TPTP Problem File: SLH0848^1.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 : Query_Optimization/0012_IKKBZ/prob_01606_074410__15518422_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1657 ( 707 unt; 354 typ; 0 def)
% Number of atoms : 3723 (1532 equ; 0 cnn)
% Maximal formula atoms : 33 ( 2 avg)
% Number of connectives : 13885 ( 477 ~; 102 |; 264 &;11711 @)
% ( 0 <=>;1331 =>; 0 <=; 0 <~>)
% Maximal formula depth : 37 ( 7 avg)
% Number of types : 60 ( 59 usr)
% Number of type conns : 966 ( 966 >; 0 *; 0 +; 0 <<)
% Number of symbols : 298 ( 295 usr; 25 con; 0-5 aty)
% Number of variables : 4315 ( 772 ^;3506 !; 37 ?;4315 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:07:12.612
%------------------------------------------------------------------------------
% Could-be-implicit typings (59)
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dtree_list_a_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_J,type,
set_dtree_list_a_b: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__b_J,type,
produc2579390649552321826st_a_b: $tType ).
thf(ty_n_t__Dtree__Odtree_Itf__b_Mt__List__Olist_Itf__a_J_J,type,
dtree_b_list_a: $tType ).
thf(ty_n_t__Dtree__Odtree_Itf__a_Mt__List__Olist_Itf__a_J_J,type,
dtree_a_list_a: $tType ).
thf(ty_n_t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J,type,
dtree_list_a_b: $tType ).
thf(ty_n_t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__a_J,type,
dtree_list_a_a: $tType ).
thf(ty_n_t__FSet__Ofset_It__List__Olist_Itf__a_J_J,type,
fset_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
set_list_a: $tType ).
thf(ty_n_t__Dtree__Odtree_Itf__b_Mtf__b_J,type,
dtree_b_b: $tType ).
thf(ty_n_t__Dtree__Odtree_Itf__b_Mtf__a_J,type,
dtree_b_a: $tType ).
thf(ty_n_t__Dtree__Odtree_Itf__a_Mtf__b_J,type,
dtree_a_b: $tType ).
thf(ty_n_t__Dtree__Odtree_Itf__a_Mtf__a_J,type,
dtree_a_a: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_t__FSet__Ofset_Itf__b_J,type,
fset_b: $tType ).
thf(ty_n_t__FSet__Ofset_Itf__a_J,type,
fset_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__b_J,type,
set_b: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_tf__b,type,
b: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (295)
thf(sy_c_Dtree_Odhead_001t__List__Olist_Itf__a_J_001tf__b,type,
dhead_list_a_b: dtree_list_a_b > ( b > list_a ) > b > list_a ).
thf(sy_c_Dtree_Odtree_ONode_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
node_list_a_list_a: list_a > fset_P6656299774166858224list_a > dtree_list_a_list_a ).
thf(sy_c_Dtree_Odtree_ONode_001t__List__Olist_Itf__a_J_001tf__a,type,
node_list_a_a: list_a > fset_P8654898996138850416_a_a_a > dtree_list_a_a ).
thf(sy_c_Dtree_Odtree_ONode_001t__List__Olist_Itf__a_J_001tf__b,type,
node_list_a_b: list_a > fset_P2153231429829016240_a_b_b > dtree_list_a_b ).
thf(sy_c_Dtree_Odtree_ONode_001tf__a_001t__List__Olist_Itf__a_J,type,
node_a_list_a: a > fset_P8006199460403381738list_a > dtree_a_list_a ).
thf(sy_c_Dtree_Odtree_ONode_001tf__a_001tf__a,type,
node_a_a: a > fset_P2559403164575059562_a_a_a > dtree_a_a ).
thf(sy_c_Dtree_Odtree_ONode_001tf__a_001tf__b,type,
node_a_b: a > fset_P5281107635120001194_a_b_b > dtree_a_b ).
thf(sy_c_Dtree_Odtree_ONode_001tf__b_001t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J,type,
node_b9108412673219173045st_a_b: b > fset_P729086328404512291st_a_b > dtree_3602950866360573648st_a_b ).
thf(sy_c_Dtree_Odtree_ONode_001tf__b_001t__List__Olist_Itf__a_J,type,
node_b_list_a: b > fset_P3762770571697108971list_a > dtree_b_list_a ).
thf(sy_c_Dtree_Odtree_ONode_001tf__b_001tf__a,type,
node_b_a: b > fset_P9004070893569812843_b_a_a > dtree_b_a ).
thf(sy_c_Dtree_Odtree_ONode_001tf__b_001tf__b,type,
node_b_b: b > fset_P2502403327259978667_b_b_b > dtree_b_b ).
thf(sy_c_Dtree_Odtree_Ocase__dtree_001t__List__Olist_Itf__a_J_001tf__b_001t__FSet__Ofset_It__Product____Type__Oprod_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_Mtf__b_J_J,type,
case_d8112115101481596194_a_b_b: ( list_a > fset_P2153231429829016240_a_b_b > fset_P2153231429829016240_a_b_b ) > dtree_list_a_b > fset_P2153231429829016240_a_b_b ).
thf(sy_c_Dtree_Odtree_Ocase__dtree_001t__List__Olist_Itf__a_J_001tf__b_001t__List__Olist_Itf__a_J,type,
case_d4425991804430049426list_a: ( list_a > fset_P2153231429829016240_a_b_b > list_a ) > dtree_list_a_b > list_a ).
thf(sy_c_Dtree_Odtree_Odarcs_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
darcs_list_a_list_a: dtree_list_a_list_a > set_list_a ).
thf(sy_c_Dtree_Odtree_Odarcs_001t__List__Olist_Itf__a_J_001tf__a,type,
darcs_list_a_a: dtree_list_a_a > set_a ).
thf(sy_c_Dtree_Odtree_Odarcs_001t__List__Olist_Itf__a_J_001tf__b,type,
darcs_list_a_b: dtree_list_a_b > set_b ).
thf(sy_c_Dtree_Odtree_Odarcs_001tf__a_001t__List__Olist_Itf__a_J,type,
darcs_a_list_a: dtree_a_list_a > set_list_a ).
thf(sy_c_Dtree_Odtree_Odarcs_001tf__a_001tf__a,type,
darcs_a_a: dtree_a_a > set_a ).
thf(sy_c_Dtree_Odtree_Odarcs_001tf__a_001tf__b,type,
darcs_a_b: dtree_a_b > set_b ).
thf(sy_c_Dtree_Odtree_Odarcs_001tf__b_001t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J,type,
darcs_7068930265994548426st_a_b: dtree_3602950866360573648st_a_b > set_dtree_list_a_b ).
thf(sy_c_Dtree_Odtree_Odarcs_001tf__b_001t__List__Olist_Itf__a_J,type,
darcs_b_list_a: dtree_b_list_a > set_list_a ).
thf(sy_c_Dtree_Odtree_Odarcs_001tf__b_001tf__a,type,
darcs_b_a: dtree_b_a > set_a ).
thf(sy_c_Dtree_Odtree_Odarcs_001tf__b_001tf__b,type,
darcs_b_b: dtree_b_b > set_b ).
thf(sy_c_Dtree_Odtree_Odverts_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
dverts_list_a_list_a: dtree_list_a_list_a > set_list_a ).
thf(sy_c_Dtree_Odtree_Odverts_001t__List__Olist_Itf__a_J_001tf__a,type,
dverts_list_a_a: dtree_list_a_a > set_list_a ).
thf(sy_c_Dtree_Odtree_Odverts_001t__List__Olist_Itf__a_J_001tf__b,type,
dverts_list_a_b: dtree_list_a_b > set_list_a ).
thf(sy_c_Dtree_Odtree_Odverts_001tf__a_001t__List__Olist_Itf__a_J,type,
dverts_a_list_a: dtree_a_list_a > set_a ).
thf(sy_c_Dtree_Odtree_Odverts_001tf__a_001tf__a,type,
dverts_a_a: dtree_a_a > set_a ).
thf(sy_c_Dtree_Odtree_Odverts_001tf__a_001tf__b,type,
dverts_a_b: dtree_a_b > set_a ).
thf(sy_c_Dtree_Odtree_Odverts_001tf__b_001t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J,type,
dverts8864706460159352569st_a_b: dtree_3602950866360573648st_a_b > set_b ).
thf(sy_c_Dtree_Odtree_Odverts_001tf__b_001t__List__Olist_Itf__a_J,type,
dverts_b_list_a: dtree_b_list_a > set_b ).
thf(sy_c_Dtree_Odtree_Odverts_001tf__b_001tf__a,type,
dverts_b_a: dtree_b_a > set_b ).
thf(sy_c_Dtree_Odtree_Odverts_001tf__b_001tf__b,type,
dverts_b_b: dtree_b_b > set_b ).
thf(sy_c_Dtree_Odtree_Oroot_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
root_list_a_list_a: dtree_list_a_list_a > list_a ).
thf(sy_c_Dtree_Odtree_Oroot_001t__List__Olist_Itf__a_J_001tf__a,type,
root_list_a_a: dtree_list_a_a > list_a ).
thf(sy_c_Dtree_Odtree_Oroot_001t__List__Olist_Itf__a_J_001tf__b,type,
root_list_a_b: dtree_list_a_b > list_a ).
thf(sy_c_Dtree_Odtree_Oroot_001tf__a_001t__List__Olist_Itf__a_J,type,
root_a_list_a: dtree_a_list_a > a ).
thf(sy_c_Dtree_Odtree_Oroot_001tf__a_001tf__a,type,
root_a_a: dtree_a_a > a ).
thf(sy_c_Dtree_Odtree_Oroot_001tf__a_001tf__b,type,
root_a_b: dtree_a_b > a ).
thf(sy_c_Dtree_Odtree_Oroot_001tf__b_001t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J,type,
root_b4214690873640572181st_a_b: dtree_3602950866360573648st_a_b > b ).
thf(sy_c_Dtree_Odtree_Oroot_001tf__b_001t__List__Olist_Itf__a_J,type,
root_b_list_a: dtree_b_list_a > b ).
thf(sy_c_Dtree_Odtree_Oroot_001tf__b_001tf__a,type,
root_b_a: dtree_b_a > b ).
thf(sy_c_Dtree_Odtree_Oroot_001tf__b_001tf__b,type,
root_b_b: dtree_b_b > b ).
thf(sy_c_Dtree_Odtree_Osucs_001t__List__Olist_Itf__a_J_001tf__b,type,
sucs_list_a_b: dtree_list_a_b > fset_P2153231429829016240_a_b_b ).
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produc4981227583692318813_a_b_b: ( list_a > b > set_Pr3443975907877334966_a_b_b ) > produc2579390649552321826st_a_b > set_Pr3443975907877334966_a_b_b ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__List__Olist_Itf__a_J_001tf__b_001t__Set__Oset_Itf__a_J,type,
produc5826231651882413559_set_a: ( list_a > b > set_a ) > produc2579390649552321826st_a_b > set_a ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__List__Olist_Itf__a_J_001tf__b_001t__Set__Oset_Itf__b_J,type,
produc5826231656185642360_set_b: ( list_a > b > set_b ) > produc2579390649552321826st_a_b > set_b ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_001tf__b,type,
produc5948858871325780166_a_b_b: produc6499617310964463488_a_b_b > dtree_list_a_b ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_001tf__b,type,
produc5719641485658034180_a_b_b: produc6499617310964463488_a_b_b > b ).
thf(sy_c_Set_OBall_001t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J,type,
ball_dtree_list_a_b: set_dtree_list_a_b > ( dtree_list_a_b > $o ) > $o ).
thf(sy_c_Set_OBall_001t__List__Olist_Itf__a_J,type,
ball_list_a: set_list_a > ( list_a > $o ) > $o ).
thf(sy_c_Set_OBall_001t__Product____Type__Oprod_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_Mtf__b_J,type,
ball_P5027410970799761494_a_b_b: set_Pr3443975907877334966_a_b_b > ( produc6499617310964463488_a_b_b > $o ) > $o ).
thf(sy_c_Set_OBall_001tf__a,type,
ball_a: set_a > ( a > $o ) > $o ).
thf(sy_c_Set_OBall_001tf__b,type,
ball_b: set_b > ( b > $o ) > $o ).
thf(sy_c_Set_OBex_001t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J,type,
bex_dtree_list_a_b: set_dtree_list_a_b > ( dtree_list_a_b > $o ) > $o ).
thf(sy_c_Set_OBex_001t__List__Olist_Itf__a_J,type,
bex_list_a: set_list_a > ( list_a > $o ) > $o ).
thf(sy_c_Set_OBex_001t__Product____Type__Oprod_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_Mtf__b_J,type,
bex_Pr2161834730871777312_a_b_b: set_Pr3443975907877334966_a_b_b > ( produc6499617310964463488_a_b_b > $o ) > $o ).
thf(sy_c_Set_OBex_001tf__a,type,
bex_a: set_a > ( a > $o ) > $o ).
thf(sy_c_Set_OBex_001tf__b,type,
bex_b: set_b > ( b > $o ) > $o ).
thf(sy_c_Set_OCollect_001t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J,type,
collec2944820760411501129st_a_b: ( dtree_list_a_b > $o ) > set_dtree_list_a_b ).
thf(sy_c_Set_OCollect_001t__List__Olist_Itf__a_J,type,
collect_list_a: ( list_a > $o ) > set_list_a ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_Mtf__b_J,type,
collec3241703154345475797_a_b_b: ( produc6499617310964463488_a_b_b > $o ) > set_Pr3443975907877334966_a_b_b ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_OCollect_001tf__b,type,
collect_b: ( b > $o ) > set_b ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_Mtf__b_J_001t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J,type,
image_5965465251548763643st_a_b: ( produc6499617310964463488_a_b_b > dtree_list_a_b ) > set_Pr3443975907877334966_a_b_b > set_dtree_list_a_b ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_Mtf__b_J_001t__Product____Type__Oprod_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_Mtf__b_J,type,
image_3365364158514602119_a_b_b: ( produc6499617310964463488_a_b_b > produc6499617310964463488_a_b_b ) > set_Pr3443975907877334966_a_b_b > set_Pr3443975907877334966_a_b_b ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_Mtf__b_J_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__b_J,type,
image_5973357989600724187st_a_b: ( produc6499617310964463488_a_b_b > produc2579390649552321826st_a_b ) > set_Pr3443975907877334966_a_b_b > set_Pr9033091269622839554st_a_b ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_Mtf__b_J_001tf__b,type,
image_4684437738885282872_b_b_b: ( produc6499617310964463488_a_b_b > b ) > set_Pr3443975907877334966_a_b_b > set_b ).
thf(sy_c_Set_Oimage_001tf__b_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__b_J,type,
image_4816005896404294376st_a_b: ( b > produc2579390649552321826st_a_b ) > set_b > set_Pr9033091269622839554st_a_b ).
thf(sy_c_Set_Oinsert_001t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J,type,
insert7065945238938772516st_a_b: dtree_list_a_b > set_dtree_list_a_b > set_dtree_list_a_b ).
thf(sy_c_Set_Oinsert_001t__List__Olist_Itf__a_J,type,
insert_list_a: list_a > set_list_a > set_list_a ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_Mtf__b_J,type,
insert8720088738084019632_a_b_b: produc6499617310964463488_a_b_b > set_Pr3443975907877334966_a_b_b > set_Pr3443975907877334966_a_b_b ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Set_Oinsert_001tf__b,type,
insert_b: b > set_b > set_b ).
thf(sy_c_Wellfounded_Oaccp_001t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J,type,
accp_dtree_list_a_b: ( dtree_list_a_b > dtree_list_a_b > $o ) > dtree_list_a_b > $o ).
thf(sy_c_member_001t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J,type,
member551035911493665803st_a_b: dtree_list_a_b > set_dtree_list_a_b > $o ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_Mt__List__Olist_Itf__a_J_J,type,
member535470554364689367list_a: produc7575571382841205696list_a > set_Pr6765974203441051894list_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__a_J_Mtf__a_J,type,
member5331345429024349271_a_a_a: produc7135266307266221376_a_a_a > set_Pr722271437332393334_a_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_Mtf__b_J,type,
member4695696432722591383_a_b_b: produc6499617310964463488_a_b_b > set_Pr3443975907877334966_a_b_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mt__List__Olist_Itf__a_J_J_Mt__List__Olist_Itf__a_J_J,type,
member1778015677931443665list_a: produc4349230196019447354list_a > set_Pr7843129071603105776list_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mtf__a_J_Mtf__a_J,type,
member5016570112408633425_a_a_a: produc5194124205918388666_a_a_a > set_Pr290715669063433840_a_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mtf__b_J_Mtf__b_J,type,
member4380921116106875537_a_b_b: produc4558475209616630778_a_b_b > set_Pr3012420139608375472_a_b_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Dtree__Odtree_Itf__b_Mt__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_J_Mt__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_J,type,
member5849886156663560074st_a_b: produc8101844211926167155st_a_b > set_Pr1226305976703325993st_a_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Dtree__Odtree_Itf__b_Mt__List__Olist_Itf__a_J_J_Mt__List__Olist_Itf__a_J_J,type,
member5717964264556279890list_a: produc8289178782644283579list_a > set_Pr3599700182896833009list_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Dtree__Odtree_Itf__b_Mtf__a_J_Mtf__a_J,type,
member1389416690741988306_b_a_a: produc1566970784251743547_b_a_a > set_Pr6735383398058187121_b_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Dtree__Odtree_Itf__b_Mtf__b_J_Mtf__b_J,type,
member753767694440230418_b_b_b: produc931321787949985659_b_b_b > set_Pr233715831748352945_b_b_b > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_c_member_001tf__b,type,
member_b: b > set_b > $o ).
thf(sy_v_e____,type,
e: b ).
thf(sy_v_r____,type,
r: list_a ).
thf(sy_v_rank,type,
rank: list_a > real ).
thf(sy_v_t,type,
t: dtree_list_a_b ).
thf(sy_v_ta____,type,
ta: dtree_list_a_b ).
% Relevant facts (1277)
thf(fact_0_normalize__rel_Ocong,axiom,
ranked1950734148545037055el_a_b = ranked1950734148545037055el_a_b ).
% normalize_rel.cong
thf(fact_1_normalize1_Ocases,axiom,
! [X: dtree_list_a_b] :
( ! [R: list_a,T1: dtree_list_a_b,E: b] :
( X
!= ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ~ ! [Xs: fset_P2153231429829016240_a_b_b] :
( ! [X2: produc6499617310964463488_a_b_b] :
( Xs
!= ( finser2303212525150181944_a_b_b @ X2 @ bot_bo2248824169281960260_a_b_b ) )
=> ! [R: list_a] :
( X
!= ( node_list_a_b @ R @ Xs ) ) ) ) ).
% normalize1.cases
thf(fact_2__C1_Oprems_C,axiom,
wf_darcs_list_a_b @ ( node_list_a_b @ r @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ).
% "1.prems"
thf(fact_3_ranked__dtree_Onormalize1_Ocong,axiom,
ranked8905849569120154423e1_a_b = ranked8905849569120154423e1_a_b ).
% ranked_dtree.normalize1.cong
thf(fact_4_normalize_Oelims,axiom,
! [X: dtree_list_a_b,Y: dtree_list_a_b] :
( ( ( ranked_normalize_a_b @ rank @ X )
= Y )
=> ( Y
= ( if_dtree_list_a_b
@ ( X
= ( ranked8905849569120154423e1_a_b @ rank @ X ) )
@ ( ranked8905849569120154423e1_a_b @ rank @ X )
@ ( ranked_normalize_a_b @ rank @ ( ranked8905849569120154423e1_a_b @ rank @ X ) ) ) ) ) ).
% normalize.elims
thf(fact_5_normalize_Osimps,axiom,
! [T12: dtree_list_a_b] :
( ( ranked_normalize_a_b @ rank @ T12 )
= ( if_dtree_list_a_b
@ ( T12
= ( ranked8905849569120154423e1_a_b @ rank @ T12 ) )
@ ( ranked8905849569120154423e1_a_b @ rank @ T12 )
@ ( ranked_normalize_a_b @ rank @ ( ranked8905849569120154423e1_a_b @ rank @ T12 ) ) ) ) ).
% normalize.simps
thf(fact_6__092_060open_062disjoint__darcs_A_123_124_Inormalize1_At_M_Ae_J_124_125_092_060close_062,axiom,
! [X2: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ X2 @ ( fset_P9138963618725001425_a_b_b @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ( ranked8905849569120154423e1_a_b @ rank @ ta ) @ e ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( produc336013011823628281_b_b_o
@ ^ [Y2: dtree_list_a_b,E1: b] :
( ~ ( member_b @ E1 @ ( darcs_list_a_b @ Y2 ) )
& ! [Z: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ Z @ ( fset_P9138963618725001425_a_b_b @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ( ranked8905849569120154423e1_a_b @ rank @ ta ) @ e ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( produc336013011823628281_b_b_o
@ ^ [Aa: dtree_list_a_b,E2: b] :
( ( ( inf_inf_set_b @ ( sup_sup_set_b @ ( darcs_list_a_b @ Y2 ) @ ( insert_b @ E1 @ bot_bot_set_b ) ) @ ( sup_sup_set_b @ ( darcs_list_a_b @ Aa ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
= bot_bot_set_b )
| ( ( produc7704165765595008946_a_b_b @ Y2 @ E1 )
= ( produc7704165765595008946_a_b_b @ Aa @ E2 ) ) )
@ Z ) ) )
@ X2 ) ) ).
% \<open>disjoint_darcs {|(normalize1 t, e)|}\<close>
thf(fact_7_root__normalize1__eq2,axiom,
! [Xs2: fset_P2153231429829016240_a_b_b,R2: list_a] :
( ! [X3: produc6499617310964463488_a_b_b] :
( Xs2
!= ( finser2303212525150181944_a_b_b @ X3 @ bot_bo2248824169281960260_a_b_b ) )
=> ( ( root_list_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ ( node_list_a_b @ R2 @ Xs2 ) ) )
= R2 ) ) ).
% root_normalize1_eq2
thf(fact_8_dtree__to__list_Ocases,axiom,
! [X: dtree_list_a_b] :
( ! [R: list_a,T: dtree_list_a_b,E: b] :
( X
!= ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T @ E ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ~ ! [Xs: fset_P2153231429829016240_a_b_b] :
( ! [X2: produc6499617310964463488_a_b_b] :
( Xs
!= ( finser2303212525150181944_a_b_b @ X2 @ bot_bo2248824169281960260_a_b_b ) )
=> ! [R: list_a] :
( X
!= ( node_list_a_b @ R @ Xs ) ) ) ) ).
% dtree_to_list.cases
thf(fact_9_singleton__uneq,axiom,
! [R2: list_a,T2: dtree_list_a_b,E3: b] :
( ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T2 @ E3 ) @ bot_bo2248824169281960260_a_b_b ) )
!= T2 ) ).
% singleton_uneq
thf(fact_10_normalize1__rel_Ocong,axiom,
ranked37597841108724548el_a_b = ranked37597841108724548el_a_b ).
% normalize1_rel.cong
thf(fact_11_finsert__absorb2,axiom,
! [X: produc6499617310964463488_a_b_b,A: fset_P2153231429829016240_a_b_b] :
( ( finser2303212525150181944_a_b_b @ X @ ( finser2303212525150181944_a_b_b @ X @ A ) )
= ( finser2303212525150181944_a_b_b @ X @ A ) ) ).
% finsert_absorb2
thf(fact_12_dtree_Oinject,axiom,
! [X1: list_a,X22: fset_P2153231429829016240_a_b_b,Y1: list_a,Y22: fset_P2153231429829016240_a_b_b] :
( ( ( node_list_a_b @ X1 @ X22 )
= ( node_list_a_b @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% dtree.inject
thf(fact_13_prod_Oinject,axiom,
! [X1: dtree_list_a_b,X22: b,Y1: dtree_list_a_b,Y22: b] :
( ( ( produc7704165765595008946_a_b_b @ X1 @ X22 )
= ( produc7704165765595008946_a_b_b @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_14_prod_Oinject,axiom,
! [X1: list_a,X22: b,Y1: list_a,Y22: b] :
( ( ( produc4781227316648555538st_a_b @ X1 @ X22 )
= ( produc4781227316648555538st_a_b @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_15_old_Oprod_Oinject,axiom,
! [A2: dtree_list_a_b,B: b,A3: dtree_list_a_b,B2: b] :
( ( ( produc7704165765595008946_a_b_b @ A2 @ B )
= ( produc7704165765595008946_a_b_b @ A3 @ B2 ) )
= ( ( A2 = A3 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_16_old_Oprod_Oinject,axiom,
! [A2: list_a,B: b,A3: list_a,B2: b] :
( ( ( produc4781227316648555538st_a_b @ A2 @ B )
= ( produc4781227316648555538st_a_b @ A3 @ B2 ) )
= ( ( A2 = A3 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_17_split__part,axiom,
! [P: $o,Q: dtree_list_a_b > b > $o] :
( ( produc336013011823628281_b_b_o
@ ^ [A4: dtree_list_a_b,B3: b] :
( P
& ( Q @ A4 @ B3 ) ) )
= ( ^ [Ab: produc6499617310964463488_a_b_b] :
( P
& ( produc336013011823628281_b_b_o @ Q @ Ab ) ) ) ) ).
% split_part
thf(fact_18_snds__neq__if__disjoint__darcs,axiom,
! [Xs2: fset_P2153231429829016240_a_b_b] :
( ! [X3: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ X3 @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( produc336013011823628281_b_b_o
@ ^ [Y2: dtree_list_a_b,E1: b] :
( ~ ( member_b @ E1 @ ( darcs_list_a_b @ Y2 ) )
& ! [Z: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ Z @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( produc336013011823628281_b_b_o
@ ^ [Aa: dtree_list_a_b,E2: b] :
( ( ( inf_inf_set_b @ ( sup_sup_set_b @ ( darcs_list_a_b @ Y2 ) @ ( insert_b @ E1 @ bot_bot_set_b ) ) @ ( sup_sup_set_b @ ( darcs_list_a_b @ Aa ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
= bot_bot_set_b )
| ( ( produc7704165765595008946_a_b_b @ Y2 @ E1 )
= ( produc7704165765595008946_a_b_b @ Aa @ E2 ) ) )
@ Z ) ) )
@ X3 ) )
=> ! [X2: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ X2 @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( produc336013011823628281_b_b_o
@ ^ [T3: dtree_list_a_b,E4: b] :
! [Y2: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ Y2 @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( produc336013011823628281_b_b_o
@ ^ [T22: dtree_list_a_b,E2: b] :
( ( E4 != E2 )
| ( ( produc7704165765595008946_a_b_b @ T3 @ E4 )
= ( produc7704165765595008946_a_b_b @ T22 @ E2 ) ) )
@ Y2 ) )
@ X2 ) ) ) ).
% snds_neq_if_disjoint_darcs
thf(fact_19_case__prod__conv,axiom,
! [F: dtree_list_a_b > b > $o,A2: dtree_list_a_b,B: b] :
( ( produc336013011823628281_b_b_o @ F @ ( produc7704165765595008946_a_b_b @ A2 @ B ) )
= ( F @ A2 @ B ) ) ).
% case_prod_conv
thf(fact_20_case__prod__conv,axiom,
! [F: dtree_list_a_b > b > produc6499617310964463488_a_b_b,A2: dtree_list_a_b,B: b] :
( ( produc4970258495817412575_a_b_b @ F @ ( produc7704165765595008946_a_b_b @ A2 @ B ) )
= ( F @ A2 @ B ) ) ).
% case_prod_conv
thf(fact_21_case__prod__conv,axiom,
! [F: dtree_list_a_b > b > produc2579390649552321826st_a_b,A2: dtree_list_a_b,B: b] :
( ( produc352076092515656579st_a_b @ F @ ( produc7704165765595008946_a_b_b @ A2 @ B ) )
= ( F @ A2 @ B ) ) ).
% case_prod_conv
thf(fact_22_case__prodI2,axiom,
! [P2: produc2579390649552321826st_a_b,C: list_a > b > $o] :
( ! [A5: list_a,B4: b] :
( ( P2
= ( produc4781227316648555538st_a_b @ A5 @ B4 ) )
=> ( C @ A5 @ B4 ) )
=> ( produc2734430842459315633_a_b_o @ C @ P2 ) ) ).
% case_prodI2
thf(fact_23_case__prodI2,axiom,
! [P2: produc6499617310964463488_a_b_b,C: dtree_list_a_b > b > $o] :
( ! [A5: dtree_list_a_b,B4: b] :
( ( P2
= ( produc7704165765595008946_a_b_b @ A5 @ B4 ) )
=> ( C @ A5 @ B4 ) )
=> ( produc336013011823628281_b_b_o @ C @ P2 ) ) ).
% case_prodI2
thf(fact_24_case__prodI,axiom,
! [F: list_a > b > $o,A2: list_a,B: b] :
( ( F @ A2 @ B )
=> ( produc2734430842459315633_a_b_o @ F @ ( produc4781227316648555538st_a_b @ A2 @ B ) ) ) ).
% case_prodI
thf(fact_25_case__prodI,axiom,
! [F: dtree_list_a_b > b > $o,A2: dtree_list_a_b,B: b] :
( ( F @ A2 @ B )
=> ( produc336013011823628281_b_b_o @ F @ ( produc7704165765595008946_a_b_b @ A2 @ B ) ) ) ).
% case_prodI
thf(fact_26_bot__fset_Orep__eq,axiom,
( ( fset_b2 @ bot_bot_fset_b )
= bot_bot_set_b ) ).
% bot_fset.rep_eq
thf(fact_27_bot__fset_Orep__eq,axiom,
( ( fset_a2 @ bot_bot_fset_a )
= bot_bot_set_a ) ).
% bot_fset.rep_eq
thf(fact_28_bot__fset_Orep__eq,axiom,
( ( fset_list_a2 @ bot_bot_fset_list_a )
= bot_bot_set_list_a ) ).
% bot_fset.rep_eq
thf(fact_29_bot__fset_Orep__eq,axiom,
( ( fset_P9138963618725001425_a_b_b @ bot_bo2248824169281960260_a_b_b )
= bot_bo4897374000430069834_a_b_b ) ).
% bot_fset.rep_eq
thf(fact_30_finsert_Orep__eq,axiom,
! [X: b,Xa: fset_b] :
( ( fset_b2 @ ( finsert_b @ X @ Xa ) )
= ( insert_b @ X @ ( fset_b2 @ Xa ) ) ) ).
% finsert.rep_eq
thf(fact_31_finsert_Orep__eq,axiom,
! [X: list_a,Xa: fset_list_a] :
( ( fset_list_a2 @ ( finsert_list_a @ X @ Xa ) )
= ( insert_list_a @ X @ ( fset_list_a2 @ Xa ) ) ) ).
% finsert.rep_eq
thf(fact_32_finsert_Orep__eq,axiom,
! [X: produc6499617310964463488_a_b_b,Xa: fset_P2153231429829016240_a_b_b] :
( ( fset_P9138963618725001425_a_b_b @ ( finser2303212525150181944_a_b_b @ X @ Xa ) )
= ( insert8720088738084019632_a_b_b @ X @ ( fset_P9138963618725001425_a_b_b @ Xa ) ) ) ).
% finsert.rep_eq
thf(fact_33_disjoint__darcs__aux2,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,E3: b,V: list_a] :
( ( wf_darcs_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
=> ( ~ ( member_b @ E3 @ ( darcs_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) )
=> ! [X2: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ X2 @ ( fset_P9138963618725001425_a_b_b @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ( node_list_a_b @ V @ bot_bo2248824169281960260_a_b_b ) @ E3 ) @ Xs2 ) ) )
=> ( produc336013011823628281_b_b_o
@ ^ [Y2: dtree_list_a_b,E1: b] :
( ~ ( member_b @ E1 @ ( darcs_list_a_b @ Y2 ) )
& ! [Z: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ Z @ ( fset_P9138963618725001425_a_b_b @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ( node_list_a_b @ V @ bot_bo2248824169281960260_a_b_b ) @ E3 ) @ Xs2 ) ) )
=> ( produc336013011823628281_b_b_o
@ ^ [Aa: dtree_list_a_b,E2: b] :
( ( ( inf_inf_set_b @ ( sup_sup_set_b @ ( darcs_list_a_b @ Y2 ) @ ( insert_b @ E1 @ bot_bot_set_b ) ) @ ( sup_sup_set_b @ ( darcs_list_a_b @ Aa ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
= bot_bot_set_b )
| ( ( produc7704165765595008946_a_b_b @ Y2 @ E1 )
= ( produc7704165765595008946_a_b_b @ Aa @ E2 ) ) )
@ Z ) ) )
@ X2 ) ) ) ) ).
% disjoint_darcs_aux2
thf(fact_34_insert__between_Ocases,axiom,
! [X: produc9008341577332299707st_a_b] :
~ ! [V2: list_a,E: b,X3: list_a,Y3: list_a,R: list_a,Xs: fset_P2153231429829016240_a_b_b] :
( X
!= ( produc1848684973559390389st_a_b @ V2 @ ( produc5064203622704112514st_a_b @ E @ ( produc673257793671328980st_a_b @ X3 @ ( produc148520996349637281st_a_b @ Y3 @ ( node_list_a_b @ R @ Xs ) ) ) ) ) ) ).
% insert_between.cases
thf(fact_35_fset__cong,axiom,
! [X: fset_P2153231429829016240_a_b_b,Y: fset_P2153231429829016240_a_b_b] :
( ( ( fset_P9138963618725001425_a_b_b @ X )
= ( fset_P9138963618725001425_a_b_b @ Y ) )
= ( X = Y ) ) ).
% fset_cong
thf(fact_36_disjoint__darcs__simp,axiom,
! [T12: dtree_list_a_b,E12: b,Xs2: fset_P2153231429829016240_a_b_b,T23: dtree_list_a_b,E22: b] :
( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E12 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T23 @ E22 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( ( produc7704165765595008946_a_b_b @ T12 @ E12 )
!= ( produc7704165765595008946_a_b_b @ T23 @ E22 ) )
=> ( ! [X3: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ X3 @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( produc336013011823628281_b_b_o
@ ^ [Y2: dtree_list_a_b,E1: b] :
( ~ ( member_b @ E1 @ ( darcs_list_a_b @ Y2 ) )
& ! [Z: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ Z @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( produc336013011823628281_b_b_o
@ ^ [Aa: dtree_list_a_b,E2: b] :
( ( ( inf_inf_set_b @ ( sup_sup_set_b @ ( darcs_list_a_b @ Y2 ) @ ( insert_b @ E1 @ bot_bot_set_b ) ) @ ( sup_sup_set_b @ ( darcs_list_a_b @ Aa ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
= bot_bot_set_b )
| ( ( produc7704165765595008946_a_b_b @ Y2 @ E1 )
= ( produc7704165765595008946_a_b_b @ Aa @ E2 ) ) )
@ Z ) ) )
@ X3 ) )
=> ( ( inf_inf_set_b @ ( sup_sup_set_b @ ( darcs_list_a_b @ T12 ) @ ( insert_b @ E12 @ bot_bot_set_b ) ) @ ( sup_sup_set_b @ ( darcs_list_a_b @ T23 ) @ ( insert_b @ E22 @ bot_bot_set_b ) ) )
= bot_bot_set_b ) ) ) ) ) ).
% disjoint_darcs_simp
thf(fact_37_ranked__dtree_Onormalize_Ocong,axiom,
ranked_normalize_a_b = ranked_normalize_a_b ).
% ranked_dtree.normalize.cong
thf(fact_38_prod_Odisc__eq__case,axiom,
! [Prod: produc6499617310964463488_a_b_b] :
( produc336013011823628281_b_b_o
@ ^ [Uu: dtree_list_a_b,Uv: b] : $true
@ Prod ) ).
% prod.disc_eq_case
thf(fact_39_prod_Ocase__distrib,axiom,
! [H: $o > $o,F: dtree_list_a_b > b > $o,Prod: produc6499617310964463488_a_b_b] :
( ( H @ ( produc336013011823628281_b_b_o @ F @ Prod ) )
= ( produc336013011823628281_b_b_o
@ ^ [X12: dtree_list_a_b,X23: b] : ( H @ ( F @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_40_prod_Ocase__distrib,axiom,
! [H: $o > produc6499617310964463488_a_b_b,F: dtree_list_a_b > b > $o,Prod: produc6499617310964463488_a_b_b] :
( ( H @ ( produc336013011823628281_b_b_o @ F @ Prod ) )
= ( produc4970258495817412575_a_b_b
@ ^ [X12: dtree_list_a_b,X23: b] : ( H @ ( F @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_41_prod_Ocase__distrib,axiom,
! [H: $o > produc2579390649552321826st_a_b,F: dtree_list_a_b > b > $o,Prod: produc6499617310964463488_a_b_b] :
( ( H @ ( produc336013011823628281_b_b_o @ F @ Prod ) )
= ( produc352076092515656579st_a_b
@ ^ [X12: dtree_list_a_b,X23: b] : ( H @ ( F @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_42_prod_Ocase__distrib,axiom,
! [H: produc6499617310964463488_a_b_b > $o,F: dtree_list_a_b > b > produc6499617310964463488_a_b_b,Prod: produc6499617310964463488_a_b_b] :
( ( H @ ( produc4970258495817412575_a_b_b @ F @ Prod ) )
= ( produc336013011823628281_b_b_o
@ ^ [X12: dtree_list_a_b,X23: b] : ( H @ ( F @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_43_prod_Ocase__distrib,axiom,
! [H: produc6499617310964463488_a_b_b > produc6499617310964463488_a_b_b,F: dtree_list_a_b > b > produc6499617310964463488_a_b_b,Prod: produc6499617310964463488_a_b_b] :
( ( H @ ( produc4970258495817412575_a_b_b @ F @ Prod ) )
= ( produc4970258495817412575_a_b_b
@ ^ [X12: dtree_list_a_b,X23: b] : ( H @ ( F @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_44_prod_Ocase__distrib,axiom,
! [H: produc6499617310964463488_a_b_b > produc2579390649552321826st_a_b,F: dtree_list_a_b > b > produc6499617310964463488_a_b_b,Prod: produc6499617310964463488_a_b_b] :
( ( H @ ( produc4970258495817412575_a_b_b @ F @ Prod ) )
= ( produc352076092515656579st_a_b
@ ^ [X12: dtree_list_a_b,X23: b] : ( H @ ( F @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_45_prod_Ocase__distrib,axiom,
! [H: produc2579390649552321826st_a_b > $o,F: dtree_list_a_b > b > produc2579390649552321826st_a_b,Prod: produc6499617310964463488_a_b_b] :
( ( H @ ( produc352076092515656579st_a_b @ F @ Prod ) )
= ( produc336013011823628281_b_b_o
@ ^ [X12: dtree_list_a_b,X23: b] : ( H @ ( F @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_46_prod_Ocase__distrib,axiom,
! [H: produc2579390649552321826st_a_b > produc6499617310964463488_a_b_b,F: dtree_list_a_b > b > produc2579390649552321826st_a_b,Prod: produc6499617310964463488_a_b_b] :
( ( H @ ( produc352076092515656579st_a_b @ F @ Prod ) )
= ( produc4970258495817412575_a_b_b
@ ^ [X12: dtree_list_a_b,X23: b] : ( H @ ( F @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_47_prod_Ocase__distrib,axiom,
! [H: produc2579390649552321826st_a_b > produc2579390649552321826st_a_b,F: dtree_list_a_b > b > produc2579390649552321826st_a_b,Prod: produc6499617310964463488_a_b_b] :
( ( H @ ( produc352076092515656579st_a_b @ F @ Prod ) )
= ( produc352076092515656579st_a_b
@ ^ [X12: dtree_list_a_b,X23: b] : ( H @ ( F @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_48_dtree_Osel_I1_J,axiom,
! [X1: list_a,X22: fset_P2153231429829016240_a_b_b] :
( ( root_list_a_b @ ( node_list_a_b @ X1 @ X22 ) )
= X1 ) ).
% dtree.sel(1)
thf(fact_49_dtail_Ocases,axiom,
! [X: produc1920479565126685823list_a] :
~ ! [R: list_a,Xs: fset_P2153231429829016240_a_b_b,Def: b > list_a] :
( X
!= ( produc2621617146629198007list_a @ ( node_list_a_b @ R @ Xs ) @ Def ) ) ).
% dtail.cases
thf(fact_50_is__subtree_Ocases,axiom,
! [X: produc1510363273921914569st_a_b] :
~ ! [X3: dtree_list_a_b,R: list_a,Xs: fset_P2153231429829016240_a_b_b] :
( X
!= ( produc783528831147138817st_a_b @ X3 @ ( node_list_a_b @ R @ Xs ) ) ) ).
% is_subtree.cases
thf(fact_51_disjoint__darcs__insert,axiom,
! [X: produc6499617310964463488_a_b_b,Xs2: fset_P2153231429829016240_a_b_b] :
( ! [X3: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ X3 @ ( fset_P9138963618725001425_a_b_b @ ( finser2303212525150181944_a_b_b @ X @ Xs2 ) ) )
=> ( produc336013011823628281_b_b_o
@ ^ [Y2: dtree_list_a_b,E1: b] :
( ~ ( member_b @ E1 @ ( darcs_list_a_b @ Y2 ) )
& ! [Z: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ Z @ ( fset_P9138963618725001425_a_b_b @ ( finser2303212525150181944_a_b_b @ X @ Xs2 ) ) )
=> ( produc336013011823628281_b_b_o
@ ^ [Aa: dtree_list_a_b,E2: b] :
( ( ( inf_inf_set_b @ ( sup_sup_set_b @ ( darcs_list_a_b @ Y2 ) @ ( insert_b @ E1 @ bot_bot_set_b ) ) @ ( sup_sup_set_b @ ( darcs_list_a_b @ Aa ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
= bot_bot_set_b )
| ( ( produc7704165765595008946_a_b_b @ Y2 @ E1 )
= ( produc7704165765595008946_a_b_b @ Aa @ E2 ) ) )
@ Z ) ) )
@ X3 ) )
=> ! [X2: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ X2 @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( produc336013011823628281_b_b_o
@ ^ [Y2: dtree_list_a_b,E1: b] :
( ~ ( member_b @ E1 @ ( darcs_list_a_b @ Y2 ) )
& ! [Z: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ Z @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( produc336013011823628281_b_b_o
@ ^ [Aa: dtree_list_a_b,E2: b] :
( ( ( inf_inf_set_b @ ( sup_sup_set_b @ ( darcs_list_a_b @ Y2 ) @ ( insert_b @ E1 @ bot_bot_set_b ) ) @ ( sup_sup_set_b @ ( darcs_list_a_b @ Aa ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
= bot_bot_set_b )
| ( ( produc7704165765595008946_a_b_b @ Y2 @ E1 )
= ( produc7704165765595008946_a_b_b @ Aa @ E2 ) ) )
@ Z ) ) )
@ X2 ) ) ) ).
% disjoint_darcs_insert
thf(fact_52_disjoint__darcs__single,axiom,
! [E3: b,T2: dtree_list_a_b] :
( ( ~ ( member_b @ E3 @ ( darcs_list_a_b @ T2 ) ) )
= ( ! [X4: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ X4 @ ( fset_P9138963618725001425_a_b_b @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T2 @ E3 ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( produc336013011823628281_b_b_o
@ ^ [Y2: dtree_list_a_b,E1: b] :
( ~ ( member_b @ E1 @ ( darcs_list_a_b @ Y2 ) )
& ! [Z: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ Z @ ( fset_P9138963618725001425_a_b_b @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T2 @ E3 ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( produc336013011823628281_b_b_o
@ ^ [Aa: dtree_list_a_b,E2: b] :
( ( ( inf_inf_set_b @ ( sup_sup_set_b @ ( darcs_list_a_b @ Y2 ) @ ( insert_b @ E1 @ bot_bot_set_b ) ) @ ( sup_sup_set_b @ ( darcs_list_a_b @ Aa ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
= bot_bot_set_b )
| ( ( produc7704165765595008946_a_b_b @ Y2 @ E1 )
= ( produc7704165765595008946_a_b_b @ Aa @ E2 ) ) )
@ Z ) ) )
@ X4 ) ) ) ) ).
% disjoint_darcs_single
thf(fact_53_disjoint__darcs__if__wf__xs,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b] :
( ( wf_darcs_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
=> ! [X2: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ X2 @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( produc336013011823628281_b_b_o
@ ^ [Y2: dtree_list_a_b,E1: b] :
( ~ ( member_b @ E1 @ ( darcs_list_a_b @ Y2 ) )
& ! [Z: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ Z @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( produc336013011823628281_b_b_o
@ ^ [Aa: dtree_list_a_b,E2: b] :
( ( ( inf_inf_set_b @ ( sup_sup_set_b @ ( darcs_list_a_b @ Y2 ) @ ( insert_b @ E1 @ bot_bot_set_b ) ) @ ( sup_sup_set_b @ ( darcs_list_a_b @ Aa ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
= bot_bot_set_b )
| ( ( produc7704165765595008946_a_b_b @ Y2 @ E1 )
= ( produc7704165765595008946_a_b_b @ Aa @ E2 ) ) )
@ Z ) ) )
@ X2 ) ) ) ).
% disjoint_darcs_if_wf_xs
thf(fact_54_wf__darcs__if__darcs_H__aux,axiom,
! [Xs2: fset_P2153231429829016240_a_b_b,R2: list_a] :
( ! [X3: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ X3 @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( produc336013011823628281_b_b_o
@ ^ [Y2: dtree_list_a_b,E4: b] : ( wf_darcs_list_a_b @ Y2 )
@ X3 ) )
=> ( ! [X3: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ X3 @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( produc336013011823628281_b_b_o
@ ^ [Y2: dtree_list_a_b,E1: b] :
( ~ ( member_b @ E1 @ ( darcs_list_a_b @ Y2 ) )
& ! [Z: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ Z @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( produc336013011823628281_b_b_o
@ ^ [Aa: dtree_list_a_b,E2: b] :
( ( ( inf_inf_set_b @ ( sup_sup_set_b @ ( darcs_list_a_b @ Y2 ) @ ( insert_b @ E1 @ bot_bot_set_b ) ) @ ( sup_sup_set_b @ ( darcs_list_a_b @ Aa ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
= bot_bot_set_b )
| ( ( produc7704165765595008946_a_b_b @ Y2 @ E1 )
= ( produc7704165765595008946_a_b_b @ Aa @ E2 ) ) )
@ Z ) ) )
@ X3 ) )
=> ( wf_darcs_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) ) ) ).
% wf_darcs_if_darcs'_aux
thf(fact_55_disjoint__darcs__if__wf__aux5,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,T12: dtree_list_a_b,E12: b,T23: dtree_list_a_b,E22: b] :
( ( wf_darcs_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E12 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T23 @ E22 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( ( produc7704165765595008946_a_b_b @ T12 @ E12 )
!= ( produc7704165765595008946_a_b_b @ T23 @ E22 ) )
=> ( ( inf_inf_set_b @ ( sup_sup_set_b @ ( darcs_list_a_b @ T12 ) @ ( insert_b @ E12 @ bot_bot_set_b ) ) @ ( sup_sup_set_b @ ( darcs_list_a_b @ T23 ) @ ( insert_b @ E22 @ bot_bot_set_b ) ) )
= bot_bot_set_b ) ) ) ) ) ).
% disjoint_darcs_if_wf_aux5
thf(fact_56_old_Oprod_Ocase,axiom,
! [F: dtree_list_a_b > b > $o,X1: dtree_list_a_b,X22: b] :
( ( produc336013011823628281_b_b_o @ F @ ( produc7704165765595008946_a_b_b @ X1 @ X22 ) )
= ( F @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_57_old_Oprod_Ocase,axiom,
! [F: dtree_list_a_b > b > produc6499617310964463488_a_b_b,X1: dtree_list_a_b,X22: b] :
( ( produc4970258495817412575_a_b_b @ F @ ( produc7704165765595008946_a_b_b @ X1 @ X22 ) )
= ( F @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_58_old_Oprod_Ocase,axiom,
! [F: dtree_list_a_b > b > produc2579390649552321826st_a_b,X1: dtree_list_a_b,X22: b] :
( ( produc352076092515656579st_a_b @ F @ ( produc7704165765595008946_a_b_b @ X1 @ X22 ) )
= ( F @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_59_case__prodE,axiom,
! [C: list_a > b > $o,P2: produc2579390649552321826st_a_b] :
( ( produc2734430842459315633_a_b_o @ C @ P2 )
=> ~ ! [X3: list_a,Y3: b] :
( ( P2
= ( produc4781227316648555538st_a_b @ X3 @ Y3 ) )
=> ~ ( C @ X3 @ Y3 ) ) ) ).
% case_prodE
thf(fact_60_case__prodE,axiom,
! [C: dtree_list_a_b > b > $o,P2: produc6499617310964463488_a_b_b] :
( ( produc336013011823628281_b_b_o @ C @ P2 )
=> ~ ! [X3: dtree_list_a_b,Y3: b] :
( ( P2
= ( produc7704165765595008946_a_b_b @ X3 @ Y3 ) )
=> ~ ( C @ X3 @ Y3 ) ) ) ).
% case_prodE
thf(fact_61_mem__Collect__eq,axiom,
! [A2: b,P: b > $o] :
( ( member_b @ A2 @ ( collect_b @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_62_mem__Collect__eq,axiom,
! [A2: produc6499617310964463488_a_b_b,P: produc6499617310964463488_a_b_b > $o] :
( ( member4695696432722591383_a_b_b @ A2 @ ( collec3241703154345475797_a_b_b @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_63_mem__Collect__eq,axiom,
! [A2: list_a,P: list_a > $o] :
( ( member_list_a @ A2 @ ( collect_list_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_64_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_65_mem__Collect__eq,axiom,
! [A2: dtree_list_a_b,P: dtree_list_a_b > $o] :
( ( member551035911493665803st_a_b @ A2 @ ( collec2944820760411501129st_a_b @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_66_Collect__mem__eq,axiom,
! [A: set_b] :
( ( collect_b
@ ^ [X4: b] : ( member_b @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_67_Collect__mem__eq,axiom,
! [A: set_Pr3443975907877334966_a_b_b] :
( ( collec3241703154345475797_a_b_b
@ ^ [X4: produc6499617310964463488_a_b_b] : ( member4695696432722591383_a_b_b @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_68_Collect__mem__eq,axiom,
! [A: set_list_a] :
( ( collect_list_a
@ ^ [X4: list_a] : ( member_list_a @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_69_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X4: a] : ( member_a @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_70_Collect__mem__eq,axiom,
! [A: set_dtree_list_a_b] :
( ( collec2944820760411501129st_a_b
@ ^ [X4: dtree_list_a_b] : ( member551035911493665803st_a_b @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_71_case__prodD,axiom,
! [F: list_a > b > $o,A2: list_a,B: b] :
( ( produc2734430842459315633_a_b_o @ F @ ( produc4781227316648555538st_a_b @ A2 @ B ) )
=> ( F @ A2 @ B ) ) ).
% case_prodD
thf(fact_72_case__prodD,axiom,
! [F: dtree_list_a_b > b > $o,A2: dtree_list_a_b,B: b] :
( ( produc336013011823628281_b_b_o @ F @ ( produc7704165765595008946_a_b_b @ A2 @ B ) )
=> ( F @ A2 @ B ) ) ).
% case_prodD
thf(fact_73_disjoint__darcs__if__wf__aux2,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,T12: dtree_list_a_b,E12: b,T23: dtree_list_a_b,E22: b] :
( ( wf_darcs_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E12 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T23 @ E22 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( ( produc7704165765595008946_a_b_b @ T12 @ E12 )
!= ( produc7704165765595008946_a_b_b @ T23 @ E22 ) )
=> ( ( inf_inf_set_b @ ( darcs_list_a_b @ T12 ) @ ( darcs_list_a_b @ T23 ) )
= bot_bot_set_b ) ) ) ) ) ).
% disjoint_darcs_if_wf_aux2
thf(fact_74_cond__case__prod__eta,axiom,
! [F: dtree_list_a_b > b > $o,G: produc6499617310964463488_a_b_b > $o] :
( ! [X3: dtree_list_a_b,Y3: b] :
( ( F @ X3 @ Y3 )
= ( G @ ( produc7704165765595008946_a_b_b @ X3 @ Y3 ) ) )
=> ( ( produc336013011823628281_b_b_o @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_75_cond__case__prod__eta,axiom,
! [F: dtree_list_a_b > b > produc6499617310964463488_a_b_b,G: produc6499617310964463488_a_b_b > produc6499617310964463488_a_b_b] :
( ! [X3: dtree_list_a_b,Y3: b] :
( ( F @ X3 @ Y3 )
= ( G @ ( produc7704165765595008946_a_b_b @ X3 @ Y3 ) ) )
=> ( ( produc4970258495817412575_a_b_b @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_76_cond__case__prod__eta,axiom,
! [F: dtree_list_a_b > b > produc2579390649552321826st_a_b,G: produc6499617310964463488_a_b_b > produc2579390649552321826st_a_b] :
( ! [X3: dtree_list_a_b,Y3: b] :
( ( F @ X3 @ Y3 )
= ( G @ ( produc7704165765595008946_a_b_b @ X3 @ Y3 ) ) )
=> ( ( produc352076092515656579st_a_b @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_77_case__prod__eta,axiom,
! [F: produc6499617310964463488_a_b_b > $o] :
( ( produc336013011823628281_b_b_o
@ ^ [X4: dtree_list_a_b,Y2: b] : ( F @ ( produc7704165765595008946_a_b_b @ X4 @ Y2 ) ) )
= F ) ).
% case_prod_eta
thf(fact_78_case__prod__eta,axiom,
! [F: produc6499617310964463488_a_b_b > produc6499617310964463488_a_b_b] :
( ( produc4970258495817412575_a_b_b
@ ^ [X4: dtree_list_a_b,Y2: b] : ( F @ ( produc7704165765595008946_a_b_b @ X4 @ Y2 ) ) )
= F ) ).
% case_prod_eta
thf(fact_79_case__prod__eta,axiom,
! [F: produc6499617310964463488_a_b_b > produc2579390649552321826st_a_b] :
( ( produc352076092515656579st_a_b
@ ^ [X4: dtree_list_a_b,Y2: b] : ( F @ ( produc7704165765595008946_a_b_b @ X4 @ Y2 ) ) )
= F ) ).
% case_prod_eta
thf(fact_80_case__prodE2,axiom,
! [Q: $o > $o,P: dtree_list_a_b > b > $o,Z2: produc6499617310964463488_a_b_b] :
( ( Q @ ( produc336013011823628281_b_b_o @ P @ Z2 ) )
=> ~ ! [X3: dtree_list_a_b,Y3: b] :
( ( Z2
= ( produc7704165765595008946_a_b_b @ X3 @ Y3 ) )
=> ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_81_case__prodE2,axiom,
! [Q: produc6499617310964463488_a_b_b > $o,P: dtree_list_a_b > b > produc6499617310964463488_a_b_b,Z2: produc6499617310964463488_a_b_b] :
( ( Q @ ( produc4970258495817412575_a_b_b @ P @ Z2 ) )
=> ~ ! [X3: dtree_list_a_b,Y3: b] :
( ( Z2
= ( produc7704165765595008946_a_b_b @ X3 @ Y3 ) )
=> ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_82_case__prodE2,axiom,
! [Q: produc2579390649552321826st_a_b > $o,P: dtree_list_a_b > b > produc2579390649552321826st_a_b,Z2: produc6499617310964463488_a_b_b] :
( ( Q @ ( produc352076092515656579st_a_b @ P @ Z2 ) )
=> ~ ! [X3: dtree_list_a_b,Y3: b] :
( ( Z2
= ( produc7704165765595008946_a_b_b @ X3 @ Y3 ) )
=> ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_83_Pair__inject,axiom,
! [A2: dtree_list_a_b,B: b,A3: dtree_list_a_b,B2: b] :
( ( ( produc7704165765595008946_a_b_b @ A2 @ B )
= ( produc7704165765595008946_a_b_b @ A3 @ B2 ) )
=> ~ ( ( A2 = A3 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_84_Pair__inject,axiom,
! [A2: list_a,B: b,A3: list_a,B2: b] :
( ( ( produc4781227316648555538st_a_b @ A2 @ B )
= ( produc4781227316648555538st_a_b @ A3 @ B2 ) )
=> ~ ( ( A2 = A3 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_85_prod__cases,axiom,
! [P: produc6499617310964463488_a_b_b > $o,P2: produc6499617310964463488_a_b_b] :
( ! [A5: dtree_list_a_b,B4: b] : ( P @ ( produc7704165765595008946_a_b_b @ A5 @ B4 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_86_prod__cases,axiom,
! [P: produc2579390649552321826st_a_b > $o,P2: produc2579390649552321826st_a_b] :
( ! [A5: list_a,B4: b] : ( P @ ( produc4781227316648555538st_a_b @ A5 @ B4 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_87_surj__pair,axiom,
! [P2: produc6499617310964463488_a_b_b] :
? [X3: dtree_list_a_b,Y3: b] :
( P2
= ( produc7704165765595008946_a_b_b @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_88_surj__pair,axiom,
! [P2: produc2579390649552321826st_a_b] :
? [X3: list_a,Y3: b] :
( P2
= ( produc4781227316648555538st_a_b @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_89_old_Oprod_Oexhaust,axiom,
! [Y: produc6499617310964463488_a_b_b] :
~ ! [A5: dtree_list_a_b,B4: b] :
( Y
!= ( produc7704165765595008946_a_b_b @ A5 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_90_old_Oprod_Oexhaust,axiom,
! [Y: produc2579390649552321826st_a_b] :
~ ! [A5: list_a,B4: b] :
( Y
!= ( produc4781227316648555538st_a_b @ A5 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_91_dverts__mset_Ocases,axiom,
! [X: dtree_list_a_b] :
~ ! [R: list_a,Xs: fset_P2153231429829016240_a_b_b] :
( X
!= ( node_list_a_b @ R @ Xs ) ) ).
% dverts_mset.cases
thf(fact_92_dtree_Oexhaust,axiom,
! [Y: dtree_list_a_b] :
~ ! [X13: list_a,X24: fset_P2153231429829016240_a_b_b] :
( Y
!= ( node_list_a_b @ X13 @ X24 ) ) ).
% dtree.exhaust
thf(fact_93_finsert__commute,axiom,
! [X: produc6499617310964463488_a_b_b,Y: produc6499617310964463488_a_b_b,A: fset_P2153231429829016240_a_b_b] :
( ( finser2303212525150181944_a_b_b @ X @ ( finser2303212525150181944_a_b_b @ Y @ A ) )
= ( finser2303212525150181944_a_b_b @ Y @ ( finser2303212525150181944_a_b_b @ X @ A ) ) ) ).
% finsert_commute
thf(fact_94_disjoint__darcs__if__wf__aux3,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,T12: dtree_list_a_b,E12: b,T23: dtree_list_a_b,E22: b] :
( ( wf_darcs_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E12 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T23 @ E22 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ~ ( member_b @ E22 @ ( darcs_list_a_b @ T12 ) ) ) ) ) ).
% disjoint_darcs_if_wf_aux3
thf(fact_95_disjoint__darcs__if__wf__aux1,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,T12: dtree_list_a_b,E12: b] :
( ( wf_darcs_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E12 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ~ ( member_b @ E12 @ ( darcs_list_a_b @ T12 ) ) ) ) ).
% disjoint_darcs_if_wf_aux1
thf(fact_96_disjoint__darcs__if__wf__aux4,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,T12: dtree_list_a_b,E12: b,T23: dtree_list_a_b,E22: b] :
( ( wf_darcs_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E12 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T23 @ E22 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( ( produc7704165765595008946_a_b_b @ T12 @ E12 )
!= ( produc7704165765595008946_a_b_b @ T23 @ E22 ) )
=> ( E12 != E22 ) ) ) ) ) ).
% disjoint_darcs_if_wf_aux4
thf(fact_97_finsert__not__fempty,axiom,
! [A2: produc6499617310964463488_a_b_b,A: fset_P2153231429829016240_a_b_b] :
( ( finser2303212525150181944_a_b_b @ A2 @ A )
!= bot_bo2248824169281960260_a_b_b ) ).
% finsert_not_fempty
thf(fact_98_fsingleton__inject,axiom,
! [A2: produc6499617310964463488_a_b_b,B: produc6499617310964463488_a_b_b] :
( ( ( finser2303212525150181944_a_b_b @ A2 @ bot_bo2248824169281960260_a_b_b )
= ( finser2303212525150181944_a_b_b @ B @ bot_bo2248824169281960260_a_b_b ) )
=> ( A2 = B ) ) ).
% fsingleton_inject
thf(fact_99_fdoubleton__eq__iff,axiom,
! [A2: produc6499617310964463488_a_b_b,B: produc6499617310964463488_a_b_b,C: produc6499617310964463488_a_b_b,D: produc6499617310964463488_a_b_b] :
( ( ( finser2303212525150181944_a_b_b @ A2 @ ( finser2303212525150181944_a_b_b @ B @ bot_bo2248824169281960260_a_b_b ) )
= ( finser2303212525150181944_a_b_b @ C @ ( finser2303212525150181944_a_b_b @ D @ bot_bo2248824169281960260_a_b_b ) ) )
= ( ( ( A2 = C )
& ( B = D ) )
| ( ( A2 = D )
& ( B = C ) ) ) ) ).
% fdoubleton_eq_iff
thf(fact_100_fset__exhaust,axiom,
! [S: fset_P2153231429829016240_a_b_b] :
( ( S != bot_bo2248824169281960260_a_b_b )
=> ~ ! [X3: produc6499617310964463488_a_b_b,S2: fset_P2153231429829016240_a_b_b] :
( S
!= ( finser2303212525150181944_a_b_b @ X3 @ S2 ) ) ) ).
% fset_exhaust
thf(fact_101_FSet_Ofset__induct,axiom,
! [P: fset_P2153231429829016240_a_b_b > $o,S: fset_P2153231429829016240_a_b_b] :
( ( P @ bot_bo2248824169281960260_a_b_b )
=> ( ! [X3: produc6499617310964463488_a_b_b,S3: fset_P2153231429829016240_a_b_b] :
( ( P @ S3 )
=> ( P @ ( finser2303212525150181944_a_b_b @ X3 @ S3 ) ) )
=> ( P @ S ) ) ) ).
% FSet.fset_induct
thf(fact_102_disjoint__insert_I2_J,axiom,
! [A: set_Pr3443975907877334966_a_b_b,B: produc6499617310964463488_a_b_b,B5: set_Pr3443975907877334966_a_b_b] :
( ( bot_bo4897374000430069834_a_b_b
= ( inf_in7962574851115124040_a_b_b @ A @ ( insert8720088738084019632_a_b_b @ B @ B5 ) ) )
= ( ~ ( member4695696432722591383_a_b_b @ B @ A )
& ( bot_bo4897374000430069834_a_b_b
= ( inf_in7962574851115124040_a_b_b @ A @ B5 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_103_disjoint__insert_I2_J,axiom,
! [A: set_dtree_list_a_b,B: dtree_list_a_b,B5: set_dtree_list_a_b] :
( ( bot_bo798015271861357502st_a_b
= ( inf_in3355993651213403836st_a_b @ A @ ( insert7065945238938772516st_a_b @ B @ B5 ) ) )
= ( ~ ( member551035911493665803st_a_b @ B @ A )
& ( bot_bo798015271861357502st_a_b
= ( inf_in3355993651213403836st_a_b @ A @ B5 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_104_disjoint__insert_I2_J,axiom,
! [A: set_b,B: b,B5: set_b] :
( ( bot_bot_set_b
= ( inf_inf_set_b @ A @ ( insert_b @ B @ B5 ) ) )
= ( ~ ( member_b @ B @ A )
& ( bot_bot_set_b
= ( inf_inf_set_b @ A @ B5 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_105_disjoint__insert_I2_J,axiom,
! [A: set_a,B: a,B5: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A @ ( insert_a @ B @ B5 ) ) )
= ( ~ ( member_a @ B @ A )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A @ B5 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_106_disjoint__insert_I2_J,axiom,
! [A: set_list_a,B: list_a,B5: set_list_a] :
( ( bot_bot_set_list_a
= ( inf_inf_set_list_a @ A @ ( insert_list_a @ B @ B5 ) ) )
= ( ~ ( member_list_a @ B @ A )
& ( bot_bot_set_list_a
= ( inf_inf_set_list_a @ A @ B5 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_107_disjoint__insert_I1_J,axiom,
! [B5: set_Pr3443975907877334966_a_b_b,A2: produc6499617310964463488_a_b_b,A: set_Pr3443975907877334966_a_b_b] :
( ( ( inf_in7962574851115124040_a_b_b @ B5 @ ( insert8720088738084019632_a_b_b @ A2 @ A ) )
= bot_bo4897374000430069834_a_b_b )
= ( ~ ( member4695696432722591383_a_b_b @ A2 @ B5 )
& ( ( inf_in7962574851115124040_a_b_b @ B5 @ A )
= bot_bo4897374000430069834_a_b_b ) ) ) ).
% disjoint_insert(1)
thf(fact_108_disjoint__insert_I1_J,axiom,
! [B5: set_dtree_list_a_b,A2: dtree_list_a_b,A: set_dtree_list_a_b] :
( ( ( inf_in3355993651213403836st_a_b @ B5 @ ( insert7065945238938772516st_a_b @ A2 @ A ) )
= bot_bo798015271861357502st_a_b )
= ( ~ ( member551035911493665803st_a_b @ A2 @ B5 )
& ( ( inf_in3355993651213403836st_a_b @ B5 @ A )
= bot_bo798015271861357502st_a_b ) ) ) ).
% disjoint_insert(1)
thf(fact_109_disjoint__insert_I1_J,axiom,
! [B5: set_b,A2: b,A: set_b] :
( ( ( inf_inf_set_b @ B5 @ ( insert_b @ A2 @ A ) )
= bot_bot_set_b )
= ( ~ ( member_b @ A2 @ B5 )
& ( ( inf_inf_set_b @ B5 @ A )
= bot_bot_set_b ) ) ) ).
% disjoint_insert(1)
thf(fact_110_disjoint__insert_I1_J,axiom,
! [B5: set_a,A2: a,A: set_a] :
( ( ( inf_inf_set_a @ B5 @ ( insert_a @ A2 @ A ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A2 @ B5 )
& ( ( inf_inf_set_a @ B5 @ A )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_111_disjoint__insert_I1_J,axiom,
! [B5: set_list_a,A2: list_a,A: set_list_a] :
( ( ( inf_inf_set_list_a @ B5 @ ( insert_list_a @ A2 @ A ) )
= bot_bot_set_list_a )
= ( ~ ( member_list_a @ A2 @ B5 )
& ( ( inf_inf_set_list_a @ B5 @ A )
= bot_bot_set_list_a ) ) ) ).
