__Diff__single,axiom,
! [A: nat,A2: set_nat] :
( ( insert_nat2 @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ A @ bot_bot_set_nat ) ) )
= ( insert_nat2 @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_489_insert__Diff__single,axiom,
! [A: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( insert7010464514620295119la_a_b @ A @ ( minus_4077726661957047470la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ A @ bot_bo4495933725496725865la_a_b ) ) )
= ( insert7010464514620295119la_a_b @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_490_fold__Cons,axiom,
! [F: relational_fmla_a_b > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b,X: relational_fmla_a_b,Xs: list_R8263082107343818799la_a_b] :
( ( fold_R3800324640475171969la_a_b @ F @ ( cons_R4161027894713053801la_a_b @ X @ Xs ) )
= ( comp_s8660012457442781028la_a_b @ ( fold_R3800324640475171969la_a_b @ F @ Xs ) @ ( F @ X ) ) ) ).
% fold_Cons
thf(fact_491_fold__Cons,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b > relational_fmla_a_b,X: relational_fmla_a_b,Xs: list_R8263082107343818799la_a_b] :
( ( fold_R509364916538903371la_a_b @ F @ ( cons_R4161027894713053801la_a_b @ X @ Xs ) )
= ( comp_R8862297837303855554la_a_b @ ( fold_R509364916538903371la_a_b @ F @ Xs ) @ ( F @ X ) ) ) ).
% fold_Cons
thf(fact_492_fold__Cons,axiom,
! [F: set_Re381260168593705685la_a_b > relational_fmla_a_b > relational_fmla_a_b,X: set_Re381260168593705685la_a_b,Xs: list_s3671366213839843941la_a_b] :
( ( fold_s6778362883525594753la_a_b @ F @ ( cons_s2104095074707585951la_a_b @ X @ Xs ) )
= ( comp_R8862297837303855554la_a_b @ ( fold_s6778362883525594753la_a_b @ F @ Xs ) @ ( F @ X ) ) ) ).
% fold_Cons
thf(fact_493_fold__Cons,axiom,
! [F: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b,X: set_Re381260168593705685la_a_b,Xs: list_s3671366213839843941la_a_b] :
( ( fold_s1495082921969268663la_a_b @ F @ ( cons_s2104095074707585951la_a_b @ X @ Xs ) )
= ( comp_s8660012457442781028la_a_b @ ( fold_s1495082921969268663la_a_b @ F @ Xs ) @ ( F @ X ) ) ) ).
% fold_Cons
thf(fact_494_fold__Cons,axiom,
! [F: nat > relational_fmla_a_b > relational_fmla_a_b,X: nat,Xs: list_nat] :
( ( fold_n2260362839379620988la_a_b @ F @ ( cons_nat @ X @ Xs ) )
= ( comp_R8862297837303855554la_a_b @ ( fold_n2260362839379620988la_a_b @ F @ Xs ) @ ( F @ X ) ) ) ).
% fold_Cons
thf(fact_495_fold__Cons,axiom,
! [F: nat > nat > nat,X: nat,Xs: list_nat] :
( ( fold_nat_nat @ F @ ( cons_nat @ X @ Xs ) )
= ( comp_nat_nat_nat @ ( fold_nat_nat @ F @ Xs ) @ ( F @ X ) ) ) ).
% fold_Cons
thf(fact_496_totalp__onD,axiom,
! [A2: set_se6865892389300016395la_a_b,R2: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b > $o,X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b] :
( ( totalp3012131419089278729la_a_b @ A2 @ R2 )
=> ( ( member3481406638322139244la_a_b @ X @ A2 )
=> ( ( member3481406638322139244la_a_b @ Y @ A2 )
=> ( ( X != Y )
=> ( ( R2 @ X @ Y )
| ( R2 @ Y @ X ) ) ) ) ) ) ).
% totalp_onD
thf(fact_497_totalp__onD,axiom,
! [A2: set_li8334890247616788453la_a_b,R2: list_R8263082107343818799la_a_b > list_R8263082107343818799la_a_b > $o,X: list_R8263082107343818799la_a_b,Y: list_R8263082107343818799la_a_b] :
( ( totalp7313527317358576227la_a_b @ A2 @ R2 )
=> ( ( member8712135081189583430la_a_b @ X @ A2 )
=> ( ( member8712135081189583430la_a_b @ Y @ A2 )
=> ( ( X != Y )
=> ( ( R2 @ X @ Y )
| ( R2 @ Y @ X ) ) ) ) ) ) ).
% totalp_onD
thf(fact_498_totalp__onD,axiom,
! [A2: set_Re381260168593705685la_a_b,R2: relational_fmla_a_b > relational_fmla_a_b > $o,X: relational_fmla_a_b,Y: relational_fmla_a_b] :
( ( totalp1437913959296628179la_a_b @ A2 @ R2 )
=> ( ( member4680049679412964150la_a_b @ X @ A2 )
=> ( ( member4680049679412964150la_a_b @ Y @ A2 )
=> ( ( X != Y )
=> ( ( R2 @ X @ Y )
| ( R2 @ Y @ X ) ) ) ) ) ) ).
% totalp_onD
thf(fact_499_totalp__onD,axiom,
! [A2: set_nat,R2: nat > nat > $o,X: nat,Y: nat] :
( ( totalp_on_nat @ A2 @ R2 )
=> ( ( member_nat2 @ X @ A2 )
=> ( ( member_nat2 @ Y @ A2 )
=> ( ( X != Y )
=> ( ( R2 @ X @ Y )
| ( R2 @ Y @ X ) ) ) ) ) ) ).
% totalp_onD
thf(fact_500_totalp__onI,axiom,
! [A2: set_se6865892389300016395la_a_b,R2: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b > $o] :
( ! [X3: set_Re381260168593705685la_a_b,Y2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X3 @ A2 )
=> ( ( member3481406638322139244la_a_b @ Y2 @ A2 )
=> ( ( X3 != Y2 )
=> ( ( R2 @ X3 @ Y2 )
| ( R2 @ Y2 @ X3 ) ) ) ) )
=> ( totalp3012131419089278729la_a_b @ A2 @ R2 ) ) ).
% totalp_onI
thf(fact_501_totalp__onI,axiom,
! [A2: set_li8334890247616788453la_a_b,R2: list_R8263082107343818799la_a_b > list_R8263082107343818799la_a_b > $o] :
( ! [X3: list_R8263082107343818799la_a_b,Y2: list_R8263082107343818799la_a_b] :
( ( member8712135081189583430la_a_b @ X3 @ A2 )
=> ( ( member8712135081189583430la_a_b @ Y2 @ A2 )
=> ( ( X3 != Y2 )
=> ( ( R2 @ X3 @ Y2 )
| ( R2 @ Y2 @ X3 ) ) ) ) )
=> ( totalp7313527317358576227la_a_b @ A2 @ R2 ) ) ).
% totalp_onI
thf(fact_502_totalp__onI,axiom,
! [A2: set_Re381260168593705685la_a_b,R2: relational_fmla_a_b > relational_fmla_a_b > $o] :
( ! [X3: relational_fmla_a_b,Y2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A2 )
=> ( ( member4680049679412964150la_a_b @ Y2 @ A2 )
=> ( ( X3 != Y2 )
=> ( ( R2 @ X3 @ Y2 )
| ( R2 @ Y2 @ X3 ) ) ) ) )
=> ( totalp1437913959296628179la_a_b @ A2 @ R2 ) ) ).
% totalp_onI
thf(fact_503_totalp__onI,axiom,
! [A2: set_nat,R2: nat > nat > $o] :
( ! [X3: nat,Y2: nat] :
( ( member_nat2 @ X3 @ A2 )
=> ( ( member_nat2 @ Y2 @ A2 )
=> ( ( X3 != Y2 )
=> ( ( R2 @ X3 @ Y2 )
| ( R2 @ Y2 @ X3 ) ) ) ) )
=> ( totalp_on_nat @ A2 @ R2 ) ) ).
% totalp_onI
thf(fact_504_totalp__on__def,axiom,
( totalp1437913959296628179la_a_b
= ( ^ [A4: set_Re381260168593705685la_a_b,R: relational_fmla_a_b > relational_fmla_a_b > $o] :
! [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ A4 )
=> ! [Y3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Y3 @ A4 )
=> ( ( X2 != Y3 )
=> ( ( R @ X2 @ Y3 )
| ( R @ Y3 @ X2 ) ) ) ) ) ) ) ).
% totalp_on_def
thf(fact_505_totalp__on__def,axiom,
( totalp_on_nat
= ( ^ [A4: set_nat,R: nat > nat > $o] :
! [X2: nat] :
( ( member_nat2 @ X2 @ A4 )
=> ! [Y3: nat] :
( ( member_nat2 @ Y3 @ A4 )
=> ( ( X2 != Y3 )
=> ( ( R @ X2 @ Y3 )
| ( R @ Y3 @ X2 ) ) ) ) ) ) ) ).
% totalp_on_def
thf(fact_506_DiffE,axiom,
! [C2: list_R8263082107343818799la_a_b,A2: set_li8334890247616788453la_a_b,B: set_li8334890247616788453la_a_b] :
( ( member8712135081189583430la_a_b @ C2 @ ( minus_4977183416030855486la_a_b @ A2 @ B ) )
=> ~ ( ( member8712135081189583430la_a_b @ C2 @ A2 )
=> ( member8712135081189583430la_a_b @ C2 @ B ) ) ) ).
% DiffE
thf(fact_507_DiffE,axiom,
! [C2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C2 @ ( minus_4077726661957047470la_a_b @ A2 @ B ) )
=> ~ ( ( member4680049679412964150la_a_b @ C2 @ A2 )
=> ( member4680049679412964150la_a_b @ C2 @ B ) ) ) ).
% DiffE
thf(fact_508_DiffE,axiom,
! [C2: set_Re381260168593705685la_a_b,A2: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ C2 @ ( minus_4705846553145473764la_a_b @ A2 @ B ) )
=> ~ ( ( member3481406638322139244la_a_b @ C2 @ A2 )
=> ( member3481406638322139244la_a_b @ C2 @ B ) ) ) ).
% DiffE
thf(fact_509_DiffE,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat2 @ C2 @ ( minus_minus_set_nat @ A2 @ B ) )
=> ~ ( ( member_nat2 @ C2 @ A2 )
=> ( member_nat2 @ C2 @ B ) ) ) ).
% DiffE
thf(fact_510_DiffD1,axiom,
! [C2: list_R8263082107343818799la_a_b,A2: set_li8334890247616788453la_a_b,B: set_li8334890247616788453la_a_b] :
( ( member8712135081189583430la_a_b @ C2 @ ( minus_4977183416030855486la_a_b @ A2 @ B ) )
=> ( member8712135081189583430la_a_b @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_511_DiffD1,axiom,
! [C2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C2 @ ( minus_4077726661957047470la_a_b @ A2 @ B ) )
=> ( member4680049679412964150la_a_b @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_512_DiffD1,axiom,
! [C2: set_Re381260168593705685la_a_b,A2: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ C2 @ ( minus_4705846553145473764la_a_b @ A2 @ B ) )
=> ( member3481406638322139244la_a_b @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_513_DiffD1,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat2 @ C2 @ ( minus_minus_set_nat @ A2 @ B ) )
=> ( member_nat2 @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_514_DiffD2,axiom,
! [C2: list_R8263082107343818799la_a_b,A2: set_li8334890247616788453la_a_b,B: set_li8334890247616788453la_a_b] :
( ( member8712135081189583430la_a_b @ C2 @ ( minus_4977183416030855486la_a_b @ A2 @ B ) )
=> ~ ( member8712135081189583430la_a_b @ C2 @ B ) ) ).
% DiffD2
thf(fact_515_DiffD2,axiom,
! [C2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C2 @ ( minus_4077726661957047470la_a_b @ A2 @ B ) )
=> ~ ( member4680049679412964150la_a_b @ C2 @ B ) ) ).
% DiffD2
thf(fact_516_DiffD2,axiom,
! [C2: set_Re381260168593705685la_a_b,A2: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ C2 @ ( minus_4705846553145473764la_a_b @ A2 @ B ) )
=> ~ ( member3481406638322139244la_a_b @ C2 @ B ) ) ).
% DiffD2
thf(fact_517_DiffD2,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat2 @ C2 @ ( minus_minus_set_nat @ A2 @ B ) )
=> ~ ( member_nat2 @ C2 @ B ) ) ).
% DiffD2
thf(fact_518_diff__right__commute,axiom,
! [A: nat,C2: nat,B2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C2 ) @ B2 )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B2 ) @ C2 ) ) ).
% diff_right_commute
thf(fact_519_not__Cons__self2,axiom,
! [X: relational_fmla_a_b,Xs: list_R8263082107343818799la_a_b] :
( ( cons_R4161027894713053801la_a_b @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_520_not__Cons__self2,axiom,
! [X: set_Re381260168593705685la_a_b,Xs: list_s3671366213839843941la_a_b] :
( ( cons_s2104095074707585951la_a_b @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_521_not__Cons__self2,axiom,
! [X: nat,Xs: list_nat] :
( ( cons_nat @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_522_foldr__Cons,axiom,
! [F: set_Re381260168593705685la_a_b > relational_fmla_a_b > relational_fmla_a_b,X: set_Re381260168593705685la_a_b,Xs: list_s3671366213839843941la_a_b] :
( ( foldr_1179560620697312285la_a_b @ F @ ( cons_s2104095074707585951la_a_b @ X @ Xs ) )
= ( comp_R8862297837303855554la_a_b @ ( F @ X ) @ ( foldr_1179560620697312285la_a_b @ F @ Xs ) ) ) ).
% foldr_Cons
thf(fact_523_foldr__Cons,axiom,
! [F: nat > relational_fmla_a_b > relational_fmla_a_b,X: nat,Xs: list_nat] :
( ( foldr_3281238248259268120la_a_b @ F @ ( cons_nat @ X @ Xs ) )
= ( comp_R8862297837303855554la_a_b @ ( F @ X ) @ ( foldr_3281238248259268120la_a_b @ F @ Xs ) ) ) ).
% foldr_Cons
thf(fact_524_foldr__Cons,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b > relational_fmla_a_b,X: relational_fmla_a_b,Xs: list_R8263082107343818799la_a_b] :
( ( foldr_789212930732525799la_a_b @ F @ ( cons_R4161027894713053801la_a_b @ X @ Xs ) )
= ( comp_R8862297837303855554la_a_b @ ( F @ X ) @ ( foldr_789212930732525799la_a_b @ F @ Xs ) ) ) ).
% foldr_Cons
thf(fact_525_insert__Diff__if,axiom,
! [X: list_R8263082107343818799la_a_b,B: set_li8334890247616788453la_a_b,A2: set_li8334890247616788453la_a_b] :
( ( ( member8712135081189583430la_a_b @ X @ B )
=> ( ( minus_4977183416030855486la_a_b @ ( insert3617074100887873119la_a_b @ X @ A2 ) @ B )
= ( minus_4977183416030855486la_a_b @ A2 @ B ) ) )
& ( ~ ( member8712135081189583430la_a_b @ X @ B )
=> ( ( minus_4977183416030855486la_a_b @ ( insert3617074100887873119la_a_b @ X @ A2 ) @ B )
= ( insert3617074100887873119la_a_b @ X @ ( minus_4977183416030855486la_a_b @ A2 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_526_insert__Diff__if,axiom,
! [X: set_Re381260168593705685la_a_b,B: set_se6865892389300016395la_a_b,A2: set_se6865892389300016395la_a_b] :
( ( ( member3481406638322139244la_a_b @ X @ B )
=> ( ( minus_4705846553145473764la_a_b @ ( insert2023870700798818565la_a_b @ X @ A2 ) @ B )
= ( minus_4705846553145473764la_a_b @ A2 @ B ) ) )
& ( ~ ( member3481406638322139244la_a_b @ X @ B )
=> ( ( minus_4705846553145473764la_a_b @ ( insert2023870700798818565la_a_b @ X @ A2 ) @ B )
= ( insert2023870700798818565la_a_b @ X @ ( minus_4705846553145473764la_a_b @ A2 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_527_insert__Diff__if,axiom,
! [X: nat,B: set_nat,A2: set_nat] :
( ( ( member_nat2 @ X @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat2 @ X @ A2 ) @ B )
= ( minus_minus_set_nat @ A2 @ B ) ) )
& ( ~ ( member_nat2 @ X @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat2 @ X @ A2 ) @ B )
= ( insert_nat2 @ X @ ( minus_minus_set_nat @ A2 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_528_insert__Diff__if,axiom,
! [X: relational_fmla_a_b,B: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b] :
( ( ( member4680049679412964150la_a_b @ X @ B )
=> ( ( minus_4077726661957047470la_a_b @ ( insert7010464514620295119la_a_b @ X @ A2 ) @ B )
= ( minus_4077726661957047470la_a_b @ A2 @ B ) ) )
& ( ~ ( member4680049679412964150la_a_b @ X @ B )
=> ( ( minus_4077726661957047470la_a_b @ ( insert7010464514620295119la_a_b @ X @ A2 ) @ B )
= ( insert7010464514620295119la_a_b @ X @ ( minus_4077726661957047470la_a_b @ A2 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_529_list_Oset__intros_I2_J,axiom,
! [Y: list_R8263082107343818799la_a_b,X22: list_l3216422893793986623la_a_b,X21: list_R8263082107343818799la_a_b] :
( ( member8712135081189583430la_a_b @ Y @ ( set_li8988895567068466906la_a_b @ X22 ) )
=> ( member8712135081189583430la_a_b @ Y @ ( set_li8988895567068466906la_a_b @ ( cons_l6716632503304403961la_a_b @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_530_list_Oset__intros_I2_J,axiom,
! [Y: set_Re381260168593705685la_a_b,X22: list_s3671366213839843941la_a_b,X21: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ Y @ ( set_se5111286776497155840la_a_b @ X22 ) )
=> ( member3481406638322139244la_a_b @ Y @ ( set_se5111286776497155840la_a_b @ ( cons_s2104095074707585951la_a_b @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_531_list_Oset__intros_I2_J,axiom,
! [Y: nat,X22: list_nat,X21: nat] :
( ( member_nat2 @ Y @ ( set_nat2 @ X22 ) )
=> ( member_nat2 @ Y @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_532_list_Oset__intros_I2_J,axiom,
! [Y: relational_fmla_a_b,X22: list_R8263082107343818799la_a_b,X21: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Y @ ( set_Re9104216502384355786la_a_b @ X22 ) )
=> ( member4680049679412964150la_a_b @ Y @ ( set_Re9104216502384355786la_a_b @ ( cons_R4161027894713053801la_a_b @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_533_list_Oset__intros_I1_J,axiom,
! [X21: list_R8263082107343818799la_a_b,X22: list_l3216422893793986623la_a_b] : ( member8712135081189583430la_a_b @ X21 @ ( set_li8988895567068466906la_a_b @ ( cons_l6716632503304403961la_a_b @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_534_list_Oset__intros_I1_J,axiom,
! [X21: set_Re381260168593705685la_a_b,X22: list_s3671366213839843941la_a_b] : ( member3481406638322139244la_a_b @ X21 @ ( set_se5111286776497155840la_a_b @ ( cons_s2104095074707585951la_a_b @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_535_list_Oset__intros_I1_J,axiom,
! [X21: nat,X22: list_nat] : ( member_nat2 @ X21 @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_536_list_Oset__intros_I1_J,axiom,
! [X21: relational_fmla_a_b,X22: list_R8263082107343818799la_a_b] : ( member4680049679412964150la_a_b @ X21 @ ( set_Re9104216502384355786la_a_b @ ( cons_R4161027894713053801la_a_b @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_537_list_Oset__cases,axiom,
! [E: list_R8263082107343818799la_a_b,A: list_l3216422893793986623la_a_b] :
( ( member8712135081189583430la_a_b @ E @ ( set_li8988895567068466906la_a_b @ A ) )
=> ( ! [Z2: list_l3216422893793986623la_a_b] :
( A
!= ( cons_l6716632503304403961la_a_b @ E @ Z2 ) )
=> ~ ! [Z1: list_R8263082107343818799la_a_b,Z2: list_l3216422893793986623la_a_b] :
( ( A
= ( cons_l6716632503304403961la_a_b @ Z1 @ Z2 ) )
=> ~ ( member8712135081189583430la_a_b @ E @ ( set_li8988895567068466906la_a_b @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_538_list_Oset__cases,axiom,
! [E: set_Re381260168593705685la_a_b,A: list_s3671366213839843941la_a_b] :
( ( member3481406638322139244la_a_b @ E @ ( set_se5111286776497155840la_a_b @ A ) )
=> ( ! [Z2: list_s3671366213839843941la_a_b] :
( A
!= ( cons_s2104095074707585951la_a_b @ E @ Z2 ) )
=> ~ ! [Z1: set_Re381260168593705685la_a_b,Z2: list_s3671366213839843941la_a_b] :
( ( A
= ( cons_s2104095074707585951la_a_b @ Z1 @ Z2 ) )
=> ~ ( member3481406638322139244la_a_b @ E @ ( set_se5111286776497155840la_a_b @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_539_list_Oset__cases,axiom,
! [E: nat,A: list_nat] :
( ( member_nat2 @ E @ ( set_nat2 @ A ) )
=> ( ! [Z2: list_nat] :
( A
!= ( cons_nat @ E @ Z2 ) )
=> ~ ! [Z1: nat,Z2: list_nat] :
( ( A
= ( cons_nat @ Z1 @ Z2 ) )
=> ~ ( member_nat2 @ E @ ( set_nat2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_540_list_Oset__cases,axiom,
! [E: relational_fmla_a_b,A: list_R8263082107343818799la_a_b] :
( ( member4680049679412964150la_a_b @ E @ ( set_Re9104216502384355786la_a_b @ A ) )
=> ( ! [Z2: list_R8263082107343818799la_a_b] :
( A
!= ( cons_R4161027894713053801la_a_b @ E @ Z2 ) )
=> ~ ! [Z1: relational_fmla_a_b,Z2: list_R8263082107343818799la_a_b] :
( ( A
= ( cons_R4161027894713053801la_a_b @ Z1 @ Z2 ) )
=> ~ ( member4680049679412964150la_a_b @ E @ ( set_Re9104216502384355786la_a_b @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_541_set__ConsD,axiom,
! [Y: list_R8263082107343818799la_a_b,X: list_R8263082107343818799la_a_b,Xs: list_l3216422893793986623la_a_b] :
( ( member8712135081189583430la_a_b @ Y @ ( set_li8988895567068466906la_a_b @ ( cons_l6716632503304403961la_a_b @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member8712135081189583430la_a_b @ Y @ ( set_li8988895567068466906la_a_b @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_542_set__ConsD,axiom,
! [Y: set_Re381260168593705685la_a_b,X: set_Re381260168593705685la_a_b,Xs: list_s3671366213839843941la_a_b] :
( ( member3481406638322139244la_a_b @ Y @ ( set_se5111286776497155840la_a_b @ ( cons_s2104095074707585951la_a_b @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member3481406638322139244la_a_b @ Y @ ( set_se5111286776497155840la_a_b @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_543_set__ConsD,axiom,
! [Y: nat,X: nat,Xs: list_nat] :
( ( member_nat2 @ Y @ ( set_nat2 @ ( cons_nat @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_nat2 @ Y @ ( set_nat2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_544_set__ConsD,axiom,
! [Y: relational_fmla_a_b,X: relational_fmla_a_b,Xs: list_R8263082107343818799la_a_b] :
( ( member4680049679412964150la_a_b @ Y @ ( set_Re9104216502384355786la_a_b @ ( cons_R4161027894713053801la_a_b @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member4680049679412964150la_a_b @ Y @ ( set_Re9104216502384355786la_a_b @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_545_can__select__def,axiom,
( can_se982673022169791512la_a_b
= ( ^ [P2: set_Re381260168593705685la_a_b > $o,A4: set_se6865892389300016395la_a_b] :
? [X2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X2 @ A4 )
& ( P2 @ X2 )
& ! [Y3: set_Re381260168593705685la_a_b] :
( ( ( member3481406638322139244la_a_b @ Y3 @ A4 )
& ( P2 @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_546_can__select__def,axiom,
( can_select_nat
= ( ^ [P2: nat > $o,A4: set_nat] :
? [X2: nat] :
( ( member_nat2 @ X2 @ A4 )
& ( P2 @ X2 )
& ! [Y3: nat] :
( ( ( member_nat2 @ Y3 @ A4 )
& ( P2 @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_547_can__select__def,axiom,
( can_se4691468443538722802la_a_b
= ( ^ [P2: list_R8263082107343818799la_a_b > $o,A4: set_li8334890247616788453la_a_b] :
? [X2: list_R8263082107343818799la_a_b] :
( ( member8712135081189583430la_a_b @ X2 @ A4 )
& ( P2 @ X2 )
& ! [Y3: list_R8263082107343818799la_a_b] :
( ( ( member8712135081189583430la_a_b @ Y3 @ A4 )
& ( P2 @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_548_can__select__def,axiom,
( can_se1623252625881650914la_a_b
= ( ^ [P2: relational_fmla_a_b > $o,A4: set_Re381260168593705685la_a_b] :
? [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ A4 )
& ( P2 @ X2 )
& ! [Y3: relational_fmla_a_b] :
( ( ( member4680049679412964150la_a_b @ Y3 @ A4 )
& ( P2 @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_549_Un__Diff,axiom,
! [A2: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b,C: set_se6865892389300016395la_a_b] :
( ( minus_4705846553145473764la_a_b @ ( sup_su4783144482993978935la_a_b @ A2 @ B ) @ C )
= ( sup_su4783144482993978935la_a_b @ ( minus_4705846553145473764la_a_b @ A2 @ C ) @ ( minus_4705846553145473764la_a_b @ B @ C ) ) ) ).
% Un_Diff
thf(fact_550_Un__Diff,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( minus_minus_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ C )
= ( sup_sup_set_nat @ ( minus_minus_set_nat @ A2 @ C ) @ ( minus_minus_set_nat @ B @ C ) ) ) ).
% Un_Diff
thf(fact_551_Un__Diff,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b] :
( ( minus_4077726661957047470la_a_b @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) @ C )
= ( sup_su5130108678486352897la_a_b @ ( minus_4077726661957047470la_a_b @ A2 @ C ) @ ( minus_4077726661957047470la_a_b @ B @ C ) ) ) ).
% Un_Diff
thf(fact_552_fold__simps_I2_J,axiom,
! [F: relational_fmla_a_b > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b,X: relational_fmla_a_b,Xs: list_R8263082107343818799la_a_b,S3: set_Re381260168593705685la_a_b] :
( ( fold_R3800324640475171969la_a_b @ F @ ( cons_R4161027894713053801la_a_b @ X @ Xs ) @ S3 )
= ( fold_R3800324640475171969la_a_b @ F @ Xs @ ( F @ X @ S3 ) ) ) ).
% fold_simps(2)
thf(fact_553_fold__simps_I2_J,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b > relational_fmla_a_b,X: relational_fmla_a_b,Xs: list_R8263082107343818799la_a_b,S3: relational_fmla_a_b] :
( ( fold_R509364916538903371la_a_b @ F @ ( cons_R4161027894713053801la_a_b @ X @ Xs ) @ S3 )
= ( fold_R509364916538903371la_a_b @ F @ Xs @ ( F @ X @ S3 ) ) ) ).
% fold_simps(2)
thf(fact_554_fold__simps_I2_J,axiom,
! [F: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b,X: set_Re381260168593705685la_a_b,Xs: list_s3671366213839843941la_a_b,S3: set_Re381260168593705685la_a_b] :
( ( fold_s1495082921969268663la_a_b @ F @ ( cons_s2104095074707585951la_a_b @ X @ Xs ) @ S3 )
= ( fold_s1495082921969268663la_a_b @ F @ Xs @ ( F @ X @ S3 ) ) ) ).
% fold_simps(2)
thf(fact_555_fold__simps_I2_J,axiom,
! [F: nat > nat > nat,X: nat,Xs: list_nat,S3: nat] :
( ( fold_nat_nat @ F @ ( cons_nat @ X @ Xs ) @ S3 )
= ( fold_nat_nat @ F @ Xs @ ( F @ X @ S3 ) ) ) ).
% fold_simps(2)
thf(fact_556_totalp__on__empty,axiom,
! [R2: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b > $o] : ( totalp3012131419089278729la_a_b @ bot_bo2891247006866115487la_a_b @ R2 ) ).
% totalp_on_empty
thf(fact_557_totalp__on__empty,axiom,
! [R2: nat > nat > $o] : ( totalp_on_nat @ bot_bot_set_nat @ R2 ) ).
% totalp_on_empty
thf(fact_558_totalp__on__empty,axiom,
! [R2: list_R8263082107343818799la_a_b > list_R8263082107343818799la_a_b > $o] : ( totalp7313527317358576227la_a_b @ bot_bo3230846185698255993la_a_b @ R2 ) ).
% totalp_on_empty
thf(fact_559_totalp__on__empty,axiom,
! [R2: relational_fmla_a_b > relational_fmla_a_b > $o] : ( totalp1437913959296628179la_a_b @ bot_bo4495933725496725865la_a_b @ R2 ) ).
% totalp_on_empty
thf(fact_560_null__rec_I1_J,axiom,
! [X: relational_fmla_a_b,Xs: list_R8263082107343818799la_a_b] :
~ ( null_R4120973431801868817la_a_b @ ( cons_R4161027894713053801la_a_b @ X @ Xs ) ) ).
% null_rec(1)
thf(fact_561_null__rec_I1_J,axiom,
! [X: set_Re381260168593705685la_a_b,Xs: list_s3671366213839843941la_a_b] :
~ ( null_s6470977797459811143la_a_b @ ( cons_s2104095074707585951la_a_b @ X @ Xs ) ) ).
% null_rec(1)
thf(fact_562_null__rec_I1_J,axiom,
! [X: nat,Xs: list_nat] :
~ ( null_nat @ ( cons_nat @ X @ Xs ) ) ).
% null_rec(1)
thf(fact_563_member__rec_I1_J,axiom,
! [X: relational_fmla_a_b,Xs: list_R8263082107343818799la_a_b,Y: relational_fmla_a_b] :
( ( member3779204850511880260la_a_b @ ( cons_R4161027894713053801la_a_b @ X @ Xs ) @ Y )
= ( ( X = Y )
| ( member3779204850511880260la_a_b @ Xs @ Y ) ) ) ).
% member_rec(1)
thf(fact_564_member__rec_I1_J,axiom,
! [X: set_Re381260168593705685la_a_b,Xs: list_s3671366213839843941la_a_b,Y: set_Re381260168593705685la_a_b] :
( ( member1547870223199108474la_a_b @ ( cons_s2104095074707585951la_a_b @ X @ Xs ) @ Y )
= ( ( X = Y )
| ( member1547870223199108474la_a_b @ Xs @ Y ) ) ) ).
% member_rec(1)
thf(fact_565_member__rec_I1_J,axiom,
! [X: nat,Xs: list_nat,Y: nat] :
( ( member_nat @ ( cons_nat @ X @ Xs ) @ Y )
= ( ( X = Y )
| ( member_nat @ Xs @ Y ) ) ) ).
% member_rec(1)
thf(fact_566_fold__commute,axiom,
! [Xs: list_nat,H: nat > nat,G: nat > nat > nat,F: nat > nat > nat] :
( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( comp_nat_nat_nat @ H @ ( G @ X3 ) )
= ( comp_nat_nat_nat @ ( F @ X3 ) @ H ) ) )
=> ( ( comp_nat_nat_nat @ H @ ( fold_nat_nat @ G @ Xs ) )
= ( comp_nat_nat_nat @ ( fold_nat_nat @ F @ Xs ) @ H ) ) ) ).
% fold_commute
thf(fact_567_fold__commute,axiom,
! [Xs: list_nat,H: relational_fmla_a_b > relational_fmla_a_b,G: nat > relational_fmla_a_b > relational_fmla_a_b,F: nat > relational_fmla_a_b > relational_fmla_a_b] :
( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( comp_R8862297837303855554la_a_b @ H @ ( G @ X3 ) )
= ( comp_R8862297837303855554la_a_b @ ( F @ X3 ) @ H ) ) )
=> ( ( comp_R8862297837303855554la_a_b @ H @ ( fold_n2260362839379620988la_a_b @ G @ Xs ) )
= ( comp_R8862297837303855554la_a_b @ ( fold_n2260362839379620988la_a_b @ F @ Xs ) @ H ) ) ) ).
% fold_commute
thf(fact_568_fold__commute,axiom,
! [Xs: list_R8263082107343818799la_a_b,H: relational_fmla_a_b > relational_fmla_a_b,G: relational_fmla_a_b > relational_fmla_a_b > relational_fmla_a_b,F: relational_fmla_a_b > relational_fmla_a_b > relational_fmla_a_b] :
( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ ( set_Re9104216502384355786la_a_b @ Xs ) )
=> ( ( comp_R8862297837303855554la_a_b @ H @ ( G @ X3 ) )
= ( comp_R8862297837303855554la_a_b @ ( F @ X3 ) @ H ) ) )
=> ( ( comp_R8862297837303855554la_a_b @ H @ ( fold_R509364916538903371la_a_b @ G @ Xs ) )
= ( comp_R8862297837303855554la_a_b @ ( fold_R509364916538903371la_a_b @ F @ Xs ) @ H ) ) ) ).
% fold_commute
thf(fact_569_fold__commute,axiom,
! [Xs: list_R8263082107343818799la_a_b,H: set_Re381260168593705685la_a_b > relational_fmla_a_b,G: relational_fmla_a_b > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b,F: relational_fmla_a_b > relational_fmla_a_b > relational_fmla_a_b] :
( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ ( set_Re9104216502384355786la_a_b @ Xs ) )
=> ( ( comp_s2334580085045555246la_a_b @ H @ ( G @ X3 ) )
= ( comp_R5892708597848006392la_a_b @ ( F @ X3 ) @ H ) ) )
=> ( ( comp_s2334580085045555246la_a_b @ H @ ( fold_R3800324640475171969la_a_b @ G @ Xs ) )
= ( comp_R5892708597848006392la_a_b @ ( fold_R509364916538903371la_a_b @ F @ Xs ) @ H ) ) ) ).
% fold_commute
thf(fact_570_fold__commute,axiom,
! [Xs: list_R8263082107343818799la_a_b,H: relational_fmla_a_b > set_Re381260168593705685la_a_b,G: relational_fmla_a_b > relational_fmla_a_b > relational_fmla_a_b,F: relational_fmla_a_b > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b] :
( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ ( set_Re9104216502384355786la_a_b @ Xs ) )
=> ( ( comp_R8870746840898429176la_a_b @ H @ ( G @ X3 ) )
= ( comp_s5312618328095978030la_a_b @ ( F @ X3 ) @ H ) ) )
=> ( ( comp_R8870746840898429176la_a_b @ H @ ( fold_R509364916538903371la_a_b @ G @ Xs ) )
= ( comp_s5312618328095978030la_a_b @ ( fold_R3800324640475171969la_a_b @ F @ Xs ) @ H ) ) ) ).
% fold_commute
thf(fact_571_fold__commute,axiom,
! [Xs: list_s3671366213839843941la_a_b,H: relational_fmla_a_b > relational_fmla_a_b,G: set_Re381260168593705685la_a_b > relational_fmla_a_b > relational_fmla_a_b,F: set_Re381260168593705685la_a_b > relational_fmla_a_b > relational_fmla_a_b] :
( ! [X3: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X3 @ ( set_se5111286776497155840la_a_b @ Xs ) )
=> ( ( comp_R8862297837303855554la_a_b @ H @ ( G @ X3 ) )
= ( comp_R8862297837303855554la_a_b @ ( F @ X3 ) @ H ) ) )
=> ( ( comp_R8862297837303855554la_a_b @ H @ ( fold_s6778362883525594753la_a_b @ G @ Xs ) )
= ( comp_R8862297837303855554la_a_b @ ( fold_s6778362883525594753la_a_b @ F @ Xs ) @ H ) ) ) ).
