TPTP Problem File: SLH0705^1.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 : FOL_Seq_Calc3/0010_Prover/prob_00036_001585__11979104_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1614 ( 696 unt; 331 typ; 0 def)
% Number of atoms : 3419 (1845 equ; 0 cnn)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 10837 ( 486 ~; 28 |; 342 &;8759 @)
% ( 0 <=>;1222 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 5 avg)
% Number of types : 39 ( 38 usr)
% Number of type conns : 1421 (1421 >; 0 *; 0 +; 0 <<)
% Number of symbols : 296 ( 293 usr; 37 con; 0-4 aty)
% Number of variables : 3560 ( 597 ^;2857 !; 106 ?;3560 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:27:41.411
%------------------------------------------------------------------------------
% Could-be-implicit typings (38)
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_J,type,
set_Pr5209464977148653383ist_fm: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
produc1754732471204517991ist_fm: $tType ).
thf(ty_n_t__Stream__Ostream_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_J,type,
stream727092118206550309m_rule: $tType ).
thf(ty_n_t__Set__Oset_It__Option__Ooption_It__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_J_J,type,
set_op6310664071589870893ist_fm: $tType ).
thf(ty_n_t__Option__Ooption_It__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_J,type,
option6029102822600979917ist_fm: $tType ).
thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_J,type,
set_op4811217697719039047ist_fm: $tType ).
thf(ty_n_t__Set__Oset_It__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_J,type,
set_fs7534060674568567527ist_fm: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_J,type,
set_se6775865681059004641ist_fm: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_M_Eo_J_J,type,
set_Pr6299178538722460242t_fm_o: $tType ).
thf(ty_n_t__Stream__Ostream_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
stream8299795917829157543ist_fm: $tType ).
thf(ty_n_t__Option__Ooption_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
option4065869692384770919ist_fm: $tType ).
thf(ty_n_t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
fset_P661503646757059847ist_fm: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
set_Pr7058068377845519745ist_fm: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__List__Olist_It__Syntax__Ofm_J_M_062_It__List__Olist_It__Syntax__Ofm_J_M_Eo_J_J_J,type,
set_li7142591310554048678t_fm_o: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
produc1996495991257130529ist_fm: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Syntax__Orule_Mt__Syntax__Orule_J_J,type,
set_Pr6543236023296505803e_rule: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Syntax__Orule_Mt__Syntax__Orule_J,type,
produc780299800278082923e_rule: $tType ).
thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Syntax__Orule_J_J,type,
set_option_rule: $tType ).
thf(ty_n_t__Option__Ooption_It__List__Olist_It__Syntax__Ofm_J_J,type,
option_list_fm: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
set_Product_prod_o_o: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Syntax__Orule_J_J,type,
set_set_rule: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Syntax__Ofm_J_J,type,
set_list_fm: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Syntax__Orule_M_Eo_J_J,type,
set_rule_o: $tType ).
thf(ty_n_t__Set__Oset_It__Option__Ooption_I_Eo_J_J,type,
set_option_o: $tType ).
thf(ty_n_t__Stream__Ostream_It__Syntax__Orule_J,type,
stream_rule: $tType ).
thf(ty_n_t__Option__Ooption_It__Syntax__Orule_J,type,
option_rule: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
product_prod_o_o: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
set_set_o: $tType ).
thf(ty_n_t__FSet__Ofset_It__Syntax__Orule_J,type,
fset_rule: $tType ).
thf(ty_n_t__Set__Oset_It__Syntax__Orule_J,type,
set_rule: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_Eo_M_Eo_J_J,type,
set_o_o: $tType ).
thf(ty_n_t__List__Olist_It__Syntax__Ofm_J,type,
list_fm: $tType ).
thf(ty_n_t__Stream__Ostream_I_Eo_J,type,
stream_o: $tType ).
thf(ty_n_t__Product____Type__Ounit,type,
product_unit: $tType ).
thf(ty_n_t__Option__Ooption_I_Eo_J,type,
option_o: $tType ).
thf(ty_n_t__FSet__Ofset_I_Eo_J,type,
fset_o: $tType ).
thf(ty_n_t__Set__Oset_I_Eo_J,type,
set_o: $tType ).
thf(ty_n_t__Syntax__Orule,type,
rule: $tType ).
% Explicit typings (293)
thf(sy_c_Abstract__Completeness_OPersistentRuleSystem_001t__Syntax__Orule_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
abstra7887161495092839066ist_fm: ( rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o ) > stream_rule > set_Pr7058068377845519745ist_fm > $o ).
thf(sy_c_Abstract__Completeness_OPersistentRuleSystem__axioms_001t__Syntax__Orule_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
abstra2778581409224112183ist_fm: ( rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o ) > stream_rule > set_Pr7058068377845519745ist_fm > $o ).
thf(sy_c_Abstract__Completeness_ORuleSystem_001_Eo_001_Eo,type,
abstra8727705743885569321em_o_o: ( $o > $o > fset_o > $o ) > stream_o > set_o > $o ).
thf(sy_c_Abstract__Completeness_ORuleSystem_001_Eo_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
abstra5822460560083087538ist_fm: ( $o > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o ) > stream_o > set_Pr7058068377845519745ist_fm > $o ).
thf(sy_c_Abstract__Completeness_ORuleSystem_001_Eo_001t__Syntax__Orule,type,
abstra5464778857772719348o_rule: ( $o > rule > fset_rule > $o ) > stream_o > set_rule > $o ).
thf(sy_c_Abstract__Completeness_ORuleSystem_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001_Eo,type,
abstra5561694101053127776t_fm_o: ( produc1996495991257130529ist_fm > $o > fset_o > $o ) > stream8299795917829157543ist_fm > set_o > $o ).
thf(sy_c_Abstract__Completeness_ORuleSystem_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
abstra4969282175937908457ist_fm: ( produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o ) > stream8299795917829157543ist_fm > set_Pr7058068377845519745ist_fm > $o ).
thf(sy_c_Abstract__Completeness_ORuleSystem_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Syntax__Orule,type,
abstra9106359784214577277m_rule: ( produc1996495991257130529ist_fm > rule > fset_rule > $o ) > stream8299795917829157543ist_fm > set_rule > $o ).
thf(sy_c_Abstract__Completeness_ORuleSystem_001t__Syntax__Orule_001_Eo,type,
abstra8374989651900038312rule_o: ( rule > $o > fset_o > $o ) > stream_rule > set_o > $o ).
thf(sy_c_Abstract__Completeness_ORuleSystem_001t__Syntax__Orule_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
abstra6455275554694158129ist_fm: ( rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o ) > stream_rule > set_Pr7058068377845519745ist_fm > $o ).
thf(sy_c_Abstract__Completeness_ORuleSystem_001t__Syntax__Orule_001t__Syntax__Orule,type,
abstra2725590884229999157e_rule: ( rule > rule > fset_rule > $o ) > stream_rule > set_rule > $o ).
thf(sy_c_Abstract__Completeness_ORuleSystem_Oper_001_Eo_001_Eo,type,
abstract_per_o_o: ( $o > $o > fset_o > $o ) > stream_o > set_o > $o > $o ).
thf(sy_c_Abstract__Completeness_ORuleSystem_Oper_001_Eo_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
abstra8404946688271621549ist_fm: ( $o > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o ) > stream_o > set_Pr7058068377845519745ist_fm > $o > $o ).
thf(sy_c_Abstract__Completeness_ORuleSystem_Oper_001_Eo_001t__Syntax__Orule,type,
abstract_per_o_rule: ( $o > rule > fset_rule > $o ) > stream_o > set_rule > $o > $o ).
thf(sy_c_Abstract__Completeness_ORuleSystem_Oper_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001_Eo,type,
abstra8144180229241661787t_fm_o: ( produc1996495991257130529ist_fm > $o > fset_o > $o ) > stream8299795917829157543ist_fm > set_o > produc1996495991257130529ist_fm > $o ).
thf(sy_c_Abstract__Completeness_ORuleSystem_Oper_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
abstra7567571716448718564ist_fm: ( produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o ) > stream8299795917829157543ist_fm > set_Pr7058068377845519745ist_fm > produc1996495991257130529ist_fm > $o ).
thf(sy_c_Abstract__Completeness_ORuleSystem_Oper_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Syntax__Orule,type,
abstra6804973471972959042m_rule: ( produc1996495991257130529ist_fm > rule > fset_rule > $o ) > stream8299795917829157543ist_fm > set_rule > produc1996495991257130529ist_fm > $o ).
thf(sy_c_Abstract__Completeness_ORuleSystem_Oper_001t__Syntax__Orule_001_Eo,type,
abstract_per_rule_o: ( rule > $o > fset_o > $o ) > stream_rule > set_o > rule > $o ).
thf(sy_c_Abstract__Completeness_ORuleSystem_Oper_001t__Syntax__Orule_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
abstra4153889242452539894ist_fm: ( rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o ) > stream_rule > set_Pr7058068377845519745ist_fm > rule > $o ).
thf(sy_c_Abstract__Completeness_ORuleSystem_Oper_001t__Syntax__Orule_001t__Syntax__Orule,type,
abstra5282173100671685808e_rule: ( rule > rule > fset_rule > $o ) > stream_rule > set_rule > rule > $o ).
thf(sy_c_Abstract__Completeness_ORuleSystem__Defs_OSaturated_001t__Syntax__Orule_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
abstra8647779336736900908ist_fm: ( rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o ) > stream_rule > stream727092118206550309m_rule > $o ).
thf(sy_c_Abstract__Completeness_ORuleSystem__Defs_Oenabled_001_Eo_001_Eo,type,
abstra8815064841987723930ed_o_o: ( $o > $o > fset_o > $o ) > $o > $o > $o ).
thf(sy_c_Abstract__Completeness_ORuleSystem__Defs_Oenabled_001_Eo_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
abstra6289332626269597475ist_fm: ( $o > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o ) > $o > produc1996495991257130529ist_fm > $o ).
thf(sy_c_Abstract__Completeness_ORuleSystem__Defs_Oenabled_001_Eo_001t__Syntax__Orule,type,
abstra7607554174847575747o_rule: ( $o > rule > fset_rule > $o ) > $o > rule > $o ).
thf(sy_c_Abstract__Completeness_ORuleSystem__Defs_Oenabled_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001_Eo,type,
abstra6028566167239637713t_fm_o: ( produc1996495991257130529ist_fm > $o > fset_o > $o ) > produc1996495991257130529ist_fm > $o > $o ).
thf(sy_c_Abstract__Completeness_ORuleSystem__Defs_Oenabled_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
abstra1009249261476546138ist_fm: ( produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o ) > produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > $o ).
thf(sy_c_Abstract__Completeness_ORuleSystem__Defs_Oenabled_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Syntax__Orule,type,
abstra5706625008861892940m_rule: ( produc1996495991257130529ist_fm > rule > fset_rule > $o ) > produc1996495991257130529ist_fm > rule > $o ).
thf(sy_c_Abstract__Completeness_ORuleSystem__Defs_Oenabled_001t__Syntax__Orule_001_Eo,type,
abstra1294392932120118903rule_o: ( rule > $o > fset_o > $o ) > rule > $o > $o ).
thf(sy_c_Abstract__Completeness_ORuleSystem__Defs_Oenabled_001t__Syntax__Orule_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
abstra3055540779341473792ist_fm: ( rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o ) > rule > produc1996495991257130529ist_fm > $o ).
thf(sy_c_Abstract__Completeness_ORuleSystem__Defs_Oenabled_001t__Syntax__Orule_001t__Syntax__Orule,type,
abstra3154316310710804774e_rule: ( rule > rule > fset_rule > $o ) > rule > rule > $o ).
thf(sy_c_Abstract__Completeness_ORuleSystem__Defs_Ofenum_001t__Syntax__Orule,type,
abstra7284221463285775110m_rule: stream_rule > stream_rule ).
thf(sy_c_Abstract__Completeness_ORuleSystem__Defs_OpickEff_001t__Syntax__Orule_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
abstra458490714785879965ist_fm: ( rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o ) > rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm ).
thf(sy_c_Abstract__Completeness_ORuleSystem__Defs_Osaturated_001t__Syntax__Orule_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
abstra7252176601000738572ist_fm: ( rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o ) > rule > stream727092118206550309m_rule > $o ).
thf(sy_c_Abstract__Completeness_Ofenum__uu_001t__Syntax__Orule,type,
abstra8121460317444261103u_rule: stream_rule > stream_rule ).
thf(sy_c_Basic__BNFs_Ofsts_001t__List__Olist_It__Syntax__Ofm_J_001t__List__Olist_It__Syntax__Ofm_J,type,
basic_2778179344159109705ist_fm: produc1996495991257130529ist_fm > set_list_fm ).
thf(sy_c_Basic__BNFs_Osnds_001t__List__Olist_It__Syntax__Ofm_J_001t__List__Olist_It__Syntax__Ofm_J,type,
basic_6053177629474610823ist_fm: produc1996495991257130529ist_fm > set_list_fm ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001_062_I_Eo_M_Eo_J,type,
complete_Inf_Inf_o_o: set_o_o > $o > $o ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001_062_It__List__Olist_It__Syntax__Ofm_J_M_062_It__List__Olist_It__Syntax__Ofm_J_M_Eo_J_J,type,
comple1444494742520622029t_fm_o: set_li7142591310554048678t_fm_o > list_fm > list_fm > $o ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001_062_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_M_Eo_J,type,
comple1695591244663615317t_fm_o: set_Pr6299178538722460242t_fm_o > produc1996495991257130529ist_fm > $o ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001_062_It__Syntax__Orule_M_Eo_J,type,
comple715424409190658129rule_o: set_rule_o > rule > $o ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001_Eo,type,
complete_Inf_Inf_o: set_o > $o ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_I_Eo_J,type,
comple3063163877087187839_set_o: set_set_o > set_o ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
comple5608129641975563656ist_fm: set_se6775865681059004641ist_fm > set_Pr7058068377845519745ist_fm ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Syntax__Orule_J,type,
comple5773694076043965236t_rule: set_set_rule > set_rule ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_I_Eo_M_Eo_J,type,
complete_Sup_Sup_o_o: set_o_o > $o > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__List__Olist_It__Syntax__Ofm_J_M_062_It__List__Olist_It__Syntax__Ofm_J_M_Eo_J_J,type,
comple4318906814866966707t_fm_o: set_li7142591310554048678t_fm_o > list_fm > list_fm > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_M_Eo_J,type,
comple5552222795444685295t_fm_o: set_Pr6299178538722460242t_fm_o > produc1996495991257130529ist_fm > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Syntax__Orule_M_Eo_J,type,
comple1826244231481717815rule_o: set_rule_o > rule > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_Eo,type,
complete_Sup_Sup_o: set_o > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_I_Eo_J,type,
comple90263536869209701_set_o: set_set_o > set_o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
comple1649742678916152942ist_fm: set_se6775865681059004641ist_fm > set_Pr7058068377845519745ist_fm ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Syntax__Orule_J,type,
comple2146307154184993742t_rule: set_set_rule > set_rule ).
thf(sy_c_FSet_Ofmember_001_Eo,type,
fmember_o: $o > fset_o > $o ).
thf(sy_c_FSet_Ofmember_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
fmembe3381613331217039976ist_fm: produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o ).
thf(sy_c_FSet_Ofmember_001t__Syntax__Orule,type,
fmember_rule: rule > fset_rule > $o ).
thf(sy_c_Fun_Ofun__upd_001_Eo_001_Eo,type,
fun_upd_o_o: ( $o > $o ) > $o > $o > $o > $o ).
thf(sy_c_Fun_Ofun__upd_001_Eo_001t__Option__Ooption_It__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_J,type,
fun_up1845550324453438770ist_fm: ( $o > option6029102822600979917ist_fm ) > $o > option6029102822600979917ist_fm > $o > option6029102822600979917ist_fm ).
thf(sy_c_Fun_Ofun__upd_001_Eo_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
fun_up141237966704039814ist_fm: ( $o > produc1996495991257130529ist_fm ) > $o > produc1996495991257130529ist_fm > $o > produc1996495991257130529ist_fm ).
thf(sy_c_Fun_Ofun__upd_001_Eo_001t__Syntax__Orule,type,
fun_upd_o_rule: ( $o > rule ) > $o > rule > $o > rule ).
thf(sy_c_Fun_Ofun__upd_001t__List__Olist_It__Syntax__Ofm_J_001t__Option__Ooption_It__List__Olist_It__Syntax__Ofm_J_J,type,
fun_up7757938787137709839ist_fm: ( list_fm > option_list_fm ) > list_fm > option_list_fm > list_fm > option_list_fm ).
thf(sy_c_Fun_Ofun__upd_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001_Eo,type,
fun_up9103843544528855860t_fm_o: ( produc1996495991257130529ist_fm > $o ) > produc1996495991257130529ist_fm > $o > produc1996495991257130529ist_fm > $o ).
thf(sy_c_Fun_Ofun__upd_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Option__Ooption_It__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_J,type,
fun_up1138168335916736873ist_fm: ( produc1996495991257130529ist_fm > option6029102822600979917ist_fm ) > produc1996495991257130529ist_fm > option6029102822600979917ist_fm > produc1996495991257130529ist_fm > option6029102822600979917ist_fm ).
thf(sy_c_Fun_Ofun__upd_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
fun_up6069510949241940413ist_fm: ( produc1996495991257130529ist_fm > produc1996495991257130529ist_fm ) > produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > produc1996495991257130529ist_fm ).
thf(sy_c_Fun_Ofun__upd_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Syntax__Orule,type,
fun_up5669935710502914729m_rule: ( produc1996495991257130529ist_fm > rule ) > produc1996495991257130529ist_fm > rule > produc1996495991257130529ist_fm > rule ).
thf(sy_c_Fun_Ofun__upd_001t__Syntax__Orule_001_Eo,type,
fun_upd_rule_o: ( rule > $o ) > rule > $o > rule > $o ).
thf(sy_c_Fun_Ofun__upd_001t__Syntax__Orule_001t__Option__Ooption_It__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_J,type,
fun_up5825503389150910729ist_fm: ( rule > option6029102822600979917ist_fm ) > rule > option6029102822600979917ist_fm > rule > option6029102822600979917ist_fm ).
thf(sy_c_Fun_Ofun__upd_001t__Syntax__Orule_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
fun_up3018851480982495581ist_fm: ( rule > produc1996495991257130529ist_fm ) > rule > produc1996495991257130529ist_fm > rule > produc1996495991257130529ist_fm ).
thf(sy_c_Fun_Ofun__upd_001t__Syntax__Orule_001t__Syntax__Orule,type,
fun_upd_rule_rule: ( rule > rule ) > rule > rule > rule > rule ).
thf(sy_c_Fun_Oinj__on_001_Eo_001_Eo,type,
inj_on_o_o: ( $o > $o ) > set_o > $o ).
thf(sy_c_Fun_Oinj__on_001_Eo_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
inj_on6870991201974782586ist_fm: ( $o > produc1996495991257130529ist_fm ) > set_o > $o ).
thf(sy_c_Fun_Oinj__on_001_Eo_001t__Set__Oset_I_Eo_J,type,
inj_on_o_set_o: ( $o > set_o ) > set_o > $o ).
thf(sy_c_Fun_Oinj__on_001_Eo_001t__Syntax__Orule,type,
inj_on_o_rule: ( $o > rule ) > set_o > $o ).
thf(sy_c_Fun_Oinj__on_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_001t__Option__Ooption_It__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_J,type,
inj_on1722708713623408887ist_fm: ( fset_P661503646757059847ist_fm > option6029102822600979917ist_fm ) > set_fs7534060674568567527ist_fm > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Syntax__Ofm_J_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
inj_on733428828453976846ist_fm: ( list_fm > produc1996495991257130529ist_fm ) > set_list_fm > $o ).
thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001_Eo,type,
inj_on6610224742944822824t_fm_o: ( produc1996495991257130529ist_fm > $o ) > set_Pr7058068377845519745ist_fm > $o ).
thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
inj_on2392985232935137457ist_fm: ( produc1996495991257130529ist_fm > produc1996495991257130529ist_fm ) > set_Pr7058068377845519745ist_fm > $o ).
thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
inj_on6475354550519685137ist_fm: ( produc1996495991257130529ist_fm > set_Pr7058068377845519745ist_fm ) > set_Pr7058068377845519745ist_fm > $o ).
thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Syntax__Orule,type,
inj_on6080897081453506741m_rule: ( produc1996495991257130529ist_fm > rule ) > set_Pr7058068377845519745ist_fm > $o ).
thf(sy_c_Fun_Oinj__on_001t__Syntax__Orule_001_Eo,type,
inj_on_rule_o: ( rule > $o ) > set_rule > $o ).
thf(sy_c_Fun_Oinj__on_001t__Syntax__Orule_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
inj_on3429812851933087593ist_fm: ( rule > produc1996495991257130529ist_fm ) > set_rule > $o ).
thf(sy_c_Fun_Oinj__on_001t__Syntax__Orule_001t__Set__Oset_It__Syntax__Orule_J,type,
inj_on_rule_set_rule: ( rule > set_rule ) > set_rule > $o ).
thf(sy_c_Fun_Oinj__on_001t__Syntax__Orule_001t__Syntax__Orule,type,
inj_on_rule_rule: ( rule > rule ) > set_rule > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_I_Eo_M_Eo_J,type,
minus_minus_o_o: ( $o > $o ) > ( $o > $o ) > $o > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_M_Eo_J,type,
minus_297532591554960245t_fm_o: ( produc1996495991257130529ist_fm > $o ) > ( produc1996495991257130529ist_fm > $o ) > produc1996495991257130529ist_fm > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Syntax__Orule_M_Eo_J,type,
minus_minus_rule_o: ( rule > $o ) > ( rule > $o ) > rule > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
minus_8437252545254675822ist_fm: fset_P661503646757059847ist_fm > fset_P661503646757059847ist_fm > fset_P661503646757059847ist_fm ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_Eo_J,type,
minus_minus_set_o: set_o > set_o > set_o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
minus_5615909706921521512ist_fm: set_Pr7058068377845519745ist_fm > set_Pr7058068377845519745ist_fm > set_Pr7058068377845519745ist_fm ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Syntax__Orule_J,type,
minus_minus_set_rule: set_rule > set_rule > set_rule ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_I_Eo_M_Eo_J,type,
uminus_uminus_o_o: ( $o > $o ) > $o > $o ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_M_Eo_J,type,
uminus8340389434899797733t_fm_o: ( produc1996495991257130529ist_fm > $o ) > produc1996495991257130529ist_fm > $o ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Syntax__Orule_M_Eo_J,type,
uminus_uminus_rule_o: ( rule > $o ) > rule > $o ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_I_Eo_J,type,
uminus_uminus_set_o: set_o > set_o ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
uminus6150396152209328632ist_fm: set_Pr7058068377845519745ist_fm > set_Pr7058068377845519745ist_fm ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Syntax__Orule_J,type,
uminus4869265918275750596t_rule: set_rule > set_rule ).
thf(sy_c_HOL_OThe_001_Eo,type,
the_o: ( $o > $o ) > $o ).
thf(sy_c_HOL_OThe_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
the_Pr8646916617578001576ist_fm: ( produc1996495991257130529ist_fm > $o ) > produc1996495991257130529ist_fm ).
thf(sy_c_HOL_OThe_001t__Syntax__Orule,type,
the_rule: ( rule > $o ) > rule ).
thf(sy_c_If_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
if_fse3714977293194272717ist_fm: $o > fset_P661503646757059847ist_fm > fset_P661503646757059847ist_fm > fset_P661503646757059847ist_fm ).
thf(sy_c_If_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
if_Pro3930376587665744871ist_fm: $o > produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > produc1996495991257130529ist_fm ).
thf(sy_c_Lattices_Osup__class_Osup_001_Eo,type,
sup_sup_o: $o > $o > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_Eo_J,type,
sup_sup_set_o: set_o > set_o > set_o ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
sup_su3073845231316762965ist_fm: set_Pr7058068377845519745ist_fm > set_Pr7058068377845519745ist_fm > set_Pr7058068377845519745ist_fm ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Syntax__Orule_J,type,
sup_sup_set_rule: set_rule > set_rule > set_rule ).
thf(sy_c_Map_Odom_001_Eo_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
dom_o_6128958874678448065ist_fm: ( $o > option6029102822600979917ist_fm ) > set_o ).
thf(sy_c_Map_Odom_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
dom_Pr1119034465550817144ist_fm: ( produc1996495991257130529ist_fm > option6029102822600979917ist_fm ) > set_Pr7058068377845519745ist_fm ).
thf(sy_c_Map_Odom_001t__Syntax__Orule_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
dom_ru6064144267908753070ist_fm: ( rule > option6029102822600979917ist_fm ) > set_rule ).
thf(sy_c_Map_Ograph_001t__List__Olist_It__Syntax__Ofm_J_001t__List__Olist_It__Syntax__Ofm_J,type,
graph_2416636321808849472ist_fm: ( list_fm > option_list_fm ) > set_Pr7058068377845519745ist_fm ).
thf(sy_c_Map_Orestrict__map_001_Eo_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
restri2688269635790113695ist_fm: ( $o > option6029102822600979917ist_fm ) > set_o > $o > option6029102822600979917ist_fm ).
thf(sy_c_Map_Orestrict__map_001t__List__Olist_It__Syntax__Ofm_J_001t__List__Olist_It__Syntax__Ofm_J,type,
restri8510528498258177010ist_fm: ( list_fm > option_list_fm ) > set_list_fm > list_fm > option_list_fm ).
thf(sy_c_Map_Orestrict__map_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
restri3026782591808725334ist_fm: ( produc1996495991257130529ist_fm > option6029102822600979917ist_fm ) > set_Pr7058068377845519745ist_fm > produc1996495991257130529ist_fm > option6029102822600979917ist_fm ).
thf(sy_c_Map_Orestrict__map_001t__Syntax__Orule_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
restri3791882098703154256ist_fm: ( rule > option6029102822600979917ist_fm ) > set_rule > rule > option6029102822600979917ist_fm ).
thf(sy_c_Option_Ooption_ONone_001_Eo,type,
none_o: option_o ).
thf(sy_c_Option_Ooption_ONone_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
none_f2701453242279119720ist_fm: option6029102822600979917ist_fm ).
thf(sy_c_Option_Ooption_ONone_001t__List__Olist_It__Syntax__Ofm_J,type,
none_list_fm: option_list_fm ).
thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
none_P1680826464017717506ist_fm: option4065869692384770919ist_fm ).
thf(sy_c_Option_Ooption_ONone_001t__Syntax__Orule,type,
none_rule: option_rule ).
thf(sy_c_Option_Ooption_OSome_001_Eo,type,
some_o: $o > option_o ).
thf(sy_c_Option_Ooption_OSome_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
some_f2162114139285431020ist_fm: fset_P661503646757059847ist_fm > option6029102822600979917ist_fm ).
thf(sy_c_Option_Ooption_OSome_001t__List__Olist_It__Syntax__Ofm_J,type,
some_list_fm: list_fm > option_list_fm ).
thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
some_P7346312016712434310ist_fm: produc1996495991257130529ist_fm > option4065869692384770919ist_fm ).
thf(sy_c_Option_Ooption_OSome_001t__Syntax__Orule,type,
some_rule: rule > option_rule ).
thf(sy_c_Option_Othese_001_Eo,type,
these_o: set_option_o > set_o ).
thf(sy_c_Option_Othese_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
these_4231445979625565508ist_fm: set_op6310664071589870893ist_fm > set_fs7534060674568567527ist_fm ).
thf(sy_c_Option_Othese_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
these_2172363061839534302ist_fm: set_op4811217697719039047ist_fm > set_Pr7058068377845519745ist_fm ).
thf(sy_c_Option_Othese_001t__Syntax__Orule,type,
these_rule: set_option_rule > set_rule ).
thf(sy_c_Order__Relation_OaboveS_001_Eo,type,
order_aboveS_o: set_Product_prod_o_o > $o > set_o ).
thf(sy_c_Order__Relation_OaboveS_001t__List__Olist_It__Syntax__Ofm_J,type,
order_aboveS_list_fm: set_Pr7058068377845519745ist_fm > list_fm > set_list_fm ).
thf(sy_c_Order__Relation_OaboveS_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
order_744099749760756108ist_fm: set_Pr5209464977148653383ist_fm > produc1996495991257130529ist_fm > set_Pr7058068377845519745ist_fm ).
thf(sy_c_Order__Relation_OaboveS_001t__Syntax__Orule,type,
order_aboveS_rule: set_Pr6543236023296505803e_rule > rule > set_rule ).
thf(sy_c_Order__Relation_Oabove_001_Eo,type,
order_above_o: set_Product_prod_o_o > $o > set_o ).
thf(sy_c_Order__Relation_Oabove_001t__List__Olist_It__Syntax__Ofm_J,type,
order_above_list_fm: set_Pr7058068377845519745ist_fm > list_fm > set_list_fm ).
thf(sy_c_Order__Relation_Oabove_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
order_5287731903943553313ist_fm: set_Pr5209464977148653383ist_fm > produc1996495991257130529ist_fm > set_Pr7058068377845519745ist_fm ).
thf(sy_c_Order__Relation_Oabove_001t__Syntax__Orule,type,
order_above_rule: set_Pr6543236023296505803e_rule > rule > set_rule ).
thf(sy_c_Order__Relation_Olinear__order__on_001_Eo,type,
order_6238756238976269133r_on_o: set_o > set_Product_prod_o_o > $o ).
thf(sy_c_Order__Relation_Olinear__order__on_001t__List__Olist_It__Syntax__Ofm_J,type,
order_4446669488389765227ist_fm: set_list_fm > set_Pr7058068377845519745ist_fm > $o ).
thf(sy_c_Order__Relation_Olinear__order__on_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
order_193637769215337174ist_fm: set_Pr7058068377845519745ist_fm > set_Pr5209464977148653383ist_fm > $o ).
thf(sy_c_Order__Relation_Olinear__order__on_001t__Syntax__Orule,type,
order_3615580131159368656n_rule: set_rule > set_Pr6543236023296505803e_rule > $o ).
thf(sy_c_Order__Relation_Opreorder__on_001_Eo,type,
order_preorder_on_o: set_o > set_Product_prod_o_o > $o ).
thf(sy_c_Order__Relation_Opreorder__on_001t__List__Olist_It__Syntax__Ofm_J,type,
order_2819943312236725001ist_fm: set_list_fm > set_Pr7058068377845519745ist_fm > $o ).
thf(sy_c_Order__Relation_Opreorder__on_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
order_678247264901640376ist_fm: set_Pr7058068377845519745ist_fm > set_Pr5209464977148653383ist_fm > $o ).
thf(sy_c_Order__Relation_Opreorder__on_001t__Syntax__Orule,type,
order_5092310374046361198n_rule: set_rule > set_Pr6543236023296505803e_rule > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_I_Eo_M_Eo_J,type,
bot_bot_o_o: $o > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__List__Olist_It__Syntax__Ofm_J_M_062_It__List__Olist_It__Syntax__Ofm_J_M_Eo_J_J,type,
bot_bo2780323056127860274t_fm_o: list_fm > list_fm > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_M_Eo_J,type,
bot_bo1394704948141485872t_fm_o: produc1996495991257130529ist_fm > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Syntax__Orule_M_Eo_J,type,
bot_bot_rule_o: rule > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_Eo,type,
bot_bot_o: $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
bot_bo2367426573206113139ist_fm: fset_P661503646757059847ist_fm ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_Eo_J,type,
bot_bot_set_o: set_o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_It__Syntax__Ofm_J_J,type,
bot_bot_set_list_fm: set_list_fm ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Option__Ooption_I_Eo_J_J,type,
bot_bot_set_option_o: set_option_o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Option__Ooption_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_J,type,
bot_bo4262429147648288435ist_fm: set_op4811217697719039047ist_fm ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Option__Ooption_It__Syntax__Orule_J_J,type,
bot_bo8966923457718485855n_rule: set_option_rule ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
bot_bo7073875226086086771od_o_o: set_Product_prod_o_o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
bot_bo4530011010606209517ist_fm: set_Pr7058068377845519745ist_fm ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_J,type,
bot_bo7330262581962992051ist_fm: set_Pr5209464977148653383ist_fm ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Syntax__Orule_Mt__Syntax__Orule_J_J,type,
bot_bo9047865283180780599e_rule: set_Pr6543236023296505803e_rule ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
bot_bot_set_set_o: set_set_o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_J,type,
bot_bo414211191905506637ist_fm: set_se6775865681059004641ist_fm ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Syntax__Orule_J_J,type,
bot_bot_set_set_rule: set_set_rule ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Syntax__Orule_J,type,
bot_bot_set_rule: set_rule ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_I_Eo_M_Eo_J,type,
top_top_o_o: $o > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__List__Olist_It__Syntax__Ofm_J_M_062_It__List__Olist_It__Syntax__Ofm_J_M_Eo_J_J,type,
top_to7326020031456794390t_fm_o: list_fm > list_fm > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_M_Eo_J,type,
top_to6798613745154499404t_fm_o: produc1996495991257130529ist_fm > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Syntax__Orule_M_Eo_J,type,
top_top_rule_o: rule > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_Eo,type,
top_top_o: $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
top_top_set_o: set_o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_J,type,
top_to6886830657606236215ist_fm: set_fs7534060674568567527ist_fm ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__List__Olist_It__Syntax__Ofm_J_J,type,
top_top_set_list_fm: set_list_fm ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_I_Eo_J_J,type,
top_top_set_option_o: set_option_o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_It__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_J_J,type,
top_to170523636175635069ist_fm: set_op6310664071589870893ist_fm ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_J,type,
top_to2023577309291471255ist_fm: set_op4811217697719039047ist_fm ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_It__Syntax__Orule_J_J,type,
top_to4271146702142796155n_rule: set_option_rule ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
top_to728902761071950545ist_fm: set_Pr7058068377845519745ist_fm ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
top_top_set_set_o: set_set_o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_J,type,
top_to7149953857590879793ist_fm: set_se6775865681059004641ist_fm ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Syntax__Orule_J_J,type,
top_top_set_set_rule: set_set_rule ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Syntax__Orule_J,type,
top_top_set_rule: set_rule ).
thf(sy_c_Product__Type_OPair_001_Eo_001_Eo,type,
product_Pair_o_o: $o > $o > product_prod_o_o ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Syntax__Ofm_J_001t__List__Olist_It__Syntax__Ofm_J,type,
produc381145313068854617ist_fm: list_fm > list_fm > produc1996495991257130529ist_fm ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
produc2592314732767339223ist_fm: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > produc1754732471204517991ist_fm ).
thf(sy_c_Product__Type_OPair_001t__Syntax__Orule_001t__Syntax__Orule,type,
produc1769597910952761123e_rule: rule > rule > produc780299800278082923e_rule ).
thf(sy_c_Product__Type_Oapsnd_001t__List__Olist_It__Syntax__Ofm_J_001t__List__Olist_It__Syntax__Ofm_J_001t__List__Olist_It__Syntax__Ofm_J,type,
produc9047611296996456864ist_fm: ( list_fm > list_fm ) > produc1996495991257130529ist_fm > produc1996495991257130529ist_fm ).
thf(sy_c_Product__Type_Ounit_OAbs__unit,type,
product_Abs_unit: $o > product_unit ).
thf(sy_c_Product__Type_Ounit_ORep__unit,type,
product_Rep_unit: product_unit > $o ).
thf(sy_c_Prover_Oeff,type,
eff: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm ).
thf(sy_c_Prover_Orules,type,
rules: stream_rule ).
thf(sy_c_Relation_ODomain_001t__List__Olist_It__Syntax__Ofm_J_001t__List__Olist_It__Syntax__Ofm_J,type,
domain2949914093644414030ist_fm: set_Pr7058068377845519745ist_fm > set_list_fm ).
thf(sy_c_Relation_OId__on_001_Eo,type,
id_on_o: set_o > set_Product_prod_o_o ).
thf(sy_c_Relation_OId__on_001t__List__Olist_It__Syntax__Ofm_J,type,
id_on_list_fm: set_list_fm > set_Pr7058068377845519745ist_fm ).
thf(sy_c_Relation_OId__on_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
id_on_4358887334071312446ist_fm: set_Pr7058068377845519745ist_fm > set_Pr5209464977148653383ist_fm ).
thf(sy_c_Relation_OId__on_001t__Syntax__Orule,type,
id_on_rule: set_rule > set_Pr6543236023296505803e_rule ).
thf(sy_c_Relation_ORange_001t__List__Olist_It__Syntax__Ofm_J_001t__List__Olist_It__Syntax__Ofm_J,type,
range_8771097359499353911ist_fm: set_Pr7058068377845519745ist_fm > set_list_fm ).
thf(sy_c_Relation_ORangep_001t__List__Olist_It__Syntax__Ofm_J_001t__List__Olist_It__Syntax__Ofm_J,type,
rangep3205624139766315645ist_fm: ( list_fm > list_fm > $o ) > list_fm > $o ).
thf(sy_c_Relation_Orefl__on_001_Eo,type,
refl_on_o: set_o > set_Product_prod_o_o > $o ).
thf(sy_c_Relation_Orefl__on_001t__List__Olist_It__Syntax__Ofm_J,type,
refl_on_list_fm: set_list_fm > set_Pr7058068377845519745ist_fm > $o ).
thf(sy_c_Relation_Orefl__on_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
refl_o4727633711164096160ist_fm: set_Pr7058068377845519745ist_fm > set_Pr5209464977148653383ist_fm > $o ).
thf(sy_c_Relation_Orefl__on_001t__Syntax__Orule,type,
refl_on_rule: set_rule > set_Pr6543236023296505803e_rule > $o ).
thf(sy_c_Relation_Otrans__on_001_Eo,type,
trans_on_o: set_o > set_Product_prod_o_o > $o ).
thf(sy_c_Relation_Otrans__on_001t__List__Olist_It__Syntax__Ofm_J,type,
trans_on_list_fm: set_list_fm > set_Pr7058068377845519745ist_fm > $o ).
thf(sy_c_Relation_Otrans__on_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
trans_4095360997890102209ist_fm: set_Pr7058068377845519745ist_fm > set_Pr5209464977148653383ist_fm > $o ).
thf(sy_c_Relation_Otrans__on_001t__Syntax__Orule,type,
trans_on_rule: set_rule > set_Pr6543236023296505803e_rule > $o ).
thf(sy_c_Set_OCollect_001_Eo,type,
collect_o: ( $o > $o ) > set_o ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Syntax__Ofm_J,type,
collect_list_fm: ( list_fm > $o ) > set_list_fm ).
thf(sy_c_Set_OCollect_001t__Option__Ooption_It__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_J,type,
collec8886999257175080888ist_fm: ( option6029102822600979917ist_fm > $o ) > set_op6310664071589870893ist_fm ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
collec7231575776666803724ist_fm: ( produc1996495991257130529ist_fm > $o ) > set_Pr7058068377845519745ist_fm ).
thf(sy_c_Set_OCollect_001t__Syntax__Orule,type,
collect_rule: ( rule > $o ) > set_rule ).
thf(sy_c_Set_Oimage_001_062_I_Eo_M_Eo_J_001t__Set__Oset_I_Eo_J,type,
image_o_o_set_o: ( ( $o > $o ) > set_o ) > set_o_o > set_set_o ).
thf(sy_c_Set_Oimage_001_062_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_M_Eo_J_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
image_3053959355298890014ist_fm: ( ( produc1996495991257130529ist_fm > $o ) > set_Pr7058068377845519745ist_fm ) > set_Pr6299178538722460242t_fm_o > set_se6775865681059004641ist_fm ).
thf(sy_c_Set_Oimage_001_062_It__Syntax__Orule_M_Eo_J_001t__Set__Oset_It__Syntax__Orule_J,type,
image_1281159361656534528t_rule: ( ( rule > $o ) > set_rule ) > set_rule_o > set_set_rule ).
thf(sy_c_Set_Oimage_001_Eo_001_Eo,type,
image_o_o: ( $o > $o ) > set_o > set_o ).
thf(sy_c_Set_Oimage_001_Eo_001t__Option__Ooption_I_Eo_J,type,
image_o_option_o: ( $o > option_o ) > set_o > set_option_o ).
thf(sy_c_Set_Oimage_001_Eo_001t__Option__Ooption_It__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_J,type,
image_7906517863533482170ist_fm: ( $o > option6029102822600979917ist_fm ) > set_o > set_op6310664071589870893ist_fm ).
thf(sy_c_Set_Oimage_001_Eo_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
image_3689732601364923150ist_fm: ( $o > produc1996495991257130529ist_fm ) > set_o > set_Pr7058068377845519745ist_fm ).
thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_I_Eo_J,type,
image_o_set_o: ( $o > set_o ) > set_o > set_set_o ).
thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
image_1666948324336172654ist_fm: ( $o > set_Pr7058068377845519745ist_fm ) > set_o > set_se6775865681059004641ist_fm ).
thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_It__Syntax__Orule_J,type,
image_o_set_rule: ( $o > set_rule ) > set_o > set_set_rule ).
thf(sy_c_Set_Oimage_001_Eo_001t__Syntax__Orule,type,
image_o_rule: ( $o > rule ) > set_o > set_rule ).
thf(sy_c_Set_Oimage_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_001t__Option__Ooption_It__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_J,type,
image_4775136921032825739ist_fm: ( fset_P661503646757059847ist_fm > option6029102822600979917ist_fm ) > set_fs7534060674568567527ist_fm > set_op6310664071589870893ist_fm ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Syntax__Ofm_J_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
image_4561957376826621658ist_fm: ( list_fm > set_Pr7058068377845519745ist_fm ) > set_list_fm > set_se6775865681059004641ist_fm ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001_Eo,type,
image_3428966142334963388t_fm_o: ( produc1996495991257130529ist_fm > $o ) > set_Pr7058068377845519745ist_fm > set_o ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Option__Ooption_It__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_J,type,
image_5753375111473792241ist_fm: ( produc1996495991257130529ist_fm > option6029102822600979917ist_fm ) > set_Pr7058068377845519745ist_fm > set_op6310664071589870893ist_fm ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Option__Ooption_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
image_4446678087167650955ist_fm: ( produc1996495991257130529ist_fm > option4065869692384770919ist_fm ) > set_Pr7058068377845519745ist_fm > set_op4811217697719039047ist_fm ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
image_6982201141141708101ist_fm: ( produc1996495991257130529ist_fm > produc1996495991257130529ist_fm ) > set_Pr7058068377845519745ist_fm > set_Pr7058068377845519745ist_fm ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Set__Oset_I_Eo_J,type,
image_7087520594942641564_set_o: ( produc1996495991257130529ist_fm > set_o ) > set_Pr7058068377845519745ist_fm > set_set_o ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
image_5852275405373602981ist_fm: ( produc1996495991257130529ist_fm > set_Pr7058068377845519745ist_fm ) > set_Pr7058068377845519745ist_fm > set_se6775865681059004641ist_fm ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Set__Oset_It__Syntax__Orule_J,type,
image_2008118025656205143t_rule: ( produc1996495991257130529ist_fm > set_rule ) > set_Pr7058068377845519745ist_fm > set_set_rule ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Syntax__Orule,type,
image_368924427985557793m_rule: ( produc1996495991257130529ist_fm > rule ) > set_Pr7058068377845519745ist_fm > set_rule ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_Eo_J_001_062_I_Eo_M_Eo_J,type,
image_set_o_o_o: ( set_o > $o > $o ) > set_set_o > set_o_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_Eo_J_001_Eo,type,
image_set_o_o: ( set_o > $o ) > set_set_o > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_Eo_J_001t__Set__Oset_I_Eo_J,type,
image_set_o_set_o: ( set_o > set_o ) > set_set_o > set_set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_001_062_It__List__Olist_It__Syntax__Ofm_J_M_062_It__List__Olist_It__Syntax__Ofm_J_M_Eo_J_J,type,
image_6629374399087694218t_fm_o: ( set_Pr7058068377845519745ist_fm > list_fm > list_fm > $o ) > set_se6775865681059004641ist_fm > set_li7142591310554048678t_fm_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_001_062_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_M_Eo_J,type,
image_1461364200386988504t_fm_o: ( set_Pr7058068377845519745ist_fm > produc1996495991257130529ist_fm > $o ) > set_se6775865681059004641ist_fm > set_Pr6299178538722460242t_fm_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_001_Eo,type,
image_6697058094847422300t_fm_o: ( set_Pr7058068377845519745ist_fm > $o ) > set_se6775865681059004641ist_fm > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
image_7201469801630334277ist_fm: ( set_Pr7058068377845519745ist_fm > set_Pr7058068377845519745ist_fm ) > set_se6775865681059004641ist_fm > set_se6775865681059004641ist_fm ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Syntax__Orule_J_001_062_It__Syntax__Orule_M_Eo_J,type,
image_5795755989128758216rule_o: ( set_rule > rule > $o ) > set_set_rule > set_rule_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Syntax__Orule_J_001_Eo,type,
image_set_rule_o: ( set_rule > $o ) > set_set_rule > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Syntax__Orule_J_001t__Set__Oset_It__Syntax__Orule_J,type,
image_2455769455774476541t_rule: ( set_rule > set_rule ) > set_set_rule > set_set_rule ).
thf(sy_c_Set_Oimage_001t__Syntax__Orule_001_Eo,type,
image_rule_o: ( rule > $o ) > set_rule > set_o ).
thf(sy_c_Set_Oimage_001t__Syntax__Orule_001t__Option__Ooption_It__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_J,type,
image_529690880457373569ist_fm: ( rule > option6029102822600979917ist_fm ) > set_rule > set_op6310664071589870893ist_fm ).
thf(sy_c_Set_Oimage_001t__Syntax__Orule_001t__Option__Ooption_It__Syntax__Orule_J,type,
image_766640639675677281n_rule: ( rule > option_rule ) > set_rule > set_option_rule ).
thf(sy_c_Set_Oimage_001t__Syntax__Orule_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
image_6941212235319914453ist_fm: ( rule > produc1996495991257130529ist_fm ) > set_rule > set_Pr7058068377845519745ist_fm ).
thf(sy_c_Set_Oimage_001t__Syntax__Orule_001t__Set__Oset_I_Eo_J,type,
image_rule_set_o: ( rule > set_o ) > set_rule > set_set_o ).
thf(sy_c_Set_Oimage_001t__Syntax__Orule_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
image_3897101447128207157ist_fm: ( rule > set_Pr7058068377845519745ist_fm ) > set_rule > set_se6775865681059004641ist_fm ).
thf(sy_c_Set_Oimage_001t__Syntax__Orule_001t__Set__Oset_It__Syntax__Orule_J,type,
image_rule_set_rule: ( rule > set_rule ) > set_rule > set_set_rule ).
thf(sy_c_Set_Oimage_001t__Syntax__Orule_001t__Syntax__Orule,type,
image_rule_rule: ( rule > rule ) > set_rule > set_rule ).
thf(sy_c_Set_Oinsert_001_Eo,type,
insert_o: $o > set_o > set_o ).
thf(sy_c_Set_Oinsert_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
insert6096777853467859671ist_fm: fset_P661503646757059847ist_fm > set_fs7534060674568567527ist_fm > set_fs7534060674568567527ist_fm ).
thf(sy_c_Set_Oinsert_001t__List__Olist_It__Syntax__Ofm_J,type,
insert_list_fm: list_fm > set_list_fm > set_list_fm ).
thf(sy_c_Set_Oinsert_001t__Option__Ooption_I_Eo_J,type,
insert_option_o: option_o > set_option_o > set_option_o ).
thf(sy_c_Set_Oinsert_001t__Option__Ooption_It__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_J,type,
insert31278938154982301ist_fm: option6029102822600979917ist_fm > set_op6310664071589870893ist_fm > set_op6310664071589870893ist_fm ).
thf(sy_c_Set_Oinsert_001t__Option__Ooption_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
insert7172606565080779063ist_fm: option4065869692384770919ist_fm > set_op4811217697719039047ist_fm > set_op4811217697719039047ist_fm ).
thf(sy_c_Set_Oinsert_001t__Option__Ooption_It__Syntax__Orule_J,type,
insert_option_rule: option_rule > set_option_rule > set_option_rule ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
insert6201435330877294327od_o_o: product_prod_o_o > set_Product_prod_o_o > set_Product_prod_o_o ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
insert589278167076416497ist_fm: produc1996495991257130529ist_fm > set_Pr7058068377845519745ist_fm > set_Pr7058068377845519745ist_fm ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
insert5410920824581998135ist_fm: produc1754732471204517991ist_fm > set_Pr5209464977148653383ist_fm > set_Pr5209464977148653383ist_fm ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Syntax__Orule_Mt__Syntax__Orule_J,type,
insert3755751280551455547e_rule: produc780299800278082923e_rule > set_Pr6543236023296505803e_rule > set_Pr6543236023296505803e_rule ).
thf(sy_c_Set_Oinsert_001t__Syntax__Orule,type,
insert_rule: rule > set_rule > set_rule ).
thf(sy_c_Set_Ois__empty_001_Eo,type,
is_empty_o: set_o > $o ).
thf(sy_c_Set_Ois__empty_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
is_emp4743207889117482483ist_fm: set_Pr7058068377845519745ist_fm > $o ).
thf(sy_c_Set_Ois__empty_001t__Syntax__Orule,type,
is_empty_rule: set_rule > $o ).
thf(sy_c_Set_Ois__singleton_001_Eo,type,
is_singleton_o: set_o > $o ).
thf(sy_c_Set_Ois__singleton_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
is_sin3852095833271400213ist_fm: set_Pr7058068377845519745ist_fm > $o ).
thf(sy_c_Set_Ois__singleton_001t__Syntax__Orule,type,
is_singleton_rule: set_rule > $o ).
thf(sy_c_Set_Opairwise_001_Eo,type,
pairwise_o: ( $o > $o > $o ) > set_o > $o ).
thf(sy_c_Set_Opairwise_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
pairwi155973253597903628ist_fm: ( produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > $o ) > set_Pr7058068377845519745ist_fm > $o ).
thf(sy_c_Set_Opairwise_001t__Syntax__Orule,type,
pairwise_rule: ( rule > rule > $o ) > set_rule > $o ).
thf(sy_c_Set_Oremove_001_Eo,type,
remove_o: $o > set_o > set_o ).
thf(sy_c_Set_Oremove_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
remove6605582569798670470ist_fm: produc1996495991257130529ist_fm > set_Pr7058068377845519745ist_fm > set_Pr7058068377845519745ist_fm ).
thf(sy_c_Set_Oremove_001t__Syntax__Orule,type,
remove_rule: rule > set_rule > set_rule ).
thf(sy_c_Set_Othe__elem_001_Eo,type,
the_elem_o: set_o > $o ).
thf(sy_c_Set_Othe__elem_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
the_el1929169733952065430ist_fm: set_Pr7058068377845519745ist_fm > produc1996495991257130529ist_fm ).
thf(sy_c_Set_Othe__elem_001t__Syntax__Orule,type,
the_elem_rule: set_rule > rule ).
thf(sy_c_Stream_Ostream_Osset_001_Eo,type,
sset_o: stream_o > set_o ).
thf(sy_c_Stream_Ostream_Osset_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
sset_P3431374123360452149ist_fm: stream8299795917829157543ist_fm > set_Pr7058068377845519745ist_fm ).
thf(sy_c_Stream_Ostream_Osset_001t__Syntax__Orule,type,
sset_rule: stream_rule > set_rule ).
thf(sy_c_Typedef_Otype__definition_001t__Product____Type__Ounit_001_Eo,type,
type_d6188575255521822967unit_o: ( product_unit > $o ) > ( $o > product_unit ) > set_o > $o ).
thf(sy_c_member_001_Eo,type,
member_o: $o > set_o > $o ).
thf(sy_c_member_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
member3493611713653719344ist_fm: fset_P661503646757059847ist_fm > set_fs7534060674568567527ist_fm > $o ).
thf(sy_c_member_001t__List__Olist_It__Syntax__Ofm_J,type,
member_list_fm: list_fm > set_list_fm > $o ).
thf(sy_c_member_001t__Option__Ooption_I_Eo_J,type,
member_option_o: option_o > set_option_o > $o ).
thf(sy_c_member_001t__Option__Ooption_It__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_J,type,
member1838329594270681206ist_fm: option6029102822600979917ist_fm > set_op6310664071589870893ist_fm > $o ).
thf(sy_c_member_001t__Option__Ooption_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
member772309659248424592ist_fm: option4065869692384770919ist_fm > set_op4811217697719039047ist_fm > $o ).
thf(sy_c_member_001t__Option__Ooption_It__Syntax__Orule_J,type,
member_option_rule: option_rule > set_option_rule > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
member7466972457876170832od_o_o: product_prod_o_o > set_Product_prod_o_o > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
member8102475879199740618ist_fm: produc1996495991257130529ist_fm > set_Pr7058068377845519745ist_fm > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
member60238275747228048ist_fm: produc1754732471204517991ist_fm > set_Pr5209464977148653383ist_fm > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Syntax__Orule_Mt__Syntax__Orule_J,type,
member3753523862196722196e_rule: produc780299800278082923e_rule > set_Pr6543236023296505803e_rule > $o ).
thf(sy_c_member_001t__Set__Oset_I_Eo_J,type,
member_set_o: set_o > set_set_o > $o ).
thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
member8436620017354071594ist_fm: set_Pr7058068377845519745ist_fm > set_se6775865681059004641ist_fm > $o ).
thf(sy_c_member_001t__Set__Oset_It__Syntax__Orule_J,type,
member_set_rule: set_rule > set_set_rule > $o ).
thf(sy_c_member_001t__Syntax__Orule,type,
member_rule: rule > set_rule > $o ).
thf(sy_v_r,type,
r: rule ).
% Relevant facts (1277)
thf(fact_0_per__rules_H,axiom,
! [R: rule,A: list_fm,B: list_fm,A2: list_fm,B2: list_fm,R2: rule,Ss: fset_P661503646757059847ist_fm] :
( ( abstra3055540779341473792ist_fm
@ ^ [R3: rule,S: produc1996495991257130529ist_fm,Ss2: fset_P661503646757059847ist_fm] :
( ( eff @ R3 @ S )
= ( some_f2162114139285431020ist_fm @ Ss2 ) )
@ R
@ ( produc381145313068854617ist_fm @ A @ B ) )
=> ( ~ ( abstra3055540779341473792ist_fm
@ ^ [R3: rule,S: produc1996495991257130529ist_fm,Ss2: fset_P661503646757059847ist_fm] :
( ( eff @ R3 @ S )
= ( some_f2162114139285431020ist_fm @ Ss2 ) )
@ R
@ ( produc381145313068854617ist_fm @ A2 @ B2 ) )
=> ( ( ( eff @ R2 @ ( produc381145313068854617ist_fm @ A @ B ) )
= ( some_f2162114139285431020ist_fm @ Ss ) )
=> ( ( fmembe3381613331217039976ist_fm @ ( produc381145313068854617ist_fm @ A2 @ B2 ) @ Ss )
=> ( R2 = R ) ) ) ) ) ).
% per_rules'
thf(fact_1_enabled__def,axiom,
! [R: rule,S2: produc1996495991257130529ist_fm] :
( ( abstra3055540779341473792ist_fm
@ ^ [R3: rule,S: produc1996495991257130529ist_fm,Ss2: fset_P661503646757059847ist_fm] :
( ( eff @ R3 @ S )
= ( some_f2162114139285431020ist_fm @ Ss2 ) )
@ R
@ S2 )
= ( ? [Sl: fset_P661503646757059847ist_fm] :
( ( eff @ R @ S2 )
= ( some_f2162114139285431020ist_fm @ Sl ) ) ) ) ).
% enabled_def
thf(fact_2_insert__Diff__single,axiom,
! [A3: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm] :
( ( insert589278167076416497ist_fm @ A3 @ ( minus_5615909706921521512ist_fm @ A @ ( insert589278167076416497ist_fm @ A3 @ bot_bo4530011010606209517ist_fm ) ) )
= ( insert589278167076416497ist_fm @ A3 @ A ) ) ).
% insert_Diff_single
thf(fact_3_insert__Diff__single,axiom,
! [A3: $o,A: set_o] :
( ( insert_o @ A3 @ ( minus_minus_set_o @ A @ ( insert_o @ A3 @ bot_bot_set_o ) ) )
= ( insert_o @ A3 @ A ) ) ).
% insert_Diff_single
thf(fact_4_insert__Diff__single,axiom,
! [A3: rule,A: set_rule] :
( ( insert_rule @ A3 @ ( minus_minus_set_rule @ A @ ( insert_rule @ A3 @ bot_bot_set_rule ) ) )
= ( insert_rule @ A3 @ A ) ) ).
% insert_Diff_single
thf(fact_5_Diff__UNIV,axiom,
! [A: set_Pr7058068377845519745ist_fm] :
( ( minus_5615909706921521512ist_fm @ A @ top_to728902761071950545ist_fm )
= bot_bo4530011010606209517ist_fm ) ).
% Diff_UNIV
thf(fact_6_Diff__UNIV,axiom,
! [A: set_o] :
( ( minus_minus_set_o @ A @ top_top_set_o )
= bot_bot_set_o ) ).
% Diff_UNIV
thf(fact_7_Diff__UNIV,axiom,
! [A: set_rule] :
( ( minus_minus_set_rule @ A @ top_top_set_rule )
= bot_bot_set_rule ) ).
% Diff_UNIV
thf(fact_8_singleton__conv,axiom,
! [A3: produc1996495991257130529ist_fm] :
( ( collec7231575776666803724ist_fm
@ ^ [X: produc1996495991257130529ist_fm] : ( X = A3 ) )
= ( insert589278167076416497ist_fm @ A3 @ bot_bo4530011010606209517ist_fm ) ) ).
% singleton_conv
thf(fact_9_singleton__conv,axiom,
! [A3: rule] :
( ( collect_rule
@ ^ [X: rule] : ( X = A3 ) )
= ( insert_rule @ A3 @ bot_bot_set_rule ) ) ).
% singleton_conv
thf(fact_10_singleton__conv,axiom,
! [A3: $o] :
( ( collect_o
@ ^ [X: $o] : ( X = A3 ) )
= ( insert_o @ A3 @ bot_bot_set_o ) ) ).
% singleton_conv
thf(fact_11_singleton__conv2,axiom,
! [A3: produc1996495991257130529ist_fm] :
( ( collec7231575776666803724ist_fm
@ ( ^ [Y: produc1996495991257130529ist_fm,Z: produc1996495991257130529ist_fm] : ( Y = Z )
@ A3 ) )
= ( insert589278167076416497ist_fm @ A3 @ bot_bo4530011010606209517ist_fm ) ) ).
% singleton_conv2
thf(fact_12_singleton__conv2,axiom,
! [A3: rule] :
( ( collect_rule
@ ( ^ [Y: rule,Z: rule] : ( Y = Z )
@ A3 ) )
= ( insert_rule @ A3 @ bot_bot_set_rule ) ) ).
% singleton_conv2
thf(fact_13_singleton__conv2,axiom,
! [A3: $o] :
( ( collect_o
@ ( ^ [Y: $o,Z: $o] : ( Y = Z )
@ A3 ) )
= ( insert_o @ A3 @ bot_bot_set_o ) ) ).
% singleton_conv2
thf(fact_14_Collect__const,axiom,
! [P: $o] :
( ( P
=> ( ( collec7231575776666803724ist_fm
@ ^ [S: produc1996495991257130529ist_fm] : P )
= top_to728902761071950545ist_fm ) )
& ( ~ P
=> ( ( collec7231575776666803724ist_fm
@ ^ [S: produc1996495991257130529ist_fm] : P )
= bot_bo4530011010606209517ist_fm ) ) ) ).
% Collect_const
thf(fact_15_Collect__const,axiom,
! [P: $o] :
( ( P
=> ( ( collect_rule
@ ^ [S: rule] : P )
= top_top_set_rule ) )
& ( ~ P
=> ( ( collect_rule
@ ^ [S: rule] : P )
= bot_bot_set_rule ) ) ) ).
% Collect_const
thf(fact_16_Collect__const,axiom,
! [P: $o] :
( ( P
=> ( ( collect_o
@ ^ [S: $o] : P )
= top_top_set_o ) )
& ( ~ P
=> ( ( collect_o
@ ^ [S: $o] : P )
= bot_bot_set_o ) ) ) ).
% Collect_const
thf(fact_17_Diff__insert0,axiom,
! [X2: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm,B: set_Pr7058068377845519745ist_fm] :
( ~ ( member8102475879199740618ist_fm @ X2 @ A )
=> ( ( minus_5615909706921521512ist_fm @ A @ ( insert589278167076416497ist_fm @ X2 @ B ) )
= ( minus_5615909706921521512ist_fm @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_18_Diff__insert0,axiom,
! [X2: $o,A: set_o,B: set_o] :
( ~ ( member_o @ X2 @ A )
=> ( ( minus_minus_set_o @ A @ ( insert_o @ X2 @ B ) )
= ( minus_minus_set_o @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_19_Diff__insert0,axiom,
! [X2: rule,A: set_rule,B: set_rule] :
( ~ ( member_rule @ X2 @ A )
=> ( ( minus_minus_set_rule @ A @ ( insert_rule @ X2 @ B ) )
= ( minus_minus_set_rule @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_20_insert__Diff1,axiom,
! [X2: produc1996495991257130529ist_fm,B: set_Pr7058068377845519745ist_fm,A: set_Pr7058068377845519745ist_fm] :
( ( member8102475879199740618ist_fm @ X2 @ B )
=> ( ( minus_5615909706921521512ist_fm @ ( insert589278167076416497ist_fm @ X2 @ A ) @ B )
= ( minus_5615909706921521512ist_fm @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_21_insert__Diff1,axiom,
! [X2: $o,B: set_o,A: set_o] :
( ( member_o @ X2 @ B )
=> ( ( minus_minus_set_o @ ( insert_o @ X2 @ A ) @ B )
= ( minus_minus_set_o @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_22_insert__Diff1,axiom,
! [X2: rule,B: set_rule,A: set_rule] :
( ( member_rule @ X2 @ B )
=> ( ( minus_minus_set_rule @ ( insert_rule @ X2 @ A ) @ B )
= ( minus_minus_set_rule @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_23_Diff__empty,axiom,
! [A: set_Pr7058068377845519745ist_fm] :
( ( minus_5615909706921521512ist_fm @ A @ bot_bo4530011010606209517ist_fm )
= A ) ).
% Diff_empty
thf(fact_24_Diff__empty,axiom,
! [A: set_o] :
( ( minus_minus_set_o @ A @ bot_bot_set_o )
= A ) ).
% Diff_empty
thf(fact_25_Diff__empty,axiom,
! [A: set_rule] :
( ( minus_minus_set_rule @ A @ bot_bot_set_rule )
= A ) ).
% Diff_empty
thf(fact_26_empty__Diff,axiom,
! [A: set_rule] :
( ( minus_minus_set_rule @ bot_bot_set_rule @ A )
= bot_bot_set_rule ) ).
% empty_Diff
thf(fact_27_empty__Diff,axiom,
! [A: set_o] :
( ( minus_minus_set_o @ bot_bot_set_o @ A )
= bot_bot_set_o ) ).
% empty_Diff
thf(fact_28_empty__Diff,axiom,
! [A: set_Pr7058068377845519745ist_fm] :
( ( minus_5615909706921521512ist_fm @ bot_bo4530011010606209517ist_fm @ A )
= bot_bo4530011010606209517ist_fm ) ).
% empty_Diff
thf(fact_29_Diff__cancel,axiom,
! [A: set_rule] :
( ( minus_minus_set_rule @ A @ A )
= bot_bot_set_rule ) ).
% Diff_cancel
thf(fact_30_Diff__cancel,axiom,
! [A: set_o] :
( ( minus_minus_set_o @ A @ A )
= bot_bot_set_o ) ).
% Diff_cancel
thf(fact_31_Diff__cancel,axiom,
! [A: set_Pr7058068377845519745ist_fm] :
( ( minus_5615909706921521512ist_fm @ A @ A )
= bot_bo4530011010606209517ist_fm ) ).
% Diff_cancel
thf(fact_32_singletonI,axiom,
! [A3: rule] : ( member_rule @ A3 @ ( insert_rule @ A3 @ bot_bot_set_rule ) ) ).
% singletonI
thf(fact_33_singletonI,axiom,
! [A3: $o] : ( member_o @ A3 @ ( insert_o @ A3 @ bot_bot_set_o ) ) ).
% singletonI
thf(fact_34_singletonI,axiom,
! [A3: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ A3 @ ( insert589278167076416497ist_fm @ A3 @ bot_bo4530011010606209517ist_fm ) ) ).
% singletonI
thf(fact_35_Diff__insert,axiom,
! [A: set_rule,A3: rule,B: set_rule] :
( ( minus_minus_set_rule @ A @ ( insert_rule @ A3 @ B ) )
= ( minus_minus_set_rule @ ( minus_minus_set_rule @ A @ B ) @ ( insert_rule @ A3 @ bot_bot_set_rule ) ) ) ).
% Diff_insert
thf(fact_36_Diff__insert,axiom,
! [A: set_o,A3: $o,B: set_o] :
( ( minus_minus_set_o @ A @ ( insert_o @ A3 @ B ) )
= ( minus_minus_set_o @ ( minus_minus_set_o @ A @ B ) @ ( insert_o @ A3 @ bot_bot_set_o ) ) ) ).
% Diff_insert
thf(fact_37_Diff__insert,axiom,
! [A: set_Pr7058068377845519745ist_fm,A3: produc1996495991257130529ist_fm,B: set_Pr7058068377845519745ist_fm] :
( ( minus_5615909706921521512ist_fm @ A @ ( insert589278167076416497ist_fm @ A3 @ B ) )
= ( minus_5615909706921521512ist_fm @ ( minus_5615909706921521512ist_fm @ A @ B ) @ ( insert589278167076416497ist_fm @ A3 @ bot_bo4530011010606209517ist_fm ) ) ) ).
% Diff_insert
thf(fact_38_UNIV__I,axiom,
! [X2: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X2 @ top_to728902761071950545ist_fm ) ).
% UNIV_I
thf(fact_39_UNIV__I,axiom,
! [X2: rule] : ( member_rule @ X2 @ top_top_set_rule ) ).
% UNIV_I
thf(fact_40_UNIV__I,axiom,
! [X2: $o] : ( member_o @ X2 @ top_top_set_o ) ).
% UNIV_I
thf(fact_41_empty__Collect__eq,axiom,
! [P: rule > $o] :
( ( bot_bot_set_rule
= ( collect_rule @ P ) )
= ( ! [X: rule] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_42_empty__Collect__eq,axiom,
! [P: $o > $o] :
( ( bot_bot_set_o
= ( collect_o @ P ) )
= ( ! [X: $o] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_43_empty__Collect__eq,axiom,
! [P: produc1996495991257130529ist_fm > $o] :
( ( bot_bo4530011010606209517ist_fm
= ( collec7231575776666803724ist_fm @ P ) )
= ( ! [X: produc1996495991257130529ist_fm] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_44_Collect__empty__eq,axiom,
! [P: rule > $o] :
( ( ( collect_rule @ P )
= bot_bot_set_rule )
= ( ! [X: rule] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_45_Collect__empty__eq,axiom,
! [P: $o > $o] :
( ( ( collect_o @ P )
= bot_bot_set_o )
= ( ! [X: $o] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_46_Collect__empty__eq,axiom,
! [P: produc1996495991257130529ist_fm > $o] :
( ( ( collec7231575776666803724ist_fm @ P )
= bot_bo4530011010606209517ist_fm )
= ( ! [X: produc1996495991257130529ist_fm] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_47_all__not__in__conv,axiom,
! [A: set_rule] :
( ( ! [X: rule] :
~ ( member_rule @ X @ A ) )
= ( A = bot_bot_set_rule ) ) ).
% all_not_in_conv
thf(fact_48_all__not__in__conv,axiom,
! [A: set_o] :
( ( ! [X: $o] :
~ ( member_o @ X @ A ) )
= ( A = bot_bot_set_o ) ) ).
% all_not_in_conv
thf(fact_49_all__not__in__conv,axiom,
! [A: set_Pr7058068377845519745ist_fm] :
( ( ! [X: produc1996495991257130529ist_fm] :
~ ( member8102475879199740618ist_fm @ X @ A ) )
= ( A = bot_bo4530011010606209517ist_fm ) ) ).
% all_not_in_conv
thf(fact_50_empty__iff,axiom,
! [C: rule] :
~ ( member_rule @ C @ bot_bot_set_rule ) ).
% empty_iff
thf(fact_51_empty__iff,axiom,
! [C: $o] :
~ ( member_o @ C @ bot_bot_set_o ) ).
% empty_iff
thf(fact_52_empty__iff,axiom,
! [C: produc1996495991257130529ist_fm] :
~ ( member8102475879199740618ist_fm @ C @ bot_bo4530011010606209517ist_fm ) ).
% empty_iff
thf(fact_53_insert__absorb2,axiom,
! [X2: rule,A: set_rule] :
( ( insert_rule @ X2 @ ( insert_rule @ X2 @ A ) )
= ( insert_rule @ X2 @ A ) ) ).
% insert_absorb2
thf(fact_54_insert__absorb2,axiom,
! [X2: $o,A: set_o] :
( ( insert_o @ X2 @ ( insert_o @ X2 @ A ) )
= ( insert_o @ X2 @ A ) ) ).
% insert_absorb2
thf(fact_55_insert__absorb2,axiom,
! [X2: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm] :
( ( insert589278167076416497ist_fm @ X2 @ ( insert589278167076416497ist_fm @ X2 @ A ) )
= ( insert589278167076416497ist_fm @ X2 @ A ) ) ).
% insert_absorb2
thf(fact_56_insert__iff,axiom,
! [A3: produc1996495991257130529ist_fm,B3: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm] :
( ( member8102475879199740618ist_fm @ A3 @ ( insert589278167076416497ist_fm @ B3 @ A ) )
= ( ( A3 = B3 )
| ( member8102475879199740618ist_fm @ A3 @ A ) ) ) ).
% insert_iff
thf(fact_57_insert__iff,axiom,
! [A3: rule,B3: rule,A: set_rule] :
( ( member_rule @ A3 @ ( insert_rule @ B3 @ A ) )
= ( ( A3 = B3 )
| ( member_rule @ A3 @ A ) ) ) ).
