TPTP Problem File: SLH0738^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Eval_FO/0005_Ailamazyan/prob_03190_130381__15936578_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1525 ( 709 unt; 238 typ; 0 def)
% Number of atoms : 3288 (1319 equ; 0 cnn)
% Maximal formula atoms : 24 ( 2 avg)
% Number of connectives : 10728 ( 188 ~; 25 |; 210 &;9103 @)
% ( 0 <=>;1202 =>; 0 <=; 0 <~>)
% Maximal formula depth : 50 ( 6 avg)
% Number of types : 28 ( 27 usr)
% Number of type conns : 1172 (1172 >; 0 *; 0 +; 0 <<)
% Number of symbols : 214 ( 211 usr; 27 con; 0-5 aty)
% Number of variables : 3856 ( 423 ^;3362 !; 71 ?;3856 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 13:11:43.261
%------------------------------------------------------------------------------
% Could-be-implicit typings (27)
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
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thf(ty_n_t__Option__Ooption_It__Nat__Onat_J,type,
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thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
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thf(ty_n_t__List__Olist_Itf__a_J,type,
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thf(ty_n_t__List__Olist_I_Eo_J,type,
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thf(ty_n_t__Nat__Onat,type,
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thf(ty_n_tf__a,type,
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% Explicit typings (211)
thf(sy_c_Ailamazyan_Oad__agr__list_001tf__a_001t__Nat__Onat,type,
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thf(sy_c_Ailamazyan_Osp__equiv_001tf__a_001t__Nat__Onat,type,
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thf(sy_c_Set_Oimage_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001tf__a,type,
image_6322530041254294468_nat_a: ( sum_sum_a_nat > a ) > set_Sum_sum_a_nat > set_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
image_7897140031816760844_a_nat: ( a > list_Sum_sum_a_nat ) > set_a > set_li6526943997496501093_a_nat ).
thf(sy_c_Set_Oimage_001tf__a_001t__Nat__Onat,type,
image_a_nat: ( a > nat ) > set_a > set_nat ).
thf(sy_c_Set_Oimage_001tf__a_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
image_7873763678140191238_a_nat: ( a > sum_sum_a_nat ) > set_a > set_Sum_sum_a_nat ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set_Ovimage_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
vimage7538740559230589179_a_nat: ( list_Sum_sum_a_nat > list_Sum_sum_a_nat ) > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat ).
thf(sy_c_Set_Ovimage_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
vimage4660099214859102426at_nat: ( list_Sum_sum_a_nat > nat ) > set_nat > set_li6526943997496501093_a_nat ).
thf(sy_c_Set_Ovimage_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001tf__a,type,
vimage7906019935533595764_nat_a: ( list_Sum_sum_a_nat > a ) > set_a > set_li6526943997496501093_a_nat ).
thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
vimage8387349081243083832_a_nat: ( nat > list_Sum_sum_a_nat ) > set_li6526943997496501093_a_nat > set_nat ).
thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Nat__Onat,type,
vimage_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
vimage3040984495076556338_a_nat: ( nat > sum_sum_a_nat ) > set_Sum_sum_a_nat > set_nat ).
thf(sy_c_Set_Ovimage_001t__Nat__Onat_001tf__a,type,
vimage_nat_a: ( nat > a ) > set_a > set_nat ).
thf(sy_c_Set_Ovimage_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
vimage8052486765430747393_a_nat: ( sum_sum_a_nat > list_Sum_sum_a_nat ) > set_li6526943997496501093_a_nat > set_Sum_sum_a_nat ).
thf(sy_c_Set_Ovimage_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Nat__Onat,type,
vimage7444966428737627988at_nat: ( sum_sum_a_nat > nat ) > set_nat > set_Sum_sum_a_nat ).
thf(sy_c_Set_Ovimage_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Sum____Type__Osum_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
vimage7613534090772763900at_nat: ( sum_sum_a_nat > sum_su4711128520861746048at_nat ) > set_Su2711871490478030048at_nat > set_Sum_sum_a_nat ).
thf(sy_c_Set_Ovimage_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
vimage6545432446589551483_a_nat: ( sum_sum_a_nat > sum_sum_a_nat ) > set_Sum_sum_a_nat > set_Sum_sum_a_nat ).
thf(sy_c_Set_Ovimage_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001tf__a,type,
vimage3056561457863698554_nat_a: ( sum_sum_a_nat > a ) > set_a > set_Sum_sum_a_nat ).
thf(sy_c_Set_Ovimage_001tf__a_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
vimage1894578532518109122_a_nat: ( a > list_Sum_sum_a_nat ) > set_li6526943997496501093_a_nat > set_a ).
thf(sy_c_Set_Ovimage_001tf__a_001t__Nat__Onat,type,
vimage_a_nat: ( a > nat ) > set_nat > set_a ).
thf(sy_c_Set_Ovimage_001tf__a_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
vimage4607795094749595324_a_nat: ( a > sum_sum_a_nat ) > set_Sum_sum_a_nat > set_a ).
thf(sy_c_Set_Ovimage_001tf__a_001tf__a,type,
vimage_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set__Impl_Oord__class_Oquicksort_001t__Nat__Onat,type,
set_or9089632773640736191rt_nat: list_nat > list_nat ).
thf(sy_c_Sum__Type_OInl_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Nat__Onat,type,
sum_In1625650605130369377at_nat: sum_sum_a_nat > sum_su4711128520861746048at_nat ).
thf(sy_c_Sum__Type_OInl_001tf__a_001t__Nat__Onat,type,
sum_Inl_a_nat: a > sum_sum_a_nat ).
thf(sy_c_Sum__Type_OInr_001t__Nat__Onat_001tf__a,type,
sum_Inr_nat_a: nat > sum_sum_a_nat ).
thf(sy_c_Sum__Type_Osum_Oisl_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Nat__Onat,type,
sum_is7876730191583103919at_nat: sum_su4711128520861746048at_nat > $o ).
thf(sy_c_Sum__Type_Osum_Oisl_001tf__a_001t__Nat__Onat,type,
sum_isl_a_nat: sum_sum_a_nat > $o ).
thf(sy_c_member_001t__List__Olist_It__Sum____Type__Osum_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
member4821912183549559215at_nat: list_S5657819638503513606at_nat > set_li4634643645628290534at_nat > $o ).
thf(sy_c_member_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
member408289922725080238_a_nat: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
member3650934779329710139_a_nat: produc5454855657571582244_a_nat > set_Pr1021281732701644634_a_nat > $o ).
thf(sy_c_member_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
member_Sum_sum_a_nat: sum_sum_a_nat > set_Sum_sum_a_nat > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_AD,type,
ad: set_a ).
thf(sy_v_R_092_060phi_062,type,
r_phi: set_na3699693778330250182_a_nat ).
thf(sy_v_R_092_060psi_062,type,
r_psi: set_na3699693778330250182_a_nat ).
thf(sy_v_X_092_060phi_062,type,
x_phi: set_li6526943997496501093_a_nat ).
thf(sy_v_X_092_060psi_062,type,
x_psi: set_li6526943997496501093_a_nat ).
thf(sy_v__092_060sigma_062xs,type,
sigma_xs: nat > sum_sum_a_nat ).
thf(sy_v__092_060sigma_062ys,type,
sigma_ys: nat > sum_sum_a_nat ).
thf(sy_v_cxs,type,
cxs: list_P5056861408695629236_a_nat ).
thf(sy_v_cys,type,
cys: list_P5056861408695629236_a_nat ).
thf(sy_v_fs_092_060phi_062,type,
fs_phi: list_Sum_sum_a_nat ).
thf(sy_v_fs_092_060psi_062,type,
fs_psi: list_Sum_sum_a_nat ).
thf(sy_v_n,type,
n: nat ).
thf(sy_v_ns_092_060phi_062,type,
ns_phi: list_nat ).
thf(sy_v_ns_092_060phi_062_H,type,
ns_phi2: list_nat ).
thf(sy_v_ns_092_060psi_062,type,
ns_psi: list_nat ).
thf(sy_v_ns_092_060psi_062_H,type,
ns_psi2: list_nat ).
thf(sy_v_nxs,type,
nxs: list_nat ).
thf(sy_v_nys,type,
nys: list_nat ).
thf(sy_v_xs,type,
xs: list_Sum_sum_a_nat ).
thf(sy_v_xs_092_060phi_062,type,
xs_phi: list_Sum_sum_a_nat ).
thf(sy_v_ys,type,
ys: list_Sum_sum_a_nat ).
thf(sy_v_ys_092_060psi_062,type,
ys_psi: list_Sum_sum_a_nat ).
% Relevant facts (1275)
thf(fact_0_xs__ys__def_I2_J,axiom,
member408289922725080238_a_nat @ ys @ x_psi ).
% xs_ys_def(2)
thf(fact_1_xs__ys__def_I1_J,axiom,
member408289922725080238_a_nat @ xs @ x_phi ).
% xs_ys_def(1)
thf(fact_2_fs_092_060psi_062__def,axiom,
( fs_psi
= ( map_na823391071729141993_a_nat @ sigma_ys @ ns_psi2 ) ) ).
% fs\<psi>_def
thf(fact_3_Inl__xs__ys_I1_J,axiom,
! [N: nat] :
( ( member_nat @ N @ ( sup_sup_set_nat @ ( set_nat2 @ ns_phi2 ) @ ( set_nat2 @ ns_psi2 ) ) )
=> ( ( sum_isl_a_nat @ ( sigma_xs @ N ) )
= ( ? [X: a] :
( ( ( sigma_xs @ N )
= ( sum_Inl_a_nat @ X ) )
& ( member_a @ X @ ad ) ) ) ) ) ).
% Inl_xs_ys(1)
thf(fact_4_Inl__xs__ys_I2_J,axiom,
! [N: nat] :
( ( member_nat @ N @ ( sup_sup_set_nat @ ( set_nat2 @ ns_phi2 ) @ ( set_nat2 @ ns_psi2 ) ) )
=> ( ( sum_isl_a_nat @ ( sigma_ys @ N ) )
= ( ? [Y: a] :
( ( ( sigma_ys @ N )
= ( sum_Inl_a_nat @ Y ) )
& ( member_a @ Y @ ad ) ) ) ) ) ).
% Inl_xs_ys(2)
thf(fact_5_fs_092_060phi_062__def,axiom,
( fs_phi
= ( map_na823391071729141993_a_nat @ sigma_xs @ ns_phi2 ) ) ).
% fs\<phi>_def
thf(fact_6_assms_I16_J,axiom,
( ys
= ( map_na823391071729141993_a_nat @ sigma_ys @ ns_psi ) ) ).
% assms(16)
thf(fact_7_assms_I13_J,axiom,
( xs
= ( map_na823391071729141993_a_nat @ sigma_xs @ ns_phi ) ) ).
% assms(13)
thf(fact_8_assms_I18_J,axiom,
( xs_phi
= ( map_na823391071729141993_a_nat @ sigma_ys @ nxs ) ) ).
% assms(18)
thf(fact_9_UnCI,axiom,
! [C: list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat] :
( ( ~ ( member408289922725080238_a_nat @ C @ B )
=> ( member408289922725080238_a_nat @ C @ A ) )
=> ( member408289922725080238_a_nat @ C @ ( sup_su4083067149120280889_a_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_10_UnCI,axiom,
! [C: a,B: set_a,A: set_a] :
( ( ~ ( member_a @ C @ B )
=> ( member_a @ C @ A ) )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnCI
thf(fact_11_UnCI,axiom,
! [C: nat,B: set_nat,A: set_nat] :
( ( ~ ( member_nat @ C @ B )
=> ( member_nat @ C @ A ) )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_12_UnCI,axiom,
! [C: sum_sum_a_nat,B: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( ~ ( member_Sum_sum_a_nat @ C @ B )
=> ( member_Sum_sum_a_nat @ C @ A ) )
=> ( member_Sum_sum_a_nat @ C @ ( sup_su6804446743777130803_a_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_13_Un__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) )
= ( ( member_nat @ C @ A )
| ( member_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_14_Un__iff,axiom,
! [C: sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C @ ( sup_su6804446743777130803_a_nat @ A @ B ) )
= ( ( member_Sum_sum_a_nat @ C @ A )
| ( member_Sum_sum_a_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_15_Un__iff,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A @ B ) )
= ( ( member_a @ C @ A )
| ( member_a @ C @ B ) ) ) ).
% Un_iff
thf(fact_16_Un__iff,axiom,
! [C: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ ( sup_su4083067149120280889_a_nat @ A @ B ) )
= ( ( member408289922725080238_a_nat @ C @ A )
| ( member408289922725080238_a_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_17_sup_Oidem,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_18_sup_Oidem,axiom,
! [A2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_19_sup_Oidem,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_20_sup_Oidem,axiom,
! [A2: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_21_sup__idem,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_22_sup__idem,axiom,
! [X2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_23_sup__idem,axiom,
! [X2: set_a] :
( ( sup_sup_set_a @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_24_sup__idem,axiom,
! [X2: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_25_sup_Oleft__idem,axiom,
! [A2: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_26_sup_Oleft__idem,axiom,
! [A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A2 @ ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) )
= ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_27_sup_Oleft__idem,axiom,
! [A2: set_a,B2: set_a] :
( ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B2 ) )
= ( sup_sup_set_a @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_28_sup_Oleft__idem,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ A2 @ ( sup_su4083067149120280889_a_nat @ A2 @ B2 ) )
= ( sup_su4083067149120280889_a_nat @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_29_sup__left__idem,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ X2 @ Y2 ) )
= ( sup_sup_set_nat @ X2 @ Y2 ) ) ).
% sup_left_idem
thf(fact_30_sup__left__idem,axiom,
! [X2: set_Sum_sum_a_nat,Y2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ X2 @ ( sup_su6804446743777130803_a_nat @ X2 @ Y2 ) )
= ( sup_su6804446743777130803_a_nat @ X2 @ Y2 ) ) ).
% sup_left_idem
thf(fact_31_sup__left__idem,axiom,
! [X2: set_a,Y2: set_a] :
( ( sup_sup_set_a @ X2 @ ( sup_sup_set_a @ X2 @ Y2 ) )
= ( sup_sup_set_a @ X2 @ Y2 ) ) ).
% sup_left_idem
thf(fact_32_sup__left__idem,axiom,
! [X2: set_li6526943997496501093_a_nat,Y2: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ X2 @ ( sup_su4083067149120280889_a_nat @ X2 @ Y2 ) )
= ( sup_su4083067149120280889_a_nat @ X2 @ Y2 ) ) ).
% sup_left_idem
thf(fact_33_sup_Oright__idem,axiom,
! [A2: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ B2 )
= ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_34_sup_Oright__idem,axiom,
! [A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) @ B2 )
= ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_35_sup_Oright__idem,axiom,
! [A2: set_a,B2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ B2 )
= ( sup_sup_set_a @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_36_sup_Oright__idem,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ ( sup_su4083067149120280889_a_nat @ A2 @ B2 ) @ B2 )
= ( sup_su4083067149120280889_a_nat @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_37_assms_I15_J,axiom,
( ys_psi
= ( map_na823391071729141993_a_nat @ sigma_xs @ nys ) ) ).
% assms(15)
thf(fact_38_sup__left__commute,axiom,
! [X2: set_nat,Y2: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ Y2 @ Z ) )
= ( sup_sup_set_nat @ Y2 @ ( sup_sup_set_nat @ X2 @ Z ) ) ) ).
% sup_left_commute
thf(fact_39_sup__left__commute,axiom,
! [X2: set_Sum_sum_a_nat,Y2: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ X2 @ ( sup_su6804446743777130803_a_nat @ Y2 @ Z ) )
= ( sup_su6804446743777130803_a_nat @ Y2 @ ( sup_su6804446743777130803_a_nat @ X2 @ Z ) ) ) ).
% sup_left_commute
thf(fact_40_sup__left__commute,axiom,
! [X2: set_a,Y2: set_a,Z: set_a] :
( ( sup_sup_set_a @ X2 @ ( sup_sup_set_a @ Y2 @ Z ) )
= ( sup_sup_set_a @ Y2 @ ( sup_sup_set_a @ X2 @ Z ) ) ) ).
% sup_left_commute
thf(fact_41_sup__left__commute,axiom,
! [X2: set_li6526943997496501093_a_nat,Y2: set_li6526943997496501093_a_nat,Z: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ X2 @ ( sup_su4083067149120280889_a_nat @ Y2 @ Z ) )
= ( sup_su4083067149120280889_a_nat @ Y2 @ ( sup_su4083067149120280889_a_nat @ X2 @ Z ) ) ) ).
% sup_left_commute
thf(fact_42_sup_Oleft__commute,axiom,
! [B2: set_nat,A2: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ B2 @ ( sup_sup_set_nat @ A2 @ C ) )
= ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_43_sup_Oleft__commute,axiom,
! [B2: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ B2 @ ( sup_su6804446743777130803_a_nat @ A2 @ C ) )
= ( sup_su6804446743777130803_a_nat @ A2 @ ( sup_su6804446743777130803_a_nat @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_44_sup_Oleft__commute,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( sup_sup_set_a @ B2 @ ( sup_sup_set_a @ A2 @ C ) )
= ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_45_sup_Oleft__commute,axiom,
! [B2: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ B2 @ ( sup_su4083067149120280889_a_nat @ A2 @ C ) )
= ( sup_su4083067149120280889_a_nat @ A2 @ ( sup_su4083067149120280889_a_nat @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_46_sup__commute,axiom,
( sup_sup_set_nat
= ( ^ [X: set_nat,Y: set_nat] : ( sup_sup_set_nat @ Y @ X ) ) ) ).
% sup_commute
thf(fact_47_sup__commute,axiom,
( sup_su6804446743777130803_a_nat
= ( ^ [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] : ( sup_su6804446743777130803_a_nat @ Y @ X ) ) ) ).
% sup_commute
thf(fact_48_sup__commute,axiom,
( sup_sup_set_a
= ( ^ [X: set_a,Y: set_a] : ( sup_sup_set_a @ Y @ X ) ) ) ).
% sup_commute
thf(fact_49_sup__commute,axiom,
( sup_su4083067149120280889_a_nat
= ( ^ [X: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat] : ( sup_su4083067149120280889_a_nat @ Y @ X ) ) ) ).
% sup_commute
thf(fact_50_sup_Ocommute,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B3: set_nat] : ( sup_sup_set_nat @ B3 @ A3 ) ) ) ).
% sup.commute
thf(fact_51_sup_Ocommute,axiom,
( sup_su6804446743777130803_a_nat
= ( ^ [A3: set_Sum_sum_a_nat,B3: set_Sum_sum_a_nat] : ( sup_su6804446743777130803_a_nat @ B3 @ A3 ) ) ) ).
% sup.commute
thf(fact_52_sup_Ocommute,axiom,
( sup_sup_set_a
= ( ^ [A3: set_a,B3: set_a] : ( sup_sup_set_a @ B3 @ A3 ) ) ) ).
% sup.commute
thf(fact_53_sup_Ocommute,axiom,
( sup_su4083067149120280889_a_nat
= ( ^ [A3: set_li6526943997496501093_a_nat,B3: set_li6526943997496501093_a_nat] : ( sup_su4083067149120280889_a_nat @ B3 @ A3 ) ) ) ).
% sup.commute
thf(fact_54_sup__assoc,axiom,
! [X2: set_nat,Y2: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X2 @ Y2 ) @ Z )
= ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ Y2 @ Z ) ) ) ).
% sup_assoc
thf(fact_55_sup__assoc,axiom,
! [X2: set_Sum_sum_a_nat,Y2: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ ( sup_su6804446743777130803_a_nat @ X2 @ Y2 ) @ Z )
= ( sup_su6804446743777130803_a_nat @ X2 @ ( sup_su6804446743777130803_a_nat @ Y2 @ Z ) ) ) ).
% sup_assoc
thf(fact_56_sup__assoc,axiom,
! [X2: set_a,Y2: set_a,Z: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X2 @ Y2 ) @ Z )
= ( sup_sup_set_a @ X2 @ ( sup_sup_set_a @ Y2 @ Z ) ) ) ).
% sup_assoc
thf(fact_57_sup__assoc,axiom,
! [X2: set_li6526943997496501093_a_nat,Y2: set_li6526943997496501093_a_nat,Z: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ ( sup_su4083067149120280889_a_nat @ X2 @ Y2 ) @ Z )
= ( sup_su4083067149120280889_a_nat @ X2 @ ( sup_su4083067149120280889_a_nat @ Y2 @ Z ) ) ) ).
% sup_assoc
thf(fact_58_sup_Oassoc,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ C )
= ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_59_sup_Oassoc,axiom,
! [A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) @ C )
= ( sup_su6804446743777130803_a_nat @ A2 @ ( sup_su6804446743777130803_a_nat @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_60_sup_Oassoc,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ C )
= ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_61_sup_Oassoc,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ ( sup_su4083067149120280889_a_nat @ A2 @ B2 ) @ C )
= ( sup_su4083067149120280889_a_nat @ A2 @ ( sup_su4083067149120280889_a_nat @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_62_inf__sup__aci_I5_J,axiom,
( sup_sup_set_nat
= ( ^ [X: set_nat,Y: set_nat] : ( sup_sup_set_nat @ Y @ X ) ) ) ).
% inf_sup_aci(5)
thf(fact_63_inf__sup__aci_I5_J,axiom,
( sup_su6804446743777130803_a_nat
= ( ^ [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] : ( sup_su6804446743777130803_a_nat @ Y @ X ) ) ) ).
% inf_sup_aci(5)
thf(fact_64_inf__sup__aci_I5_J,axiom,
( sup_sup_set_a
= ( ^ [X: set_a,Y: set_a] : ( sup_sup_set_a @ Y @ X ) ) ) ).
% inf_sup_aci(5)
thf(fact_65_inf__sup__aci_I5_J,axiom,
( sup_su4083067149120280889_a_nat
= ( ^ [X: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat] : ( sup_su4083067149120280889_a_nat @ Y @ X ) ) ) ).
% inf_sup_aci(5)
thf(fact_66_inf__sup__aci_I6_J,axiom,
! [X2: set_nat,Y2: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X2 @ Y2 ) @ Z )
= ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ Y2 @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_67_inf__sup__aci_I6_J,axiom,
! [X2: set_Sum_sum_a_nat,Y2: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ ( sup_su6804446743777130803_a_nat @ X2 @ Y2 ) @ Z )
= ( sup_su6804446743777130803_a_nat @ X2 @ ( sup_su6804446743777130803_a_nat @ Y2 @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_68_inf__sup__aci_I6_J,axiom,
! [X2: set_a,Y2: set_a,Z: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X2 @ Y2 ) @ Z )
= ( sup_sup_set_a @ X2 @ ( sup_sup_set_a @ Y2 @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_69_inf__sup__aci_I6_J,axiom,
! [X2: set_li6526943997496501093_a_nat,Y2: set_li6526943997496501093_a_nat,Z: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ ( sup_su4083067149120280889_a_nat @ X2 @ Y2 ) @ Z )
= ( sup_su4083067149120280889_a_nat @ X2 @ ( sup_su4083067149120280889_a_nat @ Y2 @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_70_inf__sup__aci_I7_J,axiom,
! [X2: set_nat,Y2: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ Y2 @ Z ) )
= ( sup_sup_set_nat @ Y2 @ ( sup_sup_set_nat @ X2 @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_71_inf__sup__aci_I7_J,axiom,
! [X2: set_Sum_sum_a_nat,Y2: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ X2 @ ( sup_su6804446743777130803_a_nat @ Y2 @ Z ) )
= ( sup_su6804446743777130803_a_nat @ Y2 @ ( sup_su6804446743777130803_a_nat @ X2 @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_72_inf__sup__aci_I7_J,axiom,
! [X2: set_a,Y2: set_a,Z: set_a] :
( ( sup_sup_set_a @ X2 @ ( sup_sup_set_a @ Y2 @ Z ) )
= ( sup_sup_set_a @ Y2 @ ( sup_sup_set_a @ X2 @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_73_inf__sup__aci_I7_J,axiom,
! [X2: set_li6526943997496501093_a_nat,Y2: set_li6526943997496501093_a_nat,Z: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ X2 @ ( sup_su4083067149120280889_a_nat @ Y2 @ Z ) )
= ( sup_su4083067149120280889_a_nat @ Y2 @ ( sup_su4083067149120280889_a_nat @ X2 @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_74_inf__sup__aci_I8_J,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ X2 @ Y2 ) )
= ( sup_sup_set_nat @ X2 @ Y2 ) ) ).
% inf_sup_aci(8)
thf(fact_75_inf__sup__aci_I8_J,axiom,
! [X2: set_Sum_sum_a_nat,Y2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ X2 @ ( sup_su6804446743777130803_a_nat @ X2 @ Y2 ) )
= ( sup_su6804446743777130803_a_nat @ X2 @ Y2 ) ) ).
% inf_sup_aci(8)
thf(fact_76_inf__sup__aci_I8_J,axiom,
! [X2: set_a,Y2: set_a] :
( ( sup_sup_set_a @ X2 @ ( sup_sup_set_a @ X2 @ Y2 ) )
= ( sup_sup_set_a @ X2 @ Y2 ) ) ).
% inf_sup_aci(8)
thf(fact_77_inf__sup__aci_I8_J,axiom,
! [X2: set_li6526943997496501093_a_nat,Y2: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ X2 @ ( sup_su4083067149120280889_a_nat @ X2 @ Y2 ) )
= ( sup_su4083067149120280889_a_nat @ X2 @ Y2 ) ) ).
% inf_sup_aci(8)
thf(fact_78_Un__left__commute,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) )
= ( sup_sup_set_nat @ B @ ( sup_sup_set_nat @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_79_Un__left__commute,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A @ ( sup_su6804446743777130803_a_nat @ B @ C2 ) )
= ( sup_su6804446743777130803_a_nat @ B @ ( sup_su6804446743777130803_a_nat @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_80_Un__left__commute,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B @ C2 ) )
= ( sup_sup_set_a @ B @ ( sup_sup_set_a @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_81_Un__left__commute,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,C2: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ A @ ( sup_su4083067149120280889_a_nat @ B @ C2 ) )
= ( sup_su4083067149120280889_a_nat @ B @ ( sup_su4083067149120280889_a_nat @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_82_Un__left__absorb,axiom,
! [A: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_sup_set_nat @ A @ B ) ) ).
% Un_left_absorb
thf(fact_83_Un__left__absorb,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A @ ( sup_su6804446743777130803_a_nat @ A @ B ) )
= ( sup_su6804446743777130803_a_nat @ A @ B ) ) ).
% Un_left_absorb
thf(fact_84_Un__left__absorb,axiom,
! [A: set_a,B: set_a] :
( ( sup_sup_set_a @ A @ ( sup_sup_set_a @ A @ B ) )
= ( sup_sup_set_a @ A @ B ) ) ).
% Un_left_absorb
thf(fact_85_Un__left__absorb,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ A @ ( sup_su4083067149120280889_a_nat @ A @ B ) )
= ( sup_su4083067149120280889_a_nat @ A @ B ) ) ).
% Un_left_absorb
thf(fact_86_Un__commute,axiom,
( sup_sup_set_nat
= ( ^ [A4: set_nat,B4: set_nat] : ( sup_sup_set_nat @ B4 @ A4 ) ) ) ).
% Un_commute
thf(fact_87_Un__commute,axiom,
( sup_su6804446743777130803_a_nat
= ( ^ [A4: set_Sum_sum_a_nat,B4: set_Sum_sum_a_nat] : ( sup_su6804446743777130803_a_nat @ B4 @ A4 ) ) ) ).
% Un_commute
thf(fact_88_Un__commute,axiom,
( sup_sup_set_a
= ( ^ [A4: set_a,B4: set_a] : ( sup_sup_set_a @ B4 @ A4 ) ) ) ).
% Un_commute
thf(fact_89_Un__commute,axiom,
( sup_su4083067149120280889_a_nat
= ( ^ [A4: set_li6526943997496501093_a_nat,B4: set_li6526943997496501093_a_nat] : ( sup_su4083067149120280889_a_nat @ B4 @ A4 ) ) ) ).
% Un_commute
thf(fact_90_Un__absorb,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ A )
= A ) ).
% Un_absorb
thf(fact_91_Un__absorb,axiom,
! [A: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A @ A )
= A ) ).
% Un_absorb
thf(fact_92_Un__absorb,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ A )
= A ) ).
% Un_absorb
thf(fact_93_Un__absorb,axiom,
! [A: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ A @ A )
= A ) ).
% Un_absorb
thf(fact_94_Un__assoc,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C2 )
= ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_95_Un__assoc,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ ( sup_su6804446743777130803_a_nat @ A @ B ) @ C2 )
= ( sup_su6804446743777130803_a_nat @ A @ ( sup_su6804446743777130803_a_nat @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_96_Un__assoc,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A @ B ) @ C2 )
= ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_97_Un__assoc,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,C2: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ ( sup_su4083067149120280889_a_nat @ A @ B ) @ C2 )
= ( sup_su4083067149120280889_a_nat @ A @ ( sup_su4083067149120280889_a_nat @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_98_ball__Un,axiom,
! [A: set_nat,B: set_nat,P: nat > $o] :
( ( ! [X: nat] :
( ( member_nat @ X @ ( sup_sup_set_nat @ A @ B ) )
=> ( P @ X ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( P @ X ) )
& ! [X: nat] :
( ( member_nat @ X @ B )
=> ( P @ X ) ) ) ) ).
% ball_Un
thf(fact_99_ball__Un,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ( ! [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ ( sup_su6804446743777130803_a_nat @ A @ B ) )
=> ( P @ X ) ) )
= ( ! [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ A )
=> ( P @ X ) )
& ! [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ B )
=> ( P @ X ) ) ) ) ).
% ball_Un
thf(fact_100_ball__Un,axiom,
! [A: set_a,B: set_a,P: a > $o] :
( ( ! [X: a] :
( ( member_a @ X @ ( sup_sup_set_a @ A @ B ) )
=> ( P @ X ) ) )
= ( ! [X: a] :
( ( member_a @ X @ A )
=> ( P @ X ) )
& ! [X: a] :
( ( member_a @ X @ B )
=> ( P @ X ) ) ) ) ).
% ball_Un
thf(fact_101_ball__Un,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,P: list_Sum_sum_a_nat > $o] :
( ( ! [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ ( sup_su4083067149120280889_a_nat @ A @ B ) )
=> ( P @ X ) ) )
= ( ! [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A )
=> ( P @ X ) )
& ! [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ B )
=> ( P @ X ) ) ) ) ).
% ball_Un
thf(fact_102_bex__Un,axiom,
! [A: set_nat,B: set_nat,P: nat > $o] :
( ( ? [X: nat] :
( ( member_nat @ X @ ( sup_sup_set_nat @ A @ B ) )
& ( P @ X ) ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A )
& ( P @ X ) )
| ? [X: nat] :
( ( member_nat @ X @ B )
& ( P @ X ) ) ) ) ).
% bex_Un
thf(fact_103_bex__Un,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ( ? [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ ( sup_su6804446743777130803_a_nat @ A @ B ) )
& ( P @ X ) ) )
= ( ? [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ A )
& ( P @ X ) )
| ? [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ B )
& ( P @ X ) ) ) ) ).
% bex_Un
thf(fact_104_bex__Un,axiom,
! [A: set_a,B: set_a,P: a > $o] :
( ( ? [X: a] :
( ( member_a @ X @ ( sup_sup_set_a @ A @ B ) )
& ( P @ X ) ) )
= ( ? [X: a] :
( ( member_a @ X @ A )
& ( P @ X ) )
| ? [X: a] :
( ( member_a @ X @ B )
& ( P @ X ) ) ) ) ).
% bex_Un
thf(fact_105_bex__Un,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,P: list_Sum_sum_a_nat > $o] :
( ( ? [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ ( sup_su4083067149120280889_a_nat @ A @ B ) )
& ( P @ X ) ) )
= ( ? [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A )
& ( P @ X ) )
| ? [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ B )
& ( P @ X ) ) ) ) ).
% bex_Un
thf(fact_106_UnI2,axiom,
! [C: nat,B: set_nat,A: set_nat] :
( ( member_nat @ C @ B )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_107_UnI2,axiom,
! [C: sum_sum_a_nat,B: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C @ B )
=> ( member_Sum_sum_a_nat @ C @ ( sup_su6804446743777130803_a_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_108_UnI2,axiom,
! [C: a,B: set_a,A: set_a] :
( ( member_a @ C @ B )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnI2
thf(fact_109_UnI2,axiom,
! [C: list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ B )
=> ( member408289922725080238_a_nat @ C @ ( sup_su4083067149120280889_a_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_110_UnI1,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ A )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_111_UnI1,axiom,
! [C: sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C @ A )
=> ( member_Sum_sum_a_nat @ C @ ( sup_su6804446743777130803_a_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_112_UnI1,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ A )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnI1
thf(fact_113_UnI1,axiom,
! [C: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ A )
=> ( member408289922725080238_a_nat @ C @ ( sup_su4083067149120280889_a_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_114_UnE,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) )
=> ( ~ ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% UnE
thf(fact_115_UnE,axiom,
! [C: sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C @ ( sup_su6804446743777130803_a_nat @ A @ B ) )
=> ( ~ ( member_Sum_sum_a_nat @ C @ A )
=> ( member_Sum_sum_a_nat @ C @ B ) ) ) ).
% UnE
thf(fact_116_UnE,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A @ B ) )
=> ( ~ ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% UnE
thf(fact_117_UnE,axiom,
! [C: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ ( sup_su4083067149120280889_a_nat @ A @ B ) )
=> ( ~ ( member408289922725080238_a_nat @ C @ A )
=> ( member408289922725080238_a_nat @ C @ B ) ) ) ).
% UnE
thf(fact_118_map__eq__conv,axiom,
! [F: produc5454855657571582244_a_nat > sum_sum_a_nat,Xs: list_P5056861408695629236_a_nat,G: produc5454855657571582244_a_nat > sum_sum_a_nat] :
( ( ( map_Pr2515564165894599445_a_nat @ F @ Xs )
= ( map_Pr2515564165894599445_a_nat @ G @ Xs ) )
= ( ! [X: produc5454855657571582244_a_nat] :
( ( member3650934779329710139_a_nat @ X @ ( set_Pr2196415342531872975_a_nat @ Xs ) )
=> ( ( F @ X )
= ( G @ X ) ) ) ) ) ).
% map_eq_conv
thf(fact_119_map__eq__conv,axiom,
! [F: produc5454855657571582244_a_nat > nat,Xs: list_P5056861408695629236_a_nat,G: produc5454855657571582244_a_nat > nat] :
( ( ( map_Pr2058079554187054650at_nat @ F @ Xs )
= ( map_Pr2058079554187054650at_nat @ G @ Xs ) )
= ( ! [X: produc5454855657571582244_a_nat] :
( ( member3650934779329710139_a_nat @ X @ ( set_Pr2196415342531872975_a_nat @ Xs ) )
=> ( ( F @ X )
= ( G @ X ) ) ) ) ) ).
% map_eq_conv
thf(fact_120_map__eq__conv,axiom,
! [F: nat > sum_sum_a_nat,Xs: list_nat,G: nat > sum_sum_a_nat] :
( ( ( map_na823391071729141993_a_nat @ F @ Xs )
= ( map_na823391071729141993_a_nat @ G @ Xs ) )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( F @ X )
= ( G @ X ) ) ) ) ) ).
% map_eq_conv
thf(fact_121_ad__agr,axiom,
( ad_agr_list_a_nat @ ad
@ ( map_na823391071729141993_a_nat @ sigma_ys
@ ( linord738340561235409698at_nat
@ ^ [X: nat] : X
@ ( append_nat @ ns_psi @ ns_psi2 ) ) )
@ ( map_na823391071729141993_a_nat @ sigma_xs
@ ( linord738340561235409698at_nat
@ ^ [X: nat] : X
@ ( append_nat @ ns_phi @ ns_phi2 ) ) ) ) ).
% ad_agr
thf(fact_122_sum_Oinject_I1_J,axiom,
! [X1: a,Y1: a] :
( ( ( sum_Inl_a_nat @ X1 )
= ( sum_Inl_a_nat @ Y1 ) )
= ( X1 = Y1 ) ) ).
% sum.inject(1)
thf(fact_123_old_Osum_Oinject_I1_J,axiom,
! [A2: a,A5: a] :
( ( ( sum_Inl_a_nat @ A2 )
= ( sum_Inl_a_nat @ A5 ) )
= ( A2 = A5 ) ) ).
% old.sum.inject(1)
thf(fact_124_isl__def,axiom,
( sum_isl_a_nat
= ( ^ [Sum: sum_sum_a_nat] :
? [X12: a] :
( Sum
= ( sum_Inl_a_nat @ X12 ) ) ) ) ).
% isl_def
thf(fact_125_sum_OdiscI_I1_J,axiom,
! [Sum2: sum_sum_a_nat,X1: a] :
( ( Sum2
= ( sum_Inl_a_nat @ X1 ) )
=> ( sum_isl_a_nat @ Sum2 ) ) ).
% sum.discI(1)
thf(fact_126_mem__Collect__eq,axiom,
! [A2: list_Sum_sum_a_nat,P: list_Sum_sum_a_nat > $o] :
( ( member408289922725080238_a_nat @ A2 @ ( collec7555443234367654128_a_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_127_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_128_mem__Collect__eq,axiom,
! [A2: sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ( member_Sum_sum_a_nat @ A2 @ ( collec7073057861543223018_a_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_129_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_130_Collect__mem__eq,axiom,
! [A: set_li6526943997496501093_a_nat] :
( ( collec7555443234367654128_a_nat
@ ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_131_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X: a] : ( member_a @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_132_Collect__mem__eq,axiom,
! [A: set_Sum_sum_a_nat] :
( ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_133_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_134_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_135_sum_Odisc_I1_J,axiom,
! [X1: a] : ( sum_isl_a_nat @ ( sum_Inl_a_nat @ X1 ) ) ).
% sum.disc(1)
thf(fact_136_ex__map__conv,axiom,
! [Ys: list_nat,F: produc5454855657571582244_a_nat > nat] :
( ( ? [Xs2: list_P5056861408695629236_a_nat] :
( Ys
= ( map_Pr2058079554187054650at_nat @ F @ Xs2 ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Ys ) )
=> ? [Y: produc5454855657571582244_a_nat] :
( X
= ( F @ Y ) ) ) ) ) ).
% ex_map_conv
thf(fact_137_ex__map__conv,axiom,
! [Ys: list_Sum_sum_a_nat,F: nat > sum_sum_a_nat] :
( ( ? [Xs2: list_nat] :
( Ys
= ( map_na823391071729141993_a_nat @ F @ Xs2 ) ) )
= ( ! [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Ys ) )
=> ? [Y: nat] :
( X
= ( F @ Y ) ) ) ) ) ).
% ex_map_conv
thf(fact_138_ex__map__conv,axiom,
! [Ys: list_Sum_sum_a_nat,F: produc5454855657571582244_a_nat > sum_sum_a_nat] :
( ( ? [Xs2: list_P5056861408695629236_a_nat] :
( Ys
= ( map_Pr2515564165894599445_a_nat @ F @ Xs2 ) ) )
= ( ! [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Ys ) )
=> ? [Y: produc5454855657571582244_a_nat] :
( X
= ( F @ Y ) ) ) ) ) ).
% ex_map_conv
thf(fact_139_map__cong,axiom,
! [Xs: list_P5056861408695629236_a_nat,Ys: list_P5056861408695629236_a_nat,F: produc5454855657571582244_a_nat > sum_sum_a_nat,G: produc5454855657571582244_a_nat > sum_sum_a_nat] :
( ( Xs = Ys )
=> ( ! [X3: produc5454855657571582244_a_nat] :
( ( member3650934779329710139_a_nat @ X3 @ ( set_Pr2196415342531872975_a_nat @ Ys ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_Pr2515564165894599445_a_nat @ F @ Xs )
= ( map_Pr2515564165894599445_a_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_140_map__cong,axiom,
! [Xs: list_P5056861408695629236_a_nat,Ys: list_P5056861408695629236_a_nat,F: produc5454855657571582244_a_nat > nat,G: produc5454855657571582244_a_nat > nat] :
( ( Xs = Ys )
=> ( ! [X3: produc5454855657571582244_a_nat] :
( ( member3650934779329710139_a_nat @ X3 @ ( set_Pr2196415342531872975_a_nat @ Ys ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_Pr2058079554187054650at_nat @ F @ Xs )
= ( map_Pr2058079554187054650at_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_141_map__cong,axiom,
! [Xs: list_nat,Ys: list_nat,F: nat > sum_sum_a_nat,G: nat > sum_sum_a_nat] :
( ( Xs = Ys )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Ys ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_na823391071729141993_a_nat @ F @ Xs )
= ( map_na823391071729141993_a_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_142_map__idI,axiom,
! [Xs: list_l4703314356710769291_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ! [X3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ ( set_li2392974972034027290_a_nat @ Xs ) )
=> ( ( F @ X3 )
= X3 ) )
=> ( ( map_li6507455427659069316_a_nat @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_143_map__idI,axiom,
! [Xs: list_a,F: a > a] :
( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( ( F @ X3 )
= X3 ) )
=> ( ( map_a_a @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_144_map__idI,axiom,
! [Xs: list_nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X3 )
= X3 ) )
=> ( ( map_nat_nat @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_145_map__idI,axiom,
! [Xs: list_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat] :
( ! [X3: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X3 @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ( ( F @ X3 )
= X3 ) )
=> ( ( map_Su2790769393171190532_a_nat @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_146_map__ext,axiom,
! [Xs: list_P5056861408695629236_a_nat,F: produc5454855657571582244_a_nat > sum_sum_a_nat,G: produc5454855657571582244_a_nat > sum_sum_a_nat] :
( ! [X3: produc5454855657571582244_a_nat] :
( ( member3650934779329710139_a_nat @ X3 @ ( set_Pr2196415342531872975_a_nat @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_Pr2515564165894599445_a_nat @ F @ Xs )
= ( map_Pr2515564165894599445_a_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_147_map__ext,axiom,
! [Xs: list_P5056861408695629236_a_nat,F: produc5454855657571582244_a_nat > nat,G: produc5454855657571582244_a_nat > nat] :
( ! [X3: produc5454855657571582244_a_nat] :
( ( member3650934779329710139_a_nat @ X3 @ ( set_Pr2196415342531872975_a_nat @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_Pr2058079554187054650at_nat @ F @ Xs )
= ( map_Pr2058079554187054650at_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_148_map__ext,axiom,
! [Xs: list_nat,F: nat > sum_sum_a_nat,G: nat > sum_sum_a_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_na823391071729141993_a_nat @ F @ Xs )
= ( map_na823391071729141993_a_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_149_list_Omap__ident__strong,axiom,
! [T: list_l4703314356710769291_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ! [Z2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ Z2 @ ( set_li2392974972034027290_a_nat @ T ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( map_li6507455427659069316_a_nat @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_150_list_Omap__ident__strong,axiom,
! [T: list_a,F: a > a] :
( ! [Z2: a] :
( ( member_a @ Z2 @ ( set_a2 @ T ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( map_a_a @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_151_list_Omap__ident__strong,axiom,
! [T: list_nat,F: nat > nat] :
( ! [Z2: nat] :
( ( member_nat @ Z2 @ ( set_nat2 @ T ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( map_nat_nat @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_152_list_Omap__ident__strong,axiom,
! [T: list_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat] :
( ! [Z2: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ Z2 @ ( set_Sum_sum_a_nat2 @ T ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( map_Su2790769393171190532_a_nat @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_153_same__append__eq,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= ( append_nat @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_154_append__same__eq,axiom,
! [Ys: list_nat,Xs: list_nat,Zs: list_nat] :
( ( ( append_nat @ Ys @ Xs )
= ( append_nat @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_155_append__assoc,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( append_nat @ ( append_nat @ Xs @ Ys ) @ Zs )
= ( append_nat @ Xs @ ( append_nat @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_156_append_Oassoc,axiom,
! [A2: list_nat,B2: list_nat,C: list_nat] :
( ( append_nat @ ( append_nat @ A2 @ B2 ) @ C )
= ( append_nat @ A2 @ ( append_nat @ B2 @ C ) ) ) ).
% append.assoc
thf(fact_157_map__append,axiom,
! [F: nat > nat,Xs: list_nat,Ys: list_nat] :
( ( map_nat_nat @ F @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( map_nat_nat @ F @ Xs ) @ ( map_nat_nat @ F @ Ys ) ) ) ).
% map_append
thf(fact_158_map__append,axiom,
! [F: nat > sum_sum_a_nat,Xs: list_nat,Ys: list_nat] :
( ( map_na823391071729141993_a_nat @ F @ ( append_nat @ Xs @ Ys ) )
= ( append_Sum_sum_a_nat @ ( map_na823391071729141993_a_nat @ F @ Xs ) @ ( map_na823391071729141993_a_nat @ F @ Ys ) ) ) ).