% fold_commute
thf(fact_572_fold__commute,axiom,
! [Xs: list_l3216422893793986623la_a_b,H: relational_fmla_a_b > relational_fmla_a_b,G: list_R8263082107343818799la_a_b > relational_fmla_a_b > relational_fmla_a_b,F: list_R8263082107343818799la_a_b > relational_fmla_a_b > relational_fmla_a_b] :
( ! [X3: list_R8263082107343818799la_a_b] :
( ( member8712135081189583430la_a_b @ X3 @ ( set_li8988895567068466906la_a_b @ Xs ) )
=> ( ( comp_R8862297837303855554la_a_b @ H @ ( G @ X3 ) )
= ( comp_R8862297837303855554la_a_b @ ( F @ X3 ) @ H ) ) )
=> ( ( comp_R8862297837303855554la_a_b @ H @ ( fold_l4061290520114711515la_a_b @ G @ Xs ) )
= ( comp_R8862297837303855554la_a_b @ ( fold_l4061290520114711515la_a_b @ F @ Xs ) @ H ) ) ) ).
% fold_commute
thf(fact_573_fold__commute,axiom,
! [Xs: list_R8263082107343818799la_a_b,H: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b,G: relational_fmla_a_b > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b,F: relational_fmla_a_b > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b] :
( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ ( set_Re9104216502384355786la_a_b @ Xs ) )
=> ( ( comp_s8660012457442781028la_a_b @ H @ ( G @ X3 ) )
= ( comp_s8660012457442781028la_a_b @ ( F @ X3 ) @ H ) ) )
=> ( ( comp_s8660012457442781028la_a_b @ H @ ( fold_R3800324640475171969la_a_b @ G @ Xs ) )
= ( comp_s8660012457442781028la_a_b @ ( fold_R3800324640475171969la_a_b @ F @ Xs ) @ H ) ) ) ).
% fold_commute
thf(fact_574_fold__commute,axiom,
! [Xs: list_nat,H: relational_fmla_a_b > relational_fmla_a_b > relational_fmla_a_b,G: nat > relational_fmla_a_b > relational_fmla_a_b,F: nat > ( relational_fmla_a_b > relational_fmla_a_b ) > relational_fmla_a_b > relational_fmla_a_b] :
( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( comp_R1525839178419564032la_a_b @ H @ ( G @ X3 ) )
= ( comp_R7625504482518399294la_a_b @ ( F @ X3 ) @ H ) ) )
=> ( ( comp_R1525839178419564032la_a_b @ H @ ( fold_n2260362839379620988la_a_b @ G @ Xs ) )
= ( comp_R7625504482518399294la_a_b @ ( fold_n4117695568252011962la_a_b @ F @ Xs ) @ H ) ) ) ).
% fold_commute
thf(fact_575_fold__commute,axiom,
! [Xs: list_s3671366213839843941la_a_b,H: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b,G: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b,F: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b] :
( ! [X3: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X3 @ ( set_se5111286776497155840la_a_b @ Xs ) )
=> ( ( comp_s8660012457442781028la_a_b @ H @ ( G @ X3 ) )
= ( comp_s8660012457442781028la_a_b @ ( F @ X3 ) @ H ) ) )
=> ( ( comp_s8660012457442781028la_a_b @ H @ ( fold_s1495082921969268663la_a_b @ G @ Xs ) )
= ( comp_s8660012457442781028la_a_b @ ( fold_s1495082921969268663la_a_b @ F @ Xs ) @ H ) ) ) ).
% fold_commute
thf(fact_576_fold__commute__apply,axiom,
! [Xs: list_nat,H: nat > nat,G: nat > nat > nat,F: nat > nat > nat,S3: nat] :
( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( comp_nat_nat_nat @ H @ ( G @ X3 ) )
= ( comp_nat_nat_nat @ ( F @ X3 ) @ H ) ) )
=> ( ( H @ ( fold_nat_nat @ G @ Xs @ S3 ) )
= ( fold_nat_nat @ F @ Xs @ ( H @ S3 ) ) ) ) ).
% fold_commute_apply
thf(fact_577_fold__commute__apply,axiom,
! [Xs: list_nat,H: relational_fmla_a_b > relational_fmla_a_b,G: nat > relational_fmla_a_b > relational_fmla_a_b,F: nat > relational_fmla_a_b > relational_fmla_a_b,S3: relational_fmla_a_b] :
( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( comp_R8862297837303855554la_a_b @ H @ ( G @ X3 ) )
= ( comp_R8862297837303855554la_a_b @ ( F @ X3 ) @ H ) ) )
=> ( ( H @ ( fold_n2260362839379620988la_a_b @ G @ Xs @ S3 ) )
= ( fold_n2260362839379620988la_a_b @ F @ Xs @ ( H @ S3 ) ) ) ) ).
% fold_commute_apply
thf(fact_578_fold__commute__apply,axiom,
! [Xs: list_R8263082107343818799la_a_b,H: relational_fmla_a_b > relational_fmla_a_b,G: relational_fmla_a_b > relational_fmla_a_b > relational_fmla_a_b,F: relational_fmla_a_b > relational_fmla_a_b > relational_fmla_a_b,S3: relational_fmla_a_b] :
( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ ( set_Re9104216502384355786la_a_b @ Xs ) )
=> ( ( comp_R8862297837303855554la_a_b @ H @ ( G @ X3 ) )
= ( comp_R8862297837303855554la_a_b @ ( F @ X3 ) @ H ) ) )
=> ( ( H @ ( fold_R509364916538903371la_a_b @ G @ Xs @ S3 ) )
= ( fold_R509364916538903371la_a_b @ F @ Xs @ ( H @ S3 ) ) ) ) ).
% fold_commute_apply
thf(fact_579_fold__commute__apply,axiom,
! [Xs: list_R8263082107343818799la_a_b,H: set_Re381260168593705685la_a_b > relational_fmla_a_b,G: relational_fmla_a_b > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b,F: relational_fmla_a_b > relational_fmla_a_b > relational_fmla_a_b,S3: set_Re381260168593705685la_a_b] :
( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ ( set_Re9104216502384355786la_a_b @ Xs ) )
=> ( ( comp_s2334580085045555246la_a_b @ H @ ( G @ X3 ) )
= ( comp_R5892708597848006392la_a_b @ ( F @ X3 ) @ H ) ) )
=> ( ( H @ ( fold_R3800324640475171969la_a_b @ G @ Xs @ S3 ) )
= ( fold_R509364916538903371la_a_b @ F @ Xs @ ( H @ S3 ) ) ) ) ).
% fold_commute_apply
thf(fact_580_fold__commute__apply,axiom,
! [Xs: list_R8263082107343818799la_a_b,H: relational_fmla_a_b > set_Re381260168593705685la_a_b,G: relational_fmla_a_b > relational_fmla_a_b > relational_fmla_a_b,F: relational_fmla_a_b > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b,S3: relational_fmla_a_b] :
( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ ( set_Re9104216502384355786la_a_b @ Xs ) )
=> ( ( comp_R8870746840898429176la_a_b @ H @ ( G @ X3 ) )
= ( comp_s5312618328095978030la_a_b @ ( F @ X3 ) @ H ) ) )
=> ( ( H @ ( fold_R509364916538903371la_a_b @ G @ Xs @ S3 ) )
= ( fold_R3800324640475171969la_a_b @ F @ Xs @ ( H @ S3 ) ) ) ) ).
% fold_commute_apply
thf(fact_581_fold__commute__apply,axiom,
! [Xs: list_s3671366213839843941la_a_b,H: relational_fmla_a_b > relational_fmla_a_b,G: set_Re381260168593705685la_a_b > relational_fmla_a_b > relational_fmla_a_b,F: set_Re381260168593705685la_a_b > relational_fmla_a_b > relational_fmla_a_b,S3: relational_fmla_a_b] :
( ! [X3: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X3 @ ( set_se5111286776497155840la_a_b @ Xs ) )
=> ( ( comp_R8862297837303855554la_a_b @ H @ ( G @ X3 ) )
= ( comp_R8862297837303855554la_a_b @ ( F @ X3 ) @ H ) ) )
=> ( ( H @ ( fold_s6778362883525594753la_a_b @ G @ Xs @ S3 ) )
= ( fold_s6778362883525594753la_a_b @ F @ Xs @ ( H @ S3 ) ) ) ) ).
% fold_commute_apply
thf(fact_582_fold__commute__apply,axiom,
! [Xs: list_l3216422893793986623la_a_b,H: relational_fmla_a_b > relational_fmla_a_b,G: list_R8263082107343818799la_a_b > relational_fmla_a_b > relational_fmla_a_b,F: list_R8263082107343818799la_a_b > relational_fmla_a_b > relational_fmla_a_b,S3: relational_fmla_a_b] :
( ! [X3: list_R8263082107343818799la_a_b] :
( ( member8712135081189583430la_a_b @ X3 @ ( set_li8988895567068466906la_a_b @ Xs ) )
=> ( ( comp_R8862297837303855554la_a_b @ H @ ( G @ X3 ) )
= ( comp_R8862297837303855554la_a_b @ ( F @ X3 ) @ H ) ) )
=> ( ( H @ ( fold_l4061290520114711515la_a_b @ G @ Xs @ S3 ) )
= ( fold_l4061290520114711515la_a_b @ F @ Xs @ ( H @ S3 ) ) ) ) ).
% fold_commute_apply
thf(fact_583_fold__commute__apply,axiom,
! [Xs: list_R8263082107343818799la_a_b,H: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b,G: relational_fmla_a_b > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b,F: relational_fmla_a_b > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b,S3: set_Re381260168593705685la_a_b] :
( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ ( set_Re9104216502384355786la_a_b @ Xs ) )
=> ( ( comp_s8660012457442781028la_a_b @ H @ ( G @ X3 ) )
= ( comp_s8660012457442781028la_a_b @ ( F @ X3 ) @ H ) ) )
=> ( ( H @ ( fold_R3800324640475171969la_a_b @ G @ Xs @ S3 ) )
= ( fold_R3800324640475171969la_a_b @ F @ Xs @ ( H @ S3 ) ) ) ) ).
% fold_commute_apply
thf(fact_584_fold__commute__apply,axiom,
! [Xs: list_nat,H: relational_fmla_a_b > relational_fmla_a_b > relational_fmla_a_b,G: nat > relational_fmla_a_b > relational_fmla_a_b,F: nat > ( relational_fmla_a_b > relational_fmla_a_b ) > relational_fmla_a_b > relational_fmla_a_b,S3: relational_fmla_a_b] :
( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( comp_R1525839178419564032la_a_b @ H @ ( G @ X3 ) )
= ( comp_R7625504482518399294la_a_b @ ( F @ X3 ) @ H ) ) )
=> ( ( H @ ( fold_n2260362839379620988la_a_b @ G @ Xs @ S3 ) )
= ( fold_n4117695568252011962la_a_b @ F @ Xs @ ( H @ S3 ) ) ) ) ).
% fold_commute_apply
thf(fact_585_fold__commute__apply,axiom,
! [Xs: list_s3671366213839843941la_a_b,H: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b,G: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b,F: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b,S3: set_Re381260168593705685la_a_b] :
( ! [X3: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X3 @ ( set_se5111286776497155840la_a_b @ Xs ) )
=> ( ( comp_s8660012457442781028la_a_b @ H @ ( G @ X3 ) )
= ( comp_s8660012457442781028la_a_b @ ( F @ X3 ) @ H ) ) )
=> ( ( H @ ( fold_s1495082921969268663la_a_b @ G @ Xs @ S3 ) )
= ( fold_s1495082921969268663la_a_b @ F @ Xs @ ( H @ S3 ) ) ) ) ).
% fold_commute_apply
thf(fact_586_Diff__insert,axiom,
! [A2: set_li8334890247616788453la_a_b,A: list_R8263082107343818799la_a_b,B: set_li8334890247616788453la_a_b] :
( ( minus_4977183416030855486la_a_b @ A2 @ ( insert3617074100887873119la_a_b @ A @ B ) )
= ( minus_4977183416030855486la_a_b @ ( minus_4977183416030855486la_a_b @ A2 @ B ) @ ( insert3617074100887873119la_a_b @ A @ bot_bo3230846185698255993la_a_b ) ) ) ).
% Diff_insert
thf(fact_587_Diff__insert,axiom,
! [A2: set_se6865892389300016395la_a_b,A: set_Re381260168593705685la_a_b,B: set_se6865892389300016395la_a_b] :
( ( minus_4705846553145473764la_a_b @ A2 @ ( insert2023870700798818565la_a_b @ A @ B ) )
= ( minus_4705846553145473764la_a_b @ ( minus_4705846553145473764la_a_b @ A2 @ B ) @ ( insert2023870700798818565la_a_b @ A @ bot_bo2891247006866115487la_a_b ) ) ) ).
% Diff_insert
thf(fact_588_Diff__insert,axiom,
! [A2: set_nat,A: nat,B: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ A @ B ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B ) @ ( insert_nat2 @ A @ bot_bot_set_nat ) ) ) ).
% Diff_insert
thf(fact_589_Diff__insert,axiom,
! [A2: set_Re381260168593705685la_a_b,A: relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ( minus_4077726661957047470la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ A @ B ) )
= ( minus_4077726661957047470la_a_b @ ( minus_4077726661957047470la_a_b @ A2 @ B ) @ ( insert7010464514620295119la_a_b @ A @ bot_bo4495933725496725865la_a_b ) ) ) ).
% Diff_insert
thf(fact_590_insert__Diff,axiom,
! [A: list_R8263082107343818799la_a_b,A2: set_li8334890247616788453la_a_b] :
( ( member8712135081189583430la_a_b @ A @ A2 )
=> ( ( insert3617074100887873119la_a_b @ A @ ( minus_4977183416030855486la_a_b @ A2 @ ( insert3617074100887873119la_a_b @ A @ bot_bo3230846185698255993la_a_b ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_591_insert__Diff,axiom,
! [A: set_Re381260168593705685la_a_b,A2: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ A @ A2 )
=> ( ( insert2023870700798818565la_a_b @ A @ ( minus_4705846553145473764la_a_b @ A2 @ ( insert2023870700798818565la_a_b @ A @ bot_bo2891247006866115487la_a_b ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_592_insert__Diff,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat2 @ A @ A2 )
=> ( ( insert_nat2 @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ A @ bot_bot_set_nat ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_593_insert__Diff,axiom,
! [A: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ A @ A2 )
=> ( ( insert7010464514620295119la_a_b @ A @ ( minus_4077726661957047470la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ A @ bot_bo4495933725496725865la_a_b ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_594_Diff__insert2,axiom,
! [A2: set_li8334890247616788453la_a_b,A: list_R8263082107343818799la_a_b,B: set_li8334890247616788453la_a_b] :
( ( minus_4977183416030855486la_a_b @ A2 @ ( insert3617074100887873119la_a_b @ A @ B ) )
= ( minus_4977183416030855486la_a_b @ ( minus_4977183416030855486la_a_b @ A2 @ ( insert3617074100887873119la_a_b @ A @ bot_bo3230846185698255993la_a_b ) ) @ B ) ) ).
% Diff_insert2
thf(fact_595_Diff__insert2,axiom,
! [A2: set_se6865892389300016395la_a_b,A: set_Re381260168593705685la_a_b,B: set_se6865892389300016395la_a_b] :
( ( minus_4705846553145473764la_a_b @ A2 @ ( insert2023870700798818565la_a_b @ A @ B ) )
= ( minus_4705846553145473764la_a_b @ ( minus_4705846553145473764la_a_b @ A2 @ ( insert2023870700798818565la_a_b @ A @ bot_bo2891247006866115487la_a_b ) ) @ B ) ) ).
% Diff_insert2
thf(fact_596_Diff__insert2,axiom,
! [A2: set_nat,A: nat,B: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ A @ B ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ A @ bot_bot_set_nat ) ) @ B ) ) ).
% Diff_insert2
thf(fact_597_Diff__insert2,axiom,
! [A2: set_Re381260168593705685la_a_b,A: relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ( minus_4077726661957047470la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ A @ B ) )
= ( minus_4077726661957047470la_a_b @ ( minus_4077726661957047470la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ A @ bot_bo4495933725496725865la_a_b ) ) @ B ) ) ).
% Diff_insert2
thf(fact_598_Diff__insert__absorb,axiom,
! [X: list_R8263082107343818799la_a_b,A2: set_li8334890247616788453la_a_b] :
( ~ ( member8712135081189583430la_a_b @ X @ A2 )
=> ( ( minus_4977183416030855486la_a_b @ ( insert3617074100887873119la_a_b @ X @ A2 ) @ ( insert3617074100887873119la_a_b @ X @ bot_bo3230846185698255993la_a_b ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_599_Diff__insert__absorb,axiom,
! [X: set_Re381260168593705685la_a_b,A2: set_se6865892389300016395la_a_b] :
( ~ ( member3481406638322139244la_a_b @ X @ A2 )
=> ( ( minus_4705846553145473764la_a_b @ ( insert2023870700798818565la_a_b @ X @ A2 ) @ ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_600_Diff__insert__absorb,axiom,
! [X: nat,A2: set_nat] :
( ~ ( member_nat2 @ X @ A2 )
=> ( ( minus_minus_set_nat @ ( insert_nat2 @ X @ A2 ) @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_601_Diff__insert__absorb,axiom,
! [X: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ~ ( member4680049679412964150la_a_b @ X @ A2 )
=> ( ( minus_4077726661957047470la_a_b @ ( insert7010464514620295119la_a_b @ X @ A2 ) @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_602_List_Oinsert__def,axiom,
( insert3209172818022656531la_a_b
= ( ^ [X2: list_R8263082107343818799la_a_b,Xs2: list_l3216422893793986623la_a_b] : ( if_lis5916486426012477945la_a_b @ ( member8712135081189583430la_a_b @ X2 @ ( set_li8988895567068466906la_a_b @ Xs2 ) ) @ Xs2 @ ( cons_l6716632503304403961la_a_b @ X2 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_603_List_Oinsert__def,axiom,
( insert443281300761040569la_a_b
= ( ^ [X2: set_Re381260168593705685la_a_b,Xs2: list_s3671366213839843941la_a_b] : ( if_lis5716551597998330143la_a_b @ ( member3481406638322139244la_a_b @ X2 @ ( set_se5111286776497155840la_a_b @ Xs2 ) ) @ Xs2 @ ( cons_s2104095074707585951la_a_b @ X2 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_604_List_Oinsert__def,axiom,
( insert_nat
= ( ^ [X2: nat,Xs2: list_nat] : ( if_list_nat @ ( member_nat2 @ X2 @ ( set_nat2 @ Xs2 ) ) @ Xs2 @ ( cons_nat @ X2 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_605_List_Oinsert__def,axiom,
( insert5052544347734373763la_a_b
= ( ^ [X2: relational_fmla_a_b,Xs2: list_R8263082107343818799la_a_b] : ( if_lis1478040310087653097la_a_b @ ( member4680049679412964150la_a_b @ X2 @ ( set_Re9104216502384355786la_a_b @ Xs2 ) ) @ Xs2 @ ( cons_R4161027894713053801la_a_b @ X2 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_606_Sup__insert,axiom,
! [A: set_se6865892389300016395la_a_b,A2: set_se1626149436229564481la_a_b] :
( ( comple7774258788699406174la_a_b @ ( insert1549238021488837435la_a_b @ A @ A2 ) )
= ( sup_su4783144482993978935la_a_b @ A @ ( comple7774258788699406174la_a_b @ A2 ) ) ) ).
% Sup_insert
thf(fact_607_Sup__insert,axiom,
! [A: set_nat,A2: set_set_nat] :
( ( comple7399068483239264473et_nat @ ( insert_set_nat @ A @ A2 ) )
= ( sup_sup_set_nat @ A @ ( comple7399068483239264473et_nat @ A2 ) ) ) ).
% Sup_insert
thf(fact_608_Sup__insert,axiom,
! [A: set_Re381260168593705685la_a_b,A2: set_se6865892389300016395la_a_b] :
( ( comple8442120529048846632la_a_b @ ( insert2023870700798818565la_a_b @ A @ A2 ) )
= ( sup_su5130108678486352897la_a_b @ A @ ( comple8442120529048846632la_a_b @ A2 ) ) ) ).
% Sup_insert
thf(fact_609_ccpo__Sup__singleton,axiom,
! [X: set_Re381260168593705685la_a_b] :
( ( comple8442120529048846632la_a_b @ ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) )
= X ) ).
% ccpo_Sup_singleton
thf(fact_610_cSup__singleton,axiom,
! [X: set_Re381260168593705685la_a_b] :
( ( comple8442120529048846632la_a_b @ ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) )
= X ) ).
% cSup_singleton
thf(fact_611_cSup__singleton,axiom,
! [X: nat] :
( ( complete_Sup_Sup_nat @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
= X ) ).
% cSup_singleton
thf(fact_612_Sup__empty,axiom,
( ( comple3423763510802972509_a_b_o @ bot_bo7939576951503870140_a_b_o )
= bot_bo8852203127187332700_a_b_o ) ).
% Sup_empty
thf(fact_613_Sup__empty,axiom,
( ( comple7774258788699406174la_a_b @ bot_bo1069171165257752277la_a_b )
= bot_bo2891247006866115487la_a_b ) ).
% Sup_empty
thf(fact_614_Sup__empty,axiom,
( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% Sup_empty
thf(fact_615_Sup__empty,axiom,
( ( comple4129898094138812600la_a_b @ bot_bo3079997351022007599la_a_b )
= bot_bo3230846185698255993la_a_b ) ).
% Sup_empty
thf(fact_616_Sup__empty,axiom,
( ( comple8442120529048846632la_a_b @ bot_bo2891247006866115487la_a_b )
= bot_bo4495933725496725865la_a_b ) ).
% Sup_empty
thf(fact_617_Sup__bot__conv_I2_J,axiom,
! [A2: set_Re4210025064442248784_a_b_o] :
( ( bot_bo8852203127187332700_a_b_o
= ( comple3423763510802972509_a_b_o @ A2 ) )
= ( ! [X2: relational_fmla_a_b > $o] :
( ( member3985631836946730265_a_b_o @ X2 @ A2 )
=> ( X2 = bot_bo8852203127187332700_a_b_o ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_618_Sup__bot__conv_I2_J,axiom,
! [A2: set_se1626149436229564481la_a_b] :
( ( bot_bo2891247006866115487la_a_b
= ( comple7774258788699406174la_a_b @ A2 ) )
= ( ! [X2: set_se6865892389300016395la_a_b] :
( ( member1693404252451860130la_a_b @ X2 @ A2 )
=> ( X2 = bot_bo2891247006866115487la_a_b ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_619_Sup__bot__conv_I2_J,axiom,
! [A2: set_set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ A2 ) )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( X2 = bot_bot_set_nat ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_620_Sup__bot__conv_I2_J,axiom,
! [A2: set_se2998258007684708507la_a_b] :
( ( bot_bo3230846185698255993la_a_b
= ( comple4129898094138812600la_a_b @ A2 ) )
= ( ! [X2: set_li8334890247616788453la_a_b] :
( ( member5650711565134853628la_a_b @ X2 @ A2 )
=> ( X2 = bot_bo3230846185698255993la_a_b ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_621_Sup__bot__conv_I2_J,axiom,
! [A2: set_se6865892389300016395la_a_b] :
( ( bot_bo4495933725496725865la_a_b
= ( comple8442120529048846632la_a_b @ A2 ) )
= ( ! [X2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X2 @ A2 )
=> ( X2 = bot_bo4495933725496725865la_a_b ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_622_Sup__bot__conv_I1_J,axiom,
! [A2: set_Re4210025064442248784_a_b_o] :
( ( ( comple3423763510802972509_a_b_o @ A2 )
= bot_bo8852203127187332700_a_b_o )
= ( ! [X2: relational_fmla_a_b > $o] :
( ( member3985631836946730265_a_b_o @ X2 @ A2 )
=> ( X2 = bot_bo8852203127187332700_a_b_o ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_623_Sup__bot__conv_I1_J,axiom,
! [A2: set_se1626149436229564481la_a_b] :
( ( ( comple7774258788699406174la_a_b @ A2 )
= bot_bo2891247006866115487la_a_b )
= ( ! [X2: set_se6865892389300016395la_a_b] :
( ( member1693404252451860130la_a_b @ X2 @ A2 )
=> ( X2 = bot_bo2891247006866115487la_a_b ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_624_Sup__bot__conv_I1_J,axiom,
! [A2: set_set_nat] :
( ( ( comple7399068483239264473et_nat @ A2 )
= bot_bot_set_nat )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( X2 = bot_bot_set_nat ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_625_Sup__bot__conv_I1_J,axiom,
! [A2: set_se2998258007684708507la_a_b] :
( ( ( comple4129898094138812600la_a_b @ A2 )
= bot_bo3230846185698255993la_a_b )
= ( ! [X2: set_li8334890247616788453la_a_b] :
( ( member5650711565134853628la_a_b @ X2 @ A2 )
=> ( X2 = bot_bo3230846185698255993la_a_b ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_626_Sup__bot__conv_I1_J,axiom,
! [A2: set_se6865892389300016395la_a_b] :
( ( ( comple8442120529048846632la_a_b @ A2 )
= bot_bo4495933725496725865la_a_b )
= ( ! [X2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X2 @ A2 )
=> ( X2 = bot_bo4495933725496725865la_a_b ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_627_Sup__union__distrib,axiom,
! [A2: set_se1626149436229564481la_a_b,B: set_se1626149436229564481la_a_b] :
( ( comple7774258788699406174la_a_b @ ( sup_su4447918665525899629la_a_b @ A2 @ B ) )
= ( sup_su4783144482993978935la_a_b @ ( comple7774258788699406174la_a_b @ A2 ) @ ( comple7774258788699406174la_a_b @ B ) ) ) ).
% Sup_union_distrib
thf(fact_628_Sup__union__distrib,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( comple7399068483239264473et_nat @ ( sup_sup_set_set_nat @ A2 @ B ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Sup_union_distrib
thf(fact_629_Sup__union__distrib,axiom,
! [A2: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ( comple8442120529048846632la_a_b @ ( sup_su4783144482993978935la_a_b @ A2 @ B ) )
= ( sup_su5130108678486352897la_a_b @ ( comple8442120529048846632la_a_b @ A2 ) @ ( comple8442120529048846632la_a_b @ B ) ) ) ).
% Sup_union_distrib
thf(fact_630_insert__remove__id,axiom,
! [X: list_R8263082107343818799la_a_b,X4: set_li8334890247616788453la_a_b] :
( ( member8712135081189583430la_a_b @ X @ X4 )
=> ( X4
= ( insert3617074100887873119la_a_b @ X @ ( minus_4977183416030855486la_a_b @ X4 @ ( insert3617074100887873119la_a_b @ X @ bot_bo3230846185698255993la_a_b ) ) ) ) ) ).
% insert_remove_id
thf(fact_631_insert__remove__id,axiom,
! [X: set_Re381260168593705685la_a_b,X4: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ X @ X4 )
=> ( X4
= ( insert2023870700798818565la_a_b @ X @ ( minus_4705846553145473764la_a_b @ X4 @ ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) ) ) ) ) ).
% insert_remove_id
thf(fact_632_insert__remove__id,axiom,
! [X: nat,X4: set_nat] :
( ( member_nat2 @ X @ X4 )
=> ( X4
= ( insert_nat2 @ X @ ( minus_minus_set_nat @ X4 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ) ).
% insert_remove_id
thf(fact_633_insert__remove__id,axiom,
! [X: relational_fmla_a_b,X4: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ X @ X4 )
=> ( X4
= ( insert7010464514620295119la_a_b @ X @ ( minus_4077726661957047470la_a_b @ X4 @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) ) ) ) ).
% insert_remove_id
thf(fact_634_Union__Un__distrib,axiom,
! [A2: set_se1626149436229564481la_a_b,B: set_se1626149436229564481la_a_b] :
( ( comple7774258788699406174la_a_b @ ( sup_su4447918665525899629la_a_b @ A2 @ B ) )
= ( sup_su4783144482993978935la_a_b @ ( comple7774258788699406174la_a_b @ A2 ) @ ( comple7774258788699406174la_a_b @ B ) ) ) ).
% Union_Un_distrib
thf(fact_635_Union__Un__distrib,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( comple7399068483239264473et_nat @ ( sup_sup_set_set_nat @ A2 @ B ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Union_Un_distrib
thf(fact_636_Union__Un__distrib,axiom,
! [A2: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ( comple8442120529048846632la_a_b @ ( sup_su4783144482993978935la_a_b @ A2 @ B ) )
= ( sup_su5130108678486352897la_a_b @ ( comple8442120529048846632la_a_b @ A2 ) @ ( comple8442120529048846632la_a_b @ B ) ) ) ).
% Union_Un_distrib
thf(fact_637_empty__Union__conv,axiom,
! [A2: set_se1626149436229564481la_a_b] :
( ( bot_bo2891247006866115487la_a_b
= ( comple7774258788699406174la_a_b @ A2 ) )
= ( ! [X2: set_se6865892389300016395la_a_b] :
( ( member1693404252451860130la_a_b @ X2 @ A2 )
=> ( X2 = bot_bo2891247006866115487la_a_b ) ) ) ) ).
% empty_Union_conv
thf(fact_638_empty__Union__conv,axiom,
! [A2: set_set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ A2 ) )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( X2 = bot_bot_set_nat ) ) ) ) ).
% empty_Union_conv
thf(fact_639_empty__Union__conv,axiom,
! [A2: set_se2998258007684708507la_a_b] :
( ( bot_bo3230846185698255993la_a_b
= ( comple4129898094138812600la_a_b @ A2 ) )
= ( ! [X2: set_li8334890247616788453la_a_b] :
( ( member5650711565134853628la_a_b @ X2 @ A2 )
=> ( X2 = bot_bo3230846185698255993la_a_b ) ) ) ) ).
% empty_Union_conv
thf(fact_640_empty__Union__conv,axiom,
! [A2: set_se6865892389300016395la_a_b] :
( ( bot_bo4495933725496725865la_a_b
= ( comple8442120529048846632la_a_b @ A2 ) )
= ( ! [X2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X2 @ A2 )
=> ( X2 = bot_bo4495933725496725865la_a_b ) ) ) ) ).
% empty_Union_conv
thf(fact_641_Union__empty__conv,axiom,
! [A2: set_se1626149436229564481la_a_b] :
( ( ( comple7774258788699406174la_a_b @ A2 )
= bot_bo2891247006866115487la_a_b )
= ( ! [X2: set_se6865892389300016395la_a_b] :
( ( member1693404252451860130la_a_b @ X2 @ A2 )
=> ( X2 = bot_bo2891247006866115487la_a_b ) ) ) ) ).
% Union_empty_conv
thf(fact_642_Union__empty__conv,axiom,
! [A2: set_set_nat] :
( ( ( comple7399068483239264473et_nat @ A2 )
= bot_bot_set_nat )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( X2 = bot_bot_set_nat ) ) ) ) ).
% Union_empty_conv
thf(fact_643_Union__empty__conv,axiom,
! [A2: set_se2998258007684708507la_a_b] :
( ( ( comple4129898094138812600la_a_b @ A2 )
= bot_bo3230846185698255993la_a_b )
= ( ! [X2: set_li8334890247616788453la_a_b] :
( ( member5650711565134853628la_a_b @ X2 @ A2 )
=> ( X2 = bot_bo3230846185698255993la_a_b ) ) ) ) ).
% Union_empty_conv
thf(fact_644_Union__empty__conv,axiom,
! [A2: set_se6865892389300016395la_a_b] :
( ( ( comple8442120529048846632la_a_b @ A2 )
= bot_bo4495933725496725865la_a_b )
= ( ! [X2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X2 @ A2 )
=> ( X2 = bot_bo4495933725496725865la_a_b ) ) ) ) ).
% Union_empty_conv
thf(fact_645_Union__empty,axiom,
( ( comple7774258788699406174la_a_b @ bot_bo1069171165257752277la_a_b )
= bot_bo2891247006866115487la_a_b ) ).
% Union_empty
thf(fact_646_Union__empty,axiom,
( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% Union_empty
thf(fact_647_Union__empty,axiom,
( ( comple4129898094138812600la_a_b @ bot_bo3079997351022007599la_a_b )
= bot_bo3230846185698255993la_a_b ) ).
% Union_empty
thf(fact_648_Union__empty,axiom,
( ( comple8442120529048846632la_a_b @ bot_bo2891247006866115487la_a_b )
= bot_bo4495933725496725865la_a_b ) ).
% Union_empty
thf(fact_649_Union__insert,axiom,
! [A: set_se6865892389300016395la_a_b,B: set_se1626149436229564481la_a_b] :
( ( comple7774258788699406174la_a_b @ ( insert1549238021488837435la_a_b @ A @ B ) )
= ( sup_su4783144482993978935la_a_b @ A @ ( comple7774258788699406174la_a_b @ B ) ) ) ).
% Union_insert
thf(fact_650_Union__insert,axiom,
! [A: set_nat,B: set_set_nat] :
( ( comple7399068483239264473et_nat @ ( insert_set_nat @ A @ B ) )
= ( sup_sup_set_nat @ A @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Union_insert
thf(fact_651_Union__insert,axiom,
! [A: set_Re381260168593705685la_a_b,B: set_se6865892389300016395la_a_b] :
( ( comple8442120529048846632la_a_b @ ( insert2023870700798818565la_a_b @ A @ B ) )
= ( sup_su5130108678486352897la_a_b @ A @ ( comple8442120529048846632la_a_b @ B ) ) ) ).
% Union_insert
thf(fact_652_comp__apply,axiom,
( comp_R8862297837303855554la_a_b
= ( ^ [F2: relational_fmla_a_b > relational_fmla_a_b,G2: relational_fmla_a_b > relational_fmla_a_b,X2: relational_fmla_a_b] : ( F2 @ ( G2 @ X2 ) ) ) ) ).
% comp_apply
thf(fact_653_comp__apply,axiom,
( comp_s8363053932693654946la_a_b
= ( ^ [F2: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b,G2: relational_fmla_a_b > set_Re381260168593705685la_a_b,X2: relational_fmla_a_b] : ( F2 @ ( G2 @ X2 ) ) ) ) ).
% comp_apply
thf(fact_654_comp__apply,axiom,
( comp_R1525839178419564032la_a_b
= ( ^ [F2: relational_fmla_a_b > relational_fmla_a_b > relational_fmla_a_b,G2: relational_fmla_a_b > relational_fmla_a_b,X2: relational_fmla_a_b] : ( F2 @ ( G2 @ X2 ) ) ) ) ).
% comp_apply
thf(fact_655_minus__set__fold,axiom,
! [A2: set_li8334890247616788453la_a_b,Xs: list_l3216422893793986623la_a_b] :
( ( minus_4977183416030855486la_a_b @ A2 @ ( set_li8988895567068466906la_a_b @ Xs ) )
= ( fold_l8426049599603076129la_a_b @ remove813535772580499850la_a_b @ Xs @ A2 ) ) ).
% minus_set_fold
thf(fact_656_minus__set__fold,axiom,
! [A2: set_se6865892389300016395la_a_b,Xs: list_s3671366213839843941la_a_b] :
( ( minus_4705846553145473764la_a_b @ A2 @ ( set_se5111286776497155840la_a_b @ Xs ) )
= ( fold_s8280270096837327341la_a_b @ remove6344774115224635824la_a_b @ Xs @ A2 ) ) ).
% minus_set_fold
thf(fact_657_minus__set__fold,axiom,
! [A2: set_nat,Xs: list_nat] :
( ( minus_minus_set_nat @ A2 @ ( set_nat2 @ Xs ) )
= ( fold_nat_set_nat @ remove_nat @ Xs @ A2 ) ) ).
% minus_set_fold
thf(fact_658_minus__set__fold,axiom,
! [A2: set_Re381260168593705685la_a_b,Xs: list_R8263082107343818799la_a_b] :
( ( minus_4077726661957047470la_a_b @ A2 @ ( set_Re9104216502384355786la_a_b @ Xs ) )
= ( fold_R3800324640475171969la_a_b @ remove4261432235257513082la_a_b @ Xs @ A2 ) ) ).
