TPTP Problem File: SLH0687^1.p
View Solutions
- 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 : VYDRA_MDL/0010_Temporal/prob_00028_001220__16559598_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1530 ( 737 unt; 242 typ; 0 def)
% Number of atoms : 3089 (1743 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 11406 ( 465 ~; 41 |; 305 &;9419 @)
% ( 0 <=>;1176 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 6 avg)
% Number of types : 16 ( 15 usr)
% Number of type conns : 1071 (1071 >; 0 *; 0 +; 0 <<)
% Number of symbols : 230 ( 227 usr; 14 con; 0-4 aty)
% Number of variables : 3681 ( 106 ^;3358 !; 217 ?;3681 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:53:26.405
%------------------------------------------------------------------------------
% Could-be-implicit typings (15)
thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J_J,type,
list_l3581718849243781313la_a_b: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J_J,type,
set_li1233427854086133095la_a_b: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J_J,type,
list_l147921316930177639la_a_b: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J_J,type,
set_se1704372200780349325la_a_b: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J_J,type,
set_li5266388670752979469la_a_b: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_It__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J_J,type,
list_s602864636976034957la_a_b: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J_J,type,
set_se3797390812436207411la_a_b: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J,type,
list_l5948067055324603441la_a_b: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J,type,
set_list_formula_a_b: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J,type,
list_set_formula_a_b: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J,type,
set_set_formula_a_b: $tType ).
thf(ty_n_t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
list_formula_a_b: $tType ).
thf(ty_n_t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
set_formula_a_b: $tType ).
thf(ty_n_t__MDL__Oformula_Itf__a_Mtf__b_J,type,
formula_a_b: $tType ).
thf(ty_n_t__MDL__Oregex_Itf__a_Mtf__b_J,type,
regex_a_b: $tType ).
% Explicit typings (227)
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
bNF_Gr4875458711411372354la_a_b: set_li1233427854086133095la_a_b > list_formula_a_b > set_li1233427854086133095la_a_b ).
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
bNF_Gr2660717057327301682la_a_b: set_list_formula_a_b > formula_a_b > set_list_formula_a_b ).
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
bNF_Gr5837620917382104168la_a_b: set_li5266388670752979469la_a_b > set_formula_a_b > set_li5266388670752979469la_a_b ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
bNF_Gr3706963439723013830la_a_b: set_li1233427854086133095la_a_b > list_l5948067055324603441la_a_b > set_list_formula_a_b ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
bNF_Gr6515907858579016630la_a_b: set_list_formula_a_b > list_formula_a_b > set_formula_a_b ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
bNF_Gr8423546037056666604la_a_b: set_li5266388670752979469la_a_b > list_set_formula_a_b > set_set_formula_a_b ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
comple2445818437193057818la_a_b: set_set_formula_a_b > set_formula_a_b ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
condit8798486063731193955la_a_b: set_set_formula_a_b > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
finite5308508654137801784la_a_b: set_list_formula_a_b > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
finite6884408037426709032la_a_b: set_formula_a_b > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
finite7542661095822535262la_a_b: set_set_formula_a_b > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J,type,
finite6039610903330827156la_a_b: set_se3797390812436207411la_a_b > $o ).
thf(sy_c_Finite__Set_Ofold_001t__MDL__Oformula_Itf__a_Mtf__b_J_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
finite2095001333592942498la_a_b: ( formula_a_b > set_formula_a_b > set_formula_a_b ) > set_formula_a_b > set_formula_a_b > set_formula_a_b ).
thf(sy_c_Finite__Set_Ofold_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
finite2548470876788853208la_a_b: ( set_formula_a_b > set_formula_a_b > set_formula_a_b ) > set_formula_a_b > set_set_formula_a_b > set_formula_a_b ).
thf(sy_c_Fun_Ocomp_001_062_It__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_Mt__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J_001_062_It__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_Mt__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
comp_s4889918555203769296la_a_b: ( ( set_formula_a_b > set_formula_a_b ) > set_formula_a_b > set_formula_a_b ) > ( formula_a_b > set_formula_a_b > set_formula_a_b ) > formula_a_b > set_formula_a_b > set_formula_a_b ).
thf(sy_c_Fun_Ocomp_001_062_It__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_Mt__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J_001_062_It__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_Mt__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
comp_s5422811073032477702la_a_b: ( ( set_formula_a_b > set_formula_a_b ) > set_formula_a_b > set_formula_a_b ) > ( set_formula_a_b > set_formula_a_b > set_formula_a_b ) > set_formula_a_b > set_formula_a_b > set_formula_a_b ).
thf(sy_c_Fun_Ocomp_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
comp_l1221189711022459592la_a_b: ( list_formula_a_b > list_formula_a_b ) > ( list_formula_a_b > list_formula_a_b ) > list_formula_a_b > list_formula_a_b ).
thf(sy_c_Fun_Ocomp_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
comp_l3822438638636892398la_a_b: ( list_formula_a_b > list_formula_a_b ) > ( set_formula_a_b > list_formula_a_b ) > set_formula_a_b > list_formula_a_b ).
thf(sy_c_Fun_Ocomp_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
comp_l3699371471522569966la_a_b: ( list_formula_a_b > set_formula_a_b ) > ( list_formula_a_b > list_formula_a_b ) > list_formula_a_b > set_formula_a_b ).
thf(sy_c_Fun_Ocomp_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
comp_l1054984425785505758la_a_b: ( list_formula_a_b > set_formula_a_b ) > ( formula_a_b > list_formula_a_b ) > formula_a_b > set_formula_a_b ).
thf(sy_c_Fun_Ocomp_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
comp_l4184407891557566484la_a_b: ( list_formula_a_b > set_formula_a_b ) > ( set_formula_a_b > list_formula_a_b ) > set_formula_a_b > set_formula_a_b ).
thf(sy_c_Fun_Ocomp_001t__MDL__Oformula_Itf__a_Mtf__b_J_001_062_It__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_Mt__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
comp_f398844620851073332la_a_b: ( formula_a_b > set_formula_a_b > set_formula_a_b ) > ( formula_a_b > formula_a_b ) > formula_a_b > set_formula_a_b > set_formula_a_b ).
thf(sy_c_Fun_Ocomp_001t__MDL__Oformula_Itf__a_Mtf__b_J_001t__MDL__Oformula_Itf__a_Mtf__b_J_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
comp_f2828989746591664536la_a_b: ( formula_a_b > formula_a_b ) > ( formula_a_b > formula_a_b ) > formula_a_b > formula_a_b ).
thf(sy_c_Fun_Ocomp_001t__MDL__Oformula_Itf__a_Mtf__b_J_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
comp_f7809688971862344414la_a_b: ( formula_a_b > set_formula_a_b ) > ( list_formula_a_b > formula_a_b ) > list_formula_a_b > set_formula_a_b ).
thf(sy_c_Fun_Ocomp_001t__MDL__Oformula_Itf__a_Mtf__b_J_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
comp_f821314402523108302la_a_b: ( formula_a_b > set_formula_a_b ) > ( formula_a_b > formula_a_b ) > formula_a_b > set_formula_a_b ).
thf(sy_c_Fun_Ocomp_001t__MDL__Oformula_Itf__a_Mtf__b_J_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
comp_f2183751631756272644la_a_b: ( formula_a_b > set_formula_a_b ) > ( set_formula_a_b > formula_a_b ) > set_formula_a_b > set_formula_a_b ).
thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_001_062_It__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_Mt__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
comp_s3812543008930346602la_a_b: ( set_formula_a_b > set_formula_a_b > set_formula_a_b ) > ( formula_a_b > set_formula_a_b ) > formula_a_b > set_formula_a_b > set_formula_a_b ).
thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_001_062_It__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_Mt__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
comp_s266308785573631648la_a_b: ( set_formula_a_b > set_formula_a_b > set_formula_a_b ) > ( set_formula_a_b > set_formula_a_b ) > set_formula_a_b > set_formula_a_b > set_formula_a_b ).
thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
comp_s1800181939013838356la_a_b: ( set_formula_a_b > list_formula_a_b ) > ( set_formula_a_b > set_formula_a_b ) > set_formula_a_b > list_formula_a_b ).
thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_001t__MDL__Oformula_Itf__a_Mtf__b_J_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
comp_s3563907018792990158la_a_b: ( set_formula_a_b > formula_a_b ) > ( formula_a_b > set_formula_a_b ) > formula_a_b > formula_a_b ).
thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_001t__MDL__Oformula_Itf__a_Mtf__b_J_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
comp_s3553664264346224132la_a_b: ( set_formula_a_b > formula_a_b ) > ( set_formula_a_b > set_formula_a_b ) > set_formula_a_b > formula_a_b ).
thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
comp_s1677114771899515924la_a_b: ( set_formula_a_b > set_formula_a_b ) > ( list_formula_a_b > set_formula_a_b ) > list_formula_a_b > set_formula_a_b ).
thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
comp_s9099485395005827076la_a_b: ( set_formula_a_b > set_formula_a_b ) > ( formula_a_b > set_formula_a_b ) > formula_a_b > set_formula_a_b ).
thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
comp_s7716128494983798842la_a_b: ( set_formula_a_b > set_formula_a_b ) > ( set_formula_a_b > set_formula_a_b ) > set_formula_a_b > set_formula_a_b ).
thf(sy_c_Fun_Ofun__upd_001t__MDL__Oformula_Itf__a_Mtf__b_J_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
fun_up6714831372235791319la_a_b: ( formula_a_b > set_formula_a_b ) > formula_a_b > set_formula_a_b > formula_a_b > set_formula_a_b ).
thf(sy_c_Fun_Oinj__on_001t__MDL__Oformula_Itf__a_Mtf__b_J_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
inj_on2935910337538581653la_a_b: ( formula_a_b > formula_a_b ) > set_formula_a_b > $o ).
thf(sy_c_Fun_Othe__inv__into_001t__MDL__Oformula_Itf__a_Mtf__b_J_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
the_in4303446237855801815la_a_b: set_formula_a_b > ( formula_a_b > formula_a_b ) > formula_a_b > formula_a_b ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J,type,
minus_8667068627093346608la_a_b: set_list_formula_a_b > set_list_formula_a_b > set_list_formula_a_b ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
minus_3283878196352184736la_a_b: set_formula_a_b > set_formula_a_b > set_formula_a_b ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J,type,
minus_4878662943024999126la_a_b: set_set_formula_a_b > set_set_formula_a_b > set_set_formula_a_b ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J_J,type,
minus_1637344976281664780la_a_b: set_se3797390812436207411la_a_b > set_se3797390812436207411la_a_b > set_se3797390812436207411la_a_b ).
thf(sy_c_If_001t__List__Olist_It__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J,type,
if_lis7719565029871594987la_a_b: $o > list_l5948067055324603441la_a_b > list_l5948067055324603441la_a_b > list_l5948067055324603441la_a_b ).
thf(sy_c_If_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
if_list_formula_a_b: $o > list_formula_a_b > list_formula_a_b > list_formula_a_b ).
thf(sy_c_If_001t__List__Olist_It__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J,type,
if_lis2278976591155604753la_a_b: $o > list_set_formula_a_b > list_set_formula_a_b > list_set_formula_a_b ).
thf(sy_c_If_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
if_set_formula_a_b: $o > set_formula_a_b > set_formula_a_b > set_formula_a_b ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J,type,
inf_in7172170795784610665la_a_b: set_list_formula_a_b > set_list_formula_a_b > set_list_formula_a_b ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__List__Olist_It__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J_J,type,
inf_in3724091778011523487la_a_b: set_li5266388670752979469la_a_b > set_li5266388670752979469la_a_b > set_li5266388670752979469la_a_b ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
inf_in5741596252272946265la_a_b: set_formula_a_b > set_formula_a_b > set_formula_a_b ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J,type,
inf_in60795025369792911la_a_b: set_set_formula_a_b > set_set_formula_a_b > set_set_formula_a_b ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J,type,
sup_su9015223001214971651la_a_b: set_list_formula_a_b > set_list_formula_a_b > set_list_formula_a_b ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__List__Olist_It__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J_J,type,
sup_su5324240277097303353la_a_b: set_li5266388670752979469la_a_b > set_li5266388670752979469la_a_b > set_li5266388670752979469la_a_b ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
sup_su8832426789560438003la_a_b: set_formula_a_b > set_formula_a_b > set_formula_a_b ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J,type,
sup_su5931044959554304553la_a_b: set_set_formula_a_b > set_set_formula_a_b > set_set_formula_a_b ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J_J,type,
sup_su1714642906130169951la_a_b: set_se3797390812436207411la_a_b > set_se3797390812436207411la_a_b > set_se3797390812436207411la_a_b ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
lattic1282556677053900650la_a_b: set_set_formula_a_b > set_formula_a_b ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
lattic2153292871565936464la_a_b: set_set_formula_a_b > set_formula_a_b ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Set__Oset_It__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J,type,
lattic7951861892593991814la_a_b: set_se3797390812436207411la_a_b > set_set_formula_a_b ).
thf(sy_c_List_OListMem_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
listMe6139171625156847023la_a_b: list_formula_a_b > list_l5948067055324603441la_a_b > $o ).
thf(sy_c_List_OListMem_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
listMem_formula_a_b: formula_a_b > list_formula_a_b > $o ).
thf(sy_c_List_OListMem_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
listMe3233215151264718677la_a_b: set_formula_a_b > list_set_formula_a_b > $o ).
thf(sy_c_List_Oappend_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
append58666856857280070la_a_b: list_l5948067055324603441la_a_b > list_l5948067055324603441la_a_b > list_l5948067055324603441la_a_b ).
thf(sy_c_List_Oappend_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
append_formula_a_b: list_formula_a_b > list_formula_a_b > list_formula_a_b ).
thf(sy_c_List_Oappend_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
append8577002312234529644la_a_b: list_set_formula_a_b > list_set_formula_a_b > list_set_formula_a_b ).
thf(sy_c_List_Obind_001t__MDL__Oformula_Itf__a_Mtf__b_J_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
bind_f552093402373037483la_a_b: list_formula_a_b > ( formula_a_b > list_formula_a_b ) > list_formula_a_b ).
thf(sy_c_List_Obind_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
bind_s2984606836244749281la_a_b: list_set_formula_a_b > ( set_formula_a_b > list_formula_a_b ) > list_formula_a_b ).
thf(sy_c_List_Obutlast_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
butlas4241030279939053135la_a_b: list_l5948067055324603441la_a_b > list_l5948067055324603441la_a_b ).
thf(sy_c_List_Obutlast_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
butlast_formula_a_b: list_formula_a_b > list_formula_a_b ).
thf(sy_c_List_Obutlast_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
butlas5801911201278550005la_a_b: list_set_formula_a_b > list_set_formula_a_b ).
thf(sy_c_List_Ocan__select_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
can_se3903203210569301220la_a_b: ( list_formula_a_b > $o ) > set_list_formula_a_b > $o ).
thf(sy_c_List_Ocan__select_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
can_se151102608688302292la_a_b: ( formula_a_b > $o ) > set_formula_a_b > $o ).
thf(sy_c_List_Ocan__select_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
can_se3565154267531510026la_a_b: ( set_formula_a_b > $o ) > set_set_formula_a_b > $o ).
thf(sy_c_List_Oconcat_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
concat6919603293682802464la_a_b: list_l3581718849243781313la_a_b > list_l5948067055324603441la_a_b ).
thf(sy_c_List_Oconcat_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
concat_formula_a_b: list_l5948067055324603441la_a_b > list_formula_a_b ).
thf(sy_c_List_Oconcat_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
concat7835744469863612742la_a_b: list_l147921316930177639la_a_b > list_set_formula_a_b ).
thf(sy_c_List_Ocoset_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
coset_845268532100101742la_a_b: list_l5948067055324603441la_a_b > set_list_formula_a_b ).
thf(sy_c_List_Ocoset_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
coset_formula_a_b: list_formula_a_b > set_formula_a_b ).
thf(sy_c_List_Ocoset_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
coset_6935746743259127700la_a_b: list_set_formula_a_b > set_set_formula_a_b ).
thf(sy_c_List_Odistinct_001t__List__Olist_It__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J,type,
distin7365129041399036280la_a_b: list_l3581718849243781313la_a_b > $o ).
thf(sy_c_List_Odistinct_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
distin27490427618170344la_a_b: list_l5948067055324603441la_a_b > $o ).
thf(sy_c_List_Odistinct_001t__List__Olist_It__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J,type,
distin6955390182775811870la_a_b: list_l147921316930177639la_a_b > $o ).
thf(sy_c_List_Odistinct_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
distinct_formula_a_b: list_formula_a_b > $o ).
thf(sy_c_List_Odistinct_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
distin5059275051008138254la_a_b: list_set_formula_a_b > $o ).
thf(sy_c_List_OdropWhile_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
dropWh8739902238574629050la_a_b: ( list_formula_a_b > $o ) > list_l5948067055324603441la_a_b > list_l5948067055324603441la_a_b ).
thf(sy_c_List_OdropWhile_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
dropWh8768975241461415850la_a_b: ( formula_a_b > $o ) > list_formula_a_b > list_formula_a_b ).
thf(sy_c_List_OdropWhile_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
dropWh5803627183431294944la_a_b: ( set_formula_a_b > $o ) > list_set_formula_a_b > list_set_formula_a_b ).
thf(sy_c_List_Ofold_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
fold_l9118537288359097167la_a_b: ( list_formula_a_b > list_formula_a_b > list_formula_a_b ) > list_l5948067055324603441la_a_b > list_formula_a_b > list_formula_a_b ).
thf(sy_c_List_Ofold_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J_001t__Set__Oset_It__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J,type,
fold_l2136769360297358853la_a_b: ( list_formula_a_b > set_list_formula_a_b > set_list_formula_a_b ) > list_l5948067055324603441la_a_b > set_list_formula_a_b > set_list_formula_a_b ).
thf(sy_c_List_Ofold_001t__MDL__Oformula_Itf__a_Mtf__b_J_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
fold_f7464356806857031615la_a_b: ( formula_a_b > list_formula_a_b > list_formula_a_b ) > list_formula_a_b > list_formula_a_b > list_formula_a_b ).
thf(sy_c_List_Ofold_001t__MDL__Oformula_Itf__a_Mtf__b_J_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
fold_f470054086739575141la_a_b: ( formula_a_b > set_formula_a_b > set_formula_a_b ) > list_formula_a_b > set_formula_a_b > set_formula_a_b ).
thf(sy_c_List_Ofold_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_001t__List__Olist_It__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J,type,
fold_s1220939906018902059la_a_b: ( set_formula_a_b > list_set_formula_a_b > list_set_formula_a_b ) > list_set_formula_a_b > list_set_formula_a_b > list_set_formula_a_b ).
thf(sy_c_List_Ofold_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
fold_s5097478484676794267la_a_b: ( set_formula_a_b > set_formula_a_b > set_formula_a_b ) > list_set_formula_a_b > set_formula_a_b > set_formula_a_b ).
thf(sy_c_List_Ofold_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_001t__Set__Oset_It__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J,type,
fold_s4424443237017773265la_a_b: ( set_formula_a_b > set_set_formula_a_b > set_set_formula_a_b ) > list_set_formula_a_b > set_set_formula_a_b > set_set_formula_a_b ).
thf(sy_c_List_Ofold_001t__Set__Oset_It__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J_001t__Set__Oset_It__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J,type,
fold_s5064541745726864391la_a_b: ( set_set_formula_a_b > set_set_formula_a_b > set_set_formula_a_b ) > list_s602864636976034957la_a_b > set_set_formula_a_b > set_set_formula_a_b ).
thf(sy_c_List_Ofoldr_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
foldr_139744914162652395la_a_b: ( list_formula_a_b > list_formula_a_b > list_formula_a_b ) > list_l5948067055324603441la_a_b > list_formula_a_b > list_formula_a_b ).
thf(sy_c_List_Ofoldr_001t__MDL__Oformula_Itf__a_Mtf__b_J_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
foldr_6622774383136399195la_a_b: ( formula_a_b > list_formula_a_b > list_formula_a_b ) > list_formula_a_b > list_formula_a_b > list_formula_a_b ).
thf(sy_c_List_Ofoldr_001t__MDL__Oformula_Itf__a_Mtf__b_J_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
foldr_2945827324002938625la_a_b: ( formula_a_b > set_formula_a_b > set_formula_a_b ) > list_formula_a_b > set_formula_a_b > set_formula_a_b ).
thf(sy_c_List_Ofoldr_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
foldr_5377326498870416695la_a_b: ( set_formula_a_b > set_formula_a_b > set_formula_a_b ) > list_set_formula_a_b > set_formula_a_b > set_formula_a_b ).
thf(sy_c_List_Oinsert_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
insert3252115831038319365la_a_b: list_formula_a_b > list_l5948067055324603441la_a_b > list_l5948067055324603441la_a_b ).
thf(sy_c_List_Oinsert_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
insert_formula_a_b: formula_a_b > list_formula_a_b > list_formula_a_b ).
thf(sy_c_List_Oinsert_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
insert6994445989384232875la_a_b: set_formula_a_b > list_set_formula_a_b > list_set_formula_a_b ).
thf(sy_c_List_Olast_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
last_l46059096318636098la_a_b: list_l5948067055324603441la_a_b > list_formula_a_b ).
thf(sy_c_List_Olast_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
last_formula_a_b: list_formula_a_b > formula_a_b ).
thf(sy_c_List_Olast_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
last_set_formula_a_b: list_set_formula_a_b > set_formula_a_b ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
cons_l2450299273141159147la_a_b: list_formula_a_b > list_l5948067055324603441la_a_b > list_l5948067055324603441la_a_b ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J,type,
cons_l7517568784372355617la_a_b: list_set_formula_a_b > list_l147921316930177639la_a_b > list_l147921316930177639la_a_b ).
thf(sy_c_List_Olist_OCons_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
cons_formula_a_b: formula_a_b > list_formula_a_b > list_formula_a_b ).
thf(sy_c_List_Olist_OCons_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
cons_set_formula_a_b: set_formula_a_b > list_set_formula_a_b > list_set_formula_a_b ).
thf(sy_c_List_Olist_OCons_001t__Set__Oset_It__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J,type,
cons_s2905031355775537607la_a_b: set_set_formula_a_b > list_s602864636976034957la_a_b > list_s602864636976034957la_a_b ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
nil_list_formula_a_b: list_l5948067055324603441la_a_b ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J,type,
nil_li5663101814674166225la_a_b: list_l147921316930177639la_a_b ).
thf(sy_c_List_Olist_ONil_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
nil_formula_a_b: list_formula_a_b ).
thf(sy_c_List_Olist_ONil_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
nil_set_formula_a_b: list_set_formula_a_b ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
map_li4767261230909057070la_a_b: ( list_formula_a_b > set_formula_a_b ) > list_l5948067055324603441la_a_b > list_set_formula_a_b ).
thf(sy_c_List_Olist_Omap_001t__MDL__Oformula_Itf__a_Mtf__b_J_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
map_fo2126546822197359848la_a_b: ( formula_a_b > formula_a_b ) > list_formula_a_b > list_formula_a_b ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J,type,
set_li405426049885584732la_a_b: list_l3581718849243781313la_a_b > set_li1233427854086133095la_a_b ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
set_list_formula_a_b2: list_l5948067055324603441la_a_b > set_list_formula_a_b ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J,type,
set_li566459811281642754la_a_b: list_l147921316930177639la_a_b > set_li5266388670752979469la_a_b ).
thf(sy_c_List_Olist_Oset_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
set_formula_a_b2: list_formula_a_b > set_formula_a_b ).
thf(sy_c_List_Olist_Oset_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
set_set_formula_a_b2: list_set_formula_a_b > set_set_formula_a_b ).
thf(sy_c_List_Olist_Oset_001t__Set__Oset_It__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J,type,
set_se5912223057565107496la_a_b: list_s602864636976034957la_a_b > set_se3797390812436207411la_a_b ).
thf(sy_c_List_Olist__ex1_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
list_e4732541319681528088la_a_b: ( list_formula_a_b > $o ) > list_l5948067055324603441la_a_b > $o ).
thf(sy_c_List_Olist__ex1_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
list_ex1_formula_a_b: ( formula_a_b > $o ) > list_formula_a_b > $o ).
thf(sy_c_List_Olist__ex1_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
list_e3834867817413275454la_a_b: ( set_formula_a_b > $o ) > list_set_formula_a_b > $o ).
thf(sy_c_List_Olists_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
lists_1922735192288930349la_a_b: set_list_formula_a_b > set_li1233427854086133095la_a_b ).
thf(sy_c_List_Olists_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
lists_formula_a_b: set_formula_a_b > set_list_formula_a_b ).
thf(sy_c_List_Olists_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
lists_4880915010259632851la_a_b: set_set_formula_a_b > set_li5266388670752979469la_a_b ).
thf(sy_c_List_Olistset_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
listset_formula_a_b: list_set_formula_a_b > set_list_formula_a_b ).
thf(sy_c_List_Omap__tailrec__rev_001t__MDL__Oformula_Itf__a_Mtf__b_J_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
map_ta6137025558560551193la_a_b: ( formula_a_b > formula_a_b ) > list_formula_a_b > list_formula_a_b > list_formula_a_b ).
thf(sy_c_List_Omap__tailrec__rev_001t__MDL__Oformula_Itf__a_Mtf__b_J_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
map_ta1969411894599079247la_a_b: ( formula_a_b > set_formula_a_b ) > list_formula_a_b > list_set_formula_a_b > list_set_formula_a_b ).
thf(sy_c_List_Omap__tailrec__rev_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
map_ta7515233025258682191la_a_b: ( set_formula_a_b > formula_a_b ) > list_set_formula_a_b > list_formula_a_b > list_formula_a_b ).
thf(sy_c_List_Omap__tailrec__rev_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
map_ta7939594666930029445la_a_b: ( set_formula_a_b > set_formula_a_b ) > list_set_formula_a_b > list_set_formula_a_b > list_set_formula_a_b ).
thf(sy_c_List_Omaps_001t__MDL__Oformula_Itf__a_Mtf__b_J_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
maps_f4007179076772103781la_a_b: ( formula_a_b > list_formula_a_b ) > list_formula_a_b > list_formula_a_b ).
thf(sy_c_List_Omaps_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
maps_s5908961544147508379la_a_b: ( set_formula_a_b > list_formula_a_b ) > list_set_formula_a_b > list_formula_a_b ).
thf(sy_c_List_Omember_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
member6172684649276339526la_a_b: list_l5948067055324603441la_a_b > list_formula_a_b > $o ).
thf(sy_c_List_Omember_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
member_formula_a_b: list_formula_a_b > formula_a_b > $o ).
thf(sy_c_List_Omember_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
member5721106492161739372la_a_b: list_set_formula_a_b > set_formula_a_b > $o ).
thf(sy_c_List_Onull_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
null_l1421712370273038483la_a_b: list_l5948067055324603441la_a_b > $o ).
thf(sy_c_List_Onull_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
null_formula_a_b: list_formula_a_b > $o ).
thf(sy_c_List_Onull_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
null_set_formula_a_b: list_set_formula_a_b > $o ).
thf(sy_c_List_Oproduct__lists_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
produc3748613542311242226la_a_b: list_l3581718849243781313la_a_b > list_l3581718849243781313la_a_b ).
thf(sy_c_List_Oproduct__lists_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
produc130815601992340706la_a_b: list_l5948067055324603441la_a_b > list_l5948067055324603441la_a_b ).
thf(sy_c_List_Oproduct__lists_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
produc7528538235918390552la_a_b: list_l147921316930177639la_a_b > list_l147921316930177639la_a_b ).
thf(sy_c_List_Oremdups_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
remdups_formula_a_b: list_formula_a_b > list_formula_a_b ).
thf(sy_c_List_Oremove1_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
remove4763865728959745157la_a_b: list_formula_a_b > list_l5948067055324603441la_a_b > list_l5948067055324603441la_a_b ).
thf(sy_c_List_Oremove1_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
remove1_formula_a_b: formula_a_b > list_formula_a_b > list_formula_a_b ).
thf(sy_c_List_Oremove1_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
remove3026683253030310187la_a_b: set_formula_a_b > list_set_formula_a_b > list_set_formula_a_b ).
thf(sy_c_List_OremoveAll_001t__List__Olist_It__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J,type,
remove2735483865023840037la_a_b: list_l5948067055324603441la_a_b > list_l3581718849243781313la_a_b > list_l3581718849243781313la_a_b ).
thf(sy_c_List_OremoveAll_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
remove9041333805894489621la_a_b: list_formula_a_b > list_l5948067055324603441la_a_b > list_l5948067055324603441la_a_b ).
thf(sy_c_List_OremoveAll_001t__List__Olist_It__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J,type,
remove7101406200717310795la_a_b: list_set_formula_a_b > list_l147921316930177639la_a_b > list_l147921316930177639la_a_b ).
thf(sy_c_List_OremoveAll_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
remove7803334386285014661la_a_b: formula_a_b > list_formula_a_b > list_formula_a_b ).
thf(sy_c_List_OremoveAll_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
remove6824600317533053627la_a_b: set_formula_a_b > list_set_formula_a_b > list_set_formula_a_b ).
thf(sy_c_List_Orev_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
rev_list_formula_a_b: list_l5948067055324603441la_a_b > list_l5948067055324603441la_a_b ).
thf(sy_c_List_Orev_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
rev_formula_a_b: list_formula_a_b > list_formula_a_b ).
thf(sy_c_List_Orev_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
rev_set_formula_a_b: list_set_formula_a_b > list_set_formula_a_b ).
thf(sy_c_List_Orotate1_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
rotate7908596367360581390la_a_b: list_l5948067055324603441la_a_b > list_l5948067055324603441la_a_b ).
thf(sy_c_List_Orotate1_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
rotate1_formula_a_b: list_formula_a_b > list_formula_a_b ).
thf(sy_c_List_Orotate1_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
rotate5784766846832941108la_a_b: list_set_formula_a_b > list_set_formula_a_b ).
thf(sy_c_List_Oshuffles_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
shuffl6594143666549603398la_a_b: list_l5948067055324603441la_a_b > list_l5948067055324603441la_a_b > set_li1233427854086133095la_a_b ).
thf(sy_c_List_Oshuffles_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
shuffles_formula_a_b: list_formula_a_b > list_formula_a_b > set_list_formula_a_b ).
thf(sy_c_List_Oshuffles_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
shuffl4965861759276814700la_a_b: list_set_formula_a_b > list_set_formula_a_b > set_li5266388670752979469la_a_b ).
thf(sy_c_List_Osorted__wrt_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
sorted287171416404617316la_a_b: ( formula_a_b > formula_a_b > $o ) > list_formula_a_b > $o ).
thf(sy_c_List_Osplice_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
splice_formula_a_b: list_formula_a_b > list_formula_a_b > list_formula_a_b ).
thf(sy_c_List_Osplice_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
splice5575443282787837998la_a_b: list_set_formula_a_b > list_set_formula_a_b > list_set_formula_a_b ).
thf(sy_c_List_Osubseqs_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
subseq7473640939082508812la_a_b: list_l5948067055324603441la_a_b > list_l3581718849243781313la_a_b ).
thf(sy_c_List_Osubseqs_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
subseqs_formula_a_b: list_formula_a_b > list_l5948067055324603441la_a_b ).
thf(sy_c_List_Osubseqs_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
subseq6158387682876310066la_a_b: list_set_formula_a_b > list_l147921316930177639la_a_b ).
thf(sy_c_List_OtakeWhile_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
takeWh2044909569185849186la_a_b: ( list_formula_a_b > $o ) > list_l5948067055324603441la_a_b > list_l5948067055324603441la_a_b ).
thf(sy_c_List_OtakeWhile_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
takeWh5313468693702967890la_a_b: ( formula_a_b > $o ) > list_formula_a_b > list_formula_a_b ).
thf(sy_c_List_OtakeWhile_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
takeWh1298768865834877576la_a_b: ( set_formula_a_b > $o ) > list_set_formula_a_b > list_set_formula_a_b ).
thf(sy_c_List_Ounion_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
union_5992773608777574823la_a_b: list_l5948067055324603441la_a_b > list_l5948067055324603441la_a_b > list_l5948067055324603441la_a_b ).
thf(sy_c_List_Ounion_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
union_formula_a_b: list_formula_a_b > list_formula_a_b > list_formula_a_b ).
thf(sy_c_List_Ounion_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
union_7611542805822840653la_a_b: list_set_formula_a_b > list_set_formula_a_b > list_set_formula_a_b ).
thf(sy_c_MDL_Oatms_001tf__a_001tf__b,type,
atms_a_b: regex_a_b > set_formula_a_b ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__MDL__Oformula_Itf__a_Mtf__b_J_M_Eo_J,type,
bot_bo2419332509030749162_a_b_o: formula_a_b > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_M_Eo_J,type,
bot_bo7837635004776446516_a_b_o: set_formula_a_b > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J,type,
bot_bo4614192416432564331la_a_b: set_list_formula_a_b ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_It__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J_J,type,
bot_bo162344608834447009la_a_b: set_li5266388670752979469la_a_b ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
bot_bo8568539672012708443la_a_b: set_formula_a_b ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J,type,
bot_bo5296870006564677521la_a_b: set_set_formula_a_b ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J_J,type,
bot_bo9046117466857082311la_a_b: set_se3797390812436207411la_a_b ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
ord_le6178842339710166055la_a_b: set_formula_a_b > set_formula_a_b > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J,type,
ord_le4913768313314656605la_a_b: set_set_formula_a_b > set_set_formula_a_b > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
top_to8846066467269685879la_a_b: set_formula_a_b ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
collec874399811774731638la_a_b: ( list_formula_a_b > $o ) > set_list_formula_a_b ).
thf(sy_c_Set_OCollect_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
collect_formula_a_b: ( formula_a_b > $o ) > set_formula_a_b ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
collec5361897267898172060la_a_b: ( set_formula_a_b > $o ) > set_set_formula_a_b ).
thf(sy_c_Set_OPow_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
pow_formula_a_b: set_formula_a_b > set_set_formula_a_b ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
image_2463093012343946057la_a_b: ( list_formula_a_b > list_formula_a_b ) > set_list_formula_a_b > set_list_formula_a_b ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
image_5057755198331993529la_a_b: ( list_formula_a_b > formula_a_b ) > set_list_formula_a_b > set_formula_a_b ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
image_4386591827427811823la_a_b: ( list_formula_a_b > set_formula_a_b ) > set_list_formula_a_b > set_set_formula_a_b ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J_001t__List__Olist_It__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J,type,
image_4866258802339552437la_a_b: ( list_set_formula_a_b > list_set_formula_a_b ) > set_li5266388670752979469la_a_b > set_li5266388670752979469la_a_b ).
thf(sy_c_Set_Oimage_001t__MDL__Oformula_Itf__a_Mtf__b_J_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
image_7669030115802156217la_a_b: ( formula_a_b > list_formula_a_b ) > set_formula_a_b > set_list_formula_a_b ).
thf(sy_c_Set_Oimage_001t__MDL__Oformula_Itf__a_Mtf__b_J_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
image_4658057545706456873la_a_b: ( formula_a_b > formula_a_b ) > set_formula_a_b > set_formula_a_b ).
thf(sy_c_Set_Oimage_001t__MDL__Oformula_Itf__a_Mtf__b_J_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
image_3063539048272112479la_a_b: ( formula_a_b > set_formula_a_b ) > set_formula_a_b > set_set_formula_a_b ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J_001t__Set__Oset_It__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J,type,
image_2898219059140093877la_a_b: ( set_list_formula_a_b > set_list_formula_a_b ) > set_se1704372200780349325la_a_b > set_se1704372200780349325la_a_b ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
image_4263524660313489391la_a_b: ( set_formula_a_b > list_formula_a_b ) > set_set_formula_a_b > set_list_formula_a_b ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
image_8609360178931715423la_a_b: ( set_formula_a_b > formula_a_b ) > set_set_formula_a_b > set_formula_a_b ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
image_2155112572986939797la_a_b: ( set_formula_a_b > set_formula_a_b ) > set_set_formula_a_b > set_set_formula_a_b ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J_001t__Set__Oset_It__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J,type,
image_1397695786084986369la_a_b: ( set_set_formula_a_b > set_set_formula_a_b ) > set_se3797390812436207411la_a_b > set_se3797390812436207411la_a_b ).
thf(sy_c_Set_Oinsert_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
insert4995524290302003025la_a_b: list_formula_a_b > set_list_formula_a_b > set_list_formula_a_b ).
thf(sy_c_Set_Oinsert_001t__List__Olist_It__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J,type,
insert4418010381955824775la_a_b: list_set_formula_a_b > set_li5266388670752979469la_a_b > set_li5266388670752979469la_a_b ).
thf(sy_c_Set_Oinsert_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
insert_formula_a_b2: formula_a_b > set_formula_a_b > set_formula_a_b ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
insert8952366156270154231la_a_b: set_formula_a_b > set_set_formula_a_b > set_set_formula_a_b ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J,type,
insert2824806981866770221la_a_b: set_set_formula_a_b > set_se3797390812436207411la_a_b > set_se3797390812436207411la_a_b ).
thf(sy_c_Set_Ois__empty_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
is_emp464069198922215119la_a_b: set_list_formula_a_b > $o ).
thf(sy_c_Set_Ois__empty_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
is_empty_formula_a_b: set_formula_a_b > $o ).
thf(sy_c_Set_Ois__empty_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
is_emp8895161027192797301la_a_b: set_set_formula_a_b > $o ).
thf(sy_c_Set_Ois__singleton_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
is_sin1449733419968980781la_a_b: set_list_formula_a_b > $o ).
thf(sy_c_Set_Ois__singleton_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
is_sin4967114517209836445la_a_b: set_formula_a_b > $o ).
thf(sy_c_Set_Ois__singleton_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
is_sin8536277385185689555la_a_b: set_set_formula_a_b > $o ).
thf(sy_c_Set_Oremove_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
remove2998734528700178044la_a_b: list_formula_a_b > set_list_formula_a_b > set_list_formula_a_b ).
thf(sy_c_Set_Oremove_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
remove_formula_a_b: formula_a_b > set_formula_a_b > set_formula_a_b ).
thf(sy_c_Set_Oremove_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
remove6203333876907372194la_a_b: set_formula_a_b > set_set_formula_a_b > set_set_formula_a_b ).
thf(sy_c_Set_Othe__elem_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
the_el407581278154503020la_a_b: set_list_formula_a_b > list_formula_a_b ).
thf(sy_c_Set_Othe__elem_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
the_elem_formula_a_b: set_formula_a_b > formula_a_b ).
thf(sy_c_Set_Othe__elem_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
the_el8292460259403742098la_a_b: set_set_formula_a_b > set_formula_a_b ).
thf(sy_c_Set_Ovimage_001t__MDL__Oformula_Itf__a_Mtf__b_J_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
vimage6809459385517368287la_a_b: ( formula_a_b > formula_a_b ) > set_formula_a_b > set_formula_a_b ).
thf(sy_c_Temporal_Ocollect__subfmlas_001tf__a_001tf__b,type,
collect_subfmlas_a_b: regex_a_b > list_formula_a_b > list_formula_a_b ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J,type,
member5770512961814848200la_a_b: list_l5948067055324603441la_a_b > set_li1233427854086133095la_a_b > $o ).
thf(sy_c_member_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
member7703986999711672120la_a_b: list_formula_a_b > set_list_formula_a_b > $o ).
thf(sy_c_member_001t__List__Olist_It__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J,type,
member289699325402759278la_a_b: list_set_formula_a_b > set_li5266388670752979469la_a_b > $o ).
thf(sy_c_member_001t__MDL__Oformula_Itf__a_Mtf__b_J,type,
member_formula_a_b2: formula_a_b > set_formula_a_b > $o ).
thf(sy_c_member_001t__Set__Oset_It__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J,type,
member7316870393999443438la_a_b: set_list_formula_a_b > set_se1704372200780349325la_a_b > $o ).
thf(sy_c_member_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J,type,
member6621951321062823262la_a_b: set_formula_a_b > set_set_formula_a_b > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J,type,
member4282342919390090900la_a_b: set_set_formula_a_b > set_se3797390812436207411la_a_b > $o ).
thf(sy_v_phis,type,
phis: list_formula_a_b ).
thf(sy_v_r,type,
r: regex_a_b ).
% Relevant facts (1278)
thf(fact_0_UnCI,axiom,
! [C: list_formula_a_b,B: set_list_formula_a_b,A: set_list_formula_a_b] :
( ( ~ ( member7703986999711672120la_a_b @ C @ B )
=> ( member7703986999711672120la_a_b @ C @ A ) )
=> ( member7703986999711672120la_a_b @ C @ ( sup_su9015223001214971651la_a_b @ A @ B ) ) ) ).
% UnCI
thf(fact_1_UnCI,axiom,
! [C: set_formula_a_b,B: set_set_formula_a_b,A: set_set_formula_a_b] :
( ( ~ ( member6621951321062823262la_a_b @ C @ B )
=> ( member6621951321062823262la_a_b @ C @ A ) )
=> ( member6621951321062823262la_a_b @ C @ ( sup_su5931044959554304553la_a_b @ A @ B ) ) ) ).
% UnCI
thf(fact_2_UnCI,axiom,
! [C: formula_a_b,B: set_formula_a_b,A: set_formula_a_b] :
( ( ~ ( member_formula_a_b2 @ C @ B )
=> ( member_formula_a_b2 @ C @ A ) )
=> ( member_formula_a_b2 @ C @ ( sup_su8832426789560438003la_a_b @ A @ B ) ) ) ).
% UnCI
thf(fact_3_Un__iff,axiom,
! [C: list_formula_a_b,A: set_list_formula_a_b,B: set_list_formula_a_b] :
( ( member7703986999711672120la_a_b @ C @ ( sup_su9015223001214971651la_a_b @ A @ B ) )
= ( ( member7703986999711672120la_a_b @ C @ A )
| ( member7703986999711672120la_a_b @ C @ B ) ) ) ).
% Un_iff
thf(fact_4_Un__iff,axiom,
! [C: set_formula_a_b,A: set_set_formula_a_b,B: set_set_formula_a_b] :
( ( member6621951321062823262la_a_b @ C @ ( sup_su5931044959554304553la_a_b @ A @ B ) )
= ( ( member6621951321062823262la_a_b @ C @ A )
| ( member6621951321062823262la_a_b @ C @ B ) ) ) ).
% Un_iff
thf(fact_5_Un__iff,axiom,
! [C: formula_a_b,A: set_formula_a_b,B: set_formula_a_b] :
( ( member_formula_a_b2 @ C @ ( sup_su8832426789560438003la_a_b @ A @ B ) )
= ( ( member_formula_a_b2 @ C @ A )
| ( member_formula_a_b2 @ C @ B ) ) ) ).
% Un_iff
thf(fact_6_sup_Oidem,axiom,
! [A2: set_set_formula_a_b] :
( ( sup_su5931044959554304553la_a_b @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_7_sup_Oidem,axiom,
! [A2: set_formula_a_b] :
( ( sup_su8832426789560438003la_a_b @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_8_sup__idem,axiom,
! [X: set_set_formula_a_b] :
( ( sup_su5931044959554304553la_a_b @ X @ X )
= X ) ).
% sup_idem
thf(fact_9_sup__idem,axiom,
! [X: set_formula_a_b] :
( ( sup_su8832426789560438003la_a_b @ X @ X )
= X ) ).
% sup_idem
thf(fact_10_sup_Oleft__idem,axiom,
! [A2: set_set_formula_a_b,B2: set_set_formula_a_b] :
( ( sup_su5931044959554304553la_a_b @ A2 @ ( sup_su5931044959554304553la_a_b @ A2 @ B2 ) )
= ( sup_su5931044959554304553la_a_b @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_11_sup_Oleft__idem,axiom,
! [A2: set_formula_a_b,B2: set_formula_a_b] :
( ( sup_su8832426789560438003la_a_b @ A2 @ ( sup_su8832426789560438003la_a_b @ A2 @ B2 ) )
= ( sup_su8832426789560438003la_a_b @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_12_sup__left__idem,axiom,
! [X: set_set_formula_a_b,Y: set_set_formula_a_b] :
( ( sup_su5931044959554304553la_a_b @ X @ ( sup_su5931044959554304553la_a_b @ X @ Y ) )
= ( sup_su5931044959554304553la_a_b @ X @ Y ) ) ).
% sup_left_idem
thf(fact_13_sup__left__idem,axiom,
! [X: set_formula_a_b,Y: set_formula_a_b] :
( ( sup_su8832426789560438003la_a_b @ X @ ( sup_su8832426789560438003la_a_b @ X @ Y ) )
= ( sup_su8832426789560438003la_a_b @ X @ Y ) ) ).
% sup_left_idem
thf(fact_14_sup_Oright__idem,axiom,
! [A2: set_set_formula_a_b,B2: set_set_formula_a_b] :
( ( sup_su5931044959554304553la_a_b @ ( sup_su5931044959554304553la_a_b @ A2 @ B2 ) @ B2 )
= ( sup_su5931044959554304553la_a_b @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_15_sup_Oright__idem,axiom,
! [A2: set_formula_a_b,B2: set_formula_a_b] :
( ( sup_su8832426789560438003la_a_b @ ( sup_su8832426789560438003la_a_b @ A2 @ B2 ) @ B2 )
= ( sup_su8832426789560438003la_a_b @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_16_set__union,axiom,
! [Xs: list_l5948067055324603441la_a_b,Ys: list_l5948067055324603441la_a_b] :
( ( set_list_formula_a_b2 @ ( union_5992773608777574823la_a_b @ Xs @ Ys ) )
= ( sup_su9015223001214971651la_a_b @ ( set_list_formula_a_b2 @ Xs ) @ ( set_list_formula_a_b2 @ Ys ) ) ) ).
% set_union
thf(fact_17_set__union,axiom,
! [Xs: list_set_formula_a_b,Ys: list_set_formula_a_b] :
( ( set_set_formula_a_b2 @ ( union_7611542805822840653la_a_b @ Xs @ Ys ) )
= ( sup_su5931044959554304553la_a_b @ ( set_set_formula_a_b2 @ Xs ) @ ( set_set_formula_a_b2 @ Ys ) ) ) ).
% set_union
thf(fact_18_set__union,axiom,
! [Xs: list_formula_a_b,Ys: list_formula_a_b] :
( ( set_formula_a_b2 @ ( union_formula_a_b @ Xs @ Ys ) )
= ( sup_su8832426789560438003la_a_b @ ( set_formula_a_b2 @ Xs ) @ ( set_formula_a_b2 @ Ys ) ) ) ).
% set_union
thf(fact_19_UnE,axiom,
! [C: list_formula_a_b,A: set_list_formula_a_b,B: set_list_formula_a_b] :
( ( member7703986999711672120la_a_b @ C @ ( sup_su9015223001214971651la_a_b @ A @ B ) )
=> ( ~ ( member7703986999711672120la_a_b @ C @ A )
=> ( member7703986999711672120la_a_b @ C @ B ) ) ) ).
% UnE
thf(fact_20_UnE,axiom,
! [C: set_formula_a_b,A: set_set_formula_a_b,B: set_set_formula_a_b] :
( ( member6621951321062823262la_a_b @ C @ ( sup_su5931044959554304553la_a_b @ A @ B ) )
=> ( ~ ( member6621951321062823262la_a_b @ C @ A )
=> ( member6621951321062823262la_a_b @ C @ B ) ) ) ).
% UnE
thf(fact_21_UnE,axiom,
! [C: formula_a_b,A: set_formula_a_b,B: set_formula_a_b] :
( ( member_formula_a_b2 @ C @ ( sup_su8832426789560438003la_a_b @ A @ B ) )
=> ( ~ ( member_formula_a_b2 @ C @ A )
=> ( member_formula_a_b2 @ C @ B ) ) ) ).
% UnE
thf(fact_22_UnI1,axiom,
! [C: list_formula_a_b,A: set_list_formula_a_b,B: set_list_formula_a_b] :
( ( member7703986999711672120la_a_b @ C @ A )
=> ( member7703986999711672120la_a_b @ C @ ( sup_su9015223001214971651la_a_b @ A @ B ) ) ) ).
% UnI1
thf(fact_23_UnI1,axiom,
! [C: set_formula_a_b,A: set_set_formula_a_b,B: set_set_formula_a_b] :
( ( member6621951321062823262la_a_b @ C @ A )
=> ( member6621951321062823262la_a_b @ C @ ( sup_su5931044959554304553la_a_b @ A @ B ) ) ) ).
% UnI1
thf(fact_24_UnI1,axiom,
! [C: formula_a_b,A: set_formula_a_b,B: set_formula_a_b] :
( ( member_formula_a_b2 @ C @ A )
=> ( member_formula_a_b2 @ C @ ( sup_su8832426789560438003la_a_b @ A @ B ) ) ) ).
% UnI1
thf(fact_25_UnI2,axiom,
! [C: list_formula_a_b,B: set_list_formula_a_b,A: set_list_formula_a_b] :
( ( member7703986999711672120la_a_b @ C @ B )
=> ( member7703986999711672120la_a_b @ C @ ( sup_su9015223001214971651la_a_b @ A @ B ) ) ) ).
% UnI2
thf(fact_26_UnI2,axiom,
! [C: set_formula_a_b,B: set_set_formula_a_b,A: set_set_formula_a_b] :
( ( member6621951321062823262la_a_b @ C @ B )
=> ( member6621951321062823262la_a_b @ C @ ( sup_su5931044959554304553la_a_b @ A @ B ) ) ) ).
% UnI2
thf(fact_27_UnI2,axiom,
! [C: formula_a_b,B: set_formula_a_b,A: set_formula_a_b] :
( ( member_formula_a_b2 @ C @ B )
=> ( member_formula_a_b2 @ C @ ( sup_su8832426789560438003la_a_b @ A @ B ) ) ) ).
% UnI2
thf(fact_28_sup__left__commute,axiom,
! [X: set_set_formula_a_b,Y: set_set_formula_a_b,Z: set_set_formula_a_b] :
( ( sup_su5931044959554304553la_a_b @ X @ ( sup_su5931044959554304553la_a_b @ Y @ Z ) )
= ( sup_su5931044959554304553la_a_b @ Y @ ( sup_su5931044959554304553la_a_b @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_29_sup__left__commute,axiom,
! [X: set_formula_a_b,Y: set_formula_a_b,Z: set_formula_a_b] :
( ( sup_su8832426789560438003la_a_b @ X @ ( sup_su8832426789560438003la_a_b @ Y @ Z ) )
= ( sup_su8832426789560438003la_a_b @ Y @ ( sup_su8832426789560438003la_a_b @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_30_sup_Oleft__commute,axiom,
! [B2: set_set_formula_a_b,A2: set_set_formula_a_b,C: set_set_formula_a_b] :
( ( sup_su5931044959554304553la_a_b @ B2 @ ( sup_su5931044959554304553la_a_b @ A2 @ C ) )
= ( sup_su5931044959554304553la_a_b @ A2 @ ( sup_su5931044959554304553la_a_b @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_31_sup_Oleft__commute,axiom,
! [B2: set_formula_a_b,A2: set_formula_a_b,C: set_formula_a_b] :
( ( sup_su8832426789560438003la_a_b @ B2 @ ( sup_su8832426789560438003la_a_b @ A2 @ C ) )
= ( sup_su8832426789560438003la_a_b @ A2 @ ( sup_su8832426789560438003la_a_b @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_32_sup__commute,axiom,
( sup_su5931044959554304553la_a_b
= ( ^ [X2: set_set_formula_a_b,Y2: set_set_formula_a_b] : ( sup_su5931044959554304553la_a_b @ Y2 @ X2 ) ) ) ).
% sup_commute
thf(fact_33_sup__commute,axiom,
( sup_su8832426789560438003la_a_b
= ( ^ [X2: set_formula_a_b,Y2: set_formula_a_b] : ( sup_su8832426789560438003la_a_b @ Y2 @ X2 ) ) ) ).
% sup_commute
thf(fact_34_sup_Ocommute,axiom,
( sup_su5931044959554304553la_a_b
= ( ^ [A3: set_set_formula_a_b,B3: set_set_formula_a_b] : ( sup_su5931044959554304553la_a_b @ B3 @ A3 ) ) ) ).
% sup.commute
thf(fact_35_sup_Ocommute,axiom,
( sup_su8832426789560438003la_a_b
= ( ^ [A3: set_formula_a_b,B3: set_formula_a_b] : ( sup_su8832426789560438003la_a_b @ B3 @ A3 ) ) ) ).
% sup.commute
thf(fact_36_sup__assoc,axiom,
! [X: set_set_formula_a_b,Y: set_set_formula_a_b,Z: set_set_formula_a_b] :
( ( sup_su5931044959554304553la_a_b @ ( sup_su5931044959554304553la_a_b @ X @ Y ) @ Z )
= ( sup_su5931044959554304553la_a_b @ X @ ( sup_su5931044959554304553la_a_b @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_37_sup__assoc,axiom,
! [X: set_formula_a_b,Y: set_formula_a_b,Z: set_formula_a_b] :
( ( sup_su8832426789560438003la_a_b @ ( sup_su8832426789560438003la_a_b @ X @ Y ) @ Z )
= ( sup_su8832426789560438003la_a_b @ X @ ( sup_su8832426789560438003la_a_b @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_38_sup_Oassoc,axiom,
! [A2: set_set_formula_a_b,B2: set_set_formula_a_b,C: set_set_formula_a_b] :
( ( sup_su5931044959554304553la_a_b @ ( sup_su5931044959554304553la_a_b @ A2 @ B2 ) @ C )
= ( sup_su5931044959554304553la_a_b @ A2 @ ( sup_su5931044959554304553la_a_b @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_39_sup_Oassoc,axiom,
! [A2: set_formula_a_b,B2: set_formula_a_b,C: set_formula_a_b] :
( ( sup_su8832426789560438003la_a_b @ ( sup_su8832426789560438003la_a_b @ A2 @ B2 ) @ C )
= ( sup_su8832426789560438003la_a_b @ A2 @ ( sup_su8832426789560438003la_a_b @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_40_inf__sup__aci_I5_J,axiom,
( sup_su5931044959554304553la_a_b
= ( ^ [X2: set_set_formula_a_b,Y2: set_set_formula_a_b] : ( sup_su5931044959554304553la_a_b @ Y2 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_41_inf__sup__aci_I5_J,axiom,
( sup_su8832426789560438003la_a_b
= ( ^ [X2: set_formula_a_b,Y2: set_formula_a_b] : ( sup_su8832426789560438003la_a_b @ Y2 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_42_inf__sup__aci_I6_J,axiom,
! [X: set_set_formula_a_b,Y: set_set_formula_a_b,Z: set_set_formula_a_b] :
( ( sup_su5931044959554304553la_a_b @ ( sup_su5931044959554304553la_a_b @ X @ Y ) @ Z )
= ( sup_su5931044959554304553la_a_b @ X @ ( sup_su5931044959554304553la_a_b @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_43_inf__sup__aci_I6_J,axiom,
! [X: set_formula_a_b,Y: set_formula_a_b,Z: set_formula_a_b] :
( ( sup_su8832426789560438003la_a_b @ ( sup_su8832426789560438003la_a_b @ X @ Y ) @ Z )
= ( sup_su8832426789560438003la_a_b @ X @ ( sup_su8832426789560438003la_a_b @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_44_inf__sup__aci_I7_J,axiom,
! [X: set_set_formula_a_b,Y: set_set_formula_a_b,Z: set_set_formula_a_b] :
( ( sup_su5931044959554304553la_a_b @ X @ ( sup_su5931044959554304553la_a_b @ Y @ Z ) )
= ( sup_su5931044959554304553la_a_b @ Y @ ( sup_su5931044959554304553la_a_b @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_45_inf__sup__aci_I7_J,axiom,
! [X: set_formula_a_b,Y: set_formula_a_b,Z: set_formula_a_b] :
( ( sup_su8832426789560438003la_a_b @ X @ ( sup_su8832426789560438003la_a_b @ Y @ Z ) )
= ( sup_su8832426789560438003la_a_b @ Y @ ( sup_su8832426789560438003la_a_b @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_46_inf__sup__aci_I8_J,axiom,
! [X: set_set_formula_a_b,Y: set_set_formula_a_b] :
( ( sup_su5931044959554304553la_a_b @ X @ ( sup_su5931044959554304553la_a_b @ X @ Y ) )
= ( sup_su5931044959554304553la_a_b @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_47_inf__sup__aci_I8_J,axiom,
! [X: set_formula_a_b,Y: set_formula_a_b] :
( ( sup_su8832426789560438003la_a_b @ X @ ( sup_su8832426789560438003la_a_b @ X @ Y ) )
= ( sup_su8832426789560438003la_a_b @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_48_Un__left__commute,axiom,
! [A: set_set_formula_a_b,B: set_set_formula_a_b,C2: set_set_formula_a_b] :
( ( sup_su5931044959554304553la_a_b @ A @ ( sup_su5931044959554304553la_a_b @ B @ C2 ) )
= ( sup_su5931044959554304553la_a_b @ B @ ( sup_su5931044959554304553la_a_b @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_49_Un__left__commute,axiom,
! [A: set_formula_a_b,B: set_formula_a_b,C2: set_formula_a_b] :
( ( sup_su8832426789560438003la_a_b @ A @ ( sup_su8832426789560438003la_a_b @ B @ C2 ) )
= ( sup_su8832426789560438003la_a_b @ B @ ( sup_su8832426789560438003la_a_b @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_50_Un__left__absorb,axiom,
! [A: set_set_formula_a_b,B: set_set_formula_a_b] :
( ( sup_su5931044959554304553la_a_b @ A @ ( sup_su5931044959554304553la_a_b @ A @ B ) )
= ( sup_su5931044959554304553la_a_b @ A @ B ) ) ).
% Un_left_absorb
thf(fact_51_Un__left__absorb,axiom,
! [A: set_formula_a_b,B: set_formula_a_b] :
( ( sup_su8832426789560438003la_a_b @ A @ ( sup_su8832426789560438003la_a_b @ A @ B ) )
= ( sup_su8832426789560438003la_a_b @ A @ B ) ) ).
% Un_left_absorb
thf(fact_52_Un__commute,axiom,
( sup_su5931044959554304553la_a_b
= ( ^ [A4: set_set_formula_a_b,B4: set_set_formula_a_b] : ( sup_su5931044959554304553la_a_b @ B4 @ A4 ) ) ) ).
% Un_commute
thf(fact_53_Un__commute,axiom,
( sup_su8832426789560438003la_a_b
= ( ^ [A4: set_formula_a_b,B4: set_formula_a_b] : ( sup_su8832426789560438003la_a_b @ B4 @ A4 ) ) ) ).
% Un_commute
thf(fact_54_Un__absorb,axiom,
! [A: set_set_formula_a_b] :
( ( sup_su5931044959554304553la_a_b @ A @ A )
= A ) ).
% Un_absorb
thf(fact_55_Un__absorb,axiom,
! [A: set_formula_a_b] :
( ( sup_su8832426789560438003la_a_b @ A @ A )
= A ) ).
% Un_absorb
thf(fact_56_Un__assoc,axiom,
! [A: set_set_formula_a_b,B: set_set_formula_a_b,C2: set_set_formula_a_b] :
( ( sup_su5931044959554304553la_a_b @ ( sup_su5931044959554304553la_a_b @ A @ B ) @ C2 )
= ( sup_su5931044959554304553la_a_b @ A @ ( sup_su5931044959554304553la_a_b @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_57_Un__assoc,axiom,
! [A: set_formula_a_b,B: set_formula_a_b,C2: set_formula_a_b] :
( ( sup_su8832426789560438003la_a_b @ ( sup_su8832426789560438003la_a_b @ A @ B ) @ C2 )
= ( sup_su8832426789560438003la_a_b @ A @ ( sup_su8832426789560438003la_a_b @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_58_ball__Un,axiom,
! [A: set_set_formula_a_b,B: set_set_formula_a_b,P: set_formula_a_b > $o] :
( ( ! [X2: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X2 @ ( sup_su5931044959554304553la_a_b @ A @ B ) )
=> ( P @ X2 ) ) )
= ( ! [X2: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X2 @ A )
=> ( P @ X2 ) )
& ! [X2: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X2 @ B )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_59_ball__Un,axiom,
! [A: set_formula_a_b,B: set_formula_a_b,P: formula_a_b > $o] :
( ( ! [X2: formula_a_b] :
( ( member_formula_a_b2 @ X2 @ ( sup_su8832426789560438003la_a_b @ A @ B ) )
=> ( P @ X2 ) ) )
= ( ! [X2: formula_a_b] :
( ( member_formula_a_b2 @ X2 @ A )
=> ( P @ X2 ) )
& ! [X2: formula_a_b] :
( ( member_formula_a_b2 @ X2 @ B )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_60_bex__Un,axiom,
! [A: set_set_formula_a_b,B: set_set_formula_a_b,P: set_formula_a_b > $o] :
( ( ? [X2: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X2 @ ( sup_su5931044959554304553la_a_b @ A @ B ) )
& ( P @ X2 ) ) )
= ( ? [X2: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X2 @ A )
& ( P @ X2 ) )
| ? [X2: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X2 @ B )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_61_bex__Un,axiom,
! [A: set_formula_a_b,B: set_formula_a_b,P: formula_a_b > $o] :
( ( ? [X2: formula_a_b] :
( ( member_formula_a_b2 @ X2 @ ( sup_su8832426789560438003la_a_b @ A @ B ) )
& ( P @ X2 ) ) )
= ( ? [X2: formula_a_b] :
( ( member_formula_a_b2 @ X2 @ A )
& ( P @ X2 ) )
| ? [X2: formula_a_b] :
( ( member_formula_a_b2 @ X2 @ B )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_62_boolean__algebra__cancel_Osup2,axiom,
! [B: set_set_formula_a_b,K: set_set_formula_a_b,B2: set_set_formula_a_b,A2: set_set_formula_a_b] :
( ( B
= ( sup_su5931044959554304553la_a_b @ K @ B2 ) )
=> ( ( sup_su5931044959554304553la_a_b @ A2 @ B )
= ( sup_su5931044959554304553la_a_b @ K @ ( sup_su5931044959554304553la_a_b @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_63_boolean__algebra__cancel_Osup2,axiom,
! [B: set_formula_a_b,K: set_formula_a_b,B2: set_formula_a_b,A2: set_formula_a_b] :
( ( B
= ( sup_su8832426789560438003la_a_b @ K @ B2 ) )
=> ( ( sup_su8832426789560438003la_a_b @ A2 @ B )
= ( sup_su8832426789560438003la_a_b @ K @ ( sup_su8832426789560438003la_a_b @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_64_boolean__algebra__cancel_Osup1,axiom,
! [A: set_set_formula_a_b,K: set_set_formula_a_b,A2: set_set_formula_a_b,B2: set_set_formula_a_b] :
( ( A
= ( sup_su5931044959554304553la_a_b @ K @ A2 ) )
=> ( ( sup_su5931044959554304553la_a_b @ A @ B2 )
= ( sup_su5931044959554304553la_a_b @ K @ ( sup_su5931044959554304553la_a_b @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_65_boolean__algebra__cancel_Osup1,axiom,
! [A: set_formula_a_b,K: set_formula_a_b,A2: set_formula_a_b,B2: set_formula_a_b] :
( ( A
= ( sup_su8832426789560438003la_a_b @ K @ A2 ) )
=> ( ( sup_su8832426789560438003la_a_b @ A @ B2 )
= ( sup_su8832426789560438003la_a_b @ K @ ( sup_su8832426789560438003la_a_b @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_66_in__set__member,axiom,
! [X: set_formula_a_b,Xs: list_set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X @ ( set_set_formula_a_b2 @ Xs ) )
= ( member5721106492161739372la_a_b @ Xs @ X ) ) ).
% in_set_member
thf(fact_67_in__set__member,axiom,
! [X: list_formula_a_b,Xs: list_l5948067055324603441la_a_b] :
( ( member7703986999711672120la_a_b @ X @ ( set_list_formula_a_b2 @ Xs ) )
= ( member6172684649276339526la_a_b @ Xs @ X ) ) ).
% in_set_member
thf(fact_68_in__set__member,axiom,
! [X: formula_a_b,Xs: list_formula_a_b] :
( ( member_formula_a_b2 @ X @ ( set_formula_a_b2 @ Xs ) )
= ( member_formula_a_b @ Xs @ X ) ) ).
% in_set_member
thf(fact_69_in__set__insert,axiom,
! [X: set_formula_a_b,Xs: list_set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X @ ( set_set_formula_a_b2 @ Xs ) )
=> ( ( insert6994445989384232875la_a_b @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_70_in__set__insert,axiom,
! [X: list_formula_a_b,Xs: list_l5948067055324603441la_a_b] :
( ( member7703986999711672120la_a_b @ X @ ( set_list_formula_a_b2 @ Xs ) )
=> ( ( insert3252115831038319365la_a_b @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_71_in__set__insert,axiom,
! [X: formula_a_b,Xs: list_formula_a_b] :
( ( member_formula_a_b2 @ X @ ( set_formula_a_b2 @ Xs ) )
=> ( ( insert_formula_a_b @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_72_list__ex1__iff,axiom,
( list_e3834867817413275454la_a_b
= ( ^ [P2: set_formula_a_b > $o,Xs2: list_set_formula_a_b] :
? [X2: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X2 @ ( set_set_formula_a_b2 @ Xs2 ) )
& ( P2 @ X2 )
& ! [Y2: set_formula_a_b] :
( ( ( member6621951321062823262la_a_b @ Y2 @ ( set_set_formula_a_b2 @ Xs2 ) )
& ( P2 @ Y2 ) )
=> ( Y2 = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_73_list__ex1__iff,axiom,
( list_e4732541319681528088la_a_b
= ( ^ [P2: list_formula_a_b > $o,Xs2: list_l5948067055324603441la_a_b] :
? [X2: list_formula_a_b] :
( ( member7703986999711672120la_a_b @ X2 @ ( set_list_formula_a_b2 @ Xs2 ) )
& ( P2 @ X2 )
& ! [Y2: list_formula_a_b] :
( ( ( member7703986999711672120la_a_b @ Y2 @ ( set_list_formula_a_b2 @ Xs2 ) )
& ( P2 @ Y2 ) )
=> ( Y2 = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_74_list__ex1__iff,axiom,
( list_ex1_formula_a_b
= ( ^ [P2: formula_a_b > $o,Xs2: list_formula_a_b] :
? [X2: formula_a_b] :
( ( member_formula_a_b2 @ X2 @ ( set_formula_a_b2 @ Xs2 ) )
& ( P2 @ X2 )
& ! [Y2: formula_a_b] :
( ( ( member_formula_a_b2 @ Y2 @ ( set_formula_a_b2 @ Xs2 ) )
& ( P2 @ Y2 ) )
=> ( Y2 = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_75_set__append,axiom,
! [Xs: list_l5948067055324603441la_a_b,Ys: list_l5948067055324603441la_a_b] :
( ( set_list_formula_a_b2 @ ( append58666856857280070la_a_b @ Xs @ Ys ) )
= ( sup_su9015223001214971651la_a_b @ ( set_list_formula_a_b2 @ Xs ) @ ( set_list_formula_a_b2 @ Ys ) ) ) ).
% set_append
thf(fact_76_set__append,axiom,
! [Xs: list_set_formula_a_b,Ys: list_set_formula_a_b] :
( ( set_set_formula_a_b2 @ ( append8577002312234529644la_a_b @ Xs @ Ys ) )
= ( sup_su5931044959554304553la_a_b @ ( set_set_formula_a_b2 @ Xs ) @ ( set_set_formula_a_b2 @ Ys ) ) ) ).
% set_append
thf(fact_77_set__append,axiom,
! [Xs: list_formula_a_b,Ys: list_formula_a_b] :
( ( set_formula_a_b2 @ ( append_formula_a_b @ Xs @ Ys ) )
= ( sup_su8832426789560438003la_a_b @ ( set_formula_a_b2 @ Xs ) @ ( set_formula_a_b2 @ Ys ) ) ) ).
% set_append
thf(fact_78_set__shuffles,axiom,
! [Zs: list_l5948067055324603441la_a_b,Xs: list_l5948067055324603441la_a_b,Ys: list_l5948067055324603441la_a_b] :
( ( member5770512961814848200la_a_b @ Zs @ ( shuffl6594143666549603398la_a_b @ Xs @ Ys ) )
=> ( ( set_list_formula_a_b2 @ Zs )
= ( sup_su9015223001214971651la_a_b @ ( set_list_formula_a_b2 @ Xs ) @ ( set_list_formula_a_b2 @ Ys ) ) ) ) ).
% set_shuffles
thf(fact_79_set__shuffles,axiom,
! [Zs: list_set_formula_a_b,Xs: list_set_formula_a_b,Ys: list_set_formula_a_b] :
( ( member289699325402759278la_a_b @ Zs @ ( shuffl4965861759276814700la_a_b @ Xs @ Ys ) )
=> ( ( set_set_formula_a_b2 @ Zs )
= ( sup_su5931044959554304553la_a_b @ ( set_set_formula_a_b2 @ Xs ) @ ( set_set_formula_a_b2 @ Ys ) ) ) ) ).
% set_shuffles
thf(fact_80_set__shuffles,axiom,
! [Zs: list_formula_a_b,Xs: list_formula_a_b,Ys: list_formula_a_b] :
( ( member7703986999711672120la_a_b @ Zs @ ( shuffles_formula_a_b @ Xs @ Ys ) )
=> ( ( set_formula_a_b2 @ Zs )
= ( sup_su8832426789560438003la_a_b @ ( set_formula_a_b2 @ Xs ) @ ( set_formula_a_b2 @ Ys ) ) ) ) ).
% set_shuffles
thf(fact_81_distinct__union,axiom,
! [Xs: list_l5948067055324603441la_a_b,Ys: list_l5948067055324603441la_a_b] :
( ( distin27490427618170344la_a_b @ ( union_5992773608777574823la_a_b @ Xs @ Ys ) )
= ( distin27490427618170344la_a_b @ Ys ) ) ).
% distinct_union
thf(fact_82_distinct__union,axiom,
! [Xs: list_set_formula_a_b,Ys: list_set_formula_a_b] :
( ( distin5059275051008138254la_a_b @ ( union_7611542805822840653la_a_b @ Xs @ Ys ) )
= ( distin5059275051008138254la_a_b @ Ys ) ) ).
% distinct_union
thf(fact_83_distinct__union,axiom,
! [Xs: list_formula_a_b,Ys: list_formula_a_b] :
( ( distinct_formula_a_b @ ( union_formula_a_b @ Xs @ Ys ) )
= ( distinct_formula_a_b @ Ys ) ) ).
% distinct_union
thf(fact_84_ListMem__iff,axiom,
( listMe3233215151264718677la_a_b
= ( ^ [X2: set_formula_a_b,Xs2: list_set_formula_a_b] : ( member6621951321062823262la_a_b @ X2 @ ( set_set_formula_a_b2 @ Xs2 ) ) ) ) ).
% ListMem_iff
thf(fact_85_ListMem__iff,axiom,
( listMe6139171625156847023la_a_b
= ( ^ [X2: list_formula_a_b,Xs2: list_l5948067055324603441la_a_b] : ( member7703986999711672120la_a_b @ X2 @ ( set_list_formula_a_b2 @ Xs2 ) ) ) ) ).
% ListMem_iff
thf(fact_86_ListMem__iff,axiom,
( listMem_formula_a_b
= ( ^ [X2: formula_a_b,Xs2: list_formula_a_b] : ( member_formula_a_b2 @ X2 @ ( set_formula_a_b2 @ Xs2 ) ) ) ) ).
% ListMem_iff
thf(fact_87_set__rotate1,axiom,
! [Xs: list_set_formula_a_b] :
( ( set_set_formula_a_b2 @ ( rotate5784766846832941108la_a_b @ Xs ) )
= ( set_set_formula_a_b2 @ Xs ) ) ).
% set_rotate1
thf(fact_88_set__rotate1,axiom,
! [Xs: list_l5948067055324603441la_a_b] :
( ( set_list_formula_a_b2 @ ( rotate7908596367360581390la_a_b @ Xs ) )
= ( set_list_formula_a_b2 @ Xs ) ) ).
% set_rotate1
thf(fact_89_set__rotate1,axiom,
! [Xs: list_formula_a_b] :
( ( set_formula_a_b2 @ ( rotate1_formula_a_b @ Xs ) )
= ( set_formula_a_b2 @ Xs ) ) ).
% set_rotate1
thf(fact_90_removeAll__id,axiom,
! [X: set_formula_a_b,Xs: list_set_formula_a_b] :
( ~ ( member6621951321062823262la_a_b @ X @ ( set_set_formula_a_b2 @ Xs ) )
=> ( ( remove6824600317533053627la_a_b @ X @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_91_removeAll__id,axiom,
! [X: list_formula_a_b,Xs: list_l5948067055324603441la_a_b] :
( ~ ( member7703986999711672120la_a_b @ X @ ( set_list_formula_a_b2 @ Xs ) )
=> ( ( remove9041333805894489621la_a_b @ X @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_92_removeAll__id,axiom,
! [X: formula_a_b,Xs: list_formula_a_b] :
( ~ ( member_formula_a_b2 @ X @ ( set_formula_a_b2 @ Xs ) )
=> ( ( remove7803334386285014661la_a_b @ X @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_93_append_Oassoc,axiom,
! [A2: list_formula_a_b,B2: list_formula_a_b,C: list_formula_a_b] :
( ( append_formula_a_b @ ( append_formula_a_b @ A2 @ B2 ) @ C )
= ( append_formula_a_b @ A2 @ ( append_formula_a_b @ B2 @ C ) ) ) ).