% map_append
thf(fact_159_map__append,axiom,
! [F: produc5454855657571582244_a_nat > sum_sum_a_nat,Xs: list_P5056861408695629236_a_nat,Ys: list_P5056861408695629236_a_nat] :
( ( map_Pr2515564165894599445_a_nat @ F @ ( append338925788367110473_a_nat @ Xs @ Ys ) )
= ( append_Sum_sum_a_nat @ ( map_Pr2515564165894599445_a_nat @ F @ Xs ) @ ( map_Pr2515564165894599445_a_nat @ F @ Ys ) ) ) ).
% map_append
thf(fact_160_map__append,axiom,
! [F: produc5454855657571582244_a_nat > nat,Xs: list_P5056861408695629236_a_nat,Ys: list_P5056861408695629236_a_nat] :
( ( map_Pr2058079554187054650at_nat @ F @ ( append338925788367110473_a_nat @ Xs @ Ys ) )
= ( append_nat @ ( map_Pr2058079554187054650at_nat @ F @ Xs ) @ ( map_Pr2058079554187054650at_nat @ F @ Ys ) ) ) ).
% map_append
thf(fact_161_set__sort,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( set_nat2 @ ( linord738340561235409698at_nat @ F @ Xs ) )
= ( set_nat2 @ Xs ) ) ).
% set_sort
thf(fact_162_sort__sort,axiom,
( ( linord738340561235409698at_nat
@ ^ [X: nat] : X
@ ( append_nat @ ns_phi @ ns_phi2 ) )
= ( linord738340561235409698at_nat
@ ^ [X: nat] : X
@ ( append_nat @ ns_psi @ ns_psi2 ) ) ) ).
% sort_sort
thf(fact_163_set__append,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( set_nat2 @ ( append_nat @ Xs @ Ys ) )
= ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) ) ) ).
% set_append
thf(fact_164_set__append,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( set_Sum_sum_a_nat2 @ ( append_Sum_sum_a_nat @ Xs @ Ys ) )
= ( sup_su6804446743777130803_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) @ ( set_Sum_sum_a_nat2 @ Ys ) ) ) ).
% set_append
thf(fact_165_set__append,axiom,
! [Xs: list_a,Ys: list_a] :
( ( set_a2 @ ( append_a @ Xs @ Ys ) )
= ( sup_sup_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys ) ) ) ).
% set_append
thf(fact_166_set__append,axiom,
! [Xs: list_l4703314356710769291_a_nat,Ys: list_l4703314356710769291_a_nat] :
( ( set_li2392974972034027290_a_nat @ ( append5415888156905520160_a_nat @ Xs @ Ys ) )
= ( sup_su4083067149120280889_a_nat @ ( set_li2392974972034027290_a_nat @ Xs ) @ ( set_li2392974972034027290_a_nat @ Ys ) ) ) ).
% set_append
thf(fact_167_ns_092_060phi_062_H__def,axiom,
( ns_phi2
= ( filter_nat
@ ^ [N2: nat] :
~ ( member_nat @ N2 @ ( set_nat2 @ ns_phi ) )
@ ns_psi ) ) ).
% ns\<phi>'_def
thf(fact_168_ns_092_060psi_062_H__def,axiom,
( ns_psi2
= ( filter_nat
@ ^ [N2: nat] :
~ ( member_nat @ N2 @ ( set_nat2 @ ns_psi ) )
@ ns_phi ) ) ).
% ns\<psi>'_def
thf(fact_169_append__eq__append__conv2,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat,Ts: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= ( append_nat @ Zs @ Ts ) )
= ( ? [Us: list_nat] :
( ( ( Xs
= ( append_nat @ Zs @ Us ) )
& ( ( append_nat @ Us @ Ys )
= Ts ) )
| ( ( ( append_nat @ Xs @ Us )
= Zs )
& ( Ys
= ( append_nat @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_170_append__eq__appendI,axiom,
! [Xs: list_nat,Xs1: list_nat,Zs: list_nat,Ys: list_nat,Us2: list_nat] :
( ( ( append_nat @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_nat @ Xs1 @ Us2 ) )
=> ( ( append_nat @ Xs @ Ys )
= ( append_nat @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_171_sort__key__const,axiom,
! [C: nat,Xs: list_nat] :
( ( linord738340561235409698at_nat
@ ^ [X: nat] : C
@ Xs )
= Xs ) ).
% sort_key_const
thf(fact_172_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_173_Collect__disj__eq,axiom,
! [P: sum_sum_a_nat > $o,Q: sum_sum_a_nat > $o] :
( ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( P @ X )
| ( Q @ X ) ) )
= ( sup_su6804446743777130803_a_nat @ ( collec7073057861543223018_a_nat @ P ) @ ( collec7073057861543223018_a_nat @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_174_Collect__disj__eq,axiom,
! [P: a > $o,Q: a > $o] :
( ( collect_a
@ ^ [X: a] :
( ( P @ X )
| ( Q @ X ) ) )
= ( sup_sup_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_175_Collect__disj__eq,axiom,
! [P: list_Sum_sum_a_nat > $o,Q: list_Sum_sum_a_nat > $o] :
( ( collec7555443234367654128_a_nat
@ ^ [X: list_Sum_sum_a_nat] :
( ( P @ X )
| ( Q @ X ) ) )
= ( sup_su4083067149120280889_a_nat @ ( collec7555443234367654128_a_nat @ P ) @ ( collec7555443234367654128_a_nat @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_176_sup__set__def,axiom,
( sup_sup_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( collect_nat
@ ( sup_sup_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A4 )
@ ^ [X: nat] : ( member_nat @ X @ B4 ) ) ) ) ) ).
% sup_set_def
thf(fact_177_sup__set__def,axiom,
( sup_su6804446743777130803_a_nat
= ( ^ [A4: set_Sum_sum_a_nat,B4: set_Sum_sum_a_nat] :
( collec7073057861543223018_a_nat
@ ( sup_su491480579010597738_nat_o
@ ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ A4 )
@ ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ B4 ) ) ) ) ) ).
% sup_set_def
thf(fact_178_sup__set__def,axiom,
( sup_sup_set_a
= ( ^ [A4: set_a,B4: set_a] :
( collect_a
@ ( sup_sup_a_o
@ ^ [X: a] : ( member_a @ X @ A4 )
@ ^ [X: a] : ( member_a @ X @ B4 ) ) ) ) ) ).
% sup_set_def
thf(fact_179_sup__set__def,axiom,
( sup_su4083067149120280889_a_nat
= ( ^ [A4: set_li6526943997496501093_a_nat,B4: set_li6526943997496501093_a_nat] :
( collec7555443234367654128_a_nat
@ ( sup_su1334248866174809316_nat_o
@ ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ A4 )
@ ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ B4 ) ) ) ) ) ).
% sup_set_def
thf(fact_180_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_181_Un__def,axiom,
( sup_su6804446743777130803_a_nat
= ( ^ [A4: set_Sum_sum_a_nat,B4: set_Sum_sum_a_nat] :
( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ A4 )
| ( member_Sum_sum_a_nat @ X @ B4 ) ) ) ) ) ).
% Un_def
thf(fact_182_Un__def,axiom,
( sup_sup_set_a
= ( ^ [A4: set_a,B4: set_a] :
( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A4 )
| ( member_a @ X @ B4 ) ) ) ) ) ).
% Un_def
thf(fact_183_Un__def,axiom,
( sup_su4083067149120280889_a_nat
= ( ^ [A4: set_li6526943997496501093_a_nat,B4: set_li6526943997496501093_a_nat] :
( collec7555443234367654128_a_nat
@ ^ [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A4 )
| ( member408289922725080238_a_nat @ X @ B4 ) ) ) ) ) ).
% Un_def
thf(fact_184_ad__agr__list__set,axiom,
! [X4: set_a,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Y2: a] :
( ( ad_agr_list_a_nat @ X4 @ Xs @ Ys )
=> ( ( member_a @ Y2 @ X4 )
=> ( ( member_Sum_sum_a_nat @ ( sum_Inl_a_nat @ Y2 ) @ ( set_Sum_sum_a_nat2 @ Ys ) )
=> ( member_Sum_sum_a_nat @ ( sum_Inl_a_nat @ Y2 ) @ ( set_Sum_sum_a_nat2 @ Xs ) ) ) ) ) ).
% ad_agr_list_set
thf(fact_185_map__eq__append__conv,axiom,
! [F: nat > nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( append_nat @ Ys @ Zs ) )
= ( ? [Us: list_nat,Vs: list_nat] :
( ( Xs
= ( append_nat @ Us @ Vs ) )
& ( Ys
= ( map_nat_nat @ F @ Us ) )
& ( Zs
= ( map_nat_nat @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_186_map__eq__append__conv,axiom,
! [F: nat > sum_sum_a_nat,Xs: list_nat,Ys: list_Sum_sum_a_nat,Zs: list_Sum_sum_a_nat] :
( ( ( map_na823391071729141993_a_nat @ F @ Xs )
= ( append_Sum_sum_a_nat @ Ys @ Zs ) )
= ( ? [Us: list_nat,Vs: list_nat] :
( ( Xs
= ( append_nat @ Us @ Vs ) )
& ( Ys
= ( map_na823391071729141993_a_nat @ F @ Us ) )
& ( Zs
= ( map_na823391071729141993_a_nat @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_187_map__eq__append__conv,axiom,
! [F: produc5454855657571582244_a_nat > sum_sum_a_nat,Xs: list_P5056861408695629236_a_nat,Ys: list_Sum_sum_a_nat,Zs: list_Sum_sum_a_nat] :
( ( ( map_Pr2515564165894599445_a_nat @ F @ Xs )
= ( append_Sum_sum_a_nat @ Ys @ Zs ) )
= ( ? [Us: list_P5056861408695629236_a_nat,Vs: list_P5056861408695629236_a_nat] :
( ( Xs
= ( append338925788367110473_a_nat @ Us @ Vs ) )
& ( Ys
= ( map_Pr2515564165894599445_a_nat @ F @ Us ) )
& ( Zs
= ( map_Pr2515564165894599445_a_nat @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_188_map__eq__append__conv,axiom,
! [F: produc5454855657571582244_a_nat > nat,Xs: list_P5056861408695629236_a_nat,Ys: list_nat,Zs: list_nat] :
( ( ( map_Pr2058079554187054650at_nat @ F @ Xs )
= ( append_nat @ Ys @ Zs ) )
= ( ? [Us: list_P5056861408695629236_a_nat,Vs: list_P5056861408695629236_a_nat] :
( ( Xs
= ( append338925788367110473_a_nat @ Us @ Vs ) )
& ( Ys
= ( map_Pr2058079554187054650at_nat @ F @ Us ) )
& ( Zs
= ( map_Pr2058079554187054650at_nat @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_189_append__eq__map__conv,axiom,
! [Ys: list_nat,Zs: list_nat,F: nat > nat,Xs: list_nat] :
( ( ( append_nat @ Ys @ Zs )
= ( map_nat_nat @ F @ Xs ) )
= ( ? [Us: list_nat,Vs: list_nat] :
( ( Xs
= ( append_nat @ Us @ Vs ) )
& ( Ys
= ( map_nat_nat @ F @ Us ) )
& ( Zs
= ( map_nat_nat @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_190_append__eq__map__conv,axiom,
! [Ys: list_Sum_sum_a_nat,Zs: list_Sum_sum_a_nat,F: nat > sum_sum_a_nat,Xs: list_nat] :
( ( ( append_Sum_sum_a_nat @ Ys @ Zs )
= ( map_na823391071729141993_a_nat @ F @ Xs ) )
= ( ? [Us: list_nat,Vs: list_nat] :
( ( Xs
= ( append_nat @ Us @ Vs ) )
& ( Ys
= ( map_na823391071729141993_a_nat @ F @ Us ) )
& ( Zs
= ( map_na823391071729141993_a_nat @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_191_append__eq__map__conv,axiom,
! [Ys: list_Sum_sum_a_nat,Zs: list_Sum_sum_a_nat,F: produc5454855657571582244_a_nat > sum_sum_a_nat,Xs: list_P5056861408695629236_a_nat] :
( ( ( append_Sum_sum_a_nat @ Ys @ Zs )
= ( map_Pr2515564165894599445_a_nat @ F @ Xs ) )
= ( ? [Us: list_P5056861408695629236_a_nat,Vs: list_P5056861408695629236_a_nat] :
( ( Xs
= ( append338925788367110473_a_nat @ Us @ Vs ) )
& ( Ys
= ( map_Pr2515564165894599445_a_nat @ F @ Us ) )
& ( Zs
= ( map_Pr2515564165894599445_a_nat @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_192_append__eq__map__conv,axiom,
! [Ys: list_nat,Zs: list_nat,F: produc5454855657571582244_a_nat > nat,Xs: list_P5056861408695629236_a_nat] :
( ( ( append_nat @ Ys @ Zs )
= ( map_Pr2058079554187054650at_nat @ F @ Xs ) )
= ( ? [Us: list_P5056861408695629236_a_nat,Vs: list_P5056861408695629236_a_nat] :
( ( Xs
= ( append338925788367110473_a_nat @ Us @ Vs ) )
& ( Ys
= ( map_Pr2058079554187054650at_nat @ F @ Us ) )
& ( Zs
= ( map_Pr2058079554187054650at_nat @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_193_ad__agr__list__comm,axiom,
! [X4: set_a,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ad_agr_list_a_nat @ X4 @ Xs @ Ys )
=> ( ad_agr_list_a_nat @ X4 @ Ys @ Xs ) ) ).
% ad_agr_list_comm
thf(fact_194_ad__agr__list__refl,axiom,
! [X4: set_a,Xs: list_Sum_sum_a_nat] : ( ad_agr_list_a_nat @ X4 @ Xs @ Xs ) ).
% ad_agr_list_refl
thf(fact_195_ad__agr__list__trans,axiom,
! [X4: set_a,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Zs: list_Sum_sum_a_nat] :
( ( ad_agr_list_a_nat @ X4 @ Xs @ Ys )
=> ( ( ad_agr_list_a_nat @ X4 @ Ys @ Zs )
=> ( ad_agr_list_a_nat @ X4 @ Xs @ Zs ) ) ) ).
% ad_agr_list_trans
thf(fact_196_Inl__inject,axiom,
! [X2: a,Y2: a] :
( ( ( sum_Inl_a_nat @ X2 )
= ( sum_Inl_a_nat @ Y2 ) )
=> ( X2 = Y2 ) ) ).
% Inl_inject
thf(fact_197_list_Omap__cong,axiom,
! [X2: list_P5056861408695629236_a_nat,Ya: list_P5056861408695629236_a_nat,F: produc5454855657571582244_a_nat > sum_sum_a_nat,G: produc5454855657571582244_a_nat > sum_sum_a_nat] :
( ( X2 = Ya )
=> ( ! [Z2: produc5454855657571582244_a_nat] :
( ( member3650934779329710139_a_nat @ Z2 @ ( set_Pr2196415342531872975_a_nat @ Ya ) )
=> ( ( F @ Z2 )
= ( G @ Z2 ) ) )
=> ( ( map_Pr2515564165894599445_a_nat @ F @ X2 )
= ( map_Pr2515564165894599445_a_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_198_list_Omap__cong,axiom,
! [X2: list_P5056861408695629236_a_nat,Ya: list_P5056861408695629236_a_nat,F: produc5454855657571582244_a_nat > nat,G: produc5454855657571582244_a_nat > nat] :
( ( X2 = Ya )
=> ( ! [Z2: produc5454855657571582244_a_nat] :
( ( member3650934779329710139_a_nat @ Z2 @ ( set_Pr2196415342531872975_a_nat @ Ya ) )
=> ( ( F @ Z2 )
= ( G @ Z2 ) ) )
=> ( ( map_Pr2058079554187054650at_nat @ F @ X2 )
= ( map_Pr2058079554187054650at_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_199_list_Omap__cong,axiom,
! [X2: list_nat,Ya: list_nat,F: nat > sum_sum_a_nat,G: nat > sum_sum_a_nat] :
( ( X2 = Ya )
=> ( ! [Z2: nat] :
( ( member_nat @ Z2 @ ( set_nat2 @ Ya ) )
=> ( ( F @ Z2 )
= ( G @ Z2 ) ) )
=> ( ( map_na823391071729141993_a_nat @ F @ X2 )
= ( map_na823391071729141993_a_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_200_list_Omap__cong0,axiom,
! [X2: list_P5056861408695629236_a_nat,F: produc5454855657571582244_a_nat > sum_sum_a_nat,G: produc5454855657571582244_a_nat > sum_sum_a_nat] :
( ! [Z2: produc5454855657571582244_a_nat] :
( ( member3650934779329710139_a_nat @ Z2 @ ( set_Pr2196415342531872975_a_nat @ X2 ) )
=> ( ( F @ Z2 )
= ( G @ Z2 ) ) )
=> ( ( map_Pr2515564165894599445_a_nat @ F @ X2 )
= ( map_Pr2515564165894599445_a_nat @ G @ X2 ) ) ) ).
% list.map_cong0
thf(fact_201_list_Omap__cong0,axiom,
! [X2: list_P5056861408695629236_a_nat,F: produc5454855657571582244_a_nat > nat,G: produc5454855657571582244_a_nat > nat] :
( ! [Z2: produc5454855657571582244_a_nat] :
( ( member3650934779329710139_a_nat @ Z2 @ ( set_Pr2196415342531872975_a_nat @ X2 ) )
=> ( ( F @ Z2 )
= ( G @ Z2 ) ) )
=> ( ( map_Pr2058079554187054650at_nat @ F @ X2 )
= ( map_Pr2058079554187054650at_nat @ G @ X2 ) ) ) ).
% list.map_cong0
thf(fact_202_list_Omap__cong0,axiom,
! [X2: list_nat,F: nat > sum_sum_a_nat,G: nat > sum_sum_a_nat] :
( ! [Z2: nat] :
( ( member_nat @ Z2 @ ( set_nat2 @ X2 ) )
=> ( ( F @ Z2 )
= ( G @ Z2 ) ) )
=> ( ( map_na823391071729141993_a_nat @ F @ X2 )
= ( map_na823391071729141993_a_nat @ G @ X2 ) ) ) ).
% list.map_cong0
thf(fact_203_list_Oinj__map__strong,axiom,
! [X2: list_P5056861408695629236_a_nat,Xa: list_P5056861408695629236_a_nat,F: produc5454855657571582244_a_nat > sum_sum_a_nat,Fa: produc5454855657571582244_a_nat > sum_sum_a_nat] :
( ! [Z2: produc5454855657571582244_a_nat,Za: produc5454855657571582244_a_nat] :
( ( member3650934779329710139_a_nat @ Z2 @ ( set_Pr2196415342531872975_a_nat @ X2 ) )
=> ( ( member3650934779329710139_a_nat @ Za @ ( set_Pr2196415342531872975_a_nat @ Xa ) )
=> ( ( ( F @ Z2 )
= ( Fa @ Za ) )
=> ( Z2 = Za ) ) ) )
=> ( ( ( map_Pr2515564165894599445_a_nat @ F @ X2 )
= ( map_Pr2515564165894599445_a_nat @ Fa @ Xa ) )
=> ( X2 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_204_list_Oinj__map__strong,axiom,
! [X2: list_P5056861408695629236_a_nat,Xa: list_P5056861408695629236_a_nat,F: produc5454855657571582244_a_nat > nat,Fa: produc5454855657571582244_a_nat > nat] :
( ! [Z2: produc5454855657571582244_a_nat,Za: produc5454855657571582244_a_nat] :
( ( member3650934779329710139_a_nat @ Z2 @ ( set_Pr2196415342531872975_a_nat @ X2 ) )
=> ( ( member3650934779329710139_a_nat @ Za @ ( set_Pr2196415342531872975_a_nat @ Xa ) )
=> ( ( ( F @ Z2 )
= ( Fa @ Za ) )
=> ( Z2 = Za ) ) ) )
=> ( ( ( map_Pr2058079554187054650at_nat @ F @ X2 )
= ( map_Pr2058079554187054650at_nat @ Fa @ Xa ) )
=> ( X2 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_205_list_Oinj__map__strong,axiom,
! [X2: list_nat,Xa: list_nat,F: nat > sum_sum_a_nat,Fa: nat > sum_sum_a_nat] :
( ! [Z2: nat,Za: nat] :
( ( member_nat @ Z2 @ ( set_nat2 @ X2 ) )
=> ( ( member_nat @ Za @ ( set_nat2 @ Xa ) )
=> ( ( ( F @ Z2 )
= ( Fa @ Za ) )
=> ( Z2 = Za ) ) ) )
=> ( ( ( map_na823391071729141993_a_nat @ F @ X2 )
= ( map_na823391071729141993_a_nat @ Fa @ Xa ) )
=> ( X2 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_206_set__union,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( set_nat2 @ ( union_nat @ Xs @ Ys ) )
= ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) ) ) ).
% set_union
thf(fact_207_set__union,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( set_Sum_sum_a_nat2 @ ( union_Sum_sum_a_nat @ Xs @ Ys ) )
= ( sup_su6804446743777130803_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) @ ( set_Sum_sum_a_nat2 @ Ys ) ) ) ).
% set_union
thf(fact_208_set__union,axiom,
! [Xs: list_a,Ys: list_a] :
( ( set_a2 @ ( union_a @ Xs @ Ys ) )
= ( sup_sup_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys ) ) ) ).
% set_union
thf(fact_209_set__union,axiom,
! [Xs: list_l4703314356710769291_a_nat,Ys: list_l4703314356710769291_a_nat] :
( ( set_li2392974972034027290_a_nat @ ( union_2406054330973375359_a_nat @ Xs @ Ys ) )
= ( sup_su4083067149120280889_a_nat @ ( set_li2392974972034027290_a_nat @ Xs ) @ ( set_li2392974972034027290_a_nat @ Ys ) ) ) ).
% set_union
thf(fact_210_sort__append__swap2,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( linord738340561235409698at_nat
@ ^ [X: nat] : X
@ ( append_nat @ Xs @ ( append_nat @ Ys @ Zs ) ) )
= ( linord738340561235409698at_nat
@ ^ [X: nat] : X
@ ( append_nat @ Ys @ ( append_nat @ Xs @ Zs ) ) ) ) ).
% sort_append_swap2
thf(fact_211_sort__append__swap,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( linord738340561235409698at_nat
@ ^ [X: nat] : X
@ ( append_nat @ Xs @ Ys ) )
= ( linord738340561235409698at_nat
@ ^ [X: nat] : X
@ ( append_nat @ Ys @ Xs ) ) ) ).
% sort_append_swap
thf(fact_212_Inl__sub__AD_I2_J,axiom,
! [Y2: a] :
( ( member_Sum_sum_a_nat @ ( sum_Inl_a_nat @ Y2 ) @ ( sup_su6804446743777130803_a_nat @ ( sup_su6804446743777130803_a_nat @ ( set_Sum_sum_a_nat2 @ ys ) @ ( set_Sum_sum_a_nat2 @ ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat @ cxs ) ) ) @ ( set_Sum_sum_a_nat2 @ xs_phi ) ) )
=> ( member_a @ Y2 @ ad ) ) ).
% Inl_sub_AD(2)
thf(fact_213_Inl__sub__AD_I1_J,axiom,
! [X2: a] :
( ( member_Sum_sum_a_nat @ ( sum_Inl_a_nat @ X2 ) @ ( sup_su6804446743777130803_a_nat @ ( sup_su6804446743777130803_a_nat @ ( set_Sum_sum_a_nat2 @ xs ) @ ( set_Sum_sum_a_nat2 @ ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat @ cys ) ) ) @ ( set_Sum_sum_a_nat2 @ ys_psi ) ) )
=> ( member_a @ X2 @ ad ) ) ).
% Inl_sub_AD(1)
thf(fact_214_set__ns_I2_J,axiom,
( ( set_nat2 @ ns_psi2 )
= ( sup_sup_set_nat @ ( set_nat2 @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ cxs ) ) @ ( set_nat2 @ nxs ) ) ) ).
% set_ns(2)
thf(fact_215_set__ns_I1_J,axiom,
( ( set_nat2 @ ns_phi2 )
= ( sup_sup_set_nat @ ( set_nat2 @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ cys ) ) @ ( set_nat2 @ nys ) ) ) ).
% set_ns(1)
thf(fact_216_ad__agr__list__link,axiom,
! [Ns: list_nat,AD: set_a,Sigma: nat > sum_sum_a_nat,Tau: nat > sum_sum_a_nat] :
( ( ad_agr_sets_a_nat @ ( set_nat2 @ Ns ) @ ( set_nat2 @ Ns ) @ AD @ Sigma @ Tau )
= ( ad_agr_list_a_nat @ AD @ ( map_na823391071729141993_a_nat @ Sigma @ Ns ) @ ( map_na823391071729141993_a_nat @ Tau @ Ns ) ) ) ).
% ad_agr_list_link
thf(fact_217_ad__agr__list__fo__nmlzd__isl,axiom,
! [X4: set_a,F: list_Sum_sum_a_nat > sum_sum_a_nat,Xs: list_l4703314356710769291_a_nat,G: list_Sum_sum_a_nat > sum_sum_a_nat,X2: list_Sum_sum_a_nat] :
( ( ad_agr_list_a_nat @ X4 @ ( map_li3106614296470775678_a_nat @ F @ Xs ) @ ( map_li3106614296470775678_a_nat @ G @ Xs ) )
=> ( ( fo_nmlzd_a @ X4 @ ( map_li3106614296470775678_a_nat @ F @ Xs ) )
=> ( ( member408289922725080238_a_nat @ X2 @ ( set_li2392974972034027290_a_nat @ Xs ) )
=> ( ( sum_isl_a_nat @ ( F @ X2 ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) ) ) ) ) ).
% ad_agr_list_fo_nmlzd_isl
thf(fact_218_ad__agr__list__fo__nmlzd__isl,axiom,
! [X4: set_a,F: a > sum_sum_a_nat,Xs: list_a,G: a > sum_sum_a_nat,X2: a] :
( ( ad_agr_list_a_nat @ X4 @ ( map_a_Sum_sum_a_nat @ F @ Xs ) @ ( map_a_Sum_sum_a_nat @ G @ Xs ) )
=> ( ( fo_nmlzd_a @ X4 @ ( map_a_Sum_sum_a_nat @ F @ Xs ) )
=> ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( ( sum_isl_a_nat @ ( F @ X2 ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) ) ) ) ) ).
% ad_agr_list_fo_nmlzd_isl
thf(fact_219_ad__agr__list__fo__nmlzd__isl,axiom,
! [X4: set_a,F: produc5454855657571582244_a_nat > sum_sum_a_nat,Xs: list_P5056861408695629236_a_nat,G: produc5454855657571582244_a_nat > sum_sum_a_nat,X2: produc5454855657571582244_a_nat] :
( ( ad_agr_list_a_nat @ X4 @ ( map_Pr2515564165894599445_a_nat @ F @ Xs ) @ ( map_Pr2515564165894599445_a_nat @ G @ Xs ) )
=> ( ( fo_nmlzd_a @ X4 @ ( map_Pr2515564165894599445_a_nat @ F @ Xs ) )
=> ( ( member3650934779329710139_a_nat @ X2 @ ( set_Pr2196415342531872975_a_nat @ Xs ) )
=> ( ( sum_isl_a_nat @ ( F @ X2 ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) ) ) ) ) ).
% ad_agr_list_fo_nmlzd_isl
thf(fact_220_ad__agr__list__fo__nmlzd__isl,axiom,
! [X4: set_a,F: nat > sum_sum_a_nat,Xs: list_nat,G: nat > sum_sum_a_nat,X2: nat] :
( ( ad_agr_list_a_nat @ X4 @ ( map_na823391071729141993_a_nat @ F @ Xs ) @ ( map_na823391071729141993_a_nat @ G @ Xs ) )
=> ( ( fo_nmlzd_a @ X4 @ ( map_na823391071729141993_a_nat @ F @ Xs ) )
=> ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( sum_isl_a_nat @ ( F @ X2 ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) ) ) ) ) ).
% ad_agr_list_fo_nmlzd_isl
thf(fact_221_ad__agr__list__fo__nmlzd__isl,axiom,
! [X4: set_a,F: sum_sum_a_nat > sum_sum_a_nat,Xs: list_Sum_sum_a_nat,G: sum_sum_a_nat > sum_sum_a_nat,X2: sum_sum_a_nat] :
( ( ad_agr_list_a_nat @ X4 @ ( map_Su2790769393171190532_a_nat @ F @ Xs ) @ ( map_Su2790769393171190532_a_nat @ G @ Xs ) )
=> ( ( fo_nmlzd_a @ X4 @ ( map_Su2790769393171190532_a_nat @ F @ Xs ) )
=> ( ( member_Sum_sum_a_nat @ X2 @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ( ( sum_isl_a_nat @ ( F @ X2 ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) ) ) ) ) ).
% ad_agr_list_fo_nmlzd_isl
thf(fact_222_stable__sort__key__sort__key,axiom,
linord227665693835759911at_nat @ linord738340561235409698at_nat ).
% stable_sort_key_sort_key
thf(fact_223_assms_I14_J,axiom,
( ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat @ cys )
= ( map_na823391071729141993_a_nat @ sigma_xs @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ cys ) ) ) ).
% assms(14)
thf(fact_224_assms_I17_J,axiom,
( ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat @ cxs )
= ( map_na823391071729141993_a_nat @ sigma_ys @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ cxs ) ) ) ).
% assms(17)
thf(fact_225_filter__filter,axiom,
! [P: nat > $o,Q: nat > $o,Xs: list_nat] :
( ( filter_nat @ P @ ( filter_nat @ Q @ Xs ) )
= ( filter_nat
@ ^ [X: nat] :
( ( Q @ X )
& ( P @ X ) )
@ Xs ) ) ).
% filter_filter
thf(fact_226_filter__filter,axiom,
! [P: produc5454855657571582244_a_nat > $o,Q: produc5454855657571582244_a_nat > $o,Xs: list_P5056861408695629236_a_nat] :
( ( filter1580576373184442087_a_nat @ P @ ( filter1580576373184442087_a_nat @ Q @ Xs ) )
= ( filter1580576373184442087_a_nat
@ ^ [X: produc5454855657571582244_a_nat] :
( ( Q @ X )
& ( P @ X ) )
@ Xs ) ) ).
% filter_filter
thf(fact_227_filter__True,axiom,
! [Xs: list_P5056861408695629236_a_nat,P: produc5454855657571582244_a_nat > $o] :
( ! [X3: produc5454855657571582244_a_nat] :
( ( member3650934779329710139_a_nat @ X3 @ ( set_Pr2196415342531872975_a_nat @ Xs ) )
=> ( P @ X3 ) )
=> ( ( filter1580576373184442087_a_nat @ P @ Xs )
= Xs ) ) ).
% filter_True
thf(fact_228_filter__True,axiom,
! [Xs: list_nat,P: nat > $o] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( P @ X3 ) )
=> ( ( filter_nat @ P @ Xs )
= Xs ) ) ).
% filter_True
thf(fact_229_filter__True,axiom,
! [Xs: list_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ! [X3: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X3 @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ( P @ X3 ) )
=> ( ( filter_Sum_sum_a_nat @ P @ Xs )
= Xs ) ) ).
% filter_True
thf(fact_230_filter__append,axiom,
! [P: nat > $o,Xs: list_nat,Ys: list_nat] :
( ( filter_nat @ P @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( filter_nat @ P @ Xs ) @ ( filter_nat @ P @ Ys ) ) ) ).
% filter_append
thf(fact_231_filter__append,axiom,
! [P: produc5454855657571582244_a_nat > $o,Xs: list_P5056861408695629236_a_nat,Ys: list_P5056861408695629236_a_nat] :
( ( filter1580576373184442087_a_nat @ P @ ( append338925788367110473_a_nat @ Xs @ Ys ) )
= ( append338925788367110473_a_nat @ ( filter1580576373184442087_a_nat @ P @ Xs ) @ ( filter1580576373184442087_a_nat @ P @ Ys ) ) ) ).
% filter_append
thf(fact_232_set__filter,axiom,
! [P: list_Sum_sum_a_nat > $o,Xs: list_l4703314356710769291_a_nat] :
( ( set_li2392974972034027290_a_nat @ ( filter5373755100216644354_a_nat @ P @ Xs ) )
= ( collec7555443234367654128_a_nat
@ ^ [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ ( set_li2392974972034027290_a_nat @ Xs ) )
& ( P @ X ) ) ) ) ).
% set_filter
thf(fact_233_set__filter,axiom,
! [P: a > $o,Xs: list_a] :
( ( set_a2 @ ( filter_a @ P @ Xs ) )
= ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
& ( P @ X ) ) ) ) ).
% set_filter
thf(fact_234_set__filter,axiom,
! [P: produc5454855657571582244_a_nat > $o,Xs: list_P5056861408695629236_a_nat] :
( ( set_Pr2196415342531872975_a_nat @ ( filter1580576373184442087_a_nat @ P @ Xs ) )
= ( collec2196941500952594553_a_nat
@ ^ [X: produc5454855657571582244_a_nat] :
( ( member3650934779329710139_a_nat @ X @ ( set_Pr2196415342531872975_a_nat @ Xs ) )
& ( P @ X ) ) ) ) ).
% set_filter
thf(fact_235_set__filter,axiom,
! [P: nat > $o,Xs: list_nat] :
( ( set_nat2 @ ( filter_nat @ P @ Xs ) )
= ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
& ( P @ X ) ) ) ) ).
% set_filter
thf(fact_236_set__filter,axiom,
! [P: sum_sum_a_nat > $o,Xs: list_Sum_sum_a_nat] :
( ( set_Sum_sum_a_nat2 @ ( filter_Sum_sum_a_nat @ P @ Xs ) )
= ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Xs ) )
& ( P @ X ) ) ) ) ).
% set_filter
thf(fact_237_pair__list__eqI,axiom,
! [Xs: list_P5056861408695629236_a_nat,Ys: list_P5056861408695629236_a_nat] :
( ( ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Xs )
= ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Ys ) )
=> ( ( ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat @ Xs )
= ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat @ Ys ) )
=> ( Xs = Ys ) ) ) ).
% pair_list_eqI
thf(fact_238_filter__id__conv,axiom,
! [P: produc5454855657571582244_a_nat > $o,Xs: list_P5056861408695629236_a_nat] :
( ( ( filter1580576373184442087_a_nat @ P @ Xs )
= Xs )
= ( ! [X: produc5454855657571582244_a_nat] :
( ( member3650934779329710139_a_nat @ X @ ( set_Pr2196415342531872975_a_nat @ Xs ) )
=> ( P @ X ) ) ) ) ).
% filter_id_conv
thf(fact_239_filter__id__conv,axiom,
! [P: nat > $o,Xs: list_nat] :
( ( ( filter_nat @ P @ Xs )
= Xs )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( P @ X ) ) ) ) ).
% filter_id_conv
thf(fact_240_filter__id__conv,axiom,
! [P: sum_sum_a_nat > $o,Xs: list_Sum_sum_a_nat] :
( ( ( filter_Sum_sum_a_nat @ P @ Xs )
= Xs )
= ( ! [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ( P @ X ) ) ) ) ).
% filter_id_conv
thf(fact_241_filter__cong,axiom,
! [Xs: list_l4703314356710769291_a_nat,Ys: list_l4703314356710769291_a_nat,P: list_Sum_sum_a_nat > $o,Q: list_Sum_sum_a_nat > $o] :
( ( Xs = Ys )
=> ( ! [X3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ ( set_li2392974972034027290_a_nat @ Ys ) )
=> ( ( P @ X3 )
= ( Q @ X3 ) ) )
=> ( ( filter5373755100216644354_a_nat @ P @ Xs )
= ( filter5373755100216644354_a_nat @ Q @ Ys ) ) ) ) ).
% filter_cong
thf(fact_242_filter__cong,axiom,
! [Xs: list_a,Ys: list_a,P: a > $o,Q: a > $o] :
( ( Xs = Ys )
=> ( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Ys ) )
=> ( ( P @ X3 )
= ( Q @ X3 ) ) )
=> ( ( filter_a @ P @ Xs )
= ( filter_a @ Q @ Ys ) ) ) ) ).
% filter_cong
thf(fact_243_filter__cong,axiom,
! [Xs: list_P5056861408695629236_a_nat,Ys: list_P5056861408695629236_a_nat,P: produc5454855657571582244_a_nat > $o,Q: produc5454855657571582244_a_nat > $o] :
( ( Xs = Ys )
=> ( ! [X3: produc5454855657571582244_a_nat] :
( ( member3650934779329710139_a_nat @ X3 @ ( set_Pr2196415342531872975_a_nat @ Ys ) )
=> ( ( P @ X3 )
= ( Q @ X3 ) ) )
=> ( ( filter1580576373184442087_a_nat @ P @ Xs )
= ( filter1580576373184442087_a_nat @ Q @ Ys ) ) ) ) ).
% filter_cong
thf(fact_244_filter__cong,axiom,
! [Xs: list_nat,Ys: list_nat,P: nat > $o,Q: nat > $o] :
( ( Xs = Ys )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Ys ) )
=> ( ( P @ X3 )
= ( Q @ X3 ) ) )
=> ( ( filter_nat @ P @ Xs )
= ( filter_nat @ Q @ Ys ) ) ) ) ).
% filter_cong
thf(fact_245_filter__cong,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,P: sum_sum_a_nat > $o,Q: sum_sum_a_nat > $o] :
( ( Xs = Ys )
=> ( ! [X3: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X3 @ ( set_Sum_sum_a_nat2 @ Ys ) )
=> ( ( P @ X3 )
= ( Q @ X3 ) ) )
=> ( ( filter_Sum_sum_a_nat @ P @ Xs )
= ( filter_Sum_sum_a_nat @ Q @ Ys ) ) ) ) ).
% filter_cong
thf(fact_246_filter__sort,axiom,
! [P: nat > $o,F: nat > nat,Xs: list_nat] :
( ( filter_nat @ P @ ( linord738340561235409698at_nat @ F @ Xs ) )
= ( linord738340561235409698at_nat @ F @ ( filter_nat @ P @ Xs ) ) ) ).
% filter_sort
thf(fact_247_sort__key__stable,axiom,
! [F: nat > nat,K: nat,Xs: list_nat] :
( ( filter_nat
@ ^ [Y: nat] :
( ( F @ Y )
= K )
@ ( linord738340561235409698at_nat @ F @ Xs ) )
= ( filter_nat
@ ^ [Y: nat] :
( ( F @ Y )
= K )
@ Xs ) ) ).
% sort_key_stable
thf(fact_248_ad__agr__list__fo__nmlzd,axiom,
! [AD: set_a,Vs2: list_Sum_sum_a_nat,Vs3: list_Sum_sum_a_nat] :
( ( ad_agr_list_a_nat @ AD @ Vs2 @ Vs3 )
=> ( ( fo_nmlzd_a @ AD @ Vs2 )
=> ( ( fo_nmlzd_a @ AD @ Vs3 )
=> ( Vs2 = Vs3 ) ) ) ) ).
% ad_agr_list_fo_nmlzd
thf(fact_249_assms_I22_J,axiom,
ord_less_eq_set_a @ ( vimage4607795094749595324_a_nat @ sum_Inl_a_nat @ ( sup_su6804446743777130803_a_nat @ ( set_Sum_sum_a_nat2 @ ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat @ cxs ) ) @ ( set_Sum_sum_a_nat2 @ xs_phi ) ) ) @ ad ).
% assms(22)
thf(fact_250_assms_I23_J,axiom,
ord_less_eq_set_a @ ( vimage4607795094749595324_a_nat @ sum_Inl_a_nat @ ( sup_su6804446743777130803_a_nat @ ( set_Sum_sum_a_nat2 @ ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat @ cys ) ) @ ( set_Sum_sum_a_nat2 @ ys_psi ) ) ) @ ad ).
% assms(23)
thf(fact_251_set___092_060sigma_062__ns_I2_J,axiom,
ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ ( image_7293268710728258664_a_nat @ sigma_ys @ ( set_nat2 @ ns_phi2 ) ) @ ( image_7293268710728258664_a_nat @ sigma_ys @ ( set_nat2 @ ns_psi2 ) ) ) @ ( sup_su6804446743777130803_a_nat @ ( sup_su6804446743777130803_a_nat @ ( set_Sum_sum_a_nat2 @ ys ) @ ( set_Sum_sum_a_nat2 @ ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat @ cxs ) ) ) @ ( set_Sum_sum_a_nat2 @ xs_phi ) ) ).
% set_\<sigma>_ns(2)
thf(fact_252_set___092_060sigma_062__ns_I1_J,axiom,
ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ ( image_7293268710728258664_a_nat @ sigma_xs @ ( set_nat2 @ ns_psi2 ) ) @ ( image_7293268710728258664_a_nat @ sigma_xs @ ( set_nat2 @ ns_phi2 ) ) ) @ ( sup_su6804446743777130803_a_nat @ ( sup_su6804446743777130803_a_nat @ ( set_Sum_sum_a_nat2 @ xs ) @ ( set_Sum_sum_a_nat2 @ ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat @ cys ) ) ) @ ( set_Sum_sum_a_nat2 @ ys_psi ) ) ).
% set_\<sigma>_ns(1)
thf(fact_253_sup__Un__eq,axiom,
! [R: set_nat,S: set_nat] :
( ( sup_sup_nat_o
@ ^ [X: nat] : ( member_nat @ X @ R )
@ ^ [X: nat] : ( member_nat @ X @ S ) )
= ( ^ [X: nat] : ( member_nat @ X @ ( sup_sup_set_nat @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_254_sup__Un__eq,axiom,
! [R: set_Sum_sum_a_nat,S: set_Sum_sum_a_nat] :
( ( sup_su491480579010597738_nat_o
@ ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ R )
@ ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ S ) )
= ( ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ ( sup_su6804446743777130803_a_nat @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_255_sup__Un__eq,axiom,
! [R: set_a,S: set_a] :
( ( sup_sup_a_o
@ ^ [X: a] : ( member_a @ X @ R )
@ ^ [X: a] : ( member_a @ X @ S ) )
= ( ^ [X: a] : ( member_a @ X @ ( sup_sup_set_a @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_256_sup__Un__eq,axiom,
! [R: set_li6526943997496501093_a_nat,S: set_li6526943997496501093_a_nat] :
( ( sup_su1334248866174809316_nat_o
@ ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ R )
@ ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ S ) )
= ( ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ ( sup_su4083067149120280889_a_nat @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_257_nxs__def,axiom,
( nxs
= ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat
@ ( filter1580576373184442087_a_nat
@ ( produc8810490045567577941_nat_o
@ ^ [N2: nat,X: sum_sum_a_nat] :
( ~ ( member_nat @ N2 @ ( set_nat2 @ ns_psi ) )
& ~ ( sum_isl_a_nat @ X ) ) )
@ ( zip_na2013496608136855606_a_nat @ ns_phi @ xs ) ) ) ) ).
% nxs_def
thf(fact_258_prod_Oexpand,axiom,
! [Prod: produc5454855657571582244_a_nat,Prod2: produc5454855657571582244_a_nat] :
( ( ( ( produc1510075366785312874_a_nat @ Prod )
= ( produc1510075366785312874_a_nat @ Prod2 ) )
& ( ( produc1280857981117566888_a_nat @ Prod )
= ( produc1280857981117566888_a_nat @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_259_prod__eqI,axiom,
! [P2: produc5454855657571582244_a_nat,Q2: produc5454855657571582244_a_nat] :
( ( ( produc1510075366785312874_a_nat @ P2 )
= ( produc1510075366785312874_a_nat @ Q2 ) )
=> ( ( ( produc1280857981117566888_a_nat @ P2 )
= ( produc1280857981117566888_a_nat @ Q2 ) )
=> ( P2 = Q2 ) ) ) ).
% prod_eqI
thf(fact_260_exE__realizer_H,axiom,
! [P: sum_sum_a_nat > nat > $o,P2: produc5454855657571582244_a_nat] :
( ( P @ ( produc1280857981117566888_a_nat @ P2 ) @ ( produc1510075366785312874_a_nat @ P2 ) )
=> ~ ! [X3: nat,Y3: sum_sum_a_nat] :
~ ( P @ Y3 @ X3 ) ) ).