% minus_set_fold
thf(fact_659_set__removeAll,axiom,
! [X: list_R8263082107343818799la_a_b,Xs: list_l3216422893793986623la_a_b] :
( ( set_li8988895567068466906la_a_b @ ( remove6300469919649359139la_a_b @ X @ Xs ) )
= ( minus_4977183416030855486la_a_b @ ( set_li8988895567068466906la_a_b @ Xs ) @ ( insert3617074100887873119la_a_b @ X @ bot_bo3230846185698255993la_a_b ) ) ) ).
% set_removeAll
thf(fact_660_set__removeAll,axiom,
! [X: set_Re381260168593705685la_a_b,Xs: list_s3671366213839843941la_a_b] :
( ( set_se5111286776497155840la_a_b @ ( remove1701277439453780425la_a_b @ X @ Xs ) )
= ( minus_4705846553145473764la_a_b @ ( set_se5111286776497155840la_a_b @ Xs ) @ ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) ) ) ).
% set_removeAll
thf(fact_661_set__removeAll,axiom,
! [X: nat,Xs: list_nat] :
( ( set_nat2 @ ( removeAll_nat @ X @ Xs ) )
= ( minus_minus_set_nat @ ( set_nat2 @ Xs ) @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ).
% set_removeAll
thf(fact_662_set__removeAll,axiom,
! [X: relational_fmla_a_b,Xs: list_R8263082107343818799la_a_b] :
( ( set_Re9104216502384355786la_a_b @ ( remove4882698675883194515la_a_b @ X @ Xs ) )
= ( minus_4077726661957047470la_a_b @ ( set_Re9104216502384355786la_a_b @ Xs ) @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) ) ).
% set_removeAll
thf(fact_663_restrict_Osimps_I1_J,axiom,
! [X: nat,A2: set_nat,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( ( member_nat2 @ X @ A2 )
=> ( ( restrict_nat_nat @ A2 @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) )
= ( cons_nat @ Y @ ( restrict_nat_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) @ Xs @ Ys ) ) ) )
& ( ~ ( member_nat2 @ X @ A2 )
=> ( ( restrict_nat_nat @ A2 @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) )
= ( restrict_nat_nat @ A2 @ Xs @ Ys ) ) ) ) ).
% restrict.simps(1)
thf(fact_664_restrict_Osimps_I1_J,axiom,
! [X: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,Xs: list_R8263082107343818799la_a_b,Y: nat,Ys: list_nat] :
( ( ( member4680049679412964150la_a_b @ X @ A2 )
=> ( ( restri353061974653101909_b_nat @ A2 @ ( cons_R4161027894713053801la_a_b @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) )
= ( cons_nat @ Y @ ( restri353061974653101909_b_nat @ ( minus_4077726661957047470la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) @ Xs @ Ys ) ) ) )
& ( ~ ( member4680049679412964150la_a_b @ X @ A2 )
=> ( ( restri353061974653101909_b_nat @ A2 @ ( cons_R4161027894713053801la_a_b @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) )
= ( restri353061974653101909_b_nat @ A2 @ Xs @ Ys ) ) ) ) ).
% restrict.simps(1)
thf(fact_665_restrict_Osimps_I1_J,axiom,
! [X: nat,A2: set_nat,Xs: list_nat,Y: relational_fmla_a_b,Ys: list_R8263082107343818799la_a_b] :
( ( ( member_nat2 @ X @ A2 )
=> ( ( restri4398310930004675157la_a_b @ A2 @ ( cons_nat @ X @ Xs ) @ ( cons_R4161027894713053801la_a_b @ Y @ Ys ) )
= ( cons_R4161027894713053801la_a_b @ Y @ ( restri4398310930004675157la_a_b @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) @ Xs @ Ys ) ) ) )
& ( ~ ( member_nat2 @ X @ A2 )
=> ( ( restri4398310930004675157la_a_b @ A2 @ ( cons_nat @ X @ Xs ) @ ( cons_R4161027894713053801la_a_b @ Y @ Ys ) )
= ( restri4398310930004675157la_a_b @ A2 @ Xs @ Ys ) ) ) ) ).
% restrict.simps(1)
thf(fact_666_restrict_Osimps_I1_J,axiom,
! [X: list_R8263082107343818799la_a_b,A2: set_li8334890247616788453la_a_b,Xs: list_l3216422893793986623la_a_b,Y: nat,Ys: list_nat] :
( ( ( member8712135081189583430la_a_b @ X @ A2 )
=> ( ( restri4831989613688677_b_nat @ A2 @ ( cons_l6716632503304403961la_a_b @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) )
= ( cons_nat @ Y @ ( restri4831989613688677_b_nat @ ( minus_4977183416030855486la_a_b @ A2 @ ( insert3617074100887873119la_a_b @ X @ bot_bo3230846185698255993la_a_b ) ) @ Xs @ Ys ) ) ) )
& ( ~ ( member8712135081189583430la_a_b @ X @ A2 )
=> ( ( restri4831989613688677_b_nat @ A2 @ ( cons_l6716632503304403961la_a_b @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) )
= ( restri4831989613688677_b_nat @ A2 @ Xs @ Ys ) ) ) ) ).
% restrict.simps(1)
thf(fact_667_restrict_Osimps_I1_J,axiom,
! [X: set_Re381260168593705685la_a_b,A2: set_se6865892389300016395la_a_b,Xs: list_s3671366213839843941la_a_b,Y: nat,Ys: list_nat] :
( ( ( member3481406638322139244la_a_b @ X @ A2 )
=> ( ( restri9062294100618341515_b_nat @ A2 @ ( cons_s2104095074707585951la_a_b @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) )
= ( cons_nat @ Y @ ( restri9062294100618341515_b_nat @ ( minus_4705846553145473764la_a_b @ A2 @ ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) ) @ Xs @ Ys ) ) ) )
& ( ~ ( member3481406638322139244la_a_b @ X @ A2 )
=> ( ( restri9062294100618341515_b_nat @ A2 @ ( cons_s2104095074707585951la_a_b @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) )
= ( restri9062294100618341515_b_nat @ A2 @ Xs @ Ys ) ) ) ) ).
% restrict.simps(1)
thf(fact_668_restrict_Osimps_I1_J,axiom,
! [X: nat,A2: set_nat,Xs: list_nat,Y: set_Re381260168593705685la_a_b,Ys: list_s3671366213839843941la_a_b] :
( ( ( member_nat2 @ X @ A2 )
=> ( ( restri9173109082640032651la_a_b @ A2 @ ( cons_nat @ X @ Xs ) @ ( cons_s2104095074707585951la_a_b @ Y @ Ys ) )
= ( cons_s2104095074707585951la_a_b @ Y @ ( restri9173109082640032651la_a_b @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) @ Xs @ Ys ) ) ) )
& ( ~ ( member_nat2 @ X @ A2 )
=> ( ( restri9173109082640032651la_a_b @ A2 @ ( cons_nat @ X @ Xs ) @ ( cons_s2104095074707585951la_a_b @ Y @ Ys ) )
= ( restri9173109082640032651la_a_b @ A2 @ Xs @ Ys ) ) ) ) ).
% restrict.simps(1)
thf(fact_669_restrict_Osimps_I1_J,axiom,
! [X: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,Xs: list_R8263082107343818799la_a_b,Y: relational_fmla_a_b,Ys: list_R8263082107343818799la_a_b] :
( ( ( member4680049679412964150la_a_b @ X @ A2 )
=> ( ( restri41315955395993636la_a_b @ A2 @ ( cons_R4161027894713053801la_a_b @ X @ Xs ) @ ( cons_R4161027894713053801la_a_b @ Y @ Ys ) )
= ( cons_R4161027894713053801la_a_b @ Y @ ( restri41315955395993636la_a_b @ ( minus_4077726661957047470la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) @ Xs @ Ys ) ) ) )
& ( ~ ( member4680049679412964150la_a_b @ X @ A2 )
=> ( ( restri41315955395993636la_a_b @ A2 @ ( cons_R4161027894713053801la_a_b @ X @ Xs ) @ ( cons_R4161027894713053801la_a_b @ Y @ Ys ) )
= ( restri41315955395993636la_a_b @ A2 @ Xs @ Ys ) ) ) ) ).
% restrict.simps(1)
thf(fact_670_restrict_Osimps_I1_J,axiom,
! [X: list_R8263082107343818799la_a_b,A2: set_li8334890247616788453la_a_b,Xs: list_l3216422893793986623la_a_b,Y: relational_fmla_a_b,Ys: list_R8263082107343818799la_a_b] :
( ( ( member8712135081189583430la_a_b @ X @ A2 )
=> ( ( restri4317776898343739444la_a_b @ A2 @ ( cons_l6716632503304403961la_a_b @ X @ Xs ) @ ( cons_R4161027894713053801la_a_b @ Y @ Ys ) )
= ( cons_R4161027894713053801la_a_b @ Y @ ( restri4317776898343739444la_a_b @ ( minus_4977183416030855486la_a_b @ A2 @ ( insert3617074100887873119la_a_b @ X @ bot_bo3230846185698255993la_a_b ) ) @ Xs @ Ys ) ) ) )
& ( ~ ( member8712135081189583430la_a_b @ X @ A2 )
=> ( ( restri4317776898343739444la_a_b @ A2 @ ( cons_l6716632503304403961la_a_b @ X @ Xs ) @ ( cons_R4161027894713053801la_a_b @ Y @ Ys ) )
= ( restri4317776898343739444la_a_b @ A2 @ Xs @ Ys ) ) ) ) ).
% restrict.simps(1)
thf(fact_671_restrict_Osimps_I1_J,axiom,
! [X: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,Xs: list_R8263082107343818799la_a_b,Y: set_Re381260168593705685la_a_b,Ys: list_s3671366213839843941la_a_b] :
( ( ( member4680049679412964150la_a_b @ X @ A2 )
=> ( ( restri5881281475243801946la_a_b @ A2 @ ( cons_R4161027894713053801la_a_b @ X @ Xs ) @ ( cons_s2104095074707585951la_a_b @ Y @ Ys ) )
= ( cons_s2104095074707585951la_a_b @ Y @ ( restri5881281475243801946la_a_b @ ( minus_4077726661957047470la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) @ Xs @ Ys ) ) ) )
& ( ~ ( member4680049679412964150la_a_b @ X @ A2 )
=> ( ( restri5881281475243801946la_a_b @ A2 @ ( cons_R4161027894713053801la_a_b @ X @ Xs ) @ ( cons_s2104095074707585951la_a_b @ Y @ Ys ) )
= ( restri5881281475243801946la_a_b @ A2 @ Xs @ Ys ) ) ) ) ).
% restrict.simps(1)
thf(fact_672_restrict_Osimps_I1_J,axiom,
! [X: set_Re381260168593705685la_a_b,A2: set_se6865892389300016395la_a_b,Xs: list_s3671366213839843941la_a_b,Y: relational_fmla_a_b,Ys: list_R8263082107343818799la_a_b] :
( ( ( member3481406638322139244la_a_b @ X @ A2 )
=> ( ( restri8859319718294224730la_a_b @ A2 @ ( cons_s2104095074707585951la_a_b @ X @ Xs ) @ ( cons_R4161027894713053801la_a_b @ Y @ Ys ) )
= ( cons_R4161027894713053801la_a_b @ Y @ ( restri8859319718294224730la_a_b @ ( minus_4705846553145473764la_a_b @ A2 @ ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) ) @ Xs @ Ys ) ) ) )
& ( ~ ( member3481406638322139244la_a_b @ X @ A2 )
=> ( ( restri8859319718294224730la_a_b @ A2 @ ( cons_s2104095074707585951la_a_b @ X @ Xs ) @ ( cons_R4161027894713053801la_a_b @ Y @ Ys ) )
= ( restri8859319718294224730la_a_b @ A2 @ Xs @ Ys ) ) ) ) ).
% restrict.simps(1)
thf(fact_673_in__set__simps_I1_J,axiom,
! [X: list_R8263082107343818799la_a_b,Y: list_R8263082107343818799la_a_b,Z: list_R8263082107343818799la_a_b,Ys: list_l3216422893793986623la_a_b] :
( ( member8712135081189583430la_a_b @ X @ ( set_li8988895567068466906la_a_b @ ( cons_l6716632503304403961la_a_b @ Y @ ( cons_l6716632503304403961la_a_b @ Z @ Ys ) ) ) )
= ( ( X = Y )
| ( member8712135081189583430la_a_b @ X @ ( set_li8988895567068466906la_a_b @ ( cons_l6716632503304403961la_a_b @ Z @ Ys ) ) ) ) ) ).
% in_set_simps(1)
thf(fact_674_in__set__simps_I1_J,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b,Z: set_Re381260168593705685la_a_b,Ys: list_s3671366213839843941la_a_b] :
( ( member3481406638322139244la_a_b @ X @ ( set_se5111286776497155840la_a_b @ ( cons_s2104095074707585951la_a_b @ Y @ ( cons_s2104095074707585951la_a_b @ Z @ Ys ) ) ) )
= ( ( X = Y )
| ( member3481406638322139244la_a_b @ X @ ( set_se5111286776497155840la_a_b @ ( cons_s2104095074707585951la_a_b @ Z @ Ys ) ) ) ) ) ).
% in_set_simps(1)
thf(fact_675_in__set__simps_I1_J,axiom,
! [X: nat,Y: nat,Z: nat,Ys: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ ( cons_nat @ Y @ ( cons_nat @ Z @ Ys ) ) ) )
= ( ( X = Y )
| ( member_nat2 @ X @ ( set_nat2 @ ( cons_nat @ Z @ Ys ) ) ) ) ) ).
% in_set_simps(1)
thf(fact_676_in__set__simps_I1_J,axiom,
! [X: relational_fmla_a_b,Y: relational_fmla_a_b,Z: relational_fmla_a_b,Ys: list_R8263082107343818799la_a_b] :
( ( member4680049679412964150la_a_b @ X @ ( set_Re9104216502384355786la_a_b @ ( cons_R4161027894713053801la_a_b @ Y @ ( cons_R4161027894713053801la_a_b @ Z @ Ys ) ) ) )
= ( ( X = Y )
| ( member4680049679412964150la_a_b @ X @ ( set_Re9104216502384355786la_a_b @ ( cons_R4161027894713053801la_a_b @ Z @ Ys ) ) ) ) ) ).
% in_set_simps(1)
thf(fact_677_remove__def,axiom,
( remove813535772580499850la_a_b
= ( ^ [X2: list_R8263082107343818799la_a_b,A4: set_li8334890247616788453la_a_b] : ( minus_4977183416030855486la_a_b @ A4 @ ( insert3617074100887873119la_a_b @ X2 @ bot_bo3230846185698255993la_a_b ) ) ) ) ).
% remove_def
thf(fact_678_remove__def,axiom,
( remove6344774115224635824la_a_b
= ( ^ [X2: set_Re381260168593705685la_a_b,A4: set_se6865892389300016395la_a_b] : ( minus_4705846553145473764la_a_b @ A4 @ ( insert2023870700798818565la_a_b @ X2 @ bot_bo2891247006866115487la_a_b ) ) ) ) ).
% remove_def
thf(fact_679_remove__def,axiom,
( remove_nat
= ( ^ [X2: nat,A4: set_nat] : ( minus_minus_set_nat @ A4 @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) ) ) ).
% remove_def
thf(fact_680_remove__def,axiom,
( remove4261432235257513082la_a_b
= ( ^ [X2: relational_fmla_a_b,A4: set_Re381260168593705685la_a_b] : ( minus_4077726661957047470la_a_b @ A4 @ ( insert7010464514620295119la_a_b @ X2 @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% remove_def
thf(fact_681_foldr1_Osimps_I1_J,axiom,
! [F: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b,X: set_Re381260168593705685la_a_b,Xs: list_s3671366213839843941la_a_b,Z: set_Re381260168593705685la_a_b] :
( ( relati7835315018957680435la_a_b @ F @ ( cons_s2104095074707585951la_a_b @ X @ Xs ) @ Z )
= ( foldr_5041855395594655059la_a_b @ F @ Xs @ X ) ) ).
% foldr1.simps(1)
thf(fact_682_foldr1_Osimps_I1_J,axiom,
! [F: nat > nat > nat,X: nat,Xs: list_nat,Z: nat] :
( ( relati2262546126839610926r1_nat @ F @ ( cons_nat @ X @ Xs ) @ Z )
= ( foldr_nat_nat @ F @ Xs @ X ) ) ).
% foldr1.simps(1)
thf(fact_683_foldr1_Osimps_I1_J,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b > relational_fmla_a_b,X: relational_fmla_a_b,Xs: list_R8263082107343818799la_a_b,Z: relational_fmla_a_b] :
( ( relati1724936043867567613la_a_b @ F @ ( cons_R4161027894713053801la_a_b @ X @ Xs ) @ Z )
= ( foldr_789212930732525799la_a_b @ F @ Xs @ X ) ) ).
% foldr1.simps(1)
thf(fact_684_SUP__set__fold,axiom,
! [F: relational_fmla_a_b > set_Re381260168593705685la_a_b,Xs: list_R8263082107343818799la_a_b] :
( ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ F @ ( set_Re9104216502384355786la_a_b @ Xs ) ) )
= ( fold_R3800324640475171969la_a_b @ ( comp_s8363053932693654946la_a_b @ sup_su5130108678486352897la_a_b @ F ) @ Xs @ bot_bo4495933725496725865la_a_b ) ) ).
% SUP_set_fold
thf(fact_685_SUP__set__fold,axiom,
! [F: nat > set_nat,Xs: list_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ ( set_nat2 @ Xs ) ) )
= ( fold_nat_set_nat @ ( comp_s2351873599094224870at_nat @ sup_sup_set_nat @ F ) @ Xs @ bot_bot_set_nat ) ) ).
% SUP_set_fold
thf(fact_686_SUP__set__fold,axiom,
! [F: relational_fmla_a_b > set_nat,Xs: list_R8263082107343818799la_a_b] :
( ( comple7399068483239264473et_nat @ ( image_8719518604786020652et_nat @ F @ ( set_Re9104216502384355786la_a_b @ Xs ) ) )
= ( fold_R246796818410179890et_nat @ ( comp_s6379547386824325045la_a_b @ sup_sup_set_nat @ F ) @ Xs @ bot_bot_set_nat ) ) ).
% SUP_set_fold
thf(fact_687_SUP__set__fold,axiom,
! [F: nat > set_Re381260168593705685la_a_b,Xs: list_nat] :
( ( comple8442120529048846632la_a_b @ ( image_654480401538864556la_a_b @ F @ ( set_nat2 @ Xs ) ) )
= ( fold_n1405130652017799602la_a_b @ ( comp_s1706329733671158483_b_nat @ sup_su5130108678486352897la_a_b @ F ) @ Xs @ bot_bo4495933725496725865la_a_b ) ) ).
% SUP_set_fold
thf(fact_688_SUP__set__fold,axiom,
! [F: nat > relational_fmla_a_b > $o,Xs: list_nat] :
( ( comple3423763510802972509_a_b_o @ ( image_616559630132021209_a_b_o @ F @ ( set_nat2 @ Xs ) ) )
= ( fold_n6542712121922774867_a_b_o @ ( comp_R2615939929001485572_o_nat @ sup_su1471977682094119364_a_b_o @ F ) @ Xs @ bot_bo8852203127187332700_a_b_o ) ) ).
% SUP_set_fold
thf(fact_689_SUP__set__fold,axiom,
! [F: nat > set_se6865892389300016395la_a_b,Xs: list_nat] :
( ( comple7774258788699406174la_a_b @ ( image_7289591248856253410la_a_b @ F @ ( set_nat2 @ Xs ) ) )
= ( fold_n8205512507173324776la_a_b @ ( comp_s5175654991236976501_b_nat @ sup_su4783144482993978935la_a_b @ F ) @ Xs @ bot_bo2891247006866115487la_a_b ) ) ).
% SUP_set_fold
thf(fact_690_SUP__set__fold,axiom,
! [F: set_Re381260168593705685la_a_b > set_nat,Xs: list_s3671366213839843941la_a_b] :
( ( comple7399068483239264473et_nat @ ( image_2112539042305366626et_nat @ F @ ( set_se5111286776497155840la_a_b @ Xs ) ) )
= ( fold_s3028460300622437992et_nat @ ( comp_s6707800551562645739la_a_b @ sup_sup_set_nat @ F ) @ Xs @ bot_bot_set_nat ) ) ).
% SUP_set_fold
thf(fact_691_SUP__set__fold,axiom,
! [F: list_R8263082107343818799la_a_b > set_nat,Xs: list_l3216422893793986623la_a_b] :
( ( comple7399068483239264473et_nat @ ( image_1345095896037326012et_nat @ F @ ( set_li8988895567068466906la_a_b @ Xs ) ) )
= ( fold_l121074154050821570et_nat @ ( comp_s6291166572922186565la_a_b @ sup_sup_set_nat @ F ) @ Xs @ bot_bot_set_nat ) ) ).
% SUP_set_fold
thf(fact_692_SUP__set__fold,axiom,
! [F: nat > set_li8334890247616788453la_a_b,Xs: list_nat] :
( ( comple4129898094138812600la_a_b @ ( image_3195463530097634108la_a_b @ F @ ( set_nat2 @ Xs ) ) )
= ( fold_n1971441788111129666la_a_b @ ( comp_s4967549954733093763_b_nat @ sup_su8392741853961112337la_a_b @ F ) @ Xs @ bot_bo3230846185698255993la_a_b ) ) ).
% SUP_set_fold
thf(fact_693_SUP__set__fold,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b > $o,Xs: list_R8263082107343818799la_a_b] :
( ( comple3423763510802972509_a_b_o @ ( image_285704734253202378_a_b_o @ F @ ( set_Re9104216502384355786la_a_b @ Xs ) ) )
= ( fold_R7577678366049538372_a_b_o @ ( comp_R6418550825408403155la_a_b @ sup_su1471977682094119364_a_b_o @ F ) @ Xs @ bot_bo8852203127187332700_a_b_o ) ) ).
% SUP_set_fold
thf(fact_694_image__eqI,axiom,
! [B2: nat,F: nat > nat,X: nat,A2: set_nat] :
( ( B2
= ( F @ X ) )
=> ( ( member_nat2 @ X @ A2 )
=> ( member_nat2 @ B2 @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_695_image__eqI,axiom,
! [B2: nat,F: relational_fmla_a_b > nat,X: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( B2
= ( F @ X ) )
=> ( ( member4680049679412964150la_a_b @ X @ A2 )
=> ( member_nat2 @ B2 @ ( image_341122591648980342_b_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_696_image__eqI,axiom,
! [B2: relational_fmla_a_b,F: nat > relational_fmla_a_b,X: nat,A2: set_nat] :
( ( B2
= ( F @ X ) )
=> ( ( member_nat2 @ X @ A2 )
=> ( member4680049679412964150la_a_b @ B2 @ ( image_4386371547000553590la_a_b @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_697_image__eqI,axiom,
! [B2: nat,F: set_Re381260168593705685la_a_b > nat,X: set_Re381260168593705685la_a_b,A2: set_se6865892389300016395la_a_b] :
( ( B2
= ( F @ X ) )
=> ( ( member3481406638322139244la_a_b @ X @ A2 )
=> ( member_nat2 @ B2 @ ( image_543665419517173420_b_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_698_image__eqI,axiom,
! [B2: set_Re381260168593705685la_a_b,F: nat > set_Re381260168593705685la_a_b,X: nat,A2: set_nat] :
( ( B2
= ( F @ X ) )
=> ( ( member_nat2 @ X @ A2 )
=> ( member3481406638322139244la_a_b @ B2 @ ( image_654480401538864556la_a_b @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_699_image__eqI,axiom,
! [B2: list_R8263082107343818799la_a_b,F: nat > list_R8263082107343818799la_a_b,X: nat,A2: set_nat] :
( ( B2
= ( F @ X ) )
=> ( ( member_nat2 @ X @ A2 )
=> ( member8712135081189583430la_a_b @ B2 @ ( image_4805200869821554566la_a_b @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_700_image__eqI,axiom,
! [B2: nat,F: list_R8263082107343818799la_a_b > nat,X: list_R8263082107343818799la_a_b,A2: set_li8334890247616788453la_a_b] :
( ( B2
= ( F @ X ) )
=> ( ( member8712135081189583430la_a_b @ X @ A2 )
=> ( member_nat2 @ B2 @ ( image_8025274900011100422_b_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_701_image__eqI,axiom,
! [B2: relational_fmla_a_b,F: relational_fmla_a_b > relational_fmla_a_b,X: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( B2
= ( F @ X ) )
=> ( ( member4680049679412964150la_a_b @ X @ A2 )
=> ( member4680049679412964150la_a_b @ B2 @ ( image_6790371041703824709la_a_b @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_702_image__eqI,axiom,
! [B2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > set_Re381260168593705685la_a_b,X: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( B2
= ( F @ X ) )
=> ( ( member4680049679412964150la_a_b @ X @ A2 )
=> ( member3481406638322139244la_a_b @ B2 @ ( image_8209480069293074043la_a_b @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_703_image__eqI,axiom,
! [B2: list_R8263082107343818799la_a_b,F: relational_fmla_a_b > list_R8263082107343818799la_a_b,X: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( B2
= ( F @ X ) )
=> ( ( member4680049679412964150la_a_b @ X @ A2 )
=> ( member8712135081189583430la_a_b @ B2 @ ( image_5997780205441267925la_a_b @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_704_Union__iff,axiom,
! [A2: set_Re381260168593705685la_a_b,C: set_se1626149436229564481la_a_b] :
( ( member3481406638322139244la_a_b @ A2 @ ( comple7774258788699406174la_a_b @ C ) )
= ( ? [X2: set_se6865892389300016395la_a_b] :
( ( member1693404252451860130la_a_b @ X2 @ C )
& ( member3481406638322139244la_a_b @ A2 @ X2 ) ) ) ) ).
% Union_iff
thf(fact_705_Union__iff,axiom,
! [A2: nat,C: set_set_nat] :
( ( member_nat2 @ A2 @ ( comple7399068483239264473et_nat @ C ) )
= ( ? [X2: set_nat] :
( ( member_set_nat @ X2 @ C )
& ( member_nat2 @ A2 @ X2 ) ) ) ) ).
% Union_iff
thf(fact_706_Union__iff,axiom,
! [A2: list_R8263082107343818799la_a_b,C: set_se2998258007684708507la_a_b] :
( ( member8712135081189583430la_a_b @ A2 @ ( comple4129898094138812600la_a_b @ C ) )
= ( ? [X2: set_li8334890247616788453la_a_b] :
( ( member5650711565134853628la_a_b @ X2 @ C )
& ( member8712135081189583430la_a_b @ A2 @ X2 ) ) ) ) ).
% Union_iff
thf(fact_707_Union__iff,axiom,
! [A2: relational_fmla_a_b,C: set_se6865892389300016395la_a_b] :
( ( member4680049679412964150la_a_b @ A2 @ ( comple8442120529048846632la_a_b @ C ) )
= ( ? [X2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X2 @ C )
& ( member4680049679412964150la_a_b @ A2 @ X2 ) ) ) ) ).
% Union_iff
thf(fact_708_ball__UN,axiom,
! [B: relational_fmla_a_b > set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b,P: relational_fmla_a_b > $o] :
( ( ! [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ B @ A2 ) ) )
=> ( P @ X2 ) ) )
= ( ! [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ A2 )
=> ! [Y3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Y3 @ ( B @ X2 ) )
=> ( P @ Y3 ) ) ) ) ) ).
% ball_UN
thf(fact_709_ball__UN,axiom,
! [B: list_R8263082107343818799la_a_b > set_Re381260168593705685la_a_b,A2: set_li8334890247616788453la_a_b,P: relational_fmla_a_b > $o] :
( ( ! [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ ( comple8442120529048846632la_a_b @ ( image_1636186708463624715la_a_b @ B @ A2 ) ) )
=> ( P @ X2 ) ) )
= ( ! [X2: list_R8263082107343818799la_a_b] :
( ( member8712135081189583430la_a_b @ X2 @ A2 )
=> ! [Y3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Y3 @ ( B @ X2 ) )
=> ( P @ Y3 ) ) ) ) ) ).
% ball_UN
thf(fact_710_ball__UN,axiom,
! [B: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b,A2: set_se6865892389300016395la_a_b,P: relational_fmla_a_b > $o] :
( ( ! [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ ( comple8442120529048846632la_a_b @ ( image_7051608999182166449la_a_b @ B @ A2 ) ) )
=> ( P @ X2 ) ) )
= ( ! [X2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X2 @ A2 )
=> ! [Y3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Y3 @ ( B @ X2 ) )
=> ( P @ Y3 ) ) ) ) ) ).
% ball_UN
thf(fact_711_bex__UN,axiom,
! [B: relational_fmla_a_b > set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b,P: relational_fmla_a_b > $o] :
( ( ? [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ B @ A2 ) ) )
& ( P @ X2 ) ) )
= ( ? [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ A2 )
& ? [Y3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Y3 @ ( B @ X2 ) )
& ( P @ Y3 ) ) ) ) ) ).
% bex_UN
thf(fact_712_bex__UN,axiom,
! [B: list_R8263082107343818799la_a_b > set_Re381260168593705685la_a_b,A2: set_li8334890247616788453la_a_b,P: relational_fmla_a_b > $o] :
( ( ? [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ ( comple8442120529048846632la_a_b @ ( image_1636186708463624715la_a_b @ B @ A2 ) ) )
& ( P @ X2 ) ) )
= ( ? [X2: list_R8263082107343818799la_a_b] :
( ( member8712135081189583430la_a_b @ X2 @ A2 )
& ? [Y3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Y3 @ ( B @ X2 ) )
& ( P @ Y3 ) ) ) ) ) ).
% bex_UN
thf(fact_713_bex__UN,axiom,
! [B: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b,A2: set_se6865892389300016395la_a_b,P: relational_fmla_a_b > $o] :
( ( ? [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ ( comple8442120529048846632la_a_b @ ( image_7051608999182166449la_a_b @ B @ A2 ) ) )
& ( P @ X2 ) ) )
= ( ? [X2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X2 @ A2 )
& ? [Y3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Y3 @ ( B @ X2 ) )
& ( P @ Y3 ) ) ) ) ) ).
% bex_UN
thf(fact_714_UnionI,axiom,
! [X4: set_se6865892389300016395la_a_b,C: set_se1626149436229564481la_a_b,A2: set_Re381260168593705685la_a_b] :
( ( member1693404252451860130la_a_b @ X4 @ C )
=> ( ( member3481406638322139244la_a_b @ A2 @ X4 )
=> ( member3481406638322139244la_a_b @ A2 @ ( comple7774258788699406174la_a_b @ C ) ) ) ) ).
% UnionI
thf(fact_715_UnionI,axiom,
! [X4: set_nat,C: set_set_nat,A2: nat] :
( ( member_set_nat @ X4 @ C )
=> ( ( member_nat2 @ A2 @ X4 )
=> ( member_nat2 @ A2 @ ( comple7399068483239264473et_nat @ C ) ) ) ) ).
% UnionI
thf(fact_716_UnionI,axiom,
! [X4: set_li8334890247616788453la_a_b,C: set_se2998258007684708507la_a_b,A2: list_R8263082107343818799la_a_b] :
( ( member5650711565134853628la_a_b @ X4 @ C )
=> ( ( member8712135081189583430la_a_b @ A2 @ X4 )
=> ( member8712135081189583430la_a_b @ A2 @ ( comple4129898094138812600la_a_b @ C ) ) ) ) ).
% UnionI
thf(fact_717_UnionI,axiom,
! [X4: set_Re381260168593705685la_a_b,C: set_se6865892389300016395la_a_b,A2: relational_fmla_a_b] :
( ( member3481406638322139244la_a_b @ X4 @ C )
=> ( ( member4680049679412964150la_a_b @ A2 @ X4 )
=> ( member4680049679412964150la_a_b @ A2 @ ( comple8442120529048846632la_a_b @ C ) ) ) ) ).
% UnionI
thf(fact_718_UN__ball__bex__simps_I1_J,axiom,
! [A2: set_se6865892389300016395la_a_b,P: relational_fmla_a_b > $o] :
( ( ! [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ ( comple8442120529048846632la_a_b @ A2 ) )
=> ( P @ X2 ) ) )
= ( ! [X2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X2 @ A2 )
=> ! [Y3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Y3 @ X2 )
=> ( P @ Y3 ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_719_UN__ball__bex__simps_I2_J,axiom,
! [B: relational_fmla_a_b > set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b,P: relational_fmla_a_b > $o] :
( ( ! [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ B @ A2 ) ) )
=> ( P @ X2 ) ) )
= ( ! [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ A2 )
=> ! [Y3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Y3 @ ( B @ X2 ) )
=> ( P @ Y3 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_720_UN__ball__bex__simps_I2_J,axiom,
! [B: list_R8263082107343818799la_a_b > set_Re381260168593705685la_a_b,A2: set_li8334890247616788453la_a_b,P: relational_fmla_a_b > $o] :
( ( ! [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ ( comple8442120529048846632la_a_b @ ( image_1636186708463624715la_a_b @ B @ A2 ) ) )
=> ( P @ X2 ) ) )
= ( ! [X2: list_R8263082107343818799la_a_b] :
( ( member8712135081189583430la_a_b @ X2 @ A2 )
=> ! [Y3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Y3 @ ( B @ X2 ) )
=> ( P @ Y3 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_721_UN__ball__bex__simps_I2_J,axiom,
! [B: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b,A2: set_se6865892389300016395la_a_b,P: relational_fmla_a_b > $o] :
( ( ! [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ ( comple8442120529048846632la_a_b @ ( image_7051608999182166449la_a_b @ B @ A2 ) ) )
=> ( P @ X2 ) ) )
= ( ! [X2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X2 @ A2 )
=> ! [Y3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Y3 @ ( B @ X2 ) )
=> ( P @ Y3 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_722_UN__ball__bex__simps_I3_J,axiom,
! [A2: set_se6865892389300016395la_a_b,P: relational_fmla_a_b > $o] :
( ( ? [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ ( comple8442120529048846632la_a_b @ A2 ) )
& ( P @ X2 ) ) )
= ( ? [X2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X2 @ A2 )
& ? [Y3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Y3 @ X2 )
& ( P @ Y3 ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_723_UN__ball__bex__simps_I4_J,axiom,
! [B: relational_fmla_a_b > set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b,P: relational_fmla_a_b > $o] :
( ( ? [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ B @ A2 ) ) )
& ( P @ X2 ) ) )
= ( ? [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ A2 )
& ? [Y3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Y3 @ ( B @ X2 ) )
& ( P @ Y3 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_724_UN__ball__bex__simps_I4_J,axiom,
! [B: list_R8263082107343818799la_a_b > set_Re381260168593705685la_a_b,A2: set_li8334890247616788453la_a_b,P: relational_fmla_a_b > $o] :
( ( ? [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ ( comple8442120529048846632la_a_b @ ( image_1636186708463624715la_a_b @ B @ A2 ) ) )
& ( P @ X2 ) ) )
= ( ? [X2: list_R8263082107343818799la_a_b] :
( ( member8712135081189583430la_a_b @ X2 @ A2 )
& ? [Y3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Y3 @ ( B @ X2 ) )
& ( P @ Y3 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_725_UN__ball__bex__simps_I4_J,axiom,
! [B: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b,A2: set_se6865892389300016395la_a_b,P: relational_fmla_a_b > $o] :
( ( ? [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ ( comple8442120529048846632la_a_b @ ( image_7051608999182166449la_a_b @ B @ A2 ) ) )
& ( P @ X2 ) ) )
= ( ? [X2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X2 @ A2 )
& ? [Y3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Y3 @ ( B @ X2 ) )
& ( P @ Y3 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_726_member__remove,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b,A2: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ X @ ( remove6344774115224635824la_a_b @ Y @ A2 ) )
= ( ( member3481406638322139244la_a_b @ X @ A2 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_727_member__remove,axiom,
! [X: nat,Y: nat,A2: set_nat] :
( ( member_nat2 @ X @ ( remove_nat @ Y @ A2 ) )
= ( ( member_nat2 @ X @ A2 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_728_member__remove,axiom,
! [X: list_R8263082107343818799la_a_b,Y: list_R8263082107343818799la_a_b,A2: set_li8334890247616788453la_a_b] :
( ( member8712135081189583430la_a_b @ X @ ( remove813535772580499850la_a_b @ Y @ A2 ) )
= ( ( member8712135081189583430la_a_b @ X @ A2 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_729_member__remove,axiom,
! [X: relational_fmla_a_b,Y: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ X @ ( remove4261432235257513082la_a_b @ Y @ A2 ) )
= ( ( member4680049679412964150la_a_b @ X @ A2 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_730_image__empty,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b] :
( ( image_6790371041703824709la_a_b @ F @ bot_bo4495933725496725865la_a_b )
= bot_bo4495933725496725865la_a_b ) ).