% append.assoc
thf(fact_94_append__assoc,axiom,
! [Xs: list_formula_a_b,Ys: list_formula_a_b,Zs: list_formula_a_b] :
( ( append_formula_a_b @ ( append_formula_a_b @ Xs @ Ys ) @ Zs )
= ( append_formula_a_b @ Xs @ ( append_formula_a_b @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_95_append__same__eq,axiom,
! [Ys: list_formula_a_b,Xs: list_formula_a_b,Zs: list_formula_a_b] :
( ( ( append_formula_a_b @ Ys @ Xs )
= ( append_formula_a_b @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_96_mem__Collect__eq,axiom,
! [A2: list_formula_a_b,P: list_formula_a_b > $o] :
( ( member7703986999711672120la_a_b @ A2 @ ( collec874399811774731638la_a_b @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_97_mem__Collect__eq,axiom,
! [A2: set_formula_a_b,P: set_formula_a_b > $o] :
( ( member6621951321062823262la_a_b @ A2 @ ( collec5361897267898172060la_a_b @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_98_mem__Collect__eq,axiom,
! [A2: formula_a_b,P: formula_a_b > $o] :
( ( member_formula_a_b2 @ A2 @ ( collect_formula_a_b @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_99_Collect__mem__eq,axiom,
! [A: set_list_formula_a_b] :
( ( collec874399811774731638la_a_b
@ ^ [X2: list_formula_a_b] : ( member7703986999711672120la_a_b @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_100_Collect__mem__eq,axiom,
! [A: set_set_formula_a_b] :
( ( collec5361897267898172060la_a_b
@ ^ [X2: set_formula_a_b] : ( member6621951321062823262la_a_b @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_101_Collect__mem__eq,axiom,
! [A: set_formula_a_b] :
( ( collect_formula_a_b
@ ^ [X2: formula_a_b] : ( member_formula_a_b2 @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_102_Collect__cong,axiom,
! [P: formula_a_b > $o,Q: formula_a_b > $o] :
( ! [X3: formula_a_b] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_formula_a_b @ P )
= ( collect_formula_a_b @ Q ) ) ) ).
% Collect_cong
thf(fact_103_same__append__eq,axiom,
! [Xs: list_formula_a_b,Ys: list_formula_a_b,Zs: list_formula_a_b] :
( ( ( append_formula_a_b @ Xs @ Ys )
= ( append_formula_a_b @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_104_removeAll__append,axiom,
! [X: formula_a_b,Xs: list_formula_a_b,Ys: list_formula_a_b] :
( ( remove7803334386285014661la_a_b @ X @ ( append_formula_a_b @ Xs @ Ys ) )
= ( append_formula_a_b @ ( remove7803334386285014661la_a_b @ X @ Xs ) @ ( remove7803334386285014661la_a_b @ X @ Ys ) ) ) ).
% removeAll_append
thf(fact_105_removeAll__append,axiom,
! [X: list_formula_a_b,Xs: list_l5948067055324603441la_a_b,Ys: list_l5948067055324603441la_a_b] :
( ( remove9041333805894489621la_a_b @ X @ ( append58666856857280070la_a_b @ Xs @ Ys ) )
= ( append58666856857280070la_a_b @ ( remove9041333805894489621la_a_b @ X @ Xs ) @ ( remove9041333805894489621la_a_b @ X @ Ys ) ) ) ).
% removeAll_append
thf(fact_106_distinct1__rotate,axiom,
! [Xs: list_formula_a_b] :
( ( distinct_formula_a_b @ ( rotate1_formula_a_b @ Xs ) )
= ( distinct_formula_a_b @ Xs ) ) ).
% distinct1_rotate
thf(fact_107_distinct1__rotate,axiom,
! [Xs: list_l5948067055324603441la_a_b] :
( ( distin27490427618170344la_a_b @ ( rotate7908596367360581390la_a_b @ Xs ) )
= ( distin27490427618170344la_a_b @ Xs ) ) ).
% distinct1_rotate
thf(fact_108_distinct1__rotate,axiom,
! [Xs: list_set_formula_a_b] :
( ( distin5059275051008138254la_a_b @ ( rotate5784766846832941108la_a_b @ Xs ) )
= ( distin5059275051008138254la_a_b @ Xs ) ) ).
% distinct1_rotate
thf(fact_109_distinct__insert,axiom,
! [X: formula_a_b,Xs: list_formula_a_b] :
( ( distinct_formula_a_b @ ( insert_formula_a_b @ X @ Xs ) )
= ( distinct_formula_a_b @ Xs ) ) ).
% distinct_insert
thf(fact_110_distinct__insert,axiom,
! [X: list_formula_a_b,Xs: list_l5948067055324603441la_a_b] :
( ( distin27490427618170344la_a_b @ ( insert3252115831038319365la_a_b @ X @ Xs ) )
= ( distin27490427618170344la_a_b @ Xs ) ) ).
% distinct_insert
thf(fact_111_distinct__insert,axiom,
! [X: set_formula_a_b,Xs: list_set_formula_a_b] :
( ( distin5059275051008138254la_a_b @ ( insert6994445989384232875la_a_b @ X @ Xs ) )
= ( distin5059275051008138254la_a_b @ Xs ) ) ).
% distinct_insert
thf(fact_112_append__eq__appendI,axiom,
! [Xs: list_formula_a_b,Xs1: list_formula_a_b,Zs: list_formula_a_b,Ys: list_formula_a_b,Us: list_formula_a_b] :
( ( ( append_formula_a_b @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_formula_a_b @ Xs1 @ Us ) )
=> ( ( append_formula_a_b @ Xs @ Ys )
= ( append_formula_a_b @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_113_shuffles__commutes,axiom,
( shuffles_formula_a_b
= ( ^ [Xs2: list_formula_a_b,Ys2: list_formula_a_b] : ( shuffles_formula_a_b @ Ys2 @ Xs2 ) ) ) ).
% shuffles_commutes
thf(fact_114_distinct__removeAll,axiom,
! [Xs: list_set_formula_a_b,X: set_formula_a_b] :
( ( distin5059275051008138254la_a_b @ Xs )
=> ( distin5059275051008138254la_a_b @ ( remove6824600317533053627la_a_b @ X @ Xs ) ) ) ).
% distinct_removeAll
thf(fact_115_distinct__removeAll,axiom,
! [Xs: list_formula_a_b,X: formula_a_b] :
( ( distinct_formula_a_b @ Xs )
=> ( distinct_formula_a_b @ ( remove7803334386285014661la_a_b @ X @ Xs ) ) ) ).
% distinct_removeAll
thf(fact_116_distinct__removeAll,axiom,
! [Xs: list_l5948067055324603441la_a_b,X: list_formula_a_b] :
( ( distin27490427618170344la_a_b @ Xs )
=> ( distin27490427618170344la_a_b @ ( remove9041333805894489621la_a_b @ X @ Xs ) ) ) ).
% distinct_removeAll
thf(fact_117_append__eq__append__conv2,axiom,
! [Xs: list_formula_a_b,Ys: list_formula_a_b,Zs: list_formula_a_b,Ts: list_formula_a_b] :
( ( ( append_formula_a_b @ Xs @ Ys )
= ( append_formula_a_b @ Zs @ Ts ) )
= ( ? [Us2: list_formula_a_b] :
( ( ( Xs
= ( append_formula_a_b @ Zs @ Us2 ) )
& ( ( append_formula_a_b @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_formula_a_b @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append_formula_a_b @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_118_can__select__set__list__ex1,axiom,
! [P: set_formula_a_b > $o,A: list_set_formula_a_b] :
( ( can_se3565154267531510026la_a_b @ P @ ( set_set_formula_a_b2 @ A ) )
= ( list_e3834867817413275454la_a_b @ P @ A ) ) ).
% can_select_set_list_ex1
thf(fact_119_can__select__set__list__ex1,axiom,
! [P: list_formula_a_b > $o,A: list_l5948067055324603441la_a_b] :
( ( can_se3903203210569301220la_a_b @ P @ ( set_list_formula_a_b2 @ A ) )
= ( list_e4732541319681528088la_a_b @ P @ A ) ) ).
% can_select_set_list_ex1
thf(fact_120_can__select__set__list__ex1,axiom,
! [P: formula_a_b > $o,A: list_formula_a_b] :
( ( can_se151102608688302292la_a_b @ P @ ( set_formula_a_b2 @ A ) )
= ( list_ex1_formula_a_b @ P @ A ) ) ).
% can_select_set_list_ex1
thf(fact_121_remove__code_I1_J,axiom,
! [X: set_formula_a_b,Xs: list_set_formula_a_b] :
( ( remove6203333876907372194la_a_b @ X @ ( set_set_formula_a_b2 @ Xs ) )
= ( set_set_formula_a_b2 @ ( remove6824600317533053627la_a_b @ X @ Xs ) ) ) ).
% remove_code(1)
thf(fact_122_remove__code_I1_J,axiom,
! [X: list_formula_a_b,Xs: list_l5948067055324603441la_a_b] :
( ( remove2998734528700178044la_a_b @ X @ ( set_list_formula_a_b2 @ Xs ) )
= ( set_list_formula_a_b2 @ ( remove9041333805894489621la_a_b @ X @ Xs ) ) ) ).
% remove_code(1)
thf(fact_123_remove__code_I1_J,axiom,
! [X: formula_a_b,Xs: list_formula_a_b] :
( ( remove_formula_a_b @ X @ ( set_formula_a_b2 @ Xs ) )
= ( set_formula_a_b2 @ ( remove7803334386285014661la_a_b @ X @ Xs ) ) ) ).
% remove_code(1)
thf(fact_124_List_Ounion__def,axiom,
( union_formula_a_b
= ( fold_f7464356806857031615la_a_b @ insert_formula_a_b ) ) ).
% List.union_def
thf(fact_125_bind__simps_I2_J,axiom,
! [X: formula_a_b,Xs: list_formula_a_b,F: formula_a_b > list_formula_a_b] :
( ( bind_f552093402373037483la_a_b @ ( cons_formula_a_b @ X @ Xs ) @ F )
= ( append_formula_a_b @ ( F @ X ) @ ( bind_f552093402373037483la_a_b @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_126_bind__simps_I2_J,axiom,
! [X: set_formula_a_b,Xs: list_set_formula_a_b,F: set_formula_a_b > list_formula_a_b] :
( ( bind_s2984606836244749281la_a_b @ ( cons_set_formula_a_b @ X @ Xs ) @ F )
= ( append_formula_a_b @ ( F @ X ) @ ( bind_s2984606836244749281la_a_b @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_127_not__distinct__conv__prefix,axiom,
! [As: list_l5948067055324603441la_a_b] :
( ( ~ ( distin27490427618170344la_a_b @ As ) )
= ( ? [Xs2: list_l5948067055324603441la_a_b,Y2: list_formula_a_b,Ys2: list_l5948067055324603441la_a_b] :
( ( member7703986999711672120la_a_b @ Y2 @ ( set_list_formula_a_b2 @ Xs2 ) )
& ( distin27490427618170344la_a_b @ Xs2 )
& ( As
= ( append58666856857280070la_a_b @ Xs2 @ ( cons_l2450299273141159147la_a_b @ Y2 @ Ys2 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_128_not__distinct__conv__prefix,axiom,
! [As: list_set_formula_a_b] :
( ( ~ ( distin5059275051008138254la_a_b @ As ) )
= ( ? [Xs2: list_set_formula_a_b,Y2: set_formula_a_b,Ys2: list_set_formula_a_b] :
( ( member6621951321062823262la_a_b @ Y2 @ ( set_set_formula_a_b2 @ Xs2 ) )
& ( distin5059275051008138254la_a_b @ Xs2 )
& ( As
= ( append8577002312234529644la_a_b @ Xs2 @ ( cons_set_formula_a_b @ Y2 @ Ys2 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_129_not__distinct__conv__prefix,axiom,
! [As: list_formula_a_b] :
( ( ~ ( distinct_formula_a_b @ As ) )
= ( ? [Xs2: list_formula_a_b,Y2: formula_a_b,Ys2: list_formula_a_b] :
( ( member_formula_a_b2 @ Y2 @ ( set_formula_a_b2 @ Xs2 ) )
& ( distinct_formula_a_b @ Xs2 )
& ( As
= ( append_formula_a_b @ Xs2 @ ( cons_formula_a_b @ Y2 @ Ys2 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_130_not__in__set__insert,axiom,
! [X: list_formula_a_b,Xs: list_l5948067055324603441la_a_b] :
( ~ ( member7703986999711672120la_a_b @ X @ ( set_list_formula_a_b2 @ Xs ) )
=> ( ( insert3252115831038319365la_a_b @ X @ Xs )
= ( cons_l2450299273141159147la_a_b @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_131_not__in__set__insert,axiom,
! [X: set_formula_a_b,Xs: list_set_formula_a_b] :
( ~ ( member6621951321062823262la_a_b @ X @ ( set_set_formula_a_b2 @ Xs ) )
=> ( ( insert6994445989384232875la_a_b @ X @ Xs )
= ( cons_set_formula_a_b @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_132_not__in__set__insert,axiom,
! [X: formula_a_b,Xs: list_formula_a_b] :
( ~ ( member_formula_a_b2 @ X @ ( set_formula_a_b2 @ Xs ) )
=> ( ( insert_formula_a_b @ X @ Xs )
= ( cons_formula_a_b @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_133_List_Oset__insert,axiom,
! [X: set_formula_a_b,Xs: list_set_formula_a_b] :
( ( set_set_formula_a_b2 @ ( insert6994445989384232875la_a_b @ X @ Xs ) )
= ( insert8952366156270154231la_a_b @ X @ ( set_set_formula_a_b2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_134_List_Oset__insert,axiom,
! [X: list_formula_a_b,Xs: list_l5948067055324603441la_a_b] :
( ( set_list_formula_a_b2 @ ( insert3252115831038319365la_a_b @ X @ Xs ) )
= ( insert4995524290302003025la_a_b @ X @ ( set_list_formula_a_b2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_135_List_Oset__insert,axiom,
! [X: formula_a_b,Xs: list_formula_a_b] :
( ( set_formula_a_b2 @ ( insert_formula_a_b @ X @ Xs ) )
= ( insert_formula_a_b2 @ X @ ( set_formula_a_b2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_136_distinct__remove1__removeAll,axiom,
! [Xs: list_set_formula_a_b,X: set_formula_a_b] :
( ( distin5059275051008138254la_a_b @ Xs )
=> ( ( remove3026683253030310187la_a_b @ X @ Xs )
= ( remove6824600317533053627la_a_b @ X @ Xs ) ) ) ).
% distinct_remove1_removeAll
thf(fact_137_distinct__remove1__removeAll,axiom,
! [Xs: list_formula_a_b,X: formula_a_b] :
( ( distinct_formula_a_b @ Xs )
=> ( ( remove1_formula_a_b @ X @ Xs )
= ( remove7803334386285014661la_a_b @ X @ Xs ) ) ) ).
% distinct_remove1_removeAll
thf(fact_138_distinct__remove1__removeAll,axiom,
! [Xs: list_l5948067055324603441la_a_b,X: list_formula_a_b] :
( ( distin27490427618170344la_a_b @ Xs )
=> ( ( remove4763865728959745157la_a_b @ X @ Xs )
= ( remove9041333805894489621la_a_b @ X @ Xs ) ) ) ).
% distinct_remove1_removeAll
thf(fact_139_splice__in__shuffles,axiom,
! [Xs: list_formula_a_b,Ys: list_formula_a_b] : ( member7703986999711672120la_a_b @ ( splice_formula_a_b @ Xs @ Ys ) @ ( shuffles_formula_a_b @ Xs @ Ys ) ) ).
% splice_in_shuffles
thf(fact_140_dropWhile__append2,axiom,
! [Xs: list_set_formula_a_b,P: set_formula_a_b > $o,Ys: list_set_formula_a_b] :
( ! [X3: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X3 @ ( set_set_formula_a_b2 @ Xs ) )
=> ( P @ X3 ) )
=> ( ( dropWh5803627183431294944la_a_b @ P @ ( append8577002312234529644la_a_b @ Xs @ Ys ) )
= ( dropWh5803627183431294944la_a_b @ P @ Ys ) ) ) ).
% dropWhile_append2
thf(fact_141_dropWhile__append2,axiom,
! [Xs: list_l5948067055324603441la_a_b,P: list_formula_a_b > $o,Ys: list_l5948067055324603441la_a_b] :
( ! [X3: list_formula_a_b] :
( ( member7703986999711672120la_a_b @ X3 @ ( set_list_formula_a_b2 @ Xs ) )
=> ( P @ X3 ) )
=> ( ( dropWh8739902238574629050la_a_b @ P @ ( append58666856857280070la_a_b @ Xs @ Ys ) )
= ( dropWh8739902238574629050la_a_b @ P @ Ys ) ) ) ).
% dropWhile_append2
thf(fact_142_dropWhile__append2,axiom,
! [Xs: list_formula_a_b,P: formula_a_b > $o,Ys: list_formula_a_b] :
( ! [X3: formula_a_b] :
( ( member_formula_a_b2 @ X3 @ ( set_formula_a_b2 @ Xs ) )
=> ( P @ X3 ) )
=> ( ( dropWh8768975241461415850la_a_b @ P @ ( append_formula_a_b @ Xs @ Ys ) )
= ( dropWh8768975241461415850la_a_b @ P @ Ys ) ) ) ).
% dropWhile_append2
thf(fact_143_dropWhile__append1,axiom,
! [X: set_formula_a_b,Xs: list_set_formula_a_b,P: set_formula_a_b > $o,Ys: list_set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X @ ( set_set_formula_a_b2 @ Xs ) )
=> ( ~ ( P @ X )
=> ( ( dropWh5803627183431294944la_a_b @ P @ ( append8577002312234529644la_a_b @ Xs @ Ys ) )
= ( append8577002312234529644la_a_b @ ( dropWh5803627183431294944la_a_b @ P @ Xs ) @ Ys ) ) ) ) ).
% dropWhile_append1
thf(fact_144_dropWhile__append1,axiom,
! [X: list_formula_a_b,Xs: list_l5948067055324603441la_a_b,P: list_formula_a_b > $o,Ys: list_l5948067055324603441la_a_b] :
( ( member7703986999711672120la_a_b @ X @ ( set_list_formula_a_b2 @ Xs ) )
=> ( ~ ( P @ X )
=> ( ( dropWh8739902238574629050la_a_b @ P @ ( append58666856857280070la_a_b @ Xs @ Ys ) )
= ( append58666856857280070la_a_b @ ( dropWh8739902238574629050la_a_b @ P @ Xs ) @ Ys ) ) ) ) ).
% dropWhile_append1
thf(fact_145_dropWhile__append1,axiom,
! [X: formula_a_b,Xs: list_formula_a_b,P: formula_a_b > $o,Ys: list_formula_a_b] :
( ( member_formula_a_b2 @ X @ ( set_formula_a_b2 @ Xs ) )
=> ( ~ ( P @ X )
=> ( ( dropWh8768975241461415850la_a_b @ P @ ( append_formula_a_b @ Xs @ Ys ) )
= ( append_formula_a_b @ ( dropWh8768975241461415850la_a_b @ P @ Xs ) @ Ys ) ) ) ) ).
% dropWhile_append1
thf(fact_146_list_Oinject,axiom,
! [X21: formula_a_b,X22: list_formula_a_b,Y21: formula_a_b,Y22: list_formula_a_b] :
( ( ( cons_formula_a_b @ X21 @ X22 )
= ( cons_formula_a_b @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_147_list_Oinject,axiom,
! [X21: set_formula_a_b,X22: list_set_formula_a_b,Y21: set_formula_a_b,Y22: list_set_formula_a_b] :
( ( ( cons_set_formula_a_b @ X21 @ X22 )
= ( cons_set_formula_a_b @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_148_insert__absorb2,axiom,
! [X: formula_a_b,A: set_formula_a_b] :
( ( insert_formula_a_b2 @ X @ ( insert_formula_a_b2 @ X @ A ) )
= ( insert_formula_a_b2 @ X @ A ) ) ).
% insert_absorb2
thf(fact_149_insert__absorb2,axiom,
! [X: set_formula_a_b,A: set_set_formula_a_b] :
( ( insert8952366156270154231la_a_b @ X @ ( insert8952366156270154231la_a_b @ X @ A ) )
= ( insert8952366156270154231la_a_b @ X @ A ) ) ).
% insert_absorb2
thf(fact_150_insert__iff,axiom,
! [A2: list_formula_a_b,B2: list_formula_a_b,A: set_list_formula_a_b] :
( ( member7703986999711672120la_a_b @ A2 @ ( insert4995524290302003025la_a_b @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member7703986999711672120la_a_b @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_151_insert__iff,axiom,
! [A2: set_formula_a_b,B2: set_formula_a_b,A: set_set_formula_a_b] :
( ( member6621951321062823262la_a_b @ A2 @ ( insert8952366156270154231la_a_b @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member6621951321062823262la_a_b @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_152_insert__iff,axiom,
! [A2: formula_a_b,B2: formula_a_b,A: set_formula_a_b] :
( ( member_formula_a_b2 @ A2 @ ( insert_formula_a_b2 @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_formula_a_b2 @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_153_insertCI,axiom,
! [A2: list_formula_a_b,B: set_list_formula_a_b,B2: list_formula_a_b] :
( ( ~ ( member7703986999711672120la_a_b @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member7703986999711672120la_a_b @ A2 @ ( insert4995524290302003025la_a_b @ B2 @ B ) ) ) ).
% insertCI
thf(fact_154_insertCI,axiom,
! [A2: set_formula_a_b,B: set_set_formula_a_b,B2: set_formula_a_b] :
( ( ~ ( member6621951321062823262la_a_b @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member6621951321062823262la_a_b @ A2 @ ( insert8952366156270154231la_a_b @ B2 @ B ) ) ) ).
% insertCI
thf(fact_155_insertCI,axiom,
! [A2: formula_a_b,B: set_formula_a_b,B2: formula_a_b] :
( ( ~ ( member_formula_a_b2 @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_formula_a_b2 @ A2 @ ( insert_formula_a_b2 @ B2 @ B ) ) ) ).
% insertCI
thf(fact_156_dropWhile__idem,axiom,
! [P: formula_a_b > $o,Xs: list_formula_a_b] :
( ( dropWh8768975241461415850la_a_b @ P @ ( dropWh8768975241461415850la_a_b @ P @ Xs ) )
= ( dropWh8768975241461415850la_a_b @ P @ Xs ) ) ).
% dropWhile_idem
thf(fact_157_member__remove,axiom,
! [X: list_formula_a_b,Y: list_formula_a_b,A: set_list_formula_a_b] :
( ( member7703986999711672120la_a_b @ X @ ( remove2998734528700178044la_a_b @ Y @ A ) )
= ( ( member7703986999711672120la_a_b @ X @ A )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_158_member__remove,axiom,
! [X: set_formula_a_b,Y: set_formula_a_b,A: set_set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X @ ( remove6203333876907372194la_a_b @ Y @ A ) )
= ( ( member6621951321062823262la_a_b @ X @ A )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_159_member__remove,axiom,
! [X: formula_a_b,Y: formula_a_b,A: set_formula_a_b] :
( ( member_formula_a_b2 @ X @ ( remove_formula_a_b @ Y @ A ) )
= ( ( member_formula_a_b2 @ X @ A )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_160_Un__insert__right,axiom,
! [A: set_set_formula_a_b,A2: set_formula_a_b,B: set_set_formula_a_b] :
( ( sup_su5931044959554304553la_a_b @ A @ ( insert8952366156270154231la_a_b @ A2 @ B ) )
= ( insert8952366156270154231la_a_b @ A2 @ ( sup_su5931044959554304553la_a_b @ A @ B ) ) ) ).
% Un_insert_right
thf(fact_161_Un__insert__right,axiom,
! [A: set_formula_a_b,A2: formula_a_b,B: set_formula_a_b] :
( ( sup_su8832426789560438003la_a_b @ A @ ( insert_formula_a_b2 @ A2 @ B ) )
= ( insert_formula_a_b2 @ A2 @ ( sup_su8832426789560438003la_a_b @ A @ B ) ) ) ).
% Un_insert_right
thf(fact_162_Un__insert__left,axiom,
! [A2: set_formula_a_b,B: set_set_formula_a_b,C2: set_set_formula_a_b] :
( ( sup_su5931044959554304553la_a_b @ ( insert8952366156270154231la_a_b @ A2 @ B ) @ C2 )
= ( insert8952366156270154231la_a_b @ A2 @ ( sup_su5931044959554304553la_a_b @ B @ C2 ) ) ) ).
% Un_insert_left
thf(fact_163_Un__insert__left,axiom,
! [A2: formula_a_b,B: set_formula_a_b,C2: set_formula_a_b] :
( ( sup_su8832426789560438003la_a_b @ ( insert_formula_a_b2 @ A2 @ B ) @ C2 )
= ( insert_formula_a_b2 @ A2 @ ( sup_su8832426789560438003la_a_b @ B @ C2 ) ) ) ).
% Un_insert_left
thf(fact_164_in__set__remove1,axiom,
! [A2: set_formula_a_b,B2: set_formula_a_b,Xs: list_set_formula_a_b] :
( ( A2 != B2 )
=> ( ( member6621951321062823262la_a_b @ A2 @ ( set_set_formula_a_b2 @ ( remove3026683253030310187la_a_b @ B2 @ Xs ) ) )
= ( member6621951321062823262la_a_b @ A2 @ ( set_set_formula_a_b2 @ Xs ) ) ) ) ).
% in_set_remove1
thf(fact_165_in__set__remove1,axiom,
! [A2: list_formula_a_b,B2: list_formula_a_b,Xs: list_l5948067055324603441la_a_b] :
( ( A2 != B2 )
=> ( ( member7703986999711672120la_a_b @ A2 @ ( set_list_formula_a_b2 @ ( remove4763865728959745157la_a_b @ B2 @ Xs ) ) )
= ( member7703986999711672120la_a_b @ A2 @ ( set_list_formula_a_b2 @ Xs ) ) ) ) ).
% in_set_remove1
thf(fact_166_in__set__remove1,axiom,
! [A2: formula_a_b,B2: formula_a_b,Xs: list_formula_a_b] :
( ( A2 != B2 )
=> ( ( member_formula_a_b2 @ A2 @ ( set_formula_a_b2 @ ( remove1_formula_a_b @ B2 @ Xs ) ) )
= ( member_formula_a_b2 @ A2 @ ( set_formula_a_b2 @ Xs ) ) ) ) ).
% in_set_remove1
thf(fact_167_list_Osimps_I15_J,axiom,
! [X21: list_formula_a_b,X22: list_l5948067055324603441la_a_b] :
( ( set_list_formula_a_b2 @ ( cons_l2450299273141159147la_a_b @ X21 @ X22 ) )
= ( insert4995524290302003025la_a_b @ X21 @ ( set_list_formula_a_b2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_168_list_Osimps_I15_J,axiom,
! [X21: set_formula_a_b,X22: list_set_formula_a_b] :
( ( set_set_formula_a_b2 @ ( cons_set_formula_a_b @ X21 @ X22 ) )
= ( insert8952366156270154231la_a_b @ X21 @ ( set_set_formula_a_b2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_169_list_Osimps_I15_J,axiom,
! [X21: formula_a_b,X22: list_formula_a_b] :
( ( set_formula_a_b2 @ ( cons_formula_a_b @ X21 @ X22 ) )
= ( insert_formula_a_b2 @ X21 @ ( set_formula_a_b2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_170_dropWhile__dropWhile1,axiom,
! [Q: formula_a_b > $o,P: formula_a_b > $o,Xs: list_formula_a_b] :
( ! [X3: formula_a_b] :
( ( Q @ X3 )
=> ( P @ X3 ) )
=> ( ( dropWh8768975241461415850la_a_b @ Q @ ( dropWh8768975241461415850la_a_b @ P @ Xs ) )
= ( dropWh8768975241461415850la_a_b @ P @ Xs ) ) ) ).
% dropWhile_dropWhile1
thf(fact_171_mk__disjoint__insert,axiom,
! [A2: list_formula_a_b,A: set_list_formula_a_b] :
( ( member7703986999711672120la_a_b @ A2 @ A )
=> ? [B5: set_list_formula_a_b] :
( ( A
= ( insert4995524290302003025la_a_b @ A2 @ B5 ) )
& ~ ( member7703986999711672120la_a_b @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_172_mk__disjoint__insert,axiom,
! [A2: set_formula_a_b,A: set_set_formula_a_b] :
( ( member6621951321062823262la_a_b @ A2 @ A )
=> ? [B5: set_set_formula_a_b] :
( ( A
= ( insert8952366156270154231la_a_b @ A2 @ B5 ) )
& ~ ( member6621951321062823262la_a_b @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_173_mk__disjoint__insert,axiom,
! [A2: formula_a_b,A: set_formula_a_b] :
( ( member_formula_a_b2 @ A2 @ A )
=> ? [B5: set_formula_a_b] :
( ( A
= ( insert_formula_a_b2 @ A2 @ B5 ) )
& ~ ( member_formula_a_b2 @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_174_remove1__commute,axiom,
! [X: formula_a_b,Y: formula_a_b,Zs: list_formula_a_b] :
( ( remove1_formula_a_b @ X @ ( remove1_formula_a_b @ Y @ Zs ) )
= ( remove1_formula_a_b @ Y @ ( remove1_formula_a_b @ X @ Zs ) ) ) ).
% remove1_commute
thf(fact_175_not__Cons__self2,axiom,
! [X: formula_a_b,Xs: list_formula_a_b] :
( ( cons_formula_a_b @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_176_not__Cons__self2,axiom,
! [X: set_formula_a_b,Xs: list_set_formula_a_b] :
( ( cons_set_formula_a_b @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_177_insert__commute,axiom,
! [X: formula_a_b,Y: formula_a_b,A: set_formula_a_b] :
( ( insert_formula_a_b2 @ X @ ( insert_formula_a_b2 @ Y @ A ) )
= ( insert_formula_a_b2 @ Y @ ( insert_formula_a_b2 @ X @ A ) ) ) ).
% insert_commute
thf(fact_178_insert__commute,axiom,
! [X: set_formula_a_b,Y: set_formula_a_b,A: set_set_formula_a_b] :
( ( insert8952366156270154231la_a_b @ X @ ( insert8952366156270154231la_a_b @ Y @ A ) )
= ( insert8952366156270154231la_a_b @ Y @ ( insert8952366156270154231la_a_b @ X @ A ) ) ) ).
% insert_commute
thf(fact_179_insert__eq__iff,axiom,
! [A2: list_formula_a_b,A: set_list_formula_a_b,B2: list_formula_a_b,B: set_list_formula_a_b] :
( ~ ( member7703986999711672120la_a_b @ A2 @ A )
=> ( ~ ( member7703986999711672120la_a_b @ B2 @ B )
=> ( ( ( insert4995524290302003025la_a_b @ A2 @ A )
= ( insert4995524290302003025la_a_b @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C3: set_list_formula_a_b] :
( ( A
= ( insert4995524290302003025la_a_b @ B2 @ C3 ) )
& ~ ( member7703986999711672120la_a_b @ B2 @ C3 )
& ( B
= ( insert4995524290302003025la_a_b @ A2 @ C3 ) )
& ~ ( member7703986999711672120la_a_b @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_180_insert__eq__iff,axiom,
! [A2: set_formula_a_b,A: set_set_formula_a_b,B2: set_formula_a_b,B: set_set_formula_a_b] :
( ~ ( member6621951321062823262la_a_b @ A2 @ A )
=> ( ~ ( member6621951321062823262la_a_b @ B2 @ B )
=> ( ( ( insert8952366156270154231la_a_b @ A2 @ A )
= ( insert8952366156270154231la_a_b @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C3: set_set_formula_a_b] :
( ( A
= ( insert8952366156270154231la_a_b @ B2 @ C3 ) )
& ~ ( member6621951321062823262la_a_b @ B2 @ C3 )
& ( B
= ( insert8952366156270154231la_a_b @ A2 @ C3 ) )
& ~ ( member6621951321062823262la_a_b @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_181_insert__eq__iff,axiom,
! [A2: formula_a_b,A: set_formula_a_b,B2: formula_a_b,B: set_formula_a_b] :
( ~ ( member_formula_a_b2 @ A2 @ A )
=> ( ~ ( member_formula_a_b2 @ B2 @ B )
=> ( ( ( insert_formula_a_b2 @ A2 @ A )
= ( insert_formula_a_b2 @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C3: set_formula_a_b] :
( ( A
= ( insert_formula_a_b2 @ B2 @ C3 ) )
& ~ ( member_formula_a_b2 @ B2 @ C3 )
& ( B
= ( insert_formula_a_b2 @ A2 @ C3 ) )
& ~ ( member_formula_a_b2 @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_182_insert__absorb,axiom,
! [A2: list_formula_a_b,A: set_list_formula_a_b] :
( ( member7703986999711672120la_a_b @ A2 @ A )
=> ( ( insert4995524290302003025la_a_b @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_183_insert__absorb,axiom,
! [A2: set_formula_a_b,A: set_set_formula_a_b] :
( ( member6621951321062823262la_a_b @ A2 @ A )
=> ( ( insert8952366156270154231la_a_b @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_184_insert__absorb,axiom,
! [A2: formula_a_b,A: set_formula_a_b] :
( ( member_formula_a_b2 @ A2 @ A )
=> ( ( insert_formula_a_b2 @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_185_insert__ident,axiom,
! [X: list_formula_a_b,A: set_list_formula_a_b,B: set_list_formula_a_b] :
( ~ ( member7703986999711672120la_a_b @ X @ A )
=> ( ~ ( member7703986999711672120la_a_b @ X @ B )
=> ( ( ( insert4995524290302003025la_a_b @ X @ A )
= ( insert4995524290302003025la_a_b @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_186_insert__ident,axiom,
! [X: set_formula_a_b,A: set_set_formula_a_b,B: set_set_formula_a_b] :
( ~ ( member6621951321062823262la_a_b @ X @ A )
=> ( ~ ( member6621951321062823262la_a_b @ X @ B )
=> ( ( ( insert8952366156270154231la_a_b @ X @ A )
= ( insert8952366156270154231la_a_b @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_187_insert__ident,axiom,
! [X: formula_a_b,A: set_formula_a_b,B: set_formula_a_b] :
( ~ ( member_formula_a_b2 @ X @ A )
=> ( ~ ( member_formula_a_b2 @ X @ B )
=> ( ( ( insert_formula_a_b2 @ X @ A )
= ( insert_formula_a_b2 @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_188_Set_Oset__insert,axiom,
! [X: list_formula_a_b,A: set_list_formula_a_b] :
( ( member7703986999711672120la_a_b @ X @ A )
=> ~ ! [B5: set_list_formula_a_b] :
( ( A
= ( insert4995524290302003025la_a_b @ X @ B5 ) )
=> ( member7703986999711672120la_a_b @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_189_Set_Oset__insert,axiom,
! [X: set_formula_a_b,A: set_set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X @ A )
=> ~ ! [B5: set_set_formula_a_b] :
( ( A
= ( insert8952366156270154231la_a_b @ X @ B5 ) )
=> ( member6621951321062823262la_a_b @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_190_Set_Oset__insert,axiom,
! [X: formula_a_b,A: set_formula_a_b] :
( ( member_formula_a_b2 @ X @ A )
=> ~ ! [B5: set_formula_a_b] :
( ( A
= ( insert_formula_a_b2 @ X @ B5 ) )
=> ( member_formula_a_b2 @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_191_insertI2,axiom,
! [A2: list_formula_a_b,B: set_list_formula_a_b,B2: list_formula_a_b] :
( ( member7703986999711672120la_a_b @ A2 @ B )
=> ( member7703986999711672120la_a_b @ A2 @ ( insert4995524290302003025la_a_b @ B2 @ B ) ) ) ).
% insertI2
thf(fact_192_insertI2,axiom,
! [A2: set_formula_a_b,B: set_set_formula_a_b,B2: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ A2 @ B )
=> ( member6621951321062823262la_a_b @ A2 @ ( insert8952366156270154231la_a_b @ B2 @ B ) ) ) ).
% insertI2
thf(fact_193_insertI2,axiom,
! [A2: formula_a_b,B: set_formula_a_b,B2: formula_a_b] :
( ( member_formula_a_b2 @ A2 @ B )
=> ( member_formula_a_b2 @ A2 @ ( insert_formula_a_b2 @ B2 @ B ) ) ) ).
% insertI2
thf(fact_194_insertI1,axiom,
! [A2: list_formula_a_b,B: set_list_formula_a_b] : ( member7703986999711672120la_a_b @ A2 @ ( insert4995524290302003025la_a_b @ A2 @ B ) ) ).
% insertI1
thf(fact_195_insertI1,axiom,
! [A2: set_formula_a_b,B: set_set_formula_a_b] : ( member6621951321062823262la_a_b @ A2 @ ( insert8952366156270154231la_a_b @ A2 @ B ) ) ).
% insertI1
thf(fact_196_insertI1,axiom,
! [A2: formula_a_b,B: set_formula_a_b] : ( member_formula_a_b2 @ A2 @ ( insert_formula_a_b2 @ A2 @ B ) ) ).
% insertI1
thf(fact_197_insertE,axiom,
! [A2: list_formula_a_b,B2: list_formula_a_b,A: set_list_formula_a_b] :
( ( member7703986999711672120la_a_b @ A2 @ ( insert4995524290302003025la_a_b @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member7703986999711672120la_a_b @ A2 @ A ) ) ) ).
% insertE
thf(fact_198_insertE,axiom,
! [A2: set_formula_a_b,B2: set_formula_a_b,A: set_set_formula_a_b] :
( ( member6621951321062823262la_a_b @ A2 @ ( insert8952366156270154231la_a_b @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member6621951321062823262la_a_b @ A2 @ A ) ) ) ).
% insertE
thf(fact_199_insertE,axiom,
! [A2: formula_a_b,B2: formula_a_b,A: set_formula_a_b] :
( ( member_formula_a_b2 @ A2 @ ( insert_formula_a_b2 @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_formula_a_b2 @ A2 @ A ) ) ) ).
% insertE
thf(fact_200_dropWhile_Osimps_I2_J,axiom,
! [P: formula_a_b > $o,X: formula_a_b,Xs: list_formula_a_b] :
( ( ( P @ X )
=> ( ( dropWh8768975241461415850la_a_b @ P @ ( cons_formula_a_b @ X @ Xs ) )
= ( dropWh8768975241461415850la_a_b @ P @ Xs ) ) )
& ( ~ ( P @ X )
=> ( ( dropWh8768975241461415850la_a_b @ P @ ( cons_formula_a_b @ X @ Xs ) )
= ( cons_formula_a_b @ X @ Xs ) ) ) ) ).
% dropWhile.simps(2)
thf(fact_201_dropWhile_Osimps_I2_J,axiom,
! [P: set_formula_a_b > $o,X: set_formula_a_b,Xs: list_set_formula_a_b] :
( ( ( P @ X )
=> ( ( dropWh5803627183431294944la_a_b @ P @ ( cons_set_formula_a_b @ X @ Xs ) )
= ( dropWh5803627183431294944la_a_b @ P @ Xs ) ) )
& ( ~ ( P @ X )
=> ( ( dropWh5803627183431294944la_a_b @ P @ ( cons_set_formula_a_b @ X @ Xs ) )
= ( cons_set_formula_a_b @ X @ Xs ) ) ) ) ).
% dropWhile.simps(2)
thf(fact_202_remove1_Osimps_I2_J,axiom,
! [X: formula_a_b,Y: formula_a_b,Xs: list_formula_a_b] :
( ( ( X = Y )
=> ( ( remove1_formula_a_b @ X @ ( cons_formula_a_b @ Y @ Xs ) )
= Xs ) )
& ( ( X != Y )
=> ( ( remove1_formula_a_b @ X @ ( cons_formula_a_b @ Y @ Xs ) )
= ( cons_formula_a_b @ Y @ ( remove1_formula_a_b @ X @ Xs ) ) ) ) ) ).
% remove1.simps(2)
thf(fact_203_remove1_Osimps_I2_J,axiom,
! [X: set_formula_a_b,Y: set_formula_a_b,Xs: list_set_formula_a_b] :
( ( ( X = Y )
=> ( ( remove3026683253030310187la_a_b @ X @ ( cons_set_formula_a_b @ Y @ Xs ) )
= Xs ) )
& ( ( X != Y )
=> ( ( remove3026683253030310187la_a_b @ X @ ( cons_set_formula_a_b @ Y @ Xs ) )
= ( cons_set_formula_a_b @ Y @ ( remove3026683253030310187la_a_b @ X @ Xs ) ) ) ) ) ).
% remove1.simps(2)
thf(fact_204_splice_Osimps_I2_J,axiom,
! [X: formula_a_b,Xs: list_formula_a_b,Ys: list_formula_a_b] :
( ( splice_formula_a_b @ ( cons_formula_a_b @ X @ Xs ) @ Ys )
= ( cons_formula_a_b @ X @ ( splice_formula_a_b @ Ys @ Xs ) ) ) ).
% splice.simps(2)
thf(fact_205_splice_Osimps_I2_J,axiom,
! [X: set_formula_a_b,Xs: list_set_formula_a_b,Ys: list_set_formula_a_b] :
( ( splice5575443282787837998la_a_b @ ( cons_set_formula_a_b @ X @ Xs ) @ Ys )
= ( cons_set_formula_a_b @ X @ ( splice5575443282787837998la_a_b @ Ys @ Xs ) ) ) ).
% splice.simps(2)
thf(fact_206_fold__simps_I2_J,axiom,
! [F: formula_a_b > list_formula_a_b > list_formula_a_b,X: formula_a_b,Xs: list_formula_a_b,S: list_formula_a_b] :
( ( fold_f7464356806857031615la_a_b @ F @ ( cons_formula_a_b @ X @ Xs ) @ S )
= ( fold_f7464356806857031615la_a_b @ F @ Xs @ ( F @ X @ S ) ) ) ).
% fold_simps(2)
thf(fact_207_fold__simps_I2_J,axiom,
! [F: formula_a_b > set_formula_a_b > set_formula_a_b,X: formula_a_b,Xs: list_formula_a_b,S: set_formula_a_b] :
( ( fold_f470054086739575141la_a_b @ F @ ( cons_formula_a_b @ X @ Xs ) @ S )
= ( fold_f470054086739575141la_a_b @ F @ Xs @ ( F @ X @ S ) ) ) ).
% fold_simps(2)
thf(fact_208_fold__simps_I2_J,axiom,
! [F: set_formula_a_b > set_formula_a_b > set_formula_a_b,X: set_formula_a_b,Xs: list_set_formula_a_b,S: set_formula_a_b] :
( ( fold_s5097478484676794267la_a_b @ F @ ( cons_set_formula_a_b @ X @ Xs ) @ S )
= ( fold_s5097478484676794267la_a_b @ F @ Xs @ ( F @ X @ S ) ) ) ).
% fold_simps(2)
thf(fact_209_dropWhile__append3,axiom,
! [P: formula_a_b > $o,Y: formula_a_b,Xs: list_formula_a_b,Ys: list_formula_a_b] :
( ~ ( P @ Y )
=> ( ( dropWh8768975241461415850la_a_b @ P @ ( append_formula_a_b @ Xs @ ( cons_formula_a_b @ Y @ Ys ) ) )
= ( append_formula_a_b @ ( dropWh8768975241461415850la_a_b @ P @ Xs ) @ ( cons_formula_a_b @ Y @ Ys ) ) ) ) ).
% dropWhile_append3
thf(fact_210_dropWhile__append3,axiom,
! [P: set_formula_a_b > $o,Y: set_formula_a_b,Xs: list_set_formula_a_b,Ys: list_set_formula_a_b] :
( ~ ( P @ Y )
=> ( ( dropWh5803627183431294944la_a_b @ P @ ( append8577002312234529644la_a_b @ Xs @ ( cons_set_formula_a_b @ Y @ Ys ) ) )
= ( append8577002312234529644la_a_b @ ( dropWh5803627183431294944la_a_b @ P @ Xs ) @ ( cons_set_formula_a_b @ Y @ Ys ) ) ) ) ).
% dropWhile_append3
thf(fact_211_remove1__split,axiom,
! [A2: list_formula_a_b,Xs: list_l5948067055324603441la_a_b,Ys: list_l5948067055324603441la_a_b] :
( ( member7703986999711672120la_a_b @ A2 @ ( set_list_formula_a_b2 @ Xs ) )
=> ( ( ( remove4763865728959745157la_a_b @ A2 @ Xs )
= Ys )
= ( ? [Ls: list_l5948067055324603441la_a_b,Rs: list_l5948067055324603441la_a_b] :
( ( Xs
= ( append58666856857280070la_a_b @ Ls @ ( cons_l2450299273141159147la_a_b @ A2 @ Rs ) ) )
& ~ ( member7703986999711672120la_a_b @ A2 @ ( set_list_formula_a_b2 @ Ls ) )
& ( Ys
= ( append58666856857280070la_a_b @ Ls @ Rs ) ) ) ) ) ) ).
% remove1_split
thf(fact_212_remove1__split,axiom,
! [A2: set_formula_a_b,Xs: list_set_formula_a_b,Ys: list_set_formula_a_b] :
( ( member6621951321062823262la_a_b @ A2 @ ( set_set_formula_a_b2 @ Xs ) )
=> ( ( ( remove3026683253030310187la_a_b @ A2 @ Xs )
= Ys )
= ( ? [Ls: list_set_formula_a_b,Rs: list_set_formula_a_b] :
( ( Xs
= ( append8577002312234529644la_a_b @ Ls @ ( cons_set_formula_a_b @ A2 @ Rs ) ) )
& ~ ( member6621951321062823262la_a_b @ A2 @ ( set_set_formula_a_b2 @ Ls ) )
& ( Ys
= ( append8577002312234529644la_a_b @ Ls @ Rs ) ) ) ) ) ) ).
% remove1_split
thf(fact_213_remove1__split,axiom,
! [A2: formula_a_b,Xs: list_formula_a_b,Ys: list_formula_a_b] :
( ( member_formula_a_b2 @ A2 @ ( set_formula_a_b2 @ Xs ) )
=> ( ( ( remove1_formula_a_b @ A2 @ Xs )
= Ys )
= ( ? [Ls: list_formula_a_b,Rs: list_formula_a_b] :
( ( Xs
= ( append_formula_a_b @ Ls @ ( cons_formula_a_b @ A2 @ Rs ) ) )
& ~ ( member_formula_a_b2 @ A2 @ ( set_formula_a_b2 @ Ls ) )
& ( Ys
= ( append_formula_a_b @ Ls @ Rs ) ) ) ) ) ) ).
% remove1_split
thf(fact_214_set__ConsD,axiom,
! [Y: list_formula_a_b,X: list_formula_a_b,Xs: list_l5948067055324603441la_a_b] :
( ( member7703986999711672120la_a_b @ Y @ ( set_list_formula_a_b2 @ ( cons_l2450299273141159147la_a_b @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member7703986999711672120la_a_b @ Y @ ( set_list_formula_a_b2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_215_set__ConsD,axiom,
! [Y: set_formula_a_b,X: set_formula_a_b,Xs: list_set_formula_a_b] :
( ( member6621951321062823262la_a_b @ Y @ ( set_set_formula_a_b2 @ ( cons_set_formula_a_b @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member6621951321062823262la_a_b @ Y @ ( set_set_formula_a_b2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_216_set__ConsD,axiom,
! [Y: formula_a_b,X: formula_a_b,Xs: list_formula_a_b] :
( ( member_formula_a_b2 @ Y @ ( set_formula_a_b2 @ ( cons_formula_a_b @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_formula_a_b2 @ Y @ ( set_formula_a_b2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_217_list_Oset__cases,axiom,
! [E: list_formula_a_b,A2: list_l5948067055324603441la_a_b] :
( ( member7703986999711672120la_a_b @ E @ ( set_list_formula_a_b2 @ A2 ) )
=> ( ! [Z2: list_l5948067055324603441la_a_b] :
( A2
!= ( cons_l2450299273141159147la_a_b @ E @ Z2 ) )
=> ~ ! [Z1: list_formula_a_b,Z2: list_l5948067055324603441la_a_b] :
( ( A2
= ( cons_l2450299273141159147la_a_b @ Z1 @ Z2 ) )
=> ~ ( member7703986999711672120la_a_b @ E @ ( set_list_formula_a_b2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_218_list_Oset__cases,axiom,
! [E: set_formula_a_b,A2: list_set_formula_a_b] :
( ( member6621951321062823262la_a_b @ E @ ( set_set_formula_a_b2 @ A2 ) )
=> ( ! [Z2: list_set_formula_a_b] :
( A2
!= ( cons_set_formula_a_b @ E @ Z2 ) )
=> ~ ! [Z1: set_formula_a_b,Z2: list_set_formula_a_b] :
( ( A2
= ( cons_set_formula_a_b @ Z1 @ Z2 ) )
=> ~ ( member6621951321062823262la_a_b @ E @ ( set_set_formula_a_b2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_219_list_Oset__cases,axiom,
! [E: formula_a_b,A2: list_formula_a_b] :
( ( member_formula_a_b2 @ E @ ( set_formula_a_b2 @ A2 ) )
=> ( ! [Z2: list_formula_a_b] :
( A2
!= ( cons_formula_a_b @ E @ Z2 ) )
=> ~ ! [Z1: formula_a_b,Z2: list_formula_a_b] :
( ( A2
= ( cons_formula_a_b @ Z1 @ Z2 ) )
=> ~ ( member_formula_a_b2 @ E @ ( set_formula_a_b2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_220_list_Oset__intros_I1_J,axiom,
! [X21: list_formula_a_b,X22: list_l5948067055324603441la_a_b] : ( member7703986999711672120la_a_b @ X21 @ ( set_list_formula_a_b2 @ ( cons_l2450299273141159147la_a_b @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_221_list_Oset__intros_I1_J,axiom,
! [X21: set_formula_a_b,X22: list_set_formula_a_b] : ( member6621951321062823262la_a_b @ X21 @ ( set_set_formula_a_b2 @ ( cons_set_formula_a_b @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_222_list_Oset__intros_I1_J,axiom,
! [X21: formula_a_b,X22: list_formula_a_b] : ( member_formula_a_b2 @ X21 @ ( set_formula_a_b2 @ ( cons_formula_a_b @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_223_list_Oset__intros_I2_J,axiom,
! [Y: list_formula_a_b,X22: list_l5948067055324603441la_a_b,X21: list_formula_a_b] :
( ( member7703986999711672120la_a_b @ Y @ ( set_list_formula_a_b2 @ X22 ) )
=> ( member7703986999711672120la_a_b @ Y @ ( set_list_formula_a_b2 @ ( cons_l2450299273141159147la_a_b @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_224_list_Oset__intros_I2_J,axiom,
! [Y: set_formula_a_b,X22: list_set_formula_a_b,X21: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ Y @ ( set_set_formula_a_b2 @ X22 ) )
=> ( member6621951321062823262la_a_b @ Y @ ( set_set_formula_a_b2 @ ( cons_set_formula_a_b @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_225_list_Oset__intros_I2_J,axiom,
! [Y: formula_a_b,X22: list_formula_a_b,X21: formula_a_b] :
( ( member_formula_a_b2 @ Y @ ( set_formula_a_b2 @ X22 ) )
=> ( member_formula_a_b2 @ Y @ ( set_formula_a_b2 @ ( cons_formula_a_b @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_226_can__select__def,axiom,
( can_se3903203210569301220la_a_b
= ( ^ [P2: list_formula_a_b > $o,A4: set_list_formula_a_b] :
? [X2: list_formula_a_b] :
( ( member7703986999711672120la_a_b @ X2 @ A4 )
& ( P2 @ X2 )
& ! [Y2: list_formula_a_b] :
( ( ( member7703986999711672120la_a_b @ Y2 @ A4 )
& ( P2 @ Y2 ) )
=> ( Y2 = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_227_can__select__def,axiom,
( can_se3565154267531510026la_a_b
= ( ^ [P2: set_formula_a_b > $o,A4: set_set_formula_a_b] :
? [X2: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X2 @ A4 )
& ( P2 @ X2 )
& ! [Y2: set_formula_a_b] :
( ( ( member6621951321062823262la_a_b @ Y2 @ A4 )
& ( P2 @ Y2 ) )
=> ( Y2 = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_228_can__select__def,axiom,
( can_se151102608688302292la_a_b
= ( ^ [P2: formula_a_b > $o,A4: set_formula_a_b] :
? [X2: formula_a_b] :
( ( member_formula_a_b2 @ X2 @ A4 )
& ( P2 @ X2 )
& ! [Y2: formula_a_b] :
( ( ( member_formula_a_b2 @ Y2 @ A4 )
& ( P2 @ Y2 ) )
=> ( Y2 = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_229_set__dropWhileD,axiom,
! [X: set_formula_a_b,P: set_formula_a_b > $o,Xs: list_set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X @ ( set_set_formula_a_b2 @ ( dropWh5803627183431294944la_a_b @ P @ Xs ) ) )
=> ( member6621951321062823262la_a_b @ X @ ( set_set_formula_a_b2 @ Xs ) ) ) ).
% set_dropWhileD
thf(fact_230_set__dropWhileD,axiom,
! [X: list_formula_a_b,P: list_formula_a_b > $o,Xs: list_l5948067055324603441la_a_b] :
( ( member7703986999711672120la_a_b @ X @ ( set_list_formula_a_b2 @ ( dropWh8739902238574629050la_a_b @ P @ Xs ) ) )
=> ( member7703986999711672120la_a_b @ X @ ( set_list_formula_a_b2 @ Xs ) ) ) ).
% set_dropWhileD
thf(fact_231_set__dropWhileD,axiom,
! [X: formula_a_b,P: formula_a_b > $o,Xs: list_formula_a_b] :
( ( member_formula_a_b2 @ X @ ( set_formula_a_b2 @ ( dropWh8768975241461415850la_a_b @ P @ Xs ) ) )
=> ( member_formula_a_b2 @ X @ ( set_formula_a_b2 @ Xs ) ) ) ).
% set_dropWhileD
thf(fact_232_dropWhile__cong,axiom,
! [L: list_set_formula_a_b,K: list_set_formula_a_b,P: set_formula_a_b > $o,Q: set_formula_a_b > $o] :
( ( L = K )
=> ( ! [X3: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X3 @ ( set_set_formula_a_b2 @ L ) )
=> ( ( P @ X3 )
= ( Q @ X3 ) ) )
=> ( ( dropWh5803627183431294944la_a_b @ P @ L )
= ( dropWh5803627183431294944la_a_b @ Q @ K ) ) ) ) ).
% dropWhile_cong
thf(fact_233_dropWhile__cong,axiom,
! [L: list_l5948067055324603441la_a_b,K: list_l5948067055324603441la_a_b,P: list_formula_a_b > $o,Q: list_formula_a_b > $o] :
( ( L = K )
=> ( ! [X3: list_formula_a_b] :
( ( member7703986999711672120la_a_b @ X3 @ ( set_list_formula_a_b2 @ L ) )
=> ( ( P @ X3 )
= ( Q @ X3 ) ) )
=> ( ( dropWh8739902238574629050la_a_b @ P @ L )
= ( dropWh8739902238574629050la_a_b @ Q @ K ) ) ) ) ).
% dropWhile_cong
thf(fact_234_dropWhile__cong,axiom,
! [L: list_formula_a_b,K: list_formula_a_b,P: formula_a_b > $o,Q: formula_a_b > $o] :
( ( L = K )
=> ( ! [X3: formula_a_b] :
( ( member_formula_a_b2 @ X3 @ ( set_formula_a_b2 @ L ) )
=> ( ( P @ X3 )
= ( Q @ X3 ) ) )
=> ( ( dropWh8768975241461415850la_a_b @ P @ L )
= ( dropWh8768975241461415850la_a_b @ Q @ K ) ) ) ) ).
% dropWhile_cong
thf(fact_235_fold__invariant,axiom,
! [Xs: list_formula_a_b,Q: formula_a_b > $o,P: list_formula_a_b > $o,S: list_formula_a_b,F: formula_a_b > list_formula_a_b > list_formula_a_b] :
( ! [X3: formula_a_b] :
( ( member_formula_a_b2 @ X3 @ ( set_formula_a_b2 @ Xs ) )
=> ( Q @ X3 ) )
=> ( ( P @ S )
=> ( ! [X3: formula_a_b,S2: list_formula_a_b] :
( ( Q @ X3 )
=> ( ( P @ S2 )
=> ( P @ ( F @ X3 @ S2 ) ) ) )
=> ( P @ ( fold_f7464356806857031615la_a_b @ F @ Xs @ S ) ) ) ) ) ).
% fold_invariant
thf(fact_236_fold__invariant,axiom,
! [Xs: list_formula_a_b,Q: formula_a_b > $o,P: set_formula_a_b > $o,S: set_formula_a_b,F: formula_a_b > set_formula_a_b > set_formula_a_b] :
( ! [X3: formula_a_b] :
( ( member_formula_a_b2 @ X3 @ ( set_formula_a_b2 @ Xs ) )
=> ( Q @ X3 ) )
=> ( ( P @ S )
=> ( ! [X3: formula_a_b,S2: set_formula_a_b] :
( ( Q @ X3 )
=> ( ( P @ S2 )
=> ( P @ ( F @ X3 @ S2 ) ) ) )
=> ( P @ ( fold_f470054086739575141la_a_b @ F @ Xs @ S ) ) ) ) ) ).
% fold_invariant
thf(fact_237_fold__invariant,axiom,
! [Xs: list_set_formula_a_b,Q: set_formula_a_b > $o,P: set_formula_a_b > $o,S: set_formula_a_b,F: set_formula_a_b > set_formula_a_b > set_formula_a_b] :
( ! [X3: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X3 @ ( set_set_formula_a_b2 @ Xs ) )
=> ( Q @ X3 ) )
=> ( ( P @ S )
=> ( ! [X3: set_formula_a_b,S2: set_formula_a_b] :
( ( Q @ X3 )
=> ( ( P @ S2 )
=> ( P @ ( F @ X3 @ S2 ) ) ) )
=> ( P @ ( fold_s5097478484676794267la_a_b @ F @ Xs @ S ) ) ) ) ) ).
% fold_invariant
thf(fact_238_List_Ofold__cong,axiom,
! [A2: list_formula_a_b,B2: list_formula_a_b,Xs: list_formula_a_b,Ys: list_formula_a_b,F: formula_a_b > list_formula_a_b > list_formula_a_b,G: formula_a_b > list_formula_a_b > list_formula_a_b] :
( ( A2 = B2 )
=> ( ( Xs = Ys )
=> ( ! [X3: formula_a_b] :
( ( member_formula_a_b2 @ X3 @ ( set_formula_a_b2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( fold_f7464356806857031615la_a_b @ F @ Xs @ A2 )
= ( fold_f7464356806857031615la_a_b @ G @ Ys @ B2 ) ) ) ) ) ).
% List.fold_cong
thf(fact_239_List_Ofold__cong,axiom,
! [A2: set_formula_a_b,B2: set_formula_a_b,Xs: list_formula_a_b,Ys: list_formula_a_b,F: formula_a_b > set_formula_a_b > set_formula_a_b,G: formula_a_b > set_formula_a_b > set_formula_a_b] :
( ( A2 = B2 )
=> ( ( Xs = Ys )
=> ( ! [X3: formula_a_b] :
( ( member_formula_a_b2 @ X3 @ ( set_formula_a_b2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( fold_f470054086739575141la_a_b @ F @ Xs @ A2 )
= ( fold_f470054086739575141la_a_b @ G @ Ys @ B2 ) ) ) ) ) ).
% List.fold_cong
thf(fact_240_List_Ofold__cong,axiom,
! [A2: set_formula_a_b,B2: set_formula_a_b,Xs: list_set_formula_a_b,Ys: list_set_formula_a_b,F: set_formula_a_b > set_formula_a_b > set_formula_a_b,G: set_formula_a_b > set_formula_a_b > set_formula_a_b] :
( ( A2 = B2 )
=> ( ( Xs = Ys )
=> ( ! [X3: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X3 @ ( set_set_formula_a_b2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( fold_s5097478484676794267la_a_b @ F @ Xs @ A2 )
= ( fold_s5097478484676794267la_a_b @ G @ Ys @ B2 ) ) ) ) ) ).
% List.fold_cong
thf(fact_241_Cons__eq__appendI,axiom,
! [X: formula_a_b,Xs1: list_formula_a_b,Ys: list_formula_a_b,Xs: list_formula_a_b,Zs: list_formula_a_b] :
( ( ( cons_formula_a_b @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_formula_a_b @ Xs1 @ Zs ) )
=> ( ( cons_formula_a_b @ X @ Xs )
= ( append_formula_a_b @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_242_Cons__eq__appendI,axiom,
! [X: set_formula_a_b,Xs1: list_set_formula_a_b,Ys: list_set_formula_a_b,Xs: list_set_formula_a_b,Zs: list_set_formula_a_b] :
( ( ( cons_set_formula_a_b @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append8577002312234529644la_a_b @ Xs1 @ Zs ) )
=> ( ( cons_set_formula_a_b @ X @ Xs )
= ( append8577002312234529644la_a_b @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_243_append__Cons,axiom,
! [X: formula_a_b,Xs: list_formula_a_b,Ys: list_formula_a_b] :
( ( append_formula_a_b @ ( cons_formula_a_b @ X @ Xs ) @ Ys )
= ( cons_formula_a_b @ X @ ( append_formula_a_b @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_244_append__Cons,axiom,
! [X: set_formula_a_b,Xs: list_set_formula_a_b,Ys: list_set_formula_a_b] :
( ( append8577002312234529644la_a_b @ ( cons_set_formula_a_b @ X @ Xs ) @ Ys )
= ( cons_set_formula_a_b @ X @ ( append8577002312234529644la_a_b @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_245_distinct__length__2__or__more,axiom,
! [A2: list_formula_a_b,B2: list_formula_a_b,Xs: list_l5948067055324603441la_a_b] :
( ( distin27490427618170344la_a_b @ ( cons_l2450299273141159147la_a_b @ A2 @ ( cons_l2450299273141159147la_a_b @ B2 @ Xs ) ) )
= ( ( A2 != B2 )
& ( distin27490427618170344la_a_b @ ( cons_l2450299273141159147la_a_b @ A2 @ Xs ) )
& ( distin27490427618170344la_a_b @ ( cons_l2450299273141159147la_a_b @ B2 @ Xs ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_246_distinct__length__2__or__more,axiom,
! [A2: formula_a_b,B2: formula_a_b,Xs: list_formula_a_b] :
( ( distinct_formula_a_b @ ( cons_formula_a_b @ A2 @ ( cons_formula_a_b @ B2 @ Xs ) ) )
= ( ( A2 != B2 )
& ( distinct_formula_a_b @ ( cons_formula_a_b @ A2 @ Xs ) )
& ( distinct_formula_a_b @ ( cons_formula_a_b @ B2 @ Xs ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_247_distinct__length__2__or__more,axiom,
! [A2: set_formula_a_b,B2: set_formula_a_b,Xs: list_set_formula_a_b] :
( ( distin5059275051008138254la_a_b @ ( cons_set_formula_a_b @ A2 @ ( cons_set_formula_a_b @ B2 @ Xs ) ) )
= ( ( A2 != B2 )
& ( distin5059275051008138254la_a_b @ ( cons_set_formula_a_b @ A2 @ Xs ) )
& ( distin5059275051008138254la_a_b @ ( cons_set_formula_a_b @ B2 @ Xs ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_248_distinct__dropWhile,axiom,
! [Xs: list_l5948067055324603441la_a_b,P: list_formula_a_b > $o] :
( ( distin27490427618170344la_a_b @ Xs )
=> ( distin27490427618170344la_a_b @ ( dropWh8739902238574629050la_a_b @ P @ Xs ) ) ) ).
% distinct_dropWhile
thf(fact_249_distinct__dropWhile,axiom,
! [Xs: list_set_formula_a_b,P: set_formula_a_b > $o] :
( ( distin5059275051008138254la_a_b @ Xs )
=> ( distin5059275051008138254la_a_b @ ( dropWh5803627183431294944la_a_b @ P @ Xs ) ) ) ).
% distinct_dropWhile
thf(fact_250_distinct__dropWhile,axiom,
! [Xs: list_formula_a_b,P: formula_a_b > $o] :
( ( distinct_formula_a_b @ Xs )
=> ( distinct_formula_a_b @ ( dropWh8768975241461415850la_a_b @ P @ Xs ) ) ) ).
% distinct_dropWhile
thf(fact_251_Cons__in__shuffles__rightI,axiom,
! [Zs: list_formula_a_b,Xs: list_formula_a_b,Ys: list_formula_a_b,Z: formula_a_b] :
( ( member7703986999711672120la_a_b @ Zs @ ( shuffles_formula_a_b @ Xs @ Ys ) )
=> ( member7703986999711672120la_a_b @ ( cons_formula_a_b @ Z @ Zs ) @ ( shuffles_formula_a_b @ Xs @ ( cons_formula_a_b @ Z @ Ys ) ) ) ) ).
% Cons_in_shuffles_rightI
thf(fact_252_Cons__in__shuffles__rightI,axiom,
! [Zs: list_set_formula_a_b,Xs: list_set_formula_a_b,Ys: list_set_formula_a_b,Z: set_formula_a_b] :
( ( member289699325402759278la_a_b @ Zs @ ( shuffl4965861759276814700la_a_b @ Xs @ Ys ) )
=> ( member289699325402759278la_a_b @ ( cons_set_formula_a_b @ Z @ Zs ) @ ( shuffl4965861759276814700la_a_b @ Xs @ ( cons_set_formula_a_b @ Z @ Ys ) ) ) ) ).
% Cons_in_shuffles_rightI
thf(fact_253_Cons__in__shuffles__leftI,axiom,
! [Zs: list_formula_a_b,Xs: list_formula_a_b,Ys: list_formula_a_b,Z: formula_a_b] :
( ( member7703986999711672120la_a_b @ Zs @ ( shuffles_formula_a_b @ Xs @ Ys ) )
=> ( member7703986999711672120la_a_b @ ( cons_formula_a_b @ Z @ Zs ) @ ( shuffles_formula_a_b @ ( cons_formula_a_b @ Z @ Xs ) @ Ys ) ) ) ).
% Cons_in_shuffles_leftI
thf(fact_254_Cons__in__shuffles__leftI,axiom,
! [Zs: list_set_formula_a_b,Xs: list_set_formula_a_b,Ys: list_set_formula_a_b,Z: set_formula_a_b] :
( ( member289699325402759278la_a_b @ Zs @ ( shuffl4965861759276814700la_a_b @ Xs @ Ys ) )
=> ( member289699325402759278la_a_b @ ( cons_set_formula_a_b @ Z @ Zs ) @ ( shuffl4965861759276814700la_a_b @ ( cons_set_formula_a_b @ Z @ Xs ) @ Ys ) ) ) ).
% Cons_in_shuffles_leftI
thf(fact_255_notin__set__remove1,axiom,
! [X: set_formula_a_b,Xs: list_set_formula_a_b,Y: set_formula_a_b] :
( ~ ( member6621951321062823262la_a_b @ X @ ( set_set_formula_a_b2 @ Xs ) )
=> ~ ( member6621951321062823262la_a_b @ X @ ( set_set_formula_a_b2 @ ( remove3026683253030310187la_a_b @ Y @ Xs ) ) ) ) ).
% notin_set_remove1
thf(fact_256_notin__set__remove1,axiom,
! [X: list_formula_a_b,Xs: list_l5948067055324603441la_a_b,Y: list_formula_a_b] :
( ~ ( member7703986999711672120la_a_b @ X @ ( set_list_formula_a_b2 @ Xs ) )
=> ~ ( member7703986999711672120la_a_b @ X @ ( set_list_formula_a_b2 @ ( remove4763865728959745157la_a_b @ Y @ Xs ) ) ) ) ).
% notin_set_remove1
thf(fact_257_notin__set__remove1,axiom,
! [X: formula_a_b,Xs: list_formula_a_b,Y: formula_a_b] :
( ~ ( member_formula_a_b2 @ X @ ( set_formula_a_b2 @ Xs ) )
=> ~ ( member_formula_a_b2 @ X @ ( set_formula_a_b2 @ ( remove1_formula_a_b @ Y @ Xs ) ) ) ) ).
% notin_set_remove1
thf(fact_258_remove1__idem,axiom,
! [X: set_formula_a_b,Xs: list_set_formula_a_b] :
( ~ ( member6621951321062823262la_a_b @ X @ ( set_set_formula_a_b2 @ Xs ) )
=> ( ( remove3026683253030310187la_a_b @ X @ Xs )
= Xs ) ) ).
% remove1_idem
thf(fact_259_remove1__idem,axiom,
! [X: list_formula_a_b,Xs: list_l5948067055324603441la_a_b] :
( ~ ( member7703986999711672120la_a_b @ X @ ( set_list_formula_a_b2 @ Xs ) )
=> ( ( remove4763865728959745157la_a_b @ X @ Xs )
= Xs ) ) ).
% remove1_idem
thf(fact_260_remove1__idem,axiom,
! [X: formula_a_b,Xs: list_formula_a_b] :
( ~ ( member_formula_a_b2 @ X @ ( set_formula_a_b2 @ Xs ) )
=> ( ( remove1_formula_a_b @ X @ Xs )
= Xs ) ) ).
% remove1_idem
thf(fact_261_removeAll_Osimps_I2_J,axiom,
! [X: set_formula_a_b,Y: set_formula_a_b,Xs: list_set_formula_a_b] :
( ( ( X = Y )
=> ( ( remove6824600317533053627la_a_b @ X @ ( cons_set_formula_a_b @ Y @ Xs ) )
= ( remove6824600317533053627la_a_b @ X @ Xs ) ) )
& ( ( X != Y )
=> ( ( remove6824600317533053627la_a_b @ X @ ( cons_set_formula_a_b @ Y @ Xs ) )
= ( cons_set_formula_a_b @ Y @ ( remove6824600317533053627la_a_b @ X @ Xs ) ) ) ) ) ).
% removeAll.simps(2)
thf(fact_262_removeAll_Osimps_I2_J,axiom,
! [X: formula_a_b,Y: formula_a_b,Xs: list_formula_a_b] :
( ( ( X = Y )
=> ( ( remove7803334386285014661la_a_b @ X @ ( cons_formula_a_b @ Y @ Xs ) )
= ( remove7803334386285014661la_a_b @ X @ Xs ) ) )
& ( ( X != Y )
=> ( ( remove7803334386285014661la_a_b @ X @ ( cons_formula_a_b @ Y @ Xs ) )
= ( cons_formula_a_b @ Y @ ( remove7803334386285014661la_a_b @ X @ Xs ) ) ) ) ) ).
% removeAll.simps(2)
thf(fact_263_removeAll_Osimps_I2_J,axiom,
! [X: list_formula_a_b,Y: list_formula_a_b,Xs: list_l5948067055324603441la_a_b] :
( ( ( X = Y )
=> ( ( remove9041333805894489621la_a_b @ X @ ( cons_l2450299273141159147la_a_b @ Y @ Xs ) )
= ( remove9041333805894489621la_a_b @ X @ Xs ) ) )
& ( ( X != Y )
=> ( ( remove9041333805894489621la_a_b @ X @ ( cons_l2450299273141159147la_a_b @ Y @ Xs ) )
= ( cons_l2450299273141159147la_a_b @ Y @ ( remove9041333805894489621la_a_b @ X @ Xs ) ) ) ) ) ).
% removeAll.simps(2)
thf(fact_264_distinct__remove1,axiom,
! [Xs: list_formula_a_b,X: formula_a_b] :
( ( distinct_formula_a_b @ Xs )
=> ( distinct_formula_a_b @ ( remove1_formula_a_b @ X @ Xs ) ) ) ).
% distinct_remove1
thf(fact_265_distinct__remove1,axiom,
! [Xs: list_l5948067055324603441la_a_b,X: list_formula_a_b] :
( ( distin27490427618170344la_a_b @ Xs )
=> ( distin27490427618170344la_a_b @ ( remove4763865728959745157la_a_b @ X @ Xs ) ) ) ).
% distinct_remove1
thf(fact_266_distinct__remove1,axiom,
! [Xs: list_set_formula_a_b,X: set_formula_a_b] :
( ( distin5059275051008138254la_a_b @ Xs )
=> ( distin5059275051008138254la_a_b @ ( remove3026683253030310187la_a_b @ X @ Xs ) ) ) ).
% distinct_remove1
thf(fact_267_insert,axiom,
! [X: formula_a_b,Xs: list_formula_a_b,Y: formula_a_b] :
( ( listMem_formula_a_b @ X @ Xs )
=> ( listMem_formula_a_b @ X @ ( cons_formula_a_b @ Y @ Xs ) ) ) ).
% insert
thf(fact_268_insert,axiom,
! [X: set_formula_a_b,Xs: list_set_formula_a_b,Y: set_formula_a_b] :
( ( listMe3233215151264718677la_a_b @ X @ Xs )
=> ( listMe3233215151264718677la_a_b @ X @ ( cons_set_formula_a_b @ Y @ Xs ) ) ) ).
% insert
thf(fact_269_ListMem_Osimps,axiom,
( listMem_formula_a_b
= ( ^ [A1: formula_a_b,A22: list_formula_a_b] :
( ? [X2: formula_a_b,Xs2: list_formula_a_b] :
( ( A1 = X2 )
& ( A22
= ( cons_formula_a_b @ X2 @ Xs2 ) ) )
| ? [X2: formula_a_b,Xs2: list_formula_a_b,Y2: formula_a_b] :
( ( A1 = X2 )
& ( A22
= ( cons_formula_a_b @ Y2 @ Xs2 ) )
& ( listMem_formula_a_b @ X2 @ Xs2 ) ) ) ) ) ).
% ListMem.simps
thf(fact_270_ListMem_Osimps,axiom,
( listMe3233215151264718677la_a_b
= ( ^ [A1: set_formula_a_b,A22: list_set_formula_a_b] :
( ? [X2: set_formula_a_b,Xs2: list_set_formula_a_b] :
( ( A1 = X2 )
& ( A22
= ( cons_set_formula_a_b @ X2 @ Xs2 ) ) )
| ? [X2: set_formula_a_b,Xs2: list_set_formula_a_b,Y2: set_formula_a_b] :
( ( A1 = X2 )
& ( A22
= ( cons_set_formula_a_b @ Y2 @ Xs2 ) )
& ( listMe3233215151264718677la_a_b @ X2 @ Xs2 ) ) ) ) ) ).
% ListMem.simps
thf(fact_271_ListMem_Ocases,axiom,
! [A12: formula_a_b,A23: list_formula_a_b] :
( ( listMem_formula_a_b @ A12 @ A23 )
=> ( ! [Xs3: list_formula_a_b] :
( A23
!= ( cons_formula_a_b @ A12 @ Xs3 ) )
=> ~ ! [Xs3: list_formula_a_b] :
( ? [Y3: formula_a_b] :
( A23
= ( cons_formula_a_b @ Y3 @ Xs3 ) )
=> ~ ( listMem_formula_a_b @ A12 @ Xs3 ) ) ) ) ).