% exE_realizer'
thf(fact_261_image__eqI,axiom,
! [B2: nat,F: nat > nat,X2: nat,A: set_nat] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A )
=> ( member_nat @ B2 @ ( image_nat_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_262_image__eqI,axiom,
! [B2: a,F: nat > a,X2: nat,A: set_nat] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A )
=> ( member_a @ B2 @ ( image_nat_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_263_image__eqI,axiom,
! [B2: nat,F: a > nat,X2: a,A: set_a] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_a @ X2 @ A )
=> ( member_nat @ B2 @ ( image_a_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_264_image__eqI,axiom,
! [B2: a,F: a > a,X2: a,A: set_a] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_a @ X2 @ A )
=> ( member_a @ B2 @ ( image_a_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_265_image__eqI,axiom,
! [B2: sum_sum_a_nat,F: nat > sum_sum_a_nat,X2: nat,A: set_nat] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A )
=> ( member_Sum_sum_a_nat @ B2 @ ( image_7293268710728258664_a_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_266_image__eqI,axiom,
! [B2: sum_sum_a_nat,F: a > sum_sum_a_nat,X2: a,A: set_a] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_a @ X2 @ A )
=> ( member_Sum_sum_a_nat @ B2 @ ( image_7873763678140191238_a_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_267_image__eqI,axiom,
! [B2: nat,F: sum_sum_a_nat > nat,X2: sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_Sum_sum_a_nat @ X2 @ A )
=> ( member_nat @ B2 @ ( image_2473878607534554506at_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_268_image__eqI,axiom,
! [B2: a,F: sum_sum_a_nat > a,X2: sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_Sum_sum_a_nat @ X2 @ A )
=> ( member_a @ B2 @ ( image_6322530041254294468_nat_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_269_image__eqI,axiom,
! [B2: list_Sum_sum_a_nat,F: nat > list_Sum_sum_a_nat,X2: nat,A: set_nat] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A )
=> ( member408289922725080238_a_nat @ B2 @ ( image_6262589752765146990_a_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_270_image__eqI,axiom,
! [B2: nat,F: list_Sum_sum_a_nat > nat,X2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( B2
= ( F @ X2 ) )
=> ( ( member408289922725080238_a_nat @ X2 @ A )
=> ( member_nat @ B2 @ ( image_2535339886381165584at_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_271_subsetI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_nat @ X3 @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% subsetI
thf(fact_272_subsetI,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ! [X3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ A )
=> ( member408289922725080238_a_nat @ X3 @ B ) )
=> ( ord_le1147066620699065093_a_nat @ A @ B ) ) ).
% subsetI
thf(fact_273_subsetI,axiom,
! [A: set_a,B: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( member_a @ X3 @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% subsetI
thf(fact_274_subsetI,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ! [X3: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X3 @ A )
=> ( member_Sum_sum_a_nat @ X3 @ B ) )
=> ( ord_le1325389633284124927_a_nat @ A @ B ) ) ).
% subsetI
thf(fact_275_subset__antisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_276_subset__antisym,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ B )
=> ( ( ord_le1325389633284124927_a_nat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_277_vimageI,axiom,
! [F: nat > nat,A2: nat,B2: nat,B: set_nat] :
( ( ( F @ A2 )
= B2 )
=> ( ( member_nat @ B2 @ B )
=> ( member_nat @ A2 @ ( vimage_nat_nat @ F @ B ) ) ) ) ).
% vimageI
thf(fact_278_vimageI,axiom,
! [F: a > nat,A2: a,B2: nat,B: set_nat] :
( ( ( F @ A2 )
= B2 )
=> ( ( member_nat @ B2 @ B )
=> ( member_a @ A2 @ ( vimage_a_nat @ F @ B ) ) ) ) ).
% vimageI
thf(fact_279_vimageI,axiom,
! [F: nat > a,A2: nat,B2: a,B: set_a] :
( ( ( F @ A2 )
= B2 )
=> ( ( member_a @ B2 @ B )
=> ( member_nat @ A2 @ ( vimage_nat_a @ F @ B ) ) ) ) ).
% vimageI
thf(fact_280_vimageI,axiom,
! [F: a > a,A2: a,B2: a,B: set_a] :
( ( ( F @ A2 )
= B2 )
=> ( ( member_a @ B2 @ B )
=> ( member_a @ A2 @ ( vimage_a_a @ F @ B ) ) ) ) ).
% vimageI
thf(fact_281_vimageI,axiom,
! [F: sum_sum_a_nat > nat,A2: sum_sum_a_nat,B2: nat,B: set_nat] :
( ( ( F @ A2 )
= B2 )
=> ( ( member_nat @ B2 @ B )
=> ( member_Sum_sum_a_nat @ A2 @ ( vimage7444966428737627988at_nat @ F @ B ) ) ) ) ).
% vimageI
thf(fact_282_vimageI,axiom,
! [F: sum_sum_a_nat > a,A2: sum_sum_a_nat,B2: a,B: set_a] :
( ( ( F @ A2 )
= B2 )
=> ( ( member_a @ B2 @ B )
=> ( member_Sum_sum_a_nat @ A2 @ ( vimage3056561457863698554_nat_a @ F @ B ) ) ) ) ).
% vimageI
thf(fact_283_vimageI,axiom,
! [F: a > sum_sum_a_nat,A2: a,B2: sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ( F @ A2 )
= B2 )
=> ( ( member_Sum_sum_a_nat @ B2 @ B )
=> ( member_a @ A2 @ ( vimage4607795094749595324_a_nat @ F @ B ) ) ) ) ).
% vimageI
thf(fact_284_vimageI,axiom,
! [F: nat > sum_sum_a_nat,A2: nat,B2: sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ( F @ A2 )
= B2 )
=> ( ( member_Sum_sum_a_nat @ B2 @ B )
=> ( member_nat @ A2 @ ( vimage3040984495076556338_a_nat @ F @ B ) ) ) ) ).
% vimageI
thf(fact_285_vimageI,axiom,
! [F: list_Sum_sum_a_nat > nat,A2: list_Sum_sum_a_nat,B2: nat,B: set_nat] :
( ( ( F @ A2 )
= B2 )
=> ( ( member_nat @ B2 @ B )
=> ( member408289922725080238_a_nat @ A2 @ ( vimage4660099214859102426at_nat @ F @ B ) ) ) ) ).
% vimageI
thf(fact_286_vimageI,axiom,
! [F: nat > list_Sum_sum_a_nat,A2: nat,B2: list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ( F @ A2 )
= B2 )
=> ( ( member408289922725080238_a_nat @ B2 @ B )
=> ( member_nat @ A2 @ ( vimage8387349081243083832_a_nat @ F @ B ) ) ) ) ).
% vimageI
thf(fact_287_vimage__eq,axiom,
! [A2: nat,F: nat > nat,B: set_nat] :
( ( member_nat @ A2 @ ( vimage_nat_nat @ F @ B ) )
= ( member_nat @ ( F @ A2 ) @ B ) ) ).
% vimage_eq
thf(fact_288_vimage__eq,axiom,
! [A2: nat,F: nat > a,B: set_a] :
( ( member_nat @ A2 @ ( vimage_nat_a @ F @ B ) )
= ( member_a @ ( F @ A2 ) @ B ) ) ).
% vimage_eq
thf(fact_289_vimage__eq,axiom,
! [A2: a,F: a > nat,B: set_nat] :
( ( member_a @ A2 @ ( vimage_a_nat @ F @ B ) )
= ( member_nat @ ( F @ A2 ) @ B ) ) ).
% vimage_eq
thf(fact_290_vimage__eq,axiom,
! [A2: a,F: a > a,B: set_a] :
( ( member_a @ A2 @ ( vimage_a_a @ F @ B ) )
= ( member_a @ ( F @ A2 ) @ B ) ) ).
% vimage_eq
thf(fact_291_vimage__eq,axiom,
! [A2: sum_sum_a_nat,F: sum_sum_a_nat > nat,B: set_nat] :
( ( member_Sum_sum_a_nat @ A2 @ ( vimage7444966428737627988at_nat @ F @ B ) )
= ( member_nat @ ( F @ A2 ) @ B ) ) ).
% vimage_eq
thf(fact_292_vimage__eq,axiom,
! [A2: sum_sum_a_nat,F: sum_sum_a_nat > a,B: set_a] :
( ( member_Sum_sum_a_nat @ A2 @ ( vimage3056561457863698554_nat_a @ F @ B ) )
= ( member_a @ ( F @ A2 ) @ B ) ) ).
% vimage_eq
thf(fact_293_vimage__eq,axiom,
! [A2: a,F: a > sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( member_a @ A2 @ ( vimage4607795094749595324_a_nat @ F @ B ) )
= ( member_Sum_sum_a_nat @ ( F @ A2 ) @ B ) ) ).
% vimage_eq
thf(fact_294_vimage__eq,axiom,
! [A2: nat,F: nat > sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( member_nat @ A2 @ ( vimage3040984495076556338_a_nat @ F @ B ) )
= ( member_Sum_sum_a_nat @ ( F @ A2 ) @ B ) ) ).
% vimage_eq
thf(fact_295_vimage__eq,axiom,
! [A2: nat,F: nat > list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member_nat @ A2 @ ( vimage8387349081243083832_a_nat @ F @ B ) )
= ( member408289922725080238_a_nat @ ( F @ A2 ) @ B ) ) ).
% vimage_eq
thf(fact_296_vimage__eq,axiom,
! [A2: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > nat,B: set_nat] :
( ( member408289922725080238_a_nat @ A2 @ ( vimage4660099214859102426at_nat @ F @ B ) )
= ( member_nat @ ( F @ A2 ) @ B ) ) ).
% vimage_eq
thf(fact_297_image__ident,axiom,
! [Y4: set_li6526943997496501093_a_nat] :
( ( image_5081948215111134021_a_nat
@ ^ [X: list_Sum_sum_a_nat] : X
@ Y4 )
= Y4 ) ).
% image_ident
thf(fact_298_split__part,axiom,
! [P: $o,Q: nat > sum_sum_a_nat > $o] :
( ( produc8810490045567577941_nat_o
@ ^ [A3: nat,B3: sum_sum_a_nat] :
( P
& ( Q @ A3 @ B3 ) ) )
= ( ^ [Ab: produc5454855657571582244_a_nat] :
( P
& ( produc8810490045567577941_nat_o @ Q @ Ab ) ) ) ) ).
% split_part
thf(fact_299_vimage__Collect__eq,axiom,
! [F: a > sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ( vimage4607795094749595324_a_nat @ F @ ( collec7073057861543223018_a_nat @ P ) )
= ( collect_a
@ ^ [Y: a] : ( P @ ( F @ Y ) ) ) ) ).
% vimage_Collect_eq
thf(fact_300_vimage__Collect__eq,axiom,
! [F: nat > sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ( vimage3040984495076556338_a_nat @ F @ ( collec7073057861543223018_a_nat @ P ) )
= ( collect_nat
@ ^ [Y: nat] : ( P @ ( F @ Y ) ) ) ) ).
% vimage_Collect_eq
thf(fact_301_vimage__Collect__eq,axiom,
! [F: nat > nat,P: nat > $o] :
( ( vimage_nat_nat @ F @ ( collect_nat @ P ) )
= ( collect_nat
@ ^ [Y: nat] : ( P @ ( F @ Y ) ) ) ) ).
% vimage_Collect_eq
thf(fact_302_le__sup__iff,axiom,
! [X2: set_nat,Y2: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X2 @ Y2 ) @ Z )
= ( ( ord_less_eq_set_nat @ X2 @ Z )
& ( ord_less_eq_set_nat @ Y2 @ Z ) ) ) ).
% le_sup_iff
thf(fact_303_le__sup__iff,axiom,
! [X2: set_li6526943997496501093_a_nat,Y2: set_li6526943997496501093_a_nat,Z: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ ( sup_su4083067149120280889_a_nat @ X2 @ Y2 ) @ Z )
= ( ( ord_le1147066620699065093_a_nat @ X2 @ Z )
& ( ord_le1147066620699065093_a_nat @ Y2 @ Z ) ) ) ).
% le_sup_iff
thf(fact_304_le__sup__iff,axiom,
! [X2: set_a,Y2: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X2 @ Y2 ) @ Z )
= ( ( ord_less_eq_set_a @ X2 @ Z )
& ( ord_less_eq_set_a @ Y2 @ Z ) ) ) ).
% le_sup_iff
thf(fact_305_le__sup__iff,axiom,
! [X2: set_Sum_sum_a_nat,Y2: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ X2 @ Y2 ) @ Z )
= ( ( ord_le1325389633284124927_a_nat @ X2 @ Z )
& ( ord_le1325389633284124927_a_nat @ Y2 @ Z ) ) ) ).
% le_sup_iff
thf(fact_306_le__sup__iff,axiom,
! [X2: nat,Y2: nat,Z: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X2 @ Y2 ) @ Z )
= ( ( ord_less_eq_nat @ X2 @ Z )
& ( ord_less_eq_nat @ Y2 @ Z ) ) ) ).
% le_sup_iff
thf(fact_307_sup_Obounded__iff,axiom,
! [B2: set_nat,C: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C ) @ A2 )
= ( ( ord_less_eq_set_nat @ B2 @ A2 )
& ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_308_sup_Obounded__iff,axiom,
! [B2: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ ( sup_su4083067149120280889_a_nat @ B2 @ C ) @ A2 )
= ( ( ord_le1147066620699065093_a_nat @ B2 @ A2 )
& ( ord_le1147066620699065093_a_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_309_sup_Obounded__iff,axiom,
! [B2: set_a,C: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C ) @ A2 )
= ( ( ord_less_eq_set_a @ B2 @ A2 )
& ( ord_less_eq_set_a @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_310_sup_Obounded__iff,axiom,
! [B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ B2 @ C ) @ A2 )
= ( ( ord_le1325389633284124927_a_nat @ B2 @ A2 )
& ( ord_le1325389633284124927_a_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_311_sup_Obounded__iff,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_312_Un__subset__iff,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C2 )
= ( ( ord_less_eq_set_nat @ A @ C2 )
& ( ord_less_eq_set_nat @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_313_Un__subset__iff,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,C2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ ( sup_su4083067149120280889_a_nat @ A @ B ) @ C2 )
= ( ( ord_le1147066620699065093_a_nat @ A @ C2 )
& ( ord_le1147066620699065093_a_nat @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_314_Un__subset__iff,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ C2 )
= ( ( ord_less_eq_set_a @ A @ C2 )
& ( ord_less_eq_set_a @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_315_Un__subset__iff,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ A @ B ) @ C2 )
= ( ( ord_le1325389633284124927_a_nat @ A @ C2 )
& ( ord_le1325389633284124927_a_nat @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_316_cys__def,axiom,
( cys
= ( filter1580576373184442087_a_nat
@ ( produc8810490045567577941_nat_o
@ ^ [N2: nat,Y: sum_sum_a_nat] :
( ~ ( member_nat @ N2 @ ( set_nat2 @ ns_phi ) )
& ( sum_isl_a_nat @ Y ) ) )
@ ( zip_na2013496608136855606_a_nat @ ns_psi @ ys ) ) ) ).
% cys_def
thf(fact_317_cxs__def,axiom,
( cxs
= ( filter1580576373184442087_a_nat
@ ( produc8810490045567577941_nat_o
@ ^ [N2: nat,X: sum_sum_a_nat] :
( ~ ( member_nat @ N2 @ ( set_nat2 @ ns_psi ) )
& ( sum_isl_a_nat @ X ) ) )
@ ( zip_na2013496608136855606_a_nat @ ns_phi @ xs ) ) ) ).
% cxs_def
thf(fact_318_vimage__Un,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( vimage_nat_nat @ F @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_sup_set_nat @ ( vimage_nat_nat @ F @ A ) @ ( vimage_nat_nat @ F @ B ) ) ) ).
% vimage_Un
thf(fact_319_vimage__Un,axiom,
! [F: a > nat,A: set_nat,B: set_nat] :
( ( vimage_a_nat @ F @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_sup_set_a @ ( vimage_a_nat @ F @ A ) @ ( vimage_a_nat @ F @ B ) ) ) ).
% vimage_Un
thf(fact_320_vimage__Un,axiom,
! [F: nat > a,A: set_a,B: set_a] :
( ( vimage_nat_a @ F @ ( sup_sup_set_a @ A @ B ) )
= ( sup_sup_set_nat @ ( vimage_nat_a @ F @ A ) @ ( vimage_nat_a @ F @ B ) ) ) ).
% vimage_Un
thf(fact_321_vimage__Un,axiom,
! [F: a > a,A: set_a,B: set_a] :
( ( vimage_a_a @ F @ ( sup_sup_set_a @ A @ B ) )
= ( sup_sup_set_a @ ( vimage_a_a @ F @ A ) @ ( vimage_a_a @ F @ B ) ) ) ).
% vimage_Un
thf(fact_322_vimage__Un,axiom,
! [F: sum_sum_a_nat > nat,A: set_nat,B: set_nat] :
( ( vimage7444966428737627988at_nat @ F @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_su6804446743777130803_a_nat @ ( vimage7444966428737627988at_nat @ F @ A ) @ ( vimage7444966428737627988at_nat @ F @ B ) ) ) ).
% vimage_Un
thf(fact_323_vimage__Un,axiom,
! [F: nat > sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( vimage3040984495076556338_a_nat @ F @ ( sup_su6804446743777130803_a_nat @ A @ B ) )
= ( sup_sup_set_nat @ ( vimage3040984495076556338_a_nat @ F @ A ) @ ( vimage3040984495076556338_a_nat @ F @ B ) ) ) ).
% vimage_Un
thf(fact_324_vimage__Un,axiom,
! [F: a > sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( vimage4607795094749595324_a_nat @ F @ ( sup_su6804446743777130803_a_nat @ A @ B ) )
= ( sup_sup_set_a @ ( vimage4607795094749595324_a_nat @ F @ A ) @ ( vimage4607795094749595324_a_nat @ F @ B ) ) ) ).
% vimage_Un
thf(fact_325_vimage__Un,axiom,
! [F: sum_sum_a_nat > a,A: set_a,B: set_a] :
( ( vimage3056561457863698554_nat_a @ F @ ( sup_sup_set_a @ A @ B ) )
= ( sup_su6804446743777130803_a_nat @ ( vimage3056561457863698554_nat_a @ F @ A ) @ ( vimage3056561457863698554_nat_a @ F @ B ) ) ) ).
% vimage_Un
thf(fact_326_vimage__Un,axiom,
! [F: list_Sum_sum_a_nat > nat,A: set_nat,B: set_nat] :
( ( vimage4660099214859102426at_nat @ F @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_su4083067149120280889_a_nat @ ( vimage4660099214859102426at_nat @ F @ A ) @ ( vimage4660099214859102426at_nat @ F @ B ) ) ) ).
% vimage_Un
thf(fact_327_vimage__Un,axiom,
! [F: list_Sum_sum_a_nat > a,A: set_a,B: set_a] :
( ( vimage7906019935533595764_nat_a @ F @ ( sup_sup_set_a @ A @ B ) )
= ( sup_su4083067149120280889_a_nat @ ( vimage7906019935533595764_nat_a @ F @ A ) @ ( vimage7906019935533595764_nat_a @ F @ B ) ) ) ).
% vimage_Un
thf(fact_328_nys__def,axiom,
( nys
= ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat
@ ( filter1580576373184442087_a_nat
@ ( produc8810490045567577941_nat_o
@ ^ [N2: nat,Y: sum_sum_a_nat] :
( ~ ( member_nat @ N2 @ ( set_nat2 @ ns_phi ) )
& ~ ( sum_isl_a_nat @ Y ) ) )
@ ( zip_na2013496608136855606_a_nat @ ns_psi @ ys ) ) ) ) ).
% nys_def
thf(fact_329_list_Oset__map,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,V: list_l4703314356710769291_a_nat] :
( ( set_li2392974972034027290_a_nat @ ( map_li6507455427659069316_a_nat @ F @ V ) )
= ( image_5081948215111134021_a_nat @ F @ ( set_li2392974972034027290_a_nat @ V ) ) ) ).
% list.set_map
thf(fact_330_list_Oset__map,axiom,
! [F: produc5454855657571582244_a_nat > nat,V: list_P5056861408695629236_a_nat] :
( ( set_nat2 @ ( map_Pr2058079554187054650at_nat @ F @ V ) )
= ( image_1677410150705809403at_nat @ F @ ( set_Pr2196415342531872975_a_nat @ V ) ) ) ).
% list.set_map
thf(fact_331_list_Oset__map,axiom,
! [F: nat > nat,V: list_nat] :
( ( set_nat2 @ ( map_nat_nat @ F @ V ) )
= ( image_nat_nat @ F @ ( set_nat2 @ V ) ) ) ).
% list.set_map
thf(fact_332_list_Oset__map,axiom,
! [F: sum_sum_a_nat > nat,V: list_Sum_sum_a_nat] :
( ( set_nat2 @ ( map_Su5227373005390213643at_nat @ F @ V ) )
= ( image_2473878607534554506at_nat @ F @ ( set_Sum_sum_a_nat2 @ V ) ) ) ).
% list.set_map
thf(fact_333_list_Oset__map,axiom,
! [F: produc5454855657571582244_a_nat > sum_sum_a_nat,V: list_P5056861408695629236_a_nat] :
( ( set_Sum_sum_a_nat2 @ ( map_Pr2515564165894599445_a_nat @ F @ V ) )
= ( image_5908522764169838868_a_nat @ F @ ( set_Pr2196415342531872975_a_nat @ V ) ) ) ).
% list.set_map
thf(fact_334_list_Oset__map,axiom,
! [F: nat > sum_sum_a_nat,V: list_nat] :
( ( set_Sum_sum_a_nat2 @ ( map_na823391071729141993_a_nat @ F @ V ) )
= ( image_7293268710728258664_a_nat @ F @ ( set_nat2 @ V ) ) ) ).
% list.set_map
thf(fact_335_list_Oset__map,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,V: list_Sum_sum_a_nat] :
( ( set_Sum_sum_a_nat2 @ ( map_Su2790769393171190532_a_nat @ F @ V ) )
= ( image_7142520692256960453_a_nat @ F @ ( set_Sum_sum_a_nat2 @ V ) ) ) ).
% list.set_map
thf(fact_336_vimage__def,axiom,
( vimage4607795094749595324_a_nat
= ( ^ [F2: a > sum_sum_a_nat,B4: set_Sum_sum_a_nat] :
( collect_a
@ ^ [X: a] : ( member_Sum_sum_a_nat @ ( F2 @ X ) @ B4 ) ) ) ) ).
% vimage_def
thf(fact_337_vimage__def,axiom,
( vimage_nat_nat
= ( ^ [F2: nat > nat,B4: set_nat] :
( collect_nat
@ ^ [X: nat] : ( member_nat @ ( F2 @ X ) @ B4 ) ) ) ) ).
% vimage_def
thf(fact_338_vimage__def,axiom,
( vimage8387349081243083832_a_nat
= ( ^ [F2: nat > list_Sum_sum_a_nat,B4: set_li6526943997496501093_a_nat] :
( collect_nat
@ ^ [X: nat] : ( member408289922725080238_a_nat @ ( F2 @ X ) @ B4 ) ) ) ) ).
% vimage_def
thf(fact_339_vimage__def,axiom,
( vimage_nat_a
= ( ^ [F2: nat > a,B4: set_a] :
( collect_nat
@ ^ [X: nat] : ( member_a @ ( F2 @ X ) @ B4 ) ) ) ) ).
% vimage_def
thf(fact_340_vimage__def,axiom,
( vimage3040984495076556338_a_nat
= ( ^ [F2: nat > sum_sum_a_nat,B4: set_Sum_sum_a_nat] :
( collect_nat
@ ^ [X: nat] : ( member_Sum_sum_a_nat @ ( F2 @ X ) @ B4 ) ) ) ) ).
% vimage_def
thf(fact_341_imageI,axiom,
! [X2: nat,A: set_nat,F: nat > nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ ( image_nat_nat @ F @ A ) ) ) ).
% imageI
thf(fact_342_imageI,axiom,
! [X2: nat,A: set_nat,F: nat > a] :
( ( member_nat @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ ( image_nat_a @ F @ A ) ) ) ).
% imageI
thf(fact_343_imageI,axiom,
! [X2: a,A: set_a,F: a > nat] :
( ( member_a @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ ( image_a_nat @ F @ A ) ) ) ).
% imageI
thf(fact_344_imageI,axiom,
! [X2: a,A: set_a,F: a > a] :
( ( member_a @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ ( image_a_a @ F @ A ) ) ) ).
% imageI
thf(fact_345_imageI,axiom,
! [X2: nat,A: set_nat,F: nat > sum_sum_a_nat] :
( ( member_nat @ X2 @ A )
=> ( member_Sum_sum_a_nat @ ( F @ X2 ) @ ( image_7293268710728258664_a_nat @ F @ A ) ) ) ).
% imageI
thf(fact_346_imageI,axiom,
! [X2: a,A: set_a,F: a > sum_sum_a_nat] :
( ( member_a @ X2 @ A )
=> ( member_Sum_sum_a_nat @ ( F @ X2 ) @ ( image_7873763678140191238_a_nat @ F @ A ) ) ) ).
% imageI
thf(fact_347_imageI,axiom,
! [X2: sum_sum_a_nat,A: set_Sum_sum_a_nat,F: sum_sum_a_nat > nat] :
( ( member_Sum_sum_a_nat @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ ( image_2473878607534554506at_nat @ F @ A ) ) ) ).
% imageI
thf(fact_348_imageI,axiom,
! [X2: sum_sum_a_nat,A: set_Sum_sum_a_nat,F: sum_sum_a_nat > a] :
( ( member_Sum_sum_a_nat @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ ( image_6322530041254294468_nat_a @ F @ A ) ) ) ).
% imageI
thf(fact_349_imageI,axiom,
! [X2: nat,A: set_nat,F: nat > list_Sum_sum_a_nat] :
( ( member_nat @ X2 @ A )
=> ( member408289922725080238_a_nat @ ( F @ X2 ) @ ( image_6262589752765146990_a_nat @ F @ A ) ) ) ).
% imageI
thf(fact_350_imageI,axiom,
! [X2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > nat] :
( ( member408289922725080238_a_nat @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ ( image_2535339886381165584at_nat @ F @ A ) ) ) ).
% imageI
thf(fact_351_in__mono,axiom,
! [A: set_nat,B: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ X2 @ A )
=> ( member_nat @ X2 @ B ) ) ) ).
% in_mono
thf(fact_352_in__mono,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,X2: list_Sum_sum_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A @ B )
=> ( ( member408289922725080238_a_nat @ X2 @ A )
=> ( member408289922725080238_a_nat @ X2 @ B ) ) ) ).
% in_mono
thf(fact_353_in__mono,axiom,
! [A: set_a,B: set_a,X2: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ X2 @ A )
=> ( member_a @ X2 @ B ) ) ) ).
% in_mono
thf(fact_354_in__mono,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,X2: sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ B )
=> ( ( member_Sum_sum_a_nat @ X2 @ A )
=> ( member_Sum_sum_a_nat @ X2 @ B ) ) ) ).
% in_mono
thf(fact_355_subsetD,axiom,
! [A: set_nat,B: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_356_subsetD,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,C: list_Sum_sum_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A @ B )
=> ( ( member408289922725080238_a_nat @ C @ A )
=> ( member408289922725080238_a_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_357_subsetD,axiom,
! [A: set_a,B: set_a,C: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% subsetD
thf(fact_358_subsetD,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,C: sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ B )
=> ( ( member_Sum_sum_a_nat @ C @ A )
=> ( member_Sum_sum_a_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_359_vimageD,axiom,
! [A2: nat,F: nat > nat,A: set_nat] :
( ( member_nat @ A2 @ ( vimage_nat_nat @ F @ A ) )
=> ( member_nat @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_360_vimageD,axiom,
! [A2: nat,F: nat > a,A: set_a] :
( ( member_nat @ A2 @ ( vimage_nat_a @ F @ A ) )
=> ( member_a @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_361_vimageD,axiom,
! [A2: a,F: a > nat,A: set_nat] :
( ( member_a @ A2 @ ( vimage_a_nat @ F @ A ) )
=> ( member_nat @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_362_vimageD,axiom,
! [A2: a,F: a > a,A: set_a] :
( ( member_a @ A2 @ ( vimage_a_a @ F @ A ) )
=> ( member_a @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_363_vimageD,axiom,
! [A2: sum_sum_a_nat,F: sum_sum_a_nat > nat,A: set_nat] :
( ( member_Sum_sum_a_nat @ A2 @ ( vimage7444966428737627988at_nat @ F @ A ) )
=> ( member_nat @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_364_vimageD,axiom,
! [A2: sum_sum_a_nat,F: sum_sum_a_nat > a,A: set_a] :
( ( member_Sum_sum_a_nat @ A2 @ ( vimage3056561457863698554_nat_a @ F @ A ) )
=> ( member_a @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_365_vimageD,axiom,
! [A2: a,F: a > sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( member_a @ A2 @ ( vimage4607795094749595324_a_nat @ F @ A ) )
=> ( member_Sum_sum_a_nat @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_366_vimageD,axiom,
! [A2: nat,F: nat > sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( member_nat @ A2 @ ( vimage3040984495076556338_a_nat @ F @ A ) )
=> ( member_Sum_sum_a_nat @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_367_vimageD,axiom,
! [A2: nat,F: nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( member_nat @ A2 @ ( vimage8387349081243083832_a_nat @ F @ A ) )
=> ( member408289922725080238_a_nat @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_368_vimageD,axiom,
! [A2: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > nat,A: set_nat] :
( ( member408289922725080238_a_nat @ A2 @ ( vimage4660099214859102426at_nat @ F @ A ) )
=> ( member_nat @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_369_vimageE,axiom,
! [A2: nat,F: nat > nat,B: set_nat] :
( ( member_nat @ A2 @ ( vimage_nat_nat @ F @ B ) )
=> ( member_nat @ ( F @ A2 ) @ B ) ) ).
% vimageE
thf(fact_370_vimageE,axiom,
! [A2: nat,F: nat > a,B: set_a] :
( ( member_nat @ A2 @ ( vimage_nat_a @ F @ B ) )
=> ( member_a @ ( F @ A2 ) @ B ) ) ).
% vimageE
thf(fact_371_vimageE,axiom,
! [A2: a,F: a > nat,B: set_nat] :
( ( member_a @ A2 @ ( vimage_a_nat @ F @ B ) )
=> ( member_nat @ ( F @ A2 ) @ B ) ) ).
% vimageE
thf(fact_372_vimageE,axiom,
! [A2: a,F: a > a,B: set_a] :
( ( member_a @ A2 @ ( vimage_a_a @ F @ B ) )
=> ( member_a @ ( F @ A2 ) @ B ) ) ).
% vimageE
thf(fact_373_vimageE,axiom,
! [A2: sum_sum_a_nat,F: sum_sum_a_nat > nat,B: set_nat] :
( ( member_Sum_sum_a_nat @ A2 @ ( vimage7444966428737627988at_nat @ F @ B ) )
=> ( member_nat @ ( F @ A2 ) @ B ) ) ).
% vimageE
thf(fact_374_vimageE,axiom,
! [A2: sum_sum_a_nat,F: sum_sum_a_nat > a,B: set_a] :
( ( member_Sum_sum_a_nat @ A2 @ ( vimage3056561457863698554_nat_a @ F @ B ) )
=> ( member_a @ ( F @ A2 ) @ B ) ) ).
% vimageE
thf(fact_375_vimageE,axiom,
! [A2: a,F: a > sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( member_a @ A2 @ ( vimage4607795094749595324_a_nat @ F @ B ) )
=> ( member_Sum_sum_a_nat @ ( F @ A2 ) @ B ) ) ).
% vimageE
thf(fact_376_vimageE,axiom,
! [A2: nat,F: nat > sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( member_nat @ A2 @ ( vimage3040984495076556338_a_nat @ F @ B ) )
=> ( member_Sum_sum_a_nat @ ( F @ A2 ) @ B ) ) ).
% vimageE
thf(fact_377_vimageE,axiom,
! [A2: nat,F: nat > list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member_nat @ A2 @ ( vimage8387349081243083832_a_nat @ F @ B ) )
=> ( member408289922725080238_a_nat @ ( F @ A2 ) @ B ) ) ).
% vimageE
thf(fact_378_vimageE,axiom,
! [A2: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > nat,B: set_nat] :
( ( member408289922725080238_a_nat @ A2 @ ( vimage4660099214859102426at_nat @ F @ B ) )
=> ( member_nat @ ( F @ A2 ) @ B ) ) ).
% vimageE
thf(fact_379_vimageI2,axiom,
! [F: nat > nat,A2: nat,A: set_nat] :
( ( member_nat @ ( F @ A2 ) @ A )
=> ( member_nat @ A2 @ ( vimage_nat_nat @ F @ A ) ) ) ).
% vimageI2
thf(fact_380_vimageI2,axiom,
! [F: a > nat,A2: a,A: set_nat] :
( ( member_nat @ ( F @ A2 ) @ A )
=> ( member_a @ A2 @ ( vimage_a_nat @ F @ A ) ) ) ).
% vimageI2
thf(fact_381_vimageI2,axiom,
! [F: nat > a,A2: nat,A: set_a] :
( ( member_a @ ( F @ A2 ) @ A )
=> ( member_nat @ A2 @ ( vimage_nat_a @ F @ A ) ) ) ).
% vimageI2
thf(fact_382_vimageI2,axiom,
! [F: a > a,A2: a,A: set_a] :
( ( member_a @ ( F @ A2 ) @ A )
=> ( member_a @ A2 @ ( vimage_a_a @ F @ A ) ) ) ).
% vimageI2
thf(fact_383_vimageI2,axiom,
! [F: sum_sum_a_nat > nat,A2: sum_sum_a_nat,A: set_nat] :
( ( member_nat @ ( F @ A2 ) @ A )
=> ( member_Sum_sum_a_nat @ A2 @ ( vimage7444966428737627988at_nat @ F @ A ) ) ) ).
% vimageI2
thf(fact_384_vimageI2,axiom,
! [F: sum_sum_a_nat > a,A2: sum_sum_a_nat,A: set_a] :
( ( member_a @ ( F @ A2 ) @ A )
=> ( member_Sum_sum_a_nat @ A2 @ ( vimage3056561457863698554_nat_a @ F @ A ) ) ) ).
% vimageI2
thf(fact_385_vimageI2,axiom,
! [F: a > sum_sum_a_nat,A2: a,A: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ ( F @ A2 ) @ A )
=> ( member_a @ A2 @ ( vimage4607795094749595324_a_nat @ F @ A ) ) ) ).
% vimageI2
thf(fact_386_vimageI2,axiom,
! [F: nat > sum_sum_a_nat,A2: nat,A: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ ( F @ A2 ) @ A )
=> ( member_nat @ A2 @ ( vimage3040984495076556338_a_nat @ F @ A ) ) ) ).
% vimageI2
thf(fact_387_vimageI2,axiom,
! [F: list_Sum_sum_a_nat > nat,A2: list_Sum_sum_a_nat,A: set_nat] :
( ( member_nat @ ( F @ A2 ) @ A )
=> ( member408289922725080238_a_nat @ A2 @ ( vimage4660099214859102426at_nat @ F @ A ) ) ) ).
% vimageI2
thf(fact_388_vimageI2,axiom,
! [F: nat > list_Sum_sum_a_nat,A2: nat,A: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ ( F @ A2 ) @ A )
=> ( member_nat @ A2 @ ( vimage8387349081243083832_a_nat @ F @ A ) ) ) ).
% vimageI2
thf(fact_389_equalityE,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_a @ A @ B )
=> ~ ( ord_less_eq_set_a @ B @ A ) ) ) ).
% equalityE
thf(fact_390_equalityE,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( A = B )
=> ~ ( ( ord_le1325389633284124927_a_nat @ A @ B )
=> ~ ( ord_le1325389633284124927_a_nat @ B @ A ) ) ) ).
% equalityE
thf(fact_391_image__iff,axiom,
! [Z: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ Z @ ( image_5081948215111134021_a_nat @ F @ A ) )
= ( ? [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A )
& ( Z
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_392_image__iff,axiom,
! [Z: sum_sum_a_nat,F: nat > sum_sum_a_nat,A: set_nat] :
( ( member_Sum_sum_a_nat @ Z @ ( image_7293268710728258664_a_nat @ F @ A ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A )
& ( Z
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_393_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_394_subset__eq,axiom,
( ord_le1147066620699065093_a_nat
= ( ^ [A4: set_li6526943997496501093_a_nat,B4: set_li6526943997496501093_a_nat] :
! [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A4 )
=> ( member408289922725080238_a_nat @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_395_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
! [X: a] :
( ( member_a @ X @ A4 )
=> ( member_a @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_396_subset__eq,axiom,
( ord_le1325389633284124927_a_nat
= ( ^ [A4: set_Sum_sum_a_nat,B4: set_Sum_sum_a_nat] :
! [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ A4 )
=> ( member_Sum_sum_a_nat @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_397_bex__imageD,axiom,
! [F: nat > sum_sum_a_nat,A: set_nat,P: sum_sum_a_nat > $o] :
( ? [X5: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X5 @ ( image_7293268710728258664_a_nat @ F @ A ) )
& ( P @ X5 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_398_bex__imageD,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,P: list_Sum_sum_a_nat > $o] :
( ? [X5: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X5 @ ( image_5081948215111134021_a_nat @ F @ A ) )
& ( P @ X5 ) )
=> ? [X3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_399_equalityD1,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% equalityD1
thf(fact_400_equalityD1,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( A = B )
=> ( ord_le1325389633284124927_a_nat @ A @ B ) ) ).
% equalityD1
thf(fact_401_Set_OequalityD2,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% Set.equalityD2
thf(fact_402_Set_OequalityD2,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( A = B )
=> ( ord_le1325389633284124927_a_nat @ B @ A ) ) ).
% Set.equalityD2
thf(fact_403_image__cong,axiom,
! [M: set_nat,N3: set_nat,F: nat > sum_sum_a_nat,G: nat > sum_sum_a_nat] :
( ( M = N3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ N3 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_7293268710728258664_a_nat @ F @ M )
= ( image_7293268710728258664_a_nat @ G @ N3 ) ) ) ) ).
% image_cong
thf(fact_404_image__cong,axiom,
! [M: set_li6526943997496501093_a_nat,N3: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,G: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( M = N3 )
=> ( ! [X3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ N3 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_5081948215111134021_a_nat @ F @ M )
= ( image_5081948215111134021_a_nat @ G @ N3 ) ) ) ) ).
% image_cong
thf(fact_405_image__mono,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A @ B )
=> ( ord_le1147066620699065093_a_nat @ ( image_5081948215111134021_a_nat @ F @ A ) @ ( image_5081948215111134021_a_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_406_image__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > sum_sum_a_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_le1325389633284124927_a_nat @ ( image_7293268710728258664_a_nat @ F @ A ) @ ( image_7293268710728258664_a_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_407_image__mono,axiom,
! [A: set_a,B: set_a,F: a > a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ ( image_a_a @ F @ B ) ) ) ).
% image_mono
thf(fact_408_image__mono,axiom,
! [A: set_a,B: set_a,F: a > sum_sum_a_nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_le1325389633284124927_a_nat @ ( image_7873763678140191238_a_nat @ F @ A ) @ ( image_7873763678140191238_a_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_409_image__mono,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,F: sum_sum_a_nat > a] :
( ( ord_le1325389633284124927_a_nat @ A @ B )
=> ( ord_less_eq_set_a @ ( image_6322530041254294468_nat_a @ F @ A ) @ ( image_6322530041254294468_nat_a @ F @ B ) ) ) ).
% image_mono
thf(fact_410_image__mono,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ B )
=> ( ord_le1325389633284124927_a_nat @ ( image_7142520692256960453_a_nat @ F @ A ) @ ( image_7142520692256960453_a_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_411_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_412_subset__iff,axiom,
( ord_le1147066620699065093_a_nat
= ( ^ [A4: set_li6526943997496501093_a_nat,B4: set_li6526943997496501093_a_nat] :
! [T2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ T2 @ A4 )
=> ( member408289922725080238_a_nat @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_413_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
! [T2: a] :
( ( member_a @ T2 @ A4 )
=> ( member_a @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_414_subset__iff,axiom,
( ord_le1325389633284124927_a_nat
= ( ^ [A4: set_Sum_sum_a_nat,B4: set_Sum_sum_a_nat] :
! [T2: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ T2 @ A4 )
=> ( member_Sum_sum_a_nat @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_415_ball__imageD,axiom,
! [F: nat > sum_sum_a_nat,A: set_nat,P: sum_sum_a_nat > $o] :
( ! [X3: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X3 @ ( image_7293268710728258664_a_nat @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X5: nat] :
( ( member_nat @ X5 @ A )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_416_ball__imageD,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,P: list_Sum_sum_a_nat > $o] :
( ! [X3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ ( image_5081948215111134021_a_nat @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X5: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X5 @ A )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_417_subset__refl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% subset_refl
thf(fact_418_subset__refl,axiom,
! [A: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ A @ A ) ).
% subset_refl
thf(fact_419_vimage__mono,axiom,
! [A: set_a,B: set_a,F: a > a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ ( vimage_a_a @ F @ A ) @ ( vimage_a_a @ F @ B ) ) ) ).
% vimage_mono
thf(fact_420_vimage__mono,axiom,
! [A: set_a,B: set_a,F: sum_sum_a_nat > a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_le1325389633284124927_a_nat @ ( vimage3056561457863698554_nat_a @ F @ A ) @ ( vimage3056561457863698554_nat_a @ F @ B ) ) ) ).
% vimage_mono
thf(fact_421_vimage__mono,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,F: nat > sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ B )
=> ( ord_less_eq_set_nat @ ( vimage3040984495076556338_a_nat @ F @ A ) @ ( vimage3040984495076556338_a_nat @ F @ B ) ) ) ).
% vimage_mono
thf(fact_422_vimage__mono,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,F: a > sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ B )
=> ( ord_less_eq_set_a @ ( vimage4607795094749595324_a_nat @ F @ A ) @ ( vimage4607795094749595324_a_nat @ F @ B ) ) ) ).
% vimage_mono
thf(fact_423_vimage__mono,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ B )
=> ( ord_le1325389633284124927_a_nat @ ( vimage6545432446589551483_a_nat @ F @ A ) @ ( vimage6545432446589551483_a_nat @ F @ B ) ) ) ).
% vimage_mono
thf(fact_424_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_425_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_426_Collect__mono,axiom,
! [P: sum_sum_a_nat > $o,Q: sum_sum_a_nat > $o] :
( ! [X3: sum_sum_a_nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le1325389633284124927_a_nat @ ( collec7073057861543223018_a_nat @ P ) @ ( collec7073057861543223018_a_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_427_subset__trans,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_428_subset__trans,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ B )
=> ( ( ord_le1325389633284124927_a_nat @ B @ C2 )
=> ( ord_le1325389633284124927_a_nat @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_429_image__subsetI,axiom,
! [A: set_nat,F: nat > nat,B: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_430_image__subsetI,axiom,
! [A: set_a,F: a > nat,B: set_nat] :
( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_a_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_431_image__subsetI,axiom,
! [A: set_nat,F: nat > a,B: set_a] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_a @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_nat_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_432_image__subsetI,axiom,
! [A: set_a,F: a > a,B: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( member_a @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_433_image__subsetI,axiom,
! [A: set_Sum_sum_a_nat,F: sum_sum_a_nat > nat,B: set_nat] :
( ! [X3: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_2473878607534554506at_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_434_image__subsetI,axiom,
! [A: set_Sum_sum_a_nat,F: sum_sum_a_nat > a,B: set_a] :
( ! [X3: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X3 @ A )
=> ( member_a @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_6322530041254294468_nat_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_435_image__subsetI,axiom,
! [A: set_nat,F: nat > sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_Sum_sum_a_nat @ ( F @ X3 ) @ B ) )
=> ( ord_le1325389633284124927_a_nat @ ( image_7293268710728258664_a_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_436_image__subsetI,axiom,
! [A: set_a,F: a > sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( member_Sum_sum_a_nat @ ( F @ X3 ) @ B ) )
=> ( ord_le1325389633284124927_a_nat @ ( image_7873763678140191238_a_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_437_image__subsetI,axiom,
! [A: set_nat,F: nat > list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member408289922725080238_a_nat @ ( F @ X3 ) @ B ) )
=> ( ord_le1147066620699065093_a_nat @ ( image_6262589752765146990_a_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_438_image__subsetI,axiom,
! [A: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > nat,B: set_nat] :
( ! [X3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_2535339886381165584at_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_439_rev__image__eqI,axiom,
! [X2: nat,A: set_nat,B2: nat,F: nat > nat] :
( ( member_nat @ X2 @ A )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_nat @ B2 @ ( image_nat_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_440_rev__image__eqI,axiom,
! [X2: nat,A: set_nat,B2: a,F: nat > a] :
( ( member_nat @ X2 @ A )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_a @ B2 @ ( image_nat_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_441_rev__image__eqI,axiom,
! [X2: a,A: set_a,B2: nat,F: a > nat] :
( ( member_a @ X2 @ A )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_nat @ B2 @ ( image_a_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_442_rev__image__eqI,axiom,
! [X2: a,A: set_a,B2: a,F: a > a] :
( ( member_a @ X2 @ A )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_a @ B2 @ ( image_a_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_443_rev__image__eqI,axiom,
! [X2: nat,A: set_nat,B2: sum_sum_a_nat,F: nat > sum_sum_a_nat] :
( ( member_nat @ X2 @ A )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_Sum_sum_a_nat @ B2 @ ( image_7293268710728258664_a_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_444_rev__image__eqI,axiom,
! [X2: a,A: set_a,B2: sum_sum_a_nat,F: a > sum_sum_a_nat] :
( ( member_a @ X2 @ A )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_Sum_sum_a_nat @ B2 @ ( image_7873763678140191238_a_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_445_rev__image__eqI,axiom,
! [X2: sum_sum_a_nat,A: set_Sum_sum_a_nat,B2: nat,F: sum_sum_a_nat > nat] :
( ( member_Sum_sum_a_nat @ X2 @ A )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_nat @ B2 @ ( image_2473878607534554506at_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_446_rev__image__eqI,axiom,
! [X2: sum_sum_a_nat,A: set_Sum_sum_a_nat,B2: a,F: sum_sum_a_nat > a] :
( ( member_Sum_sum_a_nat @ X2 @ A )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_a @ B2 @ ( image_6322530041254294468_nat_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_447_rev__image__eqI,axiom,
! [X2: nat,A: set_nat,B2: list_Sum_sum_a_nat,F: nat > list_Sum_sum_a_nat] :
( ( member_nat @ X2 @ A )
=> ( ( B2
= ( F @ X2 ) )
=> ( member408289922725080238_a_nat @ B2 @ ( image_6262589752765146990_a_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_448_rev__image__eqI,axiom,
! [X2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B2: nat,F: list_Sum_sum_a_nat > nat] :
( ( member408289922725080238_a_nat @ X2 @ A )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_nat @ B2 @ ( image_2535339886381165584at_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_449_set__eq__subset,axiom,
( ( ^ [Y5: set_a,Z3: set_a] : ( Y5 = Z3 ) )
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
& ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_450_set__eq__subset,axiom,
( ( ^ [Y5: set_Sum_sum_a_nat,Z3: set_Sum_sum_a_nat] : ( Y5 = Z3 ) )
= ( ^ [A4: set_Sum_sum_a_nat,B4: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A4 @ B4 )
& ( ord_le1325389633284124927_a_nat @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_451_subset__imageE,axiom,
! [B: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ B @ ( image_5081948215111134021_a_nat @ F @ A ) )
=> ~ ! [C3: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ C3 @ A )
=> ( B
!= ( image_5081948215111134021_a_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_452_subset__imageE,axiom,
! [B: set_a,F: a > a,A: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A )
=> ( B
!= ( image_a_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_453_subset__imageE,axiom,
! [B: set_a,F: sum_sum_a_nat > a,A: set_Sum_sum_a_nat] :
( ( ord_less_eq_set_a @ B @ ( image_6322530041254294468_nat_a @ F @ A ) )
=> ~ ! [C3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ C3 @ A )
=> ( B
!= ( image_6322530041254294468_nat_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_454_subset__imageE,axiom,
! [B: set_Sum_sum_a_nat,F: nat > sum_sum_a_nat,A: set_nat] :
( ( ord_le1325389633284124927_a_nat @ B @ ( image_7293268710728258664_a_nat @ F @ A ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
=> ( B
!= ( image_7293268710728258664_a_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_455_subset__imageE,axiom,
! [B: set_Sum_sum_a_nat,F: a > sum_sum_a_nat,A: set_a] :
( ( ord_le1325389633284124927_a_nat @ B @ ( image_7873763678140191238_a_nat @ F @ A ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A )
=> ( B
!= ( image_7873763678140191238_a_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_456_subset__imageE,axiom,
! [B: set_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B @ ( image_7142520692256960453_a_nat @ F @ A ) )
=> ~ ! [C3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ C3 @ A )
=> ( B
!= ( image_7142520692256960453_a_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_457_vimage__Collect,axiom,
! [P: sum_sum_a_nat > $o,F: a > sum_sum_a_nat,Q: a > $o] :
( ! [X3: a] :
( ( P @ ( F @ X3 ) )
= ( Q @ X3 ) )
=> ( ( vimage4607795094749595324_a_nat @ F @ ( collec7073057861543223018_a_nat @ P ) )
= ( collect_a @ Q ) ) ) ).