% image_empty
thf(fact_731_image__empty,axiom,
! [F: nat > nat] :
( ( image_nat_nat @ F @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_732_image__empty,axiom,
! [F: relational_fmla_a_b > nat] :
( ( image_341122591648980342_b_nat @ F @ bot_bo4495933725496725865la_a_b )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_733_image__empty,axiom,
! [F: nat > relational_fmla_a_b] :
( ( image_4386371547000553590la_a_b @ F @ bot_bot_set_nat )
= bot_bo4495933725496725865la_a_b ) ).
% image_empty
thf(fact_734_image__empty,axiom,
! [F: set_Re381260168593705685la_a_b > nat] :
( ( image_543665419517173420_b_nat @ F @ bot_bo2891247006866115487la_a_b )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_735_image__empty,axiom,
! [F: nat > set_Re381260168593705685la_a_b] :
( ( image_654480401538864556la_a_b @ F @ bot_bot_set_nat )
= bot_bo2891247006866115487la_a_b ) ).
% image_empty
thf(fact_736_image__empty,axiom,
! [F: nat > list_R8263082107343818799la_a_b] :
( ( image_4805200869821554566la_a_b @ F @ bot_bot_set_nat )
= bot_bo3230846185698255993la_a_b ) ).
% image_empty
thf(fact_737_image__empty,axiom,
! [F: list_R8263082107343818799la_a_b > nat] :
( ( image_8025274900011100422_b_nat @ F @ bot_bo3230846185698255993la_a_b )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_738_image__empty,axiom,
! [F: relational_fmla_a_b > set_Re381260168593705685la_a_b] :
( ( image_8209480069293074043la_a_b @ F @ bot_bo4495933725496725865la_a_b )
= bot_bo2891247006866115487la_a_b ) ).
% image_empty
thf(fact_739_image__empty,axiom,
! [F: relational_fmla_a_b > list_R8263082107343818799la_a_b] :
( ( image_5997780205441267925la_a_b @ F @ bot_bo4495933725496725865la_a_b )
= bot_bo3230846185698255993la_a_b ) ).
% image_empty
thf(fact_740_empty__is__image,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( bot_bo4495933725496725865la_a_b
= ( image_6790371041703824709la_a_b @ F @ A2 ) )
= ( A2 = bot_bo4495933725496725865la_a_b ) ) ).
% empty_is_image
thf(fact_741_empty__is__image,axiom,
! [F: nat > nat,A2: set_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat @ F @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_742_empty__is__image,axiom,
! [F: nat > relational_fmla_a_b,A2: set_nat] :
( ( bot_bo4495933725496725865la_a_b
= ( image_4386371547000553590la_a_b @ F @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_743_empty__is__image,axiom,
! [F: relational_fmla_a_b > nat,A2: set_Re381260168593705685la_a_b] :
( ( bot_bot_set_nat
= ( image_341122591648980342_b_nat @ F @ A2 ) )
= ( A2 = bot_bo4495933725496725865la_a_b ) ) ).
% empty_is_image
thf(fact_744_empty__is__image,axiom,
! [F: nat > set_Re381260168593705685la_a_b,A2: set_nat] :
( ( bot_bo2891247006866115487la_a_b
= ( image_654480401538864556la_a_b @ F @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_745_empty__is__image,axiom,
! [F: set_Re381260168593705685la_a_b > nat,A2: set_se6865892389300016395la_a_b] :
( ( bot_bot_set_nat
= ( image_543665419517173420_b_nat @ F @ A2 ) )
= ( A2 = bot_bo2891247006866115487la_a_b ) ) ).
% empty_is_image
thf(fact_746_empty__is__image,axiom,
! [F: list_R8263082107343818799la_a_b > nat,A2: set_li8334890247616788453la_a_b] :
( ( bot_bot_set_nat
= ( image_8025274900011100422_b_nat @ F @ A2 ) )
= ( A2 = bot_bo3230846185698255993la_a_b ) ) ).
% empty_is_image
thf(fact_747_empty__is__image,axiom,
! [F: nat > list_R8263082107343818799la_a_b,A2: set_nat] :
( ( bot_bo3230846185698255993la_a_b
= ( image_4805200869821554566la_a_b @ F @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_748_empty__is__image,axiom,
! [F: set_Re381260168593705685la_a_b > relational_fmla_a_b,A2: set_se6865892389300016395la_a_b] :
( ( bot_bo4495933725496725865la_a_b
= ( image_1964146275488721019la_a_b @ F @ A2 ) )
= ( A2 = bot_bo2891247006866115487la_a_b ) ) ).
% empty_is_image
thf(fact_749_empty__is__image,axiom,
! [F: list_R8263082107343818799la_a_b > relational_fmla_a_b,A2: set_li8334890247616788453la_a_b] :
( ( bot_bo4495933725496725865la_a_b
= ( image_3184217606439897301la_a_b @ F @ A2 ) )
= ( A2 = bot_bo3230846185698255993la_a_b ) ) ).
% empty_is_image
thf(fact_750_image__is__empty,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( ( image_6790371041703824709la_a_b @ F @ A2 )
= bot_bo4495933725496725865la_a_b )
= ( A2 = bot_bo4495933725496725865la_a_b ) ) ).
% image_is_empty
thf(fact_751_image__is__empty,axiom,
! [F: nat > nat,A2: set_nat] :
( ( ( image_nat_nat @ F @ A2 )
= bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_752_image__is__empty,axiom,
! [F: nat > relational_fmla_a_b,A2: set_nat] :
( ( ( image_4386371547000553590la_a_b @ F @ A2 )
= bot_bo4495933725496725865la_a_b )
= ( A2 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_753_image__is__empty,axiom,
! [F: relational_fmla_a_b > nat,A2: set_Re381260168593705685la_a_b] :
( ( ( image_341122591648980342_b_nat @ F @ A2 )
= bot_bot_set_nat )
= ( A2 = bot_bo4495933725496725865la_a_b ) ) ).
% image_is_empty
thf(fact_754_image__is__empty,axiom,
! [F: nat > set_Re381260168593705685la_a_b,A2: set_nat] :
( ( ( image_654480401538864556la_a_b @ F @ A2 )
= bot_bo2891247006866115487la_a_b )
= ( A2 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_755_image__is__empty,axiom,
! [F: set_Re381260168593705685la_a_b > nat,A2: set_se6865892389300016395la_a_b] :
( ( ( image_543665419517173420_b_nat @ F @ A2 )
= bot_bot_set_nat )
= ( A2 = bot_bo2891247006866115487la_a_b ) ) ).
% image_is_empty
thf(fact_756_image__is__empty,axiom,
! [F: list_R8263082107343818799la_a_b > nat,A2: set_li8334890247616788453la_a_b] :
( ( ( image_8025274900011100422_b_nat @ F @ A2 )
= bot_bot_set_nat )
= ( A2 = bot_bo3230846185698255993la_a_b ) ) ).
% image_is_empty
thf(fact_757_image__is__empty,axiom,
! [F: nat > list_R8263082107343818799la_a_b,A2: set_nat] :
( ( ( image_4805200869821554566la_a_b @ F @ A2 )
= bot_bo3230846185698255993la_a_b )
= ( A2 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_758_image__is__empty,axiom,
! [F: set_Re381260168593705685la_a_b > relational_fmla_a_b,A2: set_se6865892389300016395la_a_b] :
( ( ( image_1964146275488721019la_a_b @ F @ A2 )
= bot_bo4495933725496725865la_a_b )
= ( A2 = bot_bo2891247006866115487la_a_b ) ) ).
% image_is_empty
thf(fact_759_image__is__empty,axiom,
! [F: list_R8263082107343818799la_a_b > relational_fmla_a_b,A2: set_li8334890247616788453la_a_b] :
( ( ( image_3184217606439897301la_a_b @ F @ A2 )
= bot_bo4495933725496725865la_a_b )
= ( A2 = bot_bo3230846185698255993la_a_b ) ) ).
% image_is_empty
thf(fact_760_image__insert,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,A: relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ( image_6790371041703824709la_a_b @ F @ ( insert7010464514620295119la_a_b @ A @ B ) )
= ( insert7010464514620295119la_a_b @ ( F @ A ) @ ( image_6790371041703824709la_a_b @ F @ B ) ) ) ).
% image_insert
thf(fact_761_image__insert,axiom,
! [F: nat > nat,A: nat,B: set_nat] :
( ( image_nat_nat @ F @ ( insert_nat2 @ A @ B ) )
= ( insert_nat2 @ ( F @ A ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_insert
thf(fact_762_image__insert,axiom,
! [F: relational_fmla_a_b > nat,A: relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ( image_341122591648980342_b_nat @ F @ ( insert7010464514620295119la_a_b @ A @ B ) )
= ( insert_nat2 @ ( F @ A ) @ ( image_341122591648980342_b_nat @ F @ B ) ) ) ).
% image_insert
thf(fact_763_image__insert,axiom,
! [F: nat > relational_fmla_a_b,A: nat,B: set_nat] :
( ( image_4386371547000553590la_a_b @ F @ ( insert_nat2 @ A @ B ) )
= ( insert7010464514620295119la_a_b @ ( F @ A ) @ ( image_4386371547000553590la_a_b @ F @ B ) ) ) ).
% image_insert
thf(fact_764_image__insert,axiom,
! [F: set_Re381260168593705685la_a_b > nat,A: set_Re381260168593705685la_a_b,B: set_se6865892389300016395la_a_b] :
( ( image_543665419517173420_b_nat @ F @ ( insert2023870700798818565la_a_b @ A @ B ) )
= ( insert_nat2 @ ( F @ A ) @ ( image_543665419517173420_b_nat @ F @ B ) ) ) ).
% image_insert
thf(fact_765_image__insert,axiom,
! [F: nat > set_Re381260168593705685la_a_b,A: nat,B: set_nat] :
( ( image_654480401538864556la_a_b @ F @ ( insert_nat2 @ A @ B ) )
= ( insert2023870700798818565la_a_b @ ( F @ A ) @ ( image_654480401538864556la_a_b @ F @ B ) ) ) ).
% image_insert
thf(fact_766_image__insert,axiom,
! [F: nat > list_R8263082107343818799la_a_b,A: nat,B: set_nat] :
( ( image_4805200869821554566la_a_b @ F @ ( insert_nat2 @ A @ B ) )
= ( insert3617074100887873119la_a_b @ ( F @ A ) @ ( image_4805200869821554566la_a_b @ F @ B ) ) ) ).
% image_insert
thf(fact_767_image__insert,axiom,
! [F: list_R8263082107343818799la_a_b > nat,A: list_R8263082107343818799la_a_b,B: set_li8334890247616788453la_a_b] :
( ( image_8025274900011100422_b_nat @ F @ ( insert3617074100887873119la_a_b @ A @ B ) )
= ( insert_nat2 @ ( F @ A ) @ ( image_8025274900011100422_b_nat @ F @ B ) ) ) ).
% image_insert
thf(fact_768_image__insert,axiom,
! [F: relational_fmla_a_b > set_Re381260168593705685la_a_b,A: relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ( image_8209480069293074043la_a_b @ F @ ( insert7010464514620295119la_a_b @ A @ B ) )
= ( insert2023870700798818565la_a_b @ ( F @ A ) @ ( image_8209480069293074043la_a_b @ F @ B ) ) ) ).
% image_insert
thf(fact_769_image__insert,axiom,
! [F: relational_fmla_a_b > list_R8263082107343818799la_a_b,A: relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ( image_5997780205441267925la_a_b @ F @ ( insert7010464514620295119la_a_b @ A @ B ) )
= ( insert3617074100887873119la_a_b @ ( F @ A ) @ ( image_5997780205441267925la_a_b @ F @ B ) ) ) ).
% image_insert
thf(fact_770_insert__image,axiom,
! [X: nat,A2: set_nat,F: nat > nat] :
( ( member_nat2 @ X @ A2 )
=> ( ( insert_nat2 @ ( F @ X ) @ ( image_nat_nat @ F @ A2 ) )
= ( image_nat_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_771_insert__image,axiom,
! [X: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > nat] :
( ( member4680049679412964150la_a_b @ X @ A2 )
=> ( ( insert_nat2 @ ( F @ X ) @ ( image_341122591648980342_b_nat @ F @ A2 ) )
= ( image_341122591648980342_b_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_772_insert__image,axiom,
! [X: nat,A2: set_nat,F: nat > relational_fmla_a_b] :
( ( member_nat2 @ X @ A2 )
=> ( ( insert7010464514620295119la_a_b @ ( F @ X ) @ ( image_4386371547000553590la_a_b @ F @ A2 ) )
= ( image_4386371547000553590la_a_b @ F @ A2 ) ) ) ).
% insert_image
thf(fact_773_insert__image,axiom,
! [X: set_Re381260168593705685la_a_b,A2: set_se6865892389300016395la_a_b,F: set_Re381260168593705685la_a_b > nat] :
( ( member3481406638322139244la_a_b @ X @ A2 )
=> ( ( insert_nat2 @ ( F @ X ) @ ( image_543665419517173420_b_nat @ F @ A2 ) )
= ( image_543665419517173420_b_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_774_insert__image,axiom,
! [X: nat,A2: set_nat,F: nat > set_Re381260168593705685la_a_b] :
( ( member_nat2 @ X @ A2 )
=> ( ( insert2023870700798818565la_a_b @ ( F @ X ) @ ( image_654480401538864556la_a_b @ F @ A2 ) )
= ( image_654480401538864556la_a_b @ F @ A2 ) ) ) ).
% insert_image
thf(fact_775_insert__image,axiom,
! [X: nat,A2: set_nat,F: nat > list_R8263082107343818799la_a_b] :
( ( member_nat2 @ X @ A2 )
=> ( ( insert3617074100887873119la_a_b @ ( F @ X ) @ ( image_4805200869821554566la_a_b @ F @ A2 ) )
= ( image_4805200869821554566la_a_b @ F @ A2 ) ) ) ).
% insert_image
thf(fact_776_insert__image,axiom,
! [X: list_R8263082107343818799la_a_b,A2: set_li8334890247616788453la_a_b,F: list_R8263082107343818799la_a_b > nat] :
( ( member8712135081189583430la_a_b @ X @ A2 )
=> ( ( insert_nat2 @ ( F @ X ) @ ( image_8025274900011100422_b_nat @ F @ A2 ) )
= ( image_8025274900011100422_b_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_777_insert__image,axiom,
! [X: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X @ A2 )
=> ( ( insert7010464514620295119la_a_b @ ( F @ X ) @ ( image_6790371041703824709la_a_b @ F @ A2 ) )
= ( image_6790371041703824709la_a_b @ F @ A2 ) ) ) ).
% insert_image
thf(fact_778_insert__image,axiom,
! [X: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ X @ A2 )
=> ( ( insert2023870700798818565la_a_b @ ( F @ X ) @ ( image_8209480069293074043la_a_b @ F @ A2 ) )
= ( image_8209480069293074043la_a_b @ F @ A2 ) ) ) ).
% insert_image
thf(fact_779_insert__image,axiom,
! [X: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > list_R8263082107343818799la_a_b] :
( ( member4680049679412964150la_a_b @ X @ A2 )
=> ( ( insert3617074100887873119la_a_b @ ( F @ X ) @ ( image_5997780205441267925la_a_b @ F @ A2 ) )
= ( image_5997780205441267925la_a_b @ F @ A2 ) ) ) ).
% insert_image
thf(fact_780_removeAll__id,axiom,
! [X: set_Re381260168593705685la_a_b,Xs: list_s3671366213839843941la_a_b] :
( ~ ( member3481406638322139244la_a_b @ X @ ( set_se5111286776497155840la_a_b @ Xs ) )
=> ( ( remove1701277439453780425la_a_b @ X @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_781_removeAll__id,axiom,
! [X: nat,Xs: list_nat] :
( ~ ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ( removeAll_nat @ X @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_782_removeAll__id,axiom,
! [X: list_R8263082107343818799la_a_b,Xs: list_l3216422893793986623la_a_b] :
( ~ ( member8712135081189583430la_a_b @ X @ ( set_li8988895567068466906la_a_b @ Xs ) )
=> ( ( remove6300469919649359139la_a_b @ X @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_783_removeAll__id,axiom,
! [X: relational_fmla_a_b,Xs: list_R8263082107343818799la_a_b] :
( ~ ( member4680049679412964150la_a_b @ X @ ( set_Re9104216502384355786la_a_b @ Xs ) )
=> ( ( remove4882698675883194515la_a_b @ X @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_784_UnionE,axiom,
! [A2: set_Re381260168593705685la_a_b,C: set_se1626149436229564481la_a_b] :
( ( member3481406638322139244la_a_b @ A2 @ ( comple7774258788699406174la_a_b @ C ) )
=> ~ ! [X5: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ A2 @ X5 )
=> ~ ( member1693404252451860130la_a_b @ X5 @ C ) ) ) ).
% UnionE
thf(fact_785_UnionE,axiom,
! [A2: nat,C: set_set_nat] :
( ( member_nat2 @ A2 @ ( comple7399068483239264473et_nat @ C ) )
=> ~ ! [X5: set_nat] :
( ( member_nat2 @ A2 @ X5 )
=> ~ ( member_set_nat @ X5 @ C ) ) ) ).
% UnionE
thf(fact_786_UnionE,axiom,
! [A2: list_R8263082107343818799la_a_b,C: set_se2998258007684708507la_a_b] :
( ( member8712135081189583430la_a_b @ A2 @ ( comple4129898094138812600la_a_b @ C ) )
=> ~ ! [X5: set_li8334890247616788453la_a_b] :
( ( member8712135081189583430la_a_b @ A2 @ X5 )
=> ~ ( member5650711565134853628la_a_b @ X5 @ C ) ) ) ).
% UnionE
thf(fact_787_UnionE,axiom,
! [A2: relational_fmla_a_b,C: set_se6865892389300016395la_a_b] :
( ( member4680049679412964150la_a_b @ A2 @ ( comple8442120529048846632la_a_b @ C ) )
=> ~ ! [X5: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ A2 @ X5 )
=> ~ ( member3481406638322139244la_a_b @ X5 @ C ) ) ) ).
% UnionE
thf(fact_788_Inf_OINF__cong,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,C: relational_fmla_a_b > set_Re381260168593705685la_a_b,D2: relational_fmla_a_b > set_Re381260168593705685la_a_b,Inf: set_se6865892389300016395la_a_b > set_Re381260168593705685la_a_b] :
( ( A2 = B )
=> ( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ B )
=> ( ( C @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( Inf @ ( image_8209480069293074043la_a_b @ C @ A2 ) )
= ( Inf @ ( image_8209480069293074043la_a_b @ D2 @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_789_Inf_OINF__cong,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,C: relational_fmla_a_b > nat,D2: relational_fmla_a_b > nat,Inf: set_nat > nat] :
( ( A2 = B )
=> ( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ B )
=> ( ( C @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( Inf @ ( image_341122591648980342_b_nat @ C @ A2 ) )
= ( Inf @ ( image_341122591648980342_b_nat @ D2 @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_790_Inf_OINF__cong,axiom,
! [A2: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b,C: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b,D2: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b,Inf: set_se6865892389300016395la_a_b > set_Re381260168593705685la_a_b] :
( ( A2 = B )
=> ( ! [X3: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X3 @ B )
=> ( ( C @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( Inf @ ( image_7051608999182166449la_a_b @ C @ A2 ) )
= ( Inf @ ( image_7051608999182166449la_a_b @ D2 @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_791_Inf_OINF__cong,axiom,
! [A2: set_nat,B: set_nat,C: nat > nat,D2: nat > nat,Inf: set_nat > nat] :
( ( A2 = B )
=> ( ! [X3: nat] :
( ( member_nat2 @ X3 @ B )
=> ( ( C @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( Inf @ ( image_nat_nat @ C @ A2 ) )
= ( Inf @ ( image_nat_nat @ D2 @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_792_Inf_OINF__cong,axiom,
! [A2: set_li8334890247616788453la_a_b,B: set_li8334890247616788453la_a_b,C: list_R8263082107343818799la_a_b > set_Re381260168593705685la_a_b,D2: list_R8263082107343818799la_a_b > set_Re381260168593705685la_a_b,Inf: set_se6865892389300016395la_a_b > set_Re381260168593705685la_a_b] :
( ( A2 = B )
=> ( ! [X3: list_R8263082107343818799la_a_b] :
( ( member8712135081189583430la_a_b @ X3 @ B )
=> ( ( C @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( Inf @ ( image_1636186708463624715la_a_b @ C @ A2 ) )
= ( Inf @ ( image_1636186708463624715la_a_b @ D2 @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_793_Sup_OSUP__cong,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,C: relational_fmla_a_b > set_Re381260168593705685la_a_b,D2: relational_fmla_a_b > set_Re381260168593705685la_a_b,Sup: set_se6865892389300016395la_a_b > set_Re381260168593705685la_a_b] :
( ( A2 = B )
=> ( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ B )
=> ( ( C @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( Sup @ ( image_8209480069293074043la_a_b @ C @ A2 ) )
= ( Sup @ ( image_8209480069293074043la_a_b @ D2 @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_794_Sup_OSUP__cong,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,C: relational_fmla_a_b > nat,D2: relational_fmla_a_b > nat,Sup: set_nat > nat] :
( ( A2 = B )
=> ( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ B )
=> ( ( C @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( Sup @ ( image_341122591648980342_b_nat @ C @ A2 ) )
= ( Sup @ ( image_341122591648980342_b_nat @ D2 @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_795_Sup_OSUP__cong,axiom,
! [A2: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b,C: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b,D2: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b,Sup: set_se6865892389300016395la_a_b > set_Re381260168593705685la_a_b] :
( ( A2 = B )
=> ( ! [X3: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X3 @ B )
=> ( ( C @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( Sup @ ( image_7051608999182166449la_a_b @ C @ A2 ) )
= ( Sup @ ( image_7051608999182166449la_a_b @ D2 @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_796_Sup_OSUP__cong,axiom,
! [A2: set_nat,B: set_nat,C: nat > nat,D2: nat > nat,Sup: set_nat > nat] :
( ( A2 = B )
=> ( ! [X3: nat] :
( ( member_nat2 @ X3 @ B )
=> ( ( C @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( Sup @ ( image_nat_nat @ C @ A2 ) )
= ( Sup @ ( image_nat_nat @ D2 @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_797_Sup_OSUP__cong,axiom,
! [A2: set_li8334890247616788453la_a_b,B: set_li8334890247616788453la_a_b,C: list_R8263082107343818799la_a_b > set_Re381260168593705685la_a_b,D2: list_R8263082107343818799la_a_b > set_Re381260168593705685la_a_b,Sup: set_se6865892389300016395la_a_b > set_Re381260168593705685la_a_b] :
( ( A2 = B )
=> ( ! [X3: list_R8263082107343818799la_a_b] :
( ( member8712135081189583430la_a_b @ X3 @ B )
=> ( ( C @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( Sup @ ( image_1636186708463624715la_a_b @ C @ A2 ) )
= ( Sup @ ( image_1636186708463624715la_a_b @ D2 @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_798_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > nat] :
( ( member_nat2 @ X @ A2 )
=> ( member_nat2 @ ( F @ X ) @ ( image_nat_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_799_imageI,axiom,
! [X: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > nat] :
( ( member4680049679412964150la_a_b @ X @ A2 )
=> ( member_nat2 @ ( F @ X ) @ ( image_341122591648980342_b_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_800_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > relational_fmla_a_b] :
( ( member_nat2 @ X @ A2 )
=> ( member4680049679412964150la_a_b @ ( F @ X ) @ ( image_4386371547000553590la_a_b @ F @ A2 ) ) ) ).
% imageI
thf(fact_801_imageI,axiom,
! [X: set_Re381260168593705685la_a_b,A2: set_se6865892389300016395la_a_b,F: set_Re381260168593705685la_a_b > nat] :
( ( member3481406638322139244la_a_b @ X @ A2 )
=> ( member_nat2 @ ( F @ X ) @ ( image_543665419517173420_b_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_802_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > set_Re381260168593705685la_a_b] :
( ( member_nat2 @ X @ A2 )
=> ( member3481406638322139244la_a_b @ ( F @ X ) @ ( image_654480401538864556la_a_b @ F @ A2 ) ) ) ).
% imageI
thf(fact_803_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > list_R8263082107343818799la_a_b] :
( ( member_nat2 @ X @ A2 )
=> ( member8712135081189583430la_a_b @ ( F @ X ) @ ( image_4805200869821554566la_a_b @ F @ A2 ) ) ) ).
% imageI
thf(fact_804_imageI,axiom,
! [X: list_R8263082107343818799la_a_b,A2: set_li8334890247616788453la_a_b,F: list_R8263082107343818799la_a_b > nat] :
( ( member8712135081189583430la_a_b @ X @ A2 )
=> ( member_nat2 @ ( F @ X ) @ ( image_8025274900011100422_b_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_805_imageI,axiom,
! [X: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X @ A2 )
=> ( member4680049679412964150la_a_b @ ( F @ X ) @ ( image_6790371041703824709la_a_b @ F @ A2 ) ) ) ).
% imageI
thf(fact_806_imageI,axiom,
! [X: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ X @ A2 )
=> ( member3481406638322139244la_a_b @ ( F @ X ) @ ( image_8209480069293074043la_a_b @ F @ A2 ) ) ) ).
% imageI
thf(fact_807_imageI,axiom,
! [X: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > list_R8263082107343818799la_a_b] :
( ( member4680049679412964150la_a_b @ X @ A2 )
=> ( member8712135081189583430la_a_b @ ( F @ X ) @ ( image_5997780205441267925la_a_b @ F @ A2 ) ) ) ).
% imageI
thf(fact_808_image__iff,axiom,
! [Z: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ Z @ ( image_8209480069293074043la_a_b @ F @ A2 ) )
= ( ? [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ A2 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_809_image__iff,axiom,
! [Z: set_Re381260168593705685la_a_b,F: list_R8263082107343818799la_a_b > set_Re381260168593705685la_a_b,A2: set_li8334890247616788453la_a_b] :
( ( member3481406638322139244la_a_b @ Z @ ( image_1636186708463624715la_a_b @ F @ A2 ) )
= ( ? [X2: list_R8263082107343818799la_a_b] :
( ( member8712135081189583430la_a_b @ X2 @ A2 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_810_image__iff,axiom,
! [Z: set_Re381260168593705685la_a_b,F: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b,A2: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ Z @ ( image_7051608999182166449la_a_b @ F @ A2 ) )
= ( ? [X2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X2 @ A2 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_811_image__iff,axiom,
! [Z: nat,F: relational_fmla_a_b > nat,A2: set_Re381260168593705685la_a_b] :
( ( member_nat2 @ Z @ ( image_341122591648980342_b_nat @ F @ A2 ) )
= ( ? [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ A2 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_812_image__iff,axiom,
! [Z: nat,F: nat > nat,A2: set_nat] :
( ( member_nat2 @ Z @ ( image_nat_nat @ F @ A2 ) )
= ( ? [X2: nat] :
( ( member_nat2 @ X2 @ A2 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_813_bex__imageD,axiom,
! [F: relational_fmla_a_b > set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b,P: set_Re381260168593705685la_a_b > $o] :
( ? [X6: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X6 @ ( image_8209480069293074043la_a_b @ F @ A2 ) )
& ( P @ X6 ) )
=> ? [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_814_bex__imageD,axiom,
! [F: relational_fmla_a_b > nat,A2: set_Re381260168593705685la_a_b,P: nat > $o] :
( ? [X6: nat] :
( ( member_nat2 @ X6 @ ( image_341122591648980342_b_nat @ F @ A2 ) )
& ( P @ X6 ) )
=> ? [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_815_bex__imageD,axiom,
! [F: list_R8263082107343818799la_a_b > set_Re381260168593705685la_a_b,A2: set_li8334890247616788453la_a_b,P: set_Re381260168593705685la_a_b > $o] :
( ? [X6: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X6 @ ( image_1636186708463624715la_a_b @ F @ A2 ) )
& ( P @ X6 ) )
=> ? [X3: list_R8263082107343818799la_a_b] :
( ( member8712135081189583430la_a_b @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_816_bex__imageD,axiom,
! [F: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b,A2: set_se6865892389300016395la_a_b,P: set_Re381260168593705685la_a_b > $o] :
( ? [X6: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X6 @ ( image_7051608999182166449la_a_b @ F @ A2 ) )
& ( P @ X6 ) )
=> ? [X3: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_817_bex__imageD,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ? [X6: nat] :
( ( member_nat2 @ X6 @ ( image_nat_nat @ F @ A2 ) )
& ( P @ X6 ) )
=> ? [X3: nat] :
( ( member_nat2 @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_818_image__cong,axiom,
! [M: set_Re381260168593705685la_a_b,N: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > set_Re381260168593705685la_a_b,G: relational_fmla_a_b > set_Re381260168593705685la_a_b] :
( ( M = N )
=> ( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_8209480069293074043la_a_b @ F @ M )
= ( image_8209480069293074043la_a_b @ G @ N ) ) ) ) ).
% image_cong
thf(fact_819_image__cong,axiom,
! [M: set_Re381260168593705685la_a_b,N: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > nat,G: relational_fmla_a_b > nat] :
( ( M = N )
=> ( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_341122591648980342_b_nat @ F @ M )
= ( image_341122591648980342_b_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_820_image__cong,axiom,
! [M: set_se6865892389300016395la_a_b,N: set_se6865892389300016395la_a_b,F: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b,G: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b] :
( ( M = N )
=> ( ! [X3: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_7051608999182166449la_a_b @ F @ M )
= ( image_7051608999182166449la_a_b @ G @ N ) ) ) ) ).
% image_cong
thf(fact_821_image__cong,axiom,
! [M: set_nat,N: set_nat,F: nat > nat,G: nat > nat] :
( ( M = N )
=> ( ! [X3: nat] :
( ( member_nat2 @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_nat_nat @ F @ M )
= ( image_nat_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_822_image__cong,axiom,
! [M: set_li8334890247616788453la_a_b,N: set_li8334890247616788453la_a_b,F: list_R8263082107343818799la_a_b > set_Re381260168593705685la_a_b,G: list_R8263082107343818799la_a_b > set_Re381260168593705685la_a_b] :
( ( M = N )
=> ( ! [X3: list_R8263082107343818799la_a_b] :
( ( member8712135081189583430la_a_b @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_1636186708463624715la_a_b @ F @ M )
= ( image_1636186708463624715la_a_b @ G @ N ) ) ) ) ).
% image_cong
thf(fact_823_ball__imageD,axiom,
! [F: relational_fmla_a_b > set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b,P: set_Re381260168593705685la_a_b > $o] :
( ! [X3: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X3 @ ( image_8209480069293074043la_a_b @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X6: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X6 @ A2 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_824_ball__imageD,axiom,
! [F: relational_fmla_a_b > nat,A2: set_Re381260168593705685la_a_b,P: nat > $o] :
( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( image_341122591648980342_b_nat @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X6: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X6 @ A2 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_825_ball__imageD,axiom,
! [F: list_R8263082107343818799la_a_b > set_Re381260168593705685la_a_b,A2: set_li8334890247616788453la_a_b,P: set_Re381260168593705685la_a_b > $o] :
( ! [X3: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X3 @ ( image_1636186708463624715la_a_b @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X6: list_R8263082107343818799la_a_b] :
( ( member8712135081189583430la_a_b @ X6 @ A2 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_826_ball__imageD,axiom,
! [F: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b,A2: set_se6865892389300016395la_a_b,P: set_Re381260168593705685la_a_b > $o] :
( ! [X3: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X3 @ ( image_7051608999182166449la_a_b @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X6: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X6 @ A2 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_827_ball__imageD,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( image_nat_nat @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X6: nat] :
( ( member_nat2 @ X6 @ A2 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_828_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B2: nat,F: nat > nat] :
( ( member_nat2 @ X @ A2 )
=> ( ( B2
= ( F @ X ) )
=> ( member_nat2 @ B2 @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_829_rev__image__eqI,axiom,
! [X: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B2: nat,F: relational_fmla_a_b > nat] :
( ( member4680049679412964150la_a_b @ X @ A2 )
=> ( ( B2
= ( F @ X ) )
=> ( member_nat2 @ B2 @ ( image_341122591648980342_b_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_830_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B2: relational_fmla_a_b,F: nat > relational_fmla_a_b] :
( ( member_nat2 @ X @ A2 )
=> ( ( B2
= ( F @ X ) )
=> ( member4680049679412964150la_a_b @ B2 @ ( image_4386371547000553590la_a_b @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_831_rev__image__eqI,axiom,
! [X: set_Re381260168593705685la_a_b,A2: set_se6865892389300016395la_a_b,B2: nat,F: set_Re381260168593705685la_a_b > nat] :
( ( member3481406638322139244la_a_b @ X @ A2 )
=> ( ( B2
= ( F @ X ) )
=> ( member_nat2 @ B2 @ ( image_543665419517173420_b_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_832_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B2: set_Re381260168593705685la_a_b,F: nat > set_Re381260168593705685la_a_b] :
( ( member_nat2 @ X @ A2 )
=> ( ( B2
= ( F @ X ) )
=> ( member3481406638322139244la_a_b @ B2 @ ( image_654480401538864556la_a_b @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_833_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B2: list_R8263082107343818799la_a_b,F: nat > list_R8263082107343818799la_a_b] :
( ( member_nat2 @ X @ A2 )
=> ( ( B2
= ( F @ X ) )
=> ( member8712135081189583430la_a_b @ B2 @ ( image_4805200869821554566la_a_b @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_834_rev__image__eqI,axiom,
! [X: list_R8263082107343818799la_a_b,A2: set_li8334890247616788453la_a_b,B2: nat,F: list_R8263082107343818799la_a_b > nat] :
( ( member8712135081189583430la_a_b @ X @ A2 )
=> ( ( B2
= ( F @ X ) )
=> ( member_nat2 @ B2 @ ( image_8025274900011100422_b_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_835_rev__image__eqI,axiom,
! [X: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B2: relational_fmla_a_b,F: relational_fmla_a_b > relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X @ A2 )
=> ( ( B2
= ( F @ X ) )
=> ( member4680049679412964150la_a_b @ B2 @ ( image_6790371041703824709la_a_b @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_836_rev__image__eqI,axiom,
! [X: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ X @ A2 )
=> ( ( B2
= ( F @ X ) )
=> ( member3481406638322139244la_a_b @ B2 @ ( image_8209480069293074043la_a_b @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_837_rev__image__eqI,axiom,
! [X: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B2: list_R8263082107343818799la_a_b,F: relational_fmla_a_b > list_R8263082107343818799la_a_b] :
( ( member4680049679412964150la_a_b @ X @ A2 )
=> ( ( B2
= ( F @ X ) )
=> ( member8712135081189583430la_a_b @ B2 @ ( image_5997780205441267925la_a_b @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_838_image__comp,axiom,
! [F: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b,G: list_R8263082107343818799la_a_b > set_Re381260168593705685la_a_b,R3: set_li8334890247616788453la_a_b] :
( ( image_7051608999182166449la_a_b @ F @ ( image_1636186708463624715la_a_b @ G @ R3 ) )
= ( image_1636186708463624715la_a_b @ ( comp_s6757191908747318590la_a_b @ F @ G ) @ R3 ) ) ).
