TPTP Problem File: SLH0567^1.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/0024_Restrict_Frees/prob_00116_004892__17795538_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1494 ( 693 unt; 211 typ; 0 def)
% Number of atoms : 3281 (1528 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 10292 ( 363 ~; 73 |; 325 &;8508 @)
% ( 0 <=>;1023 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 6 avg)
% Number of types : 23 ( 22 usr)
% Number of type conns : 772 ( 772 >; 0 *; 0 +; 0 <<)
% Number of symbols : 192 ( 189 usr; 25 con; 0-4 aty)
% Number of variables : 3480 ( 285 ^;3112 !; 83 ?;3480 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:27:14.594
%------------------------------------------------------------------------------
% Could-be-implicit typings (22)
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% Explicit typings (189)
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thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
image_2298502056602587059at_nat: ( product_prod_nat_nat > produc5825016348098550007at_nat ) > set_Pr1261947904930325089at_nat > set_Pr2645174627780777389at_nat ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
image_5019456213502855771la_a_b: ( product_prod_nat_nat > relational_fmla_a_b ) > set_Pr1261947904930325089at_nat > set_Re381260168593705685la_a_b ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_001t__Nat__Onat,type,
image_8316665354072716238at_nat: ( produc5825016348098550007at_nat > nat ) > set_Pr2645174627780777389at_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_2384848872172141443at_nat: ( produc5825016348098550007at_nat > product_prod_nat_nat ) > set_Pr2645174627780777389at_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_001t__Product____Type__Oprod_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
image_6979785008349797877at_nat: ( produc5825016348098550007at_nat > produc5825016348098550007at_nat ) > set_Pr2645174627780777389at_nat > set_Pr2645174627780777389at_nat ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
image_7406987224865000349la_a_b: ( produc5825016348098550007at_nat > relational_fmla_a_b ) > set_Pr2645174627780777389at_nat > set_Re381260168593705685la_a_b ).
thf(sy_c_Set_Oimage_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_001t__List__Olist_Itf__a_J,type,
image_4103308382209252958list_a: ( relational_fmla_a_b > list_a ) > set_Re381260168593705685la_a_b > set_list_a ).
thf(sy_c_Set_Oimage_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_001t__Nat__Onat,type,
image_341122591648980342_b_nat: ( relational_fmla_a_b > nat ) > set_Re381260168593705685la_a_b > set_nat ).
thf(sy_c_Set_Oimage_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_5852719071363227611at_nat: ( relational_fmla_a_b > product_prod_nat_nat ) > set_Re381260168593705685la_a_b > set_Pr1261947904930325089at_nat ).
thf(sy_c_Set_Oimage_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_001t__Product____Type__Oprod_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
image_5430539854921360285at_nat: ( relational_fmla_a_b > produc5825016348098550007at_nat ) > set_Re381260168593705685la_a_b > set_Pr2645174627780777389at_nat ).
thf(sy_c_Set_Oimage_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
image_6790371041703824709la_a_b: ( relational_fmla_a_b > relational_fmla_a_b ) > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b ).
thf(sy_c_Set_Oimage_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
image_130269618930851390list_a: ( relational_fmla_a_b > set_list_a ) > set_Re381260168593705685la_a_b > set_set_list_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
image_5749939591322298757list_a: ( set_list_a > set_list_a ) > set_set_list_a > set_set_list_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7916887816326733075et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
image_3684629450409544005at_nat: ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ) > set_se7855581050983116737at_nat > set_se7855581050983116737at_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Product____Type__Oprod_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
image_3124536445484363873at_nat: ( set_Pr2645174627780777389at_nat > set_Pr2645174627780777389at_nat ) > set_se7774124317125585763at_nat > set_se7774124317125585763at_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
image_7051608999182166449la_a_b: ( set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b ) > set_se6865892389300016395la_a_b > set_se6865892389300016395la_a_b ).
thf(sy_c_Set_Oinsert_001t__List__Olist_Itf__a_J,type,
insert_list_a: list_a > set_list_a > set_list_a ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
insert8211810215607154385at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
insert3260075854425521959at_nat: produc5825016348098550007at_nat > set_Pr2645174627780777389at_nat > set_Pr2645174627780777389at_nat ).
thf(sy_c_Set_Oinsert_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
insert7010464514620295119la_a_b: relational_fmla_a_b > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
insert_set_list_a: set_list_a > set_set_list_a > set_set_list_a ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001t__Multiset__Omultiset_It__Set__Oset_It__Nat__Onat_J_J,type,
member9214616765234488813et_nat: multiset_set_nat > set_multiset_set_nat > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
member6956540099943067662at_nat: produc5825016348098550007at_nat > set_Pr2645174627780777389at_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_Itf__a_J_J,type,
member_set_list_a: set_list_a > set_set_list_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member2643936169264416010at_nat: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
member734533627512885444at_nat: set_Pr2645174627780777389at_nat > set_se7774124317125585763at_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_v_F,type,
f: set_nat ).
thf(sy_v_G,type,
g: set_Re381260168593705685la_a_b ).
thf(sy_v_Qeq,type,
qeq: set_Pr1261947904930325089at_nat ).
thf(sy_v_Qfin,type,
qfin: relational_fmla_a_b ).
thf(sy_v_X____,type,
x: set_Pr2645174627780777389at_nat ).
thf(sy_v_Y____,type,
y: set_Pr2645174627780777389at_nat ).
thf(sy_v__092_060Q_062fin,type,
q_fin: set_Pr2645174627780777389at_nat ).
thf(sy_v_simp,type,
simp: relational_fmla_a_b > relational_fmla_a_b ).
thf(sy_v_simplified,type,
simplified: relational_fmla_a_b > $o ).
thf(sy_v_x,type,
x2: nat ).
% Relevant facts (1275)
thf(fact_0_assms_I2_J,axiom,
finite6790245451575510286at_nat @ q_fin ).
% assms(2)
thf(fact_1_XY_I4_J,axiom,
finite6790245451575510286at_nat @ x ).
% XY(4)
thf(fact_2_XY_I5_J,axiom,
finite6790245451575510286at_nat @ y ).
% XY(5)
thf(fact_3_XY_I2_J,axiom,
ord_le8520675249591772685at_nat @ x @ q_fin ).
% XY(2)
thf(fact_4_assms_I3_J,axiom,
member_nat @ x2 @ ( relati62690040636126068ns_a_b @ qfin ) ).
% assms(3)
thf(fact_5_assms_I1_J,axiom,
member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ qfin @ qeq ) @ q_fin ).
% assms(1)
thf(fact_6_XY_I3_J,axiom,
( ( inf_in8776938414804536127at_nat @ ( minus_6698950876951835142at_nat @ q_fin @ x ) @ y )
= bot_bo1973934853101755969at_nat ) ).
% XY(3)
thf(fact_7_X__def,axiom,
( x
= ( insert3260075854425521959at_nat @ ( produc760948067230524457at_nat @ qfin @ qeq ) @ bot_bo1973934853101755969at_nat ) ) ).
% X_def
thf(fact_8_XY_I1_J,axiom,
( ( insert3260075854425521959at_nat @ ( produc760948067230524457at_nat @ ( simp @ ( relational_Conj_a_b @ qfin @ ( relational_DISJ_a_b @ ( relational_qps_a_b @ g ) ) ) ) @ qeq )
@ ( sup_su6099978769595272409at_nat @ ( minus_6698950876951835142at_nat @ q_fin @ ( insert3260075854425521959at_nat @ ( produc760948067230524457at_nat @ qfin @ qeq ) @ bot_bo1973934853101755969at_nat ) )
@ ( image_1371346403869992270at_nat
@ ^ [Y: nat] : ( produc760948067230524457at_nat @ ( relational_cp_a_b @ ( relational_subst_a_b @ qfin @ x2 @ Y ) ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ x2 @ Y ) @ qeq ) )
@ ( relational_eqs_a_b @ x2 @ g ) ) ) )
= ( sup_su6099978769595272409at_nat @ ( minus_6698950876951835142at_nat @ q_fin @ x ) @ y ) ) ).
% XY(1)
thf(fact_9_Y__def,axiom,
( y
= ( minus_6698950876951835142at_nat
@ ( insert3260075854425521959at_nat @ ( produc760948067230524457at_nat @ ( simp @ ( relational_Conj_a_b @ qfin @ ( relational_DISJ_a_b @ ( relational_qps_a_b @ g ) ) ) ) @ qeq )
@ ( image_1371346403869992270at_nat
@ ^ [Y: nat] : ( produc760948067230524457at_nat @ ( relational_cp_a_b @ ( relational_subst_a_b @ qfin @ x2 @ Y ) ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ x2 @ Y ) @ qeq ) )
@ ( relational_eqs_a_b @ x2 @ g ) ) )
@ ( inf_in8776938414804536127at_nat @ ( minus_6698950876951835142at_nat @ q_fin @ ( insert3260075854425521959at_nat @ ( produc760948067230524457at_nat @ qfin @ qeq ) @ bot_bo1973934853101755969at_nat ) )
@ ( insert3260075854425521959at_nat @ ( produc760948067230524457at_nat @ ( simp @ ( relational_Conj_a_b @ qfin @ ( relational_DISJ_a_b @ ( relational_qps_a_b @ g ) ) ) ) @ qeq )
@ ( image_1371346403869992270at_nat
@ ^ [Y: nat] : ( produc760948067230524457at_nat @ ( relational_cp_a_b @ ( relational_subst_a_b @ qfin @ x2 @ Y ) ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ x2 @ Y ) @ qeq ) )
@ ( relational_eqs_a_b @ x2 @ g ) ) ) ) ) ) ).
% Y_def
thf(fact_10_assms_I4_J,axiom,
relational_cov_a_b @ x2 @ qfin @ g ).
% assms(4)
thf(fact_11_calculation_I1_J,axiom,
( ( image_7702178775022393786et_nat @ ( comp_R1256417402204451841at_nat @ relati62690040636126068ns_a_b @ produc8992968103413169469at_nat ) @ ( mset_s3030574779765502096at_nat @ x ) )
!= zero_z3157962936165190495et_nat ) ).
% calculation(1)
thf(fact_12_calculation_I2_J,axiom,
subset6078030600694693471et_nat @ ( image_7702178775022393786et_nat @ ( comp_R1256417402204451841at_nat @ relati62690040636126068ns_a_b @ produc8992968103413169469at_nat ) @ ( mset_s3030574779765502096at_nat @ x ) ) @ ( image_7702178775022393786et_nat @ ( comp_R1256417402204451841at_nat @ relati62690040636126068ns_a_b @ produc8992968103413169469at_nat ) @ ( mset_s3030574779765502096at_nat @ q_fin ) ) ).
% calculation(2)
thf(fact_13_insert__Diff__single,axiom,
! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
( ( insert8211810215607154385at_nat @ A @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
= ( insert8211810215607154385at_nat @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_14_insert__Diff__single,axiom,
! [A: list_a,A2: set_list_a] :
( ( insert_list_a @ A @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) )
= ( insert_list_a @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_15_insert__Diff__single,axiom,
! [A: nat,A2: set_nat] :
( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( insert_nat @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_16_insert__Diff__single,axiom,
! [A: produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat] :
( ( insert3260075854425521959at_nat @ A @ ( minus_6698950876951835142at_nat @ A2 @ ( insert3260075854425521959at_nat @ A @ bot_bo1973934853101755969at_nat ) ) )
= ( insert3260075854425521959at_nat @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_17_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_18_singleton__conv,axiom,
! [A: product_prod_nat_nat] :
( ( collec3392354462482085612at_nat
@ ^ [X: product_prod_nat_nat] : ( X = A ) )
= ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ).
% singleton_conv
thf(fact_19_singleton__conv,axiom,
! [A: produc5825016348098550007at_nat] :
( ( collec7697611494764367948at_nat
@ ^ [X: produc5825016348098550007at_nat] : ( X = A ) )
= ( insert3260075854425521959at_nat @ A @ bot_bo1973934853101755969at_nat ) ) ).
% singleton_conv
thf(fact_20_singleton__conv,axiom,
! [A: relational_fmla_a_b] :
( ( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] : ( X = A ) )
= ( insert7010464514620295119la_a_b @ A @ bot_bo4495933725496725865la_a_b ) ) ).
% singleton_conv
thf(fact_21_singleton__conv,axiom,
! [A: list_a] :
( ( collect_list_a
@ ^ [X: list_a] : ( X = A ) )
= ( insert_list_a @ A @ bot_bot_set_list_a ) ) ).
% singleton_conv
thf(fact_22_singleton__conv,axiom,
! [A: nat] :
( ( collect_nat
@ ^ [X: nat] : ( X = A ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singleton_conv
thf(fact_23_singleton__conv2,axiom,
! [A: product_prod_nat_nat] :
( ( collec3392354462482085612at_nat
@ ( ^ [Y2: product_prod_nat_nat,Z: product_prod_nat_nat] : ( Y2 = Z )
@ A ) )
= ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ).
% singleton_conv2
thf(fact_24_singleton__conv2,axiom,
! [A: produc5825016348098550007at_nat] :
( ( collec7697611494764367948at_nat
@ ( ^ [Y2: produc5825016348098550007at_nat,Z: produc5825016348098550007at_nat] : ( Y2 = Z )
@ A ) )
= ( insert3260075854425521959at_nat @ A @ bot_bo1973934853101755969at_nat ) ) ).
% singleton_conv2
thf(fact_25_singleton__conv2,axiom,
! [A: relational_fmla_a_b] :
( ( collec3419995626248312948la_a_b
@ ( ^ [Y2: relational_fmla_a_b,Z: relational_fmla_a_b] : ( Y2 = Z )
@ A ) )
= ( insert7010464514620295119la_a_b @ A @ bot_bo4495933725496725865la_a_b ) ) ).
% singleton_conv2
thf(fact_26_singleton__conv2,axiom,
! [A: list_a] :
( ( collect_list_a
@ ( ^ [Y2: list_a,Z: list_a] : ( Y2 = Z )
@ A ) )
= ( insert_list_a @ A @ bot_bot_set_list_a ) ) ).
% singleton_conv2
thf(fact_27_singleton__conv2,axiom,
! [A: nat] :
( ( collect_nat
@ ( ^ [Y2: nat,Z: nat] : ( Y2 = Z )
@ A ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singleton_conv2
thf(fact_28_diff__diff__add__mset,axiom,
! [M: multiset_set_nat,N: multiset_set_nat,P: multiset_set_nat] :
( ( minus_7237264121398869807et_nat @ ( minus_7237264121398869807et_nat @ M @ N ) @ P )
= ( minus_7237264121398869807et_nat @ M @ ( plus_p8712254050562127327et_nat @ N @ P ) ) ) ).
% diff_diff_add_mset
thf(fact_29_image__mset__union,axiom,
! [F: produc5825016348098550007at_nat > set_nat,M: multis4094885785038667591at_nat,N: multis4094885785038667591at_nat] :
( ( image_7702178775022393786et_nat @ F @ ( plus_p1586715138884553680at_nat @ M @ N ) )
= ( plus_p8712254050562127327et_nat @ ( image_7702178775022393786et_nat @ F @ M ) @ ( image_7702178775022393786et_nat @ F @ N ) ) ) ).
% image_mset_union
thf(fact_30_image__mset__union,axiom,
! [F: set_nat > set_nat,M: multiset_set_nat,N: multiset_set_nat] :
( ( image_1110420595810377289et_nat @ F @ ( plus_p8712254050562127327et_nat @ M @ N ) )
= ( plus_p8712254050562127327et_nat @ ( image_1110420595810377289et_nat @ F @ M ) @ ( image_1110420595810377289et_nat @ F @ N ) ) ) ).
% image_mset_union
thf(fact_31_Un__Diff__cancel,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( sup_su6099978769595272409at_nat @ A2 @ ( minus_6698950876951835142at_nat @ B @ A2 ) )
= ( sup_su6099978769595272409at_nat @ A2 @ B ) ) ).
% Un_Diff_cancel
thf(fact_32_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_33_Un__Diff__cancel2,axiom,
! [B: set_Pr2645174627780777389at_nat,A2: set_Pr2645174627780777389at_nat] :
( ( sup_su6099978769595272409at_nat @ ( minus_6698950876951835142at_nat @ B @ A2 ) @ A2 )
= ( sup_su6099978769595272409at_nat @ B @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_34_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_35_Un__insert__left,axiom,
! [A: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
( ( sup_su6327502436637775413at_nat @ ( insert8211810215607154385at_nat @ A @ B ) @ C )
= ( insert8211810215607154385at_nat @ A @ ( sup_su6327502436637775413at_nat @ B @ C ) ) ) ).
% Un_insert_left
thf(fact_36_Un__insert__left,axiom,
! [A: produc5825016348098550007at_nat,B: set_Pr2645174627780777389at_nat,C: set_Pr2645174627780777389at_nat] :
( ( sup_su6099978769595272409at_nat @ ( insert3260075854425521959at_nat @ A @ B ) @ C )
= ( insert3260075854425521959at_nat @ A @ ( sup_su6099978769595272409at_nat @ B @ C ) ) ) ).
% Un_insert_left
thf(fact_37_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_38_Un__insert__right,axiom,
! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat] :
( ( sup_su6327502436637775413at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ B ) )
= ( insert8211810215607154385at_nat @ A @ ( sup_su6327502436637775413at_nat @ A2 @ B ) ) ) ).
% Un_insert_right
thf(fact_39_Un__insert__right,axiom,
! [A2: set_Pr2645174627780777389at_nat,A: produc5825016348098550007at_nat,B: set_Pr2645174627780777389at_nat] :
( ( sup_su6099978769595272409at_nat @ A2 @ ( insert3260075854425521959at_nat @ A @ B ) )
= ( insert3260075854425521959at_nat @ A @ ( sup_su6099978769595272409at_nat @ A2 @ B ) ) ) ).
% Un_insert_right
thf(fact_40_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_41_Un__empty,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( ( sup_su6099978769595272409at_nat @ A2 @ B )
= bot_bo1973934853101755969at_nat )
= ( ( A2 = bot_bo1973934853101755969at_nat )
& ( B = bot_bo1973934853101755969at_nat ) ) ) ).
% Un_empty
thf(fact_42_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_43_Un__empty,axiom,
! [A2: set_list_a,B: set_list_a] :
( ( ( sup_sup_set_list_a @ A2 @ B )
= bot_bot_set_list_a )
= ( ( A2 = bot_bot_set_list_a )
& ( B = bot_bot_set_list_a ) ) ) ).
% Un_empty
thf(fact_44_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_45_empty__Collect__eq,axiom,
! [P: produc5825016348098550007at_nat > $o] :
( ( bot_bo1973934853101755969at_nat
= ( collec7697611494764367948at_nat @ P ) )
= ( ! [X: produc5825016348098550007at_nat] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_46_empty__Collect__eq,axiom,
! [P: relational_fmla_a_b > $o] :
( ( bot_bo4495933725496725865la_a_b
= ( collec3419995626248312948la_a_b @ P ) )
= ( ! [X: relational_fmla_a_b] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_47_empty__Collect__eq,axiom,
! [P: list_a > $o] :
( ( bot_bot_set_list_a
= ( collect_list_a @ P ) )
= ( ! [X: list_a] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_48_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X: nat] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_49_Collect__empty__eq,axiom,
! [P: produc5825016348098550007at_nat > $o] :
( ( ( collec7697611494764367948at_nat @ P )
= bot_bo1973934853101755969at_nat )
= ( ! [X: produc5825016348098550007at_nat] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_50_Collect__empty__eq,axiom,
! [P: relational_fmla_a_b > $o] :
( ( ( collec3419995626248312948la_a_b @ P )
= bot_bo4495933725496725865la_a_b )
= ( ! [X: relational_fmla_a_b] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_51_Collect__empty__eq,axiom,
! [P: list_a > $o] :
( ( ( collect_list_a @ P )
= bot_bot_set_list_a )
= ( ! [X: list_a] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_52_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X: nat] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_53_all__not__in__conv,axiom,
! [A2: set_Pr2645174627780777389at_nat] :
( ( ! [X: produc5825016348098550007at_nat] :
~ ( member6956540099943067662at_nat @ X @ A2 ) )
= ( A2 = bot_bo1973934853101755969at_nat ) ) ).
% all_not_in_conv
thf(fact_54_all__not__in__conv,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( ! [X: relational_fmla_a_b] :
~ ( member4680049679412964150la_a_b @ X @ A2 ) )
= ( A2 = bot_bo4495933725496725865la_a_b ) ) ).
% all_not_in_conv
thf(fact_55_all__not__in__conv,axiom,
! [A2: set_list_a] :
( ( ! [X: list_a] :
~ ( member_list_a @ X @ A2 ) )
= ( A2 = bot_bot_set_list_a ) ) ).
% all_not_in_conv
thf(fact_56_all__not__in__conv,axiom,
! [A2: set_nat] :
( ( ! [X: nat] :
~ ( member_nat @ X @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_57_empty__iff,axiom,
! [C2: produc5825016348098550007at_nat] :
~ ( member6956540099943067662at_nat @ C2 @ bot_bo1973934853101755969at_nat ) ).
% empty_iff
thf(fact_58_empty__iff,axiom,
! [C2: relational_fmla_a_b] :
~ ( member4680049679412964150la_a_b @ C2 @ bot_bo4495933725496725865la_a_b ) ).
% empty_iff
thf(fact_59_empty__iff,axiom,
! [C2: list_a] :
~ ( member_list_a @ C2 @ bot_bot_set_list_a ) ).
% empty_iff
thf(fact_60_empty__iff,axiom,
! [C2: nat] :
~ ( member_nat @ C2 @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_61_image__eqI,axiom,
! [B2: nat,F: nat > nat,X2: nat,A2: set_nat] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A2 )
=> ( member_nat @ B2 @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_62_image__eqI,axiom,
! [B2: produc5825016348098550007at_nat,F: nat > produc5825016348098550007at_nat,X2: nat,A2: set_nat] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A2 )
=> ( member6956540099943067662at_nat @ B2 @ ( image_1371346403869992270at_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_63_image__eqI,axiom,
! [B2: relational_fmla_a_b,F: nat > relational_fmla_a_b,X2: nat,A2: set_nat] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A2 )
=> ( member4680049679412964150la_a_b @ B2 @ ( image_4386371547000553590la_a_b @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_64_image__eqI,axiom,
! [B2: nat,F: produc5825016348098550007at_nat > nat,X2: produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat] :
( ( B2
= ( F @ X2 ) )
=> ( ( member6956540099943067662at_nat @ X2 @ A2 )
=> ( member_nat @ B2 @ ( image_8316665354072716238at_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_65_image__eqI,axiom,
! [B2: produc5825016348098550007at_nat,F: produc5825016348098550007at_nat > produc5825016348098550007at_nat,X2: produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat] :
( ( B2
= ( F @ X2 ) )
=> ( ( member6956540099943067662at_nat @ X2 @ A2 )
=> ( member6956540099943067662at_nat @ B2 @ ( image_6979785008349797877at_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_66_image__eqI,axiom,
! [B2: relational_fmla_a_b,F: produc5825016348098550007at_nat > relational_fmla_a_b,X2: produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat] :
( ( B2
= ( F @ X2 ) )
=> ( ( member6956540099943067662at_nat @ X2 @ A2 )
=> ( member4680049679412964150la_a_b @ B2 @ ( image_7406987224865000349la_a_b @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_67_image__eqI,axiom,
! [B2: set_list_a,F: relational_fmla_a_b > set_list_a,X2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( B2
= ( F @ X2 ) )
=> ( ( member4680049679412964150la_a_b @ X2 @ A2 )
=> ( member_set_list_a @ B2 @ ( image_130269618930851390list_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_68_image__eqI,axiom,
! [B2: nat,F: relational_fmla_a_b > nat,X2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( B2
= ( F @ X2 ) )
=> ( ( member4680049679412964150la_a_b @ X2 @ A2 )
=> ( member_nat @ B2 @ ( image_341122591648980342_b_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_69_image__eqI,axiom,
! [B2: produc5825016348098550007at_nat,F: relational_fmla_a_b > produc5825016348098550007at_nat,X2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( B2
= ( F @ X2 ) )
=> ( ( member4680049679412964150la_a_b @ X2 @ A2 )
=> ( member6956540099943067662at_nat @ B2 @ ( image_5430539854921360285at_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_70_image__eqI,axiom,
! [B2: relational_fmla_a_b,F: relational_fmla_a_b > relational_fmla_a_b,X2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( B2
= ( F @ X2 ) )
=> ( ( member4680049679412964150la_a_b @ X2 @ A2 )
=> ( member4680049679412964150la_a_b @ B2 @ ( image_6790371041703824709la_a_b @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_71_subsetI,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 ) )
=> ( ord_le4112832032246704949la_a_b @ A2 @ B ) ) ).
% subsetI
thf(fact_72_subsetI,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ! [X3: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X3 @ A2 )
=> ( member6956540099943067662at_nat @ X3 @ B ) )
=> ( ord_le8520675249591772685at_nat @ A2 @ B ) ) ).
% subsetI
thf(fact_73_subsetI,axiom,
! [A2: set_nat,B: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat @ X3 @ B ) )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% subsetI
thf(fact_74_subset__antisym,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( ord_le8520675249591772685at_nat @ A2 @ B )
=> ( ( ord_le8520675249591772685at_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_75_subset__antisym,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_76_insert__absorb2,axiom,
! [X2: produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat] :
( ( insert3260075854425521959at_nat @ X2 @ ( insert3260075854425521959at_nat @ X2 @ A2 ) )
= ( insert3260075854425521959at_nat @ X2 @ A2 ) ) ).
% insert_absorb2
thf(fact_77_insert__absorb2,axiom,
! [X2: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
( ( insert8211810215607154385at_nat @ X2 @ ( insert8211810215607154385at_nat @ X2 @ A2 ) )
= ( insert8211810215607154385at_nat @ X2 @ A2 ) ) ).
% insert_absorb2
thf(fact_78_insert__absorb2,axiom,
! [X2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( insert7010464514620295119la_a_b @ X2 @ ( insert7010464514620295119la_a_b @ X2 @ A2 ) )
= ( insert7010464514620295119la_a_b @ X2 @ A2 ) ) ).
% insert_absorb2
thf(fact_79_insert__iff,axiom,
! [A: product_prod_nat_nat,B2: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member8440522571783428010at_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_80_insert__iff,axiom,
! [A: nat,B2: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_81_insert__iff,axiom,
! [A: produc5825016348098550007at_nat,B2: produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ A @ ( insert3260075854425521959at_nat @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member6956540099943067662at_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_82_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_83_insertCI,axiom,
! [A: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat,B2: product_prod_nat_nat] :
( ( ~ ( member8440522571783428010at_nat @ A @ B )
=> ( A = B2 ) )
=> ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B2 @ B ) ) ) ).
% insertCI
thf(fact_84_insertCI,axiom,
! [A: nat,B: set_nat,B2: nat] :
( ( ~ ( member_nat @ A @ B )
=> ( A = B2 ) )
=> ( member_nat @ A @ ( insert_nat @ B2 @ B ) ) ) ).
% insertCI
thf(fact_85_insertCI,axiom,
! [A: produc5825016348098550007at_nat,B: set_Pr2645174627780777389at_nat,B2: produc5825016348098550007at_nat] :
( ( ~ ( member6956540099943067662at_nat @ A @ B )
=> ( A = B2 ) )
=> ( member6956540099943067662at_nat @ A @ ( insert3260075854425521959at_nat @ B2 @ B ) ) ) ).
% insertCI
thf(fact_86_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_87_Diff__idemp,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( minus_6698950876951835142at_nat @ ( minus_6698950876951835142at_nat @ A2 @ B ) @ B )
= ( minus_6698950876951835142at_nat @ A2 @ B ) ) ).
% Diff_idemp
thf(fact_88_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_89_Diff__iff,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A2 @ B ) )
= ( ( member_nat @ C2 @ A2 )
& ~ ( member_nat @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_90_Diff__iff,axiom,
! [C2: produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ C2 @ ( minus_6698950876951835142at_nat @ A2 @ B ) )
= ( ( member6956540099943067662at_nat @ C2 @ A2 )
& ~ ( member6956540099943067662at_nat @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_91_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_92_DiffI,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ A2 )
=> ( ~ ( member_nat @ C2 @ B )
=> ( member_nat @ C2 @ ( minus_minus_set_nat @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_93_DiffI,axiom,
! [C2: produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ C2 @ A2 )
=> ( ~ ( member6956540099943067662at_nat @ C2 @ B )
=> ( member6956540099943067662at_nat @ C2 @ ( minus_6698950876951835142at_nat @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_94_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_95_subset__mset_Oorder__refl,axiom,
! [X2: multiset_set_nat] : ( subset6078030600694693471et_nat @ X2 @ X2 ) ).
% subset_mset.order_refl
thf(fact_96_subset__mset_Odual__order_Orefl,axiom,
! [A: multiset_set_nat] : ( subset6078030600694693471et_nat @ A @ A ) ).
% subset_mset.dual_order.refl
thf(fact_97_IntI,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ A2 )
=> ( ( member_nat @ C2 @ B )
=> ( member_nat @ C2 @ ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_98_IntI,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 @ ( inf_in8483230781156617063la_a_b @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_99_IntI,axiom,
! [C2: produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ C2 @ A2 )
=> ( ( member6956540099943067662at_nat @ C2 @ B )
=> ( member6956540099943067662at_nat @ C2 @ ( inf_in8776938414804536127at_nat @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_100_Int__iff,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A2 @ B ) )
= ( ( member_nat @ C2 @ A2 )
& ( member_nat @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_101_Int__iff,axiom,
! [C2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C2 @ ( inf_in8483230781156617063la_a_b @ A2 @ B ) )
= ( ( member4680049679412964150la_a_b @ C2 @ A2 )
& ( member4680049679412964150la_a_b @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_102_Int__iff,axiom,
! [C2: produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ C2 @ ( inf_in8776938414804536127at_nat @ A2 @ B ) )
= ( ( member6956540099943067662at_nat @ C2 @ A2 )
& ( member6956540099943067662at_nat @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_103_Un__iff,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B ) )
= ( ( member_nat @ C2 @ A2 )
| ( member_nat @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_104_Un__iff,axiom,
! [C2: produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ C2 @ ( sup_su6099978769595272409at_nat @ A2 @ B ) )
= ( ( member6956540099943067662at_nat @ C2 @ A2 )
| ( member6956540099943067662at_nat @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_105_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_106_UnCI,axiom,
! [C2: nat,B: set_nat,A2: set_nat] :
( ( ~ ( member_nat @ C2 @ B )
=> ( member_nat @ C2 @ A2 ) )
=> ( member_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% UnCI
thf(fact_107_UnCI,axiom,
! [C2: produc5825016348098550007at_nat,B: set_Pr2645174627780777389at_nat,A2: set_Pr2645174627780777389at_nat] :
( ( ~ ( member6956540099943067662at_nat @ C2 @ B )
=> ( member6956540099943067662at_nat @ C2 @ A2 ) )
=> ( member6956540099943067662at_nat @ C2 @ ( sup_su6099978769595272409at_nat @ A2 @ B ) ) ) ).
% UnCI
thf(fact_108_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_109_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_110_image__is__empty,axiom,
! [F: nat > list_a,A2: set_nat] :
( ( ( image_nat_list_a @ F @ A2 )
= bot_bot_set_list_a )
= ( A2 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_111_image__is__empty,axiom,
! [F: list_a > nat,A2: set_list_a] :
( ( ( image_list_a_nat @ F @ A2 )
= bot_bot_set_nat )
= ( A2 = bot_bot_set_list_a ) ) ).
% image_is_empty
thf(fact_112_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_113_image__is__empty,axiom,
! [F: list_a > list_a,A2: set_list_a] :
( ( ( image_list_a_list_a @ F @ A2 )
= bot_bot_set_list_a )
= ( A2 = bot_bot_set_list_a ) ) ).
% image_is_empty
thf(fact_114_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_115_image__is__empty,axiom,
! [F: list_a > relational_fmla_a_b,A2: set_list_a] :
( ( ( image_286667382342926814la_a_b @ F @ A2 )
= bot_bo4495933725496725865la_a_b )
= ( A2 = bot_bot_set_list_a ) ) ).
% image_is_empty
thf(fact_116_image__is__empty,axiom,
! [F: relational_fmla_a_b > list_a,A2: set_Re381260168593705685la_a_b] :
( ( ( image_4103308382209252958list_a @ F @ A2 )
= bot_bot_set_list_a )
= ( A2 = bot_bo4495933725496725865la_a_b ) ) ).
% image_is_empty
thf(fact_117_image__is__empty,axiom,
! [F: relational_fmla_a_b > set_list_a,A2: set_Re381260168593705685la_a_b] :
( ( ( image_130269618930851390list_a @ F @ A2 )
= bot_bo3186585308812441520list_a )
= ( A2 = bot_bo4495933725496725865la_a_b ) ) ).
% image_is_empty
thf(fact_118_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_119_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_120_empty__is__image,axiom,
! [F: nat > list_a,A2: set_nat] :
( ( bot_bot_set_list_a
= ( image_nat_list_a @ F @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_121_empty__is__image,axiom,
! [F: list_a > nat,A2: set_list_a] :
( ( bot_bot_set_nat
= ( image_list_a_nat @ F @ A2 ) )
= ( A2 = bot_bot_set_list_a ) ) ).
% empty_is_image
thf(fact_122_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_123_empty__is__image,axiom,
! [F: list_a > list_a,A2: set_list_a] :
( ( bot_bot_set_list_a
= ( image_list_a_list_a @ F @ A2 ) )
= ( A2 = bot_bot_set_list_a ) ) ).
% empty_is_image
thf(fact_124_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_125_empty__is__image,axiom,
! [F: list_a > relational_fmla_a_b,A2: set_list_a] :
( ( bot_bo4495933725496725865la_a_b
= ( image_286667382342926814la_a_b @ F @ A2 ) )
= ( A2 = bot_bot_set_list_a ) ) ).
% empty_is_image
thf(fact_126_empty__is__image,axiom,
! [F: relational_fmla_a_b > list_a,A2: set_Re381260168593705685la_a_b] :
( ( bot_bot_set_list_a
= ( image_4103308382209252958list_a @ F @ A2 ) )
= ( A2 = bot_bo4495933725496725865la_a_b ) ) ).
% empty_is_image
thf(fact_127_empty__is__image,axiom,
! [F: relational_fmla_a_b > set_list_a,A2: set_Re381260168593705685la_a_b] :
( ( bot_bo3186585308812441520list_a
= ( image_130269618930851390list_a @ F @ A2 ) )
= ( A2 = bot_bo4495933725496725865la_a_b ) ) ).
% empty_is_image
thf(fact_128_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_129_mem__Collect__eq,axiom,
! [A: produc5825016348098550007at_nat,P: produc5825016348098550007at_nat > $o] :
( ( member6956540099943067662at_nat @ A @ ( collec7697611494764367948at_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_130_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_131_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_132_Collect__mem__eq,axiom,
! [A2: set_Pr2645174627780777389at_nat] :
( ( collec7697611494764367948at_nat
@ ^ [X: produc5825016348098550007at_nat] : ( member6956540099943067662at_nat @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_133_Collect__mem__eq,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_134_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_135_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_136_image__empty,axiom,
! [F: nat > nat] :
( ( image_nat_nat @ F @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_137_image__empty,axiom,
! [F: list_a > nat] :
( ( image_list_a_nat @ F @ bot_bot_set_list_a )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_138_image__empty,axiom,
! [F: nat > list_a] :
( ( image_nat_list_a @ F @ bot_bot_set_nat )
= bot_bot_set_list_a ) ).
% image_empty
thf(fact_139_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_140_image__empty,axiom,
! [F: list_a > list_a] :
( ( image_list_a_list_a @ F @ bot_bot_set_list_a )
= bot_bot_set_list_a ) ).
% image_empty
thf(fact_141_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_142_image__empty,axiom,
! [F: relational_fmla_a_b > list_a] :
( ( image_4103308382209252958list_a @ F @ bot_bo4495933725496725865la_a_b )
= bot_bot_set_list_a ) ).
% image_empty
thf(fact_143_image__empty,axiom,
! [F: list_a > relational_fmla_a_b] :
( ( image_286667382342926814la_a_b @ F @ bot_bot_set_list_a )
= bot_bo4495933725496725865la_a_b ) ).
% image_empty
thf(fact_144_image__empty,axiom,
! [F: relational_fmla_a_b > set_list_a] :
( ( image_130269618930851390list_a @ F @ bot_bo4495933725496725865la_a_b )
= bot_bo3186585308812441520list_a ) ).
% image_empty
thf(fact_145_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_146_empty__subsetI,axiom,
! [A2: set_Re381260168593705685la_a_b] : ( ord_le4112832032246704949la_a_b @ bot_bo4495933725496725865la_a_b @ A2 ) ).
% empty_subsetI
thf(fact_147_empty__subsetI,axiom,
! [A2: set_list_a] : ( ord_le8861187494160871172list_a @ bot_bot_set_list_a @ A2 ) ).
% empty_subsetI
thf(fact_148_empty__subsetI,axiom,
! [A2: set_Pr2645174627780777389at_nat] : ( ord_le8520675249591772685at_nat @ bot_bo1973934853101755969at_nat @ A2 ) ).
% empty_subsetI
thf(fact_149_empty__subsetI,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% empty_subsetI
thf(fact_150_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_151_subset__empty,axiom,
! [A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ bot_bot_set_list_a )
= ( A2 = bot_bot_set_list_a ) ) ).
% subset_empty
thf(fact_152_subset__empty,axiom,
! [A2: set_Pr2645174627780777389at_nat] :
( ( ord_le8520675249591772685at_nat @ A2 @ bot_bo1973934853101755969at_nat )
= ( A2 = bot_bo1973934853101755969at_nat ) ) ).
% subset_empty
thf(fact_153_subset__empty,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_154_singletonI,axiom,
! [A: product_prod_nat_nat] : ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ).
% singletonI
thf(fact_155_singletonI,axiom,
! [A: produc5825016348098550007at_nat] : ( member6956540099943067662at_nat @ A @ ( insert3260075854425521959at_nat @ A @ bot_bo1973934853101755969at_nat ) ) ).
% singletonI
thf(fact_156_singletonI,axiom,
! [A: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ A @ ( insert7010464514620295119la_a_b @ A @ bot_bo4495933725496725865la_a_b ) ) ).
% singletonI
thf(fact_157_singletonI,axiom,
! [A: list_a] : ( member_list_a @ A @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) ).
% singletonI
thf(fact_158_singletonI,axiom,
! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_159_insert__image,axiom,
! [X2: nat,A2: set_nat,F: nat > produc5825016348098550007at_nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( insert3260075854425521959at_nat @ ( F @ X2 ) @ ( image_1371346403869992270at_nat @ F @ A2 ) )
= ( image_1371346403869992270at_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_160_insert__image,axiom,
! [X2: nat,A2: set_nat,F: nat > product_prod_nat_nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( insert8211810215607154385at_nat @ ( F @ X2 ) @ ( image_5846123807819985514at_nat @ F @ A2 ) )
= ( image_5846123807819985514at_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_161_insert__image,axiom,
! [X2: nat,A2: set_nat,F: nat > relational_fmla_a_b] :
( ( member_nat @ X2 @ A2 )
=> ( ( insert7010464514620295119la_a_b @ ( F @ X2 ) @ ( image_4386371547000553590la_a_b @ F @ A2 ) )
= ( image_4386371547000553590la_a_b @ F @ A2 ) ) ) ).
% insert_image
thf(fact_162_insert__image,axiom,
! [X2: produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat,F: produc5825016348098550007at_nat > produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X2 @ A2 )
=> ( ( insert3260075854425521959at_nat @ ( F @ X2 ) @ ( image_6979785008349797877at_nat @ F @ A2 ) )
= ( image_6979785008349797877at_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_163_insert__image,axiom,
! [X2: produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat,F: produc5825016348098550007at_nat > product_prod_nat_nat] :
( ( member6956540099943067662at_nat @ X2 @ A2 )
=> ( ( insert8211810215607154385at_nat @ ( F @ X2 ) @ ( image_2384848872172141443at_nat @ F @ A2 ) )
= ( image_2384848872172141443at_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_164_insert__image,axiom,
! [X2: produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat,F: produc5825016348098550007at_nat > relational_fmla_a_b] :
( ( member6956540099943067662at_nat @ X2 @ A2 )
=> ( ( insert7010464514620295119la_a_b @ ( F @ X2 ) @ ( image_7406987224865000349la_a_b @ F @ A2 ) )
= ( image_7406987224865000349la_a_b @ F @ A2 ) ) ) ).
% insert_image
thf(fact_165_insert__image,axiom,
! [X2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > set_list_a] :
( ( member4680049679412964150la_a_b @ X2 @ A2 )
=> ( ( insert_set_list_a @ ( F @ X2 ) @ ( image_130269618930851390list_a @ F @ A2 ) )
= ( image_130269618930851390list_a @ F @ A2 ) ) ) ).
% insert_image
thf(fact_166_insert__image,axiom,
! [X2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > produc5825016348098550007at_nat] :
( ( member4680049679412964150la_a_b @ X2 @ A2 )
=> ( ( insert3260075854425521959at_nat @ ( F @ X2 ) @ ( image_5430539854921360285at_nat @ F @ A2 ) )
= ( image_5430539854921360285at_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_167_insert__image,axiom,
! [X2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > product_prod_nat_nat] :
( ( member4680049679412964150la_a_b @ X2 @ A2 )
=> ( ( insert8211810215607154385at_nat @ ( F @ X2 ) @ ( image_5852719071363227611at_nat @ F @ A2 ) )
= ( image_5852719071363227611at_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_168_insert__image,axiom,
! [X2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ A2 )
=> ( ( insert7010464514620295119la_a_b @ ( F @ X2 ) @ ( image_6790371041703824709la_a_b @ F @ A2 ) )
= ( image_6790371041703824709la_a_b @ F @ A2 ) ) ) ).
% insert_image
thf(fact_169_image__insert,axiom,
! [F: nat > relational_fmla_a_b,A: nat,B: set_nat] :
( ( image_4386371547000553590la_a_b @ F @ ( insert_nat @ A @ B ) )
= ( insert7010464514620295119la_a_b @ ( F @ A ) @ ( image_4386371547000553590la_a_b @ F @ B ) ) ) ).
% image_insert
thf(fact_170_image__insert,axiom,
! [F: product_prod_nat_nat > product_prod_nat_nat,A: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat] :
( ( image_5168914502847457605at_nat @ F @ ( insert8211810215607154385at_nat @ A @ B ) )
= ( insert8211810215607154385at_nat @ ( F @ A ) @ ( image_5168914502847457605at_nat @ F @ B ) ) ) ).
% image_insert
thf(fact_171_image__insert,axiom,
! [F: product_prod_nat_nat > relational_fmla_a_b,A: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat] :
( ( image_5019456213502855771la_a_b @ F @ ( insert8211810215607154385at_nat @ A @ B ) )
= ( insert7010464514620295119la_a_b @ ( F @ A ) @ ( image_5019456213502855771la_a_b @ F @ B ) ) ) ).
% image_insert
thf(fact_172_image__insert,axiom,
! [F: relational_fmla_a_b > set_list_a,A: relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ( image_130269618930851390list_a @ F @ ( insert7010464514620295119la_a_b @ A @ B ) )
= ( insert_set_list_a @ ( F @ A ) @ ( image_130269618930851390list_a @ F @ B ) ) ) ).
% image_insert
thf(fact_173_image__insert,axiom,
! [F: relational_fmla_a_b > product_prod_nat_nat,A: relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ( image_5852719071363227611at_nat @ F @ ( insert7010464514620295119la_a_b @ A @ B ) )
= ( insert8211810215607154385at_nat @ ( F @ A ) @ ( image_5852719071363227611at_nat @ F @ B ) ) ) ).
% image_insert
thf(fact_174_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_175_image__insert,axiom,
! [F: nat > produc5825016348098550007at_nat,A: nat,B: set_nat] :
( ( image_1371346403869992270at_nat @ F @ ( insert_nat @ A @ B ) )
= ( insert3260075854425521959at_nat @ ( F @ A ) @ ( image_1371346403869992270at_nat @ F @ B ) ) ) ).
% image_insert
thf(fact_176_image__insert,axiom,
! [F: produc5825016348098550007at_nat > product_prod_nat_nat,A: produc5825016348098550007at_nat,B: set_Pr2645174627780777389at_nat] :
( ( image_2384848872172141443at_nat @ F @ ( insert3260075854425521959at_nat @ A @ B ) )
= ( insert8211810215607154385at_nat @ ( F @ A ) @ ( image_2384848872172141443at_nat @ F @ B ) ) ) ).
% image_insert
thf(fact_177_image__insert,axiom,
! [F: produc5825016348098550007at_nat > relational_fmla_a_b,A: produc5825016348098550007at_nat,B: set_Pr2645174627780777389at_nat] :
( ( image_7406987224865000349la_a_b @ F @ ( insert3260075854425521959at_nat @ A @ B ) )
= ( insert7010464514620295119la_a_b @ ( F @ A ) @ ( image_7406987224865000349la_a_b @ F @ B ) ) ) ).
% image_insert
thf(fact_178_image__insert,axiom,
! [F: product_prod_nat_nat > produc5825016348098550007at_nat,A: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat] :
( ( image_2298502056602587059at_nat @ F @ ( insert8211810215607154385at_nat @ A @ B ) )
= ( insert3260075854425521959at_nat @ ( F @ A ) @ ( image_2298502056602587059at_nat @ F @ B ) ) ) ).