% vimage_Collect
thf(fact_458_vimage__Collect,axiom,
! [P: sum_sum_a_nat > $o,F: nat > sum_sum_a_nat,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ ( F @ X3 ) )
= ( Q @ X3 ) )
=> ( ( vimage3040984495076556338_a_nat @ F @ ( collec7073057861543223018_a_nat @ P ) )
= ( collect_nat @ Q ) ) ) ).
% vimage_Collect
thf(fact_459_vimage__Collect,axiom,
! [P: nat > $o,F: nat > nat,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ ( F @ X3 ) )
= ( Q @ X3 ) )
=> ( ( vimage_nat_nat @ F @ ( collect_nat @ P ) )
= ( collect_nat @ Q ) ) ) ).
% vimage_Collect
thf(fact_460_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_461_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X: a] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_462_Collect__mono__iff,axiom,
! [P: sum_sum_a_nat > $o,Q: sum_sum_a_nat > $o] :
( ( ord_le1325389633284124927_a_nat @ ( collec7073057861543223018_a_nat @ P ) @ ( collec7073057861543223018_a_nat @ Q ) )
= ( ! [X: sum_sum_a_nat] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_463_image__subset__iff,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ ( image_5081948215111134021_a_nat @ F @ A ) @ B )
= ( ! [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A )
=> ( member408289922725080238_a_nat @ ( F @ X ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_464_image__subset__iff,axiom,
! [F: nat > sum_sum_a_nat,A: set_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( image_7293268710728258664_a_nat @ F @ A ) @ B )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_Sum_sum_a_nat @ ( F @ X ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_465_subset__image__iff,axiom,
! [B: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ B @ ( image_5081948215111134021_a_nat @ F @ A ) )
= ( ? [AA: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ AA @ A )
& ( B
= ( image_5081948215111134021_a_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_466_subset__image__iff,axiom,
! [B: set_a,F: a > a,A: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A )
& ( B
= ( image_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_467_subset__image__iff,axiom,
! [B: set_a,F: sum_sum_a_nat > a,A: set_Sum_sum_a_nat] :
( ( ord_less_eq_set_a @ B @ ( image_6322530041254294468_nat_a @ F @ A ) )
= ( ? [AA: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ AA @ A )
& ( B
= ( image_6322530041254294468_nat_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_468_subset__image__iff,axiom,
! [B: set_Sum_sum_a_nat,F: nat > sum_sum_a_nat,A: set_nat] :
( ( ord_le1325389633284124927_a_nat @ B @ ( image_7293268710728258664_a_nat @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B
= ( image_7293268710728258664_a_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_469_subset__image__iff,axiom,
! [B: set_Sum_sum_a_nat,F: a > sum_sum_a_nat,A: set_a] :
( ( ord_le1325389633284124927_a_nat @ B @ ( image_7873763678140191238_a_nat @ F @ A ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A )
& ( B
= ( image_7873763678140191238_a_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_470_subset__image__iff,axiom,
! [B: set_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B @ ( image_7142520692256960453_a_nat @ F @ A ) )
= ( ? [AA: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ AA @ A )
& ( B
= ( image_7142520692256960453_a_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_471_subset__vimage__iff,axiom,
! [A: set_nat,F: nat > sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ord_less_eq_set_nat @ A @ ( vimage3040984495076556338_a_nat @ F @ B ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_Sum_sum_a_nat @ ( F @ X ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_472_subset__vimage__iff,axiom,
! [A: set_a,F: a > nat,B: set_nat] :
( ( ord_less_eq_set_a @ A @ ( vimage_a_nat @ F @ B ) )
= ( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_nat @ ( F @ X ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_473_subset__vimage__iff,axiom,
! [A: set_a,F: a > list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ord_less_eq_set_a @ A @ ( vimage1894578532518109122_a_nat @ F @ B ) )
= ( ! [X: a] :
( ( member_a @ X @ A )
=> ( member408289922725080238_a_nat @ ( F @ X ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_474_subset__vimage__iff,axiom,
! [A: set_a,F: a > a,B: set_a] :
( ( ord_less_eq_set_a @ A @ ( vimage_a_a @ F @ B ) )
= ( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_a @ ( F @ X ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_475_subset__vimage__iff,axiom,
! [A: set_a,F: a > sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ord_less_eq_set_a @ A @ ( vimage4607795094749595324_a_nat @ F @ B ) )
= ( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_Sum_sum_a_nat @ ( F @ X ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_476_subset__vimage__iff,axiom,
! [A: set_Sum_sum_a_nat,F: sum_sum_a_nat > nat,B: set_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ ( vimage7444966428737627988at_nat @ F @ B ) )
= ( ! [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ A )
=> ( member_nat @ ( F @ X ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_477_subset__vimage__iff,axiom,
! [A: set_Sum_sum_a_nat,F: sum_sum_a_nat > list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ ( vimage8052486765430747393_a_nat @ F @ B ) )
= ( ! [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ A )
=> ( member408289922725080238_a_nat @ ( F @ X ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_478_subset__vimage__iff,axiom,
! [A: set_Sum_sum_a_nat,F: sum_sum_a_nat > a,B: set_a] :
( ( ord_le1325389633284124927_a_nat @ A @ ( vimage3056561457863698554_nat_a @ F @ B ) )
= ( ! [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ A )
=> ( member_a @ ( F @ X ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_479_subset__vimage__iff,axiom,
! [A: set_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ ( vimage6545432446589551483_a_nat @ F @ B ) )
= ( ! [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ A )
=> ( member_Sum_sum_a_nat @ ( F @ X ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_480_image__vimage__subset,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ ( image_5081948215111134021_a_nat @ F @ ( vimage7538740559230589179_a_nat @ F @ A ) ) @ A ) ).
% image_vimage_subset
thf(fact_481_image__vimage__subset,axiom,
! [F: a > sum_sum_a_nat,A: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( image_7873763678140191238_a_nat @ F @ ( vimage4607795094749595324_a_nat @ F @ A ) ) @ A ) ).
% image_vimage_subset
thf(fact_482_image__vimage__subset,axiom,
! [F: nat > sum_sum_a_nat,A: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( image_7293268710728258664_a_nat @ F @ ( vimage3040984495076556338_a_nat @ F @ A ) ) @ A ) ).
% image_vimage_subset
thf(fact_483_image__subset__iff__subset__vimage,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ ( image_5081948215111134021_a_nat @ F @ A ) @ B )
= ( ord_le1147066620699065093_a_nat @ A @ ( vimage7538740559230589179_a_nat @ F @ B ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_484_image__subset__iff__subset__vimage,axiom,
! [F: a > a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ B )
= ( ord_less_eq_set_a @ A @ ( vimage_a_a @ F @ B ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_485_image__subset__iff__subset__vimage,axiom,
! [F: sum_sum_a_nat > a,A: set_Sum_sum_a_nat,B: set_a] :
( ( ord_less_eq_set_a @ ( image_6322530041254294468_nat_a @ F @ A ) @ B )
= ( ord_le1325389633284124927_a_nat @ A @ ( vimage3056561457863698554_nat_a @ F @ B ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_486_image__subset__iff__subset__vimage,axiom,
! [F: nat > sum_sum_a_nat,A: set_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( image_7293268710728258664_a_nat @ F @ A ) @ B )
= ( ord_less_eq_set_nat @ A @ ( vimage3040984495076556338_a_nat @ F @ B ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_487_image__subset__iff__subset__vimage,axiom,
! [F: a > sum_sum_a_nat,A: set_a,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( image_7873763678140191238_a_nat @ F @ A ) @ B )
= ( ord_less_eq_set_a @ A @ ( vimage4607795094749595324_a_nat @ F @ B ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_488_image__subset__iff__subset__vimage,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( image_7142520692256960453_a_nat @ F @ A ) @ B )
= ( ord_le1325389633284124927_a_nat @ A @ ( vimage6545432446589551483_a_nat @ F @ B ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_489_prod_Ocase__distrib,axiom,
! [H: $o > $o,F: nat > sum_sum_a_nat > $o,Prod: produc5454855657571582244_a_nat] :
( ( H @ ( produc8810490045567577941_nat_o @ F @ Prod ) )
= ( produc8810490045567577941_nat_o
@ ^ [X12: nat,X22: sum_sum_a_nat] : ( H @ ( F @ X12 @ X22 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_490_prod_Odisc__eq__case,axiom,
! [Prod: produc5454855657571582244_a_nat] :
( produc8810490045567577941_nat_o
@ ^ [Uu: nat,Uv: sum_sum_a_nat] : $true
@ Prod ) ).
% prod.disc_eq_case
thf(fact_491_imageE,axiom,
! [B2: nat,F: nat > nat,A: set_nat] :
( ( member_nat @ B2 @ ( image_nat_nat @ F @ A ) )
=> ~ ! [X3: nat] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A ) ) ) ).
% imageE
thf(fact_492_imageE,axiom,
! [B2: nat,F: a > nat,A: set_a] :
( ( member_nat @ B2 @ ( image_a_nat @ F @ A ) )
=> ~ ! [X3: a] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_a @ X3 @ A ) ) ) ).
% imageE
thf(fact_493_imageE,axiom,
! [B2: a,F: nat > a,A: set_nat] :
( ( member_a @ B2 @ ( image_nat_a @ F @ A ) )
=> ~ ! [X3: nat] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A ) ) ) ).
% imageE
thf(fact_494_imageE,axiom,
! [B2: a,F: a > a,A: set_a] :
( ( member_a @ B2 @ ( image_a_a @ F @ A ) )
=> ~ ! [X3: a] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_a @ X3 @ A ) ) ) ).
% imageE
thf(fact_495_imageE,axiom,
! [B2: nat,F: sum_sum_a_nat > nat,A: set_Sum_sum_a_nat] :
( ( member_nat @ B2 @ ( image_2473878607534554506at_nat @ F @ A ) )
=> ~ ! [X3: sum_sum_a_nat] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_Sum_sum_a_nat @ X3 @ A ) ) ) ).
% imageE
thf(fact_496_imageE,axiom,
! [B2: a,F: sum_sum_a_nat > a,A: set_Sum_sum_a_nat] :
( ( member_a @ B2 @ ( image_6322530041254294468_nat_a @ F @ A ) )
=> ~ ! [X3: sum_sum_a_nat] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_Sum_sum_a_nat @ X3 @ A ) ) ) ).
% imageE
thf(fact_497_imageE,axiom,
! [B2: sum_sum_a_nat,F: nat > sum_sum_a_nat,A: set_nat] :
( ( member_Sum_sum_a_nat @ B2 @ ( image_7293268710728258664_a_nat @ F @ A ) )
=> ~ ! [X3: nat] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A ) ) ) ).
% imageE
thf(fact_498_imageE,axiom,
! [B2: sum_sum_a_nat,F: a > sum_sum_a_nat,A: set_a] :
( ( member_Sum_sum_a_nat @ B2 @ ( image_7873763678140191238_a_nat @ F @ A ) )
=> ~ ! [X3: a] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_a @ X3 @ A ) ) ) ).
% imageE
thf(fact_499_imageE,axiom,
! [B2: nat,F: list_Sum_sum_a_nat > nat,A: set_li6526943997496501093_a_nat] :
( ( member_nat @ B2 @ ( image_2535339886381165584at_nat @ F @ A ) )
=> ~ ! [X3: list_Sum_sum_a_nat] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member408289922725080238_a_nat @ X3 @ A ) ) ) ).
% imageE
thf(fact_500_imageE,axiom,
! [B2: list_Sum_sum_a_nat,F: nat > list_Sum_sum_a_nat,A: set_nat] :
( ( member408289922725080238_a_nat @ B2 @ ( image_6262589752765146990_a_nat @ F @ A ) )
=> ~ ! [X3: nat] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A ) ) ) ).
% imageE
thf(fact_501_image__image,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,G: nat > sum_sum_a_nat,A: set_nat] :
( ( image_7142520692256960453_a_nat @ F @ ( image_7293268710728258664_a_nat @ G @ A ) )
= ( image_7293268710728258664_a_nat
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ A ) ) ).
% image_image
thf(fact_502_image__image,axiom,
! [F: nat > sum_sum_a_nat,G: nat > nat,A: set_nat] :
( ( image_7293268710728258664_a_nat @ F @ ( image_nat_nat @ G @ A ) )
= ( image_7293268710728258664_a_nat
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ A ) ) ).
% image_image
thf(fact_503_image__image,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,G: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( image_5081948215111134021_a_nat @ F @ ( image_5081948215111134021_a_nat @ G @ A ) )
= ( image_5081948215111134021_a_nat
@ ^ [X: list_Sum_sum_a_nat] : ( F @ ( G @ X ) )
@ A ) ) ).
% image_image
thf(fact_504_Compr__image__eq,axiom,
! [F: a > a,A: set_a,P: a > $o] :
( ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( image_a_a @ F @ A ) )
& ( P @ X ) ) )
= ( image_a_a @ F
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_505_Compr__image__eq,axiom,
! [F: nat > a,A: set_nat,P: a > $o] :
( ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( image_nat_a @ F @ A ) )
& ( P @ X ) ) )
= ( image_nat_a @ F
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_506_Compr__image__eq,axiom,
! [F: a > nat,A: set_a,P: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( image_a_nat @ F @ A ) )
& ( P @ X ) ) )
= ( image_a_nat @ F
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_507_Compr__image__eq,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( image_nat_nat @ F @ A ) )
& ( P @ X ) ) )
= ( image_nat_nat @ F
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_508_Compr__image__eq,axiom,
! [F: sum_sum_a_nat > a,A: set_Sum_sum_a_nat,P: a > $o] :
( ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( image_6322530041254294468_nat_a @ F @ A ) )
& ( P @ X ) ) )
= ( image_6322530041254294468_nat_a @ F
@ ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_509_Compr__image__eq,axiom,
! [F: a > sum_sum_a_nat,A: set_a,P: sum_sum_a_nat > $o] :
( ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ ( image_7873763678140191238_a_nat @ F @ A ) )
& ( P @ X ) ) )
= ( image_7873763678140191238_a_nat @ F
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_510_Compr__image__eq,axiom,
! [F: nat > sum_sum_a_nat,A: set_nat,P: sum_sum_a_nat > $o] :
( ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ ( image_7293268710728258664_a_nat @ F @ A ) )
& ( P @ X ) ) )
= ( image_7293268710728258664_a_nat @ F
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_511_Compr__image__eq,axiom,
! [F: sum_sum_a_nat > nat,A: set_Sum_sum_a_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( image_2473878607534554506at_nat @ F @ A ) )
& ( P @ X ) ) )
= ( image_2473878607534554506at_nat @ F
@ ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_512_Compr__image__eq,axiom,
! [F: a > list_Sum_sum_a_nat,A: set_a,P: list_Sum_sum_a_nat > $o] :
( ( collec7555443234367654128_a_nat
@ ^ [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ ( image_7897140031816760844_a_nat @ F @ A ) )
& ( P @ X ) ) )
= ( image_7897140031816760844_a_nat @ F
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_513_Compr__image__eq,axiom,
! [F: list_Sum_sum_a_nat > a,A: set_li6526943997496501093_a_nat,P: a > $o] :
( ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( image_4685209397977471678_nat_a @ F @ A ) )
& ( P @ X ) ) )
= ( image_4685209397977471678_nat_a @ F
@ ( collec7555443234367654128_a_nat
@ ^ [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_514_Collect__subset,axiom,
! [A: set_li6526943997496501093_a_nat,P: list_Sum_sum_a_nat > $o] :
( ord_le1147066620699065093_a_nat
@ ( collec7555443234367654128_a_nat
@ ^ [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_515_Collect__subset,axiom,
! [A: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_516_Collect__subset,axiom,
! [A: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_517_Collect__subset,axiom,
! [A: set_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ord_le1325389633284124927_a_nat
@ ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_518_fst__def,axiom,
( produc1510075366785312874_a_nat
= ( produc2606280072924899667at_nat
@ ^ [X12: nat,X22: sum_sum_a_nat] : X12 ) ) ).
% fst_def
thf(fact_519_snd__def,axiom,
( produc1280857981117566888_a_nat
= ( produc2370069609027602876_a_nat
@ ^ [X12: nat,X22: sum_sum_a_nat] : X22 ) ) ).
% snd_def
thf(fact_520_image__Un,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( image_nat_nat @ F @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_sup_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_Un
thf(fact_521_image__Un,axiom,
! [F: nat > a,A: set_nat,B: set_nat] :
( ( image_nat_a @ F @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_sup_set_a @ ( image_nat_a @ F @ A ) @ ( image_nat_a @ F @ B ) ) ) ).
% image_Un
thf(fact_522_image__Un,axiom,
! [F: a > nat,A: set_a,B: set_a] :
( ( image_a_nat @ F @ ( sup_sup_set_a @ A @ B ) )
= ( sup_sup_set_nat @ ( image_a_nat @ F @ A ) @ ( image_a_nat @ F @ B ) ) ) ).
% image_Un
thf(fact_523_image__Un,axiom,
! [F: a > a,A: set_a,B: set_a] :
( ( image_a_a @ F @ ( sup_sup_set_a @ A @ B ) )
= ( sup_sup_set_a @ ( image_a_a @ F @ A ) @ ( image_a_a @ F @ B ) ) ) ).
% image_Un
thf(fact_524_image__Un,axiom,
! [F: nat > sum_sum_a_nat,A: set_nat,B: set_nat] :
( ( image_7293268710728258664_a_nat @ F @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_su6804446743777130803_a_nat @ ( image_7293268710728258664_a_nat @ F @ A ) @ ( image_7293268710728258664_a_nat @ F @ B ) ) ) ).
% image_Un
thf(fact_525_image__Un,axiom,
! [F: sum_sum_a_nat > nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( image_2473878607534554506at_nat @ F @ ( sup_su6804446743777130803_a_nat @ A @ B ) )
= ( sup_sup_set_nat @ ( image_2473878607534554506at_nat @ F @ A ) @ ( image_2473878607534554506at_nat @ F @ B ) ) ) ).
% image_Un
thf(fact_526_image__Un,axiom,
! [F: sum_sum_a_nat > a,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( image_6322530041254294468_nat_a @ F @ ( sup_su6804446743777130803_a_nat @ A @ B ) )
= ( sup_sup_set_a @ ( image_6322530041254294468_nat_a @ F @ A ) @ ( image_6322530041254294468_nat_a @ F @ B ) ) ) ).
% image_Un
thf(fact_527_image__Un,axiom,
! [F: a > sum_sum_a_nat,A: set_a,B: set_a] :
( ( image_7873763678140191238_a_nat @ F @ ( sup_sup_set_a @ A @ B ) )
= ( sup_su6804446743777130803_a_nat @ ( image_7873763678140191238_a_nat @ F @ A ) @ ( image_7873763678140191238_a_nat @ F @ B ) ) ) ).
% image_Un
thf(fact_528_image__Un,axiom,
! [F: nat > list_Sum_sum_a_nat,A: set_nat,B: set_nat] :
( ( image_6262589752765146990_a_nat @ F @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_su4083067149120280889_a_nat @ ( image_6262589752765146990_a_nat @ F @ A ) @ ( image_6262589752765146990_a_nat @ F @ B ) ) ) ).
% image_Un
thf(fact_529_image__Un,axiom,
! [F: a > list_Sum_sum_a_nat,A: set_a,B: set_a] :
( ( image_7897140031816760844_a_nat @ F @ ( sup_sup_set_a @ A @ B ) )
= ( sup_su4083067149120280889_a_nat @ ( image_7897140031816760844_a_nat @ F @ A ) @ ( image_7897140031816760844_a_nat @ F @ B ) ) ) ).
% image_Un
thf(fact_530_map2__map__map,axiom,
! [H: nat > sum_sum_a_nat > nat,F: nat > nat,Xs: list_nat,G: nat > sum_sum_a_nat] :
( ( map_Pr2058079554187054650at_nat @ ( produc2606280072924899667at_nat @ H ) @ ( zip_na2013496608136855606_a_nat @ ( map_nat_nat @ F @ Xs ) @ ( map_na823391071729141993_a_nat @ G @ Xs ) ) )
= ( map_nat_nat
@ ^ [X: nat] : ( H @ ( F @ X ) @ ( G @ X ) )
@ Xs ) ) ).
% map2_map_map
thf(fact_531_map2__map__map,axiom,
! [H: nat > sum_sum_a_nat > $o,F: nat > nat,Xs: list_nat,G: nat > sum_sum_a_nat] :
( ( map_Pr4279255274729099950_nat_o @ ( produc8810490045567577941_nat_o @ H ) @ ( zip_na2013496608136855606_a_nat @ ( map_nat_nat @ F @ Xs ) @ ( map_na823391071729141993_a_nat @ G @ Xs ) ) )
= ( map_nat_o
@ ^ [X: nat] : ( H @ ( F @ X ) @ ( G @ X ) )
@ Xs ) ) ).
% map2_map_map
thf(fact_532_map2__map__map,axiom,
! [H: nat > nat > nat,F: produc5454855657571582244_a_nat > nat,Xs: list_P5056861408695629236_a_nat,G: produc5454855657571582244_a_nat > nat] :
( ( map_Pr3938374229010428429at_nat @ ( produc6842872674320459806at_nat @ H ) @ ( zip_nat_nat @ ( map_Pr2058079554187054650at_nat @ F @ Xs ) @ ( map_Pr2058079554187054650at_nat @ G @ Xs ) ) )
= ( map_Pr2058079554187054650at_nat
@ ^ [X: produc5454855657571582244_a_nat] : ( H @ ( F @ X ) @ ( G @ X ) )
@ Xs ) ) ).
% map2_map_map
thf(fact_533_map2__map__map,axiom,
! [H: nat > sum_sum_a_nat > sum_sum_a_nat,F: nat > nat,Xs: list_nat,G: nat > sum_sum_a_nat] :
( ( map_Pr2515564165894599445_a_nat @ ( produc2370069609027602876_a_nat @ H ) @ ( zip_na2013496608136855606_a_nat @ ( map_nat_nat @ F @ Xs ) @ ( map_na823391071729141993_a_nat @ G @ Xs ) ) )
= ( map_na823391071729141993_a_nat
@ ^ [X: nat] : ( H @ ( F @ X ) @ ( G @ X ) )
@ Xs ) ) ).
% map2_map_map
thf(fact_534_map2__map__map,axiom,
! [H: sum_sum_a_nat > sum_sum_a_nat > sum_sum_a_nat,F: nat > sum_sum_a_nat,Xs: list_nat,G: nat > sum_sum_a_nat] :
( ( map_Pr3527087583225239196_a_nat @ ( produc906896977214093203_a_nat @ H ) @ ( zip_Su7355543910597222519_a_nat @ ( map_na823391071729141993_a_nat @ F @ Xs ) @ ( map_na823391071729141993_a_nat @ G @ Xs ) ) )
= ( map_na823391071729141993_a_nat
@ ^ [X: nat] : ( H @ ( F @ X ) @ ( G @ X ) )
@ Xs ) ) ).
% map2_map_map
thf(fact_535_map2__map__map,axiom,
! [H: sum_sum_a_nat > nat > nat,F: produc5454855657571582244_a_nat > sum_sum_a_nat,Xs: list_P5056861408695629236_a_nat,G: produc5454855657571582244_a_nat > nat] :
( ( map_Pr8434518024483680788at_nat @ ( produc8957381051445135989at_nat @ H ) @ ( zip_Su6417478541797927256at_nat @ ( map_Pr2515564165894599445_a_nat @ F @ Xs ) @ ( map_Pr2058079554187054650at_nat @ G @ Xs ) ) )
= ( map_Pr2058079554187054650at_nat
@ ^ [X: produc5454855657571582244_a_nat] : ( H @ ( F @ X ) @ ( G @ X ) )
@ Xs ) ) ).
% map2_map_map
thf(fact_536_map2__map__map,axiom,
! [H: nat > nat > sum_sum_a_nat,F: produc5454855657571582244_a_nat > nat,Xs: list_P5056861408695629236_a_nat,G: produc5454855657571582244_a_nat > nat] :
( ( map_Pr8085477292072239746_a_nat @ ( produc7425670176118603825_a_nat @ H ) @ ( zip_nat_nat @ ( map_Pr2058079554187054650at_nat @ F @ Xs ) @ ( map_Pr2058079554187054650at_nat @ G @ Xs ) ) )
= ( map_Pr2515564165894599445_a_nat
@ ^ [X: produc5454855657571582244_a_nat] : ( H @ ( F @ X ) @ ( G @ X ) )
@ Xs ) ) ).
% map2_map_map
thf(fact_537_map2__map__map,axiom,
! [H: nat > sum_sum_a_nat > nat,F: produc5454855657571582244_a_nat > nat,Xs: list_P5056861408695629236_a_nat,G: produc5454855657571582244_a_nat > sum_sum_a_nat] :
( ( map_Pr2058079554187054650at_nat @ ( produc2606280072924899667at_nat @ H ) @ ( zip_na2013496608136855606_a_nat @ ( map_Pr2058079554187054650at_nat @ F @ Xs ) @ ( map_Pr2515564165894599445_a_nat @ G @ Xs ) ) )
= ( map_Pr2058079554187054650at_nat
@ ^ [X: produc5454855657571582244_a_nat] : ( H @ ( F @ X ) @ ( G @ X ) )
@ Xs ) ) ).
% map2_map_map
thf(fact_538_map2__map__map,axiom,
! [H: nat > sum_sum_a_nat > $o,F: produc5454855657571582244_a_nat > nat,Xs: list_P5056861408695629236_a_nat,G: produc5454855657571582244_a_nat > sum_sum_a_nat] :
( ( map_Pr4279255274729099950_nat_o @ ( produc8810490045567577941_nat_o @ H ) @ ( zip_na2013496608136855606_a_nat @ ( map_Pr2058079554187054650at_nat @ F @ Xs ) @ ( map_Pr2515564165894599445_a_nat @ G @ Xs ) ) )
= ( map_Pr4279255274729099950_nat_o
@ ^ [X: produc5454855657571582244_a_nat] : ( H @ ( F @ X ) @ ( G @ X ) )
@ Xs ) ) ).
% map2_map_map
thf(fact_539_map2__map__map,axiom,
! [H: sum_sum_a_nat > sum_sum_a_nat > nat,F: produc5454855657571582244_a_nat > sum_sum_a_nat,Xs: list_P5056861408695629236_a_nat,G: produc5454855657571582244_a_nat > sum_sum_a_nat] :
( ( map_Pr321637745973761139at_nat @ ( produc4523541316010391868at_nat @ H ) @ ( zip_Su7355543910597222519_a_nat @ ( map_Pr2515564165894599445_a_nat @ F @ Xs ) @ ( map_Pr2515564165894599445_a_nat @ G @ Xs ) ) )
= ( map_Pr2058079554187054650at_nat
@ ^ [X: produc5454855657571582244_a_nat] : ( H @ ( F @ X ) @ ( G @ X ) )
@ Xs ) ) ).
% map2_map_map
thf(fact_540_inf__sup__ord_I4_J,axiom,
! [Y2: set_nat,X2: set_nat] : ( ord_less_eq_set_nat @ Y2 @ ( sup_sup_set_nat @ X2 @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_541_inf__sup__ord_I4_J,axiom,
! [Y2: set_li6526943997496501093_a_nat,X2: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ Y2 @ ( sup_su4083067149120280889_a_nat @ X2 @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_542_inf__sup__ord_I4_J,axiom,
! [Y2: set_a,X2: set_a] : ( ord_less_eq_set_a @ Y2 @ ( sup_sup_set_a @ X2 @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_543_inf__sup__ord_I4_J,axiom,
! [Y2: set_Sum_sum_a_nat,X2: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ Y2 @ ( sup_su6804446743777130803_a_nat @ X2 @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_544_inf__sup__ord_I4_J,axiom,
! [Y2: nat,X2: nat] : ( ord_less_eq_nat @ Y2 @ ( sup_sup_nat @ X2 @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_545_inf__sup__ord_I3_J,axiom,
! [X2: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ X2 @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_546_inf__sup__ord_I3_J,axiom,
! [X2: set_li6526943997496501093_a_nat,Y2: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ X2 @ ( sup_su4083067149120280889_a_nat @ X2 @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_547_inf__sup__ord_I3_J,axiom,
! [X2: set_a,Y2: set_a] : ( ord_less_eq_set_a @ X2 @ ( sup_sup_set_a @ X2 @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_548_inf__sup__ord_I3_J,axiom,
! [X2: set_Sum_sum_a_nat,Y2: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ X2 @ ( sup_su6804446743777130803_a_nat @ X2 @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_549_inf__sup__ord_I3_J,axiom,
! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ X2 @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_550_le__supE,axiom,
! [A2: set_nat,B2: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ X2 )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ X2 )
=> ~ ( ord_less_eq_set_nat @ B2 @ X2 ) ) ) ).
% le_supE
thf(fact_551_le__supE,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,X2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ ( sup_su4083067149120280889_a_nat @ A2 @ B2 ) @ X2 )
=> ~ ( ( ord_le1147066620699065093_a_nat @ A2 @ X2 )
=> ~ ( ord_le1147066620699065093_a_nat @ B2 @ X2 ) ) ) ).
% le_supE
thf(fact_552_le__supE,axiom,
! [A2: set_a,B2: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ X2 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ X2 )
=> ~ ( ord_less_eq_set_a @ B2 @ X2 ) ) ) ).
% le_supE
thf(fact_553_le__supE,axiom,
! [A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,X2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) @ X2 )
=> ~ ( ( ord_le1325389633284124927_a_nat @ A2 @ X2 )
=> ~ ( ord_le1325389633284124927_a_nat @ B2 @ X2 ) ) ) ).
% le_supE
thf(fact_554_le__supE,axiom,
! [A2: nat,B2: nat,X2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ X2 )
=> ~ ( ( ord_less_eq_nat @ A2 @ X2 )
=> ~ ( ord_less_eq_nat @ B2 @ X2 ) ) ) ).
% le_supE
thf(fact_555_le__supI,axiom,
! [A2: set_nat,X2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ X2 )
=> ( ( ord_less_eq_set_nat @ B2 @ X2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ X2 ) ) ) ).
% le_supI
thf(fact_556_le__supI,axiom,
! [A2: set_li6526943997496501093_a_nat,X2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A2 @ X2 )
=> ( ( ord_le1147066620699065093_a_nat @ B2 @ X2 )
=> ( ord_le1147066620699065093_a_nat @ ( sup_su4083067149120280889_a_nat @ A2 @ B2 ) @ X2 ) ) ) ).
% le_supI
thf(fact_557_le__supI,axiom,
! [A2: set_a,X2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ X2 )
=> ( ( ord_less_eq_set_a @ B2 @ X2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ X2 ) ) ) ).
% le_supI
thf(fact_558_le__supI,axiom,
! [A2: set_Sum_sum_a_nat,X2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A2 @ X2 )
=> ( ( ord_le1325389633284124927_a_nat @ B2 @ X2 )
=> ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) @ X2 ) ) ) ).
% le_supI
thf(fact_559_le__supI,axiom,
! [A2: nat,X2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ X2 )
=> ( ( ord_less_eq_nat @ B2 @ X2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ X2 ) ) ) ).
% le_supI
thf(fact_560_sup__ge1,axiom,
! [X2: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ X2 @ Y2 ) ) ).
% sup_ge1
thf(fact_561_sup__ge1,axiom,
! [X2: set_li6526943997496501093_a_nat,Y2: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ X2 @ ( sup_su4083067149120280889_a_nat @ X2 @ Y2 ) ) ).
% sup_ge1
thf(fact_562_sup__ge1,axiom,
! [X2: set_a,Y2: set_a] : ( ord_less_eq_set_a @ X2 @ ( sup_sup_set_a @ X2 @ Y2 ) ) ).
% sup_ge1
thf(fact_563_sup__ge1,axiom,
! [X2: set_Sum_sum_a_nat,Y2: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ X2 @ ( sup_su6804446743777130803_a_nat @ X2 @ Y2 ) ) ).
% sup_ge1
thf(fact_564_sup__ge1,axiom,
! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ X2 @ Y2 ) ) ).
% sup_ge1
thf(fact_565_sup__ge2,axiom,
! [Y2: set_nat,X2: set_nat] : ( ord_less_eq_set_nat @ Y2 @ ( sup_sup_set_nat @ X2 @ Y2 ) ) ).
% sup_ge2
thf(fact_566_sup__ge2,axiom,
! [Y2: set_li6526943997496501093_a_nat,X2: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ Y2 @ ( sup_su4083067149120280889_a_nat @ X2 @ Y2 ) ) ).
% sup_ge2
thf(fact_567_sup__ge2,axiom,
! [Y2: set_a,X2: set_a] : ( ord_less_eq_set_a @ Y2 @ ( sup_sup_set_a @ X2 @ Y2 ) ) ).
% sup_ge2
thf(fact_568_sup__ge2,axiom,
! [Y2: set_Sum_sum_a_nat,X2: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ Y2 @ ( sup_su6804446743777130803_a_nat @ X2 @ Y2 ) ) ).
% sup_ge2
thf(fact_569_sup__ge2,axiom,
! [Y2: nat,X2: nat] : ( ord_less_eq_nat @ Y2 @ ( sup_sup_nat @ X2 @ Y2 ) ) ).
% sup_ge2
thf(fact_570_le__supI1,axiom,
! [X2: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_571_le__supI1,axiom,
! [X2: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ X2 @ A2 )
=> ( ord_le1147066620699065093_a_nat @ X2 @ ( sup_su4083067149120280889_a_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_572_le__supI1,axiom,
! [X2: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ X2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_573_le__supI1,axiom,
! [X2: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X2 @ A2 )
=> ( ord_le1325389633284124927_a_nat @ X2 @ ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_574_le__supI1,axiom,
! [X2: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X2 @ A2 )
=> ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_575_le__supI2,axiom,
! [X2: set_nat,B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ B2 )
=> ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_576_le__supI2,axiom,
! [X2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ X2 @ B2 )
=> ( ord_le1147066620699065093_a_nat @ X2 @ ( sup_su4083067149120280889_a_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_577_le__supI2,axiom,
! [X2: set_a,B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ X2 @ B2 )
=> ( ord_less_eq_set_a @ X2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_578_le__supI2,axiom,
! [X2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X2 @ B2 )
=> ( ord_le1325389633284124927_a_nat @ X2 @ ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_579_le__supI2,axiom,
! [X2: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ X2 @ B2 )
=> ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_580_sup_Omono,axiom,
! [C: set_nat,A2: set_nat,D: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C @ A2 )
=> ( ( ord_less_eq_set_nat @ D @ B2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ C @ D ) @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_581_sup_Omono,axiom,
! [C: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat,D: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ C @ A2 )
=> ( ( ord_le1147066620699065093_a_nat @ D @ B2 )
=> ( ord_le1147066620699065093_a_nat @ ( sup_su4083067149120280889_a_nat @ C @ D ) @ ( sup_su4083067149120280889_a_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_582_sup_Omono,axiom,
! [C: set_a,A2: set_a,D: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C @ A2 )
=> ( ( ord_less_eq_set_a @ D @ B2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ C @ D ) @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_583_sup_Omono,axiom,
! [C: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat,D: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ C @ A2 )
=> ( ( ord_le1325389633284124927_a_nat @ D @ B2 )
=> ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ C @ D ) @ ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_584_sup_Omono,axiom,
! [C: nat,A2: nat,D: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ( ord_less_eq_nat @ D @ B2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D ) @ ( sup_sup_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_585_sup__mono,axiom,
! [A2: set_nat,C: set_nat,B2: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ( ord_less_eq_set_nat @ B2 @ D )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ ( sup_sup_set_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_586_sup__mono,axiom,
! [A2: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,D: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A2 @ C )
=> ( ( ord_le1147066620699065093_a_nat @ B2 @ D )
=> ( ord_le1147066620699065093_a_nat @ ( sup_su4083067149120280889_a_nat @ A2 @ B2 ) @ ( sup_su4083067149120280889_a_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_587_sup__mono,axiom,
! [A2: set_a,C: set_a,B2: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B2 @ D )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ ( sup_sup_set_a @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_588_sup__mono,axiom,
! [A2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,D: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A2 @ C )
=> ( ( ord_le1325389633284124927_a_nat @ B2 @ D )
=> ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) @ ( sup_su6804446743777130803_a_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_589_sup__mono,axiom,
! [A2: nat,C: nat,B2: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B2 @ D )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ ( sup_sup_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_590_sup__least,axiom,
! [Y2: set_nat,X2: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_set_nat @ Z @ X2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ Y2 @ Z ) @ X2 ) ) ) ).
% sup_least
thf(fact_591_sup__least,axiom,
! [Y2: set_li6526943997496501093_a_nat,X2: set_li6526943997496501093_a_nat,Z: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ Y2 @ X2 )
=> ( ( ord_le1147066620699065093_a_nat @ Z @ X2 )
=> ( ord_le1147066620699065093_a_nat @ ( sup_su4083067149120280889_a_nat @ Y2 @ Z ) @ X2 ) ) ) ).
% sup_least
thf(fact_592_sup__least,axiom,
! [Y2: set_a,X2: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X2 )
=> ( ( ord_less_eq_set_a @ Z @ X2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ Y2 @ Z ) @ X2 ) ) ) ).
% sup_least
thf(fact_593_sup__least,axiom,
! [Y2: set_Sum_sum_a_nat,X2: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ Y2 @ X2 )
=> ( ( ord_le1325389633284124927_a_nat @ Z @ X2 )
=> ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ Y2 @ Z ) @ X2 ) ) ) ).
% sup_least
thf(fact_594_sup__least,axiom,
! [Y2: nat,X2: nat,Z: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_nat @ Z @ X2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y2 @ Z ) @ X2 ) ) ) ).
% sup_least
thf(fact_595_le__iff__sup,axiom,
( ord_less_eq_set_nat
= ( ^ [X: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ X @ Y )
= Y ) ) ) ).
% le_iff_sup
thf(fact_596_le__iff__sup,axiom,
( ord_le1147066620699065093_a_nat
= ( ^ [X: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ X @ Y )
= Y ) ) ) ).
% le_iff_sup
thf(fact_597_le__iff__sup,axiom,
( ord_less_eq_set_a
= ( ^ [X: set_a,Y: set_a] :
( ( sup_sup_set_a @ X @ Y )
= Y ) ) ) ).
% le_iff_sup
thf(fact_598_le__iff__sup,axiom,
( ord_le1325389633284124927_a_nat
= ( ^ [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ X @ Y )
= Y ) ) ) ).
% le_iff_sup
thf(fact_599_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y: nat] :
( ( sup_sup_nat @ X @ Y )
= Y ) ) ) ).
% le_iff_sup
thf(fact_600_sup_OorderE,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( A2
= ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_601_sup_OorderE,axiom,
! [B2: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ B2 @ A2 )
=> ( A2
= ( sup_su4083067149120280889_a_nat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_602_sup_OorderE,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( A2
= ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_603_sup_OorderE,axiom,
! [B2: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B2 @ A2 )
=> ( A2
= ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_604_sup_OorderE,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2
= ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_605_sup_OorderI,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2
= ( sup_sup_set_nat @ A2 @ B2 ) )
=> ( ord_less_eq_set_nat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_606_sup_OorderI,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( A2
= ( sup_su4083067149120280889_a_nat @ A2 @ B2 ) )
=> ( ord_le1147066620699065093_a_nat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_607_sup_OorderI,axiom,
! [A2: set_a,B2: set_a] :
( ( A2
= ( sup_sup_set_a @ A2 @ B2 ) )
=> ( ord_less_eq_set_a @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_608_sup_OorderI,axiom,
! [A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( A2
= ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) )
=> ( ord_le1325389633284124927_a_nat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_609_sup_OorderI,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( sup_sup_nat @ A2 @ B2 ) )
=> ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_610_sup__unique,axiom,
! [F: set_nat > set_nat > set_nat,X2: set_nat,Y2: set_nat] :
( ! [X3: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ X3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ Y3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ Y3 @ X3 )
=> ( ( ord_less_eq_set_nat @ Z2 @ X3 )
=> ( ord_less_eq_set_nat @ ( F @ Y3 @ Z2 ) @ X3 ) ) )
=> ( ( sup_sup_set_nat @ X2 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_611_sup__unique,axiom,
! [F: set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,X2: set_li6526943997496501093_a_nat,Y2: set_li6526943997496501093_a_nat] :
( ! [X3: set_li6526943997496501093_a_nat,Y3: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ X3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: set_li6526943997496501093_a_nat,Y3: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ Y3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: set_li6526943997496501093_a_nat,Y3: set_li6526943997496501093_a_nat,Z2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ Y3 @ X3 )
=> ( ( ord_le1147066620699065093_a_nat @ Z2 @ X3 )
=> ( ord_le1147066620699065093_a_nat @ ( F @ Y3 @ Z2 ) @ X3 ) ) )
=> ( ( sup_su4083067149120280889_a_nat @ X2 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_612_sup__unique,axiom,
! [F: set_a > set_a > set_a,X2: set_a,Y2: set_a] :
( ! [X3: set_a,Y3: set_a] : ( ord_less_eq_set_a @ X3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: set_a,Y3: set_a] : ( ord_less_eq_set_a @ Y3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: set_a,Y3: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X3 )
=> ( ( ord_less_eq_set_a @ Z2 @ X3 )
=> ( ord_less_eq_set_a @ ( F @ Y3 @ Z2 ) @ X3 ) ) )
=> ( ( sup_sup_set_a @ X2 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_613_sup__unique,axiom,
! [F: set_Sum_sum_a_nat > set_Sum_sum_a_nat > set_Sum_sum_a_nat,X2: set_Sum_sum_a_nat,Y2: set_Sum_sum_a_nat] :
( ! [X3: set_Sum_sum_a_nat,Y3: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ X3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: set_Sum_sum_a_nat,Y3: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ Y3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: set_Sum_sum_a_nat,Y3: set_Sum_sum_a_nat,Z2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ Y3 @ X3 )
=> ( ( ord_le1325389633284124927_a_nat @ Z2 @ X3 )
=> ( ord_le1325389633284124927_a_nat @ ( F @ Y3 @ Z2 ) @ X3 ) ) )
=> ( ( sup_su6804446743777130803_a_nat @ X2 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_614_sup__unique,axiom,
! [F: nat > nat > nat,X2: nat,Y2: nat] :
( ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ X3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: nat,Y3: nat,Z2: nat] :
( ( ord_less_eq_nat @ Y3 @ X3 )
=> ( ( ord_less_eq_nat @ Z2 @ X3 )
=> ( ord_less_eq_nat @ ( F @ Y3 @ Z2 ) @ X3 ) ) )
=> ( ( sup_sup_nat @ X2 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_615_sup_Oabsorb1,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( sup_sup_set_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_616_sup_Oabsorb1,axiom,
! [B2: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ B2 @ A2 )
=> ( ( sup_su4083067149120280889_a_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_617_sup_Oabsorb1,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( sup_sup_set_a @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_618_sup_Oabsorb1,axiom,
! [B2: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B2 @ A2 )
=> ( ( sup_su6804446743777130803_a_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_619_sup_Oabsorb1,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_620_sup_Oabsorb2,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( sup_sup_set_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_621_sup_Oabsorb2,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A2 @ B2 )
=> ( ( sup_su4083067149120280889_a_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_622_sup_Oabsorb2,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( sup_sup_set_a @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_623_sup_Oabsorb2,axiom,
! [A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A2 @ B2 )
=> ( ( sup_su6804446743777130803_a_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_624_sup_Oabsorb2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_625_sup__absorb1,axiom,
! [Y2: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X2 )
=> ( ( sup_sup_set_nat @ X2 @ Y2 )
= X2 ) ) ).