% disjoint_insert(1)
thf(fact_112_insert__disjoint_I2_J,axiom,
! [A2: produc6499617310964463488_a_b_b,A: set_Pr3443975907877334966_a_b_b,B5: set_Pr3443975907877334966_a_b_b] :
( ( bot_bo4897374000430069834_a_b_b
= ( inf_in7962574851115124040_a_b_b @ ( insert8720088738084019632_a_b_b @ A2 @ A ) @ B5 ) )
= ( ~ ( member4695696432722591383_a_b_b @ A2 @ B5 )
& ( bot_bo4897374000430069834_a_b_b
= ( inf_in7962574851115124040_a_b_b @ A @ B5 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_113_insert__disjoint_I2_J,axiom,
! [A2: dtree_list_a_b,A: set_dtree_list_a_b,B5: set_dtree_list_a_b] :
( ( bot_bo798015271861357502st_a_b
= ( inf_in3355993651213403836st_a_b @ ( insert7065945238938772516st_a_b @ A2 @ A ) @ B5 ) )
= ( ~ ( member551035911493665803st_a_b @ A2 @ B5 )
& ( bot_bo798015271861357502st_a_b
= ( inf_in3355993651213403836st_a_b @ A @ B5 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_114_insert__disjoint_I2_J,axiom,
! [A2: b,A: set_b,B5: set_b] :
( ( bot_bot_set_b
= ( inf_inf_set_b @ ( insert_b @ A2 @ A ) @ B5 ) )
= ( ~ ( member_b @ A2 @ B5 )
& ( bot_bot_set_b
= ( inf_inf_set_b @ A @ B5 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_115_insert__disjoint_I2_J,axiom,
! [A2: a,A: set_a,B5: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ B5 ) )
= ( ~ ( member_a @ A2 @ B5 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A @ B5 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_116_insert__disjoint_I2_J,axiom,
! [A2: list_a,A: set_list_a,B5: set_list_a] :
( ( bot_bot_set_list_a
= ( inf_inf_set_list_a @ ( insert_list_a @ A2 @ A ) @ B5 ) )
= ( ~ ( member_list_a @ A2 @ B5 )
& ( bot_bot_set_list_a
= ( inf_inf_set_list_a @ A @ B5 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_117_insert__disjoint_I1_J,axiom,
! [A2: produc6499617310964463488_a_b_b,A: set_Pr3443975907877334966_a_b_b,B5: set_Pr3443975907877334966_a_b_b] :
( ( ( inf_in7962574851115124040_a_b_b @ ( insert8720088738084019632_a_b_b @ A2 @ A ) @ B5 )
= bot_bo4897374000430069834_a_b_b )
= ( ~ ( member4695696432722591383_a_b_b @ A2 @ B5 )
& ( ( inf_in7962574851115124040_a_b_b @ A @ B5 )
= bot_bo4897374000430069834_a_b_b ) ) ) ).
% insert_disjoint(1)
thf(fact_118_insert__disjoint_I1_J,axiom,
! [A2: dtree_list_a_b,A: set_dtree_list_a_b,B5: set_dtree_list_a_b] :
( ( ( inf_in3355993651213403836st_a_b @ ( insert7065945238938772516st_a_b @ A2 @ A ) @ B5 )
= bot_bo798015271861357502st_a_b )
= ( ~ ( member551035911493665803st_a_b @ A2 @ B5 )
& ( ( inf_in3355993651213403836st_a_b @ A @ B5 )
= bot_bo798015271861357502st_a_b ) ) ) ).
% insert_disjoint(1)
thf(fact_119_insert__disjoint_I1_J,axiom,
! [A2: b,A: set_b,B5: set_b] :
( ( ( inf_inf_set_b @ ( insert_b @ A2 @ A ) @ B5 )
= bot_bot_set_b )
= ( ~ ( member_b @ A2 @ B5 )
& ( ( inf_inf_set_b @ A @ B5 )
= bot_bot_set_b ) ) ) ).
% insert_disjoint(1)
thf(fact_120_insert__disjoint_I1_J,axiom,
! [A2: a,A: set_a,B5: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ B5 )
= bot_bot_set_a )
= ( ~ ( member_a @ A2 @ B5 )
& ( ( inf_inf_set_a @ A @ B5 )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_121_insert__disjoint_I1_J,axiom,
! [A2: list_a,A: set_list_a,B5: set_list_a] :
( ( ( inf_inf_set_list_a @ ( insert_list_a @ A2 @ A ) @ B5 )
= bot_bot_set_list_a )
= ( ~ ( member_list_a @ A2 @ B5 )
& ( ( inf_inf_set_list_a @ A @ B5 )
= bot_bot_set_list_a ) ) ) ).
% insert_disjoint(1)
thf(fact_122_singleton__conv2,axiom,
! [A2: b] :
( ( collect_b
@ ( ^ [Y4: b,Z3: b] : ( Y4 = Z3 )
@ A2 ) )
= ( insert_b @ A2 @ bot_bot_set_b ) ) ).
% singleton_conv2
thf(fact_123_singleton__conv2,axiom,
! [A2: a] :
( ( collect_a
@ ( ^ [Y4: a,Z3: a] : ( Y4 = Z3 )
@ A2 ) )
= ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% singleton_conv2
thf(fact_124_singleton__conv2,axiom,
! [A2: list_a] :
( ( collect_list_a
@ ( ^ [Y4: list_a,Z3: list_a] : ( Y4 = Z3 )
@ A2 ) )
= ( insert_list_a @ A2 @ bot_bot_set_list_a ) ) ).
% singleton_conv2
thf(fact_125_singleton__conv,axiom,
! [A2: b] :
( ( collect_b
@ ^ [X4: b] : ( X4 = A2 ) )
= ( insert_b @ A2 @ bot_bot_set_b ) ) ).
% singleton_conv
thf(fact_126_singleton__conv,axiom,
! [A2: a] :
( ( collect_a
@ ^ [X4: a] : ( X4 = A2 ) )
= ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% singleton_conv
thf(fact_127_singleton__conv,axiom,
! [A2: list_a] :
( ( collect_list_a
@ ^ [X4: list_a] : ( X4 = A2 ) )
= ( insert_list_a @ A2 @ bot_bot_set_list_a ) ) ).
% singleton_conv
thf(fact_128_child__in__darcs,axiom,
! [Y: dtree_list_a_b,E22: b,Xs2: fset_P2153231429829016240_a_b_b,R2: list_a] :
( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ Y @ E22 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ord_less_eq_set_b @ ( sup_sup_set_b @ ( darcs_list_a_b @ Y ) @ ( insert_b @ E22 @ bot_bot_set_b ) ) @ ( darcs_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) ) ) ).
% child_in_darcs
thf(fact_129_Int__Un__eq_I4_J,axiom,
! [T4: set_a,S: set_a] :
( ( sup_sup_set_a @ T4 @ ( inf_inf_set_a @ S @ T4 ) )
= T4 ) ).
% Int_Un_eq(4)
thf(fact_130_Int__Un__eq_I4_J,axiom,
! [T4: set_list_a,S: set_list_a] :
( ( sup_sup_set_list_a @ T4 @ ( inf_inf_set_list_a @ S @ T4 ) )
= T4 ) ).
% Int_Un_eq(4)
thf(fact_131_Int__Un__eq_I4_J,axiom,
! [T4: set_b,S: set_b] :
( ( sup_sup_set_b @ T4 @ ( inf_inf_set_b @ S @ T4 ) )
= T4 ) ).
% Int_Un_eq(4)
thf(fact_132_Int__Un__eq_I3_J,axiom,
! [S: set_a,T4: set_a] :
( ( sup_sup_set_a @ S @ ( inf_inf_set_a @ S @ T4 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_133_Int__Un__eq_I3_J,axiom,
! [S: set_list_a,T4: set_list_a] :
( ( sup_sup_set_list_a @ S @ ( inf_inf_set_list_a @ S @ T4 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_134_Int__Un__eq_I3_J,axiom,
! [S: set_b,T4: set_b] :
( ( sup_sup_set_b @ S @ ( inf_inf_set_b @ S @ T4 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_135_Int__Un__eq_I2_J,axiom,
! [S: set_a,T4: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T4 ) @ T4 )
= T4 ) ).
% Int_Un_eq(2)
thf(fact_136_Int__Un__eq_I2_J,axiom,
! [S: set_list_a,T4: set_list_a] :
( ( sup_sup_set_list_a @ ( inf_inf_set_list_a @ S @ T4 ) @ T4 )
= T4 ) ).
% Int_Un_eq(2)
thf(fact_137_Int__Un__eq_I2_J,axiom,
! [S: set_b,T4: set_b] :
( ( sup_sup_set_b @ ( inf_inf_set_b @ S @ T4 ) @ T4 )
= T4 ) ).
% Int_Un_eq(2)
thf(fact_138_Int__Un__eq_I1_J,axiom,
! [S: set_a,T4: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T4 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_139_Int__Un__eq_I1_J,axiom,
! [S: set_list_a,T4: set_list_a] :
( ( sup_sup_set_list_a @ ( inf_inf_set_list_a @ S @ T4 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_140_Int__Un__eq_I1_J,axiom,
! [S: set_b,T4: set_b] :
( ( sup_sup_set_b @ ( inf_inf_set_b @ S @ T4 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_141_Un__Int__eq_I4_J,axiom,
! [T4: set_a,S: set_a] :
( ( inf_inf_set_a @ T4 @ ( sup_sup_set_a @ S @ T4 ) )
= T4 ) ).
% Un_Int_eq(4)
thf(fact_142_Un__Int__eq_I4_J,axiom,
! [T4: set_list_a,S: set_list_a] :
( ( inf_inf_set_list_a @ T4 @ ( sup_sup_set_list_a @ S @ T4 ) )
= T4 ) ).
% Un_Int_eq(4)
thf(fact_143_Un__Int__eq_I4_J,axiom,
! [T4: set_b,S: set_b] :
( ( inf_inf_set_b @ T4 @ ( sup_sup_set_b @ S @ T4 ) )
= T4 ) ).
% Un_Int_eq(4)
thf(fact_144_normalize1__darcs__sub,axiom,
! [T12: dtree_list_a_b] : ( ord_less_eq_set_b @ ( darcs_list_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ T12 ) ) @ ( darcs_list_a_b @ T12 ) ) ).
% normalize1_darcs_sub
thf(fact_145_order__refl,axiom,
! [X: set_b] : ( ord_less_eq_set_b @ X @ X ) ).
% order_refl
thf(fact_146_order__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% order_refl
thf(fact_147_order__refl,axiom,
! [X: fset_P2153231429829016240_a_b_b] : ( ord_le8870638447146015504_a_b_b @ X @ X ) ).
% order_refl
thf(fact_148_dual__order_Orefl,axiom,
! [A2: set_b] : ( ord_less_eq_set_b @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_149_dual__order_Orefl,axiom,
! [A2: real] : ( ord_less_eq_real @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_150_dual__order_Orefl,axiom,
! [A2: fset_P2153231429829016240_a_b_b] : ( ord_le8870638447146015504_a_b_b @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_151_empty__Collect__eq,axiom,
! [P: b > $o] :
( ( bot_bot_set_b
= ( collect_b @ P ) )
= ( ! [X4: b] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_152_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X4: a] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_153_empty__Collect__eq,axiom,
! [P: list_a > $o] :
( ( bot_bot_set_list_a
= ( collect_list_a @ P ) )
= ( ! [X4: list_a] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_154_Collect__empty__eq,axiom,
! [P: b > $o] :
( ( ( collect_b @ P )
= bot_bot_set_b )
= ( ! [X4: b] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_155_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X4: a] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_156_Collect__empty__eq,axiom,
! [P: list_a > $o] :
( ( ( collect_list_a @ P )
= bot_bot_set_list_a )
= ( ! [X4: list_a] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_157_all__not__in__conv,axiom,
! [A: set_Pr3443975907877334966_a_b_b] :
( ( ! [X4: produc6499617310964463488_a_b_b] :
~ ( member4695696432722591383_a_b_b @ X4 @ A ) )
= ( A = bot_bo4897374000430069834_a_b_b ) ) ).
% all_not_in_conv
thf(fact_158_all__not__in__conv,axiom,
! [A: set_dtree_list_a_b] :
( ( ! [X4: dtree_list_a_b] :
~ ( member551035911493665803st_a_b @ X4 @ A ) )
= ( A = bot_bo798015271861357502st_a_b ) ) ).
% all_not_in_conv
thf(fact_159_all__not__in__conv,axiom,
! [A: set_b] :
( ( ! [X4: b] :
~ ( member_b @ X4 @ A ) )
= ( A = bot_bot_set_b ) ) ).
% all_not_in_conv
thf(fact_160_all__not__in__conv,axiom,
! [A: set_a] :
( ( ! [X4: a] :
~ ( member_a @ X4 @ A ) )
= ( A = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_161_all__not__in__conv,axiom,
! [A: set_list_a] :
( ( ! [X4: list_a] :
~ ( member_list_a @ X4 @ A ) )
= ( A = bot_bot_set_list_a ) ) ).
% all_not_in_conv
thf(fact_162_empty__iff,axiom,
! [C: produc6499617310964463488_a_b_b] :
~ ( member4695696432722591383_a_b_b @ C @ bot_bo4897374000430069834_a_b_b ) ).
% empty_iff
thf(fact_163_empty__iff,axiom,
! [C: dtree_list_a_b] :
~ ( member551035911493665803st_a_b @ C @ bot_bo798015271861357502st_a_b ) ).
% empty_iff
thf(fact_164_empty__iff,axiom,
! [C: b] :
~ ( member_b @ C @ bot_bot_set_b ) ).
% empty_iff
thf(fact_165_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_166_empty__iff,axiom,
! [C: list_a] :
~ ( member_list_a @ C @ bot_bot_set_list_a ) ).
% empty_iff
thf(fact_167_insert__absorb2,axiom,
! [X: b,A: set_b] :
( ( insert_b @ X @ ( insert_b @ X @ A ) )
= ( insert_b @ X @ A ) ) ).
% insert_absorb2
thf(fact_168_insert__absorb2,axiom,
! [X: list_a,A: set_list_a] :
( ( insert_list_a @ X @ ( insert_list_a @ X @ A ) )
= ( insert_list_a @ X @ A ) ) ).
% insert_absorb2
thf(fact_169_insert__iff,axiom,
! [A2: b,B: b,A: set_b] :
( ( member_b @ A2 @ ( insert_b @ B @ A ) )
= ( ( A2 = B )
| ( member_b @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_170_insert__iff,axiom,
! [A2: produc6499617310964463488_a_b_b,B: produc6499617310964463488_a_b_b,A: set_Pr3443975907877334966_a_b_b] :
( ( member4695696432722591383_a_b_b @ A2 @ ( insert8720088738084019632_a_b_b @ B @ A ) )
= ( ( A2 = B )
| ( member4695696432722591383_a_b_b @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_171_insert__iff,axiom,
! [A2: list_a,B: list_a,A: set_list_a] :
( ( member_list_a @ A2 @ ( insert_list_a @ B @ A ) )
= ( ( A2 = B )
| ( member_list_a @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_172_insert__iff,axiom,
! [A2: a,B: a,A: set_a] :
( ( member_a @ A2 @ ( insert_a @ B @ A ) )
= ( ( A2 = B )
| ( member_a @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_173_insert__iff,axiom,
! [A2: dtree_list_a_b,B: dtree_list_a_b,A: set_dtree_list_a_b] :
( ( member551035911493665803st_a_b @ A2 @ ( insert7065945238938772516st_a_b @ B @ A ) )
= ( ( A2 = B )
| ( member551035911493665803st_a_b @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_174_insertCI,axiom,
! [A2: b,B5: set_b,B: b] :
( ( ~ ( member_b @ A2 @ B5 )
=> ( A2 = B ) )
=> ( member_b @ A2 @ ( insert_b @ B @ B5 ) ) ) ).
% insertCI
thf(fact_175_insertCI,axiom,
! [A2: produc6499617310964463488_a_b_b,B5: set_Pr3443975907877334966_a_b_b,B: produc6499617310964463488_a_b_b] :
( ( ~ ( member4695696432722591383_a_b_b @ A2 @ B5 )
=> ( A2 = B ) )
=> ( member4695696432722591383_a_b_b @ A2 @ ( insert8720088738084019632_a_b_b @ B @ B5 ) ) ) ).
% insertCI
thf(fact_176_insertCI,axiom,
! [A2: list_a,B5: set_list_a,B: list_a] :
( ( ~ ( member_list_a @ A2 @ B5 )
=> ( A2 = B ) )
=> ( member_list_a @ A2 @ ( insert_list_a @ B @ B5 ) ) ) ).
% insertCI
thf(fact_177_insertCI,axiom,
! [A2: a,B5: set_a,B: a] :
( ( ~ ( member_a @ A2 @ B5 )
=> ( A2 = B ) )
=> ( member_a @ A2 @ ( insert_a @ B @ B5 ) ) ) ).
% insertCI
thf(fact_178_insertCI,axiom,
! [A2: dtree_list_a_b,B5: set_dtree_list_a_b,B: dtree_list_a_b] :
( ( ~ ( member551035911493665803st_a_b @ A2 @ B5 )
=> ( A2 = B ) )
=> ( member551035911493665803st_a_b @ A2 @ ( insert7065945238938772516st_a_b @ B @ B5 ) ) ) ).
% insertCI
thf(fact_179_subset__antisym,axiom,
! [A: set_b,B5: set_b] :
( ( ord_less_eq_set_b @ A @ B5 )
=> ( ( ord_less_eq_set_b @ B5 @ A )
=> ( A = B5 ) ) ) ).
% subset_antisym
thf(fact_180_subsetI,axiom,
! [A: set_Pr3443975907877334966_a_b_b,B5: set_Pr3443975907877334966_a_b_b] :
( ! [X3: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ X3 @ A )
=> ( member4695696432722591383_a_b_b @ X3 @ B5 ) )
=> ( ord_le1619362961161175062_a_b_b @ A @ B5 ) ) ).
% subsetI
thf(fact_181_subsetI,axiom,
! [A: set_list_a,B5: set_list_a] :
( ! [X3: list_a] :
( ( member_list_a @ X3 @ A )
=> ( member_list_a @ X3 @ B5 ) )
=> ( ord_le8861187494160871172list_a @ A @ B5 ) ) ).
% subsetI
thf(fact_182_subsetI,axiom,
! [A: set_a,B5: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( member_a @ X3 @ B5 ) )
=> ( ord_less_eq_set_a @ A @ B5 ) ) ).
% subsetI
thf(fact_183_subsetI,axiom,
! [A: set_dtree_list_a_b,B5: set_dtree_list_a_b] :
( ! [X3: dtree_list_a_b] :
( ( member551035911493665803st_a_b @ X3 @ A )
=> ( member551035911493665803st_a_b @ X3 @ B5 ) )
=> ( ord_le7599451563663638410st_a_b @ A @ B5 ) ) ).
% subsetI
thf(fact_184_subsetI,axiom,
! [A: set_b,B5: set_b] :
( ! [X3: b] :
( ( member_b @ X3 @ A )
=> ( member_b @ X3 @ B5 ) )
=> ( ord_less_eq_set_b @ A @ B5 ) ) ).
% subsetI
thf(fact_185_Int__iff,axiom,
! [C: produc6499617310964463488_a_b_b,A: set_Pr3443975907877334966_a_b_b,B5: set_Pr3443975907877334966_a_b_b] :
( ( member4695696432722591383_a_b_b @ C @ ( inf_in7962574851115124040_a_b_b @ A @ B5 ) )
= ( ( member4695696432722591383_a_b_b @ C @ A )
& ( member4695696432722591383_a_b_b @ C @ B5 ) ) ) ).
% Int_iff
thf(fact_186_Int__iff,axiom,
! [C: dtree_list_a_b,A: set_dtree_list_a_b,B5: set_dtree_list_a_b] :
( ( member551035911493665803st_a_b @ C @ ( inf_in3355993651213403836st_a_b @ A @ B5 ) )
= ( ( member551035911493665803st_a_b @ C @ A )
& ( member551035911493665803st_a_b @ C @ B5 ) ) ) ).
% Int_iff
thf(fact_187_Int__iff,axiom,
! [C: b,A: set_b,B5: set_b] :
( ( member_b @ C @ ( inf_inf_set_b @ A @ B5 ) )
= ( ( member_b @ C @ A )
& ( member_b @ C @ B5 ) ) ) ).
% Int_iff
thf(fact_188_Int__iff,axiom,
! [C: a,A: set_a,B5: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B5 ) )
= ( ( member_a @ C @ A )
& ( member_a @ C @ B5 ) ) ) ).
% Int_iff
thf(fact_189_Int__iff,axiom,
! [C: list_a,A: set_list_a,B5: set_list_a] :
( ( member_list_a @ C @ ( inf_inf_set_list_a @ A @ B5 ) )
= ( ( member_list_a @ C @ A )
& ( member_list_a @ C @ B5 ) ) ) ).
% Int_iff
thf(fact_190_IntI,axiom,
! [C: produc6499617310964463488_a_b_b,A: set_Pr3443975907877334966_a_b_b,B5: set_Pr3443975907877334966_a_b_b] :
( ( member4695696432722591383_a_b_b @ C @ A )
=> ( ( member4695696432722591383_a_b_b @ C @ B5 )
=> ( member4695696432722591383_a_b_b @ C @ ( inf_in7962574851115124040_a_b_b @ A @ B5 ) ) ) ) ).
% IntI
thf(fact_191_IntI,axiom,
! [C: dtree_list_a_b,A: set_dtree_list_a_b,B5: set_dtree_list_a_b] :
( ( member551035911493665803st_a_b @ C @ A )
=> ( ( member551035911493665803st_a_b @ C @ B5 )
=> ( member551035911493665803st_a_b @ C @ ( inf_in3355993651213403836st_a_b @ A @ B5 ) ) ) ) ).
% IntI
thf(fact_192_IntI,axiom,
! [C: b,A: set_b,B5: set_b] :
( ( member_b @ C @ A )
=> ( ( member_b @ C @ B5 )
=> ( member_b @ C @ ( inf_inf_set_b @ A @ B5 ) ) ) ) ).
% IntI
thf(fact_193_IntI,axiom,
! [C: a,A: set_a,B5: set_a] :
( ( member_a @ C @ A )
=> ( ( member_a @ C @ B5 )
=> ( member_a @ C @ ( inf_inf_set_a @ A @ B5 ) ) ) ) ).
% IntI
thf(fact_194_IntI,axiom,
! [C: list_a,A: set_list_a,B5: set_list_a] :
( ( member_list_a @ C @ A )
=> ( ( member_list_a @ C @ B5 )
=> ( member_list_a @ C @ ( inf_inf_set_list_a @ A @ B5 ) ) ) ) ).
% IntI
thf(fact_195_Un__iff,axiom,
! [C: produc6499617310964463488_a_b_b,A: set_Pr3443975907877334966_a_b_b,B5: set_Pr3443975907877334966_a_b_b] :
( ( member4695696432722591383_a_b_b @ C @ ( sup_su6988119916188116706_a_b_b @ A @ B5 ) )
= ( ( member4695696432722591383_a_b_b @ C @ A )
| ( member4695696432722591383_a_b_b @ C @ B5 ) ) ) ).
% Un_iff
thf(fact_196_Un__iff,axiom,
! [C: list_a,A: set_list_a,B5: set_list_a] :
( ( member_list_a @ C @ ( sup_sup_set_list_a @ A @ B5 ) )
= ( ( member_list_a @ C @ A )
| ( member_list_a @ C @ B5 ) ) ) ).
% Un_iff
thf(fact_197_Un__iff,axiom,
! [C: a,A: set_a,B5: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A @ B5 ) )
= ( ( member_a @ C @ A )
| ( member_a @ C @ B5 ) ) ) ).
% Un_iff
thf(fact_198_Un__iff,axiom,
! [C: dtree_list_a_b,A: set_dtree_list_a_b,B5: set_dtree_list_a_b] :
( ( member551035911493665803st_a_b @ C @ ( sup_su5199045856643764822st_a_b @ A @ B5 ) )
= ( ( member551035911493665803st_a_b @ C @ A )
| ( member551035911493665803st_a_b @ C @ B5 ) ) ) ).
% Un_iff
thf(fact_199_Un__iff,axiom,
! [C: b,A: set_b,B5: set_b] :
( ( member_b @ C @ ( sup_sup_set_b @ A @ B5 ) )
= ( ( member_b @ C @ A )
| ( member_b @ C @ B5 ) ) ) ).
% Un_iff
thf(fact_200_UnCI,axiom,
! [C: produc6499617310964463488_a_b_b,B5: set_Pr3443975907877334966_a_b_b,A: set_Pr3443975907877334966_a_b_b] :
( ( ~ ( member4695696432722591383_a_b_b @ C @ B5 )
=> ( member4695696432722591383_a_b_b @ C @ A ) )
=> ( member4695696432722591383_a_b_b @ C @ ( sup_su6988119916188116706_a_b_b @ A @ B5 ) ) ) ).
% UnCI
thf(fact_201_UnCI,axiom,
! [C: list_a,B5: set_list_a,A: set_list_a] :
( ( ~ ( member_list_a @ C @ B5 )
=> ( member_list_a @ C @ A ) )
=> ( member_list_a @ C @ ( sup_sup_set_list_a @ A @ B5 ) ) ) ).
% UnCI
thf(fact_202_UnCI,axiom,
! [C: a,B5: set_a,A: set_a] :
( ( ~ ( member_a @ C @ B5 )
=> ( member_a @ C @ A ) )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B5 ) ) ) ).
% UnCI
thf(fact_203_UnCI,axiom,
! [C: dtree_list_a_b,B5: set_dtree_list_a_b,A: set_dtree_list_a_b] :
( ( ~ ( member551035911493665803st_a_b @ C @ B5 )
=> ( member551035911493665803st_a_b @ C @ A ) )
=> ( member551035911493665803st_a_b @ C @ ( sup_su5199045856643764822st_a_b @ A @ B5 ) ) ) ).
% UnCI
thf(fact_204_UnCI,axiom,
! [C: b,B5: set_b,A: set_b] :
( ( ~ ( member_b @ C @ B5 )
=> ( member_b @ C @ A ) )
=> ( member_b @ C @ ( sup_sup_set_b @ A @ B5 ) ) ) ).
% UnCI
thf(fact_205_mem__case__prodI,axiom,
! [Z2: b,C: dtree_list_a_b > b > set_b,A2: dtree_list_a_b,B: b] :
( ( member_b @ Z2 @ ( C @ A2 @ B ) )
=> ( member_b @ Z2 @ ( produc6416049939707734976_set_b @ C @ ( produc7704165765595008946_a_b_b @ A2 @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_206_mem__case__prodI,axiom,
! [Z2: produc6499617310964463488_a_b_b,C: dtree_list_a_b > b > set_Pr3443975907877334966_a_b_b,A2: dtree_list_a_b,B: b] :
( ( member4695696432722591383_a_b_b @ Z2 @ ( C @ A2 @ B ) )
=> ( member4695696432722591383_a_b_b @ Z2 @ ( produc7845617512930307093_a_b_b @ C @ ( produc7704165765595008946_a_b_b @ A2 @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_207_mem__case__prodI,axiom,
! [Z2: list_a,C: dtree_list_a_b > b > set_list_a,A2: dtree_list_a_b,B: b] :
( ( member_list_a @ Z2 @ ( C @ A2 @ B ) )
=> ( member_list_a @ Z2 @ ( produc7409682806061929157list_a @ C @ ( produc7704165765595008946_a_b_b @ A2 @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_208_mem__case__prodI,axiom,
! [Z2: a,C: dtree_list_a_b > b > set_a,A2: dtree_list_a_b,B: b] :
( ( member_a @ Z2 @ ( C @ A2 @ B ) )
=> ( member_a @ Z2 @ ( produc6416049935404506175_set_a @ C @ ( produc7704165765595008946_a_b_b @ A2 @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_209_mem__case__prodI,axiom,
! [Z2: dtree_list_a_b,C: dtree_list_a_b > b > set_dtree_list_a_b,A2: dtree_list_a_b,B: b] :
( ( member551035911493665803st_a_b @ Z2 @ ( C @ A2 @ B ) )
=> ( member551035911493665803st_a_b @ Z2 @ ( produc4962517376730874505st_a_b @ C @ ( produc7704165765595008946_a_b_b @ A2 @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_210_mem__case__prodI,axiom,
! [Z2: b,C: list_a > b > set_b,A2: list_a,B: b] :
( ( member_b @ Z2 @ ( C @ A2 @ B ) )
=> ( member_b @ Z2 @ ( produc5826231656185642360_set_b @ C @ ( produc4781227316648555538st_a_b @ A2 @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_211_mem__case__prodI,axiom,
! [Z2: produc6499617310964463488_a_b_b,C: list_a > b > set_Pr3443975907877334966_a_b_b,A2: list_a,B: b] :
( ( member4695696432722591383_a_b_b @ Z2 @ ( C @ A2 @ B ) )
=> ( member4695696432722591383_a_b_b @ Z2 @ ( produc4981227583692318813_a_b_b @ C @ ( produc4781227316648555538st_a_b @ A2 @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_212_mem__case__prodI,axiom,
! [Z2: list_a,C: list_a > b > set_list_a,A2: list_a,B: b] :
( ( member_list_a @ Z2 @ ( C @ A2 @ B ) )
=> ( member_list_a @ Z2 @ ( produc621021178981935229list_a @ C @ ( produc4781227316648555538st_a_b @ A2 @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_213_mem__case__prodI,axiom,
! [Z2: a,C: list_a > b > set_a,A2: list_a,B: b] :
( ( member_a @ Z2 @ ( C @ A2 @ B ) )
=> ( member_a @ Z2 @ ( produc5826231651882413559_set_a @ C @ ( produc4781227316648555538st_a_b @ A2 @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_214_mem__case__prodI,axiom,
! [Z2: dtree_list_a_b,C: list_a > b > set_dtree_list_a_b,A2: list_a,B: b] :
( ( member551035911493665803st_a_b @ Z2 @ ( C @ A2 @ B ) )
=> ( member551035911493665803st_a_b @ Z2 @ ( produc3410578764579669713st_a_b @ C @ ( produc4781227316648555538st_a_b @ A2 @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_215_mem__case__prodI2,axiom,
! [P2: produc6499617310964463488_a_b_b,Z2: b,C: dtree_list_a_b > b > set_b] :
( ! [A5: dtree_list_a_b,B4: b] :
( ( P2
= ( produc7704165765595008946_a_b_b @ A5 @ B4 ) )
=> ( member_b @ Z2 @ ( C @ A5 @ B4 ) ) )
=> ( member_b @ Z2 @ ( produc6416049939707734976_set_b @ C @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_216_mem__case__prodI2,axiom,
! [P2: produc6499617310964463488_a_b_b,Z2: produc6499617310964463488_a_b_b,C: dtree_list_a_b > b > set_Pr3443975907877334966_a_b_b] :
( ! [A5: dtree_list_a_b,B4: b] :
( ( P2
= ( produc7704165765595008946_a_b_b @ A5 @ B4 ) )
=> ( member4695696432722591383_a_b_b @ Z2 @ ( C @ A5 @ B4 ) ) )
=> ( member4695696432722591383_a_b_b @ Z2 @ ( produc7845617512930307093_a_b_b @ C @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_217_mem__case__prodI2,axiom,
! [P2: produc6499617310964463488_a_b_b,Z2: list_a,C: dtree_list_a_b > b > set_list_a] :
( ! [A5: dtree_list_a_b,B4: b] :
( ( P2
= ( produc7704165765595008946_a_b_b @ A5 @ B4 ) )
=> ( member_list_a @ Z2 @ ( C @ A5 @ B4 ) ) )
=> ( member_list_a @ Z2 @ ( produc7409682806061929157list_a @ C @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_218_mem__case__prodI2,axiom,
! [P2: produc6499617310964463488_a_b_b,Z2: a,C: dtree_list_a_b > b > set_a] :
( ! [A5: dtree_list_a_b,B4: b] :
( ( P2
= ( produc7704165765595008946_a_b_b @ A5 @ B4 ) )
=> ( member_a @ Z2 @ ( C @ A5 @ B4 ) ) )
=> ( member_a @ Z2 @ ( produc6416049935404506175_set_a @ C @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_219_mem__case__prodI2,axiom,
! [P2: produc6499617310964463488_a_b_b,Z2: dtree_list_a_b,C: dtree_list_a_b > b > set_dtree_list_a_b] :
( ! [A5: dtree_list_a_b,B4: b] :
( ( P2
= ( produc7704165765595008946_a_b_b @ A5 @ B4 ) )
=> ( member551035911493665803st_a_b @ Z2 @ ( C @ A5 @ B4 ) ) )
=> ( member551035911493665803st_a_b @ Z2 @ ( produc4962517376730874505st_a_b @ C @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_220_mem__case__prodI2,axiom,
! [P2: produc2579390649552321826st_a_b,Z2: b,C: list_a > b > set_b] :
( ! [A5: list_a,B4: b] :
( ( P2
= ( produc4781227316648555538st_a_b @ A5 @ B4 ) )
=> ( member_b @ Z2 @ ( C @ A5 @ B4 ) ) )
=> ( member_b @ Z2 @ ( produc5826231656185642360_set_b @ C @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_221_mem__case__prodI2,axiom,
! [P2: produc2579390649552321826st_a_b,Z2: produc6499617310964463488_a_b_b,C: list_a > b > set_Pr3443975907877334966_a_b_b] :
( ! [A5: list_a,B4: b] :
( ( P2
= ( produc4781227316648555538st_a_b @ A5 @ B4 ) )
=> ( member4695696432722591383_a_b_b @ Z2 @ ( C @ A5 @ B4 ) ) )
=> ( member4695696432722591383_a_b_b @ Z2 @ ( produc4981227583692318813_a_b_b @ C @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_222_mem__case__prodI2,axiom,
! [P2: produc2579390649552321826st_a_b,Z2: list_a,C: list_a > b > set_list_a] :
( ! [A5: list_a,B4: b] :
( ( P2
= ( produc4781227316648555538st_a_b @ A5 @ B4 ) )
=> ( member_list_a @ Z2 @ ( C @ A5 @ B4 ) ) )
=> ( member_list_a @ Z2 @ ( produc621021178981935229list_a @ C @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_223_mem__case__prodI2,axiom,
! [P2: produc2579390649552321826st_a_b,Z2: a,C: list_a > b > set_a] :
( ! [A5: list_a,B4: b] :
( ( P2
= ( produc4781227316648555538st_a_b @ A5 @ B4 ) )
=> ( member_a @ Z2 @ ( C @ A5 @ B4 ) ) )
=> ( member_a @ Z2 @ ( produc5826231651882413559_set_a @ C @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_224_mem__case__prodI2,axiom,
! [P2: produc2579390649552321826st_a_b,Z2: dtree_list_a_b,C: list_a > b > set_dtree_list_a_b] :
( ! [A5: list_a,B4: b] :
( ( P2
= ( produc4781227316648555538st_a_b @ A5 @ B4 ) )
=> ( member551035911493665803st_a_b @ Z2 @ ( C @ A5 @ B4 ) ) )
=> ( member551035911493665803st_a_b @ Z2 @ ( produc3410578764579669713st_a_b @ C @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_225_singletonI,axiom,
! [A2: produc6499617310964463488_a_b_b] : ( member4695696432722591383_a_b_b @ A2 @ ( insert8720088738084019632_a_b_b @ A2 @ bot_bo4897374000430069834_a_b_b ) ) ).
% singletonI
thf(fact_226_singletonI,axiom,
! [A2: dtree_list_a_b] : ( member551035911493665803st_a_b @ A2 @ ( insert7065945238938772516st_a_b @ A2 @ bot_bo798015271861357502st_a_b ) ) ).
% singletonI
thf(fact_227_singletonI,axiom,
! [A2: b] : ( member_b @ A2 @ ( insert_b @ A2 @ bot_bot_set_b ) ) ).
% singletonI
thf(fact_228_singletonI,axiom,
! [A2: a] : ( member_a @ A2 @ ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_229_singletonI,axiom,
! [A2: list_a] : ( member_list_a @ A2 @ ( insert_list_a @ A2 @ bot_bot_set_list_a ) ) ).
% singletonI
thf(fact_230_empty__subsetI,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% empty_subsetI
thf(fact_231_empty__subsetI,axiom,
! [A: set_list_a] : ( ord_le8861187494160871172list_a @ bot_bot_set_list_a @ A ) ).
% empty_subsetI
thf(fact_232_empty__subsetI,axiom,
! [A: set_b] : ( ord_less_eq_set_b @ bot_bot_set_b @ A ) ).
% empty_subsetI
thf(fact_233_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_234_subset__empty,axiom,
! [A: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ bot_bot_set_list_a )
= ( A = bot_bot_set_list_a ) ) ).
% subset_empty
thf(fact_235_subset__empty,axiom,
! [A: set_b] :
( ( ord_less_eq_set_b @ A @ bot_bot_set_b )
= ( A = bot_bot_set_b ) ) ).
% subset_empty
thf(fact_236_ball__empty,axiom,
! [P: b > $o,X2: b] :
( ( member_b @ X2 @ bot_bot_set_b )
=> ( P @ X2 ) ) ).
% ball_empty
thf(fact_237_ball__empty,axiom,
! [P: a > $o,X2: a] :
( ( member_a @ X2 @ bot_bot_set_a )
=> ( P @ X2 ) ) ).
% ball_empty
thf(fact_238_ball__empty,axiom,
! [P: list_a > $o,X2: list_a] :
( ( member_list_a @ X2 @ bot_bot_set_list_a )
=> ( P @ X2 ) ) ).
% ball_empty
thf(fact_239_insert__subset,axiom,
! [X: produc6499617310964463488_a_b_b,A: set_Pr3443975907877334966_a_b_b,B5: set_Pr3443975907877334966_a_b_b] :
( ( ord_le1619362961161175062_a_b_b @ ( insert8720088738084019632_a_b_b @ X @ A ) @ B5 )
= ( ( member4695696432722591383_a_b_b @ X @ B5 )
& ( ord_le1619362961161175062_a_b_b @ A @ B5 ) ) ) ).
% insert_subset
thf(fact_240_insert__subset,axiom,
! [X: list_a,A: set_list_a,B5: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( insert_list_a @ X @ A ) @ B5 )
= ( ( member_list_a @ X @ B5 )
& ( ord_le8861187494160871172list_a @ A @ B5 ) ) ) ).
% insert_subset
thf(fact_241_insert__subset,axiom,
! [X: a,A: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A ) @ B5 )
= ( ( member_a @ X @ B5 )
& ( ord_less_eq_set_a @ A @ B5 ) ) ) ).
% insert_subset
thf(fact_242_insert__subset,axiom,
! [X: dtree_list_a_b,A: set_dtree_list_a_b,B5: set_dtree_list_a_b] :
( ( ord_le7599451563663638410st_a_b @ ( insert7065945238938772516st_a_b @ X @ A ) @ B5 )
= ( ( member551035911493665803st_a_b @ X @ B5 )
& ( ord_le7599451563663638410st_a_b @ A @ B5 ) ) ) ).
% insert_subset
thf(fact_243_insert__subset,axiom,
! [X: b,A: set_b,B5: set_b] :
( ( ord_less_eq_set_b @ ( insert_b @ X @ A ) @ B5 )
= ( ( member_b @ X @ B5 )
& ( ord_less_eq_set_b @ A @ B5 ) ) ) ).
% insert_subset
thf(fact_244_Un__empty,axiom,
! [A: set_b,B5: set_b] :
( ( ( sup_sup_set_b @ A @ B5 )
= bot_bot_set_b )
= ( ( A = bot_bot_set_b )
& ( B5 = bot_bot_set_b ) ) ) ).
% Un_empty
thf(fact_245_Un__empty,axiom,
! [A: set_a,B5: set_a] :
( ( ( sup_sup_set_a @ A @ B5 )
= bot_bot_set_a )
= ( ( A = bot_bot_set_a )
& ( B5 = bot_bot_set_a ) ) ) ).
% Un_empty
thf(fact_246_Un__empty,axiom,
! [A: set_list_a,B5: set_list_a] :
( ( ( sup_sup_set_list_a @ A @ B5 )
= bot_bot_set_list_a )
= ( ( A = bot_bot_set_list_a )
& ( B5 = bot_bot_set_list_a ) ) ) ).
% Un_empty
thf(fact_247_Int__insert__right__if1,axiom,
! [A2: produc6499617310964463488_a_b_b,A: set_Pr3443975907877334966_a_b_b,B5: set_Pr3443975907877334966_a_b_b] :
( ( member4695696432722591383_a_b_b @ A2 @ A )
=> ( ( inf_in7962574851115124040_a_b_b @ A @ ( insert8720088738084019632_a_b_b @ A2 @ B5 ) )
= ( insert8720088738084019632_a_b_b @ A2 @ ( inf_in7962574851115124040_a_b_b @ A @ B5 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_248_Int__insert__right__if1,axiom,
! [A2: dtree_list_a_b,A: set_dtree_list_a_b,B5: set_dtree_list_a_b] :
( ( member551035911493665803st_a_b @ A2 @ A )
=> ( ( inf_in3355993651213403836st_a_b @ A @ ( insert7065945238938772516st_a_b @ A2 @ B5 ) )
= ( insert7065945238938772516st_a_b @ A2 @ ( inf_in3355993651213403836st_a_b @ A @ B5 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_249_Int__insert__right__if1,axiom,
! [A2: b,A: set_b,B5: set_b] :
( ( member_b @ A2 @ A )
=> ( ( inf_inf_set_b @ A @ ( insert_b @ A2 @ B5 ) )
= ( insert_b @ A2 @ ( inf_inf_set_b @ A @ B5 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_250_Int__insert__right__if1,axiom,
! [A2: a,A: set_a,B5: set_a] :
( ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B5 ) )
= ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B5 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_251_Int__insert__right__if1,axiom,
! [A2: list_a,A: set_list_a,B5: set_list_a] :
( ( member_list_a @ A2 @ A )
=> ( ( inf_inf_set_list_a @ A @ ( insert_list_a @ A2 @ B5 ) )
= ( insert_list_a @ A2 @ ( inf_inf_set_list_a @ A @ B5 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_252_Int__insert__right__if0,axiom,
! [A2: produc6499617310964463488_a_b_b,A: set_Pr3443975907877334966_a_b_b,B5: set_Pr3443975907877334966_a_b_b] :
( ~ ( member4695696432722591383_a_b_b @ A2 @ A )
=> ( ( inf_in7962574851115124040_a_b_b @ A @ ( insert8720088738084019632_a_b_b @ A2 @ B5 ) )
= ( inf_in7962574851115124040_a_b_b @ A @ B5 ) ) ) ).
% Int_insert_right_if0
thf(fact_253_Int__insert__right__if0,axiom,
! [A2: dtree_list_a_b,A: set_dtree_list_a_b,B5: set_dtree_list_a_b] :
( ~ ( member551035911493665803st_a_b @ A2 @ A )
=> ( ( inf_in3355993651213403836st_a_b @ A @ ( insert7065945238938772516st_a_b @ A2 @ B5 ) )
= ( inf_in3355993651213403836st_a_b @ A @ B5 ) ) ) ).
% Int_insert_right_if0
thf(fact_254_Int__insert__right__if0,axiom,
! [A2: b,A: set_b,B5: set_b] :
( ~ ( member_b @ A2 @ A )
=> ( ( inf_inf_set_b @ A @ ( insert_b @ A2 @ B5 ) )
= ( inf_inf_set_b @ A @ B5 ) ) ) ).
% Int_insert_right_if0
thf(fact_255_Int__insert__right__if0,axiom,
! [A2: a,A: set_a,B5: set_a] :
( ~ ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B5 ) )
= ( inf_inf_set_a @ A @ B5 ) ) ) ).
% Int_insert_right_if0
thf(fact_256_Int__insert__right__if0,axiom,
! [A2: list_a,A: set_list_a,B5: set_list_a] :
( ~ ( member_list_a @ A2 @ A )
=> ( ( inf_inf_set_list_a @ A @ ( insert_list_a @ A2 @ B5 ) )
= ( inf_inf_set_list_a @ A @ B5 ) ) ) ).
% Int_insert_right_if0
thf(fact_257_insert__inter__insert,axiom,
! [A2: b,A: set_b,B5: set_b] :
( ( inf_inf_set_b @ ( insert_b @ A2 @ A ) @ ( insert_b @ A2 @ B5 ) )
= ( insert_b @ A2 @ ( inf_inf_set_b @ A @ B5 ) ) ) ).
% insert_inter_insert
thf(fact_258_insert__inter__insert,axiom,
! [A2: a,A: set_a,B5: set_a] :
( ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ ( insert_a @ A2 @ B5 ) )
= ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B5 ) ) ) ).
% insert_inter_insert
thf(fact_259_insert__inter__insert,axiom,
! [A2: list_a,A: set_list_a,B5: set_list_a] :
( ( inf_inf_set_list_a @ ( insert_list_a @ A2 @ A ) @ ( insert_list_a @ A2 @ B5 ) )
= ( insert_list_a @ A2 @ ( inf_inf_set_list_a @ A @ B5 ) ) ) ).
% insert_inter_insert
thf(fact_260_Int__insert__left__if1,axiom,
! [A2: produc6499617310964463488_a_b_b,C2: set_Pr3443975907877334966_a_b_b,B5: set_Pr3443975907877334966_a_b_b] :
( ( member4695696432722591383_a_b_b @ A2 @ C2 )
=> ( ( inf_in7962574851115124040_a_b_b @ ( insert8720088738084019632_a_b_b @ A2 @ B5 ) @ C2 )
= ( insert8720088738084019632_a_b_b @ A2 @ ( inf_in7962574851115124040_a_b_b @ B5 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_261_Int__insert__left__if1,axiom,
! [A2: dtree_list_a_b,C2: set_dtree_list_a_b,B5: set_dtree_list_a_b] :
( ( member551035911493665803st_a_b @ A2 @ C2 )
=> ( ( inf_in3355993651213403836st_a_b @ ( insert7065945238938772516st_a_b @ A2 @ B5 ) @ C2 )
= ( insert7065945238938772516st_a_b @ A2 @ ( inf_in3355993651213403836st_a_b @ B5 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_262_Int__insert__left__if1,axiom,
! [A2: b,C2: set_b,B5: set_b] :
( ( member_b @ A2 @ C2 )
=> ( ( inf_inf_set_b @ ( insert_b @ A2 @ B5 ) @ C2 )
= ( insert_b @ A2 @ ( inf_inf_set_b @ B5 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_263_Int__insert__left__if1,axiom,
! [A2: a,C2: set_a,B5: set_a] :
( ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B5 ) @ C2 )
= ( insert_a @ A2 @ ( inf_inf_set_a @ B5 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_264_Int__insert__left__if1,axiom,
! [A2: list_a,C2: set_list_a,B5: set_list_a] :
( ( member_list_a @ A2 @ C2 )
=> ( ( inf_inf_set_list_a @ ( insert_list_a @ A2 @ B5 ) @ C2 )
= ( insert_list_a @ A2 @ ( inf_inf_set_list_a @ B5 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_265_Int__insert__left__if0,axiom,
! [A2: produc6499617310964463488_a_b_b,C2: set_Pr3443975907877334966_a_b_b,B5: set_Pr3443975907877334966_a_b_b] :
( ~ ( member4695696432722591383_a_b_b @ A2 @ C2 )
=> ( ( inf_in7962574851115124040_a_b_b @ ( insert8720088738084019632_a_b_b @ A2 @ B5 ) @ C2 )
= ( inf_in7962574851115124040_a_b_b @ B5 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_266_Int__insert__left__if0,axiom,
! [A2: dtree_list_a_b,C2: set_dtree_list_a_b,B5: set_dtree_list_a_b] :
( ~ ( member551035911493665803st_a_b @ A2 @ C2 )
=> ( ( inf_in3355993651213403836st_a_b @ ( insert7065945238938772516st_a_b @ A2 @ B5 ) @ C2 )
= ( inf_in3355993651213403836st_a_b @ B5 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_267_Int__insert__left__if0,axiom,
! [A2: b,C2: set_b,B5: set_b] :
( ~ ( member_b @ A2 @ C2 )
=> ( ( inf_inf_set_b @ ( insert_b @ A2 @ B5 ) @ C2 )
= ( inf_inf_set_b @ B5 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_268_Int__insert__left__if0,axiom,
! [A2: a,C2: set_a,B5: set_a] :
( ~ ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B5 ) @ C2 )
= ( inf_inf_set_a @ B5 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_269_Int__insert__left__if0,axiom,
! [A2: list_a,C2: set_list_a,B5: set_list_a] :
( ~ ( member_list_a @ A2 @ C2 )
=> ( ( inf_inf_set_list_a @ ( insert_list_a @ A2 @ B5 ) @ C2 )
= ( inf_inf_set_list_a @ B5 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_270_Int__subset__iff,axiom,
! [C2: set_a,A: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A @ B5 ) )
= ( ( ord_less_eq_set_a @ C2 @ A )
& ( ord_less_eq_set_a @ C2 @ B5 ) ) ) ).
% Int_subset_iff
thf(fact_271_Int__subset__iff,axiom,
! [C2: set_list_a,A: set_list_a,B5: set_list_a] :
( ( ord_le8861187494160871172list_a @ C2 @ ( inf_inf_set_list_a @ A @ B5 ) )
= ( ( ord_le8861187494160871172list_a @ C2 @ A )
& ( ord_le8861187494160871172list_a @ C2 @ B5 ) ) ) ).
% Int_subset_iff
thf(fact_272_Int__subset__iff,axiom,
! [C2: set_b,A: set_b,B5: set_b] :
( ( ord_less_eq_set_b @ C2 @ ( inf_inf_set_b @ A @ B5 ) )
= ( ( ord_less_eq_set_b @ C2 @ A )
& ( ord_less_eq_set_b @ C2 @ B5 ) ) ) ).
% Int_subset_iff
thf(fact_273_Un__insert__right,axiom,
! [A: set_list_a,A2: list_a,B5: set_list_a] :
( ( sup_sup_set_list_a @ A @ ( insert_list_a @ A2 @ B5 ) )
= ( insert_list_a @ A2 @ ( sup_sup_set_list_a @ A @ B5 ) ) ) ).
% Un_insert_right
thf(fact_274_Un__insert__right,axiom,
! [A: set_b,A2: b,B5: set_b] :
( ( sup_sup_set_b @ A @ ( insert_b @ A2 @ B5 ) )
= ( insert_b @ A2 @ ( sup_sup_set_b @ A @ B5 ) ) ) ).
% Un_insert_right
thf(fact_275_Un__insert__left,axiom,
! [A2: list_a,B5: set_list_a,C2: set_list_a] :
( ( sup_sup_set_list_a @ ( insert_list_a @ A2 @ B5 ) @ C2 )
= ( insert_list_a @ A2 @ ( sup_sup_set_list_a @ B5 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_276_Un__insert__left,axiom,
! [A2: b,B5: set_b,C2: set_b] :
( ( sup_sup_set_b @ ( insert_b @ A2 @ B5 ) @ C2 )
= ( insert_b @ A2 @ ( sup_sup_set_b @ B5 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_277_Un__subset__iff,axiom,
! [A: set_b,B5: set_b,C2: set_b] :
( ( ord_less_eq_set_b @ ( sup_sup_set_b @ A @ B5 ) @ C2 )
= ( ( ord_less_eq_set_b @ A @ C2 )
& ( ord_less_eq_set_b @ B5 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_278_Un__Int__eq_I1_J,axiom,
! [S: set_a,T4: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T4 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_279_Un__Int__eq_I1_J,axiom,
! [S: set_list_a,T4: set_list_a] :
( ( inf_inf_set_list_a @ ( sup_sup_set_list_a @ S @ T4 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_280_Un__Int__eq_I1_J,axiom,
! [S: set_b,T4: set_b] :
( ( inf_inf_set_b @ ( sup_sup_set_b @ S @ T4 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_281_Un__Int__eq_I2_J,axiom,
! [S: set_a,T4: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T4 ) @ T4 )
= T4 ) ).
% Un_Int_eq(2)
thf(fact_282_Un__Int__eq_I2_J,axiom,
! [S: set_list_a,T4: set_list_a] :
( ( inf_inf_set_list_a @ ( sup_sup_set_list_a @ S @ T4 ) @ T4 )
= T4 ) ).
% Un_Int_eq(2)
thf(fact_283_Un__Int__eq_I2_J,axiom,
! [S: set_b,T4: set_b] :
( ( inf_inf_set_b @ ( sup_sup_set_b @ S @ T4 ) @ T4 )
= T4 ) ).
% Un_Int_eq(2)
thf(fact_284_Un__Int__eq_I3_J,axiom,
! [S: set_a,T4: set_a] :
( ( inf_inf_set_a @ S @ ( sup_sup_set_a @ S @ T4 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_285_Un__Int__eq_I3_J,axiom,
! [S: set_list_a,T4: set_list_a] :
( ( inf_inf_set_list_a @ S @ ( sup_sup_set_list_a @ S @ T4 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_286_Un__Int__eq_I3_J,axiom,
! [S: set_b,T4: set_b] :
( ( inf_inf_set_b @ S @ ( sup_sup_set_b @ S @ T4 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_287_singleton__insert__inj__eq_H,axiom,
! [A2: a,A: set_a,B: a] :
( ( ( insert_a @ A2 @ A )
= ( insert_a @ B @ bot_bot_set_a ) )
= ( ( A2 = B )
& ( ord_less_eq_set_a @ A @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_288_singleton__insert__inj__eq_H,axiom,
! [A2: list_a,A: set_list_a,B: list_a] :
( ( ( insert_list_a @ A2 @ A )
= ( insert_list_a @ B @ bot_bot_set_list_a ) )
= ( ( A2 = B )
& ( ord_le8861187494160871172list_a @ A @ ( insert_list_a @ B @ bot_bot_set_list_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_289_singleton__insert__inj__eq_H,axiom,
! [A2: b,A: set_b,B: b] :
( ( ( insert_b @ A2 @ A )
= ( insert_b @ B @ bot_bot_set_b ) )
= ( ( A2 = B )
& ( ord_less_eq_set_b @ A @ ( insert_b @ B @ bot_bot_set_b ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_290_singleton__insert__inj__eq,axiom,
! [B: a,A2: a,A: set_a] :
( ( ( insert_a @ B @ bot_bot_set_a )
= ( insert_a @ A2 @ A ) )
= ( ( A2 = B )
& ( ord_less_eq_set_a @ A @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_291_singleton__insert__inj__eq,axiom,
! [B: list_a,A2: list_a,A: set_list_a] :
( ( ( insert_list_a @ B @ bot_bot_set_list_a )
= ( insert_list_a @ A2 @ A ) )
= ( ( A2 = B )
& ( ord_le8861187494160871172list_a @ A @ ( insert_list_a @ B @ bot_bot_set_list_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_292_singleton__insert__inj__eq,axiom,
! [B: b,A2: b,A: set_b] :
( ( ( insert_b @ B @ bot_bot_set_b )
= ( insert_b @ A2 @ A ) )
= ( ( A2 = B )
& ( ord_less_eq_set_b @ A @ ( insert_b @ B @ bot_bot_set_b ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_293_nle__le,axiom,
! [A2: real,B: real] :
( ( ~ ( ord_less_eq_real @ A2 @ B ) )
= ( ( ord_less_eq_real @ B @ A2 )
& ( B != A2 ) ) ) ).
% nle_le
thf(fact_294_le__cases3,axiom,
! [X: real,Y: real,Z2: real] :
( ( ( ord_less_eq_real @ X @ Y )
=> ~ ( ord_less_eq_real @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_eq_real @ X @ Z2 ) )
=> ( ( ( ord_less_eq_real @ X @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_real @ Z2 @ Y )
=> ~ ( ord_less_eq_real @ Y @ X ) )
=> ( ( ( ord_less_eq_real @ Y @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_real @ Z2 @ X )
=> ~ ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_295_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_b,Z3: set_b] : ( Y4 = Z3 ) )
= ( ^ [X4: set_b,Y2: set_b] :
( ( ord_less_eq_set_b @ X4 @ Y2 )
& ( ord_less_eq_set_b @ Y2 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_296_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: real,Z3: real] : ( Y4 = Z3 ) )
= ( ^ [X4: real,Y2: real] :
( ( ord_less_eq_real @ X4 @ Y2 )
& ( ord_less_eq_real @ Y2 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_297_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: fset_P2153231429829016240_a_b_b,Z3: fset_P2153231429829016240_a_b_b] : ( Y4 = Z3 ) )
= ( ^ [X4: fset_P2153231429829016240_a_b_b,Y2: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ X4 @ Y2 )
& ( ord_le8870638447146015504_a_b_b @ Y2 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_298_ord__eq__le__trans,axiom,
! [A2: set_b,B: set_b,C: set_b] :
( ( A2 = B )
=> ( ( ord_less_eq_set_b @ B @ C )
=> ( ord_less_eq_set_b @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_299_ord__eq__le__trans,axiom,
! [A2: real,B: real,C: real] :
( ( A2 = B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_300_ord__eq__le__trans,axiom,
! [A2: fset_P2153231429829016240_a_b_b,B: fset_P2153231429829016240_a_b_b,C: fset_P2153231429829016240_a_b_b] :
( ( A2 = B )
=> ( ( ord_le8870638447146015504_a_b_b @ B @ C )
=> ( ord_le8870638447146015504_a_b_b @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_301_ord__le__eq__trans,axiom,
! [A2: set_b,B: set_b,C: set_b] :
( ( ord_less_eq_set_b @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_b @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_302_ord__le__eq__trans,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_real @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_303_ord__le__eq__trans,axiom,
! [A2: fset_P2153231429829016240_a_b_b,B: fset_P2153231429829016240_a_b_b,C: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ A2 @ B )
=> ( ( B = C )
=> ( ord_le8870638447146015504_a_b_b @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_304_order__antisym,axiom,
! [X: set_b,Y: set_b] :
( ( ord_less_eq_set_b @ X @ Y )
=> ( ( ord_less_eq_set_b @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_305_order__antisym,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_306_order__antisym,axiom,
! [X: fset_P2153231429829016240_a_b_b,Y: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ X @ Y )
=> ( ( ord_le8870638447146015504_a_b_b @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_307_order_Otrans,axiom,
! [A2: set_b,B: set_b,C: set_b] :
( ( ord_less_eq_set_b @ A2 @ B )
=> ( ( ord_less_eq_set_b @ B @ C )
=> ( ord_less_eq_set_b @ A2 @ C ) ) ) ).
% order.trans
thf(fact_308_order_Otrans,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A2 @ C ) ) ) ).
% order.trans
thf(fact_309_order_Otrans,axiom,
! [A2: fset_P2153231429829016240_a_b_b,B: fset_P2153231429829016240_a_b_b,C: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ A2 @ B )
=> ( ( ord_le8870638447146015504_a_b_b @ B @ C )
=> ( ord_le8870638447146015504_a_b_b @ A2 @ C ) ) ) ).
% order.trans
thf(fact_310_order__trans,axiom,
! [X: set_b,Y: set_b,Z2: set_b] :
( ( ord_less_eq_set_b @ X @ Y )
=> ( ( ord_less_eq_set_b @ Y @ Z2 )
=> ( ord_less_eq_set_b @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_311_order__trans,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z2 )
=> ( ord_less_eq_real @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_312_order__trans,axiom,
! [X: fset_P2153231429829016240_a_b_b,Y: fset_P2153231429829016240_a_b_b,Z2: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ X @ Y )
=> ( ( ord_le8870638447146015504_a_b_b @ Y @ Z2 )
=> ( ord_le8870638447146015504_a_b_b @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_313_linorder__wlog,axiom,
! [P: real > real > $o,A2: real,B: real] :
( ! [A5: real,B4: real] :
( ( ord_less_eq_real @ A5 @ B4 )
=> ( P @ A5 @ B4 ) )
=> ( ! [A5: real,B4: real] :
( ( P @ B4 @ A5 )
=> ( P @ A5 @ B4 ) )
=> ( P @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_314_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_b,Z3: set_b] : ( Y4 = Z3 ) )
= ( ^ [A4: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ B3 @ A4 )
& ( ord_less_eq_set_b @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_315_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: real,Z3: real] : ( Y4 = Z3 ) )
= ( ^ [A4: real,B3: real] :
( ( ord_less_eq_real @ B3 @ A4 )
& ( ord_less_eq_real @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_316_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: fset_P2153231429829016240_a_b_b,Z3: fset_P2153231429829016240_a_b_b] : ( Y4 = Z3 ) )
= ( ^ [A4: fset_P2153231429829016240_a_b_b,B3: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ B3 @ A4 )
& ( ord_le8870638447146015504_a_b_b @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_317_dual__order_Oantisym,axiom,
! [B: set_b,A2: set_b] :
( ( ord_less_eq_set_b @ B @ A2 )
=> ( ( ord_less_eq_set_b @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_318_dual__order_Oantisym,axiom,
! [B: real,A2: real] :
( ( ord_less_eq_real @ B @ A2 )
=> ( ( ord_less_eq_real @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_319_dual__order_Oantisym,axiom,
! [B: fset_P2153231429829016240_a_b_b,A2: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ B @ A2 )
=> ( ( ord_le8870638447146015504_a_b_b @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_320_dual__order_Otrans,axiom,
! [B: set_b,A2: set_b,C: set_b] :
( ( ord_less_eq_set_b @ B @ A2 )
=> ( ( ord_less_eq_set_b @ C @ B )
=> ( ord_less_eq_set_b @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_321_dual__order_Otrans,axiom,
! [B: real,A2: real,C: real] :
( ( ord_less_eq_real @ B @ A2 )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_322_dual__order_Otrans,axiom,
! [B: fset_P2153231429829016240_a_b_b,A2: fset_P2153231429829016240_a_b_b,C: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ B @ A2 )
=> ( ( ord_le8870638447146015504_a_b_b @ C @ B )
=> ( ord_le8870638447146015504_a_b_b @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_323_antisym,axiom,
! [A2: set_b,B: set_b] :
( ( ord_less_eq_set_b @ A2 @ B )
=> ( ( ord_less_eq_set_b @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_324_antisym,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_325_antisym,axiom,
! [A2: fset_P2153231429829016240_a_b_b,B: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ A2 @ B )
=> ( ( ord_le8870638447146015504_a_b_b @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_326_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_b,Z3: set_b] : ( Y4 = Z3 ) )
= ( ^ [A4: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ A4 @ B3 )
& ( ord_less_eq_set_b @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_327_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: real,Z3: real] : ( Y4 = Z3 ) )
= ( ^ [A4: real,B3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
& ( ord_less_eq_real @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_328_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: fset_P2153231429829016240_a_b_b,Z3: fset_P2153231429829016240_a_b_b] : ( Y4 = Z3 ) )
= ( ^ [A4: fset_P2153231429829016240_a_b_b,B3: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ A4 @ B3 )
& ( ord_le8870638447146015504_a_b_b @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_329_order__subst1,axiom,
! [A2: set_b,F: set_b > set_b,B: set_b,C: set_b] :
( ( ord_less_eq_set_b @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_set_b @ B @ C )
=> ( ! [X3: set_b,Y3: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y3 )
=> ( ord_less_eq_set_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_b @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_330_order__subst1,axiom,
! [A2: set_b,F: real > set_b,B: real,C: real] :
( ( ord_less_eq_set_b @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_set_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_b @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_331_order__subst1,axiom,
! [A2: set_b,F: fset_P2153231429829016240_a_b_b > set_b,B: fset_P2153231429829016240_a_b_b,C: fset_P2153231429829016240_a_b_b] :
( ( ord_less_eq_set_b @ A2 @ ( F @ B ) )
=> ( ( ord_le8870638447146015504_a_b_b @ B @ C )
=> ( ! [X3: fset_P2153231429829016240_a_b_b,Y3: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ X3 @ Y3 )
=> ( ord_less_eq_set_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_b @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_332_order__subst1,axiom,
! [A2: real,F: set_b > real,B: set_b,C: set_b] :
( ( ord_less_eq_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_set_b @ B @ C )
=> ( ! [X3: set_b,Y3: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_333_order__subst1,axiom,
! [A2: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_334_order__subst1,axiom,
! [A2: real,F: fset_P2153231429829016240_a_b_b > real,B: fset_P2153231429829016240_a_b_b,C: fset_P2153231429829016240_a_b_b] :
( ( ord_less_eq_real @ A2 @ ( F @ B ) )
=> ( ( ord_le8870638447146015504_a_b_b @ B @ C )
=> ( ! [X3: fset_P2153231429829016240_a_b_b,Y3: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_335_order__subst1,axiom,
! [A2: fset_P2153231429829016240_a_b_b,F: set_b > fset_P2153231429829016240_a_b_b,B: set_b,C: set_b] :
( ( ord_le8870638447146015504_a_b_b @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_set_b @ B @ C )
=> ( ! [X3: set_b,Y3: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y3 )
=> ( ord_le8870638447146015504_a_b_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le8870638447146015504_a_b_b @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_336_order__subst1,axiom,
! [A2: fset_P2153231429829016240_a_b_b,F: real > fset_P2153231429829016240_a_b_b,B: real,C: real] :
( ( ord_le8870638447146015504_a_b_b @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_le8870638447146015504_a_b_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le8870638447146015504_a_b_b @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_337_order__subst1,axiom,
! [A2: fset_P2153231429829016240_a_b_b,F: fset_P2153231429829016240_a_b_b > fset_P2153231429829016240_a_b_b,B: fset_P2153231429829016240_a_b_b,C: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ A2 @ ( F @ B ) )
=> ( ( ord_le8870638447146015504_a_b_b @ B @ C )
=> ( ! [X3: fset_P2153231429829016240_a_b_b,Y3: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ X3 @ Y3 )
=> ( ord_le8870638447146015504_a_b_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le8870638447146015504_a_b_b @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_338_order__subst2,axiom,
! [A2: set_b,B: set_b,F: set_b > set_b,C: set_b] :
( ( ord_less_eq_set_b @ A2 @ B )
=> ( ( ord_less_eq_set_b @ ( F @ B ) @ C )
=> ( ! [X3: set_b,Y3: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y3 )
=> ( ord_less_eq_set_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_b @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_339_order__subst2,axiom,
! [A2: set_b,B: set_b,F: set_b > real,C: real] :
( ( ord_less_eq_set_b @ A2 @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: set_b,Y3: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_340_order__subst2,axiom,
! [A2: set_b,B: set_b,F: set_b > fset_P2153231429829016240_a_b_b,C: fset_P2153231429829016240_a_b_b] :
( ( ord_less_eq_set_b @ A2 @ B )
=> ( ( ord_le8870638447146015504_a_b_b @ ( F @ B ) @ C )
=> ( ! [X3: set_b,Y3: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y3 )
=> ( ord_le8870638447146015504_a_b_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le8870638447146015504_a_b_b @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_341_order__subst2,axiom,
! [A2: real,B: real,F: real > set_b,C: set_b] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_set_b @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_set_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_b @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_342_order__subst2,axiom,
! [A2: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_343_order__subst2,axiom,
! [A2: real,B: real,F: real > fset_P2153231429829016240_a_b_b,C: fset_P2153231429829016240_a_b_b] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_le8870638447146015504_a_b_b @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_le8870638447146015504_a_b_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le8870638447146015504_a_b_b @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_344_order__subst2,axiom,
! [A2: fset_P2153231429829016240_a_b_b,B: fset_P2153231429829016240_a_b_b,F: fset_P2153231429829016240_a_b_b > set_b,C: set_b] :
( ( ord_le8870638447146015504_a_b_b @ A2 @ B )
=> ( ( ord_less_eq_set_b @ ( F @ B ) @ C )
=> ( ! [X3: fset_P2153231429829016240_a_b_b,Y3: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ X3 @ Y3 )
=> ( ord_less_eq_set_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_b @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_345_order__subst2,axiom,
! [A2: fset_P2153231429829016240_a_b_b,B: fset_P2153231429829016240_a_b_b,F: fset_P2153231429829016240_a_b_b > real,C: real] :
( ( ord_le8870638447146015504_a_b_b @ A2 @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: fset_P2153231429829016240_a_b_b,Y3: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_346_order__subst2,axiom,
! [A2: fset_P2153231429829016240_a_b_b,B: fset_P2153231429829016240_a_b_b,F: fset_P2153231429829016240_a_b_b > fset_P2153231429829016240_a_b_b,C: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ A2 @ B )
=> ( ( ord_le8870638447146015504_a_b_b @ ( F @ B ) @ C )
=> ( ! [X3: fset_P2153231429829016240_a_b_b,Y3: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ X3 @ Y3 )
=> ( ord_le8870638447146015504_a_b_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le8870638447146015504_a_b_b @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_347_order__eq__refl,axiom,
! [X: set_b,Y: set_b] :
( ( X = Y )
=> ( ord_less_eq_set_b @ X @ Y ) ) ).