% image_insert
thf(fact_179_insert__subset,axiom,
! [X2: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ ( insert8211810215607154385at_nat @ X2 @ A2 ) @ B )
= ( ( member8440522571783428010at_nat @ X2 @ B )
& ( ord_le3146513528884898305at_nat @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_180_insert__subset,axiom,
! [X2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ ( insert7010464514620295119la_a_b @ X2 @ A2 ) @ B )
= ( ( member4680049679412964150la_a_b @ X2 @ B )
& ( ord_le4112832032246704949la_a_b @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_181_insert__subset,axiom,
! [X2: produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( ord_le8520675249591772685at_nat @ ( insert3260075854425521959at_nat @ X2 @ A2 ) @ B )
= ( ( member6956540099943067662at_nat @ X2 @ B )
& ( ord_le8520675249591772685at_nat @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_182_insert__subset,axiom,
! [X2: nat,A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X2 @ A2 ) @ B )
= ( ( member_nat @ X2 @ B )
& ( ord_less_eq_set_nat @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_183_Diff__cancel,axiom,
! [A2: set_list_a] :
( ( minus_646659088055828811list_a @ A2 @ A2 )
= bot_bot_set_list_a ) ).
% Diff_cancel
thf(fact_184_Diff__cancel,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ A2 )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_185_Diff__cancel,axiom,
! [A2: set_Pr2645174627780777389at_nat] :
( ( minus_6698950876951835142at_nat @ A2 @ A2 )
= bot_bo1973934853101755969at_nat ) ).
% Diff_cancel
thf(fact_186_Diff__cancel,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( minus_4077726661957047470la_a_b @ A2 @ A2 )
= bot_bo4495933725496725865la_a_b ) ).
% Diff_cancel
thf(fact_187_empty__Diff,axiom,
! [A2: set_list_a] :
( ( minus_646659088055828811list_a @ bot_bot_set_list_a @ A2 )
= bot_bot_set_list_a ) ).
% empty_Diff
thf(fact_188_empty__Diff,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A2 )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_189_empty__Diff,axiom,
! [A2: set_Pr2645174627780777389at_nat] :
( ( minus_6698950876951835142at_nat @ bot_bo1973934853101755969at_nat @ A2 )
= bot_bo1973934853101755969at_nat ) ).
% empty_Diff
thf(fact_190_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_191_Diff__empty,axiom,
! [A2: set_list_a] :
( ( minus_646659088055828811list_a @ A2 @ bot_bot_set_list_a )
= A2 ) ).
% Diff_empty
thf(fact_192_Diff__empty,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% Diff_empty
thf(fact_193_Diff__empty,axiom,
! [A2: set_Pr2645174627780777389at_nat] :
( ( minus_6698950876951835142at_nat @ A2 @ bot_bo1973934853101755969at_nat )
= A2 ) ).
% Diff_empty
thf(fact_194_Diff__empty,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( minus_4077726661957047470la_a_b @ A2 @ bot_bo4495933725496725865la_a_b )
= A2 ) ).
% Diff_empty
thf(fact_195_insert__Diff1,axiom,
! [X2: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ X2 @ B )
=> ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X2 @ A2 ) @ B )
= ( minus_1356011639430497352at_nat @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_196_insert__Diff1,axiom,
! [X2: nat,B: set_nat,A2: set_nat] :
( ( member_nat @ X2 @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A2 ) @ B )
= ( minus_minus_set_nat @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_197_insert__Diff1,axiom,
! [X2: produc5825016348098550007at_nat,B: set_Pr2645174627780777389at_nat,A2: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ X2 @ B )
=> ( ( minus_6698950876951835142at_nat @ ( insert3260075854425521959at_nat @ X2 @ A2 ) @ B )
= ( minus_6698950876951835142at_nat @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_198_insert__Diff1,axiom,
! [X2: relational_fmla_a_b,B: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ B )
=> ( ( minus_4077726661957047470la_a_b @ ( insert7010464514620295119la_a_b @ X2 @ A2 ) @ B )
= ( minus_4077726661957047470la_a_b @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_199_Diff__insert0,axiom,
! [X2: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ~ ( member8440522571783428010at_nat @ X2 @ A2 )
=> ( ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ X2 @ B ) )
= ( minus_1356011639430497352at_nat @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_200_Diff__insert0,axiom,
! [X2: nat,A2: set_nat,B: set_nat] :
( ~ ( member_nat @ X2 @ A2 )
=> ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ B ) )
= ( minus_minus_set_nat @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_201_Diff__insert0,axiom,
! [X2: produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ~ ( member6956540099943067662at_nat @ X2 @ A2 )
=> ( ( minus_6698950876951835142at_nat @ A2 @ ( insert3260075854425521959at_nat @ X2 @ B ) )
= ( minus_6698950876951835142at_nat @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_202_Diff__insert0,axiom,
! [X2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ~ ( member4680049679412964150la_a_b @ X2 @ A2 )
=> ( ( minus_4077726661957047470la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ X2 @ B ) )
= ( minus_4077726661957047470la_a_b @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_203_Int__subset__iff,axiom,
! [C: set_Pr2645174627780777389at_nat,A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( ord_le8520675249591772685at_nat @ C @ ( inf_in8776938414804536127at_nat @ A2 @ B ) )
= ( ( ord_le8520675249591772685at_nat @ C @ A2 )
& ( ord_le8520675249591772685at_nat @ C @ B ) ) ) ).
% Int_subset_iff
thf(fact_204_Int__subset__iff,axiom,
! [C: set_nat,A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C @ ( inf_inf_set_nat @ A2 @ B ) )
= ( ( ord_less_eq_set_nat @ C @ A2 )
& ( ord_less_eq_set_nat @ C @ B ) ) ) ).
% Int_subset_iff
thf(fact_205_Int__insert__right__if1,axiom,
! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ A @ A2 )
=> ( ( inf_in2572325071724192079at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ B ) )
= ( insert8211810215607154385at_nat @ A @ ( inf_in2572325071724192079at_nat @ A2 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_206_Int__insert__right__if1,axiom,
! [A: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B ) )
= ( insert_nat @ A @ ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_207_Int__insert__right__if1,axiom,
! [A: 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_208_Int__insert__right__if1,axiom,
! [A: produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ A @ A2 )
=> ( ( inf_in8776938414804536127at_nat @ A2 @ ( insert3260075854425521959at_nat @ A @ B ) )
= ( insert3260075854425521959at_nat @ A @ ( inf_in8776938414804536127at_nat @ A2 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_209_Int__insert__right__if0,axiom,
! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ~ ( member8440522571783428010at_nat @ A @ A2 )
=> ( ( inf_in2572325071724192079at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ B ) )
= ( inf_in2572325071724192079at_nat @ A2 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_210_Int__insert__right__if0,axiom,
! [A: nat,A2: set_nat,B: set_nat] :
( ~ ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B ) )
= ( inf_inf_set_nat @ A2 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_211_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_212_Int__insert__right__if0,axiom,
! [A: produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ~ ( member6956540099943067662at_nat @ A @ A2 )
=> ( ( inf_in8776938414804536127at_nat @ A2 @ ( insert3260075854425521959at_nat @ A @ B ) )
= ( inf_in8776938414804536127at_nat @ A2 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_213_insert__inter__insert,axiom,
! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A @ A2 ) @ ( insert8211810215607154385at_nat @ A @ B ) )
= ( insert8211810215607154385at_nat @ A @ ( inf_in2572325071724192079at_nat @ A2 @ B ) ) ) ).
% insert_inter_insert
thf(fact_214_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_215_insert__inter__insert,axiom,
! [A: produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( inf_in8776938414804536127at_nat @ ( insert3260075854425521959at_nat @ A @ A2 ) @ ( insert3260075854425521959at_nat @ A @ B ) )
= ( insert3260075854425521959at_nat @ A @ ( inf_in8776938414804536127at_nat @ A2 @ B ) ) ) ).
% insert_inter_insert
thf(fact_216_Int__insert__left__if1,axiom,
! [A: product_prod_nat_nat,C: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ A @ C )
=> ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A @ B ) @ C )
= ( insert8211810215607154385at_nat @ A @ ( inf_in2572325071724192079at_nat @ B @ C ) ) ) ) ).
% Int_insert_left_if1
thf(fact_217_Int__insert__left__if1,axiom,
! [A: nat,C: set_nat,B: set_nat] :
( ( member_nat @ A @ C )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B ) @ C )
= ( insert_nat @ A @ ( inf_inf_set_nat @ B @ C ) ) ) ) ).
% Int_insert_left_if1
thf(fact_218_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_219_Int__insert__left__if1,axiom,
! [A: produc5825016348098550007at_nat,C: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ A @ C )
=> ( ( inf_in8776938414804536127at_nat @ ( insert3260075854425521959at_nat @ A @ B ) @ C )
= ( insert3260075854425521959at_nat @ A @ ( inf_in8776938414804536127at_nat @ B @ C ) ) ) ) ).
% Int_insert_left_if1
thf(fact_220_Int__insert__left__if0,axiom,
! [A: product_prod_nat_nat,C: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ~ ( member8440522571783428010at_nat @ A @ C )
=> ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A @ B ) @ C )
= ( inf_in2572325071724192079at_nat @ B @ C ) ) ) ).
% Int_insert_left_if0
thf(fact_221_Int__insert__left__if0,axiom,
! [A: nat,C: set_nat,B: set_nat] :
( ~ ( member_nat @ A @ C )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B ) @ C )
= ( inf_inf_set_nat @ B @ C ) ) ) ).
% Int_insert_left_if0
thf(fact_222_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_223_Int__insert__left__if0,axiom,
! [A: produc5825016348098550007at_nat,C: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ~ ( member6956540099943067662at_nat @ A @ C )
=> ( ( inf_in8776938414804536127at_nat @ ( insert3260075854425521959at_nat @ A @ B ) @ C )
= ( inf_in8776938414804536127at_nat @ B @ C ) ) ) ).
% Int_insert_left_if0
thf(fact_224_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_225_Un__subset__iff,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat,C: set_Pr2645174627780777389at_nat] :
( ( ord_le8520675249591772685at_nat @ ( sup_su6099978769595272409at_nat @ A2 @ B ) @ C )
= ( ( ord_le8520675249591772685at_nat @ A2 @ C )
& ( ord_le8520675249591772685at_nat @ B @ C ) ) ) ).
% Un_subset_iff
thf(fact_226_Un__subset__iff,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ C )
= ( ( ord_less_eq_set_nat @ A2 @ C )
& ( ord_less_eq_set_nat @ B @ C ) ) ) ).
% Un_subset_iff
thf(fact_227_Un__Int__eq_I1_J,axiom,
! [S: set_Pr2645174627780777389at_nat,T: set_Pr2645174627780777389at_nat] :
( ( inf_in8776938414804536127at_nat @ ( sup_su6099978769595272409at_nat @ S @ T ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_228_Un__Int__eq_I1_J,axiom,
! [S: set_Re381260168593705685la_a_b,T: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ ( sup_su5130108678486352897la_a_b @ S @ T ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_229_Un__Int__eq_I2_J,axiom,
! [S: set_Pr2645174627780777389at_nat,T: set_Pr2645174627780777389at_nat] :
( ( inf_in8776938414804536127at_nat @ ( sup_su6099978769595272409at_nat @ S @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_230_Un__Int__eq_I2_J,axiom,
! [S: set_Re381260168593705685la_a_b,T: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ ( sup_su5130108678486352897la_a_b @ S @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_231_Un__Int__eq_I3_J,axiom,
! [S: set_Pr2645174627780777389at_nat,T: set_Pr2645174627780777389at_nat] :
( ( inf_in8776938414804536127at_nat @ S @ ( sup_su6099978769595272409at_nat @ S @ T ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_232_Un__Int__eq_I3_J,axiom,
! [S: set_Re381260168593705685la_a_b,T: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ S @ ( sup_su5130108678486352897la_a_b @ S @ T ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_233_Un__Int__eq_I4_J,axiom,
! [T: set_Pr2645174627780777389at_nat,S: set_Pr2645174627780777389at_nat] :
( ( inf_in8776938414804536127at_nat @ T @ ( sup_su6099978769595272409at_nat @ S @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_234_Un__Int__eq_I4_J,axiom,
! [T: set_Re381260168593705685la_a_b,S: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ T @ ( sup_su5130108678486352897la_a_b @ S @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_235_Int__Un__eq_I1_J,axiom,
! [S: set_Pr2645174627780777389at_nat,T: set_Pr2645174627780777389at_nat] :
( ( sup_su6099978769595272409at_nat @ ( inf_in8776938414804536127at_nat @ S @ T ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_236_Int__Un__eq_I1_J,axiom,
! [S: set_Re381260168593705685la_a_b,T: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ ( inf_in8483230781156617063la_a_b @ S @ T ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_237_Int__Un__eq_I2_J,axiom,
! [S: set_Pr2645174627780777389at_nat,T: set_Pr2645174627780777389at_nat] :
( ( sup_su6099978769595272409at_nat @ ( inf_in8776938414804536127at_nat @ S @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_238_Int__Un__eq_I2_J,axiom,
! [S: set_Re381260168593705685la_a_b,T: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ ( inf_in8483230781156617063la_a_b @ S @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_239_Int__Un__eq_I3_J,axiom,
! [S: set_Pr2645174627780777389at_nat,T: set_Pr2645174627780777389at_nat] :
( ( sup_su6099978769595272409at_nat @ S @ ( inf_in8776938414804536127at_nat @ S @ T ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_240_Int__Un__eq_I3_J,axiom,
! [S: set_Re381260168593705685la_a_b,T: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ S @ ( inf_in8483230781156617063la_a_b @ S @ T ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_241_Int__Un__eq_I4_J,axiom,
! [T: set_Pr2645174627780777389at_nat,S: set_Pr2645174627780777389at_nat] :
( ( sup_su6099978769595272409at_nat @ T @ ( inf_in8776938414804536127at_nat @ S @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_242_Int__Un__eq_I4_J,axiom,
! [T: set_Re381260168593705685la_a_b,S: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ T @ ( inf_in8483230781156617063la_a_b @ S @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_243_subset__mset_Ole__zero__eq,axiom,
! [N2: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ N2 @ zero_z3157962936165190495et_nat )
= ( N2 = zero_z3157962936165190495et_nat ) ) ).
% subset_mset.le_zero_eq
thf(fact_244_subset__mset_Oextremum__unique,axiom,
! [A: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ A @ zero_z3157962936165190495et_nat )
= ( A = zero_z3157962936165190495et_nat ) ) ).
% subset_mset.extremum_unique
thf(fact_245_mset__subset__eq__mono__add__right__cancel,axiom,
! [A2: multiset_set_nat,C: multiset_set_nat,B: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ ( plus_p8712254050562127327et_nat @ A2 @ C ) @ ( plus_p8712254050562127327et_nat @ B @ C ) )
= ( subset6078030600694693471et_nat @ A2 @ B ) ) ).
% mset_subset_eq_mono_add_right_cancel
thf(fact_246_mset__subset__eq__mono__add__left__cancel,axiom,
! [C: multiset_set_nat,A2: multiset_set_nat,B: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ ( plus_p8712254050562127327et_nat @ C @ A2 ) @ ( plus_p8712254050562127327et_nat @ C @ B ) )
= ( subset6078030600694693471et_nat @ A2 @ B ) ) ).
% mset_subset_eq_mono_add_left_cancel
thf(fact_247_subset__mset_Oadd__le__cancel__right,axiom,
! [A: multiset_set_nat,C2: multiset_set_nat,B2: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ ( plus_p8712254050562127327et_nat @ A @ C2 ) @ ( plus_p8712254050562127327et_nat @ B2 @ C2 ) )
= ( subset6078030600694693471et_nat @ A @ B2 ) ) ).
% subset_mset.add_le_cancel_right
thf(fact_248_subset__mset_Oadd__le__cancel__left,axiom,
! [C2: multiset_set_nat,A: multiset_set_nat,B2: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ ( plus_p8712254050562127327et_nat @ C2 @ A ) @ ( plus_p8712254050562127327et_nat @ C2 @ B2 ) )
= ( subset6078030600694693471et_nat @ A @ B2 ) ) ).
% subset_mset.add_le_cancel_left
thf(fact_249_union__eq__empty,axiom,
! [M: multiset_set_nat,N: multiset_set_nat] :
( ( ( plus_p8712254050562127327et_nat @ M @ N )
= zero_z3157962936165190495et_nat )
= ( ( M = zero_z3157962936165190495et_nat )
& ( N = zero_z3157962936165190495et_nat ) ) ) ).
% union_eq_empty
thf(fact_250_empty__eq__union,axiom,
! [M: multiset_set_nat,N: multiset_set_nat] :
( ( zero_z3157962936165190495et_nat
= ( plus_p8712254050562127327et_nat @ M @ N ) )
= ( ( M = zero_z3157962936165190495et_nat )
& ( N = zero_z3157962936165190495et_nat ) ) ) ).
% empty_eq_union
thf(fact_251_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
! [X2: multiset_set_nat,Y3: multiset_set_nat] :
( ( zero_z3157962936165190495et_nat
= ( plus_p8712254050562127327et_nat @ X2 @ Y3 ) )
= ( ( X2 = zero_z3157962936165190495et_nat )
& ( Y3 = zero_z3157962936165190495et_nat ) ) ) ).
% subset_mset.zero_eq_add_iff_both_eq_0
thf(fact_252_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
! [X2: multiset_set_nat,Y3: multiset_set_nat] :
( ( ( plus_p8712254050562127327et_nat @ X2 @ Y3 )
= zero_z3157962936165190495et_nat )
= ( ( X2 = zero_z3157962936165190495et_nat )
& ( Y3 = zero_z3157962936165190495et_nat ) ) ) ).
% subset_mset.add_eq_0_iff_both_eq_0
thf(fact_253_image__mset__is__empty__iff,axiom,
! [F: produc5825016348098550007at_nat > set_nat,M: multis4094885785038667591at_nat] :
( ( ( image_7702178775022393786et_nat @ F @ M )
= zero_z3157962936165190495et_nat )
= ( M = zero_z1389175455191277904at_nat ) ) ).
% image_mset_is_empty_iff
thf(fact_254_image__mset__is__empty__iff,axiom,
! [F: set_nat > set_nat,M: multiset_set_nat] :
( ( ( image_1110420595810377289et_nat @ F @ M )
= zero_z3157962936165190495et_nat )
= ( M = zero_z3157962936165190495et_nat ) ) ).
% image_mset_is_empty_iff
thf(fact_255_image__mset__empty,axiom,
! [F: produc5825016348098550007at_nat > set_nat] :
( ( image_7702178775022393786et_nat @ F @ zero_z1389175455191277904at_nat )
= zero_z3157962936165190495et_nat ) ).
% image_mset_empty
thf(fact_256_image__mset__empty,axiom,
! [F: set_nat > set_nat] :
( ( image_1110420595810377289et_nat @ F @ zero_z3157962936165190495et_nat )
= zero_z3157962936165190495et_nat ) ).
% image_mset_empty
thf(fact_257_mset__set__eq__iff,axiom,
! [A2: set_nat,B: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( ( ( mset_set_nat @ A2 )
= ( mset_set_nat @ B ) )
= ( A2 = B ) ) ) ) ).
% mset_set_eq_iff
thf(fact_258_mset__set__eq__iff,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ A2 )
=> ( ( finite5600759454172676150la_a_b @ B )
=> ( ( ( mset_s7504067419864366776la_a_b @ A2 )
= ( mset_s7504067419864366776la_a_b @ B ) )
= ( A2 = B ) ) ) ) ).
% mset_set_eq_iff
thf(fact_259_mset__set__eq__iff,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( finite6790245451575510286at_nat @ A2 )
=> ( ( finite6790245451575510286at_nat @ B )
=> ( ( ( mset_s3030574779765502096at_nat @ A2 )
= ( mset_s3030574779765502096at_nat @ B ) )
= ( A2 = B ) ) ) ) ).
% mset_set_eq_iff
thf(fact_260_image__add__0,axiom,
! [S: set_multiset_set_nat] :
( ( image_2961941931136280627et_nat @ ( plus_p8712254050562127327et_nat @ zero_z3157962936165190495et_nat ) @ S )
= S ) ).
% image_add_0
thf(fact_261_image__add__0,axiom,
! [S: set_nat] :
( ( image_nat_nat @ ( plus_plus_nat @ zero_zero_nat ) @ S )
= S ) ).
% image_add_0
thf(fact_262_singleton__insert__inj__eq_H,axiom,
! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: product_prod_nat_nat] :
( ( ( insert8211810215607154385at_nat @ A @ A2 )
= ( insert8211810215607154385at_nat @ B2 @ bot_bo2099793752762293965at_nat ) )
= ( ( A = B2 )
& ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ B2 @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_263_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_264_singleton__insert__inj__eq_H,axiom,
! [A: list_a,A2: set_list_a,B2: list_a] :
( ( ( insert_list_a @ A @ A2 )
= ( insert_list_a @ B2 @ bot_bot_set_list_a ) )
= ( ( A = B2 )
& ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ B2 @ bot_bot_set_list_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_265_singleton__insert__inj__eq_H,axiom,
! [A: produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat,B2: produc5825016348098550007at_nat] :
( ( ( insert3260075854425521959at_nat @ A @ A2 )
= ( insert3260075854425521959at_nat @ B2 @ bot_bo1973934853101755969at_nat ) )
= ( ( A = B2 )
& ( ord_le8520675249591772685at_nat @ A2 @ ( insert3260075854425521959at_nat @ B2 @ bot_bo1973934853101755969at_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_266_singleton__insert__inj__eq_H,axiom,
! [A: nat,A2: set_nat,B2: nat] :
( ( ( insert_nat @ A @ A2 )
= ( insert_nat @ B2 @ bot_bot_set_nat ) )
= ( ( A = B2 )
& ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_267_singleton__insert__inj__eq,axiom,
! [B2: product_prod_nat_nat,A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
( ( ( insert8211810215607154385at_nat @ B2 @ bot_bo2099793752762293965at_nat )
= ( insert8211810215607154385at_nat @ A @ A2 ) )
= ( ( A = B2 )
& ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ B2 @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_268_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_269_singleton__insert__inj__eq,axiom,
! [B2: list_a,A: list_a,A2: set_list_a] :
( ( ( insert_list_a @ B2 @ bot_bot_set_list_a )
= ( insert_list_a @ A @ A2 ) )
= ( ( A = B2 )
& ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ B2 @ bot_bot_set_list_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_270_singleton__insert__inj__eq,axiom,
! [B2: produc5825016348098550007at_nat,A: produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat] :
( ( ( insert3260075854425521959at_nat @ B2 @ bot_bo1973934853101755969at_nat )
= ( insert3260075854425521959at_nat @ A @ A2 ) )
= ( ( A = B2 )
& ( ord_le8520675249591772685at_nat @ A2 @ ( insert3260075854425521959at_nat @ B2 @ bot_bo1973934853101755969at_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_271_singleton__insert__inj__eq,axiom,
! [B2: nat,A: nat,A2: set_nat] :
( ( ( insert_nat @ B2 @ bot_bot_set_nat )
= ( insert_nat @ A @ A2 ) )
= ( ( A = B2 )
& ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_272_Diff__eq__empty__iff,axiom,
! [A2: set_list_a,B: set_list_a] :
( ( ( minus_646659088055828811list_a @ A2 @ B )
= bot_bot_set_list_a )
= ( ord_le8861187494160871172list_a @ A2 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_273_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_274_Diff__eq__empty__iff,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( ( minus_6698950876951835142at_nat @ A2 @ B )
= bot_bo1973934853101755969at_nat )
= ( ord_le8520675249591772685at_nat @ A2 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_275_Diff__eq__empty__iff,axiom,
! [A2: set_nat,B: set_nat] :
( ( ( minus_minus_set_nat @ A2 @ B )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_276_insert__disjoint_I1_J,axiom,
! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A @ A2 ) @ B )
= bot_bo2099793752762293965at_nat )
= ( ~ ( member8440522571783428010at_nat @ A @ B )
& ( ( inf_in2572325071724192079at_nat @ A2 @ B )
= bot_bo2099793752762293965at_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_277_insert__disjoint_I1_J,axiom,
! [A: produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( ( inf_in8776938414804536127at_nat @ ( insert3260075854425521959at_nat @ A @ A2 ) @ B )
= bot_bo1973934853101755969at_nat )
= ( ~ ( member6956540099943067662at_nat @ A @ B )
& ( ( inf_in8776938414804536127at_nat @ A2 @ B )
= bot_bo1973934853101755969at_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_278_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_279_insert__disjoint_I1_J,axiom,
! [A: list_a,A2: set_list_a,B: set_list_a] :
( ( ( inf_inf_set_list_a @ ( insert_list_a @ A @ A2 ) @ B )
= bot_bot_set_list_a )
= ( ~ ( member_list_a @ A @ B )
& ( ( inf_inf_set_list_a @ A2 @ B )
= bot_bot_set_list_a ) ) ) ).
% insert_disjoint(1)
thf(fact_280_insert__disjoint_I1_J,axiom,
! [A: nat,A2: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ ( insert_nat @ A @ A2 ) @ B )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A @ B )
& ( ( inf_inf_set_nat @ A2 @ B )
= bot_bot_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_281_insert__disjoint_I2_J,axiom,
! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ( bot_bo2099793752762293965at_nat
= ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A @ A2 ) @ B ) )
= ( ~ ( member8440522571783428010at_nat @ A @ B )
& ( bot_bo2099793752762293965at_nat
= ( inf_in2572325071724192079at_nat @ A2 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_282_insert__disjoint_I2_J,axiom,
! [A: produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( bot_bo1973934853101755969at_nat
= ( inf_in8776938414804536127at_nat @ ( insert3260075854425521959at_nat @ A @ A2 ) @ B ) )
= ( ~ ( member6956540099943067662at_nat @ A @ B )
& ( bot_bo1973934853101755969at_nat
= ( inf_in8776938414804536127at_nat @ A2 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_283_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_284_insert__disjoint_I2_J,axiom,
! [A: list_a,A2: set_list_a,B: set_list_a] :
( ( bot_bot_set_list_a
= ( inf_inf_set_list_a @ ( insert_list_a @ A @ A2 ) @ B ) )
= ( ~ ( member_list_a @ A @ B )
& ( bot_bot_set_list_a
= ( inf_inf_set_list_a @ A2 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_285_insert__disjoint_I2_J,axiom,
! [A: nat,A2: set_nat,B: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ ( insert_nat @ A @ A2 ) @ B ) )
= ( ~ ( member_nat @ A @ B )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_286_disjoint__insert_I1_J,axiom,
! [B: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
( ( ( inf_in2572325071724192079at_nat @ B @ ( insert8211810215607154385at_nat @ A @ A2 ) )
= bot_bo2099793752762293965at_nat )
= ( ~ ( member8440522571783428010at_nat @ A @ B )
& ( ( inf_in2572325071724192079at_nat @ B @ A2 )
= bot_bo2099793752762293965at_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_287_disjoint__insert_I1_J,axiom,
! [B: set_Pr2645174627780777389at_nat,A: produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat] :
( ( ( inf_in8776938414804536127at_nat @ B @ ( insert3260075854425521959at_nat @ A @ A2 ) )
= bot_bo1973934853101755969at_nat )
= ( ~ ( member6956540099943067662at_nat @ A @ B )
& ( ( inf_in8776938414804536127at_nat @ B @ A2 )
= bot_bo1973934853101755969at_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_288_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_289_disjoint__insert_I1_J,axiom,
! [B: set_list_a,A: list_a,A2: set_list_a] :
( ( ( inf_inf_set_list_a @ B @ ( insert_list_a @ A @ A2 ) )
= bot_bot_set_list_a )
= ( ~ ( member_list_a @ A @ B )
& ( ( inf_inf_set_list_a @ B @ A2 )
= bot_bot_set_list_a ) ) ) ).
% disjoint_insert(1)
thf(fact_290_disjoint__insert_I1_J,axiom,
! [B: set_nat,A: nat,A2: set_nat] :
( ( ( inf_inf_set_nat @ B @ ( insert_nat @ A @ A2 ) )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A @ B )
& ( ( inf_inf_set_nat @ B @ A2 )
= bot_bot_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_291_disjoint__insert_I2_J,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat] :
( ( bot_bo2099793752762293965at_nat
= ( inf_in2572325071724192079at_nat @ A2 @ ( insert8211810215607154385at_nat @ B2 @ B ) ) )
= ( ~ ( member8440522571783428010at_nat @ B2 @ A2 )
& ( bot_bo2099793752762293965at_nat
= ( inf_in2572325071724192079at_nat @ A2 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_292_disjoint__insert_I2_J,axiom,
! [A2: set_Pr2645174627780777389at_nat,B2: produc5825016348098550007at_nat,B: set_Pr2645174627780777389at_nat] :
( ( bot_bo1973934853101755969at_nat
= ( inf_in8776938414804536127at_nat @ A2 @ ( insert3260075854425521959at_nat @ B2 @ B ) ) )
= ( ~ ( member6956540099943067662at_nat @ B2 @ A2 )
& ( bot_bo1973934853101755969at_nat
= ( inf_in8776938414804536127at_nat @ A2 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_293_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_294_disjoint__insert_I2_J,axiom,
! [A2: set_list_a,B2: list_a,B: set_list_a] :
( ( bot_bot_set_list_a
= ( inf_inf_set_list_a @ A2 @ ( insert_list_a @ B2 @ B ) ) )
= ( ~ ( member_list_a @ B2 @ A2 )
& ( bot_bot_set_list_a
= ( inf_inf_set_list_a @ A2 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_295_disjoint__insert_I2_J,axiom,
! [A2: set_nat,B2: nat,B: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ A2 @ ( insert_nat @ B2 @ B ) ) )
= ( ~ ( member_nat @ B2 @ A2 )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_296_Diff__disjoint,axiom,
! [A2: set_list_a,B: set_list_a] :
( ( inf_inf_set_list_a @ A2 @ ( minus_646659088055828811list_a @ B @ A2 ) )
= bot_bot_set_list_a ) ).
% Diff_disjoint
thf(fact_297_Diff__disjoint,axiom,
! [A2: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ A2 @ ( minus_minus_set_nat @ B @ A2 ) )
= bot_bot_set_nat ) ).
% Diff_disjoint
thf(fact_298_Diff__disjoint,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( inf_in8776938414804536127at_nat @ A2 @ ( minus_6698950876951835142at_nat @ B @ A2 ) )
= bot_bo1973934853101755969at_nat ) ).
% Diff_disjoint
thf(fact_299_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_300_subset__mset_Oadd__le__same__cancel1,axiom,
! [B2: multiset_set_nat,A: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ ( plus_p8712254050562127327et_nat @ B2 @ A ) @ B2 )
= ( subset6078030600694693471et_nat @ A @ zero_z3157962936165190495et_nat ) ) ).
% subset_mset.add_le_same_cancel1
thf(fact_301_subset__mset_Oadd__le__same__cancel2,axiom,
! [A: multiset_set_nat,B2: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ ( plus_p8712254050562127327et_nat @ A @ B2 ) @ B2 )
= ( subset6078030600694693471et_nat @ A @ zero_z3157962936165190495et_nat ) ) ).
% subset_mset.add_le_same_cancel2
thf(fact_302_subset__mset_Ole__add__same__cancel1,axiom,
! [A: multiset_set_nat,B2: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ A @ ( plus_p8712254050562127327et_nat @ A @ B2 ) )
= ( subset6078030600694693471et_nat @ zero_z3157962936165190495et_nat @ B2 ) ) ).
% subset_mset.le_add_same_cancel1
thf(fact_303_subset__mset_Ole__add__same__cancel2,axiom,
! [A: multiset_set_nat,B2: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ A @ ( plus_p8712254050562127327et_nat @ B2 @ A ) )
= ( subset6078030600694693471et_nat @ zero_z3157962936165190495et_nat @ B2 ) ) ).
% subset_mset.le_add_same_cancel2
thf(fact_304_mset__subset__eq__multiset__union__diff__commute,axiom,
! [B: multiset_set_nat,A2: multiset_set_nat,C: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ B @ A2 )
=> ( ( plus_p8712254050562127327et_nat @ ( minus_7237264121398869807et_nat @ A2 @ B ) @ C )
= ( minus_7237264121398869807et_nat @ ( plus_p8712254050562127327et_nat @ A2 @ C ) @ B ) ) ) ).
% mset_subset_eq_multiset_union_diff_commute
thf(fact_305_subset__mset_Oadd__diff__assoc2,axiom,
! [A: multiset_set_nat,B2: multiset_set_nat,C2: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ A @ B2 )
=> ( ( plus_p8712254050562127327et_nat @ ( minus_7237264121398869807et_nat @ B2 @ A ) @ C2 )
= ( minus_7237264121398869807et_nat @ ( plus_p8712254050562127327et_nat @ B2 @ C2 ) @ A ) ) ) ).
% subset_mset.add_diff_assoc2
thf(fact_306_subset__mset_Oadd__diff__assoc,axiom,
! [A: multiset_set_nat,B2: multiset_set_nat,C2: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ A @ B2 )
=> ( ( plus_p8712254050562127327et_nat @ C2 @ ( minus_7237264121398869807et_nat @ B2 @ A ) )
= ( minus_7237264121398869807et_nat @ ( plus_p8712254050562127327et_nat @ C2 @ B2 ) @ A ) ) ) ).
% subset_mset.add_diff_assoc
thf(fact_307_subset__mset_Odiff__add,axiom,
! [A: multiset_set_nat,B2: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ A @ B2 )
=> ( ( plus_p8712254050562127327et_nat @ ( minus_7237264121398869807et_nat @ B2 @ A ) @ A )
= B2 ) ) ).
% subset_mset.diff_add
thf(fact_308_mset__set_Oempty,axiom,
( ( mset_set_set_nat @ bot_bot_set_set_nat )
= zero_z3157962936165190495et_nat ) ).
% mset_set.empty
thf(fact_309_mset__set_Oempty,axiom,
( ( mset_s3030574779765502096at_nat @ bot_bo1973934853101755969at_nat )
= zero_z1389175455191277904at_nat ) ).
% mset_set.empty
thf(fact_310_mset__set_Oempty,axiom,
( ( mset_s7504067419864366776la_a_b @ bot_bo4495933725496725865la_a_b )
= zero_z3857620524502647544la_a_b ) ).
% mset_set.empty
thf(fact_311_mset__set_Oempty,axiom,
( ( mset_set_list_a @ bot_bot_set_list_a )
= zero_z4454100511807792257list_a ) ).
% mset_set.empty
thf(fact_312_mset__set_Oempty,axiom,
( ( mset_set_nat @ bot_bot_set_nat )
= zero_z7348594199698428585et_nat ) ).
% mset_set.empty
thf(fact_313_mset__set_Oinfinite,axiom,
! [A2: set_nat] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( mset_set_nat @ A2 )
= zero_z7348594199698428585et_nat ) ) ).
% mset_set.infinite
thf(fact_314_mset__set_Oinfinite,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ~ ( finite5600759454172676150la_a_b @ A2 )
=> ( ( mset_s7504067419864366776la_a_b @ A2 )
= zero_z3857620524502647544la_a_b ) ) ).
% mset_set.infinite
thf(fact_315_mset__set_Oinfinite,axiom,
! [A2: set_Pr2645174627780777389at_nat] :
( ~ ( finite6790245451575510286at_nat @ A2 )
=> ( ( mset_s3030574779765502096at_nat @ A2 )
= zero_z1389175455191277904at_nat ) ) ).
% mset_set.infinite
thf(fact_316_mset__set_Oinfinite,axiom,
! [A2: set_set_nat] :
( ~ ( finite1152437895449049373et_nat @ A2 )
=> ( ( mset_set_set_nat @ A2 )
= zero_z3157962936165190495et_nat ) ) ).
% mset_set.infinite
thf(fact_317_if__image__distrib,axiom,
! [P: relational_fmla_a_b > $o,F: relational_fmla_a_b > set_list_a,G: relational_fmla_a_b > set_list_a,S: set_Re381260168593705685la_a_b] :
( ( image_130269618930851390list_a
@ ^ [X: relational_fmla_a_b] : ( if_set_list_a @ ( P @ X ) @ ( F @ X ) @ ( G @ X ) )
@ S )
= ( sup_su4537662296134749976list_a @ ( image_130269618930851390list_a @ F @ ( inf_in8483230781156617063la_a_b @ S @ ( collec3419995626248312948la_a_b @ P ) ) )
@ ( image_130269618930851390list_a @ G
@ ( inf_in8483230781156617063la_a_b @ S
@ ( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] :
~ ( P @ X ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_318_if__image__distrib,axiom,
! [P: nat > $o,F: nat > produc5825016348098550007at_nat,G: nat > produc5825016348098550007at_nat,S: set_nat] :
( ( image_1371346403869992270at_nat
@ ^ [X: nat] : ( if_Pro1064994777914121649at_nat @ ( P @ X ) @ ( F @ X ) @ ( G @ X ) )
@ S )
= ( sup_su6099978769595272409at_nat @ ( image_1371346403869992270at_nat @ F @ ( inf_inf_set_nat @ S @ ( collect_nat @ P ) ) )
@ ( image_1371346403869992270at_nat @ G
@ ( inf_inf_set_nat @ S
@ ( collect_nat
@ ^ [X: nat] :
~ ( P @ X ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_319_if__image__distrib,axiom,
! [P: produc5825016348098550007at_nat > $o,F: produc5825016348098550007at_nat > produc5825016348098550007at_nat,G: produc5825016348098550007at_nat > produc5825016348098550007at_nat,S: set_Pr2645174627780777389at_nat] :
( ( image_6979785008349797877at_nat
@ ^ [X: produc5825016348098550007at_nat] : ( if_Pro1064994777914121649at_nat @ ( P @ X ) @ ( F @ X ) @ ( G @ X ) )
@ S )
= ( sup_su6099978769595272409at_nat @ ( image_6979785008349797877at_nat @ F @ ( inf_in8776938414804536127at_nat @ S @ ( collec7697611494764367948at_nat @ P ) ) )
@ ( image_6979785008349797877at_nat @ G
@ ( inf_in8776938414804536127at_nat @ S
@ ( collec7697611494764367948at_nat
@ ^ [X: produc5825016348098550007at_nat] :
~ ( P @ X ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_320_if__image__distrib,axiom,
! [P: nat > $o,F: nat > relational_fmla_a_b,G: nat > relational_fmla_a_b,S: set_nat] :
( ( image_4386371547000553590la_a_b
@ ^ [X: nat] : ( if_Rel1279876242545935705la_a_b @ ( P @ X ) @ ( F @ X ) @ ( G @ X ) )
@ S )
= ( sup_su5130108678486352897la_a_b @ ( image_4386371547000553590la_a_b @ F @ ( inf_inf_set_nat @ S @ ( collect_nat @ P ) ) )
@ ( image_4386371547000553590la_a_b @ G
@ ( inf_inf_set_nat @ S
@ ( collect_nat
@ ^ [X: nat] :
~ ( P @ X ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_321_if__image__distrib,axiom,
! [P: produc5825016348098550007at_nat > $o,F: produc5825016348098550007at_nat > relational_fmla_a_b,G: produc5825016348098550007at_nat > relational_fmla_a_b,S: set_Pr2645174627780777389at_nat] :
( ( image_7406987224865000349la_a_b
@ ^ [X: produc5825016348098550007at_nat] : ( if_Rel1279876242545935705la_a_b @ ( P @ X ) @ ( F @ X ) @ ( G @ X ) )
@ S )
= ( sup_su5130108678486352897la_a_b @ ( image_7406987224865000349la_a_b @ F @ ( inf_in8776938414804536127at_nat @ S @ ( collec7697611494764367948at_nat @ P ) ) )
@ ( image_7406987224865000349la_a_b @ G
@ ( inf_in8776938414804536127at_nat @ S
@ ( collec7697611494764367948at_nat
@ ^ [X: produc5825016348098550007at_nat] :
~ ( P @ X ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_322_msubset__mset__set__iff,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ A2 )
=> ( ( finite5600759454172676150la_a_b @ B )
=> ( ( subset1288168260673010552la_a_b @ ( mset_s7504067419864366776la_a_b @ A2 ) @ ( mset_s7504067419864366776la_a_b @ B ) )
= ( ord_le4112832032246704949la_a_b @ A2 @ B ) ) ) ) ).
% msubset_mset_set_iff
thf(fact_323_msubset__mset__set__iff,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ B )
=> ( ( subset6078030600694693471et_nat @ ( mset_set_set_nat @ A2 ) @ ( mset_set_set_nat @ B ) )
= ( ord_le6893508408891458716et_nat @ A2 @ B ) ) ) ) ).
% msubset_mset_set_iff
thf(fact_324_msubset__mset__set__iff,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( finite6790245451575510286at_nat @ A2 )
=> ( ( finite6790245451575510286at_nat @ B )
=> ( ( subset435230573610732880at_nat @ ( mset_s3030574779765502096at_nat @ A2 ) @ ( mset_s3030574779765502096at_nat @ B ) )
= ( ord_le8520675249591772685at_nat @ A2 @ B ) ) ) ) ).
% msubset_mset_set_iff
thf(fact_325_msubset__mset__set__iff,axiom,
! [A2: set_nat,B: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( ( subseteq_mset_nat @ ( mset_set_nat @ A2 ) @ ( mset_set_nat @ B ) )
= ( ord_less_eq_set_nat @ A2 @ B ) ) ) ) ).
% msubset_mset_set_iff
thf(fact_326_subset__mset_Otrans,axiom,
! [A: multiset_set_nat,B2: multiset_set_nat,C2: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ A @ B2 )
=> ( ( subset6078030600694693471et_nat @ B2 @ C2 )
=> ( subset6078030600694693471et_nat @ A @ C2 ) ) ) ).
% subset_mset.trans
thf(fact_327_subset__mset_Oeq__iff,axiom,
( ( ^ [Y2: multiset_set_nat,Z: multiset_set_nat] : ( Y2 = Z ) )
= ( ^ [A3: multiset_set_nat,B3: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ A3 @ B3 )
& ( subset6078030600694693471et_nat @ B3 @ A3 ) ) ) ) ).
% subset_mset.eq_iff
thf(fact_328_subset__mset_Oantisym,axiom,
! [A: multiset_set_nat,B2: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ A @ B2 )
=> ( ( subset6078030600694693471et_nat @ B2 @ A )
=> ( A = B2 ) ) ) ).
% subset_mset.antisym
thf(fact_329_subset__mset_Oeq__refl,axiom,
! [X2: multiset_set_nat,Y3: multiset_set_nat] :
( ( X2 = Y3 )
=> ( subset6078030600694693471et_nat @ X2 @ Y3 ) ) ).
% subset_mset.eq_refl
thf(fact_330_subset__mset_Ozero__le,axiom,
! [X2: multiset_set_nat] : ( subset6078030600694693471et_nat @ zero_z3157962936165190495et_nat @ X2 ) ).
% subset_mset.zero_le
thf(fact_331_subset__mset_Obot__least,axiom,
! [A: multiset_set_nat] : ( subset6078030600694693471et_nat @ zero_z3157962936165190495et_nat @ A ) ).
% subset_mset.bot_least
thf(fact_332_subset__mset_Oorder__trans,axiom,
! [X2: multiset_set_nat,Y3: multiset_set_nat,Z2: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ X2 @ Y3 )
=> ( ( subset6078030600694693471et_nat @ Y3 @ Z2 )
=> ( subset6078030600694693471et_nat @ X2 @ Z2 ) ) ) ).
% subset_mset.order_trans
thf(fact_333_subset__mset_Oantisym__conv,axiom,
! [Y3: multiset_set_nat,X2: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ Y3 @ X2 )
=> ( ( subset6078030600694693471et_nat @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% subset_mset.antisym_conv
thf(fact_334_subset__mset_Oorder__eq__iff,axiom,
( ( ^ [Y2: multiset_set_nat,Z: multiset_set_nat] : ( Y2 = Z ) )
= ( ^ [X: multiset_set_nat,Y: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ X @ Y )
& ( subset6078030600694693471et_nat @ Y @ X ) ) ) ) ).
% subset_mset.order_eq_iff
thf(fact_335_subset__mset_Oorder__antisym,axiom,
! [X2: multiset_set_nat,Y3: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ X2 @ Y3 )
=> ( ( subset6078030600694693471et_nat @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% subset_mset.order_antisym
thf(fact_336_subset__mset_Oadd__decreasing,axiom,
! [A: multiset_set_nat,C2: multiset_set_nat,B2: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ A @ zero_z3157962936165190495et_nat )
=> ( ( subset6078030600694693471et_nat @ C2 @ B2 )
=> ( subset6078030600694693471et_nat @ ( plus_p8712254050562127327et_nat @ A @ C2 ) @ B2 ) ) ) ).
% subset_mset.add_decreasing
thf(fact_337_subset__mset_Oadd__increasing,axiom,
! [A: multiset_set_nat,B2: multiset_set_nat,C2: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ zero_z3157962936165190495et_nat @ A )
=> ( ( subset6078030600694693471et_nat @ B2 @ C2 )
=> ( subset6078030600694693471et_nat @ B2 @ ( plus_p8712254050562127327et_nat @ A @ C2 ) ) ) ) ).
% subset_mset.add_increasing
thf(fact_338_subset__mset_Oadd__decreasing2,axiom,
! [C2: multiset_set_nat,A: multiset_set_nat,B2: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ C2 @ zero_z3157962936165190495et_nat )
=> ( ( subset6078030600694693471et_nat @ A @ B2 )
=> ( subset6078030600694693471et_nat @ ( plus_p8712254050562127327et_nat @ A @ C2 ) @ B2 ) ) ) ).
% subset_mset.add_decreasing2
thf(fact_339_subset__mset_Oadd__increasing2,axiom,
! [C2: multiset_set_nat,B2: multiset_set_nat,A: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ zero_z3157962936165190495et_nat @ C2 )
=> ( ( subset6078030600694693471et_nat @ B2 @ A )
=> ( subset6078030600694693471et_nat @ B2 @ ( plus_p8712254050562127327et_nat @ A @ C2 ) ) ) ) ).