% sup_absorb1
thf(fact_626_sup__absorb1,axiom,
! [Y2: set_li6526943997496501093_a_nat,X2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ Y2 @ X2 )
=> ( ( sup_su4083067149120280889_a_nat @ X2 @ Y2 )
= X2 ) ) ).
% sup_absorb1
thf(fact_627_sup__absorb1,axiom,
! [Y2: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X2 )
=> ( ( sup_sup_set_a @ X2 @ Y2 )
= X2 ) ) ).
% sup_absorb1
thf(fact_628_sup__absorb1,axiom,
! [Y2: set_Sum_sum_a_nat,X2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ Y2 @ X2 )
=> ( ( sup_su6804446743777130803_a_nat @ X2 @ Y2 )
= X2 ) ) ).
% sup_absorb1
thf(fact_629_sup__absorb1,axiom,
! [Y2: nat,X2: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ( sup_sup_nat @ X2 @ Y2 )
= X2 ) ) ).
% sup_absorb1
thf(fact_630_sup__absorb2,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( sup_sup_set_nat @ X2 @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_631_sup__absorb2,axiom,
! [X2: set_li6526943997496501093_a_nat,Y2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ X2 @ Y2 )
=> ( ( sup_su4083067149120280889_a_nat @ X2 @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_632_sup__absorb2,axiom,
! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ( sup_sup_set_a @ X2 @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_633_sup__absorb2,axiom,
! [X2: set_Sum_sum_a_nat,Y2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X2 @ Y2 )
=> ( ( sup_su6804446743777130803_a_nat @ X2 @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_634_sup__absorb2,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( sup_sup_nat @ X2 @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_635_sup_OboundedE,axiom,
! [B2: set_nat,C: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ~ ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_636_sup_OboundedE,axiom,
! [B2: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ ( sup_su4083067149120280889_a_nat @ B2 @ C ) @ A2 )
=> ~ ( ( ord_le1147066620699065093_a_nat @ B2 @ A2 )
=> ~ ( ord_le1147066620699065093_a_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_637_sup_OboundedE,axiom,
! [B2: set_a,C: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ~ ( ord_less_eq_set_a @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_638_sup_OboundedE,axiom,
! [B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ B2 @ C ) @ A2 )
=> ~ ( ( ord_le1325389633284124927_a_nat @ B2 @ A2 )
=> ~ ( ord_le1325389633284124927_a_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_639_sup_OboundedE,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_nat @ B2 @ A2 )
=> ~ ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_640_sup_OboundedI,axiom,
! [B2: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( ord_less_eq_set_nat @ C @ A2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_641_sup_OboundedI,axiom,
! [B2: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ B2 @ A2 )
=> ( ( ord_le1147066620699065093_a_nat @ C @ A2 )
=> ( ord_le1147066620699065093_a_nat @ ( sup_su4083067149120280889_a_nat @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_642_sup_OboundedI,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( ord_less_eq_set_a @ C @ A2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_643_sup_OboundedI,axiom,
! [B2: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B2 @ A2 )
=> ( ( ord_le1325389633284124927_a_nat @ C @ A2 )
=> ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_644_sup_OboundedI,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_645_sup_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [B3: set_nat,A3: set_nat] :
( A3
= ( sup_sup_set_nat @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_646_sup_Oorder__iff,axiom,
( ord_le1147066620699065093_a_nat
= ( ^ [B3: set_li6526943997496501093_a_nat,A3: set_li6526943997496501093_a_nat] :
( A3
= ( sup_su4083067149120280889_a_nat @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_647_sup_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A3: set_a] :
( A3
= ( sup_sup_set_a @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_648_sup_Oorder__iff,axiom,
( ord_le1325389633284124927_a_nat
= ( ^ [B3: set_Sum_sum_a_nat,A3: set_Sum_sum_a_nat] :
( A3
= ( sup_su6804446743777130803_a_nat @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_649_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( A3
= ( sup_sup_nat @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_650_sup_Ocobounded1,axiom,
! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_651_sup_Ocobounded1,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ A2 @ ( sup_su4083067149120280889_a_nat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_652_sup_Ocobounded1,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_653_sup_Ocobounded1,axiom,
! [A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ A2 @ ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_654_sup_Ocobounded1,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ A2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_655_sup_Ocobounded2,axiom,
! [B2: set_nat,A2: set_nat] : ( ord_less_eq_set_nat @ B2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_656_sup_Ocobounded2,axiom,
! [B2: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ B2 @ ( sup_su4083067149120280889_a_nat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_657_sup_Ocobounded2,axiom,
! [B2: set_a,A2: set_a] : ( ord_less_eq_set_a @ B2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_658_sup_Ocobounded2,axiom,
! [B2: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ B2 @ ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_659_sup_Ocobounded2,axiom,
! [B2: nat,A2: nat] : ( ord_less_eq_nat @ B2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_660_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [B3: set_nat,A3: set_nat] :
( ( sup_sup_set_nat @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_661_sup_Oabsorb__iff1,axiom,
( ord_le1147066620699065093_a_nat
= ( ^ [B3: set_li6526943997496501093_a_nat,A3: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_662_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A3: set_a] :
( ( sup_sup_set_a @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_663_sup_Oabsorb__iff1,axiom,
( ord_le1325389633284124927_a_nat
= ( ^ [B3: set_Sum_sum_a_nat,A3: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_664_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( ( sup_sup_nat @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_665_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( sup_sup_set_nat @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_666_sup_Oabsorb__iff2,axiom,
( ord_le1147066620699065093_a_nat
= ( ^ [A3: set_li6526943997496501093_a_nat,B3: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_667_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( sup_sup_set_a @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_668_sup_Oabsorb__iff2,axiom,
( ord_le1325389633284124927_a_nat
= ( ^ [A3: set_Sum_sum_a_nat,B3: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_669_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( ( sup_sup_nat @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_670_sup_OcoboundedI1,axiom,
! [C: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C @ A2 )
=> ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_671_sup_OcoboundedI1,axiom,
! [C: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ C @ A2 )
=> ( ord_le1147066620699065093_a_nat @ C @ ( sup_su4083067149120280889_a_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_672_sup_OcoboundedI1,axiom,
! [C: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C @ A2 )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_673_sup_OcoboundedI1,axiom,
! [C: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ C @ A2 )
=> ( ord_le1325389633284124927_a_nat @ C @ ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_674_sup_OcoboundedI1,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_675_sup_OcoboundedI2,axiom,
! [C: set_nat,B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ C @ B2 )
=> ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_676_sup_OcoboundedI2,axiom,
! [C: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ C @ B2 )
=> ( ord_le1147066620699065093_a_nat @ C @ ( sup_su4083067149120280889_a_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_677_sup_OcoboundedI2,axiom,
! [C: set_a,B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ C @ B2 )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_678_sup_OcoboundedI2,axiom,
! [C: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ C @ B2 )
=> ( ord_le1325389633284124927_a_nat @ C @ ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_679_sup_OcoboundedI2,axiom,
! [C: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_680_subset__code_I1_J,axiom,
! [Xs: list_l4703314356710769291_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ ( set_li2392974972034027290_a_nat @ Xs ) @ B )
= ( ! [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ ( set_li2392974972034027290_a_nat @ Xs ) )
=> ( member408289922725080238_a_nat @ X @ B ) ) ) ) ).
% subset_code(1)
thf(fact_681_subset__code_I1_J,axiom,
! [Xs: list_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( member_nat @ X @ B ) ) ) ) ).
% subset_code(1)
thf(fact_682_subset__code_I1_J,axiom,
! [Xs: list_a,B: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ B )
= ( ! [X: a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( member_a @ X @ B ) ) ) ) ).
% subset_code(1)
thf(fact_683_subset__code_I1_J,axiom,
! [Xs: list_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) @ B )
= ( ! [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ( member_Sum_sum_a_nat @ X @ B ) ) ) ) ).
% subset_code(1)
thf(fact_684_Un__mono,axiom,
! [A: set_nat,C2: set_nat,B: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B @ D2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sup_sup_set_nat @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_685_Un__mono,axiom,
! [A: set_li6526943997496501093_a_nat,C2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,D2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A @ C2 )
=> ( ( ord_le1147066620699065093_a_nat @ B @ D2 )
=> ( ord_le1147066620699065093_a_nat @ ( sup_su4083067149120280889_a_nat @ A @ B ) @ ( sup_su4083067149120280889_a_nat @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_686_Un__mono,axiom,
! [A: set_a,C2: set_a,B: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B @ D2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ ( sup_sup_set_a @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_687_Un__mono,axiom,
! [A: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,D2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ C2 )
=> ( ( ord_le1325389633284124927_a_nat @ B @ D2 )
=> ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ A @ B ) @ ( sup_su6804446743777130803_a_nat @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_688_Un__least,axiom,
! [A: set_nat,C2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_689_Un__least,axiom,
! [A: set_li6526943997496501093_a_nat,C2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A @ C2 )
=> ( ( ord_le1147066620699065093_a_nat @ B @ C2 )
=> ( ord_le1147066620699065093_a_nat @ ( sup_su4083067149120280889_a_nat @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_690_Un__least,axiom,
! [A: set_a,C2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_691_Un__least,axiom,
! [A: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ C2 )
=> ( ( ord_le1325389633284124927_a_nat @ B @ C2 )
=> ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_692_Un__upper1,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) ) ).
% Un_upper1
thf(fact_693_Un__upper1,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ A @ ( sup_su4083067149120280889_a_nat @ A @ B ) ) ).
% Un_upper1
thf(fact_694_Un__upper1,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ A @ B ) ) ).
% Un_upper1
thf(fact_695_Un__upper1,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ A @ ( sup_su6804446743777130803_a_nat @ A @ B ) ) ).
% Un_upper1
thf(fact_696_Un__upper2,axiom,
! [B: set_nat,A: set_nat] : ( ord_less_eq_set_nat @ B @ ( sup_sup_set_nat @ A @ B ) ) ).
% Un_upper2
thf(fact_697_Un__upper2,axiom,
! [B: set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ B @ ( sup_su4083067149120280889_a_nat @ A @ B ) ) ).
% Un_upper2
thf(fact_698_Un__upper2,axiom,
! [B: set_a,A: set_a] : ( ord_less_eq_set_a @ B @ ( sup_sup_set_a @ A @ B ) ) ).
% Un_upper2
thf(fact_699_Un__upper2,axiom,
! [B: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ B @ ( sup_su6804446743777130803_a_nat @ A @ B ) ) ).
% Un_upper2
thf(fact_700_Un__absorb1,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( sup_sup_set_nat @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_701_Un__absorb1,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A @ B )
=> ( ( sup_su4083067149120280889_a_nat @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_702_Un__absorb1,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( sup_sup_set_a @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_703_Un__absorb1,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ B )
=> ( ( sup_su6804446743777130803_a_nat @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_704_Un__absorb2,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( sup_sup_set_nat @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_705_Un__absorb2,axiom,
! [B: set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ B @ A )
=> ( ( sup_su4083067149120280889_a_nat @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_706_Un__absorb2,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( sup_sup_set_a @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_707_Un__absorb2,axiom,
! [B: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B @ A )
=> ( ( sup_su6804446743777130803_a_nat @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_708_subset__UnE,axiom,
! [C2: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) )
=> ~ ! [A6: set_nat] :
( ( ord_less_eq_set_nat @ A6 @ A )
=> ! [B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ B )
=> ( C2
!= ( sup_sup_set_nat @ A6 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_709_subset__UnE,axiom,
! [C2: set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ C2 @ ( sup_su4083067149120280889_a_nat @ A @ B ) )
=> ~ ! [A6: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A6 @ A )
=> ! [B5: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ B5 @ B )
=> ( C2
!= ( sup_su4083067149120280889_a_nat @ A6 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_710_subset__UnE,axiom,
! [C2: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A @ B ) )
=> ~ ! [A6: set_a] :
( ( ord_less_eq_set_a @ A6 @ A )
=> ! [B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ B )
=> ( C2
!= ( sup_sup_set_a @ A6 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_711_subset__UnE,axiom,
! [C2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ C2 @ ( sup_su6804446743777130803_a_nat @ A @ B ) )
=> ~ ! [A6: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A6 @ A )
=> ! [B5: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B5 @ B )
=> ( C2
!= ( sup_su6804446743777130803_a_nat @ A6 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_712_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_713_subset__Un__eq,axiom,
( ord_le1147066620699065093_a_nat
= ( ^ [A4: set_li6526943997496501093_a_nat,B4: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ A4 @ B4 )
= B4 ) ) ) ).
% subset_Un_eq
thf(fact_714_subset__Un__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( sup_sup_set_a @ A4 @ B4 )
= B4 ) ) ) ).
% subset_Un_eq
thf(fact_715_subset__Un__eq,axiom,
( ord_le1325389633284124927_a_nat
= ( ^ [A4: set_Sum_sum_a_nat,B4: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A4 @ B4 )
= B4 ) ) ) ).
% subset_Un_eq
thf(fact_716_Product__Type_OCollect__case__prodD,axiom,
! [X2: produc5454855657571582244_a_nat,A: nat > sum_sum_a_nat > $o] :
( ( member3650934779329710139_a_nat @ X2 @ ( collec2196941500952594553_a_nat @ ( produc8810490045567577941_nat_o @ A ) ) )
=> ( A @ ( produc1510075366785312874_a_nat @ X2 ) @ ( produc1280857981117566888_a_nat @ X2 ) ) ) ).
% Product_Type.Collect_case_prodD
thf(fact_717_case__prod__beta,axiom,
( produc8810490045567577941_nat_o
= ( ^ [F2: nat > sum_sum_a_nat > $o,P3: produc5454855657571582244_a_nat] : ( F2 @ ( produc1510075366785312874_a_nat @ P3 ) @ ( produc1280857981117566888_a_nat @ P3 ) ) ) ) ).
% case_prod_beta
thf(fact_718_split__beta,axiom,
( produc8810490045567577941_nat_o
= ( ^ [F2: nat > sum_sum_a_nat > $o,Prod3: produc5454855657571582244_a_nat] : ( F2 @ ( produc1510075366785312874_a_nat @ Prod3 ) @ ( produc1280857981117566888_a_nat @ Prod3 ) ) ) ) ).
% split_beta
thf(fact_719_ad__agr__list__mono,axiom,
! [X4: set_a,Y4: set_a,Ys: list_Sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( ord_less_eq_set_a @ X4 @ Y4 )
=> ( ( ad_agr_list_a_nat @ Y4 @ Ys @ Xs )
=> ( ad_agr_list_a_nat @ X4 @ Ys @ Xs ) ) ) ).
% ad_agr_list_mono
thf(fact_720_ad__agr__list__rev__mono,axiom,
! [Y4: set_a,X4: set_a,Ys: list_Sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( ord_less_eq_set_a @ Y4 @ X4 )
=> ( ( ad_agr_list_a_nat @ Y4 @ Ys @ Xs )
=> ( ( ord_less_eq_set_a @ ( vimage4607795094749595324_a_nat @ sum_Inl_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) ) @ Y4 )
=> ( ( ord_less_eq_set_a @ ( vimage4607795094749595324_a_nat @ sum_Inl_a_nat @ ( set_Sum_sum_a_nat2 @ Ys ) ) @ Y4 )
=> ( ad_agr_list_a_nat @ X4 @ Ys @ Xs ) ) ) ) ) ).
% ad_agr_list_rev_mono
thf(fact_721_fo__nmlzd__mono,axiom,
! [Xs: list_Sum_sum_a_nat,AD: set_a,AD2: set_a] :
( ( ord_less_eq_set_a @ ( vimage4607795094749595324_a_nat @ sum_Inl_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) ) @ AD )
=> ( ( fo_nmlzd_a @ AD2 @ Xs )
=> ( fo_nmlzd_a @ AD @ Xs ) ) ) ).
% fo_nmlzd_mono
thf(fact_722_fo__nmlzd__mono,axiom,
! [Xs: list_S5657819638503513606at_nat,AD: set_Sum_sum_a_nat,AD2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( vimage7613534090772763900at_nat @ sum_In1625650605130369377at_nat @ ( set_Su2720755619057964565at_nat @ Xs ) ) @ AD )
=> ( ( fo_nml5488872492918501081_a_nat @ AD2 @ Xs )
=> ( fo_nml5488872492918501081_a_nat @ AD @ Xs ) ) ) ).
% fo_nmlzd_mono
thf(fact_723_exE__realizer,axiom,
! [P: sum_sum_a_nat > nat > $o,P2: produc5454855657571582244_a_nat,Q: $o > $o,F: nat > sum_sum_a_nat > $o] :
( ( P @ ( produc1280857981117566888_a_nat @ P2 ) @ ( produc1510075366785312874_a_nat @ P2 ) )
=> ( ! [X3: nat,Y3: sum_sum_a_nat] :
( ( P @ Y3 @ X3 )
=> ( Q @ ( F @ X3 @ Y3 ) ) )
=> ( Q @ ( produc8810490045567577941_nat_o @ F @ P2 ) ) ) ) ).
% exE_realizer
thf(fact_724_case__prod__beta_H,axiom,
( produc8810490045567577941_nat_o
= ( ^ [F2: nat > sum_sum_a_nat > $o,X: produc5454855657571582244_a_nat] : ( F2 @ ( produc1510075366785312874_a_nat @ X ) @ ( produc1280857981117566888_a_nat @ X ) ) ) ) ).
% case_prod_beta'
thf(fact_725_case__prod__unfold,axiom,
( produc8810490045567577941_nat_o
= ( ^ [C4: nat > sum_sum_a_nat > $o,P3: produc5454855657571582244_a_nat] : ( C4 @ ( produc1510075366785312874_a_nat @ P3 ) @ ( produc1280857981117566888_a_nat @ P3 ) ) ) ) ).
% case_prod_unfold
thf(fact_726_fo__nmlzd__all__AD,axiom,
! [Xs: list_Sum_sum_a_nat,AD: set_a] :
( ( ord_le1325389633284124927_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) @ ( image_7873763678140191238_a_nat @ sum_Inl_a_nat @ AD ) )
=> ( fo_nmlzd_a @ AD @ Xs ) ) ).
% fo_nmlzd_all_AD
thf(fact_727_image__set,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,Xs: list_l4703314356710769291_a_nat] :
( ( image_5081948215111134021_a_nat @ F @ ( set_li2392974972034027290_a_nat @ Xs ) )
= ( set_li2392974972034027290_a_nat @ ( map_li6507455427659069316_a_nat @ F @ Xs ) ) ) ).
% image_set
thf(fact_728_image__set,axiom,
! [F: produc5454855657571582244_a_nat > nat,Xs: list_P5056861408695629236_a_nat] :
( ( image_1677410150705809403at_nat @ F @ ( set_Pr2196415342531872975_a_nat @ Xs ) )
= ( set_nat2 @ ( map_Pr2058079554187054650at_nat @ F @ Xs ) ) ) ).
% image_set
thf(fact_729_image__set,axiom,
! [F: produc5454855657571582244_a_nat > sum_sum_a_nat,Xs: list_P5056861408695629236_a_nat] :
( ( image_5908522764169838868_a_nat @ F @ ( set_Pr2196415342531872975_a_nat @ Xs ) )
= ( set_Sum_sum_a_nat2 @ ( map_Pr2515564165894599445_a_nat @ F @ Xs ) ) ) ).
% image_set
thf(fact_730_image__set,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( image_nat_nat @ F @ ( set_nat2 @ Xs ) )
= ( set_nat2 @ ( map_nat_nat @ F @ Xs ) ) ) ).
% image_set
thf(fact_731_image__set,axiom,
! [F: nat > sum_sum_a_nat,Xs: list_nat] :
( ( image_7293268710728258664_a_nat @ F @ ( set_nat2 @ Xs ) )
= ( set_Sum_sum_a_nat2 @ ( map_na823391071729141993_a_nat @ F @ Xs ) ) ) ).
% image_set
thf(fact_732_image__set,axiom,
! [F: sum_sum_a_nat > nat,Xs: list_Sum_sum_a_nat] :
( ( image_2473878607534554506at_nat @ F @ ( set_Sum_sum_a_nat2 @ Xs ) )
= ( set_nat2 @ ( map_Su5227373005390213643at_nat @ F @ Xs ) ) ) ).
% image_set
thf(fact_733_image__set,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( image_7142520692256960453_a_nat @ F @ ( set_Sum_sum_a_nat2 @ Xs ) )
= ( set_Sum_sum_a_nat2 @ ( map_Su2790769393171190532_a_nat @ F @ Xs ) ) ) ).
% image_set
thf(fact_734_filter__is__subset,axiom,
! [P: produc5454855657571582244_a_nat > $o,Xs: list_P5056861408695629236_a_nat] : ( ord_le8420040822840260538_a_nat @ ( set_Pr2196415342531872975_a_nat @ ( filter1580576373184442087_a_nat @ P @ Xs ) ) @ ( set_Pr2196415342531872975_a_nat @ Xs ) ) ).
% filter_is_subset
thf(fact_735_filter__is__subset,axiom,
! [P: nat > $o,Xs: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( filter_nat @ P @ Xs ) ) @ ( set_nat2 @ Xs ) ) ).
% filter_is_subset
thf(fact_736_filter__is__subset,axiom,
! [P: a > $o,Xs: list_a] : ( ord_less_eq_set_a @ ( set_a2 @ ( filter_a @ P @ Xs ) ) @ ( set_a2 @ Xs ) ) ).
% filter_is_subset
thf(fact_737_filter__is__subset,axiom,
! [P: sum_sum_a_nat > $o,Xs: list_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( set_Sum_sum_a_nat2 @ ( filter_Sum_sum_a_nat @ P @ Xs ) ) @ ( set_Sum_sum_a_nat2 @ Xs ) ) ).
% filter_is_subset
thf(fact_738_zip__map__fst__snd,axiom,
! [Zs: list_P5056861408695629236_a_nat] :
( ( zip_na2013496608136855606_a_nat @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Zs ) @ ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat @ Zs ) )
= Zs ) ).
% zip_map_fst_snd
thf(fact_739_boolean__algebra__cancel_Osup1,axiom,
! [A: set_nat,K: set_nat,A2: set_nat,B2: set_nat] :
( ( A
= ( sup_sup_set_nat @ K @ A2 ) )
=> ( ( sup_sup_set_nat @ A @ B2 )
= ( sup_sup_set_nat @ K @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_740_boolean__algebra__cancel_Osup1,axiom,
! [A: set_Sum_sum_a_nat,K: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( A
= ( sup_su6804446743777130803_a_nat @ K @ A2 ) )
=> ( ( sup_su6804446743777130803_a_nat @ A @ B2 )
= ( sup_su6804446743777130803_a_nat @ K @ ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_741_boolean__algebra__cancel_Osup1,axiom,
! [A: set_a,K: set_a,A2: set_a,B2: set_a] :
( ( A
= ( sup_sup_set_a @ K @ A2 ) )
=> ( ( sup_sup_set_a @ A @ B2 )
= ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_742_boolean__algebra__cancel_Osup1,axiom,
! [A: set_li6526943997496501093_a_nat,K: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( A
= ( sup_su4083067149120280889_a_nat @ K @ A2 ) )
=> ( ( sup_su4083067149120280889_a_nat @ A @ B2 )
= ( sup_su4083067149120280889_a_nat @ K @ ( sup_su4083067149120280889_a_nat @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_743_boolean__algebra__cancel_Osup2,axiom,
! [B: set_nat,K: set_nat,B2: set_nat,A2: set_nat] :
( ( B
= ( sup_sup_set_nat @ K @ B2 ) )
=> ( ( sup_sup_set_nat @ A2 @ B )
= ( sup_sup_set_nat @ K @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_744_boolean__algebra__cancel_Osup2,axiom,
! [B: set_Sum_sum_a_nat,K: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
( ( B
= ( sup_su6804446743777130803_a_nat @ K @ B2 ) )
=> ( ( sup_su6804446743777130803_a_nat @ A2 @ B )
= ( sup_su6804446743777130803_a_nat @ K @ ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_745_boolean__algebra__cancel_Osup2,axiom,
! [B: set_a,K: set_a,B2: set_a,A2: set_a] :
( ( B
= ( sup_sup_set_a @ K @ B2 ) )
=> ( ( sup_sup_set_a @ A2 @ B )
= ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_746_boolean__algebra__cancel_Osup2,axiom,
! [B: set_li6526943997496501093_a_nat,K: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( B
= ( sup_su4083067149120280889_a_nat @ K @ B2 ) )
=> ( ( sup_su4083067149120280889_a_nat @ A2 @ B )
= ( sup_su4083067149120280889_a_nat @ K @ ( sup_su4083067149120280889_a_nat @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_747_ad__agr__list__subset,axiom,
! [Ms: list_P5056861408695629236_a_nat,Ns: list_P5056861408695629236_a_nat,X4: set_a,Sigma: produc5454855657571582244_a_nat > sum_sum_a_nat,Sigma2: produc5454855657571582244_a_nat > sum_sum_a_nat] :
( ( ord_le8420040822840260538_a_nat @ ( set_Pr2196415342531872975_a_nat @ Ms ) @ ( set_Pr2196415342531872975_a_nat @ Ns ) )
=> ( ( ad_agr_list_a_nat @ X4 @ ( map_Pr2515564165894599445_a_nat @ Sigma @ Ns ) @ ( map_Pr2515564165894599445_a_nat @ Sigma2 @ Ns ) )
=> ( ad_agr_list_a_nat @ X4 @ ( map_Pr2515564165894599445_a_nat @ Sigma @ Ms ) @ ( map_Pr2515564165894599445_a_nat @ Sigma2 @ Ms ) ) ) ) ).
% ad_agr_list_subset
thf(fact_748_ad__agr__list__subset,axiom,
! [Ms: list_nat,Ns: list_nat,X4: set_a,Sigma: nat > sum_sum_a_nat,Sigma2: nat > sum_sum_a_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Ms ) @ ( set_nat2 @ Ns ) )
=> ( ( ad_agr_list_a_nat @ X4 @ ( map_na823391071729141993_a_nat @ Sigma @ Ns ) @ ( map_na823391071729141993_a_nat @ Sigma2 @ Ns ) )
=> ( ad_agr_list_a_nat @ X4 @ ( map_na823391071729141993_a_nat @ Sigma @ Ms ) @ ( map_na823391071729141993_a_nat @ Sigma2 @ Ms ) ) ) ) ).
% ad_agr_list_subset
thf(fact_749_ad__agr__list__subset,axiom,
! [Ms: list_a,Ns: list_a,X4: set_a,Sigma: a > sum_sum_a_nat,Sigma2: a > sum_sum_a_nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ Ms ) @ ( set_a2 @ Ns ) )
=> ( ( ad_agr_list_a_nat @ X4 @ ( map_a_Sum_sum_a_nat @ Sigma @ Ns ) @ ( map_a_Sum_sum_a_nat @ Sigma2 @ Ns ) )
=> ( ad_agr_list_a_nat @ X4 @ ( map_a_Sum_sum_a_nat @ Sigma @ Ms ) @ ( map_a_Sum_sum_a_nat @ Sigma2 @ Ms ) ) ) ) ).
% ad_agr_list_subset
thf(fact_750_ad__agr__list__subset,axiom,
! [Ms: list_Sum_sum_a_nat,Ns: list_Sum_sum_a_nat,X4: set_a,Sigma: sum_sum_a_nat > sum_sum_a_nat,Sigma2: sum_sum_a_nat > sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( set_Sum_sum_a_nat2 @ Ms ) @ ( set_Sum_sum_a_nat2 @ Ns ) )
=> ( ( ad_agr_list_a_nat @ X4 @ ( map_Su2790769393171190532_a_nat @ Sigma @ Ns ) @ ( map_Su2790769393171190532_a_nat @ Sigma2 @ Ns ) )
=> ( ad_agr_list_a_nat @ X4 @ ( map_Su2790769393171190532_a_nat @ Sigma @ Ms ) @ ( map_Su2790769393171190532_a_nat @ Sigma2 @ Ms ) ) ) ) ).
% ad_agr_list_subset
thf(fact_751_ad__agr__list__eq,axiom,
! [Ys: list_a,AD: set_a,Xs: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Ys ) @ AD )
=> ( ( ad_agr_list_a_nat @ AD @ ( map_a_Sum_sum_a_nat @ sum_Inl_a_nat @ Xs ) @ ( map_a_Sum_sum_a_nat @ sum_Inl_a_nat @ Ys ) )
=> ( Xs = Ys ) ) ) ).
% ad_agr_list_eq
thf(fact_752_prod__eq__iff,axiom,
( ( ^ [Y5: produc5454855657571582244_a_nat,Z3: produc5454855657571582244_a_nat] : ( Y5 = Z3 ) )
= ( ^ [S2: produc5454855657571582244_a_nat,T2: produc5454855657571582244_a_nat] :
( ( ( produc1510075366785312874_a_nat @ S2 )
= ( produc1510075366785312874_a_nat @ T2 ) )
& ( ( produc1280857981117566888_a_nat @ S2 )
= ( produc1280857981117566888_a_nat @ T2 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_753_dual__order_Orefl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_754_dual__order_Orefl,axiom,
! [A2: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_755_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_756_order__refl,axiom,
! [X2: set_a] : ( ord_less_eq_set_a @ X2 @ X2 ) ).
% order_refl
thf(fact_757_order__refl,axiom,
! [X2: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_758_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_759_image__Collect__subsetI,axiom,
! [P: list_Sum_sum_a_nat > $o,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat] :
( ! [X3: list_Sum_sum_a_nat] :
( ( P @ X3 )
=> ( member408289922725080238_a_nat @ ( F @ X3 ) @ B ) )
=> ( ord_le1147066620699065093_a_nat @ ( image_5081948215111134021_a_nat @ F @ ( collec7555443234367654128_a_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_760_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > nat,B: set_nat] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( member_nat @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ ( collect_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_761_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( member408289922725080238_a_nat @ ( F @ X3 ) @ B ) )
=> ( ord_le1147066620699065093_a_nat @ ( image_6262589752765146990_a_nat @ F @ ( collect_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_762_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > a,B: set_a] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( member_a @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_nat_a @ F @ ( collect_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_763_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( member_Sum_sum_a_nat @ ( F @ X3 ) @ B ) )
=> ( ord_le1325389633284124927_a_nat @ ( image_7293268710728258664_a_nat @ F @ ( collect_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_764_all__subset__image,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,P: set_li6526943997496501093_a_nat > $o] :
( ( ! [B4: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ B4 @ ( image_5081948215111134021_a_nat @ F @ A ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ B4 @ A )
=> ( P @ ( image_5081948215111134021_a_nat @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_765_all__subset__image,axiom,
! [F: a > a,A: set_a,P: set_a > $o] :
( ( ! [B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ ( image_a_a @ F @ A ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( P @ ( image_a_a @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_766_all__subset__image,axiom,
! [F: sum_sum_a_nat > a,A: set_Sum_sum_a_nat,P: set_a > $o] :
( ( ! [B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ ( image_6322530041254294468_nat_a @ F @ A ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B4 @ A )
=> ( P @ ( image_6322530041254294468_nat_a @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_767_all__subset__image,axiom,
! [F: nat > sum_sum_a_nat,A: set_nat,P: set_Sum_sum_a_nat > $o] :
( ( ! [B4: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B4 @ ( image_7293268710728258664_a_nat @ F @ A ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A )
=> ( P @ ( image_7293268710728258664_a_nat @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_768_all__subset__image,axiom,
! [F: a > sum_sum_a_nat,A: set_a,P: set_Sum_sum_a_nat > $o] :
( ( ! [B4: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B4 @ ( image_7873763678140191238_a_nat @ F @ A ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( P @ ( image_7873763678140191238_a_nat @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_769_all__subset__image,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,A: set_Sum_sum_a_nat,P: set_Sum_sum_a_nat > $o] :
( ( ! [B4: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B4 @ ( image_7142520692256960453_a_nat @ F @ A ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B4 @ A )
=> ( P @ ( image_7142520692256960453_a_nat @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_770_nall__tuples__rec__Inl,axiom,
! [Vs2: list_Sum_sum_a_nat,AD: set_a,I: nat,N: nat] :
( ( member408289922725080238_a_nat @ Vs2 @ ( nall_tuples_rec_a @ AD @ I @ N ) )
=> ( ord_less_eq_set_a @ ( vimage4607795094749595324_a_nat @ sum_Inl_a_nat @ ( set_Sum_sum_a_nat2 @ Vs2 ) ) @ AD ) ) ).
% nall_tuples_rec_Inl
thf(fact_771_nall__tuples__rec__Inl,axiom,
! [Vs2: list_S5657819638503513606at_nat,AD: set_Sum_sum_a_nat,I: nat,N: nat] :
( ( member4821912183549559215at_nat @ Vs2 @ ( nall_t5818786088032561781_a_nat @ AD @ I @ N ) )
=> ( ord_le1325389633284124927_a_nat @ ( vimage7613534090772763900at_nat @ sum_In1625650605130369377at_nat @ ( set_Su2720755619057964565at_nat @ Vs2 ) ) @ AD ) ) ).
% nall_tuples_rec_Inl
thf(fact_772_proj__tuple__idle,axiom,
! [Nxs: list_P5056861408695629236_a_nat] :
( ( proj_tuple_a @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Nxs ) @ Nxs )
= ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat @ Nxs ) ) ).
% proj_tuple_idle
thf(fact_773_sp__equiv__list__subset,axiom,
! [Ms: list_P5056861408695629236_a_nat,Ns: list_P5056861408695629236_a_nat,Sigma: produc5454855657571582244_a_nat > sum_sum_a_nat,Sigma2: produc5454855657571582244_a_nat > sum_sum_a_nat] :
( ( ord_le8420040822840260538_a_nat @ ( set_Pr2196415342531872975_a_nat @ Ms ) @ ( set_Pr2196415342531872975_a_nat @ Ns ) )
=> ( ( sp_equiv_list_a_nat @ ( map_Pr2515564165894599445_a_nat @ Sigma @ Ns ) @ ( map_Pr2515564165894599445_a_nat @ Sigma2 @ Ns ) )
=> ( sp_equiv_list_a_nat @ ( map_Pr2515564165894599445_a_nat @ Sigma @ Ms ) @ ( map_Pr2515564165894599445_a_nat @ Sigma2 @ Ms ) ) ) ) ).
% sp_equiv_list_subset
thf(fact_774_sp__equiv__list__subset,axiom,
! [Ms: list_nat,Ns: list_nat,Sigma: nat > sum_sum_a_nat,Sigma2: nat > sum_sum_a_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Ms ) @ ( set_nat2 @ Ns ) )
=> ( ( sp_equiv_list_a_nat @ ( map_na823391071729141993_a_nat @ Sigma @ Ns ) @ ( map_na823391071729141993_a_nat @ Sigma2 @ Ns ) )
=> ( sp_equiv_list_a_nat @ ( map_na823391071729141993_a_nat @ Sigma @ Ms ) @ ( map_na823391071729141993_a_nat @ Sigma2 @ Ms ) ) ) ) ).
% sp_equiv_list_subset
thf(fact_775_fo__nmlz__rec__set__rev,axiom,
! [I: nat,M2: sum_sum_a_nat > option_nat,AD: set_a,Xs: list_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( set_Sum_sum_a_nat2 @ ( fo_nmlz_rec_a @ I @ M2 @ AD @ Xs ) ) @ ( image_7873763678140191238_a_nat @ sum_Inl_a_nat @ AD ) )
=> ( ord_le1325389633284124927_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) @ ( image_7873763678140191238_a_nat @ sum_Inl_a_nat @ AD ) ) ) ).
% fo_nmlz_rec_set_rev
thf(fact_776_set__quicksort,axiom,
! [Xs: list_nat] :
( ( set_nat2 @ ( set_or9089632773640736191rt_nat @ Xs ) )
= ( set_nat2 @ Xs ) ) ).
% set_quicksort
thf(fact_777_Collect__case__prod__mono,axiom,
! [A: nat > sum_sum_a_nat > $o,B: nat > sum_sum_a_nat > $o] :
( ( ord_le759605360153541901_nat_o @ A @ B )
=> ( ord_le8420040822840260538_a_nat @ ( collec2196941500952594553_a_nat @ ( produc8810490045567577941_nat_o @ A ) ) @ ( collec2196941500952594553_a_nat @ ( produc8810490045567577941_nat_o @ B ) ) ) ) ).
% Collect_case_prod_mono
thf(fact_778_Collect__split__mono__strong,axiom,
! [X4: set_nat,A: set_Pr1021281732701644634_a_nat,Y4: set_Sum_sum_a_nat,P: nat > sum_sum_a_nat > $o,Q: nat > sum_sum_a_nat > $o] :
( ( X4
= ( image_1677410150705809403at_nat @ produc1510075366785312874_a_nat @ A ) )
=> ( ( Y4
= ( image_5908522764169838868_a_nat @ produc1280857981117566888_a_nat @ A ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X4 )
=> ! [Xa2: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ Xa2 @ Y4 )
=> ( ( P @ X3 @ Xa2 )
=> ( Q @ X3 @ Xa2 ) ) ) )
=> ( ( ord_le8420040822840260538_a_nat @ A @ ( collec2196941500952594553_a_nat @ ( produc8810490045567577941_nat_o @ P ) ) )
=> ( ord_le8420040822840260538_a_nat @ A @ ( collec2196941500952594553_a_nat @ ( produc8810490045567577941_nat_o @ Q ) ) ) ) ) ) ) ).
% Collect_split_mono_strong
thf(fact_779_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_780_less__eq__set__def,axiom,
( ord_le1147066620699065093_a_nat
= ( ^ [A4: set_li6526943997496501093_a_nat,B4: set_li6526943997496501093_a_nat] :
( ord_le8737610411969296920_nat_o
@ ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ A4 )
@ ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_781_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ord_less_eq_a_o
@ ^ [X: a] : ( member_a @ X @ A4 )
@ ^ [X: a] : ( member_a @ X @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_782_less__eq__set__def,axiom,
( ord_le1325389633284124927_a_nat
= ( ^ [A4: set_Sum_sum_a_nat,B4: set_Sum_sum_a_nat] :
( ord_le1477630214076318366_nat_o
@ ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ A4 )
@ ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_783_pred__subset__eq,axiom,
! [R: set_nat,S: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X: nat] : ( member_nat @ X @ R )
@ ^ [X: nat] : ( member_nat @ X @ S ) )
= ( ord_less_eq_set_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_784_pred__subset__eq,axiom,
! [R: set_li6526943997496501093_a_nat,S: set_li6526943997496501093_a_nat] :
( ( ord_le8737610411969296920_nat_o
@ ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ R )
@ ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ S ) )
= ( ord_le1147066620699065093_a_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_785_pred__subset__eq,axiom,
! [R: set_a,S: set_a] :
( ( ord_less_eq_a_o
@ ^ [X: a] : ( member_a @ X @ R )
@ ^ [X: a] : ( member_a @ X @ S ) )
= ( ord_less_eq_set_a @ R @ S ) ) ).