% insert_iff
thf(fact_58_insert__iff,axiom,
! [A3: $o,B3: $o,A: set_o] :
( ( member_o @ A3 @ ( insert_o @ B3 @ A ) )
= ( ( A3 = B3 )
| ( member_o @ A3 @ A ) ) ) ).
% insert_iff
thf(fact_59_insertCI,axiom,
! [A3: produc1996495991257130529ist_fm,B: set_Pr7058068377845519745ist_fm,B3: produc1996495991257130529ist_fm] :
( ( ~ ( member8102475879199740618ist_fm @ A3 @ B )
=> ( A3 = B3 ) )
=> ( member8102475879199740618ist_fm @ A3 @ ( insert589278167076416497ist_fm @ B3 @ B ) ) ) ).
% insertCI
thf(fact_60_insertCI,axiom,
! [A3: rule,B: set_rule,B3: rule] :
( ( ~ ( member_rule @ A3 @ B )
=> ( A3 = B3 ) )
=> ( member_rule @ A3 @ ( insert_rule @ B3 @ B ) ) ) ).
% insertCI
thf(fact_61_insertCI,axiom,
! [A3: $o,B: set_o,B3: $o] :
( ( ~ ( member_o @ A3 @ B )
=> ( A3 = B3 ) )
=> ( member_o @ A3 @ ( insert_o @ B3 @ B ) ) ) ).
% insertCI
thf(fact_62_Diff__idemp,axiom,
! [A: set_rule,B: set_rule] :
( ( minus_minus_set_rule @ ( minus_minus_set_rule @ A @ B ) @ B )
= ( minus_minus_set_rule @ A @ B ) ) ).
% Diff_idemp
thf(fact_63_Diff__idemp,axiom,
! [A: set_o,B: set_o] :
( ( minus_minus_set_o @ ( minus_minus_set_o @ A @ B ) @ B )
= ( minus_minus_set_o @ A @ B ) ) ).
% Diff_idemp
thf(fact_64_Diff__idemp,axiom,
! [A: set_Pr7058068377845519745ist_fm,B: set_Pr7058068377845519745ist_fm] :
( ( minus_5615909706921521512ist_fm @ ( minus_5615909706921521512ist_fm @ A @ B ) @ B )
= ( minus_5615909706921521512ist_fm @ A @ B ) ) ).
% Diff_idemp
thf(fact_65_Diff__iff,axiom,
! [C: rule,A: set_rule,B: set_rule] :
( ( member_rule @ C @ ( minus_minus_set_rule @ A @ B ) )
= ( ( member_rule @ C @ A )
& ~ ( member_rule @ C @ B ) ) ) ).
% Diff_iff
thf(fact_66_Diff__iff,axiom,
! [C: $o,A: set_o,B: set_o] :
( ( member_o @ C @ ( minus_minus_set_o @ A @ B ) )
= ( ( member_o @ C @ A )
& ~ ( member_o @ C @ B ) ) ) ).
% Diff_iff
thf(fact_67_Diff__iff,axiom,
! [C: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm,B: set_Pr7058068377845519745ist_fm] :
( ( member8102475879199740618ist_fm @ C @ ( minus_5615909706921521512ist_fm @ A @ B ) )
= ( ( member8102475879199740618ist_fm @ C @ A )
& ~ ( member8102475879199740618ist_fm @ C @ B ) ) ) ).
% Diff_iff
thf(fact_68_DiffI,axiom,
! [C: rule,A: set_rule,B: set_rule] :
( ( member_rule @ C @ A )
=> ( ~ ( member_rule @ C @ B )
=> ( member_rule @ C @ ( minus_minus_set_rule @ A @ B ) ) ) ) ).
% DiffI
thf(fact_69_DiffI,axiom,
! [C: $o,A: set_o,B: set_o] :
( ( member_o @ C @ A )
=> ( ~ ( member_o @ C @ B )
=> ( member_o @ C @ ( minus_minus_set_o @ A @ B ) ) ) ) ).
% DiffI
thf(fact_70_DiffI,axiom,
! [C: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm,B: set_Pr7058068377845519745ist_fm] :
( ( member8102475879199740618ist_fm @ C @ A )
=> ( ~ ( member8102475879199740618ist_fm @ C @ B )
=> ( member8102475879199740618ist_fm @ C @ ( minus_5615909706921521512ist_fm @ A @ B ) ) ) ) ).
% DiffI
thf(fact_71_UNIV__witness,axiom,
? [X3: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X3 @ top_to728902761071950545ist_fm ) ).
% UNIV_witness
thf(fact_72_UNIV__witness,axiom,
? [X3: rule] : ( member_rule @ X3 @ top_top_set_rule ) ).
% UNIV_witness
thf(fact_73_UNIV__witness,axiom,
? [X3: $o] : ( member_o @ X3 @ top_top_set_o ) ).
% UNIV_witness
thf(fact_74_UNIV__eq__I,axiom,
! [A: set_Pr7058068377845519745ist_fm] :
( ! [X3: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X3 @ A )
=> ( top_to728902761071950545ist_fm = A ) ) ).
% UNIV_eq_I
thf(fact_75_UNIV__eq__I,axiom,
! [A: set_rule] :
( ! [X3: rule] : ( member_rule @ X3 @ A )
=> ( top_top_set_rule = A ) ) ).
% UNIV_eq_I
thf(fact_76_UNIV__eq__I,axiom,
! [A: set_o] :
( ! [X3: $o] : ( member_o @ X3 @ A )
=> ( top_top_set_o = A ) ) ).
% UNIV_eq_I
thf(fact_77_ex__in__conv,axiom,
! [A: set_rule] :
( ( ? [X: rule] : ( member_rule @ X @ A ) )
= ( A != bot_bot_set_rule ) ) ).
% ex_in_conv
thf(fact_78_ex__in__conv,axiom,
! [A: set_o] :
( ( ? [X: $o] : ( member_o @ X @ A ) )
= ( A != bot_bot_set_o ) ) ).
% ex_in_conv
thf(fact_79_ex__in__conv,axiom,
! [A: set_Pr7058068377845519745ist_fm] :
( ( ? [X: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X @ A ) )
= ( A != bot_bo4530011010606209517ist_fm ) ) ).
% ex_in_conv
thf(fact_80_equals0I,axiom,
! [A: set_rule] :
( ! [Y2: rule] :
~ ( member_rule @ Y2 @ A )
=> ( A = bot_bot_set_rule ) ) ).
% equals0I
thf(fact_81_equals0I,axiom,
! [A: set_o] :
( ! [Y2: $o] :
~ ( member_o @ Y2 @ A )
=> ( A = bot_bot_set_o ) ) ).
% equals0I
thf(fact_82_equals0I,axiom,
! [A: set_Pr7058068377845519745ist_fm] :
( ! [Y2: produc1996495991257130529ist_fm] :
~ ( member8102475879199740618ist_fm @ Y2 @ A )
=> ( A = bot_bo4530011010606209517ist_fm ) ) ).
% equals0I
thf(fact_83_equals0D,axiom,
! [A: set_rule,A3: rule] :
( ( A = bot_bot_set_rule )
=> ~ ( member_rule @ A3 @ A ) ) ).
% equals0D
thf(fact_84_equals0D,axiom,
! [A: set_o,A3: $o] :
( ( A = bot_bot_set_o )
=> ~ ( member_o @ A3 @ A ) ) ).
% equals0D
thf(fact_85_equals0D,axiom,
! [A: set_Pr7058068377845519745ist_fm,A3: produc1996495991257130529ist_fm] :
( ( A = bot_bo4530011010606209517ist_fm )
=> ~ ( member8102475879199740618ist_fm @ A3 @ A ) ) ).
% equals0D
thf(fact_86_emptyE,axiom,
! [A3: rule] :
~ ( member_rule @ A3 @ bot_bot_set_rule ) ).
% emptyE
thf(fact_87_emptyE,axiom,
! [A3: $o] :
~ ( member_o @ A3 @ bot_bot_set_o ) ).
% emptyE
thf(fact_88_emptyE,axiom,
! [A3: produc1996495991257130529ist_fm] :
~ ( member8102475879199740618ist_fm @ A3 @ bot_bo4530011010606209517ist_fm ) ).
% emptyE
thf(fact_89_mk__disjoint__insert,axiom,
! [A3: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm] :
( ( member8102475879199740618ist_fm @ A3 @ A )
=> ? [B4: set_Pr7058068377845519745ist_fm] :
( ( A
= ( insert589278167076416497ist_fm @ A3 @ B4 ) )
& ~ ( member8102475879199740618ist_fm @ A3 @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_90_mk__disjoint__insert,axiom,
! [A3: rule,A: set_rule] :
( ( member_rule @ A3 @ A )
=> ? [B4: set_rule] :
( ( A
= ( insert_rule @ A3 @ B4 ) )
& ~ ( member_rule @ A3 @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_91_mk__disjoint__insert,axiom,
! [A3: $o,A: set_o] :
( ( member_o @ A3 @ A )
=> ? [B4: set_o] :
( ( A
= ( insert_o @ A3 @ B4 ) )
& ~ ( member_o @ A3 @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_92_insert__commute,axiom,
! [X2: rule,Y3: rule,A: set_rule] :
( ( insert_rule @ X2 @ ( insert_rule @ Y3 @ A ) )
= ( insert_rule @ Y3 @ ( insert_rule @ X2 @ A ) ) ) ).
% insert_commute
thf(fact_93_insert__commute,axiom,
! [X2: $o,Y3: $o,A: set_o] :
( ( insert_o @ X2 @ ( insert_o @ Y3 @ A ) )
= ( insert_o @ Y3 @ ( insert_o @ X2 @ A ) ) ) ).
% insert_commute
thf(fact_94_insert__commute,axiom,
! [X2: produc1996495991257130529ist_fm,Y3: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm] :
( ( insert589278167076416497ist_fm @ X2 @ ( insert589278167076416497ist_fm @ Y3 @ A ) )
= ( insert589278167076416497ist_fm @ Y3 @ ( insert589278167076416497ist_fm @ X2 @ A ) ) ) ).
% insert_commute
thf(fact_95_insert__eq__iff,axiom,
! [A3: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm,B3: produc1996495991257130529ist_fm,B: set_Pr7058068377845519745ist_fm] :
( ~ ( member8102475879199740618ist_fm @ A3 @ A )
=> ( ~ ( member8102475879199740618ist_fm @ B3 @ B )
=> ( ( ( insert589278167076416497ist_fm @ A3 @ A )
= ( insert589278167076416497ist_fm @ B3 @ B ) )
= ( ( ( A3 = B3 )
=> ( A = B ) )
& ( ( A3 != B3 )
=> ? [C2: set_Pr7058068377845519745ist_fm] :
( ( A
= ( insert589278167076416497ist_fm @ B3 @ C2 ) )
& ~ ( member8102475879199740618ist_fm @ B3 @ C2 )
& ( B
= ( insert589278167076416497ist_fm @ A3 @ C2 ) )
& ~ ( member8102475879199740618ist_fm @ A3 @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_96_insert__eq__iff,axiom,
! [A3: rule,A: set_rule,B3: rule,B: set_rule] :
( ~ ( member_rule @ A3 @ A )
=> ( ~ ( member_rule @ B3 @ B )
=> ( ( ( insert_rule @ A3 @ A )
= ( insert_rule @ B3 @ B ) )
= ( ( ( A3 = B3 )
=> ( A = B ) )
& ( ( A3 != B3 )
=> ? [C2: set_rule] :
( ( A
= ( insert_rule @ B3 @ C2 ) )
& ~ ( member_rule @ B3 @ C2 )
& ( B
= ( insert_rule @ A3 @ C2 ) )
& ~ ( member_rule @ A3 @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_97_insert__eq__iff,axiom,
! [A3: $o,A: set_o,B3: $o,B: set_o] :
( ~ ( member_o @ A3 @ A )
=> ( ~ ( member_o @ B3 @ B )
=> ( ( ( insert_o @ A3 @ A )
= ( insert_o @ B3 @ B ) )
= ( ( ( A3 = B3 )
=> ( A = B ) )
& ( ( A3 = ~ B3 )
=> ? [C2: set_o] :
( ( A
= ( insert_o @ B3 @ C2 ) )
& ~ ( member_o @ B3 @ C2 )
& ( B
= ( insert_o @ A3 @ C2 ) )
& ~ ( member_o @ A3 @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_98_insert__absorb,axiom,
! [A3: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm] :
( ( member8102475879199740618ist_fm @ A3 @ A )
=> ( ( insert589278167076416497ist_fm @ A3 @ A )
= A ) ) ).
% insert_absorb
thf(fact_99_insert__absorb,axiom,
! [A3: rule,A: set_rule] :
( ( member_rule @ A3 @ A )
=> ( ( insert_rule @ A3 @ A )
= A ) ) ).
% insert_absorb
thf(fact_100_insert__absorb,axiom,
! [A3: $o,A: set_o] :
( ( member_o @ A3 @ A )
=> ( ( insert_o @ A3 @ A )
= A ) ) ).
% insert_absorb
thf(fact_101_insert__ident,axiom,
! [X2: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm,B: set_Pr7058068377845519745ist_fm] :
( ~ ( member8102475879199740618ist_fm @ X2 @ A )
=> ( ~ ( member8102475879199740618ist_fm @ X2 @ B )
=> ( ( ( insert589278167076416497ist_fm @ X2 @ A )
= ( insert589278167076416497ist_fm @ X2 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_102_insert__ident,axiom,
! [X2: rule,A: set_rule,B: set_rule] :
( ~ ( member_rule @ X2 @ A )
=> ( ~ ( member_rule @ X2 @ B )
=> ( ( ( insert_rule @ X2 @ A )
= ( insert_rule @ X2 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_103_insert__ident,axiom,
! [X2: $o,A: set_o,B: set_o] :
( ~ ( member_o @ X2 @ A )
=> ( ~ ( member_o @ X2 @ B )
=> ( ( ( insert_o @ X2 @ A )
= ( insert_o @ X2 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_104_Set_Oset__insert,axiom,
! [X2: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm] :
( ( member8102475879199740618ist_fm @ X2 @ A )
=> ~ ! [B4: set_Pr7058068377845519745ist_fm] :
( ( A
= ( insert589278167076416497ist_fm @ X2 @ B4 ) )
=> ( member8102475879199740618ist_fm @ X2 @ B4 ) ) ) ).
% Set.set_insert
thf(fact_105_Set_Oset__insert,axiom,
! [X2: rule,A: set_rule] :
( ( member_rule @ X2 @ A )
=> ~ ! [B4: set_rule] :
( ( A
= ( insert_rule @ X2 @ B4 ) )
=> ( member_rule @ X2 @ B4 ) ) ) ).
% Set.set_insert
thf(fact_106_Set_Oset__insert,axiom,
! [X2: $o,A: set_o] :
( ( member_o @ X2 @ A )
=> ~ ! [B4: set_o] :
( ( A
= ( insert_o @ X2 @ B4 ) )
=> ( member_o @ X2 @ B4 ) ) ) ).
% Set.set_insert
thf(fact_107_insertI2,axiom,
! [A3: produc1996495991257130529ist_fm,B: set_Pr7058068377845519745ist_fm,B3: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ A3 @ B )
=> ( member8102475879199740618ist_fm @ A3 @ ( insert589278167076416497ist_fm @ B3 @ B ) ) ) ).
% insertI2
thf(fact_108_insertI2,axiom,
! [A3: rule,B: set_rule,B3: rule] :
( ( member_rule @ A3 @ B )
=> ( member_rule @ A3 @ ( insert_rule @ B3 @ B ) ) ) ).
% insertI2
thf(fact_109_insertI2,axiom,
! [A3: $o,B: set_o,B3: $o] :
( ( member_o @ A3 @ B )
=> ( member_o @ A3 @ ( insert_o @ B3 @ B ) ) ) ).
% insertI2
thf(fact_110_insertI1,axiom,
! [A3: produc1996495991257130529ist_fm,B: set_Pr7058068377845519745ist_fm] : ( member8102475879199740618ist_fm @ A3 @ ( insert589278167076416497ist_fm @ A3 @ B ) ) ).
% insertI1
thf(fact_111_insertI1,axiom,
! [A3: rule,B: set_rule] : ( member_rule @ A3 @ ( insert_rule @ A3 @ B ) ) ).
% insertI1
thf(fact_112_insertI1,axiom,
! [A3: $o,B: set_o] : ( member_o @ A3 @ ( insert_o @ A3 @ B ) ) ).
% insertI1
thf(fact_113_insertE,axiom,
! [A3: produc1996495991257130529ist_fm,B3: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm] :
( ( member8102475879199740618ist_fm @ A3 @ ( insert589278167076416497ist_fm @ B3 @ A ) )
=> ( ( A3 != B3 )
=> ( member8102475879199740618ist_fm @ A3 @ A ) ) ) ).
% insertE
thf(fact_114_insertE,axiom,
! [A3: rule,B3: rule,A: set_rule] :
( ( member_rule @ A3 @ ( insert_rule @ B3 @ A ) )
=> ( ( A3 != B3 )
=> ( member_rule @ A3 @ A ) ) ) ).
% insertE
thf(fact_115_insertE,axiom,
! [A3: $o,B3: $o,A: set_o] :
( ( member_o @ A3 @ ( insert_o @ B3 @ A ) )
=> ( ( A3 = ~ B3 )
=> ( member_o @ A3 @ A ) ) ) ).
% insertE
thf(fact_116_DiffD2,axiom,
! [C: rule,A: set_rule,B: set_rule] :
( ( member_rule @ C @ ( minus_minus_set_rule @ A @ B ) )
=> ~ ( member_rule @ C @ B ) ) ).
% DiffD2
thf(fact_117_DiffD2,axiom,
! [C: $o,A: set_o,B: set_o] :
( ( member_o @ C @ ( minus_minus_set_o @ A @ B ) )
=> ~ ( member_o @ C @ B ) ) ).
% DiffD2
thf(fact_118_DiffD2,axiom,
! [C: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm,B: set_Pr7058068377845519745ist_fm] :
( ( member8102475879199740618ist_fm @ C @ ( minus_5615909706921521512ist_fm @ A @ B ) )
=> ~ ( member8102475879199740618ist_fm @ C @ B ) ) ).
% DiffD2
thf(fact_119_DiffD1,axiom,
! [C: rule,A: set_rule,B: set_rule] :
( ( member_rule @ C @ ( minus_minus_set_rule @ A @ B ) )
=> ( member_rule @ C @ A ) ) ).
% DiffD1
thf(fact_120_DiffD1,axiom,
! [C: $o,A: set_o,B: set_o] :
( ( member_o @ C @ ( minus_minus_set_o @ A @ B ) )
=> ( member_o @ C @ A ) ) ).
% DiffD1
thf(fact_121_DiffD1,axiom,
! [C: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm,B: set_Pr7058068377845519745ist_fm] :
( ( member8102475879199740618ist_fm @ C @ ( minus_5615909706921521512ist_fm @ A @ B ) )
=> ( member8102475879199740618ist_fm @ C @ A ) ) ).
% DiffD1
thf(fact_122_DiffE,axiom,
! [C: rule,A: set_rule,B: set_rule] :
( ( member_rule @ C @ ( minus_minus_set_rule @ A @ B ) )
=> ~ ( ( member_rule @ C @ A )
=> ( member_rule @ C @ B ) ) ) ).
% DiffE
thf(fact_123_DiffE,axiom,
! [C: $o,A: set_o,B: set_o] :
( ( member_o @ C @ ( minus_minus_set_o @ A @ B ) )
=> ~ ( ( member_o @ C @ A )
=> ( member_o @ C @ B ) ) ) ).
% DiffE
thf(fact_124_DiffE,axiom,
! [C: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm,B: set_Pr7058068377845519745ist_fm] :
( ( member8102475879199740618ist_fm @ C @ ( minus_5615909706921521512ist_fm @ A @ B ) )
=> ~ ( ( member8102475879199740618ist_fm @ C @ A )
=> ( member8102475879199740618ist_fm @ C @ B ) ) ) ).
% DiffE
thf(fact_125_UNIV__def,axiom,
( top_to728902761071950545ist_fm
= ( collec7231575776666803724ist_fm
@ ^ [X: produc1996495991257130529ist_fm] : $true ) ) ).
% UNIV_def
thf(fact_126_UNIV__def,axiom,
( top_top_set_rule
= ( collect_rule
@ ^ [X: rule] : $true ) ) ).
% UNIV_def
thf(fact_127_UNIV__def,axiom,
( top_top_set_o
= ( collect_o
@ ^ [X: $o] : $true ) ) ).
% UNIV_def
thf(fact_128_empty__def,axiom,
( bot_bot_set_rule
= ( collect_rule
@ ^ [X: rule] : $false ) ) ).
% empty_def
thf(fact_129_empty__def,axiom,
( bot_bot_set_o
= ( collect_o
@ ^ [X: $o] : $false ) ) ).
% empty_def
thf(fact_130_empty__def,axiom,
( bot_bo4530011010606209517ist_fm
= ( collec7231575776666803724ist_fm
@ ^ [X: produc1996495991257130529ist_fm] : $false ) ) ).
% empty_def
thf(fact_131_mem__Collect__eq,axiom,
! [A3: $o,P: $o > $o] :
( ( member_o @ A3 @ ( collect_o @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_132_mem__Collect__eq,axiom,
! [A3: rule,P: rule > $o] :
( ( member_rule @ A3 @ ( collect_rule @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_133_mem__Collect__eq,axiom,
! [A3: produc1996495991257130529ist_fm,P: produc1996495991257130529ist_fm > $o] :
( ( member8102475879199740618ist_fm @ A3 @ ( collec7231575776666803724ist_fm @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_134_Collect__mem__eq,axiom,
! [A: set_o] :
( ( collect_o
@ ^ [X: $o] : ( member_o @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_135_Collect__mem__eq,axiom,
! [A: set_rule] :
( ( collect_rule
@ ^ [X: rule] : ( member_rule @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_136_Collect__mem__eq,axiom,
! [A: set_Pr7058068377845519745ist_fm] :
( ( collec7231575776666803724ist_fm
@ ^ [X: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_137_Collect__cong,axiom,
! [P: $o > $o,Q: $o > $o] :
( ! [X3: $o] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_o @ P )
= ( collect_o @ Q ) ) ) ).
% Collect_cong
thf(fact_138_Collect__cong,axiom,
! [P: rule > $o,Q: rule > $o] :
( ! [X3: rule] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_rule @ P )
= ( collect_rule @ Q ) ) ) ).
% Collect_cong
thf(fact_139_Collect__cong,axiom,
! [P: produc1996495991257130529ist_fm > $o,Q: produc1996495991257130529ist_fm > $o] :
( ! [X3: produc1996495991257130529ist_fm] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collec7231575776666803724ist_fm @ P )
= ( collec7231575776666803724ist_fm @ Q ) ) ) ).
% Collect_cong
thf(fact_140_insert__Collect,axiom,
! [A3: $o,P: $o > $o] :
( ( insert_o @ A3 @ ( collect_o @ P ) )
= ( collect_o
@ ^ [U: $o] :
( ( U != A3 )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_141_insert__Collect,axiom,
! [A3: rule,P: rule > $o] :
( ( insert_rule @ A3 @ ( collect_rule @ P ) )
= ( collect_rule
@ ^ [U: rule] :
( ( U != A3 )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_142_insert__Collect,axiom,
! [A3: produc1996495991257130529ist_fm,P: produc1996495991257130529ist_fm > $o] :
( ( insert589278167076416497ist_fm @ A3 @ ( collec7231575776666803724ist_fm @ P ) )
= ( collec7231575776666803724ist_fm
@ ^ [U: produc1996495991257130529ist_fm] :
( ( U != A3 )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_143_insert__compr,axiom,
( insert_o
= ( ^ [A4: $o,B5: set_o] :
( collect_o
@ ^ [X: $o] :
( ( X = A4 )
| ( member_o @ X @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_144_insert__compr,axiom,
( insert_rule
= ( ^ [A4: rule,B5: set_rule] :
( collect_rule
@ ^ [X: rule] :
( ( X = A4 )
| ( member_rule @ X @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_145_insert__compr,axiom,
( insert589278167076416497ist_fm
= ( ^ [A4: produc1996495991257130529ist_fm,B5: set_Pr7058068377845519745ist_fm] :
( collec7231575776666803724ist_fm
@ ^ [X: produc1996495991257130529ist_fm] :
( ( X = A4 )
| ( member8102475879199740618ist_fm @ X @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_146_set__diff__eq,axiom,
( minus_minus_set_rule
= ( ^ [A5: set_rule,B5: set_rule] :
( collect_rule
@ ^ [X: rule] :
( ( member_rule @ X @ A5 )
& ~ ( member_rule @ X @ B5 ) ) ) ) ) ).
% set_diff_eq
thf(fact_147_set__diff__eq,axiom,
( minus_minus_set_o
= ( ^ [A5: set_o,B5: set_o] :
( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A5 )
& ~ ( member_o @ X @ B5 ) ) ) ) ) ).
% set_diff_eq
thf(fact_148_set__diff__eq,axiom,
( minus_5615909706921521512ist_fm
= ( ^ [A5: set_Pr7058068377845519745ist_fm,B5: set_Pr7058068377845519745ist_fm] :
( collec7231575776666803724ist_fm
@ ^ [X: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ X @ A5 )
& ~ ( member8102475879199740618ist_fm @ X @ B5 ) ) ) ) ) ).
% set_diff_eq
thf(fact_149_empty__not__UNIV,axiom,
bot_bo4530011010606209517ist_fm != top_to728902761071950545ist_fm ).
% empty_not_UNIV
thf(fact_150_empty__not__UNIV,axiom,
bot_bot_set_rule != top_top_set_rule ).
% empty_not_UNIV
thf(fact_151_empty__not__UNIV,axiom,
bot_bot_set_o != top_top_set_o ).
% empty_not_UNIV
thf(fact_152_insert__UNIV,axiom,
! [X2: produc1996495991257130529ist_fm] :
( ( insert589278167076416497ist_fm @ X2 @ top_to728902761071950545ist_fm )
= top_to728902761071950545ist_fm ) ).
% insert_UNIV
thf(fact_153_insert__UNIV,axiom,
! [X2: rule] :
( ( insert_rule @ X2 @ top_top_set_rule )
= top_top_set_rule ) ).
% insert_UNIV
thf(fact_154_insert__UNIV,axiom,
! [X2: $o] :
( ( insert_o @ X2 @ top_top_set_o )
= top_top_set_o ) ).
% insert_UNIV
thf(fact_155_singleton__inject,axiom,
! [A3: rule,B3: rule] :
( ( ( insert_rule @ A3 @ bot_bot_set_rule )
= ( insert_rule @ B3 @ bot_bot_set_rule ) )
=> ( A3 = B3 ) ) ).
% singleton_inject
thf(fact_156_singleton__inject,axiom,
! [A3: $o,B3: $o] :
( ( ( insert_o @ A3 @ bot_bot_set_o )
= ( insert_o @ B3 @ bot_bot_set_o ) )
=> ( A3 = B3 ) ) ).
% singleton_inject
thf(fact_157_singleton__inject,axiom,
! [A3: produc1996495991257130529ist_fm,B3: produc1996495991257130529ist_fm] :
( ( ( insert589278167076416497ist_fm @ A3 @ bot_bo4530011010606209517ist_fm )
= ( insert589278167076416497ist_fm @ B3 @ bot_bo4530011010606209517ist_fm ) )
=> ( A3 = B3 ) ) ).
% singleton_inject
thf(fact_158_insert__not__empty,axiom,
! [A3: rule,A: set_rule] :
( ( insert_rule @ A3 @ A )
!= bot_bot_set_rule ) ).
% insert_not_empty
thf(fact_159_insert__not__empty,axiom,
! [A3: $o,A: set_o] :
( ( insert_o @ A3 @ A )
!= bot_bot_set_o ) ).
% insert_not_empty
thf(fact_160_insert__not__empty,axiom,
! [A3: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm] :
( ( insert589278167076416497ist_fm @ A3 @ A )
!= bot_bo4530011010606209517ist_fm ) ).
% insert_not_empty
thf(fact_161_doubleton__eq__iff,axiom,
! [A3: rule,B3: rule,C: rule,D: rule] :
( ( ( insert_rule @ A3 @ ( insert_rule @ B3 @ bot_bot_set_rule ) )
= ( insert_rule @ C @ ( insert_rule @ D @ bot_bot_set_rule ) ) )
= ( ( ( A3 = C )
& ( B3 = D ) )
| ( ( A3 = D )
& ( B3 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_162_doubleton__eq__iff,axiom,
! [A3: $o,B3: $o,C: $o,D: $o] :
( ( ( insert_o @ A3 @ ( insert_o @ B3 @ bot_bot_set_o ) )
= ( insert_o @ C @ ( insert_o @ D @ bot_bot_set_o ) ) )
= ( ( ( A3 = C )
& ( B3 = D ) )
| ( ( A3 = D )
& ( B3 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_163_doubleton__eq__iff,axiom,
! [A3: produc1996495991257130529ist_fm,B3: produc1996495991257130529ist_fm,C: produc1996495991257130529ist_fm,D: produc1996495991257130529ist_fm] :
( ( ( insert589278167076416497ist_fm @ A3 @ ( insert589278167076416497ist_fm @ B3 @ bot_bo4530011010606209517ist_fm ) )
= ( insert589278167076416497ist_fm @ C @ ( insert589278167076416497ist_fm @ D @ bot_bo4530011010606209517ist_fm ) ) )
= ( ( ( A3 = C )
& ( B3 = D ) )
| ( ( A3 = D )
& ( B3 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_164_singleton__iff,axiom,
! [B3: rule,A3: rule] :
( ( member_rule @ B3 @ ( insert_rule @ A3 @ bot_bot_set_rule ) )
= ( B3 = A3 ) ) ).
% singleton_iff
thf(fact_165_singleton__iff,axiom,
! [B3: $o,A3: $o] :
( ( member_o @ B3 @ ( insert_o @ A3 @ bot_bot_set_o ) )
= ( B3 = A3 ) ) ).
% singleton_iff
thf(fact_166_singleton__iff,axiom,
! [B3: produc1996495991257130529ist_fm,A3: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ B3 @ ( insert589278167076416497ist_fm @ A3 @ bot_bo4530011010606209517ist_fm ) )
= ( B3 = A3 ) ) ).
% singleton_iff
thf(fact_167_singletonD,axiom,
! [B3: rule,A3: rule] :
( ( member_rule @ B3 @ ( insert_rule @ A3 @ bot_bot_set_rule ) )
=> ( B3 = A3 ) ) ).
% singletonD
thf(fact_168_singletonD,axiom,
! [B3: $o,A3: $o] :
( ( member_o @ B3 @ ( insert_o @ A3 @ bot_bot_set_o ) )
=> ( B3 = A3 ) ) ).
% singletonD
thf(fact_169_singletonD,axiom,
! [B3: produc1996495991257130529ist_fm,A3: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ B3 @ ( insert589278167076416497ist_fm @ A3 @ bot_bo4530011010606209517ist_fm ) )
=> ( B3 = A3 ) ) ).
% singletonD
thf(fact_170_insert__Diff__if,axiom,
! [X2: rule,B: set_rule,A: set_rule] :
( ( ( member_rule @ X2 @ B )
=> ( ( minus_minus_set_rule @ ( insert_rule @ X2 @ A ) @ B )
= ( minus_minus_set_rule @ A @ B ) ) )
& ( ~ ( member_rule @ X2 @ B )
=> ( ( minus_minus_set_rule @ ( insert_rule @ X2 @ A ) @ B )
= ( insert_rule @ X2 @ ( minus_minus_set_rule @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_171_insert__Diff__if,axiom,
! [X2: $o,B: set_o,A: set_o] :
( ( ( member_o @ X2 @ B )
=> ( ( minus_minus_set_o @ ( insert_o @ X2 @ A ) @ B )
= ( minus_minus_set_o @ A @ B ) ) )
& ( ~ ( member_o @ X2 @ B )
=> ( ( minus_minus_set_o @ ( insert_o @ X2 @ A ) @ B )
= ( insert_o @ X2 @ ( minus_minus_set_o @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_172_insert__Diff__if,axiom,
! [X2: produc1996495991257130529ist_fm,B: set_Pr7058068377845519745ist_fm,A: set_Pr7058068377845519745ist_fm] :
( ( ( member8102475879199740618ist_fm @ X2 @ B )
=> ( ( minus_5615909706921521512ist_fm @ ( insert589278167076416497ist_fm @ X2 @ A ) @ B )
= ( minus_5615909706921521512ist_fm @ A @ B ) ) )
& ( ~ ( member8102475879199740618ist_fm @ X2 @ B )
=> ( ( minus_5615909706921521512ist_fm @ ( insert589278167076416497ist_fm @ X2 @ A ) @ B )
= ( insert589278167076416497ist_fm @ X2 @ ( minus_5615909706921521512ist_fm @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_173_Collect__conv__if2,axiom,
! [P: rule > $o,A3: rule] :
( ( ( P @ A3 )
=> ( ( collect_rule
@ ^ [X: rule] :
( ( A3 = X )
& ( P @ X ) ) )
= ( insert_rule @ A3 @ bot_bot_set_rule ) ) )
& ( ~ ( P @ A3 )
=> ( ( collect_rule
@ ^ [X: rule] :
( ( A3 = X )
& ( P @ X ) ) )
= bot_bot_set_rule ) ) ) ).
% Collect_conv_if2
thf(fact_174_Collect__conv__if2,axiom,
! [P: $o > $o,A3: $o] :
( ( ( P @ A3 )
=> ( ( collect_o
@ ^ [X: $o] :
( ( A3 = X )
& ( P @ X ) ) )
= ( insert_o @ A3 @ bot_bot_set_o ) ) )
& ( ~ ( P @ A3 )
=> ( ( collect_o
@ ^ [X: $o] :
( ( A3 = X )
& ( P @ X ) ) )
= bot_bot_set_o ) ) ) ).
% Collect_conv_if2
thf(fact_175_Collect__conv__if2,axiom,
! [P: produc1996495991257130529ist_fm > $o,A3: produc1996495991257130529ist_fm] :
( ( ( P @ A3 )
=> ( ( collec7231575776666803724ist_fm
@ ^ [X: produc1996495991257130529ist_fm] :
( ( A3 = X )
& ( P @ X ) ) )
= ( insert589278167076416497ist_fm @ A3 @ bot_bo4530011010606209517ist_fm ) ) )
& ( ~ ( P @ A3 )
=> ( ( collec7231575776666803724ist_fm
@ ^ [X: produc1996495991257130529ist_fm] :
( ( A3 = X )
& ( P @ X ) ) )
= bot_bo4530011010606209517ist_fm ) ) ) ).
% Collect_conv_if2
thf(fact_176_Collect__conv__if,axiom,
! [P: rule > $o,A3: rule] :
( ( ( P @ A3 )
=> ( ( collect_rule
@ ^ [X: rule] :
( ( X = A3 )
& ( P @ X ) ) )
= ( insert_rule @ A3 @ bot_bot_set_rule ) ) )
& ( ~ ( P @ A3 )
=> ( ( collect_rule
@ ^ [X: rule] :
( ( X = A3 )
& ( P @ X ) ) )
= bot_bot_set_rule ) ) ) ).
% Collect_conv_if
thf(fact_177_Collect__conv__if,axiom,
! [P: $o > $o,A3: $o] :
( ( ( P @ A3 )
=> ( ( collect_o
@ ^ [X: $o] :
( ( X = A3 )
& ( P @ X ) ) )
= ( insert_o @ A3 @ bot_bot_set_o ) ) )
& ( ~ ( P @ A3 )
=> ( ( collect_o
@ ^ [X: $o] :
( ( X = A3 )
& ( P @ X ) ) )
= bot_bot_set_o ) ) ) ).
% Collect_conv_if
thf(fact_178_Collect__conv__if,axiom,
! [P: produc1996495991257130529ist_fm > $o,A3: produc1996495991257130529ist_fm] :
( ( ( P @ A3 )
=> ( ( collec7231575776666803724ist_fm
@ ^ [X: produc1996495991257130529ist_fm] :
( ( X = A3 )
& ( P @ X ) ) )
= ( insert589278167076416497ist_fm @ A3 @ bot_bo4530011010606209517ist_fm ) ) )
& ( ~ ( P @ A3 )
=> ( ( collec7231575776666803724ist_fm
@ ^ [X: produc1996495991257130529ist_fm] :
( ( X = A3 )
& ( P @ X ) ) )
= bot_bo4530011010606209517ist_fm ) ) ) ).
% Collect_conv_if
thf(fact_179_Diff__insert__absorb,axiom,
! [X2: rule,A: set_rule] :
( ~ ( member_rule @ X2 @ A )
=> ( ( minus_minus_set_rule @ ( insert_rule @ X2 @ A ) @ ( insert_rule @ X2 @ bot_bot_set_rule ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_180_Diff__insert__absorb,axiom,
! [X2: $o,A: set_o] :
( ~ ( member_o @ X2 @ A )
=> ( ( minus_minus_set_o @ ( insert_o @ X2 @ A ) @ ( insert_o @ X2 @ bot_bot_set_o ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_181_Diff__insert__absorb,axiom,
! [X2: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm] :
( ~ ( member8102475879199740618ist_fm @ X2 @ A )
=> ( ( minus_5615909706921521512ist_fm @ ( insert589278167076416497ist_fm @ X2 @ A ) @ ( insert589278167076416497ist_fm @ X2 @ bot_bo4530011010606209517ist_fm ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_182_Diff__insert2,axiom,
! [A: set_rule,A3: rule,B: set_rule] :
( ( minus_minus_set_rule @ A @ ( insert_rule @ A3 @ B ) )
= ( minus_minus_set_rule @ ( minus_minus_set_rule @ A @ ( insert_rule @ A3 @ bot_bot_set_rule ) ) @ B ) ) ).
% Diff_insert2
thf(fact_183_Diff__insert2,axiom,
! [A: set_o,A3: $o,B: set_o] :
( ( minus_minus_set_o @ A @ ( insert_o @ A3 @ B ) )
= ( minus_minus_set_o @ ( minus_minus_set_o @ A @ ( insert_o @ A3 @ bot_bot_set_o ) ) @ B ) ) ).
% Diff_insert2
thf(fact_184_Diff__insert2,axiom,
! [A: set_Pr7058068377845519745ist_fm,A3: produc1996495991257130529ist_fm,B: set_Pr7058068377845519745ist_fm] :
( ( minus_5615909706921521512ist_fm @ A @ ( insert589278167076416497ist_fm @ A3 @ B ) )
= ( minus_5615909706921521512ist_fm @ ( minus_5615909706921521512ist_fm @ A @ ( insert589278167076416497ist_fm @ A3 @ bot_bo4530011010606209517ist_fm ) ) @ B ) ) ).
% Diff_insert2
thf(fact_185_insert__Diff,axiom,
! [A3: rule,A: set_rule] :
( ( member_rule @ A3 @ A )
=> ( ( insert_rule @ A3 @ ( minus_minus_set_rule @ A @ ( insert_rule @ A3 @ bot_bot_set_rule ) ) )
= A ) ) ).
% insert_Diff
thf(fact_186_insert__Diff,axiom,
! [A3: $o,A: set_o] :
( ( member_o @ A3 @ A )
=> ( ( insert_o @ A3 @ ( minus_minus_set_o @ A @ ( insert_o @ A3 @ bot_bot_set_o ) ) )
= A ) ) ).
% insert_Diff
thf(fact_187_insert__Diff,axiom,
! [A3: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm] :
( ( member8102475879199740618ist_fm @ A3 @ A )
=> ( ( insert589278167076416497ist_fm @ A3 @ ( minus_5615909706921521512ist_fm @ A @ ( insert589278167076416497ist_fm @ A3 @ bot_bo4530011010606209517ist_fm ) ) )
= A ) ) ).
% insert_Diff
thf(fact_188_option_Oinject,axiom,
! [X22: fset_P661503646757059847ist_fm,Y22: fset_P661503646757059847ist_fm] :
( ( ( some_f2162114139285431020ist_fm @ X22 )
= ( some_f2162114139285431020ist_fm @ Y22 ) )
= ( X22 = Y22 ) ) ).
% option.inject
thf(fact_189_iso__tuple__UNIV__I,axiom,
! [X2: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X2 @ top_to728902761071950545ist_fm ) ).
% iso_tuple_UNIV_I
thf(fact_190_iso__tuple__UNIV__I,axiom,
! [X2: rule] : ( member_rule @ X2 @ top_top_set_rule ) ).
% iso_tuple_UNIV_I
thf(fact_191_iso__tuple__UNIV__I,axiom,
! [X2: $o] : ( member_o @ X2 @ top_top_set_o ) ).
% iso_tuple_UNIV_I
thf(fact_192_prod_Oinject,axiom,
! [X1: list_fm,X22: list_fm,Y1: list_fm,Y22: list_fm] :
( ( ( produc381145313068854617ist_fm @ X1 @ X22 )
= ( produc381145313068854617ist_fm @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_193_old_Oprod_Oinject,axiom,
! [A3: list_fm,B3: list_fm,A6: list_fm,B6: list_fm] :
( ( ( produc381145313068854617ist_fm @ A3 @ B3 )
= ( produc381145313068854617ist_fm @ A6 @ B6 ) )
= ( ( A3 = A6 )
& ( B3 = B6 ) ) ) ).
% old.prod.inject
thf(fact_194_is__singletonI,axiom,
! [X2: rule] : ( is_singleton_rule @ ( insert_rule @ X2 @ bot_bot_set_rule ) ) ).
% is_singletonI
thf(fact_195_is__singletonI,axiom,
! [X2: $o] : ( is_singleton_o @ ( insert_o @ X2 @ bot_bot_set_o ) ) ).
% is_singletonI
thf(fact_196_is__singletonI,axiom,
! [X2: produc1996495991257130529ist_fm] : ( is_sin3852095833271400213ist_fm @ ( insert589278167076416497ist_fm @ X2 @ bot_bo4530011010606209517ist_fm ) ) ).
% is_singletonI
thf(fact_197_remove__def,axiom,
( remove_rule
= ( ^ [X: rule,A5: set_rule] : ( minus_minus_set_rule @ A5 @ ( insert_rule @ X @ bot_bot_set_rule ) ) ) ) ).
% remove_def
thf(fact_198_remove__def,axiom,
( remove_o
= ( ^ [X: $o,A5: set_o] : ( minus_minus_set_o @ A5 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ).
% remove_def
thf(fact_199_remove__def,axiom,
( remove6605582569798670470ist_fm
= ( ^ [X: produc1996495991257130529ist_fm,A5: set_Pr7058068377845519745ist_fm] : ( minus_5615909706921521512ist_fm @ A5 @ ( insert589278167076416497ist_fm @ X @ bot_bo4530011010606209517ist_fm ) ) ) ) ).
% remove_def
thf(fact_200_per__def,axiom,
! [R: rule] :
( ( abstra4153889242452539894ist_fm
@ ^ [R3: rule,S: produc1996495991257130529ist_fm,Ss2: fset_P661503646757059847ist_fm] :
( ( eff @ R3 @ S )
= ( some_f2162114139285431020ist_fm @ Ss2 ) )
@ rules
@ top_to728902761071950545ist_fm
@ R )
= ( ! [S: produc1996495991257130529ist_fm,R1: rule,Sl2: fset_P661503646757059847ist_fm,S3: produc1996495991257130529ist_fm] :
( ( ( member8102475879199740618ist_fm @ S @ top_to728902761071950545ist_fm )
& ( abstra3055540779341473792ist_fm
@ ^ [R3: rule,T: produc1996495991257130529ist_fm,Ss2: fset_P661503646757059847ist_fm] :
( ( eff @ R3 @ T )
= ( some_f2162114139285431020ist_fm @ Ss2 ) )
@ R
@ S )
& ( member_rule @ R1 @ ( minus_minus_set_rule @ ( sset_rule @ rules ) @ ( insert_rule @ R @ bot_bot_set_rule ) ) )
& ( ( eff @ R1 @ S )
= ( some_f2162114139285431020ist_fm @ Sl2 ) )
& ( fmembe3381613331217039976ist_fm @ S3 @ Sl2 ) )
=> ( abstra3055540779341473792ist_fm
@ ^ [R3: rule,T: produc1996495991257130529ist_fm,Ss2: fset_P661503646757059847ist_fm] :
( ( eff @ R3 @ T )
= ( some_f2162114139285431020ist_fm @ Ss2 ) )
@ R
@ S3 ) ) ) ) ).
% per_def
thf(fact_201_pickEff,axiom,
! [R: rule,S2: produc1996495991257130529ist_fm] :
( ( abstra3055540779341473792ist_fm
@ ^ [R3: rule,S: produc1996495991257130529ist_fm,Ss2: fset_P661503646757059847ist_fm] :
( ( eff @ R3 @ S )
= ( some_f2162114139285431020ist_fm @ Ss2 ) )
@ R
@ S2 )
=> ( ( eff @ R @ S2 )
= ( some_f2162114139285431020ist_fm
@ ( abstra458490714785879965ist_fm
@ ^ [R3: rule,S: produc1996495991257130529ist_fm,Ss2: fset_P661503646757059847ist_fm] :
( ( eff @ R3 @ S )
= ( some_f2162114139285431020ist_fm @ Ss2 ) )
@ R
@ S2 ) ) ) ) ).
% pickEff
thf(fact_202_member__remove,axiom,
! [X2: produc1996495991257130529ist_fm,Y3: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm] :
( ( member8102475879199740618ist_fm @ X2 @ ( remove6605582569798670470ist_fm @ Y3 @ A ) )
= ( ( member8102475879199740618ist_fm @ X2 @ A )
& ( X2 != Y3 ) ) ) ).
% member_remove
thf(fact_203_member__remove,axiom,
! [X2: rule,Y3: rule,A: set_rule] :
( ( member_rule @ X2 @ ( remove_rule @ Y3 @ A ) )
= ( ( member_rule @ X2 @ A )
& ( X2 != Y3 ) ) ) ).
% member_remove
thf(fact_204_member__remove,axiom,
! [X2: $o,Y3: $o,A: set_o] :
( ( member_o @ X2 @ ( remove_o @ Y3 @ A ) )
= ( ( member_o @ X2 @ A )
& ( X2 != Y3 ) ) ) ).
% member_remove
thf(fact_205_NE__R,axiom,
( ( sset_rule @ rules )
!= bot_bot_set_rule ) ).
% NE_R
thf(fact_206_UNIV__rules,axiom,
( ( sset_rule @ rules )
= top_top_set_rule ) ).
% UNIV_rules
thf(fact_207_top__set__def,axiom,
( top_to728902761071950545ist_fm
= ( collec7231575776666803724ist_fm @ top_to6798613745154499404t_fm_o ) ) ).
% top_set_def
thf(fact_208_top__set__def,axiom,
( top_top_set_rule
= ( collect_rule @ top_top_rule_o ) ) ).
% top_set_def
thf(fact_209_top__set__def,axiom,
( top_top_set_o
= ( collect_o @ top_top_o_o ) ) ).
% top_set_def
thf(fact_210_bot__set__def,axiom,
( bot_bot_set_rule
= ( collect_rule @ bot_bot_rule_o ) ) ).
% bot_set_def
thf(fact_211_bot__set__def,axiom,
( bot_bot_set_o
= ( collect_o @ bot_bot_o_o ) ) ).
% bot_set_def
thf(fact_212_bot__set__def,axiom,
( bot_bo4530011010606209517ist_fm
= ( collec7231575776666803724ist_fm @ bot_bo1394704948141485872t_fm_o ) ) ).
% bot_set_def
thf(fact_213_minus__set__def,axiom,
( minus_minus_set_rule
= ( ^ [A5: set_rule,B5: set_rule] :
( collect_rule
@ ( minus_minus_rule_o
@ ^ [X: rule] : ( member_rule @ X @ A5 )
@ ^ [X: rule] : ( member_rule @ X @ B5 ) ) ) ) ) ).
% minus_set_def
thf(fact_214_minus__set__def,axiom,
( minus_minus_set_o
= ( ^ [A5: set_o,B5: set_o] :
( collect_o
@ ( minus_minus_o_o
@ ^ [X: $o] : ( member_o @ X @ A5 )
@ ^ [X: $o] : ( member_o @ X @ B5 ) ) ) ) ) ).
% minus_set_def
thf(fact_215_minus__set__def,axiom,
( minus_5615909706921521512ist_fm
= ( ^ [A5: set_Pr7058068377845519745ist_fm,B5: set_Pr7058068377845519745ist_fm] :
( collec7231575776666803724ist_fm
@ ( minus_297532591554960245t_fm_o
@ ^ [X: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X @ A5 )
@ ^ [X: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X @ B5 ) ) ) ) ) ).
% minus_set_def
thf(fact_216_enabled__R,axiom,
! [S2: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ S2 @ top_to728902761071950545ist_fm )
=> ? [X3: rule] :
( ( member_rule @ X3 @ ( sset_rule @ rules ) )
& ? [Sl3: fset_P661503646757059847ist_fm] :
( ( eff @ X3 @ S2 )
= ( some_f2162114139285431020ist_fm @ Sl3 ) ) ) ) ).
% enabled_R
thf(fact_217_eff__S,axiom,
! [S2: produc1996495991257130529ist_fm,R: rule,Sl4: fset_P661503646757059847ist_fm,S4: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ S2 @ top_to728902761071950545ist_fm )
=> ( ( member_rule @ R @ ( sset_rule @ rules ) )
=> ( ( ( eff @ R @ S2 )
= ( some_f2162114139285431020ist_fm @ Sl4 ) )
=> ( ( fmembe3381613331217039976ist_fm @ S4 @ Sl4 )
=> ( member8102475879199740618ist_fm @ S4 @ top_to728902761071950545ist_fm ) ) ) ) ) ).
% eff_S
thf(fact_218_is__singletonI_H,axiom,
! [A: set_rule] :
( ( A != bot_bot_set_rule )
=> ( ! [X3: rule,Y2: rule] :
( ( member_rule @ X3 @ A )
=> ( ( member_rule @ Y2 @ A )
=> ( X3 = Y2 ) ) )
=> ( is_singleton_rule @ A ) ) ) ).
% is_singletonI'
thf(fact_219_is__singletonI_H,axiom,
! [A: set_o] :
( ( A != bot_bot_set_o )
=> ( ! [X3: $o,Y2: $o] :
( ( member_o @ X3 @ A )
=> ( ( member_o @ Y2 @ A )
=> ( X3 = Y2 ) ) )
=> ( is_singleton_o @ A ) ) ) ).
% is_singletonI'
thf(fact_220_is__singletonI_H,axiom,
! [A: set_Pr7058068377845519745ist_fm] :
( ( A != bot_bo4530011010606209517ist_fm )
=> ( ! [X3: produc1996495991257130529ist_fm,Y2: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ X3 @ A )
=> ( ( member8102475879199740618ist_fm @ Y2 @ A )
=> ( X3 = Y2 ) ) )
=> ( is_sin3852095833271400213ist_fm @ A ) ) ) ).
% is_singletonI'
thf(fact_221_Pair__inject,axiom,
! [A3: list_fm,B3: list_fm,A6: list_fm,B6: list_fm] :
( ( ( produc381145313068854617ist_fm @ A3 @ B3 )
= ( produc381145313068854617ist_fm @ A6 @ B6 ) )
=> ~ ( ( A3 = A6 )
=> ( B3 != B6 ) ) ) ).
% Pair_inject
thf(fact_222_prod__cases,axiom,
! [P: produc1996495991257130529ist_fm > $o,P2: produc1996495991257130529ist_fm] :
( ! [A7: list_fm,B7: list_fm] : ( P @ ( produc381145313068854617ist_fm @ A7 @ B7 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_223_surj__pair,axiom,
! [P2: produc1996495991257130529ist_fm] :
? [X3: list_fm,Y2: list_fm] :
( P2
= ( produc381145313068854617ist_fm @ X3 @ Y2 ) ) ).
% surj_pair
thf(fact_224_old_Oprod_Oexhaust,axiom,
! [Y3: produc1996495991257130529ist_fm] :
~ ! [A7: list_fm,B7: list_fm] :
( Y3
!= ( produc381145313068854617ist_fm @ A7 @ B7 ) ) ).
% old.prod.exhaust
thf(fact_225_is__singletonE,axiom,
! [A: set_rule] :
( ( is_singleton_rule @ A )
=> ~ ! [X3: rule] :
( A
!= ( insert_rule @ X3 @ bot_bot_set_rule ) ) ) ).
% is_singletonE
thf(fact_226_is__singletonE,axiom,
! [A: set_o] :
( ( is_singleton_o @ A )
=> ~ ! [X3: $o] :
( A
!= ( insert_o @ X3 @ bot_bot_set_o ) ) ) ).
% is_singletonE
thf(fact_227_is__singletonE,axiom,
! [A: set_Pr7058068377845519745ist_fm] :
( ( is_sin3852095833271400213ist_fm @ A )
=> ~ ! [X3: produc1996495991257130529ist_fm] :
( A
!= ( insert589278167076416497ist_fm @ X3 @ bot_bo4530011010606209517ist_fm ) ) ) ).
% is_singletonE
thf(fact_228_is__singleton__def,axiom,
( is_singleton_rule
= ( ^ [A5: set_rule] :
? [X: rule] :
( A5
= ( insert_rule @ X @ bot_bot_set_rule ) ) ) ) ).
% is_singleton_def
thf(fact_229_is__singleton__def,axiom,
( is_singleton_o
= ( ^ [A5: set_o] :
? [X: $o] :
( A5
= ( insert_o @ X @ bot_bot_set_o ) ) ) ) ).
% is_singleton_def
thf(fact_230_is__singleton__def,axiom,
( is_sin3852095833271400213ist_fm
= ( ^ [A5: set_Pr7058068377845519745ist_fm] :
? [X: produc1996495991257130529ist_fm] :
( A5
= ( insert589278167076416497ist_fm @ X @ bot_bo4530011010606209517ist_fm ) ) ) ) ).
% is_singleton_def
thf(fact_231_fminusI,axiom,
! [C: produc1996495991257130529ist_fm,A: fset_P661503646757059847ist_fm,B: fset_P661503646757059847ist_fm] :
( ( fmembe3381613331217039976ist_fm @ C @ A )
=> ( ~ ( fmembe3381613331217039976ist_fm @ C @ B )
=> ( fmembe3381613331217039976ist_fm @ C @ ( minus_8437252545254675822ist_fm @ A @ B ) ) ) ) ).
% fminusI
thf(fact_232_fminus__iff,axiom,
! [C: produc1996495991257130529ist_fm,A: fset_P661503646757059847ist_fm,B: fset_P661503646757059847ist_fm] :
( ( fmembe3381613331217039976ist_fm @ C @ ( minus_8437252545254675822ist_fm @ A @ B ) )
= ( ( fmembe3381613331217039976ist_fm @ C @ A )
& ~ ( fmembe3381613331217039976ist_fm @ C @ B ) ) ) ).
% fminus_iff
thf(fact_233_fempty__iff,axiom,
! [C: produc1996495991257130529ist_fm] :
~ ( fmembe3381613331217039976ist_fm @ C @ bot_bo2367426573206113139ist_fm ) ).
% fempty_iff
thf(fact_234_all__not__fin__conv,axiom,
! [A: fset_P661503646757059847ist_fm] :
( ( ! [X: produc1996495991257130529ist_fm] :
~ ( fmembe3381613331217039976ist_fm @ X @ A ) )
= ( A = bot_bo2367426573206113139ist_fm ) ) ).
% all_not_fin_conv
thf(fact_235_top__empty__eq2,axiom,
( top_to7326020031456794390t_fm_o
= ( ^ [X: list_fm,Y4: list_fm] : ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X @ Y4 ) @ top_to728902761071950545ist_fm ) ) ) ).
% top_empty_eq2
thf(fact_236_RuleSystem__Defs_OpickEff,axiom,
! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,R: rule,S2: produc1996495991257130529ist_fm] :
( ( abstra3055540779341473792ist_fm @ Eff @ R @ S2 )
=> ( Eff @ R @ S2 @ ( abstra458490714785879965ist_fm @ Eff @ R @ S2 ) ) ) ).
% RuleSystem_Defs.pickEff
thf(fact_237_i_ONE__R,axiom,
! [Rules: stream_rule] :
( ( sset_rule @ Rules )
!= bot_bot_set_rule ) ).
% i.NE_R
thf(fact_238_i_ONE__R,axiom,
! [Rules: stream_o] :
( ( sset_o @ Rules )
!= bot_bot_set_o ) ).
% i.NE_R
thf(fact_239_i_ONE__R,axiom,
! [Rules: stream8299795917829157543ist_fm] :
( ( sset_P3431374123360452149ist_fm @ Rules )
!= bot_bo4530011010606209517ist_fm ) ).
% i.NE_R
thf(fact_240_RuleSystem__Defs_ONE__R,axiom,
! [Rules: stream_rule] :
( ( sset_rule @ Rules )
!= bot_bot_set_rule ) ).
% RuleSystem_Defs.NE_R
thf(fact_241_RuleSystem__Defs_ONE__R,axiom,
! [Rules: stream_o] :
( ( sset_o @ Rules )
!= bot_bot_set_o ) ).
% RuleSystem_Defs.NE_R
thf(fact_242_RuleSystem__Defs_ONE__R,axiom,
! [Rules: stream8299795917829157543ist_fm] :
( ( sset_P3431374123360452149ist_fm @ Rules )
!= bot_bo4530011010606209517ist_fm ) ).
% RuleSystem_Defs.NE_R
thf(fact_243_RuleSystem_Oper__def,axiom,
! [Eff: rule > rule > fset_rule > $o,Rules: stream_rule,S5: set_rule,R: rule] :
( ( abstra2725590884229999157e_rule @ Eff @ Rules @ S5 )
=> ( ( abstra5282173100671685808e_rule @ Eff @ Rules @ S5 @ R )
= ( ! [S: rule,R1: rule,Sl2: fset_rule,S3: rule] :
( ( ( member_rule @ S @ S5 )
& ( abstra3154316310710804774e_rule @ Eff @ R @ S )
& ( member_rule @ R1 @ ( minus_minus_set_rule @ ( sset_rule @ Rules ) @ ( insert_rule @ R @ bot_bot_set_rule ) ) )
& ( Eff @ R1 @ S @ Sl2 )
& ( fmember_rule @ S3 @ Sl2 ) )
=> ( abstra3154316310710804774e_rule @ Eff @ R @ S3 ) ) ) ) ) ).
% RuleSystem.per_def
thf(fact_244_RuleSystem_Oper__def,axiom,
! [Eff: rule > $o > fset_o > $o,Rules: stream_rule,S5: set_o,R: rule] :
( ( abstra8374989651900038312rule_o @ Eff @ Rules @ S5 )
=> ( ( abstract_per_rule_o @ Eff @ Rules @ S5 @ R )
= ( ! [S: $o,R1: rule,Sl2: fset_o,S3: $o] :
( ( ( member_o @ S @ S5 )
& ( abstra1294392932120118903rule_o @ Eff @ R @ S )
& ( member_rule @ R1 @ ( minus_minus_set_rule @ ( sset_rule @ Rules ) @ ( insert_rule @ R @ bot_bot_set_rule ) ) )
& ( Eff @ R1 @ S @ Sl2 )
& ( fmember_o @ S3 @ Sl2 ) )
=> ( abstra1294392932120118903rule_o @ Eff @ R @ S3 ) ) ) ) ) ).
% RuleSystem.per_def
thf(fact_245_RuleSystem_Oper__def,axiom,
! [Eff: $o > rule > fset_rule > $o,Rules: stream_o,S5: set_rule,R: $o] :
( ( abstra5464778857772719348o_rule @ Eff @ Rules @ S5 )
=> ( ( abstract_per_o_rule @ Eff @ Rules @ S5 @ R )
= ( ! [S: rule,R1: $o,Sl2: fset_rule,S3: rule] :
( ( ( member_rule @ S @ S5 )
& ( abstra7607554174847575747o_rule @ Eff @ R @ S )
& ( member_o @ R1 @ ( minus_minus_set_o @ ( sset_o @ Rules ) @ ( insert_o @ R @ bot_bot_set_o ) ) )
& ( Eff @ R1 @ S @ Sl2 )
& ( fmember_rule @ S3 @ Sl2 ) )
=> ( abstra7607554174847575747o_rule @ Eff @ R @ S3 ) ) ) ) ) ).
% RuleSystem.per_def
thf(fact_246_RuleSystem_Oper__def,axiom,
! [Eff: $o > $o > fset_o > $o,Rules: stream_o,S5: set_o,R: $o] :
( ( abstra8727705743885569321em_o_o @ Eff @ Rules @ S5 )
=> ( ( abstract_per_o_o @ Eff @ Rules @ S5 @ R )
= ( ! [S: $o,R1: $o,Sl2: fset_o,S3: $o] :
( ( ( member_o @ S @ S5 )
& ( abstra8815064841987723930ed_o_o @ Eff @ R @ S )
& ( member_o @ R1 @ ( minus_minus_set_o @ ( sset_o @ Rules ) @ ( insert_o @ R @ bot_bot_set_o ) ) )
& ( Eff @ R1 @ S @ Sl2 )
& ( fmember_o @ S3 @ Sl2 ) )
=> ( abstra8815064841987723930ed_o_o @ Eff @ R @ S3 ) ) ) ) ) ).
% RuleSystem.per_def
thf(fact_247_RuleSystem_Oper__def,axiom,
! [Eff: $o > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_o,S5: set_Pr7058068377845519745ist_fm,R: $o] :
( ( abstra5822460560083087538ist_fm @ Eff @ Rules @ S5 )
=> ( ( abstra8404946688271621549ist_fm @ Eff @ Rules @ S5 @ R )
= ( ! [S: produc1996495991257130529ist_fm,R1: $o,Sl2: fset_P661503646757059847ist_fm,S3: produc1996495991257130529ist_fm] :
( ( ( member8102475879199740618ist_fm @ S @ S5 )
& ( abstra6289332626269597475ist_fm @ Eff @ R @ S )
& ( member_o @ R1 @ ( minus_minus_set_o @ ( sset_o @ Rules ) @ ( insert_o @ R @ bot_bot_set_o ) ) )
& ( Eff @ R1 @ S @ Sl2 )
& ( fmembe3381613331217039976ist_fm @ S3 @ Sl2 ) )
=> ( abstra6289332626269597475ist_fm @ Eff @ R @ S3 ) ) ) ) ) ).
% RuleSystem.per_def
thf(fact_248_RuleSystem_Oper__def,axiom,
! [Eff: produc1996495991257130529ist_fm > rule > fset_rule > $o,Rules: stream8299795917829157543ist_fm,S5: set_rule,R: produc1996495991257130529ist_fm] :
( ( abstra9106359784214577277m_rule @ Eff @ Rules @ S5 )
=> ( ( abstra6804973471972959042m_rule @ Eff @ Rules @ S5 @ R )
= ( ! [S: rule,R1: produc1996495991257130529ist_fm,Sl2: fset_rule,S3: rule] :
( ( ( member_rule @ S @ S5 )
& ( abstra5706625008861892940m_rule @ Eff @ R @ S )
& ( member8102475879199740618ist_fm @ R1 @ ( minus_5615909706921521512ist_fm @ ( sset_P3431374123360452149ist_fm @ Rules ) @ ( insert589278167076416497ist_fm @ R @ bot_bo4530011010606209517ist_fm ) ) )
& ( Eff @ R1 @ S @ Sl2 )
& ( fmember_rule @ S3 @ Sl2 ) )
=> ( abstra5706625008861892940m_rule @ Eff @ R @ S3 ) ) ) ) ) ).
% RuleSystem.per_def
thf(fact_249_RuleSystem_Oper__def,axiom,
! [Eff: produc1996495991257130529ist_fm > $o > fset_o > $o,Rules: stream8299795917829157543ist_fm,S5: set_o,R: produc1996495991257130529ist_fm] :
( ( abstra5561694101053127776t_fm_o @ Eff @ Rules @ S5 )
=> ( ( abstra8144180229241661787t_fm_o @ Eff @ Rules @ S5 @ R )
= ( ! [S: $o,R1: produc1996495991257130529ist_fm,Sl2: fset_o,S3: $o] :
( ( ( member_o @ S @ S5 )
& ( abstra6028566167239637713t_fm_o @ Eff @ R @ S )
& ( member8102475879199740618ist_fm @ R1 @ ( minus_5615909706921521512ist_fm @ ( sset_P3431374123360452149ist_fm @ Rules ) @ ( insert589278167076416497ist_fm @ R @ bot_bo4530011010606209517ist_fm ) ) )
& ( Eff @ R1 @ S @ Sl2 )
& ( fmember_o @ S3 @ Sl2 ) )
=> ( abstra6028566167239637713t_fm_o @ Eff @ R @ S3 ) ) ) ) ) ).
% RuleSystem.per_def
thf(fact_250_RuleSystem_Oper__def,axiom,
! [Eff: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream8299795917829157543ist_fm,S5: set_Pr7058068377845519745ist_fm,R: produc1996495991257130529ist_fm] :
( ( abstra4969282175937908457ist_fm @ Eff @ Rules @ S5 )
=> ( ( abstra7567571716448718564ist_fm @ Eff @ Rules @ S5 @ R )
= ( ! [S: produc1996495991257130529ist_fm,R1: produc1996495991257130529ist_fm,Sl2: fset_P661503646757059847ist_fm,S3: produc1996495991257130529ist_fm] :
( ( ( member8102475879199740618ist_fm @ S @ S5 )
& ( abstra1009249261476546138ist_fm @ Eff @ R @ S )
& ( member8102475879199740618ist_fm @ R1 @ ( minus_5615909706921521512ist_fm @ ( sset_P3431374123360452149ist_fm @ Rules ) @ ( insert589278167076416497ist_fm @ R @ bot_bo4530011010606209517ist_fm ) ) )
& ( Eff @ R1 @ S @ Sl2 )
& ( fmembe3381613331217039976ist_fm @ S3 @ Sl2 ) )
=> ( abstra1009249261476546138ist_fm @ Eff @ R @ S3 ) ) ) ) ) ).
% RuleSystem.per_def
thf(fact_251_RuleSystem_Oper__def,axiom,
! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S5: set_Pr7058068377845519745ist_fm,R: rule] :
( ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S5 )
=> ( ( abstra4153889242452539894ist_fm @ Eff @ Rules @ S5 @ R )
= ( ! [S: produc1996495991257130529ist_fm,R1: rule,Sl2: fset_P661503646757059847ist_fm,S3: produc1996495991257130529ist_fm] :
( ( ( member8102475879199740618ist_fm @ S @ S5 )
& ( abstra3055540779341473792ist_fm @ Eff @ R @ S )
& ( member_rule @ R1 @ ( minus_minus_set_rule @ ( sset_rule @ Rules ) @ ( insert_rule @ R @ bot_bot_set_rule ) ) )
& ( Eff @ R1 @ S @ Sl2 )
& ( fmembe3381613331217039976ist_fm @ S3 @ Sl2 ) )
=> ( abstra3055540779341473792ist_fm @ Eff @ R @ S3 ) ) ) ) ) ).
% RuleSystem.per_def
thf(fact_252_RuleSystem_Oenabled__R,axiom,
! [Eff: rule > rule > fset_rule > $o,Rules: stream_rule,S5: set_rule,S2: rule] :
( ( abstra2725590884229999157e_rule @ Eff @ Rules @ S5 )
=> ( ( member_rule @ S2 @ S5 )
=> ? [X3: rule] :
( ( member_rule @ X3 @ ( sset_rule @ Rules ) )
& ? [X_1: fset_rule] : ( Eff @ X3 @ S2 @ X_1 ) ) ) ) ).
% RuleSystem.enabled_R
thf(fact_253_RuleSystem_Oenabled__R,axiom,
! [Eff: rule > $o > fset_o > $o,Rules: stream_rule,S5: set_o,S2: $o] :
( ( abstra8374989651900038312rule_o @ Eff @ Rules @ S5 )
=> ( ( member_o @ S2 @ S5 )
=> ? [X3: rule] :
( ( member_rule @ X3 @ ( sset_rule @ Rules ) )
& ? [X_1: fset_o] : ( Eff @ X3 @ S2 @ X_1 ) ) ) ) ).
% RuleSystem.enabled_R
thf(fact_254_RuleSystem_Oenabled__R,axiom,
! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S5: set_Pr7058068377845519745ist_fm,S2: produc1996495991257130529ist_fm] :
( ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S5 )
=> ( ( member8102475879199740618ist_fm @ S2 @ S5 )
=> ? [X3: rule] :
( ( member_rule @ X3 @ ( sset_rule @ Rules ) )
& ? [X_1: fset_P661503646757059847ist_fm] : ( Eff @ X3 @ S2 @ X_1 ) ) ) ) ).
% RuleSystem.enabled_R
thf(fact_255_RuleSystem_Oeff__S,axiom,
! [Eff: produc1996495991257130529ist_fm > rule > fset_rule > $o,Rules: stream8299795917829157543ist_fm,S5: set_rule,S2: rule,R: produc1996495991257130529ist_fm,Sl4: fset_rule,S4: rule] :
( ( abstra9106359784214577277m_rule @ Eff @ Rules @ S5 )
=> ( ( member_rule @ S2 @ S5 )
=> ( ( member8102475879199740618ist_fm @ R @ ( sset_P3431374123360452149ist_fm @ Rules ) )
=> ( ( Eff @ R @ S2 @ Sl4 )
=> ( ( fmember_rule @ S4 @ Sl4 )
=> ( member_rule @ S4 @ S5 ) ) ) ) ) ) ).
% RuleSystem.eff_S
thf(fact_256_RuleSystem_Oeff__S,axiom,
! [Eff: $o > rule > fset_rule > $o,Rules: stream_o,S5: set_rule,S2: rule,R: $o,Sl4: fset_rule,S4: rule] :
( ( abstra5464778857772719348o_rule @ Eff @ Rules @ S5 )
=> ( ( member_rule @ S2 @ S5 )
=> ( ( member_o @ R @ ( sset_o @ Rules ) )
=> ( ( Eff @ R @ S2 @ Sl4 )
=> ( ( fmember_rule @ S4 @ Sl4 )
=> ( member_rule @ S4 @ S5 ) ) ) ) ) ) ).