% subset_mset.add_increasing2
thf(fact_340_subset__mset_Oord__eq__le__trans,axiom,
! [A: multiset_set_nat,B2: multiset_set_nat,C2: multiset_set_nat] :
( ( A = B2 )
=> ( ( subset6078030600694693471et_nat @ B2 @ C2 )
=> ( subset6078030600694693471et_nat @ A @ C2 ) ) ) ).
% subset_mset.ord_eq_le_trans
thf(fact_341_subset__mset_Oord__le__eq__trans,axiom,
! [A: multiset_set_nat,B2: multiset_set_nat,C2: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ A @ B2 )
=> ( ( B2 = C2 )
=> ( subset6078030600694693471et_nat @ A @ C2 ) ) ) ).
% subset_mset.ord_le_eq_trans
thf(fact_342_subset__mset_Odual__order_Otrans,axiom,
! [B2: multiset_set_nat,A: multiset_set_nat,C2: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ B2 @ A )
=> ( ( subset6078030600694693471et_nat @ C2 @ B2 )
=> ( subset6078030600694693471et_nat @ C2 @ A ) ) ) ).
% subset_mset.dual_order.trans
thf(fact_343_subset__mset_Oextremum__uniqueI,axiom,
! [A: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ A @ zero_z3157962936165190495et_nat )
=> ( A = zero_z3157962936165190495et_nat ) ) ).
% subset_mset.extremum_uniqueI
thf(fact_344_subset__mset_Oadd__nonneg__nonneg,axiom,
! [A: multiset_set_nat,B2: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ zero_z3157962936165190495et_nat @ A )
=> ( ( subset6078030600694693471et_nat @ zero_z3157962936165190495et_nat @ B2 )
=> ( subset6078030600694693471et_nat @ zero_z3157962936165190495et_nat @ ( plus_p8712254050562127327et_nat @ A @ B2 ) ) ) ) ).
% subset_mset.add_nonneg_nonneg
thf(fact_345_subset__mset_Oadd__nonpos__nonpos,axiom,
! [A: multiset_set_nat,B2: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ A @ zero_z3157962936165190495et_nat )
=> ( ( subset6078030600694693471et_nat @ B2 @ zero_z3157962936165190495et_nat )
=> ( subset6078030600694693471et_nat @ ( plus_p8712254050562127327et_nat @ A @ B2 ) @ zero_z3157962936165190495et_nat ) ) ) ).
% subset_mset.add_nonpos_nonpos
thf(fact_346_subset__mset_Odual__order_Oeq__iff,axiom,
( ( ^ [Y2: multiset_set_nat,Z: multiset_set_nat] : ( Y2 = Z ) )
= ( ^ [A3: multiset_set_nat,B3: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ B3 @ A3 )
& ( subset6078030600694693471et_nat @ A3 @ B3 ) ) ) ) ).
% subset_mset.dual_order.eq_iff
thf(fact_347_subset__mset_Odual__order_Oantisym,axiom,
! [B2: multiset_set_nat,A: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ B2 @ A )
=> ( ( subset6078030600694693471et_nat @ A @ B2 )
=> ( A = B2 ) ) ) ).
% subset_mset.dual_order.antisym
thf(fact_348_subset__mset_Oadd__nonneg__eq__0__iff,axiom,
! [X2: multiset_set_nat,Y3: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ zero_z3157962936165190495et_nat @ X2 )
=> ( ( subset6078030600694693471et_nat @ zero_z3157962936165190495et_nat @ Y3 )
=> ( ( ( plus_p8712254050562127327et_nat @ X2 @ Y3 )
= zero_z3157962936165190495et_nat )
= ( ( X2 = zero_z3157962936165190495et_nat )
& ( Y3 = zero_z3157962936165190495et_nat ) ) ) ) ) ).
% subset_mset.add_nonneg_eq_0_iff
thf(fact_349_subset__mset_Oadd__nonpos__eq__0__iff,axiom,
! [X2: multiset_set_nat,Y3: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ X2 @ zero_z3157962936165190495et_nat )
=> ( ( subset6078030600694693471et_nat @ Y3 @ zero_z3157962936165190495et_nat )
=> ( ( ( plus_p8712254050562127327et_nat @ X2 @ Y3 )
= zero_z3157962936165190495et_nat )
= ( ( X2 = zero_z3157962936165190495et_nat )
& ( Y3 = zero_z3157962936165190495et_nat ) ) ) ) ) ).
% subset_mset.add_nonpos_eq_0_iff
thf(fact_350_IntE,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A2 @ B ) )
=> ~ ( ( member_nat @ C2 @ A2 )
=> ~ ( member_nat @ C2 @ B ) ) ) ).
% IntE
thf(fact_351_IntE,axiom,
! [C2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C2 @ ( inf_in8483230781156617063la_a_b @ A2 @ B ) )
=> ~ ( ( member4680049679412964150la_a_b @ C2 @ A2 )
=> ~ ( member4680049679412964150la_a_b @ C2 @ B ) ) ) ).
% IntE
thf(fact_352_IntE,axiom,
! [C2: produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ C2 @ ( inf_in8776938414804536127at_nat @ A2 @ B ) )
=> ~ ( ( member6956540099943067662at_nat @ C2 @ A2 )
=> ~ ( member6956540099943067662at_nat @ C2 @ B ) ) ) ).
% IntE
thf(fact_353_IntD1,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A2 @ B ) )
=> ( member_nat @ C2 @ A2 ) ) ).
% IntD1
thf(fact_354_IntD1,axiom,
! [C2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C2 @ ( inf_in8483230781156617063la_a_b @ A2 @ B ) )
=> ( member4680049679412964150la_a_b @ C2 @ A2 ) ) ).
% IntD1
thf(fact_355_IntD1,axiom,
! [C2: produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ C2 @ ( inf_in8776938414804536127at_nat @ A2 @ B ) )
=> ( member6956540099943067662at_nat @ C2 @ A2 ) ) ).
% IntD1
thf(fact_356_IntD2,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A2 @ B ) )
=> ( member_nat @ C2 @ B ) ) ).
% IntD2
thf(fact_357_IntD2,axiom,
! [C2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C2 @ ( inf_in8483230781156617063la_a_b @ A2 @ B ) )
=> ( member4680049679412964150la_a_b @ C2 @ B ) ) ).
% IntD2
thf(fact_358_IntD2,axiom,
! [C2: produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ C2 @ ( inf_in8776938414804536127at_nat @ A2 @ B ) )
=> ( member6956540099943067662at_nat @ C2 @ B ) ) ).
% IntD2
thf(fact_359_in__mono,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,X2: relational_fmla_a_b] :
( ( ord_le4112832032246704949la_a_b @ A2 @ B )
=> ( ( member4680049679412964150la_a_b @ X2 @ A2 )
=> ( member4680049679412964150la_a_b @ X2 @ B ) ) ) ).
% in_mono
thf(fact_360_in__mono,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat,X2: produc5825016348098550007at_nat] :
( ( ord_le8520675249591772685at_nat @ A2 @ B )
=> ( ( member6956540099943067662at_nat @ X2 @ A2 )
=> ( member6956540099943067662at_nat @ X2 @ B ) ) ) ).
% in_mono
thf(fact_361_in__mono,axiom,
! [A2: set_nat,B: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( member_nat @ X2 @ A2 )
=> ( member_nat @ X2 @ B ) ) ) ).
% in_mono
thf(fact_362_subsetD,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,C2: relational_fmla_a_b] :
( ( ord_le4112832032246704949la_a_b @ A2 @ B )
=> ( ( member4680049679412964150la_a_b @ C2 @ A2 )
=> ( member4680049679412964150la_a_b @ C2 @ B ) ) ) ).
% subsetD
thf(fact_363_subsetD,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat,C2: produc5825016348098550007at_nat] :
( ( ord_le8520675249591772685at_nat @ A2 @ B )
=> ( ( member6956540099943067662at_nat @ C2 @ A2 )
=> ( member6956540099943067662at_nat @ C2 @ B ) ) ) ).
% subsetD
thf(fact_364_subsetD,axiom,
! [A2: set_nat,B: set_nat,C2: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( member_nat @ C2 @ A2 )
=> ( member_nat @ C2 @ B ) ) ) ).
% subsetD
thf(fact_365_Int__mono,axiom,
! [A2: set_Pr2645174627780777389at_nat,C: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat,D: set_Pr2645174627780777389at_nat] :
( ( ord_le8520675249591772685at_nat @ A2 @ C )
=> ( ( ord_le8520675249591772685at_nat @ B @ D )
=> ( ord_le8520675249591772685at_nat @ ( inf_in8776938414804536127at_nat @ A2 @ B ) @ ( inf_in8776938414804536127at_nat @ C @ D ) ) ) ) ).
% Int_mono
thf(fact_366_Int__mono,axiom,
! [A2: set_nat,C: set_nat,B: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ( ord_less_eq_set_nat @ B @ D )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ ( inf_inf_set_nat @ C @ D ) ) ) ) ).
% Int_mono
thf(fact_367_Int__assoc,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat,C: set_Pr2645174627780777389at_nat] :
( ( inf_in8776938414804536127at_nat @ ( inf_in8776938414804536127at_nat @ A2 @ B ) @ C )
= ( inf_in8776938414804536127at_nat @ A2 @ ( inf_in8776938414804536127at_nat @ B @ C ) ) ) ).
% Int_assoc
thf(fact_368_equalityE,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( A2 = B )
=> ~ ( ( ord_le8520675249591772685at_nat @ A2 @ B )
=> ~ ( ord_le8520675249591772685at_nat @ B @ A2 ) ) ) ).
% equalityE
thf(fact_369_equalityE,axiom,
! [A2: set_nat,B: set_nat] :
( ( A2 = B )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ B )
=> ~ ( ord_less_eq_set_nat @ B @ A2 ) ) ) ).
% equalityE
thf(fact_370_subset__eq,axiom,
( ord_le4112832032246704949la_a_b
= ( ^ [A4: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b] :
! [X: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X @ A4 )
=> ( member4680049679412964150la_a_b @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_371_subset__eq,axiom,
( ord_le8520675249591772685at_nat
= ( ^ [A4: set_Pr2645174627780777389at_nat,B4: set_Pr2645174627780777389at_nat] :
! [X: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X @ A4 )
=> ( member6956540099943067662at_nat @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_372_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( member_nat @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_373_Int__absorb,axiom,
! [A2: set_Pr2645174627780777389at_nat] :
( ( inf_in8776938414804536127at_nat @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_374_Int__lower1,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] : ( ord_le8520675249591772685at_nat @ ( inf_in8776938414804536127at_nat @ A2 @ B ) @ A2 ) ).
% Int_lower1
thf(fact_375_Int__lower1,axiom,
! [A2: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ A2 ) ).
% Int_lower1
thf(fact_376_Int__lower2,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] : ( ord_le8520675249591772685at_nat @ ( inf_in8776938414804536127at_nat @ A2 @ B ) @ B ) ).
% Int_lower2
thf(fact_377_Int__lower2,axiom,
! [A2: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ B ) ).
% Int_lower2
thf(fact_378_equalityD1,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( A2 = B )
=> ( ord_le8520675249591772685at_nat @ A2 @ B ) ) ).
% equalityD1
thf(fact_379_equalityD1,axiom,
! [A2: set_nat,B: set_nat] :
( ( A2 = B )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% equalityD1
thf(fact_380_Set_OequalityD2,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( A2 = B )
=> ( ord_le8520675249591772685at_nat @ B @ A2 ) ) ).
% Set.equalityD2
thf(fact_381_Set_OequalityD2,axiom,
! [A2: set_nat,B: set_nat] :
( ( A2 = B )
=> ( ord_less_eq_set_nat @ B @ A2 ) ) ).
% Set.equalityD2
thf(fact_382_subset__iff,axiom,
( ord_le4112832032246704949la_a_b
= ( ^ [A4: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b] :
! [T2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ T2 @ A4 )
=> ( member4680049679412964150la_a_b @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_383_subset__iff,axiom,
( ord_le8520675249591772685at_nat
= ( ^ [A4: set_Pr2645174627780777389at_nat,B4: set_Pr2645174627780777389at_nat] :
! [T2: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ T2 @ A4 )
=> ( member6956540099943067662at_nat @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_384_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A4 )
=> ( member_nat @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_385_Int__absorb1,axiom,
! [B: set_Pr2645174627780777389at_nat,A2: set_Pr2645174627780777389at_nat] :
( ( ord_le8520675249591772685at_nat @ B @ A2 )
=> ( ( inf_in8776938414804536127at_nat @ A2 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_386_Int__absorb1,axiom,
! [B: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_387_Int__absorb2,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( ord_le8520675249591772685at_nat @ A2 @ B )
=> ( ( inf_in8776938414804536127at_nat @ A2 @ B )
= A2 ) ) ).
% Int_absorb2
thf(fact_388_Int__absorb2,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( inf_inf_set_nat @ A2 @ B )
= A2 ) ) ).
% Int_absorb2
thf(fact_389_Int__commute,axiom,
( inf_in8776938414804536127at_nat
= ( ^ [A4: set_Pr2645174627780777389at_nat,B4: set_Pr2645174627780777389at_nat] : ( inf_in8776938414804536127at_nat @ B4 @ A4 ) ) ) ).
% Int_commute
thf(fact_390_subset__refl,axiom,
! [A2: set_Pr2645174627780777389at_nat] : ( ord_le8520675249591772685at_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_391_subset__refl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_392_Collect__mono,axiom,
! [P: produc5825016348098550007at_nat > $o,Q: produc5825016348098550007at_nat > $o] :
( ! [X3: produc5825016348098550007at_nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le8520675249591772685at_nat @ ( collec7697611494764367948at_nat @ P ) @ ( collec7697611494764367948at_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_393_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_394_Int__greatest,axiom,
! [C: set_Pr2645174627780777389at_nat,A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( ord_le8520675249591772685at_nat @ C @ A2 )
=> ( ( ord_le8520675249591772685at_nat @ C @ B )
=> ( ord_le8520675249591772685at_nat @ C @ ( inf_in8776938414804536127at_nat @ A2 @ B ) ) ) ) ).
% Int_greatest
thf(fact_395_Int__greatest,axiom,
! [C: set_nat,A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C @ A2 )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_eq_set_nat @ C @ ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% Int_greatest
thf(fact_396_subset__trans,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat,C: set_Pr2645174627780777389at_nat] :
( ( ord_le8520675249591772685at_nat @ A2 @ B )
=> ( ( ord_le8520675249591772685at_nat @ B @ C )
=> ( ord_le8520675249591772685at_nat @ A2 @ C ) ) ) ).
% subset_trans
thf(fact_397_subset__trans,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% subset_trans
thf(fact_398_empty__le,axiom,
! [A2: multiset_set_nat] : ( subset6078030600694693471et_nat @ zero_z3157962936165190495et_nat @ A2 ) ).
% empty_le
thf(fact_399_set__eq__subset,axiom,
( ( ^ [Y2: set_Pr2645174627780777389at_nat,Z: set_Pr2645174627780777389at_nat] : ( Y2 = Z ) )
= ( ^ [A4: set_Pr2645174627780777389at_nat,B4: set_Pr2645174627780777389at_nat] :
( ( ord_le8520675249591772685at_nat @ A4 @ B4 )
& ( ord_le8520675249591772685at_nat @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_400_set__eq__subset,axiom,
( ( ^ [Y2: set_nat,Z: set_nat] : ( Y2 = Z ) )
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_401_Int__left__absorb,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( inf_in8776938414804536127at_nat @ A2 @ ( inf_in8776938414804536127at_nat @ A2 @ B ) )
= ( inf_in8776938414804536127at_nat @ A2 @ B ) ) ).
% Int_left_absorb
thf(fact_402_Collect__mono__iff,axiom,
! [P: produc5825016348098550007at_nat > $o,Q: produc5825016348098550007at_nat > $o] :
( ( ord_le8520675249591772685at_nat @ ( collec7697611494764367948at_nat @ P ) @ ( collec7697611494764367948at_nat @ Q ) )
= ( ! [X: produc5825016348098550007at_nat] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_403_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X: nat] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_404_Int__Collect__mono,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,P: relational_fmla_a_b > $o,Q: relational_fmla_a_b > $o] :
( ( ord_le4112832032246704949la_a_b @ A2 @ B )
=> ( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A2 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le4112832032246704949la_a_b @ ( inf_in8483230781156617063la_a_b @ A2 @ ( collec3419995626248312948la_a_b @ P ) ) @ ( inf_in8483230781156617063la_a_b @ B @ ( collec3419995626248312948la_a_b @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_405_Int__Collect__mono,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat,P: produc5825016348098550007at_nat > $o,Q: produc5825016348098550007at_nat > $o] :
( ( ord_le8520675249591772685at_nat @ A2 @ B )
=> ( ! [X3: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X3 @ A2 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le8520675249591772685at_nat @ ( inf_in8776938414804536127at_nat @ A2 @ ( collec7697611494764367948at_nat @ P ) ) @ ( inf_in8776938414804536127at_nat @ B @ ( collec7697611494764367948at_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_406_Int__Collect__mono,axiom,
! [A2: set_nat,B: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_407_Int__left__commute,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat,C: set_Pr2645174627780777389at_nat] :
( ( inf_in8776938414804536127at_nat @ A2 @ ( inf_in8776938414804536127at_nat @ B @ C ) )
= ( inf_in8776938414804536127at_nat @ B @ ( inf_in8776938414804536127at_nat @ A2 @ C ) ) ) ).
% Int_left_commute
thf(fact_408_Diff__eq__empty__iff__mset,axiom,
! [A2: multiset_set_nat,B: multiset_set_nat] :
( ( ( minus_7237264121398869807et_nat @ A2 @ B )
= zero_z3157962936165190495et_nat )
= ( subset6078030600694693471et_nat @ A2 @ B ) ) ).
% Diff_eq_empty_iff_mset
thf(fact_409_subset__imp__msubset__mset__set,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A2 @ B )
=> ( ( finite5600759454172676150la_a_b @ B )
=> ( subset1288168260673010552la_a_b @ ( mset_s7504067419864366776la_a_b @ A2 ) @ ( mset_s7504067419864366776la_a_b @ B ) ) ) ) ).
% subset_imp_msubset_mset_set
thf(fact_410_subset__imp__msubset__mset__set,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B )
=> ( ( finite1152437895449049373et_nat @ B )
=> ( subset6078030600694693471et_nat @ ( mset_set_set_nat @ A2 ) @ ( mset_set_set_nat @ B ) ) ) ) ).
% subset_imp_msubset_mset_set
thf(fact_411_subset__imp__msubset__mset__set,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( ord_le8520675249591772685at_nat @ A2 @ B )
=> ( ( finite6790245451575510286at_nat @ B )
=> ( subset435230573610732880at_nat @ ( mset_s3030574779765502096at_nat @ A2 ) @ ( mset_s3030574779765502096at_nat @ B ) ) ) ) ).
% subset_imp_msubset_mset_set
thf(fact_412_subset__imp__msubset__mset__set,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( finite_finite_nat @ B )
=> ( subseteq_mset_nat @ ( mset_set_nat @ A2 ) @ ( mset_set_nat @ B ) ) ) ) ).
% subset_imp_msubset_mset_set
thf(fact_413_Collect__conj__eq,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( P @ X )
& ( Q @ X ) ) )
= ( inf_inf_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_414_Collect__conj__eq,axiom,
! [P: produc5825016348098550007at_nat > $o,Q: produc5825016348098550007at_nat > $o] :
( ( collec7697611494764367948at_nat
@ ^ [X: produc5825016348098550007at_nat] :
( ( P @ X )
& ( Q @ X ) ) )
= ( inf_in8776938414804536127at_nat @ ( collec7697611494764367948at_nat @ P ) @ ( collec7697611494764367948at_nat @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_415_Int__Collect,axiom,
! [X2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,P: relational_fmla_a_b > $o] :
( ( member4680049679412964150la_a_b @ X2 @ ( inf_in8483230781156617063la_a_b @ A2 @ ( collec3419995626248312948la_a_b @ P ) ) )
= ( ( member4680049679412964150la_a_b @ X2 @ A2 )
& ( P @ X2 ) ) ) ).
% Int_Collect
thf(fact_416_Int__Collect,axiom,
! [X2: nat,A2: set_nat,P: nat > $o] :
( ( member_nat @ X2 @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) )
= ( ( member_nat @ X2 @ A2 )
& ( P @ X2 ) ) ) ).
% Int_Collect
thf(fact_417_Int__Collect,axiom,
! [X2: produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat,P: produc5825016348098550007at_nat > $o] :
( ( member6956540099943067662at_nat @ X2 @ ( inf_in8776938414804536127at_nat @ A2 @ ( collec7697611494764367948at_nat @ P ) ) )
= ( ( member6956540099943067662at_nat @ X2 @ A2 )
& ( P @ X2 ) ) ) ).
% Int_Collect
thf(fact_418_Int__def,axiom,
( inf_in8483230781156617063la_a_b
= ( ^ [A4: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b] :
( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X @ A4 )
& ( member4680049679412964150la_a_b @ X @ B4 ) ) ) ) ) ).
% Int_def
thf(fact_419_Int__def,axiom,
( inf_inf_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A4 )
& ( member_nat @ X @ B4 ) ) ) ) ) ).
% Int_def
thf(fact_420_Int__def,axiom,
( inf_in8776938414804536127at_nat
= ( ^ [A4: set_Pr2645174627780777389at_nat,B4: set_Pr2645174627780777389at_nat] :
( collec7697611494764367948at_nat
@ ^ [X: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X @ A4 )
& ( member6956540099943067662at_nat @ X @ B4 ) ) ) ) ) ).
% Int_def
thf(fact_421_Collect__subset,axiom,
! [A2: set_Re381260168593705685la_a_b,P: relational_fmla_a_b > $o] :
( ord_le4112832032246704949la_a_b
@ ( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_422_Collect__subset,axiom,
! [A2: set_Pr2645174627780777389at_nat,P: produc5825016348098550007at_nat > $o] :
( ord_le8520675249591772685at_nat
@ ( collec7697611494764367948at_nat
@ ^ [X: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_423_Collect__subset,axiom,
! [A2: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_424_mset__subset__eq__exists__conv,axiom,
( subset6078030600694693471et_nat
= ( ^ [A4: multiset_set_nat,B4: multiset_set_nat] :
? [C3: multiset_set_nat] :
( B4
= ( plus_p8712254050562127327et_nat @ A4 @ C3 ) ) ) ) ).
% mset_subset_eq_exists_conv
thf(fact_425_mset__subset__eq__add__right,axiom,
! [B: multiset_set_nat,A2: multiset_set_nat] : ( subset6078030600694693471et_nat @ B @ ( plus_p8712254050562127327et_nat @ A2 @ B ) ) ).
% mset_subset_eq_add_right
thf(fact_426_mset__subset__eq__mono__add,axiom,
! [A2: multiset_set_nat,B: multiset_set_nat,C: multiset_set_nat,D: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ A2 @ B )
=> ( ( subset6078030600694693471et_nat @ C @ D )
=> ( subset6078030600694693471et_nat @ ( plus_p8712254050562127327et_nat @ A2 @ C ) @ ( plus_p8712254050562127327et_nat @ B @ D ) ) ) ) ).
% mset_subset_eq_mono_add
thf(fact_427_mset__subset__eq__add__left,axiom,
! [A2: multiset_set_nat,B: multiset_set_nat] : ( subset6078030600694693471et_nat @ A2 @ ( plus_p8712254050562127327et_nat @ A2 @ B ) ) ).
% mset_subset_eq_add_left
thf(fact_428_subset__mset_Oadd__le__imp__le__right,axiom,
! [A: multiset_set_nat,C2: multiset_set_nat,B2: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ ( plus_p8712254050562127327et_nat @ A @ C2 ) @ ( plus_p8712254050562127327et_nat @ B2 @ C2 ) )
=> ( subset6078030600694693471et_nat @ A @ B2 ) ) ).
% subset_mset.add_le_imp_le_right
thf(fact_429_subset__mset_Oadd__le__imp__le__left,axiom,
! [C2: multiset_set_nat,A: multiset_set_nat,B2: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ ( plus_p8712254050562127327et_nat @ C2 @ A ) @ ( plus_p8712254050562127327et_nat @ C2 @ B2 ) )
=> ( subset6078030600694693471et_nat @ A @ B2 ) ) ).
% subset_mset.add_le_imp_le_left
thf(fact_430_subset__mset_Oadd__right__mono,axiom,
! [A: multiset_set_nat,B2: multiset_set_nat,C2: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ A @ B2 )
=> ( subset6078030600694693471et_nat @ ( plus_p8712254050562127327et_nat @ A @ C2 ) @ ( plus_p8712254050562127327et_nat @ B2 @ C2 ) ) ) ).
% subset_mset.add_right_mono
thf(fact_431_subset__mset_Oadd__left__mono,axiom,
! [A: multiset_set_nat,B2: multiset_set_nat,C2: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ A @ B2 )
=> ( subset6078030600694693471et_nat @ ( plus_p8712254050562127327et_nat @ C2 @ A ) @ ( plus_p8712254050562127327et_nat @ C2 @ B2 ) ) ) ).
% subset_mset.add_left_mono
thf(fact_432_subset__mset_Ole__iff__add,axiom,
( subset6078030600694693471et_nat
= ( ^ [A3: multiset_set_nat,B3: multiset_set_nat] :
? [C4: multiset_set_nat] :
( B3
= ( plus_p8712254050562127327et_nat @ A3 @ C4 ) ) ) ) ).
% subset_mset.le_iff_add
thf(fact_433_subset__mset_Oless__eqE,axiom,
! [A: multiset_set_nat,B2: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ A @ B2 )
=> ~ ! [C5: multiset_set_nat] :
( B2
!= ( plus_p8712254050562127327et_nat @ A @ C5 ) ) ) ).
% subset_mset.less_eqE
thf(fact_434_subset__mset_Oadd__mono,axiom,
! [A: multiset_set_nat,B2: multiset_set_nat,C2: multiset_set_nat,D2: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ A @ B2 )
=> ( ( subset6078030600694693471et_nat @ C2 @ D2 )
=> ( subset6078030600694693471et_nat @ ( plus_p8712254050562127327et_nat @ A @ C2 ) @ ( plus_p8712254050562127327et_nat @ B2 @ D2 ) ) ) ) ).
% subset_mset.add_mono
thf(fact_435_image__mset__subseteq__mono,axiom,
! [A2: multis4094885785038667591at_nat,B: multis4094885785038667591at_nat,F: produc5825016348098550007at_nat > set_nat] :
( ( subset435230573610732880at_nat @ A2 @ B )
=> ( subset6078030600694693471et_nat @ ( image_7702178775022393786et_nat @ F @ A2 ) @ ( image_7702178775022393786et_nat @ F @ B ) ) ) ).
% image_mset_subseteq_mono
thf(fact_436_image__mset__subseteq__mono,axiom,
! [A2: multiset_set_nat,B: multiset_set_nat,F: set_nat > set_nat] :
( ( subset6078030600694693471et_nat @ A2 @ B )
=> ( subset6078030600694693471et_nat @ ( image_1110420595810377289et_nat @ F @ A2 ) @ ( image_1110420595810377289et_nat @ F @ B ) ) ) ).
% image_mset_subseteq_mono
thf(fact_437_diff__subset__eq__self,axiom,
! [M: multiset_set_nat,N: multiset_set_nat] : ( subset6078030600694693471et_nat @ ( minus_7237264121398869807et_nat @ M @ N ) @ M ) ).
% diff_subset_eq_self
thf(fact_438_empty__neutral_I1_J,axiom,
! [X2: multiset_set_nat] :
( ( plus_p8712254050562127327et_nat @ zero_z3157962936165190495et_nat @ X2 )
= X2 ) ).
% empty_neutral(1)
thf(fact_439_empty__neutral_I2_J,axiom,
! [X2: multiset_set_nat] :
( ( plus_p8712254050562127327et_nat @ X2 @ zero_z3157962936165190495et_nat )
= X2 ) ).
% empty_neutral(2)
thf(fact_440_Multiset_Odiff__cancel,axiom,
! [A2: multiset_set_nat] :
( ( minus_7237264121398869807et_nat @ A2 @ A2 )
= zero_z3157962936165190495et_nat ) ).
% Multiset.diff_cancel
thf(fact_441_diff__empty,axiom,
! [M: multiset_set_nat] :
( ( ( minus_7237264121398869807et_nat @ M @ zero_z3157962936165190495et_nat )
= M )
& ( ( minus_7237264121398869807et_nat @ zero_z3157962936165190495et_nat @ M )
= zero_z3157962936165190495et_nat ) ) ).
% diff_empty
thf(fact_442_image__Int__subset,axiom,
! [F: nat > relational_fmla_a_b,A2: set_nat,B: set_nat] : ( ord_le4112832032246704949la_a_b @ ( image_4386371547000553590la_a_b @ F @ ( inf_inf_set_nat @ A2 @ B ) ) @ ( inf_in8483230781156617063la_a_b @ ( image_4386371547000553590la_a_b @ F @ A2 ) @ ( image_4386371547000553590la_a_b @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_443_image__Int__subset,axiom,
! [F: relational_fmla_a_b > set_list_a,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] : ( ord_le8877086941679407844list_a @ ( image_130269618930851390list_a @ F @ ( inf_in8483230781156617063la_a_b @ A2 @ B ) ) @ ( inf_in4657809108759609906list_a @ ( image_130269618930851390list_a @ F @ A2 ) @ ( image_130269618930851390list_a @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_444_image__Int__subset,axiom,
! [F: nat > produc5825016348098550007at_nat,A2: set_nat,B: set_nat] : ( ord_le8520675249591772685at_nat @ ( image_1371346403869992270at_nat @ F @ ( inf_inf_set_nat @ A2 @ B ) ) @ ( inf_in8776938414804536127at_nat @ ( image_1371346403869992270at_nat @ F @ A2 ) @ ( image_1371346403869992270at_nat @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_445_image__Int__subset,axiom,
! [F: produc5825016348098550007at_nat > produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] : ( ord_le8520675249591772685at_nat @ ( image_6979785008349797877at_nat @ F @ ( inf_in8776938414804536127at_nat @ A2 @ B ) ) @ ( inf_in8776938414804536127at_nat @ ( image_6979785008349797877at_nat @ F @ A2 ) @ ( image_6979785008349797877at_nat @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_446_image__Int__subset,axiom,
! [F: produc5825016348098550007at_nat > nat,A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] : ( ord_less_eq_set_nat @ ( image_8316665354072716238at_nat @ F @ ( inf_in8776938414804536127at_nat @ A2 @ B ) ) @ ( inf_inf_set_nat @ ( image_8316665354072716238at_nat @ F @ A2 ) @ ( image_8316665354072716238at_nat @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_447_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_448_Un__Int__assoc__eq,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat,C: set_Pr2645174627780777389at_nat] :
( ( ( sup_su6099978769595272409at_nat @ ( inf_in8776938414804536127at_nat @ A2 @ B ) @ C )
= ( inf_in8776938414804536127at_nat @ A2 @ ( sup_su6099978769595272409at_nat @ B @ C ) ) )
= ( ord_le8520675249591772685at_nat @ C @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_449_Un__Int__assoc__eq,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ C )
= ( inf_inf_set_nat @ A2 @ ( sup_sup_set_nat @ B @ C ) ) )
= ( ord_less_eq_set_nat @ C @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_450_mset__set__empty__iff,axiom,
! [A2: set_set_nat] :
( ( ( mset_set_set_nat @ A2 )
= zero_z3157962936165190495et_nat )
= ( ( A2 = bot_bot_set_set_nat )
| ~ ( finite1152437895449049373et_nat @ A2 ) ) ) ).
% mset_set_empty_iff
thf(fact_451_mset__set__empty__iff,axiom,
! [A2: set_Pr2645174627780777389at_nat] :
( ( ( mset_s3030574779765502096at_nat @ A2 )
= zero_z1389175455191277904at_nat )
= ( ( A2 = bot_bo1973934853101755969at_nat )
| ~ ( finite6790245451575510286at_nat @ A2 ) ) ) ).
% mset_set_empty_iff
thf(fact_452_mset__set__empty__iff,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( ( mset_s7504067419864366776la_a_b @ A2 )
= zero_z3857620524502647544la_a_b )
= ( ( A2 = bot_bo4495933725496725865la_a_b )
| ~ ( finite5600759454172676150la_a_b @ A2 ) ) ) ).
% mset_set_empty_iff
thf(fact_453_mset__set__empty__iff,axiom,
! [A2: set_list_a] :
( ( ( mset_set_list_a @ A2 )
= zero_z4454100511807792257list_a )
= ( ( A2 = bot_bot_set_list_a )
| ~ ( finite_finite_list_a @ A2 ) ) ) ).
% mset_set_empty_iff
thf(fact_454_mset__set__empty__iff,axiom,
! [A2: set_nat] :
( ( ( mset_set_nat @ A2 )
= zero_z7348594199698428585et_nat )
= ( ( A2 = bot_bot_set_nat )
| ~ ( finite_finite_nat @ A2 ) ) ) ).
% mset_set_empty_iff
thf(fact_455_multiset__diff__union__assoc,axiom,
! [C: multiset_set_nat,B: multiset_set_nat,A2: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ C @ B )
=> ( ( minus_7237264121398869807et_nat @ ( plus_p8712254050562127327et_nat @ A2 @ B ) @ C )
= ( plus_p8712254050562127327et_nat @ A2 @ ( minus_7237264121398869807et_nat @ B @ C ) ) ) ) ).
% multiset_diff_union_assoc
thf(fact_456_subset__eq__diff__conv,axiom,
! [A2: multiset_set_nat,C: multiset_set_nat,B: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ ( minus_7237264121398869807et_nat @ A2 @ C ) @ B )
= ( subset6078030600694693471et_nat @ A2 @ ( plus_p8712254050562127327et_nat @ B @ C ) ) ) ).
% subset_eq_diff_conv
thf(fact_457_subset__mset_Ole__imp__diff__is__add,axiom,
! [A: multiset_set_nat,B2: multiset_set_nat,C2: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ A @ B2 )
=> ( ( subset6078030600694693471et_nat @ A @ B2 )
=> ( ( ( minus_7237264121398869807et_nat @ B2 @ A )
= C2 )
= ( B2
= ( plus_p8712254050562127327et_nat @ C2 @ A ) ) ) ) ) ).
% subset_mset.le_imp_diff_is_add
thf(fact_458_subset__mset_Oadd__diff__inverse,axiom,
! [A: multiset_set_nat,B2: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ A @ B2 )
=> ( ( plus_p8712254050562127327et_nat @ A @ ( minus_7237264121398869807et_nat @ B2 @ A ) )
= B2 ) ) ).
% subset_mset.add_diff_inverse
thf(fact_459_subset__mset_Odiff__diff__right,axiom,
! [A: multiset_set_nat,B2: multiset_set_nat,C2: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ A @ B2 )
=> ( ( minus_7237264121398869807et_nat @ C2 @ ( minus_7237264121398869807et_nat @ B2 @ A ) )
= ( minus_7237264121398869807et_nat @ ( plus_p8712254050562127327et_nat @ C2 @ A ) @ B2 ) ) ) ).
% subset_mset.diff_diff_right
thf(fact_460_subset__mset_Odiff__add__assoc2,axiom,
! [A: multiset_set_nat,B2: multiset_set_nat,C2: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ A @ B2 )
=> ( ( minus_7237264121398869807et_nat @ ( plus_p8712254050562127327et_nat @ B2 @ C2 ) @ A )
= ( plus_p8712254050562127327et_nat @ ( minus_7237264121398869807et_nat @ B2 @ A ) @ C2 ) ) ) ).
% subset_mset.diff_add_assoc2
thf(fact_461_subset__mset_Odiff__add__assoc,axiom,
! [A: multiset_set_nat,B2: multiset_set_nat,C2: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ A @ B2 )
=> ( ( minus_7237264121398869807et_nat @ ( plus_p8712254050562127327et_nat @ C2 @ B2 ) @ A )
= ( plus_p8712254050562127327et_nat @ C2 @ ( minus_7237264121398869807et_nat @ B2 @ A ) ) ) ) ).
% subset_mset.diff_add_assoc
thf(fact_462_subset__mset_Ole__diff__conv2,axiom,
! [A: multiset_set_nat,B2: multiset_set_nat,C2: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ A @ B2 )
=> ( ( subset6078030600694693471et_nat @ C2 @ ( minus_7237264121398869807et_nat @ B2 @ A ) )
= ( subset6078030600694693471et_nat @ ( plus_p8712254050562127327et_nat @ C2 @ A ) @ B2 ) ) ) ).
% subset_mset.le_diff_conv2
thf(fact_463_subset__mset_Ole__add__diff,axiom,
! [A: multiset_set_nat,B2: multiset_set_nat,C2: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ A @ B2 )
=> ( subset6078030600694693471et_nat @ C2 @ ( minus_7237264121398869807et_nat @ ( plus_p8712254050562127327et_nat @ B2 @ C2 ) @ A ) ) ) ).
% subset_mset.le_add_diff
thf(fact_464_image__mset__Diff,axiom,
! [B: multis4094885785038667591at_nat,A2: multis4094885785038667591at_nat,F: produc5825016348098550007at_nat > set_nat] :
( ( subset435230573610732880at_nat @ B @ A2 )
=> ( ( image_7702178775022393786et_nat @ F @ ( minus_1138467844667854112at_nat @ A2 @ B ) )
= ( minus_7237264121398869807et_nat @ ( image_7702178775022393786et_nat @ F @ A2 ) @ ( image_7702178775022393786et_nat @ F @ B ) ) ) ) ).
% image_mset_Diff
thf(fact_465_image__mset__Diff,axiom,
! [B: multiset_set_nat,A2: multiset_set_nat,F: set_nat > set_nat] :
( ( subset6078030600694693471et_nat @ B @ A2 )
=> ( ( image_1110420595810377289et_nat @ F @ ( minus_7237264121398869807et_nat @ A2 @ B ) )
= ( minus_7237264121398869807et_nat @ ( image_1110420595810377289et_nat @ F @ A2 ) @ ( image_1110420595810377289et_nat @ F @ B ) ) ) ) ).
% image_mset_Diff
thf(fact_466_disjoint__iff__not__equal,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( ( inf_in8776938414804536127at_nat @ A2 @ B )
= bot_bo1973934853101755969at_nat )
= ( ! [X: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X @ A2 )
=> ! [Y: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ Y @ B )
=> ( X != Y ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_467_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 )
= ( ! [X: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X @ A2 )
=> ! [Y: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Y @ B )
=> ( X != Y ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_468_disjoint__iff__not__equal,axiom,
! [A2: set_list_a,B: set_list_a] :
( ( ( inf_inf_set_list_a @ A2 @ B )
= bot_bot_set_list_a )
= ( ! [X: list_a] :
( ( member_list_a @ X @ A2 )
=> ! [Y: list_a] :
( ( member_list_a @ Y @ B )
=> ( X != Y ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_469_disjoint__iff__not__equal,axiom,
! [A2: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B )
= bot_bot_set_nat )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ! [Y: nat] :
( ( member_nat @ Y @ B )
=> ( X != Y ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_470_Int__empty__right,axiom,
! [A2: set_Pr2645174627780777389at_nat] :
( ( inf_in8776938414804536127at_nat @ A2 @ bot_bo1973934853101755969at_nat )
= bot_bo1973934853101755969at_nat ) ).
% Int_empty_right
thf(fact_471_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_472_Int__empty__right,axiom,
! [A2: set_list_a] :
( ( inf_inf_set_list_a @ A2 @ bot_bot_set_list_a )
= bot_bot_set_list_a ) ).
% Int_empty_right
thf(fact_473_Int__empty__right,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ A2 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% Int_empty_right
thf(fact_474_Int__empty__left,axiom,
! [B: set_Pr2645174627780777389at_nat] :
( ( inf_in8776938414804536127at_nat @ bot_bo1973934853101755969at_nat @ B )
= bot_bo1973934853101755969at_nat ) ).
% Int_empty_left
thf(fact_475_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_476_Int__empty__left,axiom,
! [B: set_list_a] :
( ( inf_inf_set_list_a @ bot_bot_set_list_a @ B )
= bot_bot_set_list_a ) ).
% Int_empty_left
thf(fact_477_Int__empty__left,axiom,
! [B: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ B )
= bot_bot_set_nat ) ).
% Int_empty_left
thf(fact_478_disjoint__iff,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( ( inf_in8776938414804536127at_nat @ A2 @ B )
= bot_bo1973934853101755969at_nat )
= ( ! [X: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X @ A2 )
=> ~ ( member6956540099943067662at_nat @ X @ B ) ) ) ) ).
% disjoint_iff
thf(fact_479_disjoint__iff,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ( inf_in8483230781156617063la_a_b @ A2 @ B )
= bot_bo4495933725496725865la_a_b )
= ( ! [X: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X @ A2 )
=> ~ ( member4680049679412964150la_a_b @ X @ B ) ) ) ) ).
% disjoint_iff
thf(fact_480_disjoint__iff,axiom,
! [A2: set_list_a,B: set_list_a] :
( ( ( inf_inf_set_list_a @ A2 @ B )
= bot_bot_set_list_a )
= ( ! [X: list_a] :
( ( member_list_a @ X @ A2 )
=> ~ ( member_list_a @ X @ B ) ) ) ) ).
% disjoint_iff
thf(fact_481_disjoint__iff,axiom,
! [A2: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B )
= bot_bot_set_nat )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ~ ( member_nat @ X @ B ) ) ) ) ).
% disjoint_iff
thf(fact_482_Int__emptyI,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ! [X3: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X3 @ A2 )
=> ~ ( member6956540099943067662at_nat @ X3 @ B ) )
=> ( ( inf_in8776938414804536127at_nat @ A2 @ B )
= bot_bo1973934853101755969at_nat ) ) ).
% Int_emptyI
thf(fact_483_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_484_Int__emptyI,axiom,
! [A2: set_list_a,B: set_list_a] :
( ! [X3: list_a] :
( ( member_list_a @ X3 @ A2 )
=> ~ ( member_list_a @ X3 @ B ) )
=> ( ( inf_inf_set_list_a @ A2 @ B )
= bot_bot_set_list_a ) ) ).
% Int_emptyI
thf(fact_485_Int__emptyI,axiom,
! [A2: set_nat,B: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ~ ( member_nat @ X3 @ B ) )
=> ( ( inf_inf_set_nat @ A2 @ B )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_486_Int__insert__right,axiom,
! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ( ( member8440522571783428010at_nat @ A @ A2 )
=> ( ( inf_in2572325071724192079at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ B ) )
= ( insert8211810215607154385at_nat @ A @ ( inf_in2572325071724192079at_nat @ A2 @ B ) ) ) )
& ( ~ ( member8440522571783428010at_nat @ A @ A2 )
=> ( ( inf_in2572325071724192079at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ B ) )
= ( inf_in2572325071724192079at_nat @ A2 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_487_Int__insert__right,axiom,
! [A: nat,A2: set_nat,B: set_nat] :
( ( ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B ) )
= ( insert_nat @ A @ ( inf_inf_set_nat @ A2 @ B ) ) ) )
& ( ~ ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B ) )
= ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_488_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_489_Int__insert__right,axiom,
! [A: produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( ( member6956540099943067662at_nat @ A @ A2 )
=> ( ( inf_in8776938414804536127at_nat @ A2 @ ( insert3260075854425521959at_nat @ A @ B ) )
= ( insert3260075854425521959at_nat @ A @ ( inf_in8776938414804536127at_nat @ A2 @ B ) ) ) )
& ( ~ ( member6956540099943067662at_nat @ A @ A2 )
=> ( ( inf_in8776938414804536127at_nat @ A2 @ ( insert3260075854425521959at_nat @ A @ B ) )
= ( inf_in8776938414804536127at_nat @ A2 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_490_Int__insert__left,axiom,
! [A: product_prod_nat_nat,C: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ( ( member8440522571783428010at_nat @ A @ C )
=> ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A @ B ) @ C )
= ( insert8211810215607154385at_nat @ A @ ( inf_in2572325071724192079at_nat @ B @ C ) ) ) )
& ( ~ ( member8440522571783428010at_nat @ A @ C )
=> ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A @ B ) @ C )
= ( inf_in2572325071724192079at_nat @ B @ C ) ) ) ) ).
% Int_insert_left
thf(fact_491_Int__insert__left,axiom,
! [A: nat,C: set_nat,B: set_nat] :
( ( ( member_nat @ A @ C )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B ) @ C )
= ( insert_nat @ A @ ( inf_inf_set_nat @ B @ C ) ) ) )
& ( ~ ( member_nat @ A @ C )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B ) @ C )
= ( inf_inf_set_nat @ B @ C ) ) ) ) ).
% Int_insert_left
thf(fact_492_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_493_Int__insert__left,axiom,
! [A: produc5825016348098550007at_nat,C: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( ( member6956540099943067662at_nat @ A @ C )
=> ( ( inf_in8776938414804536127at_nat @ ( insert3260075854425521959at_nat @ A @ B ) @ C )
= ( insert3260075854425521959at_nat @ A @ ( inf_in8776938414804536127at_nat @ B @ C ) ) ) )
& ( ~ ( member6956540099943067662at_nat @ A @ C )
=> ( ( inf_in8776938414804536127at_nat @ ( insert3260075854425521959at_nat @ A @ B ) @ C )
= ( inf_in8776938414804536127at_nat @ B @ C ) ) ) ) ).
% Int_insert_left
thf(fact_494_Diff__Int__distrib2,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat,C: set_Pr2645174627780777389at_nat] :
( ( inf_in8776938414804536127at_nat @ ( minus_6698950876951835142at_nat @ A2 @ B ) @ C )
= ( minus_6698950876951835142at_nat @ ( inf_in8776938414804536127at_nat @ A2 @ C ) @ ( inf_in8776938414804536127at_nat @ B @ C ) ) ) ).