% order_eq_refl
thf(fact_348_order__eq__refl,axiom,
! [X: real,Y: real] :
( ( X = Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_eq_refl
thf(fact_349_order__eq__refl,axiom,
! [X: fset_P2153231429829016240_a_b_b,Y: fset_P2153231429829016240_a_b_b] :
( ( X = Y )
=> ( ord_le8870638447146015504_a_b_b @ X @ Y ) ) ).
% order_eq_refl
thf(fact_350_linorder__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_linear
thf(fact_351_ord__eq__le__subst,axiom,
! [A2: set_b,F: set_b > set_b,B: set_b,C: set_b] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_set_b @ B @ C )
=> ( ! [X3: set_b,Y3: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y3 )
=> ( ord_less_eq_set_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_b @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_352_ord__eq__le__subst,axiom,
! [A2: real,F: set_b > real,B: set_b,C: set_b] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_set_b @ B @ C )
=> ( ! [X3: set_b,Y3: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_353_ord__eq__le__subst,axiom,
! [A2: fset_P2153231429829016240_a_b_b,F: set_b > fset_P2153231429829016240_a_b_b,B: set_b,C: set_b] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_set_b @ B @ C )
=> ( ! [X3: set_b,Y3: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y3 )
=> ( ord_le8870638447146015504_a_b_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le8870638447146015504_a_b_b @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_354_ord__eq__le__subst,axiom,
! [A2: set_b,F: real > set_b,B: real,C: real] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_set_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_b @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_355_ord__eq__le__subst,axiom,
! [A2: real,F: real > real,B: real,C: real] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_356_ord__eq__le__subst,axiom,
! [A2: fset_P2153231429829016240_a_b_b,F: real > fset_P2153231429829016240_a_b_b,B: real,C: real] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_le8870638447146015504_a_b_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le8870638447146015504_a_b_b @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_357_ord__eq__le__subst,axiom,
! [A2: set_b,F: fset_P2153231429829016240_a_b_b > set_b,B: fset_P2153231429829016240_a_b_b,C: fset_P2153231429829016240_a_b_b] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le8870638447146015504_a_b_b @ B @ C )
=> ( ! [X3: fset_P2153231429829016240_a_b_b,Y3: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ X3 @ Y3 )
=> ( ord_less_eq_set_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_b @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_358_ord__eq__le__subst,axiom,
! [A2: real,F: fset_P2153231429829016240_a_b_b > real,B: fset_P2153231429829016240_a_b_b,C: fset_P2153231429829016240_a_b_b] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le8870638447146015504_a_b_b @ B @ C )
=> ( ! [X3: fset_P2153231429829016240_a_b_b,Y3: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_359_ord__eq__le__subst,axiom,
! [A2: fset_P2153231429829016240_a_b_b,F: fset_P2153231429829016240_a_b_b > fset_P2153231429829016240_a_b_b,B: fset_P2153231429829016240_a_b_b,C: fset_P2153231429829016240_a_b_b] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le8870638447146015504_a_b_b @ B @ C )
=> ( ! [X3: fset_P2153231429829016240_a_b_b,Y3: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ X3 @ Y3 )
=> ( ord_le8870638447146015504_a_b_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le8870638447146015504_a_b_b @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_360_ord__le__eq__subst,axiom,
! [A2: set_b,B: set_b,F: set_b > set_b,C: set_b] :
( ( ord_less_eq_set_b @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_b,Y3: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y3 )
=> ( ord_less_eq_set_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_b @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_361_ord__le__eq__subst,axiom,
! [A2: set_b,B: set_b,F: set_b > real,C: real] :
( ( ord_less_eq_set_b @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_b,Y3: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_362_ord__le__eq__subst,axiom,
! [A2: set_b,B: set_b,F: set_b > fset_P2153231429829016240_a_b_b,C: fset_P2153231429829016240_a_b_b] :
( ( ord_less_eq_set_b @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_b,Y3: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y3 )
=> ( ord_le8870638447146015504_a_b_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le8870638447146015504_a_b_b @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_363_ord__le__eq__subst,axiom,
! [A2: real,B: real,F: real > set_b,C: set_b] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_set_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_b @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_364_ord__le__eq__subst,axiom,
! [A2: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_365_ord__le__eq__subst,axiom,
! [A2: real,B: real,F: real > fset_P2153231429829016240_a_b_b,C: fset_P2153231429829016240_a_b_b] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_le8870638447146015504_a_b_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le8870638447146015504_a_b_b @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_366_ord__le__eq__subst,axiom,
! [A2: fset_P2153231429829016240_a_b_b,B: fset_P2153231429829016240_a_b_b,F: fset_P2153231429829016240_a_b_b > set_b,C: set_b] :
( ( ord_le8870638447146015504_a_b_b @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: fset_P2153231429829016240_a_b_b,Y3: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ X3 @ Y3 )
=> ( ord_less_eq_set_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_b @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_367_ord__le__eq__subst,axiom,
! [A2: fset_P2153231429829016240_a_b_b,B: fset_P2153231429829016240_a_b_b,F: fset_P2153231429829016240_a_b_b > real,C: real] :
( ( ord_le8870638447146015504_a_b_b @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: fset_P2153231429829016240_a_b_b,Y3: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_368_ord__le__eq__subst,axiom,
! [A2: fset_P2153231429829016240_a_b_b,B: fset_P2153231429829016240_a_b_b,F: fset_P2153231429829016240_a_b_b > fset_P2153231429829016240_a_b_b,C: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: fset_P2153231429829016240_a_b_b,Y3: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ X3 @ Y3 )
=> ( ord_le8870638447146015504_a_b_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le8870638447146015504_a_b_b @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_369_linorder__le__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_370_order__antisym__conv,axiom,
! [Y: set_b,X: set_b] :
( ( ord_less_eq_set_b @ Y @ X )
=> ( ( ord_less_eq_set_b @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_371_order__antisym__conv,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_372_order__antisym__conv,axiom,
! [Y: fset_P2153231429829016240_a_b_b,X: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ Y @ X )
=> ( ( ord_le8870638447146015504_a_b_b @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_373_Collect__mono__iff,axiom,
! [P: b > $o,Q: b > $o] :
( ( ord_less_eq_set_b @ ( collect_b @ P ) @ ( collect_b @ Q ) )
= ( ! [X4: b] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_374_Collect__subset,axiom,
! [A: set_Pr3443975907877334966_a_b_b,P: produc6499617310964463488_a_b_b > $o] :
( ord_le1619362961161175062_a_b_b
@ ( collec3241703154345475797_a_b_b
@ ^ [X4: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ X4 @ A )
& ( P @ X4 ) ) )
@ A ) ).
% Collect_subset
thf(fact_375_Collect__subset,axiom,
! [A: set_list_a,P: list_a > $o] :
( ord_le8861187494160871172list_a
@ ( collect_list_a
@ ^ [X4: list_a] :
( ( member_list_a @ X4 @ A )
& ( P @ X4 ) ) )
@ A ) ).
% Collect_subset
thf(fact_376_Collect__subset,axiom,
! [A: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ A )
& ( P @ X4 ) ) )
@ A ) ).
% Collect_subset
thf(fact_377_Collect__subset,axiom,
! [A: set_dtree_list_a_b,P: dtree_list_a_b > $o] :
( ord_le7599451563663638410st_a_b
@ ( collec2944820760411501129st_a_b
@ ^ [X4: dtree_list_a_b] :
( ( member551035911493665803st_a_b @ X4 @ A )
& ( P @ X4 ) ) )
@ A ) ).
% Collect_subset
thf(fact_378_Collect__subset,axiom,
! [A: set_b,P: b > $o] :
( ord_less_eq_set_b
@ ( collect_b
@ ^ [X4: b] :
( ( member_b @ X4 @ A )
& ( P @ X4 ) ) )
@ A ) ).
% Collect_subset
thf(fact_379_set__eq__subset,axiom,
( ( ^ [Y4: set_b,Z3: set_b] : ( Y4 = Z3 ) )
= ( ^ [A6: set_b,B6: set_b] :
( ( ord_less_eq_set_b @ A6 @ B6 )
& ( ord_less_eq_set_b @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_380_subset__trans,axiom,
! [A: set_b,B5: set_b,C2: set_b] :
( ( ord_less_eq_set_b @ A @ B5 )
=> ( ( ord_less_eq_set_b @ B5 @ C2 )
=> ( ord_less_eq_set_b @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_381_Collect__mono,axiom,
! [P: b > $o,Q: b > $o] :
( ! [X3: b] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_b @ ( collect_b @ P ) @ ( collect_b @ Q ) ) ) ).
% Collect_mono
thf(fact_382_subset__refl,axiom,
! [A: set_b] : ( ord_less_eq_set_b @ A @ A ) ).
% subset_refl
thf(fact_383_subset__iff,axiom,
( ord_le1619362961161175062_a_b_b
= ( ^ [A6: set_Pr3443975907877334966_a_b_b,B6: set_Pr3443975907877334966_a_b_b] :
! [T3: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ T3 @ A6 )
=> ( member4695696432722591383_a_b_b @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_384_subset__iff,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A6: set_list_a,B6: set_list_a] :
! [T3: list_a] :
( ( member_list_a @ T3 @ A6 )
=> ( member_list_a @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_385_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
! [T3: a] :
( ( member_a @ T3 @ A6 )
=> ( member_a @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_386_subset__iff,axiom,
( ord_le7599451563663638410st_a_b
= ( ^ [A6: set_dtree_list_a_b,B6: set_dtree_list_a_b] :
! [T3: dtree_list_a_b] :
( ( member551035911493665803st_a_b @ T3 @ A6 )
=> ( member551035911493665803st_a_b @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_387_subset__iff,axiom,
( ord_less_eq_set_b
= ( ^ [A6: set_b,B6: set_b] :
! [T3: b] :
( ( member_b @ T3 @ A6 )
=> ( member_b @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_388_equalityD2,axiom,
! [A: set_b,B5: set_b] :
( ( A = B5 )
=> ( ord_less_eq_set_b @ B5 @ A ) ) ).
% equalityD2
thf(fact_389_equalityD1,axiom,
! [A: set_b,B5: set_b] :
( ( A = B5 )
=> ( ord_less_eq_set_b @ A @ B5 ) ) ).
% equalityD1
thf(fact_390_subset__eq,axiom,
( ord_le1619362961161175062_a_b_b
= ( ^ [A6: set_Pr3443975907877334966_a_b_b,B6: set_Pr3443975907877334966_a_b_b] :
! [X4: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ X4 @ A6 )
=> ( member4695696432722591383_a_b_b @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_391_subset__eq,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A6: set_list_a,B6: set_list_a] :
! [X4: list_a] :
( ( member_list_a @ X4 @ A6 )
=> ( member_list_a @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_392_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
! [X4: a] :
( ( member_a @ X4 @ A6 )
=> ( member_a @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_393_subset__eq,axiom,
( ord_le7599451563663638410st_a_b
= ( ^ [A6: set_dtree_list_a_b,B6: set_dtree_list_a_b] :
! [X4: dtree_list_a_b] :
( ( member551035911493665803st_a_b @ X4 @ A6 )
=> ( member551035911493665803st_a_b @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_394_subset__eq,axiom,
( ord_less_eq_set_b
= ( ^ [A6: set_b,B6: set_b] :
! [X4: b] :
( ( member_b @ X4 @ A6 )
=> ( member_b @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_395_equalityE,axiom,
! [A: set_b,B5: set_b] :
( ( A = B5 )
=> ~ ( ( ord_less_eq_set_b @ A @ B5 )
=> ~ ( ord_less_eq_set_b @ B5 @ A ) ) ) ).
% equalityE
thf(fact_396_subsetD,axiom,
! [A: set_Pr3443975907877334966_a_b_b,B5: set_Pr3443975907877334966_a_b_b,C: produc6499617310964463488_a_b_b] :
( ( ord_le1619362961161175062_a_b_b @ A @ B5 )
=> ( ( member4695696432722591383_a_b_b @ C @ A )
=> ( member4695696432722591383_a_b_b @ C @ B5 ) ) ) ).
% subsetD
thf(fact_397_subsetD,axiom,
! [A: set_list_a,B5: set_list_a,C: list_a] :
( ( ord_le8861187494160871172list_a @ A @ B5 )
=> ( ( member_list_a @ C @ A )
=> ( member_list_a @ C @ B5 ) ) ) ).
% subsetD
thf(fact_398_subsetD,axiom,
! [A: set_a,B5: set_a,C: a] :
( ( ord_less_eq_set_a @ A @ B5 )
=> ( ( member_a @ C @ A )
=> ( member_a @ C @ B5 ) ) ) ).
% subsetD
thf(fact_399_subsetD,axiom,
! [A: set_dtree_list_a_b,B5: set_dtree_list_a_b,C: dtree_list_a_b] :
( ( ord_le7599451563663638410st_a_b @ A @ B5 )
=> ( ( member551035911493665803st_a_b @ C @ A )
=> ( member551035911493665803st_a_b @ C @ B5 ) ) ) ).
% subsetD
thf(fact_400_subsetD,axiom,
! [A: set_b,B5: set_b,C: b] :
( ( ord_less_eq_set_b @ A @ B5 )
=> ( ( member_b @ C @ A )
=> ( member_b @ C @ B5 ) ) ) ).
% subsetD
thf(fact_401_in__mono,axiom,
! [A: set_Pr3443975907877334966_a_b_b,B5: set_Pr3443975907877334966_a_b_b,X: produc6499617310964463488_a_b_b] :
( ( ord_le1619362961161175062_a_b_b @ A @ B5 )
=> ( ( member4695696432722591383_a_b_b @ X @ A )
=> ( member4695696432722591383_a_b_b @ X @ B5 ) ) ) ).
% in_mono
thf(fact_402_in__mono,axiom,
! [A: set_list_a,B5: set_list_a,X: list_a] :
( ( ord_le8861187494160871172list_a @ A @ B5 )
=> ( ( member_list_a @ X @ A )
=> ( member_list_a @ X @ B5 ) ) ) ).
% in_mono
thf(fact_403_in__mono,axiom,
! [A: set_a,B5: set_a,X: a] :
( ( ord_less_eq_set_a @ A @ B5 )
=> ( ( member_a @ X @ A )
=> ( member_a @ X @ B5 ) ) ) ).
% in_mono
thf(fact_404_in__mono,axiom,
! [A: set_dtree_list_a_b,B5: set_dtree_list_a_b,X: dtree_list_a_b] :
( ( ord_le7599451563663638410st_a_b @ A @ B5 )
=> ( ( member551035911493665803st_a_b @ X @ A )
=> ( member551035911493665803st_a_b @ X @ B5 ) ) ) ).
% in_mono
thf(fact_405_in__mono,axiom,
! [A: set_b,B5: set_b,X: b] :
( ( ord_less_eq_set_b @ A @ B5 )
=> ( ( member_b @ X @ A )
=> ( member_b @ X @ B5 ) ) ) ).
% in_mono
thf(fact_406_subset__insertI2,axiom,
! [A: set_list_a,B5: set_list_a,B: list_a] :
( ( ord_le8861187494160871172list_a @ A @ B5 )
=> ( ord_le8861187494160871172list_a @ A @ ( insert_list_a @ B @ B5 ) ) ) ).
% subset_insertI2
thf(fact_407_subset__insertI2,axiom,
! [A: set_b,B5: set_b,B: b] :
( ( ord_less_eq_set_b @ A @ B5 )
=> ( ord_less_eq_set_b @ A @ ( insert_b @ B @ B5 ) ) ) ).
% subset_insertI2
thf(fact_408_subset__insertI,axiom,
! [B5: set_list_a,A2: list_a] : ( ord_le8861187494160871172list_a @ B5 @ ( insert_list_a @ A2 @ B5 ) ) ).
% subset_insertI
thf(fact_409_subset__insertI,axiom,
! [B5: set_b,A2: b] : ( ord_less_eq_set_b @ B5 @ ( insert_b @ A2 @ B5 ) ) ).
% subset_insertI
thf(fact_410_subset__insert,axiom,
! [X: produc6499617310964463488_a_b_b,A: set_Pr3443975907877334966_a_b_b,B5: set_Pr3443975907877334966_a_b_b] :
( ~ ( member4695696432722591383_a_b_b @ X @ A )
=> ( ( ord_le1619362961161175062_a_b_b @ A @ ( insert8720088738084019632_a_b_b @ X @ B5 ) )
= ( ord_le1619362961161175062_a_b_b @ A @ B5 ) ) ) ).
% subset_insert
thf(fact_411_subset__insert,axiom,
! [X: list_a,A: set_list_a,B5: set_list_a] :
( ~ ( member_list_a @ X @ A )
=> ( ( ord_le8861187494160871172list_a @ A @ ( insert_list_a @ X @ B5 ) )
= ( ord_le8861187494160871172list_a @ A @ B5 ) ) ) ).
% subset_insert
thf(fact_412_subset__insert,axiom,
! [X: a,A: set_a,B5: set_a] :
( ~ ( member_a @ X @ A )
=> ( ( ord_less_eq_set_a @ A @ ( insert_a @ X @ B5 ) )
= ( ord_less_eq_set_a @ A @ B5 ) ) ) ).
% subset_insert
thf(fact_413_subset__insert,axiom,
! [X: dtree_list_a_b,A: set_dtree_list_a_b,B5: set_dtree_list_a_b] :
( ~ ( member551035911493665803st_a_b @ X @ A )
=> ( ( ord_le7599451563663638410st_a_b @ A @ ( insert7065945238938772516st_a_b @ X @ B5 ) )
= ( ord_le7599451563663638410st_a_b @ A @ B5 ) ) ) ).
% subset_insert
thf(fact_414_subset__insert,axiom,
! [X: b,A: set_b,B5: set_b] :
( ~ ( member_b @ X @ A )
=> ( ( ord_less_eq_set_b @ A @ ( insert_b @ X @ B5 ) )
= ( ord_less_eq_set_b @ A @ B5 ) ) ) ).
% subset_insert
thf(fact_415_insert__mono,axiom,
! [C2: set_list_a,D2: set_list_a,A2: list_a] :
( ( ord_le8861187494160871172list_a @ C2 @ D2 )
=> ( ord_le8861187494160871172list_a @ ( insert_list_a @ A2 @ C2 ) @ ( insert_list_a @ A2 @ D2 ) ) ) ).
% insert_mono
thf(fact_416_insert__mono,axiom,
! [C2: set_b,D2: set_b,A2: b] :
( ( ord_less_eq_set_b @ C2 @ D2 )
=> ( ord_less_eq_set_b @ ( insert_b @ A2 @ C2 ) @ ( insert_b @ A2 @ D2 ) ) ) ).
% insert_mono
thf(fact_417_Int__Collect__mono,axiom,
! [A: set_Pr3443975907877334966_a_b_b,B5: set_Pr3443975907877334966_a_b_b,P: produc6499617310964463488_a_b_b > $o,Q: produc6499617310964463488_a_b_b > $o] :
( ( ord_le1619362961161175062_a_b_b @ A @ B5 )
=> ( ! [X3: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le1619362961161175062_a_b_b @ ( inf_in7962574851115124040_a_b_b @ A @ ( collec3241703154345475797_a_b_b @ P ) ) @ ( inf_in7962574851115124040_a_b_b @ B5 @ ( collec3241703154345475797_a_b_b @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_418_Int__Collect__mono,axiom,
! [A: set_dtree_list_a_b,B5: set_dtree_list_a_b,P: dtree_list_a_b > $o,Q: dtree_list_a_b > $o] :
( ( ord_le7599451563663638410st_a_b @ A @ B5 )
=> ( ! [X3: dtree_list_a_b] :
( ( member551035911493665803st_a_b @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le7599451563663638410st_a_b @ ( inf_in3355993651213403836st_a_b @ A @ ( collec2944820760411501129st_a_b @ P ) ) @ ( inf_in3355993651213403836st_a_b @ B5 @ ( collec2944820760411501129st_a_b @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_419_Int__Collect__mono,axiom,
! [A: set_a,B5: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A @ B5 )
=> ( ! [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 @ B5 @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_420_Int__Collect__mono,axiom,
! [A: set_list_a,B5: set_list_a,P: list_a > $o,Q: list_a > $o] :
( ( ord_le8861187494160871172list_a @ A @ B5 )
=> ( ! [X3: list_a] :
( ( member_list_a @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A @ ( collect_list_a @ P ) ) @ ( inf_inf_set_list_a @ B5 @ ( collect_list_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_421_Int__Collect__mono,axiom,
! [A: set_b,B5: set_b,P: b > $o,Q: b > $o] :
( ( ord_less_eq_set_b @ A @ B5 )
=> ( ! [X3: b] :
( ( member_b @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A @ ( collect_b @ P ) ) @ ( inf_inf_set_b @ B5 @ ( collect_b @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_422_Int__greatest,axiom,
! [C2: set_a,A: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ C2 @ A )
=> ( ( ord_less_eq_set_a @ C2 @ B5 )
=> ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A @ B5 ) ) ) ) ).
% Int_greatest
thf(fact_423_Int__greatest,axiom,
! [C2: set_list_a,A: set_list_a,B5: set_list_a] :
( ( ord_le8861187494160871172list_a @ C2 @ A )
=> ( ( ord_le8861187494160871172list_a @ C2 @ B5 )
=> ( ord_le8861187494160871172list_a @ C2 @ ( inf_inf_set_list_a @ A @ B5 ) ) ) ) ).
% Int_greatest
thf(fact_424_Int__greatest,axiom,
! [C2: set_b,A: set_b,B5: set_b] :
( ( ord_less_eq_set_b @ C2 @ A )
=> ( ( ord_less_eq_set_b @ C2 @ B5 )
=> ( ord_less_eq_set_b @ C2 @ ( inf_inf_set_b @ A @ B5 ) ) ) ) ).
% Int_greatest
thf(fact_425_Int__absorb2,axiom,
! [A: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A @ B5 )
=> ( ( inf_inf_set_a @ A @ B5 )
= A ) ) ).
% Int_absorb2
thf(fact_426_Int__absorb2,axiom,
! [A: set_list_a,B5: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B5 )
=> ( ( inf_inf_set_list_a @ A @ B5 )
= A ) ) ).
% Int_absorb2
thf(fact_427_Int__absorb2,axiom,
! [A: set_b,B5: set_b] :
( ( ord_less_eq_set_b @ A @ B5 )
=> ( ( inf_inf_set_b @ A @ B5 )
= A ) ) ).
% Int_absorb2
thf(fact_428_Int__absorb1,axiom,
! [B5: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B5 @ A )
=> ( ( inf_inf_set_a @ A @ B5 )
= B5 ) ) ).
% Int_absorb1
thf(fact_429_Int__absorb1,axiom,
! [B5: set_list_a,A: set_list_a] :
( ( ord_le8861187494160871172list_a @ B5 @ A )
=> ( ( inf_inf_set_list_a @ A @ B5 )
= B5 ) ) ).
% Int_absorb1
thf(fact_430_Int__absorb1,axiom,
! [B5: set_b,A: set_b] :
( ( ord_less_eq_set_b @ B5 @ A )
=> ( ( inf_inf_set_b @ A @ B5 )
= B5 ) ) ).
% Int_absorb1
thf(fact_431_Int__lower2,axiom,
! [A: set_a,B5: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B5 ) @ B5 ) ).
% Int_lower2
thf(fact_432_Int__lower2,axiom,
! [A: set_list_a,B5: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A @ B5 ) @ B5 ) ).
% Int_lower2
thf(fact_433_Int__lower2,axiom,
! [A: set_b,B5: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ A @ B5 ) @ B5 ) ).
% Int_lower2
thf(fact_434_Int__lower1,axiom,
! [A: set_a,B5: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B5 ) @ A ) ).
% Int_lower1
thf(fact_435_Int__lower1,axiom,
! [A: set_list_a,B5: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A @ B5 ) @ A ) ).
% Int_lower1
thf(fact_436_Int__lower1,axiom,
! [A: set_b,B5: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ A @ B5 ) @ A ) ).
% Int_lower1
thf(fact_437_Int__mono,axiom,
! [A: set_a,C2: set_a,B5: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B5 @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B5 ) @ ( inf_inf_set_a @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_438_Int__mono,axiom,
! [A: set_list_a,C2: set_list_a,B5: set_list_a,D2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ C2 )
=> ( ( ord_le8861187494160871172list_a @ B5 @ D2 )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A @ B5 ) @ ( inf_inf_set_list_a @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_439_Int__mono,axiom,
! [A: set_b,C2: set_b,B5: set_b,D2: set_b] :
( ( ord_less_eq_set_b @ A @ C2 )
=> ( ( ord_less_eq_set_b @ B5 @ D2 )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A @ B5 ) @ ( inf_inf_set_b @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_440_subset__Un__eq,axiom,
( ord_less_eq_set_b
= ( ^ [A6: set_b,B6: set_b] :
( ( sup_sup_set_b @ A6 @ B6 )
= B6 ) ) ) ).
% subset_Un_eq
thf(fact_441_subset__UnE,axiom,
! [C2: set_b,A: set_b,B5: set_b] :
( ( ord_less_eq_set_b @ C2 @ ( sup_sup_set_b @ A @ B5 ) )
=> ~ ! [A7: set_b] :
( ( ord_less_eq_set_b @ A7 @ A )
=> ! [B7: set_b] :
( ( ord_less_eq_set_b @ B7 @ B5 )
=> ( C2
!= ( sup_sup_set_b @ A7 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_442_Un__absorb2,axiom,
! [B5: set_b,A: set_b] :
( ( ord_less_eq_set_b @ B5 @ A )
=> ( ( sup_sup_set_b @ A @ B5 )
= A ) ) ).
% Un_absorb2
thf(fact_443_Un__absorb1,axiom,
! [A: set_b,B5: set_b] :
( ( ord_less_eq_set_b @ A @ B5 )
=> ( ( sup_sup_set_b @ A @ B5 )
= B5 ) ) ).
% Un_absorb1
thf(fact_444_Un__upper2,axiom,
! [B5: set_b,A: set_b] : ( ord_less_eq_set_b @ B5 @ ( sup_sup_set_b @ A @ B5 ) ) ).
% Un_upper2
thf(fact_445_Un__upper1,axiom,
! [A: set_b,B5: set_b] : ( ord_less_eq_set_b @ A @ ( sup_sup_set_b @ A @ B5 ) ) ).
% Un_upper1
thf(fact_446_Un__least,axiom,
! [A: set_b,C2: set_b,B5: set_b] :
( ( ord_less_eq_set_b @ A @ C2 )
=> ( ( ord_less_eq_set_b @ B5 @ C2 )
=> ( ord_less_eq_set_b @ ( sup_sup_set_b @ A @ B5 ) @ C2 ) ) ) ).
% Un_least
thf(fact_447_Un__mono,axiom,
! [A: set_b,C2: set_b,B5: set_b,D2: set_b] :
( ( ord_less_eq_set_b @ A @ C2 )
=> ( ( ord_less_eq_set_b @ B5 @ D2 )
=> ( ord_less_eq_set_b @ ( sup_sup_set_b @ A @ B5 ) @ ( sup_sup_set_b @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_448_mem__case__prodE,axiom,
! [Z2: b,C: dtree_list_a_b > b > set_b,P2: produc6499617310964463488_a_b_b] :
( ( member_b @ Z2 @ ( produc6416049939707734976_set_b @ C @ P2 ) )
=> ~ ! [X3: dtree_list_a_b,Y3: b] :
( ( P2
= ( produc7704165765595008946_a_b_b @ X3 @ Y3 ) )
=> ~ ( member_b @ Z2 @ ( C @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_449_mem__case__prodE,axiom,
! [Z2: produc6499617310964463488_a_b_b,C: dtree_list_a_b > b > set_Pr3443975907877334966_a_b_b,P2: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ Z2 @ ( produc7845617512930307093_a_b_b @ C @ P2 ) )
=> ~ ! [X3: dtree_list_a_b,Y3: b] :
( ( P2
= ( produc7704165765595008946_a_b_b @ X3 @ Y3 ) )
=> ~ ( member4695696432722591383_a_b_b @ Z2 @ ( C @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_450_mem__case__prodE,axiom,
! [Z2: list_a,C: dtree_list_a_b > b > set_list_a,P2: produc6499617310964463488_a_b_b] :
( ( member_list_a @ Z2 @ ( produc7409682806061929157list_a @ C @ P2 ) )
=> ~ ! [X3: dtree_list_a_b,Y3: b] :
( ( P2
= ( produc7704165765595008946_a_b_b @ X3 @ Y3 ) )
=> ~ ( member_list_a @ Z2 @ ( C @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_451_mem__case__prodE,axiom,
! [Z2: a,C: dtree_list_a_b > b > set_a,P2: produc6499617310964463488_a_b_b] :
( ( member_a @ Z2 @ ( produc6416049935404506175_set_a @ C @ P2 ) )
=> ~ ! [X3: dtree_list_a_b,Y3: b] :
( ( P2
= ( produc7704165765595008946_a_b_b @ X3 @ Y3 ) )
=> ~ ( member_a @ Z2 @ ( C @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_452_mem__case__prodE,axiom,
! [Z2: dtree_list_a_b,C: dtree_list_a_b > b > set_dtree_list_a_b,P2: produc6499617310964463488_a_b_b] :
( ( member551035911493665803st_a_b @ Z2 @ ( produc4962517376730874505st_a_b @ C @ P2 ) )
=> ~ ! [X3: dtree_list_a_b,Y3: b] :
( ( P2
= ( produc7704165765595008946_a_b_b @ X3 @ Y3 ) )
=> ~ ( member551035911493665803st_a_b @ Z2 @ ( C @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_453_mem__case__prodE,axiom,
! [Z2: b,C: list_a > b > set_b,P2: produc2579390649552321826st_a_b] :
( ( member_b @ Z2 @ ( produc5826231656185642360_set_b @ C @ P2 ) )
=> ~ ! [X3: list_a,Y3: b] :
( ( P2
= ( produc4781227316648555538st_a_b @ X3 @ Y3 ) )
=> ~ ( member_b @ Z2 @ ( C @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_454_mem__case__prodE,axiom,
! [Z2: produc6499617310964463488_a_b_b,C: list_a > b > set_Pr3443975907877334966_a_b_b,P2: produc2579390649552321826st_a_b] :
( ( member4695696432722591383_a_b_b @ Z2 @ ( produc4981227583692318813_a_b_b @ C @ P2 ) )
=> ~ ! [X3: list_a,Y3: b] :
( ( P2
= ( produc4781227316648555538st_a_b @ X3 @ Y3 ) )
=> ~ ( member4695696432722591383_a_b_b @ Z2 @ ( C @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_455_mem__case__prodE,axiom,
! [Z2: list_a,C: list_a > b > set_list_a,P2: produc2579390649552321826st_a_b] :
( ( member_list_a @ Z2 @ ( produc621021178981935229list_a @ C @ P2 ) )
=> ~ ! [X3: list_a,Y3: b] :
( ( P2
= ( produc4781227316648555538st_a_b @ X3 @ Y3 ) )
=> ~ ( member_list_a @ Z2 @ ( C @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_456_mem__case__prodE,axiom,
! [Z2: a,C: list_a > b > set_a,P2: produc2579390649552321826st_a_b] :
( ( member_a @ Z2 @ ( produc5826231651882413559_set_a @ C @ P2 ) )
=> ~ ! [X3: list_a,Y3: b] :
( ( P2
= ( produc4781227316648555538st_a_b @ X3 @ Y3 ) )
=> ~ ( member_a @ Z2 @ ( C @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_457_mem__case__prodE,axiom,
! [Z2: dtree_list_a_b,C: list_a > b > set_dtree_list_a_b,P2: produc2579390649552321826st_a_b] :
( ( member551035911493665803st_a_b @ Z2 @ ( produc3410578764579669713st_a_b @ C @ P2 ) )
=> ~ ! [X3: list_a,Y3: b] :
( ( P2
= ( produc4781227316648555538st_a_b @ X3 @ Y3 ) )
=> ~ ( member551035911493665803st_a_b @ Z2 @ ( C @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_458_bot_Oextremum,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% bot.extremum
thf(fact_459_bot_Oextremum,axiom,
! [A2: set_list_a] : ( ord_le8861187494160871172list_a @ bot_bot_set_list_a @ A2 ) ).
% bot.extremum
thf(fact_460_bot_Oextremum,axiom,
! [A2: set_b] : ( ord_less_eq_set_b @ bot_bot_set_b @ A2 ) ).
% bot.extremum
thf(fact_461_bot_Oextremum,axiom,
! [A2: fset_P2153231429829016240_a_b_b] : ( ord_le8870638447146015504_a_b_b @ bot_bo2248824169281960260_a_b_b @ A2 ) ).
% bot.extremum
thf(fact_462_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_463_bot_Oextremum__unique,axiom,
! [A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ bot_bot_set_list_a )
= ( A2 = bot_bot_set_list_a ) ) ).
% bot.extremum_unique
thf(fact_464_bot_Oextremum__unique,axiom,
! [A2: set_b] :
( ( ord_less_eq_set_b @ A2 @ bot_bot_set_b )
= ( A2 = bot_bot_set_b ) ) ).
% bot.extremum_unique
thf(fact_465_bot_Oextremum__unique,axiom,
! [A2: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ A2 @ bot_bo2248824169281960260_a_b_b )
= ( A2 = bot_bo2248824169281960260_a_b_b ) ) ).
% bot.extremum_unique
thf(fact_466_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_467_bot_Oextremum__uniqueI,axiom,
! [A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ bot_bot_set_list_a )
=> ( A2 = bot_bot_set_list_a ) ) ).
% bot.extremum_uniqueI
thf(fact_468_bot_Oextremum__uniqueI,axiom,
! [A2: set_b] :
( ( ord_less_eq_set_b @ A2 @ bot_bot_set_b )
=> ( A2 = bot_bot_set_b ) ) ).
% bot.extremum_uniqueI
thf(fact_469_bot_Oextremum__uniqueI,axiom,
! [A2: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ A2 @ bot_bo2248824169281960260_a_b_b )
=> ( A2 = bot_bo2248824169281960260_a_b_b ) ) ).
% bot.extremum_uniqueI
thf(fact_470_subset__singleton__iff,axiom,
! [X5: set_a,A2: a] :
( ( ord_less_eq_set_a @ X5 @ ( insert_a @ A2 @ bot_bot_set_a ) )
= ( ( X5 = bot_bot_set_a )
| ( X5
= ( insert_a @ A2 @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_471_subset__singleton__iff,axiom,
! [X5: set_list_a,A2: list_a] :
( ( ord_le8861187494160871172list_a @ X5 @ ( insert_list_a @ A2 @ bot_bot_set_list_a ) )
= ( ( X5 = bot_bot_set_list_a )
| ( X5
= ( insert_list_a @ A2 @ bot_bot_set_list_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_472_subset__singleton__iff,axiom,
! [X5: set_b,A2: b] :
( ( ord_less_eq_set_b @ X5 @ ( insert_b @ A2 @ bot_bot_set_b ) )
= ( ( X5 = bot_bot_set_b )
| ( X5
= ( insert_b @ A2 @ bot_bot_set_b ) ) ) ) ).
% subset_singleton_iff
thf(fact_473_subset__singletonD,axiom,
! [A: set_a,X: a] :
( ( ord_less_eq_set_a @ A @ ( insert_a @ X @ bot_bot_set_a ) )
=> ( ( A = bot_bot_set_a )
| ( A
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_474_subset__singletonD,axiom,
! [A: set_list_a,X: list_a] :
( ( ord_le8861187494160871172list_a @ A @ ( insert_list_a @ X @ bot_bot_set_list_a ) )
=> ( ( A = bot_bot_set_list_a )
| ( A
= ( insert_list_a @ X @ bot_bot_set_list_a ) ) ) ) ).
% subset_singletonD
thf(fact_475_subset__singletonD,axiom,
! [A: set_b,X: b] :
( ( ord_less_eq_set_b @ A @ ( insert_b @ X @ bot_bot_set_b ) )
=> ( ( A = bot_bot_set_b )
| ( A
= ( insert_b @ X @ bot_bot_set_b ) ) ) ) ).
% subset_singletonD
thf(fact_476_Un__Int__assoc__eq,axiom,
! [A: set_a,B5: set_a,C2: set_a] :
( ( ( sup_sup_set_a @ ( inf_inf_set_a @ A @ B5 ) @ C2 )
= ( inf_inf_set_a @ A @ ( sup_sup_set_a @ B5 @ C2 ) ) )
= ( ord_less_eq_set_a @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_477_Un__Int__assoc__eq,axiom,
! [A: set_list_a,B5: set_list_a,C2: set_list_a] :
( ( ( sup_sup_set_list_a @ ( inf_inf_set_list_a @ A @ B5 ) @ C2 )
= ( inf_inf_set_list_a @ A @ ( sup_sup_set_list_a @ B5 @ C2 ) ) )
= ( ord_le8861187494160871172list_a @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_478_Un__Int__assoc__eq,axiom,
! [A: set_b,B5: set_b,C2: set_b] :
( ( ( sup_sup_set_b @ ( inf_inf_set_b @ A @ B5 ) @ C2 )
= ( inf_inf_set_b @ A @ ( sup_sup_set_b @ B5 @ C2 ) ) )
= ( ord_less_eq_set_b @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_479_bot__set__def,axiom,
( bot_bot_set_b
= ( collect_b @ bot_bot_b_o ) ) ).
% bot_set_def
thf(fact_480_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_481_bot__set__def,axiom,
( bot_bot_set_list_a
= ( collect_list_a @ bot_bot_list_a_o ) ) ).
% bot_set_def
thf(fact_482_ex__in__conv,axiom,
! [A: set_Pr3443975907877334966_a_b_b] :
( ( ? [X4: produc6499617310964463488_a_b_b] : ( member4695696432722591383_a_b_b @ X4 @ A ) )
= ( A != bot_bo4897374000430069834_a_b_b ) ) ).
% ex_in_conv
thf(fact_483_ex__in__conv,axiom,
! [A: set_dtree_list_a_b] :
( ( ? [X4: dtree_list_a_b] : ( member551035911493665803st_a_b @ X4 @ A ) )
= ( A != bot_bo798015271861357502st_a_b ) ) ).
% ex_in_conv
thf(fact_484_ex__in__conv,axiom,
! [A: set_b] :
( ( ? [X4: b] : ( member_b @ X4 @ A ) )
= ( A != bot_bot_set_b ) ) ).
% ex_in_conv
thf(fact_485_ex__in__conv,axiom,
! [A: set_a] :
( ( ? [X4: a] : ( member_a @ X4 @ A ) )
= ( A != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_486_ex__in__conv,axiom,
! [A: set_list_a] :
( ( ? [X4: list_a] : ( member_list_a @ X4 @ A ) )
= ( A != bot_bot_set_list_a ) ) ).
% ex_in_conv
thf(fact_487_equals0I,axiom,
! [A: set_Pr3443975907877334966_a_b_b] :
( ! [Y3: produc6499617310964463488_a_b_b] :
~ ( member4695696432722591383_a_b_b @ Y3 @ A )
=> ( A = bot_bo4897374000430069834_a_b_b ) ) ).
% equals0I
thf(fact_488_equals0I,axiom,
! [A: set_dtree_list_a_b] :
( ! [Y3: dtree_list_a_b] :
~ ( member551035911493665803st_a_b @ Y3 @ A )
=> ( A = bot_bo798015271861357502st_a_b ) ) ).
% equals0I
thf(fact_489_equals0I,axiom,
! [A: set_b] :
( ! [Y3: b] :
~ ( member_b @ Y3 @ A )
=> ( A = bot_bot_set_b ) ) ).
% equals0I
thf(fact_490_equals0I,axiom,
! [A: set_a] :
( ! [Y3: a] :
~ ( member_a @ Y3 @ A )
=> ( A = bot_bot_set_a ) ) ).
% equals0I
thf(fact_491_equals0I,axiom,
! [A: set_list_a] :
( ! [Y3: list_a] :
~ ( member_list_a @ Y3 @ A )
=> ( A = bot_bot_set_list_a ) ) ).
% equals0I
thf(fact_492_equals0D,axiom,
! [A: set_Pr3443975907877334966_a_b_b,A2: produc6499617310964463488_a_b_b] :
( ( A = bot_bo4897374000430069834_a_b_b )
=> ~ ( member4695696432722591383_a_b_b @ A2 @ A ) ) ).
% equals0D
thf(fact_493_equals0D,axiom,
! [A: set_dtree_list_a_b,A2: dtree_list_a_b] :
( ( A = bot_bo798015271861357502st_a_b )
=> ~ ( member551035911493665803st_a_b @ A2 @ A ) ) ).
% equals0D
thf(fact_494_equals0D,axiom,
! [A: set_b,A2: b] :
( ( A = bot_bot_set_b )
=> ~ ( member_b @ A2 @ A ) ) ).
% equals0D
thf(fact_495_equals0D,axiom,
! [A: set_a,A2: a] :
( ( A = bot_bot_set_a )
=> ~ ( member_a @ A2 @ A ) ) ).
% equals0D
thf(fact_496_equals0D,axiom,
! [A: set_list_a,A2: list_a] :
( ( A = bot_bot_set_list_a )
=> ~ ( member_list_a @ A2 @ A ) ) ).
% equals0D
thf(fact_497_emptyE,axiom,
! [A2: produc6499617310964463488_a_b_b] :
~ ( member4695696432722591383_a_b_b @ A2 @ bot_bo4897374000430069834_a_b_b ) ).
% emptyE
thf(fact_498_emptyE,axiom,
! [A2: dtree_list_a_b] :
~ ( member551035911493665803st_a_b @ A2 @ bot_bo798015271861357502st_a_b ) ).
% emptyE
thf(fact_499_emptyE,axiom,
! [A2: b] :
~ ( member_b @ A2 @ bot_bot_set_b ) ).
% emptyE
thf(fact_500_emptyE,axiom,
! [A2: a] :
~ ( member_a @ A2 @ bot_bot_set_a ) ).
% emptyE
thf(fact_501_emptyE,axiom,
! [A2: list_a] :
~ ( member_list_a @ A2 @ bot_bot_set_list_a ) ).
% emptyE
thf(fact_502_mk__disjoint__insert,axiom,
! [A2: b,A: set_b] :
( ( member_b @ A2 @ A )
=> ? [B8: set_b] :
( ( A
= ( insert_b @ A2 @ B8 ) )
& ~ ( member_b @ A2 @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_503_mk__disjoint__insert,axiom,
! [A2: produc6499617310964463488_a_b_b,A: set_Pr3443975907877334966_a_b_b] :
( ( member4695696432722591383_a_b_b @ A2 @ A )
=> ? [B8: set_Pr3443975907877334966_a_b_b] :
( ( A
= ( insert8720088738084019632_a_b_b @ A2 @ B8 ) )
& ~ ( member4695696432722591383_a_b_b @ A2 @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_504_mk__disjoint__insert,axiom,
! [A2: list_a,A: set_list_a] :
( ( member_list_a @ A2 @ A )
=> ? [B8: set_list_a] :
( ( A
= ( insert_list_a @ A2 @ B8 ) )
& ~ ( member_list_a @ A2 @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_505_mk__disjoint__insert,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ? [B8: set_a] :
( ( A
= ( insert_a @ A2 @ B8 ) )
& ~ ( member_a @ A2 @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_506_mk__disjoint__insert,axiom,
! [A2: dtree_list_a_b,A: set_dtree_list_a_b] :
( ( member551035911493665803st_a_b @ A2 @ A )
=> ? [B8: set_dtree_list_a_b] :
( ( A
= ( insert7065945238938772516st_a_b @ A2 @ B8 ) )
& ~ ( member551035911493665803st_a_b @ A2 @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_507_insert__commute,axiom,
! [X: b,Y: b,A: set_b] :
( ( insert_b @ X @ ( insert_b @ Y @ A ) )
= ( insert_b @ Y @ ( insert_b @ X @ A ) ) ) ).
% insert_commute
thf(fact_508_insert__commute,axiom,
! [X: list_a,Y: list_a,A: set_list_a] :
( ( insert_list_a @ X @ ( insert_list_a @ Y @ A ) )
= ( insert_list_a @ Y @ ( insert_list_a @ X @ A ) ) ) ).
% insert_commute
thf(fact_509_insert__eq__iff,axiom,
! [A2: b,A: set_b,B: b,B5: set_b] :
( ~ ( member_b @ A2 @ A )
=> ( ~ ( member_b @ B @ B5 )
=> ( ( ( insert_b @ A2 @ A )
= ( insert_b @ B @ B5 ) )
= ( ( ( A2 = B )
=> ( A = B5 ) )
& ( ( A2 != B )
=> ? [C3: set_b] :
( ( A
= ( insert_b @ B @ C3 ) )
& ~ ( member_b @ B @ C3 )
& ( B5
= ( insert_b @ A2 @ C3 ) )
& ~ ( member_b @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_510_insert__eq__iff,axiom,
! [A2: produc6499617310964463488_a_b_b,A: set_Pr3443975907877334966_a_b_b,B: produc6499617310964463488_a_b_b,B5: set_Pr3443975907877334966_a_b_b] :
( ~ ( member4695696432722591383_a_b_b @ A2 @ A )
=> ( ~ ( member4695696432722591383_a_b_b @ B @ B5 )
=> ( ( ( insert8720088738084019632_a_b_b @ A2 @ A )
= ( insert8720088738084019632_a_b_b @ B @ B5 ) )
= ( ( ( A2 = B )
=> ( A = B5 ) )
& ( ( A2 != B )
=> ? [C3: set_Pr3443975907877334966_a_b_b] :
( ( A
= ( insert8720088738084019632_a_b_b @ B @ C3 ) )
& ~ ( member4695696432722591383_a_b_b @ B @ C3 )
& ( B5
= ( insert8720088738084019632_a_b_b @ A2 @ C3 ) )
& ~ ( member4695696432722591383_a_b_b @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_511_insert__eq__iff,axiom,
! [A2: list_a,A: set_list_a,B: list_a,B5: set_list_a] :
( ~ ( member_list_a @ A2 @ A )
=> ( ~ ( member_list_a @ B @ B5 )
=> ( ( ( insert_list_a @ A2 @ A )
= ( insert_list_a @ B @ B5 ) )
= ( ( ( A2 = B )
=> ( A = B5 ) )
& ( ( A2 != B )
=> ? [C3: set_list_a] :
( ( A
= ( insert_list_a @ B @ C3 ) )
& ~ ( member_list_a @ B @ C3 )
& ( B5
= ( insert_list_a @ A2 @ C3 ) )
& ~ ( member_list_a @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_512_insert__eq__iff,axiom,
! [A2: a,A: set_a,B: a,B5: set_a] :
( ~ ( member_a @ A2 @ A )
=> ( ~ ( member_a @ B @ B5 )
=> ( ( ( insert_a @ A2 @ A )
= ( insert_a @ B @ B5 ) )
= ( ( ( A2 = B )
=> ( A = B5 ) )
& ( ( A2 != B )
=> ? [C3: set_a] :
( ( A
= ( insert_a @ B @ C3 ) )
& ~ ( member_a @ B @ C3 )
& ( B5
= ( insert_a @ A2 @ C3 ) )
& ~ ( member_a @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_513_insert__eq__iff,axiom,
! [A2: dtree_list_a_b,A: set_dtree_list_a_b,B: dtree_list_a_b,B5: set_dtree_list_a_b] :
( ~ ( member551035911493665803st_a_b @ A2 @ A )
=> ( ~ ( member551035911493665803st_a_b @ B @ B5 )
=> ( ( ( insert7065945238938772516st_a_b @ A2 @ A )
= ( insert7065945238938772516st_a_b @ B @ B5 ) )
= ( ( ( A2 = B )
=> ( A = B5 ) )
& ( ( A2 != B )
=> ? [C3: set_dtree_list_a_b] :
( ( A
= ( insert7065945238938772516st_a_b @ B @ C3 ) )
& ~ ( member551035911493665803st_a_b @ B @ C3 )
& ( B5
= ( insert7065945238938772516st_a_b @ A2 @ C3 ) )
& ~ ( member551035911493665803st_a_b @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_514_insert__absorb,axiom,
! [A2: b,A: set_b] :
( ( member_b @ A2 @ A )
=> ( ( insert_b @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_515_insert__absorb,axiom,
! [A2: produc6499617310964463488_a_b_b,A: set_Pr3443975907877334966_a_b_b] :
( ( member4695696432722591383_a_b_b @ A2 @ A )
=> ( ( insert8720088738084019632_a_b_b @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_516_insert__absorb,axiom,
! [A2: list_a,A: set_list_a] :
( ( member_list_a @ A2 @ A )
=> ( ( insert_list_a @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_517_insert__absorb,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ( ( insert_a @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_518_insert__absorb,axiom,
! [A2: dtree_list_a_b,A: set_dtree_list_a_b] :
( ( member551035911493665803st_a_b @ A2 @ A )
=> ( ( insert7065945238938772516st_a_b @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_519_insert__ident,axiom,
! [X: b,A: set_b,B5: set_b] :
( ~ ( member_b @ X @ A )
=> ( ~ ( member_b @ X @ B5 )
=> ( ( ( insert_b @ X @ A )
= ( insert_b @ X @ B5 ) )
= ( A = B5 ) ) ) ) ).
% insert_ident
thf(fact_520_insert__ident,axiom,
! [X: produc6499617310964463488_a_b_b,A: set_Pr3443975907877334966_a_b_b,B5: set_Pr3443975907877334966_a_b_b] :
( ~ ( member4695696432722591383_a_b_b @ X @ A )
=> ( ~ ( member4695696432722591383_a_b_b @ X @ B5 )
=> ( ( ( insert8720088738084019632_a_b_b @ X @ A )
= ( insert8720088738084019632_a_b_b @ X @ B5 ) )
= ( A = B5 ) ) ) ) ).
% insert_ident
thf(fact_521_insert__ident,axiom,
! [X: list_a,A: set_list_a,B5: set_list_a] :
( ~ ( member_list_a @ X @ A )
=> ( ~ ( member_list_a @ X @ B5 )
=> ( ( ( insert_list_a @ X @ A )
= ( insert_list_a @ X @ B5 ) )
= ( A = B5 ) ) ) ) ).
% insert_ident
thf(fact_522_insert__ident,axiom,
! [X: a,A: set_a,B5: set_a] :
( ~ ( member_a @ X @ A )
=> ( ~ ( member_a @ X @ B5 )
=> ( ( ( insert_a @ X @ A )
= ( insert_a @ X @ B5 ) )
= ( A = B5 ) ) ) ) ).
% insert_ident
thf(fact_523_insert__ident,axiom,
! [X: dtree_list_a_b,A: set_dtree_list_a_b,B5: set_dtree_list_a_b] :
( ~ ( member551035911493665803st_a_b @ X @ A )
=> ( ~ ( member551035911493665803st_a_b @ X @ B5 )
=> ( ( ( insert7065945238938772516st_a_b @ X @ A )
= ( insert7065945238938772516st_a_b @ X @ B5 ) )
= ( A = B5 ) ) ) ) ).
% insert_ident
thf(fact_524_Set_Oset__insert,axiom,
! [X: b,A: set_b] :
( ( member_b @ X @ A )
=> ~ ! [B8: set_b] :
( ( A
= ( insert_b @ X @ B8 ) )
=> ( member_b @ X @ B8 ) ) ) ).
% Set.set_insert
thf(fact_525_Set_Oset__insert,axiom,
! [X: produc6499617310964463488_a_b_b,A: set_Pr3443975907877334966_a_b_b] :
( ( member4695696432722591383_a_b_b @ X @ A )
=> ~ ! [B8: set_Pr3443975907877334966_a_b_b] :
( ( A
= ( insert8720088738084019632_a_b_b @ X @ B8 ) )
=> ( member4695696432722591383_a_b_b @ X @ B8 ) ) ) ).
% Set.set_insert
thf(fact_526_Set_Oset__insert,axiom,
! [X: list_a,A: set_list_a] :
( ( member_list_a @ X @ A )
=> ~ ! [B8: set_list_a] :
( ( A
= ( insert_list_a @ X @ B8 ) )
=> ( member_list_a @ X @ B8 ) ) ) ).
% Set.set_insert
thf(fact_527_Set_Oset__insert,axiom,
! [X: a,A: set_a] :
( ( member_a @ X @ A )
=> ~ ! [B8: set_a] :
( ( A
= ( insert_a @ X @ B8 ) )
=> ( member_a @ X @ B8 ) ) ) ).
% Set.set_insert
thf(fact_528_Set_Oset__insert,axiom,
! [X: dtree_list_a_b,A: set_dtree_list_a_b] :
( ( member551035911493665803st_a_b @ X @ A )
=> ~ ! [B8: set_dtree_list_a_b] :
( ( A
= ( insert7065945238938772516st_a_b @ X @ B8 ) )
=> ( member551035911493665803st_a_b @ X @ B8 ) ) ) ).
% Set.set_insert
thf(fact_529_insertI2,axiom,
! [A2: b,B5: set_b,B: b] :
( ( member_b @ A2 @ B5 )
=> ( member_b @ A2 @ ( insert_b @ B @ B5 ) ) ) ).
% insertI2
thf(fact_530_insertI2,axiom,
! [A2: produc6499617310964463488_a_b_b,B5: set_Pr3443975907877334966_a_b_b,B: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ A2 @ B5 )
=> ( member4695696432722591383_a_b_b @ A2 @ ( insert8720088738084019632_a_b_b @ B @ B5 ) ) ) ).
% insertI2
thf(fact_531_insertI2,axiom,
! [A2: list_a,B5: set_list_a,B: list_a] :
( ( member_list_a @ A2 @ B5 )
=> ( member_list_a @ A2 @ ( insert_list_a @ B @ B5 ) ) ) ).
% insertI2
thf(fact_532_insertI2,axiom,
! [A2: a,B5: set_a,B: a] :
( ( member_a @ A2 @ B5 )
=> ( member_a @ A2 @ ( insert_a @ B @ B5 ) ) ) ).
% insertI2
thf(fact_533_insertI2,axiom,
! [A2: dtree_list_a_b,B5: set_dtree_list_a_b,B: dtree_list_a_b] :
( ( member551035911493665803st_a_b @ A2 @ B5 )
=> ( member551035911493665803st_a_b @ A2 @ ( insert7065945238938772516st_a_b @ B @ B5 ) ) ) ).
% insertI2
thf(fact_534_insertI1,axiom,
! [A2: b,B5: set_b] : ( member_b @ A2 @ ( insert_b @ A2 @ B5 ) ) ).
% insertI1
thf(fact_535_insertI1,axiom,
! [A2: produc6499617310964463488_a_b_b,B5: set_Pr3443975907877334966_a_b_b] : ( member4695696432722591383_a_b_b @ A2 @ ( insert8720088738084019632_a_b_b @ A2 @ B5 ) ) ).
% insertI1
thf(fact_536_insertI1,axiom,
! [A2: list_a,B5: set_list_a] : ( member_list_a @ A2 @ ( insert_list_a @ A2 @ B5 ) ) ).
% insertI1
thf(fact_537_insertI1,axiom,
! [A2: a,B5: set_a] : ( member_a @ A2 @ ( insert_a @ A2 @ B5 ) ) ).
% insertI1
thf(fact_538_insertI1,axiom,
! [A2: dtree_list_a_b,B5: set_dtree_list_a_b] : ( member551035911493665803st_a_b @ A2 @ ( insert7065945238938772516st_a_b @ A2 @ B5 ) ) ).
% insertI1
thf(fact_539_insertE,axiom,
! [A2: b,B: b,A: set_b] :
( ( member_b @ A2 @ ( insert_b @ B @ A ) )
=> ( ( A2 != B )
=> ( member_b @ A2 @ A ) ) ) ).
% insertE
thf(fact_540_insertE,axiom,
! [A2: produc6499617310964463488_a_b_b,B: produc6499617310964463488_a_b_b,A: set_Pr3443975907877334966_a_b_b] :
( ( member4695696432722591383_a_b_b @ A2 @ ( insert8720088738084019632_a_b_b @ B @ A ) )
=> ( ( A2 != B )
=> ( member4695696432722591383_a_b_b @ A2 @ A ) ) ) ).
% insertE
thf(fact_541_insertE,axiom,
! [A2: list_a,B: list_a,A: set_list_a] :
( ( member_list_a @ A2 @ ( insert_list_a @ B @ A ) )
=> ( ( A2 != B )
=> ( member_list_a @ A2 @ A ) ) ) ).
% insertE
thf(fact_542_insertE,axiom,
! [A2: a,B: a,A: set_a] :
( ( member_a @ A2 @ ( insert_a @ B @ A ) )
=> ( ( A2 != B )
=> ( member_a @ A2 @ A ) ) ) ).
% insertE
thf(fact_543_insertE,axiom,
! [A2: dtree_list_a_b,B: dtree_list_a_b,A: set_dtree_list_a_b] :
( ( member551035911493665803st_a_b @ A2 @ ( insert7065945238938772516st_a_b @ B @ A ) )
=> ( ( A2 != B )
=> ( member551035911493665803st_a_b @ A2 @ A ) ) ) ).
% insertE
thf(fact_544_Ball__def,axiom,
( ball_b
= ( ^ [A6: set_b,P3: b > $o] :
! [X4: b] :
( ( member_b @ X4 @ A6 )
=> ( P3 @ X4 ) ) ) ) ).
% Ball_def
thf(fact_545_Ball__def,axiom,
( ball_P5027410970799761494_a_b_b
= ( ^ [A6: set_Pr3443975907877334966_a_b_b,P3: produc6499617310964463488_a_b_b > $o] :
! [X4: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ X4 @ A6 )
=> ( P3 @ X4 ) ) ) ) ).
% Ball_def
thf(fact_546_Ball__def,axiom,
( ball_list_a
= ( ^ [A6: set_list_a,P3: list_a > $o] :
! [X4: list_a] :
( ( member_list_a @ X4 @ A6 )
=> ( P3 @ X4 ) ) ) ) ).
% Ball_def
thf(fact_547_Ball__def,axiom,
( ball_a
= ( ^ [A6: set_a,P3: a > $o] :
! [X4: a] :
( ( member_a @ X4 @ A6 )
=> ( P3 @ X4 ) ) ) ) ).
% Ball_def
thf(fact_548_Ball__def,axiom,
( ball_dtree_list_a_b
= ( ^ [A6: set_dtree_list_a_b,P3: dtree_list_a_b > $o] :
! [X4: dtree_list_a_b] :
( ( member551035911493665803st_a_b @ X4 @ A6 )
=> ( P3 @ X4 ) ) ) ) ).
% Ball_def
thf(fact_549_Int__left__commute,axiom,
! [A: set_b,B5: set_b,C2: set_b] :
( ( inf_inf_set_b @ A @ ( inf_inf_set_b @ B5 @ C2 ) )
= ( inf_inf_set_b @ B5 @ ( inf_inf_set_b @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_550_Int__left__commute,axiom,
! [A: set_a,B5: set_a,C2: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B5 @ C2 ) )
= ( inf_inf_set_a @ B5 @ ( inf_inf_set_a @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_551_Int__left__commute,axiom,
! [A: set_list_a,B5: set_list_a,C2: set_list_a] :
( ( inf_inf_set_list_a @ A @ ( inf_inf_set_list_a @ B5 @ C2 ) )
= ( inf_inf_set_list_a @ B5 @ ( inf_inf_set_list_a @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_552_Int__left__absorb,axiom,
! [A: set_b,B5: set_b] :
( ( inf_inf_set_b @ A @ ( inf_inf_set_b @ A @ B5 ) )
= ( inf_inf_set_b @ A @ B5 ) ) ).
% Int_left_absorb
thf(fact_553_Int__left__absorb,axiom,
! [A: set_a,B5: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ A @ B5 ) )
= ( inf_inf_set_a @ A @ B5 ) ) ).
% Int_left_absorb
thf(fact_554_Int__left__absorb,axiom,
! [A: set_list_a,B5: set_list_a] :
( ( inf_inf_set_list_a @ A @ ( inf_inf_set_list_a @ A @ B5 ) )
= ( inf_inf_set_list_a @ A @ B5 ) ) ).
% Int_left_absorb
thf(fact_555_Int__commute,axiom,
( inf_inf_set_b
= ( ^ [A6: set_b,B6: set_b] : ( inf_inf_set_b @ B6 @ A6 ) ) ) ).
% Int_commute
thf(fact_556_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A6: set_a,B6: set_a] : ( inf_inf_set_a @ B6 @ A6 ) ) ) ).
% Int_commute
thf(fact_557_Int__commute,axiom,
( inf_inf_set_list_a
= ( ^ [A6: set_list_a,B6: set_list_a] : ( inf_inf_set_list_a @ B6 @ A6 ) ) ) ).
% Int_commute
thf(fact_558_Int__absorb,axiom,
! [A: set_b] :
( ( inf_inf_set_b @ A @ A )
= A ) ).
% Int_absorb
thf(fact_559_Int__absorb,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ A )
= A ) ).
% Int_absorb
thf(fact_560_Int__absorb,axiom,
! [A: set_list_a] :
( ( inf_inf_set_list_a @ A @ A )
= A ) ).
% Int_absorb
thf(fact_561_Int__assoc,axiom,
! [A: set_b,B5: set_b,C2: set_b] :
( ( inf_inf_set_b @ ( inf_inf_set_b @ A @ B5 ) @ C2 )
= ( inf_inf_set_b @ A @ ( inf_inf_set_b @ B5 @ C2 ) ) ) ).
% Int_assoc
thf(fact_562_Int__assoc,axiom,
! [A: set_a,B5: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B5 ) @ C2 )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B5 @ C2 ) ) ) ).
% Int_assoc
thf(fact_563_Int__assoc,axiom,
! [A: set_list_a,B5: set_list_a,C2: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ A @ B5 ) @ C2 )
= ( inf_inf_set_list_a @ A @ ( inf_inf_set_list_a @ B5 @ C2 ) ) ) ).
% Int_assoc
thf(fact_564_IntD2,axiom,
! [C: produc6499617310964463488_a_b_b,A: set_Pr3443975907877334966_a_b_b,B5: set_Pr3443975907877334966_a_b_b] :
( ( member4695696432722591383_a_b_b @ C @ ( inf_in7962574851115124040_a_b_b @ A @ B5 ) )
=> ( member4695696432722591383_a_b_b @ C @ B5 ) ) ).
% IntD2
thf(fact_565_IntD2,axiom,
! [C: dtree_list_a_b,A: set_dtree_list_a_b,B5: set_dtree_list_a_b] :
( ( member551035911493665803st_a_b @ C @ ( inf_in3355993651213403836st_a_b @ A @ B5 ) )
=> ( member551035911493665803st_a_b @ C @ B5 ) ) ).
% IntD2
thf(fact_566_IntD2,axiom,
! [C: b,A: set_b,B5: set_b] :
( ( member_b @ C @ ( inf_inf_set_b @ A @ B5 ) )
=> ( member_b @ C @ B5 ) ) ).
% IntD2
thf(fact_567_IntD2,axiom,
! [C: a,A: set_a,B5: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B5 ) )
=> ( member_a @ C @ B5 ) ) ).
% IntD2
thf(fact_568_IntD2,axiom,
! [C: list_a,A: set_list_a,B5: set_list_a] :
( ( member_list_a @ C @ ( inf_inf_set_list_a @ A @ B5 ) )
=> ( member_list_a @ C @ B5 ) ) ).
% IntD2
thf(fact_569_IntD1,axiom,
! [C: produc6499617310964463488_a_b_b,A: set_Pr3443975907877334966_a_b_b,B5: set_Pr3443975907877334966_a_b_b] :
( ( member4695696432722591383_a_b_b @ C @ ( inf_in7962574851115124040_a_b_b @ A @ B5 ) )
=> ( member4695696432722591383_a_b_b @ C @ A ) ) ).