% image_comp
thf(fact_839_image__comp,axiom,
! [F: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b,G: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b,R3: set_se6865892389300016395la_a_b] :
( ( image_7051608999182166449la_a_b @ F @ ( image_7051608999182166449la_a_b @ G @ R3 ) )
= ( image_7051608999182166449la_a_b @ ( comp_s8660012457442781028la_a_b @ F @ G ) @ R3 ) ) ).
% image_comp
thf(fact_840_image__comp,axiom,
! [F: nat > nat,G: relational_fmla_a_b > nat,R3: set_Re381260168593705685la_a_b] :
( ( image_nat_nat @ F @ ( image_341122591648980342_b_nat @ G @ R3 ) )
= ( image_341122591648980342_b_nat @ ( comp_n6926603950940120612la_a_b @ F @ G ) @ R3 ) ) ).
% image_comp
thf(fact_841_image__comp,axiom,
! [F: nat > nat,G: nat > nat,R3: set_nat] :
( ( image_nat_nat @ F @ ( image_nat_nat @ G @ R3 ) )
= ( image_nat_nat @ ( comp_nat_nat_nat @ F @ G ) @ R3 ) ) ).
% image_comp
thf(fact_842_remove__code_I1_J,axiom,
! [X: relational_fmla_a_b,Xs: list_R8263082107343818799la_a_b] :
( ( remove4261432235257513082la_a_b @ X @ ( set_Re9104216502384355786la_a_b @ Xs ) )
= ( set_Re9104216502384355786la_a_b @ ( remove4882698675883194515la_a_b @ X @ Xs ) ) ) ).
% remove_code(1)
thf(fact_843_image__Un,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( image_6790371041703824709la_a_b @ F @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) )
= ( sup_su5130108678486352897la_a_b @ ( image_6790371041703824709la_a_b @ F @ A2 ) @ ( image_6790371041703824709la_a_b @ F @ B ) ) ) ).
% image_Un
thf(fact_844_in__image__insert__iff,axiom,
! [B: set_se6865892389300016395la_a_b,X: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ! [C4: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ C4 @ B )
=> ~ ( member4680049679412964150la_a_b @ X @ C4 ) )
=> ( ( member3481406638322139244la_a_b @ A2 @ ( image_7051608999182166449la_a_b @ ( insert7010464514620295119la_a_b @ X ) @ B ) )
= ( ( member4680049679412964150la_a_b @ X @ A2 )
& ( member3481406638322139244la_a_b @ ( minus_4077726661957047470la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_845_Union__image__insert,axiom,
! [F: relational_fmla_a_b > set_Re381260168593705685la_a_b,A: relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ F @ ( insert7010464514620295119la_a_b @ A @ B ) ) )
= ( sup_su5130108678486352897la_a_b @ ( F @ A ) @ ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ F @ B ) ) ) ) ).
% Union_image_insert
thf(fact_846_Union__image__empty,axiom,
! [A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ A2 @ ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ F @ bot_bo4495933725496725865la_a_b ) ) )
= A2 ) ).
% Union_image_empty
thf(fact_847_UNION__fun__upd,axiom,
! [A2: relational_fmla_a_b > set_Re381260168593705685la_a_b,I2: relational_fmla_a_b,B: set_Re381260168593705685la_a_b,J: set_Re381260168593705685la_a_b] :
( ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ ( fun_up452696389778429683la_a_b @ A2 @ I2 @ B ) @ J ) )
= ( sup_su5130108678486352897la_a_b @ ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ A2 @ ( minus_4077726661957047470la_a_b @ J @ ( insert7010464514620295119la_a_b @ I2 @ bot_bo4495933725496725865la_a_b ) ) ) ) @ ( if_set2835548578466827919la_a_b @ ( member4680049679412964150la_a_b @ I2 @ J ) @ B @ bot_bo4495933725496725865la_a_b ) ) ) ).
% UNION_fun_upd
thf(fact_848_cSUP__union,axiom,
! [A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( A2 != bot_bo4495933725496725865la_a_b )
=> ( ( condit454741301069713265la_a_b @ ( image_8209480069293074043la_a_b @ F @ A2 ) )
=> ( ( B != bot_bo4495933725496725865la_a_b )
=> ( ( condit454741301069713265la_a_b @ ( image_8209480069293074043la_a_b @ F @ B ) )
=> ( ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ F @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) ) )
= ( sup_su5130108678486352897la_a_b @ ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ F @ A2 ) ) @ ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ F @ B ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_849_cSUP__insert,axiom,
! [A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > set_Re381260168593705685la_a_b,A: relational_fmla_a_b] :
( ( A2 != bot_bo4495933725496725865la_a_b )
=> ( ( condit454741301069713265la_a_b @ ( image_8209480069293074043la_a_b @ F @ A2 ) )
=> ( ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ F @ ( insert7010464514620295119la_a_b @ A @ A2 ) ) )
= ( sup_su5130108678486352897la_a_b @ ( F @ A ) @ ( comple8442120529048846632la_a_b @ ( image_8209480069293074043la_a_b @ F @ A2 ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_850_Sup__fin_Oset__eq__fold,axiom,
! [X: set_Re381260168593705685la_a_b,Xs: list_s3671366213839843941la_a_b] :
( ( lattic7150925611526040158la_a_b @ ( set_se5111286776497155840la_a_b @ ( cons_s2104095074707585951la_a_b @ X @ Xs ) ) )
= ( fold_s1495082921969268663la_a_b @ sup_su5130108678486352897la_a_b @ Xs @ X ) ) ).
% Sup_fin.set_eq_fold
thf(fact_851_set__empty2,axiom,
! [Xs: list_R8263082107343818799la_a_b] :
( ( bot_bo4495933725496725865la_a_b
= ( set_Re9104216502384355786la_a_b @ Xs ) )
= ( Xs = nil_Re6358386334527539737la_a_b ) ) ).
% set_empty2
thf(fact_852_set__empty,axiom,
! [Xs: list_R8263082107343818799la_a_b] :
( ( ( set_Re9104216502384355786la_a_b @ Xs )
= bot_bo4495933725496725865la_a_b )
= ( Xs = nil_Re6358386334527539737la_a_b ) ) ).
% set_empty
thf(fact_853_in__set__simps_I3_J,axiom,
! [X: relational_fmla_a_b] :
~ ( member4680049679412964150la_a_b @ X @ ( set_Re9104216502384355786la_a_b @ nil_Re6358386334527539737la_a_b ) ) ).
% in_set_simps(3)
thf(fact_854_empty__set,axiom,
( bot_bo4495933725496725865la_a_b
= ( set_Re9104216502384355786la_a_b @ nil_Re6358386334527539737la_a_b ) ) ).
% empty_set
thf(fact_855_in__set__simps_I2_J,axiom,
! [X: relational_fmla_a_b,Y: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X @ ( set_Re9104216502384355786la_a_b @ ( cons_R4161027894713053801la_a_b @ Y @ nil_Re6358386334527539737la_a_b ) ) )
= ( X = Y ) ) ).
% in_set_simps(2)
thf(fact_856_insert__code_I2_J,axiom,
! [X: relational_fmla_a_b,Xs: list_R8263082107343818799la_a_b] :
( ( insert7010464514620295119la_a_b @ X @ ( coset_4993845101609268588la_a_b @ Xs ) )
= ( coset_4993845101609268588la_a_b @ ( remove4882698675883194515la_a_b @ X @ Xs ) ) ) ).
% insert_code(2)
thf(fact_857_the__elem__set,axiom,
! [X: relational_fmla_a_b] :
( ( the_el6350558617753882986la_a_b @ ( set_Re9104216502384355786la_a_b @ ( cons_R4161027894713053801la_a_b @ X @ nil_Re6358386334527539737la_a_b ) ) )
= X ) ).
% the_elem_set
thf(fact_858_cSup__insert__If,axiom,
! [X4: set_se6865892389300016395la_a_b,A: set_Re381260168593705685la_a_b] :
( ( condit454741301069713265la_a_b @ X4 )
=> ( ( ( X4 = bot_bo2891247006866115487la_a_b )
=> ( ( comple8442120529048846632la_a_b @ ( insert2023870700798818565la_a_b @ A @ X4 ) )
= A ) )
& ( ( X4 != bot_bo2891247006866115487la_a_b )
=> ( ( comple8442120529048846632la_a_b @ ( insert2023870700798818565la_a_b @ A @ X4 ) )
= ( sup_su5130108678486352897la_a_b @ A @ ( comple8442120529048846632la_a_b @ X4 ) ) ) ) ) ) ).
% cSup_insert_If
thf(fact_859_cSup__insert,axiom,
! [X4: set_se6865892389300016395la_a_b,A: set_Re381260168593705685la_a_b] :
( ( X4 != bot_bo2891247006866115487la_a_b )
=> ( ( condit454741301069713265la_a_b @ X4 )
=> ( ( comple8442120529048846632la_a_b @ ( insert2023870700798818565la_a_b @ A @ X4 ) )
= ( sup_su5130108678486352897la_a_b @ A @ ( comple8442120529048846632la_a_b @ X4 ) ) ) ) ) ).
% cSup_insert
thf(fact_860_cSup__union__distrib,axiom,
! [A2: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ( A2 != bot_bo2891247006866115487la_a_b )
=> ( ( condit454741301069713265la_a_b @ A2 )
=> ( ( B != bot_bo2891247006866115487la_a_b )
=> ( ( condit454741301069713265la_a_b @ B )
=> ( ( comple8442120529048846632la_a_b @ ( sup_su4783144482993978935la_a_b @ A2 @ B ) )
= ( sup_su5130108678486352897la_a_b @ ( comple8442120529048846632la_a_b @ A2 ) @ ( comple8442120529048846632la_a_b @ B ) ) ) ) ) ) ) ).
% cSup_union_distrib
thf(fact_861_foldr1_Oelims,axiom,
! [X: relational_fmla_a_b > relational_fmla_a_b > relational_fmla_a_b,Xa: list_R8263082107343818799la_a_b,Xb: relational_fmla_a_b,Y: relational_fmla_a_b] :
( ( ( relati1724936043867567613la_a_b @ X @ Xa @ Xb )
= Y )
=> ( ! [X3: relational_fmla_a_b,Xs3: list_R8263082107343818799la_a_b] :
( ( Xa
= ( cons_R4161027894713053801la_a_b @ X3 @ Xs3 ) )
=> ( Y
!= ( foldr_789212930732525799la_a_b @ X @ Xs3 @ X3 ) ) )
=> ~ ( ( Xa = nil_Re6358386334527539737la_a_b )
=> ( Y != Xb ) ) ) ) ).
% foldr1.elims
thf(fact_862_fun__upd__image,axiom,
! [X: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > relational_fmla_a_b,Y: relational_fmla_a_b] :
( ( ( member4680049679412964150la_a_b @ X @ A2 )
=> ( ( image_6790371041703824709la_a_b @ ( fun_up2215209898182204861la_a_b @ F @ X @ Y ) @ A2 )
= ( insert7010464514620295119la_a_b @ Y @ ( image_6790371041703824709la_a_b @ F @ ( minus_4077726661957047470la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) ) ) ) )
& ( ~ ( member4680049679412964150la_a_b @ X @ A2 )
=> ( ( image_6790371041703824709la_a_b @ ( fun_up2215209898182204861la_a_b @ F @ X @ Y ) @ A2 )
= ( image_6790371041703824709la_a_b @ F @ A2 ) ) ) ) ).
% fun_upd_image
thf(fact_863_Pow__set_I1_J,axiom,
( ( pow_Re5871172025630691010la_a_b @ ( set_Re9104216502384355786la_a_b @ nil_Re6358386334527539737la_a_b ) )
= ( insert2023870700798818565la_a_b @ bot_bo4495933725496725865la_a_b @ bot_bo2891247006866115487la_a_b ) ) ).
% Pow_set(1)
thf(fact_864_Pow__empty,axiom,
( ( pow_Re5871172025630691010la_a_b @ bot_bo4495933725496725865la_a_b )
= ( insert2023870700798818565la_a_b @ bot_bo4495933725496725865la_a_b @ bot_bo2891247006866115487la_a_b ) ) ).
% Pow_empty
thf(fact_865_Pow__singleton__iff,axiom,
! [X4: set_Re381260168593705685la_a_b,Y4: set_Re381260168593705685la_a_b] :
( ( ( pow_Re5871172025630691010la_a_b @ X4 )
= ( insert2023870700798818565la_a_b @ Y4 @ bot_bo2891247006866115487la_a_b ) )
= ( ( X4 = bot_bo4495933725496725865la_a_b )
& ( Y4 = bot_bo4495933725496725865la_a_b ) ) ) ).
% Pow_singleton_iff
thf(fact_866_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_867_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_868_subset__empty,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A2 @ bot_bo4495933725496725865la_a_b )
= ( A2 = bot_bo4495933725496725865la_a_b ) ) ).
% subset_empty
thf(fact_869_empty__subsetI,axiom,
! [A2: set_Re381260168593705685la_a_b] : ( ord_le4112832032246704949la_a_b @ bot_bo4495933725496725865la_a_b @ A2 ) ).
% empty_subsetI
thf(fact_870_insert__subset,axiom,
! [X: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ ( insert7010464514620295119la_a_b @ X @ A2 ) @ B )
= ( ( member4680049679412964150la_a_b @ X @ B )
& ( ord_le4112832032246704949la_a_b @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_871_Un__subset__iff,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) @ C )
= ( ( ord_le4112832032246704949la_a_b @ A2 @ C )
& ( ord_le4112832032246704949la_a_b @ B @ C ) ) ) ).
% Un_subset_iff
thf(fact_872_le__inf__iff,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
= ( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_873_inf_Obounded__iff,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B2 @ C2 ) )
= ( ( ord_less_eq_nat @ A @ B2 )
& ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_874_le__sup__iff,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b,Z: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ ( sup_su5130108678486352897la_a_b @ X @ Y ) @ Z )
= ( ( ord_le4112832032246704949la_a_b @ X @ Z )
& ( ord_le4112832032246704949la_a_b @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_875_le__sup__iff,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z )
= ( ( ord_less_eq_nat @ X @ Z )
& ( ord_less_eq_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_876_sup_Obounded__iff,axiom,
! [B2: set_Re381260168593705685la_a_b,C2: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ ( sup_su5130108678486352897la_a_b @ B2 @ C2 ) @ A )
= ( ( ord_le4112832032246704949la_a_b @ B2 @ A )
& ( ord_le4112832032246704949la_a_b @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_877_sup_Obounded__iff,axiom,
! [B2: nat,C2: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C2 ) @ A )
= ( ( ord_less_eq_nat @ B2 @ A )
& ( ord_less_eq_nat @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_878_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ bot_bo4495933725496725865la_a_b @ X )
= bot_bo4495933725496725865la_a_b ) ).
% boolean_algebra.conj_zero_left
thf(fact_879_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ X @ bot_bo4495933725496725865la_a_b )
= bot_bo4495933725496725865la_a_b ) ).
% boolean_algebra.conj_zero_right
thf(fact_880_inf__bot__left,axiom,
! [X: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ bot_bo4495933725496725865la_a_b @ X )
= bot_bo4495933725496725865la_a_b ) ).
% inf_bot_left
thf(fact_881_inf__bot__right,axiom,
! [X: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ X @ bot_bo4495933725496725865la_a_b )
= bot_bo4495933725496725865la_a_b ) ).
% inf_bot_right
thf(fact_882_singleton__insert__inj__eq,axiom,
! [B2: relational_fmla_a_b,A: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( ( insert7010464514620295119la_a_b @ B2 @ bot_bo4495933725496725865la_a_b )
= ( insert7010464514620295119la_a_b @ A @ A2 ) )
= ( ( A = B2 )
& ( ord_le4112832032246704949la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ B2 @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_883_singleton__insert__inj__eq_H,axiom,
! [A: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B2: relational_fmla_a_b] :
( ( ( insert7010464514620295119la_a_b @ A @ A2 )
= ( insert7010464514620295119la_a_b @ B2 @ bot_bo4495933725496725865la_a_b ) )
= ( ( A = B2 )
& ( ord_le4112832032246704949la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ B2 @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_884_inf__sup__absorb,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ X @ ( sup_su5130108678486352897la_a_b @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_885_sup__inf__absorb,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ X @ ( inf_in8483230781156617063la_a_b @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_886_Diff__eq__empty__iff,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ( minus_4077726661957047470la_a_b @ A2 @ B )
= bot_bo4495933725496725865la_a_b )
= ( ord_le4112832032246704949la_a_b @ A2 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_887_Int__insert__left__if0,axiom,
! [A: relational_fmla_a_b,C: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ~ ( member4680049679412964150la_a_b @ A @ C )
=> ( ( inf_in8483230781156617063la_a_b @ ( insert7010464514620295119la_a_b @ A @ B ) @ C )
= ( inf_in8483230781156617063la_a_b @ B @ C ) ) ) ).
% Int_insert_left_if0
thf(fact_888_Int__insert__left__if1,axiom,
! [A: relational_fmla_a_b,C: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ A @ C )
=> ( ( inf_in8483230781156617063la_a_b @ ( insert7010464514620295119la_a_b @ A @ B ) @ C )
= ( insert7010464514620295119la_a_b @ A @ ( inf_in8483230781156617063la_a_b @ B @ C ) ) ) ) ).
% Int_insert_left_if1
thf(fact_889_insert__inter__insert,axiom,
! [A: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ ( insert7010464514620295119la_a_b @ A @ A2 ) @ ( insert7010464514620295119la_a_b @ A @ B ) )
= ( insert7010464514620295119la_a_b @ A @ ( inf_in8483230781156617063la_a_b @ A2 @ B ) ) ) ).
% insert_inter_insert
thf(fact_890_Int__insert__right__if0,axiom,
! [A: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ~ ( member4680049679412964150la_a_b @ A @ A2 )
=> ( ( inf_in8483230781156617063la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ A @ B ) )
= ( inf_in8483230781156617063la_a_b @ A2 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_891_Int__insert__right__if1,axiom,
! [A: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ A @ A2 )
=> ( ( inf_in8483230781156617063la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ A @ B ) )
= ( insert7010464514620295119la_a_b @ A @ ( inf_in8483230781156617063la_a_b @ A2 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_892_Int__Un__eq_I4_J,axiom,
! [T: set_Re381260168593705685la_a_b,S2: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ T @ ( inf_in8483230781156617063la_a_b @ S2 @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_893_Int__Un__eq_I3_J,axiom,
! [S2: set_Re381260168593705685la_a_b,T: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ S2 @ ( inf_in8483230781156617063la_a_b @ S2 @ T ) )
= S2 ) ).
% Int_Un_eq(3)
thf(fact_894_Int__Un__eq_I2_J,axiom,
! [S2: set_Re381260168593705685la_a_b,T: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ ( inf_in8483230781156617063la_a_b @ S2 @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_895_Int__Un__eq_I1_J,axiom,
! [S2: set_Re381260168593705685la_a_b,T: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ ( inf_in8483230781156617063la_a_b @ S2 @ T ) @ S2 )
= S2 ) ).
% Int_Un_eq(1)
thf(fact_896_Un__Int__eq_I4_J,axiom,
! [T: set_Re381260168593705685la_a_b,S2: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ T @ ( sup_su5130108678486352897la_a_b @ S2 @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_897_Un__Int__eq_I3_J,axiom,
! [S2: set_Re381260168593705685la_a_b,T: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ S2 @ ( sup_su5130108678486352897la_a_b @ S2 @ T ) )
= S2 ) ).
% Un_Int_eq(3)
thf(fact_898_Un__Int__eq_I2_J,axiom,
! [S2: set_Re381260168593705685la_a_b,T: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ ( sup_su5130108678486352897la_a_b @ S2 @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_899_Un__Int__eq_I1_J,axiom,
! [S2: set_Re381260168593705685la_a_b,T: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ ( sup_su5130108678486352897la_a_b @ S2 @ T ) @ S2 )
= S2 ) ).
% Un_Int_eq(1)
thf(fact_900_bdd__above_OI,axiom,
! [A2: set_nat,M: nat] :
( ! [X3: nat] :
( ( member_nat2 @ X3 @ A2 )
=> ( ord_less_eq_nat @ X3 @ M ) )
=> ( condit2214826472909112428ve_nat @ A2 ) ) ).
% bdd_above.I
thf(fact_901_disjoint__insert_I2_J,axiom,
! [A2: set_Re381260168593705685la_a_b,B2: relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ( bot_bo4495933725496725865la_a_b
= ( inf_in8483230781156617063la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ B2 @ B ) ) )
= ( ~ ( member4680049679412964150la_a_b @ B2 @ A2 )
& ( bot_bo4495933725496725865la_a_b
= ( inf_in8483230781156617063la_a_b @ A2 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_902_disjoint__insert_I1_J,axiom,
! [B: set_Re381260168593705685la_a_b,A: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( ( inf_in8483230781156617063la_a_b @ B @ ( insert7010464514620295119la_a_b @ A @ A2 ) )
= bot_bo4495933725496725865la_a_b )
= ( ~ ( member4680049679412964150la_a_b @ A @ B )
& ( ( inf_in8483230781156617063la_a_b @ B @ A2 )
= bot_bo4495933725496725865la_a_b ) ) ) ).
% disjoint_insert(1)
thf(fact_903_insert__disjoint_I2_J,axiom,
! [A: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( bot_bo4495933725496725865la_a_b
= ( inf_in8483230781156617063la_a_b @ ( insert7010464514620295119la_a_b @ A @ A2 ) @ B ) )
= ( ~ ( member4680049679412964150la_a_b @ A @ B )
& ( bot_bo4495933725496725865la_a_b
= ( inf_in8483230781156617063la_a_b @ A2 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_904_insert__disjoint_I1_J,axiom,
! [A: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ( inf_in8483230781156617063la_a_b @ ( insert7010464514620295119la_a_b @ A @ A2 ) @ B )
= bot_bo4495933725496725865la_a_b )
= ( ~ ( member4680049679412964150la_a_b @ A @ B )
& ( ( inf_in8483230781156617063la_a_b @ A2 @ B )
= bot_bo4495933725496725865la_a_b ) ) ) ).
% insert_disjoint(1)
thf(fact_905_Diff__disjoint,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ A2 @ ( minus_4077726661957047470la_a_b @ B @ A2 ) )
= bot_bo4495933725496725865la_a_b ) ).
% Diff_disjoint
thf(fact_906_inf__sup__ord_I2_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_907_inf__sup__ord_I1_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_908_inf__le1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_909_inf__le2,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_910_le__infE,axiom,
! [X: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B2 ) )
=> ~ ( ( ord_less_eq_nat @ X @ A )
=> ~ ( ord_less_eq_nat @ X @ B2 ) ) ) ).
% le_infE
thf(fact_911_le__infI,axiom,
! [X: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ X @ A )
=> ( ( ord_less_eq_nat @ X @ B2 )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B2 ) ) ) ) ).
% le_infI
thf(fact_912_inf__mono,axiom,
! [A: nat,C2: nat,B2: nat,D: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B2 @ D )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B2 ) @ ( inf_inf_nat @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_913_le__infI1,axiom,
! [A: nat,X: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_914_le__infI2,axiom,
! [B2: nat,X: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_915_inf_OorderE,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( A
= ( inf_inf_nat @ A @ B2 ) ) ) ).
% inf.orderE
thf(fact_916_inf_OorderI,axiom,
! [A: nat,B2: nat] :
( ( A
= ( inf_inf_nat @ A @ B2 ) )
=> ( ord_less_eq_nat @ A @ B2 ) ) ).
% inf.orderI
thf(fact_917_inf__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ ( F @ X3 @ Y2 ) @ X3 )
=> ( ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ ( F @ X3 @ Y2 ) @ Y2 )
=> ( ! [X3: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ( ord_less_eq_nat @ X3 @ Z3 )
=> ( ord_less_eq_nat @ X3 @ ( F @ Y2 @ Z3 ) ) ) )
=> ( ( inf_inf_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_918_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y3: nat] :
( ( inf_inf_nat @ X2 @ Y3 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_919_inf_Oabsorb1,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( inf_inf_nat @ A @ B2 )
= A ) ) ).
% inf.absorb1
thf(fact_920_inf_Oabsorb2,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( inf_inf_nat @ A @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_921_inf__absorb1,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( inf_inf_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_922_inf__absorb2,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( inf_inf_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_923_inf_OboundedE,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B2 @ C2 ) )
=> ~ ( ( ord_less_eq_nat @ A @ B2 )
=> ~ ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% inf.boundedE
thf(fact_924_inf_OboundedI,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ A @ C2 )
=> ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B2 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_925_inf__greatest,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Z )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_926_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B4: nat] :
( A3
= ( inf_inf_nat @ A3 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_927_inf_Ocobounded1,axiom,
! [A: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B2 ) @ A ) ).
% inf.cobounded1
thf(fact_928_inf_Ocobounded2,axiom,
! [A: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_929_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B4: nat] :
( ( inf_inf_nat @ A3 @ B4 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_930_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A3: nat] :
( ( inf_inf_nat @ A3 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_931_inf_OcoboundedI1,axiom,
! [A: nat,C2: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B2 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_932_inf_OcoboundedI2,axiom,
! [B2: nat,C2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B2 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_933_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_934_le__cases3,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z ) )
=> ( ( ( ord_less_eq_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_935_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
& ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_936_ord__eq__le__trans,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( A = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_937_ord__le__eq__trans,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_938_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_939_order_Otrans,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% order.trans
thf(fact_940_order__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_941_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B2: nat] :
( ! [A5: nat,B6: nat] :
( ( ord_less_eq_nat @ A5 @ B6 )
=> ( P @ A5 @ B6 ) )
=> ( ! [A5: nat,B6: nat] :
( ( P @ B6 @ A5 )
=> ( P @ A5 @ B6 ) )
=> ( P @ A @ B2 ) ) ) ).
% linorder_wlog
thf(fact_942_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
= ( ^ [A3: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A3 )
& ( ord_less_eq_nat @ A3 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_943_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_944_dual__order_Otrans,axiom,
! [B2: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ C2 @ B2 )
=> ( ord_less_eq_nat @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_945_antisym,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_946_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
= ( ^ [A3: nat,B4: nat] :
( ( ord_less_eq_nat @ A3 @ B4 )
& ( ord_less_eq_nat @ B4 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_947_order__subst1,axiom,
! [A: nat,F: nat > nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_948_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_949_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_950_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_951_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B2: nat,C2: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_952_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_953_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_954_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_955_distrib__sup__le,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b,Z: set_Re381260168593705685la_a_b] : ( ord_le4112832032246704949la_a_b @ ( sup_su5130108678486352897la_a_b @ X @ ( inf_in8483230781156617063la_a_b @ Y @ Z ) ) @ ( inf_in8483230781156617063la_a_b @ ( sup_su5130108678486352897la_a_b @ X @ Y ) @ ( sup_su5130108678486352897la_a_b @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_956_distrib__sup__le,axiom,
! [X: nat,Y: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_957_distrib__inf__le,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b,Z: set_Re381260168593705685la_a_b] : ( ord_le4112832032246704949la_a_b @ ( sup_su5130108678486352897la_a_b @ ( inf_in8483230781156617063la_a_b @ X @ Y ) @ ( inf_in8483230781156617063la_a_b @ X @ Z ) ) @ ( inf_in8483230781156617063la_a_b @ X @ ( sup_su5130108678486352897la_a_b @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_958_distrib__inf__le,axiom,
! [X: nat,Y: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z ) ) @ ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_959_Un__Int__assoc__eq,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b] :
( ( ( sup_su5130108678486352897la_a_b @ ( inf_in8483230781156617063la_a_b @ A2 @ B ) @ C )
= ( inf_in8483230781156617063la_a_b @ A2 @ ( sup_su5130108678486352897la_a_b @ B @ C ) ) )
= ( ord_le4112832032246704949la_a_b @ C @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_960_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b,Z: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ X @ ( sup_su5130108678486352897la_a_b @ Y @ Z ) )
= ( sup_su5130108678486352897la_a_b @ ( inf_in8483230781156617063la_a_b @ X @ Y ) @ ( inf_in8483230781156617063la_a_b @ X @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_961_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b,Z: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ X @ ( inf_in8483230781156617063la_a_b @ Y @ Z ) )
= ( inf_in8483230781156617063la_a_b @ ( sup_su5130108678486352897la_a_b @ X @ Y ) @ ( sup_su5130108678486352897la_a_b @ X @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_962_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_Re381260168593705685la_a_b,Z: set_Re381260168593705685la_a_b,X: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ ( sup_su5130108678486352897la_a_b @ Y @ Z ) @ X )
= ( sup_su5130108678486352897la_a_b @ ( inf_in8483230781156617063la_a_b @ Y @ X ) @ ( inf_in8483230781156617063la_a_b @ Z @ X ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_963_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_Re381260168593705685la_a_b,Z: set_Re381260168593705685la_a_b,X: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ ( inf_in8483230781156617063la_a_b @ Y @ Z ) @ X )
= ( inf_in8483230781156617063la_a_b @ ( sup_su5130108678486352897la_a_b @ Y @ X ) @ ( sup_su5130108678486352897la_a_b @ Z @ X ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_964_distrib__imp1,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b,Z: set_Re381260168593705685la_a_b] :
( ! [X3: set_Re381260168593705685la_a_b,Y2: set_Re381260168593705685la_a_b,Z3: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ X3 @ ( sup_su5130108678486352897la_a_b @ Y2 @ Z3 ) )
= ( sup_su5130108678486352897la_a_b @ ( inf_in8483230781156617063la_a_b @ X3 @ Y2 ) @ ( inf_in8483230781156617063la_a_b @ X3 @ Z3 ) ) )
=> ( ( sup_su5130108678486352897la_a_b @ X @ ( inf_in8483230781156617063la_a_b @ Y @ Z ) )
= ( inf_in8483230781156617063la_a_b @ ( sup_su5130108678486352897la_a_b @ X @ Y ) @ ( sup_su5130108678486352897la_a_b @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_965_distrib__imp2,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b,Z: set_Re381260168593705685la_a_b] :
( ! [X3: set_Re381260168593705685la_a_b,Y2: set_Re381260168593705685la_a_b,Z3: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ X3 @ ( inf_in8483230781156617063la_a_b @ Y2 @ Z3 ) )
= ( inf_in8483230781156617063la_a_b @ ( sup_su5130108678486352897la_a_b @ X3 @ Y2 ) @ ( sup_su5130108678486352897la_a_b @ X3 @ Z3 ) ) )
=> ( ( inf_in8483230781156617063la_a_b @ X @ ( sup_su5130108678486352897la_a_b @ Y @ Z ) )
= ( sup_su5130108678486352897la_a_b @ ( inf_in8483230781156617063la_a_b @ X @ Y ) @ ( inf_in8483230781156617063la_a_b @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_966_inf__sup__distrib1,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b,Z: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ X @ ( sup_su5130108678486352897la_a_b @ Y @ Z ) )
= ( sup_su5130108678486352897la_a_b @ ( inf_in8483230781156617063la_a_b @ X @ Y ) @ ( inf_in8483230781156617063la_a_b @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_967_inf__sup__distrib2,axiom,
! [Y: set_Re381260168593705685la_a_b,Z: set_Re381260168593705685la_a_b,X: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ ( sup_su5130108678486352897la_a_b @ Y @ Z ) @ X )
= ( sup_su5130108678486352897la_a_b @ ( inf_in8483230781156617063la_a_b @ Y @ X ) @ ( inf_in8483230781156617063la_a_b @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_968_sup__inf__distrib1,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b,Z: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ X @ ( inf_in8483230781156617063la_a_b @ Y @ Z ) )
= ( inf_in8483230781156617063la_a_b @ ( sup_su5130108678486352897la_a_b @ X @ Y ) @ ( sup_su5130108678486352897la_a_b @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_969_sup__inf__distrib2,axiom,
! [Y: set_Re381260168593705685la_a_b,Z: set_Re381260168593705685la_a_b,X: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ ( inf_in8483230781156617063la_a_b @ Y @ Z ) @ X )
= ( inf_in8483230781156617063la_a_b @ ( sup_su5130108678486352897la_a_b @ Y @ X ) @ ( sup_su5130108678486352897la_a_b @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_970_Int__emptyI,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A2 )
=> ~ ( member4680049679412964150la_a_b @ X3 @ B ) )
=> ( ( inf_in8483230781156617063la_a_b @ A2 @ B )
= bot_bo4495933725496725865la_a_b ) ) ).
% Int_emptyI
thf(fact_971_disjoint__iff,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ( inf_in8483230781156617063la_a_b @ A2 @ B )
= bot_bo4495933725496725865la_a_b )
= ( ! [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ A2 )
=> ~ ( member4680049679412964150la_a_b @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_972_Int__empty__left,axiom,
! [B: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ bot_bo4495933725496725865la_a_b @ B )
= bot_bo4495933725496725865la_a_b ) ).
% Int_empty_left
thf(fact_973_Int__empty__right,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ A2 @ bot_bo4495933725496725865la_a_b )
= bot_bo4495933725496725865la_a_b ) ).
% Int_empty_right
thf(fact_974_disjoint__iff__not__equal,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ( inf_in8483230781156617063la_a_b @ A2 @ B )
= bot_bo4495933725496725865la_a_b )
= ( ! [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ A2 )
=> ! [Y3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Y3 @ B )
=> ( X2 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_975_Int__insert__left,axiom,
! [A: relational_fmla_a_b,C: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ( member4680049679412964150la_a_b @ A @ C )
=> ( ( inf_in8483230781156617063la_a_b @ ( insert7010464514620295119la_a_b @ A @ B ) @ C )
= ( insert7010464514620295119la_a_b @ A @ ( inf_in8483230781156617063la_a_b @ B @ C ) ) ) )
& ( ~ ( member4680049679412964150la_a_b @ A @ C )
=> ( ( inf_in8483230781156617063la_a_b @ ( insert7010464514620295119la_a_b @ A @ B ) @ C )
= ( inf_in8483230781156617063la_a_b @ B @ C ) ) ) ) ).
% Int_insert_left
thf(fact_976_Int__insert__right,axiom,
! [A: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ( member4680049679412964150la_a_b @ A @ A2 )
=> ( ( inf_in8483230781156617063la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ A @ B ) )
= ( insert7010464514620295119la_a_b @ A @ ( inf_in8483230781156617063la_a_b @ A2 @ B ) ) ) )
& ( ~ ( member4680049679412964150la_a_b @ A @ A2 )
=> ( ( inf_in8483230781156617063la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ A @ B ) )
= ( inf_in8483230781156617063la_a_b @ A2 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_977_Un__Int__crazy,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ ( sup_su5130108678486352897la_a_b @ ( inf_in8483230781156617063la_a_b @ A2 @ B ) @ ( inf_in8483230781156617063la_a_b @ B @ C ) ) @ ( inf_in8483230781156617063la_a_b @ C @ A2 ) )
= ( inf_in8483230781156617063la_a_b @ ( inf_in8483230781156617063la_a_b @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) @ ( sup_su5130108678486352897la_a_b @ B @ C ) ) @ ( sup_su5130108678486352897la_a_b @ C @ A2 ) ) ) ).
% Un_Int_crazy
thf(fact_978_Int__Un__distrib,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ A2 @ ( sup_su5130108678486352897la_a_b @ B @ C ) )
= ( sup_su5130108678486352897la_a_b @ ( inf_in8483230781156617063la_a_b @ A2 @ B ) @ ( inf_in8483230781156617063la_a_b @ A2 @ C ) ) ) ).
% Int_Un_distrib
thf(fact_979_Un__Int__distrib,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ A2 @ ( inf_in8483230781156617063la_a_b @ B @ C ) )
= ( inf_in8483230781156617063la_a_b @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) @ ( sup_su5130108678486352897la_a_b @ A2 @ C ) ) ) ).
% Un_Int_distrib
thf(fact_980_Int__Un__distrib2,axiom,
! [B: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ ( sup_su5130108678486352897la_a_b @ B @ C ) @ A2 )
= ( sup_su5130108678486352897la_a_b @ ( inf_in8483230781156617063la_a_b @ B @ A2 ) @ ( inf_in8483230781156617063la_a_b @ C @ A2 ) ) ) ).