% Diff_Int_distrib2
thf(fact_495_Diff__Int__distrib2,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ ( minus_4077726661957047470la_a_b @ A2 @ B ) @ C )
= ( minus_4077726661957047470la_a_b @ ( inf_in8483230781156617063la_a_b @ A2 @ C ) @ ( inf_in8483230781156617063la_a_b @ B @ C ) ) ) ).
% Diff_Int_distrib2
thf(fact_496_Diff__Int__distrib,axiom,
! [C: set_Pr2645174627780777389at_nat,A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( inf_in8776938414804536127at_nat @ C @ ( minus_6698950876951835142at_nat @ A2 @ B ) )
= ( minus_6698950876951835142at_nat @ ( inf_in8776938414804536127at_nat @ C @ A2 ) @ ( inf_in8776938414804536127at_nat @ C @ B ) ) ) ).
% Diff_Int_distrib
thf(fact_497_Diff__Int__distrib,axiom,
! [C: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( inf_in8483230781156617063la_a_b @ C @ ( minus_4077726661957047470la_a_b @ A2 @ B ) )
= ( minus_4077726661957047470la_a_b @ ( inf_in8483230781156617063la_a_b @ C @ A2 ) @ ( inf_in8483230781156617063la_a_b @ C @ B ) ) ) ).
% Diff_Int_distrib
thf(fact_498_Diff__Diff__Int,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( minus_6698950876951835142at_nat @ A2 @ ( minus_6698950876951835142at_nat @ A2 @ B ) )
= ( inf_in8776938414804536127at_nat @ A2 @ B ) ) ).
% Diff_Diff_Int
thf(fact_499_Diff__Diff__Int,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( minus_4077726661957047470la_a_b @ A2 @ ( minus_4077726661957047470la_a_b @ A2 @ B ) )
= ( inf_in8483230781156617063la_a_b @ A2 @ B ) ) ).
% Diff_Diff_Int
thf(fact_500_Diff__Int2,axiom,
! [A2: set_Pr2645174627780777389at_nat,C: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( minus_6698950876951835142at_nat @ ( inf_in8776938414804536127at_nat @ A2 @ C ) @ ( inf_in8776938414804536127at_nat @ B @ C ) )
= ( minus_6698950876951835142at_nat @ ( inf_in8776938414804536127at_nat @ A2 @ C ) @ B ) ) ).
% Diff_Int2
thf(fact_501_Diff__Int2,axiom,
! [A2: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( minus_4077726661957047470la_a_b @ ( inf_in8483230781156617063la_a_b @ A2 @ C ) @ ( inf_in8483230781156617063la_a_b @ B @ C ) )
= ( minus_4077726661957047470la_a_b @ ( inf_in8483230781156617063la_a_b @ A2 @ C ) @ B ) ) ).
% Diff_Int2
thf(fact_502_Int__Diff,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat,C: set_Pr2645174627780777389at_nat] :
( ( minus_6698950876951835142at_nat @ ( inf_in8776938414804536127at_nat @ A2 @ B ) @ C )
= ( inf_in8776938414804536127at_nat @ A2 @ ( minus_6698950876951835142at_nat @ B @ C ) ) ) ).
% Int_Diff
thf(fact_503_Int__Diff,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b] :
( ( minus_4077726661957047470la_a_b @ ( inf_in8483230781156617063la_a_b @ A2 @ B ) @ C )
= ( inf_in8483230781156617063la_a_b @ A2 @ ( minus_4077726661957047470la_a_b @ B @ C ) ) ) ).
% Int_Diff
thf(fact_504_subset__image__iff,axiom,
! [B: set_set_list_a,F: relational_fmla_a_b > set_list_a,A2: set_Re381260168593705685la_a_b] :
( ( ord_le8877086941679407844list_a @ B @ ( image_130269618930851390list_a @ F @ A2 ) )
= ( ? [AA: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ AA @ A2 )
& ( B
= ( image_130269618930851390list_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_505_subset__image__iff,axiom,
! [B: set_Re381260168593705685la_a_b,F: nat > relational_fmla_a_b,A2: set_nat] :
( ( ord_le4112832032246704949la_a_b @ B @ ( image_4386371547000553590la_a_b @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B
= ( image_4386371547000553590la_a_b @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_506_subset__image__iff,axiom,
! [B: set_Pr2645174627780777389at_nat,F: produc5825016348098550007at_nat > produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat] :
( ( ord_le8520675249591772685at_nat @ B @ ( image_6979785008349797877at_nat @ F @ A2 ) )
= ( ? [AA: set_Pr2645174627780777389at_nat] :
( ( ord_le8520675249591772685at_nat @ AA @ A2 )
& ( B
= ( image_6979785008349797877at_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_507_subset__image__iff,axiom,
! [B: set_Pr2645174627780777389at_nat,F: nat > produc5825016348098550007at_nat,A2: set_nat] :
( ( ord_le8520675249591772685at_nat @ B @ ( image_1371346403869992270at_nat @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B
= ( image_1371346403869992270at_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_508_subset__image__iff,axiom,
! [B: set_nat,F: produc5825016348098550007at_nat > nat,A2: set_Pr2645174627780777389at_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_8316665354072716238at_nat @ F @ A2 ) )
= ( ? [AA: set_Pr2645174627780777389at_nat] :
( ( ord_le8520675249591772685at_nat @ AA @ A2 )
& ( B
= ( image_8316665354072716238at_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_509_subset__image__iff,axiom,
! [B: set_nat,F: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_510_image__subset__iff,axiom,
! [F: relational_fmla_a_b > set_list_a,A2: set_Re381260168593705685la_a_b,B: set_set_list_a] :
( ( ord_le8877086941679407844list_a @ ( image_130269618930851390list_a @ F @ A2 ) @ B )
= ( ! [X: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X @ A2 )
=> ( member_set_list_a @ ( F @ X ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_511_image__subset__iff,axiom,
! [F: nat > relational_fmla_a_b,A2: set_nat,B: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ ( image_4386371547000553590la_a_b @ F @ A2 ) @ B )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member4680049679412964150la_a_b @ ( F @ X ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_512_image__subset__iff,axiom,
! [F: nat > produc5825016348098550007at_nat,A2: set_nat,B: set_Pr2645174627780777389at_nat] :
( ( ord_le8520675249591772685at_nat @ ( image_1371346403869992270at_nat @ F @ A2 ) @ B )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member6956540099943067662at_nat @ ( F @ X ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_513_subset__imageE,axiom,
! [B: set_set_list_a,F: relational_fmla_a_b > set_list_a,A2: set_Re381260168593705685la_a_b] :
( ( ord_le8877086941679407844list_a @ B @ ( image_130269618930851390list_a @ F @ A2 ) )
=> ~ ! [C6: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ C6 @ A2 )
=> ( B
!= ( image_130269618930851390list_a @ F @ C6 ) ) ) ) ).
% subset_imageE
thf(fact_514_subset__imageE,axiom,
! [B: set_Re381260168593705685la_a_b,F: nat > relational_fmla_a_b,A2: set_nat] :
( ( ord_le4112832032246704949la_a_b @ B @ ( image_4386371547000553590la_a_b @ F @ A2 ) )
=> ~ ! [C6: set_nat] :
( ( ord_less_eq_set_nat @ C6 @ A2 )
=> ( B
!= ( image_4386371547000553590la_a_b @ F @ C6 ) ) ) ) ).
% subset_imageE
thf(fact_515_subset__imageE,axiom,
! [B: set_Pr2645174627780777389at_nat,F: produc5825016348098550007at_nat > produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat] :
( ( ord_le8520675249591772685at_nat @ B @ ( image_6979785008349797877at_nat @ F @ A2 ) )
=> ~ ! [C6: set_Pr2645174627780777389at_nat] :
( ( ord_le8520675249591772685at_nat @ C6 @ A2 )
=> ( B
!= ( image_6979785008349797877at_nat @ F @ C6 ) ) ) ) ).
% subset_imageE
thf(fact_516_subset__imageE,axiom,
! [B: set_Pr2645174627780777389at_nat,F: nat > produc5825016348098550007at_nat,A2: set_nat] :
( ( ord_le8520675249591772685at_nat @ B @ ( image_1371346403869992270at_nat @ F @ A2 ) )
=> ~ ! [C6: set_nat] :
( ( ord_less_eq_set_nat @ C6 @ A2 )
=> ( B
!= ( image_1371346403869992270at_nat @ F @ C6 ) ) ) ) ).
% subset_imageE
thf(fact_517_subset__imageE,axiom,
! [B: set_nat,F: produc5825016348098550007at_nat > nat,A2: set_Pr2645174627780777389at_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_8316665354072716238at_nat @ F @ A2 ) )
=> ~ ! [C6: set_Pr2645174627780777389at_nat] :
( ( ord_le8520675249591772685at_nat @ C6 @ A2 )
=> ( B
!= ( image_8316665354072716238at_nat @ F @ C6 ) ) ) ) ).
% subset_imageE
thf(fact_518_subset__imageE,axiom,
! [B: set_nat,F: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A2 ) )
=> ~ ! [C6: set_nat] :
( ( ord_less_eq_set_nat @ C6 @ A2 )
=> ( B
!= ( image_nat_nat @ F @ C6 ) ) ) ) ).
% subset_imageE
thf(fact_519_image__subsetI,axiom,
! [A2: set_nat,F: nat > relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member4680049679412964150la_a_b @ ( F @ X3 ) @ B ) )
=> ( ord_le4112832032246704949la_a_b @ ( image_4386371547000553590la_a_b @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_520_image__subsetI,axiom,
! [A2: set_Pr2645174627780777389at_nat,F: produc5825016348098550007at_nat > relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ! [X3: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X3 @ A2 )
=> ( member4680049679412964150la_a_b @ ( F @ X3 ) @ B ) )
=> ( ord_le4112832032246704949la_a_b @ ( image_7406987224865000349la_a_b @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_521_image__subsetI,axiom,
! [A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > set_list_a,B: set_set_list_a] :
( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A2 )
=> ( member_set_list_a @ ( F @ X3 ) @ B ) )
=> ( ord_le8877086941679407844list_a @ ( image_130269618930851390list_a @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_522_image__subsetI,axiom,
! [A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A2 )
=> ( member4680049679412964150la_a_b @ ( F @ X3 ) @ B ) )
=> ( ord_le4112832032246704949la_a_b @ ( image_6790371041703824709la_a_b @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_523_image__subsetI,axiom,
! [A2: set_nat,F: nat > produc5825016348098550007at_nat,B: set_Pr2645174627780777389at_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member6956540099943067662at_nat @ ( F @ X3 ) @ B ) )
=> ( ord_le8520675249591772685at_nat @ ( image_1371346403869992270at_nat @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_524_image__subsetI,axiom,
! [A2: set_Pr2645174627780777389at_nat,F: produc5825016348098550007at_nat > produc5825016348098550007at_nat,B: set_Pr2645174627780777389at_nat] :
( ! [X3: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X3 @ A2 )
=> ( member6956540099943067662at_nat @ ( F @ X3 ) @ B ) )
=> ( ord_le8520675249591772685at_nat @ ( image_6979785008349797877at_nat @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_525_image__subsetI,axiom,
! [A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > produc5825016348098550007at_nat,B: set_Pr2645174627780777389at_nat] :
( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A2 )
=> ( member6956540099943067662at_nat @ ( F @ X3 ) @ B ) )
=> ( ord_le8520675249591772685at_nat @ ( image_5430539854921360285at_nat @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_526_image__subsetI,axiom,
! [A2: set_nat,F: nat > nat,B: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_527_image__subsetI,axiom,
! [A2: set_Pr2645174627780777389at_nat,F: produc5825016348098550007at_nat > nat,B: set_nat] :
( ! [X3: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X3 @ A2 )
=> ( member_nat @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_8316665354072716238at_nat @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_528_image__subsetI,axiom,
! [A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > nat,B: set_nat] :
( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A2 )
=> ( member_nat @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_341122591648980342_b_nat @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_529_image__mono,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > set_list_a] :
( ( ord_le4112832032246704949la_a_b @ A2 @ B )
=> ( ord_le8877086941679407844list_a @ ( image_130269618930851390list_a @ F @ A2 ) @ ( image_130269618930851390list_a @ F @ B ) ) ) ).
% image_mono
thf(fact_530_image__mono,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat,F: produc5825016348098550007at_nat > produc5825016348098550007at_nat] :
( ( ord_le8520675249591772685at_nat @ A2 @ B )
=> ( ord_le8520675249591772685at_nat @ ( image_6979785008349797877at_nat @ F @ A2 ) @ ( image_6979785008349797877at_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_531_image__mono,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat,F: produc5825016348098550007at_nat > nat] :
( ( ord_le8520675249591772685at_nat @ A2 @ B )
=> ( ord_less_eq_set_nat @ ( image_8316665354072716238at_nat @ F @ A2 ) @ ( image_8316665354072716238at_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_532_image__mono,axiom,
! [A2: set_nat,B: set_nat,F: nat > relational_fmla_a_b] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ord_le4112832032246704949la_a_b @ ( image_4386371547000553590la_a_b @ F @ A2 ) @ ( image_4386371547000553590la_a_b @ F @ B ) ) ) ).
% image_mono
thf(fact_533_image__mono,axiom,
! [A2: set_nat,B: set_nat,F: nat > produc5825016348098550007at_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ord_le8520675249591772685at_nat @ ( image_1371346403869992270at_nat @ F @ A2 ) @ ( image_1371346403869992270at_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_534_image__mono,axiom,
! [A2: set_nat,B: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_535_subset__insertI2,axiom,
! [A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,B2: product_prod_nat_nat] :
( ( ord_le3146513528884898305at_nat @ A2 @ B )
=> ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_536_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_537_subset__insertI2,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat,B2: produc5825016348098550007at_nat] :
( ( ord_le8520675249591772685at_nat @ A2 @ B )
=> ( ord_le8520675249591772685at_nat @ A2 @ ( insert3260075854425521959at_nat @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_538_subset__insertI2,axiom,
! [A2: set_nat,B: set_nat,B2: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_539_subset__insertI,axiom,
! [B: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ B @ ( insert8211810215607154385at_nat @ A @ B ) ) ).
% subset_insertI
thf(fact_540_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_541_subset__insertI,axiom,
! [B: set_Pr2645174627780777389at_nat,A: produc5825016348098550007at_nat] : ( ord_le8520675249591772685at_nat @ B @ ( insert3260075854425521959at_nat @ A @ B ) ) ).
% subset_insertI
thf(fact_542_subset__insertI,axiom,
! [B: set_nat,A: nat] : ( ord_less_eq_set_nat @ B @ ( insert_nat @ A @ B ) ) ).
% subset_insertI
thf(fact_543_subset__insert,axiom,
! [X2: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ~ ( member8440522571783428010at_nat @ X2 @ A2 )
=> ( ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ X2 @ B ) )
= ( ord_le3146513528884898305at_nat @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_544_subset__insert,axiom,
! [X2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ~ ( member4680049679412964150la_a_b @ X2 @ A2 )
=> ( ( ord_le4112832032246704949la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ X2 @ B ) )
= ( ord_le4112832032246704949la_a_b @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_545_subset__insert,axiom,
! [X2: produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ~ ( member6956540099943067662at_nat @ X2 @ A2 )
=> ( ( ord_le8520675249591772685at_nat @ A2 @ ( insert3260075854425521959at_nat @ X2 @ B ) )
= ( ord_le8520675249591772685at_nat @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_546_subset__insert,axiom,
! [X2: nat,A2: set_nat,B: set_nat] :
( ~ ( member_nat @ X2 @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X2 @ B ) )
= ( ord_less_eq_set_nat @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_547_insert__mono,axiom,
! [C: set_Pr1261947904930325089at_nat,D: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
( ( ord_le3146513528884898305at_nat @ C @ D )
=> ( ord_le3146513528884898305at_nat @ ( insert8211810215607154385at_nat @ A @ C ) @ ( insert8211810215607154385at_nat @ A @ D ) ) ) ).
% insert_mono
thf(fact_548_insert__mono,axiom,
! [C: set_Re381260168593705685la_a_b,D: set_Re381260168593705685la_a_b,A: relational_fmla_a_b] :
( ( ord_le4112832032246704949la_a_b @ C @ D )
=> ( ord_le4112832032246704949la_a_b @ ( insert7010464514620295119la_a_b @ A @ C ) @ ( insert7010464514620295119la_a_b @ A @ D ) ) ) ).
% insert_mono
thf(fact_549_insert__mono,axiom,
! [C: set_Pr2645174627780777389at_nat,D: set_Pr2645174627780777389at_nat,A: produc5825016348098550007at_nat] :
( ( ord_le8520675249591772685at_nat @ C @ D )
=> ( ord_le8520675249591772685at_nat @ ( insert3260075854425521959at_nat @ A @ C ) @ ( insert3260075854425521959at_nat @ A @ D ) ) ) ).
% insert_mono
thf(fact_550_insert__mono,axiom,
! [C: set_nat,D: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ C @ D )
=> ( ord_less_eq_set_nat @ ( insert_nat @ A @ C ) @ ( insert_nat @ A @ D ) ) ) ).
% insert_mono
thf(fact_551_Un__Int__distrib2,axiom,
! [B: set_Pr2645174627780777389at_nat,C: set_Pr2645174627780777389at_nat,A2: set_Pr2645174627780777389at_nat] :
( ( sup_su6099978769595272409at_nat @ ( inf_in8776938414804536127at_nat @ B @ C ) @ A2 )
= ( inf_in8776938414804536127at_nat @ ( sup_su6099978769595272409at_nat @ B @ A2 ) @ ( sup_su6099978769595272409at_nat @ C @ A2 ) ) ) ).
% Un_Int_distrib2
thf(fact_552_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_553_Int__Un__distrib2,axiom,
! [B: set_Pr2645174627780777389at_nat,C: set_Pr2645174627780777389at_nat,A2: set_Pr2645174627780777389at_nat] :
( ( inf_in8776938414804536127at_nat @ ( sup_su6099978769595272409at_nat @ B @ C ) @ A2 )
= ( sup_su6099978769595272409at_nat @ ( inf_in8776938414804536127at_nat @ B @ A2 ) @ ( inf_in8776938414804536127at_nat @ C @ A2 ) ) ) ).
% Int_Un_distrib2
thf(fact_554_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_555_Un__Int__distrib,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat,C: set_Pr2645174627780777389at_nat] :
( ( sup_su6099978769595272409at_nat @ A2 @ ( inf_in8776938414804536127at_nat @ B @ C ) )
= ( inf_in8776938414804536127at_nat @ ( sup_su6099978769595272409at_nat @ A2 @ B ) @ ( sup_su6099978769595272409at_nat @ A2 @ C ) ) ) ).
% Un_Int_distrib
thf(fact_556_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_557_Int__Un__distrib,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat,C: set_Pr2645174627780777389at_nat] :
( ( inf_in8776938414804536127at_nat @ A2 @ ( sup_su6099978769595272409at_nat @ B @ C ) )
= ( sup_su6099978769595272409at_nat @ ( inf_in8776938414804536127at_nat @ A2 @ B ) @ ( inf_in8776938414804536127at_nat @ A2 @ C ) ) ) ).
% Int_Un_distrib
thf(fact_558_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_559_Un__Int__crazy,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat,C: set_Pr2645174627780777389at_nat] :
( ( sup_su6099978769595272409at_nat @ ( sup_su6099978769595272409at_nat @ ( inf_in8776938414804536127at_nat @ A2 @ B ) @ ( inf_in8776938414804536127at_nat @ B @ C ) ) @ ( inf_in8776938414804536127at_nat @ C @ A2 ) )
= ( inf_in8776938414804536127at_nat @ ( inf_in8776938414804536127at_nat @ ( sup_su6099978769595272409at_nat @ A2 @ B ) @ ( sup_su6099978769595272409at_nat @ B @ C ) ) @ ( sup_su6099978769595272409at_nat @ C @ A2 ) ) ) ).
% Un_Int_crazy
thf(fact_560_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_561_double__diff,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A2 @ B )
=> ( ( ord_le4112832032246704949la_a_b @ B @ C )
=> ( ( minus_4077726661957047470la_a_b @ B @ ( minus_4077726661957047470la_a_b @ C @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_562_double__diff,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat,C: set_Pr2645174627780777389at_nat] :
( ( ord_le8520675249591772685at_nat @ A2 @ B )
=> ( ( ord_le8520675249591772685at_nat @ B @ C )
=> ( ( minus_6698950876951835142at_nat @ B @ ( minus_6698950876951835142at_nat @ C @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_563_double__diff,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ( minus_minus_set_nat @ B @ ( minus_minus_set_nat @ C @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_564_Diff__subset,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] : ( ord_le4112832032246704949la_a_b @ ( minus_4077726661957047470la_a_b @ A2 @ B ) @ A2 ) ).
% Diff_subset
thf(fact_565_Diff__subset,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] : ( ord_le8520675249591772685at_nat @ ( minus_6698950876951835142at_nat @ A2 @ B ) @ A2 ) ).
% Diff_subset
thf(fact_566_Diff__subset,axiom,
! [A2: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B ) @ A2 ) ).
% Diff_subset
thf(fact_567_Diff__mono,axiom,
! [A2: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b,D: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A2 @ C )
=> ( ( ord_le4112832032246704949la_a_b @ D @ B )
=> ( ord_le4112832032246704949la_a_b @ ( minus_4077726661957047470la_a_b @ A2 @ B ) @ ( minus_4077726661957047470la_a_b @ C @ D ) ) ) ) ).
% Diff_mono
thf(fact_568_Diff__mono,axiom,
! [A2: set_Pr2645174627780777389at_nat,C: set_Pr2645174627780777389at_nat,D: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( ord_le8520675249591772685at_nat @ A2 @ C )
=> ( ( ord_le8520675249591772685at_nat @ D @ B )
=> ( ord_le8520675249591772685at_nat @ ( minus_6698950876951835142at_nat @ A2 @ B ) @ ( minus_6698950876951835142at_nat @ C @ D ) ) ) ) ).
% Diff_mono
thf(fact_569_Diff__mono,axiom,
! [A2: set_nat,C: set_nat,D: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ( ord_less_eq_set_nat @ D @ B )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B ) @ ( minus_minus_set_nat @ C @ D ) ) ) ) ).
% Diff_mono
thf(fact_570_subset__Un__eq,axiom,
( ord_le4112832032246704949la_a_b
= ( ^ [A4: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ A4 @ B4 )
= B4 ) ) ) ).
% subset_Un_eq
thf(fact_571_subset__Un__eq,axiom,
( ord_le8520675249591772685at_nat
= ( ^ [A4: set_Pr2645174627780777389at_nat,B4: set_Pr2645174627780777389at_nat] :
( ( sup_su6099978769595272409at_nat @ A4 @ B4 )
= B4 ) ) ) ).
% subset_Un_eq
thf(fact_572_subset__Un__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ( sup_sup_set_nat @ A4 @ B4 )
= B4 ) ) ) ).
% subset_Un_eq
thf(fact_573_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 ) )
=> ~ ! [A5: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A5 @ A2 )
=> ! [B5: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ B5 @ B )
=> ( C
!= ( sup_su5130108678486352897la_a_b @ A5 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_574_subset__UnE,axiom,
! [C: set_Pr2645174627780777389at_nat,A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( ord_le8520675249591772685at_nat @ C @ ( sup_su6099978769595272409at_nat @ A2 @ B ) )
=> ~ ! [A5: set_Pr2645174627780777389at_nat] :
( ( ord_le8520675249591772685at_nat @ A5 @ A2 )
=> ! [B5: set_Pr2645174627780777389at_nat] :
( ( ord_le8520675249591772685at_nat @ B5 @ B )
=> ( C
!= ( sup_su6099978769595272409at_nat @ A5 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_575_subset__UnE,axiom,
! [C: set_nat,A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) )
=> ~ ! [A5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ A2 )
=> ! [B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ B )
=> ( C
!= ( sup_sup_set_nat @ A5 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_576_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_577_Un__absorb2,axiom,
! [B: set_Pr2645174627780777389at_nat,A2: set_Pr2645174627780777389at_nat] :
( ( ord_le8520675249591772685at_nat @ B @ A2 )
=> ( ( sup_su6099978769595272409at_nat @ A2 @ B )
= A2 ) ) ).
% Un_absorb2
thf(fact_578_Un__absorb2,axiom,
! [B: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( sup_sup_set_nat @ A2 @ B )
= A2 ) ) ).
% Un_absorb2
thf(fact_579_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_580_Un__absorb1,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( ord_le8520675249591772685at_nat @ A2 @ B )
=> ( ( sup_su6099978769595272409at_nat @ A2 @ B )
= B ) ) ).
% Un_absorb1
thf(fact_581_Un__absorb1,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( sup_sup_set_nat @ A2 @ B )
= B ) ) ).
% Un_absorb1
thf(fact_582_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_583_Un__upper2,axiom,
! [B: set_Pr2645174627780777389at_nat,A2: set_Pr2645174627780777389at_nat] : ( ord_le8520675249591772685at_nat @ B @ ( sup_su6099978769595272409at_nat @ A2 @ B ) ) ).
% Un_upper2
thf(fact_584_Un__upper2,axiom,
! [B: set_nat,A2: set_nat] : ( ord_less_eq_set_nat @ B @ ( sup_sup_set_nat @ A2 @ B ) ) ).
% Un_upper2
thf(fact_585_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_586_Un__upper1,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] : ( ord_le8520675249591772685at_nat @ A2 @ ( sup_su6099978769595272409at_nat @ A2 @ B ) ) ).
% Un_upper1
thf(fact_587_Un__upper1,axiom,
! [A2: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B ) ) ).
% Un_upper1
thf(fact_588_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_589_Un__least,axiom,
! [A2: set_Pr2645174627780777389at_nat,C: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( ord_le8520675249591772685at_nat @ A2 @ C )
=> ( ( ord_le8520675249591772685at_nat @ B @ C )
=> ( ord_le8520675249591772685at_nat @ ( sup_su6099978769595272409at_nat @ A2 @ B ) @ C ) ) ) ).
% Un_least
thf(fact_590_Un__least,axiom,
! [A2: set_nat,C: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ C ) ) ) ).
% Un_least
thf(fact_591_Un__mono,axiom,
! [A2: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,D: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A2 @ C )
=> ( ( ord_le4112832032246704949la_a_b @ B @ D )
=> ( ord_le4112832032246704949la_a_b @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) @ ( sup_su5130108678486352897la_a_b @ C @ D ) ) ) ) ).
% Un_mono
thf(fact_592_Un__mono,axiom,
! [A2: set_Pr2645174627780777389at_nat,C: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat,D: set_Pr2645174627780777389at_nat] :
( ( ord_le8520675249591772685at_nat @ A2 @ C )
=> ( ( ord_le8520675249591772685at_nat @ B @ D )
=> ( ord_le8520675249591772685at_nat @ ( sup_su6099978769595272409at_nat @ A2 @ B ) @ ( sup_su6099978769595272409at_nat @ C @ D ) ) ) ) ).
% Un_mono
thf(fact_593_Un__mono,axiom,
! [A2: set_nat,C: set_nat,B: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ( ord_less_eq_set_nat @ B @ D )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ ( sup_sup_set_nat @ C @ D ) ) ) ) ).
% Un_mono
thf(fact_594_ex__in__conv,axiom,
! [A2: set_Pr2645174627780777389at_nat] :
( ( ? [X: produc5825016348098550007at_nat] : ( member6956540099943067662at_nat @ X @ A2 ) )
= ( A2 != bot_bo1973934853101755969at_nat ) ) ).
% ex_in_conv
thf(fact_595_ex__in__conv,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( ? [X: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X @ A2 ) )
= ( A2 != bot_bo4495933725496725865la_a_b ) ) ).
% ex_in_conv
thf(fact_596_ex__in__conv,axiom,
! [A2: set_list_a] :
( ( ? [X: list_a] : ( member_list_a @ X @ A2 ) )
= ( A2 != bot_bot_set_list_a ) ) ).
% ex_in_conv
thf(fact_597_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X: nat] : ( member_nat @ X @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_598_equals0I,axiom,
! [A2: set_Pr2645174627780777389at_nat] :
( ! [Y4: produc5825016348098550007at_nat] :
~ ( member6956540099943067662at_nat @ Y4 @ A2 )
=> ( A2 = bot_bo1973934853101755969at_nat ) ) ).
% equals0I
thf(fact_599_equals0I,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ! [Y4: relational_fmla_a_b] :
~ ( member4680049679412964150la_a_b @ Y4 @ A2 )
=> ( A2 = bot_bo4495933725496725865la_a_b ) ) ).
% equals0I
thf(fact_600_equals0I,axiom,
! [A2: set_list_a] :
( ! [Y4: list_a] :
~ ( member_list_a @ Y4 @ A2 )
=> ( A2 = bot_bot_set_list_a ) ) ).
% equals0I
thf(fact_601_equals0I,axiom,
! [A2: set_nat] :
( ! [Y4: nat] :
~ ( member_nat @ Y4 @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_602_equals0D,axiom,
! [A2: set_Pr2645174627780777389at_nat,A: produc5825016348098550007at_nat] :
( ( A2 = bot_bo1973934853101755969at_nat )
=> ~ ( member6956540099943067662at_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_603_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_604_equals0D,axiom,
! [A2: set_list_a,A: list_a] :
( ( A2 = bot_bot_set_list_a )
=> ~ ( member_list_a @ A @ A2 ) ) ).
% equals0D
thf(fact_605_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_606_emptyE,axiom,
! [A: produc5825016348098550007at_nat] :
~ ( member6956540099943067662at_nat @ A @ bot_bo1973934853101755969at_nat ) ).
% emptyE
thf(fact_607_emptyE,axiom,
! [A: relational_fmla_a_b] :
~ ( member4680049679412964150la_a_b @ A @ bot_bo4495933725496725865la_a_b ) ).
% emptyE
thf(fact_608_emptyE,axiom,
! [A: list_a] :
~ ( member_list_a @ A @ bot_bot_set_list_a ) ).
% emptyE
thf(fact_609_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_610_rev__image__eqI,axiom,
! [X2: nat,A2: set_nat,B2: nat,F: nat > nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_nat @ B2 @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_611_rev__image__eqI,axiom,
! [X2: nat,A2: set_nat,B2: produc5825016348098550007at_nat,F: nat > produc5825016348098550007at_nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member6956540099943067662at_nat @ B2 @ ( image_1371346403869992270at_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_612_rev__image__eqI,axiom,
! [X2: nat,A2: set_nat,B2: relational_fmla_a_b,F: nat > relational_fmla_a_b] :
( ( member_nat @ X2 @ A2 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member4680049679412964150la_a_b @ B2 @ ( image_4386371547000553590la_a_b @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_613_rev__image__eqI,axiom,
! [X2: produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat,B2: nat,F: produc5825016348098550007at_nat > nat] :
( ( member6956540099943067662at_nat @ X2 @ A2 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_nat @ B2 @ ( image_8316665354072716238at_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_614_rev__image__eqI,axiom,
! [X2: produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat,B2: produc5825016348098550007at_nat,F: produc5825016348098550007at_nat > produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X2 @ A2 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member6956540099943067662at_nat @ B2 @ ( image_6979785008349797877at_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_615_rev__image__eqI,axiom,
! [X2: produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat,B2: relational_fmla_a_b,F: produc5825016348098550007at_nat > relational_fmla_a_b] :
( ( member6956540099943067662at_nat @ X2 @ A2 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member4680049679412964150la_a_b @ B2 @ ( image_7406987224865000349la_a_b @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_616_rev__image__eqI,axiom,
! [X2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B2: set_list_a,F: relational_fmla_a_b > set_list_a] :
( ( member4680049679412964150la_a_b @ X2 @ A2 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_set_list_a @ B2 @ ( image_130269618930851390list_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_617_rev__image__eqI,axiom,
! [X2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B2: nat,F: relational_fmla_a_b > nat] :
( ( member4680049679412964150la_a_b @ X2 @ A2 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_nat @ B2 @ ( image_341122591648980342_b_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_618_rev__image__eqI,axiom,
! [X2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B2: produc5825016348098550007at_nat,F: relational_fmla_a_b > produc5825016348098550007at_nat] :
( ( member4680049679412964150la_a_b @ X2 @ A2 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member6956540099943067662at_nat @ B2 @ ( image_5430539854921360285at_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_619_rev__image__eqI,axiom,
! [X2: 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 @ X2 @ A2 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member4680049679412964150la_a_b @ B2 @ ( image_6790371041703824709la_a_b @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_620_ball__imageD,axiom,
! [F: nat > produc5825016348098550007at_nat,A2: set_nat,P: produc5825016348098550007at_nat > $o] :
( ! [X3: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X3 @ ( image_1371346403869992270at_nat @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_621_ball__imageD,axiom,
! [F: nat > relational_fmla_a_b,A2: set_nat,P: relational_fmla_a_b > $o] :
( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ ( image_4386371547000553590la_a_b @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_622_ball__imageD,axiom,
! [F: relational_fmla_a_b > set_list_a,A2: set_Re381260168593705685la_a_b,P: set_list_a > $o] :
( ! [X3: set_list_a] :
( ( member_set_list_a @ X3 @ ( image_130269618930851390list_a @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X4: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X4 @ A2 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_623_image__cong,axiom,
! [M: set_nat,N: set_nat,F: nat > produc5825016348098550007at_nat,G: nat > produc5825016348098550007at_nat] :
( ( M = N )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_1371346403869992270at_nat @ F @ M )
= ( image_1371346403869992270at_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_624_image__cong,axiom,
! [M: set_nat,N: set_nat,F: nat > relational_fmla_a_b,G: nat > relational_fmla_a_b] :
( ( M = N )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_4386371547000553590la_a_b @ F @ M )
= ( image_4386371547000553590la_a_b @ G @ N ) ) ) ) ).
% image_cong
thf(fact_625_image__cong,axiom,
! [M: set_Re381260168593705685la_a_b,N: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > set_list_a,G: relational_fmla_a_b > set_list_a] :
( ( M = N )
=> ( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_130269618930851390list_a @ F @ M )
= ( image_130269618930851390list_a @ G @ N ) ) ) ) ).
% image_cong
thf(fact_626_bex__imageD,axiom,
! [F: nat > produc5825016348098550007at_nat,A2: set_nat,P: produc5825016348098550007at_nat > $o] :
( ? [X4: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X4 @ ( image_1371346403869992270at_nat @ F @ A2 ) )
& ( P @ X4 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_627_bex__imageD,axiom,
! [F: nat > relational_fmla_a_b,A2: set_nat,P: relational_fmla_a_b > $o] :
( ? [X4: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X4 @ ( image_4386371547000553590la_a_b @ F @ A2 ) )
& ( P @ X4 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_628_bex__imageD,axiom,
! [F: relational_fmla_a_b > set_list_a,A2: set_Re381260168593705685la_a_b,P: set_list_a > $o] :
( ? [X4: set_list_a] :
( ( member_set_list_a @ X4 @ ( image_130269618930851390list_a @ F @ A2 ) )
& ( P @ X4 ) )
=> ? [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_629_image__iff,axiom,
! [Z2: set_list_a,F: relational_fmla_a_b > set_list_a,A2: set_Re381260168593705685la_a_b] :
( ( member_set_list_a @ Z2 @ ( image_130269618930851390list_a @ F @ A2 ) )
= ( ? [X: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X @ A2 )
& ( Z2
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_630_image__iff,axiom,
! [Z2: produc5825016348098550007at_nat,F: nat > produc5825016348098550007at_nat,A2: set_nat] :
( ( member6956540099943067662at_nat @ Z2 @ ( image_1371346403869992270at_nat @ F @ A2 ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A2 )
& ( Z2
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_631_image__iff,axiom,
! [Z2: relational_fmla_a_b,F: nat > relational_fmla_a_b,A2: set_nat] :
( ( member4680049679412964150la_a_b @ Z2 @ ( image_4386371547000553590la_a_b @ F @ A2 ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A2 )
& ( Z2
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_632_imageI,axiom,
! [X2: nat,A2: set_nat,F: nat > nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ ( image_nat_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_633_imageI,axiom,
! [X2: nat,A2: set_nat,F: nat > produc5825016348098550007at_nat] :
( ( member_nat @ X2 @ A2 )
=> ( member6956540099943067662at_nat @ ( F @ X2 ) @ ( image_1371346403869992270at_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_634_imageI,axiom,
! [X2: nat,A2: set_nat,F: nat > relational_fmla_a_b] :
( ( member_nat @ X2 @ A2 )
=> ( member4680049679412964150la_a_b @ ( F @ X2 ) @ ( image_4386371547000553590la_a_b @ F @ A2 ) ) ) ).
% imageI
thf(fact_635_imageI,axiom,
! [X2: produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat,F: produc5825016348098550007at_nat > nat] :
( ( member6956540099943067662at_nat @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ ( image_8316665354072716238at_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_636_imageI,axiom,
! [X2: produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat,F: produc5825016348098550007at_nat > produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X2 @ A2 )
=> ( member6956540099943067662at_nat @ ( F @ X2 ) @ ( image_6979785008349797877at_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_637_imageI,axiom,
! [X2: produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat,F: produc5825016348098550007at_nat > relational_fmla_a_b] :
( ( member6956540099943067662at_nat @ X2 @ A2 )
=> ( member4680049679412964150la_a_b @ ( F @ X2 ) @ ( image_7406987224865000349la_a_b @ F @ A2 ) ) ) ).
% imageI
thf(fact_638_imageI,axiom,
! [X2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > set_list_a] :
( ( member4680049679412964150la_a_b @ X2 @ A2 )
=> ( member_set_list_a @ ( F @ X2 ) @ ( image_130269618930851390list_a @ F @ A2 ) ) ) ).
% imageI
thf(fact_639_imageI,axiom,
! [X2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > nat] :
( ( member4680049679412964150la_a_b @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ ( image_341122591648980342_b_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_640_imageI,axiom,
! [X2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > produc5825016348098550007at_nat] :
( ( member4680049679412964150la_a_b @ X2 @ A2 )
=> ( member6956540099943067662at_nat @ ( F @ X2 ) @ ( image_5430539854921360285at_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_641_imageI,axiom,
! [X2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ A2 )
=> ( member4680049679412964150la_a_b @ ( F @ X2 ) @ ( image_6790371041703824709la_a_b @ F @ A2 ) ) ) ).
% imageI
thf(fact_642_mk__disjoint__insert,axiom,
! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ A @ A2 )
=> ? [B6: set_Pr1261947904930325089at_nat] :
( ( A2
= ( insert8211810215607154385at_nat @ A @ B6 ) )
& ~ ( member8440522571783428010at_nat @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_643_mk__disjoint__insert,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ? [B6: set_nat] :
( ( A2
= ( insert_nat @ A @ B6 ) )
& ~ ( member_nat @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_644_mk__disjoint__insert,axiom,
! [A: produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ A @ A2 )
=> ? [B6: set_Pr2645174627780777389at_nat] :
( ( A2
= ( insert3260075854425521959at_nat @ A @ B6 ) )
& ~ ( member6956540099943067662at_nat @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_645_mk__disjoint__insert,axiom,
! [A: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ A @ A2 )
=> ? [B6: set_Re381260168593705685la_a_b] :
( ( A2
= ( insert7010464514620295119la_a_b @ A @ B6 ) )
& ~ ( member4680049679412964150la_a_b @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_646_insert__commute,axiom,
! [X2: produc5825016348098550007at_nat,Y3: produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat] :
( ( insert3260075854425521959at_nat @ X2 @ ( insert3260075854425521959at_nat @ Y3 @ A2 ) )
= ( insert3260075854425521959at_nat @ Y3 @ ( insert3260075854425521959at_nat @ X2 @ A2 ) ) ) ).
% insert_commute
thf(fact_647_insert__commute,axiom,
! [X2: product_prod_nat_nat,Y3: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
( ( insert8211810215607154385at_nat @ X2 @ ( insert8211810215607154385at_nat @ Y3 @ A2 ) )
= ( insert8211810215607154385at_nat @ Y3 @ ( insert8211810215607154385at_nat @ X2 @ A2 ) ) ) ).
% insert_commute
thf(fact_648_insert__commute,axiom,
! [X2: relational_fmla_a_b,Y3: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( insert7010464514620295119la_a_b @ X2 @ ( insert7010464514620295119la_a_b @ Y3 @ A2 ) )
= ( insert7010464514620295119la_a_b @ Y3 @ ( insert7010464514620295119la_a_b @ X2 @ A2 ) ) ) ).
% insert_commute
thf(fact_649_insert__eq__iff,axiom,
! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat] :
( ~ ( member8440522571783428010at_nat @ A @ A2 )
=> ( ~ ( member8440522571783428010at_nat @ B2 @ B )
=> ( ( ( insert8211810215607154385at_nat @ A @ A2 )
= ( insert8211810215607154385at_nat @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C3: set_Pr1261947904930325089at_nat] :
( ( A2
= ( insert8211810215607154385at_nat @ B2 @ C3 ) )
& ~ ( member8440522571783428010at_nat @ B2 @ C3 )
& ( B
= ( insert8211810215607154385at_nat @ A @ C3 ) )
& ~ ( member8440522571783428010at_nat @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_650_insert__eq__iff,axiom,
! [A: nat,A2: set_nat,B2: nat,B: set_nat] :
( ~ ( member_nat @ A @ A2 )
=> ( ~ ( member_nat @ B2 @ B )
=> ( ( ( insert_nat @ A @ A2 )
= ( insert_nat @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C3: set_nat] :
( ( A2
= ( insert_nat @ B2 @ C3 ) )
& ~ ( member_nat @ B2 @ C3 )
& ( B
= ( insert_nat @ A @ C3 ) )
& ~ ( member_nat @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_651_insert__eq__iff,axiom,
! [A: produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat,B2: produc5825016348098550007at_nat,B: set_Pr2645174627780777389at_nat] :
( ~ ( member6956540099943067662at_nat @ A @ A2 )
=> ( ~ ( member6956540099943067662at_nat @ B2 @ B )
=> ( ( ( insert3260075854425521959at_nat @ A @ A2 )
= ( insert3260075854425521959at_nat @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C3: set_Pr2645174627780777389at_nat] :
( ( A2
= ( insert3260075854425521959at_nat @ B2 @ C3 ) )
& ~ ( member6956540099943067662at_nat @ B2 @ C3 )
& ( B
= ( insert3260075854425521959at_nat @ A @ C3 ) )
& ~ ( member6956540099943067662at_nat @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_652_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_653_insert__absorb,axiom,
! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ A @ A2 )
=> ( ( insert8211810215607154385at_nat @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_654_insert__absorb,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( insert_nat @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_655_insert__absorb,axiom,
! [A: produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ A @ A2 )
=> ( ( insert3260075854425521959at_nat @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_656_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_657_insert__ident,axiom,
! [X2: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ~ ( member8440522571783428010at_nat @ X2 @ A2 )
=> ( ~ ( member8440522571783428010at_nat @ X2 @ B )
=> ( ( ( insert8211810215607154385at_nat @ X2 @ A2 )
= ( insert8211810215607154385at_nat @ X2 @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_658_insert__ident,axiom,
! [X2: nat,A2: set_nat,B: set_nat] :
( ~ ( member_nat @ X2 @ A2 )
=> ( ~ ( member_nat @ X2 @ B )
=> ( ( ( insert_nat @ X2 @ A2 )
= ( insert_nat @ X2 @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_659_insert__ident,axiom,
! [X2: produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ~ ( member6956540099943067662at_nat @ X2 @ A2 )
=> ( ~ ( member6956540099943067662at_nat @ X2 @ B )
=> ( ( ( insert3260075854425521959at_nat @ X2 @ A2 )
= ( insert3260075854425521959at_nat @ X2 @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_660_insert__ident,axiom,
! [X2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ~ ( member4680049679412964150la_a_b @ X2 @ A2 )
=> ( ~ ( member4680049679412964150la_a_b @ X2 @ B )
=> ( ( ( insert7010464514620295119la_a_b @ X2 @ A2 )
= ( insert7010464514620295119la_a_b @ X2 @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_661_Set_Oset__insert,axiom,
! [X2: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ X2 @ A2 )
=> ~ ! [B6: set_Pr1261947904930325089at_nat] :
( ( A2
= ( insert8211810215607154385at_nat @ X2 @ B6 ) )
=> ( member8440522571783428010at_nat @ X2 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_662_Set_Oset__insert,axiom,
! [X2: nat,A2: set_nat] :
( ( member_nat @ X2 @ A2 )
=> ~ ! [B6: set_nat] :
( ( A2
= ( insert_nat @ X2 @ B6 ) )
=> ( member_nat @ X2 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_663_Set_Oset__insert,axiom,
! [X2: produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ X2 @ A2 )
=> ~ ! [B6: set_Pr2645174627780777389at_nat] :
( ( A2
= ( insert3260075854425521959at_nat @ X2 @ B6 ) )
=> ( member6956540099943067662at_nat @ X2 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_664_Set_Oset__insert,axiom,
! [X2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ A2 )
=> ~ ! [B6: set_Re381260168593705685la_a_b] :
( ( A2
= ( insert7010464514620295119la_a_b @ X2 @ B6 ) )
=> ( member4680049679412964150la_a_b @ X2 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_665_insertI2,axiom,
! [A: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat,B2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ A @ B )
=> ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B2 @ B ) ) ) ).
% insertI2
thf(fact_666_insertI2,axiom,
! [A: nat,B: set_nat,B2: nat] :
( ( member_nat @ A @ B )
=> ( member_nat @ A @ ( insert_nat @ B2 @ B ) ) ) ).
% insertI2
thf(fact_667_insertI2,axiom,
! [A: produc5825016348098550007at_nat,B: set_Pr2645174627780777389at_nat,B2: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ A @ B )
=> ( member6956540099943067662at_nat @ A @ ( insert3260075854425521959at_nat @ B2 @ B ) ) ) ).