% IntD1
thf(fact_570_IntD1,axiom,
! [C: dtree_list_a_b,A: set_dtree_list_a_b,B5: set_dtree_list_a_b] :
( ( member551035911493665803st_a_b @ C @ ( inf_in3355993651213403836st_a_b @ A @ B5 ) )
=> ( member551035911493665803st_a_b @ C @ A ) ) ).
% IntD1
thf(fact_571_IntD1,axiom,
! [C: b,A: set_b,B5: set_b] :
( ( member_b @ C @ ( inf_inf_set_b @ A @ B5 ) )
=> ( member_b @ C @ A ) ) ).
% IntD1
thf(fact_572_IntD1,axiom,
! [C: a,A: set_a,B5: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B5 ) )
=> ( member_a @ C @ A ) ) ).
% IntD1
thf(fact_573_IntD1,axiom,
! [C: list_a,A: set_list_a,B5: set_list_a] :
( ( member_list_a @ C @ ( inf_inf_set_list_a @ A @ B5 ) )
=> ( member_list_a @ C @ A ) ) ).
% IntD1
thf(fact_574_IntE,axiom,
! [C: produc6499617310964463488_a_b_b,A: set_Pr3443975907877334966_a_b_b,B5: set_Pr3443975907877334966_a_b_b] :
( ( member4695696432722591383_a_b_b @ C @ ( inf_in7962574851115124040_a_b_b @ A @ B5 ) )
=> ~ ( ( member4695696432722591383_a_b_b @ C @ A )
=> ~ ( member4695696432722591383_a_b_b @ C @ B5 ) ) ) ).
% IntE
thf(fact_575_IntE,axiom,
! [C: dtree_list_a_b,A: set_dtree_list_a_b,B5: set_dtree_list_a_b] :
( ( member551035911493665803st_a_b @ C @ ( inf_in3355993651213403836st_a_b @ A @ B5 ) )
=> ~ ( ( member551035911493665803st_a_b @ C @ A )
=> ~ ( member551035911493665803st_a_b @ C @ B5 ) ) ) ).
% IntE
thf(fact_576_IntE,axiom,
! [C: b,A: set_b,B5: set_b] :
( ( member_b @ C @ ( inf_inf_set_b @ A @ B5 ) )
=> ~ ( ( member_b @ C @ A )
=> ~ ( member_b @ C @ B5 ) ) ) ).
% IntE
thf(fact_577_IntE,axiom,
! [C: a,A: set_a,B5: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B5 ) )
=> ~ ( ( member_a @ C @ A )
=> ~ ( member_a @ C @ B5 ) ) ) ).
% IntE
thf(fact_578_IntE,axiom,
! [C: list_a,A: set_list_a,B5: set_list_a] :
( ( member_list_a @ C @ ( inf_inf_set_list_a @ A @ B5 ) )
=> ~ ( ( member_list_a @ C @ A )
=> ~ ( member_list_a @ C @ B5 ) ) ) ).
% IntE
thf(fact_579_Un__left__commute,axiom,
! [A: set_b,B5: set_b,C2: set_b] :
( ( sup_sup_set_b @ A @ ( sup_sup_set_b @ B5 @ C2 ) )
= ( sup_sup_set_b @ B5 @ ( sup_sup_set_b @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_580_Un__left__absorb,axiom,
! [A: set_b,B5: set_b] :
( ( sup_sup_set_b @ A @ ( sup_sup_set_b @ A @ B5 ) )
= ( sup_sup_set_b @ A @ B5 ) ) ).
% Un_left_absorb
thf(fact_581_Un__commute,axiom,
( sup_sup_set_b
= ( ^ [A6: set_b,B6: set_b] : ( sup_sup_set_b @ B6 @ A6 ) ) ) ).
% Un_commute
thf(fact_582_Un__absorb,axiom,
! [A: set_b] :
( ( sup_sup_set_b @ A @ A )
= A ) ).
% Un_absorb
thf(fact_583_Un__assoc,axiom,
! [A: set_b,B5: set_b,C2: set_b] :
( ( sup_sup_set_b @ ( sup_sup_set_b @ A @ B5 ) @ C2 )
= ( sup_sup_set_b @ A @ ( sup_sup_set_b @ B5 @ C2 ) ) ) ).
% Un_assoc
thf(fact_584_ball__Un,axiom,
! [A: set_b,B5: set_b,P: b > $o] :
( ( ! [X4: b] :
( ( member_b @ X4 @ ( sup_sup_set_b @ A @ B5 ) )
=> ( P @ X4 ) ) )
= ( ! [X4: b] :
( ( member_b @ X4 @ A )
=> ( P @ X4 ) )
& ! [X4: b] :
( ( member_b @ X4 @ B5 )
=> ( P @ X4 ) ) ) ) ).
% ball_Un
thf(fact_585_bex__Un,axiom,
! [A: set_b,B5: set_b,P: b > $o] :
( ( ? [X4: b] :
( ( member_b @ X4 @ ( sup_sup_set_b @ A @ B5 ) )
& ( P @ X4 ) ) )
= ( ? [X4: b] :
( ( member_b @ X4 @ A )
& ( P @ X4 ) )
| ? [X4: b] :
( ( member_b @ X4 @ B5 )
& ( P @ X4 ) ) ) ) ).
% bex_Un
thf(fact_586_UnI2,axiom,
! [C: produc6499617310964463488_a_b_b,B5: set_Pr3443975907877334966_a_b_b,A: set_Pr3443975907877334966_a_b_b] :
( ( member4695696432722591383_a_b_b @ C @ B5 )
=> ( member4695696432722591383_a_b_b @ C @ ( sup_su6988119916188116706_a_b_b @ A @ B5 ) ) ) ).
% UnI2
thf(fact_587_UnI2,axiom,
! [C: list_a,B5: set_list_a,A: set_list_a] :
( ( member_list_a @ C @ B5 )
=> ( member_list_a @ C @ ( sup_sup_set_list_a @ A @ B5 ) ) ) ).
% UnI2
thf(fact_588_UnI2,axiom,
! [C: a,B5: set_a,A: set_a] :
( ( member_a @ C @ B5 )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B5 ) ) ) ).
% UnI2
thf(fact_589_UnI2,axiom,
! [C: dtree_list_a_b,B5: set_dtree_list_a_b,A: set_dtree_list_a_b] :
( ( member551035911493665803st_a_b @ C @ B5 )
=> ( member551035911493665803st_a_b @ C @ ( sup_su5199045856643764822st_a_b @ A @ B5 ) ) ) ).
% UnI2
thf(fact_590_UnI2,axiom,
! [C: b,B5: set_b,A: set_b] :
( ( member_b @ C @ B5 )
=> ( member_b @ C @ ( sup_sup_set_b @ A @ B5 ) ) ) ).
% UnI2
thf(fact_591_UnI1,axiom,
! [C: produc6499617310964463488_a_b_b,A: set_Pr3443975907877334966_a_b_b,B5: set_Pr3443975907877334966_a_b_b] :
( ( member4695696432722591383_a_b_b @ C @ A )
=> ( member4695696432722591383_a_b_b @ C @ ( sup_su6988119916188116706_a_b_b @ A @ B5 ) ) ) ).
% UnI1
thf(fact_592_UnI1,axiom,
! [C: list_a,A: set_list_a,B5: set_list_a] :
( ( member_list_a @ C @ A )
=> ( member_list_a @ C @ ( sup_sup_set_list_a @ A @ B5 ) ) ) ).
% UnI1
thf(fact_593_UnI1,axiom,
! [C: a,A: set_a,B5: set_a] :
( ( member_a @ C @ A )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B5 ) ) ) ).
% UnI1
thf(fact_594_UnI1,axiom,
! [C: dtree_list_a_b,A: set_dtree_list_a_b,B5: set_dtree_list_a_b] :
( ( member551035911493665803st_a_b @ C @ A )
=> ( member551035911493665803st_a_b @ C @ ( sup_su5199045856643764822st_a_b @ A @ B5 ) ) ) ).
% UnI1
thf(fact_595_UnI1,axiom,
! [C: b,A: set_b,B5: set_b] :
( ( member_b @ C @ A )
=> ( member_b @ C @ ( sup_sup_set_b @ A @ B5 ) ) ) ).
% UnI1
thf(fact_596_UnE,axiom,
! [C: produc6499617310964463488_a_b_b,A: set_Pr3443975907877334966_a_b_b,B5: set_Pr3443975907877334966_a_b_b] :
( ( member4695696432722591383_a_b_b @ C @ ( sup_su6988119916188116706_a_b_b @ A @ B5 ) )
=> ( ~ ( member4695696432722591383_a_b_b @ C @ A )
=> ( member4695696432722591383_a_b_b @ C @ B5 ) ) ) ).
% UnE
thf(fact_597_UnE,axiom,
! [C: list_a,A: set_list_a,B5: set_list_a] :
( ( member_list_a @ C @ ( sup_sup_set_list_a @ A @ B5 ) )
=> ( ~ ( member_list_a @ C @ A )
=> ( member_list_a @ C @ B5 ) ) ) ).
% UnE
thf(fact_598_UnE,axiom,
! [C: a,A: set_a,B5: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A @ B5 ) )
=> ( ~ ( member_a @ C @ A )
=> ( member_a @ C @ B5 ) ) ) ).
% UnE
thf(fact_599_UnE,axiom,
! [C: dtree_list_a_b,A: set_dtree_list_a_b,B5: set_dtree_list_a_b] :
( ( member551035911493665803st_a_b @ C @ ( sup_su5199045856643764822st_a_b @ A @ B5 ) )
=> ( ~ ( member551035911493665803st_a_b @ C @ A )
=> ( member551035911493665803st_a_b @ C @ B5 ) ) ) ).
% UnE
thf(fact_600_UnE,axiom,
! [C: b,A: set_b,B5: set_b] :
( ( member_b @ C @ ( sup_sup_set_b @ A @ B5 ) )
=> ( ~ ( member_b @ C @ A )
=> ( member_b @ C @ B5 ) ) ) ).
% UnE
thf(fact_601_empty__def,axiom,
( bot_bot_set_b
= ( collect_b
@ ^ [X4: b] : $false ) ) ).
% empty_def
thf(fact_602_empty__def,axiom,
( bot_bot_set_a
= ( collect_a
@ ^ [X4: a] : $false ) ) ).
% empty_def
thf(fact_603_empty__def,axiom,
( bot_bot_set_list_a
= ( collect_list_a
@ ^ [X4: list_a] : $false ) ) ).
% empty_def
thf(fact_604_insert__compr,axiom,
( insert_b
= ( ^ [A4: b,B6: set_b] :
( collect_b
@ ^ [X4: b] :
( ( X4 = A4 )
| ( member_b @ X4 @ B6 ) ) ) ) ) ).
% insert_compr
thf(fact_605_insert__compr,axiom,
( insert8720088738084019632_a_b_b
= ( ^ [A4: produc6499617310964463488_a_b_b,B6: set_Pr3443975907877334966_a_b_b] :
( collec3241703154345475797_a_b_b
@ ^ [X4: produc6499617310964463488_a_b_b] :
( ( X4 = A4 )
| ( member4695696432722591383_a_b_b @ X4 @ B6 ) ) ) ) ) ).
% insert_compr
thf(fact_606_insert__compr,axiom,
( insert_list_a
= ( ^ [A4: list_a,B6: set_list_a] :
( collect_list_a
@ ^ [X4: list_a] :
( ( X4 = A4 )
| ( member_list_a @ X4 @ B6 ) ) ) ) ) ).
% insert_compr
thf(fact_607_insert__compr,axiom,
( insert_a
= ( ^ [A4: a,B6: set_a] :
( collect_a
@ ^ [X4: a] :
( ( X4 = A4 )
| ( member_a @ X4 @ B6 ) ) ) ) ) ).
% insert_compr
thf(fact_608_insert__compr,axiom,
( insert7065945238938772516st_a_b
= ( ^ [A4: dtree_list_a_b,B6: set_dtree_list_a_b] :
( collec2944820760411501129st_a_b
@ ^ [X4: dtree_list_a_b] :
( ( X4 = A4 )
| ( member551035911493665803st_a_b @ X4 @ B6 ) ) ) ) ) ).
% insert_compr
thf(fact_609_insert__Collect,axiom,
! [A2: b,P: b > $o] :
( ( insert_b @ A2 @ ( collect_b @ P ) )
= ( collect_b
@ ^ [U: b] :
( ( U != A2 )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_610_insert__Collect,axiom,
! [A2: list_a,P: list_a > $o] :
( ( insert_list_a @ A2 @ ( collect_list_a @ P ) )
= ( collect_list_a
@ ^ [U: list_a] :
( ( U != A2 )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_611_Int__def,axiom,
( inf_in7962574851115124040_a_b_b
= ( ^ [A6: set_Pr3443975907877334966_a_b_b,B6: set_Pr3443975907877334966_a_b_b] :
( collec3241703154345475797_a_b_b
@ ^ [X4: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ X4 @ A6 )
& ( member4695696432722591383_a_b_b @ X4 @ B6 ) ) ) ) ) ).
% Int_def
thf(fact_612_Int__def,axiom,
( inf_in3355993651213403836st_a_b
= ( ^ [A6: set_dtree_list_a_b,B6: set_dtree_list_a_b] :
( collec2944820760411501129st_a_b
@ ^ [X4: dtree_list_a_b] :
( ( member551035911493665803st_a_b @ X4 @ A6 )
& ( member551035911493665803st_a_b @ X4 @ B6 ) ) ) ) ) ).
% Int_def
thf(fact_613_Int__def,axiom,
( inf_inf_set_b
= ( ^ [A6: set_b,B6: set_b] :
( collect_b
@ ^ [X4: b] :
( ( member_b @ X4 @ A6 )
& ( member_b @ X4 @ B6 ) ) ) ) ) ).
% Int_def
thf(fact_614_Int__def,axiom,
( inf_inf_set_a
= ( ^ [A6: set_a,B6: set_a] :
( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ A6 )
& ( member_a @ X4 @ B6 ) ) ) ) ) ).
% Int_def
thf(fact_615_Int__def,axiom,
( inf_inf_set_list_a
= ( ^ [A6: set_list_a,B6: set_list_a] :
( collect_list_a
@ ^ [X4: list_a] :
( ( member_list_a @ X4 @ A6 )
& ( member_list_a @ X4 @ B6 ) ) ) ) ) ).
% Int_def
thf(fact_616_Int__Collect,axiom,
! [X: produc6499617310964463488_a_b_b,A: set_Pr3443975907877334966_a_b_b,P: produc6499617310964463488_a_b_b > $o] :
( ( member4695696432722591383_a_b_b @ X @ ( inf_in7962574851115124040_a_b_b @ A @ ( collec3241703154345475797_a_b_b @ P ) ) )
= ( ( member4695696432722591383_a_b_b @ X @ A )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_617_Int__Collect,axiom,
! [X: dtree_list_a_b,A: set_dtree_list_a_b,P: dtree_list_a_b > $o] :
( ( member551035911493665803st_a_b @ X @ ( inf_in3355993651213403836st_a_b @ A @ ( collec2944820760411501129st_a_b @ P ) ) )
= ( ( member551035911493665803st_a_b @ X @ A )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_618_Int__Collect,axiom,
! [X: b,A: set_b,P: b > $o] :
( ( member_b @ X @ ( inf_inf_set_b @ A @ ( collect_b @ P ) ) )
= ( ( member_b @ X @ A )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_619_Int__Collect,axiom,
! [X: a,A: set_a,P: a > $o] :
( ( member_a @ X @ ( inf_inf_set_a @ A @ ( collect_a @ P ) ) )
= ( ( member_a @ X @ A )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_620_Int__Collect,axiom,
! [X: list_a,A: set_list_a,P: list_a > $o] :
( ( member_list_a @ X @ ( inf_inf_set_list_a @ A @ ( collect_list_a @ P ) ) )
= ( ( member_list_a @ X @ A )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_621_Collect__conj__eq,axiom,
! [P: b > $o,Q: b > $o] :
( ( collect_b
@ ^ [X4: b] :
( ( P @ X4 )
& ( Q @ X4 ) ) )
= ( inf_inf_set_b @ ( collect_b @ P ) @ ( collect_b @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_622_Collect__conj__eq,axiom,
! [P: a > $o,Q: a > $o] :
( ( collect_a
@ ^ [X4: a] :
( ( P @ X4 )
& ( Q @ X4 ) ) )
= ( inf_inf_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_623_Collect__conj__eq,axiom,
! [P: list_a > $o,Q: list_a > $o] :
( ( collect_list_a
@ ^ [X4: list_a] :
( ( P @ X4 )
& ( Q @ X4 ) ) )
= ( inf_inf_set_list_a @ ( collect_list_a @ P ) @ ( collect_list_a @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_624_Un__def,axiom,
( sup_su6988119916188116706_a_b_b
= ( ^ [A6: set_Pr3443975907877334966_a_b_b,B6: set_Pr3443975907877334966_a_b_b] :
( collec3241703154345475797_a_b_b
@ ^ [X4: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ X4 @ A6 )
| ( member4695696432722591383_a_b_b @ X4 @ B6 ) ) ) ) ) ).
% Un_def
thf(fact_625_Un__def,axiom,
( sup_sup_set_list_a
= ( ^ [A6: set_list_a,B6: set_list_a] :
( collect_list_a
@ ^ [X4: list_a] :
( ( member_list_a @ X4 @ A6 )
| ( member_list_a @ X4 @ B6 ) ) ) ) ) ).
% Un_def
thf(fact_626_Un__def,axiom,
( sup_sup_set_a
= ( ^ [A6: set_a,B6: set_a] :
( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ A6 )
| ( member_a @ X4 @ B6 ) ) ) ) ) ).
% Un_def
thf(fact_627_Un__def,axiom,
( sup_su5199045856643764822st_a_b
= ( ^ [A6: set_dtree_list_a_b,B6: set_dtree_list_a_b] :
( collec2944820760411501129st_a_b
@ ^ [X4: dtree_list_a_b] :
( ( member551035911493665803st_a_b @ X4 @ A6 )
| ( member551035911493665803st_a_b @ X4 @ B6 ) ) ) ) ) ).
% Un_def
thf(fact_628_Un__def,axiom,
( sup_sup_set_b
= ( ^ [A6: set_b,B6: set_b] :
( collect_b
@ ^ [X4: b] :
( ( member_b @ X4 @ A6 )
| ( member_b @ X4 @ B6 ) ) ) ) ) ).
% Un_def
thf(fact_629_Collect__disj__eq,axiom,
! [P: b > $o,Q: b > $o] :
( ( collect_b
@ ^ [X4: b] :
( ( P @ X4 )
| ( Q @ X4 ) ) )
= ( sup_sup_set_b @ ( collect_b @ P ) @ ( collect_b @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_630_singleton__inject,axiom,
! [A2: b,B: b] :
( ( ( insert_b @ A2 @ bot_bot_set_b )
= ( insert_b @ B @ bot_bot_set_b ) )
=> ( A2 = B ) ) ).
% singleton_inject
thf(fact_631_singleton__inject,axiom,
! [A2: a,B: a] :
( ( ( insert_a @ A2 @ bot_bot_set_a )
= ( insert_a @ B @ bot_bot_set_a ) )
=> ( A2 = B ) ) ).
% singleton_inject
thf(fact_632_singleton__inject,axiom,
! [A2: list_a,B: list_a] :
( ( ( insert_list_a @ A2 @ bot_bot_set_list_a )
= ( insert_list_a @ B @ bot_bot_set_list_a ) )
=> ( A2 = B ) ) ).
% singleton_inject
thf(fact_633_insert__not__empty,axiom,
! [A2: b,A: set_b] :
( ( insert_b @ A2 @ A )
!= bot_bot_set_b ) ).
% insert_not_empty
thf(fact_634_insert__not__empty,axiom,
! [A2: a,A: set_a] :
( ( insert_a @ A2 @ A )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_635_insert__not__empty,axiom,
! [A2: list_a,A: set_list_a] :
( ( insert_list_a @ A2 @ A )
!= bot_bot_set_list_a ) ).
% insert_not_empty
thf(fact_636_doubleton__eq__iff,axiom,
! [A2: b,B: b,C: b,D: b] :
( ( ( insert_b @ A2 @ ( insert_b @ B @ bot_bot_set_b ) )
= ( insert_b @ C @ ( insert_b @ D @ bot_bot_set_b ) ) )
= ( ( ( A2 = C )
& ( B = D ) )
| ( ( A2 = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_637_doubleton__eq__iff,axiom,
! [A2: a,B: a,C: a,D: a] :
( ( ( insert_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) )
= ( insert_a @ C @ ( insert_a @ D @ bot_bot_set_a ) ) )
= ( ( ( A2 = C )
& ( B = D ) )
| ( ( A2 = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_638_doubleton__eq__iff,axiom,
! [A2: list_a,B: list_a,C: list_a,D: list_a] :
( ( ( insert_list_a @ A2 @ ( insert_list_a @ B @ bot_bot_set_list_a ) )
= ( insert_list_a @ C @ ( insert_list_a @ D @ bot_bot_set_list_a ) ) )
= ( ( ( A2 = C )
& ( B = D ) )
| ( ( A2 = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_639_singleton__iff,axiom,
! [B: produc6499617310964463488_a_b_b,A2: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ B @ ( insert8720088738084019632_a_b_b @ A2 @ bot_bo4897374000430069834_a_b_b ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_640_singleton__iff,axiom,
! [B: dtree_list_a_b,A2: dtree_list_a_b] :
( ( member551035911493665803st_a_b @ B @ ( insert7065945238938772516st_a_b @ A2 @ bot_bo798015271861357502st_a_b ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_641_singleton__iff,axiom,
! [B: b,A2: b] :
( ( member_b @ B @ ( insert_b @ A2 @ bot_bot_set_b ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_642_singleton__iff,axiom,
! [B: a,A2: a] :
( ( member_a @ B @ ( insert_a @ A2 @ bot_bot_set_a ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_643_singleton__iff,axiom,
! [B: list_a,A2: list_a] :
( ( member_list_a @ B @ ( insert_list_a @ A2 @ bot_bot_set_list_a ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_644_singletonD,axiom,
! [B: produc6499617310964463488_a_b_b,A2: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ B @ ( insert8720088738084019632_a_b_b @ A2 @ bot_bo4897374000430069834_a_b_b ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_645_singletonD,axiom,
! [B: dtree_list_a_b,A2: dtree_list_a_b] :
( ( member551035911493665803st_a_b @ B @ ( insert7065945238938772516st_a_b @ A2 @ bot_bo798015271861357502st_a_b ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_646_singletonD,axiom,
! [B: b,A2: b] :
( ( member_b @ B @ ( insert_b @ A2 @ bot_bot_set_b ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_647_singletonD,axiom,
! [B: a,A2: a] :
( ( member_a @ B @ ( insert_a @ A2 @ bot_bot_set_a ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_648_singletonD,axiom,
! [B: list_a,A2: list_a] :
( ( member_list_a @ B @ ( insert_list_a @ A2 @ bot_bot_set_list_a ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_649_disjoint__iff__not__equal,axiom,
! [A: set_b,B5: set_b] :
( ( ( inf_inf_set_b @ A @ B5 )
= bot_bot_set_b )
= ( ! [X4: b] :
( ( member_b @ X4 @ A )
=> ! [Y2: b] :
( ( member_b @ Y2 @ B5 )
=> ( X4 != Y2 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_650_disjoint__iff__not__equal,axiom,
! [A: set_a,B5: set_a] :
( ( ( inf_inf_set_a @ A @ B5 )
= bot_bot_set_a )
= ( ! [X4: a] :
( ( member_a @ X4 @ A )
=> ! [Y2: a] :
( ( member_a @ Y2 @ B5 )
=> ( X4 != Y2 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_651_disjoint__iff__not__equal,axiom,
! [A: set_list_a,B5: set_list_a] :
( ( ( inf_inf_set_list_a @ A @ B5 )
= bot_bot_set_list_a )
= ( ! [X4: list_a] :
( ( member_list_a @ X4 @ A )
=> ! [Y2: list_a] :
( ( member_list_a @ Y2 @ B5 )
=> ( X4 != Y2 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_652_Int__empty__right,axiom,
! [A: set_b] :
( ( inf_inf_set_b @ A @ bot_bot_set_b )
= bot_bot_set_b ) ).
% Int_empty_right
thf(fact_653_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_654_Int__empty__right,axiom,
! [A: set_list_a] :
( ( inf_inf_set_list_a @ A @ bot_bot_set_list_a )
= bot_bot_set_list_a ) ).
% Int_empty_right
thf(fact_655_Int__empty__left,axiom,
! [B5: set_b] :
( ( inf_inf_set_b @ bot_bot_set_b @ B5 )
= bot_bot_set_b ) ).
% Int_empty_left
thf(fact_656_Int__empty__left,axiom,
! [B5: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ B5 )
= bot_bot_set_a ) ).
% Int_empty_left
thf(fact_657_Int__empty__left,axiom,
! [B5: set_list_a] :
( ( inf_inf_set_list_a @ bot_bot_set_list_a @ B5 )
= bot_bot_set_list_a ) ).
% Int_empty_left
thf(fact_658_disjoint__iff,axiom,
! [A: set_Pr3443975907877334966_a_b_b,B5: set_Pr3443975907877334966_a_b_b] :
( ( ( inf_in7962574851115124040_a_b_b @ A @ B5 )
= bot_bo4897374000430069834_a_b_b )
= ( ! [X4: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ X4 @ A )
=> ~ ( member4695696432722591383_a_b_b @ X4 @ B5 ) ) ) ) ).
% disjoint_iff
thf(fact_659_disjoint__iff,axiom,
! [A: set_dtree_list_a_b,B5: set_dtree_list_a_b] :
( ( ( inf_in3355993651213403836st_a_b @ A @ B5 )
= bot_bo798015271861357502st_a_b )
= ( ! [X4: dtree_list_a_b] :
( ( member551035911493665803st_a_b @ X4 @ A )
=> ~ ( member551035911493665803st_a_b @ X4 @ B5 ) ) ) ) ).
% disjoint_iff
thf(fact_660_disjoint__iff,axiom,
! [A: set_b,B5: set_b] :
( ( ( inf_inf_set_b @ A @ B5 )
= bot_bot_set_b )
= ( ! [X4: b] :
( ( member_b @ X4 @ A )
=> ~ ( member_b @ X4 @ B5 ) ) ) ) ).
% disjoint_iff
thf(fact_661_disjoint__iff,axiom,
! [A: set_a,B5: set_a] :
( ( ( inf_inf_set_a @ A @ B5 )
= bot_bot_set_a )
= ( ! [X4: a] :
( ( member_a @ X4 @ A )
=> ~ ( member_a @ X4 @ B5 ) ) ) ) ).
% disjoint_iff
thf(fact_662_disjoint__iff,axiom,
! [A: set_list_a,B5: set_list_a] :
( ( ( inf_inf_set_list_a @ A @ B5 )
= bot_bot_set_list_a )
= ( ! [X4: list_a] :
( ( member_list_a @ X4 @ A )
=> ~ ( member_list_a @ X4 @ B5 ) ) ) ) ).
% disjoint_iff
thf(fact_663_Int__emptyI,axiom,
! [A: set_Pr3443975907877334966_a_b_b,B5: set_Pr3443975907877334966_a_b_b] :
( ! [X3: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ X3 @ A )
=> ~ ( member4695696432722591383_a_b_b @ X3 @ B5 ) )
=> ( ( inf_in7962574851115124040_a_b_b @ A @ B5 )
= bot_bo4897374000430069834_a_b_b ) ) ).
% Int_emptyI
thf(fact_664_Int__emptyI,axiom,
! [A: set_dtree_list_a_b,B5: set_dtree_list_a_b] :
( ! [X3: dtree_list_a_b] :
( ( member551035911493665803st_a_b @ X3 @ A )
=> ~ ( member551035911493665803st_a_b @ X3 @ B5 ) )
=> ( ( inf_in3355993651213403836st_a_b @ A @ B5 )
= bot_bo798015271861357502st_a_b ) ) ).
% Int_emptyI
thf(fact_665_Int__emptyI,axiom,
! [A: set_b,B5: set_b] :
( ! [X3: b] :
( ( member_b @ X3 @ A )
=> ~ ( member_b @ X3 @ B5 ) )
=> ( ( inf_inf_set_b @ A @ B5 )
= bot_bot_set_b ) ) ).
% Int_emptyI
thf(fact_666_Int__emptyI,axiom,
! [A: set_a,B5: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ~ ( member_a @ X3 @ B5 ) )
=> ( ( inf_inf_set_a @ A @ B5 )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_667_Int__emptyI,axiom,
! [A: set_list_a,B5: set_list_a] :
( ! [X3: list_a] :
( ( member_list_a @ X3 @ A )
=> ~ ( member_list_a @ X3 @ B5 ) )
=> ( ( inf_inf_set_list_a @ A @ B5 )
= bot_bot_set_list_a ) ) ).
% Int_emptyI
thf(fact_668_Un__empty__right,axiom,
! [A: set_b] :
( ( sup_sup_set_b @ A @ bot_bot_set_b )
= A ) ).
% Un_empty_right
thf(fact_669_Un__empty__right,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ bot_bot_set_a )
= A ) ).
% Un_empty_right
thf(fact_670_Un__empty__right,axiom,
! [A: set_list_a] :
( ( sup_sup_set_list_a @ A @ bot_bot_set_list_a )
= A ) ).
% Un_empty_right
thf(fact_671_Un__empty__left,axiom,
! [B5: set_b] :
( ( sup_sup_set_b @ bot_bot_set_b @ B5 )
= B5 ) ).
% Un_empty_left
thf(fact_672_Un__empty__left,axiom,
! [B5: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ B5 )
= B5 ) ).
% Un_empty_left
thf(fact_673_Un__empty__left,axiom,
! [B5: set_list_a] :
( ( sup_sup_set_list_a @ bot_bot_set_list_a @ B5 )
= B5 ) ).
% Un_empty_left
thf(fact_674_Int__insert__right,axiom,
! [A2: produc6499617310964463488_a_b_b,A: set_Pr3443975907877334966_a_b_b,B5: set_Pr3443975907877334966_a_b_b] :
( ( ( member4695696432722591383_a_b_b @ A2 @ A )
=> ( ( inf_in7962574851115124040_a_b_b @ A @ ( insert8720088738084019632_a_b_b @ A2 @ B5 ) )
= ( insert8720088738084019632_a_b_b @ A2 @ ( inf_in7962574851115124040_a_b_b @ A @ B5 ) ) ) )
& ( ~ ( member4695696432722591383_a_b_b @ A2 @ A )
=> ( ( inf_in7962574851115124040_a_b_b @ A @ ( insert8720088738084019632_a_b_b @ A2 @ B5 ) )
= ( inf_in7962574851115124040_a_b_b @ A @ B5 ) ) ) ) ).
% Int_insert_right
thf(fact_675_Int__insert__right,axiom,
! [A2: dtree_list_a_b,A: set_dtree_list_a_b,B5: set_dtree_list_a_b] :
( ( ( member551035911493665803st_a_b @ A2 @ A )
=> ( ( inf_in3355993651213403836st_a_b @ A @ ( insert7065945238938772516st_a_b @ A2 @ B5 ) )
= ( insert7065945238938772516st_a_b @ A2 @ ( inf_in3355993651213403836st_a_b @ A @ B5 ) ) ) )
& ( ~ ( member551035911493665803st_a_b @ A2 @ A )
=> ( ( inf_in3355993651213403836st_a_b @ A @ ( insert7065945238938772516st_a_b @ A2 @ B5 ) )
= ( inf_in3355993651213403836st_a_b @ A @ B5 ) ) ) ) ).
% Int_insert_right
thf(fact_676_Int__insert__right,axiom,
! [A2: b,A: set_b,B5: set_b] :
( ( ( member_b @ A2 @ A )
=> ( ( inf_inf_set_b @ A @ ( insert_b @ A2 @ B5 ) )
= ( insert_b @ A2 @ ( inf_inf_set_b @ A @ B5 ) ) ) )
& ( ~ ( member_b @ A2 @ A )
=> ( ( inf_inf_set_b @ A @ ( insert_b @ A2 @ B5 ) )
= ( inf_inf_set_b @ A @ B5 ) ) ) ) ).
% Int_insert_right
thf(fact_677_Int__insert__right,axiom,
! [A2: a,A: set_a,B5: set_a] :
( ( ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B5 ) )
= ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B5 ) ) ) )
& ( ~ ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B5 ) )
= ( inf_inf_set_a @ A @ B5 ) ) ) ) ).
% Int_insert_right
thf(fact_678_Int__insert__right,axiom,
! [A2: list_a,A: set_list_a,B5: set_list_a] :
( ( ( member_list_a @ A2 @ A )
=> ( ( inf_inf_set_list_a @ A @ ( insert_list_a @ A2 @ B5 ) )
= ( insert_list_a @ A2 @ ( inf_inf_set_list_a @ A @ B5 ) ) ) )
& ( ~ ( member_list_a @ A2 @ A )
=> ( ( inf_inf_set_list_a @ A @ ( insert_list_a @ A2 @ B5 ) )
= ( inf_inf_set_list_a @ A @ B5 ) ) ) ) ).
% Int_insert_right
thf(fact_679_Int__insert__left,axiom,
! [A2: produc6499617310964463488_a_b_b,C2: set_Pr3443975907877334966_a_b_b,B5: set_Pr3443975907877334966_a_b_b] :
( ( ( member4695696432722591383_a_b_b @ A2 @ C2 )
=> ( ( inf_in7962574851115124040_a_b_b @ ( insert8720088738084019632_a_b_b @ A2 @ B5 ) @ C2 )
= ( insert8720088738084019632_a_b_b @ A2 @ ( inf_in7962574851115124040_a_b_b @ B5 @ C2 ) ) ) )
& ( ~ ( member4695696432722591383_a_b_b @ A2 @ C2 )
=> ( ( inf_in7962574851115124040_a_b_b @ ( insert8720088738084019632_a_b_b @ A2 @ B5 ) @ C2 )
= ( inf_in7962574851115124040_a_b_b @ B5 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_680_Int__insert__left,axiom,
! [A2: dtree_list_a_b,C2: set_dtree_list_a_b,B5: set_dtree_list_a_b] :
( ( ( member551035911493665803st_a_b @ A2 @ C2 )
=> ( ( inf_in3355993651213403836st_a_b @ ( insert7065945238938772516st_a_b @ A2 @ B5 ) @ C2 )
= ( insert7065945238938772516st_a_b @ A2 @ ( inf_in3355993651213403836st_a_b @ B5 @ C2 ) ) ) )
& ( ~ ( member551035911493665803st_a_b @ A2 @ C2 )
=> ( ( inf_in3355993651213403836st_a_b @ ( insert7065945238938772516st_a_b @ A2 @ B5 ) @ C2 )
= ( inf_in3355993651213403836st_a_b @ B5 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_681_Int__insert__left,axiom,
! [A2: b,C2: set_b,B5: set_b] :
( ( ( member_b @ A2 @ C2 )
=> ( ( inf_inf_set_b @ ( insert_b @ A2 @ B5 ) @ C2 )
= ( insert_b @ A2 @ ( inf_inf_set_b @ B5 @ C2 ) ) ) )
& ( ~ ( member_b @ A2 @ C2 )
=> ( ( inf_inf_set_b @ ( insert_b @ A2 @ B5 ) @ C2 )
= ( inf_inf_set_b @ B5 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_682_Int__insert__left,axiom,
! [A2: a,C2: set_a,B5: set_a] :
( ( ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B5 ) @ C2 )
= ( insert_a @ A2 @ ( inf_inf_set_a @ B5 @ C2 ) ) ) )
& ( ~ ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B5 ) @ C2 )
= ( inf_inf_set_a @ B5 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_683_Int__insert__left,axiom,
! [A2: list_a,C2: set_list_a,B5: set_list_a] :
( ( ( member_list_a @ A2 @ C2 )
=> ( ( inf_inf_set_list_a @ ( insert_list_a @ A2 @ B5 ) @ C2 )
= ( insert_list_a @ A2 @ ( inf_inf_set_list_a @ B5 @ C2 ) ) ) )
& ( ~ ( member_list_a @ A2 @ C2 )
=> ( ( inf_inf_set_list_a @ ( insert_list_a @ A2 @ B5 ) @ C2 )
= ( inf_inf_set_list_a @ B5 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_684_Un__Int__distrib2,axiom,
! [B5: set_a,C2: set_a,A: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ B5 @ C2 ) @ A )
= ( inf_inf_set_a @ ( sup_sup_set_a @ B5 @ A ) @ ( sup_sup_set_a @ C2 @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_685_Un__Int__distrib2,axiom,
! [B5: set_list_a,C2: set_list_a,A: set_list_a] :
( ( sup_sup_set_list_a @ ( inf_inf_set_list_a @ B5 @ C2 ) @ A )
= ( inf_inf_set_list_a @ ( sup_sup_set_list_a @ B5 @ A ) @ ( sup_sup_set_list_a @ C2 @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_686_Un__Int__distrib2,axiom,
! [B5: set_b,C2: set_b,A: set_b] :
( ( sup_sup_set_b @ ( inf_inf_set_b @ B5 @ C2 ) @ A )
= ( inf_inf_set_b @ ( sup_sup_set_b @ B5 @ A ) @ ( sup_sup_set_b @ C2 @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_687_Int__Un__distrib2,axiom,
! [B5: set_a,C2: set_a,A: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ B5 @ C2 ) @ A )
= ( sup_sup_set_a @ ( inf_inf_set_a @ B5 @ A ) @ ( inf_inf_set_a @ C2 @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_688_Int__Un__distrib2,axiom,
! [B5: set_list_a,C2: set_list_a,A: set_list_a] :
( ( inf_inf_set_list_a @ ( sup_sup_set_list_a @ B5 @ C2 ) @ A )
= ( sup_sup_set_list_a @ ( inf_inf_set_list_a @ B5 @ A ) @ ( inf_inf_set_list_a @ C2 @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_689_Int__Un__distrib2,axiom,
! [B5: set_b,C2: set_b,A: set_b] :
( ( inf_inf_set_b @ ( sup_sup_set_b @ B5 @ C2 ) @ A )
= ( sup_sup_set_b @ ( inf_inf_set_b @ B5 @ A ) @ ( inf_inf_set_b @ C2 @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_690_Un__Int__distrib,axiom,
! [A: set_a,B5: set_a,C2: set_a] :
( ( sup_sup_set_a @ A @ ( inf_inf_set_a @ B5 @ C2 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ A @ B5 ) @ ( sup_sup_set_a @ A @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_691_Un__Int__distrib,axiom,
! [A: set_list_a,B5: set_list_a,C2: set_list_a] :
( ( sup_sup_set_list_a @ A @ ( inf_inf_set_list_a @ B5 @ C2 ) )
= ( inf_inf_set_list_a @ ( sup_sup_set_list_a @ A @ B5 ) @ ( sup_sup_set_list_a @ A @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_692_Un__Int__distrib,axiom,
! [A: set_b,B5: set_b,C2: set_b] :
( ( sup_sup_set_b @ A @ ( inf_inf_set_b @ B5 @ C2 ) )
= ( inf_inf_set_b @ ( sup_sup_set_b @ A @ B5 ) @ ( sup_sup_set_b @ A @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_693_Int__Un__distrib,axiom,
! [A: set_a,B5: set_a,C2: set_a] :
( ( inf_inf_set_a @ A @ ( sup_sup_set_a @ B5 @ C2 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ A @ B5 ) @ ( inf_inf_set_a @ A @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_694_Int__Un__distrib,axiom,
! [A: set_list_a,B5: set_list_a,C2: set_list_a] :
( ( inf_inf_set_list_a @ A @ ( sup_sup_set_list_a @ B5 @ C2 ) )
= ( sup_sup_set_list_a @ ( inf_inf_set_list_a @ A @ B5 ) @ ( inf_inf_set_list_a @ A @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_695_Int__Un__distrib,axiom,
! [A: set_b,B5: set_b,C2: set_b] :
( ( inf_inf_set_b @ A @ ( sup_sup_set_b @ B5 @ C2 ) )
= ( sup_sup_set_b @ ( inf_inf_set_b @ A @ B5 ) @ ( inf_inf_set_b @ A @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_696_Un__Int__crazy,axiom,
! [A: set_a,B5: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ A @ B5 ) @ ( inf_inf_set_a @ B5 @ C2 ) ) @ ( inf_inf_set_a @ C2 @ A ) )
= ( inf_inf_set_a @ ( inf_inf_set_a @ ( sup_sup_set_a @ A @ B5 ) @ ( sup_sup_set_a @ B5 @ C2 ) ) @ ( sup_sup_set_a @ C2 @ A ) ) ) ).
% Un_Int_crazy
thf(fact_697_Un__Int__crazy,axiom,
! [A: set_list_a,B5: set_list_a,C2: set_list_a] :
( ( sup_sup_set_list_a @ ( sup_sup_set_list_a @ ( inf_inf_set_list_a @ A @ B5 ) @ ( inf_inf_set_list_a @ B5 @ C2 ) ) @ ( inf_inf_set_list_a @ C2 @ A ) )
= ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ ( sup_sup_set_list_a @ A @ B5 ) @ ( sup_sup_set_list_a @ B5 @ C2 ) ) @ ( sup_sup_set_list_a @ C2 @ A ) ) ) ).
% Un_Int_crazy
thf(fact_698_Un__Int__crazy,axiom,
! [A: set_b,B5: set_b,C2: set_b] :
( ( sup_sup_set_b @ ( sup_sup_set_b @ ( inf_inf_set_b @ A @ B5 ) @ ( inf_inf_set_b @ B5 @ C2 ) ) @ ( inf_inf_set_b @ C2 @ A ) )
= ( inf_inf_set_b @ ( inf_inf_set_b @ ( sup_sup_set_b @ A @ B5 ) @ ( sup_sup_set_b @ B5 @ C2 ) ) @ ( sup_sup_set_b @ C2 @ A ) ) ) ).
% Un_Int_crazy
thf(fact_699_Collect__conv__if2,axiom,
! [P: b > $o,A2: b] :
( ( ( P @ A2 )
=> ( ( collect_b
@ ^ [X4: b] :
( ( A2 = X4 )
& ( P @ X4 ) ) )
= ( insert_b @ A2 @ bot_bot_set_b ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect_b
@ ^ [X4: b] :
( ( A2 = X4 )
& ( P @ X4 ) ) )
= bot_bot_set_b ) ) ) ).
% Collect_conv_if2
thf(fact_700_Collect__conv__if2,axiom,
! [P: a > $o,A2: a] :
( ( ( P @ A2 )
=> ( ( collect_a
@ ^ [X4: a] :
( ( A2 = X4 )
& ( P @ X4 ) ) )
= ( insert_a @ A2 @ bot_bot_set_a ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect_a
@ ^ [X4: a] :
( ( A2 = X4 )
& ( P @ X4 ) ) )
= bot_bot_set_a ) ) ) ).
% Collect_conv_if2
thf(fact_701_Collect__conv__if2,axiom,
! [P: list_a > $o,A2: list_a] :
( ( ( P @ A2 )
=> ( ( collect_list_a
@ ^ [X4: list_a] :
( ( A2 = X4 )
& ( P @ X4 ) ) )
= ( insert_list_a @ A2 @ bot_bot_set_list_a ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect_list_a
@ ^ [X4: list_a] :
( ( A2 = X4 )
& ( P @ X4 ) ) )
= bot_bot_set_list_a ) ) ) ).
% Collect_conv_if2
thf(fact_702_Collect__conv__if,axiom,
! [P: b > $o,A2: b] :
( ( ( P @ A2 )
=> ( ( collect_b
@ ^ [X4: b] :
( ( X4 = A2 )
& ( P @ X4 ) ) )
= ( insert_b @ A2 @ bot_bot_set_b ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect_b
@ ^ [X4: b] :
( ( X4 = A2 )
& ( P @ X4 ) ) )
= bot_bot_set_b ) ) ) ).
% Collect_conv_if
thf(fact_703_Collect__conv__if,axiom,
! [P: a > $o,A2: a] :
( ( ( P @ A2 )
=> ( ( collect_a
@ ^ [X4: a] :
( ( X4 = A2 )
& ( P @ X4 ) ) )
= ( insert_a @ A2 @ bot_bot_set_a ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect_a
@ ^ [X4: a] :
( ( X4 = A2 )
& ( P @ X4 ) ) )
= bot_bot_set_a ) ) ) ).
% Collect_conv_if
thf(fact_704_Collect__conv__if,axiom,
! [P: list_a > $o,A2: list_a] :
( ( ( P @ A2 )
=> ( ( collect_list_a
@ ^ [X4: list_a] :
( ( X4 = A2 )
& ( P @ X4 ) ) )
= ( insert_list_a @ A2 @ bot_bot_set_list_a ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect_list_a
@ ^ [X4: list_a] :
( ( X4 = A2 )
& ( P @ X4 ) ) )
= bot_bot_set_list_a ) ) ) ).
% Collect_conv_if
thf(fact_705_insert__def,axiom,
( insert_list_a
= ( ^ [A4: list_a] :
( sup_sup_set_list_a
@ ( collect_list_a
@ ^ [X4: list_a] : ( X4 = A4 ) ) ) ) ) ).
% insert_def
thf(fact_706_insert__def,axiom,
( insert_b
= ( ^ [A4: b] :
( sup_sup_set_b
@ ( collect_b
@ ^ [X4: b] : ( X4 = A4 ) ) ) ) ) ).
% insert_def
thf(fact_707_singleton__Un__iff,axiom,
! [X: b,A: set_b,B5: set_b] :
( ( ( insert_b @ X @ bot_bot_set_b )
= ( sup_sup_set_b @ A @ B5 ) )
= ( ( ( A = bot_bot_set_b )
& ( B5
= ( insert_b @ X @ bot_bot_set_b ) ) )
| ( ( A
= ( insert_b @ X @ bot_bot_set_b ) )
& ( B5 = bot_bot_set_b ) )
| ( ( A
= ( insert_b @ X @ bot_bot_set_b ) )
& ( B5
= ( insert_b @ X @ bot_bot_set_b ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_708_singleton__Un__iff,axiom,
! [X: a,A: set_a,B5: set_a] :
( ( ( insert_a @ X @ bot_bot_set_a )
= ( sup_sup_set_a @ A @ B5 ) )
= ( ( ( A = bot_bot_set_a )
& ( B5
= ( insert_a @ X @ bot_bot_set_a ) ) )
| ( ( A
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B5 = bot_bot_set_a ) )
| ( ( A
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B5
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_709_singleton__Un__iff,axiom,
! [X: list_a,A: set_list_a,B5: set_list_a] :
( ( ( insert_list_a @ X @ bot_bot_set_list_a )
= ( sup_sup_set_list_a @ A @ B5 ) )
= ( ( ( A = bot_bot_set_list_a )
& ( B5
= ( insert_list_a @ X @ bot_bot_set_list_a ) ) )
| ( ( A
= ( insert_list_a @ X @ bot_bot_set_list_a ) )
& ( B5 = bot_bot_set_list_a ) )
| ( ( A
= ( insert_list_a @ X @ bot_bot_set_list_a ) )
& ( B5
= ( insert_list_a @ X @ bot_bot_set_list_a ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_710_Un__singleton__iff,axiom,
! [A: set_b,B5: set_b,X: b] :
( ( ( sup_sup_set_b @ A @ B5 )
= ( insert_b @ X @ bot_bot_set_b ) )
= ( ( ( A = bot_bot_set_b )
& ( B5
= ( insert_b @ X @ bot_bot_set_b ) ) )
| ( ( A
= ( insert_b @ X @ bot_bot_set_b ) )
& ( B5 = bot_bot_set_b ) )
| ( ( A
= ( insert_b @ X @ bot_bot_set_b ) )
& ( B5
= ( insert_b @ X @ bot_bot_set_b ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_711_Un__singleton__iff,axiom,
! [A: set_a,B5: set_a,X: a] :
( ( ( sup_sup_set_a @ A @ B5 )
= ( insert_a @ X @ bot_bot_set_a ) )
= ( ( ( A = bot_bot_set_a )
& ( B5
= ( insert_a @ X @ bot_bot_set_a ) ) )
| ( ( A
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B5 = bot_bot_set_a ) )
| ( ( A
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B5
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_712_Un__singleton__iff,axiom,
! [A: set_list_a,B5: set_list_a,X: list_a] :
( ( ( sup_sup_set_list_a @ A @ B5 )
= ( insert_list_a @ X @ bot_bot_set_list_a ) )
= ( ( ( A = bot_bot_set_list_a )
& ( B5
= ( insert_list_a @ X @ bot_bot_set_list_a ) ) )
| ( ( A
= ( insert_list_a @ X @ bot_bot_set_list_a ) )
& ( B5 = bot_bot_set_list_a ) )
| ( ( A
= ( insert_list_a @ X @ bot_bot_set_list_a ) )
& ( B5
= ( insert_list_a @ X @ bot_bot_set_list_a ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_713_insert__is__Un,axiom,
( insert_b
= ( ^ [A4: b] : ( sup_sup_set_b @ ( insert_b @ A4 @ bot_bot_set_b ) ) ) ) ).
% insert_is_Un
thf(fact_714_insert__is__Un,axiom,
( insert_a
= ( ^ [A4: a] : ( sup_sup_set_a @ ( insert_a @ A4 @ bot_bot_set_a ) ) ) ) ).
% insert_is_Un
thf(fact_715_insert__is__Un,axiom,
( insert_list_a
= ( ^ [A4: list_a] : ( sup_sup_set_list_a @ ( insert_list_a @ A4 @ bot_bot_set_list_a ) ) ) ) ).
% insert_is_Un
thf(fact_716_disjoint__darcs__aux1__aux2,axiom,
! [Xs2: fset_P2153231429829016240_a_b_b,E3: b,R2: list_a,T12: dtree_list_a_b,E12: b,Y: list_a,V: list_a] :
( ! [X3: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ X3 @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( produc336013011823628281_b_b_o
@ ^ [Y2: dtree_list_a_b,E1: b] :
( ~ ( member_b @ E1 @ ( darcs_list_a_b @ Y2 ) )
& ! [Z: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ Z @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( produc336013011823628281_b_b_o
@ ^ [Aa: dtree_list_a_b,E2: b] :
( ( ( inf_inf_set_b @ ( sup_sup_set_b @ ( darcs_list_a_b @ Y2 ) @ ( insert_b @ E1 @ bot_bot_set_b ) ) @ ( sup_sup_set_b @ ( darcs_list_a_b @ Aa ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
= bot_bot_set_b )
| ( ( produc7704165765595008946_a_b_b @ Y2 @ E1 )
= ( produc7704165765595008946_a_b_b @ Aa @ E2 ) ) )
@ Z ) ) )
@ X3 ) )
=> ( ~ ( member_b @ E3 @ ( darcs_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E12 )
@ ( fset_P9138963618725001425_a_b_b
@ ( ffold_6356110402769621009_a_b_b
@ ( produc7216629644754684766_a_b_b
@ ^ [T13: dtree_list_a_b,E1: b] :
( finser2303212525150181944_a_b_b
@ ( if_Pro807711176916573242_a_b_b
@ ( ( root_list_a_b @ T13 )
= Y )
@ ( produc7704165765595008946_a_b_b @ ( node_list_a_b @ V @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T13 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ E3 )
@ ( produc7704165765595008946_a_b_b @ T13 @ E1 ) ) ) )
@ bot_bo2248824169281960260_a_b_b
@ Xs2 ) ) )
=> ~ ( member_b @ E12 @ ( darcs_list_a_b @ T12 ) ) ) ) ) ).
% disjoint_darcs_aux1_aux2
thf(fact_717_sup__inf__absorb,axiom,
! [X: set_a,Y: set_a] :
( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_718_sup__inf__absorb,axiom,
! [X: set_list_a,Y: set_list_a] :
( ( sup_sup_set_list_a @ X @ ( inf_inf_set_list_a @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_719_sup__inf__absorb,axiom,
! [X: set_b,Y: set_b] :
( ( sup_sup_set_b @ X @ ( inf_inf_set_b @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_720_inf__sup__absorb,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_721_inf__sup__absorb,axiom,
! [X: set_list_a,Y: set_list_a] :
( ( inf_inf_set_list_a @ X @ ( sup_sup_set_list_a @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_722_inf__sup__absorb,axiom,
! [X: set_b,Y: set_b] :
( ( inf_inf_set_b @ X @ ( sup_sup_set_b @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_723_insert__before__commute,axiom,
! [Y1: list_a,V: list_a,E3: b] :
( finite6455031580399024771_a_b_b
@ ( produc7216629644754684766_a_b_b
@ ^ [T13: dtree_list_a_b,E1: b] :
( finser2303212525150181944_a_b_b
@ ( if_Pro807711176916573242_a_b_b
@ ( ( root_list_a_b @ T13 )
= Y1 )
@ ( produc7704165765595008946_a_b_b @ ( node_list_a_b @ V @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T13 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ E3 )
@ ( produc7704165765595008946_a_b_b @ T13 @ E1 ) ) ) ) ) ).
% insert_before_commute
thf(fact_724_sup__bot__left,axiom,
! [X: fset_P2153231429829016240_a_b_b] :
( ( sup_su5028961100972612572_a_b_b @ bot_bo2248824169281960260_a_b_b @ X )
= X ) ).
% sup_bot_left
thf(fact_725_sup__bot__left,axiom,
! [X: set_b] :
( ( sup_sup_set_b @ bot_bot_set_b @ X )
= X ) ).
% sup_bot_left
thf(fact_726_sup__bot__left,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ X )
= X ) ).
% sup_bot_left
thf(fact_727_sup__bot__left,axiom,
! [X: set_list_a] :
( ( sup_sup_set_list_a @ bot_bot_set_list_a @ X )
= X ) ).
% sup_bot_left
thf(fact_728_sup__bot__right,axiom,
! [X: fset_P2153231429829016240_a_b_b] :
( ( sup_su5028961100972612572_a_b_b @ X @ bot_bo2248824169281960260_a_b_b )
= X ) ).
% sup_bot_right
thf(fact_729_sup__bot__right,axiom,
! [X: set_b] :
( ( sup_sup_set_b @ X @ bot_bot_set_b )
= X ) ).
% sup_bot_right
thf(fact_730_sup__bot__right,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ bot_bot_set_a )
= X ) ).
% sup_bot_right
thf(fact_731_sup__bot__right,axiom,
! [X: set_list_a] :
( ( sup_sup_set_list_a @ X @ bot_bot_set_list_a )
= X ) ).
% sup_bot_right
thf(fact_732_bot__eq__sup__iff,axiom,
! [X: fset_P2153231429829016240_a_b_b,Y: fset_P2153231429829016240_a_b_b] :
( ( bot_bo2248824169281960260_a_b_b
= ( sup_su5028961100972612572_a_b_b @ X @ Y ) )
= ( ( X = bot_bo2248824169281960260_a_b_b )
& ( Y = bot_bo2248824169281960260_a_b_b ) ) ) ).
% bot_eq_sup_iff
thf(fact_733_bot__eq__sup__iff,axiom,
! [X: set_b,Y: set_b] :
( ( bot_bot_set_b
= ( sup_sup_set_b @ X @ Y ) )
= ( ( X = bot_bot_set_b )
& ( Y = bot_bot_set_b ) ) ) ).
% bot_eq_sup_iff
thf(fact_734_bot__eq__sup__iff,axiom,
! [X: set_a,Y: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ X @ Y ) )
= ( ( X = bot_bot_set_a )
& ( Y = bot_bot_set_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_735_bot__eq__sup__iff,axiom,
! [X: set_list_a,Y: set_list_a] :
( ( bot_bot_set_list_a
= ( sup_sup_set_list_a @ X @ Y ) )
= ( ( X = bot_bot_set_list_a )
& ( Y = bot_bot_set_list_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_736_sup__eq__bot__iff,axiom,
! [X: fset_P2153231429829016240_a_b_b,Y: fset_P2153231429829016240_a_b_b] :
( ( ( sup_su5028961100972612572_a_b_b @ X @ Y )
= bot_bo2248824169281960260_a_b_b )
= ( ( X = bot_bo2248824169281960260_a_b_b )
& ( Y = bot_bo2248824169281960260_a_b_b ) ) ) ).
% sup_eq_bot_iff
thf(fact_737_sup__eq__bot__iff,axiom,
! [X: set_b,Y: set_b] :
( ( ( sup_sup_set_b @ X @ Y )
= bot_bot_set_b )
= ( ( X = bot_bot_set_b )
& ( Y = bot_bot_set_b ) ) ) ).
% sup_eq_bot_iff
thf(fact_738_sup__eq__bot__iff,axiom,
! [X: set_a,Y: set_a] :
( ( ( sup_sup_set_a @ X @ Y )
= bot_bot_set_a )
= ( ( X = bot_bot_set_a )
& ( Y = bot_bot_set_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_739_sup__eq__bot__iff,axiom,
! [X: set_list_a,Y: set_list_a] :
( ( ( sup_sup_set_list_a @ X @ Y )
= bot_bot_set_list_a )
= ( ( X = bot_bot_set_list_a )
& ( Y = bot_bot_set_list_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_740_sup__bot_Oeq__neutr__iff,axiom,
! [A2: fset_P2153231429829016240_a_b_b,B: fset_P2153231429829016240_a_b_b] :
( ( ( sup_su5028961100972612572_a_b_b @ A2 @ B )
= bot_bo2248824169281960260_a_b_b )
= ( ( A2 = bot_bo2248824169281960260_a_b_b )
& ( B = bot_bo2248824169281960260_a_b_b ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_741_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_b,B: set_b] :
( ( ( sup_sup_set_b @ A2 @ B )
= bot_bot_set_b )
= ( ( A2 = bot_bot_set_b )
& ( B = bot_bot_set_b ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_742_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_a,B: set_a] :
( ( ( sup_sup_set_a @ A2 @ B )
= bot_bot_set_a )
= ( ( A2 = bot_bot_set_a )
& ( B = bot_bot_set_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_743_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_list_a,B: set_list_a] :
( ( ( sup_sup_set_list_a @ A2 @ B )
= bot_bot_set_list_a )
= ( ( A2 = bot_bot_set_list_a )
& ( B = bot_bot_set_list_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_744_sup__bot_Oleft__neutral,axiom,
! [A2: fset_P2153231429829016240_a_b_b] :
( ( sup_su5028961100972612572_a_b_b @ bot_bo2248824169281960260_a_b_b @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_745_sup__bot_Oleft__neutral,axiom,
! [A2: set_b] :
( ( sup_sup_set_b @ bot_bot_set_b @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_746_sup__bot_Oleft__neutral,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_747_sup__bot_Oleft__neutral,axiom,
! [A2: set_list_a] :
( ( sup_sup_set_list_a @ bot_bot_set_list_a @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_748_darcs__sub__if__children__sub,axiom,
! [Xs2: fset_P2153231429829016240_a_b_b,Ys: fset_P2153231429829016240_a_b_b,R2: list_a,V: list_a] :
( ( ord_le8870638447146015504_a_b_b @ Xs2 @ Ys )
=> ( ord_less_eq_set_b @ ( darcs_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) @ ( darcs_list_a_b @ ( node_list_a_b @ V @ Ys ) ) ) ) ).
% darcs_sub_if_children_sub
thf(fact_749_darcs__split,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,Ys: fset_P2153231429829016240_a_b_b] :
( ( darcs_list_a_b @ ( node_list_a_b @ R2 @ ( sup_su5028961100972612572_a_b_b @ Xs2 @ Ys ) ) )
= ( sup_sup_set_b @ ( darcs_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) @ ( darcs_list_a_b @ ( node_list_a_b @ R2 @ Ys ) ) ) ) ).
% darcs_split
thf(fact_750_inf_Oidem,axiom,
! [A2: set_b] :
( ( inf_inf_set_b @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_751_inf_Oidem,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_752_inf_Oidem,axiom,
! [A2: set_list_a] :
( ( inf_inf_set_list_a @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_753_inf__idem,axiom,
! [X: set_b] :
( ( inf_inf_set_b @ X @ X )
= X ) ).
% inf_idem
thf(fact_754_inf__idem,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ X )
= X ) ).
% inf_idem
thf(fact_755_inf__idem,axiom,
! [X: set_list_a] :
( ( inf_inf_set_list_a @ X @ X )
= X ) ).
% inf_idem
thf(fact_756_inf_Oleft__idem,axiom,
! [A2: set_b,B: set_b] :
( ( inf_inf_set_b @ A2 @ ( inf_inf_set_b @ A2 @ B ) )
= ( inf_inf_set_b @ A2 @ B ) ) ).
% inf.left_idem
thf(fact_757_inf_Oleft__idem,axiom,
! [A2: set_a,B: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ A2 @ B ) )
= ( inf_inf_set_a @ A2 @ B ) ) ).
% inf.left_idem
thf(fact_758_inf_Oleft__idem,axiom,
! [A2: set_list_a,B: set_list_a] :
( ( inf_inf_set_list_a @ A2 @ ( inf_inf_set_list_a @ A2 @ B ) )
= ( inf_inf_set_list_a @ A2 @ B ) ) ).
% inf.left_idem
thf(fact_759_inf__left__idem,axiom,
! [X: set_b,Y: set_b] :
( ( inf_inf_set_b @ X @ ( inf_inf_set_b @ X @ Y ) )
= ( inf_inf_set_b @ X @ Y ) ) ).
% inf_left_idem
thf(fact_760_inf__left__idem,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ X @ Y ) )
= ( inf_inf_set_a @ X @ Y ) ) ).
% inf_left_idem
thf(fact_761_inf__left__idem,axiom,
! [X: set_list_a,Y: set_list_a] :
( ( inf_inf_set_list_a @ X @ ( inf_inf_set_list_a @ X @ Y ) )
= ( inf_inf_set_list_a @ X @ Y ) ) ).
% inf_left_idem
thf(fact_762_inf_Oright__idem,axiom,
! [A2: set_b,B: set_b] :
( ( inf_inf_set_b @ ( inf_inf_set_b @ A2 @ B ) @ B )
= ( inf_inf_set_b @ A2 @ B ) ) ).
% inf.right_idem
thf(fact_763_inf_Oright__idem,axiom,
! [A2: set_a,B: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B ) @ B )
= ( inf_inf_set_a @ A2 @ B ) ) ).
% inf.right_idem
thf(fact_764_inf_Oright__idem,axiom,
! [A2: set_list_a,B: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ A2 @ B ) @ B )
= ( inf_inf_set_list_a @ A2 @ B ) ) ).
% inf.right_idem
thf(fact_765_inf__right__idem,axiom,
! [X: set_b,Y: set_b] :
( ( inf_inf_set_b @ ( inf_inf_set_b @ X @ Y ) @ Y )
= ( inf_inf_set_b @ X @ Y ) ) ).
% inf_right_idem
thf(fact_766_inf__right__idem,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y )
= ( inf_inf_set_a @ X @ Y ) ) ).
% inf_right_idem
thf(fact_767_inf__right__idem,axiom,
! [X: set_list_a,Y: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ X @ Y ) @ Y )
= ( inf_inf_set_list_a @ X @ Y ) ) ).
% inf_right_idem
thf(fact_768_sup_Oidem,axiom,
! [A2: set_b] :
( ( sup_sup_set_b @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_769_sup__idem,axiom,
! [X: set_b] :
( ( sup_sup_set_b @ X @ X )
= X ) ).
% sup_idem
thf(fact_770_sup_Oleft__idem,axiom,
! [A2: set_b,B: set_b] :
( ( sup_sup_set_b @ A2 @ ( sup_sup_set_b @ A2 @ B ) )
= ( sup_sup_set_b @ A2 @ B ) ) ).
% sup.left_idem
thf(fact_771_sup__left__idem,axiom,
! [X: set_b,Y: set_b] :
( ( sup_sup_set_b @ X @ ( sup_sup_set_b @ X @ Y ) )
= ( sup_sup_set_b @ X @ Y ) ) ).
% sup_left_idem
thf(fact_772_sup_Oright__idem,axiom,
! [A2: set_b,B: set_b] :
( ( sup_sup_set_b @ ( sup_sup_set_b @ A2 @ B ) @ B )
= ( sup_sup_set_b @ A2 @ B ) ) ).
% sup.right_idem
thf(fact_773_funion__fempty,axiom,
! [A: fset_P2153231429829016240_a_b_b,B5: fset_P2153231429829016240_a_b_b] :
( ( ( sup_su5028961100972612572_a_b_b @ A @ B5 )
= bot_bo2248824169281960260_a_b_b )
= ( ( A = bot_bo2248824169281960260_a_b_b )
& ( B5 = bot_bo2248824169281960260_a_b_b ) ) ) ).
% funion_fempty
thf(fact_774_fempty__fsubsetI,axiom,
! [X: fset_P2153231429829016240_a_b_b] : ( ord_le8870638447146015504_a_b_b @ bot_bo2248824169281960260_a_b_b @ X ) ).
% fempty_fsubsetI
thf(fact_775_fsubset__fempty,axiom,
! [A: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ A @ bot_bo2248824169281960260_a_b_b )
= ( A = bot_bo2248824169281960260_a_b_b ) ) ).
% fsubset_fempty
thf(fact_776_funion__finsert__right,axiom,
! [A: fset_P2153231429829016240_a_b_b,A2: produc6499617310964463488_a_b_b,B5: fset_P2153231429829016240_a_b_b] :
( ( sup_su5028961100972612572_a_b_b @ A @ ( finser2303212525150181944_a_b_b @ A2 @ B5 ) )
= ( finser2303212525150181944_a_b_b @ A2 @ ( sup_su5028961100972612572_a_b_b @ A @ B5 ) ) ) ).
% funion_finsert_right
thf(fact_777_funion__finsert__left,axiom,
! [A2: produc6499617310964463488_a_b_b,B5: fset_P2153231429829016240_a_b_b,C2: fset_P2153231429829016240_a_b_b] :
( ( sup_su5028961100972612572_a_b_b @ ( finser2303212525150181944_a_b_b @ A2 @ B5 ) @ C2 )
= ( finser2303212525150181944_a_b_b @ A2 @ ( sup_su5028961100972612572_a_b_b @ B5 @ C2 ) ) ) ).