% RuleSystem.eff_S
thf(fact_257_RuleSystem_Oeff__S,axiom,
! [Eff: produc1996495991257130529ist_fm > $o > fset_o > $o,Rules: stream8299795917829157543ist_fm,S5: set_o,S2: $o,R: produc1996495991257130529ist_fm,Sl4: fset_o,S4: $o] :
( ( abstra5561694101053127776t_fm_o @ Eff @ Rules @ S5 )
=> ( ( member_o @ S2 @ S5 )
=> ( ( member8102475879199740618ist_fm @ R @ ( sset_P3431374123360452149ist_fm @ Rules ) )
=> ( ( Eff @ R @ S2 @ Sl4 )
=> ( ( fmember_o @ S4 @ Sl4 )
=> ( member_o @ S4 @ S5 ) ) ) ) ) ) ).
% RuleSystem.eff_S
thf(fact_258_RuleSystem_Oeff__S,axiom,
! [Eff: $o > $o > fset_o > $o,Rules: stream_o,S5: set_o,S2: $o,R: $o,Sl4: fset_o,S4: $o] :
( ( abstra8727705743885569321em_o_o @ Eff @ Rules @ S5 )
=> ( ( member_o @ S2 @ S5 )
=> ( ( member_o @ R @ ( sset_o @ Rules ) )
=> ( ( Eff @ R @ S2 @ Sl4 )
=> ( ( fmember_o @ S4 @ Sl4 )
=> ( member_o @ S4 @ S5 ) ) ) ) ) ) ).
% RuleSystem.eff_S
thf(fact_259_RuleSystem_Oeff__S,axiom,
! [Eff: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream8299795917829157543ist_fm,S5: set_Pr7058068377845519745ist_fm,S2: produc1996495991257130529ist_fm,R: produc1996495991257130529ist_fm,Sl4: fset_P661503646757059847ist_fm,S4: produc1996495991257130529ist_fm] :
( ( abstra4969282175937908457ist_fm @ Eff @ Rules @ S5 )
=> ( ( member8102475879199740618ist_fm @ S2 @ S5 )
=> ( ( member8102475879199740618ist_fm @ R @ ( sset_P3431374123360452149ist_fm @ Rules ) )
=> ( ( Eff @ R @ S2 @ Sl4 )
=> ( ( fmembe3381613331217039976ist_fm @ S4 @ Sl4 )
=> ( member8102475879199740618ist_fm @ S4 @ S5 ) ) ) ) ) ) ).
% RuleSystem.eff_S
thf(fact_260_RuleSystem_Oeff__S,axiom,
! [Eff: $o > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_o,S5: set_Pr7058068377845519745ist_fm,S2: produc1996495991257130529ist_fm,R: $o,Sl4: fset_P661503646757059847ist_fm,S4: produc1996495991257130529ist_fm] :
( ( abstra5822460560083087538ist_fm @ Eff @ Rules @ S5 )
=> ( ( member8102475879199740618ist_fm @ S2 @ S5 )
=> ( ( member_o @ R @ ( sset_o @ Rules ) )
=> ( ( Eff @ R @ S2 @ Sl4 )
=> ( ( fmembe3381613331217039976ist_fm @ S4 @ Sl4 )
=> ( member8102475879199740618ist_fm @ S4 @ S5 ) ) ) ) ) ) ).
% RuleSystem.eff_S
thf(fact_261_RuleSystem_Oeff__S,axiom,
! [Eff: rule > rule > fset_rule > $o,Rules: stream_rule,S5: set_rule,S2: rule,R: rule,Sl4: fset_rule,S4: rule] :
( ( abstra2725590884229999157e_rule @ Eff @ Rules @ S5 )
=> ( ( member_rule @ S2 @ S5 )
=> ( ( member_rule @ R @ ( sset_rule @ Rules ) )
=> ( ( Eff @ R @ S2 @ Sl4 )
=> ( ( fmember_rule @ S4 @ Sl4 )
=> ( member_rule @ S4 @ S5 ) ) ) ) ) ) ).
% RuleSystem.eff_S
thf(fact_262_RuleSystem_Oeff__S,axiom,
! [Eff: rule > $o > fset_o > $o,Rules: stream_rule,S5: set_o,S2: $o,R: rule,Sl4: fset_o,S4: $o] :
( ( abstra8374989651900038312rule_o @ Eff @ Rules @ S5 )
=> ( ( member_o @ S2 @ S5 )
=> ( ( member_rule @ R @ ( sset_rule @ Rules ) )
=> ( ( Eff @ R @ S2 @ Sl4 )
=> ( ( fmember_o @ S4 @ Sl4 )
=> ( member_o @ S4 @ S5 ) ) ) ) ) ) ).
% RuleSystem.eff_S
thf(fact_263_RuleSystem_Oeff__S,axiom,
! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S5: set_Pr7058068377845519745ist_fm,S2: produc1996495991257130529ist_fm,R: rule,Sl4: fset_P661503646757059847ist_fm,S4: produc1996495991257130529ist_fm] :
( ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S5 )
=> ( ( member8102475879199740618ist_fm @ S2 @ S5 )
=> ( ( member_rule @ R @ ( sset_rule @ Rules ) )
=> ( ( Eff @ R @ S2 @ Sl4 )
=> ( ( fmembe3381613331217039976ist_fm @ S4 @ Sl4 )
=> ( member8102475879199740618ist_fm @ S4 @ S5 ) ) ) ) ) ) ).
% RuleSystem.eff_S
thf(fact_264_RuleSystem_Ointro,axiom,
! [S5: set_rule,Rules: stream8299795917829157543ist_fm,Eff: produc1996495991257130529ist_fm > rule > fset_rule > $o] :
( ! [S6: rule] :
( ( member_rule @ S6 @ S5 )
=> ! [R4: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ R4 @ ( sset_P3431374123360452149ist_fm @ Rules ) )
=> ! [Sl3: fset_rule] :
( ( Eff @ R4 @ S6 @ Sl3 )
=> ! [S7: rule] :
( ( fmember_rule @ S7 @ Sl3 )
=> ( member_rule @ S7 @ S5 ) ) ) ) )
=> ( ! [S6: rule] :
( ( member_rule @ S6 @ S5 )
=> ? [X4: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ X4 @ ( sset_P3431374123360452149ist_fm @ Rules ) )
& ? [X_12: fset_rule] : ( Eff @ X4 @ S6 @ X_12 ) ) )
=> ( abstra9106359784214577277m_rule @ Eff @ Rules @ S5 ) ) ) ).
% RuleSystem.intro
thf(fact_265_RuleSystem_Ointro,axiom,
! [S5: set_rule,Rules: stream_o,Eff: $o > rule > fset_rule > $o] :
( ! [S6: rule] :
( ( member_rule @ S6 @ S5 )
=> ! [R4: $o] :
( ( member_o @ R4 @ ( sset_o @ Rules ) )
=> ! [Sl3: fset_rule] :
( ( Eff @ R4 @ S6 @ Sl3 )
=> ! [S7: rule] :
( ( fmember_rule @ S7 @ Sl3 )
=> ( member_rule @ S7 @ S5 ) ) ) ) )
=> ( ! [S6: rule] :
( ( member_rule @ S6 @ S5 )
=> ? [X4: $o] :
( ( member_o @ X4 @ ( sset_o @ Rules ) )
& ? [X_12: fset_rule] : ( Eff @ X4 @ S6 @ X_12 ) ) )
=> ( abstra5464778857772719348o_rule @ Eff @ Rules @ S5 ) ) ) ).
% RuleSystem.intro
thf(fact_266_RuleSystem_Ointro,axiom,
! [S5: set_o,Rules: stream8299795917829157543ist_fm,Eff: produc1996495991257130529ist_fm > $o > fset_o > $o] :
( ! [S6: $o] :
( ( member_o @ S6 @ S5 )
=> ! [R4: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ R4 @ ( sset_P3431374123360452149ist_fm @ Rules ) )
=> ! [Sl3: fset_o] :
( ( Eff @ R4 @ S6 @ Sl3 )
=> ! [S7: $o] :
( ( fmember_o @ S7 @ Sl3 )
=> ( member_o @ S7 @ S5 ) ) ) ) )
=> ( ! [S6: $o] :
( ( member_o @ S6 @ S5 )
=> ? [X4: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ X4 @ ( sset_P3431374123360452149ist_fm @ Rules ) )
& ? [X_12: fset_o] : ( Eff @ X4 @ S6 @ X_12 ) ) )
=> ( abstra5561694101053127776t_fm_o @ Eff @ Rules @ S5 ) ) ) ).
% RuleSystem.intro
thf(fact_267_RuleSystem_Ointro,axiom,
! [S5: set_o,Rules: stream_o,Eff: $o > $o > fset_o > $o] :
( ! [S6: $o] :
( ( member_o @ S6 @ S5 )
=> ! [R4: $o] :
( ( member_o @ R4 @ ( sset_o @ Rules ) )
=> ! [Sl3: fset_o] :
( ( Eff @ R4 @ S6 @ Sl3 )
=> ! [S7: $o] :
( ( fmember_o @ S7 @ Sl3 )
=> ( member_o @ S7 @ S5 ) ) ) ) )
=> ( ! [S6: $o] :
( ( member_o @ S6 @ S5 )
=> ? [X4: $o] :
( ( member_o @ X4 @ ( sset_o @ Rules ) )
& ? [X_12: fset_o] : ( Eff @ X4 @ S6 @ X_12 ) ) )
=> ( abstra8727705743885569321em_o_o @ Eff @ Rules @ S5 ) ) ) ).
% RuleSystem.intro
thf(fact_268_RuleSystem_Ointro,axiom,
! [S5: set_Pr7058068377845519745ist_fm,Rules: stream8299795917829157543ist_fm,Eff: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
( ! [S6: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ S6 @ S5 )
=> ! [R4: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ R4 @ ( sset_P3431374123360452149ist_fm @ Rules ) )
=> ! [Sl3: fset_P661503646757059847ist_fm] :
( ( Eff @ R4 @ S6 @ Sl3 )
=> ! [S7: produc1996495991257130529ist_fm] :
( ( fmembe3381613331217039976ist_fm @ S7 @ Sl3 )
=> ( member8102475879199740618ist_fm @ S7 @ S5 ) ) ) ) )
=> ( ! [S6: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ S6 @ S5 )
=> ? [X4: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ X4 @ ( sset_P3431374123360452149ist_fm @ Rules ) )
& ? [X_12: fset_P661503646757059847ist_fm] : ( Eff @ X4 @ S6 @ X_12 ) ) )
=> ( abstra4969282175937908457ist_fm @ Eff @ Rules @ S5 ) ) ) ).
% RuleSystem.intro
thf(fact_269_RuleSystem_Ointro,axiom,
! [S5: set_Pr7058068377845519745ist_fm,Rules: stream_o,Eff: $o > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
( ! [S6: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ S6 @ S5 )
=> ! [R4: $o] :
( ( member_o @ R4 @ ( sset_o @ Rules ) )
=> ! [Sl3: fset_P661503646757059847ist_fm] :
( ( Eff @ R4 @ S6 @ Sl3 )
=> ! [S7: produc1996495991257130529ist_fm] :
( ( fmembe3381613331217039976ist_fm @ S7 @ Sl3 )
=> ( member8102475879199740618ist_fm @ S7 @ S5 ) ) ) ) )
=> ( ! [S6: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ S6 @ S5 )
=> ? [X4: $o] :
( ( member_o @ X4 @ ( sset_o @ Rules ) )
& ? [X_12: fset_P661503646757059847ist_fm] : ( Eff @ X4 @ S6 @ X_12 ) ) )
=> ( abstra5822460560083087538ist_fm @ Eff @ Rules @ S5 ) ) ) ).
% RuleSystem.intro
thf(fact_270_RuleSystem_Ointro,axiom,
! [S5: set_rule,Rules: stream_rule,Eff: rule > rule > fset_rule > $o] :
( ! [S6: rule] :
( ( member_rule @ S6 @ S5 )
=> ! [R4: rule] :
( ( member_rule @ R4 @ ( sset_rule @ Rules ) )
=> ! [Sl3: fset_rule] :
( ( Eff @ R4 @ S6 @ Sl3 )
=> ! [S7: rule] :
( ( fmember_rule @ S7 @ Sl3 )
=> ( member_rule @ S7 @ S5 ) ) ) ) )
=> ( ! [S6: rule] :
( ( member_rule @ S6 @ S5 )
=> ? [X4: rule] :
( ( member_rule @ X4 @ ( sset_rule @ Rules ) )
& ? [X_12: fset_rule] : ( Eff @ X4 @ S6 @ X_12 ) ) )
=> ( abstra2725590884229999157e_rule @ Eff @ Rules @ S5 ) ) ) ).
% RuleSystem.intro
thf(fact_271_RuleSystem_Ointro,axiom,
! [S5: set_o,Rules: stream_rule,Eff: rule > $o > fset_o > $o] :
( ! [S6: $o] :
( ( member_o @ S6 @ S5 )
=> ! [R4: rule] :
( ( member_rule @ R4 @ ( sset_rule @ Rules ) )
=> ! [Sl3: fset_o] :
( ( Eff @ R4 @ S6 @ Sl3 )
=> ! [S7: $o] :
( ( fmember_o @ S7 @ Sl3 )
=> ( member_o @ S7 @ S5 ) ) ) ) )
=> ( ! [S6: $o] :
( ( member_o @ S6 @ S5 )
=> ? [X4: rule] :
( ( member_rule @ X4 @ ( sset_rule @ Rules ) )
& ? [X_12: fset_o] : ( Eff @ X4 @ S6 @ X_12 ) ) )
=> ( abstra8374989651900038312rule_o @ Eff @ Rules @ S5 ) ) ) ).
% RuleSystem.intro
thf(fact_272_RuleSystem_Ointro,axiom,
! [S5: set_Pr7058068377845519745ist_fm,Rules: stream_rule,Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
( ! [S6: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ S6 @ S5 )
=> ! [R4: rule] :
( ( member_rule @ R4 @ ( sset_rule @ Rules ) )
=> ! [Sl3: fset_P661503646757059847ist_fm] :
( ( Eff @ R4 @ S6 @ Sl3 )
=> ! [S7: produc1996495991257130529ist_fm] :
( ( fmembe3381613331217039976ist_fm @ S7 @ Sl3 )
=> ( member8102475879199740618ist_fm @ S7 @ S5 ) ) ) ) )
=> ( ! [S6: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ S6 @ S5 )
=> ? [X4: rule] :
( ( member_rule @ X4 @ ( sset_rule @ Rules ) )
& ? [X_12: fset_P661503646757059847ist_fm] : ( Eff @ X4 @ S6 @ X_12 ) ) )
=> ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S5 ) ) ) ).
% RuleSystem.intro
thf(fact_273_RuleSystem__def,axiom,
( abstra9106359784214577277m_rule
= ( ^ [Eff2: produc1996495991257130529ist_fm > rule > fset_rule > $o,Rules2: stream8299795917829157543ist_fm,S8: set_rule] :
( ! [S: rule] :
( ( member_rule @ S @ S8 )
=> ! [R3: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ R3 @ ( sset_P3431374123360452149ist_fm @ Rules2 ) )
=> ! [Sl: fset_rule] :
( ( Eff2 @ R3 @ S @ Sl )
=> ! [S3: rule] :
( ( fmember_rule @ S3 @ Sl )
=> ( member_rule @ S3 @ S8 ) ) ) ) )
& ! [S: rule] :
( ( member_rule @ S @ S8 )
=> ? [X: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ X @ ( sset_P3431374123360452149ist_fm @ Rules2 ) )
& ? [X5: fset_rule] : ( Eff2 @ X @ S @ X5 ) ) ) ) ) ) ).
% RuleSystem_def
thf(fact_274_RuleSystem__def,axiom,
( abstra5464778857772719348o_rule
= ( ^ [Eff2: $o > rule > fset_rule > $o,Rules2: stream_o,S8: set_rule] :
( ! [S: rule] :
( ( member_rule @ S @ S8 )
=> ! [R3: $o] :
( ( member_o @ R3 @ ( sset_o @ Rules2 ) )
=> ! [Sl: fset_rule] :
( ( Eff2 @ R3 @ S @ Sl )
=> ! [S3: rule] :
( ( fmember_rule @ S3 @ Sl )
=> ( member_rule @ S3 @ S8 ) ) ) ) )
& ! [S: rule] :
( ( member_rule @ S @ S8 )
=> ? [X: $o] :
( ( member_o @ X @ ( sset_o @ Rules2 ) )
& ? [X5: fset_rule] : ( Eff2 @ X @ S @ X5 ) ) ) ) ) ) ).
% RuleSystem_def
thf(fact_275_RuleSystem__def,axiom,
( abstra5561694101053127776t_fm_o
= ( ^ [Eff2: produc1996495991257130529ist_fm > $o > fset_o > $o,Rules2: stream8299795917829157543ist_fm,S8: set_o] :
( ! [S: $o] :
( ( member_o @ S @ S8 )
=> ! [R3: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ R3 @ ( sset_P3431374123360452149ist_fm @ Rules2 ) )
=> ! [Sl: fset_o] :
( ( Eff2 @ R3 @ S @ Sl )
=> ! [S3: $o] :
( ( fmember_o @ S3 @ Sl )
=> ( member_o @ S3 @ S8 ) ) ) ) )
& ! [S: $o] :
( ( member_o @ S @ S8 )
=> ? [X: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ X @ ( sset_P3431374123360452149ist_fm @ Rules2 ) )
& ? [X5: fset_o] : ( Eff2 @ X @ S @ X5 ) ) ) ) ) ) ).
% RuleSystem_def
thf(fact_276_RuleSystem__def,axiom,
( abstra8727705743885569321em_o_o
= ( ^ [Eff2: $o > $o > fset_o > $o,Rules2: stream_o,S8: set_o] :
( ! [S: $o] :
( ( member_o @ S @ S8 )
=> ! [R3: $o] :
( ( member_o @ R3 @ ( sset_o @ Rules2 ) )
=> ! [Sl: fset_o] :
( ( Eff2 @ R3 @ S @ Sl )
=> ! [S3: $o] :
( ( fmember_o @ S3 @ Sl )
=> ( member_o @ S3 @ S8 ) ) ) ) )
& ! [S: $o] :
( ( member_o @ S @ S8 )
=> ? [X: $o] :
( ( member_o @ X @ ( sset_o @ Rules2 ) )
& ? [X5: fset_o] : ( Eff2 @ X @ S @ X5 ) ) ) ) ) ) ).
% RuleSystem_def
thf(fact_277_RuleSystem__def,axiom,
( abstra4969282175937908457ist_fm
= ( ^ [Eff2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream8299795917829157543ist_fm,S8: set_Pr7058068377845519745ist_fm] :
( ! [S: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ S @ S8 )
=> ! [R3: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ R3 @ ( sset_P3431374123360452149ist_fm @ Rules2 ) )
=> ! [Sl: fset_P661503646757059847ist_fm] :
( ( Eff2 @ R3 @ S @ Sl )
=> ! [S3: produc1996495991257130529ist_fm] :
( ( fmembe3381613331217039976ist_fm @ S3 @ Sl )
=> ( member8102475879199740618ist_fm @ S3 @ S8 ) ) ) ) )
& ! [S: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ S @ S8 )
=> ? [X: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ X @ ( sset_P3431374123360452149ist_fm @ Rules2 ) )
& ? [X5: fset_P661503646757059847ist_fm] : ( Eff2 @ X @ S @ X5 ) ) ) ) ) ) ).
% RuleSystem_def
thf(fact_278_RuleSystem__def,axiom,
( abstra5822460560083087538ist_fm
= ( ^ [Eff2: $o > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_o,S8: set_Pr7058068377845519745ist_fm] :
( ! [S: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ S @ S8 )
=> ! [R3: $o] :
( ( member_o @ R3 @ ( sset_o @ Rules2 ) )
=> ! [Sl: fset_P661503646757059847ist_fm] :
( ( Eff2 @ R3 @ S @ Sl )
=> ! [S3: produc1996495991257130529ist_fm] :
( ( fmembe3381613331217039976ist_fm @ S3 @ Sl )
=> ( member8102475879199740618ist_fm @ S3 @ S8 ) ) ) ) )
& ! [S: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ S @ S8 )
=> ? [X: $o] :
( ( member_o @ X @ ( sset_o @ Rules2 ) )
& ? [X5: fset_P661503646757059847ist_fm] : ( Eff2 @ X @ S @ X5 ) ) ) ) ) ) ).
% RuleSystem_def
thf(fact_279_RuleSystem__def,axiom,
( abstra2725590884229999157e_rule
= ( ^ [Eff2: rule > rule > fset_rule > $o,Rules2: stream_rule,S8: set_rule] :
( ! [S: rule] :
( ( member_rule @ S @ S8 )
=> ! [R3: rule] :
( ( member_rule @ R3 @ ( sset_rule @ Rules2 ) )
=> ! [Sl: fset_rule] :
( ( Eff2 @ R3 @ S @ Sl )
=> ! [S3: rule] :
( ( fmember_rule @ S3 @ Sl )
=> ( member_rule @ S3 @ S8 ) ) ) ) )
& ! [S: rule] :
( ( member_rule @ S @ S8 )
=> ? [X: rule] :
( ( member_rule @ X @ ( sset_rule @ Rules2 ) )
& ? [X5: fset_rule] : ( Eff2 @ X @ S @ X5 ) ) ) ) ) ) ).
% RuleSystem_def
thf(fact_280_RuleSystem__def,axiom,
( abstra8374989651900038312rule_o
= ( ^ [Eff2: rule > $o > fset_o > $o,Rules2: stream_rule,S8: set_o] :
( ! [S: $o] :
( ( member_o @ S @ S8 )
=> ! [R3: rule] :
( ( member_rule @ R3 @ ( sset_rule @ Rules2 ) )
=> ! [Sl: fset_o] :
( ( Eff2 @ R3 @ S @ Sl )
=> ! [S3: $o] :
( ( fmember_o @ S3 @ Sl )
=> ( member_o @ S3 @ S8 ) ) ) ) )
& ! [S: $o] :
( ( member_o @ S @ S8 )
=> ? [X: rule] :
( ( member_rule @ X @ ( sset_rule @ Rules2 ) )
& ? [X5: fset_o] : ( Eff2 @ X @ S @ X5 ) ) ) ) ) ) ).
% RuleSystem_def
thf(fact_281_RuleSystem__def,axiom,
( abstra6455275554694158129ist_fm
= ( ^ [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,S8: set_Pr7058068377845519745ist_fm] :
( ! [S: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ S @ S8 )
=> ! [R3: rule] :
( ( member_rule @ R3 @ ( sset_rule @ Rules2 ) )
=> ! [Sl: fset_P661503646757059847ist_fm] :
( ( Eff2 @ R3 @ S @ Sl )
=> ! [S3: produc1996495991257130529ist_fm] :
( ( fmembe3381613331217039976ist_fm @ S3 @ Sl )
=> ( member8102475879199740618ist_fm @ S3 @ S8 ) ) ) ) )
& ! [S: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ S @ S8 )
=> ? [X: rule] :
( ( member_rule @ X @ ( sset_rule @ Rules2 ) )
& ? [X5: fset_P661503646757059847ist_fm] : ( Eff2 @ X @ S @ X5 ) ) ) ) ) ) ).
% RuleSystem_def
thf(fact_282_top__empty__eq,axiom,
( top_to6798613745154499404t_fm_o
= ( ^ [X: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X @ top_to728902761071950545ist_fm ) ) ) ).
% top_empty_eq
thf(fact_283_top__empty__eq,axiom,
( top_top_rule_o
= ( ^ [X: rule] : ( member_rule @ X @ top_top_set_rule ) ) ) ).
% top_empty_eq
thf(fact_284_top__empty__eq,axiom,
( top_top_o_o
= ( ^ [X: $o] : ( member_o @ X @ top_top_set_o ) ) ) ).
% top_empty_eq
thf(fact_285_bot__empty__eq,axiom,
( bot_bot_rule_o
= ( ^ [X: rule] : ( member_rule @ X @ bot_bot_set_rule ) ) ) ).
% bot_empty_eq
thf(fact_286_bot__empty__eq,axiom,
( bot_bot_o_o
= ( ^ [X: $o] : ( member_o @ X @ bot_bot_set_o ) ) ) ).
% bot_empty_eq
thf(fact_287_bot__empty__eq,axiom,
( bot_bo1394704948141485872t_fm_o
= ( ^ [X: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X @ bot_bo4530011010606209517ist_fm ) ) ) ).
% bot_empty_eq
thf(fact_288_eqfelem__imp__iff,axiom,
! [X2: produc1996495991257130529ist_fm,Y3: produc1996495991257130529ist_fm,A: fset_P661503646757059847ist_fm] :
( ( X2 = Y3 )
=> ( ( fmembe3381613331217039976ist_fm @ X2 @ A )
= ( fmembe3381613331217039976ist_fm @ Y3 @ A ) ) ) ).
% eqfelem_imp_iff
thf(fact_289_if__split__fmem2,axiom,
! [A3: produc1996495991257130529ist_fm,Q: $o,X2: fset_P661503646757059847ist_fm,Y3: fset_P661503646757059847ist_fm] :
( ( fmembe3381613331217039976ist_fm @ A3 @ ( if_fse3714977293194272717ist_fm @ Q @ X2 @ Y3 ) )
= ( ( Q
=> ( fmembe3381613331217039976ist_fm @ A3 @ X2 ) )
& ( ~ Q
=> ( fmembe3381613331217039976ist_fm @ A3 @ Y3 ) ) ) ) ).
% if_split_fmem2
thf(fact_290_if__split__fmem1,axiom,
! [Q: $o,X2: produc1996495991257130529ist_fm,Y3: produc1996495991257130529ist_fm,B3: fset_P661503646757059847ist_fm] :
( ( fmembe3381613331217039976ist_fm @ ( if_Pro3930376587665744871ist_fm @ Q @ X2 @ Y3 ) @ B3 )
= ( ( Q
=> ( fmembe3381613331217039976ist_fm @ X2 @ B3 ) )
& ( ~ Q
=> ( fmembe3381613331217039976ist_fm @ Y3 @ B3 ) ) ) ) ).
% if_split_fmem1
thf(fact_291_eqfset__imp__iff,axiom,
! [A: fset_P661503646757059847ist_fm,B: fset_P661503646757059847ist_fm,X2: produc1996495991257130529ist_fm] :
( ( A = B )
=> ( ( fmembe3381613331217039976ist_fm @ X2 @ A )
= ( fmembe3381613331217039976ist_fm @ X2 @ B ) ) ) ).
% eqfset_imp_iff
thf(fact_292_eq__fmem__trans,axiom,
! [A3: produc1996495991257130529ist_fm,B3: produc1996495991257130529ist_fm,A: fset_P661503646757059847ist_fm] :
( ( A3 = B3 )
=> ( ( fmembe3381613331217039976ist_fm @ B3 @ A )
=> ( fmembe3381613331217039976ist_fm @ A3 @ A ) ) ) ).
% eq_fmem_trans
thf(fact_293_fequalityCE,axiom,
! [A: fset_P661503646757059847ist_fm,B: fset_P661503646757059847ist_fm,C: produc1996495991257130529ist_fm] :
( ( A = B )
=> ( ( ( fmembe3381613331217039976ist_fm @ C @ A )
=> ~ ( fmembe3381613331217039976ist_fm @ C @ B ) )
=> ~ ( ~ ( fmembe3381613331217039976ist_fm @ C @ A )
=> ( fmembe3381613331217039976ist_fm @ C @ B ) ) ) ) ).
% fequalityCE
thf(fact_294_fset__eqI,axiom,
! [A: fset_P661503646757059847ist_fm,B: fset_P661503646757059847ist_fm] :
( ! [X3: produc1996495991257130529ist_fm] :
( ( fmembe3381613331217039976ist_fm @ X3 @ A )
= ( fmembe3381613331217039976ist_fm @ X3 @ B ) )
=> ( A = B ) ) ).
% fset_eqI
thf(fact_295_equalsffemptyI,axiom,
! [A: fset_P661503646757059847ist_fm] :
( ! [Y2: produc1996495991257130529ist_fm] :
~ ( fmembe3381613331217039976ist_fm @ Y2 @ A )
=> ( A = bot_bo2367426573206113139ist_fm ) ) ).
% equalsffemptyI
thf(fact_296_equalsffemptyD,axiom,
! [A: fset_P661503646757059847ist_fm,A3: produc1996495991257130529ist_fm] :
( ( A = bot_bo2367426573206113139ist_fm )
=> ~ ( fmembe3381613331217039976ist_fm @ A3 @ A ) ) ).
% equalsffemptyD
thf(fact_297_ex__fin__conv,axiom,
! [A: fset_P661503646757059847ist_fm] :
( ( ? [X: produc1996495991257130529ist_fm] : ( fmembe3381613331217039976ist_fm @ X @ A ) )
= ( A != bot_bo2367426573206113139ist_fm ) ) ).
% ex_fin_conv
thf(fact_298_femptyE,axiom,
! [A3: produc1996495991257130529ist_fm] :
~ ( fmembe3381613331217039976ist_fm @ A3 @ bot_bo2367426573206113139ist_fm ) ).
% femptyE
thf(fact_299_fminusD2,axiom,
! [C: produc1996495991257130529ist_fm,A: fset_P661503646757059847ist_fm,B: fset_P661503646757059847ist_fm] :
( ( fmembe3381613331217039976ist_fm @ C @ ( minus_8437252545254675822ist_fm @ A @ B ) )
=> ~ ( fmembe3381613331217039976ist_fm @ C @ B ) ) ).
% fminusD2
thf(fact_300_fminusD1,axiom,
! [C: produc1996495991257130529ist_fm,A: fset_P661503646757059847ist_fm,B: fset_P661503646757059847ist_fm] :
( ( fmembe3381613331217039976ist_fm @ C @ ( minus_8437252545254675822ist_fm @ A @ B ) )
=> ( fmembe3381613331217039976ist_fm @ C @ A ) ) ).
% fminusD1
thf(fact_301_fminusE,axiom,
! [C: produc1996495991257130529ist_fm,A: fset_P661503646757059847ist_fm,B: fset_P661503646757059847ist_fm] :
( ( fmembe3381613331217039976ist_fm @ C @ ( minus_8437252545254675822ist_fm @ A @ B ) )
=> ~ ( ( fmembe3381613331217039976ist_fm @ C @ A )
=> ( fmembe3381613331217039976ist_fm @ C @ B ) ) ) ).
% fminusE
thf(fact_302_RuleSystem__Defs_Oenabled_Ocong,axiom,
abstra3055540779341473792ist_fm = abstra3055540779341473792ist_fm ).
% RuleSystem_Defs.enabled.cong
thf(fact_303_RuleSystem__Defs_Oenabled__def,axiom,
( abstra3055540779341473792ist_fm
= ( ^ [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,R3: rule,S: produc1996495991257130529ist_fm] :
? [X5: fset_P661503646757059847ist_fm] : ( Eff2 @ R3 @ S @ X5 ) ) ) ).
% RuleSystem_Defs.enabled_def
thf(fact_304_RuleSystem__Defs_OpickEff_Ocong,axiom,
abstra458490714785879965ist_fm = abstra458490714785879965ist_fm ).
% RuleSystem_Defs.pickEff.cong
thf(fact_305_pred__equals__eq2,axiom,
! [R5: set_Pr7058068377845519745ist_fm,S5: set_Pr7058068377845519745ist_fm] :
( ( ( ^ [X: list_fm,Y4: list_fm] : ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X @ Y4 ) @ R5 ) )
= ( ^ [X: list_fm,Y4: list_fm] : ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X @ Y4 ) @ S5 ) ) )
= ( R5 = S5 ) ) ).
% pred_equals_eq2
thf(fact_306_bot__empty__eq2,axiom,
( bot_bo2780323056127860274t_fm_o
= ( ^ [X: list_fm,Y4: list_fm] : ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X @ Y4 ) @ bot_bo4530011010606209517ist_fm ) ) ) ).
% bot_empty_eq2
thf(fact_307_RuleSystem_Oper_Ocong,axiom,
abstra4153889242452539894ist_fm = abstra4153889242452539894ist_fm ).
% RuleSystem.per.cong
thf(fact_308_RuleSystem__axioms,axiom,
( abstra6455275554694158129ist_fm
@ ^ [R3: rule,S: produc1996495991257130529ist_fm,Ss2: fset_P661503646757059847ist_fm] :
( ( eff @ R3 @ S )
= ( some_f2162114139285431020ist_fm @ Ss2 ) )
@ rules
@ top_to728902761071950545ist_fm ) ).
% RuleSystem_axioms
thf(fact_309_Collect__empty__eq__bot,axiom,
! [P: rule > $o] :
( ( ( collect_rule @ P )
= bot_bot_set_rule )
= ( P = bot_bot_rule_o ) ) ).
% Collect_empty_eq_bot
thf(fact_310_Collect__empty__eq__bot,axiom,
! [P: $o > $o] :
( ( ( collect_o @ P )
= bot_bot_set_o )
= ( P = bot_bot_o_o ) ) ).
% Collect_empty_eq_bot
thf(fact_311_Collect__empty__eq__bot,axiom,
! [P: produc1996495991257130529ist_fm > $o] :
( ( ( collec7231575776666803724ist_fm @ P )
= bot_bo4530011010606209517ist_fm )
= ( P = bot_bo1394704948141485872t_fm_o ) ) ).
% Collect_empty_eq_bot
thf(fact_312_is__singleton__the__elem,axiom,
( is_singleton_rule
= ( ^ [A5: set_rule] :
( A5
= ( insert_rule @ ( the_elem_rule @ A5 ) @ bot_bot_set_rule ) ) ) ) ).
% is_singleton_the_elem
thf(fact_313_is__singleton__the__elem,axiom,
( is_singleton_o
= ( ^ [A5: set_o] :
( A5
= ( insert_o @ ( the_elem_o @ A5 ) @ bot_bot_set_o ) ) ) ) ).
% is_singleton_the_elem
thf(fact_314_is__singleton__the__elem,axiom,
( is_sin3852095833271400213ist_fm
= ( ^ [A5: set_Pr7058068377845519745ist_fm] :
( A5
= ( insert589278167076416497ist_fm @ ( the_el1929169733952065430ist_fm @ A5 ) @ bot_bo4530011010606209517ist_fm ) ) ) ) ).
% is_singleton_the_elem
thf(fact_315_PersistentRuleSystem__axioms_Ointro,axiom,
! [Rules: stream_rule,Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,S5: set_Pr7058068377845519745ist_fm] :
( ! [R4: rule] :
( ( member_rule @ R4 @ ( sset_rule @ Rules ) )
=> ( abstra4153889242452539894ist_fm @ Eff @ Rules @ S5 @ R4 ) )
=> ( abstra2778581409224112183ist_fm @ Eff @ Rules @ S5 ) ) ).
% PersistentRuleSystem_axioms.intro
thf(fact_316_PersistentRuleSystem__axioms__def,axiom,
( abstra2778581409224112183ist_fm
= ( ^ [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,S8: set_Pr7058068377845519745ist_fm] :
! [R3: rule] :
( ( member_rule @ R3 @ ( sset_rule @ Rules2 ) )
=> ( abstra4153889242452539894ist_fm @ Eff2 @ Rules2 @ S8 @ R3 ) ) ) ) ).
% PersistentRuleSystem_axioms_def
thf(fact_317_these__insert__Some,axiom,
! [X2: rule,A: set_option_rule] :
( ( these_rule @ ( insert_option_rule @ ( some_rule @ X2 ) @ A ) )
= ( insert_rule @ X2 @ ( these_rule @ A ) ) ) ).
% these_insert_Some
thf(fact_318_these__insert__Some,axiom,
! [X2: $o,A: set_option_o] :
( ( these_o @ ( insert_option_o @ ( some_o @ X2 ) @ A ) )
= ( insert_o @ X2 @ ( these_o @ A ) ) ) ).
% these_insert_Some
thf(fact_319_these__insert__Some,axiom,
! [X2: produc1996495991257130529ist_fm,A: set_op4811217697719039047ist_fm] :
( ( these_2172363061839534302ist_fm @ ( insert7172606565080779063ist_fm @ ( some_P7346312016712434310ist_fm @ X2 ) @ A ) )
= ( insert589278167076416497ist_fm @ X2 @ ( these_2172363061839534302ist_fm @ A ) ) ) ).
% these_insert_Some
thf(fact_320_these__insert__Some,axiom,
! [X2: fset_P661503646757059847ist_fm,A: set_op6310664071589870893ist_fm] :
( ( these_4231445979625565508ist_fm @ ( insert31278938154982301ist_fm @ ( some_f2162114139285431020ist_fm @ X2 ) @ A ) )
= ( insert6096777853467859671ist_fm @ X2 @ ( these_4231445979625565508ist_fm @ A ) ) ) ).
% these_insert_Some
thf(fact_321_the__elem__eq,axiom,
! [X2: rule] :
( ( the_elem_rule @ ( insert_rule @ X2 @ bot_bot_set_rule ) )
= X2 ) ).
% the_elem_eq
thf(fact_322_the__elem__eq,axiom,
! [X2: $o] :
( ( the_elem_o @ ( insert_o @ X2 @ bot_bot_set_o ) )
= X2 ) ).
% the_elem_eq
thf(fact_323_the__elem__eq,axiom,
! [X2: produc1996495991257130529ist_fm] :
( ( the_el1929169733952065430ist_fm @ ( insert589278167076416497ist_fm @ X2 @ bot_bo4530011010606209517ist_fm ) )
= X2 ) ).
% the_elem_eq
thf(fact_324_PersistentRuleSystem_Oper,axiom,
! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S5: set_Pr7058068377845519745ist_fm,R: rule] :
( ( abstra7887161495092839066ist_fm @ Eff @ Rules @ S5 )
=> ( ( member_rule @ R @ ( sset_rule @ Rules ) )
=> ( abstra4153889242452539894ist_fm @ Eff @ Rules @ S5 @ R ) ) ) ).
% PersistentRuleSystem.per
thf(fact_325_these__empty,axiom,
( ( these_rule @ bot_bo8966923457718485855n_rule )
= bot_bot_set_rule ) ).
% these_empty
thf(fact_326_these__empty,axiom,
( ( these_o @ bot_bot_set_option_o )
= bot_bot_set_o ) ).
% these_empty
thf(fact_327_these__empty,axiom,
( ( these_2172363061839534302ist_fm @ bot_bo4262429147648288435ist_fm )
= bot_bo4530011010606209517ist_fm ) ).
% these_empty
thf(fact_328_PersistentRuleSystem__def,axiom,
( abstra7887161495092839066ist_fm
= ( ^ [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,S8: set_Pr7058068377845519745ist_fm] :
( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules2 @ S8 )
& ( abstra2778581409224112183ist_fm @ Eff2 @ Rules2 @ S8 ) ) ) ) ).
% PersistentRuleSystem_def
thf(fact_329_PersistentRuleSystem_Ointro,axiom,
! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S5: set_Pr7058068377845519745ist_fm] :
( ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S5 )
=> ( ( abstra2778581409224112183ist_fm @ Eff @ Rules @ S5 )
=> ( abstra7887161495092839066ist_fm @ Eff @ Rules @ S5 ) ) ) ).
% PersistentRuleSystem.intro
thf(fact_330_in__these__eq,axiom,
! [X2: produc1996495991257130529ist_fm,A: set_op4811217697719039047ist_fm] :
( ( member8102475879199740618ist_fm @ X2 @ ( these_2172363061839534302ist_fm @ A ) )
= ( member772309659248424592ist_fm @ ( some_P7346312016712434310ist_fm @ X2 ) @ A ) ) ).
% in_these_eq
thf(fact_331_in__these__eq,axiom,
! [X2: rule,A: set_option_rule] :
( ( member_rule @ X2 @ ( these_rule @ A ) )
= ( member_option_rule @ ( some_rule @ X2 ) @ A ) ) ).
% in_these_eq
thf(fact_332_in__these__eq,axiom,
! [X2: $o,A: set_option_o] :
( ( member_o @ X2 @ ( these_o @ A ) )
= ( member_option_o @ ( some_o @ X2 ) @ A ) ) ).
% in_these_eq
thf(fact_333_in__these__eq,axiom,
! [X2: fset_P661503646757059847ist_fm,A: set_op6310664071589870893ist_fm] :
( ( member3493611713653719344ist_fm @ X2 @ ( these_4231445979625565508ist_fm @ A ) )
= ( member1838329594270681206ist_fm @ ( some_f2162114139285431020ist_fm @ X2 ) @ A ) ) ).
% in_these_eq
thf(fact_334_PersistentRuleSystem_Oaxioms_I1_J,axiom,
! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S5: set_Pr7058068377845519745ist_fm] :
( ( abstra7887161495092839066ist_fm @ Eff @ Rules @ S5 )
=> ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S5 ) ) ).
% PersistentRuleSystem.axioms(1)
thf(fact_335_prod__set__simps_I2_J,axiom,
! [X2: list_fm,Y3: list_fm] :
( ( basic_6053177629474610823ist_fm @ ( produc381145313068854617ist_fm @ X2 @ Y3 ) )
= ( insert_list_fm @ Y3 @ bot_bot_set_list_fm ) ) ).
% prod_set_simps(2)
thf(fact_336_prod__set__simps_I1_J,axiom,
! [X2: list_fm,Y3: list_fm] :
( ( basic_2778179344159109705ist_fm @ ( produc381145313068854617ist_fm @ X2 @ Y3 ) )
= ( insert_list_fm @ X2 @ bot_bot_set_list_fm ) ) ).
% prod_set_simps(1)
thf(fact_337_Set_Ois__empty__def,axiom,
( is_empty_rule
= ( ^ [A5: set_rule] : ( A5 = bot_bot_set_rule ) ) ) ).
% Set.is_empty_def
thf(fact_338_Set_Ois__empty__def,axiom,
( is_empty_o
= ( ^ [A5: set_o] : ( A5 = bot_bot_set_o ) ) ) ).
% Set.is_empty_def
thf(fact_339_Set_Ois__empty__def,axiom,
( is_emp4743207889117482483ist_fm
= ( ^ [A5: set_Pr7058068377845519745ist_fm] : ( A5 = bot_bo4530011010606209517ist_fm ) ) ) ).
% Set.is_empty_def
thf(fact_340_refl__on__singleton,axiom,
! [X2: list_fm] : ( refl_on_list_fm @ ( insert_list_fm @ X2 @ bot_bot_set_list_fm ) @ ( insert589278167076416497ist_fm @ ( produc381145313068854617ist_fm @ X2 @ X2 ) @ bot_bo4530011010606209517ist_fm ) ) ).
% refl_on_singleton
thf(fact_341_refl__on__singleton,axiom,
! [X2: rule] : ( refl_on_rule @ ( insert_rule @ X2 @ bot_bot_set_rule ) @ ( insert3755751280551455547e_rule @ ( produc1769597910952761123e_rule @ X2 @ X2 ) @ bot_bo9047865283180780599e_rule ) ) ).
% refl_on_singleton
thf(fact_342_refl__on__singleton,axiom,
! [X2: $o] : ( refl_on_o @ ( insert_o @ X2 @ bot_bot_set_o ) @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ X2 @ X2 ) @ bot_bo7073875226086086771od_o_o ) ) ).
% refl_on_singleton
thf(fact_343_refl__on__singleton,axiom,
! [X2: produc1996495991257130529ist_fm] : ( refl_o4727633711164096160ist_fm @ ( insert589278167076416497ist_fm @ X2 @ bot_bo4530011010606209517ist_fm ) @ ( insert5410920824581998135ist_fm @ ( produc2592314732767339223ist_fm @ X2 @ X2 ) @ bot_bo7330262581962992051ist_fm ) ) ).
% refl_on_singleton
thf(fact_344_Saturated__def,axiom,
! [Steps: stream727092118206550309m_rule] :
( ( abstra8647779336736900908ist_fm
@ ^ [R3: rule,S: produc1996495991257130529ist_fm,Ss2: fset_P661503646757059847ist_fm] :
( ( eff @ R3 @ S )
= ( some_f2162114139285431020ist_fm @ Ss2 ) )
@ rules
@ Steps )
= ( ! [X: rule] :
( ( member_rule @ X @ ( sset_rule @ rules ) )
=> ( abstra7252176601000738572ist_fm
@ ^ [R3: rule,S: produc1996495991257130529ist_fm,Ss2: fset_P661503646757059847ist_fm] :
( ( eff @ R3 @ S )
= ( some_f2162114139285431020ist_fm @ Ss2 ) )
@ X
@ Steps ) ) ) ) ).
% Saturated_def
thf(fact_345_pairwise__alt,axiom,
( pairwise_rule
= ( ^ [R6: rule > rule > $o,S8: set_rule] :
! [X: rule] :
( ( member_rule @ X @ S8 )
=> ! [Y4: rule] :
( ( member_rule @ Y4 @ ( minus_minus_set_rule @ S8 @ ( insert_rule @ X @ bot_bot_set_rule ) ) )
=> ( R6 @ X @ Y4 ) ) ) ) ) ).
% pairwise_alt
thf(fact_346_pairwise__alt,axiom,
( pairwise_o
= ( ^ [R6: $o > $o > $o,S8: set_o] :
! [X: $o] :
( ( member_o @ X @ S8 )
=> ! [Y4: $o] :
( ( member_o @ Y4 @ ( minus_minus_set_o @ S8 @ ( insert_o @ X @ bot_bot_set_o ) ) )
=> ( R6 @ X @ Y4 ) ) ) ) ) ).
% pairwise_alt
thf(fact_347_pairwise__alt,axiom,
( pairwi155973253597903628ist_fm
= ( ^ [R6: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > $o,S8: set_Pr7058068377845519745ist_fm] :
! [X: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ X @ S8 )
=> ! [Y4: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ Y4 @ ( minus_5615909706921521512ist_fm @ S8 @ ( insert589278167076416497ist_fm @ X @ bot_bo4530011010606209517ist_fm ) ) )
=> ( R6 @ X @ Y4 ) ) ) ) ) ).
% pairwise_alt
thf(fact_348_sset__fenum,axiom,
( ( sset_rule @ ( abstra7284221463285775110m_rule @ rules ) )
= ( sset_rule @ rules ) ) ).
% sset_fenum
thf(fact_349_the__elem__def,axiom,
( the_elem_rule
= ( ^ [X5: set_rule] :
( the_rule
@ ^ [X: rule] :
( X5
= ( insert_rule @ X @ bot_bot_set_rule ) ) ) ) ) ).
% the_elem_def
thf(fact_350_the__elem__def,axiom,
( the_elem_o
= ( ^ [X5: set_o] :
( the_o
@ ^ [X: $o] :
( X5
= ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ).
% the_elem_def
thf(fact_351_the__elem__def,axiom,
( the_el1929169733952065430ist_fm
= ( ^ [X5: set_Pr7058068377845519745ist_fm] :
( the_Pr8646916617578001576ist_fm
@ ^ [X: produc1996495991257130529ist_fm] :
( X5
= ( insert589278167076416497ist_fm @ X @ bot_bo4530011010606209517ist_fm ) ) ) ) ) ).
% the_elem_def
thf(fact_352_pairwiseI,axiom,
! [S5: set_Pr7058068377845519745ist_fm,R5: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > $o] :
( ! [X3: produc1996495991257130529ist_fm,Y2: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ X3 @ S5 )
=> ( ( member8102475879199740618ist_fm @ Y2 @ S5 )
=> ( ( X3 != Y2 )
=> ( R5 @ X3 @ Y2 ) ) ) )
=> ( pairwi155973253597903628ist_fm @ R5 @ S5 ) ) ).
% pairwiseI
thf(fact_353_pairwiseI,axiom,
! [S5: set_rule,R5: rule > rule > $o] :
( ! [X3: rule,Y2: rule] :
( ( member_rule @ X3 @ S5 )
=> ( ( member_rule @ Y2 @ S5 )
=> ( ( X3 != Y2 )
=> ( R5 @ X3 @ Y2 ) ) ) )
=> ( pairwise_rule @ R5 @ S5 ) ) ).
% pairwiseI
thf(fact_354_pairwiseI,axiom,
! [S5: set_o,R5: $o > $o > $o] :
( ! [X3: $o,Y2: $o] :
( ( member_o @ X3 @ S5 )
=> ( ( member_o @ Y2 @ S5 )
=> ( ( X3 != Y2 )
=> ( R5 @ X3 @ Y2 ) ) ) )
=> ( pairwise_o @ R5 @ S5 ) ) ).
% pairwiseI
thf(fact_355_pairwiseD,axiom,
! [R5: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > $o,S5: set_Pr7058068377845519745ist_fm,X2: produc1996495991257130529ist_fm,Y3: produc1996495991257130529ist_fm] :
( ( pairwi155973253597903628ist_fm @ R5 @ S5 )
=> ( ( member8102475879199740618ist_fm @ X2 @ S5 )
=> ( ( member8102475879199740618ist_fm @ Y3 @ S5 )
=> ( ( X2 != Y3 )
=> ( R5 @ X2 @ Y3 ) ) ) ) ) ).
% pairwiseD
thf(fact_356_pairwiseD,axiom,
! [R5: rule > rule > $o,S5: set_rule,X2: rule,Y3: rule] :
( ( pairwise_rule @ R5 @ S5 )
=> ( ( member_rule @ X2 @ S5 )
=> ( ( member_rule @ Y3 @ S5 )
=> ( ( X2 != Y3 )
=> ( R5 @ X2 @ Y3 ) ) ) ) ) ).
% pairwiseD
thf(fact_357_pairwiseD,axiom,
! [R5: $o > $o > $o,S5: set_o,X2: $o,Y3: $o] :
( ( pairwise_o @ R5 @ S5 )
=> ( ( member_o @ X2 @ S5 )
=> ( ( member_o @ Y3 @ S5 )
=> ( ( X2 != Y3 )
=> ( R5 @ X2 @ Y3 ) ) ) ) ) ).
% pairwiseD
thf(fact_358_RuleSystem__Defs_Osaturated_Ocong,axiom,
abstra7252176601000738572ist_fm = abstra7252176601000738572ist_fm ).
% RuleSystem_Defs.saturated.cong
thf(fact_359_RuleSystem__Defs_OSaturated_Ocong,axiom,
abstra8647779336736900908ist_fm = abstra8647779336736900908ist_fm ).
% RuleSystem_Defs.Saturated.cong
thf(fact_360_RuleSystem__Defs_Ofenum_Ocong,axiom,
abstra7284221463285775110m_rule = abstra7284221463285775110m_rule ).
% RuleSystem_Defs.fenum.cong
thf(fact_361_RuleSystem__Defs_OSaturated__def,axiom,
( abstra8647779336736900908ist_fm
= ( ^ [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,Steps2: stream727092118206550309m_rule] :
! [X: rule] :
( ( member_rule @ X @ ( sset_rule @ Rules2 ) )
=> ( abstra7252176601000738572ist_fm @ Eff2 @ X @ Steps2 ) ) ) ) ).
% RuleSystem_Defs.Saturated_def
thf(fact_362_refl__onD,axiom,
! [A: set_Pr7058068377845519745ist_fm,R: set_Pr5209464977148653383ist_fm,A3: produc1996495991257130529ist_fm] :
( ( refl_o4727633711164096160ist_fm @ A @ R )
=> ( ( member8102475879199740618ist_fm @ A3 @ A )
=> ( member60238275747228048ist_fm @ ( produc2592314732767339223ist_fm @ A3 @ A3 ) @ R ) ) ) ).
% refl_onD
thf(fact_363_refl__onD,axiom,
! [A: set_rule,R: set_Pr6543236023296505803e_rule,A3: rule] :
( ( refl_on_rule @ A @ R )
=> ( ( member_rule @ A3 @ A )
=> ( member3753523862196722196e_rule @ ( produc1769597910952761123e_rule @ A3 @ A3 ) @ R ) ) ) ).
% refl_onD
thf(fact_364_refl__onD,axiom,
! [A: set_o,R: set_Product_prod_o_o,A3: $o] :
( ( refl_on_o @ A @ R )
=> ( ( member_o @ A3 @ A )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A3 @ A3 ) @ R ) ) ) ).
% refl_onD
thf(fact_365_refl__onD,axiom,
! [A: set_list_fm,R: set_Pr7058068377845519745ist_fm,A3: list_fm] :
( ( refl_on_list_fm @ A @ R )
=> ( ( member_list_fm @ A3 @ A )
=> ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ A3 @ A3 ) @ R ) ) ) ).
% refl_onD
thf(fact_366_refl__onD1,axiom,
! [A: set_Pr7058068377845519745ist_fm,R: set_Pr5209464977148653383ist_fm,X2: produc1996495991257130529ist_fm,Y3: produc1996495991257130529ist_fm] :
( ( refl_o4727633711164096160ist_fm @ A @ R )
=> ( ( member60238275747228048ist_fm @ ( produc2592314732767339223ist_fm @ X2 @ Y3 ) @ R )
=> ( member8102475879199740618ist_fm @ X2 @ A ) ) ) ).
% refl_onD1
thf(fact_367_refl__onD1,axiom,
! [A: set_rule,R: set_Pr6543236023296505803e_rule,X2: rule,Y3: rule] :
( ( refl_on_rule @ A @ R )
=> ( ( member3753523862196722196e_rule @ ( produc1769597910952761123e_rule @ X2 @ Y3 ) @ R )
=> ( member_rule @ X2 @ A ) ) ) ).
% refl_onD1
thf(fact_368_refl__onD1,axiom,
! [A: set_o,R: set_Product_prod_o_o,X2: $o,Y3: $o] :
( ( refl_on_o @ A @ R )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y3 ) @ R )
=> ( member_o @ X2 @ A ) ) ) ).
% refl_onD1
thf(fact_369_refl__onD1,axiom,
! [A: set_list_fm,R: set_Pr7058068377845519745ist_fm,X2: list_fm,Y3: list_fm] :
( ( refl_on_list_fm @ A @ R )
=> ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X2 @ Y3 ) @ R )
=> ( member_list_fm @ X2 @ A ) ) ) ).
% refl_onD1
thf(fact_370_refl__onD2,axiom,
! [A: set_Pr7058068377845519745ist_fm,R: set_Pr5209464977148653383ist_fm,X2: produc1996495991257130529ist_fm,Y3: produc1996495991257130529ist_fm] :
( ( refl_o4727633711164096160ist_fm @ A @ R )
=> ( ( member60238275747228048ist_fm @ ( produc2592314732767339223ist_fm @ X2 @ Y3 ) @ R )
=> ( member8102475879199740618ist_fm @ Y3 @ A ) ) ) ).
% refl_onD2
thf(fact_371_refl__onD2,axiom,
! [A: set_rule,R: set_Pr6543236023296505803e_rule,X2: rule,Y3: rule] :
( ( refl_on_rule @ A @ R )
=> ( ( member3753523862196722196e_rule @ ( produc1769597910952761123e_rule @ X2 @ Y3 ) @ R )
=> ( member_rule @ Y3 @ A ) ) ) ).
% refl_onD2
thf(fact_372_refl__onD2,axiom,
! [A: set_o,R: set_Product_prod_o_o,X2: $o,Y3: $o] :
( ( refl_on_o @ A @ R )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y3 ) @ R )
=> ( member_o @ Y3 @ A ) ) ) ).
% refl_onD2
thf(fact_373_refl__onD2,axiom,
! [A: set_list_fm,R: set_Pr7058068377845519745ist_fm,X2: list_fm,Y3: list_fm] :
( ( refl_on_list_fm @ A @ R )
=> ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X2 @ Y3 ) @ R )
=> ( member_list_fm @ Y3 @ A ) ) ) ).
% refl_onD2
thf(fact_374_pairwise__empty,axiom,
! [P: rule > rule > $o] : ( pairwise_rule @ P @ bot_bot_set_rule ) ).
% pairwise_empty
thf(fact_375_pairwise__empty,axiom,
! [P: $o > $o > $o] : ( pairwise_o @ P @ bot_bot_set_o ) ).
% pairwise_empty
thf(fact_376_pairwise__empty,axiom,
! [P: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > $o] : ( pairwi155973253597903628ist_fm @ P @ bot_bo4530011010606209517ist_fm ) ).
% pairwise_empty
thf(fact_377_pairwise__insert,axiom,
! [R: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > $o,X2: produc1996495991257130529ist_fm,S2: set_Pr7058068377845519745ist_fm] :
( ( pairwi155973253597903628ist_fm @ R @ ( insert589278167076416497ist_fm @ X2 @ S2 ) )
= ( ! [Y4: produc1996495991257130529ist_fm] :
( ( ( member8102475879199740618ist_fm @ Y4 @ S2 )
& ( Y4 != X2 ) )
=> ( ( R @ X2 @ Y4 )
& ( R @ Y4 @ X2 ) ) )
& ( pairwi155973253597903628ist_fm @ R @ S2 ) ) ) ).
% pairwise_insert
thf(fact_378_pairwise__insert,axiom,
! [R: rule > rule > $o,X2: rule,S2: set_rule] :
( ( pairwise_rule @ R @ ( insert_rule @ X2 @ S2 ) )
= ( ! [Y4: rule] :
( ( ( member_rule @ Y4 @ S2 )
& ( Y4 != X2 ) )
=> ( ( R @ X2 @ Y4 )
& ( R @ Y4 @ X2 ) ) )
& ( pairwise_rule @ R @ S2 ) ) ) ).
% pairwise_insert
thf(fact_379_pairwise__insert,axiom,
! [R: $o > $o > $o,X2: $o,S2: set_o] :
( ( pairwise_o @ R @ ( insert_o @ X2 @ S2 ) )
= ( ! [Y4: $o] :
( ( ( member_o @ Y4 @ S2 )
& ( Y4 != X2 ) )
=> ( ( R @ X2 @ Y4 )
& ( R @ Y4 @ X2 ) ) )
& ( pairwise_o @ R @ S2 ) ) ) ).
% pairwise_insert
thf(fact_380_reflI,axiom,
! [R: set_Pr7058068377845519745ist_fm] :
( ! [X3: list_fm] : ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X3 @ X3 ) @ R )
=> ( refl_on_list_fm @ top_top_set_list_fm @ R ) ) ).
% reflI
thf(fact_381_reflI,axiom,
! [R: set_Pr5209464977148653383ist_fm] :
( ! [X3: produc1996495991257130529ist_fm] : ( member60238275747228048ist_fm @ ( produc2592314732767339223ist_fm @ X3 @ X3 ) @ R )
=> ( refl_o4727633711164096160ist_fm @ top_to728902761071950545ist_fm @ R ) ) ).
% reflI
thf(fact_382_reflI,axiom,
! [R: set_Pr6543236023296505803e_rule] :
( ! [X3: rule] : ( member3753523862196722196e_rule @ ( produc1769597910952761123e_rule @ X3 @ X3 ) @ R )
=> ( refl_on_rule @ top_top_set_rule @ R ) ) ).
% reflI
thf(fact_383_reflI,axiom,
! [R: set_Product_prod_o_o] :
( ! [X3: $o] : ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ X3 ) @ R )
=> ( refl_on_o @ top_top_set_o @ R ) ) ).
% reflI
thf(fact_384_reflD,axiom,
! [R: set_Pr7058068377845519745ist_fm,A3: list_fm] :
( ( refl_on_list_fm @ top_top_set_list_fm @ R )
=> ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ A3 @ A3 ) @ R ) ) ).
% reflD
thf(fact_385_reflD,axiom,
! [R: set_Pr5209464977148653383ist_fm,A3: produc1996495991257130529ist_fm] :
( ( refl_o4727633711164096160ist_fm @ top_to728902761071950545ist_fm @ R )
=> ( member60238275747228048ist_fm @ ( produc2592314732767339223ist_fm @ A3 @ A3 ) @ R ) ) ).
% reflD
thf(fact_386_reflD,axiom,
! [R: set_Pr6543236023296505803e_rule,A3: rule] :
( ( refl_on_rule @ top_top_set_rule @ R )
=> ( member3753523862196722196e_rule @ ( produc1769597910952761123e_rule @ A3 @ A3 ) @ R ) ) ).
% reflD
thf(fact_387_reflD,axiom,
! [R: set_Product_prod_o_o,A3: $o] :
( ( refl_on_o @ top_top_set_o @ R )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A3 @ A3 ) @ R ) ) ).
% reflD
thf(fact_388_refl__on__empty,axiom,
refl_on_list_fm @ bot_bot_set_list_fm @ bot_bo4530011010606209517ist_fm ).
% refl_on_empty
thf(fact_389_refl__on__empty,axiom,
refl_on_rule @ bot_bot_set_rule @ bot_bo9047865283180780599e_rule ).
% refl_on_empty
thf(fact_390_refl__on__empty,axiom,
refl_on_o @ bot_bot_set_o @ bot_bo7073875226086086771od_o_o ).
% refl_on_empty
thf(fact_391_refl__on__empty,axiom,
refl_o4727633711164096160ist_fm @ bot_bo4530011010606209517ist_fm @ bot_bo7330262581962992051ist_fm ).
% refl_on_empty
thf(fact_392_i_Osset__fenum,axiom,
! [Rules: stream_rule] :
( ( sset_rule @ ( abstra7284221463285775110m_rule @ Rules ) )
= ( sset_rule @ Rules ) ) ).
% i.sset_fenum
thf(fact_393_RuleSystem__Defs_Osset__fenum,axiom,
! [Rules: stream_rule] :
( ( sset_rule @ ( abstra7284221463285775110m_rule @ Rules ) )
= ( sset_rule @ Rules ) ) ).
% RuleSystem_Defs.sset_fenum
thf(fact_394_pairwise__singleton,axiom,
! [P: rule > rule > $o,A: rule] : ( pairwise_rule @ P @ ( insert_rule @ A @ bot_bot_set_rule ) ) ).
% pairwise_singleton
thf(fact_395_pairwise__singleton,axiom,
! [P: $o > $o > $o,A: $o] : ( pairwise_o @ P @ ( insert_o @ A @ bot_bot_set_o ) ) ).
% pairwise_singleton
thf(fact_396_pairwise__singleton,axiom,
! [P: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > $o,A: produc1996495991257130529ist_fm] : ( pairwi155973253597903628ist_fm @ P @ ( insert589278167076416497ist_fm @ A @ bot_bo4530011010606209517ist_fm ) ) ).
% pairwise_singleton
thf(fact_397_fenum__uu__def,axiom,
abstra8121460317444261103u_rule = abstra7284221463285775110m_rule ).
% fenum_uu_def
thf(fact_398_refl__on__domain,axiom,
! [A: set_Pr7058068377845519745ist_fm,R: set_Pr5209464977148653383ist_fm,A3: produc1996495991257130529ist_fm,B3: produc1996495991257130529ist_fm] :
( ( refl_o4727633711164096160ist_fm @ A @ R )
=> ( ( member60238275747228048ist_fm @ ( produc2592314732767339223ist_fm @ A3 @ B3 ) @ R )
=> ( ( member8102475879199740618ist_fm @ A3 @ A )
& ( member8102475879199740618ist_fm @ B3 @ A ) ) ) ) ).
% refl_on_domain
thf(fact_399_refl__on__domain,axiom,
! [A: set_rule,R: set_Pr6543236023296505803e_rule,A3: rule,B3: rule] :
( ( refl_on_rule @ A @ R )
=> ( ( member3753523862196722196e_rule @ ( produc1769597910952761123e_rule @ A3 @ B3 ) @ R )
=> ( ( member_rule @ A3 @ A )
& ( member_rule @ B3 @ A ) ) ) ) ).
% refl_on_domain
thf(fact_400_refl__on__domain,axiom,
! [A: set_o,R: set_Product_prod_o_o,A3: $o,B3: $o] :
( ( refl_on_o @ A @ R )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A3 @ B3 ) @ R )
=> ( ( member_o @ A3 @ A )
& ( member_o @ B3 @ A ) ) ) ) ).
% refl_on_domain
thf(fact_401_refl__on__domain,axiom,
! [A: set_list_fm,R: set_Pr7058068377845519745ist_fm,A3: list_fm,B3: list_fm] :
( ( refl_on_list_fm @ A @ R )
=> ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ A3 @ B3 ) @ R )
=> ( ( member_list_fm @ A3 @ A )
& ( member_list_fm @ B3 @ A ) ) ) ) ).
% refl_on_domain
thf(fact_402_these__not__empty__eq,axiom,
! [B: set_option_rule] :
( ( ( these_rule @ B )
!= bot_bot_set_rule )
= ( ( B != bot_bo8966923457718485855n_rule )
& ( B
!= ( insert_option_rule @ none_rule @ bot_bo8966923457718485855n_rule ) ) ) ) ).
% these_not_empty_eq
thf(fact_403_these__not__empty__eq,axiom,
! [B: set_option_o] :
( ( ( these_o @ B )
!= bot_bot_set_o )
= ( ( B != bot_bot_set_option_o )
& ( B
!= ( insert_option_o @ none_o @ bot_bot_set_option_o ) ) ) ) ).
% these_not_empty_eq
thf(fact_404_these__not__empty__eq,axiom,
! [B: set_op4811217697719039047ist_fm] :
( ( ( these_2172363061839534302ist_fm @ B )
!= bot_bo4530011010606209517ist_fm )
= ( ( B != bot_bo4262429147648288435ist_fm )
& ( B
!= ( insert7172606565080779063ist_fm @ none_P1680826464017717506ist_fm @ bot_bo4262429147648288435ist_fm ) ) ) ) ).
% these_not_empty_eq
thf(fact_405_these__empty__eq,axiom,
! [B: set_option_rule] :
( ( ( these_rule @ B )
= bot_bot_set_rule )
= ( ( B = bot_bo8966923457718485855n_rule )
| ( B
= ( insert_option_rule @ none_rule @ bot_bo8966923457718485855n_rule ) ) ) ) ).
% these_empty_eq
thf(fact_406_these__empty__eq,axiom,
! [B: set_option_o] :
( ( ( these_o @ B )
= bot_bot_set_o )
= ( ( B = bot_bot_set_option_o )
| ( B
= ( insert_option_o @ none_o @ bot_bot_set_option_o ) ) ) ) ).
% these_empty_eq
thf(fact_407_these__empty__eq,axiom,
! [B: set_op4811217697719039047ist_fm] :
( ( ( these_2172363061839534302ist_fm @ B )
= bot_bo4530011010606209517ist_fm )
= ( ( B = bot_bo4262429147648288435ist_fm )
| ( B
= ( insert7172606565080779063ist_fm @ none_P1680826464017717506ist_fm @ bot_bo4262429147648288435ist_fm ) ) ) ) ).
% these_empty_eq
thf(fact_408_not__None__eq,axiom,
! [X2: option6029102822600979917ist_fm] :
( ( X2 != none_f2701453242279119720ist_fm )
= ( ? [Y4: fset_P661503646757059847ist_fm] :
( X2
= ( some_f2162114139285431020ist_fm @ Y4 ) ) ) ) ).
% not_None_eq
thf(fact_409_not__Some__eq,axiom,
! [X2: option6029102822600979917ist_fm] :
( ( ! [Y4: fset_P661503646757059847ist_fm] :
( X2
!= ( some_f2162114139285431020ist_fm @ Y4 ) ) )
= ( X2 = none_f2701453242279119720ist_fm ) ) ).
% not_Some_eq
thf(fact_410_option_Odistinct_I1_J,axiom,
! [X22: fset_P661503646757059847ist_fm] :
( none_f2701453242279119720ist_fm
!= ( some_f2162114139285431020ist_fm @ X22 ) ) ).
% option.distinct(1)
thf(fact_411_option_OdiscI,axiom,
! [Option: option6029102822600979917ist_fm,X22: fset_P661503646757059847ist_fm] :
( ( Option
= ( some_f2162114139285431020ist_fm @ X22 ) )
=> ( Option != none_f2701453242279119720ist_fm ) ) ).
% option.discI
thf(fact_412_option_Oexhaust,axiom,
! [Y3: option6029102822600979917ist_fm] :
( ( Y3 != none_f2701453242279119720ist_fm )
=> ~ ! [X23: fset_P661503646757059847ist_fm] :
( Y3
!= ( some_f2162114139285431020ist_fm @ X23 ) ) ) ).
% option.exhaust
thf(fact_413_split__option__ex,axiom,
( ( ^ [P3: option6029102822600979917ist_fm > $o] :
? [X6: option6029102822600979917ist_fm] : ( P3 @ X6 ) )
= ( ^ [P4: option6029102822600979917ist_fm > $o] :
( ( P4 @ none_f2701453242279119720ist_fm )
| ? [X: fset_P661503646757059847ist_fm] : ( P4 @ ( some_f2162114139285431020ist_fm @ X ) ) ) ) ) ).
% split_option_ex
thf(fact_414_split__option__all,axiom,
( ( ^ [P3: option6029102822600979917ist_fm > $o] :
! [X6: option6029102822600979917ist_fm] : ( P3 @ X6 ) )
= ( ^ [P4: option6029102822600979917ist_fm > $o] :
( ( P4 @ none_f2701453242279119720ist_fm )
& ! [X: fset_P661503646757059847ist_fm] : ( P4 @ ( some_f2162114139285431020ist_fm @ X ) ) ) ) ) ).
% split_option_all
thf(fact_415_combine__options__cases,axiom,
! [X2: option6029102822600979917ist_fm,P: option6029102822600979917ist_fm > option6029102822600979917ist_fm > $o,Y3: option6029102822600979917ist_fm] :
( ( ( X2 = none_f2701453242279119720ist_fm )
=> ( P @ X2 @ Y3 ) )
=> ( ( ( Y3 = none_f2701453242279119720ist_fm )
=> ( P @ X2 @ Y3 ) )
=> ( ! [A7: fset_P661503646757059847ist_fm,B7: fset_P661503646757059847ist_fm] :
( ( X2
= ( some_f2162114139285431020ist_fm @ A7 ) )
=> ( ( Y3
= ( some_f2162114139285431020ist_fm @ B7 ) )
=> ( P @ X2 @ Y3 ) ) )
=> ( P @ X2 @ Y3 ) ) ) ) ).
% combine_options_cases
thf(fact_416_linear__order__on__singleton,axiom,
! [X2: list_fm] : ( order_4446669488389765227ist_fm @ ( insert_list_fm @ X2 @ bot_bot_set_list_fm ) @ ( insert589278167076416497ist_fm @ ( produc381145313068854617ist_fm @ X2 @ X2 ) @ bot_bo4530011010606209517ist_fm ) ) ).
% linear_order_on_singleton
thf(fact_417_linear__order__on__singleton,axiom,
! [X2: rule] : ( order_3615580131159368656n_rule @ ( insert_rule @ X2 @ bot_bot_set_rule ) @ ( insert3755751280551455547e_rule @ ( produc1769597910952761123e_rule @ X2 @ X2 ) @ bot_bo9047865283180780599e_rule ) ) ).