% insertI2
thf(fact_668_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_669_insertI1,axiom,
! [A: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat] : ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ A @ B ) ) ).
% insertI1
thf(fact_670_insertI1,axiom,
! [A: nat,B: set_nat] : ( member_nat @ A @ ( insert_nat @ A @ B ) ) ).
% insertI1
thf(fact_671_insertI1,axiom,
! [A: produc5825016348098550007at_nat,B: set_Pr2645174627780777389at_nat] : ( member6956540099943067662at_nat @ A @ ( insert3260075854425521959at_nat @ A @ B ) ) ).
% insertI1
thf(fact_672_insertI1,axiom,
! [A: relational_fmla_a_b,B: set_Re381260168593705685la_a_b] : ( member4680049679412964150la_a_b @ A @ ( insert7010464514620295119la_a_b @ A @ B ) ) ).
% insertI1
thf(fact_673_insertE,axiom,
! [A: product_prod_nat_nat,B2: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member8440522571783428010at_nat @ A @ A2 ) ) ) ).
% insertE
thf(fact_674_insertE,axiom,
! [A: nat,B2: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member_nat @ A @ A2 ) ) ) ).
% insertE
thf(fact_675_insertE,axiom,
! [A: produc5825016348098550007at_nat,B2: produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ A @ ( insert3260075854425521959at_nat @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member6956540099943067662at_nat @ A @ A2 ) ) ) ).
% insertE
thf(fact_676_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_677_DiffD2,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A2 @ B ) )
=> ~ ( member_nat @ C2 @ B ) ) ).
% DiffD2
thf(fact_678_DiffD2,axiom,
! [C2: produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ C2 @ ( minus_6698950876951835142at_nat @ A2 @ B ) )
=> ~ ( member6956540099943067662at_nat @ C2 @ B ) ) ).
% DiffD2
thf(fact_679_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_680_DiffD1,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A2 @ B ) )
=> ( member_nat @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_681_DiffD1,axiom,
! [C2: produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ C2 @ ( minus_6698950876951835142at_nat @ A2 @ B ) )
=> ( member6956540099943067662at_nat @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_682_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_683_DiffE,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A2 @ B ) )
=> ~ ( ( member_nat @ C2 @ A2 )
=> ( member_nat @ C2 @ B ) ) ) ).
% DiffE
thf(fact_684_DiffE,axiom,
! [C2: produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ C2 @ ( minus_6698950876951835142at_nat @ A2 @ B ) )
=> ~ ( ( member6956540099943067662at_nat @ C2 @ A2 )
=> ( member6956540099943067662at_nat @ C2 @ B ) ) ) ).
% DiffE
thf(fact_685_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_686_Un__left__commute,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat,C: set_Pr2645174627780777389at_nat] :
( ( sup_su6099978769595272409at_nat @ A2 @ ( sup_su6099978769595272409at_nat @ B @ C ) )
= ( sup_su6099978769595272409at_nat @ B @ ( sup_su6099978769595272409at_nat @ A2 @ C ) ) ) ).
% Un_left_commute
thf(fact_687_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_688_Un__left__absorb,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( sup_su6099978769595272409at_nat @ A2 @ ( sup_su6099978769595272409at_nat @ A2 @ B ) )
= ( sup_su6099978769595272409at_nat @ A2 @ B ) ) ).
% Un_left_absorb
thf(fact_689_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_690_Un__commute,axiom,
( sup_su6099978769595272409at_nat
= ( ^ [A4: set_Pr2645174627780777389at_nat,B4: set_Pr2645174627780777389at_nat] : ( sup_su6099978769595272409at_nat @ B4 @ A4 ) ) ) ).
% Un_commute
thf(fact_691_Un__commute,axiom,
( sup_su5130108678486352897la_a_b
= ( ^ [A4: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b] : ( sup_su5130108678486352897la_a_b @ B4 @ A4 ) ) ) ).
% Un_commute
thf(fact_692_Un__absorb,axiom,
! [A2: set_Pr2645174627780777389at_nat] :
( ( sup_su6099978769595272409at_nat @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_693_Un__absorb,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_694_Un__assoc,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat,C: set_Pr2645174627780777389at_nat] :
( ( sup_su6099978769595272409at_nat @ ( sup_su6099978769595272409at_nat @ A2 @ B ) @ C )
= ( sup_su6099978769595272409at_nat @ A2 @ ( sup_su6099978769595272409at_nat @ B @ C ) ) ) ).
% Un_assoc
thf(fact_695_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_696_ball__Un,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat,P: produc5825016348098550007at_nat > $o] :
( ( ! [X: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X @ ( sup_su6099978769595272409at_nat @ A2 @ B ) )
=> ( P @ X ) ) )
= ( ! [X: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X @ A2 )
=> ( P @ X ) )
& ! [X: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X @ B )
=> ( P @ X ) ) ) ) ).
% ball_Un
thf(fact_697_ball__Un,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,P: relational_fmla_a_b > $o] :
( ( ! [X: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) )
=> ( P @ X ) ) )
= ( ! [X: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X @ A2 )
=> ( P @ X ) )
& ! [X: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X @ B )
=> ( P @ X ) ) ) ) ).
% ball_Un
thf(fact_698_bex__Un,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat,P: produc5825016348098550007at_nat > $o] :
( ( ? [X: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X @ ( sup_su6099978769595272409at_nat @ A2 @ B ) )
& ( P @ X ) ) )
= ( ? [X: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X @ A2 )
& ( P @ X ) )
| ? [X: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X @ B )
& ( P @ X ) ) ) ) ).
% bex_Un
thf(fact_699_bex__Un,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,P: relational_fmla_a_b > $o] :
( ( ? [X: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) )
& ( P @ X ) ) )
= ( ? [X: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X @ A2 )
& ( P @ X ) )
| ? [X: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X @ B )
& ( P @ X ) ) ) ) ).
% bex_Un
thf(fact_700_UnI2,axiom,
! [C2: nat,B: set_nat,A2: set_nat] :
( ( member_nat @ C2 @ B )
=> ( member_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% UnI2
thf(fact_701_UnI2,axiom,
! [C2: produc5825016348098550007at_nat,B: set_Pr2645174627780777389at_nat,A2: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ C2 @ B )
=> ( member6956540099943067662at_nat @ C2 @ ( sup_su6099978769595272409at_nat @ A2 @ B ) ) ) ).
% UnI2
thf(fact_702_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_703_UnI1,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ A2 )
=> ( member_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% UnI1
thf(fact_704_UnI1,axiom,
! [C2: produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ C2 @ A2 )
=> ( member6956540099943067662at_nat @ C2 @ ( sup_su6099978769595272409at_nat @ A2 @ B ) ) ) ).
% UnI1
thf(fact_705_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_706_UnE,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B ) )
=> ( ~ ( member_nat @ C2 @ A2 )
=> ( member_nat @ C2 @ B ) ) ) ).
% UnE
thf(fact_707_UnE,axiom,
! [C2: produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ C2 @ ( sup_su6099978769595272409at_nat @ A2 @ B ) )
=> ( ~ ( member6956540099943067662at_nat @ C2 @ A2 )
=> ( member6956540099943067662at_nat @ C2 @ B ) ) ) ).
% UnE
thf(fact_708_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_709_multi__union__self__other__eq,axiom,
! [A2: multiset_set_nat,X5: multiset_set_nat,Y5: multiset_set_nat] :
( ( ( plus_p8712254050562127327et_nat @ A2 @ X5 )
= ( plus_p8712254050562127327et_nat @ A2 @ Y5 ) )
=> ( X5 = Y5 ) ) ).
% multi_union_self_other_eq
thf(fact_710_union__right__cancel,axiom,
! [M: multiset_set_nat,K: multiset_set_nat,N: multiset_set_nat] :
( ( ( plus_p8712254050562127327et_nat @ M @ K )
= ( plus_p8712254050562127327et_nat @ N @ K ) )
= ( M = N ) ) ).
% union_right_cancel
thf(fact_711_union__left__cancel,axiom,
! [K: multiset_set_nat,M: multiset_set_nat,N: multiset_set_nat] :
( ( ( plus_p8712254050562127327et_nat @ K @ M )
= ( plus_p8712254050562127327et_nat @ K @ N ) )
= ( M = N ) ) ).
% union_left_cancel
thf(fact_712_union__commute,axiom,
( plus_p8712254050562127327et_nat
= ( ^ [M2: multiset_set_nat,N3: multiset_set_nat] : ( plus_p8712254050562127327et_nat @ N3 @ M2 ) ) ) ).
% union_commute
thf(fact_713_union__lcomm,axiom,
! [M: multiset_set_nat,N: multiset_set_nat,K: multiset_set_nat] :
( ( plus_p8712254050562127327et_nat @ M @ ( plus_p8712254050562127327et_nat @ N @ K ) )
= ( plus_p8712254050562127327et_nat @ N @ ( plus_p8712254050562127327et_nat @ M @ K ) ) ) ).
% union_lcomm
thf(fact_714_union__assoc,axiom,
! [M: multiset_set_nat,N: multiset_set_nat,K: multiset_set_nat] :
( ( plus_p8712254050562127327et_nat @ ( plus_p8712254050562127327et_nat @ M @ N ) @ K )
= ( plus_p8712254050562127327et_nat @ M @ ( plus_p8712254050562127327et_nat @ N @ K ) ) ) ).
% union_assoc
thf(fact_715_Int__Diff__disjoint,axiom,
! [A2: set_list_a,B: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ A2 @ B ) @ ( minus_646659088055828811list_a @ A2 @ B ) )
= bot_bot_set_list_a ) ).
% Int_Diff_disjoint
thf(fact_716_Int__Diff__disjoint,axiom,
! [A2: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ ( minus_minus_set_nat @ A2 @ B ) )
= bot_bot_set_nat ) ).
% Int_Diff_disjoint
thf(fact_717_Int__Diff__disjoint,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( inf_in8776938414804536127at_nat @ ( inf_in8776938414804536127at_nat @ A2 @ B ) @ ( minus_6698950876951835142at_nat @ A2 @ B ) )
= bot_bo1973934853101755969at_nat ) ).
% Int_Diff_disjoint
thf(fact_718_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_719_Diff__triv,axiom,
! [A2: set_list_a,B: set_list_a] :
( ( ( inf_inf_set_list_a @ A2 @ B )
= bot_bot_set_list_a )
=> ( ( minus_646659088055828811list_a @ A2 @ B )
= A2 ) ) ).
% Diff_triv
thf(fact_720_Diff__triv,axiom,
! [A2: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B )
= bot_bot_set_nat )
=> ( ( minus_minus_set_nat @ A2 @ B )
= A2 ) ) ).
% Diff_triv
thf(fact_721_Diff__triv,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( ( inf_in8776938414804536127at_nat @ A2 @ B )
= bot_bo1973934853101755969at_nat )
=> ( ( minus_6698950876951835142at_nat @ A2 @ B )
= A2 ) ) ).
% Diff_triv
thf(fact_722_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_723_subset__singleton__iff,axiom,
! [X5: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
( ( ord_le3146513528884898305at_nat @ X5 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) )
= ( ( X5 = bot_bo2099793752762293965at_nat )
| ( X5
= ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_724_subset__singleton__iff,axiom,
! [X5: set_Re381260168593705685la_a_b,A: relational_fmla_a_b] :
( ( ord_le4112832032246704949la_a_b @ X5 @ ( insert7010464514620295119la_a_b @ A @ bot_bo4495933725496725865la_a_b ) )
= ( ( X5 = bot_bo4495933725496725865la_a_b )
| ( X5
= ( insert7010464514620295119la_a_b @ A @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% subset_singleton_iff
thf(fact_725_subset__singleton__iff,axiom,
! [X5: set_list_a,A: list_a] :
( ( ord_le8861187494160871172list_a @ X5 @ ( insert_list_a @ A @ bot_bot_set_list_a ) )
= ( ( X5 = bot_bot_set_list_a )
| ( X5
= ( insert_list_a @ A @ bot_bot_set_list_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_726_subset__singleton__iff,axiom,
! [X5: set_Pr2645174627780777389at_nat,A: produc5825016348098550007at_nat] :
( ( ord_le8520675249591772685at_nat @ X5 @ ( insert3260075854425521959at_nat @ A @ bot_bo1973934853101755969at_nat ) )
= ( ( X5 = bot_bo1973934853101755969at_nat )
| ( X5
= ( insert3260075854425521959at_nat @ A @ bot_bo1973934853101755969at_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_727_subset__singleton__iff,axiom,
! [X5: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ X5 @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( ( X5 = bot_bot_set_nat )
| ( X5
= ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_728_subset__singletonD,axiom,
! [A2: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat] :
( ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) )
=> ( ( A2 = bot_bo2099793752762293965at_nat )
| ( A2
= ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% subset_singletonD
thf(fact_729_subset__singletonD,axiom,
! [A2: set_Re381260168593705685la_a_b,X2: relational_fmla_a_b] :
( ( ord_le4112832032246704949la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ X2 @ bot_bo4495933725496725865la_a_b ) )
=> ( ( A2 = bot_bo4495933725496725865la_a_b )
| ( A2
= ( insert7010464514620295119la_a_b @ X2 @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% subset_singletonD
thf(fact_730_subset__singletonD,axiom,
! [A2: set_list_a,X2: list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ X2 @ bot_bot_set_list_a ) )
=> ( ( A2 = bot_bot_set_list_a )
| ( A2
= ( insert_list_a @ X2 @ bot_bot_set_list_a ) ) ) ) ).
% subset_singletonD
thf(fact_731_subset__singletonD,axiom,
! [A2: set_Pr2645174627780777389at_nat,X2: produc5825016348098550007at_nat] :
( ( ord_le8520675249591772685at_nat @ A2 @ ( insert3260075854425521959at_nat @ X2 @ bot_bo1973934853101755969at_nat ) )
=> ( ( A2 = bot_bo1973934853101755969at_nat )
| ( A2
= ( insert3260075854425521959at_nat @ X2 @ bot_bo1973934853101755969at_nat ) ) ) ) ).
% subset_singletonD
thf(fact_732_subset__singletonD,axiom,
! [A2: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
=> ( ( A2 = bot_bot_set_nat )
| ( A2
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_733_Un__Diff__Int,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( sup_su6099978769595272409at_nat @ ( minus_6698950876951835142at_nat @ A2 @ B ) @ ( inf_in8776938414804536127at_nat @ A2 @ B ) )
= A2 ) ).
% Un_Diff_Int
thf(fact_734_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_735_Int__Diff__Un,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( sup_su6099978769595272409at_nat @ ( inf_in8776938414804536127at_nat @ A2 @ B ) @ ( minus_6698950876951835142at_nat @ A2 @ B ) )
= A2 ) ).
% Int_Diff_Un
thf(fact_736_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_737_Diff__Int,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat,C: set_Pr2645174627780777389at_nat] :
( ( minus_6698950876951835142at_nat @ A2 @ ( inf_in8776938414804536127at_nat @ B @ C ) )
= ( sup_su6099978769595272409at_nat @ ( minus_6698950876951835142at_nat @ A2 @ B ) @ ( minus_6698950876951835142at_nat @ A2 @ C ) ) ) ).
% Diff_Int
thf(fact_738_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_739_Diff__Un,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat,C: set_Pr2645174627780777389at_nat] :
( ( minus_6698950876951835142at_nat @ A2 @ ( sup_su6099978769595272409at_nat @ B @ C ) )
= ( inf_in8776938414804536127at_nat @ ( minus_6698950876951835142at_nat @ A2 @ B ) @ ( minus_6698950876951835142at_nat @ A2 @ C ) ) ) ).
% Diff_Un
thf(fact_740_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_741_image__diff__subset,axiom,
! [F: relational_fmla_a_b > set_list_a,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] : ( ord_le8877086941679407844list_a @ ( minus_4782336368215558443list_a @ ( image_130269618930851390list_a @ F @ A2 ) @ ( image_130269618930851390list_a @ F @ B ) ) @ ( image_130269618930851390list_a @ F @ ( minus_4077726661957047470la_a_b @ A2 @ B ) ) ) ).
% image_diff_subset
thf(fact_742_image__diff__subset,axiom,
! [F: nat > relational_fmla_a_b,A2: set_nat,B: set_nat] : ( ord_le4112832032246704949la_a_b @ ( minus_4077726661957047470la_a_b @ ( image_4386371547000553590la_a_b @ F @ A2 ) @ ( image_4386371547000553590la_a_b @ F @ B ) ) @ ( image_4386371547000553590la_a_b @ F @ ( minus_minus_set_nat @ A2 @ B ) ) ) ).
% image_diff_subset
thf(fact_743_image__diff__subset,axiom,
! [F: produc5825016348098550007at_nat > relational_fmla_a_b,A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] : ( ord_le4112832032246704949la_a_b @ ( minus_4077726661957047470la_a_b @ ( image_7406987224865000349la_a_b @ F @ A2 ) @ ( image_7406987224865000349la_a_b @ F @ B ) ) @ ( image_7406987224865000349la_a_b @ F @ ( minus_6698950876951835142at_nat @ A2 @ B ) ) ) ).
% image_diff_subset
thf(fact_744_image__diff__subset,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] : ( ord_le4112832032246704949la_a_b @ ( minus_4077726661957047470la_a_b @ ( image_6790371041703824709la_a_b @ F @ A2 ) @ ( image_6790371041703824709la_a_b @ F @ B ) ) @ ( image_6790371041703824709la_a_b @ F @ ( minus_4077726661957047470la_a_b @ A2 @ B ) ) ) ).
% image_diff_subset
thf(fact_745_image__diff__subset,axiom,
! [F: nat > produc5825016348098550007at_nat,A2: set_nat,B: set_nat] : ( ord_le8520675249591772685at_nat @ ( minus_6698950876951835142at_nat @ ( image_1371346403869992270at_nat @ F @ A2 ) @ ( image_1371346403869992270at_nat @ F @ B ) ) @ ( image_1371346403869992270at_nat @ F @ ( minus_minus_set_nat @ A2 @ B ) ) ) ).
% image_diff_subset
thf(fact_746_image__diff__subset,axiom,
! [F: produc5825016348098550007at_nat > produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] : ( ord_le8520675249591772685at_nat @ ( minus_6698950876951835142at_nat @ ( image_6979785008349797877at_nat @ F @ A2 ) @ ( image_6979785008349797877at_nat @ F @ B ) ) @ ( image_6979785008349797877at_nat @ F @ ( minus_6698950876951835142at_nat @ A2 @ B ) ) ) ).
% image_diff_subset
thf(fact_747_image__diff__subset,axiom,
! [F: relational_fmla_a_b > produc5825016348098550007at_nat,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] : ( ord_le8520675249591772685at_nat @ ( minus_6698950876951835142at_nat @ ( image_5430539854921360285at_nat @ F @ A2 ) @ ( image_5430539854921360285at_nat @ F @ B ) ) @ ( image_5430539854921360285at_nat @ F @ ( minus_4077726661957047470la_a_b @ A2 @ B ) ) ) ).
% image_diff_subset
thf(fact_748_image__diff__subset,axiom,
! [F: produc5825016348098550007at_nat > nat,A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_8316665354072716238at_nat @ F @ A2 ) @ ( image_8316665354072716238at_nat @ F @ B ) ) @ ( image_8316665354072716238at_nat @ F @ ( minus_6698950876951835142at_nat @ A2 @ B ) ) ) ).
% image_diff_subset
thf(fact_749_image__diff__subset,axiom,
! [F: relational_fmla_a_b > nat,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_341122591648980342_b_nat @ F @ A2 ) @ ( image_341122591648980342_b_nat @ F @ B ) ) @ ( image_341122591648980342_b_nat @ F @ ( minus_4077726661957047470la_a_b @ A2 @ B ) ) ) ).
% image_diff_subset
thf(fact_750_subset__Diff__insert,axiom,
! [A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat,C: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ A2 @ ( minus_1356011639430497352at_nat @ B @ ( insert8211810215607154385at_nat @ X2 @ C ) ) )
= ( ( ord_le3146513528884898305at_nat @ A2 @ ( minus_1356011639430497352at_nat @ B @ C ) )
& ~ ( member8440522571783428010at_nat @ X2 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_751_subset__Diff__insert,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,X2: relational_fmla_a_b,C: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A2 @ ( minus_4077726661957047470la_a_b @ B @ ( insert7010464514620295119la_a_b @ X2 @ C ) ) )
= ( ( ord_le4112832032246704949la_a_b @ A2 @ ( minus_4077726661957047470la_a_b @ B @ C ) )
& ~ ( member4680049679412964150la_a_b @ X2 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_752_subset__Diff__insert,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat,X2: produc5825016348098550007at_nat,C: set_Pr2645174627780777389at_nat] :
( ( ord_le8520675249591772685at_nat @ A2 @ ( minus_6698950876951835142at_nat @ B @ ( insert3260075854425521959at_nat @ X2 @ C ) ) )
= ( ( ord_le8520675249591772685at_nat @ A2 @ ( minus_6698950876951835142at_nat @ B @ C ) )
& ~ ( member6956540099943067662at_nat @ X2 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_753_subset__Diff__insert,axiom,
! [A2: set_nat,B: set_nat,X2: nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B @ ( insert_nat @ X2 @ C ) ) )
= ( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B @ C ) )
& ~ ( member_nat @ X2 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_754_Multiset_Odiff__right__commute,axiom,
! [M: multiset_set_nat,N: multiset_set_nat,Q: multiset_set_nat] :
( ( minus_7237264121398869807et_nat @ ( minus_7237264121398869807et_nat @ M @ N ) @ Q )
= ( minus_7237264121398869807et_nat @ ( minus_7237264121398869807et_nat @ M @ Q ) @ N ) ) ).
% Multiset.diff_right_commute
thf(fact_755_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_756_Diff__subset__conv,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat,C: set_Pr2645174627780777389at_nat] :
( ( ord_le8520675249591772685at_nat @ ( minus_6698950876951835142at_nat @ A2 @ B ) @ C )
= ( ord_le8520675249591772685at_nat @ A2 @ ( sup_su6099978769595272409at_nat @ B @ C ) ) ) ).
% Diff_subset_conv
thf(fact_757_Diff__subset__conv,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B ) @ C )
= ( ord_less_eq_set_nat @ A2 @ ( sup_sup_set_nat @ B @ C ) ) ) ).
% Diff_subset_conv
thf(fact_758_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_759_Diff__partition,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( ord_le8520675249591772685at_nat @ A2 @ B )
=> ( ( sup_su6099978769595272409at_nat @ A2 @ ( minus_6698950876951835142at_nat @ B @ A2 ) )
= B ) ) ).
% Diff_partition
thf(fact_760_Diff__partition,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( sup_sup_set_nat @ A2 @ ( minus_minus_set_nat @ B @ A2 ) )
= B ) ) ).
% Diff_partition
thf(fact_761_mset__set__Diff,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ B @ A2 )
=> ( ( mset_set_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ B ) )
= ( minus_7237264121398869807et_nat @ ( mset_set_set_nat @ A2 ) @ ( mset_set_set_nat @ B ) ) ) ) ) ).
% mset_set_Diff
thf(fact_762_mset__set__Diff,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ A2 )
=> ( ( ord_le4112832032246704949la_a_b @ B @ A2 )
=> ( ( mset_s7504067419864366776la_a_b @ ( minus_4077726661957047470la_a_b @ A2 @ B ) )
= ( minus_650714806544656072la_a_b @ ( mset_s7504067419864366776la_a_b @ A2 ) @ ( mset_s7504067419864366776la_a_b @ B ) ) ) ) ) ).
% mset_set_Diff
thf(fact_763_mset__set__Diff,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( finite6790245451575510286at_nat @ A2 )
=> ( ( ord_le8520675249591772685at_nat @ B @ A2 )
=> ( ( mset_s3030574779765502096at_nat @ ( minus_6698950876951835142at_nat @ A2 @ B ) )
= ( minus_1138467844667854112at_nat @ ( mset_s3030574779765502096at_nat @ A2 ) @ ( mset_s3030574779765502096at_nat @ B ) ) ) ) ) ).
% mset_set_Diff
thf(fact_764_mset__set__Diff,axiom,
! [A2: set_nat,B: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( mset_set_nat @ ( minus_minus_set_nat @ A2 @ B ) )
= ( minus_8522176038001411705et_nat @ ( mset_set_nat @ A2 ) @ ( mset_set_nat @ B ) ) ) ) ) ).
% mset_set_Diff
thf(fact_765_Set_Oempty__def,axiom,
( bot_bo1973934853101755969at_nat
= ( collec7697611494764367948at_nat
@ ^ [X: produc5825016348098550007at_nat] : $false ) ) ).
% Set.empty_def
thf(fact_766_Set_Oempty__def,axiom,
( bot_bo4495933725496725865la_a_b
= ( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] : $false ) ) ).
% Set.empty_def
thf(fact_767_Set_Oempty__def,axiom,
( bot_bot_set_list_a
= ( collect_list_a
@ ^ [X: list_a] : $false ) ) ).
% Set.empty_def
thf(fact_768_Set_Oempty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X: nat] : $false ) ) ).
% Set.empty_def
thf(fact_769_Compr__image__eq,axiom,
! [F: relational_fmla_a_b > set_list_a,A2: set_Re381260168593705685la_a_b,P: set_list_a > $o] :
( ( collect_set_list_a
@ ^ [X: set_list_a] :
( ( member_set_list_a @ X @ ( image_130269618930851390list_a @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_130269618930851390list_a @ F
@ ( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_770_Compr__image__eq,axiom,
! [F: produc5825016348098550007at_nat > produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat,P: produc5825016348098550007at_nat > $o] :
( ( collec7697611494764367948at_nat
@ ^ [X: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X @ ( image_6979785008349797877at_nat @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_6979785008349797877at_nat @ F
@ ( collec7697611494764367948at_nat
@ ^ [X: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_771_Compr__image__eq,axiom,
! [F: relational_fmla_a_b > produc5825016348098550007at_nat,A2: set_Re381260168593705685la_a_b,P: produc5825016348098550007at_nat > $o] :
( ( collec7697611494764367948at_nat
@ ^ [X: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X @ ( image_5430539854921360285at_nat @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_5430539854921360285at_nat @ F
@ ( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_772_Compr__image__eq,axiom,
! [F: produc5825016348098550007at_nat > relational_fmla_a_b,A2: set_Pr2645174627780777389at_nat,P: relational_fmla_a_b > $o] :
( ( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X @ ( image_7406987224865000349la_a_b @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_7406987224865000349la_a_b @ F
@ ( collec7697611494764367948at_nat
@ ^ [X: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_773_Compr__image__eq,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,P: relational_fmla_a_b > $o] :
( ( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X @ ( image_6790371041703824709la_a_b @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_6790371041703824709la_a_b @ F
@ ( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_774_Compr__image__eq,axiom,
! [F: nat > produc5825016348098550007at_nat,A2: set_nat,P: produc5825016348098550007at_nat > $o] :
( ( collec7697611494764367948at_nat
@ ^ [X: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X @ ( image_1371346403869992270at_nat @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_1371346403869992270at_nat @ F
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_775_Compr__image__eq,axiom,
! [F: nat > relational_fmla_a_b,A2: set_nat,P: relational_fmla_a_b > $o] :
( ( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X @ ( image_4386371547000553590la_a_b @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_4386371547000553590la_a_b @ F
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_776_Compr__image__eq,axiom,
! [F: produc5825016348098550007at_nat > nat,A2: set_Pr2645174627780777389at_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( image_8316665354072716238at_nat @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_8316665354072716238at_nat @ F
@ ( collec7697611494764367948at_nat
@ ^ [X: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_777_Compr__image__eq,axiom,
! [F: relational_fmla_a_b > nat,A2: set_Re381260168593705685la_a_b,P: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( image_341122591648980342_b_nat @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_341122591648980342_b_nat @ F
@ ( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_778_Compr__image__eq,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( image_nat_nat @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_nat_nat @ F
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_779_image__image,axiom,
! [F: produc5825016348098550007at_nat > produc5825016348098550007at_nat,G: nat > produc5825016348098550007at_nat,A2: set_nat] :
( ( image_6979785008349797877at_nat @ F @ ( image_1371346403869992270at_nat @ G @ A2 ) )
= ( image_1371346403869992270at_nat
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_780_image__image,axiom,
! [F: produc5825016348098550007at_nat > relational_fmla_a_b,G: nat > produc5825016348098550007at_nat,A2: set_nat] :
( ( image_7406987224865000349la_a_b @ F @ ( image_1371346403869992270at_nat @ G @ A2 ) )
= ( image_4386371547000553590la_a_b
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_781_image__image,axiom,
! [F: relational_fmla_a_b > produc5825016348098550007at_nat,G: nat > relational_fmla_a_b,A2: set_nat] :
( ( image_5430539854921360285at_nat @ F @ ( image_4386371547000553590la_a_b @ G @ A2 ) )
= ( image_1371346403869992270at_nat
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_782_image__image,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,G: nat > relational_fmla_a_b,A2: set_nat] :
( ( image_6790371041703824709la_a_b @ F @ ( image_4386371547000553590la_a_b @ G @ A2 ) )
= ( image_4386371547000553590la_a_b
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_783_image__image,axiom,
! [F: set_list_a > set_list_a,G: relational_fmla_a_b > set_list_a,A2: set_Re381260168593705685la_a_b] :
( ( image_5749939591322298757list_a @ F @ ( image_130269618930851390list_a @ G @ A2 ) )
= ( image_130269618930851390list_a
@ ^ [X: relational_fmla_a_b] : ( F @ ( G @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_784_image__image,axiom,
! [F: nat > produc5825016348098550007at_nat,G: nat > nat,A2: set_nat] :
( ( image_1371346403869992270at_nat @ F @ ( image_nat_nat @ G @ A2 ) )
= ( image_1371346403869992270at_nat
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_785_image__image,axiom,
! [F: nat > relational_fmla_a_b,G: nat > nat,A2: set_nat] :
( ( image_4386371547000553590la_a_b @ F @ ( image_nat_nat @ G @ A2 ) )
= ( image_4386371547000553590la_a_b
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_786_image__image,axiom,
! [F: relational_fmla_a_b > set_list_a,G: relational_fmla_a_b > relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( image_130269618930851390list_a @ F @ ( image_6790371041703824709la_a_b @ G @ A2 ) )
= ( image_130269618930851390list_a
@ ^ [X: relational_fmla_a_b] : ( F @ ( G @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_787_image__image,axiom,
! [F: relational_fmla_a_b > set_list_a,G: nat > relational_fmla_a_b,A2: set_nat] :
( ( image_130269618930851390list_a @ F @ ( image_4386371547000553590la_a_b @ G @ A2 ) )
= ( image_nat_set_list_a
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_788_imageE,axiom,
! [B2: set_list_a,F: relational_fmla_a_b > set_list_a,A2: set_Re381260168593705685la_a_b] :
( ( member_set_list_a @ B2 @ ( image_130269618930851390list_a @ F @ A2 ) )
=> ~ ! [X3: relational_fmla_a_b] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member4680049679412964150la_a_b @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_789_imageE,axiom,
! [B2: nat,F: nat > nat,A2: set_nat] :
( ( member_nat @ B2 @ ( image_nat_nat @ F @ A2 ) )
=> ~ ! [X3: nat] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_790_imageE,axiom,
! [B2: nat,F: produc5825016348098550007at_nat > nat,A2: set_Pr2645174627780777389at_nat] :
( ( member_nat @ B2 @ ( image_8316665354072716238at_nat @ F @ A2 ) )
=> ~ ! [X3: produc5825016348098550007at_nat] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member6956540099943067662at_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_791_imageE,axiom,
! [B2: nat,F: relational_fmla_a_b > nat,A2: set_Re381260168593705685la_a_b] :
( ( member_nat @ B2 @ ( image_341122591648980342_b_nat @ F @ A2 ) )
=> ~ ! [X3: relational_fmla_a_b] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member4680049679412964150la_a_b @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_792_imageE,axiom,
! [B2: produc5825016348098550007at_nat,F: nat > produc5825016348098550007at_nat,A2: set_nat] :
( ( member6956540099943067662at_nat @ B2 @ ( image_1371346403869992270at_nat @ F @ A2 ) )
=> ~ ! [X3: nat] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_793_imageE,axiom,
! [B2: produc5825016348098550007at_nat,F: produc5825016348098550007at_nat > produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ B2 @ ( image_6979785008349797877at_nat @ F @ A2 ) )
=> ~ ! [X3: produc5825016348098550007at_nat] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member6956540099943067662at_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_794_imageE,axiom,
! [B2: produc5825016348098550007at_nat,F: relational_fmla_a_b > produc5825016348098550007at_nat,A2: set_Re381260168593705685la_a_b] :
( ( member6956540099943067662at_nat @ B2 @ ( image_5430539854921360285at_nat @ F @ A2 ) )
=> ~ ! [X3: relational_fmla_a_b] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member4680049679412964150la_a_b @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_795_imageE,axiom,
! [B2: relational_fmla_a_b,F: nat > relational_fmla_a_b,A2: set_nat] :
( ( member4680049679412964150la_a_b @ B2 @ ( image_4386371547000553590la_a_b @ F @ A2 ) )
=> ~ ! [X3: nat] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_796_imageE,axiom,
! [B2: relational_fmla_a_b,F: produc5825016348098550007at_nat > relational_fmla_a_b,A2: set_Pr2645174627780777389at_nat] :
( ( member4680049679412964150la_a_b @ B2 @ ( image_7406987224865000349la_a_b @ F @ A2 ) )
=> ~ ! [X3: produc5825016348098550007at_nat] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member6956540099943067662at_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_797_imageE,axiom,
! [B2: relational_fmla_a_b,F: relational_fmla_a_b > relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ B2 @ ( image_6790371041703824709la_a_b @ F @ A2 ) )
=> ~ ! [X3: relational_fmla_a_b] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member4680049679412964150la_a_b @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_798_insert__Collect,axiom,
! [A: produc5825016348098550007at_nat,P: produc5825016348098550007at_nat > $o] :
( ( insert3260075854425521959at_nat @ A @ ( collec7697611494764367948at_nat @ P ) )
= ( collec7697611494764367948at_nat
@ ^ [U: produc5825016348098550007at_nat] :
( ( U != A )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_799_insert__Collect,axiom,
! [A: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
( ( insert8211810215607154385at_nat @ A @ ( collec3392354462482085612at_nat @ P ) )
= ( collec3392354462482085612at_nat
@ ^ [U: product_prod_nat_nat] :
( ( U != A )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_800_insert__Collect,axiom,
! [A: relational_fmla_a_b,P: relational_fmla_a_b > $o] :
( ( insert7010464514620295119la_a_b @ A @ ( collec3419995626248312948la_a_b @ P ) )
= ( collec3419995626248312948la_a_b
@ ^ [U: relational_fmla_a_b] :
( ( U != A )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_801_insert__Collect,axiom,
! [A: nat,P: nat > $o] :
( ( insert_nat @ A @ ( collect_nat @ P ) )
= ( collect_nat
@ ^ [U: nat] :
( ( U != A )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_802_insert__compr,axiom,
( insert8211810215607154385at_nat
= ( ^ [A3: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] :
( collec3392354462482085612at_nat
@ ^ [X: product_prod_nat_nat] :
( ( X = A3 )
| ( member8440522571783428010at_nat @ X @ B4 ) ) ) ) ) ).
% insert_compr
thf(fact_803_insert__compr,axiom,
( insert3260075854425521959at_nat
= ( ^ [A3: produc5825016348098550007at_nat,B4: set_Pr2645174627780777389at_nat] :
( collec7697611494764367948at_nat
@ ^ [X: produc5825016348098550007at_nat] :
( ( X = A3 )
| ( member6956540099943067662at_nat @ X @ B4 ) ) ) ) ) ).
% insert_compr
thf(fact_804_insert__compr,axiom,
( insert7010464514620295119la_a_b
= ( ^ [A3: relational_fmla_a_b,B4: set_Re381260168593705685la_a_b] :
( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] :
( ( X = A3 )
| ( member4680049679412964150la_a_b @ X @ B4 ) ) ) ) ) ).
% insert_compr
thf(fact_805_insert__compr,axiom,
( insert_nat
= ( ^ [A3: nat,B4: set_nat] :
( collect_nat
@ ^ [X: nat] :
( ( X = A3 )
| ( member_nat @ X @ B4 ) ) ) ) ) ).
% insert_compr
thf(fact_806_set__diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A4 )
& ~ ( member_nat @ X @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_807_set__diff__eq,axiom,
( minus_6698950876951835142at_nat
= ( ^ [A4: set_Pr2645174627780777389at_nat,B4: set_Pr2645174627780777389at_nat] :
( collec7697611494764367948at_nat
@ ^ [X: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X @ A4 )
& ~ ( member6956540099943067662at_nat @ X @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_808_set__diff__eq,axiom,
( minus_4077726661957047470la_a_b
= ( ^ [A4: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b] :
( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X @ A4 )
& ~ ( member4680049679412964150la_a_b @ X @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_809_Collect__disj__eq,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( P @ X )
| ( Q @ X ) ) )
= ( sup_sup_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_810_Collect__disj__eq,axiom,
! [P: produc5825016348098550007at_nat > $o,Q: produc5825016348098550007at_nat > $o] :
( ( collec7697611494764367948at_nat
@ ^ [X: produc5825016348098550007at_nat] :
( ( P @ X )
| ( Q @ X ) ) )
= ( sup_su6099978769595272409at_nat @ ( collec7697611494764367948at_nat @ P ) @ ( collec7697611494764367948at_nat @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_811_Collect__disj__eq,axiom,
! [P: relational_fmla_a_b > $o,Q: relational_fmla_a_b > $o] :
( ( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] :
( ( P @ X )
| ( Q @ X ) ) )
= ( sup_su5130108678486352897la_a_b @ ( collec3419995626248312948la_a_b @ P ) @ ( collec3419995626248312948la_a_b @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_812_Un__def,axiom,
( sup_sup_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A4 )
| ( member_nat @ X @ B4 ) ) ) ) ) ).
% Un_def
thf(fact_813_Un__def,axiom,
( sup_su6099978769595272409at_nat
= ( ^ [A4: set_Pr2645174627780777389at_nat,B4: set_Pr2645174627780777389at_nat] :
( collec7697611494764367948at_nat
@ ^ [X: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X @ A4 )
| ( member6956540099943067662at_nat @ X @ B4 ) ) ) ) ) ).
% Un_def
thf(fact_814_Un__def,axiom,
( sup_su5130108678486352897la_a_b
= ( ^ [A4: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b] :
( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X @ A4 )
| ( member4680049679412964150la_a_b @ X @ B4 ) ) ) ) ) ).
% Un_def
thf(fact_815_mset__set__Union,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ B )
=> ( ( ( inf_inf_set_set_nat @ A2 @ B )
= bot_bot_set_set_nat )
=> ( ( mset_set_set_nat @ ( sup_sup_set_set_nat @ A2 @ B ) )
= ( plus_p8712254050562127327et_nat @ ( mset_set_set_nat @ A2 ) @ ( mset_set_set_nat @ B ) ) ) ) ) ) ).
% mset_set_Union
thf(fact_816_mset__set__Union,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( finite6790245451575510286at_nat @ A2 )
=> ( ( finite6790245451575510286at_nat @ B )
=> ( ( ( inf_in8776938414804536127at_nat @ A2 @ B )
= bot_bo1973934853101755969at_nat )
=> ( ( mset_s3030574779765502096at_nat @ ( sup_su6099978769595272409at_nat @ A2 @ B ) )
= ( plus_p1586715138884553680at_nat @ ( mset_s3030574779765502096at_nat @ A2 ) @ ( mset_s3030574779765502096at_nat @ B ) ) ) ) ) ) ).
% mset_set_Union
thf(fact_817_mset__set__Union,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ A2 )
=> ( ( finite5600759454172676150la_a_b @ B )
=> ( ( ( inf_in8483230781156617063la_a_b @ A2 @ B )
= bot_bo4495933725496725865la_a_b )
=> ( ( mset_s7504067419864366776la_a_b @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) )
= ( plus_p5268770533993848la_a_b @ ( mset_s7504067419864366776la_a_b @ A2 ) @ ( mset_s7504067419864366776la_a_b @ B ) ) ) ) ) ) ).
% mset_set_Union
thf(fact_818_mset__set__Union,axiom,
! [A2: set_list_a,B: set_list_a] :
( ( finite_finite_list_a @ A2 )
=> ( ( finite_finite_list_a @ B )
=> ( ( ( inf_inf_set_list_a @ A2 @ B )
= bot_bot_set_list_a )
=> ( ( mset_set_list_a @ ( sup_sup_set_list_a @ A2 @ B ) )
= ( plus_p690419498615200257list_a @ ( mset_set_list_a @ A2 ) @ ( mset_set_list_a @ B ) ) ) ) ) ) ).
% mset_set_Union
thf(fact_819_mset__set__Union,axiom,
! [A2: set_nat,B: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( ( ( inf_inf_set_nat @ A2 @ B )
= bot_bot_set_nat )
=> ( ( mset_set_nat @ ( sup_sup_set_nat @ A2 @ B ) )
= ( plus_p6334493942879108393et_nat @ ( mset_set_nat @ A2 ) @ ( mset_set_nat @ B ) ) ) ) ) ) ).
% mset_set_Union
thf(fact_820_Diff__single__insert,axiom,
! [A2: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) @ B )
=> ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ X2 @ B ) ) ) ).
% Diff_single_insert
thf(fact_821_Diff__single__insert,axiom,
! [A2: set_list_a,X2: list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ X2 @ bot_bot_set_list_a ) ) @ B )
=> ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ X2 @ B ) ) ) ).
% Diff_single_insert
thf(fact_822_Diff__single__insert,axiom,
! [A2: set_Re381260168593705685la_a_b,X2: relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ ( minus_4077726661957047470la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ X2 @ bot_bo4495933725496725865la_a_b ) ) @ B )
=> ( ord_le4112832032246704949la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ X2 @ B ) ) ) ).
% Diff_single_insert
thf(fact_823_Diff__single__insert,axiom,
! [A2: set_Pr2645174627780777389at_nat,X2: produc5825016348098550007at_nat,B: set_Pr2645174627780777389at_nat] :
( ( ord_le8520675249591772685at_nat @ ( minus_6698950876951835142at_nat @ A2 @ ( insert3260075854425521959at_nat @ X2 @ bot_bo1973934853101755969at_nat ) ) @ B )
=> ( ord_le8520675249591772685at_nat @ A2 @ ( insert3260075854425521959at_nat @ X2 @ B ) ) ) ).
% Diff_single_insert
thf(fact_824_Diff__single__insert,axiom,
! [A2: set_nat,X2: nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ B )
=> ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X2 @ B ) ) ) ).
% Diff_single_insert
thf(fact_825_subset__insert__iff,axiom,
! [A2: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ X2 @ B ) )
= ( ( ( member8440522571783428010at_nat @ X2 @ A2 )
=> ( ord_le3146513528884898305at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) @ B ) )
& ( ~ ( member8440522571783428010at_nat @ X2 @ A2 )
=> ( ord_le3146513528884898305at_nat @ A2 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_826_subset__insert__iff,axiom,
! [A2: set_list_a,X2: list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ X2 @ B ) )
= ( ( ( member_list_a @ X2 @ A2 )
=> ( ord_le8861187494160871172list_a @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ X2 @ bot_bot_set_list_a ) ) @ B ) )
& ( ~ ( member_list_a @ X2 @ A2 )
=> ( ord_le8861187494160871172list_a @ A2 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_827_subset__insert__iff,axiom,
! [A2: set_Re381260168593705685la_a_b,X2: relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ X2 @ B ) )
= ( ( ( member4680049679412964150la_a_b @ X2 @ A2 )
=> ( ord_le4112832032246704949la_a_b @ ( minus_4077726661957047470la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ X2 @ bot_bo4495933725496725865la_a_b ) ) @ B ) )
& ( ~ ( member4680049679412964150la_a_b @ X2 @ A2 )
=> ( ord_le4112832032246704949la_a_b @ A2 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_828_subset__insert__iff,axiom,
! [A2: set_Pr2645174627780777389at_nat,X2: produc5825016348098550007at_nat,B: set_Pr2645174627780777389at_nat] :
( ( ord_le8520675249591772685at_nat @ A2 @ ( insert3260075854425521959at_nat @ X2 @ B ) )
= ( ( ( member6956540099943067662at_nat @ X2 @ A2 )
=> ( ord_le8520675249591772685at_nat @ ( minus_6698950876951835142at_nat @ A2 @ ( insert3260075854425521959at_nat @ X2 @ bot_bo1973934853101755969at_nat ) ) @ B ) )
& ( ~ ( member6956540099943067662at_nat @ X2 @ A2 )
=> ( ord_le8520675249591772685at_nat @ A2 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_829_subset__insert__iff,axiom,
! [A2: set_nat,X2: nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X2 @ B ) )
= ( ( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ B ) )
& ( ~ ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_830_singleton__inject,axiom,
! [A: product_prod_nat_nat,B2: product_prod_nat_nat] :
( ( ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat )
= ( insert8211810215607154385at_nat @ B2 @ bot_bo2099793752762293965at_nat ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_831_singleton__inject,axiom,
! [A: produc5825016348098550007at_nat,B2: produc5825016348098550007at_nat] :
( ( ( insert3260075854425521959at_nat @ A @ bot_bo1973934853101755969at_nat )
= ( insert3260075854425521959at_nat @ B2 @ bot_bo1973934853101755969at_nat ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_832_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_833_singleton__inject,axiom,
! [A: list_a,B2: list_a] :
( ( ( insert_list_a @ A @ bot_bot_set_list_a )
= ( insert_list_a @ B2 @ bot_bot_set_list_a ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_834_singleton__inject,axiom,
! [A: nat,B2: nat] :
( ( ( insert_nat @ A @ bot_bot_set_nat )
= ( insert_nat @ B2 @ bot_bot_set_nat ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_835_insert__not__empty,axiom,
! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
( ( insert8211810215607154385at_nat @ A @ A2 )
!= bot_bo2099793752762293965at_nat ) ).