% funion_finsert_left
thf(fact_778_finsert__inter__finsert,axiom,
! [A2: produc6499617310964463488_a_b_b,A: fset_P2153231429829016240_a_b_b,B5: fset_P2153231429829016240_a_b_b] :
( ( inf_in525273840840435522_a_b_b @ ( finser2303212525150181944_a_b_b @ A2 @ A ) @ ( finser2303212525150181944_a_b_b @ A2 @ B5 ) )
= ( finser2303212525150181944_a_b_b @ A2 @ ( inf_in525273840840435522_a_b_b @ A @ B5 ) ) ) ).
% finsert_inter_finsert
thf(fact_779_le__inf__iff,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) )
= ( ( ord_less_eq_set_a @ X @ Y )
& ( ord_less_eq_set_a @ X @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_780_le__inf__iff,axiom,
! [X: set_list_a,Y: set_list_a,Z2: set_list_a] :
( ( ord_le8861187494160871172list_a @ X @ ( inf_inf_set_list_a @ Y @ Z2 ) )
= ( ( ord_le8861187494160871172list_a @ X @ Y )
& ( ord_le8861187494160871172list_a @ X @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_781_le__inf__iff,axiom,
! [X: set_b,Y: set_b,Z2: set_b] :
( ( ord_less_eq_set_b @ X @ ( inf_inf_set_b @ Y @ Z2 ) )
= ( ( ord_less_eq_set_b @ X @ Y )
& ( ord_less_eq_set_b @ X @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_782_le__inf__iff,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_eq_real @ X @ ( inf_inf_real @ Y @ Z2 ) )
= ( ( ord_less_eq_real @ X @ Y )
& ( ord_less_eq_real @ X @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_783_le__inf__iff,axiom,
! [X: fset_P2153231429829016240_a_b_b,Y: fset_P2153231429829016240_a_b_b,Z2: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ X @ ( inf_in525273840840435522_a_b_b @ Y @ Z2 ) )
= ( ( ord_le8870638447146015504_a_b_b @ X @ Y )
& ( ord_le8870638447146015504_a_b_b @ X @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_784_inf_Obounded__iff,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B @ C ) )
= ( ( ord_less_eq_set_a @ A2 @ B )
& ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_785_inf_Obounded__iff,axiom,
! [A2: set_list_a,B: set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ ( inf_inf_set_list_a @ B @ C ) )
= ( ( ord_le8861187494160871172list_a @ A2 @ B )
& ( ord_le8861187494160871172list_a @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_786_inf_Obounded__iff,axiom,
! [A2: set_b,B: set_b,C: set_b] :
( ( ord_less_eq_set_b @ A2 @ ( inf_inf_set_b @ B @ C ) )
= ( ( ord_less_eq_set_b @ A2 @ B )
& ( ord_less_eq_set_b @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_787_inf_Obounded__iff,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_eq_real @ A2 @ ( inf_inf_real @ B @ C ) )
= ( ( ord_less_eq_real @ A2 @ B )
& ( ord_less_eq_real @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_788_inf_Obounded__iff,axiom,
! [A2: fset_P2153231429829016240_a_b_b,B: fset_P2153231429829016240_a_b_b,C: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ A2 @ ( inf_in525273840840435522_a_b_b @ B @ C ) )
= ( ( ord_le8870638447146015504_a_b_b @ A2 @ B )
& ( ord_le8870638447146015504_a_b_b @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_789_le__sup__iff,axiom,
! [X: set_b,Y: set_b,Z2: set_b] :
( ( ord_less_eq_set_b @ ( sup_sup_set_b @ X @ Y ) @ Z2 )
= ( ( ord_less_eq_set_b @ X @ Z2 )
& ( ord_less_eq_set_b @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_790_le__sup__iff,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_eq_real @ ( sup_sup_real @ X @ Y ) @ Z2 )
= ( ( ord_less_eq_real @ X @ Z2 )
& ( ord_less_eq_real @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_791_le__sup__iff,axiom,
! [X: fset_P2153231429829016240_a_b_b,Y: fset_P2153231429829016240_a_b_b,Z2: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ ( sup_su5028961100972612572_a_b_b @ X @ Y ) @ Z2 )
= ( ( ord_le8870638447146015504_a_b_b @ X @ Z2 )
& ( ord_le8870638447146015504_a_b_b @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_792_sup_Obounded__iff,axiom,
! [B: set_b,C: set_b,A2: set_b] :
( ( ord_less_eq_set_b @ ( sup_sup_set_b @ B @ C ) @ A2 )
= ( ( ord_less_eq_set_b @ B @ A2 )
& ( ord_less_eq_set_b @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_793_sup_Obounded__iff,axiom,
! [B: real,C: real,A2: real] :
( ( ord_less_eq_real @ ( sup_sup_real @ B @ C ) @ A2 )
= ( ( ord_less_eq_real @ B @ A2 )
& ( ord_less_eq_real @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_794_sup_Obounded__iff,axiom,
! [B: fset_P2153231429829016240_a_b_b,C: fset_P2153231429829016240_a_b_b,A2: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ ( sup_su5028961100972612572_a_b_b @ B @ C ) @ A2 )
= ( ( ord_le8870638447146015504_a_b_b @ B @ A2 )
& ( ord_le8870638447146015504_a_b_b @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_795_inf__bot__right,axiom,
! [X: fset_P2153231429829016240_a_b_b] :
( ( inf_in525273840840435522_a_b_b @ X @ bot_bo2248824169281960260_a_b_b )
= bot_bo2248824169281960260_a_b_b ) ).
% inf_bot_right
thf(fact_796_inf__bot__right,axiom,
! [X: set_b] :
( ( inf_inf_set_b @ X @ bot_bot_set_b )
= bot_bot_set_b ) ).
% inf_bot_right
thf(fact_797_inf__bot__right,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ bot_bot_set_a )
= bot_bot_set_a ) ).
% inf_bot_right
thf(fact_798_inf__bot__right,axiom,
! [X: set_list_a] :
( ( inf_inf_set_list_a @ X @ bot_bot_set_list_a )
= bot_bot_set_list_a ) ).
% inf_bot_right
thf(fact_799_inf__bot__left,axiom,
! [X: fset_P2153231429829016240_a_b_b] :
( ( inf_in525273840840435522_a_b_b @ bot_bo2248824169281960260_a_b_b @ X )
= bot_bo2248824169281960260_a_b_b ) ).
% inf_bot_left
thf(fact_800_inf__bot__left,axiom,
! [X: set_b] :
( ( inf_inf_set_b @ bot_bot_set_b @ X )
= bot_bot_set_b ) ).
% inf_bot_left
thf(fact_801_inf__bot__left,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X )
= bot_bot_set_a ) ).
% inf_bot_left
thf(fact_802_inf__bot__left,axiom,
! [X: set_list_a] :
( ( inf_inf_set_list_a @ bot_bot_set_list_a @ X )
= bot_bot_set_list_a ) ).
% inf_bot_left
thf(fact_803_sup__bot_Oright__neutral,axiom,
! [A2: fset_P2153231429829016240_a_b_b] :
( ( sup_su5028961100972612572_a_b_b @ A2 @ bot_bo2248824169281960260_a_b_b )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_804_sup__bot_Oright__neutral,axiom,
! [A2: set_b] :
( ( sup_sup_set_b @ A2 @ bot_bot_set_b )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_805_sup__bot_Oright__neutral,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_806_sup__bot_Oright__neutral,axiom,
! [A2: set_list_a] :
( ( sup_sup_set_list_a @ A2 @ bot_bot_set_list_a )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_807_sup__bot_Oneutr__eq__iff,axiom,
! [A2: fset_P2153231429829016240_a_b_b,B: fset_P2153231429829016240_a_b_b] :
( ( bot_bo2248824169281960260_a_b_b
= ( sup_su5028961100972612572_a_b_b @ A2 @ B ) )
= ( ( A2 = bot_bo2248824169281960260_a_b_b )
& ( B = bot_bo2248824169281960260_a_b_b ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_808_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_b,B: set_b] :
( ( bot_bot_set_b
= ( sup_sup_set_b @ A2 @ B ) )
= ( ( A2 = bot_bot_set_b )
& ( B = bot_bot_set_b ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_809_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_a,B: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ A2 @ B ) )
= ( ( A2 = bot_bot_set_a )
& ( B = bot_bot_set_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_810_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_list_a,B: set_list_a] :
( ( bot_bot_set_list_a
= ( sup_sup_set_list_a @ A2 @ B ) )
= ( ( A2 = bot_bot_set_list_a )
& ( B = bot_bot_set_list_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_811_inter__fset,axiom,
! [X: fset_P2153231429829016240_a_b_b,Xa: fset_P2153231429829016240_a_b_b] :
( ( fset_P9138963618725001425_a_b_b @ ( inf_in525273840840435522_a_b_b @ X @ Xa ) )
= ( inf_in7962574851115124040_a_b_b @ ( fset_P9138963618725001425_a_b_b @ X ) @ ( fset_P9138963618725001425_a_b_b @ Xa ) ) ) ).
% inter_fset
thf(fact_812_inter__fset,axiom,
! [X: fset_b,Xa: fset_b] :
( ( fset_b2 @ ( inf_inf_fset_b @ X @ Xa ) )
= ( inf_inf_set_b @ ( fset_b2 @ X ) @ ( fset_b2 @ Xa ) ) ) ).
% inter_fset
thf(fact_813_inter__fset,axiom,
! [X: fset_a,Xa: fset_a] :
( ( fset_a2 @ ( inf_inf_fset_a @ X @ Xa ) )
= ( inf_inf_set_a @ ( fset_a2 @ X ) @ ( fset_a2 @ Xa ) ) ) ).
% inter_fset
thf(fact_814_inter__fset,axiom,
! [X: fset_list_a,Xa: fset_list_a] :
( ( fset_list_a2 @ ( inf_inf_fset_list_a @ X @ Xa ) )
= ( inf_inf_set_list_a @ ( fset_list_a2 @ X ) @ ( fset_list_a2 @ Xa ) ) ) ).
% inter_fset
thf(fact_815_union__fset,axiom,
! [X: fset_P2153231429829016240_a_b_b,Xa: fset_P2153231429829016240_a_b_b] :
( ( fset_P9138963618725001425_a_b_b @ ( sup_su5028961100972612572_a_b_b @ X @ Xa ) )
= ( sup_su6988119916188116706_a_b_b @ ( fset_P9138963618725001425_a_b_b @ X ) @ ( fset_P9138963618725001425_a_b_b @ Xa ) ) ) ).
% union_fset
thf(fact_816_union__fset,axiom,
! [X: fset_b,Xa: fset_b] :
( ( fset_b2 @ ( sup_sup_fset_b @ X @ Xa ) )
= ( sup_sup_set_b @ ( fset_b2 @ X ) @ ( fset_b2 @ Xa ) ) ) ).
% union_fset
thf(fact_817_root__noty__if__in__insert__before,axiom,
! [T12: dtree_list_a_b,E12: b,Y: list_a,V: list_a,E3: b,Xs2: fset_P2153231429829016240_a_b_b] :
( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E12 )
@ ( fset_P9138963618725001425_a_b_b
@ ( ffold_6356110402769621009_a_b_b
@ ( produc7216629644754684766_a_b_b
@ ^ [T13: dtree_list_a_b,E1: b] :
( finser2303212525150181944_a_b_b
@ ( if_Pro807711176916573242_a_b_b
@ ( ( root_list_a_b @ T13 )
= Y )
@ ( produc7704165765595008946_a_b_b @ ( node_list_a_b @ V @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T13 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ E3 )
@ ( produc7704165765595008946_a_b_b @ T13 @ E1 ) ) ) )
@ bot_bo2248824169281960260_a_b_b
@ Xs2 ) ) )
=> ( ( V != Y )
=> ( ( root_list_a_b @ T12 )
!= Y ) ) ) ).
% root_noty_if_in_insert_before
thf(fact_818_root__not__y__in__new,axiom,
! [T12: dtree_list_a_b,E12: b,Xs2: fset_P2153231429829016240_a_b_b,Y: list_a,V: list_a,E3: b] :
( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E12 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( ( root_list_a_b @ T12 )
!= Y )
=> ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E12 )
@ ( fset_P9138963618725001425_a_b_b
@ ( ffold_6356110402769621009_a_b_b
@ ( produc7216629644754684766_a_b_b
@ ^ [T13: dtree_list_a_b,E1: b] :
( finser2303212525150181944_a_b_b
@ ( if_Pro807711176916573242_a_b_b
@ ( ( root_list_a_b @ T13 )
= Y )
@ ( produc7704165765595008946_a_b_b @ ( node_list_a_b @ V @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T13 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ E3 )
@ ( produc7704165765595008946_a_b_b @ T13 @ E1 ) ) ) )
@ bot_bo2248824169281960260_a_b_b
@ Xs2 ) ) ) ) ) ).
% root_not_y_in_new
thf(fact_819_root__not__new__in__orig,axiom,
! [T12: dtree_list_a_b,E12: b,Y: list_a,V: list_a,E3: b,Xs2: fset_P2153231429829016240_a_b_b] :
( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E12 )
@ ( fset_P9138963618725001425_a_b_b
@ ( ffold_6356110402769621009_a_b_b
@ ( produc7216629644754684766_a_b_b
@ ^ [T13: dtree_list_a_b,E1: b] :
( finser2303212525150181944_a_b_b
@ ( if_Pro807711176916573242_a_b_b
@ ( ( root_list_a_b @ T13 )
= Y )
@ ( produc7704165765595008946_a_b_b @ ( node_list_a_b @ V @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T13 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ E3 )
@ ( produc7704165765595008946_a_b_b @ T13 @ E1 ) ) ) )
@ bot_bo2248824169281960260_a_b_b
@ Xs2 ) ) )
=> ( ( ( root_list_a_b @ T12 )
!= V )
=> ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E12 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) ) ) ).
% root_not_new_in_orig
thf(fact_820_insert__before__wf__darcs,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,E3: b,T12: dtree_list_a_b,E12: b,Y: list_a,V: list_a] :
( ( wf_darcs_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
=> ( ~ ( member_b @ E3 @ ( darcs_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E12 )
@ ( fset_P9138963618725001425_a_b_b
@ ( ffold_6356110402769621009_a_b_b
@ ( produc7216629644754684766_a_b_b
@ ^ [T13: dtree_list_a_b,E1: b] :
( finser2303212525150181944_a_b_b
@ ( if_Pro807711176916573242_a_b_b
@ ( ( root_list_a_b @ T13 )
= Y )
@ ( produc7704165765595008946_a_b_b @ ( node_list_a_b @ V @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T13 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ E3 )
@ ( produc7704165765595008946_a_b_b @ T13 @ E1 ) ) ) )
@ bot_bo2248824169281960260_a_b_b
@ Xs2 ) ) )
=> ( wf_darcs_list_a_b @ T12 ) ) ) ) ).
% insert_before_wf_darcs
thf(fact_821_funion__fempty__right,axiom,
! [A: fset_P2153231429829016240_a_b_b] :
( ( sup_su5028961100972612572_a_b_b @ A @ bot_bo2248824169281960260_a_b_b )
= A ) ).
% funion_fempty_right
thf(fact_822_funion__fempty__left,axiom,
! [B5: fset_P2153231429829016240_a_b_b] :
( ( sup_su5028961100972612572_a_b_b @ bot_bo2248824169281960260_a_b_b @ B5 )
= B5 ) ).
% funion_fempty_left
thf(fact_823_finter__fempty__right,axiom,
! [A: fset_P2153231429829016240_a_b_b] :
( ( inf_in525273840840435522_a_b_b @ A @ bot_bo2248824169281960260_a_b_b )
= bot_bo2248824169281960260_a_b_b ) ).
% finter_fempty_right
thf(fact_824_finter__fempty__left,axiom,
! [B5: fset_P2153231429829016240_a_b_b] :
( ( inf_in525273840840435522_a_b_b @ bot_bo2248824169281960260_a_b_b @ B5 )
= bot_bo2248824169281960260_a_b_b ) ).
% finter_fempty_left
thf(fact_825_Collect__case__prod__mono,axiom,
! [A: dtree_list_a_b > b > $o,B5: dtree_list_a_b > b > $o] :
( ( ord_le4372168669586844721_b_b_o @ A @ B5 )
=> ( ord_le1619362961161175062_a_b_b @ ( collec3241703154345475797_a_b_b @ ( produc336013011823628281_b_b_o @ A ) ) @ ( collec3241703154345475797_a_b_b @ ( produc336013011823628281_b_b_o @ B5 ) ) ) ) ).
% Collect_case_prod_mono
thf(fact_826_fsubset__finsertI2,axiom,
! [A: fset_P2153231429829016240_a_b_b,B5: fset_P2153231429829016240_a_b_b,B: produc6499617310964463488_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ A @ B5 )
=> ( ord_le8870638447146015504_a_b_b @ A @ ( finser2303212525150181944_a_b_b @ B @ B5 ) ) ) ).
% fsubset_finsertI2
thf(fact_827_fsubset__finsertI,axiom,
! [B5: fset_P2153231429829016240_a_b_b,A2: produc6499617310964463488_a_b_b] : ( ord_le8870638447146015504_a_b_b @ B5 @ ( finser2303212525150181944_a_b_b @ A2 @ B5 ) ) ).
% fsubset_finsertI
thf(fact_828_finsert__mono,axiom,
! [C2: fset_P2153231429829016240_a_b_b,D2: fset_P2153231429829016240_a_b_b,A2: produc6499617310964463488_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ C2 @ D2 )
=> ( ord_le8870638447146015504_a_b_b @ ( finser2303212525150181944_a_b_b @ A2 @ C2 ) @ ( finser2303212525150181944_a_b_b @ A2 @ D2 ) ) ) ).
% finsert_mono
thf(fact_829_sup__set__def,axiom,
( sup_su6988119916188116706_a_b_b
= ( ^ [A6: set_Pr3443975907877334966_a_b_b,B6: set_Pr3443975907877334966_a_b_b] :
( collec3241703154345475797_a_b_b
@ ( sup_su1217491931452879395_b_b_o
@ ^ [X4: produc6499617310964463488_a_b_b] : ( member4695696432722591383_a_b_b @ X4 @ A6 )
@ ^ [X4: produc6499617310964463488_a_b_b] : ( member4695696432722591383_a_b_b @ X4 @ B6 ) ) ) ) ) ).
% sup_set_def
thf(fact_830_sup__set__def,axiom,
( sup_sup_set_list_a
= ( ^ [A6: set_list_a,B6: set_list_a] :
( collect_list_a
@ ( sup_sup_list_a_o
@ ^ [X4: list_a] : ( member_list_a @ X4 @ A6 )
@ ^ [X4: list_a] : ( member_list_a @ X4 @ B6 ) ) ) ) ) ).
% sup_set_def
thf(fact_831_sup__set__def,axiom,
( sup_sup_set_a
= ( ^ [A6: set_a,B6: set_a] :
( collect_a
@ ( sup_sup_a_o
@ ^ [X4: a] : ( member_a @ X4 @ A6 )
@ ^ [X4: a] : ( member_a @ X4 @ B6 ) ) ) ) ) ).
% sup_set_def
thf(fact_832_sup__set__def,axiom,
( sup_su5199045856643764822st_a_b
= ( ^ [A6: set_dtree_list_a_b,B6: set_dtree_list_a_b] :
( collec2944820760411501129st_a_b
@ ( sup_su1287617999525858223_a_b_o
@ ^ [X4: dtree_list_a_b] : ( member551035911493665803st_a_b @ X4 @ A6 )
@ ^ [X4: dtree_list_a_b] : ( member551035911493665803st_a_b @ X4 @ B6 ) ) ) ) ) ).
% sup_set_def
thf(fact_833_sup__set__def,axiom,
( sup_sup_set_b
= ( ^ [A6: set_b,B6: set_b] :
( collect_b
@ ( sup_sup_b_o
@ ^ [X4: b] : ( member_b @ X4 @ A6 )
@ ^ [X4: b] : ( member_b @ X4 @ B6 ) ) ) ) ) ).
% sup_set_def
thf(fact_834_inf__set__def,axiom,
( inf_in7962574851115124040_a_b_b
= ( ^ [A6: set_Pr3443975907877334966_a_b_b,B6: set_Pr3443975907877334966_a_b_b] :
( collec3241703154345475797_a_b_b
@ ( inf_in9196368829404727101_b_b_o
@ ^ [X4: produc6499617310964463488_a_b_b] : ( member4695696432722591383_a_b_b @ X4 @ A6 )
@ ^ [X4: produc6499617310964463488_a_b_b] : ( member4695696432722591383_a_b_b @ X4 @ B6 ) ) ) ) ) ).
% inf_set_def
thf(fact_835_inf__set__def,axiom,
( inf_in3355993651213403836st_a_b
= ( ^ [A6: set_dtree_list_a_b,B6: set_dtree_list_a_b] :
( collec2944820760411501129st_a_b
@ ( inf_in579736399448705737_a_b_o
@ ^ [X4: dtree_list_a_b] : ( member551035911493665803st_a_b @ X4 @ A6 )
@ ^ [X4: dtree_list_a_b] : ( member551035911493665803st_a_b @ X4 @ B6 ) ) ) ) ) ).
% inf_set_def
thf(fact_836_inf__set__def,axiom,
( inf_inf_set_b
= ( ^ [A6: set_b,B6: set_b] :
( collect_b
@ ( inf_inf_b_o
@ ^ [X4: b] : ( member_b @ X4 @ A6 )
@ ^ [X4: b] : ( member_b @ X4 @ B6 ) ) ) ) ) ).
% inf_set_def
thf(fact_837_inf__set__def,axiom,
( inf_inf_set_a
= ( ^ [A6: set_a,B6: set_a] :
( collect_a
@ ( inf_inf_a_o
@ ^ [X4: a] : ( member_a @ X4 @ A6 )
@ ^ [X4: a] : ( member_a @ X4 @ B6 ) ) ) ) ) ).
% inf_set_def
thf(fact_838_inf__set__def,axiom,
( inf_inf_set_list_a
= ( ^ [A6: set_list_a,B6: set_list_a] :
( collect_list_a
@ ( inf_inf_list_a_o
@ ^ [X4: list_a] : ( member_list_a @ X4 @ A6 )
@ ^ [X4: list_a] : ( member_list_a @ X4 @ B6 ) ) ) ) ) ).
% inf_set_def
thf(fact_839_less__eq__set__def,axiom,
( ord_le1619362961161175062_a_b_b
= ( ^ [A6: set_Pr3443975907877334966_a_b_b,B6: set_Pr3443975907877334966_a_b_b] :
( ord_le5434074902312169967_b_b_o
@ ^ [X4: produc6499617310964463488_a_b_b] : ( member4695696432722591383_a_b_b @ X4 @ A6 )
@ ^ [X4: produc6499617310964463488_a_b_b] : ( member4695696432722591383_a_b_b @ X4 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_840_less__eq__set__def,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A6: set_list_a,B6: set_list_a] :
( ord_less_eq_list_a_o
@ ^ [X4: list_a] : ( member_list_a @ X4 @ A6 )
@ ^ [X4: list_a] : ( member_list_a @ X4 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_841_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
( ord_less_eq_a_o
@ ^ [X4: a] : ( member_a @ X4 @ A6 )
@ ^ [X4: a] : ( member_a @ X4 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_842_less__eq__set__def,axiom,
( ord_le7599451563663638410st_a_b
= ( ^ [A6: set_dtree_list_a_b,B6: set_dtree_list_a_b] :
( ord_le739112815111206779_a_b_o
@ ^ [X4: dtree_list_a_b] : ( member551035911493665803st_a_b @ X4 @ A6 )
@ ^ [X4: dtree_list_a_b] : ( member551035911493665803st_a_b @ X4 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_843_less__eq__set__def,axiom,
( ord_less_eq_set_b
= ( ^ [A6: set_b,B6: set_b] :
( ord_less_eq_b_o
@ ^ [X4: b] : ( member_b @ X4 @ A6 )
@ ^ [X4: b] : ( member_b @ X4 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_844_funion__fsingleton__iff,axiom,
! [A: fset_P2153231429829016240_a_b_b,B5: fset_P2153231429829016240_a_b_b,X: produc6499617310964463488_a_b_b] :
( ( ( sup_su5028961100972612572_a_b_b @ A @ B5 )
= ( finser2303212525150181944_a_b_b @ X @ bot_bo2248824169281960260_a_b_b ) )
= ( ( ( A = bot_bo2248824169281960260_a_b_b )
& ( B5
= ( finser2303212525150181944_a_b_b @ X @ bot_bo2248824169281960260_a_b_b ) ) )
| ( ( A
= ( finser2303212525150181944_a_b_b @ X @ bot_bo2248824169281960260_a_b_b ) )
& ( B5 = bot_bo2248824169281960260_a_b_b ) )
| ( ( A
= ( finser2303212525150181944_a_b_b @ X @ bot_bo2248824169281960260_a_b_b ) )
& ( B5
= ( finser2303212525150181944_a_b_b @ X @ bot_bo2248824169281960260_a_b_b ) ) ) ) ) ).
% funion_fsingleton_iff
thf(fact_845_fsingleton__funion__iff,axiom,
! [X: produc6499617310964463488_a_b_b,A: fset_P2153231429829016240_a_b_b,B5: fset_P2153231429829016240_a_b_b] :
( ( ( finser2303212525150181944_a_b_b @ X @ bot_bo2248824169281960260_a_b_b )
= ( sup_su5028961100972612572_a_b_b @ A @ B5 ) )
= ( ( ( A = bot_bo2248824169281960260_a_b_b )
& ( B5
= ( finser2303212525150181944_a_b_b @ X @ bot_bo2248824169281960260_a_b_b ) ) )
| ( ( A
= ( finser2303212525150181944_a_b_b @ X @ bot_bo2248824169281960260_a_b_b ) )
& ( B5 = bot_bo2248824169281960260_a_b_b ) )
| ( ( A
= ( finser2303212525150181944_a_b_b @ X @ bot_bo2248824169281960260_a_b_b ) )
& ( B5
= ( finser2303212525150181944_a_b_b @ X @ bot_bo2248824169281960260_a_b_b ) ) ) ) ) ).
% fsingleton_funion_iff
thf(fact_846_finsert__is__funion,axiom,
( finser2303212525150181944_a_b_b
= ( ^ [A4: produc6499617310964463488_a_b_b] : ( sup_su5028961100972612572_a_b_b @ ( finser2303212525150181944_a_b_b @ A4 @ bot_bo2248824169281960260_a_b_b ) ) ) ) ).
% finsert_is_funion
thf(fact_847_less__eq__fset_Orep__eq,axiom,
( ord_less_eq_fset_b
= ( ^ [X4: fset_b,Xa2: fset_b] : ( ord_less_eq_set_b @ ( fset_b2 @ X4 ) @ ( fset_b2 @ Xa2 ) ) ) ) ).
% less_eq_fset.rep_eq
thf(fact_848_less__eq__fset_Orep__eq,axiom,
( ord_le8870638447146015504_a_b_b
= ( ^ [X4: fset_P2153231429829016240_a_b_b,Xa2: fset_P2153231429829016240_a_b_b] : ( ord_le1619362961161175062_a_b_b @ ( fset_P9138963618725001425_a_b_b @ X4 ) @ ( fset_P9138963618725001425_a_b_b @ Xa2 ) ) ) ) ).
% less_eq_fset.rep_eq
thf(fact_849_fsubset__fsingletonD,axiom,
! [A: fset_P2153231429829016240_a_b_b,X: produc6499617310964463488_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ A @ ( finser2303212525150181944_a_b_b @ X @ bot_bo2248824169281960260_a_b_b ) )
=> ( ( A = bot_bo2248824169281960260_a_b_b )
| ( A
= ( finser2303212525150181944_a_b_b @ X @ bot_bo2248824169281960260_a_b_b ) ) ) ) ).
% fsubset_fsingletonD
thf(fact_850_wf__darcs__sub,axiom,
! [Xs2: fset_P2153231429829016240_a_b_b,Ys: fset_P2153231429829016240_a_b_b,R3: list_a,R2: list_a] :
( ( ord_le8870638447146015504_a_b_b @ Xs2 @ Ys )
=> ( ( wf_darcs_list_a_b @ ( node_list_a_b @ R3 @ Ys ) )
=> ( wf_darcs_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) ) ) ).
% wf_darcs_sub
thf(fact_851_inf__sup__aci_I4_J,axiom,
! [X: set_b,Y: set_b] :
( ( inf_inf_set_b @ X @ ( inf_inf_set_b @ X @ Y ) )
= ( inf_inf_set_b @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_852_inf__sup__aci_I4_J,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ X @ Y ) )
= ( inf_inf_set_a @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_853_inf__sup__aci_I4_J,axiom,
! [X: set_list_a,Y: set_list_a] :
( ( inf_inf_set_list_a @ X @ ( inf_inf_set_list_a @ X @ Y ) )
= ( inf_inf_set_list_a @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_854_inf__sup__aci_I3_J,axiom,
! [X: set_b,Y: set_b,Z2: set_b] :
( ( inf_inf_set_b @ X @ ( inf_inf_set_b @ Y @ Z2 ) )
= ( inf_inf_set_b @ Y @ ( inf_inf_set_b @ X @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_855_inf__sup__aci_I3_J,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) )
= ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_856_inf__sup__aci_I3_J,axiom,
! [X: set_list_a,Y: set_list_a,Z2: set_list_a] :
( ( inf_inf_set_list_a @ X @ ( inf_inf_set_list_a @ Y @ Z2 ) )
= ( inf_inf_set_list_a @ Y @ ( inf_inf_set_list_a @ X @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_857_inf__sup__aci_I2_J,axiom,
! [X: set_b,Y: set_b,Z2: set_b] :
( ( inf_inf_set_b @ ( inf_inf_set_b @ X @ Y ) @ Z2 )
= ( inf_inf_set_b @ X @ ( inf_inf_set_b @ Y @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_858_inf__sup__aci_I2_J,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z2 )
= ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_859_inf__sup__aci_I2_J,axiom,
! [X: set_list_a,Y: set_list_a,Z2: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ X @ Y ) @ Z2 )
= ( inf_inf_set_list_a @ X @ ( inf_inf_set_list_a @ Y @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_860_inf__sup__aci_I1_J,axiom,
( inf_inf_set_b
= ( ^ [X4: set_b,Y2: set_b] : ( inf_inf_set_b @ Y2 @ X4 ) ) ) ).
% inf_sup_aci(1)
thf(fact_861_inf__sup__aci_I1_J,axiom,
( inf_inf_set_a
= ( ^ [X4: set_a,Y2: set_a] : ( inf_inf_set_a @ Y2 @ X4 ) ) ) ).
% inf_sup_aci(1)
thf(fact_862_inf__sup__aci_I1_J,axiom,
( inf_inf_set_list_a
= ( ^ [X4: set_list_a,Y2: set_list_a] : ( inf_inf_set_list_a @ Y2 @ X4 ) ) ) ).
% inf_sup_aci(1)
thf(fact_863_inf_Oassoc,axiom,
! [A2: set_b,B: set_b,C: set_b] :
( ( inf_inf_set_b @ ( inf_inf_set_b @ A2 @ B ) @ C )
= ( inf_inf_set_b @ A2 @ ( inf_inf_set_b @ B @ C ) ) ) ).
% inf.assoc
thf(fact_864_inf_Oassoc,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B ) @ C )
= ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B @ C ) ) ) ).
% inf.assoc
thf(fact_865_inf_Oassoc,axiom,
! [A2: set_list_a,B: set_list_a,C: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ A2 @ B ) @ C )
= ( inf_inf_set_list_a @ A2 @ ( inf_inf_set_list_a @ B @ C ) ) ) ).
% inf.assoc
thf(fact_866_inf__assoc,axiom,
! [X: set_b,Y: set_b,Z2: set_b] :
( ( inf_inf_set_b @ ( inf_inf_set_b @ X @ Y ) @ Z2 )
= ( inf_inf_set_b @ X @ ( inf_inf_set_b @ Y @ Z2 ) ) ) ).
% inf_assoc
thf(fact_867_inf__assoc,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z2 )
= ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) ) ) ).
% inf_assoc
thf(fact_868_inf__assoc,axiom,
! [X: set_list_a,Y: set_list_a,Z2: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ X @ Y ) @ Z2 )
= ( inf_inf_set_list_a @ X @ ( inf_inf_set_list_a @ Y @ Z2 ) ) ) ).
% inf_assoc
thf(fact_869_inf_Ocommute,axiom,
( inf_inf_set_b
= ( ^ [A4: set_b,B3: set_b] : ( inf_inf_set_b @ B3 @ A4 ) ) ) ).
% inf.commute
thf(fact_870_inf_Ocommute,axiom,
( inf_inf_set_a
= ( ^ [A4: set_a,B3: set_a] : ( inf_inf_set_a @ B3 @ A4 ) ) ) ).
% inf.commute
thf(fact_871_inf_Ocommute,axiom,
( inf_inf_set_list_a
= ( ^ [A4: set_list_a,B3: set_list_a] : ( inf_inf_set_list_a @ B3 @ A4 ) ) ) ).
% inf.commute
thf(fact_872_inf__commute,axiom,
( inf_inf_set_b
= ( ^ [X4: set_b,Y2: set_b] : ( inf_inf_set_b @ Y2 @ X4 ) ) ) ).
% inf_commute
thf(fact_873_inf__commute,axiom,
( inf_inf_set_a
= ( ^ [X4: set_a,Y2: set_a] : ( inf_inf_set_a @ Y2 @ X4 ) ) ) ).
% inf_commute
thf(fact_874_inf__commute,axiom,
( inf_inf_set_list_a
= ( ^ [X4: set_list_a,Y2: set_list_a] : ( inf_inf_set_list_a @ Y2 @ X4 ) ) ) ).
% inf_commute
thf(fact_875_inf_Oleft__commute,axiom,
! [B: set_b,A2: set_b,C: set_b] :
( ( inf_inf_set_b @ B @ ( inf_inf_set_b @ A2 @ C ) )
= ( inf_inf_set_b @ A2 @ ( inf_inf_set_b @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_876_inf_Oleft__commute,axiom,
! [B: set_a,A2: set_a,C: set_a] :
( ( inf_inf_set_a @ B @ ( inf_inf_set_a @ A2 @ C ) )
= ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_877_inf_Oleft__commute,axiom,
! [B: set_list_a,A2: set_list_a,C: set_list_a] :
( ( inf_inf_set_list_a @ B @ ( inf_inf_set_list_a @ A2 @ C ) )
= ( inf_inf_set_list_a @ A2 @ ( inf_inf_set_list_a @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_878_inf__left__commute,axiom,
! [X: set_b,Y: set_b,Z2: set_b] :
( ( inf_inf_set_b @ X @ ( inf_inf_set_b @ Y @ Z2 ) )
= ( inf_inf_set_b @ Y @ ( inf_inf_set_b @ X @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_879_inf__left__commute,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) )
= ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_880_inf__left__commute,axiom,
! [X: set_list_a,Y: set_list_a,Z2: set_list_a] :
( ( inf_inf_set_list_a @ X @ ( inf_inf_set_list_a @ Y @ Z2 ) )
= ( inf_inf_set_list_a @ Y @ ( inf_inf_set_list_a @ X @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_881_inf__sup__aci_I8_J,axiom,
! [X: set_b,Y: set_b] :
( ( sup_sup_set_b @ X @ ( sup_sup_set_b @ X @ Y ) )
= ( sup_sup_set_b @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_882_inf__sup__aci_I7_J,axiom,
! [X: set_b,Y: set_b,Z2: set_b] :
( ( sup_sup_set_b @ X @ ( sup_sup_set_b @ Y @ Z2 ) )
= ( sup_sup_set_b @ Y @ ( sup_sup_set_b @ X @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_883_inf__sup__aci_I6_J,axiom,
! [X: set_b,Y: set_b,Z2: set_b] :
( ( sup_sup_set_b @ ( sup_sup_set_b @ X @ Y ) @ Z2 )
= ( sup_sup_set_b @ X @ ( sup_sup_set_b @ Y @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_884_inf__sup__aci_I5_J,axiom,
( sup_sup_set_b
= ( ^ [X4: set_b,Y2: set_b] : ( sup_sup_set_b @ Y2 @ X4 ) ) ) ).
% inf_sup_aci(5)
thf(fact_885_sup_Oassoc,axiom,
! [A2: set_b,B: set_b,C: set_b] :
( ( sup_sup_set_b @ ( sup_sup_set_b @ A2 @ B ) @ C )
= ( sup_sup_set_b @ A2 @ ( sup_sup_set_b @ B @ C ) ) ) ).
% sup.assoc
thf(fact_886_sup__assoc,axiom,
! [X: set_b,Y: set_b,Z2: set_b] :
( ( sup_sup_set_b @ ( sup_sup_set_b @ X @ Y ) @ Z2 )
= ( sup_sup_set_b @ X @ ( sup_sup_set_b @ Y @ Z2 ) ) ) ).
% sup_assoc
thf(fact_887_sup_Ocommute,axiom,
( sup_sup_set_b
= ( ^ [A4: set_b,B3: set_b] : ( sup_sup_set_b @ B3 @ A4 ) ) ) ).
% sup.commute
thf(fact_888_sup__commute,axiom,
( sup_sup_set_b
= ( ^ [X4: set_b,Y2: set_b] : ( sup_sup_set_b @ Y2 @ X4 ) ) ) ).
% sup_commute
thf(fact_889_sup_Oleft__commute,axiom,
! [B: set_b,A2: set_b,C: set_b] :
( ( sup_sup_set_b @ B @ ( sup_sup_set_b @ A2 @ C ) )
= ( sup_sup_set_b @ A2 @ ( sup_sup_set_b @ B @ C ) ) ) ).
% sup.left_commute
thf(fact_890_sup__left__commute,axiom,
! [X: set_b,Y: set_b,Z2: set_b] :
( ( sup_sup_set_b @ X @ ( sup_sup_set_b @ Y @ Z2 ) )
= ( sup_sup_set_b @ Y @ ( sup_sup_set_b @ X @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_891_inf__sup__ord_I2_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_892_inf__sup__ord_I2_J,axiom,
! [X: set_list_a,Y: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_893_inf__sup__ord_I2_J,axiom,
! [X: set_b,Y: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_894_inf__sup__ord_I2_J,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ ( inf_inf_real @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_895_inf__sup__ord_I2_J,axiom,
! [X: fset_P2153231429829016240_a_b_b,Y: fset_P2153231429829016240_a_b_b] : ( ord_le8870638447146015504_a_b_b @ ( inf_in525273840840435522_a_b_b @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_896_inf__sup__ord_I1_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_897_inf__sup__ord_I1_J,axiom,
! [X: set_list_a,Y: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_898_inf__sup__ord_I1_J,axiom,
! [X: set_b,Y: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_899_inf__sup__ord_I1_J,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ ( inf_inf_real @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_900_inf__sup__ord_I1_J,axiom,
! [X: fset_P2153231429829016240_a_b_b,Y: fset_P2153231429829016240_a_b_b] : ( ord_le8870638447146015504_a_b_b @ ( inf_in525273840840435522_a_b_b @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_901_inf__le1,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_902_inf__le1,axiom,
! [X: set_list_a,Y: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_903_inf__le1,axiom,
! [X: set_b,Y: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_904_inf__le1,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ ( inf_inf_real @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_905_inf__le1,axiom,
! [X: fset_P2153231429829016240_a_b_b,Y: fset_P2153231429829016240_a_b_b] : ( ord_le8870638447146015504_a_b_b @ ( inf_in525273840840435522_a_b_b @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_906_inf__le2,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_907_inf__le2,axiom,
! [X: set_list_a,Y: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_908_inf__le2,axiom,
! [X: set_b,Y: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_909_inf__le2,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ ( inf_inf_real @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_910_inf__le2,axiom,
! [X: fset_P2153231429829016240_a_b_b,Y: fset_P2153231429829016240_a_b_b] : ( ord_le8870638447146015504_a_b_b @ ( inf_in525273840840435522_a_b_b @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_911_le__infE,axiom,
! [X: set_a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A2 @ B ) )
=> ~ ( ( ord_less_eq_set_a @ X @ A2 )
=> ~ ( ord_less_eq_set_a @ X @ B ) ) ) ).
% le_infE
thf(fact_912_le__infE,axiom,
! [X: set_list_a,A2: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ X @ ( inf_inf_set_list_a @ A2 @ B ) )
=> ~ ( ( ord_le8861187494160871172list_a @ X @ A2 )
=> ~ ( ord_le8861187494160871172list_a @ X @ B ) ) ) ).
% le_infE
thf(fact_913_le__infE,axiom,
! [X: set_b,A2: set_b,B: set_b] :
( ( ord_less_eq_set_b @ X @ ( inf_inf_set_b @ A2 @ B ) )
=> ~ ( ( ord_less_eq_set_b @ X @ A2 )
=> ~ ( ord_less_eq_set_b @ X @ B ) ) ) ).
% le_infE
thf(fact_914_le__infE,axiom,
! [X: real,A2: real,B: real] :
( ( ord_less_eq_real @ X @ ( inf_inf_real @ A2 @ B ) )
=> ~ ( ( ord_less_eq_real @ X @ A2 )
=> ~ ( ord_less_eq_real @ X @ B ) ) ) ).
% le_infE
thf(fact_915_le__infE,axiom,
! [X: fset_P2153231429829016240_a_b_b,A2: fset_P2153231429829016240_a_b_b,B: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ X @ ( inf_in525273840840435522_a_b_b @ A2 @ B ) )
=> ~ ( ( ord_le8870638447146015504_a_b_b @ X @ A2 )
=> ~ ( ord_le8870638447146015504_a_b_b @ X @ B ) ) ) ).
% le_infE
thf(fact_916_le__infI,axiom,
! [X: set_a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ X @ A2 )
=> ( ( ord_less_eq_set_a @ X @ B )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% le_infI
thf(fact_917_le__infI,axiom,
! [X: set_list_a,A2: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ X @ A2 )
=> ( ( ord_le8861187494160871172list_a @ X @ B )
=> ( ord_le8861187494160871172list_a @ X @ ( inf_inf_set_list_a @ A2 @ B ) ) ) ) ).
% le_infI
thf(fact_918_le__infI,axiom,
! [X: set_b,A2: set_b,B: set_b] :
( ( ord_less_eq_set_b @ X @ A2 )
=> ( ( ord_less_eq_set_b @ X @ B )
=> ( ord_less_eq_set_b @ X @ ( inf_inf_set_b @ A2 @ B ) ) ) ) ).
% le_infI
thf(fact_919_le__infI,axiom,
! [X: real,A2: real,B: real] :
( ( ord_less_eq_real @ X @ A2 )
=> ( ( ord_less_eq_real @ X @ B )
=> ( ord_less_eq_real @ X @ ( inf_inf_real @ A2 @ B ) ) ) ) ).
% le_infI
thf(fact_920_le__infI,axiom,
! [X: fset_P2153231429829016240_a_b_b,A2: fset_P2153231429829016240_a_b_b,B: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ X @ A2 )
=> ( ( ord_le8870638447146015504_a_b_b @ X @ B )
=> ( ord_le8870638447146015504_a_b_b @ X @ ( inf_in525273840840435522_a_b_b @ A2 @ B ) ) ) ) ).
% le_infI
thf(fact_921_inf__mono,axiom,
! [A2: set_a,C: set_a,B: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ ( inf_inf_set_a @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_922_inf__mono,axiom,
! [A2: set_list_a,C: set_list_a,B: set_list_a,D: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ C )
=> ( ( ord_le8861187494160871172list_a @ B @ D )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A2 @ B ) @ ( inf_inf_set_list_a @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_923_inf__mono,axiom,
! [A2: set_b,C: set_b,B: set_b,D: set_b] :
( ( ord_less_eq_set_b @ A2 @ C )
=> ( ( ord_less_eq_set_b @ B @ D )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B ) @ ( inf_inf_set_b @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_924_inf__mono,axiom,
! [A2: real,C: real,B: real,D: real] :
( ( ord_less_eq_real @ A2 @ C )
=> ( ( ord_less_eq_real @ B @ D )
=> ( ord_less_eq_real @ ( inf_inf_real @ A2 @ B ) @ ( inf_inf_real @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_925_inf__mono,axiom,
! [A2: fset_P2153231429829016240_a_b_b,C: fset_P2153231429829016240_a_b_b,B: fset_P2153231429829016240_a_b_b,D: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ A2 @ C )
=> ( ( ord_le8870638447146015504_a_b_b @ B @ D )
=> ( ord_le8870638447146015504_a_b_b @ ( inf_in525273840840435522_a_b_b @ A2 @ B ) @ ( inf_in525273840840435522_a_b_b @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_926_le__infI1,axiom,
! [A2: set_a,X: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ X ) ) ).
% le_infI1
thf(fact_927_le__infI1,axiom,
! [A2: set_list_a,X: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ X )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A2 @ B ) @ X ) ) ).
% le_infI1
thf(fact_928_le__infI1,axiom,
! [A2: set_b,X: set_b,B: set_b] :
( ( ord_less_eq_set_b @ A2 @ X )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B ) @ X ) ) ).
% le_infI1
thf(fact_929_le__infI1,axiom,
! [A2: real,X: real,B: real] :
( ( ord_less_eq_real @ A2 @ X )
=> ( ord_less_eq_real @ ( inf_inf_real @ A2 @ B ) @ X ) ) ).
% le_infI1
thf(fact_930_le__infI1,axiom,
! [A2: fset_P2153231429829016240_a_b_b,X: fset_P2153231429829016240_a_b_b,B: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ A2 @ X )
=> ( ord_le8870638447146015504_a_b_b @ ( inf_in525273840840435522_a_b_b @ A2 @ B ) @ X ) ) ).
% le_infI1
thf(fact_931_le__infI2,axiom,
! [B: set_a,X: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ X ) ) ).
% le_infI2
thf(fact_932_le__infI2,axiom,
! [B: set_list_a,X: set_list_a,A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ B @ X )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A2 @ B ) @ X ) ) ).
% le_infI2
thf(fact_933_le__infI2,axiom,
! [B: set_b,X: set_b,A2: set_b] :
( ( ord_less_eq_set_b @ B @ X )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B ) @ X ) ) ).
% le_infI2
thf(fact_934_le__infI2,axiom,
! [B: real,X: real,A2: real] :
( ( ord_less_eq_real @ B @ X )
=> ( ord_less_eq_real @ ( inf_inf_real @ A2 @ B ) @ X ) ) ).
% le_infI2
thf(fact_935_le__infI2,axiom,
! [B: fset_P2153231429829016240_a_b_b,X: fset_P2153231429829016240_a_b_b,A2: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ B @ X )
=> ( ord_le8870638447146015504_a_b_b @ ( inf_in525273840840435522_a_b_b @ A2 @ B ) @ X ) ) ).
% le_infI2
thf(fact_936_inf_OorderE,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( A2
= ( inf_inf_set_a @ A2 @ B ) ) ) ).
% inf.orderE
thf(fact_937_inf_OorderE,axiom,
! [A2: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B )
=> ( A2
= ( inf_inf_set_list_a @ A2 @ B ) ) ) ).
% inf.orderE
thf(fact_938_inf_OorderE,axiom,
! [A2: set_b,B: set_b] :
( ( ord_less_eq_set_b @ A2 @ B )
=> ( A2
= ( inf_inf_set_b @ A2 @ B ) ) ) ).
% inf.orderE
thf(fact_939_inf_OorderE,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( A2
= ( inf_inf_real @ A2 @ B ) ) ) ).
% inf.orderE
thf(fact_940_inf_OorderE,axiom,
! [A2: fset_P2153231429829016240_a_b_b,B: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ A2 @ B )
=> ( A2
= ( inf_in525273840840435522_a_b_b @ A2 @ B ) ) ) ).
% inf.orderE
thf(fact_941_inf_OorderI,axiom,
! [A2: set_a,B: set_a] :
( ( A2
= ( inf_inf_set_a @ A2 @ B ) )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ).
% inf.orderI
thf(fact_942_inf_OorderI,axiom,
! [A2: set_list_a,B: set_list_a] :
( ( A2
= ( inf_inf_set_list_a @ A2 @ B ) )
=> ( ord_le8861187494160871172list_a @ A2 @ B ) ) ).
% inf.orderI
thf(fact_943_inf_OorderI,axiom,
! [A2: set_b,B: set_b] :
( ( A2
= ( inf_inf_set_b @ A2 @ B ) )
=> ( ord_less_eq_set_b @ A2 @ B ) ) ).
% inf.orderI
thf(fact_944_inf_OorderI,axiom,
! [A2: real,B: real] :
( ( A2
= ( inf_inf_real @ A2 @ B ) )
=> ( ord_less_eq_real @ A2 @ B ) ) ).
% inf.orderI
thf(fact_945_inf_OorderI,axiom,
! [A2: fset_P2153231429829016240_a_b_b,B: fset_P2153231429829016240_a_b_b] :
( ( A2
= ( inf_in525273840840435522_a_b_b @ A2 @ B ) )
=> ( ord_le8870638447146015504_a_b_b @ A2 @ B ) ) ).
% inf.orderI
thf(fact_946_inf__unique,axiom,
! [F: set_a > set_a > set_a,X: set_a,Y: set_a] :
( ! [X3: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( F @ X3 @ Y3 ) @ X3 )
=> ( ! [X3: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( F @ X3 @ Y3 ) @ Y3 )
=> ( ! [X3: set_a,Y3: set_a,Z4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ( ord_less_eq_set_a @ X3 @ Z4 )
=> ( ord_less_eq_set_a @ X3 @ ( F @ Y3 @ Z4 ) ) ) )
=> ( ( inf_inf_set_a @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_947_inf__unique,axiom,
! [F: set_list_a > set_list_a > set_list_a,X: set_list_a,Y: set_list_a] :
( ! [X3: set_list_a,Y3: set_list_a] : ( ord_le8861187494160871172list_a @ ( F @ X3 @ Y3 ) @ X3 )
=> ( ! [X3: set_list_a,Y3: set_list_a] : ( ord_le8861187494160871172list_a @ ( F @ X3 @ Y3 ) @ Y3 )
=> ( ! [X3: set_list_a,Y3: set_list_a,Z4: set_list_a] :
( ( ord_le8861187494160871172list_a @ X3 @ Y3 )
=> ( ( ord_le8861187494160871172list_a @ X3 @ Z4 )
=> ( ord_le8861187494160871172list_a @ X3 @ ( F @ Y3 @ Z4 ) ) ) )
=> ( ( inf_inf_set_list_a @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_948_inf__unique,axiom,
! [F: set_b > set_b > set_b,X: set_b,Y: set_b] :
( ! [X3: set_b,Y3: set_b] : ( ord_less_eq_set_b @ ( F @ X3 @ Y3 ) @ X3 )
=> ( ! [X3: set_b,Y3: set_b] : ( ord_less_eq_set_b @ ( F @ X3 @ Y3 ) @ Y3 )
=> ( ! [X3: set_b,Y3: set_b,Z4: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y3 )
=> ( ( ord_less_eq_set_b @ X3 @ Z4 )
=> ( ord_less_eq_set_b @ X3 @ ( F @ Y3 @ Z4 ) ) ) )
=> ( ( inf_inf_set_b @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_949_inf__unique,axiom,
! [F: real > real > real,X: real,Y: real] :
( ! [X3: real,Y3: real] : ( ord_less_eq_real @ ( F @ X3 @ Y3 ) @ X3 )
=> ( ! [X3: real,Y3: real] : ( ord_less_eq_real @ ( F @ X3 @ Y3 ) @ Y3 )
=> ( ! [X3: real,Y3: real,Z4: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ( ord_less_eq_real @ X3 @ Z4 )
=> ( ord_less_eq_real @ X3 @ ( F @ Y3 @ Z4 ) ) ) )
=> ( ( inf_inf_real @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_950_inf__unique,axiom,
! [F: fset_P2153231429829016240_a_b_b > fset_P2153231429829016240_a_b_b > fset_P2153231429829016240_a_b_b,X: fset_P2153231429829016240_a_b_b,Y: fset_P2153231429829016240_a_b_b] :
( ! [X3: fset_P2153231429829016240_a_b_b,Y3: fset_P2153231429829016240_a_b_b] : ( ord_le8870638447146015504_a_b_b @ ( F @ X3 @ Y3 ) @ X3 )
=> ( ! [X3: fset_P2153231429829016240_a_b_b,Y3: fset_P2153231429829016240_a_b_b] : ( ord_le8870638447146015504_a_b_b @ ( F @ X3 @ Y3 ) @ Y3 )
=> ( ! [X3: fset_P2153231429829016240_a_b_b,Y3: fset_P2153231429829016240_a_b_b,Z4: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ X3 @ Y3 )
=> ( ( ord_le8870638447146015504_a_b_b @ X3 @ Z4 )
=> ( ord_le8870638447146015504_a_b_b @ X3 @ ( F @ Y3 @ Z4 ) ) ) )
=> ( ( inf_in525273840840435522_a_b_b @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_951_le__iff__inf,axiom,
( ord_less_eq_set_a
= ( ^ [X4: set_a,Y2: set_a] :
( ( inf_inf_set_a @ X4 @ Y2 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_952_le__iff__inf,axiom,
( ord_le8861187494160871172list_a
= ( ^ [X4: set_list_a,Y2: set_list_a] :
( ( inf_inf_set_list_a @ X4 @ Y2 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_953_le__iff__inf,axiom,
( ord_less_eq_set_b
= ( ^ [X4: set_b,Y2: set_b] :
( ( inf_inf_set_b @ X4 @ Y2 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_954_le__iff__inf,axiom,
( ord_less_eq_real
= ( ^ [X4: real,Y2: real] :
( ( inf_inf_real @ X4 @ Y2 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_955_le__iff__inf,axiom,
( ord_le8870638447146015504_a_b_b
= ( ^ [X4: fset_P2153231429829016240_a_b_b,Y2: fset_P2153231429829016240_a_b_b] :
( ( inf_in525273840840435522_a_b_b @ X4 @ Y2 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_956_inf_Oabsorb1,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( inf_inf_set_a @ A2 @ B )
= A2 ) ) ).
% inf.absorb1
thf(fact_957_inf_Oabsorb1,axiom,
! [A2: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B )
=> ( ( inf_inf_set_list_a @ A2 @ B )
= A2 ) ) ).
% inf.absorb1
thf(fact_958_inf_Oabsorb1,axiom,
! [A2: set_b,B: set_b] :
( ( ord_less_eq_set_b @ A2 @ B )
=> ( ( inf_inf_set_b @ A2 @ B )
= A2 ) ) ).
% inf.absorb1
thf(fact_959_inf_Oabsorb1,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( inf_inf_real @ A2 @ B )
= A2 ) ) ).
% inf.absorb1
thf(fact_960_inf_Oabsorb1,axiom,
! [A2: fset_P2153231429829016240_a_b_b,B: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ A2 @ B )
=> ( ( inf_in525273840840435522_a_b_b @ A2 @ B )
= A2 ) ) ).
% inf.absorb1
thf(fact_961_inf_Oabsorb2,axiom,
! [B: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B @ A2 )
=> ( ( inf_inf_set_a @ A2 @ B )
= B ) ) ).
% inf.absorb2
thf(fact_962_inf_Oabsorb2,axiom,
! [B: set_list_a,A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ B @ A2 )
=> ( ( inf_inf_set_list_a @ A2 @ B )
= B ) ) ).
% inf.absorb2
thf(fact_963_inf_Oabsorb2,axiom,
! [B: set_b,A2: set_b] :
( ( ord_less_eq_set_b @ B @ A2 )
=> ( ( inf_inf_set_b @ A2 @ B )
= B ) ) ).
% inf.absorb2
thf(fact_964_inf_Oabsorb2,axiom,
! [B: real,A2: real] :
( ( ord_less_eq_real @ B @ A2 )
=> ( ( inf_inf_real @ A2 @ B )
= B ) ) ).
% inf.absorb2
thf(fact_965_inf_Oabsorb2,axiom,
! [B: fset_P2153231429829016240_a_b_b,A2: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ B @ A2 )
=> ( ( inf_in525273840840435522_a_b_b @ A2 @ B )
= B ) ) ).
% inf.absorb2
thf(fact_966_inf__absorb1,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( inf_inf_set_a @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_967_inf__absorb1,axiom,
! [X: set_list_a,Y: set_list_a] :
( ( ord_le8861187494160871172list_a @ X @ Y )
=> ( ( inf_inf_set_list_a @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_968_inf__absorb1,axiom,
! [X: set_b,Y: set_b] :
( ( ord_less_eq_set_b @ X @ Y )
=> ( ( inf_inf_set_b @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_969_inf__absorb1,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( inf_inf_real @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_970_inf__absorb1,axiom,
! [X: fset_P2153231429829016240_a_b_b,Y: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ X @ Y )
=> ( ( inf_in525273840840435522_a_b_b @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_971_inf__absorb2,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( inf_inf_set_a @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_972_inf__absorb2,axiom,
! [Y: set_list_a,X: set_list_a] :
( ( ord_le8861187494160871172list_a @ Y @ X )
=> ( ( inf_inf_set_list_a @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_973_inf__absorb2,axiom,
! [Y: set_b,X: set_b] :
( ( ord_less_eq_set_b @ Y @ X )
=> ( ( inf_inf_set_b @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_974_inf__absorb2,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( inf_inf_real @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_975_inf__absorb2,axiom,
! [Y: fset_P2153231429829016240_a_b_b,X: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ Y @ X )
=> ( ( inf_in525273840840435522_a_b_b @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_976_inf_OboundedE,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B @ C ) )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B )
=> ~ ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_977_inf_OboundedE,axiom,
! [A2: set_list_a,B: set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ ( inf_inf_set_list_a @ B @ C ) )
=> ~ ( ( ord_le8861187494160871172list_a @ A2 @ B )
=> ~ ( ord_le8861187494160871172list_a @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_978_inf_OboundedE,axiom,
! [A2: set_b,B: set_b,C: set_b] :
( ( ord_less_eq_set_b @ A2 @ ( inf_inf_set_b @ B @ C ) )
=> ~ ( ( ord_less_eq_set_b @ A2 @ B )
=> ~ ( ord_less_eq_set_b @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_979_inf_OboundedE,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_eq_real @ A2 @ ( inf_inf_real @ B @ C ) )
=> ~ ( ( ord_less_eq_real @ A2 @ B )
=> ~ ( ord_less_eq_real @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_980_inf_OboundedE,axiom,
! [A2: fset_P2153231429829016240_a_b_b,B: fset_P2153231429829016240_a_b_b,C: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ A2 @ ( inf_in525273840840435522_a_b_b @ B @ C ) )
=> ~ ( ( ord_le8870638447146015504_a_b_b @ A2 @ B )
=> ~ ( ord_le8870638447146015504_a_b_b @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_981_inf_OboundedI,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ A2 @ C )
=> ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_982_inf_OboundedI,axiom,
! [A2: set_list_a,B: set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B )
=> ( ( ord_le8861187494160871172list_a @ A2 @ C )
=> ( ord_le8861187494160871172list_a @ A2 @ ( inf_inf_set_list_a @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_983_inf_OboundedI,axiom,
! [A2: set_b,B: set_b,C: set_b] :
( ( ord_less_eq_set_b @ A2 @ B )
=> ( ( ord_less_eq_set_b @ A2 @ C )
=> ( ord_less_eq_set_b @ A2 @ ( inf_inf_set_b @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_984_inf_OboundedI,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ A2 @ C )
=> ( ord_less_eq_real @ A2 @ ( inf_inf_real @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_985_inf_OboundedI,axiom,
! [A2: fset_P2153231429829016240_a_b_b,B: fset_P2153231429829016240_a_b_b,C: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ A2 @ B )
=> ( ( ord_le8870638447146015504_a_b_b @ A2 @ C )
=> ( ord_le8870638447146015504_a_b_b @ A2 @ ( inf_in525273840840435522_a_b_b @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_986_inf__greatest,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ X @ Z2 )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_987_inf__greatest,axiom,
! [X: set_list_a,Y: set_list_a,Z2: set_list_a] :
( ( ord_le8861187494160871172list_a @ X @ Y )
=> ( ( ord_le8861187494160871172list_a @ X @ Z2 )
=> ( ord_le8861187494160871172list_a @ X @ ( inf_inf_set_list_a @ Y @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_988_inf__greatest,axiom,
! [X: set_b,Y: set_b,Z2: set_b] :
( ( ord_less_eq_set_b @ X @ Y )
=> ( ( ord_less_eq_set_b @ X @ Z2 )
=> ( ord_less_eq_set_b @ X @ ( inf_inf_set_b @ Y @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_989_inf__greatest,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ X @ Z2 )
=> ( ord_less_eq_real @ X @ ( inf_inf_real @ Y @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_990_inf__greatest,axiom,
! [X: fset_P2153231429829016240_a_b_b,Y: fset_P2153231429829016240_a_b_b,Z2: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ X @ Y )
=> ( ( ord_le8870638447146015504_a_b_b @ X @ Z2 )
=> ( ord_le8870638447146015504_a_b_b @ X @ ( inf_in525273840840435522_a_b_b @ Y @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_991_inf_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B3: set_a] :
( A4
= ( inf_inf_set_a @ A4 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_992_inf_Oorder__iff,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A4: set_list_a,B3: set_list_a] :
( A4
= ( inf_inf_set_list_a @ A4 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_993_inf_Oorder__iff,axiom,
( ord_less_eq_set_b
= ( ^ [A4: set_b,B3: set_b] :
( A4
= ( inf_inf_set_b @ A4 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_994_inf_Oorder__iff,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B3: real] :
( A4
= ( inf_inf_real @ A4 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_995_inf_Oorder__iff,axiom,
( ord_le8870638447146015504_a_b_b
= ( ^ [A4: fset_P2153231429829016240_a_b_b,B3: fset_P2153231429829016240_a_b_b] :
( A4
= ( inf_in525273840840435522_a_b_b @ A4 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_996_inf_Ocobounded1,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ A2 ) ).
% inf.cobounded1
thf(fact_997_inf_Ocobounded1,axiom,
! [A2: set_list_a,B: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A2 @ B ) @ A2 ) ).
% inf.cobounded1
thf(fact_998_inf_Ocobounded1,axiom,
! [A2: set_b,B: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B ) @ A2 ) ).
% inf.cobounded1
thf(fact_999_inf_Ocobounded1,axiom,
! [A2: real,B: real] : ( ord_less_eq_real @ ( inf_inf_real @ A2 @ B ) @ A2 ) ).
% inf.cobounded1
thf(fact_1000_inf_Ocobounded1,axiom,
! [A2: fset_P2153231429829016240_a_b_b,B: fset_P2153231429829016240_a_b_b] : ( ord_le8870638447146015504_a_b_b @ ( inf_in525273840840435522_a_b_b @ A2 @ B ) @ A2 ) ).
% inf.cobounded1
thf(fact_1001_inf_Ocobounded2,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ B ) ).
% inf.cobounded2
thf(fact_1002_inf_Ocobounded2,axiom,
! [A2: set_list_a,B: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A2 @ B ) @ B ) ).
% inf.cobounded2
thf(fact_1003_inf_Ocobounded2,axiom,
! [A2: set_b,B: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B ) @ B ) ).
% inf.cobounded2
thf(fact_1004_inf_Ocobounded2,axiom,
! [A2: real,B: real] : ( ord_less_eq_real @ ( inf_inf_real @ A2 @ B ) @ B ) ).
% inf.cobounded2
thf(fact_1005_inf_Ocobounded2,axiom,
! [A2: fset_P2153231429829016240_a_b_b,B: fset_P2153231429829016240_a_b_b] : ( ord_le8870638447146015504_a_b_b @ ( inf_in525273840840435522_a_b_b @ A2 @ B ) @ B ) ).
% inf.cobounded2
thf(fact_1006_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( inf_inf_set_a @ A4 @ B3 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_1007_inf_Oabsorb__iff1,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A4: set_list_a,B3: set_list_a] :
( ( inf_inf_set_list_a @ A4 @ B3 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_1008_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_b
= ( ^ [A4: set_b,B3: set_b] :
( ( inf_inf_set_b @ A4 @ B3 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_1009_inf_Oabsorb__iff1,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B3: real] :
( ( inf_inf_real @ A4 @ B3 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_1010_inf_Oabsorb__iff1,axiom,
( ord_le8870638447146015504_a_b_b
= ( ^ [A4: fset_P2153231429829016240_a_b_b,B3: fset_P2153231429829016240_a_b_b] :
( ( inf_in525273840840435522_a_b_b @ A4 @ B3 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_1011_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A4: set_a] :
( ( inf_inf_set_a @ A4 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_1012_inf_Oabsorb__iff2,axiom,
( ord_le8861187494160871172list_a
= ( ^ [B3: set_list_a,A4: set_list_a] :
( ( inf_inf_set_list_a @ A4 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_1013_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_b
= ( ^ [B3: set_b,A4: set_b] :
( ( inf_inf_set_b @ A4 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_1014_inf_Oabsorb__iff2,axiom,
( ord_less_eq_real
= ( ^ [B3: real,A4: real] :
( ( inf_inf_real @ A4 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_1015_inf_Oabsorb__iff2,axiom,
( ord_le8870638447146015504_a_b_b
= ( ^ [B3: fset_P2153231429829016240_a_b_b,A4: fset_P2153231429829016240_a_b_b] :
( ( inf_in525273840840435522_a_b_b @ A4 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_1016_inf_OcoboundedI1,axiom,
! [A2: set_a,C: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_1017_inf_OcoboundedI1,axiom,
! [A2: set_list_a,C: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ C )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A2 @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_1018_inf_OcoboundedI1,axiom,
! [A2: set_b,C: set_b,B: set_b] :
( ( ord_less_eq_set_b @ A2 @ C )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_1019_inf_OcoboundedI1,axiom,
! [A2: real,C: real,B: real] :
( ( ord_less_eq_real @ A2 @ C )
=> ( ord_less_eq_real @ ( inf_inf_real @ A2 @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_1020_inf_OcoboundedI1,axiom,
! [A2: fset_P2153231429829016240_a_b_b,C: fset_P2153231429829016240_a_b_b,B: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ A2 @ C )
=> ( ord_le8870638447146015504_a_b_b @ ( inf_in525273840840435522_a_b_b @ A2 @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_1021_inf_OcoboundedI2,axiom,
! [B: set_a,C: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_1022_inf_OcoboundedI2,axiom,
! [B: set_list_a,C: set_list_a,A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ B @ C )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A2 @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_1023_inf_OcoboundedI2,axiom,
! [B: set_b,C: set_b,A2: set_b] :
( ( ord_less_eq_set_b @ B @ C )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_1024_inf_OcoboundedI2,axiom,
! [B: real,C: real,A2: real] :
( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ ( inf_inf_real @ A2 @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_1025_inf_OcoboundedI2,axiom,
! [B: fset_P2153231429829016240_a_b_b,C: fset_P2153231429829016240_a_b_b,A2: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ B @ C )
=> ( ord_le8870638447146015504_a_b_b @ ( inf_in525273840840435522_a_b_b @ A2 @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_1026_inf__sup__ord_I4_J,axiom,
! [Y: set_b,X: set_b] : ( ord_less_eq_set_b @ Y @ ( sup_sup_set_b @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_1027_inf__sup__ord_I4_J,axiom,
! [Y: real,X: real] : ( ord_less_eq_real @ Y @ ( sup_sup_real @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_1028_inf__sup__ord_I4_J,axiom,
! [Y: fset_P2153231429829016240_a_b_b,X: fset_P2153231429829016240_a_b_b] : ( ord_le8870638447146015504_a_b_b @ Y @ ( sup_su5028961100972612572_a_b_b @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_1029_inf__sup__ord_I3_J,axiom,
! [X: set_b,Y: set_b] : ( ord_less_eq_set_b @ X @ ( sup_sup_set_b @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_1030_inf__sup__ord_I3_J,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ X @ ( sup_sup_real @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_1031_inf__sup__ord_I3_J,axiom,
! [X: fset_P2153231429829016240_a_b_b,Y: fset_P2153231429829016240_a_b_b] : ( ord_le8870638447146015504_a_b_b @ X @ ( sup_su5028961100972612572_a_b_b @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_1032_le__supE,axiom,
! [A2: set_b,B: set_b,X: set_b] :
( ( ord_less_eq_set_b @ ( sup_sup_set_b @ A2 @ B ) @ X )
=> ~ ( ( ord_less_eq_set_b @ A2 @ X )
=> ~ ( ord_less_eq_set_b @ B @ X ) ) ) ).