% ListMem.cases
thf(fact_272_ListMem_Ocases,axiom,
! [A12: set_formula_a_b,A23: list_set_formula_a_b] :
( ( listMe3233215151264718677la_a_b @ A12 @ A23 )
=> ( ! [Xs3: list_set_formula_a_b] :
( A23
!= ( cons_set_formula_a_b @ A12 @ Xs3 ) )
=> ~ ! [Xs3: list_set_formula_a_b] :
( ? [Y3: set_formula_a_b] :
( A23
= ( cons_set_formula_a_b @ Y3 @ Xs3 ) )
=> ~ ( listMe3233215151264718677la_a_b @ A12 @ Xs3 ) ) ) ) ).
% ListMem.cases
thf(fact_273_elem,axiom,
! [X: formula_a_b,Xs: list_formula_a_b] : ( listMem_formula_a_b @ X @ ( cons_formula_a_b @ X @ Xs ) ) ).
% elem
thf(fact_274_elem,axiom,
! [X: set_formula_a_b,Xs: list_set_formula_a_b] : ( listMe3233215151264718677la_a_b @ X @ ( cons_set_formula_a_b @ X @ Xs ) ) ).
% elem
thf(fact_275_member__rec_I1_J,axiom,
! [X: formula_a_b,Xs: list_formula_a_b,Y: formula_a_b] :
( ( member_formula_a_b @ ( cons_formula_a_b @ X @ Xs ) @ Y )
= ( ( X = Y )
| ( member_formula_a_b @ Xs @ Y ) ) ) ).
% member_rec(1)
thf(fact_276_member__rec_I1_J,axiom,
! [X: set_formula_a_b,Xs: list_set_formula_a_b,Y: set_formula_a_b] :
( ( member5721106492161739372la_a_b @ ( cons_set_formula_a_b @ X @ Xs ) @ Y )
= ( ( X = Y )
| ( member5721106492161739372la_a_b @ Xs @ Y ) ) ) ).
% member_rec(1)
thf(fact_277_split__list,axiom,
! [X: list_formula_a_b,Xs: list_l5948067055324603441la_a_b] :
( ( member7703986999711672120la_a_b @ X @ ( set_list_formula_a_b2 @ Xs ) )
=> ? [Ys3: list_l5948067055324603441la_a_b,Zs2: list_l5948067055324603441la_a_b] :
( Xs
= ( append58666856857280070la_a_b @ Ys3 @ ( cons_l2450299273141159147la_a_b @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_278_split__list,axiom,
! [X: set_formula_a_b,Xs: list_set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X @ ( set_set_formula_a_b2 @ Xs ) )
=> ? [Ys3: list_set_formula_a_b,Zs2: list_set_formula_a_b] :
( Xs
= ( append8577002312234529644la_a_b @ Ys3 @ ( cons_set_formula_a_b @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_279_split__list,axiom,
! [X: formula_a_b,Xs: list_formula_a_b] :
( ( member_formula_a_b2 @ X @ ( set_formula_a_b2 @ Xs ) )
=> ? [Ys3: list_formula_a_b,Zs2: list_formula_a_b] :
( Xs
= ( append_formula_a_b @ Ys3 @ ( cons_formula_a_b @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_280_split__list__last,axiom,
! [X: list_formula_a_b,Xs: list_l5948067055324603441la_a_b] :
( ( member7703986999711672120la_a_b @ X @ ( set_list_formula_a_b2 @ Xs ) )
=> ? [Ys3: list_l5948067055324603441la_a_b,Zs2: list_l5948067055324603441la_a_b] :
( ( Xs
= ( append58666856857280070la_a_b @ Ys3 @ ( cons_l2450299273141159147la_a_b @ X @ Zs2 ) ) )
& ~ ( member7703986999711672120la_a_b @ X @ ( set_list_formula_a_b2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_281_split__list__last,axiom,
! [X: set_formula_a_b,Xs: list_set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X @ ( set_set_formula_a_b2 @ Xs ) )
=> ? [Ys3: list_set_formula_a_b,Zs2: list_set_formula_a_b] :
( ( Xs
= ( append8577002312234529644la_a_b @ Ys3 @ ( cons_set_formula_a_b @ X @ Zs2 ) ) )
& ~ ( member6621951321062823262la_a_b @ X @ ( set_set_formula_a_b2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_282_split__list__last,axiom,
! [X: formula_a_b,Xs: list_formula_a_b] :
( ( member_formula_a_b2 @ X @ ( set_formula_a_b2 @ Xs ) )
=> ? [Ys3: list_formula_a_b,Zs2: list_formula_a_b] :
( ( Xs
= ( append_formula_a_b @ Ys3 @ ( cons_formula_a_b @ X @ Zs2 ) ) )
& ~ ( member_formula_a_b2 @ X @ ( set_formula_a_b2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_283_split__list__prop,axiom,
! [Xs: list_l5948067055324603441la_a_b,P: list_formula_a_b > $o] :
( ? [X4: list_formula_a_b] :
( ( member7703986999711672120la_a_b @ X4 @ ( set_list_formula_a_b2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys3: list_l5948067055324603441la_a_b,X3: list_formula_a_b] :
( ? [Zs2: list_l5948067055324603441la_a_b] :
( Xs
= ( append58666856857280070la_a_b @ Ys3 @ ( cons_l2450299273141159147la_a_b @ X3 @ Zs2 ) ) )
& ( P @ X3 ) ) ) ).
% split_list_prop
thf(fact_284_split__list__prop,axiom,
! [Xs: list_set_formula_a_b,P: set_formula_a_b > $o] :
( ? [X4: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X4 @ ( set_set_formula_a_b2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys3: list_set_formula_a_b,X3: set_formula_a_b] :
( ? [Zs2: list_set_formula_a_b] :
( Xs
= ( append8577002312234529644la_a_b @ Ys3 @ ( cons_set_formula_a_b @ X3 @ Zs2 ) ) )
& ( P @ X3 ) ) ) ).
% split_list_prop
thf(fact_285_split__list__prop,axiom,
! [Xs: list_formula_a_b,P: formula_a_b > $o] :
( ? [X4: formula_a_b] :
( ( member_formula_a_b2 @ X4 @ ( set_formula_a_b2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys3: list_formula_a_b,X3: formula_a_b] :
( ? [Zs2: list_formula_a_b] :
( Xs
= ( append_formula_a_b @ Ys3 @ ( cons_formula_a_b @ X3 @ Zs2 ) ) )
& ( P @ X3 ) ) ) ).
% split_list_prop
thf(fact_286_split__list__first,axiom,
! [X: list_formula_a_b,Xs: list_l5948067055324603441la_a_b] :
( ( member7703986999711672120la_a_b @ X @ ( set_list_formula_a_b2 @ Xs ) )
=> ? [Ys3: list_l5948067055324603441la_a_b,Zs2: list_l5948067055324603441la_a_b] :
( ( Xs
= ( append58666856857280070la_a_b @ Ys3 @ ( cons_l2450299273141159147la_a_b @ X @ Zs2 ) ) )
& ~ ( member7703986999711672120la_a_b @ X @ ( set_list_formula_a_b2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_287_split__list__first,axiom,
! [X: set_formula_a_b,Xs: list_set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X @ ( set_set_formula_a_b2 @ Xs ) )
=> ? [Ys3: list_set_formula_a_b,Zs2: list_set_formula_a_b] :
( ( Xs
= ( append8577002312234529644la_a_b @ Ys3 @ ( cons_set_formula_a_b @ X @ Zs2 ) ) )
& ~ ( member6621951321062823262la_a_b @ X @ ( set_set_formula_a_b2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_288_split__list__first,axiom,
! [X: formula_a_b,Xs: list_formula_a_b] :
( ( member_formula_a_b2 @ X @ ( set_formula_a_b2 @ Xs ) )
=> ? [Ys3: list_formula_a_b,Zs2: list_formula_a_b] :
( ( Xs
= ( append_formula_a_b @ Ys3 @ ( cons_formula_a_b @ X @ Zs2 ) ) )
& ~ ( member_formula_a_b2 @ X @ ( set_formula_a_b2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_289_split__list__propE,axiom,
! [Xs: list_l5948067055324603441la_a_b,P: list_formula_a_b > $o] :
( ? [X4: list_formula_a_b] :
( ( member7703986999711672120la_a_b @ X4 @ ( set_list_formula_a_b2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys3: list_l5948067055324603441la_a_b,X3: list_formula_a_b] :
( ? [Zs2: list_l5948067055324603441la_a_b] :
( Xs
= ( append58666856857280070la_a_b @ Ys3 @ ( cons_l2450299273141159147la_a_b @ X3 @ Zs2 ) ) )
=> ~ ( P @ X3 ) ) ) ).
% split_list_propE
thf(fact_290_split__list__propE,axiom,
! [Xs: list_set_formula_a_b,P: set_formula_a_b > $o] :
( ? [X4: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X4 @ ( set_set_formula_a_b2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys3: list_set_formula_a_b,X3: set_formula_a_b] :
( ? [Zs2: list_set_formula_a_b] :
( Xs
= ( append8577002312234529644la_a_b @ Ys3 @ ( cons_set_formula_a_b @ X3 @ Zs2 ) ) )
=> ~ ( P @ X3 ) ) ) ).
% split_list_propE
thf(fact_291_split__list__propE,axiom,
! [Xs: list_formula_a_b,P: formula_a_b > $o] :
( ? [X4: formula_a_b] :
( ( member_formula_a_b2 @ X4 @ ( set_formula_a_b2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys3: list_formula_a_b,X3: formula_a_b] :
( ? [Zs2: list_formula_a_b] :
( Xs
= ( append_formula_a_b @ Ys3 @ ( cons_formula_a_b @ X3 @ Zs2 ) ) )
=> ~ ( P @ X3 ) ) ) ).
% split_list_propE
thf(fact_292_append__Cons__eq__iff,axiom,
! [X: list_formula_a_b,Xs: list_l5948067055324603441la_a_b,Ys: list_l5948067055324603441la_a_b,Xs4: list_l5948067055324603441la_a_b,Ys4: list_l5948067055324603441la_a_b] :
( ~ ( member7703986999711672120la_a_b @ X @ ( set_list_formula_a_b2 @ Xs ) )
=> ( ~ ( member7703986999711672120la_a_b @ X @ ( set_list_formula_a_b2 @ Ys ) )
=> ( ( ( append58666856857280070la_a_b @ Xs @ ( cons_l2450299273141159147la_a_b @ X @ Ys ) )
= ( append58666856857280070la_a_b @ Xs4 @ ( cons_l2450299273141159147la_a_b @ X @ Ys4 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys4 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_293_append__Cons__eq__iff,axiom,
! [X: set_formula_a_b,Xs: list_set_formula_a_b,Ys: list_set_formula_a_b,Xs4: list_set_formula_a_b,Ys4: list_set_formula_a_b] :
( ~ ( member6621951321062823262la_a_b @ X @ ( set_set_formula_a_b2 @ Xs ) )
=> ( ~ ( member6621951321062823262la_a_b @ X @ ( set_set_formula_a_b2 @ Ys ) )
=> ( ( ( append8577002312234529644la_a_b @ Xs @ ( cons_set_formula_a_b @ X @ Ys ) )
= ( append8577002312234529644la_a_b @ Xs4 @ ( cons_set_formula_a_b @ X @ Ys4 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys4 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_294_append__Cons__eq__iff,axiom,
! [X: formula_a_b,Xs: list_formula_a_b,Ys: list_formula_a_b,Xs4: list_formula_a_b,Ys4: list_formula_a_b] :
( ~ ( member_formula_a_b2 @ X @ ( set_formula_a_b2 @ Xs ) )
=> ( ~ ( member_formula_a_b2 @ X @ ( set_formula_a_b2 @ Ys ) )
=> ( ( ( append_formula_a_b @ Xs @ ( cons_formula_a_b @ X @ Ys ) )
= ( append_formula_a_b @ Xs4 @ ( cons_formula_a_b @ X @ Ys4 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys4 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_295_in__set__conv__decomp,axiom,
! [X: list_formula_a_b,Xs: list_l5948067055324603441la_a_b] :
( ( member7703986999711672120la_a_b @ X @ ( set_list_formula_a_b2 @ Xs ) )
= ( ? [Ys2: list_l5948067055324603441la_a_b,Zs3: list_l5948067055324603441la_a_b] :
( Xs
= ( append58666856857280070la_a_b @ Ys2 @ ( cons_l2450299273141159147la_a_b @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_296_in__set__conv__decomp,axiom,
! [X: set_formula_a_b,Xs: list_set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X @ ( set_set_formula_a_b2 @ Xs ) )
= ( ? [Ys2: list_set_formula_a_b,Zs3: list_set_formula_a_b] :
( Xs
= ( append8577002312234529644la_a_b @ Ys2 @ ( cons_set_formula_a_b @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_297_in__set__conv__decomp,axiom,
! [X: formula_a_b,Xs: list_formula_a_b] :
( ( member_formula_a_b2 @ X @ ( set_formula_a_b2 @ Xs ) )
= ( ? [Ys2: list_formula_a_b,Zs3: list_formula_a_b] :
( Xs
= ( append_formula_a_b @ Ys2 @ ( cons_formula_a_b @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_298_split__list__last__prop,axiom,
! [Xs: list_l5948067055324603441la_a_b,P: list_formula_a_b > $o] :
( ? [X4: list_formula_a_b] :
( ( member7703986999711672120la_a_b @ X4 @ ( set_list_formula_a_b2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys3: list_l5948067055324603441la_a_b,X3: list_formula_a_b,Zs2: list_l5948067055324603441la_a_b] :
( ( Xs
= ( append58666856857280070la_a_b @ Ys3 @ ( cons_l2450299273141159147la_a_b @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Xa: list_formula_a_b] :
( ( member7703986999711672120la_a_b @ Xa @ ( set_list_formula_a_b2 @ Zs2 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_299_split__list__last__prop,axiom,
! [Xs: list_set_formula_a_b,P: set_formula_a_b > $o] :
( ? [X4: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X4 @ ( set_set_formula_a_b2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys3: list_set_formula_a_b,X3: set_formula_a_b,Zs2: list_set_formula_a_b] :
( ( Xs
= ( append8577002312234529644la_a_b @ Ys3 @ ( cons_set_formula_a_b @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Xa: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ Xa @ ( set_set_formula_a_b2 @ Zs2 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_300_split__list__last__prop,axiom,
! [Xs: list_formula_a_b,P: formula_a_b > $o] :
( ? [X4: formula_a_b] :
( ( member_formula_a_b2 @ X4 @ ( set_formula_a_b2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys3: list_formula_a_b,X3: formula_a_b,Zs2: list_formula_a_b] :
( ( Xs
= ( append_formula_a_b @ Ys3 @ ( cons_formula_a_b @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Xa: formula_a_b] :
( ( member_formula_a_b2 @ Xa @ ( set_formula_a_b2 @ Zs2 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_301_split__list__first__prop,axiom,
! [Xs: list_l5948067055324603441la_a_b,P: list_formula_a_b > $o] :
( ? [X4: list_formula_a_b] :
( ( member7703986999711672120la_a_b @ X4 @ ( set_list_formula_a_b2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys3: list_l5948067055324603441la_a_b,X3: list_formula_a_b] :
( ? [Zs2: list_l5948067055324603441la_a_b] :
( Xs
= ( append58666856857280070la_a_b @ Ys3 @ ( cons_l2450299273141159147la_a_b @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Xa: list_formula_a_b] :
( ( member7703986999711672120la_a_b @ Xa @ ( set_list_formula_a_b2 @ Ys3 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_302_split__list__first__prop,axiom,
! [Xs: list_set_formula_a_b,P: set_formula_a_b > $o] :
( ? [X4: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X4 @ ( set_set_formula_a_b2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys3: list_set_formula_a_b,X3: set_formula_a_b] :
( ? [Zs2: list_set_formula_a_b] :
( Xs
= ( append8577002312234529644la_a_b @ Ys3 @ ( cons_set_formula_a_b @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Xa: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ Xa @ ( set_set_formula_a_b2 @ Ys3 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_303_split__list__first__prop,axiom,
! [Xs: list_formula_a_b,P: formula_a_b > $o] :
( ? [X4: formula_a_b] :
( ( member_formula_a_b2 @ X4 @ ( set_formula_a_b2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys3: list_formula_a_b,X3: formula_a_b] :
( ? [Zs2: list_formula_a_b] :
( Xs
= ( append_formula_a_b @ Ys3 @ ( cons_formula_a_b @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Xa: formula_a_b] :
( ( member_formula_a_b2 @ Xa @ ( set_formula_a_b2 @ Ys3 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_304_split__list__last__propE,axiom,
! [Xs: list_l5948067055324603441la_a_b,P: list_formula_a_b > $o] :
( ? [X4: list_formula_a_b] :
( ( member7703986999711672120la_a_b @ X4 @ ( set_list_formula_a_b2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys3: list_l5948067055324603441la_a_b,X3: list_formula_a_b,Zs2: list_l5948067055324603441la_a_b] :
( ( Xs
= ( append58666856857280070la_a_b @ Ys3 @ ( cons_l2450299273141159147la_a_b @ X3 @ Zs2 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa: list_formula_a_b] :
( ( member7703986999711672120la_a_b @ Xa @ ( set_list_formula_a_b2 @ Zs2 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_305_split__list__last__propE,axiom,
! [Xs: list_set_formula_a_b,P: set_formula_a_b > $o] :
( ? [X4: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X4 @ ( set_set_formula_a_b2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys3: list_set_formula_a_b,X3: set_formula_a_b,Zs2: list_set_formula_a_b] :
( ( Xs
= ( append8577002312234529644la_a_b @ Ys3 @ ( cons_set_formula_a_b @ X3 @ Zs2 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ Xa @ ( set_set_formula_a_b2 @ Zs2 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_306_split__list__last__propE,axiom,
! [Xs: list_formula_a_b,P: formula_a_b > $o] :
( ? [X4: formula_a_b] :
( ( member_formula_a_b2 @ X4 @ ( set_formula_a_b2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys3: list_formula_a_b,X3: formula_a_b,Zs2: list_formula_a_b] :
( ( Xs
= ( append_formula_a_b @ Ys3 @ ( cons_formula_a_b @ X3 @ Zs2 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa: formula_a_b] :
( ( member_formula_a_b2 @ Xa @ ( set_formula_a_b2 @ Zs2 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_307_split__list__first__propE,axiom,
! [Xs: list_l5948067055324603441la_a_b,P: list_formula_a_b > $o] :
( ? [X4: list_formula_a_b] :
( ( member7703986999711672120la_a_b @ X4 @ ( set_list_formula_a_b2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys3: list_l5948067055324603441la_a_b,X3: list_formula_a_b] :
( ? [Zs2: list_l5948067055324603441la_a_b] :
( Xs
= ( append58666856857280070la_a_b @ Ys3 @ ( cons_l2450299273141159147la_a_b @ X3 @ Zs2 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa: list_formula_a_b] :
( ( member7703986999711672120la_a_b @ Xa @ ( set_list_formula_a_b2 @ Ys3 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_308_split__list__first__propE,axiom,
! [Xs: list_set_formula_a_b,P: set_formula_a_b > $o] :
( ? [X4: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X4 @ ( set_set_formula_a_b2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys3: list_set_formula_a_b,X3: set_formula_a_b] :
( ? [Zs2: list_set_formula_a_b] :
( Xs
= ( append8577002312234529644la_a_b @ Ys3 @ ( cons_set_formula_a_b @ X3 @ Zs2 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ Xa @ ( set_set_formula_a_b2 @ Ys3 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_309_split__list__first__propE,axiom,
! [Xs: list_formula_a_b,P: formula_a_b > $o] :
( ? [X4: formula_a_b] :
( ( member_formula_a_b2 @ X4 @ ( set_formula_a_b2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys3: list_formula_a_b,X3: formula_a_b] :
( ? [Zs2: list_formula_a_b] :
( Xs
= ( append_formula_a_b @ Ys3 @ ( cons_formula_a_b @ X3 @ Zs2 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa: formula_a_b] :
( ( member_formula_a_b2 @ Xa @ ( set_formula_a_b2 @ Ys3 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_310_in__set__conv__decomp__last,axiom,
! [X: list_formula_a_b,Xs: list_l5948067055324603441la_a_b] :
( ( member7703986999711672120la_a_b @ X @ ( set_list_formula_a_b2 @ Xs ) )
= ( ? [Ys2: list_l5948067055324603441la_a_b,Zs3: list_l5948067055324603441la_a_b] :
( ( Xs
= ( append58666856857280070la_a_b @ Ys2 @ ( cons_l2450299273141159147la_a_b @ X @ Zs3 ) ) )
& ~ ( member7703986999711672120la_a_b @ X @ ( set_list_formula_a_b2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_311_in__set__conv__decomp__last,axiom,
! [X: set_formula_a_b,Xs: list_set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X @ ( set_set_formula_a_b2 @ Xs ) )
= ( ? [Ys2: list_set_formula_a_b,Zs3: list_set_formula_a_b] :
( ( Xs
= ( append8577002312234529644la_a_b @ Ys2 @ ( cons_set_formula_a_b @ X @ Zs3 ) ) )
& ~ ( member6621951321062823262la_a_b @ X @ ( set_set_formula_a_b2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_312_in__set__conv__decomp__last,axiom,
! [X: formula_a_b,Xs: list_formula_a_b] :
( ( member_formula_a_b2 @ X @ ( set_formula_a_b2 @ Xs ) )
= ( ? [Ys2: list_formula_a_b,Zs3: list_formula_a_b] :
( ( Xs
= ( append_formula_a_b @ Ys2 @ ( cons_formula_a_b @ X @ Zs3 ) ) )
& ~ ( member_formula_a_b2 @ X @ ( set_formula_a_b2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_313_in__set__conv__decomp__first,axiom,
! [X: list_formula_a_b,Xs: list_l5948067055324603441la_a_b] :
( ( member7703986999711672120la_a_b @ X @ ( set_list_formula_a_b2 @ Xs ) )
= ( ? [Ys2: list_l5948067055324603441la_a_b,Zs3: list_l5948067055324603441la_a_b] :
( ( Xs
= ( append58666856857280070la_a_b @ Ys2 @ ( cons_l2450299273141159147la_a_b @ X @ Zs3 ) ) )
& ~ ( member7703986999711672120la_a_b @ X @ ( set_list_formula_a_b2 @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_314_in__set__conv__decomp__first,axiom,
! [X: set_formula_a_b,Xs: list_set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X @ ( set_set_formula_a_b2 @ Xs ) )
= ( ? [Ys2: list_set_formula_a_b,Zs3: list_set_formula_a_b] :
( ( Xs
= ( append8577002312234529644la_a_b @ Ys2 @ ( cons_set_formula_a_b @ X @ Zs3 ) ) )
& ~ ( member6621951321062823262la_a_b @ X @ ( set_set_formula_a_b2 @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_315_in__set__conv__decomp__first,axiom,
! [X: formula_a_b,Xs: list_formula_a_b] :
( ( member_formula_a_b2 @ X @ ( set_formula_a_b2 @ Xs ) )
= ( ? [Ys2: list_formula_a_b,Zs3: list_formula_a_b] :
( ( Xs
= ( append_formula_a_b @ Ys2 @ ( cons_formula_a_b @ X @ Zs3 ) ) )
& ~ ( member_formula_a_b2 @ X @ ( set_formula_a_b2 @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_316_split__list__last__prop__iff,axiom,
! [Xs: list_l5948067055324603441la_a_b,P: list_formula_a_b > $o] :
( ( ? [X2: list_formula_a_b] :
( ( member7703986999711672120la_a_b @ X2 @ ( set_list_formula_a_b2 @ Xs ) )
& ( P @ X2 ) ) )
= ( ? [Ys2: list_l5948067055324603441la_a_b,X2: list_formula_a_b,Zs3: list_l5948067055324603441la_a_b] :
( ( Xs
= ( append58666856857280070la_a_b @ Ys2 @ ( cons_l2450299273141159147la_a_b @ X2 @ Zs3 ) ) )
& ( P @ X2 )
& ! [Y2: list_formula_a_b] :
( ( member7703986999711672120la_a_b @ Y2 @ ( set_list_formula_a_b2 @ Zs3 ) )
=> ~ ( P @ Y2 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_317_split__list__last__prop__iff,axiom,
! [Xs: list_set_formula_a_b,P: set_formula_a_b > $o] :
( ( ? [X2: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X2 @ ( set_set_formula_a_b2 @ Xs ) )
& ( P @ X2 ) ) )
= ( ? [Ys2: list_set_formula_a_b,X2: set_formula_a_b,Zs3: list_set_formula_a_b] :
( ( Xs
= ( append8577002312234529644la_a_b @ Ys2 @ ( cons_set_formula_a_b @ X2 @ Zs3 ) ) )
& ( P @ X2 )
& ! [Y2: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ Y2 @ ( set_set_formula_a_b2 @ Zs3 ) )
=> ~ ( P @ Y2 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_318_split__list__last__prop__iff,axiom,
! [Xs: list_formula_a_b,P: formula_a_b > $o] :
( ( ? [X2: formula_a_b] :
( ( member_formula_a_b2 @ X2 @ ( set_formula_a_b2 @ Xs ) )
& ( P @ X2 ) ) )
= ( ? [Ys2: list_formula_a_b,X2: formula_a_b,Zs3: list_formula_a_b] :
( ( Xs
= ( append_formula_a_b @ Ys2 @ ( cons_formula_a_b @ X2 @ Zs3 ) ) )
& ( P @ X2 )
& ! [Y2: formula_a_b] :
( ( member_formula_a_b2 @ Y2 @ ( set_formula_a_b2 @ Zs3 ) )
=> ~ ( P @ Y2 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_319_split__list__first__prop__iff,axiom,
! [Xs: list_l5948067055324603441la_a_b,P: list_formula_a_b > $o] :
( ( ? [X2: list_formula_a_b] :
( ( member7703986999711672120la_a_b @ X2 @ ( set_list_formula_a_b2 @ Xs ) )
& ( P @ X2 ) ) )
= ( ? [Ys2: list_l5948067055324603441la_a_b,X2: list_formula_a_b] :
( ? [Zs3: list_l5948067055324603441la_a_b] :
( Xs
= ( append58666856857280070la_a_b @ Ys2 @ ( cons_l2450299273141159147la_a_b @ X2 @ Zs3 ) ) )
& ( P @ X2 )
& ! [Y2: list_formula_a_b] :
( ( member7703986999711672120la_a_b @ Y2 @ ( set_list_formula_a_b2 @ Ys2 ) )
=> ~ ( P @ Y2 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_320_split__list__first__prop__iff,axiom,
! [Xs: list_set_formula_a_b,P: set_formula_a_b > $o] :
( ( ? [X2: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X2 @ ( set_set_formula_a_b2 @ Xs ) )
& ( P @ X2 ) ) )
= ( ? [Ys2: list_set_formula_a_b,X2: set_formula_a_b] :
( ? [Zs3: list_set_formula_a_b] :
( Xs
= ( append8577002312234529644la_a_b @ Ys2 @ ( cons_set_formula_a_b @ X2 @ Zs3 ) ) )
& ( P @ X2 )
& ! [Y2: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ Y2 @ ( set_set_formula_a_b2 @ Ys2 ) )
=> ~ ( P @ Y2 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_321_split__list__first__prop__iff,axiom,
! [Xs: list_formula_a_b,P: formula_a_b > $o] :
( ( ? [X2: formula_a_b] :
( ( member_formula_a_b2 @ X2 @ ( set_formula_a_b2 @ Xs ) )
& ( P @ X2 ) ) )
= ( ? [Ys2: list_formula_a_b,X2: formula_a_b] :
( ? [Zs3: list_formula_a_b] :
( Xs
= ( append_formula_a_b @ Ys2 @ ( cons_formula_a_b @ X2 @ Zs3 ) ) )
& ( P @ X2 )
& ! [Y2: formula_a_b] :
( ( member_formula_a_b2 @ Y2 @ ( set_formula_a_b2 @ Ys2 ) )
=> ~ ( P @ Y2 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_322_distinct_Osimps_I2_J,axiom,
! [X: list_formula_a_b,Xs: list_l5948067055324603441la_a_b] :
( ( distin27490427618170344la_a_b @ ( cons_l2450299273141159147la_a_b @ X @ Xs ) )
= ( ~ ( member7703986999711672120la_a_b @ X @ ( set_list_formula_a_b2 @ Xs ) )
& ( distin27490427618170344la_a_b @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_323_distinct_Osimps_I2_J,axiom,
! [X: set_formula_a_b,Xs: list_set_formula_a_b] :
( ( distin5059275051008138254la_a_b @ ( cons_set_formula_a_b @ X @ Xs ) )
= ( ~ ( member6621951321062823262la_a_b @ X @ ( set_set_formula_a_b2 @ Xs ) )
& ( distin5059275051008138254la_a_b @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_324_distinct_Osimps_I2_J,axiom,
! [X: formula_a_b,Xs: list_formula_a_b] :
( ( distinct_formula_a_b @ ( cons_formula_a_b @ X @ Xs ) )
= ( ~ ( member_formula_a_b2 @ X @ ( set_formula_a_b2 @ Xs ) )
& ( distinct_formula_a_b @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_325_List_Oinsert__def,axiom,
( insert3252115831038319365la_a_b
= ( ^ [X2: list_formula_a_b,Xs2: list_l5948067055324603441la_a_b] : ( if_lis7719565029871594987la_a_b @ ( member7703986999711672120la_a_b @ X2 @ ( set_list_formula_a_b2 @ Xs2 ) ) @ Xs2 @ ( cons_l2450299273141159147la_a_b @ X2 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_326_List_Oinsert__def,axiom,
( insert6994445989384232875la_a_b
= ( ^ [X2: set_formula_a_b,Xs2: list_set_formula_a_b] : ( if_lis2278976591155604753la_a_b @ ( member6621951321062823262la_a_b @ X2 @ ( set_set_formula_a_b2 @ Xs2 ) ) @ Xs2 @ ( cons_set_formula_a_b @ X2 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_327_List_Oinsert__def,axiom,
( insert_formula_a_b
= ( ^ [X2: formula_a_b,Xs2: list_formula_a_b] : ( if_list_formula_a_b @ ( member_formula_a_b2 @ X2 @ ( set_formula_a_b2 @ Xs2 ) ) @ Xs2 @ ( cons_formula_a_b @ X2 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_328_remove1__append,axiom,
! [X: set_formula_a_b,Xs: list_set_formula_a_b,Ys: list_set_formula_a_b] :
( ( ( member6621951321062823262la_a_b @ X @ ( set_set_formula_a_b2 @ Xs ) )
=> ( ( remove3026683253030310187la_a_b @ X @ ( append8577002312234529644la_a_b @ Xs @ Ys ) )
= ( append8577002312234529644la_a_b @ ( remove3026683253030310187la_a_b @ X @ Xs ) @ Ys ) ) )
& ( ~ ( member6621951321062823262la_a_b @ X @ ( set_set_formula_a_b2 @ Xs ) )
=> ( ( remove3026683253030310187la_a_b @ X @ ( append8577002312234529644la_a_b @ Xs @ Ys ) )
= ( append8577002312234529644la_a_b @ Xs @ ( remove3026683253030310187la_a_b @ X @ Ys ) ) ) ) ) ).
% remove1_append
thf(fact_329_remove1__append,axiom,
! [X: list_formula_a_b,Xs: list_l5948067055324603441la_a_b,Ys: list_l5948067055324603441la_a_b] :
( ( ( member7703986999711672120la_a_b @ X @ ( set_list_formula_a_b2 @ Xs ) )
=> ( ( remove4763865728959745157la_a_b @ X @ ( append58666856857280070la_a_b @ Xs @ Ys ) )
= ( append58666856857280070la_a_b @ ( remove4763865728959745157la_a_b @ X @ Xs ) @ Ys ) ) )
& ( ~ ( member7703986999711672120la_a_b @ X @ ( set_list_formula_a_b2 @ Xs ) )
=> ( ( remove4763865728959745157la_a_b @ X @ ( append58666856857280070la_a_b @ Xs @ Ys ) )
= ( append58666856857280070la_a_b @ Xs @ ( remove4763865728959745157la_a_b @ X @ Ys ) ) ) ) ) ).
% remove1_append
thf(fact_330_remove1__append,axiom,
! [X: formula_a_b,Xs: list_formula_a_b,Ys: list_formula_a_b] :
( ( ( member_formula_a_b2 @ X @ ( set_formula_a_b2 @ Xs ) )
=> ( ( remove1_formula_a_b @ X @ ( append_formula_a_b @ Xs @ Ys ) )
= ( append_formula_a_b @ ( remove1_formula_a_b @ X @ Xs ) @ Ys ) ) )
& ( ~ ( member_formula_a_b2 @ X @ ( set_formula_a_b2 @ Xs ) )
=> ( ( remove1_formula_a_b @ X @ ( append_formula_a_b @ Xs @ Ys ) )
= ( append_formula_a_b @ Xs @ ( remove1_formula_a_b @ X @ Ys ) ) ) ) ) ).
% remove1_append
thf(fact_331_maps__simps_I1_J,axiom,
! [F: formula_a_b > list_formula_a_b,X: formula_a_b,Xs: list_formula_a_b] :
( ( maps_f4007179076772103781la_a_b @ F @ ( cons_formula_a_b @ X @ Xs ) )
= ( append_formula_a_b @ ( F @ X ) @ ( maps_f4007179076772103781la_a_b @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_332_maps__simps_I1_J,axiom,
! [F: set_formula_a_b > list_formula_a_b,X: set_formula_a_b,Xs: list_set_formula_a_b] :
( ( maps_s5908961544147508379la_a_b @ F @ ( cons_set_formula_a_b @ X @ Xs ) )
= ( append_formula_a_b @ ( F @ X ) @ ( maps_s5908961544147508379la_a_b @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_333_remove__code_I2_J,axiom,
! [X: formula_a_b,Xs: list_formula_a_b] :
( ( remove_formula_a_b @ X @ ( coset_formula_a_b @ Xs ) )
= ( coset_formula_a_b @ ( insert_formula_a_b @ X @ Xs ) ) ) ).
% remove_code(2)
thf(fact_334_Sup__fin_Oset__eq__fold,axiom,
! [X: set_set_formula_a_b,Xs: list_s602864636976034957la_a_b] :
( ( lattic7951861892593991814la_a_b @ ( set_se5912223057565107496la_a_b @ ( cons_s2905031355775537607la_a_b @ X @ Xs ) ) )
= ( fold_s5064541745726864391la_a_b @ sup_su5931044959554304553la_a_b @ Xs @ X ) ) ).
% Sup_fin.set_eq_fold
thf(fact_335_Sup__fin_Oset__eq__fold,axiom,
! [X: set_formula_a_b,Xs: list_set_formula_a_b] :
( ( lattic2153292871565936464la_a_b @ ( set_set_formula_a_b2 @ ( cons_set_formula_a_b @ X @ Xs ) ) )
= ( fold_s5097478484676794267la_a_b @ sup_su8832426789560438003la_a_b @ Xs @ X ) ) ).
% Sup_fin.set_eq_fold
thf(fact_336_insert__code_I2_J,axiom,
! [X: set_formula_a_b,Xs: list_set_formula_a_b] :
( ( insert8952366156270154231la_a_b @ X @ ( coset_6935746743259127700la_a_b @ Xs ) )
= ( coset_6935746743259127700la_a_b @ ( remove6824600317533053627la_a_b @ X @ Xs ) ) ) ).
% insert_code(2)
thf(fact_337_insert__code_I2_J,axiom,
! [X: formula_a_b,Xs: list_formula_a_b] :
( ( insert_formula_a_b2 @ X @ ( coset_formula_a_b @ Xs ) )
= ( coset_formula_a_b @ ( remove7803334386285014661la_a_b @ X @ Xs ) ) ) ).
% insert_code(2)
thf(fact_338_insert__code_I2_J,axiom,
! [X: list_formula_a_b,Xs: list_l5948067055324603441la_a_b] :
( ( insert4995524290302003025la_a_b @ X @ ( coset_845268532100101742la_a_b @ Xs ) )
= ( coset_845268532100101742la_a_b @ ( remove9041333805894489621la_a_b @ X @ Xs ) ) ) ).
% insert_code(2)
thf(fact_339_rotate1_Osimps_I2_J,axiom,
! [X: formula_a_b,Xs: list_formula_a_b] :
( ( rotate1_formula_a_b @ ( cons_formula_a_b @ X @ Xs ) )
= ( append_formula_a_b @ Xs @ ( cons_formula_a_b @ X @ nil_formula_a_b ) ) ) ).
% rotate1.simps(2)
thf(fact_340_rotate1_Osimps_I2_J,axiom,
! [X: set_formula_a_b,Xs: list_set_formula_a_b] :
( ( rotate5784766846832941108la_a_b @ ( cons_set_formula_a_b @ X @ Xs ) )
= ( append8577002312234529644la_a_b @ Xs @ ( cons_set_formula_a_b @ X @ nil_set_formula_a_b ) ) ) ).
% rotate1.simps(2)
thf(fact_341_fold__Cons__rev,axiom,
! [Xs: list_formula_a_b] :
( ( fold_f7464356806857031615la_a_b @ cons_formula_a_b @ Xs )
= ( append_formula_a_b @ ( rev_formula_a_b @ Xs ) ) ) ).
% fold_Cons_rev
thf(fact_342_fold__Cons__rev,axiom,
! [Xs: list_set_formula_a_b] :
( ( fold_s1220939906018902059la_a_b @ cons_set_formula_a_b @ Xs )
= ( append8577002312234529644la_a_b @ ( rev_set_formula_a_b @ Xs ) ) ) ).
% fold_Cons_rev
thf(fact_343_not__distinct__decomp,axiom,
! [Ws: list_l5948067055324603441la_a_b] :
( ~ ( distin27490427618170344la_a_b @ Ws )
=> ? [Xs3: list_l5948067055324603441la_a_b,Ys3: list_l5948067055324603441la_a_b,Zs2: list_l5948067055324603441la_a_b,Y3: list_formula_a_b] :
( Ws
= ( append58666856857280070la_a_b @ Xs3 @ ( append58666856857280070la_a_b @ ( cons_l2450299273141159147la_a_b @ Y3 @ nil_list_formula_a_b ) @ ( append58666856857280070la_a_b @ Ys3 @ ( append58666856857280070la_a_b @ ( cons_l2450299273141159147la_a_b @ Y3 @ nil_list_formula_a_b ) @ Zs2 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_344_not__distinct__decomp,axiom,
! [Ws: list_formula_a_b] :
( ~ ( distinct_formula_a_b @ Ws )
=> ? [Xs3: list_formula_a_b,Ys3: list_formula_a_b,Zs2: list_formula_a_b,Y3: formula_a_b] :
( Ws
= ( append_formula_a_b @ Xs3 @ ( append_formula_a_b @ ( cons_formula_a_b @ Y3 @ nil_formula_a_b ) @ ( append_formula_a_b @ Ys3 @ ( append_formula_a_b @ ( cons_formula_a_b @ Y3 @ nil_formula_a_b ) @ Zs2 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_345_not__distinct__decomp,axiom,
! [Ws: list_set_formula_a_b] :
( ~ ( distin5059275051008138254la_a_b @ Ws )
=> ? [Xs3: list_set_formula_a_b,Ys3: list_set_formula_a_b,Zs2: list_set_formula_a_b,Y3: set_formula_a_b] :
( Ws
= ( append8577002312234529644la_a_b @ Xs3 @ ( append8577002312234529644la_a_b @ ( cons_set_formula_a_b @ Y3 @ nil_set_formula_a_b ) @ ( append8577002312234529644la_a_b @ Ys3 @ ( append8577002312234529644la_a_b @ ( cons_set_formula_a_b @ Y3 @ nil_set_formula_a_b ) @ Zs2 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_346_dropWhile__eq__Cons__conv,axiom,
! [P: set_formula_a_b > $o,Xs: list_set_formula_a_b,Y: set_formula_a_b,Ys: list_set_formula_a_b] :
( ( ( dropWh5803627183431294944la_a_b @ P @ Xs )
= ( cons_set_formula_a_b @ Y @ Ys ) )
= ( ( Xs
= ( append8577002312234529644la_a_b @ ( takeWh1298768865834877576la_a_b @ P @ Xs ) @ ( cons_set_formula_a_b @ Y @ Ys ) ) )
& ~ ( P @ Y ) ) ) ).
% dropWhile_eq_Cons_conv
thf(fact_347_dropWhile__eq__Cons__conv,axiom,
! [P: formula_a_b > $o,Xs: list_formula_a_b,Y: formula_a_b,Ys: list_formula_a_b] :
( ( ( dropWh8768975241461415850la_a_b @ P @ Xs )
= ( cons_formula_a_b @ Y @ Ys ) )
= ( ( Xs
= ( append_formula_a_b @ ( takeWh5313468693702967890la_a_b @ P @ Xs ) @ ( cons_formula_a_b @ Y @ Ys ) ) )
& ~ ( P @ Y ) ) ) ).
% dropWhile_eq_Cons_conv
thf(fact_348_insert__Nil,axiom,
! [X: formula_a_b] :
( ( insert_formula_a_b @ X @ nil_formula_a_b )
= ( cons_formula_a_b @ X @ nil_formula_a_b ) ) ).
% insert_Nil
thf(fact_349_insert__Nil,axiom,
! [X: set_formula_a_b] :
( ( insert6994445989384232875la_a_b @ X @ nil_set_formula_a_b )
= ( cons_set_formula_a_b @ X @ nil_set_formula_a_b ) ) ).
% insert_Nil
thf(fact_350_fold__remove1__split,axiom,
! [Xs: list_formula_a_b,F: formula_a_b > list_formula_a_b > list_formula_a_b,X: formula_a_b] :
( ! [X3: formula_a_b,Y3: formula_a_b] :
( ( member_formula_a_b2 @ X3 @ ( set_formula_a_b2 @ Xs ) )
=> ( ( member_formula_a_b2 @ Y3 @ ( set_formula_a_b2 @ Xs ) )
=> ( ( comp_l1221189711022459592la_a_b @ ( F @ X3 ) @ ( F @ Y3 ) )
= ( comp_l1221189711022459592la_a_b @ ( F @ Y3 ) @ ( F @ X3 ) ) ) ) )
=> ( ( member_formula_a_b2 @ X @ ( set_formula_a_b2 @ Xs ) )
=> ( ( fold_f7464356806857031615la_a_b @ F @ Xs )
= ( comp_l1221189711022459592la_a_b @ ( fold_f7464356806857031615la_a_b @ F @ ( remove1_formula_a_b @ X @ Xs ) ) @ ( F @ X ) ) ) ) ) ).
% fold_remove1_split
thf(fact_351_fold__remove1__split,axiom,
! [Xs: list_formula_a_b,F: formula_a_b > set_formula_a_b > set_formula_a_b,X: formula_a_b] :
( ! [X3: formula_a_b,Y3: formula_a_b] :
( ( member_formula_a_b2 @ X3 @ ( set_formula_a_b2 @ Xs ) )
=> ( ( member_formula_a_b2 @ Y3 @ ( set_formula_a_b2 @ Xs ) )
=> ( ( comp_s7716128494983798842la_a_b @ ( F @ X3 ) @ ( F @ Y3 ) )
= ( comp_s7716128494983798842la_a_b @ ( F @ Y3 ) @ ( F @ X3 ) ) ) ) )
=> ( ( member_formula_a_b2 @ X @ ( set_formula_a_b2 @ Xs ) )
=> ( ( fold_f470054086739575141la_a_b @ F @ Xs )
= ( comp_s7716128494983798842la_a_b @ ( fold_f470054086739575141la_a_b @ F @ ( remove1_formula_a_b @ X @ Xs ) ) @ ( F @ X ) ) ) ) ) ).
% fold_remove1_split
thf(fact_352_fold__remove1__split,axiom,
! [Xs: list_set_formula_a_b,F: set_formula_a_b > set_formula_a_b > set_formula_a_b,X: set_formula_a_b] :
( ! [X3: set_formula_a_b,Y3: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X3 @ ( set_set_formula_a_b2 @ Xs ) )
=> ( ( member6621951321062823262la_a_b @ Y3 @ ( set_set_formula_a_b2 @ Xs ) )
=> ( ( comp_s7716128494983798842la_a_b @ ( F @ X3 ) @ ( F @ Y3 ) )
= ( comp_s7716128494983798842la_a_b @ ( F @ Y3 ) @ ( F @ X3 ) ) ) ) )
=> ( ( member6621951321062823262la_a_b @ X @ ( set_set_formula_a_b2 @ Xs ) )
=> ( ( fold_s5097478484676794267la_a_b @ F @ Xs )
= ( comp_s7716128494983798842la_a_b @ ( fold_s5097478484676794267la_a_b @ F @ ( remove3026683253030310187la_a_b @ X @ Xs ) ) @ ( F @ X ) ) ) ) ) ).
% fold_remove1_split
thf(fact_353_rev__rev__ident,axiom,
! [Xs: list_formula_a_b] :
( ( rev_formula_a_b @ ( rev_formula_a_b @ Xs ) )
= Xs ) ).
% rev_rev_ident
thf(fact_354_rev__is__rev__conv,axiom,
! [Xs: list_formula_a_b,Ys: list_formula_a_b] :
( ( ( rev_formula_a_b @ Xs )
= ( rev_formula_a_b @ Ys ) )
= ( Xs = Ys ) ) ).
% rev_is_rev_conv
thf(fact_355_takeWhile__idem,axiom,
! [P: formula_a_b > $o,Xs: list_formula_a_b] :
( ( takeWh5313468693702967890la_a_b @ P @ ( takeWh5313468693702967890la_a_b @ P @ Xs ) )
= ( takeWh5313468693702967890la_a_b @ P @ Xs ) ) ).
% takeWhile_idem
thf(fact_356_append_Oright__neutral,axiom,
! [A2: list_formula_a_b] :
( ( append_formula_a_b @ A2 @ nil_formula_a_b )
= A2 ) ).
% append.right_neutral
thf(fact_357_append__Nil2,axiom,
! [Xs: list_formula_a_b] :
( ( append_formula_a_b @ Xs @ nil_formula_a_b )
= Xs ) ).
% append_Nil2
thf(fact_358_append__self__conv,axiom,
! [Xs: list_formula_a_b,Ys: list_formula_a_b] :
( ( ( append_formula_a_b @ Xs @ Ys )
= Xs )
= ( Ys = nil_formula_a_b ) ) ).
% append_self_conv
thf(fact_359_self__append__conv,axiom,
! [Y: list_formula_a_b,Ys: list_formula_a_b] :
( ( Y
= ( append_formula_a_b @ Y @ Ys ) )
= ( Ys = nil_formula_a_b ) ) ).
% self_append_conv
thf(fact_360_append__self__conv2,axiom,
! [Xs: list_formula_a_b,Ys: list_formula_a_b] :
( ( ( append_formula_a_b @ Xs @ Ys )
= Ys )
= ( Xs = nil_formula_a_b ) ) ).
% append_self_conv2
thf(fact_361_self__append__conv2,axiom,
! [Y: list_formula_a_b,Xs: list_formula_a_b] :
( ( Y
= ( append_formula_a_b @ Xs @ Y ) )
= ( Xs = nil_formula_a_b ) ) ).
% self_append_conv2
thf(fact_362_Nil__is__append__conv,axiom,
! [Xs: list_formula_a_b,Ys: list_formula_a_b] :
( ( nil_formula_a_b
= ( append_formula_a_b @ Xs @ Ys ) )
= ( ( Xs = nil_formula_a_b )
& ( Ys = nil_formula_a_b ) ) ) ).
% Nil_is_append_conv
thf(fact_363_append__is__Nil__conv,axiom,
! [Xs: list_formula_a_b,Ys: list_formula_a_b] :
( ( ( append_formula_a_b @ Xs @ Ys )
= nil_formula_a_b )
= ( ( Xs = nil_formula_a_b )
& ( Ys = nil_formula_a_b ) ) ) ).
% append_is_Nil_conv
thf(fact_364_Nil__is__rev__conv,axiom,
! [Xs: list_formula_a_b] :
( ( nil_formula_a_b
= ( rev_formula_a_b @ Xs ) )
= ( Xs = nil_formula_a_b ) ) ).
% Nil_is_rev_conv
thf(fact_365_rev__is__Nil__conv,axiom,
! [Xs: list_formula_a_b] :
( ( ( rev_formula_a_b @ Xs )
= nil_formula_a_b )
= ( Xs = nil_formula_a_b ) ) ).
% rev_is_Nil_conv
thf(fact_366_set__rev,axiom,
! [Xs: list_set_formula_a_b] :
( ( set_set_formula_a_b2 @ ( rev_set_formula_a_b @ Xs ) )
= ( set_set_formula_a_b2 @ Xs ) ) ).
% set_rev
thf(fact_367_set__rev,axiom,
! [Xs: list_l5948067055324603441la_a_b] :
( ( set_list_formula_a_b2 @ ( rev_list_formula_a_b @ Xs ) )
= ( set_list_formula_a_b2 @ Xs ) ) ).
% set_rev
thf(fact_368_set__rev,axiom,
! [Xs: list_formula_a_b] :
( ( set_formula_a_b2 @ ( rev_formula_a_b @ Xs ) )
= ( set_formula_a_b2 @ Xs ) ) ).
% set_rev
thf(fact_369_rev__append,axiom,
! [Xs: list_formula_a_b,Ys: list_formula_a_b] :
( ( rev_formula_a_b @ ( append_formula_a_b @ Xs @ Ys ) )
= ( append_formula_a_b @ ( rev_formula_a_b @ Ys ) @ ( rev_formula_a_b @ Xs ) ) ) ).
% rev_append
thf(fact_370_takeWhile__eq__all__conv,axiom,
! [P: set_formula_a_b > $o,Xs: list_set_formula_a_b] :
( ( ( takeWh1298768865834877576la_a_b @ P @ Xs )
= Xs )
= ( ! [X2: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X2 @ ( set_set_formula_a_b2 @ Xs ) )
=> ( P @ X2 ) ) ) ) ).
% takeWhile_eq_all_conv
thf(fact_371_takeWhile__eq__all__conv,axiom,
! [P: list_formula_a_b > $o,Xs: list_l5948067055324603441la_a_b] :
( ( ( takeWh2044909569185849186la_a_b @ P @ Xs )
= Xs )
= ( ! [X2: list_formula_a_b] :
( ( member7703986999711672120la_a_b @ X2 @ ( set_list_formula_a_b2 @ Xs ) )
=> ( P @ X2 ) ) ) ) ).
% takeWhile_eq_all_conv
thf(fact_372_takeWhile__eq__all__conv,axiom,
! [P: formula_a_b > $o,Xs: list_formula_a_b] :
( ( ( takeWh5313468693702967890la_a_b @ P @ Xs )
= Xs )
= ( ! [X2: formula_a_b] :
( ( member_formula_a_b2 @ X2 @ ( set_formula_a_b2 @ Xs ) )
=> ( P @ X2 ) ) ) ) ).
% takeWhile_eq_all_conv
thf(fact_373_Nil__in__shuffles,axiom,
! [Xs: list_formula_a_b,Ys: list_formula_a_b] :
( ( member7703986999711672120la_a_b @ nil_formula_a_b @ ( shuffles_formula_a_b @ Xs @ Ys ) )
= ( ( Xs = nil_formula_a_b )
& ( Ys = nil_formula_a_b ) ) ) ).
% Nil_in_shuffles
thf(fact_374_distinct__rev,axiom,
! [Xs: list_formula_a_b] :
( ( distinct_formula_a_b @ ( rev_formula_a_b @ Xs ) )
= ( distinct_formula_a_b @ Xs ) ) ).
% distinct_rev
thf(fact_375_distinct__rev,axiom,
! [Xs: list_l5948067055324603441la_a_b] :
( ( distin27490427618170344la_a_b @ ( rev_list_formula_a_b @ Xs ) )
= ( distin27490427618170344la_a_b @ Xs ) ) ).
% distinct_rev
thf(fact_376_distinct__rev,axiom,
! [Xs: list_set_formula_a_b] :
( ( distin5059275051008138254la_a_b @ ( rev_set_formula_a_b @ Xs ) )
= ( distin5059275051008138254la_a_b @ Xs ) ) ).
% distinct_rev
thf(fact_377_rotate1__is__Nil__conv,axiom,
! [Xs: list_formula_a_b] :
( ( ( rotate1_formula_a_b @ Xs )
= nil_formula_a_b )
= ( Xs = nil_formula_a_b ) ) ).
% rotate1_is_Nil_conv
thf(fact_378_splice__Nil2,axiom,
! [Xs: list_formula_a_b] :
( ( splice_formula_a_b @ Xs @ nil_formula_a_b )
= Xs ) ).
% splice_Nil2
thf(fact_379_split__Nil__iff,axiom,
! [Xs: list_formula_a_b,Ys: list_formula_a_b] :
( ( ( splice_formula_a_b @ Xs @ Ys )
= nil_formula_a_b )
= ( ( Xs = nil_formula_a_b )
& ( Ys = nil_formula_a_b ) ) ) ).
% split_Nil_iff
thf(fact_380_bind__simps_I1_J,axiom,
! [F: formula_a_b > list_formula_a_b] :
( ( bind_f552093402373037483la_a_b @ nil_formula_a_b @ F )
= nil_formula_a_b ) ).
% bind_simps(1)
thf(fact_381_list__ex1__simps_I1_J,axiom,
! [P: formula_a_b > $o] :
~ ( list_ex1_formula_a_b @ P @ nil_formula_a_b ) ).
% list_ex1_simps(1)
thf(fact_382_append1__eq__conv,axiom,
! [Xs: list_formula_a_b,X: formula_a_b,Ys: list_formula_a_b,Y: formula_a_b] :
( ( ( append_formula_a_b @ Xs @ ( cons_formula_a_b @ X @ nil_formula_a_b ) )
= ( append_formula_a_b @ Ys @ ( cons_formula_a_b @ Y @ nil_formula_a_b ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_383_append1__eq__conv,axiom,
! [Xs: list_set_formula_a_b,X: set_formula_a_b,Ys: list_set_formula_a_b,Y: set_formula_a_b] :
( ( ( append8577002312234529644la_a_b @ Xs @ ( cons_set_formula_a_b @ X @ nil_set_formula_a_b ) )
= ( append8577002312234529644la_a_b @ Ys @ ( cons_set_formula_a_b @ Y @ nil_set_formula_a_b ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_384_rev__singleton__conv,axiom,
! [Xs: list_formula_a_b,X: formula_a_b] :
( ( ( rev_formula_a_b @ Xs )
= ( cons_formula_a_b @ X @ nil_formula_a_b ) )
= ( Xs
= ( cons_formula_a_b @ X @ nil_formula_a_b ) ) ) ).
% rev_singleton_conv
thf(fact_385_rev__singleton__conv,axiom,
! [Xs: list_set_formula_a_b,X: set_formula_a_b] :
( ( ( rev_set_formula_a_b @ Xs )
= ( cons_set_formula_a_b @ X @ nil_set_formula_a_b ) )
= ( Xs
= ( cons_set_formula_a_b @ X @ nil_set_formula_a_b ) ) ) ).
% rev_singleton_conv
thf(fact_386_singleton__rev__conv,axiom,
! [X: formula_a_b,Xs: list_formula_a_b] :
( ( ( cons_formula_a_b @ X @ nil_formula_a_b )
= ( rev_formula_a_b @ Xs ) )
= ( ( cons_formula_a_b @ X @ nil_formula_a_b )
= Xs ) ) ).
% singleton_rev_conv
thf(fact_387_singleton__rev__conv,axiom,
! [X: set_formula_a_b,Xs: list_set_formula_a_b] :
( ( ( cons_set_formula_a_b @ X @ nil_set_formula_a_b )
= ( rev_set_formula_a_b @ Xs ) )
= ( ( cons_set_formula_a_b @ X @ nil_set_formula_a_b )
= Xs ) ) ).
% singleton_rev_conv
thf(fact_388_dropWhile__eq__Nil__conv,axiom,
! [P: set_formula_a_b > $o,Xs: list_set_formula_a_b] :
( ( ( dropWh5803627183431294944la_a_b @ P @ Xs )
= nil_set_formula_a_b )
= ( ! [X2: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X2 @ ( set_set_formula_a_b2 @ Xs ) )
=> ( P @ X2 ) ) ) ) ).
% dropWhile_eq_Nil_conv
thf(fact_389_dropWhile__eq__Nil__conv,axiom,
! [P: list_formula_a_b > $o,Xs: list_l5948067055324603441la_a_b] :
( ( ( dropWh8739902238574629050la_a_b @ P @ Xs )
= nil_list_formula_a_b )
= ( ! [X2: list_formula_a_b] :
( ( member7703986999711672120la_a_b @ X2 @ ( set_list_formula_a_b2 @ Xs ) )
=> ( P @ X2 ) ) ) ) ).
% dropWhile_eq_Nil_conv
thf(fact_390_dropWhile__eq__Nil__conv,axiom,
! [P: formula_a_b > $o,Xs: list_formula_a_b] :
( ( ( dropWh8768975241461415850la_a_b @ P @ Xs )
= nil_formula_a_b )
= ( ! [X2: formula_a_b] :
( ( member_formula_a_b2 @ X2 @ ( set_formula_a_b2 @ Xs ) )
=> ( P @ X2 ) ) ) ) ).
% dropWhile_eq_Nil_conv
thf(fact_391_fold__append,axiom,
! [F: set_formula_a_b > set_formula_a_b > set_formula_a_b,Xs: list_set_formula_a_b,Ys: list_set_formula_a_b] :
( ( fold_s5097478484676794267la_a_b @ F @ ( append8577002312234529644la_a_b @ Xs @ Ys ) )
= ( comp_s7716128494983798842la_a_b @ ( fold_s5097478484676794267la_a_b @ F @ Ys ) @ ( fold_s5097478484676794267la_a_b @ F @ Xs ) ) ) ).
% fold_append
thf(fact_392_fold__append,axiom,
! [F: formula_a_b > list_formula_a_b > list_formula_a_b,Xs: list_formula_a_b,Ys: list_formula_a_b] :
( ( fold_f7464356806857031615la_a_b @ F @ ( append_formula_a_b @ Xs @ Ys ) )
= ( comp_l1221189711022459592la_a_b @ ( fold_f7464356806857031615la_a_b @ F @ Ys ) @ ( fold_f7464356806857031615la_a_b @ F @ Xs ) ) ) ).
% fold_append
thf(fact_393_fold__append,axiom,
! [F: formula_a_b > set_formula_a_b > set_formula_a_b,Xs: list_formula_a_b,Ys: list_formula_a_b] :
( ( fold_f470054086739575141la_a_b @ F @ ( append_formula_a_b @ Xs @ Ys ) )
= ( comp_s7716128494983798842la_a_b @ ( fold_f470054086739575141la_a_b @ F @ Ys ) @ ( fold_f470054086739575141la_a_b @ F @ Xs ) ) ) ).
% fold_append
thf(fact_394_takeWhile__append2,axiom,
! [Xs: list_set_formula_a_b,P: set_formula_a_b > $o,Ys: list_set_formula_a_b] :
( ! [X3: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X3 @ ( set_set_formula_a_b2 @ Xs ) )
=> ( P @ X3 ) )
=> ( ( takeWh1298768865834877576la_a_b @ P @ ( append8577002312234529644la_a_b @ Xs @ Ys ) )
= ( append8577002312234529644la_a_b @ Xs @ ( takeWh1298768865834877576la_a_b @ P @ Ys ) ) ) ) ).
% takeWhile_append2
thf(fact_395_takeWhile__append2,axiom,
! [Xs: list_l5948067055324603441la_a_b,P: list_formula_a_b > $o,Ys: list_l5948067055324603441la_a_b] :
( ! [X3: list_formula_a_b] :
( ( member7703986999711672120la_a_b @ X3 @ ( set_list_formula_a_b2 @ Xs ) )
=> ( P @ X3 ) )
=> ( ( takeWh2044909569185849186la_a_b @ P @ ( append58666856857280070la_a_b @ Xs @ Ys ) )
= ( append58666856857280070la_a_b @ Xs @ ( takeWh2044909569185849186la_a_b @ P @ Ys ) ) ) ) ).
% takeWhile_append2
thf(fact_396_takeWhile__append2,axiom,
! [Xs: list_formula_a_b,P: formula_a_b > $o,Ys: list_formula_a_b] :
( ! [X3: formula_a_b] :
( ( member_formula_a_b2 @ X3 @ ( set_formula_a_b2 @ Xs ) )
=> ( P @ X3 ) )
=> ( ( takeWh5313468693702967890la_a_b @ P @ ( append_formula_a_b @ Xs @ Ys ) )
= ( append_formula_a_b @ Xs @ ( takeWh5313468693702967890la_a_b @ P @ Ys ) ) ) ) ).
% takeWhile_append2
thf(fact_397_takeWhile__append1,axiom,
! [X: set_formula_a_b,Xs: list_set_formula_a_b,P: set_formula_a_b > $o,Ys: list_set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X @ ( set_set_formula_a_b2 @ Xs ) )
=> ( ~ ( P @ X )
=> ( ( takeWh1298768865834877576la_a_b @ P @ ( append8577002312234529644la_a_b @ Xs @ Ys ) )
= ( takeWh1298768865834877576la_a_b @ P @ Xs ) ) ) ) ).
% takeWhile_append1
thf(fact_398_takeWhile__append1,axiom,
! [X: list_formula_a_b,Xs: list_l5948067055324603441la_a_b,P: list_formula_a_b > $o,Ys: list_l5948067055324603441la_a_b] :
( ( member7703986999711672120la_a_b @ X @ ( set_list_formula_a_b2 @ Xs ) )
=> ( ~ ( P @ X )
=> ( ( takeWh2044909569185849186la_a_b @ P @ ( append58666856857280070la_a_b @ Xs @ Ys ) )
= ( takeWh2044909569185849186la_a_b @ P @ Xs ) ) ) ) ).
% takeWhile_append1
thf(fact_399_takeWhile__append1,axiom,
! [X: formula_a_b,Xs: list_formula_a_b,P: formula_a_b > $o,Ys: list_formula_a_b] :
( ( member_formula_a_b2 @ X @ ( set_formula_a_b2 @ Xs ) )
=> ( ~ ( P @ X )
=> ( ( takeWh5313468693702967890la_a_b @ P @ ( append_formula_a_b @ Xs @ Ys ) )
= ( takeWh5313468693702967890la_a_b @ P @ Xs ) ) ) ) ).
% takeWhile_append1
thf(fact_400_takeWhile__dropWhile__id,axiom,
! [P: formula_a_b > $o,Xs: list_formula_a_b] :
( ( append_formula_a_b @ ( takeWh5313468693702967890la_a_b @ P @ Xs ) @ ( dropWh8768975241461415850la_a_b @ P @ Xs ) )
= Xs ) ).
% takeWhile_dropWhile_id
thf(fact_401_rev__eq__Cons__iff,axiom,
! [Xs: list_formula_a_b,Y: formula_a_b,Ys: list_formula_a_b] :
( ( ( rev_formula_a_b @ Xs )
= ( cons_formula_a_b @ Y @ Ys ) )
= ( Xs
= ( append_formula_a_b @ ( rev_formula_a_b @ Ys ) @ ( cons_formula_a_b @ Y @ nil_formula_a_b ) ) ) ) ).
% rev_eq_Cons_iff
thf(fact_402_rev__eq__Cons__iff,axiom,
! [Xs: list_set_formula_a_b,Y: set_formula_a_b,Ys: list_set_formula_a_b] :
( ( ( rev_set_formula_a_b @ Xs )
= ( cons_set_formula_a_b @ Y @ Ys ) )
= ( Xs
= ( append8577002312234529644la_a_b @ ( rev_set_formula_a_b @ Ys ) @ ( cons_set_formula_a_b @ Y @ nil_set_formula_a_b ) ) ) ) ).
% rev_eq_Cons_iff
thf(fact_403_rev_Osimps_I1_J,axiom,
( ( rev_formula_a_b @ nil_formula_a_b )
= nil_formula_a_b ) ).
% rev.simps(1)
thf(fact_404_maps__simps_I2_J,axiom,
! [F: formula_a_b > list_formula_a_b] :
( ( maps_f4007179076772103781la_a_b @ F @ nil_formula_a_b )
= nil_formula_a_b ) ).
% maps_simps(2)
thf(fact_405_takeWhile_Osimps_I1_J,axiom,
! [P: formula_a_b > $o] :
( ( takeWh5313468693702967890la_a_b @ P @ nil_formula_a_b )
= nil_formula_a_b ) ).
% takeWhile.simps(1)
thf(fact_406_rev__swap,axiom,
! [Xs: list_formula_a_b,Ys: list_formula_a_b] :
( ( ( rev_formula_a_b @ Xs )
= Ys )
= ( Xs
= ( rev_formula_a_b @ Ys ) ) ) ).
% rev_swap
thf(fact_407_dropWhile__dropWhile2,axiom,
! [P: formula_a_b > $o,Q: formula_a_b > $o,Xs: list_formula_a_b] :
( ! [X3: formula_a_b] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ( takeWh5313468693702967890la_a_b @ P @ ( takeWh5313468693702967890la_a_b @ Q @ Xs ) )
= ( takeWh5313468693702967890la_a_b @ P @ Xs ) ) ) ).
% dropWhile_dropWhile2
thf(fact_408_takeWhile_Osimps_I2_J,axiom,
! [P: set_formula_a_b > $o,X: set_formula_a_b,Xs: list_set_formula_a_b] :
( ( ( P @ X )
=> ( ( takeWh1298768865834877576la_a_b @ P @ ( cons_set_formula_a_b @ X @ Xs ) )
= ( cons_set_formula_a_b @ X @ ( takeWh1298768865834877576la_a_b @ P @ Xs ) ) ) )
& ( ~ ( P @ X )
=> ( ( takeWh1298768865834877576la_a_b @ P @ ( cons_set_formula_a_b @ X @ Xs ) )
= nil_set_formula_a_b ) ) ) ).
% takeWhile.simps(2)
thf(fact_409_takeWhile_Osimps_I2_J,axiom,
! [P: formula_a_b > $o,X: formula_a_b,Xs: list_formula_a_b] :
( ( ( P @ X )
=> ( ( takeWh5313468693702967890la_a_b @ P @ ( cons_formula_a_b @ X @ Xs ) )
= ( cons_formula_a_b @ X @ ( takeWh5313468693702967890la_a_b @ P @ Xs ) ) ) )
& ( ~ ( P @ X )
=> ( ( takeWh5313468693702967890la_a_b @ P @ ( cons_formula_a_b @ X @ Xs ) )
= nil_formula_a_b ) ) ) ).
% takeWhile.simps(2)
thf(fact_410_transpose_Ocases,axiom,
! [X: list_l5948067055324603441la_a_b] :
( ( X != nil_list_formula_a_b )
=> ( ! [Xss: list_l5948067055324603441la_a_b] :
( X
!= ( cons_l2450299273141159147la_a_b @ nil_formula_a_b @ Xss ) )
=> ~ ! [X3: formula_a_b,Xs3: list_formula_a_b,Xss: list_l5948067055324603441la_a_b] :
( X
!= ( cons_l2450299273141159147la_a_b @ ( cons_formula_a_b @ X3 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_411_transpose_Ocases,axiom,
! [X: list_l147921316930177639la_a_b] :
( ( X != nil_li5663101814674166225la_a_b )
=> ( ! [Xss: list_l147921316930177639la_a_b] :
( X
!= ( cons_l7517568784372355617la_a_b @ nil_set_formula_a_b @ Xss ) )
=> ~ ! [X3: set_formula_a_b,Xs3: list_set_formula_a_b,Xss: list_l147921316930177639la_a_b] :
( X
!= ( cons_l7517568784372355617la_a_b @ ( cons_set_formula_a_b @ X3 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_412_fold__rev,axiom,
! [Xs: list_formula_a_b,F: formula_a_b > list_formula_a_b > list_formula_a_b] :
( ! [X3: formula_a_b,Y3: formula_a_b] :
( ( member_formula_a_b2 @ X3 @ ( set_formula_a_b2 @ Xs ) )
=> ( ( member_formula_a_b2 @ Y3 @ ( set_formula_a_b2 @ Xs ) )
=> ( ( comp_l1221189711022459592la_a_b @ ( F @ Y3 ) @ ( F @ X3 ) )
= ( comp_l1221189711022459592la_a_b @ ( F @ X3 ) @ ( F @ Y3 ) ) ) ) )
=> ( ( fold_f7464356806857031615la_a_b @ F @ ( rev_formula_a_b @ Xs ) )
= ( fold_f7464356806857031615la_a_b @ F @ Xs ) ) ) ).
% fold_rev
thf(fact_413_fold__rev,axiom,
! [Xs: list_formula_a_b,F: formula_a_b > set_formula_a_b > set_formula_a_b] :
( ! [X3: formula_a_b,Y3: formula_a_b] :
( ( member_formula_a_b2 @ X3 @ ( set_formula_a_b2 @ Xs ) )
=> ( ( member_formula_a_b2 @ Y3 @ ( set_formula_a_b2 @ Xs ) )
=> ( ( comp_s7716128494983798842la_a_b @ ( F @ Y3 ) @ ( F @ X3 ) )
= ( comp_s7716128494983798842la_a_b @ ( F @ X3 ) @ ( F @ Y3 ) ) ) ) )
=> ( ( fold_f470054086739575141la_a_b @ F @ ( rev_formula_a_b @ Xs ) )
= ( fold_f470054086739575141la_a_b @ F @ Xs ) ) ) ).
% fold_rev
thf(fact_414_fold__rev,axiom,
! [Xs: list_set_formula_a_b,F: set_formula_a_b > set_formula_a_b > set_formula_a_b] :
( ! [X3: set_formula_a_b,Y3: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X3 @ ( set_set_formula_a_b2 @ Xs ) )
=> ( ( member6621951321062823262la_a_b @ Y3 @ ( set_set_formula_a_b2 @ Xs ) )
=> ( ( comp_s7716128494983798842la_a_b @ ( F @ Y3 ) @ ( F @ X3 ) )
= ( comp_s7716128494983798842la_a_b @ ( F @ X3 ) @ ( F @ Y3 ) ) ) ) )
=> ( ( fold_s5097478484676794267la_a_b @ F @ ( rev_set_formula_a_b @ Xs ) )
= ( fold_s5097478484676794267la_a_b @ F @ Xs ) ) ) ).
% fold_rev
thf(fact_415_rev_Osimps_I2_J,axiom,
! [X: formula_a_b,Xs: list_formula_a_b] :
( ( rev_formula_a_b @ ( cons_formula_a_b @ X @ Xs ) )
= ( append_formula_a_b @ ( rev_formula_a_b @ Xs ) @ ( cons_formula_a_b @ X @ nil_formula_a_b ) ) ) ).
% rev.simps(2)
thf(fact_416_rev_Osimps_I2_J,axiom,
! [X: set_formula_a_b,Xs: list_set_formula_a_b] :
( ( rev_set_formula_a_b @ ( cons_set_formula_a_b @ X @ Xs ) )
= ( append8577002312234529644la_a_b @ ( rev_set_formula_a_b @ Xs ) @ ( cons_set_formula_a_b @ X @ nil_set_formula_a_b ) ) ) ).
% rev.simps(2)
thf(fact_417_set__takeWhileD,axiom,
! [X: set_formula_a_b,P: set_formula_a_b > $o,Xs: list_set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X @ ( set_set_formula_a_b2 @ ( takeWh1298768865834877576la_a_b @ P @ Xs ) ) )
=> ( ( member6621951321062823262la_a_b @ X @ ( set_set_formula_a_b2 @ Xs ) )
& ( P @ X ) ) ) ).
% set_takeWhileD
thf(fact_418_set__takeWhileD,axiom,
! [X: list_formula_a_b,P: list_formula_a_b > $o,Xs: list_l5948067055324603441la_a_b] :
( ( member7703986999711672120la_a_b @ X @ ( set_list_formula_a_b2 @ ( takeWh2044909569185849186la_a_b @ P @ Xs ) ) )
=> ( ( member7703986999711672120la_a_b @ X @ ( set_list_formula_a_b2 @ Xs ) )
& ( P @ X ) ) ) ).
% set_takeWhileD
thf(fact_419_set__takeWhileD,axiom,
! [X: formula_a_b,P: formula_a_b > $o,Xs: list_formula_a_b] :
( ( member_formula_a_b2 @ X @ ( set_formula_a_b2 @ ( takeWh5313468693702967890la_a_b @ P @ Xs ) ) )
=> ( ( member_formula_a_b2 @ X @ ( set_formula_a_b2 @ Xs ) )
& ( P @ X ) ) ) ).
% set_takeWhileD
thf(fact_420_takeWhile__cong,axiom,
! [L: list_set_formula_a_b,K: list_set_formula_a_b,P: set_formula_a_b > $o,Q: set_formula_a_b > $o] :
( ( L = K )
=> ( ! [X3: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X3 @ ( set_set_formula_a_b2 @ L ) )
=> ( ( P @ X3 )
= ( Q @ X3 ) ) )
=> ( ( takeWh1298768865834877576la_a_b @ P @ L )
= ( takeWh1298768865834877576la_a_b @ Q @ K ) ) ) ) ).
% takeWhile_cong
thf(fact_421_takeWhile__cong,axiom,
! [L: list_l5948067055324603441la_a_b,K: list_l5948067055324603441la_a_b,P: list_formula_a_b > $o,Q: list_formula_a_b > $o] :
( ( L = K )
=> ( ! [X3: list_formula_a_b] :
( ( member7703986999711672120la_a_b @ X3 @ ( set_list_formula_a_b2 @ L ) )
=> ( ( P @ X3 )
= ( Q @ X3 ) ) )
=> ( ( takeWh2044909569185849186la_a_b @ P @ L )
= ( takeWh2044909569185849186la_a_b @ Q @ K ) ) ) ) ).
% takeWhile_cong
thf(fact_422_takeWhile__cong,axiom,
! [L: list_formula_a_b,K: list_formula_a_b,P: formula_a_b > $o,Q: formula_a_b > $o] :
( ( L = K )
=> ( ! [X3: formula_a_b] :
( ( member_formula_a_b2 @ X3 @ ( set_formula_a_b2 @ L ) )
=> ( ( P @ X3 )
= ( Q @ X3 ) ) )
=> ( ( takeWh5313468693702967890la_a_b @ P @ L )
= ( takeWh5313468693702967890la_a_b @ Q @ K ) ) ) ) ).