% linear_order_on_singleton
thf(fact_418_linear__order__on__singleton,axiom,
! [X2: $o] : ( order_6238756238976269133r_on_o @ ( insert_o @ X2 @ bot_bot_set_o ) @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ X2 @ X2 ) @ bot_bo7073875226086086771od_o_o ) ) ).
% linear_order_on_singleton
thf(fact_419_linear__order__on__singleton,axiom,
! [X2: produc1996495991257130529ist_fm] : ( order_193637769215337174ist_fm @ ( insert589278167076416497ist_fm @ X2 @ bot_bo4530011010606209517ist_fm ) @ ( insert5410920824581998135ist_fm @ ( produc2592314732767339223ist_fm @ X2 @ X2 ) @ bot_bo7330262581962992051ist_fm ) ) ).
% linear_order_on_singleton
thf(fact_420_aboveS__def,axiom,
( order_aboveS_o
= ( ^ [R3: set_Product_prod_o_o,A4: $o] :
( collect_o
@ ^ [B8: $o] :
( ( B8 != A4 )
& ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A4 @ B8 ) @ R3 ) ) ) ) ) ).
% aboveS_def
thf(fact_421_aboveS__def,axiom,
( order_aboveS_rule
= ( ^ [R3: set_Pr6543236023296505803e_rule,A4: rule] :
( collect_rule
@ ^ [B8: rule] :
( ( B8 != A4 )
& ( member3753523862196722196e_rule @ ( produc1769597910952761123e_rule @ A4 @ B8 ) @ R3 ) ) ) ) ) ).
% aboveS_def
thf(fact_422_aboveS__def,axiom,
( order_744099749760756108ist_fm
= ( ^ [R3: set_Pr5209464977148653383ist_fm,A4: produc1996495991257130529ist_fm] :
( collec7231575776666803724ist_fm
@ ^ [B8: produc1996495991257130529ist_fm] :
( ( B8 != A4 )
& ( member60238275747228048ist_fm @ ( produc2592314732767339223ist_fm @ A4 @ B8 ) @ R3 ) ) ) ) ) ).
% aboveS_def
thf(fact_423_aboveS__def,axiom,
( order_aboveS_list_fm
= ( ^ [R3: set_Pr7058068377845519745ist_fm,A4: list_fm] :
( collect_list_fm
@ ^ [B8: list_fm] :
( ( B8 != A4 )
& ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ A4 @ B8 ) @ R3 ) ) ) ) ) ).
% aboveS_def
thf(fact_424_UNIV__option__conv,axiom,
( top_to170523636175635069ist_fm
= ( insert31278938154982301ist_fm @ none_f2701453242279119720ist_fm @ ( image_4775136921032825739ist_fm @ some_f2162114139285431020ist_fm @ top_to6886830657606236215ist_fm ) ) ) ).
% UNIV_option_conv
thf(fact_425_UNIV__option__conv,axiom,
( top_to2023577309291471255ist_fm
= ( insert7172606565080779063ist_fm @ none_P1680826464017717506ist_fm @ ( image_4446678087167650955ist_fm @ some_P7346312016712434310ist_fm @ top_to728902761071950545ist_fm ) ) ) ).
% UNIV_option_conv
thf(fact_426_UNIV__option__conv,axiom,
( top_to4271146702142796155n_rule
= ( insert_option_rule @ none_rule @ ( image_766640639675677281n_rule @ some_rule @ top_top_set_rule ) ) ) ).
% UNIV_option_conv
thf(fact_427_UNIV__option__conv,axiom,
( top_top_set_option_o
= ( insert_option_o @ none_o @ ( image_o_option_o @ some_o @ top_top_set_o ) ) ) ).
% UNIV_option_conv
thf(fact_428_Range__insert,axiom,
! [A3: list_fm,B3: list_fm,R: set_Pr7058068377845519745ist_fm] :
( ( range_8771097359499353911ist_fm @ ( insert589278167076416497ist_fm @ ( produc381145313068854617ist_fm @ A3 @ B3 ) @ R ) )
= ( insert_list_fm @ B3 @ ( range_8771097359499353911ist_fm @ R ) ) ) ).
% Range_insert
thf(fact_429_Domain__insert,axiom,
! [A3: list_fm,B3: list_fm,R: set_Pr7058068377845519745ist_fm] :
( ( domain2949914093644414030ist_fm @ ( insert589278167076416497ist_fm @ ( produc381145313068854617ist_fm @ A3 @ B3 ) @ R ) )
= ( insert_list_fm @ A3 @ ( domain2949914093644414030ist_fm @ R ) ) ) ).
% Domain_insert
thf(fact_430_trans__singleton,axiom,
! [A3: list_fm] : ( trans_on_list_fm @ top_top_set_list_fm @ ( insert589278167076416497ist_fm @ ( produc381145313068854617ist_fm @ A3 @ A3 ) @ bot_bo4530011010606209517ist_fm ) ) ).
% trans_singleton
thf(fact_431_trans__singleton,axiom,
! [A3: produc1996495991257130529ist_fm] : ( trans_4095360997890102209ist_fm @ top_to728902761071950545ist_fm @ ( insert5410920824581998135ist_fm @ ( produc2592314732767339223ist_fm @ A3 @ A3 ) @ bot_bo7330262581962992051ist_fm ) ) ).
% trans_singleton
thf(fact_432_trans__singleton,axiom,
! [A3: rule] : ( trans_on_rule @ top_top_set_rule @ ( insert3755751280551455547e_rule @ ( produc1769597910952761123e_rule @ A3 @ A3 ) @ bot_bo9047865283180780599e_rule ) ) ).
% trans_singleton
thf(fact_433_trans__singleton,axiom,
! [A3: $o] : ( trans_on_o @ top_top_set_o @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ A3 @ A3 ) @ bot_bo7073875226086086771od_o_o ) ) ).
% trans_singleton
thf(fact_434_above__def,axiom,
( order_above_o
= ( ^ [R3: set_Product_prod_o_o,A4: $o] :
( collect_o
@ ^ [B8: $o] : ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A4 @ B8 ) @ R3 ) ) ) ) ).
% above_def
thf(fact_435_above__def,axiom,
( order_above_rule
= ( ^ [R3: set_Pr6543236023296505803e_rule,A4: rule] :
( collect_rule
@ ^ [B8: rule] : ( member3753523862196722196e_rule @ ( produc1769597910952761123e_rule @ A4 @ B8 ) @ R3 ) ) ) ) ).
% above_def
thf(fact_436_above__def,axiom,
( order_5287731903943553313ist_fm
= ( ^ [R3: set_Pr5209464977148653383ist_fm,A4: produc1996495991257130529ist_fm] :
( collec7231575776666803724ist_fm
@ ^ [B8: produc1996495991257130529ist_fm] : ( member60238275747228048ist_fm @ ( produc2592314732767339223ist_fm @ A4 @ B8 ) @ R3 ) ) ) ) ).
% above_def
thf(fact_437_above__def,axiom,
( order_above_list_fm
= ( ^ [R3: set_Pr7058068377845519745ist_fm,A4: list_fm] :
( collect_list_fm
@ ^ [B8: list_fm] : ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ A4 @ B8 ) @ R3 ) ) ) ) ).
% above_def
thf(fact_438_image__eqI,axiom,
! [B3: produc1996495991257130529ist_fm,F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,X2: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm] :
( ( B3
= ( F @ X2 ) )
=> ( ( member8102475879199740618ist_fm @ X2 @ A )
=> ( member8102475879199740618ist_fm @ B3 @ ( image_6982201141141708101ist_fm @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_439_image__eqI,axiom,
! [B3: rule,F: produc1996495991257130529ist_fm > rule,X2: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm] :
( ( B3
= ( F @ X2 ) )
=> ( ( member8102475879199740618ist_fm @ X2 @ A )
=> ( member_rule @ B3 @ ( image_368924427985557793m_rule @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_440_image__eqI,axiom,
! [B3: $o,F: produc1996495991257130529ist_fm > $o,X2: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm] :
( ( B3
= ( F @ X2 ) )
=> ( ( member8102475879199740618ist_fm @ X2 @ A )
=> ( member_o @ B3 @ ( image_3428966142334963388t_fm_o @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_441_image__eqI,axiom,
! [B3: produc1996495991257130529ist_fm,F: rule > produc1996495991257130529ist_fm,X2: rule,A: set_rule] :
( ( B3
= ( F @ X2 ) )
=> ( ( member_rule @ X2 @ A )
=> ( member8102475879199740618ist_fm @ B3 @ ( image_6941212235319914453ist_fm @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_442_image__eqI,axiom,
! [B3: rule,F: rule > rule,X2: rule,A: set_rule] :
( ( B3
= ( F @ X2 ) )
=> ( ( member_rule @ X2 @ A )
=> ( member_rule @ B3 @ ( image_rule_rule @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_443_image__eqI,axiom,
! [B3: $o,F: rule > $o,X2: rule,A: set_rule] :
( ( B3
= ( F @ X2 ) )
=> ( ( member_rule @ X2 @ A )
=> ( member_o @ B3 @ ( image_rule_o @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_444_image__eqI,axiom,
! [B3: produc1996495991257130529ist_fm,F: $o > produc1996495991257130529ist_fm,X2: $o,A: set_o] :
( ( B3
= ( F @ X2 ) )
=> ( ( member_o @ X2 @ A )
=> ( member8102475879199740618ist_fm @ B3 @ ( image_3689732601364923150ist_fm @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_445_image__eqI,axiom,
! [B3: rule,F: $o > rule,X2: $o,A: set_o] :
( ( B3
= ( F @ X2 ) )
=> ( ( member_o @ X2 @ A )
=> ( member_rule @ B3 @ ( image_o_rule @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_446_image__eqI,axiom,
! [B3: $o,F: $o > $o,X2: $o,A: set_o] :
( ( B3
= ( F @ X2 ) )
=> ( ( member_o @ X2 @ A )
=> ( member_o @ B3 @ ( image_o_o @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_447_image__is__empty,axiom,
! [F: rule > rule,A: set_rule] :
( ( ( image_rule_rule @ F @ A )
= bot_bot_set_rule )
= ( A = bot_bot_set_rule ) ) ).
% image_is_empty
thf(fact_448_image__is__empty,axiom,
! [F: $o > rule,A: set_o] :
( ( ( image_o_rule @ F @ A )
= bot_bot_set_rule )
= ( A = bot_bot_set_o ) ) ).
% image_is_empty
thf(fact_449_image__is__empty,axiom,
! [F: produc1996495991257130529ist_fm > rule,A: set_Pr7058068377845519745ist_fm] :
( ( ( image_368924427985557793m_rule @ F @ A )
= bot_bot_set_rule )
= ( A = bot_bo4530011010606209517ist_fm ) ) ).
% image_is_empty
thf(fact_450_image__is__empty,axiom,
! [F: rule > $o,A: set_rule] :
( ( ( image_rule_o @ F @ A )
= bot_bot_set_o )
= ( A = bot_bot_set_rule ) ) ).
% image_is_empty
thf(fact_451_image__is__empty,axiom,
! [F: $o > $o,A: set_o] :
( ( ( image_o_o @ F @ A )
= bot_bot_set_o )
= ( A = bot_bot_set_o ) ) ).
% image_is_empty
thf(fact_452_image__is__empty,axiom,
! [F: produc1996495991257130529ist_fm > $o,A: set_Pr7058068377845519745ist_fm] :
( ( ( image_3428966142334963388t_fm_o @ F @ A )
= bot_bot_set_o )
= ( A = bot_bo4530011010606209517ist_fm ) ) ).
% image_is_empty
thf(fact_453_image__is__empty,axiom,
! [F: rule > produc1996495991257130529ist_fm,A: set_rule] :
( ( ( image_6941212235319914453ist_fm @ F @ A )
= bot_bo4530011010606209517ist_fm )
= ( A = bot_bot_set_rule ) ) ).
% image_is_empty
thf(fact_454_image__is__empty,axiom,
! [F: $o > produc1996495991257130529ist_fm,A: set_o] :
( ( ( image_3689732601364923150ist_fm @ F @ A )
= bot_bo4530011010606209517ist_fm )
= ( A = bot_bot_set_o ) ) ).
% image_is_empty
thf(fact_455_image__is__empty,axiom,
! [F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm] :
( ( ( image_6982201141141708101ist_fm @ F @ A )
= bot_bo4530011010606209517ist_fm )
= ( A = bot_bo4530011010606209517ist_fm ) ) ).
% image_is_empty
thf(fact_456_empty__is__image,axiom,
! [F: rule > rule,A: set_rule] :
( ( bot_bot_set_rule
= ( image_rule_rule @ F @ A ) )
= ( A = bot_bot_set_rule ) ) ).
% empty_is_image
thf(fact_457_empty__is__image,axiom,
! [F: $o > rule,A: set_o] :
( ( bot_bot_set_rule
= ( image_o_rule @ F @ A ) )
= ( A = bot_bot_set_o ) ) ).
% empty_is_image
thf(fact_458_empty__is__image,axiom,
! [F: produc1996495991257130529ist_fm > rule,A: set_Pr7058068377845519745ist_fm] :
( ( bot_bot_set_rule
= ( image_368924427985557793m_rule @ F @ A ) )
= ( A = bot_bo4530011010606209517ist_fm ) ) ).
% empty_is_image
thf(fact_459_empty__is__image,axiom,
! [F: rule > $o,A: set_rule] :
( ( bot_bot_set_o
= ( image_rule_o @ F @ A ) )
= ( A = bot_bot_set_rule ) ) ).
% empty_is_image
thf(fact_460_empty__is__image,axiom,
! [F: $o > $o,A: set_o] :
( ( bot_bot_set_o
= ( image_o_o @ F @ A ) )
= ( A = bot_bot_set_o ) ) ).
% empty_is_image
thf(fact_461_empty__is__image,axiom,
! [F: produc1996495991257130529ist_fm > $o,A: set_Pr7058068377845519745ist_fm] :
( ( bot_bot_set_o
= ( image_3428966142334963388t_fm_o @ F @ A ) )
= ( A = bot_bo4530011010606209517ist_fm ) ) ).
% empty_is_image
thf(fact_462_empty__is__image,axiom,
! [F: rule > produc1996495991257130529ist_fm,A: set_rule] :
( ( bot_bo4530011010606209517ist_fm
= ( image_6941212235319914453ist_fm @ F @ A ) )
= ( A = bot_bot_set_rule ) ) ).
% empty_is_image
thf(fact_463_empty__is__image,axiom,
! [F: $o > produc1996495991257130529ist_fm,A: set_o] :
( ( bot_bo4530011010606209517ist_fm
= ( image_3689732601364923150ist_fm @ F @ A ) )
= ( A = bot_bot_set_o ) ) ).
% empty_is_image
thf(fact_464_empty__is__image,axiom,
! [F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm] :
( ( bot_bo4530011010606209517ist_fm
= ( image_6982201141141708101ist_fm @ F @ A ) )
= ( A = bot_bo4530011010606209517ist_fm ) ) ).
% empty_is_image
thf(fact_465_image__empty,axiom,
! [F: rule > rule] :
( ( image_rule_rule @ F @ bot_bot_set_rule )
= bot_bot_set_rule ) ).
% image_empty
thf(fact_466_image__empty,axiom,
! [F: rule > $o] :
( ( image_rule_o @ F @ bot_bot_set_rule )
= bot_bot_set_o ) ).
% image_empty
thf(fact_467_image__empty,axiom,
! [F: rule > produc1996495991257130529ist_fm] :
( ( image_6941212235319914453ist_fm @ F @ bot_bot_set_rule )
= bot_bo4530011010606209517ist_fm ) ).
% image_empty
thf(fact_468_image__empty,axiom,
! [F: $o > rule] :
( ( image_o_rule @ F @ bot_bot_set_o )
= bot_bot_set_rule ) ).
% image_empty
thf(fact_469_image__empty,axiom,
! [F: $o > $o] :
( ( image_o_o @ F @ bot_bot_set_o )
= bot_bot_set_o ) ).
% image_empty
thf(fact_470_image__empty,axiom,
! [F: $o > produc1996495991257130529ist_fm] :
( ( image_3689732601364923150ist_fm @ F @ bot_bot_set_o )
= bot_bo4530011010606209517ist_fm ) ).
% image_empty
thf(fact_471_image__empty,axiom,
! [F: produc1996495991257130529ist_fm > rule] :
( ( image_368924427985557793m_rule @ F @ bot_bo4530011010606209517ist_fm )
= bot_bot_set_rule ) ).
% image_empty
thf(fact_472_image__empty,axiom,
! [F: produc1996495991257130529ist_fm > $o] :
( ( image_3428966142334963388t_fm_o @ F @ bot_bo4530011010606209517ist_fm )
= bot_bot_set_o ) ).
% image_empty
thf(fact_473_image__empty,axiom,
! [F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm] :
( ( image_6982201141141708101ist_fm @ F @ bot_bo4530011010606209517ist_fm )
= bot_bo4530011010606209517ist_fm ) ).
% image_empty
thf(fact_474_insert__image,axiom,
! [X2: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm,F: produc1996495991257130529ist_fm > rule] :
( ( member8102475879199740618ist_fm @ X2 @ A )
=> ( ( insert_rule @ ( F @ X2 ) @ ( image_368924427985557793m_rule @ F @ A ) )
= ( image_368924427985557793m_rule @ F @ A ) ) ) ).
% insert_image
thf(fact_475_insert__image,axiom,
! [X2: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm,F: produc1996495991257130529ist_fm > $o] :
( ( member8102475879199740618ist_fm @ X2 @ A )
=> ( ( insert_o @ ( F @ X2 ) @ ( image_3428966142334963388t_fm_o @ F @ A ) )
= ( image_3428966142334963388t_fm_o @ F @ A ) ) ) ).
% insert_image
thf(fact_476_insert__image,axiom,
! [X2: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm,F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ X2 @ A )
=> ( ( insert589278167076416497ist_fm @ ( F @ X2 ) @ ( image_6982201141141708101ist_fm @ F @ A ) )
= ( image_6982201141141708101ist_fm @ F @ A ) ) ) ).
% insert_image
thf(fact_477_insert__image,axiom,
! [X2: rule,A: set_rule,F: rule > rule] :
( ( member_rule @ X2 @ A )
=> ( ( insert_rule @ ( F @ X2 ) @ ( image_rule_rule @ F @ A ) )
= ( image_rule_rule @ F @ A ) ) ) ).
% insert_image
thf(fact_478_insert__image,axiom,
! [X2: rule,A: set_rule,F: rule > $o] :
( ( member_rule @ X2 @ A )
=> ( ( insert_o @ ( F @ X2 ) @ ( image_rule_o @ F @ A ) )
= ( image_rule_o @ F @ A ) ) ) ).
% insert_image
thf(fact_479_insert__image,axiom,
! [X2: rule,A: set_rule,F: rule > produc1996495991257130529ist_fm] :
( ( member_rule @ X2 @ A )
=> ( ( insert589278167076416497ist_fm @ ( F @ X2 ) @ ( image_6941212235319914453ist_fm @ F @ A ) )
= ( image_6941212235319914453ist_fm @ F @ A ) ) ) ).
% insert_image
thf(fact_480_insert__image,axiom,
! [X2: $o,A: set_o,F: $o > rule] :
( ( member_o @ X2 @ A )
=> ( ( insert_rule @ ( F @ X2 ) @ ( image_o_rule @ F @ A ) )
= ( image_o_rule @ F @ A ) ) ) ).
% insert_image
thf(fact_481_insert__image,axiom,
! [X2: $o,A: set_o,F: $o > $o] :
( ( member_o @ X2 @ A )
=> ( ( insert_o @ ( F @ X2 ) @ ( image_o_o @ F @ A ) )
= ( image_o_o @ F @ A ) ) ) ).
% insert_image
thf(fact_482_insert__image,axiom,
! [X2: $o,A: set_o,F: $o > produc1996495991257130529ist_fm] :
( ( member_o @ X2 @ A )
=> ( ( insert589278167076416497ist_fm @ ( F @ X2 ) @ ( image_3689732601364923150ist_fm @ F @ A ) )
= ( image_3689732601364923150ist_fm @ F @ A ) ) ) ).
% insert_image
thf(fact_483_image__insert,axiom,
! [F: rule > rule,A3: rule,B: set_rule] :
( ( image_rule_rule @ F @ ( insert_rule @ A3 @ B ) )
= ( insert_rule @ ( F @ A3 ) @ ( image_rule_rule @ F @ B ) ) ) ).
% image_insert
thf(fact_484_image__insert,axiom,
! [F: rule > $o,A3: rule,B: set_rule] :
( ( image_rule_o @ F @ ( insert_rule @ A3 @ B ) )
= ( insert_o @ ( F @ A3 ) @ ( image_rule_o @ F @ B ) ) ) ).
% image_insert
thf(fact_485_image__insert,axiom,
! [F: rule > produc1996495991257130529ist_fm,A3: rule,B: set_rule] :
( ( image_6941212235319914453ist_fm @ F @ ( insert_rule @ A3 @ B ) )
= ( insert589278167076416497ist_fm @ ( F @ A3 ) @ ( image_6941212235319914453ist_fm @ F @ B ) ) ) ).
% image_insert
thf(fact_486_image__insert,axiom,
! [F: $o > rule,A3: $o,B: set_o] :
( ( image_o_rule @ F @ ( insert_o @ A3 @ B ) )
= ( insert_rule @ ( F @ A3 ) @ ( image_o_rule @ F @ B ) ) ) ).
% image_insert
thf(fact_487_image__insert,axiom,
! [F: $o > $o,A3: $o,B: set_o] :
( ( image_o_o @ F @ ( insert_o @ A3 @ B ) )
= ( insert_o @ ( F @ A3 ) @ ( image_o_o @ F @ B ) ) ) ).
% image_insert
thf(fact_488_image__insert,axiom,
! [F: $o > produc1996495991257130529ist_fm,A3: $o,B: set_o] :
( ( image_3689732601364923150ist_fm @ F @ ( insert_o @ A3 @ B ) )
= ( insert589278167076416497ist_fm @ ( F @ A3 ) @ ( image_3689732601364923150ist_fm @ F @ B ) ) ) ).
% image_insert
thf(fact_489_image__insert,axiom,
! [F: produc1996495991257130529ist_fm > rule,A3: produc1996495991257130529ist_fm,B: set_Pr7058068377845519745ist_fm] :
( ( image_368924427985557793m_rule @ F @ ( insert589278167076416497ist_fm @ A3 @ B ) )
= ( insert_rule @ ( F @ A3 ) @ ( image_368924427985557793m_rule @ F @ B ) ) ) ).
% image_insert
thf(fact_490_image__insert,axiom,
! [F: produc1996495991257130529ist_fm > $o,A3: produc1996495991257130529ist_fm,B: set_Pr7058068377845519745ist_fm] :
( ( image_3428966142334963388t_fm_o @ F @ ( insert589278167076416497ist_fm @ A3 @ B ) )
= ( insert_o @ ( F @ A3 ) @ ( image_3428966142334963388t_fm_o @ F @ B ) ) ) ).
% image_insert
thf(fact_491_image__insert,axiom,
! [F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,A3: produc1996495991257130529ist_fm,B: set_Pr7058068377845519745ist_fm] :
( ( image_6982201141141708101ist_fm @ F @ ( insert589278167076416497ist_fm @ A3 @ B ) )
= ( insert589278167076416497ist_fm @ ( F @ A3 ) @ ( image_6982201141141708101ist_fm @ F @ B ) ) ) ).
% image_insert
thf(fact_492_Domain__empty,axiom,
( ( domain2949914093644414030ist_fm @ bot_bo4530011010606209517ist_fm )
= bot_bot_set_list_fm ) ).
% Domain_empty
thf(fact_493_Range__empty,axiom,
( ( range_8771097359499353911ist_fm @ bot_bo4530011010606209517ist_fm )
= bot_bot_set_list_fm ) ).
% Range_empty
thf(fact_494_these__image__Some__eq,axiom,
! [A: set_fs7534060674568567527ist_fm] :
( ( these_4231445979625565508ist_fm @ ( image_4775136921032825739ist_fm @ some_f2162114139285431020ist_fm @ A ) )
= A ) ).
% these_image_Some_eq
thf(fact_495_range__constant,axiom,
! [X2: rule] :
( ( image_368924427985557793m_rule
@ ^ [Uu: produc1996495991257130529ist_fm] : X2
@ top_to728902761071950545ist_fm )
= ( insert_rule @ X2 @ bot_bot_set_rule ) ) ).
% range_constant
thf(fact_496_range__constant,axiom,
! [X2: $o] :
( ( image_3428966142334963388t_fm_o
@ ^ [Uu: produc1996495991257130529ist_fm] : X2
@ top_to728902761071950545ist_fm )
= ( insert_o @ X2 @ bot_bot_set_o ) ) ).
% range_constant
thf(fact_497_range__constant,axiom,
! [X2: produc1996495991257130529ist_fm] :
( ( image_6982201141141708101ist_fm
@ ^ [Uu: produc1996495991257130529ist_fm] : X2
@ top_to728902761071950545ist_fm )
= ( insert589278167076416497ist_fm @ X2 @ bot_bo4530011010606209517ist_fm ) ) ).
% range_constant
thf(fact_498_range__constant,axiom,
! [X2: rule] :
( ( image_rule_rule
@ ^ [Uu: rule] : X2
@ top_top_set_rule )
= ( insert_rule @ X2 @ bot_bot_set_rule ) ) ).
% range_constant
thf(fact_499_range__constant,axiom,
! [X2: $o] :
( ( image_rule_o
@ ^ [Uu: rule] : X2
@ top_top_set_rule )
= ( insert_o @ X2 @ bot_bot_set_o ) ) ).
% range_constant
thf(fact_500_range__constant,axiom,
! [X2: produc1996495991257130529ist_fm] :
( ( image_6941212235319914453ist_fm
@ ^ [Uu: rule] : X2
@ top_top_set_rule )
= ( insert589278167076416497ist_fm @ X2 @ bot_bo4530011010606209517ist_fm ) ) ).
% range_constant
thf(fact_501_range__constant,axiom,
! [X2: rule] :
( ( image_o_rule
@ ^ [Uu: $o] : X2
@ top_top_set_o )
= ( insert_rule @ X2 @ bot_bot_set_rule ) ) ).
% range_constant
thf(fact_502_range__constant,axiom,
! [X2: $o] :
( ( image_o_o
@ ^ [Uu: $o] : X2
@ top_top_set_o )
= ( insert_o @ X2 @ bot_bot_set_o ) ) ).
% range_constant
thf(fact_503_range__constant,axiom,
! [X2: produc1996495991257130529ist_fm] :
( ( image_3689732601364923150ist_fm
@ ^ [Uu: $o] : X2
@ top_top_set_o )
= ( insert589278167076416497ist_fm @ X2 @ bot_bo4530011010606209517ist_fm ) ) ).
% range_constant
thf(fact_504_rev__image__eqI,axiom,
! [X2: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm,B3: produc1996495991257130529ist_fm,F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ X2 @ A )
=> ( ( B3
= ( F @ X2 ) )
=> ( member8102475879199740618ist_fm @ B3 @ ( image_6982201141141708101ist_fm @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_505_rev__image__eqI,axiom,
! [X2: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm,B3: rule,F: produc1996495991257130529ist_fm > rule] :
( ( member8102475879199740618ist_fm @ X2 @ A )
=> ( ( B3
= ( F @ X2 ) )
=> ( member_rule @ B3 @ ( image_368924427985557793m_rule @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_506_rev__image__eqI,axiom,
! [X2: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm,B3: $o,F: produc1996495991257130529ist_fm > $o] :
( ( member8102475879199740618ist_fm @ X2 @ A )
=> ( ( B3
= ( F @ X2 ) )
=> ( member_o @ B3 @ ( image_3428966142334963388t_fm_o @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_507_rev__image__eqI,axiom,
! [X2: rule,A: set_rule,B3: produc1996495991257130529ist_fm,F: rule > produc1996495991257130529ist_fm] :
( ( member_rule @ X2 @ A )
=> ( ( B3
= ( F @ X2 ) )
=> ( member8102475879199740618ist_fm @ B3 @ ( image_6941212235319914453ist_fm @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_508_rev__image__eqI,axiom,
! [X2: rule,A: set_rule,B3: rule,F: rule > rule] :
( ( member_rule @ X2 @ A )
=> ( ( B3
= ( F @ X2 ) )
=> ( member_rule @ B3 @ ( image_rule_rule @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_509_rev__image__eqI,axiom,
! [X2: rule,A: set_rule,B3: $o,F: rule > $o] :
( ( member_rule @ X2 @ A )
=> ( ( B3
= ( F @ X2 ) )
=> ( member_o @ B3 @ ( image_rule_o @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_510_rev__image__eqI,axiom,
! [X2: $o,A: set_o,B3: produc1996495991257130529ist_fm,F: $o > produc1996495991257130529ist_fm] :
( ( member_o @ X2 @ A )
=> ( ( B3
= ( F @ X2 ) )
=> ( member8102475879199740618ist_fm @ B3 @ ( image_3689732601364923150ist_fm @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_511_rev__image__eqI,axiom,
! [X2: $o,A: set_o,B3: rule,F: $o > rule] :
( ( member_o @ X2 @ A )
=> ( ( B3
= ( F @ X2 ) )
=> ( member_rule @ B3 @ ( image_o_rule @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_512_rev__image__eqI,axiom,
! [X2: $o,A: set_o,B3: $o,F: $o > $o] :
( ( member_o @ X2 @ A )
=> ( ( B3
= ( F @ X2 ) )
=> ( member_o @ B3 @ ( image_o_o @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_513_imageI,axiom,
! [X2: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm,F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ X2 @ A )
=> ( member8102475879199740618ist_fm @ ( F @ X2 ) @ ( image_6982201141141708101ist_fm @ F @ A ) ) ) ).
% imageI
thf(fact_514_imageI,axiom,
! [X2: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm,F: produc1996495991257130529ist_fm > rule] :
( ( member8102475879199740618ist_fm @ X2 @ A )
=> ( member_rule @ ( F @ X2 ) @ ( image_368924427985557793m_rule @ F @ A ) ) ) ).
% imageI
thf(fact_515_imageI,axiom,
! [X2: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm,F: produc1996495991257130529ist_fm > $o] :
( ( member8102475879199740618ist_fm @ X2 @ A )
=> ( member_o @ ( F @ X2 ) @ ( image_3428966142334963388t_fm_o @ F @ A ) ) ) ).
% imageI
thf(fact_516_imageI,axiom,
! [X2: rule,A: set_rule,F: rule > produc1996495991257130529ist_fm] :
( ( member_rule @ X2 @ A )
=> ( member8102475879199740618ist_fm @ ( F @ X2 ) @ ( image_6941212235319914453ist_fm @ F @ A ) ) ) ).
% imageI
thf(fact_517_imageI,axiom,
! [X2: rule,A: set_rule,F: rule > rule] :
( ( member_rule @ X2 @ A )
=> ( member_rule @ ( F @ X2 ) @ ( image_rule_rule @ F @ A ) ) ) ).
% imageI
thf(fact_518_imageI,axiom,
! [X2: rule,A: set_rule,F: rule > $o] :
( ( member_rule @ X2 @ A )
=> ( member_o @ ( F @ X2 ) @ ( image_rule_o @ F @ A ) ) ) ).
% imageI
thf(fact_519_imageI,axiom,
! [X2: $o,A: set_o,F: $o > produc1996495991257130529ist_fm] :
( ( member_o @ X2 @ A )
=> ( member8102475879199740618ist_fm @ ( F @ X2 ) @ ( image_3689732601364923150ist_fm @ F @ A ) ) ) ).
% imageI
thf(fact_520_imageI,axiom,
! [X2: $o,A: set_o,F: $o > rule] :
( ( member_o @ X2 @ A )
=> ( member_rule @ ( F @ X2 ) @ ( image_o_rule @ F @ A ) ) ) ).
% imageI
thf(fact_521_imageI,axiom,
! [X2: $o,A: set_o,F: $o > $o] :
( ( member_o @ X2 @ A )
=> ( member_o @ ( F @ X2 ) @ ( image_o_o @ F @ A ) ) ) ).
% imageI
thf(fact_522_Compr__image__eq,axiom,
! [F: $o > $o,A: set_o,P: $o > $o] :
( ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ ( image_o_o @ F @ A ) )
& ( P @ X ) ) )
= ( image_o_o @ F
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_523_Compr__image__eq,axiom,
! [F: rule > $o,A: set_rule,P: $o > $o] :
( ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ ( image_rule_o @ F @ A ) )
& ( P @ X ) ) )
= ( image_rule_o @ F
@ ( collect_rule
@ ^ [X: rule] :
( ( member_rule @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_524_Compr__image__eq,axiom,
! [F: produc1996495991257130529ist_fm > $o,A: set_Pr7058068377845519745ist_fm,P: $o > $o] :
( ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ ( image_3428966142334963388t_fm_o @ F @ A ) )
& ( P @ X ) ) )
= ( image_3428966142334963388t_fm_o @ F
@ ( collec7231575776666803724ist_fm
@ ^ [X: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_525_Compr__image__eq,axiom,
! [F: $o > rule,A: set_o,P: rule > $o] :
( ( collect_rule
@ ^ [X: rule] :
( ( member_rule @ X @ ( image_o_rule @ F @ A ) )
& ( P @ X ) ) )
= ( image_o_rule @ F
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_526_Compr__image__eq,axiom,
! [F: rule > rule,A: set_rule,P: rule > $o] :
( ( collect_rule
@ ^ [X: rule] :
( ( member_rule @ X @ ( image_rule_rule @ F @ A ) )
& ( P @ X ) ) )
= ( image_rule_rule @ F
@ ( collect_rule
@ ^ [X: rule] :
( ( member_rule @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_527_Compr__image__eq,axiom,
! [F: produc1996495991257130529ist_fm > rule,A: set_Pr7058068377845519745ist_fm,P: rule > $o] :
( ( collect_rule
@ ^ [X: rule] :
( ( member_rule @ X @ ( image_368924427985557793m_rule @ F @ A ) )
& ( P @ X ) ) )
= ( image_368924427985557793m_rule @ F
@ ( collec7231575776666803724ist_fm
@ ^ [X: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_528_Compr__image__eq,axiom,
! [F: $o > produc1996495991257130529ist_fm,A: set_o,P: produc1996495991257130529ist_fm > $o] :
( ( collec7231575776666803724ist_fm
@ ^ [X: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ X @ ( image_3689732601364923150ist_fm @ F @ A ) )
& ( P @ X ) ) )
= ( image_3689732601364923150ist_fm @ F
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_529_Compr__image__eq,axiom,
! [F: rule > produc1996495991257130529ist_fm,A: set_rule,P: produc1996495991257130529ist_fm > $o] :
( ( collec7231575776666803724ist_fm
@ ^ [X: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ X @ ( image_6941212235319914453ist_fm @ F @ A ) )
& ( P @ X ) ) )
= ( image_6941212235319914453ist_fm @ F
@ ( collect_rule
@ ^ [X: rule] :
( ( member_rule @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_530_Compr__image__eq,axiom,
! [F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm,P: produc1996495991257130529ist_fm > $o] :
( ( collec7231575776666803724ist_fm
@ ^ [X: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ X @ ( image_6982201141141708101ist_fm @ F @ A ) )
& ( P @ X ) ) )
= ( image_6982201141141708101ist_fm @ F
@ ( collec7231575776666803724ist_fm
@ ^ [X: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_531_imageE,axiom,
! [B3: produc1996495991257130529ist_fm,F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm] :
( ( member8102475879199740618ist_fm @ B3 @ ( image_6982201141141708101ist_fm @ F @ A ) )
=> ~ ! [X3: produc1996495991257130529ist_fm] :
( ( B3
= ( F @ X3 ) )
=> ~ ( member8102475879199740618ist_fm @ X3 @ A ) ) ) ).
% imageE
thf(fact_532_imageE,axiom,
! [B3: produc1996495991257130529ist_fm,F: rule > produc1996495991257130529ist_fm,A: set_rule] :
( ( member8102475879199740618ist_fm @ B3 @ ( image_6941212235319914453ist_fm @ F @ A ) )
=> ~ ! [X3: rule] :
( ( B3
= ( F @ X3 ) )
=> ~ ( member_rule @ X3 @ A ) ) ) ).
% imageE
thf(fact_533_imageE,axiom,
! [B3: produc1996495991257130529ist_fm,F: $o > produc1996495991257130529ist_fm,A: set_o] :
( ( member8102475879199740618ist_fm @ B3 @ ( image_3689732601364923150ist_fm @ F @ A ) )
=> ~ ! [X3: $o] :
( ( B3
= ( F @ X3 ) )
=> ~ ( member_o @ X3 @ A ) ) ) ).
% imageE
thf(fact_534_imageE,axiom,
! [B3: rule,F: produc1996495991257130529ist_fm > rule,A: set_Pr7058068377845519745ist_fm] :
( ( member_rule @ B3 @ ( image_368924427985557793m_rule @ F @ A ) )
=> ~ ! [X3: produc1996495991257130529ist_fm] :
( ( B3
= ( F @ X3 ) )
=> ~ ( member8102475879199740618ist_fm @ X3 @ A ) ) ) ).
% imageE
thf(fact_535_imageE,axiom,
! [B3: rule,F: rule > rule,A: set_rule] :
( ( member_rule @ B3 @ ( image_rule_rule @ F @ A ) )
=> ~ ! [X3: rule] :
( ( B3
= ( F @ X3 ) )
=> ~ ( member_rule @ X3 @ A ) ) ) ).
% imageE
thf(fact_536_imageE,axiom,
! [B3: rule,F: $o > rule,A: set_o] :
( ( member_rule @ B3 @ ( image_o_rule @ F @ A ) )
=> ~ ! [X3: $o] :
( ( B3
= ( F @ X3 ) )
=> ~ ( member_o @ X3 @ A ) ) ) ).
% imageE
thf(fact_537_imageE,axiom,
! [B3: $o,F: produc1996495991257130529ist_fm > $o,A: set_Pr7058068377845519745ist_fm] :
( ( member_o @ B3 @ ( image_3428966142334963388t_fm_o @ F @ A ) )
=> ~ ! [X3: produc1996495991257130529ist_fm] :
( ( B3
= ( F @ X3 ) )
=> ~ ( member8102475879199740618ist_fm @ X3 @ A ) ) ) ).
% imageE
thf(fact_538_imageE,axiom,
! [B3: $o,F: rule > $o,A: set_rule] :
( ( member_o @ B3 @ ( image_rule_o @ F @ A ) )
=> ~ ! [X3: rule] :
( ( B3
= ( F @ X3 ) )
=> ~ ( member_rule @ X3 @ A ) ) ) ).
% imageE
thf(fact_539_imageE,axiom,
! [B3: $o,F: $o > $o,A: set_o] :
( ( member_o @ B3 @ ( image_o_o @ F @ A ) )
=> ~ ! [X3: $o] :
( ( B3
= ( F @ X3 ) )
=> ~ ( member_o @ X3 @ A ) ) ) ).
% imageE
thf(fact_540_range__eqI,axiom,
! [B3: produc1996495991257130529ist_fm,F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,X2: produc1996495991257130529ist_fm] :
( ( B3
= ( F @ X2 ) )
=> ( member8102475879199740618ist_fm @ B3 @ ( image_6982201141141708101ist_fm @ F @ top_to728902761071950545ist_fm ) ) ) ).
% range_eqI
thf(fact_541_range__eqI,axiom,
! [B3: rule,F: produc1996495991257130529ist_fm > rule,X2: produc1996495991257130529ist_fm] :
( ( B3
= ( F @ X2 ) )
=> ( member_rule @ B3 @ ( image_368924427985557793m_rule @ F @ top_to728902761071950545ist_fm ) ) ) ).
% range_eqI
thf(fact_542_range__eqI,axiom,
! [B3: $o,F: produc1996495991257130529ist_fm > $o,X2: produc1996495991257130529ist_fm] :
( ( B3
= ( F @ X2 ) )
=> ( member_o @ B3 @ ( image_3428966142334963388t_fm_o @ F @ top_to728902761071950545ist_fm ) ) ) ).
% range_eqI
thf(fact_543_range__eqI,axiom,
! [B3: produc1996495991257130529ist_fm,F: rule > produc1996495991257130529ist_fm,X2: rule] :
( ( B3
= ( F @ X2 ) )
=> ( member8102475879199740618ist_fm @ B3 @ ( image_6941212235319914453ist_fm @ F @ top_top_set_rule ) ) ) ).
% range_eqI
thf(fact_544_range__eqI,axiom,
! [B3: rule,F: rule > rule,X2: rule] :
( ( B3
= ( F @ X2 ) )
=> ( member_rule @ B3 @ ( image_rule_rule @ F @ top_top_set_rule ) ) ) ).
% range_eqI
thf(fact_545_range__eqI,axiom,
! [B3: $o,F: rule > $o,X2: rule] :
( ( B3
= ( F @ X2 ) )
=> ( member_o @ B3 @ ( image_rule_o @ F @ top_top_set_rule ) ) ) ).
% range_eqI
thf(fact_546_range__eqI,axiom,
! [B3: produc1996495991257130529ist_fm,F: $o > produc1996495991257130529ist_fm,X2: $o] :
( ( B3
= ( F @ X2 ) )
=> ( member8102475879199740618ist_fm @ B3 @ ( image_3689732601364923150ist_fm @ F @ top_top_set_o ) ) ) ).
% range_eqI
thf(fact_547_range__eqI,axiom,
! [B3: rule,F: $o > rule,X2: $o] :
( ( B3
= ( F @ X2 ) )
=> ( member_rule @ B3 @ ( image_o_rule @ F @ top_top_set_o ) ) ) ).
% range_eqI
thf(fact_548_range__eqI,axiom,
! [B3: $o,F: $o > $o,X2: $o] :
( ( B3
= ( F @ X2 ) )
=> ( member_o @ B3 @ ( image_o_o @ F @ top_top_set_o ) ) ) ).
% range_eqI
thf(fact_549_rangeI,axiom,
! [F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,X2: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ ( F @ X2 ) @ ( image_6982201141141708101ist_fm @ F @ top_to728902761071950545ist_fm ) ) ).
% rangeI
thf(fact_550_rangeI,axiom,
! [F: produc1996495991257130529ist_fm > rule,X2: produc1996495991257130529ist_fm] : ( member_rule @ ( F @ X2 ) @ ( image_368924427985557793m_rule @ F @ top_to728902761071950545ist_fm ) ) ).
% rangeI
thf(fact_551_rangeI,axiom,
! [F: produc1996495991257130529ist_fm > $o,X2: produc1996495991257130529ist_fm] : ( member_o @ ( F @ X2 ) @ ( image_3428966142334963388t_fm_o @ F @ top_to728902761071950545ist_fm ) ) ).
% rangeI
thf(fact_552_rangeI,axiom,
! [F: rule > produc1996495991257130529ist_fm,X2: rule] : ( member8102475879199740618ist_fm @ ( F @ X2 ) @ ( image_6941212235319914453ist_fm @ F @ top_top_set_rule ) ) ).
% rangeI
thf(fact_553_rangeI,axiom,
! [F: rule > rule,X2: rule] : ( member_rule @ ( F @ X2 ) @ ( image_rule_rule @ F @ top_top_set_rule ) ) ).
% rangeI
thf(fact_554_rangeI,axiom,
! [F: rule > $o,X2: rule] : ( member_o @ ( F @ X2 ) @ ( image_rule_o @ F @ top_top_set_rule ) ) ).
% rangeI
thf(fact_555_rangeI,axiom,
! [F: $o > produc1996495991257130529ist_fm,X2: $o] : ( member8102475879199740618ist_fm @ ( F @ X2 ) @ ( image_3689732601364923150ist_fm @ F @ top_top_set_o ) ) ).
% rangeI
thf(fact_556_rangeI,axiom,
! [F: $o > rule,X2: $o] : ( member_rule @ ( F @ X2 ) @ ( image_o_rule @ F @ top_top_set_o ) ) ).
% rangeI
thf(fact_557_rangeI,axiom,
! [F: $o > $o,X2: $o] : ( member_o @ ( F @ X2 ) @ ( image_o_o @ F @ top_top_set_o ) ) ).
% rangeI
thf(fact_558_trans__onD,axiom,
! [A: set_Pr7058068377845519745ist_fm,R: set_Pr5209464977148653383ist_fm,X2: produc1996495991257130529ist_fm,Y3: produc1996495991257130529ist_fm,Z2: produc1996495991257130529ist_fm] :
( ( trans_4095360997890102209ist_fm @ A @ R )
=> ( ( member8102475879199740618ist_fm @ X2 @ A )
=> ( ( member8102475879199740618ist_fm @ Y3 @ A )
=> ( ( member8102475879199740618ist_fm @ Z2 @ A )
=> ( ( member60238275747228048ist_fm @ ( produc2592314732767339223ist_fm @ X2 @ Y3 ) @ R )
=> ( ( member60238275747228048ist_fm @ ( produc2592314732767339223ist_fm @ Y3 @ Z2 ) @ R )
=> ( member60238275747228048ist_fm @ ( produc2592314732767339223ist_fm @ X2 @ Z2 ) @ R ) ) ) ) ) ) ) ).
% trans_onD
thf(fact_559_trans__onD,axiom,
! [A: set_rule,R: set_Pr6543236023296505803e_rule,X2: rule,Y3: rule,Z2: rule] :
( ( trans_on_rule @ A @ R )
=> ( ( member_rule @ X2 @ A )
=> ( ( member_rule @ Y3 @ A )
=> ( ( member_rule @ Z2 @ A )
=> ( ( member3753523862196722196e_rule @ ( produc1769597910952761123e_rule @ X2 @ Y3 ) @ R )
=> ( ( member3753523862196722196e_rule @ ( produc1769597910952761123e_rule @ Y3 @ Z2 ) @ R )
=> ( member3753523862196722196e_rule @ ( produc1769597910952761123e_rule @ X2 @ Z2 ) @ R ) ) ) ) ) ) ) ).
% trans_onD
thf(fact_560_trans__onD,axiom,
! [A: set_o,R: set_Product_prod_o_o,X2: $o,Y3: $o,Z2: $o] :
( ( trans_on_o @ A @ R )
=> ( ( member_o @ X2 @ A )
=> ( ( member_o @ Y3 @ A )
=> ( ( member_o @ Z2 @ A )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y3 ) @ R )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y3 @ Z2 ) @ R )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Z2 ) @ R ) ) ) ) ) ) ) ).
% trans_onD
thf(fact_561_trans__onD,axiom,
! [A: set_list_fm,R: set_Pr7058068377845519745ist_fm,X2: list_fm,Y3: list_fm,Z2: list_fm] :
( ( trans_on_list_fm @ A @ R )
=> ( ( member_list_fm @ X2 @ A )
=> ( ( member_list_fm @ Y3 @ A )
=> ( ( member_list_fm @ Z2 @ A )
=> ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X2 @ Y3 ) @ R )
=> ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ Y3 @ Z2 ) @ R )
=> ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X2 @ Z2 ) @ R ) ) ) ) ) ) ) ).
% trans_onD
thf(fact_562_trans__onI,axiom,
! [A: set_Pr7058068377845519745ist_fm,R: set_Pr5209464977148653383ist_fm] :
( ! [X3: produc1996495991257130529ist_fm,Y2: produc1996495991257130529ist_fm,Z3: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ X3 @ A )
=> ( ( member8102475879199740618ist_fm @ Y2 @ A )
=> ( ( member8102475879199740618ist_fm @ Z3 @ A )
=> ( ( member60238275747228048ist_fm @ ( produc2592314732767339223ist_fm @ X3 @ Y2 ) @ R )
=> ( ( member60238275747228048ist_fm @ ( produc2592314732767339223ist_fm @ Y2 @ Z3 ) @ R )
=> ( member60238275747228048ist_fm @ ( produc2592314732767339223ist_fm @ X3 @ Z3 ) @ R ) ) ) ) ) )
=> ( trans_4095360997890102209ist_fm @ A @ R ) ) ).
% trans_onI
thf(fact_563_trans__onI,axiom,
! [A: set_rule,R: set_Pr6543236023296505803e_rule] :
( ! [X3: rule,Y2: rule,Z3: rule] :
( ( member_rule @ X3 @ A )
=> ( ( member_rule @ Y2 @ A )
=> ( ( member_rule @ Z3 @ A )
=> ( ( member3753523862196722196e_rule @ ( produc1769597910952761123e_rule @ X3 @ Y2 ) @ R )
=> ( ( member3753523862196722196e_rule @ ( produc1769597910952761123e_rule @ Y2 @ Z3 ) @ R )
=> ( member3753523862196722196e_rule @ ( produc1769597910952761123e_rule @ X3 @ Z3 ) @ R ) ) ) ) ) )
=> ( trans_on_rule @ A @ R ) ) ).
% trans_onI
thf(fact_564_trans__onI,axiom,
! [A: set_o,R: set_Product_prod_o_o] :
( ! [X3: $o,Y2: $o,Z3: $o] :
( ( member_o @ X3 @ A )
=> ( ( member_o @ Y2 @ A )
=> ( ( member_o @ Z3 @ A )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ Y2 ) @ R )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y2 @ Z3 ) @ R )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ Z3 ) @ R ) ) ) ) ) )
=> ( trans_on_o @ A @ R ) ) ).
% trans_onI
thf(fact_565_trans__onI,axiom,
! [A: set_list_fm,R: set_Pr7058068377845519745ist_fm] :
( ! [X3: list_fm,Y2: list_fm,Z3: list_fm] :
( ( member_list_fm @ X3 @ A )
=> ( ( member_list_fm @ Y2 @ A )
=> ( ( member_list_fm @ Z3 @ A )
=> ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X3 @ Y2 ) @ R )
=> ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ Y2 @ Z3 ) @ R )
=> ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X3 @ Z3 ) @ R ) ) ) ) ) )
=> ( trans_on_list_fm @ A @ R ) ) ).
% trans_onI
thf(fact_566_trans__on__def,axiom,
( trans_on_list_fm
= ( ^ [A5: set_list_fm,R3: set_Pr7058068377845519745ist_fm] :
! [X: list_fm] :
( ( member_list_fm @ X @ A5 )
=> ! [Y4: list_fm] :
( ( member_list_fm @ Y4 @ A5 )
=> ! [Z4: list_fm] :
( ( member_list_fm @ Z4 @ A5 )
=> ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X @ Y4 ) @ R3 )
=> ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ Y4 @ Z4 ) @ R3 )
=> ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X @ Z4 ) @ R3 ) ) ) ) ) ) ) ) ).
% trans_on_def
thf(fact_567_Domain_Ocases,axiom,
! [A3: list_fm,R: set_Pr7058068377845519745ist_fm] :
( ( member_list_fm @ A3 @ ( domain2949914093644414030ist_fm @ R ) )
=> ~ ! [B7: list_fm] :
~ ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ A3 @ B7 ) @ R ) ) ).
% Domain.cases
thf(fact_568_Domain_Osimps,axiom,
! [A3: list_fm,R: set_Pr7058068377845519745ist_fm] :
( ( member_list_fm @ A3 @ ( domain2949914093644414030ist_fm @ R ) )
= ( ? [A4: list_fm,B8: list_fm] :
( ( A3 = A4 )
& ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ A4 @ B8 ) @ R ) ) ) ) ).
% Domain.simps
thf(fact_569_Domain_ODomainI,axiom,
! [A3: list_fm,B3: list_fm,R: set_Pr7058068377845519745ist_fm] :
( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ A3 @ B3 ) @ R )
=> ( member_list_fm @ A3 @ ( domain2949914093644414030ist_fm @ R ) ) ) ).
% Domain.DomainI
thf(fact_570_DomainE,axiom,
! [A3: list_fm,R: set_Pr7058068377845519745ist_fm] :
( ( member_list_fm @ A3 @ ( domain2949914093644414030ist_fm @ R ) )
=> ~ ! [B7: list_fm] :
~ ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ A3 @ B7 ) @ R ) ) ).
% DomainE
thf(fact_571_Domain__iff,axiom,
! [A3: list_fm,R: set_Pr7058068377845519745ist_fm] :
( ( member_list_fm @ A3 @ ( domain2949914093644414030ist_fm @ R ) )
= ( ? [Y4: list_fm] : ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ A3 @ Y4 ) @ R ) ) ) ).
% Domain_iff
thf(fact_572_Range_Ocases,axiom,
! [A3: list_fm,R: set_Pr7058068377845519745ist_fm] :
( ( member_list_fm @ A3 @ ( range_8771097359499353911ist_fm @ R ) )
=> ~ ! [A7: list_fm] :
~ ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ A7 @ A3 ) @ R ) ) ).
% Range.cases
thf(fact_573_Range_Osimps,axiom,
! [A3: list_fm,R: set_Pr7058068377845519745ist_fm] :
( ( member_list_fm @ A3 @ ( range_8771097359499353911ist_fm @ R ) )
= ( ? [A4: list_fm,B8: list_fm] :
( ( A3 = B8 )
& ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ A4 @ B8 ) @ R ) ) ) ) ).
% Range.simps
thf(fact_574_Range_Ointros,axiom,
! [A3: list_fm,B3: list_fm,R: set_Pr7058068377845519745ist_fm] :
( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ A3 @ B3 ) @ R )
=> ( member_list_fm @ B3 @ ( range_8771097359499353911ist_fm @ R ) ) ) ).
% Range.intros
thf(fact_575_RangeE,axiom,
! [B3: list_fm,R: set_Pr7058068377845519745ist_fm] :
( ( member_list_fm @ B3 @ ( range_8771097359499353911ist_fm @ R ) )
=> ~ ! [A7: list_fm] :
~ ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ A7 @ B3 ) @ R ) ) ).
% RangeE
thf(fact_576_Range__iff,axiom,
! [A3: list_fm,R: set_Pr7058068377845519745ist_fm] :
( ( member_list_fm @ A3 @ ( range_8771097359499353911ist_fm @ R ) )
= ( ? [Y4: list_fm] : ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ Y4 @ A3 ) @ R ) ) ) ).
% Range_iff
thf(fact_577_rangeE,axiom,
! [B3: produc1996495991257130529ist_fm,F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ B3 @ ( image_6982201141141708101ist_fm @ F @ top_to728902761071950545ist_fm ) )
=> ~ ! [X3: produc1996495991257130529ist_fm] :
( B3
!= ( F @ X3 ) ) ) ).
% rangeE
thf(fact_578_rangeE,axiom,
! [B3: rule,F: produc1996495991257130529ist_fm > rule] :
( ( member_rule @ B3 @ ( image_368924427985557793m_rule @ F @ top_to728902761071950545ist_fm ) )
=> ~ ! [X3: produc1996495991257130529ist_fm] :
( B3
!= ( F @ X3 ) ) ) ).
% rangeE
thf(fact_579_rangeE,axiom,
! [B3: $o,F: produc1996495991257130529ist_fm > $o] :
( ( member_o @ B3 @ ( image_3428966142334963388t_fm_o @ F @ top_to728902761071950545ist_fm ) )
=> ~ ! [X3: produc1996495991257130529ist_fm] :
( B3
= ( ~ ( F @ X3 ) ) ) ) ).
% rangeE
thf(fact_580_rangeE,axiom,
! [B3: produc1996495991257130529ist_fm,F: rule > produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ B3 @ ( image_6941212235319914453ist_fm @ F @ top_top_set_rule ) )
=> ~ ! [X3: rule] :
( B3
!= ( F @ X3 ) ) ) ).
% rangeE
thf(fact_581_rangeE,axiom,
! [B3: rule,F: rule > rule] :
( ( member_rule @ B3 @ ( image_rule_rule @ F @ top_top_set_rule ) )
=> ~ ! [X3: rule] :
( B3
!= ( F @ X3 ) ) ) ).
% rangeE
thf(fact_582_rangeE,axiom,
! [B3: $o,F: rule > $o] :
( ( member_o @ B3 @ ( image_rule_o @ F @ top_top_set_rule ) )
=> ~ ! [X3: rule] :
( B3
= ( ~ ( F @ X3 ) ) ) ) ).
% rangeE
thf(fact_583_rangeE,axiom,
! [B3: produc1996495991257130529ist_fm,F: $o > produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ B3 @ ( image_3689732601364923150ist_fm @ F @ top_top_set_o ) )
=> ~ ! [X3: $o] :
( B3
!= ( F @ X3 ) ) ) ).
% rangeE
thf(fact_584_rangeE,axiom,
! [B3: rule,F: $o > rule] :
( ( member_rule @ B3 @ ( image_o_rule @ F @ top_top_set_o ) )
=> ~ ! [X3: $o] :
( B3
!= ( F @ X3 ) ) ) ).
% rangeE
thf(fact_585_rangeE,axiom,
! [B3: $o,F: $o > $o] :
( ( member_o @ B3 @ ( image_o_o @ F @ top_top_set_o ) )
=> ~ ! [X3: $o] :
( B3
= ( ~ ( F @ X3 ) ) ) ) ).
% rangeE
thf(fact_586_transI,axiom,
! [R: set_Pr7058068377845519745ist_fm] :
( ! [X3: list_fm,Y2: list_fm,Z3: list_fm] :
( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X3 @ Y2 ) @ R )
=> ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ Y2 @ Z3 ) @ R )
=> ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X3 @ Z3 ) @ R ) ) )
=> ( trans_on_list_fm @ top_top_set_list_fm @ R ) ) ).
% transI
thf(fact_587_transI,axiom,
! [R: set_Pr5209464977148653383ist_fm] :
( ! [X3: produc1996495991257130529ist_fm,Y2: produc1996495991257130529ist_fm,Z3: produc1996495991257130529ist_fm] :
( ( member60238275747228048ist_fm @ ( produc2592314732767339223ist_fm @ X3 @ Y2 ) @ R )
=> ( ( member60238275747228048ist_fm @ ( produc2592314732767339223ist_fm @ Y2 @ Z3 ) @ R )
=> ( member60238275747228048ist_fm @ ( produc2592314732767339223ist_fm @ X3 @ Z3 ) @ R ) ) )
=> ( trans_4095360997890102209ist_fm @ top_to728902761071950545ist_fm @ R ) ) ).
% transI
thf(fact_588_transI,axiom,
! [R: set_Pr6543236023296505803e_rule] :
( ! [X3: rule,Y2: rule,Z3: rule] :
( ( member3753523862196722196e_rule @ ( produc1769597910952761123e_rule @ X3 @ Y2 ) @ R )
=> ( ( member3753523862196722196e_rule @ ( produc1769597910952761123e_rule @ Y2 @ Z3 ) @ R )
=> ( member3753523862196722196e_rule @ ( produc1769597910952761123e_rule @ X3 @ Z3 ) @ R ) ) )
=> ( trans_on_rule @ top_top_set_rule @ R ) ) ).
% transI
thf(fact_589_transI,axiom,
! [R: set_Product_prod_o_o] :
( ! [X3: $o,Y2: $o,Z3: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ Y2 ) @ R )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y2 @ Z3 ) @ R )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ Z3 ) @ R ) ) )
=> ( trans_on_o @ top_top_set_o @ R ) ) ).
% transI
thf(fact_590_transE,axiom,
! [R: set_Pr7058068377845519745ist_fm,X2: list_fm,Y3: list_fm,Z2: list_fm] :
( ( trans_on_list_fm @ top_top_set_list_fm @ R )
=> ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X2 @ Y3 ) @ R )
=> ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ Y3 @ Z2 ) @ R )
=> ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X2 @ Z2 ) @ R ) ) ) ) ).
% transE
thf(fact_591_transE,axiom,
! [R: set_Pr5209464977148653383ist_fm,X2: produc1996495991257130529ist_fm,Y3: produc1996495991257130529ist_fm,Z2: produc1996495991257130529ist_fm] :
( ( trans_4095360997890102209ist_fm @ top_to728902761071950545ist_fm @ R )
=> ( ( member60238275747228048ist_fm @ ( produc2592314732767339223ist_fm @ X2 @ Y3 ) @ R )
=> ( ( member60238275747228048ist_fm @ ( produc2592314732767339223ist_fm @ Y3 @ Z2 ) @ R )
=> ( member60238275747228048ist_fm @ ( produc2592314732767339223ist_fm @ X2 @ Z2 ) @ R ) ) ) ) ).
% transE
thf(fact_592_transE,axiom,
! [R: set_Pr6543236023296505803e_rule,X2: rule,Y3: rule,Z2: rule] :
( ( trans_on_rule @ top_top_set_rule @ R )
=> ( ( member3753523862196722196e_rule @ ( produc1769597910952761123e_rule @ X2 @ Y3 ) @ R )
=> ( ( member3753523862196722196e_rule @ ( produc1769597910952761123e_rule @ Y3 @ Z2 ) @ R )
=> ( member3753523862196722196e_rule @ ( produc1769597910952761123e_rule @ X2 @ Z2 ) @ R ) ) ) ) ).
% transE
thf(fact_593_transE,axiom,
! [R: set_Product_prod_o_o,X2: $o,Y3: $o,Z2: $o] :
( ( trans_on_o @ top_top_set_o @ R )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y3 ) @ R )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y3 @ Z2 ) @ R )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Z2 ) @ R ) ) ) ) ).
% transE
thf(fact_594_transD,axiom,
! [R: set_Pr7058068377845519745ist_fm,X2: list_fm,Y3: list_fm,Z2: list_fm] :
( ( trans_on_list_fm @ top_top_set_list_fm @ R )
=> ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X2 @ Y3 ) @ R )
=> ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ Y3 @ Z2 ) @ R )
=> ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X2 @ Z2 ) @ R ) ) ) ) ).
% transD
thf(fact_595_transD,axiom,
! [R: set_Pr5209464977148653383ist_fm,X2: produc1996495991257130529ist_fm,Y3: produc1996495991257130529ist_fm,Z2: produc1996495991257130529ist_fm] :
( ( trans_4095360997890102209ist_fm @ top_to728902761071950545ist_fm @ R )
=> ( ( member60238275747228048ist_fm @ ( produc2592314732767339223ist_fm @ X2 @ Y3 ) @ R )
=> ( ( member60238275747228048ist_fm @ ( produc2592314732767339223ist_fm @ Y3 @ Z2 ) @ R )
=> ( member60238275747228048ist_fm @ ( produc2592314732767339223ist_fm @ X2 @ Z2 ) @ R ) ) ) ) ).
% transD
thf(fact_596_transD,axiom,
! [R: set_Pr6543236023296505803e_rule,X2: rule,Y3: rule,Z2: rule] :
( ( trans_on_rule @ top_top_set_rule @ R )
=> ( ( member3753523862196722196e_rule @ ( produc1769597910952761123e_rule @ X2 @ Y3 ) @ R )
=> ( ( member3753523862196722196e_rule @ ( produc1769597910952761123e_rule @ Y3 @ Z2 ) @ R )
=> ( member3753523862196722196e_rule @ ( produc1769597910952761123e_rule @ X2 @ Z2 ) @ R ) ) ) ) ).
% transD
thf(fact_597_transD,axiom,
! [R: set_Product_prod_o_o,X2: $o,Y3: $o,Z2: $o] :
( ( trans_on_o @ top_top_set_o @ R )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y3 ) @ R )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y3 @ Z2 ) @ R )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Z2 ) @ R ) ) ) ) ).
% transD
thf(fact_598_None__notin__image__Some,axiom,
! [A: set_fs7534060674568567527ist_fm] :
~ ( member1838329594270681206ist_fm @ none_f2701453242279119720ist_fm @ ( image_4775136921032825739ist_fm @ some_f2162114139285431020ist_fm @ A ) ) ).
% None_notin_image_Some
thf(fact_599_trans__empty,axiom,
trans_on_list_fm @ top_top_set_list_fm @ bot_bo4530011010606209517ist_fm ).
% trans_empty
thf(fact_600_trans__empty,axiom,
trans_4095360997890102209ist_fm @ top_to728902761071950545ist_fm @ bot_bo7330262581962992051ist_fm ).
% trans_empty
thf(fact_601_trans__empty,axiom,
trans_on_rule @ top_top_set_rule @ bot_bo9047865283180780599e_rule ).
% trans_empty
thf(fact_602_trans__empty,axiom,
trans_on_o @ top_top_set_o @ bot_bo7073875226086086771od_o_o ).
% trans_empty
thf(fact_603_image__constant__conv,axiom,
! [A: set_rule,C: rule] :
( ( ( A = bot_bot_set_rule )
=> ( ( image_rule_rule
@ ^ [X: rule] : C
@ A )
= bot_bot_set_rule ) )
& ( ( A != bot_bot_set_rule )
=> ( ( image_rule_rule
@ ^ [X: rule] : C
@ A )
= ( insert_rule @ C @ bot_bot_set_rule ) ) ) ) ).
% image_constant_conv
thf(fact_604_image__constant__conv,axiom,
! [A: set_rule,C: $o] :
( ( ( A = bot_bot_set_rule )
=> ( ( image_rule_o
@ ^ [X: rule] : C
@ A )
= bot_bot_set_o ) )
& ( ( A != bot_bot_set_rule )
=> ( ( image_rule_o
@ ^ [X: rule] : C
@ A )
= ( insert_o @ C @ bot_bot_set_o ) ) ) ) ).
% image_constant_conv
thf(fact_605_image__constant__conv,axiom,
! [A: set_rule,C: produc1996495991257130529ist_fm] :
( ( ( A = bot_bot_set_rule )
=> ( ( image_6941212235319914453ist_fm
@ ^ [X: rule] : C
@ A )
= bot_bo4530011010606209517ist_fm ) )
& ( ( A != bot_bot_set_rule )
=> ( ( image_6941212235319914453ist_fm
@ ^ [X: rule] : C
@ A )
= ( insert589278167076416497ist_fm @ C @ bot_bo4530011010606209517ist_fm ) ) ) ) ).
% image_constant_conv
thf(fact_606_image__constant__conv,axiom,
! [A: set_o,C: rule] :
( ( ( A = bot_bot_set_o )
=> ( ( image_o_rule
@ ^ [X: $o] : C
@ A )
= bot_bot_set_rule ) )
& ( ( A != bot_bot_set_o )
=> ( ( image_o_rule
@ ^ [X: $o] : C
@ A )
= ( insert_rule @ C @ bot_bot_set_rule ) ) ) ) ).
% image_constant_conv
thf(fact_607_image__constant__conv,axiom,
! [A: set_o,C: $o] :
( ( ( A = bot_bot_set_o )
=> ( ( image_o_o
@ ^ [X: $o] : C
@ A )
= bot_bot_set_o ) )
& ( ( A != bot_bot_set_o )
=> ( ( image_o_o
@ ^ [X: $o] : C
@ A )
= ( insert_o @ C @ bot_bot_set_o ) ) ) ) ).
% image_constant_conv
thf(fact_608_image__constant__conv,axiom,
! [A: set_o,C: produc1996495991257130529ist_fm] :
( ( ( A = bot_bot_set_o )
=> ( ( image_3689732601364923150ist_fm
@ ^ [X: $o] : C
@ A )
= bot_bo4530011010606209517ist_fm ) )
& ( ( A != bot_bot_set_o )
=> ( ( image_3689732601364923150ist_fm
@ ^ [X: $o] : C
@ A )
= ( insert589278167076416497ist_fm @ C @ bot_bo4530011010606209517ist_fm ) ) ) ) ).
% image_constant_conv
thf(fact_609_image__constant__conv,axiom,
! [A: set_Pr7058068377845519745ist_fm,C: rule] :
( ( ( A = bot_bo4530011010606209517ist_fm )
=> ( ( image_368924427985557793m_rule
@ ^ [X: produc1996495991257130529ist_fm] : C
@ A )
= bot_bot_set_rule ) )
& ( ( A != bot_bo4530011010606209517ist_fm )
=> ( ( image_368924427985557793m_rule
@ ^ [X: produc1996495991257130529ist_fm] : C
@ A )
= ( insert_rule @ C @ bot_bot_set_rule ) ) ) ) ).
% image_constant_conv
thf(fact_610_image__constant__conv,axiom,
! [A: set_Pr7058068377845519745ist_fm,C: $o] :
( ( ( A = bot_bo4530011010606209517ist_fm )
=> ( ( image_3428966142334963388t_fm_o
@ ^ [X: produc1996495991257130529ist_fm] : C
@ A )
= bot_bot_set_o ) )
& ( ( A != bot_bo4530011010606209517ist_fm )
=> ( ( image_3428966142334963388t_fm_o
@ ^ [X: produc1996495991257130529ist_fm] : C
@ A )
= ( insert_o @ C @ bot_bot_set_o ) ) ) ) ).
% image_constant_conv
thf(fact_611_image__constant__conv,axiom,
! [A: set_Pr7058068377845519745ist_fm,C: produc1996495991257130529ist_fm] :
( ( ( A = bot_bo4530011010606209517ist_fm )
=> ( ( image_6982201141141708101ist_fm
@ ^ [X: produc1996495991257130529ist_fm] : C
@ A )
= bot_bo4530011010606209517ist_fm ) )
& ( ( A != bot_bo4530011010606209517ist_fm )
=> ( ( image_6982201141141708101ist_fm
@ ^ [X: produc1996495991257130529ist_fm] : C
@ A )
= ( insert589278167076416497ist_fm @ C @ bot_bo4530011010606209517ist_fm ) ) ) ) ).
% image_constant_conv
thf(fact_612_image__constant,axiom,
! [X2: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm,C: rule] :
( ( member8102475879199740618ist_fm @ X2 @ A )
=> ( ( image_368924427985557793m_rule
@ ^ [X: produc1996495991257130529ist_fm] : C
@ A )
= ( insert_rule @ C @ bot_bot_set_rule ) ) ) ).
% image_constant
thf(fact_613_image__constant,axiom,
! [X2: rule,A: set_rule,C: rule] :
( ( member_rule @ X2 @ A )
=> ( ( image_rule_rule
@ ^ [X: rule] : C
@ A )
= ( insert_rule @ C @ bot_bot_set_rule ) ) ) ).
% image_constant
thf(fact_614_image__constant,axiom,
! [X2: $o,A: set_o,C: rule] :
( ( member_o @ X2 @ A )
=> ( ( image_o_rule
@ ^ [X: $o] : C
@ A )
= ( insert_rule @ C @ bot_bot_set_rule ) ) ) ).
% image_constant
thf(fact_615_image__constant,axiom,
! [X2: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm,C: $o] :
( ( member8102475879199740618ist_fm @ X2 @ A )
=> ( ( image_3428966142334963388t_fm_o
@ ^ [X: produc1996495991257130529ist_fm] : C
@ A )
= ( insert_o @ C @ bot_bot_set_o ) ) ) ).