% le_supE
thf(fact_1033_le__supE,axiom,
! [A2: real,B: real,X: real] :
( ( ord_less_eq_real @ ( sup_sup_real @ A2 @ B ) @ X )
=> ~ ( ( ord_less_eq_real @ A2 @ X )
=> ~ ( ord_less_eq_real @ B @ X ) ) ) ).
% le_supE
thf(fact_1034_le__supE,axiom,
! [A2: fset_P2153231429829016240_a_b_b,B: fset_P2153231429829016240_a_b_b,X: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ ( sup_su5028961100972612572_a_b_b @ A2 @ B ) @ X )
=> ~ ( ( ord_le8870638447146015504_a_b_b @ A2 @ X )
=> ~ ( ord_le8870638447146015504_a_b_b @ B @ X ) ) ) ).
% le_supE
thf(fact_1035_le__supI,axiom,
! [A2: set_b,X: set_b,B: set_b] :
( ( ord_less_eq_set_b @ A2 @ X )
=> ( ( ord_less_eq_set_b @ B @ X )
=> ( ord_less_eq_set_b @ ( sup_sup_set_b @ A2 @ B ) @ X ) ) ) ).
% le_supI
thf(fact_1036_le__supI,axiom,
! [A2: real,X: real,B: real] :
( ( ord_less_eq_real @ A2 @ X )
=> ( ( ord_less_eq_real @ B @ X )
=> ( ord_less_eq_real @ ( sup_sup_real @ A2 @ B ) @ X ) ) ) ).
% le_supI
thf(fact_1037_le__supI,axiom,
! [A2: fset_P2153231429829016240_a_b_b,X: fset_P2153231429829016240_a_b_b,B: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ A2 @ X )
=> ( ( ord_le8870638447146015504_a_b_b @ B @ X )
=> ( ord_le8870638447146015504_a_b_b @ ( sup_su5028961100972612572_a_b_b @ A2 @ B ) @ X ) ) ) ).
% le_supI
thf(fact_1038_sup__ge1,axiom,
! [X: set_b,Y: set_b] : ( ord_less_eq_set_b @ X @ ( sup_sup_set_b @ X @ Y ) ) ).
% sup_ge1
thf(fact_1039_sup__ge1,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ X @ ( sup_sup_real @ X @ Y ) ) ).
% sup_ge1
thf(fact_1040_sup__ge1,axiom,
! [X: fset_P2153231429829016240_a_b_b,Y: fset_P2153231429829016240_a_b_b] : ( ord_le8870638447146015504_a_b_b @ X @ ( sup_su5028961100972612572_a_b_b @ X @ Y ) ) ).
% sup_ge1
thf(fact_1041_sup__ge2,axiom,
! [Y: set_b,X: set_b] : ( ord_less_eq_set_b @ Y @ ( sup_sup_set_b @ X @ Y ) ) ).
% sup_ge2
thf(fact_1042_sup__ge2,axiom,
! [Y: real,X: real] : ( ord_less_eq_real @ Y @ ( sup_sup_real @ X @ Y ) ) ).
% sup_ge2
thf(fact_1043_sup__ge2,axiom,
! [Y: fset_P2153231429829016240_a_b_b,X: fset_P2153231429829016240_a_b_b] : ( ord_le8870638447146015504_a_b_b @ Y @ ( sup_su5028961100972612572_a_b_b @ X @ Y ) ) ).
% sup_ge2
thf(fact_1044_le__supI1,axiom,
! [X: set_b,A2: set_b,B: set_b] :
( ( ord_less_eq_set_b @ X @ A2 )
=> ( ord_less_eq_set_b @ X @ ( sup_sup_set_b @ A2 @ B ) ) ) ).
% le_supI1
thf(fact_1045_le__supI1,axiom,
! [X: real,A2: real,B: real] :
( ( ord_less_eq_real @ X @ A2 )
=> ( ord_less_eq_real @ X @ ( sup_sup_real @ A2 @ B ) ) ) ).
% le_supI1
thf(fact_1046_le__supI1,axiom,
! [X: fset_P2153231429829016240_a_b_b,A2: fset_P2153231429829016240_a_b_b,B: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ X @ A2 )
=> ( ord_le8870638447146015504_a_b_b @ X @ ( sup_su5028961100972612572_a_b_b @ A2 @ B ) ) ) ).
% le_supI1
thf(fact_1047_le__supI2,axiom,
! [X: set_b,B: set_b,A2: set_b] :
( ( ord_less_eq_set_b @ X @ B )
=> ( ord_less_eq_set_b @ X @ ( sup_sup_set_b @ A2 @ B ) ) ) ).
% le_supI2
thf(fact_1048_le__supI2,axiom,
! [X: real,B: real,A2: real] :
( ( ord_less_eq_real @ X @ B )
=> ( ord_less_eq_real @ X @ ( sup_sup_real @ A2 @ B ) ) ) ).
% le_supI2
thf(fact_1049_le__supI2,axiom,
! [X: fset_P2153231429829016240_a_b_b,B: fset_P2153231429829016240_a_b_b,A2: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ X @ B )
=> ( ord_le8870638447146015504_a_b_b @ X @ ( sup_su5028961100972612572_a_b_b @ A2 @ B ) ) ) ).
% le_supI2
thf(fact_1050_sup_Omono,axiom,
! [C: set_b,A2: set_b,D: set_b,B: set_b] :
( ( ord_less_eq_set_b @ C @ A2 )
=> ( ( ord_less_eq_set_b @ D @ B )
=> ( ord_less_eq_set_b @ ( sup_sup_set_b @ C @ D ) @ ( sup_sup_set_b @ A2 @ B ) ) ) ) ).
% sup.mono
thf(fact_1051_sup_Omono,axiom,
! [C: real,A2: real,D: real,B: real] :
( ( ord_less_eq_real @ C @ A2 )
=> ( ( ord_less_eq_real @ D @ B )
=> ( ord_less_eq_real @ ( sup_sup_real @ C @ D ) @ ( sup_sup_real @ A2 @ B ) ) ) ) ).
% sup.mono
thf(fact_1052_sup_Omono,axiom,
! [C: fset_P2153231429829016240_a_b_b,A2: fset_P2153231429829016240_a_b_b,D: fset_P2153231429829016240_a_b_b,B: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ C @ A2 )
=> ( ( ord_le8870638447146015504_a_b_b @ D @ B )
=> ( ord_le8870638447146015504_a_b_b @ ( sup_su5028961100972612572_a_b_b @ C @ D ) @ ( sup_su5028961100972612572_a_b_b @ A2 @ B ) ) ) ) ).
% sup.mono
thf(fact_1053_sup__mono,axiom,
! [A2: set_b,C: set_b,B: set_b,D: set_b] :
( ( ord_less_eq_set_b @ A2 @ C )
=> ( ( ord_less_eq_set_b @ B @ D )
=> ( ord_less_eq_set_b @ ( sup_sup_set_b @ A2 @ B ) @ ( sup_sup_set_b @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_1054_sup__mono,axiom,
! [A2: real,C: real,B: real,D: real] :
( ( ord_less_eq_real @ A2 @ C )
=> ( ( ord_less_eq_real @ B @ D )
=> ( ord_less_eq_real @ ( sup_sup_real @ A2 @ B ) @ ( sup_sup_real @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_1055_sup__mono,axiom,
! [A2: fset_P2153231429829016240_a_b_b,C: fset_P2153231429829016240_a_b_b,B: fset_P2153231429829016240_a_b_b,D: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ A2 @ C )
=> ( ( ord_le8870638447146015504_a_b_b @ B @ D )
=> ( ord_le8870638447146015504_a_b_b @ ( sup_su5028961100972612572_a_b_b @ A2 @ B ) @ ( sup_su5028961100972612572_a_b_b @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_1056_sup__least,axiom,
! [Y: set_b,X: set_b,Z2: set_b] :
( ( ord_less_eq_set_b @ Y @ X )
=> ( ( ord_less_eq_set_b @ Z2 @ X )
=> ( ord_less_eq_set_b @ ( sup_sup_set_b @ Y @ Z2 ) @ X ) ) ) ).
% sup_least
thf(fact_1057_sup__least,axiom,
! [Y: real,X: real,Z2: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( ord_less_eq_real @ Z2 @ X )
=> ( ord_less_eq_real @ ( sup_sup_real @ Y @ Z2 ) @ X ) ) ) ).
% sup_least
thf(fact_1058_sup__least,axiom,
! [Y: fset_P2153231429829016240_a_b_b,X: fset_P2153231429829016240_a_b_b,Z2: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ Y @ X )
=> ( ( ord_le8870638447146015504_a_b_b @ Z2 @ X )
=> ( ord_le8870638447146015504_a_b_b @ ( sup_su5028961100972612572_a_b_b @ Y @ Z2 ) @ X ) ) ) ).
% sup_least
thf(fact_1059_le__iff__sup,axiom,
( ord_less_eq_set_b
= ( ^ [X4: set_b,Y2: set_b] :
( ( sup_sup_set_b @ X4 @ Y2 )
= Y2 ) ) ) ).
% le_iff_sup
thf(fact_1060_le__iff__sup,axiom,
( ord_less_eq_real
= ( ^ [X4: real,Y2: real] :
( ( sup_sup_real @ X4 @ Y2 )
= Y2 ) ) ) ).
% le_iff_sup
thf(fact_1061_le__iff__sup,axiom,
( ord_le8870638447146015504_a_b_b
= ( ^ [X4: fset_P2153231429829016240_a_b_b,Y2: fset_P2153231429829016240_a_b_b] :
( ( sup_su5028961100972612572_a_b_b @ X4 @ Y2 )
= Y2 ) ) ) ).
% le_iff_sup
thf(fact_1062_sup_OorderE,axiom,
! [B: set_b,A2: set_b] :
( ( ord_less_eq_set_b @ B @ A2 )
=> ( A2
= ( sup_sup_set_b @ A2 @ B ) ) ) ).
% sup.orderE
thf(fact_1063_sup_OorderE,axiom,
! [B: real,A2: real] :
( ( ord_less_eq_real @ B @ A2 )
=> ( A2
= ( sup_sup_real @ A2 @ B ) ) ) ).
% sup.orderE
thf(fact_1064_sup_OorderE,axiom,
! [B: fset_P2153231429829016240_a_b_b,A2: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ B @ A2 )
=> ( A2
= ( sup_su5028961100972612572_a_b_b @ A2 @ B ) ) ) ).
% sup.orderE
thf(fact_1065_sup_OorderI,axiom,
! [A2: set_b,B: set_b] :
( ( A2
= ( sup_sup_set_b @ A2 @ B ) )
=> ( ord_less_eq_set_b @ B @ A2 ) ) ).
% sup.orderI
thf(fact_1066_sup_OorderI,axiom,
! [A2: real,B: real] :
( ( A2
= ( sup_sup_real @ A2 @ B ) )
=> ( ord_less_eq_real @ B @ A2 ) ) ).
% sup.orderI
thf(fact_1067_sup_OorderI,axiom,
! [A2: fset_P2153231429829016240_a_b_b,B: fset_P2153231429829016240_a_b_b] :
( ( A2
= ( sup_su5028961100972612572_a_b_b @ A2 @ B ) )
=> ( ord_le8870638447146015504_a_b_b @ B @ A2 ) ) ).
% sup.orderI
thf(fact_1068_sup__unique,axiom,
! [F: set_b > set_b > set_b,X: set_b,Y: set_b] :
( ! [X3: set_b,Y3: set_b] : ( ord_less_eq_set_b @ X3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: set_b,Y3: set_b] : ( ord_less_eq_set_b @ Y3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: set_b,Y3: set_b,Z4: set_b] :
( ( ord_less_eq_set_b @ Y3 @ X3 )
=> ( ( ord_less_eq_set_b @ Z4 @ X3 )
=> ( ord_less_eq_set_b @ ( F @ Y3 @ Z4 ) @ X3 ) ) )
=> ( ( sup_sup_set_b @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_1069_sup__unique,axiom,
! [F: real > real > real,X: real,Y: real] :
( ! [X3: real,Y3: real] : ( ord_less_eq_real @ X3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: real,Y3: real] : ( ord_less_eq_real @ Y3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: real,Y3: real,Z4: real] :
( ( ord_less_eq_real @ Y3 @ X3 )
=> ( ( ord_less_eq_real @ Z4 @ X3 )
=> ( ord_less_eq_real @ ( F @ Y3 @ Z4 ) @ X3 ) ) )
=> ( ( sup_sup_real @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_1070_sup__unique,axiom,
! [F: fset_P2153231429829016240_a_b_b > fset_P2153231429829016240_a_b_b > fset_P2153231429829016240_a_b_b,X: fset_P2153231429829016240_a_b_b,Y: fset_P2153231429829016240_a_b_b] :
( ! [X3: fset_P2153231429829016240_a_b_b,Y3: fset_P2153231429829016240_a_b_b] : ( ord_le8870638447146015504_a_b_b @ X3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: fset_P2153231429829016240_a_b_b,Y3: fset_P2153231429829016240_a_b_b] : ( ord_le8870638447146015504_a_b_b @ Y3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: fset_P2153231429829016240_a_b_b,Y3: fset_P2153231429829016240_a_b_b,Z4: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ Y3 @ X3 )
=> ( ( ord_le8870638447146015504_a_b_b @ Z4 @ X3 )
=> ( ord_le8870638447146015504_a_b_b @ ( F @ Y3 @ Z4 ) @ X3 ) ) )
=> ( ( sup_su5028961100972612572_a_b_b @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_1071_sup_Oabsorb1,axiom,
! [B: set_b,A2: set_b] :
( ( ord_less_eq_set_b @ B @ A2 )
=> ( ( sup_sup_set_b @ A2 @ B )
= A2 ) ) ).
% sup.absorb1
thf(fact_1072_sup_Oabsorb1,axiom,
! [B: real,A2: real] :
( ( ord_less_eq_real @ B @ A2 )
=> ( ( sup_sup_real @ A2 @ B )
= A2 ) ) ).
% sup.absorb1
thf(fact_1073_sup_Oabsorb1,axiom,
! [B: fset_P2153231429829016240_a_b_b,A2: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ B @ A2 )
=> ( ( sup_su5028961100972612572_a_b_b @ A2 @ B )
= A2 ) ) ).
% sup.absorb1
thf(fact_1074_sup_Oabsorb2,axiom,
! [A2: set_b,B: set_b] :
( ( ord_less_eq_set_b @ A2 @ B )
=> ( ( sup_sup_set_b @ A2 @ B )
= B ) ) ).
% sup.absorb2
thf(fact_1075_sup_Oabsorb2,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( sup_sup_real @ A2 @ B )
= B ) ) ).
% sup.absorb2
thf(fact_1076_sup_Oabsorb2,axiom,
! [A2: fset_P2153231429829016240_a_b_b,B: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ A2 @ B )
=> ( ( sup_su5028961100972612572_a_b_b @ A2 @ B )
= B ) ) ).
% sup.absorb2
thf(fact_1077_sup__absorb1,axiom,
! [Y: set_b,X: set_b] :
( ( ord_less_eq_set_b @ Y @ X )
=> ( ( sup_sup_set_b @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_1078_sup__absorb1,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( sup_sup_real @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_1079_sup__absorb1,axiom,
! [Y: fset_P2153231429829016240_a_b_b,X: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ Y @ X )
=> ( ( sup_su5028961100972612572_a_b_b @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_1080_sup__absorb2,axiom,
! [X: set_b,Y: set_b] :
( ( ord_less_eq_set_b @ X @ Y )
=> ( ( sup_sup_set_b @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_1081_sup__absorb2,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( sup_sup_real @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_1082_sup__absorb2,axiom,
! [X: fset_P2153231429829016240_a_b_b,Y: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ X @ Y )
=> ( ( sup_su5028961100972612572_a_b_b @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_1083_sup_OboundedE,axiom,
! [B: set_b,C: set_b,A2: set_b] :
( ( ord_less_eq_set_b @ ( sup_sup_set_b @ B @ C ) @ A2 )
=> ~ ( ( ord_less_eq_set_b @ B @ A2 )
=> ~ ( ord_less_eq_set_b @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_1084_sup_OboundedE,axiom,
! [B: real,C: real,A2: real] :
( ( ord_less_eq_real @ ( sup_sup_real @ B @ C ) @ A2 )
=> ~ ( ( ord_less_eq_real @ B @ A2 )
=> ~ ( ord_less_eq_real @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_1085_sup_OboundedE,axiom,
! [B: fset_P2153231429829016240_a_b_b,C: fset_P2153231429829016240_a_b_b,A2: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ ( sup_su5028961100972612572_a_b_b @ B @ C ) @ A2 )
=> ~ ( ( ord_le8870638447146015504_a_b_b @ B @ A2 )
=> ~ ( ord_le8870638447146015504_a_b_b @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_1086_sup_OboundedI,axiom,
! [B: set_b,A2: set_b,C: set_b] :
( ( ord_less_eq_set_b @ B @ A2 )
=> ( ( ord_less_eq_set_b @ C @ A2 )
=> ( ord_less_eq_set_b @ ( sup_sup_set_b @ B @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_1087_sup_OboundedI,axiom,
! [B: real,A2: real,C: real] :
( ( ord_less_eq_real @ B @ A2 )
=> ( ( ord_less_eq_real @ C @ A2 )
=> ( ord_less_eq_real @ ( sup_sup_real @ B @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_1088_sup_OboundedI,axiom,
! [B: fset_P2153231429829016240_a_b_b,A2: fset_P2153231429829016240_a_b_b,C: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ B @ A2 )
=> ( ( ord_le8870638447146015504_a_b_b @ C @ A2 )
=> ( ord_le8870638447146015504_a_b_b @ ( sup_su5028961100972612572_a_b_b @ B @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_1089_sup_Oorder__iff,axiom,
( ord_less_eq_set_b
= ( ^ [B3: set_b,A4: set_b] :
( A4
= ( sup_sup_set_b @ A4 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_1090_sup_Oorder__iff,axiom,
( ord_less_eq_real
= ( ^ [B3: real,A4: real] :
( A4
= ( sup_sup_real @ A4 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_1091_sup_Oorder__iff,axiom,
( ord_le8870638447146015504_a_b_b
= ( ^ [B3: fset_P2153231429829016240_a_b_b,A4: fset_P2153231429829016240_a_b_b] :
( A4
= ( sup_su5028961100972612572_a_b_b @ A4 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_1092_sup_Ocobounded1,axiom,
! [A2: set_b,B: set_b] : ( ord_less_eq_set_b @ A2 @ ( sup_sup_set_b @ A2 @ B ) ) ).
% sup.cobounded1
thf(fact_1093_sup_Ocobounded1,axiom,
! [A2: real,B: real] : ( ord_less_eq_real @ A2 @ ( sup_sup_real @ A2 @ B ) ) ).
% sup.cobounded1
thf(fact_1094_sup_Ocobounded1,axiom,
! [A2: fset_P2153231429829016240_a_b_b,B: fset_P2153231429829016240_a_b_b] : ( ord_le8870638447146015504_a_b_b @ A2 @ ( sup_su5028961100972612572_a_b_b @ A2 @ B ) ) ).
% sup.cobounded1
thf(fact_1095_sup_Ocobounded2,axiom,
! [B: set_b,A2: set_b] : ( ord_less_eq_set_b @ B @ ( sup_sup_set_b @ A2 @ B ) ) ).
% sup.cobounded2
thf(fact_1096_sup_Ocobounded2,axiom,
! [B: real,A2: real] : ( ord_less_eq_real @ B @ ( sup_sup_real @ A2 @ B ) ) ).
% sup.cobounded2
thf(fact_1097_sup_Ocobounded2,axiom,
! [B: fset_P2153231429829016240_a_b_b,A2: fset_P2153231429829016240_a_b_b] : ( ord_le8870638447146015504_a_b_b @ B @ ( sup_su5028961100972612572_a_b_b @ A2 @ B ) ) ).
% sup.cobounded2
thf(fact_1098_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_b
= ( ^ [B3: set_b,A4: set_b] :
( ( sup_sup_set_b @ A4 @ B3 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_1099_sup_Oabsorb__iff1,axiom,
( ord_less_eq_real
= ( ^ [B3: real,A4: real] :
( ( sup_sup_real @ A4 @ B3 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_1100_sup_Oabsorb__iff1,axiom,
( ord_le8870638447146015504_a_b_b
= ( ^ [B3: fset_P2153231429829016240_a_b_b,A4: fset_P2153231429829016240_a_b_b] :
( ( sup_su5028961100972612572_a_b_b @ A4 @ B3 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_1101_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_b
= ( ^ [A4: set_b,B3: set_b] :
( ( sup_sup_set_b @ A4 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_1102_sup_Oabsorb__iff2,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B3: real] :
( ( sup_sup_real @ A4 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_1103_sup_Oabsorb__iff2,axiom,
( ord_le8870638447146015504_a_b_b
= ( ^ [A4: fset_P2153231429829016240_a_b_b,B3: fset_P2153231429829016240_a_b_b] :
( ( sup_su5028961100972612572_a_b_b @ A4 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_1104_sup_OcoboundedI1,axiom,
! [C: set_b,A2: set_b,B: set_b] :
( ( ord_less_eq_set_b @ C @ A2 )
=> ( ord_less_eq_set_b @ C @ ( sup_sup_set_b @ A2 @ B ) ) ) ).
% sup.coboundedI1
thf(fact_1105_sup_OcoboundedI1,axiom,
! [C: real,A2: real,B: real] :
( ( ord_less_eq_real @ C @ A2 )
=> ( ord_less_eq_real @ C @ ( sup_sup_real @ A2 @ B ) ) ) ).
% sup.coboundedI1
thf(fact_1106_sup_OcoboundedI1,axiom,
! [C: fset_P2153231429829016240_a_b_b,A2: fset_P2153231429829016240_a_b_b,B: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ C @ A2 )
=> ( ord_le8870638447146015504_a_b_b @ C @ ( sup_su5028961100972612572_a_b_b @ A2 @ B ) ) ) ).
% sup.coboundedI1
thf(fact_1107_sup_OcoboundedI2,axiom,
! [C: set_b,B: set_b,A2: set_b] :
( ( ord_less_eq_set_b @ C @ B )
=> ( ord_less_eq_set_b @ C @ ( sup_sup_set_b @ A2 @ B ) ) ) ).
% sup.coboundedI2
thf(fact_1108_sup_OcoboundedI2,axiom,
! [C: real,B: real,A2: real] :
( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ C @ ( sup_sup_real @ A2 @ B ) ) ) ).
% sup.coboundedI2
thf(fact_1109_sup_OcoboundedI2,axiom,
! [C: fset_P2153231429829016240_a_b_b,B: fset_P2153231429829016240_a_b_b,A2: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ C @ B )
=> ( ord_le8870638447146015504_a_b_b @ C @ ( sup_su5028961100972612572_a_b_b @ A2 @ B ) ) ) ).
% sup.coboundedI2
thf(fact_1110_distrib__imp1,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ! [X3: set_a,Y3: set_a,Z4: set_a] :
( ( inf_inf_set_a @ X3 @ ( sup_sup_set_a @ Y3 @ Z4 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X3 @ Y3 ) @ ( inf_inf_set_a @ X3 @ Z4 ) ) )
=> ( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z2 ) ) ) ) ).
% distrib_imp1
thf(fact_1111_distrib__imp1,axiom,
! [X: set_list_a,Y: set_list_a,Z2: set_list_a] :
( ! [X3: set_list_a,Y3: set_list_a,Z4: set_list_a] :
( ( inf_inf_set_list_a @ X3 @ ( sup_sup_set_list_a @ Y3 @ Z4 ) )
= ( sup_sup_set_list_a @ ( inf_inf_set_list_a @ X3 @ Y3 ) @ ( inf_inf_set_list_a @ X3 @ Z4 ) ) )
=> ( ( sup_sup_set_list_a @ X @ ( inf_inf_set_list_a @ Y @ Z2 ) )
= ( inf_inf_set_list_a @ ( sup_sup_set_list_a @ X @ Y ) @ ( sup_sup_set_list_a @ X @ Z2 ) ) ) ) ).
% distrib_imp1
thf(fact_1112_distrib__imp1,axiom,
! [X: set_b,Y: set_b,Z2: set_b] :
( ! [X3: set_b,Y3: set_b,Z4: set_b] :
( ( inf_inf_set_b @ X3 @ ( sup_sup_set_b @ Y3 @ Z4 ) )
= ( sup_sup_set_b @ ( inf_inf_set_b @ X3 @ Y3 ) @ ( inf_inf_set_b @ X3 @ Z4 ) ) )
=> ( ( sup_sup_set_b @ X @ ( inf_inf_set_b @ Y @ Z2 ) )
= ( inf_inf_set_b @ ( sup_sup_set_b @ X @ Y ) @ ( sup_sup_set_b @ X @ Z2 ) ) ) ) ).
% distrib_imp1
thf(fact_1113_distrib__imp2,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ! [X3: set_a,Y3: set_a,Z4: set_a] :
( ( sup_sup_set_a @ X3 @ ( inf_inf_set_a @ Y3 @ Z4 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X3 @ Y3 ) @ ( sup_sup_set_a @ X3 @ Z4 ) ) )
=> ( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z2 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z2 ) ) ) ) ).
% distrib_imp2
thf(fact_1114_distrib__imp2,axiom,
! [X: set_list_a,Y: set_list_a,Z2: set_list_a] :
( ! [X3: set_list_a,Y3: set_list_a,Z4: set_list_a] :
( ( sup_sup_set_list_a @ X3 @ ( inf_inf_set_list_a @ Y3 @ Z4 ) )
= ( inf_inf_set_list_a @ ( sup_sup_set_list_a @ X3 @ Y3 ) @ ( sup_sup_set_list_a @ X3 @ Z4 ) ) )
=> ( ( inf_inf_set_list_a @ X @ ( sup_sup_set_list_a @ Y @ Z2 ) )
= ( sup_sup_set_list_a @ ( inf_inf_set_list_a @ X @ Y ) @ ( inf_inf_set_list_a @ X @ Z2 ) ) ) ) ).
% distrib_imp2
thf(fact_1115_distrib__imp2,axiom,
! [X: set_b,Y: set_b,Z2: set_b] :
( ! [X3: set_b,Y3: set_b,Z4: set_b] :
( ( sup_sup_set_b @ X3 @ ( inf_inf_set_b @ Y3 @ Z4 ) )
= ( inf_inf_set_b @ ( sup_sup_set_b @ X3 @ Y3 ) @ ( sup_sup_set_b @ X3 @ Z4 ) ) )
=> ( ( inf_inf_set_b @ X @ ( sup_sup_set_b @ Y @ Z2 ) )
= ( sup_sup_set_b @ ( inf_inf_set_b @ X @ Y ) @ ( inf_inf_set_b @ X @ Z2 ) ) ) ) ).
% distrib_imp2
thf(fact_1116_inf__sup__distrib1,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z2 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z2 ) ) ) ).
% inf_sup_distrib1
thf(fact_1117_inf__sup__distrib1,axiom,
! [X: set_list_a,Y: set_list_a,Z2: set_list_a] :
( ( inf_inf_set_list_a @ X @ ( sup_sup_set_list_a @ Y @ Z2 ) )
= ( sup_sup_set_list_a @ ( inf_inf_set_list_a @ X @ Y ) @ ( inf_inf_set_list_a @ X @ Z2 ) ) ) ).
% inf_sup_distrib1
thf(fact_1118_inf__sup__distrib1,axiom,
! [X: set_b,Y: set_b,Z2: set_b] :
( ( inf_inf_set_b @ X @ ( sup_sup_set_b @ Y @ Z2 ) )
= ( sup_sup_set_b @ ( inf_inf_set_b @ X @ Y ) @ ( inf_inf_set_b @ X @ Z2 ) ) ) ).
% inf_sup_distrib1
thf(fact_1119_inf__sup__distrib2,axiom,
! [Y: set_a,Z2: set_a,X: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ Z2 ) @ X )
= ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ X ) @ ( inf_inf_set_a @ Z2 @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_1120_inf__sup__distrib2,axiom,
! [Y: set_list_a,Z2: set_list_a,X: set_list_a] :
( ( inf_inf_set_list_a @ ( sup_sup_set_list_a @ Y @ Z2 ) @ X )
= ( sup_sup_set_list_a @ ( inf_inf_set_list_a @ Y @ X ) @ ( inf_inf_set_list_a @ Z2 @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_1121_inf__sup__distrib2,axiom,
! [Y: set_b,Z2: set_b,X: set_b] :
( ( inf_inf_set_b @ ( sup_sup_set_b @ Y @ Z2 ) @ X )
= ( sup_sup_set_b @ ( inf_inf_set_b @ Y @ X ) @ ( inf_inf_set_b @ Z2 @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_1122_sup__inf__distrib1,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z2 ) ) ) ).
% sup_inf_distrib1
thf(fact_1123_sup__inf__distrib1,axiom,
! [X: set_list_a,Y: set_list_a,Z2: set_list_a] :
( ( sup_sup_set_list_a @ X @ ( inf_inf_set_list_a @ Y @ Z2 ) )
= ( inf_inf_set_list_a @ ( sup_sup_set_list_a @ X @ Y ) @ ( sup_sup_set_list_a @ X @ Z2 ) ) ) ).
% sup_inf_distrib1
thf(fact_1124_sup__inf__distrib1,axiom,
! [X: set_b,Y: set_b,Z2: set_b] :
( ( sup_sup_set_b @ X @ ( inf_inf_set_b @ Y @ Z2 ) )
= ( inf_inf_set_b @ ( sup_sup_set_b @ X @ Y ) @ ( sup_sup_set_b @ X @ Z2 ) ) ) ).
% sup_inf_distrib1
thf(fact_1125_sup__inf__distrib2,axiom,
! [Y: set_a,Z2: set_a,X: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ Z2 ) @ X )
= ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ X ) @ ( sup_sup_set_a @ Z2 @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_1126_sup__inf__distrib2,axiom,
! [Y: set_list_a,Z2: set_list_a,X: set_list_a] :
( ( sup_sup_set_list_a @ ( inf_inf_set_list_a @ Y @ Z2 ) @ X )
= ( inf_inf_set_list_a @ ( sup_sup_set_list_a @ Y @ X ) @ ( sup_sup_set_list_a @ Z2 @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_1127_sup__inf__distrib2,axiom,
! [Y: set_b,Z2: set_b,X: set_b] :
( ( sup_sup_set_b @ ( inf_inf_set_b @ Y @ Z2 ) @ X )
= ( inf_inf_set_b @ ( sup_sup_set_b @ Y @ X ) @ ( sup_sup_set_b @ Z2 @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_1128_disjoint__darcs__subset,axiom,
! [Xs2: fset_P2153231429829016240_a_b_b,Ys: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ Xs2 @ Ys )
=> ( ! [X3: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ X3 @ ( fset_P9138963618725001425_a_b_b @ Ys ) )
=> ( produc336013011823628281_b_b_o
@ ^ [Y2: dtree_list_a_b,E1: b] :
( ~ ( member_b @ E1 @ ( darcs_list_a_b @ Y2 ) )
& ! [Z: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ Z @ ( fset_P9138963618725001425_a_b_b @ Ys ) )
=> ( produc336013011823628281_b_b_o
@ ^ [Aa: dtree_list_a_b,E2: b] :
( ( ( inf_inf_set_b @ ( sup_sup_set_b @ ( darcs_list_a_b @ Y2 ) @ ( insert_b @ E1 @ bot_bot_set_b ) ) @ ( sup_sup_set_b @ ( darcs_list_a_b @ Aa ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
= bot_bot_set_b )
| ( ( produc7704165765595008946_a_b_b @ Y2 @ E1 )
= ( produc7704165765595008946_a_b_b @ Aa @ E2 ) ) )
@ Z ) ) )
@ X3 ) )
=> ! [X2: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ X2 @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( produc336013011823628281_b_b_o
@ ^ [Y2: dtree_list_a_b,E1: b] :
( ~ ( member_b @ E1 @ ( darcs_list_a_b @ Y2 ) )
& ! [Z: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ Z @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( produc336013011823628281_b_b_o
@ ^ [Aa: dtree_list_a_b,E2: b] :
( ( ( inf_inf_set_b @ ( sup_sup_set_b @ ( darcs_list_a_b @ Y2 ) @ ( insert_b @ E1 @ bot_bot_set_b ) ) @ ( sup_sup_set_b @ ( darcs_list_a_b @ Aa ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
= bot_bot_set_b )
| ( ( produc7704165765595008946_a_b_b @ Y2 @ E1 )
= ( produc7704165765595008946_a_b_b @ Aa @ E2 ) ) )
@ Z ) ) )
@ X2 ) ) ) ) ).
% disjoint_darcs_subset
thf(fact_1129_distrib__inf__le,axiom,
! [X: set_a,Y: set_a,Z2: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z2 ) ) @ ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_1130_distrib__inf__le,axiom,
! [X: set_list_a,Y: set_list_a,Z2: set_list_a] : ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ ( inf_inf_set_list_a @ X @ Y ) @ ( inf_inf_set_list_a @ X @ Z2 ) ) @ ( inf_inf_set_list_a @ X @ ( sup_sup_set_list_a @ Y @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_1131_distrib__inf__le,axiom,
! [X: set_b,Y: set_b,Z2: set_b] : ( ord_less_eq_set_b @ ( sup_sup_set_b @ ( inf_inf_set_b @ X @ Y ) @ ( inf_inf_set_b @ X @ Z2 ) ) @ ( inf_inf_set_b @ X @ ( sup_sup_set_b @ Y @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_1132_distrib__inf__le,axiom,
! [X: real,Y: real,Z2: real] : ( ord_less_eq_real @ ( sup_sup_real @ ( inf_inf_real @ X @ Y ) @ ( inf_inf_real @ X @ Z2 ) ) @ ( inf_inf_real @ X @ ( sup_sup_real @ Y @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_1133_distrib__inf__le,axiom,
! [X: fset_P2153231429829016240_a_b_b,Y: fset_P2153231429829016240_a_b_b,Z2: fset_P2153231429829016240_a_b_b] : ( ord_le8870638447146015504_a_b_b @ ( sup_su5028961100972612572_a_b_b @ ( inf_in525273840840435522_a_b_b @ X @ Y ) @ ( inf_in525273840840435522_a_b_b @ X @ Z2 ) ) @ ( inf_in525273840840435522_a_b_b @ X @ ( sup_su5028961100972612572_a_b_b @ Y @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_1134_distrib__sup__le,axiom,
! [X: set_a,Y: set_a,Z2: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) ) @ ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_1135_distrib__sup__le,axiom,
! [X: set_list_a,Y: set_list_a,Z2: set_list_a] : ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ X @ ( inf_inf_set_list_a @ Y @ Z2 ) ) @ ( inf_inf_set_list_a @ ( sup_sup_set_list_a @ X @ Y ) @ ( sup_sup_set_list_a @ X @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_1136_distrib__sup__le,axiom,
! [X: set_b,Y: set_b,Z2: set_b] : ( ord_less_eq_set_b @ ( sup_sup_set_b @ X @ ( inf_inf_set_b @ Y @ Z2 ) ) @ ( inf_inf_set_b @ ( sup_sup_set_b @ X @ Y ) @ ( sup_sup_set_b @ X @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_1137_distrib__sup__le,axiom,
! [X: real,Y: real,Z2: real] : ( ord_less_eq_real @ ( sup_sup_real @ X @ ( inf_inf_real @ Y @ Z2 ) ) @ ( inf_inf_real @ ( sup_sup_real @ X @ Y ) @ ( sup_sup_real @ X @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_1138_distrib__sup__le,axiom,
! [X: fset_P2153231429829016240_a_b_b,Y: fset_P2153231429829016240_a_b_b,Z2: fset_P2153231429829016240_a_b_b] : ( ord_le8870638447146015504_a_b_b @ ( sup_su5028961100972612572_a_b_b @ X @ ( inf_in525273840840435522_a_b_b @ Y @ Z2 ) ) @ ( inf_in525273840840435522_a_b_b @ ( sup_su5028961100972612572_a_b_b @ X @ Y ) @ ( sup_su5028961100972612572_a_b_b @ X @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_1139_insert__before__only1__new,axiom,
! [Xs2: fset_P2153231429829016240_a_b_b,T12: dtree_list_a_b,E12: b,T23: dtree_list_a_b,E22: b,Y: list_a,V: list_a,E3: b] :
( ! [X3: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ X3 @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( produc336013011823628281_b_b_o
@ ^ [Y2: dtree_list_a_b,E1: b] :
! [Z: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ Z @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( produc336013011823628281_b_b_o
@ ^ [Aa: dtree_list_a_b,E2: b] :
( ( ( inf_inf_set_list_a @ ( dverts_list_a_b @ Y2 ) @ ( dverts_list_a_b @ Aa ) )
= bot_bot_set_list_a )
| ( ( produc7704165765595008946_a_b_b @ Y2 @ E1 )
= ( produc7704165765595008946_a_b_b @ Aa @ E2 ) ) )
@ Z ) )
@ X3 ) )
=> ( ( ( produc7704165765595008946_a_b_b @ T12 @ E12 )
!= ( produc7704165765595008946_a_b_b @ T23 @ E22 ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E12 )
@ ( fset_P9138963618725001425_a_b_b
@ ( ffold_6356110402769621009_a_b_b
@ ( produc7216629644754684766_a_b_b
@ ^ [T13: dtree_list_a_b,E1: b] :
( finser2303212525150181944_a_b_b
@ ( if_Pro807711176916573242_a_b_b
@ ( ( root_list_a_b @ T13 )
= Y )
@ ( produc7704165765595008946_a_b_b @ ( node_list_a_b @ V @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T13 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ E3 )
@ ( produc7704165765595008946_a_b_b @ T13 @ E1 ) ) ) )
@ bot_bo2248824169281960260_a_b_b
@ Xs2 ) ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T23 @ E22 )
@ ( fset_P9138963618725001425_a_b_b
@ ( ffold_6356110402769621009_a_b_b
@ ( produc7216629644754684766_a_b_b
@ ^ [T13: dtree_list_a_b,E1: b] :
( finser2303212525150181944_a_b_b
@ ( if_Pro807711176916573242_a_b_b
@ ( ( root_list_a_b @ T13 )
= Y )
@ ( produc7704165765595008946_a_b_b @ ( node_list_a_b @ V @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T13 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ E3 )
@ ( produc7704165765595008946_a_b_b @ T13 @ E1 ) ) ) )
@ bot_bo2248824169281960260_a_b_b
@ Xs2 ) ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E12 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
| ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T23 @ E22 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) ) ) ) ) ) ).
% insert_before_only1_new
thf(fact_1140_disjoint__dverts__aux1,axiom,
! [Xs2: fset_P2153231429829016240_a_b_b,V: list_a,R2: list_a,T12: dtree_list_a_b,E12: b,Y: list_a,E3: b,T23: dtree_list_a_b,E22: b] :
( ! [X3: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ X3 @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( produc336013011823628281_b_b_o
@ ^ [T13: dtree_list_a_b,E1: b] :
! [Y2: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ Y2 @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( produc336013011823628281_b_b_o
@ ^ [T22: dtree_list_a_b,E2: b] :
( ( ( inf_inf_set_list_a @ ( dverts_list_a_b @ T13 ) @ ( dverts_list_a_b @ T22 ) )
= bot_bot_set_list_a )
| ( ( produc7704165765595008946_a_b_b @ T13 @ E1 )
= ( produc7704165765595008946_a_b_b @ T22 @ E2 ) ) )
@ Y2 ) )
@ X3 ) )
=> ( ~ ( member_list_a @ V @ ( dverts_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E12 )
@ ( fset_P9138963618725001425_a_b_b
@ ( ffold_6356110402769621009_a_b_b
@ ( produc7216629644754684766_a_b_b
@ ^ [T13: dtree_list_a_b,E1: b] :
( finser2303212525150181944_a_b_b
@ ( if_Pro807711176916573242_a_b_b
@ ( ( root_list_a_b @ T13 )
= Y )
@ ( produc7704165765595008946_a_b_b @ ( node_list_a_b @ V @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T13 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ E3 )
@ ( produc7704165765595008946_a_b_b @ T13 @ E1 ) ) ) )
@ bot_bo2248824169281960260_a_b_b
@ Xs2 ) ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T23 @ E22 )
@ ( fset_P9138963618725001425_a_b_b
@ ( ffold_6356110402769621009_a_b_b
@ ( produc7216629644754684766_a_b_b
@ ^ [T13: dtree_list_a_b,E1: b] :
( finser2303212525150181944_a_b_b
@ ( if_Pro807711176916573242_a_b_b
@ ( ( root_list_a_b @ T13 )
= Y )
@ ( produc7704165765595008946_a_b_b @ ( node_list_a_b @ V @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T13 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ E3 )
@ ( produc7704165765595008946_a_b_b @ T13 @ E1 ) ) ) )
@ bot_bo2248824169281960260_a_b_b
@ Xs2 ) ) )
=> ( ( ( produc7704165765595008946_a_b_b @ T12 @ E12 )
!= ( produc7704165765595008946_a_b_b @ T23 @ E22 ) )
=> ( ( inf_inf_set_list_a @ ( dverts_list_a_b @ T12 ) @ ( dverts_list_a_b @ T23 ) )
= bot_bot_set_list_a ) ) ) ) ) ) ).
% disjoint_dverts_aux1
thf(fact_1141_insert__before__root__nin__verts,axiom,
! [Xs2: fset_P2153231429829016240_a_b_b,R2: list_a,V: list_a,T12: dtree_list_a_b,E12: b,Y: list_a,E3: b] :
( ! [X3: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ X3 @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( produc336013011823628281_b_b_o
@ ^ [T3: dtree_list_a_b,E1: b] :
~ ( member_list_a @ R2 @ ( dverts_list_a_b @ T3 ) )
@ X3 ) )
=> ( ~ ( member_list_a @ V @ ( dverts_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E12 )
@ ( fset_P9138963618725001425_a_b_b
@ ( ffold_6356110402769621009_a_b_b
@ ( produc7216629644754684766_a_b_b
@ ^ [T13: dtree_list_a_b,E1: b] :
( finser2303212525150181944_a_b_b
@ ( if_Pro807711176916573242_a_b_b
@ ( ( root_list_a_b @ T13 )
= Y )
@ ( produc7704165765595008946_a_b_b @ ( node_list_a_b @ V @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T13 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ E3 )
@ ( produc7704165765595008946_a_b_b @ T13 @ E1 ) ) ) )
@ bot_bo2248824169281960260_a_b_b
@ Xs2 ) ) )
=> ~ ( member_list_a @ R2 @ ( dverts_list_a_b @ T12 ) ) ) ) ) ).
% insert_before_root_nin_verts
thf(fact_1142_darcs__child2__sub,axiom,
! [X: list_a,Y: list_a,Xs2: fset_P2153231429829016240_a_b_b,Ys: fset_P2153231429829016240_a_b_b,R2: list_a,R3: list_a] :
( ord_less_eq_set_b
@ ( darcs_list_a_b
@ ( node_list_a_b @ X
@ ( ffold_6356110402769621009_a_b_b
@ ( produc7216629644754684766_a_b_b
@ ^ [T3: dtree_list_a_b,Uu: b,B3: fset_P2153231429829016240_a_b_b] :
( case_d8112115101481596194_a_b_b
@ ^ [R4: list_a,Ys2: fset_P2153231429829016240_a_b_b] : ( if_fse36834823175207530_a_b_b @ ( R4 = Y ) @ ( sup_su5028961100972612572_a_b_b @ Ys2 @ B3 ) @ B3 )
@ T3 ) )
@ Xs2
@ Ys ) ) )
@ ( sup_sup_set_b @ ( darcs_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) @ ( darcs_list_a_b @ ( node_list_a_b @ R3 @ Ys ) ) ) ) ).
% darcs_child2_sub
thf(fact_1143_in__insert__before__child__in__orig,axiom,
! [T12: dtree_list_a_b,E12: b,Y: list_a,V: list_a,E3: b,Xs2: fset_P2153231429829016240_a_b_b] :
( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E12 )
@ ( fset_P9138963618725001425_a_b_b
@ ( ffold_6356110402769621009_a_b_b
@ ( produc7216629644754684766_a_b_b
@ ^ [T13: dtree_list_a_b,E1: b] :
( finser2303212525150181944_a_b_b
@ ( if_Pro807711176916573242_a_b_b
@ ( ( root_list_a_b @ T13 )
= Y )
@ ( produc7704165765595008946_a_b_b @ ( node_list_a_b @ V @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T13 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ E3 )
@ ( produc7704165765595008946_a_b_b @ T13 @ E1 ) ) ) )
@ bot_bo2248824169281960260_a_b_b
@ Xs2 ) ) )
=> ( ~ ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E12 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ? [X3: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ X3 @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
& ( produc336013011823628281_b_b_o
@ ^ [T22: dtree_list_a_b,E2: b] :
( ( ( node_list_a_b @ V @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T22 @ E2 ) @ bot_bo2248824169281960260_a_b_b ) )
= T12 )
& ( ( root_list_a_b @ T22 )
= Y )
& ( E12 = E3 ) )
@ X3 ) ) ) ) ).
% in_insert_before_child_in_orig
thf(fact_1144_root__normalize1__eq1_H,axiom,
! [T12: dtree_list_a_b,R2: list_a,E12: b] :
( ~ ( ord_less_eq_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T12 ) ) ) @ ( rank @ ( rev_a @ R2 ) ) )
=> ( ( root_list_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
= R2 ) ) ).
% root_normalize1_eq1'
thf(fact_1145_child_H__in__child2,axiom,
! [Y: list_a,Rs1: fset_P2153231429829016240_a_b_b,E12: b,Xs2: fset_P2153231429829016240_a_b_b,T23: dtree_list_a_b,E22: b,Ys: fset_P2153231429829016240_a_b_b] :
( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ ( node_list_a_b @ Y @ Rs1 ) @ E12 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T23 @ E22 ) @ ( fset_P9138963618725001425_a_b_b @ Rs1 ) )
=> ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T23 @ E22 )
@ ( fset_P9138963618725001425_a_b_b
@ ( ffold_6356110402769621009_a_b_b
@ ( produc7216629644754684766_a_b_b
@ ^ [T3: dtree_list_a_b,Uu: b,B3: fset_P2153231429829016240_a_b_b] :
( case_d8112115101481596194_a_b_b
@ ^ [R4: list_a,Ys2: fset_P2153231429829016240_a_b_b] : ( if_fse36834823175207530_a_b_b @ ( R4 = Y ) @ ( sup_su5028961100972612572_a_b_b @ Ys2 @ B3 ) @ B3 )
@ T3 ) )
@ Ys
@ Xs2 ) ) ) ) ) ).
% child'_in_child2
thf(fact_1146_fsubset__antisym,axiom,
! [A: fset_P2153231429829016240_a_b_b,B5: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ A @ B5 )
=> ( ( ord_le8870638447146015504_a_b_b @ B5 @ A )
=> ( A = B5 ) ) ) ).
% fsubset_antisym
thf(fact_1147_input__in__child2,axiom,
! [Zs: fset_P2153231429829016240_a_b_b,X: list_a,Ys: fset_P2153231429829016240_a_b_b] :
( ord_le8870638447146015504_a_b_b @ Zs
@ ( ffold_6356110402769621009_a_b_b
@ ( produc7216629644754684766_a_b_b
@ ^ [T3: dtree_list_a_b,Uu: b,B3: fset_P2153231429829016240_a_b_b] :
( case_d8112115101481596194_a_b_b
@ ^ [R4: list_a,Ys2: fset_P2153231429829016240_a_b_b] : ( if_fse36834823175207530_a_b_b @ ( R4 = X ) @ ( sup_su5028961100972612572_a_b_b @ Ys2 @ B3 ) @ B3 )
@ T3 ) )
@ Zs
@ Ys ) ) ).
% input_in_child2
thf(fact_1148_child2__subset__if__input2,axiom,
! [Ys3: fset_P2153231429829016240_a_b_b,Ys: fset_P2153231429829016240_a_b_b,X: list_a,Xs2: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ Ys3 @ Ys )
=> ( ord_le8870638447146015504_a_b_b
@ ( ffold_6356110402769621009_a_b_b
@ ( produc7216629644754684766_a_b_b
@ ^ [T3: dtree_list_a_b,Uu: b,B3: fset_P2153231429829016240_a_b_b] :
( case_d8112115101481596194_a_b_b
@ ^ [R4: list_a,Ys2: fset_P2153231429829016240_a_b_b] : ( if_fse36834823175207530_a_b_b @ ( R4 = X ) @ ( sup_su5028961100972612572_a_b_b @ Ys2 @ B3 ) @ B3 )
@ T3 ) )
@ Xs2
@ Ys3 )
@ ( ffold_6356110402769621009_a_b_b
@ ( produc7216629644754684766_a_b_b
@ ^ [T3: dtree_list_a_b,Uu: b,B3: fset_P2153231429829016240_a_b_b] :
( case_d8112115101481596194_a_b_b
@ ^ [R4: list_a,Ys2: fset_P2153231429829016240_a_b_b] : ( if_fse36834823175207530_a_b_b @ ( R4 = X ) @ ( sup_su5028961100972612572_a_b_b @ Ys2 @ B3 ) @ B3 )
@ T3 ) )
@ Xs2
@ Ys ) ) ) ).
% child2_subset_if_input2
thf(fact_1149_child2__subset__if__input1,axiom,
! [Zs2: fset_P2153231429829016240_a_b_b,Zs: fset_P2153231429829016240_a_b_b,X: list_a,Ys: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ Zs2 @ Zs )
=> ( ord_le8870638447146015504_a_b_b
@ ( ffold_6356110402769621009_a_b_b
@ ( produc7216629644754684766_a_b_b
@ ^ [T3: dtree_list_a_b,Uu: b,B3: fset_P2153231429829016240_a_b_b] :
( case_d8112115101481596194_a_b_b
@ ^ [R4: list_a,Ys2: fset_P2153231429829016240_a_b_b] : ( if_fse36834823175207530_a_b_b @ ( R4 = X ) @ ( sup_su5028961100972612572_a_b_b @ Ys2 @ B3 ) @ B3 )
@ T3 ) )
@ Zs2
@ Ys )
@ ( ffold_6356110402769621009_a_b_b
@ ( produc7216629644754684766_a_b_b
@ ^ [T3: dtree_list_a_b,Uu: b,B3: fset_P2153231429829016240_a_b_b] :
( case_d8112115101481596194_a_b_b
@ ^ [R4: list_a,Ys2: fset_P2153231429829016240_a_b_b] : ( if_fse36834823175207530_a_b_b @ ( R4 = X ) @ ( sup_su5028961100972612572_a_b_b @ Ys2 @ B3 ) @ B3 )
@ T3 ) )
@ Zs
@ Ys ) ) ) ).
% child2_subset_if_input1
thf(fact_1150_darc__in__child2__snd__if__nin__fst,axiom,
! [E3: b,X: list_a,A2: list_a,Xs2: fset_P2153231429829016240_a_b_b,Ys: fset_P2153231429829016240_a_b_b,V: list_a,R2: list_a] :
( ( member_b @ E3
@ ( darcs_list_a_b
@ ( node_list_a_b @ X
@ ( ffold_6356110402769621009_a_b_b
@ ( produc7216629644754684766_a_b_b
@ ^ [T3: dtree_list_a_b,Uu: b,B3: fset_P2153231429829016240_a_b_b] :
( case_d8112115101481596194_a_b_b
@ ^ [R4: list_a,Ys2: fset_P2153231429829016240_a_b_b] : ( if_fse36834823175207530_a_b_b @ ( R4 = A2 ) @ ( sup_su5028961100972612572_a_b_b @ Ys2 @ B3 ) @ B3 )
@ T3 ) )
@ Xs2
@ Ys ) ) ) )
=> ( ~ ( member_b @ E3 @ ( darcs_list_a_b @ ( node_list_a_b @ V @ Ys ) ) )
=> ( member_b @ E3 @ ( darcs_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) ) ) ) ).
% darc_in_child2_snd_if_nin_fst
thf(fact_1151_darc__in__child2__fst__if__nin__snd,axiom,
! [E3: b,X: list_a,A2: list_a,Xs2: fset_P2153231429829016240_a_b_b,Ys: fset_P2153231429829016240_a_b_b,V: list_a,R2: list_a] :
( ( member_b @ E3
@ ( darcs_list_a_b
@ ( node_list_a_b @ X
@ ( ffold_6356110402769621009_a_b_b
@ ( produc7216629644754684766_a_b_b
@ ^ [T3: dtree_list_a_b,Uu: b,B3: fset_P2153231429829016240_a_b_b] :
( case_d8112115101481596194_a_b_b
@ ^ [R4: list_a,Ys2: fset_P2153231429829016240_a_b_b] : ( if_fse36834823175207530_a_b_b @ ( R4 = A2 ) @ ( sup_su5028961100972612572_a_b_b @ Ys2 @ B3 ) @ B3 )
@ T3 ) )
@ Xs2
@ Ys ) ) ) )
=> ( ~ ( member_b @ E3 @ ( darcs_list_a_b @ ( node_list_a_b @ V @ Xs2 ) ) )
=> ( member_b @ E3 @ ( darcs_list_a_b @ ( node_list_a_b @ R2 @ Ys ) ) ) ) ) ).
% darc_in_child2_fst_if_nin_snd
thf(fact_1152_bex__empty,axiom,
! [P: b > $o] :
~ ? [X2: b] :
( ( member_b @ X2 @ bot_bot_set_b )
& ( P @ X2 ) ) ).
% bex_empty
thf(fact_1153_bex__empty,axiom,
! [P: a > $o] :
~ ? [X2: a] :
( ( member_a @ X2 @ bot_bot_set_a )
& ( P @ X2 ) ) ).
% bex_empty
thf(fact_1154_bex__empty,axiom,
! [P: list_a > $o] :
~ ? [X2: list_a] :
( ( member_list_a @ X2 @ bot_bot_set_list_a )
& ( P @ X2 ) ) ).
% bex_empty
thf(fact_1155_False,axiom,
~ ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ ta ) ) ) @ ( rank @ ( rev_a @ r ) ) ) ).
% False
thf(fact_1156__C1_OIH_C,axiom,
( ~ ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ ta ) ) ) @ ( rank @ ( rev_a @ r ) ) )
=> ( ( wf_darcs_list_a_b @ ta )
=> ( wf_darcs_list_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ ta ) ) ) ) ).
% "1.IH"
thf(fact_1157_fequalityE,axiom,
! [A: fset_P2153231429829016240_a_b_b,B5: fset_P2153231429829016240_a_b_b] :
( ( A = B5 )
=> ~ ( ( ord_le8870638447146015504_a_b_b @ A @ B5 )
=> ~ ( ord_le8870638447146015504_a_b_b @ B5 @ A ) ) ) ).
% fequalityE
thf(fact_1158_fequalityD1,axiom,
! [A: fset_P2153231429829016240_a_b_b,B5: fset_P2153231429829016240_a_b_b] :
( ( A = B5 )
=> ( ord_le8870638447146015504_a_b_b @ A @ B5 ) ) ).
% fequalityD1
thf(fact_1159_fequalityD2,axiom,
! [A: fset_P2153231429829016240_a_b_b,B5: fset_P2153231429829016240_a_b_b] :
( ( A = B5 )
=> ( ord_le8870638447146015504_a_b_b @ B5 @ A ) ) ).
% fequalityD2
thf(fact_1160_finter__mono,axiom,
! [A: fset_P2153231429829016240_a_b_b,C2: fset_P2153231429829016240_a_b_b,B5: fset_P2153231429829016240_a_b_b,D2: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ A @ C2 )
=> ( ( ord_le8870638447146015504_a_b_b @ B5 @ D2 )
=> ( ord_le8870638447146015504_a_b_b @ ( inf_in525273840840435522_a_b_b @ A @ B5 ) @ ( inf_in525273840840435522_a_b_b @ C2 @ D2 ) ) ) ) ).
% finter_mono
thf(fact_1161_funion__mono,axiom,
! [A: fset_P2153231429829016240_a_b_b,C2: fset_P2153231429829016240_a_b_b,B5: fset_P2153231429829016240_a_b_b,D2: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ A @ C2 )
=> ( ( ord_le8870638447146015504_a_b_b @ B5 @ D2 )
=> ( ord_le8870638447146015504_a_b_b @ ( sup_su5028961100972612572_a_b_b @ A @ B5 ) @ ( sup_su5028961100972612572_a_b_b @ C2 @ D2 ) ) ) ) ).
% funion_mono
thf(fact_1162_fsubset__refl,axiom,
! [A: fset_P2153231429829016240_a_b_b] : ( ord_le8870638447146015504_a_b_b @ A @ A ) ).
% fsubset_refl
thf(fact_1163_funion__least,axiom,
! [A: fset_P2153231429829016240_a_b_b,C2: fset_P2153231429829016240_a_b_b,B5: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ A @ C2 )
=> ( ( ord_le8870638447146015504_a_b_b @ B5 @ C2 )
=> ( ord_le8870638447146015504_a_b_b @ ( sup_su5028961100972612572_a_b_b @ A @ B5 ) @ C2 ) ) ) ).
% funion_least
thf(fact_1164_finter__lower1,axiom,
! [A: fset_P2153231429829016240_a_b_b,B5: fset_P2153231429829016240_a_b_b] : ( ord_le8870638447146015504_a_b_b @ ( inf_in525273840840435522_a_b_b @ A @ B5 ) @ A ) ).
% finter_lower1
thf(fact_1165_finter__lower2,axiom,
! [A: fset_P2153231429829016240_a_b_b,B5: fset_P2153231429829016240_a_b_b] : ( ord_le8870638447146015504_a_b_b @ ( inf_in525273840840435522_a_b_b @ A @ B5 ) @ B5 ) ).
% finter_lower2
thf(fact_1166_fsubset__trans,axiom,
! [A: fset_P2153231429829016240_a_b_b,B5: fset_P2153231429829016240_a_b_b,C2: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ A @ B5 )
=> ( ( ord_le8870638447146015504_a_b_b @ B5 @ C2 )
=> ( ord_le8870638447146015504_a_b_b @ A @ C2 ) ) ) ).
% fsubset_trans
thf(fact_1167_funion__upper1,axiom,
! [A: fset_P2153231429829016240_a_b_b,B5: fset_P2153231429829016240_a_b_b] : ( ord_le8870638447146015504_a_b_b @ A @ ( sup_su5028961100972612572_a_b_b @ A @ B5 ) ) ).
% funion_upper1
thf(fact_1168_funion__upper2,axiom,
! [B5: fset_P2153231429829016240_a_b_b,A: fset_P2153231429829016240_a_b_b] : ( ord_le8870638447146015504_a_b_b @ B5 @ ( sup_su5028961100972612572_a_b_b @ A @ B5 ) ) ).
% funion_upper2
thf(fact_1169_finter__absorb1,axiom,
! [B5: fset_P2153231429829016240_a_b_b,A: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ B5 @ A )
=> ( ( inf_in525273840840435522_a_b_b @ A @ B5 )
= B5 ) ) ).
% finter_absorb1
thf(fact_1170_finter__absorb2,axiom,
! [A: fset_P2153231429829016240_a_b_b,B5: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ A @ B5 )
=> ( ( inf_in525273840840435522_a_b_b @ A @ B5 )
= A ) ) ).
% finter_absorb2
thf(fact_1171_funion__absorb1,axiom,
! [A: fset_P2153231429829016240_a_b_b,B5: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ A @ B5 )
=> ( ( sup_su5028961100972612572_a_b_b @ A @ B5 )
= B5 ) ) ).
% funion_absorb1
thf(fact_1172_funion__absorb2,axiom,
! [B5: fset_P2153231429829016240_a_b_b,A: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ B5 @ A )
=> ( ( sup_su5028961100972612572_a_b_b @ A @ B5 )
= A ) ) ).
% funion_absorb2
thf(fact_1173_finter__greatest,axiom,
! [C2: fset_P2153231429829016240_a_b_b,A: fset_P2153231429829016240_a_b_b,B5: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ C2 @ A )
=> ( ( ord_le8870638447146015504_a_b_b @ C2 @ B5 )
=> ( ord_le8870638447146015504_a_b_b @ C2 @ ( inf_in525273840840435522_a_b_b @ A @ B5 ) ) ) ) ).
% finter_greatest
thf(fact_1174_fset__eq__fsubset,axiom,
( ( ^ [Y4: fset_P2153231429829016240_a_b_b,Z3: fset_P2153231429829016240_a_b_b] : ( Y4 = Z3 ) )
= ( ^ [A6: fset_P2153231429829016240_a_b_b,B6: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ A6 @ B6 )
& ( ord_le8870638447146015504_a_b_b @ B6 @ A6 ) ) ) ) ).
% fset_eq_fsubset
thf(fact_1175_fsubset__funion__eq,axiom,
( ord_le8870638447146015504_a_b_b
= ( ^ [A6: fset_P2153231429829016240_a_b_b,B6: fset_P2153231429829016240_a_b_b] :
( ( sup_su5028961100972612572_a_b_b @ A6 @ B6 )
= B6 ) ) ) ).
% fsubset_funion_eq
thf(fact_1176_funion__finter__assoc__eq,axiom,
! [A: fset_P2153231429829016240_a_b_b,B5: fset_P2153231429829016240_a_b_b,C2: fset_P2153231429829016240_a_b_b] :
( ( ( sup_su5028961100972612572_a_b_b @ ( inf_in525273840840435522_a_b_b @ A @ B5 ) @ C2 )
= ( inf_in525273840840435522_a_b_b @ A @ ( sup_su5028961100972612572_a_b_b @ B5 @ C2 ) ) )
= ( ord_le8870638447146015504_a_b_b @ C2 @ A ) ) ).
% funion_finter_assoc_eq
thf(fact_1177_Bex__def,axiom,
( bex_b
= ( ^ [A6: set_b,P3: b > $o] :
? [X4: b] :
( ( member_b @ X4 @ A6 )
& ( P3 @ X4 ) ) ) ) ).
% Bex_def
thf(fact_1178_Bex__def,axiom,
( bex_Pr2161834730871777312_a_b_b
= ( ^ [A6: set_Pr3443975907877334966_a_b_b,P3: produc6499617310964463488_a_b_b > $o] :
? [X4: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ X4 @ A6 )
& ( P3 @ X4 ) ) ) ) ).
% Bex_def
thf(fact_1179_Bex__def,axiom,
( bex_list_a
= ( ^ [A6: set_list_a,P3: list_a > $o] :
? [X4: list_a] :
( ( member_list_a @ X4 @ A6 )
& ( P3 @ X4 ) ) ) ) ).
% Bex_def
thf(fact_1180_Bex__def,axiom,
( bex_a
= ( ^ [A6: set_a,P3: a > $o] :
? [X4: a] :
( ( member_a @ X4 @ A6 )
& ( P3 @ X4 ) ) ) ) ).
% Bex_def
thf(fact_1181_Bex__def,axiom,
( bex_dtree_list_a_b
= ( ^ [A6: set_dtree_list_a_b,P3: dtree_list_a_b > $o] :
? [X4: dtree_list_a_b] :
( ( member551035911493665803st_a_b @ X4 @ A6 )
& ( P3 @ X4 ) ) ) ) ).
% Bex_def
thf(fact_1182_dtree_Oset__intros_I1_J,axiom,
! [X1: list_a,X22: fset_P2153231429829016240_a_b_b] : ( member_list_a @ X1 @ ( dverts_list_a_b @ ( node_list_a_b @ X1 @ X22 ) ) ) ).
% dtree.set_intros(1)
thf(fact_1183_dtree_Oset__sel_I1_J,axiom,
! [A2: dtree_list_a_b] : ( member_list_a @ ( root_list_a_b @ A2 ) @ ( dverts_list_a_b @ A2 ) ) ).
% dtree.set_sel(1)
thf(fact_1184_dtree_Oroot__def,axiom,
( root_list_a_b
= ( case_d4425991804430049426list_a
@ ^ [X12: list_a,X23: fset_P2153231429829016240_a_b_b] : X12 ) ) ).