% takeWhile_cong
thf(fact_423_distinct__takeWhile,axiom,
! [Xs: list_l5948067055324603441la_a_b,P: list_formula_a_b > $o] :
( ( distin27490427618170344la_a_b @ Xs )
=> ( distin27490427618170344la_a_b @ ( takeWh2044909569185849186la_a_b @ P @ Xs ) ) ) ).
% distinct_takeWhile
thf(fact_424_distinct__takeWhile,axiom,
! [Xs: list_set_formula_a_b,P: set_formula_a_b > $o] :
( ( distin5059275051008138254la_a_b @ Xs )
=> ( distin5059275051008138254la_a_b @ ( takeWh1298768865834877576la_a_b @ P @ Xs ) ) ) ).
% distinct_takeWhile
thf(fact_425_distinct__takeWhile,axiom,
! [Xs: list_formula_a_b,P: formula_a_b > $o] :
( ( distinct_formula_a_b @ Xs )
=> ( distinct_formula_a_b @ ( takeWh5313468693702967890la_a_b @ P @ Xs ) ) ) ).
% distinct_takeWhile
thf(fact_426_rev__conv__fold,axiom,
( rev_formula_a_b
= ( ^ [Xs2: list_formula_a_b] : ( fold_f7464356806857031615la_a_b @ cons_formula_a_b @ Xs2 @ nil_formula_a_b ) ) ) ).
% rev_conv_fold
thf(fact_427_rev__conv__fold,axiom,
( rev_set_formula_a_b
= ( ^ [Xs2: list_set_formula_a_b] : ( fold_s1220939906018902059la_a_b @ cons_set_formula_a_b @ Xs2 @ nil_set_formula_a_b ) ) ) ).
% rev_conv_fold
thf(fact_428_list_Odistinct_I1_J,axiom,
! [X21: formula_a_b,X22: list_formula_a_b] :
( nil_formula_a_b
!= ( cons_formula_a_b @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_429_list_Odistinct_I1_J,axiom,
! [X21: set_formula_a_b,X22: list_set_formula_a_b] :
( nil_set_formula_a_b
!= ( cons_set_formula_a_b @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_430_list_OdiscI,axiom,
! [List: list_formula_a_b,X21: formula_a_b,X22: list_formula_a_b] :
( ( List
= ( cons_formula_a_b @ X21 @ X22 ) )
=> ( List != nil_formula_a_b ) ) ).
% list.discI
thf(fact_431_list_OdiscI,axiom,
! [List: list_set_formula_a_b,X21: set_formula_a_b,X22: list_set_formula_a_b] :
( ( List
= ( cons_set_formula_a_b @ X21 @ X22 ) )
=> ( List != nil_set_formula_a_b ) ) ).
% list.discI
thf(fact_432_list_Oexhaust,axiom,
! [Y: list_formula_a_b] :
( ( Y != nil_formula_a_b )
=> ~ ! [X212: formula_a_b,X222: list_formula_a_b] :
( Y
!= ( cons_formula_a_b @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_433_list_Oexhaust,axiom,
! [Y: list_set_formula_a_b] :
( ( Y != nil_set_formula_a_b )
=> ~ ! [X212: set_formula_a_b,X222: list_set_formula_a_b] :
( Y
!= ( cons_set_formula_a_b @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_434_min__list_Ocases,axiom,
! [X: list_set_formula_a_b] :
( ! [X3: set_formula_a_b,Xs3: list_set_formula_a_b] :
( X
!= ( cons_set_formula_a_b @ X3 @ Xs3 ) )
=> ( X = nil_set_formula_a_b ) ) ).
% min_list.cases
thf(fact_435_remdups__adj_Ocases,axiom,
! [X: list_formula_a_b] :
( ( X != nil_formula_a_b )
=> ( ! [X3: formula_a_b] :
( X
!= ( cons_formula_a_b @ X3 @ nil_formula_a_b ) )
=> ~ ! [X3: formula_a_b,Y3: formula_a_b,Xs3: list_formula_a_b] :
( X
!= ( cons_formula_a_b @ X3 @ ( cons_formula_a_b @ Y3 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_436_remdups__adj_Ocases,axiom,
! [X: list_set_formula_a_b] :
( ( X != nil_set_formula_a_b )
=> ( ! [X3: set_formula_a_b] :
( X
!= ( cons_set_formula_a_b @ X3 @ nil_set_formula_a_b ) )
=> ~ ! [X3: set_formula_a_b,Y3: set_formula_a_b,Xs3: list_set_formula_a_b] :
( X
!= ( cons_set_formula_a_b @ X3 @ ( cons_set_formula_a_b @ Y3 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_437_neq__Nil__conv,axiom,
! [Xs: list_formula_a_b] :
( ( Xs != nil_formula_a_b )
= ( ? [Y2: formula_a_b,Ys2: list_formula_a_b] :
( Xs
= ( cons_formula_a_b @ Y2 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_438_neq__Nil__conv,axiom,
! [Xs: list_set_formula_a_b] :
( ( Xs != nil_set_formula_a_b )
= ( ? [Y2: set_formula_a_b,Ys2: list_set_formula_a_b] :
( Xs
= ( cons_set_formula_a_b @ Y2 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_439_list__induct2_H,axiom,
! [P: list_formula_a_b > list_formula_a_b > $o,Xs: list_formula_a_b,Ys: list_formula_a_b] :
( ( P @ nil_formula_a_b @ nil_formula_a_b )
=> ( ! [X3: formula_a_b,Xs3: list_formula_a_b] : ( P @ ( cons_formula_a_b @ X3 @ Xs3 ) @ nil_formula_a_b )
=> ( ! [Y3: formula_a_b,Ys3: list_formula_a_b] : ( P @ nil_formula_a_b @ ( cons_formula_a_b @ Y3 @ Ys3 ) )
=> ( ! [X3: formula_a_b,Xs3: list_formula_a_b,Y3: formula_a_b,Ys3: list_formula_a_b] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_formula_a_b @ X3 @ Xs3 ) @ ( cons_formula_a_b @ Y3 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_440_list__induct2_H,axiom,
! [P: list_formula_a_b > list_set_formula_a_b > $o,Xs: list_formula_a_b,Ys: list_set_formula_a_b] :
( ( P @ nil_formula_a_b @ nil_set_formula_a_b )
=> ( ! [X3: formula_a_b,Xs3: list_formula_a_b] : ( P @ ( cons_formula_a_b @ X3 @ Xs3 ) @ nil_set_formula_a_b )
=> ( ! [Y3: set_formula_a_b,Ys3: list_set_formula_a_b] : ( P @ nil_formula_a_b @ ( cons_set_formula_a_b @ Y3 @ Ys3 ) )
=> ( ! [X3: formula_a_b,Xs3: list_formula_a_b,Y3: set_formula_a_b,Ys3: list_set_formula_a_b] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_formula_a_b @ X3 @ Xs3 ) @ ( cons_set_formula_a_b @ Y3 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_441_list__induct2_H,axiom,
! [P: list_set_formula_a_b > list_formula_a_b > $o,Xs: list_set_formula_a_b,Ys: list_formula_a_b] :
( ( P @ nil_set_formula_a_b @ nil_formula_a_b )
=> ( ! [X3: set_formula_a_b,Xs3: list_set_formula_a_b] : ( P @ ( cons_set_formula_a_b @ X3 @ Xs3 ) @ nil_formula_a_b )
=> ( ! [Y3: formula_a_b,Ys3: list_formula_a_b] : ( P @ nil_set_formula_a_b @ ( cons_formula_a_b @ Y3 @ Ys3 ) )
=> ( ! [X3: set_formula_a_b,Xs3: list_set_formula_a_b,Y3: formula_a_b,Ys3: list_formula_a_b] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_set_formula_a_b @ X3 @ Xs3 ) @ ( cons_formula_a_b @ Y3 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_442_list__induct2_H,axiom,
! [P: list_set_formula_a_b > list_set_formula_a_b > $o,Xs: list_set_formula_a_b,Ys: list_set_formula_a_b] :
( ( P @ nil_set_formula_a_b @ nil_set_formula_a_b )
=> ( ! [X3: set_formula_a_b,Xs3: list_set_formula_a_b] : ( P @ ( cons_set_formula_a_b @ X3 @ Xs3 ) @ nil_set_formula_a_b )
=> ( ! [Y3: set_formula_a_b,Ys3: list_set_formula_a_b] : ( P @ nil_set_formula_a_b @ ( cons_set_formula_a_b @ Y3 @ Ys3 ) )
=> ( ! [X3: set_formula_a_b,Xs3: list_set_formula_a_b,Y3: set_formula_a_b,Ys3: list_set_formula_a_b] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_set_formula_a_b @ X3 @ Xs3 ) @ ( cons_set_formula_a_b @ Y3 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_443_list__nonempty__induct,axiom,
! [Xs: list_formula_a_b,P: list_formula_a_b > $o] :
( ( Xs != nil_formula_a_b )
=> ( ! [X3: formula_a_b] : ( P @ ( cons_formula_a_b @ X3 @ nil_formula_a_b ) )
=> ( ! [X3: formula_a_b,Xs3: list_formula_a_b] :
( ( Xs3 != nil_formula_a_b )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_formula_a_b @ X3 @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_444_list__nonempty__induct,axiom,
! [Xs: list_set_formula_a_b,P: list_set_formula_a_b > $o] :
( ( Xs != nil_set_formula_a_b )
=> ( ! [X3: set_formula_a_b] : ( P @ ( cons_set_formula_a_b @ X3 @ nil_set_formula_a_b ) )
=> ( ! [X3: set_formula_a_b,Xs3: list_set_formula_a_b] :
( ( Xs3 != nil_set_formula_a_b )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_set_formula_a_b @ X3 @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_445_append__Nil,axiom,
! [Ys: list_formula_a_b] :
( ( append_formula_a_b @ nil_formula_a_b @ Ys )
= Ys ) ).
% append_Nil
thf(fact_446_append_Oleft__neutral,axiom,
! [A2: list_formula_a_b] :
( ( append_formula_a_b @ nil_formula_a_b @ A2 )
= A2 ) ).
% append.left_neutral
thf(fact_447_eq__Nil__appendI,axiom,
! [Xs: list_formula_a_b,Ys: list_formula_a_b] :
( ( Xs = Ys )
=> ( Xs
= ( append_formula_a_b @ nil_formula_a_b @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_448_distinct_Osimps_I1_J,axiom,
distin27490427618170344la_a_b @ nil_list_formula_a_b ).
% distinct.simps(1)
thf(fact_449_distinct_Osimps_I1_J,axiom,
distin5059275051008138254la_a_b @ nil_set_formula_a_b ).
% distinct.simps(1)
thf(fact_450_distinct_Osimps_I1_J,axiom,
distinct_formula_a_b @ nil_formula_a_b ).
% distinct.simps(1)
thf(fact_451_dropWhile__takeWhile,axiom,
! [P: formula_a_b > $o,Q: formula_a_b > $o,Xs: list_formula_a_b] :
( ! [X3: formula_a_b] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ( dropWh8768975241461415850la_a_b @ P @ ( takeWh5313468693702967890la_a_b @ Q @ Xs ) )
= ( takeWh5313468693702967890la_a_b @ Q @ ( dropWh8768975241461415850la_a_b @ P @ Xs ) ) ) ) ).
% dropWhile_takeWhile
thf(fact_452_fold__simps_I1_J,axiom,
! [F: set_formula_a_b > set_formula_a_b > set_formula_a_b,S: set_formula_a_b] :
( ( fold_s5097478484676794267la_a_b @ F @ nil_set_formula_a_b @ S )
= S ) ).
% fold_simps(1)
thf(fact_453_fold__simps_I1_J,axiom,
! [F: formula_a_b > list_formula_a_b > list_formula_a_b,S: list_formula_a_b] :
( ( fold_f7464356806857031615la_a_b @ F @ nil_formula_a_b @ S )
= S ) ).
% fold_simps(1)
thf(fact_454_fold__simps_I1_J,axiom,
! [F: formula_a_b > set_formula_a_b > set_formula_a_b,S: set_formula_a_b] :
( ( fold_f470054086739575141la_a_b @ F @ nil_formula_a_b @ S )
= S ) ).
% fold_simps(1)
thf(fact_455_Nil__in__shufflesI,axiom,
! [Xs: list_formula_a_b,Ys: list_formula_a_b] :
( ( Xs = nil_formula_a_b )
=> ( ( Ys = nil_formula_a_b )
=> ( member7703986999711672120la_a_b @ nil_formula_a_b @ ( shuffles_formula_a_b @ Xs @ Ys ) ) ) ) ).
% Nil_in_shufflesI
thf(fact_456_dropWhile_Osimps_I1_J,axiom,
! [P: formula_a_b > $o] :
( ( dropWh8768975241461415850la_a_b @ P @ nil_formula_a_b )
= nil_formula_a_b ) ).
% dropWhile.simps(1)
thf(fact_457_remove1_Osimps_I1_J,axiom,
! [X: formula_a_b] :
( ( remove1_formula_a_b @ X @ nil_formula_a_b )
= nil_formula_a_b ) ).
% remove1.simps(1)
thf(fact_458_removeAll_Osimps_I1_J,axiom,
! [X: formula_a_b] :
( ( remove7803334386285014661la_a_b @ X @ nil_formula_a_b )
= nil_formula_a_b ) ).
% removeAll.simps(1)
thf(fact_459_removeAll_Osimps_I1_J,axiom,
! [X: list_formula_a_b] :
( ( remove9041333805894489621la_a_b @ X @ nil_list_formula_a_b )
= nil_list_formula_a_b ) ).
% removeAll.simps(1)
thf(fact_460_rotate1_Osimps_I1_J,axiom,
( ( rotate1_formula_a_b @ nil_formula_a_b )
= nil_formula_a_b ) ).
% rotate1.simps(1)
thf(fact_461_splice_Osimps_I1_J,axiom,
! [Ys: list_formula_a_b] :
( ( splice_formula_a_b @ nil_formula_a_b @ Ys )
= Ys ) ).
% splice.simps(1)
thf(fact_462_member__rec_I2_J,axiom,
! [Y: formula_a_b] :
~ ( member_formula_a_b @ nil_formula_a_b @ Y ) ).
% member_rec(2)
thf(fact_463_fold__Cons,axiom,
! [F: formula_a_b > list_formula_a_b > list_formula_a_b,X: formula_a_b,Xs: list_formula_a_b] :
( ( fold_f7464356806857031615la_a_b @ F @ ( cons_formula_a_b @ X @ Xs ) )
= ( comp_l1221189711022459592la_a_b @ ( fold_f7464356806857031615la_a_b @ F @ Xs ) @ ( F @ X ) ) ) ).
% fold_Cons
thf(fact_464_fold__Cons,axiom,
! [F: formula_a_b > set_formula_a_b > set_formula_a_b,X: formula_a_b,Xs: list_formula_a_b] :
( ( fold_f470054086739575141la_a_b @ F @ ( cons_formula_a_b @ X @ Xs ) )
= ( comp_s7716128494983798842la_a_b @ ( fold_f470054086739575141la_a_b @ F @ Xs ) @ ( F @ X ) ) ) ).
% fold_Cons
thf(fact_465_fold__Cons,axiom,
! [F: set_formula_a_b > set_formula_a_b > set_formula_a_b,X: set_formula_a_b,Xs: list_set_formula_a_b] :
( ( fold_s5097478484676794267la_a_b @ F @ ( cons_set_formula_a_b @ X @ Xs ) )
= ( comp_s7716128494983798842la_a_b @ ( fold_s5097478484676794267la_a_b @ F @ Xs ) @ ( F @ X ) ) ) ).
% fold_Cons
thf(fact_466_fold__commute__apply,axiom,
! [Xs: list_formula_a_b,H: list_formula_a_b > list_formula_a_b,G: formula_a_b > list_formula_a_b > list_formula_a_b,F: formula_a_b > list_formula_a_b > list_formula_a_b,S: list_formula_a_b] :
( ! [X3: formula_a_b] :
( ( member_formula_a_b2 @ X3 @ ( set_formula_a_b2 @ Xs ) )
=> ( ( comp_l1221189711022459592la_a_b @ H @ ( G @ X3 ) )
= ( comp_l1221189711022459592la_a_b @ ( F @ X3 ) @ H ) ) )
=> ( ( H @ ( fold_f7464356806857031615la_a_b @ G @ Xs @ S ) )
= ( fold_f7464356806857031615la_a_b @ F @ Xs @ ( H @ S ) ) ) ) ).
% fold_commute_apply
thf(fact_467_fold__commute__apply,axiom,
! [Xs: list_formula_a_b,H: list_formula_a_b > set_formula_a_b,G: formula_a_b > list_formula_a_b > list_formula_a_b,F: formula_a_b > set_formula_a_b > set_formula_a_b,S: list_formula_a_b] :
( ! [X3: formula_a_b] :
( ( member_formula_a_b2 @ X3 @ ( set_formula_a_b2 @ Xs ) )
=> ( ( comp_l3699371471522569966la_a_b @ H @ ( G @ X3 ) )
= ( comp_s1677114771899515924la_a_b @ ( F @ X3 ) @ H ) ) )
=> ( ( H @ ( fold_f7464356806857031615la_a_b @ G @ Xs @ S ) )
= ( fold_f470054086739575141la_a_b @ F @ Xs @ ( H @ S ) ) ) ) ).
% fold_commute_apply
thf(fact_468_fold__commute__apply,axiom,
! [Xs: list_formula_a_b,H: set_formula_a_b > list_formula_a_b,G: formula_a_b > set_formula_a_b > set_formula_a_b,F: formula_a_b > list_formula_a_b > list_formula_a_b,S: set_formula_a_b] :
( ! [X3: formula_a_b] :
( ( member_formula_a_b2 @ X3 @ ( set_formula_a_b2 @ Xs ) )
=> ( ( comp_s1800181939013838356la_a_b @ H @ ( G @ X3 ) )
= ( comp_l3822438638636892398la_a_b @ ( F @ X3 ) @ H ) ) )
=> ( ( H @ ( fold_f470054086739575141la_a_b @ G @ Xs @ S ) )
= ( fold_f7464356806857031615la_a_b @ F @ Xs @ ( H @ S ) ) ) ) ).
% fold_commute_apply
thf(fact_469_fold__commute__apply,axiom,
! [Xs: list_formula_a_b,H: set_formula_a_b > set_formula_a_b,G: formula_a_b > set_formula_a_b > set_formula_a_b,F: formula_a_b > set_formula_a_b > set_formula_a_b,S: set_formula_a_b] :
( ! [X3: formula_a_b] :
( ( member_formula_a_b2 @ X3 @ ( set_formula_a_b2 @ Xs ) )
=> ( ( comp_s7716128494983798842la_a_b @ H @ ( G @ X3 ) )
= ( comp_s7716128494983798842la_a_b @ ( F @ X3 ) @ H ) ) )
=> ( ( H @ ( fold_f470054086739575141la_a_b @ G @ Xs @ S ) )
= ( fold_f470054086739575141la_a_b @ F @ Xs @ ( H @ S ) ) ) ) ).
% fold_commute_apply
thf(fact_470_fold__commute__apply,axiom,
! [Xs: list_set_formula_a_b,H: set_formula_a_b > set_formula_a_b,G: set_formula_a_b > set_formula_a_b > set_formula_a_b,F: set_formula_a_b > set_formula_a_b > set_formula_a_b,S: set_formula_a_b] :
( ! [X3: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X3 @ ( set_set_formula_a_b2 @ Xs ) )
=> ( ( comp_s7716128494983798842la_a_b @ H @ ( G @ X3 ) )
= ( comp_s7716128494983798842la_a_b @ ( F @ X3 ) @ H ) ) )
=> ( ( H @ ( fold_s5097478484676794267la_a_b @ G @ Xs @ S ) )
= ( fold_s5097478484676794267la_a_b @ F @ Xs @ ( H @ S ) ) ) ) ).
% fold_commute_apply
thf(fact_471_fold__commute,axiom,
! [Xs: list_formula_a_b,H: list_formula_a_b > list_formula_a_b,G: formula_a_b > list_formula_a_b > list_formula_a_b,F: formula_a_b > list_formula_a_b > list_formula_a_b] :
( ! [X3: formula_a_b] :
( ( member_formula_a_b2 @ X3 @ ( set_formula_a_b2 @ Xs ) )
=> ( ( comp_l1221189711022459592la_a_b @ H @ ( G @ X3 ) )
= ( comp_l1221189711022459592la_a_b @ ( F @ X3 ) @ H ) ) )
=> ( ( comp_l1221189711022459592la_a_b @ H @ ( fold_f7464356806857031615la_a_b @ G @ Xs ) )
= ( comp_l1221189711022459592la_a_b @ ( fold_f7464356806857031615la_a_b @ F @ Xs ) @ H ) ) ) ).
% fold_commute
thf(fact_472_fold__commute,axiom,
! [Xs: list_formula_a_b,H: list_formula_a_b > set_formula_a_b,G: formula_a_b > list_formula_a_b > list_formula_a_b,F: formula_a_b > set_formula_a_b > set_formula_a_b] :
( ! [X3: formula_a_b] :
( ( member_formula_a_b2 @ X3 @ ( set_formula_a_b2 @ Xs ) )
=> ( ( comp_l3699371471522569966la_a_b @ H @ ( G @ X3 ) )
= ( comp_s1677114771899515924la_a_b @ ( F @ X3 ) @ H ) ) )
=> ( ( comp_l3699371471522569966la_a_b @ H @ ( fold_f7464356806857031615la_a_b @ G @ Xs ) )
= ( comp_s1677114771899515924la_a_b @ ( fold_f470054086739575141la_a_b @ F @ Xs ) @ H ) ) ) ).
% fold_commute
thf(fact_473_fold__commute,axiom,
! [Xs: list_formula_a_b,H: set_formula_a_b > list_formula_a_b,G: formula_a_b > set_formula_a_b > set_formula_a_b,F: formula_a_b > list_formula_a_b > list_formula_a_b] :
( ! [X3: formula_a_b] :
( ( member_formula_a_b2 @ X3 @ ( set_formula_a_b2 @ Xs ) )
=> ( ( comp_s1800181939013838356la_a_b @ H @ ( G @ X3 ) )
= ( comp_l3822438638636892398la_a_b @ ( F @ X3 ) @ H ) ) )
=> ( ( comp_s1800181939013838356la_a_b @ H @ ( fold_f470054086739575141la_a_b @ G @ Xs ) )
= ( comp_l3822438638636892398la_a_b @ ( fold_f7464356806857031615la_a_b @ F @ Xs ) @ H ) ) ) ).
% fold_commute
thf(fact_474_fold__commute,axiom,
! [Xs: list_formula_a_b,H: set_formula_a_b > set_formula_a_b,G: formula_a_b > set_formula_a_b > set_formula_a_b,F: formula_a_b > set_formula_a_b > set_formula_a_b] :
( ! [X3: formula_a_b] :
( ( member_formula_a_b2 @ X3 @ ( set_formula_a_b2 @ Xs ) )
=> ( ( comp_s7716128494983798842la_a_b @ H @ ( G @ X3 ) )
= ( comp_s7716128494983798842la_a_b @ ( F @ X3 ) @ H ) ) )
=> ( ( comp_s7716128494983798842la_a_b @ H @ ( fold_f470054086739575141la_a_b @ G @ Xs ) )
= ( comp_s7716128494983798842la_a_b @ ( fold_f470054086739575141la_a_b @ F @ Xs ) @ H ) ) ) ).
% fold_commute
thf(fact_475_fold__commute,axiom,
! [Xs: list_set_formula_a_b,H: set_formula_a_b > set_formula_a_b,G: set_formula_a_b > set_formula_a_b > set_formula_a_b,F: set_formula_a_b > set_formula_a_b > set_formula_a_b] :
( ! [X3: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X3 @ ( set_set_formula_a_b2 @ Xs ) )
=> ( ( comp_s7716128494983798842la_a_b @ H @ ( G @ X3 ) )
= ( comp_s7716128494983798842la_a_b @ ( F @ X3 ) @ H ) ) )
=> ( ( comp_s7716128494983798842la_a_b @ H @ ( fold_s5097478484676794267la_a_b @ G @ Xs ) )
= ( comp_s7716128494983798842la_a_b @ ( fold_s5097478484676794267la_a_b @ F @ Xs ) @ H ) ) ) ).
% fold_commute
thf(fact_476_takeWhile__tail,axiom,
! [P: set_formula_a_b > $o,X: set_formula_a_b,Xs: list_set_formula_a_b,L: list_set_formula_a_b] :
( ~ ( P @ X )
=> ( ( takeWh1298768865834877576la_a_b @ P @ ( append8577002312234529644la_a_b @ Xs @ ( cons_set_formula_a_b @ X @ L ) ) )
= ( takeWh1298768865834877576la_a_b @ P @ Xs ) ) ) ).
% takeWhile_tail
thf(fact_477_takeWhile__tail,axiom,
! [P: formula_a_b > $o,X: formula_a_b,Xs: list_formula_a_b,L: list_formula_a_b] :
( ~ ( P @ X )
=> ( ( takeWh5313468693702967890la_a_b @ P @ ( append_formula_a_b @ Xs @ ( cons_formula_a_b @ X @ L ) ) )
= ( takeWh5313468693702967890la_a_b @ P @ Xs ) ) ) ).
% takeWhile_tail
thf(fact_478_rev__induct,axiom,
! [P: list_formula_a_b > $o,Xs: list_formula_a_b] :
( ( P @ nil_formula_a_b )
=> ( ! [X3: formula_a_b,Xs3: list_formula_a_b] :
( ( P @ Xs3 )
=> ( P @ ( append_formula_a_b @ Xs3 @ ( cons_formula_a_b @ X3 @ nil_formula_a_b ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_479_rev__induct,axiom,
! [P: list_set_formula_a_b > $o,Xs: list_set_formula_a_b] :
( ( P @ nil_set_formula_a_b )
=> ( ! [X3: set_formula_a_b,Xs3: list_set_formula_a_b] :
( ( P @ Xs3 )
=> ( P @ ( append8577002312234529644la_a_b @ Xs3 @ ( cons_set_formula_a_b @ X3 @ nil_set_formula_a_b ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_480_rev__exhaust,axiom,
! [Xs: list_formula_a_b] :
( ( Xs != nil_formula_a_b )
=> ~ ! [Ys3: list_formula_a_b,Y3: formula_a_b] :
( Xs
!= ( append_formula_a_b @ Ys3 @ ( cons_formula_a_b @ Y3 @ nil_formula_a_b ) ) ) ) ).
% rev_exhaust
thf(fact_481_rev__exhaust,axiom,
! [Xs: list_set_formula_a_b] :
( ( Xs != nil_set_formula_a_b )
=> ~ ! [Ys3: list_set_formula_a_b,Y3: set_formula_a_b] :
( Xs
!= ( append8577002312234529644la_a_b @ Ys3 @ ( cons_set_formula_a_b @ Y3 @ nil_set_formula_a_b ) ) ) ) ).
% rev_exhaust
thf(fact_482_Cons__eq__append__conv,axiom,
! [X: formula_a_b,Xs: list_formula_a_b,Ys: list_formula_a_b,Zs: list_formula_a_b] :
( ( ( cons_formula_a_b @ X @ Xs )
= ( append_formula_a_b @ Ys @ Zs ) )
= ( ( ( Ys = nil_formula_a_b )
& ( ( cons_formula_a_b @ X @ Xs )
= Zs ) )
| ? [Ys5: list_formula_a_b] :
( ( ( cons_formula_a_b @ X @ Ys5 )
= Ys )
& ( Xs
= ( append_formula_a_b @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_483_Cons__eq__append__conv,axiom,
! [X: set_formula_a_b,Xs: list_set_formula_a_b,Ys: list_set_formula_a_b,Zs: list_set_formula_a_b] :
( ( ( cons_set_formula_a_b @ X @ Xs )
= ( append8577002312234529644la_a_b @ Ys @ Zs ) )
= ( ( ( Ys = nil_set_formula_a_b )
& ( ( cons_set_formula_a_b @ X @ Xs )
= Zs ) )
| ? [Ys5: list_set_formula_a_b] :
( ( ( cons_set_formula_a_b @ X @ Ys5 )
= Ys )
& ( Xs
= ( append8577002312234529644la_a_b @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_484_append__eq__Cons__conv,axiom,
! [Ys: list_formula_a_b,Zs: list_formula_a_b,X: formula_a_b,Xs: list_formula_a_b] :
( ( ( append_formula_a_b @ Ys @ Zs )
= ( cons_formula_a_b @ X @ Xs ) )
= ( ( ( Ys = nil_formula_a_b )
& ( Zs
= ( cons_formula_a_b @ X @ Xs ) ) )
| ? [Ys5: list_formula_a_b] :
( ( Ys
= ( cons_formula_a_b @ X @ Ys5 ) )
& ( ( append_formula_a_b @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_485_append__eq__Cons__conv,axiom,
! [Ys: list_set_formula_a_b,Zs: list_set_formula_a_b,X: set_formula_a_b,Xs: list_set_formula_a_b] :
( ( ( append8577002312234529644la_a_b @ Ys @ Zs )
= ( cons_set_formula_a_b @ X @ Xs ) )
= ( ( ( Ys = nil_set_formula_a_b )
& ( Zs
= ( cons_set_formula_a_b @ X @ Xs ) ) )
| ? [Ys5: list_set_formula_a_b] :
( ( Ys
= ( cons_set_formula_a_b @ X @ Ys5 ) )
& ( ( append8577002312234529644la_a_b @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_486_rev__nonempty__induct,axiom,
! [Xs: list_formula_a_b,P: list_formula_a_b > $o] :
( ( Xs != nil_formula_a_b )
=> ( ! [X3: formula_a_b] : ( P @ ( cons_formula_a_b @ X3 @ nil_formula_a_b ) )
=> ( ! [X3: formula_a_b,Xs3: list_formula_a_b] :
( ( Xs3 != nil_formula_a_b )
=> ( ( P @ Xs3 )
=> ( P @ ( append_formula_a_b @ Xs3 @ ( cons_formula_a_b @ X3 @ nil_formula_a_b ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_487_rev__nonempty__induct,axiom,
! [Xs: list_set_formula_a_b,P: list_set_formula_a_b > $o] :
( ( Xs != nil_set_formula_a_b )
=> ( ! [X3: set_formula_a_b] : ( P @ ( cons_set_formula_a_b @ X3 @ nil_set_formula_a_b ) )
=> ( ! [X3: set_formula_a_b,Xs3: list_set_formula_a_b] :
( ( Xs3 != nil_set_formula_a_b )
=> ( ( P @ Xs3 )
=> ( P @ ( append8577002312234529644la_a_b @ Xs3 @ ( cons_set_formula_a_b @ X3 @ nil_set_formula_a_b ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_488_distinct__singleton,axiom,
! [X: list_formula_a_b] : ( distin27490427618170344la_a_b @ ( cons_l2450299273141159147la_a_b @ X @ nil_list_formula_a_b ) ) ).
% distinct_singleton
thf(fact_489_distinct__singleton,axiom,
! [X: formula_a_b] : ( distinct_formula_a_b @ ( cons_formula_a_b @ X @ nil_formula_a_b ) ) ).
% distinct_singleton
thf(fact_490_distinct__singleton,axiom,
! [X: set_formula_a_b] : ( distin5059275051008138254la_a_b @ ( cons_set_formula_a_b @ X @ nil_set_formula_a_b ) ) ).
% distinct_singleton
thf(fact_491_shufflesE,axiom,
! [Zs: list_formula_a_b,Xs: list_formula_a_b,Ys: list_formula_a_b] :
( ( member7703986999711672120la_a_b @ Zs @ ( shuffles_formula_a_b @ Xs @ Ys ) )
=> ( ( ( Zs = Xs )
=> ( Ys != nil_formula_a_b ) )
=> ( ( ( Zs = Ys )
=> ( Xs != nil_formula_a_b ) )
=> ( ! [X3: formula_a_b,Xs5: list_formula_a_b] :
( ( Xs
= ( cons_formula_a_b @ X3 @ Xs5 ) )
=> ! [Z3: formula_a_b,Zs4: list_formula_a_b] :
( ( Zs
= ( cons_formula_a_b @ Z3 @ Zs4 ) )
=> ( ( X3 = Z3 )
=> ~ ( member7703986999711672120la_a_b @ Zs4 @ ( shuffles_formula_a_b @ Xs5 @ Ys ) ) ) ) )
=> ~ ! [Y3: formula_a_b,Ys6: list_formula_a_b] :
( ( Ys
= ( cons_formula_a_b @ Y3 @ Ys6 ) )
=> ! [Z3: formula_a_b,Zs4: list_formula_a_b] :
( ( Zs
= ( cons_formula_a_b @ Z3 @ Zs4 ) )
=> ( ( Y3 = Z3 )
=> ~ ( member7703986999711672120la_a_b @ Zs4 @ ( shuffles_formula_a_b @ Xs @ Ys6 ) ) ) ) ) ) ) ) ) ).
% shufflesE
thf(fact_492_shufflesE,axiom,
! [Zs: list_set_formula_a_b,Xs: list_set_formula_a_b,Ys: list_set_formula_a_b] :
( ( member289699325402759278la_a_b @ Zs @ ( shuffl4965861759276814700la_a_b @ Xs @ Ys ) )
=> ( ( ( Zs = Xs )
=> ( Ys != nil_set_formula_a_b ) )
=> ( ( ( Zs = Ys )
=> ( Xs != nil_set_formula_a_b ) )
=> ( ! [X3: set_formula_a_b,Xs5: list_set_formula_a_b] :
( ( Xs
= ( cons_set_formula_a_b @ X3 @ Xs5 ) )
=> ! [Z3: set_formula_a_b,Zs4: list_set_formula_a_b] :
( ( Zs
= ( cons_set_formula_a_b @ Z3 @ Zs4 ) )
=> ( ( X3 = Z3 )
=> ~ ( member289699325402759278la_a_b @ Zs4 @ ( shuffl4965861759276814700la_a_b @ Xs5 @ Ys ) ) ) ) )
=> ~ ! [Y3: set_formula_a_b,Ys6: list_set_formula_a_b] :
( ( Ys
= ( cons_set_formula_a_b @ Y3 @ Ys6 ) )
=> ! [Z3: set_formula_a_b,Zs4: list_set_formula_a_b] :
( ( Zs
= ( cons_set_formula_a_b @ Z3 @ Zs4 ) )
=> ( ( Y3 = Z3 )
=> ~ ( member289699325402759278la_a_b @ Zs4 @ ( shuffl4965861759276814700la_a_b @ Xs @ Ys6 ) ) ) ) ) ) ) ) ) ).
% shufflesE
thf(fact_493_splice_Oelims,axiom,
! [X: list_formula_a_b,Xa2: list_formula_a_b,Y: list_formula_a_b] :
( ( ( splice_formula_a_b @ X @ Xa2 )
= Y )
=> ( ( ( X = nil_formula_a_b )
=> ( Y != Xa2 ) )
=> ~ ! [X3: formula_a_b,Xs3: list_formula_a_b] :
( ( X
= ( cons_formula_a_b @ X3 @ Xs3 ) )
=> ( Y
!= ( cons_formula_a_b @ X3 @ ( splice_formula_a_b @ Xa2 @ Xs3 ) ) ) ) ) ) ).
% splice.elims
thf(fact_494_splice_Oelims,axiom,
! [X: list_set_formula_a_b,Xa2: list_set_formula_a_b,Y: list_set_formula_a_b] :
( ( ( splice5575443282787837998la_a_b @ X @ Xa2 )
= Y )
=> ( ( ( X = nil_set_formula_a_b )
=> ( Y != Xa2 ) )
=> ~ ! [X3: set_formula_a_b,Xs3: list_set_formula_a_b] :
( ( X
= ( cons_set_formula_a_b @ X3 @ Xs3 ) )
=> ( Y
!= ( cons_set_formula_a_b @ X3 @ ( splice5575443282787837998la_a_b @ Xa2 @ Xs3 ) ) ) ) ) ) ).
% splice.elims
thf(fact_495_union__set__fold,axiom,
! [Xs: list_l5948067055324603441la_a_b,A: set_list_formula_a_b] :
( ( sup_su9015223001214971651la_a_b @ ( set_list_formula_a_b2 @ Xs ) @ A )
= ( fold_l2136769360297358853la_a_b @ insert4995524290302003025la_a_b @ Xs @ A ) ) ).
% union_set_fold
thf(fact_496_union__set__fold,axiom,
! [Xs: list_set_formula_a_b,A: set_set_formula_a_b] :
( ( sup_su5931044959554304553la_a_b @ ( set_set_formula_a_b2 @ Xs ) @ A )
= ( fold_s4424443237017773265la_a_b @ insert8952366156270154231la_a_b @ Xs @ A ) ) ).
% union_set_fold
thf(fact_497_union__set__fold,axiom,
! [Xs: list_formula_a_b,A: set_formula_a_b] :
( ( sup_su8832426789560438003la_a_b @ ( set_formula_a_b2 @ Xs ) @ A )
= ( fold_f470054086739575141la_a_b @ insert_formula_a_b2 @ Xs @ A ) ) ).
% union_set_fold
thf(fact_498_comp__apply,axiom,
( comp_s3812543008930346602la_a_b
= ( ^ [F2: set_formula_a_b > set_formula_a_b > set_formula_a_b,G2: formula_a_b > set_formula_a_b,X2: formula_a_b] : ( F2 @ ( G2 @ X2 ) ) ) ) ).
% comp_apply
thf(fact_499_the__elem__set,axiom,
! [X: list_formula_a_b] :
( ( the_el407581278154503020la_a_b @ ( set_list_formula_a_b2 @ ( cons_l2450299273141159147la_a_b @ X @ nil_list_formula_a_b ) ) )
= X ) ).
% the_elem_set
thf(fact_500_the__elem__set,axiom,
! [X: set_formula_a_b] :
( ( the_el8292460259403742098la_a_b @ ( set_set_formula_a_b2 @ ( cons_set_formula_a_b @ X @ nil_set_formula_a_b ) ) )
= X ) ).
% the_elem_set
thf(fact_501_the__elem__set,axiom,
! [X: formula_a_b] :
( ( the_elem_formula_a_b @ ( set_formula_a_b2 @ ( cons_formula_a_b @ X @ nil_formula_a_b ) ) )
= X ) ).
% the_elem_set
thf(fact_502_list__split_Ocases,axiom,
! [X: list_formula_a_b] :
( ( X != nil_formula_a_b )
=> ~ ! [X3: formula_a_b,Xs3: list_formula_a_b] :
( X
!= ( cons_formula_a_b @ X3 @ Xs3 ) ) ) ).
% list_split.cases
thf(fact_503_list__split_Ocases,axiom,
! [X: list_set_formula_a_b] :
( ( X != nil_set_formula_a_b )
=> ~ ! [X3: set_formula_a_b,Xs3: list_set_formula_a_b] :
( X
!= ( cons_set_formula_a_b @ X3 @ Xs3 ) ) ) ).
% list_split.cases
thf(fact_504_butlast__snoc,axiom,
! [Xs: list_formula_a_b,X: formula_a_b] :
( ( butlast_formula_a_b @ ( append_formula_a_b @ Xs @ ( cons_formula_a_b @ X @ nil_formula_a_b ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_505_butlast__snoc,axiom,
! [Xs: list_set_formula_a_b,X: set_formula_a_b] :
( ( butlas5801911201278550005la_a_b @ ( append8577002312234529644la_a_b @ Xs @ ( cons_set_formula_a_b @ X @ nil_set_formula_a_b ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_506_last__snoc,axiom,
! [Xs: list_formula_a_b,X: formula_a_b] :
( ( last_formula_a_b @ ( append_formula_a_b @ Xs @ ( cons_formula_a_b @ X @ nil_formula_a_b ) ) )
= X ) ).
% last_snoc
thf(fact_507_last__snoc,axiom,
! [Xs: list_set_formula_a_b,X: set_formula_a_b] :
( ( last_set_formula_a_b @ ( append8577002312234529644la_a_b @ Xs @ ( cons_set_formula_a_b @ X @ nil_set_formula_a_b ) ) )
= X ) ).
% last_snoc
thf(fact_508_foldr__fold,axiom,
! [Xs: list_formula_a_b,F: formula_a_b > list_formula_a_b > list_formula_a_b] :
( ! [X3: formula_a_b,Y3: formula_a_b] :
( ( member_formula_a_b2 @ X3 @ ( set_formula_a_b2 @ Xs ) )
=> ( ( member_formula_a_b2 @ Y3 @ ( set_formula_a_b2 @ Xs ) )
=> ( ( comp_l1221189711022459592la_a_b @ ( F @ Y3 ) @ ( F @ X3 ) )
= ( comp_l1221189711022459592la_a_b @ ( F @ X3 ) @ ( F @ Y3 ) ) ) ) )
=> ( ( foldr_6622774383136399195la_a_b @ F @ Xs )
= ( fold_f7464356806857031615la_a_b @ F @ Xs ) ) ) ).
% foldr_fold
thf(fact_509_foldr__fold,axiom,
! [Xs: list_formula_a_b,F: formula_a_b > set_formula_a_b > set_formula_a_b] :
( ! [X3: formula_a_b,Y3: formula_a_b] :
( ( member_formula_a_b2 @ X3 @ ( set_formula_a_b2 @ Xs ) )
=> ( ( member_formula_a_b2 @ Y3 @ ( set_formula_a_b2 @ Xs ) )
=> ( ( comp_s7716128494983798842la_a_b @ ( F @ Y3 ) @ ( F @ X3 ) )
= ( comp_s7716128494983798842la_a_b @ ( F @ X3 ) @ ( F @ Y3 ) ) ) ) )
=> ( ( foldr_2945827324002938625la_a_b @ F @ Xs )
= ( fold_f470054086739575141la_a_b @ F @ Xs ) ) ) ).
% foldr_fold
thf(fact_510_foldr__fold,axiom,
! [Xs: list_set_formula_a_b,F: set_formula_a_b > set_formula_a_b > set_formula_a_b] :
( ! [X3: set_formula_a_b,Y3: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X3 @ ( set_set_formula_a_b2 @ Xs ) )
=> ( ( member6621951321062823262la_a_b @ Y3 @ ( set_set_formula_a_b2 @ Xs ) )
=> ( ( comp_s7716128494983798842la_a_b @ ( F @ Y3 ) @ ( F @ X3 ) )
= ( comp_s7716128494983798842la_a_b @ ( F @ X3 ) @ ( F @ Y3 ) ) ) ) )
=> ( ( foldr_5377326498870416695la_a_b @ F @ Xs )
= ( fold_s5097478484676794267la_a_b @ F @ Xs ) ) ) ).
% foldr_fold
thf(fact_511_last__appendR,axiom,
! [Ys: list_formula_a_b,Xs: list_formula_a_b] :
( ( Ys != nil_formula_a_b )
=> ( ( last_formula_a_b @ ( append_formula_a_b @ Xs @ Ys ) )
= ( last_formula_a_b @ Ys ) ) ) ).
% last_appendR
thf(fact_512_last__appendL,axiom,
! [Ys: list_formula_a_b,Xs: list_formula_a_b] :
( ( Ys = nil_formula_a_b )
=> ( ( last_formula_a_b @ ( append_formula_a_b @ Xs @ Ys ) )
= ( last_formula_a_b @ Xs ) ) ) ).
% last_appendL
thf(fact_513_append__butlast__last__id,axiom,
! [Xs: list_formula_a_b] :
( ( Xs != nil_formula_a_b )
=> ( ( append_formula_a_b @ ( butlast_formula_a_b @ Xs ) @ ( cons_formula_a_b @ ( last_formula_a_b @ Xs ) @ nil_formula_a_b ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_514_append__butlast__last__id,axiom,
! [Xs: list_set_formula_a_b] :
( ( Xs != nil_set_formula_a_b )
=> ( ( append8577002312234529644la_a_b @ ( butlas5801911201278550005la_a_b @ Xs ) @ ( cons_set_formula_a_b @ ( last_set_formula_a_b @ Xs ) @ nil_set_formula_a_b ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_515_butlast_Osimps_I1_J,axiom,
( ( butlast_formula_a_b @ nil_formula_a_b )
= nil_formula_a_b ) ).
% butlast.simps(1)
thf(fact_516_in__set__butlastD,axiom,
! [X: set_formula_a_b,Xs: list_set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X @ ( set_set_formula_a_b2 @ ( butlas5801911201278550005la_a_b @ Xs ) ) )
=> ( member6621951321062823262la_a_b @ X @ ( set_set_formula_a_b2 @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_517_in__set__butlastD,axiom,
! [X: list_formula_a_b,Xs: list_l5948067055324603441la_a_b] :
( ( member7703986999711672120la_a_b @ X @ ( set_list_formula_a_b2 @ ( butlas4241030279939053135la_a_b @ Xs ) ) )
=> ( member7703986999711672120la_a_b @ X @ ( set_list_formula_a_b2 @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_518_in__set__butlastD,axiom,
! [X: formula_a_b,Xs: list_formula_a_b] :
( ( member_formula_a_b2 @ X @ ( set_formula_a_b2 @ ( butlast_formula_a_b @ Xs ) ) )
=> ( member_formula_a_b2 @ X @ ( set_formula_a_b2 @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_519_distinct__butlast,axiom,
! [Xs: list_formula_a_b] :
( ( distinct_formula_a_b @ Xs )
=> ( distinct_formula_a_b @ ( butlast_formula_a_b @ Xs ) ) ) ).
% distinct_butlast
thf(fact_520_distinct__butlast,axiom,
! [Xs: list_l5948067055324603441la_a_b] :
( ( distin27490427618170344la_a_b @ Xs )
=> ( distin27490427618170344la_a_b @ ( butlas4241030279939053135la_a_b @ Xs ) ) ) ).
% distinct_butlast
thf(fact_521_distinct__butlast,axiom,
! [Xs: list_set_formula_a_b] :
( ( distin5059275051008138254la_a_b @ Xs )
=> ( distin5059275051008138254la_a_b @ ( butlas5801911201278550005la_a_b @ Xs ) ) ) ).
% distinct_butlast
thf(fact_522_snoc__eq__iff__butlast,axiom,
! [Xs: list_formula_a_b,X: formula_a_b,Ys: list_formula_a_b] :
( ( ( append_formula_a_b @ Xs @ ( cons_formula_a_b @ X @ nil_formula_a_b ) )
= Ys )
= ( ( Ys != nil_formula_a_b )
& ( ( butlast_formula_a_b @ Ys )
= Xs )
& ( ( last_formula_a_b @ Ys )
= X ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_523_snoc__eq__iff__butlast,axiom,
! [Xs: list_set_formula_a_b,X: set_formula_a_b,Ys: list_set_formula_a_b] :
( ( ( append8577002312234529644la_a_b @ Xs @ ( cons_set_formula_a_b @ X @ nil_set_formula_a_b ) )
= Ys )
= ( ( Ys != nil_set_formula_a_b )
& ( ( butlas5801911201278550005la_a_b @ Ys )
= Xs )
& ( ( last_set_formula_a_b @ Ys )
= X ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_524_last__ConsR,axiom,
! [Xs: list_formula_a_b,X: formula_a_b] :
( ( Xs != nil_formula_a_b )
=> ( ( last_formula_a_b @ ( cons_formula_a_b @ X @ Xs ) )
= ( last_formula_a_b @ Xs ) ) ) ).
% last_ConsR
thf(fact_525_last__ConsR,axiom,
! [Xs: list_set_formula_a_b,X: set_formula_a_b] :
( ( Xs != nil_set_formula_a_b )
=> ( ( last_set_formula_a_b @ ( cons_set_formula_a_b @ X @ Xs ) )
= ( last_set_formula_a_b @ Xs ) ) ) ).
% last_ConsR
thf(fact_526_last__ConsL,axiom,
! [Xs: list_formula_a_b,X: formula_a_b] :
( ( Xs = nil_formula_a_b )
=> ( ( last_formula_a_b @ ( cons_formula_a_b @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_527_last__ConsL,axiom,
! [Xs: list_set_formula_a_b,X: set_formula_a_b] :
( ( Xs = nil_set_formula_a_b )
=> ( ( last_set_formula_a_b @ ( cons_set_formula_a_b @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_528_last_Osimps,axiom,
! [Xs: list_formula_a_b,X: formula_a_b] :
( ( ( Xs = nil_formula_a_b )
=> ( ( last_formula_a_b @ ( cons_formula_a_b @ X @ Xs ) )
= X ) )
& ( ( Xs != nil_formula_a_b )
=> ( ( last_formula_a_b @ ( cons_formula_a_b @ X @ Xs ) )
= ( last_formula_a_b @ Xs ) ) ) ) ).
% last.simps
thf(fact_529_last_Osimps,axiom,
! [Xs: list_set_formula_a_b,X: set_formula_a_b] :
( ( ( Xs = nil_set_formula_a_b )
=> ( ( last_set_formula_a_b @ ( cons_set_formula_a_b @ X @ Xs ) )
= X ) )
& ( ( Xs != nil_set_formula_a_b )
=> ( ( last_set_formula_a_b @ ( cons_set_formula_a_b @ X @ Xs ) )
= ( last_set_formula_a_b @ Xs ) ) ) ) ).
% last.simps
thf(fact_530_last__in__set,axiom,
! [As: list_set_formula_a_b] :
( ( As != nil_set_formula_a_b )
=> ( member6621951321062823262la_a_b @ ( last_set_formula_a_b @ As ) @ ( set_set_formula_a_b2 @ As ) ) ) ).
% last_in_set
thf(fact_531_last__in__set,axiom,
! [As: list_l5948067055324603441la_a_b] :
( ( As != nil_list_formula_a_b )
=> ( member7703986999711672120la_a_b @ ( last_l46059096318636098la_a_b @ As ) @ ( set_list_formula_a_b2 @ As ) ) ) ).
% last_in_set
thf(fact_532_last__in__set,axiom,
! [As: list_formula_a_b] :
( ( As != nil_formula_a_b )
=> ( member_formula_a_b2 @ ( last_formula_a_b @ As ) @ ( set_formula_a_b2 @ As ) ) ) ).
% last_in_set
thf(fact_533_longest__common__suffix,axiom,
! [Xs: list_formula_a_b,Ys: list_formula_a_b] :
? [Ss: list_formula_a_b,Xs5: list_formula_a_b,Ys6: list_formula_a_b] :
( ( Xs
= ( append_formula_a_b @ Xs5 @ Ss ) )
& ( Ys
= ( append_formula_a_b @ Ys6 @ Ss ) )
& ( ( Xs5 = nil_formula_a_b )
| ( Ys6 = nil_formula_a_b )
| ( ( last_formula_a_b @ Xs5 )
!= ( last_formula_a_b @ Ys6 ) ) ) ) ).
% longest_common_suffix
thf(fact_534_last__append,axiom,
! [Ys: list_formula_a_b,Xs: list_formula_a_b] :
( ( ( Ys = nil_formula_a_b )
=> ( ( last_formula_a_b @ ( append_formula_a_b @ Xs @ Ys ) )
= ( last_formula_a_b @ Xs ) ) )
& ( ( Ys != nil_formula_a_b )
=> ( ( last_formula_a_b @ ( append_formula_a_b @ Xs @ Ys ) )
= ( last_formula_a_b @ Ys ) ) ) ) ).
% last_append
thf(fact_535_butlast_Osimps_I2_J,axiom,
! [Xs: list_formula_a_b,X: formula_a_b] :
( ( ( Xs = nil_formula_a_b )
=> ( ( butlast_formula_a_b @ ( cons_formula_a_b @ X @ Xs ) )
= nil_formula_a_b ) )
& ( ( Xs != nil_formula_a_b )
=> ( ( butlast_formula_a_b @ ( cons_formula_a_b @ X @ Xs ) )
= ( cons_formula_a_b @ X @ ( butlast_formula_a_b @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_536_butlast_Osimps_I2_J,axiom,
! [Xs: list_set_formula_a_b,X: set_formula_a_b] :
( ( ( Xs = nil_set_formula_a_b )
=> ( ( butlas5801911201278550005la_a_b @ ( cons_set_formula_a_b @ X @ Xs ) )
= nil_set_formula_a_b ) )
& ( ( Xs != nil_set_formula_a_b )
=> ( ( butlas5801911201278550005la_a_b @ ( cons_set_formula_a_b @ X @ Xs ) )
= ( cons_set_formula_a_b @ X @ ( butlas5801911201278550005la_a_b @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_537_butlast__append,axiom,
! [Ys: list_formula_a_b,Xs: list_formula_a_b] :
( ( ( Ys = nil_formula_a_b )
=> ( ( butlast_formula_a_b @ ( append_formula_a_b @ Xs @ Ys ) )
= ( butlast_formula_a_b @ Xs ) ) )
& ( ( Ys != nil_formula_a_b )
=> ( ( butlast_formula_a_b @ ( append_formula_a_b @ Xs @ Ys ) )
= ( append_formula_a_b @ Xs @ ( butlast_formula_a_b @ Ys ) ) ) ) ) ).
% butlast_append
thf(fact_538_in__set__butlast__appendI,axiom,
! [X: set_formula_a_b,Xs: list_set_formula_a_b,Ys: list_set_formula_a_b] :
( ( ( member6621951321062823262la_a_b @ X @ ( set_set_formula_a_b2 @ ( butlas5801911201278550005la_a_b @ Xs ) ) )
| ( member6621951321062823262la_a_b @ X @ ( set_set_formula_a_b2 @ ( butlas5801911201278550005la_a_b @ Ys ) ) ) )
=> ( member6621951321062823262la_a_b @ X @ ( set_set_formula_a_b2 @ ( butlas5801911201278550005la_a_b @ ( append8577002312234529644la_a_b @ Xs @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_539_in__set__butlast__appendI,axiom,
! [X: list_formula_a_b,Xs: list_l5948067055324603441la_a_b,Ys: list_l5948067055324603441la_a_b] :
( ( ( member7703986999711672120la_a_b @ X @ ( set_list_formula_a_b2 @ ( butlas4241030279939053135la_a_b @ Xs ) ) )
| ( member7703986999711672120la_a_b @ X @ ( set_list_formula_a_b2 @ ( butlas4241030279939053135la_a_b @ Ys ) ) ) )
=> ( member7703986999711672120la_a_b @ X @ ( set_list_formula_a_b2 @ ( butlas4241030279939053135la_a_b @ ( append58666856857280070la_a_b @ Xs @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_540_in__set__butlast__appendI,axiom,
! [X: formula_a_b,Xs: list_formula_a_b,Ys: list_formula_a_b] :
( ( ( member_formula_a_b2 @ X @ ( set_formula_a_b2 @ ( butlast_formula_a_b @ Xs ) ) )
| ( member_formula_a_b2 @ X @ ( set_formula_a_b2 @ ( butlast_formula_a_b @ Ys ) ) ) )
=> ( member_formula_a_b2 @ X @ ( set_formula_a_b2 @ ( butlast_formula_a_b @ ( append_formula_a_b @ Xs @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_541_dropWhile__last,axiom,
! [X: set_formula_a_b,Xs: list_set_formula_a_b,P: set_formula_a_b > $o] :
( ( member6621951321062823262la_a_b @ X @ ( set_set_formula_a_b2 @ Xs ) )
=> ( ~ ( P @ X )
=> ( ( last_set_formula_a_b @ ( dropWh5803627183431294944la_a_b @ P @ Xs ) )
= ( last_set_formula_a_b @ Xs ) ) ) ) ).
% dropWhile_last
thf(fact_542_dropWhile__last,axiom,
! [X: list_formula_a_b,Xs: list_l5948067055324603441la_a_b,P: list_formula_a_b > $o] :
( ( member7703986999711672120la_a_b @ X @ ( set_list_formula_a_b2 @ Xs ) )
=> ( ~ ( P @ X )
=> ( ( last_l46059096318636098la_a_b @ ( dropWh8739902238574629050la_a_b @ P @ Xs ) )
= ( last_l46059096318636098la_a_b @ Xs ) ) ) ) ).
% dropWhile_last
thf(fact_543_dropWhile__last,axiom,
! [X: formula_a_b,Xs: list_formula_a_b,P: formula_a_b > $o] :
( ( member_formula_a_b2 @ X @ ( set_formula_a_b2 @ Xs ) )
=> ( ~ ( P @ X )
=> ( ( last_formula_a_b @ ( dropWh8768975241461415850la_a_b @ P @ Xs ) )
= ( last_formula_a_b @ Xs ) ) ) ) ).
% dropWhile_last
thf(fact_544_foldr__conv__fold,axiom,
( foldr_6622774383136399195la_a_b
= ( ^ [F2: formula_a_b > list_formula_a_b > list_formula_a_b,Xs2: list_formula_a_b] : ( fold_f7464356806857031615la_a_b @ F2 @ ( rev_formula_a_b @ Xs2 ) ) ) ) ).
% foldr_conv_fold
thf(fact_545_foldr__conv__fold,axiom,
( foldr_5377326498870416695la_a_b
= ( ^ [F2: set_formula_a_b > set_formula_a_b > set_formula_a_b,Xs2: list_set_formula_a_b] : ( fold_s5097478484676794267la_a_b @ F2 @ ( rev_set_formula_a_b @ Xs2 ) ) ) ) ).
% foldr_conv_fold
thf(fact_546_foldr__conv__fold,axiom,
( foldr_2945827324002938625la_a_b
= ( ^ [F2: formula_a_b > set_formula_a_b > set_formula_a_b,Xs2: list_formula_a_b] : ( fold_f470054086739575141la_a_b @ F2 @ ( rev_formula_a_b @ Xs2 ) ) ) ) ).
% foldr_conv_fold
thf(fact_547_comp__def,axiom,
( comp_s3812543008930346602la_a_b
= ( ^ [F2: set_formula_a_b > set_formula_a_b > set_formula_a_b,G2: formula_a_b > set_formula_a_b,X2: formula_a_b] : ( F2 @ ( G2 @ X2 ) ) ) ) ).
% comp_def
thf(fact_548_comp__assoc,axiom,
! [F: set_formula_a_b > set_formula_a_b > set_formula_a_b,G: formula_a_b > set_formula_a_b,H: formula_a_b > formula_a_b] :
( ( comp_f398844620851073332la_a_b @ ( comp_s3812543008930346602la_a_b @ F @ G ) @ H )
= ( comp_s3812543008930346602la_a_b @ F @ ( comp_f821314402523108302la_a_b @ G @ H ) ) ) ).
% comp_assoc
thf(fact_549_comp__assoc,axiom,
! [F: ( set_formula_a_b > set_formula_a_b ) > set_formula_a_b > set_formula_a_b,G: set_formula_a_b > set_formula_a_b > set_formula_a_b,H: formula_a_b > set_formula_a_b] :
( ( comp_s3812543008930346602la_a_b @ ( comp_s5422811073032477702la_a_b @ F @ G ) @ H )
= ( comp_s4889918555203769296la_a_b @ F @ ( comp_s3812543008930346602la_a_b @ G @ H ) ) ) ).
% comp_assoc
thf(fact_550_comp__assoc,axiom,
! [F: set_formula_a_b > set_formula_a_b > set_formula_a_b,G: set_formula_a_b > set_formula_a_b,H: formula_a_b > set_formula_a_b] :
( ( comp_s3812543008930346602la_a_b @ ( comp_s266308785573631648la_a_b @ F @ G ) @ H )
= ( comp_s3812543008930346602la_a_b @ F @ ( comp_s9099485395005827076la_a_b @ G @ H ) ) ) ).
% comp_assoc
thf(fact_551_comp__eq__dest,axiom,
! [A2: set_formula_a_b > set_formula_a_b > set_formula_a_b,B2: formula_a_b > set_formula_a_b,C: set_formula_a_b > set_formula_a_b > set_formula_a_b,D: formula_a_b > set_formula_a_b,V: formula_a_b] :
( ( ( comp_s3812543008930346602la_a_b @ A2 @ B2 )
= ( comp_s3812543008930346602la_a_b @ C @ D ) )
=> ( ( A2 @ ( B2 @ V ) )
= ( C @ ( D @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_552_comp__eq__elim,axiom,
! [A2: set_formula_a_b > set_formula_a_b > set_formula_a_b,B2: formula_a_b > set_formula_a_b,C: set_formula_a_b > set_formula_a_b > set_formula_a_b,D: formula_a_b > set_formula_a_b] :
( ( ( comp_s3812543008930346602la_a_b @ A2 @ B2 )
= ( comp_s3812543008930346602la_a_b @ C @ D ) )
=> ! [V2: formula_a_b] :
( ( A2 @ ( B2 @ V2 ) )
= ( C @ ( D @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_553_comp__eq__dest__lhs,axiom,
! [A2: set_formula_a_b > set_formula_a_b > set_formula_a_b,B2: formula_a_b > set_formula_a_b,C: formula_a_b > set_formula_a_b > set_formula_a_b,V: formula_a_b] :
( ( ( comp_s3812543008930346602la_a_b @ A2 @ B2 )
= C )
=> ( ( A2 @ ( B2 @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_554_product__lists_Osimps_I1_J,axiom,
( ( produc130815601992340706la_a_b @ nil_list_formula_a_b )
= ( cons_l2450299273141159147la_a_b @ nil_formula_a_b @ nil_list_formula_a_b ) ) ).
% product_lists.simps(1)
thf(fact_555_subseqs_Osimps_I1_J,axiom,
( ( subseqs_formula_a_b @ nil_formula_a_b )
= ( cons_l2450299273141159147la_a_b @ nil_formula_a_b @ nil_list_formula_a_b ) ) ).
% subseqs.simps(1)
thf(fact_556_inter__coset__fold,axiom,
! [A: set_formula_a_b,Xs: list_formula_a_b] :
( ( inf_in5741596252272946265la_a_b @ A @ ( coset_formula_a_b @ Xs ) )
= ( fold_f470054086739575141la_a_b @ remove_formula_a_b @ Xs @ A ) ) ).
% inter_coset_fold
thf(fact_557_inter__coset__fold,axiom,
! [A: set_set_formula_a_b,Xs: list_set_formula_a_b] :
( ( inf_in60795025369792911la_a_b @ A @ ( coset_6935746743259127700la_a_b @ Xs ) )
= ( fold_s4424443237017773265la_a_b @ remove6203333876907372194la_a_b @ Xs @ A ) ) ).
% inter_coset_fold
thf(fact_558_concat__eq__append__conv,axiom,
! [Xss2: list_l5948067055324603441la_a_b,Ys: list_formula_a_b,Zs: list_formula_a_b] :
( ( ( concat_formula_a_b @ Xss2 )
= ( append_formula_a_b @ Ys @ Zs ) )
= ( ( ( Xss2 = nil_list_formula_a_b )
=> ( ( Ys = nil_formula_a_b )
& ( Zs = nil_formula_a_b ) ) )
& ( ( Xss2 != nil_list_formula_a_b )
=> ? [Xss1: list_l5948067055324603441la_a_b,Xs2: list_formula_a_b,Xs6: list_formula_a_b,Xss22: list_l5948067055324603441la_a_b] :
( ( Xss2
= ( append58666856857280070la_a_b @ Xss1 @ ( cons_l2450299273141159147la_a_b @ ( append_formula_a_b @ Xs2 @ Xs6 ) @ Xss22 ) ) )
& ( Ys
= ( append_formula_a_b @ ( concat_formula_a_b @ Xss1 ) @ Xs2 ) )
& ( Zs
= ( append_formula_a_b @ Xs6 @ ( concat_formula_a_b @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_559_minus__set__fold,axiom,
! [A: set_list_formula_a_b,Xs: list_l5948067055324603441la_a_b] :
( ( minus_8667068627093346608la_a_b @ A @ ( set_list_formula_a_b2 @ Xs ) )
= ( fold_l2136769360297358853la_a_b @ remove2998734528700178044la_a_b @ Xs @ A ) ) ).
% minus_set_fold
thf(fact_560_minus__set__fold,axiom,
! [A: set_set_formula_a_b,Xs: list_set_formula_a_b] :
( ( minus_4878662943024999126la_a_b @ A @ ( set_set_formula_a_b2 @ Xs ) )
= ( fold_s4424443237017773265la_a_b @ remove6203333876907372194la_a_b @ Xs @ A ) ) ).
% minus_set_fold
thf(fact_561_minus__set__fold,axiom,
! [A: set_formula_a_b,Xs: list_formula_a_b] :
( ( minus_3283878196352184736la_a_b @ A @ ( set_formula_a_b2 @ Xs ) )
= ( fold_f470054086739575141la_a_b @ remove_formula_a_b @ Xs @ A ) ) ).
% minus_set_fold
thf(fact_562_map__tailrec__rev_Oelims,axiom,
! [X: formula_a_b > formula_a_b,Xa2: list_formula_a_b,Xb: list_formula_a_b,Y: list_formula_a_b] :
( ( ( map_ta6137025558560551193la_a_b @ X @ Xa2 @ Xb )
= Y )
=> ( ( ( Xa2 = nil_formula_a_b )
=> ( Y != Xb ) )
=> ~ ! [A5: formula_a_b,As2: list_formula_a_b] :
( ( Xa2
= ( cons_formula_a_b @ A5 @ As2 ) )
=> ( Y
!= ( map_ta6137025558560551193la_a_b @ X @ As2 @ ( cons_formula_a_b @ ( X @ A5 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_563_map__tailrec__rev_Oelims,axiom,
! [X: formula_a_b > set_formula_a_b,Xa2: list_formula_a_b,Xb: list_set_formula_a_b,Y: list_set_formula_a_b] :
( ( ( map_ta1969411894599079247la_a_b @ X @ Xa2 @ Xb )
= Y )
=> ( ( ( Xa2 = nil_formula_a_b )
=> ( Y != Xb ) )
=> ~ ! [A5: formula_a_b,As2: list_formula_a_b] :
( ( Xa2
= ( cons_formula_a_b @ A5 @ As2 ) )
=> ( Y
!= ( map_ta1969411894599079247la_a_b @ X @ As2 @ ( cons_set_formula_a_b @ ( X @ A5 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_564_map__tailrec__rev_Oelims,axiom,
! [X: set_formula_a_b > formula_a_b,Xa2: list_set_formula_a_b,Xb: list_formula_a_b,Y: list_formula_a_b] :
( ( ( map_ta7515233025258682191la_a_b @ X @ Xa2 @ Xb )
= Y )
=> ( ( ( Xa2 = nil_set_formula_a_b )
=> ( Y != Xb ) )
=> ~ ! [A5: set_formula_a_b,As2: list_set_formula_a_b] :
( ( Xa2
= ( cons_set_formula_a_b @ A5 @ As2 ) )
=> ( Y
!= ( map_ta7515233025258682191la_a_b @ X @ As2 @ ( cons_formula_a_b @ ( X @ A5 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_565_map__tailrec__rev_Oelims,axiom,
! [X: set_formula_a_b > set_formula_a_b,Xa2: list_set_formula_a_b,Xb: list_set_formula_a_b,Y: list_set_formula_a_b] :
( ( ( map_ta7939594666930029445la_a_b @ X @ Xa2 @ Xb )
= Y )
=> ( ( ( Xa2 = nil_set_formula_a_b )
=> ( Y != Xb ) )
=> ~ ! [A5: set_formula_a_b,As2: list_set_formula_a_b] :
( ( Xa2
= ( cons_set_formula_a_b @ A5 @ As2 ) )
=> ( Y
!= ( map_ta7939594666930029445la_a_b @ X @ As2 @ ( cons_set_formula_a_b @ ( X @ A5 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_566_SuccI,axiom,
! [Kl: list_l5948067055324603441la_a_b,K: list_formula_a_b,Kl2: set_li1233427854086133095la_a_b] :
( ( member5770512961814848200la_a_b @ ( append58666856857280070la_a_b @ Kl @ ( cons_l2450299273141159147la_a_b @ K @ nil_list_formula_a_b ) ) @ Kl2 )
=> ( member7703986999711672120la_a_b @ K @ ( bNF_Gr3706963439723013830la_a_b @ Kl2 @ Kl ) ) ) ).
% SuccI
thf(fact_567_SuccI,axiom,
! [Kl: list_set_formula_a_b,K: set_formula_a_b,Kl2: set_li5266388670752979469la_a_b] :
( ( member289699325402759278la_a_b @ ( append8577002312234529644la_a_b @ Kl @ ( cons_set_formula_a_b @ K @ nil_set_formula_a_b ) ) @ Kl2 )
=> ( member6621951321062823262la_a_b @ K @ ( bNF_Gr8423546037056666604la_a_b @ Kl2 @ Kl ) ) ) ).
% SuccI
thf(fact_568_SuccI,axiom,
! [Kl: list_formula_a_b,K: formula_a_b,Kl2: set_list_formula_a_b] :
( ( member7703986999711672120la_a_b @ ( append_formula_a_b @ Kl @ ( cons_formula_a_b @ K @ nil_formula_a_b ) ) @ Kl2 )
=> ( member_formula_a_b2 @ K @ ( bNF_Gr6515907858579016630la_a_b @ Kl2 @ Kl ) ) ) ).
% SuccI
thf(fact_569_inf_Oidem,axiom,
! [A2: set_formula_a_b] :
( ( inf_in5741596252272946265la_a_b @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_570_inf_Oidem,axiom,
! [A2: set_set_formula_a_b] :
( ( inf_in60795025369792911la_a_b @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_571_inf__idem,axiom,
! [X: set_formula_a_b] :
( ( inf_in5741596252272946265la_a_b @ X @ X )
= X ) ).
% inf_idem
thf(fact_572_inf__idem,axiom,
! [X: set_set_formula_a_b] :
( ( inf_in60795025369792911la_a_b @ X @ X )
= X ) ).
% inf_idem
thf(fact_573_inf_Oleft__idem,axiom,
! [A2: set_formula_a_b,B2: set_formula_a_b] :
( ( inf_in5741596252272946265la_a_b @ A2 @ ( inf_in5741596252272946265la_a_b @ A2 @ B2 ) )
= ( inf_in5741596252272946265la_a_b @ A2 @ B2 ) ) ).
% inf.left_idem
thf(fact_574_inf_Oleft__idem,axiom,
! [A2: set_set_formula_a_b,B2: set_set_formula_a_b] :
( ( inf_in60795025369792911la_a_b @ A2 @ ( inf_in60795025369792911la_a_b @ A2 @ B2 ) )
= ( inf_in60795025369792911la_a_b @ A2 @ B2 ) ) ).
% inf.left_idem
thf(fact_575_inf__left__idem,axiom,
! [X: set_formula_a_b,Y: set_formula_a_b] :
( ( inf_in5741596252272946265la_a_b @ X @ ( inf_in5741596252272946265la_a_b @ X @ Y ) )
= ( inf_in5741596252272946265la_a_b @ X @ Y ) ) ).
% inf_left_idem
thf(fact_576_inf__left__idem,axiom,
! [X: set_set_formula_a_b,Y: set_set_formula_a_b] :
( ( inf_in60795025369792911la_a_b @ X @ ( inf_in60795025369792911la_a_b @ X @ Y ) )
= ( inf_in60795025369792911la_a_b @ X @ Y ) ) ).
% inf_left_idem
thf(fact_577_inf_Oright__idem,axiom,
! [A2: set_formula_a_b,B2: set_formula_a_b] :
( ( inf_in5741596252272946265la_a_b @ ( inf_in5741596252272946265la_a_b @ A2 @ B2 ) @ B2 )
= ( inf_in5741596252272946265la_a_b @ A2 @ B2 ) ) ).
% inf.right_idem
thf(fact_578_inf_Oright__idem,axiom,
! [A2: set_set_formula_a_b,B2: set_set_formula_a_b] :
( ( inf_in60795025369792911la_a_b @ ( inf_in60795025369792911la_a_b @ A2 @ B2 ) @ B2 )
= ( inf_in60795025369792911la_a_b @ A2 @ B2 ) ) ).
% inf.right_idem
thf(fact_579_inf__right__idem,axiom,
! [X: set_formula_a_b,Y: set_formula_a_b] :
( ( inf_in5741596252272946265la_a_b @ ( inf_in5741596252272946265la_a_b @ X @ Y ) @ Y )
= ( inf_in5741596252272946265la_a_b @ X @ Y ) ) ).
% inf_right_idem
thf(fact_580_inf__right__idem,axiom,
! [X: set_set_formula_a_b,Y: set_set_formula_a_b] :
( ( inf_in60795025369792911la_a_b @ ( inf_in60795025369792911la_a_b @ X @ Y ) @ Y )
= ( inf_in60795025369792911la_a_b @ X @ Y ) ) ).
% inf_right_idem
thf(fact_581_IntI,axiom,
! [C: list_formula_a_b,A: set_list_formula_a_b,B: set_list_formula_a_b] :
( ( member7703986999711672120la_a_b @ C @ A )
=> ( ( member7703986999711672120la_a_b @ C @ B )
=> ( member7703986999711672120la_a_b @ C @ ( inf_in7172170795784610665la_a_b @ A @ B ) ) ) ) ).
% IntI
thf(fact_582_IntI,axiom,
! [C: set_formula_a_b,A: set_set_formula_a_b,B: set_set_formula_a_b] :
( ( member6621951321062823262la_a_b @ C @ A )
=> ( ( member6621951321062823262la_a_b @ C @ B )
=> ( member6621951321062823262la_a_b @ C @ ( inf_in60795025369792911la_a_b @ A @ B ) ) ) ) ).
% IntI
thf(fact_583_IntI,axiom,
! [C: formula_a_b,A: set_formula_a_b,B: set_formula_a_b] :
( ( member_formula_a_b2 @ C @ A )
=> ( ( member_formula_a_b2 @ C @ B )
=> ( member_formula_a_b2 @ C @ ( inf_in5741596252272946265la_a_b @ A @ B ) ) ) ) ).
% IntI
thf(fact_584_Int__iff,axiom,
! [C: list_formula_a_b,A: set_list_formula_a_b,B: set_list_formula_a_b] :
( ( member7703986999711672120la_a_b @ C @ ( inf_in7172170795784610665la_a_b @ A @ B ) )
= ( ( member7703986999711672120la_a_b @ C @ A )
& ( member7703986999711672120la_a_b @ C @ B ) ) ) ).
% Int_iff
thf(fact_585_Int__iff,axiom,
! [C: set_formula_a_b,A: set_set_formula_a_b,B: set_set_formula_a_b] :
( ( member6621951321062823262la_a_b @ C @ ( inf_in60795025369792911la_a_b @ A @ B ) )
= ( ( member6621951321062823262la_a_b @ C @ A )
& ( member6621951321062823262la_a_b @ C @ B ) ) ) ).