% insert_not_empty
thf(fact_836_insert__not__empty,axiom,
! [A: produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat] :
( ( insert3260075854425521959at_nat @ A @ A2 )
!= bot_bo1973934853101755969at_nat ) ).
% insert_not_empty
thf(fact_837_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_838_insert__not__empty,axiom,
! [A: list_a,A2: set_list_a] :
( ( insert_list_a @ A @ A2 )
!= bot_bot_set_list_a ) ).
% insert_not_empty
thf(fact_839_insert__not__empty,axiom,
! [A: nat,A2: set_nat] :
( ( insert_nat @ A @ A2 )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_840_doubleton__eq__iff,axiom,
! [A: product_prod_nat_nat,B2: product_prod_nat_nat,C2: product_prod_nat_nat,D2: product_prod_nat_nat] :
( ( ( insert8211810215607154385at_nat @ A @ ( insert8211810215607154385at_nat @ B2 @ bot_bo2099793752762293965at_nat ) )
= ( insert8211810215607154385at_nat @ C2 @ ( insert8211810215607154385at_nat @ D2 @ bot_bo2099793752762293965at_nat ) ) )
= ( ( ( A = C2 )
& ( B2 = D2 ) )
| ( ( A = D2 )
& ( B2 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_841_doubleton__eq__iff,axiom,
! [A: produc5825016348098550007at_nat,B2: produc5825016348098550007at_nat,C2: produc5825016348098550007at_nat,D2: produc5825016348098550007at_nat] :
( ( ( insert3260075854425521959at_nat @ A @ ( insert3260075854425521959at_nat @ B2 @ bot_bo1973934853101755969at_nat ) )
= ( insert3260075854425521959at_nat @ C2 @ ( insert3260075854425521959at_nat @ D2 @ bot_bo1973934853101755969at_nat ) ) )
= ( ( ( A = C2 )
& ( B2 = D2 ) )
| ( ( A = D2 )
& ( B2 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_842_doubleton__eq__iff,axiom,
! [A: relational_fmla_a_b,B2: relational_fmla_a_b,C2: relational_fmla_a_b,D2: relational_fmla_a_b] :
( ( ( insert7010464514620295119la_a_b @ A @ ( insert7010464514620295119la_a_b @ B2 @ bot_bo4495933725496725865la_a_b ) )
= ( insert7010464514620295119la_a_b @ C2 @ ( insert7010464514620295119la_a_b @ D2 @ bot_bo4495933725496725865la_a_b ) ) )
= ( ( ( A = C2 )
& ( B2 = D2 ) )
| ( ( A = D2 )
& ( B2 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_843_doubleton__eq__iff,axiom,
! [A: list_a,B2: list_a,C2: list_a,D2: list_a] :
( ( ( insert_list_a @ A @ ( insert_list_a @ B2 @ bot_bot_set_list_a ) )
= ( insert_list_a @ C2 @ ( insert_list_a @ D2 @ bot_bot_set_list_a ) ) )
= ( ( ( A = C2 )
& ( B2 = D2 ) )
| ( ( A = D2 )
& ( B2 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_844_doubleton__eq__iff,axiom,
! [A: nat,B2: nat,C2: nat,D2: nat] :
( ( ( insert_nat @ A @ ( insert_nat @ B2 @ bot_bot_set_nat ) )
= ( insert_nat @ C2 @ ( insert_nat @ D2 @ bot_bot_set_nat ) ) )
= ( ( ( A = C2 )
& ( B2 = D2 ) )
| ( ( A = D2 )
& ( B2 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_845_singleton__iff,axiom,
! [B2: product_prod_nat_nat,A: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ B2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_846_singleton__iff,axiom,
! [B2: produc5825016348098550007at_nat,A: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ B2 @ ( insert3260075854425521959at_nat @ A @ bot_bo1973934853101755969at_nat ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_847_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_848_singleton__iff,axiom,
! [B2: list_a,A: list_a] :
( ( member_list_a @ B2 @ ( insert_list_a @ A @ bot_bot_set_list_a ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_849_singleton__iff,axiom,
! [B2: nat,A: nat] :
( ( member_nat @ B2 @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_850_singletonD,axiom,
! [B2: product_prod_nat_nat,A: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ B2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_851_singletonD,axiom,
! [B2: produc5825016348098550007at_nat,A: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ B2 @ ( insert3260075854425521959at_nat @ A @ bot_bo1973934853101755969at_nat ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_852_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_853_singletonD,axiom,
! [B2: list_a,A: list_a] :
( ( member_list_a @ B2 @ ( insert_list_a @ A @ bot_bot_set_list_a ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_854_singletonD,axiom,
! [B2: nat,A: nat] :
( ( member_nat @ B2 @ ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_855_insert__Diff__if,axiom,
! [X2: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
( ( ( member8440522571783428010at_nat @ X2 @ B )
=> ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X2 @ A2 ) @ B )
= ( minus_1356011639430497352at_nat @ A2 @ B ) ) )
& ( ~ ( member8440522571783428010at_nat @ X2 @ B )
=> ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X2 @ A2 ) @ B )
= ( insert8211810215607154385at_nat @ X2 @ ( minus_1356011639430497352at_nat @ A2 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_856_insert__Diff__if,axiom,
! [X2: nat,B: set_nat,A2: set_nat] :
( ( ( member_nat @ X2 @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A2 ) @ B )
= ( minus_minus_set_nat @ A2 @ B ) ) )
& ( ~ ( member_nat @ X2 @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A2 ) @ B )
= ( insert_nat @ X2 @ ( minus_minus_set_nat @ A2 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_857_insert__Diff__if,axiom,
! [X2: produc5825016348098550007at_nat,B: set_Pr2645174627780777389at_nat,A2: set_Pr2645174627780777389at_nat] :
( ( ( member6956540099943067662at_nat @ X2 @ B )
=> ( ( minus_6698950876951835142at_nat @ ( insert3260075854425521959at_nat @ X2 @ A2 ) @ B )
= ( minus_6698950876951835142at_nat @ A2 @ B ) ) )
& ( ~ ( member6956540099943067662at_nat @ X2 @ B )
=> ( ( minus_6698950876951835142at_nat @ ( insert3260075854425521959at_nat @ X2 @ A2 ) @ B )
= ( insert3260075854425521959at_nat @ X2 @ ( minus_6698950876951835142at_nat @ A2 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_858_insert__Diff__if,axiom,
! [X2: relational_fmla_a_b,B: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b] :
( ( ( member4680049679412964150la_a_b @ X2 @ B )
=> ( ( minus_4077726661957047470la_a_b @ ( insert7010464514620295119la_a_b @ X2 @ A2 ) @ B )
= ( minus_4077726661957047470la_a_b @ A2 @ B ) ) )
& ( ~ ( member4680049679412964150la_a_b @ X2 @ B )
=> ( ( minus_4077726661957047470la_a_b @ ( insert7010464514620295119la_a_b @ X2 @ A2 ) @ B )
= ( insert7010464514620295119la_a_b @ X2 @ ( minus_4077726661957047470la_a_b @ A2 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_859_Un__empty__right,axiom,
! [A2: set_Pr2645174627780777389at_nat] :
( ( sup_su6099978769595272409at_nat @ A2 @ bot_bo1973934853101755969at_nat )
= A2 ) ).
% Un_empty_right
thf(fact_860_Un__empty__right,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ A2 @ bot_bo4495933725496725865la_a_b )
= A2 ) ).
% Un_empty_right
thf(fact_861_Un__empty__right,axiom,
! [A2: set_list_a] :
( ( sup_sup_set_list_a @ A2 @ bot_bot_set_list_a )
= A2 ) ).
% Un_empty_right
thf(fact_862_Un__empty__right,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% Un_empty_right
thf(fact_863_Un__empty__left,axiom,
! [B: set_Pr2645174627780777389at_nat] :
( ( sup_su6099978769595272409at_nat @ bot_bo1973934853101755969at_nat @ B )
= B ) ).
% Un_empty_left
thf(fact_864_Un__empty__left,axiom,
! [B: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ bot_bo4495933725496725865la_a_b @ B )
= B ) ).
% Un_empty_left
thf(fact_865_Un__empty__left,axiom,
! [B: set_list_a] :
( ( sup_sup_set_list_a @ bot_bot_set_list_a @ B )
= B ) ).
% Un_empty_left
thf(fact_866_Un__empty__left,axiom,
! [B: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ B )
= B ) ).
% Un_empty_left
thf(fact_867_image__Un,axiom,
! [F: nat > produc5825016348098550007at_nat,A2: set_nat,B: set_nat] :
( ( image_1371346403869992270at_nat @ F @ ( sup_sup_set_nat @ A2 @ B ) )
= ( sup_su6099978769595272409at_nat @ ( image_1371346403869992270at_nat @ F @ A2 ) @ ( image_1371346403869992270at_nat @ F @ B ) ) ) ).
% image_Un
thf(fact_868_image__Un,axiom,
! [F: nat > relational_fmla_a_b,A2: set_nat,B: set_nat] :
( ( image_4386371547000553590la_a_b @ F @ ( sup_sup_set_nat @ A2 @ B ) )
= ( sup_su5130108678486352897la_a_b @ ( image_4386371547000553590la_a_b @ F @ A2 ) @ ( image_4386371547000553590la_a_b @ F @ B ) ) ) ).
% image_Un
thf(fact_869_image__Un,axiom,
! [F: produc5825016348098550007at_nat > produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( image_6979785008349797877at_nat @ F @ ( sup_su6099978769595272409at_nat @ A2 @ B ) )
= ( sup_su6099978769595272409at_nat @ ( image_6979785008349797877at_nat @ F @ A2 ) @ ( image_6979785008349797877at_nat @ F @ B ) ) ) ).
% image_Un
thf(fact_870_image__Un,axiom,
! [F: produc5825016348098550007at_nat > relational_fmla_a_b,A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( image_7406987224865000349la_a_b @ F @ ( sup_su6099978769595272409at_nat @ A2 @ B ) )
= ( sup_su5130108678486352897la_a_b @ ( image_7406987224865000349la_a_b @ F @ A2 ) @ ( image_7406987224865000349la_a_b @ F @ B ) ) ) ).
% image_Un
thf(fact_871_image__Un,axiom,
! [F: relational_fmla_a_b > set_list_a,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( image_130269618930851390list_a @ F @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) )
= ( sup_su4537662296134749976list_a @ ( image_130269618930851390list_a @ F @ A2 ) @ ( image_130269618930851390list_a @ F @ B ) ) ) ).
% image_Un
thf(fact_872_image__Un,axiom,
! [F: relational_fmla_a_b > produc5825016348098550007at_nat,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( image_5430539854921360285at_nat @ F @ ( sup_su5130108678486352897la_a_b @ A2 @ B ) )
= ( sup_su6099978769595272409at_nat @ ( image_5430539854921360285at_nat @ F @ A2 ) @ ( image_5430539854921360285at_nat @ F @ B ) ) ) ).
% image_Un
thf(fact_873_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_874_Un__Diff,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat,C: set_Pr2645174627780777389at_nat] :
( ( minus_6698950876951835142at_nat @ ( sup_su6099978769595272409at_nat @ A2 @ B ) @ C )
= ( sup_su6099978769595272409at_nat @ ( minus_6698950876951835142at_nat @ A2 @ C ) @ ( minus_6698950876951835142at_nat @ B @ C ) ) ) ).
% Un_Diff
thf(fact_875_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_876_image__mset_Ocompositionality,axiom,
! [F: relational_fmla_a_b > set_nat,G: produc5825016348098550007at_nat > relational_fmla_a_b,Multiset: multis4094885785038667591at_nat] :
( ( image_3750641176000701538et_nat @ F @ ( image_287164480633417683la_a_b @ G @ Multiset ) )
= ( image_7702178775022393786et_nat @ ( comp_R1256417402204451841at_nat @ F @ G ) @ Multiset ) ) ).
% image_mset.compositionality
thf(fact_877_image__mset_Ocompositionality,axiom,
! [F: set_nat > set_nat,G: produc5825016348098550007at_nat > set_nat,Multiset: multis4094885785038667591at_nat] :
( ( image_1110420595810377289et_nat @ F @ ( image_7702178775022393786et_nat @ G @ Multiset ) )
= ( image_7702178775022393786et_nat @ ( comp_s7157870221967811432at_nat @ F @ G ) @ Multiset ) ) ).
% image_mset.compositionality
thf(fact_878_image__mset_Ocompositionality,axiom,
! [F: produc5825016348098550007at_nat > set_nat,G: produc5825016348098550007at_nat > produc5825016348098550007at_nat,Multiset: multis4094885785038667591at_nat] :
( ( image_7702178775022393786et_nat @ F @ ( image_4422302805329903915at_nat @ G @ Multiset ) )
= ( image_7702178775022393786et_nat @ ( comp_P5042154492879702745at_nat @ F @ G ) @ Multiset ) ) ).
% image_mset.compositionality
thf(fact_879_multiset_Omap__comp,axiom,
! [G: relational_fmla_a_b > set_nat,F: produc5825016348098550007at_nat > relational_fmla_a_b,V: multis4094885785038667591at_nat] :
( ( image_3750641176000701538et_nat @ G @ ( image_287164480633417683la_a_b @ F @ V ) )
= ( image_7702178775022393786et_nat @ ( comp_R1256417402204451841at_nat @ G @ F ) @ V ) ) ).
% multiset.map_comp
thf(fact_880_multiset_Omap__comp,axiom,
! [G: set_nat > set_nat,F: produc5825016348098550007at_nat > set_nat,V: multis4094885785038667591at_nat] :
( ( image_1110420595810377289et_nat @ G @ ( image_7702178775022393786et_nat @ F @ V ) )
= ( image_7702178775022393786et_nat @ ( comp_s7157870221967811432at_nat @ G @ F ) @ V ) ) ).
% multiset.map_comp
thf(fact_881_multiset_Omap__comp,axiom,
! [G: produc5825016348098550007at_nat > set_nat,F: produc5825016348098550007at_nat > produc5825016348098550007at_nat,V: multis4094885785038667591at_nat] :
( ( image_7702178775022393786et_nat @ G @ ( image_4422302805329903915at_nat @ F @ V ) )
= ( image_7702178775022393786et_nat @ ( comp_P5042154492879702745at_nat @ G @ F ) @ V ) ) ).
% multiset.map_comp
thf(fact_882_image__mset_Ocomp,axiom,
! [F: relational_fmla_a_b > set_nat,G: produc5825016348098550007at_nat > relational_fmla_a_b] :
( ( comp_m1639590883043443569at_nat @ ( image_3750641176000701538et_nat @ F ) @ ( image_287164480633417683la_a_b @ G ) )
= ( image_7702178775022393786et_nat @ ( comp_R1256417402204451841at_nat @ F @ G ) ) ) ).
% image_mset.comp
thf(fact_883_image__mset_Ocomp,axiom,
! [F: set_nat > set_nat,G: produc5825016348098550007at_nat > set_nat] :
( ( comp_m5667757267881127256at_nat @ ( image_1110420595810377289et_nat @ F ) @ ( image_7702178775022393786et_nat @ G ) )
= ( image_7702178775022393786et_nat @ ( comp_s7157870221967811432at_nat @ F @ G ) ) ) ).
% image_mset.comp
thf(fact_884_image__mset_Ocomp,axiom,
! [F: produc5825016348098550007at_nat > set_nat,G: produc5825016348098550007at_nat > produc5825016348098550007at_nat] :
( ( comp_m465251289197524681at_nat @ ( image_7702178775022393786et_nat @ F ) @ ( image_4422302805329903915at_nat @ G ) )
= ( image_7702178775022393786et_nat @ ( comp_P5042154492879702745at_nat @ F @ G ) ) ) ).
% image_mset.comp
thf(fact_885_image__mset__eq__plusD,axiom,
! [F: produc5825016348098550007at_nat > set_nat,A2: multis4094885785038667591at_nat,B: multiset_set_nat,C: multiset_set_nat] :
( ( ( image_7702178775022393786et_nat @ F @ A2 )
= ( plus_p8712254050562127327et_nat @ B @ C ) )
=> ? [B5: multis4094885785038667591at_nat,C7: multis4094885785038667591at_nat] :
( ( A2
= ( plus_p1586715138884553680at_nat @ B5 @ C7 ) )
& ( B
= ( image_7702178775022393786et_nat @ F @ B5 ) )
& ( C
= ( image_7702178775022393786et_nat @ F @ C7 ) ) ) ) ).
% image_mset_eq_plusD
thf(fact_886_image__mset__eq__plusD,axiom,
! [F: set_nat > set_nat,A2: multiset_set_nat,B: multiset_set_nat,C: multiset_set_nat] :
( ( ( image_1110420595810377289et_nat @ F @ A2 )
= ( plus_p8712254050562127327et_nat @ B @ C ) )
=> ? [B5: multiset_set_nat,C7: multiset_set_nat] :
( ( A2
= ( plus_p8712254050562127327et_nat @ B5 @ C7 ) )
& ( B
= ( image_1110420595810377289et_nat @ F @ B5 ) )
& ( C
= ( image_1110420595810377289et_nat @ F @ C7 ) ) ) ) ).
% image_mset_eq_plusD
thf(fact_887_diff__union__cancelR,axiom,
! [M: multiset_set_nat,N: multiset_set_nat] :
( ( minus_7237264121398869807et_nat @ ( plus_p8712254050562127327et_nat @ M @ N ) @ N )
= M ) ).
% diff_union_cancelR
thf(fact_888_diff__union__cancelL,axiom,
! [N: multiset_set_nat,M: multiset_set_nat] :
( ( minus_7237264121398869807et_nat @ ( plus_p8712254050562127327et_nat @ N @ M ) @ N )
= M ) ).
% diff_union_cancelL
thf(fact_889_Multiset_Odiff__add,axiom,
! [M: multiset_set_nat,N: multiset_set_nat,Q: multiset_set_nat] :
( ( minus_7237264121398869807et_nat @ M @ ( plus_p8712254050562127327et_nat @ N @ Q ) )
= ( minus_7237264121398869807et_nat @ ( minus_7237264121398869807et_nat @ M @ N ) @ Q ) ) ).
% Multiset.diff_add
thf(fact_890_Collect__conv__if2,axiom,
! [P: product_prod_nat_nat > $o,A: product_prod_nat_nat] :
( ( ( P @ A )
=> ( ( collec3392354462482085612at_nat
@ ^ [X: product_prod_nat_nat] :
( ( A = X )
& ( P @ X ) ) )
= ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collec3392354462482085612at_nat
@ ^ [X: product_prod_nat_nat] :
( ( A = X )
& ( P @ X ) ) )
= bot_bo2099793752762293965at_nat ) ) ) ).
% Collect_conv_if2
thf(fact_891_Collect__conv__if2,axiom,
! [P: produc5825016348098550007at_nat > $o,A: produc5825016348098550007at_nat] :
( ( ( P @ A )
=> ( ( collec7697611494764367948at_nat
@ ^ [X: produc5825016348098550007at_nat] :
( ( A = X )
& ( P @ X ) ) )
= ( insert3260075854425521959at_nat @ A @ bot_bo1973934853101755969at_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collec7697611494764367948at_nat
@ ^ [X: produc5825016348098550007at_nat] :
( ( A = X )
& ( P @ X ) ) )
= bot_bo1973934853101755969at_nat ) ) ) ).
% Collect_conv_if2
thf(fact_892_Collect__conv__if2,axiom,
! [P: relational_fmla_a_b > $o,A: relational_fmla_a_b] :
( ( ( P @ A )
=> ( ( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] :
( ( A = X )
& ( P @ X ) ) )
= ( insert7010464514620295119la_a_b @ A @ bot_bo4495933725496725865la_a_b ) ) )
& ( ~ ( P @ A )
=> ( ( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] :
( ( A = X )
& ( P @ X ) ) )
= bot_bo4495933725496725865la_a_b ) ) ) ).
% Collect_conv_if2
thf(fact_893_Collect__conv__if2,axiom,
! [P: list_a > $o,A: list_a] :
( ( ( P @ A )
=> ( ( collect_list_a
@ ^ [X: list_a] :
( ( A = X )
& ( P @ X ) ) )
= ( insert_list_a @ A @ bot_bot_set_list_a ) ) )
& ( ~ ( P @ A )
=> ( ( collect_list_a
@ ^ [X: list_a] :
( ( A = X )
& ( P @ X ) ) )
= bot_bot_set_list_a ) ) ) ).
% Collect_conv_if2
thf(fact_894_Collect__conv__if2,axiom,
! [P: nat > $o,A: nat] :
( ( ( P @ A )
=> ( ( collect_nat
@ ^ [X: nat] :
( ( A = X )
& ( P @ X ) ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collect_nat
@ ^ [X: nat] :
( ( A = X )
& ( P @ X ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if2
thf(fact_895_Collect__conv__if,axiom,
! [P: product_prod_nat_nat > $o,A: product_prod_nat_nat] :
( ( ( P @ A )
=> ( ( collec3392354462482085612at_nat
@ ^ [X: product_prod_nat_nat] :
( ( X = A )
& ( P @ X ) ) )
= ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collec3392354462482085612at_nat
@ ^ [X: product_prod_nat_nat] :
( ( X = A )
& ( P @ X ) ) )
= bot_bo2099793752762293965at_nat ) ) ) ).
% Collect_conv_if
thf(fact_896_Collect__conv__if,axiom,
! [P: produc5825016348098550007at_nat > $o,A: produc5825016348098550007at_nat] :
( ( ( P @ A )
=> ( ( collec7697611494764367948at_nat
@ ^ [X: produc5825016348098550007at_nat] :
( ( X = A )
& ( P @ X ) ) )
= ( insert3260075854425521959at_nat @ A @ bot_bo1973934853101755969at_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collec7697611494764367948at_nat
@ ^ [X: produc5825016348098550007at_nat] :
( ( X = A )
& ( P @ X ) ) )
= bot_bo1973934853101755969at_nat ) ) ) ).
% Collect_conv_if
thf(fact_897_Collect__conv__if,axiom,
! [P: relational_fmla_a_b > $o,A: relational_fmla_a_b] :
( ( ( P @ A )
=> ( ( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] :
( ( X = A )
& ( P @ X ) ) )
= ( insert7010464514620295119la_a_b @ A @ bot_bo4495933725496725865la_a_b ) ) )
& ( ~ ( P @ A )
=> ( ( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] :
( ( X = A )
& ( P @ X ) ) )
= bot_bo4495933725496725865la_a_b ) ) ) ).
% Collect_conv_if
thf(fact_898_Collect__conv__if,axiom,
! [P: list_a > $o,A: list_a] :
( ( ( P @ A )
=> ( ( collect_list_a
@ ^ [X: list_a] :
( ( X = A )
& ( P @ X ) ) )
= ( insert_list_a @ A @ bot_bot_set_list_a ) ) )
& ( ~ ( P @ A )
=> ( ( collect_list_a
@ ^ [X: list_a] :
( ( X = A )
& ( P @ X ) ) )
= bot_bot_set_list_a ) ) ) ).
% Collect_conv_if
thf(fact_899_Collect__conv__if,axiom,
! [P: nat > $o,A: nat] :
( ( ( P @ A )
=> ( ( collect_nat
@ ^ [X: nat] :
( ( X = A )
& ( P @ X ) ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collect_nat
@ ^ [X: nat] :
( ( X = A )
& ( P @ X ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if
thf(fact_900_insert__def,axiom,
( insert8211810215607154385at_nat
= ( ^ [A3: product_prod_nat_nat] :
( sup_su6327502436637775413at_nat
@ ( collec3392354462482085612at_nat
@ ^ [X: product_prod_nat_nat] : ( X = A3 ) ) ) ) ) ).
% insert_def
thf(fact_901_insert__def,axiom,
( insert_nat
= ( ^ [A3: nat] :
( sup_sup_set_nat
@ ( collect_nat
@ ^ [X: nat] : ( X = A3 ) ) ) ) ) ).
% insert_def
thf(fact_902_insert__def,axiom,
( insert3260075854425521959at_nat
= ( ^ [A3: produc5825016348098550007at_nat] :
( sup_su6099978769595272409at_nat
@ ( collec7697611494764367948at_nat
@ ^ [X: produc5825016348098550007at_nat] : ( X = A3 ) ) ) ) ) ).
% insert_def
thf(fact_903_insert__def,axiom,
( insert7010464514620295119la_a_b
= ( ^ [A3: relational_fmla_a_b] :
( sup_su5130108678486352897la_a_b
@ ( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] : ( X = A3 ) ) ) ) ) ).
% insert_def
thf(fact_904_Diff__insert__absorb,axiom,
! [X2: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
( ~ ( member8440522571783428010at_nat @ X2 @ A2 )
=> ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X2 @ A2 ) @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_905_Diff__insert__absorb,axiom,
! [X2: list_a,A2: set_list_a] :
( ~ ( member_list_a @ X2 @ A2 )
=> ( ( minus_646659088055828811list_a @ ( insert_list_a @ X2 @ A2 ) @ ( insert_list_a @ X2 @ bot_bot_set_list_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_906_Diff__insert__absorb,axiom,
! [X2: nat,A2: set_nat] :
( ~ ( member_nat @ X2 @ A2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A2 ) @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_907_Diff__insert__absorb,axiom,
! [X2: produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat] :
( ~ ( member6956540099943067662at_nat @ X2 @ A2 )
=> ( ( minus_6698950876951835142at_nat @ ( insert3260075854425521959at_nat @ X2 @ A2 ) @ ( insert3260075854425521959at_nat @ X2 @ bot_bo1973934853101755969at_nat ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_908_Diff__insert__absorb,axiom,
! [X2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ~ ( member4680049679412964150la_a_b @ X2 @ A2 )
=> ( ( minus_4077726661957047470la_a_b @ ( insert7010464514620295119la_a_b @ X2 @ A2 ) @ ( insert7010464514620295119la_a_b @ X2 @ bot_bo4495933725496725865la_a_b ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_909_Diff__insert2,axiom,
! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat] :
( ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ B ) )
= ( minus_1356011639430497352at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) @ B ) ) ).
% Diff_insert2
thf(fact_910_Diff__insert2,axiom,
! [A2: set_list_a,A: list_a,B: set_list_a] :
( ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A @ B ) )
= ( minus_646659088055828811list_a @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) @ B ) ) ).
% Diff_insert2
thf(fact_911_Diff__insert2,axiom,
! [A2: set_nat,A: nat,B: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ B ) ) ).
% Diff_insert2
thf(fact_912_Diff__insert2,axiom,
! [A2: set_Pr2645174627780777389at_nat,A: produc5825016348098550007at_nat,B: set_Pr2645174627780777389at_nat] :
( ( minus_6698950876951835142at_nat @ A2 @ ( insert3260075854425521959at_nat @ A @ B ) )
= ( minus_6698950876951835142at_nat @ ( minus_6698950876951835142at_nat @ A2 @ ( insert3260075854425521959at_nat @ A @ bot_bo1973934853101755969at_nat ) ) @ B ) ) ).
% Diff_insert2
thf(fact_913_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_914_insert__Diff,axiom,
! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ A @ A2 )
=> ( ( insert8211810215607154385at_nat @ A @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_915_insert__Diff,axiom,
! [A: list_a,A2: set_list_a] :
( ( member_list_a @ A @ A2 )
=> ( ( insert_list_a @ A @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_916_insert__Diff,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_917_insert__Diff,axiom,
! [A: produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ A @ A2 )
=> ( ( insert3260075854425521959at_nat @ A @ ( minus_6698950876951835142at_nat @ A2 @ ( insert3260075854425521959at_nat @ A @ bot_bo1973934853101755969at_nat ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_918_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_919_Diff__insert,axiom,
! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat] :
( ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ B ) )
= ( minus_1356011639430497352at_nat @ ( minus_1356011639430497352at_nat @ A2 @ B ) @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ) ).
% Diff_insert
thf(fact_920_Diff__insert,axiom,
! [A2: set_list_a,A: list_a,B: set_list_a] :
( ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A @ B ) )
= ( minus_646659088055828811list_a @ ( minus_646659088055828811list_a @ A2 @ B ) @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) ) ).
% Diff_insert
thf(fact_921_Diff__insert,axiom,
! [A2: set_nat,A: nat,B: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B ) @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).
% Diff_insert
thf(fact_922_Diff__insert,axiom,
! [A2: set_Pr2645174627780777389at_nat,A: produc5825016348098550007at_nat,B: set_Pr2645174627780777389at_nat] :
( ( minus_6698950876951835142at_nat @ A2 @ ( insert3260075854425521959at_nat @ A @ B ) )
= ( minus_6698950876951835142at_nat @ ( minus_6698950876951835142at_nat @ A2 @ B ) @ ( insert3260075854425521959at_nat @ A @ bot_bo1973934853101755969at_nat ) ) ) ).
% Diff_insert
thf(fact_923_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_924_singleton__Un__iff,axiom,
! [X2: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ( ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat )
= ( sup_su6327502436637775413at_nat @ A2 @ B ) )
= ( ( ( A2 = bot_bo2099793752762293965at_nat )
& ( B
= ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) )
| ( ( A2
= ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) )
& ( B = bot_bo2099793752762293965at_nat ) )
| ( ( A2
= ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) )
& ( B
= ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_925_singleton__Un__iff,axiom,
! [X2: produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( ( insert3260075854425521959at_nat @ X2 @ bot_bo1973934853101755969at_nat )
= ( sup_su6099978769595272409at_nat @ A2 @ B ) )
= ( ( ( A2 = bot_bo1973934853101755969at_nat )
& ( B
= ( insert3260075854425521959at_nat @ X2 @ bot_bo1973934853101755969at_nat ) ) )
| ( ( A2
= ( insert3260075854425521959at_nat @ X2 @ bot_bo1973934853101755969at_nat ) )
& ( B = bot_bo1973934853101755969at_nat ) )
| ( ( A2
= ( insert3260075854425521959at_nat @ X2 @ bot_bo1973934853101755969at_nat ) )
& ( B
= ( insert3260075854425521959at_nat @ X2 @ bot_bo1973934853101755969at_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_926_singleton__Un__iff,axiom,
! [X2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ( insert7010464514620295119la_a_b @ X2 @ bot_bo4495933725496725865la_a_b )
= ( sup_su5130108678486352897la_a_b @ A2 @ B ) )
= ( ( ( A2 = bot_bo4495933725496725865la_a_b )
& ( B
= ( insert7010464514620295119la_a_b @ X2 @ bot_bo4495933725496725865la_a_b ) ) )
| ( ( A2
= ( insert7010464514620295119la_a_b @ X2 @ bot_bo4495933725496725865la_a_b ) )
& ( B = bot_bo4495933725496725865la_a_b ) )
| ( ( A2
= ( insert7010464514620295119la_a_b @ X2 @ bot_bo4495933725496725865la_a_b ) )
& ( B
= ( insert7010464514620295119la_a_b @ X2 @ bot_bo4495933725496725865la_a_b ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_927_singleton__Un__iff,axiom,
! [X2: list_a,A2: set_list_a,B: set_list_a] :
( ( ( insert_list_a @ X2 @ bot_bot_set_list_a )
= ( sup_sup_set_list_a @ A2 @ B ) )
= ( ( ( A2 = bot_bot_set_list_a )
& ( B
= ( insert_list_a @ X2 @ bot_bot_set_list_a ) ) )
| ( ( A2
= ( insert_list_a @ X2 @ bot_bot_set_list_a ) )
& ( B = bot_bot_set_list_a ) )
| ( ( A2
= ( insert_list_a @ X2 @ bot_bot_set_list_a ) )
& ( B
= ( insert_list_a @ X2 @ bot_bot_set_list_a ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_928_singleton__Un__iff,axiom,
! [X2: nat,A2: set_nat,B: set_nat] :
( ( ( insert_nat @ X2 @ bot_bot_set_nat )
= ( sup_sup_set_nat @ A2 @ B ) )
= ( ( ( A2 = bot_bot_set_nat )
& ( B
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
| ( ( A2
= ( insert_nat @ X2 @ bot_bot_set_nat ) )
& ( B = bot_bot_set_nat ) )
| ( ( A2
= ( insert_nat @ X2 @ bot_bot_set_nat ) )
& ( B
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_929_Un__singleton__iff,axiom,
! [A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat] :
( ( ( sup_su6327502436637775413at_nat @ A2 @ B )
= ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) )
= ( ( ( A2 = bot_bo2099793752762293965at_nat )
& ( B
= ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) )
| ( ( A2
= ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) )
& ( B = bot_bo2099793752762293965at_nat ) )
| ( ( A2
= ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) )
& ( B
= ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_930_Un__singleton__iff,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat,X2: produc5825016348098550007at_nat] :
( ( ( sup_su6099978769595272409at_nat @ A2 @ B )
= ( insert3260075854425521959at_nat @ X2 @ bot_bo1973934853101755969at_nat ) )
= ( ( ( A2 = bot_bo1973934853101755969at_nat )
& ( B
= ( insert3260075854425521959at_nat @ X2 @ bot_bo1973934853101755969at_nat ) ) )
| ( ( A2
= ( insert3260075854425521959at_nat @ X2 @ bot_bo1973934853101755969at_nat ) )
& ( B = bot_bo1973934853101755969at_nat ) )
| ( ( A2
= ( insert3260075854425521959at_nat @ X2 @ bot_bo1973934853101755969at_nat ) )
& ( B
= ( insert3260075854425521959at_nat @ X2 @ bot_bo1973934853101755969at_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_931_Un__singleton__iff,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,X2: relational_fmla_a_b] :
( ( ( sup_su5130108678486352897la_a_b @ A2 @ B )
= ( insert7010464514620295119la_a_b @ X2 @ bot_bo4495933725496725865la_a_b ) )
= ( ( ( A2 = bot_bo4495933725496725865la_a_b )
& ( B
= ( insert7010464514620295119la_a_b @ X2 @ bot_bo4495933725496725865la_a_b ) ) )
| ( ( A2
= ( insert7010464514620295119la_a_b @ X2 @ bot_bo4495933725496725865la_a_b ) )
& ( B = bot_bo4495933725496725865la_a_b ) )
| ( ( A2
= ( insert7010464514620295119la_a_b @ X2 @ bot_bo4495933725496725865la_a_b ) )
& ( B
= ( insert7010464514620295119la_a_b @ X2 @ bot_bo4495933725496725865la_a_b ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_932_Un__singleton__iff,axiom,
! [A2: set_list_a,B: set_list_a,X2: list_a] :
( ( ( sup_sup_set_list_a @ A2 @ B )
= ( insert_list_a @ X2 @ bot_bot_set_list_a ) )
= ( ( ( A2 = bot_bot_set_list_a )
& ( B
= ( insert_list_a @ X2 @ bot_bot_set_list_a ) ) )
| ( ( A2
= ( insert_list_a @ X2 @ bot_bot_set_list_a ) )
& ( B = bot_bot_set_list_a ) )
| ( ( A2
= ( insert_list_a @ X2 @ bot_bot_set_list_a ) )
& ( B
= ( insert_list_a @ X2 @ bot_bot_set_list_a ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_933_Un__singleton__iff,axiom,
! [A2: set_nat,B: set_nat,X2: nat] :
( ( ( sup_sup_set_nat @ A2 @ B )
= ( insert_nat @ X2 @ bot_bot_set_nat ) )
= ( ( ( A2 = bot_bot_set_nat )
& ( B
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
| ( ( A2
= ( insert_nat @ X2 @ bot_bot_set_nat ) )
& ( B = bot_bot_set_nat ) )
| ( ( A2
= ( insert_nat @ X2 @ bot_bot_set_nat ) )
& ( B
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_934_insert__is__Un,axiom,
( insert8211810215607154385at_nat
= ( ^ [A3: product_prod_nat_nat] : ( sup_su6327502436637775413at_nat @ ( insert8211810215607154385at_nat @ A3 @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% insert_is_Un
thf(fact_935_insert__is__Un,axiom,
( insert3260075854425521959at_nat
= ( ^ [A3: produc5825016348098550007at_nat] : ( sup_su6099978769595272409at_nat @ ( insert3260075854425521959at_nat @ A3 @ bot_bo1973934853101755969at_nat ) ) ) ) ).
% insert_is_Un
thf(fact_936_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_937_insert__is__Un,axiom,
( insert_list_a
= ( ^ [A3: list_a] : ( sup_sup_set_list_a @ ( insert_list_a @ A3 @ bot_bot_set_list_a ) ) ) ) ).
% insert_is_Un
thf(fact_938_insert__is__Un,axiom,
( insert_nat
= ( ^ [A3: nat] : ( sup_sup_set_nat @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) ) ) ).
% insert_is_Un
thf(fact_939_image__constant__conv,axiom,
! [A2: set_nat,C2: nat] :
( ( ( A2 = bot_bot_set_nat )
=> ( ( image_nat_nat
@ ^ [X: nat] : C2
@ A2 )
= bot_bot_set_nat ) )
& ( ( A2 != bot_bot_set_nat )
=> ( ( image_nat_nat
@ ^ [X: nat] : C2
@ A2 )
= ( insert_nat @ C2 @ bot_bot_set_nat ) ) ) ) ).
% image_constant_conv
thf(fact_940_image__constant__conv,axiom,
! [A2: set_list_a,C2: nat] :
( ( ( A2 = bot_bot_set_list_a )
=> ( ( image_list_a_nat
@ ^ [X: list_a] : C2
@ A2 )
= bot_bot_set_nat ) )
& ( ( A2 != bot_bot_set_list_a )
=> ( ( image_list_a_nat
@ ^ [X: list_a] : C2
@ A2 )
= ( insert_nat @ C2 @ bot_bot_set_nat ) ) ) ) ).
% image_constant_conv
thf(fact_941_image__constant__conv,axiom,
! [A2: set_nat,C2: list_a] :
( ( ( A2 = bot_bot_set_nat )
=> ( ( image_nat_list_a
@ ^ [X: nat] : C2
@ A2 )
= bot_bot_set_list_a ) )
& ( ( A2 != bot_bot_set_nat )
=> ( ( image_nat_list_a
@ ^ [X: nat] : C2
@ A2 )
= ( insert_list_a @ C2 @ bot_bot_set_list_a ) ) ) ) ).
% image_constant_conv
thf(fact_942_image__constant__conv,axiom,
! [A2: set_Re381260168593705685la_a_b,C2: nat] :
( ( ( A2 = bot_bo4495933725496725865la_a_b )
=> ( ( image_341122591648980342_b_nat
@ ^ [X: relational_fmla_a_b] : C2
@ A2 )
= bot_bot_set_nat ) )
& ( ( A2 != bot_bo4495933725496725865la_a_b )
=> ( ( image_341122591648980342_b_nat
@ ^ [X: relational_fmla_a_b] : C2
@ A2 )
= ( insert_nat @ C2 @ bot_bot_set_nat ) ) ) ) ).
% image_constant_conv
thf(fact_943_image__constant__conv,axiom,
! [A2: set_list_a,C2: list_a] :
( ( ( A2 = bot_bot_set_list_a )
=> ( ( image_list_a_list_a
@ ^ [X: list_a] : C2
@ A2 )
= bot_bot_set_list_a ) )
& ( ( A2 != bot_bot_set_list_a )
=> ( ( image_list_a_list_a
@ ^ [X: list_a] : C2
@ A2 )
= ( insert_list_a @ C2 @ bot_bot_set_list_a ) ) ) ) ).
% image_constant_conv
thf(fact_944_image__constant__conv,axiom,
! [A2: set_nat,C2: product_prod_nat_nat] :
( ( ( A2 = bot_bot_set_nat )
=> ( ( image_5846123807819985514at_nat
@ ^ [X: nat] : C2
@ A2 )
= bot_bo2099793752762293965at_nat ) )
& ( ( A2 != bot_bot_set_nat )
=> ( ( image_5846123807819985514at_nat
@ ^ [X: nat] : C2
@ A2 )
= ( insert8211810215607154385at_nat @ C2 @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% image_constant_conv
thf(fact_945_image__constant__conv,axiom,
! [A2: set_nat,C2: relational_fmla_a_b] :
( ( ( A2 = bot_bot_set_nat )
=> ( ( image_4386371547000553590la_a_b
@ ^ [X: nat] : C2
@ A2 )
= bot_bo4495933725496725865la_a_b ) )
& ( ( A2 != bot_bot_set_nat )
=> ( ( image_4386371547000553590la_a_b
@ ^ [X: nat] : C2
@ A2 )
= ( insert7010464514620295119la_a_b @ C2 @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% image_constant_conv
thf(fact_946_image__constant__conv,axiom,
! [A2: set_Re381260168593705685la_a_b,C2: list_a] :
( ( ( A2 = bot_bo4495933725496725865la_a_b )
=> ( ( image_4103308382209252958list_a
@ ^ [X: relational_fmla_a_b] : C2
@ A2 )
= bot_bot_set_list_a ) )
& ( ( A2 != bot_bo4495933725496725865la_a_b )
=> ( ( image_4103308382209252958list_a
@ ^ [X: relational_fmla_a_b] : C2
@ A2 )
= ( insert_list_a @ C2 @ bot_bot_set_list_a ) ) ) ) ).
% image_constant_conv
thf(fact_947_image__constant__conv,axiom,
! [A2: set_list_a,C2: product_prod_nat_nat] :
( ( ( A2 = bot_bot_set_list_a )
=> ( ( image_3288686587095672578at_nat
@ ^ [X: list_a] : C2
@ A2 )
= bot_bo2099793752762293965at_nat ) )
& ( ( A2 != bot_bot_set_list_a )
=> ( ( image_3288686587095672578at_nat
@ ^ [X: list_a] : C2
@ A2 )
= ( insert8211810215607154385at_nat @ C2 @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% image_constant_conv
thf(fact_948_image__constant__conv,axiom,
! [A2: set_list_a,C2: relational_fmla_a_b] :
( ( ( A2 = bot_bot_set_list_a )
=> ( ( image_286667382342926814la_a_b
@ ^ [X: list_a] : C2
@ A2 )
= bot_bo4495933725496725865la_a_b ) )
& ( ( A2 != bot_bot_set_list_a )
=> ( ( image_286667382342926814la_a_b
@ ^ [X: list_a] : C2
@ A2 )
= ( insert7010464514620295119la_a_b @ C2 @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% image_constant_conv
thf(fact_949_image__constant,axiom,
! [X2: nat,A2: set_nat,C2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( image_nat_nat
@ ^ [X: nat] : C2
@ A2 )
= ( insert_nat @ C2 @ bot_bot_set_nat ) ) ) ).
% image_constant
thf(fact_950_image__constant,axiom,
! [X2: nat,A2: set_nat,C2: list_a] :
( ( member_nat @ X2 @ A2 )
=> ( ( image_nat_list_a
@ ^ [X: nat] : C2
@ A2 )
= ( insert_list_a @ C2 @ bot_bot_set_list_a ) ) ) ).
% image_constant
thf(fact_951_image__constant,axiom,
! [X2: nat,A2: set_nat,C2: product_prod_nat_nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( image_5846123807819985514at_nat
@ ^ [X: nat] : C2
@ A2 )
= ( insert8211810215607154385at_nat @ C2 @ bot_bo2099793752762293965at_nat ) ) ) ).
% image_constant
thf(fact_952_image__constant,axiom,
! [X2: nat,A2: set_nat,C2: relational_fmla_a_b] :
( ( member_nat @ X2 @ A2 )
=> ( ( image_4386371547000553590la_a_b
@ ^ [X: nat] : C2
@ A2 )
= ( insert7010464514620295119la_a_b @ C2 @ bot_bo4495933725496725865la_a_b ) ) ) ).
% image_constant
thf(fact_953_image__constant,axiom,
! [X2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,C2: nat] :
( ( member4680049679412964150la_a_b @ X2 @ A2 )
=> ( ( image_341122591648980342_b_nat
@ ^ [X: relational_fmla_a_b] : C2
@ A2 )
= ( insert_nat @ C2 @ bot_bot_set_nat ) ) ) ).
% image_constant
thf(fact_954_image__constant,axiom,
! [X2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,C2: list_a] :
( ( member4680049679412964150la_a_b @ X2 @ A2 )
=> ( ( image_4103308382209252958list_a
@ ^ [X: relational_fmla_a_b] : C2
@ A2 )
= ( insert_list_a @ C2 @ bot_bot_set_list_a ) ) ) ).
% image_constant
thf(fact_955_image__constant,axiom,
! [X2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,C2: set_list_a] :
( ( member4680049679412964150la_a_b @ X2 @ A2 )
=> ( ( image_130269618930851390list_a
@ ^ [X: relational_fmla_a_b] : C2
@ A2 )
= ( insert_set_list_a @ C2 @ bot_bo3186585308812441520list_a ) ) ) ).
% image_constant
thf(fact_956_image__constant,axiom,
! [X2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,C2: product_prod_nat_nat] :
( ( member4680049679412964150la_a_b @ X2 @ A2 )
=> ( ( image_5852719071363227611at_nat
@ ^ [X: relational_fmla_a_b] : C2
@ A2 )
= ( insert8211810215607154385at_nat @ C2 @ bot_bo2099793752762293965at_nat ) ) ) ).
% image_constant
thf(fact_957_image__constant,axiom,
! [X2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,C2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ A2 )
=> ( ( image_6790371041703824709la_a_b
@ ^ [X: relational_fmla_a_b] : C2
@ A2 )
= ( insert7010464514620295119la_a_b @ C2 @ bot_bo4495933725496725865la_a_b ) ) ) ).
% image_constant
thf(fact_958_image__constant,axiom,
! [X2: nat,A2: set_nat,C2: produc5825016348098550007at_nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( image_1371346403869992270at_nat
@ ^ [X: nat] : C2
@ A2 )
= ( insert3260075854425521959at_nat @ C2 @ bot_bo1973934853101755969at_nat ) ) ) ).