% Int_Un_distrib2
thf(fact_981_Un__Int__distrib2,axiom,
! [B: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ ( inf_in8483230781156617063la_a_b @ B @ C ) @ A2 )
= ( inf_in8483230781156617063la_a_b @ ( sup_su5130108678486352897la_a_b @ B @ A2 ) @ ( sup_su5130108678486352897la_a_b @ C @ A2 ) ) ) ).
% Un_Int_distrib2
thf(fact_982_bot_Oextremum,axiom,
! [A: set_Re381260168593705685la_a_b] : ( ord_le4112832032246704949la_a_b @ bot_bo4495933725496725865la_a_b @ A ) ).
% bot.extremum
thf(fact_983_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_984_bot_Oextremum__unique,axiom,
! [A: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A @ bot_bo4495933725496725865la_a_b )
= ( A = bot_bo4495933725496725865la_a_b ) ) ).
% bot.extremum_unique
thf(fact_985_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_986_bot_Oextremum__uniqueI,axiom,
! [A: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A @ bot_bo4495933725496725865la_a_b )
=> ( A = bot_bo4495933725496725865la_a_b ) ) ).
% bot.extremum_uniqueI
thf(fact_987_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_988_inf__sup__ord_I4_J,axiom,
! [Y: set_Re381260168593705685la_a_b,X: set_Re381260168593705685la_a_b] : ( ord_le4112832032246704949la_a_b @ Y @ ( sup_su5130108678486352897la_a_b @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_989_inf__sup__ord_I4_J,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_990_inf__sup__ord_I3_J,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b] : ( ord_le4112832032246704949la_a_b @ X @ ( sup_su5130108678486352897la_a_b @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_991_inf__sup__ord_I3_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_992_le__supE,axiom,
! [A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b,X: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ ( sup_su5130108678486352897la_a_b @ A @ B2 ) @ X )
=> ~ ( ( ord_le4112832032246704949la_a_b @ A @ X )
=> ~ ( ord_le4112832032246704949la_a_b @ B2 @ X ) ) ) ).
% le_supE
thf(fact_993_le__supE,axiom,
! [A: nat,B2: nat,X: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B2 ) @ X )
=> ~ ( ( ord_less_eq_nat @ A @ X )
=> ~ ( ord_less_eq_nat @ B2 @ X ) ) ) ).
% le_supE
thf(fact_994_le__supI,axiom,
! [A: set_Re381260168593705685la_a_b,X: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A @ X )
=> ( ( ord_le4112832032246704949la_a_b @ B2 @ X )
=> ( ord_le4112832032246704949la_a_b @ ( sup_su5130108678486352897la_a_b @ A @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_995_le__supI,axiom,
! [A: nat,X: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ X )
=> ( ( ord_less_eq_nat @ B2 @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_996_sup__ge1,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b] : ( ord_le4112832032246704949la_a_b @ X @ ( sup_su5130108678486352897la_a_b @ X @ Y ) ) ).
% sup_ge1
thf(fact_997_sup__ge1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_998_sup__ge2,axiom,
! [Y: set_Re381260168593705685la_a_b,X: set_Re381260168593705685la_a_b] : ( ord_le4112832032246704949la_a_b @ Y @ ( sup_su5130108678486352897la_a_b @ X @ Y ) ) ).
% sup_ge2
thf(fact_999_sup__ge2,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_1000_le__supI1,axiom,
! [X: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ X @ A )
=> ( ord_le4112832032246704949la_a_b @ X @ ( sup_su5130108678486352897la_a_b @ A @ B2 ) ) ) ).
% le_supI1
thf(fact_1001_le__supI1,axiom,
! [X: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ X @ A )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B2 ) ) ) ).
% le_supI1
thf(fact_1002_le__supI2,axiom,
! [X: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ X @ B2 )
=> ( ord_le4112832032246704949la_a_b @ X @ ( sup_su5130108678486352897la_a_b @ A @ B2 ) ) ) ).
% le_supI2
thf(fact_1003_le__supI2,axiom,
! [X: nat,B2: nat,A: nat] :
( ( ord_less_eq_nat @ X @ B2 )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B2 ) ) ) ).
% le_supI2
thf(fact_1004_sup_Omono,axiom,
! [C2: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b,D: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ C2 @ A )
=> ( ( ord_le4112832032246704949la_a_b @ D @ B2 )
=> ( ord_le4112832032246704949la_a_b @ ( sup_su5130108678486352897la_a_b @ C2 @ D ) @ ( sup_su5130108678486352897la_a_b @ A @ B2 ) ) ) ) ).
% sup.mono
thf(fact_1005_sup_Omono,axiom,
! [C2: nat,A: nat,D: nat,B2: nat] :
( ( ord_less_eq_nat @ C2 @ A )
=> ( ( ord_less_eq_nat @ D @ B2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C2 @ D ) @ ( sup_sup_nat @ A @ B2 ) ) ) ) ).
% sup.mono
thf(fact_1006_sup__mono,axiom,
! [A: set_Re381260168593705685la_a_b,C2: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b,D: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A @ C2 )
=> ( ( ord_le4112832032246704949la_a_b @ B2 @ D )
=> ( ord_le4112832032246704949la_a_b @ ( sup_su5130108678486352897la_a_b @ A @ B2 ) @ ( sup_su5130108678486352897la_a_b @ C2 @ D ) ) ) ) ).
% sup_mono
thf(fact_1007_sup__mono,axiom,
! [A: nat,C2: nat,B2: nat,D: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B2 @ D )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B2 ) @ ( sup_sup_nat @ C2 @ D ) ) ) ) ).
% sup_mono
thf(fact_1008_sup__least,axiom,
! [Y: set_Re381260168593705685la_a_b,X: set_Re381260168593705685la_a_b,Z: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ Y @ X )
=> ( ( ord_le4112832032246704949la_a_b @ Z @ X )
=> ( ord_le4112832032246704949la_a_b @ ( sup_su5130108678486352897la_a_b @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_1009_sup__least,axiom,
! [Y: nat,X: nat,Z: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ Z @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_1010_le__iff__sup,axiom,
( ord_le4112832032246704949la_a_b
= ( ^ [X2: set_Re381260168593705685la_a_b,Y3: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ X2 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_1011_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y3: nat] :
( ( sup_sup_nat @ X2 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_1012_sup_OorderE,axiom,
! [B2: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ B2 @ A )
=> ( A
= ( sup_su5130108678486352897la_a_b @ A @ B2 ) ) ) ).
% sup.orderE
thf(fact_1013_sup_OorderE,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( A
= ( sup_sup_nat @ A @ B2 ) ) ) ).
% sup.orderE
thf(fact_1014_sup_OorderI,axiom,
! [A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( A
= ( sup_su5130108678486352897la_a_b @ A @ B2 ) )
=> ( ord_le4112832032246704949la_a_b @ B2 @ A ) ) ).
% sup.orderI
thf(fact_1015_sup_OorderI,axiom,
! [A: nat,B2: nat] :
( ( A
= ( sup_sup_nat @ A @ B2 ) )
=> ( ord_less_eq_nat @ B2 @ A ) ) ).
% sup.orderI
thf(fact_1016_sup__unique,axiom,
! [F: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b,X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b] :
( ! [X3: set_Re381260168593705685la_a_b,Y2: set_Re381260168593705685la_a_b] : ( ord_le4112832032246704949la_a_b @ X3 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: set_Re381260168593705685la_a_b,Y2: set_Re381260168593705685la_a_b] : ( ord_le4112832032246704949la_a_b @ Y2 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: set_Re381260168593705685la_a_b,Y2: set_Re381260168593705685la_a_b,Z3: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ Y2 @ X3 )
=> ( ( ord_le4112832032246704949la_a_b @ Z3 @ X3 )
=> ( ord_le4112832032246704949la_a_b @ ( F @ Y2 @ Z3 ) @ X3 ) ) )
=> ( ( sup_su5130108678486352897la_a_b @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_1017_sup__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ X3 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ Y2 @ X3 )
=> ( ( ord_less_eq_nat @ Z3 @ X3 )
=> ( ord_less_eq_nat @ ( F @ Y2 @ Z3 ) @ X3 ) ) )
=> ( ( sup_sup_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_1018_sup_Oabsorb1,axiom,
! [B2: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ B2 @ A )
=> ( ( sup_su5130108678486352897la_a_b @ A @ B2 )
= A ) ) ).
% sup.absorb1
thf(fact_1019_sup_Oabsorb1,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( sup_sup_nat @ A @ B2 )
= A ) ) ).
% sup.absorb1
thf(fact_1020_sup_Oabsorb2,axiom,
! [A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A @ B2 )
=> ( ( sup_su5130108678486352897la_a_b @ A @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_1021_sup_Oabsorb2,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( sup_sup_nat @ A @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_1022_sup__absorb1,axiom,
! [Y: set_Re381260168593705685la_a_b,X: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ Y @ X )
=> ( ( sup_su5130108678486352897la_a_b @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_1023_sup__absorb1,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( sup_sup_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_1024_sup__absorb2,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ X @ Y )
=> ( ( sup_su5130108678486352897la_a_b @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_1025_sup__absorb2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( sup_sup_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_1026_sup_OboundedE,axiom,
! [B2: set_Re381260168593705685la_a_b,C2: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ ( sup_su5130108678486352897la_a_b @ B2 @ C2 ) @ A )
=> ~ ( ( ord_le4112832032246704949la_a_b @ B2 @ A )
=> ~ ( ord_le4112832032246704949la_a_b @ C2 @ A ) ) ) ).
% sup.boundedE
thf(fact_1027_sup_OboundedE,axiom,
! [B2: nat,C2: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C2 ) @ A )
=> ~ ( ( ord_less_eq_nat @ B2 @ A )
=> ~ ( ord_less_eq_nat @ C2 @ A ) ) ) ).
% sup.boundedE
thf(fact_1028_sup_OboundedI,axiom,
! [B2: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b,C2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ B2 @ A )
=> ( ( ord_le4112832032246704949la_a_b @ C2 @ A )
=> ( ord_le4112832032246704949la_a_b @ ( sup_su5130108678486352897la_a_b @ B2 @ C2 ) @ A ) ) ) ).
% sup.boundedI
thf(fact_1029_sup_OboundedI,axiom,
! [B2: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ C2 @ A )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C2 ) @ A ) ) ) ).
% sup.boundedI
thf(fact_1030_sup_Oorder__iff,axiom,
( ord_le4112832032246704949la_a_b
= ( ^ [B4: set_Re381260168593705685la_a_b,A3: set_Re381260168593705685la_a_b] :
( A3
= ( sup_su5130108678486352897la_a_b @ A3 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_1031_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A3: nat] :
( A3
= ( sup_sup_nat @ A3 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_1032_sup_Ocobounded1,axiom,
! [A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] : ( ord_le4112832032246704949la_a_b @ A @ ( sup_su5130108678486352897la_a_b @ A @ B2 ) ) ).
% sup.cobounded1
thf(fact_1033_sup_Ocobounded1,axiom,
! [A: nat,B2: nat] : ( ord_less_eq_nat @ A @ ( sup_sup_nat @ A @ B2 ) ) ).
% sup.cobounded1
thf(fact_1034_sup_Ocobounded2,axiom,
! [B2: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b] : ( ord_le4112832032246704949la_a_b @ B2 @ ( sup_su5130108678486352897la_a_b @ A @ B2 ) ) ).
% sup.cobounded2
thf(fact_1035_sup_Ocobounded2,axiom,
! [B2: nat,A: nat] : ( ord_less_eq_nat @ B2 @ ( sup_sup_nat @ A @ B2 ) ) ).
% sup.cobounded2
thf(fact_1036_sup_Oabsorb__iff1,axiom,
( ord_le4112832032246704949la_a_b
= ( ^ [B4: set_Re381260168593705685la_a_b,A3: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ A3 @ B4 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_1037_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A3: nat] :
( ( sup_sup_nat @ A3 @ B4 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_1038_sup_Oabsorb__iff2,axiom,
( ord_le4112832032246704949la_a_b
= ( ^ [A3: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ A3 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_1039_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B4: nat] :
( ( sup_sup_nat @ A3 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_1040_sup_OcoboundedI1,axiom,
! [C2: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ C2 @ A )
=> ( ord_le4112832032246704949la_a_b @ C2 @ ( sup_su5130108678486352897la_a_b @ A @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_1041_sup_OcoboundedI1,axiom,
! [C2: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ C2 @ A )
=> ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_1042_sup_OcoboundedI2,axiom,
! [C2: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ C2 @ B2 )
=> ( ord_le4112832032246704949la_a_b @ C2 @ ( sup_su5130108678486352897la_a_b @ A @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_1043_sup_OcoboundedI2,axiom,
! [C2: nat,B2: nat,A: nat] :
( ( ord_less_eq_nat @ C2 @ B2 )
=> ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_1044_cSup__eq__maximum,axiom,
! [Z: nat,X4: set_nat] :
( ( member_nat2 @ Z @ X4 )
=> ( ! [X3: nat] :
( ( member_nat2 @ X3 @ X4 )
=> ( ord_less_eq_nat @ X3 @ Z ) )
=> ( ( complete_Sup_Sup_nat @ X4 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_1045_insert__mono,axiom,
! [C: set_Re381260168593705685la_a_b,D2: set_Re381260168593705685la_a_b,A: relational_fmla_a_b] :
( ( ord_le4112832032246704949la_a_b @ C @ D2 )
=> ( ord_le4112832032246704949la_a_b @ ( insert7010464514620295119la_a_b @ A @ C ) @ ( insert7010464514620295119la_a_b @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_1046_subset__insert,axiom,
! [X: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ~ ( member4680049679412964150la_a_b @ X @ A2 )
=> ( ( ord_le4112832032246704949la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ X @ B ) )
= ( ord_le4112832032246704949la_a_b @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_1047_subset__insertI,axiom,
! [B: set_Re381260168593705685la_a_b,A: relational_fmla_a_b] : ( ord_le4112832032246704949la_a_b @ B @ ( insert7010464514620295119la_a_b @ A @ B ) ) ).
% subset_insertI
thf(fact_1048_subset__insertI2,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,B2: relational_fmla_a_b] :
( ( ord_le4112832032246704949la_a_b @ A2 @ B )
=> ( ord_le4112832032246704949la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_1049_Un__Pow__subset,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] : ( ord_le1577343677690852715la_a_b @ ( sup_su4783144482993978935la_a_b @ ( pow_Re5871172025630691010la_a_b @ A2 ) @ ( pow_Re5871172025630691010la_a_b @ B ) ) @ ( pow_Re5871172025630691010la_a_b @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) ) ) ).
% Un_Pow_subset
thf(fact_1050_subset__code_I1_J,axiom,
! [Xs: list_R8263082107343818799la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ ( set_Re9104216502384355786la_a_b @ Xs ) @ B )
= ( ! [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ ( set_Re9104216502384355786la_a_b @ Xs ) )
=> ( member4680049679412964150la_a_b @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_1051_Pow__bottom,axiom,
! [B: set_Re381260168593705685la_a_b] : ( member3481406638322139244la_a_b @ bot_bo4495933725496725865la_a_b @ ( pow_Re5871172025630691010la_a_b @ B ) ) ).
% Pow_bottom
thf(fact_1052_Un__mono,axiom,
! [A2: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,D2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A2 @ C )
=> ( ( ord_le4112832032246704949la_a_b @ B @ D2 )
=> ( ord_le4112832032246704949la_a_b @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) @ ( sup_su5130108678486352897la_a_b @ C @ D2 ) ) ) ) ).
% Un_mono
thf(fact_1053_Un__least,axiom,
! [A2: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A2 @ C )
=> ( ( ord_le4112832032246704949la_a_b @ B @ C )
=> ( ord_le4112832032246704949la_a_b @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) @ C ) ) ) ).
% Un_least
thf(fact_1054_Un__upper1,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] : ( ord_le4112832032246704949la_a_b @ A2 @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) ) ).
% Un_upper1
thf(fact_1055_Un__upper2,axiom,
! [B: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b] : ( ord_le4112832032246704949la_a_b @ B @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) ) ).
% Un_upper2
thf(fact_1056_Un__absorb1,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A2 @ B )
=> ( ( sup_su5130108678486352897la_a_b @ A2 @ B )
= B ) ) ).
% Un_absorb1
thf(fact_1057_Un__absorb2,axiom,
! [B: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ B @ A2 )
=> ( ( sup_su5130108678486352897la_a_b @ A2 @ B )
= A2 ) ) ).
% Un_absorb2
thf(fact_1058_subset__UnE,axiom,
! [C: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ C @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) )
=> ~ ! [A6: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A6 @ A2 )
=> ! [B7: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ B7 @ B )
=> ( C
!= ( sup_su5130108678486352897la_a_b @ A6 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_1059_subset__Un__eq,axiom,
( ord_le4112832032246704949la_a_b
= ( ^ [A4: set_Re381260168593705685la_a_b,B5: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ A4 @ B5 )
= B5 ) ) ) ).
% subset_Un_eq
thf(fact_1060_bdd__above_OE,axiom,
! [A2: set_nat] :
( ( condit2214826472909112428ve_nat @ A2 )
=> ~ ! [M2: nat] :
~ ! [X6: nat] :
( ( member_nat2 @ X6 @ A2 )
=> ( ord_less_eq_nat @ X6 @ M2 ) ) ) ).
% bdd_above.E
thf(fact_1061_bdd__above_Ounfold,axiom,
( condit2214826472909112428ve_nat
= ( ^ [A4: set_nat] :
? [M3: nat] :
! [X2: nat] :
( ( member_nat2 @ X2 @ A4 )
=> ( ord_less_eq_nat @ X2 @ M3 ) ) ) ) ).
% bdd_above.unfold
thf(fact_1062_totalp__on__le,axiom,
! [A2: set_nat] : ( totalp_on_nat @ A2 @ ord_less_eq_nat ) ).
% totalp_on_le
thf(fact_1063_cSup__subset__mono,axiom,
! [A2: set_nat,B: set_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ A2 ) @ ( complete_Sup_Sup_nat @ B ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_1064_inj__on__Un,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( inj_on4502624401354443441la_a_b @ F @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) )
= ( ( inj_on4502624401354443441la_a_b @ F @ A2 )
& ( inj_on4502624401354443441la_a_b @ F @ B )
& ( ( inf_in8483230781156617063la_a_b @ ( image_6790371041703824709la_a_b @ F @ ( minus_4077726661957047470la_a_b @ A2 @ B ) ) @ ( image_6790371041703824709la_a_b @ F @ ( minus_4077726661957047470la_a_b @ B @ A2 ) ) )
= bot_bo4495933725496725865la_a_b ) ) ) ).
% inj_on_Un
thf(fact_1065_inj__img__insertE,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,X: relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ( inj_on4502624401354443441la_a_b @ F @ A2 )
=> ( ~ ( member4680049679412964150la_a_b @ X @ B )
=> ( ( ( insert7010464514620295119la_a_b @ X @ B )
= ( image_6790371041703824709la_a_b @ F @ A2 ) )
=> ~ ! [X7: relational_fmla_a_b,A6: set_Re381260168593705685la_a_b] :
( ~ ( member4680049679412964150la_a_b @ X7 @ A6 )
=> ( ( A2
= ( insert7010464514620295119la_a_b @ X7 @ A6 ) )
=> ( ( X
= ( F @ X7 ) )
=> ( B
!= ( image_6790371041703824709la_a_b @ F @ A6 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_1066_cSup__inter__less__eq,axiom,
! [A2: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ( condit454741301069713265la_a_b @ A2 )
=> ( ( condit454741301069713265la_a_b @ B )
=> ( ( ( inf_in1353227996657692061la_a_b @ A2 @ B )
!= bot_bo2891247006866115487la_a_b )
=> ( ord_le4112832032246704949la_a_b @ ( comple8442120529048846632la_a_b @ ( inf_in1353227996657692061la_a_b @ A2 @ B ) ) @ ( sup_su5130108678486352897la_a_b @ ( comple8442120529048846632la_a_b @ A2 ) @ ( comple8442120529048846632la_a_b @ B ) ) ) ) ) ) ).
% cSup_inter_less_eq
thf(fact_1067_cSup__inter__less__eq,axiom,
! [A2: set_nat,B: set_nat] :
( ( condit2214826472909112428ve_nat @ A2 )
=> ( ( condit2214826472909112428ve_nat @ B )
=> ( ( ( inf_inf_set_nat @ A2 @ B )
!= bot_bot_set_nat )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( inf_inf_set_nat @ A2 @ B ) ) @ ( sup_sup_nat @ ( complete_Sup_Sup_nat @ A2 ) @ ( complete_Sup_Sup_nat @ B ) ) ) ) ) ) ).
% cSup_inter_less_eq
thf(fact_1068_cSUP__subset__mono,axiom,
! [A2: set_Re381260168593705685la_a_b,G: relational_fmla_a_b > nat,B: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > nat] :
( ( A2 != bot_bo4495933725496725865la_a_b )
=> ( ( condit2214826472909112428ve_nat @ ( image_341122591648980342_b_nat @ G @ B ) )
=> ( ( ord_le4112832032246704949la_a_b @ A2 @ B )
=> ( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_341122591648980342_b_nat @ F @ A2 ) ) @ ( complete_Sup_Sup_nat @ ( image_341122591648980342_b_nat @ G @ B ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_1069_Diff__triv,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ( inf_in8483230781156617063la_a_b @ A2 @ B )
= bot_bo4495933725496725865la_a_b )
=> ( ( minus_4077726661957047470la_a_b @ A2 @ B )
= A2 ) ) ).
% Diff_triv
thf(fact_1070_Int__Diff__disjoint,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ ( inf_in8483230781156617063la_a_b @ A2 @ B ) @ ( minus_4077726661957047470la_a_b @ A2 @ B ) )
= bot_bo4495933725496725865la_a_b ) ).
% Int_Diff_disjoint
thf(fact_1071_Union__disjoint,axiom,
! [C: set_se6865892389300016395la_a_b,A2: set_Re381260168593705685la_a_b] :
( ( ( inf_in8483230781156617063la_a_b @ ( comple8442120529048846632la_a_b @ C ) @ A2 )
= bot_bo4495933725496725865la_a_b )
= ( ! [X2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X2 @ C )
=> ( ( inf_in8483230781156617063la_a_b @ X2 @ A2 )
= bot_bo4495933725496725865la_a_b ) ) ) ) ).
% Union_disjoint
thf(fact_1072_diff__shunt__var,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b] :
( ( ( minus_4077726661957047470la_a_b @ X @ Y )
= bot_bo4495933725496725865la_a_b )
= ( ord_le4112832032246704949la_a_b @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_1073_Diff__Un,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b] :
( ( minus_4077726661957047470la_a_b @ A2 @ ( sup_su5130108678486352897la_a_b @ B @ C ) )
= ( inf_in8483230781156617063la_a_b @ ( minus_4077726661957047470la_a_b @ A2 @ B ) @ ( minus_4077726661957047470la_a_b @ A2 @ C ) ) ) ).
% Diff_Un
thf(fact_1074_Diff__Int,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b] :
( ( minus_4077726661957047470la_a_b @ A2 @ ( inf_in8483230781156617063la_a_b @ B @ C ) )
= ( sup_su5130108678486352897la_a_b @ ( minus_4077726661957047470la_a_b @ A2 @ B ) @ ( minus_4077726661957047470la_a_b @ A2 @ C ) ) ) ).
% Diff_Int
thf(fact_1075_Int__Diff__Un,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ ( inf_in8483230781156617063la_a_b @ A2 @ B ) @ ( minus_4077726661957047470la_a_b @ A2 @ B ) )
= A2 ) ).
% Int_Diff_Un
thf(fact_1076_Un__Diff__Int,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ ( minus_4077726661957047470la_a_b @ A2 @ B ) @ ( inf_in8483230781156617063la_a_b @ A2 @ B ) )
= A2 ) ).
% Un_Diff_Int
thf(fact_1077_cSup__eq__non__empty,axiom,
! [X4: set_nat,A: nat] :
( ( X4 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat2 @ X3 @ X4 )
=> ( ord_less_eq_nat @ X3 @ A ) )
=> ( ! [Y2: nat] :
( ! [X6: nat] :
( ( member_nat2 @ X6 @ X4 )
=> ( ord_less_eq_nat @ X6 @ Y2 ) )
=> ( ord_less_eq_nat @ A @ Y2 ) )
=> ( ( complete_Sup_Sup_nat @ X4 )
= A ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_1078_cSup__least,axiom,
! [X4: set_nat,Z: nat] :
( ( X4 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat2 @ X3 @ X4 )
=> ( ord_less_eq_nat @ X3 @ Z ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ X4 ) @ Z ) ) ) ).
% cSup_least
thf(fact_1079_subset__singletonD,axiom,
! [A2: set_Re381260168593705685la_a_b,X: relational_fmla_a_b] :
( ( ord_le4112832032246704949la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) )
=> ( ( A2 = bot_bo4495933725496725865la_a_b )
| ( A2
= ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% subset_singletonD
thf(fact_1080_subset__singleton__iff,axiom,
! [X4: set_Re381260168593705685la_a_b,A: relational_fmla_a_b] :
( ( ord_le4112832032246704949la_a_b @ X4 @ ( insert7010464514620295119la_a_b @ A @ bot_bo4495933725496725865la_a_b ) )
= ( ( X4 = bot_bo4495933725496725865la_a_b )
| ( X4
= ( insert7010464514620295119la_a_b @ A @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% subset_singleton_iff
thf(fact_1081_set__subset__Cons,axiom,
! [Xs: list_R8263082107343818799la_a_b,X: relational_fmla_a_b] : ( ord_le4112832032246704949la_a_b @ ( set_Re9104216502384355786la_a_b @ Xs ) @ ( set_Re9104216502384355786la_a_b @ ( cons_R4161027894713053801la_a_b @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_1082_subset__Diff__insert,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,X: relational_fmla_a_b,C: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A2 @ ( minus_4077726661957047470la_a_b @ B @ ( insert7010464514620295119la_a_b @ X @ C ) ) )
= ( ( ord_le4112832032246704949la_a_b @ A2 @ ( minus_4077726661957047470la_a_b @ B @ C ) )
& ~ ( member4680049679412964150la_a_b @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_1083_Diff__partition,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A2 @ B )
=> ( ( sup_su5130108678486352897la_a_b @ A2 @ ( minus_4077726661957047470la_a_b @ B @ A2 ) )
= B ) ) ).
% Diff_partition
thf(fact_1084_Diff__subset__conv,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ ( minus_4077726661957047470la_a_b @ A2 @ B ) @ C )
= ( ord_le4112832032246704949la_a_b @ A2 @ ( sup_su5130108678486352897la_a_b @ B @ C ) ) ) ).
% Diff_subset_conv
thf(fact_1085_cSup__upper,axiom,
! [X: nat,X4: set_nat] :
( ( member_nat2 @ X @ X4 )
=> ( ( condit2214826472909112428ve_nat @ X4 )
=> ( ord_less_eq_nat @ X @ ( complete_Sup_Sup_nat @ X4 ) ) ) ) ).
% cSup_upper
thf(fact_1086_cSup__upper2,axiom,
! [X: nat,X4: set_nat,Y: nat] :
( ( member_nat2 @ X @ X4 )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( ( condit2214826472909112428ve_nat @ X4 )
=> ( ord_less_eq_nat @ Y @ ( complete_Sup_Sup_nat @ X4 ) ) ) ) ) ).
% cSup_upper2
thf(fact_1087_subset__code_I2_J,axiom,
! [A2: set_Re381260168593705685la_a_b,Ys: list_R8263082107343818799la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A2 @ ( coset_4993845101609268588la_a_b @ Ys ) )
= ( ! [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ ( set_Re9104216502384355786la_a_b @ Ys ) )
=> ~ ( member4680049679412964150la_a_b @ X2 @ A2 ) ) ) ) ).
% subset_code(2)
thf(fact_1088_cSUP__least,axiom,
! [A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > nat,M: nat] :
( ( A2 != bot_bo4495933725496725865la_a_b )
=> ( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ M ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_341122591648980342_b_nat @ F @ A2 ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_1089_subset__insert__iff,axiom,
! [A2: set_Re381260168593705685la_a_b,X: relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ X @ B ) )
= ( ( ( member4680049679412964150la_a_b @ X @ A2 )
=> ( ord_le4112832032246704949la_a_b @ ( minus_4077726661957047470la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) @ B ) )
& ( ~ ( member4680049679412964150la_a_b @ X @ A2 )
=> ( ord_le4112832032246704949la_a_b @ A2 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1090_Diff__single__insert,axiom,
! [A2: set_Re381260168593705685la_a_b,X: relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ ( minus_4077726661957047470la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) @ B )
=> ( ord_le4112832032246704949la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ X @ B ) ) ) ).
% Diff_single_insert
thf(fact_1091_cSup__le__iff,axiom,
! [S2: set_nat,A: nat] :
( ( S2 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ S2 )
=> ( ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ S2 ) @ A )
= ( ! [X2: nat] :
( ( member_nat2 @ X2 @ S2 )
=> ( ord_less_eq_nat @ X2 @ A ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_1092_cSup__mono,axiom,
! [B: set_nat,A2: set_nat] :
( ( B != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ A2 )
=> ( ! [B6: nat] :
( ( member_nat2 @ B6 @ B )
=> ? [X6: nat] :
( ( member_nat2 @ X6 @ A2 )
& ( ord_less_eq_nat @ B6 @ X6 ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ B ) @ ( complete_Sup_Sup_nat @ A2 ) ) ) ) ) ).
% cSup_mono
thf(fact_1093_subset__code_I3_J,axiom,
~ ( ord_le4112832032246704949la_a_b @ ( coset_4993845101609268588la_a_b @ nil_Re6358386334527539737la_a_b ) @ ( set_Re9104216502384355786la_a_b @ nil_Re6358386334527539737la_a_b ) ) ).
% subset_code(3)
thf(fact_1094_Pow__insert,axiom,
! [A: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( pow_Re5871172025630691010la_a_b @ ( insert7010464514620295119la_a_b @ A @ A2 ) )
= ( sup_su4783144482993978935la_a_b @ ( pow_Re5871172025630691010la_a_b @ A2 ) @ ( image_7051608999182166449la_a_b @ ( insert7010464514620295119la_a_b @ A ) @ ( pow_Re5871172025630691010la_a_b @ A2 ) ) ) ) ).
% Pow_insert
thf(fact_1095_cSUP__le__iff,axiom,
! [A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > nat,U: nat] :
( ( A2 != bot_bo4495933725496725865la_a_b )
=> ( ( condit2214826472909112428ve_nat @ ( image_341122591648980342_b_nat @ F @ A2 ) )
=> ( ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_341122591648980342_b_nat @ F @ A2 ) ) @ U )
= ( ! [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ U ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_1096_insert__partition,axiom,
! [X: set_Re381260168593705685la_a_b,F3: set_se6865892389300016395la_a_b] :
( ~ ( member3481406638322139244la_a_b @ X @ F3 )
=> ( ! [X3: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X3 @ ( insert2023870700798818565la_a_b @ X @ F3 ) )
=> ! [Xa2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ Xa2 @ ( insert2023870700798818565la_a_b @ X @ F3 ) )
=> ( ( X3 != Xa2 )
=> ( ( inf_in8483230781156617063la_a_b @ X3 @ Xa2 )
= bot_bo4495933725496725865la_a_b ) ) ) )
=> ( ( inf_in8483230781156617063la_a_b @ X @ ( comple8442120529048846632la_a_b @ F3 ) )
= bot_bo4495933725496725865la_a_b ) ) ) ).
% insert_partition
thf(fact_1097_Sup__inf__eq__bot__iff,axiom,
! [B: set_se6865892389300016395la_a_b,A: set_Re381260168593705685la_a_b] :
( ( ( inf_in8483230781156617063la_a_b @ ( comple8442120529048846632la_a_b @ B ) @ A )
= bot_bo4495933725496725865la_a_b )
= ( ! [X2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X2 @ B )
=> ( ( inf_in8483230781156617063la_a_b @ X2 @ A )
= bot_bo4495933725496725865la_a_b ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_1098_subset__subseqs,axiom,
! [X4: set_Re381260168593705685la_a_b,Xs: list_R8263082107343818799la_a_b] :
( ( ord_le4112832032246704949la_a_b @ X4 @ ( set_Re9104216502384355786la_a_b @ Xs ) )
=> ( member3481406638322139244la_a_b @ X4 @ ( image_1636186708463624715la_a_b @ set_Re9104216502384355786la_a_b @ ( set_li8988895567068466906la_a_b @ ( subseq4216486041226450954la_a_b @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_1099_subseqs__powset,axiom,
! [Xs: list_R8263082107343818799la_a_b] :
( ( image_1636186708463624715la_a_b @ set_Re9104216502384355786la_a_b @ ( set_li8988895567068466906la_a_b @ ( subseq4216486041226450954la_a_b @ Xs ) ) )
= ( pow_Re5871172025630691010la_a_b @ ( set_Re9104216502384355786la_a_b @ Xs ) ) ) ).
% subseqs_powset
thf(fact_1100_insert__subsetI,axiom,
! [X: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,X4: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ X @ A2 )
=> ( ( ord_le4112832032246704949la_a_b @ X4 @ A2 )
=> ( ord_le4112832032246704949la_a_b @ ( insert7010464514620295119la_a_b @ X @ X4 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_1101_subset__emptyI,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ! [X3: relational_fmla_a_b] :
~ ( member4680049679412964150la_a_b @ X3 @ A2 )
=> ( ord_le4112832032246704949la_a_b @ A2 @ bot_bo4495933725496725865la_a_b ) ) ).
% subset_emptyI
thf(fact_1102_Sup__fin_Oinsert__remove,axiom,
! [A2: set_se6865892389300016395la_a_b,X: set_Re381260168593705685la_a_b] :
( ( finite5238674622262875500la_a_b @ A2 )
=> ( ( ( ( minus_4705846553145473764la_a_b @ A2 @ ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) )
= bot_bo2891247006866115487la_a_b )
=> ( ( lattic7150925611526040158la_a_b @ ( insert2023870700798818565la_a_b @ X @ A2 ) )
= X ) )
& ( ( ( minus_4705846553145473764la_a_b @ A2 @ ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) )
!= bot_bo2891247006866115487la_a_b )
=> ( ( lattic7150925611526040158la_a_b @ ( insert2023870700798818565la_a_b @ X @ A2 ) )
= ( sup_su5130108678486352897la_a_b @ X @ ( lattic7150925611526040158la_a_b @ ( minus_4705846553145473764la_a_b @ A2 @ ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_1103_finite__insert,axiom,
! [A: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ ( insert7010464514620295119la_a_b @ A @ A2 ) )
= ( finite5600759454172676150la_a_b @ A2 ) ) ).
% finite_insert
thf(fact_1104_List_Ofinite__set,axiom,
! [Xs: list_R8263082107343818799la_a_b] : ( finite5600759454172676150la_a_b @ ( set_Re9104216502384355786la_a_b @ Xs ) ) ).
% List.finite_set
thf(fact_1105_finite__Un,axiom,
! [F3: set_Re381260168593705685la_a_b,G3: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ ( sup_su5130108678486352897la_a_b @ F3 @ G3 ) )
= ( ( finite5600759454172676150la_a_b @ F3 )
& ( finite5600759454172676150la_a_b @ G3 ) ) ) ).
% finite_Un
thf(fact_1106_finite__Diff__insert,axiom,
! [A2: set_Re381260168593705685la_a_b,A: relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ ( minus_4077726661957047470la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ A @ B ) ) )
= ( finite5600759454172676150la_a_b @ ( minus_4077726661957047470la_a_b @ A2 @ B ) ) ) ).
% finite_Diff_insert
thf(fact_1107_Sup__fin_Oinsert,axiom,
! [A2: set_se6865892389300016395la_a_b,X: set_Re381260168593705685la_a_b] :
( ( finite5238674622262875500la_a_b @ A2 )
=> ( ( A2 != bot_bo2891247006866115487la_a_b )
=> ( ( lattic7150925611526040158la_a_b @ ( insert2023870700798818565la_a_b @ X @ A2 ) )
= ( sup_su5130108678486352897la_a_b @ X @ ( lattic7150925611526040158la_a_b @ A2 ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_1108_infinite__imp__nonempty,axiom,
! [S2: set_Re381260168593705685la_a_b] :
( ~ ( finite5600759454172676150la_a_b @ S2 )
=> ( S2 != bot_bo4495933725496725865la_a_b ) ) ).