% pred_subset_eq
thf(fact_786_pred__subset__eq,axiom,
! [R: set_Sum_sum_a_nat,S: set_Sum_sum_a_nat] :
( ( ord_le1477630214076318366_nat_o
@ ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ R )
@ ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ S ) )
= ( ord_le1325389633284124927_a_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_787_nle__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_788_le__cases3,axiom,
! [X2: nat,Y2: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y2 ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_789_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_a,Z3: set_a] : ( Y5 = Z3 ) )
= ( ^ [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
& ( ord_less_eq_set_a @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_790_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_Sum_sum_a_nat,Z3: set_Sum_sum_a_nat] : ( Y5 = Z3 ) )
= ( ^ [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X @ Y )
& ( ord_le1325389633284124927_a_nat @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_791_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
= ( ^ [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_792_ord__eq__le__trans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( A2 = B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_793_ord__eq__le__trans,axiom,
! [A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( A2 = B2 )
=> ( ( ord_le1325389633284124927_a_nat @ B2 @ C )
=> ( ord_le1325389633284124927_a_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_794_ord__eq__le__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_795_ord__le__eq__trans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_796_ord__le__eq__trans,axiom,
! [A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_le1325389633284124927_a_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_797_ord__le__eq__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_798_order__antisym,axiom,
! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ( ord_less_eq_set_a @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_799_order__antisym,axiom,
! [X2: set_Sum_sum_a_nat,Y2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X2 @ Y2 )
=> ( ( ord_le1325389633284124927_a_nat @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_800_order__antisym,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_801_order_Otrans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% order.trans
thf(fact_802_order_Otrans,axiom,
! [A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A2 @ B2 )
=> ( ( ord_le1325389633284124927_a_nat @ B2 @ C )
=> ( ord_le1325389633284124927_a_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_803_order_Otrans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_804_order__trans,axiom,
! [X2: set_a,Y2: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ( ord_less_eq_set_a @ Y2 @ Z )
=> ( ord_less_eq_set_a @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_805_order__trans,axiom,
! [X2: set_Sum_sum_a_nat,Y2: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X2 @ Y2 )
=> ( ( ord_le1325389633284124927_a_nat @ Y2 @ Z )
=> ( ord_le1325389633284124927_a_nat @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_806_order__trans,axiom,
! [X2: nat,Y2: nat,Z: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z )
=> ( ord_less_eq_nat @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_807_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A7: nat,B6: nat] :
( ( ord_less_eq_nat @ A7 @ B6 )
=> ( P @ A7 @ B6 ) )
=> ( ! [A7: nat,B6: nat] :
( ( P @ B6 @ A7 )
=> ( P @ A7 @ B6 ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_808_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_a,Z3: set_a] : ( Y5 = Z3 ) )
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A3 )
& ( ord_less_eq_set_a @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_809_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_Sum_sum_a_nat,Z3: set_Sum_sum_a_nat] : ( Y5 = Z3 ) )
= ( ^ [A3: set_Sum_sum_a_nat,B3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B3 @ A3 )
& ( ord_le1325389633284124927_a_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_810_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_811_dual__order_Oantisym,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_812_dual__order_Oantisym,axiom,
! [B2: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B2 @ A2 )
=> ( ( ord_le1325389633284124927_a_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_813_dual__order_Oantisym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_814_dual__order_Otrans,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( ord_less_eq_set_a @ C @ B2 )
=> ( ord_less_eq_set_a @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_815_dual__order_Otrans,axiom,
! [B2: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B2 @ A2 )
=> ( ( ord_le1325389633284124927_a_nat @ C @ B2 )
=> ( ord_le1325389633284124927_a_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_816_dual__order_Otrans,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_817_antisym,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_818_antisym,axiom,
! [A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A2 @ B2 )
=> ( ( ord_le1325389633284124927_a_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_819_antisym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_820_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_a,Z3: set_a] : ( Y5 = Z3 ) )
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
& ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_821_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_Sum_sum_a_nat,Z3: set_Sum_sum_a_nat] : ( Y5 = Z3 ) )
= ( ^ [A3: set_Sum_sum_a_nat,B3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A3 @ B3 )
& ( ord_le1325389633284124927_a_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_822_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_823_order__subst1,axiom,
! [A2: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_824_order__subst1,axiom,
! [A2: set_a,F: set_Sum_sum_a_nat > set_a,B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_le1325389633284124927_a_nat @ B2 @ C )
=> ( ! [X3: set_Sum_sum_a_nat,Y3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_825_order__subst1,axiom,
! [A2: set_a,F: nat > set_a,B2: nat,C: nat] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_826_order__subst1,axiom,
! [A2: set_Sum_sum_a_nat,F: set_a > set_Sum_sum_a_nat,B2: set_a,C: set_a] :
( ( ord_le1325389633284124927_a_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_le1325389633284124927_a_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le1325389633284124927_a_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_827_order__subst1,axiom,
! [A2: set_Sum_sum_a_nat,F: set_Sum_sum_a_nat > set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_le1325389633284124927_a_nat @ B2 @ C )
=> ( ! [X3: set_Sum_sum_a_nat,Y3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X3 @ Y3 )
=> ( ord_le1325389633284124927_a_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le1325389633284124927_a_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_828_order__subst1,axiom,
! [A2: set_Sum_sum_a_nat,F: nat > set_Sum_sum_a_nat,B2: nat,C: nat] :
( ( ord_le1325389633284124927_a_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le1325389633284124927_a_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le1325389633284124927_a_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_829_order__subst1,axiom,
! [A2: nat,F: set_a > nat,B2: set_a,C: set_a] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_830_order__subst1,axiom,
! [A2: nat,F: set_Sum_sum_a_nat > nat,B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_le1325389633284124927_a_nat @ B2 @ C )
=> ( ! [X3: set_Sum_sum_a_nat,Y3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_831_order__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_832_order__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_833_order__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_le1325389633284124927_a_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_le1325389633284124927_a_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le1325389633284124927_a_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_834_order__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_835_order__subst2,axiom,
! [A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,F: set_Sum_sum_a_nat > set_a,C: set_a] :
( ( ord_le1325389633284124927_a_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
=> ( ! [X3: set_Sum_sum_a_nat,Y3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_836_order__subst2,axiom,
! [A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,F: set_Sum_sum_a_nat > set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A2 @ B2 )
=> ( ( ord_le1325389633284124927_a_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: set_Sum_sum_a_nat,Y3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X3 @ Y3 )
=> ( ord_le1325389633284124927_a_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le1325389633284124927_a_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_837_order__subst2,axiom,
! [A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,F: set_Sum_sum_a_nat > nat,C: nat] :
( ( ord_le1325389633284124927_a_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: set_Sum_sum_a_nat,Y3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_838_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_839_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_le1325389633284124927_a_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le1325389633284124927_a_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le1325389633284124927_a_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_840_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_841_order__eq__refl,axiom,
! [X2: set_a,Y2: set_a] :
( ( X2 = Y2 )
=> ( ord_less_eq_set_a @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_842_order__eq__refl,axiom,
! [X2: set_Sum_sum_a_nat,Y2: set_Sum_sum_a_nat] :
( ( X2 = Y2 )
=> ( ord_le1325389633284124927_a_nat @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_843_order__eq__refl,axiom,
! [X2: nat,Y2: nat] :
( ( X2 = Y2 )
=> ( ord_less_eq_nat @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_844_linorder__linear,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
| ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% linorder_linear
thf(fact_845_ord__eq__le__subst,axiom,
! [A2: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_846_ord__eq__le__subst,axiom,
! [A2: set_Sum_sum_a_nat,F: set_a > set_Sum_sum_a_nat,B2: set_a,C: set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_le1325389633284124927_a_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le1325389633284124927_a_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_847_ord__eq__le__subst,axiom,
! [A2: nat,F: set_a > nat,B2: set_a,C: set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_848_ord__eq__le__subst,axiom,
! [A2: set_a,F: set_Sum_sum_a_nat > set_a,B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le1325389633284124927_a_nat @ B2 @ C )
=> ( ! [X3: set_Sum_sum_a_nat,Y3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_849_ord__eq__le__subst,axiom,
! [A2: set_Sum_sum_a_nat,F: set_Sum_sum_a_nat > set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le1325389633284124927_a_nat @ B2 @ C )
=> ( ! [X3: set_Sum_sum_a_nat,Y3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X3 @ Y3 )
=> ( ord_le1325389633284124927_a_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le1325389633284124927_a_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_850_ord__eq__le__subst,axiom,
! [A2: nat,F: set_Sum_sum_a_nat > nat,B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le1325389633284124927_a_nat @ B2 @ C )
=> ( ! [X3: set_Sum_sum_a_nat,Y3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_851_ord__eq__le__subst,axiom,
! [A2: set_a,F: nat > set_a,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_852_ord__eq__le__subst,axiom,
! [A2: set_Sum_sum_a_nat,F: nat > set_Sum_sum_a_nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le1325389633284124927_a_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le1325389633284124927_a_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_853_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_854_ord__le__eq__subst,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_855_ord__le__eq__subst,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_le1325389633284124927_a_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le1325389633284124927_a_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_856_ord__le__eq__subst,axiom,
! [A2: set_a,B2: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_857_ord__le__eq__subst,axiom,
! [A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,F: set_Sum_sum_a_nat > set_a,C: set_a] :
( ( ord_le1325389633284124927_a_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_Sum_sum_a_nat,Y3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_858_ord__le__eq__subst,axiom,
! [A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,F: set_Sum_sum_a_nat > set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_Sum_sum_a_nat,Y3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X3 @ Y3 )
=> ( ord_le1325389633284124927_a_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le1325389633284124927_a_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_859_ord__le__eq__subst,axiom,
! [A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,F: set_Sum_sum_a_nat > nat,C: nat] :
( ( ord_le1325389633284124927_a_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_Sum_sum_a_nat,Y3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_860_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_861_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le1325389633284124927_a_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le1325389633284124927_a_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_862_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_863_linorder__le__cases,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% linorder_le_cases
thf(fact_864_order__antisym__conv,axiom,
! [Y2: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X2 )
=> ( ( ord_less_eq_set_a @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_865_order__antisym__conv,axiom,
! [Y2: set_Sum_sum_a_nat,X2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ Y2 @ X2 )
=> ( ( ord_le1325389633284124927_a_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_866_order__antisym__conv,axiom,
! [Y2: nat,X2: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_867_quicksort__conv__sort,axiom,
( set_or9089632773640736191rt_nat
= ( linord738340561235409698at_nat
@ ^ [X: nat] : X ) ) ).
% quicksort_conv_sort
thf(fact_868_Collect__restrict,axiom,
! [X4: set_li6526943997496501093_a_nat,P: list_Sum_sum_a_nat > $o] :
( ord_le1147066620699065093_a_nat
@ ( collec7555443234367654128_a_nat
@ ^ [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ X4 )
& ( P @ X ) ) )
@ X4 ) ).
% Collect_restrict
thf(fact_869_Collect__restrict,axiom,
! [X4: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ X4 )
& ( P @ X ) ) )
@ X4 ) ).
% Collect_restrict
thf(fact_870_Collect__restrict,axiom,
! [X4: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ X4 )
& ( P @ X ) ) )
@ X4 ) ).
% Collect_restrict
thf(fact_871_Collect__restrict,axiom,
! [X4: set_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ord_le1325389633284124927_a_nat
@ ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ X4 )
& ( P @ X ) ) )
@ X4 ) ).
% Collect_restrict
thf(fact_872_prop__restrict,axiom,
! [X2: list_Sum_sum_a_nat,Z4: set_li6526943997496501093_a_nat,X4: set_li6526943997496501093_a_nat,P: list_Sum_sum_a_nat > $o] :
( ( member408289922725080238_a_nat @ X2 @ Z4 )
=> ( ( ord_le1147066620699065093_a_nat @ Z4
@ ( collec7555443234367654128_a_nat
@ ^ [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ X4 )
& ( P @ X ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_873_prop__restrict,axiom,
! [X2: nat,Z4: set_nat,X4: set_nat,P: nat > $o] :
( ( member_nat @ X2 @ Z4 )
=> ( ( ord_less_eq_set_nat @ Z4
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ X4 )
& ( P @ X ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_874_prop__restrict,axiom,
! [X2: a,Z4: set_a,X4: set_a,P: a > $o] :
( ( member_a @ X2 @ Z4 )
=> ( ( ord_less_eq_set_a @ Z4
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ X4 )
& ( P @ X ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_875_prop__restrict,axiom,
! [X2: sum_sum_a_nat,Z4: set_Sum_sum_a_nat,X4: set_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ( member_Sum_sum_a_nat @ X2 @ Z4 )
=> ( ( ord_le1325389633284124927_a_nat @ Z4
@ ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ X4 )
& ( P @ X ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_876_X_092_060psi_062__def,axiom,
( x_psi
= ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ ad ) @ ( proj_v3643391342904276326_a_nat @ r_psi @ ns_psi ) ) ) ).
% X\<psi>_def
thf(fact_877_X_092_060phi_062__def,axiom,
( x_phi
= ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ ad ) @ ( proj_v3643391342904276326_a_nat @ r_phi @ ns_phi ) ) ) ).
% X\<phi>_def
thf(fact_878_sp__equiv__list__link,axiom,
! [Sigma: nat > sum_sum_a_nat,Ns: list_nat,Tau: nat > sum_sum_a_nat] :
( ( sp_equiv_list_a_nat @ ( map_na823391071729141993_a_nat @ Sigma @ Ns ) @ ( map_na823391071729141993_a_nat @ Tau @ Ns ) )
= ( sp_equiv_a_nat @ Sigma @ Tau @ ( set_nat2 @ Ns ) ) ) ).
% sp_equiv_list_link
thf(fact_879_proj__vals__def,axiom,
( proj_v3643391342904276326_a_nat
= ( ^ [R2: set_na3699693778330250182_a_nat,Ns2: list_nat] :
( image_6721470456781115300_a_nat
@ ^ [Tau2: nat > sum_sum_a_nat] : ( map_na823391071729141993_a_nat @ Tau2 @ Ns2 )
@ R2 ) ) ) ).
% proj_vals_def
thf(fact_880_ys_092_060psi_062__def,axiom,
( member408289922725080238_a_nat @ ( map_na823391071729141993_a_nat @ sigma_xs @ nys )
@ ( nall_tuples_rec_a @ bot_bot_set_a
@ ( finite_card_nat
@ ( vimage3040984495076556338_a_nat @ sum_Inr_nat_a
@ ( set_Sum_sum_a_nat2
@ ( map_na823391071729141993_a_nat @ sigma_xs
@ ( linord738340561235409698at_nat
@ ^ [X: nat] : X
@ ( append_nat @ ns_phi @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ cys ) ) ) ) ) ) )
@ ( size_size_list_nat @ nys ) ) ) ).
% ys\<psi>_def
thf(fact_881_map__snd__enumerate,axiom,
! [N: nat,Xs: list_Sum_sum_a_nat] :
( ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat @ ( enumer3164015132978342500_a_nat @ N @ Xs ) )
= Xs ) ).
% map_snd_enumerate
thf(fact_882_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_883_empty__iff,axiom,
! [C: list_Sum_sum_a_nat] :
~ ( member408289922725080238_a_nat @ C @ bot_bo1033123847703346641_a_nat ) ).
% empty_iff
thf(fact_884_empty__iff,axiom,
! [C: sum_sum_a_nat] :
~ ( member_Sum_sum_a_nat @ C @ bot_bo3438331934148233675_a_nat ) ).
% empty_iff
thf(fact_885_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_886_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X: nat] :
~ ( member_nat @ X @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_887_all__not__in__conv,axiom,
! [A: set_li6526943997496501093_a_nat] :
( ( ! [X: list_Sum_sum_a_nat] :
~ ( member408289922725080238_a_nat @ X @ A ) )
= ( A = bot_bo1033123847703346641_a_nat ) ) ).
% all_not_in_conv
thf(fact_888_all__not__in__conv,axiom,
! [A: set_Sum_sum_a_nat] :
( ( ! [X: sum_sum_a_nat] :
~ ( member_Sum_sum_a_nat @ X @ A ) )
= ( A = bot_bo3438331934148233675_a_nat ) ) ).
% all_not_in_conv
thf(fact_889_all__not__in__conv,axiom,
! [A: set_a] :
( ( ! [X: a] :
~ ( member_a @ X @ A ) )
= ( A = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_890_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X: nat] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_891_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X: a] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_892_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X: nat] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_893_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X: a] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_894_sum_Oinject_I2_J,axiom,
! [X23: nat,Y22: nat] :
( ( ( sum_Inr_nat_a @ X23 )
= ( sum_Inr_nat_a @ Y22 ) )
= ( X23 = Y22 ) ) ).
% sum.inject(2)
thf(fact_895_old_Osum_Oinject_I2_J,axiom,
! [B2: nat,B7: nat] :
( ( ( sum_Inr_nat_a @ B2 )
= ( sum_Inr_nat_a @ B7 ) )
= ( B2 = B7 ) ) ).
% old.sum.inject(2)
thf(fact_896_image__empty,axiom,
! [F: nat > sum_sum_a_nat] :
( ( image_7293268710728258664_a_nat @ F @ bot_bot_set_nat )
= bot_bo3438331934148233675_a_nat ) ).
% image_empty
thf(fact_897_image__empty,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( image_5081948215111134021_a_nat @ F @ bot_bo1033123847703346641_a_nat )
= bot_bo1033123847703346641_a_nat ) ).
% image_empty
thf(fact_898_image__empty,axiom,
! [F: a > a] :
( ( image_a_a @ F @ bot_bot_set_a )
= bot_bot_set_a ) ).
% image_empty
thf(fact_899_empty__is__image,axiom,
! [F: nat > sum_sum_a_nat,A: set_nat] :
( ( bot_bo3438331934148233675_a_nat
= ( image_7293268710728258664_a_nat @ F @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_900_empty__is__image,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( bot_bo1033123847703346641_a_nat
= ( image_5081948215111134021_a_nat @ F @ A ) )
= ( A = bot_bo1033123847703346641_a_nat ) ) ).
% empty_is_image
thf(fact_901_empty__is__image,axiom,
! [F: a > a,A: set_a] :
( ( bot_bot_set_a
= ( image_a_a @ F @ A ) )
= ( A = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_902_image__is__empty,axiom,
! [F: nat > sum_sum_a_nat,A: set_nat] :
( ( ( image_7293268710728258664_a_nat @ F @ A )
= bot_bo3438331934148233675_a_nat )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_903_image__is__empty,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( ( image_5081948215111134021_a_nat @ F @ A )
= bot_bo1033123847703346641_a_nat )
= ( A = bot_bo1033123847703346641_a_nat ) ) ).
% image_is_empty
thf(fact_904_image__is__empty,axiom,
! [F: a > a,A: set_a] :
( ( ( image_a_a @ F @ A )
= bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_905_subset__empty,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_906_subset__empty,axiom,
! [A: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ bot_bo3438331934148233675_a_nat )
= ( A = bot_bo3438331934148233675_a_nat ) ) ).
% subset_empty
thf(fact_907_empty__subsetI,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% empty_subsetI
thf(fact_908_empty__subsetI,axiom,
! [A: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ bot_bo3438331934148233675_a_nat @ A ) ).
% empty_subsetI
thf(fact_909_sup__bot__left,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ X2 )
= X2 ) ).
% sup_bot_left
thf(fact_910_sup__bot__left,axiom,
! [X2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ bot_bo3438331934148233675_a_nat @ X2 )
= X2 ) ).
% sup_bot_left
thf(fact_911_sup__bot__left,axiom,
! [X2: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ bot_bo1033123847703346641_a_nat @ X2 )
= X2 ) ).
% sup_bot_left
thf(fact_912_sup__bot__left,axiom,
! [X2: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ X2 )
= X2 ) ).
% sup_bot_left
thf(fact_913_sup__bot__right,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ X2 @ bot_bot_set_nat )
= X2 ) ).
% sup_bot_right
thf(fact_914_sup__bot__right,axiom,
! [X2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ X2 @ bot_bo3438331934148233675_a_nat )
= X2 ) ).
% sup_bot_right
thf(fact_915_sup__bot__right,axiom,
! [X2: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ X2 @ bot_bo1033123847703346641_a_nat )
= X2 ) ).
% sup_bot_right
thf(fact_916_sup__bot__right,axiom,
! [X2: set_a] :
( ( sup_sup_set_a @ X2 @ bot_bot_set_a )
= X2 ) ).
% sup_bot_right
thf(fact_917_bot__eq__sup__iff,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ X2 @ Y2 ) )
= ( ( X2 = bot_bot_set_nat )
& ( Y2 = bot_bot_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_918_bot__eq__sup__iff,axiom,
! [X2: set_Sum_sum_a_nat,Y2: set_Sum_sum_a_nat] :
( ( bot_bo3438331934148233675_a_nat
= ( sup_su6804446743777130803_a_nat @ X2 @ Y2 ) )
= ( ( X2 = bot_bo3438331934148233675_a_nat )
& ( Y2 = bot_bo3438331934148233675_a_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_919_bot__eq__sup__iff,axiom,
! [X2: set_li6526943997496501093_a_nat,Y2: set_li6526943997496501093_a_nat] :
( ( bot_bo1033123847703346641_a_nat
= ( sup_su4083067149120280889_a_nat @ X2 @ Y2 ) )
= ( ( X2 = bot_bo1033123847703346641_a_nat )
& ( Y2 = bot_bo1033123847703346641_a_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_920_bot__eq__sup__iff,axiom,
! [X2: set_a,Y2: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ X2 @ Y2 ) )
= ( ( X2 = bot_bot_set_a )
& ( Y2 = bot_bot_set_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_921_sup__eq__bot__iff,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( ( sup_sup_set_nat @ X2 @ Y2 )
= bot_bot_set_nat )
= ( ( X2 = bot_bot_set_nat )
& ( Y2 = bot_bot_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_922_sup__eq__bot__iff,axiom,
! [X2: set_Sum_sum_a_nat,Y2: set_Sum_sum_a_nat] :
( ( ( sup_su6804446743777130803_a_nat @ X2 @ Y2 )
= bot_bo3438331934148233675_a_nat )
= ( ( X2 = bot_bo3438331934148233675_a_nat )
& ( Y2 = bot_bo3438331934148233675_a_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_923_sup__eq__bot__iff,axiom,
! [X2: set_li6526943997496501093_a_nat,Y2: set_li6526943997496501093_a_nat] :
( ( ( sup_su4083067149120280889_a_nat @ X2 @ Y2 )
= bot_bo1033123847703346641_a_nat )
= ( ( X2 = bot_bo1033123847703346641_a_nat )
& ( Y2 = bot_bo1033123847703346641_a_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_924_sup__eq__bot__iff,axiom,
! [X2: set_a,Y2: set_a] :
( ( ( sup_sup_set_a @ X2 @ Y2 )
= bot_bot_set_a )
= ( ( X2 = bot_bot_set_a )
& ( Y2 = bot_bot_set_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_925_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( sup_sup_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ( A2 = bot_bot_set_nat )
& ( B2 = bot_bot_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_926_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ( sup_su6804446743777130803_a_nat @ A2 @ B2 )
= bot_bo3438331934148233675_a_nat )
= ( ( A2 = bot_bo3438331934148233675_a_nat )
& ( B2 = bot_bo3438331934148233675_a_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_927_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ( sup_su4083067149120280889_a_nat @ A2 @ B2 )
= bot_bo1033123847703346641_a_nat )
= ( ( A2 = bot_bo1033123847703346641_a_nat )
& ( B2 = bot_bo1033123847703346641_a_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_928_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( ( sup_sup_set_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ( A2 = bot_bot_set_a )
& ( B2 = bot_bot_set_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_929_sup__bot_Oleft__neutral,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_930_sup__bot_Oleft__neutral,axiom,
! [A2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ bot_bo3438331934148233675_a_nat @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_931_sup__bot_Oleft__neutral,axiom,
! [A2: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ bot_bo1033123847703346641_a_nat @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_932_sup__bot_Oleft__neutral,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_933_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ A2 @ B2 ) )
= ( ( A2 = bot_bot_set_nat )
& ( B2 = bot_bot_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_934_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( bot_bo3438331934148233675_a_nat
= ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) )
= ( ( A2 = bot_bo3438331934148233675_a_nat )
& ( B2 = bot_bo3438331934148233675_a_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_935_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( bot_bo1033123847703346641_a_nat
= ( sup_su4083067149120280889_a_nat @ A2 @ B2 ) )
= ( ( A2 = bot_bo1033123847703346641_a_nat )
& ( B2 = bot_bo1033123847703346641_a_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_936_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ A2 @ B2 ) )
= ( ( A2 = bot_bot_set_a )
& ( B2 = bot_bot_set_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_937_sup__bot_Oright__neutral,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_938_sup__bot_Oright__neutral,axiom,
! [A2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A2 @ bot_bo3438331934148233675_a_nat )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_939_sup__bot_Oright__neutral,axiom,
! [A2: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ A2 @ bot_bo1033123847703346641_a_nat )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_940_sup__bot_Oright__neutral,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_941_Un__empty,axiom,
! [A: set_nat,B: set_nat] :
( ( ( sup_sup_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ( A = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% Un_empty
thf(fact_942_Un__empty,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ( sup_su6804446743777130803_a_nat @ A @ B )
= bot_bo3438331934148233675_a_nat )
= ( ( A = bot_bo3438331934148233675_a_nat )
& ( B = bot_bo3438331934148233675_a_nat ) ) ) ).
% Un_empty
thf(fact_943_Un__empty,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ( sup_su4083067149120280889_a_nat @ A @ B )
= bot_bo1033123847703346641_a_nat )
= ( ( A = bot_bo1033123847703346641_a_nat )
& ( B = bot_bo1033123847703346641_a_nat ) ) ) ).
% Un_empty
thf(fact_944_Un__empty,axiom,
! [A: set_a,B: set_a] :
( ( ( sup_sup_set_a @ A @ B )
= bot_bot_set_a )
= ( ( A = bot_bot_set_a )
& ( B = bot_bot_set_a ) ) ) ).
% Un_empty
thf(fact_945_length__map,axiom,
! [F: produc5454855657571582244_a_nat > nat,Xs: list_P5056861408695629236_a_nat] :
( ( size_size_list_nat @ ( map_Pr2058079554187054650at_nat @ F @ Xs ) )
= ( size_s4076174644546656840_a_nat @ Xs ) ) ).
% length_map
thf(fact_946_length__map,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( size_size_list_nat @ ( map_nat_nat @ F @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_map
thf(fact_947_length__map,axiom,
! [F: sum_sum_a_nat > nat,Xs: list_Sum_sum_a_nat] :
( ( size_size_list_nat @ ( map_Su5227373005390213643at_nat @ F @ Xs ) )
= ( size_s5283204784079214577_a_nat @ Xs ) ) ).
% length_map
thf(fact_948_length__map,axiom,
! [F: produc5454855657571582244_a_nat > sum_sum_a_nat,Xs: list_P5056861408695629236_a_nat] :
( ( size_s5283204784079214577_a_nat @ ( map_Pr2515564165894599445_a_nat @ F @ Xs ) )
= ( size_s4076174644546656840_a_nat @ Xs ) ) ).
% length_map
thf(fact_949_length__map,axiom,
! [F: nat > sum_sum_a_nat,Xs: list_nat] :
( ( size_s5283204784079214577_a_nat @ ( map_na823391071729141993_a_nat @ F @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_map
thf(fact_950_length__map,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( size_s5283204784079214577_a_nat @ ( map_Su2790769393171190532_a_nat @ F @ Xs ) )
= ( size_s5283204784079214577_a_nat @ Xs ) ) ).
% length_map
thf(fact_951_append__eq__append__conv,axiom,
! [Xs: list_nat,Ys: list_nat,Us2: list_nat,Vs2: list_nat] :
( ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
| ( ( size_size_list_nat @ Us2 )
= ( size_size_list_nat @ Vs2 ) ) )
=> ( ( ( append_nat @ Xs @ Us2 )
= ( append_nat @ Ys @ Vs2 ) )
= ( ( Xs = Ys )
& ( Us2 = Vs2 ) ) ) ) ).
% append_eq_append_conv
thf(fact_952_append__eq__append__conv,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Us2: list_Sum_sum_a_nat,Vs2: list_Sum_sum_a_nat] :
( ( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
| ( ( size_s5283204784079214577_a_nat @ Us2 )
= ( size_s5283204784079214577_a_nat @ Vs2 ) ) )
=> ( ( ( append_Sum_sum_a_nat @ Xs @ Us2 )
= ( append_Sum_sum_a_nat @ Ys @ Vs2 ) )
= ( ( Xs = Ys )
& ( Us2 = Vs2 ) ) ) ) ).
% append_eq_append_conv
thf(fact_953_vimage__empty,axiom,
! [F: nat > sum_sum_a_nat] :
( ( vimage3040984495076556338_a_nat @ F @ bot_bo3438331934148233675_a_nat )
= bot_bot_set_nat ) ).
% vimage_empty
thf(fact_954_vimage__empty,axiom,
! [F: a > sum_sum_a_nat] :
( ( vimage4607795094749595324_a_nat @ F @ bot_bo3438331934148233675_a_nat )
= bot_bot_set_a ) ).
% vimage_empty
thf(fact_955_vimage__empty,axiom,
! [F: a > a] :
( ( vimage_a_a @ F @ bot_bot_set_a )
= bot_bot_set_a ) ).
% vimage_empty
thf(fact_956_length__sort,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( size_size_list_nat @ ( linord738340561235409698at_nat @ F @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_sort
thf(fact_957_length__enumerate,axiom,
! [N: nat,Xs: list_nat] :
( ( size_s5460976970255530739at_nat @ ( enumerate_nat @ N @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_enumerate
thf(fact_958_length__enumerate,axiom,
! [N: nat,Xs: list_Sum_sum_a_nat] :
( ( size_s4076174644546656840_a_nat @ ( enumer3164015132978342500_a_nat @ N @ Xs ) )
= ( size_s5283204784079214577_a_nat @ Xs ) ) ).
% length_enumerate
thf(fact_959_zip__append,axiom,
! [Xs: list_nat,Us2: list_nat,Ys: list_nat,Vs2: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Us2 ) )
=> ( ( zip_nat_nat @ ( append_nat @ Xs @ Ys ) @ ( append_nat @ Us2 @ Vs2 ) )
= ( append985823374593552924at_nat @ ( zip_nat_nat @ Xs @ Us2 ) @ ( zip_nat_nat @ Ys @ Vs2 ) ) ) ) ).
% zip_append
thf(fact_960_zip__append,axiom,
! [Xs: list_nat,Us2: list_Sum_sum_a_nat,Ys: list_nat,Vs2: list_Sum_sum_a_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Us2 ) )
=> ( ( zip_na2013496608136855606_a_nat @ ( append_nat @ Xs @ Ys ) @ ( append_Sum_sum_a_nat @ Us2 @ Vs2 ) )
= ( append338925788367110473_a_nat @ ( zip_na2013496608136855606_a_nat @ Xs @ Us2 ) @ ( zip_na2013496608136855606_a_nat @ Ys @ Vs2 ) ) ) ) ).
% zip_append
thf(fact_961_zip__append,axiom,
! [Xs: list_Sum_sum_a_nat,Us2: list_nat,Ys: list_Sum_sum_a_nat,Vs2: list_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_size_list_nat @ Us2 ) )
=> ( ( zip_Su6417478541797927256at_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) @ ( append_nat @ Us2 @ Vs2 ) )
= ( append2142653904031976739at_nat @ ( zip_Su6417478541797927256at_nat @ Xs @ Us2 ) @ ( zip_Su6417478541797927256at_nat @ Ys @ Vs2 ) ) ) ) ).
% zip_append
thf(fact_962_zip__append,axiom,
! [Xs: list_Sum_sum_a_nat,Us2: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Vs2: list_Sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Us2 ) )
=> ( ( zip_Su7355543910597222519_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) @ ( append_Sum_sum_a_nat @ Us2 @ Vs2 ) )
= ( append1996214168388709506_a_nat @ ( zip_Su7355543910597222519_a_nat @ Xs @ Us2 ) @ ( zip_Su7355543910597222519_a_nat @ Ys @ Vs2 ) ) ) ) ).
% zip_append
thf(fact_963_map__fst__zip,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( map_Pr3938374229010428429at_nat @ product_fst_nat_nat @ ( zip_nat_nat @ Xs @ Ys ) )
= Xs ) ) ).
% map_fst_zip
thf(fact_964_map__fst__zip,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( map_Pr6349035483858946235_a_nat @ produc5914057300446384524at_nat @ ( zip_Su6417478541797927256at_nat @ Xs @ Ys ) )
= Xs ) ) ).
% map_fst_zip
thf(fact_965_map__fst__zip,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( map_Pr3527087583225239196_a_nat @ produc3940427395981429571_a_nat @ ( zip_Su7355543910597222519_a_nat @ Xs @ Ys ) )
= Xs ) ) ).
% map_fst_zip
thf(fact_966_map__fst__zip,axiom,
! [Xs: list_nat,Ys: list_Sum_sum_a_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ ( zip_na2013496608136855606_a_nat @ Xs @ Ys ) )
= Xs ) ) ).
% map_fst_zip
thf(fact_967_map__snd__zip,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( map_Pr3938374229010428429at_nat @ product_snd_nat_nat @ ( zip_nat_nat @ Xs @ Ys ) )
= Ys ) ) ).
% map_snd_zip
thf(fact_968_map__snd__zip,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( map_Pr8434518024483680788at_nat @ produc5684839914778638538at_nat @ ( zip_Su6417478541797927256at_nat @ Xs @ Ys ) )
= Ys ) ) ).
% map_snd_zip
thf(fact_969_map__snd__zip,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( map_Pr3527087583225239196_a_nat @ produc8028804885397038213_a_nat @ ( zip_Su7355543910597222519_a_nat @ Xs @ Ys ) )
= Ys ) ) ).
% map_snd_zip
thf(fact_970_map__snd__zip,axiom,
! [Xs: list_nat,Ys: list_Sum_sum_a_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat @ ( zip_na2013496608136855606_a_nat @ Xs @ Ys ) )
= Ys ) ) ).
% map_snd_zip
thf(fact_971_card__length,axiom,
! [Xs: list_nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( set_nat2 @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).
% card_length
thf(fact_972_card__length,axiom,
! [Xs: list_Sum_sum_a_nat] : ( ord_less_eq_nat @ ( finite6080979521523705895_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) ) @ ( size_s5283204784079214577_a_nat @ Xs ) ) ).
% card_length
thf(fact_973_Set_Oempty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X: nat] : $false ) ) ).
% Set.empty_def
thf(fact_974_Set_Oempty__def,axiom,
( bot_bot_set_a
= ( collect_a
@ ^ [X: a] : $false ) ) ).
% Set.empty_def
thf(fact_975_emptyE,axiom,
! [A2: nat] :
~ ( member_nat @ A2 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_976_emptyE,axiom,
! [A2: list_Sum_sum_a_nat] :
~ ( member408289922725080238_a_nat @ A2 @ bot_bo1033123847703346641_a_nat ) ).
% emptyE
thf(fact_977_emptyE,axiom,
! [A2: sum_sum_a_nat] :
~ ( member_Sum_sum_a_nat @ A2 @ bot_bo3438331934148233675_a_nat ) ).
% emptyE
thf(fact_978_emptyE,axiom,
! [A2: a] :
~ ( member_a @ A2 @ bot_bot_set_a ) ).
% emptyE
thf(fact_979_equals0D,axiom,
! [A: set_nat,A2: nat] :
( ( A = bot_bot_set_nat )
=> ~ ( member_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_980_equals0D,axiom,
! [A: set_li6526943997496501093_a_nat,A2: list_Sum_sum_a_nat] :
( ( A = bot_bo1033123847703346641_a_nat )
=> ~ ( member408289922725080238_a_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_981_equals0D,axiom,
! [A: set_Sum_sum_a_nat,A2: sum_sum_a_nat] :
( ( A = bot_bo3438331934148233675_a_nat )
=> ~ ( member_Sum_sum_a_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_982_equals0D,axiom,
! [A: set_a,A2: a] :
( ( A = bot_bot_set_a )
=> ~ ( member_a @ A2 @ A ) ) ).
% equals0D
thf(fact_983_equals0I,axiom,
! [A: set_nat] :
( ! [Y3: nat] :
~ ( member_nat @ Y3 @ A )
=> ( A = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_984_equals0I,axiom,
! [A: set_li6526943997496501093_a_nat] :
( ! [Y3: list_Sum_sum_a_nat] :
~ ( member408289922725080238_a_nat @ Y3 @ A )
=> ( A = bot_bo1033123847703346641_a_nat ) ) ).
% equals0I
thf(fact_985_equals0I,axiom,
! [A: set_Sum_sum_a_nat] :
( ! [Y3: sum_sum_a_nat] :
~ ( member_Sum_sum_a_nat @ Y3 @ A )
=> ( A = bot_bo3438331934148233675_a_nat ) ) ).
% equals0I
thf(fact_986_equals0I,axiom,
! [A: set_a] :
( ! [Y3: a] :
~ ( member_a @ Y3 @ A )
=> ( A = bot_bot_set_a ) ) ).
% equals0I
thf(fact_987_ex__in__conv,axiom,
! [A: set_nat] :
( ( ? [X: nat] : ( member_nat @ X @ A ) )
= ( A != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_988_ex__in__conv,axiom,
! [A: set_li6526943997496501093_a_nat] :
( ( ? [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ A ) )
= ( A != bot_bo1033123847703346641_a_nat ) ) ).
% ex_in_conv
thf(fact_989_ex__in__conv,axiom,
! [A: set_Sum_sum_a_nat] :
( ( ? [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ A ) )
= ( A != bot_bo3438331934148233675_a_nat ) ) ).
% ex_in_conv
thf(fact_990_ex__in__conv,axiom,
! [A: set_a] :
( ( ? [X: a] : ( member_a @ X @ A ) )
= ( A != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_991_Inr__inject,axiom,
! [X2: nat,Y2: nat] :
( ( ( sum_Inr_nat_a @ X2 )
= ( sum_Inr_nat_a @ Y2 ) )
=> ( X2 = Y2 ) ) ).
% Inr_inject
thf(fact_992_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_nat] :
( ( size_size_list_nat @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_993_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_Sum_sum_a_nat] :
( ( size_s5283204784079214577_a_nat @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_994_neq__if__length__neq,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
!= ( size_size_list_nat @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_995_neq__if__length__neq,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
!= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_996_bot_Oextremum,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% bot.extremum
thf(fact_997_bot_Oextremum,axiom,
! [A2: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ bot_bo3438331934148233675_a_nat @ A2 ) ).
% bot.extremum
thf(fact_998_bot_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).
% bot.extremum
thf(fact_999_bot_Oextremum__unique,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_1000_bot_Oextremum__unique,axiom,
! [A2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A2 @ bot_bo3438331934148233675_a_nat )
= ( A2 = bot_bo3438331934148233675_a_nat ) ) ).
% bot.extremum_unique
thf(fact_1001_bot_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
= ( A2 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_1002_bot_Oextremum__uniqueI,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
=> ( A2 = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_1003_bot_Oextremum__uniqueI,axiom,
! [A2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A2 @ bot_bo3438331934148233675_a_nat )
=> ( A2 = bot_bo3438331934148233675_a_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_1004_bot_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
=> ( A2 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_1005_boolean__algebra_Odisj__zero__right,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ X2 @ bot_bot_set_nat )
= X2 ) ).
% boolean_algebra.disj_zero_right
thf(fact_1006_boolean__algebra_Odisj__zero__right,axiom,
! [X2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ X2 @ bot_bo3438331934148233675_a_nat )
= X2 ) ).
% boolean_algebra.disj_zero_right
thf(fact_1007_boolean__algebra_Odisj__zero__right,axiom,
! [X2: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ X2 @ bot_bo1033123847703346641_a_nat )
= X2 ) ).
% boolean_algebra.disj_zero_right
thf(fact_1008_boolean__algebra_Odisj__zero__right,axiom,
! [X2: set_a] :
( ( sup_sup_set_a @ X2 @ bot_bot_set_a )
= X2 ) ).
% boolean_algebra.disj_zero_right
thf(fact_1009_split__sum__all,axiom,
( ( ^ [P4: sum_sum_a_nat > $o] :
! [X6: sum_sum_a_nat] : ( P4 @ X6 ) )
= ( ^ [P5: sum_sum_a_nat > $o] :
( ! [X: a] : ( P5 @ ( sum_Inl_a_nat @ X ) )
& ! [X: nat] : ( P5 @ ( sum_Inr_nat_a @ X ) ) ) ) ) ).
% split_sum_all
thf(fact_1010_split__sum__ex,axiom,
( ( ^ [P4: sum_sum_a_nat > $o] :
? [X6: sum_sum_a_nat] : ( P4 @ X6 ) )
= ( ^ [P5: sum_sum_a_nat > $o] :
( ? [X: a] : ( P5 @ ( sum_Inl_a_nat @ X ) )
| ? [X: nat] : ( P5 @ ( sum_Inr_nat_a @ X ) ) ) ) ) ).
% split_sum_ex
thf(fact_1011_Inr__not__Inl,axiom,
! [B2: nat,A2: a] :
( ( sum_Inr_nat_a @ B2 )
!= ( sum_Inl_a_nat @ A2 ) ) ).
% Inr_not_Inl
thf(fact_1012_sumE,axiom,
! [S3: sum_sum_a_nat] :
( ! [X3: a] :
( S3
!= ( sum_Inl_a_nat @ X3 ) )
=> ~ ! [Y3: nat] :
( S3
!= ( sum_Inr_nat_a @ Y3 ) ) ) ).
% sumE
thf(fact_1013_old_Osum_Oexhaust,axiom,
! [Y2: sum_sum_a_nat] :
( ! [A7: a] :
( Y2
!= ( sum_Inl_a_nat @ A7 ) )
=> ~ ! [B6: nat] :
( Y2
!= ( sum_Inr_nat_a @ B6 ) ) ) ).
% old.sum.exhaust
thf(fact_1014_old_Osum_Odistinct_I1_J,axiom,
! [A2: a,B7: nat] :
( ( sum_Inl_a_nat @ A2 )
!= ( sum_Inr_nat_a @ B7 ) ) ).
% old.sum.distinct(1)
thf(fact_1015_old_Osum_Odistinct_I2_J,axiom,
! [B7: nat,A2: a] :
( ( sum_Inr_nat_a @ B7 )
!= ( sum_Inl_a_nat @ A2 ) ) ).
% old.sum.distinct(2)
thf(fact_1016_sum_Odistinct_I1_J,axiom,
! [X1: a,X23: nat] :
( ( sum_Inl_a_nat @ X1 )
!= ( sum_Inr_nat_a @ X23 ) ) ).
% sum.distinct(1)
thf(fact_1017_map__eq__imp__length__eq,axiom,
! [F: nat > sum_sum_a_nat,Xs: list_nat,G: nat > sum_sum_a_nat,Ys: list_nat] :
( ( ( map_na823391071729141993_a_nat @ F @ Xs )
= ( map_na823391071729141993_a_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_1018_map__eq__imp__length__eq,axiom,
! [F: produc5454855657571582244_a_nat > nat,Xs: list_P5056861408695629236_a_nat,G: nat > nat,Ys: list_nat] :
( ( ( map_Pr2058079554187054650at_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys ) )
=> ( ( size_s4076174644546656840_a_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_1019_map__eq__imp__length__eq,axiom,
! [F: nat > nat,Xs: list_nat,G: produc5454855657571582244_a_nat > nat,Ys: list_P5056861408695629236_a_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_Pr2058079554187054650at_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_s4076174644546656840_a_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_1020_map__eq__imp__length__eq,axiom,
! [F: nat > sum_sum_a_nat,Xs: list_nat,G: sum_sum_a_nat > sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ( map_na823391071729141993_a_nat @ F @ Xs )
= ( map_Su2790769393171190532_a_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_1021_map__eq__imp__length__eq,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,Xs: list_Sum_sum_a_nat,G: nat > sum_sum_a_nat,Ys: list_nat] :
( ( ( map_Su2790769393171190532_a_nat @ F @ Xs )
= ( map_na823391071729141993_a_nat @ G @ Ys ) )
=> ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_1022_map__eq__imp__length__eq,axiom,
! [F: produc5454855657571582244_a_nat > sum_sum_a_nat,Xs: list_P5056861408695629236_a_nat,G: nat > sum_sum_a_nat,Ys: list_nat] :
( ( ( map_Pr2515564165894599445_a_nat @ F @ Xs )
= ( map_na823391071729141993_a_nat @ G @ Ys ) )
=> ( ( size_s4076174644546656840_a_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_1023_map__eq__imp__length__eq,axiom,
! [F: produc5454855657571582244_a_nat > nat,Xs: list_P5056861408695629236_a_nat,G: sum_sum_a_nat > nat,Ys: list_Sum_sum_a_nat] :
( ( ( map_Pr2058079554187054650at_nat @ F @ Xs )
= ( map_Su5227373005390213643at_nat @ G @ Ys ) )
=> ( ( size_s4076174644546656840_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_1024_map__eq__imp__length__eq,axiom,
! [F: nat > sum_sum_a_nat,Xs: list_nat,G: produc5454855657571582244_a_nat > sum_sum_a_nat,Ys: list_P5056861408695629236_a_nat] :
( ( ( map_na823391071729141993_a_nat @ F @ Xs )
= ( map_Pr2515564165894599445_a_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_s4076174644546656840_a_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_1025_map__eq__imp__length__eq,axiom,
! [F: sum_sum_a_nat > nat,Xs: list_Sum_sum_a_nat,G: produc5454855657571582244_a_nat > nat,Ys: list_P5056861408695629236_a_nat] :
( ( ( map_Su5227373005390213643at_nat @ F @ Xs )
= ( map_Pr2058079554187054650at_nat @ G @ Ys ) )
=> ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_s4076174644546656840_a_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_1026_map__eq__imp__length__eq,axiom,
! [F: produc5454855657571582244_a_nat > nat,Xs: list_P5056861408695629236_a_nat,G: produc5454855657571582244_a_nat > nat,Ys: list_P5056861408695629236_a_nat] :
( ( ( map_Pr2058079554187054650at_nat @ F @ Xs )
= ( map_Pr2058079554187054650at_nat @ G @ Ys ) )
=> ( ( size_s4076174644546656840_a_nat @ Xs )
= ( size_s4076174644546656840_a_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_1027_length__filter__le,axiom,
! [P: produc5454855657571582244_a_nat > $o,Xs: list_P5056861408695629236_a_nat] : ( ord_less_eq_nat @ ( size_s4076174644546656840_a_nat @ ( filter1580576373184442087_a_nat @ P @ Xs ) ) @ ( size_s4076174644546656840_a_nat @ Xs ) ) ).
% length_filter_le
thf(fact_1028_length__filter__le,axiom,
! [P: nat > $o,Xs: list_nat] : ( ord_less_eq_nat @ ( size_size_list_nat @ ( filter_nat @ P @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).
% length_filter_le
thf(fact_1029_length__filter__le,axiom,
! [P: sum_sum_a_nat > $o,Xs: list_Sum_sum_a_nat] : ( ord_less_eq_nat @ ( size_s5283204784079214577_a_nat @ ( filter_Sum_sum_a_nat @ P @ Xs ) ) @ ( size_s5283204784079214577_a_nat @ Xs ) ) ).
% length_filter_le
thf(fact_1030_fo__nmlz__map,axiom,
! [AD: set_a,Sigma: nat > sum_sum_a_nat,Ns: list_nat] :
? [Tau3: nat > sum_sum_a_nat] :
( ( fo_nmlz_a @ AD @ ( map_na823391071729141993_a_nat @ Sigma @ Ns ) )
= ( map_na823391071729141993_a_nat @ Tau3 @ Ns ) ) ).
% fo_nmlz_map
thf(fact_1031_fo__nmlz__map,axiom,
! [AD: set_a,Sigma: produc5454855657571582244_a_nat > sum_sum_a_nat,Ns: list_P5056861408695629236_a_nat] :
? [Tau3: produc5454855657571582244_a_nat > sum_sum_a_nat] :
( ( fo_nmlz_a @ AD @ ( map_Pr2515564165894599445_a_nat @ Sigma @ Ns ) )
= ( map_Pr2515564165894599445_a_nat @ Tau3 @ Ns ) ) ).
% fo_nmlz_map
thf(fact_1032_sum_Odisc_I2_J,axiom,
! [X23: nat] :
~ ( sum_isl_a_nat @ ( sum_Inr_nat_a @ X23 ) ) ).
% sum.disc(2)
thf(fact_1033_sum_OdiscI_I2_J,axiom,
! [Sum2: sum_sum_a_nat,X23: nat] :
( ( Sum2
= ( sum_Inr_nat_a @ X23 ) )
=> ~ ( sum_isl_a_nat @ Sum2 ) ) ).
% sum.discI(2)
thf(fact_1034_subset__emptyI,axiom,
! [A: set_nat] :
( ! [X3: nat] :
~ ( member_nat @ X3 @ A )
=> ( ord_less_eq_set_nat @ A @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_1035_subset__emptyI,axiom,
! [A: set_li6526943997496501093_a_nat] :
( ! [X3: list_Sum_sum_a_nat] :
~ ( member408289922725080238_a_nat @ X3 @ A )
=> ( ord_le1147066620699065093_a_nat @ A @ bot_bo1033123847703346641_a_nat ) ) ).