% image_constant
thf(fact_959_finite__Diff__insert,axiom,
! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ B ) ) )
= ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A2 @ B ) ) ) ).
% finite_Diff_insert
thf(fact_960_finite__Diff__insert,axiom,
! [A2: set_nat,A: nat,B: set_nat] :
( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B ) ) )
= ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B ) ) ) ).
% finite_Diff_insert
thf(fact_961_finite__Diff__insert,axiom,
! [A2: set_Pr2645174627780777389at_nat,A: produc5825016348098550007at_nat,B: set_Pr2645174627780777389at_nat] :
( ( finite6790245451575510286at_nat @ ( minus_6698950876951835142at_nat @ A2 @ ( insert3260075854425521959at_nat @ A @ B ) ) )
= ( finite6790245451575510286at_nat @ ( minus_6698950876951835142at_nat @ A2 @ B ) ) ) ).
% finite_Diff_insert
thf(fact_962_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_963_diff__add__zero,axiom,
! [A: multiset_set_nat,B2: multiset_set_nat] :
( ( minus_7237264121398869807et_nat @ A @ ( plus_p8712254050562127327et_nat @ A @ B2 ) )
= zero_z3157962936165190495et_nat ) ).
% diff_add_zero
thf(fact_964_diff__add__zero,axiom,
! [A: nat,B2: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_965_le__add__diff__inverse2,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B2 ) @ B2 )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_966_le__add__diff__inverse,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( plus_plus_nat @ B2 @ ( minus_minus_nat @ A @ B2 ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_967_le__add__same__cancel2,axiom,
! [A: multiset_set_nat,B2: multiset_set_nat] :
( ( ord_le4034546139768944438et_nat @ A @ ( plus_p8712254050562127327et_nat @ B2 @ A ) )
= ( ord_le4034546139768944438et_nat @ zero_z3157962936165190495et_nat @ B2 ) ) ).
% le_add_same_cancel2
thf(fact_968_le__add__same__cancel2,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B2 @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% le_add_same_cancel2
thf(fact_969_le__add__same__cancel1,axiom,
! [A: multiset_set_nat,B2: multiset_set_nat] :
( ( ord_le4034546139768944438et_nat @ A @ ( plus_p8712254050562127327et_nat @ A @ B2 ) )
= ( ord_le4034546139768944438et_nat @ zero_z3157962936165190495et_nat @ B2 ) ) ).
% le_add_same_cancel1
thf(fact_970_le__add__same__cancel1,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% le_add_same_cancel1
thf(fact_971_add__le__same__cancel2,axiom,
! [A: multiset_set_nat,B2: multiset_set_nat] :
( ( ord_le4034546139768944438et_nat @ ( plus_p8712254050562127327et_nat @ A @ B2 ) @ B2 )
= ( ord_le4034546139768944438et_nat @ A @ zero_z3157962936165190495et_nat ) ) ).
% add_le_same_cancel2
thf(fact_972_add__le__same__cancel2,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_973_nongens__simp,axiom,
! [Q: relational_fmla_a_b] : ( ord_less_eq_set_nat @ ( relati62690040636126068ns_a_b @ ( simp @ Q ) ) @ ( relati62690040636126068ns_a_b @ Q ) ) ).
% nongens_simp
thf(fact_974_add__left__cancel,axiom,
! [A: multiset_set_nat,B2: multiset_set_nat,C2: multiset_set_nat] :
( ( ( plus_p8712254050562127327et_nat @ A @ B2 )
= ( plus_p8712254050562127327et_nat @ A @ C2 ) )
= ( B2 = C2 ) ) ).
% add_left_cancel
thf(fact_975_add__left__cancel,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ A @ C2 ) )
= ( B2 = C2 ) ) ).
% add_left_cancel
thf(fact_976_add__right__cancel,axiom,
! [B2: multiset_set_nat,A: multiset_set_nat,C2: multiset_set_nat] :
( ( ( plus_p8712254050562127327et_nat @ B2 @ A )
= ( plus_p8712254050562127327et_nat @ C2 @ A ) )
= ( B2 = C2 ) ) ).
% add_right_cancel
thf(fact_977_add__right__cancel,axiom,
! [B2: nat,A: nat,C2: nat] :
( ( ( plus_plus_nat @ B2 @ A )
= ( plus_plus_nat @ C2 @ A ) )
= ( B2 = C2 ) ) ).
% add_right_cancel
thf(fact_978_finite__Collect__conjI,axiom,
! [P: produc5825016348098550007at_nat > $o,Q: produc5825016348098550007at_nat > $o] :
( ( ( finite6790245451575510286at_nat @ ( collec7697611494764367948at_nat @ P ) )
| ( finite6790245451575510286at_nat @ ( collec7697611494764367948at_nat @ Q ) ) )
=> ( finite6790245451575510286at_nat
@ ( collec7697611494764367948at_nat
@ ^ [X: produc5825016348098550007at_nat] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_979_finite__Collect__conjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P ) )
| ( finite_finite_nat @ ( collect_nat @ Q ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_980_finite__Collect__conjI,axiom,
! [P: relational_fmla_a_b > $o,Q: relational_fmla_a_b > $o] :
( ( ( finite5600759454172676150la_a_b @ ( collec3419995626248312948la_a_b @ P ) )
| ( finite5600759454172676150la_a_b @ ( collec3419995626248312948la_a_b @ Q ) ) )
=> ( finite5600759454172676150la_a_b
@ ( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_981_finite__Collect__disjI,axiom,
! [P: produc5825016348098550007at_nat > $o,Q: produc5825016348098550007at_nat > $o] :
( ( finite6790245451575510286at_nat
@ ( collec7697611494764367948at_nat
@ ^ [X: produc5825016348098550007at_nat] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite6790245451575510286at_nat @ ( collec7697611494764367948at_nat @ P ) )
& ( finite6790245451575510286at_nat @ ( collec7697611494764367948at_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_982_finite__Collect__disjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P ) )
& ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_983_finite__Collect__disjI,axiom,
! [P: relational_fmla_a_b > $o,Q: relational_fmla_a_b > $o] :
( ( finite5600759454172676150la_a_b
@ ( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite5600759454172676150la_a_b @ ( collec3419995626248312948la_a_b @ P ) )
& ( finite5600759454172676150la_a_b @ ( collec3419995626248312948la_a_b @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_984_le__zero__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_985_add__le__cancel__left,axiom,
! [C2: multiset_set_nat,A: multiset_set_nat,B2: multiset_set_nat] :
( ( ord_le4034546139768944438et_nat @ ( plus_p8712254050562127327et_nat @ C2 @ A ) @ ( plus_p8712254050562127327et_nat @ C2 @ B2 ) )
= ( ord_le4034546139768944438et_nat @ A @ B2 ) ) ).
% add_le_cancel_left
thf(fact_986_add__le__cancel__left,axiom,
! [C2: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B2 ) )
= ( ord_less_eq_nat @ A @ B2 ) ) ).
% add_le_cancel_left
thf(fact_987_add__le__cancel__right,axiom,
! [A: multiset_set_nat,C2: multiset_set_nat,B2: multiset_set_nat] :
( ( ord_le4034546139768944438et_nat @ ( plus_p8712254050562127327et_nat @ A @ C2 ) @ ( plus_p8712254050562127327et_nat @ B2 @ C2 ) )
= ( ord_le4034546139768944438et_nat @ A @ B2 ) ) ).
% add_le_cancel_right
thf(fact_988_add__le__cancel__right,axiom,
! [A: nat,C2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B2 @ C2 ) )
= ( ord_less_eq_nat @ A @ B2 ) ) ).
% add_le_cancel_right
thf(fact_989_add_Oright__neutral,axiom,
! [A: multiset_set_nat] :
( ( plus_p8712254050562127327et_nat @ A @ zero_z3157962936165190495et_nat )
= A ) ).
% add.right_neutral
thf(fact_990_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_991_add__cancel__left__left,axiom,
! [B2: multiset_set_nat,A: multiset_set_nat] :
( ( ( plus_p8712254050562127327et_nat @ B2 @ A )
= A )
= ( B2 = zero_z3157962936165190495et_nat ) ) ).
% add_cancel_left_left
thf(fact_992_add__cancel__left__left,axiom,
! [B2: nat,A: nat] :
( ( ( plus_plus_nat @ B2 @ A )
= A )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_993_add__cancel__left__right,axiom,
! [A: multiset_set_nat,B2: multiset_set_nat] :
( ( ( plus_p8712254050562127327et_nat @ A @ B2 )
= A )
= ( B2 = zero_z3157962936165190495et_nat ) ) ).
% add_cancel_left_right
thf(fact_994_add__cancel__left__right,axiom,
! [A: nat,B2: nat] :
( ( ( plus_plus_nat @ A @ B2 )
= A )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_995_add__cancel__right__left,axiom,
! [A: multiset_set_nat,B2: multiset_set_nat] :
( ( A
= ( plus_p8712254050562127327et_nat @ B2 @ A ) )
= ( B2 = zero_z3157962936165190495et_nat ) ) ).
% add_cancel_right_left
thf(fact_996_add__cancel__right__left,axiom,
! [A: nat,B2: nat] :
( ( A
= ( plus_plus_nat @ B2 @ A ) )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_997_add__cancel__right__right,axiom,
! [A: multiset_set_nat,B2: multiset_set_nat] :
( ( A
= ( plus_p8712254050562127327et_nat @ A @ B2 ) )
= ( B2 = zero_z3157962936165190495et_nat ) ) ).
% add_cancel_right_right
thf(fact_998_add__cancel__right__right,axiom,
! [A: nat,B2: nat] :
( ( A
= ( plus_plus_nat @ A @ B2 ) )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_999_add__eq__0__iff__both__eq__0,axiom,
! [X2: nat,Y3: nat] :
( ( ( plus_plus_nat @ X2 @ Y3 )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_1000_zero__eq__add__iff__both__eq__0,axiom,
! [X2: nat,Y3: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X2 @ Y3 ) )
= ( ( X2 = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_1001_add__0,axiom,
! [A: multiset_set_nat] :
( ( plus_p8712254050562127327et_nat @ zero_z3157962936165190495et_nat @ A )
= A ) ).
% add_0
thf(fact_1002_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_1003_zero__diff,axiom,
! [A: multiset_set_nat] :
( ( minus_7237264121398869807et_nat @ zero_z3157962936165190495et_nat @ A )
= zero_z3157962936165190495et_nat ) ).
% zero_diff
thf(fact_1004_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_1005_diff__zero,axiom,
! [A: multiset_set_nat] :
( ( minus_7237264121398869807et_nat @ A @ zero_z3157962936165190495et_nat )
= A ) ).
% diff_zero
thf(fact_1006_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_1007_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: multiset_set_nat] :
( ( minus_7237264121398869807et_nat @ A @ A )
= zero_z3157962936165190495et_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1008_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1009_add__diff__cancel__left,axiom,
! [C2: multiset_set_nat,A: multiset_set_nat,B2: multiset_set_nat] :
( ( minus_7237264121398869807et_nat @ ( plus_p8712254050562127327et_nat @ C2 @ A ) @ ( plus_p8712254050562127327et_nat @ C2 @ B2 ) )
= ( minus_7237264121398869807et_nat @ A @ B2 ) ) ).
% add_diff_cancel_left
thf(fact_1010_add__diff__cancel__left,axiom,
! [C2: nat,A: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B2 ) )
= ( minus_minus_nat @ A @ B2 ) ) ).
% add_diff_cancel_left
thf(fact_1011_add__diff__cancel__left_H,axiom,
! [A: multiset_set_nat,B2: multiset_set_nat] :
( ( minus_7237264121398869807et_nat @ ( plus_p8712254050562127327et_nat @ A @ B2 ) @ A )
= B2 ) ).
% add_diff_cancel_left'
thf(fact_1012_add__diff__cancel__left_H,axiom,
! [A: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B2 ) @ A )
= B2 ) ).
% add_diff_cancel_left'
thf(fact_1013_add__diff__cancel__right,axiom,
! [A: multiset_set_nat,C2: multiset_set_nat,B2: multiset_set_nat] :
( ( minus_7237264121398869807et_nat @ ( plus_p8712254050562127327et_nat @ A @ C2 ) @ ( plus_p8712254050562127327et_nat @ B2 @ C2 ) )
= ( minus_7237264121398869807et_nat @ A @ B2 ) ) ).
% add_diff_cancel_right
thf(fact_1014_add__diff__cancel__right,axiom,
! [A: nat,C2: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B2 @ C2 ) )
= ( minus_minus_nat @ A @ B2 ) ) ).
% add_diff_cancel_right
thf(fact_1015_add__diff__cancel__right_H,axiom,
! [A: multiset_set_nat,B2: multiset_set_nat] :
( ( minus_7237264121398869807et_nat @ ( plus_p8712254050562127327et_nat @ A @ B2 ) @ B2 )
= A ) ).
% add_diff_cancel_right'
thf(fact_1016_add__diff__cancel__right_H,axiom,
! [A: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
= A ) ).
% add_diff_cancel_right'
thf(fact_1017_img__fst,axiom,
! [A: nat,B2: nat,S: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B2 ) @ S )
=> ( member_nat @ A @ ( image_2486076414777270412at_nat @ product_fst_nat_nat @ S ) ) ) ).
% img_fst
thf(fact_1018_img__fst,axiom,
! [A: relational_fmla_a_b,B2: set_Pr1261947904930325089at_nat,S: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ A @ B2 ) @ S )
=> ( member4680049679412964150la_a_b @ A @ ( image_7406987224865000349la_a_b @ produc8992968103413169469at_nat @ S ) ) ) ).
% img_fst
thf(fact_1019_finite__imageI,axiom,
! [F2: set_Pr2645174627780777389at_nat,H: produc5825016348098550007at_nat > produc5825016348098550007at_nat] :
( ( finite6790245451575510286at_nat @ F2 )
=> ( finite6790245451575510286at_nat @ ( image_6979785008349797877at_nat @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_1020_finite__imageI,axiom,
! [F2: set_Pr2645174627780777389at_nat,H: produc5825016348098550007at_nat > nat] :
( ( finite6790245451575510286at_nat @ F2 )
=> ( finite_finite_nat @ ( image_8316665354072716238at_nat @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_1021_finite__imageI,axiom,
! [F2: set_Pr2645174627780777389at_nat,H: produc5825016348098550007at_nat > relational_fmla_a_b] :
( ( finite6790245451575510286at_nat @ F2 )
=> ( finite5600759454172676150la_a_b @ ( image_7406987224865000349la_a_b @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_1022_finite__imageI,axiom,
! [F2: set_nat,H: nat > produc5825016348098550007at_nat] :
( ( finite_finite_nat @ F2 )
=> ( finite6790245451575510286at_nat @ ( image_1371346403869992270at_nat @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_1023_finite__imageI,axiom,
! [F2: set_nat,H: nat > nat] :
( ( finite_finite_nat @ F2 )
=> ( finite_finite_nat @ ( image_nat_nat @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_1024_finite__imageI,axiom,
! [F2: set_nat,H: nat > relational_fmla_a_b] :
( ( finite_finite_nat @ F2 )
=> ( finite5600759454172676150la_a_b @ ( image_4386371547000553590la_a_b @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_1025_finite__imageI,axiom,
! [F2: set_Re381260168593705685la_a_b,H: relational_fmla_a_b > set_list_a] :
( ( finite5600759454172676150la_a_b @ F2 )
=> ( finite5282473924520328461list_a @ ( image_130269618930851390list_a @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_1026_finite__imageI,axiom,
! [F2: set_Re381260168593705685la_a_b,H: relational_fmla_a_b > produc5825016348098550007at_nat] :
( ( finite5600759454172676150la_a_b @ F2 )
=> ( finite6790245451575510286at_nat @ ( image_5430539854921360285at_nat @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_1027_finite__imageI,axiom,
! [F2: set_Re381260168593705685la_a_b,H: relational_fmla_a_b > nat] :
( ( finite5600759454172676150la_a_b @ F2 )
=> ( finite_finite_nat @ ( image_341122591648980342_b_nat @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_1028_finite__imageI,axiom,
! [F2: set_Re381260168593705685la_a_b,H: relational_fmla_a_b > relational_fmla_a_b] :
( ( finite5600759454172676150la_a_b @ F2 )
=> ( finite5600759454172676150la_a_b @ ( image_6790371041703824709la_a_b @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_1029_finite__insert,axiom,
! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ ( insert8211810215607154385at_nat @ A @ A2 ) )
= ( finite6177210948735845034at_nat @ A2 ) ) ).
% finite_insert
thf(fact_1030_finite__insert,axiom,
! [A: produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat] :
( ( finite6790245451575510286at_nat @ ( insert3260075854425521959at_nat @ A @ A2 ) )
= ( finite6790245451575510286at_nat @ A2 ) ) ).
% finite_insert
thf(fact_1031_finite__insert,axiom,
! [A: nat,A2: set_nat] :
( ( finite_finite_nat @ ( insert_nat @ A @ A2 ) )
= ( finite_finite_nat @ A2 ) ) ).
% finite_insert
thf(fact_1032_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_1033_finite__Diff,axiom,
! [A2: set_nat,B: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B ) ) ) ).
% finite_Diff
thf(fact_1034_finite__Diff,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( finite6790245451575510286at_nat @ A2 )
=> ( finite6790245451575510286at_nat @ ( minus_6698950876951835142at_nat @ A2 @ B ) ) ) ).
% finite_Diff
thf(fact_1035_finite__Diff,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ A2 )
=> ( finite5600759454172676150la_a_b @ ( minus_4077726661957047470la_a_b @ A2 @ B ) ) ) ).
% finite_Diff
thf(fact_1036_finite__Diff2,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B ) )
= ( finite_finite_nat @ A2 ) ) ) ).
% finite_Diff2
thf(fact_1037_finite__Diff2,axiom,
! [B: set_Pr2645174627780777389at_nat,A2: set_Pr2645174627780777389at_nat] :
( ( finite6790245451575510286at_nat @ B )
=> ( ( finite6790245451575510286at_nat @ ( minus_6698950876951835142at_nat @ A2 @ B ) )
= ( finite6790245451575510286at_nat @ A2 ) ) ) ).
% finite_Diff2
thf(fact_1038_finite__Diff2,axiom,
! [B: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ B )
=> ( ( finite5600759454172676150la_a_b @ ( minus_4077726661957047470la_a_b @ A2 @ B ) )
= ( finite5600759454172676150la_a_b @ A2 ) ) ) ).
% finite_Diff2
thf(fact_1039_finite__Int,axiom,
! [F2: set_nat,G2: set_nat] :
( ( ( finite_finite_nat @ F2 )
| ( finite_finite_nat @ G2 ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ F2 @ G2 ) ) ) ).
% finite_Int
thf(fact_1040_finite__Int,axiom,
! [F2: set_Re381260168593705685la_a_b,G2: set_Re381260168593705685la_a_b] :
( ( ( finite5600759454172676150la_a_b @ F2 )
| ( finite5600759454172676150la_a_b @ G2 ) )
=> ( finite5600759454172676150la_a_b @ ( inf_in8483230781156617063la_a_b @ F2 @ G2 ) ) ) ).
% finite_Int
thf(fact_1041_finite__Int,axiom,
! [F2: set_Pr2645174627780777389at_nat,G2: set_Pr2645174627780777389at_nat] :
( ( ( finite6790245451575510286at_nat @ F2 )
| ( finite6790245451575510286at_nat @ G2 ) )
=> ( finite6790245451575510286at_nat @ ( inf_in8776938414804536127at_nat @ F2 @ G2 ) ) ) ).
% finite_Int
thf(fact_1042_finite__Un,axiom,
! [F2: set_nat,G2: set_nat] :
( ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G2 ) )
= ( ( finite_finite_nat @ F2 )
& ( finite_finite_nat @ G2 ) ) ) ).
% finite_Un
thf(fact_1043_finite__Un,axiom,
! [F2: set_Pr2645174627780777389at_nat,G2: set_Pr2645174627780777389at_nat] :
( ( finite6790245451575510286at_nat @ ( sup_su6099978769595272409at_nat @ F2 @ G2 ) )
= ( ( finite6790245451575510286at_nat @ F2 )
& ( finite6790245451575510286at_nat @ G2 ) ) ) ).
% finite_Un
thf(fact_1044_finite__Un,axiom,
! [F2: set_Re381260168593705685la_a_b,G2: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ ( sup_su5130108678486352897la_a_b @ F2 @ G2 ) )
= ( ( finite5600759454172676150la_a_b @ F2 )
& ( finite5600759454172676150la_a_b @ G2 ) ) ) ).
% finite_Un
thf(fact_1045_finite__Collect__subsets,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ A2 )
=> ( finite5238674622262875500la_a_b
@ ( collec2099942116761351594la_a_b
@ ^ [B4: set_Re381260168593705685la_a_b] : ( ord_le4112832032246704949la_a_b @ B4 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_1046_finite__Collect__subsets,axiom,
! [A2: set_Pr2645174627780777389at_nat] :
( ( finite6790245451575510286at_nat @ A2 )
=> ( finite2560455063271695812at_nat
@ ( collec5061078246732527874at_nat
@ ^ [B4: set_Pr2645174627780777389at_nat] : ( ord_le8520675249591772685at_nat @ B4 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_1047_finite__Collect__subsets,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B4: set_nat] : ( ord_less_eq_set_nat @ B4 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_1048_add__le__same__cancel1,axiom,
! [B2: multiset_set_nat,A: multiset_set_nat] :
( ( ord_le4034546139768944438et_nat @ ( plus_p8712254050562127327et_nat @ B2 @ A ) @ B2 )
= ( ord_le4034546139768944438et_nat @ A @ zero_z3157962936165190495et_nat ) ) ).
% add_le_same_cancel1
thf(fact_1049_add__le__same__cancel1,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B2 @ A ) @ B2 )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_1050_subset__mset_Ofinite__has__minimal2,axiom,
! [A2: set_multiset_set_nat,A: multiset_set_nat] :
( ( finite5272093019064105709et_nat @ A2 )
=> ( ( member9214616765234488813et_nat @ A @ A2 )
=> ? [X3: multiset_set_nat] :
( ( member9214616765234488813et_nat @ X3 @ A2 )
& ( subset6078030600694693471et_nat @ X3 @ A )
& ! [Xa: multiset_set_nat] :
( ( member9214616765234488813et_nat @ Xa @ A2 )
=> ( ( subset6078030600694693471et_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% subset_mset.finite_has_minimal2
thf(fact_1051_subset__mset_Ofinite__has__maximal2,axiom,
! [A2: set_multiset_set_nat,A: multiset_set_nat] :
( ( finite5272093019064105709et_nat @ A2 )
=> ( ( member9214616765234488813et_nat @ A @ A2 )
=> ? [X3: multiset_set_nat] :
( ( member9214616765234488813et_nat @ X3 @ A2 )
& ( subset6078030600694693471et_nat @ A @ X3 )
& ! [Xa: multiset_set_nat] :
( ( member9214616765234488813et_nat @ Xa @ A2 )
=> ( ( subset6078030600694693471et_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% subset_mset.finite_has_maximal2
thf(fact_1052_subset__mset_Ofinite__has__minimal,axiom,
! [A2: set_multiset_set_nat] :
( ( finite5272093019064105709et_nat @ A2 )
=> ( ( A2 != bot_bo7575690689620579488et_nat )
=> ? [X3: multiset_set_nat] :
( ( member9214616765234488813et_nat @ X3 @ A2 )
& ! [Xa: multiset_set_nat] :
( ( member9214616765234488813et_nat @ Xa @ A2 )
=> ( ( subset6078030600694693471et_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% subset_mset.finite_has_minimal
thf(fact_1053_subset__mset_Ofinite__has__maximal,axiom,
! [A2: set_multiset_set_nat] :
( ( finite5272093019064105709et_nat @ A2 )
=> ( ( A2 != bot_bo7575690689620579488et_nat )
=> ? [X3: multiset_set_nat] :
( ( member9214616765234488813et_nat @ X3 @ A2 )
& ! [Xa: multiset_set_nat] :
( ( member9214616765234488813et_nat @ Xa @ A2 )
=> ( ( subset6078030600694693471et_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% subset_mset.finite_has_maximal
thf(fact_1054_in__fst__imageE,axiom,
! [X2: nat,S: set_Pr1261947904930325089at_nat] :
( ( member_nat @ X2 @ ( image_2486076414777270412at_nat @ product_fst_nat_nat @ S ) )
=> ~ ! [Y4: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y4 ) @ S ) ) ).
% in_fst_imageE
thf(fact_1055_in__fst__imageE,axiom,
! [X2: relational_fmla_a_b,S: set_Pr2645174627780777389at_nat] :
( ( member4680049679412964150la_a_b @ X2 @ ( image_7406987224865000349la_a_b @ produc8992968103413169469at_nat @ S ) )
=> ~ ! [Y4: set_Pr1261947904930325089at_nat] :
~ ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ X2 @ Y4 ) @ S ) ) ).
% in_fst_imageE
thf(fact_1056_minus__set__def,axiom,
( minus_minus_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( collect_nat
@ ( minus_minus_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A4 )
@ ^ [X: nat] : ( member_nat @ X @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_1057_minus__set__def,axiom,
( minus_6698950876951835142at_nat
= ( ^ [A4: set_Pr2645174627780777389at_nat,B4: set_Pr2645174627780777389at_nat] :
( collec7697611494764367948at_nat
@ ( minus_8035352901537135103_nat_o
@ ^ [X: produc5825016348098550007at_nat] : ( member6956540099943067662at_nat @ X @ A4 )
@ ^ [X: produc5825016348098550007at_nat] : ( member6956540099943067662at_nat @ X @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_1058_minus__set__def,axiom,
( minus_4077726661957047470la_a_b
= ( ^ [A4: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b] :
( collec3419995626248312948la_a_b
@ ( minus_9215201808853403479_a_b_o
@ ^ [X: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X @ A4 )
@ ^ [X: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_1059_less__eq__set__def,axiom,
( ord_le4112832032246704949la_a_b
= ( ^ [A4: set_Re381260168593705685la_a_b,B4: set_Re381260168593705685la_a_b] :
( ord_le7191224889845164944_a_b_o
@ ^ [X: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X @ A4 )
@ ^ [X: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_1060_less__eq__set__def,axiom,
( ord_le8520675249591772685at_nat
= ( ^ [A4: set_Pr2645174627780777389at_nat,B4: set_Pr2645174627780777389at_nat] :
( ord_le4411885398519701688_nat_o
@ ^ [X: produc5825016348098550007at_nat] : ( member6956540099943067662at_nat @ X @ A4 )
@ ^ [X: produc5825016348098550007at_nat] : ( member6956540099943067662at_nat @ X @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_1061_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ord_less_eq_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A4 )
@ ^ [X: nat] : ( member_nat @ X @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_1062_fst__image__mp,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Re381260168593705685la_a_b,X2: relational_fmla_a_b,Y3: set_Pr1261947904930325089at_nat] :
( ( ord_le4112832032246704949la_a_b @ ( image_7406987224865000349la_a_b @ produc8992968103413169469at_nat @ A2 ) @ B )
=> ( ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ X2 @ Y3 ) @ A2 )
=> ( member4680049679412964150la_a_b @ X2 @ B ) ) ) ).
% fst_image_mp
thf(fact_1063_fst__image__mp,axiom,
! [A2: set_Pr1261947904930325089at_nat,B: set_nat,X2: nat,Y3: nat] :
( ( ord_less_eq_set_nat @ ( image_2486076414777270412at_nat @ product_fst_nat_nat @ A2 ) @ B )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ A2 )
=> ( member_nat @ X2 @ B ) ) ) ).
% fst_image_mp
thf(fact_1064_in__image__insert__iff,axiom,
! [B: set_se7855581050983116737at_nat,X2: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
( ! [C6: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ C6 @ B )
=> ~ ( member8440522571783428010at_nat @ X2 @ C6 ) )
=> ( ( member2643936169264416010at_nat @ A2 @ ( image_3684629450409544005at_nat @ ( insert8211810215607154385at_nat @ X2 ) @ B ) )
= ( ( member8440522571783428010at_nat @ X2 @ A2 )
& ( member2643936169264416010at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_1065_in__image__insert__iff,axiom,
! [B: set_set_list_a,X2: list_a,A2: set_list_a] :
( ! [C6: set_list_a] :
( ( member_set_list_a @ C6 @ B )
=> ~ ( member_list_a @ X2 @ C6 ) )
=> ( ( member_set_list_a @ A2 @ ( image_5749939591322298757list_a @ ( insert_list_a @ X2 ) @ B ) )
= ( ( member_list_a @ X2 @ A2 )
& ( member_set_list_a @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ X2 @ bot_bot_set_list_a ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_1066_in__image__insert__iff,axiom,
! [B: set_set_nat,X2: nat,A2: set_nat] :
( ! [C6: set_nat] :
( ( member_set_nat @ C6 @ B )
=> ~ ( member_nat @ X2 @ C6 ) )
=> ( ( member_set_nat @ A2 @ ( image_7916887816326733075et_nat @ ( insert_nat @ X2 ) @ B ) )
= ( ( member_nat @ X2 @ A2 )
& ( member_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_1067_in__image__insert__iff,axiom,
! [B: set_se7774124317125585763at_nat,X2: produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat] :
( ! [C6: set_Pr2645174627780777389at_nat] :
( ( member734533627512885444at_nat @ C6 @ B )
=> ~ ( member6956540099943067662at_nat @ X2 @ C6 ) )
=> ( ( member734533627512885444at_nat @ A2 @ ( image_3124536445484363873at_nat @ ( insert3260075854425521959at_nat @ X2 ) @ B ) )
= ( ( member6956540099943067662at_nat @ X2 @ A2 )
& ( member734533627512885444at_nat @ ( minus_6698950876951835142at_nat @ A2 @ ( insert3260075854425521959at_nat @ X2 @ bot_bo1973934853101755969at_nat ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_1068_in__image__insert__iff,axiom,
! [B: set_se6865892389300016395la_a_b,X2: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
( ! [C6: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ C6 @ B )
=> ~ ( member4680049679412964150la_a_b @ X2 @ C6 ) )
=> ( ( member3481406638322139244la_a_b @ A2 @ ( image_7051608999182166449la_a_b @ ( insert7010464514620295119la_a_b @ X2 ) @ B ) )
= ( ( member4680049679412964150la_a_b @ X2 @ A2 )
& ( member3481406638322139244la_a_b @ ( minus_4077726661957047470la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ X2 @ bot_bo4495933725496725865la_a_b ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_1069_bex2I,axiom,
! [A: relational_fmla_a_b,B2: set_Pr1261947904930325089at_nat,S: set_Pr2645174627780777389at_nat,P: relational_fmla_a_b > set_Pr1261947904930325089at_nat > $o] :
( ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ A @ B2 ) @ S )
=> ( ( ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ A @ B2 ) @ S )
=> ( P @ A @ B2 ) )
=> ? [A6: relational_fmla_a_b,B7: set_Pr1261947904930325089at_nat] :
( ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ A6 @ B7 ) @ S )
& ( P @ A6 @ B7 ) ) ) ) ).
% bex2I
thf(fact_1070_bex2I,axiom,
! [A: nat,B2: nat,S: set_Pr1261947904930325089at_nat,P: nat > nat > $o] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B2 ) @ S )
=> ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B2 ) @ S )
=> ( P @ A @ B2 ) )
=> ? [A6: nat,B7: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B7 ) @ S )
& ( P @ A6 @ B7 ) ) ) ) ).
% bex2I
thf(fact_1071_ord__eq__le__eq__trans,axiom,
! [A: set_Pr2645174627780777389at_nat,B2: set_Pr2645174627780777389at_nat,C2: set_Pr2645174627780777389at_nat,D2: set_Pr2645174627780777389at_nat] :
( ( A = B2 )
=> ( ( ord_le8520675249591772685at_nat @ B2 @ C2 )
=> ( ( C2 = D2 )
=> ( ord_le8520675249591772685at_nat @ A @ D2 ) ) ) ) ).
% ord_eq_le_eq_trans
thf(fact_1072_ord__eq__le__eq__trans,axiom,
! [A: set_nat,B2: set_nat,C2: set_nat,D2: set_nat] :
( ( A = B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ( C2 = D2 )
=> ( ord_less_eq_set_nat @ A @ D2 ) ) ) ) ).
% ord_eq_le_eq_trans
thf(fact_1073_ord__eq__le__eq__trans,axiom,
! [A: nat,B2: nat,C2: nat,D2: nat] :
( ( A = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ( C2 = D2 )
=> ( ord_less_eq_nat @ A @ D2 ) ) ) ) ).
% ord_eq_le_eq_trans
thf(fact_1074_zero__reorient,axiom,
! [X2: multiset_set_nat] :
( ( zero_z3157962936165190495et_nat = X2 )
= ( X2 = zero_z3157962936165190495et_nat ) ) ).
% zero_reorient
thf(fact_1075_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_1076_comp__cong__left,axiom,
! [X2: relational_fmla_a_b > set_nat,Y3: relational_fmla_a_b > set_nat,F: produc5825016348098550007at_nat > relational_fmla_a_b] :
( ( X2 = Y3 )
=> ( ( comp_R1256417402204451841at_nat @ X2 @ F )
= ( comp_R1256417402204451841at_nat @ Y3 @ F ) ) ) ).
% comp_cong_left
thf(fact_1077_comp__cong__right,axiom,
! [X2: produc5825016348098550007at_nat > relational_fmla_a_b,Y3: produc5825016348098550007at_nat > relational_fmla_a_b,F: relational_fmla_a_b > set_nat] :
( ( X2 = Y3 )
=> ( ( comp_R1256417402204451841at_nat @ F @ X2 )
= ( comp_R1256417402204451841at_nat @ F @ Y3 ) ) ) ).
% comp_cong_right
thf(fact_1078_fun__comp__eq__conv,axiom,
! [F: relational_fmla_a_b > set_nat,G: produc5825016348098550007at_nat > relational_fmla_a_b,Fg: produc5825016348098550007at_nat > set_nat] :
( ( ( comp_R1256417402204451841at_nat @ F @ G )
= Fg )
= ( ! [X: produc5825016348098550007at_nat] :
( ( F @ ( G @ X ) )
= ( Fg @ X ) ) ) ) ).
% fun_comp_eq_conv
thf(fact_1079_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: multiset_set_nat,B2: multiset_set_nat,C2: multiset_set_nat] :
( ( plus_p8712254050562127327et_nat @ ( plus_p8712254050562127327et_nat @ A @ B2 ) @ C2 )
= ( plus_p8712254050562127327et_nat @ A @ ( plus_p8712254050562127327et_nat @ B2 @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_1080_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B2 ) @ C2 )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_1081_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: multiset_set_nat,J: multiset_set_nat,K2: multiset_set_nat,L: multiset_set_nat] :
( ( ( I = J )
& ( K2 = L ) )
=> ( ( plus_p8712254050562127327et_nat @ I @ K2 )
= ( plus_p8712254050562127327et_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_1082_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( I = J )
& ( K2 = L ) )
=> ( ( plus_plus_nat @ I @ K2 )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_1083_group__cancel_Oadd1,axiom,
! [A2: multiset_set_nat,K2: multiset_set_nat,A: multiset_set_nat,B2: multiset_set_nat] :
( ( A2
= ( plus_p8712254050562127327et_nat @ K2 @ A ) )
=> ( ( plus_p8712254050562127327et_nat @ A2 @ B2 )
= ( plus_p8712254050562127327et_nat @ K2 @ ( plus_p8712254050562127327et_nat @ A @ B2 ) ) ) ) ).
% group_cancel.add1
thf(fact_1084_group__cancel_Oadd1,axiom,
! [A2: nat,K2: nat,A: nat,B2: nat] :
( ( A2
= ( plus_plus_nat @ K2 @ A ) )
=> ( ( plus_plus_nat @ A2 @ B2 )
= ( plus_plus_nat @ K2 @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% group_cancel.add1
thf(fact_1085_group__cancel_Oadd2,axiom,
! [B: multiset_set_nat,K2: multiset_set_nat,B2: multiset_set_nat,A: multiset_set_nat] :
( ( B
= ( plus_p8712254050562127327et_nat @ K2 @ B2 ) )
=> ( ( plus_p8712254050562127327et_nat @ A @ B )
= ( plus_p8712254050562127327et_nat @ K2 @ ( plus_p8712254050562127327et_nat @ A @ B2 ) ) ) ) ).
% group_cancel.add2
thf(fact_1086_group__cancel_Oadd2,axiom,
! [B: nat,K2: nat,B2: nat,A: nat] :
( ( B
= ( plus_plus_nat @ K2 @ B2 ) )
=> ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ K2 @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% group_cancel.add2
thf(fact_1087_add_Oassoc,axiom,
! [A: multiset_set_nat,B2: multiset_set_nat,C2: multiset_set_nat] :
( ( plus_p8712254050562127327et_nat @ ( plus_p8712254050562127327et_nat @ A @ B2 ) @ C2 )
= ( plus_p8712254050562127327et_nat @ A @ ( plus_p8712254050562127327et_nat @ B2 @ C2 ) ) ) ).
% add.assoc
thf(fact_1088_add_Oassoc,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B2 ) @ C2 )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C2 ) ) ) ).
% add.assoc
thf(fact_1089_add_Ocommute,axiom,
( plus_p8712254050562127327et_nat
= ( ^ [A3: multiset_set_nat,B3: multiset_set_nat] : ( plus_p8712254050562127327et_nat @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_1090_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B3: nat] : ( plus_plus_nat @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_1091_add_Oleft__commute,axiom,
! [B2: multiset_set_nat,A: multiset_set_nat,C2: multiset_set_nat] :
( ( plus_p8712254050562127327et_nat @ B2 @ ( plus_p8712254050562127327et_nat @ A @ C2 ) )
= ( plus_p8712254050562127327et_nat @ A @ ( plus_p8712254050562127327et_nat @ B2 @ C2 ) ) ) ).
% add.left_commute
thf(fact_1092_add_Oleft__commute,axiom,
! [B2: nat,A: nat,C2: nat] :
( ( plus_plus_nat @ B2 @ ( plus_plus_nat @ A @ C2 ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C2 ) ) ) ).
% add.left_commute
thf(fact_1093_add__left__imp__eq,axiom,
! [A: multiset_set_nat,B2: multiset_set_nat,C2: multiset_set_nat] :
( ( ( plus_p8712254050562127327et_nat @ A @ B2 )
= ( plus_p8712254050562127327et_nat @ A @ C2 ) )
=> ( B2 = C2 ) ) ).
% add_left_imp_eq
thf(fact_1094_add__left__imp__eq,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ A @ C2 ) )
=> ( B2 = C2 ) ) ).
% add_left_imp_eq
thf(fact_1095_add__right__imp__eq,axiom,
! [B2: multiset_set_nat,A: multiset_set_nat,C2: multiset_set_nat] :
( ( ( plus_p8712254050562127327et_nat @ B2 @ A )
= ( plus_p8712254050562127327et_nat @ C2 @ A ) )
=> ( B2 = C2 ) ) ).
% add_right_imp_eq
thf(fact_1096_add__right__imp__eq,axiom,
! [B2: nat,A: nat,C2: nat] :
( ( ( plus_plus_nat @ B2 @ A )
= ( plus_plus_nat @ C2 @ A ) )
=> ( B2 = C2 ) ) ).
% add_right_imp_eq
thf(fact_1097_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: multiset_set_nat,C2: multiset_set_nat,B2: multiset_set_nat] :
( ( minus_7237264121398869807et_nat @ ( minus_7237264121398869807et_nat @ A @ C2 ) @ B2 )
= ( minus_7237264121398869807et_nat @ ( minus_7237264121398869807et_nat @ A @ B2 ) @ C2 ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_1098_cancel__ab__semigroup__add__class_Odiff__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 ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_1099_set__notEmptyE,axiom,
! [S: set_Pr2645174627780777389at_nat] :
( ( S != bot_bo1973934853101755969at_nat )
=> ~ ! [X3: produc5825016348098550007at_nat] :
~ ( member6956540099943067662at_nat @ X3 @ S ) ) ).
% set_notEmptyE
thf(fact_1100_set__notEmptyE,axiom,
! [S: set_Re381260168593705685la_a_b] :
( ( S != bot_bo4495933725496725865la_a_b )
=> ~ ! [X3: relational_fmla_a_b] :
~ ( member4680049679412964150la_a_b @ X3 @ S ) ) ).
% set_notEmptyE
thf(fact_1101_set__notEmptyE,axiom,
! [S: set_list_a] :
( ( S != bot_bot_set_list_a )
=> ~ ! [X3: list_a] :
~ ( member_list_a @ X3 @ S ) ) ).
% set_notEmptyE
thf(fact_1102_set__notEmptyE,axiom,
! [S: set_nat] :
( ( S != bot_bot_set_nat )
=> ~ ! [X3: nat] :
~ ( member_nat @ X3 @ S ) ) ).
% set_notEmptyE
thf(fact_1103_memb__imp__not__empty,axiom,
! [X2: produc5825016348098550007at_nat,S: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ X2 @ S )
=> ( S != bot_bo1973934853101755969at_nat ) ) ).
% memb_imp_not_empty
thf(fact_1104_memb__imp__not__empty,axiom,
! [X2: relational_fmla_a_b,S: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ S )
=> ( S != bot_bo4495933725496725865la_a_b ) ) ).
% memb_imp_not_empty
thf(fact_1105_memb__imp__not__empty,axiom,
! [X2: list_a,S: set_list_a] :
( ( member_list_a @ X2 @ S )
=> ( S != bot_bot_set_list_a ) ) ).
% memb_imp_not_empty
thf(fact_1106_memb__imp__not__empty,axiom,
! [X2: nat,S: set_nat] :
( ( member_nat @ X2 @ S )
=> ( S != bot_bot_set_nat ) ) ).
% memb_imp_not_empty
thf(fact_1107_subset__Collect__conv,axiom,
! [S: set_Pr2645174627780777389at_nat,P: produc5825016348098550007at_nat > $o] :
( ( ord_le8520675249591772685at_nat @ S @ ( collec7697611494764367948at_nat @ P ) )
= ( ! [X: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X @ S )
=> ( P @ X ) ) ) ) ).
% subset_Collect_conv
thf(fact_1108_subset__Collect__conv,axiom,
! [S: set_nat,P: nat > $o] :
( ( ord_less_eq_set_nat @ S @ ( collect_nat @ P ) )
= ( ! [X: nat] :
( ( member_nat @ X @ S )
=> ( P @ X ) ) ) ) ).
% subset_Collect_conv
thf(fact_1109_mset__distrib,axiom,
! [A2: multiset_set_nat,B: multiset_set_nat,M: multiset_set_nat,N: multiset_set_nat] :
( ( ( plus_p8712254050562127327et_nat @ A2 @ B )
= ( plus_p8712254050562127327et_nat @ M @ N ) )
=> ~ ! [Am: multiset_set_nat,An: multiset_set_nat] :
( ( A2
= ( plus_p8712254050562127327et_nat @ Am @ An ) )
=> ! [Bm: multiset_set_nat,Bn: multiset_set_nat] :
( ( B
= ( plus_p8712254050562127327et_nat @ Bm @ Bn ) )
=> ( ( M
= ( plus_p8712254050562127327et_nat @ Am @ Bm ) )
=> ( N
!= ( plus_p8712254050562127327et_nat @ An @ Bn ) ) ) ) ) ) ).
% mset_distrib
thf(fact_1110_mset__map__id,axiom,
! [F: set_nat > produc5825016348098550007at_nat,G: produc5825016348098550007at_nat > set_nat,X5: multis4094885785038667591at_nat] :
( ! [X3: produc5825016348098550007at_nat] :
( ( F @ ( G @ X3 ) )
= X3 )
=> ( ( image_3074168642782391354at_nat @ F @ ( image_7702178775022393786et_nat @ G @ X5 ) )
= X5 ) ) ).
% mset_map_id
thf(fact_1111_mset__map__id,axiom,
! [F: produc5825016348098550007at_nat > set_nat,G: set_nat > produc5825016348098550007at_nat,X5: multiset_set_nat] :
( ! [X3: set_nat] :
( ( F @ ( G @ X3 ) )
= X3 )
=> ( ( image_7702178775022393786et_nat @ F @ ( image_3074168642782391354at_nat @ G @ X5 ) )
= X5 ) ) ).