% infinite_imp_nonempty
thf(fact_1109_finite_OemptyI,axiom,
finite5600759454172676150la_a_b @ bot_bo4495933725496725865la_a_b ).
% finite.emptyI
thf(fact_1110_finite_OinsertI,axiom,
! [A2: set_Re381260168593705685la_a_b,A: relational_fmla_a_b] :
( ( finite5600759454172676150la_a_b @ A2 )
=> ( finite5600759454172676150la_a_b @ ( insert7010464514620295119la_a_b @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_1111_infinite__Un,axiom,
! [S2: set_Re381260168593705685la_a_b,T: set_Re381260168593705685la_a_b] :
( ( ~ ( finite5600759454172676150la_a_b @ ( sup_su5130108678486352897la_a_b @ S2 @ T ) ) )
= ( ~ ( finite5600759454172676150la_a_b @ S2 )
| ~ ( finite5600759454172676150la_a_b @ T ) ) ) ).
% infinite_Un
thf(fact_1112_Un__infinite,axiom,
! [S2: set_Re381260168593705685la_a_b,T: set_Re381260168593705685la_a_b] :
( ~ ( finite5600759454172676150la_a_b @ S2 )
=> ~ ( finite5600759454172676150la_a_b @ ( sup_su5130108678486352897la_a_b @ S2 @ T ) ) ) ).
% Un_infinite
thf(fact_1113_finite__UnI,axiom,
! [F3: set_Re381260168593705685la_a_b,G3: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ F3 )
=> ( ( finite5600759454172676150la_a_b @ G3 )
=> ( finite5600759454172676150la_a_b @ ( sup_su5130108678486352897la_a_b @ F3 @ G3 ) ) ) ) ).
% finite_UnI
thf(fact_1114_finite__list,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ A2 )
=> ? [Xs3: list_R8263082107343818799la_a_b] :
( ( set_Re9104216502384355786la_a_b @ Xs3 )
= A2 ) ) ).
% finite_list
thf(fact_1115_finite__has__maximal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat2 @ X3 @ A2 )
& ! [Xa3: nat] :
( ( member_nat2 @ Xa3 @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ Xa3 )
=> ( X3 = Xa3 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1116_finite__has__minimal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat2 @ X3 @ A2 )
& ! [Xa3: nat] :
( ( member_nat2 @ Xa3 @ A2 )
=> ( ( ord_less_eq_nat @ Xa3 @ X3 )
=> ( X3 = Xa3 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1117_le__cSup__finite,axiom,
! [X4: set_nat,X: nat] :
( ( finite_finite_nat @ X4 )
=> ( ( member_nat2 @ X @ X4 )
=> ( ord_less_eq_nat @ X @ ( complete_Sup_Sup_nat @ X4 ) ) ) ) ).
% le_cSup_finite
thf(fact_1118_finite_Ocases,axiom,
! [A: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ A )
=> ( ( A != bot_bo4495933725496725865la_a_b )
=> ~ ! [A7: set_Re381260168593705685la_a_b] :
( ? [A5: relational_fmla_a_b] :
( A
= ( insert7010464514620295119la_a_b @ A5 @ A7 ) )
=> ~ ( finite5600759454172676150la_a_b @ A7 ) ) ) ) ).
% finite.cases
thf(fact_1119_finite_Osimps,axiom,
( finite5600759454172676150la_a_b
= ( ^ [A3: set_Re381260168593705685la_a_b] :
( ( A3 = bot_bo4495933725496725865la_a_b )
| ? [A4: set_Re381260168593705685la_a_b,B4: relational_fmla_a_b] :
( ( A3
= ( insert7010464514620295119la_a_b @ B4 @ A4 ) )
& ( finite5600759454172676150la_a_b @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_1120_finite__induct,axiom,
! [F3: set_Re381260168593705685la_a_b,P: set_Re381260168593705685la_a_b > $o] :
( ( finite5600759454172676150la_a_b @ F3 )
=> ( ( P @ bot_bo4495933725496725865la_a_b )
=> ( ! [X3: relational_fmla_a_b,F4: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ F4 )
=> ( ~ ( member4680049679412964150la_a_b @ X3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert7010464514620295119la_a_b @ X3 @ F4 ) ) ) ) )
=> ( P @ F3 ) ) ) ) ).
% finite_induct
thf(fact_1121_finite__ne__induct,axiom,
! [F3: set_Re381260168593705685la_a_b,P: set_Re381260168593705685la_a_b > $o] :
( ( finite5600759454172676150la_a_b @ F3 )
=> ( ( F3 != bot_bo4495933725496725865la_a_b )
=> ( ! [X3: relational_fmla_a_b] : ( P @ ( insert7010464514620295119la_a_b @ X3 @ bot_bo4495933725496725865la_a_b ) )
=> ( ! [X3: relational_fmla_a_b,F4: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ F4 )
=> ( ( F4 != bot_bo4495933725496725865la_a_b )
=> ( ~ ( member4680049679412964150la_a_b @ X3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert7010464514620295119la_a_b @ X3 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_1122_infinite__finite__induct,axiom,
! [P: set_Re381260168593705685la_a_b > $o,A2: set_Re381260168593705685la_a_b] :
( ! [A7: set_Re381260168593705685la_a_b] :
( ~ ( finite5600759454172676150la_a_b @ A7 )
=> ( P @ A7 ) )
=> ( ( P @ bot_bo4495933725496725865la_a_b )
=> ( ! [X3: relational_fmla_a_b,F4: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ F4 )
=> ( ~ ( member4680049679412964150la_a_b @ X3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert7010464514620295119la_a_b @ X3 @ F4 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_1123_Sup__fin_Oin__idem,axiom,
! [A2: set_se6865892389300016395la_a_b,X: set_Re381260168593705685la_a_b] :
( ( finite5238674622262875500la_a_b @ A2 )
=> ( ( member3481406638322139244la_a_b @ X @ A2 )
=> ( ( sup_su5130108678486352897la_a_b @ X @ ( lattic7150925611526040158la_a_b @ A2 ) )
= ( lattic7150925611526040158la_a_b @ A2 ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_1124_finite__ranking__induct,axiom,
! [S2: set_Re381260168593705685la_a_b,P: set_Re381260168593705685la_a_b > $o,F: relational_fmla_a_b > nat] :
( ( finite5600759454172676150la_a_b @ S2 )
=> ( ( P @ bot_bo4495933725496725865la_a_b )
=> ( ! [X3: relational_fmla_a_b,S5: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ S5 )
=> ( ! [Y6: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Y6 @ S5 )
=> ( ord_less_eq_nat @ ( F @ Y6 ) @ ( F @ X3 ) ) )
=> ( ( P @ S5 )
=> ( P @ ( insert7010464514620295119la_a_b @ X3 @ S5 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_ranking_induct
thf(fact_1125_finite__subset__induct_H,axiom,
! [F3: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b,P: set_Re381260168593705685la_a_b > $o] :
( ( finite5600759454172676150la_a_b @ F3 )
=> ( ( ord_le4112832032246704949la_a_b @ F3 @ A2 )
=> ( ( P @ bot_bo4495933725496725865la_a_b )
=> ( ! [A5: relational_fmla_a_b,F4: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ F4 )
=> ( ( member4680049679412964150la_a_b @ A5 @ A2 )
=> ( ( ord_le4112832032246704949la_a_b @ F4 @ A2 )
=> ( ~ ( member4680049679412964150la_a_b @ A5 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert7010464514620295119la_a_b @ A5 @ F4 ) ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1126_finite__subset__induct,axiom,
! [F3: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b,P: set_Re381260168593705685la_a_b > $o] :
( ( finite5600759454172676150la_a_b @ F3 )
=> ( ( ord_le4112832032246704949la_a_b @ F3 @ A2 )
=> ( ( P @ bot_bo4495933725496725865la_a_b )
=> ( ! [A5: relational_fmla_a_b,F4: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ F4 )
=> ( ( member4680049679412964150la_a_b @ A5 @ A2 )
=> ( ~ ( member4680049679412964150la_a_b @ A5 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert7010464514620295119la_a_b @ A5 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1127_finite__Sup__in,axiom,
! [A2: set_se6865892389300016395la_a_b] :
( ( finite5238674622262875500la_a_b @ A2 )
=> ( ( A2 != bot_bo2891247006866115487la_a_b )
=> ( ! [X3: set_Re381260168593705685la_a_b,Y2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X3 @ A2 )
=> ( ( member3481406638322139244la_a_b @ Y2 @ A2 )
=> ( member3481406638322139244la_a_b @ ( sup_su5130108678486352897la_a_b @ X3 @ Y2 ) @ A2 ) ) )
=> ( member3481406638322139244la_a_b @ ( comple8442120529048846632la_a_b @ A2 ) @ A2 ) ) ) ) ).
% finite_Sup_in
thf(fact_1128_infinite__remove,axiom,
! [S2: set_Re381260168593705685la_a_b,A: relational_fmla_a_b] :
( ~ ( finite5600759454172676150la_a_b @ S2 )
=> ~ ( finite5600759454172676150la_a_b @ ( minus_4077726661957047470la_a_b @ S2 @ ( insert7010464514620295119la_a_b @ A @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% infinite_remove
thf(fact_1129_infinite__coinduct,axiom,
! [X4: set_Re381260168593705685la_a_b > $o,A2: set_Re381260168593705685la_a_b] :
( ( X4 @ A2 )
=> ( ! [A7: set_Re381260168593705685la_a_b] :
( ( X4 @ A7 )
=> ? [X6: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X6 @ A7 )
& ( ( X4 @ ( minus_4077726661957047470la_a_b @ A7 @ ( insert7010464514620295119la_a_b @ X6 @ bot_bo4495933725496725865la_a_b ) ) )
| ~ ( finite5600759454172676150la_a_b @ ( minus_4077726661957047470la_a_b @ A7 @ ( insert7010464514620295119la_a_b @ X6 @ bot_bo4495933725496725865la_a_b ) ) ) ) ) )
=> ~ ( finite5600759454172676150la_a_b @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_1130_finite__empty__induct,axiom,
! [A2: set_Re381260168593705685la_a_b,P: set_Re381260168593705685la_a_b > $o] :
( ( finite5600759454172676150la_a_b @ A2 )
=> ( ( P @ A2 )
=> ( ! [A5: relational_fmla_a_b,A7: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ A7 )
=> ( ( member4680049679412964150la_a_b @ A5 @ A7 )
=> ( ( P @ A7 )
=> ( P @ ( minus_4077726661957047470la_a_b @ A7 @ ( insert7010464514620295119la_a_b @ A5 @ bot_bo4495933725496725865la_a_b ) ) ) ) ) )
=> ( P @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% finite_empty_induct
thf(fact_1131_Sup__fin_Obounded__iff,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A2 ) @ X )
= ( ! [X2: nat] :
( ( member_nat2 @ X2 @ A2 )
=> ( ord_less_eq_nat @ X2 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_1132_Sup__fin_OboundedI,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [A5: nat] :
( ( member_nat2 @ A5 @ A2 )
=> ( ord_less_eq_nat @ A5 @ X ) )
=> ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A2 ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_1133_Sup__fin_OboundedE,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A2 ) @ X )
=> ! [A8: nat] :
( ( member_nat2 @ A8 @ A2 )
=> ( ord_less_eq_nat @ A8 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_1134_finite__remove__induct,axiom,
! [B: set_Re381260168593705685la_a_b,P: set_Re381260168593705685la_a_b > $o] :
( ( finite5600759454172676150la_a_b @ B )
=> ( ( P @ bot_bo4495933725496725865la_a_b )
=> ( ! [A7: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ A7 )
=> ( ( A7 != bot_bo4495933725496725865la_a_b )
=> ( ( ord_le4112832032246704949la_a_b @ A7 @ B )
=> ( ! [X6: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X6 @ A7 )
=> ( P @ ( minus_4077726661957047470la_a_b @ A7 @ ( insert7010464514620295119la_a_b @ X6 @ bot_bo4495933725496725865la_a_b ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_1135_remove__induct,axiom,
! [P: set_Re381260168593705685la_a_b > $o,B: set_Re381260168593705685la_a_b] :
( ( P @ bot_bo4495933725496725865la_a_b )
=> ( ( ~ ( finite5600759454172676150la_a_b @ B )
=> ( P @ B ) )
=> ( ! [A7: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ A7 )
=> ( ( A7 != bot_bo4495933725496725865la_a_b )
=> ( ( ord_le4112832032246704949la_a_b @ A7 @ B )
=> ( ! [X6: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X6 @ A7 )
=> ( P @ ( minus_4077726661957047470la_a_b @ A7 @ ( insert7010464514620295119la_a_b @ X6 @ bot_bo4495933725496725865la_a_b ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_1136_Sup__fin_Osubset__imp,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A2 ) @ ( lattic1093996805478795353in_nat @ B ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_1137_Sup__fin_Ohom__commute,axiom,
! [H: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b,N: set_se6865892389300016395la_a_b] :
( ! [X3: set_Re381260168593705685la_a_b,Y2: set_Re381260168593705685la_a_b] :
( ( H @ ( sup_su5130108678486352897la_a_b @ X3 @ Y2 ) )
= ( sup_su5130108678486352897la_a_b @ ( H @ X3 ) @ ( H @ Y2 ) ) )
=> ( ( finite5238674622262875500la_a_b @ N )
=> ( ( N != bot_bo2891247006866115487la_a_b )
=> ( ( H @ ( lattic7150925611526040158la_a_b @ N ) )
= ( lattic7150925611526040158la_a_b @ ( image_7051608999182166449la_a_b @ H @ N ) ) ) ) ) ) ).
% Sup_fin.hom_commute
thf(fact_1138_Sup__fin_Osubset,axiom,
! [A2: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ( finite5238674622262875500la_a_b @ A2 )
=> ( ( B != bot_bo2891247006866115487la_a_b )
=> ( ( ord_le1577343677690852715la_a_b @ B @ A2 )
=> ( ( sup_su5130108678486352897la_a_b @ ( lattic7150925611526040158la_a_b @ B ) @ ( lattic7150925611526040158la_a_b @ A2 ) )
= ( lattic7150925611526040158la_a_b @ A2 ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_1139_Sup__fin_Oclosed,axiom,
! [A2: set_se6865892389300016395la_a_b] :
( ( finite5238674622262875500la_a_b @ A2 )
=> ( ( A2 != bot_bo2891247006866115487la_a_b )
=> ( ! [X3: set_Re381260168593705685la_a_b,Y2: set_Re381260168593705685la_a_b] : ( member3481406638322139244la_a_b @ ( sup_su5130108678486352897la_a_b @ X3 @ Y2 ) @ ( insert2023870700798818565la_a_b @ X3 @ ( insert2023870700798818565la_a_b @ Y2 @ bot_bo2891247006866115487la_a_b ) ) )
=> ( member3481406638322139244la_a_b @ ( lattic7150925611526040158la_a_b @ A2 ) @ A2 ) ) ) ) ).
% Sup_fin.closed
thf(fact_1140_Sup__fin_Oinsert__not__elem,axiom,
! [A2: set_se6865892389300016395la_a_b,X: set_Re381260168593705685la_a_b] :
( ( finite5238674622262875500la_a_b @ A2 )
=> ( ~ ( member3481406638322139244la_a_b @ X @ A2 )
=> ( ( A2 != bot_bo2891247006866115487la_a_b )
=> ( ( lattic7150925611526040158la_a_b @ ( insert2023870700798818565la_a_b @ X @ A2 ) )
= ( sup_su5130108678486352897la_a_b @ X @ ( lattic7150925611526040158la_a_b @ A2 ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_1141_Sup__fin_Ounion,axiom,
! [A2: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ( finite5238674622262875500la_a_b @ A2 )
=> ( ( A2 != bot_bo2891247006866115487la_a_b )
=> ( ( finite5238674622262875500la_a_b @ B )
=> ( ( B != bot_bo2891247006866115487la_a_b )
=> ( ( lattic7150925611526040158la_a_b @ ( sup_su4783144482993978935la_a_b @ A2 @ B ) )
= ( sup_su5130108678486352897la_a_b @ ( lattic7150925611526040158la_a_b @ A2 ) @ ( lattic7150925611526040158la_a_b @ B ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_1142_Sup__fin_Oremove,axiom,
! [A2: set_se6865892389300016395la_a_b,X: set_Re381260168593705685la_a_b] :
( ( finite5238674622262875500la_a_b @ A2 )
=> ( ( member3481406638322139244la_a_b @ X @ A2 )
=> ( ( ( ( minus_4705846553145473764la_a_b @ A2 @ ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) )
= bot_bo2891247006866115487la_a_b )
=> ( ( lattic7150925611526040158la_a_b @ A2 )
= X ) )
& ( ( ( minus_4705846553145473764la_a_b @ A2 @ ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) )
!= bot_bo2891247006866115487la_a_b )
=> ( ( lattic7150925611526040158la_a_b @ A2 )
= ( sup_su5130108678486352897la_a_b @ X @ ( lattic7150925611526040158la_a_b @ ( minus_4705846553145473764la_a_b @ A2 @ ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_1143_arg__min__least,axiom,
! [S2: set_Re381260168593705685la_a_b,Y: relational_fmla_a_b,F: relational_fmla_a_b > nat] :
( ( finite5600759454172676150la_a_b @ S2 )
=> ( ( S2 != bot_bo4495933725496725865la_a_b )
=> ( ( member4680049679412964150la_a_b @ Y @ S2 )
=> ( ord_less_eq_nat @ ( F @ ( lattic5380700691367270794_b_nat @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_1144_sup__Inf__absorb,axiom,
! [A2: set_se6865892389300016395la_a_b,A: set_Re381260168593705685la_a_b] :
( ( finite5238674622262875500la_a_b @ A2 )
=> ( ( member3481406638322139244la_a_b @ A @ A2 )
=> ( ( sup_su5130108678486352897la_a_b @ ( lattic5196940547480887928la_a_b @ A2 ) @ A )
= A ) ) ) ).
% sup_Inf_absorb
thf(fact_1145_union__fold__insert,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ A2 )
=> ( ( sup_su5130108678486352897la_a_b @ A2 @ B )
= ( finite7097687351672081086la_a_b @ insert7010464514620295119la_a_b @ B @ A2 ) ) ) ).
% union_fold_insert
thf(fact_1146_sup__Sup__fold__sup,axiom,
! [A2: set_se6865892389300016395la_a_b,B: set_Re381260168593705685la_a_b] :
( ( finite5238674622262875500la_a_b @ A2 )
=> ( ( sup_su5130108678486352897la_a_b @ ( comple8442120529048846632la_a_b @ A2 ) @ B )
= ( finite8209461368462410228la_a_b @ sup_su5130108678486352897la_a_b @ B @ A2 ) ) ) ).
% sup_Sup_fold_sup
thf(fact_1147_Inf__fin_OboundedE,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A2 ) )
=> ! [A8: nat] :
( ( member_nat2 @ A8 @ A2 )
=> ( ord_less_eq_nat @ X @ A8 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_1148_Inf__fin_OboundedI,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [A5: nat] :
( ( member_nat2 @ A5 @ A2 )
=> ( ord_less_eq_nat @ X @ A5 ) )
=> ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A2 ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_1149_Inf__fin_Obounded__iff,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A2 ) )
= ( ! [X2: nat] :
( ( member_nat2 @ X2 @ A2 )
=> ( ord_less_eq_nat @ X @ X2 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_1150_Sup__fold__sup,axiom,
! [A2: set_se6865892389300016395la_a_b] :
( ( finite5238674622262875500la_a_b @ A2 )
=> ( ( comple8442120529048846632la_a_b @ A2 )
= ( finite8209461368462410228la_a_b @ sup_su5130108678486352897la_a_b @ bot_bo4495933725496725865la_a_b @ A2 ) ) ) ).
% Sup_fold_sup
thf(fact_1151_Sup__fin_Oeq__fold,axiom,
! [A2: set_se6865892389300016395la_a_b,X: set_Re381260168593705685la_a_b] :
( ( finite5238674622262875500la_a_b @ A2 )
=> ( ( lattic7150925611526040158la_a_b @ ( insert2023870700798818565la_a_b @ X @ A2 ) )
= ( finite8209461368462410228la_a_b @ sup_su5130108678486352897la_a_b @ X @ A2 ) ) ) ).
% Sup_fin.eq_fold
thf(fact_1152_Inf__fin_Osubset__imp,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ B ) @ ( lattic5238388535129920115in_nat @ A2 ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_1153_Inf__fin__le__Sup__fin,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A2 ) @ ( lattic1093996805478795353in_nat @ A2 ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_1154_set__remdups,axiom,
! [Xs: list_R8263082107343818799la_a_b] :
( ( set_Re9104216502384355786la_a_b @ ( remdup8406424999117436228la_a_b @ Xs ) )
= ( set_Re9104216502384355786la_a_b @ Xs ) ) ).
% set_remdups
thf(fact_1155_list_Oset__map,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,V: list_R8263082107343818799la_a_b] :
( ( set_Re9104216502384355786la_a_b @ ( map_Re5057026939262599940la_a_b @ F @ V ) )
= ( image_6790371041703824709la_a_b @ F @ ( set_Re9104216502384355786la_a_b @ V ) ) ) ).
% list.set_map
thf(fact_1156_image__set,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,Xs: list_R8263082107343818799la_a_b] :
( ( image_6790371041703824709la_a_b @ F @ ( set_Re9104216502384355786la_a_b @ Xs ) )
= ( set_Re9104216502384355786la_a_b @ ( map_Re5057026939262599940la_a_b @ F @ Xs ) ) ) ).
% image_set
thf(fact_1157_list_Omap__ident__strong,axiom,
! [T2: list_R8263082107343818799la_a_b,F: relational_fmla_a_b > relational_fmla_a_b] :
( ! [Z3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Z3 @ ( set_Re9104216502384355786la_a_b @ T2 ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( map_Re5057026939262599940la_a_b @ F @ T2 )
= T2 ) ) ).
% list.map_ident_strong
thf(fact_1158_map__idI,axiom,
! [Xs: list_R8263082107343818799la_a_b,F: relational_fmla_a_b > relational_fmla_a_b] :
( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ ( set_Re9104216502384355786la_a_b @ Xs ) )
=> ( ( F @ X3 )
= X3 ) )
=> ( ( map_Re5057026939262599940la_a_b @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_1159_foldr__map,axiom,
! [G: relational_fmla_a_b > relational_fmla_a_b > relational_fmla_a_b,F: relational_fmla_a_b > relational_fmla_a_b,Xs: list_R8263082107343818799la_a_b,A: relational_fmla_a_b] :
( ( foldr_789212930732525799la_a_b @ G @ ( map_Re5057026939262599940la_a_b @ F @ Xs ) @ A )
= ( foldr_789212930732525799la_a_b @ ( comp_R1525839178419564032la_a_b @ G @ F ) @ Xs @ A ) ) ).
% foldr_map
thf(fact_1160_remdups_Osimps_I2_J,axiom,
! [X: relational_fmla_a_b,Xs: list_R8263082107343818799la_a_b] :
( ( ( member4680049679412964150la_a_b @ X @ ( set_Re9104216502384355786la_a_b @ Xs ) )
=> ( ( remdup8406424999117436228la_a_b @ ( cons_R4161027894713053801la_a_b @ X @ Xs ) )
= ( remdup8406424999117436228la_a_b @ Xs ) ) )
& ( ~ ( member4680049679412964150la_a_b @ X @ ( set_Re9104216502384355786la_a_b @ Xs ) )
=> ( ( remdup8406424999117436228la_a_b @ ( cons_R4161027894713053801la_a_b @ X @ Xs ) )
= ( cons_R4161027894713053801la_a_b @ X @ ( remdup8406424999117436228la_a_b @ Xs ) ) ) ) ) ).
% remdups.simps(2)
thf(fact_1161_in__set__remove1,axiom,
! [A: relational_fmla_a_b,B2: relational_fmla_a_b,Xs: list_R8263082107343818799la_a_b] :
( ( A != B2 )
=> ( ( member4680049679412964150la_a_b @ A @ ( set_Re9104216502384355786la_a_b @ ( remove1084781611380451075la_a_b @ B2 @ Xs ) ) )
= ( member4680049679412964150la_a_b @ A @ ( set_Re9104216502384355786la_a_b @ Xs ) ) ) ) ).
% in_set_remove1
thf(fact_1162_Min_Obounded__iff,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic8721135487736765967in_nat @ A2 ) )
= ( ! [X2: nat] :
( ( member_nat2 @ X2 @ A2 )
=> ( ord_less_eq_nat @ X @ X2 ) ) ) ) ) ) ).
% Min.bounded_iff
thf(fact_1163_remove1__idem,axiom,
! [X: relational_fmla_a_b,Xs: list_R8263082107343818799la_a_b] :
( ~ ( member4680049679412964150la_a_b @ X @ ( set_Re9104216502384355786la_a_b @ Xs ) )
=> ( ( remove1084781611380451075la_a_b @ X @ Xs )
= Xs ) ) ).
% remove1_idem
thf(fact_1164_notin__set__remove1,axiom,
! [X: relational_fmla_a_b,Xs: list_R8263082107343818799la_a_b,Y: relational_fmla_a_b] :
( ~ ( member4680049679412964150la_a_b @ X @ ( set_Re9104216502384355786la_a_b @ Xs ) )
=> ~ ( member4680049679412964150la_a_b @ X @ ( set_Re9104216502384355786la_a_b @ ( remove1084781611380451075la_a_b @ Y @ Xs ) ) ) ) ).
% notin_set_remove1
thf(fact_1165_in__set__remove__cases,axiom,
! [P: relational_fmla_a_b > $o,Z: relational_fmla_a_b,Xs: list_R8263082107343818799la_a_b,X: relational_fmla_a_b] :
( ( P @ Z )
=> ( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ ( set_Re9104216502384355786la_a_b @ ( remove1084781611380451075la_a_b @ Z @ Xs ) ) )
=> ( P @ X3 ) )
=> ( ( member4680049679412964150la_a_b @ X @ ( set_Re9104216502384355786la_a_b @ Xs ) )
=> ( P @ X ) ) ) ) ).
% in_set_remove_cases
thf(fact_1166_set__remove1__subset,axiom,
! [X: relational_fmla_a_b,Xs: list_R8263082107343818799la_a_b] : ( ord_le4112832032246704949la_a_b @ ( set_Re9104216502384355786la_a_b @ ( remove1084781611380451075la_a_b @ X @ Xs ) ) @ ( set_Re9104216502384355786la_a_b @ Xs ) ) ).
% set_remove1_subset
thf(fact_1167_Min_OboundedI,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [A5: nat] :
( ( member_nat2 @ A5 @ A2 )
=> ( ord_less_eq_nat @ X @ A5 ) )
=> ( ord_less_eq_nat @ X @ ( lattic8721135487736765967in_nat @ A2 ) ) ) ) ) ).
% Min.boundedI
thf(fact_1168_Min_OboundedE,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic8721135487736765967in_nat @ A2 ) )
=> ! [A8: nat] :
( ( member_nat2 @ A8 @ A2 )
=> ( ord_less_eq_nat @ X @ A8 ) ) ) ) ) ).
% Min.boundedE
thf(fact_1169_eq__Min__iff,axiom,
! [A2: set_nat,M4: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( M4
= ( lattic8721135487736765967in_nat @ A2 ) )
= ( ( member_nat2 @ M4 @ A2 )
& ! [X2: nat] :
( ( member_nat2 @ X2 @ A2 )
=> ( ord_less_eq_nat @ M4 @ X2 ) ) ) ) ) ) ).
% eq_Min_iff
thf(fact_1170_Min__le__iff,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A2 ) @ X )
= ( ? [X2: nat] :
( ( member_nat2 @ X2 @ A2 )
& ( ord_less_eq_nat @ X2 @ X ) ) ) ) ) ) ).
% Min_le_iff
thf(fact_1171_Min__eq__iff,axiom,
! [A2: set_nat,M4: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ( lattic8721135487736765967in_nat @ A2 )
= M4 )
= ( ( member_nat2 @ M4 @ A2 )
& ! [X2: nat] :
( ( member_nat2 @ X2 @ A2 )
=> ( ord_less_eq_nat @ M4 @ X2 ) ) ) ) ) ) ).
% Min_eq_iff
thf(fact_1172_Min__insert2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ! [B6: nat] :
( ( member_nat2 @ B6 @ A2 )
=> ( ord_less_eq_nat @ A @ B6 ) )
=> ( ( lattic8721135487736765967in_nat @ ( insert_nat2 @ A @ A2 ) )
= A ) ) ) ).
% Min_insert2
thf(fact_1173_Min_Osubset__imp,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ B ) @ ( lattic8721135487736765967in_nat @ A2 ) ) ) ) ) ).
% Min.subset_imp
thf(fact_1174_Min__antimono,axiom,
! [M: set_nat,N: set_nat] :
( ( ord_less_eq_set_nat @ M @ N )
=> ( ( M != bot_bot_set_nat )
=> ( ( finite_finite_nat @ N )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ N ) @ ( lattic8721135487736765967in_nat @ M ) ) ) ) ) ).
% Min_antimono
thf(fact_1175_set__remove1__eq,axiom,
! [Xs: list_R8263082107343818799la_a_b,X: relational_fmla_a_b] :
( ( distin3117373409358279142la_a_b @ Xs )
=> ( ( set_Re9104216502384355786la_a_b @ ( remove1084781611380451075la_a_b @ X @ Xs ) )
= ( minus_4077726661957047470la_a_b @ ( set_Re9104216502384355786la_a_b @ Xs ) @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% set_remove1_eq
thf(fact_1176_Min_Oinsert__remove,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic8721135487736765967in_nat @ ( insert_nat2 @ X @ A2 ) )
= X ) )
& ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic8721135487736765967in_nat @ ( insert_nat2 @ X @ A2 ) )
= ( ord_min_nat @ X @ ( lattic8721135487736765967in_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Min.insert_remove
thf(fact_1177_min_Oidem,axiom,
! [A: nat] :
( ( ord_min_nat @ A @ A )
= A ) ).
% min.idem
thf(fact_1178_min_Oleft__idem,axiom,
! [A: nat,B2: nat] :
( ( ord_min_nat @ A @ ( ord_min_nat @ A @ B2 ) )
= ( ord_min_nat @ A @ B2 ) ) ).
% min.left_idem
thf(fact_1179_min_Oright__idem,axiom,
! [A: nat,B2: nat] :
( ( ord_min_nat @ ( ord_min_nat @ A @ B2 ) @ B2 )
= ( ord_min_nat @ A @ B2 ) ) ).
% min.right_idem
thf(fact_1180_min_Obounded__iff,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( ord_min_nat @ B2 @ C2 ) )
= ( ( ord_less_eq_nat @ A @ B2 )
& ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% min.bounded_iff
thf(fact_1181_min_Oabsorb2,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_min_nat @ A @ B2 )
= B2 ) ) ).
% min.absorb2
thf(fact_1182_min_Oabsorb1,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_min_nat @ A @ B2 )
= A ) ) ).
% min.absorb1
thf(fact_1183_min__bot2,axiom,
! [X: set_Re381260168593705685la_a_b] :
( ( ord_mi4211582443342766510la_a_b @ X @ bot_bo4495933725496725865la_a_b )
= bot_bo4495933725496725865la_a_b ) ).
% min_bot2
thf(fact_1184_min__bot2,axiom,
! [X: nat] :
( ( ord_min_nat @ X @ bot_bot_nat )
= bot_bot_nat ) ).
% min_bot2
thf(fact_1185_min__bot,axiom,
! [X: set_Re381260168593705685la_a_b] :
( ( ord_mi4211582443342766510la_a_b @ bot_bo4495933725496725865la_a_b @ X )
= bot_bo4495933725496725865la_a_b ) ).
% min_bot
thf(fact_1186_min__bot,axiom,
! [X: nat] :
( ( ord_min_nat @ bot_bot_nat @ X )
= bot_bot_nat ) ).
% min_bot
thf(fact_1187_Min__insert,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( lattic8721135487736765967in_nat @ ( insert_nat2 @ X @ A2 ) )
= ( ord_min_nat @ X @ ( lattic8721135487736765967in_nat @ A2 ) ) ) ) ) ).
% Min_insert
thf(fact_1188_distinct_Osimps_I2_J,axiom,
! [X: relational_fmla_a_b,Xs: list_R8263082107343818799la_a_b] :
( ( distin3117373409358279142la_a_b @ ( cons_R4161027894713053801la_a_b @ X @ Xs ) )
= ( ~ ( member4680049679412964150la_a_b @ X @ ( set_Re9104216502384355786la_a_b @ Xs ) )
& ( distin3117373409358279142la_a_b @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_1189_finite__distinct__list,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ A2 )
=> ? [Xs3: list_R8263082107343818799la_a_b] :
( ( ( set_Re9104216502384355786la_a_b @ Xs3 )
= A2 )
& ( distin3117373409358279142la_a_b @ Xs3 ) ) ) ).
% finite_distinct_list
thf(fact_1190_min__absorb2,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_min_nat @ X @ Y )
= Y ) ) ).
% min_absorb2
thf(fact_1191_min__absorb1,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_min_nat @ X @ Y )
= X ) ) ).
% min_absorb1
thf(fact_1192_min__def,axiom,
( ord_min_nat
= ( ^ [A3: nat,B4: nat] : ( if_nat @ ( ord_less_eq_nat @ A3 @ B4 ) @ A3 @ B4 ) ) ) ).
% min_def
thf(fact_1193_min__le__iff__disj,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ ( ord_min_nat @ X @ Y ) @ Z )
= ( ( ord_less_eq_nat @ X @ Z )
| ( ord_less_eq_nat @ Y @ Z ) ) ) ).
% min_le_iff_disj
thf(fact_1194_min_OcoboundedI2,axiom,
! [B2: nat,C2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A @ B2 ) @ C2 ) ) ).
% min.coboundedI2
thf(fact_1195_min_OcoboundedI1,axiom,
! [A: nat,C2: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A @ B2 ) @ C2 ) ) ).
% min.coboundedI1
thf(fact_1196_min_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A3: nat] :
( ( ord_min_nat @ A3 @ B4 )
= B4 ) ) ) ).
% min.absorb_iff2
thf(fact_1197_min_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B4: nat] :
( ( ord_min_nat @ A3 @ B4 )
= A3 ) ) ) ).
% min.absorb_iff1
thf(fact_1198_min_Ocobounded2,axiom,
! [A: nat,B2: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ A @ B2 ) @ B2 ) ).
% min.cobounded2
thf(fact_1199_min_Ocobounded1,axiom,
! [A: nat,B2: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ A @ B2 ) @ A ) ).
% min.cobounded1
thf(fact_1200_min_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B4: nat] :
( A3
= ( ord_min_nat @ A3 @ B4 ) ) ) ) ).
% min.order_iff
thf(fact_1201_min_OboundedI,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ A @ C2 )
=> ( ord_less_eq_nat @ A @ ( ord_min_nat @ B2 @ C2 ) ) ) ) ).
% min.boundedI
thf(fact_1202_min_OboundedE,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( ord_min_nat @ B2 @ C2 ) )
=> ~ ( ( ord_less_eq_nat @ A @ B2 )
=> ~ ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% min.boundedE
thf(fact_1203_min_OorderI,axiom,
! [A: nat,B2: nat] :
( ( A
= ( ord_min_nat @ A @ B2 ) )
=> ( ord_less_eq_nat @ A @ B2 ) ) ).
% min.orderI
thf(fact_1204_min_OorderE,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( A
= ( ord_min_nat @ A @ B2 ) ) ) ).
% min.orderE
thf(fact_1205_min_Omono,axiom,
! [A: nat,C2: nat,B2: nat,D: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B2 @ D )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A @ B2 ) @ ( ord_min_nat @ C2 @ D ) ) ) ) ).