% image_constant
thf(fact_616_image__constant,axiom,
! [X2: rule,A: set_rule,C: $o] :
( ( member_rule @ X2 @ A )
=> ( ( image_rule_o
@ ^ [X: rule] : C
@ A )
= ( insert_o @ C @ bot_bot_set_o ) ) ) ).
% image_constant
thf(fact_617_image__constant,axiom,
! [X2: $o,A: set_o,C: $o] :
( ( member_o @ X2 @ A )
=> ( ( image_o_o
@ ^ [X: $o] : C
@ A )
= ( insert_o @ C @ bot_bot_set_o ) ) ) ).
% image_constant
thf(fact_618_image__constant,axiom,
! [X2: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm,C: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ X2 @ A )
=> ( ( image_6982201141141708101ist_fm
@ ^ [X: produc1996495991257130529ist_fm] : C
@ A )
= ( insert589278167076416497ist_fm @ C @ bot_bo4530011010606209517ist_fm ) ) ) ).
% image_constant
thf(fact_619_image__constant,axiom,
! [X2: rule,A: set_rule,C: produc1996495991257130529ist_fm] :
( ( member_rule @ X2 @ A )
=> ( ( image_6941212235319914453ist_fm
@ ^ [X: rule] : C
@ A )
= ( insert589278167076416497ist_fm @ C @ bot_bo4530011010606209517ist_fm ) ) ) ).
% image_constant
thf(fact_620_image__constant,axiom,
! [X2: $o,A: set_o,C: produc1996495991257130529ist_fm] :
( ( member_o @ X2 @ A )
=> ( ( image_3689732601364923150ist_fm
@ ^ [X: $o] : C
@ A )
= ( insert589278167076416497ist_fm @ C @ bot_bo4530011010606209517ist_fm ) ) ) ).
% image_constant
thf(fact_621_Domain__empty__iff,axiom,
! [R: set_Pr7058068377845519745ist_fm] :
( ( ( domain2949914093644414030ist_fm @ R )
= bot_bot_set_list_fm )
= ( R = bot_bo4530011010606209517ist_fm ) ) ).
% Domain_empty_iff
thf(fact_622_Range__empty__iff,axiom,
! [R: set_Pr7058068377845519745ist_fm] :
( ( ( range_8771097359499353911ist_fm @ R )
= bot_bot_set_list_fm )
= ( R = bot_bo4530011010606209517ist_fm ) ) ).
% Range_empty_iff
thf(fact_623_range__eq__singletonD,axiom,
! [F: produc1996495991257130529ist_fm > rule,A3: rule,X2: produc1996495991257130529ist_fm] :
( ( ( image_368924427985557793m_rule @ F @ top_to728902761071950545ist_fm )
= ( insert_rule @ A3 @ bot_bot_set_rule ) )
=> ( ( F @ X2 )
= A3 ) ) ).
% range_eq_singletonD
thf(fact_624_range__eq__singletonD,axiom,
! [F: produc1996495991257130529ist_fm > $o,A3: $o,X2: produc1996495991257130529ist_fm] :
( ( ( image_3428966142334963388t_fm_o @ F @ top_to728902761071950545ist_fm )
= ( insert_o @ A3 @ bot_bot_set_o ) )
=> ( ( F @ X2 )
= A3 ) ) ).
% range_eq_singletonD
thf(fact_625_range__eq__singletonD,axiom,
! [F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,A3: produc1996495991257130529ist_fm,X2: produc1996495991257130529ist_fm] :
( ( ( image_6982201141141708101ist_fm @ F @ top_to728902761071950545ist_fm )
= ( insert589278167076416497ist_fm @ A3 @ bot_bo4530011010606209517ist_fm ) )
=> ( ( F @ X2 )
= A3 ) ) ).
% range_eq_singletonD
thf(fact_626_range__eq__singletonD,axiom,
! [F: rule > rule,A3: rule,X2: rule] :
( ( ( image_rule_rule @ F @ top_top_set_rule )
= ( insert_rule @ A3 @ bot_bot_set_rule ) )
=> ( ( F @ X2 )
= A3 ) ) ).
% range_eq_singletonD
thf(fact_627_range__eq__singletonD,axiom,
! [F: rule > $o,A3: $o,X2: rule] :
( ( ( image_rule_o @ F @ top_top_set_rule )
= ( insert_o @ A3 @ bot_bot_set_o ) )
=> ( ( F @ X2 )
= A3 ) ) ).
% range_eq_singletonD
thf(fact_628_range__eq__singletonD,axiom,
! [F: rule > produc1996495991257130529ist_fm,A3: produc1996495991257130529ist_fm,X2: rule] :
( ( ( image_6941212235319914453ist_fm @ F @ top_top_set_rule )
= ( insert589278167076416497ist_fm @ A3 @ bot_bo4530011010606209517ist_fm ) )
=> ( ( F @ X2 )
= A3 ) ) ).
% range_eq_singletonD
thf(fact_629_range__eq__singletonD,axiom,
! [F: $o > rule,A3: rule,X2: $o] :
( ( ( image_o_rule @ F @ top_top_set_o )
= ( insert_rule @ A3 @ bot_bot_set_rule ) )
=> ( ( F @ X2 )
= A3 ) ) ).
% range_eq_singletonD
thf(fact_630_range__eq__singletonD,axiom,
! [F: $o > $o,A3: $o,X2: $o] :
( ( ( image_o_o @ F @ top_top_set_o )
= ( insert_o @ A3 @ bot_bot_set_o ) )
=> ( ( F @ X2 )
= A3 ) ) ).
% range_eq_singletonD
thf(fact_631_range__eq__singletonD,axiom,
! [F: $o > produc1996495991257130529ist_fm,A3: produc1996495991257130529ist_fm,X2: $o] :
( ( ( image_3689732601364923150ist_fm @ F @ top_top_set_o )
= ( insert589278167076416497ist_fm @ A3 @ bot_bo4530011010606209517ist_fm ) )
=> ( ( F @ X2 )
= A3 ) ) ).
% range_eq_singletonD
thf(fact_632_notin__range__Some,axiom,
! [X2: option6029102822600979917ist_fm] :
( ( ~ ( member1838329594270681206ist_fm @ X2 @ ( image_4775136921032825739ist_fm @ some_f2162114139285431020ist_fm @ top_to6886830657606236215ist_fm ) ) )
= ( X2 = none_f2701453242279119720ist_fm ) ) ).
% notin_range_Some
thf(fact_633_notin__range__Some,axiom,
! [X2: option4065869692384770919ist_fm] :
( ( ~ ( member772309659248424592ist_fm @ X2 @ ( image_4446678087167650955ist_fm @ some_P7346312016712434310ist_fm @ top_to728902761071950545ist_fm ) ) )
= ( X2 = none_P1680826464017717506ist_fm ) ) ).
% notin_range_Some
thf(fact_634_notin__range__Some,axiom,
! [X2: option_rule] :
( ( ~ ( member_option_rule @ X2 @ ( image_766640639675677281n_rule @ some_rule @ top_top_set_rule ) ) )
= ( X2 = none_rule ) ) ).
% notin_range_Some
thf(fact_635_notin__range__Some,axiom,
! [X2: option_o] :
( ( ~ ( member_option_o @ X2 @ ( image_o_option_o @ some_o @ top_top_set_o ) ) )
= ( X2 = none_o ) ) ).
% notin_range_Some
thf(fact_636_lnear__order__on__empty,axiom,
order_4446669488389765227ist_fm @ bot_bot_set_list_fm @ bot_bo4530011010606209517ist_fm ).
% lnear_order_on_empty
thf(fact_637_lnear__order__on__empty,axiom,
order_3615580131159368656n_rule @ bot_bot_set_rule @ bot_bo9047865283180780599e_rule ).
% lnear_order_on_empty
thf(fact_638_lnear__order__on__empty,axiom,
order_6238756238976269133r_on_o @ bot_bot_set_o @ bot_bo7073875226086086771od_o_o ).
% lnear_order_on_empty
thf(fact_639_lnear__order__on__empty,axiom,
order_193637769215337174ist_fm @ bot_bo4530011010606209517ist_fm @ bot_bo7330262581962992051ist_fm ).
% lnear_order_on_empty
thf(fact_640_Some__image__these__eq,axiom,
! [A: set_op6310664071589870893ist_fm] :
( ( image_4775136921032825739ist_fm @ some_f2162114139285431020ist_fm @ ( these_4231445979625565508ist_fm @ A ) )
= ( collec8886999257175080888ist_fm
@ ^ [X: option6029102822600979917ist_fm] :
( ( member1838329594270681206ist_fm @ X @ A )
& ( X != none_f2701453242279119720ist_fm ) ) ) ) ).
% Some_image_these_eq
thf(fact_641_surjD,axiom,
! [F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,Y3: produc1996495991257130529ist_fm] :
( ( ( image_6982201141141708101ist_fm @ F @ top_to728902761071950545ist_fm )
= top_to728902761071950545ist_fm )
=> ? [X3: produc1996495991257130529ist_fm] :
( Y3
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_642_surjD,axiom,
! [F: produc1996495991257130529ist_fm > rule,Y3: rule] :
( ( ( image_368924427985557793m_rule @ F @ top_to728902761071950545ist_fm )
= top_top_set_rule )
=> ? [X3: produc1996495991257130529ist_fm] :
( Y3
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_643_surjD,axiom,
! [F: produc1996495991257130529ist_fm > $o,Y3: $o] :
( ( ( image_3428966142334963388t_fm_o @ F @ top_to728902761071950545ist_fm )
= top_top_set_o )
=> ? [X3: produc1996495991257130529ist_fm] :
( Y3
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_644_surjD,axiom,
! [F: rule > produc1996495991257130529ist_fm,Y3: produc1996495991257130529ist_fm] :
( ( ( image_6941212235319914453ist_fm @ F @ top_top_set_rule )
= top_to728902761071950545ist_fm )
=> ? [X3: rule] :
( Y3
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_645_surjD,axiom,
! [F: rule > rule,Y3: rule] :
( ( ( image_rule_rule @ F @ top_top_set_rule )
= top_top_set_rule )
=> ? [X3: rule] :
( Y3
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_646_surjD,axiom,
! [F: rule > $o,Y3: $o] :
( ( ( image_rule_o @ F @ top_top_set_rule )
= top_top_set_o )
=> ? [X3: rule] :
( Y3
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_647_surjD,axiom,
! [F: $o > produc1996495991257130529ist_fm,Y3: produc1996495991257130529ist_fm] :
( ( ( image_3689732601364923150ist_fm @ F @ top_top_set_o )
= top_to728902761071950545ist_fm )
=> ? [X3: $o] :
( Y3
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_648_surjD,axiom,
! [F: $o > rule,Y3: rule] :
( ( ( image_o_rule @ F @ top_top_set_o )
= top_top_set_rule )
=> ? [X3: $o] :
( Y3
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_649_surjD,axiom,
! [F: $o > $o,Y3: $o] :
( ( ( image_o_o @ F @ top_top_set_o )
= top_top_set_o )
=> ? [X3: $o] :
( Y3
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_650_surjE,axiom,
! [F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,Y3: produc1996495991257130529ist_fm] :
( ( ( image_6982201141141708101ist_fm @ F @ top_to728902761071950545ist_fm )
= top_to728902761071950545ist_fm )
=> ~ ! [X3: produc1996495991257130529ist_fm] :
( Y3
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_651_surjE,axiom,
! [F: produc1996495991257130529ist_fm > rule,Y3: rule] :
( ( ( image_368924427985557793m_rule @ F @ top_to728902761071950545ist_fm )
= top_top_set_rule )
=> ~ ! [X3: produc1996495991257130529ist_fm] :
( Y3
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_652_surjE,axiom,
! [F: produc1996495991257130529ist_fm > $o,Y3: $o] :
( ( ( image_3428966142334963388t_fm_o @ F @ top_to728902761071950545ist_fm )
= top_top_set_o )
=> ~ ! [X3: produc1996495991257130529ist_fm] :
( Y3
= ( ~ ( F @ X3 ) ) ) ) ).
% surjE
thf(fact_653_surjE,axiom,
! [F: rule > produc1996495991257130529ist_fm,Y3: produc1996495991257130529ist_fm] :
( ( ( image_6941212235319914453ist_fm @ F @ top_top_set_rule )
= top_to728902761071950545ist_fm )
=> ~ ! [X3: rule] :
( Y3
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_654_surjE,axiom,
! [F: rule > rule,Y3: rule] :
( ( ( image_rule_rule @ F @ top_top_set_rule )
= top_top_set_rule )
=> ~ ! [X3: rule] :
( Y3
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_655_surjE,axiom,
! [F: rule > $o,Y3: $o] :
( ( ( image_rule_o @ F @ top_top_set_rule )
= top_top_set_o )
=> ~ ! [X3: rule] :
( Y3
= ( ~ ( F @ X3 ) ) ) ) ).
% surjE
thf(fact_656_surjE,axiom,
! [F: $o > produc1996495991257130529ist_fm,Y3: produc1996495991257130529ist_fm] :
( ( ( image_3689732601364923150ist_fm @ F @ top_top_set_o )
= top_to728902761071950545ist_fm )
=> ~ ! [X3: $o] :
( Y3
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_657_surjE,axiom,
! [F: $o > rule,Y3: rule] :
( ( ( image_o_rule @ F @ top_top_set_o )
= top_top_set_rule )
=> ~ ! [X3: $o] :
( Y3
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_658_surjE,axiom,
! [F: $o > $o,Y3: $o] :
( ( ( image_o_o @ F @ top_top_set_o )
= top_top_set_o )
=> ~ ! [X3: $o] :
( Y3
= ( ~ ( F @ X3 ) ) ) ) ).
% surjE
thf(fact_659_surjI,axiom,
! [G: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm] :
( ! [X3: produc1996495991257130529ist_fm] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_6982201141141708101ist_fm @ G @ top_to728902761071950545ist_fm )
= top_to728902761071950545ist_fm ) ) ).
% surjI
thf(fact_660_surjI,axiom,
! [G: produc1996495991257130529ist_fm > rule,F: rule > produc1996495991257130529ist_fm] :
( ! [X3: rule] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_368924427985557793m_rule @ G @ top_to728902761071950545ist_fm )
= top_top_set_rule ) ) ).
% surjI
thf(fact_661_surjI,axiom,
! [G: produc1996495991257130529ist_fm > $o,F: $o > produc1996495991257130529ist_fm] :
( ! [X3: $o] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_3428966142334963388t_fm_o @ G @ top_to728902761071950545ist_fm )
= top_top_set_o ) ) ).
% surjI
thf(fact_662_surjI,axiom,
! [G: rule > produc1996495991257130529ist_fm,F: produc1996495991257130529ist_fm > rule] :
( ! [X3: produc1996495991257130529ist_fm] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_6941212235319914453ist_fm @ G @ top_top_set_rule )
= top_to728902761071950545ist_fm ) ) ).
% surjI
thf(fact_663_surjI,axiom,
! [G: rule > rule,F: rule > rule] :
( ! [X3: rule] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_rule_rule @ G @ top_top_set_rule )
= top_top_set_rule ) ) ).
% surjI
thf(fact_664_surjI,axiom,
! [G: rule > $o,F: $o > rule] :
( ! [X3: $o] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_rule_o @ G @ top_top_set_rule )
= top_top_set_o ) ) ).
% surjI
thf(fact_665_surjI,axiom,
! [G: $o > produc1996495991257130529ist_fm,F: produc1996495991257130529ist_fm > $o] :
( ! [X3: produc1996495991257130529ist_fm] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_3689732601364923150ist_fm @ G @ top_top_set_o )
= top_to728902761071950545ist_fm ) ) ).
% surjI
thf(fact_666_surjI,axiom,
! [G: $o > rule,F: rule > $o] :
( ! [X3: rule] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_o_rule @ G @ top_top_set_o )
= top_top_set_rule ) ) ).
% surjI
thf(fact_667_surjI,axiom,
! [G: $o > $o,F: $o > $o] :
( ! [X3: $o] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_o_o @ G @ top_top_set_o )
= top_top_set_o ) ) ).
% surjI
thf(fact_668_surj__def,axiom,
! [F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm] :
( ( ( image_6982201141141708101ist_fm @ F @ top_to728902761071950545ist_fm )
= top_to728902761071950545ist_fm )
= ( ! [Y4: produc1996495991257130529ist_fm] :
? [X: produc1996495991257130529ist_fm] :
( Y4
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_669_surj__def,axiom,
! [F: produc1996495991257130529ist_fm > rule] :
( ( ( image_368924427985557793m_rule @ F @ top_to728902761071950545ist_fm )
= top_top_set_rule )
= ( ! [Y4: rule] :
? [X: produc1996495991257130529ist_fm] :
( Y4
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_670_surj__def,axiom,
! [F: produc1996495991257130529ist_fm > $o] :
( ( ( image_3428966142334963388t_fm_o @ F @ top_to728902761071950545ist_fm )
= top_top_set_o )
= ( ! [Y4: $o] :
? [X: produc1996495991257130529ist_fm] :
( Y4
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_671_surj__def,axiom,
! [F: rule > produc1996495991257130529ist_fm] :
( ( ( image_6941212235319914453ist_fm @ F @ top_top_set_rule )
= top_to728902761071950545ist_fm )
= ( ! [Y4: produc1996495991257130529ist_fm] :
? [X: rule] :
( Y4
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_672_surj__def,axiom,
! [F: rule > rule] :
( ( ( image_rule_rule @ F @ top_top_set_rule )
= top_top_set_rule )
= ( ! [Y4: rule] :
? [X: rule] :
( Y4
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_673_surj__def,axiom,
! [F: rule > $o] :
( ( ( image_rule_o @ F @ top_top_set_rule )
= top_top_set_o )
= ( ! [Y4: $o] :
? [X: rule] :
( Y4
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_674_surj__def,axiom,
! [F: $o > produc1996495991257130529ist_fm] :
( ( ( image_3689732601364923150ist_fm @ F @ top_top_set_o )
= top_to728902761071950545ist_fm )
= ( ! [Y4: produc1996495991257130529ist_fm] :
? [X: $o] :
( Y4
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_675_surj__def,axiom,
! [F: $o > rule] :
( ( ( image_o_rule @ F @ top_top_set_o )
= top_top_set_rule )
= ( ! [Y4: rule] :
? [X: $o] :
( Y4
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_676_surj__def,axiom,
! [F: $o > $o] :
( ( ( image_o_o @ F @ top_top_set_o )
= top_top_set_o )
= ( ! [Y4: $o] :
? [X: $o] :
( Y4
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_677_in__image__insert__iff,axiom,
! [B: set_set_rule,X2: rule,A: set_rule] :
( ! [C3: set_rule] :
( ( member_set_rule @ C3 @ B )
=> ~ ( member_rule @ X2 @ C3 ) )
=> ( ( member_set_rule @ A @ ( image_2455769455774476541t_rule @ ( insert_rule @ X2 ) @ B ) )
= ( ( member_rule @ X2 @ A )
& ( member_set_rule @ ( minus_minus_set_rule @ A @ ( insert_rule @ X2 @ bot_bot_set_rule ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_678_in__image__insert__iff,axiom,
! [B: set_set_o,X2: $o,A: set_o] :
( ! [C3: set_o] :
( ( member_set_o @ C3 @ B )
=> ~ ( member_o @ X2 @ C3 ) )
=> ( ( member_set_o @ A @ ( image_set_o_set_o @ ( insert_o @ X2 ) @ B ) )
= ( ( member_o @ X2 @ A )
& ( member_set_o @ ( minus_minus_set_o @ A @ ( insert_o @ X2 @ bot_bot_set_o ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_679_in__image__insert__iff,axiom,
! [B: set_se6775865681059004641ist_fm,X2: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm] :
( ! [C3: set_Pr7058068377845519745ist_fm] :
( ( member8436620017354071594ist_fm @ C3 @ B )
=> ~ ( member8102475879199740618ist_fm @ X2 @ C3 ) )
=> ( ( member8436620017354071594ist_fm @ A @ ( image_7201469801630334277ist_fm @ ( insert589278167076416497ist_fm @ X2 ) @ B ) )
= ( ( member8102475879199740618ist_fm @ X2 @ A )
& ( member8436620017354071594ist_fm @ ( minus_5615909706921521512ist_fm @ A @ ( insert589278167076416497ist_fm @ X2 @ bot_bo4530011010606209517ist_fm ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_680_preorder__on__def,axiom,
( order_678247264901640376ist_fm
= ( ^ [A5: set_Pr7058068377845519745ist_fm,R3: set_Pr5209464977148653383ist_fm] :
( ( refl_o4727633711164096160ist_fm @ A5 @ R3 )
& ( trans_4095360997890102209ist_fm @ top_to728902761071950545ist_fm @ R3 ) ) ) ) ).
% preorder_on_def
thf(fact_681_preorder__on__def,axiom,
( order_5092310374046361198n_rule
= ( ^ [A5: set_rule,R3: set_Pr6543236023296505803e_rule] :
( ( refl_on_rule @ A5 @ R3 )
& ( trans_on_rule @ top_top_set_rule @ R3 ) ) ) ) ).
% preorder_on_def
thf(fact_682_preorder__on__def,axiom,
( order_preorder_on_o
= ( ^ [A5: set_o,R3: set_Product_prod_o_o] :
( ( refl_on_o @ A5 @ R3 )
& ( trans_on_o @ top_top_set_o @ R3 ) ) ) ) ).
% preorder_on_def
thf(fact_683_Rangep__Range__eq,axiom,
! [R: set_Pr7058068377845519745ist_fm] :
( ( rangep3205624139766315645ist_fm
@ ^ [X: list_fm,Y4: list_fm] : ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X @ Y4 ) @ R ) )
= ( ^ [X: list_fm] : ( member_list_fm @ X @ ( range_8771097359499353911ist_fm @ R ) ) ) ) ).
% Rangep_Range_eq
thf(fact_684_Range__def,axiom,
( range_8771097359499353911ist_fm
= ( ^ [R3: set_Pr7058068377845519745ist_fm] :
( collect_list_fm
@ ( rangep3205624139766315645ist_fm
@ ^ [X: list_fm,Y4: list_fm] : ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X @ Y4 ) @ R3 ) ) ) ) ) ).
% Range_def
thf(fact_685_fun__upd__image,axiom,
! [X2: rule,A: set_rule,F: rule > rule,Y3: rule] :
( ( ( member_rule @ X2 @ A )
=> ( ( image_rule_rule @ ( fun_upd_rule_rule @ F @ X2 @ Y3 ) @ A )
= ( insert_rule @ Y3 @ ( image_rule_rule @ F @ ( minus_minus_set_rule @ A @ ( insert_rule @ X2 @ bot_bot_set_rule ) ) ) ) ) )
& ( ~ ( member_rule @ X2 @ A )
=> ( ( image_rule_rule @ ( fun_upd_rule_rule @ F @ X2 @ Y3 ) @ A )
= ( image_rule_rule @ F @ A ) ) ) ) ).
% fun_upd_image
thf(fact_686_fun__upd__image,axiom,
! [X2: rule,A: set_rule,F: rule > $o,Y3: $o] :
( ( ( member_rule @ X2 @ A )
=> ( ( image_rule_o @ ( fun_upd_rule_o @ F @ X2 @ Y3 ) @ A )
= ( insert_o @ Y3 @ ( image_rule_o @ F @ ( minus_minus_set_rule @ A @ ( insert_rule @ X2 @ bot_bot_set_rule ) ) ) ) ) )
& ( ~ ( member_rule @ X2 @ A )
=> ( ( image_rule_o @ ( fun_upd_rule_o @ F @ X2 @ Y3 ) @ A )
= ( image_rule_o @ F @ A ) ) ) ) ).
% fun_upd_image
thf(fact_687_fun__upd__image,axiom,
! [X2: rule,A: set_rule,F: rule > produc1996495991257130529ist_fm,Y3: produc1996495991257130529ist_fm] :
( ( ( member_rule @ X2 @ A )
=> ( ( image_6941212235319914453ist_fm @ ( fun_up3018851480982495581ist_fm @ F @ X2 @ Y3 ) @ A )
= ( insert589278167076416497ist_fm @ Y3 @ ( image_6941212235319914453ist_fm @ F @ ( minus_minus_set_rule @ A @ ( insert_rule @ X2 @ bot_bot_set_rule ) ) ) ) ) )
& ( ~ ( member_rule @ X2 @ A )
=> ( ( image_6941212235319914453ist_fm @ ( fun_up3018851480982495581ist_fm @ F @ X2 @ Y3 ) @ A )
= ( image_6941212235319914453ist_fm @ F @ A ) ) ) ) ).
% fun_upd_image
thf(fact_688_fun__upd__image,axiom,
! [X2: $o,A: set_o,F: $o > rule,Y3: rule] :
( ( ( member_o @ X2 @ A )
=> ( ( image_o_rule @ ( fun_upd_o_rule @ F @ X2 @ Y3 ) @ A )
= ( insert_rule @ Y3 @ ( image_o_rule @ F @ ( minus_minus_set_o @ A @ ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ) )
& ( ~ ( member_o @ X2 @ A )
=> ( ( image_o_rule @ ( fun_upd_o_rule @ F @ X2 @ Y3 ) @ A )
= ( image_o_rule @ F @ A ) ) ) ) ).
% fun_upd_image
thf(fact_689_fun__upd__image,axiom,
! [X2: $o,A: set_o,F: $o > $o,Y3: $o] :
( ( ( member_o @ X2 @ A )
=> ( ( image_o_o @ ( fun_upd_o_o @ F @ X2 @ Y3 ) @ A )
= ( insert_o @ Y3 @ ( image_o_o @ F @ ( minus_minus_set_o @ A @ ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ) )
& ( ~ ( member_o @ X2 @ A )
=> ( ( image_o_o @ ( fun_upd_o_o @ F @ X2 @ Y3 ) @ A )
= ( image_o_o @ F @ A ) ) ) ) ).
% fun_upd_image
thf(fact_690_fun__upd__image,axiom,
! [X2: $o,A: set_o,F: $o > produc1996495991257130529ist_fm,Y3: produc1996495991257130529ist_fm] :
( ( ( member_o @ X2 @ A )
=> ( ( image_3689732601364923150ist_fm @ ( fun_up141237966704039814ist_fm @ F @ X2 @ Y3 ) @ A )
= ( insert589278167076416497ist_fm @ Y3 @ ( image_3689732601364923150ist_fm @ F @ ( minus_minus_set_o @ A @ ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ) )
& ( ~ ( member_o @ X2 @ A )
=> ( ( image_3689732601364923150ist_fm @ ( fun_up141237966704039814ist_fm @ F @ X2 @ Y3 ) @ A )
= ( image_3689732601364923150ist_fm @ F @ A ) ) ) ) ).
% fun_upd_image
thf(fact_691_fun__upd__image,axiom,
! [X2: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm,F: produc1996495991257130529ist_fm > rule,Y3: rule] :
( ( ( member8102475879199740618ist_fm @ X2 @ A )
=> ( ( image_368924427985557793m_rule @ ( fun_up5669935710502914729m_rule @ F @ X2 @ Y3 ) @ A )
= ( insert_rule @ Y3 @ ( image_368924427985557793m_rule @ F @ ( minus_5615909706921521512ist_fm @ A @ ( insert589278167076416497ist_fm @ X2 @ bot_bo4530011010606209517ist_fm ) ) ) ) ) )
& ( ~ ( member8102475879199740618ist_fm @ X2 @ A )
=> ( ( image_368924427985557793m_rule @ ( fun_up5669935710502914729m_rule @ F @ X2 @ Y3 ) @ A )
= ( image_368924427985557793m_rule @ F @ A ) ) ) ) ).
% fun_upd_image
thf(fact_692_fun__upd__image,axiom,
! [X2: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm,F: produc1996495991257130529ist_fm > $o,Y3: $o] :
( ( ( member8102475879199740618ist_fm @ X2 @ A )
=> ( ( image_3428966142334963388t_fm_o @ ( fun_up9103843544528855860t_fm_o @ F @ X2 @ Y3 ) @ A )
= ( insert_o @ Y3 @ ( image_3428966142334963388t_fm_o @ F @ ( minus_5615909706921521512ist_fm @ A @ ( insert589278167076416497ist_fm @ X2 @ bot_bo4530011010606209517ist_fm ) ) ) ) ) )
& ( ~ ( member8102475879199740618ist_fm @ X2 @ A )
=> ( ( image_3428966142334963388t_fm_o @ ( fun_up9103843544528855860t_fm_o @ F @ X2 @ Y3 ) @ A )
= ( image_3428966142334963388t_fm_o @ F @ A ) ) ) ) ).
% fun_upd_image
thf(fact_693_fun__upd__image,axiom,
! [X2: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm,F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,Y3: produc1996495991257130529ist_fm] :
( ( ( member8102475879199740618ist_fm @ X2 @ A )
=> ( ( image_6982201141141708101ist_fm @ ( fun_up6069510949241940413ist_fm @ F @ X2 @ Y3 ) @ A )
= ( insert589278167076416497ist_fm @ Y3 @ ( image_6982201141141708101ist_fm @ F @ ( minus_5615909706921521512ist_fm @ A @ ( insert589278167076416497ist_fm @ X2 @ bot_bo4530011010606209517ist_fm ) ) ) ) ) )
& ( ~ ( member8102475879199740618ist_fm @ X2 @ A )
=> ( ( image_6982201141141708101ist_fm @ ( fun_up6069510949241940413ist_fm @ F @ X2 @ Y3 ) @ A )
= ( image_6982201141141708101ist_fm @ F @ A ) ) ) ) ).
% fun_upd_image
thf(fact_694_preorder__on__empty,axiom,
order_2819943312236725001ist_fm @ bot_bot_set_list_fm @ bot_bo4530011010606209517ist_fm ).
% preorder_on_empty
thf(fact_695_preorder__on__empty,axiom,
order_5092310374046361198n_rule @ bot_bot_set_rule @ bot_bo9047865283180780599e_rule ).
% preorder_on_empty
thf(fact_696_preorder__on__empty,axiom,
order_preorder_on_o @ bot_bot_set_o @ bot_bo7073875226086086771od_o_o ).
% preorder_on_empty
thf(fact_697_preorder__on__empty,axiom,
order_678247264901640376ist_fm @ bot_bo4530011010606209517ist_fm @ bot_bo7330262581962992051ist_fm ).
% preorder_on_empty
thf(fact_698_image__map__upd,axiom,
! [X2: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm,M: produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Y3: fset_P661503646757059847ist_fm] :
( ~ ( member8102475879199740618ist_fm @ X2 @ A )
=> ( ( image_5753375111473792241ist_fm @ ( fun_up1138168335916736873ist_fm @ M @ X2 @ ( some_f2162114139285431020ist_fm @ Y3 ) ) @ A )
= ( image_5753375111473792241ist_fm @ M @ A ) ) ) ).
% image_map_upd
thf(fact_699_image__map__upd,axiom,
! [X2: rule,A: set_rule,M: rule > option6029102822600979917ist_fm,Y3: fset_P661503646757059847ist_fm] :
( ~ ( member_rule @ X2 @ A )
=> ( ( image_529690880457373569ist_fm @ ( fun_up5825503389150910729ist_fm @ M @ X2 @ ( some_f2162114139285431020ist_fm @ Y3 ) ) @ A )
= ( image_529690880457373569ist_fm @ M @ A ) ) ) ).
% image_map_upd
thf(fact_700_image__map__upd,axiom,
! [X2: $o,A: set_o,M: $o > option6029102822600979917ist_fm,Y3: fset_P661503646757059847ist_fm] :
( ~ ( member_o @ X2 @ A )
=> ( ( image_7906517863533482170ist_fm @ ( fun_up1845550324453438770ist_fm @ M @ X2 @ ( some_f2162114139285431020ist_fm @ Y3 ) ) @ A )
= ( image_7906517863533482170ist_fm @ M @ A ) ) ) ).
% image_map_upd
thf(fact_701_INT__simps_I3_J,axiom,
! [C4: set_rule,A: rule > set_rule,B: set_rule] :
( ( ( C4 = bot_bot_set_rule )
=> ( ( comple5773694076043965236t_rule
@ ( image_rule_set_rule
@ ^ [X: rule] : ( minus_minus_set_rule @ ( A @ X ) @ B )
@ C4 ) )
= top_top_set_rule ) )
& ( ( C4 != bot_bot_set_rule )
=> ( ( comple5773694076043965236t_rule
@ ( image_rule_set_rule
@ ^ [X: rule] : ( minus_minus_set_rule @ ( A @ X ) @ B )
@ C4 ) )
= ( minus_minus_set_rule @ ( comple5773694076043965236t_rule @ ( image_rule_set_rule @ A @ C4 ) ) @ B ) ) ) ) ).
% INT_simps(3)
thf(fact_702_INT__simps_I3_J,axiom,
! [C4: set_o,A: $o > set_rule,B: set_rule] :
( ( ( C4 = bot_bot_set_o )
=> ( ( comple5773694076043965236t_rule
@ ( image_o_set_rule
@ ^ [X: $o] : ( minus_minus_set_rule @ ( A @ X ) @ B )
@ C4 ) )
= top_top_set_rule ) )
& ( ( C4 != bot_bot_set_o )
=> ( ( comple5773694076043965236t_rule
@ ( image_o_set_rule
@ ^ [X: $o] : ( minus_minus_set_rule @ ( A @ X ) @ B )
@ C4 ) )
= ( minus_minus_set_rule @ ( comple5773694076043965236t_rule @ ( image_o_set_rule @ A @ C4 ) ) @ B ) ) ) ) ).
% INT_simps(3)
thf(fact_703_INT__simps_I3_J,axiom,
! [C4: set_Pr7058068377845519745ist_fm,A: produc1996495991257130529ist_fm > set_rule,B: set_rule] :
( ( ( C4 = bot_bo4530011010606209517ist_fm )
=> ( ( comple5773694076043965236t_rule
@ ( image_2008118025656205143t_rule
@ ^ [X: produc1996495991257130529ist_fm] : ( minus_minus_set_rule @ ( A @ X ) @ B )
@ C4 ) )
= top_top_set_rule ) )
& ( ( C4 != bot_bo4530011010606209517ist_fm )
=> ( ( comple5773694076043965236t_rule
@ ( image_2008118025656205143t_rule
@ ^ [X: produc1996495991257130529ist_fm] : ( minus_minus_set_rule @ ( A @ X ) @ B )
@ C4 ) )
= ( minus_minus_set_rule @ ( comple5773694076043965236t_rule @ ( image_2008118025656205143t_rule @ A @ C4 ) ) @ B ) ) ) ) ).
% INT_simps(3)
thf(fact_704_INT__simps_I3_J,axiom,
! [C4: set_rule,A: rule > set_o,B: set_o] :
( ( ( C4 = bot_bot_set_rule )
=> ( ( comple3063163877087187839_set_o
@ ( image_rule_set_o
@ ^ [X: rule] : ( minus_minus_set_o @ ( A @ X ) @ B )
@ C4 ) )
= top_top_set_o ) )
& ( ( C4 != bot_bot_set_rule )
=> ( ( comple3063163877087187839_set_o
@ ( image_rule_set_o
@ ^ [X: rule] : ( minus_minus_set_o @ ( A @ X ) @ B )
@ C4 ) )
= ( minus_minus_set_o @ ( comple3063163877087187839_set_o @ ( image_rule_set_o @ A @ C4 ) ) @ B ) ) ) ) ).
% INT_simps(3)
thf(fact_705_INT__simps_I3_J,axiom,
! [C4: set_o,A: $o > set_o,B: set_o] :
( ( ( C4 = bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [X: $o] : ( minus_minus_set_o @ ( A @ X ) @ B )
@ C4 ) )
= top_top_set_o ) )
& ( ( C4 != bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [X: $o] : ( minus_minus_set_o @ ( A @ X ) @ B )
@ C4 ) )
= ( minus_minus_set_o @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ A @ C4 ) ) @ B ) ) ) ) ).
% INT_simps(3)
thf(fact_706_INT__simps_I3_J,axiom,
! [C4: set_Pr7058068377845519745ist_fm,A: produc1996495991257130529ist_fm > set_o,B: set_o] :
( ( ( C4 = bot_bo4530011010606209517ist_fm )
=> ( ( comple3063163877087187839_set_o
@ ( image_7087520594942641564_set_o
@ ^ [X: produc1996495991257130529ist_fm] : ( minus_minus_set_o @ ( A @ X ) @ B )
@ C4 ) )
= top_top_set_o ) )
& ( ( C4 != bot_bo4530011010606209517ist_fm )
=> ( ( comple3063163877087187839_set_o
@ ( image_7087520594942641564_set_o
@ ^ [X: produc1996495991257130529ist_fm] : ( minus_minus_set_o @ ( A @ X ) @ B )
@ C4 ) )
= ( minus_minus_set_o @ ( comple3063163877087187839_set_o @ ( image_7087520594942641564_set_o @ A @ C4 ) ) @ B ) ) ) ) ).
% INT_simps(3)
thf(fact_707_INT__simps_I3_J,axiom,
! [C4: set_rule,A: rule > set_Pr7058068377845519745ist_fm,B: set_Pr7058068377845519745ist_fm] :
( ( ( C4 = bot_bot_set_rule )
=> ( ( comple5608129641975563656ist_fm
@ ( image_3897101447128207157ist_fm
@ ^ [X: rule] : ( minus_5615909706921521512ist_fm @ ( A @ X ) @ B )
@ C4 ) )
= top_to728902761071950545ist_fm ) )
& ( ( C4 != bot_bot_set_rule )
=> ( ( comple5608129641975563656ist_fm
@ ( image_3897101447128207157ist_fm
@ ^ [X: rule] : ( minus_5615909706921521512ist_fm @ ( A @ X ) @ B )
@ C4 ) )
= ( minus_5615909706921521512ist_fm @ ( comple5608129641975563656ist_fm @ ( image_3897101447128207157ist_fm @ A @ C4 ) ) @ B ) ) ) ) ).
% INT_simps(3)
thf(fact_708_INT__simps_I3_J,axiom,
! [C4: set_o,A: $o > set_Pr7058068377845519745ist_fm,B: set_Pr7058068377845519745ist_fm] :
( ( ( C4 = bot_bot_set_o )
=> ( ( comple5608129641975563656ist_fm
@ ( image_1666948324336172654ist_fm
@ ^ [X: $o] : ( minus_5615909706921521512ist_fm @ ( A @ X ) @ B )
@ C4 ) )
= top_to728902761071950545ist_fm ) )
& ( ( C4 != bot_bot_set_o )
=> ( ( comple5608129641975563656ist_fm
@ ( image_1666948324336172654ist_fm
@ ^ [X: $o] : ( minus_5615909706921521512ist_fm @ ( A @ X ) @ B )
@ C4 ) )
= ( minus_5615909706921521512ist_fm @ ( comple5608129641975563656ist_fm @ ( image_1666948324336172654ist_fm @ A @ C4 ) ) @ B ) ) ) ) ).
% INT_simps(3)
thf(fact_709_INT__simps_I3_J,axiom,
! [C4: set_Pr7058068377845519745ist_fm,A: produc1996495991257130529ist_fm > set_Pr7058068377845519745ist_fm,B: set_Pr7058068377845519745ist_fm] :
( ( ( C4 = bot_bo4530011010606209517ist_fm )
=> ( ( comple5608129641975563656ist_fm
@ ( image_5852275405373602981ist_fm
@ ^ [X: produc1996495991257130529ist_fm] : ( minus_5615909706921521512ist_fm @ ( A @ X ) @ B )
@ C4 ) )
= top_to728902761071950545ist_fm ) )
& ( ( C4 != bot_bo4530011010606209517ist_fm )
=> ( ( comple5608129641975563656ist_fm
@ ( image_5852275405373602981ist_fm
@ ^ [X: produc1996495991257130529ist_fm] : ( minus_5615909706921521512ist_fm @ ( A @ X ) @ B )
@ C4 ) )
= ( minus_5615909706921521512ist_fm @ ( comple5608129641975563656ist_fm @ ( image_5852275405373602981ist_fm @ A @ C4 ) ) @ B ) ) ) ) ).
% INT_simps(3)
thf(fact_710_inj__on__insert,axiom,
! [F: rule > produc1996495991257130529ist_fm,A3: rule,A: set_rule] :
( ( inj_on3429812851933087593ist_fm @ F @ ( insert_rule @ A3 @ A ) )
= ( ( inj_on3429812851933087593ist_fm @ F @ A )
& ~ ( member8102475879199740618ist_fm @ ( F @ A3 ) @ ( image_6941212235319914453ist_fm @ F @ ( minus_minus_set_rule @ A @ ( insert_rule @ A3 @ bot_bot_set_rule ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_711_inj__on__insert,axiom,
! [F: rule > rule,A3: rule,A: set_rule] :
( ( inj_on_rule_rule @ F @ ( insert_rule @ A3 @ A ) )
= ( ( inj_on_rule_rule @ F @ A )
& ~ ( member_rule @ ( F @ A3 ) @ ( image_rule_rule @ F @ ( minus_minus_set_rule @ A @ ( insert_rule @ A3 @ bot_bot_set_rule ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_712_inj__on__insert,axiom,
! [F: rule > $o,A3: rule,A: set_rule] :
( ( inj_on_rule_o @ F @ ( insert_rule @ A3 @ A ) )
= ( ( inj_on_rule_o @ F @ A )
& ~ ( member_o @ ( F @ A3 ) @ ( image_rule_o @ F @ ( minus_minus_set_rule @ A @ ( insert_rule @ A3 @ bot_bot_set_rule ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_713_inj__on__insert,axiom,
! [F: $o > produc1996495991257130529ist_fm,A3: $o,A: set_o] :
( ( inj_on6870991201974782586ist_fm @ F @ ( insert_o @ A3 @ A ) )
= ( ( inj_on6870991201974782586ist_fm @ F @ A )
& ~ ( member8102475879199740618ist_fm @ ( F @ A3 ) @ ( image_3689732601364923150ist_fm @ F @ ( minus_minus_set_o @ A @ ( insert_o @ A3 @ bot_bot_set_o ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_714_inj__on__insert,axiom,
! [F: $o > rule,A3: $o,A: set_o] :
( ( inj_on_o_rule @ F @ ( insert_o @ A3 @ A ) )
= ( ( inj_on_o_rule @ F @ A )
& ~ ( member_rule @ ( F @ A3 ) @ ( image_o_rule @ F @ ( minus_minus_set_o @ A @ ( insert_o @ A3 @ bot_bot_set_o ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_715_inj__on__insert,axiom,
! [F: $o > $o,A3: $o,A: set_o] :
( ( inj_on_o_o @ F @ ( insert_o @ A3 @ A ) )
= ( ( inj_on_o_o @ F @ A )
& ~ ( member_o @ ( F @ A3 ) @ ( image_o_o @ F @ ( minus_minus_set_o @ A @ ( insert_o @ A3 @ bot_bot_set_o ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_716_inj__on__insert,axiom,
! [F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,A3: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm] :
( ( inj_on2392985232935137457ist_fm @ F @ ( insert589278167076416497ist_fm @ A3 @ A ) )
= ( ( inj_on2392985232935137457ist_fm @ F @ A )
& ~ ( member8102475879199740618ist_fm @ ( F @ A3 ) @ ( image_6982201141141708101ist_fm @ F @ ( minus_5615909706921521512ist_fm @ A @ ( insert589278167076416497ist_fm @ A3 @ bot_bo4530011010606209517ist_fm ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_717_inj__on__insert,axiom,
! [F: produc1996495991257130529ist_fm > rule,A3: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm] :
( ( inj_on6080897081453506741m_rule @ F @ ( insert589278167076416497ist_fm @ A3 @ A ) )
= ( ( inj_on6080897081453506741m_rule @ F @ A )
& ~ ( member_rule @ ( F @ A3 ) @ ( image_368924427985557793m_rule @ F @ ( minus_5615909706921521512ist_fm @ A @ ( insert589278167076416497ist_fm @ A3 @ bot_bo4530011010606209517ist_fm ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_718_inj__on__insert,axiom,
! [F: produc1996495991257130529ist_fm > $o,A3: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm] :
( ( inj_on6610224742944822824t_fm_o @ F @ ( insert589278167076416497ist_fm @ A3 @ A ) )
= ( ( inj_on6610224742944822824t_fm_o @ F @ A )
& ~ ( member_o @ ( F @ A3 ) @ ( image_3428966142334963388t_fm_o @ F @ ( minus_5615909706921521512ist_fm @ A @ ( insert589278167076416497ist_fm @ A3 @ bot_bo4530011010606209517ist_fm ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_719_Sup__bot__conv_I1_J,axiom,
! [A: set_set_rule] :
( ( ( comple2146307154184993742t_rule @ A )
= bot_bot_set_rule )
= ( ! [X: set_rule] :
( ( member_set_rule @ X @ A )
=> ( X = bot_bot_set_rule ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_720_Sup__bot__conv_I1_J,axiom,
! [A: set_set_o] :
( ( ( comple90263536869209701_set_o @ A )
= bot_bot_set_o )
= ( ! [X: set_o] :
( ( member_set_o @ X @ A )
=> ( X = bot_bot_set_o ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_721_Sup__bot__conv_I1_J,axiom,
! [A: set_se6775865681059004641ist_fm] :
( ( ( comple1649742678916152942ist_fm @ A )
= bot_bo4530011010606209517ist_fm )
= ( ! [X: set_Pr7058068377845519745ist_fm] :
( ( member8436620017354071594ist_fm @ X @ A )
=> ( X = bot_bo4530011010606209517ist_fm ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_722_Sup__bot__conv_I2_J,axiom,
! [A: set_set_rule] :
( ( bot_bot_set_rule
= ( comple2146307154184993742t_rule @ A ) )
= ( ! [X: set_rule] :
( ( member_set_rule @ X @ A )
=> ( X = bot_bot_set_rule ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_723_Sup__bot__conv_I2_J,axiom,
! [A: set_set_o] :
( ( bot_bot_set_o
= ( comple90263536869209701_set_o @ A ) )
= ( ! [X: set_o] :
( ( member_set_o @ X @ A )
=> ( X = bot_bot_set_o ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_724_Sup__bot__conv_I2_J,axiom,
! [A: set_se6775865681059004641ist_fm] :
( ( bot_bo4530011010606209517ist_fm
= ( comple1649742678916152942ist_fm @ A ) )
= ( ! [X: set_Pr7058068377845519745ist_fm] :
( ( member8436620017354071594ist_fm @ X @ A )
=> ( X = bot_bo4530011010606209517ist_fm ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_725_Inf__top__conv_I2_J,axiom,
! [A: set_se6775865681059004641ist_fm] :
( ( top_to728902761071950545ist_fm
= ( comple5608129641975563656ist_fm @ A ) )
= ( ! [X: set_Pr7058068377845519745ist_fm] :
( ( member8436620017354071594ist_fm @ X @ A )
=> ( X = top_to728902761071950545ist_fm ) ) ) ) ).
% Inf_top_conv(2)
thf(fact_726_Inf__top__conv_I2_J,axiom,
! [A: set_set_rule] :
( ( top_top_set_rule
= ( comple5773694076043965236t_rule @ A ) )
= ( ! [X: set_rule] :
( ( member_set_rule @ X @ A )
=> ( X = top_top_set_rule ) ) ) ) ).
% Inf_top_conv(2)
thf(fact_727_Inf__top__conv_I2_J,axiom,
! [A: set_set_o] :
( ( top_top_set_o
= ( comple3063163877087187839_set_o @ A ) )
= ( ! [X: set_o] :
( ( member_set_o @ X @ A )
=> ( X = top_top_set_o ) ) ) ) ).
% Inf_top_conv(2)
thf(fact_728_Inf__top__conv_I1_J,axiom,
! [A: set_se6775865681059004641ist_fm] :
( ( ( comple5608129641975563656ist_fm @ A )
= top_to728902761071950545ist_fm )
= ( ! [X: set_Pr7058068377845519745ist_fm] :
( ( member8436620017354071594ist_fm @ X @ A )
=> ( X = top_to728902761071950545ist_fm ) ) ) ) ).
% Inf_top_conv(1)
thf(fact_729_Inf__top__conv_I1_J,axiom,
! [A: set_set_rule] :
( ( ( comple5773694076043965236t_rule @ A )
= top_top_set_rule )
= ( ! [X: set_rule] :
( ( member_set_rule @ X @ A )
=> ( X = top_top_set_rule ) ) ) ) ).
% Inf_top_conv(1)
thf(fact_730_Inf__top__conv_I1_J,axiom,
! [A: set_set_o] :
( ( ( comple3063163877087187839_set_o @ A )
= top_top_set_o )
= ( ! [X: set_o] :
( ( member_set_o @ X @ A )
=> ( X = top_top_set_o ) ) ) ) ).
% Inf_top_conv(1)
thf(fact_731_Inter__UNIV__conv_I2_J,axiom,
! [A: set_se6775865681059004641ist_fm] :
( ( top_to728902761071950545ist_fm
= ( comple5608129641975563656ist_fm @ A ) )
= ( ! [X: set_Pr7058068377845519745ist_fm] :
( ( member8436620017354071594ist_fm @ X @ A )
=> ( X = top_to728902761071950545ist_fm ) ) ) ) ).
% Inter_UNIV_conv(2)
thf(fact_732_Inter__UNIV__conv_I2_J,axiom,
! [A: set_set_rule] :
( ( top_top_set_rule
= ( comple5773694076043965236t_rule @ A ) )
= ( ! [X: set_rule] :
( ( member_set_rule @ X @ A )
=> ( X = top_top_set_rule ) ) ) ) ).
% Inter_UNIV_conv(2)
thf(fact_733_Inter__UNIV__conv_I2_J,axiom,
! [A: set_set_o] :
( ( top_top_set_o
= ( comple3063163877087187839_set_o @ A ) )
= ( ! [X: set_o] :
( ( member_set_o @ X @ A )
=> ( X = top_top_set_o ) ) ) ) ).
% Inter_UNIV_conv(2)
thf(fact_734_Inter__UNIV__conv_I1_J,axiom,
! [A: set_se6775865681059004641ist_fm] :
( ( ( comple5608129641975563656ist_fm @ A )
= top_to728902761071950545ist_fm )
= ( ! [X: set_Pr7058068377845519745ist_fm] :
( ( member8436620017354071594ist_fm @ X @ A )
=> ( X = top_to728902761071950545ist_fm ) ) ) ) ).
% Inter_UNIV_conv(1)
thf(fact_735_Inter__UNIV__conv_I1_J,axiom,
! [A: set_set_rule] :
( ( ( comple5773694076043965236t_rule @ A )
= top_top_set_rule )
= ( ! [X: set_rule] :
( ( member_set_rule @ X @ A )
=> ( X = top_top_set_rule ) ) ) ) ).
% Inter_UNIV_conv(1)
thf(fact_736_Inter__UNIV__conv_I1_J,axiom,
! [A: set_set_o] :
( ( ( comple3063163877087187839_set_o @ A )
= top_top_set_o )
= ( ! [X: set_o] :
( ( member_set_o @ X @ A )
=> ( X = top_top_set_o ) ) ) ) ).
% Inter_UNIV_conv(1)
thf(fact_737_UN__I,axiom,
! [A3: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm,B3: produc1996495991257130529ist_fm,B: produc1996495991257130529ist_fm > set_Pr7058068377845519745ist_fm] :
( ( member8102475879199740618ist_fm @ A3 @ A )
=> ( ( member8102475879199740618ist_fm @ B3 @ ( B @ A3 ) )
=> ( member8102475879199740618ist_fm @ B3 @ ( comple1649742678916152942ist_fm @ ( image_5852275405373602981ist_fm @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_738_UN__I,axiom,
! [A3: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm,B3: rule,B: produc1996495991257130529ist_fm > set_rule] :
( ( member8102475879199740618ist_fm @ A3 @ A )
=> ( ( member_rule @ B3 @ ( B @ A3 ) )
=> ( member_rule @ B3 @ ( comple2146307154184993742t_rule @ ( image_2008118025656205143t_rule @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_739_UN__I,axiom,
! [A3: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm,B3: $o,B: produc1996495991257130529ist_fm > set_o] :
( ( member8102475879199740618ist_fm @ A3 @ A )
=> ( ( member_o @ B3 @ ( B @ A3 ) )
=> ( member_o @ B3 @ ( comple90263536869209701_set_o @ ( image_7087520594942641564_set_o @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_740_UN__I,axiom,
! [A3: rule,A: set_rule,B3: produc1996495991257130529ist_fm,B: rule > set_Pr7058068377845519745ist_fm] :
( ( member_rule @ A3 @ A )
=> ( ( member8102475879199740618ist_fm @ B3 @ ( B @ A3 ) )
=> ( member8102475879199740618ist_fm @ B3 @ ( comple1649742678916152942ist_fm @ ( image_3897101447128207157ist_fm @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_741_UN__I,axiom,
! [A3: rule,A: set_rule,B3: rule,B: rule > set_rule] :
( ( member_rule @ A3 @ A )
=> ( ( member_rule @ B3 @ ( B @ A3 ) )
=> ( member_rule @ B3 @ ( comple2146307154184993742t_rule @ ( image_rule_set_rule @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_742_UN__I,axiom,
! [A3: rule,A: set_rule,B3: $o,B: rule > set_o] :
( ( member_rule @ A3 @ A )
=> ( ( member_o @ B3 @ ( B @ A3 ) )
=> ( member_o @ B3 @ ( comple90263536869209701_set_o @ ( image_rule_set_o @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_743_UN__I,axiom,
! [A3: $o,A: set_o,B3: produc1996495991257130529ist_fm,B: $o > set_Pr7058068377845519745ist_fm] :
( ( member_o @ A3 @ A )
=> ( ( member8102475879199740618ist_fm @ B3 @ ( B @ A3 ) )
=> ( member8102475879199740618ist_fm @ B3 @ ( comple1649742678916152942ist_fm @ ( image_1666948324336172654ist_fm @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_744_UN__I,axiom,
! [A3: $o,A: set_o,B3: rule,B: $o > set_rule] :
( ( member_o @ A3 @ A )
=> ( ( member_rule @ B3 @ ( B @ A3 ) )
=> ( member_rule @ B3 @ ( comple2146307154184993742t_rule @ ( image_o_set_rule @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_745_UN__I,axiom,
! [A3: $o,A: set_o,B3: $o,B: $o > set_o] :
( ( member_o @ A3 @ A )
=> ( ( member_o @ B3 @ ( B @ A3 ) )
=> ( member_o @ B3 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_746_INT__I,axiom,
! [A: set_Pr7058068377845519745ist_fm,B3: produc1996495991257130529ist_fm,B: produc1996495991257130529ist_fm > set_Pr7058068377845519745ist_fm] :
( ! [X3: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ X3 @ A )
=> ( member8102475879199740618ist_fm @ B3 @ ( B @ X3 ) ) )
=> ( member8102475879199740618ist_fm @ B3 @ ( comple5608129641975563656ist_fm @ ( image_5852275405373602981ist_fm @ B @ A ) ) ) ) ).
% INT_I
thf(fact_747_INT__I,axiom,
! [A: set_Pr7058068377845519745ist_fm,B3: rule,B: produc1996495991257130529ist_fm > set_rule] :
( ! [X3: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ X3 @ A )
=> ( member_rule @ B3 @ ( B @ X3 ) ) )
=> ( member_rule @ B3 @ ( comple5773694076043965236t_rule @ ( image_2008118025656205143t_rule @ B @ A ) ) ) ) ).
% INT_I
thf(fact_748_INT__I,axiom,
! [A: set_Pr7058068377845519745ist_fm,B3: $o,B: produc1996495991257130529ist_fm > set_o] :
( ! [X3: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ X3 @ A )
=> ( member_o @ B3 @ ( B @ X3 ) ) )
=> ( member_o @ B3 @ ( comple3063163877087187839_set_o @ ( image_7087520594942641564_set_o @ B @ A ) ) ) ) ).
% INT_I
thf(fact_749_INT__I,axiom,
! [A: set_rule,B3: produc1996495991257130529ist_fm,B: rule > set_Pr7058068377845519745ist_fm] :
( ! [X3: rule] :
( ( member_rule @ X3 @ A )
=> ( member8102475879199740618ist_fm @ B3 @ ( B @ X3 ) ) )
=> ( member8102475879199740618ist_fm @ B3 @ ( comple5608129641975563656ist_fm @ ( image_3897101447128207157ist_fm @ B @ A ) ) ) ) ).
% INT_I
thf(fact_750_INT__I,axiom,
! [A: set_rule,B3: rule,B: rule > set_rule] :
( ! [X3: rule] :
( ( member_rule @ X3 @ A )
=> ( member_rule @ B3 @ ( B @ X3 ) ) )
=> ( member_rule @ B3 @ ( comple5773694076043965236t_rule @ ( image_rule_set_rule @ B @ A ) ) ) ) ).
% INT_I
thf(fact_751_INT__I,axiom,
! [A: set_rule,B3: $o,B: rule > set_o] :
( ! [X3: rule] :
( ( member_rule @ X3 @ A )
=> ( member_o @ B3 @ ( B @ X3 ) ) )
=> ( member_o @ B3 @ ( comple3063163877087187839_set_o @ ( image_rule_set_o @ B @ A ) ) ) ) ).
% INT_I
thf(fact_752_INT__I,axiom,
! [A: set_o,B3: produc1996495991257130529ist_fm,B: $o > set_Pr7058068377845519745ist_fm] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( member8102475879199740618ist_fm @ B3 @ ( B @ X3 ) ) )
=> ( member8102475879199740618ist_fm @ B3 @ ( comple5608129641975563656ist_fm @ ( image_1666948324336172654ist_fm @ B @ A ) ) ) ) ).
% INT_I
thf(fact_753_INT__I,axiom,
! [A: set_o,B3: rule,B: $o > set_rule] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( member_rule @ B3 @ ( B @ X3 ) ) )
=> ( member_rule @ B3 @ ( comple5773694076043965236t_rule @ ( image_o_set_rule @ B @ A ) ) ) ) ).
% INT_I
thf(fact_754_INT__I,axiom,
! [A: set_o,B3: $o,B: $o > set_o] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( member_o @ B3 @ ( B @ X3 ) ) )
=> ( member_o @ B3 @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ B @ A ) ) ) ) ).
% INT_I
thf(fact_755_Sup__UNIV,axiom,
( ( comple1649742678916152942ist_fm @ top_to7149953857590879793ist_fm )
= top_to728902761071950545ist_fm ) ).
% Sup_UNIV
thf(fact_756_Sup__UNIV,axiom,
( ( comple2146307154184993742t_rule @ top_top_set_set_rule )
= top_top_set_rule ) ).
% Sup_UNIV
thf(fact_757_Sup__UNIV,axiom,
( ( comple90263536869209701_set_o @ top_top_set_set_o )
= top_top_set_o ) ).
% Sup_UNIV
thf(fact_758_Sup__UNIV,axiom,
( ( complete_Sup_Sup_o @ top_top_set_o )
= top_top_o ) ).
% Sup_UNIV
thf(fact_759_Sup__empty,axiom,
( ( comple2146307154184993742t_rule @ bot_bot_set_set_rule )
= bot_bot_set_rule ) ).
% Sup_empty
thf(fact_760_Sup__empty,axiom,
( ( comple90263536869209701_set_o @ bot_bot_set_set_o )
= bot_bot_set_o ) ).
% Sup_empty
thf(fact_761_Sup__empty,axiom,
( ( comple1649742678916152942ist_fm @ bot_bo414211191905506637ist_fm )
= bot_bo4530011010606209517ist_fm ) ).
% Sup_empty
thf(fact_762_Sup__empty,axiom,
( ( complete_Sup_Sup_o @ bot_bot_set_o )
= bot_bot_o ) ).
% Sup_empty
thf(fact_763_Inf__UNIV,axiom,
( ( comple5773694076043965236t_rule @ top_top_set_set_rule )
= bot_bot_set_rule ) ).
% Inf_UNIV
thf(fact_764_Inf__UNIV,axiom,
( ( comple3063163877087187839_set_o @ top_top_set_set_o )
= bot_bot_set_o ) ).
% Inf_UNIV
thf(fact_765_Inf__UNIV,axiom,
( ( comple5608129641975563656ist_fm @ top_to7149953857590879793ist_fm )
= bot_bo4530011010606209517ist_fm ) ).
% Inf_UNIV
thf(fact_766_Inf__UNIV,axiom,
( ( complete_Inf_Inf_o @ top_top_set_o )
= bot_bot_o ) ).
% Inf_UNIV
thf(fact_767_Inf__empty,axiom,
( ( complete_Inf_Inf_o @ bot_bot_set_o )
= top_top_o ) ).
% Inf_empty
thf(fact_768_Inf__empty,axiom,
( ( comple5608129641975563656ist_fm @ bot_bo414211191905506637ist_fm )
= top_to728902761071950545ist_fm ) ).
% Inf_empty
thf(fact_769_Inf__empty,axiom,
( ( comple5773694076043965236t_rule @ bot_bot_set_set_rule )
= top_top_set_rule ) ).
% Inf_empty
thf(fact_770_Inf__empty,axiom,
( ( comple3063163877087187839_set_o @ bot_bot_set_set_o )
= top_top_set_o ) ).
% Inf_empty
thf(fact_771_UN__constant,axiom,
! [A: set_rule,C: set_rule] :
( ( ( A = bot_bot_set_rule )
=> ( ( comple2146307154184993742t_rule
@ ( image_rule_set_rule
@ ^ [Y4: rule] : C
@ A ) )
= bot_bot_set_rule ) )
& ( ( A != bot_bot_set_rule )
=> ( ( comple2146307154184993742t_rule
@ ( image_rule_set_rule
@ ^ [Y4: rule] : C
@ A ) )
= C ) ) ) ).
% UN_constant
thf(fact_772_UN__constant,axiom,
! [A: set_rule,C: set_o] :
( ( ( A = bot_bot_set_rule )
=> ( ( comple90263536869209701_set_o
@ ( image_rule_set_o
@ ^ [Y4: rule] : C
@ A ) )
= bot_bot_set_o ) )
& ( ( A != bot_bot_set_rule )
=> ( ( comple90263536869209701_set_o
@ ( image_rule_set_o
@ ^ [Y4: rule] : C
@ A ) )
= C ) ) ) ).
% UN_constant
thf(fact_773_UN__constant,axiom,
! [A: set_rule,C: set_Pr7058068377845519745ist_fm] :
( ( ( A = bot_bot_set_rule )
=> ( ( comple1649742678916152942ist_fm
@ ( image_3897101447128207157ist_fm
@ ^ [Y4: rule] : C
@ A ) )
= bot_bo4530011010606209517ist_fm ) )
& ( ( A != bot_bot_set_rule )
=> ( ( comple1649742678916152942ist_fm
@ ( image_3897101447128207157ist_fm
@ ^ [Y4: rule] : C
@ A ) )
= C ) ) ) ).
% UN_constant
thf(fact_774_UN__constant,axiom,
! [A: set_o,C: set_rule] :
( ( ( A = bot_bot_set_o )
=> ( ( comple2146307154184993742t_rule
@ ( image_o_set_rule
@ ^ [Y4: $o] : C
@ A ) )
= bot_bot_set_rule ) )
& ( ( A != bot_bot_set_o )
=> ( ( comple2146307154184993742t_rule
@ ( image_o_set_rule
@ ^ [Y4: $o] : C
@ A ) )
= C ) ) ) ).
% UN_constant
thf(fact_775_UN__constant,axiom,
! [A: set_o,C: set_o] :
( ( ( A = bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [Y4: $o] : C
@ A ) )
= bot_bot_set_o ) )
& ( ( A != bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [Y4: $o] : C
@ A ) )
= C ) ) ) ).
% UN_constant
thf(fact_776_UN__constant,axiom,
! [A: set_o,C: set_Pr7058068377845519745ist_fm] :
( ( ( A = bot_bot_set_o )
=> ( ( comple1649742678916152942ist_fm
@ ( image_1666948324336172654ist_fm
@ ^ [Y4: $o] : C
@ A ) )
= bot_bo4530011010606209517ist_fm ) )
& ( ( A != bot_bot_set_o )
=> ( ( comple1649742678916152942ist_fm
@ ( image_1666948324336172654ist_fm
@ ^ [Y4: $o] : C
@ A ) )
= C ) ) ) ).
% UN_constant
thf(fact_777_UN__constant,axiom,
! [A: set_Pr7058068377845519745ist_fm,C: set_rule] :
( ( ( A = bot_bo4530011010606209517ist_fm )
=> ( ( comple2146307154184993742t_rule
@ ( image_2008118025656205143t_rule
@ ^ [Y4: produc1996495991257130529ist_fm] : C
@ A ) )
= bot_bot_set_rule ) )
& ( ( A != bot_bo4530011010606209517ist_fm )
=> ( ( comple2146307154184993742t_rule
@ ( image_2008118025656205143t_rule
@ ^ [Y4: produc1996495991257130529ist_fm] : C
@ A ) )
= C ) ) ) ).
% UN_constant
thf(fact_778_UN__constant,axiom,
! [A: set_Pr7058068377845519745ist_fm,C: set_o] :
( ( ( A = bot_bo4530011010606209517ist_fm )
=> ( ( comple90263536869209701_set_o
@ ( image_7087520594942641564_set_o
@ ^ [Y4: produc1996495991257130529ist_fm] : C
@ A ) )
= bot_bot_set_o ) )
& ( ( A != bot_bo4530011010606209517ist_fm )
=> ( ( comple90263536869209701_set_o
@ ( image_7087520594942641564_set_o
@ ^ [Y4: produc1996495991257130529ist_fm] : C
@ A ) )
= C ) ) ) ).
% UN_constant
thf(fact_779_UN__constant,axiom,
! [A: set_Pr7058068377845519745ist_fm,C: set_Pr7058068377845519745ist_fm] :
( ( ( A = bot_bo4530011010606209517ist_fm )
=> ( ( comple1649742678916152942ist_fm
@ ( image_5852275405373602981ist_fm
@ ^ [Y4: produc1996495991257130529ist_fm] : C
@ A ) )
= bot_bo4530011010606209517ist_fm ) )
& ( ( A != bot_bo4530011010606209517ist_fm )
=> ( ( comple1649742678916152942ist_fm
@ ( image_5852275405373602981ist_fm
@ ^ [Y4: produc1996495991257130529ist_fm] : C
@ A ) )
= C ) ) ) ).
% UN_constant
thf(fact_780_UN__simps_I1_J,axiom,
! [C4: set_rule,A3: rule,B: rule > set_rule] :
( ( ( C4 = bot_bot_set_rule )
=> ( ( comple2146307154184993742t_rule
@ ( image_rule_set_rule
@ ^ [X: rule] : ( insert_rule @ A3 @ ( B @ X ) )
@ C4 ) )
= bot_bot_set_rule ) )
& ( ( C4 != bot_bot_set_rule )
=> ( ( comple2146307154184993742t_rule
@ ( image_rule_set_rule
@ ^ [X: rule] : ( insert_rule @ A3 @ ( B @ X ) )
@ C4 ) )
= ( insert_rule @ A3 @ ( comple2146307154184993742t_rule @ ( image_rule_set_rule @ B @ C4 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_781_UN__simps_I1_J,axiom,
! [C4: set_rule,A3: $o,B: rule > set_o] :
( ( ( C4 = bot_bot_set_rule )
=> ( ( comple90263536869209701_set_o
@ ( image_rule_set_o
@ ^ [X: rule] : ( insert_o @ A3 @ ( B @ X ) )
@ C4 ) )
= bot_bot_set_o ) )
& ( ( C4 != bot_bot_set_rule )
=> ( ( comple90263536869209701_set_o
@ ( image_rule_set_o
@ ^ [X: rule] : ( insert_o @ A3 @ ( B @ X ) )
@ C4 ) )
= ( insert_o @ A3 @ ( comple90263536869209701_set_o @ ( image_rule_set_o @ B @ C4 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_782_UN__simps_I1_J,axiom,
! [C4: set_rule,A3: produc1996495991257130529ist_fm,B: rule > set_Pr7058068377845519745ist_fm] :
( ( ( C4 = bot_bot_set_rule )
=> ( ( comple1649742678916152942ist_fm
@ ( image_3897101447128207157ist_fm
@ ^ [X: rule] : ( insert589278167076416497ist_fm @ A3 @ ( B @ X ) )
@ C4 ) )
= bot_bo4530011010606209517ist_fm ) )
& ( ( C4 != bot_bot_set_rule )
=> ( ( comple1649742678916152942ist_fm
@ ( image_3897101447128207157ist_fm
@ ^ [X: rule] : ( insert589278167076416497ist_fm @ A3 @ ( B @ X ) )
@ C4 ) )
= ( insert589278167076416497ist_fm @ A3 @ ( comple1649742678916152942ist_fm @ ( image_3897101447128207157ist_fm @ B @ C4 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_783_UN__simps_I1_J,axiom,
! [C4: set_o,A3: rule,B: $o > set_rule] :
( ( ( C4 = bot_bot_set_o )
=> ( ( comple2146307154184993742t_rule
@ ( image_o_set_rule
@ ^ [X: $o] : ( insert_rule @ A3 @ ( B @ X ) )
@ C4 ) )
= bot_bot_set_rule ) )
& ( ( C4 != bot_bot_set_o )
=> ( ( comple2146307154184993742t_rule
@ ( image_o_set_rule
@ ^ [X: $o] : ( insert_rule @ A3 @ ( B @ X ) )
@ C4 ) )
= ( insert_rule @ A3 @ ( comple2146307154184993742t_rule @ ( image_o_set_rule @ B @ C4 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_784_UN__simps_I1_J,axiom,
! [C4: set_o,A3: $o,B: $o > set_o] :
( ( ( C4 = bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X: $o] : ( insert_o @ A3 @ ( B @ X ) )
@ C4 ) )
= bot_bot_set_o ) )
& ( ( C4 != bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X: $o] : ( insert_o @ A3 @ ( B @ X ) )
@ C4 ) )
= ( insert_o @ A3 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B @ C4 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_785_UN__simps_I1_J,axiom,
! [C4: set_o,A3: produc1996495991257130529ist_fm,B: $o > set_Pr7058068377845519745ist_fm] :
( ( ( C4 = bot_bot_set_o )
=> ( ( comple1649742678916152942ist_fm
@ ( image_1666948324336172654ist_fm
@ ^ [X: $o] : ( insert589278167076416497ist_fm @ A3 @ ( B @ X ) )
@ C4 ) )
= bot_bo4530011010606209517ist_fm ) )
& ( ( C4 != bot_bot_set_o )
=> ( ( comple1649742678916152942ist_fm
@ ( image_1666948324336172654ist_fm
@ ^ [X: $o] : ( insert589278167076416497ist_fm @ A3 @ ( B @ X ) )
@ C4 ) )
= ( insert589278167076416497ist_fm @ A3 @ ( comple1649742678916152942ist_fm @ ( image_1666948324336172654ist_fm @ B @ C4 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_786_UN__simps_I1_J,axiom,
! [C4: set_Pr7058068377845519745ist_fm,A3: rule,B: produc1996495991257130529ist_fm > set_rule] :
( ( ( C4 = bot_bo4530011010606209517ist_fm )
=> ( ( comple2146307154184993742t_rule
@ ( image_2008118025656205143t_rule
@ ^ [X: produc1996495991257130529ist_fm] : ( insert_rule @ A3 @ ( B @ X ) )
@ C4 ) )
= bot_bot_set_rule ) )
& ( ( C4 != bot_bo4530011010606209517ist_fm )
=> ( ( comple2146307154184993742t_rule
@ ( image_2008118025656205143t_rule
@ ^ [X: produc1996495991257130529ist_fm] : ( insert_rule @ A3 @ ( B @ X ) )
@ C4 ) )
= ( insert_rule @ A3 @ ( comple2146307154184993742t_rule @ ( image_2008118025656205143t_rule @ B @ C4 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_787_UN__simps_I1_J,axiom,
! [C4: set_Pr7058068377845519745ist_fm,A3: $o,B: produc1996495991257130529ist_fm > set_o] :
( ( ( C4 = bot_bo4530011010606209517ist_fm )
=> ( ( comple90263536869209701_set_o
@ ( image_7087520594942641564_set_o
@ ^ [X: produc1996495991257130529ist_fm] : ( insert_o @ A3 @ ( B @ X ) )
@ C4 ) )
= bot_bot_set_o ) )
& ( ( C4 != bot_bo4530011010606209517ist_fm )
=> ( ( comple90263536869209701_set_o
@ ( image_7087520594942641564_set_o
@ ^ [X: produc1996495991257130529ist_fm] : ( insert_o @ A3 @ ( B @ X ) )
@ C4 ) )
= ( insert_o @ A3 @ ( comple90263536869209701_set_o @ ( image_7087520594942641564_set_o @ B @ C4 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_788_UN__simps_I1_J,axiom,
! [C4: set_Pr7058068377845519745ist_fm,A3: produc1996495991257130529ist_fm,B: produc1996495991257130529ist_fm > set_Pr7058068377845519745ist_fm] :
( ( ( C4 = bot_bo4530011010606209517ist_fm )
=> ( ( comple1649742678916152942ist_fm
@ ( image_5852275405373602981ist_fm
@ ^ [X: produc1996495991257130529ist_fm] : ( insert589278167076416497ist_fm @ A3 @ ( B @ X ) )
@ C4 ) )
= bot_bo4530011010606209517ist_fm ) )
& ( ( C4 != bot_bo4530011010606209517ist_fm )
=> ( ( comple1649742678916152942ist_fm
@ ( image_5852275405373602981ist_fm
@ ^ [X: produc1996495991257130529ist_fm] : ( insert589278167076416497ist_fm @ A3 @ ( B @ X ) )
@ C4 ) )
= ( insert589278167076416497ist_fm @ A3 @ ( comple1649742678916152942ist_fm @ ( image_5852275405373602981ist_fm @ B @ C4 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_789_UN__singleton,axiom,
! [A: set_rule] :
( ( comple2146307154184993742t_rule
@ ( image_rule_set_rule
@ ^ [X: rule] : ( insert_rule @ X @ bot_bot_set_rule )
@ A ) )
= A ) ).