% Int_iff
thf(fact_586_Int__iff,axiom,
! [C: formula_a_b,A: set_formula_a_b,B: set_formula_a_b] :
( ( member_formula_a_b2 @ C @ ( inf_in5741596252272946265la_a_b @ A @ B ) )
= ( ( member_formula_a_b2 @ C @ A )
& ( member_formula_a_b2 @ C @ B ) ) ) ).
% Int_iff
thf(fact_587_DiffI,axiom,
! [C: list_formula_a_b,A: set_list_formula_a_b,B: set_list_formula_a_b] :
( ( member7703986999711672120la_a_b @ C @ A )
=> ( ~ ( member7703986999711672120la_a_b @ C @ B )
=> ( member7703986999711672120la_a_b @ C @ ( minus_8667068627093346608la_a_b @ A @ B ) ) ) ) ).
% DiffI
thf(fact_588_DiffI,axiom,
! [C: set_formula_a_b,A: set_set_formula_a_b,B: set_set_formula_a_b] :
( ( member6621951321062823262la_a_b @ C @ A )
=> ( ~ ( member6621951321062823262la_a_b @ C @ B )
=> ( member6621951321062823262la_a_b @ C @ ( minus_4878662943024999126la_a_b @ A @ B ) ) ) ) ).
% DiffI
thf(fact_589_DiffI,axiom,
! [C: formula_a_b,A: set_formula_a_b,B: set_formula_a_b] :
( ( member_formula_a_b2 @ C @ A )
=> ( ~ ( member_formula_a_b2 @ C @ B )
=> ( member_formula_a_b2 @ C @ ( minus_3283878196352184736la_a_b @ A @ B ) ) ) ) ).
% DiffI
thf(fact_590_Diff__iff,axiom,
! [C: list_formula_a_b,A: set_list_formula_a_b,B: set_list_formula_a_b] :
( ( member7703986999711672120la_a_b @ C @ ( minus_8667068627093346608la_a_b @ A @ B ) )
= ( ( member7703986999711672120la_a_b @ C @ A )
& ~ ( member7703986999711672120la_a_b @ C @ B ) ) ) ).
% Diff_iff
thf(fact_591_Diff__iff,axiom,
! [C: set_formula_a_b,A: set_set_formula_a_b,B: set_set_formula_a_b] :
( ( member6621951321062823262la_a_b @ C @ ( minus_4878662943024999126la_a_b @ A @ B ) )
= ( ( member6621951321062823262la_a_b @ C @ A )
& ~ ( member6621951321062823262la_a_b @ C @ B ) ) ) ).
% Diff_iff
thf(fact_592_Diff__iff,axiom,
! [C: formula_a_b,A: set_formula_a_b,B: set_formula_a_b] :
( ( member_formula_a_b2 @ C @ ( minus_3283878196352184736la_a_b @ A @ B ) )
= ( ( member_formula_a_b2 @ C @ A )
& ~ ( member_formula_a_b2 @ C @ B ) ) ) ).
% Diff_iff
thf(fact_593_Diff__idemp,axiom,
! [A: set_formula_a_b,B: set_formula_a_b] :
( ( minus_3283878196352184736la_a_b @ ( minus_3283878196352184736la_a_b @ A @ B ) @ B )
= ( minus_3283878196352184736la_a_b @ A @ B ) ) ).
% Diff_idemp
thf(fact_594_Diff__idemp,axiom,
! [A: set_set_formula_a_b,B: set_set_formula_a_b] :
( ( minus_4878662943024999126la_a_b @ ( minus_4878662943024999126la_a_b @ A @ B ) @ B )
= ( minus_4878662943024999126la_a_b @ A @ B ) ) ).
% Diff_idemp
thf(fact_595_inf__sup__absorb,axiom,
! [X: set_set_formula_a_b,Y: set_set_formula_a_b] :
( ( inf_in60795025369792911la_a_b @ X @ ( sup_su5931044959554304553la_a_b @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_596_inf__sup__absorb,axiom,
! [X: set_formula_a_b,Y: set_formula_a_b] :
( ( inf_in5741596252272946265la_a_b @ X @ ( sup_su8832426789560438003la_a_b @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_597_sup__inf__absorb,axiom,
! [X: set_set_formula_a_b,Y: set_set_formula_a_b] :
( ( sup_su5931044959554304553la_a_b @ X @ ( inf_in60795025369792911la_a_b @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_598_sup__inf__absorb,axiom,
! [X: set_formula_a_b,Y: set_formula_a_b] :
( ( sup_su8832426789560438003la_a_b @ X @ ( inf_in5741596252272946265la_a_b @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_599_Int__insert__left__if0,axiom,
! [A2: list_formula_a_b,C2: set_list_formula_a_b,B: set_list_formula_a_b] :
( ~ ( member7703986999711672120la_a_b @ A2 @ C2 )
=> ( ( inf_in7172170795784610665la_a_b @ ( insert4995524290302003025la_a_b @ A2 @ B ) @ C2 )
= ( inf_in7172170795784610665la_a_b @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_600_Int__insert__left__if0,axiom,
! [A2: set_formula_a_b,C2: set_set_formula_a_b,B: set_set_formula_a_b] :
( ~ ( member6621951321062823262la_a_b @ A2 @ C2 )
=> ( ( inf_in60795025369792911la_a_b @ ( insert8952366156270154231la_a_b @ A2 @ B ) @ C2 )
= ( inf_in60795025369792911la_a_b @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_601_Int__insert__left__if0,axiom,
! [A2: formula_a_b,C2: set_formula_a_b,B: set_formula_a_b] :
( ~ ( member_formula_a_b2 @ A2 @ C2 )
=> ( ( inf_in5741596252272946265la_a_b @ ( insert_formula_a_b2 @ A2 @ B ) @ C2 )
= ( inf_in5741596252272946265la_a_b @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_602_Int__insert__left__if1,axiom,
! [A2: list_formula_a_b,C2: set_list_formula_a_b,B: set_list_formula_a_b] :
( ( member7703986999711672120la_a_b @ A2 @ C2 )
=> ( ( inf_in7172170795784610665la_a_b @ ( insert4995524290302003025la_a_b @ A2 @ B ) @ C2 )
= ( insert4995524290302003025la_a_b @ A2 @ ( inf_in7172170795784610665la_a_b @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_603_Int__insert__left__if1,axiom,
! [A2: set_formula_a_b,C2: set_set_formula_a_b,B: set_set_formula_a_b] :
( ( member6621951321062823262la_a_b @ A2 @ C2 )
=> ( ( inf_in60795025369792911la_a_b @ ( insert8952366156270154231la_a_b @ A2 @ B ) @ C2 )
= ( insert8952366156270154231la_a_b @ A2 @ ( inf_in60795025369792911la_a_b @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_604_Int__insert__left__if1,axiom,
! [A2: formula_a_b,C2: set_formula_a_b,B: set_formula_a_b] :
( ( member_formula_a_b2 @ A2 @ C2 )
=> ( ( inf_in5741596252272946265la_a_b @ ( insert_formula_a_b2 @ A2 @ B ) @ C2 )
= ( insert_formula_a_b2 @ A2 @ ( inf_in5741596252272946265la_a_b @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_605_insert__inter__insert,axiom,
! [A2: formula_a_b,A: set_formula_a_b,B: set_formula_a_b] :
( ( inf_in5741596252272946265la_a_b @ ( insert_formula_a_b2 @ A2 @ A ) @ ( insert_formula_a_b2 @ A2 @ B ) )
= ( insert_formula_a_b2 @ A2 @ ( inf_in5741596252272946265la_a_b @ A @ B ) ) ) ).
% insert_inter_insert
thf(fact_606_insert__inter__insert,axiom,
! [A2: set_formula_a_b,A: set_set_formula_a_b,B: set_set_formula_a_b] :
( ( inf_in60795025369792911la_a_b @ ( insert8952366156270154231la_a_b @ A2 @ A ) @ ( insert8952366156270154231la_a_b @ A2 @ B ) )
= ( insert8952366156270154231la_a_b @ A2 @ ( inf_in60795025369792911la_a_b @ A @ B ) ) ) ).
% insert_inter_insert
thf(fact_607_Int__insert__right__if0,axiom,
! [A2: list_formula_a_b,A: set_list_formula_a_b,B: set_list_formula_a_b] :
( ~ ( member7703986999711672120la_a_b @ A2 @ A )
=> ( ( inf_in7172170795784610665la_a_b @ A @ ( insert4995524290302003025la_a_b @ A2 @ B ) )
= ( inf_in7172170795784610665la_a_b @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_608_Int__insert__right__if0,axiom,
! [A2: set_formula_a_b,A: set_set_formula_a_b,B: set_set_formula_a_b] :
( ~ ( member6621951321062823262la_a_b @ A2 @ A )
=> ( ( inf_in60795025369792911la_a_b @ A @ ( insert8952366156270154231la_a_b @ A2 @ B ) )
= ( inf_in60795025369792911la_a_b @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_609_Int__insert__right__if0,axiom,
! [A2: formula_a_b,A: set_formula_a_b,B: set_formula_a_b] :
( ~ ( member_formula_a_b2 @ A2 @ A )
=> ( ( inf_in5741596252272946265la_a_b @ A @ ( insert_formula_a_b2 @ A2 @ B ) )
= ( inf_in5741596252272946265la_a_b @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_610_Int__insert__right__if1,axiom,
! [A2: list_formula_a_b,A: set_list_formula_a_b,B: set_list_formula_a_b] :
( ( member7703986999711672120la_a_b @ A2 @ A )
=> ( ( inf_in7172170795784610665la_a_b @ A @ ( insert4995524290302003025la_a_b @ A2 @ B ) )
= ( insert4995524290302003025la_a_b @ A2 @ ( inf_in7172170795784610665la_a_b @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_611_Int__insert__right__if1,axiom,
! [A2: set_formula_a_b,A: set_set_formula_a_b,B: set_set_formula_a_b] :
( ( member6621951321062823262la_a_b @ A2 @ A )
=> ( ( inf_in60795025369792911la_a_b @ A @ ( insert8952366156270154231la_a_b @ A2 @ B ) )
= ( insert8952366156270154231la_a_b @ A2 @ ( inf_in60795025369792911la_a_b @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_612_Int__insert__right__if1,axiom,
! [A2: formula_a_b,A: set_formula_a_b,B: set_formula_a_b] :
( ( member_formula_a_b2 @ A2 @ A )
=> ( ( inf_in5741596252272946265la_a_b @ A @ ( insert_formula_a_b2 @ A2 @ B ) )
= ( insert_formula_a_b2 @ A2 @ ( inf_in5741596252272946265la_a_b @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_613_Diff__insert0,axiom,
! [X: list_formula_a_b,A: set_list_formula_a_b,B: set_list_formula_a_b] :
( ~ ( member7703986999711672120la_a_b @ X @ A )
=> ( ( minus_8667068627093346608la_a_b @ A @ ( insert4995524290302003025la_a_b @ X @ B ) )
= ( minus_8667068627093346608la_a_b @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_614_Diff__insert0,axiom,
! [X: set_formula_a_b,A: set_set_formula_a_b,B: set_set_formula_a_b] :
( ~ ( member6621951321062823262la_a_b @ X @ A )
=> ( ( minus_4878662943024999126la_a_b @ A @ ( insert8952366156270154231la_a_b @ X @ B ) )
= ( minus_4878662943024999126la_a_b @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_615_Diff__insert0,axiom,
! [X: formula_a_b,A: set_formula_a_b,B: set_formula_a_b] :
( ~ ( member_formula_a_b2 @ X @ A )
=> ( ( minus_3283878196352184736la_a_b @ A @ ( insert_formula_a_b2 @ X @ B ) )
= ( minus_3283878196352184736la_a_b @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_616_insert__Diff1,axiom,
! [X: list_formula_a_b,B: set_list_formula_a_b,A: set_list_formula_a_b] :
( ( member7703986999711672120la_a_b @ X @ B )
=> ( ( minus_8667068627093346608la_a_b @ ( insert4995524290302003025la_a_b @ X @ A ) @ B )
= ( minus_8667068627093346608la_a_b @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_617_insert__Diff1,axiom,
! [X: set_formula_a_b,B: set_set_formula_a_b,A: set_set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X @ B )
=> ( ( minus_4878662943024999126la_a_b @ ( insert8952366156270154231la_a_b @ X @ A ) @ B )
= ( minus_4878662943024999126la_a_b @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_618_insert__Diff1,axiom,
! [X: formula_a_b,B: set_formula_a_b,A: set_formula_a_b] :
( ( member_formula_a_b2 @ X @ B )
=> ( ( minus_3283878196352184736la_a_b @ ( insert_formula_a_b2 @ X @ A ) @ B )
= ( minus_3283878196352184736la_a_b @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_619_Int__Un__eq_I4_J,axiom,
! [T: set_set_formula_a_b,S3: set_set_formula_a_b] :
( ( sup_su5931044959554304553la_a_b @ T @ ( inf_in60795025369792911la_a_b @ S3 @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_620_Int__Un__eq_I4_J,axiom,
! [T: set_formula_a_b,S3: set_formula_a_b] :
( ( sup_su8832426789560438003la_a_b @ T @ ( inf_in5741596252272946265la_a_b @ S3 @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_621_Int__Un__eq_I3_J,axiom,
! [S3: set_set_formula_a_b,T: set_set_formula_a_b] :
( ( sup_su5931044959554304553la_a_b @ S3 @ ( inf_in60795025369792911la_a_b @ S3 @ T ) )
= S3 ) ).
% Int_Un_eq(3)
thf(fact_622_Int__Un__eq_I3_J,axiom,
! [S3: set_formula_a_b,T: set_formula_a_b] :
( ( sup_su8832426789560438003la_a_b @ S3 @ ( inf_in5741596252272946265la_a_b @ S3 @ T ) )
= S3 ) ).
% Int_Un_eq(3)
thf(fact_623_Int__Un__eq_I2_J,axiom,
! [S3: set_set_formula_a_b,T: set_set_formula_a_b] :
( ( sup_su5931044959554304553la_a_b @ ( inf_in60795025369792911la_a_b @ S3 @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_624_Int__Un__eq_I2_J,axiom,
! [S3: set_formula_a_b,T: set_formula_a_b] :
( ( sup_su8832426789560438003la_a_b @ ( inf_in5741596252272946265la_a_b @ S3 @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_625_Int__Un__eq_I1_J,axiom,
! [S3: set_set_formula_a_b,T: set_set_formula_a_b] :
( ( sup_su5931044959554304553la_a_b @ ( inf_in60795025369792911la_a_b @ S3 @ T ) @ S3 )
= S3 ) ).
% Int_Un_eq(1)
thf(fact_626_Int__Un__eq_I1_J,axiom,
! [S3: set_formula_a_b,T: set_formula_a_b] :
( ( sup_su8832426789560438003la_a_b @ ( inf_in5741596252272946265la_a_b @ S3 @ T ) @ S3 )
= S3 ) ).
% Int_Un_eq(1)
thf(fact_627_Un__Int__eq_I4_J,axiom,
! [T: set_set_formula_a_b,S3: set_set_formula_a_b] :
( ( inf_in60795025369792911la_a_b @ T @ ( sup_su5931044959554304553la_a_b @ S3 @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_628_Un__Int__eq_I4_J,axiom,
! [T: set_formula_a_b,S3: set_formula_a_b] :
( ( inf_in5741596252272946265la_a_b @ T @ ( sup_su8832426789560438003la_a_b @ S3 @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_629_Un__Int__eq_I3_J,axiom,
! [S3: set_set_formula_a_b,T: set_set_formula_a_b] :
( ( inf_in60795025369792911la_a_b @ S3 @ ( sup_su5931044959554304553la_a_b @ S3 @ T ) )
= S3 ) ).
% Un_Int_eq(3)
thf(fact_630_Un__Int__eq_I3_J,axiom,
! [S3: set_formula_a_b,T: set_formula_a_b] :
( ( inf_in5741596252272946265la_a_b @ S3 @ ( sup_su8832426789560438003la_a_b @ S3 @ T ) )
= S3 ) ).
% Un_Int_eq(3)
thf(fact_631_Un__Int__eq_I2_J,axiom,
! [S3: set_set_formula_a_b,T: set_set_formula_a_b] :
( ( inf_in60795025369792911la_a_b @ ( sup_su5931044959554304553la_a_b @ S3 @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_632_Un__Int__eq_I2_J,axiom,
! [S3: set_formula_a_b,T: set_formula_a_b] :
( ( inf_in5741596252272946265la_a_b @ ( sup_su8832426789560438003la_a_b @ S3 @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_633_Un__Int__eq_I1_J,axiom,
! [S3: set_set_formula_a_b,T: set_set_formula_a_b] :
( ( inf_in60795025369792911la_a_b @ ( sup_su5931044959554304553la_a_b @ S3 @ T ) @ S3 )
= S3 ) ).
% Un_Int_eq(1)
thf(fact_634_Un__Int__eq_I1_J,axiom,
! [S3: set_formula_a_b,T: set_formula_a_b] :
( ( inf_in5741596252272946265la_a_b @ ( sup_su8832426789560438003la_a_b @ S3 @ T ) @ S3 )
= S3 ) ).
% Un_Int_eq(1)
thf(fact_635_Un__Diff__cancel,axiom,
! [A: set_set_formula_a_b,B: set_set_formula_a_b] :
( ( sup_su5931044959554304553la_a_b @ A @ ( minus_4878662943024999126la_a_b @ B @ A ) )
= ( sup_su5931044959554304553la_a_b @ A @ B ) ) ).
% Un_Diff_cancel
thf(fact_636_Un__Diff__cancel,axiom,
! [A: set_formula_a_b,B: set_formula_a_b] :
( ( sup_su8832426789560438003la_a_b @ A @ ( minus_3283878196352184736la_a_b @ B @ A ) )
= ( sup_su8832426789560438003la_a_b @ A @ B ) ) ).
% Un_Diff_cancel
thf(fact_637_Un__Diff__cancel2,axiom,
! [B: set_set_formula_a_b,A: set_set_formula_a_b] :
( ( sup_su5931044959554304553la_a_b @ ( minus_4878662943024999126la_a_b @ B @ A ) @ A )
= ( sup_su5931044959554304553la_a_b @ B @ A ) ) ).
% Un_Diff_cancel2
thf(fact_638_Un__Diff__cancel2,axiom,
! [B: set_formula_a_b,A: set_formula_a_b] :
( ( sup_su8832426789560438003la_a_b @ ( minus_3283878196352184736la_a_b @ B @ A ) @ A )
= ( sup_su8832426789560438003la_a_b @ B @ A ) ) ).
% Un_Diff_cancel2
thf(fact_639_Nil__eq__concat__conv,axiom,
! [Xss2: list_l5948067055324603441la_a_b] :
( ( nil_formula_a_b
= ( concat_formula_a_b @ Xss2 ) )
= ( ! [X2: list_formula_a_b] :
( ( member7703986999711672120la_a_b @ X2 @ ( set_list_formula_a_b2 @ Xss2 ) )
=> ( X2 = nil_formula_a_b ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_640_concat__eq__Nil__conv,axiom,
! [Xss2: list_l5948067055324603441la_a_b] :
( ( ( concat_formula_a_b @ Xss2 )
= nil_formula_a_b )
= ( ! [X2: list_formula_a_b] :
( ( member7703986999711672120la_a_b @ X2 @ ( set_list_formula_a_b2 @ Xss2 ) )
=> ( X2 = nil_formula_a_b ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_641_concat__append,axiom,
! [Xs: list_l5948067055324603441la_a_b,Ys: list_l5948067055324603441la_a_b] :
( ( concat_formula_a_b @ ( append58666856857280070la_a_b @ Xs @ Ys ) )
= ( append_formula_a_b @ ( concat_formula_a_b @ Xs ) @ ( concat_formula_a_b @ Ys ) ) ) ).
% concat_append
thf(fact_642_inf__sup__aci_I4_J,axiom,
! [X: set_formula_a_b,Y: set_formula_a_b] :
( ( inf_in5741596252272946265la_a_b @ X @ ( inf_in5741596252272946265la_a_b @ X @ Y ) )
= ( inf_in5741596252272946265la_a_b @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_643_inf__sup__aci_I4_J,axiom,
! [X: set_set_formula_a_b,Y: set_set_formula_a_b] :
( ( inf_in60795025369792911la_a_b @ X @ ( inf_in60795025369792911la_a_b @ X @ Y ) )
= ( inf_in60795025369792911la_a_b @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_644_inf__sup__aci_I3_J,axiom,
! [X: set_formula_a_b,Y: set_formula_a_b,Z: set_formula_a_b] :
( ( inf_in5741596252272946265la_a_b @ X @ ( inf_in5741596252272946265la_a_b @ Y @ Z ) )
= ( inf_in5741596252272946265la_a_b @ Y @ ( inf_in5741596252272946265la_a_b @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_645_inf__sup__aci_I3_J,axiom,
! [X: set_set_formula_a_b,Y: set_set_formula_a_b,Z: set_set_formula_a_b] :
( ( inf_in60795025369792911la_a_b @ X @ ( inf_in60795025369792911la_a_b @ Y @ Z ) )
= ( inf_in60795025369792911la_a_b @ Y @ ( inf_in60795025369792911la_a_b @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_646_inf__sup__aci_I2_J,axiom,
! [X: set_formula_a_b,Y: set_formula_a_b,Z: set_formula_a_b] :
( ( inf_in5741596252272946265la_a_b @ ( inf_in5741596252272946265la_a_b @ X @ Y ) @ Z )
= ( inf_in5741596252272946265la_a_b @ X @ ( inf_in5741596252272946265la_a_b @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_647_inf__sup__aci_I2_J,axiom,
! [X: set_set_formula_a_b,Y: set_set_formula_a_b,Z: set_set_formula_a_b] :
( ( inf_in60795025369792911la_a_b @ ( inf_in60795025369792911la_a_b @ X @ Y ) @ Z )
= ( inf_in60795025369792911la_a_b @ X @ ( inf_in60795025369792911la_a_b @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_648_inf__sup__aci_I1_J,axiom,
( inf_in5741596252272946265la_a_b
= ( ^ [X2: set_formula_a_b,Y2: set_formula_a_b] : ( inf_in5741596252272946265la_a_b @ Y2 @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_649_inf__sup__aci_I1_J,axiom,
( inf_in60795025369792911la_a_b
= ( ^ [X2: set_set_formula_a_b,Y2: set_set_formula_a_b] : ( inf_in60795025369792911la_a_b @ Y2 @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_650_inf_Oassoc,axiom,
! [A2: set_formula_a_b,B2: set_formula_a_b,C: set_formula_a_b] :
( ( inf_in5741596252272946265la_a_b @ ( inf_in5741596252272946265la_a_b @ A2 @ B2 ) @ C )
= ( inf_in5741596252272946265la_a_b @ A2 @ ( inf_in5741596252272946265la_a_b @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_651_inf_Oassoc,axiom,
! [A2: set_set_formula_a_b,B2: set_set_formula_a_b,C: set_set_formula_a_b] :
( ( inf_in60795025369792911la_a_b @ ( inf_in60795025369792911la_a_b @ A2 @ B2 ) @ C )
= ( inf_in60795025369792911la_a_b @ A2 @ ( inf_in60795025369792911la_a_b @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_652_inf__assoc,axiom,
! [X: set_formula_a_b,Y: set_formula_a_b,Z: set_formula_a_b] :
( ( inf_in5741596252272946265la_a_b @ ( inf_in5741596252272946265la_a_b @ X @ Y ) @ Z )
= ( inf_in5741596252272946265la_a_b @ X @ ( inf_in5741596252272946265la_a_b @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_653_inf__assoc,axiom,
! [X: set_set_formula_a_b,Y: set_set_formula_a_b,Z: set_set_formula_a_b] :
( ( inf_in60795025369792911la_a_b @ ( inf_in60795025369792911la_a_b @ X @ Y ) @ Z )
= ( inf_in60795025369792911la_a_b @ X @ ( inf_in60795025369792911la_a_b @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_654_inf_Ocommute,axiom,
( inf_in5741596252272946265la_a_b
= ( ^ [A3: set_formula_a_b,B3: set_formula_a_b] : ( inf_in5741596252272946265la_a_b @ B3 @ A3 ) ) ) ).
% inf.commute
thf(fact_655_inf_Ocommute,axiom,
( inf_in60795025369792911la_a_b
= ( ^ [A3: set_set_formula_a_b,B3: set_set_formula_a_b] : ( inf_in60795025369792911la_a_b @ B3 @ A3 ) ) ) ).
% inf.commute
thf(fact_656_inf__commute,axiom,
( inf_in5741596252272946265la_a_b
= ( ^ [X2: set_formula_a_b,Y2: set_formula_a_b] : ( inf_in5741596252272946265la_a_b @ Y2 @ X2 ) ) ) ).
% inf_commute
thf(fact_657_inf__commute,axiom,
( inf_in60795025369792911la_a_b
= ( ^ [X2: set_set_formula_a_b,Y2: set_set_formula_a_b] : ( inf_in60795025369792911la_a_b @ Y2 @ X2 ) ) ) ).
% inf_commute
thf(fact_658_inf_Oleft__commute,axiom,
! [B2: set_formula_a_b,A2: set_formula_a_b,C: set_formula_a_b] :
( ( inf_in5741596252272946265la_a_b @ B2 @ ( inf_in5741596252272946265la_a_b @ A2 @ C ) )
= ( inf_in5741596252272946265la_a_b @ A2 @ ( inf_in5741596252272946265la_a_b @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_659_inf_Oleft__commute,axiom,
! [B2: set_set_formula_a_b,A2: set_set_formula_a_b,C: set_set_formula_a_b] :
( ( inf_in60795025369792911la_a_b @ B2 @ ( inf_in60795025369792911la_a_b @ A2 @ C ) )
= ( inf_in60795025369792911la_a_b @ A2 @ ( inf_in60795025369792911la_a_b @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_660_inf__left__commute,axiom,
! [X: set_formula_a_b,Y: set_formula_a_b,Z: set_formula_a_b] :
( ( inf_in5741596252272946265la_a_b @ X @ ( inf_in5741596252272946265la_a_b @ Y @ Z ) )
= ( inf_in5741596252272946265la_a_b @ Y @ ( inf_in5741596252272946265la_a_b @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_661_inf__left__commute,axiom,
! [X: set_set_formula_a_b,Y: set_set_formula_a_b,Z: set_set_formula_a_b] :
( ( inf_in60795025369792911la_a_b @ X @ ( inf_in60795025369792911la_a_b @ Y @ Z ) )
= ( inf_in60795025369792911la_a_b @ Y @ ( inf_in60795025369792911la_a_b @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_662_IntE,axiom,
! [C: list_formula_a_b,A: set_list_formula_a_b,B: set_list_formula_a_b] :
( ( member7703986999711672120la_a_b @ C @ ( inf_in7172170795784610665la_a_b @ A @ B ) )
=> ~ ( ( member7703986999711672120la_a_b @ C @ A )
=> ~ ( member7703986999711672120la_a_b @ C @ B ) ) ) ).
% IntE
thf(fact_663_IntE,axiom,
! [C: set_formula_a_b,A: set_set_formula_a_b,B: set_set_formula_a_b] :
( ( member6621951321062823262la_a_b @ C @ ( inf_in60795025369792911la_a_b @ A @ B ) )
=> ~ ( ( member6621951321062823262la_a_b @ C @ A )
=> ~ ( member6621951321062823262la_a_b @ C @ B ) ) ) ).
% IntE
thf(fact_664_IntE,axiom,
! [C: formula_a_b,A: set_formula_a_b,B: set_formula_a_b] :
( ( member_formula_a_b2 @ C @ ( inf_in5741596252272946265la_a_b @ A @ B ) )
=> ~ ( ( member_formula_a_b2 @ C @ A )
=> ~ ( member_formula_a_b2 @ C @ B ) ) ) ).
% IntE
thf(fact_665_DiffE,axiom,
! [C: list_formula_a_b,A: set_list_formula_a_b,B: set_list_formula_a_b] :
( ( member7703986999711672120la_a_b @ C @ ( minus_8667068627093346608la_a_b @ A @ B ) )
=> ~ ( ( member7703986999711672120la_a_b @ C @ A )
=> ( member7703986999711672120la_a_b @ C @ B ) ) ) ).
% DiffE
thf(fact_666_DiffE,axiom,
! [C: set_formula_a_b,A: set_set_formula_a_b,B: set_set_formula_a_b] :
( ( member6621951321062823262la_a_b @ C @ ( minus_4878662943024999126la_a_b @ A @ B ) )
=> ~ ( ( member6621951321062823262la_a_b @ C @ A )
=> ( member6621951321062823262la_a_b @ C @ B ) ) ) ).
% DiffE
thf(fact_667_DiffE,axiom,
! [C: formula_a_b,A: set_formula_a_b,B: set_formula_a_b] :
( ( member_formula_a_b2 @ C @ ( minus_3283878196352184736la_a_b @ A @ B ) )
=> ~ ( ( member_formula_a_b2 @ C @ A )
=> ( member_formula_a_b2 @ C @ B ) ) ) ).
% DiffE
thf(fact_668_IntD1,axiom,
! [C: list_formula_a_b,A: set_list_formula_a_b,B: set_list_formula_a_b] :
( ( member7703986999711672120la_a_b @ C @ ( inf_in7172170795784610665la_a_b @ A @ B ) )
=> ( member7703986999711672120la_a_b @ C @ A ) ) ).
% IntD1
thf(fact_669_IntD1,axiom,
! [C: set_formula_a_b,A: set_set_formula_a_b,B: set_set_formula_a_b] :
( ( member6621951321062823262la_a_b @ C @ ( inf_in60795025369792911la_a_b @ A @ B ) )
=> ( member6621951321062823262la_a_b @ C @ A ) ) ).
% IntD1
thf(fact_670_IntD1,axiom,
! [C: formula_a_b,A: set_formula_a_b,B: set_formula_a_b] :
( ( member_formula_a_b2 @ C @ ( inf_in5741596252272946265la_a_b @ A @ B ) )
=> ( member_formula_a_b2 @ C @ A ) ) ).
% IntD1
thf(fact_671_IntD2,axiom,
! [C: list_formula_a_b,A: set_list_formula_a_b,B: set_list_formula_a_b] :
( ( member7703986999711672120la_a_b @ C @ ( inf_in7172170795784610665la_a_b @ A @ B ) )
=> ( member7703986999711672120la_a_b @ C @ B ) ) ).
% IntD2
thf(fact_672_IntD2,axiom,
! [C: set_formula_a_b,A: set_set_formula_a_b,B: set_set_formula_a_b] :
( ( member6621951321062823262la_a_b @ C @ ( inf_in60795025369792911la_a_b @ A @ B ) )
=> ( member6621951321062823262la_a_b @ C @ B ) ) ).
% IntD2
thf(fact_673_IntD2,axiom,
! [C: formula_a_b,A: set_formula_a_b,B: set_formula_a_b] :
( ( member_formula_a_b2 @ C @ ( inf_in5741596252272946265la_a_b @ A @ B ) )
=> ( member_formula_a_b2 @ C @ B ) ) ).
% IntD2
thf(fact_674_DiffD1,axiom,
! [C: list_formula_a_b,A: set_list_formula_a_b,B: set_list_formula_a_b] :
( ( member7703986999711672120la_a_b @ C @ ( minus_8667068627093346608la_a_b @ A @ B ) )
=> ( member7703986999711672120la_a_b @ C @ A ) ) ).
% DiffD1
thf(fact_675_DiffD1,axiom,
! [C: set_formula_a_b,A: set_set_formula_a_b,B: set_set_formula_a_b] :
( ( member6621951321062823262la_a_b @ C @ ( minus_4878662943024999126la_a_b @ A @ B ) )
=> ( member6621951321062823262la_a_b @ C @ A ) ) ).
% DiffD1
thf(fact_676_DiffD1,axiom,
! [C: formula_a_b,A: set_formula_a_b,B: set_formula_a_b] :
( ( member_formula_a_b2 @ C @ ( minus_3283878196352184736la_a_b @ A @ B ) )
=> ( member_formula_a_b2 @ C @ A ) ) ).
% DiffD1
thf(fact_677_DiffD2,axiom,
! [C: list_formula_a_b,A: set_list_formula_a_b,B: set_list_formula_a_b] :
( ( member7703986999711672120la_a_b @ C @ ( minus_8667068627093346608la_a_b @ A @ B ) )
=> ~ ( member7703986999711672120la_a_b @ C @ B ) ) ).
% DiffD2
thf(fact_678_DiffD2,axiom,
! [C: set_formula_a_b,A: set_set_formula_a_b,B: set_set_formula_a_b] :
( ( member6621951321062823262la_a_b @ C @ ( minus_4878662943024999126la_a_b @ A @ B ) )
=> ~ ( member6621951321062823262la_a_b @ C @ B ) ) ).
% DiffD2
thf(fact_679_DiffD2,axiom,
! [C: formula_a_b,A: set_formula_a_b,B: set_formula_a_b] :
( ( member_formula_a_b2 @ C @ ( minus_3283878196352184736la_a_b @ A @ B ) )
=> ~ ( member_formula_a_b2 @ C @ B ) ) ).
% DiffD2
thf(fact_680_Int__Diff,axiom,
! [A: set_formula_a_b,B: set_formula_a_b,C2: set_formula_a_b] :
( ( minus_3283878196352184736la_a_b @ ( inf_in5741596252272946265la_a_b @ A @ B ) @ C2 )
= ( inf_in5741596252272946265la_a_b @ A @ ( minus_3283878196352184736la_a_b @ B @ C2 ) ) ) ).
% Int_Diff
thf(fact_681_Int__Diff,axiom,
! [A: set_set_formula_a_b,B: set_set_formula_a_b,C2: set_set_formula_a_b] :
( ( minus_4878662943024999126la_a_b @ ( inf_in60795025369792911la_a_b @ A @ B ) @ C2 )
= ( inf_in60795025369792911la_a_b @ A @ ( minus_4878662943024999126la_a_b @ B @ C2 ) ) ) ).
% Int_Diff
thf(fact_682_Diff__Int2,axiom,
! [A: set_formula_a_b,C2: set_formula_a_b,B: set_formula_a_b] :
( ( minus_3283878196352184736la_a_b @ ( inf_in5741596252272946265la_a_b @ A @ C2 ) @ ( inf_in5741596252272946265la_a_b @ B @ C2 ) )
= ( minus_3283878196352184736la_a_b @ ( inf_in5741596252272946265la_a_b @ A @ C2 ) @ B ) ) ).
% Diff_Int2
thf(fact_683_Diff__Int2,axiom,
! [A: set_set_formula_a_b,C2: set_set_formula_a_b,B: set_set_formula_a_b] :
( ( minus_4878662943024999126la_a_b @ ( inf_in60795025369792911la_a_b @ A @ C2 ) @ ( inf_in60795025369792911la_a_b @ B @ C2 ) )
= ( minus_4878662943024999126la_a_b @ ( inf_in60795025369792911la_a_b @ A @ C2 ) @ B ) ) ).
% Diff_Int2
thf(fact_684_Int__assoc,axiom,
! [A: set_formula_a_b,B: set_formula_a_b,C2: set_formula_a_b] :
( ( inf_in5741596252272946265la_a_b @ ( inf_in5741596252272946265la_a_b @ A @ B ) @ C2 )
= ( inf_in5741596252272946265la_a_b @ A @ ( inf_in5741596252272946265la_a_b @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_685_Int__assoc,axiom,
! [A: set_set_formula_a_b,B: set_set_formula_a_b,C2: set_set_formula_a_b] :
( ( inf_in60795025369792911la_a_b @ ( inf_in60795025369792911la_a_b @ A @ B ) @ C2 )
= ( inf_in60795025369792911la_a_b @ A @ ( inf_in60795025369792911la_a_b @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_686_Int__absorb,axiom,
! [A: set_formula_a_b] :
( ( inf_in5741596252272946265la_a_b @ A @ A )
= A ) ).
% Int_absorb
thf(fact_687_Int__absorb,axiom,
! [A: set_set_formula_a_b] :
( ( inf_in60795025369792911la_a_b @ A @ A )
= A ) ).
% Int_absorb
thf(fact_688_Int__commute,axiom,
( inf_in5741596252272946265la_a_b
= ( ^ [A4: set_formula_a_b,B4: set_formula_a_b] : ( inf_in5741596252272946265la_a_b @ B4 @ A4 ) ) ) ).
% Int_commute
thf(fact_689_Int__commute,axiom,
( inf_in60795025369792911la_a_b
= ( ^ [A4: set_set_formula_a_b,B4: set_set_formula_a_b] : ( inf_in60795025369792911la_a_b @ B4 @ A4 ) ) ) ).
% Int_commute
thf(fact_690_Diff__Diff__Int,axiom,
! [A: set_formula_a_b,B: set_formula_a_b] :
( ( minus_3283878196352184736la_a_b @ A @ ( minus_3283878196352184736la_a_b @ A @ B ) )
= ( inf_in5741596252272946265la_a_b @ A @ B ) ) ).
% Diff_Diff_Int
thf(fact_691_Diff__Diff__Int,axiom,
! [A: set_set_formula_a_b,B: set_set_formula_a_b] :
( ( minus_4878662943024999126la_a_b @ A @ ( minus_4878662943024999126la_a_b @ A @ B ) )
= ( inf_in60795025369792911la_a_b @ A @ B ) ) ).
% Diff_Diff_Int
thf(fact_692_Int__left__absorb,axiom,
! [A: set_formula_a_b,B: set_formula_a_b] :
( ( inf_in5741596252272946265la_a_b @ A @ ( inf_in5741596252272946265la_a_b @ A @ B ) )
= ( inf_in5741596252272946265la_a_b @ A @ B ) ) ).
% Int_left_absorb
thf(fact_693_Int__left__absorb,axiom,
! [A: set_set_formula_a_b,B: set_set_formula_a_b] :
( ( inf_in60795025369792911la_a_b @ A @ ( inf_in60795025369792911la_a_b @ A @ B ) )
= ( inf_in60795025369792911la_a_b @ A @ B ) ) ).
% Int_left_absorb
thf(fact_694_Diff__Int__distrib,axiom,
! [C2: set_formula_a_b,A: set_formula_a_b,B: set_formula_a_b] :
( ( inf_in5741596252272946265la_a_b @ C2 @ ( minus_3283878196352184736la_a_b @ A @ B ) )
= ( minus_3283878196352184736la_a_b @ ( inf_in5741596252272946265la_a_b @ C2 @ A ) @ ( inf_in5741596252272946265la_a_b @ C2 @ B ) ) ) ).
% Diff_Int_distrib
thf(fact_695_Diff__Int__distrib,axiom,
! [C2: set_set_formula_a_b,A: set_set_formula_a_b,B: set_set_formula_a_b] :
( ( inf_in60795025369792911la_a_b @ C2 @ ( minus_4878662943024999126la_a_b @ A @ B ) )
= ( minus_4878662943024999126la_a_b @ ( inf_in60795025369792911la_a_b @ C2 @ A ) @ ( inf_in60795025369792911la_a_b @ C2 @ B ) ) ) ).
% Diff_Int_distrib
thf(fact_696_Int__left__commute,axiom,
! [A: set_formula_a_b,B: set_formula_a_b,C2: set_formula_a_b] :
( ( inf_in5741596252272946265la_a_b @ A @ ( inf_in5741596252272946265la_a_b @ B @ C2 ) )
= ( inf_in5741596252272946265la_a_b @ B @ ( inf_in5741596252272946265la_a_b @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_697_Int__left__commute,axiom,
! [A: set_set_formula_a_b,B: set_set_formula_a_b,C2: set_set_formula_a_b] :
( ( inf_in60795025369792911la_a_b @ A @ ( inf_in60795025369792911la_a_b @ B @ C2 ) )
= ( inf_in60795025369792911la_a_b @ B @ ( inf_in60795025369792911la_a_b @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_698_Diff__Int__distrib2,axiom,
! [A: set_formula_a_b,B: set_formula_a_b,C2: set_formula_a_b] :
( ( inf_in5741596252272946265la_a_b @ ( minus_3283878196352184736la_a_b @ A @ B ) @ C2 )
= ( minus_3283878196352184736la_a_b @ ( inf_in5741596252272946265la_a_b @ A @ C2 ) @ ( inf_in5741596252272946265la_a_b @ B @ C2 ) ) ) ).
% Diff_Int_distrib2
thf(fact_699_Diff__Int__distrib2,axiom,
! [A: set_set_formula_a_b,B: set_set_formula_a_b,C2: set_set_formula_a_b] :
( ( inf_in60795025369792911la_a_b @ ( minus_4878662943024999126la_a_b @ A @ B ) @ C2 )
= ( minus_4878662943024999126la_a_b @ ( inf_in60795025369792911la_a_b @ A @ C2 ) @ ( inf_in60795025369792911la_a_b @ B @ C2 ) ) ) ).
% Diff_Int_distrib2
thf(fact_700_subseqs__refl,axiom,
! [Xs: list_formula_a_b] : ( member7703986999711672120la_a_b @ Xs @ ( set_list_formula_a_b2 @ ( subseqs_formula_a_b @ Xs ) ) ) ).
% subseqs_refl
thf(fact_701_boolean__algebra__cancel_Oinf1,axiom,
! [A: set_formula_a_b,K: set_formula_a_b,A2: set_formula_a_b,B2: set_formula_a_b] :
( ( A
= ( inf_in5741596252272946265la_a_b @ K @ A2 ) )
=> ( ( inf_in5741596252272946265la_a_b @ A @ B2 )
= ( inf_in5741596252272946265la_a_b @ K @ ( inf_in5741596252272946265la_a_b @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_702_boolean__algebra__cancel_Oinf1,axiom,
! [A: set_set_formula_a_b,K: set_set_formula_a_b,A2: set_set_formula_a_b,B2: set_set_formula_a_b] :
( ( A
= ( inf_in60795025369792911la_a_b @ K @ A2 ) )
=> ( ( inf_in60795025369792911la_a_b @ A @ B2 )
= ( inf_in60795025369792911la_a_b @ K @ ( inf_in60795025369792911la_a_b @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_703_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_formula_a_b,K: set_formula_a_b,B2: set_formula_a_b,A2: set_formula_a_b] :
( ( B
= ( inf_in5741596252272946265la_a_b @ K @ B2 ) )
=> ( ( inf_in5741596252272946265la_a_b @ A2 @ B )
= ( inf_in5741596252272946265la_a_b @ K @ ( inf_in5741596252272946265la_a_b @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_704_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_set_formula_a_b,K: set_set_formula_a_b,B2: set_set_formula_a_b,A2: set_set_formula_a_b] :
( ( B
= ( inf_in60795025369792911la_a_b @ K @ B2 ) )
=> ( ( inf_in60795025369792911la_a_b @ A2 @ B )
= ( inf_in60795025369792911la_a_b @ K @ ( inf_in60795025369792911la_a_b @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_705_Diff__Un,axiom,
! [A: set_set_formula_a_b,B: set_set_formula_a_b,C2: set_set_formula_a_b] :
( ( minus_4878662943024999126la_a_b @ A @ ( sup_su5931044959554304553la_a_b @ B @ C2 ) )
= ( inf_in60795025369792911la_a_b @ ( minus_4878662943024999126la_a_b @ A @ B ) @ ( minus_4878662943024999126la_a_b @ A @ C2 ) ) ) ).
% Diff_Un
thf(fact_706_Diff__Un,axiom,
! [A: set_formula_a_b,B: set_formula_a_b,C2: set_formula_a_b] :
( ( minus_3283878196352184736la_a_b @ A @ ( sup_su8832426789560438003la_a_b @ B @ C2 ) )
= ( inf_in5741596252272946265la_a_b @ ( minus_3283878196352184736la_a_b @ A @ B ) @ ( minus_3283878196352184736la_a_b @ A @ C2 ) ) ) ).
% Diff_Un
thf(fact_707_Diff__Int,axiom,
! [A: set_set_formula_a_b,B: set_set_formula_a_b,C2: set_set_formula_a_b] :
( ( minus_4878662943024999126la_a_b @ A @ ( inf_in60795025369792911la_a_b @ B @ C2 ) )
= ( sup_su5931044959554304553la_a_b @ ( minus_4878662943024999126la_a_b @ A @ B ) @ ( minus_4878662943024999126la_a_b @ A @ C2 ) ) ) ).
% Diff_Int
thf(fact_708_Diff__Int,axiom,
! [A: set_formula_a_b,B: set_formula_a_b,C2: set_formula_a_b] :
( ( minus_3283878196352184736la_a_b @ A @ ( inf_in5741596252272946265la_a_b @ B @ C2 ) )
= ( sup_su8832426789560438003la_a_b @ ( minus_3283878196352184736la_a_b @ A @ B ) @ ( minus_3283878196352184736la_a_b @ A @ C2 ) ) ) ).
% Diff_Int
thf(fact_709_Int__Diff__Un,axiom,
! [A: set_set_formula_a_b,B: set_set_formula_a_b] :
( ( sup_su5931044959554304553la_a_b @ ( inf_in60795025369792911la_a_b @ A @ B ) @ ( minus_4878662943024999126la_a_b @ A @ B ) )
= A ) ).
% Int_Diff_Un
thf(fact_710_Int__Diff__Un,axiom,
! [A: set_formula_a_b,B: set_formula_a_b] :
( ( sup_su8832426789560438003la_a_b @ ( inf_in5741596252272946265la_a_b @ A @ B ) @ ( minus_3283878196352184736la_a_b @ A @ B ) )
= A ) ).
% Int_Diff_Un
thf(fact_711_Un__Diff__Int,axiom,
! [A: set_set_formula_a_b,B: set_set_formula_a_b] :
( ( sup_su5931044959554304553la_a_b @ ( minus_4878662943024999126la_a_b @ A @ B ) @ ( inf_in60795025369792911la_a_b @ A @ B ) )
= A ) ).
% Un_Diff_Int
thf(fact_712_Un__Diff__Int,axiom,
! [A: set_formula_a_b,B: set_formula_a_b] :
( ( sup_su8832426789560438003la_a_b @ ( minus_3283878196352184736la_a_b @ A @ B ) @ ( inf_in5741596252272946265la_a_b @ A @ B ) )
= A ) ).
% Un_Diff_Int
thf(fact_713_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: set_set_formula_a_b,Y: set_set_formula_a_b,Z: set_set_formula_a_b] :
( ( inf_in60795025369792911la_a_b @ X @ ( sup_su5931044959554304553la_a_b @ Y @ Z ) )
= ( sup_su5931044959554304553la_a_b @ ( inf_in60795025369792911la_a_b @ X @ Y ) @ ( inf_in60795025369792911la_a_b @ X @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_714_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: set_formula_a_b,Y: set_formula_a_b,Z: set_formula_a_b] :
( ( inf_in5741596252272946265la_a_b @ X @ ( sup_su8832426789560438003la_a_b @ Y @ Z ) )
= ( sup_su8832426789560438003la_a_b @ ( inf_in5741596252272946265la_a_b @ X @ Y ) @ ( inf_in5741596252272946265la_a_b @ X @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_715_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: set_set_formula_a_b,Y: set_set_formula_a_b,Z: set_set_formula_a_b] :
( ( sup_su5931044959554304553la_a_b @ X @ ( inf_in60795025369792911la_a_b @ Y @ Z ) )
= ( inf_in60795025369792911la_a_b @ ( sup_su5931044959554304553la_a_b @ X @ Y ) @ ( sup_su5931044959554304553la_a_b @ X @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_716_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: set_formula_a_b,Y: set_formula_a_b,Z: set_formula_a_b] :
( ( sup_su8832426789560438003la_a_b @ X @ ( inf_in5741596252272946265la_a_b @ Y @ Z ) )
= ( inf_in5741596252272946265la_a_b @ ( sup_su8832426789560438003la_a_b @ X @ Y ) @ ( sup_su8832426789560438003la_a_b @ X @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_717_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_set_formula_a_b,Z: set_set_formula_a_b,X: set_set_formula_a_b] :
( ( inf_in60795025369792911la_a_b @ ( sup_su5931044959554304553la_a_b @ Y @ Z ) @ X )
= ( sup_su5931044959554304553la_a_b @ ( inf_in60795025369792911la_a_b @ Y @ X ) @ ( inf_in60795025369792911la_a_b @ Z @ X ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_718_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_formula_a_b,Z: set_formula_a_b,X: set_formula_a_b] :
( ( inf_in5741596252272946265la_a_b @ ( sup_su8832426789560438003la_a_b @ Y @ Z ) @ X )
= ( sup_su8832426789560438003la_a_b @ ( inf_in5741596252272946265la_a_b @ Y @ X ) @ ( inf_in5741596252272946265la_a_b @ Z @ X ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_719_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_set_formula_a_b,Z: set_set_formula_a_b,X: set_set_formula_a_b] :
( ( sup_su5931044959554304553la_a_b @ ( inf_in60795025369792911la_a_b @ Y @ Z ) @ X )
= ( inf_in60795025369792911la_a_b @ ( sup_su5931044959554304553la_a_b @ Y @ X ) @ ( sup_su5931044959554304553la_a_b @ Z @ X ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_720_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_formula_a_b,Z: set_formula_a_b,X: set_formula_a_b] :
( ( sup_su8832426789560438003la_a_b @ ( inf_in5741596252272946265la_a_b @ Y @ Z ) @ X )
= ( inf_in5741596252272946265la_a_b @ ( sup_su8832426789560438003la_a_b @ Y @ X ) @ ( sup_su8832426789560438003la_a_b @ Z @ X ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_721_distrib__imp1,axiom,
! [X: set_set_formula_a_b,Y: set_set_formula_a_b,Z: set_set_formula_a_b] :
( ! [X3: set_set_formula_a_b,Y3: set_set_formula_a_b,Z3: set_set_formula_a_b] :
( ( inf_in60795025369792911la_a_b @ X3 @ ( sup_su5931044959554304553la_a_b @ Y3 @ Z3 ) )
= ( sup_su5931044959554304553la_a_b @ ( inf_in60795025369792911la_a_b @ X3 @ Y3 ) @ ( inf_in60795025369792911la_a_b @ X3 @ Z3 ) ) )
=> ( ( sup_su5931044959554304553la_a_b @ X @ ( inf_in60795025369792911la_a_b @ Y @ Z ) )
= ( inf_in60795025369792911la_a_b @ ( sup_su5931044959554304553la_a_b @ X @ Y ) @ ( sup_su5931044959554304553la_a_b @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_722_distrib__imp1,axiom,
! [X: set_formula_a_b,Y: set_formula_a_b,Z: set_formula_a_b] :
( ! [X3: set_formula_a_b,Y3: set_formula_a_b,Z3: set_formula_a_b] :
( ( inf_in5741596252272946265la_a_b @ X3 @ ( sup_su8832426789560438003la_a_b @ Y3 @ Z3 ) )
= ( sup_su8832426789560438003la_a_b @ ( inf_in5741596252272946265la_a_b @ X3 @ Y3 ) @ ( inf_in5741596252272946265la_a_b @ X3 @ Z3 ) ) )
=> ( ( sup_su8832426789560438003la_a_b @ X @ ( inf_in5741596252272946265la_a_b @ Y @ Z ) )
= ( inf_in5741596252272946265la_a_b @ ( sup_su8832426789560438003la_a_b @ X @ Y ) @ ( sup_su8832426789560438003la_a_b @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_723_distrib__imp2,axiom,
! [X: set_set_formula_a_b,Y: set_set_formula_a_b,Z: set_set_formula_a_b] :
( ! [X3: set_set_formula_a_b,Y3: set_set_formula_a_b,Z3: set_set_formula_a_b] :
( ( sup_su5931044959554304553la_a_b @ X3 @ ( inf_in60795025369792911la_a_b @ Y3 @ Z3 ) )
= ( inf_in60795025369792911la_a_b @ ( sup_su5931044959554304553la_a_b @ X3 @ Y3 ) @ ( sup_su5931044959554304553la_a_b @ X3 @ Z3 ) ) )
=> ( ( inf_in60795025369792911la_a_b @ X @ ( sup_su5931044959554304553la_a_b @ Y @ Z ) )
= ( sup_su5931044959554304553la_a_b @ ( inf_in60795025369792911la_a_b @ X @ Y ) @ ( inf_in60795025369792911la_a_b @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_724_distrib__imp2,axiom,
! [X: set_formula_a_b,Y: set_formula_a_b,Z: set_formula_a_b] :
( ! [X3: set_formula_a_b,Y3: set_formula_a_b,Z3: set_formula_a_b] :
( ( sup_su8832426789560438003la_a_b @ X3 @ ( inf_in5741596252272946265la_a_b @ Y3 @ Z3 ) )
= ( inf_in5741596252272946265la_a_b @ ( sup_su8832426789560438003la_a_b @ X3 @ Y3 ) @ ( sup_su8832426789560438003la_a_b @ X3 @ Z3 ) ) )
=> ( ( inf_in5741596252272946265la_a_b @ X @ ( sup_su8832426789560438003la_a_b @ Y @ Z ) )
= ( sup_su8832426789560438003la_a_b @ ( inf_in5741596252272946265la_a_b @ X @ Y ) @ ( inf_in5741596252272946265la_a_b @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_725_inf__sup__distrib1,axiom,
! [X: set_set_formula_a_b,Y: set_set_formula_a_b,Z: set_set_formula_a_b] :
( ( inf_in60795025369792911la_a_b @ X @ ( sup_su5931044959554304553la_a_b @ Y @ Z ) )
= ( sup_su5931044959554304553la_a_b @ ( inf_in60795025369792911la_a_b @ X @ Y ) @ ( inf_in60795025369792911la_a_b @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_726_inf__sup__distrib1,axiom,
! [X: set_formula_a_b,Y: set_formula_a_b,Z: set_formula_a_b] :
( ( inf_in5741596252272946265la_a_b @ X @ ( sup_su8832426789560438003la_a_b @ Y @ Z ) )
= ( sup_su8832426789560438003la_a_b @ ( inf_in5741596252272946265la_a_b @ X @ Y ) @ ( inf_in5741596252272946265la_a_b @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_727_inf__sup__distrib2,axiom,
! [Y: set_set_formula_a_b,Z: set_set_formula_a_b,X: set_set_formula_a_b] :
( ( inf_in60795025369792911la_a_b @ ( sup_su5931044959554304553la_a_b @ Y @ Z ) @ X )
= ( sup_su5931044959554304553la_a_b @ ( inf_in60795025369792911la_a_b @ Y @ X ) @ ( inf_in60795025369792911la_a_b @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_728_inf__sup__distrib2,axiom,
! [Y: set_formula_a_b,Z: set_formula_a_b,X: set_formula_a_b] :
( ( inf_in5741596252272946265la_a_b @ ( sup_su8832426789560438003la_a_b @ Y @ Z ) @ X )
= ( sup_su8832426789560438003la_a_b @ ( inf_in5741596252272946265la_a_b @ Y @ X ) @ ( inf_in5741596252272946265la_a_b @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_729_sup__inf__distrib1,axiom,
! [X: set_set_formula_a_b,Y: set_set_formula_a_b,Z: set_set_formula_a_b] :
( ( sup_su5931044959554304553la_a_b @ X @ ( inf_in60795025369792911la_a_b @ Y @ Z ) )
= ( inf_in60795025369792911la_a_b @ ( sup_su5931044959554304553la_a_b @ X @ Y ) @ ( sup_su5931044959554304553la_a_b @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_730_sup__inf__distrib1,axiom,
! [X: set_formula_a_b,Y: set_formula_a_b,Z: set_formula_a_b] :
( ( sup_su8832426789560438003la_a_b @ X @ ( inf_in5741596252272946265la_a_b @ Y @ Z ) )
= ( inf_in5741596252272946265la_a_b @ ( sup_su8832426789560438003la_a_b @ X @ Y ) @ ( sup_su8832426789560438003la_a_b @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_731_sup__inf__distrib2,axiom,
! [Y: set_set_formula_a_b,Z: set_set_formula_a_b,X: set_set_formula_a_b] :
( ( sup_su5931044959554304553la_a_b @ ( inf_in60795025369792911la_a_b @ Y @ Z ) @ X )
= ( inf_in60795025369792911la_a_b @ ( sup_su5931044959554304553la_a_b @ Y @ X ) @ ( sup_su5931044959554304553la_a_b @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_732_sup__inf__distrib2,axiom,
! [Y: set_formula_a_b,Z: set_formula_a_b,X: set_formula_a_b] :
( ( sup_su8832426789560438003la_a_b @ ( inf_in5741596252272946265la_a_b @ Y @ Z ) @ X )
= ( inf_in5741596252272946265la_a_b @ ( sup_su8832426789560438003la_a_b @ Y @ X ) @ ( sup_su8832426789560438003la_a_b @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_733_Int__insert__left,axiom,
! [A2: list_formula_a_b,C2: set_list_formula_a_b,B: set_list_formula_a_b] :
( ( ( member7703986999711672120la_a_b @ A2 @ C2 )
=> ( ( inf_in7172170795784610665la_a_b @ ( insert4995524290302003025la_a_b @ A2 @ B ) @ C2 )
= ( insert4995524290302003025la_a_b @ A2 @ ( inf_in7172170795784610665la_a_b @ B @ C2 ) ) ) )
& ( ~ ( member7703986999711672120la_a_b @ A2 @ C2 )
=> ( ( inf_in7172170795784610665la_a_b @ ( insert4995524290302003025la_a_b @ A2 @ B ) @ C2 )
= ( inf_in7172170795784610665la_a_b @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_734_Int__insert__left,axiom,
! [A2: set_formula_a_b,C2: set_set_formula_a_b,B: set_set_formula_a_b] :
( ( ( member6621951321062823262la_a_b @ A2 @ C2 )
=> ( ( inf_in60795025369792911la_a_b @ ( insert8952366156270154231la_a_b @ A2 @ B ) @ C2 )
= ( insert8952366156270154231la_a_b @ A2 @ ( inf_in60795025369792911la_a_b @ B @ C2 ) ) ) )
& ( ~ ( member6621951321062823262la_a_b @ A2 @ C2 )
=> ( ( inf_in60795025369792911la_a_b @ ( insert8952366156270154231la_a_b @ A2 @ B ) @ C2 )
= ( inf_in60795025369792911la_a_b @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_735_Int__insert__left,axiom,
! [A2: formula_a_b,C2: set_formula_a_b,B: set_formula_a_b] :
( ( ( member_formula_a_b2 @ A2 @ C2 )
=> ( ( inf_in5741596252272946265la_a_b @ ( insert_formula_a_b2 @ A2 @ B ) @ C2 )
= ( insert_formula_a_b2 @ A2 @ ( inf_in5741596252272946265la_a_b @ B @ C2 ) ) ) )
& ( ~ ( member_formula_a_b2 @ A2 @ C2 )
=> ( ( inf_in5741596252272946265la_a_b @ ( insert_formula_a_b2 @ A2 @ B ) @ C2 )
= ( inf_in5741596252272946265la_a_b @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_736_Int__insert__right,axiom,
! [A2: list_formula_a_b,A: set_list_formula_a_b,B: set_list_formula_a_b] :
( ( ( member7703986999711672120la_a_b @ A2 @ A )
=> ( ( inf_in7172170795784610665la_a_b @ A @ ( insert4995524290302003025la_a_b @ A2 @ B ) )
= ( insert4995524290302003025la_a_b @ A2 @ ( inf_in7172170795784610665la_a_b @ A @ B ) ) ) )
& ( ~ ( member7703986999711672120la_a_b @ A2 @ A )
=> ( ( inf_in7172170795784610665la_a_b @ A @ ( insert4995524290302003025la_a_b @ A2 @ B ) )
= ( inf_in7172170795784610665la_a_b @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_737_Int__insert__right,axiom,
! [A2: set_formula_a_b,A: set_set_formula_a_b,B: set_set_formula_a_b] :
( ( ( member6621951321062823262la_a_b @ A2 @ A )
=> ( ( inf_in60795025369792911la_a_b @ A @ ( insert8952366156270154231la_a_b @ A2 @ B ) )
= ( insert8952366156270154231la_a_b @ A2 @ ( inf_in60795025369792911la_a_b @ A @ B ) ) ) )
& ( ~ ( member6621951321062823262la_a_b @ A2 @ A )
=> ( ( inf_in60795025369792911la_a_b @ A @ ( insert8952366156270154231la_a_b @ A2 @ B ) )
= ( inf_in60795025369792911la_a_b @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_738_Int__insert__right,axiom,
! [A2: formula_a_b,A: set_formula_a_b,B: set_formula_a_b] :
( ( ( member_formula_a_b2 @ A2 @ A )
=> ( ( inf_in5741596252272946265la_a_b @ A @ ( insert_formula_a_b2 @ A2 @ B ) )
= ( insert_formula_a_b2 @ A2 @ ( inf_in5741596252272946265la_a_b @ A @ B ) ) ) )
& ( ~ ( member_formula_a_b2 @ A2 @ A )
=> ( ( inf_in5741596252272946265la_a_b @ A @ ( insert_formula_a_b2 @ A2 @ B ) )
= ( inf_in5741596252272946265la_a_b @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_739_insert__Diff__if,axiom,
! [X: list_formula_a_b,B: set_list_formula_a_b,A: set_list_formula_a_b] :
( ( ( member7703986999711672120la_a_b @ X @ B )
=> ( ( minus_8667068627093346608la_a_b @ ( insert4995524290302003025la_a_b @ X @ A ) @ B )
= ( minus_8667068627093346608la_a_b @ A @ B ) ) )
& ( ~ ( member7703986999711672120la_a_b @ X @ B )
=> ( ( minus_8667068627093346608la_a_b @ ( insert4995524290302003025la_a_b @ X @ A ) @ B )
= ( insert4995524290302003025la_a_b @ X @ ( minus_8667068627093346608la_a_b @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_740_insert__Diff__if,axiom,
! [X: set_formula_a_b,B: set_set_formula_a_b,A: set_set_formula_a_b] :
( ( ( member6621951321062823262la_a_b @ X @ B )
=> ( ( minus_4878662943024999126la_a_b @ ( insert8952366156270154231la_a_b @ X @ A ) @ B )
= ( minus_4878662943024999126la_a_b @ A @ B ) ) )
& ( ~ ( member6621951321062823262la_a_b @ X @ B )
=> ( ( minus_4878662943024999126la_a_b @ ( insert8952366156270154231la_a_b @ X @ A ) @ B )
= ( insert8952366156270154231la_a_b @ X @ ( minus_4878662943024999126la_a_b @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_741_insert__Diff__if,axiom,
! [X: formula_a_b,B: set_formula_a_b,A: set_formula_a_b] :
( ( ( member_formula_a_b2 @ X @ B )
=> ( ( minus_3283878196352184736la_a_b @ ( insert_formula_a_b2 @ X @ A ) @ B )
= ( minus_3283878196352184736la_a_b @ A @ B ) ) )
& ( ~ ( member_formula_a_b2 @ X @ B )
=> ( ( minus_3283878196352184736la_a_b @ ( insert_formula_a_b2 @ X @ A ) @ B )
= ( insert_formula_a_b2 @ X @ ( minus_3283878196352184736la_a_b @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_742_Un__Int__crazy,axiom,
! [A: set_set_formula_a_b,B: set_set_formula_a_b,C2: set_set_formula_a_b] :
( ( sup_su5931044959554304553la_a_b @ ( sup_su5931044959554304553la_a_b @ ( inf_in60795025369792911la_a_b @ A @ B ) @ ( inf_in60795025369792911la_a_b @ B @ C2 ) ) @ ( inf_in60795025369792911la_a_b @ C2 @ A ) )
= ( inf_in60795025369792911la_a_b @ ( inf_in60795025369792911la_a_b @ ( sup_su5931044959554304553la_a_b @ A @ B ) @ ( sup_su5931044959554304553la_a_b @ B @ C2 ) ) @ ( sup_su5931044959554304553la_a_b @ C2 @ A ) ) ) ).
% Un_Int_crazy
thf(fact_743_Un__Int__crazy,axiom,
! [A: set_formula_a_b,B: set_formula_a_b,C2: set_formula_a_b] :
( ( sup_su8832426789560438003la_a_b @ ( sup_su8832426789560438003la_a_b @ ( inf_in5741596252272946265la_a_b @ A @ B ) @ ( inf_in5741596252272946265la_a_b @ B @ C2 ) ) @ ( inf_in5741596252272946265la_a_b @ C2 @ A ) )
= ( inf_in5741596252272946265la_a_b @ ( inf_in5741596252272946265la_a_b @ ( sup_su8832426789560438003la_a_b @ A @ B ) @ ( sup_su8832426789560438003la_a_b @ B @ C2 ) ) @ ( sup_su8832426789560438003la_a_b @ C2 @ A ) ) ) ).
% Un_Int_crazy
thf(fact_744_Int__Un__distrib,axiom,
! [A: set_set_formula_a_b,B: set_set_formula_a_b,C2: set_set_formula_a_b] :
( ( inf_in60795025369792911la_a_b @ A @ ( sup_su5931044959554304553la_a_b @ B @ C2 ) )
= ( sup_su5931044959554304553la_a_b @ ( inf_in60795025369792911la_a_b @ A @ B ) @ ( inf_in60795025369792911la_a_b @ A @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_745_Int__Un__distrib,axiom,
! [A: set_formula_a_b,B: set_formula_a_b,C2: set_formula_a_b] :
( ( inf_in5741596252272946265la_a_b @ A @ ( sup_su8832426789560438003la_a_b @ B @ C2 ) )
= ( sup_su8832426789560438003la_a_b @ ( inf_in5741596252272946265la_a_b @ A @ B ) @ ( inf_in5741596252272946265la_a_b @ A @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_746_Un__Int__distrib,axiom,
! [A: set_set_formula_a_b,B: set_set_formula_a_b,C2: set_set_formula_a_b] :
( ( sup_su5931044959554304553la_a_b @ A @ ( inf_in60795025369792911la_a_b @ B @ C2 ) )
= ( inf_in60795025369792911la_a_b @ ( sup_su5931044959554304553la_a_b @ A @ B ) @ ( sup_su5931044959554304553la_a_b @ A @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_747_Un__Int__distrib,axiom,
! [A: set_formula_a_b,B: set_formula_a_b,C2: set_formula_a_b] :
( ( sup_su8832426789560438003la_a_b @ A @ ( inf_in5741596252272946265la_a_b @ B @ C2 ) )
= ( inf_in5741596252272946265la_a_b @ ( sup_su8832426789560438003la_a_b @ A @ B ) @ ( sup_su8832426789560438003la_a_b @ A @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_748_Int__Un__distrib2,axiom,
! [B: set_set_formula_a_b,C2: set_set_formula_a_b,A: set_set_formula_a_b] :
( ( inf_in60795025369792911la_a_b @ ( sup_su5931044959554304553la_a_b @ B @ C2 ) @ A )
= ( sup_su5931044959554304553la_a_b @ ( inf_in60795025369792911la_a_b @ B @ A ) @ ( inf_in60795025369792911la_a_b @ C2 @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_749_Int__Un__distrib2,axiom,
! [B: set_formula_a_b,C2: set_formula_a_b,A: set_formula_a_b] :
( ( inf_in5741596252272946265la_a_b @ ( sup_su8832426789560438003la_a_b @ B @ C2 ) @ A )
= ( sup_su8832426789560438003la_a_b @ ( inf_in5741596252272946265la_a_b @ B @ A ) @ ( inf_in5741596252272946265la_a_b @ C2 @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_750_Un__Int__distrib2,axiom,
! [B: set_set_formula_a_b,C2: set_set_formula_a_b,A: set_set_formula_a_b] :
( ( sup_su5931044959554304553la_a_b @ ( inf_in60795025369792911la_a_b @ B @ C2 ) @ A )
= ( inf_in60795025369792911la_a_b @ ( sup_su5931044959554304553la_a_b @ B @ A ) @ ( sup_su5931044959554304553la_a_b @ C2 @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_751_Un__Int__distrib2,axiom,
! [B: set_formula_a_b,C2: set_formula_a_b,A: set_formula_a_b] :
( ( sup_su8832426789560438003la_a_b @ ( inf_in5741596252272946265la_a_b @ B @ C2 ) @ A )
= ( inf_in5741596252272946265la_a_b @ ( sup_su8832426789560438003la_a_b @ B @ A ) @ ( sup_su8832426789560438003la_a_b @ C2 @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_752_Un__Diff,axiom,
! [A: set_set_formula_a_b,B: set_set_formula_a_b,C2: set_set_formula_a_b] :
( ( minus_4878662943024999126la_a_b @ ( sup_su5931044959554304553la_a_b @ A @ B ) @ C2 )
= ( sup_su5931044959554304553la_a_b @ ( minus_4878662943024999126la_a_b @ A @ C2 ) @ ( minus_4878662943024999126la_a_b @ B @ C2 ) ) ) ).
% Un_Diff
thf(fact_753_Un__Diff,axiom,
! [A: set_formula_a_b,B: set_formula_a_b,C2: set_formula_a_b] :
( ( minus_3283878196352184736la_a_b @ ( sup_su8832426789560438003la_a_b @ A @ B ) @ C2 )
= ( sup_su8832426789560438003la_a_b @ ( minus_3283878196352184736la_a_b @ A @ C2 ) @ ( minus_3283878196352184736la_a_b @ B @ C2 ) ) ) ).
% Un_Diff
thf(fact_754_fold__append__concat__rev,axiom,
! [Xss2: list_l5948067055324603441la_a_b] :
( ( fold_l9118537288359097167la_a_b @ append_formula_a_b @ Xss2 )
= ( append_formula_a_b @ ( concat_formula_a_b @ ( rev_list_formula_a_b @ Xss2 ) ) ) ) ).
% fold_append_concat_rev
thf(fact_755_Cons__in__subseqsD,axiom,
! [Y: formula_a_b,Ys: list_formula_a_b,Xs: list_formula_a_b] :
( ( member7703986999711672120la_a_b @ ( cons_formula_a_b @ Y @ Ys ) @ ( set_list_formula_a_b2 @ ( subseqs_formula_a_b @ Xs ) ) )
=> ( member7703986999711672120la_a_b @ Ys @ ( set_list_formula_a_b2 @ ( subseqs_formula_a_b @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_756_Cons__in__subseqsD,axiom,
! [Y: set_formula_a_b,Ys: list_set_formula_a_b,Xs: list_set_formula_a_b] :
( ( member289699325402759278la_a_b @ ( cons_set_formula_a_b @ Y @ Ys ) @ ( set_li566459811281642754la_a_b @ ( subseq6158387682876310066la_a_b @ Xs ) ) )
=> ( member289699325402759278la_a_b @ Ys @ ( set_li566459811281642754la_a_b @ ( subseq6158387682876310066la_a_b @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_757_subseqs__distinctD,axiom,
! [Ys: list_l5948067055324603441la_a_b,Xs: list_l5948067055324603441la_a_b] :
( ( member5770512961814848200la_a_b @ Ys @ ( set_li405426049885584732la_a_b @ ( subseq7473640939082508812la_a_b @ Xs ) ) )
=> ( ( distin27490427618170344la_a_b @ Xs )
=> ( distin27490427618170344la_a_b @ Ys ) ) ) ).
% subseqs_distinctD
thf(fact_758_subseqs__distinctD,axiom,
! [Ys: list_set_formula_a_b,Xs: list_set_formula_a_b] :
( ( member289699325402759278la_a_b @ Ys @ ( set_li566459811281642754la_a_b @ ( subseq6158387682876310066la_a_b @ Xs ) ) )
=> ( ( distin5059275051008138254la_a_b @ Xs )
=> ( distin5059275051008138254la_a_b @ Ys ) ) ) ).
% subseqs_distinctD
thf(fact_759_subseqs__distinctD,axiom,
! [Ys: list_formula_a_b,Xs: list_formula_a_b] :
( ( member7703986999711672120la_a_b @ Ys @ ( set_list_formula_a_b2 @ ( subseqs_formula_a_b @ Xs ) ) )
=> ( ( distinct_formula_a_b @ Xs )
=> ( distinct_formula_a_b @ Ys ) ) ) ).
% subseqs_distinctD
thf(fact_760_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: formula_a_b > formula_a_b,A2: formula_a_b,As: list_formula_a_b,Bs: list_formula_a_b] :
( ( map_ta6137025558560551193la_a_b @ F @ ( cons_formula_a_b @ A2 @ As ) @ Bs )
= ( map_ta6137025558560551193la_a_b @ F @ As @ ( cons_formula_a_b @ ( F @ A2 ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_761_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: formula_a_b > set_formula_a_b,A2: formula_a_b,As: list_formula_a_b,Bs: list_set_formula_a_b] :
( ( map_ta1969411894599079247la_a_b @ F @ ( cons_formula_a_b @ A2 @ As ) @ Bs )
= ( map_ta1969411894599079247la_a_b @ F @ As @ ( cons_set_formula_a_b @ ( F @ A2 ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_762_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: set_formula_a_b > formula_a_b,A2: set_formula_a_b,As: list_set_formula_a_b,Bs: list_formula_a_b] :
( ( map_ta7515233025258682191la_a_b @ F @ ( cons_set_formula_a_b @ A2 @ As ) @ Bs )
= ( map_ta7515233025258682191la_a_b @ F @ As @ ( cons_formula_a_b @ ( F @ A2 ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_763_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: set_formula_a_b > set_formula_a_b,A2: set_formula_a_b,As: list_set_formula_a_b,Bs: list_set_formula_a_b] :
( ( map_ta7939594666930029445la_a_b @ F @ ( cons_set_formula_a_b @ A2 @ As ) @ Bs )
= ( map_ta7939594666930029445la_a_b @ F @ As @ ( cons_set_formula_a_b @ ( F @ A2 ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_764_concat__conv__foldr,axiom,
( concat_formula_a_b
= ( ^ [Xss3: list_l5948067055324603441la_a_b] : ( foldr_139744914162652395la_a_b @ append_formula_a_b @ Xss3 @ nil_formula_a_b ) ) ) ).
% concat_conv_foldr
thf(fact_765_distinct__product__lists,axiom,
! [Xss2: list_l3581718849243781313la_a_b] :
( ! [X3: list_l5948067055324603441la_a_b] :
( ( member5770512961814848200la_a_b @ X3 @ ( set_li405426049885584732la_a_b @ Xss2 ) )
=> ( distin27490427618170344la_a_b @ X3 ) )
=> ( distin7365129041399036280la_a_b @ ( produc3748613542311242226la_a_b @ Xss2 ) ) ) ).