% dtree.root_def
thf(fact_1185_disjoint__darcs__child2,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,A2: list_a] :
( ( wf_darcs_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
=> ! [X2: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ X2
@ ( fset_P9138963618725001425_a_b_b
@ ( ffold_6356110402769621009_a_b_b
@ ( produc7216629644754684766_a_b_b
@ ^ [T3: dtree_list_a_b,Uu: b,B3: fset_P2153231429829016240_a_b_b] :
( case_d8112115101481596194_a_b_b
@ ^ [R4: list_a,Ys2: fset_P2153231429829016240_a_b_b] : ( if_fse36834823175207530_a_b_b @ ( R4 = A2 ) @ ( sup_su5028961100972612572_a_b_b @ Ys2 @ B3 ) @ B3 )
@ T3 ) )
@ ( ffilte1050691675278058007_a_b_b
@ ( produc336013011823628281_b_b_o
@ ^ [T3: dtree_list_a_b,E4: b] :
( ( root_list_a_b @ T3 )
!= A2 ) )
@ Xs2 )
@ Xs2 ) ) )
=> ( produc336013011823628281_b_b_o
@ ^ [Y2: dtree_list_a_b,E1: b] :
( ~ ( member_b @ E1 @ ( darcs_list_a_b @ Y2 ) )
& ! [Z: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ Z
@ ( fset_P9138963618725001425_a_b_b
@ ( ffold_6356110402769621009_a_b_b
@ ( produc7216629644754684766_a_b_b
@ ^ [T3: dtree_list_a_b,Uu: b,B3: fset_P2153231429829016240_a_b_b] :
( case_d8112115101481596194_a_b_b
@ ^ [R4: list_a,Ys2: fset_P2153231429829016240_a_b_b] : ( if_fse36834823175207530_a_b_b @ ( R4 = A2 ) @ ( sup_su5028961100972612572_a_b_b @ Ys2 @ B3 ) @ B3 )
@ T3 ) )
@ ( ffilte1050691675278058007_a_b_b
@ ( produc336013011823628281_b_b_o
@ ^ [T3: dtree_list_a_b,E4: b] :
( ( root_list_a_b @ T3 )
!= A2 ) )
@ Xs2 )
@ Xs2 ) ) )
=> ( produc336013011823628281_b_b_o
@ ^ [Aa: dtree_list_a_b,E2: b] :
( ( ( inf_inf_set_b @ ( sup_sup_set_b @ ( darcs_list_a_b @ Y2 ) @ ( insert_b @ E1 @ bot_bot_set_b ) ) @ ( sup_sup_set_b @ ( darcs_list_a_b @ Aa ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
= bot_bot_set_b )
| ( ( produc7704165765595008946_a_b_b @ Y2 @ E1 )
= ( produc7704165765595008946_a_b_b @ Aa @ E2 ) ) )
@ Z ) ) )
@ X2 ) ) ) ).
% disjoint_darcs_child2
thf(fact_1186_wf__darcs__child2,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: dtree_list_a_b,E3: b,A2: list_a] :
( ( wf_darcs_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E3 )
@ ( fset_P9138963618725001425_a_b_b
@ ( ffold_6356110402769621009_a_b_b
@ ( produc7216629644754684766_a_b_b
@ ^ [T3: dtree_list_a_b,Uu: b,B3: fset_P2153231429829016240_a_b_b] :
( case_d8112115101481596194_a_b_b
@ ^ [R4: list_a,Ys2: fset_P2153231429829016240_a_b_b] : ( if_fse36834823175207530_a_b_b @ ( R4 = A2 ) @ ( sup_su5028961100972612572_a_b_b @ Ys2 @ B3 ) @ B3 )
@ T3 ) )
@ ( ffilte1050691675278058007_a_b_b
@ ( produc336013011823628281_b_b_o
@ ^ [T3: dtree_list_a_b,E4: b] :
( ( root_list_a_b @ T3 )
!= A2 ) )
@ Xs2 )
@ Xs2 ) ) )
=> ( wf_darcs_list_a_b @ X ) ) ) ).
% wf_darcs_child2
thf(fact_1187_disjoint__darcs__aux1,axiom,
! [R2: b,Xs2: fset_P2502403327259978667_b_b_b,V: b,E3: b,Y: b] :
( ( wf_dverts_b_b @ ( node_b_b @ R2 @ Xs2 ) )
=> ( ~ ( member_b @ V @ ( dverts_b_b @ ( node_b_b @ R2 @ Xs2 ) ) )
=> ( ( wf_darcs_b_b @ ( node_b_b @ R2 @ Xs2 ) )
=> ( ~ ( member_b @ E3 @ ( darcs_b_b @ ( node_b_b @ R2 @ Xs2 ) ) )
=> ! [X2: produc931321787949985659_b_b_b] :
( ( member753767694440230418_b_b_b @ X2
@ ( fset_P6379472244080315724_b_b_b
@ ( ffold_5200765406727462407_b_b_b
@ ( produc1404479840988494799_b_b_b
@ ^ [T13: dtree_b_b,E1: b] :
( finser4810365818013240883_b_b_b
@ ( if_Pro2702819762496977205_b_b_b
@ ( ( root_b_b @ T13 )
= Y )
@ ( produc5927820332525640749_b_b_b @ ( node_b_b @ V @ ( finser4810365818013240883_b_b_b @ ( produc5927820332525640749_b_b_b @ T13 @ E1 ) @ bot_bo117012103628883007_b_b_b ) ) @ E3 )
@ ( produc5927820332525640749_b_b_b @ T13 @ E1 ) ) ) )
@ bot_bo117012103628883007_b_b_b
@ Xs2 ) ) )
=> ( produc2669682129855673534_b_b_o
@ ^ [Y2: dtree_b_b,E1: b] :
( ~ ( member_b @ E1 @ ( darcs_b_b @ Y2 ) )
& ! [Z: produc931321787949985659_b_b_b] :
( ( member753767694440230418_b_b_b @ Z
@ ( fset_P6379472244080315724_b_b_b
@ ( ffold_5200765406727462407_b_b_b
@ ( produc1404479840988494799_b_b_b
@ ^ [T13: dtree_b_b,E1a: b] :
( finser4810365818013240883_b_b_b
@ ( if_Pro2702819762496977205_b_b_b
@ ( ( root_b_b @ T13 )
= Y )
@ ( produc5927820332525640749_b_b_b @ ( node_b_b @ V @ ( finser4810365818013240883_b_b_b @ ( produc5927820332525640749_b_b_b @ T13 @ E1a ) @ bot_bo117012103628883007_b_b_b ) ) @ E3 )
@ ( produc5927820332525640749_b_b_b @ T13 @ E1a ) ) ) )
@ bot_bo117012103628883007_b_b_b
@ Xs2 ) ) )
=> ( produc2669682129855673534_b_b_o
@ ^ [Aa: dtree_b_b,E2: b] :
( ( ( inf_inf_set_b @ ( sup_sup_set_b @ ( darcs_b_b @ Y2 ) @ ( insert_b @ E1 @ bot_bot_set_b ) ) @ ( sup_sup_set_b @ ( darcs_b_b @ Aa ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
= bot_bot_set_b )
| ( ( produc5927820332525640749_b_b_b @ Y2 @ E1 )
= ( produc5927820332525640749_b_b_b @ Aa @ E2 ) ) )
@ Z ) ) )
@ X2 ) ) ) ) ) ) ).
% disjoint_darcs_aux1
thf(fact_1188_disjoint__darcs__aux1,axiom,
! [R2: a,Xs2: fset_P5281107635120001194_a_b_b,V: a,E3: b,Y: a] :
( ( wf_dverts_a_b @ ( node_a_b @ R2 @ Xs2 ) )
=> ( ~ ( member_a @ V @ ( dverts_a_b @ ( node_a_b @ R2 @ Xs2 ) ) )
=> ( ( wf_darcs_a_b @ ( node_a_b @ R2 @ Xs2 ) )
=> ( ~ ( member_b @ E3 @ ( darcs_a_b @ ( node_a_b @ R2 @ Xs2 ) ) )
=> ! [X2: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X2
@ ( fset_P783253628892185035_a_b_b
@ ( ffold_1002666695659912709_a_b_b
@ ( produc7149527399694995404_a_b_b
@ ^ [T13: dtree_a_b,E1: b] :
( finser8437519239679886002_a_b_b
@ ( if_Pro6329973184163622324_a_b_b
@ ( ( root_a_b @ T13 )
= Y )
@ ( produc331601717337510060_a_b_b @ ( node_a_b @ V @ ( finser8437519239679886002_a_b_b @ ( produc331601717337510060_a_b_b @ T13 @ E1 ) @ bot_bo2895716411488905534_a_b_b ) ) @ E3 )
@ ( produc331601717337510060_a_b_b @ T13 @ E1 ) ) ) )
@ bot_bo2895716411488905534_a_b_b
@ Xs2 ) ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Y2: dtree_a_b,E1: b] :
( ~ ( member_b @ E1 @ ( darcs_a_b @ Y2 ) )
& ! [Z: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ Z
@ ( fset_P783253628892185035_a_b_b
@ ( ffold_1002666695659912709_a_b_b
@ ( produc7149527399694995404_a_b_b
@ ^ [T13: dtree_a_b,E1a: b] :
( finser8437519239679886002_a_b_b
@ ( if_Pro6329973184163622324_a_b_b
@ ( ( root_a_b @ T13 )
= Y )
@ ( produc331601717337510060_a_b_b @ ( node_a_b @ V @ ( finser8437519239679886002_a_b_b @ ( produc331601717337510060_a_b_b @ T13 @ E1a ) @ bot_bo2895716411488905534_a_b_b ) ) @ E3 )
@ ( produc331601717337510060_a_b_b @ T13 @ E1a ) ) ) )
@ bot_bo2895716411488905534_a_b_b
@ Xs2 ) ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Aa: dtree_a_b,E2: b] :
( ( ( inf_inf_set_b @ ( sup_sup_set_b @ ( darcs_a_b @ Y2 ) @ ( insert_b @ E1 @ bot_bot_set_b ) ) @ ( sup_sup_set_b @ ( darcs_a_b @ Aa ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
= bot_bot_set_b )
| ( ( produc331601717337510060_a_b_b @ Y2 @ E1 )
= ( produc331601717337510060_a_b_b @ Aa @ E2 ) ) )
@ Z ) ) )
@ X2 ) ) ) ) ) ) ).
% disjoint_darcs_aux1
thf(fact_1189_disjoint__darcs__aux1,axiom,
! [R2: b,Xs2: fset_P9004070893569812843_b_a_a,V: b,E3: a,Y: b] :
( ( wf_dverts_b_a @ ( node_b_a @ R2 @ Xs2 ) )
=> ( ~ ( member_b @ V @ ( dverts_b_a @ ( node_b_a @ R2 @ Xs2 ) ) )
=> ( ( wf_darcs_b_a @ ( node_b_a @ R2 @ Xs2 ) )
=> ( ~ ( member_a @ E3 @ ( darcs_b_a @ ( node_b_a @ R2 @ Xs2 ) ) )
=> ! [X2: produc1566970784251743547_b_a_a] :
( ( member1389416690741988306_b_a_a @ X2
@ ( fset_P7015121240382073612_b_a_a
@ ( ffold_8028070260251416967_b_a_a
@ ( produc966445439450875023_b_a_a
@ ^ [T13: dtree_b_a,E1: a] :
( finser5446014814314998771_b_a_a
@ ( if_Pro3338468758798735093_b_a_a
@ ( ( root_b_a @ T13 )
= Y )
@ ( produc6563469333130627437_b_a_a @ ( node_b_a @ V @ ( finser5446014814314998771_b_a_a @ ( produc6563469333130627437_b_a_a @ T13 @ E1 ) @ bot_bo6618679669938717183_b_a_a ) ) @ E3 )
@ ( produc6563469333130627437_b_a_a @ T13 @ E1 ) ) ) )
@ bot_bo6618679669938717183_b_a_a
@ Xs2 ) ) )
=> ( produc9196057950986712958_a_a_o
@ ^ [Y2: dtree_b_a,E1: a] :
( ~ ( member_a @ E1 @ ( darcs_b_a @ Y2 ) )
& ! [Z: produc1566970784251743547_b_a_a] :
( ( member1389416690741988306_b_a_a @ Z
@ ( fset_P7015121240382073612_b_a_a
@ ( ffold_8028070260251416967_b_a_a
@ ( produc966445439450875023_b_a_a
@ ^ [T13: dtree_b_a,E1a: a] :
( finser5446014814314998771_b_a_a
@ ( if_Pro3338468758798735093_b_a_a
@ ( ( root_b_a @ T13 )
= Y )
@ ( produc6563469333130627437_b_a_a @ ( node_b_a @ V @ ( finser5446014814314998771_b_a_a @ ( produc6563469333130627437_b_a_a @ T13 @ E1a ) @ bot_bo6618679669938717183_b_a_a ) ) @ E3 )
@ ( produc6563469333130627437_b_a_a @ T13 @ E1a ) ) ) )
@ bot_bo6618679669938717183_b_a_a
@ Xs2 ) ) )
=> ( produc9196057950986712958_a_a_o
@ ^ [Aa: dtree_b_a,E2: a] :
( ( ( inf_inf_set_a @ ( sup_sup_set_a @ ( darcs_b_a @ Y2 ) @ ( insert_a @ E1 @ bot_bot_set_a ) ) @ ( sup_sup_set_a @ ( darcs_b_a @ Aa ) @ ( insert_a @ E2 @ bot_bot_set_a ) ) )
= bot_bot_set_a )
| ( ( produc6563469333130627437_b_a_a @ Y2 @ E1 )
= ( produc6563469333130627437_b_a_a @ Aa @ E2 ) ) )
@ Z ) ) )
@ X2 ) ) ) ) ) ) ).
% disjoint_darcs_aux1
thf(fact_1190_disjoint__darcs__aux1,axiom,
! [R2: a,Xs2: fset_P2559403164575059562_a_a_a,V: a,E3: a,Y: a] :
( ( wf_dverts_a_a @ ( node_a_a @ R2 @ Xs2 ) )
=> ( ~ ( member_a @ V @ ( dverts_a_a @ ( node_a_a @ R2 @ Xs2 ) ) )
=> ( ( wf_darcs_a_a @ ( node_a_a @ R2 @ Xs2 ) )
=> ( ~ ( member_a @ E3 @ ( darcs_a_a @ ( node_a_a @ R2 @ Xs2 ) ) )
=> ! [X2: produc5194124205918388666_a_a_a] :
( ( member5016570112408633425_a_a_a @ X2
@ ( fset_P1418902625193942923_a_a_a
@ ( ffold_3829971549183867269_a_a_a
@ ( produc6711492998157375628_a_a_a
@ ^ [T13: dtree_a_a,E1: a] :
( finser9073168235981643890_a_a_a
@ ( if_Pro6965622180465380212_a_a_a
@ ( ( root_a_a @ T13 )
= Y )
@ ( produc967250717942496748_a_a_a @ ( node_a_a @ V @ ( finser9073168235981643890_a_a_a @ ( produc967250717942496748_a_a_a @ T13 @ E1 ) @ bot_bo174011940943963902_a_a_a ) ) @ E3 )
@ ( produc967250717942496748_a_a_a @ T13 @ E1 ) ) ) )
@ bot_bo174011940943963902_a_a_a
@ Xs2 ) ) )
=> ( produc7851592914177225023_a_a_o
@ ^ [Y2: dtree_a_a,E1: a] :
( ~ ( member_a @ E1 @ ( darcs_a_a @ Y2 ) )
& ! [Z: produc5194124205918388666_a_a_a] :
( ( member5016570112408633425_a_a_a @ Z
@ ( fset_P1418902625193942923_a_a_a
@ ( ffold_3829971549183867269_a_a_a
@ ( produc6711492998157375628_a_a_a
@ ^ [T13: dtree_a_a,E1a: a] :
( finser9073168235981643890_a_a_a
@ ( if_Pro6965622180465380212_a_a_a
@ ( ( root_a_a @ T13 )
= Y )
@ ( produc967250717942496748_a_a_a @ ( node_a_a @ V @ ( finser9073168235981643890_a_a_a @ ( produc967250717942496748_a_a_a @ T13 @ E1a ) @ bot_bo174011940943963902_a_a_a ) ) @ E3 )
@ ( produc967250717942496748_a_a_a @ T13 @ E1a ) ) ) )
@ bot_bo174011940943963902_a_a_a
@ Xs2 ) ) )
=> ( produc7851592914177225023_a_a_o
@ ^ [Aa: dtree_a_a,E2: a] :
( ( ( inf_inf_set_a @ ( sup_sup_set_a @ ( darcs_a_a @ Y2 ) @ ( insert_a @ E1 @ bot_bot_set_a ) ) @ ( sup_sup_set_a @ ( darcs_a_a @ Aa ) @ ( insert_a @ E2 @ bot_bot_set_a ) ) )
= bot_bot_set_a )
| ( ( produc967250717942496748_a_a_a @ Y2 @ E1 )
= ( produc967250717942496748_a_a_a @ Aa @ E2 ) ) )
@ Z ) ) )
@ X2 ) ) ) ) ) ) ).
% disjoint_darcs_aux1
thf(fact_1191_disjoint__darcs__aux1,axiom,
! [R2: list_a,Xs2: fset_P8654898996138850416_a_a_a,V: list_a,E3: a,Y: list_a] :
( ( wf_dverts_list_a_a @ ( node_list_a_a @ R2 @ Xs2 ) )
=> ( ~ ( member_list_a @ V @ ( dverts_list_a_a @ ( node_list_a_a @ R2 @ Xs2 ) ) )
=> ( ( wf_darcs_list_a_a @ ( node_list_a_a @ R2 @ Xs2 ) )
=> ( ~ ( member_a @ E3 @ ( darcs_list_a_a @ ( node_list_a_a @ R2 @ Xs2 ) ) )
=> ! [X2: produc7135266307266221376_a_a_a] :
( ( member5331345429024349271_a_a_a @ X2
@ ( fset_P551240578171983505_a_a_a
@ ( ffold_1963405522066353041_a_a_a
@ ( produc8847821649887753246_a_a_a
@ ^ [T13: dtree_list_a_a,E1: a] :
( finser2938861521451939832_a_a_a
@ ( if_Pro1443360173218331130_a_a_a
@ ( ( root_list_a_a @ T13 )
= Y )
@ ( produc8339814766199995634_a_a_a @ ( node_list_a_a @ V @ ( finser2938861521451939832_a_a_a @ ( produc8339814766199995634_a_a_a @ T13 @ E1 ) @ bot_bo8750491735591794436_a_a_a ) ) @ E3 )
@ ( produc8339814766199995634_a_a_a @ T13 @ E1 ) ) ) )
@ bot_bo8750491735591794436_a_a_a
@ Xs2 ) ) )
=> ( produc6862388832954667705_a_a_o
@ ^ [Y2: dtree_list_a_a,E1: a] :
( ~ ( member_a @ E1 @ ( darcs_list_a_a @ Y2 ) )
& ! [Z: produc7135266307266221376_a_a_a] :
( ( member5331345429024349271_a_a_a @ Z
@ ( fset_P551240578171983505_a_a_a
@ ( ffold_1963405522066353041_a_a_a
@ ( produc8847821649887753246_a_a_a
@ ^ [T13: dtree_list_a_a,E1a: a] :
( finser2938861521451939832_a_a_a
@ ( if_Pro1443360173218331130_a_a_a
@ ( ( root_list_a_a @ T13 )
= Y )
@ ( produc8339814766199995634_a_a_a @ ( node_list_a_a @ V @ ( finser2938861521451939832_a_a_a @ ( produc8339814766199995634_a_a_a @ T13 @ E1a ) @ bot_bo8750491735591794436_a_a_a ) ) @ E3 )
@ ( produc8339814766199995634_a_a_a @ T13 @ E1a ) ) ) )
@ bot_bo8750491735591794436_a_a_a
@ Xs2 ) ) )
=> ( produc6862388832954667705_a_a_o
@ ^ [Aa: dtree_list_a_a,E2: a] :
( ( ( inf_inf_set_a @ ( sup_sup_set_a @ ( darcs_list_a_a @ Y2 ) @ ( insert_a @ E1 @ bot_bot_set_a ) ) @ ( sup_sup_set_a @ ( darcs_list_a_a @ Aa ) @ ( insert_a @ E2 @ bot_bot_set_a ) ) )
= bot_bot_set_a )
| ( ( produc8339814766199995634_a_a_a @ Y2 @ E1 )
= ( produc8339814766199995634_a_a_a @ Aa @ E2 ) ) )
@ Z ) ) )
@ X2 ) ) ) ) ) ) ).
% disjoint_darcs_aux1
thf(fact_1192_disjoint__darcs__aux1,axiom,
! [R2: b,Xs2: fset_P3762770571697108971list_a,V: b,E3: list_a,Y: b] :
( ( wf_dverts_b_list_a @ ( node_b_list_a @ R2 @ Xs2 ) )
=> ( ~ ( member_b @ V @ ( dverts_b_list_a @ ( node_b_list_a @ R2 @ Xs2 ) ) )
=> ( ( wf_darcs_b_list_a @ ( node_b_list_a @ R2 @ Xs2 ) )
=> ( ~ ( member_list_a @ E3 @ ( darcs_b_list_a @ ( node_b_list_a @ R2 @ Xs2 ) ) )
=> ! [X2: produc8289178782644283579list_a] :
( ( member5717964264556279890list_a @ X2
@ ( fset_P7818209297497886092list_a
@ ( ffold_5967413713360652167list_a
@ ( produc1264271959818302223list_a
@ ^ [T13: dtree_b_list_a,E1: list_a] :
( finser4647321697179225971list_a
@ ( if_Pro6172782175990474869list_a
@ ( ( root_b_list_a @ T13 )
= Y )
@ ( produc8734963857942192365list_a @ ( node_b_list_a @ V @ ( finser4647321697179225971list_a @ ( produc8734963857942192365list_a @ T13 @ E1 ) @ bot_bo2431243398884449407list_a ) ) @ E3 )
@ ( produc8734963857942192365list_a @ T13 @ E1 ) ) ) )
@ bot_bo2431243398884449407list_a
@ Xs2 ) ) )
=> ( produc5249043932001496062st_a_o
@ ^ [Y2: dtree_b_list_a,E1: list_a] :
( ~ ( member_list_a @ E1 @ ( darcs_b_list_a @ Y2 ) )
& ! [Z: produc8289178782644283579list_a] :
( ( member5717964264556279890list_a @ Z
@ ( fset_P7818209297497886092list_a
@ ( ffold_5967413713360652167list_a
@ ( produc1264271959818302223list_a
@ ^ [T13: dtree_b_list_a,E1a: list_a] :
( finser4647321697179225971list_a
@ ( if_Pro6172782175990474869list_a
@ ( ( root_b_list_a @ T13 )
= Y )
@ ( produc8734963857942192365list_a @ ( node_b_list_a @ V @ ( finser4647321697179225971list_a @ ( produc8734963857942192365list_a @ T13 @ E1a ) @ bot_bo2431243398884449407list_a ) ) @ E3 )
@ ( produc8734963857942192365list_a @ T13 @ E1a ) ) ) )
@ bot_bo2431243398884449407list_a
@ Xs2 ) ) )
=> ( produc5249043932001496062st_a_o
@ ^ [Aa: dtree_b_list_a,E2: list_a] :
( ( ( inf_inf_set_list_a @ ( sup_sup_set_list_a @ ( darcs_b_list_a @ Y2 ) @ ( insert_list_a @ E1 @ bot_bot_set_list_a ) ) @ ( sup_sup_set_list_a @ ( darcs_b_list_a @ Aa ) @ ( insert_list_a @ E2 @ bot_bot_set_list_a ) ) )
= bot_bot_set_list_a )
| ( ( produc8734963857942192365list_a @ Y2 @ E1 )
= ( produc8734963857942192365list_a @ Aa @ E2 ) ) )
@ Z ) ) )
@ X2 ) ) ) ) ) ) ).
% disjoint_darcs_aux1
thf(fact_1193_disjoint__darcs__aux1,axiom,
! [R2: a,Xs2: fset_P8006199460403381738list_a,V: a,E3: list_a,Y: a] :
( ( wf_dverts_a_list_a @ ( node_a_list_a @ R2 @ Xs2 ) )
=> ( ~ ( member_a @ V @ ( dverts_a_list_a @ ( node_a_list_a @ R2 @ Xs2 ) ) )
=> ( ( wf_darcs_a_list_a @ ( node_a_list_a @ R2 @ Xs2 ) )
=> ( ~ ( member_list_a @ E3 @ ( darcs_a_list_a @ ( node_a_list_a @ R2 @ Xs2 ) ) )
=> ! [X2: produc4349230196019447354list_a] :
( ( member1778015677931443665list_a @ X2
@ ( fset_P3878260710873049867list_a
@ ( ffold_2487888024171782789list_a
@ ( produc3833109818020232460list_a
@ ^ [T13: dtree_a_list_a,E1: list_a] :
( finser707373110554389746list_a
@ ( if_Pro2232833589365638644list_a
@ ( ( root_a_list_a @ T13 )
= Y )
@ ( produc4795015271317356140list_a @ ( node_a_list_a @ V @ ( finser707373110554389746list_a @ ( produc4795015271317356140list_a @ T13 @ E1 ) @ bot_bo6674672287590722174list_a ) ) @ E3 )
@ ( produc4795015271317356140list_a @ T13 @ E1 ) ) ) )
@ bot_bo6674672287590722174list_a
@ Xs2 ) ) )
=> ( produc4007918758503627967st_a_o
@ ^ [Y2: dtree_a_list_a,E1: list_a] :
( ~ ( member_list_a @ E1 @ ( darcs_a_list_a @ Y2 ) )
& ! [Z: produc4349230196019447354list_a] :
( ( member1778015677931443665list_a @ Z
@ ( fset_P3878260710873049867list_a
@ ( ffold_2487888024171782789list_a
@ ( produc3833109818020232460list_a
@ ^ [T13: dtree_a_list_a,E1a: list_a] :
( finser707373110554389746list_a
@ ( if_Pro2232833589365638644list_a
@ ( ( root_a_list_a @ T13 )
= Y )
@ ( produc4795015271317356140list_a @ ( node_a_list_a @ V @ ( finser707373110554389746list_a @ ( produc4795015271317356140list_a @ T13 @ E1a ) @ bot_bo6674672287590722174list_a ) ) @ E3 )
@ ( produc4795015271317356140list_a @ T13 @ E1a ) ) ) )
@ bot_bo6674672287590722174list_a
@ Xs2 ) ) )
=> ( produc4007918758503627967st_a_o
@ ^ [Aa: dtree_a_list_a,E2: list_a] :
( ( ( inf_inf_set_list_a @ ( sup_sup_set_list_a @ ( darcs_a_list_a @ Y2 ) @ ( insert_list_a @ E1 @ bot_bot_set_list_a ) ) @ ( sup_sup_set_list_a @ ( darcs_a_list_a @ Aa ) @ ( insert_list_a @ E2 @ bot_bot_set_list_a ) ) )
= bot_bot_set_list_a )
| ( ( produc4795015271317356140list_a @ Y2 @ E1 )
= ( produc4795015271317356140list_a @ Aa @ E2 ) ) )
@ Z ) ) )
@ X2 ) ) ) ) ) ) ).
% disjoint_darcs_aux1
thf(fact_1194_disjoint__darcs__aux1,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,V: list_a,E3: b,Y: list_a] :
( ( wf_dverts_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
=> ( ~ ( member_list_a @ V @ ( dverts_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) )
=> ( ( wf_darcs_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
=> ( ~ ( member_b @ E3 @ ( darcs_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) )
=> ! [X2: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ X2
@ ( fset_P9138963618725001425_a_b_b
@ ( ffold_6356110402769621009_a_b_b
@ ( produc7216629644754684766_a_b_b
@ ^ [T13: dtree_list_a_b,E1: b] :
( finser2303212525150181944_a_b_b
@ ( if_Pro807711176916573242_a_b_b
@ ( ( root_list_a_b @ T13 )
= Y )
@ ( produc7704165765595008946_a_b_b @ ( node_list_a_b @ V @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T13 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ E3 )
@ ( produc7704165765595008946_a_b_b @ T13 @ E1 ) ) ) )
@ bot_bo2248824169281960260_a_b_b
@ Xs2 ) ) )
=> ( produc336013011823628281_b_b_o
@ ^ [Y2: dtree_list_a_b,E1: b] :
( ~ ( member_b @ E1 @ ( darcs_list_a_b @ Y2 ) )
& ! [Z: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ Z
@ ( fset_P9138963618725001425_a_b_b
@ ( ffold_6356110402769621009_a_b_b
@ ( produc7216629644754684766_a_b_b
@ ^ [T13: dtree_list_a_b,E1a: b] :
( finser2303212525150181944_a_b_b
@ ( if_Pro807711176916573242_a_b_b
@ ( ( root_list_a_b @ T13 )
= Y )
@ ( produc7704165765595008946_a_b_b @ ( node_list_a_b @ V @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T13 @ E1a ) @ bot_bo2248824169281960260_a_b_b ) ) @ E3 )
@ ( produc7704165765595008946_a_b_b @ T13 @ E1a ) ) ) )
@ bot_bo2248824169281960260_a_b_b
@ Xs2 ) ) )
=> ( produc336013011823628281_b_b_o
@ ^ [Aa: dtree_list_a_b,E2: b] :
( ( ( inf_inf_set_b @ ( sup_sup_set_b @ ( darcs_list_a_b @ Y2 ) @ ( insert_b @ E1 @ bot_bot_set_b ) ) @ ( sup_sup_set_b @ ( darcs_list_a_b @ Aa ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
= bot_bot_set_b )
| ( ( produc7704165765595008946_a_b_b @ Y2 @ E1 )
= ( produc7704165765595008946_a_b_b @ Aa @ E2 ) ) )
@ Z ) ) )
@ X2 ) ) ) ) ) ) ).
% disjoint_darcs_aux1
thf(fact_1195_disjoint__darcs__aux1,axiom,
! [R2: list_a,Xs2: fset_P6656299774166858224list_a,V: list_a,E3: list_a,Y: list_a] :
( ( wf_dve6709000387044496771list_a @ ( node_list_a_list_a @ R2 @ Xs2 ) )
=> ( ~ ( member_list_a @ V @ ( dverts_list_a_list_a @ ( node_list_a_list_a @ R2 @ Xs2 ) ) )
=> ( ( wf_dar2483771551534525728list_a @ ( node_list_a_list_a @ R2 @ Xs2 ) )
=> ( ~ ( member_list_a @ E3 @ ( darcs_list_a_list_a @ ( node_list_a_list_a @ R2 @ Xs2 ) ) )
=> ! [X2: produc7575571382841205696list_a] :
( ( member535470554364689367list_a @ X2
@ ( fset_P2575872134735801361list_a
@ ( ffold_1751350808669368977list_a
@ ( produc2593633274636522654list_a
@ ^ [T13: dtree_list_a_list_a,E1: list_a] :
( finser5913853959220252280list_a
@ ( if_Pro2239956646901841530list_a
@ ( ( root_list_a_list_a @ T13 )
= Y )
@ ( produc6276725351364364146list_a @ ( node_list_a_list_a @ V @ ( finser5913853959220252280list_a @ ( produc6276725351364364146list_a @ T13 @ E1 ) @ bot_bo7860097226086663300list_a ) ) @ E3 )
@ ( produc6276725351364364146list_a @ T13 @ E1 ) ) ) )
@ bot_bo7860097226086663300list_a
@ Xs2 ) ) )
=> ( produc121743164402242617st_a_o
@ ^ [Y2: dtree_list_a_list_a,E1: list_a] :
( ~ ( member_list_a @ E1 @ ( darcs_list_a_list_a @ Y2 ) )
& ! [Z: produc7575571382841205696list_a] :
( ( member535470554364689367list_a @ Z
@ ( fset_P2575872134735801361list_a
@ ( ffold_1751350808669368977list_a
@ ( produc2593633274636522654list_a
@ ^ [T13: dtree_list_a_list_a,E1a: list_a] :
( finser5913853959220252280list_a
@ ( if_Pro2239956646901841530list_a
@ ( ( root_list_a_list_a @ T13 )
= Y )
@ ( produc6276725351364364146list_a @ ( node_list_a_list_a @ V @ ( finser5913853959220252280list_a @ ( produc6276725351364364146list_a @ T13 @ E1a ) @ bot_bo7860097226086663300list_a ) ) @ E3 )
@ ( produc6276725351364364146list_a @ T13 @ E1a ) ) ) )
@ bot_bo7860097226086663300list_a
@ Xs2 ) ) )
=> ( produc121743164402242617st_a_o
@ ^ [Aa: dtree_list_a_list_a,E2: list_a] :
( ( ( inf_inf_set_list_a @ ( sup_sup_set_list_a @ ( darcs_list_a_list_a @ Y2 ) @ ( insert_list_a @ E1 @ bot_bot_set_list_a ) ) @ ( sup_sup_set_list_a @ ( darcs_list_a_list_a @ Aa ) @ ( insert_list_a @ E2 @ bot_bot_set_list_a ) ) )
= bot_bot_set_list_a )
| ( ( produc6276725351364364146list_a @ Y2 @ E1 )
= ( produc6276725351364364146list_a @ Aa @ E2 ) ) )
@ Z ) ) )
@ X2 ) ) ) ) ) ) ).
% disjoint_darcs_aux1
thf(fact_1196_disjoint__darcs__aux1,axiom,
! [R2: b,Xs2: fset_P729086328404512291st_a_b,V: b,E3: dtree_list_a_b,Y: b] :
( ( wf_dve4513435013532653104st_a_b @ ( node_b9108412673219173045st_a_b @ R2 @ Xs2 ) )
=> ( ~ ( member_b @ V @ ( dverts8864706460159352569st_a_b @ ( node_b9108412673219173045st_a_b @ R2 @ Xs2 ) ) )
=> ( ( wf_dar722496709096415187st_a_b @ ( node_b9108412673219173045st_a_b @ R2 @ Xs2 ) )
=> ( ~ ( member551035911493665803st_a_b @ E3 @ ( darcs_7068930265994548426st_a_b @ ( node_b9108412673219173045st_a_b @ R2 @ Xs2 ) ) )
=> ! [X2: produc8101844211926167155st_a_b] :
( ( member5849886156663560074st_a_b @ X2
@ ( fset_P1687456239673687492st_a_b
@ ( ffold_3232036597505749367st_a_b
@ ( produc1804618233457100159st_a_b
@ ^ [T13: dtree_3602950866360573648st_a_b,E1: dtree_list_a_b] :
( finser141567303011885355st_a_b
@ ( if_Pro2199488368625827629st_a_b
@ ( ( root_b4214690873640572181st_a_b @ T13 )
= Y )
@ ( produc3051495980888433645st_a_b @ ( node_b9108412673219173045st_a_b @ V @ ( finser141567303011885355st_a_b @ ( produc3051495980888433645st_a_b @ T13 @ E1 ) @ bot_bo6210586517881791671st_a_b ) ) @ E3 )
@ ( produc3051495980888433645st_a_b @ T13 @ E1 ) ) ) )
@ bot_bo6210586517881791671st_a_b
@ Xs2 ) ) )
=> ( produc3786481918722493182_a_b_o
@ ^ [Y2: dtree_3602950866360573648st_a_b,E1: dtree_list_a_b] :
( ~ ( member551035911493665803st_a_b @ E1 @ ( darcs_7068930265994548426st_a_b @ Y2 ) )
& ! [Z: produc8101844211926167155st_a_b] :
( ( member5849886156663560074st_a_b @ Z
@ ( fset_P1687456239673687492st_a_b
@ ( ffold_3232036597505749367st_a_b
@ ( produc1804618233457100159st_a_b
@ ^ [T13: dtree_3602950866360573648st_a_b,E1a: dtree_list_a_b] :
( finser141567303011885355st_a_b
@ ( if_Pro2199488368625827629st_a_b
@ ( ( root_b4214690873640572181st_a_b @ T13 )
= Y )
@ ( produc3051495980888433645st_a_b @ ( node_b9108412673219173045st_a_b @ V @ ( finser141567303011885355st_a_b @ ( produc3051495980888433645st_a_b @ T13 @ E1a ) @ bot_bo6210586517881791671st_a_b ) ) @ E3 )
@ ( produc3051495980888433645st_a_b @ T13 @ E1a ) ) ) )
@ bot_bo6210586517881791671st_a_b
@ Xs2 ) ) )
=> ( produc3786481918722493182_a_b_o
@ ^ [Aa: dtree_3602950866360573648st_a_b,E2: dtree_list_a_b] :
( ( ( inf_in3355993651213403836st_a_b @ ( sup_su5199045856643764822st_a_b @ ( darcs_7068930265994548426st_a_b @ Y2 ) @ ( insert7065945238938772516st_a_b @ E1 @ bot_bo798015271861357502st_a_b ) ) @ ( sup_su5199045856643764822st_a_b @ ( darcs_7068930265994548426st_a_b @ Aa ) @ ( insert7065945238938772516st_a_b @ E2 @ bot_bo798015271861357502st_a_b ) ) )
= bot_bo798015271861357502st_a_b )
| ( ( produc3051495980888433645st_a_b @ Y2 @ E1 )
= ( produc3051495980888433645st_a_b @ Aa @ E2 ) ) )
@ Z ) ) )
@ X2 ) ) ) ) ) ) ).
% disjoint_darcs_aux1
thf(fact_1197_disjoint__darcs__aux1__aux1,axiom,
! [Xs2: fset_P2502403327259978667_b_b_b,R2: b,V: b,E3: b,T12: dtree_b_b,E12: b,Y: b,T23: dtree_b_b,E22: b] :
( ! [X3: produc931321787949985659_b_b_b] :
( ( member753767694440230418_b_b_b @ X3 @ ( fset_P6379472244080315724_b_b_b @ Xs2 ) )
=> ( produc2669682129855673534_b_b_o
@ ^ [Y2: dtree_b_b,E1: b] :
( ~ ( member_b @ E1 @ ( darcs_b_b @ Y2 ) )
& ! [Z: produc931321787949985659_b_b_b] :
( ( member753767694440230418_b_b_b @ Z @ ( fset_P6379472244080315724_b_b_b @ Xs2 ) )
=> ( produc2669682129855673534_b_b_o
@ ^ [Aa: dtree_b_b,E2: b] :
( ( ( inf_inf_set_b @ ( sup_sup_set_b @ ( darcs_b_b @ Y2 ) @ ( insert_b @ E1 @ bot_bot_set_b ) ) @ ( sup_sup_set_b @ ( darcs_b_b @ Aa ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
= bot_bot_set_b )
| ( ( produc5927820332525640749_b_b_b @ Y2 @ E1 )
= ( produc5927820332525640749_b_b_b @ Aa @ E2 ) ) )
@ Z ) ) )
@ X3 ) )
=> ( ( wf_dverts_b_b @ ( node_b_b @ R2 @ Xs2 ) )
=> ( ~ ( member_b @ V @ ( dverts_b_b @ ( node_b_b @ R2 @ Xs2 ) ) )
=> ( ~ ( member_b @ E3 @ ( darcs_b_b @ ( node_b_b @ R2 @ Xs2 ) ) )
=> ( ( member753767694440230418_b_b_b @ ( produc5927820332525640749_b_b_b @ T12 @ E12 )
@ ( fset_P6379472244080315724_b_b_b
@ ( ffold_5200765406727462407_b_b_b
@ ( produc1404479840988494799_b_b_b
@ ^ [T13: dtree_b_b,E1: b] :
( finser4810365818013240883_b_b_b
@ ( if_Pro2702819762496977205_b_b_b
@ ( ( root_b_b @ T13 )
= Y )
@ ( produc5927820332525640749_b_b_b @ ( node_b_b @ V @ ( finser4810365818013240883_b_b_b @ ( produc5927820332525640749_b_b_b @ T13 @ E1 ) @ bot_bo117012103628883007_b_b_b ) ) @ E3 )
@ ( produc5927820332525640749_b_b_b @ T13 @ E1 ) ) ) )
@ bot_bo117012103628883007_b_b_b
@ Xs2 ) ) )
=> ( ( member753767694440230418_b_b_b @ ( produc5927820332525640749_b_b_b @ T23 @ E22 )
@ ( fset_P6379472244080315724_b_b_b
@ ( ffold_5200765406727462407_b_b_b
@ ( produc1404479840988494799_b_b_b
@ ^ [T13: dtree_b_b,E1: b] :
( finser4810365818013240883_b_b_b
@ ( if_Pro2702819762496977205_b_b_b
@ ( ( root_b_b @ T13 )
= Y )
@ ( produc5927820332525640749_b_b_b @ ( node_b_b @ V @ ( finser4810365818013240883_b_b_b @ ( produc5927820332525640749_b_b_b @ T13 @ E1 ) @ bot_bo117012103628883007_b_b_b ) ) @ E3 )
@ ( produc5927820332525640749_b_b_b @ T13 @ E1 ) ) ) )
@ bot_bo117012103628883007_b_b_b
@ Xs2 ) ) )
=> ( ( ( produc5927820332525640749_b_b_b @ T12 @ E12 )
!= ( produc5927820332525640749_b_b_b @ T23 @ E22 ) )
=> ( ( inf_inf_set_b @ ( sup_sup_set_b @ ( darcs_b_b @ T12 ) @ ( insert_b @ E12 @ bot_bot_set_b ) ) @ ( sup_sup_set_b @ ( darcs_b_b @ T23 ) @ ( insert_b @ E22 @ bot_bot_set_b ) ) )
= bot_bot_set_b ) ) ) ) ) ) ) ) ).
% disjoint_darcs_aux1_aux1
thf(fact_1198_disjoint__darcs__aux1__aux1,axiom,
! [Xs2: fset_P5281107635120001194_a_b_b,R2: a,V: a,E3: b,T12: dtree_a_b,E12: b,Y: a,T23: dtree_a_b,E22: b] :
( ! [X3: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ X3 @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Y2: dtree_a_b,E1: b] :
( ~ ( member_b @ E1 @ ( darcs_a_b @ Y2 ) )
& ! [Z: produc4558475209616630778_a_b_b] :
( ( member4380921116106875537_a_b_b @ Z @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
=> ( produc1325217093046185599_b_b_o
@ ^ [Aa: dtree_a_b,E2: b] :
( ( ( inf_inf_set_b @ ( sup_sup_set_b @ ( darcs_a_b @ Y2 ) @ ( insert_b @ E1 @ bot_bot_set_b ) ) @ ( sup_sup_set_b @ ( darcs_a_b @ Aa ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
= bot_bot_set_b )
| ( ( produc331601717337510060_a_b_b @ Y2 @ E1 )
= ( produc331601717337510060_a_b_b @ Aa @ E2 ) ) )
@ Z ) ) )
@ X3 ) )
=> ( ( wf_dverts_a_b @ ( node_a_b @ R2 @ Xs2 ) )
=> ( ~ ( member_a @ V @ ( dverts_a_b @ ( node_a_b @ R2 @ Xs2 ) ) )
=> ( ~ ( member_b @ E3 @ ( darcs_a_b @ ( node_a_b @ R2 @ Xs2 ) ) )
=> ( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ T12 @ E12 )
@ ( fset_P783253628892185035_a_b_b
@ ( ffold_1002666695659912709_a_b_b
@ ( produc7149527399694995404_a_b_b
@ ^ [T13: dtree_a_b,E1: b] :
( finser8437519239679886002_a_b_b
@ ( if_Pro6329973184163622324_a_b_b
@ ( ( root_a_b @ T13 )
= Y )
@ ( produc331601717337510060_a_b_b @ ( node_a_b @ V @ ( finser8437519239679886002_a_b_b @ ( produc331601717337510060_a_b_b @ T13 @ E1 ) @ bot_bo2895716411488905534_a_b_b ) ) @ E3 )
@ ( produc331601717337510060_a_b_b @ T13 @ E1 ) ) ) )
@ bot_bo2895716411488905534_a_b_b
@ Xs2 ) ) )
=> ( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ T23 @ E22 )
@ ( fset_P783253628892185035_a_b_b
@ ( ffold_1002666695659912709_a_b_b
@ ( produc7149527399694995404_a_b_b
@ ^ [T13: dtree_a_b,E1: b] :
( finser8437519239679886002_a_b_b
@ ( if_Pro6329973184163622324_a_b_b
@ ( ( root_a_b @ T13 )
= Y )
@ ( produc331601717337510060_a_b_b @ ( node_a_b @ V @ ( finser8437519239679886002_a_b_b @ ( produc331601717337510060_a_b_b @ T13 @ E1 ) @ bot_bo2895716411488905534_a_b_b ) ) @ E3 )
@ ( produc331601717337510060_a_b_b @ T13 @ E1 ) ) ) )
@ bot_bo2895716411488905534_a_b_b
@ Xs2 ) ) )
=> ( ( ( produc331601717337510060_a_b_b @ T12 @ E12 )
!= ( produc331601717337510060_a_b_b @ T23 @ E22 ) )
=> ( ( inf_inf_set_b @ ( sup_sup_set_b @ ( darcs_a_b @ T12 ) @ ( insert_b @ E12 @ bot_bot_set_b ) ) @ ( sup_sup_set_b @ ( darcs_a_b @ T23 ) @ ( insert_b @ E22 @ bot_bot_set_b ) ) )
= bot_bot_set_b ) ) ) ) ) ) ) ) ).
% disjoint_darcs_aux1_aux1
thf(fact_1199_disjoint__darcs__aux1__aux1,axiom,
! [Xs2: fset_P9004070893569812843_b_a_a,R2: b,V: b,E3: a,T12: dtree_b_a,E12: a,Y: b,T23: dtree_b_a,E22: a] :
( ! [X3: produc1566970784251743547_b_a_a] :
( ( member1389416690741988306_b_a_a @ X3 @ ( fset_P7015121240382073612_b_a_a @ Xs2 ) )
=> ( produc9196057950986712958_a_a_o
@ ^ [Y2: dtree_b_a,E1: a] :
( ~ ( member_a @ E1 @ ( darcs_b_a @ Y2 ) )
& ! [Z: produc1566970784251743547_b_a_a] :
( ( member1389416690741988306_b_a_a @ Z @ ( fset_P7015121240382073612_b_a_a @ Xs2 ) )
=> ( produc9196057950986712958_a_a_o
@ ^ [Aa: dtree_b_a,E2: a] :
( ( ( inf_inf_set_a @ ( sup_sup_set_a @ ( darcs_b_a @ Y2 ) @ ( insert_a @ E1 @ bot_bot_set_a ) ) @ ( sup_sup_set_a @ ( darcs_b_a @ Aa ) @ ( insert_a @ E2 @ bot_bot_set_a ) ) )
= bot_bot_set_a )
| ( ( produc6563469333130627437_b_a_a @ Y2 @ E1 )
= ( produc6563469333130627437_b_a_a @ Aa @ E2 ) ) )
@ Z ) ) )
@ X3 ) )
=> ( ( wf_dverts_b_a @ ( node_b_a @ R2 @ Xs2 ) )
=> ( ~ ( member_b @ V @ ( dverts_b_a @ ( node_b_a @ R2 @ Xs2 ) ) )
=> ( ~ ( member_a @ E3 @ ( darcs_b_a @ ( node_b_a @ R2 @ Xs2 ) ) )
=> ( ( member1389416690741988306_b_a_a @ ( produc6563469333130627437_b_a_a @ T12 @ E12 )
@ ( fset_P7015121240382073612_b_a_a
@ ( ffold_8028070260251416967_b_a_a
@ ( produc966445439450875023_b_a_a
@ ^ [T13: dtree_b_a,E1: a] :
( finser5446014814314998771_b_a_a
@ ( if_Pro3338468758798735093_b_a_a
@ ( ( root_b_a @ T13 )
= Y )
@ ( produc6563469333130627437_b_a_a @ ( node_b_a @ V @ ( finser5446014814314998771_b_a_a @ ( produc6563469333130627437_b_a_a @ T13 @ E1 ) @ bot_bo6618679669938717183_b_a_a ) ) @ E3 )
@ ( produc6563469333130627437_b_a_a @ T13 @ E1 ) ) ) )
@ bot_bo6618679669938717183_b_a_a
@ Xs2 ) ) )
=> ( ( member1389416690741988306_b_a_a @ ( produc6563469333130627437_b_a_a @ T23 @ E22 )
@ ( fset_P7015121240382073612_b_a_a
@ ( ffold_8028070260251416967_b_a_a
@ ( produc966445439450875023_b_a_a
@ ^ [T13: dtree_b_a,E1: a] :
( finser5446014814314998771_b_a_a
@ ( if_Pro3338468758798735093_b_a_a
@ ( ( root_b_a @ T13 )
= Y )
@ ( produc6563469333130627437_b_a_a @ ( node_b_a @ V @ ( finser5446014814314998771_b_a_a @ ( produc6563469333130627437_b_a_a @ T13 @ E1 ) @ bot_bo6618679669938717183_b_a_a ) ) @ E3 )
@ ( produc6563469333130627437_b_a_a @ T13 @ E1 ) ) ) )
@ bot_bo6618679669938717183_b_a_a
@ Xs2 ) ) )
=> ( ( ( produc6563469333130627437_b_a_a @ T12 @ E12 )
!= ( produc6563469333130627437_b_a_a @ T23 @ E22 ) )
=> ( ( inf_inf_set_a @ ( sup_sup_set_a @ ( darcs_b_a @ T12 ) @ ( insert_a @ E12 @ bot_bot_set_a ) ) @ ( sup_sup_set_a @ ( darcs_b_a @ T23 ) @ ( insert_a @ E22 @ bot_bot_set_a ) ) )
= bot_bot_set_a ) ) ) ) ) ) ) ) ).
% disjoint_darcs_aux1_aux1
thf(fact_1200_disjoint__darcs__aux1__aux1,axiom,
! [Xs2: fset_P2559403164575059562_a_a_a,R2: a,V: a,E3: a,T12: dtree_a_a,E12: a,Y: a,T23: dtree_a_a,E22: a] :
( ! [X3: produc5194124205918388666_a_a_a] :
( ( member5016570112408633425_a_a_a @ X3 @ ( fset_P1418902625193942923_a_a_a @ Xs2 ) )
=> ( produc7851592914177225023_a_a_o
@ ^ [Y2: dtree_a_a,E1: a] :
( ~ ( member_a @ E1 @ ( darcs_a_a @ Y2 ) )
& ! [Z: produc5194124205918388666_a_a_a] :
( ( member5016570112408633425_a_a_a @ Z @ ( fset_P1418902625193942923_a_a_a @ Xs2 ) )
=> ( produc7851592914177225023_a_a_o
@ ^ [Aa: dtree_a_a,E2: a] :
( ( ( inf_inf_set_a @ ( sup_sup_set_a @ ( darcs_a_a @ Y2 ) @ ( insert_a @ E1 @ bot_bot_set_a ) ) @ ( sup_sup_set_a @ ( darcs_a_a @ Aa ) @ ( insert_a @ E2 @ bot_bot_set_a ) ) )
= bot_bot_set_a )
| ( ( produc967250717942496748_a_a_a @ Y2 @ E1 )
= ( produc967250717942496748_a_a_a @ Aa @ E2 ) ) )
@ Z ) ) )
@ X3 ) )
=> ( ( wf_dverts_a_a @ ( node_a_a @ R2 @ Xs2 ) )
=> ( ~ ( member_a @ V @ ( dverts_a_a @ ( node_a_a @ R2 @ Xs2 ) ) )
=> ( ~ ( member_a @ E3 @ ( darcs_a_a @ ( node_a_a @ R2 @ Xs2 ) ) )
=> ( ( member5016570112408633425_a_a_a @ ( produc967250717942496748_a_a_a @ T12 @ E12 )
@ ( fset_P1418902625193942923_a_a_a
@ ( ffold_3829971549183867269_a_a_a
@ ( produc6711492998157375628_a_a_a
@ ^ [T13: dtree_a_a,E1: a] :
( finser9073168235981643890_a_a_a
@ ( if_Pro6965622180465380212_a_a_a
@ ( ( root_a_a @ T13 )
= Y )
@ ( produc967250717942496748_a_a_a @ ( node_a_a @ V @ ( finser9073168235981643890_a_a_a @ ( produc967250717942496748_a_a_a @ T13 @ E1 ) @ bot_bo174011940943963902_a_a_a ) ) @ E3 )
@ ( produc967250717942496748_a_a_a @ T13 @ E1 ) ) ) )
@ bot_bo174011940943963902_a_a_a
@ Xs2 ) ) )
=> ( ( member5016570112408633425_a_a_a @ ( produc967250717942496748_a_a_a @ T23 @ E22 )
@ ( fset_P1418902625193942923_a_a_a
@ ( ffold_3829971549183867269_a_a_a
@ ( produc6711492998157375628_a_a_a
@ ^ [T13: dtree_a_a,E1: a] :
( finser9073168235981643890_a_a_a
@ ( if_Pro6965622180465380212_a_a_a
@ ( ( root_a_a @ T13 )
= Y )
@ ( produc967250717942496748_a_a_a @ ( node_a_a @ V @ ( finser9073168235981643890_a_a_a @ ( produc967250717942496748_a_a_a @ T13 @ E1 ) @ bot_bo174011940943963902_a_a_a ) ) @ E3 )
@ ( produc967250717942496748_a_a_a @ T13 @ E1 ) ) ) )
@ bot_bo174011940943963902_a_a_a
@ Xs2 ) ) )
=> ( ( ( produc967250717942496748_a_a_a @ T12 @ E12 )
!= ( produc967250717942496748_a_a_a @ T23 @ E22 ) )
=> ( ( inf_inf_set_a @ ( sup_sup_set_a @ ( darcs_a_a @ T12 ) @ ( insert_a @ E12 @ bot_bot_set_a ) ) @ ( sup_sup_set_a @ ( darcs_a_a @ T23 ) @ ( insert_a @ E22 @ bot_bot_set_a ) ) )
= bot_bot_set_a ) ) ) ) ) ) ) ) ).
% disjoint_darcs_aux1_aux1
thf(fact_1201_disjoint__darcs__aux1__aux1,axiom,
! [Xs2: fset_P8654898996138850416_a_a_a,R2: list_a,V: list_a,E3: a,T12: dtree_list_a_a,E12: a,Y: list_a,T23: dtree_list_a_a,E22: a] :
( ! [X3: produc7135266307266221376_a_a_a] :
( ( member5331345429024349271_a_a_a @ X3 @ ( fset_P551240578171983505_a_a_a @ Xs2 ) )
=> ( produc6862388832954667705_a_a_o
@ ^ [Y2: dtree_list_a_a,E1: a] :
( ~ ( member_a @ E1 @ ( darcs_list_a_a @ Y2 ) )
& ! [Z: produc7135266307266221376_a_a_a] :
( ( member5331345429024349271_a_a_a @ Z @ ( fset_P551240578171983505_a_a_a @ Xs2 ) )
=> ( produc6862388832954667705_a_a_o
@ ^ [Aa: dtree_list_a_a,E2: a] :
( ( ( inf_inf_set_a @ ( sup_sup_set_a @ ( darcs_list_a_a @ Y2 ) @ ( insert_a @ E1 @ bot_bot_set_a ) ) @ ( sup_sup_set_a @ ( darcs_list_a_a @ Aa ) @ ( insert_a @ E2 @ bot_bot_set_a ) ) )
= bot_bot_set_a )
| ( ( produc8339814766199995634_a_a_a @ Y2 @ E1 )
= ( produc8339814766199995634_a_a_a @ Aa @ E2 ) ) )
@ Z ) ) )
@ X3 ) )
=> ( ( wf_dverts_list_a_a @ ( node_list_a_a @ R2 @ Xs2 ) )
=> ( ~ ( member_list_a @ V @ ( dverts_list_a_a @ ( node_list_a_a @ R2 @ Xs2 ) ) )
=> ( ~ ( member_a @ E3 @ ( darcs_list_a_a @ ( node_list_a_a @ R2 @ Xs2 ) ) )
=> ( ( member5331345429024349271_a_a_a @ ( produc8339814766199995634_a_a_a @ T12 @ E12 )
@ ( fset_P551240578171983505_a_a_a
@ ( ffold_1963405522066353041_a_a_a
@ ( produc8847821649887753246_a_a_a
@ ^ [T13: dtree_list_a_a,E1: a] :
( finser2938861521451939832_a_a_a
@ ( if_Pro1443360173218331130_a_a_a
@ ( ( root_list_a_a @ T13 )
= Y )
@ ( produc8339814766199995634_a_a_a @ ( node_list_a_a @ V @ ( finser2938861521451939832_a_a_a @ ( produc8339814766199995634_a_a_a @ T13 @ E1 ) @ bot_bo8750491735591794436_a_a_a ) ) @ E3 )
@ ( produc8339814766199995634_a_a_a @ T13 @ E1 ) ) ) )
@ bot_bo8750491735591794436_a_a_a
@ Xs2 ) ) )
=> ( ( member5331345429024349271_a_a_a @ ( produc8339814766199995634_a_a_a @ T23 @ E22 )
@ ( fset_P551240578171983505_a_a_a
@ ( ffold_1963405522066353041_a_a_a
@ ( produc8847821649887753246_a_a_a
@ ^ [T13: dtree_list_a_a,E1: a] :
( finser2938861521451939832_a_a_a
@ ( if_Pro1443360173218331130_a_a_a
@ ( ( root_list_a_a @ T13 )
= Y )
@ ( produc8339814766199995634_a_a_a @ ( node_list_a_a @ V @ ( finser2938861521451939832_a_a_a @ ( produc8339814766199995634_a_a_a @ T13 @ E1 ) @ bot_bo8750491735591794436_a_a_a ) ) @ E3 )
@ ( produc8339814766199995634_a_a_a @ T13 @ E1 ) ) ) )
@ bot_bo8750491735591794436_a_a_a
@ Xs2 ) ) )
=> ( ( ( produc8339814766199995634_a_a_a @ T12 @ E12 )
!= ( produc8339814766199995634_a_a_a @ T23 @ E22 ) )
=> ( ( inf_inf_set_a @ ( sup_sup_set_a @ ( darcs_list_a_a @ T12 ) @ ( insert_a @ E12 @ bot_bot_set_a ) ) @ ( sup_sup_set_a @ ( darcs_list_a_a @ T23 ) @ ( insert_a @ E22 @ bot_bot_set_a ) ) )
= bot_bot_set_a ) ) ) ) ) ) ) ) ).
% disjoint_darcs_aux1_aux1
thf(fact_1202_disjoint__darcs__aux1__aux1,axiom,
! [Xs2: fset_P3762770571697108971list_a,R2: b,V: b,E3: list_a,T12: dtree_b_list_a,E12: list_a,Y: b,T23: dtree_b_list_a,E22: list_a] :
( ! [X3: produc8289178782644283579list_a] :
( ( member5717964264556279890list_a @ X3 @ ( fset_P7818209297497886092list_a @ Xs2 ) )
=> ( produc5249043932001496062st_a_o
@ ^ [Y2: dtree_b_list_a,E1: list_a] :
( ~ ( member_list_a @ E1 @ ( darcs_b_list_a @ Y2 ) )
& ! [Z: produc8289178782644283579list_a] :
( ( member5717964264556279890list_a @ Z @ ( fset_P7818209297497886092list_a @ Xs2 ) )
=> ( produc5249043932001496062st_a_o
@ ^ [Aa: dtree_b_list_a,E2: list_a] :
( ( ( inf_inf_set_list_a @ ( sup_sup_set_list_a @ ( darcs_b_list_a @ Y2 ) @ ( insert_list_a @ E1 @ bot_bot_set_list_a ) ) @ ( sup_sup_set_list_a @ ( darcs_b_list_a @ Aa ) @ ( insert_list_a @ E2 @ bot_bot_set_list_a ) ) )
= bot_bot_set_list_a )
| ( ( produc8734963857942192365list_a @ Y2 @ E1 )
= ( produc8734963857942192365list_a @ Aa @ E2 ) ) )
@ Z ) ) )
@ X3 ) )
=> ( ( wf_dverts_b_list_a @ ( node_b_list_a @ R2 @ Xs2 ) )
=> ( ~ ( member_b @ V @ ( dverts_b_list_a @ ( node_b_list_a @ R2 @ Xs2 ) ) )
=> ( ~ ( member_list_a @ E3 @ ( darcs_b_list_a @ ( node_b_list_a @ R2 @ Xs2 ) ) )
=> ( ( member5717964264556279890list_a @ ( produc8734963857942192365list_a @ T12 @ E12 )
@ ( fset_P7818209297497886092list_a
@ ( ffold_5967413713360652167list_a
@ ( produc1264271959818302223list_a
@ ^ [T13: dtree_b_list_a,E1: list_a] :
( finser4647321697179225971list_a
@ ( if_Pro6172782175990474869list_a
@ ( ( root_b_list_a @ T13 )
= Y )
@ ( produc8734963857942192365list_a @ ( node_b_list_a @ V @ ( finser4647321697179225971list_a @ ( produc8734963857942192365list_a @ T13 @ E1 ) @ bot_bo2431243398884449407list_a ) ) @ E3 )
@ ( produc8734963857942192365list_a @ T13 @ E1 ) ) ) )
@ bot_bo2431243398884449407list_a
@ Xs2 ) ) )
=> ( ( member5717964264556279890list_a @ ( produc8734963857942192365list_a @ T23 @ E22 )
@ ( fset_P7818209297497886092list_a
@ ( ffold_5967413713360652167list_a
@ ( produc1264271959818302223list_a
@ ^ [T13: dtree_b_list_a,E1: list_a] :
( finser4647321697179225971list_a
@ ( if_Pro6172782175990474869list_a
@ ( ( root_b_list_a @ T13 )
= Y )
@ ( produc8734963857942192365list_a @ ( node_b_list_a @ V @ ( finser4647321697179225971list_a @ ( produc8734963857942192365list_a @ T13 @ E1 ) @ bot_bo2431243398884449407list_a ) ) @ E3 )
@ ( produc8734963857942192365list_a @ T13 @ E1 ) ) ) )
@ bot_bo2431243398884449407list_a
@ Xs2 ) ) )
=> ( ( ( produc8734963857942192365list_a @ T12 @ E12 )
!= ( produc8734963857942192365list_a @ T23 @ E22 ) )
=> ( ( inf_inf_set_list_a @ ( sup_sup_set_list_a @ ( darcs_b_list_a @ T12 ) @ ( insert_list_a @ E12 @ bot_bot_set_list_a ) ) @ ( sup_sup_set_list_a @ ( darcs_b_list_a @ T23 ) @ ( insert_list_a @ E22 @ bot_bot_set_list_a ) ) )
= bot_bot_set_list_a ) ) ) ) ) ) ) ) ).
% disjoint_darcs_aux1_aux1
thf(fact_1203_disjoint__darcs__aux1__aux1,axiom,
! [Xs2: fset_P8006199460403381738list_a,R2: a,V: a,E3: list_a,T12: dtree_a_list_a,E12: list_a,Y: a,T23: dtree_a_list_a,E22: list_a] :
( ! [X3: produc4349230196019447354list_a] :
( ( member1778015677931443665list_a @ X3 @ ( fset_P3878260710873049867list_a @ Xs2 ) )
=> ( produc4007918758503627967st_a_o
@ ^ [Y2: dtree_a_list_a,E1: list_a] :
( ~ ( member_list_a @ E1 @ ( darcs_a_list_a @ Y2 ) )
& ! [Z: produc4349230196019447354list_a] :
( ( member1778015677931443665list_a @ Z @ ( fset_P3878260710873049867list_a @ Xs2 ) )
=> ( produc4007918758503627967st_a_o
@ ^ [Aa: dtree_a_list_a,E2: list_a] :
( ( ( inf_inf_set_list_a @ ( sup_sup_set_list_a @ ( darcs_a_list_a @ Y2 ) @ ( insert_list_a @ E1 @ bot_bot_set_list_a ) ) @ ( sup_sup_set_list_a @ ( darcs_a_list_a @ Aa ) @ ( insert_list_a @ E2 @ bot_bot_set_list_a ) ) )
= bot_bot_set_list_a )
| ( ( produc4795015271317356140list_a @ Y2 @ E1 )
= ( produc4795015271317356140list_a @ Aa @ E2 ) ) )
@ Z ) ) )
@ X3 ) )
=> ( ( wf_dverts_a_list_a @ ( node_a_list_a @ R2 @ Xs2 ) )
=> ( ~ ( member_a @ V @ ( dverts_a_list_a @ ( node_a_list_a @ R2 @ Xs2 ) ) )
=> ( ~ ( member_list_a @ E3 @ ( darcs_a_list_a @ ( node_a_list_a @ R2 @ Xs2 ) ) )
=> ( ( member1778015677931443665list_a @ ( produc4795015271317356140list_a @ T12 @ E12 )
@ ( fset_P3878260710873049867list_a
@ ( ffold_2487888024171782789list_a
@ ( produc3833109818020232460list_a
@ ^ [T13: dtree_a_list_a,E1: list_a] :
( finser707373110554389746list_a
@ ( if_Pro2232833589365638644list_a
@ ( ( root_a_list_a @ T13 )
= Y )
@ ( produc4795015271317356140list_a @ ( node_a_list_a @ V @ ( finser707373110554389746list_a @ ( produc4795015271317356140list_a @ T13 @ E1 ) @ bot_bo6674672287590722174list_a ) ) @ E3 )
@ ( produc4795015271317356140list_a @ T13 @ E1 ) ) ) )
@ bot_bo6674672287590722174list_a
@ Xs2 ) ) )
=> ( ( member1778015677931443665list_a @ ( produc4795015271317356140list_a @ T23 @ E22 )
@ ( fset_P3878260710873049867list_a
@ ( ffold_2487888024171782789list_a
@ ( produc3833109818020232460list_a
@ ^ [T13: dtree_a_list_a,E1: list_a] :
( finser707373110554389746list_a
@ ( if_Pro2232833589365638644list_a
@ ( ( root_a_list_a @ T13 )
= Y )
@ ( produc4795015271317356140list_a @ ( node_a_list_a @ V @ ( finser707373110554389746list_a @ ( produc4795015271317356140list_a @ T13 @ E1 ) @ bot_bo6674672287590722174list_a ) ) @ E3 )
@ ( produc4795015271317356140list_a @ T13 @ E1 ) ) ) )
@ bot_bo6674672287590722174list_a
@ Xs2 ) ) )
=> ( ( ( produc4795015271317356140list_a @ T12 @ E12 )
!= ( produc4795015271317356140list_a @ T23 @ E22 ) )
=> ( ( inf_inf_set_list_a @ ( sup_sup_set_list_a @ ( darcs_a_list_a @ T12 ) @ ( insert_list_a @ E12 @ bot_bot_set_list_a ) ) @ ( sup_sup_set_list_a @ ( darcs_a_list_a @ T23 ) @ ( insert_list_a @ E22 @ bot_bot_set_list_a ) ) )
= bot_bot_set_list_a ) ) ) ) ) ) ) ) ).