% UN_singleton
thf(fact_790_UN__singleton,axiom,
! [A: set_o] :
( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X: $o] : ( insert_o @ X @ bot_bot_set_o )
@ A ) )
= A ) ).
% UN_singleton
thf(fact_791_UN__singleton,axiom,
! [A: set_Pr7058068377845519745ist_fm] :
( ( comple1649742678916152942ist_fm
@ ( image_5852275405373602981ist_fm
@ ^ [X: produc1996495991257130529ist_fm] : ( insert589278167076416497ist_fm @ X @ bot_bo4530011010606209517ist_fm )
@ A ) )
= A ) ).
% UN_singleton
thf(fact_792_INT__constant,axiom,
! [A: set_rule,C: set_Pr7058068377845519745ist_fm] :
( ( ( A = bot_bot_set_rule )
=> ( ( comple5608129641975563656ist_fm
@ ( image_3897101447128207157ist_fm
@ ^ [Y4: rule] : C
@ A ) )
= top_to728902761071950545ist_fm ) )
& ( ( A != bot_bot_set_rule )
=> ( ( comple5608129641975563656ist_fm
@ ( image_3897101447128207157ist_fm
@ ^ [Y4: rule] : C
@ A ) )
= C ) ) ) ).
% INT_constant
thf(fact_793_INT__constant,axiom,
! [A: set_o,C: set_Pr7058068377845519745ist_fm] :
( ( ( A = bot_bot_set_o )
=> ( ( comple5608129641975563656ist_fm
@ ( image_1666948324336172654ist_fm
@ ^ [Y4: $o] : C
@ A ) )
= top_to728902761071950545ist_fm ) )
& ( ( A != bot_bot_set_o )
=> ( ( comple5608129641975563656ist_fm
@ ( image_1666948324336172654ist_fm
@ ^ [Y4: $o] : C
@ A ) )
= C ) ) ) ).
% INT_constant
thf(fact_794_INT__constant,axiom,
! [A: set_Pr7058068377845519745ist_fm,C: set_Pr7058068377845519745ist_fm] :
( ( ( A = bot_bo4530011010606209517ist_fm )
=> ( ( comple5608129641975563656ist_fm
@ ( image_5852275405373602981ist_fm
@ ^ [Y4: produc1996495991257130529ist_fm] : C
@ A ) )
= top_to728902761071950545ist_fm ) )
& ( ( A != bot_bo4530011010606209517ist_fm )
=> ( ( comple5608129641975563656ist_fm
@ ( image_5852275405373602981ist_fm
@ ^ [Y4: produc1996495991257130529ist_fm] : C
@ A ) )
= C ) ) ) ).
% INT_constant
thf(fact_795_INT__constant,axiom,
! [A: set_rule,C: set_rule] :
( ( ( A = bot_bot_set_rule )
=> ( ( comple5773694076043965236t_rule
@ ( image_rule_set_rule
@ ^ [Y4: rule] : C
@ A ) )
= top_top_set_rule ) )
& ( ( A != bot_bot_set_rule )
=> ( ( comple5773694076043965236t_rule
@ ( image_rule_set_rule
@ ^ [Y4: rule] : C
@ A ) )
= C ) ) ) ).
% INT_constant
thf(fact_796_INT__constant,axiom,
! [A: set_o,C: set_rule] :
( ( ( A = bot_bot_set_o )
=> ( ( comple5773694076043965236t_rule
@ ( image_o_set_rule
@ ^ [Y4: $o] : C
@ A ) )
= top_top_set_rule ) )
& ( ( A != bot_bot_set_o )
=> ( ( comple5773694076043965236t_rule
@ ( image_o_set_rule
@ ^ [Y4: $o] : C
@ A ) )
= C ) ) ) ).
% INT_constant
thf(fact_797_INT__constant,axiom,
! [A: set_Pr7058068377845519745ist_fm,C: set_rule] :
( ( ( A = bot_bo4530011010606209517ist_fm )
=> ( ( comple5773694076043965236t_rule
@ ( image_2008118025656205143t_rule
@ ^ [Y4: produc1996495991257130529ist_fm] : C
@ A ) )
= top_top_set_rule ) )
& ( ( A != bot_bo4530011010606209517ist_fm )
=> ( ( comple5773694076043965236t_rule
@ ( image_2008118025656205143t_rule
@ ^ [Y4: produc1996495991257130529ist_fm] : C
@ A ) )
= C ) ) ) ).
% INT_constant
thf(fact_798_INT__constant,axiom,
! [A: set_rule,C: set_o] :
( ( ( A = bot_bot_set_rule )
=> ( ( comple3063163877087187839_set_o
@ ( image_rule_set_o
@ ^ [Y4: rule] : C
@ A ) )
= top_top_set_o ) )
& ( ( A != bot_bot_set_rule )
=> ( ( comple3063163877087187839_set_o
@ ( image_rule_set_o
@ ^ [Y4: rule] : C
@ A ) )
= C ) ) ) ).
% INT_constant
thf(fact_799_INT__constant,axiom,
! [A: set_o,C: set_o] :
( ( ( A = bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [Y4: $o] : C
@ A ) )
= top_top_set_o ) )
& ( ( A != bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [Y4: $o] : C
@ A ) )
= C ) ) ) ).
% INT_constant
thf(fact_800_INT__constant,axiom,
! [A: set_Pr7058068377845519745ist_fm,C: set_o] :
( ( ( A = bot_bo4530011010606209517ist_fm )
=> ( ( comple3063163877087187839_set_o
@ ( image_7087520594942641564_set_o
@ ^ [Y4: produc1996495991257130529ist_fm] : C
@ A ) )
= top_top_set_o ) )
& ( ( A != bot_bo4530011010606209517ist_fm )
=> ( ( comple3063163877087187839_set_o
@ ( image_7087520594942641564_set_o
@ ^ [Y4: produc1996495991257130529ist_fm] : C
@ A ) )
= C ) ) ) ).
% INT_constant
thf(fact_801_INT__simps_I4_J,axiom,
! [C4: set_rule,A: set_rule,B: rule > set_rule] :
( ( ( C4 = bot_bot_set_rule )
=> ( ( comple5773694076043965236t_rule
@ ( image_rule_set_rule
@ ^ [X: rule] : ( minus_minus_set_rule @ A @ ( B @ X ) )
@ C4 ) )
= top_top_set_rule ) )
& ( ( C4 != bot_bot_set_rule )
=> ( ( comple5773694076043965236t_rule
@ ( image_rule_set_rule
@ ^ [X: rule] : ( minus_minus_set_rule @ A @ ( B @ X ) )
@ C4 ) )
= ( minus_minus_set_rule @ A @ ( comple2146307154184993742t_rule @ ( image_rule_set_rule @ B @ C4 ) ) ) ) ) ) ).
% INT_simps(4)
thf(fact_802_INT__simps_I4_J,axiom,
! [C4: set_o,A: set_rule,B: $o > set_rule] :
( ( ( C4 = bot_bot_set_o )
=> ( ( comple5773694076043965236t_rule
@ ( image_o_set_rule
@ ^ [X: $o] : ( minus_minus_set_rule @ A @ ( B @ X ) )
@ C4 ) )
= top_top_set_rule ) )
& ( ( C4 != bot_bot_set_o )
=> ( ( comple5773694076043965236t_rule
@ ( image_o_set_rule
@ ^ [X: $o] : ( minus_minus_set_rule @ A @ ( B @ X ) )
@ C4 ) )
= ( minus_minus_set_rule @ A @ ( comple2146307154184993742t_rule @ ( image_o_set_rule @ B @ C4 ) ) ) ) ) ) ).
% INT_simps(4)
thf(fact_803_INT__simps_I4_J,axiom,
! [C4: set_Pr7058068377845519745ist_fm,A: set_rule,B: produc1996495991257130529ist_fm > set_rule] :
( ( ( C4 = bot_bo4530011010606209517ist_fm )
=> ( ( comple5773694076043965236t_rule
@ ( image_2008118025656205143t_rule
@ ^ [X: produc1996495991257130529ist_fm] : ( minus_minus_set_rule @ A @ ( B @ X ) )
@ C4 ) )
= top_top_set_rule ) )
& ( ( C4 != bot_bo4530011010606209517ist_fm )
=> ( ( comple5773694076043965236t_rule
@ ( image_2008118025656205143t_rule
@ ^ [X: produc1996495991257130529ist_fm] : ( minus_minus_set_rule @ A @ ( B @ X ) )
@ C4 ) )
= ( minus_minus_set_rule @ A @ ( comple2146307154184993742t_rule @ ( image_2008118025656205143t_rule @ B @ C4 ) ) ) ) ) ) ).
% INT_simps(4)
thf(fact_804_INT__simps_I4_J,axiom,
! [C4: set_rule,A: set_o,B: rule > set_o] :
( ( ( C4 = bot_bot_set_rule )
=> ( ( comple3063163877087187839_set_o
@ ( image_rule_set_o
@ ^ [X: rule] : ( minus_minus_set_o @ A @ ( B @ X ) )
@ C4 ) )
= top_top_set_o ) )
& ( ( C4 != bot_bot_set_rule )
=> ( ( comple3063163877087187839_set_o
@ ( image_rule_set_o
@ ^ [X: rule] : ( minus_minus_set_o @ A @ ( B @ X ) )
@ C4 ) )
= ( minus_minus_set_o @ A @ ( comple90263536869209701_set_o @ ( image_rule_set_o @ B @ C4 ) ) ) ) ) ) ).
% INT_simps(4)
thf(fact_805_INT__simps_I4_J,axiom,
! [C4: set_o,A: set_o,B: $o > set_o] :
( ( ( C4 = bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [X: $o] : ( minus_minus_set_o @ A @ ( B @ X ) )
@ C4 ) )
= top_top_set_o ) )
& ( ( C4 != bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [X: $o] : ( minus_minus_set_o @ A @ ( B @ X ) )
@ C4 ) )
= ( minus_minus_set_o @ A @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B @ C4 ) ) ) ) ) ) ).
% INT_simps(4)
thf(fact_806_INT__simps_I4_J,axiom,
! [C4: set_Pr7058068377845519745ist_fm,A: set_o,B: produc1996495991257130529ist_fm > set_o] :
( ( ( C4 = bot_bo4530011010606209517ist_fm )
=> ( ( comple3063163877087187839_set_o
@ ( image_7087520594942641564_set_o
@ ^ [X: produc1996495991257130529ist_fm] : ( minus_minus_set_o @ A @ ( B @ X ) )
@ C4 ) )
= top_top_set_o ) )
& ( ( C4 != bot_bo4530011010606209517ist_fm )
=> ( ( comple3063163877087187839_set_o
@ ( image_7087520594942641564_set_o
@ ^ [X: produc1996495991257130529ist_fm] : ( minus_minus_set_o @ A @ ( B @ X ) )
@ C4 ) )
= ( minus_minus_set_o @ A @ ( comple90263536869209701_set_o @ ( image_7087520594942641564_set_o @ B @ C4 ) ) ) ) ) ) ).
% INT_simps(4)
thf(fact_807_INT__simps_I4_J,axiom,
! [C4: set_rule,A: set_Pr7058068377845519745ist_fm,B: rule > set_Pr7058068377845519745ist_fm] :
( ( ( C4 = bot_bot_set_rule )
=> ( ( comple5608129641975563656ist_fm
@ ( image_3897101447128207157ist_fm
@ ^ [X: rule] : ( minus_5615909706921521512ist_fm @ A @ ( B @ X ) )
@ C4 ) )
= top_to728902761071950545ist_fm ) )
& ( ( C4 != bot_bot_set_rule )
=> ( ( comple5608129641975563656ist_fm
@ ( image_3897101447128207157ist_fm
@ ^ [X: rule] : ( minus_5615909706921521512ist_fm @ A @ ( B @ X ) )
@ C4 ) )
= ( minus_5615909706921521512ist_fm @ A @ ( comple1649742678916152942ist_fm @ ( image_3897101447128207157ist_fm @ B @ C4 ) ) ) ) ) ) ).
% INT_simps(4)
thf(fact_808_INT__simps_I4_J,axiom,
! [C4: set_o,A: set_Pr7058068377845519745ist_fm,B: $o > set_Pr7058068377845519745ist_fm] :
( ( ( C4 = bot_bot_set_o )
=> ( ( comple5608129641975563656ist_fm
@ ( image_1666948324336172654ist_fm
@ ^ [X: $o] : ( minus_5615909706921521512ist_fm @ A @ ( B @ X ) )
@ C4 ) )
= top_to728902761071950545ist_fm ) )
& ( ( C4 != bot_bot_set_o )
=> ( ( comple5608129641975563656ist_fm
@ ( image_1666948324336172654ist_fm
@ ^ [X: $o] : ( minus_5615909706921521512ist_fm @ A @ ( B @ X ) )
@ C4 ) )
= ( minus_5615909706921521512ist_fm @ A @ ( comple1649742678916152942ist_fm @ ( image_1666948324336172654ist_fm @ B @ C4 ) ) ) ) ) ) ).
% INT_simps(4)
thf(fact_809_INT__simps_I4_J,axiom,
! [C4: set_Pr7058068377845519745ist_fm,A: set_Pr7058068377845519745ist_fm,B: produc1996495991257130529ist_fm > set_Pr7058068377845519745ist_fm] :
( ( ( C4 = bot_bo4530011010606209517ist_fm )
=> ( ( comple5608129641975563656ist_fm
@ ( image_5852275405373602981ist_fm
@ ^ [X: produc1996495991257130529ist_fm] : ( minus_5615909706921521512ist_fm @ A @ ( B @ X ) )
@ C4 ) )
= top_to728902761071950545ist_fm ) )
& ( ( C4 != bot_bo4530011010606209517ist_fm )
=> ( ( comple5608129641975563656ist_fm
@ ( image_5852275405373602981ist_fm
@ ^ [X: produc1996495991257130529ist_fm] : ( minus_5615909706921521512ist_fm @ A @ ( B @ X ) )
@ C4 ) )
= ( minus_5615909706921521512ist_fm @ A @ ( comple1649742678916152942ist_fm @ ( image_5852275405373602981ist_fm @ B @ C4 ) ) ) ) ) ) ).
% INT_simps(4)
thf(fact_810_inj__Some,axiom,
! [A: set_fs7534060674568567527ist_fm] : ( inj_on1722708713623408887ist_fm @ some_f2162114139285431020ist_fm @ A ) ).
% inj_Some
thf(fact_811_Inter__empty,axiom,
( ( comple5608129641975563656ist_fm @ bot_bo414211191905506637ist_fm )
= top_to728902761071950545ist_fm ) ).
% Inter_empty
thf(fact_812_Inter__empty,axiom,
( ( comple5773694076043965236t_rule @ bot_bot_set_set_rule )
= top_top_set_rule ) ).
% Inter_empty
thf(fact_813_Inter__empty,axiom,
( ( comple3063163877087187839_set_o @ bot_bot_set_set_o )
= top_top_set_o ) ).
% Inter_empty
thf(fact_814_Inter__UNIV,axiom,
( ( comple5773694076043965236t_rule @ top_top_set_set_rule )
= bot_bot_set_rule ) ).
% Inter_UNIV
thf(fact_815_Inter__UNIV,axiom,
( ( comple3063163877087187839_set_o @ top_top_set_set_o )
= bot_bot_set_o ) ).
% Inter_UNIV
thf(fact_816_Inter__UNIV,axiom,
( ( comple5608129641975563656ist_fm @ top_to7149953857590879793ist_fm )
= bot_bo4530011010606209517ist_fm ) ).
% Inter_UNIV
thf(fact_817_Union__UNIV,axiom,
( ( comple1649742678916152942ist_fm @ top_to7149953857590879793ist_fm )
= top_to728902761071950545ist_fm ) ).
% Union_UNIV
thf(fact_818_Union__UNIV,axiom,
( ( comple2146307154184993742t_rule @ top_top_set_set_rule )
= top_top_set_rule ) ).
% Union_UNIV
thf(fact_819_Union__UNIV,axiom,
( ( comple90263536869209701_set_o @ top_top_set_set_o )
= top_top_set_o ) ).
% Union_UNIV
thf(fact_820_Union__empty,axiom,
( ( comple2146307154184993742t_rule @ bot_bot_set_set_rule )
= bot_bot_set_rule ) ).
% Union_empty
thf(fact_821_Union__empty,axiom,
( ( comple90263536869209701_set_o @ bot_bot_set_set_o )
= bot_bot_set_o ) ).
% Union_empty
thf(fact_822_Union__empty,axiom,
( ( comple1649742678916152942ist_fm @ bot_bo414211191905506637ist_fm )
= bot_bo4530011010606209517ist_fm ) ).
% Union_empty
thf(fact_823_Union__empty__conv,axiom,
! [A: set_set_rule] :
( ( ( comple2146307154184993742t_rule @ A )
= bot_bot_set_rule )
= ( ! [X: set_rule] :
( ( member_set_rule @ X @ A )
=> ( X = bot_bot_set_rule ) ) ) ) ).
% Union_empty_conv
thf(fact_824_Union__empty__conv,axiom,
! [A: set_set_o] :
( ( ( comple90263536869209701_set_o @ A )
= bot_bot_set_o )
= ( ! [X: set_o] :
( ( member_set_o @ X @ A )
=> ( X = bot_bot_set_o ) ) ) ) ).
% Union_empty_conv
thf(fact_825_Union__empty__conv,axiom,
! [A: set_se6775865681059004641ist_fm] :
( ( ( comple1649742678916152942ist_fm @ A )
= bot_bo4530011010606209517ist_fm )
= ( ! [X: set_Pr7058068377845519745ist_fm] :
( ( member8436620017354071594ist_fm @ X @ A )
=> ( X = bot_bo4530011010606209517ist_fm ) ) ) ) ).
% Union_empty_conv
thf(fact_826_empty__Union__conv,axiom,
! [A: set_set_rule] :
( ( bot_bot_set_rule
= ( comple2146307154184993742t_rule @ A ) )
= ( ! [X: set_rule] :
( ( member_set_rule @ X @ A )
=> ( X = bot_bot_set_rule ) ) ) ) ).
% empty_Union_conv
thf(fact_827_empty__Union__conv,axiom,
! [A: set_set_o] :
( ( bot_bot_set_o
= ( comple90263536869209701_set_o @ A ) )
= ( ! [X: set_o] :
( ( member_set_o @ X @ A )
=> ( X = bot_bot_set_o ) ) ) ) ).
% empty_Union_conv
thf(fact_828_empty__Union__conv,axiom,
! [A: set_se6775865681059004641ist_fm] :
( ( bot_bo4530011010606209517ist_fm
= ( comple1649742678916152942ist_fm @ A ) )
= ( ! [X: set_Pr7058068377845519745ist_fm] :
( ( member8436620017354071594ist_fm @ X @ A )
=> ( X = bot_bo4530011010606209517ist_fm ) ) ) ) ).
% empty_Union_conv
thf(fact_829_inj__on__convol__ident,axiom,
! [F: list_fm > list_fm,X7: set_list_fm] :
( inj_on733428828453976846ist_fm
@ ^ [X: list_fm] : ( produc381145313068854617ist_fm @ X @ ( F @ X ) )
@ X7 ) ).
% inj_on_convol_ident
thf(fact_830_INT__E,axiom,
! [B3: produc1996495991257130529ist_fm,B: produc1996495991257130529ist_fm > set_Pr7058068377845519745ist_fm,A: set_Pr7058068377845519745ist_fm,A3: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ B3 @ ( comple5608129641975563656ist_fm @ ( image_5852275405373602981ist_fm @ B @ A ) ) )
=> ( ~ ( member8102475879199740618ist_fm @ B3 @ ( B @ A3 ) )
=> ~ ( member8102475879199740618ist_fm @ A3 @ A ) ) ) ).
% INT_E
thf(fact_831_INT__E,axiom,
! [B3: produc1996495991257130529ist_fm,B: rule > set_Pr7058068377845519745ist_fm,A: set_rule,A3: rule] :
( ( member8102475879199740618ist_fm @ B3 @ ( comple5608129641975563656ist_fm @ ( image_3897101447128207157ist_fm @ B @ A ) ) )
=> ( ~ ( member8102475879199740618ist_fm @ B3 @ ( B @ A3 ) )
=> ~ ( member_rule @ A3 @ A ) ) ) ).
% INT_E
thf(fact_832_INT__E,axiom,
! [B3: produc1996495991257130529ist_fm,B: $o > set_Pr7058068377845519745ist_fm,A: set_o,A3: $o] :
( ( member8102475879199740618ist_fm @ B3 @ ( comple5608129641975563656ist_fm @ ( image_1666948324336172654ist_fm @ B @ A ) ) )
=> ( ~ ( member8102475879199740618ist_fm @ B3 @ ( B @ A3 ) )
=> ~ ( member_o @ A3 @ A ) ) ) ).
% INT_E
thf(fact_833_INT__E,axiom,
! [B3: rule,B: produc1996495991257130529ist_fm > set_rule,A: set_Pr7058068377845519745ist_fm,A3: produc1996495991257130529ist_fm] :
( ( member_rule @ B3 @ ( comple5773694076043965236t_rule @ ( image_2008118025656205143t_rule @ B @ A ) ) )
=> ( ~ ( member_rule @ B3 @ ( B @ A3 ) )
=> ~ ( member8102475879199740618ist_fm @ A3 @ A ) ) ) ).
% INT_E
thf(fact_834_INT__E,axiom,
! [B3: rule,B: rule > set_rule,A: set_rule,A3: rule] :
( ( member_rule @ B3 @ ( comple5773694076043965236t_rule @ ( image_rule_set_rule @ B @ A ) ) )
=> ( ~ ( member_rule @ B3 @ ( B @ A3 ) )
=> ~ ( member_rule @ A3 @ A ) ) ) ).
% INT_E
thf(fact_835_INT__E,axiom,
! [B3: rule,B: $o > set_rule,A: set_o,A3: $o] :
( ( member_rule @ B3 @ ( comple5773694076043965236t_rule @ ( image_o_set_rule @ B @ A ) ) )
=> ( ~ ( member_rule @ B3 @ ( B @ A3 ) )
=> ~ ( member_o @ A3 @ A ) ) ) ).
% INT_E
thf(fact_836_INT__E,axiom,
! [B3: $o,B: produc1996495991257130529ist_fm > set_o,A: set_Pr7058068377845519745ist_fm,A3: produc1996495991257130529ist_fm] :
( ( member_o @ B3 @ ( comple3063163877087187839_set_o @ ( image_7087520594942641564_set_o @ B @ A ) ) )
=> ( ~ ( member_o @ B3 @ ( B @ A3 ) )
=> ~ ( member8102475879199740618ist_fm @ A3 @ A ) ) ) ).
% INT_E
thf(fact_837_INT__E,axiom,
! [B3: $o,B: rule > set_o,A: set_rule,A3: rule] :
( ( member_o @ B3 @ ( comple3063163877087187839_set_o @ ( image_rule_set_o @ B @ A ) ) )
=> ( ~ ( member_o @ B3 @ ( B @ A3 ) )
=> ~ ( member_rule @ A3 @ A ) ) ) ).
% INT_E
thf(fact_838_INT__E,axiom,
! [B3: $o,B: $o > set_o,A: set_o,A3: $o] :
( ( member_o @ B3 @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ B @ A ) ) )
=> ( ~ ( member_o @ B3 @ ( B @ A3 ) )
=> ~ ( member_o @ A3 @ A ) ) ) ).
% INT_E
thf(fact_839_INT__D,axiom,
! [B3: produc1996495991257130529ist_fm,B: produc1996495991257130529ist_fm > set_Pr7058068377845519745ist_fm,A: set_Pr7058068377845519745ist_fm,A3: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ B3 @ ( comple5608129641975563656ist_fm @ ( image_5852275405373602981ist_fm @ B @ A ) ) )
=> ( ( member8102475879199740618ist_fm @ A3 @ A )
=> ( member8102475879199740618ist_fm @ B3 @ ( B @ A3 ) ) ) ) ).
% INT_D
thf(fact_840_INT__D,axiom,
! [B3: produc1996495991257130529ist_fm,B: rule > set_Pr7058068377845519745ist_fm,A: set_rule,A3: rule] :
( ( member8102475879199740618ist_fm @ B3 @ ( comple5608129641975563656ist_fm @ ( image_3897101447128207157ist_fm @ B @ A ) ) )
=> ( ( member_rule @ A3 @ A )
=> ( member8102475879199740618ist_fm @ B3 @ ( B @ A3 ) ) ) ) ).
% INT_D
thf(fact_841_INT__D,axiom,
! [B3: produc1996495991257130529ist_fm,B: $o > set_Pr7058068377845519745ist_fm,A: set_o,A3: $o] :
( ( member8102475879199740618ist_fm @ B3 @ ( comple5608129641975563656ist_fm @ ( image_1666948324336172654ist_fm @ B @ A ) ) )
=> ( ( member_o @ A3 @ A )
=> ( member8102475879199740618ist_fm @ B3 @ ( B @ A3 ) ) ) ) ).
% INT_D
thf(fact_842_INT__D,axiom,
! [B3: rule,B: produc1996495991257130529ist_fm > set_rule,A: set_Pr7058068377845519745ist_fm,A3: produc1996495991257130529ist_fm] :
( ( member_rule @ B3 @ ( comple5773694076043965236t_rule @ ( image_2008118025656205143t_rule @ B @ A ) ) )
=> ( ( member8102475879199740618ist_fm @ A3 @ A )
=> ( member_rule @ B3 @ ( B @ A3 ) ) ) ) ).
% INT_D
thf(fact_843_INT__D,axiom,
! [B3: rule,B: rule > set_rule,A: set_rule,A3: rule] :
( ( member_rule @ B3 @ ( comple5773694076043965236t_rule @ ( image_rule_set_rule @ B @ A ) ) )
=> ( ( member_rule @ A3 @ A )
=> ( member_rule @ B3 @ ( B @ A3 ) ) ) ) ).
% INT_D
thf(fact_844_INT__D,axiom,
! [B3: rule,B: $o > set_rule,A: set_o,A3: $o] :
( ( member_rule @ B3 @ ( comple5773694076043965236t_rule @ ( image_o_set_rule @ B @ A ) ) )
=> ( ( member_o @ A3 @ A )
=> ( member_rule @ B3 @ ( B @ A3 ) ) ) ) ).
% INT_D
thf(fact_845_INT__D,axiom,
! [B3: $o,B: produc1996495991257130529ist_fm > set_o,A: set_Pr7058068377845519745ist_fm,A3: produc1996495991257130529ist_fm] :
( ( member_o @ B3 @ ( comple3063163877087187839_set_o @ ( image_7087520594942641564_set_o @ B @ A ) ) )
=> ( ( member8102475879199740618ist_fm @ A3 @ A )
=> ( member_o @ B3 @ ( B @ A3 ) ) ) ) ).
% INT_D
thf(fact_846_INT__D,axiom,
! [B3: $o,B: rule > set_o,A: set_rule,A3: rule] :
( ( member_o @ B3 @ ( comple3063163877087187839_set_o @ ( image_rule_set_o @ B @ A ) ) )
=> ( ( member_rule @ A3 @ A )
=> ( member_o @ B3 @ ( B @ A3 ) ) ) ) ).
% INT_D
thf(fact_847_INT__D,axiom,
! [B3: $o,B: $o > set_o,A: set_o,A3: $o] :
( ( member_o @ B3 @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ B @ A ) ) )
=> ( ( member_o @ A3 @ A )
=> ( member_o @ B3 @ ( B @ A3 ) ) ) ) ).
% INT_D
thf(fact_848_UN__E,axiom,
! [B3: produc1996495991257130529ist_fm,B: produc1996495991257130529ist_fm > set_Pr7058068377845519745ist_fm,A: set_Pr7058068377845519745ist_fm] :
( ( member8102475879199740618ist_fm @ B3 @ ( comple1649742678916152942ist_fm @ ( image_5852275405373602981ist_fm @ B @ A ) ) )
=> ~ ! [X3: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ X3 @ A )
=> ~ ( member8102475879199740618ist_fm @ B3 @ ( B @ X3 ) ) ) ) ).
% UN_E
thf(fact_849_UN__E,axiom,
! [B3: produc1996495991257130529ist_fm,B: rule > set_Pr7058068377845519745ist_fm,A: set_rule] :
( ( member8102475879199740618ist_fm @ B3 @ ( comple1649742678916152942ist_fm @ ( image_3897101447128207157ist_fm @ B @ A ) ) )
=> ~ ! [X3: rule] :
( ( member_rule @ X3 @ A )
=> ~ ( member8102475879199740618ist_fm @ B3 @ ( B @ X3 ) ) ) ) ).
% UN_E
thf(fact_850_UN__E,axiom,
! [B3: produc1996495991257130529ist_fm,B: $o > set_Pr7058068377845519745ist_fm,A: set_o] :
( ( member8102475879199740618ist_fm @ B3 @ ( comple1649742678916152942ist_fm @ ( image_1666948324336172654ist_fm @ B @ A ) ) )
=> ~ ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ~ ( member8102475879199740618ist_fm @ B3 @ ( B @ X3 ) ) ) ) ).
% UN_E
thf(fact_851_UN__E,axiom,
! [B3: rule,B: produc1996495991257130529ist_fm > set_rule,A: set_Pr7058068377845519745ist_fm] :
( ( member_rule @ B3 @ ( comple2146307154184993742t_rule @ ( image_2008118025656205143t_rule @ B @ A ) ) )
=> ~ ! [X3: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ X3 @ A )
=> ~ ( member_rule @ B3 @ ( B @ X3 ) ) ) ) ).
% UN_E
thf(fact_852_UN__E,axiom,
! [B3: rule,B: rule > set_rule,A: set_rule] :
( ( member_rule @ B3 @ ( comple2146307154184993742t_rule @ ( image_rule_set_rule @ B @ A ) ) )
=> ~ ! [X3: rule] :
( ( member_rule @ X3 @ A )
=> ~ ( member_rule @ B3 @ ( B @ X3 ) ) ) ) ).
% UN_E
thf(fact_853_UN__E,axiom,
! [B3: rule,B: $o > set_rule,A: set_o] :
( ( member_rule @ B3 @ ( comple2146307154184993742t_rule @ ( image_o_set_rule @ B @ A ) ) )
=> ~ ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ~ ( member_rule @ B3 @ ( B @ X3 ) ) ) ) ).
% UN_E
thf(fact_854_UN__E,axiom,
! [B3: $o,B: produc1996495991257130529ist_fm > set_o,A: set_Pr7058068377845519745ist_fm] :
( ( member_o @ B3 @ ( comple90263536869209701_set_o @ ( image_7087520594942641564_set_o @ B @ A ) ) )
=> ~ ! [X3: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ X3 @ A )
=> ~ ( member_o @ B3 @ ( B @ X3 ) ) ) ) ).
% UN_E
thf(fact_855_UN__E,axiom,
! [B3: $o,B: rule > set_o,A: set_rule] :
( ( member_o @ B3 @ ( comple90263536869209701_set_o @ ( image_rule_set_o @ B @ A ) ) )
=> ~ ! [X3: rule] :
( ( member_rule @ X3 @ A )
=> ~ ( member_o @ B3 @ ( B @ X3 ) ) ) ) ).
% UN_E
thf(fact_856_UN__E,axiom,
! [B3: $o,B: $o > set_o,A: set_o] :
( ( member_o @ B3 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B @ A ) ) )
=> ~ ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ~ ( member_o @ B3 @ ( B @ X3 ) ) ) ) ).
% UN_E
thf(fact_857_INT__extend__simps_I4_J,axiom,
! [C4: set_rule,A: set_rule,B: rule > set_rule] :
( ( ( C4 = bot_bot_set_rule )
=> ( ( minus_minus_set_rule @ A @ ( comple2146307154184993742t_rule @ ( image_rule_set_rule @ B @ C4 ) ) )
= A ) )
& ( ( C4 != bot_bot_set_rule )
=> ( ( minus_minus_set_rule @ A @ ( comple2146307154184993742t_rule @ ( image_rule_set_rule @ B @ C4 ) ) )
= ( comple5773694076043965236t_rule
@ ( image_rule_set_rule
@ ^ [X: rule] : ( minus_minus_set_rule @ A @ ( B @ X ) )
@ C4 ) ) ) ) ) ).
% INT_extend_simps(4)
thf(fact_858_INT__extend__simps_I4_J,axiom,
! [C4: set_o,A: set_rule,B: $o > set_rule] :
( ( ( C4 = bot_bot_set_o )
=> ( ( minus_minus_set_rule @ A @ ( comple2146307154184993742t_rule @ ( image_o_set_rule @ B @ C4 ) ) )
= A ) )
& ( ( C4 != bot_bot_set_o )
=> ( ( minus_minus_set_rule @ A @ ( comple2146307154184993742t_rule @ ( image_o_set_rule @ B @ C4 ) ) )
= ( comple5773694076043965236t_rule
@ ( image_o_set_rule
@ ^ [X: $o] : ( minus_minus_set_rule @ A @ ( B @ X ) )
@ C4 ) ) ) ) ) ).
% INT_extend_simps(4)
thf(fact_859_INT__extend__simps_I4_J,axiom,
! [C4: set_Pr7058068377845519745ist_fm,A: set_rule,B: produc1996495991257130529ist_fm > set_rule] :
( ( ( C4 = bot_bo4530011010606209517ist_fm )
=> ( ( minus_minus_set_rule @ A @ ( comple2146307154184993742t_rule @ ( image_2008118025656205143t_rule @ B @ C4 ) ) )
= A ) )
& ( ( C4 != bot_bo4530011010606209517ist_fm )
=> ( ( minus_minus_set_rule @ A @ ( comple2146307154184993742t_rule @ ( image_2008118025656205143t_rule @ B @ C4 ) ) )
= ( comple5773694076043965236t_rule
@ ( image_2008118025656205143t_rule
@ ^ [X: produc1996495991257130529ist_fm] : ( minus_minus_set_rule @ A @ ( B @ X ) )
@ C4 ) ) ) ) ) ).
% INT_extend_simps(4)
thf(fact_860_INT__extend__simps_I4_J,axiom,
! [C4: set_rule,A: set_o,B: rule > set_o] :
( ( ( C4 = bot_bot_set_rule )
=> ( ( minus_minus_set_o @ A @ ( comple90263536869209701_set_o @ ( image_rule_set_o @ B @ C4 ) ) )
= A ) )
& ( ( C4 != bot_bot_set_rule )
=> ( ( minus_minus_set_o @ A @ ( comple90263536869209701_set_o @ ( image_rule_set_o @ B @ C4 ) ) )
= ( comple3063163877087187839_set_o
@ ( image_rule_set_o
@ ^ [X: rule] : ( minus_minus_set_o @ A @ ( B @ X ) )
@ C4 ) ) ) ) ) ).
% INT_extend_simps(4)
thf(fact_861_INT__extend__simps_I4_J,axiom,
! [C4: set_o,A: set_o,B: $o > set_o] :
( ( ( C4 = bot_bot_set_o )
=> ( ( minus_minus_set_o @ A @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B @ C4 ) ) )
= A ) )
& ( ( C4 != bot_bot_set_o )
=> ( ( minus_minus_set_o @ A @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B @ C4 ) ) )
= ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [X: $o] : ( minus_minus_set_o @ A @ ( B @ X ) )
@ C4 ) ) ) ) ) ).
% INT_extend_simps(4)
thf(fact_862_INT__extend__simps_I4_J,axiom,
! [C4: set_Pr7058068377845519745ist_fm,A: set_o,B: produc1996495991257130529ist_fm > set_o] :
( ( ( C4 = bot_bo4530011010606209517ist_fm )
=> ( ( minus_minus_set_o @ A @ ( comple90263536869209701_set_o @ ( image_7087520594942641564_set_o @ B @ C4 ) ) )
= A ) )
& ( ( C4 != bot_bo4530011010606209517ist_fm )
=> ( ( minus_minus_set_o @ A @ ( comple90263536869209701_set_o @ ( image_7087520594942641564_set_o @ B @ C4 ) ) )
= ( comple3063163877087187839_set_o
@ ( image_7087520594942641564_set_o
@ ^ [X: produc1996495991257130529ist_fm] : ( minus_minus_set_o @ A @ ( B @ X ) )
@ C4 ) ) ) ) ) ).
% INT_extend_simps(4)
thf(fact_863_INT__extend__simps_I4_J,axiom,
! [C4: set_rule,A: set_Pr7058068377845519745ist_fm,B: rule > set_Pr7058068377845519745ist_fm] :
( ( ( C4 = bot_bot_set_rule )
=> ( ( minus_5615909706921521512ist_fm @ A @ ( comple1649742678916152942ist_fm @ ( image_3897101447128207157ist_fm @ B @ C4 ) ) )
= A ) )
& ( ( C4 != bot_bot_set_rule )
=> ( ( minus_5615909706921521512ist_fm @ A @ ( comple1649742678916152942ist_fm @ ( image_3897101447128207157ist_fm @ B @ C4 ) ) )
= ( comple5608129641975563656ist_fm
@ ( image_3897101447128207157ist_fm
@ ^ [X: rule] : ( minus_5615909706921521512ist_fm @ A @ ( B @ X ) )
@ C4 ) ) ) ) ) ).
% INT_extend_simps(4)
thf(fact_864_INT__extend__simps_I4_J,axiom,
! [C4: set_o,A: set_Pr7058068377845519745ist_fm,B: $o > set_Pr7058068377845519745ist_fm] :
( ( ( C4 = bot_bot_set_o )
=> ( ( minus_5615909706921521512ist_fm @ A @ ( comple1649742678916152942ist_fm @ ( image_1666948324336172654ist_fm @ B @ C4 ) ) )
= A ) )
& ( ( C4 != bot_bot_set_o )
=> ( ( minus_5615909706921521512ist_fm @ A @ ( comple1649742678916152942ist_fm @ ( image_1666948324336172654ist_fm @ B @ C4 ) ) )
= ( comple5608129641975563656ist_fm
@ ( image_1666948324336172654ist_fm
@ ^ [X: $o] : ( minus_5615909706921521512ist_fm @ A @ ( B @ X ) )
@ C4 ) ) ) ) ) ).
% INT_extend_simps(4)
thf(fact_865_INT__extend__simps_I4_J,axiom,
! [C4: set_Pr7058068377845519745ist_fm,A: set_Pr7058068377845519745ist_fm,B: produc1996495991257130529ist_fm > set_Pr7058068377845519745ist_fm] :
( ( ( C4 = bot_bo4530011010606209517ist_fm )
=> ( ( minus_5615909706921521512ist_fm @ A @ ( comple1649742678916152942ist_fm @ ( image_5852275405373602981ist_fm @ B @ C4 ) ) )
= A ) )
& ( ( C4 != bot_bo4530011010606209517ist_fm )
=> ( ( minus_5615909706921521512ist_fm @ A @ ( comple1649742678916152942ist_fm @ ( image_5852275405373602981ist_fm @ B @ C4 ) ) )
= ( comple5608129641975563656ist_fm
@ ( image_5852275405373602981ist_fm
@ ^ [X: produc1996495991257130529ist_fm] : ( minus_5615909706921521512ist_fm @ A @ ( B @ X ) )
@ C4 ) ) ) ) ) ).
% INT_extend_simps(4)
thf(fact_866_range__ex1__eq,axiom,
! [F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,B3: produc1996495991257130529ist_fm] :
( ( inj_on2392985232935137457ist_fm @ F @ top_to728902761071950545ist_fm )
=> ( ( member8102475879199740618ist_fm @ B3 @ ( image_6982201141141708101ist_fm @ F @ top_to728902761071950545ist_fm ) )
= ( ? [X: produc1996495991257130529ist_fm] :
( ( B3
= ( F @ X ) )
& ! [Y4: produc1996495991257130529ist_fm] :
( ( B3
= ( F @ Y4 ) )
=> ( Y4 = X ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_867_range__ex1__eq,axiom,
! [F: produc1996495991257130529ist_fm > rule,B3: rule] :
( ( inj_on6080897081453506741m_rule @ F @ top_to728902761071950545ist_fm )
=> ( ( member_rule @ B3 @ ( image_368924427985557793m_rule @ F @ top_to728902761071950545ist_fm ) )
= ( ? [X: produc1996495991257130529ist_fm] :
( ( B3
= ( F @ X ) )
& ! [Y4: produc1996495991257130529ist_fm] :
( ( B3
= ( F @ Y4 ) )
=> ( Y4 = X ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_868_range__ex1__eq,axiom,
! [F: produc1996495991257130529ist_fm > $o,B3: $o] :
( ( inj_on6610224742944822824t_fm_o @ F @ top_to728902761071950545ist_fm )
=> ( ( member_o @ B3 @ ( image_3428966142334963388t_fm_o @ F @ top_to728902761071950545ist_fm ) )
= ( ? [X: produc1996495991257130529ist_fm] :
( ( B3
= ( F @ X ) )
& ! [Y4: produc1996495991257130529ist_fm] :
( ( B3
= ( F @ Y4 ) )
=> ( Y4 = X ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_869_range__ex1__eq,axiom,
! [F: rule > produc1996495991257130529ist_fm,B3: produc1996495991257130529ist_fm] :
( ( inj_on3429812851933087593ist_fm @ F @ top_top_set_rule )
=> ( ( member8102475879199740618ist_fm @ B3 @ ( image_6941212235319914453ist_fm @ F @ top_top_set_rule ) )
= ( ? [X: rule] :
( ( B3
= ( F @ X ) )
& ! [Y4: rule] :
( ( B3
= ( F @ Y4 ) )
=> ( Y4 = X ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_870_range__ex1__eq,axiom,
! [F: rule > rule,B3: rule] :
( ( inj_on_rule_rule @ F @ top_top_set_rule )
=> ( ( member_rule @ B3 @ ( image_rule_rule @ F @ top_top_set_rule ) )
= ( ? [X: rule] :
( ( B3
= ( F @ X ) )
& ! [Y4: rule] :
( ( B3
= ( F @ Y4 ) )
=> ( Y4 = X ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_871_range__ex1__eq,axiom,
! [F: rule > $o,B3: $o] :
( ( inj_on_rule_o @ F @ top_top_set_rule )
=> ( ( member_o @ B3 @ ( image_rule_o @ F @ top_top_set_rule ) )
= ( ? [X: rule] :
( ( B3
= ( F @ X ) )
& ! [Y4: rule] :
( ( B3
= ( F @ Y4 ) )
=> ( Y4 = X ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_872_range__ex1__eq,axiom,
! [F: $o > produc1996495991257130529ist_fm,B3: produc1996495991257130529ist_fm] :
( ( inj_on6870991201974782586ist_fm @ F @ top_top_set_o )
=> ( ( member8102475879199740618ist_fm @ B3 @ ( image_3689732601364923150ist_fm @ F @ top_top_set_o ) )
= ( ? [X: $o] :
( ( B3
= ( F @ X ) )
& ! [Y4: $o] :
( ( B3
= ( F @ Y4 ) )
=> ( Y4 = X ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_873_range__ex1__eq,axiom,
! [F: $o > rule,B3: rule] :
( ( inj_on_o_rule @ F @ top_top_set_o )
=> ( ( member_rule @ B3 @ ( image_o_rule @ F @ top_top_set_o ) )
= ( ? [X: $o] :
( ( B3
= ( F @ X ) )
& ! [Y4: $o] :
( ( B3
= ( F @ Y4 ) )
=> ( Y4 = X ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_874_range__ex1__eq,axiom,
! [F: $o > $o,B3: $o] :
( ( inj_on_o_o @ F @ top_top_set_o )
=> ( ( member_o @ B3 @ ( image_o_o @ F @ top_top_set_o ) )
= ( ? [X: $o] :
( ( B3
= ( F @ X ) )
& ! [Y4: $o] :
( ( B3
= ( F @ Y4 ) )
=> ( Y4 = X ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_875_inj__image__mem__iff,axiom,
! [F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,A3: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm] :
( ( inj_on2392985232935137457ist_fm @ F @ top_to728902761071950545ist_fm )
=> ( ( member8102475879199740618ist_fm @ ( F @ A3 ) @ ( image_6982201141141708101ist_fm @ F @ A ) )
= ( member8102475879199740618ist_fm @ A3 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_876_inj__image__mem__iff,axiom,
! [F: produc1996495991257130529ist_fm > rule,A3: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm] :
( ( inj_on6080897081453506741m_rule @ F @ top_to728902761071950545ist_fm )
=> ( ( member_rule @ ( F @ A3 ) @ ( image_368924427985557793m_rule @ F @ A ) )
= ( member8102475879199740618ist_fm @ A3 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_877_inj__image__mem__iff,axiom,
! [F: produc1996495991257130529ist_fm > $o,A3: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm] :
( ( inj_on6610224742944822824t_fm_o @ F @ top_to728902761071950545ist_fm )
=> ( ( member_o @ ( F @ A3 ) @ ( image_3428966142334963388t_fm_o @ F @ A ) )
= ( member8102475879199740618ist_fm @ A3 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_878_inj__image__mem__iff,axiom,
! [F: rule > produc1996495991257130529ist_fm,A3: rule,A: set_rule] :
( ( inj_on3429812851933087593ist_fm @ F @ top_top_set_rule )
=> ( ( member8102475879199740618ist_fm @ ( F @ A3 ) @ ( image_6941212235319914453ist_fm @ F @ A ) )
= ( member_rule @ A3 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_879_inj__image__mem__iff,axiom,
! [F: rule > rule,A3: rule,A: set_rule] :
( ( inj_on_rule_rule @ F @ top_top_set_rule )
=> ( ( member_rule @ ( F @ A3 ) @ ( image_rule_rule @ F @ A ) )
= ( member_rule @ A3 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_880_inj__image__mem__iff,axiom,
! [F: rule > $o,A3: rule,A: set_rule] :
( ( inj_on_rule_o @ F @ top_top_set_rule )
=> ( ( member_o @ ( F @ A3 ) @ ( image_rule_o @ F @ A ) )
= ( member_rule @ A3 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_881_inj__image__mem__iff,axiom,
! [F: $o > produc1996495991257130529ist_fm,A3: $o,A: set_o] :
( ( inj_on6870991201974782586ist_fm @ F @ top_top_set_o )
=> ( ( member8102475879199740618ist_fm @ ( F @ A3 ) @ ( image_3689732601364923150ist_fm @ F @ A ) )
= ( member_o @ A3 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_882_inj__image__mem__iff,axiom,
! [F: $o > rule,A3: $o,A: set_o] :
( ( inj_on_o_rule @ F @ top_top_set_o )
=> ( ( member_rule @ ( F @ A3 ) @ ( image_o_rule @ F @ A ) )
= ( member_o @ A3 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_883_inj__image__mem__iff,axiom,
! [F: $o > $o,A3: $o,A: set_o] :
( ( inj_on_o_o @ F @ top_top_set_o )
=> ( ( member_o @ ( F @ A3 ) @ ( image_o_o @ F @ A ) )
= ( member_o @ A3 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_884_inj__img__insertE,axiom,
! [F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm,X2: produc1996495991257130529ist_fm,B: set_Pr7058068377845519745ist_fm] :
( ( inj_on2392985232935137457ist_fm @ F @ A )
=> ( ~ ( member8102475879199740618ist_fm @ X2 @ B )
=> ( ( ( insert589278167076416497ist_fm @ X2 @ B )
= ( image_6982201141141708101ist_fm @ F @ A ) )
=> ~ ! [X8: produc1996495991257130529ist_fm,A8: set_Pr7058068377845519745ist_fm] :
( ~ ( member8102475879199740618ist_fm @ X8 @ A8 )
=> ( ( A
= ( insert589278167076416497ist_fm @ X8 @ A8 ) )
=> ( ( X2
= ( F @ X8 ) )
=> ( B
!= ( image_6982201141141708101ist_fm @ F @ A8 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_885_inj__img__insertE,axiom,
! [F: rule > produc1996495991257130529ist_fm,A: set_rule,X2: produc1996495991257130529ist_fm,B: set_Pr7058068377845519745ist_fm] :
( ( inj_on3429812851933087593ist_fm @ F @ A )
=> ( ~ ( member8102475879199740618ist_fm @ X2 @ B )
=> ( ( ( insert589278167076416497ist_fm @ X2 @ B )
= ( image_6941212235319914453ist_fm @ F @ A ) )
=> ~ ! [X8: rule,A8: set_rule] :
( ~ ( member_rule @ X8 @ A8 )
=> ( ( A
= ( insert_rule @ X8 @ A8 ) )
=> ( ( X2
= ( F @ X8 ) )
=> ( B
!= ( image_6941212235319914453ist_fm @ F @ A8 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_886_inj__img__insertE,axiom,
! [F: $o > produc1996495991257130529ist_fm,A: set_o,X2: produc1996495991257130529ist_fm,B: set_Pr7058068377845519745ist_fm] :
( ( inj_on6870991201974782586ist_fm @ F @ A )
=> ( ~ ( member8102475879199740618ist_fm @ X2 @ B )
=> ( ( ( insert589278167076416497ist_fm @ X2 @ B )
= ( image_3689732601364923150ist_fm @ F @ A ) )
=> ~ ! [X8: $o,A8: set_o] :
( ~ ( member_o @ X8 @ A8 )
=> ( ( A
= ( insert_o @ X8 @ A8 ) )
=> ( ( X2
= ( F @ X8 ) )
=> ( B
!= ( image_3689732601364923150ist_fm @ F @ A8 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_887_inj__img__insertE,axiom,
! [F: produc1996495991257130529ist_fm > rule,A: set_Pr7058068377845519745ist_fm,X2: rule,B: set_rule] :
( ( inj_on6080897081453506741m_rule @ F @ A )
=> ( ~ ( member_rule @ X2 @ B )
=> ( ( ( insert_rule @ X2 @ B )
= ( image_368924427985557793m_rule @ F @ A ) )
=> ~ ! [X8: produc1996495991257130529ist_fm,A8: set_Pr7058068377845519745ist_fm] :
( ~ ( member8102475879199740618ist_fm @ X8 @ A8 )
=> ( ( A
= ( insert589278167076416497ist_fm @ X8 @ A8 ) )
=> ( ( X2
= ( F @ X8 ) )
=> ( B
!= ( image_368924427985557793m_rule @ F @ A8 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_888_inj__img__insertE,axiom,
! [F: rule > rule,A: set_rule,X2: rule,B: set_rule] :
( ( inj_on_rule_rule @ F @ A )
=> ( ~ ( member_rule @ X2 @ B )
=> ( ( ( insert_rule @ X2 @ B )
= ( image_rule_rule @ F @ A ) )
=> ~ ! [X8: rule,A8: set_rule] :
( ~ ( member_rule @ X8 @ A8 )
=> ( ( A
= ( insert_rule @ X8 @ A8 ) )
=> ( ( X2
= ( F @ X8 ) )
=> ( B
!= ( image_rule_rule @ F @ A8 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_889_inj__img__insertE,axiom,
! [F: $o > rule,A: set_o,X2: rule,B: set_rule] :
( ( inj_on_o_rule @ F @ A )
=> ( ~ ( member_rule @ X2 @ B )
=> ( ( ( insert_rule @ X2 @ B )
= ( image_o_rule @ F @ A ) )
=> ~ ! [X8: $o,A8: set_o] :
( ~ ( member_o @ X8 @ A8 )
=> ( ( A
= ( insert_o @ X8 @ A8 ) )
=> ( ( X2
= ( F @ X8 ) )
=> ( B
!= ( image_o_rule @ F @ A8 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_890_inj__img__insertE,axiom,
! [F: produc1996495991257130529ist_fm > $o,A: set_Pr7058068377845519745ist_fm,X2: $o,B: set_o] :
( ( inj_on6610224742944822824t_fm_o @ F @ A )
=> ( ~ ( member_o @ X2 @ B )
=> ( ( ( insert_o @ X2 @ B )
= ( image_3428966142334963388t_fm_o @ F @ A ) )
=> ~ ! [X8: produc1996495991257130529ist_fm,A8: set_Pr7058068377845519745ist_fm] :
( ~ ( member8102475879199740618ist_fm @ X8 @ A8 )
=> ( ( A
= ( insert589278167076416497ist_fm @ X8 @ A8 ) )
=> ( ( X2
= ( F @ X8 ) )
=> ( B
!= ( image_3428966142334963388t_fm_o @ F @ A8 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_891_inj__img__insertE,axiom,
! [F: rule > $o,A: set_rule,X2: $o,B: set_o] :
( ( inj_on_rule_o @ F @ A )
=> ( ~ ( member_o @ X2 @ B )
=> ( ( ( insert_o @ X2 @ B )
= ( image_rule_o @ F @ A ) )
=> ~ ! [X8: rule,A8: set_rule] :
( ~ ( member_rule @ X8 @ A8 )
=> ( ( A
= ( insert_rule @ X8 @ A8 ) )
=> ( ( X2
= ( F @ X8 ) )
=> ( B
!= ( image_rule_o @ F @ A8 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_892_inj__img__insertE,axiom,
! [F: $o > $o,A: set_o,X2: $o,B: set_o] :
( ( inj_on_o_o @ F @ A )
=> ( ~ ( member_o @ X2 @ B )
=> ( ( ( insert_o @ X2 @ B )
= ( image_o_o @ F @ A ) )
=> ~ ! [X8: $o,A8: set_o] :
( ~ ( member_o @ X8 @ A8 )
=> ( ( A
= ( insert_o @ X8 @ A8 ) )
=> ( ( X2
= ( F @ X8 ) )
=> ( B
!= ( image_o_o @ F @ A8 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_893_inj__singleton,axiom,
! [A: set_rule] :
( inj_on_rule_set_rule
@ ^ [X: rule] : ( insert_rule @ X @ bot_bot_set_rule )
@ A ) ).
% inj_singleton
thf(fact_894_inj__singleton,axiom,
! [A: set_o] :
( inj_on_o_set_o
@ ^ [X: $o] : ( insert_o @ X @ bot_bot_set_o )
@ A ) ).
% inj_singleton
thf(fact_895_inj__singleton,axiom,
! [A: set_Pr7058068377845519745ist_fm] :
( inj_on6475354550519685137ist_fm
@ ^ [X: produc1996495991257130529ist_fm] : ( insert589278167076416497ist_fm @ X @ bot_bo4530011010606209517ist_fm )
@ A ) ).
% inj_singleton
thf(fact_896_INT__insert__distrib,axiom,
! [U2: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm,A3: rule,B: produc1996495991257130529ist_fm > set_rule] :
( ( member8102475879199740618ist_fm @ U2 @ A )
=> ( ( comple5773694076043965236t_rule
@ ( image_2008118025656205143t_rule
@ ^ [X: produc1996495991257130529ist_fm] : ( insert_rule @ A3 @ ( B @ X ) )
@ A ) )
= ( insert_rule @ A3 @ ( comple5773694076043965236t_rule @ ( image_2008118025656205143t_rule @ B @ A ) ) ) ) ) ).
% INT_insert_distrib
thf(fact_897_INT__insert__distrib,axiom,
! [U2: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm,A3: $o,B: produc1996495991257130529ist_fm > set_o] :
( ( member8102475879199740618ist_fm @ U2 @ A )
=> ( ( comple3063163877087187839_set_o
@ ( image_7087520594942641564_set_o
@ ^ [X: produc1996495991257130529ist_fm] : ( insert_o @ A3 @ ( B @ X ) )
@ A ) )
= ( insert_o @ A3 @ ( comple3063163877087187839_set_o @ ( image_7087520594942641564_set_o @ B @ A ) ) ) ) ) ).
% INT_insert_distrib
thf(fact_898_INT__insert__distrib,axiom,
! [U2: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm,A3: produc1996495991257130529ist_fm,B: produc1996495991257130529ist_fm > set_Pr7058068377845519745ist_fm] :
( ( member8102475879199740618ist_fm @ U2 @ A )
=> ( ( comple5608129641975563656ist_fm
@ ( image_5852275405373602981ist_fm
@ ^ [X: produc1996495991257130529ist_fm] : ( insert589278167076416497ist_fm @ A3 @ ( B @ X ) )
@ A ) )
= ( insert589278167076416497ist_fm @ A3 @ ( comple5608129641975563656ist_fm @ ( image_5852275405373602981ist_fm @ B @ A ) ) ) ) ) ).
% INT_insert_distrib
thf(fact_899_INT__insert__distrib,axiom,
! [U2: rule,A: set_rule,A3: rule,B: rule > set_rule] :
( ( member_rule @ U2 @ A )
=> ( ( comple5773694076043965236t_rule
@ ( image_rule_set_rule
@ ^ [X: rule] : ( insert_rule @ A3 @ ( B @ X ) )
@ A ) )
= ( insert_rule @ A3 @ ( comple5773694076043965236t_rule @ ( image_rule_set_rule @ B @ A ) ) ) ) ) ).
% INT_insert_distrib
thf(fact_900_INT__insert__distrib,axiom,
! [U2: rule,A: set_rule,A3: $o,B: rule > set_o] :
( ( member_rule @ U2 @ A )
=> ( ( comple3063163877087187839_set_o
@ ( image_rule_set_o
@ ^ [X: rule] : ( insert_o @ A3 @ ( B @ X ) )
@ A ) )
= ( insert_o @ A3 @ ( comple3063163877087187839_set_o @ ( image_rule_set_o @ B @ A ) ) ) ) ) ).
% INT_insert_distrib
thf(fact_901_INT__insert__distrib,axiom,
! [U2: rule,A: set_rule,A3: produc1996495991257130529ist_fm,B: rule > set_Pr7058068377845519745ist_fm] :
( ( member_rule @ U2 @ A )
=> ( ( comple5608129641975563656ist_fm
@ ( image_3897101447128207157ist_fm
@ ^ [X: rule] : ( insert589278167076416497ist_fm @ A3 @ ( B @ X ) )
@ A ) )
= ( insert589278167076416497ist_fm @ A3 @ ( comple5608129641975563656ist_fm @ ( image_3897101447128207157ist_fm @ B @ A ) ) ) ) ) ).
% INT_insert_distrib
thf(fact_902_INT__insert__distrib,axiom,
! [U2: $o,A: set_o,A3: rule,B: $o > set_rule] :
( ( member_o @ U2 @ A )
=> ( ( comple5773694076043965236t_rule
@ ( image_o_set_rule
@ ^ [X: $o] : ( insert_rule @ A3 @ ( B @ X ) )
@ A ) )
= ( insert_rule @ A3 @ ( comple5773694076043965236t_rule @ ( image_o_set_rule @ B @ A ) ) ) ) ) ).
% INT_insert_distrib
thf(fact_903_INT__insert__distrib,axiom,
! [U2: $o,A: set_o,A3: $o,B: $o > set_o] :
( ( member_o @ U2 @ A )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [X: $o] : ( insert_o @ A3 @ ( B @ X ) )
@ A ) )
= ( insert_o @ A3 @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ B @ A ) ) ) ) ) ).
% INT_insert_distrib
thf(fact_904_INT__insert__distrib,axiom,
! [U2: $o,A: set_o,A3: produc1996495991257130529ist_fm,B: $o > set_Pr7058068377845519745ist_fm] :
( ( member_o @ U2 @ A )
=> ( ( comple5608129641975563656ist_fm
@ ( image_1666948324336172654ist_fm
@ ^ [X: $o] : ( insert589278167076416497ist_fm @ A3 @ ( B @ X ) )
@ A ) )
= ( insert589278167076416497ist_fm @ A3 @ ( comple5608129641975563656ist_fm @ ( image_1666948324336172654ist_fm @ B @ A ) ) ) ) ) ).
% INT_insert_distrib
thf(fact_905_UN__empty,axiom,
! [B: rule > set_rule] :
( ( comple2146307154184993742t_rule @ ( image_rule_set_rule @ B @ bot_bot_set_rule ) )
= bot_bot_set_rule ) ).
% UN_empty
thf(fact_906_UN__empty,axiom,
! [B: rule > set_o] :
( ( comple90263536869209701_set_o @ ( image_rule_set_o @ B @ bot_bot_set_rule ) )
= bot_bot_set_o ) ).
% UN_empty
thf(fact_907_UN__empty,axiom,
! [B: rule > set_Pr7058068377845519745ist_fm] :
( ( comple1649742678916152942ist_fm @ ( image_3897101447128207157ist_fm @ B @ bot_bot_set_rule ) )
= bot_bo4530011010606209517ist_fm ) ).
% UN_empty
thf(fact_908_UN__empty,axiom,
! [B: $o > set_rule] :
( ( comple2146307154184993742t_rule @ ( image_o_set_rule @ B @ bot_bot_set_o ) )
= bot_bot_set_rule ) ).
% UN_empty
thf(fact_909_UN__empty,axiom,
! [B: $o > set_o] :
( ( comple90263536869209701_set_o @ ( image_o_set_o @ B @ bot_bot_set_o ) )
= bot_bot_set_o ) ).
% UN_empty
thf(fact_910_UN__empty,axiom,
! [B: $o > set_Pr7058068377845519745ist_fm] :
( ( comple1649742678916152942ist_fm @ ( image_1666948324336172654ist_fm @ B @ bot_bot_set_o ) )
= bot_bo4530011010606209517ist_fm ) ).
% UN_empty
thf(fact_911_UN__empty,axiom,
! [B: produc1996495991257130529ist_fm > set_rule] :
( ( comple2146307154184993742t_rule @ ( image_2008118025656205143t_rule @ B @ bot_bo4530011010606209517ist_fm ) )
= bot_bot_set_rule ) ).
% UN_empty
thf(fact_912_UN__empty,axiom,
! [B: produc1996495991257130529ist_fm > set_o] :
( ( comple90263536869209701_set_o @ ( image_7087520594942641564_set_o @ B @ bot_bo4530011010606209517ist_fm ) )
= bot_bot_set_o ) ).
% UN_empty
thf(fact_913_UN__empty,axiom,
! [B: produc1996495991257130529ist_fm > set_Pr7058068377845519745ist_fm] :
( ( comple1649742678916152942ist_fm @ ( image_5852275405373602981ist_fm @ B @ bot_bo4530011010606209517ist_fm ) )
= bot_bo4530011010606209517ist_fm ) ).
% UN_empty
thf(fact_914_UN__insert__distrib,axiom,
! [U2: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm,A3: rule,B: produc1996495991257130529ist_fm > set_rule] :
( ( member8102475879199740618ist_fm @ U2 @ A )
=> ( ( comple2146307154184993742t_rule
@ ( image_2008118025656205143t_rule
@ ^ [X: produc1996495991257130529ist_fm] : ( insert_rule @ A3 @ ( B @ X ) )
@ A ) )
= ( insert_rule @ A3 @ ( comple2146307154184993742t_rule @ ( image_2008118025656205143t_rule @ B @ A ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_915_UN__insert__distrib,axiom,
! [U2: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm,A3: $o,B: produc1996495991257130529ist_fm > set_o] :
( ( member8102475879199740618ist_fm @ U2 @ A )
=> ( ( comple90263536869209701_set_o
@ ( image_7087520594942641564_set_o
@ ^ [X: produc1996495991257130529ist_fm] : ( insert_o @ A3 @ ( B @ X ) )
@ A ) )
= ( insert_o @ A3 @ ( comple90263536869209701_set_o @ ( image_7087520594942641564_set_o @ B @ A ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_916_UN__insert__distrib,axiom,
! [U2: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm,A3: produc1996495991257130529ist_fm,B: produc1996495991257130529ist_fm > set_Pr7058068377845519745ist_fm] :
( ( member8102475879199740618ist_fm @ U2 @ A )
=> ( ( comple1649742678916152942ist_fm
@ ( image_5852275405373602981ist_fm
@ ^ [X: produc1996495991257130529ist_fm] : ( insert589278167076416497ist_fm @ A3 @ ( B @ X ) )
@ A ) )
= ( insert589278167076416497ist_fm @ A3 @ ( comple1649742678916152942ist_fm @ ( image_5852275405373602981ist_fm @ B @ A ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_917_UN__insert__distrib,axiom,
! [U2: rule,A: set_rule,A3: rule,B: rule > set_rule] :
( ( member_rule @ U2 @ A )
=> ( ( comple2146307154184993742t_rule
@ ( image_rule_set_rule
@ ^ [X: rule] : ( insert_rule @ A3 @ ( B @ X ) )
@ A ) )
= ( insert_rule @ A3 @ ( comple2146307154184993742t_rule @ ( image_rule_set_rule @ B @ A ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_918_UN__insert__distrib,axiom,
! [U2: rule,A: set_rule,A3: $o,B: rule > set_o] :
( ( member_rule @ U2 @ A )
=> ( ( comple90263536869209701_set_o
@ ( image_rule_set_o
@ ^ [X: rule] : ( insert_o @ A3 @ ( B @ X ) )
@ A ) )
= ( insert_o @ A3 @ ( comple90263536869209701_set_o @ ( image_rule_set_o @ B @ A ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_919_UN__insert__distrib,axiom,
! [U2: rule,A: set_rule,A3: produc1996495991257130529ist_fm,B: rule > set_Pr7058068377845519745ist_fm] :
( ( member_rule @ U2 @ A )
=> ( ( comple1649742678916152942ist_fm
@ ( image_3897101447128207157ist_fm
@ ^ [X: rule] : ( insert589278167076416497ist_fm @ A3 @ ( B @ X ) )
@ A ) )
= ( insert589278167076416497ist_fm @ A3 @ ( comple1649742678916152942ist_fm @ ( image_3897101447128207157ist_fm @ B @ A ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_920_UN__insert__distrib,axiom,
! [U2: $o,A: set_o,A3: rule,B: $o > set_rule] :
( ( member_o @ U2 @ A )
=> ( ( comple2146307154184993742t_rule
@ ( image_o_set_rule
@ ^ [X: $o] : ( insert_rule @ A3 @ ( B @ X ) )
@ A ) )
= ( insert_rule @ A3 @ ( comple2146307154184993742t_rule @ ( image_o_set_rule @ B @ A ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_921_UN__insert__distrib,axiom,
! [U2: $o,A: set_o,A3: $o,B: $o > set_o] :
( ( member_o @ U2 @ A )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X: $o] : ( insert_o @ A3 @ ( B @ X ) )
@ A ) )
= ( insert_o @ A3 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B @ A ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_922_UN__insert__distrib,axiom,
! [U2: $o,A: set_o,A3: produc1996495991257130529ist_fm,B: $o > set_Pr7058068377845519745ist_fm] :
( ( member_o @ U2 @ A )
=> ( ( comple1649742678916152942ist_fm
@ ( image_1666948324336172654ist_fm
@ ^ [X: $o] : ( insert589278167076416497ist_fm @ A3 @ ( B @ X ) )
@ A ) )
= ( insert589278167076416497ist_fm @ A3 @ ( comple1649742678916152942ist_fm @ ( image_1666948324336172654ist_fm @ B @ A ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_923_image__set__diff,axiom,
! [F: rule > rule,A: set_rule,B: set_rule] :
( ( inj_on_rule_rule @ F @ top_top_set_rule )
=> ( ( image_rule_rule @ F @ ( minus_minus_set_rule @ A @ B ) )
= ( minus_minus_set_rule @ ( image_rule_rule @ F @ A ) @ ( image_rule_rule @ F @ B ) ) ) ) ).