% distinct_product_lists
thf(fact_766_distinct__product__lists,axiom,
! [Xss2: list_l147921316930177639la_a_b] :
( ! [X3: list_set_formula_a_b] :
( ( member289699325402759278la_a_b @ X3 @ ( set_li566459811281642754la_a_b @ Xss2 ) )
=> ( distin5059275051008138254la_a_b @ X3 ) )
=> ( distin6955390182775811870la_a_b @ ( produc7528538235918390552la_a_b @ Xss2 ) ) ) ).
% distinct_product_lists
thf(fact_767_distinct__product__lists,axiom,
! [Xss2: list_l5948067055324603441la_a_b] :
( ! [X3: list_formula_a_b] :
( ( member7703986999711672120la_a_b @ X3 @ ( set_list_formula_a_b2 @ Xss2 ) )
=> ( distinct_formula_a_b @ X3 ) )
=> ( distin27490427618170344la_a_b @ ( produc130815601992340706la_a_b @ Xss2 ) ) ) ).
% distinct_product_lists
thf(fact_768_concat_Osimps_I1_J,axiom,
( ( concat_formula_a_b @ nil_list_formula_a_b )
= nil_formula_a_b ) ).
% concat.simps(1)
thf(fact_769_concat_Osimps_I2_J,axiom,
! [X: list_formula_a_b,Xs: list_l5948067055324603441la_a_b] :
( ( concat_formula_a_b @ ( cons_l2450299273141159147la_a_b @ X @ Xs ) )
= ( append_formula_a_b @ X @ ( concat_formula_a_b @ Xs ) ) ) ).
% concat.simps(2)
thf(fact_770_concat__eq__appendD,axiom,
! [Xss2: list_l5948067055324603441la_a_b,Ys: list_formula_a_b,Zs: list_formula_a_b] :
( ( ( concat_formula_a_b @ Xss2 )
= ( append_formula_a_b @ Ys @ Zs ) )
=> ( ( Xss2 != nil_list_formula_a_b )
=> ? [Xss12: list_l5948067055324603441la_a_b,Xs3: list_formula_a_b,Xs5: list_formula_a_b,Xss23: list_l5948067055324603441la_a_b] :
( ( Xss2
= ( append58666856857280070la_a_b @ Xss12 @ ( cons_l2450299273141159147la_a_b @ ( append_formula_a_b @ Xs3 @ Xs5 ) @ Xss23 ) ) )
& ( Ys
= ( append_formula_a_b @ ( concat_formula_a_b @ Xss12 ) @ Xs3 ) )
& ( Zs
= ( append_formula_a_b @ Xs5 @ ( concat_formula_a_b @ Xss23 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_771_SuccD,axiom,
! [K: list_formula_a_b,Kl2: set_li1233427854086133095la_a_b,Kl: list_l5948067055324603441la_a_b] :
( ( member7703986999711672120la_a_b @ K @ ( bNF_Gr3706963439723013830la_a_b @ Kl2 @ Kl ) )
=> ( member5770512961814848200la_a_b @ ( append58666856857280070la_a_b @ Kl @ ( cons_l2450299273141159147la_a_b @ K @ nil_list_formula_a_b ) ) @ Kl2 ) ) ).
% SuccD
thf(fact_772_SuccD,axiom,
! [K: set_formula_a_b,Kl2: set_li5266388670752979469la_a_b,Kl: list_set_formula_a_b] :
( ( member6621951321062823262la_a_b @ K @ ( bNF_Gr8423546037056666604la_a_b @ Kl2 @ Kl ) )
=> ( member289699325402759278la_a_b @ ( append8577002312234529644la_a_b @ Kl @ ( cons_set_formula_a_b @ K @ nil_set_formula_a_b ) ) @ Kl2 ) ) ).
% SuccD
thf(fact_773_SuccD,axiom,
! [K: formula_a_b,Kl2: set_list_formula_a_b,Kl: list_formula_a_b] :
( ( member_formula_a_b2 @ K @ ( bNF_Gr6515907858579016630la_a_b @ Kl2 @ Kl ) )
=> ( member7703986999711672120la_a_b @ ( append_formula_a_b @ Kl @ ( cons_formula_a_b @ K @ nil_formula_a_b ) ) @ Kl2 ) ) ).
% SuccD
thf(fact_774_empty__Shift,axiom,
! [Kl2: set_li1233427854086133095la_a_b,K: list_formula_a_b] :
( ( member5770512961814848200la_a_b @ nil_list_formula_a_b @ Kl2 )
=> ( ( member7703986999711672120la_a_b @ K @ ( bNF_Gr3706963439723013830la_a_b @ Kl2 @ nil_list_formula_a_b ) )
=> ( member5770512961814848200la_a_b @ nil_list_formula_a_b @ ( bNF_Gr4875458711411372354la_a_b @ Kl2 @ K ) ) ) ) ).
% empty_Shift
thf(fact_775_empty__Shift,axiom,
! [Kl2: set_li5266388670752979469la_a_b,K: set_formula_a_b] :
( ( member289699325402759278la_a_b @ nil_set_formula_a_b @ Kl2 )
=> ( ( member6621951321062823262la_a_b @ K @ ( bNF_Gr8423546037056666604la_a_b @ Kl2 @ nil_set_formula_a_b ) )
=> ( member289699325402759278la_a_b @ nil_set_formula_a_b @ ( bNF_Gr5837620917382104168la_a_b @ Kl2 @ K ) ) ) ) ).
% empty_Shift
thf(fact_776_empty__Shift,axiom,
! [Kl2: set_list_formula_a_b,K: formula_a_b] :
( ( member7703986999711672120la_a_b @ nil_formula_a_b @ Kl2 )
=> ( ( member_formula_a_b2 @ K @ ( bNF_Gr6515907858579016630la_a_b @ Kl2 @ nil_formula_a_b ) )
=> ( member7703986999711672120la_a_b @ nil_formula_a_b @ ( bNF_Gr2660717057327301682la_a_b @ Kl2 @ K ) ) ) ) ).
% empty_Shift
thf(fact_777_Succ__Shift,axiom,
! [Kl2: set_li5266388670752979469la_a_b,K: set_formula_a_b,Kl: list_set_formula_a_b] :
( ( bNF_Gr8423546037056666604la_a_b @ ( bNF_Gr5837620917382104168la_a_b @ Kl2 @ K ) @ Kl )
= ( bNF_Gr8423546037056666604la_a_b @ Kl2 @ ( cons_set_formula_a_b @ K @ Kl ) ) ) ).
% Succ_Shift
thf(fact_778_Succ__Shift,axiom,
! [Kl2: set_list_formula_a_b,K: formula_a_b,Kl: list_formula_a_b] :
( ( bNF_Gr6515907858579016630la_a_b @ ( bNF_Gr2660717057327301682la_a_b @ Kl2 @ K ) @ Kl )
= ( bNF_Gr6515907858579016630la_a_b @ Kl2 @ ( cons_formula_a_b @ K @ Kl ) ) ) ).
% Succ_Shift
thf(fact_779_set__remove1__eq,axiom,
! [Xs: list_l5948067055324603441la_a_b,X: list_formula_a_b] :
( ( distin27490427618170344la_a_b @ Xs )
=> ( ( set_list_formula_a_b2 @ ( remove4763865728959745157la_a_b @ X @ Xs ) )
= ( minus_8667068627093346608la_a_b @ ( set_list_formula_a_b2 @ Xs ) @ ( insert4995524290302003025la_a_b @ X @ bot_bo4614192416432564331la_a_b ) ) ) ) ).
% set_remove1_eq
thf(fact_780_set__remove1__eq,axiom,
! [Xs: list_set_formula_a_b,X: set_formula_a_b] :
( ( distin5059275051008138254la_a_b @ Xs )
=> ( ( set_set_formula_a_b2 @ ( remove3026683253030310187la_a_b @ X @ Xs ) )
= ( minus_4878662943024999126la_a_b @ ( set_set_formula_a_b2 @ Xs ) @ ( insert8952366156270154231la_a_b @ X @ bot_bo5296870006564677521la_a_b ) ) ) ) ).
% set_remove1_eq
thf(fact_781_set__remove1__eq,axiom,
! [Xs: list_formula_a_b,X: formula_a_b] :
( ( distinct_formula_a_b @ Xs )
=> ( ( set_formula_a_b2 @ ( remove1_formula_a_b @ X @ Xs ) )
= ( minus_3283878196352184736la_a_b @ ( set_formula_a_b2 @ Xs ) @ ( insert_formula_a_b2 @ X @ bot_bo8568539672012708443la_a_b ) ) ) ) ).
% set_remove1_eq
thf(fact_782_set__removeAll,axiom,
! [X: list_formula_a_b,Xs: list_l5948067055324603441la_a_b] :
( ( set_list_formula_a_b2 @ ( remove9041333805894489621la_a_b @ X @ Xs ) )
= ( minus_8667068627093346608la_a_b @ ( set_list_formula_a_b2 @ Xs ) @ ( insert4995524290302003025la_a_b @ X @ bot_bo4614192416432564331la_a_b ) ) ) ).
% set_removeAll
thf(fact_783_set__removeAll,axiom,
! [X: set_formula_a_b,Xs: list_set_formula_a_b] :
( ( set_set_formula_a_b2 @ ( remove6824600317533053627la_a_b @ X @ Xs ) )
= ( minus_4878662943024999126la_a_b @ ( set_set_formula_a_b2 @ Xs ) @ ( insert8952366156270154231la_a_b @ X @ bot_bo5296870006564677521la_a_b ) ) ) ).
% set_removeAll
thf(fact_784_set__removeAll,axiom,
! [X: formula_a_b,Xs: list_formula_a_b] :
( ( set_formula_a_b2 @ ( remove7803334386285014661la_a_b @ X @ Xs ) )
= ( minus_3283878196352184736la_a_b @ ( set_formula_a_b2 @ Xs ) @ ( insert_formula_a_b2 @ X @ bot_bo8568539672012708443la_a_b ) ) ) ).
% set_removeAll
thf(fact_785_distinct__concat__iff,axiom,
! [Xs: list_l3581718849243781313la_a_b] :
( ( distin27490427618170344la_a_b @ ( concat6919603293682802464la_a_b @ Xs ) )
= ( ( distin7365129041399036280la_a_b @ ( remove2735483865023840037la_a_b @ nil_list_formula_a_b @ Xs ) )
& ! [Ys2: list_l5948067055324603441la_a_b] :
( ( member5770512961814848200la_a_b @ Ys2 @ ( set_li405426049885584732la_a_b @ Xs ) )
=> ( distin27490427618170344la_a_b @ Ys2 ) )
& ! [Ys2: list_l5948067055324603441la_a_b,Zs3: list_l5948067055324603441la_a_b] :
( ( ( member5770512961814848200la_a_b @ Ys2 @ ( set_li405426049885584732la_a_b @ Xs ) )
& ( member5770512961814848200la_a_b @ Zs3 @ ( set_li405426049885584732la_a_b @ Xs ) )
& ( Ys2 != Zs3 ) )
=> ( ( inf_in7172170795784610665la_a_b @ ( set_list_formula_a_b2 @ Ys2 ) @ ( set_list_formula_a_b2 @ Zs3 ) )
= bot_bo4614192416432564331la_a_b ) ) ) ) ).
% distinct_concat_iff
thf(fact_786_distinct__concat__iff,axiom,
! [Xs: list_l147921316930177639la_a_b] :
( ( distin5059275051008138254la_a_b @ ( concat7835744469863612742la_a_b @ Xs ) )
= ( ( distin6955390182775811870la_a_b @ ( remove7101406200717310795la_a_b @ nil_set_formula_a_b @ Xs ) )
& ! [Ys2: list_set_formula_a_b] :
( ( member289699325402759278la_a_b @ Ys2 @ ( set_li566459811281642754la_a_b @ Xs ) )
=> ( distin5059275051008138254la_a_b @ Ys2 ) )
& ! [Ys2: list_set_formula_a_b,Zs3: list_set_formula_a_b] :
( ( ( member289699325402759278la_a_b @ Ys2 @ ( set_li566459811281642754la_a_b @ Xs ) )
& ( member289699325402759278la_a_b @ Zs3 @ ( set_li566459811281642754la_a_b @ Xs ) )
& ( Ys2 != Zs3 ) )
=> ( ( inf_in60795025369792911la_a_b @ ( set_set_formula_a_b2 @ Ys2 ) @ ( set_set_formula_a_b2 @ Zs3 ) )
= bot_bo5296870006564677521la_a_b ) ) ) ) ).
% distinct_concat_iff
thf(fact_787_distinct__concat__iff,axiom,
! [Xs: list_l5948067055324603441la_a_b] :
( ( distinct_formula_a_b @ ( concat_formula_a_b @ Xs ) )
= ( ( distin27490427618170344la_a_b @ ( remove9041333805894489621la_a_b @ nil_formula_a_b @ Xs ) )
& ! [Ys2: list_formula_a_b] :
( ( member7703986999711672120la_a_b @ Ys2 @ ( set_list_formula_a_b2 @ Xs ) )
=> ( distinct_formula_a_b @ Ys2 ) )
& ! [Ys2: list_formula_a_b,Zs3: list_formula_a_b] :
( ( ( member7703986999711672120la_a_b @ Ys2 @ ( set_list_formula_a_b2 @ Xs ) )
& ( member7703986999711672120la_a_b @ Zs3 @ ( set_list_formula_a_b2 @ Xs ) )
& ( Ys2 != Zs3 ) )
=> ( ( inf_in5741596252272946265la_a_b @ ( set_formula_a_b2 @ Ys2 ) @ ( set_formula_a_b2 @ Zs3 ) )
= bot_bo8568539672012708443la_a_b ) ) ) ) ).
% distinct_concat_iff
thf(fact_788_empty__Collect__eq,axiom,
! [P: formula_a_b > $o] :
( ( bot_bo8568539672012708443la_a_b
= ( collect_formula_a_b @ P ) )
= ( ! [X2: formula_a_b] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_789_empty__Collect__eq,axiom,
! [P: set_formula_a_b > $o] :
( ( bot_bo5296870006564677521la_a_b
= ( collec5361897267898172060la_a_b @ P ) )
= ( ! [X2: set_formula_a_b] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_790_Collect__empty__eq,axiom,
! [P: formula_a_b > $o] :
( ( ( collect_formula_a_b @ P )
= bot_bo8568539672012708443la_a_b )
= ( ! [X2: formula_a_b] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_791_Collect__empty__eq,axiom,
! [P: set_formula_a_b > $o] :
( ( ( collec5361897267898172060la_a_b @ P )
= bot_bo5296870006564677521la_a_b )
= ( ! [X2: set_formula_a_b] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_792_all__not__in__conv,axiom,
! [A: set_list_formula_a_b] :
( ( ! [X2: list_formula_a_b] :
~ ( member7703986999711672120la_a_b @ X2 @ A ) )
= ( A = bot_bo4614192416432564331la_a_b ) ) ).
% all_not_in_conv
thf(fact_793_all__not__in__conv,axiom,
! [A: set_set_formula_a_b] :
( ( ! [X2: set_formula_a_b] :
~ ( member6621951321062823262la_a_b @ X2 @ A ) )
= ( A = bot_bo5296870006564677521la_a_b ) ) ).
% all_not_in_conv
thf(fact_794_all__not__in__conv,axiom,
! [A: set_formula_a_b] :
( ( ! [X2: formula_a_b] :
~ ( member_formula_a_b2 @ X2 @ A ) )
= ( A = bot_bo8568539672012708443la_a_b ) ) ).
% all_not_in_conv
thf(fact_795_empty__iff,axiom,
! [C: list_formula_a_b] :
~ ( member7703986999711672120la_a_b @ C @ bot_bo4614192416432564331la_a_b ) ).
% empty_iff
thf(fact_796_empty__iff,axiom,
! [C: set_formula_a_b] :
~ ( member6621951321062823262la_a_b @ C @ bot_bo5296870006564677521la_a_b ) ).
% empty_iff
thf(fact_797_empty__iff,axiom,
! [C: formula_a_b] :
~ ( member_formula_a_b2 @ C @ bot_bo8568539672012708443la_a_b ) ).
% empty_iff
thf(fact_798_singletonI,axiom,
! [A2: list_formula_a_b] : ( member7703986999711672120la_a_b @ A2 @ ( insert4995524290302003025la_a_b @ A2 @ bot_bo4614192416432564331la_a_b ) ) ).
% singletonI
thf(fact_799_singletonI,axiom,
! [A2: set_formula_a_b] : ( member6621951321062823262la_a_b @ A2 @ ( insert8952366156270154231la_a_b @ A2 @ bot_bo5296870006564677521la_a_b ) ) ).
% singletonI
thf(fact_800_singletonI,axiom,
! [A2: formula_a_b] : ( member_formula_a_b2 @ A2 @ ( insert_formula_a_b2 @ A2 @ bot_bo8568539672012708443la_a_b ) ) ).
% singletonI
thf(fact_801_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_formula_a_b] :
( ( inf_in5741596252272946265la_a_b @ X @ bot_bo8568539672012708443la_a_b )
= bot_bo8568539672012708443la_a_b ) ).
% boolean_algebra.conj_zero_right
thf(fact_802_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_set_formula_a_b] :
( ( inf_in60795025369792911la_a_b @ X @ bot_bo5296870006564677521la_a_b )
= bot_bo5296870006564677521la_a_b ) ).
% boolean_algebra.conj_zero_right
thf(fact_803_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_formula_a_b] :
( ( inf_in5741596252272946265la_a_b @ bot_bo8568539672012708443la_a_b @ X )
= bot_bo8568539672012708443la_a_b ) ).
% boolean_algebra.conj_zero_left
thf(fact_804_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_set_formula_a_b] :
( ( inf_in60795025369792911la_a_b @ bot_bo5296870006564677521la_a_b @ X )
= bot_bo5296870006564677521la_a_b ) ).
% boolean_algebra.conj_zero_left
thf(fact_805_inf__bot__right,axiom,
! [X: set_formula_a_b] :
( ( inf_in5741596252272946265la_a_b @ X @ bot_bo8568539672012708443la_a_b )
= bot_bo8568539672012708443la_a_b ) ).
% inf_bot_right
thf(fact_806_inf__bot__right,axiom,
! [X: set_set_formula_a_b] :
( ( inf_in60795025369792911la_a_b @ X @ bot_bo5296870006564677521la_a_b )
= bot_bo5296870006564677521la_a_b ) ).
% inf_bot_right
thf(fact_807_inf__bot__left,axiom,
! [X: set_formula_a_b] :
( ( inf_in5741596252272946265la_a_b @ bot_bo8568539672012708443la_a_b @ X )
= bot_bo8568539672012708443la_a_b ) ).
% inf_bot_left
thf(fact_808_inf__bot__left,axiom,
! [X: set_set_formula_a_b] :
( ( inf_in60795025369792911la_a_b @ bot_bo5296870006564677521la_a_b @ X )
= bot_bo5296870006564677521la_a_b ) ).
% inf_bot_left
thf(fact_809_sup__bot_Oright__neutral,axiom,
! [A2: set_set_formula_a_b] :
( ( sup_su5931044959554304553la_a_b @ A2 @ bot_bo5296870006564677521la_a_b )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_810_sup__bot_Oright__neutral,axiom,
! [A2: set_formula_a_b] :
( ( sup_su8832426789560438003la_a_b @ A2 @ bot_bo8568539672012708443la_a_b )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_811_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_set_formula_a_b,B2: set_set_formula_a_b] :
( ( bot_bo5296870006564677521la_a_b
= ( sup_su5931044959554304553la_a_b @ A2 @ B2 ) )
= ( ( A2 = bot_bo5296870006564677521la_a_b )
& ( B2 = bot_bo5296870006564677521la_a_b ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_812_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_formula_a_b,B2: set_formula_a_b] :
( ( bot_bo8568539672012708443la_a_b
= ( sup_su8832426789560438003la_a_b @ A2 @ B2 ) )
= ( ( A2 = bot_bo8568539672012708443la_a_b )
& ( B2 = bot_bo8568539672012708443la_a_b ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_813_sup__bot_Oleft__neutral,axiom,
! [A2: set_set_formula_a_b] :
( ( sup_su5931044959554304553la_a_b @ bot_bo5296870006564677521la_a_b @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_814_sup__bot_Oleft__neutral,axiom,
! [A2: set_formula_a_b] :
( ( sup_su8832426789560438003la_a_b @ bot_bo8568539672012708443la_a_b @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_815_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_set_formula_a_b,B2: set_set_formula_a_b] :
( ( ( sup_su5931044959554304553la_a_b @ A2 @ B2 )
= bot_bo5296870006564677521la_a_b )
= ( ( A2 = bot_bo5296870006564677521la_a_b )
& ( B2 = bot_bo5296870006564677521la_a_b ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_816_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_formula_a_b,B2: set_formula_a_b] :
( ( ( sup_su8832426789560438003la_a_b @ A2 @ B2 )
= bot_bo8568539672012708443la_a_b )
= ( ( A2 = bot_bo8568539672012708443la_a_b )
& ( B2 = bot_bo8568539672012708443la_a_b ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_817_sup__eq__bot__iff,axiom,
! [X: set_set_formula_a_b,Y: set_set_formula_a_b] :
( ( ( sup_su5931044959554304553la_a_b @ X @ Y )
= bot_bo5296870006564677521la_a_b )
= ( ( X = bot_bo5296870006564677521la_a_b )
& ( Y = bot_bo5296870006564677521la_a_b ) ) ) ).
% sup_eq_bot_iff
thf(fact_818_sup__eq__bot__iff,axiom,
! [X: set_formula_a_b,Y: set_formula_a_b] :
( ( ( sup_su8832426789560438003la_a_b @ X @ Y )
= bot_bo8568539672012708443la_a_b )
= ( ( X = bot_bo8568539672012708443la_a_b )
& ( Y = bot_bo8568539672012708443la_a_b ) ) ) ).
% sup_eq_bot_iff
thf(fact_819_bot__eq__sup__iff,axiom,
! [X: set_set_formula_a_b,Y: set_set_formula_a_b] :
( ( bot_bo5296870006564677521la_a_b
= ( sup_su5931044959554304553la_a_b @ X @ Y ) )
= ( ( X = bot_bo5296870006564677521la_a_b )
& ( Y = bot_bo5296870006564677521la_a_b ) ) ) ).
% bot_eq_sup_iff
thf(fact_820_bot__eq__sup__iff,axiom,
! [X: set_formula_a_b,Y: set_formula_a_b] :
( ( bot_bo8568539672012708443la_a_b
= ( sup_su8832426789560438003la_a_b @ X @ Y ) )
= ( ( X = bot_bo8568539672012708443la_a_b )
& ( Y = bot_bo8568539672012708443la_a_b ) ) ) ).
% bot_eq_sup_iff
thf(fact_821_sup__bot__right,axiom,
! [X: set_set_formula_a_b] :
( ( sup_su5931044959554304553la_a_b @ X @ bot_bo5296870006564677521la_a_b )
= X ) ).
% sup_bot_right
thf(fact_822_sup__bot__right,axiom,
! [X: set_formula_a_b] :
( ( sup_su8832426789560438003la_a_b @ X @ bot_bo8568539672012708443la_a_b )
= X ) ).
% sup_bot_right
thf(fact_823_sup__bot__left,axiom,
! [X: set_set_formula_a_b] :
( ( sup_su5931044959554304553la_a_b @ bot_bo5296870006564677521la_a_b @ X )
= X ) ).
% sup_bot_left
thf(fact_824_sup__bot__left,axiom,
! [X: set_formula_a_b] :
( ( sup_su8832426789560438003la_a_b @ bot_bo8568539672012708443la_a_b @ X )
= X ) ).
% sup_bot_left
thf(fact_825_Un__empty,axiom,
! [A: set_set_formula_a_b,B: set_set_formula_a_b] :
( ( ( sup_su5931044959554304553la_a_b @ A @ B )
= bot_bo5296870006564677521la_a_b )
= ( ( A = bot_bo5296870006564677521la_a_b )
& ( B = bot_bo5296870006564677521la_a_b ) ) ) ).
% Un_empty
thf(fact_826_Un__empty,axiom,
! [A: set_formula_a_b,B: set_formula_a_b] :
( ( ( sup_su8832426789560438003la_a_b @ A @ B )
= bot_bo8568539672012708443la_a_b )
= ( ( A = bot_bo8568539672012708443la_a_b )
& ( B = bot_bo8568539672012708443la_a_b ) ) ) ).
% Un_empty
thf(fact_827_Diff__cancel,axiom,
! [A: set_formula_a_b] :
( ( minus_3283878196352184736la_a_b @ A @ A )
= bot_bo8568539672012708443la_a_b ) ).
% Diff_cancel
thf(fact_828_Diff__cancel,axiom,
! [A: set_set_formula_a_b] :
( ( minus_4878662943024999126la_a_b @ A @ A )
= bot_bo5296870006564677521la_a_b ) ).
% Diff_cancel
thf(fact_829_empty__Diff,axiom,
! [A: set_formula_a_b] :
( ( minus_3283878196352184736la_a_b @ bot_bo8568539672012708443la_a_b @ A )
= bot_bo8568539672012708443la_a_b ) ).
% empty_Diff
thf(fact_830_empty__Diff,axiom,
! [A: set_set_formula_a_b] :
( ( minus_4878662943024999126la_a_b @ bot_bo5296870006564677521la_a_b @ A )
= bot_bo5296870006564677521la_a_b ) ).
% empty_Diff
thf(fact_831_Diff__empty,axiom,
! [A: set_formula_a_b] :
( ( minus_3283878196352184736la_a_b @ A @ bot_bo8568539672012708443la_a_b )
= A ) ).
% Diff_empty
thf(fact_832_Diff__empty,axiom,
! [A: set_set_formula_a_b] :
( ( minus_4878662943024999126la_a_b @ A @ bot_bo5296870006564677521la_a_b )
= A ) ).
% Diff_empty
thf(fact_833_set__empty,axiom,
! [Xs: list_l5948067055324603441la_a_b] :
( ( ( set_list_formula_a_b2 @ Xs )
= bot_bo4614192416432564331la_a_b )
= ( Xs = nil_list_formula_a_b ) ) ).
% set_empty
thf(fact_834_set__empty,axiom,
! [Xs: list_set_formula_a_b] :
( ( ( set_set_formula_a_b2 @ Xs )
= bot_bo5296870006564677521la_a_b )
= ( Xs = nil_set_formula_a_b ) ) ).
% set_empty
thf(fact_835_set__empty,axiom,
! [Xs: list_formula_a_b] :
( ( ( set_formula_a_b2 @ Xs )
= bot_bo8568539672012708443la_a_b )
= ( Xs = nil_formula_a_b ) ) ).
% set_empty
thf(fact_836_set__empty2,axiom,
! [Xs: list_l5948067055324603441la_a_b] :
( ( bot_bo4614192416432564331la_a_b
= ( set_list_formula_a_b2 @ Xs ) )
= ( Xs = nil_list_formula_a_b ) ) ).
% set_empty2
thf(fact_837_set__empty2,axiom,
! [Xs: list_set_formula_a_b] :
( ( bot_bo5296870006564677521la_a_b
= ( set_set_formula_a_b2 @ Xs ) )
= ( Xs = nil_set_formula_a_b ) ) ).
% set_empty2
thf(fact_838_set__empty2,axiom,
! [Xs: list_formula_a_b] :
( ( bot_bo8568539672012708443la_a_b
= ( set_formula_a_b2 @ Xs ) )
= ( Xs = nil_formula_a_b ) ) ).
% set_empty2
thf(fact_839_disjoint__insert_I2_J,axiom,
! [A: set_list_formula_a_b,B2: list_formula_a_b,B: set_list_formula_a_b] :
( ( bot_bo4614192416432564331la_a_b
= ( inf_in7172170795784610665la_a_b @ A @ ( insert4995524290302003025la_a_b @ B2 @ B ) ) )
= ( ~ ( member7703986999711672120la_a_b @ B2 @ A )
& ( bot_bo4614192416432564331la_a_b
= ( inf_in7172170795784610665la_a_b @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_840_disjoint__insert_I2_J,axiom,
! [A: set_set_formula_a_b,B2: set_formula_a_b,B: set_set_formula_a_b] :
( ( bot_bo5296870006564677521la_a_b
= ( inf_in60795025369792911la_a_b @ A @ ( insert8952366156270154231la_a_b @ B2 @ B ) ) )
= ( ~ ( member6621951321062823262la_a_b @ B2 @ A )
& ( bot_bo5296870006564677521la_a_b
= ( inf_in60795025369792911la_a_b @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_841_disjoint__insert_I2_J,axiom,
! [A: set_formula_a_b,B2: formula_a_b,B: set_formula_a_b] :
( ( bot_bo8568539672012708443la_a_b
= ( inf_in5741596252272946265la_a_b @ A @ ( insert_formula_a_b2 @ B2 @ B ) ) )
= ( ~ ( member_formula_a_b2 @ B2 @ A )
& ( bot_bo8568539672012708443la_a_b
= ( inf_in5741596252272946265la_a_b @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_842_disjoint__insert_I1_J,axiom,
! [B: set_list_formula_a_b,A2: list_formula_a_b,A: set_list_formula_a_b] :
( ( ( inf_in7172170795784610665la_a_b @ B @ ( insert4995524290302003025la_a_b @ A2 @ A ) )
= bot_bo4614192416432564331la_a_b )
= ( ~ ( member7703986999711672120la_a_b @ A2 @ B )
& ( ( inf_in7172170795784610665la_a_b @ B @ A )
= bot_bo4614192416432564331la_a_b ) ) ) ).
% disjoint_insert(1)
thf(fact_843_disjoint__insert_I1_J,axiom,
! [B: set_set_formula_a_b,A2: set_formula_a_b,A: set_set_formula_a_b] :
( ( ( inf_in60795025369792911la_a_b @ B @ ( insert8952366156270154231la_a_b @ A2 @ A ) )
= bot_bo5296870006564677521la_a_b )
= ( ~ ( member6621951321062823262la_a_b @ A2 @ B )
& ( ( inf_in60795025369792911la_a_b @ B @ A )
= bot_bo5296870006564677521la_a_b ) ) ) ).
% disjoint_insert(1)
thf(fact_844_disjoint__insert_I1_J,axiom,
! [B: set_formula_a_b,A2: formula_a_b,A: set_formula_a_b] :
( ( ( inf_in5741596252272946265la_a_b @ B @ ( insert_formula_a_b2 @ A2 @ A ) )
= bot_bo8568539672012708443la_a_b )
= ( ~ ( member_formula_a_b2 @ A2 @ B )
& ( ( inf_in5741596252272946265la_a_b @ B @ A )
= bot_bo8568539672012708443la_a_b ) ) ) ).
% disjoint_insert(1)
thf(fact_845_insert__disjoint_I2_J,axiom,
! [A2: list_formula_a_b,A: set_list_formula_a_b,B: set_list_formula_a_b] :
( ( bot_bo4614192416432564331la_a_b
= ( inf_in7172170795784610665la_a_b @ ( insert4995524290302003025la_a_b @ A2 @ A ) @ B ) )
= ( ~ ( member7703986999711672120la_a_b @ A2 @ B )
& ( bot_bo4614192416432564331la_a_b
= ( inf_in7172170795784610665la_a_b @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_846_insert__disjoint_I2_J,axiom,
! [A2: set_formula_a_b,A: set_set_formula_a_b,B: set_set_formula_a_b] :
( ( bot_bo5296870006564677521la_a_b
= ( inf_in60795025369792911la_a_b @ ( insert8952366156270154231la_a_b @ A2 @ A ) @ B ) )
= ( ~ ( member6621951321062823262la_a_b @ A2 @ B )
& ( bot_bo5296870006564677521la_a_b
= ( inf_in60795025369792911la_a_b @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_847_insert__disjoint_I2_J,axiom,
! [A2: formula_a_b,A: set_formula_a_b,B: set_formula_a_b] :
( ( bot_bo8568539672012708443la_a_b
= ( inf_in5741596252272946265la_a_b @ ( insert_formula_a_b2 @ A2 @ A ) @ B ) )
= ( ~ ( member_formula_a_b2 @ A2 @ B )
& ( bot_bo8568539672012708443la_a_b
= ( inf_in5741596252272946265la_a_b @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_848_insert__disjoint_I1_J,axiom,
! [A2: list_formula_a_b,A: set_list_formula_a_b,B: set_list_formula_a_b] :
( ( ( inf_in7172170795784610665la_a_b @ ( insert4995524290302003025la_a_b @ A2 @ A ) @ B )
= bot_bo4614192416432564331la_a_b )
= ( ~ ( member7703986999711672120la_a_b @ A2 @ B )
& ( ( inf_in7172170795784610665la_a_b @ A @ B )
= bot_bo4614192416432564331la_a_b ) ) ) ).
% insert_disjoint(1)
thf(fact_849_insert__disjoint_I1_J,axiom,
! [A2: set_formula_a_b,A: set_set_formula_a_b,B: set_set_formula_a_b] :
( ( ( inf_in60795025369792911la_a_b @ ( insert8952366156270154231la_a_b @ A2 @ A ) @ B )
= bot_bo5296870006564677521la_a_b )
= ( ~ ( member6621951321062823262la_a_b @ A2 @ B )
& ( ( inf_in60795025369792911la_a_b @ A @ B )
= bot_bo5296870006564677521la_a_b ) ) ) ).
% insert_disjoint(1)
thf(fact_850_insert__disjoint_I1_J,axiom,
! [A2: formula_a_b,A: set_formula_a_b,B: set_formula_a_b] :
( ( ( inf_in5741596252272946265la_a_b @ ( insert_formula_a_b2 @ A2 @ A ) @ B )
= bot_bo8568539672012708443la_a_b )
= ( ~ ( member_formula_a_b2 @ A2 @ B )
& ( ( inf_in5741596252272946265la_a_b @ A @ B )
= bot_bo8568539672012708443la_a_b ) ) ) ).
% insert_disjoint(1)
thf(fact_851_insert__Diff__single,axiom,
! [A2: formula_a_b,A: set_formula_a_b] :
( ( insert_formula_a_b2 @ A2 @ ( minus_3283878196352184736la_a_b @ A @ ( insert_formula_a_b2 @ A2 @ bot_bo8568539672012708443la_a_b ) ) )
= ( insert_formula_a_b2 @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_852_insert__Diff__single,axiom,
! [A2: set_formula_a_b,A: set_set_formula_a_b] :
( ( insert8952366156270154231la_a_b @ A2 @ ( minus_4878662943024999126la_a_b @ A @ ( insert8952366156270154231la_a_b @ A2 @ bot_bo5296870006564677521la_a_b ) ) )
= ( insert8952366156270154231la_a_b @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_853_Diff__disjoint,axiom,
! [A: set_formula_a_b,B: set_formula_a_b] :
( ( inf_in5741596252272946265la_a_b @ A @ ( minus_3283878196352184736la_a_b @ B @ A ) )
= bot_bo8568539672012708443la_a_b ) ).
% Diff_disjoint
thf(fact_854_Diff__disjoint,axiom,
! [A: set_set_formula_a_b,B: set_set_formula_a_b] :
( ( inf_in60795025369792911la_a_b @ A @ ( minus_4878662943024999126la_a_b @ B @ A ) )
= bot_bo5296870006564677521la_a_b ) ).
% Diff_disjoint
thf(fact_855_the__elem__eq,axiom,
! [X: formula_a_b] :
( ( the_elem_formula_a_b @ ( insert_formula_a_b2 @ X @ bot_bo8568539672012708443la_a_b ) )
= X ) ).
% the_elem_eq
thf(fact_856_the__elem__eq,axiom,
! [X: set_formula_a_b] :
( ( the_el8292460259403742098la_a_b @ ( insert8952366156270154231la_a_b @ X @ bot_bo5296870006564677521la_a_b ) )
= X ) ).
% the_elem_eq
thf(fact_857_distinct__append,axiom,
! [Xs: list_l5948067055324603441la_a_b,Ys: list_l5948067055324603441la_a_b] :
( ( distin27490427618170344la_a_b @ ( append58666856857280070la_a_b @ Xs @ Ys ) )
= ( ( distin27490427618170344la_a_b @ Xs )
& ( distin27490427618170344la_a_b @ Ys )
& ( ( inf_in7172170795784610665la_a_b @ ( set_list_formula_a_b2 @ Xs ) @ ( set_list_formula_a_b2 @ Ys ) )
= bot_bo4614192416432564331la_a_b ) ) ) ).
% distinct_append
thf(fact_858_distinct__append,axiom,
! [Xs: list_set_formula_a_b,Ys: list_set_formula_a_b] :
( ( distin5059275051008138254la_a_b @ ( append8577002312234529644la_a_b @ Xs @ Ys ) )
= ( ( distin5059275051008138254la_a_b @ Xs )
& ( distin5059275051008138254la_a_b @ Ys )
& ( ( inf_in60795025369792911la_a_b @ ( set_set_formula_a_b2 @ Xs ) @ ( set_set_formula_a_b2 @ Ys ) )
= bot_bo5296870006564677521la_a_b ) ) ) ).
% distinct_append
thf(fact_859_distinct__append,axiom,
! [Xs: list_formula_a_b,Ys: list_formula_a_b] :
( ( distinct_formula_a_b @ ( append_formula_a_b @ Xs @ Ys ) )
= ( ( distinct_formula_a_b @ Xs )
& ( distinct_formula_a_b @ Ys )
& ( ( inf_in5741596252272946265la_a_b @ ( set_formula_a_b2 @ Xs ) @ ( set_formula_a_b2 @ Ys ) )
= bot_bo8568539672012708443la_a_b ) ) ) ).
% distinct_append
thf(fact_860_ex__in__conv,axiom,
! [A: set_list_formula_a_b] :
( ( ? [X2: list_formula_a_b] : ( member7703986999711672120la_a_b @ X2 @ A ) )
= ( A != bot_bo4614192416432564331la_a_b ) ) ).
% ex_in_conv
thf(fact_861_ex__in__conv,axiom,
! [A: set_set_formula_a_b] :
( ( ? [X2: set_formula_a_b] : ( member6621951321062823262la_a_b @ X2 @ A ) )
= ( A != bot_bo5296870006564677521la_a_b ) ) ).
% ex_in_conv
thf(fact_862_ex__in__conv,axiom,
! [A: set_formula_a_b] :
( ( ? [X2: formula_a_b] : ( member_formula_a_b2 @ X2 @ A ) )
= ( A != bot_bo8568539672012708443la_a_b ) ) ).
% ex_in_conv
thf(fact_863_equals0I,axiom,
! [A: set_list_formula_a_b] :
( ! [Y3: list_formula_a_b] :
~ ( member7703986999711672120la_a_b @ Y3 @ A )
=> ( A = bot_bo4614192416432564331la_a_b ) ) ).
% equals0I
thf(fact_864_equals0I,axiom,
! [A: set_set_formula_a_b] :
( ! [Y3: set_formula_a_b] :
~ ( member6621951321062823262la_a_b @ Y3 @ A )
=> ( A = bot_bo5296870006564677521la_a_b ) ) ).
% equals0I
thf(fact_865_equals0I,axiom,
! [A: set_formula_a_b] :
( ! [Y3: formula_a_b] :
~ ( member_formula_a_b2 @ Y3 @ A )
=> ( A = bot_bo8568539672012708443la_a_b ) ) ).
% equals0I
thf(fact_866_equals0D,axiom,
! [A: set_list_formula_a_b,A2: list_formula_a_b] :
( ( A = bot_bo4614192416432564331la_a_b )
=> ~ ( member7703986999711672120la_a_b @ A2 @ A ) ) ).
% equals0D
thf(fact_867_equals0D,axiom,
! [A: set_set_formula_a_b,A2: set_formula_a_b] :
( ( A = bot_bo5296870006564677521la_a_b )
=> ~ ( member6621951321062823262la_a_b @ A2 @ A ) ) ).
% equals0D
thf(fact_868_equals0D,axiom,
! [A: set_formula_a_b,A2: formula_a_b] :
( ( A = bot_bo8568539672012708443la_a_b )
=> ~ ( member_formula_a_b2 @ A2 @ A ) ) ).
% equals0D
thf(fact_869_emptyE,axiom,
! [A2: list_formula_a_b] :
~ ( member7703986999711672120la_a_b @ A2 @ bot_bo4614192416432564331la_a_b ) ).
% emptyE
thf(fact_870_emptyE,axiom,
! [A2: set_formula_a_b] :
~ ( member6621951321062823262la_a_b @ A2 @ bot_bo5296870006564677521la_a_b ) ).
% emptyE
thf(fact_871_emptyE,axiom,
! [A2: formula_a_b] :
~ ( member_formula_a_b2 @ A2 @ bot_bo8568539672012708443la_a_b ) ).
% emptyE
thf(fact_872_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_set_formula_a_b] :
( ( sup_su5931044959554304553la_a_b @ X @ bot_bo5296870006564677521la_a_b )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_873_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_formula_a_b] :
( ( sup_su8832426789560438003la_a_b @ X @ bot_bo8568539672012708443la_a_b )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_874_singleton__inject,axiom,
! [A2: formula_a_b,B2: formula_a_b] :
( ( ( insert_formula_a_b2 @ A2 @ bot_bo8568539672012708443la_a_b )
= ( insert_formula_a_b2 @ B2 @ bot_bo8568539672012708443la_a_b ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_875_singleton__inject,axiom,
! [A2: set_formula_a_b,B2: set_formula_a_b] :
( ( ( insert8952366156270154231la_a_b @ A2 @ bot_bo5296870006564677521la_a_b )
= ( insert8952366156270154231la_a_b @ B2 @ bot_bo5296870006564677521la_a_b ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_876_insert__not__empty,axiom,
! [A2: formula_a_b,A: set_formula_a_b] :
( ( insert_formula_a_b2 @ A2 @ A )
!= bot_bo8568539672012708443la_a_b ) ).
% insert_not_empty
thf(fact_877_insert__not__empty,axiom,
! [A2: set_formula_a_b,A: set_set_formula_a_b] :
( ( insert8952366156270154231la_a_b @ A2 @ A )
!= bot_bo5296870006564677521la_a_b ) ).
% insert_not_empty
thf(fact_878_doubleton__eq__iff,axiom,
! [A2: formula_a_b,B2: formula_a_b,C: formula_a_b,D: formula_a_b] :
( ( ( insert_formula_a_b2 @ A2 @ ( insert_formula_a_b2 @ B2 @ bot_bo8568539672012708443la_a_b ) )
= ( insert_formula_a_b2 @ C @ ( insert_formula_a_b2 @ D @ bot_bo8568539672012708443la_a_b ) ) )
= ( ( ( A2 = C )
& ( B2 = D ) )
| ( ( A2 = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_879_doubleton__eq__iff,axiom,
! [A2: set_formula_a_b,B2: set_formula_a_b,C: set_formula_a_b,D: set_formula_a_b] :
( ( ( insert8952366156270154231la_a_b @ A2 @ ( insert8952366156270154231la_a_b @ B2 @ bot_bo5296870006564677521la_a_b ) )
= ( insert8952366156270154231la_a_b @ C @ ( insert8952366156270154231la_a_b @ D @ bot_bo5296870006564677521la_a_b ) ) )
= ( ( ( A2 = C )
& ( B2 = D ) )
| ( ( A2 = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_880_singleton__iff,axiom,
! [B2: list_formula_a_b,A2: list_formula_a_b] :
( ( member7703986999711672120la_a_b @ B2 @ ( insert4995524290302003025la_a_b @ A2 @ bot_bo4614192416432564331la_a_b ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_881_singleton__iff,axiom,
! [B2: set_formula_a_b,A2: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ B2 @ ( insert8952366156270154231la_a_b @ A2 @ bot_bo5296870006564677521la_a_b ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_882_singleton__iff,axiom,
! [B2: formula_a_b,A2: formula_a_b] :
( ( member_formula_a_b2 @ B2 @ ( insert_formula_a_b2 @ A2 @ bot_bo8568539672012708443la_a_b ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_883_singletonD,axiom,
! [B2: list_formula_a_b,A2: list_formula_a_b] :
( ( member7703986999711672120la_a_b @ B2 @ ( insert4995524290302003025la_a_b @ A2 @ bot_bo4614192416432564331la_a_b ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_884_singletonD,axiom,
! [B2: set_formula_a_b,A2: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ B2 @ ( insert8952366156270154231la_a_b @ A2 @ bot_bo5296870006564677521la_a_b ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_885_singletonD,axiom,
! [B2: formula_a_b,A2: formula_a_b] :
( ( member_formula_a_b2 @ B2 @ ( insert_formula_a_b2 @ A2 @ bot_bo8568539672012708443la_a_b ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_886_disjoint__iff__not__equal,axiom,
! [A: set_formula_a_b,B: set_formula_a_b] :
( ( ( inf_in5741596252272946265la_a_b @ A @ B )
= bot_bo8568539672012708443la_a_b )
= ( ! [X2: formula_a_b] :
( ( member_formula_a_b2 @ X2 @ A )
=> ! [Y2: formula_a_b] :
( ( member_formula_a_b2 @ Y2 @ B )
=> ( X2 != Y2 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_887_disjoint__iff__not__equal,axiom,
! [A: set_set_formula_a_b,B: set_set_formula_a_b] :
( ( ( inf_in60795025369792911la_a_b @ A @ B )
= bot_bo5296870006564677521la_a_b )
= ( ! [X2: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X2 @ A )
=> ! [Y2: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ Y2 @ B )
=> ( X2 != Y2 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_888_Int__empty__right,axiom,
! [A: set_formula_a_b] :
( ( inf_in5741596252272946265la_a_b @ A @ bot_bo8568539672012708443la_a_b )
= bot_bo8568539672012708443la_a_b ) ).
% Int_empty_right
thf(fact_889_Int__empty__right,axiom,
! [A: set_set_formula_a_b] :
( ( inf_in60795025369792911la_a_b @ A @ bot_bo5296870006564677521la_a_b )
= bot_bo5296870006564677521la_a_b ) ).
% Int_empty_right
thf(fact_890_Int__empty__left,axiom,
! [B: set_formula_a_b] :
( ( inf_in5741596252272946265la_a_b @ bot_bo8568539672012708443la_a_b @ B )
= bot_bo8568539672012708443la_a_b ) ).
% Int_empty_left
thf(fact_891_Int__empty__left,axiom,
! [B: set_set_formula_a_b] :
( ( inf_in60795025369792911la_a_b @ bot_bo5296870006564677521la_a_b @ B )
= bot_bo5296870006564677521la_a_b ) ).
% Int_empty_left
thf(fact_892_disjoint__iff,axiom,
! [A: set_list_formula_a_b,B: set_list_formula_a_b] :
( ( ( inf_in7172170795784610665la_a_b @ A @ B )
= bot_bo4614192416432564331la_a_b )
= ( ! [X2: list_formula_a_b] :
( ( member7703986999711672120la_a_b @ X2 @ A )
=> ~ ( member7703986999711672120la_a_b @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_893_disjoint__iff,axiom,
! [A: set_set_formula_a_b,B: set_set_formula_a_b] :
( ( ( inf_in60795025369792911la_a_b @ A @ B )
= bot_bo5296870006564677521la_a_b )
= ( ! [X2: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X2 @ A )
=> ~ ( member6621951321062823262la_a_b @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_894_disjoint__iff,axiom,
! [A: set_formula_a_b,B: set_formula_a_b] :
( ( ( inf_in5741596252272946265la_a_b @ A @ B )
= bot_bo8568539672012708443la_a_b )
= ( ! [X2: formula_a_b] :
( ( member_formula_a_b2 @ X2 @ A )
=> ~ ( member_formula_a_b2 @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_895_Int__emptyI,axiom,
! [A: set_list_formula_a_b,B: set_list_formula_a_b] :
( ! [X3: list_formula_a_b] :
( ( member7703986999711672120la_a_b @ X3 @ A )
=> ~ ( member7703986999711672120la_a_b @ X3 @ B ) )
=> ( ( inf_in7172170795784610665la_a_b @ A @ B )
= bot_bo4614192416432564331la_a_b ) ) ).
% Int_emptyI
thf(fact_896_Int__emptyI,axiom,
! [A: set_set_formula_a_b,B: set_set_formula_a_b] :
( ! [X3: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X3 @ A )
=> ~ ( member6621951321062823262la_a_b @ X3 @ B ) )
=> ( ( inf_in60795025369792911la_a_b @ A @ B )
= bot_bo5296870006564677521la_a_b ) ) ).
% Int_emptyI
thf(fact_897_Int__emptyI,axiom,
! [A: set_formula_a_b,B: set_formula_a_b] :
( ! [X3: formula_a_b] :
( ( member_formula_a_b2 @ X3 @ A )
=> ~ ( member_formula_a_b2 @ X3 @ B ) )
=> ( ( inf_in5741596252272946265la_a_b @ A @ B )
= bot_bo8568539672012708443la_a_b ) ) ).
% Int_emptyI
thf(fact_898_Un__empty__right,axiom,
! [A: set_set_formula_a_b] :
( ( sup_su5931044959554304553la_a_b @ A @ bot_bo5296870006564677521la_a_b )
= A ) ).
% Un_empty_right
thf(fact_899_Un__empty__right,axiom,
! [A: set_formula_a_b] :
( ( sup_su8832426789560438003la_a_b @ A @ bot_bo8568539672012708443la_a_b )
= A ) ).
% Un_empty_right
thf(fact_900_Un__empty__left,axiom,
! [B: set_set_formula_a_b] :
( ( sup_su5931044959554304553la_a_b @ bot_bo5296870006564677521la_a_b @ B )
= B ) ).
% Un_empty_left
thf(fact_901_Un__empty__left,axiom,
! [B: set_formula_a_b] :
( ( sup_su8832426789560438003la_a_b @ bot_bo8568539672012708443la_a_b @ B )
= B ) ).
% Un_empty_left
thf(fact_902_empty__set,axiom,
( bot_bo4614192416432564331la_a_b
= ( set_list_formula_a_b2 @ nil_list_formula_a_b ) ) ).
% empty_set
thf(fact_903_empty__set,axiom,
( bot_bo5296870006564677521la_a_b
= ( set_set_formula_a_b2 @ nil_set_formula_a_b ) ) ).
% empty_set
thf(fact_904_empty__set,axiom,
( bot_bo8568539672012708443la_a_b
= ( set_formula_a_b2 @ nil_formula_a_b ) ) ).
% empty_set
thf(fact_905_insert__is__Un,axiom,
( insert8952366156270154231la_a_b
= ( ^ [A3: set_formula_a_b] : ( sup_su5931044959554304553la_a_b @ ( insert8952366156270154231la_a_b @ A3 @ bot_bo5296870006564677521la_a_b ) ) ) ) ).
% insert_is_Un
thf(fact_906_insert__is__Un,axiom,
( insert_formula_a_b2
= ( ^ [A3: formula_a_b] : ( sup_su8832426789560438003la_a_b @ ( insert_formula_a_b2 @ A3 @ bot_bo8568539672012708443la_a_b ) ) ) ) ).
% insert_is_Un
thf(fact_907_Un__singleton__iff,axiom,
! [A: set_set_formula_a_b,B: set_set_formula_a_b,X: set_formula_a_b] :
( ( ( sup_su5931044959554304553la_a_b @ A @ B )
= ( insert8952366156270154231la_a_b @ X @ bot_bo5296870006564677521la_a_b ) )
= ( ( ( A = bot_bo5296870006564677521la_a_b )
& ( B
= ( insert8952366156270154231la_a_b @ X @ bot_bo5296870006564677521la_a_b ) ) )
| ( ( A
= ( insert8952366156270154231la_a_b @ X @ bot_bo5296870006564677521la_a_b ) )
& ( B = bot_bo5296870006564677521la_a_b ) )
| ( ( A
= ( insert8952366156270154231la_a_b @ X @ bot_bo5296870006564677521la_a_b ) )
& ( B
= ( insert8952366156270154231la_a_b @ X @ bot_bo5296870006564677521la_a_b ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_908_Un__singleton__iff,axiom,
! [A: set_formula_a_b,B: set_formula_a_b,X: formula_a_b] :
( ( ( sup_su8832426789560438003la_a_b @ A @ B )
= ( insert_formula_a_b2 @ X @ bot_bo8568539672012708443la_a_b ) )
= ( ( ( A = bot_bo8568539672012708443la_a_b )
& ( B
= ( insert_formula_a_b2 @ X @ bot_bo8568539672012708443la_a_b ) ) )
| ( ( A
= ( insert_formula_a_b2 @ X @ bot_bo8568539672012708443la_a_b ) )
& ( B = bot_bo8568539672012708443la_a_b ) )
| ( ( A
= ( insert_formula_a_b2 @ X @ bot_bo8568539672012708443la_a_b ) )
& ( B
= ( insert_formula_a_b2 @ X @ bot_bo8568539672012708443la_a_b ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_909_singleton__Un__iff,axiom,
! [X: set_formula_a_b,A: set_set_formula_a_b,B: set_set_formula_a_b] :
( ( ( insert8952366156270154231la_a_b @ X @ bot_bo5296870006564677521la_a_b )
= ( sup_su5931044959554304553la_a_b @ A @ B ) )
= ( ( ( A = bot_bo5296870006564677521la_a_b )
& ( B
= ( insert8952366156270154231la_a_b @ X @ bot_bo5296870006564677521la_a_b ) ) )
| ( ( A
= ( insert8952366156270154231la_a_b @ X @ bot_bo5296870006564677521la_a_b ) )
& ( B = bot_bo5296870006564677521la_a_b ) )
| ( ( A
= ( insert8952366156270154231la_a_b @ X @ bot_bo5296870006564677521la_a_b ) )
& ( B
= ( insert8952366156270154231la_a_b @ X @ bot_bo5296870006564677521la_a_b ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_910_singleton__Un__iff,axiom,
! [X: formula_a_b,A: set_formula_a_b,B: set_formula_a_b] :
( ( ( insert_formula_a_b2 @ X @ bot_bo8568539672012708443la_a_b )
= ( sup_su8832426789560438003la_a_b @ A @ B ) )
= ( ( ( A = bot_bo8568539672012708443la_a_b )
& ( B
= ( insert_formula_a_b2 @ X @ bot_bo8568539672012708443la_a_b ) ) )
| ( ( A
= ( insert_formula_a_b2 @ X @ bot_bo8568539672012708443la_a_b ) )
& ( B = bot_bo8568539672012708443la_a_b ) )
| ( ( A
= ( insert_formula_a_b2 @ X @ bot_bo8568539672012708443la_a_b ) )
& ( B
= ( insert_formula_a_b2 @ X @ bot_bo8568539672012708443la_a_b ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_911_Diff__insert,axiom,
! [A: set_formula_a_b,A2: formula_a_b,B: set_formula_a_b] :
( ( minus_3283878196352184736la_a_b @ A @ ( insert_formula_a_b2 @ A2 @ B ) )
= ( minus_3283878196352184736la_a_b @ ( minus_3283878196352184736la_a_b @ A @ B ) @ ( insert_formula_a_b2 @ A2 @ bot_bo8568539672012708443la_a_b ) ) ) ).
% Diff_insert
thf(fact_912_Diff__insert,axiom,
! [A: set_set_formula_a_b,A2: set_formula_a_b,B: set_set_formula_a_b] :
( ( minus_4878662943024999126la_a_b @ A @ ( insert8952366156270154231la_a_b @ A2 @ B ) )
= ( minus_4878662943024999126la_a_b @ ( minus_4878662943024999126la_a_b @ A @ B ) @ ( insert8952366156270154231la_a_b @ A2 @ bot_bo5296870006564677521la_a_b ) ) ) ).
% Diff_insert
thf(fact_913_insert__Diff,axiom,
! [A2: list_formula_a_b,A: set_list_formula_a_b] :
( ( member7703986999711672120la_a_b @ A2 @ A )
=> ( ( insert4995524290302003025la_a_b @ A2 @ ( minus_8667068627093346608la_a_b @ A @ ( insert4995524290302003025la_a_b @ A2 @ bot_bo4614192416432564331la_a_b ) ) )
= A ) ) ).
% insert_Diff
thf(fact_914_insert__Diff,axiom,
! [A2: set_formula_a_b,A: set_set_formula_a_b] :
( ( member6621951321062823262la_a_b @ A2 @ A )
=> ( ( insert8952366156270154231la_a_b @ A2 @ ( minus_4878662943024999126la_a_b @ A @ ( insert8952366156270154231la_a_b @ A2 @ bot_bo5296870006564677521la_a_b ) ) )
= A ) ) ).
% insert_Diff
thf(fact_915_insert__Diff,axiom,
! [A2: formula_a_b,A: set_formula_a_b] :
( ( member_formula_a_b2 @ A2 @ A )
=> ( ( insert_formula_a_b2 @ A2 @ ( minus_3283878196352184736la_a_b @ A @ ( insert_formula_a_b2 @ A2 @ bot_bo8568539672012708443la_a_b ) ) )
= A ) ) ).
% insert_Diff
thf(fact_916_Diff__insert2,axiom,
! [A: set_formula_a_b,A2: formula_a_b,B: set_formula_a_b] :
( ( minus_3283878196352184736la_a_b @ A @ ( insert_formula_a_b2 @ A2 @ B ) )
= ( minus_3283878196352184736la_a_b @ ( minus_3283878196352184736la_a_b @ A @ ( insert_formula_a_b2 @ A2 @ bot_bo8568539672012708443la_a_b ) ) @ B ) ) ).
% Diff_insert2
thf(fact_917_Diff__insert2,axiom,
! [A: set_set_formula_a_b,A2: set_formula_a_b,B: set_set_formula_a_b] :
( ( minus_4878662943024999126la_a_b @ A @ ( insert8952366156270154231la_a_b @ A2 @ B ) )
= ( minus_4878662943024999126la_a_b @ ( minus_4878662943024999126la_a_b @ A @ ( insert8952366156270154231la_a_b @ A2 @ bot_bo5296870006564677521la_a_b ) ) @ B ) ) ).
% Diff_insert2
thf(fact_918_Diff__insert__absorb,axiom,
! [X: list_formula_a_b,A: set_list_formula_a_b] :
( ~ ( member7703986999711672120la_a_b @ X @ A )
=> ( ( minus_8667068627093346608la_a_b @ ( insert4995524290302003025la_a_b @ X @ A ) @ ( insert4995524290302003025la_a_b @ X @ bot_bo4614192416432564331la_a_b ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_919_Diff__insert__absorb,axiom,
! [X: set_formula_a_b,A: set_set_formula_a_b] :
( ~ ( member6621951321062823262la_a_b @ X @ A )
=> ( ( minus_4878662943024999126la_a_b @ ( insert8952366156270154231la_a_b @ X @ A ) @ ( insert8952366156270154231la_a_b @ X @ bot_bo5296870006564677521la_a_b ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_920_Diff__insert__absorb,axiom,
! [X: formula_a_b,A: set_formula_a_b] :
( ~ ( member_formula_a_b2 @ X @ A )
=> ( ( minus_3283878196352184736la_a_b @ ( insert_formula_a_b2 @ X @ A ) @ ( insert_formula_a_b2 @ X @ bot_bo8568539672012708443la_a_b ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_921_Int__Diff__disjoint,axiom,
! [A: set_formula_a_b,B: set_formula_a_b] :
( ( inf_in5741596252272946265la_a_b @ ( inf_in5741596252272946265la_a_b @ A @ B ) @ ( minus_3283878196352184736la_a_b @ A @ B ) )
= bot_bo8568539672012708443la_a_b ) ).
% Int_Diff_disjoint
thf(fact_922_Int__Diff__disjoint,axiom,
! [A: set_set_formula_a_b,B: set_set_formula_a_b] :
( ( inf_in60795025369792911la_a_b @ ( inf_in60795025369792911la_a_b @ A @ B ) @ ( minus_4878662943024999126la_a_b @ A @ B ) )
= bot_bo5296870006564677521la_a_b ) ).
% Int_Diff_disjoint
thf(fact_923_Diff__triv,axiom,
! [A: set_formula_a_b,B: set_formula_a_b] :
( ( ( inf_in5741596252272946265la_a_b @ A @ B )
= bot_bo8568539672012708443la_a_b )
=> ( ( minus_3283878196352184736la_a_b @ A @ B )
= A ) ) ).
% Diff_triv
thf(fact_924_Diff__triv,axiom,
! [A: set_set_formula_a_b,B: set_set_formula_a_b] :
( ( ( inf_in60795025369792911la_a_b @ A @ B )
= bot_bo5296870006564677521la_a_b )
=> ( ( minus_4878662943024999126la_a_b @ A @ B )
= A ) ) ).
% Diff_triv
thf(fact_925_shuffles_Osimps_I2_J,axiom,
! [Xs: list_formula_a_b] :
( ( shuffles_formula_a_b @ Xs @ nil_formula_a_b )
= ( insert4995524290302003025la_a_b @ Xs @ bot_bo4614192416432564331la_a_b ) ) ).
% shuffles.simps(2)
thf(fact_926_shuffles_Osimps_I1_J,axiom,
! [Ys: list_formula_a_b] :
( ( shuffles_formula_a_b @ nil_formula_a_b @ Ys )
= ( insert4995524290302003025la_a_b @ Ys @ bot_bo4614192416432564331la_a_b ) ) ).
% shuffles.simps(1)
thf(fact_927_ShiftD,axiom,
! [Kl: list_set_formula_a_b,Kl2: set_li5266388670752979469la_a_b,K: set_formula_a_b] :
( ( member289699325402759278la_a_b @ Kl @ ( bNF_Gr5837620917382104168la_a_b @ Kl2 @ K ) )
=> ( member289699325402759278la_a_b @ ( cons_set_formula_a_b @ K @ Kl ) @ Kl2 ) ) ).
% ShiftD
thf(fact_928_ShiftD,axiom,
! [Kl: list_formula_a_b,Kl2: set_list_formula_a_b,K: formula_a_b] :
( ( member7703986999711672120la_a_b @ Kl @ ( bNF_Gr2660717057327301682la_a_b @ Kl2 @ K ) )
=> ( member7703986999711672120la_a_b @ ( cons_formula_a_b @ K @ Kl ) @ Kl2 ) ) ).
% ShiftD
thf(fact_929_remove__def,axiom,
( remove_formula_a_b
= ( ^ [X2: formula_a_b,A4: set_formula_a_b] : ( minus_3283878196352184736la_a_b @ A4 @ ( insert_formula_a_b2 @ X2 @ bot_bo8568539672012708443la_a_b ) ) ) ) ).
% remove_def
thf(fact_930_remove__def,axiom,
( remove6203333876907372194la_a_b
= ( ^ [X2: set_formula_a_b,A4: set_set_formula_a_b] : ( minus_4878662943024999126la_a_b @ A4 @ ( insert8952366156270154231la_a_b @ X2 @ bot_bo5296870006564677521la_a_b ) ) ) ) ).
% remove_def
thf(fact_931_distinct__concat,axiom,
! [Xs: list_l3581718849243781313la_a_b] :
( ( distin7365129041399036280la_a_b @ Xs )
=> ( ! [Ys3: list_l5948067055324603441la_a_b] :
( ( member5770512961814848200la_a_b @ Ys3 @ ( set_li405426049885584732la_a_b @ Xs ) )
=> ( distin27490427618170344la_a_b @ Ys3 ) )
=> ( ! [Ys3: list_l5948067055324603441la_a_b,Zs2: list_l5948067055324603441la_a_b] :
( ( member5770512961814848200la_a_b @ Ys3 @ ( set_li405426049885584732la_a_b @ Xs ) )
=> ( ( member5770512961814848200la_a_b @ Zs2 @ ( set_li405426049885584732la_a_b @ Xs ) )
=> ( ( Ys3 != Zs2 )
=> ( ( inf_in7172170795784610665la_a_b @ ( set_list_formula_a_b2 @ Ys3 ) @ ( set_list_formula_a_b2 @ Zs2 ) )
= bot_bo4614192416432564331la_a_b ) ) ) )
=> ( distin27490427618170344la_a_b @ ( concat6919603293682802464la_a_b @ Xs ) ) ) ) ) ).
% distinct_concat
thf(fact_932_distinct__concat,axiom,
! [Xs: list_l147921316930177639la_a_b] :
( ( distin6955390182775811870la_a_b @ Xs )
=> ( ! [Ys3: list_set_formula_a_b] :
( ( member289699325402759278la_a_b @ Ys3 @ ( set_li566459811281642754la_a_b @ Xs ) )
=> ( distin5059275051008138254la_a_b @ Ys3 ) )
=> ( ! [Ys3: list_set_formula_a_b,Zs2: list_set_formula_a_b] :
( ( member289699325402759278la_a_b @ Ys3 @ ( set_li566459811281642754la_a_b @ Xs ) )
=> ( ( member289699325402759278la_a_b @ Zs2 @ ( set_li566459811281642754la_a_b @ Xs ) )
=> ( ( Ys3 != Zs2 )
=> ( ( inf_in60795025369792911la_a_b @ ( set_set_formula_a_b2 @ Ys3 ) @ ( set_set_formula_a_b2 @ Zs2 ) )
= bot_bo5296870006564677521la_a_b ) ) ) )
=> ( distin5059275051008138254la_a_b @ ( concat7835744469863612742la_a_b @ Xs ) ) ) ) ) ).
% distinct_concat
thf(fact_933_distinct__concat,axiom,
! [Xs: list_l5948067055324603441la_a_b] :
( ( distin27490427618170344la_a_b @ Xs )
=> ( ! [Ys3: list_formula_a_b] :
( ( member7703986999711672120la_a_b @ Ys3 @ ( set_list_formula_a_b2 @ Xs ) )
=> ( distinct_formula_a_b @ Ys3 ) )
=> ( ! [Ys3: list_formula_a_b,Zs2: list_formula_a_b] :
( ( member7703986999711672120la_a_b @ Ys3 @ ( set_list_formula_a_b2 @ Xs ) )
=> ( ( member7703986999711672120la_a_b @ Zs2 @ ( set_list_formula_a_b2 @ Xs ) )
=> ( ( Ys3 != Zs2 )
=> ( ( inf_in5741596252272946265la_a_b @ ( set_formula_a_b2 @ Ys3 ) @ ( set_formula_a_b2 @ Zs2 ) )
= bot_bo8568539672012708443la_a_b ) ) ) )
=> ( distinct_formula_a_b @ ( concat_formula_a_b @ Xs ) ) ) ) ) ).
% distinct_concat
thf(fact_934_distinct__disjoint__shuffles,axiom,
! [Xs: list_l5948067055324603441la_a_b,Ys: list_l5948067055324603441la_a_b,Zs: list_l5948067055324603441la_a_b] :
( ( distin27490427618170344la_a_b @ Xs )
=> ( ( distin27490427618170344la_a_b @ Ys )
=> ( ( ( inf_in7172170795784610665la_a_b @ ( set_list_formula_a_b2 @ Xs ) @ ( set_list_formula_a_b2 @ Ys ) )
= bot_bo4614192416432564331la_a_b )
=> ( ( member5770512961814848200la_a_b @ Zs @ ( shuffl6594143666549603398la_a_b @ Xs @ Ys ) )
=> ( distin27490427618170344la_a_b @ Zs ) ) ) ) ) ).
% distinct_disjoint_shuffles
thf(fact_935_distinct__disjoint__shuffles,axiom,
! [Xs: list_set_formula_a_b,Ys: list_set_formula_a_b,Zs: list_set_formula_a_b] :
( ( distin5059275051008138254la_a_b @ Xs )
=> ( ( distin5059275051008138254la_a_b @ Ys )
=> ( ( ( inf_in60795025369792911la_a_b @ ( set_set_formula_a_b2 @ Xs ) @ ( set_set_formula_a_b2 @ Ys ) )
= bot_bo5296870006564677521la_a_b )
=> ( ( member289699325402759278la_a_b @ Zs @ ( shuffl4965861759276814700la_a_b @ Xs @ Ys ) )
=> ( distin5059275051008138254la_a_b @ Zs ) ) ) ) ) ).
% distinct_disjoint_shuffles
thf(fact_936_distinct__disjoint__shuffles,axiom,
! [Xs: list_formula_a_b,Ys: list_formula_a_b,Zs: list_formula_a_b] :
( ( distinct_formula_a_b @ Xs )
=> ( ( distinct_formula_a_b @ Ys )
=> ( ( ( inf_in5741596252272946265la_a_b @ ( set_formula_a_b2 @ Xs ) @ ( set_formula_a_b2 @ Ys ) )
= bot_bo8568539672012708443la_a_b )
=> ( ( member7703986999711672120la_a_b @ Zs @ ( shuffles_formula_a_b @ Xs @ Ys ) )
=> ( distinct_formula_a_b @ Zs ) ) ) ) ) ).
% distinct_disjoint_shuffles
thf(fact_937_Sup__fin_Osingleton,axiom,
! [X: set_formula_a_b] :
( ( lattic2153292871565936464la_a_b @ ( insert8952366156270154231la_a_b @ X @ bot_bo5296870006564677521la_a_b ) )
= X ) ).
% Sup_fin.singleton
thf(fact_938_is__singleton__the__elem,axiom,
( is_sin4967114517209836445la_a_b
= ( ^ [A4: set_formula_a_b] :
( A4
= ( insert_formula_a_b2 @ ( the_elem_formula_a_b @ A4 ) @ bot_bo8568539672012708443la_a_b ) ) ) ) ).
% is_singleton_the_elem
thf(fact_939_is__singleton__the__elem,axiom,
( is_sin8536277385185689555la_a_b
= ( ^ [A4: set_set_formula_a_b] :
( A4
= ( insert8952366156270154231la_a_b @ ( the_el8292460259403742098la_a_b @ A4 ) @ bot_bo5296870006564677521la_a_b ) ) ) ) ).
% is_singleton_the_elem
thf(fact_940_is__singletonI,axiom,
! [X: formula_a_b] : ( is_sin4967114517209836445la_a_b @ ( insert_formula_a_b2 @ X @ bot_bo8568539672012708443la_a_b ) ) ).
% is_singletonI
thf(fact_941_is__singletonI,axiom,
! [X: set_formula_a_b] : ( is_sin8536277385185689555la_a_b @ ( insert8952366156270154231la_a_b @ X @ bot_bo5296870006564677521la_a_b ) ) ).
% is_singletonI
thf(fact_942_Set_Ois__empty__def,axiom,
( is_empty_formula_a_b
= ( ^ [A4: set_formula_a_b] : ( A4 = bot_bo8568539672012708443la_a_b ) ) ) ).
% Set.is_empty_def
thf(fact_943_Set_Ois__empty__def,axiom,
( is_emp8895161027192797301la_a_b
= ( ^ [A4: set_set_formula_a_b] : ( A4 = bot_bo5296870006564677521la_a_b ) ) ) ).
% Set.is_empty_def
thf(fact_944_is__singletonE,axiom,
! [A: set_formula_a_b] :
( ( is_sin4967114517209836445la_a_b @ A )
=> ~ ! [X3: formula_a_b] :
( A
!= ( insert_formula_a_b2 @ X3 @ bot_bo8568539672012708443la_a_b ) ) ) ).
% is_singletonE
thf(fact_945_is__singletonE,axiom,
! [A: set_set_formula_a_b] :
( ( is_sin8536277385185689555la_a_b @ A )
=> ~ ! [X3: set_formula_a_b] :
( A
!= ( insert8952366156270154231la_a_b @ X3 @ bot_bo5296870006564677521la_a_b ) ) ) ).
% is_singletonE
thf(fact_946_bot__set__def,axiom,
( bot_bo8568539672012708443la_a_b
= ( collect_formula_a_b @ bot_bo2419332509030749162_a_b_o ) ) ).
% bot_set_def
thf(fact_947_bot__set__def,axiom,
( bot_bo5296870006564677521la_a_b
= ( collec5361897267898172060la_a_b @ bot_bo7837635004776446516_a_b_o ) ) ).
% bot_set_def
thf(fact_948_is__singletonI_H,axiom,
! [A: set_list_formula_a_b] :
( ( A != bot_bo4614192416432564331la_a_b )
=> ( ! [X3: list_formula_a_b,Y3: list_formula_a_b] :
( ( member7703986999711672120la_a_b @ X3 @ A )
=> ( ( member7703986999711672120la_a_b @ Y3 @ A )
=> ( X3 = Y3 ) ) )
=> ( is_sin1449733419968980781la_a_b @ A ) ) ) ).
% is_singletonI'
thf(fact_949_is__singletonI_H,axiom,
! [A: set_set_formula_a_b] :
( ( A != bot_bo5296870006564677521la_a_b )
=> ( ! [X3: set_formula_a_b,Y3: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X3 @ A )
=> ( ( member6621951321062823262la_a_b @ Y3 @ A )
=> ( X3 = Y3 ) ) )
=> ( is_sin8536277385185689555la_a_b @ A ) ) ) ).
% is_singletonI'
thf(fact_950_is__singletonI_H,axiom,
! [A: set_formula_a_b] :
( ( A != bot_bo8568539672012708443la_a_b )
=> ( ! [X3: formula_a_b,Y3: formula_a_b] :
( ( member_formula_a_b2 @ X3 @ A )
=> ( ( member_formula_a_b2 @ Y3 @ A )
=> ( X3 = Y3 ) ) )
=> ( is_sin4967114517209836445la_a_b @ A ) ) ) ).
% is_singletonI'
thf(fact_951_is__singleton__def,axiom,
( is_sin4967114517209836445la_a_b
= ( ^ [A4: set_formula_a_b] :
? [X2: formula_a_b] :
( A4
= ( insert_formula_a_b2 @ X2 @ bot_bo8568539672012708443la_a_b ) ) ) ) ).
% is_singleton_def
thf(fact_952_is__singleton__def,axiom,
( is_sin8536277385185689555la_a_b
= ( ^ [A4: set_set_formula_a_b] :
? [X2: set_formula_a_b] :
( A4
= ( insert8952366156270154231la_a_b @ X2 @ bot_bo5296870006564677521la_a_b ) ) ) ) ).
% is_singleton_def
thf(fact_953_listset_Osimps_I1_J,axiom,
( ( listset_formula_a_b @ nil_set_formula_a_b )
= ( insert4995524290302003025la_a_b @ nil_formula_a_b @ bot_bo4614192416432564331la_a_b ) ) ).
% listset.simps(1)
thf(fact_954_is__empty__set,axiom,
! [Xs: list_set_formula_a_b] :
( ( is_emp8895161027192797301la_a_b @ ( set_set_formula_a_b2 @ Xs ) )
= ( null_set_formula_a_b @ Xs ) ) ).
% is_empty_set
thf(fact_955_is__empty__set,axiom,
! [Xs: list_l5948067055324603441la_a_b] :
( ( is_emp464069198922215119la_a_b @ ( set_list_formula_a_b2 @ Xs ) )
= ( null_l1421712370273038483la_a_b @ Xs ) ) ).