% subset_emptyI
thf(fact_1036_subset__emptyI,axiom,
! [A: set_a] :
( ! [X3: a] :
~ ( member_a @ X3 @ A )
=> ( ord_less_eq_set_a @ A @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_1037_subset__emptyI,axiom,
! [A: set_Sum_sum_a_nat] :
( ! [X3: sum_sum_a_nat] :
~ ( member_Sum_sum_a_nat @ X3 @ A )
=> ( ord_le1325389633284124927_a_nat @ A @ bot_bo3438331934148233675_a_nat ) ) ).
% subset_emptyI
thf(fact_1038_Un__empty__left,axiom,
! [B: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ B )
= B ) ).
% Un_empty_left
thf(fact_1039_Un__empty__left,axiom,
! [B: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ bot_bo3438331934148233675_a_nat @ B )
= B ) ).
% Un_empty_left
thf(fact_1040_Un__empty__left,axiom,
! [B: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ bot_bo1033123847703346641_a_nat @ B )
= B ) ).
% Un_empty_left
thf(fact_1041_Un__empty__left,axiom,
! [B: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ B )
= B ) ).
% Un_empty_left
thf(fact_1042_Un__empty__right,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% Un_empty_right
thf(fact_1043_Un__empty__right,axiom,
! [A: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A @ bot_bo3438331934148233675_a_nat )
= A ) ).
% Un_empty_right
thf(fact_1044_Un__empty__right,axiom,
! [A: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ A @ bot_bo1033123847703346641_a_nat )
= A ) ).
% Un_empty_right
thf(fact_1045_Un__empty__right,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ bot_bot_set_a )
= A ) ).
% Un_empty_right
thf(fact_1046_fo__nmlzd__code,axiom,
( fo_nmlzd_a
= ( ^ [AD3: set_a,Xs2: list_Sum_sum_a_nat] :
( ( fo_nmlz_a @ AD3 @ Xs2 )
= Xs2 ) ) ) ).
% fo_nmlzd_code
thf(fact_1047_fo__nmlz__sound,axiom,
! [AD: set_a,Xs: list_Sum_sum_a_nat] : ( fo_nmlzd_a @ AD @ ( fo_nmlz_a @ AD @ Xs ) ) ).
% fo_nmlz_sound
thf(fact_1048_fo__nmlz__idem,axiom,
! [AD: set_a,Ys: list_Sum_sum_a_nat] :
( ( fo_nmlzd_a @ AD @ Ys )
=> ( ( fo_nmlz_a @ AD @ Ys )
= Ys ) ) ).
% fo_nmlz_idem
thf(fact_1049_fo__nmlz__ad__agr,axiom,
! [AD: set_a,Xs: list_Sum_sum_a_nat] : ( ad_agr_list_a_nat @ AD @ Xs @ ( fo_nmlz_a @ AD @ Xs ) ) ).
% fo_nmlz_ad_agr
thf(fact_1050_fo__nmlz__eqI,axiom,
! [AD: set_a,Vs2: list_Sum_sum_a_nat,Vs3: list_Sum_sum_a_nat] :
( ( ad_agr_list_a_nat @ AD @ Vs2 @ Vs3 )
=> ( ( fo_nmlz_a @ AD @ Vs2 )
= ( fo_nmlz_a @ AD @ Vs3 ) ) ) ).
% fo_nmlz_eqI
thf(fact_1051_fo__nmlz__eqD,axiom,
! [AD: set_a,Vs2: list_Sum_sum_a_nat,Vs3: list_Sum_sum_a_nat] :
( ( ( fo_nmlz_a @ AD @ Vs2 )
= ( fo_nmlz_a @ AD @ Vs3 ) )
=> ( ad_agr_list_a_nat @ AD @ Vs2 @ Vs3 ) ) ).
% fo_nmlz_eqD
thf(fact_1052_fo__nmlz__eq,axiom,
! [AD: set_a,Vs2: list_Sum_sum_a_nat,Vs3: list_Sum_sum_a_nat] :
( ( ( fo_nmlz_a @ AD @ Vs2 )
= ( fo_nmlz_a @ AD @ Vs3 ) )
= ( ad_agr_list_a_nat @ AD @ Vs2 @ Vs3 ) ) ).
% fo_nmlz_eq
thf(fact_1053_zip__eq__conv,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_P6011104703257516679at_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( zip_nat_nat @ Xs @ Ys )
= Zs )
= ( ( ( map_Pr3938374229010428429at_nat @ product_fst_nat_nat @ Zs )
= Xs )
& ( ( map_Pr3938374229010428429at_nat @ product_snd_nat_nat @ Zs )
= Ys ) ) ) ) ).
% zip_eq_conv
thf(fact_1054_zip__eq__conv,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_nat,Zs: list_P6164600145584960654at_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( zip_Su6417478541797927256at_nat @ Xs @ Ys )
= Zs )
= ( ( ( map_Pr6349035483858946235_a_nat @ produc5914057300446384524at_nat @ Zs )
= Xs )
& ( ( map_Pr8434518024483680788at_nat @ produc5684839914778638538at_nat @ Zs )
= Ys ) ) ) ) ).
% zip_eq_conv
thf(fact_1055_zip__eq__conv,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Zs: list_P1195027771636113901_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( ( zip_Su7355543910597222519_a_nat @ Xs @ Ys )
= Zs )
= ( ( ( map_Pr3527087583225239196_a_nat @ produc3940427395981429571_a_nat @ Zs )
= Xs )
& ( ( map_Pr3527087583225239196_a_nat @ produc8028804885397038213_a_nat @ Zs )
= Ys ) ) ) ) ).
% zip_eq_conv
thf(fact_1056_zip__eq__conv,axiom,
! [Xs: list_nat,Ys: list_Sum_sum_a_nat,Zs: list_P5056861408695629236_a_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( ( zip_na2013496608136855606_a_nat @ Xs @ Ys )
= Zs )
= ( ( ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Zs )
= Xs )
& ( ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat @ Zs )
= Ys ) ) ) ) ).
% zip_eq_conv
thf(fact_1057_fo__nmlz__mono,axiom,
! [AD: set_a,AD2: set_a,Xs: list_Sum_sum_a_nat] :
( ( ord_less_eq_set_a @ AD @ AD2 )
=> ( ( ord_less_eq_set_a @ ( vimage4607795094749595324_a_nat @ sum_Inl_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) ) @ AD )
=> ( ( fo_nmlz_a @ AD2 @ Xs )
= ( fo_nmlz_a @ AD @ Xs ) ) ) ) ).
% fo_nmlz_mono
thf(fact_1058_fo__nmlz__mono,axiom,
! [AD: set_Sum_sum_a_nat,AD2: set_Sum_sum_a_nat,Xs: list_S5657819638503513606at_nat] :
( ( ord_le1325389633284124927_a_nat @ AD @ AD2 )
=> ( ( ord_le1325389633284124927_a_nat @ ( vimage7613534090772763900at_nat @ sum_In1625650605130369377at_nat @ ( set_Su2720755619057964565at_nat @ Xs ) ) @ AD )
=> ( ( fo_nml8672979447642383935_a_nat @ AD2 @ Xs )
= ( fo_nml8672979447642383935_a_nat @ AD @ Xs ) ) ) ) ).
% fo_nmlz_mono
thf(fact_1059_fo__nmlz__set__rev,axiom,
! [AD: set_a,Xs: list_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( set_Sum_sum_a_nat2 @ ( fo_nmlz_a @ AD @ Xs ) ) @ ( image_7873763678140191238_a_nat @ sum_Inl_a_nat @ AD ) )
=> ( ord_le1325389633284124927_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) @ ( image_7873763678140191238_a_nat @ sum_Inl_a_nat @ AD ) ) ) ).
% fo_nmlz_set_rev
thf(fact_1060_Inl__Inr__False,axiom,
! [X2: a,Y2: nat] :
( ( sum_Inl_a_nat @ X2 )
!= ( sum_Inr_nat_a @ Y2 ) ) ).
% Inl_Inr_False
thf(fact_1061_Inr__Inl__False,axiom,
! [X2: nat,Y2: a] :
( ( sum_Inr_nat_a @ X2 )
!= ( sum_Inl_a_nat @ Y2 ) ) ).
% Inr_Inl_False
thf(fact_1062_len__fs_092_060phi_062,axiom,
( ( size_s5283204784079214577_a_nat @ fs_phi )
= ( size_size_list_nat @ ns_phi2 ) ) ).
% len_fs\<phi>
thf(fact_1063_len__xs__ys_I2_J,axiom,
( ( size_s5283204784079214577_a_nat @ ys )
= ( size_size_list_nat @ ns_psi ) ) ).
% len_xs_ys(2)
thf(fact_1064_len__xs__ys_I1_J,axiom,
( ( size_s5283204784079214577_a_nat @ xs )
= ( size_size_list_nat @ ns_phi ) ) ).
% len_xs_ys(1)
thf(fact_1065_eval__conj__set__aux_I3_J,axiom,
! [Ns_phi: list_nat,Ns_phi2: list_nat,Ns_psi: list_nat,Ns_psi2: list_nat,X_phi: set_li6526943997496501093_a_nat,AD: set_a,R_phi: set_na3699693778330250182_a_nat,X_psi: set_li6526943997496501093_a_nat,R_psi: set_na3699693778330250182_a_nat,Cxs: list_P5056861408695629236_a_nat,Xs: list_Sum_sum_a_nat,Nxs: list_nat,Cys: list_P5056861408695629236_a_nat,Ys: list_Sum_sum_a_nat,Nys: list_nat,Sigma_xs: nat > sum_sum_a_nat,Fs_phi: list_Sum_sum_a_nat,Sigma_ys: nat > sum_sum_a_nat,Fs_psi: list_Sum_sum_a_nat] :
( ( Ns_phi
= ( filter_nat
@ ^ [N2: nat] :
~ ( member_nat @ N2 @ ( set_nat2 @ Ns_phi2 ) )
@ Ns_psi ) )
=> ( ( Ns_psi2
= ( filter_nat
@ ^ [N2: nat] :
~ ( member_nat @ N2 @ ( set_nat2 @ Ns_psi ) )
@ Ns_phi2 ) )
=> ( ( X_phi
= ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ AD ) @ ( proj_v3643391342904276326_a_nat @ R_phi @ Ns_phi2 ) ) )
=> ( ( X_psi
= ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ AD ) @ ( proj_v3643391342904276326_a_nat @ R_psi @ Ns_psi ) ) )
=> ( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ns_phi2 )
& ( distinct_nat @ Ns_phi2 ) )
=> ( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ns_psi )
& ( distinct_nat @ Ns_psi ) )
=> ( ( Cxs
= ( filter1580576373184442087_a_nat
@ ( produc8810490045567577941_nat_o
@ ^ [N2: nat,X: sum_sum_a_nat] :
( ~ ( member_nat @ N2 @ ( set_nat2 @ Ns_psi ) )
& ( sum_isl_a_nat @ X ) ) )
@ ( zip_na2013496608136855606_a_nat @ Ns_phi2 @ Xs ) ) )
=> ( ( Nxs
= ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat
@ ( filter1580576373184442087_a_nat
@ ( produc8810490045567577941_nat_o
@ ^ [N2: nat,X: sum_sum_a_nat] :
( ~ ( member_nat @ N2 @ ( set_nat2 @ Ns_psi ) )
& ~ ( sum_isl_a_nat @ X ) ) )
@ ( zip_na2013496608136855606_a_nat @ Ns_phi2 @ Xs ) ) ) )
=> ( ( Cys
= ( filter1580576373184442087_a_nat
@ ( produc8810490045567577941_nat_o
@ ^ [N2: nat,Y: sum_sum_a_nat] :
( ~ ( member_nat @ N2 @ ( set_nat2 @ Ns_phi2 ) )
& ( sum_isl_a_nat @ Y ) ) )
@ ( zip_na2013496608136855606_a_nat @ Ns_psi @ Ys ) ) )
=> ( ( Nys
= ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat
@ ( filter1580576373184442087_a_nat
@ ( produc8810490045567577941_nat_o
@ ^ [N2: nat,Y: sum_sum_a_nat] :
( ~ ( member_nat @ N2 @ ( set_nat2 @ Ns_phi2 ) )
& ~ ( sum_isl_a_nat @ Y ) ) )
@ ( zip_na2013496608136855606_a_nat @ Ns_psi @ Ys ) ) ) )
=> ( ( member408289922725080238_a_nat @ Xs @ X_phi )
=> ( ( member408289922725080238_a_nat @ Ys @ X_psi )
=> ( ( Xs
= ( map_na823391071729141993_a_nat @ Sigma_xs @ Ns_phi2 ) )
=> ( ( Fs_phi
= ( map_na823391071729141993_a_nat @ Sigma_xs @ Ns_phi ) )
=> ( ( Ys
= ( map_na823391071729141993_a_nat @ Sigma_ys @ Ns_psi ) )
=> ( ( Fs_psi
= ( map_na823391071729141993_a_nat @ Sigma_ys @ Ns_psi2 ) )
=> ( ( member408289922725080238_a_nat @ Fs_phi @ ( nall_tuples_rec_a @ AD @ ( finite_card_nat @ ( vimage3040984495076556338_a_nat @ sum_Inr_nat_a @ ( set_Sum_sum_a_nat2 @ Xs ) ) ) @ ( size_size_list_nat @ Ns_phi ) ) )
=> ( ( member408289922725080238_a_nat @ Fs_psi @ ( nall_tuples_rec_a @ AD @ ( finite_card_nat @ ( vimage3040984495076556338_a_nat @ sum_Inr_nat_a @ ( set_Sum_sum_a_nat2 @ Ys ) ) ) @ ( size_size_list_nat @ Ns_psi2 ) ) )
=> ( ( ad_agr_list_a_nat @ AD
@ ( map_na823391071729141993_a_nat @ Sigma_ys
@ ( linord738340561235409698at_nat
@ ^ [X: nat] : X
@ ( append_nat @ Ns_psi @ Ns_psi2 ) ) )
@ ( map_na823391071729141993_a_nat @ Sigma_xs
@ ( linord738340561235409698at_nat
@ ^ [X: nat] : X
@ ( append_nat @ Ns_phi2 @ Ns_phi ) ) ) )
=> ( member408289922725080238_a_nat @ ( map_na823391071729141993_a_nat @ Sigma_xs @ Nys )
@ ( nall_tuples_rec_a @ bot_bot_set_a
@ ( finite_card_nat
@ ( vimage3040984495076556338_a_nat @ sum_Inr_nat_a
@ ( set_Sum_sum_a_nat2
@ ( map_na823391071729141993_a_nat @ Sigma_xs
@ ( linord738340561235409698at_nat
@ ^ [X: nat] : X
@ ( append_nat @ Ns_phi2 @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Cys ) ) ) ) ) ) )
@ ( size_size_list_nat @ Nys ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% eval_conj_set_aux(3)
thf(fact_1066_distinct_I1_J,axiom,
( ( sorted_wrt_nat @ ord_less_eq_nat @ ns_phi )
& ( distinct_nat @ ns_phi ) ) ).
% distinct(1)
thf(fact_1067_distinct_I2_J,axiom,
( ( sorted_wrt_nat @ ord_less_eq_nat @ ns_psi )
& ( distinct_nat @ ns_psi ) ) ).
% distinct(2)
thf(fact_1068_distinct__sort,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( distinct_nat @ ( linord738340561235409698at_nat @ F @ Xs ) )
= ( distinct_nat @ Xs ) ) ).
% distinct_sort
thf(fact_1069_distinct__quicksort,axiom,
! [Xs: list_nat] :
( ( distinct_nat @ ( set_or9089632773640736191rt_nat @ Xs ) )
= ( distinct_nat @ Xs ) ) ).
% distinct_quicksort
thf(fact_1070_distinct__union,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( distinct_nat @ ( union_nat @ Xs @ Ys ) )
= ( distinct_nat @ Ys ) ) ).
% distinct_union
thf(fact_1071_sorted__wrt__true,axiom,
! [Xs: list_nat] :
( sorted_wrt_nat
@ ^ [Uu: nat,Uv: nat] : $true
@ Xs ) ).
% sorted_wrt_true
thf(fact_1072_sorted__distinct__set__unique,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( ( distinct_nat @ Xs )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ys )
=> ( ( distinct_nat @ Ys )
=> ( ( ( set_nat2 @ Xs )
= ( set_nat2 @ Ys ) )
=> ( Xs = Ys ) ) ) ) ) ) ).
% sorted_distinct_set_unique
thf(fact_1073_sorted__list__of__set_Odistinct__if__distinct__map,axiom,
! [Xs: list_nat] :
( ( distinct_nat @ Xs )
=> ( distinct_nat @ Xs ) ) ).
% sorted_list_of_set.distinct_if_distinct_map
thf(fact_1074_distinct__zipI2,axiom,
! [Ys: list_Sum_sum_a_nat,Xs: list_nat] :
( ( distin2701893636801681144_a_nat @ Ys )
=> ( distin5548549501492119275_a_nat @ ( zip_na2013496608136855606_a_nat @ Xs @ Ys ) ) ) ).
% distinct_zipI2
thf(fact_1075_distinct__zipI1,axiom,
! [Xs: list_nat,Ys: list_Sum_sum_a_nat] :
( ( distinct_nat @ Xs )
=> ( distin5548549501492119275_a_nat @ ( zip_na2013496608136855606_a_nat @ Xs @ Ys ) ) ) ).
% distinct_zipI1
thf(fact_1076_sorted__wrt__mono__rel,axiom,
! [Xs: list_l4703314356710769291_a_nat,P: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o,Q: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o] :
( ! [X3: list_Sum_sum_a_nat,Y3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ ( set_li2392974972034027290_a_nat @ Xs ) )
=> ( ( member408289922725080238_a_nat @ Y3 @ ( set_li2392974972034027290_a_nat @ Xs ) )
=> ( ( P @ X3 @ Y3 )
=> ( Q @ X3 @ Y3 ) ) ) )
=> ( ( sorted1521004414301201522_a_nat @ P @ Xs )
=> ( sorted1521004414301201522_a_nat @ Q @ Xs ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_1077_sorted__wrt__mono__rel,axiom,
! [Xs: list_a,P: a > a > $o,Q: a > a > $o] :
( ! [X3: a,Y3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( ( member_a @ Y3 @ ( set_a2 @ Xs ) )
=> ( ( P @ X3 @ Y3 )
=> ( Q @ X3 @ Y3 ) ) ) )
=> ( ( sorted_wrt_a @ P @ Xs )
=> ( sorted_wrt_a @ Q @ Xs ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_1078_sorted__wrt__mono__rel,axiom,
! [Xs: list_Sum_sum_a_nat,P: sum_sum_a_nat > sum_sum_a_nat > $o,Q: sum_sum_a_nat > sum_sum_a_nat > $o] :
( ! [X3: sum_sum_a_nat,Y3: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X3 @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ( ( member_Sum_sum_a_nat @ Y3 @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ( ( P @ X3 @ Y3 )
=> ( Q @ X3 @ Y3 ) ) ) )
=> ( ( sorted6245805940552704876_a_nat @ P @ Xs )
=> ( sorted6245805940552704876_a_nat @ Q @ Xs ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_1079_sorted__wrt__mono__rel,axiom,
! [Xs: list_nat,P: nat > nat > $o,Q: nat > nat > $o] :
( ! [X3: nat,Y3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( member_nat @ Y3 @ ( set_nat2 @ Xs ) )
=> ( ( P @ X3 @ Y3 )
=> ( Q @ X3 @ Y3 ) ) ) )
=> ( ( sorted_wrt_nat @ P @ Xs )
=> ( sorted_wrt_nat @ Q @ Xs ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_1080_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X: nat] : ( member_nat @ X @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_1081_bot__empty__eq,axiom,
( bot_bo9042073657639083596_nat_o
= ( ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ bot_bo1033123847703346641_a_nat ) ) ) ).
% bot_empty_eq
thf(fact_1082_bot__empty__eq,axiom,
( bot_bo7797463397293707474_nat_o
= ( ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ bot_bo3438331934148233675_a_nat ) ) ) ).
% bot_empty_eq
thf(fact_1083_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X: a] : ( member_a @ X @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_1084_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_1085_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_1086_sorted__wrt__filter,axiom,
! [F: produc5454855657571582244_a_nat > produc5454855657571582244_a_nat > $o,Xs: list_P5056861408695629236_a_nat,P: produc5454855657571582244_a_nat > $o] :
( ( sorted7320194442031503735_a_nat @ F @ Xs )
=> ( sorted7320194442031503735_a_nat @ F @ ( filter1580576373184442087_a_nat @ P @ Xs ) ) ) ).
% sorted_wrt_filter
thf(fact_1087_sorted__wrt__filter,axiom,
! [F: nat > nat > $o,Xs: list_nat,P: nat > $o] :
( ( sorted_wrt_nat @ F @ Xs )
=> ( sorted_wrt_nat @ F @ ( filter_nat @ P @ Xs ) ) ) ).
% sorted_wrt_filter
thf(fact_1088_distinct__filter,axiom,
! [Xs: list_P5056861408695629236_a_nat,P: produc5454855657571582244_a_nat > $o] :
( ( distin5548549501492119275_a_nat @ Xs )
=> ( distin5548549501492119275_a_nat @ ( filter1580576373184442087_a_nat @ P @ Xs ) ) ) ).
% distinct_filter
thf(fact_1089_distinct__filter,axiom,
! [Xs: list_nat,P: nat > $o] :
( ( distinct_nat @ Xs )
=> ( distinct_nat @ ( filter_nat @ P @ Xs ) ) ) ).
% distinct_filter
thf(fact_1090_fo__nmlz__length,axiom,
! [AD: set_a,Xs: list_Sum_sum_a_nat] :
( ( size_s5283204784079214577_a_nat @ ( fo_nmlz_a @ AD @ Xs ) )
= ( size_s5283204784079214577_a_nat @ Xs ) ) ).
% fo_nmlz_length
thf(fact_1091_ad__agr__list__length,axiom,
! [X4: set_a,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ad_agr_list_a_nat @ X4 @ Xs @ Ys )
=> ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) ) ) ).
% ad_agr_list_length
thf(fact_1092_fo__nmlz__rec__length,axiom,
! [I: nat,M2: sum_sum_a_nat > option_nat,AD: set_a,Xs: list_Sum_sum_a_nat] :
( ( size_s5283204784079214577_a_nat @ ( fo_nmlz_rec_a @ I @ M2 @ AD @ Xs ) )
= ( size_s5283204784079214577_a_nat @ Xs ) ) ).
% fo_nmlz_rec_length
thf(fact_1093_sorted__wrt__map,axiom,
! [R: sum_sum_a_nat > sum_sum_a_nat > $o,F: produc5454855657571582244_a_nat > sum_sum_a_nat,Xs: list_P5056861408695629236_a_nat] :
( ( sorted6245805940552704876_a_nat @ R @ ( map_Pr2515564165894599445_a_nat @ F @ Xs ) )
= ( sorted7320194442031503735_a_nat
@ ^ [X: produc5454855657571582244_a_nat,Y: produc5454855657571582244_a_nat] : ( R @ ( F @ X ) @ ( F @ Y ) )
@ Xs ) ) ).
% sorted_wrt_map
thf(fact_1094_sorted__wrt__map,axiom,
! [R: sum_sum_a_nat > sum_sum_a_nat > $o,F: nat > sum_sum_a_nat,Xs: list_nat] :
( ( sorted6245805940552704876_a_nat @ R @ ( map_na823391071729141993_a_nat @ F @ Xs ) )
= ( sorted_wrt_nat
@ ^ [X: nat,Y: nat] : ( R @ ( F @ X ) @ ( F @ Y ) )
@ Xs ) ) ).
% sorted_wrt_map
thf(fact_1095_sorted__wrt__map,axiom,
! [R: nat > nat > $o,F: produc5454855657571582244_a_nat > nat,Xs: list_P5056861408695629236_a_nat] :
( ( sorted_wrt_nat @ R @ ( map_Pr2058079554187054650at_nat @ F @ Xs ) )
= ( sorted7320194442031503735_a_nat
@ ^ [X: produc5454855657571582244_a_nat,Y: produc5454855657571582244_a_nat] : ( R @ ( F @ X ) @ ( F @ Y ) )
@ Xs ) ) ).
% sorted_wrt_map
thf(fact_1096_sorted__wrt__map,axiom,
! [R: nat > nat > $o,F: nat > nat,Xs: list_nat] :
( ( sorted_wrt_nat @ R @ ( map_nat_nat @ F @ Xs ) )
= ( sorted_wrt_nat
@ ^ [X: nat,Y: nat] : ( R @ ( F @ X ) @ ( F @ Y ) )
@ Xs ) ) ).
% sorted_wrt_map
thf(fact_1097_nall__tuples__rec__length,axiom,
! [Xs: list_Sum_sum_a_nat,AD: set_a,I: nat,N: nat] :
( ( member408289922725080238_a_nat @ Xs @ ( nall_tuples_rec_a @ AD @ I @ N ) )
=> ( ( size_s5283204784079214577_a_nat @ Xs )
= N ) ) ).
% nall_tuples_rec_length
thf(fact_1098_proj__tuple__length,axiom,
! [Ns: list_nat,Ms: list_nat,Xs: list_Sum_sum_a_nat] :
( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ns )
& ( distinct_nat @ Ns ) )
=> ( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ms )
& ( distinct_nat @ Ms ) )
=> ( ( ord_less_eq_set_nat @ ( set_nat2 @ Ns ) @ ( set_nat2 @ Ms ) )
=> ( ( ( size_size_list_nat @ Ms )
= ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( ( size_s5283204784079214577_a_nat @ ( proj_tuple_a @ Ns @ ( zip_na2013496608136855606_a_nat @ Ms @ Xs ) ) )
= ( size_size_list_nat @ Ns ) ) ) ) ) ) ).
% proj_tuple_length
thf(fact_1099_sorted__wrt__append,axiom,
! [P: sum_sum_a_nat > sum_sum_a_nat > $o,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( sorted6245805940552704876_a_nat @ P @ ( append_Sum_sum_a_nat @ Xs @ Ys ) )
= ( ( sorted6245805940552704876_a_nat @ P @ Xs )
& ( sorted6245805940552704876_a_nat @ P @ Ys )
& ! [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ! [Y: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ Y @ ( set_Sum_sum_a_nat2 @ Ys ) )
=> ( P @ X @ Y ) ) ) ) ) ).
% sorted_wrt_append
thf(fact_1100_sorted__wrt__append,axiom,
! [P: nat > nat > $o,Xs: list_nat,Ys: list_nat] :
( ( sorted_wrt_nat @ P @ ( append_nat @ Xs @ Ys ) )
= ( ( sorted_wrt_nat @ P @ Xs )
& ( sorted_wrt_nat @ P @ Ys )
& ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ! [Y: nat] :
( ( member_nat @ Y @ ( set_nat2 @ Ys ) )
=> ( P @ X @ Y ) ) ) ) ) ).
% sorted_wrt_append
thf(fact_1101_exists__map,axiom,
! [Xs: list_P5056861408695629236_a_nat,Ys: list_nat] :
( ( ( size_s4076174644546656840_a_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( distin5548549501492119275_a_nat @ Xs )
=> ? [F3: produc5454855657571582244_a_nat > nat] :
( Ys
= ( map_Pr2058079554187054650at_nat @ F3 @ Xs ) ) ) ) ).
% exists_map
thf(fact_1102_exists__map,axiom,
! [Xs: list_P5056861408695629236_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ( size_s4076174644546656840_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( distin5548549501492119275_a_nat @ Xs )
=> ? [F3: produc5454855657571582244_a_nat > sum_sum_a_nat] :
( Ys
= ( map_Pr2515564165894599445_a_nat @ F3 @ Xs ) ) ) ) ).
% exists_map
thf(fact_1103_exists__map,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( distinct_nat @ Xs )
=> ? [F3: nat > nat] :
( Ys
= ( map_nat_nat @ F3 @ Xs ) ) ) ) ).
% exists_map
thf(fact_1104_exists__map,axiom,
! [Xs: list_nat,Ys: list_Sum_sum_a_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( distinct_nat @ Xs )
=> ? [F3: nat > sum_sum_a_nat] :
( Ys
= ( map_na823391071729141993_a_nat @ F3 @ Xs ) ) ) ) ).
% exists_map
thf(fact_1105_exists__map,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( distin2701893636801681144_a_nat @ Xs )
=> ? [F3: sum_sum_a_nat > nat] :
( Ys
= ( map_Su5227373005390213643at_nat @ F3 @ Xs ) ) ) ) ).
% exists_map
thf(fact_1106_exists__map,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( distin2701893636801681144_a_nat @ Xs )
=> ? [F3: sum_sum_a_nat > sum_sum_a_nat] :
( Ys
= ( map_Su2790769393171190532_a_nat @ F3 @ Xs ) ) ) ) ).
% exists_map
thf(fact_1107_sorted__map,axiom,
! [F: produc5454855657571582244_a_nat > nat,Xs: list_P5056861408695629236_a_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_Pr2058079554187054650at_nat @ F @ Xs ) )
= ( sorted7320194442031503735_a_nat
@ ^ [X: produc5454855657571582244_a_nat,Y: produc5454855657571582244_a_nat] : ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) )
@ Xs ) ) ).
% sorted_map
thf(fact_1108_sorted__map,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F @ Xs ) )
= ( sorted_wrt_nat
@ ^ [X: nat,Y: nat] : ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) )
@ Xs ) ) ).
% sorted_map
thf(fact_1109_sorted__same,axiom,
! [G: list_nat > nat,Xs: list_nat] :
( sorted_wrt_nat @ ord_less_eq_nat
@ ( filter_nat
@ ^ [X: nat] :
( X
= ( G @ Xs ) )
@ Xs ) ) ).
% sorted_same
thf(fact_1110_sorted__sort,axiom,
! [Xs: list_nat] :
( sorted_wrt_nat @ ord_less_eq_nat
@ ( linord738340561235409698at_nat
@ ^ [X: nat] : X
@ Xs ) ) ).
% sorted_sort
thf(fact_1111_sorted__sort__id,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( ( linord738340561235409698at_nat
@ ^ [X: nat] : X
@ Xs )
= Xs ) ) ).
% sorted_sort_id
thf(fact_1112_sorted__quicksort,axiom,
! [Xs: list_nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( set_or9089632773640736191rt_nat @ Xs ) ) ).
% sorted_quicksort
thf(fact_1113_proj__tuple__map,axiom,
! [Ns: list_nat,Ms: list_nat,Sigma: nat > sum_sum_a_nat] :
( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ns )
& ( distinct_nat @ Ns ) )
=> ( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ms )
& ( distinct_nat @ Ms ) )
=> ( ( ord_less_eq_set_nat @ ( set_nat2 @ Ns ) @ ( set_nat2 @ Ms ) )
=> ( ( proj_tuple_a @ Ns @ ( zip_na2013496608136855606_a_nat @ Ms @ ( map_na823391071729141993_a_nat @ Sigma @ Ms ) ) )
= ( map_na823391071729141993_a_nat @ Sigma @ Ns ) ) ) ) ) ).
% proj_tuple_map
thf(fact_1114_sorted__append,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( append_nat @ Xs @ Ys ) )
= ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
& ( sorted_wrt_nat @ ord_less_eq_nat @ Ys )
& ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ! [Y: nat] :
( ( member_nat @ Y @ ( set_nat2 @ Ys ) )
=> ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ).
% sorted_append
thf(fact_1115_sorted__filter,axiom,
! [F: nat > nat,Xs: list_nat,P: nat > $o] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F @ Xs ) )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F @ ( filter_nat @ P @ Xs ) ) ) ) ).
% sorted_filter
thf(fact_1116_sorted__filter,axiom,
! [F: produc5454855657571582244_a_nat > nat,Xs: list_P5056861408695629236_a_nat,P: produc5454855657571582244_a_nat > $o] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_Pr2058079554187054650at_nat @ F @ Xs ) )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_Pr2058079554187054650at_nat @ F @ ( filter1580576373184442087_a_nat @ P @ Xs ) ) ) ) ).
% sorted_filter
thf(fact_1117_sort__key__id__if__sorted,axiom,
! [F: produc5454855657571582244_a_nat > nat,Xs: list_P5056861408695629236_a_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_Pr2058079554187054650at_nat @ F @ Xs ) )
=> ( ( linord4234369348993689078at_nat @ F @ Xs )
= Xs ) ) ).
% sort_key_id_if_sorted
thf(fact_1118_sort__key__id__if__sorted,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F @ Xs ) )
=> ( ( linord738340561235409698at_nat @ F @ Xs )
= Xs ) ) ).
% sort_key_id_if_sorted
thf(fact_1119_sorted__sort__key,axiom,
! [F: produc5454855657571582244_a_nat > nat,Xs: list_P5056861408695629236_a_nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_Pr2058079554187054650at_nat @ F @ ( linord4234369348993689078at_nat @ F @ Xs ) ) ) ).
% sorted_sort_key
thf(fact_1120_sorted__sort__key,axiom,
! [F: nat > nat,Xs: list_nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F @ ( linord738340561235409698at_nat @ F @ Xs ) ) ) ).
% sorted_sort_key
thf(fact_1121_distinct__card,axiom,
! [Xs: list_nat] :
( ( distinct_nat @ Xs )
=> ( ( finite_card_nat @ ( set_nat2 @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ) ).
% distinct_card
thf(fact_1122_distinct__card,axiom,
! [Xs: list_Sum_sum_a_nat] :
( ( distin2701893636801681144_a_nat @ Xs )
=> ( ( finite6080979521523705895_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) )
= ( size_s5283204784079214577_a_nat @ Xs ) ) ) ).
% distinct_card
thf(fact_1123_card__distinct,axiom,
! [Xs: list_nat] :
( ( ( finite_card_nat @ ( set_nat2 @ Xs ) )
= ( size_size_list_nat @ Xs ) )
=> ( distinct_nat @ Xs ) ) ).
% card_distinct
thf(fact_1124_card__distinct,axiom,
! [Xs: list_Sum_sum_a_nat] :
( ( ( finite6080979521523705895_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) )
= ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( distin2701893636801681144_a_nat @ Xs ) ) ).
% card_distinct
thf(fact_1125_sorted__map__same,axiom,
! [F: nat > nat,G: list_nat > nat,Xs: list_nat] :
( sorted_wrt_nat @ ord_less_eq_nat
@ ( map_nat_nat @ F
@ ( filter_nat
@ ^ [X: nat] :
( ( F @ X )
= ( G @ Xs ) )
@ Xs ) ) ) ).
% sorted_map_same
thf(fact_1126_sorted__map__same,axiom,
! [F: produc5454855657571582244_a_nat > nat,G: list_P5056861408695629236_a_nat > nat,Xs: list_P5056861408695629236_a_nat] :
( sorted_wrt_nat @ ord_less_eq_nat
@ ( map_Pr2058079554187054650at_nat @ F
@ ( filter1580576373184442087_a_nat
@ ^ [X: produc5454855657571582244_a_nat] :
( ( F @ X )
= ( G @ Xs ) )
@ Xs ) ) ) ).
% sorted_map_same
thf(fact_1127_sorted__enumerate,axiom,
! [N: nat,Xs: list_Sum_sum_a_nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ ( enumer3164015132978342500_a_nat @ N @ Xs ) ) ) ).
% sorted_enumerate
thf(fact_1128_nall__tuples__rec__filter,axiom,
! [Xs: list_Sum_sum_a_nat,AD: set_a,N: nat,Ys: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ Xs @ ( nall_tuples_rec_a @ AD @ N @ ( size_s5283204784079214577_a_nat @ Xs ) ) )
=> ( ( Ys
= ( filter_Sum_sum_a_nat
@ ^ [X: sum_sum_a_nat] :
~ ( sum_isl_a_nat @ X )
@ Xs ) )
=> ( member408289922725080238_a_nat @ Ys @ ( nall_tuples_rec_a @ bot_bot_set_a @ N @ ( size_s5283204784079214577_a_nat @ Ys ) ) ) ) ) ).
% nall_tuples_rec_filter
thf(fact_1129_exists__fo__nmlzd,axiom,
! [Xs: list_P5056861408695629236_a_nat,Ys: list_Sum_sum_a_nat,AD: set_a] :
( ( ( size_s4076174644546656840_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( distin5548549501492119275_a_nat @ Xs )
=> ( ( fo_nmlzd_a @ AD @ Ys )
=> ? [F3: produc5454855657571582244_a_nat > sum_sum_a_nat] :
( Ys
= ( fo_nmlz_a @ AD @ ( map_Pr2515564165894599445_a_nat @ F3 @ Xs ) ) ) ) ) ) ).
% exists_fo_nmlzd
thf(fact_1130_exists__fo__nmlzd,axiom,
! [Xs: list_nat,Ys: list_Sum_sum_a_nat,AD: set_a] :
( ( ( size_size_list_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( distinct_nat @ Xs )
=> ( ( fo_nmlzd_a @ AD @ Ys )
=> ? [F3: nat > sum_sum_a_nat] :
( Ys
= ( fo_nmlz_a @ AD @ ( map_na823391071729141993_a_nat @ F3 @ Xs ) ) ) ) ) ) ).
% exists_fo_nmlzd
thf(fact_1131_exists__fo__nmlzd,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,AD: set_a] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( distin2701893636801681144_a_nat @ Xs )
=> ( ( fo_nmlzd_a @ AD @ Ys )
=> ? [F3: sum_sum_a_nat > sum_sum_a_nat] :
( Ys
= ( fo_nmlz_a @ AD @ ( map_Su2790769393171190532_a_nat @ F3 @ Xs ) ) ) ) ) ) ).
% exists_fo_nmlzd
thf(fact_1132_card__Inr__vimage__le__length,axiom,
! [Xs: list_Sum_sum_a_nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( vimage3040984495076556338_a_nat @ sum_Inr_nat_a @ ( set_Sum_sum_a_nat2 @ Xs ) ) ) @ ( size_s5283204784079214577_a_nat @ Xs ) ) ).
% card_Inr_vimage_le_length
thf(fact_1133_nall__tuples__rec__filter__rev,axiom,
! [Ys: list_Sum_sum_a_nat,N: nat,Xs: list_Sum_sum_a_nat,AD: set_a] :
( ( member408289922725080238_a_nat @ Ys @ ( nall_tuples_rec_a @ bot_bot_set_a @ N @ ( size_s5283204784079214577_a_nat @ Ys ) ) )
=> ( ( Ys
= ( filter_Sum_sum_a_nat
@ ^ [X: sum_sum_a_nat] :
~ ( sum_isl_a_nat @ X )
@ Xs ) )
=> ( ( ord_less_eq_set_a @ ( vimage4607795094749595324_a_nat @ sum_Inl_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) ) @ AD )
=> ( member408289922725080238_a_nat @ Xs @ ( nall_tuples_rec_a @ AD @ N @ ( size_s5283204784079214577_a_nat @ Xs ) ) ) ) ) ) ).
% nall_tuples_rec_filter_rev
thf(fact_1134_nall__tuples__rec__filter__rev,axiom,
! [Ys: list_S5657819638503513606at_nat,N: nat,Xs: list_S5657819638503513606at_nat,AD: set_Sum_sum_a_nat] :
( ( member4821912183549559215at_nat @ Ys @ ( nall_t5818786088032561781_a_nat @ bot_bo3438331934148233675_a_nat @ N @ ( size_s4096723861602926706at_nat @ Ys ) ) )
=> ( ( Ys
= ( filter1151248133354082813at_nat
@ ^ [X: sum_su4711128520861746048at_nat] :
~ ( sum_is7876730191583103919at_nat @ X )
@ Xs ) )
=> ( ( ord_le1325389633284124927_a_nat @ ( vimage7613534090772763900at_nat @ sum_In1625650605130369377at_nat @ ( set_Su2720755619057964565at_nat @ Xs ) ) @ AD )
=> ( member4821912183549559215at_nat @ Xs @ ( nall_t5818786088032561781_a_nat @ AD @ N @ ( size_s4096723861602926706at_nat @ Xs ) ) ) ) ) ) ).