% image_set_diff
thf(fact_924_image__set__diff,axiom,
! [F: rule > $o,A: set_rule,B: set_rule] :
( ( inj_on_rule_o @ F @ top_top_set_rule )
=> ( ( image_rule_o @ F @ ( minus_minus_set_rule @ A @ B ) )
= ( minus_minus_set_o @ ( image_rule_o @ F @ A ) @ ( image_rule_o @ F @ B ) ) ) ) ).
% image_set_diff
thf(fact_925_image__set__diff,axiom,
! [F: rule > produc1996495991257130529ist_fm,A: set_rule,B: set_rule] :
( ( inj_on3429812851933087593ist_fm @ F @ top_top_set_rule )
=> ( ( image_6941212235319914453ist_fm @ F @ ( minus_minus_set_rule @ A @ B ) )
= ( minus_5615909706921521512ist_fm @ ( image_6941212235319914453ist_fm @ F @ A ) @ ( image_6941212235319914453ist_fm @ F @ B ) ) ) ) ).
% image_set_diff
thf(fact_926_image__set__diff,axiom,
! [F: $o > rule,A: set_o,B: set_o] :
( ( inj_on_o_rule @ F @ top_top_set_o )
=> ( ( image_o_rule @ F @ ( minus_minus_set_o @ A @ B ) )
= ( minus_minus_set_rule @ ( image_o_rule @ F @ A ) @ ( image_o_rule @ F @ B ) ) ) ) ).
% image_set_diff
thf(fact_927_image__set__diff,axiom,
! [F: $o > $o,A: set_o,B: set_o] :
( ( inj_on_o_o @ F @ top_top_set_o )
=> ( ( image_o_o @ F @ ( minus_minus_set_o @ A @ B ) )
= ( minus_minus_set_o @ ( image_o_o @ F @ A ) @ ( image_o_o @ F @ B ) ) ) ) ).
% image_set_diff
thf(fact_928_image__set__diff,axiom,
! [F: $o > produc1996495991257130529ist_fm,A: set_o,B: set_o] :
( ( inj_on6870991201974782586ist_fm @ F @ top_top_set_o )
=> ( ( image_3689732601364923150ist_fm @ F @ ( minus_minus_set_o @ A @ B ) )
= ( minus_5615909706921521512ist_fm @ ( image_3689732601364923150ist_fm @ F @ A ) @ ( image_3689732601364923150ist_fm @ F @ B ) ) ) ) ).
% image_set_diff
thf(fact_929_image__set__diff,axiom,
! [F: produc1996495991257130529ist_fm > rule,A: set_Pr7058068377845519745ist_fm,B: set_Pr7058068377845519745ist_fm] :
( ( inj_on6080897081453506741m_rule @ F @ top_to728902761071950545ist_fm )
=> ( ( image_368924427985557793m_rule @ F @ ( minus_5615909706921521512ist_fm @ A @ B ) )
= ( minus_minus_set_rule @ ( image_368924427985557793m_rule @ F @ A ) @ ( image_368924427985557793m_rule @ F @ B ) ) ) ) ).
% image_set_diff
thf(fact_930_image__set__diff,axiom,
! [F: produc1996495991257130529ist_fm > $o,A: set_Pr7058068377845519745ist_fm,B: set_Pr7058068377845519745ist_fm] :
( ( inj_on6610224742944822824t_fm_o @ F @ top_to728902761071950545ist_fm )
=> ( ( image_3428966142334963388t_fm_o @ F @ ( minus_5615909706921521512ist_fm @ A @ B ) )
= ( minus_minus_set_o @ ( image_3428966142334963388t_fm_o @ F @ A ) @ ( image_3428966142334963388t_fm_o @ F @ B ) ) ) ) ).
% image_set_diff
thf(fact_931_image__set__diff,axiom,
! [F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm,B: set_Pr7058068377845519745ist_fm] :
( ( inj_on2392985232935137457ist_fm @ F @ top_to728902761071950545ist_fm )
=> ( ( image_6982201141141708101ist_fm @ F @ ( minus_5615909706921521512ist_fm @ A @ B ) )
= ( minus_5615909706921521512ist_fm @ ( image_6982201141141708101ist_fm @ F @ A ) @ ( image_6982201141141708101ist_fm @ F @ B ) ) ) ) ).
% image_set_diff
thf(fact_932_INF__empty,axiom,
! [F: rule > set_Pr7058068377845519745ist_fm] :
( ( comple5608129641975563656ist_fm @ ( image_3897101447128207157ist_fm @ F @ bot_bot_set_rule ) )
= top_to728902761071950545ist_fm ) ).
% INF_empty
thf(fact_933_INF__empty,axiom,
! [F: $o > set_Pr7058068377845519745ist_fm] :
( ( comple5608129641975563656ist_fm @ ( image_1666948324336172654ist_fm @ F @ bot_bot_set_o ) )
= top_to728902761071950545ist_fm ) ).
% INF_empty
thf(fact_934_INF__empty,axiom,
! [F: produc1996495991257130529ist_fm > set_Pr7058068377845519745ist_fm] :
( ( comple5608129641975563656ist_fm @ ( image_5852275405373602981ist_fm @ F @ bot_bo4530011010606209517ist_fm ) )
= top_to728902761071950545ist_fm ) ).
% INF_empty
thf(fact_935_INF__empty,axiom,
! [F: rule > set_rule] :
( ( comple5773694076043965236t_rule @ ( image_rule_set_rule @ F @ bot_bot_set_rule ) )
= top_top_set_rule ) ).
% INF_empty
thf(fact_936_INF__empty,axiom,
! [F: $o > set_rule] :
( ( comple5773694076043965236t_rule @ ( image_o_set_rule @ F @ bot_bot_set_o ) )
= top_top_set_rule ) ).
% INF_empty
thf(fact_937_INF__empty,axiom,
! [F: produc1996495991257130529ist_fm > set_rule] :
( ( comple5773694076043965236t_rule @ ( image_2008118025656205143t_rule @ F @ bot_bo4530011010606209517ist_fm ) )
= top_top_set_rule ) ).
% INF_empty
thf(fact_938_INF__empty,axiom,
! [F: rule > set_o] :
( ( comple3063163877087187839_set_o @ ( image_rule_set_o @ F @ bot_bot_set_rule ) )
= top_top_set_o ) ).
% INF_empty
thf(fact_939_INF__empty,axiom,
! [F: $o > set_o] :
( ( comple3063163877087187839_set_o @ ( image_o_set_o @ F @ bot_bot_set_o ) )
= top_top_set_o ) ).
% INF_empty
thf(fact_940_INF__empty,axiom,
! [F: produc1996495991257130529ist_fm > set_o] :
( ( comple3063163877087187839_set_o @ ( image_7087520594942641564_set_o @ F @ bot_bo4530011010606209517ist_fm ) )
= top_top_set_o ) ).
% INF_empty
thf(fact_941_INF__constant,axiom,
! [A: set_rule,C: set_Pr7058068377845519745ist_fm] :
( ( ( A = bot_bot_set_rule )
=> ( ( comple5608129641975563656ist_fm
@ ( image_3897101447128207157ist_fm
@ ^ [Y4: rule] : C
@ A ) )
= top_to728902761071950545ist_fm ) )
& ( ( A != bot_bot_set_rule )
=> ( ( comple5608129641975563656ist_fm
@ ( image_3897101447128207157ist_fm
@ ^ [Y4: rule] : C
@ A ) )
= C ) ) ) ).
% INF_constant
thf(fact_942_INF__constant,axiom,
! [A: set_o,C: set_Pr7058068377845519745ist_fm] :
( ( ( A = bot_bot_set_o )
=> ( ( comple5608129641975563656ist_fm
@ ( image_1666948324336172654ist_fm
@ ^ [Y4: $o] : C
@ A ) )
= top_to728902761071950545ist_fm ) )
& ( ( A != bot_bot_set_o )
=> ( ( comple5608129641975563656ist_fm
@ ( image_1666948324336172654ist_fm
@ ^ [Y4: $o] : C
@ A ) )
= C ) ) ) ).
% INF_constant
thf(fact_943_INF__constant,axiom,
! [A: set_Pr7058068377845519745ist_fm,C: set_Pr7058068377845519745ist_fm] :
( ( ( A = bot_bo4530011010606209517ist_fm )
=> ( ( comple5608129641975563656ist_fm
@ ( image_5852275405373602981ist_fm
@ ^ [Y4: produc1996495991257130529ist_fm] : C
@ A ) )
= top_to728902761071950545ist_fm ) )
& ( ( A != bot_bo4530011010606209517ist_fm )
=> ( ( comple5608129641975563656ist_fm
@ ( image_5852275405373602981ist_fm
@ ^ [Y4: produc1996495991257130529ist_fm] : C
@ A ) )
= C ) ) ) ).
% INF_constant
thf(fact_944_INF__constant,axiom,
! [A: set_rule,C: set_rule] :
( ( ( A = bot_bot_set_rule )
=> ( ( comple5773694076043965236t_rule
@ ( image_rule_set_rule
@ ^ [Y4: rule] : C
@ A ) )
= top_top_set_rule ) )
& ( ( A != bot_bot_set_rule )
=> ( ( comple5773694076043965236t_rule
@ ( image_rule_set_rule
@ ^ [Y4: rule] : C
@ A ) )
= C ) ) ) ).
% INF_constant
thf(fact_945_INF__constant,axiom,
! [A: set_o,C: set_rule] :
( ( ( A = bot_bot_set_o )
=> ( ( comple5773694076043965236t_rule
@ ( image_o_set_rule
@ ^ [Y4: $o] : C
@ A ) )
= top_top_set_rule ) )
& ( ( A != bot_bot_set_o )
=> ( ( comple5773694076043965236t_rule
@ ( image_o_set_rule
@ ^ [Y4: $o] : C
@ A ) )
= C ) ) ) ).
% INF_constant
thf(fact_946_INF__constant,axiom,
! [A: set_Pr7058068377845519745ist_fm,C: set_rule] :
( ( ( A = bot_bo4530011010606209517ist_fm )
=> ( ( comple5773694076043965236t_rule
@ ( image_2008118025656205143t_rule
@ ^ [Y4: produc1996495991257130529ist_fm] : C
@ A ) )
= top_top_set_rule ) )
& ( ( A != bot_bo4530011010606209517ist_fm )
=> ( ( comple5773694076043965236t_rule
@ ( image_2008118025656205143t_rule
@ ^ [Y4: produc1996495991257130529ist_fm] : C
@ A ) )
= C ) ) ) ).
% INF_constant
thf(fact_947_INF__constant,axiom,
! [A: set_rule,C: set_o] :
( ( ( A = bot_bot_set_rule )
=> ( ( comple3063163877087187839_set_o
@ ( image_rule_set_o
@ ^ [Y4: rule] : C
@ A ) )
= top_top_set_o ) )
& ( ( A != bot_bot_set_rule )
=> ( ( comple3063163877087187839_set_o
@ ( image_rule_set_o
@ ^ [Y4: rule] : C
@ A ) )
= C ) ) ) ).
% INF_constant
thf(fact_948_INF__constant,axiom,
! [A: set_o,C: set_o] :
( ( ( A = bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [Y4: $o] : C
@ A ) )
= top_top_set_o ) )
& ( ( A != bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [Y4: $o] : C
@ A ) )
= C ) ) ) ).
% INF_constant
thf(fact_949_INF__constant,axiom,
! [A: set_Pr7058068377845519745ist_fm,C: set_o] :
( ( ( A = bot_bo4530011010606209517ist_fm )
=> ( ( comple3063163877087187839_set_o
@ ( image_7087520594942641564_set_o
@ ^ [Y4: produc1996495991257130529ist_fm] : C
@ A ) )
= top_top_set_o ) )
& ( ( A != bot_bo4530011010606209517ist_fm )
=> ( ( comple3063163877087187839_set_o
@ ( image_7087520594942641564_set_o
@ ^ [Y4: produc1996495991257130529ist_fm] : C
@ A ) )
= C ) ) ) ).
% INF_constant
thf(fact_950_SUP__empty,axiom,
! [F: rule > set_rule] :
( ( comple2146307154184993742t_rule @ ( image_rule_set_rule @ F @ bot_bot_set_rule ) )
= bot_bot_set_rule ) ).
% SUP_empty
thf(fact_951_SUP__empty,axiom,
! [F: rule > set_o] :
( ( comple90263536869209701_set_o @ ( image_rule_set_o @ F @ bot_bot_set_rule ) )
= bot_bot_set_o ) ).
% SUP_empty
thf(fact_952_SUP__empty,axiom,
! [F: rule > set_Pr7058068377845519745ist_fm] :
( ( comple1649742678916152942ist_fm @ ( image_3897101447128207157ist_fm @ F @ bot_bot_set_rule ) )
= bot_bo4530011010606209517ist_fm ) ).
% SUP_empty
thf(fact_953_SUP__empty,axiom,
! [F: $o > set_rule] :
( ( comple2146307154184993742t_rule @ ( image_o_set_rule @ F @ bot_bot_set_o ) )
= bot_bot_set_rule ) ).
% SUP_empty
thf(fact_954_SUP__empty,axiom,
! [F: $o > set_o] :
( ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ bot_bot_set_o ) )
= bot_bot_set_o ) ).
% SUP_empty
thf(fact_955_SUP__empty,axiom,
! [F: $o > set_Pr7058068377845519745ist_fm] :
( ( comple1649742678916152942ist_fm @ ( image_1666948324336172654ist_fm @ F @ bot_bot_set_o ) )
= bot_bo4530011010606209517ist_fm ) ).
% SUP_empty
thf(fact_956_SUP__empty,axiom,
! [F: produc1996495991257130529ist_fm > set_rule] :
( ( comple2146307154184993742t_rule @ ( image_2008118025656205143t_rule @ F @ bot_bo4530011010606209517ist_fm ) )
= bot_bot_set_rule ) ).
% SUP_empty
thf(fact_957_SUP__empty,axiom,
! [F: produc1996495991257130529ist_fm > set_o] :
( ( comple90263536869209701_set_o @ ( image_7087520594942641564_set_o @ F @ bot_bo4530011010606209517ist_fm ) )
= bot_bot_set_o ) ).
% SUP_empty
thf(fact_958_SUP__empty,axiom,
! [F: produc1996495991257130529ist_fm > set_Pr7058068377845519745ist_fm] :
( ( comple1649742678916152942ist_fm @ ( image_5852275405373602981ist_fm @ F @ bot_bo4530011010606209517ist_fm ) )
= bot_bo4530011010606209517ist_fm ) ).
% SUP_empty
thf(fact_959_SUP__constant,axiom,
! [A: set_rule,C: set_rule] :
( ( ( A = bot_bot_set_rule )
=> ( ( comple2146307154184993742t_rule
@ ( image_rule_set_rule
@ ^ [Y4: rule] : C
@ A ) )
= bot_bot_set_rule ) )
& ( ( A != bot_bot_set_rule )
=> ( ( comple2146307154184993742t_rule
@ ( image_rule_set_rule
@ ^ [Y4: rule] : C
@ A ) )
= C ) ) ) ).
% SUP_constant
thf(fact_960_SUP__constant,axiom,
! [A: set_rule,C: set_o] :
( ( ( A = bot_bot_set_rule )
=> ( ( comple90263536869209701_set_o
@ ( image_rule_set_o
@ ^ [Y4: rule] : C
@ A ) )
= bot_bot_set_o ) )
& ( ( A != bot_bot_set_rule )
=> ( ( comple90263536869209701_set_o
@ ( image_rule_set_o
@ ^ [Y4: rule] : C
@ A ) )
= C ) ) ) ).
% SUP_constant
thf(fact_961_SUP__constant,axiom,
! [A: set_rule,C: set_Pr7058068377845519745ist_fm] :
( ( ( A = bot_bot_set_rule )
=> ( ( comple1649742678916152942ist_fm
@ ( image_3897101447128207157ist_fm
@ ^ [Y4: rule] : C
@ A ) )
= bot_bo4530011010606209517ist_fm ) )
& ( ( A != bot_bot_set_rule )
=> ( ( comple1649742678916152942ist_fm
@ ( image_3897101447128207157ist_fm
@ ^ [Y4: rule] : C
@ A ) )
= C ) ) ) ).
% SUP_constant
thf(fact_962_SUP__constant,axiom,
! [A: set_o,C: set_rule] :
( ( ( A = bot_bot_set_o )
=> ( ( comple2146307154184993742t_rule
@ ( image_o_set_rule
@ ^ [Y4: $o] : C
@ A ) )
= bot_bot_set_rule ) )
& ( ( A != bot_bot_set_o )
=> ( ( comple2146307154184993742t_rule
@ ( image_o_set_rule
@ ^ [Y4: $o] : C
@ A ) )
= C ) ) ) ).
% SUP_constant
thf(fact_963_SUP__constant,axiom,
! [A: set_o,C: set_o] :
( ( ( A = bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [Y4: $o] : C
@ A ) )
= bot_bot_set_o ) )
& ( ( A != bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [Y4: $o] : C
@ A ) )
= C ) ) ) ).
% SUP_constant
thf(fact_964_SUP__constant,axiom,
! [A: set_o,C: set_Pr7058068377845519745ist_fm] :
( ( ( A = bot_bot_set_o )
=> ( ( comple1649742678916152942ist_fm
@ ( image_1666948324336172654ist_fm
@ ^ [Y4: $o] : C
@ A ) )
= bot_bo4530011010606209517ist_fm ) )
& ( ( A != bot_bot_set_o )
=> ( ( comple1649742678916152942ist_fm
@ ( image_1666948324336172654ist_fm
@ ^ [Y4: $o] : C
@ A ) )
= C ) ) ) ).
% SUP_constant
thf(fact_965_SUP__constant,axiom,
! [A: set_Pr7058068377845519745ist_fm,C: set_rule] :
( ( ( A = bot_bo4530011010606209517ist_fm )
=> ( ( comple2146307154184993742t_rule
@ ( image_2008118025656205143t_rule
@ ^ [Y4: produc1996495991257130529ist_fm] : C
@ A ) )
= bot_bot_set_rule ) )
& ( ( A != bot_bo4530011010606209517ist_fm )
=> ( ( comple2146307154184993742t_rule
@ ( image_2008118025656205143t_rule
@ ^ [Y4: produc1996495991257130529ist_fm] : C
@ A ) )
= C ) ) ) ).
% SUP_constant
thf(fact_966_SUP__constant,axiom,
! [A: set_Pr7058068377845519745ist_fm,C: set_o] :
( ( ( A = bot_bo4530011010606209517ist_fm )
=> ( ( comple90263536869209701_set_o
@ ( image_7087520594942641564_set_o
@ ^ [Y4: produc1996495991257130529ist_fm] : C
@ A ) )
= bot_bot_set_o ) )
& ( ( A != bot_bo4530011010606209517ist_fm )
=> ( ( comple90263536869209701_set_o
@ ( image_7087520594942641564_set_o
@ ^ [Y4: produc1996495991257130529ist_fm] : C
@ A ) )
= C ) ) ) ).
% SUP_constant
thf(fact_967_SUP__constant,axiom,
! [A: set_Pr7058068377845519745ist_fm,C: set_Pr7058068377845519745ist_fm] :
( ( ( A = bot_bo4530011010606209517ist_fm )
=> ( ( comple1649742678916152942ist_fm
@ ( image_5852275405373602981ist_fm
@ ^ [Y4: produc1996495991257130529ist_fm] : C
@ A ) )
= bot_bo4530011010606209517ist_fm ) )
& ( ( A != bot_bo4530011010606209517ist_fm )
=> ( ( comple1649742678916152942ist_fm
@ ( image_5852275405373602981ist_fm
@ ^ [Y4: produc1996495991257130529ist_fm] : C
@ A ) )
= C ) ) ) ).
% SUP_constant
thf(fact_968_INT__empty,axiom,
! [B: rule > set_Pr7058068377845519745ist_fm] :
( ( comple5608129641975563656ist_fm @ ( image_3897101447128207157ist_fm @ B @ bot_bot_set_rule ) )
= top_to728902761071950545ist_fm ) ).
% INT_empty
thf(fact_969_INT__empty,axiom,
! [B: $o > set_Pr7058068377845519745ist_fm] :
( ( comple5608129641975563656ist_fm @ ( image_1666948324336172654ist_fm @ B @ bot_bot_set_o ) )
= top_to728902761071950545ist_fm ) ).
% INT_empty
thf(fact_970_INT__empty,axiom,
! [B: produc1996495991257130529ist_fm > set_Pr7058068377845519745ist_fm] :
( ( comple5608129641975563656ist_fm @ ( image_5852275405373602981ist_fm @ B @ bot_bo4530011010606209517ist_fm ) )
= top_to728902761071950545ist_fm ) ).
% INT_empty
thf(fact_971_INT__empty,axiom,
! [B: rule > set_rule] :
( ( comple5773694076043965236t_rule @ ( image_rule_set_rule @ B @ bot_bot_set_rule ) )
= top_top_set_rule ) ).
% INT_empty
thf(fact_972_INT__empty,axiom,
! [B: $o > set_rule] :
( ( comple5773694076043965236t_rule @ ( image_o_set_rule @ B @ bot_bot_set_o ) )
= top_top_set_rule ) ).
% INT_empty
thf(fact_973_INT__empty,axiom,
! [B: produc1996495991257130529ist_fm > set_rule] :
( ( comple5773694076043965236t_rule @ ( image_2008118025656205143t_rule @ B @ bot_bo4530011010606209517ist_fm ) )
= top_top_set_rule ) ).
% INT_empty
thf(fact_974_INT__empty,axiom,
! [B: rule > set_o] :
( ( comple3063163877087187839_set_o @ ( image_rule_set_o @ B @ bot_bot_set_rule ) )
= top_top_set_o ) ).
% INT_empty
thf(fact_975_INT__empty,axiom,
! [B: $o > set_o] :
( ( comple3063163877087187839_set_o @ ( image_o_set_o @ B @ bot_bot_set_o ) )
= top_top_set_o ) ).
% INT_empty
thf(fact_976_INT__empty,axiom,
! [B: produc1996495991257130529ist_fm > set_o] :
( ( comple3063163877087187839_set_o @ ( image_7087520594942641564_set_o @ B @ bot_bo4530011010606209517ist_fm ) )
= top_top_set_o ) ).
% INT_empty
thf(fact_977_UN__extend__simps_I1_J,axiom,
! [C4: set_rule,A3: rule,B: rule > set_rule] :
( ( ( C4 = bot_bot_set_rule )
=> ( ( insert_rule @ A3 @ ( comple2146307154184993742t_rule @ ( image_rule_set_rule @ B @ C4 ) ) )
= ( insert_rule @ A3 @ bot_bot_set_rule ) ) )
& ( ( C4 != bot_bot_set_rule )
=> ( ( insert_rule @ A3 @ ( comple2146307154184993742t_rule @ ( image_rule_set_rule @ B @ C4 ) ) )
= ( comple2146307154184993742t_rule
@ ( image_rule_set_rule
@ ^ [X: rule] : ( insert_rule @ A3 @ ( B @ X ) )
@ C4 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_978_UN__extend__simps_I1_J,axiom,
! [C4: set_rule,A3: $o,B: rule > set_o] :
( ( ( C4 = bot_bot_set_rule )
=> ( ( insert_o @ A3 @ ( comple90263536869209701_set_o @ ( image_rule_set_o @ B @ C4 ) ) )
= ( insert_o @ A3 @ bot_bot_set_o ) ) )
& ( ( C4 != bot_bot_set_rule )
=> ( ( insert_o @ A3 @ ( comple90263536869209701_set_o @ ( image_rule_set_o @ B @ C4 ) ) )
= ( comple90263536869209701_set_o
@ ( image_rule_set_o
@ ^ [X: rule] : ( insert_o @ A3 @ ( B @ X ) )
@ C4 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_979_UN__extend__simps_I1_J,axiom,
! [C4: set_rule,A3: produc1996495991257130529ist_fm,B: rule > set_Pr7058068377845519745ist_fm] :
( ( ( C4 = bot_bot_set_rule )
=> ( ( insert589278167076416497ist_fm @ A3 @ ( comple1649742678916152942ist_fm @ ( image_3897101447128207157ist_fm @ B @ C4 ) ) )
= ( insert589278167076416497ist_fm @ A3 @ bot_bo4530011010606209517ist_fm ) ) )
& ( ( C4 != bot_bot_set_rule )
=> ( ( insert589278167076416497ist_fm @ A3 @ ( comple1649742678916152942ist_fm @ ( image_3897101447128207157ist_fm @ B @ C4 ) ) )
= ( comple1649742678916152942ist_fm
@ ( image_3897101447128207157ist_fm
@ ^ [X: rule] : ( insert589278167076416497ist_fm @ A3 @ ( B @ X ) )
@ C4 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_980_UN__extend__simps_I1_J,axiom,
! [C4: set_o,A3: rule,B: $o > set_rule] :
( ( ( C4 = bot_bot_set_o )
=> ( ( insert_rule @ A3 @ ( comple2146307154184993742t_rule @ ( image_o_set_rule @ B @ C4 ) ) )
= ( insert_rule @ A3 @ bot_bot_set_rule ) ) )
& ( ( C4 != bot_bot_set_o )
=> ( ( insert_rule @ A3 @ ( comple2146307154184993742t_rule @ ( image_o_set_rule @ B @ C4 ) ) )
= ( comple2146307154184993742t_rule
@ ( image_o_set_rule
@ ^ [X: $o] : ( insert_rule @ A3 @ ( B @ X ) )
@ C4 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_981_UN__extend__simps_I1_J,axiom,
! [C4: set_o,A3: $o,B: $o > set_o] :
( ( ( C4 = bot_bot_set_o )
=> ( ( insert_o @ A3 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B @ C4 ) ) )
= ( insert_o @ A3 @ bot_bot_set_o ) ) )
& ( ( C4 != bot_bot_set_o )
=> ( ( insert_o @ A3 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B @ C4 ) ) )
= ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X: $o] : ( insert_o @ A3 @ ( B @ X ) )
@ C4 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_982_UN__extend__simps_I1_J,axiom,
! [C4: set_o,A3: produc1996495991257130529ist_fm,B: $o > set_Pr7058068377845519745ist_fm] :
( ( ( C4 = bot_bot_set_o )
=> ( ( insert589278167076416497ist_fm @ A3 @ ( comple1649742678916152942ist_fm @ ( image_1666948324336172654ist_fm @ B @ C4 ) ) )
= ( insert589278167076416497ist_fm @ A3 @ bot_bo4530011010606209517ist_fm ) ) )
& ( ( C4 != bot_bot_set_o )
=> ( ( insert589278167076416497ist_fm @ A3 @ ( comple1649742678916152942ist_fm @ ( image_1666948324336172654ist_fm @ B @ C4 ) ) )
= ( comple1649742678916152942ist_fm
@ ( image_1666948324336172654ist_fm
@ ^ [X: $o] : ( insert589278167076416497ist_fm @ A3 @ ( B @ X ) )
@ C4 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_983_UN__extend__simps_I1_J,axiom,
! [C4: set_Pr7058068377845519745ist_fm,A3: rule,B: produc1996495991257130529ist_fm > set_rule] :
( ( ( C4 = bot_bo4530011010606209517ist_fm )
=> ( ( insert_rule @ A3 @ ( comple2146307154184993742t_rule @ ( image_2008118025656205143t_rule @ B @ C4 ) ) )
= ( insert_rule @ A3 @ bot_bot_set_rule ) ) )
& ( ( C4 != bot_bo4530011010606209517ist_fm )
=> ( ( insert_rule @ A3 @ ( comple2146307154184993742t_rule @ ( image_2008118025656205143t_rule @ B @ C4 ) ) )
= ( comple2146307154184993742t_rule
@ ( image_2008118025656205143t_rule
@ ^ [X: produc1996495991257130529ist_fm] : ( insert_rule @ A3 @ ( B @ X ) )
@ C4 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_984_UN__extend__simps_I1_J,axiom,
! [C4: set_Pr7058068377845519745ist_fm,A3: $o,B: produc1996495991257130529ist_fm > set_o] :
( ( ( C4 = bot_bo4530011010606209517ist_fm )
=> ( ( insert_o @ A3 @ ( comple90263536869209701_set_o @ ( image_7087520594942641564_set_o @ B @ C4 ) ) )
= ( insert_o @ A3 @ bot_bot_set_o ) ) )
& ( ( C4 != bot_bo4530011010606209517ist_fm )
=> ( ( insert_o @ A3 @ ( comple90263536869209701_set_o @ ( image_7087520594942641564_set_o @ B @ C4 ) ) )
= ( comple90263536869209701_set_o
@ ( image_7087520594942641564_set_o
@ ^ [X: produc1996495991257130529ist_fm] : ( insert_o @ A3 @ ( B @ X ) )
@ C4 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_985_UN__extend__simps_I1_J,axiom,
! [C4: set_Pr7058068377845519745ist_fm,A3: produc1996495991257130529ist_fm,B: produc1996495991257130529ist_fm > set_Pr7058068377845519745ist_fm] :
( ( ( C4 = bot_bo4530011010606209517ist_fm )
=> ( ( insert589278167076416497ist_fm @ A3 @ ( comple1649742678916152942ist_fm @ ( image_5852275405373602981ist_fm @ B @ C4 ) ) )
= ( insert589278167076416497ist_fm @ A3 @ bot_bo4530011010606209517ist_fm ) ) )
& ( ( C4 != bot_bo4530011010606209517ist_fm )
=> ( ( insert589278167076416497ist_fm @ A3 @ ( comple1649742678916152942ist_fm @ ( image_5852275405373602981ist_fm @ B @ C4 ) ) )
= ( comple1649742678916152942ist_fm
@ ( image_5852275405373602981ist_fm
@ ^ [X: produc1996495991257130529ist_fm] : ( insert589278167076416497ist_fm @ A3 @ ( B @ X ) )
@ C4 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_986_INT__extend__simps_I3_J,axiom,
! [C4: set_rule,A: rule > set_rule,B: set_rule] :
( ( ( C4 = bot_bot_set_rule )
=> ( ( minus_minus_set_rule @ ( comple5773694076043965236t_rule @ ( image_rule_set_rule @ A @ C4 ) ) @ B )
= ( minus_minus_set_rule @ top_top_set_rule @ B ) ) )
& ( ( C4 != bot_bot_set_rule )
=> ( ( minus_minus_set_rule @ ( comple5773694076043965236t_rule @ ( image_rule_set_rule @ A @ C4 ) ) @ B )
= ( comple5773694076043965236t_rule
@ ( image_rule_set_rule
@ ^ [X: rule] : ( minus_minus_set_rule @ ( A @ X ) @ B )
@ C4 ) ) ) ) ) ).
% INT_extend_simps(3)
thf(fact_987_INT__extend__simps_I3_J,axiom,
! [C4: set_o,A: $o > set_rule,B: set_rule] :
( ( ( C4 = bot_bot_set_o )
=> ( ( minus_minus_set_rule @ ( comple5773694076043965236t_rule @ ( image_o_set_rule @ A @ C4 ) ) @ B )
= ( minus_minus_set_rule @ top_top_set_rule @ B ) ) )
& ( ( C4 != bot_bot_set_o )
=> ( ( minus_minus_set_rule @ ( comple5773694076043965236t_rule @ ( image_o_set_rule @ A @ C4 ) ) @ B )
= ( comple5773694076043965236t_rule
@ ( image_o_set_rule
@ ^ [X: $o] : ( minus_minus_set_rule @ ( A @ X ) @ B )
@ C4 ) ) ) ) ) ).
% INT_extend_simps(3)
thf(fact_988_INT__extend__simps_I3_J,axiom,
! [C4: set_Pr7058068377845519745ist_fm,A: produc1996495991257130529ist_fm > set_rule,B: set_rule] :
( ( ( C4 = bot_bo4530011010606209517ist_fm )
=> ( ( minus_minus_set_rule @ ( comple5773694076043965236t_rule @ ( image_2008118025656205143t_rule @ A @ C4 ) ) @ B )
= ( minus_minus_set_rule @ top_top_set_rule @ B ) ) )
& ( ( C4 != bot_bo4530011010606209517ist_fm )
=> ( ( minus_minus_set_rule @ ( comple5773694076043965236t_rule @ ( image_2008118025656205143t_rule @ A @ C4 ) ) @ B )
= ( comple5773694076043965236t_rule
@ ( image_2008118025656205143t_rule
@ ^ [X: produc1996495991257130529ist_fm] : ( minus_minus_set_rule @ ( A @ X ) @ B )
@ C4 ) ) ) ) ) ).
% INT_extend_simps(3)
thf(fact_989_INT__extend__simps_I3_J,axiom,
! [C4: set_rule,A: rule > set_o,B: set_o] :
( ( ( C4 = bot_bot_set_rule )
=> ( ( minus_minus_set_o @ ( comple3063163877087187839_set_o @ ( image_rule_set_o @ A @ C4 ) ) @ B )
= ( minus_minus_set_o @ top_top_set_o @ B ) ) )
& ( ( C4 != bot_bot_set_rule )
=> ( ( minus_minus_set_o @ ( comple3063163877087187839_set_o @ ( image_rule_set_o @ A @ C4 ) ) @ B )
= ( comple3063163877087187839_set_o
@ ( image_rule_set_o
@ ^ [X: rule] : ( minus_minus_set_o @ ( A @ X ) @ B )
@ C4 ) ) ) ) ) ).
% INT_extend_simps(3)
thf(fact_990_INT__extend__simps_I3_J,axiom,
! [C4: set_o,A: $o > set_o,B: set_o] :
( ( ( C4 = bot_bot_set_o )
=> ( ( minus_minus_set_o @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ A @ C4 ) ) @ B )
= ( minus_minus_set_o @ top_top_set_o @ B ) ) )
& ( ( C4 != bot_bot_set_o )
=> ( ( minus_minus_set_o @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ A @ C4 ) ) @ B )
= ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [X: $o] : ( minus_minus_set_o @ ( A @ X ) @ B )
@ C4 ) ) ) ) ) ).
% INT_extend_simps(3)
thf(fact_991_INT__extend__simps_I3_J,axiom,
! [C4: set_Pr7058068377845519745ist_fm,A: produc1996495991257130529ist_fm > set_o,B: set_o] :
( ( ( C4 = bot_bo4530011010606209517ist_fm )
=> ( ( minus_minus_set_o @ ( comple3063163877087187839_set_o @ ( image_7087520594942641564_set_o @ A @ C4 ) ) @ B )
= ( minus_minus_set_o @ top_top_set_o @ B ) ) )
& ( ( C4 != bot_bo4530011010606209517ist_fm )
=> ( ( minus_minus_set_o @ ( comple3063163877087187839_set_o @ ( image_7087520594942641564_set_o @ A @ C4 ) ) @ B )
= ( comple3063163877087187839_set_o
@ ( image_7087520594942641564_set_o
@ ^ [X: produc1996495991257130529ist_fm] : ( minus_minus_set_o @ ( A @ X ) @ B )
@ C4 ) ) ) ) ) ).
% INT_extend_simps(3)
thf(fact_992_INT__extend__simps_I3_J,axiom,
! [C4: set_rule,A: rule > set_Pr7058068377845519745ist_fm,B: set_Pr7058068377845519745ist_fm] :
( ( ( C4 = bot_bot_set_rule )
=> ( ( minus_5615909706921521512ist_fm @ ( comple5608129641975563656ist_fm @ ( image_3897101447128207157ist_fm @ A @ C4 ) ) @ B )
= ( minus_5615909706921521512ist_fm @ top_to728902761071950545ist_fm @ B ) ) )
& ( ( C4 != bot_bot_set_rule )
=> ( ( minus_5615909706921521512ist_fm @ ( comple5608129641975563656ist_fm @ ( image_3897101447128207157ist_fm @ A @ C4 ) ) @ B )
= ( comple5608129641975563656ist_fm
@ ( image_3897101447128207157ist_fm
@ ^ [X: rule] : ( minus_5615909706921521512ist_fm @ ( A @ X ) @ B )
@ C4 ) ) ) ) ) ).
% INT_extend_simps(3)
thf(fact_993_INT__extend__simps_I3_J,axiom,
! [C4: set_o,A: $o > set_Pr7058068377845519745ist_fm,B: set_Pr7058068377845519745ist_fm] :
( ( ( C4 = bot_bot_set_o )
=> ( ( minus_5615909706921521512ist_fm @ ( comple5608129641975563656ist_fm @ ( image_1666948324336172654ist_fm @ A @ C4 ) ) @ B )
= ( minus_5615909706921521512ist_fm @ top_to728902761071950545ist_fm @ B ) ) )
& ( ( C4 != bot_bot_set_o )
=> ( ( minus_5615909706921521512ist_fm @ ( comple5608129641975563656ist_fm @ ( image_1666948324336172654ist_fm @ A @ C4 ) ) @ B )
= ( comple5608129641975563656ist_fm
@ ( image_1666948324336172654ist_fm
@ ^ [X: $o] : ( minus_5615909706921521512ist_fm @ ( A @ X ) @ B )
@ C4 ) ) ) ) ) ).
% INT_extend_simps(3)
thf(fact_994_INT__extend__simps_I3_J,axiom,
! [C4: set_Pr7058068377845519745ist_fm,A: produc1996495991257130529ist_fm > set_Pr7058068377845519745ist_fm,B: set_Pr7058068377845519745ist_fm] :
( ( ( C4 = bot_bo4530011010606209517ist_fm )
=> ( ( minus_5615909706921521512ist_fm @ ( comple5608129641975563656ist_fm @ ( image_5852275405373602981ist_fm @ A @ C4 ) ) @ B )
= ( minus_5615909706921521512ist_fm @ top_to728902761071950545ist_fm @ B ) ) )
& ( ( C4 != bot_bo4530011010606209517ist_fm )
=> ( ( minus_5615909706921521512ist_fm @ ( comple5608129641975563656ist_fm @ ( image_5852275405373602981ist_fm @ A @ C4 ) ) @ B )
= ( comple5608129641975563656ist_fm
@ ( image_5852275405373602981ist_fm
@ ^ [X: produc1996495991257130529ist_fm] : ( minus_5615909706921521512ist_fm @ ( A @ X ) @ B )
@ C4 ) ) ) ) ) ).
% INT_extend_simps(3)
thf(fact_995_ccINF__empty,axiom,
! [F: rule > set_Pr7058068377845519745ist_fm] :
( ( comple5608129641975563656ist_fm @ ( image_3897101447128207157ist_fm @ F @ bot_bot_set_rule ) )
= top_to728902761071950545ist_fm ) ).
% ccINF_empty
thf(fact_996_ccINF__empty,axiom,
! [F: $o > set_Pr7058068377845519745ist_fm] :
( ( comple5608129641975563656ist_fm @ ( image_1666948324336172654ist_fm @ F @ bot_bot_set_o ) )
= top_to728902761071950545ist_fm ) ).
% ccINF_empty
thf(fact_997_ccINF__empty,axiom,
! [F: produc1996495991257130529ist_fm > set_Pr7058068377845519745ist_fm] :
( ( comple5608129641975563656ist_fm @ ( image_5852275405373602981ist_fm @ F @ bot_bo4530011010606209517ist_fm ) )
= top_to728902761071950545ist_fm ) ).
% ccINF_empty
thf(fact_998_ccINF__empty,axiom,
! [F: rule > set_rule] :
( ( comple5773694076043965236t_rule @ ( image_rule_set_rule @ F @ bot_bot_set_rule ) )
= top_top_set_rule ) ).
% ccINF_empty
thf(fact_999_ccINF__empty,axiom,
! [F: $o > set_rule] :
( ( comple5773694076043965236t_rule @ ( image_o_set_rule @ F @ bot_bot_set_o ) )
= top_top_set_rule ) ).
% ccINF_empty
thf(fact_1000_ccINF__empty,axiom,
! [F: produc1996495991257130529ist_fm > set_rule] :
( ( comple5773694076043965236t_rule @ ( image_2008118025656205143t_rule @ F @ bot_bo4530011010606209517ist_fm ) )
= top_top_set_rule ) ).
% ccINF_empty
thf(fact_1001_ccINF__empty,axiom,
! [F: rule > set_o] :
( ( comple3063163877087187839_set_o @ ( image_rule_set_o @ F @ bot_bot_set_rule ) )
= top_top_set_o ) ).
% ccINF_empty
thf(fact_1002_ccINF__empty,axiom,
! [F: $o > set_o] :
( ( comple3063163877087187839_set_o @ ( image_o_set_o @ F @ bot_bot_set_o ) )
= top_top_set_o ) ).
% ccINF_empty
thf(fact_1003_ccINF__empty,axiom,
! [F: produc1996495991257130529ist_fm > set_o] :
( ( comple3063163877087187839_set_o @ ( image_7087520594942641564_set_o @ F @ bot_bo4530011010606209517ist_fm ) )
= top_top_set_o ) ).
% ccINF_empty
thf(fact_1004_ccSUP__empty,axiom,
! [F: rule > set_rule] :
( ( comple2146307154184993742t_rule @ ( image_rule_set_rule @ F @ bot_bot_set_rule ) )
= bot_bot_set_rule ) ).
% ccSUP_empty
thf(fact_1005_ccSUP__empty,axiom,
! [F: rule > set_o] :
( ( comple90263536869209701_set_o @ ( image_rule_set_o @ F @ bot_bot_set_rule ) )
= bot_bot_set_o ) ).
% ccSUP_empty
thf(fact_1006_ccSUP__empty,axiom,
! [F: rule > set_Pr7058068377845519745ist_fm] :
( ( comple1649742678916152942ist_fm @ ( image_3897101447128207157ist_fm @ F @ bot_bot_set_rule ) )
= bot_bo4530011010606209517ist_fm ) ).
% ccSUP_empty
thf(fact_1007_ccSUP__empty,axiom,
! [F: $o > set_rule] :
( ( comple2146307154184993742t_rule @ ( image_o_set_rule @ F @ bot_bot_set_o ) )
= bot_bot_set_rule ) ).
% ccSUP_empty
thf(fact_1008_ccSUP__empty,axiom,
! [F: $o > set_o] :
( ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ bot_bot_set_o ) )
= bot_bot_set_o ) ).
% ccSUP_empty
thf(fact_1009_ccSUP__empty,axiom,
! [F: $o > set_Pr7058068377845519745ist_fm] :
( ( comple1649742678916152942ist_fm @ ( image_1666948324336172654ist_fm @ F @ bot_bot_set_o ) )
= bot_bo4530011010606209517ist_fm ) ).
% ccSUP_empty
thf(fact_1010_ccSUP__empty,axiom,
! [F: produc1996495991257130529ist_fm > set_rule] :
( ( comple2146307154184993742t_rule @ ( image_2008118025656205143t_rule @ F @ bot_bo4530011010606209517ist_fm ) )
= bot_bot_set_rule ) ).
% ccSUP_empty
thf(fact_1011_ccSUP__empty,axiom,
! [F: produc1996495991257130529ist_fm > set_o] :
( ( comple90263536869209701_set_o @ ( image_7087520594942641564_set_o @ F @ bot_bo4530011010606209517ist_fm ) )
= bot_bot_set_o ) ).
% ccSUP_empty
thf(fact_1012_ccSUP__empty,axiom,
! [F: produc1996495991257130529ist_fm > set_Pr7058068377845519745ist_fm] :
( ( comple1649742678916152942ist_fm @ ( image_5852275405373602981ist_fm @ F @ bot_bo4530011010606209517ist_fm ) )
= bot_bo4530011010606209517ist_fm ) ).
% ccSUP_empty
thf(fact_1013_ccSup__empty,axiom,
( ( comple2146307154184993742t_rule @ bot_bot_set_set_rule )
= bot_bot_set_rule ) ).
% ccSup_empty
thf(fact_1014_ccSup__empty,axiom,
( ( comple90263536869209701_set_o @ bot_bot_set_set_o )
= bot_bot_set_o ) ).
% ccSup_empty
thf(fact_1015_ccSup__empty,axiom,
( ( comple1649742678916152942ist_fm @ bot_bo414211191905506637ist_fm )
= bot_bo4530011010606209517ist_fm ) ).
% ccSup_empty
thf(fact_1016_ccSup__empty,axiom,
( ( complete_Sup_Sup_o @ bot_bot_set_o )
= bot_bot_o ) ).
% ccSup_empty
thf(fact_1017_ccInf__empty,axiom,
( ( complete_Inf_Inf_o @ bot_bot_set_o )
= top_top_o ) ).
% ccInf_empty
thf(fact_1018_ccInf__empty,axiom,
( ( comple5608129641975563656ist_fm @ bot_bo414211191905506637ist_fm )
= top_to728902761071950545ist_fm ) ).
% ccInf_empty
thf(fact_1019_ccInf__empty,axiom,
( ( comple5773694076043965236t_rule @ bot_bot_set_set_rule )
= top_top_set_rule ) ).
% ccInf_empty
thf(fact_1020_ccInf__empty,axiom,
( ( comple3063163877087187839_set_o @ bot_bot_set_set_o )
= top_top_set_o ) ).
% ccInf_empty
thf(fact_1021_cSup__singleton,axiom,
! [X2: $o] :
( ( complete_Sup_Sup_o @ ( insert_o @ X2 @ bot_bot_set_o ) )
= X2 ) ).
% cSup_singleton
thf(fact_1022_cInf__singleton,axiom,
! [X2: $o] :
( ( complete_Inf_Inf_o @ ( insert_o @ X2 @ bot_bot_set_o ) )
= X2 ) ).
% cInf_singleton
thf(fact_1023_Sup__SUP__eq,axiom,
( complete_Sup_Sup_o_o
= ( ^ [S8: set_o_o,X: $o] : ( member_o @ X @ ( comple90263536869209701_set_o @ ( image_o_o_set_o @ collect_o @ S8 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_1024_Sup__SUP__eq,axiom,
( comple1826244231481717815rule_o
= ( ^ [S8: set_rule_o,X: rule] : ( member_rule @ X @ ( comple2146307154184993742t_rule @ ( image_1281159361656534528t_rule @ collect_rule @ S8 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_1025_Sup__SUP__eq,axiom,
( comple5552222795444685295t_fm_o
= ( ^ [S8: set_Pr6299178538722460242t_fm_o,X: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X @ ( comple1649742678916152942ist_fm @ ( image_3053959355298890014ist_fm @ collec7231575776666803724ist_fm @ S8 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_1026_Inf__INT__eq,axiom,
( complete_Inf_Inf_o_o
= ( ^ [S8: set_o_o,X: $o] : ( member_o @ X @ ( comple3063163877087187839_set_o @ ( image_o_o_set_o @ collect_o @ S8 ) ) ) ) ) ).
% Inf_INT_eq
thf(fact_1027_Inf__INT__eq,axiom,
( comple715424409190658129rule_o
= ( ^ [S8: set_rule_o,X: rule] : ( member_rule @ X @ ( comple5773694076043965236t_rule @ ( image_1281159361656534528t_rule @ collect_rule @ S8 ) ) ) ) ) ).
% Inf_INT_eq
thf(fact_1028_Inf__INT__eq,axiom,
( comple1695591244663615317t_fm_o
= ( ^ [S8: set_Pr6299178538722460242t_fm_o,X: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X @ ( comple5608129641975563656ist_fm @ ( image_3053959355298890014ist_fm @ collec7231575776666803724ist_fm @ S8 ) ) ) ) ) ).
% Inf_INT_eq
thf(fact_1029_INF__Int__eq2,axiom,
! [S5: set_se6775865681059004641ist_fm] :
( ( comple1444494742520622029t_fm_o
@ ( image_6629374399087694218t_fm_o
@ ^ [I: set_Pr7058068377845519745ist_fm,X: list_fm,Y4: list_fm] : ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X @ Y4 ) @ I )
@ S5 ) )
= ( ^ [X: list_fm,Y4: list_fm] : ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X @ Y4 ) @ ( comple5608129641975563656ist_fm @ S5 ) ) ) ) ).
% INF_Int_eq2
thf(fact_1030_SUP__Sup__eq2,axiom,
! [S5: set_se6775865681059004641ist_fm] :
( ( comple4318906814866966707t_fm_o
@ ( image_6629374399087694218t_fm_o
@ ^ [I: set_Pr7058068377845519745ist_fm,X: list_fm,Y4: list_fm] : ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X @ Y4 ) @ I )
@ S5 ) )
= ( ^ [X: list_fm,Y4: list_fm] : ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X @ Y4 ) @ ( comple1649742678916152942ist_fm @ S5 ) ) ) ) ).
% SUP_Sup_eq2
thf(fact_1031_SUP__Sup__eq,axiom,
! [S5: set_se6775865681059004641ist_fm] :
( ( comple5552222795444685295t_fm_o
@ ( image_1461364200386988504t_fm_o
@ ^ [I: set_Pr7058068377845519745ist_fm,X: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X @ I )
@ S5 ) )
= ( ^ [X: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X @ ( comple1649742678916152942ist_fm @ S5 ) ) ) ) ).
% SUP_Sup_eq
thf(fact_1032_SUP__Sup__eq,axiom,
! [S5: set_set_rule] :
( ( comple1826244231481717815rule_o
@ ( image_5795755989128758216rule_o
@ ^ [I: set_rule,X: rule] : ( member_rule @ X @ I )
@ S5 ) )
= ( ^ [X: rule] : ( member_rule @ X @ ( comple2146307154184993742t_rule @ S5 ) ) ) ) ).
% SUP_Sup_eq
thf(fact_1033_SUP__Sup__eq,axiom,
! [S5: set_set_o] :
( ( complete_Sup_Sup_o_o
@ ( image_set_o_o_o
@ ^ [I: set_o,X: $o] : ( member_o @ X @ I )
@ S5 ) )
= ( ^ [X: $o] : ( member_o @ X @ ( comple90263536869209701_set_o @ S5 ) ) ) ) ).
% SUP_Sup_eq
thf(fact_1034_INF__Int__eq,axiom,
! [S5: set_se6775865681059004641ist_fm] :
( ( comple1695591244663615317t_fm_o
@ ( image_1461364200386988504t_fm_o
@ ^ [I: set_Pr7058068377845519745ist_fm,X: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X @ I )
@ S5 ) )
= ( ^ [X: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X @ ( comple5608129641975563656ist_fm @ S5 ) ) ) ) ).
% INF_Int_eq
thf(fact_1035_INF__Int__eq,axiom,
! [S5: set_set_rule] :
( ( comple715424409190658129rule_o
@ ( image_5795755989128758216rule_o
@ ^ [I: set_rule,X: rule] : ( member_rule @ X @ I )
@ S5 ) )
= ( ^ [X: rule] : ( member_rule @ X @ ( comple5773694076043965236t_rule @ S5 ) ) ) ) ).
% INF_Int_eq
thf(fact_1036_INF__Int__eq,axiom,
! [S5: set_set_o] :
( ( complete_Inf_Inf_o_o
@ ( image_set_o_o_o
@ ^ [I: set_o,X: $o] : ( member_o @ X @ I )
@ S5 ) )
= ( ^ [X: $o] : ( member_o @ X @ ( comple3063163877087187839_set_o @ S5 ) ) ) ) ).
% INF_Int_eq
thf(fact_1037_Sup__set__def,axiom,
( comple90263536869209701_set_o
= ( ^ [A5: set_set_o] :
( collect_o
@ ^ [X: $o] : ( complete_Sup_Sup_o @ ( image_set_o_o @ ( member_o @ X ) @ A5 ) ) ) ) ) ).
% Sup_set_def
thf(fact_1038_Sup__set__def,axiom,
( comple2146307154184993742t_rule
= ( ^ [A5: set_set_rule] :
( collect_rule
@ ^ [X: rule] : ( complete_Sup_Sup_o @ ( image_set_rule_o @ ( member_rule @ X ) @ A5 ) ) ) ) ) ).
% Sup_set_def
thf(fact_1039_Sup__set__def,axiom,
( comple1649742678916152942ist_fm
= ( ^ [A5: set_se6775865681059004641ist_fm] :
( collec7231575776666803724ist_fm
@ ^ [X: produc1996495991257130529ist_fm] : ( complete_Sup_Sup_o @ ( image_6697058094847422300t_fm_o @ ( member8102475879199740618ist_fm @ X ) @ A5 ) ) ) ) ) ).
% Sup_set_def
thf(fact_1040_Inf__set__def,axiom,
( comple3063163877087187839_set_o
= ( ^ [A5: set_set_o] :
( collect_o
@ ^ [X: $o] : ( complete_Inf_Inf_o @ ( image_set_o_o @ ( member_o @ X ) @ A5 ) ) ) ) ) ).
% Inf_set_def
thf(fact_1041_Inf__set__def,axiom,
( comple5773694076043965236t_rule
= ( ^ [A5: set_set_rule] :
( collect_rule
@ ^ [X: rule] : ( complete_Inf_Inf_o @ ( image_set_rule_o @ ( member_rule @ X ) @ A5 ) ) ) ) ) ).
% Inf_set_def
thf(fact_1042_Inf__set__def,axiom,
( comple5608129641975563656ist_fm
= ( ^ [A5: set_se6775865681059004641ist_fm] :
( collec7231575776666803724ist_fm
@ ^ [X: produc1996495991257130529ist_fm] : ( complete_Inf_Inf_o @ ( image_6697058094847422300t_fm_o @ ( member8102475879199740618ist_fm @ X ) @ A5 ) ) ) ) ) ).
% Inf_set_def
thf(fact_1043_ccpo__Sup__singleton,axiom,
! [X2: $o] :
( ( complete_Sup_Sup_o @ ( insert_o @ X2 @ bot_bot_set_o ) )
= X2 ) ).
% ccpo_Sup_singleton
thf(fact_1044_graph__map__upd,axiom,
! [M: list_fm > option_list_fm,K: list_fm,V: list_fm] :
( ( graph_2416636321808849472ist_fm @ ( fun_up7757938787137709839ist_fm @ M @ K @ ( some_list_fm @ V ) ) )
= ( insert589278167076416497ist_fm @ ( produc381145313068854617ist_fm @ K @ V ) @ ( graph_2416636321808849472ist_fm @ ( fun_up7757938787137709839ist_fm @ M @ K @ none_list_fm ) ) ) ) ).
% graph_map_upd
thf(fact_1045_sorted__list__of__set_Oinj__on,axiom,
( inj_on_o_o
@ ^ [X: $o] : X
@ top_top_set_o ) ).
% sorted_list_of_set.inj_on
thf(fact_1046_Id__on__def,axiom,
( id_on_list_fm
= ( ^ [A5: set_list_fm] :
( comple1649742678916152942ist_fm
@ ( image_4561957376826621658ist_fm
@ ^ [X: list_fm] : ( insert589278167076416497ist_fm @ ( produc381145313068854617ist_fm @ X @ X ) @ bot_bo4530011010606209517ist_fm )
@ A5 ) ) ) ) ).
% Id_on_def
thf(fact_1047_Id__onI,axiom,
! [A3: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm] :
( ( member8102475879199740618ist_fm @ A3 @ A )
=> ( member60238275747228048ist_fm @ ( produc2592314732767339223ist_fm @ A3 @ A3 ) @ ( id_on_4358887334071312446ist_fm @ A ) ) ) ).
% Id_onI
thf(fact_1048_Id__onI,axiom,
! [A3: rule,A: set_rule] :
( ( member_rule @ A3 @ A )
=> ( member3753523862196722196e_rule @ ( produc1769597910952761123e_rule @ A3 @ A3 ) @ ( id_on_rule @ A ) ) ) ).
% Id_onI
thf(fact_1049_Id__onI,axiom,
! [A3: $o,A: set_o] :
( ( member_o @ A3 @ A )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A3 @ A3 ) @ ( id_on_o @ A ) ) ) ).
% Id_onI
thf(fact_1050_Id__onI,axiom,
! [A3: list_fm,A: set_list_fm] :
( ( member_list_fm @ A3 @ A )
=> ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ A3 @ A3 ) @ ( id_on_list_fm @ A ) ) ) ).
% Id_onI
thf(fact_1051_Id__on__empty,axiom,
( ( id_on_list_fm @ bot_bot_set_list_fm )
= bot_bo4530011010606209517ist_fm ) ).
% Id_on_empty
thf(fact_1052_Id__on__empty,axiom,
( ( id_on_rule @ bot_bot_set_rule )
= bot_bo9047865283180780599e_rule ) ).
% Id_on_empty
thf(fact_1053_Id__on__empty,axiom,
( ( id_on_o @ bot_bot_set_o )
= bot_bo7073875226086086771od_o_o ) ).
% Id_on_empty
thf(fact_1054_Id__on__empty,axiom,
( ( id_on_4358887334071312446ist_fm @ bot_bo4530011010606209517ist_fm )
= bot_bo7330262581962992051ist_fm ) ).
% Id_on_empty
thf(fact_1055_graph__empty,axiom,
( ( graph_2416636321808849472ist_fm
@ ^ [X: list_fm] : none_list_fm )
= bot_bo4530011010606209517ist_fm ) ).
% graph_empty
thf(fact_1056_graph__restrictD_I1_J,axiom,
! [K: list_fm,V: list_fm,M: list_fm > option_list_fm,A: set_list_fm] :
( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ K @ V ) @ ( graph_2416636321808849472ist_fm @ ( restri8510528498258177010ist_fm @ M @ A ) ) )
=> ( member_list_fm @ K @ A ) ) ).
% graph_restrictD(1)
thf(fact_1057_graph__restrictD_I2_J,axiom,
! [K: list_fm,V: list_fm,M: list_fm > option_list_fm,A: set_list_fm] :
( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ K @ V ) @ ( graph_2416636321808849472ist_fm @ ( restri8510528498258177010ist_fm @ M @ A ) ) )
=> ( ( M @ K )
= ( some_list_fm @ V ) ) ) ).
% graph_restrictD(2)
thf(fact_1058_Id__onE,axiom,
! [C: produc1754732471204517991ist_fm,A: set_Pr7058068377845519745ist_fm] :
( ( member60238275747228048ist_fm @ C @ ( id_on_4358887334071312446ist_fm @ A ) )
=> ~ ! [X3: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ X3 @ A )
=> ( C
!= ( produc2592314732767339223ist_fm @ X3 @ X3 ) ) ) ) ).
% Id_onE
thf(fact_1059_Id__onE,axiom,
! [C: produc780299800278082923e_rule,A: set_rule] :
( ( member3753523862196722196e_rule @ C @ ( id_on_rule @ A ) )
=> ~ ! [X3: rule] :
( ( member_rule @ X3 @ A )
=> ( C
!= ( produc1769597910952761123e_rule @ X3 @ X3 ) ) ) ) ).
% Id_onE
thf(fact_1060_Id__onE,axiom,
! [C: product_prod_o_o,A: set_o] :
( ( member7466972457876170832od_o_o @ C @ ( id_on_o @ A ) )
=> ~ ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( C
!= ( product_Pair_o_o @ X3 @ X3 ) ) ) ) ).
% Id_onE
thf(fact_1061_Id__onE,axiom,
! [C: produc1996495991257130529ist_fm,A: set_list_fm] :
( ( member8102475879199740618ist_fm @ C @ ( id_on_list_fm @ A ) )
=> ~ ! [X3: list_fm] :
( ( member_list_fm @ X3 @ A )
=> ( C
!= ( produc381145313068854617ist_fm @ X3 @ X3 ) ) ) ) ).
% Id_onE
thf(fact_1062_Id__on__eqI,axiom,
! [A3: produc1996495991257130529ist_fm,B3: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm] :
( ( A3 = B3 )
=> ( ( member8102475879199740618ist_fm @ A3 @ A )
=> ( member60238275747228048ist_fm @ ( produc2592314732767339223ist_fm @ A3 @ B3 ) @ ( id_on_4358887334071312446ist_fm @ A ) ) ) ) ).
% Id_on_eqI
thf(fact_1063_Id__on__eqI,axiom,
! [A3: rule,B3: rule,A: set_rule] :
( ( A3 = B3 )
=> ( ( member_rule @ A3 @ A )
=> ( member3753523862196722196e_rule @ ( produc1769597910952761123e_rule @ A3 @ B3 ) @ ( id_on_rule @ A ) ) ) ) ).
% Id_on_eqI
thf(fact_1064_Id__on__eqI,axiom,
! [A3: $o,B3: $o,A: set_o] :
( ( A3 = B3 )
=> ( ( member_o @ A3 @ A )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A3 @ B3 ) @ ( id_on_o @ A ) ) ) ) ).
% Id_on_eqI
thf(fact_1065_Id__on__eqI,axiom,
! [A3: list_fm,B3: list_fm,A: set_list_fm] :
( ( A3 = B3 )
=> ( ( member_list_fm @ A3 @ A )
=> ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ A3 @ B3 ) @ ( id_on_list_fm @ A ) ) ) ) ).
% Id_on_eqI
thf(fact_1066_Id__on__iff,axiom,
! [X2: produc1996495991257130529ist_fm,Y3: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm] :
( ( member60238275747228048ist_fm @ ( produc2592314732767339223ist_fm @ X2 @ Y3 ) @ ( id_on_4358887334071312446ist_fm @ A ) )
= ( ( X2 = Y3 )
& ( member8102475879199740618ist_fm @ X2 @ A ) ) ) ).
% Id_on_iff
thf(fact_1067_Id__on__iff,axiom,
! [X2: rule,Y3: rule,A: set_rule] :
( ( member3753523862196722196e_rule @ ( produc1769597910952761123e_rule @ X2 @ Y3 ) @ ( id_on_rule @ A ) )
= ( ( X2 = Y3 )
& ( member_rule @ X2 @ A ) ) ) ).
% Id_on_iff
thf(fact_1068_Id__on__iff,axiom,
! [X2: $o,Y3: $o,A: set_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y3 ) @ ( id_on_o @ A ) )
= ( ( X2 = Y3 )
& ( member_o @ X2 @ A ) ) ) ).
% Id_on_iff
thf(fact_1069_Id__on__iff,axiom,
! [X2: list_fm,Y3: list_fm,A: set_list_fm] :
( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X2 @ Y3 ) @ ( id_on_list_fm @ A ) )
= ( ( X2 = Y3 )
& ( member_list_fm @ X2 @ A ) ) ) ).
% Id_on_iff
thf(fact_1070_in__graphI,axiom,
! [M: list_fm > option_list_fm,K: list_fm,V: list_fm] :
( ( ( M @ K )
= ( some_list_fm @ V ) )
=> ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ K @ V ) @ ( graph_2416636321808849472ist_fm @ M ) ) ) ).
% in_graphI
thf(fact_1071_in__graphD,axiom,
! [K: list_fm,V: list_fm,M: list_fm > option_list_fm] :
( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ K @ V ) @ ( graph_2416636321808849472ist_fm @ M ) )
=> ( ( M @ K )
= ( some_list_fm @ V ) ) ) ).
% in_graphD
thf(fact_1072_trans__Id__on,axiom,
! [A: set_Pr7058068377845519745ist_fm] : ( trans_4095360997890102209ist_fm @ top_to728902761071950545ist_fm @ ( id_on_4358887334071312446ist_fm @ A ) ) ).
% trans_Id_on
thf(fact_1073_trans__Id__on,axiom,
! [A: set_rule] : ( trans_on_rule @ top_top_set_rule @ ( id_on_rule @ A ) ) ).
% trans_Id_on
thf(fact_1074_trans__Id__on,axiom,
! [A: set_o] : ( trans_on_o @ top_top_set_o @ ( id_on_o @ A ) ) ).
% trans_Id_on
thf(fact_1075_restrict__upd__same,axiom,
! [M: rule > option6029102822600979917ist_fm,X2: rule,Y3: fset_P661503646757059847ist_fm] :
( ( restri3791882098703154256ist_fm @ ( fun_up5825503389150910729ist_fm @ M @ X2 @ ( some_f2162114139285431020ist_fm @ Y3 ) ) @ ( uminus4869265918275750596t_rule @ ( insert_rule @ X2 @ bot_bot_set_rule ) ) )
= ( restri3791882098703154256ist_fm @ M @ ( uminus4869265918275750596t_rule @ ( insert_rule @ X2 @ bot_bot_set_rule ) ) ) ) ).
% restrict_upd_same
thf(fact_1076_restrict__upd__same,axiom,
! [M: $o > option6029102822600979917ist_fm,X2: $o,Y3: fset_P661503646757059847ist_fm] :
( ( restri2688269635790113695ist_fm @ ( fun_up1845550324453438770ist_fm @ M @ X2 @ ( some_f2162114139285431020ist_fm @ Y3 ) ) @ ( uminus_uminus_set_o @ ( insert_o @ X2 @ bot_bot_set_o ) ) )
= ( restri2688269635790113695ist_fm @ M @ ( uminus_uminus_set_o @ ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ).
% restrict_upd_same
thf(fact_1077_restrict__upd__same,axiom,
! [M: produc1996495991257130529ist_fm > option6029102822600979917ist_fm,X2: produc1996495991257130529ist_fm,Y3: fset_P661503646757059847ist_fm] :
( ( restri3026782591808725334ist_fm @ ( fun_up1138168335916736873ist_fm @ M @ X2 @ ( some_f2162114139285431020ist_fm @ Y3 ) ) @ ( uminus6150396152209328632ist_fm @ ( insert589278167076416497ist_fm @ X2 @ bot_bo4530011010606209517ist_fm ) ) )
= ( restri3026782591808725334ist_fm @ M @ ( uminus6150396152209328632ist_fm @ ( insert589278167076416497ist_fm @ X2 @ bot_bo4530011010606209517ist_fm ) ) ) ) ).
% restrict_upd_same
thf(fact_1078_ComplI,axiom,
! [C: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm] :
( ~ ( member8102475879199740618ist_fm @ C @ A )
=> ( member8102475879199740618ist_fm @ C @ ( uminus6150396152209328632ist_fm @ A ) ) ) ).
% ComplI
thf(fact_1079_ComplI,axiom,
! [C: rule,A: set_rule] :
( ~ ( member_rule @ C @ A )
=> ( member_rule @ C @ ( uminus4869265918275750596t_rule @ A ) ) ) ).
% ComplI
thf(fact_1080_ComplI,axiom,
! [C: $o,A: set_o] :
( ~ ( member_o @ C @ A )
=> ( member_o @ C @ ( uminus_uminus_set_o @ A ) ) ) ).
% ComplI
thf(fact_1081_Compl__iff,axiom,
! [C: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm] :
( ( member8102475879199740618ist_fm @ C @ ( uminus6150396152209328632ist_fm @ A ) )
= ( ~ ( member8102475879199740618ist_fm @ C @ A ) ) ) ).
% Compl_iff
thf(fact_1082_Compl__iff,axiom,
! [C: rule,A: set_rule] :
( ( member_rule @ C @ ( uminus4869265918275750596t_rule @ A ) )
= ( ~ ( member_rule @ C @ A ) ) ) ).
% Compl_iff
thf(fact_1083_Compl__iff,axiom,
! [C: $o,A: set_o] :
( ( member_o @ C @ ( uminus_uminus_set_o @ A ) )
= ( ~ ( member_o @ C @ A ) ) ) ).
% Compl_iff
thf(fact_1084_apsnd__conv,axiom,
! [F: list_fm > list_fm,X2: list_fm,Y3: list_fm] :
( ( produc9047611296996456864ist_fm @ F @ ( produc381145313068854617ist_fm @ X2 @ Y3 ) )
= ( produc381145313068854617ist_fm @ X2 @ ( F @ Y3 ) ) ) ).
% apsnd_conv
thf(fact_1085_dom__const,axiom,
! [F: produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm] :
( ( dom_Pr1119034465550817144ist_fm
@ ^ [X: produc1996495991257130529ist_fm] : ( some_f2162114139285431020ist_fm @ ( F @ X ) ) )
= top_to728902761071950545ist_fm ) ).
% dom_const
thf(fact_1086_dom__const,axiom,
! [F: rule > fset_P661503646757059847ist_fm] :
( ( dom_ru6064144267908753070ist_fm
@ ^ [X: rule] : ( some_f2162114139285431020ist_fm @ ( F @ X ) ) )
= top_top_set_rule ) ).
% dom_const
thf(fact_1087_dom__const,axiom,
! [F: $o > fset_P661503646757059847ist_fm] :
( ( dom_o_6128958874678448065ist_fm
@ ^ [X: $o] : ( some_f2162114139285431020ist_fm @ ( F @ X ) ) )
= top_top_set_o ) ).
% dom_const
thf(fact_1088_Collect__neg__eq,axiom,
! [P: $o > $o] :
( ( collect_o
@ ^ [X: $o] :
~ ( P @ X ) )
= ( uminus_uminus_set_o @ ( collect_o @ P ) ) ) ).
% Collect_neg_eq
thf(fact_1089_Collect__neg__eq,axiom,
! [P: rule > $o] :
( ( collect_rule
@ ^ [X: rule] :
~ ( P @ X ) )
= ( uminus4869265918275750596t_rule @ ( collect_rule @ P ) ) ) ).
% Collect_neg_eq
thf(fact_1090_Collect__neg__eq,axiom,
! [P: produc1996495991257130529ist_fm > $o] :
( ( collec7231575776666803724ist_fm
@ ^ [X: produc1996495991257130529ist_fm] :
~ ( P @ X ) )
= ( uminus6150396152209328632ist_fm @ ( collec7231575776666803724ist_fm @ P ) ) ) ).
% Collect_neg_eq
thf(fact_1091_Compl__eq,axiom,
( uminus_uminus_set_o
= ( ^ [A5: set_o] :
( collect_o
@ ^ [X: $o] :
~ ( member_o @ X @ A5 ) ) ) ) ).
% Compl_eq
thf(fact_1092_Compl__eq,axiom,
( uminus4869265918275750596t_rule
= ( ^ [A5: set_rule] :
( collect_rule
@ ^ [X: rule] :
~ ( member_rule @ X @ A5 ) ) ) ) ).
% Compl_eq
thf(fact_1093_Compl__eq,axiom,
( uminus6150396152209328632ist_fm
= ( ^ [A5: set_Pr7058068377845519745ist_fm] :
( collec7231575776666803724ist_fm
@ ^ [X: produc1996495991257130529ist_fm] :
~ ( member8102475879199740618ist_fm @ X @ A5 ) ) ) ) ).