% is_empty_set
thf(fact_956_is__empty__set,axiom,
! [Xs: list_formula_a_b] :
( ( is_empty_formula_a_b @ ( set_formula_a_b2 @ Xs ) )
= ( null_formula_a_b @ Xs ) ) ).
% is_empty_set
thf(fact_957_lists__empty,axiom,
( ( lists_formula_a_b @ bot_bo8568539672012708443la_a_b )
= ( insert4995524290302003025la_a_b @ nil_formula_a_b @ bot_bo4614192416432564331la_a_b ) ) ).
% lists_empty
thf(fact_958_lists__empty,axiom,
( ( lists_4880915010259632851la_a_b @ bot_bo5296870006564677521la_a_b )
= ( insert4418010381955824775la_a_b @ nil_set_formula_a_b @ bot_bo162344608834447009la_a_b ) ) ).
% lists_empty
thf(fact_959_shuffles_Oelims,axiom,
! [X: list_formula_a_b,Xa2: list_formula_a_b,Y: set_list_formula_a_b] :
( ( ( shuffles_formula_a_b @ X @ Xa2 )
= Y )
=> ( ( ( X = nil_formula_a_b )
=> ( Y
!= ( insert4995524290302003025la_a_b @ Xa2 @ bot_bo4614192416432564331la_a_b ) ) )
=> ( ( ( Xa2 = nil_formula_a_b )
=> ( Y
!= ( insert4995524290302003025la_a_b @ X @ bot_bo4614192416432564331la_a_b ) ) )
=> ~ ! [X3: formula_a_b,Xs3: list_formula_a_b] :
( ( X
= ( cons_formula_a_b @ X3 @ Xs3 ) )
=> ! [Y3: formula_a_b,Ys3: list_formula_a_b] :
( ( Xa2
= ( cons_formula_a_b @ Y3 @ Ys3 ) )
=> ( Y
!= ( sup_su9015223001214971651la_a_b @ ( image_2463093012343946057la_a_b @ ( cons_formula_a_b @ X3 ) @ ( shuffles_formula_a_b @ Xs3 @ ( cons_formula_a_b @ Y3 @ Ys3 ) ) ) @ ( image_2463093012343946057la_a_b @ ( cons_formula_a_b @ Y3 ) @ ( shuffles_formula_a_b @ ( cons_formula_a_b @ X3 @ Xs3 ) @ Ys3 ) ) ) ) ) ) ) ) ) ).
% shuffles.elims
thf(fact_960_shuffles_Oelims,axiom,
! [X: list_set_formula_a_b,Xa2: list_set_formula_a_b,Y: set_li5266388670752979469la_a_b] :
( ( ( shuffl4965861759276814700la_a_b @ X @ Xa2 )
= Y )
=> ( ( ( X = nil_set_formula_a_b )
=> ( Y
!= ( insert4418010381955824775la_a_b @ Xa2 @ bot_bo162344608834447009la_a_b ) ) )
=> ( ( ( Xa2 = nil_set_formula_a_b )
=> ( Y
!= ( insert4418010381955824775la_a_b @ X @ bot_bo162344608834447009la_a_b ) ) )
=> ~ ! [X3: set_formula_a_b,Xs3: list_set_formula_a_b] :
( ( X
= ( cons_set_formula_a_b @ X3 @ Xs3 ) )
=> ! [Y3: set_formula_a_b,Ys3: list_set_formula_a_b] :
( ( Xa2
= ( cons_set_formula_a_b @ Y3 @ Ys3 ) )
=> ( Y
!= ( sup_su5324240277097303353la_a_b @ ( image_4866258802339552437la_a_b @ ( cons_set_formula_a_b @ X3 ) @ ( shuffl4965861759276814700la_a_b @ Xs3 @ ( cons_set_formula_a_b @ Y3 @ Ys3 ) ) ) @ ( image_4866258802339552437la_a_b @ ( cons_set_formula_a_b @ Y3 ) @ ( shuffl4965861759276814700la_a_b @ ( cons_set_formula_a_b @ X3 @ Xs3 ) @ Ys3 ) ) ) ) ) ) ) ) ) ).
% shuffles.elims
thf(fact_961_image__eqI,axiom,
! [B2: list_formula_a_b,F: formula_a_b > list_formula_a_b,X: formula_a_b,A: set_formula_a_b] :
( ( B2
= ( F @ X ) )
=> ( ( member_formula_a_b2 @ X @ A )
=> ( member7703986999711672120la_a_b @ B2 @ ( image_7669030115802156217la_a_b @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_962_image__eqI,axiom,
! [B2: set_formula_a_b,F: formula_a_b > set_formula_a_b,X: formula_a_b,A: set_formula_a_b] :
( ( B2
= ( F @ X ) )
=> ( ( member_formula_a_b2 @ X @ A )
=> ( member6621951321062823262la_a_b @ B2 @ ( image_3063539048272112479la_a_b @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_963_image__eqI,axiom,
! [B2: formula_a_b,F: list_formula_a_b > formula_a_b,X: list_formula_a_b,A: set_list_formula_a_b] :
( ( B2
= ( F @ X ) )
=> ( ( member7703986999711672120la_a_b @ X @ A )
=> ( member_formula_a_b2 @ B2 @ ( image_5057755198331993529la_a_b @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_964_image__eqI,axiom,
! [B2: list_formula_a_b,F: list_formula_a_b > list_formula_a_b,X: list_formula_a_b,A: set_list_formula_a_b] :
( ( B2
= ( F @ X ) )
=> ( ( member7703986999711672120la_a_b @ X @ A )
=> ( member7703986999711672120la_a_b @ B2 @ ( image_2463093012343946057la_a_b @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_965_image__eqI,axiom,
! [B2: set_formula_a_b,F: list_formula_a_b > set_formula_a_b,X: list_formula_a_b,A: set_list_formula_a_b] :
( ( B2
= ( F @ X ) )
=> ( ( member7703986999711672120la_a_b @ X @ A )
=> ( member6621951321062823262la_a_b @ B2 @ ( image_4386591827427811823la_a_b @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_966_image__eqI,axiom,
! [B2: formula_a_b,F: set_formula_a_b > formula_a_b,X: set_formula_a_b,A: set_set_formula_a_b] :
( ( B2
= ( F @ X ) )
=> ( ( member6621951321062823262la_a_b @ X @ A )
=> ( member_formula_a_b2 @ B2 @ ( image_8609360178931715423la_a_b @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_967_image__eqI,axiom,
! [B2: list_formula_a_b,F: set_formula_a_b > list_formula_a_b,X: set_formula_a_b,A: set_set_formula_a_b] :
( ( B2
= ( F @ X ) )
=> ( ( member6621951321062823262la_a_b @ X @ A )
=> ( member7703986999711672120la_a_b @ B2 @ ( image_4263524660313489391la_a_b @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_968_image__eqI,axiom,
! [B2: set_formula_a_b,F: set_formula_a_b > set_formula_a_b,X: set_formula_a_b,A: set_set_formula_a_b] :
( ( B2
= ( F @ X ) )
=> ( ( member6621951321062823262la_a_b @ X @ A )
=> ( member6621951321062823262la_a_b @ B2 @ ( image_2155112572986939797la_a_b @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_969_image__eqI,axiom,
! [B2: formula_a_b,F: formula_a_b > formula_a_b,X: formula_a_b,A: set_formula_a_b] :
( ( B2
= ( F @ X ) )
=> ( ( member_formula_a_b2 @ X @ A )
=> ( member_formula_a_b2 @ B2 @ ( image_4658057545706456873la_a_b @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_970_image__is__empty,axiom,
! [F: formula_a_b > formula_a_b,A: set_formula_a_b] :
( ( ( image_4658057545706456873la_a_b @ F @ A )
= bot_bo8568539672012708443la_a_b )
= ( A = bot_bo8568539672012708443la_a_b ) ) ).
% image_is_empty
thf(fact_971_image__is__empty,axiom,
! [F: set_formula_a_b > formula_a_b,A: set_set_formula_a_b] :
( ( ( image_8609360178931715423la_a_b @ F @ A )
= bot_bo8568539672012708443la_a_b )
= ( A = bot_bo5296870006564677521la_a_b ) ) ).
% image_is_empty
thf(fact_972_image__is__empty,axiom,
! [F: list_formula_a_b > set_formula_a_b,A: set_list_formula_a_b] :
( ( ( image_4386591827427811823la_a_b @ F @ A )
= bot_bo5296870006564677521la_a_b )
= ( A = bot_bo4614192416432564331la_a_b ) ) ).
% image_is_empty
thf(fact_973_image__is__empty,axiom,
! [F: formula_a_b > set_formula_a_b,A: set_formula_a_b] :
( ( ( image_3063539048272112479la_a_b @ F @ A )
= bot_bo5296870006564677521la_a_b )
= ( A = bot_bo8568539672012708443la_a_b ) ) ).
% image_is_empty
thf(fact_974_image__is__empty,axiom,
! [F: set_formula_a_b > set_formula_a_b,A: set_set_formula_a_b] :
( ( ( image_2155112572986939797la_a_b @ F @ A )
= bot_bo5296870006564677521la_a_b )
= ( A = bot_bo5296870006564677521la_a_b ) ) ).
% image_is_empty
thf(fact_975_empty__is__image,axiom,
! [F: formula_a_b > formula_a_b,A: set_formula_a_b] :
( ( bot_bo8568539672012708443la_a_b
= ( image_4658057545706456873la_a_b @ F @ A ) )
= ( A = bot_bo8568539672012708443la_a_b ) ) ).
% empty_is_image
thf(fact_976_empty__is__image,axiom,
! [F: set_formula_a_b > formula_a_b,A: set_set_formula_a_b] :
( ( bot_bo8568539672012708443la_a_b
= ( image_8609360178931715423la_a_b @ F @ A ) )
= ( A = bot_bo5296870006564677521la_a_b ) ) ).
% empty_is_image
thf(fact_977_empty__is__image,axiom,
! [F: list_formula_a_b > set_formula_a_b,A: set_list_formula_a_b] :
( ( bot_bo5296870006564677521la_a_b
= ( image_4386591827427811823la_a_b @ F @ A ) )
= ( A = bot_bo4614192416432564331la_a_b ) ) ).
% empty_is_image
thf(fact_978_empty__is__image,axiom,
! [F: formula_a_b > set_formula_a_b,A: set_formula_a_b] :
( ( bot_bo5296870006564677521la_a_b
= ( image_3063539048272112479la_a_b @ F @ A ) )
= ( A = bot_bo8568539672012708443la_a_b ) ) ).
% empty_is_image
thf(fact_979_empty__is__image,axiom,
! [F: set_formula_a_b > set_formula_a_b,A: set_set_formula_a_b] :
( ( bot_bo5296870006564677521la_a_b
= ( image_2155112572986939797la_a_b @ F @ A ) )
= ( A = bot_bo5296870006564677521la_a_b ) ) ).
% empty_is_image
thf(fact_980_image__empty,axiom,
! [F: list_formula_a_b > set_formula_a_b] :
( ( image_4386591827427811823la_a_b @ F @ bot_bo4614192416432564331la_a_b )
= bot_bo5296870006564677521la_a_b ) ).
% image_empty
thf(fact_981_image__empty,axiom,
! [F: formula_a_b > formula_a_b] :
( ( image_4658057545706456873la_a_b @ F @ bot_bo8568539672012708443la_a_b )
= bot_bo8568539672012708443la_a_b ) ).
% image_empty
thf(fact_982_image__empty,axiom,
! [F: formula_a_b > set_formula_a_b] :
( ( image_3063539048272112479la_a_b @ F @ bot_bo8568539672012708443la_a_b )
= bot_bo5296870006564677521la_a_b ) ).
% image_empty
thf(fact_983_image__empty,axiom,
! [F: set_formula_a_b > formula_a_b] :
( ( image_8609360178931715423la_a_b @ F @ bot_bo5296870006564677521la_a_b )
= bot_bo8568539672012708443la_a_b ) ).
% image_empty
thf(fact_984_image__empty,axiom,
! [F: set_formula_a_b > set_formula_a_b] :
( ( image_2155112572986939797la_a_b @ F @ bot_bo5296870006564677521la_a_b )
= bot_bo5296870006564677521la_a_b ) ).
% image_empty
thf(fact_985_insert__image,axiom,
! [X: formula_a_b,A: set_formula_a_b,F: formula_a_b > formula_a_b] :
( ( member_formula_a_b2 @ X @ A )
=> ( ( insert_formula_a_b2 @ ( F @ X ) @ ( image_4658057545706456873la_a_b @ F @ A ) )
= ( image_4658057545706456873la_a_b @ F @ A ) ) ) ).
% insert_image
thf(fact_986_insert__image,axiom,
! [X: formula_a_b,A: set_formula_a_b,F: formula_a_b > set_formula_a_b] :
( ( member_formula_a_b2 @ X @ A )
=> ( ( insert8952366156270154231la_a_b @ ( F @ X ) @ ( image_3063539048272112479la_a_b @ F @ A ) )
= ( image_3063539048272112479la_a_b @ F @ A ) ) ) ).
% insert_image
thf(fact_987_insert__image,axiom,
! [X: list_formula_a_b,A: set_list_formula_a_b,F: list_formula_a_b > formula_a_b] :
( ( member7703986999711672120la_a_b @ X @ A )
=> ( ( insert_formula_a_b2 @ ( F @ X ) @ ( image_5057755198331993529la_a_b @ F @ A ) )
= ( image_5057755198331993529la_a_b @ F @ A ) ) ) ).
% insert_image
thf(fact_988_insert__image,axiom,
! [X: list_formula_a_b,A: set_list_formula_a_b,F: list_formula_a_b > set_formula_a_b] :
( ( member7703986999711672120la_a_b @ X @ A )
=> ( ( insert8952366156270154231la_a_b @ ( F @ X ) @ ( image_4386591827427811823la_a_b @ F @ A ) )
= ( image_4386591827427811823la_a_b @ F @ A ) ) ) ).
% insert_image
thf(fact_989_insert__image,axiom,
! [X: set_formula_a_b,A: set_set_formula_a_b,F: set_formula_a_b > formula_a_b] :
( ( member6621951321062823262la_a_b @ X @ A )
=> ( ( insert_formula_a_b2 @ ( F @ X ) @ ( image_8609360178931715423la_a_b @ F @ A ) )
= ( image_8609360178931715423la_a_b @ F @ A ) ) ) ).
% insert_image
thf(fact_990_insert__image,axiom,
! [X: set_formula_a_b,A: set_set_formula_a_b,F: set_formula_a_b > set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X @ A )
=> ( ( insert8952366156270154231la_a_b @ ( F @ X ) @ ( image_2155112572986939797la_a_b @ F @ A ) )
= ( image_2155112572986939797la_a_b @ F @ A ) ) ) ).
% insert_image
thf(fact_991_image__insert,axiom,
! [F: list_formula_a_b > set_formula_a_b,A2: list_formula_a_b,B: set_list_formula_a_b] :
( ( image_4386591827427811823la_a_b @ F @ ( insert4995524290302003025la_a_b @ A2 @ B ) )
= ( insert8952366156270154231la_a_b @ ( F @ A2 ) @ ( image_4386591827427811823la_a_b @ F @ B ) ) ) ).
% image_insert
thf(fact_992_image__insert,axiom,
! [F: formula_a_b > formula_a_b,A2: formula_a_b,B: set_formula_a_b] :
( ( image_4658057545706456873la_a_b @ F @ ( insert_formula_a_b2 @ A2 @ B ) )
= ( insert_formula_a_b2 @ ( F @ A2 ) @ ( image_4658057545706456873la_a_b @ F @ B ) ) ) ).
% image_insert
thf(fact_993_image__insert,axiom,
! [F: formula_a_b > set_formula_a_b,A2: formula_a_b,B: set_formula_a_b] :
( ( image_3063539048272112479la_a_b @ F @ ( insert_formula_a_b2 @ A2 @ B ) )
= ( insert8952366156270154231la_a_b @ ( F @ A2 ) @ ( image_3063539048272112479la_a_b @ F @ B ) ) ) ).
% image_insert
thf(fact_994_image__insert,axiom,
! [F: set_formula_a_b > formula_a_b,A2: set_formula_a_b,B: set_set_formula_a_b] :
( ( image_8609360178931715423la_a_b @ F @ ( insert8952366156270154231la_a_b @ A2 @ B ) )
= ( insert_formula_a_b2 @ ( F @ A2 ) @ ( image_8609360178931715423la_a_b @ F @ B ) ) ) ).
% image_insert
thf(fact_995_image__insert,axiom,
! [F: set_formula_a_b > set_formula_a_b,A2: set_formula_a_b,B: set_set_formula_a_b] :
( ( image_2155112572986939797la_a_b @ F @ ( insert8952366156270154231la_a_b @ A2 @ B ) )
= ( insert8952366156270154231la_a_b @ ( F @ A2 ) @ ( image_2155112572986939797la_a_b @ F @ B ) ) ) ).
% image_insert
thf(fact_996_Cons__in__lists__iff,axiom,
! [X: list_formula_a_b,Xs: list_l5948067055324603441la_a_b,A: set_list_formula_a_b] :
( ( member5770512961814848200la_a_b @ ( cons_l2450299273141159147la_a_b @ X @ Xs ) @ ( lists_1922735192288930349la_a_b @ A ) )
= ( ( member7703986999711672120la_a_b @ X @ A )
& ( member5770512961814848200la_a_b @ Xs @ ( lists_1922735192288930349la_a_b @ A ) ) ) ) ).
% Cons_in_lists_iff
thf(fact_997_Cons__in__lists__iff,axiom,
! [X: set_formula_a_b,Xs: list_set_formula_a_b,A: set_set_formula_a_b] :
( ( member289699325402759278la_a_b @ ( cons_set_formula_a_b @ X @ Xs ) @ ( lists_4880915010259632851la_a_b @ A ) )
= ( ( member6621951321062823262la_a_b @ X @ A )
& ( member289699325402759278la_a_b @ Xs @ ( lists_4880915010259632851la_a_b @ A ) ) ) ) ).
% Cons_in_lists_iff
thf(fact_998_Cons__in__lists__iff,axiom,
! [X: formula_a_b,Xs: list_formula_a_b,A: set_formula_a_b] :
( ( member7703986999711672120la_a_b @ ( cons_formula_a_b @ X @ Xs ) @ ( lists_formula_a_b @ A ) )
= ( ( member_formula_a_b2 @ X @ A )
& ( member7703986999711672120la_a_b @ Xs @ ( lists_formula_a_b @ A ) ) ) ) ).
% Cons_in_lists_iff
thf(fact_999_in__listsI,axiom,
! [Xs: list_set_formula_a_b,A: set_set_formula_a_b] :
( ! [X3: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X3 @ ( set_set_formula_a_b2 @ Xs ) )
=> ( member6621951321062823262la_a_b @ X3 @ A ) )
=> ( member289699325402759278la_a_b @ Xs @ ( lists_4880915010259632851la_a_b @ A ) ) ) ).
% in_listsI
thf(fact_1000_in__listsI,axiom,
! [Xs: list_l5948067055324603441la_a_b,A: set_list_formula_a_b] :
( ! [X3: list_formula_a_b] :
( ( member7703986999711672120la_a_b @ X3 @ ( set_list_formula_a_b2 @ Xs ) )
=> ( member7703986999711672120la_a_b @ X3 @ A ) )
=> ( member5770512961814848200la_a_b @ Xs @ ( lists_1922735192288930349la_a_b @ A ) ) ) ).
% in_listsI
thf(fact_1001_in__listsI,axiom,
! [Xs: list_formula_a_b,A: set_formula_a_b] :
( ! [X3: formula_a_b] :
( ( member_formula_a_b2 @ X3 @ ( set_formula_a_b2 @ Xs ) )
=> ( member_formula_a_b2 @ X3 @ A ) )
=> ( member7703986999711672120la_a_b @ Xs @ ( lists_formula_a_b @ A ) ) ) ).
% in_listsI
thf(fact_1002_lists__Int__eq,axiom,
! [A: set_formula_a_b,B: set_formula_a_b] :
( ( lists_formula_a_b @ ( inf_in5741596252272946265la_a_b @ A @ B ) )
= ( inf_in7172170795784610665la_a_b @ ( lists_formula_a_b @ A ) @ ( lists_formula_a_b @ B ) ) ) ).
% lists_Int_eq
thf(fact_1003_lists__Int__eq,axiom,
! [A: set_set_formula_a_b,B: set_set_formula_a_b] :
( ( lists_4880915010259632851la_a_b @ ( inf_in60795025369792911la_a_b @ A @ B ) )
= ( inf_in3724091778011523487la_a_b @ ( lists_4880915010259632851la_a_b @ A ) @ ( lists_4880915010259632851la_a_b @ B ) ) ) ).
% lists_Int_eq
thf(fact_1004_append__in__lists__conv,axiom,
! [Xs: list_formula_a_b,Ys: list_formula_a_b,A: set_formula_a_b] :
( ( member7703986999711672120la_a_b @ ( append_formula_a_b @ Xs @ Ys ) @ ( lists_formula_a_b @ A ) )
= ( ( member7703986999711672120la_a_b @ Xs @ ( lists_formula_a_b @ A ) )
& ( member7703986999711672120la_a_b @ Ys @ ( lists_formula_a_b @ A ) ) ) ) ).
% append_in_lists_conv
thf(fact_1005_lists_OCons,axiom,
! [A2: list_formula_a_b,A: set_list_formula_a_b,L: list_l5948067055324603441la_a_b] :
( ( member7703986999711672120la_a_b @ A2 @ A )
=> ( ( member5770512961814848200la_a_b @ L @ ( lists_1922735192288930349la_a_b @ A ) )
=> ( member5770512961814848200la_a_b @ ( cons_l2450299273141159147la_a_b @ A2 @ L ) @ ( lists_1922735192288930349la_a_b @ A ) ) ) ) ).
% lists.Cons
thf(fact_1006_lists_OCons,axiom,
! [A2: set_formula_a_b,A: set_set_formula_a_b,L: list_set_formula_a_b] :
( ( member6621951321062823262la_a_b @ A2 @ A )
=> ( ( member289699325402759278la_a_b @ L @ ( lists_4880915010259632851la_a_b @ A ) )
=> ( member289699325402759278la_a_b @ ( cons_set_formula_a_b @ A2 @ L ) @ ( lists_4880915010259632851la_a_b @ A ) ) ) ) ).
% lists.Cons
thf(fact_1007_lists_OCons,axiom,
! [A2: formula_a_b,A: set_formula_a_b,L: list_formula_a_b] :
( ( member_formula_a_b2 @ A2 @ A )
=> ( ( member7703986999711672120la_a_b @ L @ ( lists_formula_a_b @ A ) )
=> ( member7703986999711672120la_a_b @ ( cons_formula_a_b @ A2 @ L ) @ ( lists_formula_a_b @ A ) ) ) ) ).
% lists.Cons
thf(fact_1008_listsE,axiom,
! [X: list_formula_a_b,L: list_l5948067055324603441la_a_b,A: set_list_formula_a_b] :
( ( member5770512961814848200la_a_b @ ( cons_l2450299273141159147la_a_b @ X @ L ) @ ( lists_1922735192288930349la_a_b @ A ) )
=> ~ ( ( member7703986999711672120la_a_b @ X @ A )
=> ~ ( member5770512961814848200la_a_b @ L @ ( lists_1922735192288930349la_a_b @ A ) ) ) ) ).
% listsE
thf(fact_1009_listsE,axiom,
! [X: set_formula_a_b,L: list_set_formula_a_b,A: set_set_formula_a_b] :
( ( member289699325402759278la_a_b @ ( cons_set_formula_a_b @ X @ L ) @ ( lists_4880915010259632851la_a_b @ A ) )
=> ~ ( ( member6621951321062823262la_a_b @ X @ A )
=> ~ ( member289699325402759278la_a_b @ L @ ( lists_4880915010259632851la_a_b @ A ) ) ) ) ).
% listsE
thf(fact_1010_listsE,axiom,
! [X: formula_a_b,L: list_formula_a_b,A: set_formula_a_b] :
( ( member7703986999711672120la_a_b @ ( cons_formula_a_b @ X @ L ) @ ( lists_formula_a_b @ A ) )
=> ~ ( ( member_formula_a_b2 @ X @ A )
=> ~ ( member7703986999711672120la_a_b @ L @ ( lists_formula_a_b @ A ) ) ) ) ).
% listsE
thf(fact_1011_lists_ONil,axiom,
! [A: set_formula_a_b] : ( member7703986999711672120la_a_b @ nil_formula_a_b @ ( lists_formula_a_b @ A ) ) ).
% lists.Nil
thf(fact_1012_in__listsD,axiom,
! [Xs: list_set_formula_a_b,A: set_set_formula_a_b] :
( ( member289699325402759278la_a_b @ Xs @ ( lists_4880915010259632851la_a_b @ A ) )
=> ! [X4: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X4 @ ( set_set_formula_a_b2 @ Xs ) )
=> ( member6621951321062823262la_a_b @ X4 @ A ) ) ) ).
% in_listsD
thf(fact_1013_in__listsD,axiom,
! [Xs: list_l5948067055324603441la_a_b,A: set_list_formula_a_b] :
( ( member5770512961814848200la_a_b @ Xs @ ( lists_1922735192288930349la_a_b @ A ) )
=> ! [X4: list_formula_a_b] :
( ( member7703986999711672120la_a_b @ X4 @ ( set_list_formula_a_b2 @ Xs ) )
=> ( member7703986999711672120la_a_b @ X4 @ A ) ) ) ).
% in_listsD
thf(fact_1014_in__listsD,axiom,
! [Xs: list_formula_a_b,A: set_formula_a_b] :
( ( member7703986999711672120la_a_b @ Xs @ ( lists_formula_a_b @ A ) )
=> ! [X4: formula_a_b] :
( ( member_formula_a_b2 @ X4 @ ( set_formula_a_b2 @ Xs ) )
=> ( member_formula_a_b2 @ X4 @ A ) ) ) ).
% in_listsD
thf(fact_1015_in__lists__conv__set,axiom,
! [Xs: list_set_formula_a_b,A: set_set_formula_a_b] :
( ( member289699325402759278la_a_b @ Xs @ ( lists_4880915010259632851la_a_b @ A ) )
= ( ! [X2: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X2 @ ( set_set_formula_a_b2 @ Xs ) )
=> ( member6621951321062823262la_a_b @ X2 @ A ) ) ) ) ).
% in_lists_conv_set
thf(fact_1016_in__lists__conv__set,axiom,
! [Xs: list_l5948067055324603441la_a_b,A: set_list_formula_a_b] :
( ( member5770512961814848200la_a_b @ Xs @ ( lists_1922735192288930349la_a_b @ A ) )
= ( ! [X2: list_formula_a_b] :
( ( member7703986999711672120la_a_b @ X2 @ ( set_list_formula_a_b2 @ Xs ) )
=> ( member7703986999711672120la_a_b @ X2 @ A ) ) ) ) ).
% in_lists_conv_set
thf(fact_1017_in__lists__conv__set,axiom,
! [Xs: list_formula_a_b,A: set_formula_a_b] :
( ( member7703986999711672120la_a_b @ Xs @ ( lists_formula_a_b @ A ) )
= ( ! [X2: formula_a_b] :
( ( member_formula_a_b2 @ X2 @ ( set_formula_a_b2 @ Xs ) )
=> ( member_formula_a_b2 @ X2 @ A ) ) ) ) ).
% in_lists_conv_set
thf(fact_1018_lists__IntI,axiom,
! [L: list_formula_a_b,A: set_formula_a_b,B: set_formula_a_b] :
( ( member7703986999711672120la_a_b @ L @ ( lists_formula_a_b @ A ) )
=> ( ( member7703986999711672120la_a_b @ L @ ( lists_formula_a_b @ B ) )
=> ( member7703986999711672120la_a_b @ L @ ( lists_formula_a_b @ ( inf_in5741596252272946265la_a_b @ A @ B ) ) ) ) ) ).
% lists_IntI
thf(fact_1019_lists__IntI,axiom,
! [L: list_set_formula_a_b,A: set_set_formula_a_b,B: set_set_formula_a_b] :
( ( member289699325402759278la_a_b @ L @ ( lists_4880915010259632851la_a_b @ A ) )
=> ( ( member289699325402759278la_a_b @ L @ ( lists_4880915010259632851la_a_b @ B ) )
=> ( member289699325402759278la_a_b @ L @ ( lists_4880915010259632851la_a_b @ ( inf_in60795025369792911la_a_b @ A @ B ) ) ) ) ) ).
% lists_IntI
thf(fact_1020_rev__image__eqI,axiom,
! [X: formula_a_b,A: set_formula_a_b,B2: list_formula_a_b,F: formula_a_b > list_formula_a_b] :
( ( member_formula_a_b2 @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member7703986999711672120la_a_b @ B2 @ ( image_7669030115802156217la_a_b @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_1021_rev__image__eqI,axiom,
! [X: formula_a_b,A: set_formula_a_b,B2: set_formula_a_b,F: formula_a_b > set_formula_a_b] :
( ( member_formula_a_b2 @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member6621951321062823262la_a_b @ B2 @ ( image_3063539048272112479la_a_b @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_1022_rev__image__eqI,axiom,
! [X: list_formula_a_b,A: set_list_formula_a_b,B2: formula_a_b,F: list_formula_a_b > formula_a_b] :
( ( member7703986999711672120la_a_b @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member_formula_a_b2 @ B2 @ ( image_5057755198331993529la_a_b @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_1023_rev__image__eqI,axiom,
! [X: list_formula_a_b,A: set_list_formula_a_b,B2: list_formula_a_b,F: list_formula_a_b > list_formula_a_b] :
( ( member7703986999711672120la_a_b @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member7703986999711672120la_a_b @ B2 @ ( image_2463093012343946057la_a_b @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_1024_rev__image__eqI,axiom,
! [X: list_formula_a_b,A: set_list_formula_a_b,B2: set_formula_a_b,F: list_formula_a_b > set_formula_a_b] :
( ( member7703986999711672120la_a_b @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member6621951321062823262la_a_b @ B2 @ ( image_4386591827427811823la_a_b @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_1025_rev__image__eqI,axiom,
! [X: set_formula_a_b,A: set_set_formula_a_b,B2: formula_a_b,F: set_formula_a_b > formula_a_b] :
( ( member6621951321062823262la_a_b @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member_formula_a_b2 @ B2 @ ( image_8609360178931715423la_a_b @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_1026_rev__image__eqI,axiom,
! [X: set_formula_a_b,A: set_set_formula_a_b,B2: list_formula_a_b,F: set_formula_a_b > list_formula_a_b] :
( ( member6621951321062823262la_a_b @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member7703986999711672120la_a_b @ B2 @ ( image_4263524660313489391la_a_b @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_1027_rev__image__eqI,axiom,
! [X: set_formula_a_b,A: set_set_formula_a_b,B2: set_formula_a_b,F: set_formula_a_b > set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member6621951321062823262la_a_b @ B2 @ ( image_2155112572986939797la_a_b @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_1028_rev__image__eqI,axiom,
! [X: formula_a_b,A: set_formula_a_b,B2: formula_a_b,F: formula_a_b > formula_a_b] :
( ( member_formula_a_b2 @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member_formula_a_b2 @ B2 @ ( image_4658057545706456873la_a_b @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_1029_ball__imageD,axiom,
! [F: formula_a_b > formula_a_b,A: set_formula_a_b,P: formula_a_b > $o] :
( ! [X3: formula_a_b] :
( ( member_formula_a_b2 @ X3 @ ( image_4658057545706456873la_a_b @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X4: formula_a_b] :
( ( member_formula_a_b2 @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_1030_ball__imageD,axiom,
! [F: set_formula_a_b > set_formula_a_b,A: set_set_formula_a_b,P: set_formula_a_b > $o] :
( ! [X3: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X3 @ ( image_2155112572986939797la_a_b @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X4: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_1031_ball__imageD,axiom,
! [F: formula_a_b > set_formula_a_b,A: set_formula_a_b,P: set_formula_a_b > $o] :
( ! [X3: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X3 @ ( image_3063539048272112479la_a_b @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X4: formula_a_b] :
( ( member_formula_a_b2 @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_1032_ball__imageD,axiom,
! [F: list_formula_a_b > set_formula_a_b,A: set_list_formula_a_b,P: set_formula_a_b > $o] :
( ! [X3: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X3 @ ( image_4386591827427811823la_a_b @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X4: list_formula_a_b] :
( ( member7703986999711672120la_a_b @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_1033_image__cong,axiom,
! [M: set_formula_a_b,N: set_formula_a_b,F: formula_a_b > formula_a_b,G: formula_a_b > formula_a_b] :
( ( M = N )
=> ( ! [X3: formula_a_b] :
( ( member_formula_a_b2 @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_4658057545706456873la_a_b @ F @ M )
= ( image_4658057545706456873la_a_b @ G @ N ) ) ) ) ).
% image_cong
thf(fact_1034_image__cong,axiom,
! [M: set_formula_a_b,N: set_formula_a_b,F: formula_a_b > set_formula_a_b,G: formula_a_b > set_formula_a_b] :
( ( M = N )
=> ( ! [X3: formula_a_b] :
( ( member_formula_a_b2 @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_3063539048272112479la_a_b @ F @ M )
= ( image_3063539048272112479la_a_b @ G @ N ) ) ) ) ).
% image_cong
thf(fact_1035_image__cong,axiom,
! [M: set_list_formula_a_b,N: set_list_formula_a_b,F: list_formula_a_b > set_formula_a_b,G: list_formula_a_b > set_formula_a_b] :
( ( M = N )
=> ( ! [X3: list_formula_a_b] :
( ( member7703986999711672120la_a_b @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_4386591827427811823la_a_b @ F @ M )
= ( image_4386591827427811823la_a_b @ G @ N ) ) ) ) ).
% image_cong
thf(fact_1036_image__cong,axiom,
! [M: set_set_formula_a_b,N: set_set_formula_a_b,F: set_formula_a_b > set_formula_a_b,G: set_formula_a_b > set_formula_a_b] :
( ( M = N )
=> ( ! [X3: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_2155112572986939797la_a_b @ F @ M )
= ( image_2155112572986939797la_a_b @ G @ N ) ) ) ) ).
% image_cong
thf(fact_1037_bex__imageD,axiom,
! [F: formula_a_b > formula_a_b,A: set_formula_a_b,P: formula_a_b > $o] :
( ? [X4: formula_a_b] :
( ( member_formula_a_b2 @ X4 @ ( image_4658057545706456873la_a_b @ F @ A ) )
& ( P @ X4 ) )
=> ? [X3: formula_a_b] :
( ( member_formula_a_b2 @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_1038_bex__imageD,axiom,
! [F: set_formula_a_b > set_formula_a_b,A: set_set_formula_a_b,P: set_formula_a_b > $o] :
( ? [X4: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X4 @ ( image_2155112572986939797la_a_b @ F @ A ) )
& ( P @ X4 ) )
=> ? [X3: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_1039_bex__imageD,axiom,
! [F: formula_a_b > set_formula_a_b,A: set_formula_a_b,P: set_formula_a_b > $o] :
( ? [X4: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X4 @ ( image_3063539048272112479la_a_b @ F @ A ) )
& ( P @ X4 ) )
=> ? [X3: formula_a_b] :
( ( member_formula_a_b2 @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_1040_bex__imageD,axiom,
! [F: list_formula_a_b > set_formula_a_b,A: set_list_formula_a_b,P: set_formula_a_b > $o] :
( ? [X4: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X4 @ ( image_4386591827427811823la_a_b @ F @ A ) )
& ( P @ X4 ) )
=> ? [X3: list_formula_a_b] :
( ( member7703986999711672120la_a_b @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_1041_image__iff,axiom,
! [Z: formula_a_b,F: formula_a_b > formula_a_b,A: set_formula_a_b] :
( ( member_formula_a_b2 @ Z @ ( image_4658057545706456873la_a_b @ F @ A ) )
= ( ? [X2: formula_a_b] :
( ( member_formula_a_b2 @ X2 @ A )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_1042_image__iff,axiom,
! [Z: set_formula_a_b,F: set_formula_a_b > set_formula_a_b,A: set_set_formula_a_b] :
( ( member6621951321062823262la_a_b @ Z @ ( image_2155112572986939797la_a_b @ F @ A ) )
= ( ? [X2: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X2 @ A )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_1043_image__iff,axiom,
! [Z: set_formula_a_b,F: formula_a_b > set_formula_a_b,A: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ Z @ ( image_3063539048272112479la_a_b @ F @ A ) )
= ( ? [X2: formula_a_b] :
( ( member_formula_a_b2 @ X2 @ A )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_1044_image__iff,axiom,
! [Z: set_formula_a_b,F: list_formula_a_b > set_formula_a_b,A: set_list_formula_a_b] :
( ( member6621951321062823262la_a_b @ Z @ ( image_4386591827427811823la_a_b @ F @ A ) )
= ( ? [X2: list_formula_a_b] :
( ( member7703986999711672120la_a_b @ X2 @ A )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_1045_imageI,axiom,
! [X: formula_a_b,A: set_formula_a_b,F: formula_a_b > list_formula_a_b] :
( ( member_formula_a_b2 @ X @ A )
=> ( member7703986999711672120la_a_b @ ( F @ X ) @ ( image_7669030115802156217la_a_b @ F @ A ) ) ) ).
% imageI
thf(fact_1046_imageI,axiom,
! [X: formula_a_b,A: set_formula_a_b,F: formula_a_b > set_formula_a_b] :
( ( member_formula_a_b2 @ X @ A )
=> ( member6621951321062823262la_a_b @ ( F @ X ) @ ( image_3063539048272112479la_a_b @ F @ A ) ) ) ).
% imageI
thf(fact_1047_imageI,axiom,
! [X: list_formula_a_b,A: set_list_formula_a_b,F: list_formula_a_b > formula_a_b] :
( ( member7703986999711672120la_a_b @ X @ A )
=> ( member_formula_a_b2 @ ( F @ X ) @ ( image_5057755198331993529la_a_b @ F @ A ) ) ) ).
% imageI
thf(fact_1048_imageI,axiom,
! [X: list_formula_a_b,A: set_list_formula_a_b,F: list_formula_a_b > list_formula_a_b] :
( ( member7703986999711672120la_a_b @ X @ A )
=> ( member7703986999711672120la_a_b @ ( F @ X ) @ ( image_2463093012343946057la_a_b @ F @ A ) ) ) ).
% imageI
thf(fact_1049_imageI,axiom,
! [X: list_formula_a_b,A: set_list_formula_a_b,F: list_formula_a_b > set_formula_a_b] :
( ( member7703986999711672120la_a_b @ X @ A )
=> ( member6621951321062823262la_a_b @ ( F @ X ) @ ( image_4386591827427811823la_a_b @ F @ A ) ) ) ).
% imageI
thf(fact_1050_imageI,axiom,
! [X: set_formula_a_b,A: set_set_formula_a_b,F: set_formula_a_b > formula_a_b] :
( ( member6621951321062823262la_a_b @ X @ A )
=> ( member_formula_a_b2 @ ( F @ X ) @ ( image_8609360178931715423la_a_b @ F @ A ) ) ) ).
% imageI
thf(fact_1051_imageI,axiom,
! [X: set_formula_a_b,A: set_set_formula_a_b,F: set_formula_a_b > list_formula_a_b] :
( ( member6621951321062823262la_a_b @ X @ A )
=> ( member7703986999711672120la_a_b @ ( F @ X ) @ ( image_4263524660313489391la_a_b @ F @ A ) ) ) ).
% imageI
thf(fact_1052_imageI,axiom,
! [X: set_formula_a_b,A: set_set_formula_a_b,F: set_formula_a_b > set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X @ A )
=> ( member6621951321062823262la_a_b @ ( F @ X ) @ ( image_2155112572986939797la_a_b @ F @ A ) ) ) ).
% imageI
thf(fact_1053_imageI,axiom,
! [X: formula_a_b,A: set_formula_a_b,F: formula_a_b > formula_a_b] :
( ( member_formula_a_b2 @ X @ A )
=> ( member_formula_a_b2 @ ( F @ X ) @ ( image_4658057545706456873la_a_b @ F @ A ) ) ) ).
% imageI
thf(fact_1054_image__Un,axiom,
! [F: list_formula_a_b > set_formula_a_b,A: set_list_formula_a_b,B: set_list_formula_a_b] :
( ( image_4386591827427811823la_a_b @ F @ ( sup_su9015223001214971651la_a_b @ A @ B ) )
= ( sup_su5931044959554304553la_a_b @ ( image_4386591827427811823la_a_b @ F @ A ) @ ( image_4386591827427811823la_a_b @ F @ B ) ) ) ).
% image_Un
thf(fact_1055_image__Un,axiom,
! [F: formula_a_b > set_formula_a_b,A: set_formula_a_b,B: set_formula_a_b] :
( ( image_3063539048272112479la_a_b @ F @ ( sup_su8832426789560438003la_a_b @ A @ B ) )
= ( sup_su5931044959554304553la_a_b @ ( image_3063539048272112479la_a_b @ F @ A ) @ ( image_3063539048272112479la_a_b @ F @ B ) ) ) ).
% image_Un
thf(fact_1056_image__Un,axiom,
! [F: set_formula_a_b > formula_a_b,A: set_set_formula_a_b,B: set_set_formula_a_b] :
( ( image_8609360178931715423la_a_b @ F @ ( sup_su5931044959554304553la_a_b @ A @ B ) )
= ( sup_su8832426789560438003la_a_b @ ( image_8609360178931715423la_a_b @ F @ A ) @ ( image_8609360178931715423la_a_b @ F @ B ) ) ) ).
% image_Un
thf(fact_1057_image__Un,axiom,
! [F: set_formula_a_b > set_formula_a_b,A: set_set_formula_a_b,B: set_set_formula_a_b] :
( ( image_2155112572986939797la_a_b @ F @ ( sup_su5931044959554304553la_a_b @ A @ B ) )
= ( sup_su5931044959554304553la_a_b @ ( image_2155112572986939797la_a_b @ F @ A ) @ ( image_2155112572986939797la_a_b @ F @ B ) ) ) ).
% image_Un
thf(fact_1058_image__Un,axiom,
! [F: formula_a_b > formula_a_b,A: set_formula_a_b,B: set_formula_a_b] :
( ( image_4658057545706456873la_a_b @ F @ ( sup_su8832426789560438003la_a_b @ A @ B ) )
= ( sup_su8832426789560438003la_a_b @ ( image_4658057545706456873la_a_b @ F @ A ) @ ( image_4658057545706456873la_a_b @ F @ B ) ) ) ).
% image_Un
thf(fact_1059_image__comp,axiom,
! [F: formula_a_b > formula_a_b,G: formula_a_b > formula_a_b,R: set_formula_a_b] :
( ( image_4658057545706456873la_a_b @ F @ ( image_4658057545706456873la_a_b @ G @ R ) )
= ( image_4658057545706456873la_a_b @ ( comp_f2828989746591664536la_a_b @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_1060_image__comp,axiom,
! [F: set_formula_a_b > formula_a_b,G: formula_a_b > set_formula_a_b,R: set_formula_a_b] :
( ( image_8609360178931715423la_a_b @ F @ ( image_3063539048272112479la_a_b @ G @ R ) )
= ( image_4658057545706456873la_a_b @ ( comp_s3563907018792990158la_a_b @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_1061_image__comp,axiom,
! [F: formula_a_b > set_formula_a_b,G: formula_a_b > formula_a_b,R: set_formula_a_b] :
( ( image_3063539048272112479la_a_b @ F @ ( image_4658057545706456873la_a_b @ G @ R ) )
= ( image_3063539048272112479la_a_b @ ( comp_f821314402523108302la_a_b @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_1062_image__comp,axiom,
! [F: set_formula_a_b > set_formula_a_b,G: formula_a_b > set_formula_a_b,R: set_formula_a_b] :
( ( image_2155112572986939797la_a_b @ F @ ( image_3063539048272112479la_a_b @ G @ R ) )
= ( image_3063539048272112479la_a_b @ ( comp_s9099485395005827076la_a_b @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_1063_image__comp,axiom,
! [F: formula_a_b > set_formula_a_b,G: set_formula_a_b > formula_a_b,R: set_set_formula_a_b] :
( ( image_3063539048272112479la_a_b @ F @ ( image_8609360178931715423la_a_b @ G @ R ) )
= ( image_2155112572986939797la_a_b @ ( comp_f2183751631756272644la_a_b @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_1064_image__comp,axiom,
! [F: formula_a_b > set_formula_a_b,G: list_formula_a_b > formula_a_b,R: set_list_formula_a_b] :
( ( image_3063539048272112479la_a_b @ F @ ( image_5057755198331993529la_a_b @ G @ R ) )
= ( image_4386591827427811823la_a_b @ ( comp_f7809688971862344414la_a_b @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_1065_image__comp,axiom,
! [F: list_formula_a_b > set_formula_a_b,G: formula_a_b > list_formula_a_b,R: set_formula_a_b] :
( ( image_4386591827427811823la_a_b @ F @ ( image_7669030115802156217la_a_b @ G @ R ) )
= ( image_3063539048272112479la_a_b @ ( comp_l1054984425785505758la_a_b @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_1066_image__comp,axiom,
! [F: set_formula_a_b > set_formula_a_b,G: set_formula_a_b > set_formula_a_b,R: set_set_formula_a_b] :
( ( image_2155112572986939797la_a_b @ F @ ( image_2155112572986939797la_a_b @ G @ R ) )
= ( image_2155112572986939797la_a_b @ ( comp_s7716128494983798842la_a_b @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_1067_image__comp,axiom,
! [F: set_formula_a_b > set_formula_a_b,G: list_formula_a_b > set_formula_a_b,R: set_list_formula_a_b] :
( ( image_2155112572986939797la_a_b @ F @ ( image_4386591827427811823la_a_b @ G @ R ) )
= ( image_4386591827427811823la_a_b @ ( comp_s1677114771899515924la_a_b @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_1068_image__comp,axiom,
! [F: list_formula_a_b > set_formula_a_b,G: set_formula_a_b > list_formula_a_b,R: set_set_formula_a_b] :
( ( image_4386591827427811823la_a_b @ F @ ( image_4263524660313489391la_a_b @ G @ R ) )
= ( image_2155112572986939797la_a_b @ ( comp_l4184407891557566484la_a_b @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_1069_image__eq__imp__comp,axiom,
! [F: formula_a_b > formula_a_b,A: set_formula_a_b,G: formula_a_b > formula_a_b,B: set_formula_a_b,H: formula_a_b > formula_a_b] :
( ( ( image_4658057545706456873la_a_b @ F @ A )
= ( image_4658057545706456873la_a_b @ G @ B ) )
=> ( ( image_4658057545706456873la_a_b @ ( comp_f2828989746591664536la_a_b @ H @ F ) @ A )
= ( image_4658057545706456873la_a_b @ ( comp_f2828989746591664536la_a_b @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_1070_image__eq__imp__comp,axiom,
! [F: formula_a_b > formula_a_b,A: set_formula_a_b,G: formula_a_b > formula_a_b,B: set_formula_a_b,H: formula_a_b > set_formula_a_b] :
( ( ( image_4658057545706456873la_a_b @ F @ A )
= ( image_4658057545706456873la_a_b @ G @ B ) )
=> ( ( image_3063539048272112479la_a_b @ ( comp_f821314402523108302la_a_b @ H @ F ) @ A )
= ( image_3063539048272112479la_a_b @ ( comp_f821314402523108302la_a_b @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_1071_image__eq__imp__comp,axiom,
! [F: formula_a_b > set_formula_a_b,A: set_formula_a_b,G: formula_a_b > set_formula_a_b,B: set_formula_a_b,H: set_formula_a_b > formula_a_b] :
( ( ( image_3063539048272112479la_a_b @ F @ A )
= ( image_3063539048272112479la_a_b @ G @ B ) )
=> ( ( image_4658057545706456873la_a_b @ ( comp_s3563907018792990158la_a_b @ H @ F ) @ A )
= ( image_4658057545706456873la_a_b @ ( comp_s3563907018792990158la_a_b @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_1072_image__eq__imp__comp,axiom,
! [F: set_formula_a_b > formula_a_b,A: set_set_formula_a_b,G: formula_a_b > formula_a_b,B: set_formula_a_b,H: formula_a_b > set_formula_a_b] :
( ( ( image_8609360178931715423la_a_b @ F @ A )
= ( image_4658057545706456873la_a_b @ G @ B ) )
=> ( ( image_2155112572986939797la_a_b @ ( comp_f2183751631756272644la_a_b @ H @ F ) @ A )
= ( image_3063539048272112479la_a_b @ ( comp_f821314402523108302la_a_b @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_1073_image__eq__imp__comp,axiom,
! [F: list_formula_a_b > formula_a_b,A: set_list_formula_a_b,G: formula_a_b > formula_a_b,B: set_formula_a_b,H: formula_a_b > set_formula_a_b] :
( ( ( image_5057755198331993529la_a_b @ F @ A )
= ( image_4658057545706456873la_a_b @ G @ B ) )
=> ( ( image_4386591827427811823la_a_b @ ( comp_f7809688971862344414la_a_b @ H @ F ) @ A )
= ( image_3063539048272112479la_a_b @ ( comp_f821314402523108302la_a_b @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_1074_image__eq__imp__comp,axiom,
! [F: formula_a_b > formula_a_b,A: set_formula_a_b,G: set_formula_a_b > formula_a_b,B: set_set_formula_a_b,H: formula_a_b > set_formula_a_b] :
( ( ( image_4658057545706456873la_a_b @ F @ A )
= ( image_8609360178931715423la_a_b @ G @ B ) )
=> ( ( image_3063539048272112479la_a_b @ ( comp_f821314402523108302la_a_b @ H @ F ) @ A )
= ( image_2155112572986939797la_a_b @ ( comp_f2183751631756272644la_a_b @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_1075_image__eq__imp__comp,axiom,
! [F: formula_a_b > formula_a_b,A: set_formula_a_b,G: list_formula_a_b > formula_a_b,B: set_list_formula_a_b,H: formula_a_b > set_formula_a_b] :
( ( ( image_4658057545706456873la_a_b @ F @ A )
= ( image_5057755198331993529la_a_b @ G @ B ) )
=> ( ( image_3063539048272112479la_a_b @ ( comp_f821314402523108302la_a_b @ H @ F ) @ A )
= ( image_4386591827427811823la_a_b @ ( comp_f7809688971862344414la_a_b @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_1076_image__eq__imp__comp,axiom,
! [F: set_formula_a_b > set_formula_a_b,A: set_set_formula_a_b,G: formula_a_b > set_formula_a_b,B: set_formula_a_b,H: set_formula_a_b > formula_a_b] :
( ( ( image_2155112572986939797la_a_b @ F @ A )
= ( image_3063539048272112479la_a_b @ G @ B ) )
=> ( ( image_8609360178931715423la_a_b @ ( comp_s3553664264346224132la_a_b @ H @ F ) @ A )
= ( image_4658057545706456873la_a_b @ ( comp_s3563907018792990158la_a_b @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_1077_image__eq__imp__comp,axiom,
! [F: formula_a_b > set_formula_a_b,A: set_formula_a_b,G: set_formula_a_b > set_formula_a_b,B: set_set_formula_a_b,H: set_formula_a_b > formula_a_b] :
( ( ( image_3063539048272112479la_a_b @ F @ A )
= ( image_2155112572986939797la_a_b @ G @ B ) )
=> ( ( image_4658057545706456873la_a_b @ ( comp_s3563907018792990158la_a_b @ H @ F ) @ A )
= ( image_8609360178931715423la_a_b @ ( comp_s3553664264346224132la_a_b @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_1078_image__eq__imp__comp,axiom,
! [F: formula_a_b > set_formula_a_b,A: set_formula_a_b,G: formula_a_b > set_formula_a_b,B: set_formula_a_b,H: set_formula_a_b > set_formula_a_b] :
( ( ( image_3063539048272112479la_a_b @ F @ A )
= ( image_3063539048272112479la_a_b @ G @ B ) )
=> ( ( image_3063539048272112479la_a_b @ ( comp_s9099485395005827076la_a_b @ H @ F ) @ A )
= ( image_3063539048272112479la_a_b @ ( comp_s9099485395005827076la_a_b @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_1079_shuffles_Osimps_I3_J,axiom,
! [X: formula_a_b,Xs: list_formula_a_b,Y: formula_a_b,Ys: list_formula_a_b] :
( ( shuffles_formula_a_b @ ( cons_formula_a_b @ X @ Xs ) @ ( cons_formula_a_b @ Y @ Ys ) )
= ( sup_su9015223001214971651la_a_b @ ( image_2463093012343946057la_a_b @ ( cons_formula_a_b @ X ) @ ( shuffles_formula_a_b @ Xs @ ( cons_formula_a_b @ Y @ Ys ) ) ) @ ( image_2463093012343946057la_a_b @ ( cons_formula_a_b @ Y ) @ ( shuffles_formula_a_b @ ( cons_formula_a_b @ X @ Xs ) @ Ys ) ) ) ) ).
% shuffles.simps(3)
thf(fact_1080_shuffles_Osimps_I3_J,axiom,
! [X: set_formula_a_b,Xs: list_set_formula_a_b,Y: set_formula_a_b,Ys: list_set_formula_a_b] :
( ( shuffl4965861759276814700la_a_b @ ( cons_set_formula_a_b @ X @ Xs ) @ ( cons_set_formula_a_b @ Y @ Ys ) )
= ( sup_su5324240277097303353la_a_b @ ( image_4866258802339552437la_a_b @ ( cons_set_formula_a_b @ X ) @ ( shuffl4965861759276814700la_a_b @ Xs @ ( cons_set_formula_a_b @ Y @ Ys ) ) ) @ ( image_4866258802339552437la_a_b @ ( cons_set_formula_a_b @ Y ) @ ( shuffl4965861759276814700la_a_b @ ( cons_set_formula_a_b @ X @ Xs ) @ Ys ) ) ) ) ).
% shuffles.simps(3)
thf(fact_1081_lists_Osimps,axiom,
! [A2: list_l5948067055324603441la_a_b,A: set_list_formula_a_b] :
( ( member5770512961814848200la_a_b @ A2 @ ( lists_1922735192288930349la_a_b @ A ) )
= ( ( A2 = nil_list_formula_a_b )
| ? [A3: list_formula_a_b,L2: list_l5948067055324603441la_a_b] :
( ( A2
= ( cons_l2450299273141159147la_a_b @ A3 @ L2 ) )
& ( member7703986999711672120la_a_b @ A3 @ A )
& ( member5770512961814848200la_a_b @ L2 @ ( lists_1922735192288930349la_a_b @ A ) ) ) ) ) ).
% lists.simps
thf(fact_1082_lists_Osimps,axiom,
! [A2: list_set_formula_a_b,A: set_set_formula_a_b] :
( ( member289699325402759278la_a_b @ A2 @ ( lists_4880915010259632851la_a_b @ A ) )
= ( ( A2 = nil_set_formula_a_b )
| ? [A3: set_formula_a_b,L2: list_set_formula_a_b] :
( ( A2
= ( cons_set_formula_a_b @ A3 @ L2 ) )
& ( member6621951321062823262la_a_b @ A3 @ A )
& ( member289699325402759278la_a_b @ L2 @ ( lists_4880915010259632851la_a_b @ A ) ) ) ) ) ).
% lists.simps
thf(fact_1083_lists_Osimps,axiom,
! [A2: list_formula_a_b,A: set_formula_a_b] :
( ( member7703986999711672120la_a_b @ A2 @ ( lists_formula_a_b @ A ) )
= ( ( A2 = nil_formula_a_b )
| ? [A3: formula_a_b,L2: list_formula_a_b] :
( ( A2
= ( cons_formula_a_b @ A3 @ L2 ) )
& ( member_formula_a_b2 @ A3 @ A )
& ( member7703986999711672120la_a_b @ L2 @ ( lists_formula_a_b @ A ) ) ) ) ) ).
% lists.simps
thf(fact_1084_lists_Ocases,axiom,
! [A2: list_l5948067055324603441la_a_b,A: set_list_formula_a_b] :
( ( member5770512961814848200la_a_b @ A2 @ ( lists_1922735192288930349la_a_b @ A ) )
=> ( ( A2 != nil_list_formula_a_b )
=> ~ ! [A5: list_formula_a_b,L3: list_l5948067055324603441la_a_b] :
( ( A2
= ( cons_l2450299273141159147la_a_b @ A5 @ L3 ) )
=> ( ( member7703986999711672120la_a_b @ A5 @ A )
=> ~ ( member5770512961814848200la_a_b @ L3 @ ( lists_1922735192288930349la_a_b @ A ) ) ) ) ) ) ).
% lists.cases
thf(fact_1085_lists_Ocases,axiom,
! [A2: list_set_formula_a_b,A: set_set_formula_a_b] :
( ( member289699325402759278la_a_b @ A2 @ ( lists_4880915010259632851la_a_b @ A ) )
=> ( ( A2 != nil_set_formula_a_b )
=> ~ ! [A5: set_formula_a_b,L3: list_set_formula_a_b] :
( ( A2
= ( cons_set_formula_a_b @ A5 @ L3 ) )
=> ( ( member6621951321062823262la_a_b @ A5 @ A )
=> ~ ( member289699325402759278la_a_b @ L3 @ ( lists_4880915010259632851la_a_b @ A ) ) ) ) ) ) ).
% lists.cases
thf(fact_1086_lists_Ocases,axiom,
! [A2: list_formula_a_b,A: set_formula_a_b] :
( ( member7703986999711672120la_a_b @ A2 @ ( lists_formula_a_b @ A ) )
=> ( ( A2 != nil_formula_a_b )
=> ~ ! [A5: formula_a_b,L3: list_formula_a_b] :
( ( A2
= ( cons_formula_a_b @ A5 @ L3 ) )
=> ( ( member_formula_a_b2 @ A5 @ A )
=> ~ ( member7703986999711672120la_a_b @ L3 @ ( lists_formula_a_b @ A ) ) ) ) ) ) ).
% lists.cases
thf(fact_1087_the__elem__image__unique,axiom,
! [A: set_list_formula_a_b,F: list_formula_a_b > set_formula_a_b,X: list_formula_a_b] :
( ( A != bot_bo4614192416432564331la_a_b )
=> ( ! [Y3: list_formula_a_b] :
( ( member7703986999711672120la_a_b @ Y3 @ A )
=> ( ( F @ Y3 )
= ( F @ X ) ) )
=> ( ( the_el8292460259403742098la_a_b @ ( image_4386591827427811823la_a_b @ F @ A ) )
= ( F @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_1088_the__elem__image__unique,axiom,
! [A: set_list_formula_a_b,F: list_formula_a_b > formula_a_b,X: list_formula_a_b] :
( ( A != bot_bo4614192416432564331la_a_b )
=> ( ! [Y3: list_formula_a_b] :
( ( member7703986999711672120la_a_b @ Y3 @ A )
=> ( ( F @ Y3 )
= ( F @ X ) ) )
=> ( ( the_elem_formula_a_b @ ( image_5057755198331993529la_a_b @ F @ A ) )
= ( F @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_1089_the__elem__image__unique,axiom,
! [A: set_formula_a_b,F: formula_a_b > set_formula_a_b,X: formula_a_b] :
( ( A != bot_bo8568539672012708443la_a_b )
=> ( ! [Y3: formula_a_b] :
( ( member_formula_a_b2 @ Y3 @ A )
=> ( ( F @ Y3 )
= ( F @ X ) ) )
=> ( ( the_el8292460259403742098la_a_b @ ( image_3063539048272112479la_a_b @ F @ A ) )
= ( F @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_1090_the__elem__image__unique,axiom,
! [A: set_formula_a_b,F: formula_a_b > formula_a_b,X: formula_a_b] :
( ( A != bot_bo8568539672012708443la_a_b )
=> ( ! [Y3: formula_a_b] :
( ( member_formula_a_b2 @ Y3 @ A )
=> ( ( F @ Y3 )
= ( F @ X ) ) )
=> ( ( the_elem_formula_a_b @ ( image_4658057545706456873la_a_b @ F @ A ) )
= ( F @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_1091_the__elem__image__unique,axiom,
! [A: set_set_formula_a_b,F: set_formula_a_b > set_formula_a_b,X: set_formula_a_b] :
( ( A != bot_bo5296870006564677521la_a_b )
=> ( ! [Y3: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ Y3 @ A )
=> ( ( F @ Y3 )
= ( F @ X ) ) )
=> ( ( the_el8292460259403742098la_a_b @ ( image_2155112572986939797la_a_b @ F @ A ) )
= ( F @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_1092_the__elem__image__unique,axiom,
! [A: set_set_formula_a_b,F: set_formula_a_b > formula_a_b,X: set_formula_a_b] :
( ( A != bot_bo5296870006564677521la_a_b )
=> ( ! [Y3: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ Y3 @ A )
=> ( ( F @ Y3 )
= ( F @ X ) ) )
=> ( ( the_elem_formula_a_b @ ( image_8609360178931715423la_a_b @ F @ A ) )
= ( F @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_1093_null__rec_I1_J,axiom,
! [X: formula_a_b,Xs: list_formula_a_b] :
~ ( null_formula_a_b @ ( cons_formula_a_b @ X @ Xs ) ) ).
% null_rec(1)
thf(fact_1094_null__rec_I1_J,axiom,
! [X: set_formula_a_b,Xs: list_set_formula_a_b] :
~ ( null_set_formula_a_b @ ( cons_set_formula_a_b @ X @ Xs ) ) ).
% null_rec(1)
thf(fact_1095_eq__Nil__null,axiom,
! [Xs: list_formula_a_b] :
( ( Xs = nil_formula_a_b )
= ( null_formula_a_b @ Xs ) ) ).
% eq_Nil_null
thf(fact_1096_null__rec_I2_J,axiom,
null_formula_a_b @ nil_formula_a_b ).
% null_rec(2)
thf(fact_1097_Sup_OSUP__image,axiom,
! [Sup: set_formula_a_b > formula_a_b,G: formula_a_b > formula_a_b,F: formula_a_b > formula_a_b,A: set_formula_a_b] :
( ( Sup @ ( image_4658057545706456873la_a_b @ G @ ( image_4658057545706456873la_a_b @ F @ A ) ) )
= ( Sup @ ( image_4658057545706456873la_a_b @ ( comp_f2828989746591664536la_a_b @ G @ F ) @ A ) ) ) ).
% Sup.SUP_image
thf(fact_1098_Sup_OSUP__image,axiom,
! [Sup: set_formula_a_b > formula_a_b,G: set_formula_a_b > formula_a_b,F: formula_a_b > set_formula_a_b,A: set_formula_a_b] :
( ( Sup @ ( image_8609360178931715423la_a_b @ G @ ( image_3063539048272112479la_a_b @ F @ A ) ) )
= ( Sup @ ( image_4658057545706456873la_a_b @ ( comp_s3563907018792990158la_a_b @ G @ F ) @ A ) ) ) ).
% Sup.SUP_image
thf(fact_1099_Sup_OSUP__image,axiom,
! [Sup: set_set_formula_a_b > set_formula_a_b,G: formula_a_b > set_formula_a_b,F: formula_a_b > formula_a_b,A: set_formula_a_b] :
( ( Sup @ ( image_3063539048272112479la_a_b @ G @ ( image_4658057545706456873la_a_b @ F @ A ) ) )
= ( Sup @ ( image_3063539048272112479la_a_b @ ( comp_f821314402523108302la_a_b @ G @ F ) @ A ) ) ) ).
% Sup.SUP_image
thf(fact_1100_Sup_OSUP__image,axiom,
! [Sup: set_set_formula_a_b > set_formula_a_b,G: set_formula_a_b > set_formula_a_b,F: formula_a_b > set_formula_a_b,A: set_formula_a_b] :
( ( Sup @ ( image_2155112572986939797la_a_b @ G @ ( image_3063539048272112479la_a_b @ F @ A ) ) )
= ( Sup @ ( image_3063539048272112479la_a_b @ ( comp_s9099485395005827076la_a_b @ G @ F ) @ A ) ) ) ).
% Sup.SUP_image
thf(fact_1101_Sup_OSUP__image,axiom,
! [Sup: set_set_formula_a_b > set_formula_a_b,G: formula_a_b > set_formula_a_b,F: set_formula_a_b > formula_a_b,A: set_set_formula_a_b] :
( ( Sup @ ( image_3063539048272112479la_a_b @ G @ ( image_8609360178931715423la_a_b @ F @ A ) ) )
= ( Sup @ ( image_2155112572986939797la_a_b @ ( comp_f2183751631756272644la_a_b @ G @ F ) @ A ) ) ) ).
% Sup.SUP_image
thf(fact_1102_Sup_OSUP__image,axiom,
! [Sup: set_set_formula_a_b > set_formula_a_b,G: formula_a_b > set_formula_a_b,F: list_formula_a_b > formula_a_b,A: set_list_formula_a_b] :
( ( Sup @ ( image_3063539048272112479la_a_b @ G @ ( image_5057755198331993529la_a_b @ F @ A ) ) )
= ( Sup @ ( image_4386591827427811823la_a_b @ ( comp_f7809688971862344414la_a_b @ G @ F ) @ A ) ) ) ).
% Sup.SUP_image
thf(fact_1103_Sup_OSUP__image,axiom,
! [Sup: set_set_formula_a_b > set_formula_a_b,G: list_formula_a_b > set_formula_a_b,F: formula_a_b > list_formula_a_b,A: set_formula_a_b] :
( ( Sup @ ( image_4386591827427811823la_a_b @ G @ ( image_7669030115802156217la_a_b @ F @ A ) ) )
= ( Sup @ ( image_3063539048272112479la_a_b @ ( comp_l1054984425785505758la_a_b @ G @ F ) @ A ) ) ) ).
% Sup.SUP_image
thf(fact_1104_Sup_OSUP__image,axiom,
! [Sup: set_set_formula_a_b > set_formula_a_b,G: set_formula_a_b > set_formula_a_b,F: set_formula_a_b > set_formula_a_b,A: set_set_formula_a_b] :
( ( Sup @ ( image_2155112572986939797la_a_b @ G @ ( image_2155112572986939797la_a_b @ F @ A ) ) )
= ( Sup @ ( image_2155112572986939797la_a_b @ ( comp_s7716128494983798842la_a_b @ G @ F ) @ A ) ) ) ).
% Sup.SUP_image
thf(fact_1105_Sup_OSUP__image,axiom,
! [Sup: set_set_formula_a_b > set_formula_a_b,G: set_formula_a_b > set_formula_a_b,F: list_formula_a_b > set_formula_a_b,A: set_list_formula_a_b] :
( ( Sup @ ( image_2155112572986939797la_a_b @ G @ ( image_4386591827427811823la_a_b @ F @ A ) ) )
= ( Sup @ ( image_4386591827427811823la_a_b @ ( comp_s1677114771899515924la_a_b @ G @ F ) @ A ) ) ) ).
% Sup.SUP_image
thf(fact_1106_Sup_OSUP__image,axiom,
! [Sup: set_set_formula_a_b > set_formula_a_b,G: list_formula_a_b > set_formula_a_b,F: set_formula_a_b > list_formula_a_b,A: set_set_formula_a_b] :
( ( Sup @ ( image_4386591827427811823la_a_b @ G @ ( image_4263524660313489391la_a_b @ F @ A ) ) )
= ( Sup @ ( image_2155112572986939797la_a_b @ ( comp_l4184407891557566484la_a_b @ G @ F ) @ A ) ) ) ).
% Sup.SUP_image
thf(fact_1107_Inf_OINF__image,axiom,
! [Inf: set_formula_a_b > formula_a_b,G: formula_a_b > formula_a_b,F: formula_a_b > formula_a_b,A: set_formula_a_b] :
( ( Inf @ ( image_4658057545706456873la_a_b @ G @ ( image_4658057545706456873la_a_b @ F @ A ) ) )
= ( Inf @ ( image_4658057545706456873la_a_b @ ( comp_f2828989746591664536la_a_b @ G @ F ) @ A ) ) ) ).
% Inf.INF_image
thf(fact_1108_Inf_OINF__image,axiom,
! [Inf: set_formula_a_b > formula_a_b,G: set_formula_a_b > formula_a_b,F: formula_a_b > set_formula_a_b,A: set_formula_a_b] :
( ( Inf @ ( image_8609360178931715423la_a_b @ G @ ( image_3063539048272112479la_a_b @ F @ A ) ) )
= ( Inf @ ( image_4658057545706456873la_a_b @ ( comp_s3563907018792990158la_a_b @ G @ F ) @ A ) ) ) ).
% Inf.INF_image
thf(fact_1109_Inf_OINF__image,axiom,
! [Inf: set_set_formula_a_b > set_formula_a_b,G: formula_a_b > set_formula_a_b,F: formula_a_b > formula_a_b,A: set_formula_a_b] :
( ( Inf @ ( image_3063539048272112479la_a_b @ G @ ( image_4658057545706456873la_a_b @ F @ A ) ) )
= ( Inf @ ( image_3063539048272112479la_a_b @ ( comp_f821314402523108302la_a_b @ G @ F ) @ A ) ) ) ).
% Inf.INF_image
thf(fact_1110_Inf_OINF__image,axiom,
! [Inf: set_set_formula_a_b > set_formula_a_b,G: set_formula_a_b > set_formula_a_b,F: formula_a_b > set_formula_a_b,A: set_formula_a_b] :
( ( Inf @ ( image_2155112572986939797la_a_b @ G @ ( image_3063539048272112479la_a_b @ F @ A ) ) )
= ( Inf @ ( image_3063539048272112479la_a_b @ ( comp_s9099485395005827076la_a_b @ G @ F ) @ A ) ) ) ).
% Inf.INF_image
thf(fact_1111_Inf_OINF__image,axiom,
! [Inf: set_set_formula_a_b > set_formula_a_b,G: formula_a_b > set_formula_a_b,F: set_formula_a_b > formula_a_b,A: set_set_formula_a_b] :
( ( Inf @ ( image_3063539048272112479la_a_b @ G @ ( image_8609360178931715423la_a_b @ F @ A ) ) )
= ( Inf @ ( image_2155112572986939797la_a_b @ ( comp_f2183751631756272644la_a_b @ G @ F ) @ A ) ) ) ).
% Inf.INF_image
thf(fact_1112_Inf_OINF__image,axiom,
! [Inf: set_set_formula_a_b > set_formula_a_b,G: formula_a_b > set_formula_a_b,F: list_formula_a_b > formula_a_b,A: set_list_formula_a_b] :
( ( Inf @ ( image_3063539048272112479la_a_b @ G @ ( image_5057755198331993529la_a_b @ F @ A ) ) )
= ( Inf @ ( image_4386591827427811823la_a_b @ ( comp_f7809688971862344414la_a_b @ G @ F ) @ A ) ) ) ).
% Inf.INF_image
thf(fact_1113_Inf_OINF__image,axiom,
! [Inf: set_set_formula_a_b > set_formula_a_b,G: list_formula_a_b > set_formula_a_b,F: formula_a_b > list_formula_a_b,A: set_formula_a_b] :
( ( Inf @ ( image_4386591827427811823la_a_b @ G @ ( image_7669030115802156217la_a_b @ F @ A ) ) )
= ( Inf @ ( image_3063539048272112479la_a_b @ ( comp_l1054984425785505758la_a_b @ G @ F ) @ A ) ) ) ).
% Inf.INF_image
thf(fact_1114_Inf_OINF__image,axiom,
! [Inf: set_set_formula_a_b > set_formula_a_b,G: set_formula_a_b > set_formula_a_b,F: set_formula_a_b > set_formula_a_b,A: set_set_formula_a_b] :
( ( Inf @ ( image_2155112572986939797la_a_b @ G @ ( image_2155112572986939797la_a_b @ F @ A ) ) )
= ( Inf @ ( image_2155112572986939797la_a_b @ ( comp_s7716128494983798842la_a_b @ G @ F ) @ A ) ) ) ).
% Inf.INF_image
thf(fact_1115_Inf_OINF__image,axiom,
! [Inf: set_set_formula_a_b > set_formula_a_b,G: set_formula_a_b > set_formula_a_b,F: list_formula_a_b > set_formula_a_b,A: set_list_formula_a_b] :
( ( Inf @ ( image_2155112572986939797la_a_b @ G @ ( image_4386591827427811823la_a_b @ F @ A ) ) )
= ( Inf @ ( image_4386591827427811823la_a_b @ ( comp_s1677114771899515924la_a_b @ G @ F ) @ A ) ) ) ).
% Inf.INF_image
thf(fact_1116_Inf_OINF__image,axiom,
! [Inf: set_set_formula_a_b > set_formula_a_b,G: list_formula_a_b > set_formula_a_b,F: set_formula_a_b > list_formula_a_b,A: set_set_formula_a_b] :
( ( Inf @ ( image_4386591827427811823la_a_b @ G @ ( image_4263524660313489391la_a_b @ F @ A ) ) )
= ( Inf @ ( image_2155112572986939797la_a_b @ ( comp_l4184407891557566484la_a_b @ G @ F ) @ A ) ) ) ).