% disjoint_darcs_aux1_aux1
thf(fact_1204_disjoint__darcs__aux1__aux1,axiom,
! [Xs2: fset_P2153231429829016240_a_b_b,R2: list_a,V: list_a,E3: b,T12: dtree_list_a_b,E12: b,Y: list_a,T23: dtree_list_a_b,E22: b] :
( ! [X3: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ X3 @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( produc336013011823628281_b_b_o
@ ^ [Y2: dtree_list_a_b,E1: b] :
( ~ ( member_b @ E1 @ ( darcs_list_a_b @ Y2 ) )
& ! [Z: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ Z @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( produc336013011823628281_b_b_o
@ ^ [Aa: dtree_list_a_b,E2: b] :
( ( ( inf_inf_set_b @ ( sup_sup_set_b @ ( darcs_list_a_b @ Y2 ) @ ( insert_b @ E1 @ bot_bot_set_b ) ) @ ( sup_sup_set_b @ ( darcs_list_a_b @ Aa ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
= bot_bot_set_b )
| ( ( produc7704165765595008946_a_b_b @ Y2 @ E1 )
= ( produc7704165765595008946_a_b_b @ Aa @ E2 ) ) )
@ Z ) ) )
@ X3 ) )
=> ( ( wf_dverts_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
=> ( ~ ( member_list_a @ V @ ( dverts_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) )
=> ( ~ ( member_b @ E3 @ ( darcs_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E12 )
@ ( fset_P9138963618725001425_a_b_b
@ ( ffold_6356110402769621009_a_b_b
@ ( produc7216629644754684766_a_b_b
@ ^ [T13: dtree_list_a_b,E1: b] :
( finser2303212525150181944_a_b_b
@ ( if_Pro807711176916573242_a_b_b
@ ( ( root_list_a_b @ T13 )
= Y )
@ ( produc7704165765595008946_a_b_b @ ( node_list_a_b @ V @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T13 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ E3 )
@ ( produc7704165765595008946_a_b_b @ T13 @ E1 ) ) ) )
@ bot_bo2248824169281960260_a_b_b
@ Xs2 ) ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T23 @ E22 )
@ ( fset_P9138963618725001425_a_b_b
@ ( ffold_6356110402769621009_a_b_b
@ ( produc7216629644754684766_a_b_b
@ ^ [T13: dtree_list_a_b,E1: b] :
( finser2303212525150181944_a_b_b
@ ( if_Pro807711176916573242_a_b_b
@ ( ( root_list_a_b @ T13 )
= Y )
@ ( produc7704165765595008946_a_b_b @ ( node_list_a_b @ V @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T13 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ E3 )
@ ( produc7704165765595008946_a_b_b @ T13 @ E1 ) ) ) )
@ bot_bo2248824169281960260_a_b_b
@ Xs2 ) ) )
=> ( ( ( produc7704165765595008946_a_b_b @ T12 @ E12 )
!= ( produc7704165765595008946_a_b_b @ T23 @ E22 ) )
=> ( ( inf_inf_set_b @ ( sup_sup_set_b @ ( darcs_list_a_b @ T12 ) @ ( insert_b @ E12 @ bot_bot_set_b ) ) @ ( sup_sup_set_b @ ( darcs_list_a_b @ T23 ) @ ( insert_b @ E22 @ bot_bot_set_b ) ) )
= bot_bot_set_b ) ) ) ) ) ) ) ) ).
% disjoint_darcs_aux1_aux1
thf(fact_1205_disjoint__darcs__aux1__aux1,axiom,
! [Xs2: fset_P6656299774166858224list_a,R2: list_a,V: list_a,E3: list_a,T12: dtree_list_a_list_a,E12: list_a,Y: list_a,T23: dtree_list_a_list_a,E22: list_a] :
( ! [X3: produc7575571382841205696list_a] :
( ( member535470554364689367list_a @ X3 @ ( fset_P2575872134735801361list_a @ Xs2 ) )
=> ( produc121743164402242617st_a_o
@ ^ [Y2: dtree_list_a_list_a,E1: list_a] :
( ~ ( member_list_a @ E1 @ ( darcs_list_a_list_a @ Y2 ) )
& ! [Z: produc7575571382841205696list_a] :
( ( member535470554364689367list_a @ Z @ ( fset_P2575872134735801361list_a @ Xs2 ) )
=> ( produc121743164402242617st_a_o
@ ^ [Aa: dtree_list_a_list_a,E2: list_a] :
( ( ( inf_inf_set_list_a @ ( sup_sup_set_list_a @ ( darcs_list_a_list_a @ Y2 ) @ ( insert_list_a @ E1 @ bot_bot_set_list_a ) ) @ ( sup_sup_set_list_a @ ( darcs_list_a_list_a @ Aa ) @ ( insert_list_a @ E2 @ bot_bot_set_list_a ) ) )
= bot_bot_set_list_a )
| ( ( produc6276725351364364146list_a @ Y2 @ E1 )
= ( produc6276725351364364146list_a @ Aa @ E2 ) ) )
@ Z ) ) )
@ X3 ) )
=> ( ( wf_dve6709000387044496771list_a @ ( node_list_a_list_a @ R2 @ Xs2 ) )
=> ( ~ ( member_list_a @ V @ ( dverts_list_a_list_a @ ( node_list_a_list_a @ R2 @ Xs2 ) ) )
=> ( ~ ( member_list_a @ E3 @ ( darcs_list_a_list_a @ ( node_list_a_list_a @ R2 @ Xs2 ) ) )
=> ( ( member535470554364689367list_a @ ( produc6276725351364364146list_a @ T12 @ E12 )
@ ( fset_P2575872134735801361list_a
@ ( ffold_1751350808669368977list_a
@ ( produc2593633274636522654list_a
@ ^ [T13: dtree_list_a_list_a,E1: list_a] :
( finser5913853959220252280list_a
@ ( if_Pro2239956646901841530list_a
@ ( ( root_list_a_list_a @ T13 )
= Y )
@ ( produc6276725351364364146list_a @ ( node_list_a_list_a @ V @ ( finser5913853959220252280list_a @ ( produc6276725351364364146list_a @ T13 @ E1 ) @ bot_bo7860097226086663300list_a ) ) @ E3 )
@ ( produc6276725351364364146list_a @ T13 @ E1 ) ) ) )
@ bot_bo7860097226086663300list_a
@ Xs2 ) ) )
=> ( ( member535470554364689367list_a @ ( produc6276725351364364146list_a @ T23 @ E22 )
@ ( fset_P2575872134735801361list_a
@ ( ffold_1751350808669368977list_a
@ ( produc2593633274636522654list_a
@ ^ [T13: dtree_list_a_list_a,E1: list_a] :
( finser5913853959220252280list_a
@ ( if_Pro2239956646901841530list_a
@ ( ( root_list_a_list_a @ T13 )
= Y )
@ ( produc6276725351364364146list_a @ ( node_list_a_list_a @ V @ ( finser5913853959220252280list_a @ ( produc6276725351364364146list_a @ T13 @ E1 ) @ bot_bo7860097226086663300list_a ) ) @ E3 )
@ ( produc6276725351364364146list_a @ T13 @ E1 ) ) ) )
@ bot_bo7860097226086663300list_a
@ Xs2 ) ) )
=> ( ( ( produc6276725351364364146list_a @ T12 @ E12 )
!= ( produc6276725351364364146list_a @ T23 @ E22 ) )
=> ( ( inf_inf_set_list_a @ ( sup_sup_set_list_a @ ( darcs_list_a_list_a @ T12 ) @ ( insert_list_a @ E12 @ bot_bot_set_list_a ) ) @ ( sup_sup_set_list_a @ ( darcs_list_a_list_a @ T23 ) @ ( insert_list_a @ E22 @ bot_bot_set_list_a ) ) )
= bot_bot_set_list_a ) ) ) ) ) ) ) ) ).
% disjoint_darcs_aux1_aux1
thf(fact_1206_disjoint__darcs__aux1__aux1,axiom,
! [Xs2: fset_P729086328404512291st_a_b,R2: b,V: b,E3: dtree_list_a_b,T12: dtree_3602950866360573648st_a_b,E12: dtree_list_a_b,Y: b,T23: dtree_3602950866360573648st_a_b,E22: dtree_list_a_b] :
( ! [X3: produc8101844211926167155st_a_b] :
( ( member5849886156663560074st_a_b @ X3 @ ( fset_P1687456239673687492st_a_b @ Xs2 ) )
=> ( produc3786481918722493182_a_b_o
@ ^ [Y2: dtree_3602950866360573648st_a_b,E1: dtree_list_a_b] :
( ~ ( member551035911493665803st_a_b @ E1 @ ( darcs_7068930265994548426st_a_b @ Y2 ) )
& ! [Z: produc8101844211926167155st_a_b] :
( ( member5849886156663560074st_a_b @ Z @ ( fset_P1687456239673687492st_a_b @ Xs2 ) )
=> ( produc3786481918722493182_a_b_o
@ ^ [Aa: dtree_3602950866360573648st_a_b,E2: dtree_list_a_b] :
( ( ( inf_in3355993651213403836st_a_b @ ( sup_su5199045856643764822st_a_b @ ( darcs_7068930265994548426st_a_b @ Y2 ) @ ( insert7065945238938772516st_a_b @ E1 @ bot_bo798015271861357502st_a_b ) ) @ ( sup_su5199045856643764822st_a_b @ ( darcs_7068930265994548426st_a_b @ Aa ) @ ( insert7065945238938772516st_a_b @ E2 @ bot_bo798015271861357502st_a_b ) ) )
= bot_bo798015271861357502st_a_b )
| ( ( produc3051495980888433645st_a_b @ Y2 @ E1 )
= ( produc3051495980888433645st_a_b @ Aa @ E2 ) ) )
@ Z ) ) )
@ X3 ) )
=> ( ( wf_dve4513435013532653104st_a_b @ ( node_b9108412673219173045st_a_b @ R2 @ Xs2 ) )
=> ( ~ ( member_b @ V @ ( dverts8864706460159352569st_a_b @ ( node_b9108412673219173045st_a_b @ R2 @ Xs2 ) ) )
=> ( ~ ( member551035911493665803st_a_b @ E3 @ ( darcs_7068930265994548426st_a_b @ ( node_b9108412673219173045st_a_b @ R2 @ Xs2 ) ) )
=> ( ( member5849886156663560074st_a_b @ ( produc3051495980888433645st_a_b @ T12 @ E12 )
@ ( fset_P1687456239673687492st_a_b
@ ( ffold_3232036597505749367st_a_b
@ ( produc1804618233457100159st_a_b
@ ^ [T13: dtree_3602950866360573648st_a_b,E1: dtree_list_a_b] :
( finser141567303011885355st_a_b
@ ( if_Pro2199488368625827629st_a_b
@ ( ( root_b4214690873640572181st_a_b @ T13 )
= Y )
@ ( produc3051495980888433645st_a_b @ ( node_b9108412673219173045st_a_b @ V @ ( finser141567303011885355st_a_b @ ( produc3051495980888433645st_a_b @ T13 @ E1 ) @ bot_bo6210586517881791671st_a_b ) ) @ E3 )
@ ( produc3051495980888433645st_a_b @ T13 @ E1 ) ) ) )
@ bot_bo6210586517881791671st_a_b
@ Xs2 ) ) )
=> ( ( member5849886156663560074st_a_b @ ( produc3051495980888433645st_a_b @ T23 @ E22 )
@ ( fset_P1687456239673687492st_a_b
@ ( ffold_3232036597505749367st_a_b
@ ( produc1804618233457100159st_a_b
@ ^ [T13: dtree_3602950866360573648st_a_b,E1: dtree_list_a_b] :
( finser141567303011885355st_a_b
@ ( if_Pro2199488368625827629st_a_b
@ ( ( root_b4214690873640572181st_a_b @ T13 )
= Y )
@ ( produc3051495980888433645st_a_b @ ( node_b9108412673219173045st_a_b @ V @ ( finser141567303011885355st_a_b @ ( produc3051495980888433645st_a_b @ T13 @ E1 ) @ bot_bo6210586517881791671st_a_b ) ) @ E3 )
@ ( produc3051495980888433645st_a_b @ T13 @ E1 ) ) ) )
@ bot_bo6210586517881791671st_a_b
@ Xs2 ) ) )
=> ( ( ( produc3051495980888433645st_a_b @ T12 @ E12 )
!= ( produc3051495980888433645st_a_b @ T23 @ E22 ) )
=> ( ( inf_in3355993651213403836st_a_b @ ( sup_su5199045856643764822st_a_b @ ( darcs_7068930265994548426st_a_b @ T12 ) @ ( insert7065945238938772516st_a_b @ E12 @ bot_bo798015271861357502st_a_b ) ) @ ( sup_su5199045856643764822st_a_b @ ( darcs_7068930265994548426st_a_b @ T23 ) @ ( insert7065945238938772516st_a_b @ E22 @ bot_bo798015271861357502st_a_b ) ) )
= bot_bo798015271861357502st_a_b ) ) ) ) ) ) ) ) ).
% disjoint_darcs_aux1_aux1
thf(fact_1207_disjoint__dverts__aux1_H,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,V: list_a,Y: list_a,E3: b] :
( ( wf_dverts_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
=> ( ~ ( member_list_a @ V @ ( dverts_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) )
=> ! [X2: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ X2
@ ( fset_P9138963618725001425_a_b_b
@ ( ffold_6356110402769621009_a_b_b
@ ( produc7216629644754684766_a_b_b
@ ^ [T13: dtree_list_a_b,E1: b] :
( finser2303212525150181944_a_b_b
@ ( if_Pro807711176916573242_a_b_b
@ ( ( root_list_a_b @ T13 )
= Y )
@ ( produc7704165765595008946_a_b_b @ ( node_list_a_b @ V @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T13 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ E3 )
@ ( produc7704165765595008946_a_b_b @ T13 @ E1 ) ) ) )
@ bot_bo2248824169281960260_a_b_b
@ Xs2 ) ) )
=> ( produc336013011823628281_b_b_o
@ ^ [Y2: dtree_list_a_b,E1: b] :
! [Z: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ Z
@ ( fset_P9138963618725001425_a_b_b
@ ( ffold_6356110402769621009_a_b_b
@ ( produc7216629644754684766_a_b_b
@ ^ [T13: dtree_list_a_b,E1a: b] :
( finser2303212525150181944_a_b_b
@ ( if_Pro807711176916573242_a_b_b
@ ( ( root_list_a_b @ T13 )
= Y )
@ ( produc7704165765595008946_a_b_b @ ( node_list_a_b @ V @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T13 @ E1a ) @ bot_bo2248824169281960260_a_b_b ) ) @ E3 )
@ ( produc7704165765595008946_a_b_b @ T13 @ E1a ) ) ) )
@ bot_bo2248824169281960260_a_b_b
@ Xs2 ) ) )
=> ( produc336013011823628281_b_b_o
@ ^ [Aa: dtree_list_a_b,E2: b] :
( ( ( inf_inf_set_list_a @ ( dverts_list_a_b @ Y2 ) @ ( dverts_list_a_b @ Aa ) )
= bot_bot_set_list_a )
| ( ( produc7704165765595008946_a_b_b @ Y2 @ E1 )
= ( produc7704165765595008946_a_b_b @ Aa @ E2 ) ) )
@ Z ) )
@ X2 ) ) ) ) ).
% disjoint_dverts_aux1'
thf(fact_1208_remove__child__sub,axiom,
! [X: list_a,Xs2: fset_P2153231429829016240_a_b_b] :
( ord_le8870638447146015504_a_b_b
@ ( ffilte1050691675278058007_a_b_b
@ ( produc336013011823628281_b_b_o
@ ^ [T3: dtree_list_a_b,E4: b] :
( ( root_list_a_b @ T3 )
!= X ) )
@ Xs2 )
@ Xs2 ) ).
% remove_child_sub
thf(fact_1209_root__normalize1__eq1,axiom,
! [T12: dtree_list_a_b,R2: list_a,E12: b] :
( ~ ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T12 ) ) ) @ ( rank @ ( rev_a @ R2 ) ) )
=> ( ( root_list_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
= R2 ) ) ).
% root_normalize1_eq1
thf(fact_1210_fempty__ffilter,axiom,
! [A: fset_P2153231429829016240_a_b_b] :
( ( ffilte1050691675278058007_a_b_b
@ ^ [Uu: produc6499617310964463488_a_b_b] : $false
@ A )
= bot_bo2248824169281960260_a_b_b ) ).
% fempty_ffilter
thf(fact_1211_disjoint__dverts__aux2,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,V: list_a,E3: b] :
( ( wf_dverts_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
=> ( ~ ( member_list_a @ V @ ( dverts_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) )
=> ! [X2: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ X2 @ ( fset_P9138963618725001425_a_b_b @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ( node_list_a_b @ V @ bot_bo2248824169281960260_a_b_b ) @ E3 ) @ Xs2 ) ) )
=> ( produc336013011823628281_b_b_o
@ ^ [Y2: dtree_list_a_b,E1: b] :
! [Z: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ Z @ ( fset_P9138963618725001425_a_b_b @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ( node_list_a_b @ V @ bot_bo2248824169281960260_a_b_b ) @ E3 ) @ Xs2 ) ) )
=> ( produc336013011823628281_b_b_o
@ ^ [Aa: dtree_list_a_b,E2: b] :
( ( ( inf_inf_set_list_a @ ( dverts_list_a_b @ Y2 ) @ ( dverts_list_a_b @ Aa ) )
= bot_bot_set_list_a )
| ( ( produc7704165765595008946_a_b_b @ Y2 @ E1 )
= ( produc7704165765595008946_a_b_b @ Aa @ E2 ) ) )
@ Z ) )
@ X2 ) ) ) ) ).
% disjoint_dverts_aux2
thf(fact_1212_filter__fset,axiom,
! [P: produc6499617310964463488_a_b_b > $o,Xs2: fset_P2153231429829016240_a_b_b] :
( ( fset_P9138963618725001425_a_b_b @ ( ffilte1050691675278058007_a_b_b @ P @ Xs2 ) )
= ( inf_in7962574851115124040_a_b_b @ ( collec3241703154345475797_a_b_b @ P ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) ) ).
% filter_fset
thf(fact_1213_filter__fset,axiom,
! [P: b > $o,Xs2: fset_b] :
( ( fset_b2 @ ( ffilter_b @ P @ Xs2 ) )
= ( inf_inf_set_b @ ( collect_b @ P ) @ ( fset_b2 @ Xs2 ) ) ) ).
% filter_fset
thf(fact_1214_filter__fset,axiom,
! [P: a > $o,Xs2: fset_a] :
( ( fset_a2 @ ( ffilter_a @ P @ Xs2 ) )
= ( inf_inf_set_a @ ( collect_a @ P ) @ ( fset_a2 @ Xs2 ) ) ) ).
% filter_fset
thf(fact_1215_filter__fset,axiom,
! [P: list_a > $o,Xs2: fset_list_a] :
( ( fset_list_a2 @ ( ffilter_list_a @ P @ Xs2 ) )
= ( inf_inf_set_list_a @ ( collect_list_a @ P ) @ ( fset_list_a2 @ Xs2 ) ) ) ).
% filter_fset
thf(fact_1216_insert__before__wf__dverts,axiom,
! [Xs2: fset_P2153231429829016240_a_b_b,V: list_a,R2: list_a,T12: dtree_list_a_b,E12: b,Y: list_a,E3: b] :
( ! [X3: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ X3 @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( produc336013011823628281_b_b_o
@ ^ [T3: dtree_list_a_b,E1: b] : ( wf_dverts_list_a_b @ T3 )
@ X3 ) )
=> ( ~ ( member_list_a @ V @ ( dverts_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E12 )
@ ( fset_P9138963618725001425_a_b_b
@ ( ffold_6356110402769621009_a_b_b
@ ( produc7216629644754684766_a_b_b
@ ^ [T13: dtree_list_a_b,E1: b] :
( finser2303212525150181944_a_b_b
@ ( if_Pro807711176916573242_a_b_b
@ ( ( root_list_a_b @ T13 )
= Y )
@ ( produc7704165765595008946_a_b_b @ ( node_list_a_b @ V @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T13 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ E3 )
@ ( produc7704165765595008946_a_b_b @ T13 @ E1 ) ) ) )
@ bot_bo2248824169281960260_a_b_b
@ Xs2 ) ) )
=> ( wf_dverts_list_a_b @ T12 ) ) ) ) ).
% insert_before_wf_dverts
thf(fact_1217_leD,axiom,
! [Y: set_list_a,X: set_list_a] :
( ( ord_le8861187494160871172list_a @ Y @ X )
=> ~ ( ord_less_set_list_a @ X @ Y ) ) ).
% leD
thf(fact_1218_leD,axiom,
! [Y: set_b,X: set_b] :
( ( ord_less_eq_set_b @ Y @ X )
=> ~ ( ord_less_set_b @ X @ Y ) ) ).
% leD
thf(fact_1219_leD,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_real @ X @ Y ) ) ).
% leD
thf(fact_1220_leD,axiom,
! [Y: fset_P2153231429829016240_a_b_b,X: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ Y @ X )
=> ~ ( ord_le6631730213922513156_a_b_b @ X @ Y ) ) ).
% leD
thf(fact_1221_leI,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% leI
thf(fact_1222_nless__le,axiom,
! [A2: set_b,B: set_b] :
( ( ~ ( ord_less_set_b @ A2 @ B ) )
= ( ~ ( ord_less_eq_set_b @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_1223_nless__le,axiom,
! [A2: real,B: real] :
( ( ~ ( ord_less_real @ A2 @ B ) )
= ( ~ ( ord_less_eq_real @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_1224_nless__le,axiom,
! [A2: fset_P2153231429829016240_a_b_b,B: fset_P2153231429829016240_a_b_b] :
( ( ~ ( ord_le6631730213922513156_a_b_b @ A2 @ B ) )
= ( ~ ( ord_le8870638447146015504_a_b_b @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_1225_contr__if__normalize1__uneq,axiom,
! [T12: dtree_list_a_b] :
( ( ( ranked8905849569120154423e1_a_b @ rank @ T12 )
!= T12 )
=> ? [V2: list_a,T24: dtree_list_a_b,E23: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ V2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T24 @ E23 ) @ bot_bo2248824169281960260_a_b_b ) ) @ T12 )
& ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T24 ) ) ) @ ( rank @ ( rev_a @ V2 ) ) ) ) ) ).
% contr_if_normalize1_uneq
thf(fact_1226_contr__before__normalize1,axiom,
! [V: list_a,T12: dtree_list_a_b,E12: b,T32: dtree_list_a_b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ V @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ( ranked8905849569120154423e1_a_b @ rank @ T32 ) )
=> ( ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T12 ) ) ) @ ( rank @ ( rev_a @ V ) ) )
=> ? [V3: list_a,T24: dtree_list_a_b,E23: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ V3 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T24 @ E23 ) @ bot_bo2248824169281960260_a_b_b ) ) @ T32 )
& ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T24 ) ) ) @ ( rank @ ( rev_a @ V3 ) ) ) ) ) ) ).
% contr_before_normalize1
thf(fact_1227_normalize1_Opinduct,axiom,
! [A0: dtree_list_a_b,P: dtree_list_a_b > $o] :
( ( accp_dtree_list_a_b @ ( ranked37597841108724548el_a_b @ rank ) @ A0 )
=> ( ! [R: list_a,T1: dtree_list_a_b,E: b] :
( ( accp_dtree_list_a_b @ ( ranked37597841108724548el_a_b @ rank ) @ ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( ( ~ ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ R ) ) )
=> ( P @ T1 ) )
=> ( P @ ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) )
=> ( ! [Xs: fset_P2153231429829016240_a_b_b,R: list_a] :
( ( accp_dtree_list_a_b @ ( ranked37597841108724548el_a_b @ rank ) @ ( node_list_a_b @ R @ Xs ) )
=> ( ! [X2: produc6499617310964463488_a_b_b] :
( Xs
!= ( finser2303212525150181944_a_b_b @ X2 @ bot_bo2248824169281960260_a_b_b ) )
=> ( ! [Z5: produc6499617310964463488_a_b_b,X2: dtree_list_a_b,Y5: b] :
( ( member4695696432722591383_a_b_b @ Z5 @ ( fset_P9138963618725001425_a_b_b @ Xs ) )
=> ( ( ( produc7704165765595008946_a_b_b @ X2 @ Y5 )
= Z5 )
=> ( P @ X2 ) ) )
=> ( P @ ( node_list_a_b @ R @ Xs ) ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% normalize1.pinduct
thf(fact_1228_normalize_Opelims,axiom,
! [X: dtree_list_a_b,Y: dtree_list_a_b] :
( ( ( ranked_normalize_a_b @ rank @ X )
= Y )
=> ( ( accp_dtree_list_a_b @ ( ranked1950734148545037055el_a_b @ rank ) @ X )
=> ~ ( ( Y
= ( if_dtree_list_a_b
@ ( X
= ( ranked8905849569120154423e1_a_b @ rank @ X ) )
@ ( ranked8905849569120154423e1_a_b @ rank @ X )
@ ( ranked_normalize_a_b @ rank @ ( ranked8905849569120154423e1_a_b @ rank @ X ) ) ) )
=> ~ ( accp_dtree_list_a_b @ ( ranked1950734148545037055el_a_b @ rank ) @ X ) ) ) ) ).
% normalize.pelims
thf(fact_1229_normalize1_Opsimps_I2_J,axiom,
! [Xs2: fset_P2153231429829016240_a_b_b,R2: list_a] :
( ! [X3: produc6499617310964463488_a_b_b] :
( Xs2
!= ( finser2303212525150181944_a_b_b @ X3 @ bot_bo2248824169281960260_a_b_b ) )
=> ( ( accp_dtree_list_a_b @ ( ranked37597841108724548el_a_b @ rank ) @ ( node_list_a_b @ R2 @ Xs2 ) )
=> ( ( ranked8905849569120154423e1_a_b @ rank @ ( node_list_a_b @ R2 @ Xs2 ) )
= ( node_list_a_b @ R2
@ ( fimage8157835862677140991_a_b_b
@ ( produc4970258495817412575_a_b_b
@ ^ [T3: dtree_list_a_b] : ( produc7704165765595008946_a_b_b @ ( ranked8905849569120154423e1_a_b @ rank @ T3 ) ) )
@ Xs2 ) ) ) ) ) ).
% normalize1.psimps(2)
thf(fact_1230_normalize1_Osimps_I2_J,axiom,
! [Xs2: fset_P2153231429829016240_a_b_b,R2: list_a] :
( ! [X3: produc6499617310964463488_a_b_b] :
( Xs2
!= ( finser2303212525150181944_a_b_b @ X3 @ bot_bo2248824169281960260_a_b_b ) )
=> ( ( ranked8905849569120154423e1_a_b @ rank @ ( node_list_a_b @ R2 @ Xs2 ) )
= ( node_list_a_b @ R2
@ ( fimage8157835862677140991_a_b_b
@ ( produc4970258495817412575_a_b_b
@ ^ [T3: dtree_list_a_b] : ( produc7704165765595008946_a_b_b @ ( ranked8905849569120154423e1_a_b @ rank @ T3 ) ) )
@ Xs2 ) ) ) ) ).
% normalize1.simps(2)
thf(fact_1231_disjoint__darcs__normalize1,axiom,
! [T12: dtree_list_a_b] :
( ( wf_darcs_list_a_b @ T12 )
=> ! [X2: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ X2
@ ( fset_P9138963618725001425_a_b_b
@ ( fimage8157835862677140991_a_b_b
@ ( produc4970258495817412575_a_b_b
@ ^ [T3: dtree_list_a_b] : ( produc7704165765595008946_a_b_b @ ( ranked8905849569120154423e1_a_b @ rank @ T3 ) ) )
@ ( sucs_list_a_b @ T12 ) ) ) )
=> ( produc336013011823628281_b_b_o
@ ^ [Y2: dtree_list_a_b,E1: b] :
( ~ ( member_b @ E1 @ ( darcs_list_a_b @ Y2 ) )
& ! [Z: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ Z
@ ( fset_P9138963618725001425_a_b_b
@ ( fimage8157835862677140991_a_b_b
@ ( produc4970258495817412575_a_b_b
@ ^ [T3: dtree_list_a_b] : ( produc7704165765595008946_a_b_b @ ( ranked8905849569120154423e1_a_b @ rank @ T3 ) ) )
@ ( sucs_list_a_b @ T12 ) ) ) )
=> ( produc336013011823628281_b_b_o
@ ^ [Aa: dtree_list_a_b,E2: b] :
( ( ( inf_inf_set_b @ ( sup_sup_set_b @ ( darcs_list_a_b @ Y2 ) @ ( insert_b @ E1 @ bot_bot_set_b ) ) @ ( sup_sup_set_b @ ( darcs_list_a_b @ Aa ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
= bot_bot_set_b )
| ( ( produc7704165765595008946_a_b_b @ Y2 @ E1 )
= ( produc7704165765595008946_a_b_b @ Aa @ E2 ) ) )
@ Z ) ) )
@ X2 ) ) ) ).
% disjoint_darcs_normalize1
thf(fact_1232_normalize1_Opelims,axiom,
! [X: dtree_list_a_b,Y: dtree_list_a_b] :
( ( ( ranked8905849569120154423e1_a_b @ rank @ X )
= Y )
=> ( ( accp_dtree_list_a_b @ ( ranked37597841108724548el_a_b @ rank ) @ X )
=> ( ! [R: list_a,T1: dtree_list_a_b,E: b] :
( ( X
= ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( ( ( ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ R ) ) )
=> ( Y
= ( node_list_a_b @ ( append_a @ R @ ( root_list_a_b @ T1 ) ) @ ( sucs_list_a_b @ T1 ) ) ) )
& ( ~ ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ R ) ) )
=> ( Y
= ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ( ranked8905849569120154423e1_a_b @ rank @ T1 ) @ E ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) )
=> ~ ( accp_dtree_list_a_b @ ( ranked37597841108724548el_a_b @ rank ) @ ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) )
=> ( ! [Xs: fset_P2153231429829016240_a_b_b,R: list_a] :
( ( accp_dtree_list_a_b @ ( ranked37597841108724548el_a_b @ rank ) @ ( node_list_a_b @ R @ Xs ) )
=> ( ! [X2: produc6499617310964463488_a_b_b] :
( Xs
!= ( finser2303212525150181944_a_b_b @ X2 @ bot_bo2248824169281960260_a_b_b ) )
=> ( ( X
= ( node_list_a_b @ R @ Xs ) )
=> ( accp_dtree_list_a_b @ ( ranked37597841108724548el_a_b @ rank ) @ ( node_list_a_b @ R @ Xs ) ) ) ) )
=> ~ ! [Xs: fset_P2153231429829016240_a_b_b] :
( ! [X2: produc6499617310964463488_a_b_b] :
( Xs
!= ( finser2303212525150181944_a_b_b @ X2 @ bot_bo2248824169281960260_a_b_b ) )
=> ! [R: list_a] :
( ( X
= ( node_list_a_b @ R @ Xs ) )
=> ( ( Y
= ( node_list_a_b @ R
@ ( fimage8157835862677140991_a_b_b
@ ( produc4970258495817412575_a_b_b
@ ^ [T3: dtree_list_a_b] : ( produc7704165765595008946_a_b_b @ ( ranked8905849569120154423e1_a_b @ rank @ T3 ) ) )
@ Xs ) ) )
=> ~ ( accp_dtree_list_a_b @ ( ranked37597841108724548el_a_b @ rank ) @ ( node_list_a_b @ R @ Xs ) ) ) ) ) ) ) ) ) ).
% normalize1.pelims
thf(fact_1233_normalize1_Oelims,axiom,
! [X: dtree_list_a_b,Y: dtree_list_a_b] :
( ( ( ranked8905849569120154423e1_a_b @ rank @ X )
= Y )
=> ( ! [R: list_a,T1: dtree_list_a_b,E: b] :
( ( X
= ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ~ ( ( ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ R ) ) )
=> ( Y
= ( node_list_a_b @ ( append_a @ R @ ( root_list_a_b @ T1 ) ) @ ( sucs_list_a_b @ T1 ) ) ) )
& ( ~ ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ R ) ) )
=> ( Y
= ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ( ranked8905849569120154423e1_a_b @ rank @ T1 ) @ E ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) ) )
=> ~ ! [Xs: fset_P2153231429829016240_a_b_b] :
( ! [X2: produc6499617310964463488_a_b_b] :
( Xs
!= ( finser2303212525150181944_a_b_b @ X2 @ bot_bo2248824169281960260_a_b_b ) )
=> ! [R: list_a] :
( ( X
= ( node_list_a_b @ R @ Xs ) )
=> ( Y
!= ( node_list_a_b @ R
@ ( fimage8157835862677140991_a_b_b
@ ( produc4970258495817412575_a_b_b
@ ^ [T3: dtree_list_a_b] : ( produc7704165765595008946_a_b_b @ ( ranked8905849569120154423e1_a_b @ rank @ T3 ) ) )
@ Xs ) ) ) ) ) ) ) ).
% normalize1.elims
thf(fact_1234_normalize1_Osimps_I1_J,axiom,
! [T12: dtree_list_a_b,R2: list_a,E3: b] :
( ( ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T12 ) ) ) @ ( rank @ ( rev_a @ R2 ) ) )
=> ( ( ranked8905849569120154423e1_a_b @ rank @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E3 ) @ bot_bo2248824169281960260_a_b_b ) ) )
= ( node_list_a_b @ ( append_a @ R2 @ ( root_list_a_b @ T12 ) ) @ ( sucs_list_a_b @ T12 ) ) ) )
& ( ~ ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T12 ) ) ) @ ( rank @ ( rev_a @ R2 ) ) )
=> ( ( ranked8905849569120154423e1_a_b @ rank @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E3 ) @ bot_bo2248824169281960260_a_b_b ) ) )
= ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ( ranked8905849569120154423e1_a_b @ rank @ T12 ) @ E3 ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) ) ).
% normalize1.simps(1)
thf(fact_1235_normalize1_Opsimps_I1_J,axiom,
! [R2: list_a,T12: dtree_list_a_b,E3: b] :
( ( accp_dtree_list_a_b @ ( ranked37597841108724548el_a_b @ rank ) @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E3 ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( ( ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T12 ) ) ) @ ( rank @ ( rev_a @ R2 ) ) )
=> ( ( ranked8905849569120154423e1_a_b @ rank @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E3 ) @ bot_bo2248824169281960260_a_b_b ) ) )
= ( node_list_a_b @ ( append_a @ R2 @ ( root_list_a_b @ T12 ) ) @ ( sucs_list_a_b @ T12 ) ) ) )
& ( ~ ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T12 ) ) ) @ ( rank @ ( rev_a @ R2 ) ) )
=> ( ( ranked8905849569120154423e1_a_b @ rank @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E3 ) @ bot_bo2248824169281960260_a_b_b ) ) )
= ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ( ranked8905849569120154423e1_a_b @ rank @ T12 ) @ E3 ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) ) ) ).
% normalize1.psimps(1)
thf(fact_1236_disjoint__sets__subset,axiom,
! [Xs2: fset_P2153231429829016240_a_b_b,Ys: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ Xs2 @ Ys )
=> ( ( ! [X3: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ X3 @ ( fset_P9138963618725001425_a_b_b @ Ys ) )
=> ( produc336013011823628281_b_b_o
@ ^ [Y2: dtree_list_a_b,E1: b] :
( ~ ( member_b @ E1 @ ( darcs_list_a_b @ Y2 ) )
& ! [Z: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ Z @ ( fset_P9138963618725001425_a_b_b @ Ys ) )
=> ( produc336013011823628281_b_b_o
@ ^ [Aa: dtree_list_a_b,E2: b] :
( ( ( inf_inf_set_b @ ( sup_sup_set_b @ ( darcs_list_a_b @ Y2 ) @ ( insert_b @ E1 @ bot_bot_set_b ) ) @ ( sup_sup_set_b @ ( darcs_list_a_b @ Aa ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
= bot_bot_set_b )
| ( ( produc7704165765595008946_a_b_b @ Y2 @ E1 )
= ( produc7704165765595008946_a_b_b @ Aa @ E2 ) ) )
@ Z ) ) )
@ X3 ) )
& ! [X3: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ X3 @ ( fset_P9138963618725001425_a_b_b @ Ys ) )
=> ( produc336013011823628281_b_b_o
@ ^ [Y2: dtree_list_a_b,E1: b] :
! [Z: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ Z @ ( fset_P9138963618725001425_a_b_b @ Ys ) )
=> ( produc336013011823628281_b_b_o
@ ^ [Aa: dtree_list_a_b,E2: b] :
( ( ( inf_inf_set_a @ ( list_dlverts_a_b @ Y2 ) @ ( list_dlverts_a_b @ Aa ) )
= bot_bot_set_a )
| ( ( produc7704165765595008946_a_b_b @ Y2 @ E1 )
= ( produc7704165765595008946_a_b_b @ Aa @ E2 ) ) )
@ Z ) )
@ X3 ) )
& ! [X3: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ X3 @ ( fset_P9138963618725001425_a_b_b @ Ys ) )
=> ( produc336013011823628281_b_b_o
@ ^ [T3: dtree_list_a_b,E4: b] :
~ ( member_list_a @ nil_a @ ( dverts_list_a_b @ T3 ) )
@ X3 ) ) )
=> ( ! [X2: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ X2 @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( produc336013011823628281_b_b_o
@ ^ [Y2: dtree_list_a_b,E1: b] :
( ~ ( member_b @ E1 @ ( darcs_list_a_b @ Y2 ) )
& ! [Z: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ Z @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( produc336013011823628281_b_b_o
@ ^ [Aa: dtree_list_a_b,E2: b] :
( ( ( inf_inf_set_b @ ( sup_sup_set_b @ ( darcs_list_a_b @ Y2 ) @ ( insert_b @ E1 @ bot_bot_set_b ) ) @ ( sup_sup_set_b @ ( darcs_list_a_b @ Aa ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
= bot_bot_set_b )
| ( ( produc7704165765595008946_a_b_b @ Y2 @ E1 )
= ( produc7704165765595008946_a_b_b @ Aa @ E2 ) ) )
@ Z ) ) )
@ X2 ) )
& ! [X2: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ X2 @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( produc336013011823628281_b_b_o
@ ^ [Y2: dtree_list_a_b,E1: b] :
! [Z: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ Z @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( produc336013011823628281_b_b_o
@ ^ [Aa: dtree_list_a_b,E2: b] :
( ( ( inf_inf_set_a @ ( list_dlverts_a_b @ Y2 ) @ ( list_dlverts_a_b @ Aa ) )
= bot_bot_set_a )
| ( ( produc7704165765595008946_a_b_b @ Y2 @ E1 )
= ( produc7704165765595008946_a_b_b @ Aa @ E2 ) ) )
@ Z ) )
@ X2 ) )
& ! [X2: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ X2 @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( produc336013011823628281_b_b_o
@ ^ [T3: dtree_list_a_b,E4: b] :
~ ( member_list_a @ nil_a @ ( dverts_list_a_b @ T3 ) )
@ X2 ) ) ) ) ) ).
% disjoint_sets_subset
thf(fact_1237_disjoint__sets__sucs,axiom,
( ! [X2: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ X2 @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ t ) ) )
=> ( produc336013011823628281_b_b_o
@ ^ [Y2: dtree_list_a_b,E1: b] :
( ~ ( member_b @ E1 @ ( darcs_list_a_b @ Y2 ) )
& ! [Z: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ Z @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ t ) ) )
=> ( produc336013011823628281_b_b_o
@ ^ [Aa: dtree_list_a_b,E2: b] :
( ( ( inf_inf_set_b @ ( sup_sup_set_b @ ( darcs_list_a_b @ Y2 ) @ ( insert_b @ E1 @ bot_bot_set_b ) ) @ ( sup_sup_set_b @ ( darcs_list_a_b @ Aa ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
= bot_bot_set_b )
| ( ( produc7704165765595008946_a_b_b @ Y2 @ E1 )
= ( produc7704165765595008946_a_b_b @ Aa @ E2 ) ) )
@ Z ) ) )
@ X2 ) )
& ! [X2: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ X2 @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ t ) ) )
=> ( produc336013011823628281_b_b_o
@ ^ [Y2: dtree_list_a_b,E1: b] :
! [Z: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ Z @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ t ) ) )
=> ( produc336013011823628281_b_b_o
@ ^ [Aa: dtree_list_a_b,E2: b] :
( ( ( inf_inf_set_a @ ( list_dlverts_a_b @ Y2 ) @ ( list_dlverts_a_b @ Aa ) )
= bot_bot_set_a )
| ( ( produc7704165765595008946_a_b_b @ Y2 @ E1 )
= ( produc7704165765595008946_a_b_b @ Aa @ E2 ) ) )
@ Z ) )
@ X2 ) )
& ! [X2: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ X2 @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ t ) ) )
=> ( produc336013011823628281_b_b_o
@ ^ [T3: dtree_list_a_b,E4: b] :
~ ( member_list_a @ nil_a @ ( dverts_list_a_b @ T3 ) )
@ X2 ) ) ) ).
% disjoint_sets_sucs
thf(fact_1238_wf__dtree__axioms,axiom,
wf_dtree_list_a_b @ t ).
% wf_dtree_axioms
thf(fact_1239_wf__verts,axiom,
wf_dverts_list_a_b @ t ).
% wf_verts
thf(fact_1240_wf__lverts,axiom,
list_wf_dlverts_a_b @ t ).
% wf_lverts
thf(fact_1241_wf__dlverts__combine,axiom,
! [X: list_a,Y: list_a] : ( list_wf_dlverts_a_b @ ( list_combine_a_b @ X @ Y @ t ) ) ).
% wf_dlverts_combine
thf(fact_1242_insert__between__wf__dverts,axiom,
! [V: list_a,E3: b,X: list_a,Y: list_a] :
( ~ ( member_list_a @ V @ ( dverts_list_a_b @ t ) )
=> ( wf_dverts_list_a_b @ ( insert1898995607788287860st_a_b @ V @ E3 @ X @ Y @ t ) ) ) ).
% insert_between_wf_dverts
thf(fact_1243_insert__between__add__v__if__x__in,axiom,
! [X: list_a,V: list_a,E3: b,Y: list_a] :
( ( member_list_a @ X @ ( dverts_list_a_b @ t ) )
=> ( ( dverts_list_a_b @ ( insert1898995607788287860st_a_b @ V @ E3 @ X @ Y @ t ) )
= ( insert_list_a @ V @ ( dverts_list_a_b @ t ) ) ) ) ).
% insert_between_add_v_if_x_in
thf(fact_1244_v__in__comb__if__in__dlverts,axiom,
! [V: a,X: list_a,Y: list_a] :
( ( member_a @ V @ ( list_dlverts_a_b @ t ) )
=> ( member_a @ V @ ( list_dlverts_a_b @ ( list_combine_a_b @ X @ Y @ t ) ) ) ) ).
% v_in_comb_if_in_dlverts
thf(fact_1245_v__in__dlverts__if__in__comb,axiom,
! [V: a,X: list_a,Y: list_a] :
( ( member_a @ V @ ( list_dlverts_a_b @ ( list_combine_a_b @ X @ Y @ t ) ) )
=> ( member_a @ V @ ( list_dlverts_a_b @ t ) ) ) ).
% v_in_dlverts_if_in_comb
thf(fact_1246_wf__dtree__sub,axiom,
! [X: dtree_list_a_b] :
( ( is_subtree_list_a_b @ X @ t )
=> ( wf_dtree_list_a_b @ X ) ) ).
% wf_dtree_sub
thf(fact_1247_wf__arcs,axiom,
wf_darcs_list_a_b @ t ).
% wf_arcs
thf(fact_1248_wf__darcs__combine,axiom,
! [X: list_a,Y: list_a] : ( wf_darcs_list_a_b @ ( list_combine_a_b @ X @ Y @ t ) ) ).
% wf_darcs_combine
thf(fact_1249_insert__between__wf__dtree,axiom,
! [E3: b,V: list_a,X: list_a,Y: list_a] :
( ~ ( member_b @ E3 @ ( darcs_list_a_b @ t ) )
=> ( ~ ( member_list_a @ V @ ( dverts_list_a_b @ t ) )
=> ( wf_dtree_list_a_b @ ( insert1898995607788287860st_a_b @ V @ E3 @ X @ Y @ t ) ) ) ) ).
% insert_between_wf_dtree
thf(fact_1250_arc__uneq__if__subtree__uneq,axiom,
! [X1: dtree_list_a_b,E12: b,Xs2: fset_P2153231429829016240_a_b_b,X22: dtree_list_a_b,E22: b,R2: list_a] :
( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X1 @ E12 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X22 @ E22 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( X1 != X22 )
=> ( ( ( node_list_a_b @ R2 @ Xs2 )
= t )
=> ( E12 != E22 ) ) ) ) ) ).
% arc_uneq_if_subtree_uneq
thf(fact_1251_subtree__uneq__if__arc__uneq,axiom,
! [X1: dtree_list_a_b,E12: b,Xs2: fset_P2153231429829016240_a_b_b,X22: dtree_list_a_b,E22: b,R2: list_a] :
( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X1 @ E12 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X22 @ E22 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( E12 != E22 )
=> ( ( ( node_list_a_b @ R2 @ Xs2 )
= t )
=> ( X1 != X22 ) ) ) ) ) ).
% subtree_uneq_if_arc_uneq
thf(fact_1252_wf__dtree__rec,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: dtree_list_a_b,E3: b] :
( ( ( node_list_a_b @ R2 @ Xs2 )
= t )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E3 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( wf_dtree_list_a_b @ X ) ) ) ).
% wf_dtree_rec
thf(fact_1253_insert__between__add__e__if__x__in,axiom,
! [X: list_a,V: list_a,E3: b,Y: list_a] :
( ( member_list_a @ X @ ( dverts_list_a_b @ t ) )
=> ( ( darcs_list_a_b @ ( insert1898995607788287860st_a_b @ V @ E3 @ X @ Y @ t ) )
= ( insert_b @ E3 @ ( darcs_list_a_b @ t ) ) ) ) ).
% insert_between_add_e_if_x_in
thf(fact_1254_insert__between__wf__darcs,axiom,
! [E3: b,V: list_a,X: list_a,Y: list_a] :
( ~ ( member_b @ E3 @ ( darcs_list_a_b @ t ) )
=> ( ~ ( member_list_a @ V @ ( dverts_list_a_b @ t ) )
=> ( wf_darcs_list_a_b @ ( insert1898995607788287860st_a_b @ V @ E3 @ X @ Y @ t ) ) ) ) ).
% insert_between_wf_darcs
thf(fact_1255_child__arc__not__subtree,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: dtree_list_a_b,E12: b] :
( ( ( node_list_a_b @ R2 @ Xs2 )
= t )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E12 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ~ ( member_b @ E12 @ ( darcs_list_a_b @ X ) ) ) ) ).
% child_arc_not_subtree
thf(fact_1256_subtree__root__not__root,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: dtree_list_a_b,E3: b] :
( ( t
= ( node_list_a_b @ R2 @ Xs2 ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E3 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( root_list_a_b @ X )
!= R2 ) ) ) ).
% subtree_root_not_root
thf(fact_1257_dlverts__comb__id,axiom,
! [X: list_a,Y: list_a] :
( ( list_dlverts_a_b @ ( list_combine_a_b @ X @ Y @ t ) )
= ( list_dlverts_a_b @ t ) ) ).
% dlverts_comb_id
thf(fact_1258_disjoint__darcs__combine,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: list_a,Y: list_a] :
( ( ( node_list_a_b @ R2 @ Xs2 )
= t )
=> ! [X2: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ X2
@ ( fset_P9138963618725001425_a_b_b
@ ( fimage8157835862677140991_a_b_b
@ ( produc4970258495817412575_a_b_b
@ ^ [T3: dtree_list_a_b] : ( produc7704165765595008946_a_b_b @ ( list_combine_a_b @ X @ Y @ T3 ) ) )
@ Xs2 ) ) )
=> ( produc336013011823628281_b_b_o
@ ^ [Y2: dtree_list_a_b,E1: b] :
( ~ ( member_b @ E1 @ ( darcs_list_a_b @ Y2 ) )
& ! [Z: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ Z
@ ( fset_P9138963618725001425_a_b_b
@ ( fimage8157835862677140991_a_b_b
@ ( produc4970258495817412575_a_b_b
@ ^ [T3: dtree_list_a_b] : ( produc7704165765595008946_a_b_b @ ( list_combine_a_b @ X @ Y @ T3 ) ) )
@ Xs2 ) ) )
=> ( produc336013011823628281_b_b_o
@ ^ [Aa: dtree_list_a_b,E2: b] :
( ( ( inf_inf_set_b @ ( sup_sup_set_b @ ( darcs_list_a_b @ Y2 ) @ ( insert_b @ E1 @ bot_bot_set_b ) ) @ ( sup_sup_set_b @ ( darcs_list_a_b @ Aa ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
= bot_bot_set_b )
| ( ( produc7704165765595008946_a_b_b @ Y2 @ E1 )
= ( produc7704165765595008946_a_b_b @ Aa @ E2 ) ) )
@ Z ) ) )
@ X2 ) ) ) ).
% disjoint_darcs_combine
thf(fact_1259_disjoint__darcs__aux3,axiom,
! [E12: b,Xs2: fset_P2153231429829016240_a_b_b,X: list_a,R2: list_a,E3: b,V: list_a,Y: list_a] :
( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ t @ E12 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( member_list_a @ X @ ( dverts_list_a_b @ t ) )
=> ( ( wf_darcs_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
=> ( ~ ( member_b @ E3 @ ( darcs_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) )
=> ( ( wf_dverts_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
=> ! [X2: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ X2
@ ( fset_P9138963618725001425_a_b_b
@ ( fimage8157835862677140991_a_b_b
@ ( produc4970258495817412575_a_b_b
@ ^ [T13: dtree_list_a_b] : ( produc7704165765595008946_a_b_b @ ( insert1898995607788287860st_a_b @ V @ E3 @ X @ Y @ T13 ) ) )
@ Xs2 ) ) )
=> ( produc336013011823628281_b_b_o
@ ^ [Y2: dtree_list_a_b,E1: b] :
( ~ ( member_b @ E1 @ ( darcs_list_a_b @ Y2 ) )
& ! [Z: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ Z
@ ( fset_P9138963618725001425_a_b_b
@ ( fimage8157835862677140991_a_b_b
@ ( produc4970258495817412575_a_b_b
@ ^ [T13: dtree_list_a_b] : ( produc7704165765595008946_a_b_b @ ( insert1898995607788287860st_a_b @ V @ E3 @ X @ Y @ T13 ) ) )
@ Xs2 ) ) )
=> ( produc336013011823628281_b_b_o
@ ^ [Aa: dtree_list_a_b,E2: b] :
( ( ( inf_inf_set_b @ ( sup_sup_set_b @ ( darcs_list_a_b @ Y2 ) @ ( insert_b @ E1 @ bot_bot_set_b ) ) @ ( sup_sup_set_b @ ( darcs_list_a_b @ Aa ) @ ( insert_b @ E2 @ bot_bot_set_b ) ) )
= bot_bot_set_b )
| ( ( produc7704165765595008946_a_b_b @ Y2 @ E1 )
= ( produc7704165765595008946_a_b_b @ Aa @ E2 ) ) )
@ Z ) ) )
@ X2 ) ) ) ) ) ) ) ).
% disjoint_darcs_aux3
thf(fact_1260_disjoint__darcs__aux3__aux1,axiom,
! [E12: b,Xs2: fset_P2153231429829016240_a_b_b,X: list_a,R2: list_a,E3: b,T23: dtree_list_a_b,E22: b,V: list_a,Y: list_a,T32: dtree_list_a_b,E32: b] :
( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ t @ E12 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( member_list_a @ X @ ( dverts_list_a_b @ t ) )
=> ( ( wf_darcs_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
=> ( ~ ( member_b @ E3 @ ( darcs_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T23 @ E22 )
@ ( image_3365364158514602119_a_b_b
@ ( produc4970258495817412575_a_b_b
@ ^ [T13: dtree_list_a_b] : ( produc7704165765595008946_a_b_b @ ( insert1898995607788287860st_a_b @ V @ E3 @ X @ Y @ T13 ) ) )
@ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T32 @ E32 )
@ ( image_3365364158514602119_a_b_b
@ ( produc4970258495817412575_a_b_b
@ ^ [T13: dtree_list_a_b] : ( produc7704165765595008946_a_b_b @ ( insert1898995607788287860st_a_b @ V @ E3 @ X @ Y @ T13 ) ) )
@ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
=> ( ( ( produc7704165765595008946_a_b_b @ T23 @ E22 )
!= ( produc7704165765595008946_a_b_b @ T32 @ E32 ) )
=> ( ( wf_dverts_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
=> ( ( inf_inf_set_b @ ( sup_sup_set_b @ ( darcs_list_a_b @ T23 ) @ ( insert_b @ E22 @ bot_bot_set_b ) ) @ ( sup_sup_set_b @ ( darcs_list_a_b @ T32 ) @ ( insert_b @ E32 @ bot_bot_set_b ) ) )
= bot_bot_set_b ) ) ) ) ) ) ) ) ) ).
% disjoint_darcs_aux3_aux1
thf(fact_1261_sucs__dverts__no__root,axiom,
! [T12: dtree_list_a_b,E12: b] :
( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E12 ) @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ t ) ) )
=> ( ( minus_646659088055828811list_a @ ( dverts_list_a_b @ ( node_list_a_b @ ( root_list_a_b @ t ) @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) ) @ ( insert_list_a @ ( root_list_a_b @ t ) @ bot_bot_set_list_a ) )
= ( dverts_list_a_b @ T12 ) ) ) ).
% sucs_dverts_no_root
thf(fact_1262_disjoint__dverts__aux3,axiom,
! [T23: dtree_list_a_b,E22: b,V: list_a,E3: b,X: list_a,Y: list_a,Xs2: fset_P2153231429829016240_a_b_b,T32: dtree_list_a_b,E32: b,E12: b,R2: list_a] :
( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T23 @ E22 )
@ ( image_3365364158514602119_a_b_b
@ ( produc4970258495817412575_a_b_b
@ ^ [T13: dtree_list_a_b] : ( produc7704165765595008946_a_b_b @ ( insert1898995607788287860st_a_b @ V @ E3 @ X @ Y @ T13 ) ) )
@ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T32 @ E32 )
@ ( image_3365364158514602119_a_b_b
@ ( produc4970258495817412575_a_b_b
@ ^ [T13: dtree_list_a_b] : ( produc7704165765595008946_a_b_b @ ( insert1898995607788287860st_a_b @ V @ E3 @ X @ Y @ T13 ) ) )
@ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
=> ( ( ( produc7704165765595008946_a_b_b @ T23 @ E22 )
!= ( produc7704165765595008946_a_b_b @ T32 @ E32 ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ t @ E12 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( member_list_a @ X @ ( dverts_list_a_b @ t ) )
=> ( ( wf_dverts_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
=> ( ~ ( member_list_a @ V @ ( dverts_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) )
=> ( ( inf_inf_set_list_a @ ( dverts_list_a_b @ T23 ) @ ( dverts_list_a_b @ T32 ) )
= bot_bot_set_list_a ) ) ) ) ) ) ) ) ).
% disjoint_dverts_aux3
thf(fact_1263_disjoint__darcs__aux3__aux2,axiom,
! [E12: b,Xs2: fset_P2153231429829016240_a_b_b,X: list_a,R2: list_a,E3: b,T23: dtree_list_a_b,E22: b,V: list_a,Y: list_a] :
( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ t @ E12 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( member_list_a @ X @ ( dverts_list_a_b @ t ) )
=> ( ( wf_darcs_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
=> ( ~ ( member_b @ E3 @ ( darcs_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T23 @ E22 )
@ ( image_3365364158514602119_a_b_b
@ ( produc4970258495817412575_a_b_b
@ ^ [T13: dtree_list_a_b] : ( produc7704165765595008946_a_b_b @ ( insert1898995607788287860st_a_b @ V @ E3 @ X @ Y @ T13 ) ) )
@ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
=> ( ( wf_dverts_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
=> ~ ( member_b @ E22 @ ( darcs_list_a_b @ T23 ) ) ) ) ) ) ) ) ).
% disjoint_darcs_aux3_aux2
thf(fact_1264_dhead__in__childverts__no__root,axiom,
! [E3: b,X: dtree_list_a_b,E5: b,Xs2: fset_P2153231429829016240_a_b_b,R2: list_a,Def2: b > list_a] :
( ( member_b @ E3 @ ( darcs_list_a_b @ X ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E5 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( ( node_list_a_b @ R2 @ Xs2 )
= t )
=> ( member_list_a @ ( dhead_list_a_b @ t @ Def2 @ E3 ) @ ( minus_646659088055828811list_a @ ( dverts_list_a_b @ X ) @ ( insert_list_a @ ( root_list_a_b @ X ) @ bot_bot_set_list_a ) ) ) ) ) ) ).
% dhead_in_childverts_no_root
thf(fact_1265_dhead__notin__subtree__wo__root,axiom,
! [X: dtree_list_a_b,E3: b,Xs2: fset_P2153231429829016240_a_b_b,P2: b,R2: list_a,Dh: b > list_a] :
( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E3 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ~ ( member_b @ P2 @ ( darcs_list_a_b @ X ) )
=> ( ( member_b @ P2 @ ( darcs_list_a_b @ t ) )
=> ( ( t
= ( node_list_a_b @ R2 @ Xs2 ) )
=> ~ ( member_list_a @ ( dhead_list_a_b @ t @ Dh @ P2 ) @ ( minus_646659088055828811list_a @ ( dverts_list_a_b @ X ) @ ( insert_list_a @ ( root_list_a_b @ X ) @ bot_bot_set_list_a ) ) ) ) ) ) ) ).
% dhead_notin_subtree_wo_root
thf(fact_1266_dhead__unique,axiom,
! [E3: b,P2: b,Dh: b > list_a] :
( ( member_b @ E3 @ ( darcs_list_a_b @ t ) )
=> ( ( member_b @ P2 @ ( darcs_list_a_b @ t ) )
=> ( ( E3 != P2 )
=> ( ( dhead_list_a_b @ t @ Dh @ E3 )
!= ( dhead_list_a_b @ t @ Dh @ P2 ) ) ) ) ) ).
% dhead_unique
thf(fact_1267_dhead__in__dverts,axiom,
! [E3: b,Def2: b > list_a] :
( ( member_b @ E3 @ ( darcs_list_a_b @ t ) )
=> ( member_list_a @ ( dhead_list_a_b @ t @ Def2 @ E3 ) @ ( dverts_list_a_b @ t ) ) ) ).
% dhead_in_dverts
thf(fact_1268_dhead__not__root,axiom,
! [E3: b,Def2: b > list_a] :
( ( member_b @ E3 @ ( darcs_list_a_b @ t ) )
=> ( ( dhead_list_a_b @ t @ Def2 @ E3 )
!= ( root_list_a_b @ t ) ) ) ).
% dhead_not_root
thf(fact_1269_dhead__in__subtree__eq__subtree,axiom,
! [T12: dtree_list_a_b,E3: b,Def2: b > list_a] :
( ( is_subtree_list_a_b @ T12 @ t )
=> ( ( member_b @ E3 @ ( darcs_list_a_b @ T12 ) )
=> ( ( dhead_list_a_b @ t @ Def2 @ E3 )
= ( dhead_list_a_b @ T12 @ Def2 @ E3 ) ) ) ) ).
% dhead_in_subtree_eq_subtree
thf(fact_1270_dhead__in__childverts,axiom,
! [E3: b,X: dtree_list_a_b,E5: b,Xs2: fset_P2153231429829016240_a_b_b,R2: list_a,Def2: b > list_a] :
( ( member_b @ E3 @ ( darcs_list_a_b @ X ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E5 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( ( node_list_a_b @ R2 @ Xs2 )
= t )
=> ( member_list_a @ ( dhead_list_a_b @ t @ Def2 @ E3 ) @ ( dverts_list_a_b @ X ) ) ) ) ) ).
% dhead_in_childverts
thf(fact_1271_dhead__in__dverts__no__root,axiom,
! [E3: b,Def2: b > list_a] :
( ( member_b @ E3 @ ( darcs_list_a_b @ t ) )
=> ( member_list_a @ ( dhead_list_a_b @ t @ Def2 @ E3 ) @ ( minus_646659088055828811list_a @ ( dverts_list_a_b @ t ) @ ( insert_list_a @ ( root_list_a_b @ t ) @ bot_bot_set_list_a ) ) ) ) ).
% dhead_in_dverts_no_root
thf(fact_1272_dhead__sub__img__eq__root__img,axiom,
! [V: list_a,Ys: fset_P2153231429829016240_a_b_b,Dh: b > list_a] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ V @ Ys ) @ t )
=> ( ( image_4816005896404294376st_a_b
@ ^ [E4: b] : ( produc4781227316648555538st_a_b @ ( dhead_list_a_b @ t @ Dh @ E4 ) @ E4 )
@ ( image_4684437738885282872_b_b_b @ produc5719641485658034180_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Ys ) ) )
= ( image_5973357989600724187st_a_b
@ ( produc352076092515656579st_a_b
@ ^ [X4: dtree_list_a_b] : ( produc4781227316648555538st_a_b @ ( root_list_a_b @ X4 ) ) )
@ ( fset_P9138963618725001425_a_b_b @ Ys ) ) ) ) ).
% dhead_sub_img_eq_root_img
thf(fact_1273_darcs__child__subset,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: dtree_list_a_b] :
( ( ( node_list_a_b @ R2 @ Xs2 )
= t )
=> ( ( member551035911493665803st_a_b @ X @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
=> ( ord_less_set_b @ ( darcs_list_a_b @ X ) @ ( darcs_list_a_b @ t ) ) ) ) ).
% darcs_child_subset
thf(fact_1274_dverts__child__subset,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: dtree_list_a_b] :
( ( ( node_list_a_b @ R2 @ Xs2 )
= t )
=> ( ( member551035911493665803st_a_b @ X @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
=> ( ord_less_set_list_a @ ( dverts_list_a_b @ X ) @ ( dverts_list_a_b @ t ) ) ) ) ).
% dverts_child_subset
thf(fact_1275_root__not__subtree,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: dtree_list_a_b] :
( ( ( node_list_a_b @ R2 @ Xs2 )
= t )
=> ( ( member551035911493665803st_a_b @ X @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
=> ~ ( member_list_a @ R2 @ ( dverts_list_a_b @ X ) ) ) ) ).
% root_not_subtree
thf(fact_1276_dhead__img__eq__root__img,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,Dh: b > list_a] :
( ( ( node_list_a_b @ R2 @ Xs2 )
= t )
=> ( ( image_4816005896404294376st_a_b
@ ^ [E4: b] : ( produc4781227316648555538st_a_b @ ( dhead_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) @ Dh @ E4 ) @ E4 )
@ ( image_4684437738885282872_b_b_b @ produc5719641485658034180_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
= ( image_5973357989600724187st_a_b
@ ( produc352076092515656579st_a_b
@ ^ [X4: dtree_list_a_b] : ( produc4781227316648555538st_a_b @ ( root_list_a_b @ X4 ) ) )
@ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) ) ) ).
% dhead_img_eq_root_img
% Helper facts (25)
thf(help_If_2_1_If_001t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_T,axiom,
! [X: dtree_list_a_b,Y: dtree_list_a_b] :
( ( if_dtree_list_a_b @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_T,axiom,
! [X: dtree_list_a_b,Y: dtree_list_a_b] :
( ( if_dtree_list_a_b @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mtf__a_J_Mtf__a_J_T,axiom,
! [X: produc5194124205918388666_a_a_a,Y: produc5194124205918388666_a_a_a] :
( ( if_Pro6965622180465380212_a_a_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mtf__a_J_Mtf__a_J_T,axiom,
! [X: produc5194124205918388666_a_a_a,Y: produc5194124205918388666_a_a_a] :
( ( if_Pro6965622180465380212_a_a_a @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mtf__b_J_Mtf__b_J_T,axiom,
! [X: produc4558475209616630778_a_b_b,Y: produc4558475209616630778_a_b_b] :
( ( if_Pro6329973184163622324_a_b_b @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mtf__b_J_Mtf__b_J_T,axiom,
! [X: produc4558475209616630778_a_b_b,Y: produc4558475209616630778_a_b_b] :
( ( if_Pro6329973184163622324_a_b_b @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Dtree__Odtree_Itf__b_Mtf__a_J_Mtf__a_J_T,axiom,
! [X: produc1566970784251743547_b_a_a,Y: produc1566970784251743547_b_a_a] :
( ( if_Pro3338468758798735093_b_a_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Dtree__Odtree_Itf__b_Mtf__a_J_Mtf__a_J_T,axiom,
! [X: produc1566970784251743547_b_a_a,Y: produc1566970784251743547_b_a_a] :
( ( if_Pro3338468758798735093_b_a_a @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Dtree__Odtree_Itf__b_Mtf__b_J_Mtf__b_J_T,axiom,
! [X: produc931321787949985659_b_b_b,Y: produc931321787949985659_b_b_b] :
( ( if_Pro2702819762496977205_b_b_b @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Dtree__Odtree_Itf__b_Mtf__b_J_Mtf__b_J_T,axiom,
! [X: produc931321787949985659_b_b_b,Y: produc931321787949985659_b_b_b] :
( ( if_Pro2702819762496977205_b_b_b @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__a_J_Mtf__a_J_T,axiom,
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( ( if_Pro1443360173218331130_a_a_a @ $true @ X @ Y )
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( ( if_Pro807711176916573242_a_b_b @ $false @ X @ Y )
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( ( if_Pro807711176916573242_a_b_b @ $true @ X @ Y )
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thf(help_If_2_1_If_001t__FSet__Ofset_It__Product____Type__Oprod_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_Mtf__b_J_J_T,axiom,
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( ( if_Pro2199488368625827629st_a_b @ $false @ X @ Y )
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! [X: produc8101844211926167155st_a_b,Y: produc8101844211926167155st_a_b] :
( ( if_Pro2199488368625827629st_a_b @ $true @ X @ Y )
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% Conjectures (1)
thf(conj_0,conjecture,
wf_darcs_list_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ ( node_list_a_b @ r @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ) ).
%------------------------------------------------------------------------------