% Compl_eq
thf(fact_1094_domD,axiom,
! [A3: produc1996495991257130529ist_fm,M: produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
( ( member8102475879199740618ist_fm @ A3 @ ( dom_Pr1119034465550817144ist_fm @ M ) )
=> ? [B7: fset_P661503646757059847ist_fm] :
( ( M @ A3 )
= ( some_f2162114139285431020ist_fm @ B7 ) ) ) ).
% domD
thf(fact_1095_domD,axiom,
! [A3: rule,M: rule > option6029102822600979917ist_fm] :
( ( member_rule @ A3 @ ( dom_ru6064144267908753070ist_fm @ M ) )
=> ? [B7: fset_P661503646757059847ist_fm] :
( ( M @ A3 )
= ( some_f2162114139285431020ist_fm @ B7 ) ) ) ).
% domD
thf(fact_1096_domD,axiom,
! [A3: $o,M: $o > option6029102822600979917ist_fm] :
( ( member_o @ A3 @ ( dom_o_6128958874678448065ist_fm @ M ) )
=> ? [B7: fset_P661503646757059847ist_fm] :
( ( M @ A3 )
= ( some_f2162114139285431020ist_fm @ B7 ) ) ) ).
% domD
thf(fact_1097_domI,axiom,
! [M: produc1996495991257130529ist_fm > option6029102822600979917ist_fm,A3: produc1996495991257130529ist_fm,B3: fset_P661503646757059847ist_fm] :
( ( ( M @ A3 )
= ( some_f2162114139285431020ist_fm @ B3 ) )
=> ( member8102475879199740618ist_fm @ A3 @ ( dom_Pr1119034465550817144ist_fm @ M ) ) ) ).
% domI
thf(fact_1098_domI,axiom,
! [M: rule > option6029102822600979917ist_fm,A3: rule,B3: fset_P661503646757059847ist_fm] :
( ( ( M @ A3 )
= ( some_f2162114139285431020ist_fm @ B3 ) )
=> ( member_rule @ A3 @ ( dom_ru6064144267908753070ist_fm @ M ) ) ) ).
% domI
thf(fact_1099_domI,axiom,
! [M: $o > option6029102822600979917ist_fm,A3: $o,B3: fset_P661503646757059847ist_fm] :
( ( ( M @ A3 )
= ( some_f2162114139285431020ist_fm @ B3 ) )
=> ( member_o @ A3 @ ( dom_o_6128958874678448065ist_fm @ M ) ) ) ).
% domI
thf(fact_1100_ComplD,axiom,
! [C: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm] :
( ( member8102475879199740618ist_fm @ C @ ( uminus6150396152209328632ist_fm @ A ) )
=> ~ ( member8102475879199740618ist_fm @ C @ A ) ) ).
% ComplD
thf(fact_1101_ComplD,axiom,
! [C: rule,A: set_rule] :
( ( member_rule @ C @ ( uminus4869265918275750596t_rule @ A ) )
=> ~ ( member_rule @ C @ A ) ) ).
% ComplD
thf(fact_1102_ComplD,axiom,
! [C: $o,A: set_o] :
( ( member_o @ C @ ( uminus_uminus_set_o @ A ) )
=> ~ ( member_o @ C @ A ) ) ).
% ComplD
thf(fact_1103_insert__dom,axiom,
! [F: rule > option6029102822600979917ist_fm,X2: rule,Y3: fset_P661503646757059847ist_fm] :
( ( ( F @ X2 )
= ( some_f2162114139285431020ist_fm @ Y3 ) )
=> ( ( insert_rule @ X2 @ ( dom_ru6064144267908753070ist_fm @ F ) )
= ( dom_ru6064144267908753070ist_fm @ F ) ) ) ).
% insert_dom
thf(fact_1104_insert__dom,axiom,
! [F: $o > option6029102822600979917ist_fm,X2: $o,Y3: fset_P661503646757059847ist_fm] :
( ( ( F @ X2 )
= ( some_f2162114139285431020ist_fm @ Y3 ) )
=> ( ( insert_o @ X2 @ ( dom_o_6128958874678448065ist_fm @ F ) )
= ( dom_o_6128958874678448065ist_fm @ F ) ) ) ).
% insert_dom
thf(fact_1105_insert__dom,axiom,
! [F: produc1996495991257130529ist_fm > option6029102822600979917ist_fm,X2: produc1996495991257130529ist_fm,Y3: fset_P661503646757059847ist_fm] :
( ( ( F @ X2 )
= ( some_f2162114139285431020ist_fm @ Y3 ) )
=> ( ( insert589278167076416497ist_fm @ X2 @ ( dom_Pr1119034465550817144ist_fm @ F ) )
= ( dom_Pr1119034465550817144ist_fm @ F ) ) ) ).
% insert_dom
thf(fact_1106_Compl__UNIV__eq,axiom,
( ( uminus6150396152209328632ist_fm @ top_to728902761071950545ist_fm )
= bot_bo4530011010606209517ist_fm ) ).
% Compl_UNIV_eq
thf(fact_1107_Compl__UNIV__eq,axiom,
( ( uminus4869265918275750596t_rule @ top_top_set_rule )
= bot_bot_set_rule ) ).
% Compl_UNIV_eq
thf(fact_1108_Compl__UNIV__eq,axiom,
( ( uminus_uminus_set_o @ top_top_set_o )
= bot_bot_set_o ) ).
% Compl_UNIV_eq
thf(fact_1109_Compl__empty__eq,axiom,
( ( uminus6150396152209328632ist_fm @ bot_bo4530011010606209517ist_fm )
= top_to728902761071950545ist_fm ) ).
% Compl_empty_eq
thf(fact_1110_Compl__empty__eq,axiom,
( ( uminus4869265918275750596t_rule @ bot_bot_set_rule )
= top_top_set_rule ) ).
% Compl_empty_eq
thf(fact_1111_Compl__empty__eq,axiom,
( ( uminus_uminus_set_o @ bot_bot_set_o )
= top_top_set_o ) ).
% Compl_empty_eq
thf(fact_1112_Compl__eq__Diff__UNIV,axiom,
( uminus4869265918275750596t_rule
= ( minus_minus_set_rule @ top_top_set_rule ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_1113_Compl__eq__Diff__UNIV,axiom,
( uminus_uminus_set_o
= ( minus_minus_set_o @ top_top_set_o ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_1114_Compl__eq__Diff__UNIV,axiom,
( uminus6150396152209328632ist_fm
= ( minus_5615909706921521512ist_fm @ top_to728902761071950545ist_fm ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_1115_Compl__insert,axiom,
! [X2: rule,A: set_rule] :
( ( uminus4869265918275750596t_rule @ ( insert_rule @ X2 @ A ) )
= ( minus_minus_set_rule @ ( uminus4869265918275750596t_rule @ A ) @ ( insert_rule @ X2 @ bot_bot_set_rule ) ) ) ).
% Compl_insert
thf(fact_1116_Compl__insert,axiom,
! [X2: $o,A: set_o] :
( ( uminus_uminus_set_o @ ( insert_o @ X2 @ A ) )
= ( minus_minus_set_o @ ( uminus_uminus_set_o @ A ) @ ( insert_o @ X2 @ bot_bot_set_o ) ) ) ).
% Compl_insert
thf(fact_1117_Compl__insert,axiom,
! [X2: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm] :
( ( uminus6150396152209328632ist_fm @ ( insert589278167076416497ist_fm @ X2 @ A ) )
= ( minus_5615909706921521512ist_fm @ ( uminus6150396152209328632ist_fm @ A ) @ ( insert589278167076416497ist_fm @ X2 @ bot_bo4530011010606209517ist_fm ) ) ) ).
% Compl_insert
thf(fact_1118_dom__eq__singleton__conv,axiom,
! [F: rule > option6029102822600979917ist_fm,X2: rule] :
( ( ( dom_ru6064144267908753070ist_fm @ F )
= ( insert_rule @ X2 @ bot_bot_set_rule ) )
= ( ? [V2: fset_P661503646757059847ist_fm] :
( F
= ( fun_up5825503389150910729ist_fm
@ ^ [X: rule] : none_f2701453242279119720ist_fm
@ X2
@ ( some_f2162114139285431020ist_fm @ V2 ) ) ) ) ) ).
% dom_eq_singleton_conv
thf(fact_1119_dom__eq__singleton__conv,axiom,
! [F: $o > option6029102822600979917ist_fm,X2: $o] :
( ( ( dom_o_6128958874678448065ist_fm @ F )
= ( insert_o @ X2 @ bot_bot_set_o ) )
= ( ? [V2: fset_P661503646757059847ist_fm] :
( F
= ( fun_up1845550324453438770ist_fm
@ ^ [X: $o] : none_f2701453242279119720ist_fm
@ X2
@ ( some_f2162114139285431020ist_fm @ V2 ) ) ) ) ) ).
% dom_eq_singleton_conv
thf(fact_1120_dom__eq__singleton__conv,axiom,
! [F: produc1996495991257130529ist_fm > option6029102822600979917ist_fm,X2: produc1996495991257130529ist_fm] :
( ( ( dom_Pr1119034465550817144ist_fm @ F )
= ( insert589278167076416497ist_fm @ X2 @ bot_bo4530011010606209517ist_fm ) )
= ( ? [V2: fset_P661503646757059847ist_fm] :
( F
= ( fun_up1138168335916736873ist_fm
@ ^ [X: produc1996495991257130529ist_fm] : none_f2701453242279119720ist_fm
@ X2
@ ( some_f2162114139285431020ist_fm @ V2 ) ) ) ) ) ).
% dom_eq_singleton_conv
thf(fact_1121_boolean__algebra_Ocompl__one,axiom,
( ( uminus6150396152209328632ist_fm @ top_to728902761071950545ist_fm )
= bot_bo4530011010606209517ist_fm ) ).
% boolean_algebra.compl_one
thf(fact_1122_boolean__algebra_Ocompl__one,axiom,
( ( uminus4869265918275750596t_rule @ top_top_set_rule )
= bot_bot_set_rule ) ).
% boolean_algebra.compl_one
thf(fact_1123_boolean__algebra_Ocompl__one,axiom,
( ( uminus_uminus_set_o @ top_top_set_o )
= bot_bot_set_o ) ).
% boolean_algebra.compl_one
thf(fact_1124_boolean__algebra_Ocompl__zero,axiom,
( ( uminus6150396152209328632ist_fm @ bot_bo4530011010606209517ist_fm )
= top_to728902761071950545ist_fm ) ).
% boolean_algebra.compl_zero
thf(fact_1125_boolean__algebra_Ocompl__zero,axiom,
( ( uminus4869265918275750596t_rule @ bot_bot_set_rule )
= top_top_set_rule ) ).
% boolean_algebra.compl_zero
thf(fact_1126_boolean__algebra_Ocompl__zero,axiom,
( ( uminus_uminus_set_o @ bot_bot_set_o )
= top_top_set_o ) ).
% boolean_algebra.compl_zero
thf(fact_1127_Un__iff,axiom,
! [C: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm,B: set_Pr7058068377845519745ist_fm] :
( ( member8102475879199740618ist_fm @ C @ ( sup_su3073845231316762965ist_fm @ A @ B ) )
= ( ( member8102475879199740618ist_fm @ C @ A )
| ( member8102475879199740618ist_fm @ C @ B ) ) ) ).
% Un_iff
thf(fact_1128_Un__iff,axiom,
! [C: rule,A: set_rule,B: set_rule] :
( ( member_rule @ C @ ( sup_sup_set_rule @ A @ B ) )
= ( ( member_rule @ C @ A )
| ( member_rule @ C @ B ) ) ) ).
% Un_iff
thf(fact_1129_Un__iff,axiom,
! [C: $o,A: set_o,B: set_o] :
( ( member_o @ C @ ( sup_sup_set_o @ A @ B ) )
= ( ( member_o @ C @ A )
| ( member_o @ C @ B ) ) ) ).
% Un_iff
thf(fact_1130_UnCI,axiom,
! [C: produc1996495991257130529ist_fm,B: set_Pr7058068377845519745ist_fm,A: set_Pr7058068377845519745ist_fm] :
( ( ~ ( member8102475879199740618ist_fm @ C @ B )
=> ( member8102475879199740618ist_fm @ C @ A ) )
=> ( member8102475879199740618ist_fm @ C @ ( sup_su3073845231316762965ist_fm @ A @ B ) ) ) ).
% UnCI
thf(fact_1131_UnCI,axiom,
! [C: rule,B: set_rule,A: set_rule] :
( ( ~ ( member_rule @ C @ B )
=> ( member_rule @ C @ A ) )
=> ( member_rule @ C @ ( sup_sup_set_rule @ A @ B ) ) ) ).
% UnCI
thf(fact_1132_UnCI,axiom,
! [C: $o,B: set_o,A: set_o] :
( ( ~ ( member_o @ C @ B )
=> ( member_o @ C @ A ) )
=> ( member_o @ C @ ( sup_sup_set_o @ A @ B ) ) ) ).
% UnCI
thf(fact_1133_sup__top__left,axiom,
! [X2: set_Pr7058068377845519745ist_fm] :
( ( sup_su3073845231316762965ist_fm @ top_to728902761071950545ist_fm @ X2 )
= top_to728902761071950545ist_fm ) ).
% sup_top_left
thf(fact_1134_sup__top__left,axiom,
! [X2: set_rule] :
( ( sup_sup_set_rule @ top_top_set_rule @ X2 )
= top_top_set_rule ) ).
% sup_top_left
thf(fact_1135_sup__top__left,axiom,
! [X2: set_o] :
( ( sup_sup_set_o @ top_top_set_o @ X2 )
= top_top_set_o ) ).
% sup_top_left
thf(fact_1136_sup__top__right,axiom,
! [X2: set_Pr7058068377845519745ist_fm] :
( ( sup_su3073845231316762965ist_fm @ X2 @ top_to728902761071950545ist_fm )
= top_to728902761071950545ist_fm ) ).
% sup_top_right
thf(fact_1137_sup__top__right,axiom,
! [X2: set_rule] :
( ( sup_sup_set_rule @ X2 @ top_top_set_rule )
= top_top_set_rule ) ).
% sup_top_right
thf(fact_1138_sup__top__right,axiom,
! [X2: set_o] :
( ( sup_sup_set_o @ X2 @ top_top_set_o )
= top_top_set_o ) ).
% sup_top_right
thf(fact_1139_boolean__algebra_Odisj__one__right,axiom,
! [X2: set_Pr7058068377845519745ist_fm] :
( ( sup_su3073845231316762965ist_fm @ X2 @ top_to728902761071950545ist_fm )
= top_to728902761071950545ist_fm ) ).
% boolean_algebra.disj_one_right
thf(fact_1140_boolean__algebra_Odisj__one__right,axiom,
! [X2: set_rule] :
( ( sup_sup_set_rule @ X2 @ top_top_set_rule )
= top_top_set_rule ) ).
% boolean_algebra.disj_one_right
thf(fact_1141_boolean__algebra_Odisj__one__right,axiom,
! [X2: set_o] :
( ( sup_sup_set_o @ X2 @ top_top_set_o )
= top_top_set_o ) ).
% boolean_algebra.disj_one_right
thf(fact_1142_boolean__algebra_Odisj__one__left,axiom,
! [X2: set_Pr7058068377845519745ist_fm] :
( ( sup_su3073845231316762965ist_fm @ top_to728902761071950545ist_fm @ X2 )
= top_to728902761071950545ist_fm ) ).
% boolean_algebra.disj_one_left
thf(fact_1143_boolean__algebra_Odisj__one__left,axiom,
! [X2: set_rule] :
( ( sup_sup_set_rule @ top_top_set_rule @ X2 )
= top_top_set_rule ) ).
% boolean_algebra.disj_one_left
thf(fact_1144_boolean__algebra_Odisj__one__left,axiom,
! [X2: set_o] :
( ( sup_sup_set_o @ top_top_set_o @ X2 )
= top_top_set_o ) ).
% boolean_algebra.disj_one_left
thf(fact_1145_sup__bot__left,axiom,
! [X2: set_rule] :
( ( sup_sup_set_rule @ bot_bot_set_rule @ X2 )
= X2 ) ).
% sup_bot_left
thf(fact_1146_sup__bot__left,axiom,
! [X2: set_o] :
( ( sup_sup_set_o @ bot_bot_set_o @ X2 )
= X2 ) ).
% sup_bot_left
thf(fact_1147_sup__bot__left,axiom,
! [X2: set_Pr7058068377845519745ist_fm] :
( ( sup_su3073845231316762965ist_fm @ bot_bo4530011010606209517ist_fm @ X2 )
= X2 ) ).
% sup_bot_left
thf(fact_1148_sup__bot__right,axiom,
! [X2: set_rule] :
( ( sup_sup_set_rule @ X2 @ bot_bot_set_rule )
= X2 ) ).
% sup_bot_right
thf(fact_1149_sup__bot__right,axiom,
! [X2: set_o] :
( ( sup_sup_set_o @ X2 @ bot_bot_set_o )
= X2 ) ).
% sup_bot_right
thf(fact_1150_sup__bot__right,axiom,
! [X2: set_Pr7058068377845519745ist_fm] :
( ( sup_su3073845231316762965ist_fm @ X2 @ bot_bo4530011010606209517ist_fm )
= X2 ) ).
% sup_bot_right
thf(fact_1151_bot__eq__sup__iff,axiom,
! [X2: set_rule,Y3: set_rule] :
( ( bot_bot_set_rule
= ( sup_sup_set_rule @ X2 @ Y3 ) )
= ( ( X2 = bot_bot_set_rule )
& ( Y3 = bot_bot_set_rule ) ) ) ).
% bot_eq_sup_iff
thf(fact_1152_bot__eq__sup__iff,axiom,
! [X2: set_o,Y3: set_o] :
( ( bot_bot_set_o
= ( sup_sup_set_o @ X2 @ Y3 ) )
= ( ( X2 = bot_bot_set_o )
& ( Y3 = bot_bot_set_o ) ) ) ).
% bot_eq_sup_iff
thf(fact_1153_bot__eq__sup__iff,axiom,
! [X2: set_Pr7058068377845519745ist_fm,Y3: set_Pr7058068377845519745ist_fm] :
( ( bot_bo4530011010606209517ist_fm
= ( sup_su3073845231316762965ist_fm @ X2 @ Y3 ) )
= ( ( X2 = bot_bo4530011010606209517ist_fm )
& ( Y3 = bot_bo4530011010606209517ist_fm ) ) ) ).
% bot_eq_sup_iff
thf(fact_1154_sup__eq__bot__iff,axiom,
! [X2: set_rule,Y3: set_rule] :
( ( ( sup_sup_set_rule @ X2 @ Y3 )
= bot_bot_set_rule )
= ( ( X2 = bot_bot_set_rule )
& ( Y3 = bot_bot_set_rule ) ) ) ).
% sup_eq_bot_iff
thf(fact_1155_sup__eq__bot__iff,axiom,
! [X2: set_o,Y3: set_o] :
( ( ( sup_sup_set_o @ X2 @ Y3 )
= bot_bot_set_o )
= ( ( X2 = bot_bot_set_o )
& ( Y3 = bot_bot_set_o ) ) ) ).
% sup_eq_bot_iff
thf(fact_1156_sup__eq__bot__iff,axiom,
! [X2: set_Pr7058068377845519745ist_fm,Y3: set_Pr7058068377845519745ist_fm] :
( ( ( sup_su3073845231316762965ist_fm @ X2 @ Y3 )
= bot_bo4530011010606209517ist_fm )
= ( ( X2 = bot_bo4530011010606209517ist_fm )
& ( Y3 = bot_bo4530011010606209517ist_fm ) ) ) ).
% sup_eq_bot_iff
thf(fact_1157_sup__bot_Oeq__neutr__iff,axiom,
! [A3: set_rule,B3: set_rule] :
( ( ( sup_sup_set_rule @ A3 @ B3 )
= bot_bot_set_rule )
= ( ( A3 = bot_bot_set_rule )
& ( B3 = bot_bot_set_rule ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_1158_sup__bot_Oeq__neutr__iff,axiom,
! [A3: set_o,B3: set_o] :
( ( ( sup_sup_set_o @ A3 @ B3 )
= bot_bot_set_o )
= ( ( A3 = bot_bot_set_o )
& ( B3 = bot_bot_set_o ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_1159_sup__bot_Oeq__neutr__iff,axiom,
! [A3: set_Pr7058068377845519745ist_fm,B3: set_Pr7058068377845519745ist_fm] :
( ( ( sup_su3073845231316762965ist_fm @ A3 @ B3 )
= bot_bo4530011010606209517ist_fm )
= ( ( A3 = bot_bo4530011010606209517ist_fm )
& ( B3 = bot_bo4530011010606209517ist_fm ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_1160_sup__bot_Oleft__neutral,axiom,
! [A3: set_rule] :
( ( sup_sup_set_rule @ bot_bot_set_rule @ A3 )
= A3 ) ).
% sup_bot.left_neutral
thf(fact_1161_sup__bot_Oleft__neutral,axiom,
! [A3: set_o] :
( ( sup_sup_set_o @ bot_bot_set_o @ A3 )
= A3 ) ).
% sup_bot.left_neutral
thf(fact_1162_sup__bot_Oleft__neutral,axiom,
! [A3: set_Pr7058068377845519745ist_fm] :
( ( sup_su3073845231316762965ist_fm @ bot_bo4530011010606209517ist_fm @ A3 )
= A3 ) ).
% sup_bot.left_neutral
thf(fact_1163_sup__bot_Oneutr__eq__iff,axiom,
! [A3: set_rule,B3: set_rule] :
( ( bot_bot_set_rule
= ( sup_sup_set_rule @ A3 @ B3 ) )
= ( ( A3 = bot_bot_set_rule )
& ( B3 = bot_bot_set_rule ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_1164_sup__bot_Oneutr__eq__iff,axiom,
! [A3: set_o,B3: set_o] :
( ( bot_bot_set_o
= ( sup_sup_set_o @ A3 @ B3 ) )
= ( ( A3 = bot_bot_set_o )
& ( B3 = bot_bot_set_o ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_1165_sup__bot_Oneutr__eq__iff,axiom,
! [A3: set_Pr7058068377845519745ist_fm,B3: set_Pr7058068377845519745ist_fm] :
( ( bot_bo4530011010606209517ist_fm
= ( sup_su3073845231316762965ist_fm @ A3 @ B3 ) )
= ( ( A3 = bot_bo4530011010606209517ist_fm )
& ( B3 = bot_bo4530011010606209517ist_fm ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_1166_sup__bot_Oright__neutral,axiom,
! [A3: set_rule] :
( ( sup_sup_set_rule @ A3 @ bot_bot_set_rule )
= A3 ) ).
% sup_bot.right_neutral
thf(fact_1167_sup__bot_Oright__neutral,axiom,
! [A3: set_o] :
( ( sup_sup_set_o @ A3 @ bot_bot_set_o )
= A3 ) ).
% sup_bot.right_neutral
thf(fact_1168_sup__bot_Oright__neutral,axiom,
! [A3: set_Pr7058068377845519745ist_fm] :
( ( sup_su3073845231316762965ist_fm @ A3 @ bot_bo4530011010606209517ist_fm )
= A3 ) ).
% sup_bot.right_neutral
thf(fact_1169_Un__empty,axiom,
! [A: set_rule,B: set_rule] :
( ( ( sup_sup_set_rule @ A @ B )
= bot_bot_set_rule )
= ( ( A = bot_bot_set_rule )
& ( B = bot_bot_set_rule ) ) ) ).
% Un_empty
thf(fact_1170_Un__empty,axiom,
! [A: set_o,B: set_o] :
( ( ( sup_sup_set_o @ A @ B )
= bot_bot_set_o )
= ( ( A = bot_bot_set_o )
& ( B = bot_bot_set_o ) ) ) ).
% Un_empty
thf(fact_1171_Un__empty,axiom,
! [A: set_Pr7058068377845519745ist_fm,B: set_Pr7058068377845519745ist_fm] :
( ( ( sup_su3073845231316762965ist_fm @ A @ B )
= bot_bo4530011010606209517ist_fm )
= ( ( A = bot_bo4530011010606209517ist_fm )
& ( B = bot_bo4530011010606209517ist_fm ) ) ) ).
% Un_empty
thf(fact_1172_Un__insert__right,axiom,
! [A: set_rule,A3: rule,B: set_rule] :
( ( sup_sup_set_rule @ A @ ( insert_rule @ A3 @ B ) )
= ( insert_rule @ A3 @ ( sup_sup_set_rule @ A @ B ) ) ) ).
% Un_insert_right
thf(fact_1173_Un__insert__right,axiom,
! [A: set_o,A3: $o,B: set_o] :
( ( sup_sup_set_o @ A @ ( insert_o @ A3 @ B ) )
= ( insert_o @ A3 @ ( sup_sup_set_o @ A @ B ) ) ) ).
% Un_insert_right
thf(fact_1174_Un__insert__right,axiom,
! [A: set_Pr7058068377845519745ist_fm,A3: produc1996495991257130529ist_fm,B: set_Pr7058068377845519745ist_fm] :
( ( sup_su3073845231316762965ist_fm @ A @ ( insert589278167076416497ist_fm @ A3 @ B ) )
= ( insert589278167076416497ist_fm @ A3 @ ( sup_su3073845231316762965ist_fm @ A @ B ) ) ) ).
% Un_insert_right
thf(fact_1175_Un__insert__left,axiom,
! [A3: rule,B: set_rule,C4: set_rule] :
( ( sup_sup_set_rule @ ( insert_rule @ A3 @ B ) @ C4 )
= ( insert_rule @ A3 @ ( sup_sup_set_rule @ B @ C4 ) ) ) ).
% Un_insert_left
thf(fact_1176_Un__insert__left,axiom,
! [A3: $o,B: set_o,C4: set_o] :
( ( sup_sup_set_o @ ( insert_o @ A3 @ B ) @ C4 )
= ( insert_o @ A3 @ ( sup_sup_set_o @ B @ C4 ) ) ) ).
% Un_insert_left
thf(fact_1177_Un__insert__left,axiom,
! [A3: produc1996495991257130529ist_fm,B: set_Pr7058068377845519745ist_fm,C4: set_Pr7058068377845519745ist_fm] :
( ( sup_su3073845231316762965ist_fm @ ( insert589278167076416497ist_fm @ A3 @ B ) @ C4 )
= ( insert589278167076416497ist_fm @ A3 @ ( sup_su3073845231316762965ist_fm @ B @ C4 ) ) ) ).
% Un_insert_left
thf(fact_1178_Un__Diff__cancel2,axiom,
! [B: set_rule,A: set_rule] :
( ( sup_sup_set_rule @ ( minus_minus_set_rule @ B @ A ) @ A )
= ( sup_sup_set_rule @ B @ A ) ) ).
% Un_Diff_cancel2
thf(fact_1179_Un__Diff__cancel2,axiom,
! [B: set_o,A: set_o] :
( ( sup_sup_set_o @ ( minus_minus_set_o @ B @ A ) @ A )
= ( sup_sup_set_o @ B @ A ) ) ).
% Un_Diff_cancel2
thf(fact_1180_Un__Diff__cancel2,axiom,
! [B: set_Pr7058068377845519745ist_fm,A: set_Pr7058068377845519745ist_fm] :
( ( sup_su3073845231316762965ist_fm @ ( minus_5615909706921521512ist_fm @ B @ A ) @ A )
= ( sup_su3073845231316762965ist_fm @ B @ A ) ) ).
% Un_Diff_cancel2
thf(fact_1181_Un__Diff__cancel,axiom,
! [A: set_rule,B: set_rule] :
( ( sup_sup_set_rule @ A @ ( minus_minus_set_rule @ B @ A ) )
= ( sup_sup_set_rule @ A @ B ) ) ).
% Un_Diff_cancel
thf(fact_1182_Un__Diff__cancel,axiom,
! [A: set_o,B: set_o] :
( ( sup_sup_set_o @ A @ ( minus_minus_set_o @ B @ A ) )
= ( sup_sup_set_o @ A @ B ) ) ).
% Un_Diff_cancel
thf(fact_1183_Un__Diff__cancel,axiom,
! [A: set_Pr7058068377845519745ist_fm,B: set_Pr7058068377845519745ist_fm] :
( ( sup_su3073845231316762965ist_fm @ A @ ( minus_5615909706921521512ist_fm @ B @ A ) )
= ( sup_su3073845231316762965ist_fm @ A @ B ) ) ).
% Un_Diff_cancel
thf(fact_1184_sup__compl__top__left1,axiom,
! [X2: set_Pr7058068377845519745ist_fm,Y3: set_Pr7058068377845519745ist_fm] :
( ( sup_su3073845231316762965ist_fm @ ( uminus6150396152209328632ist_fm @ X2 ) @ ( sup_su3073845231316762965ist_fm @ X2 @ Y3 ) )
= top_to728902761071950545ist_fm ) ).
% sup_compl_top_left1
thf(fact_1185_sup__compl__top__left1,axiom,
! [X2: set_rule,Y3: set_rule] :
( ( sup_sup_set_rule @ ( uminus4869265918275750596t_rule @ X2 ) @ ( sup_sup_set_rule @ X2 @ Y3 ) )
= top_top_set_rule ) ).
% sup_compl_top_left1
thf(fact_1186_sup__compl__top__left1,axiom,
! [X2: set_o,Y3: set_o] :
( ( sup_sup_set_o @ ( uminus_uminus_set_o @ X2 ) @ ( sup_sup_set_o @ X2 @ Y3 ) )
= top_top_set_o ) ).
% sup_compl_top_left1
thf(fact_1187_sup__compl__top__left2,axiom,
! [X2: set_Pr7058068377845519745ist_fm,Y3: set_Pr7058068377845519745ist_fm] :
( ( sup_su3073845231316762965ist_fm @ X2 @ ( sup_su3073845231316762965ist_fm @ ( uminus6150396152209328632ist_fm @ X2 ) @ Y3 ) )
= top_to728902761071950545ist_fm ) ).
% sup_compl_top_left2
thf(fact_1188_sup__compl__top__left2,axiom,
! [X2: set_rule,Y3: set_rule] :
( ( sup_sup_set_rule @ X2 @ ( sup_sup_set_rule @ ( uminus4869265918275750596t_rule @ X2 ) @ Y3 ) )
= top_top_set_rule ) ).
% sup_compl_top_left2
thf(fact_1189_sup__compl__top__left2,axiom,
! [X2: set_o,Y3: set_o] :
( ( sup_sup_set_o @ X2 @ ( sup_sup_set_o @ ( uminus_uminus_set_o @ X2 ) @ Y3 ) )
= top_top_set_o ) ).
% sup_compl_top_left2
thf(fact_1190_boolean__algebra_Odisj__cancel__left,axiom,
! [X2: set_Pr7058068377845519745ist_fm] :
( ( sup_su3073845231316762965ist_fm @ ( uminus6150396152209328632ist_fm @ X2 ) @ X2 )
= top_to728902761071950545ist_fm ) ).
% boolean_algebra.disj_cancel_left
thf(fact_1191_boolean__algebra_Odisj__cancel__left,axiom,
! [X2: set_rule] :
( ( sup_sup_set_rule @ ( uminus4869265918275750596t_rule @ X2 ) @ X2 )
= top_top_set_rule ) ).
% boolean_algebra.disj_cancel_left
thf(fact_1192_boolean__algebra_Odisj__cancel__left,axiom,
! [X2: set_o] :
( ( sup_sup_set_o @ ( uminus_uminus_set_o @ X2 ) @ X2 )
= top_top_set_o ) ).
% boolean_algebra.disj_cancel_left
thf(fact_1193_boolean__algebra_Odisj__cancel__right,axiom,
! [X2: set_Pr7058068377845519745ist_fm] :
( ( sup_su3073845231316762965ist_fm @ X2 @ ( uminus6150396152209328632ist_fm @ X2 ) )
= top_to728902761071950545ist_fm ) ).
% boolean_algebra.disj_cancel_right
thf(fact_1194_boolean__algebra_Odisj__cancel__right,axiom,
! [X2: set_rule] :
( ( sup_sup_set_rule @ X2 @ ( uminus4869265918275750596t_rule @ X2 ) )
= top_top_set_rule ) ).
% boolean_algebra.disj_cancel_right
thf(fact_1195_boolean__algebra_Odisj__cancel__right,axiom,
! [X2: set_o] :
( ( sup_sup_set_o @ X2 @ ( uminus_uminus_set_o @ X2 ) )
= top_top_set_o ) ).
% boolean_algebra.disj_cancel_right
thf(fact_1196_Sup__insert,axiom,
! [A3: $o,A: set_o] :
( ( complete_Sup_Sup_o @ ( insert_o @ A3 @ A ) )
= ( sup_sup_o @ A3 @ ( complete_Sup_Sup_o @ A ) ) ) ).
% Sup_insert
thf(fact_1197_Compl__Diff__eq,axiom,
! [A: set_rule,B: set_rule] :
( ( uminus4869265918275750596t_rule @ ( minus_minus_set_rule @ A @ B ) )
= ( sup_sup_set_rule @ ( uminus4869265918275750596t_rule @ A ) @ B ) ) ).
% Compl_Diff_eq
thf(fact_1198_Compl__Diff__eq,axiom,
! [A: set_o,B: set_o] :
( ( uminus_uminus_set_o @ ( minus_minus_set_o @ A @ B ) )
= ( sup_sup_set_o @ ( uminus_uminus_set_o @ A ) @ B ) ) ).
% Compl_Diff_eq
thf(fact_1199_Compl__Diff__eq,axiom,
! [A: set_Pr7058068377845519745ist_fm,B: set_Pr7058068377845519745ist_fm] :
( ( uminus6150396152209328632ist_fm @ ( minus_5615909706921521512ist_fm @ A @ B ) )
= ( sup_su3073845231316762965ist_fm @ ( uminus6150396152209328632ist_fm @ A ) @ B ) ) ).
% Compl_Diff_eq
thf(fact_1200_UN__simps_I2_J,axiom,
! [C4: set_rule,A: rule > set_rule,B: set_rule] :
( ( ( C4 = bot_bot_set_rule )
=> ( ( comple2146307154184993742t_rule
@ ( image_rule_set_rule
@ ^ [X: rule] : ( sup_sup_set_rule @ ( A @ X ) @ B )
@ C4 ) )
= bot_bot_set_rule ) )
& ( ( C4 != bot_bot_set_rule )
=> ( ( comple2146307154184993742t_rule
@ ( image_rule_set_rule
@ ^ [X: rule] : ( sup_sup_set_rule @ ( A @ X ) @ B )
@ C4 ) )
= ( sup_sup_set_rule @ ( comple2146307154184993742t_rule @ ( image_rule_set_rule @ A @ C4 ) ) @ B ) ) ) ) ).
% UN_simps(2)
thf(fact_1201_UN__simps_I2_J,axiom,
! [C4: set_rule,A: rule > set_o,B: set_o] :
( ( ( C4 = bot_bot_set_rule )
=> ( ( comple90263536869209701_set_o
@ ( image_rule_set_o
@ ^ [X: rule] : ( sup_sup_set_o @ ( A @ X ) @ B )
@ C4 ) )
= bot_bot_set_o ) )
& ( ( C4 != bot_bot_set_rule )
=> ( ( comple90263536869209701_set_o
@ ( image_rule_set_o
@ ^ [X: rule] : ( sup_sup_set_o @ ( A @ X ) @ B )
@ C4 ) )
= ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_rule_set_o @ A @ C4 ) ) @ B ) ) ) ) ).
% UN_simps(2)
thf(fact_1202_UN__simps_I2_J,axiom,
! [C4: set_rule,A: rule > set_Pr7058068377845519745ist_fm,B: set_Pr7058068377845519745ist_fm] :
( ( ( C4 = bot_bot_set_rule )
=> ( ( comple1649742678916152942ist_fm
@ ( image_3897101447128207157ist_fm
@ ^ [X: rule] : ( sup_su3073845231316762965ist_fm @ ( A @ X ) @ B )
@ C4 ) )
= bot_bo4530011010606209517ist_fm ) )
& ( ( C4 != bot_bot_set_rule )
=> ( ( comple1649742678916152942ist_fm
@ ( image_3897101447128207157ist_fm
@ ^ [X: rule] : ( sup_su3073845231316762965ist_fm @ ( A @ X ) @ B )
@ C4 ) )
= ( sup_su3073845231316762965ist_fm @ ( comple1649742678916152942ist_fm @ ( image_3897101447128207157ist_fm @ A @ C4 ) ) @ B ) ) ) ) ).
% UN_simps(2)
thf(fact_1203_UN__simps_I2_J,axiom,
! [C4: set_o,A: $o > set_rule,B: set_rule] :
( ( ( C4 = bot_bot_set_o )
=> ( ( comple2146307154184993742t_rule
@ ( image_o_set_rule
@ ^ [X: $o] : ( sup_sup_set_rule @ ( A @ X ) @ B )
@ C4 ) )
= bot_bot_set_rule ) )
& ( ( C4 != bot_bot_set_o )
=> ( ( comple2146307154184993742t_rule
@ ( image_o_set_rule
@ ^ [X: $o] : ( sup_sup_set_rule @ ( A @ X ) @ B )
@ C4 ) )
= ( sup_sup_set_rule @ ( comple2146307154184993742t_rule @ ( image_o_set_rule @ A @ C4 ) ) @ B ) ) ) ) ).
% UN_simps(2)
thf(fact_1204_UN__simps_I2_J,axiom,
! [C4: set_o,A: $o > set_o,B: set_o] :
( ( ( C4 = bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X: $o] : ( sup_sup_set_o @ ( A @ X ) @ B )
@ C4 ) )
= bot_bot_set_o ) )
& ( ( C4 != bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X: $o] : ( sup_sup_set_o @ ( A @ X ) @ B )
@ C4 ) )
= ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_o_set_o @ A @ C4 ) ) @ B ) ) ) ) ).
% UN_simps(2)
thf(fact_1205_UN__simps_I2_J,axiom,
! [C4: set_o,A: $o > set_Pr7058068377845519745ist_fm,B: set_Pr7058068377845519745ist_fm] :
( ( ( C4 = bot_bot_set_o )
=> ( ( comple1649742678916152942ist_fm
@ ( image_1666948324336172654ist_fm
@ ^ [X: $o] : ( sup_su3073845231316762965ist_fm @ ( A @ X ) @ B )
@ C4 ) )
= bot_bo4530011010606209517ist_fm ) )
& ( ( C4 != bot_bot_set_o )
=> ( ( comple1649742678916152942ist_fm
@ ( image_1666948324336172654ist_fm
@ ^ [X: $o] : ( sup_su3073845231316762965ist_fm @ ( A @ X ) @ B )
@ C4 ) )
= ( sup_su3073845231316762965ist_fm @ ( comple1649742678916152942ist_fm @ ( image_1666948324336172654ist_fm @ A @ C4 ) ) @ B ) ) ) ) ).
% UN_simps(2)
thf(fact_1206_UN__simps_I2_J,axiom,
! [C4: set_Pr7058068377845519745ist_fm,A: produc1996495991257130529ist_fm > set_rule,B: set_rule] :
( ( ( C4 = bot_bo4530011010606209517ist_fm )
=> ( ( comple2146307154184993742t_rule
@ ( image_2008118025656205143t_rule
@ ^ [X: produc1996495991257130529ist_fm] : ( sup_sup_set_rule @ ( A @ X ) @ B )
@ C4 ) )
= bot_bot_set_rule ) )
& ( ( C4 != bot_bo4530011010606209517ist_fm )
=> ( ( comple2146307154184993742t_rule
@ ( image_2008118025656205143t_rule
@ ^ [X: produc1996495991257130529ist_fm] : ( sup_sup_set_rule @ ( A @ X ) @ B )
@ C4 ) )
= ( sup_sup_set_rule @ ( comple2146307154184993742t_rule @ ( image_2008118025656205143t_rule @ A @ C4 ) ) @ B ) ) ) ) ).
% UN_simps(2)
thf(fact_1207_UN__simps_I2_J,axiom,
! [C4: set_Pr7058068377845519745ist_fm,A: produc1996495991257130529ist_fm > set_o,B: set_o] :
( ( ( C4 = bot_bo4530011010606209517ist_fm )
=> ( ( comple90263536869209701_set_o
@ ( image_7087520594942641564_set_o
@ ^ [X: produc1996495991257130529ist_fm] : ( sup_sup_set_o @ ( A @ X ) @ B )
@ C4 ) )
= bot_bot_set_o ) )
& ( ( C4 != bot_bo4530011010606209517ist_fm )
=> ( ( comple90263536869209701_set_o
@ ( image_7087520594942641564_set_o
@ ^ [X: produc1996495991257130529ist_fm] : ( sup_sup_set_o @ ( A @ X ) @ B )
@ C4 ) )
= ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_7087520594942641564_set_o @ A @ C4 ) ) @ B ) ) ) ) ).
% UN_simps(2)
thf(fact_1208_UN__simps_I2_J,axiom,
! [C4: set_Pr7058068377845519745ist_fm,A: produc1996495991257130529ist_fm > set_Pr7058068377845519745ist_fm,B: set_Pr7058068377845519745ist_fm] :
( ( ( C4 = bot_bo4530011010606209517ist_fm )
=> ( ( comple1649742678916152942ist_fm
@ ( image_5852275405373602981ist_fm
@ ^ [X: produc1996495991257130529ist_fm] : ( sup_su3073845231316762965ist_fm @ ( A @ X ) @ B )
@ C4 ) )
= bot_bo4530011010606209517ist_fm ) )
& ( ( C4 != bot_bo4530011010606209517ist_fm )
=> ( ( comple1649742678916152942ist_fm
@ ( image_5852275405373602981ist_fm
@ ^ [X: produc1996495991257130529ist_fm] : ( sup_su3073845231316762965ist_fm @ ( A @ X ) @ B )
@ C4 ) )
= ( sup_su3073845231316762965ist_fm @ ( comple1649742678916152942ist_fm @ ( image_5852275405373602981ist_fm @ A @ C4 ) ) @ B ) ) ) ) ).
% UN_simps(2)
thf(fact_1209_UN__simps_I3_J,axiom,
! [C4: set_rule,A: set_rule,B: rule > set_rule] :
( ( ( C4 = bot_bot_set_rule )
=> ( ( comple2146307154184993742t_rule
@ ( image_rule_set_rule
@ ^ [X: rule] : ( sup_sup_set_rule @ A @ ( B @ X ) )
@ C4 ) )
= bot_bot_set_rule ) )
& ( ( C4 != bot_bot_set_rule )
=> ( ( comple2146307154184993742t_rule
@ ( image_rule_set_rule
@ ^ [X: rule] : ( sup_sup_set_rule @ A @ ( B @ X ) )
@ C4 ) )
= ( sup_sup_set_rule @ A @ ( comple2146307154184993742t_rule @ ( image_rule_set_rule @ B @ C4 ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_1210_UN__simps_I3_J,axiom,
! [C4: set_rule,A: set_o,B: rule > set_o] :
( ( ( C4 = bot_bot_set_rule )
=> ( ( comple90263536869209701_set_o
@ ( image_rule_set_o
@ ^ [X: rule] : ( sup_sup_set_o @ A @ ( B @ X ) )
@ C4 ) )
= bot_bot_set_o ) )
& ( ( C4 != bot_bot_set_rule )
=> ( ( comple90263536869209701_set_o
@ ( image_rule_set_o
@ ^ [X: rule] : ( sup_sup_set_o @ A @ ( B @ X ) )
@ C4 ) )
= ( sup_sup_set_o @ A @ ( comple90263536869209701_set_o @ ( image_rule_set_o @ B @ C4 ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_1211_UN__simps_I3_J,axiom,
! [C4: set_rule,A: set_Pr7058068377845519745ist_fm,B: rule > set_Pr7058068377845519745ist_fm] :
( ( ( C4 = bot_bot_set_rule )
=> ( ( comple1649742678916152942ist_fm
@ ( image_3897101447128207157ist_fm
@ ^ [X: rule] : ( sup_su3073845231316762965ist_fm @ A @ ( B @ X ) )
@ C4 ) )
= bot_bo4530011010606209517ist_fm ) )
& ( ( C4 != bot_bot_set_rule )
=> ( ( comple1649742678916152942ist_fm
@ ( image_3897101447128207157ist_fm
@ ^ [X: rule] : ( sup_su3073845231316762965ist_fm @ A @ ( B @ X ) )
@ C4 ) )
= ( sup_su3073845231316762965ist_fm @ A @ ( comple1649742678916152942ist_fm @ ( image_3897101447128207157ist_fm @ B @ C4 ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_1212_UN__simps_I3_J,axiom,
! [C4: set_o,A: set_rule,B: $o > set_rule] :
( ( ( C4 = bot_bot_set_o )
=> ( ( comple2146307154184993742t_rule
@ ( image_o_set_rule
@ ^ [X: $o] : ( sup_sup_set_rule @ A @ ( B @ X ) )
@ C4 ) )
= bot_bot_set_rule ) )
& ( ( C4 != bot_bot_set_o )
=> ( ( comple2146307154184993742t_rule
@ ( image_o_set_rule
@ ^ [X: $o] : ( sup_sup_set_rule @ A @ ( B @ X ) )
@ C4 ) )
= ( sup_sup_set_rule @ A @ ( comple2146307154184993742t_rule @ ( image_o_set_rule @ B @ C4 ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_1213_UN__simps_I3_J,axiom,
! [C4: set_o,A: set_o,B: $o > set_o] :
( ( ( C4 = bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X: $o] : ( sup_sup_set_o @ A @ ( B @ X ) )
@ C4 ) )
= bot_bot_set_o ) )
& ( ( C4 != bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X: $o] : ( sup_sup_set_o @ A @ ( B @ X ) )
@ C4 ) )
= ( sup_sup_set_o @ A @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B @ C4 ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_1214_UN__simps_I3_J,axiom,
! [C4: set_o,A: set_Pr7058068377845519745ist_fm,B: $o > set_Pr7058068377845519745ist_fm] :
( ( ( C4 = bot_bot_set_o )
=> ( ( comple1649742678916152942ist_fm
@ ( image_1666948324336172654ist_fm
@ ^ [X: $o] : ( sup_su3073845231316762965ist_fm @ A @ ( B @ X ) )
@ C4 ) )
= bot_bo4530011010606209517ist_fm ) )
& ( ( C4 != bot_bot_set_o )
=> ( ( comple1649742678916152942ist_fm
@ ( image_1666948324336172654ist_fm
@ ^ [X: $o] : ( sup_su3073845231316762965ist_fm @ A @ ( B @ X ) )
@ C4 ) )
= ( sup_su3073845231316762965ist_fm @ A @ ( comple1649742678916152942ist_fm @ ( image_1666948324336172654ist_fm @ B @ C4 ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_1215_UN__simps_I3_J,axiom,
! [C4: set_Pr7058068377845519745ist_fm,A: set_rule,B: produc1996495991257130529ist_fm > set_rule] :
( ( ( C4 = bot_bo4530011010606209517ist_fm )
=> ( ( comple2146307154184993742t_rule
@ ( image_2008118025656205143t_rule
@ ^ [X: produc1996495991257130529ist_fm] : ( sup_sup_set_rule @ A @ ( B @ X ) )
@ C4 ) )
= bot_bot_set_rule ) )
& ( ( C4 != bot_bo4530011010606209517ist_fm )
=> ( ( comple2146307154184993742t_rule
@ ( image_2008118025656205143t_rule
@ ^ [X: produc1996495991257130529ist_fm] : ( sup_sup_set_rule @ A @ ( B @ X ) )
@ C4 ) )
= ( sup_sup_set_rule @ A @ ( comple2146307154184993742t_rule @ ( image_2008118025656205143t_rule @ B @ C4 ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_1216_UN__simps_I3_J,axiom,
! [C4: set_Pr7058068377845519745ist_fm,A: set_o,B: produc1996495991257130529ist_fm > set_o] :
( ( ( C4 = bot_bo4530011010606209517ist_fm )
=> ( ( comple90263536869209701_set_o
@ ( image_7087520594942641564_set_o
@ ^ [X: produc1996495991257130529ist_fm] : ( sup_sup_set_o @ A @ ( B @ X ) )
@ C4 ) )
= bot_bot_set_o ) )
& ( ( C4 != bot_bo4530011010606209517ist_fm )
=> ( ( comple90263536869209701_set_o
@ ( image_7087520594942641564_set_o
@ ^ [X: produc1996495991257130529ist_fm] : ( sup_sup_set_o @ A @ ( B @ X ) )
@ C4 ) )
= ( sup_sup_set_o @ A @ ( comple90263536869209701_set_o @ ( image_7087520594942641564_set_o @ B @ C4 ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_1217_UN__simps_I3_J,axiom,
! [C4: set_Pr7058068377845519745ist_fm,A: set_Pr7058068377845519745ist_fm,B: produc1996495991257130529ist_fm > set_Pr7058068377845519745ist_fm] :
( ( ( C4 = bot_bo4530011010606209517ist_fm )
=> ( ( comple1649742678916152942ist_fm
@ ( image_5852275405373602981ist_fm
@ ^ [X: produc1996495991257130529ist_fm] : ( sup_su3073845231316762965ist_fm @ A @ ( B @ X ) )
@ C4 ) )
= bot_bo4530011010606209517ist_fm ) )
& ( ( C4 != bot_bo4530011010606209517ist_fm )
=> ( ( comple1649742678916152942ist_fm
@ ( image_5852275405373602981ist_fm
@ ^ [X: produc1996495991257130529ist_fm] : ( sup_su3073845231316762965ist_fm @ A @ ( B @ X ) )
@ C4 ) )
= ( sup_su3073845231316762965ist_fm @ A @ ( comple1649742678916152942ist_fm @ ( image_5852275405373602981ist_fm @ B @ C4 ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_1218_sup__cancel__left2,axiom,
! [X2: set_Pr7058068377845519745ist_fm,A3: set_Pr7058068377845519745ist_fm,B3: set_Pr7058068377845519745ist_fm] :
( ( sup_su3073845231316762965ist_fm @ ( sup_su3073845231316762965ist_fm @ ( uminus6150396152209328632ist_fm @ X2 ) @ A3 ) @ ( sup_su3073845231316762965ist_fm @ X2 @ B3 ) )
= top_to728902761071950545ist_fm ) ).
% sup_cancel_left2
thf(fact_1219_sup__cancel__left2,axiom,
! [X2: set_rule,A3: set_rule,B3: set_rule] :
( ( sup_sup_set_rule @ ( sup_sup_set_rule @ ( uminus4869265918275750596t_rule @ X2 ) @ A3 ) @ ( sup_sup_set_rule @ X2 @ B3 ) )
= top_top_set_rule ) ).
% sup_cancel_left2
thf(fact_1220_sup__cancel__left2,axiom,
! [X2: set_o,A3: set_o,B3: set_o] :
( ( sup_sup_set_o @ ( sup_sup_set_o @ ( uminus_uminus_set_o @ X2 ) @ A3 ) @ ( sup_sup_set_o @ X2 @ B3 ) )
= top_top_set_o ) ).
% sup_cancel_left2
thf(fact_1221_sup__cancel__left1,axiom,
! [X2: set_Pr7058068377845519745ist_fm,A3: set_Pr7058068377845519745ist_fm,B3: set_Pr7058068377845519745ist_fm] :
( ( sup_su3073845231316762965ist_fm @ ( sup_su3073845231316762965ist_fm @ X2 @ A3 ) @ ( sup_su3073845231316762965ist_fm @ ( uminus6150396152209328632ist_fm @ X2 ) @ B3 ) )
= top_to728902761071950545ist_fm ) ).
% sup_cancel_left1
thf(fact_1222_sup__cancel__left1,axiom,
! [X2: set_rule,A3: set_rule,B3: set_rule] :
( ( sup_sup_set_rule @ ( sup_sup_set_rule @ X2 @ A3 ) @ ( sup_sup_set_rule @ ( uminus4869265918275750596t_rule @ X2 ) @ B3 ) )
= top_top_set_rule ) ).
% sup_cancel_left1
thf(fact_1223_sup__cancel__left1,axiom,
! [X2: set_o,A3: set_o,B3: set_o] :
( ( sup_sup_set_o @ ( sup_sup_set_o @ X2 @ A3 ) @ ( sup_sup_set_o @ ( uminus_uminus_set_o @ X2 ) @ B3 ) )
= top_top_set_o ) ).
% sup_cancel_left1
thf(fact_1224_Un__UNIV__right,axiom,
! [A: set_Pr7058068377845519745ist_fm] :
( ( sup_su3073845231316762965ist_fm @ A @ top_to728902761071950545ist_fm )
= top_to728902761071950545ist_fm ) ).
% Un_UNIV_right
thf(fact_1225_Un__UNIV__right,axiom,
! [A: set_rule] :
( ( sup_sup_set_rule @ A @ top_top_set_rule )
= top_top_set_rule ) ).
% Un_UNIV_right
thf(fact_1226_Un__UNIV__right,axiom,
! [A: set_o] :
( ( sup_sup_set_o @ A @ top_top_set_o )
= top_top_set_o ) ).
% Un_UNIV_right
thf(fact_1227_Un__UNIV__left,axiom,
! [B: set_Pr7058068377845519745ist_fm] :
( ( sup_su3073845231316762965ist_fm @ top_to728902761071950545ist_fm @ B )
= top_to728902761071950545ist_fm ) ).
% Un_UNIV_left
thf(fact_1228_Un__UNIV__left,axiom,
! [B: set_rule] :
( ( sup_sup_set_rule @ top_top_set_rule @ B )
= top_top_set_rule ) ).
% Un_UNIV_left
thf(fact_1229_Un__UNIV__left,axiom,
! [B: set_o] :
( ( sup_sup_set_o @ top_top_set_o @ B )
= top_top_set_o ) ).
% Un_UNIV_left
thf(fact_1230_UnI2,axiom,
! [C: produc1996495991257130529ist_fm,B: set_Pr7058068377845519745ist_fm,A: set_Pr7058068377845519745ist_fm] :
( ( member8102475879199740618ist_fm @ C @ B )
=> ( member8102475879199740618ist_fm @ C @ ( sup_su3073845231316762965ist_fm @ A @ B ) ) ) ).
% UnI2
thf(fact_1231_UnI2,axiom,
! [C: rule,B: set_rule,A: set_rule] :
( ( member_rule @ C @ B )
=> ( member_rule @ C @ ( sup_sup_set_rule @ A @ B ) ) ) ).
% UnI2
thf(fact_1232_UnI2,axiom,
! [C: $o,B: set_o,A: set_o] :
( ( member_o @ C @ B )
=> ( member_o @ C @ ( sup_sup_set_o @ A @ B ) ) ) ).
% UnI2
thf(fact_1233_UnI1,axiom,
! [C: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm,B: set_Pr7058068377845519745ist_fm] :
( ( member8102475879199740618ist_fm @ C @ A )
=> ( member8102475879199740618ist_fm @ C @ ( sup_su3073845231316762965ist_fm @ A @ B ) ) ) ).
% UnI1
thf(fact_1234_UnI1,axiom,
! [C: rule,A: set_rule,B: set_rule] :
( ( member_rule @ C @ A )
=> ( member_rule @ C @ ( sup_sup_set_rule @ A @ B ) ) ) ).
% UnI1
thf(fact_1235_UnI1,axiom,
! [C: $o,A: set_o,B: set_o] :
( ( member_o @ C @ A )
=> ( member_o @ C @ ( sup_sup_set_o @ A @ B ) ) ) ).
% UnI1
thf(fact_1236_UnE,axiom,
! [C: produc1996495991257130529ist_fm,A: set_Pr7058068377845519745ist_fm,B: set_Pr7058068377845519745ist_fm] :
( ( member8102475879199740618ist_fm @ C @ ( sup_su3073845231316762965ist_fm @ A @ B ) )
=> ( ~ ( member8102475879199740618ist_fm @ C @ A )
=> ( member8102475879199740618ist_fm @ C @ B ) ) ) ).
% UnE
thf(fact_1237_UnE,axiom,
! [C: rule,A: set_rule,B: set_rule] :
( ( member_rule @ C @ ( sup_sup_set_rule @ A @ B ) )
=> ( ~ ( member_rule @ C @ A )
=> ( member_rule @ C @ B ) ) ) ).
% UnE
thf(fact_1238_UnE,axiom,
! [C: $o,A: set_o,B: set_o] :
( ( member_o @ C @ ( sup_sup_set_o @ A @ B ) )
=> ( ~ ( member_o @ C @ A )
=> ( member_o @ C @ B ) ) ) ).
% UnE
thf(fact_1239_Un__Diff,axiom,
! [A: set_rule,B: set_rule,C4: set_rule] :
( ( minus_minus_set_rule @ ( sup_sup_set_rule @ A @ B ) @ C4 )
= ( sup_sup_set_rule @ ( minus_minus_set_rule @ A @ C4 ) @ ( minus_minus_set_rule @ B @ C4 ) ) ) ).
% Un_Diff
thf(fact_1240_Un__Diff,axiom,
! [A: set_o,B: set_o,C4: set_o] :
( ( minus_minus_set_o @ ( sup_sup_set_o @ A @ B ) @ C4 )
= ( sup_sup_set_o @ ( minus_minus_set_o @ A @ C4 ) @ ( minus_minus_set_o @ B @ C4 ) ) ) ).
% Un_Diff
thf(fact_1241_Un__Diff,axiom,
! [A: set_Pr7058068377845519745ist_fm,B: set_Pr7058068377845519745ist_fm,C4: set_Pr7058068377845519745ist_fm] :
( ( minus_5615909706921521512ist_fm @ ( sup_su3073845231316762965ist_fm @ A @ B ) @ C4 )
= ( sup_su3073845231316762965ist_fm @ ( minus_5615909706921521512ist_fm @ A @ C4 ) @ ( minus_5615909706921521512ist_fm @ B @ C4 ) ) ) ).
% Un_Diff
thf(fact_1242_Un__empty__left,axiom,
! [B: set_rule] :
( ( sup_sup_set_rule @ bot_bot_set_rule @ B )
= B ) ).
% Un_empty_left
thf(fact_1243_Un__empty__left,axiom,
! [B: set_o] :
( ( sup_sup_set_o @ bot_bot_set_o @ B )
= B ) ).
% Un_empty_left
thf(fact_1244_Un__empty__left,axiom,
! [B: set_Pr7058068377845519745ist_fm] :
( ( sup_su3073845231316762965ist_fm @ bot_bo4530011010606209517ist_fm @ B )
= B ) ).
% Un_empty_left
thf(fact_1245_Un__empty__right,axiom,
! [A: set_rule] :
( ( sup_sup_set_rule @ A @ bot_bot_set_rule )
= A ) ).
% Un_empty_right
thf(fact_1246_Un__empty__right,axiom,
! [A: set_o] :
( ( sup_sup_set_o @ A @ bot_bot_set_o )
= A ) ).
% Un_empty_right
thf(fact_1247_Un__empty__right,axiom,
! [A: set_Pr7058068377845519745ist_fm] :
( ( sup_su3073845231316762965ist_fm @ A @ bot_bo4530011010606209517ist_fm )
= A ) ).
% Un_empty_right
thf(fact_1248_boolean__algebra_Odisj__zero__right,axiom,
! [X2: set_rule] :
( ( sup_sup_set_rule @ X2 @ bot_bot_set_rule )
= X2 ) ).
% boolean_algebra.disj_zero_right
thf(fact_1249_boolean__algebra_Odisj__zero__right,axiom,
! [X2: set_o] :
( ( sup_sup_set_o @ X2 @ bot_bot_set_o )
= X2 ) ).
% boolean_algebra.disj_zero_right
thf(fact_1250_boolean__algebra_Odisj__zero__right,axiom,
! [X2: set_Pr7058068377845519745ist_fm] :
( ( sup_su3073845231316762965ist_fm @ X2 @ bot_bo4530011010606209517ist_fm )
= X2 ) ).
% boolean_algebra.disj_zero_right
thf(fact_1251_Un__def,axiom,
( sup_sup_set_o
= ( ^ [A5: set_o,B5: set_o] :
( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A5 )
| ( member_o @ X @ B5 ) ) ) ) ) ).
% Un_def
thf(fact_1252_Un__def,axiom,
( sup_sup_set_rule
= ( ^ [A5: set_rule,B5: set_rule] :
( collect_rule
@ ^ [X: rule] :
( ( member_rule @ X @ A5 )
| ( member_rule @ X @ B5 ) ) ) ) ) ).
% Un_def
thf(fact_1253_Un__def,axiom,
( sup_su3073845231316762965ist_fm
= ( ^ [A5: set_Pr7058068377845519745ist_fm,B5: set_Pr7058068377845519745ist_fm] :
( collec7231575776666803724ist_fm
@ ^ [X: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ X @ A5 )
| ( member8102475879199740618ist_fm @ X @ B5 ) ) ) ) ) ).
% Un_def
thf(fact_1254_Collect__disj__eq,axiom,
! [P: $o > $o,Q: $o > $o] :
( ( collect_o
@ ^ [X: $o] :
( ( P @ X )
| ( Q @ X ) ) )
= ( sup_sup_set_o @ ( collect_o @ P ) @ ( collect_o @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_1255_Collect__disj__eq,axiom,
! [P: rule > $o,Q: rule > $o] :
( ( collect_rule
@ ^ [X: rule] :
( ( P @ X )
| ( Q @ X ) ) )
= ( sup_sup_set_rule @ ( collect_rule @ P ) @ ( collect_rule @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_1256_Collect__disj__eq,axiom,
! [P: produc1996495991257130529ist_fm > $o,Q: produc1996495991257130529ist_fm > $o] :
( ( collec7231575776666803724ist_fm
@ ^ [X: produc1996495991257130529ist_fm] :
( ( P @ X )
| ( Q @ X ) ) )
= ( sup_su3073845231316762965ist_fm @ ( collec7231575776666803724ist_fm @ P ) @ ( collec7231575776666803724ist_fm @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_1257_insert__def,axiom,
( insert_o
= ( ^ [A4: $o] :
( sup_sup_set_o
@ ( collect_o
@ ^ [X: $o] : ( X = A4 ) ) ) ) ) ).
% insert_def
thf(fact_1258_insert__def,axiom,
( insert_rule
= ( ^ [A4: rule] :
( sup_sup_set_rule
@ ( collect_rule
@ ^ [X: rule] : ( X = A4 ) ) ) ) ) ).
% insert_def
thf(fact_1259_insert__def,axiom,
( insert589278167076416497ist_fm
= ( ^ [A4: produc1996495991257130529ist_fm] :
( sup_su3073845231316762965ist_fm
@ ( collec7231575776666803724ist_fm
@ ^ [X: produc1996495991257130529ist_fm] : ( X = A4 ) ) ) ) ) ).
% insert_def
thf(fact_1260_uminus__set__def,axiom,
( uminus_uminus_set_o
= ( ^ [A5: set_o] :
( collect_o
@ ( uminus_uminus_o_o
@ ^ [X: $o] : ( member_o @ X @ A5 ) ) ) ) ) ).
% uminus_set_def
thf(fact_1261_uminus__set__def,axiom,
( uminus4869265918275750596t_rule
= ( ^ [A5: set_rule] :
( collect_rule
@ ( uminus_uminus_rule_o
@ ^ [X: rule] : ( member_rule @ X @ A5 ) ) ) ) ) ).
% uminus_set_def
thf(fact_1262_uminus__set__def,axiom,
( uminus6150396152209328632ist_fm
= ( ^ [A5: set_Pr7058068377845519745ist_fm] :
( collec7231575776666803724ist_fm
@ ( uminus8340389434899797733t_fm_o
@ ^ [X: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X @ A5 ) ) ) ) ) ).
% uminus_set_def
thf(fact_1263_Collect__imp__eq,axiom,
! [P: $o > $o,Q: $o > $o] :
( ( collect_o
@ ^ [X: $o] :
( ( P @ X )
=> ( Q @ X ) ) )
= ( sup_sup_set_o @ ( uminus_uminus_set_o @ ( collect_o @ P ) ) @ ( collect_o @ Q ) ) ) ).
% Collect_imp_eq
thf(fact_1264_Collect__imp__eq,axiom,
! [P: rule > $o,Q: rule > $o] :
( ( collect_rule
@ ^ [X: rule] :
( ( P @ X )
=> ( Q @ X ) ) )
= ( sup_sup_set_rule @ ( uminus4869265918275750596t_rule @ ( collect_rule @ P ) ) @ ( collect_rule @ Q ) ) ) ).
% Collect_imp_eq
thf(fact_1265_Collect__imp__eq,axiom,
! [P: produc1996495991257130529ist_fm > $o,Q: produc1996495991257130529ist_fm > $o] :
( ( collec7231575776666803724ist_fm
@ ^ [X: produc1996495991257130529ist_fm] :
( ( P @ X )
=> ( Q @ X ) ) )
= ( sup_su3073845231316762965ist_fm @ ( uminus6150396152209328632ist_fm @ ( collec7231575776666803724ist_fm @ P ) ) @ ( collec7231575776666803724ist_fm @ Q ) ) ) ).
% Collect_imp_eq
thf(fact_1266_UNIV__bool,axiom,
( top_top_set_o
= ( insert_o @ $false @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).
% UNIV_bool
thf(fact_1267_Rep__unit,axiom,
! [X2: product_unit] : ( member_o @ ( product_Rep_unit @ X2 ) @ ( insert_o @ $true @ bot_bot_set_o ) ) ).
% Rep_unit
thf(fact_1268_Abs__unit__cases,axiom,
! [X2: product_unit] :
~ ! [Y2: $o] :
( ( X2
= ( product_Abs_unit @ Y2 ) )
=> ~ ( member_o @ Y2 @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).
% Abs_unit_cases
thf(fact_1269_Rep__unit__cases,axiom,
! [Y3: $o] :
( ( member_o @ Y3 @ ( insert_o @ $true @ bot_bot_set_o ) )
=> ~ ! [X3: product_unit] :
( Y3
= ( ~ ( product_Rep_unit @ X3 ) ) ) ) ).
% Rep_unit_cases
thf(fact_1270_Rep__unit__inject,axiom,
! [X2: product_unit,Y3: product_unit] :
( ( ( product_Rep_unit @ X2 )
= ( product_Rep_unit @ Y3 ) )
= ( X2 = Y3 ) ) ).
% Rep_unit_inject
thf(fact_1271_Rep__unit__inverse,axiom,
! [X2: product_unit] :
( ( product_Abs_unit @ ( product_Rep_unit @ X2 ) )
= X2 ) ).
% Rep_unit_inverse
thf(fact_1272_Abs__unit__inverse,axiom,
! [Y3: $o] :
( ( member_o @ Y3 @ ( insert_o @ $true @ bot_bot_set_o ) )
=> ( ( product_Rep_unit @ ( product_Abs_unit @ Y3 ) )
= Y3 ) ) ).
% Abs_unit_inverse
thf(fact_1273_Rep__unit__induct,axiom,
! [Y3: $o,P: $o > $o] :
( ( member_o @ Y3 @ ( insert_o @ $true @ bot_bot_set_o ) )
=> ( ! [X3: product_unit] : ( P @ ( product_Rep_unit @ X3 ) )
=> ( P @ Y3 ) ) ) ).
% Rep_unit_induct
thf(fact_1274_Abs__unit__inject,axiom,
! [X2: $o,Y3: $o] :
( ( member_o @ X2 @ ( insert_o @ $true @ bot_bot_set_o ) )
=> ( ( member_o @ Y3 @ ( insert_o @ $true @ bot_bot_set_o ) )
=> ( ( ( product_Abs_unit @ X2 )
= ( product_Abs_unit @ Y3 ) )
= ( X2 = Y3 ) ) ) ) ).
% Abs_unit_inject
thf(fact_1275_Abs__unit__induct,axiom,
! [P: product_unit > $o,X2: product_unit] :
( ! [Y2: $o] :
( ( member_o @ Y2 @ ( insert_o @ $true @ bot_bot_set_o ) )
=> ( P @ ( product_Abs_unit @ Y2 ) ) )
=> ( P @ X2 ) ) ).
% Abs_unit_induct
thf(fact_1276_type__definition__unit,axiom,
type_d6188575255521822967unit_o @ product_Rep_unit @ product_Abs_unit @ ( insert_o @ $true @ bot_bot_set_o ) ).
% type_definition_unit
% Helper facts (5)
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_T,axiom,
! [X2: produc1996495991257130529ist_fm,Y3: produc1996495991257130529ist_fm] :
( ( if_Pro3930376587665744871ist_fm @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_T,axiom,
! [X2: produc1996495991257130529ist_fm,Y3: produc1996495991257130529ist_fm] :
( ( if_Pro3930376587665744871ist_fm @ $true @ X2 @ Y3 )
= X2 ) ).
thf(help_If_3_1_If_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_T,axiom,
! [X2: fset_P661503646757059847ist_fm,Y3: fset_P661503646757059847ist_fm] :
( ( if_fse3714977293194272717ist_fm @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_T,axiom,
! [X2: fset_P661503646757059847ist_fm,Y3: fset_P661503646757059847ist_fm] :
( ( if_fse3714977293194272717ist_fm @ $true @ X2 @ Y3 )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
! [S6: produc1996495991257130529ist_fm,R12: rule,Sl5: fset_P661503646757059847ist_fm,S7: produc1996495991257130529ist_fm] :
( ~ ( member8102475879199740618ist_fm @ S6 @ top_to728902761071950545ist_fm )
| ~ ( abstra3055540779341473792ist_fm
@ ^ [R3: rule,T: produc1996495991257130529ist_fm,Ss2: fset_P661503646757059847ist_fm] :
( ( eff @ R3 @ T )
= ( some_f2162114139285431020ist_fm @ Ss2 ) )
@ r
@ S6 )
| ~ ( member_rule @ R12 @ ( minus_minus_set_rule @ top_top_set_rule @ ( insert_rule @ r @ bot_bot_set_rule ) ) )
| ( ( eff @ R12 @ S6 )
!= ( some_f2162114139285431020ist_fm @ Sl5 ) )
| ~ ( fmembe3381613331217039976ist_fm @ S7 @ Sl5 )
| ( abstra3055540779341473792ist_fm
@ ^ [R3: rule,T: produc1996495991257130529ist_fm,Ss2: fset_P661503646757059847ist_fm] :
( ( eff @ R3 @ T )
= ( some_f2162114139285431020ist_fm @ Ss2 ) )
@ r
@ S7 ) ) ).
%------------------------------------------------------------------------------