% Inf.INF_image
thf(fact_1117_Sup__fin_Oinsert__remove,axiom,
! [A: set_se3797390812436207411la_a_b,X: set_set_formula_a_b] :
( ( finite6039610903330827156la_a_b @ A )
=> ( ( ( ( minus_1637344976281664780la_a_b @ A @ ( insert2824806981866770221la_a_b @ X @ bot_bo9046117466857082311la_a_b ) )
= bot_bo9046117466857082311la_a_b )
=> ( ( lattic7951861892593991814la_a_b @ ( insert2824806981866770221la_a_b @ X @ A ) )
= X ) )
& ( ( ( minus_1637344976281664780la_a_b @ A @ ( insert2824806981866770221la_a_b @ X @ bot_bo9046117466857082311la_a_b ) )
!= bot_bo9046117466857082311la_a_b )
=> ( ( lattic7951861892593991814la_a_b @ ( insert2824806981866770221la_a_b @ X @ A ) )
= ( sup_su5931044959554304553la_a_b @ X @ ( lattic7951861892593991814la_a_b @ ( minus_1637344976281664780la_a_b @ A @ ( insert2824806981866770221la_a_b @ X @ bot_bo9046117466857082311la_a_b ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_1118_Sup__fin_Oinsert__remove,axiom,
! [A: set_set_formula_a_b,X: set_formula_a_b] :
( ( finite7542661095822535262la_a_b @ A )
=> ( ( ( ( minus_4878662943024999126la_a_b @ A @ ( insert8952366156270154231la_a_b @ X @ bot_bo5296870006564677521la_a_b ) )
= bot_bo5296870006564677521la_a_b )
=> ( ( lattic2153292871565936464la_a_b @ ( insert8952366156270154231la_a_b @ X @ A ) )
= X ) )
& ( ( ( minus_4878662943024999126la_a_b @ A @ ( insert8952366156270154231la_a_b @ X @ bot_bo5296870006564677521la_a_b ) )
!= bot_bo5296870006564677521la_a_b )
=> ( ( lattic2153292871565936464la_a_b @ ( insert8952366156270154231la_a_b @ X @ A ) )
= ( sup_su8832426789560438003la_a_b @ X @ ( lattic2153292871565936464la_a_b @ ( minus_4878662943024999126la_a_b @ A @ ( insert8952366156270154231la_a_b @ X @ bot_bo5296870006564677521la_a_b ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_1119_Sup__fin_Oremove,axiom,
! [A: set_se3797390812436207411la_a_b,X: set_set_formula_a_b] :
( ( finite6039610903330827156la_a_b @ A )
=> ( ( member4282342919390090900la_a_b @ X @ A )
=> ( ( ( ( minus_1637344976281664780la_a_b @ A @ ( insert2824806981866770221la_a_b @ X @ bot_bo9046117466857082311la_a_b ) )
= bot_bo9046117466857082311la_a_b )
=> ( ( lattic7951861892593991814la_a_b @ A )
= X ) )
& ( ( ( minus_1637344976281664780la_a_b @ A @ ( insert2824806981866770221la_a_b @ X @ bot_bo9046117466857082311la_a_b ) )
!= bot_bo9046117466857082311la_a_b )
=> ( ( lattic7951861892593991814la_a_b @ A )
= ( sup_su5931044959554304553la_a_b @ X @ ( lattic7951861892593991814la_a_b @ ( minus_1637344976281664780la_a_b @ A @ ( insert2824806981866770221la_a_b @ X @ bot_bo9046117466857082311la_a_b ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_1120_Sup__fin_Oremove,axiom,
! [A: set_set_formula_a_b,X: set_formula_a_b] :
( ( finite7542661095822535262la_a_b @ A )
=> ( ( member6621951321062823262la_a_b @ X @ A )
=> ( ( ( ( minus_4878662943024999126la_a_b @ A @ ( insert8952366156270154231la_a_b @ X @ bot_bo5296870006564677521la_a_b ) )
= bot_bo5296870006564677521la_a_b )
=> ( ( lattic2153292871565936464la_a_b @ A )
= X ) )
& ( ( ( minus_4878662943024999126la_a_b @ A @ ( insert8952366156270154231la_a_b @ X @ bot_bo5296870006564677521la_a_b ) )
!= bot_bo5296870006564677521la_a_b )
=> ( ( lattic2153292871565936464la_a_b @ A )
= ( sup_su8832426789560438003la_a_b @ X @ ( lattic2153292871565936464la_a_b @ ( minus_4878662943024999126la_a_b @ A @ ( insert8952366156270154231la_a_b @ X @ bot_bo5296870006564677521la_a_b ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_1121_finite__shuffles,axiom,
! [Xs: list_formula_a_b,Ys: list_formula_a_b] : ( finite5308508654137801784la_a_b @ ( shuffles_formula_a_b @ Xs @ Ys ) ) ).
% finite_shuffles
thf(fact_1122_List_Ofinite__set,axiom,
! [Xs: list_l5948067055324603441la_a_b] : ( finite5308508654137801784la_a_b @ ( set_list_formula_a_b2 @ Xs ) ) ).
% List.finite_set
thf(fact_1123_List_Ofinite__set,axiom,
! [Xs: list_set_formula_a_b] : ( finite7542661095822535262la_a_b @ ( set_set_formula_a_b2 @ Xs ) ) ).
% List.finite_set
thf(fact_1124_List_Ofinite__set,axiom,
! [Xs: list_formula_a_b] : ( finite6884408037426709032la_a_b @ ( set_formula_a_b2 @ Xs ) ) ).
% List.finite_set
thf(fact_1125_inf__Sup__absorb,axiom,
! [A: set_se3797390812436207411la_a_b,A2: set_set_formula_a_b] :
( ( finite6039610903330827156la_a_b @ A )
=> ( ( member4282342919390090900la_a_b @ A2 @ A )
=> ( ( inf_in60795025369792911la_a_b @ A2 @ ( lattic7951861892593991814la_a_b @ A ) )
= A2 ) ) ) ).
% inf_Sup_absorb
thf(fact_1126_inf__Sup__absorb,axiom,
! [A: set_set_formula_a_b,A2: set_formula_a_b] :
( ( finite7542661095822535262la_a_b @ A )
=> ( ( member6621951321062823262la_a_b @ A2 @ A )
=> ( ( inf_in5741596252272946265la_a_b @ A2 @ ( lattic2153292871565936464la_a_b @ A ) )
= A2 ) ) ) ).
% inf_Sup_absorb
thf(fact_1127_Sup__fin_Oinsert,axiom,
! [A: set_se3797390812436207411la_a_b,X: set_set_formula_a_b] :
( ( finite6039610903330827156la_a_b @ A )
=> ( ( A != bot_bo9046117466857082311la_a_b )
=> ( ( lattic7951861892593991814la_a_b @ ( insert2824806981866770221la_a_b @ X @ A ) )
= ( sup_su5931044959554304553la_a_b @ X @ ( lattic7951861892593991814la_a_b @ A ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_1128_Sup__fin_Oinsert,axiom,
! [A: set_set_formula_a_b,X: set_formula_a_b] :
( ( finite7542661095822535262la_a_b @ A )
=> ( ( A != bot_bo5296870006564677521la_a_b )
=> ( ( lattic2153292871565936464la_a_b @ ( insert8952366156270154231la_a_b @ X @ A ) )
= ( sup_su8832426789560438003la_a_b @ X @ ( lattic2153292871565936464la_a_b @ A ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_1129_finite__list,axiom,
! [A: set_list_formula_a_b] :
( ( finite5308508654137801784la_a_b @ A )
=> ? [Xs3: list_l5948067055324603441la_a_b] :
( ( set_list_formula_a_b2 @ Xs3 )
= A ) ) ).
% finite_list
thf(fact_1130_finite__list,axiom,
! [A: set_set_formula_a_b] :
( ( finite7542661095822535262la_a_b @ A )
=> ? [Xs3: list_set_formula_a_b] :
( ( set_set_formula_a_b2 @ Xs3 )
= A ) ) ).
% finite_list
thf(fact_1131_finite__list,axiom,
! [A: set_formula_a_b] :
( ( finite6884408037426709032la_a_b @ A )
=> ? [Xs3: list_formula_a_b] :
( ( set_formula_a_b2 @ Xs3 )
= A ) ) ).
% finite_list
thf(fact_1132_finite__distinct__list,axiom,
! [A: set_list_formula_a_b] :
( ( finite5308508654137801784la_a_b @ A )
=> ? [Xs3: list_l5948067055324603441la_a_b] :
( ( ( set_list_formula_a_b2 @ Xs3 )
= A )
& ( distin27490427618170344la_a_b @ Xs3 ) ) ) ).
% finite_distinct_list
thf(fact_1133_finite__distinct__list,axiom,
! [A: set_set_formula_a_b] :
( ( finite7542661095822535262la_a_b @ A )
=> ? [Xs3: list_set_formula_a_b] :
( ( ( set_set_formula_a_b2 @ Xs3 )
= A )
& ( distin5059275051008138254la_a_b @ Xs3 ) ) ) ).
% finite_distinct_list
thf(fact_1134_finite__distinct__list,axiom,
! [A: set_formula_a_b] :
( ( finite6884408037426709032la_a_b @ A )
=> ? [Xs3: list_formula_a_b] :
( ( ( set_formula_a_b2 @ Xs3 )
= A )
& ( distinct_formula_a_b @ Xs3 ) ) ) ).
% finite_distinct_list
thf(fact_1135_Sup__fin_Oin__idem,axiom,
! [A: set_se3797390812436207411la_a_b,X: set_set_formula_a_b] :
( ( finite6039610903330827156la_a_b @ A )
=> ( ( member4282342919390090900la_a_b @ X @ A )
=> ( ( sup_su5931044959554304553la_a_b @ X @ ( lattic7951861892593991814la_a_b @ A ) )
= ( lattic7951861892593991814la_a_b @ A ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_1136_Sup__fin_Oin__idem,axiom,
! [A: set_set_formula_a_b,X: set_formula_a_b] :
( ( finite7542661095822535262la_a_b @ A )
=> ( ( member6621951321062823262la_a_b @ X @ A )
=> ( ( sup_su8832426789560438003la_a_b @ X @ ( lattic2153292871565936464la_a_b @ A ) )
= ( lattic2153292871565936464la_a_b @ A ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_1137_in__image__insert__iff,axiom,
! [B: set_se1704372200780349325la_a_b,X: list_formula_a_b,A: set_list_formula_a_b] :
( ! [C4: set_list_formula_a_b] :
( ( member7316870393999443438la_a_b @ C4 @ B )
=> ~ ( member7703986999711672120la_a_b @ X @ C4 ) )
=> ( ( member7316870393999443438la_a_b @ A @ ( image_2898219059140093877la_a_b @ ( insert4995524290302003025la_a_b @ X ) @ B ) )
= ( ( member7703986999711672120la_a_b @ X @ A )
& ( member7316870393999443438la_a_b @ ( minus_8667068627093346608la_a_b @ A @ ( insert4995524290302003025la_a_b @ X @ bot_bo4614192416432564331la_a_b ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_1138_in__image__insert__iff,axiom,
! [B: set_se3797390812436207411la_a_b,X: set_formula_a_b,A: set_set_formula_a_b] :
( ! [C4: set_set_formula_a_b] :
( ( member4282342919390090900la_a_b @ C4 @ B )
=> ~ ( member6621951321062823262la_a_b @ X @ C4 ) )
=> ( ( member4282342919390090900la_a_b @ A @ ( image_1397695786084986369la_a_b @ ( insert8952366156270154231la_a_b @ X ) @ B ) )
= ( ( member6621951321062823262la_a_b @ X @ A )
& ( member4282342919390090900la_a_b @ ( minus_4878662943024999126la_a_b @ A @ ( insert8952366156270154231la_a_b @ X @ bot_bo5296870006564677521la_a_b ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_1139_in__image__insert__iff,axiom,
! [B: set_set_formula_a_b,X: formula_a_b,A: set_formula_a_b] :
( ! [C4: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ C4 @ B )
=> ~ ( member_formula_a_b2 @ X @ C4 ) )
=> ( ( member6621951321062823262la_a_b @ A @ ( image_2155112572986939797la_a_b @ ( insert_formula_a_b2 @ X ) @ B ) )
= ( ( member_formula_a_b2 @ X @ A )
& ( member6621951321062823262la_a_b @ ( minus_3283878196352184736la_a_b @ A @ ( insert_formula_a_b2 @ X @ bot_bo8568539672012708443la_a_b ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_1140_Sup__fin_Ohom__commute,axiom,
! [H: set_set_formula_a_b > set_set_formula_a_b,N: set_se3797390812436207411la_a_b] :
( ! [X3: set_set_formula_a_b,Y3: set_set_formula_a_b] :
( ( H @ ( sup_su5931044959554304553la_a_b @ X3 @ Y3 ) )
= ( sup_su5931044959554304553la_a_b @ ( H @ X3 ) @ ( H @ Y3 ) ) )
=> ( ( finite6039610903330827156la_a_b @ N )
=> ( ( N != bot_bo9046117466857082311la_a_b )
=> ( ( H @ ( lattic7951861892593991814la_a_b @ N ) )
= ( lattic7951861892593991814la_a_b @ ( image_1397695786084986369la_a_b @ H @ N ) ) ) ) ) ) ).
% Sup_fin.hom_commute
thf(fact_1141_Sup__fin_Ohom__commute,axiom,
! [H: set_formula_a_b > set_formula_a_b,N: set_set_formula_a_b] :
( ! [X3: set_formula_a_b,Y3: set_formula_a_b] :
( ( H @ ( sup_su8832426789560438003la_a_b @ X3 @ Y3 ) )
= ( sup_su8832426789560438003la_a_b @ ( H @ X3 ) @ ( H @ Y3 ) ) )
=> ( ( finite7542661095822535262la_a_b @ N )
=> ( ( N != bot_bo5296870006564677521la_a_b )
=> ( ( H @ ( lattic2153292871565936464la_a_b @ N ) )
= ( lattic2153292871565936464la_a_b @ ( image_2155112572986939797la_a_b @ H @ N ) ) ) ) ) ) ).
% Sup_fin.hom_commute
thf(fact_1142_Sup__fin_Oinsert__not__elem,axiom,
! [A: set_se3797390812436207411la_a_b,X: set_set_formula_a_b] :
( ( finite6039610903330827156la_a_b @ A )
=> ( ~ ( member4282342919390090900la_a_b @ X @ A )
=> ( ( A != bot_bo9046117466857082311la_a_b )
=> ( ( lattic7951861892593991814la_a_b @ ( insert2824806981866770221la_a_b @ X @ A ) )
= ( sup_su5931044959554304553la_a_b @ X @ ( lattic7951861892593991814la_a_b @ A ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_1143_Sup__fin_Oinsert__not__elem,axiom,
! [A: set_set_formula_a_b,X: set_formula_a_b] :
( ( finite7542661095822535262la_a_b @ A )
=> ( ~ ( member6621951321062823262la_a_b @ X @ A )
=> ( ( A != bot_bo5296870006564677521la_a_b )
=> ( ( lattic2153292871565936464la_a_b @ ( insert8952366156270154231la_a_b @ X @ A ) )
= ( sup_su8832426789560438003la_a_b @ X @ ( lattic2153292871565936464la_a_b @ A ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_1144_Sup__fin_Oclosed,axiom,
! [A: set_se3797390812436207411la_a_b] :
( ( finite6039610903330827156la_a_b @ A )
=> ( ( A != bot_bo9046117466857082311la_a_b )
=> ( ! [X3: set_set_formula_a_b,Y3: set_set_formula_a_b] : ( member4282342919390090900la_a_b @ ( sup_su5931044959554304553la_a_b @ X3 @ Y3 ) @ ( insert2824806981866770221la_a_b @ X3 @ ( insert2824806981866770221la_a_b @ Y3 @ bot_bo9046117466857082311la_a_b ) ) )
=> ( member4282342919390090900la_a_b @ ( lattic7951861892593991814la_a_b @ A ) @ A ) ) ) ) ).
% Sup_fin.closed
thf(fact_1145_Sup__fin_Oclosed,axiom,
! [A: set_set_formula_a_b] :
( ( finite7542661095822535262la_a_b @ A )
=> ( ( A != bot_bo5296870006564677521la_a_b )
=> ( ! [X3: set_formula_a_b,Y3: set_formula_a_b] : ( member6621951321062823262la_a_b @ ( sup_su8832426789560438003la_a_b @ X3 @ Y3 ) @ ( insert8952366156270154231la_a_b @ X3 @ ( insert8952366156270154231la_a_b @ Y3 @ bot_bo5296870006564677521la_a_b ) ) )
=> ( member6621951321062823262la_a_b @ ( lattic2153292871565936464la_a_b @ A ) @ A ) ) ) ) ).
% Sup_fin.closed
thf(fact_1146_Sup__fin_Ounion,axiom,
! [A: set_se3797390812436207411la_a_b,B: set_se3797390812436207411la_a_b] :
( ( finite6039610903330827156la_a_b @ A )
=> ( ( A != bot_bo9046117466857082311la_a_b )
=> ( ( finite6039610903330827156la_a_b @ B )
=> ( ( B != bot_bo9046117466857082311la_a_b )
=> ( ( lattic7951861892593991814la_a_b @ ( sup_su1714642906130169951la_a_b @ A @ B ) )
= ( sup_su5931044959554304553la_a_b @ ( lattic7951861892593991814la_a_b @ A ) @ ( lattic7951861892593991814la_a_b @ B ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_1147_Sup__fin_Ounion,axiom,
! [A: set_set_formula_a_b,B: set_set_formula_a_b] :
( ( finite7542661095822535262la_a_b @ A )
=> ( ( A != bot_bo5296870006564677521la_a_b )
=> ( ( finite7542661095822535262la_a_b @ B )
=> ( ( B != bot_bo5296870006564677521la_a_b )
=> ( ( lattic2153292871565936464la_a_b @ ( sup_su5931044959554304553la_a_b @ A @ B ) )
= ( sup_su8832426789560438003la_a_b @ ( lattic2153292871565936464la_a_b @ A ) @ ( lattic2153292871565936464la_a_b @ B ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_1148_finite__Diff__insert,axiom,
! [A: set_formula_a_b,A2: formula_a_b,B: set_formula_a_b] :
( ( finite6884408037426709032la_a_b @ ( minus_3283878196352184736la_a_b @ A @ ( insert_formula_a_b2 @ A2 @ B ) ) )
= ( finite6884408037426709032la_a_b @ ( minus_3283878196352184736la_a_b @ A @ B ) ) ) ).
% finite_Diff_insert
thf(fact_1149_finite__Diff__insert,axiom,
! [A: set_set_formula_a_b,A2: set_formula_a_b,B: set_set_formula_a_b] :
( ( finite7542661095822535262la_a_b @ ( minus_4878662943024999126la_a_b @ A @ ( insert8952366156270154231la_a_b @ A2 @ B ) ) )
= ( finite7542661095822535262la_a_b @ ( minus_4878662943024999126la_a_b @ A @ B ) ) ) ).
% finite_Diff_insert
thf(fact_1150_sup__fin__closed,axiom,
! [A: set_se3797390812436207411la_a_b] :
( ( finite6039610903330827156la_a_b @ A )
=> ( ( A != bot_bo9046117466857082311la_a_b )
=> ( ! [X3: set_set_formula_a_b,Y3: set_set_formula_a_b] :
( ( member4282342919390090900la_a_b @ X3 @ A )
=> ( ( member4282342919390090900la_a_b @ Y3 @ A )
=> ( member4282342919390090900la_a_b @ ( sup_su5931044959554304553la_a_b @ X3 @ Y3 ) @ ( insert2824806981866770221la_a_b @ X3 @ ( insert2824806981866770221la_a_b @ Y3 @ bot_bo9046117466857082311la_a_b ) ) ) ) )
=> ( member4282342919390090900la_a_b @ ( lattic7951861892593991814la_a_b @ A ) @ A ) ) ) ) ).
% sup_fin_closed
thf(fact_1151_sup__fin__closed,axiom,
! [A: set_set_formula_a_b] :
( ( finite7542661095822535262la_a_b @ A )
=> ( ( A != bot_bo5296870006564677521la_a_b )
=> ( ! [X3: set_formula_a_b,Y3: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X3 @ A )
=> ( ( member6621951321062823262la_a_b @ Y3 @ A )
=> ( member6621951321062823262la_a_b @ ( sup_su8832426789560438003la_a_b @ X3 @ Y3 ) @ ( insert8952366156270154231la_a_b @ X3 @ ( insert8952366156270154231la_a_b @ Y3 @ bot_bo5296870006564677521la_a_b ) ) ) ) )
=> ( member6621951321062823262la_a_b @ ( lattic2153292871565936464la_a_b @ A ) @ A ) ) ) ) ).
% sup_fin_closed
thf(fact_1152_finite__Un,axiom,
! [F3: set_set_formula_a_b,G3: set_set_formula_a_b] :
( ( finite7542661095822535262la_a_b @ ( sup_su5931044959554304553la_a_b @ F3 @ G3 ) )
= ( ( finite7542661095822535262la_a_b @ F3 )
& ( finite7542661095822535262la_a_b @ G3 ) ) ) ).
% finite_Un
thf(fact_1153_finite__Un,axiom,
! [F3: set_formula_a_b,G3: set_formula_a_b] :
( ( finite6884408037426709032la_a_b @ ( sup_su8832426789560438003la_a_b @ F3 @ G3 ) )
= ( ( finite6884408037426709032la_a_b @ F3 )
& ( finite6884408037426709032la_a_b @ G3 ) ) ) ).
% finite_Un
thf(fact_1154_finite__insert,axiom,
! [A2: set_formula_a_b,A: set_set_formula_a_b] :
( ( finite7542661095822535262la_a_b @ ( insert8952366156270154231la_a_b @ A2 @ A ) )
= ( finite7542661095822535262la_a_b @ A ) ) ).
% finite_insert
thf(fact_1155_finite__insert,axiom,
! [A2: formula_a_b,A: set_formula_a_b] :
( ( finite6884408037426709032la_a_b @ ( insert_formula_a_b2 @ A2 @ A ) )
= ( finite6884408037426709032la_a_b @ A ) ) ).
% finite_insert
thf(fact_1156_finite_OinsertI,axiom,
! [A: set_set_formula_a_b,A2: set_formula_a_b] :
( ( finite7542661095822535262la_a_b @ A )
=> ( finite7542661095822535262la_a_b @ ( insert8952366156270154231la_a_b @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_1157_finite_OinsertI,axiom,
! [A: set_formula_a_b,A2: formula_a_b] :
( ( finite6884408037426709032la_a_b @ A )
=> ( finite6884408037426709032la_a_b @ ( insert_formula_a_b2 @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_1158_infinite__Un,axiom,
! [S3: set_set_formula_a_b,T: set_set_formula_a_b] :
( ( ~ ( finite7542661095822535262la_a_b @ ( sup_su5931044959554304553la_a_b @ S3 @ T ) ) )
= ( ~ ( finite7542661095822535262la_a_b @ S3 )
| ~ ( finite7542661095822535262la_a_b @ T ) ) ) ).
% infinite_Un
thf(fact_1159_infinite__Un,axiom,
! [S3: set_formula_a_b,T: set_formula_a_b] :
( ( ~ ( finite6884408037426709032la_a_b @ ( sup_su8832426789560438003la_a_b @ S3 @ T ) ) )
= ( ~ ( finite6884408037426709032la_a_b @ S3 )
| ~ ( finite6884408037426709032la_a_b @ T ) ) ) ).
% infinite_Un
thf(fact_1160_Un__infinite,axiom,
! [S3: set_formula_a_b,T: set_formula_a_b] :
( ~ ( finite6884408037426709032la_a_b @ S3 )
=> ~ ( finite6884408037426709032la_a_b @ ( sup_su8832426789560438003la_a_b @ S3 @ T ) ) ) ).
% Un_infinite
thf(fact_1161_finite__UnI,axiom,
! [F3: set_formula_a_b,G3: set_formula_a_b] :
( ( finite6884408037426709032la_a_b @ F3 )
=> ( ( finite6884408037426709032la_a_b @ G3 )
=> ( finite6884408037426709032la_a_b @ ( sup_su8832426789560438003la_a_b @ F3 @ G3 ) ) ) ) ).
% finite_UnI
thf(fact_1162_infinite__finite__induct,axiom,
! [P: set_formula_a_b > $o,A: set_formula_a_b] :
( ! [A6: set_formula_a_b] :
( ~ ( finite6884408037426709032la_a_b @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bo8568539672012708443la_a_b )
=> ( ! [X3: formula_a_b,F4: set_formula_a_b] :
( ( finite6884408037426709032la_a_b @ F4 )
=> ( ~ ( member_formula_a_b2 @ X3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_formula_a_b2 @ X3 @ F4 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_1163_finite__ne__induct,axiom,
! [F3: set_formula_a_b,P: set_formula_a_b > $o] :
( ( finite6884408037426709032la_a_b @ F3 )
=> ( ( F3 != bot_bo8568539672012708443la_a_b )
=> ( ! [X3: formula_a_b] : ( P @ ( insert_formula_a_b2 @ X3 @ bot_bo8568539672012708443la_a_b ) )
=> ( ! [X3: formula_a_b,F4: set_formula_a_b] :
( ( finite6884408037426709032la_a_b @ F4 )
=> ( ( F4 != bot_bo8568539672012708443la_a_b )
=> ( ~ ( member_formula_a_b2 @ X3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_formula_a_b2 @ X3 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_1164_finite__induct,axiom,
! [F3: set_formula_a_b,P: set_formula_a_b > $o] :
( ( finite6884408037426709032la_a_b @ F3 )
=> ( ( P @ bot_bo8568539672012708443la_a_b )
=> ( ! [X3: formula_a_b,F4: set_formula_a_b] :
( ( finite6884408037426709032la_a_b @ F4 )
=> ( ~ ( member_formula_a_b2 @ X3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_formula_a_b2 @ X3 @ F4 ) ) ) ) )
=> ( P @ F3 ) ) ) ) ).
% finite_induct
thf(fact_1165_finite__empty__induct,axiom,
! [A: set_formula_a_b,P: set_formula_a_b > $o] :
( ( finite6884408037426709032la_a_b @ A )
=> ( ( P @ A )
=> ( ! [A5: formula_a_b,A6: set_formula_a_b] :
( ( finite6884408037426709032la_a_b @ A6 )
=> ( ( member_formula_a_b2 @ A5 @ A6 )
=> ( ( P @ A6 )
=> ( P @ ( minus_3283878196352184736la_a_b @ A6 @ ( insert_formula_a_b2 @ A5 @ bot_bo8568539672012708443la_a_b ) ) ) ) ) )
=> ( P @ bot_bo8568539672012708443la_a_b ) ) ) ) ).
% finite_empty_induct
thf(fact_1166_sup__Inf__absorb,axiom,
! [A: set_set_formula_a_b,A2: set_formula_a_b] :
( ( finite7542661095822535262la_a_b @ A )
=> ( ( member6621951321062823262la_a_b @ A2 @ A )
=> ( ( sup_su8832426789560438003la_a_b @ ( lattic1282556677053900650la_a_b @ A ) @ A2 )
= A2 ) ) ) ).
% sup_Inf_absorb
thf(fact_1167_SUP__set__fold,axiom,
! [F: formula_a_b > set_formula_a_b,Xs: list_formula_a_b] :
( ( comple2445818437193057818la_a_b @ ( image_3063539048272112479la_a_b @ F @ ( set_formula_a_b2 @ Xs ) ) )
= ( fold_f470054086739575141la_a_b @ ( comp_s3812543008930346602la_a_b @ sup_su8832426789560438003la_a_b @ F ) @ Xs @ bot_bo8568539672012708443la_a_b ) ) ).
% SUP_set_fold
thf(fact_1168_Union__Un__distrib,axiom,
! [A: set_set_formula_a_b,B: set_set_formula_a_b] :
( ( comple2445818437193057818la_a_b @ ( sup_su5931044959554304553la_a_b @ A @ B ) )
= ( sup_su8832426789560438003la_a_b @ ( comple2445818437193057818la_a_b @ A ) @ ( comple2445818437193057818la_a_b @ B ) ) ) ).
% Union_Un_distrib
thf(fact_1169_Sup__insert,axiom,
! [A2: set_formula_a_b,A: set_set_formula_a_b] :
( ( comple2445818437193057818la_a_b @ ( insert8952366156270154231la_a_b @ A2 @ A ) )
= ( sup_su8832426789560438003la_a_b @ A2 @ ( comple2445818437193057818la_a_b @ A ) ) ) ).
% Sup_insert
thf(fact_1170_Union__insert,axiom,
! [A2: set_formula_a_b,B: set_set_formula_a_b] :
( ( comple2445818437193057818la_a_b @ ( insert8952366156270154231la_a_b @ A2 @ B ) )
= ( sup_su8832426789560438003la_a_b @ A2 @ ( comple2445818437193057818la_a_b @ B ) ) ) ).
% Union_insert
thf(fact_1171_inj__img__insertE,axiom,
! [F: formula_a_b > formula_a_b,A: set_formula_a_b,X: formula_a_b,B: set_formula_a_b] :
( ( inj_on2935910337538581653la_a_b @ F @ A )
=> ( ~ ( member_formula_a_b2 @ X @ B )
=> ( ( ( insert_formula_a_b2 @ X @ B )
= ( image_4658057545706456873la_a_b @ F @ A ) )
=> ~ ! [X5: formula_a_b,A7: set_formula_a_b] :
( ~ ( member_formula_a_b2 @ X5 @ A7 )
=> ( ( A
= ( insert_formula_a_b2 @ X5 @ A7 ) )
=> ( ( X
= ( F @ X5 ) )
=> ( B
!= ( image_4658057545706456873la_a_b @ F @ A7 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_1172_Sup__union__distrib,axiom,
! [A: set_set_formula_a_b,B: set_set_formula_a_b] :
( ( comple2445818437193057818la_a_b @ ( sup_su5931044959554304553la_a_b @ A @ B ) )
= ( sup_su8832426789560438003la_a_b @ ( comple2445818437193057818la_a_b @ A ) @ ( comple2445818437193057818la_a_b @ B ) ) ) ).
% Sup_union_distrib
thf(fact_1173_UNION__fun__upd,axiom,
! [A: formula_a_b > set_formula_a_b,I: formula_a_b,B: set_formula_a_b,J: set_formula_a_b] :
( ( comple2445818437193057818la_a_b @ ( image_3063539048272112479la_a_b @ ( fun_up6714831372235791319la_a_b @ A @ I @ B ) @ J ) )
= ( sup_su8832426789560438003la_a_b @ ( comple2445818437193057818la_a_b @ ( image_3063539048272112479la_a_b @ A @ ( minus_3283878196352184736la_a_b @ J @ ( insert_formula_a_b2 @ I @ bot_bo8568539672012708443la_a_b ) ) ) ) @ ( if_set_formula_a_b @ ( member_formula_a_b2 @ I @ J ) @ B @ bot_bo8568539672012708443la_a_b ) ) ) ).
% UNION_fun_upd
thf(fact_1174_finite__Sup__in,axiom,
! [A: set_set_formula_a_b] :
( ( finite7542661095822535262la_a_b @ A )
=> ( ( A != bot_bo5296870006564677521la_a_b )
=> ( ! [X3: set_formula_a_b,Y3: set_formula_a_b] :
( ( member6621951321062823262la_a_b @ X3 @ A )
=> ( ( member6621951321062823262la_a_b @ Y3 @ A )
=> ( member6621951321062823262la_a_b @ ( sup_su8832426789560438003la_a_b @ X3 @ Y3 ) @ A ) ) )
=> ( member6621951321062823262la_a_b @ ( comple2445818437193057818la_a_b @ A ) @ A ) ) ) ) ).
% finite_Sup_in
thf(fact_1175_Sup__set__fold,axiom,
! [Xs: list_set_formula_a_b] :
( ( comple2445818437193057818la_a_b @ ( set_set_formula_a_b2 @ Xs ) )
= ( fold_s5097478484676794267la_a_b @ sup_su8832426789560438003la_a_b @ Xs @ bot_bo8568539672012708443la_a_b ) ) ).
% Sup_set_fold
thf(fact_1176_union__fold__insert,axiom,
! [A: set_formula_a_b,B: set_formula_a_b] :
( ( finite6884408037426709032la_a_b @ A )
=> ( ( sup_su8832426789560438003la_a_b @ A @ B )
= ( finite2095001333592942498la_a_b @ insert_formula_a_b2 @ B @ A ) ) ) ).
% union_fold_insert
thf(fact_1177_sup__Sup__fold__sup,axiom,
! [A: set_set_formula_a_b,B: set_formula_a_b] :
( ( finite7542661095822535262la_a_b @ A )
=> ( ( sup_su8832426789560438003la_a_b @ ( comple2445818437193057818la_a_b @ A ) @ B )
= ( finite2548470876788853208la_a_b @ sup_su8832426789560438003la_a_b @ B @ A ) ) ) ).
% sup_Sup_fold_sup
thf(fact_1178_Sup__fold__sup,axiom,
! [A: set_set_formula_a_b] :
( ( finite7542661095822535262la_a_b @ A )
=> ( ( comple2445818437193057818la_a_b @ A )
= ( finite2548470876788853208la_a_b @ sup_su8832426789560438003la_a_b @ bot_bo8568539672012708443la_a_b @ A ) ) ) ).
% Sup_fold_sup
thf(fact_1179_Sup__fin_Oeq__fold,axiom,
! [A: set_set_formula_a_b,X: set_formula_a_b] :
( ( finite7542661095822535262la_a_b @ A )
=> ( ( lattic2153292871565936464la_a_b @ ( insert8952366156270154231la_a_b @ X @ A ) )
= ( finite2548470876788853208la_a_b @ sup_su8832426789560438003la_a_b @ X @ A ) ) ) ).
% Sup_fin.eq_fold
thf(fact_1180_cSUP__union,axiom,
! [A: set_formula_a_b,F: formula_a_b > set_formula_a_b,B: set_formula_a_b] :
( ( A != bot_bo8568539672012708443la_a_b )
=> ( ( condit8798486063731193955la_a_b @ ( image_3063539048272112479la_a_b @ F @ A ) )
=> ( ( B != bot_bo8568539672012708443la_a_b )
=> ( ( condit8798486063731193955la_a_b @ ( image_3063539048272112479la_a_b @ F @ B ) )
=> ( ( comple2445818437193057818la_a_b @ ( image_3063539048272112479la_a_b @ F @ ( sup_su8832426789560438003la_a_b @ A @ B ) ) )
= ( sup_su8832426789560438003la_a_b @ ( comple2445818437193057818la_a_b @ ( image_3063539048272112479la_a_b @ F @ A ) ) @ ( comple2445818437193057818la_a_b @ ( image_3063539048272112479la_a_b @ F @ B ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_1181_cSup__insert,axiom,
! [X6: set_set_formula_a_b,A2: set_formula_a_b] :
( ( X6 != bot_bo5296870006564677521la_a_b )
=> ( ( condit8798486063731193955la_a_b @ X6 )
=> ( ( comple2445818437193057818la_a_b @ ( insert8952366156270154231la_a_b @ A2 @ X6 ) )
= ( sup_su8832426789560438003la_a_b @ A2 @ ( comple2445818437193057818la_a_b @ X6 ) ) ) ) ) ).
% cSup_insert
thf(fact_1182_cSup__insert__If,axiom,
! [X6: set_set_formula_a_b,A2: set_formula_a_b] :
( ( condit8798486063731193955la_a_b @ X6 )
=> ( ( ( X6 = bot_bo5296870006564677521la_a_b )
=> ( ( comple2445818437193057818la_a_b @ ( insert8952366156270154231la_a_b @ A2 @ X6 ) )
= A2 ) )
& ( ( X6 != bot_bo5296870006564677521la_a_b )
=> ( ( comple2445818437193057818la_a_b @ ( insert8952366156270154231la_a_b @ A2 @ X6 ) )
= ( sup_su8832426789560438003la_a_b @ A2 @ ( comple2445818437193057818la_a_b @ X6 ) ) ) ) ) ) ).
% cSup_insert_If
thf(fact_1183_cSup__union__distrib,axiom,
! [A: set_set_formula_a_b,B: set_set_formula_a_b] :
( ( A != bot_bo5296870006564677521la_a_b )
=> ( ( condit8798486063731193955la_a_b @ A )
=> ( ( B != bot_bo5296870006564677521la_a_b )
=> ( ( condit8798486063731193955la_a_b @ B )
=> ( ( comple2445818437193057818la_a_b @ ( sup_su5931044959554304553la_a_b @ A @ B ) )
= ( sup_su8832426789560438003la_a_b @ ( comple2445818437193057818la_a_b @ A ) @ ( comple2445818437193057818la_a_b @ B ) ) ) ) ) ) ) ).
% cSup_union_distrib
thf(fact_1184_cSup__inter__less__eq,axiom,
! [A: set_set_formula_a_b,B: set_set_formula_a_b] :
( ( condit8798486063731193955la_a_b @ A )
=> ( ( condit8798486063731193955la_a_b @ B )
=> ( ( ( inf_in60795025369792911la_a_b @ A @ B )
!= bot_bo5296870006564677521la_a_b )
=> ( ord_le6178842339710166055la_a_b @ ( comple2445818437193057818la_a_b @ ( inf_in60795025369792911la_a_b @ A @ B ) ) @ ( sup_su8832426789560438003la_a_b @ ( comple2445818437193057818la_a_b @ A ) @ ( comple2445818437193057818la_a_b @ B ) ) ) ) ) ) ).
% cSup_inter_less_eq
thf(fact_1185_Pow__set_I1_J,axiom,
( ( pow_formula_a_b @ ( set_formula_a_b2 @ nil_formula_a_b ) )
= ( insert8952366156270154231la_a_b @ bot_bo8568539672012708443la_a_b @ bot_bo5296870006564677521la_a_b ) ) ).
% Pow_set(1)
thf(fact_1186_insert__subset,axiom,
! [X: formula_a_b,A: set_formula_a_b,B: set_formula_a_b] :
( ( ord_le6178842339710166055la_a_b @ ( insert_formula_a_b2 @ X @ A ) @ B )
= ( ( member_formula_a_b2 @ X @ B )
& ( ord_le6178842339710166055la_a_b @ A @ B ) ) ) ).
% insert_subset
thf(fact_1187_Un__subset__iff,axiom,
! [A: set_formula_a_b,B: set_formula_a_b,C2: set_formula_a_b] :
( ( ord_le6178842339710166055la_a_b @ ( sup_su8832426789560438003la_a_b @ A @ B ) @ C2 )
= ( ( ord_le6178842339710166055la_a_b @ A @ C2 )
& ( ord_le6178842339710166055la_a_b @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_1188_sup_Obounded__iff,axiom,
! [B2: set_formula_a_b,C: set_formula_a_b,A2: set_formula_a_b] :
( ( ord_le6178842339710166055la_a_b @ ( sup_su8832426789560438003la_a_b @ B2 @ C ) @ A2 )
= ( ( ord_le6178842339710166055la_a_b @ B2 @ A2 )
& ( ord_le6178842339710166055la_a_b @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_1189_le__sup__iff,axiom,
! [X: set_formula_a_b,Y: set_formula_a_b,Z: set_formula_a_b] :
( ( ord_le6178842339710166055la_a_b @ ( sup_su8832426789560438003la_a_b @ X @ Y ) @ Z )
= ( ( ord_le6178842339710166055la_a_b @ X @ Z )
& ( ord_le6178842339710166055la_a_b @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_1190_image__subsetI,axiom,
! [A: set_formula_a_b,F: formula_a_b > formula_a_b,B: set_formula_a_b] :
( ! [X3: formula_a_b] :
( ( member_formula_a_b2 @ X3 @ A )
=> ( member_formula_a_b2 @ ( F @ X3 ) @ B ) )
=> ( ord_le6178842339710166055la_a_b @ ( image_4658057545706456873la_a_b @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_1191_inf__sup__ord_I4_J,axiom,
! [Y: set_formula_a_b,X: set_formula_a_b] : ( ord_le6178842339710166055la_a_b @ Y @ ( sup_su8832426789560438003la_a_b @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_1192_inf__sup__ord_I3_J,axiom,
! [X: set_formula_a_b,Y: set_formula_a_b] : ( ord_le6178842339710166055la_a_b @ X @ ( sup_su8832426789560438003la_a_b @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_1193_le__supE,axiom,
! [A2: set_formula_a_b,B2: set_formula_a_b,X: set_formula_a_b] :
( ( ord_le6178842339710166055la_a_b @ ( sup_su8832426789560438003la_a_b @ A2 @ B2 ) @ X )
=> ~ ( ( ord_le6178842339710166055la_a_b @ A2 @ X )
=> ~ ( ord_le6178842339710166055la_a_b @ B2 @ X ) ) ) ).
% le_supE
thf(fact_1194_le__supI,axiom,
! [A2: set_formula_a_b,X: set_formula_a_b,B2: set_formula_a_b] :
( ( ord_le6178842339710166055la_a_b @ A2 @ X )
=> ( ( ord_le6178842339710166055la_a_b @ B2 @ X )
=> ( ord_le6178842339710166055la_a_b @ ( sup_su8832426789560438003la_a_b @ A2 @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_1195_sup__ge1,axiom,
! [X: set_formula_a_b,Y: set_formula_a_b] : ( ord_le6178842339710166055la_a_b @ X @ ( sup_su8832426789560438003la_a_b @ X @ Y ) ) ).
% sup_ge1
thf(fact_1196_sup__ge2,axiom,
! [Y: set_formula_a_b,X: set_formula_a_b] : ( ord_le6178842339710166055la_a_b @ Y @ ( sup_su8832426789560438003la_a_b @ X @ Y ) ) ).
% sup_ge2
thf(fact_1197_le__supI1,axiom,
! [X: set_formula_a_b,A2: set_formula_a_b,B2: set_formula_a_b] :
( ( ord_le6178842339710166055la_a_b @ X @ A2 )
=> ( ord_le6178842339710166055la_a_b @ X @ ( sup_su8832426789560438003la_a_b @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_1198_le__supI2,axiom,
! [X: set_formula_a_b,B2: set_formula_a_b,A2: set_formula_a_b] :
( ( ord_le6178842339710166055la_a_b @ X @ B2 )
=> ( ord_le6178842339710166055la_a_b @ X @ ( sup_su8832426789560438003la_a_b @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_1199_sup_Omono,axiom,
! [C: set_formula_a_b,A2: set_formula_a_b,D: set_formula_a_b,B2: set_formula_a_b] :
( ( ord_le6178842339710166055la_a_b @ C @ A2 )
=> ( ( ord_le6178842339710166055la_a_b @ D @ B2 )
=> ( ord_le6178842339710166055la_a_b @ ( sup_su8832426789560438003la_a_b @ C @ D ) @ ( sup_su8832426789560438003la_a_b @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_1200_sup__mono,axiom,
! [A2: set_formula_a_b,C: set_formula_a_b,B2: set_formula_a_b,D: set_formula_a_b] :
( ( ord_le6178842339710166055la_a_b @ A2 @ C )
=> ( ( ord_le6178842339710166055la_a_b @ B2 @ D )
=> ( ord_le6178842339710166055la_a_b @ ( sup_su8832426789560438003la_a_b @ A2 @ B2 ) @ ( sup_su8832426789560438003la_a_b @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_1201_sup__least,axiom,
! [Y: set_formula_a_b,X: set_formula_a_b,Z: set_formula_a_b] :
( ( ord_le6178842339710166055la_a_b @ Y @ X )
=> ( ( ord_le6178842339710166055la_a_b @ Z @ X )
=> ( ord_le6178842339710166055la_a_b @ ( sup_su8832426789560438003la_a_b @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_1202_le__iff__sup,axiom,
( ord_le6178842339710166055la_a_b
= ( ^ [X2: set_formula_a_b,Y2: set_formula_a_b] :
( ( sup_su8832426789560438003la_a_b @ X2 @ Y2 )
= Y2 ) ) ) ).
% le_iff_sup
thf(fact_1203_sup_OorderE,axiom,
! [B2: set_formula_a_b,A2: set_formula_a_b] :
( ( ord_le6178842339710166055la_a_b @ B2 @ A2 )
=> ( A2
= ( sup_su8832426789560438003la_a_b @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_1204_sup_OorderI,axiom,
! [A2: set_formula_a_b,B2: set_formula_a_b] :
( ( A2
= ( sup_su8832426789560438003la_a_b @ A2 @ B2 ) )
=> ( ord_le6178842339710166055la_a_b @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_1205_sup__unique,axiom,
! [F: set_formula_a_b > set_formula_a_b > set_formula_a_b,X: set_formula_a_b,Y: set_formula_a_b] :
( ! [X3: set_formula_a_b,Y3: set_formula_a_b] : ( ord_le6178842339710166055la_a_b @ X3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: set_formula_a_b,Y3: set_formula_a_b] : ( ord_le6178842339710166055la_a_b @ Y3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: set_formula_a_b,Y3: set_formula_a_b,Z3: set_formula_a_b] :
( ( ord_le6178842339710166055la_a_b @ Y3 @ X3 )
=> ( ( ord_le6178842339710166055la_a_b @ Z3 @ X3 )
=> ( ord_le6178842339710166055la_a_b @ ( F @ Y3 @ Z3 ) @ X3 ) ) )
=> ( ( sup_su8832426789560438003la_a_b @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_1206_sup_Oabsorb1,axiom,
! [B2: set_formula_a_b,A2: set_formula_a_b] :
( ( ord_le6178842339710166055la_a_b @ B2 @ A2 )
=> ( ( sup_su8832426789560438003la_a_b @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_1207_sup_Oabsorb2,axiom,
! [A2: set_formula_a_b,B2: set_formula_a_b] :
( ( ord_le6178842339710166055la_a_b @ A2 @ B2 )
=> ( ( sup_su8832426789560438003la_a_b @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_1208_sup__absorb1,axiom,
! [Y: set_formula_a_b,X: set_formula_a_b] :
( ( ord_le6178842339710166055la_a_b @ Y @ X )
=> ( ( sup_su8832426789560438003la_a_b @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_1209_sup__absorb2,axiom,
! [X: set_formula_a_b,Y: set_formula_a_b] :
( ( ord_le6178842339710166055la_a_b @ X @ Y )
=> ( ( sup_su8832426789560438003la_a_b @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_1210_sup_OboundedE,axiom,
! [B2: set_formula_a_b,C: set_formula_a_b,A2: set_formula_a_b] :
( ( ord_le6178842339710166055la_a_b @ ( sup_su8832426789560438003la_a_b @ B2 @ C ) @ A2 )
=> ~ ( ( ord_le6178842339710166055la_a_b @ B2 @ A2 )
=> ~ ( ord_le6178842339710166055la_a_b @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_1211_sup_OboundedI,axiom,
! [B2: set_formula_a_b,A2: set_formula_a_b,C: set_formula_a_b] :
( ( ord_le6178842339710166055la_a_b @ B2 @ A2 )
=> ( ( ord_le6178842339710166055la_a_b @ C @ A2 )
=> ( ord_le6178842339710166055la_a_b @ ( sup_su8832426789560438003la_a_b @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_1212_sup_Oorder__iff,axiom,
( ord_le6178842339710166055la_a_b
= ( ^ [B3: set_formula_a_b,A3: set_formula_a_b] :
( A3
= ( sup_su8832426789560438003la_a_b @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_1213_sup_Ocobounded1,axiom,
! [A2: set_formula_a_b,B2: set_formula_a_b] : ( ord_le6178842339710166055la_a_b @ A2 @ ( sup_su8832426789560438003la_a_b @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_1214_sup_Ocobounded2,axiom,
! [B2: set_formula_a_b,A2: set_formula_a_b] : ( ord_le6178842339710166055la_a_b @ B2 @ ( sup_su8832426789560438003la_a_b @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_1215_sup_Oabsorb__iff1,axiom,
( ord_le6178842339710166055la_a_b
= ( ^ [B3: set_formula_a_b,A3: set_formula_a_b] :
( ( sup_su8832426789560438003la_a_b @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_1216_sup_Oabsorb__iff2,axiom,
( ord_le6178842339710166055la_a_b
= ( ^ [A3: set_formula_a_b,B3: set_formula_a_b] :
( ( sup_su8832426789560438003la_a_b @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_1217_sup_OcoboundedI1,axiom,
! [C: set_formula_a_b,A2: set_formula_a_b,B2: set_formula_a_b] :
( ( ord_le6178842339710166055la_a_b @ C @ A2 )
=> ( ord_le6178842339710166055la_a_b @ C @ ( sup_su8832426789560438003la_a_b @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_1218_sup_OcoboundedI2,axiom,
! [C: set_formula_a_b,B2: set_formula_a_b,A2: set_formula_a_b] :
( ( ord_le6178842339710166055la_a_b @ C @ B2 )
=> ( ord_le6178842339710166055la_a_b @ C @ ( sup_su8832426789560438003la_a_b @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_1219_Un__mono,axiom,
! [A: set_formula_a_b,C2: set_formula_a_b,B: set_formula_a_b,D2: set_formula_a_b] :
( ( ord_le6178842339710166055la_a_b @ A @ C2 )
=> ( ( ord_le6178842339710166055la_a_b @ B @ D2 )
=> ( ord_le6178842339710166055la_a_b @ ( sup_su8832426789560438003la_a_b @ A @ B ) @ ( sup_su8832426789560438003la_a_b @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_1220_Un__least,axiom,
! [A: set_formula_a_b,C2: set_formula_a_b,B: set_formula_a_b] :
( ( ord_le6178842339710166055la_a_b @ A @ C2 )
=> ( ( ord_le6178842339710166055la_a_b @ B @ C2 )
=> ( ord_le6178842339710166055la_a_b @ ( sup_su8832426789560438003la_a_b @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_1221_Un__upper1,axiom,
! [A: set_formula_a_b,B: set_formula_a_b] : ( ord_le6178842339710166055la_a_b @ A @ ( sup_su8832426789560438003la_a_b @ A @ B ) ) ).
% Un_upper1
thf(fact_1222_Un__upper2,axiom,
! [B: set_formula_a_b,A: set_formula_a_b] : ( ord_le6178842339710166055la_a_b @ B @ ( sup_su8832426789560438003la_a_b @ A @ B ) ) ).
% Un_upper2
thf(fact_1223_Un__absorb1,axiom,
! [A: set_formula_a_b,B: set_formula_a_b] :
( ( ord_le6178842339710166055la_a_b @ A @ B )
=> ( ( sup_su8832426789560438003la_a_b @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_1224_Un__absorb2,axiom,
! [B: set_formula_a_b,A: set_formula_a_b] :
( ( ord_le6178842339710166055la_a_b @ B @ A )
=> ( ( sup_su8832426789560438003la_a_b @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_1225_subset__UnE,axiom,
! [C2: set_formula_a_b,A: set_formula_a_b,B: set_formula_a_b] :
( ( ord_le6178842339710166055la_a_b @ C2 @ ( sup_su8832426789560438003la_a_b @ A @ B ) )
=> ~ ! [A7: set_formula_a_b] :
( ( ord_le6178842339710166055la_a_b @ A7 @ A )
=> ! [B6: set_formula_a_b] :
( ( ord_le6178842339710166055la_a_b @ B6 @ B )
=> ( C2
!= ( sup_su8832426789560438003la_a_b @ A7 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_1226_subset__Un__eq,axiom,
( ord_le6178842339710166055la_a_b
= ( ^ [A4: set_formula_a_b,B4: set_formula_a_b] :
( ( sup_su8832426789560438003la_a_b @ A4 @ B4 )
= B4 ) ) ) ).
% subset_Un_eq
thf(fact_1227_Un__Pow__subset,axiom,
! [A: set_formula_a_b,B: set_formula_a_b] : ( ord_le4913768313314656605la_a_b @ ( sup_su5931044959554304553la_a_b @ ( pow_formula_a_b @ A ) @ ( pow_formula_a_b @ B ) ) @ ( pow_formula_a_b @ ( sup_su8832426789560438003la_a_b @ A @ B ) ) ) ).
% Un_Pow_subset
thf(fact_1228_subset__insert,axiom,
! [X: formula_a_b,A: set_formula_a_b,B: set_formula_a_b] :
( ~ ( member_formula_a_b2 @ X @ A )
=> ( ( ord_le6178842339710166055la_a_b @ A @ ( insert_formula_a_b2 @ X @ B ) )
= ( ord_le6178842339710166055la_a_b @ A @ B ) ) ) ).
% subset_insert
thf(fact_1229_Int__Collect__mono,axiom,
! [A: set_formula_a_b,B: set_formula_a_b,P: formula_a_b > $o,Q: formula_a_b > $o] :
( ( ord_le6178842339710166055la_a_b @ A @ B )
=> ( ! [X3: formula_a_b] :
( ( member_formula_a_b2 @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le6178842339710166055la_a_b @ ( inf_in5741596252272946265la_a_b @ A @ ( collect_formula_a_b @ P ) ) @ ( inf_in5741596252272946265la_a_b @ B @ ( collect_formula_a_b @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1230_subset__code_I1_J,axiom,
! [Xs: list_formula_a_b,B: set_formula_a_b] :
( ( ord_le6178842339710166055la_a_b @ ( set_formula_a_b2 @ Xs ) @ B )
= ( ! [X2: formula_a_b] :
( ( member_formula_a_b2 @ X2 @ ( set_formula_a_b2 @ Xs ) )
=> ( member_formula_a_b2 @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_1231_distrib__sup__le,axiom,
! [X: set_formula_a_b,Y: set_formula_a_b,Z: set_formula_a_b] : ( ord_le6178842339710166055la_a_b @ ( sup_su8832426789560438003la_a_b @ X @ ( inf_in5741596252272946265la_a_b @ Y @ Z ) ) @ ( inf_in5741596252272946265la_a_b @ ( sup_su8832426789560438003la_a_b @ X @ Y ) @ ( sup_su8832426789560438003la_a_b @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_1232_distrib__inf__le,axiom,
! [X: set_formula_a_b,Y: set_formula_a_b,Z: set_formula_a_b] : ( ord_le6178842339710166055la_a_b @ ( sup_su8832426789560438003la_a_b @ ( inf_in5741596252272946265la_a_b @ X @ Y ) @ ( inf_in5741596252272946265la_a_b @ X @ Z ) ) @ ( inf_in5741596252272946265la_a_b @ X @ ( sup_su8832426789560438003la_a_b @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_1233_set__subset__Cons,axiom,
! [Xs: list_formula_a_b,X: formula_a_b] : ( ord_le6178842339710166055la_a_b @ ( set_formula_a_b2 @ Xs ) @ ( set_formula_a_b2 @ ( cons_formula_a_b @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_1234_inj__on__image__mem__iff,axiom,
! [F: formula_a_b > formula_a_b,B: set_formula_a_b,A2: formula_a_b,A: set_formula_a_b] :
( ( inj_on2935910337538581653la_a_b @ F @ B )
=> ( ( member_formula_a_b2 @ A2 @ B )
=> ( ( ord_le6178842339710166055la_a_b @ A @ B )
=> ( ( member_formula_a_b2 @ ( F @ A2 ) @ ( image_4658057545706456873la_a_b @ F @ A ) )
= ( member_formula_a_b2 @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1235_subset__Diff__insert,axiom,
! [A: set_formula_a_b,B: set_formula_a_b,X: formula_a_b,C2: set_formula_a_b] :
( ( ord_le6178842339710166055la_a_b @ A @ ( minus_3283878196352184736la_a_b @ B @ ( insert_formula_a_b2 @ X @ C2 ) ) )
= ( ( ord_le6178842339710166055la_a_b @ A @ ( minus_3283878196352184736la_a_b @ B @ C2 ) )
& ~ ( member_formula_a_b2 @ X @ A ) ) ) ).
% subset_Diff_insert
thf(fact_1236_Un__Int__assoc__eq,axiom,
! [A: set_formula_a_b,B: set_formula_a_b,C2: set_formula_a_b] :
( ( ( sup_su8832426789560438003la_a_b @ ( inf_in5741596252272946265la_a_b @ A @ B ) @ C2 )
= ( inf_in5741596252272946265la_a_b @ A @ ( sup_su8832426789560438003la_a_b @ B @ C2 ) ) )
= ( ord_le6178842339710166055la_a_b @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_1237_Diff__subset__conv,axiom,
! [A: set_formula_a_b,B: set_formula_a_b,C2: set_formula_a_b] :
( ( ord_le6178842339710166055la_a_b @ ( minus_3283878196352184736la_a_b @ A @ B ) @ C2 )
= ( ord_le6178842339710166055la_a_b @ A @ ( sup_su8832426789560438003la_a_b @ B @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_1238_Diff__partition,axiom,
! [A: set_formula_a_b,B: set_formula_a_b] :
( ( ord_le6178842339710166055la_a_b @ A @ B )
=> ( ( sup_su8832426789560438003la_a_b @ A @ ( minus_3283878196352184736la_a_b @ B @ A ) )
= B ) ) ).
% Diff_partition
thf(fact_1239_set__remove1__subset,axiom,
! [X: formula_a_b,Xs: list_formula_a_b] : ( ord_le6178842339710166055la_a_b @ ( set_formula_a_b2 @ ( remove1_formula_a_b @ X @ Xs ) ) @ ( set_formula_a_b2 @ Xs ) ) ).
% set_remove1_subset
thf(fact_1240_subset__code_I2_J,axiom,
! [A: set_formula_a_b,Ys: list_formula_a_b] :
( ( ord_le6178842339710166055la_a_b @ A @ ( coset_formula_a_b @ Ys ) )
= ( ! [X2: formula_a_b] :
( ( member_formula_a_b2 @ X2 @ ( set_formula_a_b2 @ Ys ) )
=> ~ ( member_formula_a_b2 @ X2 @ A ) ) ) ) ).
% subset_code(2)
thf(fact_1241_finite__subset__induct,axiom,
! [F3: set_formula_a_b,A: set_formula_a_b,P: set_formula_a_b > $o] :
( ( finite6884408037426709032la_a_b @ F3 )
=> ( ( ord_le6178842339710166055la_a_b @ F3 @ A )
=> ( ( P @ bot_bo8568539672012708443la_a_b )
=> ( ! [A5: formula_a_b,F4: set_formula_a_b] :
( ( finite6884408037426709032la_a_b @ F4 )
=> ( ( member_formula_a_b2 @ A5 @ A )
=> ( ~ ( member_formula_a_b2 @ A5 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_formula_a_b2 @ A5 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1242_finite__subset__induct_H,axiom,
! [F3: set_formula_a_b,A: set_formula_a_b,P: set_formula_a_b > $o] :
( ( finite6884408037426709032la_a_b @ F3 )
=> ( ( ord_le6178842339710166055la_a_b @ F3 @ A )
=> ( ( P @ bot_bo8568539672012708443la_a_b )
=> ( ! [A5: formula_a_b,F4: set_formula_a_b] :
( ( finite6884408037426709032la_a_b @ F4 )
=> ( ( member_formula_a_b2 @ A5 @ A )
=> ( ( ord_le6178842339710166055la_a_b @ F4 @ A )
=> ( ~ ( member_formula_a_b2 @ A5 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_formula_a_b2 @ A5 @ F4 ) ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1243_subset__insert__iff,axiom,
! [A: set_formula_a_b,X: formula_a_b,B: set_formula_a_b] :
( ( ord_le6178842339710166055la_a_b @ A @ ( insert_formula_a_b2 @ X @ B ) )
= ( ( ( member_formula_a_b2 @ X @ A )
=> ( ord_le6178842339710166055la_a_b @ ( minus_3283878196352184736la_a_b @ A @ ( insert_formula_a_b2 @ X @ bot_bo8568539672012708443la_a_b ) ) @ B ) )
& ( ~ ( member_formula_a_b2 @ X @ A )
=> ( ord_le6178842339710166055la_a_b @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1244_subset__code_I3_J,axiom,
~ ( ord_le6178842339710166055la_a_b @ ( coset_formula_a_b @ nil_formula_a_b ) @ ( set_formula_a_b2 @ nil_formula_a_b ) ) ).
% subset_code(3)
thf(fact_1245_subset__subseqs,axiom,
! [X6: set_formula_a_b,Xs: list_formula_a_b] :
( ( ord_le6178842339710166055la_a_b @ X6 @ ( set_formula_a_b2 @ Xs ) )
=> ( member6621951321062823262la_a_b @ X6 @ ( image_4386591827427811823la_a_b @ set_formula_a_b2 @ ( set_list_formula_a_b2 @ ( subseqs_formula_a_b @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_1246_subseqs__powset,axiom,
! [Xs: list_formula_a_b] :
( ( image_4386591827427811823la_a_b @ set_formula_a_b2 @ ( set_list_formula_a_b2 @ ( subseqs_formula_a_b @ Xs ) ) )
= ( pow_formula_a_b @ ( set_formula_a_b2 @ Xs ) ) ) ).
% subseqs_powset
thf(fact_1247_finite__remove__induct,axiom,
! [B: set_formula_a_b,P: set_formula_a_b > $o] :
( ( finite6884408037426709032la_a_b @ B )
=> ( ( P @ bot_bo8568539672012708443la_a_b )
=> ( ! [A6: set_formula_a_b] :
( ( finite6884408037426709032la_a_b @ A6 )
=> ( ( A6 != bot_bo8568539672012708443la_a_b )
=> ( ( ord_le6178842339710166055la_a_b @ A6 @ B )
=> ( ! [X4: formula_a_b] :
( ( member_formula_a_b2 @ X4 @ A6 )
=> ( P @ ( minus_3283878196352184736la_a_b @ A6 @ ( insert_formula_a_b2 @ X4 @ bot_bo8568539672012708443la_a_b ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_1248_remove__induct,axiom,
! [P: set_formula_a_b > $o,B: set_formula_a_b] :
( ( P @ bot_bo8568539672012708443la_a_b )
=> ( ( ~ ( finite6884408037426709032la_a_b @ B )
=> ( P @ B ) )
=> ( ! [A6: set_formula_a_b] :
( ( finite6884408037426709032la_a_b @ A6 )
=> ( ( A6 != bot_bo8568539672012708443la_a_b )
=> ( ( ord_le6178842339710166055la_a_b @ A6 @ B )
=> ( ! [X4: formula_a_b] :
( ( member_formula_a_b2 @ X4 @ A6 )
=> ( P @ ( minus_3283878196352184736la_a_b @ A6 @ ( insert_formula_a_b2 @ X4 @ bot_bo8568539672012708443la_a_b ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_1249_Sup__fin_Osubset,axiom,
! [A: set_set_formula_a_b,B: set_set_formula_a_b] :
( ( finite7542661095822535262la_a_b @ A )
=> ( ( B != bot_bo5296870006564677521la_a_b )
=> ( ( ord_le4913768313314656605la_a_b @ B @ A )
=> ( ( sup_su8832426789560438003la_a_b @ ( lattic2153292871565936464la_a_b @ B ) @ ( lattic2153292871565936464la_a_b @ A ) )
= ( lattic2153292871565936464la_a_b @ A ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_1250_subsetI,axiom,
! [A: set_formula_a_b,B: set_formula_a_b] :
( ! [X3: formula_a_b] :
( ( member_formula_a_b2 @ X3 @ A )
=> ( member_formula_a_b2 @ X3 @ B ) )
=> ( ord_le6178842339710166055la_a_b @ A @ B ) ) ).
% subsetI
thf(fact_1251_subset__iff,axiom,
( ord_le6178842339710166055la_a_b
= ( ^ [A4: set_formula_a_b,B4: set_formula_a_b] :
! [T2: formula_a_b] :
( ( member_formula_a_b2 @ T2 @ A4 )
=> ( member_formula_a_b2 @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_1252_subset__eq,axiom,
( ord_le6178842339710166055la_a_b
= ( ^ [A4: set_formula_a_b,B4: set_formula_a_b] :
! [X2: formula_a_b] :
( ( member_formula_a_b2 @ X2 @ A4 )
=> ( member_formula_a_b2 @ X2 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_1253_subsetD,axiom,
! [A: set_formula_a_b,B: set_formula_a_b,C: formula_a_b] :
( ( ord_le6178842339710166055la_a_b @ A @ B )
=> ( ( member_formula_a_b2 @ C @ A )
=> ( member_formula_a_b2 @ C @ B ) ) ) ).
% subsetD
thf(fact_1254_in__mono,axiom,
! [A: set_formula_a_b,B: set_formula_a_b,X: formula_a_b] :
( ( ord_le6178842339710166055la_a_b @ A @ B )
=> ( ( member_formula_a_b2 @ X @ A )
=> ( member_formula_a_b2 @ X @ B ) ) ) ).
% in_mono
thf(fact_1255_insert__subsetI,axiom,
! [X: formula_a_b,A: set_formula_a_b,X6: set_formula_a_b] :
( ( member_formula_a_b2 @ X @ A )
=> ( ( ord_le6178842339710166055la_a_b @ X6 @ A )
=> ( ord_le6178842339710166055la_a_b @ ( insert_formula_a_b2 @ X @ X6 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_1256_set__remdups,axiom,
! [Xs: list_formula_a_b] :
( ( set_formula_a_b2 @ ( remdups_formula_a_b @ Xs ) )
= ( set_formula_a_b2 @ Xs ) ) ).
% set_remdups
thf(fact_1257_remdups_Osimps_I2_J,axiom,
! [X: formula_a_b,Xs: list_formula_a_b] :
( ( ( member_formula_a_b2 @ X @ ( set_formula_a_b2 @ Xs ) )
=> ( ( remdups_formula_a_b @ ( cons_formula_a_b @ X @ Xs ) )
= ( remdups_formula_a_b @ Xs ) ) )
& ( ~ ( member_formula_a_b2 @ X @ ( set_formula_a_b2 @ Xs ) )
=> ( ( remdups_formula_a_b @ ( cons_formula_a_b @ X @ Xs ) )
= ( cons_formula_a_b @ X @ ( remdups_formula_a_b @ Xs ) ) ) ) ) ).
% remdups.simps(2)
thf(fact_1258_list_Oset__map,axiom,
! [F: formula_a_b > formula_a_b,V: list_formula_a_b] :
( ( set_formula_a_b2 @ ( map_fo2126546822197359848la_a_b @ F @ V ) )
= ( image_4658057545706456873la_a_b @ F @ ( set_formula_a_b2 @ V ) ) ) ).
% list.set_map
thf(fact_1259_image__set,axiom,
! [F: formula_a_b > formula_a_b,Xs: list_formula_a_b] :
( ( image_4658057545706456873la_a_b @ F @ ( set_formula_a_b2 @ Xs ) )
= ( set_formula_a_b2 @ ( map_fo2126546822197359848la_a_b @ F @ Xs ) ) ) ).
% image_set
thf(fact_1260_list_Omap__ident__strong,axiom,
! [T3: list_formula_a_b,F: formula_a_b > formula_a_b] :
( ! [Z3: formula_a_b] :
( ( member_formula_a_b2 @ Z3 @ ( set_formula_a_b2 @ T3 ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( map_fo2126546822197359848la_a_b @ F @ T3 )
= T3 ) ) ).
% list.map_ident_strong
thf(fact_1261_map__idI,axiom,
! [Xs: list_formula_a_b,F: formula_a_b > formula_a_b] :
( ! [X3: formula_a_b] :
( ( member_formula_a_b2 @ X3 @ ( set_formula_a_b2 @ Xs ) )
=> ( ( F @ X3 )
= X3 ) )
=> ( ( map_fo2126546822197359848la_a_b @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_1262_distinct__set__subseqs,axiom,
! [Xs: list_formula_a_b] :
( ( distinct_formula_a_b @ Xs )
=> ( distin5059275051008138254la_a_b @ ( map_li4767261230909057070la_a_b @ set_formula_a_b2 @ ( subseqs_formula_a_b @ Xs ) ) ) ) ).
% distinct_set_subseqs
thf(fact_1263_sorted__wrt__mono__rel,axiom,
! [Xs: list_formula_a_b,P: formula_a_b > formula_a_b > $o,Q: formula_a_b > formula_a_b > $o] :
( ! [X3: formula_a_b,Y3: formula_a_b] :
( ( member_formula_a_b2 @ X3 @ ( set_formula_a_b2 @ Xs ) )
=> ( ( member_formula_a_b2 @ Y3 @ ( set_formula_a_b2 @ Xs ) )
=> ( ( P @ X3 @ Y3 )
=> ( Q @ X3 @ Y3 ) ) ) )
=> ( ( sorted287171416404617316la_a_b @ P @ Xs )
=> ( sorted287171416404617316la_a_b @ Q @ Xs ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_1264_sorted__wrt__append,axiom,
! [P: formula_a_b > formula_a_b > $o,Xs: list_formula_a_b,Ys: list_formula_a_b] :
( ( sorted287171416404617316la_a_b @ P @ ( append_formula_a_b @ Xs @ Ys ) )
= ( ( sorted287171416404617316la_a_b @ P @ Xs )
& ( sorted287171416404617316la_a_b @ P @ Ys )
& ! [X2: formula_a_b] :
( ( member_formula_a_b2 @ X2 @ ( set_formula_a_b2 @ Xs ) )
=> ! [Y2: formula_a_b] :
( ( member_formula_a_b2 @ Y2 @ ( set_formula_a_b2 @ Ys ) )
=> ( P @ X2 @ Y2 ) ) ) ) ) ).
% sorted_wrt_append
thf(fact_1265_the__inv__into__into,axiom,
! [F: formula_a_b > formula_a_b,A: set_formula_a_b,X: formula_a_b,B: set_formula_a_b] :
( ( inj_on2935910337538581653la_a_b @ F @ A )
=> ( ( member_formula_a_b2 @ X @ ( image_4658057545706456873la_a_b @ F @ A ) )
=> ( ( ord_le6178842339710166055la_a_b @ A @ B )
=> ( member_formula_a_b2 @ ( the_in4303446237855801815la_a_b @ A @ F @ X ) @ B ) ) ) ) ).
% the_inv_into_into
thf(fact_1266_vimageI,axiom,
! [F: formula_a_b > formula_a_b,A2: formula_a_b,B2: formula_a_b,B: set_formula_a_b] :
( ( ( F @ A2 )
= B2 )
=> ( ( member_formula_a_b2 @ B2 @ B )
=> ( member_formula_a_b2 @ A2 @ ( vimage6809459385517368287la_a_b @ F @ B ) ) ) ) ).
% vimageI
thf(fact_1267_vimage__eq,axiom,
! [A2: formula_a_b,F: formula_a_b > formula_a_b,B: set_formula_a_b] :
( ( member_formula_a_b2 @ A2 @ ( vimage6809459385517368287la_a_b @ F @ B ) )
= ( member_formula_a_b2 @ ( F @ A2 ) @ B ) ) ).
% vimage_eq
thf(fact_1268_vimage__Un,axiom,
! [F: formula_a_b > formula_a_b,A: set_formula_a_b,B: set_formula_a_b] :
( ( vimage6809459385517368287la_a_b @ F @ ( sup_su8832426789560438003la_a_b @ A @ B ) )
= ( sup_su8832426789560438003la_a_b @ ( vimage6809459385517368287la_a_b @ F @ A ) @ ( vimage6809459385517368287la_a_b @ F @ B ) ) ) ).
% vimage_Un
thf(fact_1269_vimageD,axiom,
! [A2: formula_a_b,F: formula_a_b > formula_a_b,A: set_formula_a_b] :
( ( member_formula_a_b2 @ A2 @ ( vimage6809459385517368287la_a_b @ F @ A ) )
=> ( member_formula_a_b2 @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_1270_vimageE,axiom,
! [A2: formula_a_b,F: formula_a_b > formula_a_b,B: set_formula_a_b] :
( ( member_formula_a_b2 @ A2 @ ( vimage6809459385517368287la_a_b @ F @ B ) )
=> ( member_formula_a_b2 @ ( F @ A2 ) @ B ) ) ).
% vimageE
thf(fact_1271_vimageI2,axiom,
! [F: formula_a_b > formula_a_b,A2: formula_a_b,A: set_formula_a_b] :
( ( member_formula_a_b2 @ ( F @ A2 ) @ A )
=> ( member_formula_a_b2 @ A2 @ ( vimage6809459385517368287la_a_b @ F @ A ) ) ) ).
% vimageI2
thf(fact_1272_UNIV__I,axiom,
! [X: formula_a_b] : ( member_formula_a_b2 @ X @ top_to8846066467269685879la_a_b ) ).
% UNIV_I
thf(fact_1273_sup__top__right,axiom,
! [X: set_formula_a_b] :
( ( sup_su8832426789560438003la_a_b @ X @ top_to8846066467269685879la_a_b )
= top_to8846066467269685879la_a_b ) ).
% sup_top_right
thf(fact_1274_sup__top__left,axiom,
! [X: set_formula_a_b] :
( ( sup_su8832426789560438003la_a_b @ top_to8846066467269685879la_a_b @ X )
= top_to8846066467269685879la_a_b ) ).
% sup_top_left
thf(fact_1275_boolean__algebra_Odisj__one__right,axiom,
! [X: set_formula_a_b] :
( ( sup_su8832426789560438003la_a_b @ X @ top_to8846066467269685879la_a_b )
= top_to8846066467269685879la_a_b ) ).
% boolean_algebra.disj_one_right
thf(fact_1276_boolean__algebra_Odisj__one__left,axiom,
! [X: set_formula_a_b] :
( ( sup_su8832426789560438003la_a_b @ top_to8846066467269685879la_a_b @ X )
= top_to8846066467269685879la_a_b ) ).
% boolean_algebra.disj_one_left
thf(fact_1277_boolean__algebra_Ocomplement__unique,axiom,
! [A2: set_formula_a_b,X: set_formula_a_b,Y: set_formula_a_b] :
( ( ( inf_in5741596252272946265la_a_b @ A2 @ X )
= bot_bo8568539672012708443la_a_b )
=> ( ( ( sup_su8832426789560438003la_a_b @ A2 @ X )
= top_to8846066467269685879la_a_b )
=> ( ( ( inf_in5741596252272946265la_a_b @ A2 @ Y )
= bot_bo8568539672012708443la_a_b )
=> ( ( ( sup_su8832426789560438003la_a_b @ A2 @ Y )
= top_to8846066467269685879la_a_b )
=> ( X = Y ) ) ) ) ) ).
% boolean_algebra.complement_unique
% Helper facts (9)
thf(help_If_2_1_If_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_T,axiom,
! [X: set_formula_a_b,Y: set_formula_a_b] :
( ( if_set_formula_a_b @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_T,axiom,
! [X: set_formula_a_b,Y: set_formula_a_b] :
( ( if_set_formula_a_b @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J_T,axiom,
! [X: list_formula_a_b,Y: list_formula_a_b] :
( ( if_list_formula_a_b @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J_T,axiom,
! [X: list_formula_a_b,Y: list_formula_a_b] :
( ( if_list_formula_a_b @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J_T,axiom,
! [X: list_set_formula_a_b,Y: list_set_formula_a_b] :
( ( if_lis2278976591155604753la_a_b @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J_T,axiom,
! [X: list_set_formula_a_b,Y: list_set_formula_a_b] :
( ( if_lis2278976591155604753la_a_b @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__List__Olist_It__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J_T,axiom,
! [X: list_l5948067055324603441la_a_b,Y: list_l5948067055324603441la_a_b] :
( ( if_lis7719565029871594987la_a_b @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__List__Olist_It__MDL__Oformula_Itf__a_Mtf__b_J_J_J_T,axiom,
! [X: list_l5948067055324603441la_a_b,Y: list_l5948067055324603441la_a_b] :
( ( if_lis7719565029871594987la_a_b @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( set_formula_a_b2 @ ( collect_subfmlas_a_b @ r @ phis ) )
= ( sup_su8832426789560438003la_a_b @ ( set_formula_a_b2 @ phis ) @ ( atms_a_b @ r ) ) ) ).
%------------------------------------------------------------------------------