% min.mono
thf(fact_1206_inf__min,axiom,
inf_inf_nat = ord_min_nat ).
% inf_min
thf(fact_1207_min_Oassoc,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( ord_min_nat @ ( ord_min_nat @ A @ B2 ) @ C2 )
= ( ord_min_nat @ A @ ( ord_min_nat @ B2 @ C2 ) ) ) ).
% min.assoc
thf(fact_1208_min_Ocommute,axiom,
( ord_min_nat
= ( ^ [A3: nat,B4: nat] : ( ord_min_nat @ B4 @ A3 ) ) ) ).
% min.commute
thf(fact_1209_min_Oleft__commute,axiom,
! [B2: nat,A: nat,C2: nat] :
( ( ord_min_nat @ B2 @ ( ord_min_nat @ A @ C2 ) )
= ( ord_min_nat @ A @ ( ord_min_nat @ B2 @ C2 ) ) ) ).
% min.left_commute
thf(fact_1210_distinct__set__subseqs,axiom,
! [Xs: list_R8263082107343818799la_a_b] :
( ( distin3117373409358279142la_a_b @ Xs )
=> ( distin7072868949307901724la_a_b @ ( map_li5359091830482513994la_a_b @ set_Re9104216502384355786la_a_b @ ( subseq4216486041226450954la_a_b @ Xs ) ) ) ) ).
% distinct_set_subseqs
thf(fact_1211_hom__Min__commute,axiom,
! [H: nat > nat,N: set_nat] :
( ! [X3: nat,Y2: nat] :
( ( H @ ( ord_min_nat @ X3 @ Y2 ) )
= ( ord_min_nat @ ( H @ X3 ) @ ( H @ Y2 ) ) )
=> ( ( finite_finite_nat @ N )
=> ( ( N != bot_bot_set_nat )
=> ( ( H @ ( lattic8721135487736765967in_nat @ N ) )
= ( lattic8721135487736765967in_nat @ ( image_nat_nat @ H @ N ) ) ) ) ) ) ).
% hom_Min_commute
thf(fact_1212_Min_Osubset,axiom,
! [A2: set_nat,B: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( B != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( ord_min_nat @ ( lattic8721135487736765967in_nat @ B ) @ ( lattic8721135487736765967in_nat @ A2 ) )
= ( lattic8721135487736765967in_nat @ A2 ) ) ) ) ) ).
% Min.subset
thf(fact_1213_Min_Oinsert__not__elem,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ~ ( member_nat2 @ X @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( lattic8721135487736765967in_nat @ ( insert_nat2 @ X @ A2 ) )
= ( ord_min_nat @ X @ ( lattic8721135487736765967in_nat @ A2 ) ) ) ) ) ) ).
% Min.insert_not_elem
thf(fact_1214_Min_Oclosed,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [X3: nat,Y2: nat] : ( member_nat2 @ ( ord_min_nat @ X3 @ Y2 ) @ ( insert_nat2 @ X3 @ ( insert_nat2 @ Y2 @ bot_bot_set_nat ) ) )
=> ( member_nat2 @ ( lattic8721135487736765967in_nat @ A2 ) @ A2 ) ) ) ) ).
% Min.closed
thf(fact_1215_Min_Ounion,axiom,
! [A2: set_nat,B: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ( B != bot_bot_set_nat )
=> ( ( lattic8721135487736765967in_nat @ ( sup_sup_set_nat @ A2 @ B ) )
= ( ord_min_nat @ ( lattic8721135487736765967in_nat @ A2 ) @ ( lattic8721135487736765967in_nat @ B ) ) ) ) ) ) ) ).
% Min.union
thf(fact_1216_Min_Oeq__fold,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( lattic8721135487736765967in_nat @ ( insert_nat2 @ X @ A2 ) )
= ( finite_fold_nat_nat @ ord_min_nat @ X @ A2 ) ) ) ).
% Min.eq_fold
thf(fact_1217_Min_Oset__eq__fold,axiom,
! [X: nat,Xs: list_nat] :
( ( lattic8721135487736765967in_nat @ ( set_nat2 @ ( cons_nat @ X @ Xs ) ) )
= ( fold_nat_nat @ ord_min_nat @ Xs @ X ) ) ).
% Min.set_eq_fold
thf(fact_1218_Min_Oremove,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat2 @ X @ A2 )
=> ( ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic8721135487736765967in_nat @ A2 )
= X ) )
& ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic8721135487736765967in_nat @ A2 )
= ( ord_min_nat @ X @ ( lattic8721135487736765967in_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Min.remove
thf(fact_1219_distinct__concat__iff,axiom,
! [Xs: list_l3216422893793986623la_a_b] :
( ( distin3117373409358279142la_a_b @ ( concat5893842828213753630la_a_b @ Xs ) )
= ( ( distin6154453901707860214la_a_b @ ( remove6300469919649359139la_a_b @ nil_Re6358386334527539737la_a_b @ Xs ) )
& ! [Ys2: list_R8263082107343818799la_a_b] :
( ( member8712135081189583430la_a_b @ Ys2 @ ( set_li8988895567068466906la_a_b @ Xs ) )
=> ( distin3117373409358279142la_a_b @ Ys2 ) )
& ! [Ys2: list_R8263082107343818799la_a_b,Zs: list_R8263082107343818799la_a_b] :
( ( ( member8712135081189583430la_a_b @ Ys2 @ ( set_li8988895567068466906la_a_b @ Xs ) )
& ( member8712135081189583430la_a_b @ Zs @ ( set_li8988895567068466906la_a_b @ Xs ) )
& ( Ys2 != Zs ) )
=> ( ( inf_in8483230781156617063la_a_b @ ( set_Re9104216502384355786la_a_b @ Ys2 ) @ ( set_Re9104216502384355786la_a_b @ Zs ) )
= bot_bo4495933725496725865la_a_b ) ) ) ) ).
% distinct_concat_iff
thf(fact_1220_distinct__concat,axiom,
! [Xs: list_l3216422893793986623la_a_b] :
( ( distin6154453901707860214la_a_b @ Xs )
=> ( ! [Ys3: list_R8263082107343818799la_a_b] :
( ( member8712135081189583430la_a_b @ Ys3 @ ( set_li8988895567068466906la_a_b @ Xs ) )
=> ( distin3117373409358279142la_a_b @ Ys3 ) )
=> ( ! [Ys3: list_R8263082107343818799la_a_b,Zs2: list_R8263082107343818799la_a_b] :
( ( member8712135081189583430la_a_b @ Ys3 @ ( set_li8988895567068466906la_a_b @ Xs ) )
=> ( ( member8712135081189583430la_a_b @ Zs2 @ ( set_li8988895567068466906la_a_b @ Xs ) )
=> ( ( Ys3 != Zs2 )
=> ( ( inf_in8483230781156617063la_a_b @ ( set_Re9104216502384355786la_a_b @ Ys3 ) @ ( set_Re9104216502384355786la_a_b @ Zs2 ) )
= bot_bo4495933725496725865la_a_b ) ) ) )
=> ( distin3117373409358279142la_a_b @ ( concat5893842828213753630la_a_b @ Xs ) ) ) ) ) ).
% distinct_concat
thf(fact_1221_map__sorted__distinct__set__unique,axiom,
! [F: relational_fmla_a_b > nat,Xs: list_R8263082107343818799la_a_b,Ys: list_R8263082107343818799la_a_b] :
( ( inj_on4637813073644824290_b_nat @ F @ ( sup_su5130108678486352897la_a_b @ ( set_Re9104216502384355786la_a_b @ Xs ) @ ( set_Re9104216502384355786la_a_b @ Ys ) ) )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_Re8813966276774865461_b_nat @ F @ Xs ) )
=> ( ( distinct_nat @ ( map_Re8813966276774865461_b_nat @ F @ Xs ) )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_Re8813966276774865461_b_nat @ F @ Ys ) )
=> ( ( distinct_nat @ ( map_Re8813966276774865461_b_nat @ F @ Ys ) )
=> ( ( ( set_Re9104216502384355786la_a_b @ Xs )
= ( set_Re9104216502384355786la_a_b @ Ys ) )
=> ( Xs = Ys ) ) ) ) ) ) ) ).
% map_sorted_distinct_set_unique
thf(fact_1222_distinct__append,axiom,
! [Xs: list_R8263082107343818799la_a_b,Ys: list_R8263082107343818799la_a_b] :
( ( distin3117373409358279142la_a_b @ ( append6635100670584670532la_a_b @ Xs @ Ys ) )
= ( ( distin3117373409358279142la_a_b @ Xs )
& ( distin3117373409358279142la_a_b @ Ys )
& ( ( inf_in8483230781156617063la_a_b @ ( set_Re9104216502384355786la_a_b @ Xs ) @ ( set_Re9104216502384355786la_a_b @ Ys ) )
= bot_bo4495933725496725865la_a_b ) ) ) ).
% distinct_append
thf(fact_1223_foldr__append,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b > relational_fmla_a_b,Xs: list_R8263082107343818799la_a_b,Ys: list_R8263082107343818799la_a_b,A: relational_fmla_a_b] :
( ( foldr_789212930732525799la_a_b @ F @ ( append6635100670584670532la_a_b @ Xs @ Ys ) @ A )
= ( foldr_789212930732525799la_a_b @ F @ Xs @ ( foldr_789212930732525799la_a_b @ F @ Ys @ A ) ) ) ).
% foldr_append
thf(fact_1224_set__append,axiom,
! [Xs: list_R8263082107343818799la_a_b,Ys: list_R8263082107343818799la_a_b] :
( ( set_Re9104216502384355786la_a_b @ ( append6635100670584670532la_a_b @ Xs @ Ys ) )
= ( sup_su5130108678486352897la_a_b @ ( set_Re9104216502384355786la_a_b @ Xs ) @ ( set_Re9104216502384355786la_a_b @ Ys ) ) ) ).
% set_append
thf(fact_1225_sorted1,axiom,
! [X: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ X @ nil_nat ) ) ).
% sorted1
thf(fact_1226_sorted__simps_I2_J,axiom,
! [X: nat,Ys: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ X @ Ys ) )
= ( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Ys ) )
=> ( ord_less_eq_nat @ X @ X2 ) )
& ( sorted_wrt_nat @ ord_less_eq_nat @ Ys ) ) ) ).
% sorted_simps(2)
thf(fact_1227_sorted__distinct__set__unique,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( ( distinct_nat @ Xs )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ys )
=> ( ( distinct_nat @ Ys )
=> ( ( ( set_nat2 @ Xs )
= ( set_nat2 @ Ys ) )
=> ( Xs = Ys ) ) ) ) ) ) ).
% sorted_distinct_set_unique
thf(fact_1228_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
! [A2: set_nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( linord2614967742042102400et_nat @ A2 ) ) ).
% sorted_list_of_set.sorted_sorted_key_list_of_set
thf(fact_1229_sorted0,axiom,
sorted_wrt_nat @ ord_less_eq_nat @ nil_nat ).
% sorted0
thf(fact_1230_sorted2,axiom,
! [X: nat,Y: nat,Zs3: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ X @ ( cons_nat @ Y @ Zs3 ) ) )
= ( ( ord_less_eq_nat @ X @ Y )
& ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ Y @ Zs3 ) ) ) ) ).
% sorted2
thf(fact_1231_sorted__wrt__append,axiom,
! [P: relational_fmla_a_b > relational_fmla_a_b > $o,Xs: list_R8263082107343818799la_a_b,Ys: list_R8263082107343818799la_a_b] :
( ( sorted6115879866346780274la_a_b @ P @ ( append6635100670584670532la_a_b @ Xs @ Ys ) )
= ( ( sorted6115879866346780274la_a_b @ P @ Xs )
& ( sorted6115879866346780274la_a_b @ P @ Ys )
& ! [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ ( set_Re9104216502384355786la_a_b @ Xs ) )
=> ! [Y3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Y3 @ ( set_Re9104216502384355786la_a_b @ Ys ) )
=> ( P @ X2 @ Y3 ) ) ) ) ) ).
% sorted_wrt_append
thf(fact_1232_sorted__wrt__mono__rel,axiom,
! [Xs: list_R8263082107343818799la_a_b,P: relational_fmla_a_b > relational_fmla_a_b > $o,Q: relational_fmla_a_b > relational_fmla_a_b > $o] :
( ! [X3: relational_fmla_a_b,Y2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ ( set_Re9104216502384355786la_a_b @ Xs ) )
=> ( ( member4680049679412964150la_a_b @ Y2 @ ( set_Re9104216502384355786la_a_b @ Xs ) )
=> ( ( P @ X3 @ Y2 )
=> ( Q @ X3 @ Y2 ) ) ) )
=> ( ( sorted6115879866346780274la_a_b @ P @ Xs )
=> ( sorted6115879866346780274la_a_b @ Q @ Xs ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_1233_sorted__append,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( append_nat @ Xs @ Ys ) )
= ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
& ( sorted_wrt_nat @ ord_less_eq_nat @ Ys )
& ! [X2: nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ! [Y3: nat] :
( ( member_nat2 @ Y3 @ ( set_nat2 @ Ys ) )
=> ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ) ) ).
% sorted_append
thf(fact_1234_split__list__first__prop__iff,axiom,
! [Xs: list_R8263082107343818799la_a_b,P: relational_fmla_a_b > $o] :
( ( ? [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ ( set_Re9104216502384355786la_a_b @ Xs ) )
& ( P @ X2 ) ) )
= ( ? [Ys2: list_R8263082107343818799la_a_b,X2: relational_fmla_a_b] :
( ? [Zs: list_R8263082107343818799la_a_b] :
( Xs
= ( append6635100670584670532la_a_b @ Ys2 @ ( cons_R4161027894713053801la_a_b @ X2 @ Zs ) ) )
& ( P @ X2 )
& ! [Y3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Y3 @ ( set_Re9104216502384355786la_a_b @ Ys2 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_1235_split__list__last__prop__iff,axiom,
! [Xs: list_R8263082107343818799la_a_b,P: relational_fmla_a_b > $o] :
( ( ? [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ ( set_Re9104216502384355786la_a_b @ Xs ) )
& ( P @ X2 ) ) )
= ( ? [Ys2: list_R8263082107343818799la_a_b,X2: relational_fmla_a_b,Zs: list_R8263082107343818799la_a_b] :
( ( Xs
= ( append6635100670584670532la_a_b @ Ys2 @ ( cons_R4161027894713053801la_a_b @ X2 @ Zs ) ) )
& ( P @ X2 )
& ! [Y3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Y3 @ ( set_Re9104216502384355786la_a_b @ Zs ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_1236_in__set__conv__decomp__first,axiom,
! [X: relational_fmla_a_b,Xs: list_R8263082107343818799la_a_b] :
( ( member4680049679412964150la_a_b @ X @ ( set_Re9104216502384355786la_a_b @ Xs ) )
= ( ? [Ys2: list_R8263082107343818799la_a_b,Zs: list_R8263082107343818799la_a_b] :
( ( Xs
= ( append6635100670584670532la_a_b @ Ys2 @ ( cons_R4161027894713053801la_a_b @ X @ Zs ) ) )
& ~ ( member4680049679412964150la_a_b @ X @ ( set_Re9104216502384355786la_a_b @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_1237_in__set__conv__decomp__last,axiom,
! [X: relational_fmla_a_b,Xs: list_R8263082107343818799la_a_b] :
( ( member4680049679412964150la_a_b @ X @ ( set_Re9104216502384355786la_a_b @ Xs ) )
= ( ? [Ys2: list_R8263082107343818799la_a_b,Zs: list_R8263082107343818799la_a_b] :
( ( Xs
= ( append6635100670584670532la_a_b @ Ys2 @ ( cons_R4161027894713053801la_a_b @ X @ Zs ) ) )
& ~ ( member4680049679412964150la_a_b @ X @ ( set_Re9104216502384355786la_a_b @ Zs ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_1238_split__list__first__propE,axiom,
! [Xs: list_R8263082107343818799la_a_b,P: relational_fmla_a_b > $o] :
( ? [X6: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X6 @ ( set_Re9104216502384355786la_a_b @ Xs ) )
& ( P @ X6 ) )
=> ~ ! [Ys3: list_R8263082107343818799la_a_b,X3: relational_fmla_a_b] :
( ? [Zs2: list_R8263082107343818799la_a_b] :
( Xs
= ( append6635100670584670532la_a_b @ Ys3 @ ( cons_R4161027894713053801la_a_b @ X3 @ Zs2 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Xa3 @ ( set_Re9104216502384355786la_a_b @ Ys3 ) )
=> ~ ( P @ Xa3 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_1239_split__list__last__propE,axiom,
! [Xs: list_R8263082107343818799la_a_b,P: relational_fmla_a_b > $o] :
( ? [X6: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X6 @ ( set_Re9104216502384355786la_a_b @ Xs ) )
& ( P @ X6 ) )
=> ~ ! [Ys3: list_R8263082107343818799la_a_b,X3: relational_fmla_a_b,Zs2: list_R8263082107343818799la_a_b] :
( ( Xs
= ( append6635100670584670532la_a_b @ Ys3 @ ( cons_R4161027894713053801la_a_b @ X3 @ Zs2 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Xa3 @ ( set_Re9104216502384355786la_a_b @ Zs2 ) )
=> ~ ( P @ Xa3 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_1240_split__list__first__prop,axiom,
! [Xs: list_R8263082107343818799la_a_b,P: relational_fmla_a_b > $o] :
( ? [X6: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X6 @ ( set_Re9104216502384355786la_a_b @ Xs ) )
& ( P @ X6 ) )
=> ? [Ys3: list_R8263082107343818799la_a_b,X3: relational_fmla_a_b] :
( ? [Zs2: list_R8263082107343818799la_a_b] :
( Xs
= ( append6635100670584670532la_a_b @ Ys3 @ ( cons_R4161027894713053801la_a_b @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Xa3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Xa3 @ ( set_Re9104216502384355786la_a_b @ Ys3 ) )
=> ~ ( P @ Xa3 ) ) ) ) ).
% split_list_first_prop
thf(fact_1241_split__list__last__prop,axiom,
! [Xs: list_R8263082107343818799la_a_b,P: relational_fmla_a_b > $o] :
( ? [X6: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X6 @ ( set_Re9104216502384355786la_a_b @ Xs ) )
& ( P @ X6 ) )
=> ? [Ys3: list_R8263082107343818799la_a_b,X3: relational_fmla_a_b,Zs2: list_R8263082107343818799la_a_b] :
( ( Xs
= ( append6635100670584670532la_a_b @ Ys3 @ ( cons_R4161027894713053801la_a_b @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Xa3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Xa3 @ ( set_Re9104216502384355786la_a_b @ Zs2 ) )
=> ~ ( P @ Xa3 ) ) ) ) ).
% split_list_last_prop
thf(fact_1242_in__set__conv__decomp,axiom,
! [X: relational_fmla_a_b,Xs: list_R8263082107343818799la_a_b] :
( ( member4680049679412964150la_a_b @ X @ ( set_Re9104216502384355786la_a_b @ Xs ) )
= ( ? [Ys2: list_R8263082107343818799la_a_b,Zs: list_R8263082107343818799la_a_b] :
( Xs
= ( append6635100670584670532la_a_b @ Ys2 @ ( cons_R4161027894713053801la_a_b @ X @ Zs ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_1243_append__Cons__eq__iff,axiom,
! [X: relational_fmla_a_b,Xs: list_R8263082107343818799la_a_b,Ys: list_R8263082107343818799la_a_b,Xs4: list_R8263082107343818799la_a_b,Ys4: list_R8263082107343818799la_a_b] :
( ~ ( member4680049679412964150la_a_b @ X @ ( set_Re9104216502384355786la_a_b @ Xs ) )
=> ( ~ ( member4680049679412964150la_a_b @ X @ ( set_Re9104216502384355786la_a_b @ Ys ) )
=> ( ( ( append6635100670584670532la_a_b @ Xs @ ( cons_R4161027894713053801la_a_b @ X @ Ys ) )
= ( append6635100670584670532la_a_b @ Xs4 @ ( cons_R4161027894713053801la_a_b @ X @ Ys4 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys4 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_1244_split__list__propE,axiom,
! [Xs: list_R8263082107343818799la_a_b,P: relational_fmla_a_b > $o] :
( ? [X6: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X6 @ ( set_Re9104216502384355786la_a_b @ Xs ) )
& ( P @ X6 ) )
=> ~ ! [Ys3: list_R8263082107343818799la_a_b,X3: relational_fmla_a_b] :
( ? [Zs2: list_R8263082107343818799la_a_b] :
( Xs
= ( append6635100670584670532la_a_b @ Ys3 @ ( cons_R4161027894713053801la_a_b @ X3 @ Zs2 ) ) )
=> ~ ( P @ X3 ) ) ) ).
% split_list_propE
thf(fact_1245_split__list__first,axiom,
! [X: relational_fmla_a_b,Xs: list_R8263082107343818799la_a_b] :
( ( member4680049679412964150la_a_b @ X @ ( set_Re9104216502384355786la_a_b @ Xs ) )
=> ? [Ys3: list_R8263082107343818799la_a_b,Zs2: list_R8263082107343818799la_a_b] :
( ( Xs
= ( append6635100670584670532la_a_b @ Ys3 @ ( cons_R4161027894713053801la_a_b @ X @ Zs2 ) ) )
& ~ ( member4680049679412964150la_a_b @ X @ ( set_Re9104216502384355786la_a_b @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_1246_split__list__prop,axiom,
! [Xs: list_R8263082107343818799la_a_b,P: relational_fmla_a_b > $o] :
( ? [X6: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X6 @ ( set_Re9104216502384355786la_a_b @ Xs ) )
& ( P @ X6 ) )
=> ? [Ys3: list_R8263082107343818799la_a_b,X3: relational_fmla_a_b] :
( ? [Zs2: list_R8263082107343818799la_a_b] :
( Xs
= ( append6635100670584670532la_a_b @ Ys3 @ ( cons_R4161027894713053801la_a_b @ X3 @ Zs2 ) ) )
& ( P @ X3 ) ) ) ).
% split_list_prop
thf(fact_1247_split__list__last,axiom,
! [X: relational_fmla_a_b,Xs: list_R8263082107343818799la_a_b] :
( ( member4680049679412964150la_a_b @ X @ ( set_Re9104216502384355786la_a_b @ Xs ) )
=> ? [Ys3: list_R8263082107343818799la_a_b,Zs2: list_R8263082107343818799la_a_b] :
( ( Xs
= ( append6635100670584670532la_a_b @ Ys3 @ ( cons_R4161027894713053801la_a_b @ X @ Zs2 ) ) )
& ~ ( member4680049679412964150la_a_b @ X @ ( set_Re9104216502384355786la_a_b @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_1248_split__list,axiom,
! [X: relational_fmla_a_b,Xs: list_R8263082107343818799la_a_b] :
( ( member4680049679412964150la_a_b @ X @ ( set_Re9104216502384355786la_a_b @ Xs ) )
=> ? [Ys3: list_R8263082107343818799la_a_b,Zs2: list_R8263082107343818799la_a_b] :
( Xs
= ( append6635100670584670532la_a_b @ Ys3 @ ( cons_R4161027894713053801la_a_b @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_1249_sorted__remdups,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( remdups_nat @ Xs ) ) ) ).
% sorted_remdups
thf(fact_1250_remove1__append,axiom,
! [X: relational_fmla_a_b,Xs: list_R8263082107343818799la_a_b,Ys: list_R8263082107343818799la_a_b] :
( ( ( member4680049679412964150la_a_b @ X @ ( set_Re9104216502384355786la_a_b @ Xs ) )
=> ( ( remove1084781611380451075la_a_b @ X @ ( append6635100670584670532la_a_b @ Xs @ Ys ) )
= ( append6635100670584670532la_a_b @ ( remove1084781611380451075la_a_b @ X @ Xs ) @ Ys ) ) )
& ( ~ ( member4680049679412964150la_a_b @ X @ ( set_Re9104216502384355786la_a_b @ Xs ) )
=> ( ( remove1084781611380451075la_a_b @ X @ ( append6635100670584670532la_a_b @ Xs @ Ys ) )
= ( append6635100670584670532la_a_b @ Xs @ ( remove1084781611380451075la_a_b @ X @ Ys ) ) ) ) ) ).
% remove1_append
thf(fact_1251_sorted__remove1,axiom,
! [Xs: list_nat,A: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( remove1_nat @ A @ Xs ) ) ) ).
% sorted_remove1
thf(fact_1252_not__distinct__conv__prefix,axiom,
! [As: list_R8263082107343818799la_a_b] :
( ( ~ ( distin3117373409358279142la_a_b @ As ) )
= ( ? [Xs2: list_R8263082107343818799la_a_b,Y3: relational_fmla_a_b,Ys2: list_R8263082107343818799la_a_b] :
( ( member4680049679412964150la_a_b @ Y3 @ ( set_Re9104216502384355786la_a_b @ Xs2 ) )
& ( distin3117373409358279142la_a_b @ Xs2 )
& ( As
= ( append6635100670584670532la_a_b @ Xs2 @ ( cons_R4161027894713053801la_a_b @ Y3 @ Ys2 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_1253_remove1__split,axiom,
! [A: relational_fmla_a_b,Xs: list_R8263082107343818799la_a_b,Ys: list_R8263082107343818799la_a_b] :
( ( member4680049679412964150la_a_b @ A @ ( set_Re9104216502384355786la_a_b @ Xs ) )
=> ( ( ( remove1084781611380451075la_a_b @ A @ Xs )
= Ys )
= ( ? [Ls: list_R8263082107343818799la_a_b,Rs: list_R8263082107343818799la_a_b] :
( ( Xs
= ( append6635100670584670532la_a_b @ Ls @ ( cons_R4161027894713053801la_a_b @ A @ Rs ) ) )
& ~ ( member4680049679412964150la_a_b @ A @ ( set_Re9104216502384355786la_a_b @ Ls ) )
& ( Ys
= ( append6635100670584670532la_a_b @ Ls @ Rs ) ) ) ) ) ) ).
% remove1_split
thf(fact_1254_finite__sorted__distinct__unique,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ? [X3: list_nat] :
( ( ( set_nat2 @ X3 )
= A2 )
& ( sorted_wrt_nat @ ord_less_eq_nat @ X3 )
& ( distinct_nat @ X3 )
& ! [Y6: list_nat] :
( ( ( ( set_nat2 @ Y6 )
= A2 )
& ( sorted_wrt_nat @ ord_less_eq_nat @ Y6 )
& ( distinct_nat @ Y6 ) )
=> ( Y6 = X3 ) ) ) ) ).
% finite_sorted_distinct_unique
thf(fact_1255_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( ( distinct_nat @ Xs )
=> ( ( linord2614967742042102400et_nat @ ( set_nat2 @ Xs ) )
= Xs ) ) ) ).
% sorted_list_of_set.idem_if_sorted_distinct
thf(fact_1256_in__set__butlast__appendI,axiom,
! [X: relational_fmla_a_b,Xs: list_R8263082107343818799la_a_b,Ys: list_R8263082107343818799la_a_b] :
( ( ( member4680049679412964150la_a_b @ X @ ( set_Re9104216502384355786la_a_b @ ( butlas3860009559628690893la_a_b @ Xs ) ) )
| ( member4680049679412964150la_a_b @ X @ ( set_Re9104216502384355786la_a_b @ ( butlas3860009559628690893la_a_b @ Ys ) ) ) )
=> ( member4680049679412964150la_a_b @ X @ ( set_Re9104216502384355786la_a_b @ ( butlas3860009559628690893la_a_b @ ( append6635100670584670532la_a_b @ Xs @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_1257_in__set__butlastD,axiom,
! [X: relational_fmla_a_b,Xs: list_R8263082107343818799la_a_b] :
( ( member4680049679412964150la_a_b @ X @ ( set_Re9104216502384355786la_a_b @ ( butlas3860009559628690893la_a_b @ Xs ) ) )
=> ( member4680049679412964150la_a_b @ X @ ( set_Re9104216502384355786la_a_b @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_1258_sorted__butlast,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( butlast_nat @ Xs ) ) ) ) ).
% sorted_butlast
thf(fact_1259_last__in__set,axiom,
! [As: list_R8263082107343818799la_a_b] :
( ( As != nil_Re6358386334527539737la_a_b )
=> ( member4680049679412964150la_a_b @ ( last_R1227609581548886336la_a_b @ As ) @ ( set_Re9104216502384355786la_a_b @ As ) ) ) ).
% last_in_set
thf(fact_1260_vimage__empty,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b] :
( ( vimage568529607348484603la_a_b @ F @ bot_bo4495933725496725865la_a_b )
= bot_bo4495933725496725865la_a_b ) ).
% vimage_empty
thf(fact_1261_vimage__Un,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( vimage568529607348484603la_a_b @ F @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) )
= ( sup_su5130108678486352897la_a_b @ ( vimage568529607348484603la_a_b @ F @ A2 ) @ ( vimage568529607348484603la_a_b @ F @ B ) ) ) ).
% vimage_Un
thf(fact_1262_vimage__insert,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,A: relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ( vimage568529607348484603la_a_b @ F @ ( insert7010464514620295119la_a_b @ A @ B ) )
= ( sup_su5130108678486352897la_a_b @ ( vimage568529607348484603la_a_b @ F @ ( insert7010464514620295119la_a_b @ A @ bot_bo4495933725496725865la_a_b ) ) @ ( vimage568529607348484603la_a_b @ F @ B ) ) ) ).
% vimage_insert
thf(fact_1263_diff__diff__cancel,axiom,
! [I2: nat,N2: nat] :
( ( ord_less_eq_nat @ I2 @ N2 )
=> ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I2 ) )
= I2 ) ) ).
% diff_diff_cancel
thf(fact_1264_distinct__disjoint__shuffles,axiom,
! [Xs: list_R8263082107343818799la_a_b,Ys: list_R8263082107343818799la_a_b,Zs3: list_R8263082107343818799la_a_b] :
( ( distin3117373409358279142la_a_b @ Xs )
=> ( ( distin3117373409358279142la_a_b @ Ys )
=> ( ( ( inf_in8483230781156617063la_a_b @ ( set_Re9104216502384355786la_a_b @ Xs ) @ ( set_Re9104216502384355786la_a_b @ Ys ) )
= bot_bo4495933725496725865la_a_b )
=> ( ( member8712135081189583430la_a_b @ Zs3 @ ( shuffl3023960117626955588la_a_b @ Xs @ Ys ) )
=> ( distin3117373409358279142la_a_b @ Zs3 ) ) ) ) ) ).
% distinct_disjoint_shuffles
thf(fact_1265_set__shuffles,axiom,
! [Zs3: list_R8263082107343818799la_a_b,Xs: list_R8263082107343818799la_a_b,Ys: list_R8263082107343818799la_a_b] :
( ( member8712135081189583430la_a_b @ Zs3 @ ( shuffl3023960117626955588la_a_b @ Xs @ Ys ) )
=> ( ( set_Re9104216502384355786la_a_b @ Zs3 )
= ( sup_su5130108678486352897la_a_b @ ( set_Re9104216502384355786la_a_b @ Xs ) @ ( set_Re9104216502384355786la_a_b @ Ys ) ) ) ) ).
% set_shuffles
thf(fact_1266_diff__commute,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J2 ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K ) @ J2 ) ) ).
% diff_commute
thf(fact_1267_min__diff,axiom,
! [M4: nat,I2: nat,N2: nat] :
( ( ord_min_nat @ ( minus_minus_nat @ M4 @ I2 ) @ ( minus_minus_nat @ N2 @ I2 ) )
= ( minus_minus_nat @ ( ord_min_nat @ M4 @ N2 ) @ I2 ) ) ).
% min_diff
thf(fact_1268_diff__le__mono2,axiom,
! [M4: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M4 ) ) ) ).
% diff_le_mono2
thf(fact_1269_le__diff__iff_H,axiom,
! [A: nat,C2: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A ) @ ( minus_minus_nat @ C2 @ B2 ) )
= ( ord_less_eq_nat @ B2 @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1270_diff__le__self,axiom,
! [M4: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M4 @ N2 ) @ M4 ) ).
% diff_le_self
thf(fact_1271_diff__le__mono,axiom,
! [M4: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M4 @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).
% diff_le_mono
thf(fact_1272_Nat_Odiff__diff__eq,axiom,
! [K: nat,M4: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M4 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M4 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( minus_minus_nat @ M4 @ N2 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1273_le__diff__iff,axiom,
! [K: nat,M4: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M4 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M4 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_eq_nat @ M4 @ N2 ) ) ) ) ).
% le_diff_iff
thf(fact_1274_eq__diff__iff,axiom,
! [K: nat,M4: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M4 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ( minus_minus_nat @ M4 @ K )
= ( minus_minus_nat @ N2 @ K ) )
= ( M4 = N2 ) ) ) ) ).
% eq_diff_iff
thf(fact_1275_min__minus,axiom,
! [M4: nat,K: nat] :
( ( ord_min_nat @ M4 @ ( minus_minus_nat @ M4 @ K ) )
= ( minus_minus_nat @ M4 @ K ) ) ).
% min_minus
thf(fact_1276_min__minus_H,axiom,
! [M4: nat,K: nat] :
( ( ord_min_nat @ ( minus_minus_nat @ M4 @ K ) @ M4 )
= ( minus_minus_nat @ M4 @ K ) ) ).
% min_minus'
thf(fact_1277_lists__empty,axiom,
( ( lists_2939013368609773739la_a_b @ bot_bo4495933725496725865la_a_b )
= ( insert3617074100887873119la_a_b @ nil_Re6358386334527539737la_a_b @ bot_bo3230846185698255993la_a_b ) ) ).
% lists_empty
thf(fact_1278_in__listsI,axiom,
! [Xs: list_R8263082107343818799la_a_b,A2: set_Re381260168593705685la_a_b] :
( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ ( set_Re9104216502384355786la_a_b @ Xs ) )
=> ( member4680049679412964150la_a_b @ X3 @ A2 ) )
=> ( member8712135081189583430la_a_b @ Xs @ ( lists_2939013368609773739la_a_b @ A2 ) ) ) ).
% in_listsI
% Helper facts (13)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y: list_nat] :
( ( if_list_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y: list_nat] :
( ( if_list_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_T,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b] :
( ( if_set2835548578466827919la_a_b @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_T,axiom,
! [X: set_Re381260168593705685la_a_b,Y: set_Re381260168593705685la_a_b] :
( ( if_set2835548578466827919la_a_b @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_T,axiom,
! [X: list_R8263082107343818799la_a_b,Y: list_R8263082107343818799la_a_b] :
( ( if_lis1478040310087653097la_a_b @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_T,axiom,
! [X: list_R8263082107343818799la_a_b,Y: list_R8263082107343818799la_a_b] :
( ( if_lis1478040310087653097la_a_b @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_J_T,axiom,
! [X: list_s3671366213839843941la_a_b,Y: list_s3671366213839843941la_a_b] :
( ( if_lis5716551597998330143la_a_b @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_J_T,axiom,
! [X: list_s3671366213839843941la_a_b,Y: list_s3671366213839843941la_a_b] :
( ( if_lis5716551597998330143la_a_b @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__List__Olist_It__List__Olist_It__Relational____Calculus__Ofmla_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__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_J_T,axiom,
! [X: list_l3216422893793986623la_a_b,Y: list_l3216422893793986623la_a_b] :
( ( if_lis5916486426012477945la_a_b @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__List__Olist_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_J_T,axiom,
! [X: list_l3216422893793986623la_a_b,Y: list_l3216422893793986623la_a_b] :
( ( if_lis5916486426012477945la_a_b @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( relational_sat_a_b @ ( foldr_789212930732525799la_a_b @ relational_Disj_a_b @ xs @ q ) @ i @ sigma )
!= ( ~ ? [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ ( sup_su5130108678486352897la_a_b @ ( set_Re9104216502384355786la_a_b @ xs ) @ ( insert7010464514620295119la_a_b @ q @ bot_bo4495933725496725865la_a_b ) ) )
& ( relational_sat_a_b @ X2 @ i @ sigma ) ) ) ) ).
%------------------------------------------------------------------------------