% nall_tuples_rec_filter_rev
thf(fact_1135_eval__conj__set__aux_I2_J,axiom,
! [Ns_phi: list_nat,Ns_phi2: list_nat,Ns_psi: list_nat,Ns_psi2: list_nat,X_phi: set_li6526943997496501093_a_nat,AD: set_a,R_phi: set_na3699693778330250182_a_nat,X_psi: set_li6526943997496501093_a_nat,R_psi: set_na3699693778330250182_a_nat,Cxs: list_P5056861408695629236_a_nat,Xs: list_Sum_sum_a_nat,Nxs: list_nat,Cys: list_P5056861408695629236_a_nat,Ys: list_Sum_sum_a_nat,Nys: list_nat,Sigma_xs: nat > sum_sum_a_nat,Fs_phi: list_Sum_sum_a_nat,Sigma_ys: nat > sum_sum_a_nat,Fs_psi: list_Sum_sum_a_nat] :
( ( Ns_phi
= ( filter_nat
@ ^ [N2: nat] :
~ ( member_nat @ N2 @ ( set_nat2 @ Ns_phi2 ) )
@ Ns_psi ) )
=> ( ( Ns_psi2
= ( filter_nat
@ ^ [N2: nat] :
~ ( member_nat @ N2 @ ( set_nat2 @ Ns_psi ) )
@ Ns_phi2 ) )
=> ( ( X_phi
= ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ AD ) @ ( proj_v3643391342904276326_a_nat @ R_phi @ Ns_phi2 ) ) )
=> ( ( X_psi
= ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ AD ) @ ( proj_v3643391342904276326_a_nat @ R_psi @ Ns_psi ) ) )
=> ( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ns_phi2 )
& ( distinct_nat @ Ns_phi2 ) )
=> ( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ns_psi )
& ( distinct_nat @ Ns_psi ) )
=> ( ( Cxs
= ( filter1580576373184442087_a_nat
@ ( produc8810490045567577941_nat_o
@ ^ [N2: nat,X: sum_sum_a_nat] :
( ~ ( member_nat @ N2 @ ( set_nat2 @ Ns_psi ) )
& ( sum_isl_a_nat @ X ) ) )
@ ( zip_na2013496608136855606_a_nat @ Ns_phi2 @ Xs ) ) )
=> ( ( Nxs
= ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat
@ ( filter1580576373184442087_a_nat
@ ( produc8810490045567577941_nat_o
@ ^ [N2: nat,X: sum_sum_a_nat] :
( ~ ( member_nat @ N2 @ ( set_nat2 @ Ns_psi ) )
& ~ ( sum_isl_a_nat @ X ) ) )
@ ( zip_na2013496608136855606_a_nat @ Ns_phi2 @ Xs ) ) ) )
=> ( ( Cys
= ( filter1580576373184442087_a_nat
@ ( produc8810490045567577941_nat_o
@ ^ [N2: nat,Y: sum_sum_a_nat] :
( ~ ( member_nat @ N2 @ ( set_nat2 @ Ns_phi2 ) )
& ( sum_isl_a_nat @ Y ) ) )
@ ( zip_na2013496608136855606_a_nat @ Ns_psi @ Ys ) ) )
=> ( ( Nys
= ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat
@ ( filter1580576373184442087_a_nat
@ ( produc8810490045567577941_nat_o
@ ^ [N2: nat,Y: sum_sum_a_nat] :
( ~ ( member_nat @ N2 @ ( set_nat2 @ Ns_phi2 ) )
& ~ ( sum_isl_a_nat @ Y ) ) )
@ ( zip_na2013496608136855606_a_nat @ Ns_psi @ Ys ) ) ) )
=> ( ( member408289922725080238_a_nat @ Xs @ X_phi )
=> ( ( member408289922725080238_a_nat @ Ys @ X_psi )
=> ( ( Xs
= ( map_na823391071729141993_a_nat @ Sigma_xs @ Ns_phi2 ) )
=> ( ( Fs_phi
= ( map_na823391071729141993_a_nat @ Sigma_xs @ Ns_phi ) )
=> ( ( Ys
= ( map_na823391071729141993_a_nat @ Sigma_ys @ Ns_psi ) )
=> ( ( Fs_psi
= ( map_na823391071729141993_a_nat @ Sigma_ys @ Ns_psi2 ) )
=> ( ( member408289922725080238_a_nat @ Fs_phi @ ( nall_tuples_rec_a @ AD @ ( finite_card_nat @ ( vimage3040984495076556338_a_nat @ sum_Inr_nat_a @ ( set_Sum_sum_a_nat2 @ Xs ) ) ) @ ( size_size_list_nat @ Ns_phi ) ) )
=> ( ( member408289922725080238_a_nat @ Fs_psi @ ( nall_tuples_rec_a @ AD @ ( finite_card_nat @ ( vimage3040984495076556338_a_nat @ sum_Inr_nat_a @ ( set_Sum_sum_a_nat2 @ Ys ) ) ) @ ( size_size_list_nat @ Ns_psi2 ) ) )
=> ( ( ad_agr_list_a_nat @ AD
@ ( map_na823391071729141993_a_nat @ Sigma_ys
@ ( linord738340561235409698at_nat
@ ^ [X: nat] : X
@ ( append_nat @ Ns_psi @ Ns_psi2 ) ) )
@ ( map_na823391071729141993_a_nat @ Sigma_xs
@ ( linord738340561235409698at_nat
@ ^ [X: nat] : X
@ ( append_nat @ Ns_phi2 @ Ns_phi ) ) ) )
=> ( ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat @ ( merge_Sum_sum_a_nat @ ( zip_na2013496608136855606_a_nat @ Ns_phi2 @ Xs ) @ Cys ) )
= ( map_na823391071729141993_a_nat @ Sigma_xs
@ ( linord738340561235409698at_nat
@ ^ [X: nat] : X
@ ( append_nat @ Ns_phi2 @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Cys ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% eval_conj_set_aux(2)
thf(fact_1136_eval__conj__set__aux_I1_J,axiom,
! [Ns_phi: list_nat,Ns_phi2: list_nat,Ns_psi: list_nat,Ns_psi2: list_nat,X_phi: set_li6526943997496501093_a_nat,AD: set_a,R_phi: set_na3699693778330250182_a_nat,X_psi: set_li6526943997496501093_a_nat,R_psi: set_na3699693778330250182_a_nat,Cxs: list_P5056861408695629236_a_nat,Xs: list_Sum_sum_a_nat,Nxs: list_nat,Cys: list_P5056861408695629236_a_nat,Ys: list_Sum_sum_a_nat,Nys: list_nat,Sigma_xs: nat > sum_sum_a_nat,Fs_phi: list_Sum_sum_a_nat,Sigma_ys: nat > sum_sum_a_nat,Fs_psi: list_Sum_sum_a_nat] :
( ( Ns_phi
= ( filter_nat
@ ^ [N2: nat] :
~ ( member_nat @ N2 @ ( set_nat2 @ Ns_phi2 ) )
@ Ns_psi ) )
=> ( ( Ns_psi2
= ( filter_nat
@ ^ [N2: nat] :
~ ( member_nat @ N2 @ ( set_nat2 @ Ns_psi ) )
@ Ns_phi2 ) )
=> ( ( X_phi
= ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ AD ) @ ( proj_v3643391342904276326_a_nat @ R_phi @ Ns_phi2 ) ) )
=> ( ( X_psi
= ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ AD ) @ ( proj_v3643391342904276326_a_nat @ R_psi @ Ns_psi ) ) )
=> ( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ns_phi2 )
& ( distinct_nat @ Ns_phi2 ) )
=> ( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ns_psi )
& ( distinct_nat @ Ns_psi ) )
=> ( ( Cxs
= ( filter1580576373184442087_a_nat
@ ( produc8810490045567577941_nat_o
@ ^ [N2: nat,X: sum_sum_a_nat] :
( ~ ( member_nat @ N2 @ ( set_nat2 @ Ns_psi ) )
& ( sum_isl_a_nat @ X ) ) )
@ ( zip_na2013496608136855606_a_nat @ Ns_phi2 @ Xs ) ) )
=> ( ( Nxs
= ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat
@ ( filter1580576373184442087_a_nat
@ ( produc8810490045567577941_nat_o
@ ^ [N2: nat,X: sum_sum_a_nat] :
( ~ ( member_nat @ N2 @ ( set_nat2 @ Ns_psi ) )
& ~ ( sum_isl_a_nat @ X ) ) )
@ ( zip_na2013496608136855606_a_nat @ Ns_phi2 @ Xs ) ) ) )
=> ( ( Cys
= ( filter1580576373184442087_a_nat
@ ( produc8810490045567577941_nat_o
@ ^ [N2: nat,Y: sum_sum_a_nat] :
( ~ ( member_nat @ N2 @ ( set_nat2 @ Ns_phi2 ) )
& ( sum_isl_a_nat @ Y ) ) )
@ ( zip_na2013496608136855606_a_nat @ Ns_psi @ Ys ) ) )
=> ( ( Nys
= ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat
@ ( filter1580576373184442087_a_nat
@ ( produc8810490045567577941_nat_o
@ ^ [N2: nat,Y: sum_sum_a_nat] :
( ~ ( member_nat @ N2 @ ( set_nat2 @ Ns_phi2 ) )
& ~ ( sum_isl_a_nat @ Y ) ) )
@ ( zip_na2013496608136855606_a_nat @ Ns_psi @ Ys ) ) ) )
=> ( ( member408289922725080238_a_nat @ Xs @ X_phi )
=> ( ( member408289922725080238_a_nat @ Ys @ X_psi )
=> ( ( Xs
= ( map_na823391071729141993_a_nat @ Sigma_xs @ Ns_phi2 ) )
=> ( ( Fs_phi
= ( map_na823391071729141993_a_nat @ Sigma_xs @ Ns_phi ) )
=> ( ( Ys
= ( map_na823391071729141993_a_nat @ Sigma_ys @ Ns_psi ) )
=> ( ( Fs_psi
= ( map_na823391071729141993_a_nat @ Sigma_ys @ Ns_psi2 ) )
=> ( ( member408289922725080238_a_nat @ Fs_phi @ ( nall_tuples_rec_a @ AD @ ( finite_card_nat @ ( vimage3040984495076556338_a_nat @ sum_Inr_nat_a @ ( set_Sum_sum_a_nat2 @ Xs ) ) ) @ ( size_size_list_nat @ Ns_phi ) ) )
=> ( ( member408289922725080238_a_nat @ Fs_psi @ ( nall_tuples_rec_a @ AD @ ( finite_card_nat @ ( vimage3040984495076556338_a_nat @ sum_Inr_nat_a @ ( set_Sum_sum_a_nat2 @ Ys ) ) ) @ ( size_size_list_nat @ Ns_psi2 ) ) )
=> ( ( ad_agr_list_a_nat @ AD
@ ( map_na823391071729141993_a_nat @ Sigma_ys
@ ( linord738340561235409698at_nat
@ ^ [X: nat] : X
@ ( append_nat @ Ns_psi @ Ns_psi2 ) ) )
@ ( map_na823391071729141993_a_nat @ Sigma_xs
@ ( linord738340561235409698at_nat
@ ^ [X: nat] : X
@ ( append_nat @ Ns_phi2 @ Ns_phi ) ) ) )
=> ( ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat @ ( merge_Sum_sum_a_nat @ ( zip_na2013496608136855606_a_nat @ Ns_phi2 @ Xs ) @ ( zip_na2013496608136855606_a_nat @ Ns_phi @ Fs_phi ) ) )
= ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat
@ ( merge_Sum_sum_a_nat
@ ( zip_na2013496608136855606_a_nat
@ ( linord738340561235409698at_nat
@ ^ [X: nat] : X
@ ( append_nat @ Ns_phi2 @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Cys ) ) )
@ ( map_na823391071729141993_a_nat @ Sigma_xs
@ ( linord738340561235409698at_nat
@ ^ [X: nat] : X
@ ( append_nat @ Ns_phi2 @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Cys ) ) ) ) )
@ ( zip_na2013496608136855606_a_nat @ Nys @ ( map_na823391071729141993_a_nat @ Sigma_xs @ Nys ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% eval_conj_set_aux(1)
thf(fact_1137_eval__conj__tuple__empty,axiom,
! [Z4: set_a,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Nsx: list_nat,Nsy: list_nat,Ns: list_nat] :
( ( fo_nmlzd_a @ Z4 @ Xs )
=> ( ( fo_nmlzd_a @ Z4 @ Ys )
=> ( ( ( size_size_list_nat @ Nsx )
= ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( ( ( size_size_list_nat @ Nsy )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Nsx )
& ( distinct_nat @ Nsx ) )
=> ( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Nsy )
& ( distinct_nat @ Nsy ) )
=> ( ( Ns
= ( filter_nat
@ ^ [N2: nat] : ( member_nat @ N2 @ ( set_nat2 @ Nsy ) )
@ Nsx ) )
=> ( ( ( fo_nmlz_a @ Z4 @ ( proj_tuple_a @ Ns @ ( zip_na2013496608136855606_a_nat @ Nsx @ Xs ) ) )
!= ( fo_nmlz_a @ Z4 @ ( proj_tuple_a @ Ns @ ( zip_na2013496608136855606_a_nat @ Nsy @ Ys ) ) ) )
=> ( ( eval_conj_tuple_a @ Z4 @ Nsx @ Nsy @ Xs @ Ys )
= bot_bo1033123847703346641_a_nat ) ) ) ) ) ) ) ) ) ).
% eval_conj_tuple_empty
thf(fact_1138_distinct__map__fst__filter,axiom,
! [Xs: list_P5056861408695629236_a_nat,P: produc5454855657571582244_a_nat > $o] :
( ( distinct_nat @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Xs ) )
=> ( distinct_nat @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ ( filter1580576373184442087_a_nat @ P @ Xs ) ) ) ) ).
% distinct_map_fst_filter
thf(fact_1139_map__fst__merge,axiom,
! [Nxs: list_P5056861408695629236_a_nat,Mys: list_P5056861408695629236_a_nat] :
( ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Nxs ) )
& ( distinct_nat @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Nxs ) ) )
=> ( ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Mys ) )
& ( distinct_nat @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Mys ) ) )
=> ( ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ ( merge_Sum_sum_a_nat @ Nxs @ Mys ) )
= ( linord738340561235409698at_nat
@ ^ [X: nat] : X
@ ( append_nat @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Nxs ) @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Mys ) ) ) ) ) ) ).
% map_fst_merge
thf(fact_1140_merge__length,axiom,
! [Nxs: list_P5056861408695629236_a_nat,Mys: list_P5056861408695629236_a_nat] :
( ( size_s4076174644546656840_a_nat @ ( merge_Sum_sum_a_nat @ Nxs @ Mys ) )
= ( size_size_list_nat @ ( append_nat @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Nxs ) @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Mys ) ) ) ) ).
% merge_length
thf(fact_1141_ext__tuple__eq,axiom,
! [Fv_sub: list_nat,As: list_Sum_sum_a_nat,AD: set_a,Fv_sub_comp: list_nat] :
( ( ( size_size_list_nat @ Fv_sub )
= ( size_s5283204784079214577_a_nat @ As ) )
=> ( ( ext_tuple_a @ AD @ Fv_sub @ Fv_sub_comp @ As )
= ( image_5081948215111134021_a_nat
@ ^ [Fs: list_Sum_sum_a_nat] : ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat @ ( merge_Sum_sum_a_nat @ ( zip_na2013496608136855606_a_nat @ Fv_sub @ As ) @ ( zip_na2013496608136855606_a_nat @ Fv_sub_comp @ Fs ) ) )
@ ( nall_tuples_rec_a @ AD @ ( finite_card_nat @ ( vimage3040984495076556338_a_nat @ sum_Inr_nat_a @ ( set_Sum_sum_a_nat2 @ As ) ) ) @ ( size_size_list_nat @ Fv_sub_comp ) ) ) ) ) ).
% ext_tuple_eq
thf(fact_1142_eval__conj__tuple__empty2,axiom,
! [Z4: set_a,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Nsx: list_nat,Nsy: list_nat,Ns: list_nat] :
( ( fo_nmlzd_a @ Z4 @ Xs )
=> ( ( fo_nmlzd_a @ Z4 @ Ys )
=> ( ( ( size_size_list_nat @ Nsx )
= ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( ( ( size_size_list_nat @ Nsy )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Nsx )
& ( distinct_nat @ Nsx ) )
=> ( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Nsy )
& ( distinct_nat @ Nsy ) )
=> ( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ns )
& ( distinct_nat @ Ns ) )
=> ( ( ord_less_eq_set_nat @ ( set_nat2 @ Ns ) @ ( inf_inf_set_nat @ ( set_nat2 @ Nsx ) @ ( set_nat2 @ Nsy ) ) )
=> ( ( ( ( fo_nmlz_a @ Z4 @ ( proj_tuple_a @ Ns @ ( zip_na2013496608136855606_a_nat @ Nsx @ Xs ) ) )
!= ( fo_nmlz_a @ Z4 @ ( proj_tuple_a @ Ns @ ( zip_na2013496608136855606_a_nat @ Nsy @ Ys ) ) ) )
| ( ( ( proj_tuple_a @ Ns @ ( zip_na2013496608136855606_a_nat @ Nsx @ Xs ) )
!= ( proj_tuple_a @ Ns @ ( zip_na2013496608136855606_a_nat @ Nsy @ Ys ) ) )
& ! [X3: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X3 @ ( set_Sum_sum_a_nat2 @ ( proj_tuple_a @ Ns @ ( zip_na2013496608136855606_a_nat @ Nsx @ Xs ) ) ) )
=> ( sum_isl_a_nat @ X3 ) )
& ! [X3: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X3 @ ( set_Sum_sum_a_nat2 @ ( proj_tuple_a @ Ns @ ( zip_na2013496608136855606_a_nat @ Nsy @ Ys ) ) ) )
=> ( sum_isl_a_nat @ X3 ) ) ) )
=> ( ( eval_conj_tuple_a @ Z4 @ Nsx @ Nsy @ Xs @ Ys )
= bot_bo1033123847703346641_a_nat ) ) ) ) ) ) ) ) ) ) ).
% eval_conj_tuple_empty2
thf(fact_1143_folding__Map__graph_Odistinct__if__distinct__map,axiom,
! [Xs: list_P5056861408695629236_a_nat] :
( ( distinct_nat @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Xs ) )
=> ( distin5548549501492119275_a_nat @ Xs ) ) ).
% folding_Map_graph.distinct_if_distinct_map
thf(fact_1144_merge__map_H,axiom,
! [Nxs: list_P5056861408695629236_a_nat,Mys: list_P5056861408695629236_a_nat,Sigma: nat > sum_sum_a_nat] :
( ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Nxs ) )
& ( distinct_nat @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Nxs ) ) )
=> ( ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Mys ) )
& ( distinct_nat @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Mys ) ) )
=> ( ( ( inf_inf_set_nat @ ( image_1677410150705809403at_nat @ produc1510075366785312874_a_nat @ ( set_Pr2196415342531872975_a_nat @ Nxs ) ) @ ( image_1677410150705809403at_nat @ produc1510075366785312874_a_nat @ ( set_Pr2196415342531872975_a_nat @ Mys ) ) )
= bot_bot_set_nat )
=> ( ( ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat @ Nxs )
= ( map_na823391071729141993_a_nat @ Sigma @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Nxs ) ) )
=> ( ( ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat @ Mys )
= ( map_na823391071729141993_a_nat @ Sigma @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Mys ) ) )
=> ( ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat @ ( merge_Sum_sum_a_nat @ Nxs @ Mys ) )
= ( map_na823391071729141993_a_nat @ Sigma
@ ( linord738340561235409698at_nat
@ ^ [X: nat] : X
@ ( append_nat @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Nxs ) @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Mys ) ) ) ) ) ) ) ) ) ) ).
% merge_map'
thf(fact_1145_merge__map,axiom,
! [Ns: list_nat,Ms: list_nat,Sigma: nat > sum_sum_a_nat] :
( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ns )
& ( distinct_nat @ Ns ) )
=> ( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ms )
& ( distinct_nat @ Ms ) )
=> ( ( ( inf_inf_set_nat @ ( set_nat2 @ Ns ) @ ( set_nat2 @ Ms ) )
= bot_bot_set_nat )
=> ( ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat @ ( merge_Sum_sum_a_nat @ ( zip_na2013496608136855606_a_nat @ Ns @ ( map_na823391071729141993_a_nat @ Sigma @ Ns ) ) @ ( zip_na2013496608136855606_a_nat @ Ms @ ( map_na823391071729141993_a_nat @ Sigma @ Ms ) ) ) )
= ( map_na823391071729141993_a_nat @ Sigma
@ ( linord738340561235409698at_nat
@ ^ [X: nat] : X
@ ( append_nat @ Ns @ Ms ) ) ) ) ) ) ) ).
% merge_map
thf(fact_1146_Int__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
= ( ( member_nat @ C @ A )
& ( member_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_1147_Int__iff,axiom,
! [C: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ ( inf_in3249246906714053971_a_nat @ A @ B ) )
= ( ( member408289922725080238_a_nat @ C @ A )
& ( member408289922725080238_a_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_1148_Int__iff,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
= ( ( member_a @ C @ A )
& ( member_a @ C @ B ) ) ) ).
% Int_iff
thf(fact_1149_Int__iff,axiom,
! [C: sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C @ ( inf_in7084830621192376909_a_nat @ A @ B ) )
= ( ( member_Sum_sum_a_nat @ C @ A )
& ( member_Sum_sum_a_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_1150_IntI,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ A )
=> ( ( member_nat @ C @ B )
=> ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_1151_IntI,axiom,
! [C: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ A )
=> ( ( member408289922725080238_a_nat @ C @ B )
=> ( member408289922725080238_a_nat @ C @ ( inf_in3249246906714053971_a_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_1152_IntI,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ A )
=> ( ( member_a @ C @ B )
=> ( member_a @ C @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% IntI
thf(fact_1153_IntI,axiom,
! [C: sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C @ A )
=> ( ( member_Sum_sum_a_nat @ C @ B )
=> ( member_Sum_sum_a_nat @ C @ ( inf_in7084830621192376909_a_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_1154_inf_Obounded__iff,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) )
= ( ( ord_less_eq_set_a @ A2 @ B2 )
& ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_1155_inf_Obounded__iff,axiom,
! [A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A2 @ ( inf_in7084830621192376909_a_nat @ B2 @ C ) )
= ( ( ord_le1325389633284124927_a_nat @ A2 @ B2 )
& ( ord_le1325389633284124927_a_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_1156_inf_Obounded__iff,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C ) )
= ( ( ord_less_eq_nat @ A2 @ B2 )
& ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_1157_le__inf__iff,axiom,
! [X2: set_a,Y2: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X2 @ ( inf_inf_set_a @ Y2 @ Z ) )
= ( ( ord_less_eq_set_a @ X2 @ Y2 )
& ( ord_less_eq_set_a @ X2 @ Z ) ) ) ).
% le_inf_iff
thf(fact_1158_le__inf__iff,axiom,
! [X2: set_Sum_sum_a_nat,Y2: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X2 @ ( inf_in7084830621192376909_a_nat @ Y2 @ Z ) )
= ( ( ord_le1325389633284124927_a_nat @ X2 @ Y2 )
& ( ord_le1325389633284124927_a_nat @ X2 @ Z ) ) ) ).
% le_inf_iff
thf(fact_1159_le__inf__iff,axiom,
! [X2: nat,Y2: nat,Z: nat] :
( ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ Y2 @ Z ) )
= ( ( ord_less_eq_nat @ X2 @ Y2 )
& ( ord_less_eq_nat @ X2 @ Z ) ) ) ).
% le_inf_iff
thf(fact_1160_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_a] :
( ( inf_inf_set_a @ X2 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_1161_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X2 )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_1162_inf__bot__right,axiom,
! [X2: set_a] :
( ( inf_inf_set_a @ X2 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% inf_bot_right
thf(fact_1163_inf__bot__left,axiom,
! [X2: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X2 )
= bot_bot_set_a ) ).
% inf_bot_left
thf(fact_1164_sup__inf__absorb,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( inf_inf_set_nat @ X2 @ Y2 ) )
= X2 ) ).
% sup_inf_absorb
thf(fact_1165_sup__inf__absorb,axiom,
! [X2: set_Sum_sum_a_nat,Y2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ X2 @ ( inf_in7084830621192376909_a_nat @ X2 @ Y2 ) )
= X2 ) ).
% sup_inf_absorb
thf(fact_1166_sup__inf__absorb,axiom,
! [X2: set_a,Y2: set_a] :
( ( sup_sup_set_a @ X2 @ ( inf_inf_set_a @ X2 @ Y2 ) )
= X2 ) ).
% sup_inf_absorb
thf(fact_1167_sup__inf__absorb,axiom,
! [X2: set_li6526943997496501093_a_nat,Y2: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ X2 @ ( inf_in3249246906714053971_a_nat @ X2 @ Y2 ) )
= X2 ) ).
% sup_inf_absorb
thf(fact_1168_inf__sup__absorb,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( inf_inf_set_nat @ X2 @ ( sup_sup_set_nat @ X2 @ Y2 ) )
= X2 ) ).
% inf_sup_absorb
thf(fact_1169_inf__sup__absorb,axiom,
! [X2: set_Sum_sum_a_nat,Y2: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ X2 @ ( sup_su6804446743777130803_a_nat @ X2 @ Y2 ) )
= X2 ) ).
% inf_sup_absorb
thf(fact_1170_inf__sup__absorb,axiom,
! [X2: set_a,Y2: set_a] :
( ( inf_inf_set_a @ X2 @ ( sup_sup_set_a @ X2 @ Y2 ) )
= X2 ) ).
% inf_sup_absorb
thf(fact_1171_inf__sup__absorb,axiom,
! [X2: set_li6526943997496501093_a_nat,Y2: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ X2 @ ( sup_su4083067149120280889_a_nat @ X2 @ Y2 ) )
= X2 ) ).
% inf_sup_absorb
thf(fact_1172_Int__subset__iff,axiom,
! [C2: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A @ B ) )
= ( ( ord_less_eq_set_a @ C2 @ A )
& ( ord_less_eq_set_a @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_1173_Int__subset__iff,axiom,
! [C2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ C2 @ ( inf_in7084830621192376909_a_nat @ A @ B ) )
= ( ( ord_le1325389633284124927_a_nat @ C2 @ A )
& ( ord_le1325389633284124927_a_nat @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_1174_Un__Int__eq_I1_J,axiom,
! [S: set_nat,T3: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S @ T3 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_1175_Un__Int__eq_I1_J,axiom,
! [S: set_Sum_sum_a_nat,T3: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ ( sup_su6804446743777130803_a_nat @ S @ T3 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_1176_Un__Int__eq_I1_J,axiom,
! [S: set_a,T3: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T3 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_1177_Un__Int__eq_I1_J,axiom,
! [S: set_li6526943997496501093_a_nat,T3: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ ( sup_su4083067149120280889_a_nat @ S @ T3 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_1178_Un__Int__eq_I2_J,axiom,
! [S: set_nat,T3: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S @ T3 ) @ T3 )
= T3 ) ).
% Un_Int_eq(2)
thf(fact_1179_Un__Int__eq_I2_J,axiom,
! [S: set_Sum_sum_a_nat,T3: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ ( sup_su6804446743777130803_a_nat @ S @ T3 ) @ T3 )
= T3 ) ).
% Un_Int_eq(2)
thf(fact_1180_Un__Int__eq_I2_J,axiom,
! [S: set_a,T3: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T3 ) @ T3 )
= T3 ) ).
% Un_Int_eq(2)
thf(fact_1181_Un__Int__eq_I2_J,axiom,
! [S: set_li6526943997496501093_a_nat,T3: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ ( sup_su4083067149120280889_a_nat @ S @ T3 ) @ T3 )
= T3 ) ).
% Un_Int_eq(2)
thf(fact_1182_Un__Int__eq_I3_J,axiom,
! [S: set_nat,T3: set_nat] :
( ( inf_inf_set_nat @ S @ ( sup_sup_set_nat @ S @ T3 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_1183_Un__Int__eq_I3_J,axiom,
! [S: set_Sum_sum_a_nat,T3: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ S @ ( sup_su6804446743777130803_a_nat @ S @ T3 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_1184_Un__Int__eq_I3_J,axiom,
! [S: set_a,T3: set_a] :
( ( inf_inf_set_a @ S @ ( sup_sup_set_a @ S @ T3 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_1185_Un__Int__eq_I3_J,axiom,
! [S: set_li6526943997496501093_a_nat,T3: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ S @ ( sup_su4083067149120280889_a_nat @ S @ T3 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_1186_Un__Int__eq_I4_J,axiom,
! [T3: set_nat,S: set_nat] :
( ( inf_inf_set_nat @ T3 @ ( sup_sup_set_nat @ S @ T3 ) )
= T3 ) ).
% Un_Int_eq(4)
thf(fact_1187_Un__Int__eq_I4_J,axiom,
! [T3: set_Sum_sum_a_nat,S: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ T3 @ ( sup_su6804446743777130803_a_nat @ S @ T3 ) )
= T3 ) ).
% Un_Int_eq(4)
thf(fact_1188_Un__Int__eq_I4_J,axiom,
! [T3: set_a,S: set_a] :
( ( inf_inf_set_a @ T3 @ ( sup_sup_set_a @ S @ T3 ) )
= T3 ) ).
% Un_Int_eq(4)
thf(fact_1189_Un__Int__eq_I4_J,axiom,
! [T3: set_li6526943997496501093_a_nat,S: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ T3 @ ( sup_su4083067149120280889_a_nat @ S @ T3 ) )
= T3 ) ).
% Un_Int_eq(4)
thf(fact_1190_Int__Un__eq_I1_J,axiom,
! [S: set_nat,T3: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S @ T3 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_1191_Int__Un__eq_I1_J,axiom,
! [S: set_Sum_sum_a_nat,T3: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ S @ T3 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_1192_Int__Un__eq_I1_J,axiom,
! [S: set_a,T3: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T3 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_1193_Int__Un__eq_I1_J,axiom,
! [S: set_li6526943997496501093_a_nat,T3: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ ( inf_in3249246906714053971_a_nat @ S @ T3 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_1194_Int__Un__eq_I2_J,axiom,
! [S: set_nat,T3: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S @ T3 ) @ T3 )
= T3 ) ).
% Int_Un_eq(2)
thf(fact_1195_Int__Un__eq_I2_J,axiom,
! [S: set_Sum_sum_a_nat,T3: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ S @ T3 ) @ T3 )
= T3 ) ).
% Int_Un_eq(2)
thf(fact_1196_Int__Un__eq_I2_J,axiom,
! [S: set_a,T3: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T3 ) @ T3 )
= T3 ) ).
% Int_Un_eq(2)
thf(fact_1197_Int__Un__eq_I2_J,axiom,
! [S: set_li6526943997496501093_a_nat,T3: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ ( inf_in3249246906714053971_a_nat @ S @ T3 ) @ T3 )
= T3 ) ).
% Int_Un_eq(2)
thf(fact_1198_Int__Un__eq_I3_J,axiom,
! [S: set_nat,T3: set_nat] :
( ( sup_sup_set_nat @ S @ ( inf_inf_set_nat @ S @ T3 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_1199_Int__Un__eq_I3_J,axiom,
! [S: set_Sum_sum_a_nat,T3: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ S @ ( inf_in7084830621192376909_a_nat @ S @ T3 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_1200_Int__Un__eq_I3_J,axiom,
! [S: set_a,T3: set_a] :
( ( sup_sup_set_a @ S @ ( inf_inf_set_a @ S @ T3 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_1201_Int__Un__eq_I3_J,axiom,
! [S: set_li6526943997496501093_a_nat,T3: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ S @ ( inf_in3249246906714053971_a_nat @ S @ T3 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_1202_Int__Un__eq_I4_J,axiom,
! [T3: set_nat,S: set_nat] :
( ( sup_sup_set_nat @ T3 @ ( inf_inf_set_nat @ S @ T3 ) )
= T3 ) ).
% Int_Un_eq(4)
thf(fact_1203_Int__Un__eq_I4_J,axiom,
! [T3: set_Sum_sum_a_nat,S: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ T3 @ ( inf_in7084830621192376909_a_nat @ S @ T3 ) )
= T3 ) ).
% Int_Un_eq(4)
thf(fact_1204_Int__Un__eq_I4_J,axiom,
! [T3: set_a,S: set_a] :
( ( sup_sup_set_a @ T3 @ ( inf_inf_set_a @ S @ T3 ) )
= T3 ) ).
% Int_Un_eq(4)
thf(fact_1205_Int__Un__eq_I4_J,axiom,
! [T3: set_li6526943997496501093_a_nat,S: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ T3 @ ( inf_in3249246906714053971_a_nat @ S @ T3 ) )
= T3 ) ).
% Int_Un_eq(4)
thf(fact_1206_vimage__Int,axiom,
! [F: a > sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( vimage4607795094749595324_a_nat @ F @ ( inf_in7084830621192376909_a_nat @ A @ B ) )
= ( inf_inf_set_a @ ( vimage4607795094749595324_a_nat @ F @ A ) @ ( vimage4607795094749595324_a_nat @ F @ B ) ) ) ).
% vimage_Int
thf(fact_1207_vimage__Int,axiom,
! [F: nat > sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( vimage3040984495076556338_a_nat @ F @ ( inf_in7084830621192376909_a_nat @ A @ B ) )
= ( inf_inf_set_nat @ ( vimage3040984495076556338_a_nat @ F @ A ) @ ( vimage3040984495076556338_a_nat @ F @ B ) ) ) ).
% vimage_Int
thf(fact_1208_if__image__distrib,axiom,
! [P: nat > $o,F: nat > nat,G: nat > nat,S: set_nat] :
( ( image_nat_nat
@ ^ [X: nat] : ( if_nat @ ( P @ X ) @ ( F @ X ) @ ( G @ X ) )
@ S )
= ( sup_sup_set_nat @ ( image_nat_nat @ F @ ( inf_inf_set_nat @ S @ ( collect_nat @ P ) ) )
@ ( image_nat_nat @ G
@ ( inf_inf_set_nat @ S
@ ( collect_nat
@ ^ [X: nat] :
~ ( P @ X ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_1209_if__image__distrib,axiom,
! [P: nat > $o,F: nat > sum_sum_a_nat,G: nat > sum_sum_a_nat,S: set_nat] :
( ( image_7293268710728258664_a_nat
@ ^ [X: nat] : ( if_Sum_sum_a_nat @ ( P @ X ) @ ( F @ X ) @ ( G @ X ) )
@ S )
= ( sup_su6804446743777130803_a_nat @ ( image_7293268710728258664_a_nat @ F @ ( inf_inf_set_nat @ S @ ( collect_nat @ P ) ) )
@ ( image_7293268710728258664_a_nat @ G
@ ( inf_inf_set_nat @ S
@ ( collect_nat
@ ^ [X: nat] :
~ ( P @ X ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_1210_if__image__distrib,axiom,
! [P: nat > $o,F: nat > a,G: nat > a,S: set_nat] :
( ( image_nat_a
@ ^ [X: nat] : ( if_a @ ( P @ X ) @ ( F @ X ) @ ( G @ X ) )
@ S )
= ( sup_sup_set_a @ ( image_nat_a @ F @ ( inf_inf_set_nat @ S @ ( collect_nat @ P ) ) )
@ ( image_nat_a @ G
@ ( inf_inf_set_nat @ S
@ ( collect_nat
@ ^ [X: nat] :
~ ( P @ X ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_1211_if__image__distrib,axiom,
! [P: list_Sum_sum_a_nat > $o,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,G: list_Sum_sum_a_nat > list_Sum_sum_a_nat,S: set_li6526943997496501093_a_nat] :
( ( image_5081948215111134021_a_nat
@ ^ [X: list_Sum_sum_a_nat] : ( if_lis4685338526944683083_a_nat @ ( P @ X ) @ ( F @ X ) @ ( G @ X ) )
@ S )
= ( sup_su4083067149120280889_a_nat @ ( image_5081948215111134021_a_nat @ F @ ( inf_in3249246906714053971_a_nat @ S @ ( collec7555443234367654128_a_nat @ P ) ) )
@ ( image_5081948215111134021_a_nat @ G
@ ( inf_in3249246906714053971_a_nat @ S
@ ( collec7555443234367654128_a_nat
@ ^ [X: list_Sum_sum_a_nat] :
~ ( P @ X ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_1212_if__image__distrib,axiom,
! [P: nat > $o,F: nat > list_Sum_sum_a_nat,G: nat > list_Sum_sum_a_nat,S: set_nat] :
( ( image_6262589752765146990_a_nat
@ ^ [X: nat] : ( if_lis4685338526944683083_a_nat @ ( P @ X ) @ ( F @ X ) @ ( G @ X ) )
@ S )
= ( sup_su4083067149120280889_a_nat @ ( image_6262589752765146990_a_nat @ F @ ( inf_inf_set_nat @ S @ ( collect_nat @ P ) ) )
@ ( image_6262589752765146990_a_nat @ G
@ ( inf_inf_set_nat @ S
@ ( collect_nat
@ ^ [X: nat] :
~ ( P @ X ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_1213_distinct__append,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( distinct_nat @ ( append_nat @ Xs @ Ys ) )
= ( ( distinct_nat @ Xs )
& ( distinct_nat @ Ys )
& ( ( inf_inf_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) )
= bot_bot_set_nat ) ) ) ).
% distinct_append
thf(fact_1214_distinct__append,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( distin2701893636801681144_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) )
= ( ( distin2701893636801681144_a_nat @ Xs )
& ( distin2701893636801681144_a_nat @ Ys )
& ( ( inf_in7084830621192376909_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) @ ( set_Sum_sum_a_nat2 @ Ys ) )
= bot_bo3438331934148233675_a_nat ) ) ) ).
% distinct_append
thf(fact_1215_distinct__append,axiom,
! [Xs: list_a,Ys: list_a] :
( ( distinct_a @ ( append_a @ Xs @ Ys ) )
= ( ( distinct_a @ Xs )
& ( distinct_a @ Ys )
& ( ( inf_inf_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys ) )
= bot_bot_set_a ) ) ) ).
% distinct_append
thf(fact_1216_Int__emptyI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ~ ( member_nat @ X3 @ B ) )
=> ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_1217_Int__emptyI,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ! [X3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ A )
=> ~ ( member408289922725080238_a_nat @ X3 @ B ) )
=> ( ( inf_in3249246906714053971_a_nat @ A @ B )
= bot_bo1033123847703346641_a_nat ) ) ).
% Int_emptyI
thf(fact_1218_Int__emptyI,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ! [X3: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X3 @ A )
=> ~ ( member_Sum_sum_a_nat @ X3 @ B ) )
=> ( ( inf_in7084830621192376909_a_nat @ A @ B )
= bot_bo3438331934148233675_a_nat ) ) ).
% Int_emptyI
thf(fact_1219_Int__emptyI,axiom,
! [A: set_a,B: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ~ ( member_a @ X3 @ B ) )
=> ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_1220_disjoint__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ~ ( member_nat @ X @ B ) ) ) ) ).
% disjoint_iff
thf(fact_1221_disjoint__iff,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ( inf_in3249246906714053971_a_nat @ A @ B )
= bot_bo1033123847703346641_a_nat )
= ( ! [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A )
=> ~ ( member408289922725080238_a_nat @ X @ B ) ) ) ) ).
% disjoint_iff
thf(fact_1222_disjoint__iff,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ( inf_in7084830621192376909_a_nat @ A @ B )
= bot_bo3438331934148233675_a_nat )
= ( ! [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ A )
=> ~ ( member_Sum_sum_a_nat @ X @ B ) ) ) ) ).
% disjoint_iff
thf(fact_1223_disjoint__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
= ( ! [X: a] :
( ( member_a @ X @ A )
=> ~ ( member_a @ X @ B ) ) ) ) ).
% disjoint_iff
thf(fact_1224_Int__empty__left,axiom,
! [B: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ B )
= bot_bot_set_a ) ).
% Int_empty_left
thf(fact_1225_Int__empty__right,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ bot_bot_set_a )
= bot_bot_set_a ) ).
% Int_empty_right
thf(fact_1226_disjoint__iff__not__equal,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
= ( ! [X: a] :
( ( member_a @ X @ A )
=> ! [Y: a] :
( ( member_a @ Y @ B )
=> ( X != Y ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1227_Int__mono,axiom,
! [A: set_a,C2: set_a,B: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_inf_set_a @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_1228_Int__mono,axiom,
! [A: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,D2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ C2 )
=> ( ( ord_le1325389633284124927_a_nat @ B @ D2 )
=> ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ A @ B ) @ ( inf_in7084830621192376909_a_nat @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_1229_Int__lower1,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_1230_Int__lower1,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_1231_Int__lower2,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_1232_Int__lower2,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_1233_Int__absorb1,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( inf_inf_set_a @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_1234_Int__absorb1,axiom,
! [B: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B @ A )
=> ( ( inf_in7084830621192376909_a_nat @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_1235_Int__absorb2,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( inf_inf_set_a @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_1236_Int__absorb2,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ B )
=> ( ( inf_in7084830621192376909_a_nat @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_1237_Int__greatest,axiom,
! [C2: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C2 @ A )
=> ( ( ord_less_eq_set_a @ C2 @ B )
=> ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_1238_Int__greatest,axiom,
! [C2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ C2 @ A )
=> ( ( ord_le1325389633284124927_a_nat @ C2 @ B )
=> ( ord_le1325389633284124927_a_nat @ C2 @ ( inf_in7084830621192376909_a_nat @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_1239_Int__Collect__mono,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,P: list_Sum_sum_a_nat > $o,Q: list_Sum_sum_a_nat > $o] :
( ( ord_le1147066620699065093_a_nat @ A @ B )
=> ( ! [X3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le1147066620699065093_a_nat @ ( inf_in3249246906714053971_a_nat @ A @ ( collec7555443234367654128_a_nat @ P ) ) @ ( inf_in3249246906714053971_a_nat @ B @ ( collec7555443234367654128_a_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1240_Int__Collect__mono,axiom,
! [A: set_nat,B: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1241_Int__Collect__mono,axiom,
! [A: set_a,B: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1242_Int__Collect__mono,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,P: sum_sum_a_nat > $o,Q: sum_sum_a_nat > $o] :
( ( ord_le1325389633284124927_a_nat @ A @ B )
=> ( ! [X3: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ A @ ( collec7073057861543223018_a_nat @ P ) ) @ ( inf_in7084830621192376909_a_nat @ B @ ( collec7073057861543223018_a_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1243_Un__Int__crazy,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( inf_inf_set_nat @ B @ C2 ) ) @ ( inf_inf_set_nat @ C2 @ A ) )
= ( inf_inf_set_nat @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sup_sup_set_nat @ B @ C2 ) ) @ ( sup_sup_set_nat @ C2 @ A ) ) ) ).
% Un_Int_crazy
thf(fact_1244_Un__Int__crazy,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ A @ B ) @ ( inf_in7084830621192376909_a_nat @ B @ C2 ) ) @ ( inf_in7084830621192376909_a_nat @ C2 @ A ) )
= ( inf_in7084830621192376909_a_nat @ ( inf_in7084830621192376909_a_nat @ ( sup_su6804446743777130803_a_nat @ A @ B ) @ ( sup_su6804446743777130803_a_nat @ B @ C2 ) ) @ ( sup_su6804446743777130803_a_nat @ C2 @ A ) ) ) ).
% Un_Int_crazy
thf(fact_1245_Un__Int__crazy,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_inf_set_a @ B @ C2 ) ) @ ( inf_inf_set_a @ C2 @ A ) )
= ( inf_inf_set_a @ ( inf_inf_set_a @ ( sup_sup_set_a @ A @ B ) @ ( sup_sup_set_a @ B @ C2 ) ) @ ( sup_sup_set_a @ C2 @ A ) ) ) ).
% Un_Int_crazy
thf(fact_1246_Un__Int__crazy,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,C2: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ ( sup_su4083067149120280889_a_nat @ ( inf_in3249246906714053971_a_nat @ A @ B ) @ ( inf_in3249246906714053971_a_nat @ B @ C2 ) ) @ ( inf_in3249246906714053971_a_nat @ C2 @ A ) )
= ( inf_in3249246906714053971_a_nat @ ( inf_in3249246906714053971_a_nat @ ( sup_su4083067149120280889_a_nat @ A @ B ) @ ( sup_su4083067149120280889_a_nat @ B @ C2 ) ) @ ( sup_su4083067149120280889_a_nat @ C2 @ A ) ) ) ).
% Un_Int_crazy
thf(fact_1247_Int__Un__distrib,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( inf_inf_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( inf_inf_set_nat @ A @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_1248_Int__Un__distrib,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ A @ ( sup_su6804446743777130803_a_nat @ B @ C2 ) )
= ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ A @ B ) @ ( inf_in7084830621192376909_a_nat @ A @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_1249_Int__Un__distrib,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( inf_inf_set_a @ A @ ( sup_sup_set_a @ B @ C2 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_inf_set_a @ A @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_1250_Int__Un__distrib,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,C2: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ A @ ( sup_su4083067149120280889_a_nat @ B @ C2 ) )
= ( sup_su4083067149120280889_a_nat @ ( inf_in3249246906714053971_a_nat @ A @ B ) @ ( inf_in3249246906714053971_a_nat @ A @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_1251_Un__Int__distrib,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ A @ ( inf_inf_set_nat @ B @ C2 ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sup_sup_set_nat @ A @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_1252_Un__Int__distrib,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A @ ( inf_in7084830621192376909_a_nat @ B @ C2 ) )
= ( inf_in7084830621192376909_a_nat @ ( sup_su6804446743777130803_a_nat @ A @ B ) @ ( sup_su6804446743777130803_a_nat @ A @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_1253_Un__Int__distrib,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( sup_sup_set_a @ A @ ( inf_inf_set_a @ B @ C2 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ A @ B ) @ ( sup_sup_set_a @ A @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_1254_Un__Int__distrib,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,C2: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ A @ ( inf_in3249246906714053971_a_nat @ B @ C2 ) )
= ( inf_in3249246906714053971_a_nat @ ( sup_su4083067149120280889_a_nat @ A @ B ) @ ( sup_su4083067149120280889_a_nat @ A @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_1255_Int__Un__distrib2,axiom,
! [B: set_nat,C2: set_nat,A: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ B @ C2 ) @ A )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ B @ A ) @ ( inf_inf_set_nat @ C2 @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_1256_Int__Un__distrib2,axiom,
! [B: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ ( sup_su6804446743777130803_a_nat @ B @ C2 ) @ A )
= ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ B @ A ) @ ( inf_in7084830621192376909_a_nat @ C2 @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_1257_Int__Un__distrib2,axiom,
! [B: set_a,C2: set_a,A: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ B @ C2 ) @ A )
= ( sup_sup_set_a @ ( inf_inf_set_a @ B @ A ) @ ( inf_inf_set_a @ C2 @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_1258_Int__Un__distrib2,axiom,
! [B: set_li6526943997496501093_a_nat,C2: set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ ( sup_su4083067149120280889_a_nat @ B @ C2 ) @ A )
= ( sup_su4083067149120280889_a_nat @ ( inf_in3249246906714053971_a_nat @ B @ A ) @ ( inf_in3249246906714053971_a_nat @ C2 @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_1259_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1260_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B2 ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1261_nat__le__linear,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
| ( ord_less_eq_nat @ N @ M2 ) ) ).
% nat_le_linear
thf(fact_1262_le__antisym,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( M2 = N ) ) ) ).
% le_antisym
thf(fact_1263_eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( M2 = N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% eq_imp_le
thf(fact_1264_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_1265_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_1266_diff__le__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_1267_le__diff__iff_H,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A2 ) @ ( minus_minus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_1268_diff__le__self,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ).
% diff_le_self
thf(fact_1269_diff__le__mono,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1270_Nat_Odiff__diff__eq,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1271_le__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1272_eq__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M2 @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M2 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1273_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 )
=> ~ ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1274_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I2: nat] : ( ord_less_nat @ I2 @ N ) ) )
= N ) ).
% card_Collect_less_nat
% Helper facts (9)
thf(help_If_2_1_If_001tf__a_T,axiom,
! [X2: a,Y2: a] :
( ( if_a @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001tf__a_T,axiom,
! [X2: a,Y2: a] :
( ( if_a @ $true @ X2 @ Y2 )
= X2 ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y2: nat] :
( ( if_nat @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y2: nat] :
( ( if_nat @ $true @ X2 @ Y2 )
= X2 ) ).
thf(help_If_2_1_If_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_T,axiom,
! [X2: sum_sum_a_nat,Y2: sum_sum_a_nat] :
( ( if_Sum_sum_a_nat @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_T,axiom,
! [X2: sum_sum_a_nat,Y2: sum_sum_a_nat] :
( ( if_Sum_sum_a_nat @ $true @ X2 @ Y2 )
= X2 ) ).
thf(help_If_3_1_If_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_T,axiom,
! [X2: list_Sum_sum_a_nat,Y2: list_Sum_sum_a_nat] :
( ( if_lis4685338526944683083_a_nat @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_T,axiom,
! [X2: list_Sum_sum_a_nat,Y2: list_Sum_sum_a_nat] :
( ( if_lis4685338526944683083_a_nat @ $true @ X2 @ Y2 )
= X2 ) ).
% Conjectures (3)
thf(conj_0,hypothesis,
member_nat @ n @ ( sup_sup_set_nat @ ( set_nat2 @ ns_phi2 ) @ ( set_nat2 @ ns_psi2 ) ) ).
thf(conj_1,hypothesis,
( ( sum_isl_a_nat @ ( sigma_xs @ n ) )
| ( sum_isl_a_nat @ ( sigma_ys @ n ) ) ) ).
thf(conj_2,conjecture,
( ( sigma_xs @ n )
= ( sigma_ys @ n ) ) ).
%------------------------------------------------------------------------------