% mset_map_id
thf(fact_1112_not__finite__existsD,axiom,
! [P: produc5825016348098550007at_nat > $o] :
( ~ ( finite6790245451575510286at_nat @ ( collec7697611494764367948at_nat @ P ) )
=> ? [X_1: produc5825016348098550007at_nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_1113_not__finite__existsD,axiom,
! [P: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
=> ? [X_1: nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_1114_not__finite__existsD,axiom,
! [P: relational_fmla_a_b > $o] :
( ~ ( finite5600759454172676150la_a_b @ ( collec3419995626248312948la_a_b @ P ) )
=> ? [X_1: relational_fmla_a_b] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_1115_pigeonhole__infinite__rel,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat,R: produc5825016348098550007at_nat > produc5825016348098550007at_nat > $o] :
( ~ ( finite6790245451575510286at_nat @ A2 )
=> ( ( finite6790245451575510286at_nat @ B )
=> ( ! [X3: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X3 @ A2 )
=> ? [Xa: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X3 @ B )
& ~ ( finite6790245451575510286at_nat
@ ( collec7697611494764367948at_nat
@ ^ [A3: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ A3 @ A2 )
& ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_1116_pigeonhole__infinite__rel,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_nat,R: produc5825016348098550007at_nat > nat > $o] :
( ~ ( finite6790245451575510286at_nat @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( ! [X3: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X3 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ~ ( finite6790245451575510286at_nat
@ ( collec7697611494764367948at_nat
@ ^ [A3: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ A3 @ A2 )
& ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_1117_pigeonhole__infinite__rel,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Re381260168593705685la_a_b,R: produc5825016348098550007at_nat > relational_fmla_a_b > $o] :
( ~ ( finite6790245451575510286at_nat @ A2 )
=> ( ( finite5600759454172676150la_a_b @ B )
=> ( ! [X3: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X3 @ A2 )
=> ? [Xa: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ B )
& ~ ( finite6790245451575510286at_nat
@ ( collec7697611494764367948at_nat
@ ^ [A3: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ A3 @ A2 )
& ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_1118_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B: set_Pr2645174627780777389at_nat,R: nat > produc5825016348098550007at_nat > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite6790245451575510286at_nat @ B )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ? [Xa: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X3 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A2 )
& ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_1119_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B: set_nat,R: nat > nat > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A2 )
& ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_1120_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B: set_Re381260168593705685la_a_b,R: nat > relational_fmla_a_b > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite5600759454172676150la_a_b @ B )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ? [Xa: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A2 )
& ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_1121_pigeonhole__infinite__rel,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Pr2645174627780777389at_nat,R: relational_fmla_a_b > produc5825016348098550007at_nat > $o] :
( ~ ( finite5600759454172676150la_a_b @ A2 )
=> ( ( finite6790245451575510286at_nat @ B )
=> ( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A2 )
=> ? [Xa: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X3 @ B )
& ~ ( finite5600759454172676150la_a_b
@ ( collec3419995626248312948la_a_b
@ ^ [A3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ A3 @ A2 )
& ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_1122_pigeonhole__infinite__rel,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_nat,R: relational_fmla_a_b > nat > $o] :
( ~ ( finite5600759454172676150la_a_b @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ~ ( finite5600759454172676150la_a_b
@ ( collec3419995626248312948la_a_b
@ ^ [A3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ A3 @ A2 )
& ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_1123_pigeonhole__infinite__rel,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,R: relational_fmla_a_b > relational_fmla_a_b > $o] :
( ~ ( finite5600759454172676150la_a_b @ A2 )
=> ( ( finite5600759454172676150la_a_b @ B )
=> ( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A2 )
=> ? [Xa: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ B )
& ~ ( finite5600759454172676150la_a_b
@ ( collec3419995626248312948la_a_b
@ ^ [A3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ A3 @ A2 )
& ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_1124_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_1125_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: multiset_set_nat,J: multiset_set_nat,K2: multiset_set_nat,L: multiset_set_nat] :
( ( ( ord_le4034546139768944438et_nat @ I @ J )
& ( K2 = L ) )
=> ( ord_le4034546139768944438et_nat @ ( plus_p8712254050562127327et_nat @ I @ K2 ) @ ( plus_p8712254050562127327et_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1126_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K2 = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1127_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: multiset_set_nat,J: multiset_set_nat,K2: multiset_set_nat,L: multiset_set_nat] :
( ( ( I = J )
& ( ord_le4034546139768944438et_nat @ K2 @ L ) )
=> ( ord_le4034546139768944438et_nat @ ( plus_p8712254050562127327et_nat @ I @ K2 ) @ ( plus_p8712254050562127327et_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1128_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1129_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: multiset_set_nat,J: multiset_set_nat,K2: multiset_set_nat,L: multiset_set_nat] :
( ( ( ord_le4034546139768944438et_nat @ I @ J )
& ( ord_le4034546139768944438et_nat @ K2 @ L ) )
=> ( ord_le4034546139768944438et_nat @ ( plus_p8712254050562127327et_nat @ I @ K2 ) @ ( plus_p8712254050562127327et_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1130_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1131_add__mono,axiom,
! [A: multiset_set_nat,B2: multiset_set_nat,C2: multiset_set_nat,D2: multiset_set_nat] :
( ( ord_le4034546139768944438et_nat @ A @ B2 )
=> ( ( ord_le4034546139768944438et_nat @ C2 @ D2 )
=> ( ord_le4034546139768944438et_nat @ ( plus_p8712254050562127327et_nat @ A @ C2 ) @ ( plus_p8712254050562127327et_nat @ B2 @ D2 ) ) ) ) ).
% add_mono
thf(fact_1132_add__mono,axiom,
! [A: nat,B2: nat,C2: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ C2 @ D2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B2 @ D2 ) ) ) ) ).
% add_mono
thf(fact_1133_add__left__mono,axiom,
! [A: multiset_set_nat,B2: multiset_set_nat,C2: multiset_set_nat] :
( ( ord_le4034546139768944438et_nat @ A @ B2 )
=> ( ord_le4034546139768944438et_nat @ ( plus_p8712254050562127327et_nat @ C2 @ A ) @ ( plus_p8712254050562127327et_nat @ C2 @ B2 ) ) ) ).
% add_left_mono
thf(fact_1134_add__left__mono,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B2 ) ) ) ).
% add_left_mono
thf(fact_1135_less__eqE,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ~ ! [C5: nat] :
( B2
!= ( plus_plus_nat @ A @ C5 ) ) ) ).
% less_eqE
thf(fact_1136_add__right__mono,axiom,
! [A: multiset_set_nat,B2: multiset_set_nat,C2: multiset_set_nat] :
( ( ord_le4034546139768944438et_nat @ A @ B2 )
=> ( ord_le4034546139768944438et_nat @ ( plus_p8712254050562127327et_nat @ A @ C2 ) @ ( plus_p8712254050562127327et_nat @ B2 @ C2 ) ) ) ).
% add_right_mono
thf(fact_1137_add__right__mono,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B2 @ C2 ) ) ) ).
% add_right_mono
thf(fact_1138_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
? [C4: nat] :
( B3
= ( plus_plus_nat @ A3 @ C4 ) ) ) ) ).
% le_iff_add
thf(fact_1139_add__le__imp__le__left,axiom,
! [C2: multiset_set_nat,A: multiset_set_nat,B2: multiset_set_nat] :
( ( ord_le4034546139768944438et_nat @ ( plus_p8712254050562127327et_nat @ C2 @ A ) @ ( plus_p8712254050562127327et_nat @ C2 @ B2 ) )
=> ( ord_le4034546139768944438et_nat @ A @ B2 ) ) ).
% add_le_imp_le_left
thf(fact_1140_add__le__imp__le__left,axiom,
! [C2: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B2 ) )
=> ( ord_less_eq_nat @ A @ B2 ) ) ).
% add_le_imp_le_left
thf(fact_1141_add__le__imp__le__right,axiom,
! [A: multiset_set_nat,C2: multiset_set_nat,B2: multiset_set_nat] :
( ( ord_le4034546139768944438et_nat @ ( plus_p8712254050562127327et_nat @ A @ C2 ) @ ( plus_p8712254050562127327et_nat @ B2 @ C2 ) )
=> ( ord_le4034546139768944438et_nat @ A @ B2 ) ) ).
% add_le_imp_le_right
thf(fact_1142_add__le__imp__le__right,axiom,
! [A: nat,C2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B2 @ C2 ) )
=> ( ord_less_eq_nat @ A @ B2 ) ) ).
% add_le_imp_le_right
thf(fact_1143_comm__monoid__add__class_Oadd__0,axiom,
! [A: multiset_set_nat] :
( ( plus_p8712254050562127327et_nat @ zero_z3157962936165190495et_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_1144_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_1145_add_Ocomm__neutral,axiom,
! [A: multiset_set_nat] :
( ( plus_p8712254050562127327et_nat @ A @ zero_z3157962936165190495et_nat )
= A ) ).
% add.comm_neutral
thf(fact_1146_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_1147_finite__has__minimal2,axiom,
! [A2: set_Pr2645174627780777389at_nat,A: produc5825016348098550007at_nat] :
( ( finite6790245451575510286at_nat @ A2 )
=> ( ( member6956540099943067662at_nat @ A @ A2 )
=> ? [X3: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X3 @ A2 )
& ( ord_le8838844241092602199at_nat @ X3 @ A )
& ! [Xa: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ Xa @ A2 )
=> ( ( ord_le8838844241092602199at_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1148_finite__has__minimal2,axiom,
! [A2: set_Re381260168593705685la_a_b,A: relational_fmla_a_b] :
( ( finite5600759454172676150la_a_b @ A2 )
=> ( ( member4680049679412964150la_a_b @ A @ A2 )
=> ? [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A2 )
& ( ord_le4236940698060306943la_a_b @ X3 @ A )
& ! [Xa: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Xa @ A2 )
=> ( ( ord_le4236940698060306943la_a_b @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1149_finite__has__minimal2,axiom,
! [A2: set_se7774124317125585763at_nat,A: set_Pr2645174627780777389at_nat] :
( ( finite2560455063271695812at_nat @ A2 )
=> ( ( member734533627512885444at_nat @ A @ A2 )
=> ? [X3: set_Pr2645174627780777389at_nat] :
( ( member734533627512885444at_nat @ X3 @ A2 )
& ( ord_le8520675249591772685at_nat @ X3 @ A )
& ! [Xa: set_Pr2645174627780777389at_nat] :
( ( member734533627512885444at_nat @ Xa @ A2 )
=> ( ( ord_le8520675249591772685at_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1150_finite__has__minimal2,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat @ A @ A2 )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
& ( ord_less_eq_set_nat @ X3 @ A )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1151_finite__has__minimal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_eq_nat @ X3 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1152_finite__has__maximal2,axiom,
! [A2: set_Pr2645174627780777389at_nat,A: produc5825016348098550007at_nat] :
( ( finite6790245451575510286at_nat @ A2 )
=> ( ( member6956540099943067662at_nat @ A @ A2 )
=> ? [X3: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X3 @ A2 )
& ( ord_le8838844241092602199at_nat @ A @ X3 )
& ! [Xa: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ Xa @ A2 )
=> ( ( ord_le8838844241092602199at_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1153_finite__has__maximal2,axiom,
! [A2: set_Re381260168593705685la_a_b,A: relational_fmla_a_b] :
( ( finite5600759454172676150la_a_b @ A2 )
=> ( ( member4680049679412964150la_a_b @ A @ A2 )
=> ? [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A2 )
& ( ord_le4236940698060306943la_a_b @ A @ X3 )
& ! [Xa: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Xa @ A2 )
=> ( ( ord_le4236940698060306943la_a_b @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1154_finite__has__maximal2,axiom,
! [A2: set_se7774124317125585763at_nat,A: set_Pr2645174627780777389at_nat] :
( ( finite2560455063271695812at_nat @ A2 )
=> ( ( member734533627512885444at_nat @ A @ A2 )
=> ? [X3: set_Pr2645174627780777389at_nat] :
( ( member734533627512885444at_nat @ X3 @ A2 )
& ( ord_le8520675249591772685at_nat @ A @ X3 )
& ! [Xa: set_Pr2645174627780777389at_nat] :
( ( member734533627512885444at_nat @ Xa @ A2 )
=> ( ( ord_le8520675249591772685at_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1155_finite__has__maximal2,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat @ A @ A2 )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
& ( ord_less_eq_set_nat @ A @ X3 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1156_finite__has__maximal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_eq_nat @ A @ X3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1157_add__implies__diff,axiom,
! [C2: multiset_set_nat,B2: multiset_set_nat,A: multiset_set_nat] :
( ( ( plus_p8712254050562127327et_nat @ C2 @ B2 )
= A )
=> ( C2
= ( minus_7237264121398869807et_nat @ A @ B2 ) ) ) ).
% add_implies_diff
thf(fact_1158_add__implies__diff,axiom,
! [C2: nat,B2: nat,A: nat] :
( ( ( plus_plus_nat @ C2 @ B2 )
= A )
=> ( C2
= ( minus_minus_nat @ A @ B2 ) ) ) ).
% add_implies_diff
thf(fact_1159_diff__diff__eq,axiom,
! [A: multiset_set_nat,B2: multiset_set_nat,C2: multiset_set_nat] :
( ( minus_7237264121398869807et_nat @ ( minus_7237264121398869807et_nat @ A @ B2 ) @ C2 )
= ( minus_7237264121398869807et_nat @ A @ ( plus_p8712254050562127327et_nat @ B2 @ C2 ) ) ) ).
% diff_diff_eq
thf(fact_1160_diff__diff__eq,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B2 ) @ C2 )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B2 @ C2 ) ) ) ).
% diff_diff_eq
thf(fact_1161_fstE,axiom,
! [X2: product_prod_nat_nat,A: nat,B2: nat,P: nat > $o] :
( ( X2
= ( product_Pair_nat_nat @ A @ B2 ) )
=> ( ( P @ ( product_fst_nat_nat @ X2 ) )
=> ( P @ A ) ) ) ).
% fstE
thf(fact_1162_fstE,axiom,
! [X2: produc5825016348098550007at_nat,A: relational_fmla_a_b,B2: set_Pr1261947904930325089at_nat,P: relational_fmla_a_b > $o] :
( ( X2
= ( produc760948067230524457at_nat @ A @ B2 ) )
=> ( ( P @ ( produc8992968103413169469at_nat @ X2 ) )
=> ( P @ A ) ) ) ).
% fstE
thf(fact_1163_finite_OemptyI,axiom,
finite6790245451575510286at_nat @ bot_bo1973934853101755969at_nat ).
% finite.emptyI
thf(fact_1164_finite_OemptyI,axiom,
finite5600759454172676150la_a_b @ bot_bo4495933725496725865la_a_b ).
% finite.emptyI
thf(fact_1165_finite_OemptyI,axiom,
finite_finite_list_a @ bot_bot_set_list_a ).
% finite.emptyI
thf(fact_1166_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_1167_infinite__imp__nonempty,axiom,
! [S: set_Pr2645174627780777389at_nat] :
( ~ ( finite6790245451575510286at_nat @ S )
=> ( S != bot_bo1973934853101755969at_nat ) ) ).
% infinite_imp_nonempty
thf(fact_1168_infinite__imp__nonempty,axiom,
! [S: set_Re381260168593705685la_a_b] :
( ~ ( finite5600759454172676150la_a_b @ S )
=> ( S != bot_bo4495933725496725865la_a_b ) ) ).
% infinite_imp_nonempty
thf(fact_1169_infinite__imp__nonempty,axiom,
! [S: set_list_a] :
( ~ ( finite_finite_list_a @ S )
=> ( S != bot_bot_set_list_a ) ) ).
% infinite_imp_nonempty
thf(fact_1170_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_1171_all__subset__image,axiom,
! [F: relational_fmla_a_b > set_list_a,A2: set_Re381260168593705685la_a_b,P: set_set_list_a > $o] :
( ( ! [B4: set_set_list_a] :
( ( ord_le8877086941679407844list_a @ B4 @ ( image_130269618930851390list_a @ F @ A2 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ B4 @ A2 )
=> ( P @ ( image_130269618930851390list_a @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_1172_all__subset__image,axiom,
! [F: nat > relational_fmla_a_b,A2: set_nat,P: set_Re381260168593705685la_a_b > $o] :
( ( ! [B4: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ B4 @ ( image_4386371547000553590la_a_b @ F @ A2 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A2 )
=> ( P @ ( image_4386371547000553590la_a_b @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_1173_all__subset__image,axiom,
! [F: produc5825016348098550007at_nat > produc5825016348098550007at_nat,A2: set_Pr2645174627780777389at_nat,P: set_Pr2645174627780777389at_nat > $o] :
( ( ! [B4: set_Pr2645174627780777389at_nat] :
( ( ord_le8520675249591772685at_nat @ B4 @ ( image_6979785008349797877at_nat @ F @ A2 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_Pr2645174627780777389at_nat] :
( ( ord_le8520675249591772685at_nat @ B4 @ A2 )
=> ( P @ ( image_6979785008349797877at_nat @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_1174_all__subset__image,axiom,
! [F: nat > produc5825016348098550007at_nat,A2: set_nat,P: set_Pr2645174627780777389at_nat > $o] :
( ( ! [B4: set_Pr2645174627780777389at_nat] :
( ( ord_le8520675249591772685at_nat @ B4 @ ( image_1371346403869992270at_nat @ F @ A2 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A2 )
=> ( P @ ( image_1371346403869992270at_nat @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_1175_all__subset__image,axiom,
! [F: produc5825016348098550007at_nat > nat,A2: set_Pr2645174627780777389at_nat,P: set_nat > $o] :
( ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ ( image_8316665354072716238at_nat @ F @ A2 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_Pr2645174627780777389at_nat] :
( ( ord_le8520675249591772685at_nat @ B4 @ A2 )
=> ( P @ ( image_8316665354072716238at_nat @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_1176_all__subset__image,axiom,
! [F: nat > nat,A2: set_nat,P: set_nat > $o] :
( ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F @ A2 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A2 )
=> ( P @ ( image_nat_nat @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_1177_rev__finite__subset,axiom,
! [B: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ B )
=> ( ( ord_le4112832032246704949la_a_b @ A2 @ B )
=> ( finite5600759454172676150la_a_b @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_1178_rev__finite__subset,axiom,
! [B: set_Pr2645174627780777389at_nat,A2: set_Pr2645174627780777389at_nat] :
( ( finite6790245451575510286at_nat @ B )
=> ( ( ord_le8520675249591772685at_nat @ A2 @ B )
=> ( finite6790245451575510286at_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_1179_rev__finite__subset,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( finite_finite_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_1180_infinite__super,axiom,
! [S: set_Re381260168593705685la_a_b,T: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ S @ T )
=> ( ~ ( finite5600759454172676150la_a_b @ S )
=> ~ ( finite5600759454172676150la_a_b @ T ) ) ) ).
% infinite_super
thf(fact_1181_infinite__super,axiom,
! [S: set_Pr2645174627780777389at_nat,T: set_Pr2645174627780777389at_nat] :
( ( ord_le8520675249591772685at_nat @ S @ T )
=> ( ~ ( finite6790245451575510286at_nat @ S )
=> ~ ( finite6790245451575510286at_nat @ T ) ) ) ).
% infinite_super
thf(fact_1182_infinite__super,axiom,
! [S: set_nat,T: set_nat] :
( ( ord_less_eq_set_nat @ S @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T ) ) ) ).
% infinite_super
thf(fact_1183_finite__subset,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A2 @ B )
=> ( ( finite5600759454172676150la_a_b @ B )
=> ( finite5600759454172676150la_a_b @ A2 ) ) ) ).
% finite_subset
thf(fact_1184_finite__subset,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( ord_le8520675249591772685at_nat @ A2 @ B )
=> ( ( finite6790245451575510286at_nat @ B )
=> ( finite6790245451575510286at_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_1185_finite__subset,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_1186_finite_OinsertI,axiom,
! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
( ( finite6177210948735845034at_nat @ A2 )
=> ( finite6177210948735845034at_nat @ ( insert8211810215607154385at_nat @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_1187_finite_OinsertI,axiom,
! [A2: set_Pr2645174627780777389at_nat,A: produc5825016348098550007at_nat] :
( ( finite6790245451575510286at_nat @ A2 )
=> ( finite6790245451575510286at_nat @ ( insert3260075854425521959at_nat @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_1188_finite_OinsertI,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( insert_nat @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_1189_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_1190_disjointI,axiom,
! [A: set_Pr2645174627780777389at_nat,B2: set_Pr2645174627780777389at_nat] :
( ! [X3: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X3 @ A )
=> ~ ( member6956540099943067662at_nat @ X3 @ B2 ) )
=> ( ( inf_in8776938414804536127at_nat @ A @ B2 )
= bot_bo1973934853101755969at_nat ) ) ).
% disjointI
thf(fact_1191_disjointI,axiom,
! [A: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A )
=> ~ ( member4680049679412964150la_a_b @ X3 @ B2 ) )
=> ( ( inf_in8483230781156617063la_a_b @ A @ B2 )
= bot_bo4495933725496725865la_a_b ) ) ).
% disjointI
thf(fact_1192_disjointI,axiom,
! [A: set_list_a,B2: set_list_a] :
( ! [X3: list_a] :
( ( member_list_a @ X3 @ A )
=> ~ ( member_list_a @ X3 @ B2 ) )
=> ( ( inf_inf_set_list_a @ A @ B2 )
= bot_bot_set_list_a ) ) ).
% disjointI
thf(fact_1193_disjointI,axiom,
! [A: set_nat,B2: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ~ ( member_nat @ X3 @ B2 ) )
=> ( ( inf_inf_set_nat @ A @ B2 )
= bot_bot_set_nat ) ) ).
% disjointI
thf(fact_1194_Diff__infinite__finite,axiom,
! [T: set_nat,S: set_nat] :
( ( finite_finite_nat @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_1195_Diff__infinite__finite,axiom,
! [T: set_Pr2645174627780777389at_nat,S: set_Pr2645174627780777389at_nat] :
( ( finite6790245451575510286at_nat @ T )
=> ( ~ ( finite6790245451575510286at_nat @ S )
=> ~ ( finite6790245451575510286at_nat @ ( minus_6698950876951835142at_nat @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_1196_Diff__infinite__finite,axiom,
! [T: set_Re381260168593705685la_a_b,S: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ T )
=> ( ~ ( finite5600759454172676150la_a_b @ S )
=> ~ ( finite5600759454172676150la_a_b @ ( minus_4077726661957047470la_a_b @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_1197_inter__eq__subsetI,axiom,
! [S: set_Pr2645174627780777389at_nat,S2: set_Pr2645174627780777389at_nat,A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( ord_le8520675249591772685at_nat @ S @ S2 )
=> ( ( ( inf_in8776938414804536127at_nat @ A2 @ S2 )
= ( inf_in8776938414804536127at_nat @ B @ S2 ) )
=> ( ( inf_in8776938414804536127at_nat @ A2 @ S )
= ( inf_in8776938414804536127at_nat @ B @ S ) ) ) ) ).
% inter_eq_subsetI
thf(fact_1198_inter__eq__subsetI,axiom,
! [S: set_nat,S2: set_nat,A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ S @ S2 )
=> ( ( ( inf_inf_set_nat @ A2 @ S2 )
= ( inf_inf_set_nat @ B @ S2 ) )
=> ( ( inf_inf_set_nat @ A2 @ S )
= ( inf_inf_set_nat @ B @ S ) ) ) ) ).
% inter_eq_subsetI
thf(fact_1199_finite__UnI,axiom,
! [F2: set_nat,G2: set_nat] :
( ( finite_finite_nat @ F2 )
=> ( ( finite_finite_nat @ G2 )
=> ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G2 ) ) ) ) ).
% finite_UnI
thf(fact_1200_finite__UnI,axiom,
! [F2: set_Pr2645174627780777389at_nat,G2: set_Pr2645174627780777389at_nat] :
( ( finite6790245451575510286at_nat @ F2 )
=> ( ( finite6790245451575510286at_nat @ G2 )
=> ( finite6790245451575510286at_nat @ ( sup_su6099978769595272409at_nat @ F2 @ G2 ) ) ) ) ).
% finite_UnI
thf(fact_1201_finite__UnI,axiom,
! [F2: set_Re381260168593705685la_a_b,G2: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ F2 )
=> ( ( finite5600759454172676150la_a_b @ G2 )
=> ( finite5600759454172676150la_a_b @ ( sup_su5130108678486352897la_a_b @ F2 @ G2 ) ) ) ) ).
% finite_UnI
thf(fact_1202_Un__infinite,axiom,
! [S: set_nat,T: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T ) ) ) ).
% Un_infinite
thf(fact_1203_Un__infinite,axiom,
! [S: set_Pr2645174627780777389at_nat,T: set_Pr2645174627780777389at_nat] :
( ~ ( finite6790245451575510286at_nat @ S )
=> ~ ( finite6790245451575510286at_nat @ ( sup_su6099978769595272409at_nat @ S @ T ) ) ) ).
% Un_infinite
thf(fact_1204_Un__infinite,axiom,
! [S: set_Re381260168593705685la_a_b,T: set_Re381260168593705685la_a_b] :
( ~ ( finite5600759454172676150la_a_b @ S )
=> ~ ( finite5600759454172676150la_a_b @ ( sup_su5130108678486352897la_a_b @ S @ T ) ) ) ).
% Un_infinite
thf(fact_1205_infinite__Un,axiom,
! [S: set_nat,T: set_nat] :
( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T ) ) )
= ( ~ ( finite_finite_nat @ S )
| ~ ( finite_finite_nat @ T ) ) ) ).
% infinite_Un
thf(fact_1206_infinite__Un,axiom,
! [S: set_Pr2645174627780777389at_nat,T: set_Pr2645174627780777389at_nat] :
( ( ~ ( finite6790245451575510286at_nat @ ( sup_su6099978769595272409at_nat @ S @ T ) ) )
= ( ~ ( finite6790245451575510286at_nat @ S )
| ~ ( finite6790245451575510286at_nat @ T ) ) ) ).
% infinite_Un
thf(fact_1207_infinite__Un,axiom,
! [S: set_Re381260168593705685la_a_b,T: set_Re381260168593705685la_a_b] :
( ( ~ ( finite5600759454172676150la_a_b @ ( sup_su5130108678486352897la_a_b @ S @ T ) ) )
= ( ~ ( finite5600759454172676150la_a_b @ S )
| ~ ( finite5600759454172676150la_a_b @ T ) ) ) ).
% infinite_Un
thf(fact_1208_set__diff__diff__left,axiom,
! [A2: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat,C: set_Pr2645174627780777389at_nat] :
( ( minus_6698950876951835142at_nat @ ( minus_6698950876951835142at_nat @ A2 @ B ) @ C )
= ( minus_6698950876951835142at_nat @ A2 @ ( sup_su6099978769595272409at_nat @ B @ C ) ) ) ).
% set_diff_diff_left
thf(fact_1209_set__diff__diff__left,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b] :
( ( minus_4077726661957047470la_a_b @ ( minus_4077726661957047470la_a_b @ A2 @ B ) @ C )
= ( minus_4077726661957047470la_a_b @ A2 @ ( sup_su5130108678486352897la_a_b @ B @ C ) ) ) ).
% set_diff_diff_left
thf(fact_1210_mset__le__addE,axiom,
! [Xs: multiset_set_nat,Ys: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ Xs @ Ys )
=> ~ ! [Zs: multiset_set_nat] :
( Ys
!= ( plus_p8712254050562127327et_nat @ Xs @ Zs ) ) ) ).
% mset_le_addE
thf(fact_1211_mset__le__distrib,axiom,
! [X5: multiset_set_nat,A2: multiset_set_nat,B: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ X5 @ ( plus_p8712254050562127327et_nat @ A2 @ B ) )
=> ~ ! [Xa2: multiset_set_nat,Xb: multiset_set_nat] :
( ( X5
= ( plus_p8712254050562127327et_nat @ Xa2 @ Xb ) )
=> ( ( subset6078030600694693471et_nat @ Xa2 @ A2 )
=> ~ ( subset6078030600694693471et_nat @ Xb @ B ) ) ) ) ).
% mset_le_distrib
thf(fact_1212_mset__union__subset,axiom,
! [A2: multiset_set_nat,B: multiset_set_nat,C: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ ( plus_p8712254050562127327et_nat @ A2 @ B ) @ C )
=> ( ( subset6078030600694693471et_nat @ A2 @ C )
& ( subset6078030600694693471et_nat @ B @ C ) ) ) ).
% mset_union_subset
thf(fact_1213_mset__le__decr__left1,axiom,
! [A: multiset_set_nat,C2: multiset_set_nat,B2: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ ( plus_p8712254050562127327et_nat @ A @ C2 ) @ B2 )
=> ( subset6078030600694693471et_nat @ A @ B2 ) ) ).
% mset_le_decr_left1
thf(fact_1214_mset__le__decr__left2,axiom,
! [C2: multiset_set_nat,A: multiset_set_nat,B2: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ ( plus_p8712254050562127327et_nat @ C2 @ A ) @ B2 )
=> ( subset6078030600694693471et_nat @ A @ B2 ) ) ).
% mset_le_decr_left2
thf(fact_1215_mset__le__incr__right1,axiom,
! [A: multiset_set_nat,B2: multiset_set_nat,C2: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ A @ B2 )
=> ( subset6078030600694693471et_nat @ A @ ( plus_p8712254050562127327et_nat @ B2 @ C2 ) ) ) ).
% mset_le_incr_right1
thf(fact_1216_assms_I5_J,axiom,
ord_less_eq_set_nat @ ( relational_fv_a_b @ qfin ) @ f ).
% assms(5)
thf(fact_1217_fv__simp,axiom,
! [Q: relational_fmla_a_b] : ( ord_less_eq_set_nat @ ( relational_fv_a_b @ ( simp @ Q ) ) @ ( relational_fv_a_b @ Q ) ) ).
% fv_simp
thf(fact_1218_finite__Collect__le__nat,axiom,
! [K2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_eq_nat @ N4 @ K2 ) ) ) ).
% finite_Collect_le_nat
thf(fact_1219_sr__simp,axiom,
! [Q: relational_fmla_a_b] :
( ( relational_sr_a_b @ Q )
=> ( relational_sr_a_b @ ( simp @ Q ) ) ) ).
% sr_simp
thf(fact_1220_cov__fixbound,axiom,
! [X2: nat,Q: relational_fmla_a_b,G2: set_Re381260168593705685la_a_b,Q2: set_Re381260168593705685la_a_b] :
( ( relational_cov_a_b @ X2 @ Q @ G2 )
=> ( ( member_nat @ X2 @ ( relational_fv_a_b @ Q ) )
=> ( ( restri6210327484173102631nd_a_b
@ ( insert7010464514620295119la_a_b @ ( relational_cp_a_b @ ( relational_erase_a_b @ Q @ X2 ) )
@ ( insert7010464514620295119la_a_b @ ( simp @ ( relational_Conj_a_b @ Q @ ( relational_DISJ_a_b @ ( relational_qps_a_b @ G2 ) ) ) )
@ ( sup_su5130108678486352897la_a_b @ ( minus_4077726661957047470la_a_b @ Q2 @ ( insert7010464514620295119la_a_b @ Q @ bot_bo4495933725496725865la_a_b ) )
@ ( image_4386371547000553590la_a_b
@ ^ [Y: nat] : ( relational_cp_a_b @ ( relational_subst_a_b @ Q @ X2 @ Y ) )
@ ( relational_eqs_a_b @ X2 @ G2 ) ) ) ) )
@ X2 )
= ( minus_4077726661957047470la_a_b @ ( restri6210327484173102631nd_a_b @ Q2 @ X2 ) @ ( insert7010464514620295119la_a_b @ Q @ bot_bo4495933725496725865la_a_b ) ) ) ) ) ).
% cov_fixbound
thf(fact_1221_rb__INV__step,axiom,
! [X2: nat,Q: relational_fmla_a_b,Q2: set_Re381260168593705685la_a_b,Q3: relational_fmla_a_b,G2: set_Re381260168593705685la_a_b] :
( ( restrict_rb_INV_a_b @ simplified @ X2 @ Q @ Q2 )
=> ( ( member4680049679412964150la_a_b @ Q3 @ ( restri6210327484173102631nd_a_b @ Q2 @ X2 ) )
=> ( ( relational_cov_a_b @ X2 @ Q3 @ G2 )
=> ( restrict_rb_INV_a_b @ simplified @ X2 @ Q
@ ( insert7010464514620295119la_a_b @ ( relational_cp_a_b @ ( relational_erase_a_b @ Q3 @ X2 ) )
@ ( insert7010464514620295119la_a_b @ ( simp @ ( relational_Conj_a_b @ Q3 @ ( relational_DISJ_a_b @ ( relational_qps_a_b @ G2 ) ) ) )
@ ( sup_su5130108678486352897la_a_b @ ( minus_4077726661957047470la_a_b @ Q2 @ ( insert7010464514620295119la_a_b @ Q3 @ bot_bo4495933725496725865la_a_b ) )
@ ( image_4386371547000553590la_a_b
@ ^ [Y: nat] : ( relational_cp_a_b @ ( relational_subst_a_b @ Q3 @ X2 @ Y ) )
@ ( relational_eqs_a_b @ X2 @ G2 ) ) ) ) ) ) ) ) ) ).
% rb_INV_step
thf(fact_1222_simplification__axioms,axiom,
relati2910603115655104169on_a_b @ simp @ simplified ).
% simplification_axioms
thf(fact_1223_simplified__cp,axiom,
! [Q: relational_fmla_a_b] : ( simplified @ ( relational_cp_a_b @ Q ) ) ).
% simplified_cp
thf(fact_1224_rb__INV__cpropagated,axiom,
! [X2: nat,Q: relational_fmla_a_b,Q2: set_Re381260168593705685la_a_b,Q3: relational_fmla_a_b] :
( ( restrict_rb_INV_a_b @ simplified @ X2 @ Q @ Q2 )
=> ( ( member4680049679412964150la_a_b @ Q3 @ Q2 )
=> ( simplified @ Q3 ) ) ) ).
% rb_INV_cpropagated
thf(fact_1225_rb__INV__finite,axiom,
! [X2: nat,Q: relational_fmla_a_b,Q2: set_Re381260168593705685la_a_b] :
( ( restrict_rb_INV_a_b @ simplified @ X2 @ Q @ Q2 )
=> ( finite5600759454172676150la_a_b @ Q2 ) ) ).
% rb_INV_finite
thf(fact_1226_simplified__simp,axiom,
! [Q: relational_fmla_a_b] : ( simplified @ ( simp @ Q ) ) ).
% simplified_simp
thf(fact_1227_rb__INV__fv,axiom,
! [X2: nat,Q: relational_fmla_a_b,Q2: set_Re381260168593705685la_a_b,Q3: relational_fmla_a_b,Z2: nat] :
( ( restrict_rb_INV_a_b @ simplified @ X2 @ Q @ Q2 )
=> ( ( member4680049679412964150la_a_b @ Q3 @ Q2 )
=> ( ( member_nat @ Z2 @ ( relational_fv_a_b @ Q3 ) )
=> ( member_nat @ Z2 @ ( relational_fv_a_b @ Q ) ) ) ) ) ).
% rb_INV_fv
thf(fact_1228_simplified__fv__simp,axiom,
! [Q: relational_fmla_a_b] :
( ( simplified @ Q )
=> ( ( relational_fv_a_b @ ( simp @ Q ) )
= ( relational_fv_a_b @ Q ) ) ) ).
% simplified_fv_simp
thf(fact_1229_fv__simp__DISJ__eq,axiom,
! [Q2: set_Re381260168593705685la_a_b,A2: set_nat] :
( ( finite5600759454172676150la_a_b @ Q2 )
=> ( ( ( simp @ ( relational_DISJ_a_b @ Q2 ) )
!= ( relational_Bool_a_b @ $false ) )
=> ( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ Q2 )
=> ( ( simplified @ X3 )
& ( ( relational_fv_a_b @ X3 )
= A2 ) ) )
=> ( ( relational_fv_a_b @ ( simp @ ( relational_DISJ_a_b @ Q2 ) ) )
= A2 ) ) ) ) ).
% fv_simp_DISJ_eq
thf(fact_1230_rb__INV__rrb,axiom,
! [X2: nat,Q: relational_fmla_a_b,Q2: set_Re381260168593705685la_a_b,Q3: relational_fmla_a_b] :
( ( restrict_rb_INV_a_b @ simplified @ X2 @ Q @ Q2 )
=> ( ( member4680049679412964150la_a_b @ Q3 @ Q2 )
=> ( relational_rrb_a_b @ Q3 ) ) ) ).
% rb_INV_rrb
thf(fact_1231_simp__False,axiom,
( ( simp @ ( relational_Bool_a_b @ $false ) )
= ( relational_Bool_a_b @ $false ) ) ).
% simp_False
thf(fact_1232_rrb__simp,axiom,
! [Q: relational_fmla_a_b] :
( ( relational_rrb_a_b @ Q )
=> ( relational_rrb_a_b @ ( simp @ Q ) ) ) ).
% rrb_simp
thf(fact_1233_rb__INV__init,axiom,
! [Q: relational_fmla_a_b,X2: nat] :
( ( simplified @ Q )
=> ( ( relational_rrb_a_b @ Q )
=> ( restrict_rb_INV_a_b @ simplified @ X2 @ Q @ ( restri569617705344514291sj_a_b @ Q ) ) ) ) ).
% rb_INV_init
thf(fact_1234_simplified__flat__DisjD,axiom,
! [Q3: relational_fmla_a_b,Q: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Q3 @ ( restri569617705344514291sj_a_b @ Q ) )
=> ( ( simplified @ Q )
=> ( simplified @ Q3 ) ) ) ).
% simplified_flat_DisjD
thf(fact_1235_eval__simp__False,axiom,
! [I2: product_prod_b_nat > set_list_a] :
( ( relational_eval_a_b @ ( simp @ ( relational_Bool_a_b @ $false ) ) @ I2 )
= bot_bot_set_list_a ) ).
% eval_simp_False
thf(fact_1236_eval__on__simp,axiom,
! [X5: set_nat,Q: relational_fmla_a_b] :
( ( relati8814510239606734169on_a_b @ X5 @ ( simp @ Q ) )
= ( relati8814510239606734169on_a_b @ X5 @ Q ) ) ).
% eval_on_simp
thf(fact_1237_eval__simp__DISJ__closed,axiom,
! [Q2: set_Re381260168593705685la_a_b,I2: product_prod_b_nat > set_list_a] :
( ( finite5600759454172676150la_a_b @ Q2 )
=> ( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ Q2 )
=> ( ( relational_fv_a_b @ X3 )
= bot_bot_set_nat ) )
=> ( ( relational_eval_a_b @ ( simp @ ( relational_DISJ_a_b @ Q2 ) ) @ I2 )
= ( comple6928918032620976721list_a
@ ( image_130269618930851390list_a
@ ^ [Q4: relational_fmla_a_b] : ( relational_eval_a_b @ Q4 @ I2 )
@ Q2 ) ) ) ) ) ).
% eval_simp_DISJ_closed
thf(fact_1238_finite__less__ub,axiom,
! [F: nat > nat,U2: nat] :
( ! [N5: nat] : ( ord_less_eq_nat @ N5 @ ( F @ N5 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ U2 ) ) ) ) ).
% finite_less_ub
thf(fact_1239_bounded__Max__nat,axiom,
! [P: nat > $o,X2: nat,M: nat] :
( ( P @ X2 )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M ) )
=> ~ ! [M3: nat] :
( ( P @ M3 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1240_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N3: set_nat] :
? [M4: nat] :
! [X: nat] :
( ( member_nat @ X @ N3 )
=> ( ord_less_eq_nat @ X @ M4 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1241_Sup__nat__empty,axiom,
( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% Sup_nat_empty
thf(fact_1242_diff__is__0__eq,axiom,
! [M5: nat,N2: nat] :
( ( ( minus_minus_nat @ M5 @ N2 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M5 @ N2 ) ) ).
% diff_is_0_eq
thf(fact_1243_diff__is__0__eq_H,axiom,
! [M5: nat,N2: nat] :
( ( ord_less_eq_nat @ M5 @ N2 )
=> ( ( minus_minus_nat @ M5 @ N2 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1244_nat__add__left__cancel__le,axiom,
! [K2: nat,M5: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K2 @ M5 ) @ ( plus_plus_nat @ K2 @ N2 ) )
= ( ord_less_eq_nat @ M5 @ N2 ) ) ).
% nat_add_left_cancel_le
thf(fact_1245_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_1246_le0,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% le0
thf(fact_1247_diff__diff__cancel,axiom,
! [I: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1248_Nat_Odiff__diff__right,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1249_Nat_Oadd__diff__assoc2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K2 ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1250_Nat_Oadd__diff__assoc,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K2 ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1251_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1252_eq__diff__iff,axiom,
! [K2: nat,M5: nat,N2: nat] :
( ( ord_less_eq_nat @ K2 @ M5 )
=> ( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ( ( minus_minus_nat @ M5 @ K2 )
= ( minus_minus_nat @ N2 @ K2 ) )
= ( M5 = N2 ) ) ) ) ).
% eq_diff_iff
thf(fact_1253_le__diff__iff,axiom,
! [K2: nat,M5: nat,N2: nat] :
( ( ord_less_eq_nat @ K2 @ M5 )
=> ( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M5 @ K2 ) @ ( minus_minus_nat @ N2 @ K2 ) )
= ( ord_less_eq_nat @ M5 @ N2 ) ) ) ) ).
% le_diff_iff
thf(fact_1254_Nat_Odiff__diff__eq,axiom,
! [K2: nat,M5: nat,N2: nat] :
( ( ord_less_eq_nat @ K2 @ M5 )
=> ( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M5 @ K2 ) @ ( minus_minus_nat @ N2 @ K2 ) )
= ( minus_minus_nat @ M5 @ N2 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1255_diff__le__mono,axiom,
! [M5: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M5 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M5 @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).
% diff_le_mono
thf(fact_1256_diff__le__self,axiom,
! [M5: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M5 @ N2 ) @ M5 ) ).
% diff_le_self
thf(fact_1257_le__diff__conv,axiom,
! [J: nat,K2: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K2 ) ) ) ).
% le_diff_conv
thf(fact_1258_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_1259_diff__le__mono2,axiom,
! [M5: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M5 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M5 ) ) ) ).
% diff_le_mono2
thf(fact_1260_Nat_Ole__diff__conv2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1261_Nat_Odiff__add__assoc,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K2 )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K2 ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1262_Nat_Odiff__add__assoc2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K2 )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K2 ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1263_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K2 )
= ( J
= ( plus_plus_nat @ K2 @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1264_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K2: nat,B2: nat] :
( ( P @ K2 )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B2 ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1265_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N4: nat] :
? [K3: nat] :
( N4
= ( plus_plus_nat @ M4 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1266_trans__le__add2,axiom,
! [I: nat,J: nat,M5: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M5 @ J ) ) ) ).
% trans_le_add2
thf(fact_1267_trans__le__add1,axiom,
! [I: nat,J: nat,M5: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M5 ) ) ) ).
% trans_le_add1
thf(fact_1268_nat__le__linear,axiom,
! [M5: nat,N2: nat] :
( ( ord_less_eq_nat @ M5 @ N2 )
| ( ord_less_eq_nat @ N2 @ M5 ) ) ).
% nat_le_linear
thf(fact_1269_add__le__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% add_le_mono1
thf(fact_1270_add__le__mono,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K2 @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1271_le__antisym,axiom,
! [M5: nat,N2: nat] :
( ( ord_less_eq_nat @ M5 @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M5 )
=> ( M5 = N2 ) ) ) ).
% le_antisym
thf(fact_1272_le__Suc__ex,axiom,
! [K2: nat,L: nat] :
( ( ord_less_eq_nat @ K2 @ L )
=> ? [N5: nat] :
( L
= ( plus_plus_nat @ K2 @ N5 ) ) ) ).
% le_Suc_ex
thf(fact_1273_eq__imp__le,axiom,
! [M5: nat,N2: nat] :
( ( M5 = N2 )
=> ( ord_less_eq_nat @ M5 @ N2 ) ) ).
% eq_imp_le
thf(fact_1274_le__trans,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K2 )
=> ( ord_less_eq_nat @ I @ K2 ) ) ) ).
% le_trans
% Helper facts (7)
thf(help_If_2_1_If_001t__Set__Oset_It__List__Olist_Itf__a_J_J_T,axiom,
! [X2: set_list_a,Y3: set_list_a] :
( ( if_set_list_a @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__List__Olist_Itf__a_J_J_T,axiom,
! [X2: set_list_a,Y3: set_list_a] :
( ( if_set_list_a @ $true @ X2 @ Y3 )
= X2 ) ).
thf(help_If_2_1_If_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_T,axiom,
! [X2: relational_fmla_a_b,Y3: relational_fmla_a_b] :
( ( if_Rel1279876242545935705la_a_b @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_T,axiom,
! [X2: relational_fmla_a_b,Y3: relational_fmla_a_b] :
( ( if_Rel1279876242545935705la_a_b @ $true @ X2 @ Y3 )
= X2 ) ).
thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_T,axiom,
! [X2: produc5825016348098550007at_nat,Y3: produc5825016348098550007at_nat] :
( ( if_Pro1064994777914121649at_nat @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_T,axiom,
! [X2: produc5825016348098550007at_nat,Y3: produc5825016348098550007at_nat] :
( ( if_Pro1064994777914121649at_nat @ $true @ X2 @ Y3 )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( image_7702178775022393786et_nat @ ( comp_R1256417402204451841at_nat @ relati62690040636126068ns_a_b @ produc8992968103413169469at_nat )
@ ( mset_s3030574779765502096at_nat
@ ( insert3260075854425521959at_nat @ ( produc760948067230524457at_nat @ ( simp @ ( relational_Conj_a_b @ qfin @ ( relational_DISJ_a_b @ ( relational_qps_a_b @ g ) ) ) ) @ qeq )
@ ( sup_su6099978769595272409at_nat @ ( minus_6698950876951835142at_nat @ q_fin @ ( insert3260075854425521959at_nat @ ( produc760948067230524457at_nat @ qfin @ qeq ) @ bot_bo1973934853101755969at_nat ) )
@ ( image_1371346403869992270at_nat
@ ^ [Y: nat] : ( produc760948067230524457at_nat @ ( relational_cp_a_b @ ( relational_subst_a_b @ qfin @ x2 @ Y ) ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ x2 @ Y ) @ qeq ) )
@ ( relational_eqs_a_b @ x2 @ g ) ) ) ) ) )
= ( plus_p8712254050562127327et_nat @ ( minus_7237264121398869807et_nat @ ( image_7702178775022393786et_nat @ ( comp_R1256417402204451841at_nat @ relati62690040636126068ns_a_b @ produc8992968103413169469at_nat ) @ ( mset_s3030574779765502096at_nat @ q_fin ) ) @ ( image_7702178775022393786et_nat @ ( comp_R1256417402204451841at_nat @ relati62690040636126068ns_a_b @ produc8992968103413169469at_nat ) @ ( mset_s3030574779765502096at_nat @ x ) ) ) @ ( image_7702178775022393786et_nat @ ( comp_R1256417402204451841at_nat @ relati62690040636126068ns_a_b @ produc8992968103413169469at_nat ) @ ( mset_s3030574779765502096at_nat @ y ) ) ) ) ).
%------------------------------------------------------------------------------